One-loop Feynman integrals with Carlson hypergeometric functions

. In this paper, we present analytic results for scalar one-loop two-, three-, four-point Feynman integrals with complex internal masses. The calculations are considered in general space-time dimension D for two- and three-point functions and D = 4 for four-point functions. The analytic results are expressed in terms of the Carlson hypergeometric functions ( R -functions) and valid for both real and complex internal masses.


Introduction
In order to confront particle physics theory with high-precision of experimental data at future colliders, theoretical predictions including high-order corrections are required. In general framework for computing high-order corrections, detailed calculations for one-loop multi-leg and higher-loop are necessary for building blocks. When we compute scattering processes which Feynman diagrams involve internal unstable particles that can be on-shell, we have to resume Feynman propagators with a complex mass term in the denominator. In other words, one has to perform the perturbative renormalization in the Complex-Mass Scheme [1]. Therefore, the calculations for Feynman loop integrals with complex internal masses are of great interest. Furthermore, within the general framework for computing two-loop or higherloop corrections scalar one-loop integrals in general space-time dimension play a crucial role for several reasons. Higher-terms in the ε-expansion from one-loop integrals are necessary for building blocks. In additional, one-loop integrals at higher space-time dimension D > 4 may be taken into account in the framework.
There have been available many calculations for scalar one-loop integrals in D = 4 − 2ε dimensions at ε 0 -expansion [2][3][4][5][6][7][8][9][10][11]. Scalar one-loop integrals in general dimension D have performed in [12][13][14][15][16]. However, not all of these calculations cover general dimension D with a general ε-expansion at general scale and complex internal masses. In this paper, based on the method in [5][6][7][8], we present analytic results for scalar one-loop two-, three-, fourpoint Feynman integrals with complex internal masses. The calculations are considered in general space-time dimension D for two-and three-point functions and D = 4 for fourpoint functions. The analytic results are expressed in terms of the Carlson hypergeometric functions.
The layout of the paper is as follows: In section 2, we present in detail the method for evaluating scalar one-loop functions. In this section, analytic results for one-loop two-, threeand four-point functions are presented. Conclusions and outlooks are devoted in section 3. Several useful formulas used in this calculation can be found in the appendix.
Based on the method introduced in Refs. [5][6][7], we present the calculations for scalar oneloop functions with complex internal masses. Scalar one-loop N-point functions are defined Where inverse Feynman propagators are given The Feynman prescription is iρ. We use momenta q k = k j=1 p j , p j are external momenta and they are inward as shown in Fig. 1. The internal masses in the Complex-Mass scheme are taken the form of The Γ k are decay widths of unstable particles. The momenta q k may take the following configuration which have J non-zero components. Here, q 10 = 0 for q 2 1 < 0 and q 11 = 0 for q 2 1 > 0. As a result, scalar product of external and internal momenta are obtained l · q k = l 0 · q k0 − l 1 · q k1 · · · − l J−1 · q k(J−1) .
In parallel space which is the linear span of the external momenta and its orthogonal space (POS) [5,6], scalar one-loop N-point functions are taken the form of: The propagators now become for k = 1, 2, · · · , N. The calculations can be summarized as follows. We first make the partition for the integrand of J N as = a lk l 0 + b lk l 1 + · · · + c lk l J−1 +d lk .
Where we have introduced the following kinematic variables Making a shift we convert all P k in (13) to P N . As a matter of this fact, the l ⊥ -integral then yields a simple form which can be taken easily as follows: We then arrive at the (J − 1)-fold integrals In this formula a lk , b lk , · · · , c lk ∈ R and d lk = (q l − q k ) 2 − (m 2 l − m 2 k ) ∈ C which is obtained fromd lk after applying the shift (18). The integrals in (20) can be carried out with the help of residue theorem. For that purpose, one first linearizes the l 0 for example, .i.e l ′ 1 = l 1 + l 0 . The result reads We close the contour integration for l 0 that the poles in (22) locate outside the contour.
with AB lk = a lk − b lk . The singularity poles of the integrand in (21) are obtained: and The pole l 0 in (22) locates upper (lower) in l 0 -complex plane if l 1 < 0 (l 1 > 0) respectively. We plan to close the contour integration for l 0 that l 0 -poles in (22) locate outside the contour, seen Fig. 2 for more detail. As a result, the poles in (23) are only taken into account to the residue contributions for l 0 -integration. The resulting reads Where the δ-function is defined as New kinematic variablesÃ mlk , · · · ,C mlk ∈ R andF mlk ∈ C are obtained from residue contributions of the poles in (23). The functions f ± lk indicate the location of the poles in (23) in the l 0 complex plane: We continue to linearize l 1 in numerator of the integrand of (24) by applying a Euler shift l 1 → l 1 + β lk l 2 . β lk can be chosen in such a way of the disappearance of l 2 1 -term. The residue theorem is applied against for l 1 -integration. At the final stage, the resulting integrals can be expressed in terms of R-functions [18] which is defined as with β = k i=1 b i . In next subsections, we present analytic results for scalar one-loop two-, three-and four-point functions. Detailed calculations for these functions have published in Ref. [17].

