Spectral Function of Fermion Coupled with Massive Vector Boson at Finite Temperature in Gauge Invariant Formalism

Spectral Function of Fermion Coupled with Massive Vector Boson at Finite Temperature in Gauge Invariant Formalism

Daisuke Satow d-sato@ruby.scphys.kyoto-u.ac.jp    Yoshimasa Hidaka hidaka@ruby.scphys.kyoto-u.ac.jp    Teiji Kunihiro kunihiro@ruby.scphys.kyoto-u.ac.jp Department of Physics, Faculty of Science, Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan
July 25, 2019
Abstract

We investigate spectral properties of a fermion coupled with a massive gauge boson with a mass at finite temperature () in the perturbation theory. The massive gauge boson is introduced as a gauge boson in the Stueckelberg formalism with a gauge parameter . We find that the fermion spectral function has a three-peak structure for irrespective of the choice of the gauge parameter, while it tends to have one faint peak at the origin and two peaks corresponding to the normal fermion and anti-plasmino excitations familiar in QED in the hard thermal loop approximation for . We show that our formalism successfully describe the fermion spectral function in the whole region with the correct high- limit except for the faint peak at the origin, although some care is needed for choice of the gauge parameter for . We clarify that for , the fermion pole is almost independent of the gauge parameter in the one-loop order, while for , the one-loop analysis is valid only for where is the fermion-boson coupling constant, implying that the one-loop analysis can not be valid for large gauge parameters as in the unitary gauge.

pacs:
11.10.Wx, 12.38.Mh
preprint: KUNS-2315

I Introduction

It is well known that for extremely high temperature () where the hard thermal loop (HTL) approximation in QED and QCD frenkel-taylor (); braaten-pisarski (); weldon:1982aq (); weldon () is valid, a fermion (quark) coupled with thermally excited gauge fields (gluons) make collective excitations, i.e., the normal fermion (particle) and the anti-plasmino excitation with distinct peaks in the fermion spectral function weldon (); this feature obtained in the HTL approximation is also known to be gauge invariant in the sense that the fermion self-energy at one-loop order does not depend on gauge braaten-pisarski (). As for lower region, a possible change in the spectral properties of the quark in association with chiral transition in QCD was investigated kitazawa-NJL (), using the Nambu-Jona-Lasinio model NJL (), and it is shown that the coupling with the chiral soft modes kunihiro () make the quark spectral function have distinct three peaks near but above the critical temperature of chiral transition. The appearance of such a novel spectral function at was later confirmed kitazawa () for a massless fermion coupled with an elementary massive boson with a mass , irrespective of the type of the massive boson. The mechanism for realizing the three-peak structure in the spectral function was also elucidated kitazawa () in terms of the Landau damping owing to the collisions of the fermion with thermally excited bosons111 This feature that the three-peak structure arises at is not altered even for a massive fermion with a mass as long as is not too large compared with mitsutani (). .

Then one may naturally ask a question if the fermion spectral function at those lower would smoothly connect with that at extremely high , i.e. the HTL result in QED/QCD: If it is not the case, it means that we do not have a unified understanding of the fermion spectral properties in the whole region. Partly to answer this question, we investigate spectral properties of a fermion coupled with a massive vector boson introduced as a gauge boson in the (generalized) Stueckelberg formalism with a gauge parameter stueckelberg-review (); stueckelberg (), and carefully examine their possible gauge dependence at , at the one-loop order as in kitazawa (). Here the spectral properties include the number of the fermion poles, the pole position in the complex energy plane and the spectral function in the momentum-energy plane. We are also interested in how the quasi-particle nature of the fermion is realized or destroyed by the coupling with a massive boson at finite .

We find that the present formalism gives a valid description of the fermion coupled with a massive vector boson for the whole temperature () region at one-loop order in a unified way; thereby we reveal the characteristics of the fermion spectral properties depending on the distinct regions, i.e. (I) ,  (II)  and (III) . Especially, we shall show that the fermion spectral function certainly tends to have a three-peak structure for in the small momentum region with supports in the positive, zero and negative energy regions.

