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Short-range nuclear forces in singlet channels

Continuing our effort to build a consistent power counting for chiral nuclear effective field theory (EFT), we discuss the subleading contact interactions, or counterterms, in the singlet channels of nucleon-nucleon scattering, with renormalization group invariance as the constraint. We argue that the rather large cutoff error of the leading amplitude requires O(Q) of the EFT expansion to be nonvanishing, contrary to Weinberg's original power counting. This, together with the ultraviolet divergences of two-pion exchanges in the distorted-wave expansion, leads to enhancement of the 1S0 counterterms and results in a pionless-theory-like power counting for the singlet channels.


I. INTRODUCTION
The power counting is one of the essential ingredients of any effective field theory (EFT), which not only keeps track of an infinite number of operators and Feynman diagrams but also estimates a priori the neglected contributions for a given order. Naive dimensional analysis (NDA), a cornerstone of Weinberg's original power counting [1] (WPC) for few-nucleon systems, is often employed to assess the size of coupling constants: each derivative on the Lagrangian terms is always suppressed by the underlying scale of chiral EFT, M hi . Though phenomenologically successful [2][3][4][5][6], WPC has been shown to be inconsistent with the principle of renormalization group (RG) invariance [7][8][9][10][11][12][13][14][15][16][17], especially in the triplet channels where the singular attraction of one pion exchange (OPE) calls for modifications to WPC at as early as leading order (LO) [10]. The issues of RG invariance are less acute in the singlet channels since the LO amplitudes of WPC for these channels are indeed RG invariant [7,10,18,19]. But a modification to WPC for the subleading counterterms in the singlet channels has been argued in Refs. [11,14,20] to be necessary. Following our investigation of the triplet channels [16,17], we use RG invariance as the guideline to study the subleading counterterms of the singlet S and P waves.
The EFT expansion of the T -matrix at low energies has the quintessential form, where Q denotes generically external momenta, n the counting index, M lo low-energy mass scales and F n (x) the non-analytic functions from loop integrals. Even though the nonperturbative unitarity requires any nonrelativistic, nonperturbative T -matrix to scale as Q −1 , we choose to label LO as O(1) so that one does not need to change the standard chiral power counting for irreducible pion exchange diagrams, e.g., OPE is O(1). Subleading orders of the EFT expansion are labeled by their relative correction to LO, i.e., next-to-leading order (NLO) by O(Q/M hi ) or O(Q) for short, and next-to-next-to-leading order (NNLO) by O(Q 2 /M 2 hi ) or O(Q 2 ), and so on. In any EFT calculations employing a ultraviolet (UV) momentum cutoff Λ, the cutoff independence-RG invariance-of the T -matrix is usually imperfect at a given order. There usually exists a residual cutoff dependence, or a cutoff error, of the T -matrix with a large but finite cutoff, which vanishes though as Λ → ∞. Seen in this light, RG invariance needs a little more careful interpretation. Because (i) the cutoff error is part of the theoretical uncertainty at a given order and (ii) the theoretical uncertainty is, by definition, of the same order as the next-order EFT correction, the cutoff error must be smaller than or of the same order as the next-order correction. This seemingly trivial statement constrains power counting in a non-trivial way when certain order is considered vanishing. Of our interest is WPC, in which the O(Q) corrections have long been deemed to be zero; thus, the theoretical uncertainty for LO is considered by WPC to be O(Q 2 ).
It follows that the LO cutoff error should vanish at least as fast as Q 2 /Λ 2 : While this is the case for the triplet channels [8,9,21], we will show that it is not for 1 S 0 , which forces us to modify WPC for 1 S 0 at subleading orders even though its LO satisfies RG invariance.
If a counterterm is not required by RG invariance at O(Q n ) but is counted O(Q n ) in NDA, we will follow NDA to power count that counterterm. In other words, we do not minimize the number of counterterms at a given order using RG invariance as the criterion [17,22]. The rationale for this is as follows. RG analysis does not study the degrees of freedom that are not built in the effective Lagrangian, such as non-Goldstone bosons and/or very heavy excited baryon states. To avoid underestimating the contributions of these degrees of freedom to counterterms, we set the minimal size of counterterms as the one given by NDA.
