Optimized perturbation theory for charged scalar fields at finite temperature and in an external magnetic field

Optimized perturbation theory for charged scalar fields at finite temperature and in an external magnetic field

D. C. Duarte dyana.c.duarte@gmail.com Departamento de Ciências Naturais, Universidade Federal de São João Del Rei, 36301-000, São João Del Rei, MG, Brazil    R. L. S. Farias ricardofarias@ufsj.edu.br Departamento de Ciências Naturais, Universidade Federal de São João Del Rei, 36301-000, São João Del Rei, MG, Brazil    Rudnei O. Ramos rudnei@uerj.br Departamento de Física Teórica, 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 temperature and in the presence of an external magnetic field is examined. The effective potential is evaluated nonperturbatively in the context of the optimized perturbation theory method. It is explicitly shown that in all ranges of the magnetic field, from weak to large fields, the phase transition is second order and that the critical temperature increases with the magnetic field. In addition, we present an efficient way to deal with the sum over the Landau levels, which is of interest especially in the case of working with weak magnetic fields.

98.80.Cq, 11.10.Wx

I Introduction

Phase transition phenomena in spontaneously broken quantum field theories have long been a subject of importance and interest due to their wide range of possible applications, going from low energy phenomena in condensed matter systems to high energy phase transitions in particle physics and cosmology (for reviews, see for example book (); rivers (); boya ()).

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 attracted considerable attention in the past linde () and received reinvigorated interest recently, mostly because of the physics associated with heavy-ion collision experiments. In heavy-ion collisions, it is supposed that large magnetic fields can be generated, and the study of their effects in the hadronic phase transition then became subject of intense interest (see e.g. recent () for a recent review). Magnetic fields can lead in particular to important changes in the chiral/deconfinement transition in quantum chromodynamics (QCD) qcd () and even the possibility of generating new phases efrain ()

As far the influence of external magnetic fields and thermal effects on phase transformations are concerned, one well known example that comes to our mind is the physics associated with superconductivity, in particular in the context of the Ginzburg-Landau theory tink (). Let us recall in that case thermal effects alone tend to produce a phase transition at a critical temperature where superconductivity is destroyed and the system goes to a normal ordered state. The phase transition in this case is second order. However, in the presence of an external magnetic field, but below some critical value, by increasing the temperature the system undergoes a first order phase transition instead. This simple example already shows that magnetic fields may have influence on the phase transition other than we would expect from thermal effects alone. There are also other examples of more complex systems where external magnetic fields may have a drastic effect on the symmetry behavior. Among these effects, besides the possibility of changing the order of the phase transition, as in the Ginzburg-Landau superconductor, it can in some circumstances strengthen the order of the phase transition, like in the electroweak phase transition in the presence of external fields Kajantie (), or there can also be dynamical effects, like delaying the phase transition Fiore (). External magnetic fields alone can also lead to dynamical symmetry breaking (magnetic catalysis) Miransky () (for an earlier account, see Ref. Klimenko ()).

Likewise, nonperturbative effects may affect the symmetry properties of a system, once the external parameters are changed, in a way different than seeing through a purely perturbative calculation, or by a mean-field leading order description. This is because perturbation theory is typically beset by problems, for example around critical points, due to infrared divergences, or at high temperatures, when powers of coupling constants can become surmounted by powers of the temperature (see e.g. the textbooks bellac (); kapusta () for extensive discussions). Thus, high temperature field theories and the study of phase transitions in general require the use of nonperturbative methods, through which large classes of terms need to be resummed . Familiar techniques used to perform these resummations include, for example, ring diagram (or daisy and superdaisy) schemes Espinosa (); sato (), composite operator methods Camelia () and field propagator dressing methods Banerjee (); Parwani (). Other methods used include also numerical lattice studies and expansions in parameters not related to a coupling constant, like the expansion and the -expansion zinn (), the use of two-particle irreducible (2PI) effective actions cjt (); berges (), hard-thermal-loop resummation htl (), variational methods, like the screened perturbation theory spt (); andersen-braaten-strickland () and the optimized perturbation theory (OPT) opt (). Of course, any resummation technique must be implemented with care so to avoid possible overcounting of terms and lack of self-consistency. Failure in not following this basic care can lead to a variety of problems, like predicting nonexistent phenomena or producing a different order for the phase transition. One classical example of this was the earlier implementations of daisy and superdaisy schemes, that at some point were giving wrong results, e.g. predicting a first order transition carr () for the 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 arnold (). These wrong results were all because of the wrong implementation of the ring-diagram summation at the level of the effective potential, as clearly explained in the first reference in arnold ()

