Two-loop QED corrections to the Altarelli-Parisi splitting functions
IICAS 09/16IFIC/15-88
Two-loop QED corrections to the Altarelli-Parisi splittingfunctions
Daniel de Florian ( a ) ∗ , Germ´an F. R. Sborlini ( b ) † and Germ´an Rodrigo ( b ) ‡ ( a ) International Center for Advanced Studies (ICAS), UNSAM, Campus Miguelete, 25 de Mayoy Francia, (1650) Buenos Aires, Argentina ( b ) Instituto de F´ısica Corpuscular, Universitat de Val`encia - Consejo Superior de InvestigacionesCient´ıficas, Parc Cient´ıfic, E-46980 Paterna, Valencia, Spain
Abstract
We compute the two-loop QED corrections to the Altarelli-Parisi (AP) splittingfunctions by using a deconstructive algorithmic Abelianization of the well-known NLOQCD corrections. We present explicit results for the full set of splitting kernelsin a basis that includes the leptonic distribution functions that, starting from thisorder in the QED coupling, couple to the partonic densities. Finally, we perform aphenomenological analysis of the impact of these corrections in the splitting functions.September 2016 ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ h e p - ph ] N ov I. INTRODUCTION
The availability of highly precise experimental data collected in the LHC Run II demands topush forward the accuracy frontier at the theoretical side. In the context of perturbative QCD,higher-order corrections have been computed for a large variety of processes, reaching even N LOaccuracy in some cases. In consequence, some contributions, that were considered sub-leading longtime ago, are starting to compete with QCD effects and might be crucial to compare theoreticalpredictions with experimental data.Besides the substantial work accomplished in the perturbative sector, it is essential to reachthe same level of accuracy on the non-perturbative side, i.e., on the parton distribution functions(PDFs). The calculation of the NNLO corrections to the splitting functions performed in [1–4]and the development of modern parton distribution analysis [5–10] allows to achieve the requiredprecision in QCD.Since α ∼ α , NLO ElectroWeak (EW) corrections compete with NNLO QCD contributions.As a result, an accurate description for many observables requires the inclusion of the correspond-ing EW effects, which might account for a few percent level correction. Recent work has beenperformed on the PDF sector to incorporate the EW effect (the dominant QED terms) in theevolution equations [11–13]. The appearance of the photon and the leptonic densities is the firstmain modification in the evolution of PDFs due to the inclusion of QED corrections. The EWcorrections to PDFs need to be carefully studied for precise predictions at the LHC, as concludedfrom modern analysis performed up to NNLO in QCD and LO in QED. In fact, it was shownthat the corrections induced are non negligible and, moreover, become crucial at higher energies[14–17].Heretofore, the evolution of parton densities was performed using only LO QED kernels. Re-cently, we presented the calculation of the NLO combined QCD–QED contributions (i.e., O ( α α S ))to the evolution kernels [18]. Also, one-loop corrections to double [19–22] and triple [23–25]collinear splitting functions with photons have been computed. With the improvement of accu-racy as the main motivation, we present for the first time, the expressions for the Altarelli-Parisi(AP) splitting functions [26] to O ( α ) that completes the full set of two-loop kernels. Followingthe algorithmic procedure developed in Ref. [18], we make use of the diagram-by-diagram clas-sification available in the original NLO QCD results presented in Refs. [27–29] and, then, wemodified consistently their color structure to account for the gluon-photon replacement. In thiscase, we explicitly concentrate on the QED corrections, without including those arising from Weakbosons, which only become relevant for very extreme