†These authors contributed equally to this work and share first authorship.
*Corresponding authors.
Introduction
Associated production of J / ψ J/\psi J / ψ [1 ] . Early colour-singlet calculations established the contributions from associated open charm and two-gluon production [2 , 3 ] , with subsequent studies examining the charm-associated rate and differential distributions [4 , 5 ] .
Measurements of prompt charmonium production by BaBar and Belle [6 , 7 ] , followed by Belle’s observation of a large double-charm component [8 ] , exposed discrepancies with leading-order (LO) predictions. Zhang and Chao calculated the next-to-leading-order (NLO) QCD correction to the total rate [9 ] ; Gong and Wang performed an independent calculation and studied differential distributions [10 ] . For commonly used inputs, these calculations found an enhancement by a factor of approximately 1.7–1.8 over LO [9 , 10 ] . Belle reported a charm-associated prompt cross section of 0.74 ± 0.08 − 0.08 + 0.09 p b 0.74 \pm0.08 {}^{+0.09}_{-0.08}\,\mathrm{pb} 0.74 ± 0.08 − 0.08 + 0.09 pb [11 ] . The NLO production-angle distribution, however, differed from the shape measured by Belle [10 ] . Further studies have examined relativistic corrections [12 ] and renormalisation-scale setting using the principle of maximum conformality [13 ] .
The complementary J / ψ + g g J/\psi+gg J / ψ + g g [11 ] . Its NLO QCD corrections were calculated independently in refs. [14 , 15 ] , and colour-singlet relativistic corrections were obtained in refs. [16 , 17 ] . Further developments include endpoint resummation [18 ] , NLO colour-octet contributions [19 ] , initial-state radiation [20 ] , and the recent mixed O ( α s v 2 ) O(\alpha_s v^2) O ( α s v 2 ) [21 ] .
NLO QCD predictions for these inclusive production channels have been available for more than a decade [9 , 10 , 14 , 15 ] . More recently, several exclusive processes have been studied at next-to-next-to-leading order (NNLO). For exclusive e + e − → J / ψ + η c e^+e^- \to J/\psi+\eta_c e + e − → J / ψ + η c [22 ] first obtained NNLO predictions, followed by Huang et al. [23 ] ; analytic two-loop amplitudes are now available as asymptotic expansions in m c 2 / s m_c^2/s m c 2 / s [24 ] . NNLO corrections to exclusive double-J / ψ J/\psi J / ψ [25 , 26 ] .
Large higher-order corrections in exclusive quarkonium processes have also motivated alternative perturbative prescriptions. An amplitude-level prescription retains the complete modulus square of the truncated amplitude and has been proposed and studied for J / ψ → 3 γ J/\psi\to3\gamma J / ψ → 3 γ [27 , 28 , 29 ] . Zhan et al. have applied this prescription at NLO to γ → J / ψ + γ \gamma\to J/\psi+\gamma γ → J / ψ + γ [30 ] . Separately, Maxia et al. have proposed a soft-improved matching scheme that includes finite soft contributions in the NNLO renormalisation of colour-singlet S-wave NRQCD matrix elements [31 ] .
In the inclusive case, NNLO results are available for the hadronic decays of η c \eta_c η c η b \eta_b η b η Q → g g \eta_Q \to gg η Q → g g [32 ] . For J / ψ J/\psi J / ψ J / ψ + c c ˉ + g J/\psi+ c\bar{c} + g J / ψ + c c ˉ + g
General subtraction methods organise this cancellation locally in phase space. At NLO, the FKS [33 ] and dipole [34 ] methods provide standard constructions, with extensions to massive partons [35 ] and NRQCD production [36 ] . NNLO approaches include antenna subtraction [37 ] , sector-improved residue subtraction [38 ] , CoLoRFulNNLO [39 ] and local analytic sector subtraction [40 ] , as well as nested soft-collinear subtraction [41 ] . Fully differential massive-pair production from a colourless initial state has been treated within the latter scheme [42 ] and CoLoRFulNNLO [43 ] .
In this work we present the first NNLO QCD calculation of direct colour-singlet J / ψ J/\psi J / ψ [42 ] . The heavy-quark mass removes collinear singularities involving the open charm quarks, but overlapping soft and light-parton collinear limits remain in the double-real channels. Soft limits are subtracted before the remaining collinear limits, and the integrated counterterms are combined with the virtual contributions on the corresponding reduced phase spaces. For the double-soft terms, we use the analytic integrals for back-to-back massive emitters of ref. [44 ] ; their extension to arbitrary relative angles has recently been studied in ref. [45 ] .
We combine the real and virtual corrections into finite phase-space integrals, with analytic cancellation of infrared poles on the single-real and Born phase spaces. The subtraction mappings keep the measured quarkonium momentum fixed, allowing us to retain the differential information. We present approximate predictions at s = 10.6 G e V \sqrt{s} = 10.6\ \mathrm{GeV} s = 10.6 GeV
§ 2 defines the process, observables and renormalisation conventions. § 3 explains how the NLO and NNLO real and virtual corrections are separated and recombined into finite contributions. The numerical implementation and the phenomenological results are discussed separately in § 4 and § 5 , followed by the conclusions in § 6 . Appendix § 8 derives the angular reconstruction, and appendix § 9 gives the integrated kernels and pole cancellations. The local counterterms and their integrals, phase-space mappings and auxiliary analytic functions are given in appendices § 10 , § 11 and § 12 , respectively.
Theoretical framework
Process definition and perturbative expansion
The process studied here is direct J / ψ J/\psi J / ψ
e − ( p − ) + e + ( p + ) ⟶ γ ∗ ( Q ) ⟶ J / ψ ( P ) + c ( p 1 ) + c ˉ ( p 2 ) + X l i g h t , Q = p − + p + , Q 2 = s . (1) e^{-}(p_{-})+e^{+}(p_{+})\longrightarrow\gamma^{*}(Q)\longrightarrow J/\psi(P)+c(p_{1})+\bar{c}(p_{2})+X_{\mathrm{light}},
\qquad Q=p_{-}+p_{+},\qquad Q^{2}=s. \tag*{(1)} e − ( p − ) + e + ( p + ) ⟶ γ ∗ ( Q ) ⟶ J / ψ ( P ) + c ( p 1 ) + c ˉ ( p 2 ) + X light , Q = p − + p + , Q 2 = s . (1) The quarkonium is produced directly in the colour-singlet c c ˉ [ 3 S 1 [ 1 ] ] c\bar{c}[{}^{3}S_{1}^{[1]}] c c ˉ [ 3 S 1 [ 1 ] ] v v v J / ψ J/\psi J / ψ P / 2 P/2 P /2 P 2 = M ψ 2 = 4 m c 2 P^{2}=M_{\psi}^{2}=4m_{c}^{2} P 2 = M ψ 2 = 4 m c 2 p 1 p_{1} p 1 p 2 p_{2} p 2 X l i g h t = ∅ , g , g g , q q ˉ X_{\mathrm{light}}=\varnothing,g,gg,q\bar{q} X light = ∅ , g , g g , q q ˉ q = u , d , s q=u,d,s q = u , d , s
NRQCD factorisation gives [1 ]
d σ = d σ ^ [ e + e − ⟶ c c ˉ [ 3 S 1 [ 1 ] ] + c c ˉ + X l i g h t ] ⟨ O 1 ⟩ J / ψ ( μ Λ ) . (2) \mathrm{d}\sigma=\mathrm{d}\hat{\sigma}\bigl[e^{+}e^{-}\longrightarrow c\bar{c}[{}^{3}S_{1}^{[1]}]+c\bar{c}+X_{\mathrm{light}}\bigr]\langle\mathcal{O}_{1}\rangle_{J/\psi}(\mu_{\Lambda}). \tag*{(2)} d σ = d σ ^ [ e + e − ⟶ c c ˉ [ 3 S 1 [ 1 ] ] + c c ˉ + X light ] ⟨ O 1 ⟩ J / ψ ( μ Λ ) . (2) Here d σ ^ \mathrm{d}\hat{\sigma} d σ ^ ⟨ O 1 ⟩ J / ψ \langle\mathcal{O}_{1}\rangle_{J/\psi} ⟨ O 1 ⟩ J / ψ ∣ R ( 0 ) ∣ 2 = ( 2 π / 3 ) ⟨ O 1 ⟩ J / ψ \lvert R(0)\rvert^{2}=(2\pi/3)\langle\mathcal{O}_{1}\rangle_{J/\psi} ∣ R ( 0 ) ∣ 2 = ( 2 π /3 ) ⟨ O 1 ⟩ J / ψ
We denote the contribution of order α 2 α s 2 + n \alpha^{2}\alpha_{s}^{2+n} α 2 α s 2 + n σ \sigma σ σ ( n ) \sigma^{(n)} σ ( n ) n = 0 , 1 , 2 n=0,1,2 n = 0 , 1 , 2
σ L O = σ ( 0 ) , \sigma_{\mathrm{LO}}=\sigma^{(0)}, σ LO = σ ( 0 ) , σ N L O = σ ( 0 ) + σ ( 1 ) , \sigma_{\mathrm{NLO}}=\sigma^{(0)}+\sigma^{(1)}, σ NLO = σ ( 0 ) + σ ( 1 ) , σ N N L O = σ ( 0 ) + σ ( 1 ) + σ ( 2 ) . \sigma_{\mathrm{NNLO}}=\sigma^{(0)}+\sigma^{(1)}+\sigma^{(2)}. σ NNLO = σ ( 0 ) + σ ( 1 ) + σ ( 2 ) . Angular reconstruction from the virtual-photon tensor
We obtain the total cross sections σ \sigma σ σ L \sigma_{L} σ L
Π μ ν = − g μ ν + Q μ Q ν s , Π L μ ν = e L μ e L ν , e L μ = P μ − ( P ⋅ Q ) Q μ / s ( P ⋅ Q ) 2 / s − M ψ 2 . (3) \Pi^{\mu\nu}=-g^{\mu\nu}+\frac{Q^{\mu}Q^{\nu}}{s},
\qquad
\Pi_{L}^{\mu\nu}=e_{L}^{\mu}e_{L}^{\nu},
\qquad
e_{L}^{\mu}=\frac{P^{\mu}-(P\cdot Q)Q^{\mu}/s}{\sqrt{(P\cdot Q)^{2}/s-M_{\psi}^{2}}.} \tag*{(3)} Π μν = − g μν + s Q μ Q ν , Π L μν = e L μ e L ν , e L μ = ( P ⋅ Q ) 2 / s − M ψ 2 . P μ − ( P ⋅ Q ) Q μ / s (3) Let θ \theta θ J / ψ J/\psi J / ψ
d σ d cos θ = 3 8 [ ( σ + σ L ) + ( σ − 3 σ L ) cos 2 θ ] , α θ = σ − 3 σ L σ + σ L . (4) \frac{\mathrm{d}\sigma}{\mathrm{d}\cos\theta}
=\frac{3}{8}\left[(\sigma+\sigma_{L})+(\sigma-3\sigma_{L})\cos^{2}\theta\right],
\qquad
\alpha_{\theta}=\frac{\sigma-3\sigma_{L}}{\sigma+\sigma_{L}}. \tag*{(4)} d cos θ d σ = 8 3 [ ( σ + σ L ) + ( σ − 3 σ L ) cos 2 θ ] , α θ = σ + σ L σ − 3 σ L . (4) The derivation is given in appendix A.
Renormalisation conventions
We use dimensional regularisation with d = 4 − 2 ϵ d = 4 - 2\epsilon d = 4 − 2 ϵ M S ‾ \overline{\mathrm{MS}} MS u , d , s u,d,s u , d , s
At relative order α s 2 \alpha_s^2 α s 2 c c ˉ [ 3 S 1 [ 1 ] ] c\bar{c}[{}^3S_1^{[1]}] c c ˉ [ 3 S 1 [ 1 ] ] M S ‾ \overline{\mathrm{MS}} MS [46 , 47 , 48 ] . Subtracting the term in (B.6) from the ultraviolet-renormalised QCD amplitude implements this matching and introduces the factorisation scale μ Λ \mu_\Lambda μ Λ μ Λ = m c \mu_\Lambda= m_c μ Λ = m c μ R \mu_R μ R
Organisation of the NLO and NNLO calculation
Let M c c ˉ + X ( ℓ ) \mathcal{M}^{(\ell)}_{c\bar{c}+X} M c c ˉ + X ( ℓ ) ℓ \ell ℓ γ ∗ → c c ˉ [ 3 S 1 [ 1 ] ] + c c ˉ + X \gamma^* \to c\bar{c}[{}^3S_1^{[1]}] + c\bar{c} + X γ ∗ → c c ˉ [ 3 S 1 [ 1 ] ] + c c ˉ + X d Φ c c ˉ + X d\Phi_{c\bar{c}+X} d Φ c c ˉ + X J / ψ + c c ˉ + X J/\psi+ c\bar{c} + X J / ψ + c c ˉ + X J / ψ J/\psi J / ψ a s = α s / ( 4 π ) a_s = \alpha_s/(4\pi) a s = α s / ( 4 π ) M = M ( 0 ) + a s M ( 1 ) + a s 2 M ( 2 ) + ⋯ \mathcal{M} = \mathcal{M}^{(0)} + a_s\mathcal{M}^{(1)} + a_s^2\mathcal{M}^{(2)} + \cdots M = M ( 0 ) + a s M ( 1 ) + a s 2 M ( 2 ) + ⋯
F c c ˉ + X ( 0 ) = ∣ M c c ˉ + X ( 0 ) ∣ 2 , F c c ˉ + X ( 1 ) = 2 a s Re [ M c c ˉ + X ( 0 ) ∗ M c c ˉ + X ( 1 ) ] , F c c ˉ + X ( 2 ) = a s 2 ( 2 Re [ M c c ˉ + X ( 0 ) ∗ M c c ˉ + X ( 2 ) ] + ∣ M c c ˉ + X ( 1 ) ∣ 2 ) . (5) \begin{aligned}
F^{(0)}_{c\bar{c}+X} &= \left|\mathcal{M}^{(0)}_{c\bar{c}+X}\right|^2,\\
F^{(1)}_{c\bar{c}+X} &= 2a_s\operatorname{Re}\left[\mathcal{M}^{(0)*}_{c\bar{c}+X}\mathcal{M}^{(1)}_{c\bar{c}+X}\right],\\
F^{(2)}_{c\bar{c}+X} &= a_s^2\left(2\operatorname{Re}\left[\mathcal{M}^{(0)*}_{c\bar{c}+X}\mathcal{M}^{(2)}_{c\bar{c}+X}\right] + \left|\mathcal{M}^{(1)}_{c\bar{c}+X}\right|^2\right). \tag*{(5)}
\end{aligned} F c c ˉ + X ( 0 ) F c c ˉ + X ( 1 ) F c c ˉ + X ( 2 ) = M c c ˉ + X ( 0 ) 2 , = 2 a s Re [ M c c ˉ + X ( 0 ) ∗ M c c ˉ + X ( 1 ) ] , = a s 2 ( 2 Re [ M c c ˉ + X ( 0 ) ∗ M c c ˉ + X ( 2 ) ] + M c c ˉ + X ( 1 ) 2 ) . (5) Final-state spin and colour sums and the photon projection in (3) are implicit. The phase-space integrals of these short-distance terms give the physical cross sections after multiplication by
N = α ∣ R ( 0 ) ∣ 2 6 m c s 2 . \mathcal{N} = \frac{\alpha|R(0)|^2}{6m_c s^2}. N = 6 m c s 2 α ∣ R ( 0 ) ∣ 2 . This factor includes the initial-state flux and the NRQCD matrix element through ∣ R ( 0 ) ∣ 2 |R(0)|^2 ∣ R ( 0 ) ∣ 2
σ ( 0 ) = N ∫ d Φ c c ˉ F c c ˉ ( 0 ) ⏟ B , σ ( 1 ) = N ∫ d Φ c c ˉ + g F c c ˉ + g ( 0 ) ⏟ R + N ∫ d Φ c c ˉ F c c ˉ ( 1 ) ⏟ V , σ ( 2 ) = N ∑ a b ∫ d Φ c c ˉ + a b F c c ˉ + a b ( 0 ) ⏟ R R + N ∫ d Φ c c ˉ + g F c c ˉ + g ( 1 ) ⏟ R V + N ∫ d Φ c c ˉ F c c ˉ ( 2 ) ⏟ V V . (6) \begin{aligned}
\sigma^{(0)} &= \mathcal{N}\underbrace{\int d\Phi_{c\bar{c}}F^{(0)}_{c\bar{c}}}_{B},\\
\sigma^{(1)} &= \mathcal{N}\underbrace{\int d\Phi_{c\bar{c}+g}F^{(0)}_{c\bar{c}+g}}_{R}+\mathcal{N}\underbrace{\int d\Phi_{c\bar{c}}F^{(1)}_{c\bar{c}}}_{V},\\
\sigma^{(2)} &= \mathcal{N}\sum_{ab}\underbrace{\int d\Phi_{c\bar{c}+ab}F^{(0)}_{c\bar{c}+ab}}_{RR}+\mathcal{N}\underbrace{\int d\Phi_{c\bar{c}+g}F^{(1)}_{c\bar{c}+g}}_{RV}+\mathcal{N}\underbrace{\int d\Phi_{c\bar{c}}F^{(2)}_{c\bar{c}}}_{VV}. \tag*{(6)}
\end{aligned} σ ( 0 ) σ ( 1 ) σ ( 2 ) = N B ∫ d Φ c c ˉ F c c ˉ ( 0 ) , = N R ∫ d Φ c c ˉ + g F c c ˉ + g ( 0 ) + N V ∫ d Φ c c ˉ F c c ˉ ( 1 ) , = N ab ∑ R R ∫ d Φ c c ˉ + ab F c c ˉ + ab ( 0 ) + N R V ∫ d Φ c c ˉ + g F c c ˉ + g ( 1 ) + N V V ∫ d Φ c c ˉ F c c ˉ ( 2 ) . (6) Here a b = g g ab = gg ab = g g u u ˉ u\bar{u} u u ˉ d d ˉ d\bar{d} d d ˉ s s ˉ s\bar{s} s s ˉ g g gg g g 1 / 2 ! 1/2! 1/2 !
Representative NNLO diagrams are shown in Figure 1 .
Figure 1. Representative NNLO diagrams for e + e − → J / ψ + c c ˉ + X e^+e^- \to J/\psi+ c\bar{c} + X e + e − → J / ψ + c c ˉ + X c c ˉ [ 3 S 1 [ 1 ] ] c\bar{c}[{}^3S_1^{[1]}] c c ˉ [ 3 S 1 [ 1 ] ] q = u , d , s q = u,d,s q = u , d , s
We use nested soft-collinear subtraction for massive partons [42 ] to turn these sums into separately finite integrals. Local soft and collinear counterterms are subtracted from the real-emission integrands and integrated analytically over the unresolved variables. Their integrals cancel the explicit infrared poles of the virtual corrections at fixed lower-multiplicity kinematics. In each finite contribution below, [ ⋅ ] ϵ 0 [\cdot]_{\epsilon}^{0} [ ⋅ ] ϵ 0 ϵ 0 \epsilon^{0} ϵ 0 f i n \mathrm{fin} fin F ( 1 , 2 ) F^{(1,2)} F ( 1 , 2 ) (5) with the loop amplitudes replaced by their infrared-finite remainders, defined in appendix B.2.
NLO: real–virtual matching
At NLO, soft subtraction and infrared factorisation split R R R V V V U U U
R ⟶ R F + R U V ⟶ V F + V U ↓ U (7) \begin{aligned}
R &\longrightarrow R_F + \boxed{R_U}\\
V &\longrightarrow V_F + \boxed{V_U}\\
&\hspace{4.2cm}\downarrow\\
&\hspace{4.2cm}U \tag*{(7)}
\end{aligned} R V ⟶ R F + R U ⟶ V F + V U ↓ U (7) The real-emission contribution R R R k k k S k S_k S k P P P
R F = N ∫ [ ( 1 − S k ) { d Φ c c ˉ + g F c c ˉ + g ( 0 ) } ] ϵ 0 . (8) R_{\mathrm{F}}=\mathcal{N}\int\left[(1-S_k)\left\{\mathrm{d}\Phi_{c\bar{c}+g}F_{c\bar{c}+g}^{(0)}\right\}\right]_{\epsilon_0}. \tag*{(8)} R F = N ∫ [ ( 1 − S k ) { d Φ c c ˉ + g F c c ˉ + g ( 0 ) } ] ϵ 0 . (8) The soft projection removes k k k F R , U F_{\mathrm{R,U}} F R , U
d Φ c c ˉ F R , U = ∫ u n r e s S k { d Φ c c ˉ + g F c c ˉ + g ( 0 ) } . (9) \mathrm{d}\Phi_{c\bar{c}}F_{\mathrm{R,U}}=\int_{\mathrm{unres}}S_k\left\{\mathrm{d}\Phi_{c\bar{c}+g}F_{c\bar{c}+g}^{(0)}\right\}. \tag*{(9)} d Φ c c ˉ F R , U = ∫ unres S k { d Φ c c ˉ + g F c c ˉ + g ( 0 ) } . (9) Here and below, ∫ u n r e s \int_{\mathrm{unres}} ∫ unres F R , U F_{\mathrm{R,U}} F R , U § 9 .§ 1 and C.1.
The virtual interference separates into its finite remainder and infrared term,
F c c ˉ ( 1 ) = F c c ˉ ( 1 ) , f i n + F V , U . (10) F_{c\bar{c}}^{(1)}=F_{c\bar{c}}^{(1),\mathrm{fin}}+F_{\mathrm{V,U}}. \tag*{(10)} F c c ˉ ( 1 ) = F c c ˉ ( 1 ) , fin + F V , U . (10) The regulated Born-phase-space integrals of F R , U F_{\mathrm{R,U}} F R , U F V , U F_{\mathrm{V,U}} F V , U N \mathcal{N} N R U R_{\mathrm{U}} R U V U V_{\mathrm{U}} V U (7) . Their poles cancel at fixed Born kinematics, giving
U = N ∫ [ d Φ c c ˉ ( F R , U + F V , U ) ] ϵ 0 , V F = N ∫ [ d Φ c c ˉ F c c ˉ ( 1 ) , f i n ] ϵ 0 . (11) \begin{aligned}
U&=\mathcal{N}\int\left[\mathrm{d}\Phi_{c\bar{c}}(F_{\mathrm{R,U}}+F_{\mathrm{V,U}})\right]_{\epsilon_0},\\
V_{\mathrm{F}}&=\mathcal{N}\int\left[\mathrm{d}\Phi_{c\bar{c}}F_{c\bar{c}}^{(1),\mathrm{fin}}\right]_{\epsilon_0}. \tag*{(11)}
\end{aligned} U V F = N ∫ [ d Φ c c ˉ ( F R , U + F V , U ) ] ϵ 0 , = N ∫ [ d Φ c c ˉ F c c ˉ ( 1 ) , fin ] ϵ 0 . (11) The explicit infrared terms and their cancellation are given in appendix § 9 .§ 3 . Thus the NLO correction is
σ ( 1 ) = R F + V F + U . (12) \sigma^{(1)}=R_{\mathrm{F}}+V_{\mathrm{F}}+U. \tag*{(12)} σ ( 1 ) = R F + V F + U . (12) The braces label the remaining phase-space integrations.
NNLO: double-real, real-virtual and double-virtual matching
At NNLO, the same separation into finite remainders and terms to be combined gives
R R ⟶ R R F + R R S U + R R D U , R V ⟶ R V F + R V S U + R V F R + R V D U , V V ⟶ V V F + V V F R + V V D U . (13) \begin{aligned}
\mathrm{RR}&\longrightarrow\mathrm{RR}_{\mathrm{F}}+\boxed{\mathrm{RR}_{\mathrm{SU}}}+\mathrm{RR}_{\mathrm{DU}},\\
\mathrm{RV}&\longrightarrow\mathrm{RV}_{\mathrm{F}}+\boxed{\mathrm{RV}_{\mathrm{SU}}}+\boxed{\mathrm{RV}_{\mathrm{FR}}}+\mathrm{RV}_{\mathrm{DU}},\\
\mathrm{VV}&\longrightarrow\mathrm{VV}_{\mathrm{F}}+\boxed{\mathrm{VV}_{\mathrm{FR}}}+\mathrm{VV}_{\mathrm{DU}}. \tag*{(13)}
\end{aligned} RR RV VV ⟶ RR F + RR SU + RR DU , ⟶ RV F + RV SU + RV FR + RV DU , ⟶ VV F + VV FR + VV DU . (13) The F terms are the locally subtracted real contributions and the finite double-virtual contribution. Each boxed column sums to the finite contribution below it: the single-unresolved contribution SU, the terms proportional to the finite one-loop interference FR, and the Born-proportional double-unresolved contribution DU.
Locally subtracted real contributions. For the two light-parton momenta k 3 k_3 k 3 k 4 k_4 k 4 S 34 \mathrm{S}_{34} S 34 C ≡ C 34 \mathrm{C} \equiv\mathrm{C}_{34} C ≡ C 34 g g gg g g E 4 ∗ < E 3 ∗ E_4^* < E_3^* E 4 ∗ < E 3 ∗ K = Q − P K = Q - P K = Q − P S 4 \mathrm{S}_4 S 4
R R F ( q q ˉ ) = N ∑ q = u , d , s ∫ [ ( 1 − C ) ( 1 − S 34 ) { d Φ c c ˉ + q q ˉ F c c ˉ + q q ˉ ( 0 ) } ] ϵ 0 (14) \mathrm{RR}_{\mathrm{F}}(q\bar{q}) = \mathcal{N} \sum_{q=u,d,s} \int\left[(1-\mathrm{C})(1-\mathrm{S}_{34})\left\{\mathrm{d}\Phi_{c\bar{c}+q\bar{q}}F_{c\bar{c}+q\bar{q}}^{(0)}\right\}\right]_{\epsilon_0} \tag*{(14)} RR F ( q q ˉ ) = N q = u , d , s ∑ ∫ [ ( 1 − C ) ( 1 − S 34 ) { d Φ c c ˉ + q q ˉ F c c ˉ + q q ˉ ( 0 ) } ] ϵ 0 (14) R R F ( g g ) = N ∫ [ ( 1 − C ) ( 1 − S 4 ) ( 1 − S 34 ) { d Φ c c ˉ + g g F c c ˉ + g g ( 0 ) } ] ϵ 0 . \mathrm{RR}_{\mathrm{F}}(gg) = \mathcal{N} \int\left[(1-\mathrm{C})(1-\mathrm{S}_4)(1-\mathrm{S}_{34})\left\{\mathrm{d}\Phi_{c\bar{c}+gg}F_{c\bar{c}+gg}^{(0)}\right\}\right]_{\epsilon_0}. RR F ( g g ) = N ∫ [ ( 1 − C ) ( 1 − S 4 ) ( 1 − S 34 ) { d Φ c c ˉ + g g F c c ˉ + g g ( 0 ) } ] ϵ 0 . The operators act from right to left on the complete expressions in braces. The g g gg g g § 10 .§ 2.1 and § 10 .§ 2.2 .
For RV, infrared factorisation first gives
F c c ˉ + g ( 1 ) = F c c ˉ + g ( 1 ) , f i n + F R V , S U . (15) F_{c\bar{c}+g}^{(1)} = F_{c\bar{c}+g}^{(1),\mathrm{fin}} + F_{\mathrm{RV},\mathrm{SU}}. \tag*{(15)} F c c ˉ + g ( 1 ) = F c c ˉ + g ( 1 ) , fin + F RV , SU . (15) The finite remainder still becomes singular as the gluon momentum vanishes. Subtracting its soft limit gives
R V F = N ∫ [ ( 1 − S k ) { d Φ c c ˉ + g F c c ˉ + g ( 1 ) , f i n } ] ϵ 0 . (16) \mathrm{RV}_{\mathrm{F}} = \mathcal{N} \int\left[(1-\mathrm{S}_k)\left\{\mathrm{d}\Phi_{c\bar{c}+g}F_{c\bar{c}+g}^{(1),\mathrm{fin}}\right\}\right]_{\epsilon_0}. \tag*{(16)} RV F = N ∫ [ ( 1 − S k ) { d Φ c c ˉ + g F c c ˉ + g ( 1 ) , fin } ] ϵ 0 . (16) The infrared term F R V , S U F_{\mathrm{RV},\mathrm{SU}} F RV , SU § 9 .§ 2 , and the soft counterterms of both RV terms in appendix § 10 .§ 3.1 .
Single-unresolved contribution. The single-unresolved terms removed in (14) are generated by C ( 1 − S 34 ) \mathrm{C}(1-\mathrm{S}_{34}) C ( 1 − S 34 ) q q ˉ q\bar{q} q q ˉ [ C + ( 1 − C ) S 4 ] ( 1 − S 34 ) [\mathrm{C}+(1-\mathrm{C})\mathrm{S}_4](1-\mathrm{S}_{34}) [ C + ( 1 − C ) S 4 ] ( 1 − S 34 ) g g gg g g S 34 \mathrm{S}_{34} S 34 F R R , S U F_{\mathrm{RR},\mathrm{SU}} F RR , SU
d Φ c c ˉ + g F R R , S U = ∑ q = u , d , s ∫ u n r e s C { d Φ c c ˉ + q q ˉ F c c ˉ + q q ˉ ( 0 ) } (17) \mathrm{d}\Phi_{c\bar{c}+g}F_{\mathrm{RR},\mathrm{SU}} =
\sum_{q=u,d,s}\int_{\mathrm{unres}}\mathrm{C}\left\{\mathrm{d}\Phi_{c\bar{c}+q\bar{q}}F_{c\bar{c}+q\bar{q}}^{(0)}\right\} \tag*{(17)} d Φ c c ˉ + g F RR , SU = q = u , d , s ∑ ∫ unres C { d Φ c c ˉ + q q ˉ F c c ˉ + q q ˉ ( 0 ) } (17) + ∫ u n r e s [ C + ( 1 − C ) S 4 ] { d Φ c c ˉ + g g F c c ˉ + g g ( 0 ) } . {}+\int_{\mathrm{unres}}\left[\mathrm{C}+(1-\mathrm{C})\mathrm{S}_4\right]\left\{\mathrm{d}\Phi_{c\bar{c}+gg}F_{c\bar{c}+gg}^{(0)}\right\}. + ∫ unres [ C + ( 1 − C ) S 4 ] { d Φ c c ˉ + g g F c c ˉ + g g ( 0 ) } . The remaining momentum k k k
The pieces containing − S 34 -\mathrm{S}_{34} − S 34 − S k { d Φ c c ˉ + g F R R , S U } -\mathrm{S}_k\{\mathrm{d}\Phi_{c\bar{c}+g}F_{\mathrm{RR},\mathrm{SU}}\} − S k { d Φ c c ˉ + g F RR , SU } c c ˉ + g c\bar{c}+g c c ˉ + g R R D U \mathrm{RR}_{\mathrm{DU}} RR DU § 9 .§ 4.2 gives this domain split and the explicit relation (B.19), including the nested single-soft overlap in g g gg g g
Combining F R R , S U F_{\mathrm{RR},\mathrm{SU}} F RR , SU F R V , S U F_{\mathrm{RV},\mathrm{SU}} F RV , SU
S U = N ∫ [ ( 1 − S k ) { d Φ c c ˉ + g ( F R R , S U + F R V , S U ) } ] ϵ 0 . (18) \mathrm{SU} = \mathcal{N} \int\left[(1-\mathrm{S}_k)\left\{\mathrm{d}\Phi_{c\bar{c}+g}\left(F_{\mathrm{RR},\mathrm{SU}}+F_{\mathrm{RV},\mathrm{SU}}\right)\right\}\right]_{\epsilon_0}. \tag*{(18)} SU = N ∫ [ ( 1 − S k ) { d Φ c c ˉ + g ( F RR , SU + F RV , SU ) } ] ϵ 0 . (18) The two terms, each with this soft subtraction, define R R S U \mathrm{RR}_{\mathrm{SU}} RR SU R V S U \mathrm{RV}_{\mathrm{SU}} RV SU (13) . Their poles cancel both before and after the soft projection; the explicit expressions and both identities are given in appendix § 9 .§ 4.2 .
Born-phase-space contributions. The doubly integrated RR counterterms define F R R , D U F_{\mathrm{RR,DU}} F RR , DU
d Φ c c ˉ F R R , D U = ∑ a b ∫ u n r e s ( S 34 − C S > ) { d Φ c c ˉ + a b F c c ˉ + a b ( 0 ) } . (19) \mathrm{d}\Phi_{c\bar c}F_{\mathrm{RR,DU}}=\sum_{ab}\int_{\mathrm{unres}}(\mathrm{S}_{34}-\mathrm{CS}_{>})\{\mathrm{d}\Phi_{c\bar c+ab}F_{c\bar c+ab}^{(0)}\}. \tag*{(19)} d Φ c c ˉ F RR , DU = ab ∑ ∫ unres ( S 34 − CS > ) { d Φ c c ˉ + ab F c c ˉ + ab ( 0 ) } . (19) Both unresolved partons are integrated here. The symbol C S > \mathrm{CS}_{>} CS > (14) .
The soft counterterms of the RV finite remainder and infrared term sum to the soft limit of the full RV interference. Soft factorisation separates its integral into a term proportional to F c c ˉ ( 1 ) , f i n F_{c\bar c}^{(1),\mathrm{fin}} F c c ˉ ( 1 ) , fin F R V , F R F_{\mathrm{RV,FR}} F RV , FR F c c ˉ ( 0 ) F_{c\bar c}^{(0)} F c c ˉ ( 0 ) F R V , D U F_{\mathrm{RV,DU}} F RV , DU
∫ u n r e s S k { d Φ c c ˉ + g F c c ˉ + g ( 1 ) } = d Φ c c ˉ ( F R V , F R + F R V , D U ) . (20) \int_{\mathrm{unres}}\mathrm{S}_{k}\{\mathrm{d}\Phi_{c\bar c+g}F_{c\bar c+g}^{(1)}\}=\mathrm{d}\Phi_{c\bar c}(F_{\mathrm{RV,FR}}+F_{\mathrm{RV,DU}}). \tag*{(20)} ∫ unres S k { d Φ c c ˉ + g F c c ˉ + g ( 1 ) } = d Φ c c ˉ ( F RV , FR + F RV , DU ) . (20) Their explicit expressions are given in appendix C.3.2. Infrared factorisation gives the corresponding decomposition of the double-virtual contribution,
F c c ˉ ( 2 ) = F c c ˉ ( 2 ) , f i n + F V V , F R + F V V , D U , (21) F_{c\bar c}^{(2)}=F_{c\bar c}^{(2),\mathrm{fin}}+F_{\mathrm{VV,FR}}+F_{\mathrm{VV,DU}}, \tag*{(21)} F c c ˉ ( 2 ) = F c c ˉ ( 2 ) , fin + F VV , FR + F VV , DU , (21) where the FR and DU terms are proportional to F c c ˉ ( 1 ) , f i n F_{c\bar c}^{(1),\mathrm{fin}} F c c ˉ ( 1 ) , fin F c c ˉ ( 0 ) F_{c\bar c}^{(0)} F c c ˉ ( 0 )
F R = N ∫ [ d Φ c c ˉ ( F R V , F R + F V V , F R ) ] ϵ 0 . (22) \mathrm{FR}=\mathcal{N}\int\left[\mathrm{d}\Phi_{c\bar c}(F_{\mathrm{RV,FR}}+F_{\mathrm{VV,FR}})\right]_{\epsilon^0}. \tag*{(22)} FR = N ∫ [ d Φ c c ˉ ( F RV , FR + F VV , FR ) ] ϵ 0 . (22) D U = N ∫ [ d Φ c c ˉ ( F R R , D U + F R V , D U + F V V , D U ) ] ϵ 0 . (23) \mathrm{DU}=\mathcal{N}\int\left[\mathrm{d}\Phi_{c\bar c}(F_{\mathrm{RR,DU}}+F_{\mathrm{RV,DU}}+F_{\mathrm{VV,DU}})\right]_{\epsilon^0}. \tag*{(23)} DU = N ∫ [ d Φ c c ˉ ( F RR , DU + F RV , DU + F VV , DU ) ] ϵ 0 . (23) The separate regulated integrals of these F X , Y F_{X,Y} F X , Y N \mathcal{N} N X Y XY X Y (13) . In FR, the same integrated soft and virtual infrared factors as in U U U
V V F = N ∫ [ d Φ c c ˉ F c c ˉ ( 2 ) , f i n ] ϵ 0 . (24) \mathrm{VV}_{\mathrm{F}}=\mathcal{N}\int\left[\mathrm{d}\Phi_{c\bar c}F_{c\bar c}^{(2),\mathrm{fin}}\right]_{\epsilon^0}. \tag*{(24)} VV F = N ∫ [ d Φ c c ˉ F c c ˉ ( 2 ) , fin ] ϵ 0 . (24) The finite NNLO correction is therefore
σ ( 2 ) = R R F ( g g ) + R R F ( q q ˉ ) ⏟ c c ˉ + g g , c c ˉ + q q ˉ + R V F + S U ⏟ c c ˉ + g + V V F + F R + D U ⏟ c c ˉ . (25) \begin{aligned}
\sigma^{(2)}={}&\underbrace{\mathrm{RR}_{\mathrm{F}}(gg)+\mathrm{RR}_{\mathrm{F}}(q\bar q)}_{c\bar c+gg,\;c\bar c+q\bar q}\\
&+\underbrace{\mathrm{RV}_{\mathrm{F}}+\mathrm{SU}}_{c\bar c+g}+\underbrace{\mathrm{VV}_{\mathrm{F}}+\mathrm{FR}+\mathrm{DU}}_{c\bar c}. \tag*{(25)}
\end{aligned} σ ( 2 ) = c c ˉ + g g , c c ˉ + q q ˉ RR F ( g g ) + RR F ( q q ˉ ) + c c ˉ + g RV F + SU + c c ˉ VV F + FR + DU . (25) The phase-space mappings in appendix D keep P P P
Numerical implementation
Input parameters and approximation of the virtual terms
The input parameters used in the numerical calculation are listed in table 1. We take α s ( 5 ) ( M Z ) = 0.1180 \alpha_s^{(5)}(M_Z) = 0.1180 α s ( 5 ) ( M Z ) = 0.1180 M Z = 91.1876 M_Z = 91.1876 M Z = 91.1876 m ‾ b ( m ‾ b ) = 4.18 \overline{m}_b(\overline{m}_b) = 4.18 m b ( m b ) = 4.18 [49 ] . The coupling is evolved with three-loop running and two-loop flavour matching at μ b = m ‾ b ( m ‾ b ) \mu_b = \overline{m}_b(\overline{m}_b) μ b = m b ( m b ) [50 ] . This gives α s ( 4 ) ( 2 m c ) = 0.2528 \alpha_s^{(4)}(2m_c) = 0.2528 α s ( 4 ) ( 2 m c ) = 0.2528
For the wave function, we adopt the central value ⟨ O 1 ⟩ J / ψ = 0.440 \langle\mathcal{O}_1 \rangle_{J/\psi} = 0.440 ⟨ O 1 ⟩ J / ψ = 0.440 3 ^3 3 [23 ] . This value was extracted in ref. [51 ] , including NLO QCD and relativistic corrections. We use the same running coupling and matrix element at all three perturbative orders to isolate the successive QCD corrections, as in ref. [23 ] .
Table 1. Central input parameters.
For the finite virtual contributions V F V_{\mathrm F} V F V V F \mathrm{VV}_{\mathrm F} VV F s V = 10.5 \sqrt{s_V} = 10.5 s V = 10.5 s = 10.6 \sqrt{s} = 10.6 s = 10.6 s V / m c 2 = 49 s_V/m_c^2 = 49 s V / m c 2 = 49 s \sqrt{s} s [10 ] within 0.5% for matched inputs (§ 4.3 ), providing a lower-order check of the approximation used here.
Matrix elements and phase-space integration
Virtual contributions. We reduce the loop integrals to master integrals with Kira 3.1 [52 ] and evaluate them with AMFlow [53 ] . Values at successive phase-space points are obtained with AMFlow’s internal differential-equation solver. The finite virtual contributions V F V_{\mathrm F} V F V V F \mathrm{VV}_{\mathrm F} VV F
Real radiation and integrated counterterms. The real-emission squared matrix elements and one-loop–tree interference are obtained with an in-house program written in FORM [54 ] . We simplify their rational expressions with the pfd.lib library [55 ] in Singular [56 ] . Common subexpression elimination allows repeated numerator and denominator factors to be evaluated once and reused, reducing the cost of the phase-space integration. The tree-level squared matrix elements agree with independent results generated with FDC [57 ] . We evaluate the scalar one-loop integrals in the real-emission calculation using the dimension-changing transformation (DCT) [58 ] . The implementation uses Arb/Acb, a C library for arbitrary-precision real and complex ball arithmetic [59 ] , to evaluate the integrals and loop interferences with adjustable precision and propagated numerical error bounds. The phase-space integrals of the Born, real-emission and integrated-counterterm contributions are evaluated with VEGAS+ [60 ] .
Energy and momentum spectra. We first compute the quarkonium energy spectrum in bins of E ψ = Q ⋅ P / s E_{\psi}=Q\cdot P/\sqrt{s} E ψ = Q ⋅ P / s d σ / d ∣ P ∣ d\sigma/d|P| d σ / d ∣ P ∣ § 11 .§ 5 .
Numerical checks and uncertainty estimates
Lower-order benchmarks. To compare with Gong and Wang [10 ] , we use their input parameters, including α s = 0.259 \alpha_s=0.259 α s = 0.259 ∣ R ( 0 ) ∣ 2 = 0.944 G e V 3 |R(0)|^2=0.944\ \mathrm{GeV}^3 ∣ R ( 0 ) ∣ 2 = 0.944 GeV 3 J / ψ + c c ˉ J/\psi+c\bar{c} J / ψ + c c ˉ
Monte Carlo uncertainties. We estimate the combined Monte Carlo uncertainty conservatively by adding the standard errors of the individual contributions linearly. At the central scale, this gives 0.00042 pb for the NNLO total cross section, or 0.13% of the total and 0.83% of the NNLO correction, with R V F \mathrm{RV}_{\mathrm{F}} RV F σ L \sigma_{\mathrm{L}} σ L
Virtual-integration errors. We estimate the phase-space integration errors in V F \mathrm{V}_{\mathrm{F}} V F V V F \mathrm{VV}_{\mathrm{F}} VV F [61 ] . Combining the inner and outer estimates and summing over the separately integrated terms gives 1.2 × 10 − 3 1.2\times10^{-3} 1.2 × 1 0 − 3 Table 3 , with their definition and sum given in appendix § 11 .§ 5 .
Table 3. Table 3. Finite contributions at μ R = 2 m c \mu_R = 2m_c μ R = 2 m c ± \pm ±
Spectrum checks. At NNLO, the relative Monte Carlo uncertainty estimates per energy bin have a median of 1.4% and a maximum of 3.0%. The integrated spectrum agrees with the total cross section within the conservative Monte Carlo uncertainty estimates. To check the sensitivity to interpolation, we replace the cubic spline for V F \mathrm{V}_{\mathrm{F}} V F V V F \mathrm{VV}_{\mathrm{F}} VV F E ψ = 3.00 E_{\psi}=3.00 E ψ = 3.00 3.10 G e V 3.10\ \mathrm{GeV} 3.10 GeV § 11 .§ 5 ).
Phenomenological results
The central scale is μ R = 2 m c \mu_R = 2m_c μ R = 2 m c m c ≤ μ R ≤ 4 m c m_c \le\mu_R \le4m_c m c ≤ μ R ≤ 4 m c μ Λ = m c \mu_\Lambda= m_c μ Λ = m c α s ( 4 ) ( μ R ) \alpha_s^{(4)}(\mu_R) α s ( 4 ) ( μ R )
Experimental samples and comparison procedure
Our prediction describes direct colour-singlet J / ψ J/\psi J / ψ J / ψ J/\psi J / ψ [11 ] . For the momentum spectrum, the main comparison uses Belle’s prompt open-charm sample, which excludes double charmonium but retains feed-down.