In this work we will analyze the phase transition for a self-interacting complex scalar field model and determine how an external magnetic field, combined with thermal effects, affects the transition. All calculations will be performed in the context of the OPT nonperturbative method. Our reasons for revisiting here the phase transition in this model are two-fold. First because this same model has been studied recently in the context of the ring-diagram resummation method ayala-ring (), where it was found that the ring-diagrams render the phase transition first order and that the effect of magnetic fields was to strengthen the order of the transition and also to lower the critical temperature for the onset of the (first order) phase transition. So in this work we want to reevaluate these findings in the context of the OPT method. We recall, from the discussion of the previous paragraph, that the ring diagram method requires special attention in its implementation and that previous works have already concluded erroneously about its effects on the transition. The OPT method has a long history of successful applications (for a far from complete list of previous works and applications see e.g. Refs. previous1 (); previous2 () and references therein). The OPT method automatically resums large classes of terms in a self-consistent way so to avoid possible dangerous overcounting of diagrams. In our implementation of the OPT here, we will see that already at the first order in the OPT it is equivalent to the daisy 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 an external magnetic field. Finally, we also want to properly treat the effects of small magnetic fields (which requires summing over very large Landau-levels) in an efficient way, particularly suitable for numerical work. This way we can evaluate 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 the Euler-Maclaurin formula and fully investigate the validity of its use as an approximation for the Landau level sums for 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 application of the OPT method for the problem we study in this paper. It is shown explicitly which terms are resummed by the OPT. We also verify that the Goldstone theorem is fulfilled in the OPT. In Sec. III we study the phase transition in the spontaneously broken self-interacting quartic complex scalar field in the OPT method, first by incorporating only thermal effects and then by including both temperature and an external magnetic field. In Sec. IV we study the advantages of using the Euler-Maclaurin formula for the sum over the Landau levels. The accuracy and convergence of the method is fully examined. We determine that the sum over the landau levels, in the regime of low magnetic fields, where typically we must sum over very large levels so to reach good accuracy, can be efficiently performed within the very few first terms in the Euler-Maclaurin formula. As an application, an analytical formula for the critical temperature for phase restoration, as a function of the magnetic field, , is derived. Our final conclusions are given in Sec. V. Finally, an appendix is included to show some of the details of the renormalization of the model in 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 symmetry and spontaneously symmetry breaking at tree-level in the potential, whose Lagrangian density is of the standard form,


where for spontaneously symmetry breaking. It is convenient to write the complex scalar field in terms of real and imaginary components, . In terms of a (real) vacuum expectation value (VEV) for the field, , we can, without loss of generality, shift the field around its VEV in the direction,


The Feynman propagators for and , in terms of the VEV then reads,


Using the tree-level VEV for the field, in Eqs. (3) and (4), is then associated with the massive Higgs mode (with mass squared at the tree level) and is the Goldstone mode of the field, remaining massless throughout the symmetry broken phase.

ii.1 Implementing the Optimized Perturbation Theory

The implementation of the OPT in the Lagrangian density is the standard one previous1 (); previous2 (), where it is implemented through an interpolation procedure,


where and is the Lagrangian density part with the renormalization counterterms needed to render the theory finite. In , the dimensionless parameter is a bookkeeping parameter used only to keep track of the order that the OPT is implemented (it is set to one at the end) and is a (mass) parameter determined variationally at any given finite order of the OPT. A popular variational criterion used to determine is known as the principle of minimal sensitivity (PMS), defined by the variational relation PMS ()


which is applied to some physical quantity , calculated up to some order- in the OPT. The optimum value , which satisfies Eq. (6), is a function of the original parameters of the theory and it is in general a nontrivial function of the couplings. It is because of the variational principle used that nonperturbative results are generated. Other variational criteria can likewise be defined differently, but they produce the same final result for the quantity , as shown recently FGR ().

