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New analytic formulas for one-loop three-point Feynman integrals in the general space-time dimension (

Future experimental programs at the High-Luminosity Large Hadron Collider (HL-LHC) [

One-loop Feynman integrals in the general space-time dimension play a crucial role for several reasons. Within the general framework for computing two-loop or higher-loop corrections, higher terms in the

The layout of the paper is as follows: In

Detailed evaluations for one-loop three-point integrals are presented in this section.

We arrive at notations for the calculations in this subsection. Feynman integrals of scalar one-loop three-point functions are defined:

Hereafter we use

One-loop triangle diagrams.

It is known that an algebraically compact expression and numerically stable representation of Feynman diagrams can be obtained by using kinematic variables such as the determinants of Caylay and Gram matrices [

In the same manner, the Cayley determinants of one-loop two-point Feynman diagrams are obtained by shrinking a propagator in the three-point integrals. The determinants are written explicitly as:

As in the definitions above, we also get the Gram determinants of two-point functions as follows:

In this work, the analytic formulas for scalar one-loop three-point integrals are expressed as functions with arguments of the ratio of the above kinematic determinants. Therefore, it is worth introducing the following index variables:

By introducing Feynman parameters, we then integrate over the loop momentum. The resulting integral after taking it over one of the Feynman parameters reads

The corresponding coefficients

A detailed calculation for

We first consider the simple case, i.e., two light-like external momenta. Without loss of generality, we can take

This shows that the denominator function of the integrand in Eq. (

Both the integrands are singularities at

This means that the residue contributions at this pole of two integrations in Eq. (

Another representation of

The results shown in Eqs. (

(1)

One-loop triangle diagrams with all massless internal lines are considered. In this case, the integral

It is important to note that we use

(2)

If the internal mass configuration takes

This result is in agreement with Eq. (B2) in Ref. [

When

(3)

We are also interested in the case of

It is easy to find out that

As in the previous explanation, the integral

(4)

This case has been calculated in Ref. [

It is confirmed that

As a result of this fact,

Following Eq. (

(5)

This case has been investigated in Ref. [

Using the Mellin–Barnes relation one has

After taking it over the

By closing the integration contour to the right side of the imaginary axis in the

We now discuss the method for the one light-like momentum case. Without any loss of generality, we can choose

We find that two integrands have the same singularity pole at

It is important to note that the results for

We can get another representation of

From Eqs. (

The results in Eqs. (

(1)

One first arrives at the case of all massless internal lines. In this case, Eq. (

We have already used

If

(2)

We are concerned with the case in which the internal masses have

We make a change in the variable like

This representation gives agreement with the result of Eq. (C6) in Ref. [

When

(3)

This case has been investigated in Ref. [

This gives perfect agreement with Ref. [

The result reads

(4)

Noting that

Here we have already performed a shift

Another representation of

From this representation, one can perform an analytic continuation of this result in the limits of

(5)

We are going to consider an interesting case,

Both terms on the right-hand side of Eq. (

The integral will be worked out by applying the Mellin–Barnes relation, which is

We are going to generalize the method for the general case in which

By choosing

It is also important to note that the final result will be independent of

The integration after shifting

To achieve a more symmetric form we make further transformations

Finally, following an idea in Ref. [

The analytic result for

Applying the formula for the master integral in

It is important to note that the results in Eqs. (

The results shown in Eqs. (

As an example, we consider the case of

The operator

Following the tensor reduction method in Ref. [

In this formula, the condition for the indices

As an example, we first take the simplest case

In the next step, the scalar integrals

In order to solve the system of equations (

If det

New analytic formulas for one-loop three-point Feynman integrals in the general space-time dimension are presented in this paper. The results are expressed in terms of Appell

This research is funded by Vietnam National University Ho Chi Minh City (VNU-HCM) under grant number C2019-18-06.

Open Access funding: SCOAP

We consider the master integral

We will discuss the method of evaluating this integral under the above conditions. In the case of

In order to work out the master integral one should write the polynomial of

We have introduced the kinematic variables

Using the Mellin–Barnes relation [

With the help of this relation, the Feynman parameter integral will be cast into a simpler form. It will be calculated in terms of Gauss hypergeometric functions. In particular, we have

One makes a shift

We apply a further transformation like

Following Eq. (

Putting this result into Eq. (

Applying the transformation for Appell

The Gauss hypergeometric series is given by (see Eq. (1.1.1.4) in Ref. [

The series of Appell

Basic linear transformation formulas for Gauss

We collect all transformations for Appell

If

Furthermore, if

Similarly,