Jin Zhang^,School of Physics and Electronic Engineering, Yuxi Normal University, Yuxi, Yunnan, 653100, China

Received:2021-05-17Revised:2021-06-13Accepted:2021-06-21Online:2021-08-19


Abstract
The scalar one-loop four-point function with one massless vertex is evaluated analytically by employing the loop regularization method. According to the method, a characteristic scale μ_s is introduced to regularize the divergent integrals. The infrared divergent parts, which take the form of ${\mathrm{ln}}^{2}({\lambda }^{2}/{\mu }_{s}^{2})$ and $\mathrm{ln}({\lambda }^{2}/{\mu }_{s}^{2})$ as μ_s → 0 where λ is a constant and expressed in terms of masses and Mandelstam variables, and the infrared stable parts are well separated. The result is shown explicitly via 44 dilogarithms in the kinematic sector in which our evaluation is valid.
Keywords： one-loop four-point integral;loop regularization;dilogarithm


PDF (378KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite
Cite this article
Jin Zhang. Scalar one-loop four-point integral with one massless vertex in loop regularization. Communications in Theoretical Physics, 2021, 73(10): 105203- doi:10.1088/1572-9494/ac0ce2

1. Introduction

The precise tests of physics within the framework of the Standard Model (SM) of particle physics and finding new physics beyond the SM always need to evaluate amplitudes of some physical process at quark level to higher orders of some coupling constant via perturbation theory. Analytic results of Feynman diagrams play a key role in investigating the infrared and ultraviolet structure of a theory but also in the ensuing numerical calculation. Methods of approaching this goal involve multi-loop or/and multi-point Feynman diagram evaluation. Up to now the particle physics community has developed powerful methods for higher-order Feynman diagram calculation, state-of-the-art techniques including integrating by parts [1, 2], evaluating by Mellin–Barnes representation [3], the differential equations method [4–10] and so on. For technical details of each approach and other rare methods, one can refer to [11–14] and the references therein. In evaluating Feynman diagrams we should realize that there is a significant difference in expressing the final results between massless and massive theories. In massless theories, the Feynman integrals can be expressed in terms of polylogarithms [15, 16]. However, the evaluation of massive multiple-loop Feynman integrals are more complicated than the massless cases since the results cannot be expressed via polylogarithms, so the elliptic generalization of polylogarithms, so-called elliptic polylogarithms, are needed. Examples of this trend can be found in [17–25]; for the mathematical ground of elliptic polylogarithms and other properties, in particular the analytic structures, see [26–31].

The evaluation of one-loop integrals of Feynman diagrams holds a prominent position from both theoretical and experimental perspectives [32–35]. It has been shown that the general N-point (N ≥ 5) scalar one-loop integrals can be recursively expressed in terms of combinations of (N − 1)-point integrals. Hence an arbitrary N-point (N ≥ 5) integral can be reduced to the sum of several scalar one-loop four-point integrals. Since tensor-type integrals can be reduced to scalar integrals by the Passarino–Veltman [36] scheme, we can obtain the desired results from scalar integrals by including appropriate tensor structures which are formed by metric tensor g_μν and external momenta. Consequently, all types of one-loop integrals will be evaluated analytically in principle. In other words, scalar one-loop four-point integrals play an intermediate role that transits the intractable N-point(N ≥ 5) integrals to accessible ones. Therefore it is helpful to investigate the scalar four-point integrals carefully.

In the pioneering work by 't Hooft and Veltman [37], scalar one-loop one-, two-, three- and four-point functions are studied generally and the scalar four-point function with real masses is expressed in terms of 24 dilogarithms, but for the case of complex masses it needs 108 dilogarithms. However, there is still a long way to go before the results can be used in practical applications. Subsequently, by using so-called projective transformation [37], it is found that the scalar one-loop four-point function can be reduced to 16 dilogarithms in some kinematical regions [38], and generalizations to tensor [39] and pentagon integrals [40, 41] are also carried out. An important application is made in [42] which employs box integrals to study some electroweak processes in the SM. A more complete work is given by [43] which calculates a set of scalar one-loop four-point integrals with massless internal lines and some massive external lines; the results obtained are convenient for analytic continuation. Scalar one-loop three- and four-point integrals for quantum chromodynamics are calculated in the space-like region in [44] where the ultraviolet, infrared and collinear divergent integrals are widely investigated. A thorough work in evaluating scalar four-point functions, which are valid for complex masses, is presented in [45], in which all the regular and soft and/or collinear singular integrals are analyzed by making use of dimensional and mass regularization.

Nearly all the scalar four-point integrals mentioned above are evaluated with the dimensional regularization method [46]. An alternative way to extract singular parts in evaluating Feynman diagrams is loop regularization [47, 48], which has been successfully applied in some practical calculations in hadronic weak decays of B mesons [49–51]. However, up to ${ \mathcal O }({\alpha }_{s}^{2})$, only contributions of the so-called type-a and type-b diagrams to the amplitudes are considered in [49–51]; the evaluation of type-c diagrams is still lacking. If we want to include the contribution of type-c to the amplitudes, we must extract a scalar integral as depicted in figure 1. In other words, based on the integral corresponding to figure 1, it is possible to complete the evaluation of ${ \mathcal O }({\alpha }_{s}^{2})$ contributions to the amplitudes such that we may determine the branching ratios and direct charge–parity violation of B → ππ up to ${ \mathcal O }({\alpha }_{s}^{2})$ via the six-quark operator effective weak Hamiltonian factorization approach, in order to be consistent with the evaluation of diagrams of type-a and type-b which are given in using loop regularization method. The advantage is that through the marriage of loop regularization and six-quark operator effective weak Hamiltonian factorization, it offers an alternative to evaluate the hadronic observables.

Figure 1.

New window|Download| PPT slide
Figure 1.Scalar one-loop four-point integral with one massless vertex. The dashed and solid lines are massless and massive, respectively. As usual all the external momenta are inward.


On the other hand, noticing the significant role played by the scalar one-loop four-point integral in reducing the tedious N-point (N ≥ 5) integrals to tractable ones, it is helpful to study this diagram by loop regularization such that it provides some reference to the popular dimensional regularization scheme. The diagram under investigation in this paper, corresponding to ‘Box 13' of [44], is collinear divergent¹(¹A analysis of soft and collinear divergence of a diagram with loop regularization via Cayley matrix will be presented in a separate publication.). We know that the divergent structure of an amplitude is independent of the regularization scheme, but the expressions of the divergent part and the stable part may be distinct for different regularization schemes. Hence, we hope that by a specific example we may show how the infrared divergent and infrared stable parts are extracted via loop regularization. As far as we know there are many types of infrared divergent one-loop four-point integrals; however, in this paper we are not as ambitious as the aforementioned works on scalar one-loop four-point integrals which try to investigate the infrared integral of every type thoroughly, and instead we only focus on the diagram depicted in figure 1. In this sense we just present a case study on scalar one-loop four-point integrals by loop regularization. We hope that the results shed some light on the evaluation of scalar one-loop four-point integrals, but are also helpful in calculating some box diagram mediated decaying processes.

Before starting our evaluation, the following comments are in order.(i)To perform the integrals over Feynman parameters, the Euler shift is adopted. Accordingly, two equations, i.e., equations (32) and (61), should be satisfied by the transforming parameters α and β. We assume that the two equations have two real roots, and one root of each equation lies in the range (0, 1) as stated in our evaluation. These requirements fix a kinematic allowed sector in the space spanned by the masses and external momentum; we call it sector I.
(ii)In our evaluation we need to factorize 12 quadratic polynomials of F-type, which are denoted by F_ij (i = 0, 1; j = 1, 2, 3), and of G-type, which are denoted by G_ij(i = 0, 1; j = 1, 2, 3), into products of their roots. The F-type functions are in the denominator and the G-type functions are arguments of logarithms; the coefficients of the two type functions are formed by on-shell masses and masses of propagators as well as invariant combination of external momentum. It is obvious that the factorization should be done carefully since it depends on if the quadratic polynomials have two real roots. In other words, the validity of each factorization determines a kinematic sector where the quadratic polynomial has two real roots. Hence, in all there are 12 kinematic sectors to be fixed, and the intersection of them is the kinematic sector where our evaluation is allowed, which we call sector II. Figuring out sector II exactly is difficult in the complicated space established by the masses and external momentum. In order to solve the dilemma, we assume that there may be some kinematic sector in which all the factorizations are valid. For a case-by-case analysis of similar integrals one can refer to the appendix of [52] and appendix D of [53].


The overlap of sector I and II is the desired kinematic part where the results obtained in this paper can be correctly applied. We assume that there is some method by which the overlapping sector can be determined, although we do not find it explicitly in this paper. It is worth emphasizing that the required kinematic sector may be unphysical or may not even exist; if this is the case, we just present a formal study on the infrared scalar one-loop four-point integral.

The paper is organized as follows. After this short introduction we display some mathematical functions which are frequently used in our evaluation in section 2. Then in section 3 details of the evaluation and results are presented. Section 4 contains our short summary. Some necessary formulae are listed in the appendix.

2. Preliminaries

We define the following massive scalar one-loop four-point integral with one massless vertex, as depicted in figure 1(1)$\begin{eqnarray}I=\int \displaystyle \frac{{{\rm{d}}}^{4}k}{{\left(2\pi \right)}^{4}}\displaystyle \frac{1}{{{D}}_{1}{{ \mathcal D }}_{2}{{ \mathcal D }}_{3}{{ \mathcal D }}_{4}},\end{eqnarray}$where(2)$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal D }}_{1} & = & {k}^{2}+{\rm{i}}\varepsilon ,\\ {{ \mathcal D }}_{2} & = & {\left(k+{p}_{1}\right)}^{2}+{\rm{i}}\varepsilon ,\\ {{ \mathcal D }}_{3} & = & {\left(k+{p}_{1}+{p}_{2}\right)}^{2}-{m}_{2}^{2}+{\rm{i}}\varepsilon ,\\ {{ \mathcal D }}_{4} & = & {\left(k+{p}_{1}+{p}_{2}+{p}_{3}\right)}^{2}-{m}_{3}^{2}+{\rm{i}}\varepsilon .\end{array}\end{eqnarray}$The iϵ will be systematically retained in our evaluation, and m₂ and m₃ are the masses of the two massive internal lines. As usual we fix all the external momenta as inward, and they are related by the energy conservation(3)$\begin{eqnarray}{p}_{1}+{p}_{2}+{p}_{3}+{p}_{4}=0.\end{eqnarray}$We assume that the four external momenta satisfy(4)$\begin{eqnarray}\begin{array}{rcl}{p}_{1}^{2} & = & 0,\quad \quad {p}_{2}^{2}={\omega }_{2}^{2},\quad \quad {p}_{3}^{2}={\omega }_{3}^{2},\\ {p}_{4}^{2} & = & {\left({p}_{1}+{p}_{2}+{p}_{3}\right)}^{2}={\omega }_{4}^{2},\end{array}\end{eqnarray}$and for brevity we define(5)$\begin{eqnarray}{s}_{{ij}}={p}_{i}+{p}_{j}.\end{eqnarray}$

The results will be expressed in terms of logarithms and dilogarithms. As usual we choose for the principal value of the logarithms to lie in the negative axis; hence, we find(6)$\begin{eqnarray}\mathrm{ln}(x\pm {\rm{i}}\varepsilon )=\mathrm{ln}| x| \pm {\rm{i}}\pi ,\quad \quad x\lt 0\end{eqnarray}$In expanding the logarithm of products one should take into account the convention [37](7)$\begin{eqnarray}\mathrm{ln}({ab})=\mathrm{ln}a+\mathrm{ln}b+\eta (a,b),\end{eqnarray}$where the η term is(8)$\begin{eqnarray}\begin{array}{rcl}\eta (a,b) & = & 2\pi {\rm{i}}\{\theta (-\mathrm{Im}\,a)\theta (-\mathrm{Im}\,b)\theta \left(\mathrm{Im}({ab})\right)\\ & & -\theta (\mathrm{Im}\,a)\theta (\mathrm{Im}\,b)\theta \left(-\mathrm{Im}({ab})\right)\},\end{array}\end{eqnarray}$and following this rule it is easy to obtain(9)$\begin{eqnarray}\begin{array}{c}\mathrm{ln}({ab})=\mathrm{ln}a+\mathrm{ln}b\\ \mathrm{if}\,\,\mathrm{Im}(a)\,\mathrm{and}\,\,\mathrm{Im}(b)\,\mathrm{have}\,\,\mathrm{different}\,\,\mathrm{signs},\\ \mathrm{ln}\frac{a}{b}=\mathrm{ln}a-\mathrm{ln}b\\ \mathrm{if}\,\,\mathrm{Im}(a)\,{and}\,\,\mathrm{Im}(b)\,\mathrm{have}\,\,\mathrm{the}\,\,\mathrm{same}\,\,\mathrm{signs}.\end{array}\end{eqnarray}$

As we know the dilogarithm develops an imaginary part for x ≥ 1, we have [43](10)$\begin{eqnarray}{\mathrm{Li}}_{2}(x\pm {\rm{i}}\varepsilon )=-{\mathrm{Li}}_{2}\left(\frac{1}{x}\right)-\frac{1}{2}{\mathrm{ln}}^{2}x+\frac{{\pi }^{2}}{3}\pm {\rm{i}}\pi \mathrm{ln}x.\end{eqnarray}$Most of our results involve analytic continuation of dilogarithms of two variables(11)$\begin{eqnarray}{\mathrm{Li}}_{2}(1-{x}_{1}{x}_{2}),\end{eqnarray}$and this can be completed by the substitution [42, 45](12)$\begin{eqnarray}\begin{array}{l}{\mathrm{Li}}_{2}(1-{x}_{1}{x}_{2})\to { \mathcal L }({x}_{1},{x}_{2})\\ \quad ={\mathrm{Li}}_{2}(1-{x}_{1}{x}_{2})+\eta ({x}_{1},{x}_{2})\mathrm{ln}(1-{x}_{1}{x}_{2}).\end{array}\end{eqnarray}$where η(x₁, x₂) is given by equation (8).