The investigation of the possible gauge dependence turns out to be involved owing to the appearance of a novel mass scale , inherent in the present formalism, as well as the boson mass and temperature . One should remark here that the Proca formalism adopted in kitazawa () is not adequate for this purpose, because this formalism corresponds to a special gauge with (unitary gauge), and does not lead to the proper high- limit, or , which should be the HTL approximation in QED at one-loop level weldon (). This is the reason why we have adopted the Stueckelberg formalism to describe the massive vector boson. We remark that although the pole position is gauge-independent in the exact calculation rebhan (), a gauge-dependence of the fermion pole may appear in the perturbation theory at finite in general. Since the Proca formalism corresponding to the limit leads to a wrong high- limit, there should exist an adequate gauge-parameter region in which the results in the perturbation theory hardly show gauge dependence: Indeed, we show that this is the case in the present work.

This paper is organized as follows. In Sec. II, we formulate the gauge theory in which the gauge boson acquires finite mass. We perform a calculation of a fermion self-energy at finite temperature. In Sec. III, the numerical results of the fermion spectral properties are shown. In Sec. IV, we discuss the gauge dependence of fermion pole appearing when in an analytic way. Section V is devoted to a summary and concluding remarks. In Appendix A, we briefly describe how the abelian Higgs model is reduced to the massive gauge theory in the Stueckelberg formalism. In Appendix B, we present detailed calculational procedures for the fermion self-energy in our model. Appendix C is devoted to making an order estimate of some terms appearing in the text.

Ii U(1) gauge theory with massive gauge boson

In this section, we formulate the gauge theory with a massive gauge boson, and introduce a propagator and a spectral function at finite temperature in the imaginary time formalism lebellac (); kapusta (). We perform a calculation of the self-energy of a fermion coupled with a massive vector boson at one-loop order.

ii.1 General formalism

First, we introduce a gauge theory with a massive gauge boson. The gauge boson acquires a mass by the Higgs mechanism, keeping the gauge symmetry. The gauge theory is one way to construct a renormalizable quantum field theory with a massive vector boson. We employ the Stueckelberg formalism stueckelberg (); stueckelberg-review () proposed long ago, which is equivalent to the abelian Higgs model with a constant absolute value of the Higgs field stueckelberg-review (); higgs-stueckelberg (). This correspondence is reviewed in Appendix A. Then our Lagrangian reads

(1)

where , and are a massive vector, a scalar and a fermion field, respectively. The scalar field is called the Stueckelberg field, which corresponds to the phase of the Higgs field in the abelian Higgs model. is a field strength, the coupling constant, the vector boson mass, and is a gauge parameter. is the gauge fixing term defined by

(2)

We work with the Minkowski metric, . We shall deal with a massless fermion assuming that the mass is neglected, which should be valid at high temperatures. Our Lagrangian is invariant under the gauge transformation except for the gauge fixing term, :

(3)
(4)
(5)

There are no interaction between the Stueckelberg field and the fermion field, and we chose the gauge fixing term so that the interaction term between the vector field and the Stueckelberg field vanishes. We can drop the Stueckelberg field as long as a correlation function is concerned, while it can not be when the thermodynamic potential is considered, where it is important to take into account the correct degrees of freedom.

The propagator of the free massive vector boson is now given by

(6)

In the limit, the propagator tends to

(7)

which is the massive vector-boson propagator in the Proca formalism222 Here we note that the propagator in the Proca formalism does not vanish but rather approaches a constant value in the limit, in contrast to that in the Stueckelberg formalism. This causes the non-renormalizability and leads to a bad behavior at high temperature proca-problem (); kitazawa (); dolan-jackiw (). .

The fermion propagator in the imaginary time formalism kapusta (); lebellac () is expressed with the self-energy as

(8)

where is the Matsubara frequency for fermion. Note that and are matrices with the spinor indices. The retarded fermion propagator is given by an analytic continuation, :

(9)

where the retarded self-energy is given by

(10)

Introducing the projection operator on the (anti-)particle sector , we can decompose the retarded propagator and self-energy into the respective sector as follows:

(11)
(12)

with .

In the particle sector, the pole satisfies the following equation:

(13)

From the analyticity of the retarded propagator, the pole is located on the real axis or the lower half-plane of complex . If the imaginary part of the pole is small, the pole is well described in terms of a quasi-particle picture, where the real part of the pole corresponds to the energy while the imaginary part to the decay width of the quasi-particle. If the imaginary part is large, then it would be meaningless to consider excitations in terms of any particle picture.

Figure 1: The diagram which contributes to the fermion self-energy at one-loop order. The solid line represents the fermion and the wavy line represents the massive vector boson.

It is known that the self-energy at zero momentum has the following symmetry,

(14)
(15)

which implies that if there exists a fermion pole at at zero momentum, there is also a pole at at zero momentum.

