Optimized perturbation theory for charged scalar fields at finite temperature and in an external magnetic field
aa r X i v : . [ h e p - ph ] O c t Optimized perturbation theory for charged scalar fields at finite temperature and inan external magnetic field
D. C. Duarte, ∗ R. L. S. Farias, † and Rudnei O. Ramos ‡ Departamento de Ciˆencias Naturais, Universidade Federal de S˜ao Jo˜ao Del Rei, 36301-000, S˜ao Jo˜ao Del Rei, MG, Brazil Departamento de F´ısica Te´orica, Universidade do Estado do Rio de Janeiro, 20550-013 Rio de Janeiro, RJ, Brazil
Symmetry restoration in a theory of a self-interacting charged scalar field at finite temperatureand in the presence of an external magnetic field is examined. The effective potential is evaluatednonperturbatively in the context of the optimized perturbation theory method. It is explicitlyshown that in all ranges of the magnetic field, from weak to large fields, the phase transition issecond order and that the critical temperature increases with the magnetic field. In addition, wepresent an efficient way to deal with the sum over the Landau levels, which is of interest especiallyin the case of working with weak magnetic fields.
PACS numbers: 98.80.Cq, 11.10.Wx
I. INTRODUCTION
Phase transition phenomena in spontaneously broken quantum field theories have long been a subject of importanceand interest due to their wide range of possible applications, going from low energy phenomena in condensed mattersystems to high energy phase transitions in particle physics and cosmology (for reviews, see for example [1–3]).In addition to thermal effects, phase transition phenomena are also known to be triggered by other external effects,like, for example, by external fields. In particular, those changes caused by external magnetic fields have attractedconsiderable attention in the past [4] and received reinvigorated interest recently, mostly because of the physicsassociated with heavy-ion collision experiments. In heavy-ion collisions, it is supposed that large magnetic fields canbe generated, and the study of their effects in the hadronic phase transition then became subject of intense interest (seee.g. [5] for a recent review). Magnetic fields can lead in particular to important changes in the chiral/deconfinementtransition in quantum chromodynamics (QCD) [6] and even the possibility of generating new phases [7]As far the influence of external magnetic fields and thermal effects on phase transformations are concerned, one wellknown example that comes to our mind is the physics associated with superconductivity, in particular in the contextof the Ginzburg-Landau theory [8]. Let us recall in that case thermal effects alone tend to produce a phase transitionat a critical temperature where superconductivity is destroyed and the system goes to a normal ordered state. Thephase transition in this case is second order. However, in the presence of an external magnetic field, but below somecritical value, by increasing the temperature the system undergoes a first order phase transition instead. This simpleexample already shows that magnetic fields may have influence on the phase transition other than we would expectfrom thermal effects alone. There are also other examples of more complex systems where external magnetic fieldsmay have a drastic effect on the symmetry behavior. Among these effects, besides the possibility of changing theorder of the phase transition, as in the Ginzburg-Landau superconductor, it can in some circumstances strengthenthe order of the phase transition, like in the electroweak phase transition in the presence of external fields [9], or therecan also be dynamical effects, like delaying the phase transition [10]. External magnetic fields alone can also lead todynamical symmetry breaking (magnetic catalysis) [11] (for an earlier account, see Ref. [12]).Likewise, nonperturbative effects may affect the symmetry properties of a system, once the external parameters arechanged, in a way different than seeing through a purely perturbative calculation, or by a mean-field leading orderdescription. This is because perturbation theory is typically beset by problems, for example around critical points, dueto infrared divergences, or at high temperatures, when powers of coupling constants can become surmounted by powersof the temperature (see e.g. the textbooks [13, 14] for extensive discussions). Thus, high temperature field theoriesand the study of phase transitions in general require the use of nonperturbative methods, through which large classesof terms need to be resummed . Familiar techniques used to perform these resummations include, for example, ringdiagram (or daisy and superdaisy) schemes [15, 16], composite operator methods [17] and field propagator dressingmethods [18, 19]. Other methods used include also numerical lattice studies and expansions in parameters not related ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] to a coupling constant, like the 1 /N expansion and the ǫ -expansion [20], the use of two-particle irreducible (2PI)effective actions [21, 22], hard-thermal-loop resummation [23], variational methods, like the screened perturbationtheory [24, 25] and the optimized perturbation theory (OPT) [26]. Of course, any resummation technique must beimplemented with care so to avoid possible overcounting of terms and lack of self-consistency. Failure in not followingthis basic care can lead to a variety of problems, like predicting nonexistent phenomena or producing a differentorder for the phase transition. One classical example of this was the earlier implementations of daisy and superdaisyschemes, that at some point were giving wrong results, e.g. predicting a first order transition [27] for the λφ theory,an unexpected result since the model belongs to the universality class of the Ising model, which is second order.These methods have also initially predicted a stronger first order phase transition in the electroweak standard model,a result soon proved to be misleading [28]. These wrong results were all because of the wrong implementation of thering-diagram summation at the level of the effective potential, as clearly explained in the first reference in [28]In this work we will analyze the phase transition for a self-interacting complex scalar field model and determine howan external magnetic field, combined with thermal effects, affects the transition. All calculations will be performed inthe context of the OPT nonperturbative method. Our reasons for revisiting here the phase transition in this modelare two-fold. First because this same model has been studied recently in the context of the ring-diagram resummationmethod [29], where it was found that the ring-diagrams render the phase transition first order and that the effectof magnetic fields was to strengthen the order of the transition and also to lower the critical temperature for theonset of the (first order) phase transition. So in this work we want to reevaluate these findings in the context ofthe OPT method. We recall, from the discussion of the previous paragraph, that the ring