Inclusive prompt photon production in electron-nucleus scattering at small x

Journal of High Energy Physics, May 2018

Kaushik Roy, Raju Venugopalan

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Inclusive prompt photon production in electron-nucleus scattering at small x

HJE Inclusive prompt photon production in electron-nucleus scattering at small x Kaushik Roy 0 1 2 3 Raju Venugopalan 0 1 2 0 Bldg. 510A, Upton, NY 11973 , U.S.A 1 Stony Brook , NY 11794 , U.S.A 2 Physics Department, Brookhaven National Laboratory 3 Department of Physics and Astronomy, Stony Brook University We compute the di erential cross-section for inclusive prompt photon production in deeply inelastic scattering (DIS) of electrons on nuclei at small x in the framework of the Color Glass Condensate (CGC) e ective theory. The leading order (LO) computation in this framework resums leading logarithms in x as well as power corrections to all orders in Qs2;A=Q2, where Qs;A(x) is the nuclear saturation scale. This LO result is proportional to universal dipole and quadrupole Wilson line correlators in the nucleus. In the soft photon limit, the Low-Burnett-Kroll theorem allows us to recover existing results on inclusive DIS dijet production. The k ? and collinearly factorized expressions for prompt photon production in DIS are also recovered in a leading twist approximation to our result. In the latter case, our result corresponds to the dominant next-to-leading order (NLO) perturbative QCD contribution at small x. We next discuss the computation of the NLO corrections to inclusive prompt photon production in the CGC framework. In particular, we emphasize the advantages for higher order computations in inclusive photon production, and for fully inclusive DIS, arising from the simple momentum space structure of the dressed quark and gluon \shock wave" propagators in the \wrong" light cone gauge A moving with P + Deep Inelastic Scattering (Phenomenology); NLO Computations - N ! 1. 1 Introduction 2.1 2.2 4.1 4.3 5.1 5.2 5.3 2 Components of the LO amplitude computation Derivation of the amplitude Classifying contributions to the amplitude 2.3 Final expression for the amplitude 3 The inclusive photon cross-section at LO 4 Properties of the photon production amplitude at LO k?-factorization and collinear factorization limits 4.2 Inclusive dijet cross-section in the soft photon limit The dressed fermion propagator 5 Components of the NLO computation Advantages of the \wrong" light cone gauge Gluon \shock wave" propagator in A = 0 gauge Processes contributing to the NLO amplitude 6 Summary and outlook A Notations and conventions B Gauge invariance and soft photon factorization at LO B.1 Ward identity B.2 Soft photon factorization C Kinematically allowed processes measured early on by the H1 and ZEUS experiments [1{3] for the case of photoproduction, where the negative 4-momentum transfer squared, Q2 = q2 of the exchanged virtual ? photon is close to zero. Subsequently, the rst measurements of prompt photon production { 1 { in e + p DIS, isolated and accompanied by jets, were performed by ZEUS and H1 for a wide range of Q2 [4{7]. Isolated photons in DIS have proven to be clean and well calibrated probes of QCD dynamics [8, 9]. We will explore here their potential for uncovering a novel gluon saturation regime of QCD [10, 11] at small Bjorken x. This regime is characterized by the manybody recombination and screening dynamics of gluons that competes with the perturbative bremsstrahlung of increasing numbers of soft gluons at small x. An emergent dynamical scale Qs;A(x) screens color charges at increasingly smaller distances with decreasing x thereby ensuring that the squared eld strengths do not exceed 1= S, where S is the QCD coupling constant. The dynamics in this regime of QCD is fully nonlinear. Nevertheless, at su ciently small x, or large enough nuclear size A, where Qs2;A(x) 1. Many-body weak coupling techniques can therefore be employed to perform systematic computations in this nonlinear QCD regime [12{14]. The rich dynamics of gluon saturation is captured in an e ective eld theory (EFT), the Color Glass Condensate (CGC) [12{18], and we will apply it here to compute the inclusive photon cross-section in DIS o nuclei. The CGC EFT is formulated in the in nite momentum frame [19{22] of the nucleus with an appropriate choice of gauge. It relies on the separation, in the longitudinal momentum fraction x, of the degrees of freedom into static color sources at large x coupled eikonally to dynamical \wee" gluon elds at small x [16]. Because a large number of large x sources couple to the wee gluons, the color charge of the sources can lie in any one of a number of higher dimensional color representations of the SU(3) algebra [12, 23]. This results in a stochastic distribution of classical color sources over a gauge invariant weight functional WYA [ A] representing the probability density of the color charge density A(x ; x?). The subscript YA = ln(x0=x) denotes the spacetime rapidity separation of small x target gluons relative to those at x x0 corresponding to the nuclear beam rapidity, Ybeam. The expectation value of any physical operator at rapidity YA is determined by the average over all possible color charge con gurations: Z hOiYA = [D A] WYA [ A]O[ A] ; (1.1) (1.2) where O[ A] is the expectation value of the operator for a particular con guration of color sources. In the problem of interest, this operator is the di erential cross-section A for inclusive photon production for a given A. Requiring that physical observables be independent of the arbitrary scale separation between sources and elds results in the JIMWLK functional renormalization group equation [24{28] = H A; A WYA [ A] : This equation describes the evolution of the distribution of color charges in the nuclear wavefunction from its fragmentation region at large x (or small YA) to the small x (or large YA) of interest as determined by the kinematics of the process. Here H is the JIMWLK Hamiltonian; explicit expressions and properties of the Hamiltonian are discussed for instance in [15, 29]. It is worthwhile to note here that the JIMWLK evolution equation { 2 { fermion line. The blobs represent the resummed eikonal interactions between the quark/antiquark HJEP05(218)3 and the classical color eld of the nucleus. and all orders in Qs2;A=kA2;? can alternatively be expressed as the Balitsky-JIMWLK hierarchy [30, 31] of equations for expectation values of n-point Wilson line correlators. In the limit of large number of colors Nc ! 1 and for large nuclei A ! 1, the closed form Balitsky-Kovchegov (BK) equation is obtained for the two-point correlator of Wilson lines [30, 32]. The BK equation is a good approximation in many practical situations to the full Balitsky-JIMWLK hierarchy [33, 34] and is therefore extremely useful in phenomenological studies at collider experiments. An attractive feature of the CGC, as in any EFT, is that there is a well de ned power counting for systematic higher order computations. As we will show explicitly later for our speci c case, this power counting allows one to match results in the CGC framework to those in the collinear factorization and k? factorization perturbative frameworks in appropriate kinematic limits. For e + A and p + A collisions, the appropriate CGC power counting is in a so-called \dilute-dense" limit [16] where dilute color source density p in the generic projectile is of order g 1 in the QCD coupling, while the density of color sources 1=g in the nuclear target. Strictly speaking, the dilute-dense power counting corresponds to computations that are lowest order in the dimensionless ratios Qs2;p=kp2;? 