We will treat subleading potentials as perturbations on top of the LO T -matrix, which is nonperturbative iteration of the LO potential. This is sometimes casually called perturbative renormalization. Underlying this approach is the point of view that power counting should be done at the level of physical observables, in our case, the on-shell scattering amplitude. Infrared enhancement due to nucleon intermediate states should be incorporated into the power counting [23,24] rather than be used as the pretext to settle for WPC. However, this is not to say that a consistent power counting for the nonperturbative approach, in which the full iteration of the whole potential is performed, cannot be found. For the development in this direction, we refer the reader to Refs. [25][26][27].
Our paper is structured as follows. In Sec. II, the subleading counterterms are classified into three categories according to the diagrams that drive their RG evolution. After a short review of LO in the singlet channels, we discuss in Sec. III the rather large cutoff error of the LO 1 S 0 amplitude and the consequence of that for power counting. We analyze in Sec. IV how the nonvanishing O(Q) counterterm and two pion exchange contribute to O(Q 2 ). This is followed by a discussion and a conclusion offered in Sec. V.

II. EVOLUTION OF SUBLEADING COUNTERTERMS
The LO T -matrix, T (0) , arises from the full iteration of OPE (in low partial waves) and a set of counterterms that ensure the RG invariance, which means that T (0) (k; Λ) is independent of Λ when Λ ≫ k, where k is the magnitude of the center-of-mass momentum. Subleading orders are given by perturbative insertions of higher-derivative counterterms and/or irreducible multiple pion exchanges into LO. Although renormalization at LO [19,28] is far more intricate, we expect to have a better visualization of renormalization at subleading orders by forming a fairly simple correspondence between a loop diagram and the counterterm to subtract its UV divergences, much like ones that exist in perturbative EFTs. In Wilson's language of RG analysis, this counterterm is the one that "evolves" most significantly when the cutoff of the loop diagram is rescaled from Λ to a smaller value, Λ ′ , but remains large in the sense Λ ′ ≫ k. It will help our discussion to classify loop diagrams and their corresponding counterterms into the following three categories.

A. Residual counterterms
The first class of diagrams are actually those of T (0) . Of course the LO contact operators, by definition, are the counterterms to renormalize T (0) ; however, in order to systematically remove the residual cutoff dependence of T (0) , one must take account of the contact operators with more derivatives than the leading one. We call those higher-derivative operators the residual counterterms for T (0) . Even though they are not as important as the LO counterterms or OPE, they might be more important than multiple pion exchanges which start to contribute at O(Q 2 ). Its important for us to find a way to estimate their sizes before calculations are carried out.
The authors of Refs. [11,20,29] have attempted to analyze both LO and residual counterterms using the Wilson RG equation, with OPE as the only long-range force. This is a very difficult task and several assumptions were made in Refs. [11,20,29]. Energy and momentum dependences of contact operators were assumed to be independent of each other. But we know that, when treated as perturbation on top of LO, they can be related by the equation of motion. The RG invariance of the off-shell T -matrix was imposed, though only the on-shell quantities need to be RG invariant.
While this excessive requirement cannot be deemed wrong, one may be concerned that the resulting power counting demands more counterterms than necessary, only to ensure the RG invariance of the off-shell part of the T -matrix. What is most debatable is perhaps the existence of a fixed-point solution to the RG equation, around which the power counting is obtained. Although this appears to be reasonable in the singlet channels, it is clearly at odds with the running of counterterms in the attractive triplet channels, where a limit cycle-like behavior was observed [8,10].
Instead of performing a comprehensive analysis of the RG equation, we impose a narrower definition for residual counterterms: the correction brought by the residual counterterm is of the same size as the cutoff error: [8,9,21]. Thus, the correction due to the 3 S 1 residual counterterm is rated as O(Q 2 M 1/2 lo Λ −5/2 ), less than the leading two pion exchange (TPE0), which is O(Q 2 /M 2 hi ). It should now become apparent that we use residual counterterm merely as a mnemonic device to reflect the order of magnitude of the LO cutoff error. With this notion we can interpret the inequality (2) as follows: WPC requires the residual counterterm to be no more important than TPE0. It is indeed true for the triplet channels, but, as we will show, not so for 1 S 0 .

