Massless sunset diagrams in finite asymmetric volumes

This paper discusses the methods and the results used in an accompanying paper describing the matching of effective chiral Lagrangians in dimensional and lattice regularizations. We present methods to compute 2-loop massless sunset diagrams in finite asymmetric volumes in the framework of these regularizations. We also consider 1-loop sums in both regularizations, extending the results of Hasenfratz and Leutwyler for the case of dimensional regularization and we introduce a new method to calculate precisely the expansion coefficients of the 1-loop lattice sums.


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1 Introduction In an accompanying paper [1] we have computed a free energy and the mass gap in the δ-regime in the framework of chiral perturbation theory without an explicit symmetry breaking term, in finite asymmetric volumes. The computation was done both in dimensional regularization (DR) and in the lattice regularization. Matching the results in two regularizations enables us to establish relations between the 4-derivative couplings appearing in these regularizations. The technical details are provided in the present paper.
In the computation of the free energy and the finite-volume mass gap we encounter one and two loop integrals over the volume. A class of 1-loop sums with dimensional regularization have been considered in detail by Hasenfratz and Leutwyler [2] in DR, we summarize some of their results and also add some more. The main part of the paper deals with the precise evaluation of the 2-loop sunset diagrams for both regularizations. For our purposes we only require diagrams with zero external momenta but the methods are more general. Massive sunset diagrams in finite volume have been treated in detail by Bijnens, Boström and Lähde [3,4], but as far as we can tell their methods are not applicable for zero masses. In appendix B we outline an alternative method of calculating sunset diagrams by introducing a mass, in addition to the finite volume. This setup allows to take the massless limit as well.
Finally in section 6 we consider 1-and 2-loop massless sums in asymmetric finite volumes with lattice regularization. For the 1-loop sums we introduce a new method to compute precisely the coefficients of their expansions in the lattice cutoff.
The motivation to consider lattice-regularized chiral perturbation theory is addressed in [1].

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2 Dimensional regularization at finite volume Our goal is to compute massless one and two loop diagrams in finite volume with periodic boundary conditions within the framework of dimensional regularization. As in [2] we start by considering diagrams in the massive theory.
The free massive propagator is where p runs over a d-dimensional momentum space infinite lattice p = 2π(n 0 /L 0 , . . . , n ds /L ds ) , where d s = d − 1 and n µ are integers and 3) The numerical evaluation of the graphs will be done for spatially cubic volumes L 1 = . . . = L ds ≡ L s , either for the hypercubic case L 0 = L s or for the elongated geometry L 0 > L s . We will dimensionally regularize by adding q extra compact dimensions of size L and analytically continue the resulting loop formulae to q = −2ǫ.

Massive propagator sums in DR
In appendices of their classic paper [2] Hasenfratz and Leutwyler considered the sums

4)
H r (p) = Γ(r)(p 2 + M 2 ) −r , (2.5) where p is now a D = d + q dimensional vector and For convenience, in this subsection we reproduce some of their results, in particular those that we need in [1]. Invoking the identity 1 where f (x) is the Fourier transform of f (p): where v k are integers, one obtains In the sum over w the term w = 0 corresponds to the infinite volume limit (2.12) In particular where γ E = −Γ ′ (1) = 0.577 . . . is the Euler constant. It is useful to separate the infinite volume term to arrive at the representation In this decomposition the volume dependence is exclusively contained in the function g r which is unambiguous for any integer value of D and is entire in r.
The sum occurring in the representation of g r converges rapidly if the sides of the box are large compared to the Compton wavelength, M L µ ≫ 1. If all L µ → ∞ then g r → 0 exponentially. Note if one considers other limits e.g. the temperature to T = 1/L 0 to zero i.e. L 0 → ∞ with L 1 , . . . , L ds fixed, then g r approaches a finite limit For r = 0 this relation reflects the fact that in the limit T → 0 the partition function is dominated by the contribution from the ground state, the quantity ln det D/2L 0 approaching the Casimir energy associated with the d s −dimensional box. For the goal of considering the massless sums we need the properties of g r in the limit M → 0 where infrared singularities occur. Introducing
The dependence on the auxiliary scale L of a given quantity (when the other variables are made dimensionless) is given by its dimension. Using this we will often use L = L s for the spatially cubic volume and denote ℓ ≡ ℓ 0 = L 0 /L s andl = L/L s .
Since the product in (2.19) often appears below we also introduce the ancillary definition (with some abuse of notation) (For the general case of D dimensions we shall also use S D (t/ℓ 2 ) and S D (ℓ 2 /t) defined analogously.) Note that the sum in (2.18) converges very fast for x > 1. The function S(x) satisfies the well known remarkable identity (2.23) Using (2.18) for x > 1 and (2.23) for x < 1 one needs only a few terms in the corresponding sums to calculate S(x) very precisely. Using (2.23) one obtains the representation In (2.25) we introduced the notation [. . .] sub which will be used often below. It is defined by The quantity A r does not contain infrared singularities and has an expansion in M 2 of the form The expansion coefficients 1 depend on d and the ratios ℓ µ : Some values for α s for ℓ µ = 1, ∀µ are given in table 1. Infrared divergences are contained in the incomplete Γ-function B s , in the form of fractional powers of M 2 : The pole in the Γ-function at r = −N , N ∈ N is canceled by a pole occurring in the piece which is analytic in M . Merging the two singularities one obtains a logarithmic contribution
We have where for infinite volume: and Denote the logarithmic derivative of S(x) by satisfying the relation With this one has Putting the parts together we have

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where D r has the expansion in z: For the totally symmetric box ℓ µ = 1 , ∀ µ we have We then have for D ∼ 3: and for D ∼ 4: In our computation [1] we need to keep in consideration order q = D − d terms in G r,1 :

Massless propagator sums in dimensional regularization
We are now in a position to give results for dimensionally regularized momentum sums which can be re-expressed in terms of the massless propagator where the sum is over momenta p = 2π(n 0 /L 0 , . . . , n D−1 /L D−1 ) , n k ∈ Z , and the prime on the sum means that the zero momentum is omitted:

4)
where we have introduced the notationG(x) = ∂ 2 G(x)/∂x 2 0 . The I nm can be obtained by taking the limit of zero mass of the massive sums considered in the previous subsections. Here we will only consider the cases m = 0, 1 and for these we have 4 Now for d = 2: and for d = 4: (3.14) The behavior of g 1 , g 2 for M → 0 is readily extracted from (2.35) and the representations (2.36), (2.40) and (2.43).

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So for the corresponding massless sums we obtain for d = 2:
So for the symmetric case we have

The free propagator with periodic bc in DR
Again we consider a D dimensional volume with L 0 ≡ L t , L 1 = · · · = L d−1 ≡ L s , and L d = . . . = L D−1 ≡ L with periodic boundary conditions in each direction. The goal of this section is to give the analytic basis for writing efficient programs for G(x) for the numerical evaluation of classes of Feynman diagrams in coordinate space. For this purpose we find it convenient to write the finite volume propagator for the massless case, with the zero mode subtracted as where ∆(x) is the infinite volume propagator and the finite volume piece g(x) will be considered in detail in subsection 3.1.2. Note that G(x) satisfies the periodic boundary conditions, while ∆(x) and g(x) do not. The singularity of G(x) is given entirely by ∆(x), accordingly g(x) is a smooth function.

Propagator in infinite volume, D-dimensions
For the infinite volume propagator one has where r = |x| and This is related to the area of the unit sphere

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The first and second "time" derivatives over t = x 0 are given bẏ In particular for D = 4 one has Ω 4 = 2π 2 and Note that by definition (by analytic continuation from D < 2) in DR one has ∆(0) = 0.