1, 1, where kp;? and kA;? correspond to the momentum transfer from the projectile and target respectively. The dilute-dense power counting was implemented recently by one of us and collaborators in computing inclusive photon production in p + A collisions to next-to-leading order (NLO) accuracy [35] extending earlier leading order (LO) computations in this context [36]. The power counting in the CGC framework for the e + A DIS case at hand is simpler than the p + A case because we don't have a power counting in p on account of the lepton probe. In general, at LO in the QCD coupling constant, we obtain the two classes of processes shown in gure 1 that contribute to the inclusive prompt photon cross-section. The Class I processes represent the bremsstrahlung of a photon from a valence quark in the wavefunction of the target nucleus. Their contribution to the di erential crosssection for inclusive photon production has been calculated [36, 37] at small x in the CGC framework albeit in the context of p + A collisions. This result can however be straightforwardly adapted to the present problem. { 3 { photon production. Both real emission and interference contributions are shown. x where the virtual photon emitted by the electron uctuates into a long lived quarkantiquark dipole [38, 39] and the dipole subsequently has a nearly instantaneous \shock wave" eikonal scattering o the gauge elds in the nuclear target [17, 40, 41]. In this dipole picture of e + A DIS, the power counting is dictated by strong color sources A 1=g in the target, and their energy evolution with respect to the quark-antiquark dipole. Attaching new sources to the diagram in gure 1 does not change the order in strong coupling constant, because g A 1. Independent powers of g arise only when a vertex is not connected to the source. From this argument, it should be clear that we are performing an all-twist expansion in g A at each order in S in this power counting scheme.1 At NLO, typical contributions2 to Class I processes are shown in gure 2. For inclusive photon production, these possess the divergence structure inherent in NLO quark production containing i) dominant contributions from the large phase space in transverse momentum available to the emitted gluon ( Sln(k?) 1), ii) subdominant contribution from logarithms sensitive to x (since valence quark distributions are peaked at x 1) and nally, iii) nite pieces that are not phase space enhanced. By an appropriate choice of scale and scheme for factorization, the nite pieces can be absorbed in the de nition of the quark distribution function. These contributions are therefore e ectively of order O( e) (where e is the QED ne structure constant) if we replace the bare valence quark distribution in the nucleus by one that absorbs the leading logarithmic contributions. These last contributions, to all orders in perturbation theory, are captured by the DGLAP [44{46] renormalization group (RG) evolution of the valence quark distributions. At small x, the Class I contributions are strongly suppressed relative to the Class II contribution. The reason for this is that the former are sensitive to the valence quark distribution in the target while the latter are sensitive to the gluon distribution. At small x, as seen in the e + p HERA DIS data [47{52], the gluon distribution clearly dominates the valence quark distribution. Since our interest is in inclusive photon production at small x, we shall not examine Class I type processes any further and shall focus our attention exclusively on Class II processes. 1Note that there is also an LO contribution analogous to that described in [42] whereby the real photon is emitted from a quark loop that the virtual photon uctuates into. Just as in the p+A case, this contribution is highly suppressed relative to the Class II processes we will discuss. 2For a recent computation of the real gluon emission contribution, see [43]. { 4 { In the rst part of this paper, we will compute the Class II LO diagrams shown in gure 1. We will subsequently discuss the structure of the NLO computation of inclusive photon production. We shall in particular discuss a simple formulation of the dressed gluon propagator in light cone gauge. The NLO computations are important for two reasons. Firstly, as noted in our discussion of Class I NLO contributions, there are logarithmically enhanced contributions of order S ln(x0=x) which can be as large as the LO contributions. Such contributions appear in each order of perturbation theory; they can be isolated and resummed in a so-called leading logarithmic (LLx) RG treatment. There are also pure S suppressed NLO contributions. These are important for precision measurements but can also lead to qualitative changes in spectra that impact the discovery potential of such measurements. In particular, they will be important for a quantitative extraction of the saturation scale Qs;A(x). These statements can be understood more clearly by considering the organization of the perturbative series for the argument of hOi in eq. (1.1) as3 Here each coe cient cn = Pj1=1 nj (g A)j , where the matrix elements nj are numbers of order unity, resums the contributions obtained by adding extra sources of magnitude A 1=g to the allowed graphs. Before we proceed any further, we must invoke the essential ingredient of the CGC e ective eld theory | the separation of large x static light cone sources from dynamical small x gauge elds | as represented by the structure of eq. (1.1). The starting point of any CGC computation therefore includes an initial cuto scale 0+ (in the `+' longitudinal momentum) or Y0 (in the rapidity) between sources and elds. This is shown schematically in gure 3. At leading order (LO), the cuto scale Y0 = ln( beam= 0+)) distinguishing soft and hard partons is arbitrary and the fast quantum 0+ (or modes with k+ 0+ or Y < Y0 are represented by the classical color source density A with the weight functional W 0+(Y0)[ A]. The quantum evolution of sources and described by the following renormalization group procedure (RG). One rst integrates out elds is quantum uctuations within the range 1+ < jk+j < + 0 . Here 1+ is chosen such that that generate such logarithms upon change of scale from the distribution of color sources: W + [ A] = 1 + ln( 0+= 1+):H 1 0 W + [ A] or equivalently, WY1 [ A] = 1 + Y:H WY0 [ A] : 3An extended discussion along these lines for eld theories with strong time dependent sources can be found in [53, 54]; for the CGC speci cally, see [55{57]. { 5 { 1, are integrated out and absorbed into the source densities at the scale renormalization group (RG) pattern is repeated successively generating the JIMWLK RG equation + 1 . This self-similar for the source densities. See text for a detailed discussion. by contributions from these classes. These contributions are actually O(1) in magnitude because of the presence of large logarithms in x leading to Sln(1=x) 1. Here H is the JIMWLK Hamiltonian we alluded to previously. Hence these particular NLO O contributions generate a classical e ective theory at this new scale expressed as Z [D A] W + [ A](OLO + 0 ONLO) = [D A] W + [ A]OLO[ A] ; 1 where the LLx contributions have been absorbed in the JIMWLK evolution of W . To leading order accuracy, the perturbative expansion in eq. (1.3) then has the coe cients: cn = X dnj (g A)j ln(1=x) ; n = 1; 2; : : : ; Z n (1.5) (1.6) At NLO however, there are S contributions that do not come accompanied with logarithms in x. These would then correspond to coe cients expressed more generally in the expansion of the perturbative series as cn = X fni j (g A)j ln(1=x) ; i n = 1; 2; : : : ; (1.7) 1 j=0 n X 1 i=0 j=0 in particular the coe cients fnnj 1. For our process of interest, some of the corresponding diagrams are shown in gure 5. These additional contributions are not rapidity ordered { 6 { should be combined with next-to-leading-log (NLLx) JIMWLK evolution to obtain the full NLO result for inclusive photon production in e + A collisions at small x. Fortunately, the NLLx JIMWLK evolution equations are known [58{60] as well as the NLLx BK equations [61{63]. Therefore with the computation of the NLO diagrams represented in gure 5, all the elements will be in place for quantitative predictions, to NLO accuracy, for inclusive photon production in e + A collisions at small x. This paper represents the rst step in this direction. While we will