# \(k_T\)-factorization approach to the Higgs boson production in \(ZZ^*\rightarrow 4\ell \) channel at the LHC

## Abstract

We calculated a differential cross section of the Higgs boson production in the \(h\rightarrow ZZ^*\rightarrow 4\ell \) decay channel within the framework of \(k_T\)-factorization. Results are obtained using an off-shell matrix element for the \(g^*g^*\rightarrow h\rightarrow ZZ^*\rightarrow 4\ell \) process together with Ciafaloni–Catani–Fiorani–Marchesini (CCFM) evolution equations for an unintegrated gluon distribution function. We have presented a comparison of our results with the latest experimental measurements at \(\sqrt{S} = 8\hbox { TeV}\) and \(\sqrt{S} = 13\ \hbox {TeV}\) from the ATLAS and CMS collaborations at the LHC. In addition to this, we have compared our results with the results from the collinear factorization formalism calculated up to next-to-next-to-leading order plus next-to-next-to-leading logarithm (NNLO + NNLL) accuracy obtained using the HRes code for the Higgs boson production in the gluon-gluon fusion process. Our estimates are consistently close to NNLO + NNLL results obtained using a collinear factorization formalism and are also in agreement with experimental measurements.

## 1 Introduction

The Higgs boson discovery at the LHC, by ATLAS and CMS collaborations [1, 2], has enabled experimental measurements to be taken to investigate its properties. The ATLAS and CMS collaborations have performed an improved measurement of the Higgs boson mass, considering an invariant mass spectra of the \(h\rightarrow \gamma \gamma \) and \(h\rightarrow ZZ^*\rightarrow 4\ell \) decay channels [3, 4]. Further studies of spin and the parity quantum number of the Higgs boson have established that it is a neutral scalar boson, with a mass equal to 125.09 GeV, rather than a pseudoscalar boson [5, 6, 7]. Its coupling strength to vector bosons and to fermions is studied by analyzing various decay modes of the Higgs boson [4, 8, 9, 10]. Establishing various aspects of the Higgs boson’s properties and coupling strength allows us to study other aspects of it.

A dominant channel for the inclusive Higgs boson production at the LHC is gluon–gluon fusion [11, 12, 13]. Hence, the Higgs boson production at the LHC can be effectively used to understand the gluon dynamics inside a proton. The gluon density \(xf_g(x,\mu ^2_F)\) in a proton is a function of the Bjorken variable *x* and the hard scale \(\mu ^2_F\). The scale evolution of parton densities, in general, is described using the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) evolution equation [14, 16, 18, 19], where large logarithmic terms proportional to \(\ln \mu ^2_F\) are resummed up to all orders.

The factorization theorem in perturbative quantum chromodynamics (pQCD) allows us to write a convolution of the matrix element of the short distance process and the universal parton distribution functions to obtain an inclusive cross section for a given scattering process [21]. The QCD collinear factorization theorem is based on the collinear approach where the parton distribution function depends on the longitudinal momentum fraction *x* and the hard scale \(\mu ^2_F\). The Higgs boson production cross section at leading order and higher order QCD corrections to it, up to next-to-next-to-next-to-leading order (\(\hbox {N}^3\hbox {LO}\)), have been computed within the collinear factorization framework [22, 23, 24, 25, 26, 27]. However, it should be noted that the NNLO and \(\hbox {N}^3\hbox {LO}\) results that were obtained so far are using an effective theory and in the heavy top quark mass limit. Study of the Higgs boson’s transverse momentum spectrum resummed at NNLL accuracy is shown in Refs. [28, 29, 30]. Recently, the state-of-the-art predictions for the Higgs boson’s transverse momentum at the LHC, at next-to-next-to-next-to-leading-logarithmic accuracy (\(\hbox {N}^3\hbox {LL}\)) matched, at NNLO is presented in Ref. [31].

