DESY 12-049 March 2012
HU-EP-12/10
LTH 940
LPN 12-042
SFB/CPP-12-16

S. Moch, P. Uwer and A. Vogt

Deutsches Elektronensynchrotron DESY

Platanenallee 6, D–15738 Zeuthen, Germany

Humboldt-Universität zu Berlin, Institut für Physik

Newtonstraße 15, D-12489 Berlin, Germany

Department of Mathematical Sciences, University of Liverpool

Liverpool L69 3BX, United Kingdom

Abstract

We study the QCD corrections at next-to-next-to-leading order (NNLO) to the cross section for the hadronic pair-production of top quarks. We present new results in the high-energy limit using the well-known framework of -factorization. We combine these findings with the known threshold corrections and present improved approximate NNLO results over the full kinematic range. This approach is employed to quantify the residual theoretical uncertainty of the approximate NNLO results which amounts to about 4% for the Tevatron and 5% for the LHC cross-section predictions. Our analytic results in the high-energy limit will provide an important check on future computations of the complete NNLO cross sections.

The cross section for top-quark pair production has been measured very precisely at the hadron colliders Tevatron and LHC with an experimental accuracy challenging the precision provided by the perturbative QCD corrections at next-to-leading order (NLO), which have been known for a long time [1, 2], see also [3, 4]. Much recent activity has been concerned with improvements of the theoretical status beyond NLO, see [5] and refs. therein. The dominant terms at higher orders have been used to derive approximate QCD corrections to next-to-next-to-leading order (NNLO) for the inclusive cross section [6]. These consist of large threshold logarithms at next-to-next-to-leading logarithmic accuracy (NNLL) which can even be resummed to all orders in perturbation theory and could provide sufficiently precise phenomenological predictions. Yet, recent phenomenological studies based on threshold resummation to NNLL [7, 8, 9, 10, 11] have reported somewhat differing predictions and, moreover, have proposed different means of estimating the residual theoretical uncertainty which is predominantly due to uncalculated higher orders (beyond NNLO) and the effects of hard radiation not accounted for by threshold enhanced logarithms.

In this letter we consider the constraints on hadronic heavy-flavor production imposed by the high-energy factorization of the cross section [12, 13]. This provides important complementary information on the hard partonic scattering processes in the limit when the center-of-mass energy is much larger than the top-quark mass. It allows to extend previous approximations of the exact (yet unknown) NNLO results to the entire kinematical range and thus to obtain a more realistic uncertainty inherent in those approximate NNLO results.

The hadronic cross section for top-quark pair production is computed by the convolution of the partonic scaling functions with the parton luminosities ,

 σh1h2→t¯tX(S,m)=αs2m2∑i,jS∫4m2dsLij(s,S,μ)fij(s,m,μ,αs), (1)

where denotes the hadronic center-of-mass energy squared, and the top-quark mass in the on-shell (pole-mass) scheme. The parton luminosities are defined as

 Lij(s,S,μ)=1SS∫sd^s^sfi/h1(^sS,μ)fj/h2(s^s,μ), (2)

with the standard parton distribution functions (PDFs) . The QCD coupling constant is evaluated at the scale in the scheme with light flavors, and the renormalization and factorization have been identified (i.e., ). Up to NNLO, the scaling functions can be expanded as

 fij=f(0)ij+4παs{f(10)ij+LMf(11)ij}+(4παs)2{f(20)ij+LMf(21)ij+L2Mf(22)ij}+O(αs3), (3)

where we abbreviate . The dependence on , included by the functions and is known exactly from [14, 7, 15].

For the high-energy factorization one considers Mellin moments with respect to ,

 fij(ω,μ)=1∫0dρρω−1fij(ρ,μ). (4)

The resummation of the high-energy logarithms in for , or, equivalently, of the singular terms in Mellin space as , is based on the framework of PDFs un-integrated in the transverse momentum and the concept of -factorization. It is often also denoted to as small- resummation referring to the context of deep-inelastic scattering (DIS). The procedure involves two steps, i.e., the computation of amplitudes with the initial particles off-shell in , and the subsequent convolution with a gluon PDF which has the corrections for small- included. For hadronic heavy-quark production, this leads to an expression for the cross section in Mellin space as a product of the (small- resummed) gluon PDF and the corresponding impact factor depending on through an anomalous dimension . which is determined by the well-known BFKL kernel.

For the purpose of this letter we are interested in the NNLO predictions of high-energy factorization in the framework of standard collinear factorization. This requires the computation of the leading terms in Mellin space as . Using the heavy-quark impact factor of [13], the analytic result for inclusive heavy flavor hadro-production at NLO [4], the FORM routines of [16, 17], and the PSLQ algorithm as implemented in Maple we arrive at the following expressions for the scaling functions at high energies for a general SU gauge theory, where we define .

