A Conventions

Fermion Interactions and Universal Behavior in Strongly Interacting Theories

Abstract

The theory of the strong interaction, Quantum Chromodynamics (QCD), describes the generation of hadronic masses and the state of hadronic matter during the early stages of the evolution of the universe. As a complement, experiments with ultracold fermionic atoms provide a clean environment to benchmark our understanding of dynamical formation of condensates and the generation of bound states in strongly interacting many-body systems.

Renormalization group (RG) techniques offer great potential for theoretical advances in both hot and dense QCD as well as many-body physics, but their connections have not yet been investigated in great detail. We aim to take a further step to bridge this gap. A cross-fertilization is indeed promising since it may eventually provide us with an ab-initio description of hadronization, condensation, and bound-state formation in strongly interacting theories. After giving a thorough introduction to the derivation and analysis of fermionic RG flows, we give an introductory review of our present understanding of universal long-range behavior in various different theories, ranging from non-relativistic many-body problems to relativistic gauge theories, with an emphasis on scaling behavior of physical observables close to quantum phase transitions (i. e. phase transitions at zero temperature) as well as thermal phase transitions.

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Fermion Interactions and Universal Behavior in Strongly Interacting Theories] Fermion Interactions and Universal Behavior
in Strongly Interacting Theories

Jens Braun]Jens Braun 1

Contents:

1 Introduction

Strongly interacting fermions play a very prominent role in nature. The dynamics of a large variety of theories close to the boundary between a phase of gapped and ungapped fermions is determined by strong fermion interactions. For instance, the chiral finite-temperature phase boundary in quantum chromodynamics (QCD), the theory of the strong interaction, is governed by strong fermionic self-interactions. In the low-temperature phase the quark sector is driven to criticality due to strong quark-gluon interactions. These strong gluon-induced quark self-interactions eventually lead to a breaking of the chiral symmetry and the quarks acquire a dynamically generated mass. The chirally symmetric high-temperature phase, on the other hand, is characterized by massless quarks. The investigation of the QCD phase boundary represents one of the major research fields in physics, both experimentally and theoretically. Since the dynamics of the quarks close to the chiral phase boundary affect the equation of state of the theory, a comprehensive understanding of the quark dynamics is of great importance for the analysis of present and future heavy-ion collision experiments at BNL, CERN and the FAIR facility [1].

While heavy-ion collision experiments provide us with information on hot and dense QCD, experiments with ultracold trapped atoms provide an accessible and controllable system where strongly-interacting quantum many-body phenomena can be investigated precisely. In contrast to the theory of strong interactions, the interaction strength can be considered a free parameter in these systems which can be tuned by hand. In fact, the interaction strength is directly proportional to the -wave scattering length and can therefore be modulated via an external magnetic field using Feshbach resonances [2]. It is therefore possible to study quantum phenomena such as superfluidity and Bose-Einstein condensation in these systems. From a theorist’s point of view, this strong degree of experimental control opens up the possibility to test non-perturbative methods for the description of strongly interacting systems.

Phases of ultracold Fermi gases at zero and finite temperature have been studied experimentally, see e. g. Refs. [3, 4, 5, 6, 7] as well as theoretically, see e. g. Refs. [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], over the past few years. In particular, studies with renormalization group (RG) methods exhibit many technical similarities to studies of QCD at finite temperature and density, see e. g. Refs. [28, 29, 30, 31, 32, 33]. Physically, in both cases the phase boundary is determined by strong interactions of the fermions. While the asymptotic limits of the phase diagram of ultracold atoms for small positive and small negative (s-wave) scattering length associated with Bose-Einstein condensation and Bardeen-Cooper-Schrieffer (BCS) superfluidity [34], respectively, are under control theoretically [35, 36, 37, 38, 39], our understanding of the finite-temperature phase diagram in the limit of large scattering length (strong-coupling limit) is still incomplete [8, 10, 11, 14, 25, 27].

Aside from phase transitions at finite temperature, experiments with ultracold fermionic atoms provide a very clean environment for studies of quantum phase transitions. Experiments with a dilute gas of atoms in two different hyperfine spin states have been carried out in a harmonic trap at a finite spin-polarization [3, 4]. Since there is effectively no spin relaxation in these experiments, in contrast to most other condensed matter systems, the polarization remains constant for long times. Deforming the system by varying the polarization allows us to gain a deep insight into BCS superfluidity and its underlying mechanisms [34]. Originally, BCS theory has been worked out for systems in which the Fermi surfaces of the two spin states are identical, i. e. the polarization of the system is zero. As a function of the polarization, a quantum phase transition occurs at which the (fully polarized) normal phase becomes energetically more favorable than the superconducting phase [17, 20, 22]. After giving a thorough introduction on the level of (advanced) graduate students to the derivation and analysis of fermionic RG flow equations2 in Sects. 2 and 3, we shall discuss aspects of symmetry breaking and condensate-formation in non-relativistic theories from a RG point of view in Sect. 4.1. For simplicity, we restrict ourselves to systems with a vanishing polarization. The generalization of our RG approach to spin-polarized gases is straightforward and has been discussed in Refs. [23, 40].

Phase separation between a superfluid core and a surrounding normal phase has been indeed observed in experiments with an imbalanced population of trapped spin-polarized atoms at unitarity at MIT and Rice University [3, 4]. The density profiles measured in these experiments prove the existence of a skin of the majority atoms. A critical polarization associated with a quantum phase transition has been found in both the MIT and Rice experiment. Aside from studies at zero temperature, finite-temperature studies of a spin-polarized gas have been performed at Rice University [4]. In these experiments the critical polarization, above which the superfluid core disappears, has been measured as a function of the temperature. In accordance with theoretical studies [15, 18, 23], the results from the Rice group suggest that a tricritical point exists in the phase diagram spanned by temperature and polarization, at which the superfluid-normal phase transition changes from second to first order as the temperature is lowered. Depending on the physical observable, it is in principle possible that finite-size and particle-number effects are visible in the experimental data. Concerning the critical polarization, such effects have been studied in Ref. [41].