3. Calculation and results

3.1. Basic formula

First, we notice that the first two denominators can be combined via Feynman parameterization:(13)$\begin{eqnarray}\begin{array}{c}\frac{1}{({k}^{2}+i\varepsilon )\left[{\left(k+{p}_{1}\right)}^{2}+{\rm{i}}\varepsilon \right]}={\int }_{0}^{1}{\rm{d}}u\,\frac{1}{{\left[{\left(k+{{up}}_{1}\right)}^{2}+{\rm{i}}\varepsilon \right]}^{2}},\\ \mathrm{if}\quad {p}_{1}^{2}=0.\end{array}\end{eqnarray}$Then the primitive integral in equation (1) becomes(14)$\begin{eqnarray}\begin{array}{c}I=\int \frac{{{\rm{d}}}^{4}k}{{\left(2\pi \right)}^{4}}{\int }_{0}^{1}{\rm{d}}u\,\frac{1}{{\left[{\left(k+{{up}}_{1}\right)}^{2}+{\rm{i}}\varepsilon \right]}^{2}}\\ \times \,\frac{1}{\left[{\left(k+{p}_{1}+{p}_{2}\right)}^{2}-{m}_{2}^{2}+{\rm{i}}\varepsilon \right]\left[{\left(k+{p}_{1}+{p}_{2}+{p}_{3}\right)}^{2}-{m}_{3}^{2}+{\rm{i}}\varepsilon \right]}.\end{array}\end{eqnarray}$Then, by using Feynman parameterization twice, equation (14) can be written in the form(15)$\begin{eqnarray}\begin{array}{rcl}I & = & 3\,!\displaystyle \int \displaystyle \frac{{{\rm{d}}}^{4}k}{{\left(2\pi \right)}^{4}}{\displaystyle \int }_{0}^{1}{\rm{d}}u{\displaystyle \int }_{0}^{1}{\rm{d}}x{\rm{d}}y{\rm{d}}z\,\delta (1-x-y-z)x\\ & & \times \left\{x\left[{\left(k+{{up}}_{1}\right)}^{2}+{\rm{i}}\varepsilon \right]+y\left[{\left(k+{p}_{1}+{p}_{2}\right)}^{2}\right.\right.\\ & & \left.-{m}_{2}^{2}+{\rm{i}}\varepsilon \right]\\ & & {\left.+z\left[{\left(k+{p}_{1}+{p}_{2}+{p}_{3}\right)}^{2}-{m}_{3}^{2}+{\rm{i}}\varepsilon \right]\right\}}^{-4}.\end{array}\end{eqnarray}$Sine there is infrared divergence in I, in order to carry out the integration over k, an appropriate regularization scheme must be employed. Instead of the most popular dimensional regularization, an alternative is loop regularization [47, 48]. According to the method, the loop momentum k transforms as(16)$\begin{eqnarray}\begin{array}{l}{k}^{2}\to {\left[{k}^{2}\right]}_{l}={k}^{2}-{M}_{l}^{2},\\ \displaystyle \int \displaystyle \frac{{{\rm{d}}}^{4}k}{{\left(2\pi \right)}^{4}}\to \displaystyle \int {\left[\displaystyle \frac{{{\rm{d}}}^{4}k}{{\left(2\pi \right)}^{4}}\right]}_{l}=\mathop{\mathrm{lim}}\limits_{N,{M}_{i}^{2}\to \infty }\sum _{l=0}^{N}{c}_{l}^{N}\displaystyle \int \displaystyle \frac{{{\rm{d}}}^{4}k}{{\left(2\pi \right)}^{4}},\end{array}\end{eqnarray}$which is constrained by(17)$\begin{eqnarray}\begin{array}{l}\mathop{\mathrm{lim}}\limits_{N,{M}_{i}^{2}\to \infty }\sum _{l=0}^{N}{c}_{l}^{N}{\left({M}_{l}^{2}\right)}^{2}=0,\quad {c}_{0}^{N}=0\\ (i=0,1,\ldots ,N\,\mathrm{and}\,n=0,1,..).\end{array}\end{eqnarray}$From equation (17) the coefficients c_l^N can be worked out:$\begin{eqnarray*}{c}_{l}^{N}={\left(-1\right)}^{l}\displaystyle \frac{N!}{l!(N-l)!},\end{eqnarray*}$and the regulator mass is given by(18)$\begin{eqnarray}{M}_{l}^{2}={\mu }_{s}^{2}+{{lM}}_{R}^{2}.\end{eqnarray}$This then leads to the desired integration form over k:(19)$\begin{eqnarray}\begin{array}{l}{k}^{2}\to {k}^{2}-{\mu }_{s}^{2}-{{lM}}_{R}^{2},\\ \displaystyle \int \displaystyle \frac{{{\rm{d}}}^{4}k}{{\left(2\pi \right)}^{4}}\to \mathop{\mathrm{lim}}\limits_{N,{M}_{R}^{2}\to \infty }\sum _{l=0}^{N}{\left(-1\right)}^{l}\displaystyle \frac{N!}{l!(N-l)!}\displaystyle \int \displaystyle \frac{{{\rm{d}}}^{4}k}{{\left(2\pi \right)}^{4}}.\end{array}\end{eqnarray}$If there is only infrared divergence, when the integration over loop momentum is completed, terms involving M_R will vanish after taking the limit; thus in this case it amounts to introduce a characteristic scale μ_s in the amplitudes. With this in mind, after the integration over k is performed, we obtain(20)$\begin{eqnarray}\begin{array}{rcl}I & = & \displaystyle \frac{{\rm{i}}}{{\left(4\pi \right)}^{2}}{\displaystyle \int }_{0}^{1}{\rm{d}}u{\displaystyle \int }_{0}^{1}{\rm{d}}x{\rm{d}}y{\rm{d}}z\,x\delta (1-x-y-z)\\ & & \times \displaystyle \frac{1}{{\left[Q(u,x,y,z)\right]}^{2}},\end{array}\end{eqnarray}$where Q is a polynomial defined as(21)$\begin{eqnarray}\begin{array}{c}Q(u,x,y,z)={s}_{12}^{2}{y}^{2}+{\omega }_{4}^{2}{z}^{2}+({s}_{12}^{2}+{\omega }_{4}^{2}-{\omega }_{3}^{2}){yz}\\ \quad +\,\left[y({s}_{12}^{2}-{\omega }_{2}^{2})+z({\omega }_{4}^{2}-{s}_{23}^{2})\right]{ux}-y({s}_{12}^{2}-{m}_{2}^{2})\\ \quad -z({\omega }_{4}^{2}-{m}_{3}^{2})-{\mu }_{s}^{2}-{\rm{i}}\varepsilon .\end{array}\end{eqnarray}$The integration over u is trivial and thus we do it first:(22)$\begin{eqnarray}\begin{array}{rcl}I & = & \displaystyle \frac{-{\rm{i}}}{{\left(4\pi \right)}^{2}}{\displaystyle \int }_{0}^{1}{\rm{d}}x{\rm{d}}y{\rm{d}}z\,\delta (1-x-y-z)\displaystyle \frac{1}{B(y,z)}\\ & & \times \left[\displaystyle \frac{1}{Q(1,x,y,z)}-\displaystyle \frac{1}{Q(0,x,y,z)}\right],\end{array}\end{eqnarray}$where(23)$\begin{eqnarray}B(y,z)=y({s}_{12}^{2}-{\omega }_{2}^{2})+z({\omega }_{4}^{2}-{s}_{23}^{2}).\end{eqnarray}$The integration over z can be performed immediately; it leads to(24)$\begin{eqnarray}\begin{array}{rcl}I & = & \displaystyle \frac{-{\rm{i}}}{{\left(4\pi \right)}^{2}}{\displaystyle \int }_{0}^{1}{\rm{d}}x{\displaystyle \int }_{0}^{1-x}{\rm{d}}y\,\displaystyle \frac{1}{B(y,1-x-y)}\\ & \times & \left[\displaystyle \frac{1}{Q(1,x,y,1-x-y)}-\displaystyle \frac{1}{Q(0,x,y,1-x-y)}\right].\end{array}\end{eqnarray}$To proceed we make the following transformation on x and y:(25)$\begin{eqnarray}x=1-x^{\prime} ,\quad \quad y=x^{\prime} -y^{\prime} ,\end{eqnarray}$which yields a convenient form for equation (24):(26)$\begin{eqnarray}\begin{array}{rcl}I & = & \displaystyle \frac{-{\rm{i}}}{{\left(4\pi \right)}^{2}}{\displaystyle \int }_{0}^{1}{\rm{d}}x^{\prime} {\displaystyle \int }_{0}^{x^{\prime} }{\rm{d}}y^{\prime} \,\displaystyle \frac{1}{C(x^{\prime} ,y^{\prime} )}\\ & & \times \left[\displaystyle \frac{1}{{W}_{1}(x^{\prime} ,y^{\prime} )}-\displaystyle \frac{1}{{W}_{0}(x^{\prime} ,y^{\prime} )}\right],\end{array}\end{eqnarray}$where(27)$\begin{eqnarray}\begin{array}{rcl}C(x^{\prime} ,y^{\prime} ) & = & ({s}_{12}^{2}-{\omega }_{2}^{2})x^{\prime} +({s}_{13}^{2}-{\omega }_{3}^{2})y^{\prime} ,\\ {W}_{1}(x^{\prime} ,y^{\prime} ) & = & {\omega }_{2}^{2}x{{\prime} }^{2}+{\omega }_{3}^{2}y{{\prime} }^{2}+({s}_{12}^{2}-{\omega }_{2}^{2}-{\omega }_{3}^{2})x^{\prime} y^{\prime} -{\omega }_{2}^{2}x^{\prime} \\ & & +({m}_{3}^{2}-{m}_{2}^{2}-{s}_{23}^{2}+{\omega }_{2}^{2})y^{\prime} -{\mu }_{s}^{2}-{\rm{i}}\varepsilon ,\\ {W}_{0}(x^{\prime} ,y^{\prime} ) & = & {s}_{12}^{2}x{{\prime} }^{2}+{\omega }_{3}^{2}y{{\prime} }^{2}+({\omega }_{4}^{2}-{s}_{12}^{2}-{\omega }_{3}^{2})x^{\prime} y^{\prime} \\ & & -({s}_{12}^{2}-{m}_{2}^{2})x^{\prime} +({s}_{12}^{2}-{m}_{2}^{2}-{\omega }_{4}^{2}+{m}_{3}^{2})y^{\prime} \\ & & -{\mu }_{s}^{2}-{\rm{i}}\varepsilon .\end{array}\end{eqnarray}$For later convenience we split I into two parts:(28)$\begin{eqnarray}I={I}_{1}+{I}_{0},\end{eqnarray}$where the two components are(29)$\begin{eqnarray}\begin{array}{rcl}{I}_{1} & = & \displaystyle \frac{-{\rm{i}}}{{\left(4\pi \right)}^{2}}{\displaystyle \int }_{0}^{1}{\rm{d}}x^{\prime} {\displaystyle \int }_{0}^{x^{\prime} }{\rm{d}}y^{\prime} \,\displaystyle \frac{1}{C(x^{\prime} ,y^{\prime} ){W}_{1}(x^{\prime} ,y^{\prime} )},\\ {I}_{0} & = & \displaystyle \frac{{\rm{i}}}{{\left(4\pi \right)}^{2}}{\displaystyle \int }_{0}^{1}{\rm{d}}x^{\prime} {\displaystyle \int }_{0}^{x^{\prime} }{\rm{d}}y^{\prime} \,\displaystyle \frac{1}{C(x^{\prime} ,y^{\prime} ){W}_{0}(x^{\prime} ,y^{\prime} )}.\end{array}\end{eqnarray}$We will evaluate I₁ and I₀ in the forthcoming two subsections, respectively.

3.2. Evaluation of I₁

Since the denominator W₁ is quadratic both in $x^{\prime} $ and $y^{\prime} $, in order to cope with terms involving x², we perform the so-called Euler shift on $y^{\prime} $:(30)$\begin{eqnarray}y^{\prime} =\rho +\alpha x^{\prime} ,\end{eqnarray}$such that(31)$\begin{eqnarray}\begin{array}{l}{\displaystyle \int }_{0}^{1}{\rm{d}}x^{\prime} {\displaystyle \int }_{0}^{x^{\prime} }{\rm{d}}y^{\prime} ={\displaystyle \int }_{0}^{1-\alpha }{\rm{d}}\rho \\ {\displaystyle \int }_{\rho /(1-\alpha )}^{1}{\rm{d}}x^{\prime} -{\displaystyle \int }_{0}^{-\alpha }{\rm{d}}\rho {\displaystyle \int }_{-\rho /\alpha }^{1}{\rm{d}}x^{\prime} ,\end{array}\end{eqnarray}$where the parameter α is chosen to obey the condition(32)$\begin{eqnarray}{\omega }_{3}^{2}{\alpha }^{2}+\alpha ({s}_{23}^{2}-{\omega }_{2}^{2}-{\omega }_{3}^{2})+{\omega }_{2}^{2}=0.\end{eqnarray}$From equation (32) we find(33)$\begin{eqnarray}\alpha =\displaystyle \frac{1}{2{\omega }_{3}^{2}}\left[-({s}_{23}^{2}-{\omega }_{2}^{2}-{\omega }_{3}^{2})\pm {\lambda }^{1/2}({s}_{23}^{2},{\omega }_{2}^{2},{\omega }_{3}^{2})\right],\end{eqnarray}$where λ(x, y, z) is the well-known Källén function(34)$\begin{eqnarray}\lambda (x,y,z)={\left(x+y+z\right)}^{2}-2{xy}-2{yz}-2{xz}.\end{eqnarray}$In our evaluation we take(35)$\begin{eqnarray}\alpha =\frac{1}{2{\omega }_{3}^{2}}\left[-({s}_{23}^{2}-{\omega }_{2}^{2}-{\omega }_{3}^{2})+{\lambda }^{1/2}({s}_{23}^{2},{\omega }_{2}^{2},{\omega }_{3}^{2})\right]\end{eqnarray}$and assume 0 < α < 1. According to equation (32), all the $x{{\prime} }^{2}$-dependent terms vanish. Now that W₁ is linear in $x^{\prime} $, we denote it as(36)$\begin{eqnarray}{Y}_{1}(x^{\prime} ,\rho )={f}_{1}(\rho )x^{\prime} +{f}_{2}(\rho ),\end{eqnarray}$where f₁ and f₂ only depend on ρ; the explicit expressions are²(²Notice that the lower indices of f and g tell us the maximum power of ρ.)(37)$\begin{eqnarray}\begin{array}{ccc}{f}_{1}(\rho ) & = & {A}_{1}\rho +{B}_{1},\\ {f}_{2}(\rho ) & = & {A}_{2}{\rho }^{2}+{B}_{2}\rho -{\mu }_{s}^{2}-{\rm{i}}\varepsilon .\end{array}\end{eqnarray}$All the coefficients in equation (37) are constants which are formed by on-shell masses in equation (4) and propagator masses m_i(i = 2, 3) as well as combinations of external momentum s_ij in equation (5):(38)$\begin{eqnarray}\begin{array}{rcl}{A}_{1} & = & {s}_{23}^{2}+(2\alpha -1){\omega }_{3}^{2}-{\omega }_{2}^{2},\\ {B}_{1} & = & ({m}_{2}^{2}-{\omega }_{2}^{2})+\alpha ({m}_{3}^{2}-{m}_{2}^{2}+{\omega }_{2}^{2}-{s}_{23}^{2}),\\ {A}_{2} & = & {\omega }_{3}^{2},\\ {B}_{2} & = & {m}_{3}^{2}-{m}_{2}^{2}+{\omega }_{2}^{2}-{s}_{23}^{2},\\ {D}_{1} & = & {s}_{13}^{2}-{\omega }_{3}^{2}.\end{array}\end{eqnarray}$Under the transformation in equation (30), the function $C(x^{\prime} ,y^{\prime} )$ becomes(39)$\begin{eqnarray}X(x^{\prime} ,\rho )={g}_{0}(\alpha )x^{\prime} +{g}_{1}(\rho ),\end{eqnarray}$where g₀(α) is constant and g₁(ρ) is linear in ρ:(40)$\begin{eqnarray}{g}_{0}(\alpha )={s}_{12}^{2}-{\omega }_{2}^{2}+\alpha ({s}_{13}^{2}-{\omega }_{3}^{2}),\quad {g}_{1}(\rho )={D}_{1}\rho .\end{eqnarray}$This yields a compact form for I₁, which reads(41)$\begin{eqnarray}\begin{array}{rcl}{I}_{1} & = & \displaystyle \frac{-{\rm{i}}}{{\left(4\pi \right)}^{2}}\left[{\displaystyle \int }_{0}^{1-\alpha }{\rm{d}}\rho {\displaystyle \int }_{\rho /(1-\alpha )}^{1}{\rm{d}}x^{\prime} \right.\\ & & \left.-{\displaystyle \int }_{0}^{-\alpha }{\rm{d}}\rho {\displaystyle \int }_{-\rho /\alpha }^{1}{\rm{d}}x^{\prime} \right]\displaystyle \frac{1}{({g}_{0}x^{\prime} +{g}_{1})({f}_{1}x^{\prime} +{f}_{2})}\\ & = & \displaystyle \frac{-{\rm{i}}}{{\left(4\pi \right)}^{2}}\left[{\displaystyle \int }_{0}^{1-\alpha }{\rm{d}}\rho {\displaystyle \int }_{\rho /(1-\alpha )}^{1}{\rm{d}}x^{\prime} \right.\\ & & \left.-{\displaystyle \int }_{0}^{-\alpha }{\rm{d}}\rho {\displaystyle \int }_{-\rho /\alpha }^{1}{\rm{d}}x^{\prime} \right]\displaystyle \frac{1}{{g}_{0}{f}_{2}-{g}_{1}{f}_{1}}\\ & \times & \left(\displaystyle \frac{{g}_{0}}{{g}_{0}x^{\prime} +{g}_{1}}-\displaystyle \frac{{f}_{1}}{{f}_{1}x^{\prime} +{f}_{2}}\right)\\ & = & {I}_{A}+{I}_{B},\end{array}\end{eqnarray}$where we have split I₁ into two parts:(42)$\begin{eqnarray}\begin{array}{rcl}{I}_{A} & = & \displaystyle \frac{-{\rm{i}}}{{\left(4\pi \right)}^{2}}{\displaystyle \int }_{0}^{1-\alpha }{\rm{d}}\rho {\displaystyle \int }_{\rho /(1-\alpha )}^{1}{\rm{d}}x^{\prime} \displaystyle \frac{1}{{g}_{0}{f}_{2}-{g}_{1}{f}_{1}}\\ & & \times \left(\displaystyle \frac{{g}_{0}}{{g}_{0}x^{\prime} +{g}_{1}}-\displaystyle \frac{{f}_{1}}{{f}_{1}x^{\prime} +{f}_{2}}\right),\end{array}\end{eqnarray}$(43)$\begin{eqnarray}\begin{array}{ccc}{I}_{B} & = & \frac{{\rm{i}}}{{\left(4\pi \right)}^{2}}{\int }_{0}^{-\alpha }{\rm{d}}\rho {\int }_{-\rho /\alpha }^{1}{\rm{d}}x^{\prime} \frac{1}{{g}_{0}{f}_{2}-{g}_{1}{f}_{1}}\\ & & \times \left(\frac{{g}_{0}}{{g}_{0}x^{\prime} +{g}_{1}}-\frac{{f}_{1}}{{f}_{1}x^{\prime} +{f}_{2}}\right).\end{array}\end{eqnarray}$The integration over $x^{\prime} $ in equations (42) and (43) is elementary and we can carry out it immediately; the results are(44)$\begin{eqnarray}\begin{array}{rcl}{I}_{A} & = & \displaystyle \frac{-{\rm{i}}}{{\left(4\pi \right)}^{2}}{\displaystyle \int }_{0}^{1-\alpha }{\rm{d}}\rho \displaystyle \frac{1}{{g}_{0}{f}_{2}-{g}_{1}{f}_{1}}\left[\mathrm{ln}\left({g}_{0}+{g}_{1}\right)\right.\\ & & -\mathrm{ln}\left(\displaystyle \frac{\rho }{1-\alpha }\,{g}_{0}+{g}_{1}\right)\\ & & \left.-\mathrm{ln}\left({f}_{1}+{f}_{2}\right)+\mathrm{ln}\left(\displaystyle \frac{\rho }{1-\alpha }\,{f}_{1}+{f}_{2}\right)\right],\end{array}\end{eqnarray}$and(45)$\begin{eqnarray}\begin{array}{rcl}{I}_{B} & = & \displaystyle \frac{{\rm{i}}}{{\left(4\pi \right)}^{2}}{\displaystyle \int }_{0}^{-\alpha }{\rm{d}}\rho \displaystyle \frac{1}{{g}_{0}{f}_{2}-{g}_{1}{f}_{1}}\left[\mathrm{ln}\left({g}_{0}+{g}_{1}\right)\right.\\ & & -\mathrm{ln}\left(-\displaystyle \frac{\rho }{\alpha }\,{g}_{0}+{g}_{1}\right)\\ & & \left.-\mathrm{ln}\left({f}_{1}+{f}_{2}\right)+\mathrm{ln}\left(-\displaystyle \frac{\rho }{\alpha }\,{f}_{1}+{f}_{2}\right)\right].\end{array}\end{eqnarray}$