We digitise the experimental points and error bars from Belle’s published plots. For the angular distribution, we convert the efficiency-corrected yields to cross sections using Belle’s measured total rate; the plotted errors are obtained by applying the same conversion factor to the yield errors.
Integrated cross sections and NNLO corrections
Table 2 gives the fixed-order results. Writing α s = α s ( 4 ) ( μ R ) \alpha_s = \alpha_s^{(4)}(\mu_R) α s = α s ( 4 ) ( μ R ) L R = ln [ μ R / ( 2 m c ) ] L_R = \ln[\mu_R/(2m_c)] L R = ln [ μ R / ( 2 m c )]
Table 2. Results for σ \sigma σ σ L \sigma_L σ L α θ \alpha_\theta α θ μ R = 2 m c \mu_R = 2m_c μ R = 2 m c m c ≤ μ R ≤ 4 m c m_c \le\mu_R \le4m_c m c ≤ μ R ≤ 4 m c
σ N N L O ≃ σ L O [ 1 + ( 2.837 + 2.621 L R ) α s + ( 4.945 + 12.49 L R + 5.165 L R 2 ) α s 2 ] , σ L , N N L O ≃ σ L , L O [ 1 + ( 2.195 + 2.650 L R ) α s + ( 1.258 + 10.017 L R + 5.241 L R 2 ) α s 2 ] . (26) \begin{aligned}
\sigma_{\mathrm{NNLO}} &\simeq\sigma_{\mathrm{LO}}\left[1 + (2.837 + 2.621L_R)\alpha_s + (4.945 + 12.49L_R + 5.165L_R^2)\alpha_s^2\right],\\
\sigma_{L,\mathrm{NNLO}} &\simeq\sigma_{L,\mathrm{LO}}\left[1 + (2.195 + 2.650L_R)\alpha_s + (1.258 + 10.017L_R + 5.241L_R^2)\alpha_s^2\right]. \tag*{(26)}
\end{aligned} σ NNLO σ L , NNLO ≃ σ LO [ 1 + ( 2.837 + 2.621 L R ) α s + ( 4.945 + 12.49 L R + 5.165 L R 2 ) α s 2 ] , ≃ σ L , LO [ 1 + ( 2.195 + 2.650 L R ) α s + ( 1.258 + 10.017 L R + 5.241 L R 2 ) α s 2 ] . (26) where σ L O ≃ 2.4893 α s 2 p b \sigma_{\mathrm{LO}} \simeq2.4893\alpha_s^2\ \mathrm{pb} σ LO ≃ 2.4893 α s 2 pb σ L , L O ≃ 0.38866 α s 2 p b \sigma_{L,\mathrm{LO}} \simeq0.38866\alpha_s^2\ \mathrm{pb} σ L , LO ≃ 0.38866 α s 2 pb
At the central scale, the NLO cross section is 1.72 times the LO value; the NNLO correction is
σ ( 2 ) ≃ 0.0503 p b , σ ( 2 ) σ N L O ≃ 18.4 % . (27) \sigma^{(2)} \simeq0.0503\ \mathrm{pb}, \qquad\frac{\sigma^{(2)}}{\sigma_{\mathrm{NLO}}} \simeq18.4\%. \tag*{(27)} σ ( 2 ) ≃ 0.0503 pb , σ NLO σ ( 2 ) ≃ 18.4%. (27) Figure 2 shows the renormalisation-scale dependence of the LO, NLO and NNLO total cross sections. The relative scale variation decreases from − 29.3 % + 48.9 % ^{+48.9\%}_{-29.3\%} − 29.3% + 48.9% − 20.3 % + 12.7 % ^{+12.7\%}_{-20.3\%} − 20.3% + 12.7%
Figure 2. Renormalisation-scale dependence of the direct J / ψ + c c ˉ + X J/\psi+c\bar{c}+X J / ψ + c c ˉ + X m c = 1.5 m_c=1.5 m c = 1.5 μ R = 2 m c \mu_R=2m_c μ R = 2 m c
The NNLO cross section reaches its maximum at μ R ≃ 1.65 \mu_R \simeq1.65 μ R ≃ 1.65
( σ L O , σ N L O , σ N N L O ) ∣ μ R = 1.65 G e V ≃ ( 0.271 , 0.385 , 0.365 ) p b . (\sigma_{\mathrm{LO}},\sigma_{\mathrm{NLO}},\sigma_{\mathrm{NNLO}})\big|_{\mu_R=1.65\ \mathrm{GeV}} \simeq(0.271,\,0.385,\,0.365)\ \mathrm{pb}. ( σ LO , σ NLO , σ NNLO ) μ R = 1.65 GeV ≃ ( 0.271 , 0.385 , 0.365 ) pb . Relative to NLO, the NNLO cross section is 5.2% lower at this scale but 18.4% higher at the central scale.
Table 3 gives the Born and finite higher-order contributions. For V V F \mathrm{VV}_{F} VV F V V 20 \mathrm{VV}_{20} VV 20 V V 11 \mathrm{VV}_{11} VV 11
Belle’s charm-associated prompt cross section is 0.74 ± 0.08 0.74 \pm0.08 0.74 ± 0.08 − 0.08 + 0.09 {}^{+0.09}_{-0.08} − 0.08 + 0.09 [11 ] . The NNLO result in Table 2 gives the direct colour-singlet open-charm contribution to this rate. A new NNLO calculation at m c = 1.4 m_c=1.4 m c = 1.4 [10 ] . We form the ratio of their NLO cross sections at the same μ R \mu_R μ R ∣ R ( 0 ) ∣ 2 \lvert R(0)\rvert^2 ∣ R ( 0 ) ∣ 2 1
K m ( μ R ) = σ N L O ( m c = 1.4 G e V ; μ R ) σ N L O ( m c = 1.5 G e V ; μ R ) , K_m(\mu_R)=\frac{\sigma_{\mathrm{NLO}}(m_c=1.4\ \mathrm{GeV};\mu_R)}{\sigma_{\mathrm{NLO}}(m_c=1.5\ \mathrm{GeV};\mu_R)}, K m ( μ R ) = σ NLO ( m c = 1.5 GeV ; μ R ) σ NLO ( m c = 1.4 GeV ; μ R ) , σ N N L O e s t ( 1.4 G e V ; μ R ) = K m ( μ R ) σ N N L O ( 1.5 G e V ; μ R ) . (28) \sigma_{\mathrm{NNLO}}^{\mathrm{est}}(1.4\ \mathrm{GeV};\mu_R)=K_m(\mu_R)\,\sigma_{\mathrm{NNLO}}(1.5\ \mathrm{GeV};\mu_R). \tag*{(28)} σ NNLO est ( 1.4 GeV ; μ R ) = K m ( μ R ) σ NNLO ( 1.5 GeV ; μ R ) . (28) The NLO results give K m ≃ 1.40 K_m\simeq1.40 K m ≃ 1.40 μ R = 3 \mu_R=3 μ R = 3
For the open-charm channel, production of ψ ( 2 S ) \psi(2S) ψ ( 2 S ) J / ψ J/\psi J / ψ [10 ] . Table 4 summarises the effects of the lower mass and this feed-down estimate. Together they raise the estimated open-charm rate to about 0.59 pb at μ R = 3 \mu_R = 3 μ R = 3 μ R = 1.65 \mu_R = 1.65 μ R = 1.65
Table 4. Table 4. NNLO open-charm cross sections in pb. The m c = 1.4 m_c = 1.4 m c = 1.4 (28) ; ψ ( 2 S ) \psi(2S) ψ ( 2 S ) [10 ] .
To illustrate these effects on the distributions, we also show the NNLO result multiplied by 1.29 and by 1.29 K m ( μ R ) 1.29K_m(\mu_R) 1.29 K m ( μ R ) m c = 1.5 m_c = 1.5 m c = 1.5 μ R \mu_R μ R K m K_m K m
Production-angle distribution
For each fixed cos θ \cos\theta cos θ (4) with σ \sigma σ σ L \sigma_L σ L μ R \mu_R μ R m c ≤ μ R ≤ 4 m c m_c \le\mu_R \le4m_c m c ≤ μ R ≤ 4 m c μ R = 2 m c \mu_R = 2m_c μ R = 2 m c
d σ N N L O d cos θ ≃ ( 0.1366 + 0.0756 cos 2 θ ) p b . (29) \frac{d\sigma_{\mathrm{NNLO}}}{d\cos\theta} \simeq\left(0.1366 + 0.0756\cos^2\theta\right)\,\mathrm{pb}. \tag*{(29)} d cos θ d σ NNLO ≃ ( 0.1366 + 0.0756 cos 2 θ ) pb . (29) The central angular coefficient α θ \alpha_\theta α θ 0.523 ≤ α θ ≤ 0.649 0.523 \le\alpha_\theta\le0.649 0.523 ≤ α θ ≤ 0.649
Figure 3 compares the angular distribution in ∣ cos θ ∣ |\cos\theta| ∣ cos θ ∣ [11 ] . Belle measures α θ = − 0.2 6 − 0.22 + 0.24 \alpha_\theta= -0.26^{+0.24}_{-0.22} α θ = − 0.2 6 − 0.22 + 0.24 α θ \alpha_\theta α θ
Figure 3. Production-angle distributions compared with Belle prompt charm-associated data [11 ] : (a) direct production at m c = 1.5 m_c = 1.5 m c = 1.5 μ R = 3 \mu_R = 3 μ R = 3
Momentum distribution
For the energy spectrum, we vary μ R \mu_R μ R m c ≤ μ R ≤ 4 m c m_c \le\mu_R \le4m_c m c ≤ μ R ≤ 4 m c
Figure 4 compares the J / ψ J/\psi J / ψ [11 ] .2
Figure 4. Momentum spectra: (a) direct production at m c = 1.5 m_c = 1.5 m c = 1.5 Figure 3 , keeping the calculated spectral shape. Filled squares are Belle prompt open-charm data; open squares also include double charmonium [11 ] . Thick lines show μ R = 3 \mu_R = 3 μ R = 3
At the central scale, the NNLO correction raises the direct cross section near the peak, bringing it closer to Belle’s prompt open-charm data. The combined mass and feed-down rescaling brings the integrated rate close to the measured open-charm rate, but leaves the direct-production shape unchanged. This shape still gives a larger fraction of high-momentum J / ψ J/\psi J / ψ J / ψ J/\psi J / ψ
Conclusions
We have calculated the NNLO QCD corrections to direct colour-singlet J / ψ J/\psi J / ψ
For m c = 1.5 m_c = 1.5 m c = 1.5 s = 10.6 \sqrt{s} = 10.6 s = 10.6 0.32 4 − 0.066 + 0.041 0.324^{+0.041}_{-0.066} 0.32 4 − 0.066 + 0.041 m c ≤ μ R ≤ 4 m c m_c \le\mu_R \le4m_c m c ≤ μ R ≤ 4 m c μ Λ = m c \mu_\Lambda= m_c μ Λ = m c μ R = 2 m c \mu_R = 2m_c μ R = 2 m c − 20.3 % + 12.7 % ^{+12.7\%}_{-20.3\%} − 20.3% + 12.7%
Assuming that the NLO mass dependence persists at NNLO, lowering m c m_c m c ψ ( 2 S ) \psi(2S) ψ ( 2 S ) μ R = 3 \mu_R = 3 μ R = 3 μ R = 1.65 \mu_R = 1.65 μ R = 1.65
At the central scale, the NNLO correction raises the absolute momentum spectrum near the peak, bringing it closer to Belle’s prompt open-charm data. The predicted spectrum still gives a larger fraction of high-momentum J / ψ J/\psi J / ψ
By contrast, the NNLO correction shifts the angular shape further away from Belle’s prompt charm-associated data. At μ R = 2 m c \mu_R = 2m_c μ R = 2 m c α θ \alpha_\theta α θ α θ = − 0.26 \alpha_\theta= -0.26 α θ = − 0.26 α θ \alpha_\theta α θ − 0.031 + 0.096 ^{+0.096}_{-0.031} − 0.031 + 0.096
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant No. 12135013. Generative AI tools assisted with code development, analytic and numerical checks, and manuscript preparation.
Derivation of the angular reconstruction
We write the quarkonium energy as E ψ = Q ⋅ P / s E_\psi= Q \cdot P/\sqrt{s} E ψ = Q ⋅ P / s
Q = ( s , 0 ) , n = P ∣ P ∣ , n − = p − ∣ p − ∣ , cos θ = n ⋅ n − . Q = (\sqrt{s},0), \qquad n = \frac{P}{\lvert P\rvert}, \qquad n_- = \frac{p_-}{\lvert p_-\rvert}, \qquad\cos\theta= n \cdot n_-. Q = ( s , 0 ) , n = ∣ P ∣ P , n − = ∣ p − ∣ p − , cos θ = n ⋅ n − . At fixed quarkonium momentum P P P H μ ν ( E ψ , n ) H^{\mu\nu}(E_{\psi},\boldsymbol{n}) H μν ( E ψ , n )
d σ d E ψ d Ω n = 3 4 π s L μ ν H μ ν , L μ ν = p − μ p + ν + p + μ p − ν − s 2 g μ ν . \frac{d\sigma}{dE_{\psi}\,d\Omega_{\boldsymbol{n}}}=\frac{3}{4\pi s}L_{\mu\nu}H^{\mu\nu},
\qquad
L^{\mu\nu}=p_-^{\mu}p_+^{\nu}+p_+^{\mu}p_-^{\nu}-\frac{s}{2}g^{\mu\nu}. d E ψ d Ω n d σ = 4 π s 3 L μν H μν , L μν = p − μ p + ν + p + μ p − ν − 2 s g μν . Here L μ ν L^{\mu\nu} L μν (3) give the covariant contractions
H = Π μ ν H μ ν , H L = Π L , μ ν H μ ν , Q ⋅ e L = 0 , e L 2 = − 1. H=\Pi_{\mu\nu}H^{\mu\nu},
\qquad
H_L=\Pi_{L,\mu\nu}H^{\mu\nu},
\qquad
Q\cdot e_L=0,
\qquad
e_L^2=-1. H = Π μν H μν , H L = Π L , μν H μν , Q ⋅ e L = 0 , e L 2 = − 1. Since both projectors depend only on Q Q Q P P P Π μ ν → Π L μ ν \Pi^{\mu\nu}\to\Pi_L^{\mu\nu} Π μν → Π L μν
In the centre-of-mass frame, current conservation, Q μ H μ ν = Q ν H μ ν = 0 Q_{\mu}H^{\mu\nu}=Q_{\nu}H^{\mu\nu}=0 Q μ H μν = Q ν H μν = 0 n \boldsymbol{n} n [10 ] ,
H i j ( E ψ , n ) = H T ( E ψ ) 2 ( δ i j − n i n j ) + H L ( E ψ ) n i n j , H^{ij}(E_{\psi},\boldsymbol{n})=\frac{H_T(E_{\psi})}{2}\left(\delta^{ij}-n^i n^j\right)+H_L(E_{\psi})n^i n^j, H ij ( E ψ , n ) = 2 H T ( E ψ ) ( δ ij − n i n j ) + H L ( E ψ ) n i n j , H ( E ψ ) = δ i j H i j = H T ( E ψ ) + H L ( E ψ ) , H L ( E ψ ) = n i n j H i j . (30) H(E_{\psi})=\delta_{ij}H^{ij}=H_T(E_{\psi})+H_L(E_{\psi}),
\qquad
H_L(E_{\psi})=n_i n_j H^{ij}. \tag*{(30)} H ( E ψ ) = δ ij H ij = H T ( E ψ ) + H L ( E ψ ) , H L ( E ψ ) = n i n j H ij . (30) The contributions from the two photon polarisations perpendicular to P P P H T H_T H T e L μ = ( 0 , n ) e_L^{\mu}=(0,\boldsymbol{n}) e L μ = ( 0 , n )
Using L i j = s 2 ( δ i j − n i n j ) L^{ij}=\frac{s}{2}(\delta^{ij}-n^i n^j) L ij = 2 s ( δ ij − n i n j )
d 2 σ d E ψ d cos θ = 3 8 ( 1 + cos 2 θ ) H T ( E ψ ) + 3 4 ( 1 − cos 2 θ ) H L ( E ψ ) . (31) \frac{d^2\sigma}{dE_{\psi}\,d\cos\theta}=\frac{3}{8}(1+\cos^2\theta)H_T(E_{\psi})+\frac{3}{4}(1-\cos^2\theta)H_L(E_{\psi}). \tag*{(31)} d E ψ d cos θ d 2 σ = 8 3 ( 1 + cos 2 θ ) H T ( E ψ ) + 4 3 ( 1 − cos 2 θ ) H L ( E ψ ) . (31) Both angular weights integrate to unity over − 1 ≤ cos θ ≤ 1 -1\le\cos\theta\le1 − 1 ≤ cos θ ≤ 1 H T + H L = d σ / d E ψ = H H_T+H_L=d\sigma/dE_{\psi}=H H T + H L = d σ / d E ψ = H H T = H − H L H_T=H-H_L H T = H − H L
d 2 σ d E ψ d cos θ = 3 8 ( 1 + cos 2 θ ) H ( E ψ ) + 3 8 ( 1 − 3 cos 2 θ ) H L ( E ψ ) . (32) \frac{d^2\sigma}{dE_{\psi}\,d\cos\theta}=\frac{3}{8}(1+\cos^2\theta)H(E_{\psi})+\frac{3}{8}(1-3\cos^2\theta)H_L(E_{\psi}). \tag*{(32)} d E ψ d cos θ d 2 σ = 8 3 ( 1 + cos 2 θ ) H ( E ψ ) + 8 3 ( 1 − 3 cos 2 θ ) H L ( E ψ ) . (32) The two projected cross sections are
σ = ∫ d E ψ H ( E ψ ) , σ L = ∫ d E ψ H L ( E ψ ) . \sigma=\int dE_{\psi}\,H(E_{\psi}),
\qquad
\sigma_L=\int dE_{\psi}\,H_L(E_{\psi}). σ = ∫ d E ψ H ( E ψ ) , σ L = ∫ d E ψ H L ( E ψ ) . Integrating (A.3) over E ψ E_{\psi} E ψ (4) .
Finite contributions and infrared pole cancellation
This appendix gives the integrated kernels and pole cancellations for the finite contributions defined in § 3 . The local counterterms and their unresolved integrals are derived in appendix § 10 .
Kinematics and phase-space conventions
We define the recoil momentum, its invariant mass and four-velocity, and the heavy-quark speed at Born kinematics by
K = Q − P , W = K 2 , u K = K / W , β = 1 − 4 m c 2 K 2 . (33) K = Q - P,\qquad W = \sqrt{K^2},\qquad u_K = K/W,\qquad\beta= \sqrt{1-\frac{4m_c^2}{K^2}}. \tag*{(33)} K = Q − P , W = K 2 , u K = K / W , β = 1 − K 2 4 m c 2 . (33) A superscript ∗ \ast ∗ K K K E k ∗ = u K ⋅ k E_k^\ast= u_K \cdot k E k ∗ = u K ⋅ k
For an emitted massless parton with momentum k k k ( K − k ) 2 = K 2 − 2 W E k ∗ ≥ 4 m c 2 (K-k)^2 = K^2 - 2W E_k^\ast\ge4m_c^2 ( K − k ) 2 = K 2 − 2 W E k ∗ ≥ 4 m c 2
E k ∗ ≤ E 0 ∗ ( K ) , E 0 ∗ ( K ) = K 2 − 4 m c 2 2 W = W β 2 2 . (34) E_k^\ast\le E_0^\ast(K),\qquad E_0^\ast(K) = \frac{K^2-4m_c^2}{2W} = \frac{W\beta^2}{2}. \tag*{(34)} E k ∗ ≤ E 0 ∗ ( K ) , E 0 ∗ ( K ) = 2 W K 2 − 4 m c 2 = 2 W β 2 . (34) We include this kinematic bound in the massless one-particle phase-space element,
[ d k ] K = d d k ( 2 π ) d − 1 δ + ( k 2 ) θ ( E 0 ∗ − u K ⋅ k ) . (35) [dk]_K = \frac{d^d k}{(2\pi)^{d-1}}\delta_+(k^2)\theta(E_0^\ast-u_K\cdot k). \tag*{(35)} [ d k ] K = ( 2 π ) d − 1 d d k δ + ( k 2 ) θ ( E 0 ∗ − u K ⋅ k ) . (35) Here δ + ( k 2 ) = θ ( k 0 ) δ ( k 2 ) \delta_+(k^2)=\theta(k^0)\delta(k^2) δ + ( k 2 ) = θ ( k 0 ) δ ( k 2 )
To separate the spectator kinematics from the coloured final state X X X K K K
d Φ J / ψ + X ( Q ) = d K 2 2 π d Φ 2 ( Q ; P , K ) d Φ X ( K ) . (36) d\Phi_{J/\psi+X}(Q)=\frac{dK^2}{2\pi}\,d\Phi_2(Q;P,K)\,d\Phi_X(K). \tag*{(36)} d Φ J / ψ + X ( Q ) = 2 π d K 2 d Φ 2 ( Q ; P , K ) d Φ X ( K ) . (36) Here d Φ X ( K ) d\Phi_X(K) d Φ X ( K ) J / ψ + X J/\psi+X J / ψ + X
d Φ B = d Φ c c ˉ , d Φ R = d Φ c c ˉ + g , d Φ R R a b = d Φ c c ˉ + a b . d\Phi_B=d\Phi_{c\bar c},\qquad d\Phi_R=d\Phi_{c\bar c+g},\qquad d\Phi_{RR}^{ab}=d\Phi_{c\bar c+ab}. d Φ B = d Φ c c ˉ , d Φ R = d Φ c c ˉ + g , d Φ R R ab = d Φ c c ˉ + ab . The soft and collinear projections act on the squared matrix elements and interferences together with their phase space, at fixed P P P K K K E 0 ∗ ( K ) E_0^\ast(K) E 0 ∗ ( K )
All counterterms use the dimensional factor C ϵ = ( μ R 2 e γ E / 4 π ) ϵ C_\epsilon=(\mu_R^2e^{\gamma_E}/4\pi)^\epsilon C ϵ = ( μ R 2 e γ E /4 π ) ϵ γ E \gamma_E γ E
Virtual amplitudes and infrared factors
The colour factors are C F = ( N c 2 − 1 ) / ( 2 N c ) = 4 / 3 C_F=(N_c^2-1)/(2N_c)=4/3 C F = ( N c 2 − 1 ) / ( 2 N c ) = 4/3 C A = N c = 3 C_A=N_c=3 C A = N c = 3 T F = 1 / 2 T_F=1/2 T F = 1/2 n l = 3 n_l=3 n l = 3 n f = 4 n_f=4 n f = 4 β 0 , Q \beta_{0,Q} β 0 , Q
β 0 ( n f ) = 11 3 C A − 4 3 T F n f , β 0 , Q = − 4 3 T F = − 2 3 , \beta_0^{(n_f)}=\frac{11}{3}C_A-\frac{4}{3}T_F n_f,\qquad\beta_{0,Q}=-\frac{4}{3}T_F=-\frac{2}{3}, β 0 ( n f ) = 3 11 C A − 3 4 T F n f , β 0 , Q = − 3 4 T F = − 3 2 , β 0 ( 3 ) = 9 , β 0 ( 4 ) = β 0 ( 3 ) + β 0 , Q = 25 3 . (37) \beta_0^{(3)}=9,\qquad\beta_0^{(4)}=\beta_0^{(3)}+\beta_{0,Q}=\frac{25}{3}. \tag*{(37)} β 0 ( 3 ) = 9 , β 0 ( 4 ) = β 0 ( 3 ) + β 0 , Q = 3 25 . (37) Coupling renormalisation uses β 0 ( 4 ) \beta_0^{(4)} β 0 ( 4 ) β 0 ( 3 ) \beta_0^{(3)} β 0 ( 3 ) β 0 , Q \beta_{0,Q} β 0 , Q Z 2 Z_2 Z 2 Z A ( 1 ) Z_A^{(1)} Z A ( 1 ) [42 ] .
NRQCD matching. The colour-singlet vector-current renormalisation gives the NRQCD subtraction term [46 , 47 , 48 ]
M N R Q C D = − α s 2 4 ϵ ( C F 2 3 + C F C A 2 ) ( μ Λ 2 e γ E 4 π μ R 2 ) − 2 ϵ M ( 0 ) . (38) \mathcal{M}_{\mathrm{NRQCD}}=-\frac{\alpha_s^2}{4\epsilon}\left(\frac{C_F^2}{3}+\frac{C_F C_A}{2}\right)\left(\frac{\mu_\Lambda^2 e^{\gamma_E}}{4\pi\mu_R^2}\right)^{-2\epsilon}\mathcal{M}^{(0)}. \tag*{(38)} M NRQCD = − 4 ϵ α s 2 ( 3 C F 2 + 2 C F C A ) ( 4 π μ R 2 μ Λ 2 e γ E ) − 2 ϵ M ( 0 ) . (38) Through NNLO, the matched amplitude is M = M Q C D , r e n − M N R Q C D \mathcal{M}=\mathcal{M}_{\mathrm{QCD,ren}}-\mathcal{M}_{\mathrm{NRQCD}} M = M QCD , ren − M NRQCD M Q C D , r e n \mathcal{M}_{\mathrm{QCD,ren}} M QCD , ren C F 2 / 3 + C F C A / 2 = 70 / 27 C_F^2/3+C_F C_A/2=70/27 C F 2 /3 + C F C A /2 = 70/27
Infrared factors and finite remainders. We separate the remaining QCD infrared poles from the finite remainders following ref. [42 ] , section 2.3. For the Born channel, we abbreviate M c c ˉ ( ℓ ) \mathcal{M}_{c\bar c}^{(\ell)} M c c ˉ ( ℓ ) M ( ℓ ) \mathcal{M}^{(\ell)} M ( ℓ )
M f i n ( 1 ) = M ( 1 ) − Z 1 M ( 0 ) , \mathcal{M}_{\mathrm{fin}}^{(1)}=\mathcal{M}^{(1)}-Z_1\mathcal{M}^{(0)}, M fin ( 1 ) = M ( 1 ) − Z 1 M ( 0 ) , M f i n ( 2 ) = M ( 2 ) − Z 1 M f i n ( 1 ) − Z 2 M ( 0 ) . (39) \mathcal{M}_{\mathrm{fin}}^{(2)}=\mathcal{M}^{(2)}-Z_1\mathcal{M}_{\mathrm{fin}}^{(1)}-Z_2\mathcal{M}^{(0)}. \tag*{(39)} M fin ( 2 ) = M ( 2 ) − Z 1 M fin ( 1 ) − Z 2 M ( 0 ) . (39) In terms of the heavy-pair anomalous-dimension coefficients Γ n = C F γ c u s p , Q ( n ) ( v 12 ) + 2 γ Q ( n ) \Gamma_n=C_F\gamma_{\mathrm{cusp},Q}^{(n)}(v_{12})+2\gamma_Q^{(n)} Γ n = C F γ cusp , Q ( n ) ( v 12 ) + 2 γ Q ( n ) v 12 = 1 − m c 4 / ( p 1 ⋅ p 2 ) 2 v_{12}=\sqrt{1-m_c^4/(p_1\cdot p_2)^2} v 12 = 1 − m c 4 / ( p 1 ⋅ p 2 ) 2 [42 , 62 , 63 ]
Z 1 = Γ 0 2 ϵ , Z_1=\frac{\Gamma_0}{2\epsilon}, Z 1 = 2 ϵ Γ 0 , Z 2 = Γ 0 ( Γ 0 − 2 β 0 ( 3 ) ) 8 ϵ 2 + Γ 1 4 ϵ + β 0 , Q Γ 0 2 ϵ ln μ R 2 m c 2 . (40) Z_2=\frac{\Gamma_0(\Gamma_0-2\beta_0^{(3)})}{8\epsilon^2}+\frac{\Gamma_1}{4\epsilon}+\frac{\beta_{0,Q}\Gamma_0}{2\epsilon}\ln\frac{\mu_R^2}{m_c^2}. \tag*{(40)} Z 2 = 8 ϵ 2 Γ 0 ( Γ 0 − 2 β 0 ( 3 ) ) + 4 ϵ Γ 1 + 2 ϵ β 0 , Q Γ 0 ln m c 2 μ R 2 . (40) The anomalous dimensions are expanded as γ i ( α s ) = ∑ n ≥ 0 a s n + 1 γ i ( n ) \gamma_i(\alpha_s)=\sum_{n\geq0}a_s^{n+1}\gamma_i^{(n)} γ i ( α s ) = ∑ n ≥ 0 a s n + 1 γ i ( n )
γ c u s p , Q ( 0 ) ( v ) = 4 v [ 1 2 ln 1 + v 1 − v − i π ] , γ Q ( 0 ) = − 2 C F , \gamma_{\mathrm{cusp},Q}^{(0)}(v)=\frac{4}{v}\left[\frac{1}{2}\ln\frac{1+v}{1-v}-i\pi\right],\qquad\gamma_Q^{(0)}=-2C_F, γ cusp , Q ( 0 ) ( v ) = v 4 [ 2 1 ln 1 − v 1 + v − iπ ] , γ Q ( 0 ) = − 2 C F , γ Q ( 1 ) = C F C A ( 2 π 2 3 − 98 9 − 4 ζ 3 ) + 40 9 C F T F n l . (41) \gamma_Q^{(1)}=C_FC_A\left(\frac{2\pi^2}{3}-\frac{98}{9}-4\zeta_3\right)+\frac{40}{9}C_FT_Fn_l. \tag*{(41)} γ Q ( 1 ) = C F C A ( 3 2 π 2 − 9 98 − 4 ζ 3 ) + 9 40 C F T F n l . (41) The two-loop massive cusp coefficient γ c u s p , Q ( 1 ) ( v ) \gamma_{\mathrm{cusp},Q}^{(1)}(v) γ cusp , Q ( 1 ) ( v ) § 12 .§ 5 . Here and below, ζ n = ζ ( n ) \zeta_n=\zeta(n) ζ n = ζ ( n )
Born and finite virtual contributions. We abbreviate F c c ˉ ( 0 ) F_{c\bar c}^{(0)} F c c ˉ ( 0 ) F c c ˉ ( 1 ) , f i n F_{c\bar c}^{(1),\mathrm{fin}} F c c ˉ ( 1 ) , fin F c c ˉ ( 2 ) , f i n F_{c\bar c}^{(2),\mathrm{fin}} F c c ˉ ( 2 ) , fin F B F_B F B F V f i n F_V^{\mathrm{fin}} F V fin F V V f i n F_{VV}^{\mathrm{fin}} F V V fin
F B = ∣ M ( 0 ) ∣ 2 , F V f i n = 2 a s Re [ M ( 0 ) ∗ M f i n ( 1 ) ] , F V V f i n = a s 2 ( 2 Re [ M ( 0 ) ∗ M f i n ( 2 ) ] + ∣ M f i n ( 1 ) ∣ 2 ) . (42) \begin{aligned}
F_B &= \lvert\mathcal{M}^{(0)}\rvert^2,\\
F_V^{\mathrm{fin}} &= 2a_s\operatorname{Re}\left[\mathcal{M}^{(0)*}\mathcal{M}_{\mathrm{fin}}^{(1)}\right],\\
F_{VV}^{\mathrm{fin}} &= a_s^2\left(2\operatorname{Re}\left[\mathcal{M}^{(0)*}\mathcal{M}_{\mathrm{fin}}^{(2)}\right]+\lvert\mathcal{M}_{\mathrm{fin}}^{(1)}\rvert^2\right). \tag*{(42)}
\end{aligned} F B F V fin F V V fin = ∣ M ( 0 ) ∣ 2 , = 2 a s Re [ M ( 0 ) ∗ M fin ( 1 ) ] , = a s 2 ( 2 Re [ M ( 0 ) ∗ M fin ( 2 ) ] + ∣ M fin ( 1 ) ∣ 2 ) . (42) The finite double-virtual contribution consists of
F V V = V V 20 + V V 11 , F_{VV}=VV_{20}+VV_{11}, F V V = V V 20 + V V 11 , V V 20 = 2 a s 2 N ∫ [ d Φ B Re ( M ( 0 ) ∗ M f i n ( 2 ) ) ] ϵ 0 , V V 11 = a s 2 N ∫ [ d Φ B ∣ M f i n ( 1 ) ∣ 2 ] ϵ 0 . (43) VV_{20}=2a_s^2\mathcal{N}\int\left[\mathrm{d}\Phi_B\operatorname{Re}\left(\mathcal{M}^{(0)*}\mathcal{M}_{\mathrm{fin}}^{(2)}\right)\right]_{\epsilon_0},\qquad VV_{11}=a_s^2\mathcal{N}\int\left[\mathrm{d}\Phi_B\lvert\mathcal{M}_{\mathrm{fin}}^{(1)}\rvert^2\right]_{\epsilon_0}. \tag*{(43)} V V 20 = 2 a s 2 N ∫ [ d Φ B Re ( M ( 0 ) ∗ M fin ( 2 ) ) ] ϵ 0 , V V 11 = a s 2 N ∫ [ d Φ B ∣ M fin ( 1 ) ∣ 2 ] ϵ 0 . (43) The two terms are the finite two-loop–tree interference and squared one-loop finite remainder. The infrared terms generated by (B.7) and (B.8) are
F V , U = 2 a s Re Z 1 F B , F_{V,U}=2a_s\operatorname{Re}Z_1F_B, F V , U = 2 a s Re Z 1 F B , F V V , F R = 2 a s Re Z 1 F V f i n , F_{VV,\mathrm{FR}}=2a_s\operatorname{Re}Z_1F_V^{\mathrm{fin}}, F V V , FR = 2 a s Re Z 1 F V fin , F V V , D U = a s 2 ( 2 Re Z 2 + ∣ Z 1 ∣ 2 ) F B . (44) F_{VV,\mathrm{DU}}=a_s^2\left(2\operatorname{Re}Z_2+\lvert Z_1\rvert^2\right)F_B. \tag*{(44)} F V V , DU = a s 2 ( 2 Re Z 2 + ∣ Z 1 ∣ 2 ) F B . (44) Real–virtual contribution. For J / ψ + c c ˉ + g J/\psi+c\bar c+g J / ψ + c c ˉ + g F R = F c c ˉ + g ( 0 ) F_R=F_{c\bar c+g}^{(0)} F R = F c c ˉ + g ( 0 ) F R V = F c c ˉ + g ( 1 ) F_{RV}=F_{c\bar c+g}^{(1)} F R V = F c c ˉ + g ( 1 ) F R V f i n = F c c ˉ + g ( 1 ) , f i n F_{RV}^{\mathrm{fin}}=F_{c\bar c+g}^{(1),\mathrm{fin}} F R V fin = F c c ˉ + g ( 1 ) , fin
F R V = 2 a s Re Z c c ˉ g ( 1 ) F R + F R V f i n . (45) F_{RV}=2a_s\operatorname{Re}Z_{c\bar cg}^{(1)}F_R+F_{RV}^{\mathrm{fin}}. \tag*{(45)} F R V = 2 a s Re Z c c ˉ g ( 1 ) F R + F R V fin . (45) The first term is F R V , S U F_{RV,\mathrm{SU}} F R V , SU (15) , with the one-loop infrared factor [42 ]
Z c c ˉ g ( 1 ) = − C A γ c u s p ( 0 ) 4 ϵ 2 + 1 2 ϵ [ γ g ( 0 ) + 2 γ Q ( 0 ) + ( C F − C A 2 ) γ c u s p , Q ( 0 ) ( v 12 ) − C A γ c u s p ( 0 ) 2 ( ln m c μ R 2 p 1 ⋅ k + ln m c μ R 2 p 2 ⋅ k + 2 i π ) ] , (46) \begin{aligned}
Z_{c\bar cg}^{(1)}={}&-\frac{C_A\gamma_{\mathrm{cusp}}^{(0)}}{4\epsilon^2}
+\frac{1}{2\epsilon}\left[\gamma_g^{(0)}+2\gamma_Q^{(0)}+\left(C_F-\frac{C_A}{2}\right)\gamma_{\mathrm{cusp},Q}^{(0)}(v_{12})\right.\\
&\left.-\frac{C_A\gamma_{\mathrm{cusp}}^{(0)}}{2}\left(\ln\frac{m_c\mu_R}{2p_1\cdot k}+\ln\frac{m_c\mu_R}{2p_2\cdot k}+2i\pi\right)\right], \tag*{(46)}
\end{aligned} Z c c ˉ g ( 1 ) = − 4 ϵ 2 C A γ cusp ( 0 ) + 2 ϵ 1 [ γ g ( 0 ) + 2 γ Q ( 0 ) + ( C F − 2 C A ) γ cusp , Q ( 0 ) ( v 12 ) − 2 C A γ cusp ( 0 ) ( ln 2 p 1 ⋅ k m c μ R + ln 2 p 2 ⋅ k m c μ R + 2 iπ ) ] , (46) where γ c u s p ( 0 ) = 4 \gamma_{\mathrm{cusp}}^{(0)}=4 γ cusp ( 0 ) = 4 γ g ( 0 ) = − β 0 ( 3 ) \gamma_g^{(0)}=-\beta_0^{(3)} γ g ( 0 ) = − β 0 ( 3 )
NLO contributions and pole cancellation
In the soft counterterm, omitting k k k d Φ B [ d k ] K \mathrm{d}\Phi_B[\mathrm{d}k]_K d Φ B [ d k ] K
R F = N ∫ [ d Φ R ( k ) F R − d Φ B [ d k ] K S k F R ] ϵ 0 . (47) R_F=\mathcal{N}\int\left[\mathrm{d}\Phi_R(k)\,F_R-\mathrm{d}\Phi_B[\mathrm{d}k]_K\,S_kF_R\right]_{\epsilon_0}. \tag*{(47)} R F = N ∫ [ d Φ R ( k ) F R − d Φ B [ d k ] K S k F R ] ϵ 0 . (47) The soft counterterm is specified in appendix C.1, and the common phase-space parametrisation of the two terms in appendix D.2. Integrating the counterterm in d d d I S I_S I S
F R , U = ∫ [ d k ] K S k F R = a s C F I S F B . F_{R,U}=\int[\mathrm{d}k]_K\,S_kF_R=a_sC_FI_SF_B. F R , U = ∫ [ d k ] K S k F R = a s C F I S F B . At Born kinematics, v 12 = 2 β / ( 1 + β 2 ) v_{12}=2\beta/(1+\beta^2) v 12 = 2 β / ( 1 + β 2 ) A ( β ) A(\beta) A ( β )
C F I S = C F A ( β ) ϵ + O ( ϵ 0 ) , 2 R e Z 1 = R e Γ 0 ϵ = − C F A ( β ) ϵ . (48) C_F I_S = \frac{C_F A(\beta)}{\epsilon}+O(\epsilon^0),
\qquad
2\mathop{\mathrm{Re}} Z_1=\frac{\mathop{\mathrm{Re}}\Gamma_0}{\epsilon}=-\frac{C_F A(\beta)}{\epsilon}. \tag*{(48)} C F I S = ϵ C F A ( β ) + O ( ϵ 0 ) , 2 Re Z 1 = ϵ Re Γ 0 = − ϵ C F A ( β ) . (48) The pole part, denoted by [ ⋅ ] p o l e s [\cdot]_{\mathrm{poles}} [ ⋅ ] poles
[ C F I S + 2 R e Z 1 ] p o l e s = 0. (49) \left[C_F I_S+2\mathop{\mathrm{Re}} Z_1\right]_{\mathrm{poles}}=0. \tag*{(49)} [ C F I S + 2 Re Z 1 ] poles = 0. (49) NNLO contributions and pole cancellation
The pole-cancellation identities below hold separately for the two photon projections.
Subtracted real contributions
For the two double-real channels we abbreviate F c c ˉ + q q ˉ ( 0 ) F_{c\bar{c}+q\bar{q}}^{(0)} F c c ˉ + q q ˉ ( 0 ) F c c ˉ + g g ( 0 ) F_{c\bar{c}+gg}^{(0)} F c c ˉ + g g ( 0 ) F q q ˉ F_{q\bar{q}} F q q ˉ F g g F_{gg} F g g (14) are specified in appendices C.2.1 and C.2.2; their phase-space parametrisations are given in appendices D.3 and D.4, respectively. The integral for R V F \mathrm{RV}F RV F F R F_R F R F R V f i n F_{\mathrm{RV}}^{\mathrm{fin}} F RV fin
Single-unresolved contribution (SU)
The integrated counterterm defined in (17) is F R R , S U = I R R , S U F R F_{\mathrm{RR,SU}}=I_{\mathrm{RR,SU}}F_R F RR , SU = I RR , SU F R
I R R , S U ( k , ϵ ) = g s 2 K S 4 + C g g ( E k ∗ , ϵ ) + n l C q q ˉ ( E k ∗ , ϵ ) − C C S 4 ( E k ∗ , ϵ ) . (50) I_{\mathrm{RR,SU}}(k,\epsilon)=g_s^2\mathcal{K}_{S_4}+C_{gg}(E_k^*,\epsilon)+n_l C_{q\bar{q}}(E_k^*,\epsilon)-C_{\mathrm{CS}_4}(E_k^*,\epsilon). \tag*{(50)} I RR , SU ( k , ϵ ) = g s 2 K S 4 + C g g ( E k ∗ , ϵ ) + n l C q q ˉ ( E k ∗ , ϵ ) − C CS 4 ( E k ∗ , ϵ ) . (50) Here g s 2 K S 4 g_s^2\mathcal{K}_{S_4} g s 2 K S 4 C g g C_{gg} C g g C q q ˉ C_{q\bar{q}} C q q ˉ C C S 4 C_{\mathrm{CS}_4} C CS 4 k k k p p p k 3 k_3 k 3
Adding the RV infrared term and subtracting the remaining soft limit gives SU in (18) . The operator S k S_k S k d Φ R [ I R R , S U + 2 a s R e Z c g ( 1 ) ] F R d\Phi_R[I_{\mathrm{RR,SU}}+2a_s\mathop{\mathrm{Re}}Z_{cg}^{(1)}]F_R d Φ R [ I RR , SU + 2 a s Re Z c g ( 1 ) ] F R d Φ B [ d k ] K d\Phi_B[dk]_K d Φ B [ d k ] K
In the double-soft counterterms, the separate bounds E 3 ∗ , E 4 ∗ < E 0 ∗ E_3^*,E_4^*<E_0^* E 3 ∗ , E 4 ∗ < E 0 ∗ E ∗ = E 3 ∗ + E 4 ∗ E^*=E_3^*+E_4^* E ∗ = E 3 ∗ + E 4 ∗ E 0 ∗ E_0^* E 0 ∗ C S 34 C S_{34} C S 34 E ∗ = E 0 ∗ E^*=E_0^* E ∗ = E 0 ∗ C S < C S_{<} C S < (18) : only the splitting variables are integrated, with the parent momentum left unintegrated in d Φ B [ d p ] K d\Phi_B[dp]_K d Φ B [ d p ] K C S > C S_{>} C S > d Φ B d\Phi_B d Φ B
1 = ( 1 − C ) ( 1 − S 34 ) + ( C − C S < ) + ( S 34 − C S > ) , q q ˉ , 1=(1-C)(1-S_{34})+(C-CS_{<})+(S_{34}-CS_{>}),\qquad q\bar{q}, 1 = ( 1 − C ) ( 1 − S 34 ) + ( C − C S < ) + ( S 34 − C S > ) , q q ˉ , 1 = ( 1 − C ) ( 1 − S 4 ) ( 1 − S 34 ) + [ C − C S < + ( 1 − C ) S 4 ( 1 − S 34 ) ] + ( S 34 − C S > ) , g g . \begin{aligned}
1={}&(1-C)(1-S_4)(1-S_{34})\\
&+[C-CS_{<}+(1-C)S_4(1-S_{34})]+(S_{34}-CS_{>}),\qquad gg.
