Renormalization-group improved predictions for Higgs boson production at large
We study the next-to-next-to-leading logarithmic order resummation for the large Higgs boson production at the LHC in the framework of soft-collinear effective theory. We find that the resummation effects reduce the scale uncertainty significantly and decrease the QCD NLO results by about in the large region. The finite top quark mass effects and the effects of the NNLO singular terms are also discussed.
In the standard model (SM), the Higgs boson is predicted by the Higgs mechanism in which the would-be Goldstones become the longitudinal components of the W and Z bosons. Although the existence of Higgs boson has been proposed for a long time, searching for this particle in the experiments has failed until the recent discovery at the LHC Chatrchyan et al. (2012); Aad et al. (2012). In general, the Higgs boson may not be responsible for the mass origin of the fermions, and the current experimental data still allow the couplings of the Higgs boson with the fermions to deviate from the SM ones, especially, the coupling of the Higgs boson to the top quark Chatrchyan et al. (2013). Therefore, precise measurements of the couplings of the Higgs boson with other SM particles will test the Higgs mechanism in the SM Harlander and Neumann (2013); Azatov and Paul (2014); Banfi et al. (2014); Grojean et al. (2014).
The global fit method with current experiment data about the Higgs boson production and decay in various channels only provides indirect information on the top quark Yukawa coupling, which suffers from ambiguities from unknown new particles propagating in the loops. The most direct process to determine the coupling of the Higgs boson to the top quark is the associated production and single top associated production . However, the current abilities to measure the coupling of Higgs boson to the top quark through associated production are still weak Chatrchyan et al. (2013); ATL (2014) because of its small production cross section and complicated final states with copious decay products. The single top associated production has even smaller cross section because of the electro-weak interactions there, and is very challenging to measure.
Recently, a complementary method to determine the coupling of the Higgs boson to the top quark has been proposed in Refs. Azatov and Paul (2014); Banfi et al. (2014); Grojean et al. (2014) by investigating the large behavior of the Higgs boson in the process with . This method is feasible because the top quark mass can not be taken to be infinity when the Higgs boson has a large . The top quark Yukawa coupling can be detected from the measurement of the variable Azatov and Paul (2014)
where is the number of events in which the Higgs boson is larger or smaller than a critical value , for example, 300 GeV. is the corresponding theoretical predictions in the SM. It is pointed out that Azatov and Paul (2014) the factor, defined as the ratio of higher order results to the LO ones, for the Higgs boson distribution is roughly independent and very large, about 2, and that the resummation effects are negligible in the range they considered. All these arguments are based on the calculation by the HqT program de Florian et al. (2011). However, the resummation scheme used in the HqT program is only valid in the small region, which is much less than 100 GeV. The resummation prediction on the Higgs boson distribution in the large region, larger than 100 GeV, is investigated using the traditional method at next-to-leading-logarithm (NLL) de Florian et al. (2006). The resumed logarithms are different in the small and large regions. When the Higgs boson is small, the threshold region is defined as , where is the center-of-mass energy of the colliding partons. In the threshold region, only the soft gluon radiation is allowed, which leads to large logarithms . In contrast, in the large regions, the large logarithms are with , where de Florian et al. (2006). It is easy to observe that does not necessarily lead to , which means that the HqT program can not resum the large logarithms in the large regions.
Notice that when the recoiling hardest jet against the Higgs boson is observed and additional jets are vetoed, there is a new kind of large Sudakov logarithms , which can be resummed Liu and Petriello (2013a, b); Boughezal et al. (2014). If the mass of the jet is also measured, denoted as , additional logarithms have been resummed up to next-to-next-to-leading-logarithm (NNLL) Jouttenus et al. (2013).
In this paper, we provide the resummed prediction for at large regions up to NNLL, without explicit observation of a jet, in contract with the case in Jouttenus et al. (2013). We will work in the soft-collinear effective theory (SCET) Bauer et al. (2000, 2001); Bauer and Stewart (2001); Bauer et al. (2002); Becher and Neubert (2006a). In the threshold limits of large Higgs boson production, the final state radiations and beam remnants are highly suppressed which leads to final states consisting of a Higgs boson and an inclusive jet, as well as the remaining soft radiations, and therefore to the appearance of the large logarithms in the cross section. Then the resummation effects should be taken into account to obtain more precise predictions. The preliminarily theoretical NNLO analyses have been perfromed in Ref. Becher et al. (2014a). The resummation formalism in SCET is different from that used in Ref. de Florian et al. (2006). In the threshold region, the partonic cross section can be factorized to a hard function times a convolution between jet and soft functions. Each part has a explicit theoretical field definition which can be calculated perturbatively. In particular, each function contains only a single energy scale so that there is no potential large logarithms in each of them. The relative scale hierarchy between different functions is alleviated by running from one to the other via renormalization group equations. As a consequence, the large logarithms of the ratio of the different scales can be resummed to all orders.
