The Higgs transverse momentum distribution in gluon fusion as a multiscale problem

Journal of High Energy Physics, Jan 2016

We consider Higgs production in gluon fusion and in particular the prediction of the Higgs transverse momentum distribution. We discuss the ambiguities affecting the matching procedure between fixed order matrix elements and the resummation to all orders of the terms enhanced by log(p T H /m H ) factors. Following a recent proposal [1], we argue that the gluon fusion process, computed considering two active quark flavors, is a multiscale problem from the point of view of the resummation of the collinear singular terms. We perform an analysis at parton level of the collinear behavior of the \( \mathcal{O}\left({\alpha}_s\right) \) real emission amplitudes; relying on the collinear singularities structure of the latter, we derive an upper limit to the range of transverse momenta where the collinear approximation is valid. This scale is then used as the value of the resummation scale in the analytic resummation framework or as the value of the h parameter in the POWHEG-BOX code. A variation of this scale can be used to generate an uncertainty band associated to the matching procedure. Finally, we provide a phenomenological analysis in the Standard Model, in the Two Higgs Doublet Model and in the Minimal Supersymmetric Standard Model. In the two latter cases, we provide an ansatz for the central value of the matching parameters not only for a Standard Model-like Higgs boson, but also for heavy scalars and in scenarios where the bottom quark may play the dominant role.

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The Higgs transverse momentum distribution in gluon fusion as a multiscale problem

Received: May The Higgs transverse momentum distribution in gluon fusion as a multiscale problem E. Bagnaschi 0 1 A. Vicini 0 1 2 0 Via Celoria 16, I-20133 Milano , Italy 1 Notkestra e 85 , D-22607 Hamburg , Germany 2 Tif lab, Dipartimento di Fisica, Universita degli Studi di Milano and INFN , Sezione di Milano We consider Higgs production in gluon fusion and in particular the prediction of the Higgs transverse momentum distribution. We discuss the ambiguities a ecting the matching procedure between xed order matrix elements and the resummation to all orders of the terms enhanced by log(pTH =mH ) factors. Following a recent proposal [1], we argue that the gluon fusion process, computed considering two active quark avors, is a multiscale problem from the point of view of the resummation of the collinear singular terms. We perform an analysis at parton level of the collinear behavior of the O( s) real emission amplitudes; relying on the collinear singularities structure of the latter, we derive an upper limit to the range of transverse momenta where the collinear approximation is valid. This scale is then used as the value of the resummation scale in the analytic resummation framework or as the value of the h parameter in the POWHEG-BOX code. A variation of this scale can be used to generate an uncertainty band associated to the matching procedure. Finally, we provide a phenomenological analysis in the Standard Model, in the Two Higgs Doublet Model and in the Minimal Supersymmetric Standard Model. In the two latter cases, we provide an ansatz for the central value of the matching parameters not only for a Standard Model-like Higgs boson, but also for heavy scalars and in scenarios where the bottom quark may play the dominant role. QCD Phenomenology; Monte Carlo Simulations - 2 Remarks on the computation of the Higgs p?H distribution Analytic resummation and the collinear limit Numerical resummation in the NLO+PS framework 2.2.1 The role of the damping factor Dh in the POWHEG-BOX framework HJEP01(26)5 2.3 The value of the SCALUP variable 3 Collinear approximation of partonic squared matrix elements Helicity amplitudes and kinematic variables Partonic analysis 3.2.1 3.2.2 Scalar Higgs Pseudoscalar Higgs 3.3 Dependence on auxiliary parameters 4 Standard Model phenomenology 4.1 Comparison of POWHEG and HRes 5 Beyond SM phenomenology 5.1 5.2 2HDM phenomenology MSSM phenomenology 6 Conclusions A Scan over the Higgs mass of the scales wt;b;i 1 Introduction 2.1 2.2 3.1 3.2 1 Introduction A new state with a mass of approximately 125 GeV has been observed at the LHC [2, 3]. Many investigations are under way to determine its properties and to test its compatibility with the Higgs scalar boson of the Standard Model (SM). The precise measurements of the total production cross section and of the branching ratios in the di erent allowed decay channels [4, 5] have shown that the new state couples to the known fermions and gauge bosons following the SM predictions. Other studies target the kinematics of the decay products to distinguish among the various spin-parity combinations [6]. Finally, further work will be necessary to clarify the structure of the scalar potential. In the SM the main production mode of the Higgs boson at hadron colliders is through the gluon fusion mechanism. The coupling of the Higgs boson to the gluons is mediated by a loop of colored particles, with the largest contribution to the process given by the { 1 { in refs. [7{9] and in refs. [10{15]. Recently, expressions for the N3LO corrections have been published in refs. [16{21]. The calculation using the complete SM Lagrangian was done up to NLO-QCD [22{25]. The exact treatment of the quark loops (mostly from top, of mH ' 125 GeV and a collision energy p bottom and charm) at NLO-QCD yields an O( 1%) correction, for an Higgs with a mass S = 14 TeV. Moreover nite top-mass e ects at NNLO-QCD have been estimated and found to be of O(1%) [26{31]. Beyond QCD corrections, also soft-gluon resummation e ects are available [32{36]. xed-order Moreover, the rst-order electroweak (EW) contributions have been evaluated in refs. [37{44] and an estimate of the mixed QCD-EW contributions has been presented in ref. [45]. The PDF and s uncertainties on the total Higgs production cross section have been studied in ref. [46]. The production cross section of a Higgs boson at large transverse momentum has been computed at LO-QCD, retaining the full quark-mass dependence, in refs. [47, 48]. The NLO-EW corrections to this observable have been considered in refs. [49, 50] in the HQEFT limit. The NLO-QCD corrections, in the HQEFT, have been computed [51{53]. An estimation of top-mass e ects at NLO-QCD has been presented in ref. [54]. The rst results towards the determination of the Higgs production at large transverse momentum, in the HQEFT, with NNLO-QCD accuracy, have been presented in ref. [55]. In this paper we want to reconsider the uncertainties that a ect the theoretical predic? tion for the Higgs boson transverse momentum pH . The transverse momentum distribution is an observable generated by QCD radiation. In the region of small pH the presence of ? terms enhanced by large log(pH =mH ) factors spoils the accuracy of the xed-order re? sults; in order to obtain a physically meaningful prediction these logarithms have to be resummed. Various techniques are available to perform the resummation. Once the latter is achieved, the resummed result has to be matched to the xed-order one. Particular care is required to avoid the double counting of those logarithmic contributions that are present in both computations. The matching procedure introduces additional unphysical variables, the matching parameters, that de ne how the spectrum is divided into a soft region, where the resummed result is indeed applied, and a hard region where the xed-order result is instead considered as the correct description of the spectrum. In the HQEFT framework, the corrections up to NLO-QCD for the Higgs transverse momentum distribution have been analytically computed and matched with the transverse momentum resummation at NNLL accuracy. The results have been originally implemented in the code HqT [56{58] and later in the parton Monte Carlo program HRes [59]. A similar discussion, in the Soft Collinear E ective Theory (SCET) approach, has been presented in refs. [60{62]. In the context of matched NLO+Parton Shower (PS) Monte Carlo event generators, which implement the resummation algorithmically in the computer