Ikda 99/04
Renormalization Group Flow
Equation of State of Quarks and Mesons

Bernd–Jochen Schaefer
Institut für Kernphysik
Technische Universität Darmstadt
D–64289 Darmstadt
Hans–Jürgen Pirner
Institut für Theoretische Physik
Universität Heidelberg
Philosophenweg 19
69120 Heidelberg, Germany
Email: [email protected]Email:

Nonperturbative flow equations within an effective linear sigma model coupled to constituent quarks for two quark flavors are derived and solved. A heat kernel regularization is employed for a renormalization group improved effective potential. We determine the initial values of the coupling constants in the effective potential at zero temperature. Solving the evolution equations with the same initial values at finite temperature in the chiral limit, we find a second order phase transition at MeV. Due to the smooth decoupling of massive modes, we can directly link the low-temperature four-dimensional theory to the three-dimensional high-temperature theory. We calculate the equation of state in the chiral limit and for finite pion masses and determine universal critical exponents.

1 Introduction

At zero temperature and zero chemical potential, the chiral symmetry of Quantum Chromodynamics (QCD) is spontaneously broken. One expects that the chiral symmetry will be restored at sufficiently high temperature and a phase transition occurs separating the low temperature and high temperature regions. This phenomenon may be realized in high energy heavy-ion collisions at RHIC (Brookhaven) and LHC (Geneva) and is therefore a subject of general interest. It is a matter of intense debate at what temperatures the phase transition would occur and what its nature is.

In this paper we study the chiral phase transition at finite temperature both in the chiral limit and for finite pion masses. We present flow equations which link the universal critical behaviour at finite temperature to the well known physics at . Calculating the effective potential in the framework of the renormalization group theory with a heat kernel regularization, we investigate the critical behaviour of the chiral phase transition for two light quark flavors and calculate the critical exponents. It is still an open question whether full QCD with two massless quarks111The influence of the strange quark is neglected in this work. and the three-dimensional Heisenberg model belong to the same universality class. We use the linear sigma model with quarks which at exhibits a spontaneous breaking of the symmetry to an symmetry. We show how the symmetry is restored at sufficiently high temperature. Lattice investigations hint that the two flavor chiral phase transition lies in the universality class [1] from which the behaviour of the condensates and the long distance properties at the critical temperature ensue. If this is confirmed the transition would be basically driven by the pions and the chiral sigma particle.

It was realized that a perturbative expansion of the effective potential breaks down near the critical temperature due to infrared divergences and the nature of the phase transition cannot be investigated in any finite order in perturbation theory. The situation was partly resolved in [2] by means of the average potential which employs renormalization group ideas. Constructing evolution equations for the dependence of the effective potential on a variable infrared cutoff this approach sums effectively the relevant class of perturbative diagrams in the complicated infrared regime. We follow this method and calculate in a nonperturbative way the effective potential using a variable heat kernel cutoff. The advantage of this cutoff method is that the evolution equations can be obtained in analytical form which makes the physics content very transparent. The aim of this paper is twofold: We demonstrate how the renormalization group improvement of the mass and coupling constant evolution give the main physics of the chiral phase transition, i. e. the equation of state away from the critical temperature. At low temperatures one sees the success of the model handling the pion and sigma degrees of freedom in a comparable accuracy to chiral perturbation theory. At intermediate temperatures the quarks take over as main agents contributing significantly to the change of the chiral condensate. The high temperature behavior of the linear sigma model has not enough degrees of freedom to model full QCD lacking in particular gluons.

By explicitly constructing the thermodynamic equation of state we think a more realistic assessment of the contents of the model is possible. We also think that our method is very pedagogical making this powerful method available to physicists who are interested in an intuitive access to an otherwise very technical subject.

The paper is organized as follows: In section 2 a general summary of our formalism for an -component theory coupled to quarks is given. After the calculation of the effective potential our ansatz for renormalization group improvement is presented. In section 3 the nature of the chiral phase transition is elucidated. The critical behaviour is discussed and critical exponents are determined. In this section the free energy density i.e. the thermodynamic potential for the transition and the equation of state is calculated in the chiral limit. Section 4 is reserved for a discussion of the same properties for finite pion mass. Section 5 presents our conclusions. In order to improve the readability of this paper we postpone extensive equations to appendices.

