1 Introduction

LTH 900
December 2010

Generalized double-logarithmic large-x resummation

[1mm] in inclusive deep-inelastic scattering

A.A. Almasy, G. Soar and A. Vogt

Department of Mathematical Sciences, University of Liverpool

Liverpool L69 3BX, United Kingdom

[2.5cm] Abstract

We present all-order results for the highest three large- logarithms of the splitting functions and and of the coefficient functions , and for structure functions in Higgs- and gauge-boson exchange DIS in massless perturbative QCD. The corresponding coefficients have been derived by studying the unfactorized partonic structure functions in dimensional regularization independently in terms of their iterative structure and in terms of the constraints imposed by the functional forms of the real- and virtual-emission contributions together with their Kinoshita–Lee-Nauenberg cancellations required by the mass-factorization theorem. The numerical resummation corrections are small for the splitting functions, but partly very large for the coefficient functions. The highest two (three for ) logarithms can be resummed in a closed form in terms of new special functions recently introduced in the context of the resummation of the leading logarithms.

1 Introduction

The splitting functions governing the scale dependence of the parton densities of hadrons and the coefficient functions for inclusive deep-inelastic scattering (DIS) are benchmark quantities of perturbative QCD [1]. At this point these are the only quantities depending on a dimensionless scaling variable, the parton momentum fraction and the Bjorken variable (both usually denoted by ), for which third-order corrections in the strong coupling constant are fully known. The corresponding three-loop calculations started with DIS sum rules [2, 3, 4] and proceeded via a series of low integer moments of the splitting functions and the coefficient functions for the most important structure functions [5, 6, 7] to the corresponding complete calculations of Refs. [8, 9, 10, 11, 12].

Such higher-order calculations do not only improve the numerical accuracy of the predictions of perturbative QCD but also help to uncover general structures, for example in the soft-gluon limit . Writing the expansion of the splitting functions in the  scheme as

 Pik(x,αs)=∞∑n=0an+1sP(n)ik(x)withas≡αs4π, (1.1)

the diagonal (quark-quark and gluon-gluon) splitting functions take the form

 P(n−1)kk(x)=A(n)k(1−x)−1++B(n)kδ(1−x)+C(n)kln(1−x)+O(1) (1.2)

with the -loop quark and gluon cusp anomalous dimensions related by at , where and are the usual SU(N) colour factors with and in QCD [13]. It was not known before Ref. [14], inspired partly by observations made for the three-loop results in Refs.[8, 9], that the third term in Eq. (1.2) is linear in at all orders , and that its coefficients (with ) are simple functions of lower-order cusp anomalous dimensions.

The form of the off-diagonal (quark-gluon and gluon-quark) splitting functions, on the other hand, is not stable under higher-order corrections but shows a double-logarithmic enhancement,

 P(n)i≠k(x)=2n−1∑ℓ=0D(n,ℓ)ikln2n−ℓ(1−x)+O(1). (1.3)

The terms with form the leading-logarithmic (LL) large- approximation, those with the next-to-leading-logarithmic (NLL) approximation etc. Recently an all-order resummation of the former contributions to Eq. (1.3) has been presented [15]. A main purpose of this article is to extend those results to the next-to-next-to-leading logarithmic (NNLL) terms.

The dominant large- contributions to the quark coefficient functions for gauge-boson exchange structure functions in DIS such as and (and the gluon coefficient function for Higgs-exchange structure function in the heavy-top limit [16, 17]) also show a double logarithmic enhancement. These terms are resummed to all orders by the soft-gluon exponentiation [18, 19, 20, 21, 22, 23, 24, 25] which presently fixes the coefficients of the highest six logarithms analytically, and the seventh term for all numerical purposes since the effect to the presently unknown four-loop cusp anomalous dimension can be neglected in this context [25, 26, 27]. A resummation of the highest three logarithms has been inferred in Ref. [28] from the properties of the corresponding flavour non-singlet physical evolution kernels. While those subleading contributions to the ‘diagonal’ coefficient functions are not the main topic of this article, we will be able to verify the DIS part of those results and fix the only missing coefficient for the fourth (NL) logarithms.

The ‘off-diagonal’ coefficient functions, such as

 Ca,k(x,αs)=∞∑n=1ansc(n)a,k(x)fora,k=2,g   or   ϕ,q, (1.4)

receive a double-logarithmic higher-order enhancement as as well,

 c(n)a,k(x)=2n−2∑ℓ=0D(n,ℓ)a,kln2n−1−ℓ(1−x)+O(1). (1.5)

Also here the LL coefficients have been determined at all orders in Ref. [15], and also here we will extend those results by deriving the corresponding NLL and NNLL results.