B. Primordial counterterms
Better known are the second class of diagrams: irreducible multiple pion exchanges evaluated in the plane, or free spherical, wave basis. Primordial counterterms [17] are the contact operators necessary to subtract the divergences of these pion exchange diagrams.
On the basis of "naturalness", the power counting of the primordial counterterm should be the same as that of the pion exchange diagram in question, which is reliably handled by WPC. For example, TPE0 is O(Q 2 ) and its primordial counterterm, a second-order polynomial in momenta, is counted O(Q 2 ) as well.

C. Distorted-wave counterterms
The third class of diagrams are insertions of irreducible multiple pion exchanges into LO, that is, pion exchanges sandwiched between distorted waves-the LO wave functions, ψ k . The counterterms to absorb the divergences of these diagrams are called distorted-wave counterterms.
Coordinate space provides the best stage for qualitative discussion of distorted-wave UV divergences. Any reasonable UV regulator will roughly separate radial coordinate into two parts: the inside, 0 < r Λ −1 , and the outside, Λ −1 r < ∞. Details of the regulator decide how sharp the separation is. When contact interactions are present at LO, such as in both S waves, the inside and outside part of the LO wave function, ψ in k and ψ out k , are respectively subject to the LO contact potential and OPE. The LO contact potential and OPE have different short-distance structures, which result in different short-distance behaviors of ψ in k and ψ out k . For instance, as shown in, e.g., Refs. [18,30], ψ out k (r) in 1 S 0 has an irregular component diverging like ∼ 1/r near r ∼ Λ −1 , which differs very much from a free spherical wave. However, since the LO contact potential is always well defined upon regularization, the inside wave function ψ in k is not drastically different from a free spherical wave.
The distorted-wave matrix element of a subleading contact potential, ψ k |V sub S |ψ k , is dominated by the integration over the inside region. Since ψ in k (r) behaves similarly to a free wave at short distance, ψ k |V sub S |ψ k is expected to be as UV singular as its free-wave counterpart, i.e., plane wave|V sub S |plane wave . On the other hand, the distorted-wave matrix element of a subleading long-range potential, ψ k |V sub L |ψ k , is mostly decided by the integration of ψ out k (r). The irregular component of ψ out k (r), if present, could make ψ k |V sub L |ψ k more divergent than its free-wave counterpart. Therefore, the primordial counterterm that renormalizes a given multiple pion exchange in the plane wave basis may no longer renormalize the same pion exchange between distorted waves. If it does not, the distorted-wave counterterm will have to be more singular than its primordial counterpart, that is, will have more derivatives. This indeed happens to a toy model considered in Ref. [31] and in [14][15][16] and 1 S 0 [14], among possibly other channels of N N scattering. We will reproduce in Sec. IV the distorted-wave enhancement of the 1 S 0 subleading counterterms, which was first shown in Ref. [14].
The distorted-wave multiple pion exchanges, ψ k |V sub L |ψ k , are power counted as the same as the free-wave matrix elements because it has been established at LO that any number of insertions of V (0) -the LO potential-does not enhance or diminish the amplitude. Based on, again, naturalness, it follows that the distorted-wave counterterm is power counted as the same as its primordial counterpart, even though one of them may have more derivatives.