Calculating g(x) for general D
We start with the representation Then following [2] we split the region according to where we have used the Poisson summation formula in the form 1 For D = 4 this gives Taking µ = αL 2 /(4π) with some length scale L ≈ L s ≈ L t and α ≈ 1 only a few terms are needed in the sums. Of course, the final result does not depend on α.
It is convenient to use the Jacobi theta function

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The first sum above converges quickly for 1 ≤ u while the second for 0 < u < 1. One has With this one obtains (using the arbitrary scale L) the following representation for g(x) suitable for numerical evaluation: and (3.56) In our computations we will need the expansion of g(0) and ∂ 2 ν g(0) in q = D − d up to first order: Using (2.31) and (2.67) one gets for d > 2 For the double derivatives one gets similarly (cf. appendix A) Some values of ∂ 2 ν g 0 (0) , ∂ 2 ν g 1 (0) for d = 4 obtained from (3.61), and (3.62) are given in tables 6, 7 at different ν, ℓ andl. Here ν = 0, 1, x denotes the time, one of the spatial coordinates, and one of the D − 4 auxiliary coordinates, respectively. The coefficients JHEP06(2016)102  Table 5.
Since g(x) satisfies the relation g(x) = 1/V D , one obtains for d = 4, Expanding in D − 4 this leads to The values given in table 6, 7 satisfy these relations. It turns out (as expected) that ∂ 2 1 g 0 (0; ℓ) approaches the ℓ → ∞ limit very fast.
In D = 4 one has for small x 3.2 Evaluation of some 1-loop integrals using the representation in subsection 3.1 As an illustration of the coordinate method that we will use to compute the sunset diagrams in section 4, we first apply them to the computation of x G(x) 2 for d = 4. The latter was Table 6. Numerical values of ∂ 2 ν g 0 (0; ℓ), ∂ 2 ν g 1 (0; ℓ,l) for ν = 0, 1. already treated in [2] using the momentum-space representation and the result presented in (3.26). We evaluate this Feynman graph in position-space using the decomposition G(x) = ∆(x) + g(x) and writing the result as a sum of different terms, each of which is calculated using appropriate methods, e.g. using subtraction, splitting the integration domain, etc. Comparing the result with the momentum space result of [2] is useful as a test of some subroutines that were used for numerical evaluation of the sunset diagram treated in section 4.  Table 7. Numerical values of ∂ 2 ν g 0 (0; ℓ), ∂ 2 ν g 1 (0; ℓ,l) for the second derivative taken along one of the "extra" dimensions. with (3.68) To separate the divergent terms we will also need integrals over the d-dimensional sphere S with radius ρ, (which will be taken to be 1/2 in the actual numerical calculations) as well as integrals over V 0 \S. Table 8. Values for Σ i and Σ for ℓ = 1 and ℓ = 2.

Massless sunset diagram with DR
In this section we shall calculate the dimensionless quantity 7 The notation W was used in eq. (3.47) of [1]. , At D = 4 only Ψ 4 and Ψ 5 have a pole, the others are finite.

Ψ 2
We split Ψ 2 in two terms Ψ 2b is zero for ℓ = 1. A direct calculation for ℓ = 2 in Cartesian coordinates is not too precise, and we get Ψ 2b = 0.001458 . . . , ℓ = 2. The reason for poor convergence is the integrable singularity at the origin. One can improve drastically the convergence by changing the variables The region η ∈ [0, 1/2], u i ∈ [−1, 1] corresponds to the pyramid with the hyper-face x 0 = +1/2 as basis. 8 The change of variables is illustrated in figure 2. Precise values of Ψ 2a , Ψ 2b for ℓ = 1, ℓ = 2 are given in table 10. 8 These variables are also convenient to describe the part of the pyramid cut out by sphere S:  .9)). This trick improves the precision of integration in the case when one has an integrable singularity at x = 0.

