JHEP02(2014)114 Springer , a February 1, 2014 February 26, 2014 : December 18, 2013 : : G

JHEP02(2014)114 Springer , a February 1, 2014 February 26, 2014 : December 18, 2013 : : G. Rodrigo 10.1007/JHEP02(2014)114 c , Accepted production are calculated Received Published doi: γ − µ + µ [email protected] , T. Riemann, Published for SISSA by b [email protected] , M. Gunia, b [email protected] [email protected] J. Gluza, and their implementation in the , , b γ . − 3 µ H. Czyż, + 1312.3610 a The Authors. µ d c B-Physics, Standard Model

KLOE and Babar have an observed discrepancy of 2% to 5% in the invariant , [email protected] → Dedicated to the memory of our late collegue and friend Jochem Fleischer (1937 - 2013). cross section predictions of the Monte Carlo event generator PHOKHARA7.0. In − γ e − µ + E-mail: [email protected] [email protected] University of Silesia, Uniwersytecka 4,Deutsches PL-40-007 Elektronen-Synchrotron, DESY, Katowice, Poland Platanenallee 6, 15738 Zeuthen,Max-Planck-Institut Germany für Physik, Föhringer Ring 6, 80805 Munich, Germany Instituto de Física Corpuscular,Consejo Universitat Superior de de Valéncia, InvestigacionesParc Científicas, Científic, 46980 Paterna, Valencia,Department Spain of Field Theory and Particle Physics, Institute of Physics, + b c d a Open Access Article funded by SCOAP origin of the observed experimental discrepancy. Keywords: ArXiv ePrint: which form a gauge-invariantmay set contribute when combined to with certaincomplemented their distributions with box two-photon at diagram final the state partners.erator percent emission PHOKHARA7.0. They contributions level. We not demonstrate included that Alsocharge-averaged in the experimental the numerical the setups, real influence gen- reaches, not emissionenergies. for more was realistic As than a 0.1% result, we at exclude KLOE the approximations and in 0.3% earlier at versions of BaBar PHOKHARA as µ this article, the completeand NLO implemented radiative in corrections thebility to Monte is Carlo guaranteed by event two generatorThe independent novel PHOKHARA9.0. approaches features to include Numerical the the contribution real relia- of and pentagon the diagrams virtual in corrections. the virtual corrections, Abstract: pion pair production cross section. These measurements are based on approximate NLO F. Campanario, and V. Yundin e event generator PHOKHARA Complete QED NLO contributions to the reaction JHEP02(2014)114 3 ]. At high energies, 5 – 1 14 γ 17 − µ

+ – 1 – 9 µ → 8 − e 18 + 20 6 6 e 8 3 10 9 1 16 17 18 2.2.2 Two-hard photon emission 2.1.1 The2.1.2 DT-method The QH-method 2.2.1 Soft photon emission One of the methods used to extract the hadronic cross section is the ”radiative return”, A.1 KLOE A.2 BaBar pion form factor measurements at BaBar and KLOE PHOKHARA9.0 2.1 Virtual corrections 2.2 Real photon emission initial state radiation at the energy lowered by the emitted photons. Due to the complexity the cross section canhas be to calculated rely using on the perturbative experimental QCD, measurements. however atexploiting low the energies fact one thatcan the be cross section factorised of into a the known process perturbative with factor initial and state the photon hadronic radiation cross section without 1 Introduction The total cross sectionphysical for electron-positron observable annihilation weighting into significantlymagnetic hadrons moment is on and a the the running veryof electromagnetic theory important the fine Standard error structure Model constant and of used its in extensions. the the For tests muon recent reviews see anomalous e.g. [ C The soft photon integrals A KLOE and BaBar event selections cuts B Two-hard photon emission 5 Impact of the radiative corrections added to6 the event generator Conclusions on the 3 Implementation of the radiative corrections in the event4 generator Tests of the code Contents 1 Introduction 2 Radiative corrections to JHEP02(2014)114 ] ] 19 13 , 12 we give a sketches the C 3 ] and by the Ka- and 16 ]. Both experiments , B 11 15 – 9 ], with the expected error scattering was measured, 6 − e Consequently, detailed Monte + 1 . The latter process is used for e and appendices ] and KLOE [ 2 ]. This enabled us to compare specific 8 , + photons 19 7 , −

]. In view of the planned improvement of the 18 µ 2 is the energy of the collider. This demands a – 2 – + s µ → are calculated, tested and implemented into the event − γ e − + µ e ] with high precision as a ﬁrst numerical test; see ref. [ + µ 16 ] using the GRACE system. The electroweak radiative corrections were , and peak and 5% when approaching the energy of 1 GeV. The origin → 14 ], in PhD theses [ 15 ρ − 17 e can appear, where + e 2 e s/m + photons − , solved in ref. [ π + ¯ ttγ π → → − e The article is organised as follows: in section In this article, the complete radiative NLO Quantum Electrodynamics (QED) correc- The most important contribution to the hadronic cross section is the pion pair produc- − Technically, the accuracy problems in the calculation of the radiative corrections are similar to the ones + 1 e e + detailed description of the calculation of the radiativein corrections. Section calculated there, but with the photon emitted at large angles only. for details. Here,Carlo environment, we PHOKHARA9.0. perform Becauseical a there instabilities, complete are and calculation several becausetwo in known we independent sources implementations the have of of no frame the numer- external QEDcomplete of virtual cross-check radiative corrections. a available, corrections, we Having realistic detailed implemented organizedare physics the Monte for presented studies here. became possible, and their results of the order of good control on theCarlo numerical accuracy studies of are our mandatory. amplitudes. towice/Zeuthen First group studies were [ presentedphase in space refs. points [ with [ most challenging. Because thereenhanced is contributions, they no are scale expectedmic available to enhancements which be can might small. be lead generated However,experimental to it in setups. is logarithmically some The known regions fact that of that logarith- reason the we poses phase can an space not additional within neglect challenge complicated the in the small calculation electron of mass the for virtual the amplitudes same since ratios above mentioned discrepancy between BaBarcalculation and KLOE, and it to is establish essential the to importance make a of full the NLO missingtions contributions. to the reaction generator PHOKHARA. From a technical point of view, the pentagon diagrams are the event generator to extractphysical the hadronic content. cross The section,e it PHOKHARA is event necessary generator to wasmonitoring check the used carefully luminosity. to its So generateincluded far, the the the version reactions dominant of next-to-leading PHOKHARA order used (NLO) by BaBar radiative and corrections. KLOE In view of the four times smaller thanmuch the as present one, possible. itusing is The the crucial pion radiative to return production pullquote method, down individual cross errors by the section at BaBar theoretical the in [ errorthem level of is as a up fraction to of 2%of a at percent, the the while discrepancy the remains discrepancy unclear. between Since both experiments use the PHOKHARA [ experimental framework can be achieved only by means oftion an channel. event Its generator. accuracy isof an issue the as muon it provides anomalous thedirect magnetic main measurement source moment of of [ error the in muon the evaluation anomalous magnetic moment [ of the experimental setup, the extraction of the hadronic cross section within a realistic JHEP02(2014)114 . → . In − e + 2.2 e , the main 4 , where a real 2 (d) FSR . γ − µ + µ → − e + e γ − (c) FSR µ + µ

