NNLL resummation for the associated production of a top pair and a Higgs boson at the LHC

Journal of High Energy Physics, Feb 2017

We study the resummation of soft gluon emission corrections to the production of a top-antitop pair in association with a Higgs boson at the Large Hadron Collider. Starting from a soft-gluon resummation formula derived in previous work, we develop a bespoke parton-level Monte Carlo program which can be used to calculate the total cross section along with differential distributions. We use this tool to study the phenomenological impact of the resummation to next-to-next-to-leading logarithmic (NNLL) accuracy, finding that these corrections increase the total cross section and the differential distributions with respect to NLO calculations of the same observables.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

https://link.springer.com/content/pdf/10.1007%2FJHEP02%282017%29126.pdf

NNLL resummation for the associated production of a top pair and a Higgs boson at the LHC

Received: November NNLL resummation for the associated production of a top pair and a Higgs boson at the LHC Alessandro Broggio 1 2 4 8 9 10 11 Andrea Ferroglia 1 2 4 6 7 9 10 11 Ben D. Pecjak 1 2 3 4 9 10 11 Li Lin Yang 0 1 2 4 5 9 10 11 g 1 2 4 9 10 11 Open Access 1 2 4 9 10 11 c The Authors. 1 2 4 9 10 11 0 Collaborative Innovation Center of Quantum Matter 1 365 Fifth Avenue , New York, NY, 10016 U.S.A 2 300 Jay Street, Brooklyn, NY, 11201 U.S.A 3 Institute for Particle Physics Phenomenology, Ogden Centre for Fundamental Physics 4 James Franck-Stra e 1, Garching , D-85748 Germany 5 School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University 6 Physics Department, New York City College of Technology, The City University of New York 7 The Graduate School and University Center, The City University of New York 8 Physik Department T31, Technische Universitat Munchen 9 No. 5 Yiheyuan Road, Beijing, 100871 China 10 South Rd , Durham, DH1 3LE United Kingdom 11 Department of Physics, University of Durham , Science Laboratories We study the resummation of soft gluon emission corrections to the production of a top-antitop pair in association with a Higgs boson at the Large Hadron Collider. Starting from a soft-gluon resummation formula derived in previous work, we develop a bespoke parton-level Monte Carlo program which can be used to calculate the total cross section along with di erential distributions. We use this tool to study the phenomenological impact of the resummation to next-to-next-to-leading logarithmic (NNLL) accuracy, nding that these corrections increase the total cross section and the di erential distributions with respect to NLO calculations of the same observables. top; pair; and; a; NLO Computations; QCD Phenomenology 1 Introduction Outline of the calculation Numerical results Scale choices Total cross section Di erential distributions Introduction The associated production of a top-quark pair and a Higgs boson can provide direct information on the Yukawa coupling of the Higgs boson to the top quark, which is crucial for verifying the origin of fermion masses and may shed light on the hierarchy problem related to the mass of the Higgs boson. For this reason, experimental collaborations at the Large Hadron Collider (LHC) are actively searching for this Higgs-boson production mode in the currently ongoing Run II. The Standard Model (SM) cross section for this process at a center-of-mass energy of 13 TeV is quite small, of the order of 0:5 pb. Di erences between the measured cross section and the corresponding SM predictions could indicate the presence of new physics which modi es the top-quark Yukawa coupling. Consequently, a large amount of work has been devoted to the study of this process beyond leading order (LO) in the SM. The LO cross section scales as s2 , where the strong coupling constant and the electromagnetic ne structure constant, respectively. The next-to-leading order (NLO) QCD corrections to this process were rst evaluated more than ten years ago [1{6]. This process also served as a benchmark for validating automated tools for NLO calculations; in [7, 8] the NLO corrections were calculated automatically and interfaced with Monte Carlo event generators, thus including parton shower and hadronization e ects. Electroweak corrections to this process were studied in [9{11]. NLO QCD and electroweak corrections were included in the POWHEG framework in [12]. In [13] the NLO corrections to the associated production of a top pair and a Higgs boson were studied by considering also the decay of the top quark and o -shell e ects. The cross section for the associated production of a top pair, a Higgs boson and an additional jet at NLO was evaluated in [14]. Perturbative calculations for the ttH production process are di cult and involved, due to the presence of ve external legs, four of which carry color charges. Consequently, it is not likely that the next-to-next-to-leading order (NNLO) QCD corrections for this process will be computed in the near future. For this reason, the impact of soft gluon emission corrections beyond NLO was the subject of recent studies. In [15] the soft gluon emission corrections to the total ttH cross section in the production threshold limit were evaluated up to next-to-leading logarithmic (NLL) accuracy; the production threshold is de ned as the kinematic region in which the partonic center-of-mass energy approaches 2mt + mH , which is the minimal energy of the nal state. In [16], on the other hand, we applied SoftCollinear E ective Theory (SCET) methods1 in order to study the impact of soft-gluon corrections to the associated production of a top pair and a Higgs boson in the partonic threshold limit,2 i.e. in the limit where the partonic center-of-mass energy approaches the invariant mass M of the ttH nal state. The mass M is bounded from above only by the hadronic center-of-mass