We develop a subtractive renormalization scheme to evaluate the P-wave NN scattering phase shifts using chiral effective theory potentials. This allows us to consider arbitrarily high cutoffs in the Lippmann-Schwinger equation (LSE). We employ NN potentials computed up to next-to-next-to-leading order (NNLO) in chiral effective theory, using both dimensional regularization and spectral-function regularization. Our results obtained from the subtracted P-wave LSE show that renormalization of the NNLO potential can be achieved by using the generalized NN scattering lengths as input--an alternative to fitting the constant that multiplies the P-wave contact interaction in the chiral effective theory NN force. However, in order to obtain a reasonable fit to the NN data at NNLO the generalized scattering lengths must be varied away from the values extracted from the so-called high-precision potentials. We investigate how the generalized scattering lengths extracted from NN data using various chiral potentials vary with the cutoff in the LSE. The cutoff-dependence of these observables, as well as of the phase shifts at $T_{lab} \approx 100$ MeV, suggests that for a chiral potential computed with dimensional regularization the highest LSE cutoff it is sensible to adopt is approximately 1 GeV. Using spectral-function regularization to compute the two-pion-exchange potentials postpones the onset of cutoff dependence in these quantities, but does not remove it.
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…
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 apply the similarity renormalization group (SRG) approach to evolve a nucleon-nucleon ($NN$) interaction in leading-order (LO) chiral effective field theory (ChEFT), renormalized within the framework of the subtracted kernel method (SKM). We derive a fixed-point interaction and show the renormalization group (RG) invariance in the SKM approach. We also compare the evolution of $NN$ potentials with the subtraction scale through a SKM RG equation in the form of a non-relativistic Callan-Symanzik (NRCS) equation and the evolution with the similarity cutoff through the SRG transformation.
We apply five subtractions in the Lippman-Schwinger (LS) equation in order to perform a non-perturbative renormalization of chiral N3LO nucleon-nucleon interactions. Here we compute the phase shifts for the uncoupled peripheral waves at renormalization scales between $0.1~ \rm{fm}^{-1}$ and $1 ~ \rm{fm}^{-1}$. In this range, the results are scale invariant and provide an overall goof agreement with the Nijmegen partial wave analysis up to at least $E_{\rm{lab}} = 150 ~ \rm{MeV}$, with a cutoff at $Λ= 30~\rm{fm}^{-1}$.
We have developed a subtractive renormalization method with which we can evaluate nucleon-nucleon (NN) scattering phase shifts produced by the NN potential obtained at leading, next-to-leading, and next-to-next-to-leading order (NNLO) in chiral effective theory ($χ$ET). In this method the low-energy constants associated with short-distance NN physics are eliminated from the Lippmann-Schwinger equation (LSE) for the NN t-matrix, in favor of physical observables. This allows us to straightforwardly compute scattering phase shifts for ultra-violet cutoffs of at least 10 GeV. We then perform detai…
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…
The leading-order nucleon-nucleon (NN) potential derived from chiral perturbation theory consists of one-pion exchange plus a short-distance contact interaction. We show that in the 1S0 and 3S1-3D1 channels renormalization of the Lippmann-Schwinger equation for this potential can be achieved by performing one subtraction. This subtraction requires as its only input knowledge of the NN scattering lengths. This procedure leads to a set of integral equations for the partial-wave NN t-matrix which give cutoff-independent results for the corresponding NN phase shifts. This reformulation of the NN s…
We extend our subtractive-renormalization method in order to evaluate the 1S0 and 3S1-3D1 NN scattering phase shifts up to next-to-next-to-leading order (NNLO) in chiral effective theory. We show that, if energy-dependent contact terms are employed in the NN potential, the 1S0 phase shift can be obtained by carrying out two subtractions on the Lippmann-Schwinger equation. These subtractions use knowledge of the the scattering length and the 1S0 phase shift at a specific energy to eliminate the low-energy constants in the contact interaction from the scattering equation. For the J=1 coupled cha…
We consider the NN interaction in pionless effective field theory (EFT) up to next-to-next-to-leading order (NNLO) and use a recursive subtractive renormalization scheme to describe NN scattering in the 1S0 channel. We fix the strengths of the contact interactions at a reference scale, chosen to be the one that provides the best fit for the phase-shifts, and then slide the renormalization scale by evolving the driving terms of the subtracted Lippmann-Schwinger equation through a non-relativistic Callan-Symanzik equation. The results show that such a systematic renormalization scheme with multi…
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.
