Aspects of breaking

in the Nambu and Jona-Lasinio model

A. A. Osipov^{1}^{1}1On leave from Joint
Institute for Nuclear Research, Laboratory of Nuclear
Problems, 141980 Dubna, Moscow Region, Russia.
Email address: ,
B. Hiller^{2}^{2}2Email address:
,
V. Bernard^{3}^{3}3
Email address: ,
A. H. Blin^{4}^{4}4Email address:

Centro de Física Teórica, Departamento de Física da Universidade de Coimbra,

3004-516 Coimbra, Portugal

Université Louis Pasteur, Laboratoire de Physique Théorique 3-5,

rue de l’Université, F-67084 Strasbourg, France

Abstract

The six-quark instanton induced ’t Hooft interaction, which breaks the unwanted symmetry of QCD, is a source of perturbative corrections to the leading order result formed by the four-quark forces with the chiral symmetry. A detailed quantitative calculation is carried out to bosonize the model by the functional integral method. We concentrate our efforts on finding ways to integrate out the auxiliary bosonic variables. The functional integral over these variables cannot be evaluated exactly. We show that the modified stationary phase approach leads to a resummation within the perturbative series and calculate the integral in the “two-loop” approximation. The result is a correction to the effective mesonic Lagrangian which may be important for the low-energy spectrum and dynamics of the scalar and pseudoscalar nonets.

PACS number(s): 12.39.Fe, 11.30.Rd, 11.30.Qc

##
1. Introduction

In the absence of a quantitative framework within QCD to deal with its large distance dynamics, the physics of hadrons is usually approached either through an effective field theory or through phenomenological parametrizations based on some simple ansatz with solid symmetry grounds. In the first case the theory is written in terms of mesonic degrees of freedom [1, 2]. In the second case the existence of underlying multi-quark interactions can be assumed. In this way one takes into account the quark structure of mesons explicitly, which may yield very important information about the structure of the QCD vacuum. Although these interactions are not renormalizable (as compared with CHPT, which is renormalizable in the sense of effective field theories), it is often possible to derive useful results in such models by introducing an ultraviolet cutoff.

Not much is known about the origin of multi-quark vertices. The semi-classical theory based on the QCD instanton vacuum [3] provides evidences in favor of these interactions: two or more quarks can scatter off the same instanton (or anti-instanton), certain correlations between quarks originate from averaging over their positions and orientations in color space, the result being an effective quark Lagrangian. Assuming a dominant role for quark zero modes in this scattering process one obtains -quark interactions ( is the number of quark flavors), which are known as ’t Hooft interactions [4]. Actually, an infinite number of multi-quark interactions, starting from the four-quark ones, has been found in the instanton-gas model considered beyond the zero mode approximation, all of them being of the same importance [5].

On the other hand, accurate lattice measurements for the realistic QCD vacuum show a hierarchy between the gluon field correlators with a dominance of the lowest one [6]. A similar hierarchy of the multi-quark interactions can be triggered after averaging over gluon fields. If this is true, there is an apparent contradiction with the instanton-gas model. This point has been stressed in [7].

The hierarchy problem of multi-quark interactions can be addressed on pure phenomenological grounds. For that we suggest to simplify the task considering only the four- and six-quark interactions. Once one knows the Lagrangian the obvious question arises: does the system possess a stable vacuum state and does this state correspond to our phenomenological expectations? If hierarchy takes place this question is pertinent for the leading four-quark interaction, because in this case the effective quark Lagrangian can be studied step by step in the hierarchy with the assumption that four-quark vertices are the most important ones. In the opposite case we must study the system as a whole to answer the question. For the best known and simplest example, to which we dedicate most of our attention here, the solution may be found analytically. As a result one can obtain definite answers to the above questions with a convincing indication in favor of a hierarchy for the considered example.

Our choice of model is not accidental. The importance of the four-fermion interactions has been recognized for many years, starting in the early sixties, when Nambu and Jona-Lasinio (NJL) [8] used it for studying dynamical breaking of chiral symmetry. Later on a modified form of this interaction, written in terms of quark fields, has been used to derive the QCD effective action at long distances [9, 10, 11].

The six-quark vertices contain additional information about the vacuum [12]. They break explicitly the symmetry and are the only source of OZI-violating effects [13]. Taken together with the NJL interactions, the ’t Hooft Lagrangian gives a good description of the pseudoscalar nonet, especially the and masses and mixing [14, 15]. In this form the model has been widely explored at the mean-field level [16].

