Top-quark pair hadroproduction at NNLO: differential predictions with the MSbar mass

We consider top-quark pair production at the LHC within the MSbar scheme for the renormalisation of the top-quark mass, and we present predictions for total and differential cross sections at next-to-next-to-leading order (NNLO) in QCD. Our state-of-the-art calculation extends the available differential results by one order in the perturbative expansion, and it is relevant for a precise determination of the top-quark mass, including possible effects of the running mass in the MSbar scheme. We consider variations of the scale at which the MSbar mass of the top quark is evaluated, extending the usual 7-point to a 15-point scale variation. This additional variation is crucial for a reliable estimate of the theoretical uncertainties, especially at low perturbative orders and close to the production threshold of the top-quark pair. We also compute, for the first time, the invariant-mass distribution of the top-quark pair by using a running mass, evaluated at a dynamic scale. Our predictions for the invariant-mass distribution in the MSbar scheme are compared with a recent measurement performed by the CMS Collaboration. We observe that the inclusion of the NNLO corrections improves the agreement with the data, and we discuss effects due to the QCD running of the MSbar mass of the top quark.


Introduction
The top-quark mass is a fundamental parameter of the Standard Model (SM). Its large size, of the order of the electroweak (EW) scale, is associated with a strong coupling to the Higgs boson, and therefore with a possible role of the top quark in EW symmetry breaking. Its precise value directly affects the stability of the EW vacuum in the SM, since top-quark contributions drive the evolution of the Higgs boson self-coupling. In addition, the top-quark mass crucially enters in the computation of radiative corrections to precision SM observables.
For the above reasons, a precise measurement of the top-quark mass is of utmost importance for the LHC and future collider facilities. The measurement and interpretation of the top-quark mass at hadron colliders is, however, quite controversial (for a review, see Ref. [1]). The topquark mass is a parameter of the underlying field theory, and this implies that, at the formal level, it has to be treated similarly to any other bare parameter of the SM. The top-quark mass has to be renormalised and, in particular, its meaning and value depend upon the adopted renormalisation scheme.
A widely used renormalisation scheme for the top-quark mass is the pole scheme. Within this scheme the renormalisation procedure fixes the pole of the quark propagator, at any order in perturbation theory, to the same value, which is the pole mass M t . Other mass renormalisation procedures, such as the MS scheme (which is the customary scheme for the renormalisation of the QCD coupling α S ), can be used. In the MS scheme, the ultraviolet divergences are renormalised by removing only the singular contributions in the dimensionally regularized formulation of the underlying field theory. In this scheme the pole of the quark propagator receives corrections at any order in perturbation theory and, therefore, the MS renormalised mass m t (µ m ) differs from the pole mass M t . Moreover, the MS mass depends on the auxiliary renormalisation scale µ m , while the pole mass is renormalisation-scale independent. The physical predictions of the theory are independent of the scheme, provided the two renormalised masses and the corresponding perturbative calculations are formally related to all orders in perturbation theory. However, the use of different mass renormalisation schemes can have a non-negligible quantitative impact in the context of fixed-order predictions, especially at low perturbative orders.
In the context of top-quark production at hadron colliders, the top quark is viewed as a 'physical', though unstable, particle. In the limit of vanishing width of the top quark, this picture directly leads to considering theoretical calculations for the production of on-shell top quarks (t) and antiquarks (t) with a definite pole mass. The main source of top-quark events at hadron colliders is the production of top-quark pairs. QCD radiative corrections to the tt total cross section are available up to next-to-next-to-leading order (NNLO) within the pole mass scheme [2][3][4][5][6]. In Refs. [7][8][9][10] the NNLO QCD calculation of tt production is extended to the computation of differential cross sections.
Considering the calculation of on-shell tt production, the pole mass of the top quark can be re-expressed at the formal level in terms of a different mass parameter, such as the MS mass of the top quark. Approximate NNLO results for the tt total cross section using the MS mass were presented in Ref. [11]. Using the results of Refs. [2][3][4][5], the calculation was later extended to complete NNLO, and first next-to-leading order (NLO) results for differential distributions were presented in Ref. [12]. A common aspect of the studies of Refs. [11,12] is that only a fixed renormalisation scale is used for the MS mass. Specifically, the renormalisation scale is set to the value µ m = m t , where the mass parameter m t is defined by the relation m t (m t ) = m t in terms of the MS mass. Eventually, in Refs. [11,12] the pole mass M t is perturbatively re-expressed in terms of the scale-independent parameter m t in the MS scheme.
In this paper we present, for the first time, a fully differential QCD calculation of the top-quark pair cross section at NNLO by using the MS mass. The results are based on our implementation of the tt production cross section presented in Refs. [6,10], where fully differential predictions were obtained in the pole scheme by using the q T -subtraction method [13]. In addition, and at variance with previous works, we do not consider only one fixed renormalisation scale for the evaluation of the MS mass. We instead study µ m variations and their effect on the estimate of theoretical uncertainties. The use of the MS scheme or, more generally, of a short-distance and scale-dependent renormalisation procedure of the top-quark mass can have potential theoretical advantages with respect to the use of a fixed (scale-independent) top-quark mass. Indeed, the scale-dependent mass can be evaluated at the physical scale that is relevant for the observable under consideration. Therefore, in this paper we also study the use of dynamic scales for the running MS mass. Specifically, we compute the invariant-mass distribution of the tt pair by using fixed and dynamic scales, and we compare our NNLO results with a recent measurement performed by the CMS Collaboration [14].
The paper is organized as follows. In Section 2 we introduce the MS scheme for the renormalisation of the top-quark mass, we discuss its relation with the pole scheme, and we present the relevant formulas for the calculation of the NNLO cross section. In Section 3 we present our numerical results for tt production at the LHC energy of 13 TeV. In Section 3.1 we consider the results for the total cross section, and we discuss their scale dependence and related scale uncertainties. In Section 3.2 we consider differential cross sections, and we perform detailed comparisons between results in the pole and MS schemes. Finally, in Section 3.3 we present new results obtained by using fixed and running values of the MS mass, and we compare them with the corresponding CMS data. Our results are summarised in Section 4.

The heavy-quark cross section in the MS scheme
Perturbative calculations of QCD scattering processes involve UV divergences. Part of the divergences are reabsorbed in the renormalisation of the QCD coupling α S , and the customary procedure is to work in the MS scheme. The removal of the UV divergences associated with quark masses also needs the choice of a renormalisation scheme. Different schemes lead to renormalised masses whose relative difference is formally of O(α S ). A 'natural' renormalisation scheme is the pole scheme. The renormalised quark mass in this scheme is determined orderby-order in the perturbative expansion by the pole of the quark propagator, and, therefore, it corresponds to the customary meaning of mass for the case of a 'physical' quark. A possible alternative scheme is the use of MS renormalisation also for the quark mass.
The top-quark mass in the pole scheme, M t , is related to the MS mass at the scale µ m , m t (µ m ), through the following perturbative relation: The first two perturbative coefficients d (1) and d (2) in Eq. (1) have the values [15,16] where The three-loop coefficient d (3) was computed in Refs. [17,18], and the numerical result for d (4) was presented in Ref. [19]. Here and in the following, α S (µ) is the QCD coupling in the MS renormalisation scheme, and its running with the scale µ is understood to be computed with n f = 5 light flavours. The light quarks are considered to be massless throughout the paper.