There is indeed direct evidence that finite-size effects can alter the phase structure of a given theory. For example, Monte-Carlo studies of the Gross-Neveu model show that the finite-temperature phase diagram of the uniform system is modified significantly due to the non-commensurability of the spatial lattice size with the intrinsic length scale of the inhomogeneous condensate [42, 43]. In particular, the phase with an inhomogeneous ground state shrinks. Such commensurability effects may be present in trapped ultracold Fermi gases as well. Since the Gross-Neveu model in is reminiscent of QCD in many ways, the existence of a stable ground state governed by an inhomogeneous condensate is subject of an ongoing debate, see e. g. Refs. [44, 45]. In any case, it is well-known that the mass spectrum and the thermodynamics of QCD has an intriguing dependence on the volume size and the boundary conditions of the fields, see e. g. Refs. [46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57].

Our theoretical understanding of the phase structure of trapped fermions is currently mostly based on Density Functional Theory (DFT) [58] in a local density approximation (LDA) in which, for example, derivatives of densities are omitted in the ansatz for the action, see e. g. Refs. [18, 16, 19, 24, 26]. From a field-theoretical point of view, DFT corresponds to a mapping of the (effective) action of a fermionic theory onto an action which depends solely on the density. The latter then plays the role of a composite degree of freedom of fermions. Thus, the underlying idea is reminiscent of the Hubbard-Stratonovich transformation [59, 60] widely used in low-energy QCD models and spin systems. In any case, the introduction of an effective degree of freedom, such as the density, turns out to be advantageous for a description of theories with an inhomogeneous ground-state. Again, experiments with ultracold atoms allow us to test different approaches and approximation schemes. In Refs. [16, 24], the equation of states of the superfluid and the normal phase of a uniform system have been employed to construct a density functional which allows to study the ground-state properties of trapped Fermi gases. Such a procedure corresponds to an LDA. While there is some evidence that Fermi gases in isotropic traps can be quantitatively understood within DFT in LDA [24], the description of atoms in a highly-elongated trap in LDA seem to fail and derivatives of the density need to be taken into account [61, 62]. In the spirit of these studies, we shall discuss a functional RG approach to DFT in Sect. 4.2 which relies on an expansion of the energy density functional in terms of correlation functions and allows to include effects beyond LDA in a systematic fashion.

Heavy nuclei combine aspects of dense and hot QCD and systems of ultracold atoms. We again need to describe strong interactions between fermions, the nucleons, which form a stable bound state depending on, e. g., the number of protons and neutrons. These interactions are repulsive at short range and attractive at long range as in the case of ultracold atomic gases. Loosely speaking, heavy nuclei can be viewed as spin-polarized systems of two fermion species comparable to those systems studied in experiments with trapped spin-polarized atoms at MIT and Rice University [3, 4]. In fact, almost all nuclei have more neutrons than protons.3 Therefore density profiles of protons and neutrons in heavy nuclei are evocative of the profiles associated with the two fermion states in experiments with ultracold atoms. For heavy nuclei, DFT remains currently to be the only feasible approach for a calculation of ground-state properties associated with inhomogeneous densities. State-of-the-art density functional approaches are essentially based on fitting the parameters of a given density functional such that one reproduces the experimentally determined values of the ground-state properties of various heavy nuclei [63, 64, 65]. These density functionals are then employed to describe ground-state properties of other heavy nuclei. As mentioned above, we shall briefly discuss an RG approach to DFT in Sect. 4.2, which opens up the possibility to study ground-state properties of heavy nuclei from the underlying nucleon-nucleon interactions. Such an ab-initio DFT approach might prove to be useful for future studies of ground-state properties of (heavy) self-bound systems.

Hot and dense QCD, ultracold atoms and nuclear physics represent just three examples for systems in which the dynamics are governed by strong fermion interactions. Of course, the list can be extended almost arbitrarily. In the context of condensed-matter theory, we encounter systems such as so-called high- superconductors. In this case the challenge is to describe reliably the dynamics of electrons at finite temperature in an ambient solid-state system. The so-called Hubbard model provides a theory to describe these superconductors [66, 67] and has been extensively studied with renormalization-group techniques, see e. g. Refs. [68, 69, 70, 71]. It is worth noting that both the mechanisms as well as the techniques are remarkably similar to the ones in renormalization-group studies of gauged fermionic systems interacting strongly via competing channels [30, 31, 32], such as QCD, and of imbalanced Fermi gases in free space-time [23]. In Sect. 5, we discuss more general aspects of (non-gauged) Gross-Neveu- and Nambu-Jona-Lasinio-type models which also exhibit technical similarities to studies of condensed-matter systems. Nambu-Jona-Lasinio-type models are widely used as effective QCD low-energy models. On the other hand, Gross-Neveu-type models have been employed as toy models to study certain aspects of the QCD phase diagram but they are also related to models in condensed-matter theory, e. g. to models of ferromagnetic (relativistic) superconductors [72, 73]. In this review, we shall use Gross-Neveu- and Nambu-Jona-Lasinio-type models to discuss dynamical chiral symmetry breaking (via competing channels) and the role of momentum dependences of fermionic interactions.

