Differential distributions for top-quark hadro-production with a running mass

We take a look at how the differential distributions for top-quark production are affected by changing to the running mass scheme. Specifically we consider the transverse momentum, rapidity and pair-invariant mass distributions at NLO for the top-quark mass in the MSbar scheme. It is found that, similar to the total cross section, the perturbative expansion converges faster and the scale dependence improves using the mass in the MSbar scheme as opposed to the on-shell scheme. We also update the analysis for the total cross section using the now available full NNLO contribution.

with the scale dependent MS mass by means of the well-known relation in perturbation theory, for the scheme change from m pole t to the running MS mass m(µ r ) taken at the renormalization scale µ r . To NNLO the coefficients d 1 and d 2 are given by [22] (see also Refs. [23,24])  with ℓ = ln µ 2 r m(µ r ) 2 and assuming vanishing masses for all lighter quarks.
Let us briefly illustrate the advantages of the running MS mass m(µ r ) for the total tt cross section. The recently completed exact NNLO QCD result [6][7][8][9] turned out to be very close, i.e., within O(1 − 2%), to previous approximations based on the combined threshold and high-energy asymptotics [26] and has been presented as a function of the pole mass m pole t . The necessary scheme transformation from m pole t to m(µ r ), i.e., the application of eq. (1), has been discussed in [18] and is implemented in the program Hathor (version 1.5) [27], a tool for the calculation of the total tt cross section in hadronic collisions.
The much improved apparent convergence of the perturbative expansion with the running mass as well as the scale stability are illustrated in Figs. 1 and 2 where we compare theory predictions for the total tt cross section as a function of the pole and the MS mass, respectively. Fig. 1 displays the increase in the cross section values from LO to NNLO, where we have taken the parton distribution functions (PDFs) to be order independent. For an on-shell mass m pole t = 173 GeV, for instance, the relative increase is σ NLO /σ LO = 1.46 and σ NNLO /σ NLO = 1.12 at the scale µ r = µ f = m pole t . This is to be compared with a much reduced increase of only σ NLO /σ LO = 1.26 and σ NNLO /σ NLO = 1.03 for m(m) = 163 GeV in the MS scheme at the scale µ r = µ f = m(m). These findings can be understood by noting that the scheme transformation of eq. (1) applied to the total tt cross section effectively shifts all parton-level corrections to the threshold region thereby improving the apparent convergence of the perturbation series, see, e.g., [28]. Fig. 2 shows the scale stability for the LHC predictions confirming earlier findings for the Tevatron, cf. [18]. The scale variation for the cross section in the on-shell scheme in the standard range µ/m pole t ∈ [1/2, 2] amounts to ∆σ NNLO = +3.8% −6.0% , whereas for the running mass we only find ∆σ NNLO = +0.1% −3.0% for the range µ/m(m) ∈ [1/2, 2]. Interestingly, for an on-shell mass the point of minimal sensitivity where σ LO ≃ σ NLO ≃ σ NNLO is located at fairly low scales, µ ≃ m pole t /4 ≃ 45 GeV, whereas for a running mass it resides at the scale µ = O(m(m)), i.e., it coincides with the natural hard scale of the process. These results imply, that experimental determinations of the running mass from the measured cross section are feasible with very good accuracy and a small residual theoretical uncertainty. For Tevatron data such analyses have already been performed in the past [17,29].
For completeness, we include here values for the full NNLO cross sections at the Tevatron ( √ S = 1.96 TeV) and at the LHC for various energies of interest.   Next we discuss the single-differential distributions in the top-quark's transverse momentum p t T and rapidity y t and in the invariant mass m tt of the tt-pair, which are all known to NLO in QCD [10,11] in the conventional pole mass scheme. As we are interested in the differential cross sections with the mass in the MS scheme, we briefly recall the kinematics of heavy-quark hadroproduction, where h 1 and h 2 are hadrons, X [Q] denotes any allowed hadronic final state containing at least the heavy anti-quark, and Q(p 1 ) is the identified heavy-quark with mass m. The hadronic invariants in this reaction are The double differential cross section for eq. (5) in terms of the hard parton cross section σ i j and PDFs f i at the factorization scale µ 2 reads and the partonic invariants are related to their hadronic counterparts through with the limits on x 1 and x 2 , In order to write the differential cross section in terms of p t T , y t and m tt , we will also need their definitions in terms of the hadronic invariants. For the case of p t T and y t , the relations are whereas for m tt , pair-invariant mass kinematics is used, in which case the requirements on the integrals are In these kinematics, the relevant partonic invariants for writing the differential cross section in terms of m tt are, with β t = 1 − 4m 2 / m tt 2 and θ the scattering angle of the top quark. Full discussions of the kinematics to NLO for one-particle inclusive and pair-invariant mass kinematics are available in [11,31] respectively.
