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Inclusive Higgs boson production at LHC within the k T -factorization approach
Maxim Malysheva 1 Artem Lipatova,b Nikolai Zotov a D. V. Skobeltsyn Institute of Nuclear Physics, M. V. Lomonosov Moscow State University Leninskie Gory 1, 119991 Moscow, Russia b Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia
a

We study the inclusive Higgs boson production with its subsequent decay to diphoton pair at LHC energies in the k T -factorization QCD approach. We take into account the off-shell gluon fusion subprocess g g H . As unintegrated (or transverse momentum dependent) gluon distributions we use densities obtained with CCFM evolution equation and KMR prescription. We evaluate the theoretical uncertainties of our calculations and compare them with the results of traditional pQCD calculations. We find good agreement between our predictions and first experimental data of the ATLAS Collaboration.

The recent discovery of the Higgs boson [1, 2] has become a triumph of the Glashow-Salam-Weinberg theory of electroweak interactions and simultaneously marks the commencement of a new era in high energy physics. Spin and relative production rates of the observed particle in different decay modes are in very good agreement with the SM expectations for the Higgs boson. The subprocess of gluon-gluon fusion, g g H , is the basic mechanism of inclusive Higgs boson production in proton-proton collisions at the LHC energy. The g g H effective interaction is mediated in the lowest order by a triangle loop of heavy (primarily t) quark. In the conventional collinear QCD approach NNLO calculations [3­8] matched mith NNLL resumation [9, 10] are performed to describe experimental data [11]. An alternative approach is based on the k T -factorization of QCD2 . One of the advantages of the method is that one can obtain good description of experimental data even in LO due to partial incorporating of higher orders corrections in unintegrated (or transverse momentum dependent) parton distributions. Investigation of Higgs boson production in the k T -factorization approach has its own history. The process was studied in [15] and p T -distributions were presented. Reasonable agreement with collinear NNLO results was achieved in the lowest perturbative order. Further, the finite top quark mass was correctly introduced in [16]. Concerning justifying the k T -factorization formula for Higgs boson production, big progress has been made in the last few years [17] (see also a review [18] and references therein). Recently the ATLAS Collaboration has reported first measurements of the Higgs boson differential cross sections in the diphoton decay mode [11]. In particular, the distributions with respect to the diphoton transverse momentum p T , rapidity y and helicity angle | cos | have been presented. The goal of this work is to describe those data in the k T -factorization QCD approach.
1 2

malyshev@theory.sinp.msu.ru Detailed description of the k T -factorization approach can be found in reviews [12­14].

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XXIInd International Workshop "High-Energy Physics and Quantum Field Theory", June 24 ­ July 1, 2015, Samara, Russia

Now we shortly describe our calculation steps. Our consideration is based on the off-shell gluon fusion subprocess g g H . In the limit of large top quark mass mt the effective Lagrangian for the Higgs boson coupling to gluons reads [19, 20]:

L

ggH

=

s ( G F 2) 12

1 /2

a Gµ G

aµ

H,

(1 )

a where GF is the Fermi constant, Gµ is the gluon field strenth tensor and H is the scalar field. Then one can easily obtaine the triangle vertex for two off-shell gluons having four-momenta k1 and k2 and color indices a and b: s µ µ , a b (2 ) Tg g H (k1 , k2 ) = i ab ( G F 2) 1 / 2 ( k 2 k - ( k 1 k 2 ) g µ ) . 1 3

The triangle vertex for H is derived analogously. One just needs to take into account also W boson loop. Then one has [19, 20]: s Lgg H = (3 ) A( GF 2)1/2 Fµ F µ H , 8 where Fµ is the electromagnetic field strength tensor. The triangle vertex for two photons with fourmomenta p1 and p2 reads: T where
µ H

( p1 , p2 ) = i

s A ( G F 2) 2

1 /2

( p2 p1 - ( p1 p2 ) g µ ),

µ

(4 )

A = AW (m2 /4m2 ) + Nc H W


f

Q 2 A f ( m 2 /4 m 2 ) , H W f
2

(5 ) (6 ) (7 ) (8 )

Here Nc is the color factor and Q f is the electric charge of the fermion f .