In addition to fermion dynamics at finite temperature, quantum phase transitions play a prominent role in condensed-matter theory, e. g., in the context of graphene. Effective theories of graphene, such as QED and the Thirring model, are expected to approach a quantum critical point when the number of fermion species, namely the number of electron species, is varied [74, 75]. RG studies of these effective theories, see e. g. Refs. [74, 76, 75], are closely related to studies of quantum phase transitions in QCD [77, 29, 30, 31, 78, 79]. Similar to the situation in QED, a quantum phase transition from a chirally broken to a conformal phase is expected in QCD when the number of (massless) quark flavors is increased. Studies of the dependence on the number of fermion species seem to be a purely academic question. Depending on the theory under consideration, however, such a deformation of the theory may allow us to gain insights into the dynamics of fermions close to a phase boundary in a controlled fashion. For example, the gauge coupling in QCD becomes small when the number of quark flavors is increased and therefore perturbative approaches in the gauge sector become meaningful. Moreover, an understanding of strongly-flavored QCD-like gauge theories is crucial for applications beyond the standard-model, namely for so-called walking technicolor scenarios for the Higgs sector [80, 81, 82, 83, 84, 85, 86, 87, 88]. In Sect. 6, we shall discuss chiral symmetry breaking in gauge theories with  fermion flavors. In particular, we shall present a detailed discussion of the scaling behavior of physical observables close to the quantum phase transition which occurs for large .

Our discussion shows that systems of strongly interacting fermions play indeed are very prominent role in nature and that their dynamics determine the behavior of a wide class of physical systems with seemingly substantial differences. However, our discussion also shows that the underlying mechanisms of symmetry breaking and the applied techniques are very similar in these different fields. Therefore a phenomenological and technical cross-fertilization offers great potential to gain a better understanding of the associated physical processes. As outlined, examples include an understanding of the dynamical generation of hadron masses as well as of the dynamical formation of condensates and bound-states in ultracold gases from first principles. The main intent of the present review is to give a general introduction to the underlying mechanisms of symmetry breaking and bound-state formation in strongly-interacting fermionic theories. In particular, we aim to give a thorough introduction into the scaling behavior of physical observables close to critical points, ranging from power-law scaling behavior to essential scaling. As a universal tool for studies of quantum field theories we employ mainly Wilsonian-type renormalization-group techniques [89, 90, 91, 92, 93, 94, 95]. For concrete calculations we shall use the so-called Wetterich equation [95] which we briefly introduce in the next section. Reviews focussing on various different aspects of renormalization-group approaches can be found in Refs. [96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110].

2 Renormalization Group - Basic Ideas

We begin with a brief introduction of the basic ideas of RG approaches including a discussion of the Wetterich equation. The latter describes the scale dependence of the quantum effective action which underlies our studies in this and the following sections.

In perturbation theory, the correlation functions of a given quantum field theory contain divergences which can be removed by a renormalization prescription. The choice of such a prescription defines a renormalization scheme and renders all (coupling) constants of a given theory scheme-dependent. Since the renormalized (coupling) constants are nothing but mathematical parameters, their values can be arbitrarily changed by changing the renormalization prescription. We stress that these renormalized constants should not be confused with physical observables such as, for example, the phase transition temperature or the physical mass of a particle. Physical observables are, of course, invariant under a variation of the renormalization prescription, provided we have not truncated the perturbation series. If we consider a truncated perturbation series, we find that there is a residual dependence on the renormalization scheme which can be controlled to some extent by the so-called ”Principle of Minimum Sensitivity” [111], see also discussion below.

At this point we are then still free to perform additional finite renormalizations. This results in different effective renormalization prescriptions. A given renormalization prescription can then be considered as a particular reordering of the perturbative expansion which expresses it in terms of new renormalized constants [112]. Let us assume that the transformations between the finite renormalizations can be parametrized by introducing an auxiliary single mass scale . This scale corresponds to a UV (cutoff) scale at which the parameters of the theory are fixed. A set of RG equations for a given theory then describes the changes of the renormalized parameters of this theory (e. g. the coupling constant) induced by a variation of the auxiliary mass scale . The set of renormalization transformations is called the renormalization group.

Let us now consider a (renormalized) microscopic theory at some large momentum scale defined by a (classical) action . Wilson’s basic idea of the renormalization group is to start with such a classical action and then to integrate out successively all fluctuations from high to low momentum scales [89, 90, 91]. This procedure results in an action which depends on an IR regulator scale, say , which plays the role of a reference scale. The values of the (scale-dependent) couplings defining this action on the different scales are related by continuous RG transformations. We shall refer to the change of a coupling under a variation of the scale  as the RG flow of the coupling. In this picture, universality means that the RG flow of the couplings is governed by a fixed point. The possibility of identifying fixed points of a theory makes the RG such a powerful tool for studying statistical field theories as well as quantum field theories. As we shall discuss below, critical behavior near phase transitions is intimately linked to the fixed-point structure of the theory under consideration.

In this review we employ a non-perturbative RG flow equation, the Wetterich equation [95], for the so-called effective average action in order to analyze critical behavior in physical systems. The effective average action  depends on an intrinsic momentum scale which parameterizes the Wilsonian RG transformations. We note that such an approach is based on the fact that an infinitesimal RG transformation (i. e. an RG step), performed by an integration over a single momentum shell of width , is finite. For this reason we are able to integrate out all quantum fluctuations through an infinite sequence of such RG steps. The flow equation for then describes the continuous trajectory from the microscopic theory at large momentum scales to the full quantum effective action (macroscopic theory) at small momentum scales. Thus, it allows us to cover physics over a wide range of scales.

Here, we only discuss briefly the derivation and the properties of the RG flow equation for the effective average action ; for details we refer to the original work by Wetterich [95]. The scale-dependent effective action is a generalization of the (quantum) effective action but only includes the effects of fluctuations with momenta . Therefore is sometimes called a coarse-grained effective action since quantum fluctuations on length scales smaller than are integrated out. The underlying idea is to calculate the generating functional of one-particle irreducible (1PI) graphs of a given theory by starting at an ultraviolet (UV) scale  with the microscopic (classical) action and then successively integrating out quantum fluctuations by lowering the scale . The quantum effective action is then obtained in the limit . In other words, the coarse-grained effective action  interpolates between the classical action at the UV scale  and the 1PI generating functional in the infrared limit (IR) . The starting point for the derivation of the flow equation of is a UV- and IR-regularized generating functional  for the Greens functions:4

(1)

where and is the scale-dependent generating functional for the connected Greens functions. The field variable as well as the source are considered as generalized vectors in field space and are defined as

(2)

Moreover, we have introduced a generalized scalar product in field space: . Here, the field  represents a Dirac spinor, and denotes a real-valued scalar field. The dots indicate that other types of fields, e. g. gauge-fields, are allowed as well. For non-relativistic theories of fermions, the generating functional can be defined accordingly. We assume that the theory is well-defined by a UV-regularized generating functional: The index indicates that we only integrate over fields with momenta , i. e. we implicitly take for . To regularize the infrared modes a cutoff term has been inserted into the path integral. It is defined as

(3)

where is a matrix-valued regulator function. Through the insertion of the cutoff term, we have defined a generating functional which now depends on the scale .