At Born level we have up to order ,

 f(0)q¯q = (5) f(0)gg = π\*(415\*NcVc−718\*1Nc\*Vc)+ω\*π\*(−781900\*NcVc+4336\*1Nc\*Vc+{815\*NcVc−79\*1Nc\*Vc}\*ln2). (6)

Note that subleading terms in , i.e.. and higher are not predicted by BFKL evolution. These terms are however required for the asymptotic behavior in NNLO.

At NLO up to order with denoting the number of light flavors the functions read,

 4πf(10)q¯q = 1915400\*Nc−8398100\*1Nc+2213240\*1Nc3−{215\*1Nc−215\*1Nc3}\*ζ2+{150−150\*1Nc2}\*nf , (7) 4πf(11)q¯q = 1190\*Nc−1190\*1Nc−{145−145\*1Nc2}\*nf, (8) 4πf(10)gq = 1ω\*(77225−41108\*1Nc2)−194893108000+13135764800\*1Nc2+{154225−4154\*1Nc2}\*ln2 , (9) 4πf(11)gq = 1ω\*(−215+736\*1Nc2)+9411800−527720\*1Nc2−{415−718\*1Nc2}\*ln2, (10) 4πf(10)gg = 1ω\*(308225\*Nc2Vc−4127\*1Vc)+36475115120\*1Vc−69711680\*1Nc2\*Vc−736427108000\*Nc2Vc (11) +{616225\*Nc2Vc−8227\*1Vc}\*ln2+815\*Nc2Vc\*ζ2−{1120\*Nc2Vc+48935\*1Vc−14135\*1Nc2\*Vc}\*ζ3 +89\*Nc2Vc\*CF4+1720\*NcVc\*nf, 4πf(11)gg = 1ω\*(−815\*Nc2Vc+79\*1Vc)+407150\*Nc2Vc−10327\*1Vc−{1615\*Nc2Vc−149\*1Vc}\*ln2. (12)

Finally, at NNLO we have up to order ,

 (4π)2f(20)q¯q = 1ω2\*1π\*(24623375\*Nc−8846381000\*1Nc+235648\*1Nc3−{115\*Nc+11360\*1Nc−772\*1Nc3}\*ζ2) (13) +1ω\*C(20)x,q¯q , (4π)2f(21)q¯q = 1ω2\*1π\*(−77225\*Nc+19492700\*1Nc−41108\*1Nc3)+1ω\*1π\*(222613108000\*Nc−708437162000\*1Nc (14) +14980764800\*1Nc3−{154225\*Nc−19491350\*1Nc+4154\*1Nc3}\*ln2−{127\*Nc−227\*1Nc +127\*1Nc3}\*nf), (4π)2f(22)q¯q = 1ω2\*1π\*(115\*Nc−59360\*1Nc+772\*1Nc3)−1ω\*1π\*(11213600\*Nc−27013600\*1Nc+79180\*1Nc3 (15) −{215\*Nc−59180\*1Nc+736\*1Nc3}\*ln2), (4π)2f(20)gq = 1ω2\*1π\*(24621125\*Nc−479324\*1Nc−{215\*Nc+736\*1Nc}\*ζ2)+1ω\*C(20)x,gq, (16) (4π)2f(21)gq = 1ω2\*1π\*(−7775\*Nc+4136\*1Nc)+1ω\*1π\*(1496933216000\*Nc−3625007226800\*1Nc+69713360\*1Nc3 (17) −{15475\*Nc−4118\*1Nc}\*ln2−415\*Nc\*ζ2+{1140\*Nc+48970\*1Nc−14170\*1Nc3}\*ζ3 −49\*Nc\*CF4−{2937200−154\*1Nc2}\*nf), (4π)2f(22)gq = 1ω2\*1π\*(15\*Nc−724\*1Nc)+1ω\*1π\*(−15411200\*Nc+78714320\*1Nc+{25\*Nc−712\*1Nc}\*ln2 (18) (4π)2f(20)gg = 1ω2\*1π\*(30892250\*NcVc+196963375\*Nc−{5990\*NcVc+415\*Nc}\*ζ2)+1ω\*C(20)x,gg, (19) (4π)2f(21)gg = 1ω2\*1π\*(−616225\*Nc3Vc+8227\*NcVc)+1ω\*1π\*(35840918000\*Nc3Vc−125210322680\*NcVc+6971840\*1Nc\*Vc (20) −169\*Nc3Vc\*CF4−{2931800\*Nc2Vc−2675\*1Vc+103324\*1Nc2\*Vc}\*nf), (4π)2f(22)gg = 1ω2\*1π\*(815\*Nc3Vc−79\*NcVc)+1ω\*1π\*(−1771450\*Nc3Vc+40372\*NcVc (21) +{1615\*Nc3Vc−149\*NcVc}\*ln2+{445\*Nc2Vc−47270\*1Vc+7108\*1Nc2\*Vc}\*nf),

where denote the values of the Riemann zeta-function and the constant is calculated from

 CF4=1∫0dρρF4(x)=−0.1333, (22)

where is given in eq. (19) of [4] and the value for has been determined numerically.