kinematical conditions, where their massesare neglected in comparison to other scales involved in the process.The structure of the paper is as follows. In Section II we recall the evolution equations for thedifferent distributions and the corresponding kernels, introducing the notation required to presentour results. Also, we present there the constraints from sum rules that determine the behaviour ofsplitting kernels in the end-point region (i.e. x = 1). In Section III we summarize the algorithmthat we use to obtain the QED corrections to the splitting functions and present the correspondingkernels. Using these formulae, we study the changes introduced in the AP kernels by both O ( α )and O ( α α S ), in Section IV. Finally, conclusions are given in Section V. II. EVOLUTION EQUATIONS AND DEFINITIONS
In the context of combined QCD–QED contributions, it is mandatory to take into accountlepton distributions. In Ref. [18], we computed the O ( α α S ) contributions to the AP kernels andwe showed that leptons decouple from the QCD sector at that accuracy. Thus, in that case, weneglected lepton distributions. Moreover, this simplification remains true for O ( α α n S ) because thequark-lepton mixing appears starting at O ( α ). Therefore, here we follow the path established inRef. [30] and obtain the exact set of evolution equations for the combined QCD–QED model in aproper basis.As usual, the first step consists in writing down the evolution equations for quark, lepton,gluon and photon distributions. These equations are obtained starting from those available inRef.[18] by adding lepton distributions and the corresponding AP kernels, P ij . Using the standarddefinition for the convolution operator, i.e.( f ⊗ g )( x ) = (cid:90) x dyy f (cid:18) xy (cid:19) g ( y ) , (1)and introducing t = ln ( µ ) as the evolution variable, we have dgdt = (cid:88) f P gf ⊗ f + (cid:88) f P g ¯ f ⊗ ¯ f + P gg ⊗ g + P gγ ⊗ γ , (2) dγdt = (cid:88) f P γf ⊗ f + (cid:88) f P γ ¯ f ⊗ ¯ f + P γg ⊗ g + P γγ ⊗ γ , (3) dq i dt = (cid:88) f P q i f ⊗ f + (cid:88) f P q i ¯ f ⊗ ¯ f + P q i g ⊗ g + P q i γ ⊗ γ , (4) dl i dt = (cid:88) f P l i f ⊗ f + (cid:88) f P l i ¯ f ⊗ ¯ f + P l i g ⊗ g + P l i γ ⊗ γ , (5)and similarly for antiparticles by using charge conjugation invariance. Here the sum over fermions f runs over all the active flavours of quarks ( n F ) and leptons ( n L ). In the previous formulae, µ represents the factorization scale.Along this work we will present the expressions for the splitting functions including QCD andQED corrections. Thus, we expand them according to P ij = a S P (1 , ij + a P (0 , ij + a P (2 , ij + a S a P (1 , ij + a P (0 , ij + ... , (6)where the upper indices indicate the (QCD,QED) order of the calculation, while a S ≡ α S π , a ≡ α π , (7)allow to set the standard normalization of the splitting functions.The presence of electromagnetic interactions introduces a charge dependence in the splittingfunctions. Moreover, due to higher-order QED corrections, a mixing among leptons and QCDpartons might take place, which leads to more complicated evolution equations. In fact, in themost general case, Eqs. (2)-(5) constitute a system of 20 ×
20 coupled first-order differentialequations. However, notable simplifications are achieved at each order of the truncated expansionby imposing physical constraints. For instance, at O ( α α n S ), the kernels depend on the electriccharge of the initiating fermions (up or down), such that in general P (1 ,n ) q ∼ e q . As we will showlater, at O ( α ), the charge content of the P ql kernels becomes non trivial due to the exchange ofa pair of photons.The quark splitting functions are decomposed as P q i q k = δ ik P Vqq + P Sqq , (8) P q i ¯ q k = δ ik P Vq ¯ q + P Sq ¯ q , (9) P ± q = P Vqq ± P Vq ¯ q , (10)which act as a definition of P Vqq and P Vq ¯ q , i.e. the non-singlet components. In a completely analogousway, we write P l i l k = δ ik P Vll + P Sll , (11) P l i ¯ l k = δ ik P Vl ¯ l + P Sl ¯ l , (12) P ± l = P Vll ± P Vl ¯ l , (13)for the lepton kernels. For mixed lepton-quark splittings we use P Slq ≡ P lq to simplify the notation.The canonical basis of distributions is given by B c = { u, ¯ u, . . . , t, ¯ t, e, ¯ e, . . . , τ, ¯ τ , g, γ } , (14)when considering the full charged-fermion content of the Standard Model. Working with a reducedamount of fermions simply involves removing them from the previous basis. As explained inRef.