code, the results in the HQEFT, for Higgs production via gluon fusion, have been presented in { 2 { refs. [63, 64]. Two shower Monte Carlo codes that retain the NNLO-QCD accuracy on the inclusive observables, in the HQEFT, have been presented in refs. [65, 66]. Despite the fact that the exact matrix elements retaining the full dependence on the quark masses were available for quite some time, they have been implemented in a NLO+PS Monte Carlo for the rst time in ref. [67], in the POWHEG approach, and later in MC@NLO [68]. A similar study, in the framework of analytic resummation, has been presented in ref. [69] and later in ref. [1]. Recently, these e ects have been implemented in the NNLOPS code [70]. Quark mass e ects have also been discussed for observables like the jet veto distribution in refs. [71, 72]. Moreover, in ref. [71], the structure of the collinear singularities and of the regular terms present in gluon fusion at O( s) is analyzed in detail. In ref. [1] it has been pointed out that the matched computation of the Higgs transverse momentum distribution is a problem with three scales, namely the Higgs mass, the internal quark mass and the transverse momentum of the Higgs boson. The matching prescription between xed-order and resummed results should account for all these scales, to avoid, as far as possible, the inclusion of spuriously large higher-order terms in the nal result. It should be noted that the presence of non-negligible interference e ects between the top and the bottom quarks assigns a simultaneous active role to both internal quarks present in the scattering amplitude. In the framework of SCET, the separation between the singular regions where a resummation is needed and the corresponding regular parts has been discussed in refs. [73, 74] with the introduction of appropriate pro le functions at the level of the hadronic cross section; this approach has been applied to Higgs studies in ref. [75]. The problem of the determination of a sensible value for the scale that separates the two transverse momentum regions, the one where the resummation is needed and the one where a xed-order description is reliable, has been discussed in QCD, at the level of the partonic cross section, in ref. [76]. Recently, in ref. [77], the determination of these scales has been realized in QCD, with an approach that exploits some general properties of the Higgs transverse momentum distribution at hadron level, to derive the largest interval of transverse momenta where the resummed expression can be applied. In this paper we elaborate the approach of ref. [76], and present a derivation at parton level of the interval of transverse momenta where the collinear approximation of the squared matrix element is accurate and the transverse momentum resummation can be safely applied. A comparison of the present results against those of refs. [77, 78] is currently ongoing [79]. Higgs production via gluon fusion may provide interesting information about possible signals of physics beyond the Standard Model (BSM), like those possible in the Two Higgs Doublets Model (2HDM) or those predicted in the Minimal Supersymmetric Standard Model (MSSM), thanks to the possible exchange in the loop of new colored particle that act as mediators of the interaction between the gluons and the Higgs boson. The total cross section for Higgs production (see ref. [80] for a recent review) and the Higgs transverse momentum distributions provide complementary information (see e.g. ref. [67]) to disentangle the SM from MSSM. The possibility of extracting sensible information from others, on the choice of the matching parameters. the data depends on the accuracy of the prediction of the p?H distribution, and, among { 3 { ing expression with xed order results; we make some comments on the analytic procedure and discuss in more detail the NLO+PS Monte Carlo formulation. In particular we discuss in both cases the role of the scales associated to the matching and we describe the di erences between the two approaches. In section 3 we discuss in detail the gg ! gH and the qg ! qH processes, with respect to their collinear behavior. The latter is used to identify an interval of transverse momenta where the collinear approximation of the squared matrix element is accurate and where it is thus safe to apply the resummation procedure; we introduce a scale w that represents the upper bound of this interval. We discuss both scalar and and in the NLO+PS Monte Carlo framework, with the code gg H quark-mass-effects present in the POWHEG-BOX; the corresponding Higgs p?H distributions are eventually compared. Finally, in section 5 we discuss the implications of the determination of the scale w in the MSSM and in the 2HDM, with the possible production of new heavy states with masses of several hundred GeV and with a possible strong coupling of the Higgs to the bottom quark, enhanced with respect to the SM case. For this study we use the generators gg H MSSM and gg H 2HDM, also present in the POWHEG-BOX. 2 2.1 Remarks on the computation of the Higgs pH distribution ? Analytic resummation and the collinear limit The Higgs boson acquires a transverse momentum p?H because of its recoil against QCD radiation. In xed-order perturbation theory the emission of initial state massless partons yields, in the collinear limit, a logarithmic divergence of the Higgs transverse momentum distribution, signaling a breakdown of the perturbative approach, with an e ective expansion parameter s(p?H ) log(pH =mH ) ? analytic resummation to all orders of the terms 1 in the phase space region of vanishing pH . The s(p?H ) log(pH =mH ) n is performed by ? ? exploiting the universal properties of QCD radiation in the collinear limit and restores an acceptable physical behavior (the Sudakov suppression) of the Higgs transverse momentum distribution in the limit pH In the collinear limit pH ? ! 0 [81{90]. and qg ! qH diverge and can be written, via a Laurent expansion, as Mexact = Mdiv=pH + ? Mreg. In this limit, the second term can be neglected with respect to the rst one and it is possible to recognize that Mdiv is proportional to the Born amplitude times the appropriate radiation term. This factorized structure of the amplitude, neglecting the contribution coming from Mreg which is assumed to be small, can be extended to all orders and it forms the basis of the resummation procedure. Indeed, we can iterate this factorization in the case of the amplitude for the emission of n additional partons. In impact parameter space, this procedure leads to a factorized expression with n divergent emission factors times a ? ! 0 the amplitude for the real emission processes gg ! gH { 4 { term proportional to the Born amplitude. The expression for the approximated amplitude describing the emission of up to n partons can be cast in the form of an exponential series, which can thus be summed to all orders. The relative contribution of Mreg to the full amplitude can be used to assess the accuracy of the collinear approximation and of the factorization hypothesis. The resummed partonic cross section has a factorized structure given by the product of a universal exponential factor, which accounts for the resummation to all orders of the logarithmically divergent terms, multiplied by a process dependent function, which describes the details of the hard scattering process. This factorization requires the introduction of a scale res, called resummation scale [56]. The latter de nes the region where the resummation is applied and it is usually set to a value between 0 and the hard-scattering scale. A customary choice in the literature, for inclusive Higgs production, is to set the central value res = mH =2 [56]. The precise choice of this value is one of the main topics of this paper and will be further discussed in the next sections. Analogously to what happens with the renormalization and factorization scales, the physical observables should not depend upon res, but the truncation at a xed order of the logarithmic expansion leaves a residual dependence on it, which can be used to estimate the uncertainty due to the missing higher-order logarithmic terms; a variation of the scale res in the interval [ res=2; 2 res] is customarily adopted. The matching procedure requires to x the integral of the Higgs transverse momentum distribution to a constant, which is conventionally set to the value of the xed order total cross section [56]. This constraint holds exactly for any choice of res, so that any variation of the resummation scale modi es the shape of the distribution but not its integral and yields thus a correlation between low- and intermediate-pH regions. ? 