QCD corrections to the production at the ILC
Abstract
The is an irreducible background process in measuring the decay width, if Higgs boson is produced in association with a boson which subsequently decays via at the ILC. In this paper we study the impact of the QCD corrections to the observables of the process in the standard model. We investigate the dependence of the leadingorder and QCD corrected cross sections on colliding energy and the additional jet veto schemes. We also present the results of the LO and QCD corrected distributions of the transverse momenta of final particles, and the invariant masses of  and pair.
PACS: 13.66.Jn, 14.65.Fy, 12.38.Bx
I. Introduction
The Higgs mechanism is an essential part of the standard model (SM) [1, 2], which gives masses to the gauge bosons and fermions. Until now the Higgs boson has not been directly detected yet in experiment. The LEP collaborations have established the lower bound of the SM Higgs mass as at the confidence level (CL) [3]. The Fermilab Tevatron experiments have excluded the SM Higgs boson with mass between and at CL [4]. Recently, the ATLAS and CMS experiments at the LHC have provided the upper limits of the SM Higgs mass as and at CL respectively, and there are several Higgs like events around the locations of (ATLAS) and (CMS) [5][6]. Further searching for Higgs boson and studying the phenomenology concerning its properties are still the important tasks for the present and upcoming high energy colliders.
After the discovery of the Higgs boson, the main tasks will be the precise measurements of its couplings with fermions and gauge bosons and its decay width [7]. The future International Linear Collider (ILC) is an ideal machine for conducting efficiently and precisely the measurements for the standard model (SM) Higgs properties. The ILC is designed with and in the first phase of operation [8]. The measurements of the Higgsstrahlung Bjorken process provide precision access to the studies of triple interactions between Higgs boson and gauge bosons ( and ) [9, 10]. As both the Higgs boson and boson are unstable particles, we can only detect their final decay products. For the boson, the main decay channel is , whose branching fraction is [11]. The Higgs coupling studies at the ILC usually can be carried out by means of (i) process [12], (ii) , and (iii) via fusion [13]. In the SM and beyond, such as the twoHiggsdoublet model (THDM) and the minimal supersymmetric standard model (MSSM), the precise ILC data for the Yukawa Higgs boson processes are significant for probing the small SM Yukawa bottom coupling and determining the ratio of the vacuum expectation values [14]. The production events can be selected by tagging both (anti)bottom jets. As for the light SM Higgs boson, its main decay is the mode with a branch fraction about , but this decay mode would be difficult to detect accurately. The rare diphoton Higgs decay channel is of great importance, since a precise measurement of its width can help us to understand the nature of the Higgs boson and may possibly provide hints for new physics beyond the SM. This requires not only the precise measurement for the diphoton Higgs decay width, but also accurate predictions for new physics signal and its background. Fortunately, the ILC instrument would provide excellent facilities in energy and geometric resolutions of the electromagnetic detectors to isolate the narrow signal from the huge continuum background. Ref.[15] provides the conclusion that a precision of on the partial decay width of can be achieved at the ILC by the help of an excellent calorimeter.
The calculations for reaction at the treelevel are given in Ref.[13], and the study for measuring the branching ratio of at a linear collider is provided in Ref.[15]. There it is demonstrated that the ability to distinguish Higgs boson signature at linear colliders, crucially depends on the understanding of the signature and the corresponding background with multiparticle final states. If we choose the production events at the ILC with the subsequent and decays, we obtain the events with final state, and the process becomes an important irreducible background of production. Our calculation shows the integrated cross section for the process can exceed at the ILC, more than thirty thousand events could be obtained in the first phase of operation, and then the statistical error could be less than . Therefore, it is necessary to provide the accurate theoretical predictions for the process in order to measure the diphoton decay width of Higgs boson at the future ILC.
In this paper, we calculate the full QCD corrections to the process . In the following section we present the analytical calculations for the process at the leadingorder (LO) and order. The numerical results and discussions are given in section III. Section IV summarizes the conclusions.