Finally we will also address the coefficient functions and for the longitudinal structure function . These quantities have a perturbative expansion of the form (1.4), and are given by

 (1.6)

at large , i.e., they are suppressed by one power of and with respect to their counterparts for the structure function . The coefficients for have been obtained already in Ref. [29], again from physical-kernel considerations. For the gluon coefficient function, however, only the LL coefficients have been determined completely in that article. Below we will verify those results and extend also the resummation of to the NNLL terms.

The remainder of this article is organized as follows: In Section 2 we derive all-order expressions for the (dimensionally regulated) Mellin- space transition functions as far as required for the mass-factorization of the structure functions at the level of the dominant and sub-dominant contributions discussed above. The NNLL resummations of the unfactorized partonic stucture functions are then derived in Section 3 for the cases (1.4), where to NLL accuract we employ two different methods to drive the same results, and in Section 4 for . The partly rather lengthy results of these three sections are then combined, and in Sections 5 and 6 we present and discuss the respective resummed expressions for the moments of the off-diagonal splitting functions – which receive rather small resummation corrections at relevant values of – and the coefficient functions (1.4) and for which these corrections are (very) large. We summarize our findings in Section 7 where we also present a brief outlook to future extensions and applications of some of the results. Closed expressions have not been found so far for the third logarithms in Eqs. (1.3) and (1.5). Numerical and symbolic tables of NNLL coefficients to high orders are therefore finally presented in Appendix A for the splitting functions and Appendix B for the coefficient functions.

2 Large-x/large-N mass factorization to all orders

The main part of our calculations is performed after transformation to Mellin- space,

 f(N)=∫10dxxN−1f(x) or f(N)=∫10dx(xN−1−1)f(x)+, (2.1)

where the ubiquitous -space Mellin convolutions are reduced to simple products. To the accuracy required below, the relations between the large- logarithms and their moment-space counterparts are given by

 (−1)k(lnk−1(1−x)1−x)+ \lx@stackrelM= 1k([S1−(N)]k+12k(k−1)ζ2[S1−(N)]k−2 {     }+16k(k−1)(k−2)ζ3[S1−(N)]k−3+O([S1−(N)]k−4)), (−1)klnk(1−x) \lx@stackrelM= 1N(lnk˜N+12k(k−1)ζ2lnk−2˜N+16k(k−1)(k−2)ζ3lnk−3˜N (2.2) {     }+O(lnk−4˜N)), (−1)k(1−x)lnk(1−x) \lx@stackrelM= 1N2(lnk˜N−klnk−1˜N+12k(k−1)ζ2lnk−2˜N+O(lnk−3˜N))

with and , i.e., with . Here indicates that the right-hand-side is the Mellin transform (2.1) of the previous expression.

The primary objects of our resummations are the dimensionally regulated unfactorized partonic structure functions or forward Compton amplitudes for the combinations of and of Eqs. (1.4) and (1.6). For brevity suppressing all functional dependences on , and the dimensional offset with , these quantities can be factorized as

 Ta,k=˜Ca,iZik. (2.3)

Here the process-dependent -dimensional coefficient functions (Wilson coefficients) include contributions with all non-negative powers of . The universal transition functions (renormalization constants) collect all negative powers of and are related to the splitting functions in Eq. (1.1) (or the anomalous dimensions ) by

 −γ=P=dZdlnQ2Z−1. (2.4)

Here and below we identify, as already in the introduction, the renormalization and factorization scale with the physical hard scale without loss of information. Using the -dimensional evolution of the coupling,

 dasdlnQ2=−ϵas+β(as) (2.5)

where denotes the usual four-dimensional beta function of QCD, with , Eq. (2.4) can be solved for order by order in .