[30], B c is not the optimal choice to reduce the mixing among the different parton distributionsin the evolution system. Thus, for n F = 5 active flavours it is more suitable to work with thefollowing set: B = { u v , d v , s v , c v , b v , e v , µ v , τ v , ∆ uc , ∆ ds , ∆ sb , ∆ l , ∆ UD , ∆ l , Σ , Σ l , g, γ } , (15)where q v = q i − ¯ q i , (16) l v = l i − ¯ l i , (17)are the valence distributions, whilst∆ uc = u + ¯ u − c − ¯ c , (18)∆ ds = d + ¯ d − s − ¯ s , (19)∆ sb = s + ¯ s − b − ¯ b , (20)∆ l = e + ¯ e − µ − ¯ µ , (21)∆ UD = u + ¯ u + c + ¯ c − d − ¯ d − s − ¯ s − b − ¯ b , (22)∆ l = e + ¯ e + µ + ¯ µ − τ + ¯ τ ) , (23)Σ = n F (cid:88) i =1 ( q i + ¯ q i ) , (24)Σ l = n L (cid:88) i =1 ( l i + ¯ l i ) , (25)are the remaining combinations (gluons and photons are treated separately). To include the topdistribution in case of a six flavour analysis, it is necessary to introduce the elements { t v , ∆ ct } andto extend the definitions of ∆ UD and Σ.As we mentioned before, QED interactions introduce a classification of the different fermionsaccording to the absolute value of their electromagnetic (EM) charges. Thus, there are threepossible fermionic sectors: up-like quarks ( u and e u = 2 / d and e d = 1 / l with e l = 1). Particles inside each sector are indistinguishable by QCD–QEDinteractions. Also, it is useful to define∆ P SfF ≡ P SfF − P Sf ¯ F , (26)¯ P SfF ≡ P SfF + P Sf ¯ F , (27)where f and F denote the possible fermion subgroups ( u , d or l ). Notice that in the context ofQCD–QED, it might occur that P lq (cid:54) = P ql due to higher-order contributions. However, at O ( α ),they are the same and the equality can be used to achieve further simplifications. Moreover, atthis order, it is verified that ∆ P SfF ≡ , (28)due to charge conjugation invariance.In the most general case, the corresponding QCD–QED combined evolution equations for thedistributions in the optimized basis are given by: dq v i dt = P − q i ⊗ q v i + n F (cid:88) j =1 ∆ P Sq i q j ⊗ q v j + ∆ P Sq i l ⊗ (cid:32) n L (cid:88) j =1 l v j (cid:33) , (29) dl v i dt = P − l ⊗ l v i + n F (cid:88) j =1 ∆ P Slq j ⊗ q v j + ∆ P Sll ⊗ (cid:32) n L (cid:88) j =1 l v j (cid:33) , (30)for valence distributions, d Σ dt = P + u + P + d ⊗ Σ + P + u − P + d ⊗ ∆ UD + n u ¯ P Suu + n d ¯ P Sdd + ( n u + n d ) ¯ P Sud ⊗ Σ+ n u ¯ P Suu − n d ¯ P Sdd − ( n u − n d ) ¯ P Sud ⊗ ∆ UD + (cid:0) n u ¯ P Sul + n d ¯ P Sdl (cid:1) ⊗ Σ l + 2( n u P ug + n d P dg ) ⊗ g + 2( n u P uγ + n d P dγ ) ⊗ γ , (31) d Σ l dt = n L ¯ P Slu + ¯ P Sld ⊗ Σ + n L ¯ P Slu − ¯ P Sld ⊗ ∆ UD + (cid:0) P + l + n L ¯ P Sll (cid:1) ⊗ Σ l + 2 n L ( P lg ⊗ g + P lγ ⊗ γ ) , (32)for the singlets and d { ∆ uc , ∆ ct } dt = P + u ⊗ { ∆ uc , ∆ ct } , (33) d { ∆ ds , ∆ sb } dt = P + d ⊗ { ∆ ds , ∆ sb } , (34) d ∆ l dt = P + l ⊗ ∆ l , (35) d ∆ UD dt = P + u + P + d ⊗ ∆ UD + P + u − P + d ⊗ Σ + n u ¯ P Suu − n d ¯ P Sdd + ( n u − n d ) ¯ P Sud ⊗ Σ+ n u ¯ P Suu + n d ¯ P Sdd − ( n u + n d ) ¯ P Sud ⊗ ∆ UD + (cid:0) n u ¯ P Sul − n d ¯ P Sdl (cid:1) ⊗ Σ l + 2( n u P ug − n d P dg ) ⊗ g + 2( n u P uγ − n d P dγ ) ⊗ γ , (36) d ∆ l dt = P + l ⊗ ∆ l , (37)for the remaining fermionic distributions. Here n u ( n d ) refers to the active number of u ( d ) type-quarks, respectively ( n F = n u + n d ), and n L is the number of leptons under consideration. Itis worth noticing that only ∆ ij and ∆ li decouple from the other distributions. Besides that, ifwe restrict to O ( α α S ), we recover the equations presented in Ref. [18]. At O ( α ), the splittingkernels are charge dependent but ∆ P S ≡