Ii. Calculations
In both the LO and QCD oneloop calculations for the process , we adopted the t’HooftFeynman gauge, if not stated otherwise. We use the FeynArts3.4 package [18] to generate Feynman diagrams and their corresponding amplitudes. The reductions of the output amplitudes are implemented by using the developed FormCalc6.0 package [20].
(1) LO cross section
The pair production associated with two photons via electronpositron collision at the treelevel is a pure electroweak process. We denote this process as , where label the fourmomenta of incoming positron, electron and outgoing final particles, respectively. Because the Yukawa coupling of Higgs/Goldstone to fermions is proportional to the fermion mass, we ignore the contributions of the Feynman diagrams which involve the Yukawa couplings between any Higgs/Goldstone boson and electrons. There are 40 generic treelevel diagrams for the process , some of them are depicted in Fig.1. The internal wavyline in Fig.1 represents  or boson.
The differential cross section for the process at the LO is expressed as
(2.1) 
where , factor comes from the two final identical photons, and is the fourbody phase space element given by
(2.2) 
The summation in Eq.(2.1) is taken over the spins of final particles, and the bar over the summation recalls averaging over initial spin states. In the calculations, the internal boson is potentially resonant, and requires to introduce the finite width in propagators. Therefore, we consider boson mass, the related boson mass and the cosine squared of Weinberg weak mixing angle () consistently being complex quantities in order to keep the gauge invariance [19]. Their complex masses and Weinberg weak mixing angle are define as
(2.3) 
where , are conventional real masses and , represent the corresponding total widths, and the propagator poles are located at on the complex plane. Since the  and boson propagators are not involved in the loops for the QCD corrections, we shall not meet the calculations of Npoint integrals with complex internal mass. In our LO and QCD oneloop level calculations for the process, we put cuts on the transverse momenta of the produced photons and (anti)bottomquarks (, ), final photonphoton resolution (), bottomantibottom resolution () and final (anti)bottomphoton resolution () (The definition of will be declared in the following section). Then the LO cross section for the process is IRfinite.
(2) QCD corrections
The full QCD corrections to the process can be divided into two parts: QCD virtual and real gluon emission corrections. The QCD virtual corrections include the contributions of the selfenergy, triangle, box, pentagon and counterterm diagrams. Since we take non zero bottomquark mass, the virtual QCD corrections do not contain any collinear infrared (IR) singularity, and only the soft IR singularities are involved in the virtual corrections. We adopt dimensional regularization scheme with to extract both UV and IR divergences which correspond to the pole located at () on the complex plane, and manipulate the matrix in dimensions by employing a naive scheme presented in Ref.[26], which keeps an anticommuting in all dimensions. The wave function of the external (anti)bottomquark field and its mass are renormalized in the onmassshell renormalization scheme.
By introducing a suitable set of counterterms, the UV singularities from oneloop diagrams can be canceled, and the total amplitude of these oneloop Feynman diagrams is UVfinite. In the renormalization procedure, we define the relevant renormalization constants of bottomquark wave functions and mass as
(2.4) 
With the onmassshell renormalization conditions we get the renormalization constants as
(2.5) 
where takes the real part of the loop integrals appearing in the selfenergies only, and the unrenormalized bottomquark selfenergies at are expressed as
(2.6) 
The IR divergences from the oneloop diagrams involving virtual gluon can be canceled by adding the real gluon emission correction. We denote the real gluon emission process as , where a real gluon radiates from the internal or external (anti)bottom quark line. We employ both the phase space slicing (PSS) method [27] and the dipole subtraction method [28] for gluon radiation to combine the real and virtual corrections in order to make a cross check. In the PSS method the phase space of gluon emission process is divided by introducing a soft gluon cutoff (). That means the real gluon emission correction can be written in the form as . In this work we take the non zero mass of bottomquark and no collinear singularity exists in the QCD calculation. Therefore, we do not need to set the collinear cut in adopting PSS method. Then the full QCD correction to the process is finite and can be expressed as
(2.7) 
We use our modified FormCalc6.0 programs [20] to simplify analytically the oneloop amplitudes involving UV and IR singularities, and extract the IRsingular terms from oneloop integrals in the amplitudes by adopting the expressions for the IR singularities in oneloop integrals [21]. The numerical evaluations of the IR safe Npoint scalar integrals are implemented by using the expressions presented in Refs.[22, 23, 24]. The tensor loop integrals are expressed in scalar integrals via PassarinoVeltman(PV) reductions [25].
