Renormalization of the 2PI-Hartree approximation in a broken phase with nonzero superflow
aa r X i v : . [ h e p - ph ] D ec Renormalization of the 2PI-Hartree approximation in a broken phasewith nonzero superflow
G. Fej˝os ∗ Theoretical Research Division, Nishina Center, RIKEN, Wako 351-0198, Japan
Nonperturbative renormalization and explicit construction of the effective potential of the Hartreeapproximation of the two-particle-irreducible formalism are carried out in an inhomogeneous fieldconfiguration describing a uniform superfluid. Based on the earlier article [G. Fej˝os et. al , Nucl.Phys.
A803 , 115 (2008)], we clarify certain aspects of renormalizability corresponding to the findingsof [M. G. Alford et. al , Phys. Rev. D , 085005 (2014)]. We show that renormalizability ofthe approximation can be ensured by regularization schemes respecting Lorentz and translationinvariance. Elimination of nonconventional superflow-dependent divergences is presented in detail,together with a discussion on the finite-temperature treatment. PACS numbers: 11.10.GhKeywords: 2PI formalism, superfluidity
I. INTRODUCTION
The two-particle-irreducible (2PI) formalism is a pop-ular functional method applied to quantum field theoriesboth in and out of equilibrium. The key quantity of theformulation is the 2PI effective action [1], which containsthe mean field and also the propagators as variables. Sta-tionary conditions of the action lead to equations for theone- and two-point functions. The advantage of the for-malism lies in the fact that, due to the self-consistent na-ture of the resulting equations, their solutions realize aninfinite resummation of the perturbative series, leading toa more accurate description compared to ordinary per-turbation theory, particularly when coupling constantsare not small.The simplest approximation of the 2PI effective ac-tion is the Hartree truncation. It leads to a momentum-independent self-energy, making the calculations particu-larly simple. It has been used extensively in different ar-eas, such as chiral symmetry restoration [2, 3] and proper-ties of bulk viscosity [4], curved spacetimes [5], nontopo-logical solitons [6] and superfluidity [7, 8]. The approxi-mation represents a valuable tool if one is to look for thethermodynamic behavior of scalar theories, even thoughit lacks in giving information e.g., on particle lifetimes,and it also violates Goldstone’s theorem. The latter canbe cured by different methods, which has been also ofimportance and interest [7, 9, 10].Renormalization of 2PI approximations has an ex-tended body of literature. The most striking featureis the observation that the consistent cancelation of in-finities cannot be achieved by equal mass and couplingcounterterms [11–13]. As first clarified in Ref. [12], thisproperty can be traced back to the fact that there are sev-eral independent representations of the propagator andthe four-point function, which coincide in the full 2PItheory, but in general not in its approximations. If these ∗ [email protected] quantities differ, their divergences also do; therefore, onlyan appropriate resummation of the perturbative seriesof the corresponding counterterms has to be taken intoaccount, leading to their inequality. Furthermore, com-plicated group structure can also extend the number ofthem, which arises from various projections of the four-point function getting resummed differently; therefore, sodo the projections of the counterterms themselves. Wenote that the splitting of the counterterms is only due tothe truncation of the 2PI effective potential; given thatone is able to include all diagrams, all the mass and cou-pling counterterms coincide. It can also be argued thatthe O ( λ n ) truncation of the effective action will lead tocounterterms that differ only at O ( λ n +1 ), where λ is thecoupling constant. Without going into details, in O ( N )-like models, the 2PI-Hartree approximation contains asingle mass and