2 Flow Equations

Wilson’s original renormalization group ideas have been used widely in perturbative and nonperturbative calculations. A nonperturbative approach has been used on lattices and also implemented in continuum Quantum Field Theory by Wegner, Houghton and Polchinski [3],[4]. Flow equations describe the average of an effective action over a volume and are the continuum analogue of a block spin transformation on the lattice. All degrees of freedom with momenta larger than an infrared scale , the coarse graining scale, are effectively integrated out. The flow equations are ultraviolet and infrared finite through the introduction of an infrared cutoff function , which obeys several requirements described in the next subsection. The solution of these equations provides the full effective potential, once the infrared cutoff is removed (). On the other side in the limit or equal to some ultraviolet cutoff the effective average potential results in the classical potential where no fluctuations are included. This ultraviolet cutoff may be associated with the highest momentum scale for which the theory parameterizes the physics adequately. All quantum fluctuations with momenta larger than are included in the potential at this scale, thus no further necessity for an ultraviolet regularization scheme arises. The effective couplings at the large scale specify the potential completely. The knowledge of the -dependence of the effective average potential then allows to interpolate from the classical ’bare’ potential at down to the effective ’renormalized’ potential for including more and more quantum fluctuations as the evolution proceeds towards zero.

This method of the flow equation has been developed intensively by Wetterich and his collaborators [2]. One important ingredient of the evolution equation approach by this group is the appearance of the exact inverse propagator in the flow equation, which makes the RG-improved one-loop equation exact and intrinsically complicated [21]. The solved truncation of this exact equation is very similar to the RG-improved one loop equations used here where the bare couplings are replaced by their corresponding full -dependent expressions. The technically difficult part in the exact RG-approach is not the establishment of an exact relation, but rather the choice of a suitable nonperturbative truncation scheme in order to solve this highly nonlinear coupled set of differential equations. We use a simple truncation scheme and analytical flow equations can be derived. We believe that a deeper connection between Wetterich’s and our approach should be possible.

The flow equations are nonperturbative, i.e. also applicable for problems with large couplings, loosely spoken, because the evolution of the different couplings is interlinked and no particular truncation in coupling constant is used. The solutions of the flow equations give an accurate picture of the high temperature phase transition in four dimensions. It is possible to study the phase transition at the critical temperature and to calculate the critical exponents with surprisingly high accuracy.

2.1 The Effective Potential

In this subsection we develop the formalism starting with the derivation of the flow equations for the linear sigma model combined with two quark flavours. The linear sigma model is a hybrid model which contains quark and mesonic – and –fields as collective degrees of freedom in order to simplify the strong interaction. The cutoff plays the role of a compositeness scale, below which the effective - and -fields can be treated as elementary. For the moment we leave the connection of this theory to QCD as an open question and see how far we get with this model. Preliminary investigations in this direction have been made in ref. [6].

The partition function at zero temperature is given by


Here we omit any external sources , because in the one loop approximation the effective potential and the generating functional for one particle irreducible Green functions do not differ. The Euclidean222We denote all Euclidean operators with an index ’E’. Lagrangian in dimensions looks like


The parameter is an explicit symmetry breaking term in -direction, which gives the Goldstone boson a finite mass. In order to avoid later any recurrences we derive the effective potential with an explicit symmetry breaking term . The chiral limit can be extracted by putting to zero in all equations.

Taking advantage of the properties of the minimum of the potential we can eliminate the mass parameter in favour of and . Then the potential can be written with as


where we introduce the vector as a shorthand notation for the meson fields.

Formal integration over the fermions yields a non-local determinant, which can be regularized by the heat kernel representation. We use the abbreviations , () and .

