Soft-photon corrections to the Bethe-Heitler process in the γp→ l + l − p reaction
SSoft-photon corrections to the Bethe-Heitler process in the γp → l + l − p reaction Matthias Heller, Oleksandr Tomalak, and Marc Vanderhaeghen Institut f¨ur Kernphysik and PRISMA Cluster of Excellence,Johannes Gutenberg Universit¨at, Mainz, Germany (Dated: April 20, 2018)We report on the calculation of first-order QED corrections for the γp → l + l − p (cid:48) process. An upcoming experiment at MAMI (Mainz) aims to compare the crosssections of muon- and electron-pair production in this reaction to test lepton univer-sality. Precise knowledge of the electromagnetic radiative corrections is needed forthese measurements. As a first step, we present the leading QED radiative correc-tions in the soft-photon approximation when accounting for the finite lepton mass.For the kinematics at MAMI, we find corrections of the percent level for muons, andof order 10% for electrons. Contents
I. Introduction II. Lepton-pair production at tree level III. Leading-order radiative corrections in the soft-photon limit
IV. Results and discussion V. Conclusions and outlook Acknowledgements a r X i v : . [ h e p - ph ] A p r References I. INTRODUCTION
Recent experiments found a significant difference in the proton charge radius, comparingmeasurements with electrons and muons. Currently, the most precise measurements withelectron scattering were performed by the A1 Collaboration in Mainz [1, 2]. The protonradius extracted from these measurements is R E = 0 . R E = 0 . γ p → l + l − p (cid:48) . Such experiment only requires a relative measurement through theratio of electron- and muon-pair production cross sections slightly above di-muon productionthreshold. According to the finding of Ref. [25], the measurement of this ratio with absoluteprecision of around 7 × − can test lepton universality at 3 σ significance level. An upcomingexperiment at MAMI is planned to perform such measurements [26].For a precise theoretical prediction, it is, however, necessary to include higher-ordercorrections to the process. In this article, we report as a first step on the calculation of thefirst-order QED corrections in the soft-photon limit when accounting for the finite leptonmass.The outline of the paper is as follows. In Sec. II, we introduce the kinematical notationsfor the process γp → l + l − p (cid:48) and give the formulas for the cross section at tree level. InSec. III, we evaluate the first-order QED corrections to the cross section in the soft-photonapproximation. This limit is defined by a soft scaling of the loop momenta. We give theanalytic expressions for the real and virtual corrections. We show that they factorize in termsof the tree level cross section, and explicitly check the cancellation of infrared divergences.In Sec. IV, we present the results of this work. We quantify how the ratio of cross sectionsof muon- and electron-pair production to electron-pair production is affected by radiativecorrections. We give our conclusions and an outlook in Sec. V. II. LEPTON-PAIR PRODUCTION AT TREE LEVEL
The Bethe-Heitler process at tree level is described by two graphs, see Fig. 1. We use p ( p (cid:48) ) for the momenta of the initial (final) proton, and p ( p ) for the momenta of leptons l − ( l + ) respectively. The initial photon has momentum p , and the virtual photon momentumin the one-photon exchange graphs of Fig. 1 is defined as p = p − p (cid:48) . The Mandelstamvariables for this process are defined as ( p + p ) = s ll , (1)( p − p ) = t ll , (2) p = ( p − p (cid:48) ) = t. (3) p p p p p ′ p p p p p p ′ p FIG. 1: The Bethe-Heitler process at tree level.
