Resummed prediction for Higgs boson production through b\bar{b} annihilation at N{}^{3}LO+N{}^{3}LL

Resummed prediction for Higgs boson production through annihilation at NLo+nLl

Ajjath A H, The Institute of Mathematical Sciences, HBNI, IV Cross Road, Taramani, Chennai 600113, IndiaTheory Group, Deutsches Elektronen-Synchrotron (DESY), Notkestrasse 85, D-22607 Hamburg, Germany    Amlan Chakraborty, The Institute of Mathematical Sciences, HBNI, IV Cross Road, Taramani, Chennai 600113, IndiaTheory Group, Deutsches Elektronen-Synchrotron (DESY), Notkestrasse 85, D-22607 Hamburg, Germany    Goutam Das, The Institute of Mathematical Sciences, HBNI, IV Cross Road, Taramani, Chennai 600113, IndiaTheory Group, Deutsches Elektronen-Synchrotron (DESY), Notkestrasse 85, D-22607 Hamburg, Germany    Pooja Mukherjee, The Institute of Mathematical Sciences, HBNI, IV Cross Road, Taramani, Chennai 600113, IndiaTheory Group, Deutsches Elektronen-Synchrotron (DESY), Notkestrasse 85, D-22607 Hamburg, Germany    V. Ravindran The Institute of Mathematical Sciences, HBNI, IV Cross Road, Taramani, Chennai 600113, IndiaTheory Group, Deutsches Elektronen-Synchrotron (DESY), Notkestrasse 85, D-22607 Hamburg, Germany

We present an accurate theoretical prediction for the production of Higgs boson through bottom quark annihilation at the LHC up to next-to-next-to-next-to leading order (NLO) plus next-to-next-to-next-to-leading logarithmic (NLL) accuracy. We determine the third order perturbative Quantum Chromodymanics (QCD) correction to the process dependent constant in the resummed expression using the three loop bottom quark form factor and third order quark soft distribution function. Thanks to the recent computation of NLO corrections to this production cross section from all the partonic channels, an accurate matching can be obtained for a consistent predictions at NLO+NLL accuracy in QCD. We have studied in detail the impact of resummed threshold contributions to inclusive cross sections at various centre of mass energy and also discussed their sensitivity to renormalisation and factorisation scales.

Resummation, Perturbative QCD

IMSC/2019/05/04, DESY 19-076

1 Introduction

Discovery of the Higgs boson in is one of the biggest achievements of ALTAS Aad:2012tfa () and CMS Chatrchyan:2012xdj () collaborations at the Large Hadron Collider (LHC). It is a mile stone in the success of the Standard Model (SM). Having understood the generation of mass for the elementary particles through spontaneous symmetry breaking, it is important to understand the properties Heinemeyer:2013tqa () of the Higgs boson such as spin, CP properties, self coupling and the couplings to the SM fermions and vector bosons. In the SM, Higgs boson is spin-0 and CP-even and it couples to SM fermions through Yukawa coupling. There exists several beyond the SM (BSM) scenarios which allow non SM spin-0 or spin-2 bosons to couple to SM particles. In addition, there exist CP mixing in the extended Higgs sectors. All these scenarios demonstrate distinct observable effects which can be studied at the LHC thereby constraining various BSM physics.

In recent years, efforts to understand the shape of the Higgs potential by measuring the Higgs self coupling in di-Higgs boson production is underway. It has important implications for the hierarchy problem, the vacuum metastability, the electroweak phase transition and baryogenesis. The measurement of the di-Higgs boson production at the LHC is plagued by the tiny cross section deFlorian:2016spz (), however, the high luminosity option can help.

Yukawa couplings of the Higgs boson to the quarks and leptons of the SM are free parameters and hence, understanding their pattern is a great deal. As the mass of the Higgs boson is close to the electroweak scale, these couplings are highly sensitive to scales of new physics and measuring them precisely can probe various high scale physics scenarios. Both ATLAS ATLAS-CONF-2019-005 () and CMS Sirunyan:2018koj () collaborations have made dedicated efforts to measure them in various decay channels of the Higgs boson. Yukawa coupling of bottom quarks to the Higgs boson is one of the most sought parameter. However, measuring this coupling in the dominant decay channel of the Higgs boson to a pair of bottom quarks is a challenging task. Associated production of Higgs boson with vector bosons or with top quarks, its subsequent decay to bottom quarks is the promising one while there are also other proposals Englert:2015dlp ().

