Measurement of additional radiation in the initial-state-radiation processes $e^+e^-\to \mu^+\mu^-\gamma$ and $e^+e^-\to \pi^+\pi^-\gamma$ at BABAR

A dedicated measurement of additional radiation in $e^+e^-\to\mu^+\mu^-\gamma$ and $e^+e^-\to\pi^+\pi^-\gamma$ initial-state-radiation events is presented using the full BABAR data sample. For the first time results are presented at next-to- and next-to-next-to-leading order, with one and two additional photons, respectively, for radiation from the initial and final states. Comparison with predictions from Phokhara and AfkQed Monte Carlo generators is performed, revealing discrepancies in the one-photon rates and angular distributions for the former. This disagreement has a negligible effect on the BABAR measurement of the $e^+e^-\to\pi^+\pi^-(\gamma)$ cross section, but could affect other measurements significantly. This study sheds a new light on the longstanding discrepancy in this channel that affects the theoretical prediction of hadronic vacuum polarization contributions to the muon magnetic moment anomaly.

Introduction.The current Standard Model prediction [1] for the muon magnetic moment anomaly a µ falls short of the direct measurement by 5.0σ [2,3], possibly indicating physics beyond the Standard Model.A leading source of uncertainty in a µ arises from imprecise knowledge of the contribution of the hadronic vac-uum polarization (HVP) evaluated from dispersion relations and measurements of the e + e − → hadrons cross section.The dominant channel in this cross section is e + e − → π + π − (γ), which furnishes 73% of the HVP contribution to a µ .However, a longstanding discrepancy persists between the two most precise results using the arXiv:2308.05233v2[hep-ex] 21 Dec 2023 initial-state radiation (ISR) technique, from BABAR [4,5] and KLOE [6][7][8][9].The result from KLOE lies 2.9σ below that of BABAR in the region of the ρ resonance.Recent results from the CMD-3 experiment [10] with the energy scan technique lie above both BABAR and KLOE, exceeding them by 2.2σ and 5.1σ, respectively.This situation prevents a meaningful update of the a µ prediction to compare with the new Fermilab measurement [3].Tension is also observed with the lattice QCD evaluation [11][12][13][14][15][16] of the HVP contribution.The need to clarify these tensions and improve the precision on a µ calls for further studies of the different approaches.
In this Letter we focus on higher-order radiative processes, likely to affect differently the various ISR experiments, namely the next-to-leading order (NLO) processes e + e − → µ + µ − γγ and π + π − γγ, and the next-to-next-toleading order (NNLO) processes e + e − → µ + µ − γγγ and π + π − γγγ.Each photon can be emitted from the initialstate e + or e − , or a final-state muon or pion (FSR).Building on our previous work [4,5], where very loose kinematical constraints were applied in order to reduce dependence on simulation, we make improved measurements of the relative contributions of events with two photons and study their kinematics.We then perform the first study of events with three photons.
Dataset and simulated samples.The analysis is based on 424.2 fb −1 (43.9 fb −1 ) of data [17] collected at the SLAC PEP-II asymmetric e + e − collider operated at (and 40 MeV below) the Υ (4S) resonance.The BABAR detector and performance are described in detail elsewhere [18,19], and its response, including the variations of beam and detector conditions over time, is simulated with GEANT4 [20].
The two Monte Carlo (MC) generators used in this analysis are described in Appendix A. Signal processes e + e − → Xγ(γ), with X = µµ or ππ, are simulated with Phokhara with ten times the data statistics.Small samples of these processes are also simulated with the AfkQed generator.In both cases, one high-energy photon is generated within the detector angular range.Phokhara generation is limited to NLO, with up to one additional ISR or FSR photon emitted with an angular distribution according to the full NLO matrix element.AfkQed provides NLO and NNLO simulated samples, within the approximation that the additional ISR photons are generated collinear to the beams.
Background ISR processes e + e − → K + K − γ(γ) and e + e − → Xγ (X = π + π − π 0 , π + π − 2π 0 , ...) are simulated with Phokhara and AfkQed, respectively.Background processes e + e − → qq (q = u, d, s, c) are generated with JETSET [21] and e + e − → τ + τ − with KK2f [22].As the models used in the generators for the background processes may be unreliable at low multiplicity or do not reproduce the measured hadronic spectra, data-driven corrections are applied to the MC estimations (Appendix B).Backgrounds mostly affect the pion channel while they are very low in the muon channel, mainly from well simulated τ τ events.
