Elimination of the Landau Ghost from Chiral Solitons
Abstract
We show a practical way based on the KällénLehmann representation for the twopoint functions to eliminate the instability of the vacuum against formation of small sized meson configurations in the chiral model.
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1 Introduction
Since QCD is presently intractable at low energies, approximation schemes based on effective models have been introduced [1]. These models in a way incorporate some important features of QCD, like chiral symmetry and its spontaneous breakdown.
Though being approximative, these effective models still cannot be treated exactly. Therefore we have essentially two levels of approximation: i) when we choose the model itself and ii) when we specify an approximation scheme in order to perform actual calculations within the model. This implies that the results may depend on the actual approximation scheme. Therefore, a failure of the model could be due to the model itself, due to the approximation scheme or both. So, if we want to test the model, we should be careful about the approximation scheme and keep the dependence on ad hoc parameters as small as possible.
One of the effective models of QCD to describe the nucleon is the chiral model [2]. The standard procedure to calculate nucleonic properties in the soliton picture is to use the loop expansion up to the one quark loop level, i.e; to integrate out the quark degrees of freedom and use the remaining purely mesonic effective action at the zero loop level in the mesonic fields [3]. One can choose some cutoff procedure to regularize the quark determinant [4], but in any case one has to renormalize by introducing appropriate counter terms in order to define the parameters appearing in the original Lagrangian of the model [5]. Eventually one can let the cutoff go to infinity as it is usually done in renormalization theory, or leave it finite. We note that in the model, however, neither the cutoff scheme itself nor the size of the cutoff can be determined uniquely.
It is known, however, that the loop expansion for asymptotically non free theories without ad hoc cutoff parameters displays an unphysical pole in the corresponding propagators, the so called Landau ghost [6]. Therefore, if one follows the procedure outlined above, one finds that the renormalized model is plagued by the Landau ghost, which leads to an instability of the usual translational invariant vacuum [7, 8]. In view of our remarks at the beginning of this section, the problem of this Landau ghost can be treated in several ways:

The effective model is considered as unphysical and is abandoned.

The approximation scheme (loop expansion) is considered appropriate, but the effective model can be used only with a regularization scheme, hereby being restricted to energies far below the unphysical pole.

