Introduction
The physics programs of the current and proposed lepton colliders [1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 ] rest on a theoretical control of QED radiation that is at least as good as the experimental systematics [12 , 13 ] . QED corrections to the initial and final states distort every observable that is sensitive to photon emissions, at the level of tens of percent, so that a description accurate to the per-mil level is a precondition for interpreting the data rather than a refinement of it.
The organizing principle for all-order QED is the theorem of Yennie, Frautschi and Suura (YFS) [14 ] . The infrared (IR) singular parts of the real and virtual corrections are extracted to all orders and exponentiated, leaving IR-finite remainders that are computed order by order and retain the full kinematic dependence of each emission. Two equivalent realizations of the theorem are in use. The first is the so-called exclusive exponentiation (EEX) where the factorization is performed on the squared amplitude and the remainders are squared quantities and the sum over which charged line emitted a given photon is incoherent, with coherent effects added as a corrective weight. While in the coherent exclusive exponentiation (CEEX) [15 , 16 ] the factorization is performed on the amplitude, the remainders are complex helicity amplitudes, and the sum over the assignments of photons to emitters is taken before squaring, so that the interference between initial-state radiation (ISR) and final-state radiation (FSR), the initial-final interference (IFI), is kept at the amplitude level. CEEX was derived and realized in KKMC [15 , 17 ] for e + e − → f f ˉ e^+e^- \to f\bar{f} e + e − → f f ˉ q q ˉ → f f ˉ q\bar{q} \to f\bar{f} q q ˉ → f f ˉ [18 ] . The same family of YFS Monte Carlos was instrumental throughout the LEP physics program, for the luminosity measurement [19 ] , the Z Z Z [20 ] , lepton-pair [21 ] and W W W [22 ] production. Other codes such as BABAYAGA [23 ] and McMule [24 ] reach comparable precision for specific final states, the former through a structure-function resummation matched to a QED parton shower and the latter at fixed NNLO order.
In general-purpose event generators [25 , 26 , 27 , 28 , 29 , 30 , 31 ] , multi-photon radiation beyond fixed order is supplied by a universal QED parton shower [32 , 33 , 34 , 35 , 36 ] , accurate to the leading logarithm. The fixed-order electroweak corrections themselves are now automated for generic processes [37 , 38 ] , but matching them to a shower in the POWHEG/MC@NLO sense [39 , 40 ] has not been realized for e + e − e^+e^- e + e − [41 ] , known to next-to-leading logarithmic accuracy [42 ] and recently to two loops [43 , 44 ] , with a short-distance partonic cross section. This inclusive picture does not by itself give the fully exclusive, coherent photon kinematics that CEEX and ACRAIC provide.
SHERPA [27 , 45 ] computes tree-level helicity amplitudes for essentially arbitrary Standard Model final states with COMIX [46 ] , a Berends–Giele recursion [47 ] , obtains one-loop amplitudes from OPENLOOPS [48 ] and RECOLA [49 , 50 ] through its interfaces [51 ] , and carries YFS machinery for decays [52 ] and for lepton colliders [53 , 54 ] . The latter provides a squared-level YFS exponentiation matched to the complete one-loop electroweak correction, denoted YFSNLOE W _{\mathrm{EW}} EW [54 ] . What has been missing is the amplitude-level form of the resummation at arbitrary multiplicity. The need shows at higher multiplicity, where the squared-level matching can become unstable while the amplitude-level alternative remains stable, as we find for four-lepton production in § 2.2 C.
In this paper we present ACRAIC2 emission stages , and requiring each stage to be separately gauge invariant fixes an admissible decomposition to be a partition of the charged legs into charge-neutral sets, which we derive from the radiating dipoles of the process, including resonance stages for Z / γ ∗ Z/\gamma^* Z / γ ∗ W W W N + 2 N+2 N + 2 N + 3 N+3 N + 3 e + e − → μ + μ − e^+e^- \to\mu^+\mu^- e + e − → μ + μ − O ( α ) \mathcal{O}(\alpha) O ( α ) γ Z \gamma Z γ Z E W _{\mathrm{EW}} EW Z Z Z § 2.2 C).
The paper is organized as follows. § 1.1 recalls the YFS form factor and the squared and amplitude-level soft factors. Section III describes the ACRAIC algorithm, namely the stages, partitions and propagator shifts, the remainders, the crude density and the weight, the treatment of soft photons, the virtual correction and the cost. Section IV states what the construction requires of an automated amplitude generator and how COMIX meets it. § 2.2 presents the validation, and Sec. VI concludes. The conventions for the soft factors are collected in Appendix A.
YFS exponentiation
This section fixes the notation and recalls the ingredients of YFS exponentiation that are used below. Standard results are quoted without derivation and for explicit details we refer to [14 , 16 , 53 ] . We follow the notation of [54 ] and every expression is written for an arbitrary number of external legs.
We consider a process with two incoming momenta p a , p b p_a,p_b p a , p b N N N N + 2 N+2 N + 2 { p i } \{p_i\} { p i } Q i Q_i Q i
θ i = { − 1 incoming , + 1 outgoing . (1) \theta_i =
\begin{cases}
-1 & \text{incoming},\\
+1 & \text{outgoing}.
\end{cases} \tag*{(1)} θ i = { − 1 + 1 incoming , outgoing . (1) Charge conservation is the single statement
∑ i Q i θ i = 0. (2) \sum_i Q_i\theta_i = 0. \tag*{(2)} i ∑ Q i θ i = 0. (2) The photons are k 1 , … , k n γ k_1,\ldots,k_{n_\gamma} k 1 , … , k n γ σ j = ± 1 \sigma_j=\pm1 σ j = ± 1 Ω \Omega Ω [54 ] they combine into the YFS form factor
Y ( Ω ) = ∑ i < j [ R e B i j + B ~ i j ( Ω ) ] , (3) Y(\Omega)=\sum_{i<j}\left[\mathop{\mathrm{Re}} B_{ij}+\widetilde{B}_{ij}(\Omega)\right], \tag*{(3)} Y ( Ω ) = i < j ∑ [ Re B ij + B ij ( Ω ) ] , (3) in which the virtual function B i j B_{ij} B ij B ~ i j ( Ω ) \widetilde{B}_{ij}(\Omega) B ij ( Ω ) 2 α 2\alpha 2 α i j ij ij (3) from its initial-initial, initial-final and final-final dipoles [52 , 53 ] , so the form factor is available at arbitrary multiplicity. The squared-level soft factor of a real photon, the eikonal factor, is written as a single current with its normalization explicit,
S ~ ( k ) = − α 4 π 2 ( ∑ i Q i θ i p i p i ⋅ k ) 2 , (4) \widetilde{S}(k)=-\frac{\alpha}{4\pi^2}\left(\sum_i Q_i\theta_i\frac{p_i}{p_i\cdot k}\right)^2, \tag*{(4)} S ( k ) = − 4 π 2 α ( i ∑ Q i θ i p i ⋅ k p i ) 2 , (4) with the photon phasespace weight d 3 k / k 0 \mathrm{d}^3k/k^0 d 3 k / k 0
d σ ( ∞ ) = ∑ n γ = 0 ∞ e Y ( Ω ) n γ ! d Φ n [ ∏ i = 1 n γ d Φ γ i S ~ ( k i ) Θ ( k i , Ω ) ] \mathrm{d}\sigma^{(\infty)}=\sum_{n_\gamma=0}^{\infty}\frac{e^{Y(\Omega)}}{n_\gamma!}\,\mathrm{d}\Phi_n\left[\prod_{i=1}^{n_\gamma}\mathrm{d}\Phi_{\gamma_i}\,\widetilde{S}(k_i)\,\Theta(k_i,\Omega)\right] d σ ( ∞ ) = n γ = 0 ∑ ∞ n γ ! e Y ( Ω ) d Φ n [ i = 1 ∏ n γ d Φ γ i S ( k i ) Θ ( k i , Ω ) ] × ( β ~ 0 ( Φ n ) + ∑ j = 1 n γ β ~ 1 ( Φ n + 1 ) S ~ ( k j ) + ∑ j , k = 1 j < k n γ β ~ 2 ( Φ n + 2 ) S ~ ( k j ) S ~ ( k k ) + ⋯ ) , (5) {}\times\left(\widetilde{\beta}_0(\Phi_n)+\sum_{j=1}^{n_\gamma}\frac{\widetilde{\beta}_1(\Phi_{n+1})}{\widetilde{S}(k_j)}
+\sum_{\substack{j,k=1\\j<k}}^{n_\gamma}\frac{\widetilde{\beta}_2(\Phi_{n+2})}{\widetilde{S}(k_j)\widetilde{S}(k_k)}+\cdots\right), \tag*{(5)} × β 0 ( Φ n ) + j = 1 ∑ n γ S ( k j ) β 1 ( Φ n + 1 ) + j , k = 1 j < k ∑ n γ S ( k j ) S ( k k ) β 2 ( Φ n + 2 ) + ⋯ , (5) where d Φ n \mathrm{d}\Phi_n d Φ n
β ~ n R = ∑ n V β ~ n R + n V n V , (6) \widetilde{\beta}_{n_R}=\sum_{n_V}\widetilde{\beta}^{n_V}_{n_R+n_V}, \tag*{(6)} β n R = n V ∑ β n R + n V n V , (6) with the superscript the order in α \alpha α
For a single real photon of momentum k k k σ \sigma σ
s ( σ ) ( k ) = e ∑ i Q i θ i ε ( σ ) ∗ ⋅ p i k ⋅ p i . (7) \mathfrak{s}^{(\sigma)}(k)=e\sum_i Q_i\theta_i\frac{\varepsilon^{(\sigma)*}\cdot p_i}{k\cdot p_i}. \tag*{(7)} s ( σ ) ( k ) = e i ∑ Q i θ i k ⋅ p i ε ( σ ) ∗ ⋅ p i . (7) It is a sum over single legs, and the pairwise structure of Eqs. (3) and (4) arises only after squaring,
S ~ ( k ) = ∑ σ ∣ s ( σ ) ( k ) ∣ 2 / ( 16 π 3 ) , (8) \widetilde{S}(k)=\sum_\sigma\left|\mathfrak{s}^{(\sigma)}(k)\right|^2/(16\pi^3), \tag*{(8)} S ( k ) = σ ∑ s ( σ ) ( k ) 2 / ( 16 π 3 ) , (8) matching Eq. (4) .
At the amplitude level, we can write the exact Feynman amplitude with n n n M n \mathcal{M}_n M n ℘ \wp ℘ e + e − → f f ˉ e^+e^-\to f\bar f e + e − → f f ˉ
M n ( { λ } ; { k j , σ j } ) = ∑ ℘ S ℘ β ^ 0 ( ℘ ) ( { λ } ) + ∑ j = 1 n ∑ ℘ j ˉ S ℘ j ˉ β ^ 1 ( ℘ j ˉ ) ( { λ } ; k j , σ j ) + ⋯ , (9) \mathcal{M}_n(\{\lambda\};\{k_j,\sigma_j\})=
\sum_{\wp}S_{\wp}\widehat{\beta}^{(\wp)}_0(\{\lambda\})
+\sum_{j=1}^{n}\sum_{\wp_{\bar j}}S_{\wp_{\bar j}}\widehat{\beta}^{(\wp_{\bar j})}_1(\{\lambda\};k_j,\sigma_j)+\cdots, \tag*{(9)} M n ({ λ } ; { k j , σ j }) = ℘ ∑ S ℘ β 0 ( ℘ ) ({ λ }) + j = 1 ∑ n ℘ j ˉ ∑ S ℘ j ˉ β 1 ( ℘ j ˉ ) ({ λ } ; k j , σ j ) + ⋯ , (9) where { λ } \{\lambda\} { λ } ℘ \wp ℘ { I , F } n \{\mathrm{I},\mathrm{F}\}^n { I , F } n ℘ j ˉ \wp_{\bar j} ℘ j ˉ j j j S ℘ = ∏ i s ℘ ( i ) ( σ i ) ( k i ) S_{\wp}=\prod_i\mathfrak{s}^{(\sigma_i)}_{\wp(i)}(k_i) S ℘ = ∏ i s ℘ ( i ) ( σ i ) ( k i ) ℘ \wp ℘ s I \mathfrak{s}_{\mathrm{I}} s I s F \mathfrak{s}_{\mathrm{F}} s F (7) from the initial and final legs, and the superscript ℘ \wp ℘ s s s
P − K I ℘ , K I ℘ = ∑ j : ℘ ( j ) = I k j , (10) P-K_{\mathrm{I}}^{\wp},\qquad K_{\mathrm{I}}^{\wp}=\sum_{j:\wp(j)=\mathrm{I}}k_j, \tag*{(10)} P − K I ℘ , K I ℘ = j : ℘ ( j ) = I ∑ k j , (10) with P P P K I ℘ K_{\mathrm{I}}^{\wp} K I ℘ s ℘ = ( P − K I ℘ ) 2 s_{\wp}=(P-K_{\mathrm{I}}^{\wp})^2 s ℘ = ( P − K I ℘ ) 2 [16 ] the authors split the remainder of photon j j j (9) carries their sum, which depends only on the assignment of the other photons and is what a generated one-photon amplitude provides. The remainders are helicity amplitudes, with the order in α \alpha α β ^ r ( n ) \widehat{\beta}^{(n)}_r β r ( n ) β ~ r n \widetilde{\beta}^n_r β r n E W _{\mathrm{EW}} EW
β ^ 0 ( 0 ) ( { λ } ) = M 0 ( { λ } ) , (11) \widehat{\beta}^{(0)}_0(\{\lambda\})=\mathcal{M}_0(\{\lambda\}), \tag*{(11)} β 0 ( 0 ) ({ λ }) = M 0 ({ λ }) , (11) β ^ 0 ( 1 ) ( { λ } ) = M 0 ( 1 ) ( { λ } ) − 1 2 B V M 0 ( { λ } ) , (12) \widehat{\beta}^{(1)}_0(\{\lambda\})=\mathcal{M}^{(1)}_0(\{\lambda\})-\frac{1}{2}B_{\mathrm{V}}\mathcal{M}_0(\{\lambda\}), \tag*{(12)} β 0 ( 1 ) ({ λ }) = M 0 ( 1 ) ({ λ }) − 2 1 B V M 0 ({ λ }) , (12) β ^ 1 ( 1 ) ( { λ } ; k , σ ) = M 1 ( { λ } ; k , σ ) − s I ( σ ) ( k ) M 0 I ( { λ } ) − s F ( σ ) ( k ) M 0 ( { λ } ) , (13) \widehat{\beta}^{(1)}_1(\{\lambda\};k,\sigma)=\mathcal{M}_1(\{\lambda\};k,\sigma)-\mathfrak{s}^{(\sigma)}_{\mathrm{I}}(k)\mathcal{M}^{\mathrm{I}}_0(\{\lambda\})
-\mathfrak{s}^{(\sigma)}_{\mathrm{F}}(k)\mathcal{M}_0(\{\lambda\}), \tag*{(13)} β 1 ( 1 ) ({ λ } ; k , σ ) = M 1 ({ λ } ; k , σ ) − s I ( σ ) ( k ) M 0 I ({ λ }) − s F ( σ ) ( k ) M 0 ({ λ }) , (13) where M 0 I \mathcal{M}^{\mathrm{I}}_0 M 0 I s s s P − k P-k P − k M 0 ( 1 ) \mathcal{M}^{(1)}_0 M 0 ( 1 ) B V B_{\mathrm{V}} B V ∑ i < j Re B i j \sum_{i<j}\operatorname{Re}B_{ij} ∑ i < j Re B ij β ~ 0 1 \widetilde{\beta}^{1}_0 β 0 1 ∑ ‾ { λ } \overline{\sum}_{\{\lambda\}} ∑ { λ } 1 4 ∑ ‾ { λ } \frac{1}{4}\overline{\sum}_{\{\lambda\}} 4 1 ∑ { λ } P 1 , 2 P_{1,2} P 1 , 2 ( 1 ± P 1 , 2 ) / 2 (1\pm P_{1,2})/2 ( 1 ± P 1 , 2 ) /2
∑ ‾ { λ } ∣ β ^ 0 ( 0 ) + β ^ 0 ( 1 ) ∣ 2 = β ~ 0 0 + β ~ 0 1 + ∑ ‾ { λ } ∣ β ^ 0 ( 1 ) ∣ 2 . (14) \overline{\sum}_{\{\lambda\}}\left|\widehat{\beta}^{(0)}_0+\widehat{\beta}^{(1)}_0\right|^2
=\widetilde{\beta}^{0}_0+\widetilde{\beta}^{1}_0+\overline{\sum}_{\{\lambda\}}\left|\widehat{\beta}^{(1)}_0\right|^2. \tag*{(14)} ∑ { λ } β 0 ( 0 ) + β 0 ( 1 ) 2 = β 0 0 + β 0 1 + ∑ { λ } β 0 ( 1 ) 2 . (14) The two matchings carry the same O ( α ) \mathcal{O}(\alpha) O ( α ) O ( α 2 ) \mathcal{O}(\alpha^2) O ( α 2 ) (9) as in Eq. (5) , with the helicity-averaged ∑ ‾ { λ } , σ ∣ M n ∣ 2 \overline{\sum}_{\{\lambda\},\sigma}|\mathcal{M}_n|^2 ∑ { λ } , σ ∣ M n ∣ 2 S ~ \widetilde{S} S
The difference between the two matchings is where the modulus is taken, as shown in Figure 1 and Figure 2 . YFSN L O E W _{\mathrm{NLOEW}} NLOEW β ^ 1 \widehat{\beta}_1 β 1
Figure 1. The squared-level matching of the O ( α ) O(\alpha) O ( α ) e + e − → N e^{+}e^{-} \to N e + e − → N 1 … N 1 \ldots N 1 … N e ± e^{\pm} e ± β ~ 1 \widetilde{\beta}_1 β 1 S ~ ( k ) ∣ M 0 ∣ 2 \widetilde{S}(k)|\mathcal{M}_0|^2 S ( k ) ∣ M 0 ∣ 2
Figure 2. The amplitude-level matching of the O ( α ) O(\alpha) O ( α ) β ^ 1 ( k ) \widehat{\beta}_1(k) β 1 ( k ) s I ( k ) \mathfrak{s}_{\mathrm{I}}(k) s I ( k ) P − k P-k P − k s F ( k ) \mathfrak{s}_{\mathrm{F}}(k) s F ( k ) P P P s I + s F = s \mathfrak{s}_{\mathrm{I}}+\mathfrak{s}_{\mathrm{F}}=\mathfrak{s} s I + s F = s ℘ \wp ℘ s ℘ ( j ) \mathfrak{s}_{\wp}(j) s ℘ ( j )
The ACRAIC algorithm
This section describes the ACRAIC algorithm and how the coherent sum of Eq. (9) is generalized to arbitrary final states, which amplitudes it requires, and how the event weight is formed. What the algorithm demands of the amplitude generator that supplies those amplitudes is the subject of Sec. IV.