Once the self-energy is obtained, the spectral function of the (anti-)particle sector is expressed as

(16)

When the peak is narrow enough, the position of the peak is given by and the width of the peak is given by .

ii.2 Calculation at one-loop order

Now let us evaluate the self-energy at one-loop order; the corresponding diagram is shown in Fig. 1. is expressed as

(17)

where is the propagator of the free fermion, , and . Some manipulations lead to

(18)

The retarded self-energy in the one-loop approximation is given by the analytic continuation from . Here, we have introduced the following loop functions:

(19)
(20)

We see that there are two kinds of mass in Eq. (18), and , the latter of which is unphysical because it depends on the gauge parameter. However, the existence of such an unphysical mass causes two different high temperature limit as will be shown in Sec. IV. We will also show that approaches the fermion self-energy in QED if we take the massless limit , which is not the case in the Proca formalism. It should be noted here that Eq. (18) shows that there is a special value of : when , the terms containing the unphysical mass are all cancelled out and only the first term remains, i.e., .

The self-energy in the (anti-)particle sector in the one-loop approximation is given by . Then, as is derived in Appendix B, we have for the imaginary part of ,

(21)

where , and .

The real part may be obtained using the dispersion relation from the imaginary part. Especially, the finite temperature part of the real part of the self-energy, , is expressed as

(22)

Here P denotes the principal value. The zero temperature part of Re is not determined by Eq. (22) because it has ultraviolet divergence. We make renormalization using twice-subtracted dispersion relation, which reads

(23)

We impose the on-shell renormalization condition, and , to determine and . The vacuum part of is obtained by taking the limit of Eq. (21);

(24)

Thus we arrive at

(25)

As mentioned before, our theory based on the Stueckelberg formalism approaches QED at high enough temperature where the masses are negligible in comparison with . Let us see this. For , the imaginary and real part of the self-energy are reduced to

(26)
(27)

respectively. Here, we have retained only the terms which are proportional to in Eq. (21). These Eqs. (26) and (27) coincide exactly with the well-known results in the HTL approximation in QED frenkel-taylor (); braaten-pisarski (); weldon:1982aq (); weldon (). There is a caveat in the above manipulation, which has been taken for granted in the usual derivation of the HTL approximation in the gauge theory: The ignored terms may become comparable to terms which are proportional to in some gauges and hence the above naive power-counting turns out to be invalid. We will analyze this possibility in Sec. IV.

Iii Numerical Results

In this section, we show numerical results of the fermion spectral function and the fermion poles at various temperatures. In the following, the coupling constant is fixed to a small value, , so that the analysis based on the one-loop calculation can be valid: Except when the coupling constant dependence of the pole is analyzed, the coupling constant will be fixed to . On the other hand, the gauge parameter, , will be varied freely in order to see the gauge-dependence of the spectral properties of the fermion calculated at one-loop level.

iii.1 Low temperature ()

In this subsection, we show numerical results at a so low temperature that dependence of the results is hardly seen, which may check our analytical and numerical calculations.

Figure 2 shows the fermion spectral function in the particle sector (with a positive particle number) at for . There appears a very narrow peak near , which is very reminiscent of zero temperature case. This is natural for , because the thermal effect is exponentially suppressed by the Boltzmann factor , and hence the breaking of Lorentz symmetry is small. This small breaking of Lorentz symmetry implies that the particle pole is almost on-shell value at , i.e., , and hence the gauge dependence of the pole hardly appears.

Figure 2: The fermion spectral function in the particle sector as a function of energy and momentum for .
Figure 3: The fermion spectral function in the particle sector as a function of energy and momentum at at various temperatures.
Figure 4: The spectral fermion function in the particle sector as a function of energy and momentum at .
Figure 5: The left panel shows the gauge dependence of the fermion pole with a positive particle number in the positive energy region for , at zero momentum: the horizontal and vertical axes denote the real and imaginary part of the energy, respectively. The right panel shows the fermion spectral function in the particle sector as a function of energy for the same , as the left panel.
Figure 6: The left panel shows the coupling constant dependence of the fermion pole with a particle number in the positive energy region for and , at zero momentum: the horizontal and the vertical axis denote the real and imaginary part of the energy, respectively. The dots from left to right correspond to , , , , , respectively. The right panel shows the coupling constant dependence of the real part of the fermion(particle) pole in the positive energy region at zero momentum for the same and . The solid line is the fitted linear function.
Figure 7: The upper figure shows the fermion spectral function in the particle sector as a function of energy for , and . The lower figure shows the real part and the imaginary part of the fermion self-energy as functions of energy for the same , , and . The dotted line denotes . The intersection points between this line and are the solutions of the dispersion relation.