In terms of the interpolated Lagrangian density, Eq. (5), the Feynman rules in the OPT method are as follows. The interaction vertex is changed from to . The quadratic terms in Eq. (5) define new vertex insertion terms. The propagators for () now become


ii.2 The effective potential in the OPT method

As in the many other previous applications previous2 (); FGR (), we apply the PMS, Eq. (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 the previous Feynman rules in the OPT method given above, the effective potential is given by the vacuum diagrams shown in Fig. 1.

Figure 1: Feynman diagrams contributing to the effective potential to first order in the OPT. Solid lines stand for the propagator, while dashed lines stand for the propagator. A black dot is an insertion of . A crossed dot is a mass renormalization insertion.

From the diagrams shown in Fig. 1 and by further expanding the propagators Eqs. (7) and (8) in , the explicit expression for the renormalized effective potential at first order in the OPT (some of the details of the renormalization in the OPT method are given in the Appendix A) becomes


where the momentum integrals are expressed in Euclidean space and the regularization is performed in scheme, where, in the finite temperature only case,


where is the Euler-Mascheroni constant, is an arbitrary mass regularization scale, is the dimension of space and are the Matsubara frequencies for bosons at temperature .

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 -space direction and use a gauge where the external electromagnetic field is . The Feynman propagator for a charged scalar field with charge becomes , where is the energy dispersion (for charged scalar bosons) in an external constant magnetic field tsai (),


where the last term denotes the Landau levels (). Likewise, the momentum integrals, taking into account the degeneracy multiplicity of the Landau levels schwinger (), are now represented by


ii.3 Optimization results and Goldstone theorem

Let us now apply the PMS condition (6) on the effective potential Eq. (9). We obtain straightforwardly that the optimum satisfies the nontrivial (renormalized) gap equation


The extrema of the effective potential are defined as usual, by requiring that the first derivative of the effective potential with respect to vanishes. This gives us the trivial solution and the solution for the minimum,


We can now verify the effective mass for the field, in particular for the component of the complex scalar field. The effective mass for in the OPT is given by


where is the (renormalized) field’s self-energy, which at first order in the OPT is trivially found to be given by


Using now Eqs. (16), (13) and (14) in Eq. (15), we obtain immediately that , which is nothing but the result expected due to Goldstone theorem for the symmetry broken complex scalar field model. It can also be verified that this result carries out at higher orders in the OPT method, in which case, at some order- in the OPT, the self-energy is the one at the respective order, , entering in the above equations. Previous demonstrations of the Goldstone theorem in the OPT were for the linear sigma model hatsuda () and for the Nambu–Jona-Lasinio model njl ().

Finally, it is noticed from the PMS equation given above, Eq. (13), that the OPT naturally resums the leading order loop terms of the field’s self-energy. So it is quite analogous, at already in the first order in the OPT approximation, to the ring-diagram resummation Espinosa (); arnold (). And that this resummation is also performed automatically in a self-consistent way, it is clear from the interpolation procedure. While enters in all propagators, thus carrying the self-energy corrections, the insertions of (the term in Eq. (5) is treated as an additional interaction vertex) subtract spurious contributions at each order, thus avoiding dangerous overcounting of diagrams (this is also similar to the procedure used in the screened perturbation theory spt (); andersen-braaten-strickland ()). Going to higher orders in the OPT resums higher order classes of diagrams. For our purposes of comparing with the ring-diagram results, it suffices then to keep up to first order in the OPT.

Iii Phase structure and symmetry restoration at finite temperature and in an external magnetic field

We are now ready to study the phase structure and the symmetry restoration in the complex scalar field model at finite temperature and in an external magnetic field. It is convenient to first investigate the case of finite temperature only, so to later compare with the case when an external magnetic field is added.

iii.1 Symmetry restoration at finite temperature

It is convenient to write the explicit expressions for each term entering in the effective potential in Eq. (9). At finite temperature and in the absence of an external magnetic field, using (10), we have that the first momentum integral in (9) becomes (recalling that )


where we have identified explicitly the divergent term and the finite term, , is given by


Likewise, for the remaining momentum integrals in (9) we obtain


where is given by


Next, there is the term coming from the mass counterterm in Eq. (9) (see Appendix A),


where we have also defined


Finally, we have the two-loop contributions shown in Fig. 1. They all sum up in the OPT expansion to give


From Eqs. (17), (19), (20), (22) and (24), we find the complete expression for the renormalized effective potential at first order in the OPT:


The phase structure at finite temperature is completely determined by Eqs. (25), (13) and (14). The optimum , determined by the PMS criterion, and the vacuum expectation value (VEV) for the scalar field, can now be both, respectively, be expressed as




In the OPT we can exactly compute the critical temperature of phase transition. Using that at the critical point the VEV of the field vanishes, , then from Eq. (26) we obtain that


which upon using it in Eq. (27), we obtain


From the definition of , Eq. (21), we then obtain that


and when using the above result back in Eq. (29), we obtain the exact result for the critical temperature at this first order in the OPT:


This result for is the same predicted before for a scalar field model with two real components jackiw (), obtained in the high temperature one-loop approximation. The result (31) is exact in our OPT approximation and obtained independent of any high temperature approximation, as usually assumed in any previous calculations.

Figure 2: Temperature dependence of the minimum of the effective potential in the OPT with the PMS optimization criterion. is the minimum of the tree-level potential. The parameters used are and .

From Eqs. (26) and (27), we obtain directly the behavior of the VEV as a function of the temperature. For comparison purposes, we use analogous parameters as adopted in Ref. ayala-ring () 111 Note that in ayala-ring () the authors used a different numerical factor for the quartic term in the tree-level potential, which gives rise to the extra factor in the numerical value for in our notation. In the approximations used in ayala-ring (), the terms involving the regularization scale were not included. Here we express all quantities in terms of the regularization scale ., where and . For these parameters, the authors in Ref. ayala-ring () find a first order phase transition. In Fig. 2 we show the result for the minimum of the effective potential, as a function of the temperature. The effective potential, for different temperatures and same parameters as in Fig. 2, is shown in Fig. 3. Note that the effective potential has an imaginary part for temperatures below , Eq. (31), since there can be values of for which becomes negative. This happens for values of in between the inflection points of the potential, which defines the spinodal region of instability in between the (degenerate) VEV of the field, determined by Eq. (27). In Fig. 3, for the case of , we show the real part of the effective potential, so to be able to show all the potential, including the spinodal region.

Figure 3: The effective potential (subtracting the vacuum energy at ), for the same parameters of Fig. 2 and for temperatures: , and at the critical temperature, .

It should be noted that the minimum of the effective potential, determined by the coupled Eqs. (26) and (27), is well defined and unique, where the only solutions for Eq. (27) are the two degenerate minima for , while for , the only solution is . Therefore, the phase transition can readily be inferred to be second order for any set of parameters. It is also clear from the results shown by Figs. 2 and 3 that the transition due to thermal effects only is of second order. The effective potential does not develop local minima and the VEV of the field varies continuously from till , where it vanishes. This is the expected behavior, since the complex scalar field model belongs to the same universality class of a Heisenberg model, for which the phase transition is second order book ().

Thus, we see that the OPT method correctly describes the phase transition in the model, capturing the correct physics. Furthermore, as an added bonus, we can exactly compute the critical temperature of transition

iii.2 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 the presence of a constant magnetic field, the Feynman rules change as shown in Subsec. II.2. The momentum integrals in Eq. (9) are given by (12) and the dispersion relation for charged particles is given by Eq. (11), which takes into account the Landau energy levels of a charged particle in a magnetic field222Note that the dispersion relation Eq. (11) only applies for the fields in the complex base , since are not appropriate eigenstates of charge. However, when re-expressing the effective potential in the complex scalar field base, the final result turns out to be the same as in Eq. (9) or (25), but with the functions and depending now also on the magnetic field..

At finite , the first momentum integral term in (9) becomes:


By performing the sum over the Matsubara frequencies in (32), we obtain two terms. One is independent of the temperature and the other is for . The term is


where to write the second line in (33), we used the analytic continuation of the Hurwitz zeta function zeta (),


and is its -derivative. In writing the final terms in Eq. (33) we have also dropped constant terms since they do not affect the phase structure.

The part of (32) is


We can now write Eq. (32) in a form analogous to Eq. (17),


where is given by


Analogous manipulations leading to Eq. (36) allow us to write the remaining momentum integral terms in Eq. (9) like




Actually, Eq. (38) is just the derivative of Eq. (36) with respect to , as can be easily checked.