In order to convert to cross section predictions with the mass in the MS scheme, we start from the on-shell description: where X denotes any of the variables p t T , y t and so on. If we now replace m pole t with m(µ r ) using eq. (1), we can expand in α s and obtain a description of the differential cross section in the MS scheme.
The only extra part required is the mass derivative of the Born contribution. This has been computed semi-analytically for the p t T , y t , and m tt distributions. To see why we also need some numerical derivatives in this calculation, consider eq. (6) for the Born contribution to the double differential cross section as a starting point: where the delta function imposes Born kinematics and can be used to carry out the integral over x 2 through its relation to s,t 1 and u 1 . Re-writing the cross section in terms of p t T and y t provides us with the form of the integrand that will need to be evaluated, where L(x 1 , x 2 , µ 2 ) = f 1 (x 1 , µ 2 ) f 2 (x 2 , µ 2 )/x 1 x 2 is the differential parton luminosity.
The most important aspect to note is that both x 2 and x − 1 depend on the top-quark mass through their relations to the Mandelstam variables. This means that the mass derivative of the PDFs needs to be done numerically using This form of the derivative is found to converge well. Aside from this, all other derivatives are known analytically. When compared with a fully numerical calculation of the derivative term, it is found that the two methods agree to less than 1%. In the case of m tt , the integration limits and variables do not depend on the top-quark mass (m) so all derivatives are computed analytically.
Using the relations presented here, we have computed the differential cross sections for ttproduction in terms of p t T , y t and m tt . We have used the program MCFM [32] for the NLO corrections [13,33]  mass derivative terms. The calculations were carried out using the ABM11 [25] and CT10 [30] PDFs at NLO. As a check, each curve was integrated to obtain a result for the full cross section. In all cases, the value agreed within less than 1% of the cross section computed using Hathor. As well, the mass derivatives were checked by computing the differential cross sections at values of the top mass ranging between 150 GeV and 180 GeV. A curve was fit to each point in the relevant spectrum to obtain the derivative at the given MS mass. Again, these values agreed within less than 1% of the (semi-)analytic derivatives used.
In Fig. 3 the rapidity distributions are shown for the MS and pole mass schemes. It is clear from these that at NLO, the convergence of the perturbative series as well as the scale dependence improves. In the pole-mass scheme, a relative increase for the cross section ratios σ NLO /σ LO = 1.50 is seen, while in the MS scheme we have σ NLO /σ LO = 1.31 at y t = 0. The scale variation in the on-shell scheme is ∆σ NLO = +9.5% −14% while in the MS scheme, we have ∆σ NLO = +4.5% −12% again at y t = 0. Fig. 4 shows the transverse momentum distributions. Again we see an improvement when moving from the pole mass scheme to the MS scheme. In this case the improvement in the NLO contribution is a bit better with σ NLO /σ LO = 1.50 for the pole mass scheme and σ NLO /σ LO = 1.25 in the MS scheme. The scale variation goes from ∆σ NLO = +13% −13% in the pole mass scheme to ∆σ NLO = +6.4% In addition to these improvements, moving from the pole mass to the MS scheme changes the overall shape of the distributions so that the peak positions generally become more pronounced. This is a consequence of the radiative corrections being shifted to the threshold region as mentioned dσ/dp  earlier. However, the peak positions in both the p t T and m tt distributions are stable against radiative corrections. At most they are seen to shift by 1%, which is unlike the case for tt-production from e + e − collisions where the position of the tt-threshold peak shifts significantly upon adding NLO and NNLO perturbative corrections to the total cross section expressed in terms of the pole mass [34].
