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Electroweak corrections to Higgs boson production at the future LHeC

Precision measurement of quark Yukawa couplings is a crucial aspect of Higgs property study. Proposed as a future upgrade of the Large Hadron Collider (LHC), the Large Hadron electron Collider (LHeC) provides opportunities to probe quark Yukawa couplings with a high precision because of relatively low rate from QCD background as compared with that of the Higgs processes at the LHC. For this purpose, it is important to have a precision prediction of the Higgs production rate at the LHeC. The leading production channel of the Higgs boson at the LHeC is via weak boson fusion (WBF) which has the unique kinematic feature of forward tagging jets. Since there is no color exchange between the forward-backward objects in WBF processes, electroweak radiative corrections become particularly important. In this paper, we focus on the electroweak radiative corrections at next-to-leading order for Higgs boson production in the WBF channel at the LHeC. In the two renormalization schemes we use, the loop corrections are respectively at the level of 9% and 18% relative to the leading order result, and the next-to-leading calculation in both schemes agree up to a truncation error expected from the perturbative method. The size of the EW correction exceeds that from QCD radiations and is shown to be important for the study of Higgs phenomenology.

corrections in both integrated cross section and differential distributions.
We note that part of the EW loop effect has been calculated in a study [18] to esimate the significance of the triple Higgs-self coupling at the LHeC. However, the calculation only involves a small subset of the loop diagrams associated with the coupling. We shall see that the background (with no Higgs self coupling) cross section in this study can be largely reduced by including all EW loop processes, which makes a negative contribution and therefore yields a better constraint on the Higgs self coupling.
In this paper we present a calculation of the full NLO EW corrections to the WBF cross section in e-p collisions. To make the discussion general we consider only the partonic scattering amplitude with an on-shell Higgs, and do not simulate the final state to which the Higgs decays. For simplicity, the focus will be on the charge current(CC) WBF, which is dominant over the neutral current(NC) process. Moreover, the outgoing electron of the NC process could be tagged in a broad kinematical region where it could be distinguished from the CC process.
The calculation proceeds in the dipole subtraction formalism [19], first designed for a unified treatment of various singularities encountered in scattering amplitudes with real and virtual QCD radiations. The method was then adapted to the calculation of QED radiative corrections and generalized to include the radiative effect that is sensitive to the mass of the initial or final fermions [20][21][22]. We shall show how to consistently include the fermion mass effect in our calculation of the WBF process using the dipole method.
The rest of the paper is organized as follows. In Sec. II, we give the description of the details for the analytical and numerical parts of the calculation. In Sec. III, we discuss the numerical significance of the NLO EW corrections in both total and differential cross sections of the WBF process, and the implication on the study of phenomenology. Then we conclude the paper in Sec. IV.

II. Detailed calculation of the process A. LO contribution
The CC WBF ep → ν e jH is depicted at the leading order by the partonic process eq → ν e q H. An example diagram is shown in Fig. 1. The cross section is given by where B(P A , p q ) denotes the born amplitude squared at the parton level with the incoming electron and quark momenta P A and p q . The partonic center-of-mass (CM) energy squared is given byŝ = (P A + p q ) 2 , and dΦ 3 is the phase space element of the three body final state. In this paper we assume no particular polarization for external particles, so that the in B(P A , p q ) and all other squared amplitudes below, we sum and average over colors and spins of the initial and final states. η q is the fraction of the proton momentum P B carried by the incoming quark q. The hadronic cross section is factorized at the scale µ F into a quark distribution function f q (η b , µ F ), convolved with a squared hard scattering amplitude depending implicitly on η q . The evolution of f q (η b , µ F ) introduces NLO QCD and QED terms to the LO cross section, which is not a problem as long as we do not double count these terms in our NLO calculation. The dominant contribution comes from the inital quark flavors u, c,d ands, while the effect ofb and t is marginal and not included in this paper.