iii.2 Intermediate temperature ()

We plot the spectral function of the particle sector at , , , for in Fig. 3. We can see that the spectral function at these temperatures have structures qualitatively different from that at low temperature: Even at , we see a split of a peak around the origin seen in Fig. 2 into two peaks with a small bump in the negative energy region, which is reminiscent of the anti-plasmino peak known in QED/QCD at high . These features are enhanced as is raised, and we see a clear three-peak structure around at with a prominent peak and a clear bump in the positive and negative energy region, respectively. We also see that the peak near the origin is attenuated as is further raised up to .

Since it is known that the details of the shape of the spectral function may be gauge-dependent in general, let us see how the three-peak structure depends on the gauge parameter. Figure 4 shows the gauge-parameter dependence of the fermion spectral function in the particle sector at ; the gauge parameter is varied as , , and . One might find only single curve of the spectral function in the figure, although this figure actually shows four curves of it with different ; thus it clearly tells us that the shape of the spectral function at with a three-peak structure is virtually independent of the gauge parameter.

The virtual gauge-independence of the shape of the spectral function implies that the pole of the propagator is also the case. We show the gauge (in)dependence of the pole in the positive energy region at with a particle number in the left panel of Fig. 5, which shows that the pole position is almost independent of the choice of the gauge parameter, as anticipated: Note that the gauge parameter is varied in a wider range than in Fig. 4, i.e., , , , , , . A remark is in order here: The pole in the negative energy region at has the same properties as that in the positive energy region, as is assured by Eqs. (14) and (15).

Such a gauge-independence of the poles necessarily reflects in that of the spectral function. The right panel of Fig. 5 shows the fermion spectral function at zero momentum for the wide range of up to , together with that obtained in the Proca formalism333 The rapid decrease of the spectral function in is caused by the exponential damping of in that region.. From this figure, we confirm that the spectral function at zero momentum is virtually gauge-independent for the wide range of .

We also note that the position and the width of the peaks coincide with the real and imaginary part of the poles, respectively, which is due to the fact that the imaginary part of the poles is small in comparison with the real part, as seen in the left panel of Fig. 5. Thus the shape of the spectral function with a three-peak structure necessarily gets to have almost no gauge-dependence.

We show the coupling constant dependence of the fermion pole at zero momentum in Fig. 6 for . The real part is almost proportional to , like that in QED in the HTL approximation. The coupling constant dependence of the imaginary part is not large.

What is the mechanism for realizing the three-peak structure of the fermion spectral function? Figure 7 shows the real and imaginary part of the self-energy for and at , together with the corresponding spectral function. A detailed analysis of the imaginary part tells us that the peaks of the imaginary part correspond to a Landau damping of the fermion by a scattering with thermally excited bosons. Since these features of the fermion self-energy is very similar to that shown in kitazawa (), the mechanism for realizing the three-peak structure found in our formalism is understood to be the same as discussed in kitazawa ().

iii.3 High temperature ()

In this subsection, we show numerical results in the high temperature () region, where the mass of the vector boson (and the fermion) can be neglected in comparison with , i.e., ; this means that itself may not be infinitely large.

We show the fermion pole in the positive energy region in the left panel of Fig. 8 at and for and . The pole in the Proca formalism and that in the HTL approximation in QED are also shown. We see that the gauge dependence of the fermion pole is no longer negligible. Since the exact pole position in the complex energy plane should be gauge-independent rebhan (), the above result suggests that the one-loop analysis is no longer valid in this high- region in contrast to the and regions, at least in some gauge. We will present a detailed discussion on how the gauge dependence arises at high region in Sec. IV.

One should notice that the pole for is located in the upper energy plane, which could be problematic because it implies a loss of the analyticity of the retarded propagator and also negativeness of the spectral function, as seen from Eq. (16).

Figure 8: The left panel shows the gauge dependence of the fermion pole at in the positive energy region at for : The horizontal and the vertical axis denote the real and imaginary part of energy. The right panel shows the fermion spectral function in the particle sector as a function of energy for the same and .

