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We review the Nambu and Jona-Lasinio model (NJL), proposed long time ago, in the sixties, as a fermion interaction theory with chiral symmetry. The theory is not renormalizable and presents a symmetry breaking due to quantum effects which depends on the strength of the coupling constant. We may associate a phase transition with this symmetry breaking, leading from fermion states to a fermion condensate which can be described effectively by a scalar field. Our purpose in this paper is to exploit the interesting properties of NJL in a different context other than particle physics by studying its cosmological dynamics. We are interested in finding whether possibly the NJL model could be used to describe the still unknown dark energy and/or dark matter, from up to 95% of the energy content of the universe at present time.

In the last years the study of our universe has received a great deal of attention since, on the one hand fundamental theoretical cosmological questions remain unanswered and, on the other hand we have now the opportunity to measure the cosmological parameters with an extraordinary precision. In the last decades, research in cosmology has revealed the presence of unexplained forms of matter and energy called Dark Energy “DE” and Dark Matter “DM” making up to 95% of the energy content of the universe at present time. The study of supernovas SNIa shows that the universe is not only expanding, but besides it is accelerating [

It has been established that our universe is flat and dominated at present time by Dark Energy “DE” and Dark Matter “DM” with

Nowadays there are a huge number of ideas aimed to explain these unknown cosmological fluids DE and DM, from the theoretical point of view, none of them being still conclusive. This situation supports and motivates our research. Given that our most successful theory of matter, the Standard Model of particle physics (SM), which is settled within the theoretical frame of Quantum Field Theory (QFT), it would be reasonable to ask a theory attempting to describe dark fluids to be based on QFT as well. In this paper we study a fermion interaction theory with a chiral symmetry, the Nambu-Jona-Lasinio (NJL) model. Though this is an old and well known model in the context of hadron physics, it has interesting properties and it is worth to consider it with a different perspective, by studying its possible relevance for Cosmological Physics. Other examples of QFT models of DE and DM have been proposed using gauge groups, similar to QCD in particle physics, and have been studied to understand the nature of Dark Energy [

We organized the present work as follows: In Section 2 we present the NJL model. In Section 3 we review the pertinent cosmological theory. Sections 4 and 5 present a study of the cosmological dynamics of a NJL fluid with a weak and strong coupling, respectively. In Section 6 we consider the addition of a cosmological constant to our NJL fluid, and analyze the different possible behaviours. In Section 7 we comment an interesting possible way to modify the original NJL model, obtaining an additional term in the effective potential which could be related with a Cosmological Constant. Finally, in Section 8 we summarize our results and present the conclusions.

Inspired by a, by then recently explained phenomenon in Superconductivity research, professors Y. Nambu and Jona-Lasinio, suggested that the mass of fermion particles (described by a Dirac equation) could be generated from a primary four-fermion self interaction, leading to a chiral symmetry breaking. The proposed Lagrangian, invariant under chiral transformations, has the form

where

with no original mass term for the fermions. Since the coupling has dimension-2 in mass units, the theory is non-renormalizable. However, we are interested in considering the NJL model as an effective theory, useful below certain energy scale. The theory (1) describes a four-fermion interaction which can be expanded following conventional perturbation theory, and represented by Feynman diagrams (

The infinite number of fermion loops can be resumed giving a non-perturbative potential. This can be easily done by introducing an auxiliary scalar field

The field

the Euler-Lagrange equations,

where

ter m with a physical dimension of mass, so that ^{1} The term ^{2}

From the equivalent Lagrangian one may read the fermion mass and the tree level scalar potential

The effect of quantum processes (represented by loop diagrams) may be taken into account through the well known Coleman-Weinberg potential

the minus sign in

and the potential becomes

Notice that the one-loop potential

For the sake of concision we also define

In this way, taking quantum corrections into account we obtain an effective potential given by

with the complete potential

As a function of

Equation (13) gives the complete NJL scalar potential, and we are interested in studying its cosmological implications. Let us determine the asymptotic behaviour of the scalar potential V in Equation (13). To analyze the potential we seek for extremum points. For the function

and for the derivative of V we have

The condition

The first one says that the origin

derivative

we see that if

define a critical value of the coupling

so that we see that for a weak coupling

Now let us determine the second (possible) extreme of the potential. Since the r.h.s of

the second equation in Equation (17) is negative (i.e.

has a solution only for a strong coupling

for the intersection between the curve of the function

second Equation (17), and the constant in the l.h.s. In this case do exist an intersection (only one, as the r.h.s. is a monotonic function), giving a solution for the x variable, leading in its turn to a non-trivial solution in ^{3} The extremum in this case corresponds to a minimum. Notice that in all cases we have at large x the limit

Therefore, we have: if

To estimate the value of the potential at the minimum for

procedures. Let us introduce a parameter s to write g in the form

In this way we make sure to have a strong coupling by taking

which provides a good idea of how

From Equation (4) the field

Now, if the field has an expectation value

Thus, we see that two different fluid phases (massless fermions or fermion condensate) are obtained depending on the strength of the coupling. Next, we investigate the cosmological dynamics of each of these fluids.

The widely accepted current standard cosmological model (the Big Bang theory) is based in Einstein’s theory of General Relativity. If conditions of spatial homogeneity and isotropy are assumed, the space-time metric adopt the well-known simple form

where the variables ^{4} to a very high precision, we will use this same theoretical framework. Because the necessary equations are well known and their deduction can be found in standard text books, in the following we limit ourselves to write them and to give only a brief explanation.