The approximation scheme is improved such that the effective model does not a priori yield unphysical results for energies up to the order of the baryon mass.
The second viewpoint implies that vacuum loops are either simply discarded, or calculated with finite cutoff. Concerning the first alternative (drop vacuum loops), we note that in some situations, like in an external field, such a treatment violates conservation laws [9] and cannot be implemented consistently. Besides this, the investigation of the change of the vacuum structure due to the finite density should be an important subject for any relativistic field theory. Concerning the second alternative (finite cutoff), we note that the Landau ghost in hardonic theories occurs at rather low energies of the order of the baryon mass (). This means that its presence is a real problem for effective quark theories, and its avoidance by the introduction of an ad hoc parameter is somewhat unsatisfactory. Since the energy scale relevant for effective quark theories is set by hadronic masses, a treatment according to the third viewpoint, which we will adopt in this paper, is a feasible alternative. We consider the Landau ghost as one of the basic infinities in effective field theories which should be brought under control without the introduction of further parameters. In a different context, this viewpoint has been taken already a long time ago by Redmond and Bogoliubov et al [10] who have shown how to construct ghost free propagators in the framework of the loop expansion based on the requirement of the KällénLehmann representation. Their propagators have several interesting features: The associated wave function renormalization constant is finite, and they have an essential singularity at ( is the coupling constant) which is expected from more intuitive physical arguments [11]. Moreover, since the associated running coupling constant has no singularity at spacelike momentum transfers, the ’zero charge problem’ is avoided. Their method has recently been formulated in the language of the effective action [12], which is suitable for effective quark theories, and also applied to infinite systems [12, 13]. The main purpose of the present paper is to show a practical way to incorporate the ghost elimination in a finite system like the soliton, and to demonstrate that with this procedure the translational invariant vacuum becomes stable with respect to decay into an array of small sized configurations. We will not construct the fully self consistent solutions including valence quarks in this paper which is left for a future work. (For another possibility to avoid the Landau ghost by introducing vector bosons, see ref. [14].)
The model has so far not yet been fully exploited to obtain self consistent solutions including the effect of the Dirac sea. One of our motivations for investigating this model is as follows: In contrast to the NambuJonaLasinio (NJL) model [15], which has been extensively used to construct solitonic solutions [16, 17, 18], it has a interaction term with variable strength (specified by the mass) which controls the deviation of the chiral radius from its vacuum value. Such a term would contribute to the ground state energy the more the farther the chiral fields are away from the chiral circle of the vacuum. Since it is known that interactions terms of the form [19] or of higher order like the t’Hooft determinant interaction [17] can prevent the collapse of the NJL soliton, there is the possibility that stable solitons exist for the linear model. The major obstacle to a full investigation of the solitonic sector of the model, however, was the instability of the translational invariant vacuum. The fact, to be established in this paper, that the method of Redmond and Bogoliubov et al leads to a stabilization of the translational invariant vacuum is a first step towards self consistent solutions in the model.
The rest of the paper is organized as follows: To be self contained and to set the formalism we state in sect. 2 some results for the ground state energy in the model obtained earlier [7], and explain the role of the Landau ghost in a way which seems most transparent to us and which is appropriate for implementing the ghost elimination procedure. In sect. 3 we construct a ghost free model and show results for the ground state energy, and in sect. 4 we summarize our results.
2 The ground state energy and the Landau ghost
We now proceed to exhibit how the Landau ghost contributes to the vacuum instability. The Euclidean Lagrangian of the model reads, after introducing the ’physical’ parameters and instead of the original ones and [2]:
(1) 
with
(2)  
(3) 
Here stands for the u and d quarks, is the effective quark mass, is the chiral field, and with the pion decay constant. The model contains the two free parameters (or ) and . is the vacuum expectation value of , and in the symmetric limit () we have [2]. Our Euclidean metric is such that with , with the upper limit of Euclidean time integration, and with the usual Dirac matrix. Defining the effective bosonic action from the generating functional as usual [3] by we obtain after integrating out the quark fields
(4) 
with [7]
(5)  
(6) 
Here where is the number of colors and refers to the Dirac and isospin indices, , and
(7) 
The integrals over the Euclidean time extend from 0 to . The counter terms (c.t.) in (5), which include the subtraction of the translational invariant vacuum contribution (), are determined such that gives no terms of the same form as those already present in . This corresponds to the mesonic mass and wave function renormalization at the renormalization point [5]. From here on we consider the chiral symmetric case ().
The vacuum instability can be seen by expanding the ground state energy in powers of a characteristic length scale where the classical meson fields are localized. We assume time independent fields with . If we use as the integration variable, every derivative gives rise to a factor , and therefore terms involving derivatives will be dominant for small R compared to those without derivatives. Therefore, up to contributes only the kinetic term. To obtain the lowest order contribution from we expand the logarithm in (5) and note that the term linear in is cancelled by a counter term. Then for small R the leading term is the one quadratic in the derivatives, i.e; quadratic in the first term of eq. (7). From the above we see that the linear and the non linear model look the same at small sizes. The complete term of order is
(8) 
Let us consider eq.(8) in a continuous and infinite plane wave (PW) basis , where is the volume. Then the matrix element equals , i.e; the Fourier transform of with . Using the dimensionless variables and and keeping for only the first term in eq.(7) we obtain for (8) in the continuum limit [7]
(9) 
with
(10) 
and is the Fourier transform of :
(11) 
Adding the kinetic term, we get the leading contribution to the effective action in the small size expansion as
(12) 
Since
To see that the Landau ghost in the meson propagators is responsible for this vacuum instability, we use the fact that the effective action, when expanded around the fields in the translational invariant vacuum (), gives the inverse and propagators in the translational invariant vacuum as the coefficients of the terms of second order in and . If we express of eq. (7) in terms of using , we see that these second order terms are completely contained in (8). Taking also the terms of second order in in the mesonic term in (6), we obtain [13]
(13) 
with the inverse Euclidean SchwingerDyson propagators
(14)  
(15) 
In fig. 1 we show the inverse pion propagator (14) for by the dashed line. The previously noted behaviour of leads to the Landau ghost pole at Euclidean , and as a consequence (13) can become more and more negative for large , i.e; for small sized fields. In this limit of small sized fields the leading contribution to (13) comes from the terms in eqs. (14) and (15), and if expressed in terms of this gives again eq. (9). [Note that with the Fourier transform of and similar for .]