For the inclusive Higgs boson production at the LHC, the longitudinal momentum fraction of the incident gluons is small (\(x_1 x_2\sim \) 0.0089–0.0175). This domain of small longitudinal momentum fraction (*x*) is still in the perturbative regime where is it expected that collinear factorization should break down because the large logarithmic term proportional to 1 / *x* becomes dominant [32, 33, 34]. The contribution from the terms proportional to 1 / *x* is taken into account in the Balitsky–Fadin–Kuraev–Lipatov (BFKL) evolution equation [35, 37, 39]. An unintegrated parton densities (uPDFs) obeying BFKL evolution, convoluted with an off-shell matrix element within a generalized factorization is called \(k_T\)-factorization [41, 42, 44, 45, 46]. The evolution equation is valid for both small *x* and large *x* is given by the Ciafaloni–Catani–Fiorani–Marchesini (CCFM) evolution equation [47, 48, 49, 50]. CCFM evolution is equivalent to BFKL evolution in the limit of very small *x* and is equivalent to the DGLAP evolution for a large *x* region.

In this work, we have not implemented the reggeized parton approach [51, 52, 53, 54, 55] based on the Lipatov effective action formalism, which ensures a gauge invariance of the off-shell amplitude [56, 57]. However, our investigation is based on the assumption that the off-shell partonic amplitudes being gauge-invariant in a small-*x* limit. The approach, which is based on Lipatov effective action formalism that has been employed recently in the calculation using \(k_T\)-factorization approach for the inclusive prompt photon production at LHC [58].

In this paper, the inclusive Higgs boson production within the \(k_T\)-factorization approach, together with CCFM evolution equations have been studied and demonstrated importance of higher order corrections included within the \(k_T\)-factorization [59]. The authors of Ref. [60] have shown that \(k_T\)-factorization gives a description of an experimental data from ATLAS experiment for the differential cross section of the Higgs boson production in the diphoton decay channel. They have calculated a leading order (LO) matrix element for the partonic subprocess \(gg\rightarrow h\rightarrow \gamma \gamma \) considering gluons to be off-shell. Inclusive Higgs boson production analysis based on off-shell gluon-gluon fusion, and considering \(H\rightarrow \gamma \gamma \), \(H\rightarrow ZZ^*\rightarrow 4\ell \) (where \(\ell = e,\mu \)) and \(H\rightarrow W^+W^- \rightarrow e^{\pm }\mu ^{\mp }\nu \bar{\nu }\) decay channel is given in Ref. [61].

The ATLAS and CMS collaboration at the LHC presented a measurement of a fiducial differential cross section of the Higgs boson decay into four-leptons at \(\sqrt{S} =8\hbox { TeV}\) [62, 63] and \(\sqrt{S} = 13\hbox { TeV}\) [64]. We compare the results obtained using the \(k_T\)-factorization approach with the recent ATLAS and CMS data. We have evaluated the off-shell matrix element for the partonic subprocess \(g^*g^*\rightarrow h\rightarrow ZZ^*\rightarrow 4\ell , \ell =e,\mu \). Convolution of the off-shell matrix element of partonic subprocess with CCFM uPDFs [65] is used to obtain a differential cross section.

This article is organized as follows: We discuss in detail the formalism behind our study and the necessary expressions for further numerical analysis in Sect. 2. In Sect. 3, we give the results of our numerical simulation. Here we also discuss the details of the analyses that have gone into the study. Finally, we conclude and draw inferences from the analysis in Sect. 4.

## 2 Formalism

*h*is the Higgs scalar field. The effective

*ggh*triangle vertex (see Eq. (11)) thus becomes

*Z*boson, respectively. \(\mathbf{k}_{i\perp }\) are the intrinsic transverse momenta of the initial gluons. \(\varphi \) is the azimuthal angle between \(\mathbf{k}_{1\perp }\) and \(\mathbf{k}_{2\perp }\). \(m_h\) and \(m_Z\) are the Higgs boson and

*Z*boson masses, respectively. The partonic center of mass energy is denoted by \(\hat{s}\). \(\theta _W\) and \(g_W\) are the weak mixing angle and the coupling of weak interaction, respectively.

^{1}(see Eq. (36))

*y*and \(\mathbf{p}_{\perp }\) are rapidities and transverse momenta of the final state leptons, respectively. The hadronic center of mass energy is denoted by

*S*.

