Study of chiral symmetry restoration in linear and nonlinear O(N) models using the auxiliary-field method

# Study of chiral symmetry restoration in linear and nonlinear O(n) models using the auxiliary-field method

Elina Seel, Stefan Strüber, Francesco Giacosa, and Dirk H. Rischke Institute for Theoretical Physics, Goethe University, Max-von-Laue-Str. 1, D–60438 Frankfurt am Main, Germany Frankfurt Institute for Advanced Studies, Goethe University, Ruth-Moufang-Str. 1, D–60438 Frankfurt am Main, Germany
###### Abstract

We consider the linear model and introduce an auxiliary field to eliminate the scalar self-interaction. Using a suitable limiting process this model can be continuously transformed into the nonlinear version of the model. We demonstrate that, up to two-loop 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 two-point functions. We numerically compute the chiral condensate and the and meson masses at nonzero temperature in the one-loop 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 space-time dimensions with orthogonal symmetry are widely used in many areas of physics. Some applications of these models are quantum dots, high-temperature superconductivity, low-dimensional 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 many-body 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 low-energy 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 non-vanishing 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 low-temperature 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 second-order 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 many-body approximation scheme is the so-called two-particle irreducible (2PI) or Cornwall-Jackiw-Tomboulis (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 one-particle irreducible (1PI) Green’s functions to that for 2PI Green’s functions , where and are the one- and two-point functions. The central quantity in this formalism is the sum of all 2PI vacuum diagrams, . Any many-body 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 two-point functions transform as rank-1 and rank-2 tensors. A disadvantage is that Ward-Takahashi identities for higher-order 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 so-called “external” resummation of random-phase approximation diagrams with internal lines given by the full propagators of the approximation used in the CJT formalism hees ().

In the literature different many-body 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 so-called “double-bubble” approximation petropoulos (); dirk (); bielich (); roder (); grinstein (); pol (); camelia (); amelino (); roh (); nemoto (); petro (); knollivanov (), in Ref. ruppert () sunset-type 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 auxiliary-field method Cooper:2005vw (); Jakovac:2008zq (); Fejos:2009dm (). The auxiliary field allows us to obtain the nonlinear version of the model by a well-defined limiting process from the linear version. We demonstrate that, to two-loop order, the effective potential is equivalent to the one of the standard linear model without auxiliary field, once the one- and two-point 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 one-loop approximation. Although we restrict our treatment to one-loop order, the condensate equation for the auxiliary field introduces self-consistently computed loops in the equations for the masses. Therefore, the one-loop approximation with auxiliary field is qualitatively similar to the standard double-bubble (Hartree-Fock) 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 counter-term 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 so-called 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 auxiliary-field method to that of the standard approach (i.e., without auxiliary field) when replacing the one- and two-point 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 non-vanishing 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 Four-vectors are denoted by capital letters, . We use the imaginary-time formalism to compute quantities at nonzero temperature, i.e., the energy is , where is the Matsubara frequency. For bosons, . Energy-momentum integrals are denoted as

 ∫Kf(K)≡T∞∑n=−∞∫d3→k(2π)3f(iωn,→k) . (1)

## Ii The O(n) model

The generating functional of the model with symmetry at nonzero temperature is given by

 ZL(ε,h)=N∫DαDΦexp(∫1/T0dτ∫Vd3→xLσ-α), (2)

with the Lagrangian

 Lσ-α=12(∂μΦ)2−U(Φ,α), U(Φ,α)=i2α(Φ2−υ20)+Nε8α2−hσ , (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 :

 ZL(ε,h)=~N∫DΦexp(∫1/T0dτ∫Vd3→xLσ), (4)

with the Lagrangian

 Lσ=12(∂μΦ)2−12Nε(Φ2−υ20)2+hσ . (5)

As one can see, the potential of the model exhibits the typical tilted Mexican-hat 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 pseudo-Goldstone 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 ,