three different coupling counterterms.The reader is referred to Refs. [13, 14] for a detaileddescription.Papers considering 2PI renormalization and the ex-plicit calculation of counterterms and the effective poten-tial itself in the broken phase of scalar theories usually as-sume that the condensate is homogeneous. Recently, the2PI-Hartree approximation was used to describe a rolereversal in first and second sound in a uniform superfluid[7], which requires the mean field to be spacetime depen-dent. Renormalization of this superflow-dependent con-densation was also discussed, but with several ambigouspoints. The authors of Ref. [7] argue that renormaliz-ability depends on the actual renormalization conditionsimposed. This peculiar statement arises from the appear-ance of unconventional superflow-dependent divergencesfound in the one-loop part of the 2PI effective potential,which seem to be able to be eliminated only when certainrenormalization conditions are imposed.In this paper, we attempt to clarify the divergencestructure of the system and show that, regarding renor-malizability, there is no restriction whatsoever on renor-malization conditions. As it will be shown, an appropri-ate choice of the regularization procedure lies in the coreof this statement. It will turn out that cancelation ofunconventional superflow-dependent subdivergences re-quires the regularization to obey a certain “phase shiftsymmetry” of the quantum effective action [16]. As a re-sult, one needs to use a Lorentz- and translation-invariantregularization, which actually raises nontrivial questionsat finite temperature. A possible resolution of these is-sues will also be presented.The paper is organized as follows. In Sec. II, we in-troduce the model, the symmetry breaking pattern andthe approximate 2PI effective potential. In Sec. III, wepresent the renormalization of the propagator equationsand the field derivative (i.e., basically the field equation).We will put particular emphasis on differences comparedto our earlier procedure described in Ref [13]. In Sec. IV,we show the finiteness of the effective potential explicitlyand discuss and resolve the aforementioned problems ofthe finite-temperature calculation. Finally, in Sec. V,the reader finds some concluding remarks. II. BASICS
Let us consider the dynamics of a complex ϕ fieldthrough the Lagrangian L ( ϕ ) = 12 ∂ µ ϕ∂ µ ϕ ∗ − m ϕϕ ∗ − λ ϕϕ ∗ ) , (1)which displays a U (1) global symmetry, with the cou-pling constant λ >
0. We are interested in a symmetry-breaking pattern in which the condensation of the ϕ fieldhas a spacetime-dependent phase: <ϕ> = ve iψ ( x ) . In thispaper, we restrict ourselves to a case in which ∂ µ ψ ( x ) =const., describing a uniform superfluid. The shifted La-grangian reads as L ( ϕ + ve iψ ) = 12 ∂ µ ϕ∂ µ ϕ ∗ − m ϕϕ ∗ − λ ϕϕ ∗ ) + v ∂ψ ) − m v − λ v + iv ( ∂ µ ϕ ∗ ∂ µ ψe iψ − ∂ µ ϕ∂ µ ψe − iψ ) − m v ϕe − iψ + ϕ ∗ e iψ ) − λv ϕe − iψ + ϕ ∗ e iψ ) − λv ϕ ∗ ϕ + v )( ϕe − iψ + ϕ ∗ e iψ ) , (2)where we used the shorthand notation ( ∂ψ ) = ∂ µ ψ∂ µ ψ .Because of the ∂ µ ψ inhomogeneity, we receive an extraterm in the classical potential, coming from the kineticterm: V [ v ; ψ ] = m v λv − v ∂ψ ) . (3)Assuming the symmetry-breaking pattern describedabove, we shall build up the 2PI-Hartree effective poten-tial of the theory and show how it is free of divergenceswith appropriately chosen counterterms, with particularemphasis on possible divergences caused by the appear-ance of the nonzero ∂ µ ψ superflow.As mentioned in the introduction, the 2PI effective po-tential has two types of variables, condensates and prop-agators. In the usual representation, it reads as V [ v, G ] = ( m + δm ) v λ + δλ ) v − v ∂ψ ) − i Z Tr ln G − − i Z Tr( G − G −
