Weak interaction corrections to muon pair production via the photon fusion at the LHC

Analytical formulas describing the correction due to the $Z$ boson exchange to the cross section of the reaction $pp\rightarrow p\mu^+\mu^- X$ are presented. When the invariant mass of the produced muon pair $W\gtrsim 150~\text{GeV}$ and its total transverse momentum is large, the correction is of the order of 20%.


Introduction
The ATLAS collaboration measured the cross section of muon pair production in ultraperipheral collisions (UPC) of protons at the Large Hadron Collider [1].In the case of UPC the lepton pair is produced in the γγ fusion and is accompanied by forward scattering of both protons.We calculated the corresponding cross section in [2].Our result agrees with the one obtained by the ATLAS collaboration within the experimental accuracy.The results obtained with the help of Monte Carlo simulations can be found in [3][4][5][6].The µ + µ − pair production occurs in γZ and ZZ fusion as well.However, for the scattered proton to remain intact the square of the 4-momentum of the emitted photon (or Z boson) Q 2 should not considerably exceed (200 MeV) 2 .For larger Q 2 the cross section is suppressed by the elastic form factor [7]. Thus the contribution of the diagram with the virtual Z boson exchange is suppressed by the factor Q 2 /M 2 Z ∼ 10 −5 and can be safely omitted.But if only one of the protons remains intact while the second one dissolves producing a hadron jet then substitution of the photon emitted by the second proton by a Z boson may lead to numerically noticeable corrections.This is so since the value of Q 2 is now bounded from above only by the invariant mass of the produced muon pair W (for Q 2 > W 2 the cross section of muon pair production is suppressed as a power of W 2 /Q 2 ).In this way the contribution from the Z boson exchange being proportional to [Q 2 /(M 2 Z + Q 2 )] 2 can become noticeable for W 2 ≳ M 2 Z .The ATLAS collaboration measured the fiducial cross section of the reaction of lepton pair (e + e − or µ + µ − ) production when one of the scattered protons is detected by the ATLAS Forward Proton Spectrometer [8].The other proton can remain intact or disintegrate.In the latter case the events were selected in which the momentum squared of the emitted photon did not exceed (5 GeV) 2 (see our recent paper [9] with formulas and numerical estimates of the cross sections measured in [8]).
Dilepton production in proton-proton collisions through γγ-fusion with the first proton scattered elastically while the second one produced a hadron jet was considered in our paper [10].Analytical formulas describing the cross section of a muon pair production were derived therein.
In the present paper we calculate weak interaction corrections to the cross section obtained in [10] which should be used for comparison of the Standard Model predictions with future experimental data.New Physics can manifest itself in interactions of muons at high energies.Thus accurate Standard Model predictions are needed to probe it.Tiny deviation of the measured value of muon magnetic moment from the SM result [11,12] may indicate considerable deviation of muon pair production cross section at large invariant masses of muon pair W ∼ 100 GeV ÷ 1 TeV.To find them the results of the present paper should be accounted for.
In Section 2 general formulas are derived.Z boson exchange contributions are discussed in Section 3. Numerical results are presented in Section 4 and in Section 5 we conclude.