While the Higgs boson dominantly decays to bottom quarks in SM, its production from bottom quark annihilation is much smaller than the gluon initiated subprocess. However, as the precise measurement of the Higgs cross sections is underway at the LHC, inclusion of bottom quark initiated channels in the theoretical predictions is unavoidable. Unlike in the SM, bottom quarks in the Minimal Supersymmetric SM (MSSM) Gunion:1992hs () couple to neutral Higgs boson with the coupling proportional to which in some parameter region can increase the production rate. Here, the angle is related to the ratio, denoted by , of the vacuum expectation values of two Higgs doublets. Hence, there is a considerable interest in studying the production mechanism of a single and a pair of Higgs bosons through bottom quark annihilation. In the literature, the production of Higgs boson(s) is studied using two approaches namely four flavours and five flavour schemes often called 4FS and 5FS respectively. In 4FS, the bottom quark distribution in the proton is set equal to zero, however they are radiatively generated through gluons in the proton allowing the possibility of their annihilation to produce the Higgs boson. Such contributions are enhanced by logarithms that are sensitive to bottom quark mass and hence they need to be resummed to get reliable predictions. The alternative approach, 5FS, avoids this enhancement through the introduction of non-zero bottom quark distributions in the proton. The origin of these distributions can be traced back to the bottom quarks resulting from gluon distributions inside the proton, thanks to the DGLAP evolution equation of parton distribution functions, which resums collinear enhanced logarithms to all orders in perturbation theory. While both these schemes should give the same result at the observable level, care is needed while comparing their predictions. Since the leading order contribution to 4FS is two to three scattering reaction, while in 5FS, it is two to one, the higher order computations in perturbative Quantum Chromodynamics (QCD) in the later scheme is lot easier. Hence, till today, only next to leading order QCD effects Dittmaier:2003ej (); Dawson:2003kb (); Wiesemann:2014ioa () are known in 4FS while in 5FS, recently the state of the art NLO prediction Duhr:2019kwi () is available. The later computation provides an opportunity to compare NLO predictions against those at NLO computed in 4FS in a consistent manner. Note that, in 5FS, the complete NLO Dicus:1998hs (); Balazs:1998sb () and NNLO Harlander:2003ai () as well as dominant threshold effects at NLO Ravindran:2006cg (); Ahmed:2014cha () are known for quiet sometime. In this article we improve the 5FS cross-section resumming threshold logarithms up to NLL accuracy. The fact that the 5FS cross-section provides dominant cross-section in a matched prediction Bonvini:2016fgf (); Forte:2015hba () is very well known for quite a long. Thus threshold improved result justifies in the context of precision study for this process. Recently the 5FS prediction has been also improved Ebert:2017uel () resumming time-like logarithms in SCET framework.

Fixed order predictions in perturbative QCD are often plagued with large logarithms resulting from certain boundaries of the phase space and hence their reliability in those regions are questionable. In the inclusive production rates, when partonic scaling variable , which corresponds to the emission of soft gluons, large logarithms are generated at every order in perturbation theory. Here, is the Higgs boson mass and is the partonic centre of mass energy. One finds a similar problem in the transverse momentum and rapidity distributions of Higgs boson when there are soft gluon emissions. This is resolved by resumming these large logs to all orders in perturbation theory through a systematic resummation approach. For inclusive rates, we refer Sterman:1986aj (); Catani:1989ne (); Catani:1990rp () to the earliest approach. Catani and Trentadue, in their seminal work Catani:1989ne (), demonstrated the resummation of leading large logs for the inclusive rates in Mellin space.

Needless to say that the inclusion of higher order QCD effects is of utmost importance to achieve precision prediction. In addition, these terms reduce the dependence on the unphysical scales such as renormalisation and factorisation at the observable level. Note that for the production of scalar Higgs boson through gluon fusion, the NLO contribution is now known Anastasiou:2015ema (); Anastasiou:2016cez (), which was further improved by the resummation of threshold logarithms, arising from soft gluon emissions, to NLL accuracy Bonvini:2014joa (); Bonvini:2016frm (); Bonvini:2014tea () (the prime denotes that in addition to the NLL terms the resummed result includes higher logarithmic order terms coming from the matching to NLO). Such an analysis lead to a precise determination of the SM Higgs cross section at the LHC with small uncertainty.

