Some Applications of Renormalization Group to Few-Body Problems in EFT

General aspects of applying renormalization group in perturbative calculations are briefly discussed. Next, an application of the Wilsonian renormalization group with multitude of cutoff parameters to halo EFT for P-wave shallow states is considered.


Introduction
In perturbative calculations of quantum field theoretical (QFT) loop diagrams it is often very convenient to apply dimensional regularization. While parametrizing logarithmic divergences in terms of poles when the space-time dimension approaches its physical value, dimensional regularization automatically removes all power-law divergences and uniquely fixes the corresponding finite parts. Because of this in (mesonic) chiral perturbation theory (ChPT) [1,2], loop diagrams of higher orders do not contribute to the renormalization of coupling constants of the lower-order Lagrangian when the dimensional regularization is used [3]. For this case, the renormalization scheme dependence of renormalized couplings corresponding to logarithmic divergences and the explicit dependence of the loop diagrams on the renormalization scale exactly cancel each other order-by-order in the chiral expansion. Motivated by the dimensionally regulated mesonic ChPT, one may be tempted to demand renormalization group invariance order-by-order of the perturbative expansion in the few-nucleon sector of baryon ChPT. Our understanding is that if this goal would really be achieved, one would lose the advantage of exploiting the freedom of the choice of the renormalization condition to improve the convergence of perturbative series.
The main idea of using the renormalization group (RG) to our advantage in perturbative calculations is simple and can be demonstrated in a toy example as follows. Let in some theory the exact expression of a physical quantity be given by where x is a parameter of the theory whileh (= 1) controls the quantum corrections. Suppose, for whatever reason (e.g. because in reality we are only able to do approximate calculations), we need to approximate this quantity with a power series expansion, order-by-order with better and better accuracy. If |x| < 1 we can expand f (x) in a convergent Taylor series around x = 0 and approximate the exact function by the sum of first N terms For |x| > 1 the expansion specified in Eq. (2) is useless as it leads to a divergent series. In this case, it is advantageous to use an alternative way by rewriting the function f (x) identically and expanding in a different way: where The exact expression of f (x) is of course μ-independent, however the sum of any finite number of terms in Eq. (3) depends on μ. Notice that while formally this dependence is of a higher order, i.e. ∼h N +1 for the sum of the first N terms, the convergence properties of the series in Eq. (3) crucially depend on the choice of μ. For example, for x = 2 this series converges only if μ > 3/4 and the convergence is best close to μ = 1. This simple example demonstrates an essential feature of the RG applied to perturbative calculations. That is, exploiting the scale-dependence of finite sums of the perturbative series one can choose such values of the scale parameter that lead to optimal convergence properties of perturbative series.
It is an essential feature of QFT perturbative calculations that the sum of any finite number of terms of perturbative series depends on the choice of the renormalization condition. If this sum were independent of the choice of the scale parameter μ,then one would not be able to improve the convergence of the perturbative series using the RG.
In QFTs, the RG ideas have been realized in the Gell-Mann and Low [4] as well as in the Wilsonian [5] formulations. The first deals with renormalized quantities and their dependences on the renormalization scales, while the second studies the dependence of bare quantities on the cutoff parameter.
Using the subtractive renormalization and the Gell-Mann and Low RG in QFTs with several coupling constants leads to a multi-dimensional space of renormalization scales. While in standard perturbative calculations in QFT one often takes all renormalization scales equal to each other, it turns out that in the few-body sector of the chiral EFT (for reviews see, e.g., Refs. [6][7][8][9][10]), it is advantageous to study the dependence on scale parameters in more details. In particular, by taking different values of various renormalization scales, a better convergence of perturbative series can be obtained, see e.g., Ref. [11]. An alternative way of dealing with the issue of renormalization in the few-body sector of chiral EFT is by applying the Wilsonian approach, which usually considers the trajectories in the space of the bare low-energy constants as functions of a single cutoff parameter [12]. This makes it more restricted compared to the Gell-Mann and Low RG flow parametrized by multiple renormalization scales. To exploit the full freedom of choosing the renormalization condition in perturbative calculations in the few-body sector of chiral EFT, the Wilsonian RG approach has been generalized by introducing a multitude of cutoff parameters in Ref. [13]. In the next section we consider an application of this new framework to halo EFT for P-wave shallow bound states.