1) + V , (4)where G and G are self-consistent and tree-level prop-agators, respectively, and V contains all two-particle-irreducible diagrams, with vertices of the shifted La-grangian (2), built up by self-consistent propagators.Note that we also indicated counterterms explicitly (fromnow on, we shall use m b := m + δm , λ := λ + δλ ).The tree-level propagator around which we build up (re-summed) perturbation theory corresponds to the real and imaginary parts of the transformed field ϕe iψ . Its ele-ments are i G − ( k ) = k − m b + ( ∂ψ ) − ( λ A + 2 λ B ) v , (5a) i G − ( k ) = k − m b + ( ∂ψ ) − λ A v , (5b) i G − ( k ) = − ik µ ∂ µ ψ, (5c) i G − ( k ) = 2 ik µ ∂ µ ψ, (5d)where λ A = λ + δλ A and λ B = λ + δλ B are different bare coupling constants corresponding to two four-indexinvariant tensors of the O (2) group [13]. We remind thereader that this is due to the splitting of the λ ( ϕϕ ∗ ) / ϕ into a two component ϕ a vector, this splitting means λ ( ϕϕ ∗ ) ≡≡ λ δ ab δ cd + δ ac δ bd + δ ad δ bc ) ϕ a ϕ b ϕ c ϕ d −→
13 [ λ A δ ab δ ad + λ B ( δ ac δ bd + δ ad δ bc )] ϕ a ϕ b ϕ c ϕ d . (6)Relation (6) basically states that different countertermshave to be associated with different invariant tensors inthe interaction term, as already announced in the Intro-duction. Note that in the classical potential [i.e. secondterm on the right-hand side of (4)] no such splitting ofthe countercouplings is necessary; there, we used a unique δλ counterterm.In the Hartree approximation, V is approximated withthe double scoop diagrams, V = λ A (cid:18)Z k Tr G ( k ) (cid:19) + λ B Z k Z p Tr (cid:0) G ( k ) G ( p ) + G ( k ) G T ( p ) (cid:1) , (7)where the same λ A and λ B bare couplings appeared asin the tree-level propagator. Equation (7) leads to amomentum-independent self-energy after differentiationwith respect to G , which represents the simplest approx-imation of the gap equations in the 2PI formalism. Notethat throughout the paper the momentum integrals con-tain (a yet undefined) regularization, and without indi-cating, they are considered at some finite temperature τ . The stationary conditions δV /δ G = 0, ∂V /∂v = 0lead to propagator and field equations. From the former,we get i G − = i G − − λ A Z k Tr G ( k ) − λ B (cid:18)Z k G ( k ) + Z G T ( k ) (cid:19) , (8)while the field derivative reads as ∂V ∂v = v (cid:16) m b + ( λ + δλ ) v − ( ∂ψ ) + ( λ A + 2 λ B ) Z k G ( k ) + λ A Z k G ( k ) (cid:17) . (9)In what follows, we shall perform renormalization onboth (8) and (9). Note that it is not necessary to re-quire the field derivative to vanish; its expression has tobe renormalizable for arbitrary values of the backgroundfield, once the solution of G is exploited. This statementdoes not hold for δV /δ G , and in (8), we deal with thepropagator equation itself. III. RENORMALIZATION
In the following, we adopt the renormalization proce-dure developed in Refs. [13, 15]. This is based on ascheme in which the divergence structure of a given loopintegral is obtained by expanding its integrand aroundan auxiliary propagator G ( k ) = i/ ( k − M ) and iden-tifying divergences via the zero-temperature quantities: T (2) d := Z τ =0 k G ( k ) , (10a) T (0) d := − i Z τ =0 k G ( k ) , (10b) where M plays the role of the renormalization scale.Furthermore, we also define T (2) ,µνd := − Z τ =0 k k µ k ν G ( k ) (cid:12)(cid:12)(cid:12) div , (11a) T (0) ,µνd := − Z τ =0 k k µ k ν G ( k ) (cid:12)(cid:12)(cid:12) div . (11b)These integrals will appear in the divergence analysis,and they can be expressed through (10) (see the Ap-pendix). Our procedure heavily relies on the fact thatoverall divergences cannot depend explicitly on the tem-perature; therefore, they can be defined through zero-temperature integrals. Note that implicit temperature-dependent subdivergences via the masses and/or the su-perflow might appear, and they have to be taken care ofseparately.First, we discuss the renormalization of the propagatorequation (8). Let us define the tadpole integrals as T ( M ; ψ ) := Z k G ( k ) , (12a) T ( M ; ψ ) := Z k G ( k ) , (12b)where G is the self-consistent propagator matrix, alreadyintroduced in the previous subsection. With