Muon pair production in γγ fusion
Diagrams for muon pair production in ultraperipheral collisions of protons are shown in Fig. 1.
Figure 1: Muon pair production in the photon fusion.The photon with momentum q 1 is radiated by the elastically scattered proton and the photon with momentum q 2 is radiated by the quark.
The cross section for two-photon production can be expressed in terms of the amplitude M µα of γγ → µ + µ − transition as follows (see [9] and the review of two-photon particle production [13]): where the leading factor 2 takes into account the symmetrical process where the other proton survives, α is the fine structure constant, Q q is the charge of quark q, ρ µν and ρ (2 µν are the density matrices of the photons, dΓ is the phase space of the muon pair, m p is the proton mass, 2 ) is the parton distribution function for quark q, x is the fraction of the momentum of the disintegrating proton carried by the quark, Q 2 2 = −q 2 2 , E ′ 1 and E ′ 2 are the energies of the proton and the quark after the collision.
For the density matrix ρ (1) originating from the elastically scattered proton we have (see Eqs. (25-27) from [9]): Here Q 2 1 = −q 2 1 , and ) are the Sachs form factors of the proton.For the latter we use the dipole approximation: where µ p = 2.79 is the proton magnetic moment and r p = 0.84 fm is the proton charge radius [14].
The density matrix of the photon emitted by the quark is It is convenient to consider the lepton pair production in the basis of virtual photon helicity states.In the center of mass system (c.m.s.) of the colliding photons, let q 1 = (ω 1 , 0, 0, q), q 2 = (ω 2 , 0, 0, −q).The standard set of orthonormal 4-vectors orthogonal to q 1 and q 2 is (q, 0, 0, ω1 ), (−q, 0, 0, ω2 ). ( Due to the conservation of vector current, the covariant density matrices ρ µν i satisfy q µ 1 ρ µν 1 = q µ 2 ρ µν 2 = 0. Thus, we can write ρ ab i = (−1) a+b e aµ i e bν i * ρ µν i , where a, b ∈ {±1, 0}, and ρ ab i are the density matrices in the helicity representation.The amplitudes of the lepton pair production in the helicity basis M ab appear from the following equation: ++ ρ (2) ++ ρ (2) ++ ρ (2) In this expression non-diagonal terms (those with a ̸ = b or c ̸ = d) originated from the interference are omitted since their contributions cancel out when one integrates over azimuthal angles of the proton and the quark in the final state [13,15].The contribution of the longitudinally polarized photon emitted by the proton is neglected since it is proportional to Q 2 1 /W 2 , where W is the invariant mass of the produced lepton pair.The reason is that due to the elastic form factors Matrix elements of the photon density matrices in the helicity representation for transverse polarizations were found in [9] (see also [13,15]).For the first photon we have: where E is the proton energy in the c.m.s. of the colliding protons, q 1⊥ is the transversal momentum of the photon and ω 1 is its energy in the same system.The function D(Q 2 1 ) is defined in (2).In the following we will calculate the cross section of the muon pair production in the parton model.For the second photon which is emitted by the quark with the initial energy xE, 0 < x < 1, we have: Changing the integration variables from d 3 p and substituting expressions ( 8), ( 9) and ( 10) in (1), we get: where σ γγ * →µ + µ − is the cross section of µ + µ − production in γγ * collision, q 1 q 2 = (W 2 − q 2 2 )/2 and p 1 p 2 = 2E 2 x.The differential cross section of muon pair production in the c.m.s. of the photons is: where M ab are the amplitudes of the process γγ * → µ + µ − in the helicity representation, θ is the scattering angle.It is convenient to express the corresponding cross section as the sum of the terms with the photon emitted by the quark polarized transversely and longitudinally: , where according to Eq. (E3) from [13] σ where m is the muon mass.Integration of (11) with respect to q 2 1⊥ factors out the equivalent photon spectrum of the proton [9, Eqs.(4,5)] It is convenient to change the integration variables in (11) from the photon energies ω 1 and ω 2 to the square of the invariant mass of the produced pair W 2 = 4ω 1 ω 2 + q 2 2 and its rapidity y = 1/2 ln ω 1 /ω 2 : dω 1 dω 2 dq 2 2⊥ = (1/4)dW 2 dydQ 2 2 .Taking into account that , where γ q = E q /m q ≈ 3xE/m p = 3xγ1 , we obtain: where • e −y /2.When we were considering this process in [9], we were working under experimental constraints that allowed us to use the approximation Q 2 2 ≪ W 2 .In that case the photon emitted by the quark is approximately real, the cross section for the photon fusion does not depend on Q 2  2 , and we could factor out the function that we loosely interpreted as the equivalent photon spectrum of quark q: where p µµ T = 5 GeV is the experimental constraint on the muon system transversal momentum imposed in [9].We can use an analogous function here to simplify Eq. ( 15) and to make it obvious how the violation of the equivalent photon approximation occurs in this problem: where s = 4E 2 is the Mandelstam variable of the pp → pµ + µ − q reaction.In this way from ( 15) we obtain: where we assume the experimental constraint on the invariant mass of the muon pair Ŵ ≳ 10 GeV.