The final expression for the renormalized effective potential at finite temperature and in a constant external magnetic field has an analogous form as Eq. (25), where by using Eqs. (36) and (38) in (9), we obtain


Finally, the PMS criterion and the minimum of the effective potential are again also of the form as Eqs. (26) and (27):




With Eqs. (40), (41) and (42), we are now in position to study the effects of an external magnetic field in the phase structure of the model, in addition to the thermal effects. In Fig. 4 we show the phase diagram for the symmetry broken complex scalar field model in the plane. It shows that the magnetic field strengths the symmetry broken phase, increasing the critical temperature for phase transition. We must note that this result for the phase diagram is very different from that seen in the superconductor or in the scalar quantum electrodynamics (QED) case scalarQED (). In contrast to our case, where we are only studying the effects of and 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 gauge symmetry, in the symmetry broken phase, because of the Higgs mechanism the gauge field becomes massive and the external magnetic field gets screened (the Meissner effect). As the magnetic field is increased, eventually above a critical field the phase is restored through a first order phase transition. In the study made in this paper, on the other hand, we consider only the case of a broken global symmetry333One of the authors (ROR) would like to thank K. G. Klimenko for discussions on this topic and for also pointing him out the possible similarity 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 pioncond ()..

The effect of the magnetic field on the global symmetry can also be seen in the next two figures. In Fig. 5 we have plotted the VEV of the field at fixed , thus starting from a symmetry restored phase, , as a function of . The critical temperature, for the parameters used, is , which fully agrees with Eq. (31), and the symmetry returns to be broken, , for a magnetic field above a critical value given by . In Fig. 6 the effective potential is plotted for the same fixed temperature of Fig. 5 and for values of below, at and above the critical value , above which the symmetry becomes broken again (as before, to show the spinodal region of the potential, we have plotted only the real part of it). It should be noticed from these results that the phase transition is once again second order. The effect of the magnetic field is to enlarge the symmetry broken region by making the critical temperature larger, but it does not change the order of the transition.

Figure 4: The phase diagram of the system in the plane. The parameters considered here are: , . The solid line separates the regions of symmetry broken (SB) and symmetry restored (SR) phases.
Figure 5: Magnetic field dependence of the VEV of the field, , for a fixed value of above . The parameters considered here are: , and .
Figure 6: The effective potential for the same parameters of Fig. 5 and for three different values of the magnetic field: , and .

Finally, we have again used for comparison a case with analogous parameters as considered in Ref. ayala-ring (), and contrary to the results found in that reference, we have found again here only a second order phase transition. For the analogous parameters used in ayala-ring () (see observations made in previous subsection), , and , the change of the critical temperature in relation to the case shown in Fig. 3 is only to produce a slight shift of the critical temperature to a value higher. In Fig. 7, we have plotted the VEV of the field for the case of zero magnetic field and for a much higher value of the magnetic field, , so to be able to visualize the difference. Even so, the difference even for such value of magnetic field is only marginal, changing the critical temperature obtained at , , to for the value of used. In all cases we have tested for a nonvanishing magnetic field and other different parameters of the potential, the transition is again verified to be second order, with the VEV varying always continuously from its value at to zero at .

Figure 7: Temperature dependence of the normalized VEV . The critical temperature for is . Including the effects of the magnetic field, with , increase to . The other parameters used here are: and .

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 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 were presented in analytical form. However, at finite temperature this is not possible in general, therefore, we either need to numerically perform the sums over the Landau levels, or find suitable approximations for the expressions. This job becomes easier in the high magnetic field regime, with , for which most of the time suffices to use only the very first terms of the sum, or even just the leading Landau level term. Higher order terms are quickly Boltzmann suppressed in this case. But this is not so in the low magnetic field regime, with , and , where most of the time requires to sum up to very large Landau levels. This has the potential to make any numerical computation to become quickly expensively, specially when there are in addition numerical integrations involved, like in our case here (note that we have not made use of high temperature approximations, but worked with the complete expressions in terms of the temperature dependent integrals). Simple approximate analytical results, which are always desirable to obtain, can also become difficult to obtain without a suitable approximation that can be used. One such approximation was discussed in Ref. ayala-ring (), but it is valid only for very small magnetic fields. Here we discuss a more natural alternative for dealing with the cases of low magnetic fields, based on the Euler-Maclaurin formula.

