Asymptotic behavior of angular integrals in the massless limit

We investigate the small-mass asymptotics of a class of massive $d$ dimensional angular integrals. These integrals arise in a wide range of perturbative quantum field theory calculations. We derive expressions characterizing their behavior in the vicinity of the massless limit for all cases with up to two denominators. The results established in this work are applicable to phase-space calculations where an integration over virtuality including the massless limit is required.

When massless particles are present, the angular integration contains collinear singularities.To regularize these divergencies, the calculations are performed in d = 4 − 2ε dimensions [33,34].
In this manuscript we investigate the asymptotic behavior of integrals of the form I (m) j1,j2 (v 12 , v 11 , v 22 ; ε) in the limit of one or both masses going to zero.In principle, the expansion of all two-denominator angular integrals with integer powers j 1 , j 2 is known to all orders in * fabian.wunder@uni-tuebingen.de the dimensional regularization parameter ε [3][4][5].However, these expansions are not always sufficient.
As an illustration of the potential issue occurring in the massless limit, let us look at the double-massive angular integral with j 1 = j 2 = 1.It has the well known ε-expansion [1, 3-5] One readily sees that the massless limit v 11 → 0 is illdefined at the level of the ε-expansion, since v 12 − √ X approaches zero.This is a problem if we were to consider an integral of the form Here, we would like to replace I 1,1 (v 12 , v 11 , v 22 ; ε) by its ε-expansion under the integral and employ the distributional identity [35][36][37] . However, we cannot use the form of Eq. (2) due to its divergence in the v 11 → 0 limit.Instead, to properly perform the integration one has to extract the asymptotic behavior of I The aim of this work is to provide ε-expansions for all massive angular integrals with up to two propagators, where the asymptotic behavior in the massless limit is manifest and which are hence suitable for usage within integrals of the form (3).
Using recursion relations derived from integration-byparts (IBP) identities, the powers j 1 and j 2 can always be reduced to the cases j 1,2 = 0 or 1 [5].Hence it suffices to consider the master integrals I 1,1 (v 12 , v 11 ; ε), and 1,1 (v 12 , v 11 , v 22 ; ε).The remainder of this manuscript is organized as follows.In section II.we recall the two-point splitting lemma which we subsequently use to establish the asymptotic behavior of the master integrals in the massless limits v 11 , v 22 → 0. Section III.concludes the paper.

II. ASYMPTOTIC BEHAVIOR IN THE MASSLESS LIMIT
The main tool for the extraction of the asymptotic behavior will be the two-point splitting lemma [5].Using the notation it states that for any two vectors v 1 and v 2 , we can choose any scalar λ and construct the linear combina- This allows us to express a given angular integral in terms of other angular integrals where a new auxiliary vector v 3 has been inserted.By choosing appropriate values for λ, the vector v 3 can be given desirable properties, most importantly being massless.This idea has been fruitfully employed in reference [5] for the calculation of the all-order ε-expansion of the double-massive integral.

A. Asymptotic form of the massive one-denominator integral
We start with the investigation of the massive onedenominator master integral Its ε-expansion is [3-5] which is singular in the v 11 → 0 limit.To extract the massless limit from Eq. ( 7) explicitly, we define the auxiliary "zero" vector We observe that v 1 indeed approaches v 2 in the limit v 11 → 0. A graphical illustration of this construction is given in Fig. 1.
The two-point splitting lemma (6) provides us with the identity Integrating Eq. ( 9) and substituting the value for λ we get Hence, we have transformed the massive onedenominator integral into the sum of a massless one-denominator integral and a single-massive twodenominator integral, where the coefficient of the latter vanishes in the massless limit.It is ).For the onedenominator kinematics we have , and ω − 12 = 0. Plugging the ε-expansions into Eq.( 10), we receive In this form the asymptotic behavior for v 11 → 0 is explicit.We observe that I 1 (v 11 ; ε) has a part constant in the massless limit and a part proportional to v −ε 11 .It is the latter that causes the logarithmic divergence in Eq. (8).Note that both parts have a 1/ε pole which cancels between the two for finite v 11 .
. Sketch illustrating the splitting of the singlemassive two-denominator integral in Eq. ( 14).The figure shows the slice x0 = 1 of Minkowski space, the blue circle indicates the intersection with the light-cone where v 2 = 0.