Finally, summing over valent u and d and sea quarks we get: 3 Z boson exchange corrections Figure 2: Weak interaction corrections to muon pair production in semi-inclusive pp → pµ + µ − X reaction originating from the diagrams with Z boson exchange.
Corrections to the described by Fig. 1 and (1) γγ-fusion reaction come from the interference with the two diagrams shown in Fig. 2 and from the square of these diagrams.In both cases the expression for the density matrix ρ (2) αβ given by (4) should be modified in order to take into account the axial coupling of Z boson to quarks.Let us designate these matrices ρ(2) and ≈ ρ (2) correspondingly.Masses of quarks can be safely neglected making matrices ρ(2) and ≈ ρ (2)  transversal: ρ(2) αβ q 2β = 0.That is why they can be expanded over the same 4-vectors e a 2µ , presented in (5).The coupling of Z boson to quarks equals: where e = √ 4πα, θ W is the electroweak mixing angle, s 2 W ≈ 0.231 [16], and T q 3 is the weak isospin of quark q, so for ρ(2) αβ and αβ we obtain: (2) Tr p′ Calculating matrix elements of ρ(2) ab in the helicity representation according to (7), we get that the correction proportional to g q A cancels in ρ(2) 00 and is proportional to xE/ω 2 in ρ(2) ++ and ρ(2) −− , so it can be neglected when compared to the contribution of the correction proportional to g q V which behaves as (xE/ω 2 ) 2 (see (10)).Therefore: ab . (25) In the case of αβ we observe similar suppression of terms originating from the terms proportional to the product g q A • g q V .Neglecting them we obtain: ab .
The amplitude that corresponds to the sum of the diagrams in Fig. 1 and Fig. 2 equals: For the γγ → µ + µ − and γZ → µ + µ − amplitudes (see Fig. 3) we have: Substituting (28) and ( 29) into (27) and separating the term proportional to M γ µα , we obtain: The following two statements are proved in Appendix: (1) the amplitudes in the square brackets which describe the vector and axial couplings to muons do not interfere and (2) the square of the amplitude with the axial coupling equals that with the vector coupling (we are working in W ≫ m domain).In this way for the square of the amplitude we get: where A γγ is the amplitude for the reaction via photon fusion only.Thus we get that the squares of the helicity amplitudes describing the γγ * → µ + µ − reaction presented in (12) should be multiplied by the same factor κ in order to take into account the diagrams with the Z boson exchange.It is convenient to calculate the helicity amplitudes M γ in the muons c.m.s.
The sum of the squares of the amplitudes of the processes with the longitudinal polarization of the vector boson (Z or photon) emitted by the quark is: and the corresponding cross section is It equals zero for Q 2 2 = 0, as it should be.For the contribution of the transversal amplitudes with opposite helicities we obtain: Since the sum of helicities of the produced leptons equals one, while that of the annihilating vector bosons equals two, the production at θ = 0, π is forbidden by the helicity conservation.The factor sin θ takes care of it.For the contribution to the cross section we obtain: where m is a lepton mass.Performing straightforward calculations, for the square of the amplitude induced by photons with helicities e + 1,2 we obtain: where v is the muon velocity in the c.m.s., v 2 = 1 − 4m 2 /W 2 .Helicity of the initial state of the two photons equals zero while for massless muons helicity of the µ + µ − pair equals ±1.That is why the amplitude M ++ should be zero at θ = 0, π when the helicity is conserved (massless muons).The factor sin 2 θ multiplying the first term in the curly braces takes care of this.In the case of muon (or antimuon) spin flip the final state will have zero helicity, and muons production at θ = 0, π is allowed.The two last terms in the curly braces are responsible for the forward and backward muon production.Though their numerators are proportional to m 2 , they make finite contribution to the cross section in the limit m → 0 due to the denominators which come from muon propagators.Integrating these terms over θ, we obtain a finite in the limit m/W → 0 contribution to the cross section which comes from the θ ∼ m/W and θ ∼ π − m/W domains.This phenomena is the essence of the chiral anomaly [17,18]: even in the limit m → 0 production of muons at θ = 0, π is still allowed.
The contribution to the cross section of muon pair production is the following: and σTT = σ+− + σ−+ + σ++ + σ−− , where σ+− + σ−+ is given by (36).Let us note that even in the case of the collision of real photons (Q 2 2 = 0) the cross section σ++ + σ−− is non-zero.As a final result we get: We have discussed γγ and γZ contributions.Let us note that the ZZ contribution should be very small for the process under consideration since it is suppressed by