0. Thus, in that case, Eqs. (29) and (30) become dq v i dt = P − q i ⊗ q v i , (38) dl v i dt = P − l ⊗ l v i , (39)and ¯ P Sij ≡ P Sij , P
Slq ≡ P Sql . (40)Moreover, if we only allow QED interactions, all splitting kernels with gluons vanish and the gluondistribution is decoupled from the other ones. A. Constraints from Sum Rules
On one side, QCD–QED interactions preserve the fermion number. In particular, this impliesthat splitting kernels must fulfil (cid:90) dx P − f = 0 , (41)because the factorization scale µ is arbitrary. On the other hand, the arbitrariness of µ also impliesthat the momentum of the proton is conserved during the evolution. Using the parton model, wehave 0 = dPdt = (cid:90) dx x (cid:32) dgdt + dγdt + (cid:88) f (cid:18) dfdt + d ¯ fdt (cid:19)(cid:33) , (42)where the sum is over all the possible fermion flavours (both quarks and leptons). If we expressEq. (42) by using the optimized basis, we impose its validity in each component. The non-trivialconstraints are: • Gluon and photon components, (cid:90) dx x (2 n d P dg + 2 n u P ug + 2 n L P lg + P γg + P gg ) = 0 , (43) (cid:90) dx x (2 n d P dγ + 2 n u P uγ + 2 n L P lγ + P gγ + P γγ ) = 0 ; (44) • ∆ UD component, (cid:90) dx x (cid:18) P + u − P + d n L ¯ P Slu − ¯ P Sld n u ¯ P Suu − n d ¯ P Sdd − ( n u − n d ) ¯ P Sud P gu − P gd P γu − P γd (cid:19) = 0 , (45) • and, finally, the singlet components (Σ and Σ l , respectively), (cid:90) dx x (cid:18) P + u + P + d n L ¯ P Slu + ¯ P Sld n u ¯ P Suu + n d ¯ P Sdd n F ¯ P Sud P gu + P gd P γu + P γd (cid:19) = 0 , (46) (cid:90) dx x (cid:0) n u ¯ P Sul + n d ¯ P Sdl + n L ¯ P Sll + P + l + P gl + P γl (cid:1) = 0 . (47)In the following sections, we will use these equations to provide a strict check of the calculationand, at the same time, fix the value of the splitting kernels in the end-point x = 1. III. SPLITTING KERNELS AT O ( α ) Let’s start by recalling some well-known results for the lowest order splitting functions. At O ( α S ), only QCD partons are involved [26]; thus, P (1 , qq ( x ) = C F (cid:20) x (1 − x ) + + 32 δ (1 − x ) (cid:21) = C F (cid:20) p qq ( x ) + 32 δ (1 − x ) (cid:21) ,P (1 , qg ( x ) = T R (cid:2) x + (1 − x ) (cid:3) = T R p qg ( x ) ,P (1 , gq ( x ) = C F (cid:20) − x ) x (cid:21) = C F p gq ( x ) ,P (1 , gg ( x ) = 2 C A (cid:20) x (1 − x ) + + 1 − xx + x (1 − x ) (cid:21) + β δ (1 − x ) , (48)with β = N C − n F T R and the plus distribution defined as (cid:90) dx f ( x )(1 − x ) + = (cid:90) dx f ( x ) − f (1)1 − x , (49)for any regular test function f . As usual, the normalization of the fundamental representation isset to T R = 1 / C A = N C , C F = N C − N C , (50)are the SU( N C ) group factors. In particular, for QCD ( N C = 3), we have C A = 3 and C F = 4 / p ij ,which will be used to simplify the presentation of higher-order corrections. At O ( α ), splittingprocesses can be described by replacing the color factors in P (1 , ij with the corresponding EMcharges. In this way, we have [30] P (0 , ff ( x ) = e f (cid:20) p qq ( x ) + 32 δ (1 − x ) (cid:21) ,P (0 , fγ ( x ) = e f p qg ( x ) ,P (0 , γf ( x ) = e f p gq ( x ) ,P (0 , γγ ( x ) = − (cid:88) f e f δ (1 − x ) , (51)where f denotes any fermion (quark or lepton) with its corresponding EM charge e f , and (cid:88) f e af = N C n F (cid:88) j =1 e aq j + n L (cid:88) j =1 e al j , (52)is the sum over fermion charges, taking into account that quark-photon interactions are degeneratedue to color degrees of freedom ( N C ). Also, in the case of P (0 , fγ an extra factor of N C has to beincluded whenever the fermion f is a colored quark.In order to obtain the pure two-loop QED corrections P (0 , ij , we follow the ideas