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NIKHEF/98-025 September 1998

Soft Gluon Resummation for

arXiv:hep-ph/9809550v1 26 Sep 1998

Heavy Quark Electroproduction

Eric Laenen and Sven-Olaf Moch

NIKHEF Theory Group P.O. Box 41882, 1009 DB Amsterdam, The Netherlands

Abstract We present the threshold resummation for the cross section for electroproduction of heavy quarks. We work to next-to-leading logarithmic accuracy, and in single-particle inclusive kinematics. We provide next-to-leading and next-to-next-to-leading order expansions of our resummed formula, and examine numerically the quality of these ?nite order approximations. For the case of charm we study their impact on the structure function F2 and its di?erential distribution with respect to the charm transverse momentum.

1

Introduction

The production of heavy quarks in deep-inelastic scattering is a reaction of great interest because it sheds light on a number of fundamental issues in the QCD-improved parton model. First, at moderate momentum transfer Q, it is a direct probe of the gluon density in the proton. Second, it allows the exploration of the relevant degrees of freedom, as a function of Q, for describing this process, the options comprising the treatment of the heavy quark as a quantum ?eld external to the proton, via an “e?ective” parton density, or as a combination thereof. Finally, detailed studies of perturbative QCD dynamics are possible using the presence of at least two hard scales: Q and the heavy quark mass m. Charm quarks have been identi?ed in deep-inelastic electron proton collisions by the EMC collaboration [1] at small Q and low ?nal state invariant mass (large x), and more recently and in much greater numbers at high Q and medium to small x by the ZEUS and H1 collaborations at HERA [2]. Considerably more charm (and bottom) data are anticipated at HERA. At the theoretical level the reaction has already been studied extensively. In the framework where the heavy quark is not treated as a parton, leading order (LO) [3] and next-to-leading order (NLO) [4] calculations of the inclusive structure functions exist. Moreover, LO [5] and NLO [6] calculations of fully di?erential distributions have been performed in recent years. So far theory agrees reasonably well with the HERA data. We shall not concern ourselves here with descriptions in which the heavy quark is (partly) treated as a parton [7]. At the HERA center of mass energy of 314 GeV, one naively expects the production of a heavy quark pair with a minimum invariant mass of a couple of GeV to be insensitive to threshold e?ects, however a closer examination [8] reveals that this is not true: the density of gluons, which controls the heavy quark electroproduction rate, is large at small momentum fractions of the proton, favoring partonic processes where the initial gluon has only somewhat more energy than needed to produce the ?nal state of interest. As we shall show, for the inclusive cross section and for x values larger than about 0.01, the higher order QCD corrections for this reaction are in fact dominated by singular distributions n (αs /n!)[ln2n?1 (w )/w ]+ at order n. The dimensionless weight w is a function of the momenta of the external partons, chosen in accordance with the kinematics [9], and vanishes at threshold. Our main purpose in this paper is to resum these singular functions to all orders in perturbation theory to next-to-leading logarithmic (NLL) accuracy, and, moreover, in single-particle inclusive (1PI) kinematics. The required technology has recently been developed in Refs. [9,10,11,12, 13]. We also provide analytic results for the resummation and ?nite order expansion for pair invariant mass kinematics (appendix B). All our analytic results are valid for either charm or bottom production, but we concentrate our numerical studies on the case of single inclusive charm production. The interest in this analysis is ?rst of all intrinsic, and lies in studying the quality of the next-to-leading logarithmic resummation for single-particle inclusive observables. Second, while charm the exactly computed O (αs ) corrections to the inclusive structure function F2 [4] are, for experimentally accessible x values, on average about 30% - 40% [14], i.e. non-negligible but not too alarming, the size of even higher order corrections bears examination. Our results, in 2 particular our estimates for the unknown exact next-to-next-to-leading order (NNLO) O (αs ) corrections, should help establish the theoretical error on observables for the reaction under study, and on the gluon density extracted from charm quark production at HERA. Finally, our results might contribute to a future global analysis for resummed parton densities. We have organized our paper as follows. Section 2 contains the derivation of the resummed 2

formula for the single-particle inclusive di?erential cross section for heavy quark electroproduction. In section 3 we study its ?nite order expansions both analytically and numerically. We examine both partonic quantities (coe?cient functions), as well as hadronic observables, viz. charm the inclusive structure function F2 and its di?erential distribution in charm transverse momentum. Section 4 contains our conclusions. In appendix A we present some useful formulae related to Laplace transformations, and in appendix B we perform the threshold resummation and NNLO expansion for the heavy quark electroproduction cross section in pair-inclusive kinematics.

2

Resummed di?erential cross section

We begin with the de?nition of the exact single-particle inclusive kinematics. We study the electron (e) – proton (P ) reaction e(l) + P (p) → e(l ? q ) + Q(p1 ) + X [Q](p′2 ) , (1)

where X [Q] denotes any allowed hadronic ?nal state containing at least the heavy antiquark, Q(p1 ) is a heavy quark, and q ? the momentum transfer of the leptonic to the hadronic sector of the scattering. After integrating over the azimuthal angle between the lepton scattering plane 2 and the tagged heavy quark plane, neglecting Z -boson exchange (we assume |q 2 | ? MZ ) and summing over X , the cross section of the reaction (1) may be written as [15] d2 FL,P (x, Q2 , m2 , T1 , U1 ) d2 F2,P (x, Q2 , m2 , T1 , U1 ) ? y2 . dT1 dU1 dT1 dU1 (2) The functions d2 Fk,P /dT1 dU1 , k = 2, L are the double-di?erential deep-inelastic heavy quark structure functions and α is the ?ne structure constant. The kinematic variables Q2 , x, y are de?ned by p·q Q2 2 2 , y= . (3) Q = ?q > 0 , x = 2p · q p·l 1 + (1 ? y )2 S = (p + q )2 ≡ S ′ ? Q2 and S4 = S + T1 + U1 + Q2 . (5) The structure functions d2 Fk,P /dT1 dU1 describe the strong interaction part of the reaction (1). They enjoy the factorization d2 Fk,P (x, S4 , T1 , U1 , Q2 , m2 ) 1 = ′2 dT1 dU1 S

1

d4 σ eP →eQX 2πα2 = dx dQ2 dT1 dU1 x Q4

We also de?ne the overall invariants ,

T1 = (p ? p1 )2 ? m2

,

U1 = (q ? p1 )2 ? m2 ,

(4)

(6) where the sum is over all massless parton ?avors, and φi/P (z, ?2 ) is the parton distribution function (PDF) for ?avor i in the proton and z the momentum fraction of parton i in the proton with z ? = ?U1 /(S ′ + T1 ). The dimensionless functions ωk,i describe the underlying hard parton scattering processes and depend on the partonic invariants s′ , t1 , u1 and s4 , to be 3

i=q,q ?,g z ?

dz x s4 t1 u1 Q2 m2 φi/P (z, ?2 ) ωk,i , 2 , 2 , 2 , 2 , 2 , αs (?) , z z ? ? ? ? ?

de?ned in Eq. (8). The factorization scale is denoted by ? and in this paper is always taken equal to the renormalization scale. As we are operating near threshold we will assume a ?xed number of light ?avors in the evolution of the φi/P and the strong coupling αs . A further simpli?cation we adopt is neglecting the contributions from quarks and antiquarks in the sum over ?avors in Eq. (6). This is welljusti?ed as their contribution at NLO was found to be a mere 5% [4,14]. We therefore only consider the gluon-initiated partonic subprocess γ ? (q ) + g (k ) ?→ Q(p1 ) + X ′ [Q](p′2 ) , where k = z p. The partonic invariants are s = (k + q )2 ≡ s′ + q 2 , t1 = (k ? p1 )2 ? m2 , u1 = (q ? p1 )2 ? m2 . (8) (7)

2 2 Note that t1 = zT1 , s′ = z (S + Q2 ) and u1 = U1 . The invariant s4 = MX ′ ? m measures the inelasticity of the reaction (7) and is given by

s′ + t1 + u1 = s4 .