The on-shell condition for external particles implies: p = p = m , (4) p = p (cid:48) = M , (5) p = 0 . (6)At leading order, the scattering amplitude M is given by M = ¯ u ( p )( ie ) (cid:20) γ ν i ( (cid:54) p − (cid:54) p + m )( p − p ) − m γ µ + γ µ i ( (cid:54) p − (cid:54) p + m )( p − p ) − m γ ν (cid:21) ( ie ) v ( p ) ×× − it ε ν ( p )¯ u ( p (cid:48) )( − ie )Γ µ ( t ) u ( p ) , (7)where the electromagnetic vertex Γ µ for the proton is expressed asΓ µ ( t ) = F D ( t ) γ µ − iF P ( t ) σ µν ( p ) ν M , (8)with the proton’s Dirac and Pauli form factors F D and F P , respectively.The corresponding unpolarized differential cross section dσ is given by (cid:32) dσdt ds ll d Ω CM l + l − ll (cid:33) = 1(2 π ) β (2 M E γ ) (cid:34)(cid:88) i (cid:88) f ( M ∗ M ) (cid:35) , (9)where E γ is the lab energy of the initial photon and Ω CM l + l − ll is the solid angle of the leptonpair in their center-of-mass frame, in which the lepton velocity is denoted by β = (cid:115) − m s ll . (10)In Eq. (9), we average over all polarizations in the initial state and sum over the polarizationsin the final state. We express the cross section as a product of hadronic and leptonic partsas (cid:32) dσdt ds ll d Ω CM l + l − ll (cid:33) = α β π (2 M E γ ) t L µν H µν , (11)where the fine-structure constant is defined as α ≡ e / π ≈ / L µν (including the average over the initial photon polarization)is given by L µν = −
12 Tr (cid:20) ( (cid:54) p + m ) (cid:18) γ α ( (cid:54) p − (cid:54) p + m )( p − p ) − m γ µ + γ µ ( (cid:54) p − (cid:54) p + m )( p − p ) − m γ α (cid:19) ( (cid:54) p − m ) (cid:18) γ ν ( (cid:54) p − (cid:54) p + m )( p − p ) − m γ α + γ α ( (cid:54) p − (cid:54) p + m )( p − p ) − m γ ν (cid:19)(cid:21) , (12)and the unpolarized hadronic tensor H µν by H µν = 12 Tr (cid:2) ( (cid:54) p (cid:48) + M ) Γ µ ( (cid:54) p + M ) (Γ † ) ν (cid:3) . (13)Using (8), the unpolarized hadronic tensor can be expressed as H µν = ( − g µν + p µ p ν p ) (cid:2) M τ G M ( t ) (cid:3) + ˜ p µ ˜ p ν
41 + τ (cid:2) G E ( t ) + τ G M ( t ) (cid:3) , (14)where ˜ p ≡ ( p + p (cid:48) ) / τ ≡ − t/ (4 M ), and where we conveniently express the hadronic tensorin terms of electric ( G E ) and magnetic ( G M ) form factors defined as G E = F D − τ F P , (15) G M = F D + F P , (16)which are functions of the spacelike momentum transfer t .For the electric and magnetic proton form factors, which enter the total cross sectionsfor lepton-pair production, we exploit the fit of Ref. [2], which is based on a global analysisof the electron-proton scattering data at Q <
10 GeV with an empirical account of TPEcorrections.In the experimental setup, when only the recoil proton is measured, one has to integrate(11) over the lepton angles: (cid:18) dσdt ds ll (cid:19) = α β π (2 M E γ ) t · ˆ d Ω CM l + l − ll L µν H µν . (17)The kinematical invariant t is in one-to-one relation with the recoiling proton lab momentum (cid:126)p (cid:48) (or energy E (cid:48) ): | (cid:126)p (cid:48) | = 2 M (cid:112) τ (1 + τ ) , (18) E (cid:48) = M (1 + 2 τ ) , (19)whereas the invariant s ll is then determined from the recoiling proton lab scattering angle:cos θ p (cid:48) = s ll + 2( s + M ) τ s − M ) (cid:112) τ (1 + τ ) , (20)where s is the squared center-of-mass energy, which can be expressed in terms of the initialphoton-beam energy E γ : s = 2 E γ M + M . (21) FIG. 2: Ratio of the cross sections in γp → ( e + e − + µ + µ − ) p vs γp → ( e + e − ) p . The blue bandcorresponds to a 3 σ band, where σ = 7 × − . In Ref. [25], the authors calculated the ratio R of cross sections between electron- andmuon-pair production: R ( s ll , s ll ) ≡ [ σ ( µ + µ − )] ( s ll ) + [ σ ( e + e − )]( s ll )[ σ ( e + e − )]( s ll ) , (22)which depends on the invariant mass of the lepton pair s ll , and a reference point s ll to whichthe measurement is normalized.The corresponding plot for the kinematical range accessible at MAMI is shown in Fig. 2.The normalization is shown for the choice s ll = s ll , i.e., at each point above the muon-pairproduction threshold the sum of the cross sections for muon- and electron-pair productionis divided by the corresponding cross section for electron-pair production. In this plot, theblue curve describes the scenario, when lepton universality holds, i.e., G µE = G eE , while thered curve corresponds to a case when lepton universality is broken by an amount of 1%.The blue band describes the 3 σ deviation if this observable is measured with an absoluteaccuracy of 7 × − . We will show in this work that radiative corrections shift this curveby more than 3 σ , making their inclusion indispensable for a comparison with experiment. III. LEADING-ORDER RADIATIVE CORRECTIONS IN THE SOFT-PHOTONLIMIT
We evaluate the first-order QED corrections to the γp → l + l − p process in the soft-photonlimit. This limit is defined by a scaling of the momenta k of virtual photons in the loopsand real photon momenta in the bremsstrahlung process, with respect to external scales, as k ∼ λ, (23)where λ is a small parameter. We calculate the diagrams at leading order in λ . This proce-dure reproduces all infrared-divergent contributions and results in a finite, gauge-invariantpiece. The resulting cross-section correction factorizes in terms of the tree-level cross sectiongiven by Eq. (11). A. Virtual corrections
We start by calculating the one-loop virtual radiative corrections. In the soft-photonapproximation, only box diagrams contribute. We list all propagators and their scaling with λ in Tab. I: propagator denominator scaling at least as( k + p ) − m λ ( k − p ) − m λk λ ( p − p + k ) − m λ . Only integralswith the first 3 propagators contribute, since these integrals have a denominator scaling as λ ,which is the scaling of the integral measure in the numerator. The integral measure d k scales as d k ∼ λ . (24) p p p p p ′ p p p p p p ′ p k k FIG. 3: QED box diagrams contributing to the radiative corrections calculation in the soft-photonapproximation.
Therefore, to obtain a contribution of order 1, we need a denominator of order λ . This isonly possible for the box diagrams when the first 3 propagators of Tab. I are present in aFeynman integral.For the box diagrams shown in Fig. 3, we obtain the following leading contribution: M box = ( ie ) 4 · ( p p ) · M µ − d ˆ d d k (2 π ) d p + k ) − m k − p ) − m k + O ( λ )= − e π (cid:0) s ll − m (cid:1) · M · C (cid:0) m , s ll , m , , m , m (cid:1) , (25)with the 3-point function C in dimensional regularization, see Ref. [27]: C (cid:0) m , s ll , m , , m , m (cid:1) = 1 s ll β (cid:26)(cid:20) (cid:15) IR − γ E + ln (cid:18) πµ m (cid:19)(cid:21) ln (cid:18) β − β + 1 (cid:19) +2 Li (cid:18) β − β (cid:19) + ln (cid:18) β − β (cid:19) −