The cutoff function has to fulfill three conditions. Since has been introduced to regularize the IR, it must fulfill

(4)

where . Second, the function must vanish in the IR-limit, i. e. for :

(5)

This condition ensures that we obtain the 1PI generating functional for . Third, the cutoff function should obey

(6)

for fixed . This property guarantees that for .

In this review, we shall always use cutoff functions which can be written in terms of a dimensionless regulator shape function . For simple relativistic scalar theories, we may choose

(7)

For studies of theories with chiral fermions, it is convenient to employ a cutoff function which preserves chiral symmetry. An appropriate choice is [113]

(8)

On the other hand, for non-relativistic fermionic many-body problems the choice of the cutoff function should respect the presence of a Fermi surface. An appropriate choice for such a cutoff function is given by [27]

(9)

where, for instance,

(10)

The chemical potential of the fermions is given by  and defines the associated Fermi surface. This choice for the regulator function arranges the momentum-shell integrations around the Fermi surface, i. e. modes with momenta remain unchanged while the momenta of modes with are cut off.

For scalar field theories, the presence of a cutoff function of the form is in general not problematic. For gauge theories, however, it causes difficulties due to condition (4) which essentially requires that the cutoff function acts like a mass term for small momenta. Therefore the cutoff function necessarily breaks gauge symmetry. We stress that this observation does by no means imply that such an approach cannot be applied to gauge theories. In fact, it is always necessary to fix the gauge in order to treat gauge theories perturbatively within a path-integral approach. This gauge-fixing procedure also breaks gauge invariance. Gauge-invariant results are then obtained by resolving Ward-Takahashi identities. Consequently, we can think of the cutoff function as an additional source of gauge-symmetry breaking. In analogy to perturbation theory, one then needs to deal with modified Ward-Takahashi identities in order to recover gauge invariance [114, 115, 116, 117, 118, 119]. In addition, there are essentially two alternatives: first, one can construct manifestly gauge-invariant flows as proposed in [120, 121, 108]. Second, we can apply special (useful) gauges, such as the background-field gauge [122, 123]. We refer the reader to Ref. [105] for a detailed introduction to RG flows in gauge theories.

The coarse-grained effective action  can in principle be obtained from the IR-regularized functional  in a standard fashion, see, e. g., the standard textbook derivation of the quantum effective action  in Refs. [123, 112]. However, we employ here a modified Legendre transformation to calculate the coarse-grained effective action:5

(11)

The so-called classical field  is implicitly defined by the supremum prescription. The modification of the Legendre transformation is necessary for the connection of with the classical action in the limit . From this definition of we find the RG flow equation of the coarse-grained effective action, the so-called Wetterich equation [95], by taking the derivative with respect to the scale :

(12)

with being the RG “time” and . The -point functions are defined as follows:

(13)

Thus, is matrix-valued in field space. The super-trace arises since contains both fermionic as well as bosonic degrees of freedom and it provides a minus sign in the fermionic subspace of the matrix. The double-line in Eq. (12) represents the full propagator of the theory which includes the complete field dependence. The solid black dot in the loop stands for the insertion of . The structure of the flow equation reveals that the regulator function specifies the Wilsonian momentum-shell integrations, such that the RG flow of is dominated by fluctuations with momenta .

The flow equation (12) has been obtained by taking the derivative of with respect to the scale . However, we have not taken into account a possible scale dependence of the classical field yielding a term  on the right-hand side of Eq. (12). We stress that the inclusion of this term is a powerful extension of the flow equation discussed here, since it allows to bridge the gap between microscopic and macroscopic degrees of freedom in the RG flow, e. g. between quarks and gluons and hadronic degrees of freedom, without any fine-tuning [124, 28, 125]. More technically speaking, this extension makes it possible to perform continuous Hubbard-Stratonovich transformations in the RG flow. We shall not employ these techniques here since they do not provide us with additional insights into the fermionic fixed-point structure to which the scope of the present review is limited. For details concerning such an extension of the flow equation (12), we refer the reader to Refs. [124, 28, 125, 104, 105, 126, 127]. In Ref. [32] these so-called re-bosonization techniques6 have been employed for a first-principles study of the QCD phase boundary.

As should be the case for an exact one loop flow [128], the Wetterich equation (12) is linear in the inverse of the full propagator. Moreover, it is a nonlinear functional differential equation, since it involves the inverse of the second functional derivative of the effective action. We stress, however, that the loop in Eq. (12) is not a simple perturbative loop since it depends on the full propagator. In fact, it can be shown that arbitrarily high loop-orders are summed up by integrating this flow equation [128]. Nonetheless it is possible and sometimes even technically convenient to rewrite (12) in a form which is reminiscent of the textbook form of the one-loop contribution to the effective action:

(14)

Here, denotes a formal derivative acting only on the -dependence of the regulator function . Replacing by the (scale-independent) second functional derivative of the classical action, , we can perform the integration over the RG scale analytically and obtain the standard one-loop expression for the effective action:

(15)

where

(16)

Here, the second term on the right-hand side corresponds to the boundary condition for the RG flow at the UV scale , which renders finite.