 δLσ-αδα−∂μδLσ-αδ∂μα=0 ⟹ iα=2Nε(Φ2−υ20) . (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,

 ZNL(h) =limε→ 0+ZL(ε,h)=limε→ 0+N∫DαDΦexp[∫1/T0dτ∫Vd3→xLσ-α] =N′∫DΦδ[Φ2−υ20]exp{∫1/T0dτ∫Vd3→x[12(∂μΦ)2+hσ]} , (7)

because can be identified with

 δ[Φ2−υ20]∼limε→ 0+∫Dαexp{−∫1/T0dτ∫Vd3→x[i2α(Φ2−υ20)+Nε8α2]}, (8)

which is the mathematically well-defined (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 Cornwall-Jackiw-Tomboulis (CJT) formalism cornwall (). In order to apply this method, we need to identify the tree-level potential, the tree-level propagators, as well as the interaction vertices from the underlying Lagrangian.

### iii.1 Tree-level potential, tree-level 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 non-vanishing 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

 U(σ+ϕ,\boldmathπ,α+α0)=i2(α0+α)(σ2+\boldmathπ2+2σϕ+ϕ2−υ20)+Nε8(α0+α)2−h(ϕ+σ) , (9)

In order to derive the Lagrangian from which we can read off the tree-level potential, the tree-level 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

 dUdϕ≡0,dUdα0≡0. (10)

The resulting expression for the Lagrangian reads

 Lσ-α =12(∂μσ)2+12(∂μ\boldmathπ)2−iα02σ2−iα02\boldmathπ2−12Nε4α2−iϕσα −i2α(σ2+\boldmathπ2)−U(ϕ,α0), (11)

where the tree-level potential is

 U(ϕ,α0)=i2α0(ϕ2−υ20)+Nε8α20−hϕ . (12)

There is a bilinear mixing term, which renders the mass matrix non-diagonal in the fields and .

We can think of two ways to treat this mixing term:

1. we keep this term and allow for a non-diagonal propagator which mutually transforms the fields and into each other.

2. we perform a shift,

 α⟶α−4iϕNεσ , (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 two-loop 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

 ¯Lσ-α =12(∂μσ)2+12(∂μ\boldmathπ)2−12(iα0+4ϕ2Nε)σ2−12(iα0)\boldmathπ2−12Nε4α2 −i2α(σ2+\boldmathπ2)−2ϕNεσ(σ2+\boldmathπ2)−U(ϕ,α0) . (14)

From this expression, we can immediately read off the inverse tree-level propagator matrix,

 ¯D−1(K;ϕ,α0)=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝¯D−1α00⋯0¯D−1σ(K;ϕ,α0)0⋯00¯D−1π(K;α0)⋮⋮⋱⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝Nε400⋯0−K2+iα0+4ϕ2Nε0⋯00−K2+iα0⋮⋮⋱⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (15)

The shift (13) has the following consequences:

1. the Jacobian associated with the transformation is unity, thus the functional integration in Eq. (7) remains unaffected.

2. it generates a term in the mass, which diverges in the limit see Eq. (15). This is expected, since the particle becomes infinitely heavy in the nonlinear version of the model.

Finally, we identify the tree-level vertices from the Lagrangian (11): there are two three-point 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 three-point 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

 Veff(ϕ,α0,G)=U(ϕ,α0)+12∫K[lnG−1α(K)+lnG−1σ(K)+(N−1)lnG−1π(K)] +12∫K[¯D−1αGα(K)+¯D−1σ(K;ϕ,α0)Gσ(K)+(N−1)¯D−1π(K;α0)Gπ(K)−(N+1)]+V2(ϕ,G), (16)

The term represents the sum of all two-particle irreducible diagrams constructed from the three-point vertices in Eq. (14). By definition, these diagrams consist of at least two loops. The one- and two-point functions are determined by the stationary conditions for the effective potential