– 3 – ] to regularize the ultraviolet (UV) and infrared → 20 − e + e (b) ISR : The tree diagrams for , the possible impact of the radiative corrections, analogous to 5 − + µ µ Figure 1 γ γ (a) ISR . contains the conclusions. at NLO QED. They can be classified into several independent gauge invariant . There are two types of contributions, those with initial state photon emission 6 γ 1 They do not constitute a class of gauge independent diagrams by themselves. Gauge At NLO QED, there are the virtual and the real corrections resulting in threeWe types use of dimensional regularization [ − − + e e µ + The first class involves loopmost corrections challenging with diagrams the are two lepton thephoton lines four is attached pentagon emitted to diagrams, the from shown loop. an in internal The figure line. invariant groups are formed when a pentagon is associated with two box diagrams where a 2.1 Virtual corrections Besides photonic self-energy corrections,µ there are 32subsets, diagrams which contributing we to will call Penta-Box, Box-Triangle-Bubble and Triangle contributions. malization counter-terms. Both theThese virtual divergences and cancel the in realcanceled corrections the and are sum both infrared for the divergent. infrared-safe virtual+realnumerically (soft) observables. integrable. and The the Details IR real of (hard) divergencesthe the corrections are following, become real we separately emission describe calculation the are method given used in to section compute the virtual amplitudes. contributions to the cross section,interference terms. the ISR and the FSR contributions, and their(IR) ISR-FSR divergences. The UV divergences of the virtual amplitude are removed by the renor- 2 Radiative corrections to The tree level diagramsfigure contributing to the(ISR) leading and order final (LO)separately state amplitude gauge photon invariant. are emission shown (FSR). in The ISR and FSR pairs of diagrams are the relevance of the NLOinvestigated. radiative In corrections, which section werethose missing studied here, in on PHOKHARA7.0, the is Section pion form factor measurements of BaBar and KLOE is discussed. implementation of the radiativetests corrections of into PHOKHARA9.0. the correctness In and section the numerical stability of the code are presented. Further, JHEP02(2014)114 . . + − . µ µ 4 . γ − . The treat- µ + 4 (d) (c) µ → + − − e e e + e + − µ µ ). The triangle contributions 4 ] and will not be discussed here. (c) 23 + − e e , will be discussed in detail in section

. (b) γ 1 – 4 – − ]. These self-energies are treated separately and + − µ µ µ + 22 µ , 21 They constitute a gauge-invariant universal correction → − 2 e (b) + e + − e e . There are two independent gauge invariant subsets, for FSR (two 4 ) and for ISR (two lower lines of figure . There, a real photon is emitted from one fermion line, and the other 4 − + 5 µ µ : The four diagrams with pentagon topology for . The contribution of these twelve Penta-Box diagram combinations, inter- 3 (a) : One of the four gauge invariant combinations of a pentagon with two boxes with . (a) Figure 2 In the present article, two independent programs using two different methods are used. Finally, we mention the diagrams with external photon emission and self-energy inser- Box-Triangle-Bubble and Triangle contributions contain corrections to one lepton line + − The sum of contributions from diagrams with real emission from fermionic self-energy insertions to the . e e 2 ment of vacuum polarisationresonance in contributions the is described PHOKHARA in event detail generator, in together ref. with [ narrow photon propagator vanishes due to the Furry theorem. tions to the photonwhich propagator. can be accountedconstant for to in the any appropriate QEDhave scale calculation been [ by omitted simply from running our the fixed-order fine loop structure amplitude definition in figure photon, connecting to thecan other be lepton found line. in The figure upper contributing lines boxes, of vertices figure andare bubbles given in figure photon (off-shell) entering a 3-point function is connected to the other fermion line. fering with the tree level diagrams of figure (electron or muon)photon and are are attached further tothe classified the loop same depending corrections lepton whether to line. the a The lepton loop Box-Triangle-Bubble line class and with contains the a all real real (on-shell) photon and a second off-shell Figure 3 external photon emission for photon is radiated from theically same in external figure (electron or muon) line. This is shown schemat- JHEP02(2014)114 . . . γ − µ + µ → (l) (p) (d) − (d) (h) e + e (o) (k) ]. We refer to it as “Double preci- (c) (c) (g) 24

– 5 – libraries [ . PJFry γ − (j) (n) (b) µ (f) (b) + µ → − e + ], and we refer to it as “Quadruple precision - Helicity method” (QH- e 25 (a) : The four triangle diagrams with external photon emission in : The set of sixteen one-loop Box, triangle and self-energy diagrams with internal (i) (e) (a) (m) . . Figure 5 numerical routines, includingsion the - Trace method”described (DT-method). in ref. The [ othermethod) one because is numerical calculations based areindependent on done implementations the partially are using helicity necessary quadruple formalism to precision. as gain Such suﬃcient numerical reliability. One is based on the trace method and the calculations are done using double precision Figure 4 photon emission in JHEP02(2014)114 ∼ GeV /s 2 e m 2000 , usage of / ], combined 1 ≈ ) = 0 38 . We used the , 1 e p 6 19 ( m · 1 p ]. 24 ] and then dressed with par- 26 . We reduce the tensor integrals ]. Next, the diagrams are passed = 3 28 GeV, one faces e.g. a ratio R 10 ] according to the QED model description − 27 ] for scalar integrals. More technical details

should be understood as generic off-shell cur- 41 = 1 – 6 – 2 V s √ and 1 , the transversality condition, e.g. V ν γ ) d ] or OneLOop [ − ] and uses the PJFry tensor reduction package [ 3 40 , ] script, which substitutes Feynman rules according to the selected ]. ]. It will be interesting to see whether this improves speed or stability of the , constitute the first building block, which we call Boxline and also 37 = (2 39 – 29 19 6 44 µ , 31 γ ν 36 γ ]. It is alternative to tensor reduction and relies instead on the direct calculation of µ dimensions to scalar 1- to 4-point functions. They depend on the reduction γ 43 , . The DT implementation of tensor integral calculation relies on the approach ]. Our choice (with a one-to-one correspondence) are the so-called chords, the 2 9 42 − , − 30 , 10 37 ]. To build the virtual amplitude, four building blocks are used. Corrections to 25 − 25 = 4 Advanced tensor integral calculations became a standard task in recent years, mainly For the calculation of the newly added one-loop pentagon contributions, one has to 7 A new approach to the treatment of pentagon diagrams is under development in the OLEC d 3 − rents which can be, insecond this lepton case, line. an The on-shell physical photon amplitude is or built an by off-shell considering photon, allproject which physical [ permutations forms the tensor contractions [ numerics. ref. [ a lepton line withternal two bosons, real figure (on-shell orinclude off-shell) the photons corresponding attached counter-termseffective in current which a approach, are fixed thus, order not of shown ex- in figure with QCDLoop/FF [ can be found in ref. [ 2.1.2 The QH-method The second implementation (QH-method) uses the helicity formalism as described in vanishing inverse Gram determinants. Anotherin one is our just physical the extreme process,as spread because of an scales we independent met cannot10 parameter. neglect the With electrondeveloped mass in refs. [ refs. [ shifts of internal momenta with respect to the looptriggered by momentum LHC [ physics. Nevertheless, ensuringfor sufficient numerical several stability is reasons. demanding An often discussed issue is the treatment (or avoidance) of small or and multiplies them byare the connected complex by conjugated the set completeness relation. of Born Then, Dirac diagrams.calculate traces 5-point The are tensor taken. fermion Feynman integrals lines up toin rank basis chosen. Often one uses as basis momenta, the external momenta of the diagram as in model. Further manipulationscations are can done be with enabled FORM.identities by like In setting configuration addition, parameters. someDirac This general equation includes simplifi- and gamma momentum algebra FORM conservation. tablebase. The We resulting use expressions it are as an written input in in the the squaring program which sums the diagrams With the DT-method, topologies areticles generated and by QGRAF momenta [ by thefile. DIANA The program resulting output [ containswhich a list is of defined Feynman diagrams by in thethrough the TML the textual representation, markup FORM [ language script [ 2.1.1 The DT-method JHEP02(2014)114 , 3 . 2 V ) d ( ], following the 1 V 47 , ] which reduces the . 46 γ 45 − µ ]. + µ 25 → 2 − V e ) ] to be able to compute diagrams + c ( e 25 1 V