## 3 Results and discussion

With all the calculational tools at our disposal, we proceed to perform a numerical calculation using Eq. (5) together with the off-shell hard scattering amplitude given in Eq. (4). We estimate the cross section of the Higgs boson production as a function of transverse momentum (\(p_T\)) and rapidity (*y*) of the Higgs boson in the four-lepton decay channel. Results are obtained using CCFM A0 the set of uPDFs [65] which is commonly used for such phenomenological studies. Recently a fit to a high precision data from deeply inelastic scattering at the HERA is performed using a \(k_T\)-factorization and CCFM evolution [69]. A transverse momentum dependent gluon density function including experimental and theoretical uncertainties were obtained. The application of these unintegrated gluon densities to vector boson + jet production processes at LHC is given in Ref. [70]. Unintegrated gluon densities including experimental and theoretical uncertainties are given in the CCFM JH2013-set in TMDlib library [71, 72]. We have used the TMDlib library to calculate our results using CCFM JH2013-set. For our phenomenological study, we have used the CCFM JH2013-set2 which is determined from the fit to both stucture functions \(F^{(charm)}_2\) and \(F_2\) data whereas CCFM JH2013-set1 is determined from the fit to inclusive \(F_2\) data only.

Total decay width and mass of the Higgs boson is set to be equal to 4.0 MeV and 125.09 GeV, respectively [7]. We have implemented kinematical cuts on the rapidity and transverse momentum of leptons used by ATLAS and CMS experiments in their measurements. For the ATLAS experiment, the absolute value of rapidity is \(|\eta |< 2.5\) and the leading transverse momentum of the lepton is \(p_T < 20\) GeV. The transverse momenta of sub-leading leptons are \(p_T < 15, 10, 7\) GeV. Similarly for the CMS experiment, the absolute value of rapidity is \(|\eta |< 2.5\) and the ordered transverse momenta of the leptons are \(p_T < 20, 10, 7, 7\) GeV. The cross section in Eq. (5) depends on the renormalization and factorization scales \(\mu _R\) and \(\mu _F\), respectively. The scale uncertainty in the cross section is estimated by varying the scale between \(\mu _R = \mu _F = m_h/2\) and \(\mu _R = \mu _F = 2 m_h\).

Total inclusive cross section (\(\sigma ^{tot}\)) for the Higgs boson production in gluon-gluon fusion channel

\(\sqrt{S}\) (TeV) | \(\sigma ^{tot}\) (pb) | |
---|---|---|

(\(k_T\)-factorization) | (Collinear factorization) | |

8 | 27.12 | 6.11 |

13 | 66.40 | 15.10 |

We have presented our results in Figs. 2, 3 and 4. Figures 2, 4 shows the result of the differential cross section for the Higgs boson production in the four-lepton decay channel at \(\sqrt{S} = 8\hbox { TeV}\) and \(\sqrt{S} = 13\hbox { TeV}\), respectively. We have compared our results of the differential cross section obtained using the \(k_T\)-factorization approach with experimental measurements from the ATLAS at \(\sqrt{S} = 8\hbox { TeV}\), 13 TeV and CMS collaboration at \(\sqrt{S} = 8\hbox { TeV}\) [62, 63]. The solid (red) and dashed (purple) histogram corresponds to our results obtained using the CCFM JH2013-set2 set of uPDFs and CCFM A0 set of uPDFs respectively. We also see that the results obtained using CCFM JH2013-set2 has a better agreement with experimetal measurements than A0 set.

Our results are plotted against state-of-the-art results for the cross section calculated up to next-to-leading order plus next-to-leading logarithm (NLO + NLL) and next-to-next-to-leading order plus next-to-next-leading logarithm (NNLO + NNLL) obtained using the HRes tool [28, 76] within the collinear factorization framework. Our results of both the differential cross section in \(p_T\) and *y* using the \(k_T\)-factorization framework with CCFM unintegrated PDFs are consistently close to NNLO + NNLL results at \(\sqrt{S} = 8\hbox { TeV}\) and \(\sqrt{S} = 13\hbox { TeV}\). This can be explained considering the fact that the main part of higher order corrections are included in the \(k_T\)-factorization approach [59, 77, 78]. For the \(p_T\) distribution, we are using the convention that NLO + NLL and NNLO + NNLL results are labelled as LO + NLL and NLO + NNLL, respectively considering that the \(p_T\) distribution is non-zero at NLO.

## 4 Conclusions

In this paper, we present a phenomenological study of the Higgs boson production in the four-lepton decay channel within the \(k_T\)-factorization framework. Here CCFM unintegrated parton densities were convoluted with the hard matrix element, considering initial gluons to be off-shell. We present a comparison of our results with experimental measurements. Our results are evaluated using same experimental conditions (i.e., the same \(p_T\) and *y* cuts were used for our estimates as given by the experimental results) for both the ATLAS and CMS at \(\sqrt{S} = 8\hbox { TeV}\) and at \(\sqrt{S} = 13\hbox { TeV}\), respectively.