For , there appear clear two peaks in the spectral function in a robust way, as shown in Fig. 9; the two peaks are found to tend to the normal fermion(particle) and the anti-plasmino of QED in the HTL approximation frenkel-taylor (); braaten-pisarski (); weldon:1982aq (); weldon (), respectively; see Sec. IV.

There persists the other peak at the origin in the energy-momentum space. One can confirm that its residue is of the order of , which is very small if we consider the case, by making power counting. Such a peak at the origin was also obtained in kitazawa (), though in the Proca formalism. One should also remark that such a peak at the zero energy is not obtained in QED in the HTL approximation, in which the vector boson mass is set to zero from the beginning, in contrast to the present case444 The absence of a peak at the vanishing energy in QED with the HTL approximation is easily understood as follows: In the HTL approximation of QED, Re behaves as in the limit. Thus at , the pole condition Eq. (13) will not be satisfied and hence there can not exist a pole at the origin.. It should be intriguing to explore whether this peak at the origin extends to a finite- region, and hence the three-peak structure of the fermion spectral function persists even in such a high- region, i.e., for . In fact, this is a challenging problem in quantum field theory at finite temperature, because a sensible analysis of such an infrared region requires a systematic method to remove the so called pinch singularities pinchSingularity (). This task is beyond the scope of the present work, and we leave such an analysis as a future work persistency ().

Our numerical calculation has shown that one can have virtually gauge-independent results even in the one-loop analysis if the gauge parameter is in the region . We shall argue that the perturbative expansion should be valid for in Sec. IV. It means that the spectral function of a fermion coupled with a massive vector boson as calculated in the Stueckelberg formalism nicely approaches that in QED in the HTL approximation at high irrespective of the choice of the gauge parameter , if the order of is confined to . This is actually already suggested by the asymptotic form Eq. (27) for .

Figure 9: The fermion spectral function in the particle sector as a function of energy and momentum at with and .

Iv Analysis of Gauge dependence of the Pole at high temperature ()

Our numerical calculation has shown that the pole position of the fermion propagator is virtually independent of the gauge parameter for the cases of and : The former case is simply because the thermal contribution due to a boson with a mass is greatly suppressed by a Boltzmann factor when . By contrast, for , the numerical results in Sec. III.3 show that the pole of the fermion propagator has a large gauge dependence for large . In this section, we discuss the gauge dependence of the pole of the fermion propagator at weak coupling at high temperature. In particular, we focus on the region . In this region, one expect that the mass of the vector boson can be neglected, and thus the self-energy approach that in the HTL approximation of QED frenkel-taylor (); braaten-pisarski (); weldon:1982aq (); weldon (), in which the fermion has the pole of order . Therefore, we analyze the pole of the fermion propagator by assuming . Here we introduce a small dimensionless parameter,

(28)

Thus we have two small dimensionless parameters, and , which are treated as independent parameters, so that the self-energy is expanded by combined powers of and . If the the power of and are both positive, the high temperature limit will be well defined and smoothly connected to that of QED. However, as will be shown below, an inverse power of appears at one loop level when the gauge parameter is large, and hence the high temperature limit becomes inevitably different from that of QED.

In the following analysis, we put for simplicity. The pole position obtained in the perturbation theory generally depends on the gauge parameter as well as , and due to the truncation of the perturbative expansion. We parametrize the pole of the fermion propagator as

(29)

where is a function of order one, and depends on the gauge parameter . If the limit,

(30)

is independent of , then the pole is independent of the gauge parameter at the order . Thus one sees that the gauge dependent part may be defined by

(31)

where . For a reference, we recall that in the case of QED weldon (). When the inequality,

(32)

is satisfied, the gauge dependence can be neglected. In reality with a finite , the region of the gauge parameter satisfying Eq. (32) will be limited. We shall call the region that the gauge parameter satisfies Eq. (32) as an adequate gauge parameter region. The purpose of this section is to find the adequate gauge parameter region.

Let us first show a numerical result of the real and the imaginary part of the pole at as functions of in Fig. 10: For a large (), the dependence of the real part of the pole is large, and especially for very large , say , the magnitude of it is no longer of , but is of a smaller order, , as will be shown later. The imaginary part of the pole for is positive and apparently problematic because it means that the analyticity of the retarded propagator is lost and the fermion spectral function will become negative. As we shall show later, however, the absolute value of the imaginary part is of and should be considered together with higher order contributions. So the negative imaginary part with a small absolute value can be ignored in this order of the coupling.