A main feature of the MS mass is that it is a scale-dependent quantity, since it depends on the arbitrary mass renormalisation scale µ m . An ensuing feature (which follows from Eq. (1) and the fact that M t does not depend on µ m ) is that the scale dependence of m t (µ m ) is perturbatively computable. The dependence of m t (µ m ) on the scale is driven by the renormalisation group equation where the mass anomalous dimension γ m (α S ) has the perturbative expansion and γ 0 = 3 4 C F = 1. The perturbative expansion of the mass anomalous dimension in Eq. (5) is explicitly known [20,21] up to O(α 4 S ). We note that comparing to the α S evolution with n f = 5 flavours, the mass evolution is roughly a factor of two slower. Indeed, the first coefficient , and, setting n f = 5 we have The actual value of m t (µ m ) at any scale µ m can be specified by relying on the knowledge of M t and, therefore, by using Eq. (1). Alternatively, m t (µ m ) can be obtained by solving its renormalisation group equation, provided the MS mass is known at some reference scale. The choice of the reference scale is arbitrary, and it can be done independently of the knowledge of M t . For instance, one can simply choose a specific value of µ m (e.g., analogously to the choice of the mass of the Z boson as a typical reference scale for the running coupling α S ). The reference scale can also be chosen through an 'intrinsic' definition, as a scale of the same order as the MS mass itself. A customary intrinsic definition corresponds to the scale m t such that m t (m t ) = m t . In this case the coefficients d (1) (m t ) and d (2) (m t ) in Eq. (2) are positive and of order unity, and, therefore, the reference mass m t is typically smaller than the pole mass M t by about 10 GeV.
We note that the reference scale m t has no special physical meaning, since an intrinsic MS reference scale can be introduced differently. For instance, one can define the scale m λ such that m t (λ m λ ) = m λ , where λ is a parameter of order unity. The choice λ = 1 leads to m λ = m t . Different choices of λ lead to scales m λ whose relative difference from m t is of O(α S ).
We also recall that the features of the MS scheme are unchanged by using any other MSlike scheme that is obtained by a perturbative redefinition of m t (µ m ) with scale-independent coefficients. The relative difference between the running masses of the two schemes is of O(α S ) (this affects Eq. (1) starting from its first-order coefficient d (1) ), and the first-order coefficient γ 0 of Eq. (5) is unchanged (the change of scheme affects Eq. (5) starting from O(α 2 S )). It is well known that the renormalised pole mass is affected by a renormalon ambiguity [22][23][24]. More precisely, at large values of k the perturbative coefficients d (k) in Eq. (1) are factorially growing with k. This implies that a non-perturbative ambiguity of O(Λ QCD ) affects the definition of the pole mass itself (no corrections of O(Λ QCD ) affect the MS mass). The knowledge of the coefficients d (k) with k ≤ 4 can be combined with that of the asymptotic factorial behavior [24] to obtain approximations [25][26][27] of the perturbative coefficients of Eq. (1) beyond the four-loop order. High-order approximations of Eq. (1) can also be used to estimate the renormalon ambiguity on the value of the pole mass. In Ref. [26] the ambiguity is estimated to be about 110 MeV, while Ref. [27] estimates it to be about 250 MeV. We note that both estimates are of the order or below the accuracy that can be reasonably achieved in LHC measurements of the top-quark mass.
The renormalon ambiguity does not only plague the pole mass definition, but it can in principle affect the perturbative expansion of tt cross sections. The total cross section expressed in terms of the MS mass is expected to be not affected by renormalons at O(Λ QCD ). However, the tt total cross section is not directly measurable in practice. Renormalon effects are unavoidable [28] in the perturbative computation of tt production observables that are defined by realistic selection cuts applied in LHC experiments. Even if LHC experimental measurements are often extrapolated to the full phase space, the extrapolation procedure is not able to (theoretically) correct for renormalon effects.
At the LHC top-quark physics can be studied either indirectly or directly. By indirectly we mean through quantities in which the top quark appears as a virtual (off-shell) particle that is not directly observed. For instance, since the Higgs boson is mostly produced by gluon fusion through the coupling to a top-quark loop, studies of Higgs boson production give information on the top quark, including the effects of its mass. For all the quantities in which the top quark appears indirectly, mass renormalisation can be carried out equivalently in any renormalisation scheme. For instance, one can introduce MS renormalisation and the MS mass without using and even mentioning the pole mass (one can also do the opposite, of course). For all these quantities the MS and pole masses can be introduced on equal footing.
The direct studies of top-quark physics at the LHC are those in which the top quark (and/or antiquark) or, more precisely, its decay products, are directly observed in the final state. These studies are based on a definite physical picture that fully relies on the concepts of pole mass, M t , and width, Γ t , of the top quark. The top quark is so heavy (M t ∼ 173 GeV from direct measurements at the Tevatron and the LHC [29]) and so unstable (Γ t ∼ 1.4 GeV [29]) that it decays by weak interactions before strong interactions come into play and form bound states. Therefore, the top quark is viewed as a 'physical', though unstable, particle (its pole mass and width gain a physical meaning) that manifests itself in a resonance peak in physical cross sections. LHC experimental data on top-quark production entirely rely on this physical picture, since the top-quark signal is extracted by the quasi-resonant behaviour of the (supposed) decay products of the top quark. No LHC experimental data on top-quark production can be obtained without referring to the pole of the propagator of the top quark (i.e., without having introduced the concept of pole mass). Therefore, the pole mass and the MS mass do not appear on equal footing in the context of top-quark production at the LHC. The pole mass has a primary role, and the MS mass has (somehow) an auxiliary role. Note that this is not only a conceptual aspect, since the difference between M t and m t (µ m ) (at scales µ m of the order of the top-quark mass) can be as large as about 10 GeV, and, hence, much larger than Γ t . Therefore, the pole and MS masses cannot be regarded as being approximately equal for practical experimental purposes. We also note that the size of the renormalon effects on M t is definitely smaller than Γ t and, hence, renormalons do not change the picture of the top quark as a 'physical' unstable particle. The main caveat to this picture is due to the fact that the top quark carries colour charge, and QCD colour is confined through hadronization and not observable in the final state. This implies that colour confinement produces quantitative corrections on the identification of the decay products of the top quark. However, such effects are not related to the difference between the pole and MS masses.
The experimental treatment of the top quark as a physical unstable particle has a direct correspondence in theoretical calculations based on the narrow-width approximation. In the limit Γ t M t , top-quark production is computed by setting the top quark on shell and with a mass equal to the pole mass M t 1 . We use the shorthand notation σ(M t ; X) to generically denote cross sections or differential cross sections for on-shell tt production 2 . The perturbative QCD calculation of σ(M t ; X) defines the perturbative function σ(α S (µ R ), µ R , µ F ; M t ; X) that is computed order-by-order as a series expansion in powers of α S (µ R ). The dependence on the auxiliary remormalization scale µ R and factorisation scale µ F is due to the MS renormalisation of α S and the MS factorisation of the parton distribution functions (PDFs) of the colliding hadrons. The perturbative expansion of σ(α S (µ R ), µ R , µ F ; M t ; X) up to NNLO can be written as where the perturbative coefficients σ (i) (M t ; µ R , µ F ; X) explicitly depend on the pole mass M t of the on-shell top quark and antiquark that are produced in the final state.