III. LO AND ITS RESIDUAL COUNTERTERMS
We consider first 1 S 0 . The LO amplitude is constructed by resumming V (0) , which is a constant counterterm C1 S 0 plus OPE, where with p ′ ( p ) the outgoing (incoming) momentum in the center-of-mass frame, q ≡ p ′ − p, g A = 1.26 and f π = 92.4 MeV. We can redefine C1 S 0 such that the point-like piece embedded in OPE is separated from the Yukawa potential and rewrite long and short-range potentials for 1 S 0 , respectively, where α π ≡ g 2 A m N /16πf 2 π ∼ (290MeV) −1 and 4π/m N is a common factor of nonrelativistic Feynman amplitudes. Here we have dropped the subscript 1 S 0 to simplify the notation. C has been expanded in anticipation that the running of C(Λ) with respect to Λ could be modified at each order, though the number of physical inputs to determine C remains one (or stated differently, the boundary condition for the RG flow of C(Λ) remains fixed). There is little dispute about the power counting of (renormalized) C [20,24,32]: where Q ∼ M lo ∼ α −1 π . The relatively large size of α −1 π , compared with m π , is crucial for the singlet-channel success of KSW scheme [23,33], in which OPE is treated perturbatively. But here we take the view point that α −1 π is an infrared mass scale because (i) it is still smaller than M hi and (ii) the perturbative Yukawa works less well when m π takes a larger value but still stays within the validity of chiral EFT.
To study the UV divergences, we find useful the elegant machinery developed in Ref. [7]. First, we define the resummed Yukawa amplitude: where E ≡ k 2 /m N is the center-of-mass energy. The superscript "Λ" only serves as a reminder that every integral is in principle regularized. While the LO P and higher wave amplitudes are given solely by the resummed Yukawa, the LO S-wave amplitude needs summing up insertions of C (0) to all orders, which is eventually given by [7]  where (11) Figure 1 shows the diagrams that, when resummed, represent I k and χ(p; k). The power counting of I k and χ(k; k) will follow, e.g., that of the first diagram of their resummation series: and χ(k; k) ∼ 1.
Although we cannot calculate analytically I k , χ(p; k) or T Y , the dominant UV divergences can be captured by noticing that every insertion of V Y -combined with the Schrödinger propagatorsuppresses the UV divergences by 1/Λ [7]. Therefore, the most significant cutoff dependences of I k are in the first two diagrams of the lower row in Fig. 1: the first is linear and the second is with 4π Here I R k is the finite part, κ π = α π m 2 π and β i are dimensionless and depend on the details of the UV regulator. While β 0 and β 1 are numbers coming out of the first two diagrams, the 1/Λ cutoff dependence, β 2 , receives contributions from all the diagrams; therefore, β 2 is a nonperturbative function of k, κ π and m 2 π . In the spirit of keeping track of 1/Λ cutoff dependences, one finds that where χ R k is the finite part and γ 2 is another dimensionless function. As for T Y , we notice that its residual cutoff is only O(κ π k 2 /Λ 3 ).
We would like to have available the coordinate space form of the LO wave function, which will be useful in analyzing the distorted-wave counterterms for TPEs. Since they are somewhat out of the main line of our discussion, the relevant expressions concerning the wave function are relegated to Appendix A.
Introducing the renormalized coupling, C R , such that we can rewrite the on-shell T -matrix as As Λ → ∞, this is no more than reproducing one of the results in Ref. [7]. However, one can go further and infer from the cutoff error some information about the subleading 1 S 0 counterterms.
Using Eqs. (12) and (14), power counting (7), I k ∼ m N Q 4π and χ(k; k) ∼ 1, one finds, as promised, that the cutoff error is O(k 2 /M lo Λ): where V (1) S denotes the short-range part of the O(Q) potential, and G is the Schrödinger propagator.
If the loop integrals in χ(k; k) (10) and I k (11) are dimensionally regularized, the LO cutoff error (17) vanishes. But it is a model-dependent thinking to count on a specific regulator to be superior and to ignore the cutoff error that may arise with other regulators.
Higher partial wave amplitudes are decided by T Y alone. The quite small cutoff error of T Y , O(κ π k 2 /Λ 3 ), means that the residual counterterm for 1 P 1 is no more important than O(Q 2 ) and that WPC does not need to change for 1 P 1 : Through the steps shown in Appendix B, one can rewrite the O(Q) 1 S 0 amplitude (18) in a more comprehensible form: where V (0) ( x ) is V (0) represented in coordinate space. Since C (1) does not incorporate any new physical information, it is at our disposal to choose the value of C (1) so long as it helps renormalization. With we choose the value of C (1) such that Now we can express T (1) in terms of renormalized quantities: It is obvious that, for T (1) to be O(Q), the scaling of D R must be in comparison with C R (7). The interpretation of T (1) becomes particularly simple at the chiral limit where T Y vanishes: the D term plays the role of the effective range.