discuss key aspects of the structure of the NLO computation here, the full computation will be presented in future publications in preparation. The paper is organized as follows. In section 2, we begin by discussing the ingredients necessary for the computation of the amplitude for inclusive photon production; a compact expression for this amplitude is given in eq. (2.40). In section 3, we will calculate the crosssection for the production of a direct photon accompanied by a quark-antiquark pair as well as the inclusive di erential cross-section for direct photon production at LO. The former provides the rate for inclusive production of a photon with a dijet pair in small x kinematics. Our results for these cross-sections are expressed respectively in eqs. (3.11) and (3.12) as a convolution of the lepton tensor L that is familiar from inclusive DIS and a hadron tensor constituted of all-twist lightlike Wilson line correlators. In section 4, which we divide into three subsections, we discuss the important properties of the photon production amplitude at leading order. We rst examine the k our computation in the limit of large transverse momentum k ? and collinear factorization limits of ? QSA. In the former case, the cross-section is proportional to the unintegrated gluon distribution within the nucleus while the latter is proportional to the usual leading twist nuclear gluon distribution in perturbative QCD, evaluated at the scale Q2 of the virtual photon. The corresponding expressions for the hadron tensor in this limit are given in eqs. (4.4) and (4.17) respectively. We next demonstrate that the Low-Burnett-Kroll theorem [64{66] is satis ed when k we see explicitly that the nonradiative DIS di erential cross-section for the inclusive dijet case in DIS matches extant results in the literature for the same [37]. In the nal subsection, we show that the amplitude derived in eq. (2.40) has a simple and e cient interpretation in terms of a modi ed fermion propagator in the classical background eld of the nucleus. ! 0: This will prove bene cial for higher order computations. { 7 { HJEP05(218)3 Section 5 outlines the machinery for the computation of the amplitude at NLO. This includes a discussion of the motivation behind choosing the \wrong" light cone gauge A = 0 for the kinematics of our process as the e cient gauge in which to perform our computations. We then derive the corresponding momentum space Feynman rules for the small uctuation gluon propagator and conclude this section with an analysis of the contributing processes at this order. We end the paper with a brief summary and an outline of work in progress. Appendices A, B and C supplement the material in the body of the paper. The notations and conventions are clari ed in appendix A. Appendix B includes the proof of gauge invariance by virtue of the Ward identity. It also contains computational detail for the subsection dealing with the soft photon factorization of the photon production amplitude. Appendix C includes a detailed discussion of the techniques used in determining kinematically allowed processes contributing to inclusive photon production at both LO and NLO. 2 Components of the LO amplitude computation In this section, we will outline the components needed for the computation of the amplitude for Class II processes at LO in the CGC framework. In the e ective theory of CGC, owing to their large occupancy A 1=g, the dynamics of small x gluons to LO is described by the classical Yang-Mills equations4 [D ; F ](x) = g + (x ) A(x?) : The nucleus is assumed to move in the positive z-direction at nearly the speed of light with large light cone longitudinal momentum P + N ! 1. (See appendix A for the conventions adopted in this work.) We also assume in this frame that the virtual photon has a large q component of its momentum. An essential element in the computation of the LO amplitude is the fermion propagator in the background strong classical color eld of the nucleus. In Lorenz gauge @ A = 0, this has the coordinate space representation [67] Z Z { 8 { (2.1) (2.2) (2.3) where ( x ) (y ) Z S0(x; y) = Z d4p (2 )4 d4z (z ) U~ y(z?) 1 S0(x; z) S0(z; y) ; e ip:(x y)S0(p); S0(p) = i(p= + m) p2 d4z (z ) U~ (z?) 1 is the free fermion propagator. This e ective propagator appears in the momentum space Feynman rules as an e ective vertex with the following factors, Tji(q; p) = (2 ) (p q ) d2x ? ei(q? p?):x? U~ ( )(x?) 1 ji ; (2.4) 4Note that henceforth the factor of g is taken out of the color charge density | this di ers from the notation in [15] for instance. indices in the fundamental representation. where the plus (minus) sign corresponds to insertions on a quark (antiquark) line respectively and the Wilson line U~ is written in the fundamental representation of SU(Nc) as in nite momentum frame.5 In Feynman diagrams, this e ective vertex insertion is drawn as depicted in gure 6. These e ective vertices do not change the order of the diagrams parametrically by any power of g. In principle therefore there can be diagrams with multiple insertions with the emitted photon sandwiched between any two such vertices. However there is a kinematic constraint that restricts many such possibilities: On the same fermion line(quark or antiquark), there cannot be a photon sandwiched between two e ective vertices. This has the physical meaning that an outgoing fermion (with a de nite sign of p ) can get scattered o of the nucleus and subsequently emit a photon but does not su er a secondary scattering because the time scale governing the scattering is of order 1=PN+, which is nearly instantaneous in the (2.5) (2.6) (2.7) (2.8) 2.1 Derivation of the amplitude The amplitude for the process where with X denoting any other particle produced in the collision, can be written as e(~l) + A(P ) ! e(l~0) + Q(k) + Q(p) + (k ) + X ; e Q2 u(l~0) M(~l; l~0; q; k; p; k ) = u(~l)M (q; k; p; k ; ) ; M (q; k; p; k ; ) = (k ; )M represents the amplitude for the hadronic subprocess and is the quantity of interest. Here (k ; ) is the polarization vector for the outgoing photon. The momentum assignments are summarized in table 1 and boldface letters stand for the corresponding 3-momenta. 5Mathematically, this is manifest in the contour integration over the `+' component of the undetermined momentum with all the poles being on the same side of the real axis. This is discussed in further detail in appendix C. { 9 { k: quark, directed outward ~l: incoming electron l: nucleus to antiquark line for multiple insertions q: exchanged virtual photon P : total momentum of nal state = p + k + k k : outgoing photon momentum labels and their directions are clearly shown. i and j stand for color indices in the fundamental representation of SU(Nc). Squaring the expression for the amplitude in eq. (2.7), and performing the necessary averaging and sum over electron spins,6 we can write (2.9) (2.10) (2.11) HJEP05(218)3 is of course di erent because it allows for the emission of a photon from the quark-antiquark pair. After these bare preliminaries, we shall proceed to compute M . 2.2 Classifying contributions to the amplitude There are a total of ten contributions, which we will group into diagrams with gluon eld insertions on the quark or antiquark line or both. The diagrams for the rst group are shown in gure 7. The contribution to the amplitude from the diagram labeled (1), 6We use here the identity X (k ; ) (k ; ) = g ; as the sum over outgoing photon polarizations. X is the lepton tensor that is identical to that obtained in inclusive DIS. The hadron tensor X spins X = M (q; k; p; k )M (1)(q; k; p; k ) = 2 (eqf )2 (P q ) e iP?:x? u(k)R(1)(P?)hU~ (x?) Z x ? 1 (antiquark) respectively and where Rx is a shorthand notation for R d2x?, i (j) represent the color indices of the quark ? to obtain the second line of eq. (2.14). Note that the P?