Study of chiral symmetry restoration in linear and nonlinear models using the auxiliaryfield method
Abstract
We consider the linear model and introduce an auxiliary field to eliminate the scalar selfinteraction. Using a suitable limiting process this model can be continuously transformed into the nonlinear version of the model. We demonstrate that, up to twoloop order in the CJT formalism, the effective potential of the model with auxiliary field is identical to the one of the standard linear model, if the auxiliary field is eliminated using the stationary values for the corresponding one and twopoint functions. We numerically compute the chiral condensate and the and meson masses at nonzero temperature in the oneloop approximation of the CJT formalism. The order of the chiral phase transition depends sensitively on the choice of the renormalization scheme. In the linear version of the model and for explicitly broken chiral symmetry, it turns from crossover to first order as the mass of the particle increases. In the nonlinear case, the order of the phase transition turns out to be of first order. In the region where the parameter space of the model allows for physical solutions, Goldstone’s theorem is always fulfilled.
I Introduction
Scalar models in spacetime dimensions with orthogonal symmetry are widely used in many areas of physics. Some applications of these models are quantum dots, hightemperature superconductivity, lowdimensional systems, polymers, organic metals, biological molecular arrays, and chains. In this paper, we focus on a physical system consisting of interacting pions and mesons at nonzero temperature . For three spatial dimensions, , an analytical solution to this model does not exist. Thus, one has to use manybody approximation schemes in order to compute quantities of interest, such as the effective potential, the order parameter, and the masses of the particles as a function of . As an approximation scheme never gives the exact solution, it is of interest to compare different schemes and assess their physical relevance.
For the symmetry group for the internal degrees of freedom is locally isomorphic to the chiral symmetry group of quantum chromodynamics (QCD) with massless quark flavors. The phenomena of lowenergy QCD are largely governed by chiral symmetry.
In the case of zero quark masses the QCD Lagrangian is invariant under transformations, being the number of quark flavors. However, the true symmetry of QCD is only , because of the axial anomaly which explicitly breaks due nontrivial topological effects hooft (). For nonzero but degenerate quark masses, the symmetry is explicitly broken, such that QCD has only a flavor symmetry. In reality, different quark flavors have different masses, reducing the symmetry of QCD to , which corresponds to baryon number conservation. In the vacuum, the axial symmetry is also spontaneously broken by a nonvanishing expectation value of the quark condensate witten (). According to Goldstone’s theorem, this leads to Goldstone bosons.
The chiral symmetry is restored at a temperature which for dimensional reasons is expected to be of the order of MeV. This scenario is indeed confirmed by lattice simulations, in which (for physical quark masses) a crossover transition at MeV has been observed 1 ().
For vanishing quark masses, the high and the lowtemperature phases of QCD have different symmetries, and therefore must be separated by a phase transition. The order of this chiral phase transition is determined by the global symmetry of the QCD Lagrangian; for , the transition is of first order if ; for the transition can be of second order if pisarski (). If the quark masses are nonzero, the secondorder phase transition becomes crossover.
The calculation of hadronic properties at nonzero temperature faces serious technical difficulties. For a nonconvex effective potential standard perturbation theory cannot be applied. Furthermore, nonzero temperature introduces an additional scale which invalidates the usual power counting in terms of the coupling constant dolan (). A consistent calculation to a given order in the coupling constant then may require a resummation of whole classes of diagrams braaten ().
A convenient technique to perform such a resummation and thus arrive at a particular manybody approximation scheme is the socalled twoparticle irreducible (2PI) or CornwallJackiwTomboulis (CJT) formalism cornwall (); kadanoff (), which is a relativistic generalization of the functional formalism luttinger (); baym (). The CJT formalism extends the concept of the generating functional for oneparticle irreducible (1PI) Green’s functions to that for 2PI Green’s functions , where and are the one and twopoint functions. The central quantity in this formalism is the sum of all 2PI vacuum diagrams, . Any manybody approximation scheme can be derived as a particular truncation of .
An advantage of the CJT formalism is that it avoids double counting and fulfills detailed balance relations and thus is thermodynamically consistent. Another advantage is that the Noether currents are conserved for an arbitrary truncation of , as long as the one and twopoint functions transform as rank1 and rank2 tensors. A disadvantage is that WardTakahashi identities for higherorder vertex functions are no longer fulfilled hees (). As a consequence, Goldstone’s theorem is violated petropoulos (); dirk (). A strategy to restore Goldstone’s theorem is to perform a socalled “external” resummation of randomphase approximation diagrams with internal lines given by the full propagators of the approximation used in the CJT formalism hees ().
