QED corrections to $e^+e^-\toπ^+π^-(γ)$ with amplitude-level resummation

Introduction

For decades, a persistent discrepancy existed between theoretical predictions [1] and experimental measurements [2, 3, 4] of the muon anomalous magnetic moment $a_\mu$ [5], motivating extensive searches for new physics [6]. The Muon $g - 2$ experiment (E989) at Fermilab [7], combining its full dataset with the earlier Brookhaven (BNL) measurement [8], recently reached a precision of 124 ppb [4, 9].

While the 2020 White Paper [1] yielded $a_\mu(\mathrm{EXP}) - a_\mu(\mathrm{SM}) = (251 \pm59) \times10^{-11}$ – a $4.2\sigma$ discrepancy, recent updates report $a_\mu(\mathrm{EXP}) - a_\mu(\mathrm{SM}) = (38 \pm63) \times10^{-11}$ [10], in agreement within current uncertainties. This shift is driven by lattice QCD calculations of the leading-order hadronic vacuum polarisation (HVP) contribution $a_\mu^{\mathrm{HVP,\,LO}}$ by the Budapest-Marseille-Wuppertal (BMW) Collaboration [11, 12], since confirmed by other lattice collaborations [13, 14, 15].

A substantial tension remains within the HVP sector itself. Lattice calculations disagree with data-driven estimates based on $e^+e^- \to$ hadrons cross sections, and the $e^+e^-$ datasets disagree among themselves [16], with CMD-3 [17, 18] yielding a higher $e^+e^- \to\pi^+\pi^-$ cross section than KLOE [19, 20, 21, 22] and BaBar [23]. BaBar has since presented a new measurement with its full initial-state-radiation data set, which separates the pion and muon pairs through their angular distributions [24, 25]. The CMD-3 result is therefore typically excluded from global data-driven averages of $a_\mu^{\mathrm{HVP,\,LO}}$. For comprehensive reviews of these HVP tensions, see [10, 26, 27]. HVP dominates the error budget of $a_\mu$, and the two-pion final state accounts for the largest part of the HVP contribution [1, 10]. Pion channels are equally crucial for the hadronic contribution to the running coupling $\alpha(s)$, obtained from a dispersion integral over $e^+e^-$ annihilation data, and for $\alpha(M_Z^2)$, a key input for precision studies of electroweak pseudo-observables at future colliders such as FCC-ee [28, 29].

At low-energy flavour factories, the $e^+e^- \to\pi^+\pi^-$ cross section is measured either by an energy scan, varying the centre-of-mass energy, or by radiative return [30, 31], where an initial-state (ISR) photon gives access to a range of energies at fixed beam energy. Both methods require careful treatment of radiative corrections, which can significantly affect the measured cross section and thus $a_\mu^{\mathrm{HVP}}$. Radiative corrections to $e^+e^-$ processes have a long history [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52], including the structure-function and parton-shower methods for QED corrections. Today such corrections are implemented in Monte Carlo (MC) event generators, essential tools for experimental analyses, whose precision is often limited by the perturbative order of the radiative corrections. A reliable extraction of $a_\mu^{\mathrm{HVP}}$ from $e^+e^- \to\pi^+\pi^-$ requires higher orders. The standard generator for the pion channel, PHOKHARA [53, 54, 55, 56, 57, 58, 59, 60], includes the next-to-leading-order (NLO) radiative corrections for both the $\pi^+\pi^-\gamma$ and $\mu^+\mu^-\gamma$ channels [61, 62], but as a fixed-order prediction it does not include multiple photon emission [63, 64]. For pions, multiple photon emission matched to NLO was first implemented in the BabaYaga@NLO event generator [65, 66] with a parton shower [67], recently extended to the $2 \to3$ radiative channel [68]. Fixed-order corrections to $e^+e^- \to\pi^+\pi^-(\gamma)$ are also available from the MC integrator McMule [69], which provides the $\pi^+\pi^-$ NLO corrections with the pion form factor inside the loops and the hadronic vacuum polarisation in the NNLO initial-state corrections [70]. Its NLO initial-state corrections to $\pi^+\pi^-\gamma$ were presented together with those of other codes in [71]. The disperon QED method [70] brings dispersive input, such as the pion vector form factor and the hadronic vacuum polarisation, into the loop amplitudes of Monte Carlo calculations, with the form factor entering the loops in form-factor scalar QED (FsQED).

A different treatment of the form factor in the loops, attaching it directly to the photon propagators via generalised vector-meson dominance (GVMD), was implemented for this process in Ref. [67] and extended to the NLO corrections to $\pi^+\pi^-\gamma$ [72, 73]. Towards full NNLO [74, 75, 76, 77, 78, 79], tensor decompositions within the GVMD framework for radiative return have provided the higher-order-in-$\epsilon$ real-virtual building blocks needed to add two-loop virtual and double-real photon corrections to Monte Carlo event generators.

In this work, we present a new calculation of the radiative corrections to $e^+e^- \to\pi^+\pi^-(\gamma)$ in the Yennie-Frautschi-Suura (YFS) formalism [42], which resums the soft photons that accompany every event to all orders, unlike fixed-order calculations. The calculation was first implemented in SHERPA [80] with the methods of [81, 82]. We present its implementation in the well-established $e^+e^-$ generator KKMC [83, 84], with the higher-order corrections constructed in the coherent exclusive exponentiation (CEEX) method [85], and in SHERPA with the newly implemented ACRAIC [86] method. The two independent implementations provide a powerful cross-check of each other. We estimate the missing higher orders with theory nuisance parameters [87, 88], as introduced for YFS resummation in Ref. [86], which give an uncertainty for every bin of every distribution tied to identified missing terms (section 2.6).

Theory

The charged pions are treated as point-like scalars, and all corrections are computed in scalar QED (sQED) dressed with the pion form factor. We model the $\pi^\pm$ as dynamical scalar particles which couple to the photon through the sQED Lagrangian,

With the momenta labelled $e^-(p_1)e^+(p_2) \to\pi^-(p_3)\pi^+(p_4)\gamma(k_1)\cdots\gamma(k_{n_\gamma})$, the Feynman rules are

with the scalar propagator $i/(p^2-m_\pi^2+i0)$ and the usual QED rules for the electrons. The form factor is evaluated in every amplitude, with resolved or unresolved photons, and $F_\pi\equiv1$ recovers point-like sQED.

By the YFS theorem the all-order differential cross-section is,

where $\widetilde{S}(k_i)$ is the eikonal factor of photon $i$, $Y(\Omega)$ the YFS form factor, the sum of the virtual and real soft-photon integrals of every pair of charged legs over the soft-photon domain $\Omega$, outside which $\Theta(k_i,\Omega)$ keeps the resolved photons, and $\widetilde{\beta}_{n_\gamma}$ the infrared-finite residuals, each summed over any number of virtual photons. These are defined as in Refs. [85, 86, 89]. Both generators implement the amplitude-level form of eq. (4).

Pion Form Factor

The pion electromagnetic form factor $F_\pi(q^2)$ is parameterised in vector meson dominance (VMD) [90, 91] as a normalised sum of resonance propagators $P_v$ with mixing factors $M_v$,

with complex amplitudes $c=|c|e^{i\varphi}$, free masses and widths, and $P_v(0)=1$, so that $F_\pi(0)=1$. We use the STRONG2020 parametrisation [71] in both generators, with $v\in\{\rho,\rho',\rho'',\rho'''\}$ in the Gounaris–Sakurai form, $\rho$–$\omega$ and $\rho$–$\phi$ mixing on the $\rho$ only, $M_{\rho'}=M_{\rho''}=M_{\rho'''}=1$, and its auxiliary functions and resonance parameters. The narrow $\omega$ and $\phi$ of the mixing factors are constant-width Breit–Wigner propagators,

The broad $\rho$ resonances, with their energy-dependent width, take the Gounaris–Sakurai (GS) form [92],

where $\Gamma(q^2,m_v,\Gamma_v)$ is the energy-dependent width of the $\pi^+\pi^-$ channel and $f$ and $d$ are the Gounaris–Sakurai corrections that keep the propagator analytic and normalise it at $q^2=0$. In COMIX the form factor is a vertex factor at the virtuality of the photon current joining the pion pair, so real photons couple point-like and the same currents serve every multiplicity.

The measured form factor fixes the coupling of one virtual photon to the pion pair, and how it enters amplitudes with more photons, in particular inside loops, is a matter of modelling, with three treatments in use. The factorised approach, F$\times$sQED [62, 71], multiplies the point-like scalar-QED amplitudes by $F_\pi$ at the virtuality of the $s$-channel photon [93, 94, 95]. In the generalised vector-meson dominance (GVMD) approach [96, 97] the form factor is a sum of vector-meson poles attached to every photon coupling to the pions, loop photons included, which turns the loops into point-like integrals with massive photons. FsQED dresses every pion vertex with form factors and evaluates the loops through dispersion relations [98, 99]. Disperon QED [70] brings FsQED to loop-level Monte Carlo by writing the form factor on a photon propagator as a spectral integral over photons of variable mass, the disperons, whose loops are standard while the spectral integral is sampled with the phase space. GVMD and FsQED agree well for the charge asymmetry, and both describe the CMD-3 measurement [17, 67, 97], which the factorised approach does not. The diagrams and the form-factor arguments of the three approaches at next-to-leading order are collected in [67].