IV. O(Q 2 )
A. where with the regularized Dirac function δ Λ ( r ) defined by Eq. (B9). For the related computational details of the above equation (and Eqs. (29) and (30)), we refer the reader to Appendix B. The first term in the bracket of Eq. (26) does not bring more information than T (1) (24); it merely restores the unitarity up to O(Q 2 ). The second term has new structure, which becomes more apparent at the chiral limit where, since V Y vanishes, it resembles the shape parameter of a contact-only theory.
With power counting (7) and (25) and Eqs. (13) and (15), one sees that the "shape parameter" term is O(Q 3 /M 2 hi (M lo + β 0 Λ)). We cannot relegate it to O(Q 3 ) even though it vanishes at Λ → ∞ because β 0 is not necessarily O(1) or larger. In fact, if dimensional regularization is used β 0 is zero, which makes the shape parameter term O(Q 2 ) [23]. To control this piece of regularization dependence, we must add a four-derivative counterterm Before considering TPE0, we remark that the other four-derivative term E (p ′ 4 + p 4 ), when treated as perturbation, is redundant: With Eqs. (24) and (29), T E can be expressed as a combination of the C and E operators. A more general argument is of course the field redefinition inspired by the nucleon equation of motion [28].
The analytic part of V (0) 2π (q) is a second-order momentum polynomial, i.e., its primordial counterterm, which, when projected onto 1 S 0 , is nothing but the D term. The insertion of the D term into T (0) is shown in Sec. III. We now consider the matrix element of the non-analytic part, which diverges as r → 0 in coordinate space: V (0) 2π (r) ∼ 1/(M 2 hi r 5 ). This was first shown in Ref. [14], though in a slightly different notation than ours. This is perhaps most readily done in coordinate space where V (0) 2π is diagonal, where ψ k (r) is the LO 1 S 0 wave function. The "outside" part of ψ k (r) (Λ −1 r) is subject to the Yukawa and is dominated by the irregular solution H k (r) (A3) at short distance, which diverges as 1/r near r ∼ Λ −1 . As a consequence, the UV divergence of T (2) 2π is captured by the integration of the outside wave function from any infrared length down to r ∼ Λ −1 , where N /C (0) is RG invariant (see Appendix A) and "F.T." refers to finite terms. ρ i are functions of κ π /Λ and have at most logarithmic dependence on Λ. While ρ 0 Λ 4 and ρ 1 k 2 Λ 2 can be respectively subtracted by C (2) (27) and D (1) (28), the divergence proportional to ρ 2 k 4 ln Λ needs E (0) (29) to cancel. That is, the 1 S 0 distorted-wave counterterm of TPE0-the E term-has two more derivatives than the primordial counterterm-the D term.
Using the fact that a multiple pion exchange with O(Q n /M n hi ) correction to TPE0 behaves as 1/r 5+n at short distance and repeating the above procedure, we can eventually conclude that, for any multiple pion exchange, the 1 S 0 distorted-wave counterterm is a momentum polynomial with two more powers than their primordial counterpart.
For the sake of consistency, we have kept track of residual cutoff dependence at each order and have assured ourselves that they are not lower-order than next-order correction. Besides being an increasingly cumbersome task, it is less important to show explicitly the cutoff error at each order; the power counting is much less demanding on the RG invariance because no certain order is considered vanishing beyond O(Q) thanks to multiple pion exchanges.
We have seen two motivations to promote the E counterterm to O(Q 2 ): (i) to control the regularization dependence of two insertions of the D term and (ii) to absorb the distorted-wave UV divergences of two pion exchanges.
Unfortunately, the integral in Eq. (32) cannot be evaluated analytically; thus we cannot express the full O(Q 2 ) amplitude in terms of the previously defined renormalized building blocks. But the structure of V S at O(Q 2 ) will suffice in the numerical calculations carried out later: B. P wave The distorted-wave counterterm for TPE0 in 1 P 1 is the same as the primordial counterterm because, without an irregular component, the LO P -outside wave solution cannot make the distorted- wave counterterm more singular than the primordial one. It follows from this, combined with the observation that the residual counterterm for 1 P 1 is not larger than TPE0, that WPC does not need to change for 1 P 1 ; a single P -wave counterterm is what is needed for O(Q 2 ) and O(Q 3 ),

C. Numerics
We compare our EFT calculations with the Nijmegen partial wave analysis (PWA) [34]. The expressions for the delta-less TPEs from Ref. [3] are adopted here. Plotted in Fig. 2  Unlike in the triplet channels, the analytical arguments for renormalizability in the singlet channels are quite solid. So it is less crucial to examine numerically the cutoff (in)dependence of the EFT amplitudes. However, it is still reassuring to see that the O(Q 2 ) curve with Λ = 1 GeV is closer to the Λ = 2 GeV curve, suggesting the cutoff independence for large Λ.
Although WPC is intact for 1 P 1 , we plot in Fig. 3 1 P 1 phase shifts, for completeness. The cutoff independence is rather trivial for 1 P 1 ; therefore, only Λ = 1.5 GeV is employed. Since going to O(Q 3 ) in 1 P 1 is much easier than in 1 S 0 , we include O(Q 3 ) results as well, in which the  [2,36,37], with the πN ∆ low-energy constants determined by the πN scattering data around the delta peak [38][39][40], will improve the convergence of chiral N N EFT. In fact, aside from the open issues of power counting counterterms, the delta-ful nuclear forces have been shown to achieve a more rapid convergence in the two-nucleon [41,42] and, on a more qualitative level, in the three-nucleon [43] sectors.

V. DISCUSSION AND CONCLUSION
We have studied how RG invariance constrains in the singlet channels the structure of subleading counterterms, with S and P waves as the examples. Our analysis shows a hierarchy of 1 S 0 counterterms that resembles the pionless theory while WPC remains unchanged in 1 P 1 and higher singlet partial waves.
To facilitate the discussion, the subleading counterterms are classified into three categories according to the loop diagrams that drive their evolution (see Sec. II). The residual counterterms eliminate the small cutoff dependence of the LO amplitude, in order to achieve the exact RG invariance. The primordial and distorted-wave counterterms are the short-range operators necessary to absorb the divergences of multiple pion exchanges sandwiched between free and LO interacting states, respectively.
We have argued that RG invariance provides two mechanisms to enhance, relative to WPC, the short-range forces in 1 S 0 . (i) As the residual counterterms for the LO amplitude, they scale similarly to the contact interactions of the pionless theory [23,44]: for a generic 1 S 0 counterterm C 2n with 2n derivatives, (ii) As the distorted-wave counterterms for multiple pion exchanges, they are enhanced by , but only starting from the four-derivative term, Since the enhancement due to the residual counterterms dominates, we power count 1 S 0 contact interaction according to (35), as if the theory were the pionless one. In particular, the O(Q 3 ) counterterms have the following structure: Because of the uncertainties associated with c i in TPE1, we refrain from actually calculating O(Q 3 ) of 1 S 0 . Summarized in Table I is our power counting for the two-nucleon sector in both singlet and triplet channels for S and P waves.
We scrutinize WPC with a more stringent interpretation of RG invariance: not only should the cutoff dependence become vanishingly small for Λ M hi , but it must vanish sufficiently fast so that the accuracy claimed by the power counting is consistent with the cutoff error. This leads to a crucial conclusion in our analysis that, contrary to WPC, O(Q) of the EFT expansion does not vanish. Instead, O(Q) is made of one insertion of the two-derivative 1 S 0 counterterm: D/2(p ′ 2 + p 2 ). Although we are not the first to propose this, our argument, that the cutoff error of the LO amplitude is one order lower than TPE0 and has to be corrected by the D term alone, provides some new insights. For instance, unlike Ref. [7] (also discussed later in Ref. [45]), our rationale is a priori and does not rely on the numerical value of D in a particular renormalization scheme.