– 7 – 2 V ) b -point diagrams and the method of [ and constitute an independent gauge group. To compute ( 4 3 1 V : Boxline contributions for ] for up to ] for higher-point tensor integrals. The scalar integrals are calculated 30 of CPU time, reducing the CPU time of the code by a factor 10. 25 Figure 6 . In addition, we use the vertex corrections to a lepton line with one real % ]. 4 2 V 48 , ) a ( 22 1 V . The third building group is formed by the Penta-Box diagrams depicted in figure We have implemented a rescue system for phase space points where the Ward identities The building blocks do not use special properties like transversality or being on-shell for We use a cache system in all the building blocks such that the information of the loop- 5 . important because the phase spaceinstabilities integration in of the the calculation virtual of contribution the shows one-loop numerical tensor integralsare [ not satisfied with anapplying accuracy of quadruple at least precision three only digits. First, to we the calculate the scalar amplitudes integrals and tensor reduction routines. with the corresponding momentum,We classified to our check contributions thefulfilled. in accuracy gauge Those of invariant subsets identities thecomputing so are computed cost, that called using amplitudes. the gauge the Ward cache testseach identities system. and gauge are They are invariant are subset checked checked for with distinguishing every a between phase space small FSR point additional and and ISR contributions. This is the evaluation of the loop-dependentwith parts, less than any additional 10 helicity amplitude is computed the real photons attached toattached the to lepton them. lines, instead, This we allows assume us external to effective use current Ward identities, by replacing an effective current convention of ref. [ as in refs. [ dependent parts are stored.different This helicity is amplitudes particularly existcombinations important corresponding of for to the this the external process different particles. since helicity up After and to the polarization 32 first helicity is computed, which include with two fermion lines. ThisCPU includes time the required use to oftime, evaluate Chisholm the this identities Penta-Box improves [ contributions the bythe stability a small of factor the electron ten. code mass.Veltman At reduction since [ the it The same makes calculation explicit of terms tensor proportional integrals to is done by using Passarino- photon attached to it.ure All possible contributions result inwhich the involve the triangle pentagon contributions of diagrams.two initial fig- The fermion lines last in building figure them, block we is generalized obtained the by crossing software the developed in ref. [ and contractions with externalsubsets currents of yielding figure the Box-Triangle-Bubble gauge invariant JHEP02(2014)114 is integrated ) 1 cross section at k γ − µ + µ → − e + e would appear, where s is the energy 2 e s/m is well below one in ten thousand. The 3 −

– 8 – = 10 to the CPU time. 10% ]. The high rank of the rescue system allows to obtain full 25 cancellations of the order of ) e p ( u e ], where subleading contributions were neglected. We distinguish between soft m 50 , ) = 49 e p ( u These two problems are reflected in a bad accuracy of the gauge test and, therefore, Despite the cache system and the use of Chisholm identities, thisWe have implementation tried is to evaluate the amplitudes only in double precision to improve the speed e /p and 2-hard photon emission. 2.2.1 Soft photon emission In the soft photonout contribution, analytically. the The phase integrals space to of be one performed, of the which factorise photons in ( front of the square of 2.2 Real photon emission The real two-photon emission,NLO which is now contributes included in to the PHOKHARA the of code refs. completely [ in contrast to the implementation problem is naturally solved usingspins the are DT-method done since the analyticallydecided summation to and and implement averaging many over the ofimplemented in DT-method these full quadruple in numerical precision PHOKHARA9.0 where cancellations theof and gauge are the test compare code. ensures avoided. the it numerical with accuracy We the code retained, and many numerical cancellation occur.of For example, in the numericalof calculation the collider, if the Dirac equation is notin applied the or large is number not of treated identified carefully. unstable points using only double precision. The second sub-Cayley determinants. Thus, full doublecoefficients. precision is not Second, achieved forcancellations within all between tensor the the integrals helicity differentprecision method helicity which amplitudes can formalism, be resulting there larger intoThese than an exist numerical the cancellations additional one extremely are due loss related to large of to the presence the of fact small that Gram the determinants. mass of the electron has to be involve the evaluation of three-following and the four-point notation functions up of to ref.double Rank [ precision 11 for and mild 9, cancellationsarise. respectively, in First, the for Gram the determinants.for failing phase However, which space two the points, problems there expansion was breaks always some down internal due combination to the presence of additional cancellations in rescue system adds an additional still seven times slower thatthe the 32 DT-method helicity which amplitudes can and be the traced parts back evaluated to in theof quadruple the evaluation precision. code of by a factor two using dedicated subroutines for small Gram determinants. These energy-momentum conservation is still fulfilledthe external at particles the on higher their numericalsystem mass-shell. accuracy evaluates If retaining the the Ward amplitude identitiesthis are using system, not quadruple satisfied, we the precision find rescue identities in that the for all proportion parts a of of requested phase the accuracy space code. of points that With do not pass the Ward This requires reconstructing the external momenta in quadruple precision, so that global JHEP02(2014)114 ] , 2 ]. 49 , q and 1 max 52 . As (2.3) (2.1) (2.2) , E , q 2 51 is given max , , ) ) , p E 1 2 − , r p µ 2 . The faster λ ( , q 1 2 1 q II 1 · u q , q 1 2 ) k 2 , p − 1 , λ p 1 1 rodrigo/phokhara/ ( λ k 2 ( ∼ · F q 2 B q ) + and the photon energy cut-off µ + λ λ ( † II 2 v p 2 · p 1 k reaction, read ) + . − − µ γ http://ific.uv.es/ λ 1 − ( max λ k

µ 1 I · E p u + 1 2 – 9 – ) p µ 2 = → , λ r 1 1 − 1 λ k k e ( 3 + E d A e ) + Z µ α λ π ( 4 − † I v are the momenta of the positron, electron, antimuon and muon, = ) = 2 q ], a very compact form for the two-photon FSR amplitude is obtained , r 2 2 50 , λ , , q 1 and 1 49 1 , λ , q q . 2 − , µ C 2 , p . The energy of one of the photons has to be bigger than the cut-off 1 p , λ p B , + ( 1 µ , which are neglected in the analytic calculation. p F λ dependence can be cast into a single parameter The only amplitude, which was missing in earlier versions of PHOKHARA was the In principle, this is a well known formula, which can be found in the literature [ PHOKHARA9.0 max M E max PHOKHARA9.0 is available fromstated the already, webpage all new partsby two of methods the and/or released groups computervirtual of code corrections, the were we authors calculated of useroutine independently this the in article. two the To independent released ensure version codes the of stability described PHOKHARA9.0 of in the is section used, which is the code based on the the sum of the∼ soft and hard contributions should not depend on3 this cut-off up to Implementation terms of the radiative corrections in the event generator similar to refs. [ where the matrices Aappendix and B and the convention used to define the spinors are given in spin summation. The conventionwas for adopted. the helicity amplitude method introduced in ref.two-photon [ FSR. Interferences betweenmatching the the coded ones from amplitudes, Penta-Box with diagrams, infrared were also divergences not included. After some algebra in appendix 2.2.2 Two-hard photon emission For the two-hard photontudes emission, was the used helicity and method cross for checked the with calculation a of dedicated the code ampli- based on the trace method of its form depends on thethe frame formulae, in which which are thislonger valid cut-off than in is the any applied. usual frame We onesgiven in found as in it which we the suitable express the frame, everything to cut-off through give whereuse is the the invariants, defined. four but cut-off momenta are They is less are defined, universal. in a The contrast bit explicit to expression the for usual expressions which E However, as the integral over the photon energy is performed only up to a given cut-off where respectively. The infrared regulator — the photon mass the full amplitude describing the JHEP02(2014)114 ]. ]. 53 50 2.2.1 mode, γ − µ and coded + µ 2.2.2 → ] is used for this − 1 0 24 e 32 event 787 521 + N e 5 5 7 9 − − − − DT 10 10 10 10 · · · /σ 0 · DT 2(2) 4(2) 2(4) ∆ . . . 6(6) σ 4 3 2 4 4 6 8 − − − − [nb] 10 10 10 10 · · · 0

· QH – 10 – ∆ 7(1) 1(1) 4(3) =1.02 GeV . . . 4(4) σ s can appear. = 6.332(1) [nb] = 6.332(1) [nb] 2 2 1 2 e √ 4 4 6 DT QH 8 − − − σ σ − s/m [nb] 10 10 10 10 · · · 0 · DT ∆ 7(1) 1(1) 4(3) . . . 4(4) σ 2 2 1 . C > | 0.1 0.01 ∆ 0.001 | 0.0001 0.00001 : Comparison between the codes based on the DT-method and the QH-method at The newly added part —accordingly. the two photon FSR is described in section PHOKHARA9.0. The soft photon emissionand is in calculated appendix using the formulae discussedThe two-hard-photon in emission section part uses the helicity amplitudes as defined in ref. [ The virtual corrections areisations calculated is in done doublepurpose with precision and the and the trace the relevant method. parts sum of over the The polar- libraries software developed PJFry there [ are distributed with We performed very detailed tests for the different gauge invariant blocks separately [ The virtual corrections constitute the most challenging part. The presence of Gram • • • =1.02 GeV. No selection cuts are applied. See text for details on the definition of the s determinants can constitute anlenging additional for source some of realistic instabilitiesfavoured, experimental resulting which selection in can cuts collinear be where photon more emissions. forward chal- photon emissions are Here, we mainly show the results of the tests concerning the sum of all contributions. They The released code was tested veryto extensively both assure for the the technical realexperimental and accuracy the measurements. of virtual contributions the The code necessitypotential of to threat retaining be of a much numericalsince better finite instabilities cancellations electron than of both mass the the for possesses one order the required of a real for and the virtual contributions 4 Tests of the code ingredients of the released code.when The it listed changes is concern running only with the the complete NLO radiative corrections: Table 1 √ table entries. DT-method. The other one can be obtained on request. We sketch here shortly the new JHEP02(2014)114 (DT- DT for muon ∆ σ 7 event 125 9599 1044 ) — the specific cuts for their definition) 42621 N 115091 4 A , one retains at least 1 3 3 4 5 − − − 6 − 3 − DT 10 10 10 10 · · · 10 · /σ · and ) and with event selections close DT 1 1(9) ∆ 2 21(4) 18(2) 10(9) . 9(9) . . . σ 1 3 1 1 and , where at least two digits are correct. invariant mass distribution. As we can 4 5 6 7 ) and BaBar (table 1 − 8 4 − − − − 3 that one can lose completely the accuracy − µ 10 [nb] 10 10 10 2 · + 10 · · ·