Cutoff regularized subleading order ${}^1S_0$ NN potential of effective field theory (EFT) is iterated using Lippmann-Schwinger equation. It is shown that the scattering amplitudes calculated in cutoff and subtractively renormalized EFT are equal up to the accuracy of performed calculations. Non-perturbative renormalization, where part of divergences are absorbed into two contact interaction coupling constants with subsequent removal of regularization is also performed. Cutoff and dimensional regularizations both lead to finite but different results within this scheme.
We study the unitarized meson-baryon scattering amplitude at leading order in the strangeness $S=-1$ sector using time-ordered perturbation theory for a manifestly Lorentz-invariant formulation of chiral effective field theory. By solving the coupled-channel integral equations with the full off-shell dependence of the effective potential and applying subtractive renormalization, we analyze the renormalized scattering amplitudes and obtain the two-pole structure of the $Λ(1405)$ resonance. We also point out the necessity of including higher-order terms.
The renormalization of singular chiral potentials as applied to NN scattering and the structure of the deuteron is discussed. It is shown how zero range theories may be implemented non-perturbatively as constrained from known long range NN forces.
The renormalization of the NN interaction with the Chiral Two Pion Exchange Potential computed using relativistic baryon chiral perturbation theory is considered. The short distance singularity reduces the number of counter-terms to about a half as those in the heavy-baryon expansion. Phase shifts and deuteron properties are evaluated and a general overall agreement is observed.
We extend the renormalization of the NN interaction with Chiral Two Pion Exchange Potential to the calculation of non-central partial wave phase shifts with total angular momentum j<=5 . The short distance singularity structure of the potential as well as the requirement of orthogonality conditions on the wave functions determines exactly the number of undetermined parameters after renormalization.
Renormalization of proton-proton fusion is studied in the framework of chiral effective field theory. Strict perturbative treatment of subleading corrections is applied in the analysis. Possible enhancement of two-nucleon contact axial current operators is the focus of the study. We find evidence that supports a previous proposal in the literature to promote one of the contact axial current operators.
The renormalization, finiteness and off-shellness of short distance power like singular interactions is discussed. We show analytically that the renormalizability of the off-shell scattering amplitude relies completely on the corresponding on-shell amplitude without proliferation of new counterterms. We illustrate the result by complementary calculations both in coordinate as well as in momentum space in the simplest 1S0 channel for chiral np interactions including Two Pion Exchange.
We are celebrating thirty-years from the seminal papers from Weinberg proposing to use the rules of Chiral Perturbation Theory to nucleon systems. His proposal was to build an Effective Field Theory (EFT) with Chiral symmetry as a key property. A big effort on this line have been made during these three decades, however the issue of renormalization of the theory is not settled. I will briefly review my path on this issue and give my personal view.
We analyse the renormalization of the of two-nucleon interaction with multiple subtractions in peripheral waves considering two chiral forces at N3LO. Phase shifts at low energies are then computed with several subtraction points below μ= 10 / fm. We show that for most peripheral waves the phase shifts have nearly no dependence on the renormalization scale. In two cases the phase shifts converge slowly as the renormalization scale approaches μ= 1 / fm and in one case the phase shifts presented oscillations with respect to the subtraction point μ.