 δVeffδϕ=0 ,  δVeffδα0=0 , δVeffδGi(K)=0 ,i=α,σ,π1,…,πN−1. (17)

This leads to the following equations for the condensates,

 h =iα0ϕ+4ϕNε∫KGσ(K)+δV2(ϕ,G)δϕ , (18) iα0 =2Nε[ϕ2−υ20+∫KGσ(K)+(N−1)∫KGπ(K)] . (19)

For the two-point functions we obtain from Eq. (17) the Dyson equations

 G−1α(K)=¯D−1α+Πα(K),G−1σ(K)=¯D−1σ(K;ϕ,α0)+Πσ(K),G−1π(K)=¯D−1π(K;α0)+Ππ(K), (20)

where the self-energies are

 Πi(K)=2δV2(ϕ,G)δGi(K),i=α,σ,π1,…,πN−1. (21)

In the following two subsections, we give the explicit expressions for the condensate and mass equations in one- and two-loop approximation, respectively.

### iii.3 One-loop approximation

In one-loop approximation, . Equation (19) remains the same while Eq. (18) simplifies to

 h=iα0ϕ+4ϕNε∫KGσ(K)=2ϕNε[ϕ2−υ20+3∫KGσ(K)+(N−1)∫KGπ(K)], (22)

where for the second equality we have used Eq. (19) to replace . For , all self-energies are zero, cf. Eq. (21), i.e., the full inverse two-point functions are identical to the inverse tree-level propagators. From Eq. (20) one immediately sees that the two-point functions for meson and pion can be written in the form

 Gi(K)=[¯D−1i(K;ϕ,α)]−1=(−K2+M2i)−1,i=σ,π, (23)

with the (squared) masses

 M2σ =iα0+4ϕ2Nε≡2Nε[3ϕ2−υ20+∫KGσ(K)+(N−1)∫KGπ(K)], (24) M2π =iα0≡2Nε[ϕ2−υ20+∫KGσ(K)+(N−1)∫KGπ(K)]. (25)

For the second equalities we have used the condensate equation (19) to replace . Note that this introduces self-consistently computed tadpole integrals into the equations for the masses.

Neglecting terms which are subleading in — an approximation commonly referred to as the large- (or Hartree) limit — Eqs. (22), (24), and (25) reduce to

 h =ϕM2π+O(N−1) , (26) M2σ =M2π+4ϕ2Nε , (27) M2π =2Nε[ϕ2−υ20+N∫kGπ(k)]+O(N−1) . (28)

Note that the condensate and the v.e.v. are i.e.,

### iii.4 Two-loop approximation

To two-loop order there are the four sunset-type diagrams shown in Fig. 1, constructed from the three-point vertices between three fields, one and two fields, as well as between one field with either two or two fields, respectively. There are no double-bubble-type diagrams, due to the absence of four-point vertices. In two-loop approximation,

 V2(ϕ,G) =14∫K∫PGα(K+P)[Gσ(K)Gσ(P)+(N−1)Gπ(K)Gπ(P)] −(2ϕNε)2∫K∫PGσ(K+P)[3Gσ(K)Gσ(P)+(N−1)Gπ(K)Gπ(P)] . (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

 h =iα0ϕ+4ϕNε∫KGσ(K)−2ϕ(2Nε)2∫K∫PGσ(K+P)[3Gσ(K)Gσ(P)+(N−1)Gπ(K)Gπ(P)] =2ϕNε{ϕ2−υ20+3∫KGσ(K)+(N−1)∫KGπ(K) −4Nε∫K∫PGσ(K+P)[3Gσ(K)Gσ(P)+(N−1)Gπ(K)Gπ(P)]} , (30)

where in the second equality we have used Eq. (19) to replace . This is identical with the condensate equation in the two-loop approximation for the usual linear model without auxiliary field, see Sec. III.5.