(9)

Our goal is to resum those higher order contributions to ωk,g that contain 1PI plus-distributions. The latter are de?ned by lnl (s4 /m2 ) s4 = lim

+

?→ 0

lnl (s4 /m2 ) 1 ? δ (s 4 ) . θ(s4 ? ?) + lnl+1 s4 l+1 m2

(10)

These distributions are the result of imperfect cancellations between soft and virtual contrii+2 butions to the cross section. At order O (αs ), i = 0, 1, . . . the leading logarithmic (LL) corrections correspond to l = 2i + 1, the NLL ones to l = 2i etc. In pursuing our goal, we follow the general principles of Ref. [9], and the methods of Refs. [10,11,12,16]. We restrict ourselves from here onwards to the structure function F2,P . The resummation of FL,P demands special attention [17]. Moreover, F2,P constitutes the bulk of the cross section in Eq. (2). By replacing the incoming proton in Eq. (6) with an incoming gluon, one may compute the hard part ω2,g in infrared-regulated perturbation theory. The resummation of ω2,g rests upon the simultaneous factorization of the dynamics and the kinematics of the observable in the threshold region of phase space. This factorization is pictured in Fig. 1. The ?gure corresponds to the partonic process (7) and represents a general partonic con?guration that produces large corrections, i.e. corrections containing the singular functions of Eq. (10). The ?gure shows the factorization of the partonic cross section into various functions, each organizing the large corrections corresponding to a particular region of phase space. Such factorizations have been discussed earlier for deep-inelastic scattering, Drell-Yan [18], heavy quark [16,19], dijet [10] and for general single-particle/jet inclusive cross sections [12]. The function ψg/g contains the full dynamics of partons moving collinearly to the incoming gluon g . It includes all leading and some next-to-leading enhancements. The function S (kS ) summarizes the dynamics of soft gluons that are not collinear to g , and includes all remaining next-toleading contributions. The function H2,g ≡ h? 2,g h2,g incorporates the e?ects of o?-shell partons, and contains no singular functions. There are large next-to-leading corrections associated with the outgoing heavy quarks. Were we to treat the heavy quarks as massless, each heavy quark would be assigned its own jet function, see e.g. Refs. [10,12,13]. In our case the heavy quark 4

mass prevents collinear singularities, so that all singular functions arising from the ?nal state heavy quarks are due to soft gluons. Hence, near threshold, all the singular behavior associated with the heavy quarks may be included in the soft function [16]. We note that S (kS ) is simply a function, and not a matrix in a space of color tensors, in contrast to heavy quark [16] or jet production [10,11] in hadronic collisions. The kinematics of reaction (7) decomposes in a manner that corresponds exactly to the factorization of Fig. 1. Momentum conservation at the parton level means q + z p = p1 + p2 + k S , (11)

where, in view of the above discussion, p1 and p2 may be interpreted as the on-shell momenta of the heavy quark and anti-quark respectively. Squaring and dividing by m2 , we have S4 2 p2 · k S 2 p · q ′ ? 2 p · p1 ? (1 ? z ) + 2 2 m m m2 s4 ?u1 ?u1 + 2 ≡ w1 + wS , (12) ? (1 ? z ) m2 m m2 2 where we have dropped terms of order S4 . Notice that the kinematics is in fact speci?ed by ? ? the dimensionless vector ζ , de?ned as ζ = p? 2 /m. In Eq. (12) we have identi?ed the overall single-particle inclusive weight w , mentioned in the introduction, with S4 /m2 . At ?xed S4 the

q h2

;g

q h2

;g

S

g=g

Figure 1:

Factorization of heavy quark electroproduction near threshold. The double lines denote eikonal propagators.

infrared-regulated, di?erential partonic structure function factorizes [9] as s′2 d2 F2,g (x, S4 , t1 , u1, Q2 , m2 ) = H2,g (t1 , u1, Q2 , m2 ) dw1 dwS (13) dt1 du1 ?u1 S4 ? wS ψg/g (w1 , p, ζ, n) S (wS , βi , ζ, n) , ? w1 ×δ 2 m m2

where the kinematics relation (12) is implemented in the δ -function. The βi are the fourvelocities of the particles in the Born approximation to Eq. (7), i.e. with X [Q](p′2 ) replaced by Q(p2 ). The various functions are computed in n · A = 0 gauge. Further manipulations are carried out most conveniently in terms of Laplace moments, de?ned by

∞

?(N ) = f

0

dw e?N w f (w ) . 5

(14)

The upper limit of this integral is not so important, and may also be put at 1, where ln w = 0. As both the gauge and kinematics vector are timelike, we may choose 1 n? = ζ ? , in which case the 1PI ψi/i are equal to the center-of-mass densities (for which ζ ? = n? = δ ?0 ) [12]. Replacing the incoming proton with an incoming gluon in Eq. (6) and comparing with Eq. (13), we derive ω ? 2,g N, t1 u1 Q2 m2 , , , ?2 ?2 ?2 ?2 = H2,g × t1 u1 Q2 m2 , , , ?2 ?2 ?2 ?2 ?( m , ζ, βi) . S N? (15)

?g/g (N (?u1 /m2 ) , (p · ζ/?)) ψ ?g/g (N (?u1 /m2 ) , ?) φ

Given the factorization in Eqs. (13) and (15), the arguments of [9,18] ensure that the N dependence in each of the functions of Eq. (15) exponentiates. We next discuss each function in Eq. (13), or (15), in turn, starting with the ψg/g wavefunction. In analogy with the center-of-mass de?nitions [18,10] it may be de?ned as an operator matrix element 2 p·ζ ψg/g (w, p, ζ, n) = 2 2(Nc ? 1) 4π (v · p)2 1 ×

∞

dy e?iy (1?w)p·ζ

?∞ n·A=0 ,

g (p)|F ? ⊥ (yζ ) v? vν F ⊥ ν (0)|g (p)

(16)

where w = 1 ? z and v? is a ?xed lightlike four-vector in the opposite direction as p? , such that v · p is of order one. The ?rst factor refers to a spin and color average. The states are normalized such that 0|A⊥ (0)|g (p) = ?⊥ (p). Expression (16) as a whole is normalized such that ψg/g (w ) = δ (w ) .

(0)

(17)

We have calculated the order αs corrections to this operator matrix element in general axial gauge n · A = 0. The O (αs ) corrections to the operator matrix element in (16) are in d = 4 ? 2? dimensions, and to next-to-leading logarithmic accuracy ψg/g (w,

(1)

αs (?) p·ζ ?1 1 , ?) = CA ? π ? ? w + 1 w ln

+

+

+

2 ln(w ) w

+

s ?2

? 1 + ln(2νg ) + O (?) ,

(18)

? s/2. Note that ln(2νg ) + ln(s/?2) = ln(4(p · ζ )2/?2 ). We have gluon, de?ned by p? = βg abbreviated 1/? ? = 1/? ? γE + ln 4π . We have veri?ed that the steps for the resummation of the Sudakov double logarithms ln2 (N ) in this function trace exactly those for the center-of-mass density introduced for the Drell-Yan process in Ref. [18]. We choose for φg/g the MS density, which does not have Sudakov double logarithms. To the same accuracy as ψg/g in Eq. (18), it is given by

? where νg = (βg · n)2 /|n|2 , with n? chosen equal to ζ ?, and βg the four-velocity of the incoming

φg/g (w, αs (?), ?) = δ (w ) +

1

αs (?) ?1 CA π ? ?

1 w

.

+

(19)

Note that for lightlike kinematics, such as in direct photon production, it may be convenient to keep both vectors separate [12]. 2 For the equivalent de?nition for (anti)quarks, see Ref. [12].

