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Infrared and extended on-mass-shell renormalization of two-loop diagrams

Using a toy model Lagrangian we demonstrate the application of both infrared and extended on-mass-shell renormalization schemes to multiloop diagrams by considering as an example a two-loop self-energy diagram. We show that in both cases the renormalized diagrams satisfy a straightforward power counting.


I. INTRODUCTION
The generalization of mesonic chiral perturbation theory [1,2,3] to the one-nucleon sector originally turned out to be problematic due to the fact that higher-order loops contribute to lower-order calculations [4] (for a recent review see, e.g., Ref. [5]). This problem was first overcome in the framework of the so-called heavy-baryon approach to chiral perturbation theory [6,7]. More recently, it has been realized that the power counting can also be restored in the original manifestly Lorentz-invariant formulation [8,9,10,11,12,13,14,15,16]. In this context, the infrared (IR) regularization of Becher and Leutwyler [10] (based on ideas of Ref. [8,9]) is the most widely used renormalization scheme 1 for one-loop diagrams [17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]. In Ref. [33] it has been suggested that the IR renormalization can also be generalized to multiloop diagrams. In a recent paper [34], we have reformulated the IR renormalization in a form analogous to the extended on-mass-shell (EOMS) scheme of Ref. [14]. In its new formulation the IR renormalization can also be directly applied to diagrams involving resonances as well as multiloop diagrams. In Ref. [35] the application of the EOMS scheme to two-loop diagrams has been illustrated for the nucleon self-energy "rainbow" diagram in the chiral limit. The aim of this work is to demonstrate the application of the reformulated IR renormalization [34] to two-loop diagrams. For comparison the results of the EOMS renormalization are also given.

II. LAGRANGIAN AND POWER COUNTING
In order to simplify the calculations and-without loss of generality-to make the renormalization procedure more transparent, we suppress the complicated spin and chiral structure of baryon chiral perturbation theory and restrict ourselves to the toy model Lagrangian where π and Ψ are scalar particles with masses M and m, respectively, M ≪ m, and g > 0.
The Lagrangian L 1 contains all interaction terms which are consistent with Lorentz invariance and the invariance under the transformations π → −π and Ψ → −Ψ. 2 Both the IR and the EOMS renormalization are constructed such that, after subtraction, Feynman diagrams have certain "chiral" orders D, which are determined by the following power counting: if Q stands for small quantities such as the π mass M, small external four momenta of π or small external three-momenta of Ψ, then the πΨ interaction explicitly shown in Eq. (1) counts as Q 0 , the Ψ propagator as Q −1 , the π propagator as Q −2 , and a loop integration in n dimensions as Q n , respectively. 1 Since the IR regularization is in fact a renormalization scheme we will further on refer to it as IR renormalization. 2 No particular physical content is meant by imposing these discrete symmetries.