All of the above formulae may be easily converted to momentum space with the replacements and , cf. eq. (4). At NNLO, the leading terms (LL) proportional to in the NNLO quantities follow directly from [13]. In addition, the new next-to-leading terms (NLL) proportional to in the scale dependent parts in have been derived using standard renormalization group methods, see [14, 7, 15]. This leaves the unknown NLL terms denoted by , and in eqs. (13), (16) and (19). It is a general feature of small- expansions that the formally subleading terms are numerically important, and that the ratio of NLL to the LL term is large, see, e.g., eqs. (14), (17) and (20). Therefore, an estimate for these unknown terms is phenomenologically required.

We estimate , and as follows. It has been observed (and also exploited constructively) [12] that the impact factors in the high energy factorization for a number of different processes with initial state hadrons are related to each other. In particular, the Abelian part of the impact factor for heavy-quark hadro-production is connected by a simple rescaling proportional to from the one for heavy-quark DIS evaluated at the scale of for the photon virtuality.

In the latter case, that is for the deep-inelastic production of a heavy-quark pair via scattering off a virtual photon off an initial quark or gluon, the NLL terms at NNLO have recently been addressed in [18]. In DIS the heavy-quark coefficient functions are subject to an exact factorization [19] in the asymptotic limit into the respective coefficient functions with massless quarks and heavy-quark operator matrix elements (OMEs). The approximate NNLO results for those heavy-quark coefficient functions are based on the three-loop results of [20, 21] and can be extended to good accuracy to all scales for the photon virtuality, in particular also to the scale , see [18] for details. We can use this information to estimate the ratios and of the NLL to the LL terms for and in eqs. (16), (19). Subsequently, we multiply these ratios with the exact LL terms of eqs. (16), (19) which assumes, of course, that the non-Abelian contributions to the NLL terms for heavy-quark hadro-production do not lead to significant deviations. This assumption is motivated by the fact that the LL terms of the scaling functions at high energy are related by simple replacements of color factors, e.g., to LL accuracy. Also, in cases where the NLL are known exactly, e.g., the three-loop splitting functions [22], such relations still hold to a good approximation. In this way we arrive at,

 C(20)x,gq = rx,gq1π\*(737813121500−251540\*ζ2)withrx,gq=−5.6,…,−7.7, (23) C(20)x,gg = rx,gg1π\*(32440318000−251240\*ζ2)withrx,gg=−4.8,…,−8.2, (24)

where the terms in brackets derive from the LL term of eqs. (16), (19) proportional to with and substituted. The uncertainty ranges in the estimates for and from [18] are mainly driven by the finite number of Mellin moments currently available for the heavy-quark OMEs [21], which limit the extrapolation to . For in eq. (24) these findings are also corroborated by the results of a Padé estimate. See e.g., [23] for definitions and the use of Padé estimates at higher orders in perturbations theory. We use eqs. (6), (11) as input for a Padé estimate of to derive the value of and we have also checked that the Padé procedure predicts the NLL terms of , , and in eqs. (17), (18), (20) and (21) even with an accuracy of 5%.

For we can neither establish directly a relation to heavy-quark DIS nor can we perform a Padé estimate due to the vanishing NLO limit. Therefore, we use the same range of values for the ratio given in eq. (24), however rescaled a factor derived from the respective ratios of the NLL to the LL terms for and in eqs. (5), (6). The motivation for this procedure is again, the above mentioned relations of the various scaling functions under simple exchange of color factors, see [12, 13]. Thus we use

 C(20)x,q¯q = rx,qq1π\*(502417273375−2511215\*ζ2)withrx,qq=−3.0,…,−5.1, (25)

where the brackets contain the LL result of eq. (13) with the substitution and . As a check, we note that this procedure, if applied to the above mentioned Padé estimate for and predicts the NLL terms in and of eqs. (14), (15) again with an accuracy of typically 5%. Therefore, we conclude that the range for quoted in eq. (25) is a rather conservative one.