Event selection.Radiative events are selected by requiring a detected photon in the laboratory polar-angle range 0.35 < θ < 2.4 rad, and with a measured energy E * γ > 4 GeV in the center-of-mass (c.m.) frame.Exactly two tracks of opposite charges are required to extrapolate to the collision region, each with: transverse momentum > 0.1 GeV/c, θ in the range 0.4−2.45rad, at least 15 hits in the drift chamber.Events can contain any number of additional photons and tracks that do not satisfy the requirements.Detected photons with E γ > 50 MeV and polar angle in the range 0.35−2.4rad are retained for the fits described below.The photon with the highest measured energy in the c.m. frame is denoted γ ISR .Events in which both tracks are tightly identified [19] as muons (pions) are assigned to the dimuon (dipion) sample.In the following, the rate of each radiative process is quoted as a fraction of the total yield in those dimuon and dipion samples, respectively.Topologies considered in the analysis are sketched in Fig. 1.NLO study.The dimuon and dipion samples are submitted to two types of kinematic fits to e + e − → π + π − γ ISR γ, with one photon in addition to γ ISR : -γ ISR γ LA fits: a photon γ LA is detected at large angle (LA) to the beam, and its measured energy and angles are used in a fit.
-γ ISR γ SA fits: an undetected small angle (SA) photon γ SA is approximated as being collinear with either beam.
Both NLO fits use the measured γ ISR energy and direction, and the parameters and covariance matrices of the two tracks.To be consistent with the cross section measurement, the pion mass is assumed for both tracks [4,5].All events undergo the γ ISR γ SA fits while only events with at least one additional detected photon with E γ > 50 MeV undergo the γ ISR γ LA fits, taking each photon in turn.The fit with the lowest χ 2 is retained.An optimized contour in the (χ 2 γISRγSA , χ 2 γISRγLA ) plane (so-called 2D-χ 2 selection) is used to reduce backgrounds.This only selection applied to the NLO samples retains 99% of the signal events, as determined in MC estimations and measured on the muon sample.Events are assigned to the NLO SA or LA samples according to the lowest χ 2 and if the additional photon energy exceeds 0.2 GeV.As γ LA photons are measured in the detector, the energy threshold is expressed in the laboratory system, while for γ SA ISR photons, it is set in the c.m. frame.Events that satisfy the 2D-χ 2 , but not the energy, requirements are categorized as leading order (LO) events.We retain dimuons in the m ππ mass range from threshold to 1.4 GeV/c 2 and dipions in the range between 0.6 and 0.9 GeV/c 2 around the ρ resonance where backgrounds are manageable (5.5% and 23% for the SA and LA pion samples, respectively, dominated by muon misidentification).For the dipion sample, the background from fake photons due to pion interactions in the calorimeter is measured in data and MC simulations and the relative excess of (21.5 ± 3.5)% in data is corrected for.
In the SA samples, the energy distributions E * γSA of the additional photon are shown in Fig. 2 for muons and pions.As expected from their ISR nature, the results of the two processes are consistent.A marked discrepancy between data and the NLO Phokhara predictions is observed, both in shape and normalization, in the dominant low-energy region.
This discrepancy is further investigated.The collinearity approximation of the γ SA photon with the beams in the γ ISR γ SA fit is found to induce energy-and polarangle-dependent biases for both the χ 2 value and the γ SA fitted energy, as determined in MC studies.However a comparison of the outputs of the γ ISR γ SA and γ ISR γ LA fits for events with a detected photon verifies with data that the E * γSA bias is correctly simulated, hence is not responsible for the observed data to Phokhara discrepancy.A further test, independent from the NLO fits, is provided by a zero-constraint (0C) calculation of the energy and angles of the additional photon.With the constraint of 4-momentum conservation in e + e − → π + π − γ ISR γ 0C , the calculation uses the two track momenta and the γ ISR direction.To avoid spurious γ 0C photons arising from shower fluctuations, the γ ISR energy measurement is ignored and γ 0C candidates within a cone of 0.5 rad around the γ ISR direction are excluded.The distributions of the polar angle θ γ0C for events with a calculated photon energy E * γ0C above 0.2 GeV, are shown in Fig. 3(a) for the data and Phokhara dimuon samples.Comparison reveals that Phokhara severely overesti- mates the NLO rate at small angles to the beams, with improving agreement at larger angles.As detailed studies demonstrate (Appendix C), this feature is not due to θ γ0C resolution.The data/Phokhara ratio of the photon energy spectra E * γ0C , shown in Fig. 3(b), reveals a mismatch between the LO and NLO rates and presents a positive slope, as in the γ ISR γ SA fit.Since no angular assumption enters the 0C calculation, the observed energy dependence of the data/Phokhara ratio is thus not an artifact caused by the collinearity approximation of the γ SA photon.In contrast to Phokhara, neither mismatch between the LO and NLO categories nor a data/MC energy slope are observed when conducting the same exercise with AfkQed.