\end{aligned} 1 = ( 1 − C ) ( 1 − S 4 ) ( 1 − S 34 ) + [ C − C S < + ( 1 − C ) S 4 ( 1 − S 34 )] + ( S 34 − C S > ) , g g . Each identity acts on the squared matrix element and phase space of the corresponding double-real channel. The three groups give R R F \mathrm{RR}_{\mathrm{F}} RR F R R S U \mathrm{RR}_{\mathrm{SU}} RR SU R R D U \mathrm{RR}_{\mathrm{DU}} RR DU − S k { d Φ R F R R , S U } -\mathbf{S}_k\{d\Phi_{\mathrm{R}}F_{\mathrm{RR},\mathrm{SU}}\} − S k { d Φ R F RR , SU }
S k { d Φ R F R R , S U } = ∑ a b ∫ u n r e s C S { d Φ R R a b F a b } + ∫ u n r e s ( 1 − C ) S 4 S 34 { d Φ R R g g F g g } . (51) \mathbf{S}_k\{d\Phi_{\mathrm{R}}F_{\mathrm{RR},\mathrm{SU}}\}=\sum_{ab}\int_{\mathrm{unres}}\mathbf{CS}\{d\Phi_{\mathrm{RR}}^{ab}F_{ab}\}+\int_{\mathrm{unres}}(1-\mathbf{C})\mathbf{S}_4\mathbf{S}_{34}\{d\Phi_{\mathrm{RR}}^{gg}F_{gg}\}. \tag*{(51)} S k { d Φ R F RR , SU } = ab ∑ ∫ unres CS { d Φ RR ab F ab } + ∫ unres ( 1 − C ) S 4 S 34 { d Φ RR g g F g g } . (51) The first term leaves the parent momentum unintegrated on d Φ B [ d p ] K d\Phi_{\mathrm{B}}[dp]_K d Φ B [ d p ] K d Φ B [ d k 3 ] K d\Phi_{\mathrm{B}}[dk_3]_K d Φ B [ d k 3 ] K
The pole parts of the integrated RR kernels are
[ g s 2 K S 4 ] p o l e s = a s { 2 C A ϵ 2 + 1 ϵ [ 4 C F − ( C F − C A 2 ) Re γ c u s p , Q ( 0 ) ( v 12 ) + 2 C A ln m c 2 μ R 2 4 ( p 1 ⋅ k ) ( p 2 ⋅ k ) ] } , (52) \left[g_s^2\mathcal{K}_{\mathrm{S}_4}\right]_{\mathrm{poles}}=a_s\left\{\frac{2C_A}{\epsilon^2}+\frac{1}{\epsilon}\left[4C_F-\left(C_F-\frac{C_A}{2}\right)\operatorname{Re}\gamma_{\mathrm{cusp},Q}^{(0)}(v_{12})+2C_A\ln\frac{m_c^2\mu_R^2}{4(p_1\cdot k)(p_2\cdot k)}\right]\right\}, \tag*{(52)} [ g s 2 K S 4 ] poles = a s { ϵ 2 2 C A + ϵ 1 [ 4 C F − ( C F − 2 C A ) Re γ cusp , Q ( 0 ) ( v 12 ) + 2 C A ln 4 ( p 1 ⋅ k ) ( p 2 ⋅ k ) m c 2 μ R 2 ] } , (52) [ C g g − C C S 4 ] p o l e s = a s ϵ 11 C A 3 , [ n l C q q ˉ ] p o l e s = − a s ϵ 4 T F n l 3 . \left[C_{gg}-C_{\mathrm{CS}_4}\right]_{\mathrm{poles}}=\frac{a_s}{\epsilon}\frac{11C_A}{3},\qquad\left[n_l C_{q\bar q}\right]_{\mathrm{poles}}=-\frac{a_s}{\epsilon}\frac{4T_Fn_l}{3}. [ C g g − C CS 4 ] poles = ϵ a s 3 11 C A , [ n l C q q ˉ ] poles = − ϵ a s 3 4 T F n l . Using the infrared factor in (B.14), we obtain
[ I R R , S U ] p o l e s = [ g s 2 K S 4 ] p o l e s + a s β 0 ( 3 ) ϵ = − 2 a s Re Z c c g ( 1 ) . (53) \left[I_{\mathrm{RR},\mathrm{SU}}\right]_{\mathrm{poles}}=\left[g_s^2\mathcal{K}_{\mathrm{S}_4}\right]_{\mathrm{poles}}+\frac{a_s\beta_0^{(3)}}{\epsilon}=-2a_s\operatorname{Re}Z_{ccg}^{(1)}. \tag*{(53)} [ I RR , SU ] poles = [ g s 2 K S 4 ] poles + ϵ a s β 0 ( 3 ) = − 2 a s Re Z cc g ( 1 ) . (53) For the remaining soft subtraction, we define I R R , S U S ( k , ϵ ) I_{\mathrm{RR},\mathrm{SU}}^{\mathrm{S}}(k,\epsilon) I RR , SU S ( k , ϵ )
S k [ I R R , S U F R ] = I R R , S U S S k F R . \mathbf{S}_k[I_{\mathrm{RR},\mathrm{SU}}F_{\mathrm{R}}]=I_{\mathrm{RR},\mathrm{SU}}^{\mathrm{S}}\mathbf{S}_kF_{\mathrm{R}}. S k [ I RR , SU F R ] = I RR , SU S S k F R . The kernel has the form of (B.18) with Born heavy-quark momenta, p 1 + p 2 = K p_1+p_2=K p 1 + p 2 = K v 12 = 2 β / ( 1 + β 2 ) v_{12}=2\beta/(1+\beta^2) v 12 = 2 β / ( 1 + β 2 ) k k k d Φ B [ d k ] K d\Phi_{\mathrm{B}}[dk]_K d Φ B [ d k ] K
[ I R R , S U S ] p o l e s = a s { 2 C A ϵ 2 + 1 ϵ [ β 0 ( 3 ) + ( C F − C A 2 ) A ( β ) + 2 C A ( 1 + ln m c 2 μ R 2 4 ( p 1 ⋅ k ) ( p 2 ⋅ k ) ) ] } . (54) \left[I_{\mathrm{RR},\mathrm{SU}}^{\mathrm{S}}\right]_{\mathrm{poles}}=a_s\left\{\frac{2C_A}{\epsilon^2}+\frac{1}{\epsilon}\left[\beta_0^{(3)}+\left(C_F-\frac{C_A}{2}\right)A(\beta)+2C_A\left(1+\ln\frac{m_c^2\mu_R^2}{4(p_1\cdot k)(p_2\cdot k)}\right)\right]\right\}. \tag*{(54)} [ I RR , SU S ] poles = a s { ϵ 2 2 C A + ϵ 1 [ β 0 ( 3 ) + ( C F − 2 C A ) A ( β ) + 2 C A ( 1 + ln 4 ( p 1 ⋅ k ) ( p 2 ⋅ k ) m c 2 μ R 2 ) ] } . (54) The soft limit of the RV infrared term follows from one-loop soft factorisation [42 ] ,
S k [ 2 a s Re Z c c g ( 1 ) F R ] = a s { 2 Re Z 1 + [ R 12 ( 1 ) ( k ) + Z a s ( 1 ) + Z A ( 1 ) ] p o l e s } S k F R . (55) \mathbf{S}_k[2a_s\operatorname{Re}Z_{ccg}^{(1)}F_{\mathrm{R}}]=a_s\left\{2\operatorname{Re}Z_1+\left[R_{12}^{(1)}(k)+Z_{a_s}^{(1)}+Z_A^{(1)}\right]_{\mathrm{poles}}\right\}\mathbf{S}_kF_{\mathrm{R}}. \tag*{(55)} S k [ 2 a s Re Z cc g ( 1 ) F R ] = a s { 2 Re Z 1 + [ R 12 ( 1 ) ( k ) + Z a s ( 1 ) + Z A ( 1 ) ] poles } S k F R . (55) Here R 12 ( 1 ) R_{12}^{(1)} R 12 ( 1 ) Z a s ( 1 ) Z_{a_s}^{(1)} Z a s ( 1 ) Z A ( 1 ) Z_A^{(1)} Z A ( 1 )
[ R 12 ( 1 ) ( k ) ] p o l e s = − 2 C A ϵ 2 − 2 C A ϵ [ 1 − A ( β ) 4 + ln m c 2 μ R 2 4 ( p 1 ⋅ k ) ( p 2 ⋅ k ) ] , (56) \left[R_{12}^{(1)}(k)\right]_{\mathrm{poles}}=-\frac{2C_A}{\epsilon^2}-\frac{2C_A}{\epsilon}\left[1-\frac{A(\beta)}{4}+\ln\frac{m_c^2\mu_R^2}{4(p_1\cdot k)(p_2\cdot k)}\right], \tag*{(56)} [ R 12 ( 1 ) ( k ) ] poles = − ϵ 2 2 C A − ϵ 2 C A [ 1 − 4 A ( β ) + ln 4 ( p 1 ⋅ k ) ( p 2 ⋅ k ) m c 2 μ R 2 ] , (56) [ Z a s ( 1 ) + Z A ( 1 ) ] p o l e s = − β 0 ( 3 ) ϵ . \left[Z_{a_s}^{(1)}+Z_A^{(1)}\right]_{\mathrm{poles}}=-\frac{\beta_0^{(3)}}{\epsilon}. [ Z a s ( 1 ) + Z A ( 1 ) ] poles = − ϵ β 0 ( 3 ) . Together with 2 Re Z 1 = − C F A ( β ) / ϵ 2\operatorname{Re} Z_1=-C_F A(\beta)/\epsilon 2 Re Z 1 = − C F A ( β ) / ϵ
[ I R R , S U S ] p o l e s + a s { 2 Re Z 1 + [ R 12 ( 1 ) ( k ) + Z α s ( 1 ) + Z A ( 1 ) ] p o l e s } = 0. \left[I_{\mathrm{RR,SU}}^{\mathrm{S}}\right]_{\mathrm{poles}}+a_s\left\{2\operatorname{Re}Z_1+\left[R_{12}^{(1)}(k)+Z_{\alpha_s}^{(1)}+Z_A^{(1)}\right]_{\mathrm{poles}}\right\}=0. [ I RR , SU S ] poles + a s { 2 Re Z 1 + [ R 12 ( 1 ) ( k ) + Z α s ( 1 ) + Z A ( 1 ) ] poles } = 0. Thus the poles cancel separately on d Φ R ( k ) d\Phi_R(k) d Φ R ( k ) d Φ B [ d k ] K d\Phi_B[dk]_K d Φ B [ d k ] K
[ ( 1 − S k ) { d Φ R ( k ) [ I R R , S U + 2 a s Re Z c c g ( 1 ) ] F R } ] p o l e s = 0. (57) \left[(1-\mathcal{S}_k)\left\{d\Phi_R(k)\left[I_{\mathrm{RR,SU}}+2a_s\operatorname{Re}Z_{\mathrm{ccg}}^{(1)}\right]F_R\right\}\right]_{\mathrm{poles}}=0. \tag*{(57)} [ ( 1 − S k ) { d Φ R ( k ) [ I RR , SU + 2 a s Re Z ccg ( 1 ) ] F R } ] poles = 0. (57) Finite-remainder contribution (FR)
The part of the integrated RV soft counterterm (C.44) proportional to F V f i n F_V^{\mathrm{fin}} F V fin F R V , F R = a s C F I S F V f i n F_{\mathrm{RV,FR}}=a_sC_F\mathcal{I}_S F_V^{\mathrm{fin}} F RV , FR = a s C F I S F V fin F V V , F R = 2 a s Re Z 1 F V f i n F_{\mathrm{VV,FR}}=2a_s\operatorname{Re}Z_1F_V^{\mathrm{fin}} F VV , FR = 2 a s Re Z 1 F V fin (22) . The poles cancel by the same identity (B.17).
Double-unresolved contribution (DU)
For the Born-proportional terms in DU, we define the kernels by
F R R , D U = a s 2 I R R , D U F B , F_{\mathrm{RR,DU}}=a_s^2 I_{\mathrm{RR,DU}}F_B, F RR , DU = a s 2 I RR , DU F B , F R V , D U = a s 2 ( I R V , c u r + I R V , Z ) F B , F_{\mathrm{RV,DU}}=a_s^2\left(I_{\mathrm{RV,cur}}+I_{\mathrm{RV,Z}}\right)F_B, F RV , DU = a s 2 ( I RV , cur + I RV , Z ) F B , F V V , D U = a s 2 I V V , D U F B . F_{\mathrm{VV,DU}}=a_s^2 I_{\mathrm{VV,DU}}F_B. F VV , DU = a s 2 I VV , DU F B . Integrating the double-soft counterterms minus the high-energy overlaps, S 34 − C S > S_{34}-\mathrm{CS}_{>} S 34 − CS >
I R R , D U = C F 2 2 I S 2 − 8 C F C A A S ( f A A g g − f A B g g ) + 16 C F T F n l A S ( f A A q q ˉ − f A B q q ˉ ) − 1 a s 2 ( K C S , g + n l K C S , q ) . (58) I_{\mathrm{RR,DU}}=\frac{C_F^2}{2}\mathcal{I}_S^2-8C_FC_A\mathcal{A}_S\left(f_{AA}^{gg}-f_{AB}^{gg}\right)
+16C_FT_Fn_l\mathcal{A}_S\left(f_{AA}^{q\bar q}-f_{AB}^{q\bar q}\right)-\frac{1}{a_s^2}\left(\mathcal{K}_{\mathrm{CS},g}+n_l\mathcal{K}_{\mathrm{CS},q}\right). \tag*{(58)} I RR , DU = 2 C F 2 I S 2 − 8 C F C A A S ( f AA g g − f A B g g ) + 16 C F T F n l A S ( f AA q q ˉ − f A B q q ˉ ) − a s 2 1 ( K CS , g + n l K CS , q ) . (58) The first term describes independent soft-gluon emissions; the functions f i j g g f_{ij}^{gg} f ij g g f i j q q ˉ f_{ij}^{q\bar q} f ij q q ˉ A S \mathcal{A}_S A S A A AA AA A B AB A B K C S , g \mathcal{K}_{\mathrm{CS},g} K CS , g K C S , q \mathcal{K}_{\mathrm{CS},q} K CS , q E ∗ > E 0 ∗ E^*>E_0^* E ∗ > E 0 ∗
I R V , c u r = 4 C F C A e 2 ϵ L S ∑ n = − 3 0 ϵ n R n ( β ) + O ( ϵ ) , I_{\mathrm{RV,cur}}=4C_FC_Ae^{2\epsilon L_S}\sum_{n=-3}^{0}\epsilon^n\mathcal{R}_n(\beta)+O(\epsilon), I RV , cur = 4 C F C A e 2 ϵ L S n = − 3 ∑ 0 ϵ n R n ( β ) + O ( ϵ ) , I R V , Z = C F I S [ Z α s ( 1 ) + Z A ( 1 ) + 2 Re Z 1 ] , I_{\mathrm{RV,Z}}=C_F\mathcal{I}_S\left[Z_{\alpha_s}^{(1)}+Z_A^{(1)}+2\operatorname{Re}Z_1\right], I RV , Z = C F I S [ Z α s ( 1 ) + Z A ( 1 ) + 2 Re Z 1 ] , I V V , D U = 2 Re Z 2 + ∣ Z 1 ∣ 2 . (59) I_{\mathrm{VV,DU}}=2\operatorname{Re}Z_2+\lvert Z_1\rvert^2. \tag*{(59)} I VV , DU = 2 Re Z 2 + ∣ Z 1 ∣ 2 . (59) The coefficients R n \mathcal{R}_n R n A ( β ) A(\beta) A ( β ) B s ( β ) B_s(\beta) B s ( β )
C F I S = − Re Γ 0 ϵ + C F [ A ( β ) L S + B s ( β ) ] + O ( ϵ ) , C_F\mathcal{I}_S=-\frac{\operatorname{Re}\Gamma_0}{\epsilon}+C_F\left[A(\beta)L_S+B_s(\beta)\right]+O(\epsilon), C F I S = − ϵ Re Γ 0 + C F [ A ( β ) L S + B s ( β ) ] + O ( ϵ ) , Z α s ( 1 ) + Z A ( 1 ) = − β 0 ( 3 ) ϵ + β 0 , Q ln μ R 2 m c 2 + O ( ϵ ) . (60) Z_{\alpha_s}^{(1)}+Z_A^{(1)}=-\frac{\beta_0^{(3)}}{\epsilon}+\beta_{0,Q}\ln\frac{\mu_R^2}{m_c^2}+O(\epsilon). \tag*{(60)} Z α s ( 1 ) + Z A ( 1 ) = − ϵ β 0 ( 3 ) + β 0 , Q ln m c 2 μ R 2 + O ( ϵ ) . (60) Expanding I R R , D U I_{\mathrm{RR,DU}} I RR , DU I R V , c u r I_{\mathrm{RV,cur}} I RV , cur [42 , 44 , 64 ]
[ I R R , D U ] ϵ − 3 = − C A Re Γ 0 , [ I R V , c u r ] ϵ − 3 = + C A Re Γ 0 , \left[I_{\mathrm{RR,DU}}\right]_{\epsilon^{-3}}=-C_A\operatorname{Re}\Gamma_0,\qquad\left[I_{\mathrm{RV,cur}}\right]_{\epsilon^{-3}}=+C_A\operatorname{Re}\Gamma_0, [ I RR , DU ] ϵ − 3 = − C A Re Γ 0 , [ I RV , cur ] ϵ − 3 = + C A Re Γ 0 , [ I R R , D U + I R V , c u r ] p o l e s = Re Γ 0 ( Re Γ 0 − β 0 ( 3 ) ) 2 ϵ 2 − Re Γ 1 2 ϵ − C F ( Re Γ 0 − β 0 ( 3 ) ) [ A ( β ) L S + B s ( β ) ] ϵ . (61) \left[I_{\mathrm{RR,DU}}+I_{\mathrm{RV,cur}}\right]_{\mathrm{poles}}=\frac{\operatorname{Re}\Gamma_0(\operatorname{Re}\Gamma_0-\beta_0^{(3)})}{2\epsilon^2}-\frac{\operatorname{Re}\Gamma_1}{2\epsilon}
-\frac{C_F(\operatorname{Re}\Gamma_0-\beta_0^{(3)})[A(\beta)L_S+B_s(\beta)]}{\epsilon}. \tag*{(61)} [ I RR , DU + I RV , cur ] poles = 2 ϵ 2 Re Γ 0 ( Re Γ 0 − β 0 ( 3 ) ) − 2 ϵ Re Γ 1 − ϵ C F ( Re Γ 0 − β 0 ( 3 ) ) [ A ( β ) L S + B s ( β )] . (61) Using (B.28) and (B.16) gives
[ I R V , Z ] p o l e s = − Re Γ 0 ( Re Γ 0 − β 0 ( 3 ) ) ϵ 2 − β 0 , Q Re Γ 0 ϵ ln μ R 2 m c 2 + C F ( Re Γ 0 − β 0 ( 3 ) ) [ A ( β ) L S + B s ( β ) ] ϵ . (62) \left[I_{\mathrm{RV,Z}}\right]_{\mathrm{poles}}=-\frac{\operatorname{Re}\Gamma_0(\operatorname{Re}\Gamma_0-\beta_0^{(3)})}{\epsilon^2}-\frac{\beta_{0,Q}\operatorname{Re}\Gamma_0}{\epsilon}\ln\frac{\mu_R^2}{m_c^2}
+\frac{C_F(\operatorname{Re}\Gamma_0-\beta_0^{(3)})[A(\beta)L_S+B_s(\beta)]}{\epsilon}. \tag*{(62)} [ I RV , Z ] poles = − ϵ 2 Re Γ 0 ( Re Γ 0 − β 0 ( 3 ) ) − ϵ β 0 , Q Re Γ 0 ln m c 2 μ R 2 + ϵ C F ( Re Γ 0 − β 0 ( 3 ) ) [ A ( β ) L S + B s ( β )] . (62) From (B.8),
I V V , D U = Re Γ 0 ( Re Γ 0 − β 0 ( 3 ) ) 2 ϵ 2 + Re Γ 1 2 ϵ + β 0 , Q Re Γ 0 ϵ ln μ R 2 m c 2 . (63) I_{\mathrm{VV,DU}}=\frac{\operatorname{Re}\Gamma_0(\operatorname{Re}\Gamma_0-\beta_0^{(3)})}{2\epsilon^2}+\frac{\operatorname{Re}\Gamma_1}{2\epsilon}+\frac{\beta_{0,Q}\operatorname{Re}\Gamma_0}{\epsilon}\ln\frac{\mu_R^2}{m_c^2}. \tag*{(63)} I VV , DU = 2 ϵ 2 Re Γ 0 ( Re Γ 0 − β 0 ( 3 ) ) + 2 ϵ Re Γ 1 + ϵ β 0 , Q Re Γ 0 ln m c 2 μ R 2 . (63) Combining (B.29)–(B.31) gives
[ I R R , D U + I R V , c u r + I R V , Z + I V V , D U ] p o l e s = 0. (64) \left[I_{\mathrm{RR,DU}}+I_{\mathrm{RV,cur}}+I_{\mathrm{RV,Z}}+I_{\mathrm{VV,DU}}\right]_{\mathrm{poles}}=0. \tag*{(64)} [ I RR , DU + I RV , cur + I RV , Z + I VV , DU ] poles = 0. (64) Local and integrated subtraction terms
The kinematics, unresolved-energy bounds and phase-space notation are defined in appendix B.1. In the locally subtracted combinations, denoted by the superscript sub, each squared matrix element or interference is integrated with its corresponding full or projected phase space.
Single-real contribution (R)
The squared matrix element F R F_R F R k k k
Local soft subtraction. The soft counterterm uses d Φ B [ d k ] K d\Phi_B[dk]_K d Φ B [ d k ] K 0 < E k ∗ < E 0 ∗ 0<E_k^*<E_0^* 0 < E k ∗ < E 0 ∗ p 1 + p 2 = K p_1+p_2=K p 1 + p 2 = K d Φ R d\Phi_R d Φ R p 1 + p 2 = K − k p_1+p_2=K-k p 1 + p 2 = K − k
S 12 ( k ) = 2 p 1 ⋅ p 2 ( p 1 ⋅ k ) ( p 2 ⋅ k ) − m c 2 ( p 1 ⋅ k ) 2 − m c 2 ( p 2 ⋅ k ) 2 . (65) S_{12}(k)=\frac{2p_1\cdot p_2}{(p_1\cdot k)(p_2\cdot k)}-\frac{m_c^2}{(p_1\cdot k)^2}-\frac{m_c^2}{(p_2\cdot k)^2}. \tag*{(65)} S 12 ( k ) = ( p 1 ⋅ k ) ( p 2 ⋅ k ) 2 p 1 ⋅ p 2 − ( p 1 ⋅ k ) 2 m c 2 − ( p 2 ⋅ k ) 2 m c 2 . (65) The corresponding soft limit is
S k F R = g s 2 C ϵ C F S 12 ( k ) F B . S_kF_R=g_s^2C_\epsilon C_FS_{12}(k)F_B. S k F R = g s 2 C ϵ C F S 12 ( k ) F B . The local subtraction is abbreviated as F R s u b = F R − S k F R F_R^{\mathrm{sub}}=F_R-S_kF_R F R sub = F R − S k F R R F R_F R F
Integrated soft counterterm. Integration over the unresolved gluon gives
F R , U = ∫ [ d k ] K S k F R = g s 2 C F ( 4 π ) 2 [ A ( β ) ϵ + A ( β ) ln μ R 2 4 ( E 0 ∗ ) 2 + B s ( β ) + O ( ϵ ) ] F B . (66) F_{R,U}=\int[dk]_K\,\mathcal{S}_kF_R=\frac{g_s^2 C_F}{(4\pi)^2}\left[\frac{A(\beta)}{\epsilon}+A(\beta)\ln\frac{\mu_R^2}{4(E_0^*)^2}+B_s(\beta)+O(\epsilon)\right]F_B. \tag*{(66)} F R , U = ∫ [ d k ] K S k F R = ( 4 π ) 2 g s 2 C F [ ϵ A ( β ) + A ( β ) ln 4 ( E 0 ∗ ) 2 μ R 2 + B s ( β ) + O ( ϵ ) ] F B . (66) where
A ( β ) = 4 + 2 ( 1 + β 2 ) β ln 1 − β 1 + β , A(\beta)=4+\frac{2(1+\beta^2)}{\beta}\ln\frac{1-\beta}{1+\beta}, A ( β ) = 4 + β 2 ( 1 + β 2 ) ln 1 + β 1 − β , B s ( β ) = − 4 β ln 1 − β 1 + β − 2 ( 1 + β 2 ) β [ Li 2 ( 2 β 1 + β ) − Li 2 ( − 2 β 1 − β ) ] . (67) B_s(\beta)=-\frac{4}{\beta}\ln\frac{1-\beta}{1+\beta}-\frac{2(1+\beta^2)}{\beta}\left[\operatorname{Li}_2\left(\frac{2\beta}{1+\beta}\right)-\operatorname{Li}_2\left(\frac{-2\beta}{1-\beta}\right)\right]. \tag*{(67)} B s ( β ) = − β 4 ln 1 + β 1 − β − β 2 ( 1 + β 2 ) [ Li 2 ( 1 + β 2 β ) − Li 2 ( 1 − β − 2 β ) ] . (67) At NNLO, the soft integral is needed beyond O ( ϵ 0 ) O(\epsilon^0) O ( ϵ 0 ) L S = ln [ μ R 2 / ( 4 ( E 0 ∗ ) 2 ) ] L_S=\ln[\mu_R^2/(4(E_0^*)^2)] L S = ln [ μ R 2 / ( 4 ( E 0 ∗ ) 2 )] F R , U = a s C F I S F B F_{R,U}=a_sC_F\mathcal{I}_SF_B F R , U = a s C F I S F B
I S ( β , ϵ , L S ) = − 4 e ϵ ( γ E + L S ) Γ ( 1 − ϵ ) ϵ Γ ( 2 − 2 ϵ ) h ( β , ϵ ) , (68) \mathcal{I}_S(\beta,\epsilon,L_S)=-\frac{4e^{\epsilon(\gamma_E+L_S)}\Gamma(1-\epsilon)}{\epsilon\Gamma(2-2\epsilon)}h(\beta,\epsilon), \tag*{(68)} I S ( β , ϵ , L S ) = − ϵ Γ ( 2 − 2 ϵ ) 4 e ϵ ( γ E + L S ) Γ ( 1 − ϵ ) h ( β , ϵ ) , (68) h ( β , ϵ ) = ( 1 + β 2 − 2 ϵ ) 2 F 1 ( 1 , 1 2 ; 3 2 − ϵ ; β 2 ) − ( 1 − 2 ϵ ) . (69) h(\beta,\epsilon)=(1+\beta^2-2\epsilon)\,{}_2F_1\left(1,\frac{1}{2};\frac{3}{2}-\epsilon;\beta^2\right)-(1-2\epsilon). \tag*{(69)} h ( β , ϵ ) = ( 1 + β 2 − 2 ϵ ) 2 F 1 ( 1 , 2 1 ; 2 3 − ϵ ; β 2 ) − ( 1 − 2 ϵ ) . (69) The integrated counterterm combines with the virtual infrared term to give U U U § 9 .§ 3 ; the same kernel enters F R F_R F R § 9 .§ 4.3 .
Double-real contribution (RR)
Light-quark-pair emission: q q ˉ q\bar q q q ˉ
For an additional q ( k 3 ) q ˉ ( k 4 ) q(k_3)\bar q(k_4) q ( k 3 ) q ˉ ( k 4 ) F q q ˉ F_{q\bar q} F q q ˉ S 34 S_{34} S 34 C C C
F q q ˉ s u b = ( 1 − C ) ( 1 − S 34 ) F q q ˉ = F q q ˉ − S 34 F q q ˉ − C F q q ˉ + C S 34 F q q ˉ . (70) \begin{aligned}
F_{q\bar q}^{\mathrm{sub}}=(1-C)(1-S_{34})F_{q\bar q} \\
&=F_{q\bar q}-S_{34}F_{q\bar q}-CF_{q\bar q}+CS_{34}F_{q\bar q}. \tag*{(70)}
\end{aligned} F q q ˉ sub = ( 1 − C ) ( 1 − S 34 ) F q q ˉ = F q q ˉ − S 34 F q q ˉ − C F q q ˉ + C S 34 F q q ˉ . (70) Double-soft subtraction. The factorised phase space is d Φ B [ d k 3 ] K [ d k 4 ] K \mathrm{d}\Phi_B[dk_3]_K[dk_4]_K d Φ B [ d k 3 ] K [ d k 4 ] K 0 < E 3 ∗ < E 0 ∗ 0<E_3^*<E_0^* 0 < E 3 ∗ < E 0 ∗ 0 < E 4 ∗ < E 0 ∗ 0<E_4^*<E_0^* 0 < E 4 ∗ < E 0 ∗ K K K
S 34 F q q ˉ = g s 4 c ϵ 2 C F T F ( I 11 − I 12 − I 21 + I 22 ) F B , (71) S_{34}F_{q\bar q}=g_s^4c_\epsilon^2C_FT_F(\mathcal{I}_{11}-\mathcal{I}_{12}-\mathcal{I}_{21}+\mathcal{I}_{22})F_B, \tag*{(71)} S 34 F q q ˉ = g s 4 c ϵ 2 C F T F ( I 11 − I 12 − I 21 + I 22 ) F B , (71) with
I i j = ( p i ⋅ k 3 ) ( p j ⋅ k 4 ) + ( p i ⋅ k 4 ) ( p j ⋅ k 3 ) − ( p i ⋅ p j ) ( k 3 ⋅ k 4 ) ( k 3 ⋅ k 4 ) 2 ( p i ⋅ k 34 ) ( p j ⋅ k 34 ) , \mathcal{I}_{ij}=\frac{(p_i\cdot k_3)(p_j\cdot k_4)+(p_i\cdot k_4)(p_j\cdot k_3)-(p_i\cdot p_j)(k_3\cdot k_4)}{(k_3\cdot k_4)^2(p_i\cdot k_{34})(p_j\cdot k_{34})}, I ij = ( k 3 ⋅ k 4 ) 2 ( p i ⋅ k 34 ) ( p j ⋅ k 34 ) ( p i ⋅ k 3 ) ( p j ⋅ k 4 ) + ( p i ⋅ k 4 ) ( p j ⋅ k 3 ) − ( p i ⋅ p j ) ( k 3 ⋅ k 4 ) , k 34 = k 3 + k 4 . (72) k_{34}=k_3+k_4. \tag*{(72)} k 34 = k 3 + k 4 . (72) Integrating over both unresolved momenta gives [44 ]
∫ [ d k 3 ] K [ d k 4 ] K S 34 F q q ˉ = g s 4 C F T F ( 4 π ) 4 16 A S ( f A A q q ˉ − f A B q q ˉ ) F B . (73) \int[dk_3]_K[dk_4]_K\,S_{34}F_{q\bar q}=\frac{g_s^4C_FT_F}{(4\pi)^4}16\mathcal{A}_S(f_{AA}^{q\bar q}-f_{AB}^{q\bar q})F_B. \tag*{(73)} ∫ [ d k 3 ] K [ d k 4 ] K S 34 F q q ˉ = ( 4 π ) 4 g s 4 C F T F 16 A S ( f AA q q ˉ − f A B q q ˉ ) F B . (73) The common double-soft normalisation is
A S ( ϵ , L S ) = e 2 ϵ ( γ E + L S ) Γ 2 ( 1 − ϵ ) Γ 2 ( 2 − 2 ϵ ) . (74) \mathcal{A}_S(\epsilon,L_S)=\frac{e^{2\epsilon(\gamma_E+L_S)}\Gamma^2(1-\epsilon)}{\Gamma^2(2-2\epsilon)}. \tag*{(74)} A S ( ϵ , L S ) = Γ 2 ( 2 − 2 ϵ ) e 2 ϵ ( γ E + L S ) Γ 2 ( 1 − ϵ ) . (74) The functions f i j q q ˉ f_{ij}^{q\bar q} f ij q q ˉ O ( ϵ 0 ) O(\epsilon^0) O ( ϵ 0 ) § 12 .§ 1 .