diagram method requiresspecial attention in its implementation and that previous works have already concluded erroneously about its effectson the transition. The OPT method has a long history of successful applications (for a far from complete list ofprevious works and applications see e.g. Refs. [30, 31] and references therein). The OPT method automaticallyresums large classes of terms in a self-consistent way so to avoid possible dangerous overcounting of diagrams. Inour implementation of the OPT here, we will see that already at the first order in the OPT it is equivalent to thedaisy and superdaisy schemes. The OPT then provides a safe comparison with these other nonperturbative schemes.In particular, to our knowledge, this is the first study of the OPT method when considering the inclusion of anexternal magnetic field. Finally, we also want to properly treat the effects of small magnetic fields (which requiressumming over very large Landau-levels) in an efficient way, particularly suitable for numerical work. This way we canevaluate in a precise way the effects of external magnetic fields ranging from very small to very large field intensities(in which case, in general, just a few Landau-levels suffice to be considered). For this study we will make use of theEuler-Maclaurin formula and fully investigate the validity of its use as an approximation for the Landau level sumsfor different ranges for the magnetic field.The remaining of this paper is organized as follows. In Sec. II we introduce the model and explain the applicationof the OPT method for the problem we study in this paper. It is shown explicitly which terms are resummed by theOPT. We also verify that the Goldstone theorem is fulfilled in the OPT. In Sec. III we study the phase transitionin the spontaneously broken self-interacting quartic complex scalar field in the OPT method, first by incorporatingonly thermal effects and then by including both temperature and an external magnetic field. In Sec. IV we study theadvantages of using the Euler-Maclaurin formula for the sum over the Landau levels. The accuracy and convergenceof the method is fully examined. We determine that the sum over the landau levels, in the regime of low magneticfields, where typically we must sum over very large levels so to reach good accuracy, can be efficiently performedwithin the very few first terms in the Euler-Maclaurin formula. As an application, an analytical formula for thecritical temperature for phase restoration, as a function of the magnetic field, T c ( B ), is derived. Our final conclusionsare given in Sec. V. Finally, an appendix is included to show some of the details of the renormalization of the modelin the context of the OPT. II. THE COMPLEX SCALAR FIELD MODEL AND THE OPT IMPLEMENTATION
In our study we will make use of a self-interacting quartic complex scalar field model with a global U (1) symmetryand spontaneously symmetry breaking at tree-level in the potential, whose Lagrangian density is of the standard form, L = | ∂ µ φ | + m | φ | − λ | φ | , (2.1)where m > φ in terms ofreal and imaginary components, φ = ( φ + iφ ) / √
2. In terms of a (real) vacuum expectation value (VEV) for thefield, h φ i ≡ ϕ/ √
2, we can, without loss of generality, shift the field φ around its VEV in the φ direction, φ → ϕ + φ φ → φ . (2.2)The Feynman propagators for φ and φ , in terms of the VEV then reads, D φ ( P ) = iP + m − λ ϕ + iε , (2.3) D φ ( P ) = iP + m − λ ϕ + iε . (2.4)Using the tree-level VEV for the field, ϕ = 6 m /λ in Eqs. (2.3) and (2.4), φ is then associated with the massiveHiggs mode (with mass squared m = 2 m at the tree level) and φ is the Goldstone mode of the field, remainingmassless throughout the symmetry broken phase. A. Implementing the Optimized Perturbation Theory
The implementation of the OPT in the Lagrangian density is the standard one [30, 31], where it is implementedthrough an interpolation procedure, L → L δ = X i =1 (cid:26)
12 ( ∂ µ φ i ) −
12 Ω φ i + δ η φ i − δ λ (cid:0) φ i (cid:1) (cid:27) + ∆ L ct ,δ , (2.5)where Ω = − m + η and ∆ L ct ,δ is the Lagrangian density part with the renormalization counterterms needed torender the theory finite. In L δ , the dimensionless parameter δ is a bookkeeping parameter used only to keep trackof the order that the OPT is implemented (it is set to one at the end) and η is a (mass) parameter determinedvariationally at any given finite order of the OPT. A popular variational criterion used to determine η is known asthe principle of minimal sensitivity (PMS), defined by the variational relation [32] d Φ ( k ) dη (cid:12)(cid:12)(cid:12) ¯ η,δ =1 = 0 , (2.6)which is applied to some physical quantity Φ ( k ) , calculated up to some order- k in the OPT. The optimum value ¯ η ,which satisfies Eq. (2.6), is a function of the original parameters of the theory and it is in general a nontrivial functionof the couplings. It is because of the variational principle used that nonperturbative results are generated. Othervariational criteria can likewise be defined differently, but they produce the same final result for the quantity Φ ( k ) , asshown recently [33].In terms of the interpolated Lagrangian density, Eq. (2.5), the Feynman rules in the OPT method are as follows.The interaction vertex is changed from − iλ to − iδλ . The quadratic terms δη φ i / φ i ( i = 1 ,
2) now become D φ ,δ ( P ) = iP − Ω − δλ ϕ + iε , (2.7) D φ ,δ ( P ) = iP − Ω − δλ ϕ + iε . (2.8) B. The effective potential in the OPT method
As in the many other previous applications [31, 33], we apply the PMS, Eq. (2.6), directly on the effective potential,which is the most convenient quantity to study the phase structure of the model. To first order in the OPT, using
X X
FIG. 1: Feynman diagrams contributing to the effective potential to first order in the OPT. Solid lines stand for the φ propagator, while dashed lines stand for the φ propagator. A black dot is an insertion of δη . A crossed dot is a massrenormalization insertion. the previous Feynman rules in the OPT method given above, the effective potential V eff ( ϕ ) is given by the vacuumdiagrams shown in Fig. 1.From the diagrams shown in Fig. 1 and by further expanding the propagators Eqs. (2.7) and (2.8) in δ , the explicitexpression for the renormalized effective potential at first order in the OPT (some of the details of the renormalizationin the OPT method are given in the Appendix A) becomes V eff ( ϕ ) = Ω ϕ − δ η ϕ + δ λ ϕ − i Z P ln( P − Ω ) − δη Z P iP − Ω + δ λ ϕ Z P iP − Ω + δ λ π ǫ Ω Z P iP − Ω + δ λ (cid:20)Z P iP − Ω (cid:21) , (2.9)where the momentum integrals are expressed in Euclidean space and the regularization is performed in MS scheme,where, in the finite temperature only case, Z P ≡ iT X P = iω n (cid:18) e γ E M π (cid:19) ǫ Z d d p (2 π ) d , (2.10)where γ E is the Euler-Mascheroni constant, M is an arbitrary mass regularization scale, d = 3 − ǫ is the dimensionof space and ω n = 2 πnT, ( n = 0 , ± , · · · ) are the Matsubara frequencies for bosons at temperature T .The inclusion of an external magnetic field presents no additional difficulty. For example, without loss of generality,we can consider a constant magnetic field in the z -space direction and use a gauge where the external electromagneticfield is A µ = (0 , , Bx, e becomes P − m →− ω n − E k ( p z , B ), where E k ( p z , B ) is the energy dispersion (for charged scalar bosons) in an external constant magneticfield [34], E k = p z + m + (2 k + 1) eB, (2.11)where the last term denotes the Landau levels ( k = 0 , , , · · · ). Likewise, the momentum integrals, taking into accountthe degeneracy multiplicity of the Landau levels [35], are now represented by Z P ≡ i eB π + ∞ X k =0 T X P = iω n (cid:18) e γ E M π (cid:19) ǫ Z d d − p z (2 π ) d − . (2.12) C. Optimization results and Goldstone theorem