energy. In [16] a resummation formula for the soft emission corrections was derived and all of the elements necessary for the evaluation of that formula to next-to-next-to-leading logarithmic (NNLL) accuracy were evaluated. By using these results, a study of the approximate NNLO corrections originating from soft gluon emission in the partonic threshold limit was carried out. In particular, an in-house parton level Monte Carlo program was developed and employed to evaluate the total cross section and several di erential distributions. However, a direct numerical evaluation of the soft gluon emission corrections to NNLL was not performed in [16]. Recently, results for the total cross section and invariant mass distribution at NLL accuracy in the partonic threshold limit were presented in [18]. From the technical point of view, the associated production of a top pair and a W boson shows several similarities to the associated production of a top pair and a Higgs However, the former process involves only one partonic production channel in the partonic threshold limit, namely the quark annihilation channel, while the latter also receives large contributions from the gluon fusion channel. For this reason some of us recently studied the resummation of the soft gluon corrections in the partonic threshold limit to ttW production [19]. In that work the resummation was carried out up to NNLL accuracy in Mellin space. An in-house parton level Monte Carlo program for the numerical evaluation of the resummation formulas was developed and employed to obtain predictions for the total cross section and several di erential distributions at the LHC operating at a center-of-mass energy of 8 and 13 TeV. (The NNLL resummation in the partonic threshold limit for ttW production in momentum space was studied in [20].) By building upon the results of [16] and [19], in this paper we study the resummation of soft gluon emission corrections to the associated production of a top-quark pair and a Higgs boson in Mellin space. We developed an in-house parton level Monte Carlo code which allows us to evaluate numerically soft emission corrections to this process up to NNLL accuracy. In this paper, we employ the expression \parton level Monte Carlo" in order to indicate a numerical program where the momenta of the incoming partons 1See [17] for an introduction to SCET. 2Often this limit is referred to as PIM kinematics. The acronym PIM stands for Pair Invariant Mass and was extensively employed in the context of top-quark pair production. While the generalization to our case is trivial, the word \pair" should not be applied to the process under study here, where the nal state invariant mass involves 3 particles. as well as the momenta of the top quark, antitop quark and Higgs boson are generated, and arbitrary kinematic distributions depending on the momenta of the nal state heavy particles can be studied in the soft emission limit. By matching these results with complete NLO calculations carried out with MadGraph5_aMC@NLO [21] (which we will indicate with MG5 aMC in the rest of this paper) we obtain predictions for the total cross section and several di erential distributions which are valid to NLO+NNLL accuracy. We also compute the observables at NLO+NLL accuracy and using NNLO approximations of the NLO+NNLL results, and show that these less precise computations miss important e ects. The paper is organized as follows: in section 2 we review the salient features of the technique employed to obtain and evaluate the relevant resummation formulas. In section 3 we present predictions, valid to NLO+NNLL accuracy, for the total cross section and several di erential distributions for the associated production of a top pair and a Higgs boson at the LHC operating at a center-of-mass energy of 13 TeV. Finally, section 4 contains our Outline of the calculation The associated production of a top quark pair and a Higgs boson receives contributions from the partonic process i(p1) + j(p2) ! t(p3) + t(p4) + H(p5) + X ; where ij 2 fqq; qq; ggg at lowest order in QCD, and X indicates the unobserved partonic nal-state radiation. Two Mandelstam invariants play a crucial role in our discussion: s^ = (p1 + p2)2 = 2p1 p2 ; M 2 = (p3 + p4 + p5)2 : The soft or partonic threshold limit is de ned as the kinematic situation in which In this region, the unobserved nal state can contain only soft radiation. The factorization formula for the QCD cross section in the partonic threshold limit was derived in [16] and reads (s; mt; mH ) = dPSttH Tr Hij (fpg; ) Sij min = (2mt + mH ) In (2.4), s indicates the square of the hadronic center-of-mass energy and We use the symbol fpg to indicate the set of external momenta p1; ; p5. The trace Tr [Hij Sij ] is proportional to the spin and color averaged squared matrix element for ttH + Xs production in the process initiated by the two partons i and j, where Xs indicates the unobserved soft gluons in the nal state. The hard functions Hij , which are matrices in color space, are obtained from the color decomposed virtual corrections to the 2 ! 