Ψ 3
We also split Ψ 3 in two parts All quantities appearing in Ψ 3a are already available, and there are no problems with numerical computation of Ψ 3b for d = 4. Values of Ψ 3a , Ψ 3b for ℓ = 1, ℓ = 2 are given in table 10. As a check, for ℓ = 1 one has Ψ 3b = 1 4 Σ 3 which gives the same value as in table 10.
Adding values for Ψ 4b we have Here Ψ 4c is the sum of Ψ 4b and the contribution from the constant term in (3.72 As for Σ 1 to separate the divergent terms we separate from the volume a D-dimensional sphere S with radius ρ: Ψ 5a is zero since by symmetry it is proportional to ∆(0). Next for ν = 0 we have The averages appearing above are .
Since S∆ (x)∆(x)r 2 = 0 by symmetry, we obtain As a check one can verify that Ψ 5b (1, 1) = 0 sinceg 0 (0; 1) = 1/4 andg 1 (0; 1, 1) = −1/16. The last two integrals are convergent. Note that in d = 4 one has The integral in Ψ 5c is convergent due to the subtraction (ĝ(x) = O r 4 ) and it is obviously zero for ℓ = 1 due to cubic symmetry. Numerical integration with increasing precision indicates convergence to zero also for ℓ > 1 althoughĝ(x) does not have the cubic symmetry in this case. A closer look shows that the angular integration at fixed r gives zero. This can be explained as follows. Since g(x) = 1/ℓ it follows from (4.21) thatĝ(x) is a harmonic function, ĝ(x) = 0. Further, sinceĝ(x) = O r 4 its expansion in powers of r contains only spherical harmonics of order larger than two. The angular dependence of∆(x) ∝ 4t 2 − x 2 in Ψ 5c is given by a spherical harmonics of order two, hence due to the orthogonality of the spherical harmonics the angular integration indeed gives zero.

Ψ 6 and Ψ 7
The integral Ψ 6 is zero Collecting all terms one gets 9 for the sunset diagram eq. (4.1) The sum of non-singular terms in Ψ(ℓ,l) are collected in W(ℓ). Its values for ℓ = 1, 2 are given in table 10. According to (3.61), (3.62) one has

Checks
For the special case ℓ =l = 1 The lhs. is given by Using (3.74) the rhs. is which agrees with (4.40) up to the 14th digit. The other contribution is Since ∆(x) = −δ(x) and using ∆(0) = 0 one has in agreement with (4.42). Also we checked the results for arbitrary ℓ by computing Ψ in a completely independent way outlined in appendix B. Doing this we obtained W(ℓ = 1) = 0.925362611856 , and W(ℓ = 2) = 0.154824638695. They differ from the results in table 10 in the 14th and 7th JHEP06(2016)102 digits, respectively. We haven't located the source of the discrepancy for ℓ = 2, but the estimate to 7 digits is at present sufficient for our purposes.
Comparing the two approaches, note than in position-space the singularity is only at x = 0, and several terms considered above have to be treated in DR to cure the singularity. In contrast to this, in the momentum-space approach the singularity of the sunset diagram appears in two loop-momentum variables, hence it is more difficult to handle it.

Other results
Ψ has no pole at D = 3, and using similar methods for numerical evaluation as described for D ∼ 4 we obtain 10 (for L = L s = 1, cf. (4.1) and (4.36)). We have also evaluated the sunset integral by similar methods.

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Next for D ∼ 4:  The free Green function G F (x, y) is determined by the following four properties: where Λ is the interior of Λ , and for x, y ∈ Λ Here the second term is due to the subtraction of the zero mode and 11 Note here T is the extent in the time direction not the temperature.

Calculation of the sunset diagram
Next we turn to the numerical computation of Ψ for D ∼ 4 in (5.11), setting initially L = 1 and recovering later the dependence on L. In analogy to the computation of Ψ in section 4 it is advantageous to first separate the infinite-volume propagator: Although h is defined for ℓ → ∞, we find it convenient to use (5.16) with a large but finite ℓ, since the deviation decreases exponentially fast. In this case

Ψ 2
We split The first integral gives For the second one we divide the volume into 8 pyramids, e.g. the one defined by (4.9) with η ∈ [0, 1/2] and u i ∈ [−1, 1]. The advantage of this is that the Jacobian, η 3 cancels the integrable singularity at x = 0. Since the integrand is even in all components x µ one can restrict the integration to u i ∈ [0, 1]. Thereby we obtain Ψ 2b = −0.0176501762 . Note that the parameterization (4.9) is also useful to calculate integrals over V 0 \S, taking 1/ 2 √ 1 + u 2 ≤ η ≤ 1/2. The integral over the whole torus can also be done this way.