Emission stages and partitions
For a general 2 → N 2\to N 2 → N emission stages , sets L g L_g L g g = 1 , … , G g=1,\ldots,G g = 1 , … , G (7) from the legs of the stage,
s g ( σ ) ( k ) = e ∑ i ∈ L g Q i θ i ε ( σ ) ∗ ⋅ p i k ⋅ p i . (15) \mathfrak{s}^{(\sigma)}_g(k)=e\sum_{i\in L_g}Q_i\theta_i\frac{\varepsilon^{(\sigma)*}\cdot p_i}{k\cdot p_i}. \tag*{(15)} s g ( σ ) ( k ) = e i ∈ L g ∑ Q i θ i k ⋅ p i ε ( σ ) ∗ ⋅ p i . (15) The coherent sum is assembled from the individual s g \mathfrak{s}_g s g ε ( σ ) → ε ( σ ) + λ k \varepsilon^{(\sigma)}\to\varepsilon^{(\sigma)}+\lambda k ε ( σ ) → ε ( σ ) + λk
δ s g ( σ ) ( k ) = e λ ∑ i ∈ L g Q i θ i . (16) \delta\mathfrak{s}^{(\sigma)}_g(k)=e\lambda\sum_{i\in L_g}Q_i\theta_i. \tag*{(16)} δ s g ( σ ) ( k ) = e λ i ∈ L g ∑ Q i θ i . (16) a partial charge sum. Gauge invariance of every term of the coherent sum therefore requires every stage to be charge neutral,
∑ i ∈ L g Q i θ i = 0 for every stage g , (17) \sum_{i\in L_g} Q_i\theta_i = 0 \quad\text{for every stage }g, \tag*{(17)} i ∈ L g ∑ Q i θ i = 0 for every stage g , (17) and the requirement that no emitter be omitted or counted twice is the closure condition
∑ g = 1 G s ℘ ( g ) ( k ) = s ( σ ) ( k ) . (18) \sum_{g=1}^{G}\mathfrak{s}^{(g)}_{\wp}(k)=\mathfrak{s}^{(\sigma)}(k). \tag*{(18)} g = 1 ∑ G s ℘ ( g ) ( k ) = s ( σ ) ( k ) . (18) The two conditions are independent and a charged pair assigned to two stages at once leaves every stage neutral and violates Eq. (18) . A subprocess bounded by resonant propagators satisfies Eq. (17) provided the resonance is counted as a leg of both the stage that produces it (θ = + 1 \theta=+1 θ = + 1 θ = − 1 \theta=-1 θ = − 1
ACRAIC derives the stages from the dipoles that the YFS generator of SHERPA radiates from. The initial state forms one stage and each final-state dipole forms one more. A candidate stage that fails Eq. (17) is merged into a single final stage, which is then neutral by Eq. (2) . For 2 → 2 2\to2 2 → 2 G = 2 G=2 G = 2 (9) is recovered term by term with s I + s F = s \mathfrak{s}_{\mathrm{I}}+\mathfrak{s}_{\mathrm{F}}=\mathfrak{s} s I + s F = s G = 1 G=1 G = 1
For a Z / γ ∗ Z/\gamma^{*} Z / γ ∗ W + W − W^{+}W^{-} W + W − { e − , e + , W − , W + } \{e^{-},e^{+},W^{-},W^{+}\} { e − , e + , W − , W + } W W W W W W W W W (18) . The stage structure is that of the double-pole treatment. of W W W [22 ] . It follows from the identity
1 D ( P ′ + k ) D ( P ′ ) = 1 2 P ′ ⋅ k [ 1 D ( P ′ ) − 1 D ( P ′ + k ) ] (19) \frac{1}{D(P' + k)D(P')} = \frac{1}{2P'\cdot k}\left[\frac{1}{D(P')} - \frac{1}{D(P' + k)}\right] \tag*{(19)} D ( P ′ + k ) D ( P ′ ) 1 = 2 P ′ ⋅ k 1 [ D ( P ′ ) 1 − D ( P ′ + k ) 1 ] (19) with D ( p ) = p 2 − M W 2 + i M W Γ W D(p)=p^2-M_W^2+iM_W\Gamma_W D ( p ) = p 2 − M W 2 + i M W Γ W W W W P ′ P' P ′ W W W P ′ P' P ′ W W W P ′ + k P'+k P ′ + k W W W (17) and (18) for the same process. The W W W O ( α ) \mathcal{O}(\alpha) O ( α ) (17) and (18) and reproduce the same O ( α ) \mathcal{O}(\alpha) O ( α ) W W W W W W [55 ] , is another possibility and we leave it to future work.
A partition ℘ \wp ℘ G G G G n G^n G n (9) becomes
M n = ∑ ℘ S ℘ β ^ 0 ℘ + ∑ j ∑ ℘ ȷ ˉ S ℘ ȷ ˉ β ^ 1 ℘ ȷ ˉ ( k j ) + ∑ j < l ∑ ℘ ȷ ˉ l ˉ S ℘ ȷ ˉ l ˉ β ^ 2 ℘ ȷ ˉ l ˉ ( k j , k l ) + ⋯ , (20) \begin{aligned}
\mathcal{M}_n={}&\sum_{\wp}S_{\wp}\,\hat{\beta}^{\wp}_0+\sum_j\sum_{\wp_{\bar{\jmath}}}S_{\wp_{\bar{\jmath}}}\,\hat{\beta}^{\wp_{\bar{\jmath}}}_1(k_j)\\
&+\sum_{j<l}\sum_{\wp_{\bar{\jmath}\bar{l}}}S_{\wp_{\bar{\jmath}\bar{l}}}\,\hat{\beta}^{\wp_{\bar{\jmath}\bar{l}}}_2(k_j,k_l)+\cdots, \tag*{(20)}
\end{aligned} M n = ℘ ∑ S ℘ β ^ 0 ℘ + j ∑ ℘ ˉ ∑ S ℘ ˉ β ^ 1 ℘ ˉ ( k j ) + j < l ∑ ℘ ˉ l ˉ ∑ S ℘ ˉ l ˉ β ^ 2 ℘ ˉ l ˉ ( k j , k l ) + ⋯ , (20) where ℘ \wp ℘ { 1 , … , G } \{1,\ldots,G\} { 1 , … , G } ℘ ȷ ˉ \wp_{\bar{\jmath}} ℘ ˉ ℘ ȷ ˉ l ˉ \wp_{\bar{\jmath}\bar{l}} ℘ ˉ l ˉ j j j j j j l l l S S S K g = ∑ j : ℘ ( j ) = g k j K_g=\sum_{j:\wp(j)=g}k_j K g = ∑ j : ℘ ( j ) = g k j g g g K I ℘ K_I^{\wp} K I ℘ (10) . The partition Born β ^ 0 ℘ \hat{\beta}^{\wp}_0 β ^ 0 ℘ s s s P − K I P-K_I P − K I K I K_I K I q g q_g q g q g + K g q_g+K_g q g + K g c c c L ( c ) L(c) L ( c ) L ( c ) L(c) L ( c ) K g K_g K g c c c K g K_g K g
The external momenta of the partition Born do not conserve momentum once the photons are removed, and this is intended. The eikonal theorem expresses the soft limit of M 1 \mathcal{M}_1 M 1 (19) shows for a resonance.
Remainders and virtual corrections
We write M 0 ( { λ } ; { Δ } ) \mathcal{M}_0(\{\lambda\};\{\Delta\}) M 0 ({ λ } ; { Δ }) M 1 ( { λ } ; k , σ ; { Δ } ) \mathcal{M}_1(\{\lambda\};k,\sigma;\{\Delta\}) M 1 ({ λ } ; k , σ ; { Δ }) (11) and (13) , with N + 2 N+2 N + 2 N + 3 N+3 N + 3 { Δ } \{\Delta\} { Δ } ℘ \wp ℘ { Δ ℘ } \{\Delta_{\wp}\} { Δ ℘ } β ^ 0 ℘ = M 0 ℘ \hat{\beta}^{\wp}_0=\mathcal{M}^{\wp}_0 β ^ 0 ℘ = M 0 ℘ M 1 ℘ ( k j ) \mathcal{M}^{\wp}_1(k_j) M 1 ℘ ( k j ) k j k_j k j ℘ \wp ℘ ℘ \wp ℘ i i i ℘ i → g \wp_{i\to g} ℘ i → g ℘ \wp ℘ i i i g g g i i i
S i X ℘ = ∑ g s g ( k i ) X ℘ i → g , (21) S_iX^{\wp}=\sum_g s_g(k_i)X^{\wp_{i\to g}}, \tag*{(21)} S i X ℘ = g ∑ s g ( k i ) X ℘ i → g , (21) where X ℘ X^{\wp} X ℘ M 0 ℘ \mathcal{M}^{\wp}_0 M 0 ℘ M 1 ℘ ( k j ) \mathcal{M}^{\wp}_1(k_j) M 1 ℘ ( k j ) P P P
∏ i ∈ P S i X = ∑ ℘ P S ℘ P X ℘ P (22) \prod_{i\in P}S_iX=\sum_{\wp_P}S_{\wp_P}X^{\wp_P} \tag*{(22)} i ∈ P ∏ S i X = ℘ P ∑ S ℘ P X ℘ P (22) with ℘ P \wp_P ℘ P P P P (20) is then
β ^ 1 ( k j ) = M 1 ( k j ) − S j β ^ 0 , (23) \hat{\beta}_1(k_j)=\mathcal{M}_1(k_j)-S_j\hat{\beta}_0, \tag*{(23)} β ^ 1 ( k j ) = M 1 ( k j ) − S j β ^ 0 , (23) the generalization of Eq. (13) in which each stage’s Born sits at that assignment’s propagator kinematics. The coherent amplitudes that ACRAIC evaluates are Eq. (20) truncated after β ^ 0 \hat{\beta}_0 β ^ 0 β ^ 1 \hat{\beta}_1 β ^ 1 C \mathcal{C} C M n \mathcal{M}_n M n C \mathcal{C} C β ^ k \hat{\beta}_k β ^ k k k k M n \mathcal{M}_n M n M n \mathcal{M}_n M n β ^ n \hat{\beta}_n β ^ n O ( α 0 ) \mathcal{O}(\alpha^0) O ( α 0 ) O ( α 1 ) \mathcal{O}(\alpha^1) O ( α 1 )
C 0 = ∑ ℘ S ℘ β ^ 0 ℘ , (24) C_0 = \sum_{\wp} S_{\wp}\,\hat{\beta}_0^{\wp}, \tag*{(24)} C 0 = ℘ ∑ S ℘ β ^ 0 ℘ , (24) C 1 = [ ∏ i S i ] β ^ 0 + ∑ j ∈ R [ ∏ i ≠ j S i ] β ^ 1 ( k j ) , (25) C_1 = \left[\prod_i S_i\right]\hat{\beta}_0+\sum_{j\in\mathcal{R}}\left[\prod_{i\ne j}S_i\right]\hat{\beta}_1(k_j), \tag*{(25)} C 1 = [ i ∏ S i ] β ^ 0 + j ∈ R ∑ i = j ∏ S i β ^ 1 ( k j ) , (25) where ℘ \wp ℘ R \mathcal{R} R R \mathcal{R} R J J J
Two properties of Eq. (25) are used below. First, with one photon it returns C 1 = M 1 ( k ) C_1=\mathcal{M}_1(k) C 1 = M 1 ( k ) S β ^ 0 S\hat{\beta}_0 S β ^ 0 O ( α ) \mathcal{O}(\alpha) O ( α ) n ≥ 2 n\ge2 n ≥ 2 β ^ 0 ℘ \hat{\beta}_0^{\wp} β ^ 0 ℘ R \mathcal{R} R ( 1 − ∣ R ∣ ) (1-|\mathcal{R}|) ( 1 − ∣ R ∣ ) (25) is written so that they are.