Collinear subtraction. We define
E ∗ = E 3 ∗ + E 4 ∗ , z = E 3 ∗ E ∗ , η = 1 − cos θ 34 ∗ 2 . (75) E^* = E_3^* + E_4^*, \qquad z = \frac{E_3^*}{E^*}, \qquad\eta= \frac{1-\cos\theta_{34}^*}{2}. \tag*{(75)} E ∗ = E 3 ∗ + E 4 ∗ , z = E ∗ E 3 ∗ , η = 2 1 − cos θ 34 ∗ . (75) The collinear projection replaces k 3 + k 4 k_3+k_4 k 3 + k 4 p = k 3 / z p=k_3/z p = k 3 / z
d K 2 2 π d Φ 2 ( Q ; P , K ) d Φ 2 ( K − k 3 − k 4 ; p 1 , p 2 ) [ d k 3 ] K [ d k 4 ] K \frac{dK^2}{2\pi}\,d\Phi_2(Q;P,K)\,d\Phi_2(K-k_3-k_4;p_1,p_2)\,[dk_3]_K[dk_4]_K 2 π d K 2 d Φ 2 ( Q ; P , K ) d Φ 2 ( K − k 3 − k 4 ; p 1 , p 2 ) [ d k 3 ] K [ d k 4 ] K → k 3 + k 4 ↦ p d K 2 2 π d Φ 2 ( Q ; P , K ) d Φ 2 ( K − p ; p 1 , p 2 ) [ d k 3 ] K [ d k 4 ] K . \xrightarrow{k_3+k_4\mapsto p}
\frac{dK^2}{2\pi}\,d\Phi_2(Q;P,K)\,d\Phi_2(K-p;p_1,p_2)\,[dk_3]_K[dk_4]_K. k 3 + k 4 ↦ p 2 π d K 2 d Φ 2 ( Q ; P , K ) d Φ 2 ( K − p ; p 1 , p 2 ) [ d k 3 ] K [ d k 4 ] K . The heavy-pair threshold in d Φ 2 ( K − p ; p 1 , p 2 ) d\Phi_2(K-p;p_1,p_2) d Φ 2 ( K − p ; p 1 , p 2 ) ( K − p ) 2 = W 2 − 2 W E ∗ ≥ 4 m c 2 (K-p)^2=W^2-2WE^*\geq4m_c^2 ( K − p ) 2 = W 2 − 2 W E ∗ ≥ 4 m c 2 E ∗ ≤ E 0 ∗ E^*\leq E_0^* E ∗ ≤ E 0 ∗
Taking the spatial direction of p p p K K K d Ω ⊥ ∗ d\Omega_\perp^* d Ω ⊥ ∗ k 4 k_4 k 4 d − 2 d-2 d − 2 d Ω ⊥ ∗ = d ϕ ⊥ ∗ d\Omega_\perp^*=d\phi_\perp^* d Ω ⊥ ∗ = d ϕ ⊥ ∗ 0 ≤ ϕ ⊥ ∗ < 2 π 0\leq\phi_\perp^*<2\pi 0 ≤ ϕ ⊥ ∗ < 2 π
d E 3 ∗ d E 4 ∗ = E ∗ d E ∗ d z , dE_3^*\,dE_4^*=E^*\,dE^*\,dz, d E 3 ∗ d E 4 ∗ = E ∗ d E ∗ d z , d Ω 4 ∗ = ( sin θ 34 ∗ ) 1 − 2 ϵ d θ 34 ∗ d Ω ⊥ ∗ = 2 1 − 2 ϵ [ η ( 1 − η ) ] − ϵ d η d Ω ⊥ ∗ . d\Omega_4^*=(\sin\theta_{34}^*)^{1-2\epsilon}d\theta_{34}^*d\Omega_\perp^*
=2^{1-2\epsilon}[\eta(1-\eta)]^{-\epsilon}d\eta\,d\Omega_\perp^*. d Ω 4 ∗ = ( sin θ 34 ∗ ) 1 − 2 ϵ d θ 34 ∗ d Ω ⊥ ∗ = 2 1 − 2 ϵ [ η ( 1 − η ) ] − ϵ d η d Ω ⊥ ∗ . Following ref. [42 ] , appendix D.1, the collinear projection sets the smooth factor ( 1 − η ) − ϵ (1-\eta)^{-\epsilon} ( 1 − η ) − ϵ η − ϵ \eta^{-\epsilon} η − ϵ [ d p ] K [dp]_K [ d p ] K d Φ R ( p ) d Φ C , u n r e s d\Phi_R(p)d\Phi_{C,\mathrm{unres}} d Φ R ( p ) d Φ C , unres 0 < z , η < 1 0<z,\eta<1 0 < z , η < 1 p p p
d Φ C , u n r e s = ( E ∗ ) 2 − 2 ϵ [ z ( 1 − z ) ] 1 − 2 ϵ 2 ( 2 π ) 3 − 2 ϵ d z 2 1 − 2 ϵ η − ϵ d η d Ω ⊥ ∗ , (76) d\Phi_{C,\mathrm{unres}}=
\frac{(E^*)^{2-2\epsilon}[z(1-z)]^{1-2\epsilon}}{2(2\pi)^{3-2\epsilon}}
\,dz\,2^{1-2\epsilon}\eta^{-\epsilon}d\eta\,d\Omega_\perp^*, \tag*{(76)} d Φ C , unres = 2 ( 2 π ) 3 − 2 ϵ ( E ∗ ) 2 − 2 ϵ [ z ( 1 − z ) ] 1 − 2 ϵ d z 2 1 − 2 ϵ η − ϵ d η d Ω ⊥ ∗ , (76) ∫ d Ω ⊥ ∗ = 2 π 1 − ϵ Γ ( 1 − ϵ ) . \int d\Omega_\perp^*=\frac{2\pi^{1-\epsilon}}{\Gamma(1-\epsilon)}. ∫ d Ω ⊥ ∗ = Γ ( 1 − ϵ ) 2 π 1 − ϵ . With n = 2 u K − p / E ∗ n=2u_K-p/E^* n = 2 u K − p / E ∗ g ⊥ μ ν = g μ ν − ( p μ n ν + n μ p ν ) / ( p ⋅ n ) g_\perp^{\mu\nu}=g^{\mu\nu}-(p^\mu n^\nu+n^\mu p^\nu)/(p\cdot n) g ⊥ μν = g μν − ( p μ n ν + n μ p ν ) / ( p ⋅ n ) k 4 k_4 k 4 k ⊥ μ = g ⊥ μ ν k 4 ν k_\perp^\mu=g_\perp^{\mu\nu}k_{4\nu} k ⊥ μ = g ⊥ μν k 4 ν
k ⊥ 2 = − ( E 4 ∗ ) 2 sin 2 θ 34 ∗ = − 4 ( 1 − z ) 2 ( E ∗ ) 2 η ( 1 − η ) . k_\perp^2=-(E_4^*)^2\sin^2\theta_{34}^*
=-4(1-z)^2(E^*)^2\eta(1-\eta). k ⊥ 2 = − ( E 4 ∗ ) 2 sin 2 θ 34 ∗ = − 4 ( 1 − z ) 2 ( E ∗ ) 2 η ( 1 − η ) . The spin-correlated splitting tensor and local counterterm are
P q q ˉ μ ν = T F [ − g ⊥ μ ν + 4 z ( 1 − z ) k ⊥ μ k ⊥ ν k ⊥ 2 ] , (77) P_{q\bar q}^{\mu\nu}=T_F\left[-g_\perp^{\mu\nu}+4z(1-z)\frac{k_\perp^\mu k_\perp^\nu}{k_\perp^2}\right], \tag*{(77)} P q q ˉ μν = T F [ − g ⊥ μν + 4 z ( 1 − z ) k ⊥ 2 k ⊥ μ k ⊥ ν ] , (77) C F q q ˉ = g s 2 c ϵ k 3 ⋅ k 4 P q q ˉ μ ν F R , μ ν . C_F{}_{q\bar q}=\frac{g_s^2c_\epsilon}{k_3\cdot k_4}P_{q\bar q}^{\mu\nu}F_{R,\mu\nu}. C F q q ˉ = k 3 ⋅ k 4 g s 2 c ϵ P q q ˉ μν F R , μν . Here F R , μ ν F_{R,\mu\nu} F R , μν F R ( p ) = − g ⊥ μ ν F R , μ ν F_R(p)=-g_\perp^{\mu\nu}F_{R,\mu\nu} F R ( p ) = − g ⊥ μν F R , μν k 3 ⋅ k 4 = 2 z ( 1 − z ) ( E ∗ ) 2 η k_3\cdot k_4=2z(1-z)(E^*)^2\eta k 3 ⋅ k 4 = 2 z ( 1 − z ) ( E ∗ ) 2 η
At fixed reduced kinematics, F R , μ ν F_{R,\mu\nu} F R , μν [42 ]
⟨ k ⊥ μ k ⊥ ν k ⊥ 2 ⟩ ⊥ ≡ ∫ d Ω ⊥ ∗ k ⊥ μ k ⊥ ν / k ⊥ 2 ∫ d Ω ⊥ ∗ = g ⊥ μ ν d − 2 . \left\langle\frac{k_\perp^\mu k_\perp^\nu}{k_\perp^2}\right\rangle_\perp
\equiv
\frac{\int d\Omega_\perp^*\,k_\perp^\mu k_\perp^\nu/k_\perp^2}{\int d\Omega_\perp^*}
=\frac{g_\perp^{\mu\nu}}{d-2}. ⟨ k ⊥ 2 k ⊥ μ k ⊥ ν ⟩ ⊥ ≡ ∫ d Ω ⊥ ∗ ∫ d Ω ⊥ ∗ k ⊥ μ k ⊥ ν / k ⊥ 2 = d − 2 g ⊥ μν . so ⟨ P q q ˉ μ ν F R , μ ν ⟩ ⊥ = T F [ 1 − 2 z ( 1 − z ) / ( 1 − ϵ ) ] F R ( p ) \langle P_{q\bar q}^{\mu\nu}F_{R,\mu\nu}\rangle_\perp=T_F[1-2z(1-z)/(1-\epsilon)]F_R(p) ⟨ P q q ˉ μν F R , μν ⟩ ⊥ = T F [ 1 − 2 z ( 1 − z ) / ( 1 − ϵ )] F R ( p ) z z z η \eta η L C = ln [ μ R 2 / ( 4 ( E ∗ ) 2 ) ] L_C=\ln[\mu_R^2/(4(E^*)^2)] L C = ln [ μ R 2 / ( 4 ( E ∗ ) 2 )]
∫ d Φ C , u n r e s C F q q ˉ = C q q ˉ ( E ∗ , ϵ ) F R ( p ) , \int d\Phi_{C,\mathrm{unres}}\,\mathcal{C}F_{q\bar q}=\mathcal{C}_{q\bar q}(E^*,\epsilon)F_R(p), ∫ d Φ C , unres C F q q ˉ = C q q ˉ ( E ∗ , ϵ ) F R ( p ) , C q q ˉ ( E ∗ , ϵ ) = g s 2 T F ( 4 π ) 2 [ − 4 3 ϵ − 4 3 L C − 46 9 + ϵ ( − 2 3 L C 2 − 46 9 L C − 562 27 + 6 ζ 2 ) + O ( ϵ 2 ) ] . (78) \mathcal{C}_{q\bar q}(E^*,\epsilon)=\frac{g_s^2T_F}{(4\pi)^2}\left[-\frac{4}{3\epsilon}-\frac{4}{3}L_C-\frac{46}{9}+\epsilon\left(-\frac{2}{3}L_C^2-\frac{46}{9}L_C-\frac{562}{27}+6\zeta_2\right)+O(\epsilon^2)\right]. \tag*{(78)} C q q ˉ ( E ∗ , ϵ ) = ( 4 π ) 2 g s 2 T F [ − 3 ϵ 4 − 3 4 L C − 9 46 + ϵ ( − 3 2 L C 2 − 9 46 L C − 27 562 + 6 ζ 2 ) + O ( ϵ 2 ) ] . (78) The expression valid to all orders in ϵ \epsilon ϵ [42 ]
C q q ˉ ( E ∗ , ϵ ) = g s 2 T F ( 4 π ) 2 ( μ R 2 4 ( E ∗ ) 2 ) ϵ 2 [ ϵ ( 10 − 8 ϵ ) − 4 ] ϵ e γ E ϵ Γ ( 1 − 2 ϵ ) Γ ( 2 − 2 ϵ ) Γ ( 4 − 4 ϵ ) Γ ( 2 − ϵ ) . (79) \mathcal{C}_{q\bar q}(E^*,\epsilon)=\frac{g_s^2T_F}{(4\pi)^2}\left(\frac{\mu_R^2}{4(E^*)^2}\right)^\epsilon\frac{2[\epsilon(10-8\epsilon)-4]}{\epsilon}\frac{e^{\gamma_E\epsilon}\Gamma(1-2\epsilon)\Gamma(2-2\epsilon)}{\Gamma(4-4\epsilon)\Gamma(2-\epsilon)}. \tag*{(79)} C q q ˉ ( E ∗ , ϵ ) = ( 4 π ) 2 g s 2 T F ( 4 ( E ∗ ) 2 μ R 2 ) ϵ ϵ 2 [ ϵ ( 10 − 8 ϵ ) − 4 ] Γ ( 4 − 4 ϵ ) Γ ( 2 − ϵ ) e γ E ϵ Γ ( 1 − 2 ϵ ) Γ ( 2 − 2 ϵ ) . (79) Soft-collinear overlap and its integration domain. To construct the overlap C S 34 \mathrm{CS}_{34} CS 34
d K 2 2 π d Φ 2 ( Q ; P , K ) d Φ 2 ( K − k 3 − k 4 ; p 1 , p 2 ) [ d k 3 ] K [ d k 4 ] K → S 34 d Φ B [ d k 3 ] K [ d k 4 ] K , \frac{dK^2}{2\pi}d\Phi_2(Q;P,K)d\Phi_2(K-k_3-k_4;p_1,p_2)[dk_3]_K[dk_4]_K
\xrightarrow{\mathrm{S}_{34}}
d\Phi_B[dk_3]_K[dk_4]_K, 2 π d K 2 d Φ 2 ( Q ; P , K ) d Φ 2 ( K − k 3 − k 4 ; p 1 , p 2 ) [ d k 3 ] K [ d k 4 ] K S 34 d Φ B [ d k 3 ] K [ d k 4 ] K , so the reduced heavy momenta satisfy p 1 + p 2 = K p_1+p_2=K p 1 + p 2 = K E 3 ∗ = z E ∗ E_3^*=zE^* E 3 ∗ = z E ∗ E 4 ∗ = ( 1 − z ) E ∗ E_4^*=(1-z)E^* E 4 ∗ = ( 1 − z ) E ∗ p = k 3 / z p=k_3/z p = k 3 / z d E 3 ∗ d E 4 ∗ = E ∗ d E ∗ d z dE_3^*dE_4^*=E^*dE^*dz d E 3 ∗ d E 4 ∗ = E ∗ d E ∗ d z d Π ( p ) = d d p δ + ( p 2 ) / ( 2 π ) d − 1 d\Pi(p)=d^dp\,\delta_+(p^2)/(2\pi)^{d-1} d Π ( p ) = d d p δ + ( p 2 ) / ( 2 π ) d − 1 ( 1 − η ) − ϵ → 1 (1-\eta)^{-\epsilon}\to1 ( 1 − η ) − ϵ → 1
d Φ B [ d k 3 ] K [ d k 4 ] K → C d Φ B d Φ C S , u n r e s , d\Phi_B[dk_3]_K[dk_4]_K\xrightarrow{\mathrm{C}}d\Phi_B\,d\Phi_{\mathrm{CS},\mathrm{unres}}, d Φ B [ d k 3 ] K [ d k 4 ] K C d Φ B d Φ CS , unres , d Φ C S , u n r e s = d Π ( p ) d Φ C , u n r e s θ ( E 0 ∗ − z E ∗ ) θ ( E 0 ∗ − ( 1 − z ) E ∗ ) . d\Phi_{\mathrm{CS},\mathrm{unres}}=d\Pi(p)\,d\Phi_{C,\mathrm{unres}}\,\theta(E_0^*-zE^*)\theta(E_0^*-(1-z)E^*). d Φ CS , unres = d Π ( p ) d Φ C , unres θ ( E 0 ∗ − z E ∗ ) θ ( E 0 ∗ − ( 1 − z ) E ∗ ) . Here d Φ C , u n r e s d\Phi_{C,\mathrm{unres}} d Φ C , unres p p p
0 < E ∗ < E max ∗ ( z ) , E max ∗ ( z ) = min ( E 0 ∗ z , E 0 ∗ 1 − z ) = E 0 ∗ max ( z , 1 − z ) . (80) 0<E^*<E_{\max}^*(z),\qquad E_{\max}^*(z)=\min\left(\frac{E_0^*}{z},\frac{E_0^*}{1-z}\right)=\frac{E_0^*}{\max(z,1-z)}. \tag*{(80)} 0 < E ∗ < E m a x ∗ ( z ) , E m a x ∗ ( z ) = min ( z E 0 ∗ , 1 − z E 0 ∗ ) = max ( z , 1 − z ) E 0 ∗ . (80) The remaining integrations run over 0 < z < 1 0<z<1 0 < z < 1 0 < η < 1 0<\eta<1 0 < η < 1 C S 34 \mathrm{CS}_{34} CS 34 E ∗ = E 0 ∗ E^*=E_0^* E ∗ = E 0 ∗ C S < \mathrm{CS}_{<} CS < C S > \mathrm{CS}_{>} CS >
With the eikonal current j μ ( p ) j^\mu(p) j μ ( p )
j μ ( p ) = p 1 μ p 1 ⋅ p − p 2 μ p 2 ⋅ p , j^\mu(p)=\frac{p_1^\mu}{p_1\cdot p}-\frac{p_2^\mu}{p_2\cdot p}, j μ ( p ) = p 1 ⋅ p p 1 μ − p 2 ⋅ p p 2 μ , C S 34 F q q ˉ = g s 4 C F k 3 ⋅ k 4 P q q ˉ μ ν j μ ( p ) j ν ( p ) F B . (81) \mathrm{CS}_{34}F_{q\bar q}=\frac{g_s^4C_F}{k_3\cdot k_4}P_{q\bar q}^{\mu\nu}j_\mu(p)j_\nu(p)F_B. \tag*{(81)} CS 34 F q q ˉ = k 3 ⋅ k 4 g s 4 C F P q q ˉ μν j μ ( p ) j ν ( p ) F B . (81) The current obeys p ⋅ j = 0 p\cdot j=0 p ⋅ j = 0 − g μ ν j μ j ν = − j 2 = S 12 ( p ) -g_{\mu\nu}j^\mu j^\nu=-j^2=S_{12}(p) − g μν j μ j ν = − j 2 = S 12 ( p ) ⟨ P q q ˉ μ ν j μ j ν ⟩ ⊥ = T F [ 1 − 2 z ( 1 − z ) / ( 1 − ϵ ) ] S 12 ( p ) \langle P_{q\bar q}^{\mu\nu}j_\mu j_\nu\rangle_\perp=T_F[1-2z(1-z)/(1-\epsilon)]S_{12}(p) ⟨ P q q ˉ μν j μ j ν ⟩ ⊥ = T F [ 1 − 2 z ( 1 − z ) / ( 1 − ϵ )] S 12 ( p ) C S < \mathrm{CS}_{<} CS < p p p 0 < E ∗ < E 0 ∗ 0 < E^{*} < E^{*}_{0} 0 < E ∗ < E 0 ∗ C S > \mathrm{CS}_{>} CS >
∫ d Φ C , u n r e s C S < F q q ˉ = g s 2 C ϵ C F C q q ˉ ( E ∗ , ϵ ) S 12 ( p ) F B , \int\mathrm{d}\Phi_{\mathrm{C,unres}}\,\mathrm{CS}_{<}F_{q\bar q}
= g_s^2 C_{\epsilon} C_F C_{q\bar q}(E^{*},\epsilon)S_{12}(p)F_B, ∫ d Φ C , unres CS < F q q ˉ = g s 2 C ϵ C F C q q ˉ ( E ∗ , ϵ ) S 12 ( p ) F B , ∫ d Φ C S , u n r e s C S > F q q ˉ = K C S , q F B , \int\mathrm{d}\Phi_{\mathrm{CS,unres}}\,\mathrm{CS}_{>}F_{q\bar q}
= \mathcal{K}_{\mathrm{CS},q}F_B, ∫ d Φ CS , unres CS > F q q ˉ = K CS , q F B , K C S , q = g s 4 C F T F ( 4 π ) 4 4 e 2 ϵ ( γ E + L S ) ϵ 2 Γ ( 2 − 2 ϵ ) h ( β , ϵ ) ( W q , m − W q , 0 ) . (82) \mathcal{K}_{\mathrm{CS},q}
= \frac{g_s^4 C_F T_F}{(4\pi)^4}
\frac{4e^{2\epsilon(\gamma_E+L_S)}}{\epsilon^2\Gamma(2-2\epsilon)}
h(\beta,\epsilon)(\mathcal{W}_{q,m}-\mathcal{W}_{q,0}). \tag*{(82)} K CS , q = ( 4 π ) 4 g s 4 C F T F ϵ 2 Γ ( 2 − 2 ϵ ) 4 e 2 ϵ ( γ E + L S ) h ( β , ϵ ) ( W q , m − W q , 0 ) . (82) Here C q q ˉ C_{q\bar q} C q q ˉ h ( β , ϵ ) h(\beta,\epsilon) h ( β , ϵ )
Overlap weights. We use the complete and incomplete beta functions
B x ( a , b ) = ∫ 0 x d t t a − 1 ( 1 − t ) b − 1 , B ( a , b ) = B 1 ( a , b ) , B_x(a,b)=\int_0^x \mathrm{d}t\,t^{a-1}(1-t)^{b-1}, \qquad B(a,b)=B_1(a,b), B x ( a , b ) = ∫ 0 x d t t a − 1 ( 1 − t ) b − 1 , B ( a , b ) = B 1 ( a , b ) , with analytic continuation where required. After factoring out T F T_F T F
w q ( z , ϵ ) = [ z ( 1 − z ) ] − 2 ϵ [ 1 − 2 z ( 1 − z ) 1 − ϵ ] . w_q(z,\epsilon)=[z(1-z)]^{-2\epsilon}\left[1-\frac{2z(1-z)}{1-\epsilon}\right]. w q ( z , ϵ ) = [ z ( 1 − z ) ] − 2 ϵ [ 1 − 1 − ϵ 2 z ( 1 − z ) ] . The two weights in (C.18) are
W q , 0 = ∫ 0 1 d z w q ( z , ϵ ) = B ( 1 − 2 ϵ , 1 − 2 ϵ ) − 2 1 − ϵ B ( 2 − 2 ϵ , 2 − 2 ϵ ) , \begin{aligned}
\mathcal{W}_{q,0}
&=\int_0^1 \mathrm{d}z\,w_q(z,\epsilon)\\
&=B(1-2\epsilon,1-2\epsilon)-\frac{2}{1-\epsilon}B(2-2\epsilon,2-2\epsilon),
\end{aligned} W q , 0 = ∫ 0 1 d z w q ( z , ϵ ) = B ( 1 − 2 ϵ , 1 − 2 ϵ ) − 1 − ϵ 2 B ( 2 − 2 ϵ , 2 − 2 ϵ ) , W q , m = ∫ 0 1 d z w q ( z , ϵ ) max ( z , 1 − z ) 4 ϵ = 2 [ B 1 / 2 ( 1 − 2 ϵ , 1 + 2 ϵ ) − 2 1 − ϵ B 1 / 2 ( 2 − 2 ϵ , 2 + 2 ϵ ) ] . (83) \begin{aligned}
\mathcal{W}_{q,m}
&=\int_0^1 \mathrm{d}z\,w_q(z,\epsilon)\max(z,1-z)^{4\epsilon}\\
&=2\left[B_{1/2}(1-2\epsilon,1+2\epsilon)-\frac{2}{1-\epsilon}B_{1/2}(2-2\epsilon,2+2\epsilon)\right]. \tag*{(83)}
\end{aligned} W q , m = ∫ 0 1 d z w q ( z , ϵ ) max ( z , 1 − z ) 4 ϵ = 2 [ B 1/2 ( 1 − 2 ϵ , 1 + 2 ϵ ) − 1 − ϵ 2 B 1/2 ( 2 − 2 ϵ , 2 + 2 ϵ ) ] . (83) Their difference is
W q , m − W q , 0 = ϵ ( 8 3 ln 2 − 23 9 ) + ϵ 2 ( 16 3 ζ 2 − 4 3 ln 2 − 245 27 ) + O ( ϵ 3 ) . (84) \mathcal{W}_{q,m}-\mathcal{W}_{q,0}
=\epsilon\left(\frac{8}{3}\ln2-\frac{23}{9}\right)
+\epsilon^2\left(\frac{16}{3}\zeta_2-\frac{4}{3}\ln2-\frac{245}{27}\right)
+O(\epsilon^3). \tag*{(84)} W q , m − W q , 0 = ϵ ( 3 8 ln 2 − 9 23 ) + ϵ 2 ( 3 16 ζ 2 − 3 4 ln 2 − 27 245 ) + O ( ϵ 3 ) . (84) Two-gluon emission: g g gg g g
For the two gluons g ( k 3 ) g ( k 4 ) g(k_3)g(k_4) g ( k 3 ) g ( k 4 ) E 3 ∗ > E 4 ∗ E^{*}_3>E^{*}_4 E 3 ∗ > E 4 ∗ F g g F_{gg} F g g S 34 S_{34} S 34 S 4 S_4 S 4 C C C
F g g s u b = ( 1 − C ) ( 1 − S 4 ) ( 1 − S 34 ) F g g = F g g − S 34 F g g − S 4 F g g − C F g g + S 4 S 34 F g g + C S 34 F g g + C S 4 F g g − C S 4 S 34 F g g . (85) \begin{aligned}
F_{gg}^{\mathrm{sub}}
&=(1-C)(1-S_4)(1-S_{34})F_{gg}\\
&=F_{gg}-S_{34}F_{gg}-S_4F_{gg}-CF_{gg}\\
&\quad+S_4S_{34}F_{gg}+CS_{34}F_{gg}+CS_4F_{gg}-CS_4S_{34}F_{gg}. \tag*{(85)}
\end{aligned} F g g sub = ( 1 − C ) ( 1 − S 4 ) ( 1 − S 34 ) F g g = F g g − S 34 F g g − S 4 F g g − C F g g + S 4 S 34 F g g + C S 34 F g g + C S 4 F g g − C S 4 S 34 F g g . (85) The factor 2 restoring the two energy-ordered sectors cancels the identical-gluon symmetry factor 1 / 2 ! 1/2! 1/2 !
Double-soft subtraction. The double-soft projection removes both gluons from the heavy-pair recoil:
d Φ R R g g = d K 2 2 π d Φ 2 ( Q ; P , K ) d Φ 2 ( K − k 3 − k 4 ; p 1 , p 2 ) [ d k 3 ] K [ d k 4 ] K θ ( E 3 ∗ − E 4 ∗ ) → S 34 d Φ B [ d k 3 ] K [ d k 4 ] K θ ( E 3 ∗ − E 4 ∗ ) . \mathrm{d}\Phi_{\mathrm{RR}}^{gg} = \frac{\mathrm{d}K^2}{2\pi}\,\mathrm{d}\Phi_2(Q;P,K)\,\mathrm{d}\Phi_2(K-k_3-k_4;p_1,p_2)[\mathrm{d}k_3]_K[\mathrm{d}k_4]_K\theta(E_3^*-E_4^*) \\
\xrightarrow{\mathcal{S}_{34}} \mathrm{d}\Phi_B[\mathrm{d}k_3]_K[\mathrm{d}k_4]_K\theta(E_3^*-E_4^*). d Φ RR g g = 2 π d K 2 d Φ 2 ( Q ; P , K ) d Φ 2 ( K − k 3 − k 4 ; p 1 , p 2 ) [ d k 3 ] K [ d k 4 ] K θ ( E 3 ∗ − E 4 ∗ ) S 34 d Φ B [ d k 3 ] K [ d k 4 ] K θ ( E 3 ∗ − E 4 ∗ ) . The unresolved energies obey 0 < E 4 ∗ < E 3 ∗ < E 0 ∗ 0 < E_4^* < E_3^* < E_0^* 0 < E 4 ∗ < E 3 ∗ < E 0 ∗
The massive double-soft current of ref. [42 ] gives
S 34 F g g = g s 4 c ϵ 2 [ C F 2 S 12 ( k 3 ) S 12 ( k 4 ) − C A C F ( S 11 g g − S 12 g g − S 21 g g + S 22 g g ) ] F B , \mathcal{S}_{34}F_{gg}=g_s^4c_\epsilon^2\left[C_F^2\mathcal{S}_{12}(k_3)\mathcal{S}_{12}(k_4)-C_AC_F\left(\mathcal{S}_{11}^{gg}-\mathcal{S}_{12}^{gg}-\mathcal{S}_{21}^{gg}+\mathcal{S}_{22}^{gg}\right)\right]F_B, S 34 F g g = g s 4 c ϵ 2 [ C F 2 S 12 ( k 3 ) S 12 ( k 4 ) − C A C F ( S 11 g g − S 12 g g − S 21 g g + S 22 g g ) ] F B , S i j g g = S i j ( 0 ) + m c 2 S i j ( m ) + m c 2 S j i ( m ) . (86) \mathcal{S}_{ij}^{gg}=\mathcal{S}_{ij}^{(0)}+m_c^2\mathcal{S}_{ij}^{(m)}+m_c^2\mathcal{S}_{ji}^{(m)}. \tag*{(86)} S ij g g = S ij ( 0 ) + m c 2 S ij ( m ) + m c 2 S j i ( m ) . (86) To specify the correlated part, set k 34 = k 3 + k 4 k_{34}=k_3+k_4 k 34 = k 3 + k 4
A i j = ( p i ⋅ k 3 ) ( p j ⋅ k 4 ) + ( p i ⋅ k 4 ) ( p j ⋅ k 3 ) , A_{ij}=(p_i\cdot k_3)(p_j\cdot k_4)+(p_i\cdot k_4)(p_j\cdot k_3), A ij = ( p i ⋅ k 3 ) ( p j ⋅ k 4 ) + ( p i ⋅ k 4 ) ( p j ⋅ k 3 ) , B i j = ( p i ⋅ k 34 ) ( p j ⋅ k 34 ) , B_{ij}=(p_i\cdot k_{34})(p_j\cdot k_{34}), B ij = ( p i ⋅ k 34 ) ( p j ⋅ k 34 ) , C i j = ( p i ⋅ k 3 ) ( p j ⋅ k 4 ) ( p i ⋅ k 4 ) ( p j ⋅ k 3 ) . (87) C_{ij}=(p_i\cdot k_3)(p_j\cdot k_4)(p_i\cdot k_4)(p_j\cdot k_3). \tag*{(87)} C ij = ( p i ⋅ k 3 ) ( p j ⋅ k 4 ) ( p i ⋅ k 4 ) ( p j ⋅ k 3 ) . (87) Then
S i j ( 0 ) = ( 1 − ϵ ) A i j ( k 3 ⋅ k 4 ) 2 B i j − ( p i ⋅ p j ) 2 2 C i j ( 2 − A i j B i j ) + 1 k 3 ⋅ k 4 [ p i ⋅ p j ( p i ⋅ k 3 ) ( p j ⋅ k 4 ) + p i ⋅ p j ( p i ⋅ k 4 ) ( p j ⋅ k 3 ) − p i ⋅ p j B i j ( 2 + A i j 2 2 C i j ) ] , \mathcal{S}_{ij}^{(0)}=\frac{(1-\epsilon)A_{ij}}{(k_3\cdot k_4)^2B_{ij}}-\frac{(p_i\cdot p_j)^2}{2C_{ij}}\left(2-\frac{A_{ij}}{B_{ij}}\right)
+\frac{1}{k_3\cdot k_4}\left[\frac{p_i\cdot p_j}{(p_i\cdot k_3)(p_j\cdot k_4)}+\frac{p_i\cdot p_j}{(p_i\cdot k_4)(p_j\cdot k_3)}-\frac{p_i\cdot p_j}{B_{ij}}\left(2+\frac{A_{ij}^2}{2C_{ij}}\right)\right], S ij ( 0 ) = ( k 3 ⋅ k 4 ) 2 B ij ( 1 − ϵ ) A ij − 2 C ij ( p i ⋅ p j ) 2 ( 2 − B ij A ij ) + k 3 ⋅ k 4 1 [ ( p i ⋅ k 3 ) ( p j ⋅ k 4 ) p i ⋅ p j + ( p i ⋅ k 4 ) ( p j ⋅ k 3 ) p i ⋅ p j − B ij p i ⋅ p j ( 2 + 2 C ij A ij 2 ) ] , S i j ( m ) = − 1 4 ( p i ⋅ k 3 ) ( p i ⋅ k 4 ) ( k 3 ⋅ k 4 ) + p i ⋅ p j 2 C i j p j ⋅ k 34 p i ⋅ k 34 − 1 2 ( k 3 ⋅ k 4 ) B i j [ ( p i ⋅ k 3 ) 2 ( p i ⋅ k 3 ) ( p j ⋅ k 4 ) + ( p i ⋅ k 4 ) 2 ( p i ⋅ k 4 ) ( p j ⋅ k 3 ) ] . (88) \mathcal{S}_{ij}^{(m)}=-\frac{1}{4(p_i\cdot k_3)(p_i\cdot k_4)(k_3\cdot k_4)}+\frac{p_i\cdot p_j}{2C_{ij}}\frac{p_j\cdot k_{34}}{p_i\cdot k_{34}}
-\frac{1}{2(k_3\cdot k_4)B_{ij}}\left[\frac{(p_i\cdot k_3)^2}{(p_i\cdot k_3)(p_j\cdot k_4)}+\frac{(p_i\cdot k_4)^2}{(p_i\cdot k_4)(p_j\cdot k_3)}\right]. \tag*{(88)} S ij ( m ) = − 4 ( p i ⋅ k 3 ) ( p i ⋅ k 4 ) ( k 3 ⋅ k 4 ) 1 + 2 C ij p i ⋅ p j p i ⋅ k 34 p j ⋅ k 34 − 2 ( k 3 ⋅ k 4 ) B ij 1 [ ( p i ⋅ k 3 ) ( p j ⋅ k 4 ) ( p i ⋅ k 3 ) 2 + ( p i ⋅ k 4 ) ( p j ⋅ k 3 ) ( p i ⋅ k 4 ) 2 ] . (88) Integrating the double-soft counterterm over the ordered gluon phase space at fixed Born kinematics gives [44 ]
∫ [ d k 3 ] K [ d k 4 ] K θ ( E 3 ∗ − E 4 ∗ ) S 34 F g g = g s 4 ( 4 π ) 4 [ C F 2 2 I S 2 − 8 C A C F A S ( f A A g g − f A B g g ) ] F B . (89) \int[\mathrm{d}k_3]_K[\mathrm{d}k_4]_K\theta(E_3^*-E_4^*)\mathcal{S}_{34}F_{gg}
=\frac{g_s^4}{(4\pi)^4}\left[\frac{C_F^2}{2}\mathcal{I}_S^2-8C_AC_F\mathcal{A}_S(f_{AA}^{gg}-f_{AB}^{gg})\right]F_B. \tag*{(89)} ∫ [ d k 3 ] K [ d k 4 ] K θ ( E 3 ∗ − E 4 ∗ ) S 34 F g g = ( 4 π ) 4 g s 4 [ 2 C F 2 I S 2 − 8 C A C F A S ( f AA g g − f A B g g ) ] F B . (89) The functions I S \mathcal{I}_S I S A S \mathcal{A}_S A S f i j g g f_{ij}^{gg} f ij g g
Single-soft subtraction and double-soft overlap. The two soft projections give
d Φ R R g g → S 4 d Φ R ( k 3 ) [ d k 4 ] K θ ( E 3 ∗ − E 4 ∗ ) , \mathrm{d}\Phi_{\mathrm{RR}}^{gg}\xrightarrow{\mathcal{S}_4}\mathrm{d}\Phi_R(k_3)[\mathrm{d}k_4]_K\theta(E_3^*-E_4^*), d Φ RR g g S 4 d Φ R ( k 3 ) [ d k 4 ] K θ ( E 3 ∗ − E 4 ∗ ) , d Φ R R g g → S 4 S 34 d Φ B [ d k 3 ] K [ d k 4 ] K θ ( E 3 ∗ − E 4 ∗ ) . \mathrm{d}\Phi_{\mathrm{RR}}^{gg}\xrightarrow{\mathcal{S}_4\mathcal{S}_{34}}\mathrm{d}\Phi_B[\mathrm{d}k_3]_K[\mathrm{d}k_4]_K\theta(E_3^*-E_4^*). d Φ RR g g S 4 S 34 d Φ B [ d k 3 ] K [ d k 4 ] K θ ( E 3 ∗ − E 4 ∗ ) . At fixed k 3 k_3 k 3
[ d k 4 ] K θ ( E 3 ∗ − E 4 ∗ ) = ( E 4 ∗ ) 1 − 2 ϵ 2 ( 2 π ) 3 − 2 ϵ d E 4 ∗ d Ω 4 ∗ , 0 < E 4 ∗ < E 3 ∗ < E 0 ∗ , [dk_4]_K\theta(E_3^* - E_4^*) = \frac{(E_4^*)^{1-2\epsilon}}{2(2\pi)^{3-2\epsilon}}dE_4^*\,d\Omega_4^*, \qquad0 < E_4^* < E_3^* < E_0^*, [ d k 4 ] K θ ( E 3 ∗ − E 4 ∗ ) = 2 ( 2 π ) 3 − 2 ϵ ( E 4 ∗ ) 1 − 2 ϵ d E 4 ∗ d Ω 4 ∗ , 0 < E 4 ∗ < E 3 ∗ < E 0 ∗ , with d Ω 4 ∗ d\Omega_4^* d Ω 4 ∗
The local single-soft term and its double-soft overlap are
S 4 F g g = g s 2 C ϵ D 4 g F R ( k 3 ) , S_4F_{gg}=g_s^2 C_\epsilon\mathcal{D}_4^g F_R(k_3), S 4 F g g = g s 2 C ϵ D 4 g F R ( k 3 ) , S 4 S 34 F g g = g s 4 C ϵ 2 D 4 g C F S 12 ( k 3 ) F B , S_4S_{34}F_{gg}=g_s^4 C_\epsilon^2 \mathcal{D}_4^g C_F S_{12}(k_3)F_B, S 4 S 34 F g g = g s 4 C ϵ 2 D 4 g C F S 12 ( k 3 ) F B , D 4 g = C F S 12 ( k 4 ) + C A [ p 1 ⋅ k 3 ( p 1 ⋅ k 4 ) ( k 3 ⋅ k 4 ) + p 2 ⋅ k 3 ( p 2 ⋅ k 4 ) ( k 3 ⋅ k 4 ) − p 1 ⋅ p 2 ( p 1 ⋅ k 4 ) ( p 2 ⋅ k 4 ) ] . (90) \mathcal{D}_4^g=C_F S_{12}(k_4)+C_A\left[\frac{p_1\cdot k_3}{(p_1\cdot k_4)(k_3\cdot k_4)}+\frac{p_2\cdot k_3}{(p_2\cdot k_4)(k_3\cdot k_4)}-\frac{p_1\cdot p_2}{(p_1\cdot k_4)(p_2\cdot k_4)}\right]. \tag*{(90)} D 4 g = C F S 12 ( k 4 ) + C A [ ( p 1 ⋅ k 4 ) ( k 3 ⋅ k 4 ) p 1 ⋅ k 3 + ( p 2 ⋅ k 4 ) ( k 3 ⋅ k 4 ) p 2 ⋅ k 3 − ( p 1 ⋅ k 4 ) ( p 2 ⋅ k 4 ) p 1 ⋅ p 2 ] . (90) The momenta in each counterterm are evaluated on its corresponding phase space.
Integrating over the softer-gluon momentum at fixed k 3 k_3 k 3
∫ [ d k 4 ] K θ ( E 3 ∗ − E 4 ∗ ) S 4 F g g = g s 2 K S 4 F R ( k 3 ) , \int[dk_4]_K\theta(E_3^*-E_4^*)S_4F_{gg}=g_s^2\mathcal{K}_{S_4}F_R(k_3), ∫ [ d k 4 ] K θ ( E 3 ∗ − E 4 ∗ ) S 4 F g g = g s 2 K S 4 F R ( k 3 ) , ∫ [ d k 4 ] K θ ( E 3 ∗ − E 4 ∗ ) S 4 S 34 F g g = g s 4 C ϵ C F K S 4 S 12 ( k 3 ) F B , \int[dk_4]_K\theta(E_3^*-E_4^*)S_4S_{34}F_{gg}=g_s^4C_\epsilon C_F\mathcal{K}_{S_4}S_{12}(k_3)F_B, ∫ [ d k 4 ] K θ ( E 3 ∗ − E 4 ∗ ) S 4 S 34 F g g = g s 4 C ϵ C F K S 4 S 12 ( k 3 ) F B , K S 4 = C F [ 2 J ( p 1 , p 2 ) − J ( p 1 , p 1 ) − J ( p 2 , p 2 ) ] + C A [ J ( p 1 , k 3 ) + J ( p 2 , k 3 ) − J ( p 1 , p 2 ) ] . (91) \mathcal{K}_{S_4}=C_F[2\mathcal{J}(p_1,p_2)-\mathcal{J}(p_1,p_1)-\mathcal{J}(p_2,p_2)]
+C_A[\mathcal{J}(p_1,k_3)+\mathcal{J}(p_2,k_3)-\mathcal{J}(p_1,p_2)]. \tag*{(91)} K S 4 = C F [ 2 J ( p 1 , p 2 ) − J ( p 1 , p 1 ) − J ( p 2 , p 2 )] + C A [ J ( p 1 , k 3 ) + J ( p 2 , k 3 ) − J ( p 1 , p 2 )] . (91) Here J ( a , b ) ≡ J ( a , b ; E 3 ∗ ) \mathcal{J}(a,b)\equiv\mathcal{J}(a,b;E_3^*) J ( a , b ) ≡ J ( a , b ; E 3 ∗ ) § 12 .§ 2 . The remaining integrations are over d Φ R ( k 3 ) d\Phi_R(k_3) d Φ R ( k 3 ) d Φ B [ d k 3 ] K d\Phi_B[dk_3]_K d Φ B [ d k 3 ] K § 9 .§ 4.2 .
Collinear subtraction. In this ordered channel we define the energy fraction of the softer gluon by z = E 4 ∗ / E ∗ z=E_4^*/E^* z = E 4 ∗ / E ∗ E ∗ = E 3 ∗ + E 4 ∗ E^*=E_3^*+E_4^* E ∗ = E 3 ∗ + E 4 ∗ 0 < z < 1 / 2 0<z<1/2 0 < z < 1/2 k 3 + k 4 k_3+k_4 k 3 + k 4 p = k 3 / ( 1 − z ) p=k_3/(1-z) p = k 3 / ( 1 − z )
d Φ R g g → k 3 + k 4 ↦ p d K 2 2 π d Φ 2 ( Q ; P , K ) d Φ 2 ( K − p ; p 1 , p 2 ) [ d k 3 ] K [ d k 4 ] K θ ( E 3 ∗ − E 4 ∗ ) . d\Phi_R^{gg}\xrightarrow{k_3+k_4\mapsto p}\frac{dK^2}{2\pi}d\Phi_2(Q;P,K)d\Phi_2(K-p;p_1,p_2)[dk_3]_K[dk_4]_K\theta(E_3^*-E_4^*). d Φ R g g k 3 + k 4 ↦ p 2 π d K 2 d Φ 2 ( Q ; P , K ) d Φ 2 ( K − p ; p 1 , p 2 ) [ d k 3 ] K [ d k 4 ] K θ ( E 3 ∗ − E 4 ∗ ) . With E 3 ∗ = ( 1 − z ) E ∗ E_3^*=(1-z)E^* E 3 ∗ = ( 1 − z ) E ∗ E 4 ∗ = z E ∗ E_4^*=zE^* E 4 ∗ = z E ∗ η = ( 1 − cos θ 34 ∗ ) / 2 \eta=(1-\cos\theta_{34}^*)/2 η = ( 1 − cos θ 34 ∗ ) /2 § 10 .§ 2.1 . Factoring out [ d p ] K [dp]_K [ d p ] K d Φ R ( p ) d Φ C , u n r e s g g d\Phi_R(p)d\Phi_{C,\mathrm{unres}}^{gg} d Φ R ( p ) d Φ C , unres g g
d Φ C , u n r e s g g = d Φ C , u n r e s θ ( 1 2 − z ) . d\Phi_{C,\mathrm{unres}}^{gg}=d\Phi_{C,\mathrm{unres}}\theta\left(\frac{1}{2}-z\right). d Φ C , unres g g = d Φ C , unres θ ( 2 1 − z ) . Thus (C.12) is integrated over 0 < z < 1 / 2 0<z<1/2 0 < z < 1/2 0 < η < 1 0<\eta<1 0 < η < 1 p p p
Using the transverse projector of appendix § 10 .§ 2.1 , we have k ⊥ μ = g ⊥ μ ν k 4 ν ∗ k_\perp^\mu=g_\perp^{\mu\nu}k_{4\nu}^* k ⊥ μ = g ⊥ μν k 4 ν ∗ k ⊥ 2 = − 4 z 2 ( E ∗ ) 2 η ( 1 − η ) k_\perp^2=-4z^2(E^*)^2\eta(1-\eta) k ⊥ 2 = − 4 z 2 ( E ∗ ) 2 η ( 1 − η )
P g g μ ν = 2 C A [ − g ⊥ μ ν ( z 1 − z + 1 − z z ) − 2 ( 1 − ϵ ) z ( 1 − z ) k ⊥ μ k ⊥ ν k ⊥ 2 ] . (92) P_{gg}^{\mu\nu}=2C_A\left[-g_\perp^{\mu\nu}\left(\frac{z}{1-z}+\frac{1-z}{z}\right)-2(1-\epsilon)z(1-z)\frac{k_\perp^\mu k_\perp^\nu}{k_\perp^2}\right]. \tag*{(92)} P g g μν = 2 C A [ − g ⊥ μν ( 1 − z z + z 1 − z ) − 2 ( 1 − ϵ ) z ( 1 − z ) k ⊥ 2 k ⊥ μ k ⊥ ν ] . (92) and the local counterterm is
C F g g = g s 2 C ϵ k 3 ⋅ k 4 P g g μ ν F R , μ ν ( p ) , (93) \mathrm{CF}_{gg}=\frac{g_s^2 C_\epsilon}{k_3\cdot k_4}P_{gg}^{\mu\nu}F_{R,\mu\nu}(p), \tag*{(93)} CF g g = k 3 ⋅ k 4 g s 2 C ϵ P g g μν F R , μν ( p ) , (93) k 3 ⋅ k 4 = 2 z ( 1 − z ) ( E ∗ ) 2 η . k_3\cdot k_4=2z(1-z)(E^*)^2\eta. k 3 ⋅ k 4 = 2 z ( 1 − z ) ( E ∗ ) 2 η . Transverse averaging gives
⟨ P g g μ ν F R , μ ν ( p ) ⟩ ⊥ = 2 C A [ z 1 − z + 1 − z z + z ( 1 − z ) ] F R ( p ) . \left\langle P_{gg}^{\mu\nu}F_{R,\mu\nu}(p)\right\rangle_\perp=2C_A\left[\frac{z}{1-z}+\frac{1-z}{z}+z(1-z)\right]F_R(p). ⟨ P g g μν F R , μν ( p ) ⟩ ⊥ = 2 C A [ 1 − z z + z 1 − z + z ( 1 − z ) ] F R ( p ) . Integrating over the ordered splitting variables gives
∫ d Φ C , u n r e s g g C F g g = C g g ( E ∗ , ϵ ) F R ( p ) , \int d\Phi_{\mathrm{C,unres}}^{gg}\,\mathrm{CF}_{gg}=C_{gg}(E^*,\epsilon)F_R(p), ∫ d Φ C , unres g g CF g g = C g g ( E ∗ , ϵ ) F R ( p ) , C g g ( E ∗ , ϵ ) = g s 2 C A ( 4 π ) 2 [ 2 ϵ 2 + 2 L C + 11 / 3 ϵ + L C 2 + 11 3 L C + 134 9 − 9 ζ 2 + O ( ϵ ) ] . (94) C_{gg}(E^*,\epsilon)=\frac{g_s^2C_A}{(4\pi)^2}\left[\frac{2}{\epsilon^2}+\frac{2L_C+11/3}{\epsilon}+L_C^2+\frac{11}{3}L_C+\frac{134}{9}-9\zeta_2+O(\epsilon)\right]. \tag*{(94)} C g g ( E ∗ , ϵ ) = ( 4 π ) 2 g s 2 C A [ ϵ 2 2 + ϵ 2 L C + 11/3 + L C 2 + 3 11 L C + 9 134 − 9 ζ 2 + O ( ϵ ) ] . (94) The higher-order terms in ϵ \epsilon ϵ
C g g ( E ∗ , ϵ ) = g s 2 C A ( 4 π ) 2 ( μ R 2 4 ( E ∗ ) 2 ) ϵ 6 ( 2 − 3 ϵ ) ϵ 2 e γ E ϵ Γ 2 ( 2 − 2 ϵ ) Γ ( 4 − 4 ϵ ) Γ ( 1 − ϵ ) . (95) C_{gg}(E^*,\epsilon)=\frac{g_s^2C_A}{(4\pi)^2}\left(\frac{\mu_R^2}{4(E^*)^2}\right)^\epsilon\frac{6(2-3\epsilon)}{\epsilon^2}\frac{e^{\gamma_E\epsilon}\Gamma^2(2-2\epsilon)}{\Gamma(4-4\epsilon)\Gamma(1-\epsilon)}. \tag*{(95)} C g g ( E ∗ , ϵ ) = ( 4 π ) 2 g s 2 C A ( 4 ( E ∗ ) 2 μ R 2 ) ϵ ϵ 2 6 ( 2 − 3 ϵ ) Γ ( 4 − 4 ϵ ) Γ ( 1 − ϵ ) e γ E ϵ Γ 2 ( 2 − 2 ϵ ) . (95) Double-soft–collinear overlap. Starting from the double-soft phase space, the same change of variables and angular projection give
d Φ B [ d k 3 ] K [ d k 4 ] K θ ( E 3 ∗ − E 4 ∗ ) → C d Φ B d Φ C S , u n r e s g g , d\Phi_B[dk_3]_K[dk_4]_K\theta(E_3^*-E_4^*)\xrightarrow{\mathrm{C}}d\Phi_B\,d\Phi_{\mathrm{CS,unres}}^{gg}, d Φ B [ d k 3 ] K [ d k 4 ] K θ ( E 3 ∗ − E 4 ∗ ) C d Φ B d Φ CS , unres g g , d Φ C S , u n r e s g g = d Π ( p ) d Φ C S , u n r e s g g θ ( E 0 ∗ − ( 1 − z ) E ∗ ) . d\Phi_{\mathrm{CS,unres}}^{gg}=d\Pi(p)\,d\Phi_{\mathrm{CS,unres}}^{gg}\theta(E_0^*-(1-z)E^*). d Φ CS , unres g g = d Π ( p ) d Φ CS , unres g g θ ( E 0 ∗ − ( 1 − z ) E ∗ ) . The reduced heavy momenta satisfy p 1 + p 2 = K p_1+p_2=K p 1 + p 2 = K 0 < z < 1 / 2 0<z<1/2 0 < z < 1/2 0 < E ∗ < E max ∗ ( z ) = E 0 ∗ / ( 1 − z ) 0<E^*<E^*_{\max}(z)=E_0^*/(1-z) 0 < E ∗ < E m a x ∗ ( z ) = E 0 ∗ / ( 1 − z ) 0 < η < 1 0<\eta<1 0 < η < 1 E ∗ = E 0 ∗ E^*=E_0^* E ∗ = E 0 ∗ C S < \mathrm{CS}_{<} CS < C S > \mathrm{CS}_{>} CS > j μ ( p ) j^\mu(p) j μ ( p )
C S 34 F g g = g s 4 C ϵ 2 C F k 3 ⋅ k 4 P g g μ ν j μ ( p ) j ν ( p ) F B . (96) \mathrm{CS}_{34}F_{gg}=\frac{g_s^4C_\epsilon^2C_F}{k_3\cdot k_4}P_{gg}^{\mu\nu}j_\mu(p)j_\nu(p)F_B. \tag*{(96)} CS 34 F g g = k 3 ⋅ k 4 g s 4 C ϵ 2 C F P g g μν j μ ( p ) j ν ( p ) F B . (96) Using − g μ ν j μ j ν = S 12 ( p ) -g^{\mu\nu}j_\mu j_\nu=S_{12}(p) − g μν j μ j ν = S 12 ( p )
⟨ P g g μ ν j μ j ν ⟩ ⊥ = 2 C A [ z 1 − z + 1 − z z + z ( 1 − z ) ] S 12 ( p ) . \left\langle P_{gg}^{\mu\nu}j_\mu j_\nu\right\rangle_\perp=2C_A\left[\frac{z}{1-z}+\frac{1-z}{z}+z(1-z)\right]S_{12}(p). ⟨ P g g μν j μ j ν ⟩ ⊥ = 2 C A [ 1 − z z + z 1 − z + z ( 1 − z ) ] S 12 ( p ) . As in the quark-pair channel, the splitting variables are integrated at fixed p p p C S < \mathrm{CS}_{<} CS < 0 < E ∗ < E 0 ∗ 0<E^*<E_0^* 0 < E ∗ < E 0 ∗ C S > \mathrm{CS}_{>} CS >