depicted inRef. [18]. We start from the results on the two-loop QCD anomalous dimensions in the light-conegauge, originally performed for the non-singlet component by Curci, Furmanski and Petronzio inRef. [27] and extended to the singlet case in Refs. [28, 29] ∗ . Then we take the correspondingAbelian limit, which involves replacing each gluon by a photon [32]. This automatically avoids thepresence of diagrams with non-Abelian vertices (at least in pure QED). The last step consists inreplacing the original color structure with the one obtained after the double replacement g → γ ,and multiplying by the EM charge of the fermions involved in the process.However, as anticipated in Sec. II, lepton distributions enter in the evolution of the systemat O ( α ), which forces us to compute also lepton-quark and lepton-photon kernels at this order.The procedure is completely analogous to the one described before. ∗ For a complete review of the previous developments in the computation of higher-order corrections to the splittingkernels and the anomalous dimension, see Ref. [31] and the references therein. n F . Once we replacegluons with photons, virtual leptons are also allowed inside the loop. Both for leptons and quarks,the QED coupling is proportional to their EM charges. In consequence, the replacement n F → (cid:88) f e f , (53)has to be implemented in all the contributions arising from quark loops in the pure QCD kernels.Another subtle point that we must carefully treat is the presence of massive EW bosons. Aswe mentioned before, we neglect their contribution in this work. This is due to the fact that theirmass is kept strictly non-vanishing, thus acting as an IR-regulator. In other terms, IR-singulardiagrams for processes involving heavy EW bosons can always be treated by making use of QCD–QED splitting functions and factorizing the massive particle into the hard scattering subprocess.So, let’s present the explicit results. In first place, kernels involving gluons vanish at this order;hence, P (0 , fg = 0 , P (0 , gf = 0 , P (0 , γg = 0 ,P (0 , gγ = 0 , P (0 , gg = 0 . (54)Then, we consider those kernels which involve quarks and photons, P (0 , qγ = C A e q (cid:26) − x − (1 − x )ln ( x ) − (1 − x )ln ( x ) + 4ln (1 − x )+ p qg ( x ) (cid:20) (cid:18) − xx (cid:19) − (cid:18) − xx (cid:19) − π (cid:21)(cid:27) , (55) P (0 , γq = e q (cid:20) − (cid:0) − x ) + ln (1 − x ) (cid:1) p gq ( x ) + (cid:18) x (cid:19) ln ( x ) − (cid:16) − x (cid:17) ln ( x ) − x ln (1 − x ) − x − (cid:21) − e q (cid:32)(cid:88) f e f (cid:33) (cid:20) x + p gq ( x ) (cid:18)
209 + 43 ln (1 − x ) (cid:19)(cid:21) , (56) P V (0 , qq = − e q (cid:20)(cid:18) x ) ln (1 − x ) + 32 ln ( x ) (cid:19) p qq ( x ) + 3 + 7 x x )+ 1 + x ( x ) + 5(1 − x ) + (cid:18) π − − ζ (cid:19) δ (1 − x ) (cid:21) − e q (cid:32)(cid:88) f e f (cid:33) (cid:20)
43 (1 − x ) + p qq ( x ) (cid:18)
23 ln ( x ) + 109 (cid:19) + (cid:18) π (cid:19) δ (1 − x ) (cid:21) , (57) P V (0 , q ¯ q = e q [4(1 − x ) + 2(1 + x )ln ( x ) + 2 p qq ( − x ) S ( x )] , (58) P S (0 , qQ = P S (0 , q ¯ Q = C A e q e Q p s ( x ) , (59)1where { q, Q } denote different quark flavours and we defined the function p s ( x ) = 209 x − x − x + (cid:18) x + 83 x (cid:19) ln ( x ) − (1 + x )ln ( x ) , (60)which appears in all the higher-order corrections to the singlet components. The function S ( x )is given by [28, 33] S ( x ) = (cid:90) xx x dzz ln (cid:18) − zz (cid:19) = ln ( x )2 − ζ − ( − x ) − x ) ln (1 + x ) . In these formulae, ζ n is the Riemann zeta function, which verifies ζ = π / ζ ≈ . P (0 , lγ = e l C A e q P (0 , qγ , (61) P (0 , γl = e l (cid:20) − (3ln (1 − x ) + ln (1 − x )) p gq ( x ) + (cid:18) x (cid:19) ln ( x ) − (cid:16) − x (cid:17) ln ( x ) − x ln (1 − x ) − x − (cid:21) − e l (cid:32)(cid:88) f e f (cid:33) (cid:20) x + p gq ( x ) (cid:18)