In the literature different manybody approximations have been applied to examine the thermodynamical behavior of the model in its linear and nonlinear versions. In Ref. chiku () optimized perturbation theory was used to compute the effective potential, spectral functions, and dilepton emission rates. The CJT formalism has been applied to study the thermodynamics of the model in the socalled “doublebubble” approximation petropoulos (); dirk (); bielich (); roder (); grinstein (); pol (); camelia (); amelino (); roh (); nemoto (); petro (); knollivanov (), in Ref. ruppert () sunsettype diagrams have been included. The expansion has also been used several times to study various properties of the model at zero coleman (); root () and nonzero meyer (); bochkarev (); warringa (); brauner () temperature.
In this paper, we derive the effective potential for the linear model within the auxiliaryfield method Cooper:2005vw (); Jakovac:2008zq (); Fejos:2009dm (). The auxiliary field allows us to obtain the nonlinear version of the model by a welldefined limiting process from the linear version. We demonstrate that, to twoloop order, the effective potential is equivalent to the one of the standard linear model without auxiliary field, once the one and twopoint functions involving the auxiliary field are replaced by their stationary values. We then calculate the masses and the condensates of the model at nonzero in oneloop approximation. Although we restrict our treatment to oneloop order, the condensate equation for the auxiliary field introduces selfconsistently computed loops in the equations for the masses. Therefore, the oneloop approximation with auxiliary field is qualitatively similar to the standard doublebubble (HartreeFock) approximation in the treatment without auxiliary field. However, since the equations for the masses differ quantitatively, they lead to different results for the order parameter and the masses of the particles as a function of .
The order of the chiral phase transition depends sensitively on the choice of renormalization scheme. In the linear version of the model and for explicitly broken chiral symmetry, it turns from crossover to first order as the mass of the particle increases. In the counterterm renormalization scheme, this transition happens for smaller values of the meson than in the case where vacuum contributions to tadpole diagrams are simply neglected (the socalled trivial regularization). In the nonlinear case the phase transition is of first order. Besides, in the region where the parameter space of the model allows for physical solutions of the mass equations, Goldstone’s theorem is always respected.
The manuscript is organized as follows: in Sec. II the linear and nonlinear versions of the model are presented and it is shown how they can be related with the help of an auxiliary field. In Sec. III the effective potential and the equations for the condensate and masses are derived. We demonstrate the equivalence of the auxiliaryfield method to that of the standard approach (i.e., without auxiliary field) when replacing the one and twopoint functions of the auxiliary field by their stationary values. In Sec. IV the results are presented for the linear and nonlinear versions of the model in the case of nonvanishing and vanishing explicit symmetry breaking. Section V concludes this paper with a summary of our results and an outlook for further studies. An Appendix contains an alternative proof of the equivalence of the treatment with and without auxiliary field, and details concerning the renormalization of tadpole integrals.
We use units The metric tensor is diag Fourvectors are denoted by capital letters, . We use the imaginarytime formalism to compute quantities at nonzero temperature, i.e., the energy is , where is the Matsubara frequency. For bosons, . Energymomentum integrals are denoted as
(1) 
Ii The model
The generating functional of the model with symmetry at nonzero temperature is given by
(2) 
with the Lagrangian
(3) 
where , and is an auxiliary field serving as a Lagrange multiplier. One can obtain the generating functional of the model in its familiar form by integrating out the field :
(4) 
with the Lagrangian
(5) 
As one can see, the potential of the model exhibits the typical tilted Mexicanhat shape, with the parameter being the coupling constant, the parameter for explicit symmetry breaking, and the vacuum expectation value (v.e.v.) of . The fields can be identified as the pseudoGoldstone fluctuations.
Another way to see the equivalence to the standard form of the model is to use the equation of motion for the auxiliary field ,
(6) 
When plugging the latter into one recovers, as expected, the familiar Lagrangian .