Ever since quantum field theory was first applied to the derivation of nuclear forces in the mid-20th century, the renormalization of pion exchange with realistic couplings has presented a challenge. The implementation of effective field theories (EFTs) in the 1990s promised a solution to this problem but unexpected obstacles were encountered. The response of the nuclear community has been to focus on "chiral potentials" with regulators chosen to produce a good description of data. Meanwhile, a successful EFT without explicit pion exchange --- Pionless EFT --- has been formulated where renorma…
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…
The renormalization of the chiral np interaction in the 1S0 channel to N3LO in Weinberg counting for the long distance potential with one single momentum and energy independent counterterm is carried out. This renormalization scheme yields finite and unique results and is free of short distance off-shell ambiguities. We observe good convergence in the entire elastic range below pion production threshold and find that there are some small physical effects missing in the purely pionic chiral NN potential with or without inclusion of explicit Delta degrees of freedom. We also study the renormaliz…
The renormalization of the chiral np interaction with Delta-excitations in intermediate states is considered at next-to-leading order (NLO) and next-to-next-to-leading order (N2LO) for central waves. The inclusion of the Delta-excitation as an explicit degree of freedom improves the convergence properties of the effective field theory results for np scattering with respect to Delta-less theory, and allows the existence of a deuteron bound state in the infinite cut-off limit. The 1S_0 singlet and 3S_1 triplet phase shifts reproduce data for p ~ m_pi. The role of spectral function regularization…
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 …
In this work we consider the renormalization of the chiral two-pion exchange potential with explicit delta-excitations for nucleon-nucleon scattering at next-to-leading (NLO) and next-to-next-to-leading order (N2LO). Due to the singular nature of the chiral potentials, correlations between different partial waves are generated. In particular we show that two-body scattering by a short distance power like singular attractive interaction can be renormalized in all partial waves with a single counterterm, provided the singularities are identical. A parallel statement holds in the presence of tens…
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…
The $K^{+}$ meson properties in the nuclear medium are investigated by considering the wavefunction renormalization as a first step to reveal the in-medium properties of the $K^{+}$ meson in the context of partial restoration of chiral symmetry. The $K^{+}N$ elastic scattering amplitude is constructed using chiral perturbation theory up to the next-to-leading order. Using the constructed amplitude, we calculate the wavefunction renormalization in the Thomas-Fermi approximation. We obtained a good description of the $K^{+}N$ elastic scattering amplitude. The obtained wavefunction renormalizatio…
We analyze the renormalization of the NN interaction at low energies and the deuteron bound state through the Chiral Two Pion Exchange Potential assumed to be valid from zero to infinity. The short distance Van der Waals singularity structure of the potential as well as the requirement of orthogonality conditions on the wave functions determines that, after renormalization, in the 1S0 singlet channel and in the 3S1-3D1 triplet channel one can use the deuteron binding energy, the asymptotic D/S ratio and the S-wave scattering lengths as well as the chiral potential parameters as independent var…
We briefly review general concepts of renormalization in quantum field theory and discuss their application to solutions of integral equations with singular potentials in the few-nucleon sector of the low-energy effective field theory of QCD. We also describe a particular subtractive renormalization scheme and consider a specific application to a toy-model with a singular potential serving as its effective field theoretical leading-order approximation.
We study the effect of one-loop corrections from nucleon together with those from boson in the massless chiral sigma model, where we perform the Coleman-Weinberg renormalization procedure. This renormalization procedure has a mechanism of spontaneous symmetry breaking due to radiative corrections in $ϕ^4$ theory. We apply it to the system of nucleon and bosons with chiral symmetry where the negative-mass term of bosons does not exist. Spontaneous chiral symmetry breaking is derived from the contribution of nucleon and boson loops which generates the masses of nucleon, scalar meson, and vector …
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, …
I discuss the progress we have made on modifying Weinberg's prescription for chiral nuclear forces, using renormalization group invariance as the guideline. Some of the published results are presented.
An exposition is made of recent developments using techniques of unitary chiral perturbation theory, $U χP T$, which allows one to extend predictions using chiral Lagrangians to higher energies than ordinary chiral perturbation theory, including the region of low lying mesonic and baryonic resonances, some of which are dynamically generated in the approach. Results for meson meson scattering, pion and kaon form factors and meson baryon scattering are shown. Applications are done for nuclear problems showing the results for the kaon and eta selfenergies, phi renormalization in a nuclear medium …
The complete renormalization of the weak Lagrangian to chiral order q^2 in heavy baryon chiral perturbation theory is performed using heat kernel techniques. The results are compared with divergences appearing in the calculation of Feynman graphs for the nonleptonic hyperon decay Lambda -> p pi^- and an estimate for the size of the counterterm contributions to the s-wave amplitudes in nonleptonic hyperon decays is given.