The tree-level O ( α 2 ) \mathcal{O}(\alpha^2) O ( α 2 ) j j j l l l
β ^ 2 ( k j , k l ) = M 2 ( k j , k l ) − S l β ^ 1 ( k j ) − S j β ^ 1 ( k l ) − S j S l β ^ 0 , (26) \hat{\beta}_2(k_j,k_l)=\mathcal{M}_2(k_j,k_l)-S_l\hat{\beta}_1(k_j)-S_j\hat{\beta}_1(k_l)-S_jS_l\hat{\beta}_0, \tag*{(26)} β ^ 2 ( k j , k l ) = M 2 ( k j , k l ) − S l β ^ 1 ( k j ) − S j β ^ 1 ( k l ) − S j S l β ^ 0 , (26) with one N + 4 N+4 N + 4 J J J K K K J J J
C J = ∑ K , ∣ K ∣ ≤ J [ ∏ i ∉ K S i ] β ^ ∣ K ∣ ( K ) = [ ∏ i S i ] β ^ 0 + ∑ j [ ∏ i ≠ j S i ] β ^ 1 ( k j ) + ∑ j < l [ ∏ i ≠ j , l S i ] β ^ 2 ( k j , k l ) + ⋯ , (27) \begin{aligned}
C_J={}&\sum_{\substack{K,\ |K|\le J}}\left[\prod_{i\notin K}S_i\right]\hat{\beta}_{|K|}(K)\\
={}&\left[\prod_iS_i\right]\hat{\beta}_0+\sum_j\left[\prod_{i\ne j}S_i\right]\hat{\beta}_1(k_j)\\
&+\sum_{j<l}\left[\prod_{i\ne j,l}S_i\right]\hat{\beta}_2(k_j,k_l)+\cdots, \tag*{(27)}
\end{aligned} C J = = K , ∣ K ∣ ≤ J ∑ i ∈ / K ∏ S i β ^ ∣ K ∣ ( K ) [ i ∏ S i ] β ^ 0 + j ∑ i = j ∏ S i β ^ 1 ( k j ) + j < l ∑ i = j , l ∏ S i β ^ 2 ( k j , k l ) + ⋯ , (27) where the sum stops at remainders of J J J (25) , so C 1 C_1 C 1 J = 1 J=1 J = 1 β ^ ∣ K ∣ \hat{\beta}_{|K|} β ^ ∣ K ∣ K K K J J J J + 1 J+1 J + 1 J J J
The IR-finite one-loop remainder enters as a per-helicity factor on the Born-level amplitude C 0 C_0 C 0 (24) , added to the C 1 C_1 C 1 (25) helicity by helicity,
C 1 V ( { λ } ) = C 1 ( { λ } ) + V ( { λ } ) C 0 ( { λ } ) . (28) C_1^{\mathrm{V}}(\{\lambda\})=C_1(\{\lambda\})+\mathcal{V}(\{\lambda\})C_0(\{\lambda\}). \tag*{(28)} C 1 V ({ λ }) = C 1 ({ λ }) + V ({ λ }) C 0 ({ λ }) . (28) With the tree and one-loop helicity amplitudes a 0 ( { λ } ) a_0(\{\lambda\}) a 0 ({ λ }) a 1 ( { λ } ) a_1(\{\lambda\}) a 1 ({ λ })
V ( { λ } ) = a 1 ( { λ } ) a 0 ( { λ } ) − 1 2 B V , (29) \mathcal{V}(\{\lambda\})=\frac{a_1(\{\lambda\})}{a_0(\{\lambda\})}-\frac{1}{2}\mathcal{B}_{\mathrm{V}}, \tag*{(29)} V ({ λ }) = a 0 ({ λ }) a 1 ({ λ }) − 2 1 B V , (29) where B V \mathcal{B}_{\mathrm{V}} B V (12) , evaluated in the coupling of the loop provider, so that V ( { λ } ) M 0 ( { λ } ) = β ^ 0 ( 1 ) ( { λ } ) \mathcal{V}(\{\lambda\})\mathcal{M}_0(\{\lambda\})=\hat{\beta}_0^{(1)}(\{\lambda\}) V ({ λ }) M 0 ({ λ }) = β ^ 0 ( 1 ) ({ λ }) a 1 / a 0 a_1/a_0 a 1 / a 0 β ^ 0 ( 1 ) = V M 0 \hat{\beta}_0^{(1)}=\mathcal{V}\mathcal{M}_0 β ^ 0 ( 1 ) = V M 0 O ( α ) \mathcal{O}(\alpha) O ( α ) (14) is the interference of the one-loop remainder with the Born,
β ~ 0 1 = ∑ { λ } 2 Re [ M 0 ∗ ( { λ } ) V ( { λ } ) M 0 ( { λ } ) ] = 2 ∑ { λ } ∣ M 0 ( { λ } ) ∣ 2 Re V ( { λ } ) . (30) \begin{aligned}
\widetilde{\beta}_0^1=\sum_{\{\lambda\}}2\operatorname{Re}\left[\mathcal{M}_0^*(\{\lambda\})\mathcal{V}(\{\lambda\})\mathcal{M}_0(\{\lambda\})\right] \\
&=2\sum_{\{\lambda\}}\left|\mathcal{M}_0(\{\lambda\})\right|^2\operatorname{Re}\mathcal{V}(\{\lambda\}). \tag*{(30)}
\end{aligned} β 0 1 = { λ } ∑ 2 Re [ M 0 ∗ ({ λ }) V ({ λ }) M 0 ({ λ }) ] = 2 { λ } ∑ ∣ M 0 ({ λ }) ∣ 2 Re V ({ λ }) . (30) in which M 0 ∗ M 0 = ∣ M 0 ∣ 2 \mathcal{M}_0^*\mathcal{M}_0=|\mathcal{M}_0|^2 M 0 ∗ M 0 = ∣ M 0 ∣ 2 Re V \operatorname{Re}\mathcal{V} Re V V ( { λ } ) \mathcal{V}(\{\lambda\}) V ({ λ }) β ^ 0 0 = ∑ { λ } ∣ M 0 ( { λ } ) ∣ 2 \hat{\beta}_0^0=\sum_{\{\lambda\}}|\mathcal{M}_0(\{\lambda\})|^2 β ^ 0 0 = ∑ { λ } ∣ M 0 ({ λ }) ∣ 2
V ‾ = ∑ { λ } ∣ M 0 ( { λ } ) ∣ 2 Re V ( { λ } ) ∑ { λ } ∣ M 0 ( { λ } ) ∣ 2 = β ~ 0 1 2 β ^ 0 0 (31) \overline{\mathcal{V}}=\frac{\sum_{\{\lambda\}}|\mathcal{M}_0(\{\lambda\})|^2\operatorname{Re}\mathcal{V}(\{\lambda\})}{\sum_{\{\lambda\}}|\mathcal{M}_0(\{\lambda\})|^2}=\frac{\widetilde{\beta}_0^1}{2\hat{\beta}_0^0} \tag*{(31)} V = ∑ { λ } ∣ M 0 ({ λ }) ∣ 2 ∑ { λ } ∣ M 0 ({ λ }) ∣ 2 Re V ({ λ }) = 2 β ^ 0 0 β 0 1 (31) is half the squared-level virtual relative to the Born, so that on photon-free events ACRAIC and YFSNLOew carry the same O ( α ) \mathcal{O}(\alpha) O ( α )
The factor V ( { λ } ) \mathcal{V}(\{\lambda\}) V ({ λ }) V ( { λ } ) = V ‾ \mathcal{V}(\{\lambda\})=\overline{\mathcal{V}} V ({ λ }) = V (28) becomes
∣ C 1 V ∣ 2 = ∣ C 1 ∣ 2 + 2 V ‾ Re ( C 0 , C 1 ) + V ‾ 2 ∣ C 0 ∣ 2 , (32) \left|C_1^{\mathrm{V}}\right|^2=\left|C_1\right|^2+2\overline{\mathcal{V}}\operatorname{Re}(C_0,C_1)+\overline{\mathcal{V}}^2\left|C_0\right|^2, \tag*{(32)} C 1 V 2 = ∣ C 1 ∣ 2 + 2 V Re ( C 0 , C 1 ) + V 2 ∣ C 0 ∣ 2 , (32) ⟨ C 0 , C 1 ⟩ = ∑ { λ } C 0 ∗ ( { λ } ) C 1 ( { λ } ) , (33) \langle C_0,C_1\rangle=\sum_{\{\lambda\}}C_0^*(\{\lambda\})C_1(\{\lambda\}), \tag*{(33)} ⟨ C 0 , C 1 ⟩ = { λ } ∑ C 0 ∗ ({ λ }) C 1 ({ λ }) , (33) This helicity-averaged form is correct at O ( α ) \mathcal{O}(\alpha) O ( α ) O ( α ) \mathcal{O}(\alpha) O ( α )
∣ C 1 V ‾ ∣ 2 = ( 1 + V ‾ ) 2 ∣ C 1 ∣ 2 , (34) \left|\overline{C_1^{\mathrm{V}}}\right|^2=(1+\overline{\mathcal{V}})^2\left|C_1\right|^2, \tag*{(34)} C 1 V 2 = ( 1 + V ) 2 ∣ C 1 ∣ 2 , (34) and the difference is of O ( α 2 ) \mathcal{O}(\alpha^2) O ( α 2 )
Near a resonance the initial-final part of V \mathcal{V} V g g g M M M Γ \Gamma Γ
I g ( q 2 ) = a g ln M ~ 2 − q 2 M ~ 2 , I_g(q^2)=a_g\ln\frac{\widetilde{M}^2-q^2}{\widetilde{M}^2}, I g ( q 2 ) = a g ln M 2 M 2 − q 2 , a g = α π ∑ i ∉ L g ∑ j ∈ L g Q i θ i Q j θ j ln ( 2 ∣ p i ⋅ p j ∣ ) , (35) a_g=\frac{\alpha}{\pi}\sum_{i\notin L_g}\sum_{j\in L_g}Q_i\theta_iQ_j\theta_j\ln(2|p_i\cdot p_j|), \tag*{(35)} a g = π α i ∈ / L g ∑ j ∈ L g ∑ Q i θ i Q j θ j ln ( 2∣ p i ⋅ p j ∣ ) , (35) with M ~ 2 = M 2 − i M Γ \widetilde{M}^{2}=M^{2}-iM\Gamma M 2 = M 2 − i M Γ q 2 q^{2} q 2 (17) . For e + e − → f f ˉ e^{+}e^{-}\to f\bar{f} e + e − → f f ˉ
a g = − 2 ( α / π ) Q e Q f ln ( t / u ) (36) a_g=-2(\alpha/\pi)Q_eQ_f\ln(t/u) \tag*{(36)} a g = − 2 ( α / π ) Q e Q f ln ( t / u ) (36) and I g I_g I g γ Z \gamma Z γ Z Z Z Z [56 ] . A real photon with energy E ≫ Γ E\gg\Gamma E ≫ Γ C 1 C_1 C 1 (28) applies the compensating virtual logarithm to C 0 C_0 C 0 I g I_g I g V \mathcal{V} V exp [ I g ( q ℘ 2 ) ] \exp[I_g(q_{\wp}^{2})] exp [ I g ( q ℘ 2 )] β ^ 0 \hat{\beta}_0 β ^ 0 M 1 \mathcal{M}_1 M 1 β ^ 2 \hat{\beta}_2 β ^ 2 q ℘ 2 q_{\wp}^{2} q ℘ 2 1 + I g 1+I_g 1 + I g
V I ( { λ } ) M 0 ( { λ } ) = I g M 0 r e s ( { λ } ) , (37) \mathcal{V}_I(\{\lambda\})\mathcal{M}_0(\{\lambda\})=I_g\mathcal{M}_0^{\mathrm{res}}(\{\lambda\}), \tag*{(37)} V I ({ λ }) M 0 ({ λ }) = I g M 0 res ({ λ }) , (37) and V → V − V I \mathcal{V}\to\mathcal{V}-\mathcal{V}_I V → V − V I e + e − → Z Z → 4 ℓ e^{+}e^{-}\to ZZ\to4\ell e + e − → Z Z → 4 ℓ [55 ] .
The structure of Eq. (35) is established at one loop. In the pole approximation the resonant soft-photon, non-factorizable, correction is universal for any number of charged or neutral resonances. It depends only on the charges, the momenta and the complex masses, with the photon momentum routed through the resonance propagators as in the stages here [57 ] . The factorizable and non-factorizable parts separate to all orders in the double-pole approximation [58 ] , but an exponentiated form has been derived only for a single resonance, at two loops [59 ] , and proposed for W W W [55 ] . Beyond O ( α ) \mathcal{O}(\alpha) O ( α ) I g I_g I g O ( α ) \mathcal{O}(\alpha) O ( α ) Γ / M \Gamma/M Γ/ M
At O ( α 2 ) \mathcal{O}(\alpha^{2}) O ( α 2 ) β ^ 2 \hat{\beta}_2 β ^ 2 (26) for every pair of photons,
C 2 = C 1 + ∑ j < l [ ∏ i ≠ j , l S i ] β ^ 2 ( k j , k l ) , (38) C_2=C_1+\sum_{j<l}\left[\prod_{i\ne j,l}S_i\right]\hat{\beta}_2(k_j,k_l), \tag*{(38)} C 2 = C 1 + j < l ∑ i = j , l ∏ S i β ^ 2 ( k j , k l ) , (38) which is Eq. (27) at J = 2 J=2 J = 2 V ‾ \overline{\mathcal{V}} V (31) , less the resonant logarithm of Eq. (35) . Together they give the full O ( α 2 ) \mathcal{O}(\alpha^{2}) O ( α 2 ) (38) once the loop is included,
∣ C ‾ 2 V ∣ 2 = ∣ C 2 + V ‾ C 1 ∣ 2 + ∑ j ∈ R Δ j Re ⟨ C 2 , C 1 ( j ) ⟩ + 2 Re ( V ( 2 ) ) ∣ C 0 ∣ 2 , (39) \left|\overline{C}_2^{\mathcal{V}}\right|^2=\left|C_2+\overline{\mathcal{V}}C_1\right|^2+\sum_{j\in\mathcal{R}}\Delta_j\,\operatorname{Re}\langle C_2,C_1^{(j)}\rangle+2\operatorname{Re}(\mathcal{V}^{(2)})|C_0|^2, \tag*{(39)} C 2 V 2 = C 2 + V C 1 2 + j ∈ R ∑ Δ j Re ⟨ C 2 , C 1 ( j ) ⟩ + 2 Re ( V ( 2 ) ) ∣ C 0 ∣ 2 , (39) where C 1 ( j ) = [ ∏ i ≠ j S i ] M 1 ( k j ) C_1^{(j)}=\left[\prod_{i\ne j}S_i\right]\mathcal{M}_1(k_j) C 1 ( j ) = [ ∏ i = j S i ] M 1 ( k j ) j j j V ( 2 ) \mathcal{V}^{(2)} V ( 2 ) V ( 2 ) \mathcal{V}^{(2)} V ( 2 ) O ( α ) \mathcal{O}(\alpha) O ( α ) O ( α ) \mathcal{O}(\alpha) O ( α ) C 1 C_1 C 1 C 0 C_0 C 0 β ^ 2 \hat{\beta}_2 β ^ 2 O ( α 3 ) \mathcal{O}(\alpha^{3}) O ( α 3 ) V ‾ \overline{\mathcal{V}} V j j j C 1 C_1 C 1 V ‾ \overline{\mathcal{V}} V j j j V j \mathcal{V}_j V j j j j
M 1 ( k j ) ( 1 + V j ) = M 1 ( k j ) ( 1 + V ) + M 1 ( k j ) ( V j − V ) , (40) \mathcal{M}_1(k_j)(1+\mathcal{V}_j)=\mathcal{M}_1(k_j)(1+\mathcal{V})+\mathcal{M}_1(k_j)(\mathcal{V}_j-\mathcal{V}), \tag*{(40)} M 1 ( k j ) ( 1 + V j ) = M 1 ( k j ) ( 1 + V ) + M 1 ( k j ) ( V j − V ) , (40) and Δ j \Delta_j Δ j 2 Re ( V j − V ) 2\operatorname{Re}(\mathcal{V}_j-\mathcal{V}) 2 Re ( V j − V ) 1 2 Δ j C 1 ( j ) \frac{1}{2}\Delta_j C_1^{(j)} 2 1 Δ j C 1 ( j ) (39) , up to terms of O ( α 3 ) \mathcal{O}(\alpha^{3}) O ( α 3 ) M 1 ( 1 ) ( k ) → S ( k ) M 0 ( 1 ) \mathcal{M}_1^{(1)}(k)\to\mathcal{S}(k)\mathcal{M}_0^{(1)} M 1 ( 1 ) ( k ) → S ( k ) M 0 ( 1 ) k → 0 k\to0 k → 0 B V \mathcal{B}_{\mathrm{V}} B V Δ j → 0 \Delta_j\to0 Δ j → 0 k j → 0 k_j\to0 k j → 0
The real-virtual is the first of a tower of one-loop residuals. The residuals are formed automatically from the one-loop amplitudes the provider returns. β ^ 2 [ 1 ] \hat{\beta}_2^{[1]} β ^ 2 [ 1 ]
Weight Distribution
We write ∣ C k ∣ 2 ‾ = ∑ { λ } ∣ C k ( { λ } ) ∣ 2 \overline{|C_k|^2}=\sum_{\{\lambda\}}|C_k(\{\lambda\})|^2 ∣ C k ∣ 2 = ∑ { λ } ∣ C k ({ λ }) ∣ 2
∣ C a p p r o x ∣ 2 = ∑ ℘ ∏ j ∣ s ℘ ( j ) ( k j ) ∣ 2 × s s ℘ ∑ { λ } ∣ M ~ 0 ( { λ } ; s ℘ ) ∣ 2 (41) \left|\mathcal{C}_{\mathrm{approx}}\right|^2
= \sum_{\wp}\prod_j\left|\mathfrak{s}_{\wp(j)}(k_j)\right|^2
\times\frac{s}{s_{\wp}}\sum_{\{\lambda\}}\left|\widetilde{\mathcal{M}}_0(\{\lambda\};s_{\wp})\right|^2 \tag*{(41)} ∣ C approx ∣ 2 = ℘ ∑ j ∏ s ℘ ( j ) ( k j ) 2 × s ℘ s { λ } ∑ M 0 ({ λ } ; s ℘ ) 2 (41) where M ~ 0 ( { λ } ; s ℘ ) \widetilde{\mathcal{M}}_0(\{\lambda\};s_{\wp}) M 0 ({ λ } ; s ℘ ) P − K ℘ 0 P-K_{\wp}^{0} P − K ℘ 0
W A C R A I C = ∣ C J V ∣ 2 ∣ C a p p r o x ∣ 2 (42) \mathcal{W}_{\mathrm{ACRAIC}}=\frac{\left|\mathcal{C}^{V}_{J}\right|^2}{\left|\mathcal{C}_{\mathrm{approx}}\right|^2} \tag*{(42)} W ACRAIC = ∣ C approx ∣ 2 C J V 2 (42) with C J V \mathcal{C}^{V}_{J} C J V O ( α J ) \mathcal{O}(\alpha^{J}) O ( α J ) ∣ C 0 ∣ 2 \left|\mathcal{C}_0\right|^2 ∣ C 0 ∣ 2
At O ( α ) \mathcal{O}(\alpha) O ( α ) (42) cannot be negative. Its numerator is a sum over helicities of squared moduli, its denominator Eq. (41) a sum of positive terms, and the form factor multiplies it as an exponential. This holds event by event, for every final state, energy and photon multiplicity, and it survives the virtual correction in both forms Eqs. (28) and (32) , since both enter inside the modulus. At O ( α 2 ) \mathcal{O}(\alpha^2) O ( α 2 ) (39) stands outside the modulus and has no definite sign, so the O ( α 2 ) \mathcal{O}(\alpha^2) O ( α 2 ) O ( α 2 ) \mathcal{O}(\alpha^2) O ( α 2 ) 1 + v + ∑ j V j 1+v+\sum_j V_j 1 + v + ∑ j V j V j V_j V j 1 + V j 1+V_j 1 + V j Table I counts the negative weights for e + e − → μ + μ − e^+e^-\to\mu^+\mu^- e + e − → μ + μ − O ( α ) \mathcal{O}(\alpha) O ( α )
Table I. Fraction of negative event weights for e + e − → μ + μ − e^+e^-\to\mu^+\mu^- e + e − → μ + μ − 66 < m μ μ < 116 66<m_{\mu\mu}<116 66 < m μμ < 116 O ( α ) \mathcal{O}(\alpha) O ( α )
For a fraction f f f ( ∑ i w i ) 2 / ∑ i w i 2 (\sum_i w_i)^2/\sum_i w_i^2 ( ∑ i w i ) 2 / ∑ i w i 2 ( 1 − 2 f ) 2 (1-2f)^2 ( 1 − 2 f ) 2 Table I the loss is about one per cent, small for this process, but the fraction grows with the photon and process multiplicity and negative weights survive unweighting as events of weight − 1 -1 − 1 O ( α ) \mathcal{O}(\alpha) O ( α )
comparisons of Sec. V test, but it means that the precision required with samples of 10 12 10^{12} 1 0 12 Z Z Z [60 , 61 , 62 , 63 , 64 , 65 , 66 , 67 ] , where the negative weights arise in the matching of next-to-leading-order calculations to parton showers. ACRAIC does not produce negative weights at O ( α ) \mathcal{O}(\alpha) O ( α )
Realization within S0.8HERPA
The algorithm of Sec. III needs, for every event, the tree amplitudes, resolved in all external helicities, at the physical legs and with partition-dependent propagator momenta, together with polarization vectors and one-loop amplitudes that are consistent with them. This section states what that requires of an automated amplitude generator, and how Comix and the one-loop providers interfaced to SHERPA meet it.