∫ d Φ C , u n r e s g g C S < F g g = g s 2 C ϵ C F C g g ( E ∗ , ϵ ) S 12 ( p ) F B , \int d\Phi_{\mathrm{C,unres}}^{gg}\,\mathrm{CS}_{<}F_{gg}=g_s^2C_\epsilon C_F C_{gg}(E^*,\epsilon)S_{12}(p)F_B, ∫ d Φ C , unres g g CS < F g g = g s 2 C ϵ C F C g g ( E ∗ , ϵ ) S 12 ( p ) F B , ∫ d Φ C S , u n r e s g g C S > F g g = K C S , g F B , \int d\Phi_{\mathrm{CS,unres}}^{gg}\,\mathrm{CS}_{>}F_{gg}=\mathcal{K}_{\mathrm{CS},g}F_B, ∫ d Φ CS , unres g g CS > F g g = K CS , g F B , K C S , g = g s 4 C F C A ( 4 π ) 4 4 e 2 ϵ ( γ E + L S ) ϵ 2 Γ ( 2 − 2 ϵ ) h ( β , ϵ ) ( W g , m − W g , 0 ) . (97) \mathcal{K}_{\mathrm{CS},g}=\frac{g_s^4C_FC_A}{(4\pi)^4}\frac{4e^{2\epsilon(\gamma_E+L_S)}}{\epsilon^2\Gamma(2-2\epsilon)}h(\beta,\epsilon)(\mathcal{W}_{g,m}-\mathcal{W}_{g,0}). \tag*{(97)} K CS , g = ( 4 π ) 4 g s 4 C F C A ϵ 2 Γ ( 2 − 2 ϵ ) 4 e 2 ϵ ( γ E + L S ) h ( β , ϵ ) ( W g , m − W g , 0 ) . (97) Here C g g \mathcal{C}_{gg} C g g h h h
For the ordered gluon pair, we factor out C A C_A C A
w g ( z , ϵ ) = 2 [ z ( 1 − z ) ] − 2 ϵ [ z 1 − z + 1 − z z + z ( 1 − z ) ] . w_g(z,\epsilon)=2[z(1-z)]^{-2\epsilon}\left[\frac{z}{1-z}+\frac{1-z}{z}+z(1-z)\right]. w g ( z , ϵ ) = 2 [ z ( 1 − z ) ] − 2 ϵ [ 1 − z z + z 1 − z + z ( 1 − z ) ] . The weights in (C.33) are
W g , 0 = ∫ 0 1 / 2 d z w g ( z , ϵ ) = 2 B ( 2 − 2 ϵ , − 2 ϵ ) + B ( 2 − 2 ϵ , 2 − 2 ϵ ) , (98) \mathcal{W}_{g,0}=\int_0^{1/2} dz\,w_g(z,\epsilon)
=2B(2-2\epsilon,-2\epsilon)+B(2-2\epsilon,2-2\epsilon), \tag*{(98)} W g , 0 = ∫ 0 1/2 d z w g ( z , ϵ ) = 2 B ( 2 − 2 ϵ , − 2 ϵ ) + B ( 2 − 2 ϵ , 2 − 2 ϵ ) , (98) W g , m = ∫ 0 1 / 2 d z w g ( z , ϵ ) ( 1 − z ) 4 ϵ = 2 [ B 1 / 2 ( 2 − 2 ϵ , 2 ϵ ) + B 1 / 2 ( − 2 ϵ , 2 + 2 ϵ ) + B 1 / 2 ( 2 − 2 ϵ , 2 + 2 ϵ ) ] . (99) \mathcal{W}_{g,m}=\int_0^{1/2} dz\,w_g(z,\epsilon)(1-z)^{4\epsilon}
=2\left[B_{1/2}(2-2\epsilon,2\epsilon)+B_{1/2}(-2\epsilon,2+2\epsilon)+B_{1/2}(2-2\epsilon,2+2\epsilon)\right]. \tag*{(99)} W g , m = ∫ 0 1/2 d z w g ( z , ϵ ) ( 1 − z ) 4 ϵ = 2 [ B 1/2 ( 2 − 2 ϵ , 2 ϵ ) + B 1/2 ( − 2 ϵ , 2 + 2 ϵ ) + B 1/2 ( 2 − 2 ϵ , 2 + 2 ϵ ) ] . (99) Their difference is
W g , m − W g , 0 = ϵ ( 131 18 − 22 3 ln 2 − 4 ζ 2 ) + ϵ 2 ( 790 27 − 44 3 ζ 2 − 16 ζ 3 ) + O ( ϵ 3 ) . (100) \mathcal{W}_{g,m}-\mathcal{W}_{g,0}
=\epsilon\left(\frac{131}{18}-\frac{22}{3}\ln2-4\zeta_2\right)
+\epsilon^2\left(\frac{790}{27}-\frac{44}{3}\zeta_2-16\zeta_3\right)+O(\epsilon^3). \tag*{(100)} W g , m − W g , 0 = ϵ ( 18 131 − 3 22 ln 2 − 4 ζ 2 ) + ϵ 2 ( 27 790 − 3 44 ζ 2 − 16 ζ 3 ) + O ( ϵ 3 ) . (100) Single-soft–collinear and iterated overlaps. For these counterterms, we keep the momentum k 3 k_3 k 3
E 4 ∗ = z 1 − z E 3 ∗ , d E 4 ∗ = E 3 ∗ ( 1 − z ) 2 d z , 0 < z < 1 2 E_4^*=\frac{z}{1-z}E_3^*,\qquad dE_4^*=\frac{E_3^*}{(1-z)^2}dz,\qquad0<z<\frac{1}{2} E 4 ∗ = 1 − z z E 3 ∗ , d E 4 ∗ = ( 1 − z ) 2 E 3 ∗ d z , 0 < z < 2 1 and the angular projection ( 1 − η ) − ϵ → 1 (1-\eta)^{-\epsilon}\to1 ( 1 − η ) − ϵ → 1
d Φ R ( k 3 ) [ d k 4 ] K θ ( E 3 ∗ − E 4 ∗ ) → C d Φ R ( k 3 ) d Φ C S 4 , u n r e s , d\Phi_R(k_3)[dk_4]_K\theta(E_3^*-E_4^*)\xrightarrow{\mathcal C}d\Phi_R(k_3)d\Phi_{\mathrm{CS4,unres}}, d Φ R ( k 3 ) [ d k 4 ] K θ ( E 3 ∗ − E 4 ∗ ) C d Φ R ( k 3 ) d Φ CS4 , unres , d Φ B [ d k 3 ] K [ d k 4 ] K θ ( E 3 ∗ − E 4 ∗ ) → C d Φ B [ d k 3 ] K d Φ C S 4 , u n r e s , d\Phi_B[dk_3]_K[dk_4]_K\theta(E_3^*-E_4^*)\xrightarrow{\mathcal C}d\Phi_B[dk_3]_Kd\Phi_{\mathrm{CS4,unres}}, d Φ B [ d k 3 ] K [ d k 4 ] K θ ( E 3 ∗ − E 4 ∗ ) C d Φ B [ d k 3 ] K d Φ CS4 , unres , where
d Φ C S 4 , u n r e s = ( E 3 ∗ ) 2 − 2 ϵ z 1 − 2 ϵ ( 1 − z ) − 3 + 2 ϵ 2 ( 2 π ) 3 − 2 ϵ d z 2 1 − 2 ϵ η − ϵ d η d Ω ⊥ ∗ . (101) d\Phi_{\mathrm{CS4,unres}}=
\frac{(E_3^*)^{2-2\epsilon}z^{1-2\epsilon}(1-z)^{-3+2\epsilon}}{2(2\pi)^{3-2\epsilon}}
\,dz\,2^{1-2\epsilon}\eta^{-\epsilon}d\eta\,d\Omega_\perp^*. \tag*{(101)} d Φ CS4 , unres = 2 ( 2 π ) 3 − 2 ϵ ( E 3 ∗ ) 2 − 2 ϵ z 1 − 2 ϵ ( 1 − z ) − 3 + 2 ϵ d z 2 1 − 2 ϵ η − ϵ d η d Ω ⊥ ∗ . (101) The integration covers 0 < z < 1 / 2 0<z<1/2 0 < z < 1/2 0 < η < 1 0<\eta<1 0 < η < 1 0 < E 4 ∗ < E 3 ∗ < E 0 ∗ 0<E_4^*<E_3^*<E_0^* 0 < E 4 ∗ < E 3 ∗ < E 0 ∗
The local counterterms are
C S 4 F g g = g s 2 c ϵ 2 C A ( 1 − z ) z k 3 ⋅ k 4 F R ( k 3 ) , \mathrm{CS}_4F_{gg}=g_s^2c_\epsilon\frac{2C_A(1-z)}{zk_3\cdot k_4}F_R(k_3), CS 4 F g g = g s 2 c ϵ z k 3 ⋅ k 4 2 C A ( 1 − z ) F R ( k 3 ) , C S 4 S 34 F g g = g s 4 c ϵ 2 2 C A ( 1 − z ) z k 3 ⋅ k 4 C F S 12 ( k 3 ) F B . (102) \mathrm{CS}_4\mathrm{S}_{34}F_{gg}=g_s^4c_\epsilon^2\frac{2C_A(1-z)}{zk_3\cdot k_4}C_F\mathrm{S}_{12}(k_3)F_B. \tag*{(102)} CS 4 S 34 F g g = g s 4 c ϵ 2 z k 3 ⋅ k 4 2 C A ( 1 − z ) C F S 12 ( k 3 ) F B . (102) Here k 3 ⋅ k 4 = 2 ( E 3 ∗ ) 2 z η / ( 1 − z ) k_3\cdot k_4=2(E_3^*)^2z\eta/(1-z) k 3 ⋅ k 4 = 2 ( E 3 ∗ ) 2 z η / ( 1 − z ) ( 1 − z ) / z = E 3 ∗ / E 4 ∗ (1-z)/z=E_3^*/E_4^* ( 1 − z ) / z = E 3 ∗ / E 4 ∗
Using ∫ 0 1 / 2 d z z − 1 − 2 ϵ ( 1 − z ) − 1 + 2 ϵ = − 1 / ( 2 ϵ ) \int_0^{1/2}dz\,z^{-1-2\epsilon}(1-z)^{-1+2\epsilon}=-1/(2\epsilon) ∫ 0 1/2 d z z − 1 − 2 ϵ ( 1 − z ) − 1 + 2 ϵ = − 1/ ( 2 ϵ ) ∫ 0 1 d η η − 1 − ϵ = − 1 / ϵ \int_0^1d\eta\,\eta^{-1-\epsilon}=-1/\epsilon ∫ 0 1 d η η − 1 − ϵ = − 1/ ϵ
∫ d Φ C S 4 , u n r e s C S 4 F g g = C C S 4 ( E 3 ∗ , ϵ ) F R ( k 3 ) , \int d\Phi_{\mathrm{CS4,unres}}\,\mathrm{CS}_4F_{gg}
=C_{\mathrm{CS4}}(E_3^*,\epsilon)F_R(k_3), ∫ d Φ CS4 , unres CS 4 F g g = C CS4 ( E 3 ∗ , ϵ ) F R ( k 3 ) , ∫ d Φ C S 4 , u n r e s C S 4 S 34 F g g = C C S 4 ( E 3 ∗ , ϵ ) g s 2 c ϵ C F S 12 ( k 3 ) F B , (103) \int d\Phi_{\mathrm{CS4,unres}}\,\mathrm{CS}_4\mathrm{S}_{34}F_{gg}
=C_{\mathrm{CS4}}(E_3^*,\epsilon)g_s^2c_\epsilon C_F\mathrm{S}_{12}(k_3)F_B, \tag*{(103)} ∫ d Φ CS4 , unres CS 4 S 34 F g g = C CS4 ( E 3 ∗ , ϵ ) g s 2 c ϵ C F S 12 ( k 3 ) F B , (103) C C S 4 ( E 3 ∗ , ϵ ) = g s 2 C A ( 4 π ) 2 [ 2 ϵ 2 + 2 L 3 ϵ + L 3 2 − ζ 2 + O ( ϵ ) ] . C_{\mathrm{CS4}}(E_3^*,\epsilon)
=\frac{g_s^2C_A}{(4\pi)^2}\left[\frac{2}{\epsilon^2}+\frac{2L_3}{\epsilon}+L_3^2-\zeta_2+O(\epsilon)\right]. C CS4 ( E 3 ∗ , ϵ ) = ( 4 π ) 2 g s 2 C A [ ϵ 2 2 + ϵ 2 L 3 + L 3 2 − ζ 2 + O ( ϵ ) ] . Here L 3 = ln [ μ R 2 / ( 4 ( E 3 ∗ ) 2 ) ] L_3 = \ln[\mu_R^2/(4(E_3^*)^2)] L 3 = ln [ μ R 2 / ( 4 ( E 3 ∗ ) 2 )]
C C S 4 ( E 3 ∗ , ϵ ) = g s 2 C A ( 4 π ) 2 ( μ R 2 4 ( E 3 ∗ ) 2 ) ϵ 2 e γ E ϵ ϵ 2 Γ ( 1 − ϵ ) . (104) \mathcal{C}_{\mathrm{CS}_4}(E_3^*,\epsilon) = \frac{g_s^2 C_A}{(4\pi)^2}\left(\frac{\mu_R^2}{4(E_3^*)^2}\right)^\epsilon\frac{2e^{\gamma_E\epsilon}}{\epsilon^2\Gamma(1-\epsilon)}. \tag*{(104)} C CS 4 ( E 3 ∗ , ϵ ) = ( 4 π ) 2 g s 2 C A ( 4 ( E 3 ∗ ) 2 μ R 2 ) ϵ ϵ 2 Γ ( 1 − ϵ ) 2 e γ E ϵ . (104) The remaining integrations are over d Φ R ( k 3 ) d\Phi_R(k_3) d Φ R ( k 3 ) d Φ B [ d k 3 ] K d\Phi_B[dk_3]_K d Φ B [ d k 3 ] K
Real-virtual contribution (RV)
The real-virtual contribution is integrated over d Φ R d\Phi_R d Φ R d Φ B [ d k ] K d\Phi_B[dk]_K d Φ B [ d k ] K 0 < E k ∗ < E 0 ∗ 0 < E_k^* < E_0^* 0 < E k ∗ < E 0 ∗
Local soft subtraction
Applying soft subtraction to the finite remainder and pole term in (B.13) gives
F R V s u b = ( 1 − S k ) F R V f i n , F_{\mathrm{RV}}^{\mathrm{sub}} = (1-\mathcal{S}_k)F_{\mathrm{RV}}^{\mathrm{fin}}, F RV sub = ( 1 − S k ) F RV fin , F R V , S U s u b = ( 1 − S k ) [ 2 a s Re Z c c ˉ g ( 1 ) F R ] . (105) F_{\mathrm{RV,SU}}^{\mathrm{sub}} = (1-\mathcal{S}_k)\left[2a_s\operatorname{Re}Z_{c\bar{c}g}^{(1)}F_R\right]. \tag*{(105)} F RV , SU sub = ( 1 − S k ) [ 2 a s Re Z c c ˉ g ( 1 ) F R ] . (105) These terms enter R V F \mathrm{RV}_F RV F S U \mathrm{SU} SU
The soft limit uses the massive one-loop soft current [65 , 66 ] . In the normalisation of ref. [42 ] ,
R 12 ( 1 ) ( k ) = 4 C A [ μ R 2 p 1 ⋅ p 2 2 ( p 1 ⋅ k ) ( p 2 ⋅ k ) ] ϵ [ ∑ n = − 2 1 ϵ n R n + O ( ϵ 2 ) ] , R n = − 1 2 r n . (106) R_{12}^{(1)}(k) = 4C_A\left[\frac{\mu_R^2p_1\cdot p_2}{2(p_1\cdot k)(p_2\cdot k)}\right]^\epsilon\left[\sum_{n=-2}^{1}\epsilon^n R_n + O(\epsilon^2)\right], \qquad R_n=-\frac{1}{2}r_n. \tag*{(106)} R 12 ( 1 ) ( k ) = 4 C A [ 2 ( p 1 ⋅ k ) ( p 2 ⋅ k ) μ R 2 p 1 ⋅ p 2 ] ϵ [ n = − 2 ∑ 1 ϵ n R n + O ( ϵ 2 ) ] , R n = − 2 1 r n . (106) Here r n r_n r n [66 ] . The coefficients R − 2 , … , R 1 R_{-2},\ldots,R_1 R − 2 , … , R 1
The renormalised interference has the soft limit [42 ]
S k F R V = g s 2 C ϵ C F S 12 ( k ) { F V f i n + a s [ R 12 ( 1 ) ( k ) + Z α s ( 1 ) + Z A ( 1 ) + 2 Re Z 1 ] F B } , \mathcal{S}_kF_{\mathrm{RV}} = g_s^2C_\epsilon C_F\mathcal{S}_{12}(k)\left\{F_V^{\mathrm{fin}}+a_s\left[R_{12}^{(1)}(k)+Z_{\alpha_s}^{(1)}+Z_A^{(1)}+2\operatorname{Re}Z_1\right]F_B\right\}, S k F RV = g s 2 C ϵ C F S 12 ( k ) { F V fin + a s [ R 12 ( 1 ) ( k ) + Z α s ( 1 ) + Z A ( 1 ) + 2 Re Z 1 ] F B } , Z α s ( 1 ) = − β 0 ( 4 ) ϵ , Z A ( 1 ) = β 0 , Q ϵ ( μ R 2 m c 2 ) ϵ Γ ( 1 + ϵ ) e γ E ϵ . (107) Z_{\alpha_s}^{(1)}=-\frac{\beta_0^{(4)}}{\epsilon}, \qquad Z_A^{(1)}=\frac{\beta_{0,Q}}{\epsilon}\left(\frac{\mu_R^2}{m_c^2}\right)^\epsilon\Gamma(1+\epsilon)e^{\gamma_E\epsilon}. \tag*{(107)} Z α s ( 1 ) = − ϵ β 0 ( 4 ) , Z A ( 1 ) = ϵ β 0 , Q ( m c 2 μ R 2 ) ϵ Γ ( 1 + ϵ ) e γ E ϵ . (107) Here Z 1 Z_1 Z 1 Z α s ( 1 ) Z_{\alpha_s}^{(1)} Z α s ( 1 ) Z A ( 1 ) Z_A^{(1)} Z A ( 1 )
Applying the soft limit to (B.13) gives the counterterm for the finite remainder:
S k F R V f i n = S k F R V − S k [ 2 a s Re Z c c ˉ g ( 1 ) F R ] . \mathcal{S}_kF_{\mathrm{RV}}^{\mathrm{fin}}=\mathcal{S}_kF_{\mathrm{RV}}-\mathcal{S}_k\left[2a_s\operatorname{Re}Z_{c\bar{c}g}^{(1)}F_R\right]. S k F RV fin = S k F RV − S k [ 2 a s Re Z c c ˉ g ( 1 ) F R ] . The two terms are combined on the same soft phase space, retaining their dimensional prefactors until the explicit poles cancel. The second term is also the counterterm subtracted in F R V , S U s u b F_{\mathrm{RV,SU}}^{\mathrm{sub}} F RV , SU sub
Integrated soft counterterm
The d d d
S i n t ( 0 ) = C ϵ ∫ [ d k ] K S 12 ( k ) = I S ( β , ϵ , L S ) ( 4 π ) 2 , \mathcal{S}_{\mathrm{int}}^{(0)}=C_{\epsilon}\int[dk]_{K}\,S_{12}(k)=\frac{\mathcal{I}_{S}(\beta,\epsilon,L_{S})}{(4\pi)^2}, S int ( 0 ) = C ϵ ∫ [ d k ] K S 12 ( k ) = ( 4 π ) 2 I S ( β , ϵ , L S ) , R i n t ( 1 ) = C ϵ ∫ [ d k ] K S 12 ( k ) R 12 ( 1 ) ( k ) = 4 C A ( 4 π ) 2 e 2 ϵ L S ∑ n = − 3 0 ϵ n R n ( β ) + O ( ϵ ) . (108) \mathcal{R}_{\mathrm{int}}^{(1)}=C_{\epsilon}\int[dk]_{K}\,S_{12}(k)R_{12}^{(1)}(k)
=\frac{4C_{A}}{(4\pi)^2}e^{2\epsilon L_{S}}\sum_{n=-3}^{0}\epsilon^n\mathcal{R}_{n}(\beta)+O(\epsilon). \tag*{(108)} R int ( 1 ) = C ϵ ∫ [ d k ] K S 12 ( k ) R 12 ( 1 ) ( k ) = ( 4 π ) 2 4 C A e 2 ϵ L S n = − 3 ∑ 0 ϵ n R n ( β ) + O ( ϵ ) . (108) The tree-level integral I S \mathcal{I}_{S} I S ϵ \epsilon ϵ R − 3 , … , R 0 \mathcal{R}_{-3},\ldots,\mathcal{R}_{0} R − 3 , … , R 0
The integrated counterterm to be added back is
∫ [ d k ] K S k F R V = F R V , F R + F R V , D U = g s 2 C F S i n t ( 0 ) F V f i n + g s 2 C F a s { R i n t ( 1 ) + [ Z α s ( 1 ) + Z A ( 1 ) + 2 Re Z 1 ] S i n t ( 0 ) } F B . (109) \begin{aligned}
\int[dk]_{K}\,S_{k}F_{\mathrm{RV}}=F_{\mathrm{RV,FR}}+F_{\mathrm{RV,DU}} \\
&=g_{s}^{2}C_{F}\mathcal{S}_{\mathrm{int}}^{(0)}F_{V}^{\mathrm{fin}}+g_{s}^{2}C_{F}a_{s}\left\{\mathcal{R}_{\mathrm{int}}^{(1)}+\left[Z_{\alpha_{s}}^{(1)}+Z_{A}^{(1)}+2\operatorname{Re}Z_{1}\right]\mathcal{S}_{\mathrm{int}}^{(0)}\right\}F_{B}. \tag*{(109)}
\end{aligned} ∫ [ d k ] K S k F RV = F RV , FR + F RV , DU = g s 2 C F S int ( 0 ) F V fin + g s 2 C F a s { R int ( 1 ) + [ Z α s ( 1 ) + Z A ( 1 ) + 2 Re Z 1 ] S int ( 0 ) } F B . (109) The terms proportional to F V f i n F_{V}^{\mathrm{fin}} F V fin F B F_{B} F B F R V , F R F_{\mathrm{RV,FR}} F RV , FR F R V , D U F_{\mathrm{RV,DU}} F RV , DU (22) and (23) .
Phase-space parametrisation and numerical integration
We derive common phase-space parametrisations for the real terms and local counterterms of appendix C, starting from the d = 4 − 2 ϵ d=4-2\epsilon d = 4 − 2 ϵ P P P
Outer phase space and heavy-pair kinematics
Our phase-space convention is
d Φ n ( d ) ( L ; q 1 , … , q n ) = ( 2 π ) d δ ( d ) ( L − ∑ i = 1 n q i ) ∏ i = 1 n d d − 1 q i ( 2 π ) d − 1 2 q i 0 , q i 0 = q i 2 + m i 2 . (110) d\Phi_{n}^{(d)}(L;q_{1},\ldots,q_{n})=(2\pi)^{d}\delta^{(d)}\left(L-\sum_{i=1}^{n}q_{i}\right)\prod_{i=1}^{n}\frac{d^{d-1}q_{i}}{(2\pi)^{d-1}2q_{i}^{0}},\qquad q_{i}^{0}=\sqrt{\mathbf{q}_{i}^{2}+m_{i}^{2}}. \tag*{(110)} d Φ n ( d ) ( L ; q 1 , … , q n ) = ( 2 π ) d δ ( d ) ( L − i = 1 ∑ n q i ) i = 1 ∏ n ( 2 π ) d − 1 2 q i 0 d d − 1 q i , q i 0 = q i 2 + m i 2 . (110) We denote the solid-angle element on the unit sphere S ν S^{\nu} S ν d Ω ν d\Omega_{\nu} d Ω ν ν \nu ν
d Ω ν = ( sin ϑ ) ν − 1 d ϑ d Ω ν − 1 , Ω ν ≡ ∫ d Ω ν = 2 π ( ν + 1 ) / 2 Γ ( ( ν + 1 ) / 2 ) . d\Omega_{\nu}=(\sin\vartheta)^{\nu-1}d\vartheta\,d\Omega_{\nu-1},\qquad\Omega_{\nu}\equiv\int d\Omega_{\nu}=\frac{2\pi^{(\nu+1)/2}}{\Gamma((\nu+1)/2)}. d Ω ν = ( sin ϑ ) ν − 1 d ϑ d Ω ν − 1 , Ω ν ≡ ∫ d Ω ν = Γ (( ν + 1 ) /2 ) 2 π ( ν + 1 ) /2 . For a two-body final state, integrating the momentum-conserving delta function in the rest frame of L L L
d Φ 2 ( d ) ( L ; q a , q b ) = λ ( L 2 , m a 2 , m b 2 ) ( d − 3 ) / 2 2 d − 1 ( 2 π ) d − 2 ( L 2 ) ( d − 2 ) / 2 d Ω d − 2 , (111) d\Phi_2^{(d)}(L;q_a,q_b)=\frac{\lambda(L^2,m_a^2,m_b^2)^{(d-3)/2}}{2^{d-1}(2\pi)^{d-2}(L^2)^{(d-2)/2}}\,d\Omega_{d-2}, \tag*{(111)} d Φ 2 ( d ) ( L ; q a , q b ) = 2 d − 1 ( 2 π ) d − 2 ( L 2 ) ( d − 2 ) /2 λ ( L 2 , m a 2 , m b 2 ) ( d − 3 ) /2 d Ω d − 2 , (111) where λ ( x , y , z ) = x 2 + y 2 + z 2 − 2 x y − 2 x z − 2 y z \lambda(x,y,z)=x^2+y^2+z^2-2xy-2xz-2yz λ ( x , y , z ) = x 2 + y 2 + z 2 − 2 x y − 2 x z − 2 y z
For the outer factor Q → P + K Q\to P+K Q → P + K
d Φ o u t ( d ) = d K 2 2 π ∫ S d − 3 d Φ 2 ( d ) ( Q ; P , K ) = Ω d − 3 λ ( s , M ψ 2 , K 2 ) ( d − 3 ) / 2 2 d − 1 ( 2 π ) d − 1 s ( d − 2 ) / 2 ( sin 2 θ ) ( d − 4 ) / 2 d K 2 d cos θ . (112) \begin{aligned}
d\Phi_{\mathrm{out}}^{(d)}=\frac{dK^2}{2\pi}\int_{S^{d-3}}d\Phi_2^{(d)}(Q;P,K) \\
&=\frac{\Omega_{d-3}\lambda(s,M_\psi^2,K^2)^{(d-3)/2}}{2^{d-1}(2\pi)^{d-1}s^{(d-2)/2}}(\sin^2\theta)^{(d-4)/2}\,dK^2\,d\cos\theta. \tag*{(112)}
\end{aligned} d Φ out ( d ) = 2 π d K 2 ∫ S d − 3 d Φ 2 ( d ) ( Q ; P , K ) = 2 d − 1 ( 2 π ) d − 1 s ( d − 2 ) /2 Ω d − 3 λ ( s , M ψ 2 , K 2 ) ( d − 3 ) /2 ( sin 2 θ ) ( d − 4 ) /2 d K 2 d cos θ . (112) The range is 4 m c 2 < K 2 < ( s − M ψ ) 2 4m_c^2<K^2<(\sqrt{s}-M_\psi)^2 4 m c 2 < K 2 < ( s − M ψ ) 2
For the heavy pair, let T = p 1 + p 2 T=p_1+p_2 T = p 1 + p 2 n T ∈ S d − 2 \boldsymbol{n}_T\in S^{d-2} n T ∈ S d − 2 T T T d Ω T d\Omega_T d Ω T
p 1 , 2 ( T ) = ( T 2 2 , ± T 2 − 4 m c 2 2 n T ) , p_{1,2}^{(T)}=\left(\frac{\sqrt{T^2}}{2},\pm\frac{\sqrt{T^2-4m_c^2}}{2}\boldsymbol{n}_T\right), p 1 , 2 ( T ) = ( 2 T 2 , ± 2 T 2 − 4 m c 2 n T ) , d Φ 2 ( d ) ( T ) ≡ d Φ 2 ( d ) ( T ; p 1 , p 2 ) = ( T 2 ) ( d − 4 ) / 2 2 d − 1 ( 2 π ) d − 2 ( 1 − 4 m c 2 T 2 ) ( d − 3 ) / 2 d Ω T . (113) d\Phi_2^{(d)}(T)\equiv d\Phi_2^{(d)}(T;p_1,p_2)=\frac{(T^2)^{(d-4)/2}}{2^{d-1}(2\pi)^{d-2}}\left(1-\frac{4m_c^2}{T^2}\right)^{(d-3)/2}d\Omega_T. \tag*{(113)} d Φ 2 ( d ) ( T ) ≡ d Φ 2 ( d ) ( T ; p 1 , p 2 ) = 2 d − 1 ( 2 π ) d − 2 ( T 2 ) ( d − 4 ) /2 ( 1 − T 2 4 m c 2 ) ( d − 3 ) /2 d Ω T . (113) We boost p 1 , 2 ( T ) p_{1,2}^{(T)} p 1 , 2 ( T ) K K K T / T ∗ 0 \boldsymbol{T}/T^{*0} T / T ∗ 0 K / K 0 \boldsymbol{K}/K^0 K / K 0 T = K T=K T = K d Φ B ( d ) = d Φ o u t ( d ) d Φ 2 ( d ) ( K ) d\Phi_B^{(d)}=d\Phi_{\mathrm{out}}^{(d)}d\Phi_2^{(d)}(K) d Φ B ( d ) = d Φ out ( d ) d Φ 2 ( d ) ( K )
For the finite Born integral, we set ϵ = 0 \epsilon=0 ϵ = 0 ( d ) (d) ( d )
d Φ o u t = d Φ o u t ( d ) ∣ d = 4 = λ ( s , M ψ 2 , K 2 ) 32 π 2 s d K 2 d cos θ . (114) d\Phi_{\mathrm{out}}=d\Phi_{\mathrm{out}}^{(d)}\big|_{d=4}=\frac{\sqrt{\lambda(s,M_\psi^2,K^2)}}{32\pi^2s}\,dK^2\,d\cos\theta. \tag*{(114)} d Φ out = d Φ out ( d ) d = 4 = 32 π 2 s λ ( s , M ψ 2 , K 2 ) d K 2 d cos θ . (114) With the beam along the z z z x z xz x z P \boldsymbol{P} P
E ψ = s + M ψ 2 − K 2 2 s , ∣ P ∣ = λ ( s , M ψ 2 , K 2 ) 2 s , E_\psi=\frac{s+M_\psi^2-K^2}{2\sqrt{s}},\qquad|\boldsymbol{P}|=\frac{\sqrt{\lambda(s,M_\psi^2,K^2)}}{2\sqrt{s}}, E ψ = 2 s s + M ψ 2 − K 2 , ∣ P ∣ = 2 s λ ( s , M ψ 2 , K 2 ) , P = ( E ψ , ∣ P ∣ sin θ , 0 , ∣ P ∣ cos θ ) , K = Q − P . (115) P=(E_\psi,|\boldsymbol{P}|\sin\theta,0,|\boldsymbol{P}|\cos\theta),\qquad K=Q-P. \tag*{(115)} P = ( E ψ , ∣ P ∣ sin θ , 0 , ∣ P ∣ cos θ ) , K = Q − P . (115) The corresponding range is M ψ < E ψ < ( s + M ψ 2 − 4 m c 2 ) / ( 2 s ) M_\psi<E_\psi<(s+M_\psi^2-4m_c^2)/(2\sqrt{s}) M ψ < E ψ < ( s + M ψ 2 − 4 m c 2 ) / ( 2 s ) s / 2 \sqrt{s}/2 s /2 M ψ = 2 m c M_\psi=2m_c M ψ = 2 m c
The heavy-quark direction and solid-angle element in the pair rest frame are
n T ≡ n ( θ T , ϕ T ) = ( sin θ T cos ϕ T , sin θ T sin ϕ T , cos θ T ) , \boldsymbol{n}_T\equiv\boldsymbol{n}(\theta_T,\phi_T)=(\sin\theta_T\cos\phi_T,\sin\theta_T\sin\phi_T,\cos\theta_T), n T ≡ n ( θ T , ϕ T ) = ( sin θ T cos ϕ T , sin θ T sin ϕ T , cos θ T ) , d Ω T = d cos θ T d ϕ T . (116) d\Omega_T=d\cos\theta_T\,d\phi_T. \tag*{(116)} d Ω T = d cos θ T d ϕ T . (116) The two-body phase space reduces to
d Φ 2 ( T ) ≡ d Φ 2 ( T ; p 1 , p 2 ) = 1 32 π 2 1 − 4 m c 2 T 2 d Ω T . (117) d\Phi_2(T) \equiv d\Phi_2(T;p_1,p_2) = \frac{1}{32\pi^2}\sqrt{1-\frac{4m_c^2}{T^2}}\,d\Omega_T. \tag*{(117)} d Φ 2 ( T ) ≡ d Φ 2 ( T ; p 1 , p 2 ) = 32 π 2 1 1 − T 2 4 m c 2 d Ω T . (117) The common outer variables are mapped to u i ∈ [ 0 , 1 ] u_i \in[0,1] u i ∈ [ 0 , 1 ]
K 2 = 4 m c 2 + Δ K u 1 , cos θ = 2 u 2 − 1 , Δ K = ( s − M ψ ) 2 − 4 m c 2 , K^2 = 4m_c^2 + \Delta_K u_1,\qquad\cos\theta= 2u_2-1,\qquad\Delta_K = \left(\sqrt{s}-M_\psi\right)^2-4m_c^2, K 2 = 4 m c 2 + Δ K u 1 , cos θ = 2 u 2 − 1 , Δ K = ( s − M ψ ) 2 − 4 m c 2 , d Φ o u t = Δ K λ ( s , M ψ 2 , K 2 ) 16 π 2 s d u 1 d u 2 . (118) d\Phi_{\mathrm{out}} = \frac{\Delta_K\sqrt{\lambda(s,M_\psi^2,K^2)}}{16\pi^2s}\,du_1\,du_2. \tag*{(118)} d Φ out = 16 π 2 s Δ K λ ( s , M ψ 2 , K 2 ) d u 1 d u 2 . (118) For the Born phase space, set T = K T=K T = K cos θ T = 2 u 3 − 1 \cos\theta_T=2u_3-1 cos θ T = 2 u 3 − 1 ϕ T = 2 π u 4 \phi_T=2\pi u_4 ϕ T = 2 π u 4
d Φ B = d Φ o u t d Φ 2 ( K ) = J B d 4 u , J B = Δ K λ ( s , M ψ 2 , K 2 ) β 128 π 3 s . (119) d\Phi_B=d\Phi_{\mathrm{out}}\,d\Phi_2(K)=J_B\,d^4u,\qquad J_B=\frac{\Delta_K\sqrt{\lambda(s,M_\psi^2,K^2)}\,\beta}{128\pi^3s}. \tag*{(119)} d Φ B = d Φ out d Φ 2 ( K ) = J B d 4 u , J B = 128 π 3 s Δ K λ ( s , M ψ 2 , K 2 ) β . (119) Single-gluon emission
For 0 < E k ∗ < E 0 ∗ 0<E_k^*<E_0^* 0 < E k ∗ < E 0 ∗
d Φ R ( d ) = d Φ o u t ( d ) d Φ 3 ( d ) ( K ; p 1 , p 2 , k ) = d Φ o u t ( d ) d Φ 2 ( d ) ( K − k ) d Π ( d ) ( k ) , d\Phi_R^{(d)}=d\Phi_{\mathrm{out}}^{(d)}d\Phi_3^{(d)}(K;p_1,p_2,k)=d\Phi_{\mathrm{out}}^{(d)}d\Phi_2^{(d)}(K-k)\,d\Pi^{(d)}(k), d Φ R ( d ) = d Φ out ( d ) d Φ 3 ( d ) ( K ; p 1 , p 2 , k ) = d Φ out ( d ) d Φ 2 ( d ) ( K − k ) d Π ( d ) ( k ) , d Φ S ( d ) ≡ d Φ B ( d ) [ d k ] K = d Φ o u t ( d ) d Φ 2 ( d ) ( K ) d Π ( d ) ( k ) , d\Phi_S^{(d)}\equiv d\Phi_B^{(d)}[dk]_K=d\Phi_{\mathrm{out}}^{(d)}d\Phi_2^{(d)}(K)\,d\Pi^{(d)}(k), d Φ S ( d ) ≡ d Φ B ( d ) [ d k ] K = d Φ out ( d ) d Φ 2 ( d ) ( K ) d Π ( d ) ( k ) , d Π ( d ) ( k ) = ( E k ∗ ) d − 3 2 ( 2 π ) d − 1 d E k ∗ d Ω k ∗ . (120) d\Pi^{(d)}(k)=\frac{(E_k^*)^{d-3}}{2(2\pi)^{d-1}}\,dE_k^*\,d\Omega_k^*. \tag*{(120)} d Π ( d ) ( k ) = 2 ( 2 π ) d − 1 ( E k ∗ ) d − 3 d E k ∗ d Ω k ∗ . (120) Here [ d k ] K = d Π ( d ) ( k ) θ ( E 0 ∗ − E k ∗ ) [dk]_K=d\Pi^{(d)}(k)\theta(E_0^*-E_k^*) [ d k ] K = d Π ( d ) ( k ) θ ( E 0 ∗ − E k ∗ ) d Ω k ∗ d\Omega_k^* d Ω k ∗ S d − 2 S^{d-2} S d − 2 K K K
For a unit direction n k ∗ \boldsymbol{n}_k^* n k ∗
k ∗ = E k ∗ ( 1 , n k ∗ ) , T R = K − k , T R 2 = W 2 − 2 W E k ∗ , T S = K , T S 2 = W 2 . (121) k^*=E_k^*(1,\boldsymbol{n}_k^*),\qquad
T_R=K-k,\qquad T_R^2=W^2-2WE_k^*,\qquad T_S=K,\qquad T_S^2=W^2. \tag*{(121)} k ∗ = E k ∗ ( 1 , n k ∗ ) , T R = K − k , T R 2 = W 2 − 2 W E k ∗ , T S = K , T S 2 = W 2 . (121) Combining (D.4) and (D.11) gives, for X = R , S X=R,S X = R , S
d Φ X ( d ) = d Φ o u t ( d ) ( E k ∗ ) d − 3 ( T X 2 ) ( d − 4 ) / 2 2 d ( 2 π ) 2 d − 3 ( 1 − 4 m c 2 T X 2 ) ( d − 3 ) / 2 d E k ∗ d Ω k ∗ d Ω T . (122) d\Phi_X^{(d)}=d\Phi_{\mathrm{out}}^{(d)}
\frac{(E_k^*)^{d-3}(T_X^2)^{(d-4)/2}}{2^d(2\pi)^{2d-3}}
\left(1-\frac{4m_c^2}{T_X^2}\right)^{(d-3)/2}
dE_k^*\,d\Omega_k^*\,d\Omega_T. \tag*{(122)} d Φ X ( d ) = d Φ out ( d ) 2 d ( 2 π ) 2 d − 3 ( E k ∗ ) d − 3 ( T X 2 ) ( d − 4 ) /2 ( 1 − T X 2 4 m c 2 ) ( d − 3 ) /2 d E k ∗ d Ω k ∗ d Ω T . (122) For the finite single-real integrals, we set ϵ = 0 \epsilon=0 ϵ = 0 d Ω k ∗ = d cos θ k ∗ d ϕ k ∗ d\Omega_k^*=d\cos\theta_k^*\,d\phi_k^* d Ω k ∗ = d cos θ k ∗ d ϕ k ∗
d Φ X = d Φ o u t E k ∗ 512 π 5 1 − 4 m c 2 T X 2 d E k ∗ d Ω k ∗ d Ω T . (123) d\Phi_X=d\Phi_{\mathrm{out}}\frac{E_k^*}{512\pi^5}\sqrt{1-\frac{4m_c^2}{T_X^2}}\,dE_k^*\,d\Omega_k^*\,d\Omega_T. \tag*{(123)} d Φ X = d Φ out 512 π 5 E k ∗ 1 − T X 2 4 m c 2 d E k ∗ d Ω k ∗ d Ω T . (123) At fixed K 2 K^2 K 2 cos θ \cos\theta cos θ E k ∗ E_k^* E k ∗ θ k ∗ \theta_k^* θ k ∗ ϕ k ∗ \phi_k^* ϕ k ∗ θ T \theta_T θ T ϕ T \phi_T ϕ T n k ∗ = n ( θ k ∗ , ϕ k ∗ ) \boldsymbol{n}_k^*=n(\theta_k^*,\phi_k^*) n k ∗ = n ( θ k ∗ , ϕ k ∗ ) T = T R T=T_R T = T R T S T_S T S
We combine the outer map (D.9) with
E k ∗ = E 0 ∗ u 3 , cos θ k ∗ = 2 u 4 − 1 , ϕ k ∗ = 2 π u 5 , E_k^*=E_0^*u_3,\qquad\cos\theta_k^*=2u_4-1,\qquad\phi_k^*=2\pi u_5, E k ∗ = E 0 ∗ u 3 , cos θ k ∗ = 2 u 4 − 1 , ϕ k ∗ = 2 π u 5 , cos θ T = 2 u 6 − 1 , ϕ T = 2 π u 7 . (124) \cos\theta_T=2u_6-1,\qquad\phi_T=2\pi u_7. \tag*{(124)} cos θ T = 2 u 6 − 1 , ϕ T = 2 π u 7 . (124) Then d Φ X = J X g d 7 u d\Phi_X = J_X^g\,d^7u d Φ X = J X g d 7 u
J 0 g = Δ K λ ( s , M ψ 2 , K 2 ) E k ∗ E 0 ∗ 512 π 5 s , J_0^g = \frac{\Delta_K\sqrt{\lambda(s,M_{\psi}^2,K^2)}E_k^*E_0^*}{512\pi^5s}, J 0 g = 512 π 5 s Δ K λ ( s , M ψ 2 , K 2 ) E k ∗ E 0 ∗ , J R g = J 0 g 1 − 4 m c 2 T R 2 , J S g = J 0 g β . (125) J_R^g = J_0^g\sqrt{1-\frac{4m_c^2}{T_R^2}}, \qquad J_S^g = J_0^g\beta. \tag*{(125)} J R g = J 0 g 1 − T R 2 4 m c 2 , J S g = J 0 g β . (125) At fixed K K K p i , R → p i , S p_{i,R}\to p_{i,S} p i , R → p i , S J R g / J S g → 1 J_R^g/J_S^g\to1 J R g / J S g → 1 E k ∗ → 0 E_k^*\to0 E k ∗ → 0
The finite real contribution is
R F = N ∫ [ 0 , 1 ] 7 d 7 u [ J R g F R ( P ; p 1 , R , p 2 , R , k ) − J S g ( S k F R ) ( P ; p 1 , S , p 2 , S , k ) ] . (126) R_F = \mathcal{N}\int_{[0,1]^7}d^7u\left[J_R^g F_R(P;p_{1,R},p_{2,R},k)-J_S^g(\mathcal{S}_kF_R)(P;p_{1,S},p_{2,S},k)\right]. \tag*{(126)} R F = N ∫ [ 0 , 1 ] 7 d 7 u [ J R g F R ( P ; p 1 , R , p 2 , R , k ) − J S g ( S k F R ) ( P ; p 1 , S , p 2 , S , k ) ] . (126) For R V F R V_F R V F F R F_R F R F R V f i n F_{RV}^{\mathrm{fin}} F R V fin (18) , the real map applies to the combined RR and RV insertion, and the soft map to its S k \mathcal{S}_k S k
Light-quark-pair emission
Separating the two massless momenta from the heavy-pair phase space gives
d Φ R R q q ˉ , ( d ) = d Φ o u t ( d ) d Φ 4 ( d ) ( K ; p 1 , p 2 , k 3 , k 4 ) = d Φ o u t ( d ) d Φ 4 ( d ) ( K − k 3 − k 4 ) d Π ( d ) ( k 3 ) d Π ( d ) ( k 4 ) . (127) \begin{aligned}
d\Phi_{\mathrm{RR}}^{q\bar q,(d)}=d\Phi_{\mathrm{out}}^{(d)}d\Phi_4^{(d)}(K;p_1,p_2,k_3,k_4) \\
&=d\Phi_{\mathrm{out}}^{(d)}d\Phi_4^{(d)}(K-k_3-k_4)d\Pi^{(d)}(k_3)d\Pi^{(d)}(k_4). \tag*{(127)}
\end{aligned} d Φ RR q q ˉ , ( d ) = d Φ out ( d ) d Φ 4 ( d ) ( K ; p 1 , p 2 , k 3 , k 4 ) = d Φ out ( d ) d Φ 4 ( d ) ( K − k 3 − k 4 ) d Π ( d ) ( k 3 ) d Π ( d ) ( k 4 ) . (127) We use E ∗ = E 3 ∗ + E 4 ∗ E^*=E_3^*+E_4^* E ∗ = E 3 ∗ + E 4 ∗ z = E 3 ∗ / E ∗ z=E_3^*/E^* z = E 3 ∗ / E ∗ η = ( 1 − cos θ 34 ∗ ) / 2 \eta=(1-\cos\theta_{34}^*)/2 η = ( 1 − cos θ 34 ∗ ) /2 n 3 ∗ ∈ S d − 2 \mathbf{n}_3^*\in S^{d-2} n 3 ∗ ∈ S d − 2 k 3 k_3 k 3 e ⊥ ∗ \mathbf{e}_\perp^* e ⊥ ∗ n 3 ∗ \mathbf{n}_3^* n 3 ∗ p = k 3 / z p=k_3/z p = k 3 / z
n 4 ∗ = ( 1 − 2 η ) n 3 ∗ + 2 η ( 1 − η ) e ⊥ ∗ , \mathbf{n}_4^*=(1-2\eta)\mathbf{n}_3^*+2\sqrt{\eta(1-\eta)}\mathbf{e}_\perp^*, n 4 ∗ = ( 1 − 2 η ) n 3 ∗ + 2 η ( 1 − η ) e ⊥ ∗ , k 3 ∗ = z E ∗ ( 1 , n 3 ∗ ) , k 4 ∗ = ( 1 − z ) E ∗ ( 1 , n 4 ∗ ) , p ∗ = E ∗ ( 1 , n 3 ∗ ) . (128) k_3^*=zE^*(1,\mathbf{n}_3^*),\qquad k_4^*=(1-z)E^*(1,\mathbf{n}_4^*),\qquad p^*=E^*(1,\mathbf{n}_3^*). \tag*{(128)} k 3 ∗ = z E ∗ ( 1 , n 3 ∗ ) , k 4 ∗ = ( 1 − z ) E ∗ ( 1 , n 4 ∗ ) , p ∗ = E ∗ ( 1 , n 3 ∗ ) . (128) The solid-angle elements d Ω 3 ∗ d\Omega_3^* d Ω 3 ∗ d Ω ⊥ ∗ d\Omega_\perp^* d Ω ⊥ ∗ S d − 2 S^{d-2} S d − 2 S d − 3 S^{d-3} S d − 3
d E 3 ∗ d E 4 ∗ = E ∗ d E ∗ d z , dE_3^*dE_4^*=E^*dE^*dz, d E 3 ∗ d E 4 ∗ = E ∗ d E ∗ d z , d Ω 4 ∗ = 2 1 − 2 ϵ [ η ( 1 − η ) ] − ϵ d η d Ω ⊥ ∗ , d\Omega_4^*=2^{1-2\epsilon}[\eta(1-\eta)]^{-\epsilon}d\eta\,d\Omega_\perp^*, d Ω 4 ∗ = 2 1 − 2 ϵ [ η ( 1 − η ) ] − ϵ d η d Ω ⊥ ∗ , d Π ( d ) ( k 3 ) d Π ( d ) ( k 4 ) = 2 1 − 2 ϵ ( E ∗ ) 3 − 4 ϵ [ z ( 1 − z ) ] 1 − 2 ϵ 4 ( 2 π ) 6 − 4 ϵ [ η ( 1 − η ) ] − ϵ d E ∗ d z d Ω 3 ∗ d η d Ω ⊥ ∗ . (129) d\Pi^{(d)}(k_3)d\Pi^{(d)}(k_4)=\frac{2^{1-2\epsilon}(E^*)^{3-4\epsilon}[z(1-z)]^{1-2\epsilon}}{4(2\pi)^{6-4\epsilon}}[\eta(1-\eta)]^{-\epsilon}dE^*\,dz\,d\Omega_3^*\,d\eta\,d\Omega_\perp^*. \tag*{(129)} d Π ( d ) ( k 3 ) d Π ( d ) ( k 4 ) = 4 ( 2 π ) 6 − 4 ϵ 2 1 − 2 ϵ ( E ∗ ) 3 − 4 ϵ [ z ( 1 − z ) ] 1 − 2 ϵ [ η ( 1 − η ) ] − ϵ d E ∗ d z d Ω 3 ∗ d η d Ω ⊥ ∗ . (129) Here 0 < z , η < 1 0<z,\eta<1 0 < z , η < 1
The daughter-energy bounds z E ∗ < E 0 ∗ zE^*<E_0^* z E ∗ < E 0 ∗ ( 1 − z ) E ∗ < E 0 ∗ (1-z)E^*<E_0^* ( 1 − z ) E ∗ < E 0 ∗ E max ∗ E_{\max}^* E m a x ∗ E p h y s ∗ E_{\mathrm{phys}}^* E phys ∗
( K − k 3 − k 4 ) 2 = W 2 − 2 W E ∗ + 4 z ( 1 − z ) ( E ∗ ) 2 η ≥ 4 m c 2 , (K-k_3-k_4)^2=W^2-2WE^*+4z(1-z)(E^*)^2\eta\geq4m_c^2, ( K − k 3 − k 4 ) 2 = W 2 − 2 W E ∗ + 4 z ( 1 − z ) ( E ∗ ) 2 η ≥ 4 m c 2 , E max ∗ ( z ) = E 0 ∗ max ( z , 1 − z ) , E 0 ∗ ≤ E p h y s ∗ ≤ E max ∗ , E_{\max}^*(z)=\frac{E_0^*}{\max(z,1-z)},\qquad E_0^*\leq E_{\mathrm{phys}}^*\leq E_{\max}^*, E m a x ∗ ( z ) = max ( z , 1 − z ) E 0 ∗ , E 0 ∗ ≤ E phys ∗ ≤ E m a x ∗ , E p h y s ∗ ( z , η ) = W β 2 1 + 1 − 4 β 2 z ( 1 − z ) η . (130) E_{\mathrm{phys}}^*(z,\eta)=\frac{W\beta^2}{1+\sqrt{1-4\beta^2z(1-z)\eta}}. \tag*{(130)} E phys ∗ ( z , η ) = 1 + 1 − 4 β 2 z ( 1 − z ) η W β 2 . (130) The physical upper bound is the smaller root of the threshold equation, for which the heavy-pair momentum is future-directed. Table 5 lists the corresponding heavy-pair momenta and energy ranges.