3 tree-level process. The soft functions Sij (which are also matrices in color space) are related to color-decomposed real emission corrections in the soft limit; they depend on plus distributions of the form as well as on the Dirac delta function of argument (1 z). The parton luminosity functions ij are de ned as the convolutions of the parton distribution functions (PDFs) for the partons i and j in the protons N1 and N2: ij (y; ) = x fi=N1 (x; ) fj=N2 x In the soft limit the indices ij 2 fqq; qq; ggg, as at LO. The hard and soft functions are two-by-two matrices for qq-initiated (quark annihilation) processes, and three-by-three matrices for gg-initiated (gluon fusion) processes. Contributions from other production channels such as qg and qg are subleading in the soft limit. We shall refer to such processes collectively as the \quark-gluon" or the \qg" channel in what follows. The hard functions satisfy renormalization group equations governed by the soft anomalous dimension matrices ij , which depend on the partonic channel considered. These anomalous dimension matrices, which are needed to carry out the resummation of soft gluon corrections, were derived in [22, 23]. The hard functions, soft functions, and soft anomalous dimensions must be computed in xed-order perturbation theory up to a given s. In this work we study the resummation up to NNLL accuracy. For this task we need to evaluate the hard functions, soft functions and soft anomalous dimensions to NLO. All of these elements were already evaluated to the order needed here [16, 22{24]. In particular, the NLO hard functions were evaluated by customizing two of the oneloop provider programs available on the market, GoSam [25{29] and Openloops [30]. The numerical evaluation of the hard functions for this work has been performed by using a modi ed version of Openloops in combination with Collier [31{35]. GoSam in combination with Ninja [29, 36, 37] was used to cross-check our results. The resummation formula for the associated production of a ttH nal state in Mellin space is similar to the one which was derived for the production of a ttW nal state in [19] (s; mt; mH ) = 1 Z c+i1 where we introduced the Mellin transform of the luminosity functions e ij , and Since the soft limit z ! 1 corresponds to the limit N ! 1 in Mellin space, we neglected The hard and soft functions in (2.8) can be evaluated in xed order perturbation theory at scales at which they are free from large logarithms. We indicate these scales with s, respectively. Subsequently, by solving the renormalization group (RG) equations for the hard and soft functions one can evolve the hard scattering kernels in (2.9) to the factorization scale f . One obtains Large logarithmic corrections depending on the ratio of the scales h and s are resummed in the channel-dependent matrix-valued evolution factors Ue . The expression for the evolution factors is which is formally identical to the expression found for the corresponding quantity in carrying out the resummation for ttW production. For the de nition of the various RG factors appearing in (2.11) we refer the reader to [19]. However, while for ttW production one needs to consider the evolution factor in the quark-annihilation channel only, for ttH production one also needs to evaluate the appropriate anomalous dimensions and evolution factor for the gluon fusion channel. The functions U in (2.11) depend on s evaluated at three di erent scales: f . In practice, it is convenient to rewrite the evolution factors in terms of s( h) only. This can be done by employing the running of s at three loops [38]. By doing this, which becomes [19] f = s (MZ ) from MMHT 2014 PDFs The leading logarithmic (LL) function g1, the NLL function g2, and the NNLL function g3 can be obtained starting from (2.11). One can see that the l.h.s. of (2.10) is independent of h and s if the evolution factors and the hard and soft functions are known to all orders in perturbation theory. This is impossible in practice, which introduces a residual dependence on the choice of the scales s in any numerical evaluation of (2.11) or (2.12). The hard and soft functions are free from large logarithms if one chooses h of N in the hard scattering kernel, whose existence is related to the Landau pole in In this work, we choose the integration path in the complex N plane when evaluating the inverse Mellin transform according to the Minimal Prescription (MP) introduced in [39]. In the numerics, we need the parton luminosity functions in Mellin space. These can be constructed using techniques described in [40, 41]. Numerical results In this section we present predictions for the total cross section and di erential distributions for the ttH production process. The main goal of this work is to obtain predictions for physical observables which are valid to NLO+NNLL accuracy. However, we also perform some systematic studies meant to provide insight into the validity of various approximations to this state-of-the-art result. In all cases, we use the input parameters listed in table 1, and MMHT 2014 PDFs [42]. We switch PDF orders as appropriate for a given perturbative approximation according to the scheme given in table 2, where we also specify the computer code used in each case. As a preliminary step we check that with our choice of scales and input parameters the NLO expansion of the NNLL resummation formula (which we refer to as \approximate NLO") provides a satisfactory approximation to the exact NLO calculation. Such an approximation of (2.10) captures the leading terms in the Mellin-space soft limit (N ! 1) of the NLO cross section, namely the single and double powers of ln N as well as N independent terms. Even though the N -independent terms depend on the Mandelstam variables, we will refer to them as \constant" terms in what follows. Analogous comparisons of approximate NLO and complete NLO calculations were carried out for ttW production in [19]. In [16], similar comparisons were also performed for ttH production, but with two di erences with respect to the current work: the renormalization and factorization scales xed (independent of M ) instead of dynamic (correlated with M ), and the leading terms were represented in momentum space instead of Mellin space. NLO no qg nNLO (Mellin) (NLO+NNLL)NNLO exp: in-house MC in-house MC +MG5 aMC in-house MC +MG5 aMC in-house MC +MG5 aMC in-house MC +MG5 aMC 378:7+18250:2:5 473:3+02:80:6 482:1+1305::91 474:8+4571::29 480:1+5175::77 486:4+2294::95 497:9+19:84:5 482:7+1201::71 s = 13 TeV and MMHT 2014 PDFs. The default value of the factorization scale is f;0 = M=2, and the uncertainties are estimated through scale variations of this (and the resummation scales s and h when applicable) as