A Berends–Giele recursion [47 ] builds an amplitude from off-shell currents. The current of a set π \pi π
J ( π ) = P ( p π ) ∑ N = 2 n ∑ { π 1 , … , π N } ∈ ℘ N ( π ) V N ( π 1 , … , π N ) J ( π 1 ) ⋯ J ( π N ) , p π = ∑ i ∈ π p i (43) J(\pi)=P(p_{\pi})\sum_{N=2}^{n}\sum_{\{\pi_1,\ldots,\pi_N\}\in\wp_N(\pi)}
V_N(\pi_1,\ldots,\pi_N)J(\pi_1)\cdots J(\pi_N),
\qquad
p_{\pi}=\sum_{i\in\pi}p_i \tag*{(43)} J ( π ) = P ( p π ) N = 2 ∑ n { π 1 , … , π N } ∈ ℘ N ( π ) ∑ V N ( π 1 , … , π N ) J ( π 1 ) ⋯ J ( π N ) , p π = i ∈ π ∑ p i (43) where ℘ N ( π ) \wp_N(\pi) ℘ N ( π ) π \pi π N N N n n n V N V_N V N N N N P ( p π ) P(p_{\pi}) P ( p π ) p π p_{\pi} p π J ( { i } ) J(\{i\}) J ({ i }) [46 ] implements Eq. (43) , and its cost for a fixed helicity configuration grows polynomially in the number of legs, which makes the ( N + 3 ) (N+3) ( N + 3 ) (25) affordable. at multiplicities where a diagrammatic expansion would not be. The remainders map onto the recursion without modification. M 0 \mathcal{M}_0 M 0 M 1 \mathcal{M}_1 M 1 (20) is therefore exactly the output of a general tree-level amplitude generator. It is also not limited to the Standard Model, and the same recursion can be used for any model that the generator supports through its UFO interface [68 , 69 , 70 ] , including effective field theories and BSM models. In particular, it applies to the QED corrections in scalar QED, which is the standard description of photon radiation off pions in e + e − → π + π − ( N γ ) e^{+}e^{-} \to\pi^{+}\pi^{-}(N\gamma) e + e − → π + π − ( N γ ) [24 , 71 , 72 , 73 , 74 , 75 , 76 , 77 , 78 ] . SHERPA’s photon emission already treats scalar emitters [52 , 54 ] . A detailed study of this process is the topic of a separate paper [79 ] . The same applies to μ ± e − → μ ± e − \mu^{\pm}e^{-} \to\mu^{\pm}e^{-} μ ± e − → μ ± e − O ( α ) \mathcal{O}(\alpha) O ( α ) O ( α 2 ) \mathcal{O}(\alpha^{2}) O ( α 2 ) [80 ] . The complete two-loop virtual is available for a massless electron and a massive muon [81 , 82 ] , and the electron mass can be restored by the massification used here, so it could replace the dominant terms.
Because every current of Eq. (43) is labelled by its leg set, the shift rule of Sec. III A is realized by replacing, per partition, P ( p π ) P(p_{\pi}) P ( p π ) P ( p π + Δ π ) P(p_{\pi}+\Delta_{\pi}) P ( p π + Δ π ) π \pi π s s s W W W W W W W W W s s s P − K P-K P − K
The coherent sum Eq. (20) is not invariant under a redefinition of the phase of an individual term and only the modulus of a single term is. The amplitudes M 0 \mathcal{M}_0 M 0 M 1 \mathcal{M}_1 M 1 (15) must therefore be built with the same gauge reference vector and the same spinor phase conventions, or β ^ 1 \hat{\beta}_1 β ^ 1 k k k (17) ), so only the common phase convention has to be fixed.
The one-loop virtual of both ACRAIC and YFSNLOEW is generally taken from SHERPA’s native interface to one-loop providers, such as OPENLOOPS [48 ] and RECOLLA [49 , 50 ] or for some select processes internally coded amplitudes. The helicity-resolved factor Eq. (29) requires the complex one-loop amplitude per helicity, which RECOLLA computes and stores inside the call that the squared-level virtual already makes, so that reading it out costs nothing further. OPENLOOPS returns squared quantities, and with it ACRAIC uses the helicity-averaged form Eq. (32) . The same holds for the real-virtual.
For one-loop corrections in YFS generators it has generally been a requirement that all masses are kept. This can lead to numerical issues that we would like to avoid. For leptons whose mass is small against every invariant of the process, the massive one-loop amplitude can be obtained from the massless one up to power corrections [83 , 84 , 85 , 86 , 87 ] ,
M ( m ) ( { p } ) = ∏ i Z i 1 / 2 M ( 0 ) ( { p ~ } ) + O ( m 2 / ∣ s i j ∣ ) (44) \mathcal{M}^{(m)}(\{p\}) = \prod_i Z_i^{1/2}\mathcal{M}^{(0)}(\{\widetilde{p}\}) + \mathcal{O}(m^2/|s_{ij}|) \tag*{(44)} M ( m ) ({ p }) = i ∏ Z i 1/2 M ( 0 ) ({ p }) + O ( m 2 /∣ s ij ∣ ) (44) where the product runs over the external leptons and, at one loop,
Z i ( 1 ) = α 4 π Q i 2 [ 2 ϵ 2 + 2 L i + 1 ϵ + L i 2 + L i + 4 + ζ 2 ] (45) Z_i^{(1)} = \frac{\alpha}{4\pi}Q_i^2\left[\frac{2}{\epsilon^2}+\frac{2L_i+1}{\epsilon}+L_i^2+L_i+4+\zeta_2\right] \tag*{(45)} Z i ( 1 ) = 4 π α Q i 2 [ ϵ 2 2 + ϵ 2 L i + 1 + L i 2 + L i + 4 + ζ 2 ] (45) with L i = ln ( μ 2 / m i 2 ) L_i=\ln(\mu^2/m_i^2) L i = ln ( μ 2 / m i 2 ) [87 ] . The relation Eq. (44) is the basis of the two-loop calculations of [84 , 86 , 87 ] and of the two-loop virtual in McMule [24 ] , which uses massive amplitudes at one loop and for e + e − → e + e − γ e^{+}e^{-}\to e^{+}e^{-}\gamma e + e − → e + e − γ [88 ] . Here the mass expansion is applied to the amplitude of any loop provider, independently of the process and helicity by helicity, inside the amplitude-level exponentiation. The loop provider is called with the external legs as massless, at the massless { p ~ } \{\widetilde{p}\} { p }
The factor of Eq. (29) becomes
V ( { λ } ) = a 0 ( 1 ) ( { λ } ) a 0 ( 0 ) ( { λ } ) + 1 2 ∑ i Z i ( 1 ) − 1 2 B V (46) \mathcal{V}(\{\lambda\}) = \frac{a_0^{(1)}(\{\lambda\})}{a_0^{(0)}(\{\lambda\})}+\frac{1}{2}\sum_i Z_i^{(1)}-\frac{1}{2}\mathcal{B}_{V} \tag*{(46)} V ({ λ }) = a 0 ( 0 ) ({ λ }) a 0 ( 1 ) ({ λ }) + 2 1 i ∑ Z i ( 1 ) − 2 1 B V (46) and the same factor multiplies the tree in the real-virtual of Sec. III B. The trees themselves remain the fully massive ones from COMIX, so that only the ratio of loop to tree is approximated and the soft and collinear structure of the real emission is unchanged. The poles of Z i ( 1 ) Z_i^{(1)} Z i ( 1 ) B V \mathcal{B}_{V} B V Δ α \Delta\alpha Δ α G μ G_\mu G μ m 2 m^2 m 2
r = V ( m ) − V ( 0 ) − ∑ i Z i ( 1 ) (47) r = \mathcal{V}^{(m)}-\mathcal{V}^{(0)}-\sum_i Z_i^{(1)} \tag*{(47)} r = V ( m ) − V ( 0 ) − i ∑ Z i ( 1 ) (47) with V \mathcal{V} V e + e − → μ + μ − e^+e^- \to\mu^+\mu^- e + e − → μ + μ − ≈ 10 − 7 \approx10^{-7} ≈ 1 0 − 7 M Z M_Z M Z 3 × 10 − 6 3\times10^{-6} 3 × 1 0 − 6 m μ 2 / s m_\mu^2/s m μ 2 / s ∣ t ∣ |t| ∣ t ∣ 3 × 10 − 4 3\times10^{-4} 3 × 1 0 − 4 10 − 4 10^{-4} 1 0 − 4 s = M Z \sqrt{s}=M_Z s = M Z 10 − 4 10^{-4} 1 0 − 4 m / E m/E m / E 10 − 3 10^{-3} 1 0 − 3 π α / ( 2 β ) \pi\alpha/(2\beta) π α / ( 2 β ) τ \tau τ e + e − → μ + μ − e + e − e^+e^- \to\mu^+\mu^-e^+e^- e + e − → μ + μ − e + e − 2 × 10 − 6 2\times10^{-6} 2 × 1 0 − 6 M Z M_Z M Z m 2 / ∣ 2 p i ⋅ p j ∣ m^2/|2p_i\cdot p_j| m 2 /∣2 p i ⋅ p j ∣
For photonic corrections, with no closed lepton loop, the factorization Eq. (44) holds at two loops with Z i = 1 + Z i ( 1 ) + Z i ( 2 ) Z_i=1+Z_i^{(1)}+Z_i^{(2)} Z i = 1 + Z i ( 1 ) + Z i ( 2 ) [84 , 86 , 87 ] . We use it for the non-factorizable part of the photonic β ^ 0 ( 2 ) \hat{\beta}_0^{(2)} β ^ 0 ( 2 ) e + e − → ℓ + ℓ − e^+e^- \to\ell^+\ell^- e + e − → ℓ + ℓ − ℓ = μ , τ \ell=\mu,\tau ℓ = μ , τ F F F
ln M ( m ) − ln F i n i ( m ) − ln F f i n ( m ) = ln M ( 0 ) − ln F i n i ( 0 ) − ln F f i n ( 0 ) + O ( m 2 / ∣ s i j ∣ ) (48) \ln\mathcal{M}^{(m)}-\ln F_{\mathrm{ini}}^{(m)}-\ln F_{\mathrm{fin}}^{(m)}
=
\ln\mathcal{M}^{(0)}-\ln F_{\mathrm{ini}}^{(0)}-\ln F_{\mathrm{fin}}^{(0)}
+\mathcal{O}(m^2/|s_{ij}|) \tag*{(48)} ln M ( m ) − ln F ini ( m ) − ln F fin ( m ) = ln M ( 0 ) − ln F ini ( 0 ) − ln F fin ( 0 ) + O ( m 2 /∣ s ij ∣ ) (48) since every lepton enters the amplitude once and one form factor once, and its Z i Z_i Z i [89 ] and the massless form factor of [84 ] . Once the one-loop infrared function of the initial-final dipoles is subtracted, its two-loop part has no infrared poles and does not depend on the infrared scheme, and for massless leptons it is a function of t / s t/s t / s e + e − → γ γ e^+e^- \to\gamma\gamma e + e − → γ γ [90 ] . For this process, the photonic β ^ 0 ( 2 ) \hat{\beta}_0^{(2)} β ^ 0 ( 2 ) e + e − e^+e^- e + e −
V ( 2 ) → V ( 2 ) + 1 2 ( α 2 π ) 2 Δ ( t / s ) (49) \mathcal{V}^{(2)}\to\mathcal{V}^{(2)}+\frac{1}{2}\left(\frac{\alpha}{2\pi}\right)^2\Delta(t/s) \tag*{(49)} V ( 2 ) → V ( 2 ) + 2 1 ( 2 π α ) 2 Δ ( t / s ) (49) Δ = 2 B s T 0 − f 0 2 − 2 r F Re r 1 + r F 2 . (50) \Delta=\frac{2B_s}{T_0}-\frac{f_0}{2}-2r_F\operatorname{Re}r_1+r_F^2. \tag*{(50)} Δ = T 0 2 B s − 2 f 0 − 2 r F Re r 1 + r F 2 . (50) Here B s B_s B s [90 ] , T 0 = 8 ( t / u + u / t ) T_0=8(t/u+u/t) T 0 = 8 ( t / u + u / t ) r 1 r_1 r 1 e + e − → γ γ e^+e^-\to\gamma\gamma e + e − → γ γ r F = − 4 r_F=-4 r F = − 4 f 0 f_0 f 0 r 1 r_1 r 1 e + e − → γ γ e^+e^-\to\gamma\gamma e + e − → γ γ Δ \Delta Δ s s s m e 2 / min ( ∣ t ∣ , ∣ u ∣ ) > 10 − 3 m_e^2/\min(|t|,|u|)>10^{-3} m e 2 / min ( ∣ t ∣ , ∣ u ∣ ) > 1 0 − 3
( 1 2 B V 2 + B V V 1 ) ∣ M 0 ∣ 2 ‾ , (51) \left(\frac{1}{2}\mathcal{B}_V^2+\mathcal{B}_V\mathcal{V}_1\right)\overline{|\mathcal{M}_0|^2}, \tag*{(51)} ( 2 1 B V 2 + B V V 1 ) ∣ M 0 ∣ 2 , (51) with V 1 \mathcal{V}_1 V 1 m e 2 / s m_e^2/s m e 2 / s 10 − 6 10^{-6} 1 0 − 6
Theory Uncertainties
We estimate the missing higher orders with theory nuisance parameters [91 , 92 ] instead of the difference to the next-lower order which is generally used for YFS. The amplitudes of Eqs. (25) and (38) are sums of classes, C = ∑ c C c C=\sum_c C_c C = ∑ c C c β ^ 0 \hat{\beta}_0 β ^ 0 β ^ 1 \hat{\beta}_1 β ^ 1 O ( α 2 ) \mathcal{O}(\alpha^2) O ( α 2 ) β ^ 2 \hat{\beta}_2 β ^ 2 T c T_c T c C c C_c C c 1 + ϵ 1+\epsilon 1 + ϵ θ t ∈ [ − 1 , 1 ] \theta_t\in[-1,1] θ t ∈ [ − 1 , 1 ] N t N_t N t θ t = ± 1 \theta_t = \pm1 θ t = ± 1