The general elements of become extremely lengthy at very high powers of . Here, however, we are interested only in the LL, NLL and NNLL contributions at order for and and for and (required for Eq. (2.3) also in the off-diagonal cases). Consequently there can be at most one off-diagonal factor per term. Moreover in Eq. (1.1) can enter only (once) at NNLL accuracy, and higher-order coefficients in Eq. (1.1) are not at all relevant at this level. Finally in Eq. (2.5) and only contribute from the NLL and NNLL terms, respectively, and would enter only at the next logarithmic accuracy. All this can be easily read off already from the well-known third-order expression for ,

 Z = 1+as1ϵγ(0)+a2s(12ϵ2(γ(0)−β0)γ(0)+12ϵγ(1)) (2.6) {     }{     }+16ϵ2[(γ(0)−2β0)γ(1)+(γ(1)−β1)2γ(0)]+16ϵγ(2))+…

together with Eqs. (1.2), (1.3) and (2) above.

The terms that do contribute to the off-diagonal entries of at the present accuracy can be grouped as follows:

 Z(k)ab=Z(k)ab∣∣0+Z(k)ab∣∣β0+Z(k)ab∣∣β20+Z(k)ab∣∣γ(1)γ(1)+Z(k)ab∣∣γ(1)γ(ℓ). (2.7)

The first term on the right-hand-side collects all contributions with at most one higher-order anomalous dimension but no contribution from the beta function. It starts at and reads

 Z(k)ab∣∣0 = 1k!ϵk{k−1∑i=0ϵik−1−i∑j=0(j+i)!j!(γ(0)aa)k−1−i−jγ(i)ab(γ(0)bb)j (2.8) {     }{     }+ϵk−3∑j=012(k−j−2)(k−j−1)(γ(0)aa)jγ(0)abγ(1)bb(γ(0)bb)k−j−3 {     }{     }+ϵk−3∑j=012(k−j−2)(k+j+1)(γ(0)aa)k−j−3γ(1)aaγ(0)ab(γ(0)bb)j}.

The first line includes, for , the LL expression used in Ref. [15]. The contributions linear in contribute from and NLL accuracy and are given by

 Z(k)ab∣∣β0 = −β021k!ϵkk−2∑i=0ϵik−2−i∑j=0(i+j)!j![k(k−1)−i(i+j+1)](γ(0)aa)k−2−i−jγ(i)ab(γ(0)bb)j,

while the corresponding NNLL term in Eq. (2.7) for is

 Z(k)ab∣∣β20 = β20241k!ϵkk−3∑i=1ϵik−3−i∑j=0(j+i)!j![k(k−1)(k−2)(3\*k−1)−6i(i+j+1)k(k−1) (2.10) {     }{     }{     }+i(3i+1)(i+j+1)(i+j+2)](γ(0)aa)k−3−i−jγ(i)ab(γ(0)bb)j.

Finally we distinguish NNLL contributions with , where one of the factors is the off-diagonal entry, and contributions with , where the latter has to be off-diagonal at the present level of accuracy (recall that every term includes one off-diagonal anomalous dimension ). The former terms contribute from order and are given by

 Z(k)ab∣∣γ(1)γ(1) = 1k!ϵ−k+2k−4∑i=0k−4−i∑j=0[(k−i−2)2−1−j(k−i−1)]

and the corresponding final contribution to Eq. (2.7) at reads

 Z(k)ab∣∣γ(1)γ(ℓ) = 1k!ϵkk−3∑ℓ=2ϵℓ+1{k−ℓ−3∑i=0k−3−i−ℓ∑j=0(k−i−1)(k−i−j−3)!(k−i−j−ℓ−3)! (2.12) {     }{     }{     }{     }{     }{     }⋅(γ(0)aa)iγ(1)aa(γ(0)aa)jγ(ℓ)ab(γ(0)bb)k−i−j−ℓ−3 {     }{     }{     }{     }+k−ℓ−3∑i=0k−3−i−ℓ∑j=0(k−1−i)!(k−ℓ−i−1)!(k−ℓ−i−j−2) {     }{     }{     }{     }{     }{     }⋅(γ(0)aa)iγ(ℓ)ab(γ(0)bb)jγ(1)bb(γ(0)bb)k−i−j−ℓ−3}.

The coefficients Eqs. (2.8) – (2.12) have been inferred by analyzing the respective first five to seven non-trivial orders and then verified to ‘all’ orders using, as for a large part of our symbolic manipulations, the programs Form and TForm [30, 31].