-dependence appearing in R(1) includes an implicit dependence on the three transverse momenta constituting P ? = k? + The contributions from diagrams (2) and (3) in gure 7 can be computed similarly; the sum of the contributions from these three diagrams can then be written as q ) Z e iP?:x? u(k)T (q)(P?)hU~ (x?) 1 representing the emission of the photon by the quark subsequent to its multiple scatterings with the nucleus, is given by where qf is the charge of a quark/antiquark of a given avor. Employing eqs. (2.3) and (2.4), and choosing a frame where q ? = 0 for the virtual photon, we can write the above expresR(1)(P?) = (k= + k= + m) k ? + p?. 3 X =1 where M with R(2)(P?) = R(3)(P?) = (q (q p (=q p (=q P= ) + 2k P= ) + 2k k )2 Similarly, the contribution from diagrams with insertions on the antiquark line are shown 7The notation here and henceforth closely follows that employed previously in the discussion of photon production in p + A collisions in the CGC framework [35]. (p= m) v(p) = 2p v(p) ; u(k) (k= + m) = 2k u(k) ; (2.14) (2.15) ij (2.16) (2.17) ; R(4)(P?) = R( 5 )(P?) = R(6)(P?) = (q (q Z S0(l i v(p) ; ij (2.19) (2.20) (2.21) (2.22) (2.23) Finally, we need to consider the diagrams shown in gure 9 with insertions on both lines. For these cases, we will need to integrate over the nuclear momentum transfer l. The contribution from the diagram labeled (7) can be written as M (7)(q; k; p; k ) = The integration over l is trivial because of the (l ) factor from one of the e ective vertices. The integration over l+ can be done using the theorem of residues. The result 6 X =4 where The momentum labels and their directions are clearly shown. i and j stand for color indices in the fundamental representation of SU(Nc). in gure 8. Their collective contribution can be written as ( )(q; k; p; k ) = 2 (eqf )2 (P q ) e iP?:y? u(k)T (q)(P?)hU~ y(y?) 1 can be again compactly written as (7)(q; k; p; k ) = 2 (eqf )2 (P q ) Z Z e iP?:x?+il?:x? il?:y? u(k)R(7)(l?; P?) R(7)(l?; P?) = l is a shorthand notation for R d2l?=(2 )2. ? compactly written as appearing in the denominator is deliberately written in a way to make contact with the expression in eq. (2.14); this will be exploited in a later part of the calculation. In the 2 + m2 stands for the squared transverse mass and p= = ipi. Finally, ? The remaining three diagrams can be worked out similarly and the combined result bels and their directions are clearly shown. i and j stand for color indices in the fundamental representation of SU(Nc). (2.24) (2.25) (2.26) (2.27) where The factor 10 X =7 where e iP?:x?+il?:x? il?:y? q ) Z Z The various R-factors obtained after the contour integration over l+ are (=q (=q P= + =l ) + 2k ? In the expressions above, the compactly written functions are S(l?) = 4(p + k )(q p k ) q + V (l?) = 8p (q p )(q M 2(p? + k ? l?)q 2(p + k )(q p ) l ) ? k ) l ) ? k ) (k= + 2p ) S(l?) =l? + 2p ) V (l?) ; =l? + 2p ) W (l?) p ) ; ; (2.28) (2.29) : (2.30) p ) ? ! + =l ; ? M 2(p? 2(q l? + k ? ) p k ) ! + =l : ? 2.3 Final expression for the amplitude this result can be expressed compactly as The net contribution from all the diagrams is obtained by adding the expressions in eqs. (2.16), (2.19) and (2.26). Introducing a dummy integration R x ? Rk e ik?:x? = 1, ? X =1 M q ) Z p ) ? M 2(p? + k ? p ) ? M 2(p? + k ? u(k)hT (qq)(l?; P?) U~ (x?)U~ y(y?) + T (q)(P?) T (qq)(l?; P?) U~ y(y?) i The factors =la and =lb in eqs. (2.29) and (2.30) are obtained by evaluating =l at the enclosed poles depending on the choice of contours: (2.31) (2.32) (2.33) (2.34) 1 (2.35) q + k + q + k + =la = =lb = This expression can be further simpli ed by observing that the following relations holds for the various R-factors given in the previous section, which leads to R(1)(P?) R(2)(P?) R(7)(0?; P?) = 0 ; R(9)(0?; P?) = 0 ; R(3)(P?) (R(8) + R(10))(0?; P?) = 0 ; Since the second term in the sum of the three terms in eq. (2.35) is independent of y?, that sets l? = 0 for this term. Hence the identity in eq. (2.37) implies that the second term in eq. (2.35) vanishes. There is an intuitive way of understanding this relation. For this, we identify that the transverse momentum kicks to the antiquark and quark lines are respectively l P ? l?. In the limit of l? going to zero, diagrams (1) and (7) give identical contributions with the transverse momentum transfer being P ? for both cases. The same argument holds for processes (2) and (9). For processes (8) and (10), the insertions on the antiquark line are on either side of the photon and hence it is their sum which equals the contribution from process (3) for vanishing l?. The same argument holds for processes (4), ( 5 ) and (6) with the corresponding limit being l kick to the quark line vanishes. One thus gets ? = P ? so that in this case the transverse momentum R( 5 )(P?) R(6)(P?) R(8)(P?; P?) = 0 ; R(10)(P?; P?) = 0 ; R(4)(P?) (R(7) + R(9))(P?; P?) = 0 ; For the same reason articulated previously, this identity implies that the third term in eq. (2.35) vanishes as well. The relations in eqs. (2.37) and (2.39) should be compared to the second line of eq. (2.46) in [35] for the p + A NLO calculation. Such qualitative similarities will appear throughout the LO discussion in this paper. Since the second and third terms in eq. (2.35) vanish, the LO amplitude therefore reduces to the expression: (2.36) (2.37) ? and (2.38) (2.39) q ) Z Z ? The inclusive photon cross-section at LO In proceeding to write down the expression for the inclusive photon di erential crosssection, we note that since eq. (2.40) contains a delta function prefactor, the squared amplitude will naively contain a squared delta function. However this potential problem is resolved by realizing that the photon plane wave hitherto considered should be replaced by a properly normalized wave packet for the incoming virtual photon.8 Following a similar procedure to that described in [36] , and working in a frame where the lepton and the nucleus are moving towards each other at near light speed, we can write the di erential probability as d dxdQ2 = where h: : :iYA is used to denote the CGC color average, de ned in eq. (1.1), of the modulus squared of the amplitude which is the prescription [69] used for inclusive processes. In this expression, s = (~l+ PN )2 = Q2=xy = 4EEN is the e + A squared center-of-mass energy, where E is the incoming electron energy and EN the energy per nucleon of the incoming nucleus. Further, x = Q2=2PN q and y = q PN =~l:PN respectively denote the familiar DIS Lorentz invariant variables Bjorken x and the inelasticity. Finally, Ek, Ep and Ek are the respective relativistic energies of the quark, antiquark and the outgoing photon. The amplitude squared, averaged over colors, spins and polarizations, can be expressed as jMj 2 YA = 16 2 2qf4Nc L X ; 1 X in [69]: e i(P? l?):x? il?:y?+i(P? l0?):x0?+il0?:y?0 qq;qq(l?; l?0jP?) (x?; y?; x0?; y?0) : In the above expression, the Dirac trace is written in the compact notation introduced q ) ; (3.1) (3.2) (3.3) (3.4) (3.5) (3.6) HJEP05(218)3 qq;qq(l?; l?0jP?) = Trh(k= + m)T (qq) (l?; P?)(m p=)^0T (qq)y(l?0; P?)^0i : The nonperturbative information on strongly correlated gluons in the nucleus is entirely contained in the term (x?; y?; x0?; y?0) = 1 D(x?; y?) D(y?0; x0 ) + Q(x?; y?; y?0; x0?) ; ? where D and Q are respectively the dipole and quadrupole Wilson line correlators de ned as 1 1 D(x?; y?) = Nc hTr U~ (x?)U~ y(y?) iYA ; Q(x?; y?; y?0; x0 ) = ? Nc hTr U~ (y?0)U~ y(x0?)U~ (x?)U~ y(y?) iYA : 8A nice discussion of this very question can be found in section IV of [36] and in pages 99-107 of [68]. These are universal, gauge invariant quantities that appear in many processes in both p + A and e + A collisions [37, 70]. Be de ning these correlators in terms of nuclear unintegrated distributions [35, 69, 71], momentum conservation, (2 )2 (2)(P? l1?). Now, de ne the expression for X can be given an explicit momentum space `look'. For this, we rst introduce a dummy integration over l1? and a -function representing overall transverse Z Z x?;y?;x0?;y?0 x?;y?;x0?;y?0 = = l 1? l 1? e i(l1? l?):x? il?:y?+i(l1? l0?):x0?+il0?:y?0 D(x?; y?) 2 SNc YDA (l1? 2 l?; l?) ; e i(l1? l?):x? il?:y?+i(l1? l0?):x0?+il0?:y?0 Q(x?; y?; y?0; x0 ) ? 2 SNc YQA (l1? 