µ · QH – 11 – ∆ 3(1) 1(2) 4(4) 24(2) . . . =10.56 GeV 2(1) with the Born contributions in figure σ . s 8 7 1 2 = 0.07004(4) [nb] = 0.07004(4) [nb] 2 √ for the 4 DT QH 5 6 8 − 7 7 σ σ − − − − . The integrated cross sections for both codes, DT-method 10 [nb] 10 10 · 10 10 · · A · · . The tests were performed for two different energies 1.02 GeV ), are in perfect agreement in all cases and the statistical errors DT 4 – ∆ 3(1) 7(6) 24(2) . . QH 7(5) 6(6) σ . 1 8 7 σ 2 (QH-method). At 1.02 GeV, tables , the predictions for the cross section disagree at the relative accuracy > | QH 0.1 ∆ 0.01 event ∆ 0.001 shows that the cross section from events for which one loses the precision | 0.0001 σ 0.00001 2 : Comparison between the codes based on the DT-method and QH-method at ) and QH-method ( The checks clearly show the control on the numerical accuracy of the virtual corrections. For ten million of examined events with one photon in the final state, we count for how . To check whether these events might have an impact on the differential cross section, =10.56 GeV. No selection cuts are applied. See text for details on the definition of the | DT s σ ∆ for KLOE and Babartribution energies, we in have this studied the article.Penta-Box relevance diagrams of defined For the in this most section angular purpose, challenging distributions con- we and in compareclearly figure the see contributions from from the one-loop muon and antimuon angular distributions, the size of the Penta-Box is small (below 0.3%), thecare released at program high energies based if onis no the recommended. event DT-method selection should is be applied used and with a cross check withAdditionally, the using QH-method the DT-method and realistic cuts (see appendix method) and digit of accuracy forirrelevant. the At 10.56 matrix GeV, one element observes(the squared in order and table the of crosshappen magnitude section for of of the the these BaBarEven results events event if is selection is table cuts, however table always the same), but that does not many events, N | we also calculate the cross section corresponding to these events for both codes are summarized in tables and 10.56 GeV without any eventto the selection ones (tables used in theapplied experiments are KLOE found (table in appendix ( are well below the per mille level. Table 2 √ table entries. JHEP02(2014)114 σ σ are | (5) ] for the G | 50 are avoided | (4) G | 2 0 event 713 0 6 5068 1852 26 event N N to the cross section to the cross section 5 5 5 ∆ ∆ 9 6 − − − σ σ 7 − DT − − DT 10 10 10 10 · · · /σ 10 0 · 10 · /σ 0 · DT DT 4(2) 5(2) 9(2) ∆ . . . 2(1) 1(4) ∆ σ 7 6 4 . 5(2) σ 2 4 4 5 9 − − 9 − 10 − − 10 10 − [nb] 10 [nb] · · 10 10 · · 0 10 · 0

· QH ], where it was done for charged pions in the ). The contribution ). The contribution QH ∆ 7(3) ∆ , we plot the Penta-Box contributions using the 17(4) 02(4) 2(2) 54 . – 12 – 2(2) σ . . . A A denote QT-method and DT-method respectively. denote QT-method and DT-method respectively. σ 7 3(1) 9 1 1 1 = 1.575(2) [nb] = 1.575(2) [nb] = 0.0005655(7) [nb] = 0.0005655(7) [nb] DT DT 4 4 9 5 DT QH 10 − − − 9 − σ σ − BaBar event selection − KLOE event selection DT QH [nb] 10 10 10 [nb] and and 10 σ σ · · · 10 package, the leading inverse Gram determinants 10 · 0 treats properly small Gram determinants, as discussed in 0 · · DT DT ∆ QH QH 2(2) ∆ 7(3) . σ 17(4) 02(4) . 3(1) 2(2) σ . . 1 7 1 1 PJFry PJFry . See text for details on the definition of the table entries. . See text for details on the definition of the table entries. ]. > 2 2 > | | 55 ∆ , | 0.0001 0.01 ∆ GeV GeV 0.00001 0.001 | . Subscripts . Subscripts 0.000001 0.0001 18 | | 0.00001 [ 0.000001 ∆ ∆ ]. With the 96) 96) 9 | | . . 0 0 19 10 , , : Comparison between the codes based on the DT-method and QH-method for : Comparison between the codes based on the DT-method and the QH-method for 34 34 . . package with (left) and without (right) using expansion for small Gram determinants. (0 (0 To appreciate these results, in figure ∈ ∈ 2 2 Table 3 KLOE event selection cutsfor (see a appendix chosen q PJFry The right panelevents reveals to discrepancies onlydetails after in increasing [ theeliminated number in the of tensor Monte reduction and Carlo small inverse Gram determinants observables. We confirm herecharge the even expectations distributions are that indeedcontributions, the small. we neglected For refer corrections a the [ classificationfinal reader of state. to the Replacing ref. charge pions odd [ with and muons even does not change the classification presented there. BaBar event selection cutsfor (see a appendix chosen q contributions can reach the percent level and they cannot be neglected for the charge odd Table 4 JHEP02(2014)114 1 1 90 8 8 . . 0 0 80 6 6 . . 0 0 70 4 4 . . is the angle of 0 0 60 θ 2 2 . . 0 0 GeV ] GeV GeV 2 50 θ θ 56 02 . 56 0 0 . . cos cos [GeV = 1 2 2 2 = 10 . . 40 = 10 s q 0 0 s s √ − − √ √ 4 4 . . 30 0 0 − − 6 6 ]. The results are completely . . 0 0 20 − − 8 8 56 . . , 0 0 10 − − polar angle, while 19 − − − − 1 1 µ µ µ µ B B θ θ θ θ pb pb + − − 0 dσ dσ dσ dσ 0 cos cos cos cos 2 B 0 µ d d d d dq 001 001 002 003 dσ . . . . + + + + 001 002 0005 0008 0006 0004 0002 0015 0018 0016 0014 0012 0025 / . . 0 0 0 0 ...... 0005 0015 0025 + + + + 0 0 . . . 0 0 0 0 0 0 0 0 0 0 0 2 µ µ µ µ pb 0 0 0 B B θ θ θ θ pb pb dq dσ dσ dσ dσ dσ cos cos cos cos d d d d is the