Then we divide the upper limit of the integral of I_A into two parts:(46)$\begin{eqnarray}\begin{array}{c}{I}_{1}=\frac{-{\rm{i}}}{{\left(4\pi \right)}^{2}}\left({\int }_{0}^{1}{\rm{d}}\rho +{\int }_{0}^{-\alpha }{\rm{d}}\rho \right)\frac{1}{{g}_{0}{f}_{2}-{g}_{1}{f}_{1}}\\ \left[\mathrm{ln}\left({g}_{0}+{g}_{1}\right)-\mathrm{ln}\left(\frac{\rho }{1-\alpha }\,{g}_{0}+{g}_{1}\right)\right.\\ \left.-\mathrm{ln}\left({f}_{1}+{f}_{2}\right)+\mathrm{ln}\left(\frac{\rho }{1-\alpha }\,{f}_{1}+{f}_{2}\right)\right].\end{array}\end{eqnarray}$Combined with equation (45), after some cancellation we find(47)$\begin{eqnarray}{I}_{1}=\frac{-{\rm{i}}}{{\left(4\pi \right)}^{2}}({I}_{11}+{I}_{12}+{I}_{13}).\end{eqnarray}$The first component I₁₁ is given by(48)$\begin{eqnarray}\begin{array}{rcl}{I}_{11} & = & {\displaystyle \int }_{0}^{1}{\rm{d}}\rho \,\displaystyle \frac{1}{{g}_{0}{f}_{2}-{g}_{1}{f}_{1}}\left[\mathrm{ln}({g}_{0}+{g}_{1})-\mathrm{ln}({f}_{1}+{f}_{2})\right]\\ & = & {\displaystyle \int }_{0}^{1}{\rm{d}}\rho \displaystyle \frac{1}{{F}_{11}(\rho )}\left[\mathrm{ln}({g}_{0}+{g}_{1})-\mathrm{ln}{G}_{11}(\rho )\right],\end{array}\end{eqnarray}$where(49)$\begin{eqnarray}\begin{array}{ccc}{F}_{11}(\rho ) & = & ({A}_{2}{g}_{0}-{A}_{1}{D}_{1}){\rho }^{2}+({B}_{2}{g}_{0}-{B}_{1}{D}_{1})\rho \\ & & -{g}_{0}({\mu }_{s}^{2}+{\rm{i}}\varepsilon ),\\ {G}_{11}(\rho ) & = & {A}_{2}{\rho }^{2}+({A}_{1}+{B}_{2})\rho +{B}_{1}-{\mu }_{s}^{2}-{\rm{i}}\varepsilon .\end{array}\end{eqnarray}$The other two components are(50)$\begin{eqnarray}\begin{array}{rcl}{I}_{12} & = & {\displaystyle \int }_{0}^{1-\alpha }{\rm{d}}\rho \,\displaystyle \frac{1}{{g}_{0}{f}_{2}-{g}_{1}{f}_{1}}\left[-\mathrm{ln}\left(\displaystyle \frac{\rho }{1-\alpha }{g}_{0}+{g}_{1}\right)\right.\\ & & \left.+\mathrm{ln}\left(\displaystyle \frac{\rho }{1-\alpha }{f}_{1}+{f}_{2}\right)\right],\end{array}\end{eqnarray}$(51)$\begin{eqnarray}\begin{array}{ccc}{I}_{13} & = & {\int }_{0}^{-\alpha }{\rm{d}}\rho \,\frac{1}{{g}_{0}{f}_{2}-{g}_{1}{f}_{1}}\left[\mathrm{ln}\left(-\frac{\rho }{\alpha }{g}_{0}+{g}_{1}\right)\right.\\ & & \left.-\mathrm{ln}\left(-\frac{\rho }{\alpha }{f}_{1}+{f}_{2}\right)\right].\end{array}\end{eqnarray}$In order to regularize the upper limit of the integral in equation (50), we make the following variable substitution:(52)$\begin{eqnarray}\rho =(1-\alpha )\xi .\end{eqnarray}$Without confusion, we relabel ξ as ρ; then the integral takes the form(53)$\begin{eqnarray}\begin{array}{rcl}{I}_{12} & = & (1-\alpha ){\displaystyle \int }_{0}^{1}{\rm{d}}\rho \,\displaystyle \frac{1}{{F}_{12}(\rho )}\left\{-\mathrm{ln}\left[\left({g}_{0}\right.\right.\right.\\ & & \left.\left.\left.+(1-\alpha ){D}_{1}\right)\rho \right]+\mathrm{ln}{G}_{12}(\rho )\right\},\end{array}\end{eqnarray}$where(54)$\begin{eqnarray}\begin{array}{ccc}{F}_{12}(\rho ) & = & {\left(1-\alpha \right)}^{2}({A}_{2}{g}_{0}-{A}_{1}{D}_{1}){\rho }^{2}\\ & & +(1-\alpha )({B}_{2}{g}_{0}-{B}_{1}{D}_{1})\rho -{g}_{0}({\mu }_{s}^{2}+{\rm{i}}\varepsilon ),\\ {G}_{12}(\rho ) & = & (1-\alpha )[{A}_{1}+(1-\alpha ){A}_{2}]{\rho }^{2}\\ & & +[{B}_{1}+(1-\alpha ){B}_{2}]\rho -{\mu }_{s}^{2}-{\rm{i}}\varepsilon .\end{array}\end{eqnarray}$Similarly, we make the following transformation in equation (51):(55)$\begin{eqnarray}\rho =-\alpha \xi ,\end{eqnarray}$and relabel ξ as ρ, which leads to(56)$\begin{eqnarray}{I}_{13}=-\alpha {\int }_{0}^{1}{\rm{d}}\rho \,\displaystyle \frac{1}{{F}_{13}(\rho )}\left\{\mathrm{ln}\left[({g}_{0}-{D}_{1}\alpha )\rho \right]-\mathrm{ln}{G}_{13}(\rho )\right\},\end{eqnarray}$where(57)$\begin{eqnarray}\begin{array}{rcl}{F}_{13}(\rho ) & = & {\alpha }^{2}({A}_{2}{g}_{0}-{A}_{1}{D}_{1}){\rho }^{2}\\ & & +\alpha ({B}_{1}{D}_{1}-{B}_{2}{g}_{0})\rho -{g}_{0}({\mu }_{s}^{2}+{\rm{i}}\varepsilon ),\\ {G}_{13}(\rho ) & = & \alpha (\alpha {A}_{2}-{A}_{1}){\rho }^{2}\\ & & -(\alpha {B}_{2}-{B}_{1})\rho -{\mu }_{s}^{2}-{\rm{i}}\varepsilon .\end{array}\end{eqnarray}$The details of evaluating of I₁₁, I₁₂ and I₁₃ are presented in appendix C.

3.3. Evaluation of I₀

The original expression of I₀ is(58)$\begin{eqnarray}{I}_{0}=\frac{{\rm{i}}}{{\left(4\pi \right)}^{2}}{\int }_{0}^{1}{\rm{d}}x^{\prime} {\int }_{0}^{x\mbox{'}}{\rm{d}}y^{\prime} \,\frac{1}{C(x^{\prime} ,y^{\prime} ){W}_{0}(x^{\prime} ,y^{\prime} )}.\end{eqnarray}$To eliminate the awkward term depending on $x{{\prime} }^{2}$, we also make the Euler shift on $y^{\prime} $:(59)$\begin{eqnarray}y^{\prime} =\rho +\beta x^{\prime} ,\end{eqnarray}$such that(60)$\begin{eqnarray}\begin{array}{l}{\displaystyle \int }_{0}^{1}{\rm{d}}x^{\prime} {\displaystyle \int }_{0}^{x^{\prime} }{\rm{d}}y^{\prime} ={\displaystyle \int }_{0}^{1-\beta }{\rm{d}}\rho {\displaystyle \int }_{\rho /(1-\beta )}^{1}{\rm{d}}x^{\prime} \\ -{\displaystyle \int }_{0}^{-\beta }{\rm{d}}\rho {\displaystyle \int }_{-\rho /\beta }^{1}{\rm{d}}x^{\prime} ,\end{array}\end{eqnarray}$where β is chosen to obey the condition(61)$\begin{eqnarray}{\omega }_{3}^{2}{\beta }^{2}+({\omega }_{4}^{2}-{s}_{12}^{2}-{\omega }_{3}^{2})\beta +{s}_{12}^{2}=0,\end{eqnarray}$which renders that all the $x{{\prime} }^{2}$-dependent terms vanish. The roots of equation (61) are(62)$\begin{eqnarray}\beta =\displaystyle \frac{1}{2{\omega }_{3}^{2}}\left[-({\omega }_{4}^{2}-{s}_{12}^{2}-{\omega }_{3}^{2})\pm {\lambda }^{1/2}({\omega }_{4}^{2},{s}_{12}^{2},{\omega }_{3}^{2})\right],\end{eqnarray}$where λ(x, y, z) is the Källén function defined in equation (34). In our evaluation we take(63)$\begin{eqnarray}\beta =\displaystyle \frac{1}{2{\omega }_{3}^{2}}\left[-({\omega }_{4}^{2}-{s}_{12}^{2}-{\omega }_{3}^{2})+{\lambda }^{1/2}({\omega }_{4}^{2},{s}_{12}^{2},{\omega }_{3}^{2})\right],\end{eqnarray}$and assume 0 < β < 1. Accordingly, W₀ is linear in $x^{\prime} $, and we denote it as(64)$\begin{eqnarray}{Y}_{0}(x^{\prime} ,\rho )={h}_{1}(\rho )x^{\prime} +{h}_{2}(\rho ),\end{eqnarray}$where the two functions h₁(ρ) and h₂(ρ) only depend on ρ; they are given by(65)$\begin{eqnarray}\begin{array}{rcl}{h}_{1}(\rho ) & = & {K}_{1}\rho +{N}_{1},\\ {h}_{2}(\rho ) & = & {K}_{2}{\rho }^{2}+{N}_{2}\rho -{\mu }_{s}^{2}-{\rm{i}}\varepsilon ,\end{array}\end{eqnarray}$where all the coefficients in equation (65) are constants:(66)$\begin{eqnarray}\begin{array}{rcl}{K}_{1} & = & 2(\beta -1){\omega }_{3}^{2}+{s}_{13}^{2}+{s}_{23}^{2}-{\omega }_{2}^{2},\\ {N}_{1} & = & (\beta -1)({s}_{12}^{2}-{m}_{2}^{2})-\beta ({\omega }_{4}^{2}-{m}_{3}^{2}),\\ {K}_{2} & = & {\omega }_{3}^{2},\\ {N}_{2} & = & {s}_{12}^{2}-{m}_{2}^{2}-{\omega }_{4}^{2}+{m}_{3}^{2},\end{array}\end{eqnarray}$while $C(x^{\prime} ,y^{\prime} )$ transforms into(67)$\begin{eqnarray}X(x^{\prime} ,\rho )={g}_{0}(\beta )x^{\prime} +{g}_{1}(\rho ),\end{eqnarray}$where g₀(β) and g₁(ρ) are given by(68)$\begin{eqnarray}\begin{array}{ccc}{g}_{0}(\beta ) & = & {s}_{12}^{2}-{\omega }_{2}^{2}+\beta ({s}_{13}^{2}-{\omega }_{3}^{2}),\\ {g}_{1}(\rho ) & = & {D}_{1}\rho .\end{array}\end{eqnarray}$Then we rewrite I₀ in a compact form:(69)$\begin{eqnarray}\begin{array}{rcl}{I}_{0} & = & \displaystyle \frac{{\rm{i}}}{{\left(4\pi \right)}^{2}}\left[{\displaystyle \int }_{0}^{1-\beta }{\rm{d}}\rho {\displaystyle \int }_{\rho /(1-\beta )}^{1}{\rm{d}}x^{\prime} -{\displaystyle \int }_{0}^{-\beta }{\rm{d}}\rho {\displaystyle \int }_{-\rho /\beta }^{1}{\rm{d}}x^{\prime} \right]\\ & & \times \displaystyle \frac{1}{({g}_{0}x^{\prime} +{g}_{1})({h}_{1}x^{\prime} +{h}_{2})}=\displaystyle \frac{-{\rm{i}}}{{\left(4\pi \right)}^{2}}\\ & & \times \left[{\displaystyle \int }_{0}^{1-\beta }{\rm{d}}\rho {\displaystyle \int }_{\rho /(1-\beta )}^{1}{\rm{d}}x^{\prime} -{\displaystyle \int }_{0}^{-\beta }{\rm{d}}\rho {\displaystyle \int }_{-\rho /\beta }^{1}{\rm{d}}x^{\prime} \right]\\ & & \times \displaystyle \frac{1}{{g}_{0}{h}_{2}-{g}_{1}{h}_{1}}\left(\displaystyle \frac{{g}_{0}}{{g}_{0}x^{\prime} +{g}_{1}}-\displaystyle \frac{{h}_{1}}{{h}_{1}x^{\prime} +{h}_{2}}\right).\end{array}\end{eqnarray}$After the trivial integration over $x^{\prime} $ is performed, we find(70)$\begin{eqnarray}{I}_{0}={I}_{C}+{I}_{D},\end{eqnarray}$where(71)$\begin{eqnarray}\begin{array}{rcl}{I}_{C} & = & \displaystyle \frac{{\rm{i}}}{{\left(4\pi \right)}^{2}}{\displaystyle \int }_{0}^{1-\beta }{\rm{d}}\rho \,\displaystyle \frac{1}{{g}_{0}{h}_{2}-{g}_{1}{h}_{1}}\left[\mathrm{ln}\left({g}_{0}+{g}_{1}\right)\right.\\ & & -\mathrm{ln}\left(\displaystyle \frac{\rho }{1-\beta }{g}_{0}+{g}_{1}\right)\\ & & \left.-\mathrm{ln}\left({h}_{1}+{h}_{2}\right)+\mathrm{ln}\left(\displaystyle \frac{\rho }{1-\beta }{h}_{1}+{h}_{2}\right)\right]\\ & = & \displaystyle \frac{-{\rm{i}}}{{\left(4\pi \right)}^{2}}\left({\displaystyle \int }_{0}^{1}{\rm{d}}\rho \,+{\displaystyle \int }_{0}^{-\beta }{\rm{d}}\rho \right)\displaystyle \frac{1}{{g}_{0}{h}_{2}-{g}_{1}{h}_{1}}\\ & & \left[\mathrm{ln}\left({g}_{0}+{g}_{1}\right)-\mathrm{ln}\left(\displaystyle \frac{\rho }{1-\beta }{g}_{0}+{g}_{1}\right)\right.\\ & & \left.-\mathrm{ln}\left({h}_{1}+{h}_{2}\right)+\mathrm{ln}\left(\displaystyle \frac{\rho }{1-\beta }{h}_{1}+{h}_{2}\right)\right],\end{array}\end{eqnarray}$and(72)$\begin{eqnarray}\begin{array}{rcl}{I}_{D} & = & \displaystyle \frac{{\rm{i}}}{{\left(4\pi \right)}^{2}}{\displaystyle \int }_{0}^{-\beta }{\rm{d}}\rho \,\displaystyle \frac{1}{{g}_{0}{h}_{2}-{g}_{1}{h}_{1}}\\ & & \times \left[\mathrm{ln}({g}_{0}+{g}_{1})-\mathrm{ln}\left(-\displaystyle \frac{\rho }{\beta }{g}_{0}+{g}_{1}\right)\right.\\ & & \left.-\mathrm{ln}({h}_{1}+{h}_{2})+\mathrm{ln}\left(-\displaystyle \frac{\rho }{\beta }{h}_{1}+{h}_{2}\right)\right].\end{array}\end{eqnarray}$After some cancellation, we have(73)$\begin{eqnarray}{I}_{0}=\frac{{\rm{i}}}{{\left(4\pi \right)}^{2}}({I}_{01}+{I}_{02}+{I}_{03}).\end{eqnarray}$The first component is(74)$\begin{eqnarray}\begin{array}{rcl}{I}_{01} & = & {\displaystyle \int }_{0}^{1}{\rm{d}}\rho \,\displaystyle \frac{1}{{g}_{0}{h}_{2}-{g}_{1}{h}_{1}}\left[\mathrm{ln}({g}_{0}+{g}_{1})-\mathrm{ln}({h}_{1}+{h}_{2})\right]\\ & = & {\displaystyle \int }_{0}^{1}{\rm{d}}u\,\displaystyle \frac{1}{{F}_{01}(\rho )}\left[\mathrm{ln}\left({D}_{1}u+{g}_{0}\right)-\mathrm{ln}{G}_{01}(\rho )\right],\end{array}\end{eqnarray}$where(75)$\begin{eqnarray}\begin{array}{rcl}{F}_{01}(\rho ) & = & ({K}_{2}{g}_{0}-{K}_{1}{D}_{1}){\rho }^{2}\\ & & +({g}_{0}{N}_{2}-{N}_{1}{D}_{1})\rho -{g}_{0}({\mu }_{s}^{2}+{\rm{i}}\varepsilon ),\\ {G}_{01}(\rho ) & = & {K}_{2}{\rho }^{2}+({K}_{1}+{N}_{2})\rho +{N}_{1}-{\mu }_{s}^{2}-{\rm{i}}\varepsilon .\end{array}\end{eqnarray}$The second component is(76)$\begin{eqnarray}\begin{array}{ccc}{I}_{02} & = & {\int }_{0}^{1-\beta }{\rm{d}}\rho \,\frac{1}{{g}_{0}{h}_{2}-{g}_{1}{h}_{1}}\left[-\mathrm{ln}\left(\frac{\rho }{1-\beta }{g}_{0}+{g}_{1}\right)\right.\\ & & \left.+\mathrm{ln}\left(\frac{\rho }{1-\beta }{h}_{1}+{h}_{2}\right)\right].\end{array}\end{eqnarray}$We make the following transform on I₀₂:(77)$\begin{eqnarray}\rho =(1-\beta )\xi ,\end{eqnarray}$and relabeling ξ as ρ, we obtain(78)$\begin{eqnarray}\begin{array}{rcl}{I}_{02} & = & (1-\beta ){\displaystyle \int }_{0}^{1}{\rm{d}}\rho \,\displaystyle \frac{1}{{F}_{02}(\rho )}\left\{-\mathrm{ln}\left[\left({g}_{0}\right.\right.\right.\\ & & \left.\left.\left.+(1-\beta ){D}_{1}\right)\rho \right]+\mathrm{ln}{G}_{02}(\rho )\right\},\end{array}\end{eqnarray}$where(79)$\begin{eqnarray}\begin{array}{rcl}{F}_{02}(\rho ) & = & {\left(1-\beta \right)}^{2}({K}_{2}{g}_{0}-{K}_{1}{D}_{1}){\rho }^{2}\\ & & -(1-\beta )({N}_{1}{D}_{1}-{N}_{2}{g}_{0})\rho -{g}_{0}({\mu }_{s}^{2}+{\rm{i}}\varepsilon ),\\ {G}_{02}(\rho ) & = & (1-\beta )[{K}_{1}+(1-\beta ){K}_{2}]{\rho }^{2}\\ & & +[{N}_{1}+(1-\beta ){N}_{2}]\rho -{\mu }_{s}^{2}-{\rm{i}}\varepsilon .\end{array}\end{eqnarray}$The last component is(80)$\begin{eqnarray}\begin{array}{ccc}{I}_{03} & = & {\int }_{0}^{-\beta }{\rm{d}}\rho \,\frac{1}{{g}_{0}{h}_{2}-{g}_{1}{h}_{1}}\left[\mathrm{ln}\left(-\frac{\rho }{\beta }{g}_{0}+{g}_{1}\right)\right.\\ & & \left.-\mathrm{ln}\left(-\frac{u}{\beta }{h}_{1}+{h}_{2}\right)\right].\end{array}\end{eqnarray}$We make the following transformation:(81)$\begin{eqnarray}\rho =-\beta \xi ,\end{eqnarray}$and relabeling ξ as ρ,(82)$\begin{eqnarray}\begin{array}{rcl}{I}_{03} & = & -\beta {\displaystyle \int }_{0}^{1}{\rm{d}}\rho \,\displaystyle \frac{1}{{F}_{03}(\rho )}\left\{\mathrm{ln}\left[\left({g}_{0}-\beta {D}_{1}\right)\rho \right]\right.\\ & & \left.-\mathrm{ln}{G}_{03}(\rho )\right\},\end{array}\end{eqnarray}$where(83)$\begin{eqnarray}\begin{array}{rcl}{F}_{03}(\rho ) & = & {\beta }^{2}({K}_{2}{g}_{0}-{K}_{1}{D}_{1}){\rho }^{2}\\ & & +\beta ({N}_{1}{D}_{1}-{N}_{2}{g}_{0})\rho -{g}_{0}({\mu }_{s}^{2}+{\rm{i}}\varepsilon ),\\ {G}_{03}(\rho ) & = & \beta (\beta {K}_{2}-{K}_{1}){\rho }^{2}\\ & & -(\beta {N}_{2}-{N}_{1})\rho -{\mu }_{s}^{2}-{\rm{i}}\varepsilon .\end{array}\end{eqnarray}$The details of evaluating of I₀₁, I₀₂ and I₀₃ are given in appendix D.