In the context of Eq. (7) (and its generalization to higher perturbative orders), we remark on the fact that M t is not simply a parameter of the QCD Lagrangian, but it is also, and importantly, a key kinematical parameter of the phase space. The final-state top quark has a mass M t , and the on-shell constraint p 2 = M t 2 affects each of the components p ν of the four momentum p of the top quark. This in turn produces a dependence on M t of all the kinematical variables of the produced final state. For instance, if the kinematical variable X is the invariant mass m tt of the tt pair, it has an implicit dependence on M t , which in particular leads to the constraint m tt > 2M t . Therefore, the differential cross section with respect to m tt has a 'physical' threshold at m tt = 2M t , and it vanishes for smaller values of m tt (this is true for the perturbative cross sections σ (i) in Eq. (7) at each perturbative order). Obviously, the kinematical/phase space dependence of the cross section on M t is the consequence of the underlying dynamical approximation, namely, the fact that the cross section in Eq. (7) deals with the production of an 'unstable' top quark with pole mass M t in the limit of vanishing width Γ t .
Throughout the paper we refer to the cross section in Eq. (7) (and its generalization to higher orders) as the on-shell cross section computed in the pole mass scheme.
Starting from the pole scheme cross section σ(α S (µ R ), µ R , µ F ; M t ; X), we theoretically define the MS cross sectionσ through a formal replacement of M t with its dependence on the MS mass m t (µ m ). Our definition is (8) where the right-hand side of the equation is the pole scheme cross section in which the pole mass M t has been expressed in terms of the MS mass through the all-order relation in Eq. (1). The cross sections σ andσ in Eq. (8) are equal if regarded as formal expressions to all orders in α S . The order-by-order expansions of σ andσ are instead different. The perturbative expansion of σ is simply obtained by considering the right-hand side of Eq. (8) and expressing d(m t (µ m ), µ m ) as a function of α S (µ R ) and m t (µ m ) (see Eq. (1)). The expression,σ NNLO , of the MS scheme cross section up to NNLO is and the explicit contributionsσ (i) arē where d (1) (µ m ) and d (2) (µ m ) are the coefficients in Eq. (2), and the term proportional to β 0 in Eq. (12) arises from expressing Eq. (1) in terms of α S (µ R ) (rather than α S (µ m )).
We comment on some main features of the definition of the cross sectionσ in the MS scheme.
The perturbative contributionsσ (i) are given in terms of the corresponding contributions σ (i) as computed for on-shell tt production in the pole mass scheme. The cross sectionsσ (i) depend on the on-shell cross sections σ (i) and their derivatives (∂ m ≡ ∂/∂m) with respect to the top-quark mass. These are partial derivatives at fixed values of the auxiliary scales µ R , µ F and fixed values of X.
The tt total cross section only depends on the top-quark mass (and auxiliary scales). Fiducial and differential cross sections depend on the additional variables X. As previously discussed, in the perturbative calculation for on-shell top quarks, the kinematical variables X of differential cross sections do depend on the pole mass M t . According to the formal definition of the MS cross section in Eq. (8), these variables X are considered to be independent of M t , and they acquire an implicit dependence on m t (µ m ).
We also note that the partial derivatives ∂ k ln σ (i) /(∂ ln m) k can be very sizeable in some cases. If this happens, the fixed-order expansion of the MS scheme cross section can become quantitatively unstable, although the all-order equality in Eq. (8) remains valid.
The perturbative contributionsσ (i) to the MS cross sections depend on the MS mass m t (µ m ) and on the mass renormalisation scale µ m . The auxiliary scale µ m is formally arbitrary. The usual procedure to obtain quantitative predictions in the presence of auxiliary scales, such as µ R , µ F and µ m , is to assign them some central values and, then, consider variations around these central values. There are no reasons to consider perturbative predictions by uniquely fixing the value of µ m (e.g., by setting µ m = m t ) without examining the effect of varying µ m .
As a final general comment on the MS scheme cross sectionσ we note that, analogously to the pole scheme cross section, it regards top-quark production in the limit of vanishing top width Γ t . The formal definition ofσ in Eq. (8) does not introduce any physical corrections due to the finite width of the top quark.
In the following, we comment on the actual structure of the perturbative coefficients in Eqs. (9)- (12). At the LO there are no mass renormalisation effects to be considered. The LO MS scheme cross sectionσ (0) in Eq. (10) is equal to the on-shell cross section σ (0) in Eq. (7), apart from the replacement of the pole mass M t with the MS mass m t (µ m ). Since we are dealing with the production of on-shell quarks, this formal replacement is questionable from a physics viewpoint. For instance, the question becomes evident by considering the differential cross sections with respect to m tt . The pole scheme differential cross section dσ/dm tt (at any perturbative order) has a physical threshold at m tt = 2M t , whereas in the MS cross section at LO the threshold is at m tt = 2m t (µ m ). If the difference between M t and m t (µ m ) is much larger than the width Γ t , such displacement of the production threshold is definitely unphysical.
At the NLO, one-loop corrections on internal quark lines require mass renormalisation, which can be carried out either in the pole or the MS scheme. However, the NLO cross section σ (1) in Eq. (7) still involves on-shell top quarks with pole mass M t . The NLO MS cross sectionσ (1) in Eq. (11) includes two contributions: one contribution is simply σ (1) with the replacement M t → m t (µ m ), and the other contribution is controlled by the mass derivative of the LO on-shell cross section σ (0) . This second contribution represents the correction that is applied toσ for having naively identified M t with m t (µ m ) at the LO. For instance, the correction is quantitatively very large for the differential cross section dσ (1) /dm tt close to the threshold region where m tt ∼ 2m t (µ m ) ∼ 2M t , since the mass derivative of dσ (0) /dm tt is very large in this region.
A similar discussion can be extended at NNLO and higher orders. In particular, the mass derivatives of the on-shell cross sections σ (j) that appear in the perturbative contributionsσ (i) (j < i) partly originate from having replaced the pole mass M t with the MS mass m t (µ m ) in the lower-order cross sections for on-shell top-quark production.
We summarise the main points of our general discussion of the MS scheme cross section. Dealing with the production of on-shell top quarks with pole mass M t , we have considered the pole scheme cross section σ(α S , M t , X) in Eq. (7). The MS scheme cross section σ(α S , m t (µ m ), X) is then introduced through the pole scheme cross section and the mass relation in Eq. (1). The cross sectionσ is defined by using the formal all-order identity in Eq. (8). In this relation, the cross section variables X (which can depend on the on-shell kinematics of the produced top quarks) are considered to be independent of the pole mass M t . The cross sections σ andσ differ order-by-order in α S . Owing to the perturbative nature of the definition in Eq. (8) and using mass scales µ m of the order of the top-quark mass (so that the coefficients d (k) (µ m ) in Eq. (1) are of order unity), we expect the following perturbative behaviour. At low perturbative orders, σ andσ can give quantitative results that are consistent (within perturbative uncertainties), and the difference can be larger for cross sections that depend on kinematical variables X that are more sensitive to kinematic thresholds related to on-shell topquark production. At higher perturbative orders, σ andσ can give very similar quantitative results, thus leading to an equivalent perturbative description. As we briefly discuss below, at such perturbative orders, the MS formulation can take advantage of the dynamical features of the running mass m t (µ m ).
In the case of cross sections that depend on physical scales X that are much larger than the mass of the top quark (e.g., if X = m tt at high values of m tt ), tt production takes place in a multiscale dynamical regime. Therefore, we can expect that an improved perturbative description can be achieved in the context of the MS scheme by using a running mass m t (µ m ) with a dynamical value of the renormalisation scale µ m . We also note that in such high-scale regime the variable X has a weaker dependence on the pole mass of the produced on-shell top quark.