The Euler-Maclaurin (EM) formula provides a connection between integrals and sums. It can be used to evaluate finite sums and infinite series using integrals, or vice-versa. In its most general form it can be written as EMidentity ():


where are the Bernoulli numbers, defined by the generating function


and are the Bernoulli polynomials, with generating function


The notation in in Eq. (43) means the fractional part of and means the -th derivative of the function. The last term in Eq. (43) 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 Landau levels and we estimate the errors involved in doing so. In the following, we will call the 0th-order in the EM formula when only the first term in the RHS of Eq. (43) is kept, the 1st-order when also the second term is kept and so on so forth. Thus, for example, the Landau sum part of in Eq. (37), up to second order in the EM formula, becomes




Likewise, the Landau sum part of in Eq. (39), up to second order in the EM formula, becomes


In Tab. 1 we assess the reliability of using the EM at each order (we have studied it up to the forth-order) for the sums in Eqs. (46) and (48), for different values for the ratio , and we compare the results with the exact ones.

EM approx. Landau Sum error () EM approx. Landau Sum error ()
0 - 688.7717 0.04 508.2575 1.67
1 - 689.0258 515.7328 0.22
2 - 689.0271 - 689.0264 517.0372 516.8633 0.03
3 - 689.0270 516.7096 0.03
4 - 689.0271 517.2016 0.07
0 - 68.6567 0.35 47.7384 5.00
1 - 68.8954 49.8997 0.70
2 - 68.8990 - 68.8989 50.3031 50.2488 0.11
3 - 68.8988 50.2007 0.10
4 - 68.8990 50.3549 0.21
0 - 6.6735 2.99 3.9418 14.07
1 - 6.8706 0.13 4.4852 2.22
2 - 6.8797 - 6.8793 4.6037 4.5873 0.36
3 - 6.8791 4.5725 0.30
4 - 6.8796 4.6199 0.70
0 - 0.5400 18.70 0.2195 34.89
1 - 0.6497 2.18 0.3128 7.21
2 - 0.6652 - 0.6642 0.15 0.3416 0.3371 1.34
3 - 0.6637 0.07 0.3329 1.25
4 - 0.6651 0.14 0.3465 2.79
0 - 0.0159 59.54 0.0034 66.99
1 - 0.0328 16.54 0.0081 21.36
2 - 0.0407 - 0.0393 3.56 0.0112 0.0103 8.74
3 - 0.0383 2.55 0.0095 7.77
4 - 0.0413 5.09 0.0124 20.39
0 - 88.48 89.52
1 - 38.52 39.54
2 - - 40.87 47.82
3 - 80.80 35.00
4 - 551.81 905.27
Table 1: The precision of the EM formula (without the remainder) at each order (up to 4th-order), compared with the result from the Landau sum. In all cases we have considered in Eqs. (46) and (48), when extending them up to 4th-order.

From the results shown in Tab. 1, we see that the EM approximation produces results that already at first order have good accuracy compared to the exact values coming from the full Landau sums. It performs particularly very well for values of , reaching convergence within just the very first few terms in the expansion. However, for values , increasing the order in the EM series, it quickly looses accuracy. This can be traced to the fact that as we go to a higher order in the derivatives appearing in the sum on the RHS of Eq. (43), higher powers of are produced, eventually spoiling the approximation (though it oscillates around the true value). This apparent runaway behavior is only cured by the introduction of the last term in Eq. (43), the remainder.

It should be noted that Eq. (43) is not an approximation to the sum, but it is actually an identity. The important term in that context is the remainder, the last term in Eq. (43). In all our numerical tests, within the numerical precision for the integrals (for convenience, we have performed all the numerical integrations in Mathematica math ()), we have verified that Eq. (43) has agreed with the results from the sum over the Landau levels in the LHS of Eqs. (46) and (48), for all values tested and at all orders (when including the remainder), with a precision of less than . This is quite impressed, recalling that we can in principle keep only the first few terms in Eqs. (46) and (48) (the first order approximation) when including the remainder. For numerical computation this is a tremendous advantage, since Eq. (43) can easily be coded, compared to having to perform sums over very large Landau levels (particularly in the cases of low magnetic fields) and the multiple integrations that may be required.

