Factorization of the soft gluon divergence from the dipole picture deep inelastic scattering cross sections at next-to-leading order

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY-NC-ND 4.0 https://creativecommons.org/licenses/by-nc-nd/4.0/ Factorization of the soft gluon divergence from the dipole picture deep inelastic scattering cross sections at next-to-leading order © the author(s) 2019 Published version Ducloue, B.; Hänninen, H.; Lappi, T.; Zhu, Y. Ducloue, B., Hänninen, H., Lappi, T., & Zhu, Y. (2019). Factorization of the soft gluon divergence from the dipole picture deep inelastic scattering cross sections at next-to-leading order. In Hard Probes 2018 : Proceedings of the 9th International Conference on Hard and Electromagnetic Probes of High-Energy Nuclear Collisions. Sissa. PoS : Proceedings of Science, 345. https://doi.org/10.22323/1.345.0116 2019 PoS(HardProbes2018)116 Factorization of the soft gluon divergence from the dipole picture deep inelastic scattering cross sections at next-to-leading order Bertrand Ducloué Institut de Physique Théorique, Université Paris-Saclay, CEA, CNRS, F-91191 Gif-sur-Yvette, France E-mail: [email protected] Henri Hänninen∗ Department of Physics, P.O. Box 35, 40014 University of Jyväskylä, Finland E-mail: [email protected] Tuomas Lappi Department of Physics, P.O. Box 35, 40014 University of Jyväskylä, Finland Helsinki Institute of Physics, P.O. Box 64, 00014 University of Helsinki, Finland E-mail: [email protected] Yan Zhu Physik-Department, Technische Universität München, D-85748 Garching, Germany E-mail: [email protected] We use a factorization scheme, analogous to the one proposed for single inclusive forward hadron production in hadronic collisions, to regularize the soft gluon divergence present in the deep inelastic scattering (DIS) cross sections in the dipole picture at next-to-leading order (NLO). We show numerically that in this carefully constructed scheme it is possible to obtain meaningful results for the DIS cross sections at NLO, and so we are able to quantitatively study the recently derived NLO corrections to the DIS cross sections. We find that the NLO corrections can be significant and sensitive to the details of the factorization scheme used for the resummation of the large logarithms into the Balitsky-Kovchegov (BK) evolution equation. In the case of an approximative factorization scheme we observe a problematic behavior of the DIS cross sections similar to what has been seen with analogously factorized single inclusive cross sections in hadronic collisions. International Conference on Hard and Electromagnetic Probes of High-Energy Nuclear Collisions 30 September - 5 October 2018 Aix-Les-Bains, Savoie, France ∗Speaker. c Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ PoS(HardProbes2018)116 Dipole Picture NLO DIS Henri Hänninen 1. Introduction Deep inelastic scattering (DIS) provides a clean process to study the partonic structure of hadrons. At small Bjorken-xit is convenient to look at the process in the dipole picture where the scattering factorizes into two parts: first the virtual photon fluctuates into a quark-antiquark pair in a QED process, and subsequently the quark dipole scatters off the target in a QCD process. Already at leading order the dipole picture has led to satisfactory fits to HERA DIS data, using the running coupling BK equation [1,2], see e.g. Refs. [3,4]. Recent progress on both NLO BK [5,6,7,8,9] and NLO DIS impact factors [10,11,12] have made full NLO cross section computations possible in the dipole picture. In this work, we construct a subtraction scheme for the resummation of the large logarithms of energy present in the NLO DIS impact factors according to the principle presented in Ref. [13], which was shown to be effective in the case of single inclusive particle production in hadronic collisions in Ref. [14]. To demonstrate the effectiveness of the subtraction procedure we computed in Ref. [15] the DIS structure functions at NLO accuracy. The numerical results allow us to evaluate the importance of the NLO contributions and to estimate the stability of the perturbative expansion for this quantity. 