Let us now apply the PMS condition (2.6) on the effective potential Eq. (2.9). We obtain straightforwardly thatthe optimum ¯ η satisfies the nontrivial (renormalized) gap equation¯ η = λ ϕ + 2 λ (cid:20)Z P iP − Ω + Ω π ǫ (cid:21)(cid:12)(cid:12)(cid:12) η =¯ η . (2.13)The extrema of the effective potential are defined as usual, by requiring that the first derivative of the effectivepotential with respect to ϕ vanishes. This gives us the trivial solution ¯ ϕ = 0 and the solution for the minimum,¯ ϕ = 6 m λ − (cid:20)Z P iP − Ω + Ω π ǫ (cid:21) . (2.14)We can now verify the effective mass for the field, in particular for the φ component of the complex scalar field.The effective mass for φ in the OPT is given by m , = − m + ¯ η + λ ϕ + Σ (¯ η ) , (2.15)where Σ (¯ η ) is the (renormalized) field’s self-energy, which at first order in the OPT is trivially found to be given byΣ (1)2 (¯ η ) = − ¯ η + 2 λ (cid:20)Z P iP − Ω + Ω π ǫ (cid:21)(cid:12)(cid:12)(cid:12) η =¯ η . (2.16)Using now Eqs. (2.16), (2.13) and (2.14) in Eq. (2.15), we obtain immediately that m , = 0, which is nothingbut the result expected due to Goldstone theorem for the symmetry broken complex scalar field model. It can also beverified that this result carries out at higher orders in the OPT method, in which case, at some order- k in the OPT,the self-energy is the one at the respective order, Σ ( k ) , entering in the above equations. Previous demonstrationsof the Goldstone theorem in the OPT were for the linear sigma model [36] and for the SU (2) Nambu–Jona-Lasiniomodel [37].Finally, it is noticed from the PMS equation given above, Eq. (2.13), that the OPT naturally resums the leadingorder loop terms of the field’s self-energy. So it is quite analogous, at already in the first order in the OPT approx-imation, to the ring-diagram resummation [15, 28]. And that this resummation is also performed automatically ina self-consistent way, it is clear from the interpolation procedure. While ¯ η enters in all propagators, thus carryingthe self-energy corrections, the insertions of ¯ η (the term δη φ i / III. PHASE STRUCTURE AND SYMMETRY RESTORATION AT FINITE TEMPERATURE AND INAN EXTERNAL MAGNETIC FIELD
We are now ready to study the phase structure and the symmetry restoration in the complex scalar field model atfinite temperature and in an external magnetic field. It is convenient to first investigate the case of finite temperatureonly, so to later compare with the case when an external magnetic field is added.
A. Symmetry restoration at finite temperature
It is convenient to write the explicit expressions for each term entering in the effective potential in Eq. (2.9). Atfinite temperature and in the absence of an external magnetic field, using (2.10), we have that the first momentumintegral in (2.9) becomes (recalling that Ω = − m + η ) − i Z P ln( P − Ω ) = − Ω π ) ǫ + Y ( T, η ) , (3.1)where we have identified explicitly the divergent term and the finite term, Y ( T, η ), is given by Y ( T, η ) = − π ) (cid:20)
32 + ln (cid:18) M Ω (cid:19)(cid:21) Ω + T π Z ∞ dz z ln h − exp (cid:16) − p z + Ω /T (cid:17)i . (3.2)Likewise, for the remaining momentum integrals in (2.9) we obtain − δη Z P iP − Ω = − δη (cid:20) − Ω (4 π ) ǫ + X ( T, η ) (cid:21) , (3.3) δ λ ϕ Z P iP − Ω = δ λ ϕ (cid:20) − Ω (4 π ) ǫ + X ( T, η ) (cid:21) , (3.4)where X ( T, η ) is given by X ( T, η ) = Ω π (cid:20) ln (cid:18) Ω M (cid:19) − (cid:21) + T π Z ∞ dz z q z + Ω T (cid:18)q z + Ω T (cid:19) − . (3.5)Next, there is the term coming from the mass counterterm in Eq. (2.9) (see Appendix A), δ λ π ǫ Ω Z P iP − Ω = − δλ π ) Ω ǫ + δλ π Ω X ( T, η ) 1 ǫ − δ λ (4 π ) W ( η ) , (3.6)where we have also defined W ( η ) = 12 (cid:20) ln (cid:18) Ω M (cid:19) − (cid:21) + 12 + π . (3.7)Finally, we have the two-loop contributions shown in Fig. 1. They all sum up in the OPT expansion to give δ λ (cid:20)Z P iP − Ω (cid:21) = δ λ (4 π ) ǫ − δλ Ω π X ( T, η ) 1 ǫ + δ λ (4 π ) W ( η ) + δ λ X ( T, η ) . (3.8)From Eqs. (3.1), (3.3), (3.4), (3.6) and (3.8), we find the complete expression for the renormalized effective potentialat first order in the OPT: V eff ( ϕ, T, η ) = − m ϕ + (1 − δ ) η ϕ + δ λ ϕ + Y ( T, η ) + δ (cid:26) − η + λ (cid:2) ϕ + X ( T, η ) (cid:3)(cid:27) X ( T, η ) . (3.9)The phase structure at finite temperature is completely determined by Eqs. (3.9), (2.13) and (2.14). The optimum¯ η , determined by the PMS criterion, and the vacuum expectation value (VEV) for the scalar field, can now be both,respectively, be expressed as ¯ η = λ ϕ + 2 λ X ( T, ¯ η ) , (3.10)and ¯ ϕ = 6 m λ − X ( T, ¯ η ) . (3.11)In the OPT we can exactly compute the critical temperature of phase transition. Using that at the critical pointthe VEV of the field vanishes, ¯ ϕ ( T c ) = 0, then from Eq. (3.10) we obtain that¯ η ( T c ) = 2 λ X ( T c , ¯ η ( T c )) , (3.12)which upon using it in Eq. (3.11), we obtain0 = 6 m λ − X ( T c , ¯ η ( T c )) ⇒ m − ¯ η ( T c ) = − Ω ( T c ) = 0 . (3.13)From the definition of X ( T, ¯ η ), Eq. (3.5), we then obtain that X ( T c , ¯ η ( T c )) = T c π Z ∞ dz z exp ( z ) − T c , (3.14)and when using the above result back in Eq. (3.13), we obtain the exact result for the critical temperature at thisfirst order in the OPT: T c = 18 m λ . (3.15)This result for T c is the same predicted before for a scalar field model with two real components [38], obtained inthe high temperature one-loop approximation. The result (3.15) is exact in our OPT approximation and obtainedindependent of any high temperature approximation, as usually assumed in any previous calculations. T/M ϕ ( T ) / ϕ FIG. 2: Temperature dependence of the minimum of the effective potential ϕ ( T ) in the OPT with the PMS optimizationcriterion. ϕ is the minimum of the tree-level potential. The parameters used are m/M = 20 and λ = 0 . From Eqs. (3.10) and (3.11), we obtain directly the behavior of the VEV ¯ ϕ as a function of the temperature. Forcomparison purposes, we use analogous parameters as adopted in Ref. [29] , where m/M = 20 and λ = 0 . Note that in [29] the authors used a different numerical factor for the quartic term in the tree-level potential, which gives rise to theextra factor 3 / λ in our notation. In the approximations used in [29], the terms involving the regularizationscale were not included. Here we express all quantities in terms of the regularization scale M . For these parameters, the authors in Ref. [29] find a first order phase transition. In Fig. 2 we show the result for theminimum of the effective potential, ¯ ϕ ≡ ϕ ( T ) as a function of the temperature. The effective potential, for differenttemperatures and same parameters as in Fig. 2, is shown in Fig. 3. Note that the effective potential has an imaginarypart for temperatures below T c , Eq. (3.15), since there can be values of ϕ for which − m + ¯ η becomes negative.This happens for values of ϕ in between the inflection points of the potential, which defines the spinodal region ofinstability in between the (degenerate) VEV of the field, determined by Eq. (3.11). In Fig. 3, for the case of T < T c ,we show the real part of the effective potential, so to be able to show all the potential, including the spinodal region. -100 -50 0 50 100 ϕ /M -20246810 [ V e ff ( ϕ , T ) - V e ff ( , T ) ] . / T c T < T c T = T c T > T c FIG. 3: The effective potential (subtracting the vacuum energy at ϕ = 0), for the same parameters of Fig. 2 and fortemperatures: T /M = 1380, T /M = 1390 and at the critical temperature, T c /M ≃ . It should be noted that the minimum of the effective potential, determined by the coupled Eqs. (3.10) and (3.11),is well defined and unique, where the only solutions for Eq. (3.11) are the two degenerate minima for
T < T c , whilefor T ≥ T c , the only solution is ¯ ϕ = 0. Therefore, the phase transition can readily be inferred to be second order forany set of parameters. It is also clear from the results shown by Figs. 2 and 3 that the transition due to thermaleffects only is of second order. The effective potential does not develop local minima and the VEV of the field variescontinuously from T = 0 till T = T c , where it vanishes. This is the expected behavior, since the complex scalar fieldmodel belongs to the same universality class of a O (2) Heisenberg model, for which the phase transition is secondorder [1].Thus, we see that the OPT method correctly describes the phase transition in the model, capturing the correctphysics. Furthermore, as an added bonus, we can exactly compute the critical temperature of transition B. The phase structure in a constant external magnetic field
Let us now investigate the phase structure when an external magnetic field is also applied to the system. In thepresence of a constant magnetic field, the Feynman rules change as shown in Subsec. II B. The momentum integralsin Eq. (2.9) are given by (2.12) and the dispersion relation for charged particles is given by Eq. (2.11), which takesinto account the Landau energy levels of a charged particle in a magnetic field .At finite B , the first momentum integral term in (2.9) becomes: − i Z P ln( P − Ω ) = eB π + ∞ X k =0 (cid:18) e γ E M π (cid:19) ǫ Z d − ǫ p z (2 π ) − ǫ T + ∞ X n = −∞ ln (cid:2) ω n + p z + Ω + (2 k + 1) eB (cid:3) . (3.16) Note that the dispersion relation Eq. (2.11) only applies for the fields in the complex base ( φ, φ ∗ ), since ( φ , φ ) are not appropriateeigenstates of charge. However, when re-expressing the effective potential in the complex scalar field base, the final result turns out tobe the same as in Eq. (2.9) or (3.9), but with the functions X and Y depending now also on the magnetic field. By performing the sum over the Matsubara frequencies in (3.16), we obtain two terms. One is independent of thetemperature and the other is for T = 0. The T = 0 term is eB π + ∞ X k =0 (cid:18) e γ E M π (cid:19) ǫ Z d − ǫ p z (2 π ) − ǫ p p z + Ω + (2 k + 1) eB = − ( eB ) π (cid:18) e γ E M πeB (cid:19) ǫ Γ( − ǫ )(4 π ) − ǫ ζ (cid:18) − ǫ, Ω + eB eB (cid:19) = − Ω π (cid:20) ǫ + 1 + ln (cid:18) M eB (cid:19) + O ( ǫ ) (cid:21) + ( eB ) π ζ ′ (cid:18) − , Ω + eB eB (cid:19) , (3.17)where to write the second line in (3.17), we used the analytic continuation of the Hurwitz zeta function [39], ζ ( s, a ) = ∞ X k =0 k + a ) s , (3.18)and ζ ′ ( s, a ) is its s -derivative. In writing the final terms in Eq. (3.17) we have also dropped constant terms since theydo not affect the phase structure.The T = 0 part of (3.16) is eBπ T ∞ X k =0 Z + ∞−∞ dz π ln ( − exp " − r z + Ω T + (2 k + 1) eBT . (3.19)We can now write Eq. (3.16) in a form analogous to Eq. (3.1), − i Z P ln( P − Ω ) = − Ω π ) ǫ + ˜ Y ( T, B, η ) , (3.20)where ˜ Y ( T, B, η ) is given by˜ Y ( T, B, η ) = − Ω π (cid:20) (cid:18) M eB (cid:19)(cid:21) + ( eB ) π ζ ′ (cid:18) − , Ω + eB eB (cid:19) + eBπ T ∞ X k =0 Z + ∞−∞ dz π ln ( − exp " − r z + Ω T + (2 k + 1) eBT . (3.21)Analogous manipulations leading to Eq. (3.20) allow us to write the remaining momentum integral terms in Eq. (2.9)like Z P iP − Ω = − Ω (4 π ) ǫ + ˜ X ( T, B, η ) , (3.22)where ˜ X ( T, B, η ) = eB π ln (cid:20) Γ (cid:18) Ω + eB eB (cid:19)(cid:21) − eB π ln(2 π ) − Ω π ln (cid:18) M eB (cid:19) + eB π + ∞ X k =0 Z + ∞−∞ dz π q z + Ω T + (2 k + 1) eBT (cid:20)q z + Ω T + (2 k + 1) eBT (cid:21) − . (3.23)Actually, Eq. (3.22) is just the derivative of Eq. (3.20) with respect to Ω , as can be easily checked.0The final expression for the renormalized effective potential at finite temperature and in a constant external magneticfield has an analogous form as Eq. (3.9), where by using Eqs. (3.20) and (3.22) in (2.9), we obtain V eff ( ϕ, T, B, η ) = − m ϕ + (1 − δ ) η ϕ + δ λ ϕ + ˜ Y ( T, B, η ) + δ (cid:26) − η + λ h ϕ + ˜ X ( T, B, η ) i(cid:27) ˜ X ( T, B, η ) . (3.24)Finally, the PMS criterion and the minimum of the effective potential are again also of the form as Eqs. (3.10)and (3.11): ¯ η = λ ϕ + 2 λ X ( T, B, ¯ η ) , (3.25)and ¯ ϕ = 6 m λ − X ( T, B, ¯ η ) . (3.26)With Eqs. (3.24), (3.25) and (3.26), we are now in position to study the effects of an external magnetic field inthe phase structure of the model, in addition to the thermal effects. In Fig. 4 we show the phase diagram for thesymmetry broken complex scalar field model in the ( B, T ) plane. It shows that the magnetic field strengths thesymmetry broken phase, increasing the critical temperature for phase transition. We must note that this result forthe phase diagram is very different from that seen in the superconductor or in the scalar quantum electrodynamics(QED) case [40]. In contrast to our case, where we are only studying the effects of B and T on the charged scalars,in the scalar QED there is also the interactions with the gauge field. In the scalar QED case with a local U (1) gaugesymmetry, in the symmetry broken phase, because of the Higgs mechanism the gauge field becomes massive and theexternal magnetic field gets screened (the Meissner effect). As the magnetic field is increased, eventually above acritical field the phase is restored through a first order phase transition. In the study made in this paper, on the otherhand, we consider only the case of a broken global U (1) symmetry .The effect of the magnetic field on the global U (1) symmetry can also be seen in the next two figures. In Fig. 5 wehave plotted the VEV of the field at fixed T > T c , thus starting from a symmetry restored phase, ¯ ϕ = 0, as a functionof B . The critical temperature, for the parameters used, is T c /M ≃ .