In the next Section we present detailed quantitative studies of the perturbative features of the MS scheme cross sections. We also discuss an implementation of the running mass m t (µ m ) in the computation of the differential cross section with respect to the tt invariant mass m tt .

Results
In this Section we present inclusive and differential results for tt production in the MS scheme up to NNLO, and we compare them with the corresponding results obtained in the pole scheme. All our results are based on the calculation of tt production up to NNLO that is reported in Refs. [6,10,30]. The calculation is carried out within the Matrix [31] framework, which features a completely automated implementation of the q T subtraction formalism [13] at the NNLO. The core of the Matrix framework is the Monte Carlo program Munich 3 , which includes a fully automated implementation of the NLO dipole subtraction method for massless [32,33] and massive [34] partons, and an efficient phase space integration. All the required (spin-and colour-correlated) tree-level and one-loop (squared) amplitudes are obtained by using OpenLoops [35][36][37]. The required two-loop amplitudes are available in a numerical form [38]. More details on the implementation of tt production in Matrix can be found in Ref. [10].
The calculation of Ref. [10] directly leads to the numerical evaluation of the perturbative contributions σ (0) , σ (1) and σ (2) (see Eq. (7)) to the cross sections for on-shell tt production in the pole scheme. According to Eqs. (9)- (12), the calculation of the MS cross sectionσ up to NNLO requires the computation of the first and second derivatives of the LO result σ (0) with respect to the mass, and of the first derivative of the NLO corrections σ (1) , computed in the pole scheme. This calculation is performed by computing the cross sections σ (0) and σ (1) for several values of the top-quark mass around m = m t (µ m ), and performing a quadratic fit of the results, from which the numerical values of the relevant derivatives are obtained. This procedure is carried out for the total cross section and, analogously, for each bin in the variable X of the considered differential distributions dσ/dX.
To present our quantitative results, we focus on pp collisions at the centre-of-mass energy √ s = 13 TeV. In Section 3.1 and 3.2 we consider perturbative calculations in the pole scheme and in the MS scheme by using values of the renormalisation scale µ m of the order of the topquark mass. We use n f = 5 massless quark flavours and the corresponding NNPDF31 sets [39] of parton distribution functions (PDFs) with α S (m Z ) = 0.118. In particular, N n LO (with n = 0, 1, 2) predictions are obtained by using PDFs at the corresponding perturbative order and the evolution of α S (µ R ) at (n + 1)-loop order, as provided by the PDF set. Our results in the pole scheme with M t = 173.3 GeV [29] are compared with the corresponding results in the MS scheme with m t = 163.7 GeV. These two values of M t and m t are numerically related by mass renormalisation at the NNLO, namely, we use the relation in Eq. (1) at three-loop order by including the coefficients d (1) , d (2) and d (3) [15][16][17][18] (we set d (k) = 0 if k ≥ 4). We note that the same value of m t is used regardless of the order of the calculation.
Unless otherwise stated, the top-quark mass (either m t or M t ) is used in our calculations as the central value µ 0 for the renormalisation (both µ R and µ m ) and factorisation scales. We use the customary procedure of performing scale variations around the central scales to estimate the uncertainties from perturbative contributions at higher orders or, more precisely, to roughly set a lower limit on such uncertainties. The scale uncertainty bands for the predictions in the pole scheme are obtained by setting µ 0 = M t and performing independent variations of µ R and µ F . We set µ i = ξ i µ 0 , and we vary the parameter ξ i according to ξ i = {1/2, 1, 2} with the constraints µ i /µ j ≤ 2 (i, j = R, F ). This prescription leads to the customary 7-point scale uncertainty. In the case of the MS scheme, we have an additional auxiliary scale, µ m , which (as discussed in Section 2) has to be varied. We perform an independent variation of the three auxiliary scales, by setting µ i = ξ i µ 0 (here µ 0 = m t ) and varying ξ i as ξ i = {1/2, 1, 2}, with the constraints µ i /µ j ≤ 2 (i, j = R, F, m). This prescription leads to a 15-point scale variation. By varying µ m in the interval 0.5 m t < µ m < 2 m t , the MS mass varies in the range 155.5 GeV ∼ < m t (µ m ) ∼ < 173.3 GeV. This dependence of m t (µ m ) on µ m is computed at NNLO accuracy 4 (i.e., we consider the evolution equation (4) with the anomalous dimension γ m (α S ) of Eq. (5) that is evaluated up to O(α 3 S )) by using the package CRunDec [40]. We note that the upper limit on m t (µ m ) is very close to the value of the pole mass M t .

Total cross section
We start the presentation of our results by considering the tt total cross section. In Table 1 we compare the results at LO, NLO and NNLO in the pole scheme with the corresponding results in the MS scheme. The scale uncertainties of the MS scheme results as evaluated in different ways (see the comments below) are also presented. We have checked that the MS scheme results at fixed µ m = m t , including µ R and µ F variations around µ 0 = m t (third column of Table 1), are in excellent quantitative agreement with those obtained by using the numerical program Hathor [11,41].
We present some comments on the scale dependence of the results in Table 1. We observe that the scale uncertainties obtained in the MS scheme by using the 15-point scale variation, by keeping µ m fixed or by keeping µ R and µ F fixed (see, correspondingly, the second, third or fourth column in Table 1) are quantitatively very similar. The reason for this similarity is that the total cross section increases as µ m increases, while it has the opposite dependence   on µ R and µ F . Indeed, if µ m increases, the value of m t (µ m ) decreases, thereby leading to an increase of the cross section. On the contrary, the cross section decreases as µ R increases, and it slightly decreases as µ F increases. This µ R dependence is due to the overall proportionality of the cross section to the factor α 2 S (µ R ) (see Eq. (9)). The µ F dependence is due the fact that the tt cross section is sensitive to relatively large momentum fractions of the colliding partons, and in this kinematical region the scaling violations of the PDFs are slightly negative. Moreover, we note that the absolute variation of the cross section that is obtained with the µ m scan is similar in size to the one obtained by varying µ R (whose dependence dominates in the 7-point variation of µ R and µ F ). These features imply that the uncertainty obtained by varying all the three scales simultaneously (µ R = µ F = µ m ) is much smaller than the one obtained with independent variations. We also note that larger scale uncertainties are obtained if the constraint 1/2 ≤ µ m /µ i ≤ 2, with i = R, F , is relaxed (the corresponding results are not shown in Table 1). This fact is also in line to the observations made above.
Using Table 1, we can compare the results in the pole and MS schemes, and we note the following main features. Order-by-order in perturbation theory, the total cross section at central scales is larger in the MS scheme: the increase is about 30% at LO, 15% at NLO and 5% at NNLO. In particular, the NNLO results in the two schemes are very similar. Referring to the full scale dependence (as given in the first two columns of Table 1), the order-by-order differences between the two schemes are comparable to the corresponding scale dependence effects. Therefore, at each perturbative order the predictions in the MS scheme are consistent with the corresponding predictions in the pole scheme within their scale uncertainties. At LO the results in the two schemes have a very similar scale dependence. At higher orders the MS scheme results have a reduced scale dependence. The reduction of the scale dependence is moderate at NLO and more sizeable at NNLO. We note, however, that the NNLO scale uncertainties of the MS scheme result are highly asymmetric: the upward variation is much smaller than the downward variation. This means that the MS scheme cross section is computed close to a local maximum (namely, close to a region of local minimal sensitivity [42]) of the scale dependence of the NNLO result. This may in turn lead to an underestimate of the perturbative uncertainty due to higher-order corrections.