The goal of the this paper is to present the prediction for the inclusive production of Higgs boson in bottom quark annihilation at the LHC taking into account the resummed threshold corrections at next-to-next-to-next to leading logarithmic accuracy. We can achieve this using the recent NLO prediction Duhr:2019kwi () and independent threshold constant computed to third order in QCD in this paper. Here, denotes the Mellin moment. The is obtained using the three loop form factor Gehrmann:2014vha () of the Higgs-bottom-anti bottom operator and the third order soft distribution function computed in Ahmed:2014cha ().

2 Theoretical Framework

The Lagrangian that describes the Yukawa interaction of the SM Higgs boson with bottom quarks is given by


where is the bottom quark field and is the SM Higgs field. is the Yukawa coupling given by , where is the vacuum expectation value of the Higgs field, is the mass of the bottom quark. Note that we will use the non-zero mass of the bottom quark only in the Yukawa coupling, elsewhere it is treated as massless quarks i.e. we use the VFS scheme throughout our analysis. The inclusive cross-section for the production of Higgs boson in proton proton collision is given by


where is the non-perturbative parton distribution function with denoting the parton type and its momentum fraction, The scaling variable where is the hadronic centre of mass energy. The renormalization and factorisation scales are denoted by and respectively. The born cross section is given by


The , mass factorised parton level cross section, is calculable order by order in strong coupling constant, in perturbative QCD:


At each order in perturbation theory we can write


with . In the above equation collects all those contributions that result from soft and collinear configurations of partons in the scattering events. They are often proportional to distributions of the kind and where


The superscript is the short form of soft plus virtual. The remaining contribution is called the regular part of the cross section denoted by . In QCD, was computed up to NNLO level in Harlander:2003ai (), at NLO level in the threshold limit it was obtained in Ravindran:2006cg (); Ahmed:2014cha () and recently the complete NLO result has been reported in Duhr:2019kwi ().

The threshold contributions to inclusive cross section originate from soft and collinear partons in the virtual and real emission subprocesses. These contributions demonstrate remarkable factorisation property through process independent cusp, soft and collinear anomalous dimensions. Consequently, the leading contributions resulting from the large threshold logarithms of the form can be summed to all orders in a systematic fashion. Following Sterman:1986aj (); Catani:1989ne (); Catani:1990rp (), the resummation of these logarithms can be efficiently achieved in Mellin -space and the resulting resummed threshold contribution takes the following form:




where we have set . The cusp anomalous dimension and the constant are process independent and hence can be obtained from the resummation result of Drell-Yan process. They are known upto third order in QCD and are listed in the Appendix C. The independent constant on the other hand is process dependent. It gets contribution from the process dependent virtual Higgs-bottom-anti bottom quark form factor Gehrmann:2014vha () and the soft distribution Ravindran:2006cg () resulting from real emission subprocesses. We followed the method described in Ravindran:2006cg () (also see Ahmed:2014cla ()) to obtain . Expanding in powers of as


we obtain up to three loop level


where and are the Casmirs of and and . In addition are the Riemann zeta functions.

Following the method described in Catani:2003zt (), we have computed to NLL accuracy. Defining , we obtain


with , the EulerMascheroni constant. The successive terms in the above equation defines the resummation accuracy LL, NLL etc. Note that in the context of resummation, is of . The resummation coefficients in Eq. (13) matches exactly with Catani:2003zt (); Moch:2005ba () and for completeness we collect those in App. A.

Thus, the resummed contribution to the Higgs production in bottom quark annihilation in Mellin -space takes the simple form


The coefficients for are listed in the Appendix. Note that except both the cusp and collinear anomalous dimensions are known to third order in . Similarly is also known to order . Since the approximate is available in the literature we can readily predict NLL contributions to inclusive Higgs production in bottom quark annihilation. In the next section, using the recently available predictions Duhr:2019kwi () for the fixed order NLO contributions, we present the resummed prediction to NLL + NLO accuracy.

3 Phenomenology

In this section, we present a detailed discussion on the numerical impact of the resummed threshold contribution to the inclusive production of the Higgs boson in bottom quark annihilation at the LHC up to NLO+NLL accuracy in perturbative QCD. The resummed part of the inclusive cross section for the Higgs production in space can be obtained by taking -th Mellin moment of Eq. (2) as




is the part of the cross section where in Eq. (2) is replaced by whose -th moment is given in Eq. (7). The resummed cross section is then added to the fixed order one after subtracting the Mellin moment of in the large limit. This is done because they are already present in the fixed order result and hence this will avoid double counting. This is achieved through a matching procedure at every order. Finally, the matched result takes the following form:


In the above equation the superscript NLL in implies that we retain up to terms in the and up to in the exponent given in Eq. (13). Similarly, NLO implies that we retain the fixed order result up to order . The subscript in the last term of the above equation means truncation of the series in to desired accuracy.