Next to leading order effective field theory calculations are performed for $ {}^1S_0$ NN scattering using subtractive renormalization procedure. One pion exchange and contact interaction potentials are iterated using Lippman-Schwinger equation. Satisfactory fit to the Nijmegen data is obtained for the momenta up to 300 MeV in the centre of mass frame. Phase shifts are also compared with the results of KSW approach where pions are included perturbatively.
The leading divergences of the generating functional for Green functions of quark currents between one--nucleon states are calculated with heat kernel techniques. The results allow for a chiral invariant renormalization of all two--nucleon Green functions of the pion--nucleon system to $O(p^3)$ in the low--energy expansion.
The complete renormalization of the generating functional for Green functions of quark currents between one--baryon states in three flavor heavy baryon chiral perturbation theory is performed to order q^3. As an example, we study the kaon loop induced divergences in neutral pion photoproduction off protons.
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 $\bar{\rm MS}$ 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. In addition to simplicity our method has the benefit that it can be easily applied to multiloop diagrams. As an applicati…
Mass- and wave-function renormalization is calculated to order $p^4$ in heavy baryon chiral perturbation theory. Two different schemes used in the literature are considered. Several technical issues like field redefinitions, non-transformation of sources as well as subtleties related to the definition of the baryon propagator are discussed. The nucleon axial-vector coupling constant $g_A$ is calculated to order $p^4$ as an illustrative example.
We are interested in the consistency between the cutoff, chiral symmetry, and the power counting. For this purpose, we apply the Wilsonian renormalization group (RG) to an operator and then decrease the Wilsonian cutoff. As an example, we study the s-wave pion production operator for NN\to dπ, derived in chiral perturbation theory. We find that the renormalized part of the RG effective operator is accurately absorbed by chiral counter terms of higher order with natural coefficients. Thus, the use of the (sharp) cutoff regularization does not require us to introduce chiral-symmetry-breaking cou…
In nuclear chiral perturbation theory (ChPT), an operator is defined in a space with a cutoff which may be varied within a certain range. The operator runs as a result of the variation of the cutoff [renormalization group (RG) running]. In order for ChPT to be useful, the operator should run in a way consistent with the counting rule; that is, the running of chiral counter terms have to be of natural size. We vary the cutoff using the Wilsonian renormalization group (WRG) equation, and examine this consistency. As an example, we study the s-wave pion production operator for NN\to d pi, derived…
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…
We analyze the sigma meson mass and width together with the pion-pion scattering parameters in the context of non-local chiral quark models with wave-function renormalization (WFR). We consider both non-local interactions based on the frequently used exponential form factor, and on fits to the mass and renormalization functions obtained in lattice calculations. In the case of the sigma properties we obtain results which are less dependent on the parameterization than in the standard local NJL model, and which are in reasonable agreement with the recently reported empirical values. We also show…
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…
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.
We show using diagramtic arguments that in some (but not all) cases, the temperature dependent part of the chiral vortical effect coefficient is independent of the coupling constant. An interpretation of this result in terms of quantization in the effective 3 dimensional Chern-Simons theory is also given. In the language of 3D dimensionally reduced theory, the value of the chiral vortical coefficient is related to the formula $\sum_{n=1}^\infty n=-1/12$. We also show that in the presence of dynamical gauge fields, the CVE coefficient is not protected from renormalization, even in the large $N$…
We analyze the chiral phase transition of the Nambu--Jona-Lasinio model in the cold and dense region on the lattice developing the Grassmann version of the anisotropic tensor renormalization group algorithm. The model is formulated with the Kogut--Susskind fermion action. We use the chiral condensate as an order parameter to investigate the restoration of the chiral symmetry. The first-order chiral phase transition is clearly observed in the dense region at vanishing temperature with $μ/T\sim O(10^3)$ on a large volume of $V=1024^4$. We also present the results for the equation of state.
The chiral nucleon-meson model, previously applied to systems with equal number of neutrons and protons, is extended to asymmetric nuclear matter. Fluctuations are included in the framework of the functional renormalization group. The equation of state for pure neutron matter is studied and compared to recent advanced many-body calculations. The chiral condensate in neutron matter is computed as a function of baryon density. It is found that, once fluctuations are incorporated, the chiral restoration transition for pure neutron matter is shifted to high densities, much beyond three times the d…
The interplay of mesonic fluctuations with an axial $U(1)_A$-symmetry breaking and resulting effects on the location of a possibly existing critical endpoint in the QCD phase diagram are investigated in a framework of the functional renormalization group within a $N_f = 2 + 1$ flavor quark-meson model truncation. The axial $U(1)_A$-symmetry breaking is imposed by a mesonic Kobayashi-Maskawa-'t Hooft determinant. The quark mass sensitivity of the chiral phase transition with and without the $U(1)_A$-symmetry breaking is studied.