ACRAIC and KKMC implement the factorised approach, Figure 1 (a)–(j), and optionally GVMD in the loops, panels (k)–(o). In both generators the form factor sits on the amplitude, separately for each partition of the photons. It is evaluated at $q^2=s$ in the Born amplitude (a), at the virtuality $s'=(p_1+p_2-k_1)^2$ of the $s$-channel photon of the partition for a photon from the beams (b), and at $q^2=s$ for a photon from the pions (c) and in the seagull (d), where the same $F_\pi(s)$ on both vertices keeps the sum gauge invariant. The one-loop corrections (e)–(j), computed for point-like pions, multiply the Born amplitude with $F_\pi(s)$ and comprise the initial- and final-state vertices (e, f), the final-state seagull triangle (g), the initial-final box and crossed box (h, i) and the two-photon seagull (j), suppressed by $m_e^2$.

Vector (PDF)

In GVMD the initial-state correction is unchanged, the form factor sitting on the $s$-channel vertex. In the final and initial-final loops $F_\pi(\ell^2)$ sits on every $\gamma\pi\pi$ and $\gamma\gamma\pi\pi$ vertex of a loop photon of virtuality $\ell^2$, so a final-state loop photon (k, l) carries $F_\pi(\ell^2)$ and the two photons of the boxes and the two-photon seagull (m)–(o) carry $F_\pi(\ell_1^2)F_\pi(\ell_2^2)$. Inside the loops we use the five-pole Breit–Wigner fit to the CMD-3 data of [67], and the tree amplitudes keep eq. (5). The loop kernels with massive photons, rederived and checked in § 2.3, enter as relative corrections to the Born amplitude.

Real Corrections

The real corrections enter through the residuals $\widetilde{\beta}_{n\gamma}$ of eq. (4), the $n_\gamma$-photon amplitudes with their soft limits removed. Both generators work with amplitudes for fermion helicities $\{\lambda\}$ and photon helicity $\sigma$, in the conventions of CEEX [85] and ACRAIC [86]. For a photon $(k,\sigma)$ the one-photon residual is the exact amplitude minus the eikonal current times the Born amplitude,

summed over the charged legs $i$ with charge $Q_i$ and $Z_i=1$ ($-1$) for incoming (outgoing) legs, so that $\sum_\sigma\lvert s^{(\sigma)}(k)\rvert^2=16\pi^3\widetilde{S}(k)$. Both $\mathfrak{s}$ and $\mathcal{M}_1$ use the polarisation vectors $\varepsilon^{(\sigma)}(k)$ of the external photon currents of COMIX. The residual is finite as $k\to0$, and the residuals of all photons and photon pairs are summed coherently before squaring, which keeps the initial-final interference at every order.

In SHERPA every tree-level amplitude in the residuals is generated by COMIX [100], from the charged scalar of the model with its $\gamma\pi\pi$ and $\gamma\gamma\pi\pi$ vertices and the form factor as a vertex factor. Both electrons and pions are kept massive and the residuals are evaluated at the physical event momenta. Assigning a photon to the initial or final stage fixes the virtuality of the $s$-channel propagator and the form factor, which COMIX applies as a shift of the propagator momentum without rebuilding the currents. For a new final state ACRAIC therefore needs only the model with its form factor and the virtual correction.

In KKMC the pion amplitudes are coded analytically in the CEEX spinor formalism [101]. With the lepton current $J^{\mu} = \bar{v}(p_{2})\gamma^{\mu}u(p_{1})$, the pion current $K = p_{4} - p_{3}$ and $s^{\prime} = P_{X}^{2}$ for the $s$-channel momentum $P_{X}$ of the partition, the Born amplitude and the initial- and final-state eikonal currents of eq. (8), with $\epsilon\equiv\epsilon^{(\sigma)*}(k)$, are

The one-photon residuals $h=\mathcal{M}_1-s\mathcal{M}_0$ are, for initial-state emission,

and for final-state emission the two pion diagrams and the seagull, contracted with $\epsilon$, reduce to

which is finite as $k\to0$ and vanishes for $\epsilon\to k$. The two-photon residuals follow likewise from the exact amplitudes $\mathcal{M}_{12}$, with all photon orderings and seagulls,

where for one initial- and one final-state photon the amplitude factorises through the virtual photon as $(s_1J+L_1)\cdot(s_2H_0+h_2)/s'$, with $H_0$ the photonless pion current and $L_1,h_2$ the hard parts of the beam and pion currents. In the CEEX sum each residual carries the eikonal factors of the other photons of its partition, and at second order the one-photon residuals also the one-loop initial- and final-state vertex factors. For a momentum $p=(E,\mathbf{p})$ with helicity $\lambda=\pm1$ and

the spinors in the Dirac basis are

Virtual Corrections

We compute the complete one-loop QED corrections to $e^+e^-\to\pi^+\pi^-$ (momenta as in § 2), also given in [67] and available in tools such as OPENLOOPS [102], in scalar QED, where point-like pions couple to the photon through the three-point and the two-photon seagull vertices, keeping the full electron-mass dependence. The amplitudes were generated with FEYNARTS [103], the Dirac and Lorentz algebra was done with FEYNCALC [104, 105], and the tensor integrals were reduced to $A_0$, $B_0$, $C_0$ and $D_0$ [106, 107], evaluated with LOOPTOOLS [108]. The poles, the on-shell renormalisation constants and the analytic finite parts that validate the numerics came from Package-X [109, 110] through FeynHelpers [111].

Independent electron and pion charges $Q_e$ and $Q_\pi$ split the interference $2\operatorname{Re}(\mathcal{M}_1\mathcal{M}_0^*)$ into initial-state (ISR), final-state (FSR) and initial-final interference (IFI) corrections, proportional to $Q_e^4Q_\pi^2$, $Q_e^2Q_\pi^4$ and $Q_e^3Q_\pi^3$ respectively. Since gauge invariance holds for any two charges, each subset is separately gauge invariant and UV finite after renormalisation. ISR is the electron vertex correction, FSR the pion vertex correction and the two diagrams with a virtual photon between the seagull vertex and an outgoing pion, and IFI the direct and crossed boxes and the diagram with both photons between the electron line and the seagull vertex. We renormalise on shell [107] with $\alpha= \alpha(0)$. As in the STRONG2020 comparisons, vacuum polarisation in the $s$-channel photon is not included anywhere in this paper, so the charge and photon-field renormalisations cancel and only the field renormalisation of the external legs remains. The wave-function renormalisation constants are

and

with $\Sigma_e(p) = \not{p}\,\Sigma_e^V(p^2) + m_e\Sigma_e^S(p^2)$ split into the vector and scalar parts of the electron self-energy and $\Sigma_\pi$ the pion self-energy. Reduced to Passarino-Veltman master integrals they read

with $B_0'$ the derivative in the external momentum squared, the electron constant reproducing the familiar $\delta Z_e=-\frac{\alpha}{4\pi}\left(\Delta_{\mathrm{UV}}+2\Delta_{\mathrm{IR}}+3\ln(\mu^2/m_e^2)+4\right)$. Mass counterterms, which sit on internal electron and pion propagators, do not appear. The only internal line of the tree is the $s$-channel photon, the loop propagators already carry the physical mass, a counterterm there being of order $\alpha^2$, and on the external legs the on-shell mass counterterms cancel the self-energies at $p^2=m^2$, leaving the residues $\delta Z_e$ and $\delta Z_\pi$, which are needed for the real-virtual correction of § 2.4. UV and IR singularities are regulated dimensionally in $D=4-2\epsilon$, and an infrared photon mass changes only the master integrals, not the coefficients below.

We write the virtual correction as

with $\delta_{\mathrm{CT}}^{\mathrm{ISR}}=2\delta Z_e$, $\delta_{\mathrm{CT}}^{\mathrm{FSR}}=2\delta Z_\pi$ and $\delta_{\mathrm{CT}}^{\mathrm{IFI}}=0$. The scalar integrals $I_k$ are normalised as in LOOPTOOLS, $I=\mu^{4-D}/(i\pi^{D/2}r_\Gamma)\int\mathrm{d}^Dq\;(\ldots)$. The results are given at $D=4$, including the rational terms left by the $O(\epsilon)$ parts of $D$-dependent coefficients of UV poles, in terms of $s=(p_1+p_2)^2$, $t=(p_1-p_3)^2$ and $u=(p_1-p_4)^2=2m_e^2+2m_\pi^2-s-t$. With $M^2=m_e^2+m_\pi^2$ the Born matrix element is proportional to

Initial-state corrections. Relative to the Born, the loop gives coefficients of $A_0(m_e^2)$, $B_0(s,m_e^2,m_e^2)$, $B_0(m_e^2,0,m_e^2)$ and the vertex triangle, with the rational term $m_e^2\rho-2$ at $D = 4$. Combining with $2\delta Z_e$ through $A_0(m_e^2) = m_e^2[B_0(m_e^2,0,m_e^2)-1]$, all constant terms cancel and

where $C_0^{e e \gamma}(s)$ is the vertex triangle with two electron and one photon propagator, and the UV poles cancel between the two $B_0$ functions.

Final-state corrections. For point-like pions the FSR correction does not depend on the scattering angle,

with $2\delta Z_\pi$ combined with the loop and $C_0^{\pi\pi\gamma}(s)$ the vertex triangle with two pion and one photon propagator. The UV poles again cancel between the two $B_0$ functions, and the structure is that of [67].