2 l?; l?; l1? l?0; l?0) : YA should be contrasted with the correlators appearing in the p + A NLO calculation (see eqs. (3.11)-(3.13) of [35]) for inclusive photon production. Eq. (3.3) where D YA and Q therefore becomes X = where Z 0 Z l1? l?;l0? qq;qq(l?; l?0jl1?) YDA (l1? l?; l?) + YDA0 (l1? l0 ; l0 ) + YQA (l1? ? ? l?; l?; l1? l0 ; l0 ) ? ? ; (3.7) (3.8) (3.9) (3.10) The form of eq. (3.9) becomes particularly helpful in the next section where we will obtain k ? and collinear factorized limits of these results and compare them to the corresponding leading twist pQCD results. Following [35], we de ne the products d6K ? = d2p?d2k?d2k ? and d = d pd kd k and write the nal form of the triple di erential cross-section for inclusive photon production at LO as d dxdQ2d6K d3 K ? = 2q4y2Nc f 1 1 512 5Q2 (2 )4 2q L X q ) ; (3.11) are given by eqs. (2.10) and (3.9) respectively. In e + A DIS data, the above results can be applied to measurements of isolated photons accompanied by two jets or a quarkonium state, such as the J= meson. Alternatively, one may integrate over the quark or antiquark to obtain the di erential cross-section for direct photon plus jet production. Such measurements were performed for e + p DIS at HERA [7] for a wide range of Q2 and transverse energy of the nal state photon E ?, and are also feasible at future electron-nucleus collider facilities. The single inclusive di erential cross-section for inclusive prompt photon production is obtained by integrating over the quark and antiquark momenta and rapidities. We obtain d dxdQ2d2k ? d k = 2q4y2Nc 1 Z +1 dk Z +1 dp Z f A(x ) = 2A ; for Gaussian random color sources in a large nucleus [12{15]. This follows from the arguments outlined in the McLerran-Venugopalan (MV) model describing the nonperturbative gluon distribution at the onset of quantum evolution. In the MV model, 2A = A=2 R2 A1=3 is the average color charge squared of the valence quarks per color and per unit transverse area of a nucleus having mass number, A. Although there is no transverse coordinate dependence in 2A in the MV model, explicit numerical solutions [33, 34] of the Balitsky-JIMWLK hierarchy demonstrate that a nonlocal 2A ! 2A(x?) well-approximates these numerical solutions when YA (l1?) = 2 NcCF g2 Z l 2 1? x ? 2A(YA; x?) ; satis es the BK equation. In the limit of large transverse momentum, YA (l1?) evolves according to the BFKL equation [72, 73]. Employing these ingredients, we obtain the following leading twist expression for the 4 where Properties of the photon production amplitude at LO k?-factorization and collinear factorization limits In this section, we shall consider the large transverse momentum, l1? limit of the unintegrated distributions de ned in eqs. (3.7){(3.8). This is equivalent to expanding the Wilson line U~ (x?) de ned in eq. (2.5), to lowest nontrivial order in A=r2? and using the relations D aA(x ; x?) bA(y ; y?)E = ab (x y ) (2)(x? ? y ) A(x ) ; hadronic tensor: where LT X = 2 S YA (P?) Nc (P?) = Trh(k= + m) T (q)(P?) is obtained using eqs. (2.37) and (2.39) respectively in the Dirac trace de ned by eq. (3.4). An alternate way to arrive at the above expression is to compute the amplitude perturbatively. This can be done straightforwardly in Lorenz gauge by expanding the Wilson q ) ; (3.12) HJEP05(218)3 (4.1) (4.2) (4.3) (4.4) ^ 0i (4.5) The nuclear source is represented by the lower bold line. lines appearing in the amplitude expressions corresponding to processes (1)-(10). However the leading twist contribution is of O( A); therefore, processes (7)-(10) containing two Wilson line insertions will not be considered here. These leading twist diagrams are shown in gure 10. The leading twist amplitude can be written as (4.6) (4.7) (4.8) (4.9) (4.10) where and The Fourier transform of the gluon eld is given by M LT(q; k; p; k ) = (k ; ) (q; 0)M LT(q; k; p; k ) ; M LT(q; k; p; k ) = Aa;A(P q)ma (q; k; p; k ) : Aa;A(P q) = 2 g + (P q ) a;A(P?) ; ? ma X m ;a (q; k; p; k ) ; corresponds to the contributions from the leading twist diagrams (LT1)-(LT6), which can be individually written as m ;a+(q; k; p; k ) = ( ie2gqf2 u(k)R( )(P?)tav(p) ; +ie2gqf2 u(k)R( )(P?)tav(p) ; = 1; 2; 3 = 4; 5; 6 ; 6 =1 Speci cally, for the transverse elds, we have a sourceless Klein-Gordon equation D?2(B)] bi(x) = 0 ; after xing the residual gauge freedom with the Gauss' law constraint D (B)b Taking = in eq. (5.8), we get another sourceless equation which is the Gauss' law constraint imposed on initial eld con gurations [55{57]. Finally, using the above constraint to write D?:b? ; and keeping in mind the discontinuity of the spatial covariant derivative operator at x = 0, we can verify that eqs. (5.9) and (5.11) are consistent with the nal small uctuation equation The transverse eld is then assumed to be a simple plane wave for x < 0 and a linear superposition of plane waves for x > 0. This is done in a manner to ensure continuity of the elds across x = 0, which again is possible because of the absence of delta function singularities in the above equations at x = 0. The expressions for the Green's functions for the small uctuations thus obtained are given in the next subsection. In contrast, in the right light cone gauge A+ = 0, the discontinuity of the elds at the origin and the corresponding singular structure of the electric elds do not allow one to simply integrate the equations of motion. Hence it is convenient to rst obtain the small uctuation Green functions in A = 0 gauge and then, if required, gauge transform the results to A+ = 0. The expressions for the propagator in A+ = 0 gauge are complicated and not of relevance in our NLO computation. Following the procedure outlined above and using the identity U~ (x?)taU~ y(x?) = tbU ba(x?) ; tc U (x?)U y(z?) Z d2q ? (2 )2 Z ca ; we can express the transverse components of the uctuation eld as [14, 88], bi(x) = eip:x i ( x )ta + (x ) d2z ? e i(q? p?):(x? z?)ei(q?2 p2?) 2xp (5.9) where i is a unit vector. Using the above expression for bi, we get b+(x) = [D?; b?] 1 1 eip:x j ( x )ipj ta + (x ) Z d2q ? (2 )2 Z d2z ? e i(q? p?):(x? z?) ei(q?2 p2?) 2xp i(pj + qj )tc U (x?)U y(z?) ca : (5.15) 12The covariant derivative, D (B) = @ igB ;aT a. The small uctuation gluon propagator in coordinate space is de ned as G ;ab(x; y) = Db ;a(x)b ;b(y)E : (5.16) The nonzero components of the propagator in this gauge are G++, G+i, Gi+ and Gij . Using the expression for Gij in [14, 88] and eq. (5.13), it is straightforward to derive the following alternate expressions useful for momentum space calculations: Gij;ab(x; y) = g(x ; x?) Gij;ab(x; y) gy(y ; y?) 0 where G 0 ;ab(x; y) = Z Z d4z (z )g(x ; x?) n (x ) ( y )[U y(z?) o cd ( x ) (y )[U (z?) gy(y ; y?) Gik;ac(x; z)( i2@z+ )Gl0j;db(z; y) kl ; 0 (5.17) is the free gluon propagator in A = 0 gauge. Further, g(x ; x?) = ( x ) + (x )U (x?) ; (5.19) is a gauge transformation matrix that transforms the background eld B (x) to Lorenz gauge in which it is a singular eld having only a `plus' component. The same matrix, albeit in the fundamental representation, appears while calculating the fermion propagator [67] in the background classical eld. The momentum space expression for the gluon propagator in A = 0 gauge with the background eld B(x) in Lorenz gauge can be expressed as Gij;ab(p; p0) = (2 )4 (4)(p p0) Gij;ab(p) + Gik;ac(p) Tkl;cd(p; p0) Glj;db(p0) ; 0 0 0 (5.20) where Tij;ab(p; p0) = 2 (p p0 ) (2p )sign(p ) d2z ? ei(p? p0?):z? [U sign(p )(z?) Z One can similarly derive the following momentum space relations for the other nonzero components of G : G++;ab(p; p0) = (2 )4 (4)(p Gi+;ab(p; p0) = (2 )4 (4)(p p0) G++;ab(p) + G+i;ac(p) Tij;cd(p; p0) Gj+;db(p0) ; 0 0 0 p0) Gi+;ab(p) + Gij;ac(p) Tjk;cd(p; p0) Gk+;db(p0) : 0 0 0 (5.22) The general form for the small uctuation propagator in momentum space and in A = 0 gauge can be compactly expressed as G ;ab(p; p0) = (2 )4 (4)(p p0) G 0 ;ab(p) + G 0 ;ac(p) T ;cd(p; p0) G 0 ;db(p0) ; (5.23) 1]ab ij : (5.21) Here, and henceforth, the `tilde' factors in eqs. (4.24) and (5.25) are dropped. where the e ective vertex of the dressed gluon shock wave is given by T ;ab(p; p0) = 2 (p p0 ) (2p )g sign(p ) d2z ? ei(p? p0?):z? [U sign(p )(z?) 1]ab ; Z a form identical to (with the substitution of the fundamental Wilson line with its adjoint counterpart) the dressed fermion e ective vertex. While the gluon shock wave propagator in A = 0 gauge was rst derived in [14, 88], this form for the propagator in momentum space is given in a later paper by Balitsky and Belitsky [92] albeit it is also implicit in [67, 93]. The discussion in section 4.3 however suggests that we should modify the e ective gluon vertex to include the \no scattering" contribution by omitting the unit matrix in eq. (5.24) above; the modi ed vertex we will use henceforth in computations is T ;ab(p; p0) = 2 (p p0 ) (2p )g sign(p ) Z d2z ? ei(p? p0?):z? Uasbign(p )(z?) : (5.24) (5.25) To summarize, higher order all-twist computations at small x can be performed with the conventional Feynman rules of covariant perturbation theory albeit with the dressed quark and gluon e ective vertices shown in gure 12. 