NLO QCD Corrections to ($\gamma^{*}\rightarrow Q \bar{Q}$)-Reggeon Vertex

Next-to-leading order QCD corrections to the $\gamma^{*} \longrightarrow Q\bar{Q}-Reggeon$ vertex are calculated, where $Q\bar{Q}$ denotes a heavy quark pair $c\bar c$ or $b\bar b$. The heavy quark mass effects on the photon impact factor are found to be significant, and hence may influence the results of high energy photon-photon scattering and heavy quark pair leptoproduction. In our NLO calculation, similar to the massless case, the ultraviolate(UV) divergences are fully renormalized in the standard procedure, while the infrared (IR) divergences are regulated by the parameter $\epsilon_{IR}$ in dimensional regularization. For the process $\gamma^* + q \to Q\bar{Q} + q$, we calculate all NLO coefficients in terms of $\epsilon_{IR}$, and find they are enhanced due to the heavy quark mass, as compared with the light quark case, and the enhancement factors increase rapidly as the quark mass increases. This might essentially indicate the quark mass effect, in spite of the absence of real corrections that are needed in a complete NLO calculation. Moreover, unlike the $\gamma^{*}$ to massless-quark-Reggeon vertex, the results in the present work may apply to the real photon case.

People noticed that the virtual photon-photon scattering is an ideal process [9,10] to testify the BFKL predictions and hence the pQCD calculation reliability, which is highly expected in the study of strong interaction physics. The cross section σ γ * γ * tot is calculable in the BFKL approach with relatively high precision and can be realized in experiment through a measurement of the reaction e + e − → e + e − + X by tagging the outgoing leptons at high energy electron-positron colliders, like LEP and more optimistically CEPC [11] or ILC [12] in the future. Moreover, the recently proposed electron-ion colliders like EIC [13] and EicC [14] may also be good places to study the single diffractive process and hence on small-x physics at lower energies.
Till now, the leading order(LO) BFKL predictions for e + e − → e + e − + X process have been confronted to the LEP data [15][16][17], in which the experimental measurement is found above the two-gluon-exchange model result, while below the leading order BFKL Pomeron prediction. Now that the next-to-leading order(NLO) corrections to BFKL kernel is ready and numerically big and negative [8,18], the higher order corrections may lower the the BFKL Pomeron exchange calculation and approach to the experimental measurement. However, in the NLO BFKL calculations, there is still a task remaining, calculating the NLO corrections to the coupling of the BFKL Pomeron and the external photons, which is called photon impact factor [19][20][21].
The photon impact factor can be obtained by calculating the discontinuity of the process γ * + reggon −→ γ * + reggon process in light of the optical theorem, which tells that the discontinuity is proportional to the scattering amplitude, γ * + reggeon −→ QQ, squared. The reggon, is identified with the elementary t-channel gluons. The NLO corrections to the photon impact factor with massless internal quarks were given in Ref. [22], while here the Q(Q) are massive quarks, charm or bottom quarks. In the NLO calculation, in analogous to Ref. [22], we calculate the NLO corrections to γ * + reggeon −→ QQ amplitude on for example left-hand side of the discontinuity line, and the leading In fact, here the calculation procedure is similar to the massless case performed in Ref. [22]. In this paper, as in [22], we give the results of the first step, in which the vertex is extracted from the scattering process γ * + q −→ QQ + q in the high energy limit.  The kinematics of the γ * + q → QQ + q process is illustrated Fig.1, where q and p are the 4-momenta of photon and incident quark respectively. In the calculation, ε represents for the polarization vector of the photon, s for the collision energy of the γ * q system, and m for the mass of the heavy quark. The Lorentz invariants appeared The momenta k and r can be decomposed in the Sudakov form, i.e, with q ′ = q + xp, q ′ 2 = p 2 = 0 and βs = The typical Feynman diagrams for NLO corrections are shown in Fig.2. Of these diagrams, except Fig.2.14, the upper part quark-antiquark exchange diagrams are implied.
For the color octet t-channel configuration, the sum of all diagrams has to be antisymmetric if we interchange quark and antiquark: k → q − k − r, λ → λ ′ , where λ, λ ′ are the helicities of the quark and antiquark respectively. In particular, the "box" graph shown in Fig. 3.14 has to be antisymmetric by itself. Throughout the calculation, Feynman gauge is employed, and for the t-channel gluons the metric tensor is decomposed into as usual. Note, in practical calculations, the transverse term is not taken into account, whose contributions are suppressed by powers of s. We use the helicity formalism as well, and then our results can be expressed in terms of the following matrix elements 5 FIG. 2: Feynman diagrams for the process γ * + q → QQ + q. 6 similar to [22], i.e., Here λ a is the generators of the color group. Note that the last four helicity matrix elements do not exist in [22], which always come up with the factor of m.
Take high energy limit in the calculation, where and do not impose any restrictions on the remaining invariants, we then obtain the following amplitude with and The only unknown piece of the NLO amplitude is T (1) , which corresponds to the processes in Fig.2.1 -2.14 and will be calculated analytically in the following.