The fixed order analytical results Harlander:2003ai () up to NNLO order have been implemented in a fortran code. We have validated our predictions to a very good accuracy with the available public code SuShi Harlander:2012pb (). For NLO+NLL analysis, we have used the numbers presented in Tab. II in the arXiv version of the paper Duhr:2019kwi () for NLO. We perform the inverse Mellin transform of the resummed -space result using an in-house fortran code. We have used the Minimal prescription Catani:1996yz () to deal with the Landau pole in the Mellin inversion routine.

Since we work in the 5FS, we take throughout. We use the MMHT2014(68cl) PDF set Harland-Lang:2014zoa () and renormalisation group (RG) running for at each order. At NLO + NLL, we have used the four loop RG running for , however for the PDF, we use NNLO set due to the unavailability of NLO pdf sets. One could in principal use the strong coupling as provided through LHAPDF Buckley:2014ana () interface, which at the NNLO level changes the cross-section by . The Yukawa coupling is also evolved using 4 loop RG with bottom mass GeV. It is well known that the optimal choice for the central scale to study the Higgs production in bottom quark annihilation is and . This choice mimics most of the higher order contributions. Hence, we have made this choice throughout. In addition, we have predictions for other central scale choice namely .

Figure 1: Resummed cross-section plotted against the hadronic centre-of-mass energy (). The band corresponds to the scale variation around the central scale choice along with the prediction for the central scale. In the lower inset the resummed K-factor has been shown along with scale uncertainties (see text).
Figure 2: Fixed order and resummed cross-sections are plotted successively at each order up to NNLO level with unphysical factorisation () scale varied in the range keeping renormalisation scale () fixed at central value . Similar variation for is done in the second panel keeping . In the last panel, the is set to and is varied in the same range.
Figure 3: The perturbative convergence is shown for fixed order and resummed order for TeV LHC. The central scale is fixed at and the asymmetric error bars are obtained by varying the scales by around their central scale.
(TeV) NNLO+NNLL (pb) NLO+NLL (pb)
7 0.174 0.172
8 0.225 0.223
13 0.535 0.536
14 0.605 0.608
Table 1: Resummed cross-sections are provided for TeV LHC with PDF4LHC2015nnlo pdf set for the central scale choice .

In Fig. 1, we present the resummed cross-section up to NNLL+NNLO level against the hadronic centre-of-mass energy for 7 TeV to 100 TeV. The bands in the plot correspond to the scale variations obtained by varying the unphysical renormalisation and factorisation scales in the range where the central scales are taken to be . In the lower inset we show the corresponding resummed -factors, defined as the ratio of the cross-section at a particular order (NLL+NLO, NNLL+NNLO) over the same at the LL+LO order. At NLL the -factor increases by and at NNLL by compared to LL for TeV LHC. We find that the uncertainty due to and scales increases with the energy of the collider, however at the current energy of the LHC it is within at NNLL. The reason for the large uncertainty at high could be due to the lack of knowledge of the PDF sets at these energies

In Fig. 2, we study the sensitivity of our predictions to the renormalisation and factorisation scales at the LHC energy TeV. For a conservative choice, we vary all the scales in the range , however in the later part of our analysis we followed the usual one in the range . For the variation (first panel), we have kept whereas for the variation (second panel), we keep the factorisation scale to . Finally in the third panel, we set and vary from to . The cross-section decreases by when is varied ten times its central value at LO. On the other hand it increases rapidly when is decreased and increases around when the scale is taken one tenth of the central value. In the resum case the corresponding numbers are and respectively. At NLO level the fixed order cross-section changes by to whereas for NLO+NLL case it improves and changes by to . At the next order the corresponding numbers are , and , respectively. We observe that the resultant uncertainty from the renormalisation and factorisation scales after adding resummed result with the fixed one do not reduce significantly. This could be due to the fact that the resummation takes into account only the bottom quark initiated channels beyond NLO accuracy. But the inclusion of the other channels can eventually lead to the reduction of the uncertainty. Moreover as we can see, the cross-section is particularly very sensitive to the variation even at NNLO level associated to the bottom quark pdf. A detailed analysis with respect to the choice of pdfs is thus required particularly how the underlying theoretical assumptions and models in the PDF fits affect the b-quark pdf determination Accardi:2016ndt ().