In Ref. [11] it was pointed out that the perturbative convergence of the total cross section in the MS scheme appears to be faster than in the pole scheme. Such behaviour is indeed visible by considering the results in Table 1 at central values of the scales. To quantify the effect we can introduce the K-factors, K (N)NLO = σ (N)NLO /σ (N)LO , namely the ratios of the cross section results at two subsequent orders. In the pole scheme we have K NLO = 1.52 and K NNLO = 1.09, while in the MS scheme we have K NLO = 1.32 and K NNLO = 1.01. At each perturbative order the K-factor in the MS scheme is smaller than that in the pole scheme, thus leading to a faster apparent convergence of the perturbative expansion. We recall that we are using an NNLO relation to obtain the value of m t from M t = 173.3 GeV. The faster apparent convergence of the MS scheme results would be partly reduced by considering a strictly formal order-by-order comparisons between the two mass schemes. Such comparison implies the use of Eq. (1) at each corresponding order, and it leads to m t = 165.8 GeV at LO and m t = 164.2 GeV at NLO. The corresponding LO and NLO cross sections in the MS scheme have the values 589.0 pb and 808.6 pb, respectively. Therefore, in this case the K-factors in the MS scheme would be K NLO = 1.37 and K NNLO = 1.03.
We are not able to offer a physical interpretation of the faster apparent convergence of the MS scheme results, but we do have a technical explanation for it. The explanation uses the structure of the pole and MS scheme cross sections in Eqs. (7)- (12), and it is discussed below.
We first consider the NLO K-factor. The LO cross sectionσ (0) in the MS scheme is simply obtained by evaluating the LO cross section σ (0) in the pole scheme with a value of the top-quark mass (m t = 163.7 GeV) that is significantly lower than the value of M t . Since the on-shell total cross section is a decreasing function of the top-quark mass, the LO cross section increases in going from the pole scheme to the MS scheme. Such increase partly contributes to a decrease of the value of K NLO . A further decrease of K NLO is produced by the NLO radiative corrections. The NLO correction σ (1) in the pole scheme is positive and sizeable. The NLO correctionσ (1) in the MS scheme receives the additional contribution of d (1) m∂ m σ (0) (see Eq. (11)) that is negative (d (1) is positive, and the derivative ∂ m σ (0) is negative since σ (0) is a decreasing function of the top-quark mass) and not small. Therefore, we haveσ (1) < σ (1) , and this effect also decreases the size of K NLO in going from the pole scheme to the MS scheme.
The discussion of the NNLO K-factor follows the same lines as at NLO. We simply notice two main effects in going from the pole scheme to the MS scheme: the NLO cross section increases and the NNLO correction decreases (because in Eq. (12) the coefficients d (1) and d (2) are positive and the mass derivatives of the cross sections are negative). Both effects contribute to decrease the value of K NNLO , thus producing a faster apparent convergence. Moreover, we point out that both effects are the consequence of two basics facts: the pole scheme cross section is a decreasing function of the top-quark mass with radiative corrections that are relatively large at lower orders; the top-quark mass m t that is used in the MS scheme calculation is significantly lower than M t . Therefore, the features of faster apparent convergence of the tt cross section with respect to the behaviour in the pole scheme are common to any renormalisation scheme that perturbatively introduces a renormalisation mass that is systematically smaller than M t .
From the above discussion one is led to conclude that the computation in the MS scheme scheme pole MS MS pole  leads to an improved perturbative stability with respect to the computation in the pole scheme. However, we find that such pattern of perturbative convergence is strongly dependent on the choice of the central values of the auxiliary scales. This fact can be observed, for instance, in Table 2 where we report the results of Table 1 at central scales and two additional sets of results obtained in the pole and MS schemes by using different central values of the auxiliary scales.
We note that the MS scheme results with µ R = µ F = m t and µ m = m t /2 have a reduced perturbative convergence (the K-factors are K NLO = 1.53 and K NNLO = 1.08), and, at each perturbative order, they are quantitatively very similar to the pole scheme results with µ R = µ F = M t . This similarity is not unexpected, since if µ m = m t /2 the value of the top-quark mass m t (µ m ) that is used in the MS scheme computation is almost equal to M t .
By contrast, if the central scales µ R = µ F = M t /2 are used within the pole scheme (as suggested in Ref. [43]), the computation shows a faster perturbative convergence (the K-factors are K NLO = 1.31 and K NNLO = 1.01) and, at each perturbative order, the results are quantitatively very similar to the MS scheme results with µ R = µ F = µ m = m t . Within the pole scheme a faster perturbative convergence is achieved by lowering the value of µ R (and µ F , to a smaller extent), thus increasing the cross section results at lower orders.
We draw some overall conclusions from our comparison between pole and MS scheme results for the total cross section up to NNLO. At each perturbative order the two schemes give results that are consistent within the corresponding scale uncertainties, and, in particular, the results are quite similar at NNLO. The pattern of slower or faster apparent perturbative convergence depends on both the mass renormalisation scheme and the actual values of the auxiliary scales. This is a consequence of the fact that at low orders the radiative corrections are relatively large and the results have a sizeable dependence on µ R and on µ m within the MS scheme. Using the central scales of Table 1, the perturbative convergence of the MS scheme is faster than that of the pole scheme. An opposite (or intermediate) pattern of perturbative convergence can be obtained by using other central scales that are within the range of scale variations that is used in Table 1. Therefore, in the case of the total cross section, the scale setting that is used in Table 1 is sufficiently representative of the scale dependence effects that can affect the comparison between the pole and MS schemes. We expect that this remains true for other tt  Table 1. production observables. Therefore, in our subsequent comparison of the pole and MS schemes for single-differential cross sections we still use the scale setting of Table 1.

Differential cross sections
We now move to consider differential results. At the centre-of-mass energy √ s = 13 TeV, we have computed the differential cross sections dσ/dX with respect to the following variables X: the invariant mass m tt (Fig. 1) and the rapidity y tt (Fig. 2) of the tt pair, the average value of the transverse momentum p T,tav (Fig. 3) and rapidity y tav (Fig. 4)  In the lower panels of Figs. 1-4 we show the ratios of the perturbative results with respect to the central NNLO result in the corresponding mass renormalisation scheme. The LO and NLO differential cross sections with respect to m tt , p T,tav and y tav in the MS scheme were computed in Ref. [12], and our LO and NLO results are consistent with them.
Comparing the MS and pole scheme results in Figs. 1-4, we observe some overall features that are fully analogous to those observed by considering the tt total cross section. The MS scheme results typically have smaller K-factors and a larger overlap between the scale variation  Table 1.   Table 1.  Table 1. bands at subsequent perturbative orders. At NLO, these features were already noticed in Ref. [12]. Going to NNLO we note, in particular, that within the MS scheme the NNLO scale uncertainty band is contained within the NLO band for most of the kinematical regions considered in Figs. 1-4. These overall comments apply to the results in Figs. 1-4 with the exception of the m tt distribution at low values of m tt . The low-m tt region deserves further specific comments.