explained in the text, see the discussion around (3.5). h = f in the NNLL resummation formula (2.10). For this reason, the matched NLO+NNLL cross section is given by NLO+NNLL = NLO + The di erence of terms in the square brackets contributes at NNLO and beyond, adding NNLL resummation onto the NLO result. In order to study the convergence of resummed perturbation theory, we will also calculate NLO+NLL results, de ned as NLO+NLL = NLO + The di erence of terms in the square brackets contributes at NNLO and beyond, adding NLL resummation onto the NLO result. However, in contrast to the approximate NLO result, the constant piece of the NLO expansion of the NLL resummation formula contains explicit dependence on the matching scales s, in addition to that on numerical dependence on these scales is formally of NNLL order (and is indeed canceled h dependence in the NLO hard and soft functions in the NNLL result), and provides an additional handle on estimating the size of NNLL corrections using the NLL resummation formula. While we are mainly interested in NNLL resummation e ects, it is also interesting to study to what extent these all-orders corrections are approximated by their NNLO truncation. To this end, we consider \approximate NNLO" calculations based on the NNLL resummation formula (2.10). Approximate NNLO calculations include all powers of ln N and part of the constant terms from a complete NNLO calculation, but neglect terms which vanish as N ! 1. Since the constant terms are not fully determined by an NNLL calculation (only their -dependence is, through the RG equations), there is some freedom as to how to construct such approximations. Here we consider two possibilities. The rst follows the procedure used in [19] for the case of ttW production. A detailed description of which constant pieces are included in that NNLO approximation can be found in section 4 of [19].3 We match these NNLO corrections, obtained in the soft limit, with the NLO ones in the usual way: nNLO = where we introduced the acronym nNLO to indicate approximate NNLO corrections matched to full NLO calculations. The second NNLO approximation we consider is based on the direct expansion of the NLO+NNLL result to NNLO. This di ers from the approximate NNLO result used above by constant terms, which are formally of N3LL order. We de ne this approximation through the matching equation NNLO exp. = h NNLL expanded to NNLO In both cases above, the di erence of terms in the square brackets is a pure NNLO correction. Contrary to the approximate NNLO result used in (3.3), which depends only on f by construction, the constant pieces of the NNLO expansion of the NNLL result in (3.4) contain explicit dependence on h and s, in addition to that on f . This scale dependence is formally of N3LL order, and can be used to estimate the size of such corrections to the NNLL results. Moreover, the NNLO approximation (3.4) di ers from the NLO+NNLL result through terms of N3LO and higher, so comparing the two results gives a direct measure of how important such terms are numerically. In fact, were an exact NNLO calculation to appear, adding to it these beyond NNLO terms would achieve NNLO+NNLL Scale choices Numerical evaluations of the resummed formulas have a residual dependence on the choice of the hard and soft scales h and s . This feature arises from the fact that the various factors in (2.10) have to be evaluated at a given order in perturbation theory. When the resummation is carried out in Mellin space the standard default choice of these scales is QCD" resummation method [39, 45, 46], and is the one we shall use here. Furthermore, both the xed-order and resummed results have a residual dependence on the factorization scale f . The factorization scale should be chosen in such a way that logarithms of the ratio threshold limit it is natural to choose a dynamical value for the factorization scale which is correlated with the nal state invariant mass M . Figure 1 shows the dependence of the total cross section calculated within various perturbative approximations on the choice of 3In [16] such approximate NNLO formulas were obtained starting from the resummation formula in momentum space, and thus di er from Mellin space results through power corrections and constant terms. However, we have checked that the two approaches lead to results which are numerically almost identical. while the NLO+NNLL and nNLO curves are obtained using MMHT 2014 NNLO PDFs. the ratio f =M at the LHC with p curves have a very di erent behavior for small values of f . In addition, gure 1 shows 0:5, as the NLO result falls rather steeply away to smaller values in that region, while the other three curves remain reasonably stable. Because of these considerations, in the following we employ two di erent default choices be advantageous because the lower-order perturbative results are larger at lower that the apparent convergence of the perturbative series is improved, but other than this numerical fact there is no obvious reason to exclude the natural hard scale M as a default choice so we study this as well. In both cases, the uncertainty associated to the choice of (i 2 fs; f; hg). The scale uncertainty above the central value of an observable O (the total cross section, or the value of a di erential cross section in a given bin) is then evaluated by combining in quadrature the quantities value can be obtained in the same way by combining in quadrature the quantities de ned as in (3.5) but with \max" replaced by \min". We use this procedure to obtain the perturbative uncertainties given in all of the tables and gures that follow. Total cross section We begin our analysis by considering the total cross section for the associated production of a top pair and a Higgs boson at the LHC operating at a center-of-mass energy of 13 TeV. NLO NLL NNLL nNLO carried out with the default factorization scale choices f;0 = M=2 (left) and f;0 = M (right). The labels \NLL" and \NNLL" on the horizontal axis indicate NLO+NLL and NLO+NNLL calgure 2, which presents a visual comparison between the main we set f;0 = M , and in results at the two di erent scales. We rst compare the approximate NLO corrections generated from NNLL soft-gluon resummation (second row of each table), with the full NLO corrections without (third row of each table) and with (fourth row of each table) the qg channel. Since the approximate NLO results include only the leading-power contributions from the gluon fusion and quarkannihilation channels in the soft limit, the di erence between these results and the NLO corrections without the qg channel gives a measure of the importance of power corrections away from this limit. The two results are seen to di er by no more than a few percent, even though the NLO corrections are large. This shows that at NLO the power corrections away from the soft limit for these channels are quite small. Comparing the NLO results with and without the qg channel reveals that this channel contributes signi cantly to the scale in the soft limit make up the bulk of the NLO correction provides a strong motivation to resum them to all orders. No information is lost by doing this, as both sources of power corrections are taken into account by matching with NLO as discussed above. Since the power corrections are treated in xed order, the perturbative uncertainties associated with them are estimated through the standard approach of scale variations. We next turn to the NLO+NLL and NLO+NNLL cross sections, which are the main results of this section. The exact numbers are given in tables 2 and 3, and a pictorial representation is given in gure 2. The results for the default scale choice f;0 = M=2 converge quite nicely. The scale uncertainties get progressively smaller when moving from NLO to NLO+NLL to NLO+NNLL, and the higher-order results are roughly within the vergence is still reasonable but not quite as good, mainly because the NLO and NLO+NLL NLO no qg nNLO (Mellin) (NLO+NNLL)NNLO exp: in-house MC in-house MC +MG5 aMC in-house MC +MG5 aMC in-house MC +MG5 aMC in-house MC +MG5 aMC 293:5+8651::27 444:7+2389::62 447:0+3450::14 423:0+5419::97 466:2+2226::98 514:3+4329::95 488:4+98::43 485:7+61:58:0 Table 3. Total cross section for ttH at the LHC with p s = 13 TeV and MMHT 2014 PDFs. The results are obtained as in table 2, but with the default value of the factorization scale chosen instead as f;0 = M . rather accidental considering its wider variation over a larger range of f , as seen in ure 1. However, one should remember that the scales h and s are kept xed at their default values in the NLO+NLL and NLO+NNLL curves of gure 1, while they are varied as explained above in order to obtain the scale uncertainty reported in the tables. In analogy to the two di erent choices for the default factorization scale considered in this work, one can wonder about the numerical impact of choosing the default hard scale which we carried out in order to study the scale uncertainty associated to the NLO+NNLL results. We keeping all the other scales equal to their default values, the total cross section increases by about 2%, irrespective of the choice of the default value of the factorization scale. Finally, we discuss the NNLO approximations to the NNLL resummation formula. The NNLO is rather small, roughly at or below the 5% level after taking scale uncertainties into approximately at the 10% level. In either case, gure 2 shows very clearly that the nNLO results display an arti cially small scale dependence compared to the NLO+NNLL results, con rming the cautionary statements made in [16] about the reliability of the nNLO scale dependence in estimating higher-order perturbative corrections. The results in this section highlight the importance of an NNLL calculation. Taken as a whole, they show that both NLO+NLL and approximate NNLO calculations are a poor proxy for the more complete NLO+NNLL calculation. We have considered two default scale choices, f;0 = M=2 and the default scale choice is arbitrary, and it would not be unreasonable to combine the envelope of results from the two choices into a single, larger perturbative uncertainty. The NLO+NNLL results quoted at either scale would not change signi cantly through such a Di erential distributions In this section we discuss results for di erential distributions. In particular, we consider: the distribution di erential with respect to the invariant mass of the top pair and Higgs boson in the nal state, M ; the distribution di erential with respect to the invariant mass of the top-quark pair, Mtt; boson, pTH ; the distribution di erential with respect to the transverse momentum of the Higgs the distribution di erential with respect to the transverse momentum of the top the comparison between complete NLO calculations and approximate NLO calculations for all of the distributions listed above. We observe that for all of the distributions the approximate NLO scale uncertainty band (in blue) is included in the NLO scale uncertainty band (bins with the red frame). However, the approximate NLO uncertainty is smaller than the NLO uncertainty in all bins. Furthermore the bin-by-bin ratio of the two distributions, found at the bottom of each panel, shows that the NLO and approximate NLO corrections have somewhat di erent shapes. As for the case of the total cross section, it is reasonable to look at how the approximate NLO distributions compare to the NLO calculations when the contribution of the qg channel is left out from the latter. This comparison can be found in gure 4. One can see that approximate NLO and NLO distributions without the qg channel agree quite well and the size of the respective uncertainty bands is very similar. As observed in the case of the total cross section, the fact that the leading terms in the soft limit make up a sizable