Iii. Numerical results and discussions
In this section we present the numerical results and discussions of the LO and QCD corrected cross sections and the kinematical distributions of the final particles for the process at the ILC by using non zero bottomquark and electron masses fixed at , . For the complex masses of  and boson in Eq.(2.3), the real parts, and , are set to be the onshell physical masses of and , i.e., and . The decay widths of and , which are the imaginary parts of the complex masses, are taken to be and , respectively [11]. The fine structure constant is set to be , and the strong coupling constant at the pole has the value of . The running strong coupling constant, , is evaluated at the threeloop level ( scheme) with five active flavors [11]. For the definitions of detectable hard photon and (anti)bottom quark we require the constraints of , (), , and , where we apply the jet algorithm presented in Ref.[16] to the final photons and (anti)bottomjets. In the jet algorithm of Ref.[16] is defined as with and denoting the separation between the two particles in azimuthal angle and pseudorapidity respectively. We set the QCD renormalization scale being in the numerical calculations if no other statement. In further numerical evaluations, we take the cuts for final particles having the values as , , and unless otherwise stated. In the calculations, we use the ’inclusive’ and ’exclusive’ selection schemes for the events including an additional gluonjet. In ’inclusive’ scheme there is no restriction to the gluonjet, but in the ’exclusive’ scheme the threejet events satisfy the conditions of and are excluded.
We investigate the LO contribution from the channel as shown in Figs.1(46), and compare that part with the contribution from all the diagrams for the process. We find that the cross section for the channel is about of the LO total cross section for the process, when the colliding energy () goes from to . It shows that the dominant contributions are from the diagrams with resonant exchanging, i.e., process, and the amplitude squared for process is approximately proportional a BreitWigner function as , where is the squared invariant mass of pair. For this kind of integration functions with large variation, an efficient and stable Monte Carlo integration program is requested. We adopted our inhouse program to implement the four and fivebody phase space integrations by applying the importance sampling for variable . In order to prevent numerical instability in tensor integral reductions, we coded the numerical calculation programs in Fortran77 with quadriprecision. With these programs the precision and efficiency of Monte Carlo integration are greatly improved. In order to verify the reliability of our numerical results, we performed the following checks:

The LO cross section for the process has been calculated by adopting two independent packages and two gauges in the conditions of with the cuts of and for final photons, and no cut for (anti)bottom quark. The numerical results are obtained as: (1) By using CompHEP4.5.1 program [17], we get (in Feynman gauge) and (in unitary gauge). (2) By using our inhouse phasespace integration routine, we obtain (in Feynman gauge) and (in unitary gauge). We can see they are all in good agreement within the statistic errors.

The independence of the full QCD correction on the soft cutoff is confirmed numerically. Fig.2(a) and Fig.2(b) demonstrate that the full QCD correction to the process at the ILC does not depend on the arbitrarily chosen small value of the cutoff within the calculation errors, where we take , , , , and . In Fig.2(a), the fourbody correction (), fivebody correction () and the full QCD correction () to the process are depicted as the functions of the soft cutoff running from to . The amplified curve for the full correction is presented in Fig.2(b) together with calculation errors. The independence of the total QCD correction to the process on the cutoff is a necessary condition that must be fulfilled for the correctness of our calculations.

We adopt also the dipole subtraction method to deal with the IR singularities for further verification. The results including statistic errors are plotted as the shadowing region in Fig.2(b). It shows the results by using both the PSS method and the dipole subtraction method are in good agreement. In further numerical calculations we adopt the dipole subtract method.

The exact cancelations of UV and IR divergencies in our QCD calculations are verified.