5.3 Processes contributing to the NLO amplitude Having outlined the necessary components for the computation of the inclusive photon amplitude to NLO, we will discuss here the relevant NLO processes. The general ideas were already presented in the introduction of this paper and we will elaborate further on that discussion. To begin with, the perturbative expansion in eq. (1.3) of the expectation value of a general operator O calculated for a particular source charge density con guration A can be expanded out to next-to-next-to-leading order (NNLO) as O[ A] = f00j (g A)j + f10j (g A)j S + f11j (g A)j S ln(1=x) {z NLO NN{zLO } | + f20j (g A) j S2 + f21j (g A) j S2 ln(1=x) + f22j (g A) j S2 ln2(1=x) + : : : ; (5.26) } where the repeated indices are assumed to be summed over. For the di erential crosssection for inclusive photon production, at each order in perturbation theory, there are contributions to the small x evolution of the random color sources (encoded in the dipole and quadrupole Wilson line correlators given by eq. (3.5)). These are enhanced by logarithms in x and when S ln(1=x) 1, they complicate the naive power counting. In addition, we will have non-log enhanced perturbative corrections to the light cone wavefunction for the virtual photon that uctuates into a quark-antiquark dipole (as well as ancillary complications due to the photon in the nal state) that should be combined with the ln(1=x) contributions systematically. It is therefore e cient to reexpress the generic expansion in eq. (5.26) by factorizing the cross-section into a coe cient term and a matrix element of Wilson line correlators between nuclear wavefunctions, at each order in d [ A] = CLO d^LO + CLO d^LLx + CNLO | + CNLO | 0S{:z 0S | 0S: S{lzn(1=x)} S: S{lzn(1=x) } | 0S: 2S{lzn(1=x) } d^LLx + CLO d^NNLx + : : : : S{:z 0S d^NLO } S, as (5.27) We computed the rst term in the r.h.s. of the above equation in the previous sections. The second term is given by contributions from the diagrams of the type depicted in gure 4. For S ln(1=x) 1, these contributions are as large as the LO contribution and can be absorbed in the LO result by a rede nition of the weight functional W . The resulting LLx RG equation (the JIMWLK equation) e ciently sums such leading contributions to all orders of perturbation theory while preserving the structure of the LO result. The third term appearing in eq. (5.27) represents genuine S suppressed contributions that do not take into account the small x evolution of the sources. In addition to the real emission diagrams shown in gures 5 and 18, there are contributions from loop graphs depicted by the second and third representative diagrams in gure 5. Since the cross-section for inclusive gluon production in DIS was computed previously [94] at small x, taking the soft photon limit to the real emission contributions will provide in principle a check of our results. The complete calculation of this term will give the next order correction CNLO to the LO coe cient function or \impact factor" CLO. It is worth mentioning here that the ^NLO has a di erent correlator structure than ^LO due to the additional adjoint Wilson lines and hence the distinct nomenclature. The overall contribution of the third term in eq. (5.27) is strongly subleading at small x. The two terms in the second line of eq. (5.27) are part of the NNLO contributions to inclusive photon production but are potentially quantitatively important because they are e ectively of order S when S ln(1=x) 1; these form the next-to-leading-log (NLLx) rapidity separation between fast sources and slow elds is shown clearly. Integration over modes with Y > Y0 is represented by the loops in the upper part of the gure. The contributions below the rapidity cut are part of the LLx evolution of the color sources. See text for more details. contribution to the cross-section. Under our RG ideology, the two terms can be considered separately. The rst term in the second line of eq. (5.27) is represented by the diagrams in gure 13. The dashed horizontal line represents the rapidity scale Y0 separating the classical elds that interact with the qq system from the static color sources RG evolved by the LLx JIMWLK equation from the target. The two diagrams shown are respectively the real and virtual contributions that are characteristic of the terms that generate the JIMWLK kernel. The computation of the NLO impact factor (above the Y0 cut) using the techniques developed here is in progress and will be reported in a follow-up paper [95]. A highly nontrivial check will be to reproduce the NLO impact factor for fully inclusive DIS [96{103] that should be recovered in the soft photon limit. It will also be important for the consistency of the framework to demonstrate JIMWLK factorization by explicit computation. nal term in eq. (5.27) is represented by the diagrams in gure 14. In this case, there are no radiative corrections above the rapidity cut and the dynamics is described by the LO impact factor. The contributions shown below the cut correspond to NLO contributions to the JIMWLK kernel. These have been computed previously [58, 62] (see also [59, 60]) and can therefore be used to construct the NLLx result for inclusive photon production in e + A collisions. We note that NLLx corrections have recently been implemented for numerical computations in fully inclusive DIS [104]. An advantage of our approach is that it is fully implemented in momentum space; many shock wave computations use a mixture of momentum and coordinate space variables that is cumbersome. This is especially so in computing running coupling contributions [61, 62, 105] where coordinate space prescriptions can be problematic as noted in [106]. In our framework, there is no need to switch back and forth; all computations can be realized fully in momentum space with our modi ed Feynman rules. These issues will be addressed in future work. 6 Summary and outlook We presented in this paper a rst computation of inclusive photon production in e+A DIS at small x within the CGC EFT. At LO, the cross-section is directly proportional to universal 1 at small x. These diagrams contribute to the NLO JIMWLK kernel. gauge invariant dipole and quadrupole Wilson line correlators which are ubiquitous in nal states that are measured in high energy p + A collisions and potentially in e + A collisions at a future Electron-Ion Collider (EIC). Indeed, since inclusive photon production in DIS has photons in both the initial and nal states, it holds promise of being a clean golden channel for unambiguous discovery of gluon saturation, complementary to other e + A DIS measurements [107, 108]. In the soft photon limit, we recover the inclusive DIS dijet crosssection derived previously in [37]. As argued there, this dijet e + A channel may provide direct access to the nuclear Weizsacker-Williams gluon distribution. We next discussed the structure of dominant small x contributions to the inclusive photon cross-section