The advantage of the representation (2) of the generating functional of the linear model is that, by taking the limit , one naturally obtains the nonlinear version of the model with the fields constrained by the condition . In fact,
(7) 
because can be identified with
(8) 
which is the mathematically welldefined (i.e., convergent) form of the usual representation of the functional function. Equation (8) ensures that the Mexican hat potential becomes infinitely steep and, consequently, the mass of the radial degree of freedom infinite.
Note that in some previous studies of the nonlinear model meyer (); bochkarev (), the dependence in Eq. (8) was not appropriately handled: there, the limit was exchanged with the functional integration, effectively setting in the exponent. This, however, is incorrect, since the additional term is essential to establish the link between the linear model and the nonlinear one. Without this term, an integration over the auxiliary field does not give the correct potential of the linear model. Thus, for a proper construction of the nonlinear limit of the model the dependence must be included.
Iii The CJT effective potential
In this work we study the thermodynamical behavior of the linear model, and in particular the temperature dependence of the masses of the modes and of the condensate. To this end one has to apply methods that go beyond the standard loop expansion which is not applicable when the effective potential is not convex rivers (), as is the case here because of spontaneous chiral symmetry breaking. A method that allows to compute quantities like the effective potential, the masses, and the order parameter at nonzero temperature is provided by the CornwallJackiwTomboulis (CJT) formalism cornwall (). In order to apply this method, we need to identify the treelevel potential, the treelevel propagators, as well as the interaction vertices from the underlying Lagrangian.
iii.1 Treelevel potential, treelevel propagators, and vertices
In our case, the fields occurring in the Lagrangian are , as well as the auxiliary field . In general, the fields and attain nonvanishing vacuum expectation values. In order to take this fact into account, we perform a shift and , respectively. This leaves the kinetic terms in the Lagrangian (3) unchanged, while the potential becomes
(9) 
In order to derive the Lagrangian from which we can read off the treelevel potential, the treelevel propagators, and the interaction vertices, we use the fact that linear terms in the fields vanish on account of the famous tadpole cancellation which utilizes the definition of the vacuum expectation values via the conditions
(10) 
The resulting expression for the Lagrangian reads
(11) 
where the treelevel potential is
(12) 
There is a bilinear mixing term, which renders the mass matrix nondiagonal in the fields and .
We can think of two ways to treat this mixing term:

we keep this term and allow for a nondiagonal propagator which mutually transforms the fields and into each other.

we perform a shift,
(13) which eliminates the bilinear term.
In the following, we discuss the construction of the CJT effective potential only for case (ii). The discussion of case (i) will be delegated to Appendix A where we explicitly demonstrate that, to twoloop order, the effective potential and the equations for the condensates and the masses are the same as for case (ii) when quantities involving the auxiliary field are replaced by their stationary values.
After the shift (13), the resulting expression for the Lagrangian reads
(14) 
From this expression, we can immediately read off the inverse treelevel propagator matrix,
(15) 
The shift (13) has the following consequences:
Finally, we identify the treelevel vertices from the Lagrangian (11): there are two threepoint vertices connecting the auxiliary field to either two or two fields, respectively. (These are the same vertices that also appear in case (i), see Appendix A.) Furthermore, there is a threepoint vertex with three fields, and one with one and two –fields. These vertices are proportional to . (These vertices arise from the shift (13); they do not appear in case (i), see Appendix A.)
iii.2 CJT effective potential
The effective potential assumes the form
(16) 
The term represents the sum of all twoparticle irreducible diagrams constructed from the threepoint vertices in Eq. (14). By definition, these diagrams consist of at least two loops. The one and twopoint functions are determined by the stationary conditions for the effective potential
(17) 
This leads to the following equations for the condensates,
(18)  
(19) 
For the twopoint functions we obtain from Eq. (17) the Dyson equations
(20) 
where the selfenergies are
(21) 
In the following two subsections, we give the explicit expressions for the condensate and mass equations in one and twoloop approximation, respectively.
iii.3 Oneloop approximation
In oneloop approximation, . Equation (19) remains the same while Eq. (18) simplifies to
(22) 
where for the second equality we have used Eq. (19) to replace . For , all selfenergies are zero, cf. Eq. (21), i.e., the full inverse twopoint functions are identical to the inverse treelevel propagators. From Eq. (20) one immediately sees that the twopoint functions for meson and pion can be written in the form
(23) 
with the (squared) masses
(24)  
(25) 
For the second equalities we have used the condensate equation (19) to replace . Note that this introduces selfconsistently computed tadpole integrals into the equations for the masses.