Now we shall show that such an order estimate of the pole can be done analytically. We start with an analysis of the self-energy, under the condition that , by decomposing the self-energy (18) to seven parts,

(33)

where we have introduced the following dimensionless functions:

(34)
(35)
(36)
(37)
(38)
(39)
(40)

Here and are obtained by performing the analytic continuation () to and . From Eq. (13), one sees that the poles satisfy the condition

(41)
Figure 10: The upper panel shows the gauge dependence of the real part of the pole in the positive energy region at zero momentum at , for . The horizontal and the vertical axes denote the gauge parameter and the real part of the pole, respectively. The data whose is smaller than the upper limit of adequate gauge parameter, , are denoted by the solid line and the data whose is larger than by the dotted line. The data which satisfy , which are expected to approach the value in the Proca formalism, are plotted with another dotted line. The upper and lower dashed lines show the thermal mass in the HTL approximation and that in the Proca formalism , respectively. The lower panel shows the gauge dependence of the imaginary part of the pole for the same and as in the upper panel.
Figure 11: The classification of the gauge parameter regions depending on the relative magnitude of to at high temperature, i.e., for or . Each number in the figures corresponds to the case with same number in the text. The light gray regions are the adequate gauge parameter regions for which Eq. (32) is satisfied, whereas the dark gray region are not.

We first note that if the following equation has a root,

(42)

there is a pole of order . Furthermore, if the happens to be independent of , the pole is gauge independent. Therefore, let us take the following function as a measure of the gauge dependence, instead of Eq. (31):

(43)

Then a criterion for the adequate gauge parameter region may be given by

(44)

We now make an order estimate of the seven terms defined in Eqs. (34) (40). One finds that this task is reduced to that of and . The following relations are shown in Appendix C:

  • When ,

    (45)
  • When ,

    (46)
  • When ,

    (47)

Here, denotes or , and the vacuum parts have been dropped. Using these relations, we will find the adequate gauge parameter region in the following subsections, and the result is summarized in Fig. 11, which shows that the adequate gauge parameter is .

, , , and , which do not depend on the gauge parameter, are estimated to be

(48)

We remark that coincides with the fermion self-energy in the HTL approximation in QED, as it should be.

We note that and are of the order of an inverse power of , which would make it impossible to take the massless limit. These ‘dangerous’ terms are found to be nicely canceled out with other ’s when is not so large, whereas for large , the cancellation does not happen, and the condition can not be satisfied, as will be shown below.

iv.1 (case 1)

When , the mass scale and are negligible in comparison with , and then the self-energy coincides with that in HTL approximation in QED in this approximation, because the non-leading terms can be neglected as we have seen in Sec. II.

On account of the order estimate Eq. (45), we obtain

(49)

The leading terms of and cancel out555As seen from Eqs. (89) and (94), and yield terms which are proportional to , so , , , and seems to yield terms which are proportional to and the massless limit () can not be taken. Actually, from Eqs. (89) and (94), we see that the terms discussed above cancel out. Therefore we can take the massless limit and the fermion self-energy approaches that in QED. with and , respectively, and the terms of the order and remain666These terms come from the imaginary part of , , , and , which can be confirmed by retaining the next-to-leading term in Eqs. (82) and (90). The contribution from the real parts are much smaller than these terms. . Thus one sees that the gauge dependent part is of the order of , which means that the pole is gauge independent in practice, which also can be confirmed from Fig. 10, provided that the inequality is satisfied; in this case, the adequate gauge parameter region is the region of which satisfies the above inequality.

Let us see that the imaginary part of the self-energy is of order in the Landau gauge (). For , the imaginary part of the self-energy is evaluated to be

(50)

where the first line is obtained by substituting and in Eq. (63). In the second line the inequality was used. Although this is positive and apparently breaks the analyticity, this order of the coupling should not be determined in the one-loop order, because the two-loop diagrams contain contributions of order . In fact, the analyticity problem can be cured by taking into account the two-loop diagrams.

iv.2 (case 2)

iv.2.1 (case 2-a)

On account of the order estimate for , , and given in Eq. (46), we obtain

(51)

which is the same as Eq. (49). The leading term of cancels out with that of and the remaining terms are of the order of 777This term comes from Im, which can be confirmed from Eq. (105). The real parts are negligible compared with this term. , while and do not. In the present case, however, is larger than and , and hence dominates the gauge-dependent terms, which should be made small. This smallness is guaranteed when , which defines the adequate gauge parameter region; this coincides with that in the case 1. Notice that the inequality must be assumed in this case, otherwise the inequality can not be satisfied.