Further comparison of our estimates with the state-of-the-art results of a differential cross section within collinear factorization up to NLO + NLL and NNLO + NNLL obtained using HRes code is presented. We have also estimated a total inclusive cross section for the Higgs boson production within both collinear factorization and \(k_T\)-factorization framework. Within \(k_T\) factorization approach, we have compared the results obtained using CCFM JH2013-set2 and CCFM A0 uPDF set. Our results for the differential cross section with CCFM JH2012-set2 are close to the NNLO + NNLL results obtained using the HRes tool.

Our results show that the observed \(p_T\) distribution of the final state can be generated at leading order subprocesses, using unintegrated gluon distributions. Moreover, gluons in the initial state have finite transverse momenta, which results in the transverse momenta of the final state. The total inclusive cross section estimated using \(k_T\)-factorization is close to the NNLO results obtained using collinear factorization. The main reason for this behavior is that the main part of higher order correction in collinear pQCD is already included in the \(k_T\)-factorization [79] framework.

The higher order corrections within \(k_T\)-factorization at the parton level would be an interesting study, to see any additional effect. The cross section for the Higgs boson production has been calculated using a mixture of LO and NLO partonic diagrams and unintegrated PDFs from the \(k_T\)-factorization. Considering the effect of the transverse momentum of the initial gluon on the transverse momentum distribution of the final state, our study, as well as further studies in this direction could impose constraints on uPDFs of gluons.

## Footnotes

- 1.
The processes contributing to 4 lepton final states are \(e^+e^-e^+e^-\), \(\mu ^+\mu ^-\mu ^+\mu ^-\), \(e^+e^-\mu ^+\mu ^-\) and \(\mu ^+\mu ^-e^+e^-\). We have added these contributions with proper weight. The first two processes have two pairs of identical particles. Hence their phase space has to be multiplied by a factor \((1/2)\times (1/2) = 1/4\) to get rid of overcounting.

## Notes

### Acknowledgements

The authors would like to extend their sincere gratitude to V. Ravindran for continuous support and fruitful discussions during the course of this work and A. Lipatov for providing their codes of a similar study for diphoton distributions. VR would like to thank Giancarlo Ferrera for useful discussions on HRes tool and related topics. RI would also like to acknowledge the hospitality provided by the Institute of Mathematical Sciences, Chennai, India where a part of the work has been done. RI would like to thank the DST-FIST grant SR/FST/PSIl-020/2009 for support