3.4. Results and discussion

Collecting the components in the appendices C and D, we arrive at the final results for I:(84)$\begin{eqnarray}I={I}_{1}+{I}_{2},\end{eqnarray}$where the manifest expressions of I₁ and I₁ are$\begin{eqnarray*}\begin{array}{l}{I}_{1}=\displaystyle \frac{-{\rm{i}}}{{\left(4\pi \right)}^{2}}({I}_{11}+{I}_{12}+{I}_{13})\\ \quad =\,\displaystyle \frac{-{\rm{i}}}{{\left(4\pi \right)}^{2}}\displaystyle \frac{1}{{a}_{1}[{\rho }_{+}^{(1)}-{\rho }_{-}^{(1)}]}\left\{\mathrm{ln}\displaystyle \frac{{\lambda }_{1}^{2}}{{\mu }_{s}^{2}}+\mathrm{ln}\displaystyle \frac{{\lambda }_{1}^{2}}{{\mu }_{s}^{2}}\left[\mathrm{ln}\alpha -\mathrm{ln}(1-\alpha )\right]\right.\\ \quad +\,\mathrm{ln}\left|1-\displaystyle \frac{{\lambda }^{2}}{{\mu }_{s}^{2}}\right|\mathrm{ln}\displaystyle \frac{{g}_{0}+{D}_{1}{\rho }_{+}^{(1)}}{{G}_{11}(1)}\\ \quad -\,\mathrm{ln}\left|1-\displaystyle \frac{(1-\alpha ){\lambda }^{2}}{{\mu }_{s}^{2}}\right|\mathrm{ln}\displaystyle \frac{{g}_{0}+(1-\alpha ){D}_{1}}{{G}_{12}(1)}\\ \quad +\,\mathrm{ln}\left|1+\displaystyle \frac{\alpha {\lambda }_{1}^{2}}{{\mu }_{s}^{2}}\right|\mathrm{ln}\displaystyle \frac{{g}_{0}-\alpha {D}_{1}}{{G}_{13}(1)}+{\rm{i}}\pi \mathrm{ln}\displaystyle \frac{{\lambda }_{1}^{2}}{{\mu }_{s}^{2}}\\ \quad +\,{\rm{i}}\pi \left[\mathrm{ln}\displaystyle \frac{{g}_{0}+{D}_{1}{\rho }_{+}^{(1)}}{{G}_{11}(1)}-\mathrm{ln}\displaystyle \frac{{g}_{0}+(1-\alpha ){D}_{1}}{{G}_{12}(1)}\right.\\ \quad \left.-\,\mathrm{ln}(1-\alpha )\right]-\displaystyle \frac{{\pi }^{2}}{6}+\displaystyle \frac{1}{2}\left[{\mathrm{ln}}^{2}\alpha -{\mathrm{ln}}^{2}(1-\alpha )\right]\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}\quad -\,\mathrm{ln}\left[1-\displaystyle \frac{1}{{\rho }_{-}^{(1)}}\right]\mathrm{ln}\displaystyle \frac{{g}_{0}+{D}_{1}{\rho }_{+}^{(1)}}{{G}_{11}(1)}\\ \quad +\,\mathrm{ln}\left[1-\displaystyle \frac{1-\alpha }{{\rho }_{-}^{(1)}}\right]\mathrm{ln}\displaystyle \frac{{g}_{0}+(1-\alpha ){D}_{1}}{{G}_{12}(1)}\\ \quad -\,\mathrm{ln}\left[1+\displaystyle \frac{\alpha }{{\rho }_{-}^{(1)}}\right]\mathrm{ln}\displaystyle \frac{{g}_{0}-\alpha {D}_{1}}{{G}_{13}(1)}+{\mathrm{Li}}_{2}\left[\displaystyle \frac{-{D}_{1}{\rho }_{+}^{(1)}}{-{D}_{1}{\rho }_{+}^{(1)}-{g}_{0}}\right]\\ \quad -\,{\mathrm{Li}}_{2}\left[\displaystyle \frac{{D}_{1}(1-{\rho }_{+}^{(1)})}{-{D}_{1}{\rho }_{+}^{(1)}-{g}_{0}}\right]-{\mathrm{Li}}_{2}\left[\displaystyle \frac{-{D}_{1}{\rho }_{-}^{(1)}}{-{D}_{1}{\rho }_{-}^{(1)}-{g}_{0}}\right]\\ \quad +\,{\mathrm{Li}}_{2}\left[\displaystyle \frac{{D}_{1}(1-{\rho }_{-}^{(1)})}{-{D}_{1}{\rho }_{-}^{(1)}-{g}_{0}}\right]+2{\mathrm{Li}}_{2}\left(\displaystyle \frac{{\mu }_{s}^{2}}{{\lambda }_{1}^{2}}\right)\\ \quad +\,{\mathrm{Li}}_{2}\left[1-\displaystyle \frac{{\rho }_{+}^{(1)}}{1-{\rho }_{+}^{(1)}}\displaystyle \frac{{\rho }_{11}^{(1)}}{1-{\rho }_{11}^{(1)}}\right]\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}+\,{\mathrm{Li}}_{2}\left[1-\displaystyle \frac{{\rho }_{+}^{(1)}}{1-{\rho }_{+}^{(1)}}\displaystyle \frac{{\rho }_{11}^{(2)}}{1-{\rho }_{11}^{(2)}}\right]\\ \quad +\,2{\mathrm{Li}}_{2}\left[\displaystyle \frac{1}{{\rho }_{-}^{(1)}}\right]-{\mathrm{Li}}_{2}\left[1-\displaystyle \frac{{\rho }_{-}^{(1)}}{1-{\rho }_{-}^{(1)}}\displaystyle \frac{{\rho }_{11}^{(1)}}{1-{\rho }_{11}^{(1)}}\right]\\ \quad -\,{\mathrm{Li}}_{2}\left[1-\displaystyle \frac{{\rho }_{-}^{(1)}}{1-{\rho }_{-}^{(1)}}\displaystyle \frac{{\rho }_{11}^{(2)}}{1-{\rho }_{11}^{(2)}}\right]\\ \quad -\,{\mathrm{Li}}_{2}\left[\displaystyle \frac{{\mu }_{s}^{2}}{(1-\alpha ){\lambda }_{1}^{2}}\right]-{\mathrm{Li}}_{2}\left[1-\displaystyle \frac{{\rho }_{+}^{(1)}}{1-\alpha -{\rho }_{+}^{(1)}}\displaystyle \frac{{\rho }_{12}^{(1)}}{1-{\rho }_{12}^{(1)}}\right]\\ \quad -\,{\mathrm{Li}}_{2}\left[1-\displaystyle \frac{{\rho }_{+}^{(1)}}{1-\alpha -{\rho }_{+}^{(1)}}\displaystyle \frac{{\rho }_{12}^{(2)}}{1-{\rho }_{12}^{(2)}}\right]\\ \quad -\,{\mathrm{Li}}_{2}\left[\displaystyle \frac{1-\alpha }{{\rho }_{-}^{(1)}}\right]+{\mathrm{Li}}_{2}\left[1-\displaystyle \frac{{\rho }_{-}^{(1)}}{1-\alpha -{\rho }_{-}^{(1)}}\displaystyle \frac{{\rho }_{12}^{(1)}}{1-{\rho }_{12}^{(1)}}\right]\end{array}\end{eqnarray*}$(85)$\begin{eqnarray}\begin{array}{l}+\,{\mathrm{Li}}_{2}\left[1-\displaystyle \frac{{\rho }_{-}^{(1)}}{1-\alpha -{\rho }_{-}^{(1)}}\displaystyle \frac{{\rho }_{12}^{(2)}}{1-{\rho }_{12}^{(2)}}\right]\\ \quad +\,{\mathrm{Li}}_{2}\left[-\displaystyle \frac{{\mu }_{s}^{2}}{\alpha {\lambda }_{1}^{2}}\right]+{\mathrm{Li}}_{2}\left[1+\displaystyle \frac{{\rho }_{+}^{(1)}}{\alpha +{\rho }_{+}^{(1)}}\displaystyle \frac{{\rho }_{13}^{(1)}}{1-{\rho }_{13}^{(1)}}\right]\\ \quad +\,{\mathrm{Li}}_{2}\left[1+\displaystyle \frac{{\rho }_{+}^{(1)}}{\alpha +{\rho }_{+}^{(1)}}\displaystyle \frac{{\rho }_{13}^{(2)}}{1-{\rho }_{13}^{(2)}}\right]\\ \quad +\,{\mathrm{Li}}_{2}\left[-\displaystyle \frac{\alpha }{{\rho }_{-}^{(1)}}\right]-{\mathrm{Li}}_{2}\left[1+\displaystyle \frac{{\rho }_{-}^{(1)}}{\alpha +{\rho }_{-}^{(1)}}\displaystyle \frac{{\rho }_{13}^{(1)}}{1-{\rho }_{13}^{(1)}}\right]\\ \quad \left.-\,{\mathrm{Li}}_{2}\left[1+\displaystyle \frac{{\rho }_{-}^{(1)}}{\alpha +{\rho }_{-}^{(1)}}\displaystyle \frac{{\rho }_{13}^{(2)}}{1-{\rho }_{13}^{(2)}}\right]\right\},\end{array}\end{eqnarray}$and$\begin{eqnarray*}\begin{array}{l}{I}_{0}=\displaystyle \frac{{\rm{i}}}{{\left(4\pi \right)}^{2}}({I}_{01}+{I}_{02}+{I}_{03})\\ \quad =\,\displaystyle \frac{{\rm{i}}}{{\left(4\pi \right)}^{2}}\displaystyle \frac{1}{{a}_{0}[{\rho }_{+}^{(0)}-{\rho }_{-}^{(0)}]}\left\{{\mathrm{ln}}^{2}\displaystyle \frac{{\lambda }_{0}^{2}}{{\mu }_{s}^{2}}+\mathrm{ln}\displaystyle \frac{{\lambda }_{0}^{2}}{{\mu }_{s}^{2}}\right.\\ \quad \left[\mathrm{ln}\beta -\mathrm{ln}(1-\beta )\right]+\mathrm{ln}\left|1-\displaystyle \frac{{\lambda }_{0}^{2}}{{\mu }_{s}^{2}}\right|\mathrm{ln}\displaystyle \frac{{g}_{0}+{D}_{1}{\rho }_{+}^{(0)}}{{G}_{01}(1)}\\ \quad -\,\mathrm{ln}\left|1-\displaystyle \frac{(1-\beta ){\lambda }_{0}^{2}}{{\mu }_{s}^{2}}\right|\mathrm{ln}\displaystyle \frac{{g}_{0}+(1-\beta ){D}_{1}}{{G}_{02}(1)}\\ \quad +\,\mathrm{ln}\left|1+\displaystyle \frac{\beta {\lambda }_{0}^{2}}{{\mu }_{s}^{2}}\right|\mathrm{ln}\displaystyle \frac{{g}_{0}-\beta {D}_{1}}{{G}_{03}(1)}\\ \quad +\,{\rm{i}}\pi \mathrm{ln}\displaystyle \frac{{\lambda }_{0}^{2}}{{\mu }_{s}^{2}}+{\rm{i}}\pi \left[\mathrm{ln}\displaystyle \frac{{g}_{0}+{D}_{1}{\rho }_{+}^{(0)}}{{G}_{01}(1)}\right.\\ \quad \left.-\,\mathrm{ln}\displaystyle \frac{{g}_{0}+(1-\beta ){D}_{1}}{{G}_{02}(1)}-\mathrm{ln}(1-\beta )\right]\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}-\,\displaystyle \frac{{\pi }^{2}}{6}+\displaystyle \frac{1}{2}\left[{\mathrm{ln}}^{2}\beta -{\mathrm{ln}}^{2}(1-\beta )\right]\\ \quad -\,\mathrm{ln}\left(1-\displaystyle \frac{1}{{\rho }_{-}^{(0)}}\right)\mathrm{ln}\displaystyle \frac{{g}_{0}+{D}_{1}{\rho }_{-}^{(0)}}{{G}_{01}(1)}\\ \quad +\,\mathrm{ln}\left(1-\displaystyle \frac{1-\beta }{{\rho }_{-}^{(0)}}\right)\mathrm{ln}\displaystyle \frac{{g}_{0}+(1-\beta ){D}_{1}}{{G}_{02}(1)}\\ \quad -\,\mathrm{ln}\left(1+\displaystyle \frac{\beta }{{\rho }_{-}^{(0)}}\right)\mathrm{ln}\displaystyle \frac{{g}_{0}-\beta {D}_{1}}{{G}_{03}(1)}\\ \quad +\,{\mathrm{Li}}_{2}\left[\displaystyle \frac{-{D}_{1}{\rho }_{+}^{(0)}}{-{D}_{1}{\rho }_{+}^{(0)}-{g}_{0}}\right]-{\mathrm{Li}}_{2}\left[\displaystyle \frac{{D}_{1}(1-{\rho }_{+}^{(0)})}{-{D}_{1}{\rho }_{+}^{(0)}-{g}_{0}}\right]\\ \quad -\,{\mathrm{Li}}_{2}\left[\displaystyle \frac{-{D}_{1}{\rho }_{-}^{(0)}}{-{D}_{1}{\rho }_{-}^{(0)}-{g}_{0}}\right]+{\mathrm{Li}}_{2}\left[\displaystyle \frac{{D}_{1}(1-{\rho }_{-}^{(0)})}{-{D}_{1}{\rho }_{-}^{(0)}-{g}_{0}}\right]\end{array}\end{eqnarray*}$$\begin{eqnarray*}\begin{array}{l}+\,2{\mathrm{Li}}_{2}\left(\displaystyle \frac{{\mu }_{s}^{2}}{{\lambda }_{0}^{2}}\right)+{\mathrm{Li}}_{2}\left[1-\displaystyle \frac{{\rho }_{+}^{(0)}}{1-{\rho }_{+}^{(0)}}\displaystyle \frac{{\rho }_{01}^{(1)}}{1-{\rho }_{01}^{(1)}}\right]\\ \quad +\,{\mathrm{Li}}_{2}\left[1-\displaystyle \frac{{\rho }_{+}^{(0)}}{1-{\rho }_{+}^{(0)}}\displaystyle \frac{{\rho }_{01}^{(2)}}{1-{\rho }_{01}^{(2)}}\right]\\ \quad +\,2{\mathrm{Li}}_{2}\left[\displaystyle \frac{1}{{\rho }_{-}^{(0)}}\right]-{\mathrm{Li}}_{2}\left[1-\displaystyle \frac{{\rho }_{-}^{(0)}}{1-{\rho }_{-}^{(0)}}\displaystyle \frac{{\rho }_{01}^{(1)}}{1-{\rho }_{01}^{(1)}}\right]\\ \quad -\,{\mathrm{Li}}_{2}\left[1-\displaystyle \frac{{\rho }_{-}^{(0)}}{1-{\rho }_{-}^{(0)}}\displaystyle \frac{{\rho }_{01}^{(2)}}{1-{\rho }_{01}^{(2)}}\right]\\ \quad -\,{\mathrm{Li}}_{2}\left[\displaystyle \frac{{\mu }_{s}^{2}}{(1-\beta ){\lambda }_{0}^{2}}\right]-{\mathrm{Li}}_{2}\left[1-\displaystyle \frac{{\rho }_{+}^{(0)}}{1-\beta -{\rho }_{+}^{(0)}}\displaystyle \frac{{\rho }_{02}^{(1)}}{1-{\rho }_{02}^{(1)}}\right]\\ \quad -\,{\mathrm{Li}}_{2}\left[1-\displaystyle \frac{{\rho }_{+}^{(0)}}{1-\beta -{\rho }_{+}^{(0)}}\displaystyle \frac{{\rho }_{02}^{(2)}}{1-{\rho }_{02}^{(2)}}\right]\end{array}\end{eqnarray*}$(86)$\begin{eqnarray}\begin{array}{l}-\,{\mathrm{Li}}_{2}\left[\displaystyle \frac{1-\beta }{{\rho }_{-}^{(0)}}\right]+{\mathrm{Li}}_{2}\left[1-\displaystyle \frac{{\rho }_{-}^{(0)}}{1-\beta -{\rho }_{-}^{(0)}}\displaystyle \frac{{\rho }_{02}^{(1)}}{1-{\rho }_{02}^{(1)}}\right]\\ \quad +\,{\mathrm{Li}}_{2}\left[1-\displaystyle \frac{{\rho }_{-}^{(0)}}{1-\beta -{\rho }_{-}^{(0)}}\displaystyle \frac{{\rho }_{02}^{(2)}}{1-{\rho }_{02}^{(2)}}\right]\\ \quad +\,{\mathrm{Li}}_{2}\left(-\displaystyle \frac{{\mu }_{s}^{2}}{\beta {\lambda }_{0}^{2}}\right)+{\mathrm{Li}}_{2}\left[1+\displaystyle \frac{{\rho }_{+}^{(0)}}{\beta +{\rho }_{+}^{(0)}}\displaystyle \frac{{\rho }_{03}^{(1)}}{1-{\rho }_{03}^{(1)}}\right]\\ \quad +\,{\mathrm{Li}}_{2}\left[1+\displaystyle \frac{{\rho }_{+}^{(0)}}{\beta +{\rho }_{+}^{(0)}}\displaystyle \frac{{\rho }_{03}^{(2)}}{1-{\rho }_{03}^{(2)}}\right]\\ \quad +\,{\mathrm{Li}}_{2}\left[-\displaystyle \frac{\beta }{{\rho }_{-}^{(0)}}\right]-{\mathrm{Li}}_{2}\left[1+\displaystyle \frac{{\rho }_{-}^{(0)}}{\beta +{\rho }_{-}^{(0)}}\displaystyle \frac{{\rho }_{03}^{(1)}}{1-{\rho }_{03}^{(1)}}\right]\\ \quad \left.-\,{\mathrm{Li}}_{2}\left[1+\displaystyle \frac{{\rho }_{-}^{(0)}}{\beta +{\rho }_{-}^{(0)}}\displaystyle \frac{{\rho }_{03}^{(2)}}{1-{\rho }_{03}^{(2)}}\right]\right\}.\end{array}\end{eqnarray}$Combining equations (85) and (86), we obtain the final result for the integral in equation (1) in loop regularization. It is obvious that the infrared divergence appears as terms proportional to ${\mathrm{ln}}^{2}({\lambda }_{i}^{2}/{\mu }_{s}^{2})$ or $\mathrm{ln}({\lambda }_{i}^{2}/{\mu }_{s}^{2})(i=0,1)$ when the characteristic scale μ_s → 0. There are also infrared divergent and finite imaginary terms, which stem from analytic continuation of logarithms and dilogarithms by equations (6) and (10), respectively. As μ_s → 0, the dilogarithms with arguments proportional to ${\mu }_{s}^{2}$, i.e., in the form of ${\mathrm{Li}}_{2}(\gamma {\mu }_{s}^{2})$ where γ is a constant, will vanish. Since we hold the iϵ systematically in the evaluation, all the dilogarithms with two arguments of the form(87)$\begin{eqnarray}{\mathrm{Li}}_{2}(1-{x}_{1}{x}_{2}),\end{eqnarray}$can be expedient to make analytic continuation via equation (12). However, for brevity we retain the original form for every term like equation (87), otherwise there are too many terms to be listed.

