Problems in the derivations of the renormalization group equation for the low momentum nucleon interactions
Koji Harada
备注：
16 pages, no figure; (ver.3) A significant revision is made; the condition under which the extra terms in the RGE cancel is clarified (ver.4) some expressions modified; it is emphasized that the numerical works in the literature are not affected by this paper
We carefully examine all the four derivations of the renormalization group equation (RGE) for the so-called $\Vlk$ potential, given by Bogner, \textit{et. al.}[nucl-th/0111042]. Two derivations based on the ``semi-group composition law'' are shown to be unjustified, while the other two based on the completeness relation of the model space must be modified if there are bound states. It is however shown that the RGE is unchanged if the bound state wavefunctions in the reduced theory are required to have the same low-momentum components as those in the original theory. Several aspects of the $\Vlk$ approach are also discussed.
We study the interplay between charge symmetry breaking and renormalization in the NN system for s-waves. We find a set of universality relations which disentangle explicitly the known long distance dynamics from low energy parameters and extend them to the Coulomb case. We analyze within such an approach the One-Boson-Exchange potential and the theoretical conditions which allow to relate the proton-neutron, proton-proton and neutron-neutron scattering observables without the introduction of extra new parameters and providing good phenomenological success.
The renormalization of EFT for nucleon-nucleon scattering in nonperturbative regimes is investigated in a compact parametrization of the $T$-matrix. The key difference between perturbative and nonperturbative renormalization is clarified. The underlying theory perspective and the 'fixing' of the prescriptions for the $T$-matrix from physical boundary conditions are stressed.
The OS and PDS renormalization schemes for the effective field theory with nucleons and pions are investigated. We explain in detail how the renormalization is implemented using local counterterms. Fits to the NN scattering data are performed in the 1S0 and 3S1 channels for different values of mu_R. An error analysis indicates that the range of the theory with perturbative pions is consistent with 500 MeV.
Fluctuations are included in a chiral nucleon-meson model within the framework of the functional renormalization group. The model, with parameters fitted to reproduce the nuclear liquid-gas phase transition, is used to study the phase diagram of QCD. We find good agreement with results from chiral effective field theory. Moreover, the results show a separation of the chemical freeze-out line and chiral symmetry restoration at large baryon chemical potentials.
The renormalization of iterated one-pion exchange (OPE) is studied in Chiral Effective Field Theory ($χ$EFT) for the antinucleon-nucleon ($\overline{N}\!N$) system. The OPE potential is cut off at a certain distance and contact interactions are represented by a complex spherical well with the same radius. We investigate the dependence on the cutoff radius of the phase shifts, inelasticities, and mixing angles for the low partial waves in $\overline{N}\!N$ scattering. We show that renormalization requires additional contact interactions compared to the expectation based on naive dimensional ana…
Decoupling via the Similarity Renormalization Group (SRG) of low-energy nuclear physics from high-energy details of the nucleon-nucleon interaction is examined for two-body observables and few-body binding energies. The universal nature of this decoupling is illustrated and errors from suppressing high-momentum modes above the decoupling scale are shown to be perturbatively small.
The similarity renormalization group (SRG) is based on unitary transformations that suppress off-diagonal matrix elements, forcing the hamiltonian towards a band-diagonal form. A simple SRG transformation applied to nucleon-nucleon interactions leads to greatly improved convergence properties while preserving observables, and provides a method to consistently evolve many-body potentials and other operators.
The renormalization of EFT for nucleon-nucleon scattering in nonperturbative regimes is investigated in a compact parametrization of the $T$-matrix. The key difference between perturbative and nonperturbative renormalization is clarified. The underlying theory perspective and the 'fixing' of the prescriptions for the $T$-matrix from physical boundary conditions are stressed.