12 ln (cid:18) β − β + 1 (cid:19) − π (cid:27) . (26)In Eqs. (25), (26), µ is a scale introduced to account for the correct energy dimensionof the integral. Physical quantities have to be independent of this scale, as well as of theinfrared regulator (cid:15) IR ≡ − d/ <
0. All other diagrams are infrared finite and scale at leastas λ . Therefore, the other graphs do not contribute in the soft-photon limit. We use the same notation for this function as in http://qcdloop.fnal.gov/ p p p p p p ′ p p ′ p p p p p p ′ p FIG. 4: Counterterm diagrams, which contribute to the γp → l + l − p process. These give riseto infrared-divergent contributions in the on-shell subtraction scheme and have therefore to beaccounted for when calculating the radiative corrections in the soft-photon approximation.FIG. 5: Diagrams for the calculation of the counterterms. The upper diagram defines the vertexcounterterm, the lower diagram corresponds to the lepton self-energy Although the box diagrams are UV finite, we have to include counterterm corrections,shown in Fig. 4, since they contain infrared-divergent parts in the on-shell subtractionscheme, which we follow here. We describe these contributions according to Ref. [28].In the on-shell subtraction scheme, the vertex counterterm is defined to fix the electron0charge e at q = 0. Considering the vertex function in Fig. 5, one can decompose thediagram into two tensor structures with corresponding form factors F and G :¯ u ( p (cid:48) )Γ µ u ( p ) = ¯ u ( p (cid:48) ) (cid:104) (1 + F ( q )) γ µ + iG ( q ) σ µν q ν m (cid:105) u ( p ) , (27)with q = p (cid:48) − p. (28)Only F ( q ) is UV divergent, and one finds at q = 0 the renormalization constant: Z = 1 − F (0) == 1 − e (4 π ) (cid:26)(cid:20) (cid:15) UV − γ E + ln (cid:18) πµ m (cid:19)(cid:21) + 2 (cid:20) (cid:15) IR − γ E + ln (cid:18) πµ m (cid:19)(cid:21) + 4 (cid:27) . (29)This leads to the renormalized vertex:˜Γ µ = Γ µ + ( Z − γ µ , (30)that in the soft-photon limit (˜Γ µs ), which corresponds to taking only the infrared-divergentpart, is expressed as ˜Γ µ s = − α π γ µ (cid:20) (cid:15) IR − γ E + ln (cid:18) πµ m (cid:19)(cid:21) . (31)The contribution of the two vertex counterterms in Fig. 4 is then given by M ctvertex = − απ (cid:20) (cid:15) IR − γ E + ln (cid:18) πµ m (cid:19)(cid:21) M . (32)The self-energy counterterm is defined from the lepton self-energy correction Σ( p ), whichis expressed in terms of the lepton propagator S : iS = iS + iS ( − i )Σ( p ) iS, (33)with free fermion propagator given by S ( p ) = (cid:54) p + mp − m . (34)Calculating up to first order, we have to include the one-loop correction: − i Σ( (cid:54) p ) = − e µ − d ˆ d d k (2 π ) d γ α ( (cid:54) p + (cid:54) k + m ) γ α (( p + k ) − m ) k . (35)1The on-shell renormalization condition fixes the pole at p = m with residue equal to one.This gives the renormalization constants Z and Z m : Z = 1 + d Σ( (cid:54) p ) d (cid:54) p (cid:12)(cid:12)(cid:12)(cid:12) (cid:54) p = m , (36)(1 − Z m ) Z m = Σ( m ) . (37)The evaluation of Σ( p ) and its derivative, results in the renormalization constants: Z = 1 − e (4 π ) (cid:26)(cid:20) (cid:15) UV − γ E + ln (cid:18) πµ m (cid:19)(cid:21) + 2 (cid:20) (cid:15) IR − γ E + ln (cid:18) πµ m (cid:19)(cid:21) + 4 (cid:27) , (38) Z Z m = 1 − e (4 π ) (cid:26) (cid:20) (cid:15) UV − γ E + ln (cid:18) πµ m (cid:19)(cid:21) + 2 (cid:20) (cid:15) IR − γ E + ln (cid:18) πµ m (cid:19)(cid:21) + 8 (cid:27) . (39)The renormalized self-energy is then given by˜Σ( p ) = Σ( p ) − ( Z − (cid:54) p + ( Z