1 1 9 + . 0 µ – 13 – 8 8 θ . . 0 0 8 . 0 6 6 . . 0 0 7 . 0 4 4 . . 0 0 6 . 0 2 2 . . 0 0 GeV GeV ] 5 + + 2 . GeV µ µ 56 0 02 . θ θ 0 0 . 02 . [GeV cos cos = 1 4 2 2 = 10 . 2 . . = 1 s 0 0 0 q s s √ − − √ √ 4 4 3 . . . distributions. 0 0 0 − − − 6 6 . . 2 . 0 0 µ 0 − − + 8 8 . . 1 µ 0 0 . 0 − − 2 + + µ µ q 1 1 θ θ B B 0 − − dσ dσ 0 0 cos cos 0 for the charge ‘blind’ observable. These deﬁnitions are the same in all the ﬁgures 02 01 01 02 02 01 02 01 : Relevance of NLO Penta-Box contributions at KLOE (left) and Babar (right) 2 d d B : Relevance of NLO Penta-Box contributions at KLOE (above) and Babar (below) ...... 015 005 015 005 025 015 005 015 005 025 0 . 0 . 0 . 0 . . 0 . 0 . . 0 . 0 . / / 001 002 dq dσ . . 0 0 0 0 0 0 0 0 0 0 0005 0015 0005 0025 + + − − − − − 0 0 . . . . / µ µ − − − − − 0 0 0 0 θ θ 2 pb pb pb µ − dq dσ dσ dσ cos cos d d or + energies for using asymptotic expansionsnumerical and stability Pade of approximants. thestable tensor and For reductions, well more see under refs. control. details [ concerning the Figure 8 energies for muon angularµ distributions: and will not be repeated in captions. Figure 7 JHEP02(2014)114 1 even if ) events 4 9 0.9 − 10 ( 10 ] was used to 0.8 10 57 10 · ] 2 0.7 4 [GeV 2 Q 0.6 . Secondly, the numerical 4 − 0.5 10 · 2 0.4 0.3 0

0.08 0.06 0.04 0.02 -0.06 -0.08 -0.02 -0.04

[nbarn] /dQ d σ 2

at 1 GeV, while at 10 GeV , it was only about 7 – 14 – − 10 . 15 , was applied. − B 10 . A perfect agreement at this level of accuracy was observed. 4 − (bottom: absolute error estimate). On right: the same calculated without 10 · : On left: muon pair distributions including 5-point functions at KLOE calculated 2 ] and in appendix PJFry in some corners of the phase space. However, since those phase space regions did not Both the soft and the real parts were tested checking the independence of the cross The biggest observed relative difference of the codes was at the level of The new contributions of the real two-photon emission were tested comparing two The soft real emission analytic formulae were also checked. Firstly, by comparing to the the pion form factor measurements at BaBar and KLOE 3 49 − 5 Impact of the radiative correctionsPHOKHARA7.0 added has to been the usedversion event by 4.0 generator BaBar to on and 7.0 KLOE the until muon quite production recently. channel In was fact, not from changed. Comparing numerics section and differential distributionswhere the of integral the over the separationpart, one where parameter photon the phase between integral space is the istest obtained performed using soft was the analytically, part, and Monte Carlo the generation. hard The accuracy of this in [ both codes were usingwas double checked. precision For only. the Additionally, T-methoda in analytically, relative both and accuracy for cases of the gauge H-method invariance numerically, obtaining affect the relevant observables (invariant masschanged and to angular cure distributions), this the behaviour code by was not means of appropriatecompletely expansions. independent codes. Inobtain one, an the analytic trace result, method in (T) the and second one, FORM [ the helicity amplitude method (H), described if the method isstability limited of in the some code cases wasof to tested the an comparing same the accuracy code. quadruple of Thecode and relative the of accuracy double the of precision generator the versions 10 double is precision at version the used level in of the released decicated routines to avoid smallhave Gram been determinants. generated. Approximately integral obtained by means of a Monte Carlo methods. A good agreement was found even Figure 9 with JHEP02(2014)114 85 . 1 0 8 8 . . 0 0 6 . 75 . 0 0 4 7 . . 0 0 2 ] . 2 65 0 . 0 θ [GeV 0 6 2 GeV - BaBar CUTS . cos q 0 56 2 . . 0 55 − . 0 = 10 10.56 GeV - BaBar CUTS 4 . s = 0 √ . 5 s . − 0 √ 6 . 0 55) . − 45 . P Hnew ) 0 0 8 . , 0 dcos PH ( − 4 ) . 2 P Hnew 0 54 dσ/ . 1 dcos 2 PH − ( 001 002 003 − distribution), from which the hadronic . . . dσ/dq 0 0005 0015 0025 (0 0 0 0 . . . PH − 01 01 dσ/ ) . . 0 0 0 008 006 004 002 006 004 002 008 2 dσ/dq 0 ...... 0 . 2 PH 0 0 0 0 0 0 0 0 q ∈ − dcos . As one can see, the radiative corrections ( − − − − 2 dσ/dq dσ/ q

10 8 1 . – 15 – 0 6 9 . . bins. For other muon pair invariant mass ranges, the 0 0 2 q 4 8 . . 0 0 2 . ] 7 . 0 2 0 θ [GeV 0 2 6 GeV - KLOE CUTS . q cos 0 02 . for different 2 . 0 = 1 5 1.02 GeV - KLOE CUTS . − s 0 12 = √ s 4 . √ 0 4 − . 0 6 P Hnew . ) 0 − 3 dcos . PH : The relative difference between differential cross sections of PHOKHARA7.0 : The relative difference between differential cross sections of PHOKHARA7.0 ( ) 0 2 P Hnew 0 8 dσ/ . dcos 001 002 001 and figure 0 2 PH − . . . ( 0005 0015 0005 0 0 0 . . . dσ/dq − 0 0 0 0 PH − − dσ/ ) 006 004 002 006 004 002 − dσ/dq ...... 11 2 PH 0 0 0 0 0 0 dcos For the charge sensitive observables, which were available at PHOKHARA7.0 only at The charge averaged angular distributions are also not very much different as shown in ( − − − dσ/dq dσ/ results are similar to the onesusing shown. the charge We can averaged conclude observables, atshould the this not missing point have radiative that affected corrections in are the the very experiments extraction small of and the hadronicLO cross (the section. ISR-FSR interference was present at LO only) the new corrections are relatively cross section is extracted,missing is in shown PHOKHARA7.0 are in small. figure and They up reach up to to 0.25% 0.1 for % the for the BaBar KLOE event event selection. selection figure with PHOKHARA9.0, which includes the completebetween NLO the corrections, one charge has average toalready distributions distinguish included for in PHOKHARA7.0 which and the7.0 the was bulk charge limited sensitive of to observables the the forof leading which NLO the order version corrections hadronic only. cross In was section, theinvariant experimental muon charge framework averaged pair for observables mass the were extraction distribution used. (i.e. The the most important Figure 11 and PHOKHARA9.0 (with subscript ‘new’). and PHOKHARA9.0 (with subscript ‘new’). Figure 10 JHEP02(2014)114 . ∈ 95) 13 . 2 8 . 0 q 1 0 , ) and 8 . 94 6 . 0 . 0 PH (0 6 . 0 4 . ∈ 0 4 . 0 2 q 2 . 2 0 . 0 with full NLO QED θ θ 0 0 γ for KLOE cuts; cos cos 2 − . 0 2 . µ − 0 1.02 GeV - KLOE CUTS 10.56 GeV - BaBar CUTS + − 95) 4 . = . = 0 µ s 0 s 4 − . √ , √ 0 — left plot; 6 . − 0 → 94 − . P Hnew 6 ) . 8 − . 0 55) (0 0 e . dcos − new PH ( − ) 0 + dσ/ , e 1 PH PH ∈ 8 dcos . − ( − 0 0 − − 54 − PH 0 01 dσ/ µ µ 2 ) . . 08 06 04 02 08 06 04 02 008 006 004 002 008 006 004 002 ...... − + 0 ...... q 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + + − dcos µ µ (0 ( − − − − − − − − dσ/