6

The resummation of the singular contributions to this function is done via the Altarelli-Parisi equation [20], see also [9,21]. We only need this result in combination with the resummed ψg/g ?g/g /φ ?g/g in Eq. (15). This ratio is ?nite, as can readily be veri?ed at one-loop as the ratio ψ from Eqs. (18) and (19). The scale dependence of these functions may be found as follows [10]. The scale dependence ? of ψg/g is governed by ? d ?g/g N, p · ζ , αs (?), ? = γψ (αs (?)) . ln ψ d? ? (20)

The anomalous dimension on the right does not depend on the moment N , as the only ultraviolet divergences are due to gluon wavefunction renormalization. Hence, γψ = 2γg , where γg is the anomalous dimension of the gluon ?eld in axial gauge. The anomalous dimension of the MS density does depend on N , via the Altarelli-Parisi equation d ?g/g (N, αs (?), ?) = 2γg/g (N, αs (?)) . (21) ? ln φ d? ?(N, ζ ). It summarizes the singular contributions arising We next discuss the soft function S from soft gluons that are not collinear to the incoming jet, and is infrared ?nite. It depends on the kinematics of the hard scattering and contributes only at next-to-leading logarithm. Threshold resummation for soft functions is treated in detail in Refs. [11,16]; a brief sketch will su?ce here. The factorization in Eq. (13) introduces ultraviolet divergences, distributed in such a way between the hard function H and the soft function S , that they cancel in the product [9,10]. The function S may be de?ned as the matrix element of a composite operator that connects Wilson lines in the directions of the external partons. Its extra ultraviolet divergences ? obeys the renormalization are cancelled by the renormalization of this operator. As a result, S group equation ? ? ? = ? (2Re {ΓS (αs (?))}) S ?, ? + β (αs ) S (22) ?? ? αs ?. In the reaction under study the anomalous dimension is a then resums the ln(N ) terms in S 1 × 1 matrix in color space, so we may solve Eq. (22) directly: ? S m ? m , αs ? , αs (?) = S N? ? N exp ? ?

?

?/N

?

d? 2Re {ΓS (αs (?′ ))}? ?. ′ ?

′

?

(23)

The resummed hard part may now be written in moment space as t1 u1 Q2 m2 , , , ?2 ?2 ?2 ?2 t1 u1 Q2 m2 , , , ?2 ?2 ?2 ?2

m

ω ? 2,g N,

= H2,g

? m , αs ? S ? N

× exp E(g)

?/N

?u1 , m2 N m2

exp

?2 .

?

d?′ ?u1 γg (αs (?′ )) ? γg/g N , αs (?′) ′ ? m2 (24)

× exp

?

d?′ 2 Re {ΓS (αs (?′ ))} ?′

7

?g/g /φ ?g/g and The ?rst exponent summarizes the N -dependence of the wave function ratio ψ is, in moment space, the same as for heavy quark and dijet production [10,11]

∞

E(g) Nu , m

2

=

0

(1 ? e?Nu w ) dw w ,

1

w2

√ dλ A(g) αs ( λm) λ (25)

1 + ν(g) [αs (w m)] 2 ν(g) = 2CA

αs 4(p · ζ )2 1 ? ln π m2

,

(26)

where Nu ≡ N (?u1 /m2 ), and A(g) is given by the expression A(g) (αs ) = CA αs αs 1 + K π 2 π

2

,

(27)

with K = CA (67/18 ? π 2 /6) ? 5/9nf [22] and nf the number of quark ?avors. ?g/g /φ ?g/g The second exponent controls the factorization scale dependence of the ratio ψ through the anomalous dimensions γg and γg/g . In axial gauge we ?nd them to be αs (?) , π αs (?) (CA ln(Nu ) ? b2 ) + O(1/N ) . γg/g (Nu , αs (?)) = ? π γg (αs (?)) = b2 (28) (29)

where b2 = (11CA ? 2nf )/12. Finally, the O (αs ) expression for the soft anomalous dimension ΓS can be inferred from the UV divergences of the eikonal Feynman graphs in Fig. 2. We obtain the result

q p1 q p1 q p1

k

q

p2

k

p1

q

p2

k

p1

p2

k

p2

k

p2

Figure 2:

One-loop corrections to the soft function S for heavy-quark production in photon-gluon fusion. The double lines denote eikonal propagators. The lines labelled k corresponds to a gluonic eikonal line, and those labelled p1 and p2 denote quark and antiquark eikonal lines, respectively.

ΓS =

αs CF (?Lβ ? 1) π CA 4(p · ζ )2 ? ln 2 m2 ? Lβ ? 1 ? ln 8 ?t1 ?u1 ? ln m2 m2 . (30)

where Lβ is given by Lβ = 1?β 1 ? 2 m2 /s ln β 1+β + iπ , β= 1 ? 4m2 /s , (31)

and, for single-particle inclusive kinematics, ?u1 2p · ζ = 2 . m m (32)

At the conclusion of this section a few remarks are in order. First, the integral in the exponent in Eq. (25) can only be interpreted in a formal sense, as some prescription should be implemented to avoid integration over the Landau pole in the running coupling. We do not address such renormalon ambiguities here. In this paper we only employ the resummed expressions as generating functionals of approximate perturbation theory. Second, although our main focus is on 1PI kinematics, it is straightforward [12] to derive from Eq. (24) the resummed expression for pair-invariant mass kinematics, i.e. the resummed cross section for the process in Eqs. (1) and (7), di?erential in the QQ invariant mass. We present this derivation in appendix B.

3

Finite order results

In this section we expand our resummed cross section to one and two loop order so as to compare with exact NLO calculations [4,6] and provide NNLO approximations. We begin by deriving NLL analytic formulae for the single-particle inclusive, partonic hard part ω2,g . Subsequently we study their behavior in the hadronic inclusive structure function charm charm F2 , and in the di?erential distribution dF2 /dpT .

3.1

Partonic results at NLO and NNLO

(0)

The Born level hard part (see e.g. [3,4]) for the process γ ? + g ?→ Q + Q is

Born ′ ω2,g (s′ , t1 , u1 ) = δ (s′ + t1 + u1 ) σ2 ,g (s , t1 , u1 ) , Born with σ2 given by ,g Born ′ σ2 ,g (s , t1 , u1 ) =

(33)

αs 2 Nc CF 2 t1 u1 m2 s′ m2 s′ + +4 1? eq 2 Q 2π Nc ? 1 u1 t1 t1 u1 t1 u1 s′ 2 12q 2 m2 s′ q4 m2 q 2 s s′ q 2 2? + ′ +2 +2 ? ′ + 2 t1 u1 t1 u1 t1 u1 t1 u1 s t1 u1 s

(34) .

We now derive the NLO soft gluons corrections to ω2,g (s′ , t1 , u1) by expanding the resummed hard part in Eq. (24) to one loop, using the explicit expressions for the various functions in Eq. (24), given in Eqs. (18), (19) and (27)-(31). We ?nd

Born ′ ω2,g (s′ , t1 , u1 ) ? K (1) σ2 ,g (s , t1 , u1 ) , (1)

(35)

9

where K (1) = = 1 (1) (1) ψg/g (w1 , ?) ? φg/g (w1 , ?) ?u1 +

+

+

w1 =s4 /(?u1 ),?→0

1 1 2 m wS

+

2 Re {ΓS }

(36)

wS =s4 /m2

ln(s4 /m2 ) αs (?) 2 CA π s4

1 s4

CA ln

+

t1 ?2 + ReLβ ? ln u1 m2 ?u1 ?2 ln m2 m2 , (37)

?2 CF (ReLβ + 1) + δ (s4 ) CA ln

with ? the MS-mass factorization scale and Lβ given in Eq. (31). The plus-distributions in s4 have been de?ned in Eq. (10). To next-to-leading logarithm, the result in Eq. (37) agrees with the exact αs corrections of Ref. [4]. This holds for the contributions both independent and dependent on the factorization scale, i.e. terms including powers of ln(?/m). The scale-dependent logarithms ln(?/m) are explicitly generated by the second exponent in Eq. (24), involving the anomalous dimensions γg and γg/g given in Eqs. (28) and (29). In deriving γg/g in Eq. (29) we have carefully determined all terms which are constant in moment space, including the ln(?u1 /m2 ) logarithm, absorbed in Nu . In this way we are able to determine all factorization scale dependent terms to NLL accuracy. The NNLO corrections may be found in a manner exactly analogous to what was done for Drell-Yan in Ref. [23] and for Higgs production in Ref. [21]. We expand Eq. (24) to second order in αs in moment space. With the table of Laplace transforms in appendix A for the 2 singular distributions in Eq. (10), we then ?nd the O(αs ) threshold-enhanced corrections in momentum space. Our result is

Born ′ ω2,g (s′ , t1 , u1 ) ? K (2) σ2 ,g (s , t1 , u1 ) , (2)

(38)

with K (2) =

2 αs (? ) ln3 (s4 /m2 ) 2 2 C A π2 s4

(39)

+

+

ln2 (s4 /m2 ) s4

2 3 CA +

ln

t1 ?2 + ReLβ ? ln u1 m2

?2 CA (b2 + 3 CF (ReLβ + 1)) + ln(s4 /m2 ) s4 ln

+

?2 m2

2 CA

?2 ln

t1 ?u1 ?2 ? 2ReLβ + 2 ln + ln u1 m2 m2

+2 CA (b2 + 2 CF (ReLβ + 1)) + 1 s4 ln2

+

?2 m2

2 ?CA ln

?u1 m2

?