We remark that the lower boundary of the integral over the proper time reflects the ultraviolet cutoff scale , while the upper limit of the proper time integral represents the infrared scale. Later, we will choose an infrared cutoff function in such a way that no ultraviolet cutoff in the proper time integration is needed. Details concerning the heat kernel representation and definitions can be found in ref. [12].

Because we are interested in the calculation of the effective potential no wave function renormalization corrections are taken into account. This means that we set the wave function renormalization constant , which simplifies further calculations because derivatives resulting from the heat kernel expression can be omitted. This is not a principal restriction of the method. On the contrary the chosen heat kernel regularization is designed for a derivative expansion and higher derivatives can be taken into account not only for flat manifolds but also for curved manifolds [5].

We find for the partition function at zero temperature the following expression which is exact up to the omission of derivatives in the fermion exponent


The fermion determinant is in general part of the functional integration over the remaining meson fields weighted by the meson Lagrangian . In the following we limit this integration over the fermions to the one-loop level. Thus we decouple the effect of modified fermions back on the meson dynamics, pull the non-local determinant in front of the meson integration and replace the meson fields with the vacuum expectation value (VEV) . The total effective action splits into two pieces and becomes a sum of fermionic and bosonic terms each depending on :


In a plane wave basis for the diagonal part of the heat kernel [12] the effective action for the fermions has the form


where the trace runs over color-, flavor- and spin-space and gives a factor for two quark flavours.

The remaining meson integration is also performed on a one loop level and is obtained by a saddle point approximation. The classical mesonic potential is expanded around the minimum, which corresponds to the VEV . Thus taking quadratic fluctuations of the mesons into account and using again a heat kernel regularization for the fluctuation determinant one finds the effective action for the bosons:


Here the trace runs over the fluctuation matrices in -space. Note the opposite sign of the bosonic and fermionic effective actions. In the -model the calculation of the second derivatives of the potential is straightforward and the trace can be evaluated explicitly for components. One has to determine the eigenvalues of the fluctuation matrix evaluated at the minimum of the potential i. e. the masses of the system. With an explicit symmetry breaking term the trace is given by


The first term on the r.h.s. of eq. (10) represents the mass squared eigenvalues associated with the three pions while the other term describes the -meson (cf. eqs. (59)-(61)). In general this trace is a sum over all exponentiated squares of mass eigenvalues times the proper time .

2.2 The Renormalization Group Improvement

The effective action , eqs. (7)-(9), contains so far only fluctuations up to the one loop level and depends on the ultraviolet heat kernel cutoff via the proper time integration. In order to cut off the fluctuations in the infrared we introduce into the proper time integrand a universal -dependent function


with a dimensionless argument which depends on the product of the proper time and the square of the infrared cutoff . This function has to satisfy three conditions:

We require that , since the action with infrared cutoff should tend to the full effective action at , i. e. in the limit the infrared cutoff is removed.

The function must satisfy , to suppress modes with fixed “virtuality” for large . Remember that the upper limit of the proper time integration represents the infrared regime of the effective action which is regularized by this condition of the function .

Thirdly, the first derivative of the cutoff function has to obey


with being any regular function in the vicinity of the origin. This condition assures that the renormalization group equations for the -dependent effective action are ultraviolet finite. The cutoff function regularizes the derivative of the effective potential with respect to the infrared parameter , therefore a further ultraviolet heat kernel cutoff is unwanted. Due the above condition the ultraviolet divergences in the effective action itself are not affected by the presence of the infrared cutoff [21, 25]. It is this condition which distinguishes the cutoff function from a simple minded mass cutoff.

One possible choice for the function which satisfy all these requirements is


The function depicts a smooth cutoff of the momentum modes and allows for an elegant and simple form of the evolution equations.

A smooth cutoff is unavoidable if one wants to have a well defined derivative expansion. For the effective potential alone also a sharp cutoff would do [4],[9],[8].

Even in this case the above choice has great advantages over a sharp cutoff, at least in the proper-time integral formalism where it may cause unpleasant oscillations (cf. e.g.[28]).