Initial-final interference. The IFI is odd under $t \leftrightarrow u$ and reads, at $D = 4$,

Here $D_0(t)$ and $D_0(u)$ are the direct and crossed boxes, with two photon, one electron and one pion propagator, and the triangles arise from their reduction. $C_0^{\gamma\gamma e}(s)$ and $C_0^{\gamma\gamma\pi}(s)$ have two photon and one electron or pion propagator, with external invariants $(s,m_e^2,m_e^2)$ and $(s,m_\pi^2,m_\pi^2)$, and $C_0(t)$ and $C_0(u)$ have one photon, one electron and one pion propagator, with $(t,m_e^2,m_\pi^2)$ and $(u,m_e^2,m_\pi^2)$. The $B_0$ term is UV finite and vanishes for $m_e \to0$, and $C_0^{\gamma\gamma e}$ is IR finite since neither photon connects two external legs, so their coefficients leave no rational term at $D = 4$. The IR singularity sits in the boxes and in $C_0(t)$ and $C_0(u)$, and the IR poles of $\delta_{\mathrm{ISR}}$, $\delta_{\mathrm{FSR}}$ and $\delta_{\mathrm{IFI}}$ cancel against the YFS form factor. The finite parts and the $1/\epsilon$ poles of each subset agree with [67] to $10^{-16}$ at every phase-space point tested, and the expressions are algebraically identical. In the GVMD treatment of the loops (§ 2.1) the final-state and initial-final corrections combine the same kernels with loop photons of complex squared mass, Eqs. (44)–(46) of [67].

Real-Virtual Corrections

The complete one-loop QED correction to $e^{+}e^{-} \to\pi^{+}\pi^{-}\gamma$, with momenta $e^{-}(p_{1})$, $e^{+}(p_{2})$, $\pi^{-}(p_{3})$, $\pi^{+}(p_{4})$ and $\gamma(k_{1})$, is computed in the same way, in scalar QED with point-like pions and the full electron mass. Its initial-state [112, 113] and final-state [56] parts and the complete sQED correction [62] are known, and it has been matched to a parton shower [68]. It provides the exact real-virtual correction to $\widetilde{\beta}_{1}$ at second order and the virtual correction for processes with a photon in the Born, such as the tagged-photon BESIII and B scenarios. FEYNARTS [103] and FEYNCALC [104, 105] give the amplitudes as tensor coefficients of one- to five-point functions, evaluated with LOOPTOOLS [108] or COLLIER [114]. The tree amplitude has five diagrams, two with the photon from the electrons, two from the pions and one seagull (loop examples in Figure 2). The on-shell counterterms in the $\alpha(0)$ scheme comprise the field counterterms $\delta Z_e$ and $\delta Z_\pi$ of section 2.3 and, since the real photon puts an electron or pion line off shell, mass counterterms on these internal propagators,

Vector (PDF)

with $m_{e,0}=m_e+\delta m_e$ and $m_{\pi,0}^2=m_\pi^2+\delta m_\pi^2$, entering the subsets with initial- and final-state emission respectively. As for the $2\to2$ correction, closed charged loops, and thus vacuum polarisation, are dropped.

Tagging every photon coupling with the charge of its line gives,

With three photon couplings in the tree and five in the loop, every term is of eighth order in the charges, and since both amplitudes couple at least once to each line, $2\le m,n\le6$, leaving five separately gauge-invariant subsets. $\mathcal{V}_{62}$ holds the initial-state loop on initial-state emission, $\mathcal{V}_{26}$ the final-state loop on final-state emission, $\mathcal{V}_{44}$ the mixed terms, including the boxes and pentagons times the Born with final-state emission, while $\mathcal{V}_{53}$ and $\mathcal{V}_{35}$, odd under $\pi^{+}\leftrightarrow\pi^{-}$, generate the charge asymmetry. The coefficients are kept as functions of $D=4-2\epsilon$, so the rational terms are exact, and the finite part can be given in conventional dimensional regularisation, with a $D$-dimensional photon polarisation sum, or in the ’t Hooft–Veltman scheme, which differ by the infrared pole times the $O(\epsilon)$ part of the Born. The YFS subtraction uses the four-dimensional eikonal factor times the Born. Each charge subset is checked at hard points at $\sqrt{s}=0.7$ and $1.02\ \mathrm{GeV}$. The ultraviolet poles cancel to $10^{-19}$ relative to the Born. A fit of one coefficient per renormalisation constant over all points and subsets returns $\pm1$, with the on-shell constants computed independently. The $1/\epsilon$ infrared pole equals the eikonal sum over the four charged legs times the tree $|\mathcal{M}_0|^2$ to $10^{-13}$. In the soft limit the finite part of every subset tends, linearly in the photon energy, to the eikonal factor times the $2\to2$ correction of section 2.3. This validates it against the $2\to2$ result and shows that $\widetilde{\beta}_1^{(2)}$ is infrared finite. Replacing the photon polarisation by its momentum gives zero to $\sim10^{-8}$ or better for the initial-state loops, the mixed subset and the final-state subsets. Unlike the ultraviolet and infrared checks, this one catches missing diagrams such as the seagulls.

Table 1 gives the correction at two hard points at $\sqrt{s}=0.7$ GeV and one at 1.02 GeV, built in the centre-of-mass frame with the electron along $+z$. The photon has energy $k_1^0=x\sqrt{s}/2$ and angles $\theta_\gamma$ and $\phi_\gamma$, and the pions are back to back in their rest frame, the $\pi^-$ along $(\theta^*,\phi^*)$ in the same axes, boosted with the photon recoil. With $m_e=0.51099895$ MeV, $m_\pi=139.57039$ MeV, $\alpha=1/137.035999139$, the integrals normalised as in section 2.3, LOOPTOOLS and COLLIER agree to $10^{-10}$. The finite parts are a sizeable fraction of the tree because, before the YFS subtraction, they contain the electron-mass collinear logarithms multiplying the infrared pole. The ancillary material gives the four-momenta to twelve digits and the finite parts in conventional dimensional regularisation.

As in the $2\to2$ case, in the factorised approach the correction to the point-like diagrams of Figure 2, taken relative to the point-like tree, multiplies the tree with the form factor placed as in Figure 1 (b)–(d), so all diagrams, the boxes and pentagons (d)–(f) included, carry the form factor of the tree they interfere with. In the GVMD option the loop photons carry $F_\pi(\ell^2)$ as in Figure 1 (k)–(o). With $F_\pi(\ell^2)=\sum_\nu c_\nu L_\nu/(L_\nu-\ell^2)$ and $\sum_\nu c_\nu=1$, the photon

caption: tbl:1: Benchmark values of the one-loop correction to $e^+e^- \to\pi^+\pi^-\gamma$ for point-like pions at three phase-space points, constructed as described in the text. For each charge subset, $\delta_{mn} = Q_e^m Q_\pi^n \mathcal{V}_{mn}/\lvert\mathcal{M}_0\rvert^2$ is given in units of $10^{-2}$: the coefficient of the infrared pole $1/\epsilon$ and the renormalised finite part in the ’t Hooft–Veltman scheme at $\mu= 1$ GeV. $\lvert\mathcal{M}_0\rvert^2$ is the spin-averaged point-like tree, summed over its three charge subsets.

:::

propagator times the form factor splits into

and a loop photon between the two pions, carrying $F_\pi^2$, adds terms with two masses and, for coinciding poles, their derivative. The resulting massive-photon loops are again taken relative to the tree with the same pole form factor and multiply the tree with the Gounaris–Sakurai form factor. These tables pass the ultraviolet and infrared checks, with the infrared pole equal to the eikonal factor times the form-factor tree, and reduce to the point-like ones for massless photons. Disperon QED [70, 115] gives the correction with the form factor inside the loops, with $\mathcal{V}_{62}$ complete, $\mathcal{V}_{53}$ in FsQED, $\mathcal{V}_{44}$ estimated and the final-state subsets left out, and is the reference for the form-factor dependence of $\mathcal{V}_{53}$, absent from our point-like loops. It has also been computed in GVMD [72] and with a dispersive form factor [73].

Numerical stability

The exact correction matters for hard initial-state photons away from the beams, its logarithmic expansion describing it well close to them. As the photon approaches a beam the box and pentagon coefficients cancel against each other, with both LOOPTOOLS and COLLIER, to a degree controlled by

which combines the photon energy with its angle to the nearer beam, so the failure region is not set by the angle alone. In double precision the finite part loses all significance for $r \lesssim10^{-6}$, reached by a photon with $x \sim10^{-2}$ already at $\sim10^{-2}\,\mathrm{rad}$. The precision is lost in the coefficient arithmetic, not the loop functions. Quadruple precision inside LOOPTOOLS or a different loop-momentum routing through the pentagons does not help, while quadruple-precision coefficients with double-precision loop functions restore the amplitude down to $r \sim10^{-7}$. An independent calculation evaluates pentagons, coefficients and loop functions alike in quadruple precision close to the beams [68]. All predictions use our own amplitude, which also provides the form-factor variants. The predictions of § 3 use the exact correction except for initial-state photons with $r < r_c = 10^{-3}$, where the exact remainder is replaced by its leading and next-to-leading logarithmic limit $\mathcal{V}_2$ [116]. This changes the fiducial cross sections by $\sim0.01\%$ at most, checked with both the independent evaluation and the quadruple-precision coefficients. The remainder is taken relative to the one-loop correction that ACRAIC applies to $\hat{\beta}_1$, with the initial-state vertex evaluated at $s$. In KKMC the initial-state loops on a beam photon, $\mathcal{V}_{62}$, multiply the one-photon initial-state CEEX amplitude for $r \ge r_c$, while the other subsets, without amplitude-level counterpart in CEEX, enter as an extra event weight, the same remainder at the hardest photon. At KLOE-I the remainder comes from $\mathcal{V}_{44}$, $\mathcal{V}_{35}$ and $\mathcal{V}_{26}$, each a few tenths of a per cent and largely cancelling, while $\mathcal{V}_{62}$ and $\mathcal{V}_{53}$ give $\sim0.01\%$, and the two generators agree on the correction. The exact remainder dominates the generation cost, by about an order of magnitude with hard photons. The correction therefore comes from a smaller sample whose events carry a second weight with $\mathcal{V}_2$ in place of the exact remainder, and the ratio of the two distributions, far more precise than either as both weights belong to the same events, is applied bin by bin to the $\mathcal{V}_2$ samples at under a tenth of the cost of new samples.