d σ ( θ ) = d σ [ 1 + ∑ t θ t N t T c ( t ) ] , (52) \mathrm{d}\sigma(\theta) = \mathrm{d}\sigma\left[1 + \sum_t \theta_t N_t T_c(t)\right], \tag*{(52)} d σ ( θ ) = d σ [ 1 + t ∑ θ t N t T c ( t ) ] , (52) T c = 2 Re ⟨ C c , C J V ⟩ ∣ C J V ∣ 2 , ∑ c T c = 2 , (53) T_c = \frac{2\operatorname{Re}\langle C_c,C_J^V\rangle}{\lvert C_J^V\rvert^2}, \qquad\sum_c T_c = 2, \tag*{(53)} T c = ∣ C J V ∣ 2 2 Re ⟨ C c , C J V ⟩ , c ∑ T c = 2 , (53) δ σ = [ ∑ t ( N t ∫ d σ T c ( t ) ) 2 ] 1 / 2 , (54) \delta\sigma= \left[\sum_t\left(N_t\int\mathrm{d}\sigma\,T_c(t)\right)^2\right]^{1/2}, \tag*{(54)} δ σ = [ t ∑ ( N t ∫ d σ T c ( t ) ) 2 ] 1/2 , (54) g e , f = α π ( ln s m e , f 2 − 1 ) , (55) g_{e,f} = \frac{\alpha}{\pi}\left(\ln\frac{s}{m_{e,f}^2}-1\right), \tag*{(55)} g e , f = π α ( ln m e , f 2 s − 1 ) , (55) N β ^ 1 = 1 2 g k k ! , N β ^ 2 = 1 2 g , (56) N_{\hat{\beta}_1} = \frac{1}{2}\frac{g^k}{k!}, \qquad N_{\hat{\beta}_2} = \frac{1}{2}g, \tag*{(56)} N β ^ 1 = 2 1 k ! g k , N β ^ 2 = 2 1 g , (56) N β ^ 0 i n i = 1 2 g e m m ! , N β ^ 0 f i n = 1 2 Q f 2 g f m m ! , (57) N_{\hat{\beta}_0}^{\mathrm{ini}} = \frac{1}{2}\frac{g_e^m}{m!}, \qquad N_{\hat{\beta}_0}^{\mathrm{fin}} = \frac{1}{2}\frac{Q_f^2g_f^m}{m!}, \tag*{(57)} N β ^ 0 ini = 2 1 m ! g e m , N β ^ 0 fin = 2 1 m ! Q f 2 g f m , (57) N β ^ 0 × = ( g e + Q f 2 g f ) m − g e m − ( Q f 2 g f ) m 2 m ! . (58) N_{\hat{\beta}_0}^{\times} = \frac{(g_e+Q_f^2g_f)^m-g_e^m-(Q_f^2g_f)^m}{2m!}. \tag*{(58)} N β ^ 0 × = 2 m ! ( g e + Q f 2 g f ) m − g e m − ( Q f 2 g f ) m . (58) Here g g g g e g_e g e g f g_f g f k k k m m m β ^ 0 \hat{\beta}_0 β ^ 0 m = 2 m=2 m = 2 O ( α ) O(\alpha) O ( α ) m = 3 m=3 m = 3 O ( α 2 ) O(\alpha^2) O ( α 2 ) β ^ 0 ( 2 ) \hat{\beta}_0^{(2)} β ^ 0 ( 2 ) β ^ 0 \hat{\beta}_0 β ^ 0 N β ^ 0 × N_{\hat{\beta}_0}^{\times} N β ^ 0 × [93 ] .
For the missing electroweak corrections at the Z Z Z [94 ] , the step from the complete one-loop corrections in the G μ G_\mu G μ [95 ] as errors on the axial-vector form factor and on the effective mixing angle of each fermion. We vary both by these errors in the GRIFFIN matrix element itself, with Γ Z \Gamma_Z Γ Z e + e − → μ + μ − e^+e^- \to\mu^+\mu^- e + e − → μ + μ − e + e − → τ + τ − e^+e^- \to\tau^+\tau^- e + e − → τ + τ − 2.4 × 10 − 4 2.4 \times10^{-4} 2.4 × 1 0 − 4 sin 2 θ e f f \sin^2\theta_{\mathrm{eff}} sin 2 θ eff 4.3 × 10 − 5 4.3 \times10^{-5} 4.3 × 1 0 − 5 7.6 × 10 − 5 7.6 \times10^{-5} 7.6 × 1 0 − 5 [95 ] by hand. It is then independent of our own analysis, and it follows any improved estimate in a later release of GRIFFIN without further work on our side. Both errors lie below the O ( α ) O(\alpha) O ( α ) Z Z Z Z Z Z [94 ] , and their missing higher orders, which matter for events whose invariant mass after initial-state radiation lies below the peak, are not in the band. We also give the difference between the α ( M Z ) \alpha(M_Z) α ( M Z ) G μ G_\mu G μ e + e − → Z H e^+e^- \to ZH e + e − → Z H 1 % 1\% 1% [96 , 97 ] and 1.5 % 1.5\% 1.5% μ + μ − H \mu^+\mu^-H μ + μ − H Z Z Z [98 ] in the α ( 0 ) \alpha(0) α ( 0 ) 0.3 % 0.3\% 0.3% G μ G_\mu G μ [97 ] . The two-loop electroweak correction with closed fermion loops are + 0.7 % +0.7\% + 0.7% α ( 0 ) \alpha(0) α ( 0 ) [99 ] , and the complete two-loop electroweak amplitudes are known only unrenormalized [100 ] .
Light-fermion pair corrections are not generated by ACRAIC and are in neither band. For e + e − → μ + μ − e^+e^- \to\mu^+\mu^- e + e − → μ + μ − s ′ > 0.5 s s' > 0.5s s ′ > 0.5 s m μ μ > 60 m_{\mu\mu} > 60 m μμ > 60 − 2.5 × 10 − 3 -2.5 \times10^{-3} − 2.5 × 1 0 − 3 10 − 4 10^{-4} 1 0 − 4 10 − 4 10^{-4} 1 0 − 4 2 → 4 2 \to4 2 → 4 O ( α 2 ) O(\alpha^2) O ( α 2 )
Validation
In this section we validate our ACRAIC implementation. We start with conditions that hold on every event, namely the internal consistency conditions of Sec. III, the event-by-event agreement with YFSNLOew, and the closure of the tree residuals against the exact multi-photon amplitudes. We then compare with KKMC, where an independent amplitude-level implementation exists, in cross sections and distributions at photonic O ( α 2 ) O(\alpha^2) O ( α 2 ) § 2.2 A turns to the step to O ( α 2 ) O(\alpha^2) O ( α 2 ) Z Z Z § 2.2 B applies the same machinery, as a further example application rather than a validation in itself, to the Higgs recoil mass at FCC-ee. Table II lists the processes, the order of the ACRAIC prediction for each, and the source of its two-loop content. Every cross section and distribution in this section is computed with RIVET [102 ] , storing the weight variations of Sec. IV A with YODA [103 ] .
Our implementation in section IV rests on statements that can be checked on every event. At G = 2 G=2 G = 2 N = 2 N=2 N = 2 2 → 2 2 \to2 2 → 2 (15) evaluates a neutral pair in a different order, and in the density matrix. Neutrality, Eq. (17) , is imposed as a precondition on the stage de composition, and closure, Eq. (18) , is checked numerically per photon. The number of partitions is checked against G G G C 1 = M 1 ( k ) C_1 = \mathcal{M}_1(k) C 1 = M 1 ( k ) e + e − → H μ + μ − e^+e^- \to H\mu^+\mu^- e + e − → H μ + μ − G = 2 G = 2 G = 2 e + e − → μ + μ − τ + τ − e^+e^- \to\mu^+\mu^-\tau^+\tau^- e + e − → μ + μ − τ + τ − G = 3 G = 3 G = 3 2 → 2 2 \to2 2 → 2 μ + μ − τ + τ − \mu^+\mu^-\tau^+\tau^- μ + μ − τ + τ − β ^ 1 \hat{\beta}_1 β ^ 1
Figure 3. One-loop massification at fixed phase-space points, shown as the residual r r r (47) relative to the Born, from OPENLOOPS. Top left: e + e − → μ + μ − e^+e^- \to\mu^+\mu^- e + e − → μ + μ − ∣ r ∣ |r| ∣ r ∣ e + e − → μ + μ − e + e − e^+e^- \to\mu^+\mu^-e^+e^- e + e − → μ + μ − e + e − ∣ r ∣ |r| ∣ r ∣ m e 2 / ∣ t ∣ m_e^2/|t| m e 2 /∣ t ∣ e + e − → μ + μ − γ e^+e^- \to\mu^+\mu^-\gamma e + e − → μ + μ − γ m μ / E μ m_\mu/E_\mu m μ / E μ
Table 2. The order of the ACRAIC prediction for each process and the source of its two-loop content. The logarithms of β ^ 0 ( 2 ) \hat{\beta}_0^{(2)} β ^ 0 ( 2 ) 1 2 δ f 2 + δ f w + δ N L L \frac{1}{2}\delta_f^2+\delta_{fw}+\delta_{\mathrm{NLL}} 2 1 δ f 2 + δ f w + δ NLL (64) , the leading- and next-to-leading-logarithmic terms of the two-loop photonic vertex form factor, and the vertex constants those of the massive two-loop form factor [84 ] . Every O ( α 2 ) \mathcal{O}(\alpha^2) O ( α 2 ) O ( α 2 ) \mathcal{O}(\alpha^2) O ( α 2 )
Where the two matchings must coincide, they can be compared event by event, since both columns are evaluated on the same events and multiply the same generator weight. We compare the real-only correction factors for ACRAIC and the Y F S N L O E W \mathrm{YFS}_{\mathrm{NLO}_{\mathrm{EW}}} YFS NLO EW ( 1 + ν ) (1+\nu) ( 1 + ν )
ACRAIC samples one helicity per photon, so per event ∣ M 1 ∣ 2 |\mathcal{M}_1|^2 ∣ M 1 ∣ 2 Y F S N L O E W \mathrm{YFS}_{\mathrm{NLO}_{\mathrm{EW}}} YFS NLO EW + 1 +1 + 1 − 1 -1 − 1 Y F S N L O E W \mathrm{YFS}_{\mathrm{NLO}_{\mathrm{EW}}} YFS NLO EW 1 + ν 1+\nu 1 + ν ν = β ~ 0 / β ~ 0 \nu=\widetilde{\beta}_0/\widetilde{\beta}_0 ν = β 0 / β 0 1 + ν 1+\nu 1 + ν V = V ‾ = v / 2 \mathcal{V}=\overline{\mathcal{V}}=v/2 V = V = v /2 (31) gives, per event,
∑ { λ } ∣ ( 1 + V ) M 0 ∣ 2 ∑ { λ } ∣ M 0 ∣ 2 = ( 1 + v / 2 ) 2 = 1 + v + v 2 4 , (59) \frac{\sum_{\{\lambda\}} |(1+\mathcal{V})\mathcal{M}_0|^2}{\sum_{\{\lambda\}}|\mathcal{M}_0|^2}=(1+v/2)^2=1+v+\frac{v^2}{4}, \tag*{(59)} ∑ { λ } ∣ M 0 ∣ 2 ∑ { λ } ∣ ( 1 + V ) M 0 ∣ 2 = ( 1 + v /2 ) 2 = 1 + v + 4 v 2 , (59) w A C R A I C w Y F S N L O E W = 1 + v 2 4 ( 1 + v ) . (60) \frac{w_{\mathrm{ACRAIC}}}{w_{\mathrm{YFSNLOEW}}}=1+\frac{v^2}{4(1+v)}. \tag*{(60)} w YFSNLOEW w ACRAIC = 1 + 4 ( 1 + v ) v 2 . (60) the ∑ { λ } ∣ β ^ 0 ( 1 ) ∣ 2 \sum_{\{\lambda\}}|\widehat{\beta}^{(1)}_0|^2 ∑ { λ } ∣ β 0 ( 1 ) ∣ 2 (14) that the squared-level matching does not carry. The term is of O ( α 2 ) O(\alpha^2) O ( α 2 ) (27) are tested point by point against the exact tree amplitude. On events with n ≤ 5 n\leq5 n ≤ 5 β ^ \widehat{\beta} β (27) are evaluated for J = 0 , … , n J=0,\ldots,n J = 0 , … , n n n n J = n J=n J = n n n n Figure 5 . Below J = n J=n J = n
Figure 5. Truncation of the residuals. Left and middle: the partition- and helicity-summed ∣ C J ∣ 2 |C_J|^2 ∣ C J ∣ 2 (27) against the exact n n n ∣ M n ∣ 2 |\mathcal{M}_n|^2 ∣ M n ∣ 2 e + e − → μ + μ − + n γ e^+e^- \to\mu^+\mu^- + n\gamma e + e − → μ + μ − + nγ s = M Z \sqrt{s}=M_Z s = M Z L J \mathcal{L}_J L J M n [ 1 ] \mathcal{M}_n^{[1]} M n [ 1 ] e + e − → ν μ ν ˉ μ + n γ e^+e^- \to\nu_\mu\bar{\nu}_\mu+ n\gamma e + e − → ν μ ν ˉ μ + nγ s = M Z \sqrt{s}=M_Z s = M Z
The one-loop residuals close in the same way. On e + e − → ν μ ν ˉ μ + n γ e^+e^-\to\nu_\mu\bar{\nu}_\mu+n\gamma e + e − → ν μ ν ˉ μ + nγ Z Z Z n = 1 n=1 n = 1 Figure 5 compares the interference of the one-loop truncation L J \mathcal{L}_J L J M n [ 1 ] \mathcal{M}^{[1]}_n M n [ 1 ] J = n J=n J = n β ^ 2 [ 1 ] \widehat{\beta}^{[1]}_2 β 2 [ 1 ] β ^ 3 [ 1 ] \widehat{\beta}^{[1]}_3 β 3 [ 1 ]
In Figure 4 we compare ACRAIC at photonic O ( α 2 ) O(\alpha^2) O ( α 2 ) β ^ 0 ( 2 ) \widehat{\beta}^{(2)}_0 β 0 ( 2 ) O ( α 2 ) C E E X O(\alpha^2)_{\mathrm{CEEX}} O ( α 2 ) CEEX Z Z Z O ( α 2 ) C E E X O(\alpha^2)_{\mathrm{CEEX}} O ( α 2 ) CEEX 0.5 × 10 − 3 0.5\times10^{-3} 0.5 × 1 0 − 3 2.6 ( 6 ) × 10 − 3 2.6(6)\times10^{-3} 2.6 ( 6 ) × 1 0 − 3 2.5 ( 5 ) × 10 − 3 2.5(5)\times10^{-3} 2.5 ( 5 ) × 1 0 − 3 Z Z Z 0.5 × 10 − 3 0.5\times10^{-3} 0.5 × 1 0 − 3 0.6 ( 2 ) × 10 − 3 0.6(2)\times10^{-3} 0.6 ( 2 ) × 1 0 − 3 Z Z Z