1 CA b2 2

.

and b2 = (11CA ? 2nf )/12. 10

3.2

Gluon coe?cient functions

charm The inclusive structure function F2 is obtained from Eq. (6) by integrating over T1 and U1 . ,P (k,l) It can be expanded in coe?cient functions c2,g as follows (dropping the P subscript) 2 αs (?) e2 cQ = dz φg/P (z, ?2 ) 4π 2 m2 ax 1 ∞ k

charm F2 (x, Q2 , m2 )

(4παs (?))k

k =0 l=0

c2,g (η, ξ ) lnl

(k,l)

?2 , (40) m2

where a = (Q2 + 4m2 )/Q2 and we recall that contributions from light initial state quarks are (k,l) neglected. The function φg/P (z, ?2 ) denotes the gluon PDF and the functions c2,g depend on the scaling variables η = s ? 1, 4m2 ξ = Q2 . m2 (41)

The variable η is a direct measure of the distance to the partonic threshold. The inclusive coe?cient functions are obtained from Eqs. (37) and (39) by

s′ (1+β )/2 (k,l) c2,g (η, ξ ) smax 4

=

s′ (1?β )/2

d(?t1 )

0

d2 c2,g (s′ , t1 , u1 ) , ds4 dt1 ds4

(k,l)

(42)

where the double di?erential coe?cient functions are in turn related to the hard part ω2,g of Eq. (6) by

k

s′2

l=0

d2 c2,g (s′ , t1 , u1 ) l ?2 4π 2 m2 (k) ′ ln 2 = ω2,g (s , t1 , u1) k 2 dt1 ds4 m αs e2 c (4παs ) Q

(k,l)

.

u1 =s′ +t1 ?s4

(43)

Also, in Eq. (42) we abbreviated [4] smax 4 s = ′ s t1 s′ (1 ? β ) t1 + 2 s′ (1 + β ) t1 + 2 . (44)

We begin our numerical studies with the gluon coe?cient functions c2,g (η, ξ ), i.e. those that are not accompanied by scale-dependent logarithms. We recall that all our results are (0,0) (1,0) derived in the MS scheme. The functions c2,g and c2,g are known exactly, see Ref. [4]. The one-loop expansion of our resummed hard part provides an approximation to the exactly known (1,0) c2,g . To judge the bene?ts of resumming to next-to-leading logarithmic as compared to leading logarithmic accuracy, we can distill also LL expressions, by keeping only the [ln(s4 /m2 )/s4 ]+ and [ln3 (s4 /m2 )/s4 ]+ terms in Eqs. (37) and (39) respectively. (k,0) In Figs. 3a and 4a we plot the functions c2,g (η, ξ ), k = 0, 1 versus η , for two values of ξ = Q2 /m2 . For the coe?cient functions only the ratio ξ matters, however, we chose those values to correspond to Q2 = 0.1 and 10 GeV2 for a charm mass 3 of m = 1.5 GeV. These ?gures reveal that, at one loop, the LL accuracy provides a good approximation for very small η , however, with signi?cant deviations from the exact result for larger η , i.e. as the distance

3

(k,0)

In the next subsection, where we study the hadronic structure function, we use the value m = 1.6 GeV.

11

1.2 1 0.8 0.6 0.4 0.2 0 -0.2 10

(a) c(0,0) 2g c(1,0) 2g ξ=.44e-1

1.2 1 0.8 0.6 0.4 0.2 0

(b) ξ=.44e-1

c(2,0) 2g

-3

10

-2

10

-1

1 10 2 η = s/(4m ) - 1

10

2

10

3

-0.2 10

-3

10

-2

10

-1

1 10 2 η = s/(4m ) - 1

10

2

10

3

Figure 3:

(1,0)

(a): The η -dependence of the coe?cient functions c2,g (η, ξ ), k = 0, 1 for Q2 = 0.1 GeV 2

(k,0) (1,0)

(k,0)

with m = 1.5 GeV. Plotted are the exact results for c2,g , k = 0, 1 (solid lines), the LL approximation to c2,g (dotted line) and the NLL approximation to c2,g (dashed line). (b): The η -dependence of the coe?cient function c2,g (η, ξ ) for Q2 = 0.1 GeV 2 with m = 1.5 GeV. Plotted are the LL approximation (dotted line) and the NLL approximation (dashed line).

(2,0)

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 10

(a) c(0,0) 2g c(1,0) 2g ξ=.44e+1

-3

10

-2

10

-1

1 10 2 η = s/(4m ) - 1

10

2

10

3

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 10

(b) ξ=.44e+1

c(2,0) 2g

-3

10

-2

10

-1

1 10 2 η = s/(4m ) - 1

10

2

10

3

Figure 4:

(a): The η -dependence of the coe?cient functions c2,g (η, ξ ), k = 0, 1 for Q2 = 10 GeV 2 with m = 1.5 GeV. The notation is the same as in Fig. 3a. (b): The η -dependence of the coe?cient (2,0) function c2,g (η, ξ ) for Q2 = 10 GeV 2 with m = 1.5 GeV. The notation is the same as in Fig. 3b.

(k,0)

12

0.4 0.3 0.2 0.1 0 -0.1 10

(a) c(1,0) 2g ξ=.44e+1

0.5 0.4 0.3 0.2 0.1 0 -0.1

(b) c(2,0) 2g ξ=.44e+1

-3

10

-2

10

-1

1 10 2 η = s/(4m ) - 1

10

2

10

3

10

-3

10

-2

10

-1

1 2 10 η = s/(4m ) - 1

10

2

10

3

(a): The NLL approximation to the coe?cient function c2,g (η, ξ ) for Q2 = 10 GeV 2 and m = 1.5 GeV with restrictions to the small η -region. Plotted are the unmodi?ed NLL result (solid line), the NLL result with a factor θ (m2 ? s4 ) included (dotted line) and the NLL result multiplied √ with a damping factor 1/ 1 + η (dashed line). (b): The η -dependence of the NLL approximation to (2,0) the coe?cient function c2,g (η, ξ ) for Q2 = 10 GeV 2 and m = 1.5 GeV. The notation is the same as in Fig. 5a.