Z m − m. (40)Taking only the infrared-divergent piece in the soft-photon limit ( ˜Σ s ), we obtain:˜Σ s ( p ) = α π ( (cid:54) p − m ) (cid:20) (cid:15) IR − γ E + ln (cid:18) πµ m (cid:19)(cid:21) . (41)The contribution of the self-energy counterterm M ctse in Fig. 4 is therefore given by M ctse = α π (cid:20) (cid:15) IR − γ E + ln (cid:18) πµ m (cid:19)(cid:21) M . (42)Adding virtual corrections of Eq. (25) and counterterms of Eqs. (32) and (42), we obtainthe virtual one-loop correction in the soft-photon limit M s;V : M s;V = − α π (cid:26)(cid:0) s ll − m (cid:1) C (cid:0) m , s ll , m , , m , m (cid:1) + (cid:20) (cid:15) IR − γ E + ln (cid:18) πµ m (cid:19)(cid:21)(cid:27) M . (43)The resulting virtual correction to the cross section is then given, to first order in α , by (cid:18) dσdt ll ds ll (cid:19) s;V = 2 Re [ M ∗ × M s;V ] . (44)It can be expressed as (cid:18) dσdt ll ds ll (cid:19) s;V = (cid:18) dσdt ll ds ll (cid:19) (cid:18) δ IRs;V + δ s;V (cid:19) , (45)with the infrared-divergent part: δ IRs;V = (cid:18) − απ (cid:19) (cid:20)(cid:18) β β (cid:19) ln (cid:18) − β β (cid:19) + 1 (cid:21) (cid:20) (cid:15) IR − γ E + ln (cid:18) πµ m (cid:19)(cid:21) , (46)and the finite part: δ s;V = (cid:18) − απ (cid:19) (cid:18) β β (cid:19) (cid:26) (cid:18) ββ + 1 (cid:19) + 12 ln (cid:18) − β β (cid:19) − π (cid:27) . (47)2 p p p p p ′ p p p p p p p p p p p ′ p p ′ p p ′ k kkk FIG. 6: Diagrams with real photon emission from the lepton lines for the Bethe-Heitler process.In the soft-photon limit, the diagram with the photon attached to the internal (off-shell) fermionline does not contribute.
B. Soft-photon bremsstrahlung
Besides the QED virtual radiative corrections, one has to account for processes withradiation of undetected photons.The diagrams contributing to the soft bremsstrahlung from the lepton side are shownin Fig. 6. Note that the diagram, where the photon is attached to the internal leptonline, vanishes for λ → k , we find the squared matrix element for this process in the3form: (cid:12)(cid:12) M ( γp → γ s l + l − p ) (cid:12)(cid:12) = (cid:12)(cid:12) M ( γp → l + l − p ) (cid:12)(cid:12) ( − e ) (cid:20) p µ p · k − p µ p · k (cid:21) · (cid:20) p µ p · k − p µ p · k (cid:21) . (48)To calculate the contribution to the cross section, one then has to integrate over the unde-tected soft-photon energy up to a small value ∆ E s , determined by the experimental resolu-tion.Due to the energy-momentum conserving δ -function, δ ( p + p − p − p − p (cid:48) − k ), theintegration domain has a complicated shape in the lab system. The integration can becarried out in the rest frame S of the real ( p ) and virtual ( p ) photons, which is also therest frame of the di-lepton pair and soft photon, defined by (cid:126)p + (cid:126)p = (cid:126)p + (cid:126)p + (cid:126)k = 0 . (49)In such frame, the dependence of the integral with respect to the soft-photon momentumbecomes isotropic. For the differential cross section, we then need to evaluate: (cid:18) dσdtds ll (cid:19) s;R = − (cid:18) dσdtds ll (cid:19) e (2 π ) ˆ | (cid:126)k | < ∆ E s d (cid:126)k k (cid:20) m ( p k ) + m ( p k ) − p p )( p k )( p k ) (cid:21) , (50)where the integration is performed in the frame S .The integrals are infrared divergent and can be carried out analytically after dimensionalregularization. They have been worked out, e.g., in Ref. [29]. For the kinematics in system S , where the