Numerical results
Replacing in (20) 2 ) with σTS + σTT given in (33) and (39) we obtain an expression for the differential cross section of a muon pair production which takes into account the Z boson exchange: where we used the equality ω 2 = (W 2 + Q 2 2 )/4ω 1 .For better convergence of the numerical integration it is convenient to change the order of integrals: In this formula we explicitly separated κ (Q 2 2 ) since it is the only source of weak interaction corrections in (41).For all charged fermions in the Standard Model g V and g A have the same sign as their charge Q, so the weak interaction correction is positive.
Let us make a couple of comments regarding formula (41) and its accuracy.The lower integration limit on Q 2 2 in (41) is much smaller than (1 GeV) 2 .So we should explain if it is consistent with the parton approximation.This lower limit comes from kinematics and does not take into account other requirements.However, there is a soft cut at small Q 2 2 in the parton distribution functions provided by LHAPDF [19] so we decided not to change this limit in (41).Numerically, the region Q 2 2 < (1 GeV) 2 gives ∼ 15% of the differential cross section for W = 20 GeV and even less for larger W .We must stress that it does not affect the absolute value of the weak interaction correction since κ (Q 2 2 ) = 1 for Q 2 2 ≲ (1 GeV) 2 with very good accuracy.
We should also note that the photon flux of the disintegrating proton is not fully described by the parton approximation.Resonance phenomena and other effects can increase the cross section by 10-15% [20][21][22].However, the corresponding region Q 2 2 ∼ (1 GeV) 2 is not relevant for the weak interaction correction, so we do not take these effects into account.
Results of our numerical calculation are shown in Fig. 4. We see that the weak interaction correction does not give a noticeable increase in the cross section.The reason for that is clear: all scales of Q 2 2 in (41) are equally important while for the weak interaction correction only the domain The weak interaction correction is more pronounced when we set the lower limit on Q 2 2 , Q 2 2 > Q2 2 , closer to the electroweak scale, M 2 Z .For Q2 ≫ 1 GeV it means a cut on the total transverse momentum of the produced pair, p µµ T > Q2 .Such a selection is both theoretically and experimentally clean.From the theory side we have no complications due to low-Q 2 physics.From the experiment side this selection means high-p T events with two distinguishable muons (at least for W ≳ Q2 since the cross section is additionally suppressed for Q 2 2 > W 2 , see (41)).The results for Q2 = 30, 50, 70 GeV are shown in Fig. 5.We see that the weak interaction correction reaches 20 %.
In the limit Q 2 2 ≫ M 2 Z the function κ(Q 2 2 ) reaches its maximum value: However, to get an idea of what the correction in principle might be, one should keep in mind that u-and c-quarks and antiquarks give ∼ 80 % of the cross section in photon-photon fusion.

Conclusions
We have calculated the weak interaction correction to the cross section of lepton pair production in semi-exclusive process.Beforehand it was not obvious if this correction is significant or not.
It turned out that it gives few percent increase of the production cross section.However, if we set the lower limit on the net transverse momentum of the produced pair, the correction goes up and can reach 20 %.Numerical calculations were performed with the help of libepa [23].For the parton distribution functions we use MMHT2014nnlo68cl [24] provided by LHAPDF [19].

Figure 4 :
Figure 4: Upper plot: differential cross section of the photon fusion only (orange dashed line) and with the weak interaction correction added (blue solid line).Lower plot: their ratio.

Figure 5 :
Figure 5: Differential sections for different lower limits on Q 2 2 .Styles and colors of the lines are the same as in Fig. 4.