We investigate the axial-vector nucleon-to-delta transition form factors in the framework of relativistic baryon chiral perturbation theory at the one-loop order using the complex-mass renormalization scheme. We determine the available six free parameters by fitting to an empirical parametrization of the form factors obtained from the BNL neutrino bubble chamber experiments. A unique feature of our calculation is the prediction of a non-vanishing form factor $C_3^A(Q^2)$. Moreover, our results show a surprising sensitivity to the coupling constant $\texttt{g}_1$ of the leading-order Lagrangian…
Regularization and renormalization is discussed in the context of low-energy effective field theory treatments of two or more heavy particles (such as nucleons). It is desirable to regulate the contact interactions from the outset by treating them as having a finite range. The low energy physical observables should be insensitive to this range provided that the range is of a similar or greater scale than that of the interaction. Alternative schemes, such as dimensional regularization, lead to paradoxical conclusions such as the impossibility of repulsive interactions for truly low energy effec…
We compare the subtractive renormalization and the Wilsonian renormalization group approaches in the context of an effective field theory for the two-nucleon system. Based on an exactly solvable model of contact interactions, we observe that the standard Wilsonian renormalization group approach with a single cutoff parameter does not cover the whole space spanned by the renormalization scale parameters of the subtractive formalism. In particular, renormalization schemes corresponding to Weinberg's power counting in the case of an unnaturally large scattering length are beyond the region covere…
The importance of imposing physical boundary conditions on the $T$-matrix to remove the nonperturbative renormalization prescription dependence is stressed and demonstrated in two diagonal channels, $^1P_1$ and $^1D_2$, with the help of Padé expansion.
We develop a method for constructing the Heavy Baryon Chiral Perturbation Theory Lagrangian (L_{HBChPT}), to a given chiral order, within HBChPT. We work within SU(2) theory, with only the pion field interacting with the nucleon. The main difficulties, which are solved, are to develop techniques for implementing charge conjugation invariance, and for taking the nucleon on shell, both within the nonrelativistic formalism. We obtain complete lists of independent terms in L_{HBChPT} through O(q^3) for off- shell nucleons. Then, eliminating equation-of-motion (eom) terms at the relativistic and no…
Some form of nonperturbative regularization is necessary if effective field theory treatments of the NN interaction are to yield finite answers. We discuss various regularization schemes used in the literature. Two of these methods involve formally iterating the divergent interaction and then regularizing and renormalizing the resultant amplitude. Either a (sharp or smooth) cutoff can be introduced, or dimensional regularization can be applied. We show that these two methods yield different results after renormalization. Furthermore, if a cutoff is used, the NN phase shift data cannot be repro…
We discuss renormalization group approaches to strongly interacting Fermi systems, in the context of Landau's theory of Fermi liquids and functional methods, and their application to neutron matter.
We construct a complete list of O(q^4) terms directly within Heavy Baryon Chiral Perturbation Theory (HBChPT) in the absence of external fields and assuming isospin symmetry. In addition to a phase rule recently developed (NPA646, 343, '99), a variety of algebraic identities and reparameterization invariance, are used to ensure linear independence of the terms and their low energy coupling constants. We first construct O(q^4) terms terms for off-shell nucleons, and then perform the on-shell reduction, again within HBChPT. We discuss an application of the O(q^4) terms to the evaluation of the O…
Using Dyson--Schwinger equations, the nucleon propagator is analyzed nonperturbatively in a field--theoretical model for the pion--nucleon interaction. Infinities are circumvented by using pion--nucleon form factors which define the physical scale. It is shown that the correct, finite, on--shell nucleon renormalization is important for the value of the mass--shift and the propagator. For physically acceptable forms of the pion--nucleon form factor the rainbow approximation together with renormalization is inconsistent. Going beyond the rainbow approximation, the full pion--nucleon vertex is mo…