soft-photon momentum: | (cid:126)k | (cid:28) | (cid:126)p | , | (cid:126)p | , (51)with the lepton momenta: p = p = √ s ll , (cid:126)p = − (cid:126)p , (52)we obtain: (cid:18) dσdtds ll (cid:19) s;R = (cid:18) dσdtds ll (cid:19) (cid:18) δ IRs;R + δ s;R (cid:19) , (53)where δ IRs;R is the infrared-divergent contribution due to real photon emission: δ IRs;R = (cid:18) − απ (cid:19) (cid:20)(cid:18) β β (cid:19) ln (cid:18) β − β (cid:19) − (cid:21) (cid:20) (cid:15) IR − γ E + ln (cid:18) πµ m (cid:19)(cid:21) , (54)4and δ s;R is the corresponding finite part: δ s;R = (cid:18) − απ (cid:19) (cid:26) ln (cid:18) E s m (cid:19) (cid:20) (cid:18) β β (cid:19) ln (cid:18) − β β (cid:19)(cid:21) + 1 β ln (cid:18) − β β (cid:19) ++ (cid:18) β β (cid:19) (cid:20) (cid:18) β β (cid:19) + 12 ln (cid:18) − β β (cid:19)(cid:21)(cid:27) . (55)The maximum value of the undetected soft-photon energy ∆ E s is defined in the system S .One can re-express it in terms of the detector resolutions. We consider the case of detectingthe recoil proton only. The energy E (cid:48) and angle θ p (cid:48) of the scattered proton are measured inthe lab frame. The missing mass M miss of the system is defined by M = ( p + p + k ) = s ll + 2 M miss E s , (56) E s = M − s ll M miss , (57)where E s denotes the soft-photon energy.The missing mass M miss is experimentally determined from the quantity: M = ( p + p − p (cid:48) ) = 4 M τ (cid:32) E γ (cid:114) ττ cos θ p (cid:48) − E γ − M (cid:33) , (58)where τ is determined from the lab proton momentum by Eq. (18), and θ p (cid:48) is the experi-mentally measured recoil proton scattering angle in the laboratory frame.For the process without radiation, this angle is given by Eq. (20), which can be equiva-lently obtained from Eq. (58) by the replacement M → s ll : s ll = 4 M τ (cid:32) E γ (cid:114) ττ cos θ p (cid:48) | no rad − E γ − M (cid:33) . (59)Combining Eqs. (58) and (59), we can express the soft-photon energy of Eq. (57)approximately as: E s = 2 M E γ (cid:112) τ (1 + τ ) √ s ll (cid:20) cos θ p (cid:48) − cos θ p (cid:48) | no rad (cid:21) . (60)Consequently, the experimental recoiling proton angular resolution, denoted as ∆ θ p (cid:48) ,determines the maximum value ∆ E s of the undetected soft-photon energy, which enters theradiative correction of Eq. (55), as∆ E s = 2 M E γ (cid:112) τ (1 + τ ) √ s ll sin θ p (cid:48) ∆ θ p (cid:48) . (61)5 C. Total result and exponentiation
Adding the real and virtual contributions of Eqs. (54) and (46), we find a cancellation ofall infrared divergences on the level of the cross section: δ IRs;R + δ IRs;V = 0 . (62)For the finite part of the first-order QED corrections in the soft-photon approximation: δ = δ s;R + δ s;V , (63)we find the result: δ = − (cid:16) απ (cid:17) (cid:26)(cid:20) ln (cid:18) E s m (cid:19) + ln (cid:18) − β β (cid:19)(cid:21) (cid:20) (cid:18) β β (cid:19) ln (cid:18) − β β (cid:19)(cid:21) + (cid:18) − ββ (cid:19) ln (cid:18) − β β (cid:19) + (cid:18) β β (cid:19) (cid:20) (cid:18) β β (cid:19) − π (cid:21)(cid:27) , (64)which reduces in the limit s ll >> m to: δ = − (cid:16) απ (cid:17) (cid:26) ln (cid:18) E s s ll (cid:19) (cid:20) (cid:18) m s ll (cid:19)(cid:21) − π (cid:27) . (65)To account for the emission of a higher amount of soft photons or higher-order virtualcorrections due to soft photons in the loop, we follow Ref. [30] and exponentiate the termsleading to double logarithmic