One-loop QCD corrections to the $e^+e^- \to W^+ W^- b\bar b$ process at the ILC

We study the full contributions at the leading order(LO) and QCD next-to-leading order(NLO) to the cross section of the \eewwbb process in the standard model(SM) at the ILC. In dealing the resonance problem we adopted the complex mass scheme in both tree-level and one-loop level perturbative calculations. Our numerical results show that the K-factor varies from 1.501 to 0.847 when $\sqrt{s}$ goes up from $360 GeV$ to $1.5 TeV$. We investigate the dependence of the LO and QCD NLO corrected cross sections of process \eewwbb on colliding energy $\sqrt{s}$ and Higgs-boson mass. We also present the results of the LO and QCD NLO corrected distributions of the transverse momenta of final particles, and the invariant masses of $Wb$-, $b\bar b$- and $WW$-pair.


I. Introduction
The Higgs boson, which gives masses to the weak vector bosons and fermions, plays an important role in the standard model(SM). Unfortunately, it has not been directly detected yet in experiments.
Searching for Higgs boson within the standard model(SM) and study the phenomenology concerning Higgs properties are the important tasks at the present and upcoming high energy colliders. LEP II experiments have provided the lower limit on the SM Higgs mass as 114.4 GeV at the 95% confidence level, which is extracted from the results of searches for e + e − → Z 0 H 0 production [1,2]. While the indirect evidences of the SM Higgs mass through electroweak precision measurements indicate the 95% C.L. upper bound as m H 182 GeV , when the lower limit on m H is used in determination of this upper limit [2]. On the other hand, the heavy top-quark practically plays a central and crucial role in probing the electroweak symmetry breaking as well as the flavor problem in all the extended models beyond the SM which address the hierarchy problem. Recently, a new datum of top-quark mass has been already presented by the CDF and D0 experiments at Fermilab, and the preliminary world average mass of the top-quark is known as m t = 172.5 ± 1.3(stat) ± 1.9(syst) GeV , which corresponds to a 20% precision improvement relative to the previous combination [3].
The future International Linear Collider (ILC) is proposed by the particle physics community with the entire colliding energy in the range of 200 GeV < √ s < 500 GeV and an integrated luminosity of around 500 (f b) −1 in four years. The machine should be upgradeable to √ s ∼ 1 T eV with an integrated luminosity of 1 (ab) −1 in three years [4]. Most of the main physics topics within the SM or its beyond at T eV energy scale can be explored at such a machine. Emphasis is given to the study of top-quark physics, electroweak physics in the SM, and the measurements in the extended SM, such as supersymmetry.
Compared with the hadron colliders, such as the Tevatron and the CERN Large Hadron Collider (LHC), the ILC can produce top and Higgs boson signal events more easily resolved from backgrounds. Therefore, the ILC is an ideal facility to study top and Higgs physics with much more precise measurement for their parameters. At the ILC we can also carry out the study of gauge boson interactions, and the delicate cancellations which are related to the gauge structure of the theory and essential to preserve unitarity. Furthermore, the ILC experiment might be able to explore the signature of the new physics, if the SM is really only an effective theory at low energy.
At the ILC, detecting the top-quark pair production process e + e − → tt is a good way to study the top-quark properties, and the associated Higgs production with Z 0 boson e + e − → H 0 Z 0 is one of the cleanest signature in discovering Higgs boson if the the b-quark trigger system has high performances except vertex detectors [5]. The former process will be followed by the subsequential decay through tt → W + W − bb [6], while the later process goes via H 0 Z 0 → W + W − bb through decays H 0 → W + W − and Z 0 → bb if the Higgs boson mass is larger than 2m W [7]. Therefore, the signature of e + e − → W + W − bb at the ILC serves as non-resonant background to both topquark pair production and associated production of Higgs boson with Z 0 boson. We can see that it is crucial to separate the top and Higgs signatures from the other W + W − bb production backgrounds in ILC experimental data analyzing. In the precise measurements of the signals of both the tt pair and H 0 Z 0 associated production processes, the relevant irreducible background from e + e − → W + W − bb should be carefully investigated.
In Refs. [6,8,9,10,11,12] the NLO electroweak and QCD corrections to the process e + e − → tt and decay t → W + b have been already extensively studied. And the non-relativistic effect near the threshold of tt production is also studied carefully in Ref. [13], which can not be reliably described with fixed QCD orders in perturbative theory. The Higgs-strahlung Bjorken process e + e − → H 0 Z 0 was investigated in Ref. [14], and the process e + e − → tt → W + W − bb → 6f with six fermion final states after W pair decays has been also calculated at the lowest order in Ref. [15]. The evaluation of the e + e − → W + W − bb process with finite width method at the tree-level is also presented in Ref. [7,16]. All those studies indicate that the precise investigations of the characteristics of top-quark and the Higgs-boson are significant for the future e + e − ILC experiments.
In this paper we present the calculations of the cross section of the process e + e − → W + W − bb at the leading order(LO) and its QCD next-to-leading order(NLO) (O(α s )) corrections. The paper is organized as follows: In the following section we present the analytical calculations for process