In principle, the top quark mass should be kept to be finite in all the theoretical predictions in the large regions Ravindran et al. (2002); Harlander et al. (2012); Azatov and Paul (2014); Grojean et al. (2014). But because of the difficulty in calculating massive loops, this is achieved only for the LO result Ellis et al. (1988); Baur and Glover (1990) and the NLO total cross section expanded in Harlander and Ozeren (2009). The differential cross section is calculated only in the large top quark limit up to NLO Schmidt (1997); Ravindran et al. (2002); de Florian et al. (1999); Glosser and Schmidt (2002); Ravindran et al. (2002). More recently, a big progress is made by computing the NNLO total cross section of the sub-process Boughezal et al. (2013). Therefore, an approximated differential cross section with finite top quark mass beyond the LO is usually used, which is obtained by multiplying the LO differential cross section with finite top quark mass with a differential factor, as done in Ref. Grazzini and Sargsyan (2013). We will take into account the finite top quark mass effects in the resummation predictions following this method.
The precision prediction on Higgs boson production at large regions can not only test the top quark Yukawa coupling discussed above, but also be a probe of the new physics. For example, in the SM the large transverse momentum spectrum of the Higgs boson produced in gluon fusion can be quite different from one of the minimal supersymmetric standard model Langenegger et al. (2006); Bagnaschi et al. (2012). Light particles beyond the SM can be probed via the ratio of the partially integrated Higgs transverse momentum distribution to the inclusive rate Arnesen et al. (2009).
This paper is organized as follows. In Sec. II, we analyze the kinematics of the Higgs boson and one jet associated production and give the definition of the threshold region. In Sec. III, we present the factorization and resummation formalism in momentum space using SCET. In Sec. IV, we present the hard function, jet function and soft functions at NLO. Then, we study the scale independence of the final result analytically. In Sec. V, we discuss the numerical results for this process at the LHC. We conclude in Sec. VI.
Ii Analysis of kinematics
First of all, we introduce the relevant kinematical variables needed in our analysis. The dominant partonic processes for the Higgs boson and one jet production are , and . The LO Feynman diagrams for the process are shown in Fig. 1.
It is convenient to define two lightlike vectors along the beam directions, and , which are related by . Then, we introduce initial collinear fields along and to describe the collinear particles in the beam directions. In the center-of-mass (c.m.) frame of the hadronic collision, the momenta of the incoming hadrons are given by
Here is the c.m. energy of the collider and we have neglected the masses of the hadrons. The momenta of the incoming partons, with a light-cone momentum fraction of the hadronic momentum, are
At the hadronic and partonic level, the momentum conservation gives
respectively, where is the momentum of the Higgs boson. We define the partonic jet with jet momentum to be the set of all final state partons except the Higgs boson in the partonic processes, while the hadronic jet with jet momentum contains all the hadrons as well as the beam remnants in the final state except the Higgs boson.
We also define the Mandelstam variables as
for hadrons, and
for partons, respectively. In terms of the Mandelstam variables, the hadronic and partonic threshold variables are defined as
where is the mass of the Higgs boson. The hadronic threshold limit is defined as Laenen et al. (1998). In this limit, the final state radiations and beam remnants are highly suppressed, which leads to final states consisting of a Higgs boson and an energetic jet, as well as the remaining soft radiations. Taking this limit requires simultaneously, and we get
where . This expression can help to check the factorization scale invariance, which is shown in detail below. Near the partonic threshold, the boson must be recoiling against a jet and there is only phase space for the jet to be nearly massless. In this case, , where is the momentum of the final state collinear partons forming the jet and is the momentum of the soft radiations.
We note that in both hadronic and partonic threshold limit, the Higgs boson is not forced to be produced at rest, it can have a large momentum. Actually, as the momentum of the Higgs boson becomes larger and larger, the final-state phase space lies more close to the threshold limit. We point that the definition of the partonic threshold limit is different from the case of de Florian et al. (2006), as discussed in the introduction. They are equivalent to each other only if the momentum component of the Higgs boson in the partonic c.m. frame vanishes.
