QCD corrections to the $\gamma\gamma b\bar b$ production at the ILC

The $e^+e^- \to \gamma\gamma b\bar b$ is an irreducible background process in measuring the $H^0 \to \gamma\gamma$ decay width, if Higgs boson is produced in association with a $Z^0$-boson which subsequently decays via $Z^0 \to b\bar b$ at the ILC. In this paper we study the impact of the ${\cal O}(\alpha_s)$ QCD corrections to the observables of the $e^+e^- \to \gamma\gamma b\bar b$ process in the standard model. We investigate the dependence of the leading-order and ${\cal O}(\alpha_s)$ QCD corrected cross sections on colliding energy and the additional jet veto schemes. We also present the results of the LO and ${\cal O}(\alpha_s)$ QCD corrected distributions of the transverse momenta of final particles, and the invariant masses of $b\bar b$- and $\gamma\gamma$-pair.


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  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 √ s = 200 ∼ 500 GeV and L = 1000 f b −1 in the first phase of operation [8]. The measurements of the Higgs-strahlung Bjorken process e + e − → H 0 Z 0 provide precision access to the studies of triple interactions between Higgs boson and gauge bosons (Z 0 Z 0 H 0 and γZ 0 H 0 ) [9,10]. As both the Higgs boson and Z 0 -boson are unstable particles, we can only detect their final decay products. For the Z 0 -boson, the main decay channel is Z 0 → bb, whose branching fraction is 15.12% [11]. The Higgs coupling studies at the ILC usually can be carried out by means of (i) e + e − → H 0 Z 0 → H 0 l + l − (l = e, µ) process [12], (ii) e + e − → H 0 Z 0 → H 0 qq, and (iii) via W W -fusion e + e − → H 0 νν [13]. In the SM and beyond, such as the two-Higgs-doublet model (THDM) and the minimal supersym- The calculations for e + e − → γγff reaction at the tree-level are given in Ref. [13], and the study for measuring the branching ratio of H 0 → γγ at a linear e + e − collider is provided in Ref. [15]. There it is demonstrated that the ability to distinguish Higgs boson signature at linear e + e − colliders, crucially depends on the understanding of the signature and the corresponding background with multi-particle final states. If we choose the Z 0 H 0 production events at the ILC with the subsequent H 0 → γγ and Z 0 → bb decays, we obtain the events with bbγγ final state, and the e + e − → bbγγ process becomes an important irreducible background of Z 0 H 0 production. Our calculation shows the integrated cross section for the e + e − → bbγγ process can exceed 30 f b at the √ s = 300 GeV ILC, more than thirty thousand bbγγ events could be obtained in the first phase of operation, and then the statistical error could be less than 1%.
Therefore, it is necessary to provide the accurate theoretical predictions for the e + e − → γγbb process in order to measure the diphoton decay width of Higgs boson at the future ILC.
In this paper, we calculate the full O(α s ) QCD corrections to the process e + e − → γγbb . In the following section we present the analytical calculations for the process at the leading-order