II. Calculations
The calculations for the process e + e − → W + W − bb are carried out in 't Hooft-Feynman gauge.
In the QCD NLO calculations, we use the dimensional regularization(DR) method to isolate the ultraviolet(UV) and infrared(IR) singularities. In order to preserve gauge invariance, we adopt the approach of the complex mass scheme to deal with the unstable particles in the calculations for the tree-level cross section and QCD NLO radiative correction [17,18] The formula for calculating the IR divergent integrals with complex internal masses in DR scheme are obtained by analytically extending the expressions in Ref. [21] to the complex plane. The numerical evaluations of IR safe one-point, two-point, three-point and four-point integrals with internal complex masses, are implemented by using the expressions analytically continued to complex plane from those presented in Refs. [22,23]. And the 5-point scalar integral can be expressed in terms of multiple scalar four-point integrals [24]. The subroutines for one-loop integrals with complex masses are coded based on the LoopTools2.1 [20] package which comes from FF library [25]. The 2 → 4 phase-space integration routine [27] is created based on the 2to3.F program in FormCalc4.1 package. The five-body phase-space integration for hard gluon radiation process e + e − → W + W − bb g is accomplished by using CompHEP-4.4p3 program [26].
Now we present the analytically calculations of the tree-level cross section for e + e − → W + W − bb and its QCD NLO radiative corrections. The notations for the process are defined as where p i (i = 1 − 6) label the four-momenta of incoming e + , e − and outgoing final particles, respectively. There are 64 generic tree-level diagrams for the process e + e − → W + W − bb presented in Fig.1, where internal wavy-line represents γ, Z 0 , or W ± and internal dash-line represents a Higgs-boson H 0 or a Goldstone G 0 (G ± ). We can easily find that in Fig.1 there includes the tree- The differential cross section for the process e + e − → W + W − bb at the tree-level is obtained by the tree-level is obtained by where dΦ 4 is the four-body phase space element given by The summation in Eq.(2.2) is taken over the spins and colors of final states, and the bar over the summation recalls averaging over initial spin states. In the calculation, the internal Z 0 and Higgs boson can be real, and the top-quark propagator can also be resonance when √ s > 2m t .
To deal with these resonant singularities, we use the so-called complex mass scheme(CMS) in our perturbative calculations [17,18]. The complex masses of W-, Z-, H-boson and top-quark are defined as In the CMS approach the complex masses for all related unstable particles should be taken everywhere in both tree-level and one-loop level calculations. Then the gauge invariance can be conserved and singularity poles of propagators are avoided.
In calculating the complete QCD NLO corrections, we should consider the contributions of  To cancel the IR soft divergency appeared in the virtual correction, we should consider the contribution of the real gluon emission process e + e − → W + W − bbg. We denote the real gluon emission process as To calculate the contribution of this process, we introduce an arbitrary small soft cutoff δ s to separate its 5-body phase-space into two regions [28], i.e., soft(E 7 ≤ δ s √ s/2) and hard(E 7 > δ s √ s/2) regions. After adopting the soft gluon approximation, the expression of σ sof t for e + e − → W + W − bbg process with soft gluon has the form as where C F = 4/3 and g 56 are defined as: In above equation, λ(s 56 , m 2 b , m 2 b ) is the kinematical function defined by: ∆E = E 7 = δ s √ s/2, s 56 = (p 5 + p 6 ) 2 and Our created 2 → 4 phase space integration routine [27], is adopted in the tree-level and one-loop level calculations for e + e − → W + W − bb process. The IR singularity part of the soft gluon emission process e + e − → W + W − bb (g) can be exactly cancelled by the IR singularity induced by the oneloop virtual gluon correction. We apply CompHEP-4.4p3 program [26] to implement the phase space integration of the hard gluon emission process e + e − → W + W − bb +g. Finally, we get the finite total cross section including complete NLO QCD corrections for the process e + e − → W + W − bb by summing up all the contribution parts, (2.10)