Ψ 3
It is convenient to decompose this into three terms   Expanding one has The terms Ψ 5b , ψ Next from (4.28)

Final result for Ψ
Collecting all terms one obtains the final result 15 Note that the terms containing ℓ cancel and one can take here the ℓ → ∞ limit. This result depends on the O (D − 4) term ofḧ(0) which in turn depends onl, the size of the extra dimensions. However, this dependence cancels from a physical quantity, like the mass gap.

Finite volume momentum sums with lattice regularization
In this section we present results for certain one and two loop momentum sums that we require for our computation of the free energy in massless χ-PT with lattice regularization [1]. We work in an asymmetric d-dimensional volume L 0 = L t , L µ = L , µ = 1, . . . , d s and periodic boundary in each direction. We work with the standard lattice action so that the free propagator is given by where the sum is over p µ = 2πn µ /L µ , n µ = 0, . . . , L µ − 1 , V = ds µ=0 L µ andp µ = 2 sin(p µ /2) . In many equations we will set the lattice spacing a to 1.
Forward and backward difference operators are defined by whereμ is the unit vector in the µ-direction, and the symmetric derivative

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Defining the lattice Laplacian by the propagator (6.1) satisfies Some useful relations involving the lattice propagator are given in appendix C.

Some 1-loop momentum sums
We define the following 1-loop momentum sums Some of these are related to each other e.g.

13)
I 01 = 2 , (6.14) I 02 = 6 , (6.15) We are interested in the expansion of these sums for N = L/a → ∞. As has been shown in ref. [7] the 1-loop sums we consider here have a cutoff dependence of the form where δ is determined by behavior of the summand, |k| −δ at small momenta. The 2loop sums, however, have a more general structure. To cover the different cases we use the notation 16 X A = r (X A;r + X A;rx ln N ) N −r . (6.19)

Leading terms
In many cases the infinite volume limit L → ∞ of the sums (provided the limit exists) can be computed to arbitrary precision using recursion relations in coordinate space. This observation was first made by Vohwinkel and described in detail for d = 4 in ref. [8] and later for d = 2 in [9] and for d = 3 in [10]. Some results for d = 4 are given in tables 11 , 12. (According to (6.19) the infinite volume limit is denoted by I nm;0 , J nm;0 , etc.)

Expansion coefficients
To determine the expansion coefficients of the 1-loop sums we have applied two methods.
One is simply to compute the sums to a high precision for a large range of N and fit the data to the expected form, inserting the precisely known leading term when available. Alternatively we obtain the coefficients analytically as integrals involving the theta function (3.52), as described in the next subsection.  Table 12. Values for J nm;0 , J nml;0 , K nm;0 for d = 4.

Lattice analogue of the theta function
Consider the Fourier transform of f (θ) defined on the interval 0 ≤ θ ≤ 2π: Multiplying the equation by (2π) −1 f (θ) and integrating over θ one obtains Using this relation one obtains wherep k = 2 sin(πk/N ) and φ n (z) = e −2z I n (2z) (6.24) where I n (z) is the modified Bessel function, which for integer n is given by For convenience of the reader we have summarized some properties of I n that we use in appendix D. For fixed z, Q N (z) approaches φ 0 (z) exponentially fast. The approach becomes slower with increasing z, but the expansion (D.7) shows that even when the argument increases slower than N 2 one still has lim N →∞ with the difference decreasing faster than any inverse power of N . This is not true for z ∝ N 2 , and for this case one obtains another scaling function. We introduce the lattice counterpart of S(x) by Note that the N → ∞ limit is not uniform in x.
Similarly to the continuum case, the first representation in (6.27) converges very fast for x ≥ 1 while the second one for x ≤ 1. In both cases one needs only a few terms in the corresponding sum. For Q N (z) in (6.23) this corresponds to z > z 0 (N ) and z < z 0 (N ) with z 0 (N ) = N 2 /(4π).
The relatively slowly convergent lattice sums I nm defined in (6.7) can be calculated using S N (x). For m = 0 one has where ℓ µ = N µ /N . (N is arbitrary and one can choose it to be the spatial size, N = N s ).