fraction of the NLO correction also in each bin of the di erential distributions provides a strong motivation to study the all order resummation of the soft emission corrections. We remind the reader that the contribution of the qg-channel at NLO is included in the NLO+NLL, NLO+NNLL and nNLO predictions discussed below through the matching The comparison between the NLO and the NLO+NNLL calculations of the di erential distributions can be found in gure 5. We see that the NLO+NNLL uncertainty band is included in the NLO scale uncertainty band in almost all bins of the distributions considered here. The exception is the bins in the far tail of the M and Mtt distributions, where the NLO+NNLL band is not completely included in the NLO one, but is higher than the NLO one. In general one can observe that the central value of the NLO+NNLL calculation is slightly larger than the central value of the NLO one in almost all bins of the distributions shown in gure 5. 500 600 700 800 900 1000 1100 100 150 200 250 300 100 150 200 250 300 NLO (red band). The default factorization scale is chosen as f;0 = M=2, and the uncertainty bands are generated through scale variations as explained in the text. Figure 6 shows a comparison between NLO+NLL and NLO+NNLL results. The central value of these two calculations is quite close in all bins. The main e ect of the corrections at NLO+NNLL is to shrink slightly the scale uncertainty bands with respect to the NLO+NLL results everywhere with the exception of the bins in the far tail of the M and Mtt distributions. We conclude our discussion of the results obtained with the choice f;0 = M=2 by comparing in gure 7 the NLO+NNLL, nNLO and NLO+NNLL expanded predictions for the various distributions. The gure shows the ratio, separately for each bin, of the band refers to NLO+NNLL calculations, the dashed red band to nNLO calculations and the NLO w/o qg NLO w/o qg NLO w/o qg 500 600 700 800 900 1000 1100 100 150 200 250 300 100 150 200 250 300 tributions without the quark-gluon channel contribution (red band). All settings are as in gure 3. dashed black band to the NNLO expansion of the NLO+NNLL resummation. The dashed black band and the blue band thus di er by NNLL resummation e ects of order N3LO and higher. Numerically, these e ects contribute roughly at the 5% level, and as for the total cross section the NNLO truncation of the NLO+NNLL resummation formula tends to underestimate the uncertainty of the all-orders resummation. The di erence between the dashed red band and the dashed black band is due to constant NNLO corrections, which are of N3LL order. Taking the envelope of the two NNLO approximations (i.e. the black and red bands) gives a more realistic estimate of the scale uncertainty, which is generally contained within the NLO+NNLL result (the exception is the high-pTH bins). NLO NLO 500 600 700 800 900 1000 1100 100 150 200 250 300 100 150 200 250 300 NLO calculation (red band). The uncertainty bands are generated through scale variations of f , s and h as explained in the text. We want at this point to study results for a di erent choice of the default factorization scale, namely However, NLO+NNLL predictions obtained with the two choices are in good agreement. For what concerns the di erential distributions studied here this can be seen by comparing and NLO+NNLL calculations with that at NLO+NLL the overlap between the distributions evaluated at f;0 = M and 500 600 700 800 900 1000 1100 100 150 200 250 300 100 150 200 250 300 NLO+NLL calculation (red band). The uncertainty bands are generated through scale variations. NLO+NNLL with The good agreement between the two bands shown in each panel of gure 9 indicates that NLO+NNLL predictions are quite stable with respect to di erent (but reasonable) choices of the standard value for the factorization scale. Conclusions In this paper we evaluated the resummation of the soft emission corrections to the associated production of a top-quark pair and a Higgs boson at the LHC in the partonic threshold limit z ! 1. The calculation is carried out to NNLL accuracy and it is matched through scale variations. to the complete NLO cross section in QCD. The numerical evaluation of observables at NLO+NNLL was carried out by means of an in-house parton level Monte Carlo code developed for this work, based on the resummation formula derived in [16]. The resummation procedure is however carried out in Mellin space, following the same approach employed in [43, 44] for the calculation of the (boosted) top-quark pair production cross section and in [19] for the calculation of the cross section for the associated production of a top-quark pair and a W boson. In the previous sections we presented predictions for the total cross section for this production process at the LHC operating at a center-of-mass energy of 13 TeV. In addition, we showed results for four di erent di erential distributions depending on the four-momenta of the massive particles in the nal state: the di erential distributions in the invariant mass of the ttH particles, in the invariant mass of the tt pair, in the transverse momentum of the Higgs boson, and in the transverse momentum of the top quark. We found that the relative size of the NNLL corrections with respect to the NLO cross section is rather sensitive to the choice of the factorization scale f . In particular, for the two choices which f;0 = M , it was found that the NNLL corrections enhance the total cross section and di erential distributions in all bins considered. The NNLL soft emission corrections expressed as a percentage of the NLO observables are f,0 = M/2) 500 600 700 800 900 1000 1100 f;0 = M=2 (blue band) compared to the scale variations. predictions obtained by setting nd compatible results. This fact shows that the NLO+NNLL predictions are quite stable with respect to the factorization scale choice. Indeed, it would not be unreasonable to