III. Checks
We have performed the following checks to prove the reliability of our calculation:   Table 1. There our results are obtained by using both CompHEP-4.4p3 program and our created 2 → 4 phase-space integration routine, and compared with the corresponding ones presented in Ref. [16]. We can see there is a good agreement between ours and those presented in Ref. [16]. The in-house 2 → 4 phase-space integration routine was also once verified in our previous work [27]. • In the following section, we shall clarify other verifications.

IV. Numerical results and discussion
In our numerical calculation we take the following input parameters [29,30]:  Due to the application of the CMS approach, we use the complex weak mixing angle defined as In our LO and NLO numerical calculations we set the QCD renormalization scale µ as µ = m W +m b , and take the strong coupling α s (µ 2 ) = 0.11885, which is obtained by using the formula at three-loop level (M S scheme) with the five active flavors [29].
Since the widths of top-quark and Higgs boson haven't been well provided or measured experimentally by now, we use their theoretical results from perturbative calculations. Considering the fact that top-quark mass is above m W + m b , and V tb ∼ 1, the decay of top-quark is dominated by undergoing two-body decay t → W + b, and the total decay width of top-quark is approximately equal to the decay width of t → W + b. Neglecting terms of order m 2 b /m 2 t , α 2 s and (α s /π)M 2 W /m 2 t , the width predicted in the SM is [31]: The reasonable physical decay width of Higgs boson is obtained by employing the program Hdecay [32], when m H = 120 GeV . As indicated in Fig.3(a), both curves for the cross sections at the LO and NLO increase quickly in the √ s region of [350 GeV, 400 GeV ] and decrease when √ s > 430 GeV . Fig.3(b) shows that the corresponding K-factor decreases slowly from 1.501 to 0.847 as √ s running from 360 GeV to 1.5 T eV . The large positive peak near the tt threshold in Fig.3(b) is due to a Coulomb singularity effect coming from the instantaneous gluon exchange between heavy quarks which has a small spatial momentum. In Table 2 we list the values of σ tree , σ N LO and K-factor at some typical √ s points, which are read out from Figs.3(a-b). Since the QCD correction to the e + e − → W + W − bb process with high colliding energy can be approximately decomposed into the QCD correction to the tt production plus the corrections to the t(t) → W + b(W −b ) decays when m H < 2m W , we make following verification to check our results. We evaluate the QCD correction to e + e − → W + W − bb process by combining the QCD corrections to e + e − → tt production and t(t) → W + b(W −b ) decays together, and get the K-factors to process e + e − → W + W − bb as 0.8562(1) for √ s = 1 T eV and 0.8433(1) for √ s = 1.5 T eV , which are coincident with the corresponding ones in Table 2 in error ranges.
In Fig.4(a)  with high colliding energy, is from top-pair production channel e + e − → tt and followed by the Here we can see that the QCD NLO correction slightly suppresses the LO differential cross section dσ LO /dM (W + b) .
As we know, if Higgs boson has a mass larger than 2m W , the e + e − → H 0 * Z 0 * → W + W − bb channel will certainly slightly increase both the LO and QCD NLO corrected cross sections for W + W − bb production due to the Higgs-boson resonant effect as shown in Fig.4(a).

V. Summary
In this paper we calculate the complete one-loop QCD corrections in the SM to the process e + e − →