Let us discuss the Lagrangian which we shall use in our analysis. On lines suggested by multicolor chromodynamics it can be argued [17] that the anomaly vanishes in the large limit, so that mesons come degenerate in mass nonets. Hence the leading order (in counting) mesonic Lagrangian and the corresponding underlying quark Lagrangian must inherit the chiral symmetry of massless QCD. In accordance with these expectations the symmetric NJL interactions,

(1) |

can be used to specify the corresponding local part of the effective quark Lagrangian in channels with quantum numbers . The Gell-Mann matrices acting in flavor space, obey the basic property .

The ’t Hooft determinantal interactions are described by the Lagrangian [4]

(2) |

where the matrices are projectors and the determinant is over flavor indices.

The coupling constant is a dimensional parameter () with the large asymptotics . The coupling , , counts as and, therefore, the Lagrangian (1) dominates over at large . It differs from the counting , which one obtains in the instanton-gas vacuum [5].

It is assumed here for simplicity that interactions between quarks can be taken in the long wavelength limit where they are effectively local. The ’t Hooft-type ansatz (2) is a frequently used approximation. Even in this essentially simplified form the determinantal interaction has all basic ingredients to describe the dynamical symmetry breaking of the hadronic vacuum and explicitly breaks the axial symmetry [18]. The effective mesonic Lagrangian, corresponding to the non-local determinantal interaction, has been found in [19].

Anticipating our result, we would like to note that if the hierarchy
of multi-quark interactions really occurs in nature, the perturbative
treatment seems adequate. The NJL interaction alone has a stable
vacuum state corresponding to spontaneously broken chiral symmetry.
But, as we shall show, the effective quark theory based on the
Lagrangian^{5}^{5}5The other multi-quark terms have been
neglected here.

(3) |

has a fatal flaw: if is comparable with , it has no stable ground state. This feature of the model is invisible in a perturbative approach in . The situation is exactly analogous to the problem of a harmonic oscillator perturbed by an term. This system has no ground state, but perturbation theory around a local minimum does not know this. In eq.(3) the current quark mass, , is a diagonal matrix with elements , which explicitly breaks the global chiral symmetry of the Lagrangian.

There is also a special problem related with the bosonization of multi-quark interactions. To bosonize the theory one introduces auxiliary bosonic variables to render fermionic vertices bilinear in the quark fields. This procedure requires twice more bosonic degrees of freedom than necessary [15]. Redundant variables must be integrated out and this integration is problematic as soon as one goes beyond the lowest order stationary phase approximation [20, 21]: the lowest order result is simply the value of the integrand taken at one definite stationary point [18]. In this paper we shall show for the first time how to extract systematically the higher order corrections which contribute to the effective mesonic Lagrangian, and what to do with infinities contained in these corrections. The model (3) is considered to illustrate our calculations. If the coupling is not too small, corrections can be much larger than one might expect from counting, and be important for the mesonic () mass spectra and their dynamics. The large value of the mass difference obtained already at leading order and totally determined by the six-quark interactions gives some credit to large corrections. Additionally, one can expect some enhancement at next to leading order by virtue of divergent factors: the cutoff scale is not known beforehand, and can be relatively large.

Our study represents a very simplified view on the matter and should be considered as a first rough estimate which can be improved if necessary. The most essential approximation made here is related with the local character of the multi-quark vertices in eq.(3). It leads to -singularities beyond the mean-field framework (see also [20, 21]). Nevertheless, if it turns out that the low-energy QCD vacuum contains the hierarchy of multi-quark interactions, our approach can be used as a basis for a more serious work in this direction.

The paper is organized as follows. In Section 2 the functional integral representation, , for a bosonized version of the NJL model with ’t Hooft interactions is derived. To make clear the approximations which will be used for the functional integration, a similar non functional integral, , is considered in Section 3. The reader who just wants to get a general idea of what we intend to study, can find it here. This section contains several clarifying points which are important in the following. In subsection 3.1 we give the exact result for . Its stationary phase asymptotics is obtained in subsection 3.2. The semi-classical method yields the same result, as shown in subsection 3.3, as does the method presented in subsection 3.4. The point of these numerous calculations is just to show that all methods considered lead to the same asymptotics for the integral which is totally determined by the number of stationary points. The perturbative treatment is given in subsection 3.5. Here we discuss why the perturbation theory result is essentially different from the asymptotics obtained in the previous subsections, even for very small value of the expansion parameter. We resum the perturbative series in subsection 3.6. Calculating the next to leading order correction we illustrate the main goal of our studies in the forthcoming sections.

In Section 4 (subsection 4.1) we show that the chiral symmetry group imposes strong constraints which are only compatible either with the perturbative approach, or the expansion in a parameter that multiplies the total Lagrangian density (the loop expansion). Otherwise, as the consistent stationary phase treatment shows (subsection 4.2), the model is unstable.