The role of dimensional regularization is discussed and compared with that of cut-off regularization in some quantum mechanical problems with ultraviolet divergence in two and three dimensions with special emphasis on the nucleon-nucleon interaction. Both types of renormalizations are performed for attractive divergent one- and two-term separable potentials, a divergent tensor potential, and the sum of a delta function and its derivatives. We allow energy-dependent couplings, and determine the form that these couplings should take if equivalence between the two regularization schemes is to be …
The Nucleon-Nucleon One Meson Exchange Potential, its wave functions and related Meson Exchange Currents are analyzed for point-like nucleons. The leading Nc contributions generate a local and energy independent potential which presents 1/r^3 singularities, requiring renormalization. We show how invoking suitable boundary conditions, neutron-proton phase shifts and deuteron properties become largely insensitive to the nucleon substructure and to the vector mesons. Actually, reasonable agreement with low energy data for realistic values of the coupling constants (e.g. SU(3) values) is found. Th…
Methods based on Wilson's renormalization group have been successfully applied in the context of nuclear physics to analyze the scale dependence of effective nucleon-nucleon ($NN$) potentials, as well as to consistently integrate out the high-momentum components of phenomenological high-precision $NN$ potentials in order to derive phase-shift equivalent softer forms, the so called $V_{low-k}$ potentials. An alternative renormalization group approach that has been applied in this context is the Similarity Renormalization Group (SRG), which is based on a series of continuous unitary transformati…
We discuss the problem of renormalization of dynamical equations which arises in an effective field theory description of nuclear forces. By using a toy model of the separable NN potential leading to logarithmic singularities in the Born series, we show that renormalization gives rise to nucleon dynamics which is governed by a generalized dynamical equation with a nonlocal-in-time interaction operator.
The role of cut-off and dimensional regularizations is discussed in the context of obtaining a renormalized nucleon-nucleon potential from the chiral Lagrangian formulation of the effective field theory due to Weinberg. Both types of renormalizations are performed for the sum of an attractive delta function and its derivatives. The equivalence between the two forms of regularizations can be established with the use of energy-dependent bare couplings and the explicit forms of these couplings are determined.
A methodology to renormalize the nucleon-nucleon interaction, using a recursive multiple subtraction approach to construct the kernel of the scattering equation, is presented. We solve the subtracted scattering equation with the next-leading-order (NLO) and next-to-next-leading-order (NNLO) interactions. The results are presented for all partial waves up to $j=2$, fitted to low-energy experimental data. In our renormalizaton group invariant method, when introducing the NLO and NNLO interactions, the subtraction energy emerges as a renormalization scale and the momentum associated with it comes…
We discuss the problem of renormalization of dynamical equations which arises in an effective field theory description of nuclear forces. By using a toy model of the separable NN potential leading to logarithmic singularities in the Born series, we show that renormalization gives rise to nucleon dynamics which is governed by a generalized dynamical equation with a nonlocal-in-time interaction operator. We show that this dynamical equation can open new possibilities for applying the EFT approach to the description of low-energy nucleon dynamics.
The criticisms in manuscript arXiv:0803.4371 do not change the renormalization group equation for the "$\vlowk$" interaction nor do they affect any results in the literature. Several other potentially misleading statements about low-momentum interactions are also addressed.
A new framework for computing the Similarity Renormalization Group (SRG) evolution of three-nucleon forces (3NF) in momentum representation is presented. The use of antisymmetric three-particle hyperspherical momentum states ensures unitary evolutions within certain basis truncations, much like antisymmetric harmonic oscillator SRG evolutions. Additionally, in each partial wave the $T_{\mbox{rel}}$-SRG regulator is exactly represented, similar to recent 3NF momentum representation evolutions. Unitary equivalence is demonstrated for the triton using several chiral two- plus three-nucleon intera…
A general expression for the $π-N$ and $π-Δ$ scattering lengths is derived in the framework of a microscopic calculation. Annihilation, negative energy wave-functions and spontaneous chiral symmetry are included consistently. The point-like limit is used to calculate the scattering lengths.