From a technical point of view, the representation (14) turns out to be a convenient starting point for our studies of the fixed-point structure of four-fermion interactions. In order to calculate flow equations of four-fermion interactions, we decompose the inverse regularized propagator on the right-hand side of the flow equation into a field-independent () and a field-dependent () part,

(17)

We can then expand the flow equation in powers of the fields according to

(18)

The powers of can be calculated by simple matrix multiplications. The flow equations for the various couplings can now be obtained by comparing the coefficients of the four-fermion operators on the right-hand side of Eq. (18) with the couplings specified in the definition of the effective action. In other words, the flow equation of higher -point functions are obtained straightforwardly from the flow equation (12) (or, equivalently, from Eq. (18)) by taking the appropriate number of functional derivatives. From this, we observe that the RG flow of the -point function depends in general on the flow of the - and -point function. This means that we obtain an infinite tower of coupled flow equations by taking functional derivatives of the flow equation (12). In most cases we are not able to solve this infinite tower of flow equations. Thus, we need to truncate the effective action and restrict it to include only correlation functions with external fields. However, such a truncation poses severe problems: first, the system of flow equations is no longer closed and, second, neglecting higher -point functions may render the flow unstable in the IR region of strongly coupled theories. For example in QCD, one would naively expect that contributions from higher -point functions are important.

Finding reliable truncations of the effective action is the most difficult step and requires a lot of physical insight. We stress that an expansion in terms of -point functions must not be confused with an expansion in some small parameter as in perturbation theory. The assumption here is that the influence of neglected operators on the already included operators is small. Once we have chosen a truncation for studying a given theory, we need to check its reliability. One possibility for such a check is to extend the truncation by including additional operators and then check if the results obtained from this new truncation are in agreement with the earlier results. If this is not the case, one must rethink the chosen truncation. However, even if the results are not sensitive to the specific set of additional operators added to the truncation, this does not necessarily mean that one has included all relevant operators in the calculation. A second possibility to assess the reliability of a given truncation is to exploit the fact that physical observables should not depend on the regularization scheme. Since the scheme is specified by the cutoff function, the physical observables should be independent of this choice. In the present approach the scheme is defined by our choice for the regulator function . Thus, we can vary and then check if the results depend on the choice of the cutoff function. If this is the case, an extension of the truncation might be required. In addition to a simple variation of regulator functions, we may actually exploit the dependence on to optimize the truncated RG flow of a given theory. For example, an optimization criterion can be based on the size of the gap induced in the effective propagator , see Refs. [129, 130, 131]. We then denote those regulators to be optimized for which the gap is maximized with respect to the cutoff scheme. In addition to such an optimization of RG flows within a given regulator class, a more general criterion has been put forward in Ref. [104]. The latter defines the optimized regulator to be the one for which the regularized theory is already closest to the full theory at , for a given gap induced in the effective propagator . This optimization criterion yields an RG trajectory which defines the shortest path in theory space between the UV theory at and the full theory at . Both optimization criteria naturally encompass the so-called “Principle of Minimum Sensitivity”. However, in contradistinction to the “Principle of Minimum Sensitivity”, the optimization of (truncated) RG flows does not rely on the existence of extremal values of physical observables which may arise from a variation of the regularization scheme. For a detailed discussion of optimization criteria and properties of optimized RG flows, we refer the reader to Refs. [130, 129, 131, 104].

Nonetheless, even an approximate solution of the flow equation (12) can describe non-perturbative physics reliably, provided the relevant degrees of freedom in the form of RG relevant operators are kept in the ansatz for the effective action.

3 RG Flow of Four-Fermion Interactions - A Simple Example

In this section we discuss a simple four-fermion theory which already allows us to gain some important insight into the mechanisms of symmetry breaking in strongly-interacting theories. A study of a simple four-fermion theory is useful for many reasons. First, it allows us to highlight various methods and technical aspects such as Fierz ambiguities, (partial) bosonization and the role of explicit symmetry breaking. Second, a confrontation of this model study with our analysis of symmetry breaking in gauge theories is instructive: To be specific, we will consider the mechanisms of chiral symmetry breaking to point out the substantial differences between these theories.

3.1 A Simple Example and the Fierz Ambiguity

In this section we discuss the basic concepts and problems in describing strongly-interacting fermionic theories, with a particular emphasis on the application of RG approaches. To this end, we employ a Nambu–Jona-Lasinio-type model. Such models play a very prominent role in theoretical physics. Originally, the Nambu–Jona-Lasinio (NJL) model has been used as an effective theory to describe spontaneous symmetry breaking in particle physics based on an analogy with superconducting materials [132, 133], see Ref. [134] for a review. RG methods have been extensively employed to study critical behavior in QCD with the aid of NJL-type models, see e. g. Refs. [113, 135, 136, 100, 137, 138, 53, 139, 33, 55]. Usually these model studies rely on a (partially) bosonized version of the action. We shall discuss aspects of bosonization in Sect. 3.2. For the sake of simplicity we start with a purely fermionic formulation of the NJL model with only one fermion species. This model has been extensively studied at zero temperature with the functional RG in Refs. [140, 77]. In particular, the ambiguities arising from Fierz transformations have been explicitly worked out and discussed. We shall follow the discussion in Refs. [140, 77] but extend it with respect to issues arising at finite temperature and for a finite (explicit) fermion mass. In addition, we exploit this model to discuss general aspects of theories with many fermion flavors as well as quantum critical behavior.

In the following we consider a simple ansatz for the effective action in Euclidean space-time dimensions:

(19)

where is the bare four-fermion coupling and is the so-called fermionic wave-function renormalization. The coupling is considered to be RG-scale dependent. Here, we consider four-fermion couplings as fundamental parameters. However, in other theories fermionic self-interactions might be fluctuation-induced. In QCD, for example, they are induced by two-gluon exchange and are therefore not fundamental as we shall discuss in Sect. 6, see also Refs. [28, 29, 30, 31, 32, 78]. We would like to add that the NJL model in  is perturbatively non-renormalizable. In the following we define it with a fixed UV cutoff . Also the regularization scheme therefore belongs to the definition of the model. We shall come back to this issue in Sects. 3.2 and 5.1.