the assump-tion of the form i G − = (cid:18) k − M + ( ∂ψ ) − ik µ ∂ µ ψ ik µ ∂ µ ψ k − M + ( ∂ψ ) (cid:19) , (13)(8) leads to the following equations for the diagonal ele-ments: M = m b + ( λ A + 2 λ B ) v + ( λ A + 2 λ B ) T ( M ; ψ ) + λ A T ( M ; ψ ) , (14a) M = m b + λ A v + ( λ A + 2 λ B ) T ( M ; ψ ) + λ A T ( M ; ψ ) . (14b)Note that, with (13) the off-diagonal elements of (8)are fulfilled automatically, since the corresponding inte-grands of the tadpoles are odd under the transformation k → − k , and therefore their integrals give zero.Following the route of Ref. [13], we now have to ana-lyze the sub- and overall divergences of the tadpole inte-grals appearing on the right-hand sides of (14). Becauseof the presence of a nonzero superflow, this procedurechanges compared to the analysis performed in Refs. [13]and [7]. After inverting (13), we get G ( k ) = ik − M + ( ∂ψ ) − (2 k µ ∂ µ ψ ) k − M +( ∂ψ ) , (15a) G ( k ) = ik − M + ( ∂ψ ) − (2 k µ ∂ µ ψ ) k − M +( ∂ψ ) , (15b) G ( k ) = − ik µ ∂ µ ψ · G ( k ) , (15c) G ( k ) = 2 ik µ ∂ µ ψ · G ( k ) . (15d)where G ( k ) = i/ [( k − M + ( ∂ψ ) )( k − M + ( ∂ψ ) ) − (2 k µ ∂ µ ψ ) ]. Using (15), the tadpoles read as T ( M ; ψ ) = Z k ik − M + ( ∂ψ ) − (2 k µ ∂ µ ψ ) k − M +( ∂ψ ) , (16a) T ( M ; ψ ) = Z k ik − M + ( ∂ψ ) − (2 k µ ∂ µ ψ ) k − M +( ∂ψ ) . (16b)After a short calculation, for the divergent parts we get T ( M ; ψ ) | div = T (2) d + ( M − ∂ µ ψ∂ µ ψ − M ) T (0) d + ∂ µ ψ∂ ν ψ · T (0) ,µνd , (17a) T ( M ; ψ ) | div = T (2) d + ( M − ∂ µ ψ∂ µ ψ − M ) T (0) d + ∂ µ ψ∂ ν ψ · T (0) ,µνd . (17b)In the Appendix, it is shown that T (0) ,µνd = g µν T (0) d ,and therefore the subdivergences related to the super-flow cancel. Note that this is a regularization-dependentstatement. Nevertheless, as long as it obeys Lorentz in-variance, the above relation remains true. We will comeback to this issue later, but at this point, one concludesthat there are no counterterms that need to be introducedcorresponding to the superflow.To obtain the mass and coupling countertems, we re-visit the “one-step” renormalization described in Ref.[13], i.e., we a priori assume the existence of finite ver-sions of (14), and insert the finite masses obtained thisway to the right-hand side of the unrenormalized equa-tions. The finite gap equations are M = m + 3 λv + λT F ( M ; ψ ) + 3 λT F ( M ; ψ ) , (18a) M = m + λv + λT F ( M ; ψ ) + 3 λT F ( M ; ψ ) , (18b)where T F ( M i ; ψ ) ≡ T ( M i ; ψ ) − T div ( M i ; ψ ) [ i = 1 , v , and the tadpole- (and there-fore environment-) dependent subdivergences to vanishindependently, one arrives at six conditions for δλ A and δλ B and two for δm . Only three of these relations areindependent, and one recovers the results of Ref. [13]: δλ B = − λT (0) d λ λT (0) d , (19a) δλ A = − λT (0) d λ + δλ B λT (0) d , (19b) δm = − λ A + λ B ) h T (2) d + ( m − M ) T (0) d i . (19c)Now, we turn to the field derivative ∂V /∂v (whichalso leads to the equation of state when one searches forits stationary point). Comparing (9) with (14a), we seethat we have to require δλ = δλ A + 2 δλ B to cancel thedivergences. The finite expression reads as ∂V ∂v = v (cid:0) M − λv − ( ∂ψ ) (cid:1) . (20) IV. EFFECTIVE POTENTIAL
The one-particle-irreducible (1PI) effective potential(up to a constant) can be obtained by substituting thesolution of the propagator equations into V . In thissection, we show that it is finite with the countertermsalready determined, and all superflow-dependent diver-gences get eliminated, if the regularization procedure ontop of Lorentz symmetry also obeys translation invari-ance.Let us first start with (8). Multiplying both sides with G ( p ) /
2, taking the trace and integrating over p , we getthe following useful relation (valid only for the solutionof the propagator equation): i Z p Tr[ G − ( p ) G ( p ) −