iv.2.2 (case 2-b)

Again, on account of the order estimate in Eq. (46), , , and are estimated to be

(52)

Though and have the same order of magnitude, they do not cancel out, which can be confirmed from Eqs. (95) and (101). Therefore the largest contribution is ; the present region for the gauge parameter () is not an adequate gauge parameter region.

iv.3 (case 3)

Here we treat the case where the gauge parameter is far larger than ; this case includes the unitary gauge (). Owing to the order estimate Eq. (47) for this case, , , and are estimated to be

(53)

Again, although and have the same order of magnitude, they do not actually cancel out on account of the difference in the coefficients, which can be seen from Eqs. (95) and (117). In this case, the largest contribution is ; therefore this region is not an adequate gauge parameter region. This also suggests that the order of the pole in the unitary gauge is not .

So let us now discuss the pole in the unitary gauge. First we assume the pole is of order instead of . Using Eqs. (47), (125), and (129), we have the following order estimates:

(54)

For , and vanish. Furthermore and can be also neglected, because . The remaining parts , and are estimated more precisely with the use of Eqs. (95) and (117), as follows:

(55)

Collecting these terms, we reach at

(56)

This is precisely the same as the result obtained in the Proca formalism in the high temperature limit kitazawa (); the thermal mass in this case reads

(57)

which can be seen from Fig. 10 also. We have confirmed numerically that the spectral function also approaches that in the Proca formalism as .

iv.4 Brief Summary

Let us summarize and discuss the results obtained so far in the preceding subsections for the gauge-parameter dependence of the fermion propagator at high temperature. The results are summarized in Fig. 11. We have found an adequate gauge parameter region as in which possible gauge dependence is of higher order of the couplings and hence can be neglected. We remark that this parameter restriction should also apply to QED.

In fact, the electron self-energy in the HTL approximation in QED at next-to-leading level in the one-loop order reads wang (); Mottola ()

(58)

If it were that and , the third term would be the same order as the leading term, which implies that the gauge-independence is badly broken. Conversely speaking, if is much smaller than , then the gauge-dependent part becomes of order and can be neglected; this gives an adequate gauge-parameter region for QED.

There is a difference between QED and the massive vector theory in the Stueckelberg formalism: The limit can not be taken in the former case because the third term in Eq. (58) diverges, while it can in the latter case. In this limit, the pole of the fermion propagator in the latter case becomes of order .

On the other hand, also small yields the problem; the imaginary part of the fermion self-energy becomes positive for , which breaks the analyticity of the self-energy.

The dependence of the self-energy can be understood intuitively as follows: In the Stueckelberg formalism, there are two masses: the physical mass and the unphysical mass . For , , the self-energy naturally approaches that of QED because both masses can be neglected. By contrast, if the unphysical mass is not smaller than , can not be neglected, although the temperature is enough high compared with the physical mass . In this case, the self-energy at one loop level does not approach that of QED.

We also note here that in the case, the boson mass can not be taken to zero from the outset, but the self-energy is approximately equal to that in the HTL approximation in QED as long as . This implies that the value, , is the upper limit of the boson mass that we can neglect.

Summarizing the situation, we see that there are three region of the gauge parameter: In the first region, the theory approaches QED, and is in adequate gauge parameter region. In the second region, the theory approaches QED, but is out of adequate gauge parameter region. In the third region, the theory does not approach QED, and is out of adequate gauge parameter region. We concludes that the gauge parameter should be chosen to be in numerical calculations in this formalism.

V Summary and concluding remarks

We have investigated the spectral properties of a fermion coupled with a massive vector boson in the whole temperature () region at one-loop order. The vector boson with a mass () is introduced as a gauge boson in the Stueckelberg formalism so that the high limit, or equivalently the massless limit in the sense that , can be taken888 As is mentioned in Introduction, the Proca formalism does not yield sensible results for proca-problem (); kitazawa (); dolan-jackiw (). : We have successfully analyzed and clarified the characteristics of the spectral properties of the fermion in the distinct three regions of , i.e. (I) ,  (II)  and (III)  regions, in a unified way. We have also carefully examined the possible gauge dependence of the spectral properties of the fermion in the respective three regions, separately.