The corresponding result for the diagonal entries of the Z-matrices are much simpler due to Eq. (1.2). Including also terms which contribute to the next-to-next-to-next-to-leading logarithmic (NLL) terms suppressed by one power of , the coefficients at order are given by

 Z(k)aa = ϵ−kk!(γ(0)aa)k+ϵ−k+12(k−2)!(γ(0)aa)k−2γ(1)aa−β02ϵ−k(k−2)!(γ(0)aa)k−1 (2.13) +β2024ϵ−k(k−3)!(3k−1)(γ(0)aa)k−2+ϵ−k+23(k−3)!(γ(0)aa)k−3γ(2)aa −β012ϵ−k+1(k−3)!(3k−5)(γ(0)aa)k−3γ(1)aa−β13ϵ−k+1(k−3)!(γ(0)aa)k−2 +ϵ−k+28(k−4)!(γ(0)aa)k−4(γ(1)aa)2+β2048ϵ−k+1(k−4)!(k−1)(3k−8)(γ(0)aa)k−4γ(1)aa −β3048ϵ−k(k−4)!k(k−1)(γ(0)aa)k−3.

Only the first four terms contribute to the NNLL expression entering the off-diagonal mass factorization (2.3). Note that, unlike at order , and are NLL and NLL quantities at order due to and in Eq. (1.2).

The unfactorized structure functions (2.3) are given by these results multiplied by

 ˜Ca,i=δaγδiq+δaϕδig+∞∑n=1ans∞∑k=0ϵkc(n,k)a,i. (2.14)

The index generically represents the gauge-boson exchange structure functions (except for ). are the -th order coefficient functions in Eq. (1.4) for any combination of and . The quantities – usually denoted by , etc in fixed-order calculations – are enhanced by factors with respect to those four-dimensional coefficient functions.

The calculation of to order and (for : ) provides the NLO (leading-order, next-to-leading-order etc) renormalization-group improved fixed-order approximation to the structure functions . It is obvious from Eqs. (2.6) – (2.14) that a full NLO result completely fixes the highest powers of to all orders in . An all-order resummation of the splitting functions and coefficient functions requires, at the logarithmic accuracy under consideration, an extension of these results to all powers of . The flavour-singlet structure functions considered here are fully known at NLO from Refs. [8, 9, 10, 16, 17] and the earlier coefficient-function calculations of Refs.[32, 33, 34, 35]. Hence a double-logarithmic resummation based on these results can be expected to predict up to the highest three logarithms at all higher orders, including the corresponding contributions to the three-loop coefficient functions for and exactly computed in Refs. [11, 17] and the large- predictions of the four-loop splitting functions in the latter article.

3 Resummation of the unfactorized expressions for F2 and Fϕ

In this section we derive all-order expressions for the leading contributions to the off-diagonal amplitudes or unfactorized structure functions and at NNLL accuracy. We will apply two approaches: first an iteration of amplitudes generalizing the leading logarithmic results of Ref. [15], and then an apparently new and more rigorous treatment which makes use of only the -dimensional structure of the unfactorized structure functions in the large- limit and the KLN cancellations [36, 37] between its real- and virtual-emission contributions.

Both calculations require the corresponding expressions for the parts of , , and which can be determined from the diagonal amplitudes and in the limit governed by the soft-gluon exponentiation. These quantities are given by

 Ta,k=exp(^as˜T(1)a,k+^a2s˜T(2)a,k+^a3s˜T(3)a,k+…) (3.1)

with

 ˜T(n)a,k=∞∑ℓ=−n−1ϵℓ(R(n,ℓ)a,kexp(nϵlnN)−V(n,ℓ)a,k). (3.2)

For the quark case the coefficients entering the highest four logarithms at all orders in and  read

 R(1,−2)2,q=4,R(1,−1)2,q=3,R(1,0)2,q=17−4ζ2,R(1,1)2,q=14−3ζ2−8ζ3, V(1,−2)2,q=4,V(1,−1)2,q=6,V(1,0)2,q=16+2ζ2,V(1,1)2,q=32−3ζ2−283ζ3, (3.3)
 R(2,−3)2,q = V(2,−3)2,q=β0, R(2,−2)2,q = (43−2ζ2)CA+196β0,V(2,−2)2,q=R(2,−2)2,q−32β0, R(2,−1)2,q = V(2,−1)2,q = (3.4) R(3,−4)2,q = V(3,−4)2,q=49β20,V(3,−3)2,q=R(3,−3)2,q+β20, R(3,−3)2,q = −149C2A−229CFCA+23CFβ0+(6227−169ζ2)CAβ0+6727β20 (3.5)

and

 R(4,−5)2,q=V(4,−5)2,q=14β30, (3.6)

where we have suppressed an obvious overall factor of and expressed the dependence of the number of effectively massless flavours in terms of . To NLL accuracy these results are converted to the renormalized coupling used elsewhere in this article via