From Eq. (21) we derive the self-energies as

 Πα =12∫P[Gσ(P)Gσ(K−P)+(N−1)Gπ(P)Gπ(K−P)], (31) Πσ(K) =∫PGσ(P)Gα(K−P)−2(2ϕNε)2∫P[9Gσ(P)Gσ(K−P)+(N−1)Gπ(P)Gπ(K−P)], (32) Ππ(K) =∫PGπ(P)Gα(K−P)−4(2ϕNε)2∫PGσ(P)Gπ(K−P). (33)

Then, the Dyson equations (20) for the full two-point functions read

 G−1α(K) =¯D−1α+Πα(K)=Nε4+12∫P[Gσ(P)Gσ(K−P)+(N−1)Gπ(P)Gπ(K−P)], (34) G−1σ(K) =¯D−1σ(K;ϕ,α0)+Πσ(K)=−K2+iα0+4ϕ2Nε+Πσ(K) =−K2+2Nε[3ϕ2−υ20+∫KGσ(K)+(N−1)∫KGπ(K)]+∫PGσ(P)Gα(K−P) −2(2ϕNε)2∫P[9Gσ(P)Gσ(K−P)+(N−1)Gπ(P)Gπ(K−P)], (35) G−1π(K) =¯D−1π(K;α0)+Ππ(K)=−K2+iα0+Ππ(K) =−K2+2Nε[ϕ2−υ20+∫KGσ(K)+(N−1)∫KGπ(K)]+∫PGπ(P)Gα(K−P) −4(2ϕNε)2∫PGσ(P)Gπ(K−P). (36)

Here, we have also made use of Eq. (19) for the auxiliary field.

### iii.5 Recovering the standard two-loop approximation

In this subsection, we demonstrate that, to two-loop 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 two-loop 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

 Vlσmeff(ϕ,G)=12Nε(ϕ2−υ20)2−hϕ+12∑i=σ,\scriptsize% \boldmathπ∫K[lnG−1i(K)+D−1i(K;ϕ)Gi(K)−1]+Vlσm2(ϕ,G), (37)

where the inverse tree-level propagators are

 D−1σ(K;ϕ)=−K2+2Nε(3ϕ2−υ20),D−1π(K;ϕ)=−K2+2Nε(ϕ2−υ20), (38)

and, to two-loop order,

 Vlσm2(ϕ,G) =32Nε[∫KGσ(K)]2+(N+1)N−12Nε[∫KGπ(K)]2+N−1Nε∫KGπ(K)∫PGσ(P) −(2ϕNε)2∫K∫PGσ(K+P)[3Gσ(K)Gσ(P)+(N−1)Gπ(K)Gπ(P)]. (39)

The first line is the contribution from double-bubble diagrams arising from the four-point 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 double-bubble contribution [this sign was missed in Ref. ruppert ()]. The equation arising from the stationarity condition (17) for reads

 h =2ϕNε{ϕ2−υ20+3∫KGσ(K)+(N−1)∫KGπ(K) −4Nε∫K∫PGσ(K+P)[3Gσ(K)Gσ(P)+(N−1)Gπ(K)Gπ(P)]} . (40)

This is identical with Eq. (30), i.e., with the equation obtained via the auxiliary-field formalism, once the auxiliary field is eliminated with the help of Eq. (19).

The self-energies for meson and pion read

 Πlσmσ(K) =2Nε[3∫KGσ(K)+(N−1)∫KGπ(K)] −2(2ϕNε)2∫P[9Gσ(P)Gσ(K−P)+(N−1)Gπ(P)Gπ(K−P)], (41) Πlσmπ(K) =2Nε[∫KGσ(K)+(N+1)∫KGπ(K)]−4(2ϕNε)2∫PGπ(P)Gπ(K−P). (42)