The first alternative is considered in Section 5. To evaluate the functional integral , we consider it like the natural infinite-dimensional limit of ordinary finite-dimensional Gaussian integrals (subsection 5.1). The perturbative treatment essentially simplifies calculations. Nevertheless, the integration over auxiliary variables leads to a special problem with -singularities. We discuss this aspect of the bosonization in subsection 5.2. Another way to obtain the result is shown in subsection 5.3. Here a more elaborate spectral representation method is used to justify our computations. This subsection contains the prescriptions for the regularization of the infinities, and a discussion of their reliability.

The second alternative (the loop expansion) is considered in Section 6. Here, in subsection 6.1, we obtain in closed form the two-loop contributions to the functional integral and give arguments to justify this result. In subsection 6.2 we end with short conclusions and suggest future applications of our result.

The summary is given and outlook is surveyed in Section 7.

The Appendix contains the definition and the main properties of Airy’s functions.

##
2. Bosonization

The many-fermion vertices of Lagrangian (3) can be linearized by introducing the functional unity [15]

(4) | |||||

in the vacuum-to-vacuum amplitude

(5) |

We consider the theory of quark fields in four-dimensional Minkowski space. It is assumed that the quark fields have color and flavor indices which range over the set . The auxiliary bosonic fields, , and, become the composite scalar and pseudoscalar mesons and the auxiliary fields, , and, , must be integrated out.

By means of the simple trick (4), it is easy to write down the amplitude (5) as

(6) |

with

(7) |

(8) |

(9) |

We assume here that , and so on for all auxiliary fields . The totally symmetric constants are related to the flavor determinant, and equal to

(10) | |||||

We use the standard definitions for antisymmetric and symmetric structure constants of flavor symmetry. One can find, for instance, the following useful relations

(11) |

At this stage it is easy to rewrite eq.(6), by changing the order of integrations, in a form appropriate to accomplish the bosonization, i.e., to calculate the integrals over quark fields and integrate out from the unphysical part associated with the auxiliary bosonic variables ()

(12) | |||||

where

(13) | |||||

(14) |

The Fermi fields enter the action bilinearly, thus one can always integrate over them, since one deals with a Gaussian integral. One should also shift the scalar fields by demanding that the vacuum expectation values of the shifted fields vanish . In other words, all tadpole graphs in the end should sum to zero, giving us the gap equation to fix the constituent quark masses corresponding to the physical vacuum state.

The functional integrals over and

(15) |

are the main subject of our study. We put here , and is chosen so that .

Let us join the auxiliary bosonic variables in one -component object where we identify and ; run from to independently. It is clear then, that . Analogously, we will use for external fields and .

Next, consider the sum . If we require

(16) |

we find after some algebra

(17) |

with the following important property to be fulfilled

(18) |

Now it is easy to see that the functional integral (15) can be written in a compact way

(19) |

where

(20) |

We have arrived at a functional integral with a cubic polynomial in the exponent.

## 3. Digression to a one dimensional case

To get a rough idea of how to evaluate the integral (19) we start with its one-dimensional analog

(21) |

where

(22) |

and are constants. This integral plays the same role as our desired functional one, but is well defined as an improper Riemann integral of the real variable .

### 3.1 The exact result

The integral (21) can be evaluated exactly. To show this let us express the polynomial in the form

(23) |

where is chosen to satisfy the following equation

(24) |

The coefficients are

(25) |

Hence the integral (21) is given by

(26) |

We can rewrite eq.(26) in terms of the new variable

(27) |

thus we arrive at

(28) |

where is defined to be

(29) |

This integral is well known. The result of integration can be represented in terms of the Airy function (see Appendix for details)

(30) |

If is real, the function is also real, and the phase of is equal to . In the following, after some generalizations made for the integral , its phase will represent the effective action of a dynamical system. The expression for this phase is the main goal of our calculations.

### 3.2 The stationary phase result

The result (30) for is exact. It can be approximated at large values of by its asymptotic series, still giving us an exact expression for the phase. To obtain asymptotics let us transform eq.(28) to a more convenient form by the replacement

(31) |

One has

(32) |

where . The integral (32)
is already in the form appropriate for the stationary phase method
to be applied. Indeed, as the term
gives a rapidly oscillating contribution to
the integrand in eq.(32) which cancels out, except at the
regions of critical points. The contribution from these regions can
be evaluated on the basis of the stationary phase method. In our case
, , , i.e.,
and large arguments^{6}^{6}6All large
arguments given in this section hint of course at the functional
integral case which will be discussed later.
can be used to
justify the limit. Both critical points
() belong to the interval of
integration. Thus we have

(33) | |||||

The last term in both exponents can be factorized and then expanded in a power series of

(34) | |||||

and integrated term by term. The corresponding integrals are evaluated exactly

(35) |

giving as a result the following asymptotics for the integral

(36) |

Let us stress that both critical points contribute to the resulting asymptotic series reproducing the well known asymptotics of the Airy’s function (155). We have considered here the case when the series oscillates. In the opposite case, , the asymptotics falls down exponentially (154), and does not have the appropriate form from the physical point of view.