We apply the subtracted kernel method (SKM), a renormalization approach based on recursive multiple subtractions performed in the kernel of the scattering equation, to the chiral nucleon-nucleon (NN) interactions up to next-to-next-to-leading-order (NNLO). We evaluate the phase-shifts in the 1S0 channel at each order in Weinberg's power counting scheme and in a modified power counting scheme which yields a systematic power-law improvement. We also explicitly demonstrate that the SKM procedure is renormalization group invariant under the change of the subtraction scale through a non-relativisti…
We start from a Lagrangian describing scalar "nucleons" and mesons which interact through a simple vertex. Okubo's method of unitary transformation is used to describe a single nucleon dressed by its meson cloud. We find an expression for the physical mass of the nucleon being correct up to second order in the coupling constant. It is then verified that this result is the same as the corresponding expression found by Feynman techniques. Finally we also express the three boost operators in terms of the physical nucleon mass. Doing so we find expressions for all the ten generators of Poincaré tr…
Effects of T- and P-odd weak nucleon interaction in nuclei: renormalizations due to residual strong interaction, matrix elements between compound states and their correlations with P-violating matrix elements
We present a detailed study of the recently conjectured infrared renormalization group limit cycle in QCD using chiral effective field theory. It was conjectured that small increases in the up and down quark masses can move QCD to the critical trajectory for an infrared limit cycle in the three-nucleon system. At the critical quark masses, the binding energies of the deuteron and its spin-singlet partner are tuned to zero and the triton has infinitely many excited states with an accumulation point at the three-nucleon threshold. We exemplify three parameter sets where this effect occurs at nex…
Naively, the "best" method of renormalization is the one where a momentum cutoff is taken to infinity while maintaining stable results due to a cutoff-dependent adjustment of counterterms. We have applied this renormalization method in the non-perturbative calculation of phase-shifts for nucleon-nucleon (NN) scattering using chiral NN potentials up to next-to-next-to-next-to-leading order (N3LO). For lower partial waves, we find that there is either no convergence with increasing order or, if convergence occurs, the results do not always converge to the empirical values. For higher partial wav…
We study the nonperturbative renormalization of the nucleon-nucleon (NN) interaction at next-to-leading order (NLO) and next-to-next-to-leading order (NNLO) of chiral effective field theory. A systematic variation of the cutoff parameter is performed for values below the chiral symmetry breaking scale of about 1 GeV. The accuracy of the predictions is determined by calculating the $χ^2$ for the reproduction of the NN data for energy intervals below pion-production threshold. At NLO, the NN data are described well up to about 100 MeV laboratory energy and, at NNLO, up to about 200 MeV---with, e…
We analyze the renormalization of the nucleon--nucleon interaction at low energies in coordinate space for both one and two pion exchange chiral potentials. The singularity structure of the long range potential and the requirement of orthogonality respectively determines, once renormalizability is imposed, the minimum and maximum number of counterterms allowed in the effective description of the nucleon--nucleon interaction in a non-perturbative context.
The Wilsonian renormalization group approach to the Lippmann-Schwinger equation with a multitude of cutoff parameters is introduced. A system of integro-differential equations for the cutoff-dependent potential is obtained. As an illustration, a perturbative solution of these equations with two cutoff parameters for a simple case of an S-wave low-energy potential in the form of a Taylor series in momenta is obtained. The relevance of the obtained results for the effective field theory approach to nucleon-nucleon scattering is discussed.