We explore the role of a holographic Wilsonian cut-off in simple probe brane models with chiral symmetry breaking/restoration phase transitions. The Wilsonian cut-off allows us to define supergravity solutions for off-shell configurations and hence to define a potential for the chiral condensate. We pay particular attention to the need for configurations whose action we are comparing to have the same IR and UV boundary conditions. We exhibit new first and second order phase transitions with changing cut-off. We derive the effective potential for the condensate including mean field and BKT type…
$S$-wave $KN$ scattering in a renormalizable chiral effective field theory
Shandong Provincial Key Laboratory of Nuclear Science, Nuclear Energy Technology and Comprehensive Utilization, & School of Nuclear Science, Energy and Power Engineering, Shandong University（山东省核科学与技术综合利用重点实验室、山东大学核科学与能源动力工程学院）
Chiral Doubling, Renormalization Group Fixed Points, and Dense Matter Equations of State
Institute of Theoretical Physics, University of Wroclaw（弗罗茨瓦夫大学理论物理研究所）；International Institute for Sustainability with Knotted Chiral Meta Matter (WPI-SKCM 2 ), Hiroshima University（广岛大学可链手性超物质可持续发展国际研究所）
We study the structure of dynamically generated bound states in the chiral unitary approach. The compositeness of a bound state is defined through the wavefunction renormalization constant in the nonrelativistic field theory. We apply this argument to the chiral unitary approach and derive the relation between compositeness of the bound state and the subtraction constant of the loop integral. The compositeness condition is fairly compatible with the natural renormalization scheme, previously introduced in a different context.
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…
We formulate the thermal renormalization group, an implementation of the Wilsonian RG in the real-time (CTP) formulation of finite temperature field theory, for fermionic fields. Using a model with scalar and fermionic degrees of freedom which should describe the two-flavor chiral phase-transition, we discuss the mechanism behind fermion decoupling and universality at second order transitions. It turns out that an effective mass-like term in the fermion propagator which is due to thermal fluctuations and does not break chiral symmetry is necessary for fermion decoupling to work. This situation…
We explicitly relate effective meson-baryon Lagrangian models, chiral bags, and Skyrmions in the following way. First, effective Lagrangians are constructed in a manner consistent with an underlying large-N_c QCD. An infinite set of graphs dress the bare Yukawa couplings at *leading* order in 1/N_c, and are summed using semiclassical techniques. What emerges is a picture of the large-N_c baryon reminiscent of the chiral bag: hedgehog pions for r > 1/Λpatched onto bare nucleon degrees of freedom for r < 1/Λ, where the ``bag radius'' 1/Λis the UV cutoff on the graphs. Next, a novel renormalizati…
We investigate the quark-meson model in a magnetic field using the exact functional renormalization group equation beyond the local-potential approximation. Our truncation of the effective action involves anisotropic wave function renormalization for mesons, which allows us to investigate how the magnetic field distorts the propagation of neutral mesons. Solving the flow equation numerically, we find that the transverse velocity of mesons decreases with the magnetic field at all temperatures, which is most prominent at zero temperature. The meson screening masses and the pion decay constants a…
We analyze the dynamical chiral symmetry breaking (D$χ$SB) in the Nambu-Jona-Lasinio (NJL) model by using the non-perturbative renormalization group (NPRG) equation. The equation takes a form of two-dimensional partial differential equation for the multi-fermion effective interactions $V(x,t)$ where $x$ is $\barψψ$ operator and $t$ is the logarithm of the renormalization scale. The D$χ$SB occurs due to the quantum corrections, which means it emerges at some finite $t_{\rm c}$ in the mid of integrating the equation with respect to $t$. At $t_{\rm c}$ some singularities suddenly appear in $V$ wh…