∈ 8 . 8 2 . – 16 – 0 0 q 6 6 . . ). 0 0 new 4 4 . . 0 0 PH 2 2 . . 0 0 θ θ 0 0 cos cos 2 2 . . 0 0 1.02 GeV - KLOE CUTS 1.02 GeV - KLOE CUTS − − = = . s s 4 4 . . √ √ 0 0 − − 14 6 P Hnew . 6 ) . 0 0 : The asymmetries given by PHOKHARA7.0 (denoted as for BaBar cuts. − dcos − new PH : The relative difference between differential cross sections of PHOKHARA7.0 ( ) 8 dσ/ . PH PH 8 dcos . 0 − ( 0 75) − 0 − − . − PH 0 dσ/ µ µ ) 05 04 03 02 01 03 02 01 05 04 002 001 006 005 004 003 002 001 0 ...... − + ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + + , dcos µ µ ( − − − − − − − 74 dσ/ . 6 Conclusions The presented studies allow forgenerator the development PHOKHARA9.0 of simulating a numerically the stable reaction Monte Carlo event For BaBar, the asymmetryAt is low naturally invariant suppressed masses, by asthe tagging compared muons the to photon to the at flyasymmetry energy large in is available angles. the at at opposite the theshown experiment, direction level in it to of figure forces few the percent photon and and it thus is the dominated suppression. by the The LO contributions as bigger and reach typicallyevent a selection few was designed percent toalso as diminish expected mostly the from killing FSR NLOcoming radiative the corrections. corrections from asymmetry and the The two as coming photon KLOE such from emission it is one was however photon surviving the emission. cuts as The shown in asymmetry figure Figure 13 PHOKHARA9.0 (denoted as — right plot. and PHOKHARA9.0 (with(0 subscript ‘new’). Figure 12 JHEP02(2014)114 γ − 75) . µ 0 1 + , ) and µ 8 74 . . 0 → PH (0 6 . − new PH 0 e ∈ PH + 4 . e 0 2 q 2 . 0 θ 0 cos 2 . 0 − 10.56 GeV - BaBar CUTS 4 . = 0 s − √ — left plot; 6 . 0 − 55) 8 . . 0 0 − , 1 ]. − 54 − − 0 . µ µ 03 02 01 03 02 01 59 ...... − + 0 0 0 0 0 0 + + (0 µ µ − − −

∈ 2 – 17 – 1 q 8 . 0 ). ◦ 6 . ]. FC is funded by a Marie Curie fellowship (PIEF- new 0 PH new 130 PH 4 . 0 PH < 2 . ± 0 µ ] to PHOKHARA8.0 [ θ 0 cos < θ 58 , 2 . ◦ 0 − 50 10.56 GeV - BaBar CUTS 50 4 . = 0 s − √ GeV 6 . 0 − 02 . 8 . : The asymmetries given by PHOKHARA7.0 (denoted as 0 − = 1 1 − s 0 − − 02 01 01 02 . . . . µ µ 015 005 005 015 √ Muon tracks: 0 . 0 . . 0 . 0 − + 0 0 0 0 + + − − µ µ − − • • http://www.lnf.infn.it/wg/sighad/ A.1 KLOE ing Group on[ Radiative CorrectionsGA-2011-298960). and MG Monte was supported Carlo by “Świder” Generators PhD for program. LowA Energies" KLOE and BaBar event selections cuts by the Spanish Government andand EU CSD2007-00042 ERDF Consolider funds Project (grants CPAN), FPA2011-23778, byDeutsche FPA2011-23596 GV Forschungsgemeinschaft (PROMETEUII/2013/007), by via the the9 Sonderforschungsbereich/Transregio Computational SFB/TR- Particlethe Physics and framework by ofMinistry the the of Alexander Sofja von Education Kovaleskaja Humboldt and Award 2010, Foundation, Research. in endowed by This the work German is Federal part of the activity of the “Work- Acknowledgments This work has been supportedUnion by under the the Research Executivethe Grant Agency Polish Agreement (REA) Ministry number of of the PITN-GA-2010-264564 Science European (LHCPhenoNet), and by High Education under grant number N N202 102638, used by the BaBarKLOE and and 0.3% KLOE for collaborationsexperiments BaBar. cannot are be attributed We aﬀected to conclude the onlyin missing that corrections PHOKHARA4.0 at for the [ the the observed reaction level discrepancies between of these 0.1% for PHOKHARA9.0 (denoted as — right plot. accuracy. The radiativegenerator corrections can reach which a few were percent. missing Though, in it was the shown previous that the versions charge blind of observables the Figure 14 JHEP02(2014)114 (B.1) (B.2) , ) ) ) ) 2 2 5 q q 6 a · ) are given by · a 1 − k 2 e − 2 k 1 e 9 2.3 + k 10 N N − )( a e + ) a + ) 1 e )( J 2 2 q + J 1 ∗− 1 q q · ∗ 1 q 11 ε · · 2 12 ε · a 7 2 2 8 k 2 a a k k ( a k ( ( ( − + + − ) 1 1 2 ) 5 6 q 3 2 N N 4 a · a a q ) ) 2 a · 1 + 1 − k 2 ∗− 1 − ∗ 1 e q q e ε k · ε · + )( − 1 + 2 2 e 2 1 + )( e a k a k J q 1 J ( ( 7 · q 8

1 a · a + k + 1 ( GeV (muon) ( 1 – 18 – 1 1 ) 3 4 MeV − − N ◦ N a a ) ) > 2 2 1 + 1 9 | ∗− 2 10 ∗ 2 q q a N 165 2 N 115 ε · ◦ ε a · ) ) q 1 1 2 1 | 2 + 2 2 ∗− > a < k a k ∗ 2 q q ( ε ( ( 160 · ε · ◦ ( ( 1 1 11 trk 12 3 3 k k < a 15 s s ( e a e ( 4 4 ± − − < + + µ < m ) four momenta; < θ ) = ) = ) four momenta; + 2 2 ) four momenta; ◦ MeV + µ − ) four momenta; , λ e , λ 20 e 1 1 80 − GeV 96) 96) . . λ λ µ ( ( 0 0 , , 56 A B . GeV (antimuon) and 34 34 . . - photon four momenta. 1 (0 (0 2 > = 10 k - electron ( - positron ( - muon ( - antimuon ( ∈ ∈ | , s 1 1 2 1 2 2 2 1 q q k p q p Minimal photon energy/missing energy 3 GeV q √ Muon tracks: Track mass: q Missing photon angle | • • • • • • • • • • • • • and The coeﬃcients of the two-hard photon emission amplitude of eq. ( B Two-hard photon emission We use here the following notation: A.2 BaBar JHEP02(2014)114 (B.9) (B.6) (B.7) (B.8) (B.3) (B.4) (B.5) , , − − e e + + , e e J J · · , ! 2 1 2 q q + k 0 2 2 · a 1 . , , , ) ) ) ) . k − − − 0 1 1 2 2 ) a , q q q q − + , e ! − 2 · · · ·

2 ∗ 1 ∗ 2 ∗ 2 ∗ 1 + p ! + I q e ε ε ε ε , II ! · v 1 0 J = v + 2 0 1 ) ! µ k

− 1 − 2

γ + 2( + 2( + 2( + 2( I p p II + ( µ u ≡ u + + − − 2 1 ! a 2 − + , ∗ 1 ∗ 2 k k e q 1) = ε ε . !

· + − 1 2) 2) + + = 1) = + ! e − − ) 1 2 1 ∗ 2 ∗ 1 − p ) ≡ − k J k k ε ε , λ θ/ θ/ ! , λ 2 2) 2) ) − 1 ! ======( = 2 p − , θ/ θ/ ( p ) ) , 3 2 2 6 8 4 p sin ( ( ia cos ( 12 10 a ( p , a/ , λ , λ ( iφ χ − 3 ± p p sin ( χ cos ( − | ,

! | | e e + ! ) λ λ − + + 0 1 2 e e 1 0 = 0 E 3 − J J − + E ia a · · ,N

are expressed in terms of the polar and azimuthal angles 1) = p

1 2 2 E E p + ) λ q q ∓ +1) = − k for a particle and an antiparticle are given by: λ 1 , , 2 2 · 0 p p − , λ 1 a v ,,,, a a a a p p a ( ) ) ) ) k p − − ( ( , µ

1 1 2 2 ( ) +1) = χ χ ∓ q q q q − + χ +1) = , e + 2 · · · · ! . : , and 1

1 ∗ 1 ∗ 1 ∗ 2 ∗ 2 + p 2 ) = ) = 2 − q µ p e + ε ε ε ε p / 0 u · ( p J = σ + 1 2 ( ) χ p, λ p, λ k ± ± χ + 1 µ µ − + 2 ( ( 0 + 2( + 2( + 2( + 2( p σ p σ v u + ( µ + + 2 1 1 − + a