1] = λ A (cid:20)Z k Tr G ( k ) (cid:21) + λ B Z k Z p Tr (cid:2) G ( k )[ G ( p ) + G T ( p )] (cid:3) . (21)If we make use of the identity (21) in V [see Eq. (4)],then the simplified expression of V can be obtained, V [ v ] = m b v + λ v − v ∂ µ ψ∂ µ ψ − i Z k Tr ln G − ( k ) − λ B Z k Z p Tr (cid:2) G ( k ) G ( p ) + G ( k ) G T ( p ) (cid:3) − λ A (cid:18)Z k Tr G ( k ) (cid:19) − N, (22)where N is a normalization factor to be determined later,which ensures that at zero field and temperature the ef-fective potential is zero. Note that in (22) the propaga-tors should not be considered as variables but substitutedsolutions of (8). After calculating the traces, we get V [ v ] = m b v + λ v − v ∂ µ ψ∂ µ ψ + L ( M , M ; ψ ) − λ A + λ B (cid:16) T ( M ; ψ ) + T ( M ; ψ ) (cid:17) − λ B (cid:16) T ( M ; ψ ) − T ( M ; ψ ) (cid:17) − N, (23)where L ( M , M ; ψ ) := − i Z k log h ( k − M + ∂ µ ψ∂ µ ψ ) × ( k − M + ∂ µ ψ∂ µ ψ ) − (2 k µ ∂ µ ψ ) i (24)is the remaining trace-log piece of the one-loop part[fourth term on the right-hand side of (4)].The divergence structure of the tadpoles is alreadyknown from the previous section, and now we have tocalculate L ( M , M ; ψ ) | div . The scheme we use is thesame as in the previous subsection: we expand the prop-agators of the integrand around the auxiliary propagator G ( k ) and identify the divergent terms through its zero-temperature integrals (10) and (11). First, we separate a quartic divergence via the term L ( M , M ; ψ ) and thenidentify the rest, which are all quadratic and logarithmic.We arrive at L ( M , M ; ψ ) | div = L τ =0 ( M , M ; ψ ) + (cid:0) M + M − M − ∂ψ ) (cid:1) T (2) d h(cid:0) M − M − ( ∂ψ ) (cid:1) + (cid:0) M − M − ( ∂ψ ) (cid:1) i T (0) d ∂ψ ) T (2) d − ( ∂ψ ) T (0) d ∂ µ ψ∂ ν ψ " T (2) ,µνd (cid:0) M + M − M − ∂ψ ) (cid:1) T (0) ,µνd − ∂ µ ψ∂ ν ψ " T (2) ,µνd − ( ∂ψ ) T (0) ,µνd . (25)With the use of the expressions of divergent quantities T (0) ,µνd and T (2) ,µνd , which are given in the Appendix,we realize that all ψ dependence cancels, except the firstterm on the right-hand side. L ( M , M ; ψ ) | div = L τ =0 ( M , M ; ψ )+ ( M + M − M ) T (2) d (cid:16) ( M − M ) + ( M − M ) (cid:17) T (0) d . (26)The normalization factor N in (23) is determined bythe condition that at zero temperature, V τ =01 P I ( v = 0) =0. Let us denote the solution of the gap equations (14)by M at zero field and temperature (the two equationscoincide in this case), M = m b + 2( λ A + λ B ) T τ =0 ( M ; ψ ) . (27)where T τ =0 ( M ; ψ ) : = i Z τ =0 k k − M + ∂ µ ψ∂ µ ψ − (2 k µ ∂ µ ψ ) k − M + ∂ µ ψ∂ µ ψ ! − . (28)The normalization factor is then N = L τ =0 ( M, M ; ψ ) − ( λ A + λ B ) T τ =0 ( M ; ψ ) . (29)The 1PI effective potential (at finite temperature in gen-eral) is therefore V [ v ] = m b v + λ v − v ∂ µ ψ∂ µ ψ + L ( M , M ; ψ ) − L τ =0 ( M, M ; ψ ) − λ A + λ B (cid:16) T ( M ; ψ ) + T ( M ; ψ ) (cid:17) − λ B (cid:16) T ( M ; ψ ) − T ( M ; ψ ) (cid:17) + ( λ A + λ B ) (cid:0) T τ =0 ( M ; ψ ) (cid:1) . (30) We have seen in the previous subsection that, if the reg-ularization obeys Lorentz invariance, the tadpoles haveno superflow-dependent overall divergence, but one stillmight be worried about the same type of divergences in L ( M , M ; ψ ) [see the first term on the right-hand side of(26)] and therefore also about the applied subtractions of N , which should be environment independent. The termin question can be also written in the form of L τ =0 ( M , M ; ψ ) = − i Z τ =0 k log h (cid:0) ( k − ∂ψ ) − M (cid:1) × (cid:0) ( k + ∂ψ ) − M (cid:1) i , (31)where, if the regularization does not break translationinvariance, we can shift the integration momenta sepa-rately to get L τ =0 ( M , M ; ψ ) = − i Z τ =0 k log( k − M ) ≡ L τ =0 ( M , M ; 0) , (32)which is ψ independent. The same argument leadsto relations L τ =0 ( M, M ; ψ ) = L τ =0 ( M, M ; 0) and T τ =0 ( M ; ψ ) = T τ =0 ( M ; 0), and therefore N is also ψ independent. (Note that, for example, any type of cut-off regularization explicitly breaks translation invariance,and in this case, depending on the