### 3.3 Semi-classical asymptotics

One can calculate the integral (21) assuming that a large parameter is already present in the exponent. For instance, the reduced Planck constant, , can be considered as such a parameter

(37) |

In this case one obtains the asymptotic expansion for at small values of . This approach is known as the semi-classical expansion.

The real function has two critical points

(38) |

Both of them are real at and, therefore, belong to the contour of integration. In the neibourhood of these points the polynomial is conveniently written in the form

(39) |

where

(40) |

Thus the integral under consideration is estimated as

(41) |

or

(42) |

Noting that

(43) |

one can see that this result coincides with our previous estimate (36).

It may be helpful to remark that if one would take the contribution of only one critical point in (41), the resulting asymptotics would be obviously different, and, what is important for us, the phase too.

### 3.4 The asymptotics

The representation (39) can be used as a first step to estimate the integral without any reference to the semi-classical expansion. Alternatively, our calculations can be based on the large asymptotics. According to the formula (39), we have identically

(44) |

This holds whether or (see eq.(38)) is chosen here.

Next, replacing the variables

(45) |

we arrive at

(46) |

where we have used the following notations

(47) |

This form of the integral is already appropriate to apply the stationary phase method at large . Large arguments can be used to justify the limit, for it is known that .

Let us obtain the leading term in this asymptotics. The critical points are given by the equation

(48) |

Therefore, the integral has the following asymptotical estimate at

(49) | |||||

The first term of the integrand is the contribution of the critical point , the second one comes due to . Noting, that

(50) |

one has

(51) | |||||

It is now obvious that the integrand does not depend on , thus

(52) | |||||

One can see finally that the obtained asymptotical series

(53) |

coincides with our previous result (42).

### 3.5 The perturbation theory approach

Our asymptotic estimate requires a further explanation in what concerns the definition of the improper Riemann integral (21). It is understood as the limit

(54) |

where we assumed that both stationary points (see eq.(38)) belong to the interval ; the asymptotics of was formed accordingly by two independent contributions: . Integrating in , we took into account that only a neighbourhood of a stationary point is important, and extended the integration limits to .

Let us suppose now that the coupling is small, as compared with , and one can consider the limit . It is easy to see that

(55) |

at small . The first solution is regular at , but the second one shows the singular behavior

(56) |

This behavior reflects the increasing importance of the stationary point in the physical problem for which the cubic term is considered as a perturbation. The second stationary point, , varying as , can finally leave the interval , changing as a result the asymptotics of the integral to

(57) |

The intuitive argumentation given above must be clarified. One should not think that the real value of is very important for the matter. Actually, one excludes the singular critical point from the phase by directly expanding the phase in the neighbourhood of a regular stationary point, as we already did it for . What is really important here is to conclude that only the regular critical point determines the perturbative regime of the system. According to this attitude, we use the term “perturbative regime” in a wide sense: the standard perturbative expansion in powers of , which we will obtain below, is merely another way of looking at the one critical point asymptotics of .

Let us separate the unperturbed part from the perturbation in the integral

(58) |

Expanding the integrand in powers of the coupling , we obtain

(59) | |||||

The new feature is the fact that the integrand of has only one stationary point, , as opposed to . The singular critical point is gone.

The evaluation of the Gaussian integral is straightforward

(60) |

This reduces our integral to the form

(61) | |||||

Up to the terms of order we have now

Noting, that

one can represent this result in the more convenient form (up to the same order of accuracy)

(63) |

where

(64) |

The perturbative result is an approximation, which can be systematically improved. In its truncated form the obtained series differs from the expansion which one can obtain from (see next subsection for details). It is clear, however, that if one sums up the series, the perturbative and asymptotic estimates will coincide, i.e., .

### 3.6 Resumming the perturbative series

Let us calculate the contribution to the integral which comes from the critical point in eq.(34). It is not difficult to find out that exactly this point represents the regular solution at . For this part of the integral we will use the symbol , as we did before for ,

(65) | |||||

It is clear that in the last sum only the even powers of contribute. For the first two terms () one obtains

(66) |

Since

(67) |

we can rewrite this result in the compact form

(68) |

where

(69) |

(70) | |||||

One can explicitly see that the series in powers of obtained
from the integral perfectly coincides^{7}^{7}7Actually, we
checked the terms up to and including order.
with the perturbative expansions (63) and (64).

The series (68) corresponds to a partial resummation of the perturbative series. In order to see what is resummed, let us introduce a parameter according to the substitution: