Processes of heavy quark pair (lepton pair) and two gluon (two photon) production in the high energy quark (electron) proton peripheral collisions

We considered the three jet production processes in the region of the incident lepton, photon, quark or gluon fragmentation. The fourth jet is created by the recoil proton. The kinematics of jet production is discussed in jets production in the fragmentation region. The non-trivial relation between the momenta of the recoil proton and the polar angle of its emission was derived. Based on this formalism the differential cross sections of QCD processes $gp \to (ggg) p; \,\,\,qp \to (q \bar{Q} Q) p; \,\,\, gp \to (g Q \bar{Q})p$ were obtained, including the distribution on transverse momentum component of jets fragments. It was shown that the role of the contribution of"non-Abelian"nature may become dominant in a particular kinematics of the final particles. The kinematics, in which the initial particle changes the direction of movement to the opposite one, was considered in the case of heavy quark-antiquark pair production. Different distributions, including spectral, azimuthal and polar angle distribution on the fragments of jets can be arranged using our results. We present besides the behavior of the ratio of non-Abelian contribution to the cross section to the total contribution. We show that it dominates for large values of the transverse momenta of jets component (gluons or quarks). Some historical introduction to the cross-sections of peripheral processes, including 2$\gamma$ creation mechanism production, including the result Brodsky-Kinoshita-Terazawa, is given.


I. INTRODUCTION
It is known [1] that the differential cross sections of small angle elastic (and inelastic) scattering processes do not fall with increasing the center of the mass total energy √ s, s = 4E 2 . The reason for this is the contribution to the cross section from the photon exchange between charged particles. Similar phenomena take place as well in the strong interaction sector, where gluons take place instead of a photon.
The simplest processes of this kind are the scattering of a charged particle in the external field of nuclei and the elastic scattering of one sort of charged particles on the other one. The total cross sections of these processes do not exist due to contributions of large impact parameters, which correspond to small scattering angles. The momentum of the virtual photon in the scattering channel (t channel) tends to the mass shell. So the virtual photon in the t channel becomes a real one. In the case of inelastic processes a + b → a + b + X, with the set of particles x belonging to one of the directions in the center of mass a or b, the cross sections are finite [2][3][4][5][6][7][8][9]. Besides, the square of 4-momentum of a virtual photon is negative and restricted from below by the magnitude of some quantity of the created set of particles, invariant mass square of (ax), (bx). The finiteness of the transfer momentum module caused the so-called Weizsacker-Williams enhancement [7]. Namely, the region of small momentum transfer is realized in the appearance of a large logarithmic factor L = ln(s 2 /(m 2 1 m 2 2 )). For modern colliders this factor is of an order of 20. It often turns out that the consideration is restricted to the WW approximation. This means the accuracy of the order 1 + O 1 L . The cross sections of inelastic peripheral processes are as usually large.
The background caused by the events of the large-angle kinematics of produced particles determines the accuracy of peripheral cross sections So the total accuracy of theoretical estimates is better than 5%. The cross sections of interaction of photons with, the target will also not fall with energy when taking into account the contributions of higher orders of perturbation theory (PT).
The main attention in our paper is paid to the double gluon emission and production of the pair of heavy quarks with subsequent jet production, in the fragmentation region of the incident particle.
Our paper is organized as follows.
First, we give an estimation of the magnitudes of the cross sections of several processes in high energy ep → (eab)p, qp → (qab)p collisions in the fragmentation region of a projectile e, q. In Part II, we give a short historical introduction to the study of the processes of lepton pair production in high energy lepton-lepton, ion-ion collisions. In Part III, the method of description of high energy processes based on the Sudakov parametrization of the four momenta of the problem is developed.The differential cross sections are expressed in terms of physically measurable energy fractions and the transverse component of the final particles. In Part IV, the simplest QCD processes with 2 jet production are presented. In Part V, we consider the process of heavy quark-anti-quark pair production in collisions of projectile with the color-less target. In Part VI, the QED process of double bremsstrahlung is studied. In Part VII, a similar QCD process of emission of two gluons is considered. In Part VIII, the specific details of jet production on a fixed target are considered.
In Conclusion, we discuss the results and pay attention to the relation of the contributions of Abelian (QED) and non-Abelian nature. It seems that in the case of large magnitudes of transversal quark momenta the role of non-Abelian contributions dominates. In Conclusion, we also discuss the "jet reflection" phenomena. It consists in the change of the direction of motion of the light projectile to the opposite one in the case of heavy pair production.
In Appendices A-C, the explicit expressions beyond the WW approximation as well as in the WW one are presented. In Appendix G, this method is applied to the problem of transmission of the longitudinal polarization of the initial electron to the positron from the pair created.
The cross section of the heavy pair production in electron-proton and quark-proton collisions can be written as We put here √ s = 3T eV , M Q = M = 1, 5GeV . In this case L q ≈ 30. For a process of two photon and two gluon production we have for typical transfer momentum squared k 2 = 1GeV 2 .

II. QED PERIPHERAL PROCESS, PAIR PRODUCTION
In 1934, the cross section of pair production in high energy lepton collisions was calculated in the so-called double-WW approximation [8] In 1937 G.Racah [9] published the total cross section of the process of pair creation in the collision of charged particles with the charges Z 1 e, Z 2 e, p 1 , m 1 , p 2 , m 2 are the 4-momenta and masses of the initial particles In papers by Baier and Fadin [11] as well as Lipatov and Kuraev [10] the total cross section of the production process of an electron-positron pair in electron-positron collisions (only two exchanged photons) was obtained In the case of production of a muon pair we obtain These formulae are in agreement with ones obtained by G. Racah [9]. The method used to obtain the cross section consists in imposing some cuts on the transverse momenta and energy fractions, which in principle can be used in experiment. Adding separate contributions we obtain the results given above. In Appendix A, we give the sketch of derivation of the LL formula and discuss the experimental cuts.
Note that in the case of production of a heavy muon pair the corrections of the order (m 2 µ /s)L n must be taken into account. Really, for √ s < 1Gev the cross section calculated theoretically is negative.
In 1970, in paper by Brodsky, Kinoshita and Terazawa a special case of production of a heavy object by two virtual photons in electron-electron collisions was investigated [12]: with 2M being the mass of a created system and The BKT formula in the modern language desribes the Drell-Yan process. Really, it consists in the probability P γ e to find the virtual photon in the electron: and the conversion of these probabilities with the cross section of a hard subprocess γγ → F (see for details Appendix E). In this step it is useful keep in mind the following integrals, The parton language can also be applied to describe the processes in the fragmentation region. In the case when the heavy object is created in the fragmentation region eē → (eF )ē we have In Appendix D, this formalism is applied to the problem of transferring the longitudinal polarization of the initial electron to the positron.
Besides, the two-photon mechanism mentioned above, the so called "bremsstrahlung" mechanism, must be taken into account. It consists in the emission of a light-like virtual photon by one of the initial particles with a subsequent conversion to the lepton pair. When calculating the differential and total cross sections the effect of the Fermi-Dirac statistic must be taken into account.
Other QED peripheral processes, single and double bremsstrahlung, take into account the radiative corrections as well as the details of calculation. Can be found in reviews [13][14][15] It results in a non-leading contribution. Really, the contribution from the diagram corresponding to the single-photon production mechanism [15] is: π 2 ln 2 + 163 108 where factor 2 takes into account both kinematic situations when a jet moves along both the initial directions. The effect of identity of final particles taken into account, contributes to the total cross section (both directions are taken into account) [17] σ int = 2 2α 4 ρ 105πm 2 [−374ξ(3) − 120π 2 ln 2 + 13591 90 π 2 − 2729 12 ] = 2 α 4 πm 2 (−0.14)ρ; (13) σ int is the contribution from the interference term associated with the identity of particles in the final state. For electron pair production and muon pair production the specific effect of the charge-odd contribution to spectral distributions takes place. It is caused by the interference of the 2-gamma mechanism and the bremsstrahlung one.
Similar effects take place in the process of bremsstrahlung and pair production by a gluon and a quark on the proton or the nuclei. We will restrict ourselves below only to the cases when a proton or nuclei remain to be a proton or nuclei. No excitation of the target is allowed.
In the case of large transverse momenta of the jet particle component the subtle effect of the double logarithmic contributions in the fragmentation region disappears.

III. KINEMATIC OF PERIPHERAL PROCESSES, SUDAKOV PARAMETRIZATION
First, we remind the general Sudakov technique to study the peripheral kinematics of the QED process e + p → (e + l +l) + p of creation of a heavy charged lepton pair in high-energy electron-proton collisions in the fragmentation region of the electron, The peripheral kinematics or the electron fragmentation region is defined as It is convenient to use the Sudakov parametrization of momenta. For this aim we introduce two light-like 4 vectors constructed from the momenta of the initial particlesp 2 = p 2 − p 1 (m 2 p /s),p 1 = p 1 − p 2 (m 2 /s) [17] q = αp 2 + βp 1 + q ⊥ , q ± = α ±p2 + x ±p1 + q ⊥± , where a is the two-dimensional vector transversal to the beam axis (direction of p 1 , center of mass reference frame implied), and x, x ± are the energy fractions of the scattered electron and the heavy lepton pair, and x + x − + x + = 1. Below, we will omit the tilde sign. According to the energy-momentum conservation law, we also have The on mass shell condition for the scattered proton p ′ 2 2 − m 2 p = 0, being written in terms of the Sudakov variables, reads (one must take into account the relation 2p 2p2 = m 2 p ) One finds for q 2 = sαβ − q 2 : We conclude that in the case s 1 = 0 a virtual photon has a space-like 4-vector and, in addition |q 2 | > q 2 min = m 2 p (s 1 /s) 2 . The quantity s 1 = 2qp 1 = (p ′ 1 + q + + q − ) 2 − q 2 − m 2 = sα in the WW approximation q = 0 coincides with the square of the invariant mass of the jet moving in the initial quark momentum direction. Using the on mass shell conditions for momenta of the scattered muon and the created pair of heavy quarks we find (in the WW approximation) The matrix element can be written as with J (e,p) being the currents associated with electron and proton blocks of the relevant Feynman diagram. The main contribution arises from the longitudinal components of the tensor g µν = g µν⊥ + (2/s)(p µ 2 p ν 1 + p ν 2 p µ 1 ): So we obtain for the squared module of the summed over spin states of the matrix element The quantities Φ (e,p) (so called impact factors) remain finite in the limit of high energies s → ∞. In particular, The electron current obeys the gauge condition Using this relation we obtain for our process where e = q/| q| can be interpreted as a polarization vector of the virtual photon. To obtain the differential cross section we must rearrange the phase volume of the final state (the electron remains to be a spectator, whereas the scattered muon is accompanied with n particles) including the additional variable q as We use the Sudakov variables: Performing the integrations over the "small" Sudakov variables α, α ± we obtain It can be noted that the cross section does not depend on s at large s and tends to zero in the limit of zero recoil momentum of the spectator electron q → 0. The last property is the consequence of gauge invariance of the theory. Once being integrated over the recoil momentum, the cross section reveals the so called WW enhancement factor where Q 2 ∼ M 2 is the scale of transfer momentum squared in the process.

IV. 2-JET QCD PROCESSES IN QUARK( GLUON) PROTON COLLISIONS
The differential cross sections of the processes differs only by the color factors from similar expressions in QED. with with and and B 1,3 are the effective Stokes parameters of a gluon. Besides p is the transverse component of the scattered electron momentum; q is the same value for the recoil proton; φ p , φ q are the azimuthal angles between the transverse component of a gluon and p, q.
For the process of quark-anti-quark pair production by a gluon on a proton we have The first factor is the color factor 1

V. PRODUCTION OF HEAVY CHARGED LEPTON (QUARK) PAIRS IN ELECTRON (QUARK) PROTON COLLISIONS
We will distinguish two mechanisms of the heavy fermion pair creation, the so called "bremsstrahlung mechanism" (see Fig. 1a) and the "two-photon" one (Fig. 1b). The matrix element of the process in the kinematic region of µ particle fragmentation can be written as Here we adopt Sudakov's parametrization of the four-vectors The first term in the square brackets contains the Compton sub-process e(p 1 ) + γ * (q) → e(p ′ 1 ) + γ(q 1 ) amplitudē u(p ′ 1 )Q µ u(p 1 ) with (we use here the on-mass shell conditions for the initial and final electrons (quarks)) where we use the notation So we obtain The two-photon amplitude contains a Dirac sub-process γ * (q 1 ) Again, with on-mass shell conditions for the heavy fermion pair it can be written as with the definitions With this notation we have The square of the matrix element summed over spin states has the form It is important to know that all the quantities entering into R qQ do not depend on s and are proportional to q 2 in the WW limit q → 0. Keeping in mind that in the combinationsp 2q andqp 2 one can replaceq →q ⊥ one may use the relations needed in calculating the traces (we neglect the contributions of an order of m 2 /M 2 compared to the ones of an order of unity) The differential cross sections have the form In the case of processes with a quark instead a muon, we must take into account the quark color degrees of freedom with C col = (N 2 − 1)/(4N 2 ), where we also take into account the averaging over the color of quarks. The explicit expressions for R br , R 2γ , R odd are given in Appendix B. In the double WW approximation the contribution to the differential cross section has the form Exact formulae are given in Appendix B (see B8). The function is presented in Fig. 4.
In Table I the charge asymmetry defined as is presented at the symmetric point x = x − = x + = 1/3, φ = π/2 for several typical values y + < y − . This quantity has a value of an order of A +− ∼ 10 −2 . For the use of y + > y − the quantity A +− changes the sign.

VI. DOUBLE BREMSSTRAHLUNG IN ELECTRON -PROTON COLLISIONS
In the lowest order of perturbation QED theory there are 20 Feynman diagrams describing the double bremsstrahlung process i.e. emission of two hard photons in collisions of the high energy electron with a charged heavy target (heavy lepton). We will restrict ourselves to the consideration of the emission from the electron line only. The set of six Feynman diagrams provides the gauge invariant set (see Fig.1a,b). With respect to the exchanged photon they split into two independent subsets of Feynman amplitudes, both gauge invariant (see . Fig 1,a; Fig.1,b). The relevant matrix element is with (see details in Appendix C, D), and Here e i = e i (k i ) are the polarization vectors of hard photons. It can be checked that the expression for M 2Y →(2γγ)Y turns to zero in replacing p 2 → q as well as e i → k i which is the consequence of gauge invariance. We adopt below the Sudakov parametrization of the relevant 4-vectors and use the notation and relations (different compared to the previous section) (1) + (2) The expression for the matrix element given above can be written in a form to display the explicit gauge invariance, which is suitable especially for investigation in the (WW) [7] approximation. For this aim we note that the combinations turn to zero in the limit q → 0. Excluding the term containing the denominator (1)(2 ′ ) we can rewrite the expression for O 12 in the form with A similar expression for the set of other Feynman diagrams (can be obtained from the first one by the replacement k 1 , e 1 ↔ k 2 , e 2 ) is with The matrix element squared summed over spin states is |M ep→(eγγ)p | 2 = 32(4πα) 4 s 2 π(q 2 ) 2 R γγ , From a topological point of view it is convenient to write down R γγ as a sum of planar and non-planar Feynman diagrams for the cross section The differential cross section has the form Factor 1/2! takes into account the identity of photons in the final state.

VII. QUARK-PROTON COLLISION: EMISSION OF TWO GLUON JETS
The matrix element of the process of two gluon jets production in a peripheral quark-colorless fermion target collision (see Fig.2) has the form where R 1 = (t a t b ) r2r1 , R 2 = (t b t a ) r2r1 , with r 2 (r 1 ) describing the color states of the scattered (initial) quark. Here the quantities O 12 , O 21 were obtained above (see (60),(66)), with the replacement k 1 → q + , k 2 → q − , where we imply e 1 → e a , e 2 → e b ) and The quantity q 2 1 is presented (26,35). It can be checked that the matrix element obeys gauge invariance, namely, it turns to zero if one replaces p 2 → q and e i (k i ) → k i . The expression for the matrix element at R 1 = R 2 = 1, coincides with the QED result [21]. Below we will use the expression for O 3 in the form To work with the irreducible color states, we use the projectors in color space These projectors obey the equations (77) Here (Ã) r1r2 = (A) r2r1 and summation over a, b is implied. In our case 1 1 As a check we have (R 1R1 ) = (R 2R2 ) = T rt a t b t b t a = N C 2 F ; (R 1R2 ) = T rt a t b t a t b = − 1 2 C F . These relations are fulfilled.
The expansion on irreducible color representations is So the matrix element squared summed over color and spin states can be written as with In the case x 1 = x 2 = x = 1/3 and φ = π/2, F non−abel and F are presented (see Tabls 2) for typical values of y 1 , y 2 . The differential cross section is The explicit expressions for F as well as for R γγ are too cumbersome. Nevertheless, they are suitable for further analytic and numerical integration when obtaining different distributions. As some probe of QCD the quantity which can be considered as a specific for QCD deviation from the QED process of the double bremsstrahlung. It is presented in Table 3 in the WW approximation for x = x 1 = x 2 = 1/3, φ = π/2, for different values y 1 , y 2 . For values q 2 1 ∼ M 2 the non-Abelian contribution dominates in A gg (see Table III). For large q 2

VIII. THREE-JET STRUCTURE IN PERIPHERAL COLLISIONS ON A FIXED TARGET
In the case of jet production with a projectile on a fixed target (nucleon), an additional wide-angle jet can be created by the recoil target particle.
Consider for simplicity the photo-production of a pair qq process on a nucleon.

IX. THE PROCESS gP → (gQQ)P
Another process where the non-Abelian structure of QCD manifests itself is the crossing process to one considered in IV The matrix element can be written as It can be checked that both contributions M (1) and M (2) obey the gauge condition.
In conclusion, we remind a remarkable property of the kinematics of processes in the fragmentation region. It is known as a "cumulation" phenomenon. It consists of events with production of a heavy quark-anti-quark pair, accompanied by the "reflected" scattered parent light particle. It was known in processes of production of a muonanti-muon pair in the fragmentation region of an electron in electron-positron collisions [24]. It turns out that the electron "accompanying" the pair created in the kinematic region near the threshold moves in the direction opposite to the initial electron direction. In the case of production of a heavy quark-anti-quark pair by one of the valence quarks from the initial proton, the parent (light) quark is effectively reflected. So the jet created by this quark corresponds in fact to two jets, one consists of the pair created and two spectator quarks from the initial proton and the other moving in the opposite direction created by the "reflected" quark. To see it, let us consider the kinematics of a peripheral process q(p 1 ) + q(p 2 ) → Q(p a ) +Q(p b ) + q(p ′ 1 ) + q(p ′ 2 ). Using the Sudakov parametrization (3) with we obtain for 4-momentum of the scattered quark Comparing its component along the z axis from the first and second terms we find that for the "reflection" phenomenon takes place. For instance, assuming p 2 ∼ M 2 >> m 2 we have x < (M/2E) ∼ 1. This situation can be realized near the threshold of the heavy pair production. The expressions for the differential cross section of two hard photon emission were obtained in the WW approximation in [22] by using the explicit expression of the double Compton scattering cross section obtained in [23].
Using the formulae given above, the energy-energy correlations of the jets in the final state can be constructed. It consists in the construction of an average of the product of the energy fractions of the heavy quarks. As well, the azimuthal angle correlation, which is the average of 2( q +i )( q −j )/ √ s, can be investigated. Energy spectra, total cross sections and sum rules for different processes in the fragmentation region can be investigated in full analogy with the QED program for colliding eY = (eX)Y [13]. The approach developed here can be used for description of jets in the fragmentation region with creation of K,K states when the heavy strange quark and anti-quark are created. The jets originated from D,D and B,B can be considered as well.
In the plot (see Fig. 4), the dependence of the QQ pair production cross section from the so called "two-photon" mechanism is presented. It has rather a large cross section and can be measured in experiment.
In Table I, the charge-asymmetry effect due to interference of the "bremsstrahlung" and "two-photon" mechanisms is presented as a ratio of the corresponding contributions to the differential cross section. This quantity can also be measured in spite of its rather small value |A +− | ∼ 0.02 − 0.03.
In Table II, the contributions from the so called "planar" and "non-planar" contribution to the differential cross section of the double bremsstrahlung process in electron-target collisions are presented. Table III gives the ratio of contributions with the QED-type gluon splitting contribution to the process of two gluon jet production in the quark-target collisions. Its rather large expected value A gg can also be measured in experiment.