A full, nonperturbative RG analysis, with OPE as the only long-range force, of the counterterms was attempted in Refs. [11,20], which also concluded that the D counterterm is more important than TPE0. Although the nonperturbative RG analysis appears to be free of any guesswork for obtaining power counting, the robustness of the conclusions of Refs. [11,20] is obscured by the assumptions made therein to derive and solve the RG equation. On the other hand, our approach can be viewed as the explicit, order-by-order examination of an ansatz-the proposed power counting-to the RG equation. If RG invariance can be shown to hold at all orders, which we could not rigorously achieve though, we cannot think of any reason why the proposed power counting could not be one of the solutions to the RG equation.
It is instructive to compare the power counting of 1 S 0 with that of the attractive triplet channels.
A non-vanishing O(Q) arising in 1 S 0 but not in the triplet channels has everything to do with the fact that OPE is regular (1/r) in 1 S 0 but singular (1/r 3 ) in the triplet channels. It is interesting that the singular attraction of OPE costs a few more LO counterterms in the attractive triplet channels (e.g., 3 P 0 and 3 P 2 − 3 F 2 ) but in the mean time it avoids the pionless theory-like proliferation of subleading counterterms.
The distorted-wave enhancement to the singlet-channel short-range forces occurs in only S wave ( 1 S 0 ), and it affects the power counting to a lesser extent than that of the residual counterterms.
In contrast, the distorted-wave enhancement in the attractive triplet channels takes place in higher partial waves ( 3 P 0 , 3 P 2 − 3 F 2 , etc.) but not in S wave, and it plays more important role in power counting than the residual counterterms.
where ξ n (x) and δ n (x) are analytic functions around x = 0 and can be further expanded to obtain the expansions of J k (r) and H k (r) in powers of r.
On the other hand, the three-dimensional (in-state) wave function is related to the LO T -matrix by Therefore, ψ k (0) is given by Using Eqs. (9), (10) and (11), we obtain N and θ in Eq. (A2) are functions of C (0) (Λ) and Λ because the inside and outside wave functions need to match near r ∼ Λ −1 when k = 0 or any other low momentum where we decide to fit C (0) (Λ). In the meantime, the behavior of ψ k (r) at r → ∞ must be RG invariant in order for extracting scattering observables, ψ k (r) = N C (0) χ(k; k) (C (0) ) −1 − I k H k (r) + θ(C (0) , Λ)J k (r) , r ≫ Λ −1 .
Recalling that χ(k; k)/[(C (0) ) −1 − I k ] is RG invariant and comparing the above wave function at r → ∞ to the form of an asymptotic state, we find that N /C (0) and θ are both RG invariant.

Appendix B: Useful integrals for subleading T -matrix
We briefly describe the computations of the integrals stemming from insertions of subleading counterterms into the LO T -matrix, such as Eqs. (21), (26), (29) and (30). Note that the rather formal results shown here are not entirely independent of the regulator and we do not use the expressions derived here in the numerical calculations. However, they have enough mathematical rigor to illustrate how renormalization is carried out.
When evaluating (1 + T (0) G)V (1) S (GT (0) + 1) (21), we need to know the following integral By differentiating with respect to x on both sides of Eq. (A4) and letting x = 0 in the end, one finds k 2 + Λ d 3 l (2π) 3 l 2 T (0) ( l, k; k) E − l 2 m N + iǫ On the other hand, from the Schrödinger equation for the LO wave function, where V (0) ( x ) is the LO potential represented in coordinate space. With the above equations and Eq. (A6), one can eventually arrive at Eq. (21).
Successive differentiation on both sides of Eq. (A4) and letting x = 0 in the end relates integrals of the following form, to higher derivatives of the wave function at r = 0, which can in turn be related by successive differentiation on Eq. (B3) to V (0) ( x ) and its derivatives at r = 0.
Integrals involving the LO interacting Green function G k are encountered in computing Eq. (26): .
Its second derivative is given by where δ (3) Again, successive differentiation with respect to x 1 and/or x 2 on both sides of Eq. (B8) and letting x 1,2 = 0 leads to the integrals (B5) with larger m and/or n.