Our ansatz (19) for the effective action can be considered as the leading order approximation in a systematic expansion in derivatives. The associated small parameter of such an expansion is the so-called anomalous dimension of the fermion fields. If this parameter is small, then such a derivative expansion is indeed justified. We shall come back to this issue below. In any case, we will drop terms in our studies which are of higher order in derivatives, such as terms .

The action (19) is clearly invariant under simple phase transformations,

(20)

but also under chiral U() transformations (axial phase transformations),

(21)

where is an arbitrary “rotation” angle. A necessary condition for the chiral symmetry of the NJL model is the absence of explicit mass terms for the fermion fields in the action, such as . As we shall discuss in more detail below, the chiral symmetry can be still broken spontaneously, if a finite vacuum expectation value is generated by loop corrections associated with (strong) fermionic self-interactions. Breaking of chiral symmetry in the ground state of the theory is then indicated by a dynamically generated mass term for the fermions. This mass term is associated, e. g., with a constituent quark mass in low-energy models of QCD and similar to the gap in condensed-matter theory. The relation between the strength of the four-fermion interactions and the symmetry properties of the ground-state are discussed in detail in Sects. 3.2 and 3.3.

We may now ask whether the action (19) is complete or whether other four-fermion couplings, such as a vector interaction , can be generated dynamically due to quantum fluctuations. We first realize that the four-fermion interaction in our ansatz (19) can be expressed in terms of a vector and axial-vector interaction term with the aid of so-called Fierz transformations, see App. B for details:

(22)

This ambiguity in the representation of a four-fermion interaction term arises due to the fact that an arbitrary -matrix can be expanded in terms of a complete and orthonormalized set of -matrices as follows:

(23)

The expansion of a combination of two matrices and then reads (say for fixed  and )

(24)

In the case of four-fermion interactions we may classify the basis elements according to the transformation properties of the corresponding interaction terms , i. e. scalar channel, vector channel, tensor channel, axial-vector channel and pseudo-scalar channel. To be specific, we choose , , , and as basis elements of the Clifford algebra defined by the matrices, see App. B for details. To obtain Eq. (22) we then simply apply Eq. (24) to the matrix products and , respectively. Thus, a Fierz transformation can be considered as a reordering of the fermion fields. We stress that this is by no means related to quantum effects but a simple algebraic operation. Nonetheless it suggests that other four-fermion couplings compatible with the underlying symmetries of our model exist and are potentially generated by quantum effects.

With our choice for the set of basis elements it is straightforward to write down the most general ansatz for the effective action  which is compatible with the underlying symmetries of the model, i. e. the symmetries with respect to U() phase transformations, U() chiral transformations and Lorentz transformations:7

(25)

Because of Eq. (22) only two of the three couplings , and are independent. Thus, it suffices to consider the following action with implicitly redefined four-fermion couplings:

(26)

Note that we could have also chosen to remove, e. g., the vector-channel interaction term with the aid of Eq. (22) at the expense of getting the axial-vector interaction. From a phenomenological point of view it is tempting to attach a physical meaning to, e. g., the vector-channel interaction and interpret it as an effective mass term for vector bosons  as done in mean-field studies of Walecka-type models [141]: . However, the present analysis shows that one has to be careful to attach such a phenomenological interpretation to this term since the Fierz transformations allow us to remove this term completely from the action, see also Sect. 3.2.

In this section we drop a possible momentum dependence of the four-fermion couplings. Thus, we only take into account the leading term of an expansion of the four-fermion couplings in powers of the dimensionless external momenta , e. g.

(27)

In momentum space, the corresponding interaction term in the expansion of the effective action (26) in terms of fermionic self-interactions then assumes the following form, see App. A for our conventions of the Fourier transformation:

(28)

where and correspondingly for the other four-fermion interaction terms in Eq. (26). Note that only three of the four four-momenta are independent due to momentum conservation. We stress that we also apply this expansion at finite temperature , see Sect. 3.5.3. In this case, it then corresponds to an expansion in powers of the dimensionless Matsubara modes and . Thus, we assume that .

The approximation (27) does not permit a study of properties of bound states of fermions, such as meson masses in QCD, in the chirally broken regime; such bound states manifest themselves as momentum singularities in the four-fermion couplings in Minkowski space. Nonetheless, the point-like limit can still be a reasonable approximation for . In the chirally symmetric regime above the chiral phase transition it allows us to gain some insight into the question how the theory approaches the regime with broken chiral symmetry in the ground state [30, 31, 32]. In Sect. 3.2 we shall discuss how the momentum dependence of the fermionic interactions can be conveniently resolved in order to gain access to the mass spectrum in the regime with broken chiral symmetry.

Let us now compute the RG flow equations, i. e. the so-called functions, for the four-fermion couplings in the point-like limit. To this end, we compute the second functional derivative of the effective action with respect to the fields

(29)

see also Eq. (13), and evaluate it for homogeneous (constant) background fields and . In momentum space this means that we evaluate at

(30)

where and on the right-hand side denote the homogeneous background fields. Following Eq. (17), we then split the resulting matrix into a field-independent part and a part which depends on and . To detail the derivation of flow equations of four-fermion interactions in a simple manner, we first restrict ourselves to the simplified ansatz (19) of the effective action. In this case, the so-called (regularized) propagator matrix and the fluctuation matrix read

(31)

and

(32)

respectively, where

Since we evaluated for constant fields, both and are diagonal in momentum space. At this point it is not yet necessary to specify the regulator function exactly.