The effective action with the cutoff function has now the following form:




This equation defines the one loop effective potential with an additional infrared cutoff function. In a further approximation we truncate the effective potential at terms . Such a procedure leads to simple evolution differential equations for few couplings (). These evolution equations are improved by replacing the bare couplings () and the constant VEV () with the running couplings () and the running VEV (). This renormalization group improvement resums the relevant infrared divergent Feynman diagrams and gives the important nonperturbative evolution of the potential. The substitution can be compared with ordinary renormalization of -point functions in perturbation theory, where one replaces on the one loop level the bare couplings with the renormalized ones. This simple strategy is pursued in this paper and has many advantages. Another possibility is to derive an evolution equation for the full potential using the heat kernel regularization. This is postponed to another paper in the future [16].

The evolution of the potential itself with virtuality leads to two phases and therefore two sets of equations. In the chiral limit the minimum of the potential (VEV) will evolve from a large value to zero in a second order phase transition. For finite pion masses this transition becomes a crossover and the two sets of flow equations merge to one set of equations, as we will show analytically. In order to simplify and clarify the following discussion we start with the chiral limit and set for the rest of this section.

For high virtuality the system is in the symmetric phase which is defined by a vanishing VEV () of the potential. Through the derivative with respect to the fields


we can define the mass and the coupling constant in the symmetric regime


The effective potential at the minimum is given by


Taking the derivatives of equation (16) with respect to the scale we find the following coupled sets of flow equations for the symmetric phase ():


In the spontaneously broken region the VEV is finite () and the mass parameter tends to negative values. We prefer to parameterize the evolution of the potential in this region in terms of and the minimum of the potential , which is defined by


This equation enables us to find the evolution of the minimum . By again taking the derivatives with respect to the scale we find the flow equation for in the broken phase ()


Here an important remark must be added, that we first evaluate the derivatives on the right hand sides for constant couplings , and constant VEV with respect to . Thus the scale derivative acts only on the infrared cutoff function inside the proper time integrand.

3 Equation of state in the chiral limit

In this section we discuss the flow equations for the two flavour linear sigma model in the chiral limit. We present the numerical solution of the flow equations with respect to the scale and generalize the equations to finite temperature.

3.1 The evolution for

In the symmetric phase at zero temperature we use the equations (20,21,22) and get the evolution equations for the free energy density, mass parameter and quartic coupling. Due to the choice of the cutoff function the resulting equations are very transparent. The heat kernel cutoff is not a simple mass cutoff, in effect the derivative cf. eq. (14) of the cutoff function is simpler than the cutoff function itself, therefore the evolution equations have such a concise form:


The factor in eq. (27) arises from the combination (4 mesons minus fermionic degrees of freedom). In the symmetric phase the pions are degenerate with the -meson and the squared mass parameter is always positive. The VEV vanishes and does not appear on the right hand sides of the evolution equations. In the broken phase the pion mass vanishes in the chiral limit. The constituent quark mass is proportional to and is zero in the symmetric phase.

Quartic coupling [MeV] The evolution of
Figure 1: The evolution of with respect to the scale for different initial values.

To solve the evolution equations we need three initial values of , and at in the UV. We start the evolution in the symmetric phase deep in the ultraviolet region of QCD at GeV and set . The other two parameters and are adjusted to get at and the transition to chiral symmetry breaking at GeV. This leads to and .

In addition we need the Yukawa coupling which we fix to for all in order to get a constituent quark mass of MeV at the end of the evolution. Because we do not take any wave function renormalization into account has only a weak scale dependence. In the work of Wetterich et al. [2] the main contribution to the strong evolution of the Yukawa coupling stems from the evolution of the meson wave function renormalization.

In fact, the initial value for the quartic coupling turns out to be not very important due to the Gaussian fix point . This behaviour can be seen in figure 1, where the evolution of the quartic coupling towards zero for different initial values is plotted. Near the evolution of is dominated by the bosonic part, the first term on the r.h.s. of eq. (29) whereas the evolution of responds to the fermion loops. When we reach chiral symmetry breaking at the scale GeV, we switch to the equations for the broken phase characterized by :


The first two terms on the r.h.s. of the above equations are again related to the boson fluctuations while the last parts are connected to the fermionic contributions to the evolution.