42, which fully agrees with Eq. (3.15), and thesymmetry returns to be broken, ¯ ϕ = 0, for a magnetic field above a critical value given by eB c /M ≃ .
93. In Fig.6 the effective potential is plotted for the same fixed temperature of Fig. 5 and for values of B below, at and abovethe critical value B c , above which the symmetry becomes broken again (as before, to show the spinodal region of thepotential, we have plotted only the real part of it). It should be noticed from these results that the phase transitionis once again second order. The effect of the magnetic field is to enlarge the symmetry broken region by making thecritical temperature larger, but it does not change the order of the transition.Finally, we have again used for comparison a case with analogous parameters as considered in Ref. [29], and contraryto the results found in that reference, we have found again here only a second order phase transition. For the analogousparameters used in [29] (see observations made in previous subsection), m/M = 20, λ = 0 . eB/M = 30,the change of the critical temperature in relation to the B = 0 case shown in Fig. 3 is only to produce a slight shiftof the critical temperature to a value 0 .
08% higher. In Fig. 7, we have plotted the VEV of the field for the caseof zero magnetic field and for a much higher value of the magnetic field, eB/M = 3000, so to be able to visualizethe difference. Even so, the difference even for such value of magnetic field is only marginal, changing the criticaltemperature obtained at B = 0, T c ( B = 0) /M ≃ .
64, to T c ( B ) /M ≃ .
87 for the value of B used. In allcases we have tested for a nonvanishing magnetic field and other different parameters of the potential, the transitionis again verified to be second order, with the VEV varying always continuously from its value at T = 0 to zero at T = T c ( B ). One of the authors (ROR) would like to thank K. G. Klimenko for discussions on this topic and for also pointing him out the possiblesimilarity with the symmetry behavior found in this paper due to the magnetic field, with pion condensation seen in the Nambu–Jona-Lasinio model with 2 flavor quarks plus baryon and isospin chemical potentials [41]. eB/M T / M SR SB
FIG. 4: The phase diagram of the system in the (
B, T ) plane. The parameters considered here are: λ = 0 . m/M = 1. Thesolid line separates the regions of symmetry broken (SB) and symmetry restored (SR) phases. eB/M ϕ ( T , B ) / M FIG. 5: Magnetic field dependence of the VEV of the field, ¯ ϕ ( T, B ), for a fixed value of T above T c . The parameters consideredhere are: λ = 0 . m/M = 1 and T /M = 15.
IV. USING THE EULER-MACLAURIN FORMULA FOR THE SUM OVER LANDAU LEVELS
When working with quantum field theories in a magnetic field, we need to deal with the sum over the Landau levels.At T = 0 this does not present a problem in general, since we can express the expressions in terms of zeta functions.We have seen this in the previous section, where all terms at T = 0 were presented in analytical form. However, atfinite temperature this is not possible in general, therefore, we either need to numerically perform the sums over theLandau levels, or find suitable approximations for the expressions. This job becomes easier in the high magnetic fieldregime, with √ eB ≫ T , for which most of the time suffices to use only the very first terms of the sum, or even just theleading Landau level term. Higher order terms are quickly Boltzmann suppressed in this case. But this is not so inthe low magnetic field regime, with √ eB . T , and Ω /T .
1, where most of the time requires to sum up to very largeLandau levels. This has the potential to make any numerical computation to become quickly expensively, speciallywhen there are in addition numerical integrations involved, like in our case here (note that we have not made use ofhigh temperature approximations, but worked with the complete expressions in terms of the temperature dependentintegrals). Simple approximate analytical results, which are always desirable to obtain, can also become difficult toobtain without a suitable approximation that can be used. One such approximation was discussed in Ref. [29], but itis valid only for very small magnetic fields. Here we discuss a more natural alternative for dealing with the cases oflow magnetic fields, based on the Euler-Maclaurin formula.2 -2 -1 0 1 2 ϕ /M [ V e ff ( ϕ , T , B ) - V e ff ( , T , B ) ] / M eB/M = 45eB = eB c eB/M = 60 FIG. 6: The effective potential for the same parameters of Fig. 5 and for three different values of the magnetic field: eB/M =45, eB c /M = 53 .
93 and eB/M = 60. T/M ϕ ( T , B ) / ϕ eB = 0eB/M = 3000 FIG. 7: Temperature dependence of the normalized VEV ¯ ϕ ( T, B ) /ϕ . The critical temperature for B = 0 is T B =0 c /M ≃ . eB/M = 3000, T c increase to T c /M ≃ .