validity of Lorentz in-variance the tadpoles might not, but (31) does contain a ψ dependent overall divergence.) The symmetry behindthis ψ independence is the invariance of the zero tempera-ture 1PI effective action (based on formal considerations)[16], Γ τ =01 P I [ ϕe − iαx ; ∂ µ ψ ] = Γ τ =01 P I [ ϕ ; ∂ µ ψ − α µ ] , (33)which shows that at zero field expectation value ψ isonly a spurious field having no physical relevance. Nev-ertheless, if one chooses a regularization that breaks thisinvariance explicitly, then ψ -dependent divergences canand will be generated.We still have to check the cancelation of environment-dependent subdivergences in (30), which appear via themasses M and M . The easiest way to show that (30)is finite is to follow the route of Ref. [17]. One exploitsthe unrenormalized equations (14) and (27), expressesthe tadpoles, and then substitutes them into (30). Aftera short calculation, one arrives at the finite expression V P I [ v ] = M v − λ v − v ∂ µ ψ∂ µ ψ + L F ( M , M ; ψ ) − L τ =0 F ( M, M ; ψ )+ (cid:16) m ( M + M − M ) − ( M − M ) − M + M − M ) (cid:17) / λ, (34)where L F ( M , M ; ψ ) = L ( M , M ; ψ ) − L ( M , M ; ψ ) | div , (35)and correspondingly L τ =0 F ( M, M ; ψ ) = L τ =0 ( M, M ; ψ ) − L ( M, M ; ψ ) | div . (36)Equation (34) shows the explicit finiteness of the effec-tive potential and that it is properly normalized. Notethat, depending on actual model parameters (and pos-sibly on the superflow itself), it might not be possibleto access v = 0 at zero temperature (due to the dis-appearance of the solution of the propagator and fieldequations). In this case, one has to choose another sub-traction point for defining the normalization factor N ,e.g., the minimum of the effective potential.Finally, let us discuss an ambiguous point of the pro-cedure described above, appearing at finite temperature.We saw that at any temperature τ counterterms (definedat zero temperature) render all sub- and overall diver-gences finite, but we have not yet addressed the questionof how the demands of regularization (i.e. Lorentz andtranslation invariance) and a finite-temperature calcula-tion can be accommodated. This issue is nontrivial dueto the following.When calculating the effective potential, one has toperform (e.g. in imaginary time formalism) Matsub-ara sums in the trace-log term L ( M , M ; ψ ) and alsoin the tadpoles T ( M ; ψ ), T ( M ; ψ ). These summationscan be done analytically, leading each term to a three-dimensional momentum integral. But after this step,Lorentz invariance is immediately broken, and if one cutsthe momentum integral with a UV cutoff, translationalinvariance will also be lost. As discussed in the previ-ous subsections, this leads to superflow-dependent diver-gences in both the one-loop and tadpole integrals. Onehas two choices at this point: ) keep track of these diver-gences and subtract them by hand, since these are onlyrelated to a “bad” choice of regularization, or ) chooseinstead a Lorentz- and translation-invariant regulariza-tion even at finite temperature. Let us follow the secondchoice. Even though, through the implicit temperature depen-dence of the masses, one cannot define finite-temperatureand vacuum parts of the diagrams properly (since τ willremain implicitly in the latter one), it is always possi-ble to separate the explicit temperature dependence fromthe implicit one. The importance of this lies in the factthat only the former, “vacuum” parts contain overall di-vergences, and therefore only these need to be regular-ized. In other words, only in these terms do we needto apply a Lorentz- and translation-invariant regulariza-tion. For example, after performing the Matsubara sum, L ( M , M ; ψ ) reads as L ( M , M ; ψ ) = X i =1 , Z d k (2 π ) (cid:16) ω i ( k ) / τ ln(1 − e ω i ( k ) /τ ) (cid:17) , (37)where ω i ( k ) [ i = 1 ,
2] is the energy of an eigenmode (de-termined by the zeros of the propagator determinant).In (37), only zero-point fluctuations (first term in thebracket) diverge, but as mentioned already, in its cur-rent form, it is not suitable for avoiding the appearanceof superflow-dependent divergences. The way out is torewrite only the zero-point fluctuations into their τ = 0original form (i.e., before performing the Matsubara sum)or to actually define the finite temperature L ( M , M ; ψ )as L ( M , M ; ψ ) = Z τ =0 k log (cid:16) ( k − M + ∂ µ ψ∂ µ ψ ) × ( k − M + ∂ µ ψ∂ µ ψ ) − (2 k µ ∂ µ ψ ) (cid:17) + X i =1 , Z d k (2 π ) τ ln(1 − e ω i ( k ) /τ ) , (38)instead of (37). In the first term of the right-hand sideof (38) now we can apply an appropriate regularization,while the second term is completely finite. We thereforesolved the problem: we obtained a form in which thedivergent integral can be Lorentz and translation invari-ant, and at the same time, it also describes the finite-temperature behavior.The same kind of procedure has to be applied also toevery tadpole integral: after separating the vacuum fromthe explicit temperature-dependent part, one rewritesthe former as a Lorentz-invariant integral, and definesits divergence using a Lorentz- (or Euclidean after Wickrotation) and translation-invariant regularization, whichleads eventually to the disappearance of all superflow-dependent divergences. Nevertheless, as already men-tioned, if one is to use a regularization breaking the previ-ous properties, then new, superflow-dependent countert-erms have to be added to the Lagrangian. An analysis ofthis type is beyond the scope of the paper, but since theprocedure described here works without any restrictions(even at finite temperature), we do not feel the necessityof such an approach. V. CONCLUSIONS
In this paper, we investigated whether the 2PI-Hartreeapproximation is renormalizable in the broken phase witha nonzero superflow in a U (1) symmetric scalar the-ory. Somewhat contrary to the findings of Ref. [7], weargued that with the counterterms already determinedin Ref. [13] there is no ambiguity of the effective po-tential; it is finite and well defined at all renormaliza-tion scales ( M ). We have found two main differencescompared to the analysis of Ref. [13]: ) in the ef-fective potential, one-loop and tadpole integrals mightcontain divergences related to the superflow, but if ) aLorentz- and translation-invariant (but otherwise com-pletely arbitrary) regularization is used, these do notappear at all. Concerning the finite-temperature treat-ment, we proposed to separate the loop integrals as sumsof the explicit and implicit temperature-dependent partsand rewrite (or actually define) the former one using aLorentz- and translation-invariant regularization, in or-der to avoid the appearance of environment-dependentdivergences.The ambiguous findings of Ref. [7] are due to theincompleteness of the divergence analysis of the 2PIeffective potential. On the one hand, the authorsmiss that the double scoop diagrams might lead tosuperflow-dependent divergences, if the regularizationbreaks Lorentz invariance, and on the other hand, theyskip the analysis of the sensitivity of the divergence struc-ture of the one-loop part with respect to the regulariza-tion used. Since they ultimately neglect all the vacuumparts, it would be interesting to see how and in whatregime these terms were of importance from the view ofthe solution of the coupled propagator and field equa-tions. The renormalization method and our explicitlyfinite representation of the effective potential given herewould allow one to perform such an investigation in astraightforward way. ACKNOWLEDGEMENTS
The author thanks Urko Reinosa for drawing atten-tion on the symmetry property of the effective actionand also for useful comments concerning the manuscript.The careful reading of the manuscript by Zsolt Sz´ep isalso greatly acknowledged, together with discussions withGergely Mark´o. This work was supported by the ForeignPostdoctoral Research program of RIKEN.