B. NLO EW corrections
The numerically dominant NLO EW corrections are from the loop diagrams of various topologies, ranging from self energy graphs to pentagons. Soft and collinear singularities may arise in some of these diagrams when photons are present in the loop. Representative loop diagrams with and without these singularities are shown in Fig. 3 and Fig. 2, respectively.
A consistent treatment of the singularities requires the inclusion of all diagrams with real emission of photons , whose momenta can potentially be either soft or collinear to the emittee. In properly defined observables, cancellation of part of the singularities takes place between real-emission and loop diagrams. Mass singularities associated with initial state radiations are removed by the collinear counter terms provided by the factorization proceedure, to be discussed below. There are also realemission processes induced by photons as constituents of the initial state protons. In this case, no corresponding loop diagrams are present and the singular terms are cancelled solely by the collinear counter terms.
Within the energy range of the LHeC, we treat u, d, s, and c as massless flavors, whose couplings with the Higgs are also neglected. The contribution from b and t is marginal and not included. The role of the electron mass is somewhat intricate and will be discussed later. With these simplifications at NLO, there are in total 628 loop diagrams produced by the program MADGRAPH5 AMC@NLO [23], and 42 real emission diagrams by FEYNARTS [24] with quark, anti-quark, and photon initial states.

Renormalization
The calculation of one-loop diagrams has been performed in the t'Hooft-Feynman gauge with the program MADLOOP [25]. Several packages are linked by MADLOOP for the reduction of tensor integrals, each based on one of two distinct procedures: the Tensor Integral Reduction [26,27] and the Ossola-Papadopoulos-Pittau method [28]. To handle the resonances of unstable particles, we employ the complex mass scheme [29][30][31]. For simplicity, we neglect quark mixing and do not renormalize the quark mixing matrix.
The mass and field strength renormalization constants are determined within the on-shell renormalization scheme, except for the field strenth Z AA of the photon that is closely related to the renormalization of the electric charge. It can be obtained by imposing a renormalization condition at some momentum scale, or equivalently, by specifying the form of the renormalized fine structure constant. In this work we adopt two schemes. The G µ scheme is defined by the choice [32] where the Fermi constant G µ is determined from the muon decay experiment, while the α(M Z ) scheme is obtained by evolving the physical fine structure constant α(0) ≈ 1/137 from zero momentum transfer to the scale M Z , yielding α(M Z ) ≈ 1/129.
The numerical value of α in both schemes differ from α(0), which is obtained by requiring that all high order corrections to the electron-photon 3-point function vanish in the Thomson limit [32,33]. The differences, albeit small in practice, are sensitive to the light fermion masses on which the loop contribution to the photon wave function depend. The sensitivity manifests as logrithmic terms of the form ln m f [32,34], indicating a collinear divergence in the on-shell photon wave function as the fermion masses m f approach 0. It is straightforward to show that in G µ and α(M Z ) schemes, Z AA , as well as the charge renormalization constant Z e , are free from the collinear divergence, such that the definition of the physical charge on-shell is maintained. As long as no external photons are present in the LO process, we are able to consistently implement the renormalization constants and neglect the small mass m f in the calculation of the loop diagrams with dimensional regularization.

Factorization
After factorizing the parton distribution functions (PDFs) of the incoming quarks, their mass can be safely set to zero in the partonic scattering amplitudes initiated by these flavors. The mass singularities from the initial state photon-quark splitting signal our ignorance of the nonperturbative interactions of the partons in the proton below some factorization scale µ F , on which the QED evolution of the PDFs depends. The corresponding poles in the hard scattering process are subtracted by adding a collinear counter term to the squared amplitudes. In this calculation we use the PDF set CT14qed inc proton [35] from the LHAPDF library [36] with the NLO QED evolution. The collinear counter term is consistently implemented in the MS factorization scheme.
On the other hand, the incoming electron has no internal structure to accommodate for the singularity from collinear radiations. In principle, one could keep m e throughout the calculation and retain the dependence of the hard scattering amplitude on ln m e . This collinear singularity is physical and would break the convergence of the perturbative calculation if m e were too small. Fortunately, the size of the fine structure constant at the electro-weak scale ensures the convergence of the perturbation series in most of the kinematic regions we will consider (see Sec. III C). However, in reducing the tensor integrals we find that some of the pentagon diagrams are sensitive to the electron mass, which leads to a significant disagreement among different reduction programs when m e takes a small but non-zero value. This numerical instability is cured if m e is neglected and the mass singularity is regulated dimensionally.
With this observation, we factor out the mass singularity from the collinear photon radiation off the incoming electron. The singular terms are then put in a lepton distribution function, with the form to O(α) in the MS factorization scheme [37] (Here and below, α can be α Gµ or α(M Z ), depending on the renormalization scheme used) This leaves a hard scattering part independent of the electron mass. Following this path, we are able to avoid the numerical problem during the reduction proceedure and the calculation of the hard scattering cross section can be greatly simplified. We should note that there is also a ln m e dependence from final state radiations, which is also physical if the radiated photon is observed. This gives a so-called non-collinear safe observable [38]. In this case m e cannot be ignored. However in this paper we treat collinear-safe jet observables (see Sec. III C) obtained by combining the collinear photon with a final state quark. Hence no sensitivity to the final state collinear radiation exists and m e = 0 can be used.