87. The other parameters usedhere are: λ = 0 . m/M = 20. The Euler-Maclaurin (EM) formula provides a connection between integrals and sums. It can be used to evaluatefinite sums and infinite series using integrals, or vice-versa. In its most general form it can be written as [42]: b X k = a f ( k ) = Z ba f ( x ) dx + 12 [ f ( a ) + f ( b )] + n X i =1 b i (2 i )! h f (2 i − ( b ) − f (2 i − ( a ) i + Z ba B n +1 ( { x } )(2 n + 1)! f (2 n +1) ( x ) dx , (4.1)where b i are the Bernoulli numbers, defined by the generating function x exp( x ) − ∞ X n =0 b n x n n ! , (4.2)and B n ( x ) are the Bernoulli polynomials, with generating function3 z exp( zx )exp( z ) − ∞ X n =0 B n ( x ) z n n ! . (4.3)The notation { x } in B k +1 ( { x } ) in Eq. (4.1) means the fractional part of x and f ( k ) ( x ) means the k -th derivative ofthe function. The last term in Eq. (4.1) is known as the remainder .Next, we study the reliability of the use of the EM formula as a suitable approximation for the sums over Landaulevels and we estimate the errors involved in doing so. In the following, we will call the 0th-order in the EM formulawhen only the first term in the RHS of Eq. (4.1) is kept, the 1st-order when also the second term is kept and so onso forth. Thus, for example, the Landau sum part of ˜ Y ( T, B, η ) in Eq. (3.21), up to second order in the EM formula,becomes L ˜ Y = ∞ X k =0 Z + ∞−∞ dz π ln h − e − E ( z, Ω ,T,B,k ) i ≃ Z + ∞−∞ dz π (cid:26)Z ∞ ln h − e − E ( z, Ω ,T,B,k ) i dk + 12 ln h − e − E ( z, Ω ,T,B, i − eB T E ( z, Ω , T, B, (cid:2) e E ( z, Ω ,T,B, − (cid:3) ) , (4.4)where E ( z, Ω , T, B, k ) = r z + Ω T + (2 k + 1) eBT . (4.5)Likewise, the Landau sum part of ˜ X ( T, B, η ) in Eq. (3.23), up to second order in the EM formula, becomes L ˜ X = ∞ X k =0 Z + ∞−∞ dz π E ( z, Ω , T, B, k ) 1 e E ( z, Ω ,T,B,k ) − ≃ Z + ∞−∞ dz π (cid:26)Z ∞ E ( z, Ω , T, B, k ) 1 e E ( z, Ω ,T,B,k ) − dk + 12 1 E ( z, Ω , T, B, (cid:2) e E ( z, Ω ,T,B, − (cid:3) + eB T E ( z, Ω , T, B, (cid:2) e E ( z, Ω ,T,B, − (cid:3) (cid:20) E ( z, Ω , T, B,
0) + e E ( z, Ω ,T,B, e E ( z, Ω ,T,B, − (cid:21)) . (4.6)In Tab. I we assess the reliability of using the EM at each order (we have studied it up to the forth-order) for thesums in Eqs. (4.4) and (4.6), for different values for the ratio eB/T , and we compare the results with the exact ones.From the results shown in Tab. I, we see that the EM approximation produces results that already at first orderhave good accuracy compared to the exact values coming from the full Landau sums. It performs particularly verywell for values of eB ≪ T , reaching convergence within just the very first few terms in the expansion. However, forvalues eB & T , increasing the order in the EM series, it quickly looses accuracy. This can be traced to the fact thatas we go to a higher order in the derivatives appearing in the sum on the RHS of Eq. (4.1), higher powers of eB/T areproduced, eventually spoiling the approximation (though it oscillates around the true value). This apparent runawaybehavior is only cured by the introduction of the last term in Eq. (4.1), the remainder.It should be noted that Eq. (4.1) is not an approximation to the sum, but it is actually an identity. The importantterm in that context is the remainder, the last term in Eq. (4.1). In all our numerical tests, within the numericalprecision for the integrals (for convenience, we have performed all the numerical integrations in Mathematica [43]),we have verified that Eq. (4.1) has agreed with the results from the sum over the Landau levels in the LHS of Eqs.(4.4) and (4.6), for all values tested and at all orders (when including the remainder), with a precision of less than0 . eB/T order L ˜ Y L ˜ X EM approx. Landau Sum error (%) EM approx. Landau Sum error (%)0 - 688.7717 0.04 508.2575 1.671 - 689.0258 1 . × − .
001 2 - 689.0271 - 689.0264 1 . × − . × − . × − . × − .
01 2 - 68.8990 - 68.8989 1 . × − . × − . × − . . × − . × − . × − . × − . × − . × − . × − . × − - 5 . × − . × − . × − . × − . × − . × − . × − magnetic fields. To our knowledge, one of the few works that we are aware of that have previously made use ofthe EM formula before were for example Refs. [44–46]. In Ref. [44] the author derived the effective potential of theabelian-Higgs model, including both thermal effects and an external magnetic field, in the one-loop approximation.The EM formula was used at its zeroth order (by just transforming the sum in an integral) to obtain analyticalresults for the effective potential in the high magnetic field region. But from the results of Tab. I, this is exactly theregion and order in the approximation that lead to the largest errors. In [45] the EM formula was used to obtain anexpression for the effective potential for vector bosons and to study pair production in an external magnetic field,while in [46] it was used to obtain analytical expressions for the internal energies for non-interacting bosons confinedwithin a harmonic oscillator potential. None of these previous works have assessed the reliability of the use of theEM formula.As an application of the EM formula to our problem, we estimate from it the dependence of the critical temperaturewith the magnetic field. Recall from Eq. (3.13) the critical temperature can be obtained by setting Ω = 0 and ¯ ϕ = 0in the equation for the VEV. Thus, from Eq. (3.26), we need to solve the equation6 m λ − X ( T c , B ) (cid:12)(cid:12)(cid:12) Ω=0 = 0 . (4.7)As we have seen from the results of Tab. I, the EM formula for low magnetic fields produce reliable results already atthe second order, and for very small magnetic fields, good precision is reached even at first order. Thus, for examplefrom the expansion in Eq. (4.6), and performing the k integral in the first term, we obtain5 L ˜ X ≃ − π T eB Z ∞ dz ln (cid:16) − e −√ z + a (cid:17) + 12 π Z ∞ dz √ z + a e √ z + a − eB πT Z ∞ dz z + a ) (cid:16) e √ z + a − (cid:17) √ z + a + e √ z + a e √ z + a − ! , (4.8)where a = eB/T c . Results for the integrals in Eq. (4.8) can be found in the low magnetic field regime, where a ≪ a can be found. In fact, the integrals in Eq. (4.8) can be easily related to the h i ( a ) Bose-Einstein integrals found inthat context (see for example the App. A in Ref. [14]), from where we obtain the leading order terms in the expansionin powers of a : L ˜ X ≃ π a − a + O ( a ) . (4.9)Going to higher order in the EM series only lead to O ( a n ) (with n ≥
0) terms, for a = √ eB/T c ≪ T c ( B ) ≃ m λ + 25 eB π r π m λ eB ! . (4.10)The result (4.10) shows the growth of the critical temperature with the magnetic field, as seen numerically from theresults obtained in the previous section. A comparison with our previous results also shows that Eq. (4.10) providesan excellent fit for T c ( B ), leading to results with less than 1% error compared to the full numerical results for T c , forvalues of field such that eB/T c .
1. For example, for the parameters λ = 0 . eB = 400 m , the approximation(4.10) gives T c /m ≃ .
97, while the full numerical calculation gives T c /m = 18 . V. CONCLUSIONS
In this paper we have revisited the phase transition problem for the self-interacting complex scalar field modelwhen thermal effects and an external magnetic field are present. We have studied this problem in the context ofthe nonperturbative method of the optimized perturbation theory. We have shown that the OPT method preservesthe Goldstone theorem and that carrying out the approximation to first order is analogous to the ring diagramresummation method. The OPT carries out the resummation of self-energy diagrams in a self-consistent way, avoidingovercounting issues, which have previously plagued the ring diagram method in its very early applications.By using the OPT method, we have demonstrated that the phase transition in the model is always second order inthe presence of thermal effects and by including an external magnetic field, there is no change to the phase transitionorder. The effect of the external magnetic field is to strengthen the symmetry broken phase, producing a larger VEVfor the field and also a larger critical temperature. The study we made in this paper considered only the effects ofthe magnetic field and temperature on the charged scalar field (the model is one with a global U (1) symmetry). Ourstudy, in a sense, is closer to the case of studying the effects of B and T in the linear sigma model [47], where nocouplings to gauge fields (except to the external field) are considered in general. The inclusion of other field degreesof freedom, or when promoting the symmetry from a global one to a local symmetry, can of course change bothqualitatively and quantitatively how the magnetic field affects the system. In particular, as concerning the possibilityof producing a first order phase transition, either due to thermal effects, or by a magnetic field or from both. This iswhat we expect in the context of the scalar QED, where due to the screening of the magnetic field in the broken phase,the phase diagram and transition are very different. The phase structure can also be very different when adding otherinteractions and having other symmetries. For example, in the context of the electroweak phase transition [9, 10, 48],it has been shown that the effect of the magnetic field is to make the first order transition stronger. By omittingvacuum energy terms from the effective potential, which may have phenomenological motivations, as in Ref. [47], mayalso lead to a first order transition, instead of a second order one.6Still comparing our results with those obtained in the abelian Higgs model, there is a tantalizing question of whetherfor very strong magnetic fields a new phase could be formed. This could be for example a phase with global vorticescondensation, thus restoring the symmetry again. This would be in analogy to the local, Nielsen-Olesen vortexconndensation that can happen in type II superconductors [40]. Vortex condensation in that case is energeticallyfavorable to happen for fields beyond a critical value and when the mass of the Higgs field becomes larger that themass of the gauge field. This is an interesting possibility to investigate in the future.As an aside, but a complementary part of this work, we have verified the reliability of the use of the Euler-Maclaurinformula as an approximation for the sum over Landau energies in a magnetic field. We have verified that it producesresults with errors of less than 0 .
1% already at the first few orders in the EM formula and that it leads to particularlysuitable approximations in the low magnetic field region ( eB/T ≪ Acknowledgments
Work partially supported by CAPES, CNPq and FAPEMIG (Brazilian agencies). R.L.S.F. would like to thank A.Ayala, E. S. Fraga, A. J. Mizher, H. C. G. Caldas and M. B. Pinto for discussions on related matters.
Appendix A: Renormalization in the OPT method
In this appendix we briefly explain the renormalization of the effective potential Eq. (2.9). First, note that theinterpolation procedure in the OPT method, Eq. (2.5), introduces only new quadratic terms, thus, it does not alterthe renormalizability of the original theory. Therefore, the counterterms needed to render the theory finite have thesame polynomial structure of the original Lagrangian [30].In obtaining the renormalized effective potential, we first note that from the order δ term Eq. (3.4), we obtain themass renormalization counterterm entering in ∆ L ct ,δ in Eq. (2.5), − A δ φ i , (A1)where A δ = δ λ Ω π ǫ . (A2)From Eq. (A1), we obtain the two mass counterterms diagrams contributing to the effective potential at first orderin the OPT and shown in Fig. 1.By collecting all the divergences from the vacuum loop terms in Eq. (2.9), we have that the divergence from thecontribution (3.4) is canceled by the counterterm Eq. (A1). A potential temperature and magnetic field dependentdivergence, proportional to either X in the two-loop vacuum term Eq. (3.8), or ˜ X in the case of a finite externalmagnetic field, is explicitly canceled against identical term coming from the mass counterterm diagrams, Eq. (3.6).The remaining divergences in the effective potential are only vacuum ones, independent of the background field, andthey can be all canceled by introducing a vacuum counterterm ∆ V ct , added to the effective potential. At the firstorder in the OPT, ∆ V ct is given by7∆ V ct ,δ = (cid:18) Ω π ) − δη π Ω (cid:19) ǫ + δλ Ω π ) ǫ . (A3)Thus, at first order in the OPT, we only require the two counterterms, Eqs. (A1) and (A3). Going to second orderin the OPT it will also require a coupling constant renormalization counterterm, C δ λφ i / δ contributions for Eqs. (A1) and (A3). Going to higherorders in the OPT only produces additional O ( δ n ), n >
2, contributions to the mass, vertex and vacuum counterterms. [1] N. Goldenfeld,
Lectures on Phase Transitions and The Renormalization Group , Frontiers in Physics, Vol. 85 (Addison-Wesley, NY, 1992).[2] R. J. Rivers, E. Kavoussanaki and G. Karra, Cond. Mat. Phys. , 133 (2000); R. J. Rivers, Patterns of Symmetry Breaking inCosmology and the Laboratory , Proceedings of the National Workshop on Cosmological Phase Transitions and TopologicalDefects (Porto, Portugal, 2003), ed. T.A. Girard (Grafitese, Edificio Ciencia, 2004), pp11-23.[3] D. Boyanovsky, H. J. de Vega and D. J. Schwarz, Ann. Rev. Nucl. Part. Sci. , 441 (2006) [arXiv:hep-ph/0602002].[4] A. D. Linde, Rep. Prog. Phys. 42, 389 (1979).[5] K. Fukushima, [arXiv:1108.2939 [hep-ph]].[6] R. Gatto and M. Ruggieri Phys. Rev. D , 034016 (2011); M. D’Elia, S. Mukherjee and F. Sanfilippo, Phys. Rev. D ,051501 (2010). A. J. Mizher, M. N. Chernodub and E. S. Fraga, Phys. Rev. D , 105016 (2010); A. Ayala, A. Bashir, A.Raya and A. Sanchez Phys. Rev. D , 036005 (2009); D. P. Menezes, M. Benghi Pinto, S. S. Avancini and C. Providˆencia,Phys. Rev. C , 065805 (2009).[7] E. J. Ferrer, V. de la Incera and C. Manuel, Phys. Rev. Lett. , 152002 (2005); B. Feng, E. J. Ferrer and V. de la Incera,arXiv:1105.2498 [nucl-th].[8] M. Tinkham, Introduction to superconductivity , (McGraw-Hill, 2nd Edition, New York, NY, 1996).[9] K. Kajantie, M. Laine, J. Peisa, K. Rummukainen and M. E. Shaposhnikov, Nucl. Phys.
B544 , 357 (1999).[10] R. Fiore, A. Tiesi, L. Masperi and A. Megevand, Mod. Phys. Lett.