2. Next-to-leading order cross sections and soft gluon divergence The total photon-proton cross sections at leading order in the dipole picture for a transversely (T) and longitudinally (L) polarized photon read σLO L,T(xB j,Q2) = 4Ncαem ∑ f e2 fZ1 0 dz1Zx0,x1 KLO L,T(z1,x0,x1,xB j),(2.1) with the notation Rx0=Rd2x0 2πand KLO L,T(z1,x0,x1,X)∼Ψγ∗ L,T→q¯q(1−S01(X)).In other words the integrands are products of the light cone wavefunctions for the γ∗→q¯qfluctuation and the q¯q dipole–color field scattering amplitude 1−S01. The scattering matrix S01 for the dipole–color field scattering is given by the two-point correlation function of Wilson lines: S01(X)≡S(x01 =x0−x1,X) = 1 Nc TrU(x0)U†(x1)X ,(2.2) where x0and x1are the transverse positions of the quark and antiquark, and Xis the momentum fraction at which the two-point correlation function is evaluated. The momentum fraction is related to the BK equation evolution variable via y=ln1/X. At next-to-leading order the virtual photon Fock state contains contributions from a gluon loop to the quark-antiquark dipole and a new parton state q¯qg with a quark-antiquark-gluon tripole, where in the former case the q¯qdipole and in the latter the q¯qg tripole scatters off the target. These NLO corrections have been calculated in mixed space by G. Beuf [10,11] using conventional dimensional regularization, and verified in the four-dimensional helicity scheme [12]. The total γ∗pscattering cross section at next-to-leading order can be written in the following "unsubtracted" form in accordance with the general idea presented in Ref. [13]: σNLO L,T=σIC L,T+σqg L,T+σdip L,T.(2.3) 1 PoS(HardProbes2018)116 Dipole Picture NLO DIS Henri Hänninen Here the first term is the lowest order contribution with an unevolved target and the last two terms contain all the corrections proportional to αs. The quark-gluon σqg L,Tand dipole σdip L,Tcontributions can be thought to emerge from the computation of the NLO real emission and gluon loop diagrams, respectively. However, the diagram computation results are separately ultraviolet (UV) divergent and so must be combined in a way that shows the cancellation of the divergences, which has been done for Eq. (2.3). This was done in Ref. [11] by introducing suitable subtraction terms. The choice of the subtraction terms is not unique and the UV subtraction was done in an alternative way in Ref. [12] yielding equivalent final results. We have written the UV finite results from Ref. [11] in the following form: σqg L,T=8Ncαem αsCF π∑ f e2 fZ1 0 dz1Z1−z1dz2 z2Zx0,x1,x2 KNLO L,T(z1,z2,x0,x1,x2,X(z2)),(2.4) σdip L,T=4Ncαem αsCF π∑ f e2 fZ1 0 dz1Zx0,x1 KLO L,T(z1,x0,x1,Xdip)1 2ln2z1 1−z1−π2 6+5 2,(2.5) where the expressions for the NLO integrands KNLO L,Tcan be found in Ref. [15], with x2being the transverse position of the gluon, zitheir longitudinal momentum fractions, and the q¯qg state Wilson line scattering operator is S012(X) = Nc 2CFS02(X)S21(X)−1 Nc2S01(X).(2.6) Note the incompletely defined logarithmic integral over z2in Eq. (2.4). Since the integrands KNLO L,T(z1,z2,x0,x1,x2,X)tend to non-zero values as z2→0 at fixed X, the qg contributions are logarithmically divergent. This large logarithm needs to be subtracted and resummed into the target BK evolution. On the other hand, the "dipole"-contribution (2.5) does not contain such a large logarithm and so has been integrated over the internal gluon loop momentum fraction z2. We will briefly review the subtraction procedure constructed in Ref. [15]. First, the lowest order term in Eq. (2.3) is identified as the leading order cross section (2.1) without leading log resummation, i.e. with the dipole scattering amplitude evaluated at the initial condition X=x0. In order to achieve a stable perturbative expansion at NLO, the dipole amplitudes in the qg-contribution σqg L,Tmust be evaluated at a rapidity that depends on the fractional momentum z2of the emitted gluon[13,6]. We argue from kinematics, since at small z2the target momentum fraction behaves as X(z2)≈k2 ⊥/(z2W2), where k⊥is the gluon transverse momentum, that it might be possible to approximate X(z2)≈xB j/z2for DIS, with similar argumentation as for the