Let us now employ the above findings. Specifically, we are interested in combining the approximations in the two kinematical regions, i.e., at threshold and at high energy (small-) in order to arrive at smoothly interpolating functional forms for the scaling functions. Whenever possible, we compare to the exact results in order to check the quality of the approach. We choose the following ansatz for at one- and two-loops,

 f(1)ij = ρlf(1)threshij+βkf(1)LLxijηγC+ηγ, (26) f(2)ij = ρlf(2)threshij+βk(−lnρf(2)LLxij+f(2)NLLxijηγC+ηγ), (27)

where is the heavy-quark velocity and is the distance from threshold. For the parton channels the parameters take the values , and for we have , . These values reflect the exact functional dependence on and in the respective kinematical limits. The well-known threshold expansions are denoted and given, e.g., in [15]. The high-energy asymptotic behavior is split in LL and NLL parts and corresponding to eqs. (7)–(21). The high- tail proportional to (or in Mellin space) is smoothly matched with a factor . The suppression parameters in eqs. (26), (27) take the following values at NLO as a best fit for ,

 γ=0.99,  C=20.9  for gqandγ=1.18,  C=97.3  for gg, (28)

and at NNLO fitted to ,

 γ=1.37,  C=47.9  for q¯q,γ=0.90,  C=16.4  for gqandγ=0.84,  C=12.6  for gg. (29)

In Fig. 1 we show the results of this procedure for the scaling functions and . In particular, we compare the exact results with the approximations of eqs. (26), (27) using the values of eqs. (28) and (29) for the parameters and . The plots in Fig. 1 show a perfect match at both end of the kinematical range with an accuracy at the per mille level and even better as and for . This is very a strong check in particular on the results of which are known numerically from renormalization group methods [7]. Some deviations between the approximations of eqs. (26), (27) and the exact results in the central range of are visible in Fig. 1. However, these have generally a small impact on cross section predictions for hadron colliders, because the necessary convolution with the parton luminosities in eq. (1) is a non-local operation and has a smoothening effect. Moreover, the parton luminosities are steeply falling functions as grows large, giving numerically the most weight to the threshold region, which is after all the rational behind phenomenology based on the threshold resummation. In summary, the plots in Fig. 1 demonstrate that the chosen approach of combining the threshold expansion and the high-energy asymptotics leads to very good approximations of the exact scaling functions.

The main object of our interest are the scaling functions . Here we aim at defining an uncertainty band which combines both, the threshold approximation and the high-energy limit, and also accounts for an error estimate due to the uncalculated next term in the expansions in either kinematical region. At large , this is achieved with the NLL terms in and which contain the values of , and with the conservatively estimated ranges given in eqs. (23)–(25). The known threshold contributions for the functions and on the other hand contain the complete tower of logarithmically enhanced terms in , where , as well as all Coulomb corrections at two loops proportional to and which dominate as . Therefore, an estimate for an additional contribution of order (and vanishing as ) to be added to and serves as check on their inherent uncertainty. A Padé estimate based on the exact NLO results and yields for these constant and in the normalization of eq. (3) the values,

 C(20)β,q¯q = f(0)q¯q(4π)4(12769−172\*ln2+2563\*ln22−863\*ζ2−209\*nf+83\*nf\*ln2)2, (30) C(20)β,gg = f(0)q¯q(4π)4(444421−21367\*ln2+192\*ln22−2837\*ζ2)2, (31)

while the default values in phenomenological studies are usually taken as , see, e.g., the discussion in [15]. We neglect the -channel in these considerations, since it is very small near threshold anyway.

Thus, on the basis of eq. (27) and the discussion above we take the following two variants for the unknown full and dependence of the two-loop scaling functions,

 f(20)A/Bij = f(20)threshij+C(20)A/Bβ,ij+β3f(2)LLxij(−lnρ+rA/Bx,ijηγC+ηγ),for ij=qq,gg, (32) f(20)A/Bgq = ρf(20)threshgq+β5f(2)LLxgq(−lnρ+rA/Bx,gqηγC+ηγ), (33)

where we take the same parameters and for the respective channel as determined for in eq. (29) and the values for and are chosen as

 C(20)Aβ,qq=0,rAx,qq=−3.0, C(20)Bβ,qq=C(20)β,q¯q,rBx,qq=−5.1, (34) rAx,gq=−5.6, C(20)Bβ,qq=C(20)β,q¯q,rBx,gq=−7.7, (35) C(20)Aβ,gg=0,rAx,gg=−4.8, C(20)Bβ,gg=C(20)β,gg,rBx,gg=−8.2. (36)

The results for eqs. (32) and (33) are displayed in Fig. 2. The above procedure leads to the bands shown which widen significantly for large center-of-mass energies and rise with the same slope as due to the known logarithmic dependence on of the LL terms. It is evident from Fig. 2 and the numerical size of the various constants, and in eqs. (30), (31) as well as , and in eqs. (23)–(25) that the uncertainty in the latter is dominating even in the range of . Therefore a more accurate determination of , and , preferably a computation from first principles, is highly desirable. To a minor extent, the bands in Fig. 2 depend on the chosen matching, i.e., on eq. (29). However, the values for and in eq. (29) are all of the same order and, as we have shown in Fig. 1 this part of our procedure leads to reasonable descriptions in all cases where exact results are available.