Fixed order and resummation

To test the convergence of the all-order resummation of soft and soft-collinear logarithms in ACRAIC and KKMC, its expansion in $\alpha$ is truncated at second and third order. Writing the bracket of eq. (4) for $n_\gamma$ photons as $R_{n_\gamma}$ and counting $Y(\Omega)$, the virtual correction $v and each photon as one power of $\alpha$, schematically

where $[\ldots]_{\alpha^k}$ keeps the terms up to $\alpha^k$. Without real photons, for example, $\left[e^Y(1+v/2+\delta^{(2)})^2\right]_{\alpha^2}=1+Y+v+Y^2/2+Yv+v^2/4+2\delta^{(2)}$, with $\delta^{(2)}$ the two-loop vertex. The strict fixed-order $\mathcal{O}(\alpha)$ prediction is $\sigma^{(1)}$ of eq. (28) with the exact one-photon matrix element and the one-loop virtual correction, the next-to-leading order of $e^+e^-\to\pi^+\pi^-$ computed by the fixed-order generators. In the tagged-photon scenarios its leading term is the tree-level $e^+e^-\to\pi^+\pi^-\gamma$, a leading-order prediction for the observed final state. The truncations $\sigma^{(2)}$ and $\sigma^{(3)}$ of the $\mathcal{O}(\alpha^2L)$ prediction measure how much of the resummed result lies beyond each order. They test only the higher orders the resummation contains, the powers of the soft and collinear logarithms from $e^{Y(\Omega)}$, the eikonal factors and the virtual correction. Corrections it does not contain, such as the hard three-photon matrix element or the three-loop virtual correction, appear in no truncation and are covered only by the theory nuisance parameters.

Theoretical Uncertainties

The uncertainty of the generators used in radiative-return and scan measurements has so far been judged from the spread between generators or from the size of the last correction included [63, 71], without a systematic estimate of their missing higher orders that we know of. Neither measure is reliable here, as generators sharing an approximation agree without being right, and in radiative return the first-order correction is of order one, so the last correction no longer estimates the next. We therefore use theory nuisance parameters [87, 88], applied to YFS resummation as in [86], each standing for an identified missing term. The amplitude of an event is a sum of corrections, $C=\sum_c C_c$, the $\widetilde{\beta}_0$ term, the $\widetilde{\beta}_1$ of initial- and final-state photons and, at second order, the $\widetilde{\beta}_2$ of photon pairs. The template of a correction is the response of the event weight to rescaling $C_c$ alone, and each $\theta_t$ multiplies one template with a normalisation $N_t$, the size of the first term the class omits,

with $\langle A,B\rangle=\sum_{\{\lambda\},\sigma}A^*B$ the helicity-averaged inner product and $C$ the coherent amplitude of [86]. Observables are linear in the parameters, and for independent parameters of unit variance the band adds the $\theta_t=\pm1$ variations in quadrature,

With $g_e=(\alpha/\pi)[\ln(s/m_e^2)-1]$ and $g_\pi=(\alpha/\pi)[\ln(s/m_\pi^2)-1]$, the normalisations of [86] are $N=g^k/(2!k!)$ for a one-photon class of a prediction truncated at order $k$, with $g=g_e$ ($g_\pi$) for initial-state (final-state) photons, and the $\widetilde{\beta}_0$ classes have $N=g^m/(2m!)$ for the first order $m$ they miss, separately for the initial and final state and their products. The factor $1/2$ converts a relative cross-section correction to the amplitude. The $\tilde{\beta}_2$ classes, absent from the first-order prediction, enter its band with $N = 1$ and one parameter for the three pair classes, on templates from the second-order prediction on the same events.

Two features of pion production need further parameters. First, the second order is incomplete for pions, the two-loop pion vertex being approximated and the two-loop initial-final boxes absent, so the $\mathcal{O}(\alpha^2 L)$ band carries one $\tilde{\beta}_0$ parameter with $N = (\alpha/\pi)^2\pi^2/2$, again halved to the amplitude. Second, in radiative return a series whose next term is $g$ times the last no longer describes the order-one first-order correction. There a radiative-return parameter takes as template in each bin the relative size $r_k = \sigma_k/\sigma_{k-1} - 1$ of the last correction included, applied to the current prediction, $r_k\sigma_k$, and as normalisation the YFS exponent $\gamma= 2(\alpha/\pi)(L - 1)$, so the next relative correction is $\gamma r_k$, one further power of the logarithm rather than $r_k$ itself. Its use and the size of every parameter are discussed in § 3.10. The templates are additional weights on the same events, so every histogram carries its band, correlated across bins and observables through the shared parameters. In KKMC the classes are the corresponding $\hat{\beta}$ terms of the CEEX amplitudes, so the bands of the two generators differ only where they assign a correction to different classes.

Numerical Results

Input parameters

Both generators use on-shell masses, the fine-structure constant at zero momentum transfer, no vacuum polarisation and the STRONG2020 pion form factor of eq. (5) with the resonance parameters of Ref. [71] (Table 2). The soft cut-offs of the YFS integrals and the thresholds below which a photon enters the hard corrections only through the eikonal factor are technical parameters, and the fiducial cross sections do not depend on the soft cut-off.

Parameter	Value
$\alpha^{-1}(0)$	137.03599976
$m_e$ [MeV]	0.51099895
$m_\pi$ [MeV]	139.57039
$F_\pi(q^2)$	STRONG2020 [71]
vacuum polarisation	off
soft cut-off	$E_{\min} = 1.75$ keV

Fiducial selection

Table 3 summarises the fiducial selections.

Scenario	$\sqrt{s}$ [GeV]	Pion selection	Photon selection	Further cuts
CMD	0.7	$\theta_{\mathrm{avg}} \in[1,\pi-1]$ rad<br>$p^{\pm} > 0.45\sqrt{s}/2$	–	$\delta\phi< 0.15$ rad<br>$\xi< 0.25$ rad
KLOE-I	1.02	$\theta^{\pm} \in[50^\circ,130^\circ]$<br>$\lvert p_z^{\pm}\rvert> 90$ MeV $\vee p_{\perp}^{\pm} > 160$ MeV	$\theta_\gamma\in[50^\circ,130^\circ] \wedge E_\gamma> 20$ MeV	$M_{\pi^+\pi^-}^2 \in[0.1,0.85]$ GeV$^2$
KLOE-II	1.02	as KLOE-I	$\theta_\gamma< 15^\circ\vee\theta_\gamma> 165^\circ$	$M_{\pi^+\pi^-}^2 \in[0.35,0.95]$ GeV$^2$
BESIII	4	$\lvert\cos\theta^{\pm}\rvert< 0.93$<br>$p_{\perp}^{\pm} > 300$ MeV	$\lvert\cos\theta_\gamma\rvert< 0.8 \wedge E_\gamma> 25$ MeV<br>$\vee0.86 < \lvert\cos\theta_\gamma\rvert< 0.92 \wedge E_\gamma> 50$ MeV	$\exists! \gamma:E_\gamma> 400$ MeV
B	10	$\theta^{\pm} \in[0.65,2.75]$ rad<br>$p^{\pm} > 1$ GeV	$\theta_\gamma\in[0.6,2.7]$ rad $\wedge E_\gamma> 3$ GeV	$\theta_{\gamma^{(h)},\gamma e} < 0.3$ rad<br>$M_{\pi^+\pi^-\gamma} > 8$ GeV

Both generators couple the pions to photons in point-like scalar QED with $F_\pi$ of § 2.1 on the $s$-channel photon, exponentiate the soft photons to all orders through the YFS form factor and the eikonal factors of all dipoles, and include the initial-final interference at amplitude level through the sum over partitions [85].

At $\mathcal{O}(\alpha)$:

• $\widetilde{\beta}_0$ with the one-loop virtual correction of § 2.3, the electron vertex, the pion vertex and the initial-final boxes, computed for point-like pions and supplemented with $F_\pi(s)$ either in F×sQED or GVMD. The pion vertex of KKMC, $F_{\mathrm{fin}}$, equals $\delta_{\mathrm{FSR}}$ of Equation (21) to $10^{-16}$, and the boxes are evaluated with the same code. Each generator removes the infrared singularity with its own YFS form factor, which its exponent puts back. KKMC evaluates the boxes once per event at the Born point and applies the two vertices as $(1+F_{\mathrm{ini}})(1+F_{\mathrm{fin}})$, so its $\mathcal{O}(\alpha)$ also contains the $\mathcal{O}(\alpha^2)$ product $F_{\mathrm{ini}}F_{\mathrm{fin}}$, which raises the CMD cross section by $\sim0.01\%$ and is optional in ACRAIC.