One-loop two-point functions
In POS, J 2 takes the form of [5,6] Here q = q(q 10 , − → 0 D−1 ) for q 2 > 0. If q 2 < 0, we refer [17] for detailed evaluations. The results in [17] have shown that the below formulas for J 2 are valid for both above cases. The R-function representation for two-point integrals is as follows [17]: We can derive other representations for J 2 by employing the transformations in appendix for R-functions from (46) to (51). For example, using Euler's transformation (50) for R-functions (50), Eq. (29) becomes It can be seen that the right hand sides of Eqs. (29,30) are symmetric under the interchange of m 2 1 ↔ m 2 2 . From Eqs. (29,30) we can take the limits of m 2 1 = m 2 2 → 0 and q 2 → 0 respectively, seen Ref.

One-loop three-point functions
The momenta q 1 , q 2 take the following configuration q 1 = q 1 (q 10 , q 11 , . Here q 10 = 0 for q 2 1 < 0 and q 11 = 0 for q 2 1 > 0. The results for J 3 in this paper cover both the above cases. The integral J 3 in POS takes the form of [5,6] Scalar one-loop three-point functions are also expressed in terms of R-functions [18] for m l. The kinematic variables appear in subsection are listed: = −AB km c lk + AB lk c km , The factor S ± lk is given We turn our attention into the analytic results for scalar one-loop four-point functions in next subsection.

One-loop four-point functions
At present, the calculations for four-point functions are performed in D = 4. We set configuration of external momenta as follows q 1 = (q 10 , q 11 , 0, 0), q 2 = (q 20 , q 21 , 0, 0), q 3 = (q 30 , q 31 , q 32 , 0). Where q 10 = 0 for q 2 1 < 0 and q 11 = 0 for q 2 1 > 0. Our result presented in this paper cover all the above cases. In POS, J 4 takes the form of with P k = (l + q k ) 2 − m 2 k + iρ for k = 1, 2, · · · , 4. Scalar one-loop four-point functions are written as one-fold integrals [17] as follows Where the related kinematic variables are given: with σ mlk = 0, −11/βmlk. The S (σ mlk , z) and G(z) are obtained: with Z mlk = D mlk β mlk − P mlk and K mlk = E mlk β mlk − Q mlk − P mlk F nmlk . The functions f ± lk (and g ± mlk ) are defined as in (26) with replacing c lk /AB lk by d lk /AC lk (and C mlk /B mlk ) respectively. The J 4 in (36) is decomposed into two basic integrals as follows: The ε-expansions for all R-functions appear in this paper have devoted in Ref.
[17]. The numerical checks for all analytic formulas in this paper and applications of this work to compute Feynman diagrams in real scattering processes have shown in [17].

Conclusions
We have presented the analytic results for scalar one-loop two-, three-, four-point Feynman integrals with complex internal masses. The analytic results in this paper are valid for both real and complex internal masses. The calculations have carried out in general space-time dimension for two-and three-point functions. At present work, the four-point functions have performed in D = 4. The analytic formulas have expressed in terms of the R-functions. In future work, we will extend this work to tensor one-loop integrals (to be published).