The invariant-mass distribution for on-shell tt production has a physical threshold at the value m tt = 2M t , and it has a sharply increasing behaviour just above the threshold. Within the pole scheme ( Fig. 1 right), this behaviour is fulfilled order-by-order in perturbation theory. As we have already mentioned in Sect. 2, this physical behaviour is spoiled by the fixed-order perturbative expansion in the MS scheme. This is the consequence of the identification of the top-quark mass with the MS mass m t (µ m ) in the computation of the perturbative on-shell cross sections σ (i) of Eqs. (10)- (12). The order-by-order computation of the m tt distribution in the MS scheme produces a threshold at the value m tt = 2m t (µ m ), and such threshold is unphysical for a twofold reason: the threshold value differs from 2M t , and it depends on the unphysical (arbitrary) auxiliary scale µ m . Besides producing an unphysical threshold, the MS scheme perturbative expansion also produces instabilities at subsequent perturbative orders since the mass derivatives (m∂ m ) k σ (i) in Eqs. (11) and (12) are very large close to the threshold region. These perturbative instabilities are clearly visible in Fig. 1 by comparing the MS and pole scheme results at low values of m tt : in the first and, partly, second m tt bins the MS scheme results have larger K-factors (at both NLO and NNLO) and wider scale uncertainty bands.
The unphysical perturbative behaviour of the MS scheme computation at low values of m tt can be alleviated by considering three related (and correlated) aspects: the use of large bin sizes, the inclusion of variations of the renormalisation scale µ m , the computation of higherorder contributions. We comment on these aspects in turn.
We recall that, in our MS scheme computation of the m tt cross section, the MS mass m t (µ m ) varies (due to µ m variations) in the range 155.5 GeV ∼ < m t (µ m ) ∼ < 173.3 GeV (we also recall that the upper value coincides with the value of M t ). The first m tt bin in Fig. 1 extends from 300 GeV to 360 GeV, so that the unphysical thresholds are always included in the first bin. The reduction of the bin size will amplify the unphysical behaviour of the MS scheme computation.
At low values of m tt the scale dependence of the MS scheme results is very large, thus leading to a sizeable perturbative uncertainty and, therefore, mitigating the effect of the unphysical features at the quantitative level. We note that at low values of m tt this scale dependence is largely dominated by the effect of µ m variations (which, through Eq. (9), changes the position of the m tt threshold), while the variations of µ R and µ F lead to much smaller quantitative effects. This is in contrast with the MS scheme results in the region of higher values of m tt , where (analogously to the total cross section results in the second and third columns of Table 1)  The unphysical features of the MS scheme computation at low m tt are due to the low-order perturbative expansion in Eqs. (10)- (12). Owing to the formal all-order equality in Eq. (8), these unphysical features tend to 'disappear' by expanding the MS scheme cross sectionσ at a 'sufficiently' high order (see the results in Fig. 5 and related accompanying comments).
Considering the behaviour of the m tt distribution near the threshold region, we note that the perturbative computation in the pole scheme also leads to enhanced radiative corrections, which are of dynamical origin. The authors of Refs. [44,45] have performed an all-order resummed calculation of Coulomb-type radiative corrections combined with effects of the finite width Γ t of the top quark. The resummed calculation of Refs. [44,45] leads to an increase of about 9% of the NNLO differential cross section integrated over the bin where 300 GeV < m tt < 380 GeV (larger resummation effects occur for the detailed shape of dσ/dm tt over a more restricted region of size ∆m tt ∼ Γ t around the on-shell threshold at m tt = 2M t ). We remark on the fact that the dynamical effects considered in Refs. [44,45] are unrelated to those produced by the change of mass renormalisation schemes from the pole to the MS scheme.
We note that our comments and discussion on the unphysical features of the invariant-mass distribution at low values of m tt similarly apply to other differential distributions in kinematical regions that are sensitive to thresholds related to on-shell tt production. For instance, this is the case for the y tt and y tav differential cross sections in the very high rapidity region (specifically, at values of |y tt | and |y tav | that are larger than those considered in the results of Figs. 2 and 4).
To perform a more direct comparison of the MS and pole scheme results for the differential cross sections in Figs. 1-4, we compute the ratio between them at each perturbative order for  each of the distributions. The ratios are presented in Fig. 5, where the lower panels show in more detail the NNLO results and their uncertainty bands in both the MS and pole schemes. We see that for all the distributions the results in the two schemes are consistent at LO, NLO and NNLO within the corresponding scale uncertainties. In particular, and in relation to our previous discussion of the low-m tt region, we note that the quantitative effect of µ m variations in the MS scheme is particularly relevant to get consistency with the pole scheme results for the invariant-mass distribution. At NNLO the scale uncertainty band of the MS scheme results is typically of the same size (see the m tt and p T,tav cross sections) or smaller (see the y tt and y tav cross sections) than the corresponding band in the pole scheme. In the cases of the y tt and y tav distributions, we also see that the NNLO scale band in the MS scheme is highly asymmetric with respect to its central value (which, therefore, is evaluated close to a region of local minimal sensitivity to scale variations), consistently with the similar behaviour of the total cross section in Table 1.
In Fig. 5 we also observe that the shape differences between the MS and pole schemes are significantly reduced by the inclusion of high-order corrections, and they are already quite small at NNLO. Moreover, and importantly, in all the kinematical regions of Fig. 5 we note a sizeable overlap between the MS and pole scheme uncertainty bands at NNLO: this fact shows the expected similarity between the two schemes once enough perturbative orders are included in the calculation.

Comparison with CMS data and running-mass effects
Up to now we have presented perturbative calculations in the MS scheme by using values of the renormalisation scale µ m that are of the order of the top-quark mass. In the following we refer to these calculations as predictions with a fixed MS mass, since the scale µ m is not necessarily related to the characteristic scale of the differential cross section under consideration. In the remaining part of this Section we also consider QCD predictions that use a running MS mass, namely, perturbative calculations in which the MS mass m t (µ m ) is evaluated at a dynamical value of µ m that is related to the hard-scattering scale of the differential cross section. Specifically, we consider QCD predictions for the invariant-mass cross section dσ/dm tt , since its characteristic scale is m tt , which can be parametrically much larger than the top-quark mass.
In Ref. [14] the CMS Collaboration performed a measurement of the invariant-mass distribution for tt production based on 35.9 fb −1 of LHC data at the centre-of-mass energy √ s = 13 TeV. The measurement was then compared with QCD predictions in the MS scheme to the purpose of performing a determination of the top-quark mass.
The procedure used in Ref. [14] by the CMS Collaboration is as follows. The theoretical results for dσ/dm tt are obtained by using the NLO QCD calculation [12] in the MS scheme with a fixed scale µ m = m t , and treating m t (m t ) = m t as a free parameter. The value of m t in each invariant-mass bin is then determined by comparing these theoretical predictions with the data point in the same bin. The fitted value m (k) t of m t in the k th bin is then used to compute m t (µ k ) at the characteristic invariant-mass scale µ k [14] of the corresponding bin. The computation of m t (µ k ) from m t (m  [14] is that the µ k dependence of the determined values of m t (µ k ) agrees (within theoretical and experimental errors) with the expectation from the evolution equation (4) at LO.
The final result of Ref. [14] implies that the fitted values of m (k) t in the various invariantmass bins are consistent (within errors) with a single common (i.e., bin-independent) value. In view of this, we conclude that the CMS data of Ref. [14] on dσ/dm tt are consistent with the NLO QCD predictions in the MS scheme as obtained by using a fixed value m t (m t ) = m t of the MS mass, namely, without introducing dynamical effects due to the running of the MS mass 5 . Therefore, the analysis performed in Ref. [14] has no direct sensitivity to running-mass effects, contrary to what is stated therein.