3 Ghost removal in a finite system
We now address the problem of removing the Landau ghost in a finite system. The procedure is described in ref. [10] and has been extended to the path integral formalism in ref. [12]. To calculate the ground state energy we have to first obtain meson propagators with the correct analytical properties. These propagators are then implanted into the effective action which is in turn used to calculate the ground state energy.
The prescription of ref. [12] to eliminate the Landau ghost from the effective action is to replace the SchwingerDyson propagators in the second order term (13) by the KällénLehmann (KL) propagators . According to ref. [10], these are constructed from the KL representation using a spectral function obtained from the oneloop meson self energy, and are free of the Landau ghost. Thus, in this method the loop approximation is is used only for the imaginary part, while the real part is calculated from the dispersion relation. Since this method avoids the Landau ghost, it clearly represents an improvement of the straight forward loop approximation for the whole propagator. The new effective action becomes
(16) 
In order to be consistent with chiral symmetry, however, this ghost subtraction has to be done under the constraint of the Ward identity [2] with the vertex where one of the external pions has zero momentum. To preserve this identity the difference of the and inverse propagators has to remain invariant under ghost subtraction [13], i.e; , and we obtain from (16) and (17)
(17) 
It can be shown [10] that the KL propagator can be obtained from the SD propagator by subtracting the ghost pole: , where is the residue of the pole at Euclidean . We show for and by the solid line in fig. 1.
The ghost free ground state energy is thus obtained by adding the piece (17) to with given by eq. (4). The exact evaluation of , given by eq. (5), requires the diagonalization of the Dirac hamiltonian , and this is done most conveniently in a basis which diagonalizes the free Dirac hamiltonian : with in a box of size D with discrete momenta . (Here labels all necessary quantum numbers except colour.) D and should be taken large enough such that all results are unchanged under further increase. We will call this discrete and finite basis the KahanaRipka (KR) basis [20]. On the other hand, the ghost subtraction is formulated most conveniently in the continuous and infinite PW basis, for which the result is given by eq. (17). Therefore, we proceed in two steps: First, in order to demonstrate that the KR basis gives the same results as the PW basis provided that and are taken large enough, we evaluate in the small size approximation both in the KR and the PW basis and compare the results. Then we evaluate exactly in the KR basis and add the ghost subtraction term (17).
The expression for in the small size approximation using the PW basis has already been given in eq. (9), and the corresponding result for the KR basis is obtained from eq. (8) as
(18) 
Since for the exact calculation one has to diagonalize , it is convenient here to treat as where is the square of a finite dimensional matrix. In fig. 2 we show the smallsize approximation to the Fermion loop energy, based on eq. (9) (solid line) and eq. (18) (dashed line) in units of as a function of , assuming a Hedgehog profile with the vacuum value for the chiral radius and winding number :
(19) 
(In this case there are no additional terms in (9) due to . Also, or , viewed as a function of , is independent of (or ). When the KR basis is used, the effective action becomes a function of , and in order to reach convergence () in the smallsize approximation we had to go up as high as for . This is in contrast to models with finite cutoff like the NJL model, where is sufficient for solitons of normal size [17]. The calculations in this paper are performed with values for ranging from (70,5.7) for small to (20,20) for large , such that is kept constant. Fig. 2 demonstrates that the results using the PW and the KR basis agree well if and also the box size are taken large enough.
Comparison with the full calculation, to be discussed below, shows that the result of fig. 2 is quite accurate up to . The fact that the ground state energy relative to the translational invariant vacuum is negative for these small R indicates the vacuum instability due to the Landau ghost discussed above.
The total ground state energy in the small size approximation is shown in fig. 3. It is obtained from eq. (12), and for the ghost free model by further adding the piece (17). We see that the ground state energy in the small size approximation becomes a positive quantity after removal of the Landau ghost.
We now return to the full effective action, eqs. (5),(6). In the KR basis can be written in the form [7]
(20) 
with as before, and are the eigenvalues of h. Since we have already demonstrated for the small size expansion that for and large enough the results with the KR basis agree well with those of the PW basis, it is feasible to use the expression in the PW basis, eq. (17), for the ghost subtraction term .
Fig. 3 shows the total ground state energy including the ghost (dashed line) and without the ghost (solid line) in units of as a function of for the Hedgehog meson profile (19). After ghost subtraction, the total ground state energy relative to the translational invariant vacuum is positive, which demonstrates that amending the twopoint functions such that they satisfy the KL representation is sufficient to stabilize the vacuum against the formation of arrays of strongly localized meson field configurations.
4 Summary
In effective quarkmeson theories, which have been designed to model QCD in the energy region of hadron masses, the Landau ghost appears in the meson propagators at rather low energies of . Its presence is a real problem for these theories and leads to an instability of the translational invariant vacuum. In this paper we have taken the viewpoint that the Landau ghost should be brought under control without the ad hoc introduction of further parameters. Based on the work of Redmond and Bogoliubov et al [10], who have shown how one can improve the loop expansion such that the propagators have the correct analytical properties, we have shown a practical way to eliminate the Landau ghost from chiral solitons in the model. We have demonstrated that the ghost free effective action formalism of ref. [12] can be applied successfully to finite systems. In particular, since the ghost subtraction term is expressed most conveniently in a continuous and infinite plane wave basis, we have investigated in detail the conditions under which it is feasible to add this term to the Fermion loop term which is evaluated in a discrete and finite basis. By considering vacuum configurations characterized by various sizes of the meson profiles we have shown that in our ghost free model the translational invariant vacuum has the lowest energy and is therefore stable with respect to decay into small sized configurations.
The method given in this paper can form the base for further investigations on self consistent soliton solutions in the model. In particular, as mentioned in the Introduction, it would be interesting to see whether self consistent solutions can be obtained with a reasonable strength of the term.
The problem of the Landau ghost and the associated vacuum instability occurs also in relativistic mesonnucleon theories for nuclear structure [21]. For example, if a finite nucleus is described in the relativistic Hartree approximation including the effect of the Dirac sea, one has to apply the same ’overall ghost subtraction’ as performed in this paper (see eq. (16)). In more sophisticated approximations like the ’1/N’ expansion [22] or the ’modified loop expansion’ [23] the Landau ghost appears also in the subgraphs (even for infinite systems), and the ghost subtraction must be applied to these subgraphs. Thus, although we used the language of effective quark theories in this paper, the method is applicable to relativistic nuclear structure physics as well.
References