A14 , 407 (1999).[11] V. P. Gusynin, V. A. Miransky and I. A. Shovkovy, Phys. Rev. Lett. , 3499 (1994); V. P. Gusynin, V. A. Miranskyand I. A. Shovkovy, Nucl. Phys. B462 , 249 (1996); K. G. Klimenko and D. Ebert, Phys. Atom. Nucl. , 124 (2005);V. C. Zhukovsky, K. G. Klimenko, V. V. Khudyakov and D. Ebert, JETP Lett. , 121 (2001). E. J. Ferrer, V. de laIncera, Nucl. Phys. B824 , 217 (2010)[12] K.G. Klimenko, Z. Phys.
C54 , 323 (1992); Theor. Math. Phys. , 1161 (1992); ibid. , 1 (1992).[13] M. Le Bellac, Thermal Field Theory , (Cambridge University Press, Cambridge, England, 1996).[14] J. I. Kapusta and C. Gale,
Finite-Temperature Field Theory: Principles and Applications , (Cambridge University Press,Cambridge, England, 2006).[15] J. R. Espinosa, M. Quir´os and F. Zwirner, Phys. Lett.
B291 , 115 (1992).[16] J. Arafune, K. Ogata and J. Sato, Prog. Theor. Phys. , 119 (1998).[17] G. Amelino-Camelia and S.-Y. Pi, Phys. Rev. D , 2356 (1993).[18] N. Banerjee and S. Mallik, Phys. Rev. D , 3368 (1991).[19] R. R. Parwani, Phys. Rev. D , 4695 (1992); erratum, Phys. Rev. D , 5965 (1993).[20] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Oxford University Press, 1996).[21] J. M. Cornwall, R. Jackiw and E. Tomboulis, Phys. Rev. D (1974) 2428.[22] J. Berges, Sz. Borsanyi, U. Reinosa and J. Serreau, Phys. Rev. D , 105004 (2005).[23] E. Braaten and R. D. Pisarski, Nucl. Phys. B337 , 569 (1990); J. Frenkel and J. C. Taylor, Nucl. Phys.
B334 , 199 (1990);J. O. Andersen, E. Braaten, and M. Strickland, Phys. Rev. Lett. , 2139 (1999); Phys. Rev. D , 014017 (1999).[24] F. Karsch, A. Patk´os, and P. Petreczky, Phys. Lett. B ,69 (1997); J. O. Andersen and M. Strickland, Phys. Rev. D ,105012 (2001); Phys. Rev. D , 025011 (2005); J. O. Andersen and L. Kyllingstad, Phys. Rev. D , 076008 (2008).[25] J.O. Andersen, E. Braaten and M. Strickland, Phys. Rev. D , 105008 (2001).[26] V. I. Yukalov, Moscow Univ. Phys. Bull. , 10 (1976); Theor. Math. Phys. , 652 (1976); R. Seznec and J. Zinn-Justin,J. Math. Phys. , 1398 (1979); J. C. Le Guillou and J. Zinn-Justin, Ann. Phys. , 57 (1983); R. P. Feynman and H.Kleinert, Phys. Rev. A , 5080 (1986); H. F. Jones and M. Moshe, Phys. Lett. B234 , 492 (1990); A. Neveu, Nucl. Phys.(Proc. Suppl.)
B18 , 242 (1991); V. Yukalov, J. Math. Phys , 1235 (1991); K. G. Klimenko, Z. Phys. C60 , 677 (1993);for a review, see H. Kleinert and V. Schulte-Frohlinde,
Critical Properties of φ -Theories , Chap. 19 (World Scientific,Singapure 2001).[27] M. E. Carrington, Phys. Rev. D , 2933 (1992).[28] M. Dine, R. G. Leigh, P. Y. Huet, A. D. Linde and D. A. Linde, Phys. Rev. D46 , 550 (1992); P. Arnold and O. Espinosa,Phys. Rev. D , 3546 (1993).[29] A. Ayala, A. Sanchez, G. Piccinelli and S. Sahu, Phys. Rev. D , 023004 (2005). [30] M. B. Pinto and R. O. Ramos, Phys. Rev. D , 105005 (1999); Phys. Rev. D , 125016 (2000); G. Krein, D.P. Menezesand M.B. Pinto, Phys. Lett. B , 5 (1996); M. C. B. Abdalla, J. A. Helayel-Neto, Daniel L. Nedel and Carlos R. Senise,Jr, Phys. Rev. D , 125020 (2008).[31] J.-L. Kneur, M. B. Pinto and R. O. Ramos, Phys. Rev. D , 125020 (2006); J.-L. Kneur, M. B. Pinto, R. O. Ramos andE. Staudt, Phys. Rev. D , 045020 (2007); Phys. Lett. B , 136 (2007); E. S. Fraga, L. F. Palhares and M. B. Pinto,Phys. Rev. D , 065026 (2009).[32] P. M. Stevenson, Phys. Rev. D , 2916 (1981).[33] R. L. S. Farias, G. Krein and R. O. Ramos, Phys. Rev. D , 065046 (2008).[34] W. -Y. Tsai, Phys. Rev. D , 1945 (1973).[35] J. Schwinger, Phys. Rev. , 664 (1951).[36] S. Chiku and T. Hatsuda, Phys. Rev. D , 076001 (1998).[37] J. -L. Kneur, M. B. Pinto and R. O. Ramos, Phys. Rev. C , 065205 (2010).[38] L. Dolan and R. Jackiw, Phys. Rev. D , 3320 (1974).[39] E. Elizalde, A. D. Odintsov and A. Romeo, Zeta Regularization Techniques with Applications , (River Edge, NJ, WorldScientific, 1994).[40] B. J. Harrington, H. K. Shepard, Nucl. Phys.
B105 , 527 (1976); G. M. Shore, Annals Phys. , 259 (1981); Y. Fujimoto,T. Garavaglia, Phys. Lett.
B148 , 220 (1984).[41] D. Ebert, K. G. Klimenko, Phys. Rev.
D80 , 125013 (2009). [arXiv:0911.1944 [hep-ph]].[42] M. Abramowitz and I. A. Stegun (Eds.),
Handbook of Mathematical Functions with Formulas, Graphs, and MathematicalTables , (New York, NJ, Dover, 1972); T. M. Apostol, Amer. Math. Monthly , 409 (1999); G. Arfken, “BernoulliNumbers, Euler-Maclaurin Formula.” sec. 5.9 in
Mathematical Methods for Physicists , (3rd ed. Orlando, FL, AcademicPress, 1985).[43] Wolfram Research, Inc.,
Mathematica , Version 7.0, Champaign, IL (2008).[44] J. Chakrabarti, Phys. Rev. D , 2232 (1981).[45] V. R. Khalilov and C.-Lin Ho, Phys. Rev. D , 033003 (1999).[46] H. Haugerud, T. Haugset and F. Ravndal, Phys. Lett. A225 , 18 (1997).[47] J. O. Andersen and R. Khan, [arXiv:1105.1290 [hep-ph]].[48] P. Elmfors, K. Enqvist and K. Kainulainen, Phys. Lett.
B440 , 269 (1998).[49] D. Grasso and H. R. Rubinstein, Phys. Rept.348