Four-particle final states
Four particle final states with an on-shell Higgs at the parton level can be produced at NLO (O(α 4 )) via e − + q → ν e + γ + q + H and e − + γ → ν e + q + q + H. Example diagrams for these processes are given in Fig. 4. Representative real-emission diagrams for CC WBF at the LHeC. The first and second lines correspond respectively to the processes e − + q → ν e + γ + q + H and e − + γ → ν e + q + q + H.
The subtraction procedure starts by constructing terms that asymptotically approach the squared amplitude for the real emission process in the phase space region where the radiated particle (the emittee) becomes soft or collinear to the emittor. The dipole structure of the subtraction term is encoded in the charge correlation between the splitting pair and a third particle (spectator). Three types of dipoles are present for our processes and the subtraction term takes the form [22] The indices i, j, k denote the emitter, emittee, and spectator, respectively, in the final state. a and b are initial state partons, where a plays the role of the spectator in D a ij , and the emitter in D a j,k and D a,b j , while b is the spectator. Here we extend the meaning of "parton" to include the inital state electron and photon that undergo the electromegnetic interaction. (QED partons) The outter sum is over all possible choices of the splitting pairs, while the inner sum runs over the spectators. The diagramatic representation of the three dipole types can be found in Ref. [19,22]. In each dipole term, the singular behavior is extracted and encoded in a splitting function factorized from the corresponding born process, whose final state coincides with the kinematics of the radiative process in the soft or collinear limit. To obtain the born kinematics one combines the emitter and emittee from the corresponding splitting process to a single particle. The momenta of this particle and of the spectator are then shifted in such a way as to respect momentum conservation and on-shell conditions for the 3-particle final state [19]. The explicit expressions for the dipole terms and shifted momenta are given in Appendix A Now we subtract these terms from the real emission part to remove any singularity in the phase space. The cross section is thereby expressed by where b denotes the parton that collides with the electron; it can be a quark, anti-quark, or photon, and plays the role of either an emitter or spectator. It should not be confused with the "b" in Eq. 4, which only denotes a spectator.
For simplicity we consider collinear-safe observables, for which we asign a jet function F (2) to the real emission process to combine quarks and photons according to a certain choice of the jet algorithm, despite in general that the formalism allows to treat collinear unsafe observables as well [38]. F (2) depends on the momenta of the 4-particle final state. Correspondingly, for each dipole term in Eq. 4, there is a function F (1) of the 3-particle phase space obtained from the real emission process via the procedure described after Eq. 4. F (1) 's are implicitly included in Eqs. 4 and 5, as well as in Eq. 1. To ensure the cancellation of singularities, F (2) must approach the corresponding F (1) in various soft and collinear limits.
At this order, only the delta function term of the electron PDF in Eq. 3 contributes, which is trivially integrated out over the electron momentum fraction. (This is also true for the LO case in Eq. 1) Since the integrand in Eq. 5 is finite, integration over the four particle phase space can be safely done in four dimensions with numerical method.