• $\widetilde{\beta}_1$ from the exact tree amplitudes for one real photon, emitted from the electrons or from the pions, with $F_\pi$ evaluated at the virtuality of the $s$-channel photon of each partition. ACRAIC takes them from COMIX, KKMC from its own scalar-QED helicity amplitudes.

At $\mathcal{O}(\alpha^2 L)$, in addition:

• $\widetilde{\beta}_2$ from the exact tree amplitudes for two real photons, both from the electrons, both from the pions or one from each, coherently.

• The exact one-loop correction to the emission of one photon from the electrons or the pions, for every charge subset.

• The square of the one-loop amplitude, in both.

• The photonic two-loop vertices in $\tilde{\beta}_0$. KKMC adds the two-loop electron vertex at leading and next-to-leading logarithmic accuracy and, for the pions, $F_{\mathrm{fin}}+F_{\mathrm{fin}}^2/2$. ACRAIC adds the leading logarithms of every pair of charged legs, including the product of the initial- and final-state vertices, the next-to-leading logarithms and the constant of the massive two-loop vertex [119].

Neither generator includes the two-loop boxes, the genuine two-loop scalar-QED pion vertex, or the photonic corrections of third order. The second order therefore contains the $\alpha^2L^2$ and the dominant $\alpha^2L$ terms, with $L=\ln(s/m_e^2)$, but not the non-logarithmic $\alpha^2$ terms, hence the label $\mathcal{O}(\alpha^2L)$, while $\mathcal{O}(\alpha^2)$ refers to the perturbative order itself. § 2.1 describes where $F_\pi$ enters the amplitudes of both generators.

CMD

At $\mathcal{O}(\alpha)$ the pion angles of KKMC CEEX and SHERPA ACRAIC agree to better than 0.1% in every bin with $|\cos\theta^\pm|<0.5$ (Figure 3), and the fiducial cross sections (Table 4) to $\sim 0.002%, within the statistical uncertainty. This is a first non-trivial cross-check between two independent amplitude-level implementations of these corrections for $e^{+}e^{-}\to\pi^{+}\pi^{-}$.

Scenario	Order	KKMC CEEX $\sigma_{\mathrm{fid}}$ [nb]	SHERPA ACRAIC $\sigma_{\mathrm{fid}}$ [nb]	Ratio ACRAIC/CEEX
CMD	$\mathcal{O}(\alpha)$	570.078(9)	570.066(5)	0.99998(2)
	$\mathcal{O}(\alpha^2L)$	570.242(9)	570.287(5)	1.00008(2)
	$\mathcal{O}(\alpha^2L)/\mathcal{O}(\alpha)$	1.00029	1.00039	–
KLOE-I	$\mathcal{O}(\alpha)$	3.2512(3)	3.2513(2)	1.00005(11)
	$\mathcal{O}(\alpha^2L)$	3.2012(3)	3.2033(2)	1.00067(11)
	$\mathcal{O}(\alpha^2L)/\mathcal{O}(\alpha)$	0.98462	0.98524	–
KLOE-II	$\mathcal{O}(\alpha)$	15.966(1)	15.965(1)	0.99995(11)
	$\mathcal{O}(\alpha^2L)$	16.020(1)	16.017(1)	0.99979(11)
	$\mathcal{O}(\alpha^2L)/\mathcal{O}(\alpha)$	1.00337	1.00322	–
BESIII	$\mathcal{O}(\alpha)$	0.088510(11)	0.088507(10)	0.99997(16)
	$\mathcal{O}(\alpha^2L)$	0.083889(13)	0.083911(9)	1.00026(19)
	$\mathcal{O}(\alpha^2L)/\mathcal{O}(\alpha)$	0.94780	0.94807	–
	$\pi^+\pi^-\gamma$ resummed NLO	–	0.083677(13)	–
B	$\mathcal{O}(\alpha)$	0.011160(1)	0.011157(2)	0.99971(17)
	$\mathcal{O}(\alpha^2L)$	0.010056(3)	0.010058(1)	1.00015(28)
	$\mathcal{O}(\alpha^2L)/\mathcal{O}(\alpha)$	0.90104	0.90143	–
	$\pi^+\pi^-\gamma$ resummed NLO	–	0.010052(1)	–

In the acollinearity and acoplanarity (Figure 4) the two calculations agree within the statistical fluctuations even in the tails, which fall by several orders of magnitude. There the $\mathcal{O}(\alpha^{2}L)$ correction, negative and a few tenths of a per cent in both generators, exceeds the $\mathcal{O}(\alpha)$ band several times. It is the one-loop correction to the one-photon events, which also changes the soft part of the one-photon amplitude, since the virtual correction differs between the radiative and the Born point, whereas the $\widetilde{\beta}_{1}$ parameters rescale only the hard part, small here.

Vector (PDF)

The charge-symmetric $\pi^{\pm}$ momentum spectra (Figure 5) agree from the radiative tail to the elastic peak. In the observables most sensitive to the radiation itself, $M_{\pi^{+}\pi^{-}}$ and $E_{\gamma}/\sqrt{s}$ (Figure 6), the two $\mathcal{O}(\alpha)$ calculations agree within the statistical precision, including the hardphoton tail.

Vector (PDF)

Vector (PDF)

The $\mathcal{O}(\alpha^{2}L)$ corrections are $\sim0.03\%$ in both generators, small because the CMD cuts select events close to the elastic peak, and flat in the angular distributions, where the two generators again agree to better than $0.1\%$ per bin. Where a hard photon is emitted, at $M_{\pi^{+}\pi^{-}} < 500$ MeV and at $E_{\gamma}/\sqrt{s} \approx0.4$, they grow to a few per cent. These tails carry less than $0.01\%$ of the fiducial cross section, and there the two generators agree within the statistical precision of ACRAIC. With the parameters of the incomplete second order for pions of section 2.6, the $\mathcal{O}(\alpha^{2}L)$ band is $\sim0.01\%$ in the angular distributions and $\sim0.1\%$ where a hard photon is emitted (third panels of Figure 3 to Figure 6). On the fiducial cross section the step reduces the band from $\sim0.05\%$, dominated by the $\widetilde{\beta}_{0}$ parameters, to $\sim0.005\%$, dominated by the incomplete second order.

Figure 7 shows the charge asymmetry

Vector (PDF)

with $\theta= \theta_{\mathrm{avg}}$, in the CMD-3 selection with momentum cuts scaled to the beam energy, at 43 energies in $0.35 < \sqrt{s} < 1.05$ GeV with finer steps around the $\omega$ (0.765–0.795 GeV) and in the $\phi$ region (1.01–1.05 GeV) [17]. With point-like loops in the factorised approach the asymmetry is $\sim10^{-2}$ over the whole range. GVMD in the loops lowers it at every energy, by $0.8 \times10^{-3}$ at $\sqrt{s} = 0.35$ GeV and by up to $1.15 \times10^{-2}$ at $0.78$ GeV, where it removes most of the asymmetry. At the $\omega$ the point-like asymmetry falls from $\sim13 \times10^{-3}$ to $\sim11 \times10^{-3}$ between $0.775$ and $0.785$ GeV in both generators, and the GVMD shift shrinks by $1.8 \times10^{-3}$. In the $\phi$ region the point-like asymmetry of both generators dips by $\sim3 \times10^{-4}$ at $1.015$ GeV and rises by $\sim10^{-3}$ above $1.02$ GeV. The ACRAIC GVMD shift shrinks from $5.4 \times10^{-3}$ at $1.01$ GeV to $4.2 \times10^{-3}$ at $1.019$ GeV and grows again to $5.7 \times10^{-3}$ at $1.025$ GeV, and the KKMC shift follows it within its errors of $10^{-3}$. The two generators agree on the shift within the statistical uncertainties, with $\chi^{2}/N = 0.54$ for $N = 43$ and a mean difference of $0.04(14) \times10^{-3}$, and on the asymmetry itself. The $\mathcal{O}(\alpha) \to\mathcal{O}(\alpha^{2}L)$ step changes it by at most $3 \times10^{-5}$ with point-like loops in both generators and, with GVMD, by up to $4 \times10^{-5}$ in KKMC and $1.8 \times10^{-4}$ in ACRAIC.