Figure 5:

(1,0)

from threshold increases. On the other hand, the NLL approximation is excellent over a much wider range in η , up to values of about 10. (2,0) We also show c2,g , obtained from Eqs. (39) and (43), in LL and NLL approximation in Figs. 3b and 4b for the same values of m and Q2 as before. We observe more structure than (1,0) in the c2,g curves. Figs. 3b and 4b show some sizable deviations from zero at large η , which leads us to consider the following issue. Kinematically, larger η values allow w = s4 /m2 ? 1, whereas, in contrast, the threshold region is de?ned by w ≈ 0. Therefore we should consider the inclusion of a factor θ(m2 ? s4 ) in the integral (42) in order to suppress spurious large corrections in the large η region. In Figs. 5a and 5b we show the e?ect of the veto θ(m2 ? s4 ) on the NLL coe?cient functions (k,0) c2,g for k = 1, 2 for√ values of Q2 = 10 GeV2 and m = 1.5 GeV. Also shown is the e?ect of a damping factor 1/ 1 + η [24], which can be included outside the integral in Eq. (42) and is therefore easier to implement. We see that the e?ect of the veto is well-mimicked by the damping factor, therefore we shall use the latter, rather than the veto, where needed in what follows. Of course, for smaller values of η the damping factor leads to a slight underestimation of the coe?cient functions compared to the veto, but these e?ects are negligible, as Fig. 5 shows. Let us now investigate those coe?cient functions c2,g (η, ξ ), l ≥ 1 in Eq. (40) that determine charm the dependence of F2 on the mass factorization scale ?. As mentioned earlier, this is of charm some relevance since the exact NLO F2 exhibits considerably more ? sensitivity at large x (near threshold), than at small x, cf. Ref. [8]. The natural question arises whether and, if so how much, our approximate NNLO results might improve the situation. (1,1) Explicitly, the coe?cient functions under consideration up to NNLO are c2,g which is (2,1) (2,2) known exactly [4] and the previously unknown functions c2,g and c2,g . We can infer the exact

(k,l)

13

0.4 0.3 0.2 0.1 0 -0.1 -0.2 10

(a) ξ=.44e-1

0.4 0.3 0.2

(b) ξ=.44e-1 c(2,1) 2g

c(1,1) 2g

0.1 0 -0.1 c(2,2) 2g

-3

10

-2

10

-1

1 10 2 η = s/(4m ) - 1

10

2

10

3

-0.2 10

-3

10

-2

10

-1

1 10 2 η = s/(4m ) - 1

10

2

10

3

(a): The η -dependence of the coe?cient function c2,g (η, ξ ) for Q2 = 0.1 GeV 2 with m = 1.5 GeV. Plotted are the exact result (solid line), the LL approximation (dotted line) and the NLL (2,l) approximation (dashed line). (b): The η -dependence of the coe?cient functions c2,g (η, ξ ), l = 1, 2 for Q2 = 0.1 GeV 2 with m = 1.5 GeV. Plotted are the exact results (solid lines), the LL approximations (dotted lines) and the NLL approximations (dashed lines).

Figure 6:

(1,1)

0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 10

(a) ξ=.44e+1 c(1,1) 2g

0.2 0.15 0.1 0.05 0 -0.05 -0.1

(b) c(2,1) 2g

ξ=.44e+1

c(2,2) 2g

-3

10

-2

10

-1

1 10 2 η = s/(4m ) - 1

10

2

10

3

-0.15 10

-3

10

-2

10

-1

1 10 2 η = s/(4m ) - 1

10

2

10

3

(a): The η -dependence of the coe?cient function c2,g (η, ξ ) for Q2 = 10 GeV 2 with m = 1.5 GeV. The notation is the same as in Fig. 6a. (b): The η -dependence of the coe?cient (2,l) functions c2,g (η, ξ ), l = 1, 2 for Q2 = 0.1 GeV 2 with m = 1.5 GeV. The notation is the same as in Fig. 6b.

Figure 7:

(1,1)

14

0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 10

(a) c(1,1) 2g ξ=.44e+1

0.4 0.3 0.2 0.1 0 -0.1

(b) ξ=.44e+1 c(2,1) 2g c(2,2) 2g

-3

10

-2

10

-1

1 10 2 η = s/(4m ) - 1

10

2

10

3

-0.2 10

-3

10

-2

10

-1

1 10 2 η = s/(4m ) - 1

10

2

10

3

(a): The NLL approximation to the coe?cient function c2,g (η, ξ ) for Q2 = 10 GeV 2 and m = 1.5 GeV with restrictions to the small η -region. The notation is the same as in Fig. 5a (b): The (2,l) NLL approximation to the coe?cient function c2,g (η, ξ ), l = 1, 2 for Q2 = 10 GeV 2 and m = 1.5 GeV with restrictions to the small η -region. The notation is the same as in Fig. 5a.

Figure 8:

(1,1)

results for these functions from renormalization group arguments. We ?nd c2,g

(1,1)

4

=

c2,g

(2,1)

1 (0,0) 1 (0,0) (0) b2 c2,g ? c2,g ? Pgg , 2 4π 2 1 1 (0,0) 1 1 (1,0) (0,0) (1,0) (1) (0) = b3 c2,g ? c2,g ? Pgg + 2 2b2 c2,g ? c2,g ? Pgg , 2 2 (4π ) 2 4π 2 = 1 3 (0,0) 1 (0,0) (0,0) (0) (0,0) b2 , 2 c2,g ? b2 c2,g ? Pgg + c2,g ? Pgg 2 2 (4π ) 4 8

(45) (46) (47)

c2,g

(2,2)

2 with β -function coe?cients b2 = (11CA ? 2nf )/12, b3 = (34CA ? 6CF nf ? 10CA nf )/48 and the (0) (1) one- and two-loop gluon-gluon splitting functions Pgg , Pgg , cf. Ref. [25]. The convolutions (i,0) involving a coe?cient function c2,g are de?ned as 1 (i,0) c2,g

?

(j ) Pgg

(η (x), ξ ) ≡

dz c2,g

ax

(i,0)

η

x (j ) , ξ Pgg (z ) , z

(48)

where a = (Q2 + 4m2 )/Q2 and η (x) is derived from Eq. (41), η (x) = ξ/4(1/x ? 1) ? 1. Finally, (0,0) the function Pgg in Eq. (47) is given by the standard convolution of two splitting functions (0) Pgg ,

1 (0,0) Pgg (x) 1

≡

dx1

0 0

(0) (0) dx2 δ (x ? x1 x2 )Pgg (x1 )Pgg (x2 ) .

(49)

With Eqs. (45)-(47) in hand, we are able to check the quality of our NLL approximation at NLO and NNLO against exact answers. Again, to fully appreciate the improvement of NLL resummation, we truncate our approximation to LL accuracy, keeping only the ln(?/m)[1/s4 ]+

4

Eq. (45) agrees with the result for c2,g

(1,1)

in Ref. [4].

15

5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 10

(a)

φg/P(x/z,? ) x = 0.1

2

-3

10

-2

10

-1

1 10 2 η = s/(4m ) - 1

10

2

10

3

200 175 150 125 100 75 50 25 0 10

(b)

φg/P(x/z,? ) x = 0.01

2

-3

10

-2

10

-1

1 10 2 η = s/(4m ) - 1

10

2

10

3

(a): The CTEQ4M gluon density φg/P (x/z, ?2 ) as a function of η for m = 1.6 GeV, Q2 = 10 GeV and x = 0.1. Plotted are the scale choice ? = m (solid line), ? = 2m (dashed line) and ? = Q2 + 4m2 (dotted line). (b): Same as Fig. 9a for x = 0.01.

Figure 9:

term in Eq. (37) and the terms ln(?/m)[ln2 (s4 /m2 )/s4 ]+ and ln2 (?/m)[ln(s4 /m2 )/s4 ]+ in Eq. (39), respectively. (1,1) In Figs. 6a and 7a we plot the function c2,g (η, ξ ) versus η , for two values of ξ = Q2 /m2 , (2,l) while Figs. 6b and 7b display the functions c2,g , l = 1, 2. The evaluation of Eq. (46) has (1,0) been performed with the help of the parametrization for c2,g from Ref. [26]. From Figs. 6 and 7, we see again that the NLL approximations are superior to the LL ones. In Figs. 8a and 8b we show the e?ect of the veto θ(m2 ? s4 ) in the integral (42) and the damping factor √ (1,1) (2,l) 1/ 1 + η on c2,g (η, ξ ) and c2,g (η, ξ ), l = 1, 2 for values of ξ corresponding to Q2 = 10 GeV2 and m = 1.5 GeV. In the following sections we shall for the sake of uniformity include a (2,l) (1,1) (1,1) damping factor for the c2,g (η, ξ ), l = 1, 2 and c2,g (η, ξ ), even though the undamped NLL c2,g (1,1) approximates the exact c2,g somewhat better, as a comparison of Figs. 7a and 8a shows. We veri?ed that the di?erence is negligible at the hadronic level.