 ^as=as−β0ϵa2s+(β20ϵ2−β12ϵ)a3s+β30ϵ3a4s+…. (3.7)

The corresponding gluonic coefficients are required only to NNLL accuracy here and read

 R(1,−2)ϕ,g = 4CA,R(1,−1)ϕ,g=β0,R(1,0)ϕ,g=(43−4ζ2)CA+53β0, V(1,−2)ϕ,g = 4CA,V(1,−1)ϕ,g=0,V(1,0)ϕ,g=2ζ2CA, (3.8) R(2,−3)ϕ,g = V(2,−3)ϕ,g=CAβ0, R(2,−2)ϕ,g = (43−2ζ2)C2A+53CAβ0+12β20,V(2,−2)ϕ,g=R(2,−2)ϕ,g−12β20, (3.9) R(3,−4)ϕ,g = V(3,−4)ϕ,g=49CAβ20. (3.10)

After the transformation to the renormalized coupling, Eqs. (3.1) and (3.2) for need to be multiplied by the renormalization constant of [38, 39],

 1−2β0ϵ−1a2s+3β20ϵ−2a3s+…. (3.11)

Note that there is no new physics content in Eqs. (3.1) – (3.10) which represent the Mellin transforms of the decomposition of and used in Refs. [40, 41] and its obvious higher- order generalizations, cf. Ref. [42], recast in an exponential all-order form. Recall also, from the same references, that of the coefficients , relevant for the -th logarithm at order only one combination is not fixed by lower-order information, and that the same holds for their virtual-correction counterparts , .

We are now ready to address the resummation of the off-diagonal amplitudes. Using the particularly simply colour structure of the identical leading-logarithmic contributions to and , their LL resummation has been inferred in Ref. [15] to which the reader is referred for a detailed discussion. The result can be written as

 (3.12)

where, of course, only the respective first terms of

 T(1)ϕ,q = −CFNϵexp(ϵlnN)(2−ϵ−(3−2ζ2)ϵ2+…), T(1)2,g = −nfNϵexp(ϵlnN)(2−2ϵ−(6−2ζ2)ϵ2+…) (3.13)

are required. The corresponding expressions for and can be read off from Eqs. (3.1), (3.2), (3) and (3) above.

Eq. (3.12) can be generalized to next-to-leading logarithmic accuracy, where running-coupling effects enter for the first time, using the natural ansatz

 T(n)a,k\raisebox−1.991693pt$\lx@stackrelNL=$1nT(1)a,k{n−1∑i=0fa,k(n,i,ϵ)T(i)ϕ,gT(n−i−1)2,q−β0ϵn−2∑i=0ga,k(n,i)T(i)ϕ,gT(n−i−2)2,q} (3.14)

where are linear functions of . It is not a priori clear that such a simple ansatz is compatible with the infinite number of constraints provided by the NLL coefficients of the four highest powers of at all orders in which are provided by the results of Refs.[8, 9, 11, 17] including the large- results for the four-loop splitting functions and in Eq. (1.1). However, all these constraints can indeed be fulfilled, and the resulting coefficients are given by

 fϕ,q(n,i,ϵ) = (n−1i)−1[1+ϵ(β08CA(i+1)(n−i)θi1−32(1−nδi0))], f2,g(n,i,ϵ) = (n−1i)−1[1+ϵ(β08CAi(n−i−3)+12(3i+1−nθn2δin−1))] (3.15)

and

 gϕ,q(n,i)=g2,g(n,i)=(ni+1)−1 (3.16)

with for and else. Eqs. (3.14) – (3.16) together with their diagonal counterparts (3.1), keeping the and contributions to and the terms of in Eq. (3.2), facilitate the extension of the all-order mass factorization to the next-to-leading logarithms.

An extension of Eq. (3.14) to the the third logarithms can be expected to become much more cumbersome, requiring at least corrections to Eqs. (3.15), corrections to Eqs. (3.16) and a new contribution, but presumably also terms respectively involving or . Instead of pursuing this approach, we now switch to our second method for the resummation of and .

For this purpose we consider the calculation of via suitably projected gauge-boson parton cross sections as performed at two loops in Refs. [32, 33, 34]. The maximal ( particles) phase space for these processes at order is [43, 44]

 (1−x)−1−nϵ∫10d(3