Three-particle final states
Evidently, the subtracted dipole terms are auxiliary and need to be put back to avoid any artifact in the calculation. The structure of these terms is simple enough such that the singularities can be isolated in a factorized form after an analytical integration over a one-particle phase space. These "integrated" dipole terms have a phase space with one fewer particle, and are added back to combine with the loop diagrams. The result contains only collinear singularities from the initial state radiations, which are then canceled by the collinear counter terms from the factorization procedure.
The dipole and collinear counter terms can be reorganized into a term I that contains all the soft and collinear divergences present in the loop diagrams, and two other terms K and P that contain finite remainders after cancellation of singularities. In particular, K and P include the finite terms from factorization of initial state collinear singularities. The terms that depend on the choice of factorization scheme are contained in K, while P gives the dependence on the factorization scale µ F . The cross section from the three particle final states can therefore be expressed in terms of the contribution from the born and loop processes together with I,K and P [19,21,22] with a labeling the electron and b the parton from the proton; a (b ) is the flavor from the splitting of a (b) that goes into the LO hard scattering amplitude. Here B is the squared born amplitude, and V the interference between born and loop processes. The beam and initial state parton momenta satisfy p a = η a P A and p b = η a P B . In the last line of Eq. 6, we use the O(α) term of the electron PDF, as is indicated by its superscript. Note the flux factor 1/2ŝ generally takes the different value in the first and last lines. The subscripts of B and V in the first and last lines show that in these terms, the flavors entering the hard scattering must be the same as those before initial state splitting. In contrast, the subcripts of K and P terms show that the flavors may change before and after splitting in these terms. The superscript of K or P labels the other incoming particle that does not go through the splitting. As in Eqs. 1, 4 and 5, we implicitly include a jet function F (1) for the phase space of each term in Eq. 6. The loop integrals in V are done in d = 4 − 2 dimensions and all singularities after renormalization manifest as single and double poles, to be canceled exactly by I. Therefore, the phase space momenta of V and B in the first line of Eq. 6 should be the same, i.e., in either 4 or d dimensions. After the poles are canceled, one takes the limit → 0 and performs the phase space integration in 4 dimensions, as indicated by the superscript of the phase space element dΦ (4) 3 . While I,K and P all contribute to the quark/anti-quark induced processes, photon induced processes only receive contribution from non-singular terms K and P .
The expressions of I,K and P for specific processes are listed in Appendix B. The integrand of all terms in Eq. 6 are finite and the phase space can be integrated over in four dimensions. Once σ N LO 3 and σ N LO 4 are obtained, we combine them to give a physical cross section at NLO.

A. Setup
To make numerical predictions for the WBF cross section, we take the energies of the incoming electron and proton to be corresponding to a CM energy √ s = 2 E p E e ≈ 1.98 TeV. The renormalization and factorization scales are set to Q 2 = M 2 W . In addition, we use the following set of parameters As discussed in Sec. II B 1 and II B 2, the dominant fermion mass effect is incorporated into the renormalized fine structure constant as well as the electron distribution. Hence we do not specify the values of fermion masses here, and set them to zero throughout the calculation of the hard scattering cross section. The only exception is the explicit use of m e in the electron distribution from Eq. 3 Also, as stated before, we use a unit CKM matrix in this work. In this calculation, the tree level Feynman diagrams and amplitudes are produced and evaluated by the FEYNARTS, FORM-CALC, and LOOPTOOLS package set [24,39]. The analytical calculation of the dipole terms are carried out with the program Mathematica. The one-loop amplitudes are computed using the program MADLOOP [25] implemented by MAD-GRAPH5 AMC@NLO package [23]. We have developed our own code to generate the 3-and 4-particle phase space, over which the numerical integration is performed with the Vegas Monte-Carlo program implemented by the CUBA library [40].

B. Consistency checks
To verify the calculation we have done several checks at different levels. First, the tree-level amplitudes for born and real emission processes computed with FORMCALC are compared with those obtained from MADGRAPH5.
In the four-particle final state part, we have made sure the real emission contribution approach the corresponding dipole terms in various soft and collinear limits, and the integrand after subtraction is stable in a Monte-Calo integration. Also, the explicit form of the dipole terms for the real emission processes is compared numerically with the dipoles generated by MADDIPOLE [41,42].
For the three particle final state terms, we have verified analytically in the case of triangular loop diagrams that all double and single poles produced in dimensional regularization are canceled among the loop graphs, dipoles, and the collinear counter terms. This verification is then carried out for all loop diagrams numerically. In the finite part, we checked that the dependence on the renormalization scale is canceled between the loop and dipole terms 1 . There is no running of the coupling and masses in the renormalization schemes we choose, and the cancellation of the scale dependence is exact at the order we are working with.