[MeV][MeV] The evolution of the minimum
of the effective potential
Figure 2: The evolution of the minimum of the effective potential for and the evolution of the mass parameter for .

The constant factor in the first part of the upper equations comes from the three massless pions while the second part of the bosonic term contains the contribution of the -meson with squared mass . In the broken phase the -meson is massive () and the -meson contribution decouples from the evolution for small values in the IR region, when ; whereas the massless pions contribute permanently to the flow in the chiral limit. The fermionic quark part of the flow equation enters with the opposite sign to the evolution and the quarks become more and more massive in the infrared. The factors have a similar origin as in the symmetric phase. The contribution from the pions is three times larger than the -meson contribution. Two flavors , two particles and anti-particles and two spin degrees of freedom yield a factor of .

The functions in squared brackets can be called threshold functions, because they describe the smooth decoupling of massive modes from the evolution towards the infrared limit of the theory given by the scale . With the heat kernel method and our choice of the function we can calculate these threshold functions analytically. They are of the form where is some positive integer power. So the quark loop gives a factor with positive which switches off the quark contribution to the evolution of the couplings for . Similarly, the square of the sigma mass over the infrared scale suppresses the massive boson loop. At small evolution parameters only zero mass pion loops drive the evolution. This is very intuitive and coincides with the ideas of chiral perturbation theory. In fact, it has been also shown [20] that this model reproduces the chiral expansions even beyond leading order. As we will see in the equation of state the suppression of the quark terms for the evolution does not mean that the quarks do not contribute to the thermodynamic functions like the pressure or energy density below the phase transition temperature. In this respect the model has the same defects as more naive NJL models without quark confinement.

We remark that in the heat kernel expression for the effective potential the inverse fermion propagator enters quadratically in the combination therefore we can use the same infrared cutoff function as in the bosonic integral without breaking chiral symmetry. If we want to take into account the running of the Yukawa coupling the fermion propagators also have to be squared. One also recognizes the signs of the bosonic and fermionic contributions to the -function. The bosons lead to an infrared stable (ultraviolet unstable) -coupling, whereas the fermions counteract this tendency. Going from high to low one sees that the mesonic self-interaction will balance at intermediate values of , whereas in the far infrared the boson term wins (cf. figure 1). The chiral symmetry changes at MeV, where the system goes over from the symmetric phase to the spontaneously broken phase. At this scale eqs. (29) and (32) become identical, i.e. the -function of is continuous (cf. figure 1).

We have argued in a separate paper [11] that this transition at zero temperature may be visible in electron scattering, where real photoproduction with virtuality corresponds to a heat kernel cutoff and inelastic scattering with higher to the equivalent scales. The transition with increasing from the constituent quark to the parton in deep inelastic scattering is a natural consequence in the linear sigma model. The vacuum expectation value stabilizes at small values of and the evolution ends with . When the heavy particles, the sigma and quarks, have decoupled, the change of the vacuum expectation value becomes proportional to , the evolution stops (see figure 2). It is interesting to investigate how a more sophisticated calculation with running Yukawa coupling and wave function renormalization relates the field theoretic model to more observables. Thereby the evolution itself may be tested in a region which perturbative QCD i. e. the Altarelli Parisi evolution equations cannot reach.

3.2 The evolution for finite

The extension of the flow equations to finite temperature for thermal equilibrium is achieved by the Matsubara technique. We integrate the momenta in equation (16), by splitting the zero momentum component from the three-dimensional spatial momentum components and convert the integration over into a summation over Matsubara frequencies for the bosons and for the quarks [7]:


In the symmetric phase the corresponding equations are written in terms of the positive mass parameter and :


The flow equation for the free energy density in the symmetric phase, eq. (35), does not depend on any coupling neither at zero nor at finite temperature. The minimum of the effective potential yields an integration constant. Each contribution to the flow equations at finite temperature is taken into account in a similar way as in the zero temperature case (see above). The modifications in the flow equations appear in the threshold functions within the squared brackets in eqs. (35) - (37).