[1]
E. Witten, Nucl. Phys. B 233 (1983) 422; 433
H.B. Nielsen and A. Pathkos, Nucl. Phys. B 195 (1982) 137
M. Betz and R. Goldflam, Phys. Rev. D 28 (1983) 2848. 
[2]
M. GellMann and M. Lévy, Nuovo Cimento 16
(1960) 705
B.W. Lee, Chiral Dynamics, Gordon and Breach, New York, 1972. 
[3]
T. Eguchi, Phys. Rev. D 14 (1976) 2755
I.J. Aitchison and C.M. Fraser, Phys. Lett. B 146(1984) 63; Phys. Rev. D 31 (1985) 2605.  [4] W. Broniowski and M. Kutschera, Phys. Lett. B 242 (1990) 133.
 [5] J. Mignaco and E. Remiddi, Nuov. Cim. 1 A (1971) 376.
 [6] L. D. Landau, in Niels Bohr and the development of physics, ed. W. Pauli (Pergamon, London, 1955).
 [7] G. Ripka and S.Kahana, Phys. Rev. D 36 (1987) 1233.

[8]
R. Perry, Phys. Lett. B 199 (1987) 489
V. Soni, Phys. Lett. B 183 (1987) 91
T.D. Cohen, M.K. Banerjee and VY. Ren, Phys. Rev. C 36 (1987) 1653
R.J. Furnstahl and C.J. Horowitz, Nucl. Phys. A 485 (1988) 632
T. Kohmura, Y. Miyama, T. Nagai, S. Ohnaka and J.Da Providência, Phys. Lett. B 226 (1989) 207
T. Mannel, T. Ohl and P. Manakos, Z. Phys. A 335 (1990) 341.  [9] S. Ichii, W. Bentz, A. Arima and T. Suzuki, Phys. Lett. 192 (1987) 11.