Small increases in the up and down quark masses of QCD would tune the theory to the critical renormalization group trajectory for an infrared limit cycle in the three-nucleon system. At critical values of the quark masses, the deuteron binding energy goes to zero and the triton has infinitely many excited states with an accumulation point at the 3-nucleon threshold. The ratio of the binding energies of successive states approaches a universal constant that is close to 515. The proximity of physical QCD to the critical trajectory for this limit cycle explains the success of an effective field t…
The renormalization of iterated one-pion exchange (OPE) has been studied in Chiral Effective Field Theory ($χ$EFT) for the antinucleon-nucleon ($\overline{N}\!N$) scattering in some partial waves (Phys. Rev. C 105, 054005 (2022)). We go further for the other higher partial waves but with total angular momenta $J\leq 3$ in this paper. Contact interactions are represented by a complex spherical well in coordinate space. Changing the radius of the spherical well means changing the cutoff. We check the cutoff dependence of the phase shifts, inelasticities, and mixing angles for the partial waves, …
We present a study of two-nucleon scattering in chiral effective field theory with a finite cutoff to next-to-leading order in the chiral expansion. In the proposed scheme, the contributions of the lowest-order interaction to the scattering amplitude are summed up to an arbitrary order, while the corrections beyond leading order are iterated only once. We consider a general form of the regulator for the leading-order potential including local and non-local structures. The main objective of the paper is to address formal aspects of renormalizability within the considered scheme. In particular, …
We use the recently proposed In-Medium Similarity Renormalization Group (IM-SRG) to carry out a systematic study of closed-shell nuclei up to $\nuc{Ni}{56}$, based on chiral two- plus three-nucleon interactions. We analyze the capabilities of the IM-SRG by comparing our results for the ground-state energy to Coupled Cluster calculations, as well as to quasi-exact results from the Importance-Truncated No-Core Shell Model. Using chiral two- plus three-nucleon Hamiltonians whose resolution scales are lowered by free-space SRG evolution, we obtain good agreement with experimental binding energies …
Effective Hamiltonians and effective electroweak operators are calculated with the Okubo-Lee-Suzuki formalism for two-nucleon systems. Working within a harmonic oscillator basis, first without and then with a confining harmonic oscillator trap, we demonstrate the effects of renormalization on observables calculated for truncated basis spaces. We illustrate the renormalization effects for the root-mean-square point-proton radius, electric quadrupole moment, magnetic dipole moment, Gamow-Teller transition and neutrinoless double-beta decay operator using nucleon-nucleon interactions from chiral …
We use standard distorted wave theory techniques and dimensional regularization to find out solutions of the nucleon-nucleon Lippman--Schwinger equation with a kernel determined by the Weinberg's next-to-leading potential, which consists of one--pion exchange and additional contact terms with derivatives. Though for simplicity, we restrict the discussion to the $^1S_0$ channel and to contact terms containing up to two derivatives, the generalization to higher waves and/or number of derivatives is straightforward. The undetermined low energy constants emerging out of the renormalization procedu…
We discuss a renormalization scheme for relativistic baryon chiral perturbation theory which provides a simple and consistent power counting for renormalized diagrams. The method involves finite subtractions of dimensionally regularized diagrams beyond the standard modified minimal subtraction scheme of chiral perturbation theory to remove contributions violating the power counting. This is achieved by a suitable renormalization of the parameters of the most general effective Lagrangian. As applications we discuss the mass of the nucleon, the $σ$ term, and the scalar and electromagnetic form f…
Weinberg's contributions to the power counting and derivation of few-nucleon forces in Chiral EFT are briefly recalled. Subsequent improvements are reviewed, concluding with the recent suggestion of a combinatorial enhancement.
We discuss conceptual aspects of renormalization in the context of effective field theories for the two-nucleon system. It is shown that, contrary to widespread belief, renormalization scheme dependence of the scattering amplitude can only be eliminated up to the order the calculations are performed. We further consider an effective theory for an exactly solvable quantum mechanical model which possesses a long- and short-range interaction to simulate pionful effective field theory. We discuss the meaning of low-energy theorems in this model and demonstrate their validity in calculations with a…
The Similarity Renormalization Group (SRG) is used to soften interactions for ab initio nuclear structure calculations by decoupling low- and high-energy Hamiltonian matrix elements. The substantial contribution of both initial and SRG-induced three-nucleon forces requires their consistent evolution in a three-particle basis space before applying them to larger nuclei. While in principle the evolved Hamiltonians are unitarily equivalent, in practice the need for basis truncation introduces deviations, which must be monitored. Here we present benchmark no-core full configuration calculations wi…
A simple class of unitary renormalization group transformations that force hamiltonians towards a band-diagonal form produce few-body interactions in which low- and high-energy states are decoupled, which can greatly simplify many-body calculations. One such transformation has been applied to phenomenological and effective field theory nucleon-nucleon interactions with success, but further progress requires consistent treatment of at least the three-nucleon interaction. In this paper we demonstrate in an extremely simple model how these renormalization group transformations consistently evolve…
The phase space structure of zero temperature Quarkyonic Matter is a Fermi sphere of Quark Matter, surrounded by a shell of Nucleonic Matter. We construct a quasi particle model of Quarkyonic Matter based on the constituent quark model, where the quark and nucleon masses are related by m_Q = m_N/N_c, and N_c is the number of quark colors. The region of occupied states is for quarks k_Q < k_F/N_c, and for nucleons k_F < k_N < k_F + Δ. We first consider the general problem of Quarkyonic Matter with hard core nucleon interactions. We then specialize to a quasi-particle model where the hard core n…
Low momentum two-nucleon interactions obtained with the renormalization group method and the similarity renormalization group method are used to study the cutoff dependence of low energy 3N and 4N scattering observables. The residual cutoff dependence arises from omitted short-ranged 3N (and higher) forces that are induced by the renormalization group transformations, and may help to estimate the sensitivity of various 3N and 4N scattering observables to short-ranged many-body forces.
Approximate analytical formulas for the self-consistent renormalization of P,T-odd and P-odd weak nuclear potentials by the residual nucleon-nucleon strong interaction are derived. The contact spin-flip nucleon-nucleon interaction reduces the constant of the P,T-odd potential 1.5 times for the proton and 1.8 times for the neutron. Renormalization of the P-odd potential is caused by the velocity dependent spin-flip component of the strong interaction. In the standard variant of $π+ ρ$-exchange, the conventional strength values lead to anomalous enhancement of the P-odd potential. Moreover, the …
Nucleon-nucleon scattering in the $^1S_0$ partial wave is considered in chiral effective field theory within the renormalizable formulation of Ref. [1] beyond the leading-order approximation. By applying subtractive renormalization, the subleading contact interaction in this channel is taken into account non-perturbatively. For a proper choice of renormalization conditions, the predicted energy dependence of the phase shift and the coefficients in the effective range expansion are found to be in a good agreement with the results of the Nijmegen partial wave analysis.
In this work, we apply the subtracted kernel method (SKM) to the chiral nucleon-nucleon ($NN$) interaction in the $^3P_0$ channel up to next-to-next-to-leading-order ($NNLO$). We demonstrate, by explicit numerical calculations, that the SKM procedure is renormalization group invariant under the change of the subtraction scale provided the driving-term of the subtracted scattering equation is evolved through a non-relativistic Callan-Symanzik (NRCS) flow-equation.
We derive the nucleon-nucleon interaction from the Skyrme model using second order perturbation theory and the dipole approximation to skyrmion dynamics. Unlike previous derivations, our derivation accounts for the non-trivial kinetic and potential parts of the skyrmion-skyrmion interaction lagrangian and how they couple in the quantum calculation. We derive the eight low energy interaction potentials and compare them with the phenomenological Paris model, finding qualitative agreement in seven cases.
We derive generic relativistic hydrodynamical equations with dissipative effects from the underlying Boltzmann equation in a mechanical and systematic way on the basis of so called the renormalization-group (RG) method. A macroscopic frame vector is introduced to specify the frame on which the macroscopic dynamics is described. Our method is so mechanical with only few ansatz that our method give a microscopic foundation of the available hydrodynamical equations, and also can be applied to make a reduction of the kinetic equations other than the simple Boltzmann equation.