It is meaningful to compare the results with some similar results obtained by the dimensional regularization scheme [44, 45]. In the dimensional regularization scheme the final result in equation (1) may be expressed in a generic form as(88)$\begin{eqnarray}\frac{A}{{\varepsilon }_{\mathrm{IR}}^{2}}+\frac{B}{{\varepsilon }_{\mathrm{IR}}}+\mathrm{infrared}\,\mathrm{finite}\,\mathrm{part},\quad {\varepsilon }_{\mathrm{IR}}=d-4,\end{eqnarray}$where d is the dimension of space-time, coefficients A and B are complex functions of kinematic variables, and the terms in the infrared finite part are proportional to ${\varepsilon }_{\mathrm{IR}}^{n}(n=0,1,2,\ldots )$. After the infrared cancellations, only terms independent of ${\varepsilon }_{\mathrm{IR}}^{n}$ are left in practical applications. In loop regularization, the infrared finite part depends on the characteristic scale μ_s but μ_s is not fixed. This means that when μ_s is sliding, the infrared finite part is variable. Thus, when we analyze several processes based on loop regularization, if the result of each process is consistent with experiments for the same μ_s, it implies that the given μ_s is reasonable for these processes.

4. Conclusions

In this paper the scalar one-loop four-point function with a massless vertex is calculated analytically by loop regularization, and the infrared divergent and infrared stable parts are well separated. The results are convenient to analytically continue to other kinematic sectors which are beyond our assumption. Following the steps proposed in this paper, one-loop tensor-type four-point integrals are easy to evaluate. In particular, this will be beneficial for calculating complete contributions up to ${ \mathcal O }({\alpha }_{s}^{2})$ to the amplitudes via the six-quark operator effective weak Hamiltonian factorization approach. However, we should realize that since the kinematic region in which our results are valid is determined by 12 equations, as a result it greatly constrains the application of the final results. If the results obtained by an other regularization method (for instance, dimensional regularization), analytically or numerically, are valid in the same kinematic region as our paper, a cross check is feasible. A practical way of checking may be to apply the results to evaluate some specific decay processes, and then compare the evaluated results with the experiments. This will be one of our research works undertaken in the near future.

Appendix A. Factorization of quadratic equation with two real roots

Suppose f(x) is a quadratic polynomial with imaginary part(89)$\begin{eqnarray}\begin{array}{rcl}f(x) & = & {{ax}}^{2}+{bx}-c({\mu }_{s}^{2}+{\rm{i}}\varepsilon ),\\ b & \gt & 0,\quad {b}^{2}-4{ac}{\mu }_{s}^{2}\gt 0,\end{array}\end{eqnarray}$where ϵ is a real, positive infinitesimal. The two zeros of equation (89) are(90)$\begin{eqnarray}\begin{array}{rcl}x & = & \displaystyle \frac{1}{2a}\left[-b\pm \sqrt{{b}^{2}+4{ac}({\mu }_{s}^{2}+{\rm{i}}\varepsilon )}\,\right]\\ & = & \displaystyle \frac{1}{2a}\left\{-b\pm \sqrt{{b}^{2}-4{ac}{\mu }_{s}^{2}}\left[1+2{ac}\cdot {\rm{i}}\varepsilon +{ \mathcal O }({\varepsilon }^{2})+...\right]\right\}\\ & \approx & \displaystyle \frac{1}{2a}\left[-b\pm \sqrt{{b}^{2}-4{ac}{\mu }_{s}^{2}}\pm 2{ac}\cdot {\rm{i}}\varepsilon \right],\end{array}\end{eqnarray}$where we have used the property that the product of any finite quantity with ϵ is still a real infinitesimal, and we still denote it by ϵ. Supposing ${\mu }_{s}^{2}\ll 1$, we may expand the two zeros as power series of ${\mu }_{s}^{2}$:(91)$\begin{eqnarray}x=\displaystyle \frac{1}{2a}\left\{-b\pm b\left[1-\displaystyle \frac{2{ac}}{{b}^{2}}{\mu }_{s}^{2}+{ \mathcal O }({\mu }_{s}^{4})\right]\pm 2{ac}\cdot {\rm{i}}\varepsilon \right\},\end{eqnarray}$and then we get two roots:(92)$\begin{eqnarray}\begin{array}{rcl}{x}^{(+)} & = & -\displaystyle \frac{c}{b}{\mu }_{s}^{2}+{\rm{i}}\varepsilon ,\\ {x}^{(-)} & = & -\displaystyle \frac{b}{a}+\displaystyle \frac{c}{b}{\mu }_{s}^{2}-{\rm{i}}\varepsilon .\end{array}\end{eqnarray}$Now we would like to factorize equation (89) as(93)$\begin{eqnarray}f(x)=a[x-{x}^{(+)}][x-{x}^{(-)}],\end{eqnarray}$and then we obtain(94)$\begin{eqnarray}a=\displaystyle \frac{f(1)}{[1-{x}^{(+)}][1-{x}^{(-)}]},\end{eqnarray}$This leads to an useful factorization on f(x):(95)$\begin{eqnarray}f(x)=\displaystyle \frac{f(1)[x-{x}^{(+)}][x-{x}^{(-)}]}{[1-{x}^{(+)}][1-{x}^{(-)}]},\end{eqnarray}$and assuming f(x) is real and f(1) > 0, according to equation (9), we find the useful expression below:(96)$\begin{eqnarray}\mathrm{ln}f(x)=\mathrm{ln}f(1)+\mathrm{ln}\displaystyle \frac{[x-{x}^{(+)}][x-{x}^{(-)}]}{[1-{x}^{(+)}][1-{x}^{(-)}]}.\end{eqnarray}$

Appendix B. Useful auxiliary integrals

In this section we list some integral formulae which are useful in the calculation; they are taken from [44, 54].(97)$\begin{eqnarray}{\int }_{0}^{1}{\rm{d}}x\displaystyle \frac{\mathrm{ln}x}{a+{bx}}=\displaystyle \frac{1}{b}{\mathrm{Li}}_{2}\left(-\displaystyle \frac{b}{a}\right),\end{eqnarray}$(98)$\begin{eqnarray}\begin{array}{l}{\displaystyle \int }_{0}^{1}{\rm{d}}x\displaystyle \frac{\mathrm{ln}(c+{ex})}{a+{bx}}=\displaystyle \frac{1}{b}\left\{\mathrm{ln}\left(\displaystyle \frac{{bc}-{ae}}{b}\right)\mathrm{ln}\displaystyle \frac{a+b}{a}\right.\\ \quad \left.-{\mathrm{Li}}_{2}\left[\displaystyle \frac{e(a+b)}{{ae}-{bc}}\right]+{\mathrm{Li}}_{2}\left(\displaystyle \frac{{ae}}{{ae}-{bc}}\right)\right\},\end{array}\end{eqnarray}$(99)$\begin{eqnarray}\begin{array}{l}{\displaystyle \int }_{0}^{1}{\rm{d}}x\displaystyle \frac{1}{x-{x}_{0}}\mathrm{ln}\left[\displaystyle \frac{(x-{x}_{1})(x-{x}_{2})}{(1-{x}_{1})(1-{x}_{2})}\right]\\ =\,-{\mathrm{Li}}_{2}\left(1-\displaystyle \frac{{x}_{0}-1}{{x}_{0}}\displaystyle \frac{{x}_{1}}{{x}_{1}-1}\right)-{\mathrm{Li}}_{2}\left(1-\displaystyle \frac{{x}_{0}-1}{{x}_{0}}\displaystyle \frac{{x}_{2}}{{x}_{2}-1}\right)\\ +\,{\mathrm{Li}}_{2}\left(\displaystyle \frac{1}{1-{x}_{1}}\right)+{\mathrm{Li}}_{2}\left(\displaystyle \frac{1}{1-{x}_{2}}\right)+2{\mathrm{Li}}_{2}\left(\displaystyle \frac{1}{{x}_{0}}\right).\end{array}\end{eqnarray}$