2.2 Numerical resummation in the NLO+PS framework Another approach to the resummation of terms enhanced by the factor log(p?H =mH ) is the one obtained in the context of PS Monte Carlo, where the multiple emission of partons is numerically simulated via the PS algorithm. The matching between the xed order NLOQCD results and the PS has been discussed in refs. [63, 91, 92] and it is implemented in several tools regularly used in the experimental analyses. In a su ciently general way we can write the matching formula as (2.1) (2.2) d = Bs( B)d B ts0 + d r NLO normalization factor. The latter is de ned as Bs( B) = B( B) + V^ n( B) + R^s( B; r)d r : Z In this formula V^ n represents the UV- and IR-regularized virtual contribution. We use the hat to indicate that an amplitude has been IR-regularized. The partonic subprocesses { 5 { with the emission of an additional real parton can be split into two groups: those that are divergent in the limit of collinear emission, called Rdiv, and the ones that are instead regular, Rreg. We can further subdivide the squared matrix elements of the divergent subprocesses in two parts: Finally, we use the symbol variable: The term Rs contains the collinear singularity of Rdiv, while Rf is a nite remainder. ts for the Sudakov form factor, with t as the shower ordering Rdiv = Rs + R : f ts = e R dtt00 RBs d r (t0 t) : The splitting of Rdiv in eq. (2.3) is de ned up to a nite part which can be reabsorbed in Rs. In the literature two di erent choices have been adopted: in POWHEG Rs = Rdiv, while the relevant Altarelli-Parisi splitting functions. sPij B is proportional to the product of the Born matrix elements times It is interesting to observe that di erent de nitions for Rs generate higher-order e ects in the matched di erential cross section. The possibility of de ning the nite part Rf in an arbitrary way can be exploited to parameterize the uncertainties related to the matching procedure. (2.3) (2.4) (2.5) (2.6) HJEP01(26)5 2.2.1 The role of the damping factor Dh in the POWHEG-BOX framework In the POWHEG-BOX framework, the separation between Rs and Rf can be achieved in a dynamical way using the damping factor Dh, de ned as Dh = h 2 h2 + (pH )2 ? : The divergent and the regular part of Rdiv = Rs + Rf are then de ned as: Rs = Dh Rdiv ; R f = (1 Dh) Rdiv : The role of the scale h is to separate the low and the high transverse-momentum regions and it therefore speci es the range of momenta for which the Sudakov form factor is possibly di erent from 1. In the limit pH ? h we obtain Rs ! Rdiv and Rf ! 0. In this limit ? when pH the Higgs p?H distribution is suppressed by the Sudakov form factor. On the other hand, h we have Rs ! 0 and Rf ! Rdiv and the Sudakov form factor tends to 1. In this latter regime the emission of a real parton is described at xed order by the matrix elements Rf = Rdiv. The di erential distribution generated according to eq. (2.1) contains higher order terms, beyond the claimed accuracy of the calculation, due to the product of B Rs. { 6 { 10 4 0 K( B) = 1 + O( s) : (2.7) Indeed in the large pH region we have ? d = B( B)d B t0 + d r 2 R 1 0 0 t Rs( ) B( B) Originally the factor Dh was introduced to damp the Rs contribution at large pH and to recover the exact xed order result in this kinematic region, at the level of the rst emission ? handled by POWHEG. By varying the scale h, it is possible to check how well the xed order distribution is recovered for large values of pH , as can be seen from gure 1. ? We observe that, while at the level of the rst emission generated by POWHEG (obtained at the level of Les Houches Event File (LHEF)) the NLO result is fully recovered, the showering of the events causes the high-pH tail of the distribution to rise over the NLO prediction. The total NLO cross section is always preserved for any value of h, as can be checked ? by integrating eq. (2.1) over the whole phase space. This property implies in turn that the low- and high-pH regions of the di erential cross section are correlated. Any increase of the distribution at low-pH translates in a decrease of the high-pH tail and vice versa. ? ? in section 2.1 for the resummation scale res: indeed, for pH < h or for pH < res the ? ? ? Sudakov suppression yields a regular behavior of the Higgs transverse momentum distribution, whereas for pH larger than these scales the xed-order description is recovered, at the level of description given by POWHEG. It should however be remarked that res and h have a completely di erent origin. The scale res is introduced as the scale at which the resummation is de ned and the factorization of the partonic cross section implemented. It necessarily appears in the arguments of the logarithmic terms that are resummed. The damping factor Dh is instead a convenient p?H -dependent parameterization of the ambiguity in the de nition of Rs. In a di erent perspective, the scale h controls the range of pH over which the rst term in eq. ( 2.1) is active in the generation of the rst real emission. Since this term contains the normalization factor B, the scale h in turn controls also how the total NLO cross section is spread over the p?H distribution. The role e ectively played by the scale h has some similarities with the one described The emission of the radiation in the POWHEG approach is described by eq. (2.1). Neglecting the contribution coming from the term Rreg (negligible in the case of the Higgs production in gluon fusion), we have two di erent categories of events. One corresponds to the terms in curly brackets (B-events), while the second one is described by the term Rf (remnant events). The latter is present only if the damping factor Dh is used. To avoid double counting of the emissions, in the POWHEG approach the PS is required to emit radiation at transverse-momentum scale lower than the one of the parton emitted by the POWHEG-BOX. More in detail, in the case of B events the PS should start to consider the possibility of an emission exactly at the scale at which the POWHEG parton was radiated. In the default POWHEG-BOX implementation the same choice is applied, for a reason of uniformity, also to the remnant events. This information is therefore computed on an eventby-event basis and the passed to the PS using the SCALUP eld in the LHE event record. It might happen that the value of SCALUP is large and that the description by the PS of real radiation at large transverse momenta is not accurate, since this approach is based on the soft/collinear approximation. It is then natural to consider as an option the possibility of setting an upper bound to the value of the SCALUP variable, close or equal to the one adopted for the h parameter.1 This choice is applied only to the events generated by the Rf part of the real matrix elements, in order to respect the POWHEG accuracy given by the rst term in eq. (2.1) and account for a higher-order e ect. Since these events are relevant only for the description of the high-pH region (in turn de ned by the scale choice h), we do not expect a modi cation of the shape of the distribution in the low-pH region. Indeed, in this way the action of the PS is restricted to the lower part of the pH spectrum, whereas the ? ? large-pH tail is described purely by the LO matrix elements. In section 4 we will compare the description of the Higgs high-pH tail with the default and with the modi ed SCALUP ? ? ? values, in the case of the SM. 1More precisely, we take min(p?H ; h). { 8 { Collinear approximation of partonic squared matrix elements ? matrix element in the collinear limit pH In the previous section we have recalled that the resummation to all orders of the terms enhanced by a log(pH =mH ) factor is possible thanks to the factorization of the squared ? ! 