209 + 43 ln (1 − x ) (cid:19)(cid:21) , (62) P V (0 , ll = − e l (cid:20)(cid:18) x ) ln (1 − x ) + 32 ln ( x ) (cid:19) p qq ( x ) + 3 + 7 x x )+ 1 + x ( x ) + 5(1 − x ) + (cid:18) π − − ζ (cid:19) δ (1 − x ) (cid:21) − e l (cid:32)(cid:88) f e f (cid:33) (cid:20)
43 (1 − x ) + p qq ( x ) (cid:18)
23 ln ( x ) + 109 (cid:19) + (cid:18) π (cid:19) δ (1 − x ) (cid:21) , (63) P V (0 , l ¯ l = e l e q P V (0 , q ¯ q , (64) P S (0 , lL = P S (0 , l ¯ L = e l e L p s ( x ) . (65)Mixed quark-lepton evolution kernels are given by P S (0 , lq = P S (0 , l ¯ q = e l e q p s ( x ) , (66) P S (0 , ql = P S (0 , q ¯ l = C A e l e q p s ( x ) , (67)and we notice that they share the same functional dependence, with the exception of the globalnormalization (influenced by the average over the quantum numbers of the initial particle). Finally,for the photon splitting kernel we have P (0 , γγ = (cid:32)(cid:88) f e f (cid:33) (cid:20) −
16 + 8 x + 203 x + 43 x − (6 + 10 x )ln ( x ) − x )ln ( x ) − δ (1 − x ) (cid:21) , (68)2that, at this order, includes both real and virtual corrections, in contrast with O ( α α S ) contribu-tions [18]. IV. PHENOMENOLOGICAL IMPACT OF QED CORRECTIONS
According to the expansion shown in Eq. (6), the weight of higher-order corrections is sup-pressed by powers of α and α S . In fact, working at µ = M Z , we have a = 1 . × − and a S = 1 . × − . Thus, we anticipate that QED contributions to the AP kernels are smallcompared to pure QCD kernels. However, it might still happen that their effects become magni-fied due to the specific shape of the different PDFs. For this reason, we perform a study of theQCD and QED contributions to the splitting kernels to anticipate the possible consequences inthe evolution of the PDFs. � �� ( ��� ) � �� ( ��� ) �� - � ( � �� ( ��� ) ∼ � �� ( ��� ) ) � �� ( ��� ) � �� ( ��� ) ��� ��� ��� ��� ��� ��� - ���� - ���������������� � � �� ( � �� � � � � ) ( % ) FIG. 1. K factors for the qq splitting functions (%). We separate among u (solid lines) and d (dashedlines) quarks to study the EM charge effects, and also among the different perturbative orders. Noticethat α terms are dominant (they are suppressed by a factor 10 in this plot) and they exhibit almostthe same behaviour for both u and d quarks. Let’s start with the analysis of the pure quark kernels P qq . We define the ratio K ( i,j ) ab = a i S a j P ( i,j ) ab ( x ) P LO ab ( x ) , (69)3 � �� ( ��� ) �� - � ( � �� ( ��� ) ∼ � �� ( ��� ) ) � �� ( ��� ) ��� ��� ��� ��� ��� ��� - ��� - ��������������� � � �� ( � �� � � � � ) ( % ) � �� ( ��� ) �� - � ( � �� ( ��� ) ∼ � �� ( ��� ) ) � �� ( ��� ) ��� ��� ��� ��� ��� ��� - ��� - ��� - ��� - ��������� � � �� ( � �� � � � � ) ( % ) FIG. 2. K factors for the qg (left) and gq (right) splitting functions (%). We include O ( α ) and O ( α α S )contributions, and we also distinguish according to the EM charge of the involved quark; solid lines areused for u quarks, whilst d quarks are displayed with dashed lines. The O ( α ) term is dominant, and wesuppress it by a factor 10 in order to improve the graphical presentation. where P LO ab ( x ) is the contribution to the evolution kernel at the lowest order in α and α S . Noticethat P LO ab = a S P (1 , ab + a P (0 , ab , (70)i.e. P LO ab is not necessarily the lowest order contribution in only one of the couplings. We extractthe O ( α ) contributions from Refs. [27, 28] and the O ( α α S ) ones from Ref. [18]; the resultingplot is given in Fig. 1. We distinguish there among quarks belonging to the up and down sector,respectively. As expected, deviations arising from QED corrections for u quarks turn out to bebigger than those for d quarks, since they are proportional to e q . P (2 , qq terms are dominant inboth cases; they represent a O (10%) correction, at least. However, the other corrections are ofthe same order of magnitude; approximately ± . x →
0, there is a positive enhancement of P (1 , qq for x ≈ . − .