The RG flow equation for  can now be computed straightforwardly by comparing the coefficients of the four-fermion interaction terms on the right-hand side of Eq. (18) with the couplings included in our ansatz (19). From the fluctuation matrix it is clear that only the term in Eq. (18) contributes to the RG flow of the four-fermion coupling . For this initial study, we simply take the four-fermion terms on the right-hand side of the flow equation “as they appear” and ignore Fierz transformations of these terms. We then find

(33)

where , i. e. . Here, we have defined the dimensionless renormalized coupling

(34)

The so-called threshold function corresponds to a one-particle irreducible (1PI) Feynman diagram, see left diagram in Fig. 1, and describes the decoupling of massive and also thermal modes in case of finite-temperature studies. Moreover, the regularization-scheme dependence is encoded in these threshold functions, see App. D for their definitions.

Figure 1: Left diagram: PI Feynman diagram associated with the -term on the right-hand side of the RG flow equation (33). Note that our (functional) RG study includes resummations of this diagram to arbitrary order in . Right diagram: PI Feynman diagram associated with the RG running of the fermionic wave-function renormalization .

In Fig. 2 we show a sketch of the -function for vanishing temperature. Apart from a Gaußian fixed point, , we find a second non-trivial fixed point :

(35)

In the present leading-order approximation of the derivative expansion we have , see below. We then find

(36)

for an optimized (linear) regulator function (for which ) and

(37)

for the sharp cutoff (for which ). It is instructive to have a closer look at Eq. (33). This flow equation represents an ordinary differential equation which can be solved analytically for . Its solution reads

(38)

where

(39)

In order to derive Eq. (38), it is convenient to expand the right-hand side of Eq. (33) about the fixed-point . The physical meaning of the so-called critical exponent  will be discussed in more detail below. In Sect. 3.4.1 we will then see that this exponent governs the scaling behavior of physical observables close to a quantum critical point.

For , we find that  does not dependent on the RG scale  as it should be: . Choosing an initial value at the initial UV scale , the solution (38) of the flow equation predicts that the theory becomes non-interacting in the infrared regime ( for ), i. e. chiral symmetry remains unbroken in this case, see Fig. 2. For , we find that the four-fermion coupling  increases rapidly and diverges eventually at a finite scale : . This behavior of the coupling and the associated fixed-point structure are tightly linked to the question whether chiral symmetry is broken in the ground state or not: The value of the non-trivial fixed-point can be considered as a critical value of the coupling which separates the chirally symmetric regime and the regime with a broken chiral symmetry in the ground state. We shall discuss this in more detail in Sects. 3.23.3 and 3.4.

In the derivation of the flow equation (33) we have dropped contributions arising from four-fermion interactions with different transformation properties, e. g. a vector-channel interaction. From the expansion (18) of the flow equation, we can indeed read off that contributions to the flow of four-fermion couplings other than might be generated, even though they have not been included in the truncation (19): the matrix multiplications on the right-hand side of Eq. (18) mix the contributions from the propagator , which is proportional to , with the contributions from the field-dependent part :

(40)

This term obviously contributes to the flow of the -coupling.8 Moreover, contributions of this type couple the flow equations of the various four-fermion interactions to one another. Thus, quantum fluctuations induce a vector-channel interaction, even though we have not included such an interaction term initially. This observation explains why we need to include a basis which is complete with respect to Fierz transformations, such as in the effective action (25). We stress that the effective action (25) is closed in the sense that no contributions to four-fermion interactions, which are not covered by the truncation, are generated in the RG flow: any other pointlike four-fermion interaction compatible with the underlying symmetries of the theory can be written in terms of the interactions included in these effective actions by means of Fierz transformations.

Figure 2: Sketch of the function of the four-fermion interaction for zero temperature (black/solid line). The arrows indicate the direction of the RG flow towards the infrared.

In the point-like limit the RG flow of the four-fermion coupling is completely decoupled from the RG flow of fermionic -point functions of higher order. For example, -fermion interactions do not contribute to the RG flow of the coupling in this limit. Using the one-loop structure of the Wetterich equation, this statement can be proven diagrammatically: it is not possible to construct a one-loop diagram with only for external legs out of fermionic -point functions () which are compatible with the underlying chiral symmetry.

Up to now we have only discussed the running of a four-fermion coupling. We have not yet discussed how to compute the running of the wave-function renormalization . In general, the flow equation for  can be obtained from evaluated for a spatially varying background field,

where denotes the external momentum.9 The -dependent second functional derivative  can still be split into a field-independent and a field-dependent part. However, the latter is no longer diagonal in momentum space. The flow equations for the wave-function renormalizations can then be computed by comparing the coefficients of the terms bilinear in fermionic fields which appear on the right-hand side of Eq. (18) with the kinetic terms in the ansatz for the effective action. In our present approximation, we find that the RG running of  is trivial, i. e.

(41)

Thus, the associated anomalous dimensions is zero. In fact, this follows immediately from the associated 1PI Feynman diagram, see diagram on the right in Fig. 1, which has only one internal fermion line.10 In the following we therefore set the wave-function renormalization to one, , which implies .

Let us now turn to the effective action (26). The flow equations of the various couplings can be derived along the same lines as the RG equation for the -coupling detailed above. We find

(42)
(43)

where the dimensionless (renormalized) couplings are defined as

(44)

In the derivation of the flow equations for  and  also terms of the type and

(45)

appear. While the latter vanishes identically, see also App. B, the former can be completely transformed into a scalar-pseudoscalar and vector-interaction channel with the aid of the Fierz transformation (22). In fact, any four-fermion interaction term appearing in the derivation of the flow equations for the present system can be unambiguously rewritten in terms of these two interaction channels. Thus, the above RG flows are closed with respect to Fierz transformations. Due to Eq. (22) we could have also used, e. g., a scalar-pseudoscalar and an axial-vector interaction to describe the properties of our simplified theory without loss of physical information. The present choice for a complete basis of four-fermion interactions is one of several possibilities.

Our flow equations for  and  agree with the results found in Refs. [142, 140]. The RG flow of the couplings  and  is governed by three fixed points which are given by11

(46)

where

(47)

These fixed-points are of phenomenological importance. First of all, they might be related to (quantum) phase transitions. Second, we can define sets of initial values for the RG flows of the couplings  and  for which we find condensate formation associated with (chiral) symmetry breaking in the IR, as we shall discuss in detail in the two subsequent sections. The existence of such sets of initial conditions is not a generic feature of fermionic models but also appears in (chiral) gauge theories. In QCD and QED, four-fermion interactions are generated dynamically due to strong quark-gluon interactions, see our discussion in Sect. 6.