## References

- 1.Georges Aad et al., Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC. Phys. Lett. B
**716**, 1–29 (2012)ADSCrossRefGoogle Scholar - 2.Serguei Chatrchyan et al., Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC. Phys. Lett. B
**716**, 30–61 (2012)ADSCrossRefGoogle Scholar - 3.Georges Aad et al., Measurement of the Higgs boson mass from the \(H\rightarrow \gamma \gamma \) and \(H \rightarrow ZZ^{*} \rightarrow 4\ell \) channels with the ATLAS detector using 25 \(\text{ fb }^{-1}\) of \(pp\) collision data. Phys. Rev. D
**90**(5), 052004 (2014)Google Scholar - 4.Vardan Khachatryan et al., Precise determination of the mass of the Higgs boson and tests of compatibility of its couplings with the standard model predictions using proton collisions at. Eur. Phys. J. C
**75**(5), 212 (2015)ADSCrossRefGoogle Scholar - 5.Serguei Chatrchyan et al., Study of the Mass and Spin-Parity of the Higgs Boson Candidate Via Its Decays to Z Boson Pairs. Phys. Rev. Lett.
**110**(8), 081803 (2013)ADSCrossRefGoogle Scholar - 6.Georges Aad et al., Evidence for the spin-0 nature of the Higgs boson using ATLAS data. Phys. Lett. B
**726**, 120–144 (2013)ADSCrossRefGoogle Scholar - 7.C. Patrignani et al., Review of Particle Physics. Chin. Phys.
**C40**(10), 100001 (2016)ADSGoogle Scholar - 8.Vardan Khachatryan et al., Constraints on the spin-parity and anomalous HVV couplings of the Higgs boson in proton collisions at 7 and 8 TeV. Phys. Rev. D
**92**(1), 012004 (2015)ADSMathSciNetCrossRefGoogle Scholar - 9.Georges Aad et al. Study of the spin and parity of the Higgs boson in diboson decays with the ATLAS detector. Eur. Phys. J., C75(10), 476, (2015). [Erratum: Eur. Phys. J.C76,no.3,152(2016)]Google Scholar
- 10.Georges Aad et al., Measurements of the Higgs boson production and decay rates and coupling strengths using pp collision data at \(\sqrt{s}=7\) and 8 TeV in the ATLAS experiment. Eur. Phys. J. C
**76**(1), 6 (2016)Google Scholar - 11.H.M. Georgi, S.L. Glashow, M.E. Machacek, Dimitri V Nanopoulos, Higgs Bosons from Two Gluon Annihilation in Proton Proton Collisions. Phys. Rev. Lett.
**40**, 692 (1978)ADSCrossRefGoogle Scholar - 12.D. Graudenz, M. Spira, P.M. Zerwas, QCD corrections to Higgs boson production at proton proton colliders. Phys. Rev. Lett.
**70**, 1372–1375 (1993)ADSCrossRefGoogle Scholar - 13.M. Spira, A. Djouadi, D. Graudenz, P.M. Zerwas, Higgs boson production at the LHC. Nucl. Phys. B
**453**, 17–82 (1995)ADSCrossRefGoogle Scholar - 14.V.N. Gribov, L.N. Lipatov, Deep inelastic e p scattering in perturbation theory. Sov. J. Nucl. Phys.
**15**, 438–450 (1972)Google Scholar - 15.V.N. Gribov, L.N. Lipatov, Deep inelastic e p scattering in perturbation theory. Yad. Fiz.
**15**, 781 (1972)Google Scholar - 16.L.N. Lipatov, The parton model and perturbation theory. Sov. J. Nucl. Phys.
**20**, 94–102 (1975)Google Scholar - 17.L.N. Lipatov, The parton model and perturbation theory. Yad. Fiz.
**20**, 181 (1974)Google Scholar - 18.G.Guido Altarelli, Parisi, Asymptotic Freedom in Parton Language. Nucl. Phys. B
**126**, 298–318 (1977)ADSCrossRefGoogle Scholar - 19.L. Yuri, Dokshitzer, Calculation of the Structure Functions for Deep Inelastic Scattering and \(e^+ e^-\) Annihilation by Perturbation Theory in Quantum Chromodynamics. Sov. Phys. JETP
**46**, 641–653 (1977)Google Scholar - 20.L. Yuri, Dokshitzer, Calculation of the Structure Functions for Deep Inelastic Scattering and \(e^+ e^-\) Annihilation by Perturbation Theory in Quantum Chromodynamics. Zh. Eksp. Teor. Fiz.