Figure 4. ACRAIC at photonic O ( α 2 ) \mathcal{O}(\alpha^2) O ( α 2 ) β ^ 0 ( 2 ) \hat{\beta}_0^{(2)} β ^ 0 ( 2 ) O ( α 2 ) C E E X \mathcal{O}(\alpha^2)_{\mathrm{CEEX}} O ( α 2 ) CEEX e + e − → μ + μ − e^+e^- \to\mu^+\mu^- e + e − → μ + μ − s = M Z \sqrt{s}=M_Z s = M Z cos θ ∗ \cos\theta^* cos θ ∗ ± 10 − 3 \pm10^{-3} ± 1 0 − 3
Table IV compares ACRAIC with YFSNLOEW for thirteen processes at the Z Z Z G μ G_\mu G μ α ( 0 ) \alpha(0) α ( 0 ) Table III in the complex-mass scheme [104 , 105 ] . Both columns carry the same complete electroweak one-loop virtual. The cross sections are fiducial, using generic cuts applied to final-state particles,
Table III. Electroweak input parameters, SHERPA’s defaults. Γ τ = 0 \Gamma_\tau=0 Γ τ = 0 Γ H = 0 \Gamma_H=0 Γ H = 0 u u ˉ u\bar{u} u u ˉ d d ˉ d\bar{d} d d ˉ
Table IV. Fiducial cross sections at O ( α ) \mathcal{O}(\alpha) O ( α ) Table III . Stable top quarks, no fiducial cuts (σ f i d = σ t o t \sigma_{\mathrm{fid}}=\sigma_{\mathrm{tot}} σ fid = σ tot
∣ cos θ ∣ < 0.9 |\cos\theta|<0.9 ∣ cos θ ∣ < 0.9
60 G e V < m f f ˉ 60\,\mathrm{GeV}<m_{f\bar f} 60 GeV < m f f ˉ Z Z Z
66 G e V < m ℓ ℓ < 116 G e V 66\,\mathrm{GeV}<m_{\ell\ell}<116\,\mathrm{GeV} 66 GeV < m ℓℓ < 116 GeV
E γ > 0.3 s , ∣ cos θ γ ∣ < 0.85 E_\gamma>0.3\sqrt{s},\ |\cos\theta_\gamma|<0.85 E γ > 0.3 s , ∣ cos θ γ ∣ < 0.85
Equation not extracted
(61) The angle θ ∗ \theta^* θ ∗ f f ˉ f\bar f f f ˉ [106 ] . Without initial-state radiation it is the angle to the electron beam. YFSLO is the Born-level of the same resummation, without O ( α ) O(\alpha) O ( α ) O ( α ) O(\alpha) O ( α )
At the Z Z Z γ γ \gamma\gamma γ γ γ γ \gamma\gamma γ γ H ν ν ˉ H\nu\bar{\nu} H ν ν ˉ H μ + μ − H\mu^+\mu^- H μ + μ − τ + τ − ν μ ν ˉ μ \tau^+\tau^-\nu_\mu\bar{\nu}_\mu τ + τ − ν μ ν ˉ μ μ + μ − τ + τ − \mu^+\mu^-\tau^+\tau^- μ + μ − τ + τ − 1.3 ( 5 ) % 1.3(5)\% 1.3 ( 5 ) % τ + τ − \tau^+\tau^- τ + τ − 2.6 ( 8 ) % 2.6(8)\% 2.6 ( 8 ) % § 2.2 , a few large weights dominate the YFSNLOEW column of these four-lepton final states, which we attribute to poor numerical convergence rather than a defect of either prediction. μ + μ − ν ν ˉ \mu^+\mu^-\nu\bar{\nu} μ + μ − ν ν ˉ W W W
The Z Z Z Table IV differ from one by up to 4 × 10 − 3 4\times10^{-3} 4 × 1 0 − 3 Z Z Z Table IV , and split the difference into three pieces (Table V ),
Table V. The difference T T T (62) of the fiducial cross sections of Table IV , split into the exponentiated resonant logarithm (G \mathcal{G} G V V V R R R 10 − 4 10^{-4} 1 0 − 4 G ( V + R ) < 0.1 × 10 − 4 \mathcal{G}(V+R)<0.1\times10^{-4} G ( V + R ) < 0.1 × 1 0 − 4
T = σ A C R A I C σ Y F S N L O E W − 1 = G + V + R . (62) T=\frac{\sigma_{\mathrm{ACRAIC}}}{\sigma_{\mathrm{YFSNLOEW}}}-1=\mathcal{G}+\mathcal{V}+\mathcal{R}. \tag*{(62)} T = σ YFSNLOEW σ ACRAIC − 1 = G + V + R . (62) G \mathcal{G} G (35) exponentiated into the propagators, the ratio of the two ACRAIC runs. V V V v 2 / 4 v^2/4 v 2 /4 (60) , which the squared-level matching does not carry. R R R G G G V V V G G G V V V O ( α 2 ) \mathcal{O}(\alpha^2) O ( α 2 ) R R R O ( α ) \mathcal{O}(\alpha) O ( α ) Z Z Z R R R 10 − 4 10^{-4} 1 0 − 4 e + e − e^+e^- e + e −
− 1.8 ( 5 ) × 10 − 4 -1.8(5) \times10^{-4} − 1.8 ( 5 ) × 1 0 − 4 2 × 10 − 4 2 \times10^{-4} 2 × 1 0 − 4 G G G V V V T T T Table IV therefore measure how far two matchings that coincide at O ( α ) \mathcal{O}(\alpha) O ( α ) O ( α 2 ) \mathcal{O}(\alpha^2) O ( α 2 ) V V V v v v Z Z Z G μ G_\mu G μ O ( α ) \mathcal{O}(\alpha) O ( α ) ( v / 2 ) 2 (v/2)^2 ( v /2 ) 2 O ( α ) \mathcal{O}(\alpha) O ( α ) O ( α ) \mathcal{O}(\alpha) O ( α ) Table VI gives the strict truncation, with the virtual on C 0 C_0 C 0 ( v / 2 ) 2 (v/2)^2 ( v /2 ) 2 Z Z Z 0.3 0.3 0.3 1.6 × 10 − 3 1.6\times10^{-3} 1.6 × 1 0 − 3 H ν ν ˉ H\nu\bar{\nu} H ν ν ˉ 2.5 × 10 − 3 2.5\times10^{-3} 2.5 × 1 0 − 3 2.5 × 10 − 4 2.5\times10^{-4} 2.5 × 1 0 − 4 ν ν ˉ \nu\bar{\nu} ν ν ˉ u u ˉ u\bar{u} u u ˉ H ν ν ˉ H\nu\bar{\nu} H ν ν ˉ d d ˉ d\bar{d} d d ˉ γ γ \gamma\gamma γ γ H μ + μ − H\mu^{+}\mu^{-} H μ + μ − 1 1 1 4 × 10 − 3 4\times10^{-3} 4 × 1 0 − 3 G \mathcal{G} G Table V .
Table VI. The strict O ( α ) O(\alpha) O ( α ) C 0 C_0 C 0 ( v / 2 ) 2 (v/2)^2 ( v /2 ) 2 O ( α ) O(\alpha) O ( α ) Table IV , which has the virtual on the whole O ( α ) O(\alpha) O ( α ) Table IV .
ACRAIC carries the interference between initial- and final-state emission in its coherent partition sum, photon by photon. The naive YFSNLOEW keeps it only in the initial-final form factor, which is integrated over the photon phase space and therefore over its direction. The interference is antisymmetric in the fermion angle, so its integral survives in the cross section but its angular dependence is lost. YFSNLOEW can instead reweight every photon by the ratio of the full eikonal factor, initial-final terms included, to the one it was generated with. We label this variant “+ IF real” and it is the one used for Table IV . Against ACRAIC on the five Z Z Z Figure 6 , with and without the reweighting.
Figure 6. The effect of initial-final interference on the muon angle, with and without the reweighting.
For processes with photons in the hard process the Born amplitude already carries a hard photon, and the residuals and the partition sum are built on top of it. Table VII compares the two columns for this family, which comprises the 2 → 3 2\to3 2 → 3 Z Z Z γ γ γ \gamma\gamma\gamma γ γ γ H f f ˉ γ Hf\bar{f}\gamma H f f ˉ γ
Table VII. Fiducial cross sections with a Born-level photon at O ( α ) O(\alpha) O ( α ) § 2.2 . † ^{\dagger} † β ˉ 0 \bar{\beta}_0 β ˉ 0
Z Z Z E γ > 0.2 s , 60 G e V < m f f \quad E_{\gamma}>0.2\sqrt{s},\quad60\,\mathrm{GeV}<m_{f f} E γ > 0.2 s , 60 GeV < m f f
240 240 240 E γ > 10 G e V , 66 < m μ μ < 116 G e V , \quad E_{\gamma}>10\,\mathrm{GeV},\quad66<m_{\mu\mu}<116\,\mathrm{GeV}, E γ > 10 GeV , 66 < m μμ < 116 GeV ,
both: ∣ cos θ γ ∣ < 0.95 , θ γ f > 0.12 , \quad|\cos\theta_{\gamma}|<0.95,\quad\theta_{\gamma f}>0.12, ∣ cos θ γ ∣ < 0.95 , θ γ f > 0.12 ,
γ γ \gamma\gamma γ γ E γ i > 0.3 s , ∣ cos θ γ i ∣ < 0.85 \quad E_{\gamma_i}>0.3\sqrt{s},\quad|\cos\theta_{\gamma_i}|<0.85 E γ i > 0.3 s , ∣ cos θ γ i ∣ < 0.85
with the pair selected as in Table IV , and ν ν ˉ γ \nu\bar{\nu}\gamma ν ν ˉ γ γ γ γ \gamma\gamma\gamma γ γ γ m f f m_{ff} m f f Table VII ).
The O ( α 2 ) O(\alpha^2) O ( α 2 )
The second-order predictions contain the photonic corrections of O ( α 2 ) O(\alpha^2) O ( α 2 ) γ Z \gamma Z γ Z O ( α 2 ) O(\alpha^2) O ( α 2 ) O ( α 2 ) O(\alpha^2) O ( α 2 ) O ( α 2 ) O(\alpha^2) O ( α 2 ) O ( α ) O(\alpha) O ( α ) O ( α 2 ) O(\alpha^2) O ( α 2 ) Table VIII gives the result for the Z Z Z γ γ \gamma\gamma γ γ + 0.36 % +0.36\% + 0.36% + 0.41 % +0.41\% + 0.41% Table VIII , footnote b), so the two steps are not contributions of quite the same physical content. For H f f ˉ Hf\bar{f} H f f ˉ Table VIII, footnote a), so the two ratios it enters, for H μ + μ − H\mu^{+}\mu^{-} H μ + μ − H ν ν ˉ H\nu\bar{\nu} H ν ν ˉ O ( α ) O(\alpha) O ( α ) O ( α ) O(\alpha) O ( α ) β ˉ 0 2 \bar{\beta}_0^2 β ˉ 0 2 2 → 2 2\to2 2 → 2 Z Z Z Z Z Z Figure 7 shows the differential distributions of ACRAIC at photonic O ( α 2 ) O(\alpha^2) O ( α 2 ) O ( α ) O(\alpha) O ( α )
Table VIII. The fiducial cross sections of YFSN L O E W _{\mathrm{NLOEW}} NLOEW O ( α ) \mathcal{O}(\alpha) O ( α ) n N L O E W _{\mathrm{nNLOEW}} nNLOEW O ( α 2 ) \mathcal{O}(\alpha^2) O ( α 2 ) 2 → 2 2 \to2 2 → 2 O ( α ) \mathcal{O}(\alpha) O ( α ) O ( α ) \mathcal{O}(\alpha) O ( α ) O ( α 2 ) \mathcal{O}(\alpha^2) O ( α 2 ) β ^ 0 ( 2 ) \hat{\beta}_0^{(2)} β ^ 0 ( 2 ) Z Z Z O ( α 2 ) \mathcal{O}(\alpha^2) O ( α 2 ) H f f ˉ Hf\bar{f} H f f ˉ a H f f ˉ ^a Hf\bar{f} a H f f ˉ n N L O E W _{\mathrm{nNLOEW}} nNLOEW b γ γ ^b \gamma\gamma b γ γ n N L O E W _{\mathrm{nNLOEW}} nNLOEW
Figure 7. ACRAIC at photonic O ( α 2 ) \mathcal{O}(\alpha^{2}) O ( α 2 ) β ^ 0 ( 2 ) \hat{\beta}_{0}^{(2)} β ^ 0 ( 2 ) O ( α ) \mathcal{O}(\alpha) O ( α ) Table VIII , as a ratio to O ( α 2 ) \mathcal{O}(\alpha^{2}) O ( α 2 ) x γ = 2 E γ / s x_{\gamma}=2E_{\gamma}/\sqrt{s} x γ = 2 E γ / s e + e − → μ + μ − e^{+}e^{-}\to\mu^{+}\mu^{-} e + e − → μ + μ −
The O ( α ) \mathcal{O}(\alpha) O ( α ) O ( α 2 ) \mathcal{O}(\alpha^2) O ( α 2 ) μ + μ − \mu^+\mu^- μ + μ − μ + μ − \mu^+\mu^- μ + μ − μ − \mu^- μ − β ^ 0 ( 2 ) \hat{\beta}_0^{(2)} β ^ 0 ( 2 ) O ( α 2 ) \mathcal{O}(\alpha^2) O ( α 2 ) Table VIII . Table IX gives its pieces for μ + μ − \mu^+\mu^- μ + μ − Z Z Z O ( α 1 ) C E E X \mathcal{O}(\alpha^1)_{\mathrm{CEEX}} O ( α 1 ) CEEX O ( α 2 ) C E E X \mathcal{O}(\alpha^2)_{\mathrm{CEEX}} O ( α 2 ) CEEX O ( α 1 ) C E E X \mathcal{O}(\alpha^1)_{\mathrm{CEEX}} O ( α 1 ) CEEX ( 1 + F i ) ( 1 + F f ) (1+F_i)(1+F_f) ( 1 + F i ) ( 1 + F f )
Table IX. The two-loop photonic β ^ 0 ( 2 ) \hat{\beta}_0^{(2)} β ^ 0 ( 2 ) e + e − → μ + μ − e^+e^- \to\mu^+\mu^- e + e − → μ + μ − Z Z Z Table VIII . Top: its pieces, each the shift of the ACRAIC fiducial cross section on the same events with and without it, against the step KKMC adds between O ( α 1 ) C E E X \mathcal{O}(\alpha^1)_{\mathrm{CEEX}} O ( α 1 ) CEEX O ( α 2 ) C E E X \mathcal{O}(\alpha^2)_{\mathrm{CEEX}} O ( α 2 ) CEEX O ( α ) → O ( α 2 ) \mathcal{O}(\alpha) \to\mathcal{O}(\alpha^2) O ( α ) → O ( α 2 ) O ( α ) \mathcal{O}(\alpha) O ( α ) O ( α 1 ) C E E X \mathcal{O}(\alpha^1)_{\mathrm{CEEX}} O ( α 1 ) CEEX O ( α ) \mathcal{O}(\alpha) O ( α )