In Sect. 3.1 we have computed the tt total cross section by fixing µ m = m t and using the corresponding fixed value, m t (m t ) = m t , of the MS mass. Such QCD predictions can be used to determine the value of m t through a comparison with data for the tt total cross section (see, e.g., Ref. [46]), but such comparison cannot be used to measure the running of the MS mass of the top quark. Analogously, in the case of differential cross sections, QCD predictions in the MS scheme that are obtained by using a fixed value of the mass renormalization scale µ m and, hence, a fixed value of the MS mass (such as the NLO calculation used in Ref. [14]) can be exploited to determine this value, but they cannot be exploited to study the scale dependence and the running of the MS mass. The investigation of this behaviour requires (at least) the use of QCD calculations with a running (i.e., not fixed at a unique value) renormalization scale µ m .
Taking this fact into account, in the following we present a comparison (see Fig. 6) of the CMS data of Ref. [14] with QCD predictions in the MS scheme up to NNLO. The QCD predictions, which refer to the same binning as in the CMS measurement, are obtained by using either a fixed (Fig. 6 left) or a running (Fig. 6 right) MS mass, as specified below. For both kinds of predictions, we exactly follow the setup employed in Ref. [14]: we use the ABMP16 PDF sets [46,47] with n f = 5 massless-quark flavours, and the corresponding values of the QCD coupling, α S (m Z ) = 0.1191 and α S (m Z ) = 0.1147 at NLO and NNLO, respectively. The value of m t is set to 161.6 GeV, which corresponds to the result obtained by the CMS Collaboration [48] from a fit of the tt total cross section (using the same data set as in Ref. [14]) based on NNLO predictions in the MS scheme computed with the ABMP16 PDFs and the corresponding α S . We note that such value of m t is lower than the one used to obtain all our previous results in the MS scheme (e.g., the results in Figs. 1 and 5). We also note that the value m t = 161.6 GeV corresponds to the pole mass M t = 170.8 GeV, by using the relation in Eq. (1) at three-loop order.
The QCD predictions with a fixed MS mass (Fig. 6 left) are computed analogously to those in Fig. 1. We use the central value µ 0 = m t for the three auxiliary scales µ R , µ F and µ m , and we consider the 15-point scale variations around this central value. At NLO this calculation corresponds to the one performed in Ref. [14], with the main difference that we include the uncertainties due to the variation of µ m by a factor of 2 around µ 0 (µ m is kept fixed to m t in  [14]. The theory uncertainty bands are obtained by performing 15-point scale variations around the central value µ 0 of the auxiliary scales µ R , µ F and µ m . In the left panel µ 0 is fixed to m t , whereas in the right panel µ 0 is dynamically set to µ k /2 (µ k is the centre of gravity of the cross section in the k th bin of m tt ). In the lower right panel the dashed lines indicate the NLO and NNLO results obtained with µ m = m t and µ R = µ F = µ k /2.
Ref. [14], though the effect of PDF uncertainties is considered therein).
The QCD predictions with a running MS mass (Fig. 6 right) are computed by performing the 15-point scale variations around values of the central scale µ 0 (for the three auxiliary scales µ R , µ F and µ m ) of the order of m tt /2, which is the characteristic hard-scattering scale of the differential cross section dσ/dm tt . Specifically, in the k th invariant-mass bin we set µ 0 = µ k /2 (setting directly µ 0 = m tt /2 in our MS scheme calculation is more challenging from a computational point of view), where µ k is the centre of gravity of the m tt cross section in the k th bin as computed by the CMS Collaboration [14]. The values of µ k range from µ 1 = 384 GeV in the 1 st bin to µ 4 = 1020 GeV in the 4 th bin (see Table 1 in Ref. [14]), and the corresponding values of the running-mass range from m t (µ 1 /2) = 159.5 GeV to m t (µ 4 /2) = 149.0 GeV (we use the evolution equation (4) at NNLO, as implemented in the package CRunDec [40]). We note that the fixed (µ 0 = m t ) and dynamic (µ 0 = µ k /2) scales substantially differ only in the high-m tt region (for instance, in the first bin µ 1 /2 = 192 GeV and m t differ by less than a factor of two, and both values are thus included within the scale variation range that we consider). Therefore, our comparison between fixed-mass and running-mass predictions has the purpose of investigating differences only at relatively high values of the invariant mass.
From the theory-data comparison in Fig. 6 we see that the NNLO results in the MS scheme with fixed or running masses are both in excellent agreement with data. We also note that the agreement improves (especially for the predictions with the running mass at high values of m tt ) in going from NLO to NNLO. As observed in Fig. 5, the NNLO predictions for dσ/dm tt in the pole and MS schemes are consistent within their scale uncertainties. Therefore, we also note that the level of agreement between data and NNLO theory is not a peculiarity of the results in the MS scheme. The CMS data are also fully compatible with the NNLO predictions in the pole scheme, provided they are obtained by using a value of pole mass M t that is consistent (according to Eq. (1) at NNLO) with the value of m t of the corresponding predictions in the MS scheme.
Inspecting the results reported in Fig. 6, we see that NNLO (and also NLO) MS scheme predictions with a fixed and a running mass are consistent with each other within their scale uncertainties (in particular the two NNLO scale variation bands have a substantial overlap). In the highest invariant-mass bin of Fig. 6, the data point agrees better with the central NNLO prediction with a fixed MS mass, which leads to a larger value of dσ/dm tt . This larger cross section is due the fact that the NNLO result for dσ/dm tt at high m tt is a decreasing function of the central scale. Indeed, the MS prediction with a fixed MS mass uses the central scale µ 0 = m t , while the prediction with the running mass uses a larger value, (µ 0 ∼ m tt /2) of the central scale. A qualitatively similar scale dependence of dσ/dm tt in the high-m tt region (e.g., m tt ∼ > 1 TeV) is observed in the NNLO results [43] in the pole scheme. Therefore, such scale dependence (both in the MS and the pole scheme) is mostly driven by the scale µ R of α S and the scale µ F of the PDFs (increasing m tt the PDFs are sensitive to the region of increasingly higher values of momentum fractions, where scaling violations are negative).
Our QCD predictions with running MS mass use the central scale µ 0 = µ k /2 ∼ m tt /2 for all the auxiliary scales µ R , µ F and µ m . To disentangle the effect of the running of the top-quark mass m t (µ m ) from the effect due to the running of α S and to the scaling violations of the PDFs, we also present (see the dashed lines in the lower panel of Fig. 6 right) the results that are obtained by keeping µ m = m t fixed, while still using the dynamic scale µ k /2 for µ R and µ F . Even though the use of the running mass m t (µ k /2) leads to a slightly better agreement with the data, the difference from the result with µ m = m t (solid lines in Fig. 6 right) is very small in comparison to the size of the theoretical and experimental uncertainties.
Comparing the QCD results at central scales in the lower panel of Fig. 6 right, we note the following features. Going from m t (µ k /2) (solid lines) to m t (m t ) (dashed lines), the MS mass m t (µ m ) of the top-quark increases and, accordingly with naive expectations, the m tt cross section decreases. The decrease of the cross section is rather uniform throughout, from the region of low to high values of m tt . This is also not unexpected. At low invariant masses the parametric dependence of the m tt cross section on m t (µ m ) is large (since the value of µ m effectively changes the position of the invariant-mass threshold), but the difference between the fixed and the dynamic scales, µ m = m t and µ m = µ k /2 ∼ m tt /2, is very small. At high invariant masses this difference becomes larger, but the parametric dependence of the m tt cross section on the mass m t (µ m ) is much smaller.