∗− 1 ∗− 2 2 = k k e q ε ε · + + 1 = = − e + + 1 2 1 λ/ ∗− 2 ∗− 1 p k k ε ε k J µ ± γ a = = = = ( = = = 3 5 7 9 1 1 11 a a a a a N a However, for incoming particles in their CMS coordinate frame with z-axis along the The helicity eigenstates The helicity spinors for electron. for positron and positron direction they simplify to: of the momentum vector Where helicity and related objects are deﬁned in the following form: For the reader’s convenience, we give here all the relevant deﬁnitions. The gamma matrices where JHEP02(2014)114 ) max λ (C.9) (C.4) (C.5) (C.6) (C.7) (C.8) (C.1) (C.2) (C.3) E 2 (B.10) ) , = ) , r . r 2 i , r , ( θ ) 2 , q r 1 λ 1 , q q q µ sin 1 · ( 1 q 1 r q ∓ ( log ( k , , , i ) ) FSR φ − r r 2 FSR , , ) F 1 j F 2 2 q k µ cos i 2 · 2 µ 2 µ p log ( log ( i q ) + 2 2 ( 1 ) + m 2 q m p q     2 ) 2 e 1 − 2 e , r · · p , r q ij 2 i 1 , which depend explicitly on 1 m 2 β m 2 4 4 k 4 k φ ) ) , q /s e , q /q 1 µ 2 − . 4 e 1 β − (1+ 4 µ − β p 2 sin 2 µ 2 , q m · m ) 1 ) 1 i , q p 2 1 j µ j (1+ k θ 2 e k 16 (1+ 1 16 2 q k q · β · i , p i p q 1 2 log 2 , p sβ 1 p p q p q 1 − j ( cos ( p p q ( / 1 i log ( log r ± 2 µ q p k 2 /s , 1 1 µ 1 2 2 1 1 j · i 2 i p m q k k p k k 1 INT 1 2 e φ INT 1 3 3 1) p m E E

p q can be split into three parts ISR,FSR,INT F d d F and the photon mass regulator 4 m − ( sin − − = – 20 – 1 − − Z Z i 1 2.1 1 1 i k k max 1 1 =1 ) + 2 ) + 2 2 α α 3 + π π     E E X d 4 4 i i,j − − q q , r , r α α for φ 2 2 π π π α 2 2 Z i = = , p , p − − − F i 1 1 α cos ) = ) = π ij p β p i 4 − β ( defined in ( θ , r , r r 2 2 ) ) = ) = ) = ISR , q , p cos , r ISR , r , r , r ) = 1 F 1 2 2 2 2 q F p ± , r ( ( , q , , q , q , p 2 1 1 1 0 1 ) = q , q p ) = ISR , q FSR ( , q ( 1 2 , r 2 r F 2 r F 1 , r 2 and read , q √ 2 , p , p 2 , q 1 ISR 1 FSR ir , q 1 p p , p 1 F F F ( ( 1 , q ) = r , q p 2 F ∓ ( 2 . , p 2 INT , p 1 = , 1 INT p F i ( p F ( , λ F r = 1 i F i k Those parts of the functions The photon polarisation vectors in the helicity basis are defined as ( µ ε and with the ratio of theare photon denoted energy by cut-off and with C The soft photonThe integrals function with JHEP02(2014)114 ) 4 t − (C.18) (C.17) (C.10) (C.11) (C.13) (C.14) (C.15) (C.16) (C.12) 4 y t t 4 : 4 t 1) ) t − i 4 ¯ 1) x − t − x , 3 t 2 3 3 − − t t log ( 3 t y t 3 , 3 3 (0) t t , t t i 4 ( y ) )) − 3 t x )) t 4 1 + 4 2 t 3 t ( 2 t 1 y t 2 t , 4 3 , p 1) , q = ( ) t t − 1 1 2 d Li i 2 − p q y 3 − 1 + − ( have the following form: x t ( I Li 1 . . Li − 3 3 ( 3 3 d t I 2 I t 3 . fin − ) x 2 − 2 t i, 1 C 2 )) 1 ( x j F − 2 − 2 log ) s 3 + Li / y ) − µ 2 3 ) t 3 , q − t 3 2 3 + c 2 t 2 2 i t then: − 3 x 3 t ) 3 3 − p 2 t q t | t 3 p t , ir I ( ) ( Li 1 t 0 ( x 4 x x ) c 1 3 1 F t t 3 t ¯ − x 3 3 − 4 4 t I I )( 3 I 3 x t 1 + | | t t < t t 1 C j 2 t 2 3 ) ) 1 = )(1 + ¯ ¯ − − x x ) + t ) − 3 1 3 ¯ | 2 x 4 π 4 + + i t 2 3 t 1) 2 f − + t 1 ) + 4 t ) + t F 2 ¯ y 3 x t x 1 (4 2 1 − | b x 1 + − 1 | t t t − ( − log Γ(1 3 (¯ p q ( − 1 + ( 2 ( ( I Li 4 1 2 1 2 . If are four momenta, log b x 1 1 t − C =1 (0) (0) + 1 2 t x 2 3 I I ( (1 + ) +

1 2 f 1 x x 2 ( ( X 2 − 2 i,j C (1 + | − 2 Li log 2 2 2 C − 2 πx π π α – 21 – ¯ , x α α Li + x 4 − π 8 1 t 4 4 4 1 2 a 1 | log ( x − log 3 4 1 3 1) − − − log ( 3 − t I t 3 4 y ¯ − x t 3 log 4 a t t | y t 3 ) I − t 3 3 4 t t 3 into ∆ = 3 − 3 | y 4 (0) t t 3 3 t t 1 t 1 t 2 3 y t 3 ¯ C x t 3 t 3 x = ) = ) = ) = t t 3 t 1 1 | − ( t t t 3 2 2 2 t πx x − − − 1 1 − t are neglected. The ‘translation’ to dimensional regularisa- 1 t and 2 − − f f 3 3 , q , q , q 1 + − ( 1 x − 2 2 1 + of the soft formula t t ) 1 1 1 1 + t 3 3 s 1 + 1 λ 1 y x t t 1 ( q q ≡ 2 λ t t ( ( max y x , q x 2 2 log fin t t ( 2 ) = ) E i, )) = fin fin 1 2 2 1 2 2 log x Li Li , , 2 I F , p ( log 1 2 1 1 , x Li Li + + + I , x p 1 ISR of FSR + log + log + + + ( 1 x F i F x ( = log fin x ( 1 = log , 3 a f 3 I a I 3 INT I F ≡ ≡ ) function depends on the sign of 2 ) 2 a 3 , x then: I 1 , x 1 x 0 ( x a ( > 3 a The The arguments The remaining parts Terms proportional to I 3 1 I f If tion results in changing JHEP02(2014)114 (C.21) (C.22) (C.23) (C.19) (C.20) 3 3 4 t t t 4 ) t 4 4 + + 3 t t 3 4 t t ) ) y x 3 t 2 4 t t 4 3 t t 3 4 t t + ) t t 4 4 3 y t 3 1 + t 1 + + 2 4 t t ) ) ) t t x ( 1 + 4 4 4 ) x x 1 1 1 t 2 3 + t t t ) log 1 + − t t ) 3 4 t 4 t ( ( 3 1 t y 4 t 1 + − ) t 2 t − + + t 1 + 1) 4 ( 1 + y 1 4 4 t 3 − Li t 1 − 2 2 1) 1 + t t t y − 1) − t 3 y 1 − 2 t )( )( 2 4 Li − Li ( )( t − 2 4 3 3 4 t 2 4 4 3 t 3 − t t t 1 1 1 ( t t 2 3 t )( Li t ) ) − + 3 Li t 2 2 3 4 3 t 2 2 1 4 2 ( t t )( t ¯ x log t + + ¯ ¯ 2 + log ( x x ( − 3 4 1 − 4 ) 1 Li t 1 1 2 t ¯ ) + 4 4 4 x 3 3 1 2 4 t 4 4 ¯ + ) x t t t t t 4 t ¯ ¯ ¯ 1 t x x x 3 t 4 3 ) ) ) t 4 4 t t − 1 2 4 ¯ 1 + x x t 4 + )( )( t 4 3 3 ) t )( − t 3 t − t ¯ 2 4 − − x ) t t t 3 4 4 3 t 3 − 1 1 t 1 y 2 t t 3 − t t )( − − t 3 2 2 − 4 | t t − | | t )(1 + x − 3 t + + + ) 2 4 1 2 4 t )( t 3 1 1 2 4 | 3 − t 1 + 4 − − t ¯ + 4 2 3 t x y y x − t 3 + t ¯ ¯ 2 t )(1 + 1 + x x ( y | 1 | (1 t t ) ) 1 + t t t − ) + log t 4 | | ) 3 3 3 t 2 3 4 2 1 1 2 ¯ + x ( 1 ( ( t ( x 4 4 t t 3 t 2 4 1 1 | t t | t t t ( ¯ − t x + t − ( ( ( 3 2 1 3 2 + | ( 4 t 2 | 2 3 2 4 2 4 t 3 ¯ 2 t x 4 x ( t t t 3 (1 + 2 1 − − + 2 2 2 t | 2 | t t 3 t ) ) ) 2 3 (0))( (0))( 2 ( Li 1 log ¯ x t