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For m > 0 It is useful to split the integral and write (for ℓ 1 = . . . = ℓ d−1 = 1, ℓ 0 = ℓ and general d) where To obtain the expansion of I nm for large N we need the asymptotic expansion of S N (x) (6.33) in the next subsection.

Asymptotic behavior for N → ∞
As discussed before, for z = z N ∝ N α with α < 2 the correction term Q N (z N ) − φ 0 (z N ) goes to zero exponentially fast for N → ∞.
Expanding (6.27) in 1/N 2 one obtains an asymptotic expansion (6.34) As mentioned above the expansion (6.33) is not uniform. We will also need the behavior of the corresponding terms at x = 0: Introducing z = yN α with 0 < α < 2 one has S as N 4πN α−2 y ∼ N φ 0 (N α y) . (6.36)

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This means that the singularity of S as N (x) at x ∼ 0 matches the asymptotic behavior of φ 0 (z) = e −2z I 0 (2z) for z → ∞. The relation can be checked using the asymptotic form (D.6) and S(x) = x −1/2 1 + O e −π/x . The relation between the x > 1 and x < 1 regions for S(x) is given in (2.23). Differentiating it one obtains the corresponding relations between the derivatives of S(x).
At x = x 0 (N ) the difference S N (x) − S(x) changes sign: for x < x 0 (N ) one has S N (x) < S(x) while for x > x 0 (N ) one has S N (x) > S(x). Interestingly, for N ≥ 8 one has with a very good precision x 0 (N ) ≈ 4πz 0 /N 2 where z 0 = 0.06447351504.
For N → ∞, N 0 = ℓN in d-dimensions one has for the bracket appearing in the integrand of (6.32) Their leading singularity for x → 0 is given by Γ(m + 1/2) Γ(1/2) (12m 2 + 20m + 4dm + 8d + d 2 ) × ℓ 2m+1 x −(m+2+d/2) . (6.43) As illustrated in the next subsection, for the case when I nm has a finite N → ∞ limit (i.e. 2(n − m) < d) one obtains 17 Expanding in powers of 1/N 2 one gets for the coefficient of N −ν Consider I 10 for d > 2 in a N 0 × N ds volume (N 0 = L 0 /a = N ℓ, N = L s /a): (6.46) One can show that for z 0 ∝ N 2 only the first term contributes to the constant piece I 10;0 and one obtains 18 To calculate the 1/N r corrections we consider the differences and For z 0 = z 0 (N ) = cN α , where 1 < α < 2 both integrals (6.48) and (6.49) go to zero for N → ∞ exponentially fast. We have Here the z 0 dependence should cancel, i.e. Q as N (z 0 ) and φ 0 (z 0 ) should have the same asymptotic behavior for z 0 (N ) = cN α and N → ∞. Note that the argument x 0 (N ) = 4πz 0 (N )/N 2 ∝ N α−2 of the functions S(x) in (6.50) goes to zero in this limit. Hence the contributions from the large-z asymptotic of φ 0 (z) and the small-x asymptotic of S as N (x)/N cancel each other. This is indeed the case, one has 19 The result of numerical integration with MAPLE for d = 4 agrees to 27 digits with the exact value given in table 11. 19 The N -dependence on Q as N (z) cancels in the asymptotic expansion.
Repeating the steps used above one gets for the expansion coefficients of I 21 : with Again, for d = 3, 4 I 21;d−2 is related to the corresponding DR shape coefficient e.g.
So far in this subsection we have only considered sums which have a finite infinite volume limit. As an example of a sum which does not have this property we consider I 20 . For d = 3 I 20 is linearly divergent (see (6.74) , (6.75)).
In the rest of this subsection we only consider the case d = 4 for which I 20 is logarithmically divergent for N → ∞. Here we will separate the ∼ log(N ) and the constant terms. 20 Restricting also to ℓ = 1 we have where x 0 = 4πz 0 /N 2 . Choosing z 0 = cN 2−ǫ with some fixed small ǫ > 0 in the first term one could replace Q N (z) by φ 0 (z) up to exponentially small corrections.
20 For our work in ref. [1] the constant term is actually needed only for the renormalization.