Appendix C. Evaluating the three components of I₁

In this section we present the details of evaluation of the three component of I₁. In appendices C and D, equation (10) is frequently employed. In principle, dilogarithms with two arguments can be analytically continued by using equation (12), but for brevity we do not make the analytic continuation and just retain their original form.• evaluation of I₁₁
The integral is(100)$\begin{eqnarray}\begin{array}{rcl}{I}_{11} & = & {\displaystyle \int }_{0}^{1}{\rm{d}}\rho \,\displaystyle \frac{1}{{g}_{0}{f}_{2}-{g}_{1}{f}_{1}}\left[\mathrm{ln}({g}_{0}+{g}_{1})-\mathrm{ln}({f}_{1}+{f}_{2})\right]\\ & = & {\displaystyle \int }_{0}^{1}{\rm{d}}\rho \,\displaystyle \frac{1}{{F}_{11}(\rho )}\left[\mathrm{ln}({D}_{1}\rho +{g}_{0})-\mathrm{ln}{G}_{11}(\rho )\right],\end{array}\end{eqnarray}$where the denominator F₁₁(ρ) is quadratic in ρ:(101)$\begin{eqnarray}\begin{array}{rcl}{F}_{11}(\rho ) & = & ({A}_{2}{g}_{0}-{A}_{1}{D}_{1}){\rho }^{2}\\ & & +({B}_{2}{g}_{0}-{B}_{1}{D}_{1})\rho -{g}_{0}({\mu }_{s}^{2}+{\rm{i}}\varepsilon )\\ & = & {a}_{11}[\rho -{\rho }_{11}^{(+)}][\rho -{\rho }_{11}^{(-)}],\end{array}\end{eqnarray}$The two zeros in equation (101) are(102)$\begin{eqnarray}\begin{array}{rcl}{a}_{11}={a}_{1} & = & {A}_{2}{g}_{0}-{A}_{1}{D}_{1},\\ {\rho }_{11}^{(+)}={\rho }_{1}^{(+)} & = & \displaystyle \frac{{\mu }_{s}^{2}}{{\lambda }_{1}^{2}}+{\rm{i}}\varepsilon ,\\ {\rho }_{11}^{(-)}={\rho }_{1}^{(-)} & = & -\left({\kappa }_{1}+\displaystyle \frac{{\mu }_{s}^{2}}{{\lambda }_{1}^{2}}\right)-{\rm{i}}\varepsilon ,\end{array}\end{eqnarray}$where in order to simplify the symbols, we have relabeled a₁₁, ${\rho }_{11}^{(+)}$ and ${\rho }_{11}^{(-)}$ as a₁, ${\rho }_{1}^{(+)}$ and ${\rho }_{1}^{(-)}$, respectively. The dimensional λ₁ and dimensionless κ₁ are defined as follows:(103)$\begin{eqnarray}{\lambda }_{1}=\displaystyle \frac{{B}_{2}{g}_{0}-{B}_{1}{D}_{1}}{{g}_{0}},\qquad \qquad {\kappa }_{1}=\displaystyle \frac{{B}_{2}{g}_{0}-{B}_{1}{D}_{1}}{{A}_{2}{g}_{0}-{A}_{1}{D}_{1}},\end{eqnarray}$The manifest expression of G₁₁ is(104)$\begin{eqnarray}{G}_{11}(\rho )={A}_{2}{\rho }^{2}+({A}_{1}+{B}_{2})\rho +{B}_{1}-{\mu }_{s}^{2}-{\rm{i}}\varepsilon .\end{eqnarray}$By employing the equations in appendix A, it is not difficult to factorize G₁₁ into the product of its roots:(105)$\begin{eqnarray}{G}_{11}(\rho )=\displaystyle \frac{{G}_{11}(1)\left[\rho -{\rho }_{11}^{(1)}\right]\left[\rho -{\rho }_{11}^{(2)}\right]}{\left[1-{\rho }_{11}^{(1)}\right]\left[1-{\rho }_{11}^{(2)}\right]},\end{eqnarray}$where(106)$\begin{eqnarray}\begin{array}{rcl}{G}_{11}(1) & = & {m}_{3}^{2}-{\omega }_{2}^{2}+\alpha ({m}_{3}^{2}-{m}_{2}^{2}-{\omega }_{2}^{2}+{s}_{23}^{2})-{\mu }_{s}^{2}-{\rm{i}}\varepsilon ,\\ {\rho }_{11}^{(1)} & = & \frac{1}{2{A}_{2}}\left[-({A}_{1}+{B}_{2})+\sqrt{{{\rm{\Delta }}}_{11}}\right],\\ {\rho }_{11}^{(2)} & = & \frac{1}{2{A}_{2}}\left[-({A}_{1}+{B}_{2})-\sqrt{{{\rm{\Delta }}}_{11}}\right],\\ {{\rm{\Delta }}}_{11} & = & {\left({A}_{1}+{B}_{2}\right)}^{2}-4{A}_{2}({B}_{1}-{\mu }_{s}^{2}-{\rm{i}}\varepsilon ).\end{array}\,\end{eqnarray}$By using the formula in appendix B, we obtain(107)$\begin{eqnarray}\begin{array}{rcl}{I}_{11} & = & {\displaystyle \int }_{0}^{1}{\rm{d}}\rho \,\displaystyle \frac{1}{{a}_{1}[\rho -{\rho }_{1}^{(+)}][\rho -{\rho }_{1}^{(-)}]}\left\{\mathrm{ln}\left({D}_{1}\rho +{g}_{0}\right)\right.\\ & & \left.-\mathrm{ln}\displaystyle \frac{{G}_{11}(1)\left[\rho -{\rho }_{11}^{(1)}\right]\left[\rho -{\rho }_{11}^{(2)}\right]}{\left[1-{\rho }_{11}^{(1)}\right]\left[1-{\rho }_{11}^{(2)}\right]}\right\}\\ & = & \displaystyle \frac{1}{{a}_{1}[{\rho }_{1}^{(+)}-{\rho }_{1}^{(-)}]}{\displaystyle \int }_{0}^{1}{\rm{d}}\rho \left[\displaystyle \frac{1}{\rho -{\rho }_{1}^{(+)}}-\displaystyle \frac{1}{\rho -{\rho }_{1}^{(-)}}\right]\\ & & \times \left\{\mathrm{ln}\left({D}_{1}\rho +{g}_{0}\right)-\mathrm{ln}\displaystyle \frac{{G}_{11}(1)}{{\mu }^{2}}\right.\\ & & \left.-\mathrm{ln}\displaystyle \frac{\left[\rho -{\rho }_{11}^{(1)}\right]\left[\rho -{\rho }_{11}^{(2)}\right]}{\left[1-{\rho }_{11}^{(1)}\right]\left[1-{\rho }_{11}^{(2)}\right]}\right\}\\ & = & \displaystyle \frac{1}{{a}_{1}[{\rho }_{1}^{(+)}-{\rho }_{1}^{(-)}]}\left\{{\mathrm{ln}}^{2}\displaystyle \frac{{\mu }_{s}^{2}}{{\lambda }_{1}^{2}}+\mathrm{ln}\left|1-\displaystyle \frac{{\lambda }_{1}^{2}}{{\mu }_{s}^{2}}\right|\right.\\ & & \mathrm{ln}\displaystyle \frac{{g}_{0}+{D}_{1}{\rho }_{1}^{(+)}}{{G}_{1}(1)}+2{\rm{i}}\pi \mathrm{ln}\displaystyle \frac{{\lambda }_{1}^{2}}{{\mu }_{s}^{2}}+{\rm{i}}\pi \mathrm{ln}\displaystyle \frac{{g}_{0}+{D}_{1}{\rho }_{1}^{(+)}}{{G}_{1}(1)}\\ & & -\displaystyle \frac{2{\pi }^{2}}{3}-\mathrm{ln}\left[1-\displaystyle \frac{1}{{\rho }_{1}^{(-)}}\right]\mathrm{ln}\displaystyle \frac{{g}_{0}+{D}_{1}{\rho }_{1}^{(-)}}{{G}_{1}(1)}\\ & & +{\mathrm{Li}}_{2}\left[\displaystyle \frac{-{D}_{1}{\rho }_{1}^{(+)}}{-{D}_{1}{\rho }_{1}^{(+)}-{g}_{0}}\right]-{\mathrm{Li}}_{2}\left[\displaystyle \frac{{D}_{1}(1-{\rho }_{1}^{(+)})}{-{D}_{1}{\rho }_{1}^{(+)}-{g}_{0}}\right]\\ & & -{\mathrm{Li}}_{2}\left[\displaystyle \frac{-{D}_{1}{\rho }_{1}^{(-)}}{-{D}_{1}{\rho }_{1}^{(-)}-{g}_{0}}\right]+{\mathrm{Li}}_{2}\left[\displaystyle \frac{{D}_{1}(1-{\rho }_{1}^{(-)})}{-{D}_{1}{\rho }_{1}^{(-)}-{g}_{0}}\right]\\ & & +2{\mathrm{Li}}_{2}\left(\displaystyle \frac{{\mu }_{s}^{2}}{{\lambda }_{1}^{2}}\right)+{\mathrm{Li}}_{2}\left[1-\displaystyle \frac{{\rho }_{1}^{(+)}}{1-{\rho }_{1}^{(+)}}\displaystyle \frac{{\rho }_{11}^{(1)}}{1-{\rho }_{11}^{(1)}}\right]\\ & & +{\mathrm{Li}}_{2}\left[1-\displaystyle \frac{{\rho }_{1}^{(+)}}{1-{\rho }_{1}^{(+)}}\displaystyle \frac{{\rho }_{11}^{(2)}}{1-{\rho }_{11}^{(2)}}\right]\\ & & +2{\mathrm{Li}}_{2}\left[\displaystyle \frac{1}{{\rho }_{1}^{(-)}}\right]-{\mathrm{Li}}_{2}\left[1-\displaystyle \frac{{\rho }_{1}^{(-)}}{1-{\rho }_{1}^{(-)}}\displaystyle \frac{{\rho }_{11}^{(1)}}{1-{\rho }_{11}^{(1)}}\right]\\ & & \left.-{\mathrm{Li}}_{2}\left[1-\displaystyle \frac{{\rho }_{1}^{(-)}}{1-{\rho }_{1}^{(-)}}\displaystyle \frac{{\rho }_{11}^{(2)}}{1-{\rho }_{11}^{(2)}}\right]\right\}.\end{array}\end{eqnarray}$• evaluation of I₁₂
After the variable substitution, the integral is(108)$\begin{eqnarray}\begin{array}{rcl}{I}_{12} & = & {\displaystyle \int }_{0}^{1-\alpha }{\rm{d}}\rho \,\displaystyle \frac{1}{{g}_{0}{f}_{2}-{g}_{1}{f}_{1}}\left[-\mathrm{ln}\left(\displaystyle \frac{\rho }{1-\alpha }{g}_{0}+{g}_{1}\right)\right.\\ & & \left.+\mathrm{ln}\left(\displaystyle \frac{\rho }{1-\alpha }{f}_{1}+{f}_{2}\right)\right]\\ & = & (1-\alpha ){\displaystyle \int }_{0}^{1}{\rm{d}}\rho \,\displaystyle \frac{1}{{F}_{12}(\rho )}\left\{-\mathrm{ln}\left[\left({g}_{0}+(1-\alpha ){D}_{1}\right)\rho \right]\right.\\ & & \left.+\mathrm{ln}{G}_{12}(\rho )\right\},\end{array}\end{eqnarray}$where F₁₂(ρ) is given by(109)$\begin{eqnarray}\begin{array}{rcl}{F}_{12}(\rho ) & = & {\left(1-\alpha \right)}^{2}({A}_{2}{g}_{0}-{A}_{1}{D}_{1}){\rho }^{2}\\ & & +(1-\alpha )({B}_{2}{g}_{0}-{B}_{1}{D}_{1})\rho -{g}_{0}({\mu }_{s}^{2}+{\rm{i}}\varepsilon )\\ & = & {a}_{12}[\rho -{\rho }_{12}^{(+)}][\rho -{\rho }_{12}^{(-)}],\end{array}\end{eqnarray}$with(110)$\begin{eqnarray}\begin{array}{ccc}{a}_{12} & = & {\left(1-\alpha \right)}^{2}({A}_{2}{g}_{0}-{A}_{1}{D}_{1}),\\ {\rho }_{12}^{(+)} & = & \frac{{\rho }_{1}^{(+)}}{1-\alpha },\quad {\rho }_{12}^{(-)}=\frac{{\rho }_{1}^{(-)}}{1-\alpha }.\end{array}\end{eqnarray}$The argument of the second logarithm in the numerator is defined as(111)$\begin{eqnarray}\begin{array}{rcl}{G}_{12}(\rho ) & = & (1-\alpha )[{A}_{1}+(1-\alpha ){A}_{2}]{\rho }^{2}\\ & & +[{B}_{1}+(1-\alpha ){B}_{2}]\rho -{\mu }_{s}^{2}-{\rm{i}}\varepsilon \\ & = & \displaystyle \frac{{G}_{12}(1)\left[\rho -{\rho }_{12}^{(1)}\right]\left[\rho -{\rho }_{12}^{(2)}\right]}{\left[1-{\rho }_{12}^{(1)}\right]\left[1-{\rho }_{12}^{(2)}\right]},\end{array}\end{eqnarray}$with(112)$\begin{eqnarray}\begin{array}{l}{G}_{12}(1)={m}_{3}^{2}-{\mu }_{s}^{2}-{\rm{i}}\varepsilon ,\\ \quad {\rho }_{12}^{(1)}=\displaystyle \frac{1}{2(1-\alpha )[{A}_{1}+(1-\alpha ){A}_{2}]}\\ \quad \times \{-[{B}_{1}+(1-\alpha ){B}_{2}]+\sqrt{{{\rm{\Delta }}}_{12}}\},\\ \quad {\rho }_{12}^{(2)}=\displaystyle \frac{1}{2(1-\alpha )[{A}_{1}+(1-\alpha ){A}_{2}]}\\ \quad \times \{-[{B}_{1}+(1-\alpha ){B}_{2}]-\sqrt{{{\rm{\Delta }}}_{12}}\},\\ \quad {{\rm{\Delta }}}_{12}={\left[{B}_{1}+(1-\alpha ){B}_{2}\right]}^{2}\\ \quad +\,4(1-\alpha )[{A}_{1}+(1-\alpha ){A}_{2}]({\mu }_{s}^{2}-{\rm{i}}\varepsilon ),\end{array}\end{eqnarray}$where equation (32) has been employed. By using the equation in appendix B, it is not difficult to obtain$\begin{eqnarray*}\begin{array}{l}{I}_{12}=(1-\alpha ){\displaystyle \int }_{0}^{1}{\rm{d}}\rho \,\displaystyle \frac{1}{{a}_{12}[\rho -{\rho }_{12}^{(+)}][\rho -{\rho }_{12}^{(-)}]}\\ \times \,\left\{-\mathrm{ln}\left[\left({g}_{0}+(1-\alpha ){D}_{1}\right)\rho \right]\right.\\ \left.+\,\mathrm{ln}\displaystyle \frac{{G}_{12}(1)[\rho -{\rho }_{12}^{(1)}][\rho -{\rho }_{12}^{(2)}]}{[1-{\rho }_{12}^{(1)}][1-{\rho }_{12}^{(2)}]}\,\right\}\\ =\,\displaystyle \frac{(1-\alpha )}{{a}_{12}[{\rho }_{12}^{(+)}-{\rho }_{12}^{(-)}]}{\displaystyle \int }_{0}^{1}{\rm{d}}\rho \\ \left[\displaystyle \frac{1}{\rho -{\rho }_{12}^{(+)}}-\displaystyle \frac{1}{\rho -{\rho }_{12}^{(-)}}\right]\\ \times \,\left\{-\mathrm{ln}\left[\left({g}_{0}+(1-\alpha ){D}_{1}\right)\rho \right]+\mathrm{ln}\displaystyle \frac{{G}_{12}}{{\mu }^{2}}\right.\\ \left.+\,\mathrm{ln}\displaystyle \frac{[\rho -{\rho }_{12}^{(1)}][\rho -{\rho }_{12}^{(2)}]}{[1-{\rho }_{12}^{(1)}][1-{\rho }_{12}^{(2)}]}\,\right\}\\ =\,\displaystyle \frac{1}{{a}_{1}[{\rho }_{1}^{(+)}-{\rho }_{1}^{(-)}]}\left\{-\displaystyle \frac{1}{2}{\mathrm{ln}}^{2}\displaystyle \frac{{\lambda }_{1}^{2}}{{\mu }_{s}^{2}}\right.\end{array}\end{eqnarray*}$(113)$\begin{eqnarray}\begin{array}{l}-\mathrm{ln}(1-\alpha )\mathrm{ln}\displaystyle \frac{{\lambda }_{1}^{2}}{{\mu }_{s}^{2}}-\displaystyle \frac{1}{2}{\mathrm{ln}}^{2}(1-\,\alpha )\\ -\,\mathrm{ln}\left|1-\displaystyle \frac{(1-\alpha ){\lambda }_{1}^{2}}{{\mu }_{s}^{2}}\right|\mathrm{ln}\displaystyle \frac{{g}_{0}+(1-\alpha ){D}_{1}}{{G}_{12}(1)}\\ -\,{\rm{i}}\pi \mathrm{ln}\displaystyle \frac{{\lambda }_{1}^{2}}{{\mu }_{s}^{2}}-{\rm{i}}\pi \mathrm{ln}(1-\alpha )\\ -\,{\rm{i}}\pi \mathrm{ln}\displaystyle \frac{{g}_{0}+(1-\alpha ){D}_{1}}{{G}_{12}(1)}+\displaystyle \frac{{\pi }^{2}}{3}\\ +\,\mathrm{ln}\left[1-\displaystyle \frac{1-\alpha }{{\rho }_{1}^{(-)}}\right]\mathrm{ln}\displaystyle \frac{{g}_{0}+(1-\alpha ){D}_{1}}{{G}_{12}(1)}\\ -\,{\mathrm{Li}}_{2}\left[\displaystyle \frac{{\mu }_{s}^{2}}{(1-\alpha ){\lambda }_{1}^{2}}\right]\\ -\,{\mathrm{Li}}_{2}\left[1-\displaystyle \frac{{\rho }_{1}^{(+)}}{1-\alpha -{\rho }_{1}^{(+)}}\displaystyle \frac{{\rho }_{12}^{(1)}}{1-{\rho }_{12}^{(1)}}\right]\\ -\,{\mathrm{Li}}_{2}\left[1-\displaystyle \frac{{\rho }_{1}^{(+)}}{1-\alpha -{\rho }_{1}^{(+)}}\displaystyle \frac{{\rho }_{12}^{(2)}}{1-{\rho }_{12}^{(2)}}\right]\\ -\,{\mathrm{Li}}_{2}\left[\displaystyle \frac{1-\alpha }{{\rho }_{1}^{(-)}}\right]\\ +\,{\mathrm{Li}}_{2}\left[1-\displaystyle \frac{{\rho }_{1}^{(-)}}{1-\alpha -{\rho }_{1}^{(-)}}\displaystyle \frac{{\rho }_{12}^{(1)}}{1-{\rho }_{12}^{(1)}}\right]\\ \left.+\,{\mathrm{Li}}_{2}\left[1-\displaystyle \frac{{\rho }_{1}^{(-)}}{1-\alpha -{\rho }_{1}^{(-)}}\displaystyle \frac{{\rho }_{12}^{(2)}}{1-{\rho }_{12}^{(2)}}\right]\right\}.\end{array}\end{eqnarray}$• evaluation of I₁₃
The integral is(114)$\begin{eqnarray}{I}_{13}=-\alpha {\int }_{0}^{1}{\rm{d}}\rho \,\displaystyle \frac{1}{{F}_{13}(\rho )}\left\{\mathrm{ln}\left[({g}_{0}-{D}_{1}\alpha )\rho \right]-\mathrm{ln}{G}_{13}(\rho )\right\},\end{eqnarray}$where(115)$\begin{eqnarray}\begin{array}{rcl}{F}_{13}(\rho ) & = & {\alpha }^{2}({A}_{2}{g}_{0}-{A}_{1}{D}_{1}){\rho }^{2}\\ & & +\alpha ({B}_{1}{D}_{1}-{B}_{2}{g}_{0})\rho -{g}_{0}({\mu }_{s}^{2}+{\rm{i}}\varepsilon )\\ & = & {a}_{13}[\rho -{\rho }_{13}^{(+)}][\rho -{\rho }_{13}^{(-)}],\end{array}\end{eqnarray}$with(116)$\begin{eqnarray}\begin{array}{ccc}{a}_{13} & = & {\alpha }^{2}({A}_{2}{g}_{0}-{A}_{1}{D}_{1}),\quad \quad {\rho }_{13}^{(+)}=-\frac{{\rho }_{1}^{(+)}}{\alpha },\\ {\rho }_{13}^{(-)} & = & -\frac{{\rho }_{1}^{(-)}}{\alpha }.\end{array}\end{eqnarray}$The argument of the second logarithm in the numerator is(117)$\begin{eqnarray}\begin{array}{rcl}{G}_{13}(u) & = & \alpha (\alpha {A}_{2}-{A}_{1}){\rho }^{2}-(\alpha {B}_{2}-{B}_{1})\rho -{\mu }_{s}^{2}-{\rm{i}}\varepsilon \\ & = & \displaystyle \frac{{G}_{13}(1)[\rho -{\rho }_{13}^{(1)}][\rho -{\rho }_{13}^{(2)}]}{[1-{\rho }_{13}^{(1)}][1-{\rho }_{13}^{(2)}]},\end{array}\end{eqnarray}$with(118)$\begin{eqnarray}\begin{array}{rcl}{G}_{13}(1) & = & {m}_{2}^{2}-{\mu }_{s}^{2}-{\rm{i}}\varepsilon ,\\ {\rho }_{13}^{(1)} & = & \displaystyle \frac{1}{2\alpha (\alpha {A}_{2}-{A}_{1})}\left[-({B}_{1}-\alpha {B}_{2})+\sqrt{{{\rm{\Delta }}}_{13}}\right],\\ {\rho }_{13}^{(2)} & = & \displaystyle \frac{1}{2\alpha (\alpha {A}_{2}-{A}_{1})}\left[-({B}_{1}-\alpha {B}_{2})-\sqrt{{{\rm{\Delta }}}_{13}}\right],\\ {{\rm{\Delta }}}_{13} & = & {\left({B}_{1}-\alpha {B}_{2}\right)}^{2}+4\alpha (\alpha {A}_{2}-{A}_{1})({\mu }_{s}^{2}+{\rm{i}}\varepsilon ),\end{array}\end{eqnarray}$where equation (32) has been employed. By using the equations listed in appendix B, it is easy to obtain$\begin{eqnarray*}\begin{array}{l}{I}_{13}=-\alpha {\displaystyle \int }_{0}^{1}{\rm{d}}\rho \,\displaystyle \frac{1}{{a}_{13}[\rho -{\rho }_{13}^{(+)}][\rho -{\rho }_{13}^{(+)}]}\left\{\mathrm{ln}\left[({g}_{0}-{D}_{1}\alpha )\rho \right]\right.\\ \left.-\,\mathrm{ln}\displaystyle \frac{{G}_{13}(1)\left[\rho -{\rho }_{13}^{(1)}\right]\left[\rho -{\rho }_{13}^{(2)}\right]}{\left[1-{\rho }_{13}^{(1)}\right]\left[1-{\rho }_{13}^{(2)}\right]}\right\}\\ =\,\displaystyle \frac{-\alpha }{{a}_{13}\left[{\rho }_{13}^{(+)}-{\rho }_{13}^{(-)}\right]}{\displaystyle \int }_{0}^{1}{\rm{d}}\rho \left[\displaystyle \frac{1}{\rho -{\rho }_{13}^{(+)}}-\displaystyle \frac{1}{\rho -{\rho }_{13}^{(-)}}\right]\\ \times \,\left\{\mathrm{ln}\left[({g}_{0}-{D}_{1}\alpha )\rho \right]-\mathrm{ln}\displaystyle \frac{{G}_{13}(1)}{{\mu }^{2}}\right.\\ \left.-\,\mathrm{ln}\displaystyle \frac{[\rho -{\rho }_{13}^{(1)}][\rho -{\rho }_{13}^{(2)}]}{[1-{\rho }_{13}^{(1)}][1-{\rho }_{13}^{(2)}]}\,\right\}\\ =\,\displaystyle \frac{1}{{a}_{1}\left[{\rho }_{1}^{(+)}-{\rho }_{1}^{(-)}\right]}\left\{\displaystyle \frac{1}{2}{\mathrm{ln}}^{2}\displaystyle \frac{{\lambda }^{2}}{{\mu }_{s}^{2}}+\mathrm{ln}\alpha \mathrm{ln}\displaystyle \frac{{\lambda }^{2}}{{\mu }_{s}^{2}}\right.\\ +\,\displaystyle \frac{1}{2}{\mathrm{ln}}^{2}\alpha +\mathrm{ln}\left(1+\displaystyle \frac{\alpha {\lambda }^{2}}{{\mu }_{s}^{2}}\right)\mathrm{ln}\displaystyle \frac{{g}_{0}-\alpha {D}_{1}}{{G}_{13}(1)}\end{array}\end{eqnarray*}$(119)$\begin{eqnarray}\begin{array}{l}+\,\displaystyle \frac{{\pi }^{2}}{6}-\mathrm{ln}\left[1+\displaystyle \frac{\alpha }{{\rho }_{1}^{(-)}}\right]\mathrm{ln}\displaystyle \frac{{g}_{0}-\alpha {D}_{1}}{{G}_{13}(1)}\\ +\,{\mathrm{Li}}_{2}\left(-\displaystyle \frac{{\mu }_{s}^{2}}{\alpha {\lambda }^{2}}\right)+{\mathrm{Li}}_{2}\left[1+\displaystyle \frac{{\rho }_{1}^{(+)}}{\alpha +{\rho }_{1}^{(+)}}\displaystyle \frac{{\rho }_{13}^{(1)}}{1-{\rho }_{13}^{(1)}}\right]\\ +\,{\mathrm{Li}}_{2}\left[1+\displaystyle \frac{{\rho }_{1}^{(+)}}{\alpha +{\rho }_{1}^{(+)}}\displaystyle \frac{{\rho }_{13}^{(2)}}{1-{\rho }_{13}^{(2)}}\right]\\ +\,{\mathrm{Li}}_{2}\left[-\displaystyle \frac{\alpha }{{\rho }_{1}^{(-)}}\right]-{\mathrm{Li}}_{2}\left[1+\displaystyle \frac{{\rho }_{1}^{(-)}}{\alpha +{\rho }_{1}^{(-)}}\displaystyle \frac{{\rho }_{13}^{(1)}}{1-{\rho }_{13}^{(1)}}\right]\\ \left.-\,{\mathrm{Li}}_{2}\left[1+\displaystyle \frac{{\rho }_{1}^{(-)}}{\alpha +{\rho }_{1}^{(-)}}\displaystyle \frac{{\rho }_{13}^{(2)}}{1-{\rho }_{13}^{(2)}}\right]\right\}.\end{array}\end{eqnarray}$