The asymmetry comes entirely from initial-final interference. None of the parameters of section 3.9 is odd in the pion charge, so their rescalings nearly cancel in the ratio, leaving only the small angular variation of their templates, and alone they fall short of the step by one to two orders of magnitude. We therefore add a parameter that rescales the initial-final interference beyond the exponent, i.e. the initial-final box and the hard one-photon interference. Its template, the part of the correction odd under a sign flip of the finalstate emissions minus the same part of $\tilde{\beta}_0$, is obtained by flipping the final-state charges in ACRAIC and from interference on minus off in KKMC, and the two agree to $\sim10^{-3}$ at 0.7 GeV. Its normalisation is $(g_e+g_\pi)^k/k!$ at $O(\alpha^k)$, with $g_f=(\alpha/\pi)(\ln(s/m_f^2)-1)$. For KKMC at $O(\alpha^2 L)$ a further parameter of size $(g_e+g_\pi)/2$ accounts for the missing product of the vertex correction and the pion box. The $O(\alpha)$ band is then $\sim10^{-4}$ in both generators with either treatment of the loops. It covers the step at every energy with point-like loops ($|\theta_t|\leq0.23$), and with GVMD the KKMC step at every energy except 0.5 GeV ($|\theta_t|=2.4$). Near 0.5 GeV the GVMD template passes through zero, as the box and the hard-photon interference cancel, and the band misses the ACRAIC step between 0.45 and 0.57 GeV, with $|\theta_t|$ up to 7.4 at 0.5 GeV. In the $\phi$ region the ACRAIC step lies just outside the again small GVMD band at six of the seven energies ($1.0<|\theta_t|<2.0$). At $O(\alpha^2 L)$ the band is $\sim10^{-5}$ for ACRAIC and up to $\sim10^{-4}$ for KKMC, whose band covers the difference between the two generators’ steps with point-like loops ($|\theta_t|\leq0.33$) but not with GVMD ($|\theta_t|>1$ at every energy above 0.4 GeV), because the hard-photon interference in its template has the opposite sign to the GVMD box and dilutes it. All bands are small against the GVMD shift, the $O(\alpha)$ band at most 6% of it and the $O(\alpha^2 L)$ band less, so they are narrower than the lines in Figure 7 and not drawn, and the form-factor modelling dominates the theoretical uncertainty of the asymmetry. In these predictions the

tree-level amplitudes use the STRONG2020 Gounaris–Sakurai form factor of eq. (5), and the CMD-3 Breit–Wigner fit enters only inside the GVMD loops.

KLOE-I

At KLOE-I every event carries a large-angle photon with $E_\gamma> 20$ MeV, a direct test of the one-photon amplitudes of $\tilde{\beta}_1$ with $F_\pi$ at the reduced virtuality of each partition. At $\mathcal{O}(\alpha)$ the fiducial cross sections agree within their statistical uncertainty of $\sim0.01\%$, and the photon and pion angles (Figure 8 to Figure 10) within the statistical precision without a slope. The $\mathcal{O}(\alpha)$ bands, which include the radiative-return parameter of § 3.10, are a few per cent on the fiducial cross section and in every $M_{\pi^+\pi^-}$ range in both generators. At $\mathcal{O}(\alpha^2 L)$ the fiducial band is $\sim0.1\%$, again mostly from the radiative-return parameter, whose $\mathcal{O}(\alpha^2)$ correction is dominated by the one-loop correction to the hard-photon emission, included exactly in both generators. They agree on the $\mathcal{O}(\alpha) \to\mathcal{O}(\alpha^{2L})$ step to $\sim0.06\%$, within the $\mathcal{O}(\alpha^2L)$ band but several times the statistical uncertainty, and bin by bin. Against BabaYaga@NLO (section A) both generators lie $\sim1\%$ above at $\mathcal{O}(\alpha)$ and below at $\mathcal{O}(\alpha^2L)$, by $\sim0.02\%$ (ACRAIC) and $\sim0.09\%$ (KKMC). In the polar angle of the $\pi^+$ (Figure 11) both follow the shape of BabaYaga@NLO and at $\mathcal{O}(\alpha^2L)$ rise relative to it by a few tenths of a per cent across $\theta^+$, more than their bands. With an uncertainty of the size of our $\mathcal{O}(\alpha^2L)$ bands, not quoted for BabaYaga@NLO, the predictions agree. For $\pi^+\pi^-\gamma$ in the same scenario the exact next-to-leading-order calculations of BabaYaga@NLO and PHOKHARA differ by a slope of a few per cent across $\theta^+$ [68], an order of magnitude more than the difference found here, so our results support the BabaYaga@NLO prediction in this distribution.

Vector (PDF)

Vector (PDF)

Vector (PDF)

KLOE-II

At KLOE-II the photon is not detected and the missing momentum must point within $15^\circ$ of a beam, so collinear initial-state radiation dominates. At $\mathcal{O}(\alpha)$ the fiducial cross sections agree within their statistical uncertainty of $\sim0.01\%$. In the $M_{\pi^+\pi^-}$ ranges below 700 MeV, 700–850 MeV and above 850 MeV (Figure 12 to Figure 14) the ratio agrees with one to $\sim0.1\%$, as does the missing-momentum angle near both beams, and the pion angles agree to $\sim0.01\%$. The $O(\alpha)$ bands, including the radiative-return parameter, are $\sim2\%$ on the fiducial cross section in both generators and shrink to $\sim0.02\%$ at $O(\alpha^2 L)$, because the radiative-return parameter scales with the step between the orders. This step, $\sim0.3\%$, comes from the real-virtual correction of collinear initial-state photons, and the generators agree on it to $\sim0.02\%$ and on the fiducial cross section within the statistical uncertainty.

Vector (PDF)

Vector (PDF)

Vector (PDF)

BESIII

At BESIII the tagged photon carries most of the beam energy, with $E_\gamma> 0.4\sqrt{s}$ for most of the fiducial cross section. At $O(\alpha)$ the fiducial cross sections agree to $\sim0.003\%$, within the statistical uncertainty of the KKMC sample. In Figure 15 the ratio agrees with one to $\sim0.1\%$ in the $M_{\pi^+\pi^-}$ ranges below 600 MeV, 600–900 MeV and 900–1200 MeV and in the photon energy, and the $\rho$–$\omega$ interference near 782 MeV is reproduced bin by bin. At $O(\alpha^2 L)$, with the exact two-photon matrix element in both generators, the fiducial cross sections agree to $\sim0.03\%$, within the statistical uncertainty, and the step, $\sim-5\%$, agrees bin by bin. It reduces the fiducial band, dominated at both orders by the radiative-return parameter, from $\sim6\%$ to $\sim0.4\%$. The step is largest at low $M_{\pi^+\pi^-}$, where the photon is hardest, and below $M_{\pi^+\pi^-} \sim500$ MeV and below the peak of the photon energy it reaches tens of per cent and leaves the $O(\alpha)$ bands. These events need a second hard photon sharing the energy [64], a leading-order effect of the $\pi^+\pi^-\gamma$ final state rather than a missing higher order (§ 3.10), which also places the $O(\alpha)$ predictions outside their bands against BabaYaga@NLO in the same regions (section A). At $O(\alpha^{2}L)$ the generators agree within their bands in every distribution.

Figure 16 compares the $\pi^+\pi^-$ predictions with BabaYaga@NLO and with the explicit $\pi^+\pi^-\gamma$ calculation of ACRAIC at next-to-leading order, which includes the tagged photon and the one-loop correction to its emission in the Born. The further photons, matched to the COMIX amplitudes, change the $\pi^+\pi^-\gamma$ cross section by at most a few tenths of a per cent and are included. This prediction agrees with BabaYaga@NLO to $\sim1\%$ in all distributions and to $\sim0.1\%$ in most bins around the $\rho$ peak, within its band in every bin. Its next-to-leading-order correction reduces the fiducial band from several per cent at leading order, dominated by the radiative-return parameter, to a few tenths of a per cent, comparable with the $O(\alpha^{2}L)$ band of the $\pi^+\pi^-$ predictions. The $\pi^+\pi^-$ and $\pi^+\pi^-\gamma$ predictions agree on the fiducial cross section to $\sim0.1\%$ at strict first order, where they are the same calculation (section A), and for ACRAIC at strict second order.

Vector (PDF)

The $\pi^{+}\pi^{-}$ predictions draw the tagged photon from the YFS crude distribution, which follows the eikonal factors and rarely gives a hard photon at a large angle, and $\sim10\%$ of the generated events of ACRAIC are effective, against a few tenths for the $\pi^{+}\pi^{-}\gamma$ calculation, which samples the tagged photon directly. For tagged-photon measurements the latter is therefore both the more natural and the more efficient description.

B

At B the photon carries more than 60% of the beam energy in every event. At $\mathcal{O}(\alpha)$ the fiducial cross sections agree to $\sim0.03\%$, within the statistical uncertainty of the KKMC sample, and the distributions (Figure 17 and Figure 18) within the statistical precision, $\sim0.1\%$ where most events lie. At $\mathcal{O}(\alpha^{2}L)$ they agree to $\sim0.02\%$, within the statistical uncertainty, and the steps, $\sim-10\%$, agree bin by bin, reducing the fiducial band, dominated at both orders by the radiative-return parameter, from $\sim9\%$ to $\sim0.9\%$. As at BESIII the step leaves the $O(\alpha)$ bands where a second hard photon matters, below $M_{\pi^+\pi^-} \sim650$ MeV, below the peak of the photon energy and in both photon angles, by up to tens of per cent in the lowest-mass bins. The angle to the direction opposite the pion pair needs a second photon. With the photon in the Born process (Figure 19), the $\pi^+\pi^-\gamma$ prediction agrees with BabaYaga@NLO to $\sim1\%$ in the distributions and to $\sim0.1\%$ around the $\rho$ peak in $M_{\pi^+\pi^-}$, and its next-to-leading-order correction reduces the fiducial band from several per cent to $\sim0.1\%$, several times smaller than the $O(\alpha^2 L)$ band of the $\pi^+\pi^-$ predictions. Those agree with BabaYaga@NLO and with the $\pi^+\pi^-\gamma$ prediction to a few tenths of a per cent down to 400 MeV and to $\lesssim1\%$ in the lowest bins. In the photon energy and angles (Figure 20) the two descriptions agree within the statistical precision away from the fiducial edges.