at next-to-leading order and next-to-next-to-leading order. The essential ingredients here are the dressed quark and gluon propagators and the corresponding e ective vertices in the shock wave classical background eld of a nucleus at high energies. These e ective vertices are proportional to the respective fundamental and adjoint Wilson lines that carry information about all-twist gluon correlations in the nucleus. The structure of the quark and gluon dressed propagators is remarkably simple in the \wrong" light cone gauge A = 0 and therefore permits e cient higher order computations. The computations are further simpli ed by exploiting the natural separation between static sources and dynamical gauge elds in the MV model and the JIMWLK RG treatment thereof. In particular, the cross-section can be factorized into impact factor contributions that are convoluted with the RG evolution of products of lightlike Wilson lines. The nontrivial ingredients that need to be computed are the NLO impact factor for inclusive photon production and the NLLx RG evolution of the Balitsky-JIMWLK hierarchy of Wilson line correlators. While the latter is known, the former needs to be determined; these computations are in progress and will be reported on separately [95]. We note that, motivated by collider experiments, small x computations in the gluon saturation regime are increasingly to next-to-leading order accuracy. Some examples of the studies being performed include, besides inclusive DIS [104], DIS di ractive dijet production [109, 110] and exclusive light vector meson production [111], single inclusive forward hadron production in p + p and p + A collisions [112{115] and more recently inclusive photon production in p + A collisions [35, 42]. Looking further ahead, we believe that the momentum space methods discussed here can exploited to make progress in these and related computations. The forms of the shock wave propagators rst derived in [14, 88], and their expressions in terms of e ective vertices [67, 92], are identical to the quark-quark-reggeon and gluongluon-reggeon propagators [116{118] in Lipatov's reggeon eld theory [119]. We have shown here that the slightly modi ed quark and gluon e ective vertices in eqs. (4.24) and (5.25) signi cantly simplify the computation of DIS inclusive photon production. Because of the form of the shock wave propagators, our results are equally valid for leading twist or alltwist computations. It would therefore be interesting to see if our modi ed Feynman rules are useful in simplifying multiloop leading twist computations in the Regge limit [120, 121] or perhaps, conversely and more interestingly, results derived in those cases applied to advance computations in the saturation regime of high parton densities. Acknowledgments R.V would like to thank Andrey Tarasov for a useful discussion. This material is based on work supported by the U.S. Department of Energy, O ce of Science, O ce of Nuclear Physics, under Contracts No. DE-SC0012704 and within the framework TMD Theory Topical Collaboration. K. R is supported by an LDRD grant from Brookhaven Science Associates. A Notations and conventions The metric used is the 2 metric, g^ = diag(+1; 1; 1; 1), where the `carat' denotes quantities in usual spacetime coordinates. The light cone coordinates are de ned as x + = x^0 + x^3 p 2 with the transverse coordinates remaining the same and transforming as in Minkowski space. The same de nition holds for the gamma matrices + and with the Dirac algebra given by f (A.1) where g+ = g + = 1 and gij = ij (i; j = 1; 2) are the nonzero entries of the metric tensor. In this convention, a:b = a+b + a b a?:b? and a+ = a , ai = ai. B Gauge invariance and soft photon factorization at LO The rst part of this appendix provides a proof of the Ward identity for the nal state real photon, thus establishing gauge invariance. In the second part, we will provide an explicit expression for the nonradiative DIS amplitude recovered using the soft photon limit, k ! 0. We will next derive the di erential cross-section for inclusive dijet production in DIS for the case of an incoming longitudinally polarized virtual photon. The case of the transversely polarized virtual photon follows identically and will therefore not be shown. HJEP05(218)3 The Ward identity for the nal state photon requires that k M (q; k; p; k ) = 0 : Towards this end, we make the replacement (k ; ) ! k in the amplitude and check the e ects of contracting k with T (qq)(l?; P?). Instead of using the nal forms for the various R-factors constituting T (qq), we use their original forms in which the contour integrations are not performed. This simpli es the calculation. For example, we have u(k)k R(7)(l?; P?)v(p) = i3 Z dl+ 2 u(k) =q p= + =l + m = l (q p + l)2 m2 + i" (l p)2 2p We can use the second identity in eq. (2.15) and the relation u(k)k= (k= + m) = (2k:k )u(k) ; to derive eq. (B.2). Likewise, we can write u(k)k R(9)(l?; P?)v(p) = i3 u(k) Z dl+ 2 (q p= + =l + m)k= (=q = l p + l 2p k )2 p= + =l + m) p + l)2 m2 + i" (l p)2 v(p) : (B.4) (B.1) u(k) 2p k )2 = k q = p= + =l k= + m m2 + i" v(p) ; Z dl+ 2 = k u(k) = l 2p (l p k )2 =q p= + =l k= + m =l p= + m (q p + l k )2 m2 + i" (l p)2 Adding eqs. (B.2) and (B.4), and using the fact that p2 m2 = (p= + m)(p= m), it is easy to show that one of the terms in square brackets in the denominator cancels out, giving u(k)k hR(7)(l?; P?) + R(9)(l?; P?)iv(p) = i3 Z dl+ 2 u(k) q = p= + =l k= + m = l m2 + i" (l p)2 2p v(p) : (B.5) Analogously, we can use the rst identity in eq. (2.15) and (p= m)k= v(p) = (2p:k )v(p) ; to write and u(k)k R(10)(l?; P?)v(p) = i3 u(k)k R(8)(l?; P?)v(p) = i3 Adding the expressions in eqs. (B.7) and (B.8), and using a similar reasoning used to simplify the previous sum, we get u(k)k hR(7)(l?; P?) + R(9)(l?; P?)iv(p) = u(k)k hR(8)(l?; P?) + R(10)(l?; P?)iv(p) : u(k)k T (qq)(l?; P?)v(p) = 0 This implies that the B.2 thereby showing that the Ward identity is indeed satis ed for the outgoing photon. It can also be shown that the same relation holds for the case of the exchanged virtual photon if we are considering the hadronic subprocess. This gives us the freedom to consider only part of the photon propagator which is implicitly assumed in writing eq. (2.7). Soft photon factorization By taking the soft photon limit k ! 0 of our LO amplitude expression in eq. (2.40), we can recover the nonradiative DIS amplitude given by M NR(q; k; p) = 2 (eqf ) (P 1 2q u(k) z(1 z)Q2 +?M 2(l? =l?) + m p ) ? q ) Z Z Z ? y ? l ? e iP?:x?+il?:x? e il?:y? (2p + =l ) ? U~ (x?)U~ y(y?) 1 where P now equals p + k. We will now show that the above expression can be used to factorize the amplitude squared in terms of products of light cone wavefunctions and a dipole scattering factor. In the soft photon limit, the amplitude for the subprocess (B.9) (B.10) (B.11) ij v(p) ; (B.12) (B.13) is given can be written as (q) ! Q(k) + Q(p) + (k ) M(q; k; p; k ; ; 0) = (q; 0)M (q; k; p; k ) ; where (q; 0) is the polarization vector for the incoming virtual photon and M by eq. (B.11). In order to identify the transverse and longitudinally polarized photon wavefunctions, it is convenient to parametrize the polarization vectors as T (q; 0 = +1) = 0; 0; T (q; 0 = 1) = 0; 0; p ; Q 2q 1 p ; 2 1 q 2 Q i p 2 i p 2 ; L(q; 0 = 0) = ; 0; 0 ; (B.14) where T and L stand for transverse and longitudinal respectively. These vectors satisfy the relations g 2T (q; 0) = 1; q q q2 X 0= 1 2L(q; 0) = 1; T (q; 0) T (q; 00) = 0; 00 ; T (q; 0) T (q; 0) + L(q; 0 = 0) L(q; 0 = 0) : (B.15) 1 2q Q 2q u(k) + Q ? + (p= ? z)Q2 + M 2(l? p ) ? ? =l ) + m + (2p + =l ) ? U~ (x?)U~ y(y?) (B.16) 1 ? ! l? and using the property (A.1) for gamma matrices along with the Dirac equation, we can simplify the above result to M NR(q; k; p; 0 = 0) = 2 (eqf ) (P e ik?:x? ip?:y?+il?:(x? y?) ? 