0. Based upon those considerations, we now explain our procedure to determine the accuracy of the collinear approximation of the full squared matrix element with respect to pH , focusing for simplicity on the channel gg ! gH. We then derive numerically the value w of the upper limit of the p?H range where the collinear approximation is accurate, for the scalar and the pseudoscalar nal ? states, considering both the gg ! gH and the qg ! qH channels. 3.1 Helicity amplitudes and kinematic variables We consider the helicity amplitudes2 M 1; 2; 3 (s; p?H ; m2H ) for the process gg ! gH, whose complete expressions can be found for example in ref. [48]. We reorganize them, via a Laurent expansion, as follows: M 1; 2; 3 (s; p?H ; m2H ) = Mdiv 1; 2; 3 (s; m2H )=p?H + Mre1g; 2; 3 (s; p?H ; m2H ) and we use this decomposition to compute the unpolarized squared matrix element exactly, We de ne the ratio C: jMj2, or its collinearly divergent part jMdiv=p?H j2. C(s; p?H ; m2H ) = jM(s; p?H ; m2H )j2 jMdiv(s; m2H )=p?H j2 ; ? which quanti es how the unpolarized exact squared matrix element di ers from its collinear approximation as a function of pH . We observe that by construction we have limp?H !0 C(s; p?H ; m2H ) = 1. In our study we also consider the behavior of the interference term between the top and the bottom quark. For this speci c case we rede ne the parameter C as Cint(s; p?H ; m2H ) = 2Re Mt(s; pH ; m2H )Mb (s; p?H ; m2H ) ? 2Re Mdiv;t(s; p?H ; m2H )Mdiv;b(s; p?H ; m2H ) = pH 2 ? We introduce the following practical criterion: the regular part of the amplitude becomes non-negligible with respect to its collinear counterpart for a value w of pH such that : ? (3.1) (3.2) (3.3) (3.4) To x the setup of our study we choose C = 0:1. This value is arbitrary, but its order of magnitude can be justi ed in the framework of a QCD calculation, since the size of the terms without a collinear logarithmic enhancement is s= times a coe cient of order 1. We do not assign any special meaning to the scale that will be found with our analysis but we rather consider it as a starting point to compute an uncertainty band. At the end of the section we analyze the dependence on the speci c value of C. 2 1 = outgoing one. 1, 2 = 1 are the helicities of the two incoming gluons and 3 = 1 is the helicity of the { 9 { minimum value for s: The amplitude of the process gg ! Hg is a function of two independent kinematic variables, e.g. s and pH . The production of a ? nal state with a de nite p?H requires a smin = m2H + 2(p?H )2 + 2p?H q ? (pH )2 + m2H : We study the behavior of the amplitude as a function of pH for s = smin + ssoft, where ssoft is necessary to avoid the soft divergence and focus only on the collinear behavior. The choice of a value of s close to smin is phenomenologically motivated by the strong PDF ? suppression in the hadronic cross section for increasing partonic s. An analogous procedure is used to determine the scale w for the qg ! qH subprocess, with the analytic expressions of ref. [25] and in the case of pseudoscalar production using the formulae in ref. [93]. We assume that the full amplitude is the sum of a top and a bottom contribution, neglecting the light quark generations. Furthermore, we do not consider the possibility of additional colored particles running in the loop, since the current LHC results hint to the fact that, if these states exist, their mass is probably much larger than the top mass. Therefore they would not a ect the shape of the Higgs transverse momentum distribution for values of pH that are phenomenologically interesting. Under the assumption that the coupling of the gluons to the quarks is the one dictated by QCD and that all the details about the coupling of the Higgs to the quarks can be factorized from the rest of the amplitude, we can consider the value of the scales wt, wb and wi, that we respectively nd in the case of squared matrix elements with only top quark diagrams, only bottom diagrams or for the top-bottom interference, as model independent. As a consequence, the determination of the scales depends only on the quark and the Higgs masses. While the scales computed with only one quark might have a physical interpretation in the BSM scenarios where that quark yields the dominant contribution to the cross section, the scale of the interference terms, while unphysical, is a necessary tool to treat accurately the full theory in scenarios where both top and bottom quarks are equally important. Since the full squared matrix element, including top and bottom quarks, factorizes in the collinear limit, the same pattern should be followed not only by the terms with the squared amplitude of one single quark, but also by the interference terms, making our treatment viable. In sections 4 and 5 we will discuss how these results can be exploited in a model speci c framework. 3.2.1 To exemplify the outcomes of our procedure, we show the results for the variable C(s; p?H ; m2H ) for a Higgs boson with mH = 125 GeV and mH = 500 GeV in in the case of the gg ! gH subprocess. We plot in red and blue the behavior of the squared matrix elements computed including only the top or only the bottom diagrams, gure 2, in green we show the behavior of the interference of the top and bottom amplitudes. In HJEP01(26)5 2:0 the gg channel. In red we show the results for the squared top quark amplitude, in blue the ones for squared bottom amplitude and in green the ones for the interference. For comparison, in the case of mH = 125 GeV, we also plot the curve for the HQEFT in orange. the same gure, for mH = 125 GeV, we plot in orange the results obtained by applying the same procedure to the HQEFT matrix elements. We rst discuss the impact of the regular terms in the case of a light Higgs. We compare the results obtained with the exact matrix elements including only the top quark ? with the ones in the HQEFT; we observe that in both models a deviation by more than 10% from the collinear approximation occurs for p?H > 55 GeV. Since it is present in both cases, this e ect should thus not be interpreted as a top mass e ect; the latter becomes visible for pH > 150 GeV. From the analysis of the helicity amplitudes, we observe that this deviation from the collinear approximation stems from M the deviation from the collinear approximation starts from pH > 19 GeV. In the case of + . For the bottom quark, the interference terms, we observe that the determination of the scale wi is dominated by the behavior of the bottom amplitude; the corresponding value, wi = 9 GeV, is smaller ? than the ones obtained in the other two cases. In the case of a heavy Higgs, with mH > mt; mb, the scale of the process is set by the mass of the boson (e.g. mH = 500 GeV) and the HQEFT approximation of the amplitude is not valid. We observe that the amplitude that includes only the top-quark diagrams deviates from its collinear approximation3 for p?H > 111 GeV. Instead, the squared matrix element that includes only the bottom-quark diagrams deviates from its collinear approximation for p?H > 63 GeV. Finally, for the interference terms we nd the bound p?H > 18 GeV. In the left section of table 1 and in gure 3 we present the values of the scales w, derived from the study of scalar Higgs production for di erent choices of mH 2 [125; 800] GeV, separately in the case of squared matrix elements computed including only the top, only for mH = 125 GeV we have r ' 1=2. terms of the ratio r = pH=mH; in the case under discussion (only top diagrams) we nd r ' 1=4 whereas ? Scalar, collinear deviation scale w (GeV) wgg b wgg i wqg t wqg b wqg i mined separately in the two partonic subprocess (left) and after their combination according to eq. (3.8) (right). for the top quark, in blue the one for the bottom and in green the one for the interference term. The dashed style represents the scales obtained in the gg channel, the dot-dashed style the ones in the qg channel. Continuous lines are used for the merged scales. We also show as a dotted line the scale choices mH =2 [56] and mH =1:2 [5]. In orange we show the results for the HQEFT. the bottom diagrams or for the interference of the top and bottom amplitudes; the results are presented separately for the two partonic subprocesses, gg ! gH and qg ! qH. We observe that in the gg ! gH channel both scales