iii.4 Twoloop approximation
To twoloop order there are the four sunsettype diagrams shown in Fig. 1, constructed from the threepoint vertices between three fields, one and two fields, as well as between one field with either two or two fields, respectively. There are no doublebubbletype diagrams, due to the absence of fourpoint vertices. In twoloop approximation,
(29) 
The overall sign follows from the fact that the effective potential has the same sign as the free energy. The combinatorial factors in front of the individual terms follow as usual from counting the possibilities of connecting lines between the vertices, with an overall factor of because there are two vertices.
The condensate equation (19) for the auxiliary field again remains unchanged while Eq. (18) becomes
(30) 
where in the second equality we have used Eq. (19) to replace . This is identical with the condensate equation in the twoloop approximation for the usual linear model without auxiliary field, see Sec. III.5.
iii.5 Recovering the standard twoloop approximation
In this subsection, we demonstrate that, to twoloop order, the results are the same as for a direct application of the CJT formalism to the original Lagrangian (5) of the linear model (a case that we term “standard twoloop approximation”), if we eliminate the field using the stationary values for the condensate and the full propagator . The effective potential for the original linear model reads
(37) 
where the inverse treelevel propagators are
(38) 
and, to twoloop order,
(39) 
The first line is the contribution from doublebubble diagrams arising from the fourpoint vertices with four fields or two and two fields in the Lagrangian (5). The second line corresponds to the sunset diagrams shown in the second row of Fig. 1. These are the same in the linear model with or without auxiliary field. Note that the sunset contribution differs in sign from the doublebubble contribution [this sign was missed in Ref. ruppert ()]. The equation arising from the stationarity condition (17) for reads
(40) 
This is identical with Eq. (30), i.e., with the equation obtained via the auxiliaryfield formalism, once the auxiliary field is eliminated with the help of Eq. (19).
The selfenergies for meson and pion read
(41)  
(42) 
Therefore, the Dyson equations for the full twopoint functions read
(43)  
(44) 
These equations are identical with the Dyson equations (35) and (36), if we replace the propagator of the auxiliary field in those equations using the Dyson equation (34). In order to see this, we formally write
(45) 
If we insert this into the respective terms in Eqs. (35) and (36), we observe that the terms for generate contributions which are at least of second order in loops (because is already a oneloop term). However, to twoloop order in the effective potential, it is sufficient to consider the 1PI selfenergies to oneloop order only. Therefore, we may neglect all contributions in Eq. (45) except for the (treelevel) term. Then, we may replace
(46) 
in Eqs. (35) and (36), i.e., they become simple tadpole contributions to the selfenergies. Combining these with the other tadpole contributions, we observe that, indeed, Eqs. (35) and (36) become identical with Eqs. (43) and (44).
Finally, we also show that the effective potential (29) in twoloop approximation for , Eq. (29), becomes identical with the effective potential for the standard linear model, Eq. (39), if we replace the expectation value and the full twopoint function for the auxiliary field by their stationary values. To this end, it is advantageous to consider the treelevel, the oneloop, and the twoloop contributions in Eq. (16) separately. The treelevel potential at the stationary value for reads
(47) 
For the oneloop contribution, we expand the logarithm of the inverse twopoint function for the auxiliary field using the Dyson equation (34) and employ the expansion (45),
(48) 
We observe that the terms linear in as well as the unit terms cancel. In the final result, the first term is a (negligible) constant. The remaining series starts with a term with two powers of . Since is (at least) of oneloop order, when integrating over , this term is (at least) of threeloop order in the effective potential. (In fact, since , one readily convinces oneself that the term in the series corresponds to the wellknown basketball diagram.) To twoloop order in the effective potential, we may therefore neglect the series in Eq. (48).
Using Eq. (19), the remaining oneloop terms in the effective potential (16) read
(49) 
where the last term arises from the tadpole contributions to Eq. (19). Multiplying them with full twopoint functions , , and integrating over , they lead to the doublebubbletype terms shown in the last line. Note that the coefficients of the full twopoint functions in the second line are just the inverse treelevel propagators in the standard linear model, cf. Eq. (38).
Finally, we consider the twoloop contribution (29). To twoloop order, it is justified to replace , and we obtain