combine the envelope of the results at the two di erent scale choices into a single result with a larger perturbative uncertainty, which for the case of the total cross section would be at about the 20% level. By taking the envelope of the corresponding NLO results, one nds instead an uncertainty larger than 30%. We also studied the total cross section and di erential distributions at NLO+NLL accuracy and with NNLO approximations of the NLO+NNLL resummation formula, and found that both of these are a poor proxy for the more complete NLO+NNLL results, especially for higher values of f;0. The study carried out in this paper is not an alternative to a calculation of the NNLO corrections to the scale variations. associated production of a top quark and a Higgs boson. The latter would greatly improve the quality of the predictions for this process and represent a major technical achievement. On the contrary, the study of the soft emission corrections to NNLL accuracy must be considered complementary to a NNLO calculation. If a NNLO calculation were to become available in the future, it would be possible to match it to the results presented in this paper in order to obtain NNLO+NNLL accuracy predictions for the total cross section and di erential distributions studied here. In the meantime, NLO+NNLL calculations allow us to obtain predictions which include in a consistent way higher order corrections and are a ected by a scale uncertainty which is smaller than the one a ecting NLO calculations. The parton level Monte Carlo developed for this paper could be extended to include the decays of the top quarks and the Higgs boson following the work done in [48]. This would allow one to impose cuts on the momenta of the detected particles. Furthermore, our code could serve as a template for the calculation of the NNLL soft emission corrections to the associated production of a top pair and a Z boson at the LHC. The latter is a process of signi cant phenomenological interest which has already been investigated experimentally at both the Run I and Run II of the LHC. We plan to study the NLO+NNLL cross section for this process in future work. Acknowledgments The in-house Monte Carlo code which we developed and employed to evaluate the (differential) cross sections presented in this paper was run on the computer cluster of the Center for Theoretical Physics at the Physics Department of New York City College of Technology. We thank A. Signer for discussions about the Monte Carlo implementation, P. Maierhofer and S. Pozzorini for their assistance with the program Openloops, and and G. Ossola for useful discussions. The work of A.F. is supported in part by the National Science Foundation under Grant No. PHY-1417354. B.P. would like to thank the ESI Vienna for hospitality and support during the completion of this work. The work of L.L.Y. was supported in part by the National Natural Science Foundation of China under Grant No. 11575004 and 11635001. Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. [hep-ph/0107081] [INSPIRE]. corrections to ttH production in hadron collisions, Nucl. Phys. B 653 (2003) 151 [hep-ph/0211352] [INSPIRE]. Tevatron, Phys. Rev. D 65 (2002) 053017 [hep-ph/0109066] [INSPIRE]. Phys. Rev. Lett. 87 (2001) 201804 [hep-ph/0107101] [INSPIRE]. production at the LHC, Phys. Rev. D 67 (2003) 071503 [hep-ph/0211438] [INSPIRE]. with top quarks at the large hadron collider: NLO QCD corrections, Phys. Rev. D 68 (2003) 034022 [hep-ph/0305087] [INSPIRE]. pseudoscalar Higgs production in association with a top-antitop pair, Phys. Lett. B 701 (2011) 427 [arXiv:1104.5613] [INSPIRE]. production in association with a top anti-top pair at NLO with parton showering, Europhys. Lett. 96 (2011) 11001 [arXiv:1108.0387] [INSPIRE]. corrections to ttH production with top quark decays at hadron collider, Phys. Lett. B 738 (2014) 1 [arXiv:1407.1110] [INSPIRE]. hadroproduction in association with a top-quark pair, JHEP 09 (2014) 065 [arXiv:1407.0823] [INSPIRE]. to top-pair hadroproduction in association with heavy bosons, JHEP 06 (2015) 184 [arXiv:1504.03446] [INSPIRE]. [INSPIRE]. [arXiv:1506.07448] [INSPIRE]. leptonic decays in association with a Higgs boson at the LHC, JHEP 11 (2015) 209 Next-to-Leading-Order QCD Corrections to Higgs Boson Production in Association with a Top Quark Pair and a Jet, Phys. Rev. Lett. 111 (2013) 171801 [arXiv:1307.8437] [INSPIRE]. ttH production at the LHC, JHEP 03 (2016) 065 [arXiv:1509.02780] [INSPIRE]. top pair and a Higgs boson beyond NLO, JHEP 03 (2016) 124 [arXiv:1510.01914] [INSPIRE]. Notes Phys. 896 (2015) 1 [arXiv:1410.1892] [INSPIRE]. invariant mass for associated ttH production at the LHC, in proceedings of the 4th Large [arXiv:1609.01619] [INSPIRE]. a W boson at next-to-next-to-leading logarithmic accuracy, JHEP 09 (2016) 089 [arXiv:1607.05303] [INSPIRE]. production at hadron colliders, Phys. Rev. D 90 (2014) 094009 [arXiv:1409.1460] [INSPIRE]. di erential cross sections and their matching to parton shower simulations, JHEP 07 (2014) 079 [arXiv:1405.0301] [INSPIRE]. amplitudes with massive partons, Phys. Rev. Lett. 103 (2009) 201601 [arXiv:0907.4791] [INSPIRE]. [INSPIRE]. Model and beyond, Eur. Phys. J. C 74 (2014) 3001 [arXiv:1404.7096] [INSPIRE]. to calculate one-loop tensor integrals with up to six external legs, Comput. Phys. Commun. 180 (2009) 2317 [arXiv:0810.0992] [INSPIRE]. unitarity-based reduction algorithm at the integrand-level, JHEP 08 (2010) 080 [arXiv:1006.0710] [INSPIRE]. [31] A. Denner and S. Dittmaier, Reduction of one loop tensor ve point integrals, Nucl. Phys. B 734 (2006) 62 [hep-ph/0509141] [INSPIRE]. [35] A. Denner, S. Dittmaier and L. Hofer, COLLIER: A fortran-based complex one-loop library in extended regularizations, Comput. Phys. Commun. 