wt;b increase with the Higgs mass, with the exception of the region of real top-pair production threshold, where the e ect on wt of additional terms that induce a deviation from the collinear approximation is visible. In the bottom-quark case such phenomenon does not show up, because for realistic values of mH the process scale is always well above the bottom-pair production threshold. The interference scale wi has a peculiar behavior: in fact, it shows a growth with mH until the top-pair production threshold and then it decreases for larger mH , until it vanishes for mH = 589 GeV, with our mt and mb choices; for even larger mH values it grows 9 14 23 26 23 17 6 24 46 again. In order to explain why wi vanishes, we should recall that the interference terms, as a function of mH and for xed mt and mb, are not positive de nite and may change sign for a speci c value of mH ; in particular, when the underlying LO (i.e. of the process gg ! H) interference terms vanish, also the collinear approximation does. In this point the interference terms of the processes gg ! gH and qg ! qH are thus collinear nite, the function C(s; p?H ; mH ) diverges for all pH and the scale wi is equal to zero, indicating that ? the p?H distribution is regular and a LL resummation is not needed. It should be noted that, for this speci c con guration, the importance of the interference term is in any case small, since it vanishes at LO. We observe that in the qg ! qH channel the scales are lower than in the previous case and that they decrease for increasing values of the Higgs mass. The basic argument to explain this di erent behavior can be found analytically in the HQEFT: we expand the ratio C(s; p?H ; m2H ) in powers of pH around p?H = 0 and we nd ? The di erent behavior with respect to mH of the scale w is due, in the gg case, to the fact that the function C receives corrections with negative powers of mH , so that for heavy Higgs masses there is a larger interval of pH where the collinear limit provides a good approximation of the full result; in the qg case instead, there are corrections quadratic in mH , such that the deviation of C from 1, for large mH , occurs at smaller p?H values. A numerical analysis with the full dependence on the top and bottom masses con rms the ? explanation derived above in the HQEFT. The interference scale vanishes, as expected, for the same mH value in the gg ! gH and the qg ! qH channel, since they factorize to the same LO term. The di erent values of the scales wt;b obtained in the two partonic channels gg and qg give rise to a practical problem, in case one wants to use at hadron level one single scale to control the e ects of multiple parton emissions; given that the w value from the gg channel is always larger than the one from the qg channel, we can expect that the nal value will lie in between; we evaluate it with a weighted average, with the relative contributions of the two channels in each bin, further adjusted to account for the shape of the physical distribution. We de ne Z wgg wqg 0 wgg+qg(mH ) dp?H @wgg d dgpg?H+qg + wqg d dgpg?H+qg A d qg 1 dp?H d gg+qg indtpe?Hrval ; where interval = Z wgg wqg dp?H d gg+qg dp?H : d gg dp?H (3.6) (3.7) (3.8) (3.9) wqg b wqg i determined separately in the two partonic subprocess (left) and after their combination according to eq. (3.8) (right). In gure 3, and in table 1, in the last three columns to the right, we show the results with p of this combination, which are our best determination for the scales to be used in the simulation of the hadronic di erential cross section.4 We have used the code SusHi [94], the evaluation of the hadronic cross section, the combined scales are dependent on the p S = 13 TeV, to compute the weights used in eq. (3.8). Since this procedure requires S value used and on the other hadronic parameters. In particular this is true for the choice of the renormalization and factorization scale, that we have assumed to be r = f = mH . However we have veri ed that the e ect on the channel-combined value for the scales is only at the of few GeVs, well within the uncertainty band that we are considering. A ner scan in the Higgs mass, is provided in tables 4 and 5, in appendix A. 3.2.2 In table 2 we present a sample of the results, analogous to the ones of the previous subsection, for the case of pseudoscalar Higgs production. The general behavior of the two partonic channels is similar to the one observed for scalar production. One di erence can be observed at the top-pair threshold, where a cusp appears in the wt prediction, re ecting the analogous feature of the total cross section. The scale wi vanishes for a di erent value of the pseudoscalar mass, mA = 445 GeV, because of the di erent LO dependence on mA, mt and mb. As for the scalar case, a more detailed scan as a function of mA is available in tables 4 and 5, in appendix A. 3.3 Dependence on auxiliary parameters The value of the resummation scale has been determined with an analysis of the partonic squared matrix element, for xed value of the partonic invariant s. For a given nal state con guration and in particular for a given value of pH , the hadronic distribution receives contributions from all the partonic cross sections with smin s S, where S is the ? 4The relative weight of the two partonic channels, as a function of the Higgs mass, is slowly varying, so wb = 0:1 wbqg + 0:9 wbgg. These relations approximate the exact combination at the 5% level. that we can approximate the result of equation (3.8) with the simpler relations wt = 0:2 wqg + 0:8 wtgg and t V eG200 350 300 )250 V eG200 w150 100 50 0 the scale determination on the choice of the value of the cut-o ssoft; on the right dependence of the scale determination on the choice of the value C 2 [0:05; 0:2]. The dashed curves represent the values obtained by enlarging the parameter whose dependence is under study while the dot-dashed curves are obtained by the rescaling of the parameter to a smaller value. hadronic Mandelstam invariant. To make an educated guess of the resummation scale, we have studied the partonic con guration which has the largest weight at hadron level; due to the PDF suppression at large x, this happens to be the smallest possible value of s. The choice s = smin satis es this requirement but introduces an additional technical problem, namely the presence of soft divergences in the amplitude. To avoid this issue when computing the curves in gure 2 we have set s = smin + ssoft with ssoft = (100 GeV)2. We have veri ed that the results are weakly dependent on the speci c value of ssoft, as shown in gure 4 (left plot) where the bands describe the results, as a function of the Higgs mass, obtained with a variation of ssoft in a range [1=10; 10] with respect to the central choice. In particular we remark that the scale prediction is stable for small values of ssoft, i.e. in the soft-emission region, phenomenologically the most relevant. In gure 4 (right plot) we show the dependence of the scale determination on the value assigned to C. The bands describe the results, as a function of the Higgs mass, obtained by varying the parameter in the interval C 2 [0:05; 0:2]. As expected, e.g. from the inspection of gure 2, there is a direct proportionality between the value of C and the resulting scale w. Due to the assumptions used in our procedure, we stress that the determination of the central value for w does not have an absolute meaning. It is rather the starting point to de ne an interval of reasonable values for the scale w that in turn should be used to compute an uncertainty band for the transverse momentum distribution. 