In the LA samples, no energy dependence is observed in the data/MC ratio of the E γLA distributions from the γ ISR γ LA fit, neither with Phokhara nor AfkQed, and the data to Phokhara discrepancy in normalization is much reduced.The relative contributions of ISR and FSR processes are measured using the minimum of the two angles in the laboratory frame between a track and the LA photon θ min(trk,γLA) [5].We observe clear peaks in the data and MC distributions of this angle below 20 • , due to FSR, over a broad distribution from LA ISR (Appendix D).
NNLO study.We then perform an NNLO analysis using three kinematic fits, γ ISR2 γ SA , γ ISR γ SA γ LA and γ ISR2 γ LA , where two photons are allowed in addition to γ ISR .The SA photons γ SA are approximated to be collinear with one of the beams and the LA photons γ LA are measured in the detector.All events undergo the γ ISR2 γ SA fit, with one SA photon along each beam.Only events with at least one or two additional detected photons with E γ > 50 MeV enter the γ ISR γ SA γ LA and γ ISR2 γ LA fits, respectively.No 2D-χ 2 selection is applied, and events are assigned to the γ ISR2 γ SA , γ ISR γ SA γ LA or γ ISR2 γ LA sample if the χ 2 of that fit is lower than all others, including the NLO fits.
In all NNLO categories, the background from NLO events with spurious additional photons is estimated with the Phokhara sample, after correcting for data and MC rate differences observed above at LO and NLO levels.The non-NLO background is very small in the dimuon samples, but the dipion samples are affected by larger background from multihadronic ISR processes, especially those containing a π 0 .This background is reduced by BDT (boosted decision tree) selections designed for each NNLO category (Appendix E).To further reduce background contributions, the fitted additional photons are required to pass energy thresholds specific to each NNLO category.After these selections, the NLO contribution to the background is dominant in all NNLO muon categories and pion γ ISR2 γ SA .In the pion γ ISR γ SA γ LA and γ ISR2 γ LA samples, the multihadronic ISR contributions dominate but do not exceed 45% of the data.The NNLO data signal yields are obtained by subtracting both the NLO contribution and the estimated non-NLO background.
Significant NNLO signals are found in data in all γ ISR2 γ SA , γ ISR γ SA γ LA and γ ISR2 γ LA categories.Results for the most abundant γ ISR2 γ SA category are summarized in Fig. 4, requiring E * γSA > 0.2 GeV for the most energetic SA photon and E * γSA > 0.1 GeV for the other.The χ 2 distributions and energy spectra of the SA photons are shown for muons (a), (c) and pions (b), (d).In both samples, the NLO background (solid/red histogram) dominates over the very low non-NLO background (dashed/blue histogram).The energy spectra of the SA photons for γ ISR2 γ SA signals in data are further compared to AfkQed predictions, where the AfkQed NNLO signals are restricted to the events with two SA photons that satisfy the energy thresholds.
Final results combining NLO and NNLO studies.Final results are obtained considering all NLO and NNLO radiative processes using a feedthrough probability matrix between true and reconstructed categories established with the Phokhara and AfkQed samples.Feedthrough contributions are estimated through an iterative procedure to correct for rate differences of each category in data and MC samples.Feedthrough yields estimated with AfkQed are corrected to account for its missing LA-ISR component.