**73**, 1216 (1977)Google Scholar - 21.John C. Collins, Davison E. Soper, George F. Sterman, Factorization of Hard Processes in QCD. Adv. Ser. Direct. High Energy Phys.
**5**, 1–91 (1989)ADSCrossRefGoogle Scholar - 22.S. Dawson, Radiative corrections to Higgs boson production. Nucl. Phys. B
**359**, 283–300 (1991)ADSCrossRefGoogle Scholar - 23.A. Djouadi, M. Spira, P.M. Zerwas, Production of Higgs bosons in proton colliders: QCD corrections. Phys. Lett. B
**264**, 440–446 (1991)ADSCrossRefGoogle Scholar - 24.Robert V. Harlander, William B. Kilgore, Next-to-next-to-leading order Higgs production at hadron colliders. Phys. Rev. Lett.
**88**, 201801 (2002)ADSCrossRefGoogle Scholar - 25.Charalampos Anastasiou, Kirill Melnikov, Higgs boson production at hadron colliders in NNLO QCD. Nucl. Phys. B
**646**, 220–256 (2002)ADSCrossRefGoogle Scholar - 26.V. Ravindran, J. Smith, W.L. van Neerven, NNLO corrections to the total cross-section for Higgs boson production in hadron hadron collisions. Nucl. Phys. B
**665**, 325–366 (2003)ADSCrossRefGoogle Scholar - 27.Charalampos Anastasiou, Claude Duhr, Falko Dulat, Franz Herzog, Bernhard Mistlberger, Higgs Boson Gluon-Fusion Production in QCD at Three Loops. Phys. Rev. Lett.
**114**, 212001 (2015)ADSCrossRefGoogle Scholar - 28.D. de Florian, G. Ferrera, M. Grazzini, D. Tommasini, Higgs boson production at the LHC: transverse momentum resummation effects in the \(H\rightarrow \gamma \gamma \), \(H \rightarrow WW \rightarrow \ell \nu \ell \nu \) and \(H \rightarrow ZZ \rightarrow 4\ell \) decay modes. JHEP
**06**, 132 (2012)Google Scholar - 29.G. Bozzi, S. Catani, D. de Florian, M. Grazzini, The q(T) spectrum of the Higgs boson at the LHC in QCD perturbation theory. Phys. Lett. B
**564**, 65–72 (2003)ADSCrossRefGoogle Scholar - 30.Giuseppe Bozzi, Stefano Catani, Daniel de Florian, Massimiliano Grazzini, Transverse-momentum resummation and the spectrum of the Higgs boson at the LHC. Nucl. Phys. B
**737**, 73–120 (2006)ADSCrossRefGoogle Scholar - 31.Wojciech Bizon, Pier Francesco Monni, Emanuele Re, Luca Rottoli, Paolo Torrielli, Momentum-space resummation for transverse observables and the Higgs \(p_\perp \) at \(\text{ N }^3\text{ LL+NNLO }\). (2017)Google Scholar
- 32.Bo Andersson et al., Small x phenomenology: Summary and status. Eur. Phys. J. C
**25**, 77–101 (2002)ADSCrossRefGoogle Scholar - 33.Jeppe R. Andersen et al., Small x phenomenology: Summary and status. Eur. Phys. J. C
**35**, 67–98 (2004)CrossRefGoogle Scholar - 34.Jeppe R. Andersen et al., Small x Phenomenology: Summary of the 3rd Lund Small x Workshop in 2004. Eur. Phys. J. C
**48**, 53–105 (2006)ADSCrossRefGoogle Scholar - 35.E.A. Kuraev, L .N. Lipatov, Victor S Fadin, Multi - Reggeon Processes in the Yang-Mills Theory. Sov. Phys. JETP
**44**, 443–450 (1976)ADSGoogle Scholar - 36.E.A. Kuraev, L .N. Lipatov, Victor S Fadin, Multi - Reggeon Processes in the Yang-Mills Theory. Zh. Eksp. Teor. Fiz. 71,840 (1976)Google Scholar
- 37.E.A. Kuraev, L.N. Lipatov, Victor S Fadin, The Pomeranchuk Singularity in Nonabelian Gauge Theories. Sov. Phys. JETP
**45**, 199–204 (1977)ADSMathSciNetGoogle Scholar - 38.E.A. Kuraev, L.N. Lipatov, Victor S. Fadin, The Pomeranchuk Singularity in Nonabelian Gauge Theories. Zh. Eksp. Teor. Fiz. 72,377 (1977)Google Scholar
- 39.I.I. Balitsky, L.N. Lipatov, The Pomeranchuk Singularity in Quantum Chromodynamics. Sov. J. Nucl. Phys.