Wilsonian RG Approach with Multitude of Cutoff Parameters Applied to Halo EFT
Halo EFT for P-wave states has been introduced in Ref. [14]. It deals with a fine-tuned system of two nonrelativistic particles with the range of interaction R ∼ 1/M hi , where M hi is some mass scale. At sufficiently low energies, the on-shell scattering amplitude for momentum k in a partial wave with orbital angular momentum l can be written in terms of the effective range expansion (ERE) [15] where δ denotes the phase shift, a stands for the scattering length, r for effective range, and v i are the shape parameters. The parameters of the ERE starting from the effective range r are expected to scale with the , etc., under condition that the effective range function k 2l+1 cot δ does not have poles in the near-threshold region. On the other hand, the scattering length a can, in principle, take any value. In the current work, following Refs. [14,16,17], we consider fine-tuned systems, for which the P-wave scattering amplitude in Eq. (4) features poles located in the validity range of the EFT, k ∼ M lo M hi . However, while in first two references the formulation of the EFT with an auxiliary spin-1 dimer field has been employed, Ref. [17] and the current work considers a formulation without auxiliary fields. We are interested in two fine tuning scenarios. In particular, first, following Ref. [14], we assume that the first two terms in the ERE are fine-tuned as follows For such systems, the two lowest-order contact interactions in the effective two-particle potential have to be iterated to all orders [14] by solving the partial-wave Lippmann-Schwinger (LS) equation Notice that we included theh (= 1) factor in the LS equation to keep track of the loop corrections. In the potential of Eq. (6) p ≡ |p | and p ≡ |p | are the center-of-mass momenta of particles in the initial and final states, respectively. The on-shell amplitude corresponds to p = p = k, i.e. it is given by T (k) ≡ T (k, k). Another, less fine-tuned scenario with has been considered in Ref. [16]. Both scenarios have been revisited in Ref. [17], the main findings of which are briefly summarized below. The potential of Eq. (6) leads to ultraviolet (UV) divergences in the LS equation and requires regularization. By imposing cutoff to the integration momentum, the regulated solution can be matched to the ERE. The condition of reproducing the first two terms of the ERE fixes the bare LECs C 2 ( ) and C 4 ( ). However, both parameters become complex if the cutoff is taken beyond the soft scale in the problem, M lo . This result agrees with the causality bounds of Ref. [18], however it apparently contradicts the conclusions obtained using the EFT with auxiliary dimer fields [14,16,19].
As detailed in Ref. [17], the above problem is caused by the inconsistent (from the EFT point of view) renormalization of the LS equation with perturbatively non-renormalizable potentials, which requires the inclusion of an infinite number of counterterms, see e.g. Ref. [20]. On the other hand, the scattering amplitude for resonant P-waves in halo EFT with no auxiliary fields can be renormalized consistently by using the usual QFT subtractive renormalization technique to all orders in the loop expansion. The resulting scattering amplitude is finite in the → ∞ limit, both perturbatively (i.e., at any finite order ofh, corresponding to iterations of Eq. (7)) and non-perturbatively. A separable form of the considered effective potential admits a closed-form expression for the counterterms, needed to absorb all divergences in the LS equation. The renormalization conditions can be chosen such that all renormalized LECs scale according to naive dimensional analysis, and the renormalized contributions of diagrams obey manifest power counting, with the residual dependence of the amplitude on the subtraction scales being beyond the actual order of the calculation. The renormalized LECs C R 2 and C R 4 can be expressed in terms of a and r regardless of their actual values, in a close analogy with the EFT formulations of Refs. [14,16].
Next, the standard Wilsonian RG analysis of P-wave scattering has been performed in Ref. [17] following the philosophy of Refs. [12,[21][22][23]. For doubly fine-tuned systems specified in Eq. (5), the unitary fixed point plays an important role. This unstable fixed point describes scale-free P-wave systems with a −1 → 0 and r → 0 and has two relevant directions [23]. All theories describing doubly fine-tuned systems in Eq. (5) get attracted by the unitary fixed point when M lo M hi . This allows one to identify a systematic and universal expansion of the scattering amplitude for such fine-tuned systems by analyzing the scaling of perturbations around the fixed point for ∼ M lo . On the other hand, the behavior of singly fine-tuned systems specified in Eq. (8) is not expected to be governed by the expansion around the unitary fixed point.
Below we apply the Wilsonian RG with multitude of cutoff parameters to the considered fine tuned P-wave systems. To do so, following Ref. [13], we rewrite the potential as and introduce two cutoffs via where it is implied that 1 ≥ 2 . Equation (10) can be represented in a separable form: and therefore the corresponding LS equation for the scattering amplitude can be straightforwardly solved analytically. Matching the solution to the ERE of Eq. (4), we fix the LECs C 2 und C 4 by demanding that the first two terms of the ERE are reproduced. This leads to two solutions. We fix the physically relevant solution by demanding that the bare couplings C 2 und C 4 possess Taylor series expansion in powers ofh. 1 After substitutingh = 1, the obtained LECs C 2 and C 4 have the form: , The LECs C 2 and C 4 are not allowed to become complex, therefore the argument of the square root has to be non-negative. This leads to the condition that the quantity in non-negative. The standard Wilsonian RG analysis corresponds to 1 = 2 = . In this case, we have from Eq. (13) which turns negative for sufficiently large values and, therefore, the LECs C 2 and C 4 become complex. For doubly fine tuned systems specified by Eq. (5), the cutoff cannot be taken larger than ∼ M lo . For two independent cutoff parameters 1 and 2 , the condition that α( 1 , 2 ) has to be non-negative can be satisfied for any values of the scattering length a and the effective range r by taking sufficiently large values of 1 .
Next we check the convergence of the ERE. To do so we subtract the first two terms − 1 a + 1 2 r k 2 from the calculated expression for k cot δ and obtain the following result for the remainder: The second term in the bracket has a convergent expansion in k 2 for 1 k and the expansion of the first term converges if the condition is satisfied. By taking sufficiently large value of 1 this condition can always be fulfilled. For both considered fine tuned systems this amounts to taking 1 ∼ M hi or larger. By taking 1 ∼ M hi and 2 ∼ M lo , we find that C 2 ∼ 1/(m M 3 hi ) and C 4 ∼ 1/(m M 5 hi ), i.e. both are of natural size, as expected according to Weinberg's power counting.

Summary
Renormalization group analysis is a powerful method that has numerous applications in a wide range of physics problems. In this work we very briefly touched upon some general issues of using RG methods in perturbative calculations. Next we considered an application of RG analysis to few-body problems in low-energy EFT of the strong interaction. In particular, we applied the Wilsonian RG method with a multitude of cutoff parameters of Ref. [13] to halo EFT of fine tuned P-wave systems. We find, in agreement with general considerations, that the new approach indeed offers a broader freedom in choosing the cutoff parameters such that the convergence of perturbative series is substantially improved as compared with the standard Wilsonian approach.
Author Contribution All authors worked on the whole manuscript.