3.3

charm Inclusive structure function F2

Having obtained encouraging results for the quality of the partonic approximations Eqs. (37) and (39) at the inclusive level, we shall now assess their e?ect at the hadronic level. Throughout we use the CTEQ4M [27] gluon density. For NLO plots we use the two-loop expression for αs with nf = 4, Λ = 0.298 GeV. Respectively, for NNLO plots, the three-loop expression [28] for αs and an adjusted Λ = 0.265 GeV. In Fig. 9a and 9b we plot the gluon density φg/P (x/z, ?2 ) as a function of η on the same (k,l) scale as the coe?cient functions c2,g (η, = 0.1 and 0.01 and three di?erent choices √ξ )2 for x 2 of the factorization scale (? = m, 2m, Q + 4m ). From now on we use a charm mass of m = 1.6 GeV and take Q2 = 10 GeV. Our heavy quark mass is always a pole mass, as in Ref. [4]. The ?gure shows that the gluon density indeed provides support for the coe?cient functions in Eq. (40) in the threshold region for x ≥ 0.01, although the ?gure also shows the extent of the support to be somewhat scale sensitive. A more revealing way to examine the support of the gluon density in Eq. (40) is to plot the

16

0.006 0.005 0.004 0.003 0.002 0.001 0 0.2

(a) charm F2

0.06 0.05 0.04 0.03 0.02 0.01 0 0.3 0.4 zmax 0.5 0.6 0.7 0.8 0.9 1

(b) charm F2

( × 0.5)

10

-1

zmax

1

Figure 10:

charm (x, Q2 , z (a): F2 max ) as a function of zmax at NLO with the CTEQ4M gluon PDF, 2 x = 0.1, m = 1.6 GeV, Q = 10 GeV and ? = m (upper three curves), ? = Q2 + 4m2 (lower three curves). Plotted are: The exact results (solid lines), the LL approximations (dotted lines) and the √ NLL approximations with the damping factor 1/ 1 + η on the scale dependent terms only (dashed lines). (b): Same as Fig. 10a for x = 0.01. The lower three curves, ? = Q2 + 4m2 , have been scaled down by a factor of 2.

integral as a function of zmax ,

charm F2 (x, Q2 , m2 , zmax ) 2 αs (?) e2 cQ ? 4π 2 m2 zmax ∞ k

dz φg/P (z, ?2 )

ax

(4παs (?))k

k =0 l=0

c2g (η, ξ ) lnl

(k,l)

?2 . (50) m2

Varying the theoretical cut-o? zmax allows us see where the integral (50) acquires most of its value. The physical structure function corresponds to zmax = 1. In Figs. 10a and 10b we show the results for the numerical evaluation of Eq. (50) and compare to the exact NLO calculation of [4] for two values of x = 0.1, 0.01 and two √ Ref. 2 choices of the factorization scale ? = m, Q + 4m2 . The scale dependent√ terms in the NLL approximation to Eq. (50) have been multiplied with a damping factor 1/ 1 + η in order to suppress spurious contributions from the large η region, i.e. far above partonic threshold.5 charm Figs. 10a and 10b show that, for the chosen kinematical range in x and Q2 , F2 as de?ned by Eq. (50) originates entirely from that region in η in which we approximate the exact results charm very well. In words, for the values of x shown, F2 is completely determined by partonic processes close to threshold. We observe that the NLL approximation to Eq. (50) is much better than the LL one at the hadronic level as well. As we shall shortly show in more detail, it is also signi?cantly more stable under variations of the factorization scale. In Figs. 11a and 11b we display the results for Eq. (50) evaluated to NNLO in NLL approximation for the same kinematical values and choices for ? as before. Again, we have multiplied (1,1) (2,l) the NLO √ scale dependent terms c2,g and also all NNLO terms c2,g , l = 0, 1, 2 with a damping factor 1/ 1 + η . Figs. 11a and 11b reveal that the NNLO corrections are numerically quite sizable. However, the variations with respect to the mass factorization scale are still large for both values of x = 0.1, 0.01. Let us therefore turn to the issue of factorization scale dependence of the hadronic structure

See the discussion in section 3.2. If we add the damping factor also to the scale independent terms, our NLL results at ? = m are only 5% smaller.

5

17

0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0.2

(a) charm F2

0.06 0.05 0.04 0.03 0.02 0.01 0 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 zmax

(b) charm F2

10

-1

zmax

1

charm (x, Q2 , z (a): F2 max ) as a function of zmax at NNLO in NLL approximation with the √ CTEQ4M gluon PDF, x = 0.1, m = 1.6 GeV, Q2 = 10 GeV and with the damping factor 1/ 1 + η (2,0) on all scale dependent terms and on c2,g . Plotted are: Scale choices ? = m (upper curve) and ? = Q2 + 4m2 (lower curve). (b): Same as Fig. 11a for x = 0.01. Plotted are: Scale choices ? = m (lower curve) and ? = Q2 + 4m2 (upper curve).

Figure 11:

charm function in more detail. The scale dependence√ of F2 , cf. Eq.(40), at NLO is exhibited in 2 Figs. 12a and 12b over a range in m ≤ ? ≤ 2 Q + 4m2 . For x = 0.1, 0.01 we compare the exact result of Ref. [4] with the LL and the NLL approximations from Eq. (37). The coe?cient √ (1,1) c2,g is in the approximations again multiplied by the damping factor 1/ 1 + η . We see from Figs. 12a and 12b that resummation to NLL accuracy is crucial to obtain a charm reliable behaviour with respect to the ?-dependence of F2 at NLO. The NLL curve traces the exact result extremely well, in particular at phenomenological interesting smaller x values, when the support of gluon PDF extends to regions of η = 1 to 10. In fact, Fig. 12b clearly shows that in this kinematical domain, the LL approximation is rather poorly behaved. Continuing, we display in Figs. 13a and 13b the e?ect of the NNLO corrections on the scale charm charm dependence of F2 . We compare an improved NLL approximation to F2 at NNLO, which charm consists of the exact result of Ref. [4] for F2 at NLO and, additionally, our NLL approximate charm NNLO results from Eq.(39), with the so-called best approximation to F2 at NNLO. This best approximation extends beyond the NLL accuracy at two-loop order as it contains the full (2,1) (2,2) (2,0) exact results for c2,g and c2,g from Eqs. (46), (47), such that only c2,g remains approximate to NLL accuracy. We observe practically no improvement, with respect to the NLO result, in the scale depencharm dence of F2 at large x near threshold (Fig. 13a), where the relative variations with respect charm to ? were largest before. We also see here that the two approximations to F2 at NNLO, the NLL improved and the best, agree very well. They are di?ering only by terms beyond the NLL accuracy, which turn out to be rather small numerically. Therefore it is also unlikely that (2,0) any corrections to c2,g beyond the NLL accuracy will help to soften the large ?-dependence of charm F2 at large x. On the other hand, for smaller x (Fig. 13b), we indeed see an indication of increased NNLO charm stability. The best NNLO approximation to F2 reduces the value of the relative variation √ 2 charm charm 2 (? = m)/F2 (? = 2 Q + 4m ) from 1.13 at NLO to 1.07 at NNLO, so that there is F2 practically no dependence on ? left. Naturally, for smaller x, where one probes regions away from threshold, terms beyond the NLL accuracy also have some numerical importance, as one

18

0.006 0.005 0.004 0.003 0.002 0.001 0 1

(a) charm F2

0.06 0.055 0.05 0.045 0.04 0.035 0.03 0.025 0.02 ? [ GeV ] 10 1

(b) charm F2

? [ GeV ]

10

charm at NLO with the CTEQ4M Figure 12: (a): The ?-dependence of the charm structure function F2

gluon PDF, x = 0.1, m = 1.6 GeV and Q2 = 10 GeV. Plotted are: The exact result (solid line), the √ LL approximation (dotted line) and the NLL approximation with the damping factor 1/ 1 + η on the scale dependent terms (dashed line). (b): Same as Fig. 12a for x = 0.01.