Table 5. Heavy-pair recoils and energy ranges for the four q q ˉ q\bar{q} q q ˉ
The soft projection retains the full radiation phase space in (D.20). For C and C S 34 \mathrm{CS}_{34} CS 34 ( 1 − η ) − ϵ → 1 (1-\eta)^{-\epsilon}\to1 ( 1 − η ) − ϵ → 1 η − ϵ \eta^{-\epsilon} η − ϵ Table 5 ,
d Φ X q q ˉ , ( d ) = d Φ o u t ( d ) d Φ 2 ( d ) ( T X ) d Π ( d ) ( k 3 ) d Π ( d ) ( k 4 ) { 1 , X = R R , S 34 , ( 1 − η ) ϵ , X = C , C S 34 . (131) d\Phi_X^{q\bar q,(d)}=d\Phi_{\mathrm{out}}^{(d)}d\Phi_2^{(d)}(T_X)d\Pi^{(d)}(k_3)d\Pi^{(d)}(k_4)
\begin{cases}
1, & X=\mathrm{RR},\mathrm{S}_{34},\\
(1-\eta)^\epsilon, & X=\mathrm{C},\mathrm{CS}_{34}.
\end{cases} \tag*{(131)} d Φ X q q ˉ , ( d ) = d Φ out ( d ) d Φ 2 ( d ) ( T X ) d Π ( d ) ( k 3 ) d Π ( d ) ( k 4 ) { 1 , ( 1 − η ) ϵ , X = RR , S 34 , X = C , CS 34 . (131) The spin-correlated kernels (C.13) and (C.17) use e ⊥ ∗ μ = ( 0 , e ⊥ ∗ ) e_\perp^{*\mu}=(0,e_\perp^*) e ⊥ ∗ μ = ( 0 , e ⊥ ∗ ) k ⊥ μ k ⊥ ν / k ⊥ 2 = − e ⊥ ∗ μ e ⊥ ∗ ν k_\perp^\mu k_\perp^\nu/k_\perp^2=-e_\perp^{*\mu}e_\perp^{*\nu} k ⊥ μ k ⊥ ν / k ⊥ 2 = − e ⊥ ∗ μ e ⊥ ∗ ν p p p
For the subtracted integral R R F ( q q ˉ ) \mathrm{RR}_{F}(q\bar q) RR F ( q q ˉ ) ϵ = 0 \epsilon=0 ϵ = 0 2 d η d ϕ ⊥ ∗ 2\,d\eta\,d\phi_\perp^* 2 d η d ϕ ⊥ ∗
d Φ X q q ˉ = d Φ o u t z ( 1 − z ) ( E ∗ ) 3 4096 π 8 1 − 4 m c 2 T X 2 d E ∗ d z d Ω 3 ∗ d η d ϕ ⊥ ∗ d Ω T . (132) d\Phi_X^{q\bar q}=d\Phi_{\mathrm{out}}\frac{z(1-z)(E^*)^3}{4096\pi^8}\sqrt{1-\frac{4m_c^2}{T_X^2}}\,dE^*\,dz\,d\Omega_3^*\,d\eta\,d\phi_\perp^*\,d\Omega_T. \tag*{(132)} d Φ X q q ˉ = d Φ out 4096 π 8 z ( 1 − z ) ( E ∗ ) 3 1 − T X 2 4 m c 2 d E ∗ d z d Ω 3 ∗ d η d ϕ ⊥ ∗ d Ω T . (132) Here d Ω 3 ∗ = d cos θ 3 ∗ d ϕ 3 ∗ d\Omega_3^*=d\cos\theta_3^*\,d\phi_3^* d Ω 3 ∗ = d cos θ 3 ∗ d ϕ 3 ∗ ϕ 3 ∗ , ϕ ⊥ ∗ ∈ [ 0 , 2 π ) \phi_3^*,\phi_\perp^*\in[0,2\pi) ϕ 3 ∗ , ϕ ⊥ ∗ ∈ [ 0 , 2 π )
To construct the four-dimensional momenta, set n 3 ∗ = n ( θ 3 ∗ , ϕ 3 ∗ ) n_3^*=n(\theta_3^*,\phi_3^*) n 3 ∗ = n ( θ 3 ∗ , ϕ 3 ∗ ) a ∗ a^* a ∗ ∣ a ∗ ⋅ n 3 ∗ ∣ |a^*\cdot n_3^*| ∣ a ∗ ⋅ n 3 ∗ ∣
e 1 ∗ = a ∗ − ( a ∗ ⋅ n 3 ∗ ) n 3 ∗ 1 − ( a ∗ ⋅ n 3 ∗ ) 2 , e 2 ∗ = n 3 ∗ × e 1 ∗ , e_1^*=\frac{a^*-(a^*\cdot n_3^*)n_3^*}{\sqrt{1-(a^*\cdot n_3^*)^2}},\qquad e_2^*=n_3^*\times e_1^*, e 1 ∗ = 1 − ( a ∗ ⋅ n 3 ∗ ) 2 a ∗ − ( a ∗ ⋅ n 3 ∗ ) n 3 ∗ , e 2 ∗ = n 3 ∗ × e 1 ∗ , e ⊥ ∗ = cos ϕ ⊥ ∗ e 1 ∗ + sin ϕ ⊥ ∗ e 2 ∗ . (133) e_\perp^*=\cos\phi_\perp^*e_1^*+\sin\phi_\perp^*e_2^*. \tag*{(133)} e ⊥ ∗ = cos ϕ ⊥ ∗ e 1 ∗ + sin ϕ ⊥ ∗ e 2 ∗ . (133) The daughter momenta follow from (D.19), and the heavy-quark momenta from (D.4) with the corresponding T X T_X T X Table 5 .
We map the common interval 0 < E ∗ < E max ∗ ( z ) 0<E^*<E^*_{\max}(z) 0 < E ∗ < E m a x ∗ ( z ) 0 < u 3 < 1 0<u_3<1 0 < u 3 < 1
E ∗ = E max ∗ ( z ) u 3 , z = u 4 , η = u 7 2 , E^*=E^*_{\max}(z)u_3,\qquad z=u_4,\qquad\eta=u_7^2, E ∗ = E m a x ∗ ( z ) u 3 , z = u 4 , η = u 7 2 , cos θ 3 ∗ = 2 u 5 − 1 , ϕ 3 ∗ = 2 π u 6 , ϕ ⊥ ∗ = 2 π u 8 , \cos\theta_3^*=2u_5-1,\qquad\phi_3^*=2\pi u_6,\qquad\phi_\perp^*=2\pi u_8, cos θ 3 ∗ = 2 u 5 − 1 , ϕ 3 ∗ = 2 π u 6 , ϕ ⊥ ∗ = 2 π u 8 , cos θ T = 2 u 9 − 1 , ϕ T = 2 π u 10 . (134) \cos\theta_T=2u_9-1,\qquad\phi_T=2\pi u_{10}. \tag*{(134)} cos θ T = 2 u 9 − 1 , ϕ T = 2 π u 10 . (134) With d E ∗ d z = E max ∗ ( z ) d u 3 d u 4 dE^{*}\,dz=E^{*}_{\max}(z)\,du_{3}\,du_{4} d E ∗ d z = E m a x ∗ ( z ) d u 3 d u 4 d η = 2 u 7 d u 7 d\eta=2u_{7}\,du_{7} d η = 2 u 7 d u 7 d Φ X q q ˉ = J X q q ˉ d 10 u d\Phi^{q\bar q}_{X}=J^{q\bar q}_{X}\,d^{10}u d Φ X q q ˉ = J X q q ˉ d 10 u
J 0 q q ˉ = Δ K λ ( s , M ψ 2 , K 2 ) 2048 π 7 s z ( 1 − z ) ( E ∗ ) 3 E max ∗ ( z ) 2 u 7 , J^{q\bar q}_{0}=\frac{\Delta_{K}\sqrt{\lambda(s,M_{\psi}^{2},K^{2})}}{2048\pi^{7}s}\,z(1-z)(E^{*})^{3}E^{*}_{\max}(z)\,2u_{7}, J 0 q q ˉ = 2048 π 7 s Δ K λ ( s , M ψ 2 , K 2 ) z ( 1 − z ) ( E ∗ ) 3 E m a x ∗ ( z ) 2 u 7 , J X q q ˉ = J 0 q q ˉ 1 − 4 m c 2 T X 2 θ ( u 3 , max X − u 3 ) , J^{q\bar q}_{X}=J^{q\bar q}_{0}\sqrt{1-\frac{4m_{c}^{2}}{T_{X}^{2}}}\,\theta(u^{X}_{3,\max}-u_{3}), J X q q ˉ = J 0 q q ˉ 1 − T X 2 4 m c 2 θ ( u 3 , m a x X − u 3 ) , u 3 , max R R = E p h y s ∗ E max ∗ , u 3 , max C = max ( z , 1 − z ) , u 3 , max S 34 = u 3 , max C S 34 = 1. (135) u^{\mathrm{RR}}_{3,\max}=\frac{E^{*}_{\mathrm{phys}}}{E^{*}_{\max}},\qquad
u^{\mathrm{C}}_{3,\max}=\max(z,1-z),\qquad
u^{\mathrm{S}_{34}}_{3,\max}=u^{\mathrm{CS}_{34}}_{3,\max}=1. \tag*{(135)} u 3 , m a x RR = E m a x ∗ E phys ∗ , u 3 , m a x C = max ( z , 1 − z ) , u 3 , m a x S 34 = u 3 , m a x CS 34 = 1. (135) Each term is evaluated only where its step function is nonzero. The common coordinates give k 3 + k 4 → p k_{3}+k_{4}\to p k 3 + k 4 → p η → 0 \eta\to0 η → 0 T X → K T_{X}\to K T X → K E ∗ → 0 E^{*}\to0 E ∗ → 0
The finite contribution is
R R F ( q q ˉ ) = N ∑ q = u , d , s ∫ [ 0 , 1 ] 10 d 10 u [ J R R q q ˉ F q q ˉ − J S 34 q q ˉ ( S 34 F q q ˉ ) − J C q q ˉ ( C F q q ˉ ) + J C S 34 q q ˉ ( C S 34 F q q ˉ ) ] . (136) \mathrm{RR}_{F}(q\bar q)=\mathcal{N}\sum_{q=u,d,s}\int_{[0,1]^{10}}d^{10}u\left[J^{q\bar q}_{\mathrm{RR}}F_{q\bar q}-J^{q\bar q}_{\mathrm{S}_{34}}(\mathrm{S}_{34}F_{q\bar q})-J^{q\bar q}_{\mathrm{C}}(\mathrm{C}F_{q\bar q})+J^{q\bar q}_{\mathrm{CS}_{34}}(\mathrm{CS}_{34}F_{q\bar q})\right]. \tag*{(136)} RR F ( q q ˉ ) = N q = u , d , s ∑ ∫ [ 0 , 1 ] 10 d 10 u [ J RR q q ˉ F q q ˉ − J S 34 q q ˉ ( S 34 F q q ˉ ) − J C q q ˉ ( C F q q ˉ ) + J CS 34 q q ˉ ( CS 34 F q q ˉ ) ] . (136) Ordered two-gluon emission
In the ordered sector E 4 ∗ < E 3 ∗ E^{*}_{4}<E^{*}_{3} E 4 ∗ < E 3 ∗ J / ψ + c c ˉ + g g J/\psi+c\bar c+gg J / ψ + c c ˉ + g g
d Φ R R g g , ( d ) = d Φ o u t ( d ) d Φ 4 ( d ) ( K ; p 1 , p 2 , k 3 , k 4 ) θ ( E 3 ∗ − E 4 ∗ ) = d Φ o u t ( d ) d Φ 2 ( d ) ( K − k 3 − k 4 ) d Π ( d ) ( k 3 ) d Π ( d ) ( k 4 ) θ ( E 3 ∗ − E 4 ∗ ) . (137) \begin{aligned}
d\Phi^{gg,(d)}_{\mathrm{RR}}=d\Phi^{(d)}_{\mathrm{out}}d\Phi^{(d)}_{4}(K;p_{1},p_{2},k_{3},k_{4})\,\theta(E^{*}_{3}-E^{*}_{4}) \\
&=d\Phi^{(d)}_{\mathrm{out}}d\Phi^{(d)}_{2}(K-k_{3}-k_{4})\,d\Pi^{(d)}(k_{3})d\Pi^{(d)}(k_{4})\,\theta(E^{*}_{3}-E^{*}_{4}). \tag*{(137)}
\end{aligned} d Φ RR g g , ( d ) = d Φ out ( d ) d Φ 4 ( d ) ( K ; p 1 , p 2 , k 3 , k 4 ) θ ( E 3 ∗ − E 4 ∗ ) = d Φ out ( d ) d Φ 2 ( d ) ( K − k 3 − k 4 ) d Π ( d ) ( k 3 ) d Π ( d ) ( k 4 ) θ ( E 3 ∗ − E 4 ∗ ) . (137) We use E ∗ = E 3 ∗ + E 4 ∗ E^{*}=E^{*}_{3}+E^{*}_{4} E ∗ = E 3 ∗ + E 4 ∗ z = E 4 ∗ / E ∗ z=E^{*}_{4}/E^{*} z = E 4 ∗ / E ∗ η = ( 1 − cos θ 34 ∗ ) / 2 \eta=(1-\cos\theta^{*}_{34})/2 η = ( 1 − cos θ 34 ∗ ) /2 d d d
k 3 ∗ = ( 1 − z ) E ∗ ( 1 , n 3 ∗ ) , k 4 ∗ = z E ∗ ( 1 , n 4 ∗ ) , p ∗ = E ∗ ( 1 , n 3 ∗ ) . (138) k^{*}_{3}=(1-z)E^{*}(1,\boldsymbol{n}^{*}_{3}),\qquad
k^{*}_{4}=zE^{*}(1,\boldsymbol{n}^{*}_{4}),\qquad
p^{*}=E^{*}(1,\boldsymbol{n}^{*}_{3}). \tag*{(138)} k 3 ∗ = ( 1 − z ) E ∗ ( 1 , n 3 ∗ ) , k 4 ∗ = z E ∗ ( 1 , n 4 ∗ ) , p ∗ = E ∗ ( 1 , n 3 ∗ ) . (138) The two-gluon phase space follows from (D.20), now restricted to 0 < z < 1 / 2 0<z<1/2 0 < z < 1/2 p = k 3 / ( 1 − z ) p=k_{3}/(1-z) p = k 3 / ( 1 − z )
The harder-gluon bound ( 1 − z ) E ∗ < E 0 ∗ (1-z)E^{*}<E^{*}_{0} ( 1 − z ) E ∗ < E 0 ∗ E max ∗ ( z ) = E 0 ∗ / ( 1 − z ) E^{*}_{\max}(z)=E^{*}_{0}/(1-z) E m a x ∗ ( z ) = E 0 ∗ / ( 1 − z ) E p h y s ∗ ( z , η ) E^{*}_{\mathrm{phys}}(z,\eta) E phys ∗ ( z , η ) Table 6 gives the heavy-pair momenta and energy ranges for all eight terms. The reduced single-real squared matrix element is evaluated at k 3 k_{3} k 3 S 4 \mathrm{S}_{4} S 4 C S 4 \mathrm{CS}_{4} CS 4 p p p C \mathrm{C} C
Table 6. Heavy-pair recoils and energy ranges for the eight ordered g g gg g g
All terms containing C \mathrm{C} C C S 4 \mathrm{CS}_{4} CS 4 [ d k 3 ] K [dk_{3}]_{K} [ d k 3 ] K E 3 ∗ = ( 1 − z ) E ∗ E_3^*=(1-z)E^* E 3 ∗ = ( 1 − z ) E ∗ ( E ∗ , z ) (E^*,z) ( E ∗ , z ) Table 6 , the d d d
d Φ X g g , ( d ) = d Φ o u t ( d ) d Φ 2 ( d ) ( T X ) d Π ( d ) ( k 3 ) d Π ( d ) ( k 4 ) { 1 , X = R R , S 4 , S 34 , S 4 S 34 , ( 1 − η ) ϵ , X = C , C S 4 , C S 34 , C S 4 S 34 . (139) d\Phi_X^{gg,(d)}=d\Phi_{\mathrm{out}}^{(d)}d\Phi_2^{(d)}(T_X)d\Pi^{(d)}(k_3)d\Pi^{(d)}(k_4)
\begin{cases}
1, & X=\mathrm{RR},\mathrm{S}_4,\mathrm{S}_{34},\mathrm{S}_4\mathrm{S}_{34},\\
(1-\eta)^\epsilon, & X=\mathrm{C},\mathrm{CS}_4,\mathrm{CS}_{34},\mathrm{CS}_4\mathrm{S}_{34}.
\end{cases} \tag*{(139)} d Φ X g g , ( d ) = d Φ out ( d ) d Φ 2 ( d ) ( T X ) d Π ( d ) ( k 3 ) d Π ( d ) ( k 4 ) { 1 , ( 1 − η ) ϵ , X = RR , S 4 , S 34 , S 4 S 34 , X = C , CS 4 , CS 34 , CS 4 S 34 . (139) For the subtracted integral R R F ( g g ) \mathrm{RR}_{\mathrm{F}}(gg) RR F ( g g ) ϵ = 0 \epsilon=0 ϵ = 0
d Φ X g g = d Φ o u t z ( 1 − z ) ( E ∗ ) 3 4096 π 8 1 − 4 m c 2 T X 2 d E ∗ d z d Ω 3 ∗ d η d ϕ ⊥ ∗ d Ω T . (140) d\Phi_X^{gg}=d\Phi_{\mathrm{out}}\frac{z(1-z)(E^*)^3}{4096\pi^8}\sqrt{1-\frac{4m_c^2}{T_X^2}}\,dE^*\,dz\,d\Omega_3^*\,d\eta\,d\phi_\perp^*\,d\Omega_T. \tag*{(140)} d Φ X g g = d Φ out 4096 π 8 z ( 1 − z ) ( E ∗ ) 3 1 − T X 2 4 m c 2 d E ∗ d z d Ω 3 ∗ d η d ϕ ⊥ ∗ d Ω T . (140) The four-dimensional directions are constructed as in (D.24), and the heavy-quark momenta follow from (D.4) with the corresponding T X T_X T X
The ordered-sector map combines (D.9) with
E ∗ = E max ∗ ( z ) u 3 , z = u 4 2 , η = u 7 2 , cos θ 3 ∗ = 2 u 5 − 1 , ϕ 3 ∗ = 2 π u 6 , ϕ ⊥ ∗ = 2 π u 8 , cos θ T = 2 u 9 − 1 , ϕ T = 2 π u 10 . (141) \begin{aligned}
E^*&=E_{\max}^*(z)u_3,\qquad z=\frac{u_4}{2},\qquad\eta=u_7^2,\\
\cos\theta_3^*&=2u_5-1,\qquad\phi_3^*=2\pi u_6,\qquad\phi_\perp^*=2\pi u_8,\\
\cos\theta_T&=2u_9-1,\qquad\phi_T=2\pi u_{10}. \tag*{(141)}
\end{aligned} E ∗ cos θ 3 ∗ cos θ T = E m a x ∗ ( z ) u 3 , z = 2 u 4 , η = u 7 2 , = 2 u 5 − 1 , ϕ 3 ∗ = 2 π u 6 , ϕ ⊥ ∗ = 2 π u 8 , = 2 u 9 − 1 , ϕ T = 2 π u 10 . (141) Here d E ∗ d z = 1 2 E max ∗ ( z ) d u 3 d u 4 dE^*\,dz=\frac{1}{2}E_{\max}^*(z)\,du_3\,du_4 d E ∗ d z = 2 1 E m a x ∗ ( z ) d u 3 d u 4 d η = 2 u 7 d u 7 d\eta=2u_7\,du_7 d η = 2 u 7 d u 7 d Φ X g g = J X g g d 10 u d\Phi_X^{gg}=J_X^{gg}d^{10}u d Φ X g g = J X g g d 10 u
J 0 g g = Δ K λ ( s , M ψ 2 , K 2 ) 4096 π 7 s z ( 1 − z ) ( E ∗ ) 3 E max ∗ ( z ) 2 u 7 , J_0^{gg}=\frac{\Delta_K\sqrt{\lambda(s,M_\psi^2,K^2)}}{4096\pi^7s}z(1-z)(E^*)^3E_{\max}^*(z)\,2u_7, J 0 g g = 4096 π 7 s Δ K λ ( s , M ψ 2 , K 2 ) z ( 1 − z ) ( E ∗ ) 3 E m a x ∗ ( z ) 2 u 7 , J X g g = J 0 g g 1 − 4 m c 2 T X 2 θ ( u 3 , max X − u 3 ) , J_X^{gg}=J_0^{gg}\sqrt{1-\frac{4m_c^2}{T_X^2}}\,\theta(u_{3,\max}^X-u_3), J X g g = J 0 g g 1 − T X 2 4 m c 2 θ ( u 3 , m a x X − u 3 ) , u 3 , max R R = E p h y s ∗ E max ∗ , u 3 , max C = 1 − z , u_{3,\max}^{\mathrm{RR}}=\frac{E_{\mathrm{phys}}^*}{E_{\max}^*},\qquad u_{3,\max}^{\mathrm{C}}=1-z, u 3 , m a x RR = E m a x ∗ E phys ∗ , u 3 , m a x C = 1 − z , u 3 , max X = 1 , X = S 4 , C S 4 , S 34 , S 4 S 34 , C S 34 , C S 4 S 34 . (142) u_{3,\max}^X=1,\qquad X=\mathrm{S}_4,\mathrm{CS}_4,\mathrm{S}_{34},\mathrm{S}_4\mathrm{S}_{34},\mathrm{CS}_{34},\mathrm{CS}_4\mathrm{S}_{34}. \tag*{(142)} u 3 , m a x X = 1 , X = S 4 , CS 4 , S 34 , S 4 S 34 , CS 34 , CS 4 S 34 . (142) The single-soft limit z → 0 z\to0 z → 0 k 4 → 0 k_4\to0 k 4 → 0 p → k 3 p\to k_3 p → k 3
Evaluating each term on the phase space specified in Table 6 and grouping terms with the same Jacobian gives
R R F ( g g ) = N ∫ [ 0 , 1 ] 10 d 10 u [ J R R g g F g g − J C g g ( C F g g ) − J S 4 g g ( S 4 F g g − C S 4 F g g ) − J S 34 g g ( S 34 F g g − S 4 S 34 F g g − C S 34 F g g + C S 4 S 34 F g g ) ] . (143) \mathrm{RR}_{\mathrm{F}}(gg)=\mathcal{N}\int_{[0,1]^{10}}d^{10}u\left[J_{\mathrm{RR}}^{gg}F_{gg}-J_{\mathrm{C}}^{gg}(\mathrm{C}F_{gg})-J_{\mathrm{S}_4}^{gg}(\mathrm{S}_4F_{gg}-\mathrm{CS}_4F_{gg})-J_{\mathrm{S}_{34}}^{gg}(\mathrm{S}_{34}F_{gg}-\mathrm{S}_4\mathrm{S}_{34}F_{gg}-\mathrm{CS}_{34}F_{gg}+\mathrm{CS}_4\mathrm{S}_{34}F_{gg})\right]. \tag*{(143)} RR F ( g g ) = N ∫ [ 0 , 1 ] 10 d 10 u [ J RR g g F g g − J C g g ( C F g g ) − J S 4 g g ( S 4 F g g − CS 4 F g g ) − J S 34 g g ( S 34 F g g − S 4 S 34 F g g − CS 34 F g g + CS 4 S 34 F g g ) ] . (143) Numerical integration and spectrum construction
Virtual quadrature. We integrate V F V_{\mathrm{F}} V F V V F \mathrm{VV}_{\mathrm{F}} VV F 15 × 15 15\times15 15 × 15 s V \sqrt{s_V} s V E ψ E_\psi E ψ s 2 = P ⋅ p 1 s_2=P\cdot p_1 s 2 = P ⋅ p 1 § 11 .§ 1 gives
d Φ B = d E ψ d s 2 32 π 3 s V , M ψ < E ψ < s V 2 , s 2 , − < s 2 < s 2 , + , d\Phi_B = \frac{dE_\psi\,ds_2}{32\pi^3\sqrt{s_V}}, \qquad M_\psi< E_\psi< \frac{\sqrt{s_V}}{2}, \qquad s_{2,-} < s_2 < s_{2,+}, d Φ B = 32 π 3 s V d E ψ d s 2 , M ψ < E ψ < 2 s V , s 2 , − < s 2 < s 2 , + , s 2 , ± ( E ψ ) = s V E ψ − M ψ 2 2 ± β 4 λ ( s V , M ψ 2 , K 2 ) (144) s_{2,\pm}(E_\psi) = \frac{\sqrt{s_V}E_\psi-M_\psi^2}{2} \pm\frac{\beta}{4}\sqrt{\lambda(s_V,M_\psi^2,K^2)} \tag*{(144)} s 2 , ± ( E ψ ) = 2 s V E ψ − M ψ 2 ± 4 β λ ( s V , M ψ 2 , K 2 ) (144) where K 2 = s V + M ψ 2 − 2 s V E ψ K^2=s_V+M_\psi^2-2\sqrt{s_V}E_\psi K 2 = s V + M ψ 2 − 2 s V E ψ E ψ / m c E_\psi/m_c E ψ / m c s 2 / m c 2 s_2/m_c^2 s 2 / m c 2
At each point, the ϵ 0 \epsilon^0 ϵ 0 ϵ = ± h \epsilon=\pm h ϵ = ± h h = 10 − 3 h=10^{-3} h = 1 0 − 3 ϵ \epsilon ϵ O ( h 2 ) O(h^2) O ( h 2 )
Quadrature error estimates. For each contribution and photon projection, let f i j f_{ij} f ij N \mathcal{N} N i , j = 1 , … , 15 i,j=1,\ldots,15 i , j = 1 , … , 15 w K , w G w^K,w^G w K , w G I 7 = { 2 , 4 , … , 14 } \mathcal{I}_7=\{2,4,\ldots,14\} I 7 = { 2 , 4 , … , 14 }
q i K = ∑ j = 1 15 w i j K f i j , q i G = ∑ j ∈ I 7 w i j G f i j , q_i^K=\sum_{j=1}^{15}w_{ij}^K f_{ij}, \qquad q_i^G=\sum_{j\in\mathcal{I}_7}w_{ij}^G f_{ij}, q i K = j = 1 ∑ 15 w ij K f ij , q i G = j ∈ I 7 ∑ w ij G f ij , I K K = ∑ i = 1 15 w i K q i K , I G K = ∑ i ∈ I 7 w i G q i K (145) I_{KK}=\sum_{i=1}^{15}w_i^Kq_i^K, \qquad I_{GK}=\sum_{i\in\mathcal{I}_7}w_i^Gq_i^K \tag*{(145)} I K K = i = 1 ∑ 15 w i K q i K , I G K = i ∈ I 7 ∑ w i G q i K (145) The embedded error estimate combines the outer difference with the weighted inner differences:
Δ e m b = ∣ I K K − I G K ∣ + ∑ i = 1 15 w i K ∣ q i K − q i G ∣ . (146) \Delta_{\mathrm{emb}}=\left|I_{KK}-I_{GK}\right|+\sum_{i=1}^{15}w_i^K\left|q_i^K-q_i^G\right|. \tag*{(146)} Δ emb = ∣ I K K − I G K ∣ + i = 1 ∑ 15 w i K q i K − q i G . (146) The stored nodes differ slightly from the standard Gauss–Kronrod nodes because of rationalisation and small shifts of several inner midpoints. Recomputing interpolatory weights at the actual nodes changes Δ e m b \Delta_{\mathrm{emb}} Δ emb
Table 7 gives the estimates for the individual virtual contributions and their linear sum.
Table 7. Embedded Gauss–Kronrod error estimates for the virtual contributions at μ R = 2 m c \mu_R=2m_c μ R = 2 m c μ Λ = m c \mu_\Lambda=m_c μ Λ = m c
Energy bins and interpolation. We evaluate bin averages of d σ / d E ψ d\sigma/dE_\psi d σ / d E ψ 3.00 ≤ E ψ ≤ 5.30 3.00 \le E_\psi\le5.30 3.00 ≤ E ψ ≤ 5.30 s 2 s_2 s 2 E ψ = 3.00 E_\psi= 3.00 E ψ = 3.00
Momentum spectrum. The quarkonium momentum magnitude in the centre-of-mass frame is ∣ P ∣ = E ψ 2 − M ψ 2 |\mathbf{P}| = \sqrt{E_\psi^2-M_\psi^2} ∣ P ∣ = E ψ 2 − M ψ 2
d σ d ∣ P ∣ = ∣ P ∣ E ψ d σ d E ψ . (147) \frac{d\sigma}{d|\mathbf{P}|}=\frac{|\mathbf{P}|}{E_\psi}\frac{d\sigma}{dE_\psi}. \tag*{(147)} d ∣ P ∣ d σ = E ψ ∣ P ∣ d E ψ d σ . (147) For the binned spectrum, we convert the energy-bin boundaries to momentum and divide the cross section in each bin by the corresponding momentum-bin width. Evaluating VF and VVF at s V \sqrt{s_V} s V ∣ P ∣ ≃ 4.31 |\mathbf{P}| \simeq4.31 ∣ P ∣ ≃ 4.31
Auxiliary functions and infrared coefficients
We give the analytic coefficients used in the counterterms of appendix § 10 and the virtual infrared factors of appendix § 9 .§ 2 .