4 Standard Model phenomenology We consider now the evaluation of the Higgs transverse momentum distribution in protonproton collisions at the LHC in the SM. We use the analytic results of [95] implemented in the public code HRes and the shower Monte Carlo implemented in the POWHEG-BOX [67]. For the former, we study the impact of di erent choices of the resummation scale res, while with the latter we vary the value h which enters the damping factor Dh. In both cases we consider the possibility of a separate treatment of the top and of the bottom quark contributions. In the numerical analysis we use mt = 172:5 GeV, mb = 4:75 GeV, the PDF sets MSTW2008nlo68cl and MSTW2008nnlo68cl [96] with their corresponding values of s(mZ ). We chose R = F = mH as the renormalization and factorization scales. We use PYTHIA8 [97, 98] with the tune AU-CT10 to shower the POWHEG events. This speci c tune The center of mass energy at the LHC has been assumed to be p S = 13 TeV. was chosen since it is the same used by the ATLAS collaboration for their Higgs analyses. The gluon fusion process, including the top and the bottom quark diagrams in the scattering amplitude, is a three-scale problem, as was already stressed in ref. [1] and as we have seen in the previous sections: the Higgs mass, the value of pH and the mass of the quark. The bottom quark contributions spoil the validity of the factorization hypothesis for pH values ? smaller than in the top quark case and require a dedicated treatment. In order to make explicit the role of the top and of the bottom quarks, the squared matrix elements can be ? rearranged as jM(top + bot)j2 = jM(top)j2 + jM(top + bot)j2 h jM(top)j2i ; (4.1) where we have put in round bracket the quarks that run in the loops of the diagrams. The square brackets contain the top-bottom interference terms and the square of the modulus of the bottom amplitude. The rationale behind this rearrangement is that in the SM the dominant contribution to the gluon fusion is due to the top quark diagrams, while the bottom quark diagrams yield a correction to the former; it is thus reasonable to make one dedicated scale choice for the top quark and a second scale choice for all the other terms, even if they still include top quark diagrams via interference terms. We recall that by construction the total cross section does not depend on the value of the resummation scale in HRes (or equivalently of the scale h in POWHEG). This fact allows us to write the following identity (top + bot) = (top; t) + [ (top + bot; b) (top; b)] ; (4.2) where here and after, with a slight abuse of notation, we have introduced the symbol (q; ) to indicate the total cross section evaluated with the quark q in the loops, using, in the numerical code, the matching parameter at the scale . The latter is the resummation scale Qi in HRes and the scale h in POWHEG. This equation is trivial for the total cross section, and represents a possible recipe for the evaluation of di erential observables, speci cally the Higgs boson transverse momentum.5 For our phenomenological analysis we use two scales, one for the squared matrix element with only the top quark and one for the other contributions, to allow a comparison 5In POWHEG, at the di erential level, the extraction of a speci c contribution by subtraction is bound to introduce spurious terms due to the fact that the Sudakov form factor is non-universal. However, due to our speci c scale choices that guarantee a good accuracy of the collinear approximation in the p where the Sudakov form factor has its major e ect, we can argue that in this region the argument in the exponent of the Sudakov factor is well approximated by the relevant universal expression R=B ' sPij=t, ?H range limiting the impact of the spurious terms. with the results presented in ref. [1]; we use a combination analogous to the one of eq. (4.2) to evaluate also the di erential distributions. In section 3.3 we have given an estimation of the uncertainty in the determination of the scales w by varying the auxiliary parameters that we have used in our computation. In theory it is possible to use the range of scales obtained with such a procedure as the range of values to be used for the matching parameter to estimate the uncertainty on the prediction for the transverse momentum distribution. However these values depend in a non-trivial manner on the Higgs mass considered. We observe that a variation by a factor of 2 of the central value widely covers the range of scales that we nd with our explicit computation, thus yielding a conservative assessment of the uncertainty. To simplify the uncertaintyestimation procedure we have then decided to compute the uncertainty bands using the following standard prescription: we consider the 9 combinations of the pairs ( t; b) of the two matching parameters, which can be obtained from the sets ( t=2; t; 2 t) and ( b=2; b; 2 b), where we called t and b the respective central values, and we take the envelope of all the predictions. We consider the three following cases and, in each of them, we compute the uncertainty band according to the rule described above: 1. we use POWHEG and we set the scale of the top quark diagrams ht = mH =2 and the scale of the bottom quark contributions hb = mb 2. we use POWHEG and follow the analysis described in section 3 and in particular the values of table 1: we set ht = wt = 48 GeV and hb = wi = 9 GeV. The wi is chosen over wb since the interference terms yield a larger contribution to the process than the bottom quark squared matrix elements. 3. we use HRes at LO+NLL accuracy and set the resummation scale of the top quark diagrams Q1 = mH =2 and the resummation scale of the bottom quark contributions Q2 = mb, following the choices of ref. [1]; ? The distributions obtained with HRes and POWHEG share the same matrix elements that describe at NLO-QCD the inclusive Higgs boson production, and di er by subleading NNLO and by higher order terms, which might nevertheless be numerically relevant. The comparison of the shape6 of the pH distribution, in gure 5, of the results of item 1 (blue dot line) and 3 (dashed red line) is meant to expose the di erences of the two codes taken with their default setup, when they are run with the same accuracy for the total cross section, NLO-QCD, and with the same value for the matching parameters. On the left we show the absolute comparison of the results, while on the right we show the ratio of the di erent predictions over the one obtained with POWHEG and the HRes scale choice (item 1). As discussed in section 2 the two basic formulae used to generate the Higgs pH spectrum di er by subleading O( s2) and higher-order terms, part of which are controlled by the resummation scale in HRes or by the h scale in POWHEG. For the above reason, even if we assign the same numerical values to the scales Q and h, we expect a certain level of ? discrepancy for the central predictions. 6With the term shape we mean that we have normalized the di erential distribution to 1. 0:00 10 1 10 2 10 3 10 4 10 5 0 as computed by HRes and POWHEG , for di erent values of the scales. On the left we show the absolute value of the shape, while on the right we normalize the results to the one obtained with our ht = mH =2 and hb = mb. In dotted blue we show the result obtained with ht = mH =2 and hb = mb; with a continuous green we show the prediction obtained with ht = wt = 48 GeV and hb = wi = 9 GeV; the dashed red line is prediction obtained with HRES at LO+NLL, with Q1 = mH =2 and Q2 = mb. For all the three curves we show the corresponding uncertainty bands using the same colors. With a continuous gray line we show the results obtained at NLO. Indeed we see that in the region where resummation e ects are relevant, the two ? in this part of the spectrum. codes behave di erently, with HRes giving a softer distribution than POWHEG. Speci cally, the shape of the distribution produced by HRes is larger than the one from POWHEG for pH ? for pH ? 