A f ( ) = 2[ + ( - 1) f ( ) ] / , arcsin2 ( ), 1; f ( ) = 2 1+ 1-1/ 1 - log - i , > 1. 4 1 - 1 -1 /

AW ( ) = - [2 2 + 3 + 3 ( 2 - 1 ) f ( ) ] / 2 ,

Using the effective vertices (2), (4) one can easily obtain the off-shell matrix element for gluon-gluon fusion subprocess g g H . The only difference with the traditional calculations comes in so-called k T -factorization prescription for summation over polarizations of incoming gluons:





µ

k k = T2 T . kT

µ

(9 )

In the collinear limit (k T 0) this expression converges to the ordinary one after averaging on the azimuthal angle. So, the matrix element squared takes the following form:

|M| =

2

^^ s2 ( s + p 2 ) 2 1 T 2 2 G 2 | A |2 SF 1152 4 ^ ( s - m2 )2 + m2 H H

2 H

cos2 ,

(1 0 )

^ where H is the Higgs boson decay width, s = (k1 + k2 )2 , the transverse momentum of the Higgs particle p T = k1T + k2T , and is the azimuthal angle between the transverse momenta of the initial gluons. The expression (10) is fully consistent with the one obtained in [15]. 2


XXIInd International Workshop "High-Energy Physics and Quantum Field Theory", June 24 ­ July 1, 2015, Samara, Russia

The cross-section for the inclusive Higgs boson production in proton-proton collision in the k T -factorization approach is calculated as convolution of the off-shell partonic cross-section with the unintegrated gluon distributions in the proton: =

|M| d d2 2 f ( x , k 2 , µ2 ) f g ( x 2 , k 2 T , µ2 ) d p 2 T d k 2 T d k 2 T d y 1 d y 2 1 , 2 1 1 2 g 1 1T 2 2 16 ( x 1 x 2 s )

2

(1 1 )

where s is the total centre-of-mass energy, y1,2 are the rapidities of the produced photons, and the 1,2 and x1,2 are the azimuthal angles and colliding proton longitudinal momenta fractions of the incoming gluons respectively. f g ( x, k2 , µ2 ) is the unintegrated gluon density in the proton. In our numerical calculations T we tested two sets of such distributions. First, CCFM A0 set [21], is obtained as a numerical solution of the CCFM gluon evolution equation, where all input parameters were fitted in a way to describe the proton structure function F2 ( x, Q2 ). The second set was obtained with Kimber-Martin-Ryskin (KMR) prescription [22, 23]. In this method the unintegrated distributions can be calculated from the conventional collinear ones. As the input we used leading order MSTW set [24]. Following [25], we also multiply the KMR based distributions by special K -factor, absorbing the main part of non-logarithmic loop corrections to the gluon fusion cross-section: s ( µ2 ) 2 K = e xp C A (1 2 ) , 2 where the color factor C rithms [26].
A

= 3, and the scale µ2 = p

4 /3 T

m

2 /3 H

allows to eliminate certain subleading loga-

Now we turn to results of our simulations [27]. We set the renormalization and factorization scales equal to µ R = µ F = m H . We vary the parameter between 1/2 and 2 about the default value = 1 in order to estimate the scale uncertainties of our calculations3 . We set m H = 126.8 GeV and H = 4.3 MeV. We use the leading order formula for the strong coupling constant S (2 ) with n f = 4 active quark flavors at QC D = 200 MeV, so that S (m2 ) = 0.1232. We also use the running QED coupling constant (µ2 ). Z The multidimensional integration in (11) was performed by the means of Monte-Carlo technique, using the routine VEGAS [28]. The results of our calculations [27] are presented in Figs. 1--3 in comparison with the ATLAS data. The ATLAS kinematical region is defined by | | < 2.37, 105 < M < 160 GeV and ET / M > 0.35(0.25) for the leading (subleading) photon, where M is the invariant mass of produced photon pair. In left panels, the solid histograms are obtained with the CCFM A0 gluon density by fixing both the factorization and renormalization scales at the default value, whereas the upper and lower dashed histograms correspond to the scale variation as described above. The dash-dotted histograms correspond to the predictions obtained with the KMR gluon distribution. We find that the ATLAS data are reasonably well described by the k T factorization approach. In right panels of Figs. 1--3 we plot the matched NNLO + NNLL pQCD predictions [9, 10] (or NLO ones for | cos | distribution) taken from [11] in comparison with our results and the ATLAS data. One can see that the measured cross sections are typically higher than the collinear QCD predictions, although no significant deviation within the theoretical and experimental uncertainties is observed. However, the k T -factorization predictions at the default scale are rather similar to upper bound of collinear QCD results, providing us better agreement with the ATLAS data. Higher order corrections are known to be large in the collinear factorization: their effect increases the leading order cross section by about 80--100% [4, 5]. So, Figs. 1--3 illustrate the main advantage of the k T -factorization approach: it is possible to obtain in a straightforward
3 In the case of CCFM parton distribution such a variation leads to the usage of separate sets of gluon distribution -- A0+ set (for = 2) and A0­ ( = 1/2) [21].