Appendix D. Evaluating the three components of I₀

• evaluation of I₀₁
The integral is(120)$\begin{eqnarray}\begin{array}{rcl}{I}_{01} & = & {\displaystyle \int }_{0}^{1}{\rm{d}}\rho \,\displaystyle \frac{1}{{g}_{0}{h}_{2}-{g}_{1}{h}_{1}}\left[\mathrm{ln}({g}_{0}+{g}_{1})-\mathrm{ln}({h}_{1}+{h}_{2})\right]\\ & = & {\displaystyle \int }_{0}^{1}{\rm{d}}\rho \,\displaystyle \frac{1}{{F}_{01}(\rho )}\left[\mathrm{ln}\left({D}_{1}\rho +{g}_{0}\right)-\mathrm{ln}{G}_{01}(\rho )\right],\end{array}\end{eqnarray}$where the denominator is(121)$\begin{eqnarray}\begin{array}{l}{F}_{01}(u)=({K}_{2}{g}_{0}-{K}_{1}{D}_{1}){\rho }^{2}\\ +({g}_{0}{N}_{2}-{N}_{1}{D}_{1})\rho -{g}_{0}({\mu }_{s}^{2}+{\rm{i}}\varepsilon )\\ ={a}_{01}\left[\rho -{\rho }_{01}^{(+)}\right]\left[\rho -{\rho }_{01}^{(-)}\right],\end{array}\end{eqnarray}$with(122)$\begin{eqnarray}\begin{array}{rcl}{a}_{01}={a}_{0} & = & {K}_{2}{g}_{0}-{K}_{1}{D}_{1}\\ {\rho }_{01}^{(+)}={\rho }_{0}^{(+)} & = & \displaystyle \frac{{\lambda }_{0}^{2}}{{\mu }_{s}^{2}}+{\rm{i}}\varepsilon ,\\ {\rho }_{01}^{(-)}={\rho }_{0}^{(-)} & = & -\left({\kappa }_{0}+\displaystyle \frac{{\lambda }_{0}^{2}}{{\mu }_{s}^{2}}\right)-{\rm{i}}\varepsilon ,\end{array}\end{eqnarray}$where in order to simplify the symbols, we have relabeled a₀₁, ${\rho }_{01}^{(+)}$ and ${\rho }_{01}^{(-)}$ as a₀, ${\rho }_{0}^{(+)}$ and ${\rho }_{0}^{(-)}$, respectively. The dimensional λ₀ and dimensionless κ₀ are defined as follows:(123)$\begin{eqnarray}\begin{array}{ccc}{\lambda }_{0}^{2} & = & \frac{{N}_{2}{g}_{0}-{N}_{1}{D}_{1}}{{g}_{0}},\\ {\kappa }_{0} & = & \frac{{N}_{2}{g}_{0}-{N}_{1}{D}_{1}}{{K}_{2}{g}_{0}-{K}_{1}{D}_{1}}.\end{array}\end{eqnarray}$The argument of the second logarithm in the numerator is(124)$\begin{eqnarray}\begin{array}{rcl}{G}_{01}(\rho ) & = & {K}_{2}{\rho }^{2}+({K}_{1}+{N}_{2})\rho +{N}_{1}-{\mu }_{s}^{2}-{\rm{i}}\varepsilon \\ & = & \displaystyle \frac{{G}_{01}(1)\left[\rho -{\rho }_{01}^{(1)}\right]\left[\rho -{\rho }_{01}^{(2)}\right]}{\left[1-{\rho }_{01}^{(1)}\right]\left[1-{\rho }_{01}^{(2)}\right]},\end{array}\end{eqnarray}$with(125)$\begin{eqnarray}\begin{array}{rcl}{G}_{01}(1) & = & {m}_{3}^{2}-{s}_{12}^{2}+\beta ({m}_{3}^{2}+2{\omega }_{3}^{2}\\ & & +{s}_{12}^{2}-{m}_{2}^{2}-{m}_{4}^{2})-{\mu }_{s}^{2}-{\rm{i}}\varepsilon ,\\ {\rho }_{01}^{(1)} & = & \frac{1}{2{K}_{2}}[-({K}_{1}+{N}_{2})+\sqrt{{{\rm{\Delta }}}_{01}}],\\ {\rho }_{01}^{(2)} & = & \frac{1}{2{K}_{2}}[-({K}_{1}+{N}_{2})-\sqrt{{{\rm{\Delta }}}_{01}}],\\ {{\rm{\Delta }}}_{01} & = & {\left({K}_{1}+{N}_{2}\right)}^{2}-4{K}_{2}({N}_{1}-{\mu }_{s}^{2}-{\rm{i}}\varepsilon ).\end{array}\end{eqnarray}$With the help of the equations listed in appendix B, we find$\begin{eqnarray*}\begin{array}{l}{I}_{01}={\displaystyle \int }_{0}^{1}{\rm{d}}\rho \,\displaystyle \frac{1}{{a}_{0}\left[\rho -{\rho }_{0}^{(+)}\right]\left[\rho -{\rho }_{0}^{(-)}\right]}\\ \left[\mathrm{ln}\left({D}_{1}\rho +{g}_{0}\right)-\mathrm{ln}{G}_{01}(\rho )\right]\\ =\,\displaystyle \frac{1}{{a}_{0}\left[{\rho }_{0}^{(+)}-{\rho }_{0}^{(-)}\right]}{\displaystyle \int }_{0}^{1}{\rm{d}}\rho \left[\displaystyle \frac{1}{\rho -{\rho }_{0}^{(+)}}-\displaystyle \frac{1}{\rho -{\rho }_{0}^{(-)}}\right]\\ \times \,\left\{\mathrm{ln}\left({D}_{1}\rho +{g}_{0}\right)-\mathrm{ln}\displaystyle \frac{{G}_{01}(1)}{{\mu }^{2}}\right.\\ \left.-\,\mathrm{ln}\displaystyle \frac{\left[\rho -{\rho }_{01}^{(1)}\right]\left[\rho -{u}_{01}^{(2)}\right]}{\left[1-{\rho }_{01}^{(1)}\right]\left[1-{\rho }_{01}^{(2)}\right]}\,\right\}\\ =\,\displaystyle \frac{1}{{a}_{0}\left[{\rho }_{0}^{(+)}-{\rho }_{0}^{(-)}\right]}\left\{{\mathrm{ln}}^{2}\displaystyle \frac{{\lambda }_{0}^{2}}{{\mu }_{s}^{2}}\right.\\ +\,\mathrm{ln}\left|1-\displaystyle \frac{{\lambda }_{0}^{2}}{{\mu }_{s}^{2}}\right|\mathrm{ln}\displaystyle \frac{{g}_{0}+{D}_{1}{\rho }_{0}^{(+)}}{{G}_{01}(1)}+2{\rm{i}}\pi \mathrm{ln}\displaystyle \frac{{\lambda }_{0}^{2}}{{\mu }_{s}^{2}}\\ +\,{\rm{i}}\pi \mathrm{ln}\displaystyle \frac{{g}_{0}+{D}_{1}{\rho }_{0}^{(+)}}{{G}_{01}(1)}-\displaystyle \frac{2{\pi }^{2}}{3}\\ -\,\mathrm{ln}\displaystyle \frac{{g}_{0}+{D}_{1}{\rho }_{0}^{(-)}}{{G}_{01}(1)}\mathrm{ln}\left[1-\displaystyle \frac{1}{{\rho }_{0}^{(-)}}\right]\\ +\,{\mathrm{Li}}_{2}\left[\displaystyle \frac{-{D}_{1}{\rho }_{0}^{(+)}}{-{D}_{1}{\rho }_{0}^{(+)}-{g}_{0}}\right]-{\mathrm{Li}}_{2}\left[\displaystyle \frac{{D}_{1}(1-{\rho }_{0}^{(+)})}{-{D}_{1}{\rho }_{0}^{(+)}-{g}_{0}}\right]\end{array}\end{eqnarray*}$(126)$\begin{eqnarray}\begin{array}{l}-\,{\mathrm{Li}}_{2}\left[\displaystyle \frac{-{D}_{1}{\rho }_{0}^{(-)}}{-{D}_{1}{\rho }_{0}^{(-)}-{g}_{0}}\right]+{\mathrm{Li}}_{2}\left[\displaystyle \frac{{D}_{1}(1-{\rho }_{0}^{(-)})}{-{D}_{1}{\rho }_{0}^{(-)}-{g}_{0}}\right]\\ +\,2{\mathrm{Li}}_{2}\left(\displaystyle \frac{{\mu }_{s}^{2}}{{\lambda }_{0}^{2}}\right)+{\mathrm{Li}}_{2}\left[1-\displaystyle \frac{{\rho }_{0}^{(+)}}{1-{\rho }_{0}^{(+)}}\displaystyle \frac{{\rho }_{01}^{(1)}}{1-{\rho }_{01}^{(1)}}\right]\\ +\,{\mathrm{Li}}_{2}\left[1-\displaystyle \frac{{\rho }_{0}^{(+)}}{1-{\rho }_{0}^{(+)}}\displaystyle \frac{{\rho }_{01}^{(2)}}{1-{\rho }_{01}^{(2)}}\right]\\ +\,2{\mathrm{Li}}_{2}\left[\displaystyle \frac{1}{{\rho }_{0}^{(-)}}\right]-{\mathrm{Li}}_{2}\left[1-\displaystyle \frac{{\rho }_{0}^{(-)}}{1-{\rho }_{0}^{(-)}}\displaystyle \frac{{\rho }_{01}^{(1)}}{1-{\rho }_{01}^{(1)}}\right]\\ \left.-\,{\mathrm{Li}}_{2}\left[1-\displaystyle \frac{{\rho }_{0}^{(-)}}{1-{\rho }_{0}^{(-)}}\displaystyle \frac{{\rho }_{01}^{(2)}}{1-{\rho }_{01}^{(2)}}\right]\right\}.\end{array}\end{eqnarray}$• evaluation of I₀₂