in our O ( α ) \mathcal{O}(\alpha) O ( α ) [84 ] beyond its logarithms and the non-factorizable two-loop boxes of photon exchange from the helicity amplitudes of [89 ] , are not in KKMC. For γ γ \gamma\gamma γ γ (50) changes the cross section by − 3 × 10 − 5 -3 \times10^{-5} − 3 × 1 0 − 5 m f f m_{f f} m f f O ( α ) \mathcal{O}(\alpha) O ( α ) C 0 C_0 C 0 β ^ 0 ( 2 ) \hat{\beta}_0^{(2)} β ^ 0 ( 2 ) O ( α ) \mathcal{O}(\alpha) O ( α ) β ^ 1 \hat{\beta}_1 β ^ 1
For ν ν ˉ \nu\bar{\nu} ν ν ˉ 1.3 1.3 1.3 1.5 × 10 − 3 1.5 \times10^{-3} 1.5 × 1 0 − 3 0.2 % 0.2\% 0.2% O ( α ) \mathcal{O}(\alpha) O ( α ) 0.16 % 0.16\% 0.16% O ( α 2 ) \mathcal{O}(\alpha^2) O ( α 2 )
Table X gives the budget of the fiducial cross sections at the Z Z Z O ( α ) \mathcal{O}(\alpha) O ( α ) 1.5 1.5 1.5 2.2 × 10 − 3 2.2 \times10^{-3} 2.2 × 1 0 − 3 μ + μ − \mu^+\mu^- μ + μ − τ + τ − \tau^+\tau^- τ + τ − 1.5 × 10 − 3 1.5 \times10^{-3} 1.5 × 1 0 − 3 ν ν ˉ \nu\bar{\nu} ν ν ˉ d d ˉ d\bar{d} d d ˉ β ^ 0 \hat{\beta}_0 β ^ 0 β ^ 0 ≃ 2 \hat{\beta}_0 \simeq2 β ^ 0 ≃ 2 4 × 10 − 3 4 \times10^{-3} 4 × 1 0 − 3 e + e − e^+e^- e + e − t t t β ^ 0 \hat{\beta}_0 β ^ 0 β ^ 1 \hat{\beta}_1 β ^ 1 1.0 1.0 1.0 1.6 × 10 − 3 1.6 \times10^{-3} 1.6 × 1 0 − 3 μ + μ − \mu^+\mu^- μ + μ − τ + τ − \tau^+\tau^- τ + τ − 1.3 × 10 − 3 1.3 \times10^{-3} 1.3 × 1 0 − 3 ν ν ˉ \nu\bar{\nu} ν ν ˉ d d ˉ d\bar{d} d d ˉ 2.9 × 10 − 3 2.9 \times10^{-3} 2.9 × 1 0 − 3 e + e − e^+e^- e + e − ∣ θ ∣ ≃ 0.65 |\theta| \simeq0.65 ∣ θ ∣ ≃ 0.65 0.9 0.9 0.9 β ^ 0 ( 2 ) \hat{\beta}_0^{(2)} β ^ 0 ( 2 ) ∣ θ ∣ ≃ 0.8 |\theta| \simeq0.8 ∣ θ ∣ ≃ 0.8 0.9 0.9 0.9 β ^ 0 \hat{\beta}_0 β ^ 0 μ + μ − \mu^+\mu^- μ + μ − τ + τ − \tau^+\tau^- τ + τ − ν ν ˉ \nu\bar{\nu} ν ν ˉ 0.65 0.65 0.65 d d ˉ d\bar{d} d d ˉ 1.05 1.05 1.05 e + e − e^+e^- e + e − ∣ θ ∣ ≲ 1 |\theta| \lesssim1 ∣ θ ∣ ≲ 1 ν ν ˉ \nu\bar{\nu} ν ν ˉ 1.4 1.4 1.4 β ^ 1 \hat{\beta}_1 β ^ 1 β ^ 1 \hat{\beta}_1 β ^ 1 10 − 6 10^{-6} 1 0 − 6 x γ x_\gamma x γ β ^ 0 ( 2 ) \hat{\beta}_0^{(2)} β ^ 0 ( 2 )
Table X. Uncertainty budget of the fiducial cross sections at the Z Z Z 10 − 3 10^{-3} 1 0 − 3 σ f i d \sigma_{\mathrm{fid}} σ fid O ( α ) \mathcal{O}(\alpha) O ( α ) O ( α 2 ) \mathcal{O}(\alpha^2) O ( α 2 ) β ^ 0 ( 2 ) \hat{\beta}_0^{(2)} β ^ 0 ( 2 ) O ( α ) → O ( α 2 ) \mathcal{O}(\alpha)\to\mathcal{O}(\alpha^2) O ( α ) → O ( α 2 ) ∣ θ ∣ |\theta| ∣ θ ∣ O ( α ) \mathcal{O}(\alpha) O ( α ) β ^ 0 ( 2 ) \hat{\beta}_0^{(2)} β ^ 0 ( 2 ) σ f i d \sigma_{\mathrm{fid}} σ fid α ( M Z ) \alpha(M_Z) α ( M Z ) G μ G_\mu G μ 2 → 2 2\to2 2 → 2
ends_mid=1 the O ( α 2 ) \mathcal{O}(\alpha^{2}) O ( α 2 ) 10 − 5 10^{-5} 1 0 − 5 1.6 × 10 − 4 1.6\times10^{-4} 1.6 × 1 0 − 4 e + e − e^{+}e^{-} e + e − β ^ 0 ( 3 ) \hat{\beta}_{0}^{(3)} β ^ 0 ( 3 ) μ + μ − \mu^{+}\mu^{-} μ + μ − β ^ 1 \hat{\beta}_{1} β ^ 1 β ^ 2 \hat{\beta}_{2} β ^ 2 10 − 5 10^{-5} 1 0 − 5 β ^ 1 \hat{\beta}_{1} β ^ 1 e + e − e^{+}e^{-} e + e − 4 × 10 − 5 4\times10^{-5} 4 × 1 0 − 5
Figure 8 repeats this across the Z Z Z 88 ≤ s ≤ 94 G e V 88\leq\sqrt{s}\leq94\ \mathrm{GeV} 88 ≤ s ≤ 94 GeV e + e − → μ + μ − e^{+}e^{-}\to\mu^{+}\mu^{-} e + e − → μ + μ − Table VIII , with β ^ 0 ( 2 ) \hat{\beta}_{0}^{(2)} β ^ 0 ( 2 ) O ( α 2 ) \mathcal{O}(\alpha^{2}) O ( α 2 ) O ( α ) \mathcal{O}(\alpha) O ( α ) 2.2 × 10 − 3 2.2\times10^{-3} 2.2 × 1 0 − 3 1.5 1.5 1.5 2.1 × 10 − 3 2.1\times10^{-3} 2.1 × 1 0 − 3 ∣ θ ∣ |\theta| ∣ θ ∣ 0.7 0.7 0.7 1 1 1 88 G e V 88\ \mathrm{GeV} 88 GeV β ^ 0 ( 2 ) \hat{\beta}_{0}^{(2)} β ^ 0 ( 2 ) β ^ 0 \hat{\beta}_{0} β ^ 0 O ( α 2 ) \mathcal{O}(\alpha^{2}) O ( α 2 ) 7 × 10 − 5 7\times10^{-5} 7 × 1 0 − 5 240 G e V 240\ \mathrm{GeV} 240 GeV Table XI ) the O ( α ) \mathcal{O}(\alpha) O ( α ) 1.7 1.7 1.7 2.7 × 10 − 3 2.7\times10^{-3} 2.7 × 1 0 − 3 10 − 3 10^{-3} 1 0 − 3 ∣ θ ∣ ≈ 0.35 |\theta|\approx0.35 ∣ θ ∣ ≈ 0.35 H μ + μ − H\mu^{+}\mu^{-} H μ + μ − 0.6 0.6 0.6 H ν ν ˉ H\nu\bar{\nu} H ν ν ˉ β ^ 0 ( 2 ) \hat{\beta}_{0}^{(2)} β ^ 0 ( 2 ) H μ + μ − H\mu^{+}\mu^{-} H μ + μ − H ν ν ˉ H\nu\bar{\nu} H ν ν ˉ O ( α 2 ) \mathcal{O}(\alpha^{2}) O ( α 2 ) 10 − 4 10^{-4} 1 0 − 4 H μ + μ − H\mu^{+}\mu^{-} H μ + μ − − 1.6 ( 4 ) × 10 − 2 -1.6(4)\times10^{-2} − 1.6 ( 4 ) × 1 0 − 2 Table XI , is about 8 × 10 − 3 8\times10^{-3} 8 × 1 0 − 3 O ( α 2 ) \mathcal{O}(\alpha^{2}) O ( α 2 ) e + e − → Z H e^{+}e^{-}\to ZH e + e − → Z H
Figure 8. The Z Z Z e + e − → μ + μ − e^+e^- \to\mu^+\mu^- e + e − → μ + μ − Table VIII : ACRAIC at photonic O ( α 2 ) \mathcal{O}(\alpha^2) O ( α 2 ) O ( α ) \mathcal{O}(\alpha) O ( α ) O ( α 2 ) \mathcal{O}(\alpha^2) O ( α 2 ) O ( α 2 ) \mathcal{O}(\alpha^2) O ( α 2 )
Table XI. Uncertainty budget of the fiducial cross sections at 240 GeV, in units of 10 − 3 10^{-3} 1 0 − 3 σ f i d \sigma_{\mathrm{fid}} σ fid O ( α ) \mathcal{O}(\alpha) O ( α ) O ( α 2 ) \mathcal{O}(\alpha^{2}) O ( α 2 ) β ^ 0 ( 2 ) \hat{\beta}_{0}^{(2)} β ^ 0 ( 2 ) O ( α ) → O ( α 2 ) \mathcal{O}(\alpha)\to\mathcal{O}(\alpha^{2}) O ( α ) → O ( α 2 ) ∣ θ ∣ |\theta| ∣ θ ∣ O ( α ) \mathcal{O}(\alpha) O ( α ) β ^ 0 ( 2 ) \hat{\beta}_{0}^{(2)} β ^ 0 ( 2 ) σ f i d \sigma_{\mathrm{fid}} σ fid α ( M Z ) \alpha(M_Z) α ( M Z ) G μ G_\mu G μ
The initial-state part of the terms the O ( α 2 ) \mathcal{O}(\alpha^{2}) O ( α 2 ) O ( α 2 ) \mathcal{O}(\alpha^{2}) O ( α 2 ) [107 ] . For photon emission alone the renormalization-group structure of the radiator [108 ] then fixes its leading ( α 3 L 3 ) (\alpha^{3}L^{3}) ( α 3 L 3 ) ( α 3 L 2 ) (\alpha^{3}L^{2}) ( α 3 L 2 ) O ( α 3 ) \mathcal{O}(\alpha^{3}) O ( α 3 ) O ( α ) \mathcal{O}(\alpha) O ( α ) O ( α 2 ) \mathcal{O}(\alpha^{2}) O ( α 2 ) L = ln ( s / m e 2 ) L=\ln(s/m_{e}^{2}) L = ln ( s / m e 2 ) O ( α 2 ) \mathcal{O}(\alpha^{2}) O ( α 2 ) Figure 9 ). The test covers initial-state radiation only. For e + e − → μ + μ − e^+e^- \to\mu^+\mu^- e + e − → μ + μ − Z Z Z 3.4 × 10 − 5 3.4 \times10^{-5} 3.4 × 1 0 − 5 4.6 × 10 − 5 4.6 \times10^{-5} 4.6 × 1 0 − 5 m μ μ m_{\mu\mu} m μμ 6.9 × 10 − 5 6.9 \times10^{-5} 6.9 × 1 0 − 5 σ f i d \sigma_{\mathrm{fid}} σ fid ∣ θ ∣ ≈ 0.5 \lvert\theta\rvert\approx0.5 ∣ θ ∣ ≈ 0.5 0.7 0.7 0.7 × \times × β ^ 0 \hat{\beta}_0 β ^ 0 e + e − → ν ν ˉ e^+e^- \to\nu\bar{\nu} e + e − → ν ν ˉ 2.6 × 10 − 5 2.6 \times10^{-5} 2.6 × 1 0 − 5 β ^ 0 ( 3 ) \hat{\beta}_0^{(3)} β ^ 0 ( 3 ) 3.4 × 10 − 5 3.4 \times10^{-5} 3.4 × 1 0 − 5 4.6 × 10 − 5 4.6 \times10^{-5} 4.6 × 1 0 − 5 ∣ θ ∣ ≈ 1.3 \lvert\theta\rvert\approx1.3 ∣ θ ∣ ≈ 1.3 1.8 1.8 1.8 β ^ 1 \hat{\beta}_1 β ^ 1 Z Z Z O ( α 2 ) \mathcal{O}(\alpha^2) O ( α 2 ) Z Z Z 1 1 1 2 × 10 − 3 2 \times10^{-3} 2 × 1 0 − 3 σ f i d \sigma_{\mathrm{fid}} σ fid 2.1 × 10 − 3 2.1 \times10^{-3} 2.1 × 1 0 − 3 μ + μ − \mu^+\mu^- μ + μ − Table X , and about 3 × 10 − 3 3 \times10^{-3} 3 × 1 0 − 3 A F B A_{\mathrm{FB}} A FB M Z M_Z M Z O ( α ) \mathcal{O}(\alpha) O ( α ) O ( α 2 ) \mathcal{O}(\alpha^2) O ( α 2 ) Figure 8 applies it as a correction, Table X does not. With the correction applied, the electroweak band of Figure 8 is the estimate of its missing higher orders, which is below the O ( α ) \mathcal{O}(\alpha) O ( α ) Table X , σ f i d \sigma_{\mathrm{fid}} σ fid α ( M Z ) \alpha(M_Z) α ( M Z ) G μ G_\mu G μ − 7.6 -7.6 − 7.6
Figure 9. The ACRAIC O ( α 2 ) \mathcal{O}(\alpha^{2}) O ( α 2 ) O ( α 3 ) \mathcal{O}(\alpha^{3}) O ( α 3 ) O ( α 2 ) \mathcal{O}(\alpha^{2}) O ( α 2 ) [107 , 108 ] , relative to the cross section at s = M Z \sqrt{s}=M_Z s = M Z e + e − → μ + μ − e^{+}e^{-}\to\mu^{+}\mu^{-} e + e − → μ + μ − m μ μ > m μ μ min m_{\mu\mu}>m_{\mu\mu}^{\min} m μμ > m μμ m i n Table VIII otherwise. Right, the fiducial cross-sections for μ + μ − \mu^{+}\mu^{-} μ + μ − ν μ ν ˉ μ \nu_{\mu}\bar{\nu}_{\mu} ν μ ν ˉ μ
to − 10 × 10 − 3 -10 \times10^{-3} − 10 × 1 0 − 3 − 2.5 × 10 − 3 -2.5 \times10^{-3} − 2.5 × 1 0 − 3 O ( α ) \mathcal{O}(\alpha) O ( α ) O ( α 2 ) \mathcal{O}(\alpha^2) O ( α 2 ) β ^ 0 ( 2 ) \hat{\beta}_0^{(2)} β ^ 0 ( 2 ) Z Z Z
The Higgs recoil mass at FCC-ee
As an example application of the theory-nuisance-parameter machinery of Sec. IV A, rather than a further validation, we consider the Higgs recoil mass, which a dedicated phenomenological study could develop into a full precision analysis. At FCC-ee the Higgs boson mass will be measured from the mass recoiling against the lepton pair in e + e − → Z ( → ℓ + ℓ − ) H e^+e^- \to Z(\to\ell^+\ell^-)H e + e − → Z ( → ℓ + ℓ − ) H 4 4 4 [109 , 110 ] . The recoil mass is computed from the nominal collision energy, so every photon that escapes, initial-state radiation above all, and the spread of the beam energies, move events out of the peak at M H M_H M H
ACRAIC describes it event by event. Every photon is generated exclusively, and initial- and final-state emission and their interference are summed coherently at amplitude level, with the soft logarithms exponentiated to all orders. For e + e − → H μ + μ − e^+e^-\to H\mu^+\mu^- e + e − → H μ + μ − O ( α 2 ) \mathcal{O}(\alpha^2) O ( α 2 )
the complete one-loop electroweak corrections to the Born (Sec. III B) applied to the whole O ( α ) \mathcal{O}(\alpha) O ( α )
the exact tree amplitudes with one and with two photons, the latter through β ^ 2 \hat{\beta}_2 β ^ 2 (26) ,
the real-virtual, the one-loop correction to the emission of one photon, averaged over helicities (Sec. III B),
the photonic two-loop β ^ 0 ( 2 ) \hat{\beta}_0^{(2)} β ^ 0 ( 2 )
the resonant initial-final logarithm of the Z Z Z (35) ),
the beam-energy spread of FCC-ee (Appendix B).