C. Phenomenology
First, we report the result of the integrated cross section computed in two renormalization schemes.   in the LO cross sections obatined in the two schemes. This is solely due to the difference in the renormalized fine structure constants. In α Gµ scheme, the NLO terms reduces the LO result by ∼ 9%, while the relative correction in α(M Z ) scheme is as large as ∼ 18%. The sum of LO and NLO result, however, agree in both schemes, up to a difference at one higher order. This in fact provides another consistency check for the calculation. In both schemes the bulk of the NLO correction come from the three particle final states. The allocation of the contribution between 3-and 4-particle final states is an artifact that relies on the choice of the subtraction terms away from the divergent region of the phase space. But this will of course not affect the sum of the two contributions. Notice that the coupling in α Gµ scheme is smaller and leads to a much faster convergence of the perturbation series. In the following discussion we shall stick with this scheme. Next, we turn to the study of distributions. As is remarked in the introduction, a distinctive feature of the WBF event in e-p collisions is an energetic jet produced in the far backward direction with large transverse momentum. (We take the direction of the incoming electron to be forward) The decay product of the Higgs tends to be well separated from this jet, with a rapidity in the central region. To make full use of this event shape, we follow the strategy in Ref. [13,17] and identify the "tagging jet"as the one with the largest transverse momentum, imposing the cuts where p j T and η j are the transverse momenta and rapidity of the tagging jet; M H,j is the invariant mass of the Higgs-tagging jet system. Further, the large momentum transfer to the neutrino from the hard scattering is reflected by the requirement with / E T denoting the transverse energy of the neutrino. Normally one also applies basic cuts on other particles, which we refrain from doing as we are not performing a full signal-background analysis in this work. Also, the subtraction formalism allows to integrate over the region collinear to the incoming beams. Each jet function in Eqs. 5 and 6 plays the role of constructing the observable with the momenta from its argument list 2 , as well as checking if the event passes the cut. Here we use k T algorithm [43][44][45] with the parameter D = 0.8 to define jet observables.
In the following, we present distributions in various observables O computed at LO and NLO (here "NLO" actually means LO plus the correction at NLO). To show the relative corrections from the EW radiations with respect to the LO result, we define the K factor as a function of the observable O. We first display the differential distribution of the tagging jet transverse momentum p j T in Fig. 5 (a), where the LO and NLO curves are shown in red and blue respectively. The cross section peaks near small value of p j T and drops apace with increasing jet transverse momentum, with an average p j T of order the W boson mass. The relative EW corrections are negative and reach -14% at small p j T . Note that the size of the relative correction becomes smaller at large p T , which is very different from the case of typical tranverse momentum distributions in p-p collisions [46], or in e-p collisions with NLO QCD corrections [17]. This behavior arises from the initial state radiation (ISR) of the photon from the electron [47,48] that results in the large logarithmic correction ∼ ln m e (See the last line of Eq. 6). Near the production threshold of the final state with a Higgs boson, this large logarithm is enhanced because virtual photon radiations are not compensated by the real emission contribution whose phase space shrinks as the electron momentum fraction approaches 1. This is exactly the case of the small transverse momentum region in Fig. 5, which corresponds to a small CM energy for the hard scattering process near the threshold. [fb] The distribution in the rapidity η j of the tagging parton is shown in Fig. 6, where the peak is clearly located in the backward region. The EW corrections vary between -3% and -20%. The substantial correction at the large negative rapidity can be similarly traced back to the threshold effect that enhances the initial state radiation. The large rapidity of the tagging jet corresponds to a small transverse momentum p j T , and thus a CM energy close to the threshold.  The Higgs observables (tansverse momentum p H T and rapidity η H ) are even more sensitive to the ISR effect. Figs. 7 and 8 show that at the threshold region (corresponding to small p H T or large negative η H ), the size of the correction can be as large as -32% (for p H T ) or -69% (for η H ). It indicates that a resummation of large logarithms to all orders may be needed in this region [47]. Also, one observes that these distributions peak at a relatively large transverse momentum (p H T ∼ 100 GeV), or moderate rapidity (η H ∼ −1), which confirms that the Higss boson is produced in a distinct region that can be well separated from the backward jet.
Finally, we present in Fig. 9 the distribution in the azimuthal angle between the tagging quark and the missing transverse energy. This distribution is sensitive to the anomalous HW W couplings [49,50], whereas in the SM the differential cross section exhibits only a steady increase with the increasing ∆φ M ET −j . The relative EW corrections also display a mild dependence on ∆φ M ET −j and change in the range between -14% to -8%, which turn out to be dominated by the loop contributions that are not sensitive to the electron mass m e . These loop graphs depend on logarithms of the momentum transfer from the incoming fermions. A large momentum transfer would disturb the fermions considerably that the the tagging jet and the neutrino in the final state are away from a back to back configuration. As a result, these logarithms become large at small ∆φ M ET −j and increase the correction by ∼ 100%. Note that the loop contribution also plays an important role in the corrections for distributions of the transverse momentum and rapidity. However, unlike the ISR effect, including the loop corrections amounts to multiplying the LO distributions of these observables by almost an overall factor, rather than distorting them.