For convenience, we can also write the threshold variable as
where , is the momentum of soft radiations, is the energy of the jet and is the lightlike vector associated with the jet direction. In the threshold limit (), incomplete cancelation of the divergences between real and virtual corrections leads to singular distributions , with . It is the purpose of threshold resummation to sum up these contributions to all orders in .
The total cross section is given by
where we have changed the integration variables into the Higgs boson transverse momentum squared , rapidity , and . The regions of the integration variables are given by
The other kinematical variables can be expressed in terms of these four integration variables.
Iii Factorization and Resummation Formalism in SCET
In the frame of SCET, we define a small expanded parameter () in the threshold limit . Here, Q is the characteristic energy of the hard scattering process. The momentum of a collinear particle scales as
and the momentum of a soft particle scales as
The soft fields scale as , , and the collinear fermion field scales as . The light-cone components of the collinear gluon field scale the same way as its momentum in covariant gauge.
The soft gluon field are multipole expanded around to maintain a consistent power counting in . Thus, the soft gluon operator depends only on at leading power, and its Fourier transform only depends on . It is needed to mention that is of order of the jet mass and is assumed to be in the perturbative region.
In the limit of the infinite top quark mass, the effective Lagrangian of Higgs boson production via gluon fusion can be written as Schmidt (1997)
where is the Wilson coefficient at order. The leading power effective operator of in SCET is as follows,
where is the effective gluon field in the frame of SCET. The corresponding hadronic operator can be written as
The generic expression of the cross section is
After redefining the field to decouple the soft interactions, the operator factorizes into a collinear and a soft part
where the collinear part has the same form as in Eq. (19) with the collinear fields replaced by those not interacting with soft gluons, and the soft part . is the soft Wilson lines defined as
where P indicates path ordering. From here, we omit the color index for simplicity, and we rewrite the squared amplitude in Eq. (22) as
Substituting the definition of the gluon jet function, soft function, parton distribution functions (PDFs) and hard function into Eq. (25),
where is the squared amplitude at LO after averaging the spins and colors.
The other channels follow the same approach to obtain the factorization formulas. By crossing symmetry, the LO cross sections in other channels are obtained by
Here, we point out that the factorization form given in Eq.(30) is only valid in the threshold limit defined by , which means that the Higgs boson should have a large . The traditional transverse momentum dependent factorization and resummation Collins et al. (1985); Collins (2011) is important when the total transverse momentum of the Higgs boson and the recoiling jet is small, that is obvious not the same threshold region as the case we have discussed in this paper. An application of the transverse momentum resummation in Higgs plus one jet production has been discussed in Liu and Petriello (2013a, b); Boughezal et al. (2014); Sun et al. (2014).
Iv The Hard, Jet and Soft Functions at NLO
The hard, jet and soft functions describe interactions at different scales, which can be calculated order by order in QCD, respectively. At the NNLL accuracy, we need the explicit expressions of the hard, jet and soft functions up to NLO. In this section, we summarize the relevant analytic results of them.
iv.1 Hard functions
The hard functions are absolute value squared of the Wilson coefficients of the operators, which can be obtained by matching the full theory onto SCET. It is obtained by subtracting the IR divergences in the scheme from the UV renormalized amplitudes of the full theory. At the LO, the hard function is normalized to . In general, it is related to the amplitudes of the full theory, using
where are obtained by subtracting the IR divergences in the scheme from the UV renormalized amplitudes of the full theory Manohar (2003); Idilbi and Ji (2005); Ahrens et al. (2010). At NLO, in practice, it is necessary to calculate the one-loop on-shell Feynman diagrams of this process, as shown in Fig. 2.
Our results of hard functions are consistent with the results in Ref Jouttenus et al. (2013). The hard functions at the other scales can be obtained by evolution of renormalization group (RG) equations. The RG equations for hard functions are governed by the anomalous-dimension matrix, which has been calculated in Refs. Becher and Neubert (2009a); Ferroglia et al. (2009a, b); Becher and Neubert (2009b, c); Ahrens et al. (2012). In our case, the RG equations for hard functions are given by
where is the universal anomalous-dimension function related to the cusp anomalous dimension of Wilson loops with lightlike segments Korchemsky and Radyushkin (1987); Korchemsky (1989); Korchemskaya and Korchemsky (1992), while and control the single-logarithmic evolution. Their explicit expressions are shown in Ref. Becher and Neubert (2009c). In the following, all anomalous dimensions are expanded in unit of , for example, .
Solving the RG equations, the hard function at an arbitrary scale are given by:
where and are defined as Becher et al. (2007)
The general hard function up to can be written as