B. Asymptotic form of the single-massive two-denominator integral
The second master integral we look at, is the singlemassive two-denominator integral We have already encountered its ε-expansion in Eq. (11).We observe that the expansion is singular in the limit v 11 → 0 because the variable ω − 12 diverges.To extract the asymptotic behavior of the singlemassive integral for small masses, we want to separate off its massless limit.To this end we define the auxiliary vector v 3 = (1 − λ) v 1 + λ v 2 .For v 3 to be massless, i.e. v 33 = 0, we set λ = v 11 /(v 11 − 2v 12 ).A graphical illustration of the splitting construction is given in Fig. 2.
Upon integration of the associated two-point splitting identity, which is of the form of Eq. ( 6), we receive with the scalar products v 13 = v 11 v 12 /(2v 12 − v 11 ) and v 23 = 2v 2 12 /(2v 12 − v 11 ).Hence, we have transformed the single-massive twodenominator integral into the sum of a massless two-denominator integral and a single-massive twodenominator angular integral, where the coefficient of the latter vanishes in the massless limit.
The ε-expansion of the massless two-denominator in-tegral reads [3-5] for the expansion of I 1,1 (v 13 , v 11 ; ε) we can again use Eq.(11).Plugging these into (14), we receive ) and the abbreviation ν = 1 − v 11 /(2v 12 ).In the massless limit ω ± 13 approaches v 12 respectively 0, and ν goes to 1. Again we have found a form of the ε-expansion, where the asymptotic behavior for v 11 → 0 is explicit.As for the one-denominator integral, we have a finite part and a part proportional to v −ε 11 .Note that Eq. ( 16) trivially reduces to Eq. ( 15) for v 11 = 0, something that could not be easily seen from Eq. (11).

C. Asymptotic form of the double-massive two-denominator integral
Finally, we consider the double-massive twodenominator master integral, We have already discussed the divergent behavior of its ε-expansion in the introduction, see Eq. ( 2).Using two-point splitting, the double-massive integral can be expressed as a sum of single-massive integrals [5].For the double-massive integral, we have to consider the cases of one or both masses approaching zero.To treat both limits together we will employ a splitting that treats v 1 and v 2 symmetrically and directly extracts the double massless limit.
We define two auxiliary vectors To make v 3 and v 4 massless as well as coinciding with v 1 respectively v 2 in the respective massless limit, we choose . Sketch illustrating the splitting of the doublemassive two-denominator integral in Eq. ( 18).The figure shows the slice x0 = 1 of Minkowski space, the blue circle indicates the intersection with the light-cone where v 2 = 0.
The scalar products of the auxiliary vectors are ).Note that both v 13 and v 24 vanish in the respective massless limits.A graphical illustration of the splitting is given in Fig. 3.
Upon integration of Eq. ( 18), we receive 1,1 (v 24 , v 22 ; ε) .(20) This identity splits the double-massive integral into two single-massive integrals and a massless integral.For these we can use the ε-expansions from eqs. (11) and (15), resulting in the representation of the doublemassive integral The asymptotics of the double-massive integral is manifest in Eq. ( 21), we have a part constant in both massless limits, a part proportional to v −ε 11 , and a part proportional to v −ε 22 .Note that the 1/ε poles cancel between the parts if we expand in ε for finite v 11 and v 22 .For v 22 = 0 we immediately recover Eq. ( 16).
The full expressions for the functions f 0,1 parametrizing the order ε 2 parts of the massless respectively singlemassive two-denominator integral can be found in the appendix.The order ε 2 is included here, since applying the expansion (4) for both v 11 and v 22 may result in a 1/ε 2 pole.In the limit v 11 → 0, we have ω + 13 → 1 and ω − 13 → 0. Analogously, in the limit v 22 → 0, it is ω + 24 → 1 and ω − 24 → 0. Hence, a double massless pole term δ(v 11 ) δ(v 22 )/ε 2 will receive a contribution from the ε 2 coefficient function in the double massless limit.The specific value required for If one is interested in only a single massless limit, say v 11 → 0 while v 22 finite, we may expand v −ε 22 for some explicit simplifications of logarithms.In this case, we find the representation The asymptotic form for v 22 → 0 at finite v 11 is the same upon interchanging v 11 ↔ v 22 .

III. CONCLUSION
We have established ε-expansions with manifest small-mass asymptotics for all massive angular master integrals with up to two denominators.The main results of this paper are the asymptotic expansions of the

• massive integral I
(1) 1 in Eq. ( 12), • single-massive integral I (1) 1,1 in Eq. ( 16), • double-massive integral I (2) 1,1 in Eq. (21).By means of recursion relations derived from IBP identities these results extend to all two-denominator angular integrals with integer coefficients.In the construction of the asymptotic expansion the two-point splitting lemma proved to be an immensely useful tool.It allowed for the extraction of the massless limits in terms of suitable massless angular integrals.
Appendix: Order ε 2 coefficients of the double-massive integral The explicit form of the order ε 2 coefficient functions f 0 and f 1 of the massless respectively single-massive two-denominator integrals appearing in Eq. ( 21) are This generalized polylogarithm of weight 3 and depth 2 can be expressed in terms of classical polylogarithms [38,39].For 0 < y < x < 1 it holds The specific value needed for the double massless limit is S 1,1,1 (1, 0) = 0, which can be trivially read off from the integral representation (A.3).

Figure 1 .
Figure 1.Sketch illustrating the splitting of the massive one-denominator integral in Eq. (9).The figure shows the slice x0 = 1 of Minkowski space, the blue circle indicates the intersection with the light-cone where v 2 = 0.
The kinematic variables X, v 13 , v 24 , and v 34 are defined in the text above; they all depend on v 12 , v 11 , and v 22 .Importantly v 11 /v 13 → 2 for v 11 → 0 and analogously v 22 /v 23 → 2 for v 22 → 0.