IV. Conclusion
We have computed the NLO EW corrections for the Higgs production process via CC WBF at the LHeC. This is the first calculation that takes into account the full EW effect at one loop for this process. To handle the singularities of various origins in the amplitudes, we have worked in the dipole subtraction formalism and factorized the collinear radiation of the initial states. We have developed our own program to implement the subtraction procedure and carry out the numerical integration over the phase space. Several checks have been done to verify the consistency of the calculation.
For the CM energy at 1.98 TeV, we have found a relative correction of 9% (18%) in the total cross section with no cuts using the G µ (α M Z ) renormalization scheme. The agreement between the two schemes are significantly improved when the LO and NLO-correction terms are added. The differential cross sections for several observables have also been computed in G µ scheme under the selection cuts for the WBF process. The corrections for these variables are within -30%, except for the case where resummation of terms in perturbation series may be needed. Large distortion of the distributions are also observed due to the ISR effect. Despite the smallness of the EW coupling α as compared with the strong coupling α S , the EW correction is considerably larger than the QCD correction at NLO both because of the large number of diagrams (compared with the number of the QCD loop diagrams), and because of the ISR effect associated with photon radiations by the electron beam. The results show that the EW corrections is very important for the study of Higgs phenomenology at the LHeC and should not be neglected in a full analysis of the WBF process.

V. Acknowledgement
The authors thank Lilin Yang, Ruibo Li, and Xiang Lv for helpful discussions. B.W. thank Zhuoni Qian for useful suggestions on histograming, and Ted Rogers for providing a very important reference. This work is supported by National Science Foundation of China (11875232, 12147103, 12105068). B.W. is also supported by Hangzhou Normal University Start-up Fund.

A. Expressions for the four particle final states
The specific form of the subtraction term for the quark-and anti-quark-initiated process e − (p a ) + q(p b ) → ν e (p 1 ) + γ(p 2 ) + q (p 3 ) + H(p 4 ) reads |M sub | 2 = D e q γ + D e γ,q + D e,q γ + D q q γ + D q γ,q + D q,e where the momentum fractions and charge-correlators are defined as As remarked in Sec. II B 3, the born amplitudes in Eqs. A1 depend on the shifted momenta obtained from the corresponding radiative processes. These momenta are shown as the arguments of the born amplitudes, where momenta from the initial/final state is on the right/left of the semicolon. (Momenta with no modification from the 4-particle final states are not shown) They are related to the momenta of the real emission processes by (P ab +P ab ) µ (P ab +P ab ) ν P 2 ab + P abPab +P µ ab P ab,ν P 2 ab p ν i .

(A3)
Note thatp i for all final state particles in D e,q γ and D q,e γ must transform according to Eq. A3. The charge correlator Q 2 ik is defined by the charges of the flavor i and k as where θ i/k is 1 (−1) if it is in the final (initial) state. For quark-initiated processes in our calculation, i.e., q = u, c, and q = d, s, the charge correlators in Eq. A1 are where each initial state fermion aquires a minus sign on its charge. The anti-quark-initiated processes, with q =d,s, and q =ū,c, differ from the corresponding quark-initiated processes only by the charge correlators, which are For the photon-induced processes e − (p a ) + γ(p b ) → ν e (p 1 ) + q(p 2 ) + q (p 3 ) + H(p 4 ) the subtraction term takes the form where in addition to the definitions made in Eq. A2, we have The "tilde" symbol in the subscript of each charge correlator denotes the flavor from the photon splitting that enters the corresponding born process, namely, b2 denotesq and b3 denotesq . The charge correlators for q =ū,c, and q = d, s are B. Expressions for the three particle final states In this appendix we list the specific form of the I,K and P terms that are factorized from the born process e − + q → ν e (p 1 ) + q (p 2 ) + H(p 3 ), where the momenta of the initial particles are not shown explicitly. For the corresponding I,K and P terms, they are given by the arguments of the born factors in the first three lines of Eq. 6. The generic expressions for the quark-and anti-quark-initiated processes are where "q" in I q can be a quark or anti-quark.
In K b f f and P b f f , "f " can be a quark, anti-quark, or electron, while "c " is the initial flavor not going through splitting ("c" without prime is reserved for the charm quark, which is only a particular case of c . See below). Inserting the charge correlators for specific flavors, we find For the photon-initiated processes, I terms do not contribute. Also the electron splitting terms are of higher order in α and do not contribute in this case either. K and P terms take the form The expressions for specific splitting processes are K e γu/c (x) = α 2π K e γd/s (x) = α 2π P e γu/c (x, µ 2 F ) = α 2π P e γd/s (x, µ 2 F ) = α 2π