The NLO γ ISR γ SA rates and E * γSA energy distributions are further corrected for a hidden NNLO feedthrough of the same size as the open γ ISR2 γ SA rate, due to the additional collinear ISR radiation: while double SA radiation from opposite beams is identified by the γ ISR2 γ SA fit, double SA radiation from the same beam cannot be distinguished from a single-photon radiation where the equivalent single photon merges the two γ SA ener- gies.The same-beam NNLO component in the measured γ ISR γ SA spectrum is thus inferred from the sum of γ SA energies in data γ ISR2 γ SA events, and subtracted.As shown in the bottom panels in Fig. 2, the NLO SA photon energy spectrum in data after the NNLO correction agrees better in shape with the Phokhara prediction, but it has a deficit in rate by more than 20%.The final fractions of all fit categories with respect to the total event sample are given in Table I.The systematic uncertainties include those from efficiency corrections, background subtraction and feedthrough corrections with the latter two being the dominant contributions.Template fits to the θ min(trk,γLA) distributions in the γ ISR γ LA , γ ISR γ SA γ LA and γ ISR2 γ LA samples are used to estimate the fractions of FSR and large-angle ISR at the NLO and NNLO levels (Appendix D).
The results are summarized below: • NNLO contributions are clearly observed with a total fraction of (3.47 ± 0.38)% for muons and (3.36 ± 0.39)% for pions.This allows the correction of NLO rates for cross feeds from NNLO categories.The corrected shape of the energy distribution for single SA photons provides further evidence, and good internal consistency.
• The NLO SA-ISR fractions in the Phokhara generator are higher than in the data, with data/MC ratios of 0.763±0.019for dimuons and 0.750±0.008for dipions, while the respective LA-ISR ratios, 0.96±0.03and 0.98±0.03,are consistent with unity.This indicates a problem in the angular distribution of the NLO photon generated by Phokhara, with a large excess at small angles to the beams.
• The AfkQed generator provides a reasonable description of the rates and energy distributions of NLO and NNLO data.The sum of SA-ISR and LA-ISR rates in data up to the cutoff at 2.3 GeV photon energy applied at generation (Appendix A) leads to slightly high data/MC ratios of 1.061 ± 0.015 for muons and 1.043 ± 0.010 for pions.
• The ratio between data and the Phokhara prediction for NLO FSR is found to be 0.86 ± 0.05 for muons and 0.76 ± 0.12 for pions.The corre- sponding data/AfkQed ratios are 1.09 ± 0.06 for muons and 1.08 ± 0.10 for pions.In both cases, the ratio pion/muon is consistent with unity and supports the pion pointlike behavior for additional FSR.This result, obtained with a fake-photon subtraction that both includes a data/MC correction and takes into account NNLO feedthrough, supersedes the previous result of an excess of (21 ± 5)% reported in the BABAR publication [4,5].
The BABAR measurements [4,5] are performed with a very loose selection that incorporates all NLO and higher order radiative processes.Therefore, no dependence on a particular event generator is introduced when handling additional radiation (Appendix F).Only the acceptance, determined using Phokhara, is affected by the generator shortcomings.However the acceptance correction of (0.3 ± 0.1) × 10 −3 is negligible compared to the estimated systematic uncertainty of 0.5% in the π + π − (γ) cross section measurement [4,5].
In contrast, other experiments using the ISR approach [6-8, 23, 24] do not measure additional radiation.They select events mostly in the LO topology, which includes virtual and soft emission, and rely on the Phokhara generator to account for the missing hard NLO contribution.While the total NLO correction in the cross section is strongly reduced by the near cancellation of the hard NLO terms by the infrared-finite sum of soft and virtual terms [25], each component is large and systematic uncertainties related to the undetected hard NLO contribution far exceed the theoretical uncer-tainty in the total NLO correction.The results of the present analysis question the validity of the procedure relying on Phokhara on two grounds: first, the hard NLO yield in Phokhara is significantly larger than our measurement, and second, NNLO contributions, absent in Phokhara, are found to be at a level larger than the systematic uncertainties quoted in Refs.[6-8, 23, 24].A quantitative evaluation of these effects requires detailed studies of the specific experimental conditions, which is beyond the scope of this Letter.However the unique measurements reported here, and their implication for ISR experiments that rely on the Phokhara generator to account for unmeasured radiative events, offer new key insights that may allow progress in the longstanding quandary that affects the theoretical prediction of HVP contributions to a µ , with respect to both the final value and its currently degraded precision.erly normalize these backgrounds; ii) to optimize the 2Dχ 2 selection; iii) to reduce backgrounds in NNLO pion samples.As an example, the BDT designed to select the γ ISR2 γ SA pion signal uses discriminant variables as the γ ISR energy and angles with respect to other photons and tracks, multiplicity and total energy of additional photons, and the pion angle with respect to the ISR photon direction in the dipion c.m. system.The signal training sample is from the NNLO AfkQed events satisfying the χ 2 and SA photon energy requirements of the γ ISR2 γ SA selection, while background is from simulation of nonππ processes.Figure 7 shows the corresponding BDT response distributions for signal and backgrounds (top), and the comparison between data and backgrounds (bottom).The good agreement of the data and non-ππ background in the far-negative BDT response region indicates that the background estimation is satisfactory.The response of the NLO background taken from Phokhara is also well simulated.The selection applied maximizes the signal-over-background ratio and results in a BDT efficiency of 0.926(2) (where the uncertainty is statistical).Appendix F: The BaBar data-driven approach.