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XXIInd International Workshop "High-Energy Physics and Quantum Field Theory", June 24 ­ July 1, 2015, Samara, Russia

d/dpT [fb/GeV]

KMR A0 ATLAS

d/dpT [fb/GeV]

2

2

NNLO + NNLL A0 ATLAS

1.5

1.5

1

1

0.5

0.5

0 0 50 100 150 200 pT [GeV]

0 0 50 100 150 200 pT [GeV]

(a)

(b)

Figure 1: The differential cross section of the Higgs boson production in p p collisions at the LHC as a function of diphoton transverse momentum. Left panel: the solid and dash-dotted histograms correspond to the CCFM A0 and KMR predictions, respectively; and the upper and lower dashed histograms correspond to the scale variations in the CCFM-based calculations, as it is described in the text. Right panel: the solid histogram corresponds to the CCFM A0 predictions, and the hatched histogram represent the NNLO + NNLL predictions obtained in the collinear QCD factorization (taken from [11]). The experimental data are from ATLAS [11].

d/d|y| [fb]

KMR A0 ATLAS

d/d|y| [fb]

50

50

NNLO + NNLL A0 ATLAS

40

40

30

30

20

20

10

10

0 0 0.5 1 1.5 2 |y|

0 0 0.5 1 1.5 2 |y|

(a)

(b)

Figure 2: The differential cross section of the Higgs boson production in p p collisions at the LHC as a function of diphoton rapidity. Notation of all histograms are the same as in Fig. 1. The NNLO + NNLL predictions are taken from [11]. The experimental data are from ATLAS [11].

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XXIInd International Workshop "High-Energy Physics and Quantum Field Theory", June 24 ­ July 1, 2015, Samara, Russia

d/d|cos *| [fb]

120 100 80 60 40 20 0 0 0.2 0.4

KMR A0 ATLAS

d/d|cos *| [fb]

140

140 120 100 80 60 40 20 0

NLO + PS A0 ATLAS

0.6

0.8

1 |cos *|

0

0.2

0.4

0.6

0.8

1 |cos *|

(a)

(b)

Figure 3: The differential cross section of the Higgs boson production in p p collisions at the LHC as a function of the helicity angle. Notation of all histograms are the same as in Fig. 1. The NLO predictions are taken from [11]. The experimental data are from ATLAS [11]. manner analytic description which reproduces the main features of rather cumbersome higher order pQCD calculations. In conclusion, the inclusive Higgs boson production with its subsequent decay to diphoton pair in the k T -factorization QCD approach at LHC energies has been studied for the first time. The off-shell matrix element for g g H subprocess has been evaluated. Reasonably good description of ATLAS data for the inclusive production of Higgs boson, decaying to diphoton pair, at LHC has been obtained. The results give the upper limit of NNLO+NNLL predictions, which shows the effective including of higher orders corrections in the k T -factorization approach. We have demonstrated that the k T -factorization approach can be used to study processes incorporating Higgs bosons decays and that the experimental data give limitations on the transverse momentum dependent. Future experimental analyses are necessary in order to discriminate between NNLO+NNLL and k T -factorization predictions. M.A.M. is very grateful to the organizing committee for their really helpful support. A.V.L. and N.P.Z. would also appreciate DESY Directorate for the support in the framework of Moscow -- DESY project on Monte-Carlo implementation for HERA -- LHC.

References
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XXIInd International Workshop "High-Energy Physics and Quantum Field Theory", June 24 ­ July 1, 2015, Samara, Russia

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