**28**, 822–829 (1978)Google Scholar - 40.I.I. Balitsky, L.N. Lipatov, The Pomeranchuk Singularity in Quantum Chromodynamics. Yad. Fiz.
**28**, 1597 (1978)Google Scholar - 41.L.V. Gribov, E.M. Levin, M.G. Ryskin, Semihard Processes in QCD. Phys. Rept.
**100**, 1–150 (1983)Google Scholar - 42.E.M. Levin, M.G. Ryskin, YuM Shabelski, A.G. Shuvaev, Heavy quark production in semihard nucleon interactions. Sov. J. Nucl. Phys.
**53**, 657 (1991)Google Scholar - 43.E.M. Levin, M.G. Ryskin, YuM Shabelski, A.G. Shuvaev, Heavy quark production in semihard nucleon interactions. Yad. Fiz.
**53**, 1059 (1991)Google Scholar - 44.S. Catani, M. Ciafaloni, F. Hautmann, GLUON CONTRIBUTIONS TO SMALL x HEAVY FLAVOR PRODUCTION. Phys. Lett. B
**242**, 97–102 (1990)ADSCrossRefGoogle Scholar - 45.S. Catani, M. Ciafaloni, F. Hautmann, High-energy factorization and small x heavy flavor production. Nucl. Phys. B
**366**, 135–188 (1991)ADSCrossRefGoogle Scholar - 46.John C Collins, R.Keith Ellis, Heavy quark production in very high-energy hadron collisions. Nucl. Phys.
**B360**, 3–30 (1991)ADSCrossRefGoogle Scholar - 47.Marcello Ciafaloni, Coherence Effects in Initial Jets at Small \(q^2 / s\). Nucl. Phys. B
**296**, 49–74 (1988)Google Scholar - 48.S. Catani, F. Fiorani, G. Marchesini, QCD Coherence in Initial State Radiation. Phys. Lett. B
**234**, 339–345 (1990)ADSCrossRefGoogle Scholar - 49.S. Catani, F. Fiorani, G. Marchesini, Small x Behavior of Initial State Radiation in Perturbative QCD. Nucl. Phys. B
**336**, 18–85 (1990)ADSCrossRefGoogle Scholar - 50.Giuseppe Marchesini, QCD coherence in the structure function and associated distributions at small x. Nucl. Phys. B
**445**, 49–80 (1995)ADSMathSciNetCrossRefGoogle Scholar - 51.L.N. Lipatov, M.I. Vyazovsky, QuasimultiRegge processes with a quark exchange in the t channel. Nucl. Phys. B
**597**, 399–409 (2001)ADSCrossRefGoogle Scholar - 52.A.V. Bogdan, V.S. Fadin, A Proof of the reggeized form of amplitudes with quark exchanges. Nucl. Phys. B
**740**, 36–57 (2006)ADSCrossRefGoogle Scholar - 53.Martin Hentschinski, Pole prescription of higher order induced vertices in Lipatov’s QCD effective action. Nucl. Phys. B
**859**, 129–142 (2012)ADSCrossRefGoogle Scholar - 54.G. Chachamis, M. Hentschinski, J.D. Madrigal Martinez, ASabio Vera, Quark contribution to the gluon Regge trajectory at NLO from the high energy effective action. Nucl. Phys.
**B861**, 133–144 (2012)ADSCrossRefGoogle Scholar - 55.Martin Hentschinski, Agustin Sabio Vera, NLO jet vertex from Lipatov’s QCD effective action. Phys. Rev. D
**85**, 056006 (2012)ADSCrossRefGoogle Scholar - 56.L.N. Lipatov, Gauge invariant effective action for high-energy processes in QCD. Nucl. Phys. B
**452**, 369–400 (1995)ADSCrossRefGoogle Scholar - 57.L.N. Lipatov, Small x physics in perturbative QCD. Phys. Rept.
**286**, 131–198 (1997)ADSCrossRefGoogle Scholar - 58.A.V. Lipatov, M.A. Malyshev, Reconsideration of the inclusive prompt photon production at the LHC with \(k_T\)-factorization. Phys. Rev. D
**94**(3), 034020 (2016)Google Scholar - 59.A.V. Lipatov, N.P. Zotov, Higgs boson production at hadron colliders in the \(k_T\)-factorization approach. Eur. Phys. J. C
**44**, 559–566 (2005)Google Scholar - 60.A.V. Lipatov, M.A. Malyshev, N.P. Zotov, Phenomenology of \(k_t\)-factorization for inclusive Higgs boson production at LHC. Phys. Lett. B
**735**, 79–83 (2014)Google Scholar - 61.N.A. Abdulov, A.V. Lipatov, M.A. Malyshev, Inclusive Higgs boson production at the LHC in the kt-factorization approach. Phys. Rev. D
**97**(5), 054017 (2018)ADSCrossRefGoogle Scholar - 62.Georges Aad et al., Fiducial and differential cross sections of Higgs boson production measured in the four-lepton decay channel in \(pp\) collisions at \(\sqrt{s}=8\) TeV