We can classify the various fixed points according to their directions in the space spanned by the couplings. To this end, we first linearize the RG flow equations of the couplings near a fixed point:

(48)

and . We refer to as the stability matrix. The two eigenvectors  and eigenvalues  (critical exponents) of this matrix essentially determine the RG evolution near a fixed point:12

(49)

The solution of the RG flow in the fixed-point regime is then given by

(50)

Here, the ’s define the initial conditions at the scale . From the solution of the linearized flow it becomes apparent that positive critical exponents, , correspond to RG relevant, i. e. infrared repulsive, directions. On the other hand, negative critical exponents  correspond to RG irrelevant, i. e. infrared attractive, directions. The classification of marginal directions associated with vanishing critical exponents requires to consider higher orders in the expansion about the fixed point.

Figure 3: Sketch of the -function of the four-fermion interaction for (black line) and  (blue/dashed line). The arrows indicate the direction of the RG flow towards the infrared.

Using the flow equations (42) and (43), we find that the Gaußian fixed point has two IR attractive directions; the eigenvalues are . The fixed points with and  with have both one IR attractive and one IR repulsive direction. We would like to add that the fixed-point values of the four-fermion couplings are not universal quantities as the dependence of their RG flows on the threshold function indicates. However, the statement about the mere existence of these fixed points is universal, because the regulator-dependent factor is a positive number for any regulator. Moreover, the critical exponents  themselves are universal. The latter can be indeed related to the exponents associated with (quantum) phase transitions, as we shall discuss in Sect. 3.4. Therefore the accuracy of the critical exponents can be used to measure the quality of a given truncation as has been done in the context of scalar field theories, see e. g. Refs. [143, 144, 145, 106, 146]. In a pragmatic sense, the computation of critical exponents allows us to estimate how well the dynamics close to a phase transition are captured within our ansatz for the effective action.

Let us conclude our discussion with a comparison of the RG flows (42) and (43) obtained from a complete basis of four-fermion interactions with the RG flow equation (33) from our single-channel approximation. We immediately observe that setting in Eq. (42) does not yield the flow equation (33). Thus, the values of the non-trivial fixed point of this coupling are not identical but differ by a factor of two.13 For a finite , we find that the vector-channel interaction is dynamically generated due to quantum fluctuations even if we have initially set the vector-channel interaction to zero. In fact, a finite -coupling shifts the parabola associated with the -function of the coupling , and vice versa, see Fig. 3. Thus, the -coupling can potentially induce critical behavior in the vector-channel, i. e. a diverging four-fermion coupling. We shall discuss this in more detail in Sect. 3.3 after we have clarified the physical meaning of diverging four-fermion couplings in the subsequent section.

3.2 Bosonization and the Momentum Dependence of Fermion Interactions

In this section we study the NJL model with one fermion species in a partially bosonized form. Partial bosonization of fermionic theories is a well-established concept which makes use of the so-called Hubbard-Stratonovich transformation [59, 60]. The advantage of a partially bosonized formulation of NJL-type models over their purely fermionic formulation is that it allows us to include the momentum dependence of four-fermion interactions in a simple manner. Therefore it opens up the possibility to study conveniently the mass spectrum of a theory which emerges from the spontaneous breakdown of its underlying symmetries, e. g. the chiral symmetry. As a bonus, it relates the Ginzburg-Landau picture of spontaneous symmetry breaking, as known from statistical physics, with dynamical bound-state formation in strongly-interacting fermionic theories.

In the following we derive the RG flow equations for the partially bosonized version of this theory and discuss dynamical chiral symmetry breaking. In particular, we explain the mapping of the (partially) bosonized equations onto the RG equations of the four-fermion couplings in the purely fermionic description of our model. This finally allows us to relate the fixed-point structure of the purely fermionic formulation to spontaneous (chiral) symmetry breaking.

The generating functional reads14

(51)

with the action

(52)

see also Eq. (25). As discussed in the previous section, this action is over-complete in the sense that only two of the three couplings , and are independent. We shall come back to this issue in the partially bosonized formulation below.

Our NJL model possesses a chiral symmetry, see Eq. (21), which can be broken dynamically, if a finite vacuum expectation value is generated. This is associated with the Nambu-Goldstone theorem [132, 133, 147, 148] which relates a spontaneously broken continuous symmetry of a given theory to the existence of massless states in the spectrum. To apply this theorem to the present model, we need to compute the vacuum expectation value of the commutator of the so-called chiral charge , which is the generator of the chiral symmetry transformations, and the composite field :

(53)

We observe that the generator does not commute with the field , if the vacuum expectation value of is finite. Thus, the chiral symmetry of our model can be indeed broken spontaneously. Following the Nambu-Goldstone theorem this implies the existence of a massless pseudo-scalar Nambu-Goldstone boson in the channel of the composite field . Since the action does not contain such a state, the massless state must be a bound state. We refer to this type of boson as a pion in the context of QCD, see Sect. 5. At this point we have traced the question of chiral symmetry breaking back to the existence of a finite expectation value of the composite field .

Formally, we may introduce auxiliary fields in the path integral by introducing an exponential factor into the integrand of the generating functional. This is known as a Hubbard-Stratonovich transformation. To bosonize the scalar-pseudoscalar interaction channel we use

(54)

where we have combined the scalar fields into the vector , where and are real-valued scalar fields.15 The fields , , and are auxiliary fields (which have no dynamics so far), is a normalization factor, and the constants , and remain arbitrary for the moment. Multiplying the integrand of the generating functional with such a factor leaves the Greens functions of the theory unchanged. We now shift the integration variables in the so-modified generating functional according to16