Double-soft functions
The functions in {(C.9)} and {(C.25)} use the normalisation A S \mathcal{A}_S A S
f i j q q ˉ ( β , ϵ ) = ∑ n = − 2 0 ϵ n f i j , n q q ˉ ( β ) + O ( ϵ ) , f_{ij}^{q\bar q}(\beta,\epsilon)=\sum_{n=-2}^{0}\epsilon^n f_{ij,n}^{q\bar q}(\beta)+O(\epsilon), f ij q q ˉ ( β , ϵ ) = n = − 2 ∑ 0 ϵ n f ij , n q q ˉ ( β ) + O ( ϵ ) , f i j g g ( β , ϵ ) = ∑ n = − 3 0 ϵ n f i j , n g g ( β ) + O ( ϵ ) . (148) f_{ij}^{gg}(\beta,\epsilon)=\sum_{n=-3}^{0}\epsilon^n f_{ij,n}^{gg}(\beta)+O(\epsilon). \tag*{(148)} f ij g g ( β , ϵ ) = n = − 3 ∑ 0 ϵ n f ij , n g g ( β ) + O ( ϵ ) . (148) The coefficients below are taken from ref. [44 ] and its ancillary file. They are expressed in terms of Goncharov polylogarithms, defined by
G ( a 1 , … , a w ; x ) = ∫ 0 x d t t − a 1 G ( a 2 , … , a w ; t ) , G(a_1,\ldots,a_w;x)=\int_0^x\frac{dt}{t-a_1}G(a_2,\ldots,a_w;t), G ( a 1 , … , a w ; x ) = ∫ 0 x t − a 1 d t G ( a 2 , … , a w ; t ) , G ( ; x ) = 1 , G ( 0 , … , 0 ⏟ w ; x ) = ln w x w ! . (149) G(;x)=1,\qquad G(\underbrace{0,\ldots,0}_{w};x)=\frac{\ln^w x}{w!}. \tag*{(149)} G ( ; x ) = 1 , G ( w 0 , … , 0 ; x ) = w ! ln w x . (149) The recursive integral definition applies when at least one index is nonzero. We use G a 1 , … , a w = G ( a 1 , … , a w ; β ) G_{a_1,\ldots,a_w}=G(a_1,\ldots,a_w;\beta) G a 1 , … , a w = G ( a 1 , … , a w ; β ) a i ∈ { 0 , ± 1 , ± 3 } a_i \in\{0,\pm1,\pm3\} a i ∈ { 0 , ± 1 , ± 3 } 0 < β < 1 0 < \beta< 1 0 < β < 1
Double-soft quark-pair coefficients
The self-correlations are
f A A , − 2 q q ˉ = − 1 4 , f^{q\bar q}_{AA,-2}=-\frac{1}{4}, f AA , − 2 q q ˉ = − 4 1 , f A A , − 1 q q ˉ = − 2 ln 2 − 3 2 + 1 4 β ( − G − 1 + G 1 ) . (150) f^{q\bar q}_{AA,-1}=-\frac{2\ln2-3}{2}+\frac{1}{4\beta}(-G_{-1}+G_1). \tag*{(150)} f AA , − 1 q q ˉ = − 2 2 ln 2 − 3 + 4 β 1 ( − G − 1 + G 1 ) . (150) f A A , 0 q q ˉ = 8 ln 2 − ζ 2 − 6 + 8 ln 2 − 11 4 β ( − G − 1 + G 1 ) + 1 2 β ( G − 1 , − 1 − G 1 , 1 ) + G − 1 , 1 + G 1 , − 1 − 2 β + 1 2 β G 0 , − 1 − 2 β − 1 2 β G 0 , 1 . (151) f^{q\bar q}_{AA,0}=8\ln2-\zeta_2-6+\frac{8\ln2-11}{4\beta}(-G_{-1}+G_1)+\frac{1}{2\beta}(G_{-1,-1}-G_{1,1})+G_{-1,1}+G_{1,-1}-\frac{2\beta+1}{2\beta}G_{0,-1}-\frac{2\beta-1}{2\beta}G_{0,1}. \tag*{(151)} f AA , 0 q q ˉ = 8 ln 2 − ζ 2 − 6 + 4 β 8 ln 2 − 11 ( − G − 1 + G 1 ) + 2 β 1 ( G − 1 , − 1 − G 1 , 1 ) + G − 1 , 1 + G 1 , − 1 − 2 β 2 β + 1 G 0 , − 1 − 2 β 2 β − 1 G 0 , 1 . (151) The correlations between distinct emitters are
f A B , − 2 q q ˉ = − 1 12 + β 2 + 1 12 β ( − G − 1 + G 1 ) , f^{q\bar q}_{AB,-2}=-\frac{1}{12}+\frac{\beta^2+1}{12\beta}(-G_{-1}+G_1), f A B , − 2 q q ˉ = − 12 1 + 12 β β 2 + 1 ( − G − 1 + G 1 ) , f A B , − 1 q q ˉ = − 12 ln 2 − 17 36 + − 37 β 2 + 24 ln 2 ( β 2 + 1 ) − 43 72 β ( − G − 1 + G 1 ) + β 2 + 1 6 β ( G − 1 , − 1 − G − 1 , 1 + G 1 , − 1 − G 1 , 1 ) + β 2 + 1 3 β ( − G 0 , − 1 + G 0 , 1 ) . (152) f^{q\bar q}_{AB,-1}=-\frac{12\ln2-17}{36}+\frac{-37\beta^2+24\ln2(\beta^2+1)-43}{72\beta}(-G_{-1}+G_1)+\frac{\beta^2+1}{6\beta}(G_{-1,-1}-G_{-1,1}+G_{1,-1}-G_{1,1})+\frac{\beta^2+1}{3\beta}(-G_{0,-1}+G_{0,1}). \tag*{(152)} f A B , − 1 q q ˉ = − 36 12 ln 2 − 17 + 72 β − 37 β 2 + 24 ln 2 ( β 2 + 1 ) − 43 ( − G − 1 + G 1 ) + 6 β β 2 + 1 ( G − 1 , − 1 − G − 1 , 1 + G 1 , − 1 − G 1 , 1 ) + 3 β β 2 + 1 ( − G 0 , − 1 + G 0 , 1 ) . (152) f A B , 0 q q ˉ = − − 192 ln 2 + 18 ζ 2 + 161 54 + G − 1 + G 1 12 + 154 β 2 − 24 ln 2 ( 10 β 2 + 7 ) + 36 ζ 2 ( β 2 + 1 ) + 115 108 β ( G 1 − G − 1 ) + − 37 β 2 + 24 ln 2 ( β 2 + 1 ) − 43 36 β ( G − 1 , − 1 − G 1 , 1 ) + − 17 β 2 + 12 ln 2 ( β 2 + 1 ) − 23 18 β ( G 1 , − 1 − G − 1 , 1 ) + − 71 β 2 + 48 ln 2 ( β 2 + 1 ) − 89 36 β ( G 0 , 1 − G 0 , − 1 ) + 1 3 ( G − 1 , 1 + G 1 , − 1 − G 0 , − 1 − G 0 , 1 ) + β 2 + 1 3 β [ − G − 1 , − 1 , − 1 + G − 1 , − 1 , 1 + G − 1 , 1 , − 1 + G − 1 , 1 , 1 − G 1 , − 1 , − 1 − G 1 , − 1 , 1 − G 1 , 1 , − 1 + G 1 , 1 , 1 − 2 G − 1 , 0 , 1 + 2 G 0 , − 1 , − 1 − 2 G 0 , 0 , − 1 + 2 G 0 , 0 , 1 − 2 G 0 , 1 , 1 + 2 G 1 , 0 , − 1 ] . (153) \begin{aligned}
f^{q\bar q}_{AB,0}={}&-\frac{-192\ln2+18\zeta_2+161}{54}+\frac{G_{-1}+G_1}{12}\\
&+\frac{154\beta^2-24\ln2(10\beta^2+7)+36\zeta_2(\beta^2+1)+115}{108\beta}(G_1-G_{-1})\\
&+\frac{-37\beta^2+24\ln2(\beta^2+1)-43}{36\beta}(G_{-1,-1}-G_{1,1})\\
&+\frac{-17\beta^2+12\ln2(\beta^2+1)-23}{18\beta}(G_{1,-1}-G_{-1,1})\\
&+\frac{-71\beta^2+48\ln2(\beta^2+1)-89}{36\beta}(G_{0,1}-G_{0,-1})\\
&+\frac{1}{3}(G_{-1,1}+G_{1,-1}-G_{0,-1}-G_{0,1})\\
&+\frac{\beta^2+1}{3\beta}\bigl[-G_{-1,-1,-1}+G_{-1,-1,1}+G_{-1,1,-1}+G_{-1,1,1}-G_{1,-1,-1}\\
&\qquad-G_{1,-1,1}-G_{1,1,-1}+G_{1,1,1}-2G_{-1,0,1}+2G_{0,-1,-1}\\
&\qquad-2G_{0,0,-1}+2G_{0,0,1}-2G_{0,1,1}+2G_{1,0,-1}\bigr]. \tag*{(153)}
\end{aligned} f A B , 0 q q ˉ = − 54 − 192 ln 2 + 18 ζ 2 + 161 + 12 G − 1 + G 1 + 108 β 154 β 2 − 24 ln 2 ( 10 β 2 + 7 ) + 36 ζ 2 ( β 2 + 1 ) + 115 ( G 1 − G − 1 ) + 36 β − 37 β 2 + 24 ln 2 ( β 2 + 1 ) − 43 ( G − 1 , − 1 − G 1 , 1 ) + 18 β − 17 β 2 + 12 ln 2 ( β 2 + 1 ) − 23 ( G 1 , − 1 − G − 1 , 1 ) + 36 β − 71 β 2 + 48 ln 2 ( β 2 + 1 ) − 89 ( G 0 , 1 − G 0 , − 1 ) + 3 1 ( G − 1 , 1 + G 1 , − 1 − G 0 , − 1 − G 0 , 1 ) + 3 β β 2 + 1 [ − G − 1 , − 1 , − 1 + G − 1 , − 1 , 1 + G − 1 , 1 , − 1 + G − 1 , 1 , 1 − G 1 , − 1 , − 1 − G 1 , − 1 , 1 − G 1 , 1 , − 1 + G 1 , 1 , 1 − 2 G − 1 , 0 , 1 + 2 G 0 , − 1 , − 1 − 2 G 0 , 0 , − 1 + 2 G 0 , 0 , 1 − 2 G 0 , 1 , 1 + 2 G 1 , 0 , − 1 ] . (153) Double-soft gluon coefficients
The self-correlations are
f A A , − 3 g g = − 1 8 , f^{gg}_{AA,-3}=-\frac{1}{8}, f AA , − 3 g g = − 8 1 , f A A , − 2 g g = 1 4 + 1 4 β ( − G − 1 + G 1 ) , f^{gg}_{AA,-2}=\frac{1}{4}+\frac{1}{4\beta}(-G_{-1}+G_1), f AA , − 2 g g = 4 1 + 4 β 1 ( − G − 1 + G 1 ) , f A A , − 1 g g = − 4 ln 2 − 1 2 + 3 4 β ( G − 1 − G 1 ) + β − 1 4 β ( − G − 1 , − 1 + G 1 , 1 ) + 1 2 β ( − G 0 , − 1 + G 0 , 1 ) + β + 1 4 β ( G 1 , − 1 − G 1 , 1 ) . (154) \begin{aligned}
f^{gg}_{AA,-1}=-\frac{4\ln2-1}{2}+\frac{3}{4\beta}(G_{-1}-G_1)+\frac{\beta-1}{4\beta}(-G_{-1,-1}+G_{1,1}) \\
&+\frac{1}{2\beta}(-G_{0,-1}+G_{0,1})+\frac{\beta+1}{4\beta}(G_{1,-1}-G_{1,1}). \tag*{(154)}
\end{aligned} f AA , − 1 g g = − 2 4 ln 2 − 1 + 4 β 3 ( G − 1 − G 1 ) + 4 β β − 1 ( − G − 1 , − 1 + G 1 , 1 ) + 2 β 1 ( − G 0 , − 1 + G 0 , 1 ) + 4 β β + 1 ( G 1 , − 1 − G 1 , 1 ) . (154) f A A , 0 g g = 12 ln 2 − 2 ζ 2 − 3 − 8 β ln 2 + β + 2 2 β 2 G − 1 + 8 β ln 2 + β − 2 2 β 2 G 1 + 3 2 ( G − 1 , − 1 + G 1 , 1 ) + β + 3 2 β G − 1 , 1 + β − 3 2 β G 1 , − 1 − 4 β − 3 2 β G 0 , − 1 − 4 β + 3 2 β G 0 , 1 + 3 β − 1 4 β ( G − 1 , − 1 , − 1 − G − 1 , − 1 , 1 ) + β − 1 2 β ( − G − 1 , 0 , − 1 + G − 1 , 0 , 1 ) + β + 1 4 β ( − G − 1 , 1 , − 1 + G − 1 , 1 , 1 ) + 2 β − 1 2 β ( − G 0 , − 1 , − 1 + G 0 , − 1 , 1 ) + 1 β ( − G 0 , 0 , − 1 + G 0 , 0 , 1 ) + 2 β + 1 2 β ( G 0 , 1 , − 1 − G 0 , 1 , 1 ) + β − 1 4 β ( G 1 , − 1 , − 1 − G 1 , − 1 , 1 ) + β + 1 2 β ( G 1 , 0 , − 1 − G 1 , 0 , 1 ) + 3 β + 1 4 β ( − G 1 , 1 , − 1 + G 1 , 1 , 1 ) . (155) \begin{aligned}
f^{gg}_{AA,0}={}&12\ln2-2\zeta_{2}-3-\frac{8\beta\ln2+\beta+2}{2\beta^{2}}G_{-1}+\frac{8\beta\ln2+\beta-2}{2\beta^{2}}G_{1}\\
&+\frac{3}{2}(G_{-1,-1}+G_{1,1})+\frac{\beta+3}{2\beta}G_{-1,1}+\frac{\beta-3}{2\beta}G_{1,-1}\\
&-\frac{4\beta-3}{2\beta}G_{0,-1}-\frac{4\beta+3}{2\beta}G_{0,1}+\frac{3\beta-1}{4\beta}(G_{-1,-1,-1}-G_{-1,-1,1})\\
&+\frac{\beta-1}{2\beta}(-G_{-1,0,-1}+G_{-1,0,1})+\frac{\beta+1}{4\beta}(-G_{-1,1,-1}+G_{-1,1,1})\\
&+\frac{2\beta-1}{2\beta}(-G_{0,-1,-1}+G_{0,-1,1})+\frac{1}{\beta}(-G_{0,0,-1}+G_{0,0,1})\\
&+\frac{2\beta+1}{2\beta}(G_{0,1,-1}-G_{0,1,1})+\frac{\beta-1}{4\beta}(G_{1,-1,-1}-G_{1,-1,1})\\
&+\frac{\beta+1}{2\beta}(G_{1,0,-1}-G_{1,0,1})+\frac{3\beta+1}{4\beta}(-G_{1,1,-1}+G_{1,1,1}). \tag*{(155)}
\end{aligned} f AA , 0 g g = 12 ln 2 − 2 ζ 2 − 3 − 2 β 2 8 β ln 2 + β + 2 G − 1 + 2 β 2 8 β ln 2 + β − 2 G 1 + 2 3 ( G − 1 , − 1 + G 1 , 1 ) + 2 β β + 3 G − 1 , 1 + 2 β β − 3 G 1 , − 1 − 2 β 4 β − 3 G 0 , − 1 − 2 β 4 β + 3 G 0 , 1 + 4 β 3 β − 1 ( G − 1 , − 1 , − 1 − G − 1 , − 1 , 1 ) + 2 β β − 1 ( − G − 1 , 0 , − 1 + G − 1 , 0 , 1 ) + 4 β β + 1 ( − G − 1 , 1 , − 1 + G − 1 , 1 , 1 ) + 2 β 2 β − 1 ( − G 0 , − 1 , − 1 + G 0 , − 1 , 1 ) + β 1 ( − G 0 , 0 , − 1 + G 0 , 0 , 1 ) + 2 β 2 β + 1 ( G 0 , 1 , − 1 − G 0 , 1 , 1 ) + 4 β β − 1 ( G 1 , − 1 , − 1 − G 1 , − 1 , 1 ) + 2 β β + 1 ( G 1 , 0 , − 1 − G 1 , 0 , 1 ) + 4 β 3 β + 1 ( − G 1 , 1 , − 1 + G 1 , 1 , 1 ) . (155) The correlations between distinct emitters are
f A B , − 3 g g = 3 8 + β 2 + 1 4 β ( − G − 1 + G 1 ) , f^{gg}_{AB,-3}=\frac{3}{8}+\frac{\beta^{2}+1}{4\beta}(-G_{-1}+G_{1}), f A B , − 3 g g = 8 3 + 4 β β 2 + 1 ( − G − 1 + G 1 ) , f A B , − 2 g g = − 4 3 + 13 β 2 + 31 24 β ( G − 1 − G 1 ) + ( β 2 + 1 ) 2 4 β 2 ( − G − 1 , − 1 + G − 1 , 1 + G 1 , − 1 − G 1 , 1 ) + β 2 + 1 2 β ( − G 0 , − 1 + G 0 , 1 ) , \begin{aligned}
f^{gg}_{AB,-2}={}&-\frac{4}{3}+\frac{13\beta^{2}+31}{24\beta}(G_{-1}-G_{1})\\
&+\frac{(\beta^{2}+1)^{2}}{4\beta^{2}}(-G_{-1,-1}+G_{-1,1}+G_{1,-1}-G_{1,1})+\frac{\beta^{2}+1}{2\beta}(-G_{0,-1}+G_{0,1}),
\end{aligned} f A B , − 2 g g = − 3 4 + 24 β 13 β 2 + 31 ( G − 1 − G 1 ) + 4 β 2 ( β 2 + 1 ) 2 ( − G − 1 , − 1 + G − 1 , 1 + G 1 , − 1 − G 1 , 1 ) + 2 β β 2 + 1 ( − G 0 , − 1 + G 0 , 1 ) , f A B , − 1 g g = 15 ln 2 − 13 9 + − 62 β 2 + 66 ln 2 ( β 2 + 1 ) + 25 36 β ( − G − 1 + G 1 ) + 12 β 4 − β 3 + 9 β 2 + 2 β + 24 12 β 2 ( G − 1 , − 1 − G − 1 , 1 ) + 4 β 2 + 13 6 β ( G 0 , − 1 − G 0 , 1 ) + 12 β 4 + β 3 + 9 β 2 − 2 β + 24 12 β 2 ( − G 1 , − 1 + G 1 , 1 ) + ( β 2 + 1 ) ( β 2 + β + 1 ) β 2 ( G − 1 , − 1 , − 1 − G − 1 , − 1 , 1 ) + 3 ( β 2 + 1 ) 2 4 β 2 ( − G − 1 , 0 , − 1 + G − 1 , 0 , 1 + G 1 , 0 , − 1 − G 1 , 0 , 1 ) + β 2 + 1 β 2 ( − G − 1 , 1 , − 1 + G − 1 , 1 , 1 − G 0 , 0 , − 1 + G 0 , 0 , 1 − G 1 , − 1 , − 1 + G 1 , − 1 , 1 ) + ( β 2 + 1 ) 2 β 2 ( − G 0 , − 1 , − 1 + G 0 , − 1 , 1 + G 0 , 1 , − 1 − G 0 , 1 , 1 ) + ( β 2 + 1 ) ( β 2 − β + 1 ) β 2 ( − G 1 , 1 , − 1 + G 1 , 1 , 1 ) . (156) \begin{aligned}
f^{gg}_{AB,-1}={}&\frac{15\ln2-13}{9}+\frac{-62\beta^{2}+66\ln2(\beta^{2}+1)+25}{36\beta}(-G_{-1}+G_{1})\\
&+\frac{12\beta^{4}-\beta^{3}+9\beta^{2}+2\beta+24}{12\beta^{2}}(G_{-1,-1}-G_{-1,1})+\frac{4\beta^{2}+13}{6\beta}(G_{0,-1}-G_{0,1})\\
&+\frac{12\beta^{4}+\beta^{3}+9\beta^{2}-2\beta+24}{12\beta^{2}}(-G_{1,-1}+G_{1,1})\\
&+\frac{(\beta^{2}+1)(\beta^{2}+\beta+1)}{\beta^{2}}(G_{-1,-1,-1}-G_{-1,-1,1})\\
&+\frac{3(\beta^{2}+1)^{2}}{4\beta^{2}}(-G_{-1,0,-1}+G_{-1,0,1}+G_{1,0,-1}-G_{1,0,1})\\
&+\frac{\beta^{2}+1}{\beta^{2}}(-G_{-1,1,-1}+G_{-1,1,1}-G_{0,0,-1}+G_{0,0,1}-G_{1,-1,-1}+G_{1,-1,1})\\
&+\frac{(\beta^{2}+1)^{2}}{\beta^{2}}(-G_{0,-1,-1}+G_{0,-1,1}+G_{0,1,-1}-G_{0,1,1})\\
&+\frac{(\beta^{2}+1)(\beta^{2}-\beta+1)}{\beta^{2}}(-G_{1,1,-1}+G_{1,1,1}). \tag*{(156)}
\end{aligned} f A B , − 1 g g = 9 15 ln 2 − 13 + 36 β − 62 β 2 + 66 ln 2 ( β 2 + 1 ) + 25 ( − G − 1 + G 1 ) + 12 β 2 12 β 4 − β 3 + 9 β 2 + 2 β + 24 ( G − 1 , − 1 − G − 1 , 1 ) + 6 β 4 β 2 + 13 ( G 0 , − 1 − G 0 , 1 ) + 12 β 2 12 β 4 + β 3 + 9 β 2 − 2 β + 24 ( − G 1 , − 1 + G 1 , 1 ) + β 2 ( β 2 + 1 ) ( β 2 + β + 1 ) ( G − 1 , − 1 , − 1 − G − 1 , − 1 , 1 ) + 4 β 2 3 ( β 2 + 1 ) 2 ( − G − 1 , 0 , − 1 + G − 1 , 0 , 1 + G 1 , 0 , − 1 − G 1 , 0 , 1 ) + β 2 β 2 + 1 ( − G − 1 , 1 , − 1 + G − 1 , 1 , 1 − G 0 , 0 , − 1 + G 0 , 0 , 1 − G 1 , − 1 , − 1 + G 1 , − 1 , 1 ) + β 2 ( β 2 + 1 ) 2 ( − G 0 , − 1 , − 1 + G 0 , − 1 , 1 + G 0 , 1 , − 1 − G 0 , 1 , 1 ) + β 2 ( β 2 + 1 ) ( β 2 − β + 1 ) ( − G 1 , 1 , − 1 + G 1 , 1 , 1 ) . (156) f A B , 0 g g = − 1740 ln 2 + 180 ζ 2 + 1787 108 − 11 β 2 + 12 12 β 2 ( G − 1 + G 1 ) + 1769 β 2 − 12 ln 2 ( 199 β 2 + 259 ) + 396 ζ 2 ( β 2 + 1 ) + 2573 216 β ( G 1 − G − 1 ) + ( β − 1 ) ( β + 1 ) 2 ( β + 3 ) 4 β 2 ( − 2 G − 3 , − 1 + G − 3 , 1 ) − β 4 − 24 β 2 + 8 ln 2 ( 3 β 2 − 1 ) + 29 4 β 2 ( G − 1 , − 1 + G 1 , 1 ) + 23 β 2 + 6 ln 2 ( β 2 + 4 ) − 55 9 β ( G − 1 , − 1 − G 1 , 1 ) + 3 β 4 − 43 β 2 + 12 ln 2 ( 3 β 2 − 1 ) + 39 6 β 2 ( G − 1 , 1 + G 1 , − 1 ) + 13 β 2 − 24 ln 2 ( β 2 + 4 ) + 43 36 β ( G − 1 , 1 − G 1 , − 1 ) + − 209 β 2 + 192 ln 2 ( β 2 + 1 ) − 47 36 β ( G 0 , 1 − G 0 , − 1 ) + 5 3 ( G 0 , − 1 + G 0 , 1 ) + ( β − 3 ) ( β − 1 ) 2 ( β + 1 ) 4 β 2 ( G 3 , − 1 − 2 G 3 , 1 ) + ( β + 1 ) 2 ( 2 β − 1 ) 2 β 2 ( − 2 G − 1 , − 3 , − 1 + G − 1 , − 3 , 1 ) + − 48 β 4 + 2 β 3 − 3 β 2 − 73 β + 12 ln 2 ( β + 1 ) 2 ( β 2 + 1 ) − 108 12 β 2 G − 1 , − 1 , − 1 − − 48 β 4 − 28 β 3 − 75 β 2 − 91 β + 12 ln 2 ( β + 1 ) 2 ( β 2 + 1 ) − 84 12 β 2 G − 1 , − 1 , 1 − − 6 β 4 + 4 β 3 − 11 β 2 + 3 β + 2 ln 2 ( β 2 + 1 ) 2 − 10 2 β 2 G − 1 , 0 , − 1 + − 18 β 4 − 10 β 3 − 33 β 2 − 13 β + 6 ln 2 ( β 2 + 1 ) 2 − 30 6 β 2 G − 1 , 0 , 1 + 40 β 3 − 75 β 2 + 97 β + 12 ln 2 ( β − 1 ) 2 ( β 2 + 1 ) + 24 12 β 2 G − 1 , 1 , − 1 − 8 β 3 − 21 β 2 + 77 β + 12 ln 2 ( β − 1 ) 2 ( β 2 + 1 ) + 6 12 β 2 G − 1 , 1 , 1 + ( β − 1 ) ( β + 1 ) 2 β ( G − 1 , 3 , − 1 − G 1 , − 3 , 1 − 2 G − 1 , 3 , 1 + 2 G 1 , − 3 , − 1 ) + 24 β 4 + 4 β 3 + 18 β 2 + 7 β + 48 6 β 2 G 0 , − 1 , − 1 − 8 β 4 − 2 β 3 + 6 β 2 − β + 16 2 β 2 G 0 , − 1 , 1 + 4 β 2 + 13 3 β ( G 0 , 0 , − 1 − G 0 , 0 , 1 ) − 8 β 4 + 2 β 3 + 6 β 2 + β + 16 2 β 2 G 0 , 1 , − 1 + 24 β 4 − 4 β 3 + 18 β 2 − 7 β + 48 6 β 2 G 0 , 1 , 1 \begin{aligned}
f_{AB,0}^{gg}={}&\frac{-1740\ln2+180\zeta_{2}+1787}{108}-\frac{11\beta^{2}+12}{12\beta^{2}}(G_{-1}+G_{1})\\
&+\frac{1769\beta^{2}-12\ln2(199\beta^{2}+259)+396\zeta_{2}(\beta^{2}+1)+2573}{216\beta}(G_{1}-G_{-1})\\
&+\frac{(\beta-1)(\beta+1)^{2}(\beta+3)}{4\beta^{2}}(-2G_{-3,-1}+G_{-3,1})\\
&-\frac{\beta^{4}-24\beta^{2}+8\ln2(3\beta^{2}-1)+29}{4\beta^{2}}(G_{-1,-1}+G_{1,1})\\
&+\frac{23\beta^{2}+6\ln2(\beta^{2}+4)-55}{9\beta}(G_{-1,-1}-G_{1,1})\\
&+\frac{3\beta^{4}-43\beta^{2}+12\ln2(3\beta^{2}-1)+39}{6\beta^{2}}(G_{-1,1}+G_{1,-1})\\
&+\frac{13\beta^{2}-24\ln2(\beta^{2}+4)+43}{36\beta}(G_{-1,1}-G_{1,-1})\\
&+\frac{-209\beta^{2}+192\ln2(\beta^{2}+1)-47}{36\beta}(G_{0,1}-G_{0,-1})+\frac{5}{3}(G_{0,-1}+G_{0,1})\\
&+\frac{(\beta-3)(\beta-1)^{2}(\beta+1)}{4\beta^{2}}(G_{3,-1}-2G_{3,1})\\
&+\frac{(\beta+1)^{2}(2\beta-1)}{2\beta^{2}}(-2G_{-1,-3,-1}+G_{-1,-3,1})\\
&+\frac{-48\beta^{4}+2\beta^{3}-3\beta^{2}-73\beta+12\ln2(\beta+1)^{2}(\beta^{2}+1)-108}{12\beta^{2}}G_{-1,-1,-1}\\
&-\frac{-48\beta^{4}-28\beta^{3}-75\beta^{2}-91\beta+12\ln2(\beta+1)^{2}(\beta^{2}+1)-84}{12\beta^{2}}G_{-1,-1,1}\\
&-\frac{-6\beta^{4}+4\beta^{3}-11\beta^{2}+3\beta+2\ln2(\beta^{2}+1)^{2}-10}{2\beta^{2}}G_{-1,0,-1}\\
&+\frac{-18\beta^{4}-10\beta^{3}-33\beta^{2}-13\beta+6\ln2(\beta^{2}+1)^{2}-30}{6\beta^{2}}G_{-1,0,1}\\
&+\frac{40\beta^{3}-75\beta^{2}+97\beta+12\ln2(\beta-1)^{2}(\beta^{2}+1)+24}{12\beta^{2}}G_{-1,1,-1}\\
&-\frac{8\beta^{3}-21\beta^{2}+77\beta+12\ln2(\beta-1)^{2}(\beta^{2}+1)+6}{12\beta^{2}}G_{-1,1,1}\\
&+\frac{(\beta-1)(\beta+1)}{2\beta}(G_{-1,3,-1}-G_{1,-3,1}-2G_{-1,3,1}+2G_{1,-3,-1})\\
&+\frac{24\beta^{4}+4\beta^{3}+18\beta^{2}+7\beta+48}{6\beta^{2}}G_{0,-1,-1}\\
&-\frac{8\beta^{4}-2\beta^{3}+6\beta^{2}-\beta+16}{2\beta^{2}}G_{0,-1,1}+\frac{4\beta^{2}+13}{3\beta}(G_{0,0,-1}-G_{0,0,1})\\
&-\frac{8\beta^{4}+2\beta^{3}+6\beta^{2}+\beta+16}{2\beta^{2}}G_{0,1,-1}\\
&+\frac{24\beta^{4}-4\beta^{3}+18\beta^{2}-7\beta+48}{6\beta^{2}}G_{0,1,1}
\end{aligned} f A B , 0 g g = 108 − 1740 ln 2 + 180 ζ 2 + 1787 − 12 β 2 11 β 2 + 12 ( G − 1 + G 1 ) + 216 β 1769 β 2 − 12 ln 2 ( 199 β 2 + 259 ) + 396 ζ 2 ( β 2 + 1 ) + 2573 ( G 1 − G − 1 ) + 4 β 2 ( β − 1 ) ( β + 1 ) 2 ( β + 3 ) ( − 2 G − 3 , − 1 + G − 3 , 1 ) − 4 β 2 β 4 − 24 β 2 + 8 ln 2 ( 3 β 2 − 1 ) + 29 ( G − 1 , − 1 + G 1 , 1 ) + 9 β 23 β 2 + 6 ln 2 ( β 2 + 4 ) − 55 ( G − 1 , − 1 − G 1 , 1 ) + 6 β 2 3 β 4 − 43 β 2 + 12 ln 2 ( 3 β 2 − 1 ) + 39 ( G − 1 , 1 + G 1 , − 1 ) + 36 β 13 β 2 − 24 ln 2 ( β 2 + 4 ) + 43 ( G − 1 , 1 − G 1 , − 1 ) + 36 β − 209 β 2 + 192 ln 2 ( β 2 + 1 ) − 47 ( G 0 , 1 − G 0 , − 1 ) + 3 5 ( G 0 , − 1 + G 0 , 1 ) + 4 β 2 ( β − 3 ) ( β − 1 ) 2 ( β + 1 ) ( G 3 , − 1 − 2 G 3 , 1 ) + 2 β 2 ( β + 1 ) 2 ( 2 β − 1 ) ( − 2 G − 1 , − 3 , − 1 + G − 1 , − 3 , 1 ) + 12 β 2 − 48 β 4 + 2 β 3 − 3 β 2 − 73 β + 12 ln 2 ( β + 1 ) 2 ( β 2 + 1 ) − 108 G − 1 , − 1 , − 1 − 12 β 2 − 48 β 4 − 28 β 3 − 75 β 2 − 91 β + 12 ln 2 ( β + 1 ) 2 ( β 2 + 1 ) − 84 G − 1 , − 1 , 1 − 2 β 2 − 6 β 4 + 4 β 3 − 11 β 2 + 3 β + 2 ln 2 ( β 2 + 1 ) 2 − 10 G − 1 , 0 , − 1 + 6 β 2 − 18 β 4 − 10 β 3 − 33 β 2 − 13 β + 6 ln 2 ( β 2 + 1 ) 2 − 30 G − 1 , 0 , 1 + 12 β 2 40 β 3 − 75 β 2 + 97 β + 12 ln 2 ( β − 1 ) 2 ( β 2 + 1 ) + 24 G − 1 , 1 , − 1 − 12 β 2 8 β 3 − 21 β 2 + 77 β + 12 ln 2 ( β − 1 ) 2 ( β 2 + 1 ) + 6 G − 1 , 1 , 1 + 2 β ( β − 1 ) ( β + 1 ) ( G − 1 , 3 , − 1 − G 1 , − 3 , 1 − 2 G − 1 , 3 , 1 + 2 G 1 , − 3 , − 1 ) + 6 β 2 24 β 4 + 4 β 3 + 18 β 2 + 7 β + 48 G 0 , − 1 , − 1 − 2 β 2 8 β 4 − 2 β 3 + 6 β 2 − β + 16 G 0 , − 1 , 1 + 3 β 4 β 2 + 13 ( G 0 , 0 , − 1 − G 0 , 0 , 1 ) − 2 β 2 8 β 4 + 2 β 3 + 6 β 2 + β + 16 G 0 , 1 , − 1 + 6 β 2 24 β 4 − 4 β 3 + 18 β 2 − 7 β + 48 G 0 , 1 , 1 − − 8 β 3 − 21 β 2 − 77 β + 12 ln 2 ( β + 1 ) 2 ( β 2 + 1 ) + 6 12 β 2 G 1 , − 1 , − 1 + − 40 β 3 − 75 β 2 − 97 β + 12 ln 2 ( β + 1 ) 2 ( β 2 + 1 ) + 24 12 β 2 G 1 , − 1 , 1 + − 18 β 4 + 10 β 3 − 33 β 2 + 13 β + 6 ln 2 ( β 2 + 1 ) 2 − 30 6 β 2 G 1 , 0 , − 1 − 6 β 4 − 4 β 3 − 11 β 2 − 3 β + 2 ln 2 ( β 2 + 1 ) 2 − 10 2 β 2 G 1 , 0 , 1 − − 48 β 4 + 28 β 3 − 75 β 2 + 91 β + 12 ln 2 ( β − 1 ) 2 ( β 2 + 1 ) − 84 12 β 2 G 1 , 1 , − 1 + − 48 β 4 − 2 β 3 − 3 β 2 + 73 β + 12 ln 2 ( β − 1 ) 2 ( β 2 + 1 ) − 108 12 β 2 G 1 , 1 , 1 + ( β − 1 ) 2 ( 2 β + 1 ) 2 β 2 ( − G 1 , 3 , − 1 + 2 G 1 , 3 , 1 ) + β 2 + 1 2 β 2 [ ( β + 1 ) 2 2 ( 2 G − 1 , − 1 , − 3 , − 1 − 2 G 1 , − 1 , − 3 , − 1 − G − 1 , − 1 , − 3 , 1 + G 1 , − 1 , − 3 , 1 ) ] − ( 7 β 2 + 10 β + 7 ) G − 1 , − 1 , − 1 , − 1 + ( 5 β 2 + 6 β + 5 ) G − 1 , − 1 , − 1 , 1 + ( 5 β 2 + 4 β + 5 ) ( G − 1 , − 1 , 0 , − 1 − G − 1 , − 1 , 0 , 1 ) + 4 β ( G − 1 , − 1 , 1 , − 1 + G − 1 , 1 , − 1 , 1 − G 0 , − 1 , 1 , − 1 + G 0 , − 1 , 1 , 1 − G 0 , 0 , 0 , − 1 + G 0 , 0 , 0 , 1 − G 0 , 1 , − 1 , − 1 + G 0 , 1 , − 1 , 1 − G 1 , − 1 , 1 , − 1 − G 1 , 1 , − 1 , 1 ) + 3 β 2 − 2 β + 3 2 ( G − 1 , − 1 , 1 , 1 − G 1 , − 1 , 1 , 1 ) + ( 7 β 2 + 8 β + 7 ) G − 1 , 0 , − 1 , − 1 − ( 5 β 2 + 8 β + 5 ) G − 1 , 0 , − 1 , 1 + 3 ( β 2 + 1 ) ( − G − 1 , 0 , 0 , − 1 + G − 1 , 0 , 0 , 1 + G 1 , 0 , 0 , − 1 − G 1 , 0 , 0 , 1 ) − ( 3 β 2 + 8 β + 3 ) G − 1 , 0 , 1 , − 1 + ( β 2 + 8 β + 1 ) G − 1 , 0 , 1 , 1 + 3 β 2 + 2 β + 3 2 ( − G − 1 , 1 , − 1 , − 1 + G 1 , 1 , − 1 , − 1 ) + ( β 2 + 4 β + 1 ) ( − G − 1 , 1 , 0 , − 1 + G − 1 , 1 , 0 , 1 ) + 3 ( β + 1 ) 2 G − 1 , 1 , 1 , − 1 − ( β 2 + 10 β + 1 ) G − 1 , 1 , 1 , 1 + ( β − 1 ) 2 2 ( G − 1 , 1 , 3 , − 1 − G 1 , 1 , 3 , − 1 ) + ( β − 1 ) 2 ( − G − 1 , 1 , 3 , 1 + G 1 , 1 , 3 , 1 ) + 4 ( 2 β 2 + β + 2 ) ( G 0 , − 1 , − 1 , − 1 − G 0 , − 1 , − 1 , 1 ) + 6 ( β 2 + 1 ) ( − G 0 , − 1 , 0 , − 1 + G 0 , − 1 , 0 , 1 + G 0 , 1 , 0 , − 1 − G 0 , 1 , 0 , 1 ) + 8 ( β 2 + 1 ) ( − G 0 , 0 , − 1 , − 1 + G 0 , 0 , − 1 , 1 + G 0 , 0 , 1 , − 1 − G 0 , 0 , 1 , 1 ) + 4 ( 2 β 2 − β + 2 ) ( − G 0 , 1 , 1 , − 1 + G 0 , 1 , 1 , 1 ) − ( β 2 − 10 β + 1 ) G 1 , − 1 , − 1 , − 1 + 3 ( β − 1 ) 2 G 1 , − 1 , − 1 , 1 + ( β 2 − 4 β + 1 ) ( G 1 , − 1 , 0 , − 1 − G 1 , − 1 , 0 , 1 ) + ( β 2 − 8 β + 1 ) G 1 , 0 , − 1 , − 1 − ( 3 β 2 − 8 β + 3 ) G 1 , 0 , − 1 , 1 − ( 5 β 2 − 8 β + 5 ) G 1 , 0 , 1 , − 1 + ( 7 β 2 − 8 β + 7 ) G 1 , 0 , 1 , 1 \begin{aligned}
&-\frac{-8\beta^{3}-21\beta^{2}-77\beta+12\ln2(\beta+1)^{2}(\beta^{2}+1)+6}{12\beta^{2}}G_{1,-1,-1}\\
&+\frac{-40\beta^{3}-75\beta^{2}-97\beta+12\ln2(\beta+1)^{2}(\beta^{2}+1)+24}{12\beta^{2}}G_{1,-1,1}\\
&+\frac{-18\beta^{4}+10\beta^{3}-33\beta^{2}+13\beta+6\ln2(\beta^{2}+1)^{2}-30}{6\beta^{2}}G_{1,0,-1}\\
&-\frac{6\beta^{4}-4\beta^{3}-11\beta^{2}-3\beta+2\ln2(\beta^{2}+1)^{2}-10}{2\beta^{2}}G_{1,0,1}\\
&-\frac{-48\beta^{4}+28\beta^{3}-75\beta^{2}+91\beta+12\ln2(\beta-1)^{2}(\beta^{2}+1)-84}{12\beta^{2}}G_{1,1,-1}\\
&+\frac{-48\beta^{4}-2\beta^{3}-3\beta^{2}+73\beta+12\ln2(\beta-1)^{2}(\beta^{2}+1)-108}{12\beta^{2}}G_{1,1,1}\\
&+\frac{(\beta-1)^{2}(2\beta+1)}{2\beta^{2}}(-G_{1,3,-1}+2G_{1,3,1})\\
&+\frac{\beta^{2}+1}{2\beta^{2}}\left[\frac{(\beta+1)^{2}}{2}(2G_{-1,-1,-3,-1}-2G_{1,-1,-3,-1}-G_{-1,-1,-3,1}+G_{1,-1,-3,1})\right]\\
&-(7\beta^{2}+10\beta+7)G_{-1,-1,-1,-1}+(5\beta^{2}+6\beta+5)G_{-1,-1,-1,1}\\
&+(5\beta^{2}+4\beta+5)(G_{-1,-1,0,-1}-G_{-1,-1,0,1})\\
&+4\beta(G_{-1,-1,1,-1}+G_{-1,1,-1,1}-G_{0,-1,1,-1}+G_{0,-1,1,1}-G_{0,0,0,-1}\\
&\qquad+G_{0,0,0,1}-G_{0,1,-1,-1}+G_{0,1,-1,1}-G_{1,-1,1,-1}-G_{1,1,-1,1})\\
&+\frac{3\beta^{2}-2\beta+3}{2}(G_{-1,-1,1,1}-G_{1,-1,1,1})\\
&+(7\beta^{2}+8\beta+7)G_{-1,0,-1,-1}-(5\beta^{2}+8\beta+5)G_{-1,0,-1,1}\\
&+3(\beta^{2}+1)(-G_{-1,0,0,-1}+G_{-1,0,0,1}+G_{1,0,0,-1}-G_{1,0,0,1})\\
&-(3\beta^{2}+8\beta+3)G_{-1,0,1,-1}+(\beta^{2}+8\beta+1)G_{-1,0,1,1}\\
&+\frac{3\beta^{2}+2\beta+3}{2}(-G_{-1,1,-1,-1}+G_{1,1,-1,-1})\\
&+(\beta^{2}+4\beta+1)(-G_{-1,1,0,-1}+G_{-1,1,0,1})+3(\beta+1)^{2}G_{-1,1,1,-1}\\
&-(\beta^{2}+10\beta+1)G_{-1,1,1,1}+\frac{(\beta-1)^{2}}{2}(G_{-1,1,3,-1}-G_{1,1,3,-1})\\
&+(\beta-1)^{2}(-G_{-1,1,3,1}+G_{1,1,3,1})\\
&+4(2\beta^{2}+\beta+2)(G_{0,-1,-1,-1}-G_{0,-1,-1,1})\\
&+6(\beta^{2}+1)(-G_{0,-1,0,-1}+G_{0,-1,0,1}+G_{0,1,0,-1}-G_{0,1,0,1})\\
&+8(\beta^{2}+1)(-G_{0,0,-1,-1}+G_{0,0,-1,1}+G_{0,0,1,-1}-G_{0,0,1,1})\\
&+4(2\beta^{2}-\beta+2)(-G_{0,1,1,-1}+G_{0,1,1,1})\\
&-(\beta^{2}-10\beta+1)G_{1,-1,-1,-1}+3(\beta-1)^{2}G_{1,-1,-1,1}\\
&+(\beta^{2}-4\beta+1)(G_{1,-1,0,-1}-G_{1,-1,0,1})\\
&+(\beta^{2}-8\beta+1)G_{1,0,-1,-1}-(3\beta^{2}-8\beta+3)G_{1,0,-1,1}\\
&-(5\beta^{2}-8\beta+5)G_{1,0,1,-1}+(7\beta^{2}-8\beta+7)G_{1,0,1,1}
\end{aligned} − 12 β 2 − 8 β 3 − 21 β 2 − 77 β + 12 ln 2 ( β + 1 ) 2 ( β 2 + 1 ) + 6 G 1 , − 1 , − 1 + 12 β 2 − 40 β 3 − 75 β 2 − 97 β + 12 ln 2 ( β + 1 ) 2 ( β 2 + 1 ) + 24 G 1 , − 1 , 1 + 6 β 2 − 18 β 4 + 10 β 3 − 33 β 2 + 13 β + 6 ln 2 ( β 2 + 1 ) 2 − 30 G 1 , 0 , − 1 − 2 β 2 6 β 4 − 4 β 3 − 11 β 2 − 3 β + 2 ln 2 ( β 2 + 1 ) 2 − 10 G 1 , 0 , 1 − 12 β 2 − 48 β 4 + 28 β 3 − 75 β 2 + 91 β + 12 ln 2 ( β − 1 ) 2 ( β 2 + 1 ) − 84 G 1 , 1 , − 1 + 12 β 2 − 48 β 4 − 2 β 3 − 3 β 2 + 73 β + 12 ln 2 ( β − 1 ) 2 ( β 2 + 1 ) − 108 G 1 , 1 , 1 + 2 β 2 ( β − 1 ) 2 ( 2 β + 1 ) ( − G 1 , 3 , − 1 + 2 G 1 , 3 , 1 ) + 2 β 2 β 2 + 1 [ 2 ( β + 1 ) 2 ( 2 G − 1 , − 1 , − 3 , − 1 − 2 G 1 , − 1 , − 3 , − 1 − G − 1 , − 1 , − 3 , 1 + G 1 , − 1 , − 3 , 1 ) ] − ( 7 β 2 + 10 β + 7 ) G − 1 , − 1 , − 1 , − 1 + ( 5 β 2 + 6 β + 5 ) G − 1 , − 1 , − 1 , 1 + ( 5 β 2 + 4 β + 5 ) ( G − 1 , − 1 , 0 , − 1 − G − 1 , − 1 , 0 , 1 ) + 4 β ( G − 1 , − 1 , 1 , − 1 + G − 1 , 1 , − 1 , 1 − G 0 , − 1 , 1 , − 1 + G 0 , − 1 , 1 , 1 − G 0 , 0 , 0 , − 1 + G 0 , 0 , 0 , 1 − G 0 , 1 , − 1 , − 1 + G 0 , 1 , − 1 , 1 − G 1 , − 1 , 1 , − 1 − G 1 , 1 , − 1 , 1 ) + 2 3 β 2 − 2 β + 3 ( G − 1 , − 1 , 1 , 1 − G 1 , − 1 , 1 , 1 ) + ( 7 β 2 + 8 β + 7 ) G − 1 , 0 , − 1 , − 1 − ( 5 β 2 + 8 β + 5 ) G − 1 , 0 , − 1 , 1 + 3 ( β 2 + 1 ) ( − G − 1 , 0 , 0 , − 1 + G − 1 , 0 , 0 , 1 + G 1 , 0 , 0 , − 1 − G 1 , 0 , 0 , 1 ) − ( 3 β 2 + 8 β + 3 ) G − 1 , 0 , 1 , − 1 + ( β 2 + 8 β + 1 ) G − 1 , 0 , 1 , 1 + 2 3 β 2 + 2 β + 3 ( − G − 1 , 1 , − 1 , − 1 + G 1 , 1 , − 1 , − 1 ) + ( β 2 + 4 β + 1 ) ( − G − 1 , 1 , 0 , − 1 + G − 1 , 1 , 0 , 1 ) + 3 ( β + 1 ) 2 G − 1 , 1 , 1 , − 1 − ( β 2 + 10 β + 1 ) G − 1 , 1 , 1 , 1 + 2 ( β − 1 ) 2 ( G − 1 , 1 , 3 , − 1 − G 1 , 1 , 3 , − 1 ) + ( β − 1 ) 2 ( − G − 1 , 1 , 3 , 1 + G 1 , 1 , 3 , 1 ) + 4 ( 2 β 2 + β + 2 ) ( G 0 , − 1 , − 1 , − 1 − G 0 , − 1 , − 1 , 1 ) + 6 ( β 2 + 1 ) ( − G 0 , − 1 , 0 , − 1 + G 0 , − 1 , 0 , 1 + G 0 , 1 , 0 , − 1 − G 0 , 1 , 0 , 1 ) + 8 ( β 2 + 1 ) ( − G 0 , 0 , − 1 , − 1 + G 0 , 0 , − 1 , 1 + G 0 , 0 , 1 , − 1 − G 0 , 0 , 1 , 1 ) + 4 ( 2 β 2 − β + 2 ) ( − G 0 , 1 , 1 , − 1 + G 0 , 1 , 1 , 1 ) − ( β 2 − 10 β + 1 ) G 1 , − 1 , − 1 , − 1 + 3 ( β − 1 ) 2 G 1 , − 1 , − 1 , 1 + ( β 2 − 4 β + 1 ) ( G 1 , − 1 , 0 , − 1 − G 1 , − 1 , 0 , 1 ) + ( β 2 − 8 β + 1 ) G 1 , 0 , − 1 , − 1 − ( 3 β 2 − 8 β + 3 ) G 1 , 0 , − 1 , 1 − ( 5 β 2 − 8 β + 5 ) G 1 , 0 , 1 , − 1 + ( 7 β 2 − 8 β + 7 ) G 1 , 0 , 1 , 1 + ( 5 β 2 − 4 β + 5 ) ( − G 1 , 1 , 0 , − 1 + G 1 , 1 , 0 , 1 ) + ( 5 β 2 − 6 β + 5 ) G 1 , 1 , 1 , − 1 − ( 7 β 2 − 10 β + 7 ) G 1 , 1 , 1 , 1 . (157) \begin{aligned}
&+ (5\beta^2-4\beta+5)(-G_{1,1,0,-1}+G_{1,1,0,1})\\
&+ (5\beta^2-6\beta+5)G_{1,1,1,-1}-(7\beta^2-10\beta+7)G_{1,1,1,1}. \tag*{(157)}
\end{aligned} + ( 5 β 2 − 4 β + 5 ) ( − G 1 , 1 , 0 , − 1 + G 1 , 1 , 0 , 1 ) + ( 5 β 2 − 6 β + 5 ) G 1 , 1 , 1 , − 1 − ( 7 β 2 − 10 β + 7 ) G 1 , 1 , 1 , 1 . (157) Eikonal integrals for single-soft subtraction
The kernel K S 4 \mathcal{K}_{\mathrm{S4}} K S4 [42 , 67 ]
J ( a , b ; E 3 ∗ ) = C ϵ ∫ [ d k ] K θ ( E 3 ∗ − u K ⋅ k ) a ⋅ b ( a ⋅ k ) ( b ⋅ k ) = − 1 ( 4 π ) 2 [ I a b ( − 1 ) ϵ 2 + I a b ( 0 ) + L 3 I a b ( − 1 ) ϵ \begin{aligned}
\mathcal{J}(a,b;E_3^*)=\mathcal{C}_{\epsilon}\int[dk]_{K}\,\theta(E_3^*-u_K\cdot k)\frac{a\cdot b}{(a\cdot k)(b\cdot k)} \\
&=-\frac{1}{(4\pi)^2}\left[\frac{I_{ab}^{(-1)}}{\epsilon^2}+\frac{I_{ab}^{(0)}+L_3I_{ab}^{(-1)}}{\epsilon}
\right.
\end{aligned} J ( a , b ; E 3 ∗ ) = C ϵ ∫ [ d k ] K θ ( E 3 ∗ − u K ⋅ k ) ( a ⋅ k ) ( b ⋅ k ) a ⋅ b = − ( 4 π ) 2 1 [ ϵ 2 I ab ( − 1 ) + ϵ I ab ( 0 ) + L 3 I ab ( − 1 ) + I a b ( 1 ) + L 3 I a b ( 0 ) + ( L 3 2 2 − 3 2 ζ 2 ) I a b ( − 1 ) ] + O ( ϵ ) , (158) \left.