15 and a negative one for x ≈ . − . P qg and P gq (Fig. 2). Pure QCD contributions tothe splittings involving gluons are dominant against mixed QCD–QED ones; in any case, thesecontributions become increasingly relevant in the low x region. Since gluon PDFs are magnifiedin that region, we expect a non-negligible effect in the evolution. The small EM charge separationobserved in the P qg kernel for α correction originates from the normalization of the K factor via4 � � γ ( ��� ) �� - � ( � � γ ( ��� ) ∼ � � γ ( ��� ) ) � � γ ( ��� ) ��� ��� ��� ��� ��� ��� - � - ��� � � � γ ( � �� � � � � ) ( % ) �� - � ( � γ � ( ��� ) ∼ � γ � ( ��� ) ) � γ � ( ��� ) ∼ � γ � ( ��� ) ��� ��� ��� ��� ��� ��� - � - ��� � � γ � ( � �� � � � � ) ( % ) FIG. 3. K factors for the qγ (left) and γq (right) splitting functions (%). α and α α S terms are included,with the last one being the dominant contribution. The EM charge distinction is enhanced in P ( i,j ) qγ splitting, around x ≈ .
65. Mixed QCD–QED contributions are suppresed by a factor 10 to improve thevisibility in the plot.
Eq. (70).On the other hand, kernels involving a photon receive larger QED corrections, as observedfor P qγ and P γq (Fig. 3). Mixed O ( α α S ) QCD–QED contributions can reach the 20% level for P qγ , while the two-loop QED terms modify the photon initiated kernel by up to 2% at small x .Furthermore, kernels involving leptons provide a non-trivial modification of QCD PDFs at O ( α ).In Fig. 4 we plot the K factors for P ll , P lγ and P γl , respectively. Again, corrections reach the2% level for the photon initiated kernels that can produce non-negligible effects to the photondistribution in a global analysis. V. CONCLUSIONS
In this paper, we have presented for the first time explicit expressions for the Altarelli-Parisisplitting kernels to O ( α ), completing the computation of the two-loop kernels needed to studythe evolution of parton distributions to the precision achievable at the LHC. The full set of kernelsincludes those related to both photon and leptonic densities, the latest being allowed to mix inthe evolution with parton distributions, mixing that starts at two-loops in QED.We have obtained the corresponding kernels from the well-known NLO QCD corrections to the5 � �� ( ��� ) � � γ ( ��� ) � γ � ( ��� ) ��� ��� ��� ��� ��� ��� - ��� - ��� - ��� - ������������ � � �� ( � � � ) ( % ) FIG. 4. K factors for the O ( α ) corrections to the splitting functions involving leptons. In the range0 . ≤ x ≤ .
95, these contributions represent less than 2%; they become more sizable near x = 0 (for K (0 , lγ and K (0 , ll ) and x = 1 (for K (0 , lγ and K (0 , γl ). splitting functions, after carefully applying a well-defined algorithm to take the Abelian limit ofthe pure QCD expressions.Finally, we have performed a phenomenological analysis to study the implications of thesecorrections in the splitting functions. We find that two-loop corrections are negligible for the purequark kernels, but become sizable for P qg and P gq at small x values (see Fig. 5). The effect ofQED corrections turns out to generate O (2%) corrections for the splitting functions initiated byphotons, which will alter the shape and size of the photon and leptonic distribution functions ina global analysis. Acknowledgments