0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.002 1

(a) charm F2

0.06 0.055 0.05 0.045 0.04 0.035 0.03 0.025 0.02 ? [ GeV ] 10 1

(b) charm F2

? [ GeV ]

10

charm at NNLO with the (a): The ?-dependence of the charm structure function F2 CTEQ4M gluon PDF, x = 0.1, m = 1.6 GeV, Q2 = 10 GeV. Plotted are the improved NLL approximation (dashed line; exact NLO result plus NLL approximate NNLO result with the damping factor √ (2,1) (2,2) 1/ 1 + η ) and the best approximation (solid line; exact results at NLO and for c2,g and c2,g plus √ (2,0) NLL approximation c2,g with the damping factor 1/ 1 + η ). (b): Same as Fig. 13a for x = 0.01.

Figure 13:

19

10 10 10 10

-1

(a) charm F2

2.75 2.5 2.25 2 1.75 1.5 1.25 1

(b) K-factor

-2

-3

-4 -3 -2 -1

10

10

x

10

1

10

-3

10

-2

x

10

-1

1

gluon PDF, ? = m = 1.6 GeV and Q2 = 10 GeV. Plotted are: The exact result (solid line), the LL approximation (dotted line) and the NLL approximation (dashed line). (b): The x-dependence charm /F charm (solid line) and F charm charm charm of the ratios F2 (N LO ) 2 (LO ) 2 (N N LO ) /F2 (N LO ) (dashed line) with F2 (N N LO ) in the improved NLL approximation (exact NLO result plus NLL approximate NNLO result with the √ damping factor 1/ 1 + η ) and parameter choices as in Fig. 14a.

charm at NLO with the CTEQ4M Figure 14: (a): The x-dependence of the charm structure function F2

10 10 10 10 10

-1 -2 -3 -4 -5

(a) charm F2

10 10 10 ( × 0.1 ) 10 10 10

-1 -2 -3 -4 -5 -6

(b) charm F2

( × 0.1 )

10

-3

10

-2

x

10

-1

1

10

-3

10

-2

x

10

-1

1

charm at NLO (lower band), scaled down by a factor of (a): The x-dependence of F2 10 and (upper band) in the improved NLL approximation at NNLO (exact NLO result plus NLL √ approximate NNLO result with the damping factor 1/ 1 + η ) with the CTEQ4M gluon PDF, ? = m, Q2 = 10 GeV. The bands correspond to the charm mass variation 1.35 GeV ≤ m ≤ 1.7 GeV. (b): Same as Fig. 10a for ? = Q2 + 4m2 .

Figure 15:

20

charm may see from the di?erence between the two approximations to F2 in Fig. 13b. Finally, we note that a fully consistent NNLO scale dependence study would require not-yet-available NNLO parton densities, and the presently also unknown three-loop anomalous dimensions.

charm The ?nal comparison in this subsection involves the size and x dependence of F2 at a ?xed value of the factorization scale ? = m. In Fig. 14a we compare the exact result of Ref. [4] with the LL and the NLL approximations from Eq. (37), over a range of x, 0.003 ≤ x ≤ 0.3. We ?nd that the deviations of the NLL approximation from the exact result are often very small, at most 10% for the value of x = 0.003. The excellence of the NLL approximation to the x dependence also holds for other values of ?. In Fig. 14b we display for the same kinematics and over the same range in x the e?ect of charm charm the NNLO corrections. We plot the K-factors F2 (N N LO ) /F2 (N LO ) and, for comparison, also charm charm 6 charm F2 (we (N LO ) /F2 (LO ) . At NNLO we have taken the improved NLL approximation to F2 recall: the exact NLO result plus the NLL approximate NNLO result with the damping factor √ 1/ 1 + η ). We see that particularly for smaller x, the size of the NNLO corrections is negligible, the K-factor being close to one, whereas for larger x, their overall size is still quite big, almost a factor of 2 at x = 0.1. charm We note furthermore that the largest error on the approximate NNLO F2 , as at NLO [14], is still due to the uncertainty in the charm mass. Varying the mass of the charm from charm 1.35 to 1.7 GeV, we found sizeable variations in the absolute value of F2 , in particular a larger x. In Figs. 15a and 15b we plot for two values of the factorization scale ? the improved charm NLL approximation to F2 as a function of x, 0.003 ≤ x ≤ 0.3. For comparison, Figs. 15a and 15b also contain the exact NLO result with same variations of the charm mass. In general, charm decreasing the charm mass increases F2 . charm This concludes our investigation of the inclusive structure function F2 .

3.4

charm dF2 /dpT

charm In this subsection we study the distribution dF2 /dpT . Soft gluon resummation might be especially fruitful here because the requirement that the detected charm quark has a ?xed transverse momentum increases the sensitivity to threshold dynamics by e?ectively increasing the heavy quark mass from m to the transverse mass mT = m2 + p2 T. 2 Eq. (6) provides us with the di?erential distribution d F2 /dT1 dU1 from which we can derive the experimentally relevant distribution d2 F2 /dydpT , with y being the rapidity in the c.m. frame of the vitual photon-proton system and pT the transverse momentum of the detected heavy quark. In terms of the Mandelstam variables of Eq. (4), we can write the energy of the outgoing heavy-quark as

E = ?

Q2 + T1 + U1 √ . 2 S

(51)

The transverse momentum pT and the longitundinal momentum pL are determined by

2 ′ 2 2 2 ′ S ′ 2 (p 2 T + m ) = S T1 U1 + Q T1 + Q S T1 , 2 2 2 p2 L = E ? m ? pT ,

(52) (53)

6

charm For F2 (LO) we used a two-loop αs and NLO gluon density.

21

10

-1

(a)

-2

charm dF2 /dpT

10

-1

(b)

charm dF2 /dpT

10

10 10

-3

-2

10 10

-4

-3

0

2

4 pT [ GeV ]

6

8

0

2.5

5 7.5 pT [ GeV ]

10

12.5

Figure 16:

charm /dp as a function of p at NLO with the (a): The di?erential distribution dF2 T T

CTEQ4M gluon PDF, x = 0.01, m = 1.6 GeV, Q2 = 10 GeV and scale choice ? = Q2 + 4(m2 + p2 T ). Plotted are: The exact result (solid line), the LL approximation (dotted line) and the NLL approxi√ mation with the damping factor 1/ 1 + η on the scale dependent terms (dashed line). (b): Same as Fig. 16a for x = 0.001.

where S ′ = S + Q2 . The rapidity y y = 1 E + pL ln 2 E ? pL . (54)

may then be expressed in terms of (T1 , U1 ) by Eqs. (51) and (53). The transformation (T1 , U1 ) → (pT , y ) is then de?ned. Neglecting all light initial state quarks, one obtains from Eq. (6) the result for the inclusive distribution dF2 /dpT , 2 pT d2 F2 (x, pT , Q2 , m2 ) ? dpT S′

y+ 1

dy

y? z?

dz x φg/P (z, ?2 ) ω2,g , s4 , t1 , u1 , Q2 , m2 , ?2 , z z

(55)

2 where 0 ≤ p2 T ≤ S/4 ? m in the physical region and

y

±

= ± cosh

?1

S √ 2 mT

√

.

(56)

With the above expressions, and using the one- and two-loop expansions of the hard part charm in section 3.1, we can construct NLL approximations to dF2 /dpT at NLO and NNLO. In charm analogy to the inclusive F2 , we can then compare the approximate NLO results with the exact ones of Ref. [15], and, where appropiate, make NNLO estimates. In Figs. 16a and 16b we show the NLO results vs. pT for x = 0.01 and 0.001, and m = 1.6 GeV and Q2 = 10 GeV. As in Ref. [15], the factorization scale has been chosen as ? = 7 Q2 + 4(m2 + p2 T ). At NLO, we compare our LL and NLL approximate results with the exact results of Ref. [15] and we see that the exact curves are reproduced well both in shape and magnitude by our NLL

charm We have checked that the distribution dF2 /dpT , when integrated over pT , reproduces the inclusive charm charm structure function F2 in section 3.3 for the choice ? = Q2 + 4m2 . 7

22

10

-1

(a)

-2

charm dF2 /dpT

10

-1

(b)

charm dF2 /dpT

10

10 10

-3

-2

10 10

-4

-3

0

2

4 pT [ GeV ]

6

8

0

2.5

5 7.5 pT [ GeV ]

10

12.5

Figure 17:

charm /dp as a function of p with the CTEQ4M (a): The di?erential distribution dF2 T T

gluon PDF, x = 0.01, m = 1.6 GeV, Q2 = 10 GeV and scale choice ? = Q2 + 4(m2 + p2 T ). Plotted are: The exact NLO result (solid line) and the improved NLL approximation at NNLO (dashed line) √ (exact NLO result plus NLL approximate NNLO result with the damping factor 1/ 1 + η ). (b): Same as Fig. 17a for x = 0.001.

approximations, whereas the curves for LL accuracy systematically underestimate the true charm result. In particular, we want to point out that for dF2 /dpT the kinematical region in which our NLL threshold approximation applies, extends well down to x = 0.001, with an error of at most 15% for small pT , see Fig. 16b. The deviation at larger pT is presumably related to ln(pT /m) logarithms, associated with heavy quark fragmentation. For the same choice of kinematics, Figs. 17a and 17b display the results for the improved charm NLL approximation to dF2 /dpT at NNLO. We see that the small pT region receives sizeable contributions, the value of the maximum increases by 40% - 50%, whereas in the large pT region there is little change.