Appendix A. DIVERGENT INTEGRALS
In the Appendix we calculate the divergent quantitiesof (11). Assuming that the regularization does not breakLorentz symmetry, both T (0) ,µνd and T (2) ,µνd have to beproportional to g µν , since this is the only two-index ten-sor that is Lorentz invariant. For T (0) ,µνd , we have T (0) ,µνd = g µν i Z τ =0 k k ( k − M ) (cid:12)(cid:12)(cid:12)(cid:12) div . (A1)Adding and subtracting M in the numerator, we imme-diately see that T (0) ,µνd = g µν T (0) d . (A2)The other integral is T (2) ,µνd = Z τ =0 k k µ k ν ( k − M ) (cid:12)(cid:12)(cid:12)(cid:12) div . (A3)Similarly to T (0) ,µνd , we exploit Lorentz symmetry andwrite T (2) ,µνd = Z τ =0 k g µν k ( k − M ) (cid:12)(cid:12)(cid:12)(cid:12) div , (A4)which is T (2) ,µνd = g µν ( T (2) d + M T (0) d ) . (A5) [1] J. M. Cornwall, R. Jackiw, and E. Tomboulis, Phys. Rev.D , 2428 (1974).[2] J. T. Lenaghan, D. H. Rischke, and J. Schaffner-Bielich,Phys. Rev. D , 085008 (2000).[3] D. R¨oder, J. Ruppert, and D. H. Rischke, Phys. Rev.D , 016003 (2003).[4] A. Dobado and J. M. Torres-Rincon, Phys. Rev. D ,074021 (2012).[5] T. Arai, Phys. Rev. D , 104064 (2012).[6] A. Tranberg and D. J. Weir, J. High Energy Phys. 04( ) 184.[7] M. G. Alford, S. K. Mallavarapu, A. Schmitt, and S.Stetina, Phys. Rev. D , 085005 (2014).[8] A. Schmitt, Phys. Rev. D , 065024 (2014). [9] Yu. B. Ivanov and F. Riek, J. Knoll, Phys. Rev. D ,105016 (2005).[10] A. Pilaftsis and D. Teresi, Nucl. Phys. B , 594 (2013).[11] H. van Hees and J. Knoll, Phys. Rev. D , 105005 (2002).[12] J. Berges, Sz. Borsanyi, U. Reinosa, and J. Serreau, Ann.Phys. (N.Y.), 320 (2005) 344.[13] G. Fej˝os, A. Patk´os, and Zs. Sz´ep, Nucl. Phys. A , 115(2008).[14] U. Reinosa and Zs. Sz´ep, Phys. Rev. D , 125026 (2011).[15] G. Fej˝os, A. Patk´os, and Zs. Sz´ep, Phys. Rev. D ,025015 (2009).[16] G. Mark´o, U. Reinosa, and Zs. Sz´ep, arXiv:1410.6998.[17] G. Mark´o, U. Reinosa, and Zs. Sz´ep, Phys. Rev. D87