enhancements as (cid:18) dσdt ds ll (cid:19) s;tot = (cid:18) dσdt ds ll (cid:19) · F exp (cid:26) − απ (cid:20) ln (cid:18) E s m (cid:19) + ln (cid:18) − β β (cid:19)(cid:21) (cid:20) (cid:18) β β (cid:19) ln (cid:18) − β β (cid:19)(cid:21)(cid:27) × (cid:26) − απ (cid:20)(cid:18) − ββ (cid:19) ln (cid:18) − β β (cid:19) + (cid:18) β β (cid:19) (cid:20) (cid:18) β β (cid:19) − π (cid:21)(cid:21)(cid:27) ≡ (cid:18) dσdt ds ll (cid:19) (1 + δ exp ) . (66)Note that in Eq. (66) terms of single logarithmic nature of order α are still missing, andrequire a full one-loop calculation. The normalization factor F in Eq. (66) is due to thephysical assumption that in an experiment the sum of all soft-photon energies is smallerthan ∆ E s , instead of requiring that each soft-photon energy is individually smaller than∆ E s . It was shown in Ref. [30] that when including the leading correction from unity, thenormalization factor F is given by: F = 1 − α (cid:20) (cid:18) β β (cid:19) ln (cid:18) − β β (cid:19)(cid:21) + ... (67)6Although we account for the factor F explicitly, its deviation from unity is quite small: for s ll = 0 .
077 GeV approximately − . × − for electrons and − . × − for muons.7 IV. RESULTS AND DISCUSSION
In Fig. 7, we show the corrections at fixed s ll = 0 .
077 GeV as a function of the soft-photon energy. We observe a logarithmic behavior of the correction factor δ which givesrise to the so-called radiative tail. We also show the exponentiated form, δ exp , given byEq. (66), which estimates higher-order effects of soft-photon corrections. Assuming a value∆ E s = 0 .
01 GeV, δ at first order differs by about 0 .
006 for electron-pair production and isindistinguishable at the level of precision for muon-pair production (the difference is around − . × − ). - - - - - FIG. 7: QED corrections to the cross section in the soft-photon limit as a function of the soft-photon energy ∆ E s , which corresponds to the integrated over angular bins ∆ θ p (cid:48) according to Eq.(61). This variation stems from the integrated over radiative tail. The external kinematics and thedi-lepton invariant mass s ll = 0 .
077 GeV are indicated on the plot. In Fig. 8, we show the radiative corrections to the cross section in the kinematical range8of s ll between 0 and 0 .
08 GeV . The muon threshold is at s ll = 4 m µ ≈ .
045 GeV (verticaldashed red line in Fig 8). We observe that the corrections for electrons are negative of order10 percent, while the corrections for muons are positive of order 1 percent. - - - FIG. 8: First-order QED corrections to the cross section in the soft-photon limit, using ∆ E s =0 .
01 GeV. The vertical dashed red line indicates the muon-pair production threshold at s ll ≈ .
045 GeV . Taking radiative corrections into account, the ratio of Eq. (22) is now given by R ( s ll , s ll ) ≡ [ σ ( µ + µ − )(1 + δ µ )] ( s ll ) + [ σ ( e + e − )(1 + δ e )]( s ll )[ σ ( e + e − )(1 + δ e )]( s ll ) , (68)which depends on the measured invariant lepton mass s ll and the reference point s ll , to whichthe cross section is normalized. δ e and δ µ are given by Eq. (66). One chooses s ll < m µ ,such that the reference measurement is below the muon-pair-production threshold, and onlyelectron pairs are created.9 FIG. 9: Ratio of cross sections between electron- and muon-pair production at tree level (bluecurve) and with account of first-order QED corrections estimated using ∆ E s = 0 .