4

Conclusions

We have performed the resummation of threshold logarithms for the electroproduction of heavy quarks, to next-to-leading logarithmic accuracy, and in single-particle inclusive and pair invariant mass kinematics. For the former, we have executed an extensive numerical investigation into the quality of the approximation for the inclusive charm structure function, and for its transverse momentum distribution. In addition we have provided both analytical and numerical results for NNLO approximations, for those kinematic con?gurations where the NLO approximation ?ts the exact results well. We found that the region of applicability of our analysis extends well into the kinematic range of the HERA experiments, a result of the reaction being largely driven by initial state gluons, which on average have not much energy. Our studies show a clear superiority of next-toleading logarithmic threshold resummation over leading logarithmic resummation. In view of the present and much larger future HERA data sample, we hope that our results will ?nd use, for example in the determination of the gluon density, and its uncertainty, and, more generally, that they provide motivation for further application and development of threshold resummation techniques. 23

Acknowledgments

We would like to thank Gianluca Oderda, Jack Smith and George Sterman for enlightning discussions. This work is part of the research program of the Foundation for Fundamental Research of Matter (FOM) and the National Organization for Scienti?c Research (NWO).

Appendix A:

Laplace transforms

In this appendix we list the Laplace transform table needed to obtains the results of section 3. De?ne ∞ n 2 s4 ?N s4 /m2 ln (s4 /m ) d e . (A.1) In (N ) = m2 s4 /m2 +

0

For the lowest four values of n this integral is, up to O (1/N ) ?, I0 (N ) = ? ln N 1 2 ? 1 ln N + ζ2 , I1 (N ) = 2 2 1 3 ? ? ? 2 ζ3 , I2 (N ) = ? ln N ? ζ2 ln N 3 3 1 4 ? 3 ? + 2ζ3 ln N ? + 3 ζ 2 + 3 ζ4 , ln N + ζ2 ln2 N I3 (N ) = 4 2 4 2 2 (A.2) (A.3) (A.4) (A.5)

? = NeγE and γE denoting the Euler constant. Note that these answers are identical to with N the ones for the Mellin transform in [29].

Appendix B:

NLL resummed heavy quark pair-inclusive cross section

Here we consider the singular behavior of the cross section for heavy quark electroproduction, in pair-invariant mass (PIM) kinematics. We present the resummed cross section, and its oneand two-loop expansions. Speci?cally, we consider the reaction γ ? (q ) + P (p) ?→ QQ(p′ ) + X (pX ) , (B.1)

We denote p′2 = (p1 + p2 )2 = M 2 and de?ne τ = M 2 /S . The de?nitions of other invariants are given in section 2. The cross section of interest is di?erential with respect to M 2 , the scattering angle θ (in the pair center of mass frame), and the rapidity y of the pair. Adopting the same approximations as in section 2, it reads d3 σ (x, M 2 , θ, y, Q2) 1 = dM 2 d cos θ dy S ′2 1 1 dx′ ′ 2 φ ( x , ? ) δ y ? ln g/P x′ 2 x′ Q2 M 2 × ω z, θ, 2 , 2 , αs (?) . ? ? dz δ z? x4m2 /Q2 x′ ? x (B.2)

We have used here the observation [30] that the inclusive partonic cross section may be used to compute the singular behavior of the hadronic cross section at ?xed (small) rapidity. 24

The kinematics of the reaction (B.1) near threshold is determined by the vector ζ ? = (1, 0). (Recall that for 1PI kinematics ζ ? = p? 2 /m). Near threshold it decomposes as w = (1 ? τ ) = (1 ? x′ ) ≡ w1 2k S · ζ 1 + √ 1?x S

1 + wS . 1?x Following the methods described in section 2, we ?nd, in moment space ω ? N, θ, Q2 , αs (M ) M2 = H θ, Q2 , αs (M ) M2

M/N

(B.3)

? 1, αs M S N . (B.4)

× exp E(g)

1 , M2 N 1?x

exp

M

d?′ 2 Re {ΓS (αs (?′ ))} ?′

For simplicity, we have set the factorization scale ? = M , however it may be easily restored following the arguments leading to Eq. (24). The various functions in this expression are given explicitly in Eqs. (25)-(27) and (30), in which u1 = ?s′ (1 + β cos θ)/2, with β = t1 = ?s′ (1 ? β cos θ)/2 , (B.5)

1 ? 4m2 /s. The lowest order hard part in Eq. (B.2), de?ned by ω (0) (s′ , M, θ) = s′ δ (1 ? τ ) σ Born (s′ , M, θ) , (B.6)

is related to its counterpart for 1PI kinematics in Eq. (34) by σ Born (s′ , M, θ) =

?s′ s′ 2π 2 αβ Born ′ σ s , t = (1 ? β cos θ ) , u = ? (1 + β cos θ) . 1 1 s′ Q2 2,g 2 2 ω (1) (s′ , θ, M ) ? K (1) σ Born (s′ , θ, M ) ,

(B.7)

The NLL approximation to the exact NLO correction with ? = M is (B.8) 1 + ReLβ 1?x (B.9) where K (1) =

αs (M ) ln(1 ? z ) 2 CA π 1?z + ln M2 t1 u1 ? ln m4 m2

+

+

1 1?z

+

CA ?2 ln ,

? 2 CF (ReLβ + 1)

where lnl (1 ? z ) 1?z

+

The NNLO corrections with ? = M are to NLL accuracy where K (2) =

lnl (1 ? z ) 1 = lim θ(1 ? z ? δ ) + lnl+1 (δ ) δ (1 ? z ) . δ →0 1?z l+1 ω (2) (s′ , θ, M ) ? K (2) σ Born (s′ , θ, M ) ,

(B.10)

(B.11) t1 u1 + ReLβ m4 . (B.12)

2 ln3 (1 ? z ) αs (M ) 2 2 C A π2 1?z

+

+

ln2 (1 ? z ) 1?z

2 ln 3 CA +

?6 ln

M2 1 ? 3 ln 1?x m2 25

? 2 CA (b2 + 3 CF (ReLβ + 1))

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27

赞助商链接

- Soft gluon resummation for heavy quark and dijet cross sections
- Soft-gluon resummation for heavy quark production in hadronic collisions
- Soft-Gluon Resummation for Heavy Quark Production in Charged-Current Deep Inelastic Scatter
- Soft-Gluon Resummation for Bottom Fragmentation in Top Quark Decay
- Joint resummation for heavy quark production
- Soft-Gluon Angular Screening in Heavy-Quark Fragmentation
- Variable number schemes for heavy quark electroproduction
- Resummation for heavy quark and jet cross sections
- Resummation for Heavy Quark Production Near Partonic Threshold
- Working Group Report Heavy-Ion Physics and Quark-Gluon Plasma
- Obtaining the nuclear gluon distribution from heavy quark decays to lepton pairs in p$A$ co
- Heavy-quark production in gluon fusion at two loops in QCD
- Resummation for heavy quark and jet cross sections
- Dissociation of Heavy Quarkonia in the Quark-Gluon Plasma
- Working Group Report Heavy-Ion Physics and Quark-Gluon Plasma

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