At LO, the cross-section varies a lot with . However, at NNLO level, the uncertainty due to variation reduces to whereas at NNLO+NNLL this goes down to .

In Fig. 3, we compare the resummed cross-section against the fixed order one for TeV LHC. The uncertainty is obtained using the 7-point variation i.e., by varying and scales around their central values and respectively in the range .

At NNLL level the uncertainty is comparable to the corresponding fixed order result. However the central value of the resummed result shows a better perturbative convergence compared to the FO result. We find that the NLO cross-section increases by compared to LO whereas NNLO increases by compared to NLO. In the resummed case, the NLO+NLL cross-section increases by compared to LO+LL result while NNLO+NNLL result increases by compared to the NLO+NLL one at the central scale.

Recently, the complete result for to third order, namely s have become available Duhr:2019kwi () and this allows us to predict the matched cross section to NLO+NLL accuracy. We have used same set of input parameters as in Duhr:2019kwi () for our study. In particular, we have used the PDF4LHC2015nnlo set with strong coupling evolved at four loops. The bottom quark mass is taken to be GeV. At the LHC energy TeV we find that the NLO SV cross-section deviates from the complete NLO result by around . Our prediction for the cross-section at the central scale at NLONLL level is pb which changes the fixed order cross-section by around . The renormalisation and factorisation scale dependence can also be studied for the matched cross section, once their code Duhr:2019kwi () is publicly available. However, we expect the scale variation at this order will be negligible and the dominant uncertainty will come from the PDF errors which can be addressed when N3LO pdfs become available.

In Table-1 we quote the prediction at NNLO+NNLL and NLO+NLL level for the central scale choice for centre-of-mass energies and TeV. We find that the central scale choice stabilises for different collider energies already at NNLO+NNLL level. At NLO+NLL level, the change is well within .

4 Conclusions

In this article we have studied the role of resummed threshold corrections to the inclusive cross section for producing the Higgs boson through bottom quark annihilation at the LHC. While this is a sub-dominant channel compared to gluon fusion subprocess, the precise measurements that are carried out at the LHC in the context of processes involving Higgs bosons demand the inclusion of this channel for the precision studies. Complete NNLO QCD corrections Harlander:2003ai () and soft plus virtual corrections Ahmed:2014cha () to NLO level to this observable are known for a while. More recently, a complete NLO contributions resulting from all the partonic channels where the Higgs boson couples to bottom quarks became available Duhr:2019kwi (). In hadronic colliders, soft gluons play important role in most of the observables. In the fixed order perturbative computations, in certain kinematic regions the soft gluons dominate. The large logarithms resulting from these soft gluons often spoil the reliability of the fixed order predictions. The resolution to this problem is to resum these logarithms to all orders in perturbation theory. The framework to resum such logarithms in a systematic fashion to all orders in perturbation theory for inclusive observables is well established and results based on this demonstrate better and reliable predictions compared to the fixed order ones. In the present article we have done a detailed study to understand the role of the soft gluons on the inclusive cross section for producing Higgs boson in bottom quark annihilation. This is achieved within the frame work of threshold resummation in Mellin-N space. We have done this to NLO+NLL accuracy. We have used the recent prediction at NLO level from Duhr:2019kwi () for the fixed order contribution and for the resummed part at NLL level, except the process dependent constant , rest of the ingredients are already known. We have computed this constant for the first time using the three loop form factor Gehrmann:2014vha () and the quark soft distribution function Ahmed:2014cha () known to third order in QCD. Our numerical result at NLO+NLL in QCD is the most precise prediction for the inclusive cross section for the production of Higgs boson through bottom quarks at the LHC. We have predicted the inclusive rates at various centre of mass energies with the corresponding -factors. In addition, we studied in detail the sensitivity of the resummed predictions to the renormalisation and factorisation scales in order to estimate the theoretical errors precisely.


We would like to thank P. K. Dhani for initial collaboration. We also thank S. Moch, F. J. Tackmann and J. K. L. Michel for useful discussions.

Appendix A Resummation constants


Appendix B Soft-Virtual coefficients in -space

Here we have collected all the large coefficients for this process. Defining they are given below:


Appendix C The Cusp and the soft anomalous dimensions

The quark cusp anomalous dimensions are given as Moch:2004pa (),


The four loops coefficient Moch:2018wjh (); Lee:2017mip (); Grozin:2018vdn (); Henn:2019rmi (); Lee:2019zop (); vonManteuffel:2019wbj () is also known numerically and the perturbative series for finally looks like



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