III. CALCULATION METHODS
The method we use in our calculation will be briefly described in the following before presenting the explicit results.
In our calculation the MATHEMATICA package FeynArts [23] is applied to generate the Feynman diagrams and amplitudes that are relevant to our process. We use FeynCalc [24] to calculate and simplify the amplitudes. Color in the t-channel is projected onto the antisymmetric octet as done in [22]. After simplifying the Dirac matrix and employing the Dirac equation of motion, we express the amplitudes as the combination of helicity matrix elements and Passarino-Veltman integrals. For integrals that contain divergences, we separate the divergence part from the finite one. The high energy limit is taken In this work an extra scale m exists in comparison to [22], and it is too lengthy and redundancy to turn all the Passarino-Veltman integrals into scalar integrals or logarithms and dilogarithms functions. For amplitudes that do not contain divergences and high energy scale s, we express them as the combination of Passarino-Veltman integrals and helicity matrix elements. The numerical values of these integrals will be evaluated by Looptools [25]. For the pentagon diagram Fig.2.13, the reduction method given in [26] is employed to reduce the corresponding integrals to box integrals. In the end we find our result agrees with that in [22] when taking the m → 0 limit.

IV. ANALYTIC RESULTS
The NLO amplitudes in T (1) of our concern can be expressed in terms of different diagrams: Here, the subscripts i denote diagrams in Fig.2, and the amplitudesĀ i represent those with quark-antiquark interchange in A i .
A. Results of two-and three-point diagrams We categorize the diagrams similar to [22]. The results of Figs.2.1 and its conjugate one are given below, which are split into divergent and finite parts in MS scheme: Here Here, The calculation procedure of Fig.2.4 is similar to that of Fig.1, and its result reads: where The result of the conjugate diagram of Fig.2

.4 is:
The and In the zero mass limit, it is obvious that these two amplitudes are just the eqs. (35) and (36) in [22].
Figs.2.7 -2.9 contribute to both the upper and lower vertices, and we find and The results of conjugate diagrams of Fig.2.5 -2.9 can readily be obtained by substi- The result of vertex correction diagram Fig.2.10 can be expressed as follows: Here 12 The NLO amplitude of the conjugate diagram of Fig.2.10 is The result of the quark self-energy diagram Fig.2.11 is 13 and the amplitude of the conjugate diagram of Fig.3.11 reads

B. The Box diagrams
Here, we present the results of box diagrams. In the calculation of diagrams Fig.2.2 and Fig.2.3, the four-point integrals have to be concerned. After taking the high energy limit and eliminating the s suppressed terms, in the end the results turn out to be quit simple, that is Here, C 0 (1) = C 0 (t a , t, m 2 , m 2 , 0, 0), as in the A 1 .
The results of diagrams conjugating to Note that there are lns terms in the amplitudes of Figs.2.2 and 2.3, and also their conjugate partners.
Next, we give the result of adjacent box as shown in Fig.2.12.
14 in which the divergent part A −1 12 is with and M 2 = (q + r) 2 , defined in the above paragraph.
The finite term A 0 12 is given in the Appendix. The amplitude of the conjugate diagram of Fig.2.12 goes as where the divergent term The finite piece is also given in the Appendix.
The opposite box diagram Fig.2.14 does not contain any divergence, its lengthy analytic expression is presented in the Appendix.