In the highest invariant-mass bin of Fig. 6, we also see that the difference between the solid and the dashed lines of Fig. 6 (right) is definitely smaller than the difference between the central predictions (solid lines) on the left-hand side and the right-hand side of Fig. 6. This fact confirms our previous conclusion that the difference between our fixed-mass and running-mass predictions of Fig. 6 is mostly driven by the scales µ R and µ F of α S and PDFs. The quantitative dependence on the mass renormalisation scale is smaller for a twofold reason: the running of the MS mass of the top quark is slower than that of the QCD coupling α S (see Eq. (6)), and the scaling violations of the PDFs increase with increasing m tt .
From our discussion of the results in Fig. 6 we conclude that the data of Ref. [14] are not able to pin down effects produced by the running of the MS mass in the NNLO predictions. This conclusion is the consequence of the relatively large theoretical uncertainties of the predictions and, partly, of the size of the experimental errors.
We complete our discussion on the invariant-mass distributions with few additional considerations.
The computation of QCD observables that depend on a single hard-scattering scale (e.g, the tt total cross section, which depends on the top-quark mass) is usually performed by setting the central scale µ 0 of the corresponding fixed-order calculation to a value of the order of the hardscattering scale. The differential cross section dσ/dm tt is not a single-scale QCD observable, since it depends on both the top-quark mass and the invariant mass of the tt pair. If these two mass scales are parametrically different, we are dealing with a two-scale QCD observable, and high-order radiative QCD corrections are expected to be quantitatively relevant. In the context of fixed-order QCD predictions for two-scale observables, a customary procedure to investigate higher-order effects is to set the auxiliary scale µ 0 of central QCD predictions at a value within the range of the two mass scales. In our predictions of Fig. 6 with fixed and running MS masses, we have considered the two 'extreme' choices µ 0 = m t and µ 0 m tt /2, and we have found relatively similar (within scale uncertainties) results for m tt ∼ < 1 TeV (i.e., the value of µ k in the 4 th bin of Fig. 6).
In the case of two-scale QCD observables, the use of fixed-order calculations with dynamic values of µ 0 is expected to be a sensible theoretical procedure, provided the two scales are parametrically not very different. The direct calculation of higher-order contributions (for instance, through all-order resummation techniques) is instead theoretically more appropriate in the kinematical region where the two mass scales are parametrically very different. In the specific case of dσ/dm tt , this is the multi-TeV invariant-mass region. In Ref. [49], the differential cross section dσ/dm tt at high (multi-TeV) values of m tt was studied by combining the NNLO calculation in the pole scheme [43] with resummed calculations [50,51] of contributions due to soft and collinear radiation. The combination of resummed calculations with the NNLO calculation in the MS scheme can be of interest to perform further investigations on the effects of the running of the top-quark mass.

Summary
This paper has been devoted to present and discuss QCD predictions for tt production at the LHC by using the MS renormalisation scheme for the definition of the top-quark mass.
We have remarked that the LHC experimental data refer to the production of 'physical' (though unstable) top quarks and antiquarks with a definite value of the pole mass M t (and width Γ t ). At the theoretical level, this physical picture directly leads (in the limit Γ t → 0) to considering perturbative calculations for on-shell tt production in the pole scheme. We have then discussed how the on-shell calculations in the pole scheme can be transformed into corresponding calculations in the MS scheme through a formal all-order perturbative replacement of the renormalised top-quark mass. We have highlighted possible unphysical features (e.g., in connection with tt production thresholds) that are produced by such formal replacement order-by-order in QCD perturbation theory. We have also discussed how running-mass effects can be introduced in the perturbative calculation within the MS scheme.
In the previous literature, QCD predictions within the MS scheme had been limited to calculations of the tt total cross section up to NNLO and of single-differential cross sections up to NLO, and these calculations used the fixed value m t (m t ) = m t of the top-quark mass.
In this work we have computed total cross sections and single-differential distributions for tt production at NNLO by using the MS scheme for the renormalisation of the top-quark mass. The NNLO results substantially increase the precision of the MS scheme theoretical results for differential distributions that were previously available only at NLO, and, therefore, our results are very relevant in the context of the experimental determination of the MS mass of the top quark.
In our computation we have consistently included variations of the renormalisation scale µ m of the MS mass m t (µ m ). We have then considered a 15-point variation (once variations of µ R and µ F are included) of the QCD auxiliary scales (µ R , µ F and µ m ) to estimate the scale uncertainties of the fixed-order QCD predictions. In particular, we find that the inclusion of µ m variations is crucial to consistently and correctly assess the size of the perturbative uncertainties of the MS scheme predictions, especially at low perturbative orders and for differential distributions (e.g., the distribution of the invariant mass, m tt , of the tt pair) in the vicinity of tt production thresholds. Moreover, we have also computed the invariant-mass distribution at high m tt by using a dynamic value of the renormalisation scale µ m , therefore effectively introducing a running value, m t (µ m ), of the top-quark mass in the MS scheme.
We have presented the results of detailed QCD calculations of the tt total cross section in both the pole and MS schemes up to NNLO by using different central values µ 0 of the auxiliary scales µ R , µ F and µ m , and including uncertainties from scale variations around the central scale. The comparison between the pole and MS schemes shows consistent results within scale uncertainties. Comparing the pole scheme results with µ 0 = M t to the MS scheme results with µ 0 = m t , we confirm previous findings in the literature: the perturbative convergence of these MS scheme results appears to be faster, with larger overlap of the scale uncertainty bands at subsequent perturbative orders, and with smaller corrections and scale uncertainties at NNLO. However, we have pointed out that the features of faster or slower apparent convergence strongly depend on both the value of µ 0 and the type of mass renormalisation scheme. Within each of the two mass renormalisation schemes, the apparent convergence can be made faster (or slower) by changing the value of the central scale µ 0 .
Considering scales µ 0 of the order of the top-quark mass, we have presented results of singledifferential distributions in the MS scheme up to NNLO, and we have also compared them with corresponding results in the pole scheme. The comparison between the results in the two schemes leads to conclusions that are similar to those that apply to the total cross section. In particular, we concluded that the shape differences between the pole and MS scheme results are significantly reduced by the inclusion of the high-order contributions, and they are quite small at NNLO. Moreover, in all the kinematical regions that we have considered, we have noted a sizeable overlap between the pole and MS scheme uncertainty bands at NNLO. The high similarity between these pole and MS scheme results at NNLO for differential distributions is a relevant feature, since it also justifies the study of running-mass effects through the introduction of dynamic values of the renormalisation scale µ m for the MS mass m t (µ m ).
We have considered a recent measurement of the m tt differential cross section performed by the CMS Collaboration. The measurement extends up to m tt ∼ < 1 TeV. We have computed corresponding QCD predictions up to NNLO within the MS scheme by using either fixed (µ m = m t ) or dynamic (µ m ∼ m tt /2) central values of µ m . We have discussed the effects that are produced by the dynamic scale. We have observed an excellent agreement between the experimental data and the theory results at NNLO. In particular, the NNLO results lead to a sizeable reduction of the scale uncertainties with respect to the corresponding results at NLO, thus paving the way to a precise determination of the top-quark mass in the MS scheme. The NNLO predictions with fixed and dynamic values of µ m are consistent within their scale uncertainties, whose size is similar to that of the experimental errors. Therefore, we have concluded that these CMS data are not able to pin down effects produced by the running of the MS mass in the NNLO predictions. Additional theoretical studies of running-mass effects in QCD predictions are left to future investigations.