Previous and ongoing measurements of the ππ cross section in BABAR rely on NLO kinematic fits that explicitly include one photon in addition to γ ISR .Although some higher order events may be rejected by the NLO fits, this loss is corrected for by the 2D-χ 2 efficiency measured on the data [4,5].It is only when calculating the event acceptance prior to the kinematic fits that generator dependence occurs.
To quantify the effect of the overestimated hard NLO contribution in Phokhara, the BABAR acceptance is computed as a function of mass at LO and NLO levels using different options of the generator.Figure 8 shows the ratio of NLO/LO acceptances, for full NLO events (top) or events with hard NLO (E * γ > 50 MeV) excluded (bottom).Final states at LO (no generated additional photon) have the same acceptance as virtual+soft NLO events (no hard additional photon) at better than 1 per mil [Fig.8(bottom)].In contrast, hard NLO radiation does affect the acceptance, with a small variation in mass within 1% [Fig.8(top)].This acceptance correction, which needs to be computed from Phokhara, is however strongly correlated between the ππγ(γ) and µµγ(γ) processes, and therefore the effect largely vanishes when taking their ratio for the ππ cross section measurement, for an overall acceptance correction of (0.9981±0.0004), constant with mass from threshold up to 1.4 GeV/c 2 .The systematic bias induced by the hard NLO excess observed in Phokhara and the missing hard NNLO component is estimated from this result to be (0.3 ± 0.1) × 10 −3 .

FIG. 2 :
FIG. 2: Distributions of the fitted energy of the SA photon in the c.m. frame for the muon (a) and pion (b) samples.The data (full dots) are compared with Phokhara (open/blue dots) in each upper plot, and their ratio is shown in the middle plot.The lower plots show the ratio after the NNLO corrections discussed later in the text.

FIG. 3 :
FIG. 3: (a) Distributions of the calculated polar angle θγ 0C for dimuon events with E * γ 0C > 0.2 GeV in the data (full dots) and Phokhara (open dots), along with the calculated θγ 0C using MC truth information (dashed histogram).(b) Ratios of data to Phokhara (full dots) and data to AfkQed (open dots) distributions of E * γ 0C in the energy range below and above 0.2 GeV corresponding to LO and NLO, respectively, in muon (upper) and pion (lower) samples.

FIG. 4 :
FIG. 4: Distributions of the χ 2 (a), (b) of the γISR2γSA kinematic fit for the muon (left) and pion (right) samples.The corresponding distributions (c), (d) of the c.m. energies of the more (full dots) and less energetic (open dots) SA photons after subtraction of backgrounds and NLO feedthrough, compared with AfkQed predictions (histograms), normalized to the data for E * γ SA < 2.3 GeV.

FIG. 7 :
FIG. 7: BDT response distributions for the training samples (top panel); BDT response distributions showing data and different background components (middle and bottom panels).

TABLE I :
Event fractions in data for the µµ and ππ processes in all fit categories.The numbers in parentheses represent uncertainties, where the first is statistical and the second systematic.The results, except for NNLO 2LA (which is not simulated by any generator currently available) are corrected using efficiencies that vary category by category between 99% and 72%, except for NLO FSR ππ (40%) and NNLO FSR ππ (22% due to BDT selection.)