01 GeV (orangecurve) with 3 σ error bands. The red curve denotes the scenario when lepton universality is brokenwith G µE /G eE = 1 .
01, including the radiative corrections in the soft-photon approximation.
In Fig. 9, we show the differential cross section ratio R of Eq. (22), including first-orderQED corrections in the soft-photon approximation with ∆ E s = 0 .
01 GeV. One sees fromthis plot, that the inclusion of radiative corrections is indispensable, since the ratio of crosssections, defined in Eq. (22), is shifted to higher values by more than the 3 σ band. Theradiative corrections to R are of the order of a few percent. The red curve in Fig. 9 showsthe scenario when lepton universality is violated by G µE /G eE = 1 .
01. Following Ref. [25],we use 3 σ bands around the curves, with the experimental resolution σ = 7 × − . Thestatement that lepton universality can be tested with a 3 σ confidence level remains true ifone adds radiative corrections as can be seen in Fig. 9.In Fig. 10, we show the corresponding ratio between the cross sections normalized toa value below the muon-pair production threshold. As a reference point, we choose s ll =0 FIG. 10: Ratio of cross sections between electron- and muon-pair production at tree level (bluecurve) and with account of first-order QED corrections estimated using ∆ E s = 0 .
01 GeV (orangecurve), normalized to the electron-pair production cross section at s ll = 0 .
02 GeV . .
02 GeV . The bands now correspond to the renormalized 3 σ bands, i.e., σ = 7 × − · σ ( e + e − )( s ll ) σ ( e + e − )( s ll ) . V. CONCLUSIONS AND OUTLOOK
In this work, we have calculated QED radiative corrections to the photoproduction ofelectron and muon pairs on a proton target in the soft-photon approximation. Only radia-tion from the produced pair and box diagrams with photon and lepton legs contribute in thisapproximation when accounting for the finite lepton mass. The resulting correction to thecross section factorizes in terms of the tree-level contribution. We expressed the proportion-ality factor in a compact analytical form. With account of radiative corrections, the ratioof photoproduction cross sections of e + e − + µ + µ − to e + e − pairs at the same beam energy1(as well when compared to the e + e − cross section at an energy below the muon-productionthreshold) increases by a percent amount comparing to the tree level result. Such changesare significantly larger than the precision needed to distinguish between the proton chargeradii extractions from experiments with muons and electrons. It makes a correct inclusionof radiative corrections paramount for the experimental realization. As a next step, we planto extend the radiative correction result in the soft-photon approximation presented in thiswork to a full one-loop QED calculation on the lepton side and to include the box diagramsresulting from the two-photon exchange between lepton and proton with an intermediateproton state using the techniques developed in Refs. [31, 32] for elastic l − p scattering. Forthe leading corrections resulting from the hadronic side we expect, from the correspondingresults for the elastic l − p scattering, to receive cross section corrections at the percent levelfor the electron case. Such anticipated corrections would translate in a change of the ratioof e + e − + µ + µ − to e + e − cross sections at the per mille level, corresponding with the 1 σ accuracy goal discussed above for this quantity. Acknowledgements
We would like to thank Dr. Aleksandrs Aleksejevs and Shihao Wu for useful discussions.This work was supported by the Deutsche Forschungsgemeinschaft DFG in part throughthe Collaborative Research Center [The Low-Energy Frontier of the Standard Model (SFB1044)], and in part through the Cluster of Excellence [Precision Physics, Fundamental Inter-actions and Structure of Matter (PRISMA)]. Matthias Heller is supported in part by GRKSymmetry Breaking (DFG/GRK 1581). Our figures were generated using
Jaxodraw [33],based on
AxoDraw [34]. For our
Mathematica plots, we use the package
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