In Figs.3(a,b) we depict the LO, QCD corrected cross sections and the corresponding Kfactors () for the process versus the colliding energy at the ILC by taking and the cut set for bquarks and photons mentioned above. The figures show the QCD corrected results by adopting the ’inclusive’ and ’exclusive’ threejet event selection schemes, separately. We list some of the data read out from these curves of Figs.3(a,b) in Table 1. We can see from the table that the Kfactor of the QCD correction varies quantitatively in the range of to for ’inclusive’ scheme, but in the range of to for ’exclusive’ scheme, when colliding energy varies from to . As we know if the colliding energy is very large, the dominant contribution for the process is from the production and followed by the real boson decay . Then the QCD Kfactor for the process is approximately equal to that for the later boson decay process. We make a comparison of the Kfactors for the and the process by using the ’inclusive’ threejet event selection scheme. We get the Kfactor of the decay with the value of , and find it is agree with the Kfactor of process at the ILC with very high colliding energy, e.g., for . From our calculations, we get the ’inclusive’ QCD relative correction of at the ILC is about , which is larger than the QCD correction estimated from the trivial QCD corrections for the decay convoluted with the production cross section for . It shows that a complete QCD calculation for process is necessary, especially in the first phase of ILC operation. We make a comparison for the renormalization scale choices: i.e., and . The former scale value is close to at the ILC running with a relative small colliding energy. The QCD corrections with the ’inclusive’ selection scheme at are obtained as , for , and , for as shown in Table 1. It demonstrates that the QCD correction to the process is not sensitive to these two renormalization scale choices.
(GeV)  200  300  400  500  800 

37.19(1)  35.86(1)  31.86(1)  26.59(1)  14.947(8)  
(I)  40.61(5)  39.20(5)  34.64(4)  28.77(3)  15.99(2) 
(I)  1.092(3)  1.093(3)  1.087(3)  1.082(3)  1.070(3) 
(II)  38.08(5)  36.77(5)  32.57(4)  27.10(3)  15.16(2) 
(II)  1.024(3)  1.025(3)  1.021(3)  1.019(3)  1.014(3) 
Due to the CPconservation, the distribution should be the same as antibottom’s (). Here we present the LO and QCD corrected distributions of the transverse momenta for the bottomquark and the leading photon with the ’inclusive’ threejet event selection scheme in Fig.4(a) and Fig.4(b) respectively, the corresponding Kfactors are also plotted there. The socalled leading photon is defined as the photon with the highest energy among the two final photons. These results are obtained by taking , and the cut set for bquarks and photons as mentioned above. From these two figures we can see that the QCD corrections enhance both the LO differential cross sections and , especially in low region. The distribution curves in Fig.4(b) drop with growing . Fig.4(a) shows that the differential cross sections (, ) have their maximal values at about , but Fig.4(b) shows the maximal values of and are located at about .
We plot the spectra of  and pair invariant masses (denoted as and ) with the ’inclusive’ threejet event selection scheme at the LO and in Figs.5(a) and (b), respectively. There we take , , , , and . We can see from Fig.5(a) that most of the events are concentrated around a peak located at the vicinity of . That shows the fact that the contribution to the cross section for the process at the ILC, is mainly from real boson production channel and followed by the subsequent real decay . Both the Figs.5(a) and (b) show that the QCD corrections enhance the LO differential cross sections and . The precise prediction for the distribution of the pair invariant mass is very significant, because it is the irreducible continuum background for the Higgsboson signature of decay in the production process.
Iv. Summary
In this paper we calculate the complete QCD corrections to the process in the SM at the ILC. We study the dependence of the LO and QCD corrected cross sections on the colliding energy , and investigate the LO and QCD corrected distributions of the transverse momenta of final particles and the spectra of the invariant masses of  and pair. The precise spectrum for the invariant mass of pair is very important, since it is the irreducible background if the Higgs boson is produced via channel. Our calculations show that the size of the QCD correction exhibits a obvious dependence on the additional gluonjet veto scheme. The numerical results show that the QCD corrections with ’inclusive’ scheme enhance the LO results by about to when we take the cut of , , , with the colliding energy running from to .
Acknowledgments: This work was supported in part by the National Natural Science Foundation of China (Contract Nos.10875112, 10675110, 11005101, 11075150), and the Specialized Research Fund for the Doctoral Program of Higher Education (SRFDP)(No.20093402110030).
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