Therefore, the Dyson equations for the full two-point functions read

 G−1σ(K) =D−1σ(K;ϕ)+Πlσmσ(K) =−K2+2Nε(3ϕ2−υ20)+2Nε[3∫KGσ(K)+(N−1)∫KGπ(K)] −2(2ϕNε)2∫P[9Gσ(P)Gσ(K−P)+(N−1)Gπ(P)Gπ(K−P)], (43) G−1π(K) =D−1π(K;ϕ)+Πlσmπ(K) =−K2+2Nε(ϕ2−υ20)+2Nε[∫KGσ(K)+(N+1)∫KGπ(K)]−4(2ϕNε)2∫PGπ(P)Gσ(K−P). (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

 Gα(K)=[G−1α(K)]−1=[¯D−1α+Πα(K)]−1=¯Dα∞∑n=0[−¯DαΠα(K)]n. (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 one-loop term). However, to two-loop order in the effective potential, it is sufficient to consider the 1PI self-energies to one-loop order only. Therefore, we may neglect all contributions in Eq. (45) except for the (tree-level) term. Then, we may replace

 ∫PGi(P)Gα(K−P)⟶∫PGi(P)¯Dα=4Nε∫PGi(P),i=σ,π, (46)

in Eqs. (35) and (36), i.e., they become simple tadpole contributions to the self-energies. 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 two-loop 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 two-point function for the auxiliary field by their stationary values. To this end, it is advantageous to consider the tree-level, the one-loop, and the two-loop contributions in Eq. (16) separately. The tree-level potential at the stationary value for reads

 U(ϕ,α0)=12(ϕ2−υ20)2Nε[ϕ2−υ20+∫KGσ(K)+(N−1)∫KGπ(K)] −Nε8(2Nε)2[ϕ2−υ20+∫KGσ(K)+(N−1)∫KGπ(K)]2−hϕ =12Nε{ϕ2−υ20−[∫KGσ(K)]2−2(N−1)∫KGσ(K)∫PGπ(P)−(N−1)2[∫KGπ(K)]2}−hϕ. (47)

For the one-loop contribution, we expand the logarithm of the inverse two-point function for the auxiliary field using the Dyson equation (34) and employ the expansion (45),

 lnG−1α(K)+¯D−1αGα(K)−1=ln¯D−1α+ln[1+¯DαΠα(K)]+¯D−1α[¯D−1α+Πα(K)]−1−1 =lnNε4+¯DαΠα(K)−∞∑n=21n[−¯DαΠα(K)]n+1−¯DαΠα(K)+∞∑n=2[−¯DαΠα(K)]n−1 =lnNε4+∞∑n=2[−¯DαΠα(K)]n(1−1n). (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 one-loop order, when integrating over , this term is (at least) of three-loop order in the effective potential. (In fact, since , one readily convinces oneself that the term in the series corresponds to the well-known basketball diagram.) To two-loop order in the effective potential, we may therefore neglect the series in Eq. (48).

Using Eq. (19), the remaining one-loop terms in the effective potential (16) read

 12∫K[lnG−1σ(K)+(N−1)lnG−1π(K)+¯D−1σ(K;ϕ,α0)Gσ(K)+(N−1)¯D−1π(K;α0)Gπ(K)−N] =12∫K{lnG−1σ(K)+(N−1)lnG−1π(K) +[−K2+2Nε(3ϕ2−υ20)]Gσ(K)+(N−1)[−K2+2Nε(ϕ2−υ20)]Gπ(K)−N} +1Nε[∫KGσ(K)+(N−1)∫KGπ(K)]2, (49)

where the last term arises from the tadpole contributions to Eq. (19). Multiplying them with full two-point functions , , and integrating over , they lead to the double-bubble-type terms shown in the last line. Note that the coefficients of the full two-point functions in the second line are just the inverse tree-level propagators in the standard linear model, cf. Eq. (38).

Finally, we consider the two-loop contribution (29). To two-loop order, it is justified to replace , and we obtain

 V2(ϕ,G) ≃1Nε{[∫KGσ(K)]2+(N−1)[∫