This work is partially supported by CONICET, ANPCyT, by the Spanish Government andEU ERDF funds (grants FPA2014-53631-C2-1-P and SEV-2014-0398) and by GV (PROME-6 K ug ( ) - ( K ug ( ) ∼ K dg ( ) ) K dg ( ) - - x K qg ( Q CD , Q E D ) ( % ) - K gu ( ) - - K gu ( ) - - K gd ( ) - K gd ( ) - - K gq ( Q CD , Q E D ) ( % ) - K uu ( ) K uu ( ) - ( K uu ( ) ∼ K dd ( ) )- K dd ( ) K dd ( ) - - K qq ( Q CD , Q E D ) ( % ) K u γ ( ) - ( K u γ ( ) ∼ K d γ ( ) ) K d γ ( ) - - K q γ ( Q CD , Q E D ) ( % ) - - ( K γ u ( ) ∼ K γ d ( ) )- K γ u ( ) ∼ - K γ d ( ) - - - K γ q ( Q CD , Q E D ) ( % ) K ll ( ) K l γ ( ) - K γ l ( ) - - x K ab ( , ) ( % ) FIG. 5. K factors in the low- x region. We plot the O ( α ), O ( α α S ) and O ( α ) corrections to the mixedQCD-QED splitting functions . In the first line, we considered K qg (left), K gq (center) and K qq (right).In the second one, we show K qγ (left), K γq (center) and the O ( α ) corrections to the kernels involvingleptons (right). TEUII/2013/007). [1] S. Moch, J. A. M. Vermaseren and A. Vogt, Nucl. Phys. B (2002) 413 [hep-ph/0110331].[2] S. Moch, J. A. M. Vermaseren and A. Vogt, Nucl. Phys. B (2004) 101 [hep-ph/0403192].[3] A. Vogt, S. Moch and J. A. M. Vermaseren, Nucl. Phys. B (2004) 129 [hep-ph/0404111].[4] A. Vogt, S. Moch and J. Vermaseren, Acta Phys. Polon. B (2006) 683 [hep-ph/0511112].[5] S. Alekhin, J. Blumlein and S. Moch, Phys. Rev. D (2014) 5, 054028 [arXiv:1310.3059 [hep-ph]].[6] S. Dulat et al. , Phys. Rev. D (2016) no.3, 033006 [arXiv:1506.07443 [hep-ph]].[7] P. Jimenez-Delgado and E. Reya, Phys. Rev. D (2014) 7, 074049 [arXiv:1403.1852 [hep-ph]].[8] H. Abramowicz et al. [H1 and ZEUS Collaborations], Eur. Phys. J. C (2015) no.12, 580[arXiv:1506.06042 [hep-ex]].[9] L. A. Harland-Lang, A. D. Martin, P. Motylinski and R. S. Thorne, Eur. Phys. J. C (2015) 5,204 [arXiv:1412.3989 [hep-ph]]. [10] R. D. Ball et al. [NNPDF Collaboration], JHEP (2015) 040 [arXiv:1410.8849 [hep-ph]].[11] A. D. Martin, R. G. Roberts, W. J. Stirling and R. S. Thorne, Eur. Phys. J. C (2005) 155[hep-ph/0411040].[12] R. D. Ball et al. [NNPDF Collaboration], Nucl. Phys. B (2013) 290 [arXiv:1308.0598 [hep-ph]].[13] V. Bertone, S. Carrazza and J. Rojo, Comput. Phys. Commun. (2014) 1647 [arXiv:1310.1394[hep-ph]].[14] C. Schmidt, J. Pumplin, D. Stump and C.-P. Yuan, arXiv:1509.02905 [hep-ph].[15] R. Sadykov, arXiv:1401.1133 [hep-ph].[16] S. Carrazza, arXiv:1509.00209 [hep-ph].[17] V. Bertone, S. Carrazza, D. Pagani and M. Zaro, JHEP (2015) 194 [arXiv:1508.07002 [hep-ph]].[18] D. de Florian, G. F. R. Sborlini and G. Rodrigo, Eur. Phys. J. C (2016) no.5, 282[arXiv:1512.00612 [hep-ph]].[19] M. Gluck and E. Reya, Phys. Rev. D (1983) 2749.[20] M. Gluck, E. Reya and A. Vogt, Phys. Rev. D (1992) 3986.[21] M. Fontannaz and E. Pilon, Phys. Rev. D (1992) 382.[22] G. F. R. Sborlini, D. de Florian and G. Rodrigo, JHEP (2014) 018 [arXiv:1310.6841 [hep-ph]].[23] G. F. R. Sborlini, arXiv:1410.1680 [hep-ph].[24] G. F. R. Sborlini, D. de Florian and G. Rodrigo, JHEP (2014) 161 [arXiv:1408.4821 [hep-ph]].[25] G. F. R. Sborlini, D. de Florian and G. Rodrigo, JHEP (2015) 021 [arXiv:1409.6137 [hep-ph]].[26] G. Altarelli and G. Parisi, Nucl. Phys. B (1977) 298.[27] G. Curci, W. Furmanski and R. Petronzio, Nucl. Phys. B (1980) 27.[28] W. Furmanski and R. Petronzio, Phys. Lett. B (1980) 437.[29] R. K. Ellis and W. Vogelsang, hep-ph/9602356.[30] M. Roth and S. Weinzierl, Phys. Lett. B (2004) 190 [hep-ph/0403200].[31] J. Blumlein, Prog. Part. Nucl. Phys. (2013) 28 [arXiv:1208.6087 [hep-ph]].[32] A. L. Kataev, Phys. Lett. B (1992) 209.[33] J. Bl¨umlein and S. Kurth, Phys. Rev. D60