II. Calculations
In both the LO and QCD one-loop calculations for the process e + e − → γγbb , we adopted  (1) LO cross section The bb-pair production associated with two photons via electron-positron collision at the tree-level is a pure electroweak process. We denote this process as e There are 40 generic tree-level diagrams for the process e + e − → γγbb , some of them are depicted in Fig.1. The internal wavy-line in Fig.1 represents γ-or Z 0 -boson.
The differential cross section for the process e + e − → γγbb at the LO is expressed as where N c = 3, factor 1 2! comes from the two final identical photons, and dΦ 4 is the four-body phase space element given by 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 Z 0 -boson is potentially resonant, and requires to introduce the finite width in propagators. Therefore, we consider Z 0 -boson mass, the related W ± -boson mass and the cosine squared of Weinberg weak mixing angle (θ W ) consistently being complex quantities in order to keep the gauge invariance [19]. Their complex masses and Weinberg weak mixing angle are define as where m W , m Z are conventional real masses and Γ W , Γ Z represent the corresponding total widths, and the propagator poles are located at µ X on the complex p 2 -plane. Since the Z 0 -and T,cut ), final photon-photon resolution (∆R cut γγ ), bottom-antibottom resolution (∆R cut bb ) and final (anti)bottom-photon resolution (∆R cut b(b)γ ) (The definition of ∆R will be declared in the following section). Then the LO cross section for the e + e − → γγbb process is IR-finite. Since we take non zero bottom-quark 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 D = 4 − 2ǫ to extract both UV and IR divergences which correspond to the pole located at D = 4 (ǫ = 0) on the complex Dplane, and manipulate the γ 5 matrix in D-dimensions by employing a naive scheme presented in Ref. [26], which keeps an anticommuting γ 5 in all dimensions. The wave function of the external (anti)bottom-quark field and its mass are renormalized in the on-mass-shell renormalization scheme.
By introducing a suitable set of counterterms, the UV singularities from one-loop diagrams can be canceled, and the total amplitude of these one-loop Feynman diagrams is UV-finite. In the renormalization procedure, we define the relevant renormalization constants of bottom-quark wave functions and mass as With the on-mass-shell renormalization conditions we get the O(α s ) renormalization constants as where Re takes the real part of the loop integrals appearing in the self-energies only, and the unrenormalized bottom-quark self-energies at O(α s ) are expressed as The IR divergences from the one-loop 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 (δ s = 2 ∆E 7 / √ s). That means the real gluon emission correction can be written in the form as ∆σ real QCD = ∆σ sof t QCD + ∆σ hard QCD .
In this work we take the non zero mass of bottom-quark and no collinear singularity exists in the O(α s ) QCD calculation. Therefore, we do not need to set the collinear cut δ c in adopting PSS method. Then the full O(α s ) QCD correction to the process e + e − → γγbb is finite and can be expressed as We use our modified FormCalc6.0 programs [20] to simplify analytically the one-loop amplitudes involving UV and IR singularities, and extract the IR-singular terms from one-loop apply the jet algorithm presented in Ref. [16] to the final photons and (anti)bottom-jets. In the jet algorithm of Ref. [16] ∆R is defined as (∆R) 2 ≡ (∆φ) 2 + (∆η) 2 with ∆φ and ∆η denoting the separation between the two particles in azimuthal angle and pseudorapidity respectively.
We set the QCD renormalization scale being µ = √ s/2 in the numerical calculations if no other statement. In further numerical evaluations, we take the cuts for final particles having the values as p T,cut = 20 GeV , ∆R cut γγ = 0.5 and ∆R cut bb = ∆R cut b(b)γ = 1 unless otherwise stated. In the calculations, we use the 'inclusive' and 'exclusive' selection schemes for the events including an additional gluon-jet. In 'inclusive' scheme there is no restriction to the gluon-jet, but in the 'exclusive' scheme the three-jet events satisfy the conditions of p T,cut = 20 GeV , ∆R cut γγ = 0.5 and ∆R cut bb = ∆R cut b(b)γ = 1. In Fig.2(a), the four-body correction (∆σ (4) ), five-body correction (∆σ (5) ) and the full O(α s ) QCD correction (∆σ QCD ) to the e + e − → γγbb process are depicted as the functions of the soft cutoff δ s running from 1 × 10 −5 to 2 × 10 −2 . The amplified curve for the full O(α s ) correction is presented in Fig.2(b) together with calculation errors. The independence of the total O(α s ) QCD correction to the e + e − → γγbb process on the cutoff δ s 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 ±1σ 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. 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 K-factor of the O(α s ) QCD correction varies quantitatively in the range of 1.092 to 1.070 for 'inclusive' scheme, but in the range of 1.024 to 1.014 for 'exclusive' scheme, when colliding energy √ s varies from 200 GeV to 800 GeV . As we know if the colliding energy is very large, the dominant contribution for the e + e − → γγbb process is from the γγZ 0 production and followed by the real Z 0 -boson decay Z 0 → bb. Then the QCD K-factor for the e + e − → γγbb process is approximately equal to that for the later Z 0 boson decay process. We make a comparison of the K-factors for the e + e − → γγZ 0 → γγbb and the e + e − → γγbb process by using the 'inclusive' three-jet event selection scheme. We get the K-factor of the Z 0 → bb decay with the value of 1.069, and find it is agree with the K-factor of e + e − → γγbb process at the ILC with very high colliding energy, e.g.,  Table 1. It demonstrates that the O(α s ) QCD correction to the e + e − → bbγγ process is not sensitive to these two renormalization scale choices. Here we present the LO and QCD corrected distributions of the transverse momenta for the bottom-quark and the leading photon with the 'inclusive' three-jet event selection scheme in Fig.4(a) and Fig.4(b) respectively, the corresponding K-factors 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 √ s = 500 GeV , µ = √ s/2 and the cut set for b-quarks and photons as mentioned above. From these two figures we can see that the O(α s ) QCD corrections enhance both the LO differential cross sections dσ LO /dp T and dσ LO /dp T distribution curves in Fig.4(b) drop with growing p (γ) T . Fig.4(a) shows that the differential cross sections (dσ LO /dp T , dσ N LO /dp the contribution to the cross section for the e + e − → γγbb process at the ILC, is mainly from real Z 0 -boson production channel e + e − → γγZ 0 and followed by the subsequent real Z 0 decay Z 0 → bb. Both the Figs.5(a) and (b) show that the QCD corrections enhance the LO differential cross sections dσ LO /dM (bb) and dσ LO /dM (γγ) . The precise prediction for the distribution of the (γγ)-pair invariant mass is very significant, because it is the irreducible continuum background for the Higgs-boson signature of H 0 → γγ decay in the γγbb production process. T,cut = 20 GeV , ∆R cut bb = ∆R cut b(b)γ = 1 with the colliding energy running from 200 GeV to 800 GeV .