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One has In the second integral of (6.63) one can replace S N (x) by S(x) up to O 1/N 2 correction.
Adding the two terms and for large N one obtains One can repeat these steps for the general case of I nm . Using the representation (6.32) one reproduces the form of the expansion given by (6.18). The log N term comes from the 1/x and 1/z terms of the corresponding integrands while in the rest one can set x 0 = 0 and z 0 = ∞, as in (6.64) and (6.65). Finally, the coefficient N 2(n−m)−d in front of the second integral in (6.32) reproduces the prefactor N δ−d of the sum in (6.18).

Expansions for d = 4
We are interested in the expansion of the lattice sums for N −1 = a/L → 0 (at a fixed aspect ratio ℓ) up to and including the O a 4 /L 4 terms. For the sums we require we have:  The coefficients I nm;ν can be calculated (at least for the cases with finite infinite-volume limit) from (6.45).
Next consider the lattice sum J 31 (see (6.8)); for d = 4 one has In tables 13 and 14 we give values of the coefficients above for ℓ = 1, 2, 3 and ℓ = 4, 5 respectively using the integral representation (6.45) with MAPLE. We checked that all results agreed to at least 12 digits in all cases with fits of the data (using the precise infinite volume results when available).  Here only I 20 , I 30 and I 31 diverge for N → ∞. However, also in these cases the coefficients I 20;1 , I 30;−1 , I 30;1 , I 31;−1 , I 31;1 are correctly given by (6.45).
The shape coefficients I 10;1 and I 20;−1 are related to β n defined in (2.42) through: Further there are relations to β n defined in [11]: In this subsection we consider only the ℓ = 1 case and concentrate on terms which do not vanish in the infinite volume limit. For the logarithmically divergent sum I 10 we have where x 0 = 4πz 0 /N 2 . Choosing z 0 = cN 2−ǫ with some fixed small ǫ > 0 in the first term one could replace Q N (z) by φ 0 (z) up to exponentially small corrections. One has In the second integral of (6.80) one can replace S N (x) by S(x) up to O 1/N 2 correction.

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Adding the two terms and for large N one obtains (6.83) Evaluating the integrals one has For ℓ = 1 we trivially have For n ≥ 2 in the large N limit the leading term of I n0 is proportional to N 2n−2 . The corresponding coefficient is given by (see [12]

Some 2-loop momentum sums
In this subsection we consider only the following 2-loop lattice sums which appear in the computations [1]: It is sometimes useful to write where where where W 3c;0 is the infinite volume sum which can be obtained for example by computing W 3c for ℓ = 1: The coefficient W 3a;2 is related to ψ, β 1 , β 1 defined in [11] through W 3a;2 = ψ + 1 3 β 1 β 1 . (6.114)

Case d = 4
Again we have obtained the expansions by fitting the sums for a range of N for fixed ℓ.  These are satisfied by the numerical values above.

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Using the Schwinger representation: We then break the t k integrations into parts (t 0 = L 2 /(4π)): and considered these separately. The advantage of this procedure is that in each case the x integration can be done first analytically. The disadvantage over the coordinate space method is that there are many terms to be considered and the integration over the t i is often quite difficult. The term W (1) is the simplest: and for x → ∞ where U 0 (p) = 1 , Eq. (D.7) can also be regarded as asymptotic expansion for x → ∞, ν = fixed.
For large x one has ρ = −1/(2x) + O x −3 . This shows that for x n n α with α < 2 for n → ∞ e −xn I n (x n ) decreases exponentially fast. Therefore  Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.