Given all the advantages of using the EM formula, it is quite surprising that it is not used frequently in physics, despite being well known in mathematics. In particular, we see that it is well suitable in problems involving external magnetic fields. To our knowledge, one of the few works that we are aware of that have previously made use of the EM formula before were for example Refs. chakra (); ho (); ravndal (). In Ref. chakra () the author derived the effective potential of the abelian-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 analytical results for the effective potential in the high magnetic field region. But from the results of Tab. 1, this is exactly the region and order in the approximation that lead to the largest errors. In ho () the EM formula was used to obtain an expression for the effective potential for vector bosons and to study pair production in an external magnetic field, while in ravndal () it was used to obtain analytical expressions for the internal energies for non-interacting bosons confined within a harmonic oscillator potential. None of these previous works have assessed the reliability of the use of the EM formula.

As an application of the EM formula to our problem, we estimate from it the dependence of the critical temperature with the magnetic field. Recall from Eq. (29) the critical temperature can be obtained by setting and in the equation for the VEV. Thus, from Eq. (42), we need to solve the equation


As we have seen from the results of Tab. 1, the EM formula for low magnetic fields produce reliable results already at the second order, and for very small magnetic fields, good precision is reached even at first order. Thus, for example from the expansion in Eq. (48), and performing the integral in the first term, we obtain


where . Results for the integrals in Eq. (50) can be found in the low magnetic field regime, where . This regime is equivalent to the high temperature regime in quantum field theory, from where a series expansion in can be found. In fact, the integrals in Eq. (50) can be easily related to the Bose-Einstein integrals found in that context (see for example the App. A in Ref. kapusta ()), from where we obtain the leading order terms in the expansion in powers of :


Going to higher order in the EM series only lead to (with ) terms, for .

Using the result (51) in Eq. (39) and then back in Eq. (49), after some algebra, we obtain the approximate result for the critical temperature as a function of the magnetic field,


The result (52) shows the growth of the critical temperature with the magnetic field, as seen numerically from the results obtained in the previous section. A comparison with our previous results also shows that Eq. (52) provides an excellent fit for , leading to results with less than error compared to the full numerical results for , for values of field such that . For example, for the parameters and , the approximation (52) gives , while the full numerical calculation gives .

V Conclusions

In this paper we have revisited the phase transition problem for the self-interacting complex scalar field model when thermal effects and an external magnetic field are present. We have studied this problem in the context of the nonperturbative method of the optimized perturbation theory. We have shown that the OPT method preserves the Goldstone theorem and that carrying out the approximation to first order is analogous to the ring diagram resummation method. The OPT carries out the resummation of self-energy diagrams in a self-consistent way, avoiding overcounting 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 in the presence of thermal effects and by including an external magnetic field, there is no change to the phase transition order. The effect of the external magnetic field is to strengthen the symmetry broken phase, producing a larger VEV for the field and also a larger critical temperature. The study we made in this paper considered only the effects of the magnetic field and temperature on the charged scalar field (the model is one with a global symmetry). Our study, in a sense, is closer to the case of studying the effects of and in the linear sigma model andersen (), where no couplings to gauge fields (except to the external field) are considered in general. The inclusion of other field degrees of freedom, or when promoting the symmetry from a global one to a local symmetry, can of course change both qualitatively and quantitatively how the magnetic field affects the system. In particular, as concerning the possibility of producing a first order phase transition, either due to thermal effects, or by a magnetic field or from both. This is what 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 other interactions and having other symmetries. For example, in the context of the electroweak phase transition Kajantie (); Fiore (); Kimmo (), it has been shown that the effect of the magnetic field is to make the first order transition stronger. By omitting vacuum energy terms from the effective potential, which may have phenomenological motivations, as in Ref. andersen (), may also lead to a first order transition, instead of a second order one.

Still comparing our results with those obtained in the abelian Higgs model, there is a tantalizing question of whether for very strong magnetic fields a new phase could be formed. This could be for example a phase with global vortices condensation, thus restoring the symmetry again. This would be in analogy to the local, Nielsen-Olesen vortex conndensation that can happen in type II superconductors scalarQED (). Vortex condensation in that case is energetically favorable to happen for fields beyond a critical value and when the mass of the Higgs field becomes larger that the mass 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-Maclaurin formula as an approximation for the sum over Landau energies in a magnetic field. We have verified that it produces results with errors of less than already at the first few orders in the EM formula and that it leads to particularly suitable approximations in the low magnetic field region (). The low magnetic field region is typically the region where we have to face the