2z(1 z)Q l2 + q2 1 Q U~ (x?)U~ y(y?) 1 u(k) v(p) ; (B.17) We will now explicitly compute the cross-section for the longitudinally polarized case. With the above choice of the 0 = 0 polarization vector, eq. (B.13) gives M NR(q; k; p; 0 = 0) = L(q; 0 = 0)M NR(q; k; p) = 2 (eqf ) (P where the (0 ) appearing in the amplitude squared is normalized as described earlier in section 3. Now using Tr (k= + m) (p= m)^0 +^0 = (2p )(2k ) ; (B.22) where q2 = z(1 z)Q2 + m2. The term proportional to 1=Q vanishes because of the y?) arising from the integration over l? and the identity U~ (x?)U~ y(x?) = 1. Finally using the formula where K0 is the modi ed Bessel function of the second kind, we can recast M NR as Z d2l ? eil?:(x? y?) l2 + q2 1 2 K0( qjx? y?j) ; M NR(q; k; p; 0) = 2 (P q )M~ NR(q; k; p; 0) ; (B.18) (B.19) where M ~ NR(q; k; p; 0 = 0) = (eqf ) q ) z(1 Z q ) Z Z Z 1 ij jM NR j 2 The di erential cross-section for inclusive dijet production in DIS is given by d d3kd3p 1 1 1 2q (2 )32Ek (2 )32Ep Z Z e ik?:x? ip?:y? ? y U~ (x?)U~ y(y?) K0( qjx? y?j) 2 2z(1 z)Q u(k) (B.20) YA q ) ; (B.21) dressed fermion lines. The momenta l1 and l2 are the momentum kicks from the nucleus to the quark and antiquark respectively. and the form of the 0 = 0 photon wavefunction given in [37] r 4 L (q ; z; r) = 2 z(1 z)QK0( qr) ; (B.23) it is a matter of straightforward algebra to show that d L = Nc qf2 (q k ) Z d2x ? d2x0? d2y ? d2y0 (2 )2 (2 )2 (2 )2 (2 )?2 e ik?:(x? x0?)e ip?:(y? y?0) X ; 1 L (q ; z; jx? y?j) L (q ; z; jx0? YA Nc 1 DTr U~ (y?0)U~ y(x0?) E YA Nc 1 DTr U~ (y?0)U~ y(x0?)U~ (x?)U~ y(y?) E YA : (B.24) This result exactly matches eq. (22) of [37] obtained for DIS dijet production at small x. The case of the transversely polarized photon proceeds in a similar fashion. C Kinematically allowed processes In this appendix, we will explicitly demonstrate the topologies of the LO and NLO Feynman diagrams that are allowed by the kinematics of the process. The techniques described in this section are quite general and can be extended to nd kinematically allowed diagrams beyond NLO. Let us rst consider the most general diagram (see gure 15) for the LO process with all dressed fermion lines and photon emission from the quark line. We will now show it is possible to nd all allowed processes starting from this generic template provided we use the modi ed Feynman rules discussed in section 4.3. An identical treatment follows for the case of photon emission from the antiquark line. The amplitude for the hadronic subprocess is given by M where and D = l 1 where Rli = R d4li=(2 )4; i = 1; 2 and T is given by eq. (4.24). Integrating out l1 and l2 using the delta functions embedded in the vertex factors, we are left with the delta function (q P ) representing the overall longitudinal momentum conservation and integrations over transverse spatial and momentum coordinates. However the quantity of interest is the integral over l1+ and l2+ given by dl2+ N D ; N = u(k) l=1 + m) (k= + k= l=1 + m) (=q + l=2 p= + m) (l=2 p= + m) S0(q + l2 p) S0(l2 p)Tnj ( p; l2 p)v(p) ; l1) (C.1) (C.2) (C.3) (C.4) i" ! 2p ; M 2(l1? p ) ? + 2(k + k ) 2(k + k ) p+ + M 2(l1? 2p p ) k+ + k+ q+ + l2+ 1 M 2(k? 2k i" ! 2k denote the numerator and denominator respectively. Examining the structure of the poles in the propagator terms of the denominator, there are two l1+ poles on the positive side of the real axis and independently, two l2+ poles on either side of the real axis. Using the property ( )2 = 0, it is easy to see that the numerator does not contain any term proportional to l1+ or l2+. Therefore the contour for the integration over l1+ can be closed below the real axis thereby giving a null result. The arguments presented thus far does not invoke the possibility of \no scattering" in our de nition of the e ective vertices or equivalently the U~ = 1 case. It can be shown easily that for either U~im = im or U~mn = mn in the vertex factors appearing in eq. (C.1), we will get a nonzero result from the above contour integration. Under these conditions, the two factors appearing in the rst line of eq. (C.4) resemble energy denominators that appear in light cone perturbation theory (LCPT) [40, 83{86] and the integration over l2+ can be done using the residue theorem. Corresponding to these conditions on the U~ 's, we get the two allowed diagrams with photon emission from the quark line as shown in gure 16; these are embedded in our generic diagram gure 15. One can therefore start with the generic case and eventually deduce the impossibility of a secondary scattering subsequent to emission of the photon by an already scattered the quark line. The remaining two diagrams can be obtained by interchanging the quark and fermion. This is a consequence of the eikonal approximation in which the nucleus moving at near light speed interacts instantaneously with the quarks thereby removing the possibility of a second scattering. Once the allowed diagrams are computed, we simply need to deduct the contribution in which both quark and antiquark propagate unscattered. The latter have the same magnitude as the allowed diagrams modulo the Wilson line factors. The net contribution is given by eq. (2.40). The same physical principle discussed above for LO also applies to the NLO diagrams which are classi ed into three broad categories. For the diagrams contributing to the LLx and NLLx JIMWLK evolution, the allowed topologies can be easily extracted from existing literature [58, 61, 62].13 Since the upper part of these diagrams has the same structure as LO diagrams, the rules discussed in the previous section apply trivially. In the following, we therefore discuss only the genuine S suppressed contributions in gures 5. We consider one such representative generic diagram (see gure 17) which represents real emission of a gluon from the quark or antiquark in addition to the nal state photon. We use the line of reasoning made for the LO case to deduce the allowed processes. The amplitude for this subprocess is given by Y4 Z k=1 lk M ;a = i(eqf )2g u(k)Tim(k l1; k)S0(k l1) l1) Tmn(k + k Tpq(q + l4 G 0 ;bc(l3)T ;ca(l3; kg) (kg) ; 1 l2; k + k l1)S0(k + k 1 l2)(tb)np S0(k + k l 1 l2 + l3) p; k + k 1 l2 + l3)S0(q + l4 p) Tqj ( p; l4 where the vertex factors for the fermion and gluon propagators are given respectively by eqs. (4.24) and (5.25). By carefully integrating out the li 's (i = 1; : : : ; 4) using the -functions, the amplitude can be cast in terms of a momentum conserving delta function (q p k k integrations over transverse spatial and momentum coordinates and the following integral 13We should mention here that our diagrammatic representation is similar in spirit to the \shock wave" approach used in these works. (C.5) kg ), where D1 = l 3 l 4 l 1 M 2(k? 2k l 2kg 3? + i" ! 2kg + kg ) M 2(l4? 2p 2(k + k I1 = Y4 Z k=1 dlk+ D1 k + + k q + + l4+ p ) l i" ! 2p : l2+ + l3+ M 2(l4? 2(k + k the momentum transfer from the nucleus to the quark and antiquark line at di erent points in the scattering process. l3 is the momentum carried by the gluon prior to scattering o the nucleus. i and j represent fundamental color indices. The Wilson line factors associated with each e ective vertex are shown on the right. The expressions above clearly demonstrate that the numerator doesn't have any term proportional to l1+, l2+ or l3+ and the poles for all these variables are on the same side of the i" ! 2k k + + k l M 2(k? + k ? 2(k + k ) + k ) + + k l M 2(k? + k ? 2(k + k ) 2(k + k ) l2? + l3?) 2(k + k + kg ) p ) + kg ) 2(k + k + kg ) N1 = u(k) [ +k l1?) + m] [ +(k + k ) ?:(k? + k ? l1?) + m] ?:(k? + k ? l2?) + m] [ +(k + k + kg ) ?:(k? + k ? l2? + l3?) + m] [ +(k + k + kg ) p?) + m] [ +p + m] (kg) + (l3: (kg))n kg is obtained using n: (kg) = 0 and ( )2 = 0 and (C.6) (C.7) (C.8) Lorentz contracted nucleus shown here by the red rectangular wall. real axis. Hence the integration contours for any such variable can be deformed in a way so as not to enclose any pole giving a result of zero for general U and U~ 's depicted in gure 17. 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Kaushik Roy, Raju Venugopalan. Inclusive prompt photon production in electron-nucleus scattering at small x, Journal of High Energy Physics, 2018, 13, DOI: 10.1007/JHEP05(2018)013