Vector (PDF)

Vector (PDF)

Vector (PDF)

Fixed order Results

Table 5 compares the strict fixed-order $O(\alpha)$ prediction of ACRAIC, defined in § 2.5, with the resummed $O(\alpha)$ and $O(\alpha^2 L)$ ones. The resummed predictions of KKMC agree with these to $\lesssim0.07\%$ (Table 4), so resummation acts the same in both generators, and the strict fixed order of KKMC agrees with that of ACRAIC at BESIII and B to $\sim0.1\%$. Resummation changes the cross section by $\sim0.5\%$ at CMD, where the cuts select elastic events, and by $\sim10\%$ at BESIII and B, where a hard photon is required. In the tagged-photon scenarios, including KLOE-I and KLOE-II, the fixed order is the leading order of $\pi^{+}\pi^{-}\gamma$, and most of this change is its next-to-leading-order correction (section A). At BESIII the second order partly undoes the first, bringing the $\mathcal{O}(\alpha^{2}L)$ result closer to the fixed order. The fixed-order $\pi^{+}\pi^{-}$ and $\pi^{+}\pi^{-}\gamma$ calculations agree with the exact fixed order of BabaYaga@NLO to $\sim0.1\%$.

Scenario	Fixed order $\mathcal{O}(\alpha)$ [nb]	Resummed $\mathcal{O}(\alpha)$ [nb]	Resummed $\mathcal{O}(\alpha^{2}L)$ [nb]	$\mathcal{O}(\alpha)$/FO	$\mathcal{O}(\alpha^{2}L)$/FO
CMD	567.42(12)	570.041(87)	570.263(87)	1.0046	1.0050
KLOE-I	2.6604(3)	3.2513(2)	3.2033(2)	1.2221	1.2041
KLOE-II	16.456(2)	15.965(1)	16.017(1)	0.9702	0.9733
BESIII	0.080115(15)	0.088507(10)	0.083911(9)	1.1048	1.0474
B	0.010416(2)	0.011157(2)	0.010058(1)	1.0712	0.9656

Vector (PDF)

For the $\pi^{+}\pi^{-}\gamma$ prediction at next-to-leading order (Figure 21 and Figure 22), resummation changes the pion-pair mass by $\lesssim1\%$, within the band of the fixed order, and the photon energy by up to $\sim10\%$. Just below its upper end the resummed prediction lies below the fixed order, beyond its band, as the phase space for further radiation closes and its logarithms, which the fixed order does not contain, grow.

Vector (PDF)

Vector (PDF)

At CMD the acollinearity $\xi$ probes the soft limit instead (Figure 23). At fixed order the events without a real photon sit at $\xi= 0$ and one real photon gives the tail, and the exponentiated soft-photon factor raises the first bin by several per cent and lowers the tail next to it by $\sim10\%$.

Vector (PDF)

Truncating the resummed result at second order changes it by up to $\sim0.7\%$ (Table 6), almost all of it the third-order term, and the remainder beyond third order lies inside the $O(\alpha^{2}L)$ band in every scenario. This tests the higher orders the resummation contains, while the third-order corrections it does not contain, such as the hard three-photon matrix element and the three-loop virtual correction, are estimated by the theory nuisance parameters alone.

Scenario	$\mathcal{O}(\alpha^{2}L)$/resummed	$\mathcal{O}(\alpha^{3})$/resummed	Third-order term	Beyond third order	TNP band
CMD	1.0001(1)	1.0000(1)	$-3 \times10^{-5}$	$-2 \times10^{-5}$	$5.4 \times10^{-5}$
KLOE-I	1.0073(1)	1.0004(1)	$-6.9 \times10^{-3}$	$-3.9 \times10^{-4}$	$9.8 \times10^{-4}$
KLOE-II	1.0024(1)	0.9998(1)	$-2.6 \times10^{-3}$	$+1.6 \times10^{-4}$	$2.3 \times10^{-4}$
BESIII	0.9981(2)	1.0001(2)	$+2.0 \times10^{-3}$	$-1.2 \times10^{-4}$	$4.1 \times10^{-3}$
B	1.0022(3)	0.9999(2)	$-2.3 \times10^{-3}$	$+0.9 \times10^{-4}$	$8.6 \times10^{-3}$

Theory-nuisance-parameter breakdown

Theory-nuisance-parameter breakdown

Table 7 breaks the fiducial bands into their parameters. The templates $T_c$ of § 2.6 of a generator sum to twice its cross section, and the two generators share this total between the corrections differently in two places. First, the one-photon residual $\widetilde{\beta}_1$, the one-photon amplitude minus the eikonal factor times the Born amplitude, depends on where this subtraction is made, at its own reduced point in KKMC and at the Born point of the stage in ACRAIC. The initial-state template of KKMC is therefore larger at CMD, KLOE-I and BESIII and about four times larger at B, with a correspondingly larger $\widetilde{\beta}_0$ template except at CMD. Second, KKMC assigns the photons of a two-photon amplitude to the initial or the final state partition by partition and ACRAIC by the emitting line. The sum of the two-photon templates is the same in both, negative and a few tenths of a per cent at KLOE-I, but its split between the initial-state and the mixed corrections is not. As independent parameters the three two-photon corrections would add pieces of opposite sign in quadrature and enlarge the KKMC band, by tens of per cent at $M_{\pi^+\pi^-} = 420$–$500$ MeV at CMD, so they share one parameter. The bands of the two generators then agree where initial-state radiation dominates, at KLOE-II and in the CMD elastic peak, and elsewhere differ by the one-photon split, by at most $\sim10\%$ of the band on the fiducial cross sections.

Scenario		$\mathcal{O}(\alpha)$ $\widetilde{\beta}_0$	$\mathcal{O}(\alpha)$ $\widetilde{\beta}_1^{\mathrm{ISR}}$	$\mathcal{O}(\alpha)$ $\widetilde{\beta}_1^{\mathrm{FSR}}$	$\mathcal{O}(\alpha)$ $\widetilde{\beta}_2$	$\mathcal{O}(\alpha)$ RR	$\mathcal{O}(\alpha)$ total	$\mathcal{O}(\alpha^2 L)$ $\widetilde{\beta}_1^{\mathrm{ISR}}$	$\mathcal{O}(\alpha^2 L)$ $\widetilde{\beta}_2^{\mathrm{L}}$	$\mathcal{O}(\alpha^2 L)$ RR	$\mathcal{O}(\alpha^2 L)$ total
CMD	KKMC	0.52	0.13	0.004	0.001	–	0.53	0.002	0.054	–	0.054
	ACRAIC	0.52	0.09	0.000	0.002	–	0.52	0.001	0.053	–	0.054
KLOE-I	KKMC	0.74	7.87	0.14	4.23	24.1	25.7	0.13	0.068	1.01	1.03
	ACRAIC	0.71	6.40	0.09	4.09	24.1	25.3	0.11	0.065	0.97	0.98
KLOE-II	KKMC	0.68	4.83	0.005	0.42	17.2	17.8	0.08	0.061	0.22	0.24
	ACRAIC	0.67	4.82	0.001	0.42	17.2	17.9	0.08	0.060	0.21	0.23
BESIII	KKMC	1.18	10.5	0.03	9.90	63.7	65.3	0.17	0.065	4.11	4.12
	ACRAIC	1.08	6.60	0.05	9.45	63.7	64.7	0.16	0.064	4.08	4.09
B	KKMC	2.01	29.4	0.67	4.87	82.9	88.2	0.20	0.065	8.63	8.63
	ACRAIC	1.34	6.88	0.79	4.71	82.9	83.4	0.18	0.063	8.59	8.59

The radiative-return parameter RR of § 2.6 is needed in the four radiative-return scenarios, where the eikonal approximation of the hard photon, all of $\mathcal{O}(\alpha^0)$, overestimates the cross section and the first order corrects it by $r_1$, a few tens of per cent at KLOE-II and KLOE-I and of order $-1$ at BESIII and B. The $\mathcal{O}(\alpha)$ prediction is then effectively the leading order of $e^+e^- \to\pi^+\pi^-\gamma$, and its missing higher orders are the full next-order correction of that process, which the other templates, $\sim1$–$3\%$ at BESIII and B, would not cover because their normalisations assume a well-behaved series in $\alpha L$. As $r_1$ agrees between the generators to four digits, RR does not depend on how they split the corrections into classes. At $\mathcal{O}(\alpha)$ it dominates the bands and covers the $\mathcal{O}(\alpha^2L)$ correction in all four scenarios, with $|\theta_t| \lesssim1$, but not at the low pion-pair masses of BESIII and B (Figure 15 and 17), whose second hard photon the $\mathcal{O}(\alpha)$ $\pi^+\pi^-$ prediction contains only in the eikonal approximation. This gap is a leading-order effect of the $\pi^+\pi^-\gamma$ final state rather than a missing higher order that a parameter could estimate. The $\pi^+\pi^-\gamma$ calculation of ACRAIC, which has this final state at Born level, shows no such gap, and in the pion-pair mass the matched photons change it by $\sim0.1\%$. At CMD $r_1 \sim+1\%$ is the soft and virtual correction of the elastic events, which the $\tilde{\beta}_0$ parameters already cover, and RR is not used.

The band of the explicit-photon $\pi^+\pi^-\gamma$ prediction (lower panels of Figure 16 and Figure 19) takes as RR parameter $\gamma$ times the first-order correction $r_1^\gamma$ of its leading order, the tree-level

$\pi^+\pi^-\gamma$ process. On the fiducial cross section $r_1^{\pi\pi\gamma}$ is $\sim+5\%$ at BESIII and $\sim-3\%$ at B (section A), so the RR band is a few tenths of a per cent in both, with the other parameters far smaller. Per bin $r_1^{\pi\pi\gamma}$ varies more and the bands follow it. Being proportional to the net correction, RR underestimates the missing higher orders wherever the real and virtual corrections cancel.