50 GeV, while for higher p?H the behavior is the opposite. In the high-p?H region, mH , we see that the HRes result coincides with the xed-order distribution: in fact, the code HRes uses the full matched expression for p?H values smaller than mH and implements a smooth transition to the pure xed-order expression, which is used in the high-pH tail; for this same reason, the HRes resummation scale uncertainty band vanishes In the high-p?H range POWHEG shows a distribution harder than the xed-order one, because of the showering e ects applied on top of the POWHEG formula for the rst emission. Since HRes does not include non-perturbative e ects, which are present in the selected tune of the PYTHIA shower, an additional problem in the comparison emerges: the non? perturbative e ects are relevant at small transverse momenta of the radiated partons. In addition, in the low-pH region, the di erent expression of the HRes and POWHEG Sudakov form factors (for the latter see equation (2.4)) has a role to determine the precise shape of the distribution. By construction, the unitarity constraint, that forces the total cross section to be always preserved, implies an anti-correlation between the low-p?H and the high-p?H parts of the spectrum. The comparison in gure 5 of the results of approximations 1 and 2 shows the sensitivity, within the POWHEG formulation of the matching, to the h scale variations. The two 0:00 10 1 10 2 10 3 10 4 10 5 0 using the default POWHEG-BOX implementation (blue) and the one with the modi ed SCALUP prescription for the remnant events (orange). central values lie in the uncertainty bands obtained with the other scale choice. The main di erence can be observed at small pH , whereas the deviation for 50 be interpreted as a consequence of the unitarity constraint. ? pH ? 150 GeV can close to each other. We observe, by using ht = wt and hb = wi, an accidental improvement of the agreement between HRes and POWHEG in the region of p?H < 100 GeV, where the two central values lie In gure 6 we present the impact in POWHEG of a di erent choice of the variable SCALUP, as discussed in section 2.3. We set SCALUP=ht, a constant value, while we keep unchanged all the other parameters and in particular the value of the scales ht;b in the damping factor D(h). The choice for the SCALUP value is in accordance with the dominant role played by the top-quark loop in the SM. We observe that the central prediction of this modi ed POWHEG version is lower than the default one for pH 200GeV and tends to recover the ? xed-order distribution at large transverse momenta. We interpret the reduction of the di erential cross section at large pH as due to the missing contribution in this region from the PS emissions. The accuracy of the latter is questionable, since the PS is based on the soft/collinear approximation and might be inadequate to describe large-pH radiation. ? ? 5 Beyond SM phenomenology The description of the Higgs transverse momentum distribution in the SM, with mH = 125 GeV, is characterized by the dominant role played by the top-quark contribution, such that the bottom-quark e ects can be treated as a correction. Moreover, with a light scalar Higgs, the HQEFT limit is a good approximation of the full SM, and the determination of the scale of validity of the collinear approximation (and hence of the applicability range of the resummation techniques) reduces to a problem involving only mH and pH . At variance with the previous case, and still in the SM, we know that with a heavy Higgs boson, the description of the p?H distribution is a multiscale problem; indeed, the minimal energy scale necessary to produce the nal state immediately probes the top-quark loop. ? In a generic BSM scenario it is possible to consider enhanced couplings of the bottom quarks to a relatively heavy Higgs boson, scalar or pseudoscalar. In these con gurations, our intuition, accustomed to a light SM-like Higgs phenomenology, may fail in the determination of the correct regime where the resummation techniques can be safely applied. Since a priori we do not know exactly how the contributions from the di erent quarks HJEP01(26)5 interplay in the full result, following ref. [77], we can generalize eq. (4.2) to (top + bot) = (top; t) + (bot; b) + [ (top + bot; i) (top; i) (bot; i)] ; (5.1) where the last term allows us to use a separate scale for the top-bottom interference term. As before, the parton level analyses discussed in section 3 provide a model independent ansatz for the three relevant scales, t;b;i: these are the scales wt,wb and wi, listed in tables 1 and 2 as a function of the Higgs boson mass. In order to illustrate the phenomenological consequences of our study, we show our predictions in the 2HDM and in the MSSM7 and we compute the uncertainty bands with an extension of the procedure described in section 4: we consider all the 27 combinations of the three matching scales and then take their envelope. The range of scales spanned represents again a conservative choice to assess the matching uncertainty. 5.1 2HDM phenomenology We consider the type-II 2HDM. We adopt a purely heuristic approach to show the impact of our study, choosing the 2HDM parameters that are relevant for the gluon fusion process by following only the requirement that they represent three di erent scenarios: one where the cross section is dominated by the top-quark; one where the contribution of the top and the bottom quark are of the same order of magnitude; and one where the process is dominated by the bottom quark matrix elements. The explicit values for the parameters are reported in table 3 for all the three scenarios. In all three cases we choose to study a heavy Higgs of mH = 500 GeV. The corresponding values for the scales are wt = 96 GeV, wb = 58 GeV and wi = 17 GeV. For the simulation we adopt the Monte Carlo generator gg H 2HDM available in the POWHEG-BOX. We now present our best predictions obtained with the three-scale procedure and check how well they are approximated by a one-scale approach. In gure 7 we show the results for the rst scenario. In this case we have that the process is dominated by top quark contribution. Indeed we notice that the three scales result is well approximated by the one scale result with the scale taken equal to the top scale. 7A detailed comparison with the approach of ref. [77] is currently ongoing [79]. 1d HpdT 10 1 10 2 10 3 10 4 10 5 0 500 50 1 pTH (GeV) pTH (GeV) computed by the gg H 2HDM generator in the 2HDM scenario A. On the left we show the absolute value of the shape, while on the right we normalize the results to the one obtained with the scales determined by our procedure, ht = wt = 96 GeV,hb = wb = 58 GeV and hi = wi = 17 GeV. In dotted blue we show the result obtained with our scale choice, its uncertainty band drawn in lighter blue; with a continuous green (dashed red line) we show the prediction obtained with h = mh=2 (h = mh=1:2). In dashed black we show the results obtained with a single run with the scale h set to wt. Finally in gray we show the NLO prediction. 2HDM scenarios Scenario A Scenario B Scenario C 2:0 1:5 R1:0 0:5 0:0 0 500 12 1 On the other hand, in the second case shown in gure 8, we have that the contributions coming from the two quarks are of the same order of magnitude. In this case we observe that the result obtained by using three scales is not recovered by simulations with just a single scale, with either the top or the bottom one. Finally, in gure 9 we see that in the bottom dominated scenario, we have a similar situation as in the top dominated case, though the scale to be used in a one scale run is wb instead of wt. uncertainty band. In all three cases we stress that the using values of the order of mH =2 or mH =1:2 for the matching parameter h yields results that are in the best case at the limit of the 10 1 10 2 10 3 10 4 10 5 0 100 10 1 10 2 10 3 10 4 10 5 0 POWHEG+PY8, ht;b;i = wt;b;i POWHEG+PY8, h=mh=2 POWHEG+PY8, h=mh=1:2 POWHEG+PY8, h=wb NLO 2:0 1:5 R1:0 0:5 0:00 2:0 1:5 R1:0 0:5 0:00 pTH (GeV) pTH (GeV) with a single run with h = wt while the continuous black line corresponds to h = wb. pTH (GeV) pTH (GeV) with a single run with h = wb. MSSM phenomenology We consider an explicit example in the MSSM by taking, in its parameter space, a point still allowed by the most recent available data, according to the analysis of ref. [99], and to the results of the code HiggsBounds [100{103]. The same point has been considered also in ref. [77]. We choose the so called mmod+ scenario de ned in ref. [99] and set MA = 500 GeV h and tan = 17 to fully specify our input parameters; as a result we obtain that the masses of the two CP-even Higgses are respectively mh = 125:6 GeV and mH = 499:9 GeV. The corresponding values of the w scales are: wt = 96 (109) GeV, wb = 58 (58) GeV and wi = 17 (14) GeV, for a scalar (pseudoscalar) boson. We use these values to set the t;b;i parameters that enter eq. (5.1). For the simulation we adopt the Monte Carlo generator gg H MSSM available in the POWHEG-BOX. In the simulation we include the full particle content of the MSSM. We do not expect an important