with the ATLAS detector. Phys. Lett. B
**738**, 234–253 (2014)Google Scholar - 63.Vardan Khachatryan et al., Measurement of differential and integrated fiducial cross sections for Higgs boson production in the four-lepton decay channel in pp collisions at \( \sqrt{s}=7 \) and 8 TeV. JHEP
**04**, 005 (2016)Google Scholar - 64.Morad Aaboud et al., Measurement of inclusive and differential cross sections in the \(H \rightarrow ZZ^* \rightarrow 4\ell \) decay channel in \(pp\) collisions at \(\sqrt{s}=13\) TeV with the ATLAS detector. JHEP
**10**, 132 (2017)Google Scholar - 65.Hannes Jung, Un-integrated PDFs in CCFM. in Deep inelastic scattering. Proceedings, 12th International Workshop, DIS 2004, Strbske Pleso, Slovakia, April 14-18, 2004. Vol. 1 + 2, pages 299–302, (2004)Google Scholar
- 66.John R. Ellis, Mary K. Gaillard, Dimitri V. Nanopoulos, A Phenomenological Profile of the Higgs Boson. Nucl. Phys. B
**106**, 292 (1976)ADSCrossRefGoogle Scholar - 67.Mikhail A Shifman, A.I. Vainshtein, M.B. Voloshin, Valentin I Zakharov, Low-Energy Theorems for Higgs Boson Couplings to Photons. Sov. J. Nucl. Phys.
**30**, 711–716 (1979)Google Scholar - 68.Mikhail A. Shifman, A. I. Vainshtein, M. B. Voloshin, Valentin I. Zakharov, Low-Energy Theorems for Higgs Boson Couplings to Photons. Yad. Fiz. 30,1368 (1979)Google Scholar
- 69.F. Hautmann, H. Jung, Transverse momentum dependent gluon density from DIS precision data. Nucl. Phys. B
**883**, 1–19 (2014)ADSCrossRefGoogle Scholar - 70.S. Dooling, F. Hautmann, H. Jung, Hadroproduction of electroweak gauge boson plus jets and TMD parton density functions. Phys. Lett. B
**736**, 293–298 (2014)ADSCrossRefGoogle Scholar - 71.F. Hautmann, H. Jung, M. Krmer, P.J. Mulders, E.R. Nocera, T.C. Rogers, A. Signori, TMDlib and TMDplotter: library and plotting tools for transverse-momentum-dependent parton distributions. Eur. Phys. J. C
**74**, 3220 (2014)ADSCrossRefGoogle Scholar - 72.Patrick L.S. Connor, Hannes Jung, Francesco Hautmann, Johannes Scheller, TMDlib 1.0.8 and TMDplotter 2.1.1. PoS
**DIS2016**, 039 (2016)Google Scholar - 73.A.D. Martin, W.J. Stirling, R.S. Thorne, G. Watt, Parton distributions for the LHC. Eur. Phys. J. C
**63**, 189–285 (2009)ADSCrossRefGoogle Scholar - 74.Charalampos Anastasiou, Stephan Buehler, Franz Herzog, Achilleas Lazopoulos, Inclusive Higgs boson cross-section for the LHC at 8 TeV. JHEP
**04**, 004 (2012)ADSCrossRefGoogle Scholar - 75.H. Jung et al., The CCFM Monte Carlo generator CASCADE version 2.2.03. Eur. Phys. J.
**C70**, 1237–1249 (2010)ADSCrossRefGoogle Scholar - 76.Massimiliano Grazzini, Hayk Sargsyan, Heavy-quark mass effects in Higgs boson production at the LHC. JHEP
**09**, 129 (2013)ADSCrossRefGoogle Scholar - 77.F. Hautmann, Heavy top limit and double logarithmic contributions to Higgs production at \(m_H^2 / s \ll 1\). Phys. Lett. B
**535**, 159–162 (2002)Google Scholar - 78.M.G. Ryskin, A.G. Shuvaev, YuM Shabelski, Charm hadroproduction in \(k_T\)-factorization approach. Phys. Atom. Nucl.
**64**, 120–131 (2001)Google Scholar - 79.G. Watt, A.D. Martin, M.G. Ryskin, Unintegrated parton distributions and electroweak boson production at hadron colliders. Phys. Rev.,
**D70**, 014012, (2004). [Erratum: Phys. Rev.D70,079902(2004)]Google Scholar - 80.Antoni Szczurek, Marta Luszczak, Rafal Maciula, Inclusive production of Higgs boson in the two-photon channel at the LHC within \(k_{t}\)-factorization approach and with the Standard Model couplings. Phys. Rev. D
**90**(9), 094023 (2014)Google Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP^{3}.