Another salient feature not shown in Fig. 5 above occurs in the MS differential cross section with respect to the invariant mass of the tt pair. Very close to the threshold of tt production the contribution reponsible for the change in the mass renormalization scheme, i.e., the derivative term in eq. (13), becomes large. This is due to the presence of a 1/β t which diverges as m tt → m, cf. eq. (11). These large corrections have the effect of causing the invariant mass spectrum to dip below zero for values of m tt > ∼ 2m t . In the full spectrum, however, this is counterbalanced by the positive contribution resulting in a cross section integrated over m tt that agrees within less than 1% with the value calculated in Hathor.
Obviously, this behavior is an indication of the breakdown of fixed-order perturbation theory.
First of all, bound-state effects in tt production at hadron colliders arise in the kinematic region m tt > ∼ 2m t , i.e., when the velocity β of the top quarks is small, β ≪ 1. In this region, the conventional perturbative expansion in α s breaks down, owing to singular terms ∼ (α s /β) n in the n-loop amplitude, which require the all-order resummation of the Coulomb corrections [35,36]. This resummation for tt dynamics close to threshold is carried out in a non-relativistic effective theory by means of a Schrödinger equation for which the pole mass definition seems to be the natural choice and which implies a certain power counting, so that all terms of order m t β 2 ∼ m t α 2 s are formally of equal size.
If the contribution for the change in the mass renormalization scheme δm sd from the pole mass to a so-called short-distance mass m sd t such as the MS mass m(µ r ) is parametrically larger than m t α 2 s that is δm sd ≡ m pole t − m sd ∼ m sd t α s , then δm sd becomes the dominant term in the kinematic region m tt > ∼ 2m t . Such situation is realized for δm sd ∼ m t α s , cf. eq. (13), and excludes the MS mass from being a useful mass near threshold. Of course, all these findings on the scheme choice for the mass definition close to the threshold are long known from studies for tt production in e + e − collisions [34]. Various solutions have been proposed, e.g., the alternative use of a so-called 1S mass [37] defined through the perturbative contribution to the mass of a hypothetical n = 1, 3 S 1 toponium bound state, cf. [38] for an application to tt hadro-production or the use of a "potentialsubtracted" (PS) mass [39], recently considered in [40] in the context of finite-width effects in unstable-particle production at hadron colliders. In any case, since the conventional perturbative expansion of the cross section breaks down for m tt > ∼ 2m t we do not display this particular kinematic region in Fig. 5. Moreover, with the currently given experimental resultion for the m tt -bins, cf. [4], it will be difficult to access this region at the LHC at all.
For completeness we also provide a table of values for the cross section at LHC with √ S = 8 TeV at binned values of y t , p t T and m tt with binning approximately equal to that of [4]. Comparing the data generated using ABM11 as compared to CT10, we see that there is an overall shift downward consistent with that observed for the total cross section, cf. Tabs. 1 and 2. The improvement of the apparent perturbative convergence and the scale stability when moving from the pole mass scheme to the MS scheme is consistent for both PDF sets.   Table 3: Values for the y t differential cross section for top-quark pair-production at LO and NLO for various y t using the PDF set CT10 [30] with √ S = 8TeV. All rates are in pb.
In summary, we have shown how treating the differential cross sections for tt production in the MS scheme for the top-quark mass has benefits as compared to the pole mass scheme. The     perturbative series shows the same improvement in convergence and scale dependence as has been observed for the total cross section. As a consequence the NLO contributions with a MS mass are expected to provide already very precise cross section predictions. An extension to NNLO  Table 7: Values for the m tt differential cross section for top-quark pair-production at LO and NLO for various m tt using the PDF set CT10 [30]. All rates are in pb/GeV.  accuracy would provide results with a still smaller theoretical uncertainty from the scale variation. Yet, the predictions at the nominal scale, i.e., µ r = m(m), are expected to remain largely unchanged. As future prospects we note that the refinement of the present phenomenological analysis to NNLO accuracy is certainly feasible once the complete NNLO QCD corrections for differential tt production are available. As a first step in this direction, one may consider approximate NNLO corrections based, e.g., on the dominant threshold logarithms. Other obvious improvements are extension to double-differential distributions and other exclusive observables, even including topquark decay.