C. The Pentagon diagram
In this subsection we deal with the pentagon diagram Fig.2.13. The calculation procedure is complicated, and is performed by computer algebra. Due to the fact that the integrals depend upon the large scale s, they can be greatly simplified in the high energy limit. The results are as follows: Here, and The finite piece A 13 in (35) is listed in the Appendix. Note, we can reproduce the massless result [22] when taking the m → 0 limit in above expressions.
We can obtain the amplitude of the conjugate diagram of Fig.2.13 by replacing s → −s in the D 0 (1), D 0 (2), D 0 (3), D 0 (4) and D i (14) → −D i (14). With there replacements one can easily find that the energy dependence ln s terms in Fig.2.13 cancel the terms in its conjugate diagram. Therefore, the energy dependence terms merely come from A 2+3 +Ā 2+3 .

V. RENORMALIZATION
The results of our concerned process contain both infrared and ultraviolet divergences. The ultraviolet divergences may be renormalized via standard procedure, i.e. canceled by counter terms, in modified minimal subtraction (MS) scheme here.
The infrared divergences may be canceled out when the soft gluon radiation process γ * + reggeon −→ QQg is taken into account.
The ultraviolet divergences exist only in the self-energy and triangle diagrams, which are and For the ultraviolet divergence discussed above, when taking the massless limit we can find it is in agreement with the result in [22]. We denote Z 2 , Z 3 , Z m , Z g as the quarkfield, gluon-field, mass, and coupling renormalization constants, respectively. Note, in our calculation the renormalization constants Z 2 and Z m are defined in on-shell Scheme, while Z 3 and Z g are given in MS scheme, which tells: and After including the counter terms, all above ultraviolet divergences and those divergences from their conjugate diagrams, are canceled out. Hence our results will be ultraviolet finite.

VI. NUMERICAL RESULTS
In the following we show the mass effect numerically in NLO corrections of the photon impact factor. Since in this work the infrared divergences still exist, the NLO amplitude squared can be expressed as We then can evaluate respectively the mass effects for the Born term |M| 2 Born , secondorder infrared divergent term |M| 2 Loop IR2 , first-order infrared divergent term |M| 2 Loop IR1 , and NLO finite term |M| 2 Loop finite .
In the numerical evaluation, we take the following inputs: From Fig.3 we can see that, for s = 1000 GeV 2 and s = 2000 GeV 2 , the curves change little. As the quark mass gets larger, the NLO amplitude squared also gets larger quickly, even for the |M| 2 Loop IR1 , the ratio is nearly 20 when the quark mass is 6 GeV. So we may conclude that, in the γ * −→ QQ − Reggeon vertex, the quark mass effects on the photon impact factor are significant, and may influence the results of high energy photon-photon scattering and heavy quark pair leptoproduction.

VII. CONCLUSIONS
In this paper, we calculated the process γ * + q −→ QQ + q at NLO with fully virtual corrections in the high energy limit for massive quark pair Q(Q, which tells the coupling of the reggeized gluon to γ * −→ QQ. This calculation is just the first step of our final purpose, to obtain the complete NLO corrections to the photon impact factor and checking the BFKL pomeron prediction for the process γ * + γ * −→ γ * + γ * at high energies.
In our calculation all contributions from loop diagrams, i.e., self-energy, triangle, box and pentagon diagrams, are regulated in dimensional regularization scheme and the ultraviolet divergences are renormalized by adding the corresponding counter terms. In the end, the infrared divergences still exist in the result, which would be canceled after taking account of the real corrections.
We find that for γ * + q −→ QQ + q process in the massive quark case, the quark mass effects are significant at the next-to-leading order of accuracy, which indicates that for heavy quark diffractive photoproduction, the quark mass is indispensable. Our result might essentially show the quark mass effect, in spite of the absence of real corrections, 19 which are needed in a complete NLO calculation. Moreover, for the photon impact factor with heavy quarks, the photon is legitimate to be real in order to guarantee the perturbative QCD calculations applicable.

The loop integrals in LoopTools are defined as
21 All the coefficients such as C ij , D ijk can be evaluated numerically by LoopTools.
D i (14) is defined in the above paragraphs. The finite parts of these four-point integrals can also be obtained from [27], or evaluated by LoopTools numerically.