III. APPLICATION TO THE TWO-LOOP SELF-ENERGY
As a specific example of a two-loop diagram, we consider the Ψ self-energy diagram of Fig. 1 (a), where 1/2 is a symmetry factor and with n denoting the number of space-time dimensions. Using the above power counting, we assign the order Q 2n−5 to the diagram of Fig. 1 (a). A dimensional counting analysis [36] (see the Appendix for an illustration) suggests that, if first M → 0 and then p 2 − m 2 → 0, the self-energy Σ Ψ can be written as The functions F (p 2 , m 2 , M 2 , n), G(p 2 , m 2 , M 2 , n), and H(p 2 , m 2 , M 2 , n) can be expanded in nonnegative integer powers of M 2 and, in the following, will be analyzed in detail. First, we consider F (p 2 , m 2 , M 2 , n). The Taylor expansion of F in M 2 is obtained by expanding the integrand of Eq. (3) in M 2 and interchanging summation and integration: .
Inserting Eq. (6) into Eq. (5) and taking into account that p 2 − m 2 counts as order Q, we see that the part which is proportional to noninteger powers of p 2 − m 2 (for noninteger n) is of order Q 2n−5 and therefore satisfies the power counting. The other part is a Taylor expansion in M 2 and (p 2 − m 2 ) and contains terms which violate the power counting. This Taylor series can be obtained by formally expanding the integrand of the original integral of Eq. (3) in M 2 and p 2 − m 2 [14] and interchanging summation and integration. Those terms in the Taylor series which do not violate the power counting contain IR divergences. In the IR renormalization, all terms of the so obtained Taylor series must be subtracted from the original integral. [These subtractions are generated by the counterterm diagrams of Fig. 1 (c)]. However, before the subtraction all IR divergences need to be removed from the subtraction terms [34]. This is necessary to ensure that the subtracted expression does not contain IR divergences. In the following, the omission of IR divergences from the subtraction terms in the IR renormalization is always implied. In the EOMS scheme we only need to subtract those terms which violate the power counting (these terms are free of infrared divergences)-in the present case all terms of order Q 2 or less [14]. Up to and including order Q 2 the subtraction terms of the EOMS renormalization scheme are given by where the coefficients are given by The corresponding subtraction terms of the IR renormalization scheme read 3 with Next, let us investigate G(p 2 , m 2 , M 2 , n) which is identified from Eq. (3) by rescaling k 1 → Mk 1 , k 2 → k 2 and k 1 → k 1 , k 2 → Mk 2 . Both cases produce an overall factor of M n−2 . One then expands the remaining integrands in M, interchanges summation and integration, and adds up the two contributions. Since the integral in Eq. (3) is invariant under the interchange of k 1 and k 2 , we only need to perform one of the above manipulations and then multiply the result with a factor of 2. As a result we obtain where r s = r! s!(r − s)! is a binomial coefficient and the integrals I ij,ab (p 2 , m 2 , n) are defined as For j + b odd, the loop integral of Eq. (10) vanishes, because in that case the integrand is an odd function of k 1 . From this one concludes that the nonvanishing terms in Eq. (9) are proportional to nonnegative integer powers of M 2 . Using a dimensional counting analysis [36], it can be shown that I ij,ab (p 2 , m 2 , n) is of the form ij,ab,l (m 2 , n).
(11) Combining this result with Eqs. (4) and (9), we see that the term nonanalytic in p 2 − m 2 is of order Q 2n−5+a (a ≥ 0) and therefore does not violate the power counting. On the other hand, the first part of Eq. (11) contains terms that are analytic in p 2 − m 2 which, when combined with Eqs. (4) and (9), give rise to contributions which are nonanalytic in M: Note that the first term in Eq. (12) violates the power counting and, since it is nonanalytic in M, cannot be directly absorbed by a counterterm. In solving this apparent puzzle, one has to keep in mind that, in order to consistently renormalize the diagram of Fig. 1 (a), all relevant contributions from counterterms in L 1 of Eq. (1) must be taken into account. In the present case, these countertems originate in the renormalization of the one-loop diagrams of Fig. 2 (a) and (b) with the corresponding expressions and respectively. Both diagrams are assigned the order Q n−3 . Evaluating Eqs. (13) and (14) we find that M 2a and M 2b contain contributions that violate the power counting. To renormalize these diagrams we follow Refs. [10,14] and, up to order Q, find the subtraction terms as and where λ(m, n) = m n−4 Γ(2 − n/2) (4π) n/2 (n − 3) .
The counterterms are then given by for the EOMS scheme and by in the IR renormalization. In a two-loop calculation these counterterms give a contribution to the self-energy as shown in Fig. 1 (b). The corresponding expressions read (20) and where In fact, the last term in Eq. (21) is of order Q 4 , and can therefore be neglected in a calculation up to and including order Q 3 . Noting that we see that in the IR renormalization the self-energy contribution of Eq. (21), which stems from the renormalization of one-loop diagrams, exactly cancels the contributions of Eq. (12) which are explicitly shown, including the part which violates the power counting. In the EOMS scheme no terms that satisfy the power counting are subtracted. Let us finally consider the last term of Eq. (4). The function H(p 2 , m 2 , M 2 , n) is obtained from Eq. (3) by simultaneously rescaling k 1 → Mk 1 and k 2 → Mk 2 , extracting a factor of M 2n−4 , expanding the remaining integrand in M, and interchanging integration and summation, yielding Noting that M/(p 2 − m 2 ) ∼ Q 0 , it is easy to see that Eq. (25) in combination with the factor M 2n−4 of Eq. (4) satisfies the power counting. Furthermore, only nonnegative integer powers of M 2 survive in H. Combining the results above, all terms violating the power counting are canceled in the sum of the diagrams in Fig. 1 in both the IR and the EOMS renormalization. Finally, we have also studied the case where first p 2 − m 2 → 0 and then M → 0, and have explicitly verified that the renormalization procedure remains exactly the same, i.e., the counterterms are the same and the renormalized diagram satisfies the power counting.

IV. CONCLUSION
In conclusion, we have demonstrated that the application of both the reformulated IR renormalization and the EOMS scheme to two-loop diagrams leads to a consistent power counting. In this context, the subtraction of one-loop sub-diagrams plays an important role in the renormalization of two-loop diagrams. The procedure can be applied iteratively to multiloop diagrams. Calculations using the Lagrangian of baryon chiral perturbation theory are more involved, but the general features of the renormalization program do not change. and expand the second factor in Eq. (30) in positive powers of M 2a i −2 . In both cases one obtains integrals of the type d nkkα as the coefficients of the expansion. However, such integrals vanish in dimensional regularization. Therefore the only nontrivial terms in the sum of Eq. (27) correspond to either a i = 0 or a i = 1. Thus we obtain where and I (1) On the other hand, the integral I πΨ (0, p) can be calculated directly without applying the dimensional counting technique. Doing so and comparing with the results given above one sees that the dimensional counting method leads to the correct expressions. While the loop integrals of Eq. (33) have a simple analytic structure in p 2 −m 2 , the technique sketched above can be directly applied to the loop integrals of Eq. (32) when p 2 − m 2 → 0, now using the change of variable k = (p 2 − m 2 ) c jk with arbitrary nonnegative real numbers c j .