single inclusive particle production in Ref. [13,14]. At small target momentum fraction X<x0this yields a lower limit z2>xB j/x0. Now we can complete the "unsubtracted" form of the NLO cross sections (2.3) by setting X(z2)≡xB j/z2and the limit z2>xB j/x0. In order to write the NLO cross section as the full LO cross section (2.1) and some αscorrections, we note that by taking the z2→0 limit of KNLO L,T(in the explicit z2dependence, not in the implicit through X(z2)) one gets an integral version of the BK evolution equation. Using this fact, we write the "subtracted" form of the NLO cross section as σNLO L,T=σLO L,T+σqg,sub. L,T+σdip L,T,where σLO L,Tis the leading order expression (2.1) with evolved target 2 PoS(HardProbes2018)116 Dipole Picture NLO DIS Henri Hänninen 1 10 100 Q2GeV2 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 FT/σ0 2GeV2 xBj = 10−3 FLO T FLO+dip T FLO+qg T FNLO T (a) 1 10 100 Q2GeV2 −0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 FT/σ0 2GeV2 xBj = 10−3 FLO T FLO+dip T FLO+qg T FNLO T (b) 1 10 100 Q2GeV2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 FNLO/F LO xBj = 10−3 L, αs= 0.2 L, αsx2 01T, αs= 0.2 T, αsx2 01 (c) Figure 1: (a):FT(Q2)LO and NLO contributions in the "unsubtracted" scheme. (b):FT(Q2)in the approximative xB j-subtracted scheme. (c): Ratios FNLO(Q2)/FLO(Q2)with fixed and running coupling. and σqg,sub. L,T=8Ncαem αsCF π∑ f e2 fZ1 0 dz1Z1 xB j/x0 dz2 z2Zx0,x1,x2θ(1−z1−z2) ×KNLO L,T(z1,z2,x0,x1,x2,X(z2))−KNLO L,T(z1,0,x0,x1,x2,X(z2)).(2.7) It might be tempting to make simplifying approximations to the subtracted scheme by neglecting the lower limit in z2as small and the z2dependence in X(z2). With these adjustments one gets the "xB j-subtracted" scheme where in σqg,sub. L,Twe take X(z2)≡xB j and xB j/x0→0. This xB j-subtracted scheme is analogous to the "CXY" scheme used in single inclusive particle production in [14] and while this approximative scheme is formally equivalent at this order of perturbation theory, the "CXY" scheme has been shown to lead to problematic results at high momentum scales. 3. Numerical results To demonstrate the behavior of the unsubtracted and xB j-subtracted schemes we present some of our results from Ref. [15]. We neglect impact parameter effects so the presented results are of FL,T/(σ0/2)where FL,T(xB j,Q2) = Q2 4π2αem σL,T(xB j,Q2).For the target evolution the LO BK equation with an MV initial condition [16] was used. To study the importance of running coupling effects, a fixed coupling αs=0.2 was compared to a parent-dipole running coupling αs=αs(x2 01). In Fig. 1a we show the effect of the σqg and σdip NLO corrections to the structure functions as functions of Q2in the unsubtracted scheme (2.7). First we note that the NLO corrections overall are moderate and yield reasonable results, similarly to the analogous scheme with single inclusive particle production [14]. Overall the NLO corrections decrease the structure functions. In Fig. 1b we show the same quantities in the xB j-subtraction scheme. The quark-gluon contribution is again negative but even larger in magnitude, increasingly so at large Q2, making the full NLO structure functions negative at Q2&10GeV2. So while the approximations made for the xB j-subtraction scheme are in principle valid in a weak coupling sense, they have a large effect in practice in this region and can lead to unphysical results. This is similar to a negativity issue seen with single inclusive particle production at high transverse momenta [14]. 3 PoS(HardProbes2018)116 Dipole Picture NLO DIS Henri Hänninen In Fig. 1c we return to the working unsubtracted scheme to demonstrate the effects of the running coupling and the magnitude of the NLO corrections. We see that the NLO corrections are of the order of a few tens of percent in the Q2range of interest. Secondly, one should note that in this scheme the choice of running coupling can have large effects: the sign of the NLO corrections to FTchanges when one uses the parent-dipole running coupling. This is due to the large cancellations between the different kinds of NLO contributions in this scheme. 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