212 (2017) 220 [arXiv:1604.06792] [INSPIRE]. amplitudes through Laurent series expansion, JHEP 06 (2012) 095 [Erratum JHEP 11 (2012) 128] [arXiv:1203.0291] [INSPIRE]. One-loop Massive Amplitudes from Integrand Reduction via Laurent Expansion, JHEP 03 (2014) 115 [arXiv:1312.6678] [INSPIRE]. resummation, Nucl. Phys. B 726 (2005) 317 [hep-ph/0506288] [INSPIRE]. hadronic collisions, Nucl. Phys. B 478 (1996) 273 [hep-ph/9604351] [INSPIRE]. [arXiv:1405.3654] [INSPIRE]. [INSPIRE]. in SCET vs. direct QCD: Higgs production as a case study, JHEP 01 (2015) 046 [arXiv:1409.0864] [INSPIRE]. [INSPIRE]. [1] W. Beenakker , S. Dittmaier , M. Kramer , B. Plumper , M. Spira and P.M. Zerwas , Higgs radiation o top quarks at the Tevatron and the LHC , Phys. Rev. Lett . 87 ( 2001 ) 201805 [2] W. Beenakker , S. Dittmaier , M. Kramer , B. Plumper , M. Spira and P.M. Zerwas , NLO QCD [3] L. Reina , S. Dawson and D. Wackeroth , QCD corrections to associated tth production at the [ 4] L. Reina and S. Dawson , Next-to-leading order results for tth production at the Tevatron , [5] S. Dawson , L.H. Orr , L. Reina and D. Wackeroth , Associated top quark Higgs boson [6] S. Dawson , C. Jackson , L.H. Orr , L. Reina and D. Wackeroth , Associated Higgs production [7] R. Frederix , S. Frixione , V. Hirschi , F. Maltoni , R. Pittau and P. Torrielli , Scalar and [9] Y. Zhang , W.-G. Ma , R.-Y. Zhang , C. Chen and L. Guo, QCD NLO and EW NLO [10] S. Frixione , V. Hirschi , D. Pagani , H.S. Shao and M. Zaro , Weak corrections to Higgs [11] S. Frixione , V. Hirschi , D. Pagani , H.S. Shao and M. Zaro , Electroweak and QCD corrections [12] H.B. Hartanto , B. Jager , L. Reina and D. Wackeroth , Higgs boson production in association with top quarks in the POWHEG BOX , Phys . Rev . D 91 ( 2015 ) 094003 [arXiv:1501.04498] [13] A. Denner and R. Feger , NLO QCD corrections to o -shell top-antitop production with [14] H. van Deurzen , G. Luisoni , P. Mastrolia , E. Mirabella , G. Ossola and T. Peraro , [15] A. Kulesza , L. Motyka , T. Stebel and V. Theeuwes , Soft gluon resummation for associated [16] A. Broggio , A. Ferroglia , B.D. Pecjak , A. Signer and L.L. Yang , Associated production of a [17] T. Becher , A. Broggio and A. Ferroglia , Introduction to Soft-Collinear E ective Theory, Lect. [18] A. Kulesza , L. Motyka , T. Stebel and V. Theeuwes , Soft gluon resummation at xed Hadron Collider Physics Conference (LHCP 2016 ), Lund, Sweden, 13 { 18 June 2016 [19] A. Broggio , A. Ferroglia , G. Ossola and B.D. Pecjak , Associated production of a top pair and [20] H.T. Li , C.S. Li and S.A. Li , Renormalization group improved predictions for ttW [21] J. Alwall et al., The automated computation of tree-level and next-to-leading order [22] A. Ferroglia , M. Neubert , B.D. Pecjak and L.L. Yang , Two-loop divergences of scattering [23] A. Ferroglia , M. Neubert , B.D. Pecjak and L.L. Yang , Two-loop divergences of massive scattering amplitudes in non-abelian gauge theories , JHEP 11 ( 2009 ) 062 [arXiv:0908.3676] [24] V. Ahrens , A. Ferroglia , M. Neubert , B.D. Pecjak and L.L. Yang , Renormalization-Group Improved Predictions for Top-Quark Pair Production at Hadron Colliders , JHEP 09 ( 2010 ) [25] G. Cullen et al., Automated One -Loop Calculations with GoSam, Eur. Phys. J. C 72 ( 2012 ) [26] G. Cullen et al., GoSam-2 . 0 : a tool for automated one-loop calculations within the Standard [27] T. Binoth , J.P. Guillet , G. Heinrich , E. Pilon and T. Reiter , Golem95 : A Numerical program [28] P. Mastrolia , G. Ossola , T. Reiter and F. Tramontano , Scattering amplitudes from [29] T. Peraro , Ninja: Automated Integrand Reduction via Laurent Expansion for One-Loop [30] F. Cascioli , P. Maierhofer and S. Pozzorini , Scattering Amplitudes with Open Loops, Phys. [32] A. Denner and S. Dittmaier , Reduction schemes for one-loop tensor integrals, Nucl . Phys . B [33] A. Denner and S. Dittmaier , Scalar one-loop 4-point integrals, Nucl . Phys . B 844 ( 2011 ) 199 [34] A. Denner , S. Dittmaier and L. Hofer , COLLIER | A fortran-library for one-loop integrals , [36] P. Mastrolia , E. Mirabella and T. Peraro , Integrand reduction of one-loop scattering [37] H. van Deurzen , G. Luisoni , P. Mastrolia , E. Mirabella , G. Ossola and T. Peraro , Multi-leg [38] S. Moch , J.A.M. Vermaseren and A. Vogt , Higher-order corrections in threshold [39] S. Catani , M.L. Mangano , P. Nason and L. Trentadue , The Resummation of soft gluons in [40] M. Bonvini , Resummation of soft and hard gluon radiation in perturbative QCD , Ph .D. Thesis , Universita degli Studi di Genova, Dipartimento di Fisica (DIFI) , Genova Italy ( 2012 ) [41] M. Bonvini and S. Marzani , Resummed Higgs cross section at N 3LL , JHEP 09 ( 2014 ) 007 [42] L.A. Harland-Lang , A.D. Martin , P. Motylinski and R.S. Thorne , Parton distributions in the LHC era: MMHT 2014 PDFs, Eur . Phys. J. C 75 ( 2015 ) 204 [arXiv:1412.3989] [INSPIRE]. [43] A. Ferroglia , B.D. Pecjak , D.J. Scott and L.L. Yang , QCD resummations for boosted top production, PoS(TOP2015)052 [arXiv:1512 .02535] [INSPIRE]. [44] B.D. Pecjak , D.J. Scott , X. Wang and L.L. Yang , Resummed di erential cross sections for top-quark pairs at the LHC , Phys. Rev. Lett . 116 ( 2016 ) 202001 [arXiv:1601.07020] [45] M. Bonvini , S. Forte , M. Ghezzi and G. Ridol , Threshold Resummation in SCET vs. Perturbative QCD : An Analytic Comparison, Nucl . Phys . B 861 ( 2012 ) 337 [46] M. Bonvini , S. Forte , G. Ridol and L. Rottoli , Resummation prescriptions and ambiguities [47] J.C. Collins , D.E. Soper and G.F. Sterman , Factorization of Hard Processes in QCD, Adv. Ser. Direct . High Energy Phys . 5 ( 1989 ) 1 [hep-ph/0409313] [INSPIRE]. [48] A. Broggio , A.S. Papanastasiou and A. Signer , Renormalization-group improved fully di erential cross sections for top pair production , JHEP 10 ( 2014 ) 098 [arXiv:1407.2532]


This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2FJHEP02%282017%29126.pdf

NNLL resummation for the associated production of a top pair and a Higgs boson at the LHC, Journal of High Energy Physics, 2017, DOI: 10.1007/JHEP02(2017)126