The integral is(127)$\begin{eqnarray}\begin{array}{rcl}{I}_{02} & = & (1-\beta ){\displaystyle \int }_{0}^{1}{\rm{d}}\rho \,\displaystyle \frac{1}{{F}_{02}(\rho )}\left\{-\mathrm{ln}\left[\left({g}_{0}\right.\right.\right.\\ & & \left.\left.\left.+(1-\beta ){D}_{1}\right)\rho \right]+\mathrm{ln}{G}_{02}(\rho )\right\},\end{array}\end{eqnarray}$where the denominator is(128)$\begin{eqnarray}\begin{array}{rcl}{F}_{02}(\rho ) & = & {\left(1-\beta \right)}^{2}({K}_{2}{g}_{0}-{K}_{1}{D}_{1}){\rho }^{2}\\ & & -(1-\beta )({N}_{1}{D}_{1}-{N}_{2}{g}_{0})\rho -{g}_{0}({\mu }_{s}^{2}+{\rm{i}}\varepsilon )\\ & = & {a}_{02}\left[\rho -{\rho }_{02}^{(+)}\right]\left[\rho -{\rho }_{02}^{(-)}\right],\end{array}\end{eqnarray}$with(129)$\begin{eqnarray}\begin{array}{rcl}{a}_{02} & = & {\left(1-\beta \right)}^{2}({K}_{2}{g}_{0}-{K}_{1}{D}_{1}),\\ {\rho }_{02}^{(+)} & = & \frac{{\rho }_{0}^{(+)}}{1-\beta },\\ {\rho }_{02}^{(-)} & = & \frac{{\rho }_{0}^{(-)}}{1-\beta }.\end{array}\end{eqnarray}$The argument of the second logarithm in the numerator is(130)$\begin{eqnarray}\begin{array}{rcl}{G}_{02}(\rho ) & = & (1-\beta )[{K}_{1}+(1-\beta ){K}_{2}]{\rho }^{2}\\ & & +[{N}_{1}+(1-\beta ){N}_{2}]\rho -{\mu }_{s}^{2}-{\rm{i}}\varepsilon \\ & = & \displaystyle \frac{{G}_{02}(1)\left[\rho -{\rho }_{02}^{(1)}\right]\left[\rho -{\rho }_{02}^{(2)}\right]}{\left[1-{\rho }_{02}^{(1)}\right]\left[1-{\rho }_{02}^{(2)}\right]},\end{array}\end{eqnarray}$with(131)$\begin{eqnarray}\begin{array}{l}{G}_{02}(1)={m}_{3}^{2}-{\mu }_{s}^{2}-{\rm{i}}\varepsilon ,\\ {\rho }_{02}^{(1)}=\displaystyle \frac{1}{2(1-\beta )[{K}_{1}+(1-\beta ){K}_{2}]}\\ \quad \left\{-[{N}_{1}+(1-\beta ){N}_{2}]+\sqrt{{{\rm{\Delta }}}_{02}}\right\},\\ {\rho }_{02}^{(2)}=\displaystyle \frac{1}{2(1-\beta )[{K}_{1}+(1-\beta ){K}_{2}]}\\ \quad \left\{-[{N}_{1}+(1-\beta ){N}_{2}]-\sqrt{{{\rm{\Delta }}}_{02}}\right\},\\ {{\rm{\Delta }}}_{02}={\left[{N}_{1}+(1-\beta ){N}_{2}\right]}^{2}\\ \quad +4(1-\beta )[{K}_{1}+(1-\beta ){K}_{2}]({\mu }_{s}^{2}+{\rm{i}}\varepsilon ),\end{array}\end{eqnarray}$where equation (61) has been employed. By using the equations in appendix B, it is not difficult to obtain the result$\begin{eqnarray*}\begin{array}{rcl}{I}_{02} & = & (1-\beta ){\displaystyle \int }_{0}^{1}{\rm{d}}\rho \,\displaystyle \frac{1}{{a}_{02}\left[\rho -{\rho }_{02}^{(+)}\right]\left[\rho -{\rho }_{02}^{(-)}\right]}\\ & & \left\{-\mathrm{ln}\left[\left({g}_{0}+(1-\beta ){D}_{1}\right)\rho \right]+\mathrm{ln}{G}_{02}(\rho )\right\}\\ & = & \displaystyle \frac{1-\beta }{{a}_{02}\left[{\rho }_{02}^{(+)}-{\rho }_{02}^{(-)}\right]}{\displaystyle \int }_{0}^{1}{\rm{d}}\rho \left[\displaystyle \frac{1}{\rho -{\rho }_{02}^{(+)}}-\displaystyle \frac{1}{\rho -{\rho }_{02}^{(-)}}\right]\\ & & \left\{-\mathrm{ln}\left[\left({g}_{0}+(1-\beta ){D}_{1}\right)\rho \right]\right.\\ & & \left.+\mathrm{ln}\displaystyle \frac{{G}_{02}(1)}{{\mu }^{2}}+\mathrm{ln}\displaystyle \frac{\left[\rho -{\rho }_{02}^{(1)}\right]\left[\rho -{\rho }_{02}^{(2)}\right]}{\left[1-{\rho }_{02}^{(1)}\right]\left[1-{\rho }_{02}^{(2)}\right]}\,\right\}\\ & = & \displaystyle \frac{1}{{a}_{0}\left[{\rho }_{0}^{(+)}-{\rho }_{0}^{(-)}\right]}\left\{-\displaystyle \frac{1}{2}{\mathrm{ln}}^{2}\displaystyle \frac{{\lambda }_{0}^{2}}{{\mu }_{s}^{2}}\right.\\ & & -\mathrm{ln}(1-\beta )\mathrm{ln}\displaystyle \frac{{\lambda }_{0}^{2}}{{\mu }_{s}^{2}}-\displaystyle \frac{1}{2}{\mathrm{ln}}^{2}(1-\beta )\\ & & -\mathrm{ln}\left|1-\displaystyle \frac{(1-\beta ){\lambda }_{0}^{2}}{{\mu }_{s}^{2}}\right|\mathrm{ln}\displaystyle \frac{{g}_{0}+(1-\beta ){D}_{1}}{{G}_{02}(1)}\\ & & -{\rm{i}}\pi \mathrm{ln}\displaystyle \frac{{\lambda }_{0}^{2}}{{\mu }_{s}^{2}}-{\rm{i}}\pi \mathrm{ln}(1-\beta )\\ & & -{\rm{i}}\pi \mathrm{ln}\displaystyle \frac{{g}_{0}+(1-\beta ){D}_{1}}{{G}_{02}(1)}+\displaystyle \frac{{\pi }^{2}}{3}+\mathrm{ln}\displaystyle \frac{{g}_{0}+(1-\beta ){D}_{1}}{{G}_{02}(1)}\end{array}\end{eqnarray*}$(132)$\begin{eqnarray}\begin{array}{rcl} & & \mathrm{ln}\left[1-\displaystyle \frac{1-\beta }{{\rho }_{0}^{(-)}}\right]\\ & & -{\mathrm{Li}}_{2}\left[\displaystyle \frac{{\mu }_{s}^{2}}{(1-\beta ){\lambda }_{0}^{2}}\right]-{\mathrm{Li}}_{2}\left[1-\displaystyle \frac{{\rho }_{0}^{(+)}}{1-\beta -{\rho }_{0}^{(+)}}\displaystyle \frac{{\rho }_{02}^{(1)}}{1-{\rho }_{02}^{(1)}}\right]\\ & & -{\mathrm{Li}}_{2}\left[1-\displaystyle \frac{{\rho }_{0}^{(+)}}{1-\beta -{\rho }_{0}^{(+)}}\displaystyle \frac{{\rho }_{02}^{(2)}}{1-{\rho }_{02}^{(2)}}\right]\\ & & -{\mathrm{Li}}_{2}\left[\displaystyle \frac{1-\beta }{{\rho }_{0}^{(-)}}\right]+{\mathrm{Li}}_{2}\\ & & \left[1-\displaystyle \frac{{\rho }_{0}^{(-)}}{1-\beta -{\rho }_{0}^{(-)}}\displaystyle \frac{{\rho }_{02}^{(1)}}{1-{\rho }_{02}^{(1)}}\right]\\ & & \left.+{\mathrm{Li}}_{2}\left[1-\displaystyle \frac{{\rho }_{0}^{(-)}}{1-\beta -{\rho }_{0}^{(-)}}\displaystyle \frac{{\rho }_{02}^{(2)}}{1-{\rho }_{02}^{(2)}}\right]\right\}.\end{array}\end{eqnarray}$• evaluation of I₀₃
The integral is(133)$\begin{eqnarray}\begin{array}{rcl}{I}_{03} & = & -\beta {\displaystyle \int }_{0}^{1}{\rm{d}}\rho \,\displaystyle \frac{1}{{F}_{03}(\rho )}\\ & & \left\{\mathrm{ln}\left[\left({g}_{0}-\beta {D}_{1}\right)\rho \right]-\mathrm{ln}{G}_{03}(\rho )\right\},\end{array}\end{eqnarray}$where we define(134)$\begin{eqnarray}\begin{array}{rcl}{F}_{03}(\rho ) & = & {\beta }^{2}({K}_{2}{g}_{0}-{K}_{1}{D}_{1}){\rho }^{2}\\ & & +\beta ({N}_{1}{D}_{1}-{N}_{2}{g}_{0})\rho -{g}_{0}({\mu }_{s}^{2}+{\rm{i}}\varepsilon )\\ & = & {a}_{03}\left[\rho -{\rho }_{03}^{(+)}\right]\left[\rho -{\rho }_{03}^{(-)}\right],\end{array}\end{eqnarray}$with(135)$\begin{eqnarray}\begin{array}{rcl}{a}_{03} & = & {\beta }^{2}({K}_{2}{g}_{0}-{K}_{1}{D}_{1}),\\ {\rho }_{03}^{(+)} & = & -\displaystyle \frac{{\rho }_{0}^{(+)}}{\beta },\\ {\rho }_{03}^{(-)} & = & -\displaystyle \frac{{\rho }_{0}^{(-)}}{\beta },\end{array}\end{eqnarray}$The argument of the second logarithm in the numerator is(136)$\begin{eqnarray}\begin{array}{rcl}{G}_{03}(\rho ) & = & \beta (\beta {K}_{2}-{K}_{1}){\rho }^{2}-(\beta {N}_{2}-{N}_{1})\rho -{\mu }_{s}^{2}-{\rm{i}}\varepsilon \\ & = & \displaystyle \frac{{G}_{03}(1)\left[\rho -{\rho }_{03}^{(1)}\right]\left[\rho -{\rho }_{03}^{(2)}\right]}{\left[1-{\rho }_{03}^{(1)}\right]\left[1-{\rho }_{03}^{(2)}\right]},\end{array}\end{eqnarray}$with(137)$\begin{eqnarray}\begin{array}{rcl}{G}_{03}(1) & = & {m}_{2}^{2}-{\mu }_{s}^{2}-{\rm{i}}\varepsilon ,\\ {\rho }_{03}^{(1)} & = & \displaystyle \frac{1}{2\beta (\beta {K}_{2}-{K}_{1})}[-({N}_{1}-\beta {N}_{2})+\sqrt{{{\rm{\Delta }}}_{03}}],\\ {\rho }_{03}^{(2)} & = & \displaystyle \frac{1}{2\beta (\beta {K}_{2}-{K}_{1})}[-({N}_{1}-\beta {N}_{2})-\sqrt{{{\rm{\Delta }}}_{03}}],\\ {{\rm{\Delta }}}_{03} & = & {\left({N}_{1}-\beta {N}_{2}\right)}^{2}+4\beta (\beta {K}_{2}-{K}_{1})({\mu }_{s}^{2}+{\rm{i}}\varepsilon ),\end{array}\end{eqnarray}$where equation (61) has been used. By employing the equations in appendix B we can obtain(138)$\begin{eqnarray}\begin{array}{rcl}{I}_{03} & = & -\beta {\displaystyle \int }_{0}^{1}{\rm{d}}\rho \,\displaystyle \frac{1}{{a}_{03}\left[\rho -{\rho }_{03}^{(+)}\right]\left[\rho -{\rho }_{03}^{(-)}\right]}\\ & & \left\{\mathrm{ln}\left[\left({g}_{0}-\beta {D}_{1}\right)\rho \right]-\mathrm{ln}{G}_{03}(\rho )\right\}\\ & = & \displaystyle \frac{-\beta }{{a}_{03}\left[{\rho }_{03}^{(+)}-{\rho }_{03}^{(-)}\right]}{\displaystyle \int }_{0}^{1}{\rm{d}}\rho \left[\displaystyle \frac{1}{\rho -{\rho }_{03}^{(+)}}-\displaystyle \frac{1}{\rho -{\rho }_{03}^{(-)}}\right]\\ & & \left\{\mathrm{ln}\left[\left({g}_{0}-\beta {D}_{1}\right)\rho \right]\right.\\ & & \left.-\mathrm{ln}\displaystyle \frac{{G}_{03}(1)}{{\mu }^{2}}-\mathrm{ln}\displaystyle \frac{\left[\rho -{\rho }_{03}^{(1)}\right]\left[\rho -{\rho }_{03}^{(1)}\right]}{\left[1-{\rho }_{03}^{(1)}\right]\left[1-{\rho }_{03}^{(2)}\right]}\right\}\\ & = & \displaystyle \frac{1}{{a}_{0}\left[{\rho }_{0}^{(+)}-{\rho }_{0}^{(-)}\right]}\left\{\displaystyle \frac{1}{2}{\mathrm{ln}}^{2}\displaystyle \frac{{\lambda }_{0}^{2}}{{\mu }_{s}^{2}}+\mathrm{ln}\beta \mathrm{ln}\displaystyle \frac{{\lambda }_{0}^{2}}{{\mu }_{s}^{2}}\right.\\ & & +\displaystyle \frac{1}{2}{\mathrm{ln}}^{2}\beta +\mathrm{ln}\left|1+\displaystyle \frac{\beta {\lambda }_{0}^{2}}{{\mu }_{s}^{2}}\right|\mathrm{ln}\displaystyle \frac{{g}_{0}-\beta {D}_{1}}{{G}_{03}(1)}\\ & & +\displaystyle \frac{{\pi }^{2}}{6}-\mathrm{ln}\left[1+\displaystyle \frac{\beta }{{\rho }_{0}^{(-)}}\right]\mathrm{ln}\displaystyle \frac{{g}_{0}-\beta {D}_{1}}{{G}_{03}(1)}+{\mathrm{Li}}_{2}\left(-\displaystyle \frac{{\mu }_{s}^{2}}{\beta {\lambda }_{0}^{2}}\right)\\ & & +{\mathrm{Li}}_{2}\left[1+\displaystyle \frac{{\rho }_{0}^{(+)}}{\beta +{\rho }_{0}^{(+)}}\ \displaystyle \frac{{\rho }_{03}^{(1)}}{1-{\rho }_{03}^{(1)}}\right]\\ & & +{\mathrm{Li}}_{2}\left[1+\displaystyle \frac{{\rho }_{0}^{(+)}}{\beta +{\rho }_{0}^{(+)}}\displaystyle \frac{{\rho }_{03}^{(2)}}{1-{\rho }_{03}^{(2)}}\right]\\ & & +{\mathrm{Li}}_{2}\left[-\displaystyle \frac{\beta }{{\rho }_{0}^{(-)}}\right]-{\mathrm{Li}}_{2}\left[1+\displaystyle \frac{{\rho }_{0}^{(-)}}{\beta +{\rho }_{0}^{(-)}}\displaystyle \frac{{\rho }_{03}^{(1)}}{1-{\rho }_{03}^{(1)}}\right]\\ & & \left.-{\mathrm{Li}}_{2}\left[1+\displaystyle \frac{{\rho }_{0}^{(-)}}{\beta +{\rho }_{0}^{(-)}}\displaystyle \frac{{\rho }_{03}^{(2)}}{1-{\rho }_{03}^{(2)}}\right]\right\}.\end{array}\end{eqnarray}$

Acknowledgments

The author thanks H E Haber for helpful discussion on the properties of dilogarithms.

Reference View Option By original order
By published year
By cited within times
By Impact factor

[1]Chetyrkin K G Tkachov F V 1981 Nucl. Phys. B 192 159
DOI:10.1016/0550-3213(81)90199-1 [Cited within: 1]

[2]Baikov P A 1996 Phys. Lett. B 385 404
DOI:10.1016/0370-2693(96)00835-0 [Cited within: 1]

[3]Boos E E Davydychev A I 1991 Theor. Math. Phys. 89 1052
DOI:10.1007/BF01016805 [Cited within: 1]

[4]Kotikov A V 1991 Phys. Lett. B 254 158
DOI:10.1016/0370-2693(91)90413-K [Cited within: 1]

[5]Kotikov A V 1991 Phys. Lett. B 267 123
DOI:10.1016/0370-2693(91)90536-Y

Kotikov A V 1992 Phys. Lett. B 295 409 erratum:
DOI:10.1016/0370-2693(91)90536-Y

[6]Kotikov A V 1991 Phys. Lett. B 259 314
DOI:10.1016/0370-2693(91)90834-D

[7]Kotikov A V 1991 Mod. Phys. Lett. A 6 677
DOI:10.1142/S0217732391000695

[8]Remiddi E 1997 Nuovo Cim. A 110 1435


[9]Argeri M Mastrolia P 2007 Int. J. Mod. Phys. A 22 4375
DOI:10.1142/S0217751X07037147

[10]Henn J M 2015 J. Phys. A 48 153001
DOI:10.1088/1751-8113/48/15/153001 [Cited within: 1]

[11]Smirnov V A 2012 Analytic Tools for Feynman Integrals Berlin Springer 83
[Cited within: 1]

[12]Buttar C et al. 2005 Standard model and Higgs working group: summary report
4th Les Houches Workshop on Physics at TeV CollidersarXiv: hep-ph/0604120


[13]Bern Z(NLO Multileg Working Group)et al. 2007 The NLO multileg working group: summary report
5th Les Houches Workshop on Physics at TeV Colliders (PhysTeV 2007)arXiv: hep-ph/0803.0494


[14]Binoth T(SM and NLO Multileg Working Group)et al. 2009 The SM and NLO multileg working group: summary report
6th Les Houches Workshop on Physics at TeV Colliders (PhysTeV 2009)arXiv: hep-ph/1003.1241
[Cited within: 1]

[15]Lewin L 1981 Polylogarithms and Associated FunctionsII ednNew York North Holland 283
[Cited within: 1]

[16]Remiddi E Vermaseren J A M 2000 Int. J. Mod. Phys. A 15 725
DOI:10.1142/S0217751X00000367 [Cited within: 1]

[17]Bloch S Vanhove P 2015 J. Number. Theor. 148 328
DOI:10.1016/j.jnt.2014.09.032 [Cited within: 1]

[18]Adams L Bogner C Weinzierl S 2014 J. Math. Phys.J. Math. Phys. 55 102301
DOI:10.1063/1.4896563

[19]Adams L Bogner C Weinzierl S 2015 J. Math. Phys.J. Math. Phys. 56 072303
DOI:10.1063/1.4926985

[20]Broedel J et al. 2018 J. High Energy Phys.JHEP05(2018)093


[21]Broedel J et al. 2018 Phys. Rev. D 97 116009
DOI:10.1103/PhysRevD.97.116009

[22]Broedel J et al. 2019 J. High Energy Phys.JHEP05(2019)120


[23]Remiddi E Tancredi L 2017 Nucl. Phys. B 925 212
DOI:10.1016/j.nuclphysb.2017.10.007

[24]Bezuglov M A Onishchenko A I Veretin O L 2021 Nucl. Phys. B 963 115302
DOI:10.1016/j.nuclphysb.2020.115302

[25]Blümlein J Schneider C Paule P 2019 Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory Switzerland Springer 93
[Cited within: 1]

[26]Beilinson A Levin A 1994 Proc. Symp. Pure. Math. 55 126
[Cited within: 1]

[27]Levin A 1997 Compos. Math. 106 267
DOI:10.1023/A:1000193320513

[28]Goncharov A B 1998 Math. Res. Lett. 5 497
DOI:10.4310/MRL.1998.v5.n4.a7

[29]Levin A Racinet G arXiv: math/0703237


[30]Brown F C S Levin Aarxiv: math/1110.6917


[31]Passarino G 2017 Eur. Phys. J. C 77 77
DOI:10.1140/epjc/s10052-017-4623-1 [Cited within: 1]

[32]Denner A 1993 Fortsch. Phys. 41 307
[Cited within: 1]

[33]Kniehl B A 1994 Phys. Rept. 240 211
DOI:10.1016/0370-1573(94)90037-X

[34]Bardin D Y Passarino G 1999 The Standard Model in The Making: Precision Study of The Electroweak Interactions Oxford Clarendon 141


[35]Denner A Dittmaier S 2020 Phys. Rept. 864 1
DOI:10.1016/j.physrep.2020.04.001 [Cited within: 1]

[36]Passarino G Veltman M J G 1979 Nucl. Phys. B 160 151
DOI:10.1016/0550-3213(79)90234-7 [Cited within: 1]

[37]'t Hooft G Veltman M J G 1979 Nucl. Phys. B 153 365
DOI:10.1016/0550-3213(79)90605-9 [Cited within: 3]

[38]Denner A Nierste U Scharf R 1991 Nucl. Phys. B 367 637
DOI:10.1016/0550-3213(91)90011-L [Cited within: 1]

[39]van Oldenborgh G J Vermaseren J A M 1990 Z. Phys. C 46 425
DOI:10.1007/BF01621031 [Cited within: 1]

[40]Bern Z Dixon L J Kosower D A 1993 Phys. Lett. B 302 299
DOI:10.1016/0370-2693(93)90400-C [Cited within: 1]

Bern Z Dixon L J Kosower D A 1993 Phys. Lett. B 318 649 erratum:
DOI:10.1016/0370-2693(93)90400-C [Cited within: 1]

[41]Bern Z Dixon L J Kosower D A 1994 Nucl. Phys. B 412 751
DOI:10.1016/0550-3213(94)90398-0 [Cited within: 1]

[42]Beenakker W Denner A 1990 Nucl. Phys. B 338 349
DOI:10.1016/0550-3213(90)90636-R [Cited within: 2]

[43]Duplancic G Nizic B 2001 Eur. Phys. J. C 20 357
DOI:10.1007/s100520100675 [Cited within: 2]

[44]Ellis R K Zanderighi G 2008 J. High Energy Phys.JHEP02(2008)002
DOI:10.1088/1126-6708/2008/02/002 [Cited within: 4]

[45]Denner A Dittmaier S 2011 Nucl. Phys. B 844 199
DOI:10.1016/j.nuclphysb.2010.11.002 [Cited within: 3]

[46]'t Hooft G Veltman M J G 1972 Nucl. Phys. B 44 189
DOI:10.1016/0550-3213(72)90279-9 [Cited within: 1]

[47]Wu Y-L 2003 Int. J. Mod. Phys. A 18 5363
DOI:10.1142/S0217751X03015222 [Cited within: 2]

[48]Wu Y-L 2004 Mod. Phys. Lett. A 19 2191
DOI:10.1142/S0217732304015361 [Cited within: 2]

[49]Su F et al. 2011 J. Phys. G 38 015006
DOI:10.1088/0954-3899/38/1/015006 [Cited within: 2]

[50]Huang D Tang Y Wu Y-L 2012 Commun. Theor. Phys. 57 427
DOI:10.1088/0253-6102/57/3/14

[51]Su F et al. 2012 Eur. Phys. J. C 72 1914
DOI:10.1140/epjc/s10052-012-1914-4 [Cited within: 2]

[52]Haber H E Kane G L 1984 Nucl. Phys. B 232 333
DOI:10.1016/0550-3213(84)90570-4 [Cited within: 1]

[53]Haber H E Wyler D 1989 Nucl. Phys. B 323 267
DOI:10.1016/0550-3213(89)90143-0 [Cited within: 1]

[54]Devoto A Duke D W 1984 Riv. Nuovo Cim. 7N6 1
DOI:10.1007/BF02724330 [Cited within: 1]