4 t log x 3 2 2 2 1 1 | 2 1 (1 + Li Li Li Li ¯ t t x p t )( ) C (1 + ¯ ¯ ¯ 1 2 log x x x | 2 x x + 2 2 C 2 − 1 1 1 )( p )( + + − − – 22 – ¯ ¯ 1 2 C x x 4 p x x x − − − 3 1 Li 1 1 1 t t (0)) 2¯ 2¯ 2¯ log ( 4 ) x x 1 1 t 4 log 3 4 3 4 (0)) + (0) (0) 2¯ 2¯ + ) x ) + + t t t t t − − − log (0) (0) 1 4 4 (0)) 4 1 1 4 ) ) ) 3 ) 4 − 2 2 t t 3 t log 2 3 1 1 4 2 2 2 t 4 x t x x 3 4 3 t − − 3 t − t | | | 3 3 − t t 4 4 x x 4 y t t t t x 4 3 4 4 t t 2 2 2 t t 2 2 t t − 1 t 1 t − y )( − | | + )( − 4 ¯ ¯ ¯ x x x 4( 4( x 3 t 3 + + + 2 2 4 + − t 3 | | | 3 4 1 t t t 1 + y (0) + t 3 − t t − + − − t 1 + ¯ ¯ x x − t t x y x x 2 | | 1 + 1 + y x x ( ( (0) t t t t 4 4 4 + + + y + + x t t t − t t t t ( ( ( (1 ( 2 + (0) 2 2 2 2 2 2 + + y y x x 2 | | | x 3 3 2( t t 1 t t t 1 1 1 t 2 2 x | | ( ( 2 2 2 2 2 2 log log log 2( ¯ ¯ ¯ x x x 1 1 | | | 4 3 2( log ( t ( t ( 1 2 1 2 1 2 ¯ ¯ Li Li Li Li Li Li x x | | ( ( − − − − + + + + − + − + log − + − − = = = = log = log = b b c c d a 3 3 3 3 3 3 I I I C C C ≡ ≡ ≡ ≡ ≡ ≡ ) ) ) ) ) ) 2 2 2 2 2 2 , x , x , x , x , x , x 1 1 1 1 1 1 x x x x x x ( ( ( ( ( ( b c d b c 3 3 a 3 3 3 3 I I I C C C JHEP02(2014)114 (C.30) (C.24) (C.25) (C.26) (C.27) (C.28) (C.29) ] ) 2 Z Erratum M ( [ α (2010) 585 and ) µ | ) ) 1 4 4 ∆ C 66 2) ) t t ¯ x 4 √ (2011) 1515 t − − − ) − arXiv:1105.3149 2 4 g (2009) 1 3 3 ∓ − [ 2 2 t ( t t ) ¯ x 3 )) | 3 4 2 )( t )( − 2 t t C 71 ¯ 4 4 x ) ) 1 1 477 ¯ t t x )( 1 2 2 (1 1 4 1 ¯ x ¯ ¯ t 2 x x ¯ x + + 1 1 ¯ x | − 3 4 ¯ ¯ 1 x x 2 + (0)) + Eur. Phys. J. − t t ¯ x 2 1 x , 4 2 | 2 − − )( t )( | x 1 ) ]. − 1 3 3 + ¯ 2 2 1 1 ¯ 1 t t x )( | | (2011) 085003 Reevaluation of the hadronic ¯ x | − 2 1 3 x 1 1 | | 1 t . ¯ 2 x ¯ ¯ 2 x x + + Phys. Rept. ) | | 2 | − || ¯ Eur. Phys. J. x , ) ) 3 4 2 (0) ) + | 2 (0)( 2 SPIRE | 4 4 2 t t , 2 (0)( 2 ¯ 2 x 2 Quest for precision in hadronic cross sections ¯ t t x ) ¯ 1 4 x | G 38 2 x 1 x IN 2 | )( t )( x 2 − ¯ 1 Z x

) x 2 2 x (0))( (0))( − − − )( ¯ (( ][ x 2 )( g (¯ ¯ ¯ + 1 1 x x | 2 M 2 − 2 3 3 1 ¯ x ( 1 1 ) ) x x t t 2 – 23 – ) + ¯ 1 ¯ x x − 2 2 | ) + 2 p x x α 2 1 2 2 x 2 ¯ )( )( || ¯ x ¯ x x 3 4 2¯ 2¯ 2 − − ¯ x ) | (¯ x | 1 ¯ 4 4 t t ¯ 1 x x 1 1 1 1 1 t t 1 2¯ ¯ J. Phys. 1 x x − − x ¯ | x (0)) x − ¯ x (¯ ¯ , (¯ x (0) (0) | 1 (0) (0) − | 2 2 + + − 2 | | 2 2 2 1 1 | + − x − | 2 − 2 2 and to − 4 3 − 2 x x 1 | x x (0) + The muon 1 | t t 2 2 2 ¯ ¯ x x 2 1 2 ¯ 2 | 1 4 4 2 x 2 − | | ¯ | x ¯ | | x | 4( 4( x ), which permits any use, distribution and reproduction in ¯ 2 )( )( | x ¯ 2 x x 2 2 1 ( 2 ]. ]. | 3 3 ¯ − x | 1 1 ¯ + + ¯ x t t ¯ ¯ x x | x | p 1 | | (0) − g − | 2 2 2 + 2 ¯ 2 x | | | | − arXiv:1010.4180 + + | 2 1 1 2 x 1 1 | ¯ 2 (0)( 4 3 ¯ ¯ x x x ¯ (0) ¯ x (0)( | x 1 SPIRE SPIRE p | t | t 1 1 | 2( | 2 1 1 ( ( ( ( ¯ x x x IN IN x | ¯ x x p ( ( | 2¯ +( − − + + ][ ][ CC-BY 4.0 collaboration, S. Actis et al., = = = = = This article is distributed under the terms of the Creative Commons 4 2 d y x , t 3 ∆ = ( t 3 C t C (2012) 1874] [ ≡ ≡ ≡ ) ) ≡ ]. 2 2 ) ) 2 2 , x , x C 72 1 1 , x , x x 1 x 1 SPIRE ( ( x x y x arXiv:0902.3360 arXiv:0912.0749 IN ( ( contributions to the muon ibid. [ at low energy: Monte[ Carlo tools vs. experimental data re-evaluated using new precise data [ Working Group on RadiativeLow Corrections Energies and Monte Carlo Generators for t t 4 d , M. Davier, A. Hoecker, B. Malaescu and Z. Zhang, F. Jegerlehner and A. Nyﬀeler, K. Hagiwara, R. Liao, A.D. Martin, D. Nomura and T. Teubner, 3 3 t [3] [4] [2] [1] C References Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. JHEP02(2014)114 ]. ] Acta cross , (2011) (2009) γ ) (2003) SPIRE − γ cross , talk held ( B and the µ IN ) Nucl. Phys. − (2009) ]. + γ ][ , )) π ( µ γ B 700 B 670 + and from threshold − C 27 ( π π ]. ) γ 103 → (2012) 032013 φ + with − − SPIRE − → π π π e IN (2009) 2957 ¯ tγ contribution to the + + − + arXiv:1212.4524 (2003) 244 t → e ][ π π e SPIRE [ D 86 − + , Humboldt-Universität − IN → π e → → e Phys. Lett. Phys. Lett. 153 + B 40 ][ − + , , − − π e e e e Eur. Phys. J. + hep-ph/0512180 + + e , e e [ , Ph.D. thesis, Uniwersytet ( ( Phys. Rev. Lett. (2013) 336 σ σ Phys. Rev. , , Ph.D. thesis , arXiv:0909.1750 The radiative return at Electron-positron annihilation into three B 720 (2006) 617 [ arXiv:1211.1112 Acta Phys. Polon. [ The new g-2 experiment at Fermilab , C 47

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