XI. ACKNOWLEDGEMENTS
One of us (E.K.) acknowledges the support of RFBR, grants 11-02-00112a. We are grateful to E.Kokoulina, A.Liptaj, Yu.Bystritskiy for taking part at the initial stage of this work. We are also grateful to Dr. Ahmed Ali, who attracted our attention to the jet program. We are grateful to the Heizenberg-Landau Fond 2013-2014 for financial support. One of us (EAK) is grateful to the Orsay theoretical department for hospitality, to Eric Voiter and Egle Tomasi-Gustafsson, V.Skalozub and V.I.Chizikov for attention.      (55)) as a function of z.
The quantity R 2g enters into the differential cross section in combination R 2g /((q 2 2 ) 2 ) with q 2 2 = −(m 2x2 + p 2 )/x. To see rather delicate compensations in the region of small q 2 2 , p 2 , we must rearrange the electron tensor as (here we use the gauge condition q 2µū (q − )Q µ v(q + ) = 0 and q 2 ≈xp 1 − p ⊥ ) In this form the compensation is clearly seen. So we have with where we remind At least Appendix B: Distributions in the WW approximation, heavy fermion pair production Differential distributions in the WW approximation are where we imply q 2 ± = q 2 ± ; φ is the azimuthal angle between two-dimensional vectors q + , q − , and R br W W = (1 + P +− )( Here we use the notation The differential distribution in the W W approximation is We put as well the contribution to the differential cross section from two gamma mechanisms integrated over both virtual photon transfer momentum variables where The restrictions are implied. The distribution as a function of the two gluon invariant mass square was considered in [25]. Appendix D: Transfer of circular polarization of the initial electron to the positron in the fragmentation region This phenomenon is similar to "handedness" when the initial polarized particle causes the polarization of the fermionic fragments of the jet created by this projectile, or reveals itself in kinematical correlations of momenta of different pions from the jet. The matrix element of the process e(p 1 , λ)ē(p 2 ) → [e(p ′ 1 )e(q − )ē(q + , λ 1 )]ē(p ′ 2 ) has the form For the summed on the spin states square of the matrix element we have Spp 1q1 γ µ γ ν γ 5 q 2 1 q 2 ((1 − β 1 ) 2 ( q 2 1 + β 2 1 m 2 ) 2 ( q 2 + m 2 α 2 2 ) 2 , with with a =x +x− x ,x = 1 − x = x + + x − . Averaging over the azimuthal angle d 2 q permits one to extract the general factor s 2 q 2 , which will be absorbed in the total expression for spectral distributions on the energy fractions of fermions in a jet. Using the expression for phase volume in the fragmentation region we first extract the leading logarithmic factor L 1 L q with L 1 = s 0 d q 2 1 q 2 1 /( q 2 1 + m 2 e β 2 1 ) 2 , L q = s 0 d q 2 q 2 /( q 2 + m 2 e α 2 2 ) 2 .
With the logarithmic accuracy we have For the unpolarized part of the cross section we obtain Integration over d q 2 − leads to Here we imply the threshold restriction (4m 2 e /s < x + + x − ). Consider now the contribution to the cross section associated with the polarized part of the matrix element. Performing the extraction of factor q 2 1 and the relevant azimuthal averaging procedure we must do a shift transformation q − = q − + (x − /x) q 1 and q + = − q − + (x − /x) q 1 .   In terms of the shifted variables the quadratic form s 1 is s 1 = (x/(x + x − )[ q 2 − + am 2 ]. In a similar way we obtain We note that compared with the unpolarized case the terms proportional to the electron mass squared do not contribute. Further integration leads to the final result The degree of polarization transferred from the initial electron to the final positron can be found as . (D10)