contribution from the squarks because in this point of the MSSM parameter space their masses are, respectively, mt~1 = 876 GeV, mt~2 = 1134 GeV, m~b1 = 1007 GeV and m~b2 = 999 GeV. With this speci c parameter choice, the light CP-even Higgs is similar to the SM scalar, not only for the total cross section, but also for the shape of the p?H distribution. The heavy CP-even scalar and the pseudoscalar bosons have instead di erent properties, because of the di erent coupling strength to the top and to the bottom quarks. In gures 10 and 11 we show the results for the shape of the transverse momentum distribution in the case of the light CP-even Higgs (top plot) and of the heavy CP-even Higgs (bottom plot). We do not show the plot for the pseudoscalar since we expect a behavior similar to the one of the heavy Higgs. Besides plotting the central values and the uncertainty band corresponding to our scale choice, we also show the results obtained with only one matching scale, with the commonly used prescriptions h = mh;H =2 and h = mh;H =1:2. We observe that the three choices yield a di erent shape of the distribution in the soft region where resummation e ects are important: the scale choices h = mh;H =2 and h = mh;H =1:2 give a suppression in the rst bins and an enhancement for pH larger than 40 GeV with respect to the distribution obtained following eq. (5.1). In the case ? of a light Higgs, we see that the central value obtained with h = mh=2 is contained in the uncertainty band of the prediction computed by using three scales, while the result corresponding to h = mh=1:2 is at the edge of the same uncertainty band. In the case of the heavy CP-even Higgs, the h = mH =2 and h = mH =1:2 curves lie outside the uncertainty band of the three-scale result; they deviate from its central value by O(40%), both in the low- and in the high-pH tails. ? 6 Conclusions The study of the Higgs transverse momentum distribution may provide important insights ? about the properties of the recently discovered scalar resonance. The theoretical prediction of this observable requires, in the region of small pH values, the resummation to all orders of terms enhanced by powers of log(pH =mH ), while at large values of pH , calculations provide the most accurate description available. The consistent matching of ? the two approaches requires the introduction of a momentum scale, that separates the soft and the hard pH regions. ? Since the validity of the resummation formalism relies on the collinear factorization of the squared matrix elements describing real parton emissions, we investigated the accuracy of the collinear approximation in the gluon fusion process, in the presence of an exact description of the top and bottom quarks running in the virtual loop. The discussion involves three scales, namely the Higgs mass, the Higgs transverse momentum and the quark masses. Relying on the collinear singularities structure of the O( s) real matrix elements, we determined, in a model independent way, as a function only of the Higgs and the quark HJEP01(26)5 h T 1d pd 10 1 1d HpdT 100 10 1 POWHEG+PY8, ht;b;i = wt;b;i POWHEG+PY8, h=mh=2 POWHEG+PY8, ht;b;i = wt;b;i POWHEG+PY8, h=mh=2 POWHEG+PY8, h=mh=1:2 NLO 2:0 1:5 R1:0 0:5 0:0 0 2:0 1:5 R1:0 0:5 0:0 0 HJEP01(26)5 phT (GeV) phT (GeV) = 17 and mA = 500 GeV in the mhmod+ scenario. On the left we show the absolute value of the shape, while on the right we normalize the results to the one obtained with our scale choice. In dotted blue we show the result obtained with our scale choice, its uncertainty band drawn in lighter blue; with a continuous green (dashed red line, we show) the prediction obtained with h = mh=2 (h = mh=1:2). pTH (GeV) masses, three scales, wt; wb and wi, associated to the terms in the full squared matrix elements containing only the top-, only the bottom-quark contributions or the top-bottom interference terms. Their values, presented in tables 1 and 2 and, with a ner scan of the Higgs mass, in appendix A, represent our main result. These scales are derived from a parton-level analysis and can be eventually used in any hadron-level computation (analytic or Monte Carlo) of the Higgs pH distribution, following eq. (5.1). They indicate the upper limit of the pH range where the resummed part of the cross section can be evaluated in a reliable way, because of the good accuracy of the collinear approximation of the full squared matrix elements. They represent an ansatz for the matching scales, whose values do not have an absolute meaning, but are rather the starting points to build an uncertainty band. The procedure to compute an uncertainty band is described in section 4 and o ers a simple but quite conservative recipe to derive this band. A more aggressive approach would exploit the scales obtained with a variation of the parameter C 2 [0:05; 0:2], as discussed in section 3. Our analysis is relevant for an accurate prediction of the Higgs pH distribution, both in ? the SM and in BSM scenarios. In the latter case, our approach allows us to decompose the di erent contributions to the pH distribution, also in the presence of a non trivial interplay between the Higgs transverse momentum and the Higgs, top and bottom masses, for any generic ratio between the strength of the couplings of the Higgs boson to the top and to ? the bottom quarks. The description of the Higgs transverse momentum distribution, based on the use of three di erent scales for the matching parameter, represents our best ansatz for this observable. We remark, however, that in various cases this result can be accurately approximated with only one run that uses one single scale, the one associated to the dominant contribution to the scattering amplitude. This conclusion is obviously possible only a posteriori. We stress the impact of the matching scale determination with one nal comment, relevant in the context of the searches for new heavy scalars, referring to the results shown in gures 10 and 11. Our procedure de nes the scales wt;b;i, whose variation in a given range is then exploited to compute an uncertainty band of the distribution. The results presented in section 5 are obtained with a conservative choice for the range of scale variation, as described at the beginning of the same section. The use of a single-scale simulation, with the matching scale set equal to the commonly adopted SM value mH=2, can lead to predictions that lie outside of this most conservative uncertainty band described above and that can di er with respect to our best central value by 30-40% both in the low- and in ? the high-pH tails of the distribution. Acknowledgments We would like to thank Giancarlo Ferrera, Stefano Frixione, Massimiliano Grazzini and Carlo Oleari for many discussions on the subject. We thank Giuseppe Degrassi, Robert Harlander, Hendrik Mantler, Pietro Slavich and Marius Wiesemann for carefully reading the manuscript. We are indebted with Paolo Nason for several suggestions and clari cations about POWHEG. E.B. was partially supported by the EU ITN grant LHCPhenoNet, PITN-GA-2010264564. A.V. is supported in part by an Italian PRIN2010 grant, by a European Investment Bank EIBURS grant, and by the European Commission through the HiggsTools Initial Training Network PITN-GA-2012-316704. A.V. thanks the LPTHE, the University \Pierre et Marie Curie" Paris VI and the Institut Lagrange in Paris for nancial support in summer 2013. Scan over the Higgs mass of the scales wt;b;i In the appendix we include two tables with the values of the combined gg-qg collineardeviation scales, for scalar and pseudoscalar masses between 100 GeV and 500 GeV, separately for the top, the bottom and the interference contribution. The top pole mass has been set to 172:5 GeV, while the bottom pole mass is equal 4:75 GeV, following the prescription by the Higgs Cross section Working Group (HXSWG). The merging of the scales was implemented by using the information on the relative importance of the two partonic subprocess as given by the code SusHi[94]. 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E. Bagnaschi, A. Vicini. The Higgs transverse momentum distribution in gluon fusion as a multiscale problem, Journal of High Energy Physics, 2016, 56, DOI: 10.1007/JHEP01(2016)056