In the region (I), the fermion spectral properties hardly change from those in the vacuum, which are gauge-independent. In the region (II), the fermion spectral function gets to have a three-peak structure in the small momentum region with supports in the positive, zero and negative energy regions; the three-peak structure becomes prominent when for . We have confirmed numerically that the fermion poles and hence the fermion spectral function shows virtually no dependence on the gauge parameter () for . It is thus natural that the similar three-peak structure of the fermion spectral function was obtained in the Proca formalism for the massive vector field for kitazawa (), since the Proca formalism exactly corresponds to the unitary gauge . Conversely speaking, the three-peak structure found in kitazawa () is not an artifact by a special choice of the gauge and is physical.

It is interesting that the spectral function of a fermion coupled with a scalar massive boson shows also a similar three-peak structure for at the one-loop level kitazawa (); mitsutani (). The present analysis has established that a fermion coupled with a massive boson with a mass has a three-peak structure for small momenta with supports in the positive, vanishing and negative energy regions at temperatures comparable with the boson mass, irrespective of the type of the boson, at the one-loop order.

For   (region (III)), the fermion spectral function tends to have distinct two peaks precisely corresponding to those seen in QED in the HTL approximation frenkel-taylor (); braaten-pisarski (); weldon:1982aq (); weldon (). It means that our formalism nicely describes the spectral function of the fermion coupled with a massive vector boson even in the high region. There is, however, a tricky point related to a possible gauge dependence. We have found that there exists an adequate region of the gauge parameter in the high- region for the perturbation theory at finite : If is of the order , the analysis at the one-loop order makes no problem and is reliable, keeping the positivity of the spectral function and so on; otherwise, however, these fundamental properties may be lost. This is because there exist two mass parameters, i.e., the vector boson mass and the ghost mass inherently in the Stueckelberg formalism. Thus the precise high- region should be defined by the two conditions, and . Our extensive analytic study has proved this observation and showed that when ( is the coupling constant), the one-loop analysis is reliable even in the region (III). Accordingly, if the unitary gauge () is adopted for the massive vector boson, the one-loop analysis can not be valid for , as is shown in a different context proca-problem (); dolan-jackiw ().

Our numerical calculation has shown that there still remains a peak at the origin in the - plane, though with a faint strength even in the region (III); this is in contrast to QED in the HTL approximation where such a peak is absent. Although it is an interesting possibility that the three peak structure persists at the high- region and even in QED, a sensible analysis of the spectral properties around such a low-energy region requires a resummed perturbation theory to deal with possible pinch singularities pinchSingularity (). Thus we leave an analysis of the spectral properties in the very low energy region as a future work and hope to report elsewhere persistency ().

We can think of some physical situations where the present analysis can be relevant, since massive vector bosons at finite appear in various physical systems. In QCD, vector bosons or vector-bosonic modes may decrease their masses in association with the restoration of chiral symmetry at finite rhomeson (). It would be not surprising if there exist vector bosonic modes even in the deconfined and chiral symmetric phase in the vicinity of the critical temperature , since the existence of other hadronic modes kunihiro (); detar (); shuryak () and bound states charmonium () are suggested in that temperature region. There may also exist a vector-type glue ball in such a system. Then the present analysis would suggest that the quark spectra can be largely affected in such a system where the boson mass is comparable to in the order of magnitude. In the electro-weak theory, the dispersion relation of neutrinos at high may possibly be affected by the weak bosons the masses of which change with boyanovsky (). One of the findings of the present analysis tells us that the one-loop analysis in the unitary gauge can not be applicable when , which includes the vicinity of the critical point.

Acknowledgments

We thank M. Kitazawa and M. Harada for helpful comments and discussions. This work was supported by the Grant-in-Aid for the Global COE Program “The Next Generation of Physics, Spun from Universality and Emergence” from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan and by a Grant-in-Aid for Scientific Research by the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan (Nos. 20540265, 1907797).

Appendix A The Correspondence Between the Stueckelberg Formalism and the Abelian Higgs Model

In this Appendix, we briefly show that the abelian Higgs model is reduced to the gauge theory with a massive gauge boson in the Stueckelberg formalism stueckelberg-review (); higgs-stueckelberg ().

The Lagrangian of the abelian Higgs model reads

(59)

with . Here and denote the vector and the Higgs field, respectively.

We fix the absolute value of the Higgs field and use the following polar representation:

(60)

with . We remark that the scalar field , which will turn to be identified with the Stueckelberg field, is introduced as the phase of