The photonic β ^ 0 ( 2 ) \hat{\beta}_0^{(2)} β ^ 0 ( 2 ) V ( 2 ) \mathcal{V}^{(2)} V ( 2 ) f i j f_{ij} f ij e + e − e^+e^- e + e − μ + μ − \mu^+\mu^- μ + μ − w w w
V f = ∑ i < j f i j , (63) \mathcal{V}_f=\sum_{i<j} f_{ij}, \tag*{(63)} V f = i < j ∑ f ij , (63) V ( 2 ) = 1 2 V f 2 + V f w + V N L L + C V . (64) \mathcal{V}^{(2)}=\frac{1}{2}\mathcal{V}_f^2+\mathcal{V}_f w+\mathcal{V}_{\mathrm{NLL}}+C_{\mathrm{V}}. \tag*{(64)} V ( 2 ) = 2 1 V f 2 + V f w + V NLL + C V . (64) The first term carries the leading logarithms of the two-loop photonic corrections, including the product of the initial- and final-state vertices. The second is the product of the photonic and the weak one-loop corrections. V N L L \mathcal{V}_{\mathrm{NLL}} V NLL Table IX . C V C_{\mathrm{V}} C V e + e − e^+e^- e + e − μ + μ − \mu^+\mu^- μ + μ − [84 ] . The prediction does not contain
μ + μ − \mu^{+}\mu^{-} μ + μ − Z Z Z σ f i d \sigma_{\mathrm{fid}} σ fid 10 − 4 10^{-4} 1 0 − 4 Table IX ),
the remaining photonic two-loop terms without large logarithms, beyond the vertex constants, and the photonic corrections of third order, which the O ( α 2 ) \mathcal{O}(\alpha^{2}) O ( α 2 ) 10 − 4 10^{-4} 1 0 − 4
the two-loop electroweak and the mixed electroweak–QCD corrections, which the electroweak band of about 1% estimates (Sec. IV A),
light-fermion pairs.
Figure 10 shows the peak for e + e − → H μ + μ − e^{+}e^{-}\to H\mu^{+}\mu^{-} e + e − → H μ + μ − Z H ZH Z H M H M_H M H Z H ZH Z H [9 ] , 0.165% per beam, smears the edge by about 0.3 GeV and moves a fifth of the events below M H M_H M H O ( α ) \mathcal{O}(\alpha) O ( α ) O ( α 2 ) \mathcal{O}(\alpha^{2}) O ( α 2 ) 10 − 2 10^{-2} 1 0 − 2 10 − 2 10^{-2} 1 0 − 2 O ( α ) \mathcal{O}(\alpha) O ( α ) 10 − 3 10^{-3} 1 0 − 3 O ( α 2 ) \mathcal{O}(\alpha^{2}) O ( α 2 ) Figure 7 ).
Figure 10. The recoil mass in e + e − → H μ + μ − e^+e^- \to H\mu^+\mu^- e + e − → H μ + μ − s = 240 \sqrt{s}=240 s = 240 O ( α 2 ) O(\alpha^2) O ( α 2 ) Table VIII . Left: without and with the beam-energy spread of FCC-ee (0.165% per beam [9 ] ). Right: with the spread, O ( α 2 ) O(\alpha^2) O ( α 2 ) O ( α ) O(\alpha) O ( α ) O ( α 2 ) O(\alpha^2) O ( α 2 )
Four-lepton production at the W W W
As an application beyond 2 → 2 2 \to2 2 → 2 2 → 3 2 \to3 2 → 3 e + e − → 4 e^+e^- \to4 e + e − → 4 s = 161 \sqrt{s}=161 s = 161 W W W W W W W W W 2 → 4 2 \to4 2 → 4 (44) , and the real-virtual is factorized. The resonant exponentiation of Eq. (35) is not applied, since it is not derived for charged resonances. ACRAIC is at O ( α ) O(\alpha) O ( α ) Table III . The fiducial region follows the tuned comparison of the LEP2 four-fermion report [111 ] . Photons within 5 ∘ 5^\circ 5 ∘ 20 ∘ 20^\circ 2 0 ∘
Table XII gives the fiducial cross sections of the six charged-current channels ℓ − ν ˉ ℓ ℓ ′ + ν ℓ ′ \ell^-\bar{\nu}_{\ell}\ell^{\prime+}\nu_{\ell^\prime} ℓ − ν ˉ ℓ ℓ ′ + ν ℓ ′ ℓ ≠ ℓ ′ \ell\ne\ell^\prime ℓ = ℓ ′ μ + μ − ν μ ν ˉ μ \mu^+\mu^-\nu_\mu\bar{\nu}_\mu μ + μ − ν μ ν ˉ μ O ( α ) O(\alpha) O ( α ) e − ν ˉ e μ + ν μ e^-\bar{\nu}_e\mu^+\nu_\mu e − ν ˉ e μ + ν μ μ − ν ˉ μ e + ν e \mu^-\bar{\nu}_\mu e^+\nu_e μ − ν ˉ μ e + ν e
Table XII. Fiducial cross sections of e + e − → 4 e^+e^- \to4 e + e − → 4 s = 161 \sqrt{s}=161 s = 161 O ( α ) O(\alpha) O ( α )
These final states are also where the squared-level matching fails, and so they motivate the amplitude-level construction. In the same samples the weights of YFSN L O _{\mathrm{NLO}} NLO 1 + ν j 1+\nu_j 1 + ν j N L O _{\mathrm{NLO}} NLO (41) , both at the same photon kinematics, and it is positive at O ( α ) O(\alpha) O ( α ) Table XII are 0.3–0.8%. An O ( α 2 ) O(\alpha^2) O ( α 2 ) β ^ 2 \hat{\beta}_2 β ^ 2 E W _{\mathrm{EW}} EW 2 → 4 2 \to4 2 → 4
Conclusions and outlook
We have presented ACRAIC, an automated amplitude-level YFS exponentiation for general final states in SHERPA. Of the ingredients for coherent exclusive exponentiation only the coherent sum depends on the final state being a fermion pair. We replace the initial/final assignment of photons by an assignment to emission stages where the gauge invariance of every term requires each stage to be charge neutral, and closure requires the stages to exhaust the charged legs exactly once. The stages are derived from the radiating dipoles of the process, with a resonance entering as a leg of both its production and its decay stage, which for W W W N + 2 N+2 N + 2 N + 3 N+3 N + 3 O ( α ) O(\alpha) O ( α )
At photonic O ( α 2 ) O(\alpha^2) O ( α 2 ) § 2.2 , with every difference between the columns accounted for class by class. The theory-nuisance-parameter bands of Sec. IV A cover the missing photonic orders for the two example applications, the Z Z Z § 2.2 A) and the Higgs recoil mass at FCC-ee (§ 2.2 B), apart from the backward fermion angle at the Z Z Z O ( α ) O(\alpha) O ( α )
We expect ACRAIC to be better suited to large-scale production at FCC-ee, where the samples are far larger than any here and the weights matter as much as the formal accuracy. Its weight is positive at O ( α ) O(\alpha) O ( α ) μ + μ − \mu^+\mu^- μ + μ − Table I ), and for four-lepton final states it avoids the single-event tails of the squared-level weight (§ 2.2 C), so a large sample is not dominated by a few events. Several extensions follow from the components described here. Radiation from the W W W W W W O ( α 2 ) O(\alpha^2) O ( α 2 ) β ^ 0 ( 2 ) \hat{\beta}_0^{(2)} β ^ 0 ( 2 ) γ Z \gamma Z γ Z β ^ 0 ( 2 ) \hat{\beta}_0^{(2)} β ^ 0 ( 2 ) ℓ + ℓ − \ell^+\ell^- ℓ + ℓ − γ γ \gamma\gamma γ γ
Figure 11 summarizes which amplitudes enter, by loop order and photon multiplicity, and the resulting logarithmic accuracy. The tree and one-loop residuals are constructed automatically for any process from the tree amplitudes of COMIX and the one-loop amplitudes of the loop provider. The β ^ 3 [ 0 ] \hat{\beta}_3^{[0]} β ^ 3 [ 0 ] β ^ 4 [ 0 ] \hat{\beta}_4^{[0]} β ^ 4 [ 0 ] β ^ 2 [ 1 ] \hat{\beta}_2^{[1]} β ^ 2 [ 1 ] O ( α 2 ) O(\alpha^2) O ( α 2 ) β ^ 3 [ 1 ] \hat{\beta}_3^{[1]} β ^ 3 [ 1 ] β ^ 0 [ 2 ] \hat{\beta}_0^{[2]} β ^ 0 [ 2 ] β ^ 0 ( 2 ) \hat{\beta}_0^{(2)} β ^ 0 ( 2 )
Figure 11. What ACRAIC includes, by loops ℓ \ell ℓ n n n β ^ n [ ℓ ] \hat{\beta}^{[\ell]}_n β ^ n [ ℓ ] (27) , contributing at O ( α n + ℓ ) \mathcal{O}(\alpha^{n+\ell}) O ( α n + ℓ ) O ( α ) \mathcal{O}(\alpha) O ( α ) O ( α 2 ) \mathcal{O}(\alpha^2) O ( α 2 ) O ( α 2 ) \mathcal{O}(\alpha^2) O ( α 2 ) α k L m \alpha^k L^m α k L m
The initial-state photons of ACRAIC are attached to the electron line, so the initial-state treatment can be combined in SHERPA with its state-of-the-art description of QCD final states. This covers next-to-leading-order QCD matrix elements matched to parton showers [112 ] , multijet merging at next-to-leading order [113 ] , including e + e − → e^+e^- \to e + e − → [114 ] , and the SHERPA parton showers [115 , 116 , 117 ] . Photon emission from the quarks, its interplay with the QCD evolution and the initial-final interference in coloured processes then need a consistent treatment.
Acknowledgments
A.P. thanks the other members of the SHERPA collaboration for their work and dedication, which made this paper possible. A.P would also like to thank Ayres Freitas for many discussions on two-loop corrections and Jonas Lindert for continued support with OPENLOOPS. The work of A.P. is supported by grant No. 2023/50/A/ST2/00224 of the National Science Centre (NCN), Poland. A.P gratefully acknowledges the Polish high-performance computing infrastructure PLGrid (HPC Center: ACK Cyfronet AGH) for providing computer facilities and support within computational grant no. PLG/2026/019132. A.P. thanks Claude (Anthropic) for assistance improving the presentation of this paper and for help testing the numerical implementation of the algorithm.
Conventions for the soft factors
The relative phases of the terms of Eq. (20) are physical, and an inconsistency between the conventions of the amplitudes and of the soft factors is invisible to every squared quantity. We collect here the conventions on which they depend.
a. Legs and charges. All momenta are physical, with positive energy. The direction label θ i \theta_i θ i (1) carries the crossing of the incoming legs, so that an incoming particle of charge Q i Q_i Q i − Q i -Q_i − Q i Z i Z_i Z i [54 ] , θ i = − Z i \theta_i=-Z_i θ i = − Z i Q i θ i = − Z i Q i Q_i\theta_i=-Z_iQ_i Q i θ i = − Z i Q i θ = + 1 \theta=+1 θ = + 1 θ = − 1 \theta=-1 θ = − 1 (18) .
b. Polarization vectors. The polarization vectors ε ( σ ) ( k ) \varepsilon^{(\sigma)}(k) ε ( σ ) ( k ) (7) and (15) are those of the external vector currents of COMIX for an outgoing massless vector, constructed from its spinors with the gauge reference vector n = ( 1 , 0 , 1 , 0 ) n=(1,0,1,0) n = ( 1 , 0 , 1 , 0 ) M 1 \mathcal{M}_1 M 1 M 0 \mathcal{M}_0 M 0 M 1 \mathcal{M}_1 M 1
c. Stage soft factors. The stage factors Eq. (15) are the restriction of the same sum to the legs of the stage, with the same signs and normalization, so that Eq. (18) holds identically when the stages exhaust the charged legs. The squared-level soft factor is Eq. (8) .
Beam energy spread
We model the beam energy spread (BES) by an independent Gaussian for each beam, the parametrization of KKMC [17 ] . Beam i i i E 0 E_0 E 0 r r r σ E = r E 0 \sigma_E=rE_0 σ E = r E 0 N N N r r r 0.13 % 0.13\% 0.13% Z Z Z 0.165 % 0.165\% 0.165% Z H ZH Z H [9 ] . The beam momenta are built at the maximum energy E max = E 0 ( 1 + N r ) E_{\max}=E_0(1+Nr) E m a x = E 0 ( 1 + N r ) x i = E i / E max x_i=E_i/E_{\max} x i = E i / E m a x
f ( x i ) = 1 2 π σ x erf ( N / 2 ) exp [ − ( x i − x 0 ) 2 2 σ x 2 ] , x 0 = 1 1 + N r , σ x = r x 0 (65) f(x_i)=\frac{1}{\sqrt{2\pi}\,\sigma_x\operatorname{erf}(N/\sqrt{2})}\exp\left[-\frac{(x_i-x_0)^2}{2\sigma_x^2}\right],
\qquad
x_0=\frac{1}{1+Nr},\qquad\sigma_x=rx_0 \tag*{(65)} f ( x i ) = 2 π σ x erf ( N / 2 ) 1 exp [ − 2 σ x 2 ( x i − x 0 ) 2 ] , x 0 = 1 + N r 1 , σ x = r x 0 (65) on ∣ x i − x 0 ∣ ≤ N σ x |x_i-x_0|\le N\sigma_x ∣ x i − x 0 ∣ ≤ N σ x E 0 E_0 E 0 σ E \sigma_E σ E f ( x 1 ) f ( x 2 ) f(x_1)f(x_2) f ( x 1 ) f ( x 2 ) ρ \rho ρ
f ( x 1 , x 2 ) = N ρ 1 − ρ 2 f ( x 1 ) f ( x 2 ) exp [ − ρ 2 ( d 1 2 + d 2 2 ) − 2 ρ d 1 d 2 2 ( 1 − ρ 2 ) ] , (66) f(x_1,x_2)=\frac{\mathcal{N}_\rho}{\sqrt{1-\rho^2}}f(x_1)f(x_2)
\exp\left[-\frac{\rho^2(d_1^2+d_2^2)-2\rho d_1d_2}{2(1-\rho^2)}\right], \tag*{(66)} f ( x 1 , x 2 ) = 1 − ρ 2 N ρ f ( x 1 ) f ( x 2 ) exp [ − 2 ( 1 − ρ 2 ) ρ 2 ( d 1 2 + d 2 2 ) − 2 ρ d 1 d 2 ] , (66) with d i = ( x i − x 0 ) / σ x d_i=(x_i-x_0)/\sigma_x d i = ( x i − x 0 ) / σ x N ρ \mathcal{N}_\rho N ρ ∣ d i ∣ ≤ N |d_i|\le N ∣ d i ∣ ≤ N N = 4 N=4 N = 4 6 ⋅ 10 − 5 6\cdot10^{-5} 6 ⋅ 1 0 − 5
An event therefore has its own centre-of-mass energy s = ( p 1 + p 2 ) 2 s=(p_1+p_2)^2 s = ( p 1 + p 2 ) 2 E 1 = E 2 E_1=E_2 E 1 = E 2 s s s 4 E 0 2 4E_0^2 4 E 0 2
References
The initial-state dipole is built on the beams of the event, and the initial-state photons, their soft cutoff ε s / 2 \varepsilon\sqrt{s}/2 ε s /2 s ′ s^{\prime} s ′
The form factors of all dipoles, the soft scale of the initial-final exponent, the final-state cutoff, the virtual subtraction and the soft cuts of the matching are evaluated at the event’s s s s
ACRAIC evaluates the partition sum and the crude density in the centre-of-mass frame of the event.
The event is generated in that frame and boosted to the laboratory along the beam axis, together with all photons. Only the analysis refers to the nominal energy.