[10]
P.J. Redmond, Phys. Rev. 112 (1958) 1404;
N.N. Bogoliubov, A.A. Logunov and D.V. Shirkov, Sov. Phys. JETP (Engl. Transl.) 37 (1960) 574.  [11] F.J. Dyson, Phys. Rev. 85 (1952) 631.

[12]
K. Tanaka, W. Bentz, A. Arima, F. Beck, Nucl. Phys. A 528
(1991) 676
K. Tanaka and W. Bentz, Nucl. Phys. A 540 (1992) 383.  [13] M. Kato, W. Bentz and K. Tanaka, Phys. Rev. C 45 (1992) 2445.
 [14] S.H. Kahana and G. Ripka, Phys. Lett. 278 (1992) 11.

[15]
Y. Nambu and G. JonaLasinio, Phys. Rev. 122 (1961) 345,
124 (1961) 246
U. Vogl and W. Weise, Prog. Part. Nucl. Phys. 27 (1991) 195. 
[16]
P. Sieber, Th. Meissner, F. Grümmer and K. Goeke, Nucl.
Phys. A547 (1992) 459
T. Watabe and H. Toki, Progr. Theor. Phys. 87 (1992) 651
J. Schlienz, H. Weigel, H. Reinhardt and R. Alkhofer, Phys. Lett. B 315 (1993) 6.  [17] M. Kato, W. Bentz, K. Yazaki and K. Tanaka, Nucl. Phys. A 551 (1993) 541.
 [18] N. Ishii, W. Bentz and K. Yazaki, Phys. Lett. B 301 (1993) 165; B 318 (1993) 26.

[19]
Th. Meissner, G. Ripka, R. Wünsch, P. Sieber,
F. Grümmer and K. Goeke, Phys. Lett. B 299 (1993) 183
C. Weiss, R. Alkhofer and H. Weigel, Mod. Phys. Lett. A 8 (1993) 79.  [20] S. Kahana and G. Ripka, Nucl. Phys. A 429 (1984) 462.

[21]
B.D. Serot and J.D. Walecka, Adv. in Nucl. Phys. 16,
ed. J.W. Negele and E. Vogt, Plenum, 1986,1
B.D. Serot, Rep. Prog. Phys. 55 (1992) 1855.  [22] K. Tanaka, W. Bentz and A. Arima, Nucl. Phys. A 555 (1993) 151.
 [23] K. Wehrberger, R. Wittman and B.D. Serot. Phys. Rev. C 42 (1990) 2680.
Figure caption

The inverse pion propagator with for Euclidean . The dashed line shows the SchwingerDyson propagator and the solid line the KällénLehmann propagator.

The total energy in units of as a function of . The dotted and dasheddotted lines refer to the small size expansion before and after the ghost subtraction, respectively, and the dashed and solid lines show the full result before and after the ghost subtraction, respectively.