Conclusion

We have presented QED radiative corrections to $e^{+}e^{-}\to\pi^{+}\pi^{-}(\gamma)$ with amplitude-level YFS resummation for the first time, implemented independently in SHERPA (via ACRAIC) and KKMC (via CEEX). Pions are point-like sQED scalars dressed with an electromagnetic form factor, either factorised ($F\times\mathrm{sQED}$) or via GVMD in the loops. Both generators match the YFS resummation to the complete first order, including exact one- and two-photon tree-level matrix elements and the initial-final boxes and interference, and to the logarithmically enhanced second order through the two-loop photonic vertices and the exact one-loop correction to one-photon emission. Theory-nuisance-parameter bands estimate the missing higher orders, extending Ref. [86] with a radiative-return parameter for scenarios in which the first-order correction is of order one and with parameters for the omitted parts of the second order. Across the five STRONG2020 scenarios (CMD, KLOE-I, KLOE-II, BESIII and B) the first-order fiducial cross sections of SHERPA and KKMC agree to $\lesssim0.03\%$ and the differential distributions within the statistical uncertainties. At $\mathcal{O}(\alpha^{2}L)$ the bands on the fiducial cross sections are $\sim0.005\%$ for the elastic CMD scenario, $\sim0.1\%$ at KLOE-I and $\sim0.02\%$ at KLOE-II, and in the tagged-photon scenarios $\sim0.4\%$ at BESIII and $\sim0.9\%$ at B, dominated by the radiative-return parameter. The GVMD form factor inside the loops, instead of point-like loops, reduces the forward-backward asymmetry for CMD-3 kinematics by up to $\sim1\%$ near the $\rho$ resonance.

Against BabaYaga@NLO the $\mathcal{O}(\alpha^{2}L)$ fiducial cross sections of CMD, KLOE-I and KLOE-II agree within $0.01\%$ to $0.1\%$. In the polar angles of the pions at KLOE-I, where the next-to-leading-order calculations of BabaYaga@NLO and PHOKHARA differ by a slope of a few per cent [68], both generators follow the shape of BabaYaga@NLO to a few tenths of a per cent and so support it.

The next steps are the two-loop boxes, the genuine two-loop scalar-QED pion vertex and the $\mathcal{O}(\alpha^{3})$ photonic corrections, sized by the theory nuisance parameters. The boxes and the two-loop pion vertex are non-logarithmic at second order, and the $\widetilde{\beta}_{0}^{2\mathrm{L}}$ parameter sizes them at $\sim0.005\%$, relevant only for the elastic CMD scenario. The $\mathcal{O}(\alpha^{3})$ photonic corrections are sized at $\sim0.02\%$ by the one-photon and $\widetilde{\beta}_{0}$ parameters, and by the radiative-return parameter in the radiative-return scenarios. The limiting factor is the calculation of the amplitudes, and once they are available the implementation in both generators is straightforward as the YFS subtraction scheme is defined to all orders.

Acknowledgements

We would like to thank Janusz Gluza for discussions and remarks. We are grateful to Fedor Ignatov for providing the CMD-3 data and for useful discussions. We are also grateful to the STRONG2020 working group for providing the scenarios and for useful discussions. The work is supported in part 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 Centers: CI TASK, ACK Cyfronet AGH) for providing computer facilities and support within computational grant no. PLG/2024/017287. F.K.’s work has been supported by STFC under grant agreement ST/P006744/1.

Comparison with BabaYaga@NLO

We compare the results of § 3 with BabaYaga@NLO [65, 66, 67], which matches the exact $\mathcal{O}(\alpha)$ corrections to a leading-logarithmic QED parton shower (NLOPS), using the RadioMonteCarLow 2 predictions [117] with the same STRONG2020 pion form factor and without vacuum polarisation (NLOPS$_{\mathrm{novp}}$). A full comparison is left to the next STRONG2020 study.

CMD. The fiducial cross sections of KKMC CEEX and ACRAIC agree with BabaYaga@NLO to $\lesssim0.01\%$ at $\mathcal{O}(\alpha)$ and lie $\sim0.04\%$ above it at $\mathcal{O}(\alpha^{2}L)$, the size of the photonic two-loop vertex corrections absent from the NLOPS prediction. The angular distributions $\cos\theta^{\pm}$ and $\theta_{\mathrm{avg}}$ agree with BabaYaga@NLO to $\sim0.01\%$ in both generators (Figure 24a).

Vector (PDF)

For the pion-pair mass (Figure 24b) we use version 2.1 of BabaYaga@NLO [68] with the CMD selection, which includes the exact two-photon tree amplitude also in the $\pi^{+}\pi^{-}$ predictions. Between 600 and 690 MeV, where the events carry one hard photon, both generators lie a few tenths of a per cent above BabaYaga@NLO at both orders, beyond their $\mathcal{O}(\alpha)$ bands. As they agree with each other, the difference lies between the YFS treatment and the parton shower. Below 520 MeV, where every event carries two hard photons, they agree with BabaYaga@NLO at $\mathcal{O}(\alpha)$ and lie $\sim2\%$ above it at $\mathcal{O}(\alpha^{2}L)$.

KLOE-I. With a tagged hard photon in every event the higher orders matter more. Both $\mathcal{O}(\alpha)$ predictions lie $\sim1\%$ above BabaYaga@NLO, and ACRAIC and KKMC at $\mathcal{O}(\alpha^{2}L)$ $\sim0.02\%$ and $\sim0.09\%$ below it (Figure 25). The $\mathcal{O}(\alpha)$ excess, flat in the photon and pion angles, grows with the photon energy from $\sim1\%$ at $M_{\pi^{+}\pi^{-}}>700$ MeV to several per cent below 500 MeV. It lies inside the $\mathcal{O}(\alpha)$ band of a few per cent set by the radiative-return parameter of § 3.10, and only slightly outside it below 500 MeV. It is therefore dominated by the $\mathcal{O}(\alpha^{2})$ corrections to hard-photon emission, which BabaYaga@NLO includes at leading-logarithmic accuracy through its parton shower and KKMC through $\widetilde{\beta}_{2}$ and its real-virtual correction. At $\mathcal{O}(\alpha^{2}L)$ KKMC and ACRAIC agree with BabaYaga@NLO to $\sim0.1\%$ across $M_{\pi^{+}\pi^{-}}$ and the photon energy, and with each other to $\sim0.07\%$ on the fiducial cross section.

Vector (PDF)

KLOE-II. With the photon along a beam the corrections are smaller. The $\mathcal{O}(\alpha)$ predictions lie $\sim0.3\%$ below BabaYaga@NLO, inside the $\mathcal{O}(\alpha)$ band, and KKMC and ACRAIC at $\mathcal{O}(\alpha^{2}L)$ are $\sim0.03\%$ above (Figure 26). The $\mathcal{O}(\alpha)$ deficit is largest, $\sim1\%$, below 700 MeV, where the $\mathcal{O}(\alpha^{2}L)$ predictions are $\sim0.1\%$ above BabaYaga@NLO. The missing-momentum angle behaves alike near both beams.

Vector (PDF)

BESIII and B. In the pion-pair mass (Figure 27) the $\pi^{+}\pi^{-}\gamma$ prediction of ACRAIC agrees with BabaYaga@NLO to $\sim0.1\%$ over the whole range. At $\mathcal{O}(\alpha^{2}L)$ the $\pi^{+}\pi^{-}$ predictions of both generators agree with it to a few tenths of a per cent, and to $\lesssim1\%$ in the lowest-mass bins at B. At $\mathcal{O}(\alpha)$ they lie far above it, outside their bands, below $M_{\pi^+\pi^-} \sim600$ MeV and the photon-energy peak, by tens of per cent at BESIII and up to several times at B. These events need a second hard photon, present in the parton shower and at $\mathcal{O}(\alpha^{2}L)$ but only eikonally at $\mathcal{O}(\alpha)$ (§ 3.10). In the photon spectra at B (Figure 28) the $\pi^+\pi^-\gamma$ prediction agrees with BabaYaga@NLO to $\sim0.1\%$ in both angles and in the energy above the lowest bins. The $\pi^+\pi^-$ predictions agree to $\sim0.1\%$ in the polar angle and lie up to $\sim1\%$ above below the photon-energy peak and at larger angles to the direction opposite the pion pair, within their bands.

Vector (PDF)

Fixed order

BabaYaga@NLO also provides the exact $\mathcal{O}(\alpha)$ corrections without resummation. The strict fixed order of ACRAIC is the next-to-leading order of $\pi^+\pi^-$ at CMD, where it agrees with BabaYaga@NLO within the $\sim0.01\%$ statistical precision, and in the tagged-photon scenarios the $\pi^+\pi^-$ and $\pi^+\pi^-\gamma$ calculations at $\mathcal{O}(\alpha)$ for the leading and next-to-leading order (Figure 29). There the cross sections agree to $\sim0.1\%$, within the statistical precision of ACRAIC, except at leading order at KLOE-I, where ACRAIC lies a few tenths of a per cent, several standard deviations, above BabaYaga@NLO for reasons not yet identified. The distributions agree to a few per cent per bin away from the tails.

Vector (PDF)