+I_{ab}^{(1)}+L_3I_{ab}^{(0)}+\left(\frac{L_3^2}{2}-\frac{3}{2}\zeta_2\right)I_{ab}^{(-1)}\right]+O(\epsilon), \tag*{(158)} + I ab ( 1 ) + L 3 I ab ( 0 ) + ( 2 L 3 2 − 2 3 ζ 2 ) I ab ( − 1 ) ] + O ( ϵ ) , (158) where L 3 = ln [ μ R 2 / ( 4 ( E 3 ∗ ) 2 ) ] L_3=\ln[\mu_R^2/(4(E_3^*)^2)] L 3 = ln [ μ R 2 / ( 4 ( E 3 ∗ ) 2 )]
For two distinct massive emitters, a 2 = b 2 = m c 2 a^2=b^2=m_c^2 a 2 = b 2 = m c 2 u ∗ = a ∗ / ( u K ⋅ a ) u^*=a^*/(u_K\cdot a) u ∗ = a ∗ / ( u K ⋅ a ) w ∗ = b ∗ / ( u K ⋅ b ) w^*=b^*/(u_K\cdot b) w ∗ = b ∗ / ( u K ⋅ b ) K K K v a b = 1 − m c 4 / ( a ⋅ b ) 2 v_{ab}=\sqrt{1-m_c^4/(a\cdot b)^2} v ab = 1 − m c 4 / ( a ⋅ b ) 2
I a b ( − 1 ) = 0 , I a b ( 0 ) = 1 v a b ln 1 + v a b 1 − v a b , I_{ab}^{(-1)}=0,\qquad I_{ab}^{(0)}=\frac{1}{v_{ab}}\ln\frac{1+v_{ab}}{1-v_{ab}}, I ab ( − 1 ) = 0 , I ab ( 0 ) = v ab 1 ln 1 − v ab 1 + v ab , I a b ( 1 ) = 1 − u ∗ ⋅ w ∗ Δ a b 2 − ρ a b 2 [ K ( z 2 ) − K ( z 1 ) ] . (159) I_{ab}^{(1)}=\frac{1-u^*\cdot w^*}{\sqrt{\Delta_{ab}^2-\rho_{ab}^2}}\left[\mathcal{K}(z_2)-\mathcal{K}(z_1)\right]. \tag*{(159)} I ab ( 1 ) = Δ ab 2 − ρ ab 2 1 − u ∗ ⋅ w ∗ [ K ( z 2 ) − K ( z 1 ) ] . (159) The auxiliary variables in I a b ( 1 ) I_{ab}^{(1)} I ab ( 1 )
Δ a b = ∣ u ∗ − w ∗ ∣ , ρ a b 2 = ( u ∗ ) 2 ( w ∗ ) 2 − ( u ∗ ⋅ w ∗ ) 2 , \Delta_{ab}=\lvert u^*-w^*\rvert,\qquad\rho_{ab}^2=(u^*)^2(w^*)^2-(u^*\cdot w^*)^2, Δ ab = ∣ u ∗ − w ∗ ∣ , ρ ab 2 = ( u ∗ ) 2 ( w ∗ ) 2 − ( u ∗ ⋅ w ∗ ) 2 , X 1 = ( u ∗ ) 2 − u ∗ ⋅ w ∗ , X 2 = ( w ∗ ) 2 − u ∗ ⋅ w ∗ , X_1=(u^*)^2-u^*\cdot w^*,\qquad X_2=(w^*)^2-u^*\cdot w^*, X 1 = ( u ∗ ) 2 − u ∗ ⋅ w ∗ , X 2 = ( w ∗ ) 2 − u ∗ ⋅ w ∗ , z 1 = X 1 2 + ρ a b 2 − X 1 , z 2 = X 2 2 + ρ a b 2 + X 2 . (160) z_1=\sqrt{X_1^2+\rho_{ab}^2}-X_1,\qquad z_2=\sqrt{X_2^2+\rho_{ab}^2}+X_2. \tag*{(160)} z 1 = X 1 2 + ρ ab 2 − X 1 , z 2 = X 2 2 + ρ ab 2 + X 2 . (160) With z ± = Δ a b ± Δ a b 2 − ρ a b 2 z_{\pm}=\Delta_{ab}\pm\sqrt{\Delta_{ab}^2-\rho_{ab}^2} z ± = Δ ab ± Δ ab 2 − ρ ab 2 K \mathcal{K} K
K ( y ) = − 2 Li 2 ( 2 z − ( z + − y ) ( z + − z − ) ( z − + y ) ) − 2 Li 2 ( − 2 z + ( z − + y ) ( z + − z − ) ( z + − y ) ) \mathcal{K}(y)=-2\operatorname{Li}_2\left(\frac{2z_-(z_+-y)}{(z_+-z_-)(z_-+y)}\right)-2\operatorname{Li}_2\left(\frac{-2z_+(z_-+y)}{(z_+-z_-)(z_+-y)}\right) K ( y ) = − 2 Li 2 ( ( z + − z − ) ( z − + y ) 2 z − ( z + − y ) ) − 2 Li 2 ( ( z + − z − ) ( z + − y ) − 2 z + ( z − + y ) ) − 1 2 ln 2 [ ( y − z − ) ( z + − y ) ( y + z − ) ( z + + y ) ] . (161) -\frac{1}{2}\ln^2\left[\frac{(y-z_-)(z_+-y)}{(y+z_-)(z_++y)}\right]. \tag*{(161)} − 2 1 ln 2 [ ( y + z − ) ( z + + y ) ( y − z − ) ( z + − y ) ] . (161) For the Born momenta p 1 + p 2 = K p_1+p_2=K p 1 + p 2 = K β = 1 − 4 m c 2 / K 2 \beta=\sqrt{1-4m_c^2/K^2} β = 1 − 4 m c 2 / K 2
I p 1 p 2 ( 1 ) = 1 + β 2 β [ Li 2 ( 2 β 1 + β ) − Li 2 ( − 2 β 1 − β ) ] . I_{p_1p_2}^{(1)}=\frac{1+\beta^2}{\beta}\left[\operatorname{Li}_2\left(\frac{2\beta}{1+\beta}\right)-\operatorname{Li}_2\left(-\frac{2\beta}{1-\beta}\right)\right]. I p 1 p 2 ( 1 ) = β 1 + β 2 [ Li 2 ( 1 + β 2 β ) − Li 2 ( − 1 − β 2 β ) ] . For a = b = p i a=b=p_i a = b = p i κ i = 1 − m c 2 / ( u K ⋅ p i ) 2 \kappa_i=\sqrt{1-m_c^2/(u_K\cdot p_i)^2} κ i = 1 − m c 2 / ( u K ⋅ p i ) 2
I p i p i ( − 1 ) = 0 , I p i p i ( 0 ) = 2 , I p i p i ( 1 ) = − 2 κ i ln 1 − κ i 1 + κ i . (162) I_{p_ip_i}^{(-1)}=0,\qquad I_{p_ip_i}^{(0)}=2,\qquad I_{p_ip_i}^{(1)}=-\frac{2}{\kappa_i}\ln\frac{1-\kappa_i}{1+\kappa_i}. \tag*{(162)} I p i p i ( − 1 ) = 0 , I p i p i ( 0 ) = 2 , I p i p i ( 1 ) = − κ i 2 ln 1 + κ i 1 − κ i . (162) For a massive–massless pair ( p i , k 3 ) (p_i,k_3) ( p i , k 3 ) ξ i = ( p i ⋅ k 3 ) / [ ( u K ⋅ p i ) ( u K ⋅ k 3 ) ] \xi_i=(p_i\cdot k_3)/[(u_K\cdot p_i)(u_K\cdot k_3)] ξ i = ( p i ⋅ k 3 ) / [( u K ⋅ p i ) ( u K ⋅ k 3 )]
I p i k 3 ( − 1 ) = − 1 , I p i k 3 ( 0 ) = ln ξ i 2 1 − κ i 2 , I_{p_i k_3}^{(-1)}=-1,\qquad I_{p_i k_3}^{(0)}=\ln\frac{\xi_i^2}{1-\kappa_i^2}, I p i k 3 ( − 1 ) = − 1 , I p i k 3 ( 0 ) = ln 1 − κ i 2 ξ i 2 , I p i k 3 ( 1 ) = − 2 [ 1 4 ln 2 1 − κ i 1 + κ i + ln ξ i 1 + κ i ln ξ i 1 − κ i + Li 2 ( 1 − ξ i 1 + κ i ) + Li 2 ( 1 − ξ i 1 − κ i ) ] . (163) I_{p_i k_3}^{(1)}=-2\left[\frac{1}{4}\ln^2\frac{1-\kappa_i}{1+\kappa_i}+\ln\frac{\xi_i}{1+\kappa_i}\ln\frac{\xi_i}{1-\kappa_i}+\operatorname{Li}_2\left(1-\frac{\xi_i}{1+\kappa_i}\right)+\operatorname{Li}_2\left(1-\frac{\xi_i}{1-\kappa_i}\right)\right]. \tag*{(163)} I p i k 3 ( 1 ) = − 2 [ 4 1 ln 2 1 + κ i 1 − κ i + ln 1 + κ i ξ i ln 1 − κ i ξ i + Li 2 ( 1 − 1 + κ i ξ i ) + Li 2 ( 1 − 1 − κ i ξ i ) ] . (163) Local one-loop soft-current coefficients
For two outgoing massive emitters, we define
α 1 = m 2 2 p 2 ⋅ k 2 ( p 1 ⋅ p 2 ) ( p 1 ⋅ k ) , α 2 = m 1 2 p 1 ⋅ k 2 ( p 1 ⋅ p 2 ) ( p 2 ⋅ k ) , \alpha_1=\frac{m_2^2p_2\cdot k}{2(p_1\cdot p_2)(p_1\cdot k)},\qquad\alpha_2=\frac{m_1^2p_1\cdot k}{2(p_1\cdot p_2)(p_2\cdot k)}, α 1 = 2 ( p 1 ⋅ p 2 ) ( p 1 ⋅ k ) m 2 2 p 2 ⋅ k , α 2 = 2 ( p 1 ⋅ p 2 ) ( p 2 ⋅ k ) m 1 2 p 1 ⋅ k , d i = 1 − 2 α i , v = 1 − 4 α 1 α 2 = v 12 , v ± = 1 ± v 2 , x = v − v + . (164) d_i=1-2\alpha_i,\qquad v=\sqrt{1-4\alpha_1\alpha_2}=v_{12},\qquad v_{\pm}=\frac{1\pm v}{2},\qquad x=\frac{v_-}{v_+}. \tag*{(164)} d i = 1 − 2 α i , v = 1 − 4 α 1 α 2 = v 12 , v ± = 2 1 ± v , x = v + v − . (164) At Born kinematics, v = 2 β / ( 1 + β 2 ) v=2\beta/(1+\beta^2) v = 2 β / ( 1 + β 2 )
L α i = ln α i v + , L ˉ α i = ln ( 1 − α i v + ) , L + = ln v + , L v = ln v . (165) L_{\alpha_i}=\ln\frac{\alpha_i}{v_+},\qquad\bar{L}_{\alpha_i}=\ln\left(1-\frac{\alpha_i}{v_+}\right),\qquad L_+=\ln v_+,\qquad L_v=\ln v. \tag*{(165)} L α i = ln v + α i , L ˉ α i = ln ( 1 − v + α i ) , L + = ln v + , L v = ln v . (165) The coefficients of the poles in (C.41) are [66 ]
R − 2 = − 1 2 , R − 1 = − 1 2 [ L + + v − v ( L α 1 + L α 2 ) ] . (166) R_{-2}=-\frac{1}{2},\qquad R_{-1}=-\frac{1}{2}\left[L_++\frac{v_-}{v}(L_{\alpha_1}+L_{\alpha_2})\right]. \tag*{(166)} R − 2 = − 2 1 , R − 1 = − 2 1 [ L + + v v − ( L α 1 + L α 2 ) ] . (166) The finite and O ( ϵ ) O(\epsilon) O ( ϵ )
R 0 = − 1 2 { 1 v [ ( α 1 v + − α 2 v − ) L α 1 2 + ( α 2 v + − α 1 v − ) L α 2 2 d 1 + d 2 + ( L α 1 + L α 2 ) ( v + L + − L v ) − Li 2 ( x ) ] + 1 2 L + 2 + ζ 2 ( 7 v − 19 2 ) } . (167) R_0=-\frac{1}{2}\left\{\frac{1}{v}\left[\frac{(\alpha_1v_+-\alpha_2v_-)L_{\alpha_1}^2+(\alpha_2v_+-\alpha_1v_-)L_{\alpha_2}^2}{d_1+d_2}+(L_{\alpha_1}+L_{\alpha_2})(v_+L_+-L_v)-\operatorname{Li}_2(x)\right]+\frac{1}{2}L_+^2+\zeta_2\left(\frac{7}{v}-\frac{19}{2}\right)\right\}. \tag*{(167)} R 0 = − 2 1 { v 1 [ d 1 + d 2 ( α 1 v + − α 2 v − ) L α 1 2 + ( α 2 v + − α 1 v − ) L α 2 2 + ( L α 1 + L α 2 ) ( v + L + − L v ) − Li 2 ( x ) ] + 2 1 L + 2 + ζ 2 ( v 7 − 2 19 ) } . (167) R 1 = − 1 2 { 1 d 1 + d 2 [ ( 1 − d 1 − d 2 ) ( L ˉ α 1 L α 1 2 + L ˉ α 2 L α 2 2 ) + 2 L α 1 Li 2 ( α 1 v + ) + 2 L α 2 Li 2 ( α 2 v + ) − Li 2 ( x ) ( L α 1 + L α 2 ) R_1=-\frac{1}{2}\left\{\frac{1}{d_1+d_2}\left[(1-d_1-d_2)(\bar{L}_{\alpha_1}L_{\alpha_1}^2+\bar{L}_{\alpha_2}L_{\alpha_2}^2)+2L_{\alpha_1}\operatorname{Li}_2\left(\frac{\alpha_1}{v_+}\right)+2L_{\alpha_2}\operatorname{Li}_2\left(\frac{\alpha_2}{v_+}\right)-\operatorname{Li}_2(x)(L_{\alpha_1}+L_{\alpha_2})\right.\right. R 1 = − 2 1 { d 1 + d 2 1 [ ( 1 − d 1 − d 2 ) ( L ˉ α 1 L α 1 2 + L ˉ α 2 L α 2 2 ) + 2 L α 1 Li 2 ( v + α 1 ) + 2 L α 2 Li 2 ( v + α 2 ) − Li 2 ( x ) ( L α 1 + L α 2 ) + 2 Li 3 ( x ) − 2 Li 3 ( α 1 v + ) − 2 Li 3 ( α 2 v + ) + 2 ζ 3 − 7 ζ 2 ( L α 1 + L α 2 ) \left.\left.+2\operatorname{Li}_3(x)-2\operatorname{Li}_3\left(\frac{\alpha_1}{v_+}\right)-2\operatorname{Li}_3\left(\frac{\alpha_2}{v_+}\right)+2\zeta_3-7\zeta_2(L_{\alpha_1}+L_{\alpha_2})\right.\right. + 2 Li 3 ( x ) − 2 Li 3 ( v + α 1 ) − 2 Li 3 ( v + α 2 ) + 2 ζ 3 − 7 ζ 2 ( L α 1 + L α 2 ) + 1 v [ ( ( α 2 v + − α 1 v − ) L α 1 2 + ( α 1 v + − α 2 v − ) L α 2 2 ) L + + ( α 1 − α 2 ) ( L α 1 2 − L α 2 2 ) L v − ( d 1 L α 1 + d 2 L α 2 ) ( Li 2 ( x ) − 7 ζ 2 ) ] ] } . (168) \left.\left.+\frac{1}{v}\left[((\alpha_2v_+-\alpha_1v_-)L_{\alpha_1}^2+(\alpha_1v_+-\alpha_2v_-)L_{\alpha_2}^2)L_++(\alpha_1-\alpha_2)(L_{\alpha_1}^2-L_{\alpha_2}^2)L_v-(d_1L_{\alpha_1}+d_2L_{\alpha_2})(\operatorname{Li}_2(x)-7\zeta_2)\right]\right]\right\}. \tag*{(168)} + v 1 [ (( α 2 v + − α 1 v − ) L α 1 2 + ( α 1 v + − α 2 v − ) L α 2 2 ) L + + ( α 1 − α 2 ) ( L α 1 2 − L α 2 2 ) L v − ( d 1 L α 1 + d 2 L α 2 ) ( Li 2 ( x ) − 7 ζ 2 ) ] ] } . (168) + 1 v [ ( L + ( 3 + v 4 L + − L v ) − 9 v 2 ζ 2 ) ( L α 1 + L α 2 ) − v 6 ( L α 1 3 + L α 2 3 ) + 2 Li 3 ( 1 − x ) + Li 3 ( x ) − ( Li 2 ( x ) + ζ 2 ( 5 + 19 2 v ) ) L + + 12 ζ 2 L v ] + 1 6 L + 3 − ( 7 3 + 1 v ) ζ 3 } . (169) +\frac{1}{v}[(L_+(\frac{3+v}{4}L_+-L_v)-\frac{9v}{2}\zeta_2)(L_{\alpha_1}+L_{\alpha_2})-\frac{v}{6}(L_{\alpha_1}^3+L_{\alpha_2}^3)
\\
+2\operatorname{Li}_3(1-x)+\operatorname{Li}_3(x)-(\operatorname{Li}_2(x)+\zeta_2(5+\frac{19}{2}v))L_++12\zeta_2L_v
]
\\
+\frac{1}{6}L_+^3-(\frac{7}{3}+\frac{1}{v})\zeta_3
\}. \tag*{(169)} + v 1 [( L + ( 4 3 + v L + − L v ) − 2 9 v ζ 2 ) ( L α 1 + L α 2 ) − 6 v ( L α 1 3 + L α 2 3 ) + 2 Li 3 ( 1 − x ) + Li 3 ( x ) − ( Li 2 ( x ) + ζ 2 ( 5 + 2 19 v )) L + + 12 ζ 2 L v ] + 6 1 L + 3 − ( 3 7 + v 1 ) ζ 3 } . (169) Integrated one-loop soft-current coefficients
The integrated coefficients R n \mathcal{R}_n R n β = 1 − 4 m c 2 / K 2 \beta=\sqrt{1-4m_c^2/K^2} β = 1 − 4 m c 2 / K 2 H a 1 , … , a w = H a 1 , … , a w ( β ) H_{a_1,\ldots,a_w}=H_{a_1,\ldots,a_w}(\beta) H a 1 , … , a w = H a 1 , … , a w ( β )
H ∅ ( t ) = 1 , H 0 , … , 0 ⏟ w ( t ) = ln w t w ! , H_{\emptyset}(t)=1,\qquad H_{\underbrace{0,\ldots,0}_{w}}(t)=\frac{\ln^w t}{w!}, H ∅ ( t ) = 1 , H w 0 , … , 0 ( t ) = w ! ln w t , H a , b ⃗ ( t ) = ∫ 0 t d u f a ( u ) H b ⃗ ( u ) , ( a , b ⃗ ) ≠ ( 0 , … , 0 ) , (170) H_{a,\vec{b}}(t)=\int_0^t du\,f_a(u)H_{\vec{b}}(u),\qquad(a,\vec{b})\ne(0,\ldots,0), \tag*{(170)} H a , b ( t ) = ∫ 0 t d u f a ( u ) H b ( u ) , ( a , b ) = ( 0 , … , 0 ) , (170) f − 1 ( u ) = 1 1 + u , f 0 ( u ) = 1 u , f 1 ( u ) = 1 1 − u . f_{-1}(u)=\frac{1}{1+u},\qquad f_0(u)=\frac{1}{u},\qquad f_1(u)=\frac{1}{1-u}. f − 1 ( u ) = 1 + u 1 , f 0 ( u ) = u 1 , f 1 ( u ) = 1 − u 1 . For a j ∈ { 0 , ± 1 } a_j\in\{0,\pm1\} a j ∈ { 0 , ± 1 } H a 1 , … , a w ( β ) = ( − 1 ) N + G ( a 1 , … , a w ; β ) H_{a_1,\ldots,a_w}(\beta)=(-1)^{N_+}G(a_1,\ldots,a_w;\beta) H a 1 , … , a w ( β ) = ( − 1 ) N + G ( a 1 , … , a w ; β ) N + N_+ N + + 1 +1 + 1
The coefficients are [42 ]
R − 3 ( β ) = − 1 + 1 + β 2 2 β ( H − 1 + H 1 ) , (171) \mathcal{R}_{-3}(\beta)=-1+\frac{1+\beta^2}{2\beta}(H_{-1}+H_1), \tag*{(171)} R − 3 ( β ) = − 1 + 2 β 1 + β 2 ( H − 1 + H 1 ) , (171) R − 2 ( β ) = 1 + ( 1 + β 2 ) 2 4 β 2 ( H − 1 + H 1 ) 2 − 2 β ( H − 1 + H 1 ) + 1 + β 2 β ( H 0 , − 1 + H 0 , 1 ) , (172) \mathcal{R}_{-2}(\beta)=1+\frac{(1+\beta^2)^2}{4\beta^2}(H_{-1}+H_1)^2-\frac{2}{\beta}(H_{-1}+H_1)
+\frac{1+\beta^2}{\beta}(H_{0,-1}+H_{0,1}), \tag*{(172)} R − 2 ( β ) = 1 + 4 β 2 ( 1 + β 2 ) 2 ( H − 1 + H 1 ) 2 − β 2 ( H − 1 + H 1 ) + β 1 + β 2 ( H 0 , − 1 + H 0 , 1 ) , (172) R − 1 ( β ) = ( 11 − 3 ( 1 + β 2 ) β ) ζ 2 − ( 1 + β 2 ) ( 3 β 2 + 2 β + 3 ) 12 β 2 H − 1 3 + ( 1 + β 2 ) ( 3 β 2 − 2 β + 3 ) 12 β 2 ( H 1 3 + 3 H − 1 H 1 2 ) − 2 + β + β 3 2 β 2 ( H 1 2 + 2 H − 1 H 1 ) + H − 1 2 [ − 2 + β + β 3 2 β 2 + ( β − 1 ) 2 ( 1 + β 2 ) 4 β 2 H 1 ] − 4 β ( H 0 , − 1 + H 0 , 1 ) + ( H − 1 + H 1 ) [ 2 β + ( 1 + β 2 ) ( 3 β 2 − 11 β + 3 ) 2 β 2 ζ 2 ] + ( 1 + β 2 ) 2 β 2 ( H 0 , − 1 + H 0 , 1 − H − 1 , 1 ) + 2 ( 1 + β 2 ) β ( H − 1 , 1 + H 0 , 0 , − 1 + H 0 , 0 , 1 ) . (173) \begin{aligned}
\mathcal{R}_{-1}(\beta)={}&\left(11-\frac{3(1+\beta^2)}{\beta}\right)\zeta_2-\frac{(1+\beta^2)(3\beta^2+2\beta+3)}{12\beta^2}H_{-1}^3\\
&+\frac{(1+\beta^2)(3\beta^2-2\beta+3)}{12\beta^2}\left(H_1^3+3H_{-1}H_1^2\right)-\frac{2+\beta+\beta^3}{2\beta^2}\left(H_1^2+2H_{-1}H_1\right)\\
&+H_{-1}^2\left[\frac{-2+\beta+\beta^3}{2\beta^2}+\frac{(\beta-1)^2(1+\beta^2)}{4\beta^2}H_1\right]-\frac{4}{\beta}(H_{0,-1}+H_{0,1})\\
&+(H_{-1}+H_1)\left[\frac{2}{\beta}+\frac{(1+\beta^2)(3\beta^2-11\beta+3)}{2\beta^2}\zeta_2\right]\\
&+\frac{(1+\beta^2)^2}{\beta^2}(H_{0,-1}+H_{0,1}-H_{-1,1})\\
&+\frac{2(1+\beta^2)}{\beta}(H_{-1,1}+H_{0,0,-1}+H_{0,0,1}). \tag*{(173)}
\end{aligned} R − 1 ( β ) = ( 11 − β 3 ( 1 + β 2 ) ) ζ 2 − 12 β 2 ( 1 + β 2 ) ( 3 β 2 + 2 β + 3 ) H − 1 3 + 12 β 2 ( 1 + β 2 ) ( 3 β 2 − 2 β + 3 ) ( H 1 3 + 3 H − 1 H 1 2 ) − 2 β 2 2 + β + β 3 ( H 1 2 + 2 H − 1 H 1 ) + H − 1 2 [ 2 β 2 − 2 + β + β 3 + 4 β 2 ( β − 1 ) 2 ( 1 + β 2 ) H 1 ] − β 4 ( H 0 , − 1 + H 0 , 1 ) + ( H − 1 + H 1 ) [ β 2 + 2 β 2 ( 1 + β 2 ) ( 3 β 2 − 11 β + 3 ) ζ 2 ] + β 2 ( 1 + β 2 ) 2 ( H 0 , − 1 + H 0 , 1 − H − 1 , 1 ) + β 2 ( 1 + β 2 ) ( H − 1 , 1 + H 0 , 0 , − 1 + H 0 , 0 , 1 ) . (173) R 0 ( β ) = − 4 + [ 6 ( 1 + β 2 ) β ( 1 − 2 ln 2 ) − 11 ] ζ 2 + 14 3 ζ 3 + 1 + β 2 12 β 2 [ ( 3 β 2 + 4 β + 3 ) H − 1 4 + ( 3 β 2 − 4 β + 3 ) H 1 4 ] + ( 13 6 − 3 2 β 2 − 2 β ) H 1 3 + H − 1 3 [ − 13 6 + 3 2 β 2 − 2 β + 4 ( 1 + β 2 ) 3 β H 1 ] + 3 ( 1 + β 2 ) ζ 2 β 2 H 0 [ ( 1 + β 2 ) ( H − 1 + H 1 ) − 2 β ] + ( 1 + β 2 ) 2 β 2 ( H 0 , − 1 + H 0 , 1 ) 2 + [ 4 β + ( 1 + β 2 ) ( 3 β 2 − 11 β + 3 ) β 2 ζ 2 − 2 ( 1 + β 2 ) 2 β 2 H − 1 , 1 ] ( H 0 , − 1 + H 0 , 1 ) + H 1 2 [ − 1 + 2 β 2 + β + ( 1 + β 2 ) ( β 2 − 12 β + 1 ) 4 β 2 ζ 2 − 2 ( 1 + β 2 ) ( β 2 − β + 1 ) β 2 H − 1 , 1 + ( 1 + β 2 ) ( 3 β 2 − 2 β + 3 ) 2 β 2 ( H 0 , − 1 + H 0 , 1 ) ] + H − 1 2 [ − 1 + 2 β 2 − β − ( 1 + β 2 ) ( 23 β 2 + 12 β + 23 ) 4 β 2 ζ 2 ] + ( 7 2 − 3 2 β 2 + 2 β − 4 β ) H 1 + ( β − 1 ) 2 ( 1 + β 2 ) 2 β 2 H 1 2 − 6 ( 1 + β 2 ) β H − 1 , 1 − ( 1 + β 2 ) ( 3 β 2 + 2 β + 3 ) 2 β 2 ( H 0 , − 1 + H 0 , 1 ) − 4 β H − 1 , 1 − 4 β [ ( β − 1 ) 2 H − 1 , − 1 , 1 + ( β + 1 ) 2 H − 1 , 1 , 1 ] + ( 8 − 2 ( 1 + β 2 ) β ) ( H 0 , − 1 , − 1 + H 0 , 1 , − 1 ) − 8 β ( H 0 , 0 , − 1 + H 0 , 0 , 1 ) + 2 ( 4 + 1 + β 2 β ) ( H 0 , − 1 , 1 + H 0 , 1 , 1 ) + ( H − 1 + H 1 ) [ ( 6 − 6 β 2 + 6 ( 1 + β 2 ) 2 ln 2 β 2 ) ζ 2 − 7 ( 1 + β 2 ) 3 β ζ 3 + ( 1 + β 2 ) ( β 2 − 4 β + 1 ) β 2 ( H 0 , − 1 , − 1 + H 0 , 1 , − 1 ) + 2 ( 1 + β 2 ) 2 β 2 ( H 0 , 0 , − 1 + H 0 , 0 , 1 ) − ( 1 + β 2 ) ( β 2 + 4 β + 1 ) β 2 ( H 0 , − 1 , 1 + H 0 , 1 , 1 ) ] + 2 H 1 β 2 [ β ( 8 − 3 β 2 ) ζ 2 + ( 2 β 3 − 3 β 2 + 2 β + 3 ) H − 1 , 1 − ( 2 + β + β 3 ) ( H 0 , − 1 + H 0 , 1 ) + ( 1 + β 2 ) ( ( β − 1 ) 2 H − 1 , − 1 , 1 + ( β + 1 ) 2 H − 1 , 1 , 1 ) ] \begin{aligned}
\mathcal{R}_0(\beta)={}&-4+\left[\frac{6(1+\beta^2)}{\beta}(1-2\ln2)-11\right]\zeta_2+\frac{14}{3}\zeta_3\\
&+\frac{1+\beta^2}{12\beta^2}\left[(3\beta^2+4\beta+3)H_{-1}^{4}+(3\beta^2-4\beta+3)H_{1}^{4}\right]\\
&+\left(\frac{13}{6}-\frac{3}{2\beta^2}-2\beta\right)H_1^3+H_{-1}^3\left[-\frac{13}{6}+\frac{3}{2\beta^2}-2\beta+\frac{4(1+\beta^2)}{3\beta}H_1\right]\\
&+\frac{3(1+\beta^2)\zeta_2}{\beta^2}H_0\left[(1+\beta^2)(H_{-1}+H_1)-2\beta\right]+\frac{(1+\beta^2)^2}{\beta^2}(H_{0,-1}+H_{0,1})^2\\
&+\left[\frac{4}{\beta}+\frac{(1+\beta^2)(3\beta^2-11\beta+3)}{\beta^2}\zeta_2-\frac{2(1+\beta^2)^2}{\beta^2}H_{-1,1}\right](H_{0,-1}+H_{0,1})\\
&+H_1^2\left[-1+\frac{2}{\beta^2}+\beta+\frac{(1+\beta^2)(\beta^2-12\beta+1)}{4\beta^2}\zeta_2-\frac{2(1+\beta^2)(\beta^2-\beta+1)}{\beta^2}H_{-1,1}\right.\\
&\qquad\left.+\frac{(1+\beta^2)(3\beta^2-2\beta+3)}{2\beta^2}(H_{0,-1}+H_{0,1})\right]\\
&+H_{-1}^2\left[-1+\frac{2}{\beta^2}-\beta-\frac{(1+\beta^2)(23\beta^2+12\beta+23)}{4\beta^2}\zeta_2\right]\\
&+\left(\frac{7}{2}-\frac{3}{2\beta^2}+\frac{2}{\beta}-4\beta\right)H_1+\frac{(\beta-1)^2(1+\beta^2)}{2\beta^2}H_1^2-\frac{6(1+\beta^2)}{\beta}H_{-1,1}\\
&-\frac{(1+\beta^2)(3\beta^2+2\beta+3)}{2\beta^2}(H_{0,-1}+H_{0,1})\\
&-4\beta H_{-1,1}-\frac{4}{\beta}\left[(\beta-1)^2H_{-1,-1,1}+(\beta+1)^2H_{-1,1,1}\right]\\
&+\left(8-\frac{2(1+\beta^2)}{\beta}\right)(H_{0,-1,-1}+H_{0,1,-1})-\frac{8}{\beta}(H_{0,0,-1}+H_{0,0,1})\\
&+2\left(4+\frac{1+\beta^2}{\beta}\right)(H_{0,-1,1}+H_{0,1,1})\\
&+(H_{-1}+H_1)\left[\left(6-\frac{6}{\beta^2}+\frac{6(1+\beta^2)^2\ln2}{\beta^2}\right)\zeta_2-\frac{7(1+\beta^2)}{3\beta}\zeta_3\right.\\
&\qquad\left.+\frac{(1+\beta^2)(\beta^2-4\beta+1)}{\beta^2}(H_{0,-1,-1}+H_{0,1,-1})+\frac{2(1+\beta^2)^2}{\beta^2}(H_{0,0,-1}+H_{0,0,1})\right.\\
&\qquad\left.-\frac{(1+\beta^2)(\beta^2+4\beta+1)}{\beta^2}(H_{0,-1,1}+H_{0,1,1})\right]\\
&+\frac{2H_1}{\beta^2}\left[\beta(8-3\beta^2)\zeta_2+(2\beta^3-3\beta^2+2\beta+3)H_{-1,1}-(2+\beta+\beta^3)(H_{0,-1}+H_{0,1})\right.\\
&\qquad\left.+(1+\beta^2)\left((\beta-1)^2H_{-1,-1,1}+(\beta+1)^2H_{-1,1,1}\right)\right]
\end{aligned} R 0 ( β ) = − 4 + [ β 6 ( 1 + β 2 ) ( 1 − 2 ln 2 ) − 11 ] ζ 2 + 3 14 ζ 3 + 12 β 2 1 + β 2 [ ( 3 β 2 + 4 β + 3 ) H − 1 4 + ( 3 β 2 − 4 β + 3 ) H 1 4 ] + ( 6 13 − 2 β 2 3 − 2 β ) H 1 3 + H − 1 3 [ − 6 13 + 2 β 2 3 − 2 β + 3 β 4 ( 1 + β 2 ) H 1 ] + β 2 3 ( 1 + β 2 ) ζ 2 H 0 [ ( 1 + β 2 ) ( H − 1 + H 1 ) − 2 β ] + β 2 ( 1 + β 2 ) 2 ( H 0 , − 1 + H 0 , 1 ) 2 + [ β 4 + β 2 ( 1 + β 2 ) ( 3 β 2 − 11 β + 3 ) ζ 2 − β 2 2 ( 1 + β 2 ) 2 H − 1 , 1 ] ( H 0 , − 1 + H 0 , 1 ) + H 1 2 [ − 1 + β 2 2 + β + 4 β 2 ( 1 + β 2 ) ( β 2 − 12 β + 1 ) ζ 2 − β 2 2 ( 1 + β 2 ) ( β 2 − β + 1 ) H − 1 , 1 + 2 β 2 ( 1 + β 2 ) ( 3 β 2 − 2 β + 3 ) ( H 0 , − 1 + H 0 , 1 ) ] + H − 1 2 [ − 1 + β 2 2 − β − 4 β 2 ( 1 + β 2 ) ( 23 β 2 + 12 β + 23 ) ζ 2 ] + ( 2 7 − 2 β 2 3 + β 2 − 4 β ) H 1 + 2 β 2 ( β − 1 ) 2 ( 1 + β 2 ) H 1 2 − β 6 ( 1 + β 2 ) H − 1 , 1 − 2 β 2 ( 1 + β 2 ) ( 3 β 2 + 2 β + 3 ) ( H 0 , − 1 + H 0 , 1 ) − 4 β H − 1 , 1 − β 4 [ ( β − 1 ) 2 H − 1 , − 1 , 1 + ( β + 1 ) 2 H − 1 , 1 , 1 ] + ( 8 − β 2 ( 1 + β 2 ) ) ( H 0 , − 1 , − 1 + H 0 , 1 , − 1 ) − β 8 ( H 0 , 0 , − 1 + H 0 , 0 , 1 ) + 2 ( 4 + β 1 + β 2 ) ( H 0 , − 1 , 1 + H 0 , 1 , 1 ) + ( H − 1 + H 1 ) [ ( 6 − β 2 6 + β 2 6 ( 1 + β 2 ) 2 ln 2 ) ζ 2 − 3 β 7 ( 1 + β 2 ) ζ 3 + β 2 ( 1 + β 2 ) ( β 2 − 4 β + 1 ) ( H 0 , − 1 , − 1 + H 0 , 1 , − 1 ) + β 2 2 ( 1 + β 2 ) 2 ( H 0 , 0 , − 1 + H 0 , 0 , 1 ) − β 2 ( 1 + β 2 ) ( β 2 + 4 β + 1 ) ( H 0 , − 1 , 1 + H 0 , 1 , 1 ) ] + β 2 2 H 1 [ β ( 8 − 3 β 2 ) ζ 2 + ( 2 β 3 − 3 β 2 + 2 β + 3 ) H − 1 , 1 − ( 2 + β + β 3 ) ( H 0 , − 1 + H 0 , 1 ) + ( 1 + β 2 ) ( ( β − 1 ) 2 H − 1 , − 1 , 1 + ( β + 1 ) 2 H − 1 , 1 , 1 ) ] + H − 1 [ ( 28 β + 6 β ) ζ 2 + ( 13 2 − 9 2 β 2 − 6 β ) H 1 2 + ( 1 + β 2 ) ( 3 β 2 − 4 β + 3 ) 3 β 2 H 1 3 + ( − 14 + 6 β 2 ) H − 1 , 1 + 2 ( − 2 + β + β 3 ) β 2 ( H 0 , − 1 + H 0 , 1 ) + 2 ( 1 + β 2 ) β 2 [ ( β 2 + 10 β + 1 ) H − 1 , − 1 , 1 + ( β 2 − 10 β + 1 ) H − 1 , 1 , 1 ] ] + H − 1 H 1 [ − 2 + 4 β 2 + 2 β − ( 1 + β 2 ) ( 11 β 2 + 12 β + 11 ) 2 β 2 ζ 2 − 2 ( 1 + β 2 ) ( β 2 − 4 β + 1 ) β 2 H − 1 , 1 + ( β − 1 ) 2 ( 1 + β 2 ) β 2 ( H 0 , − 1 + H 0 , 1 ) ] + 24 ( 1 + β 2 ) β ( − H − 1 , − 1 , − 1 , 1 + H − 1 , − 1 , 1 , 1 − H − 1 , 1 , 1 , 1 ) + 16 ( 1 + β 2 ) β ( H 0 , − 1 , − 1 , − 1 + H 0 , − 1 , − 1 , 1 + H 0 , − 1 , 1 , − 1 + H 0 , − 1 , 1 , 1 + H 0 , 1 , − 1 , − 1 + H 0 , 1 , − 1 , 1 + H 0 , 1 , 1 , − 1 + H 0 , 1 , 1 , 1 ) + 4 ( 1 + β 2 ) β ( H 0 , 0 , 0 , − 1 + H 0 , 0 , 0 , 1 ) . (174) \begin{aligned}
{}+H_{-1}\Biggl[&\left(\frac{28}{\beta}+6\beta\right)\zeta_2+\left(\frac{13}{2}-\frac{9}{2\beta^2}-6\beta\right)H_1^2+\frac{(1+\beta^2)(3\beta^2-4\beta+3)}{3\beta^2}H_1^3\\
&+\left(-14+\frac{6}{\beta^2}\right)H_{-1,1}+\frac{2(-2+\beta+\beta^3)}{\beta^2}(H_{0,-1}+H_{0,1})\\
&+\frac{2(1+\beta^2)}{\beta^2}\left[(\beta^2+10\beta+1)H_{-1,-1,1}+(\beta^2-10\beta+1)H_{-1,1,1}\right]\Biggr]\\
{}+H_{-1}H_1\Biggl[&-2+\frac{4}{\beta^2}+2\beta-\frac{(1+\beta^2)(11\beta^2+12\beta+11)}{2\beta^2}\zeta_2\\
&-\frac{2(1+\beta^2)(\beta^2-4\beta+1)}{\beta^2}H_{-1,1}+\frac{(\beta-1)^2(1+\beta^2)}{\beta^2}(H_{0,-1}+H_{0,1})\Biggr]\\
{}+{}&\frac{24(1+\beta^2)}{\beta}(-H_{-1,-1,-1,1}+H_{-1,-1,1,1}-H_{-1,1,1,1})\\
{}+{}&\frac{16(1+\beta^2)}{\beta}(H_{0,-1,-1,-1}+H_{0,-1,-1,1}+H_{0,-1,1,-1}+H_{0,-1,1,1}\\
&\qquad+H_{0,1,-1,-1}+H_{0,1,-1,1}+H_{0,1,1,-1}+H_{0,1,1,1})\\
{}+{}&\frac{4(1+\beta^2)}{\beta}(H_{0,0,0,-1}+H_{0,0,0,1}). \tag*{(174)}
\end{aligned} + H − 1 [ + H − 1 H 1 [ + + + ( β 28 + 6 β ) ζ 2 + ( 2 13 − 2 β 2 9 − 6 β ) H 1 2 + 3 β 2 ( 1 + β 2 ) ( 3 β 2 − 4 β + 3 ) H 1 3 + ( − 14 + β 2 6 ) H − 1 , 1 + β 2 2 ( − 2 + β + β 3 ) ( H 0 , − 1 + H 0 , 1 ) + β 2 2 ( 1 + β 2 ) [ ( β 2 + 10 β + 1 ) H − 1 , − 1 , 1 + ( β 2 − 10 β + 1 ) H − 1 , 1 , 1 ] ] − 2 + β 2 4 + 2 β − 2 β 2 ( 1 + β 2 ) ( 11 β 2 + 12 β + 11 ) ζ 2 − β 2 2 ( 1 + β 2 ) ( β 2 − 4 β + 1 ) H − 1 , 1 + β 2 ( β − 1 ) 2 ( 1 + β 2 ) ( H 0 , − 1 + H 0 , 1 ) ] β 24 ( 1 + β 2 ) ( − H − 1 , − 1 , − 1 , 1 + H − 1 , − 1 , 1 , 1 − H − 1 , 1 , 1 , 1 ) β 16 ( 1 + β 2 ) ( H 0 , − 1 , − 1 , − 1 + H 0 , − 1 , − 1 , 1 + H 0 , − 1 , 1 , − 1 + H 0 , − 1 , 1 , 1 + H 0 , 1 , − 1 , − 1 + H 0 , 1 , − 1 , 1 + H 0 , 1 , 1 , − 1 + H 0 , 1 , 1 , 1 ) β 4 ( 1 + β 2 ) ( H 0 , 0 , 0 , − 1 + H 0 , 0 , 0 , 1 ) . (174) Two-loop massive cusp anomalous dimension
For the outgoing heavy-quark pair with relative velocity v = v 12 v=v_{12} v = v 12 χ = 1 2 ln [ ( 1 + v ) / ( 1 − v ) ] \chi=\frac{1}{2}\ln[(1+v)/(1-v)] χ = 2 1 ln [( 1 + v ) / ( 1 − v )] Z 2 Z_2 Z 2 [42 , 62 , 63 ]
γ c u s p , Q ( 1 ) ( v ) = γ c u s p ( 1 ) v ( χ − i π ) + 8 C A { ζ 3 − 5 π 2 6 + χ 2 + 1 v 2 [ χ 3 3 + χ Li 2 ( e − 2 χ ) − 5 π 2 χ 6 + Li 3 ( e − 2 χ ) − ζ 3 ] + 1 v [ 5 π 2 6 ( 1 + χ ) − χ 2 − 2 χ ln ( 1 − e − 2 χ ) − χ 3 3 + Li 2 ( e − 2 χ ) ] + i π [ 1 v 2 ( π 2 6 − χ 2 − Li 2 ( e − 2 χ ) ) − 2 χ + 1 v ( − π 2 6 + 2 ln ( 1 − e − 2 χ ) + 2 χ + χ 2 ) ] } . (175) \begin{aligned}
\gamma_{\mathrm{cusp},Q}^{(1)}(v)={}&\frac{\gamma_{\mathrm{cusp}}^{(1)}}{v}(\chi-i\pi)+8C_A\Biggl\{\zeta_3-\frac{5\pi^2}{6}+\chi^2\\
&+\frac{1}{v^2}\left[\frac{\chi^3}{3}+\chi\operatorname{Li}_2(e^{-2\chi})-\frac{5\pi^2\chi}{6}+\operatorname{Li}_3(e^{-2\chi})-\zeta_3\right]\\
&+\frac{1}{v}\left[\frac{5\pi^2}{6}(1+\chi)-\chi^2-2\chi\ln(1-e^{-2\chi})-\frac{\chi^3}{3}+\operatorname{Li}_2(e^{-2\chi})\right]\\
&+i\pi\left[\frac{1}{v^2}\left(\frac{\pi^2}{6}-\chi^2-\operatorname{Li}_2(e^{-2\chi})\right)-2\chi\right.\\
&\left.\qquad+\frac{1}{v}\left(-\frac{\pi^2}{6}+2\ln(1-e^{-2\chi})+2\chi+\chi^2\right)\right]\Biggr\}. \tag*{(175)}
\end{aligned} γ cusp , Q ( 1 ) ( v ) = v γ cusp ( 1 ) ( χ − iπ ) + 8 C A { ζ 3 − 6 5 π 2 + χ 2 + v 2 1 [ 3 χ 3 + χ Li 2 ( e − 2 χ ) − 6 5 π 2 χ + Li 3 ( e − 2 χ ) − ζ 3 ] + v 1 [ 6 5 π 2 ( 1 + χ ) − χ 2 − 2 χ ln ( 1 − e − 2 χ ) − 3 χ 3 + Li 2 ( e − 2 χ ) ] + iπ [ v 2 1 ( 6 π 2 − χ 2 − Li 2 ( e − 2 χ ) ) − 2 χ + v 1 ( − 6 π 2 + 2 ln ( 1 − e − 2 χ ) + 2 χ + χ 2 ) ] } . (175) where the massless cusp coefficient is
γ c u s p ( 1 ) = ( 268 9 − 4 π 2 3 ) C A − 80 9 T F n l , n l = 3. \gamma_{\mathrm{cusp}}^{(1)}=\left(\frac{268}{9}-\frac{4\pi^2}{3}\right)C_A-\frac{80}{9}T_F n_l,\qquad n_l=3. γ cusp ( 1 ) = ( 9 268 − 3 4 π 2 ) C A − 9 80 T F n l , n l = 3. 