The equation

relates the expansion rate (in time) of the scale factor a, and the curvature k of the universe, to the total energy density

Introducing the usual definition relating the Hubble parameter H with the rate of change in time of the scale factor a

Equation (23) (with

The continuity equation for a fluid with energy density

For a perfect fluid “a” satisfying a barotropic equation of state

A scalar field

where we have also defined the kinetic energy

For a given component fluid “a”, it is useful to know its relative density, defined as the ratio of its energy density to the total energy density:

where we have used Equation (25) in the second equality. In a flat universe one has the condition

It is interesting to note that while Equation (31) remains valid even when we have a negative

Taking the time derivative in Equation (25) and using Equation (26), it can be found

Note that the r.h.s. in Equation (32) is always negative. The equation of motion for a spatially homogeneous scalar field, (a modified Klein-Gordon equation) is given by

It is also useful an equation for the acceleration of the scale factor:

Differential Equations (24), (32), (33), together with (27) constitute a complete set which can be solved numerically (since we cannot always write an analytical solution). Nevertheless, it is convenient to attempt to outline the general behaviour of the dynamical system. Thus, before going to solve for our NJL potential, let us point out the following generic facts:

The evolution of the scalar field is such that it will minimize the scalar potential

is satisfied. Thus, Equation (25) says that

Nevertheless, it is interesting o note that there is no known physical principle forbidding the existence of a fluid with a negative potential

As we have seen in Section 2, for a weak coupling

is a condition to be satisfied. This, of course, in not always the case: we could take an initial field amplitude

an accelerating universe, though it would be an “early” acceleration, as it would be present an initial times, i.e. before letting the fluid densities to dilute and field to evolve. As time passes, the field rolls down minimizing the potential, and eventually acquires some value

Given that the densities of matter and radiation never reach a null value in a finite time, and that the field amplitude tends to be stabilized around the minimum (i.e.

We show an example of numerical solution in the figures. In

where the whole coefficient multiplying on

Since at late times, when the field oscillates around its minimum with a quadratic potential, the average value is

Within the context of Early Cosmic Inflation theory, the so called Slow Roll parameters are defined as follows:

which have to satisfy the conditions

The strong coupling case leads to a fermion condensate and therefore to a negative potential V at its minimum. The potential has at the origin

Now, while the expanding phase is taking place, the field is rolling down, eventually entering in a damped oscillatory regime nearly the minimum, where the potential has become negative,

As we mention before, in Section 3, a similar circumstance arises in dealing with the relative densities

Consider now a universe containing matter and radiation in addition to our NJL fluid. An interesting question is, may the presence of these fluids prevent the universe to collapse? Remember that the condition for an increasing scale factor can be reduced to the inequality (35). If the scale factor is supposed to grow forever, this condition must be hold always. Now, according to the explanations given above, initially the scale factor is growing indeed. Thus, from Equation (27) we see that the densities of both barotropic fluids (matter and radiation) must be decreasing. At the same time, because the field is stabilizing in the minimum of the potential, the kinetic energy of the field

The previous qualitative generic analysis is verified by the numerical solution for our NJL potential in particular (Figures 7-11). By observing the graphics, we found an unpredicted interesting non-trivial behaviour of the field amplitude: while the scale factor undergoes the expanding, and contracting phases successively, an damped oscillating phase around ^{5} Is this an acceptable result? Intuitively, as a is decreasing, it is reasonable to expect all densities to be growing. In particular, if the field density

explained observing Equation (26). The energy evolution of a barotropic fluid

and for a scalar field with energy density

We can see from Equations (40) and (41) that for a positive barotropic fluid

Due to its theoretical properties and observational requirements, a Cosmological Constant is a very usual and useful ingredient included in cosmological models, and it is worth to consider such contribution in our model. Its defining property is an energy density

a) Free Fermions (

Given that the left hand side in this inequality is diminishing in time, whereas the right hand side remains constant, we have that eventually this inequality cannot hold anymore, and becomes an equality, meaning

Thus, we see that for a free fermions NJL fluid with a Cosmological Constant, the universe necessarily accelerate, the precise moment depending on the amount of energy densities

b) Fermion Condensate (^{6}

Remember that the potential take positive values as well as negative ones, so both possibilities must be taken into account. Certainly one can find such set of values of V for a given

From definitions (30) it can be found that

If a positive acceleration eventually come up, the above expression is expected to become an equality. Now, suppose

What about a collapse in the future? May the presence of a cosmological constant prevent a decreasing scale factor (time going forward)? For a growing scale factor we have^{7} As we explained before, if the scale factor is to reach a maximum

If we want to keep our analysis as simple as possible, we may ignore the contribution from radiation,

Now, nothing forbids to exist a potential sufficiently deep

After

It is interesting to observe that a Cosmological Constant may be seen as a particular case of a scalar field evolving under a potential stabilized with a positive minimum. As we have seen, the NJL model has two different behaviours depending on the value of the coupling constant g. For weak coupling

with

corresponds to a massive scalar field with energy density

presence of a cosmological constant

As we have seen until now, the original NJL model has interesting cosmological consequences. However, the model by itself does not reproduce the observed feature of an accelerated expansion of the universe, and it is not desirable to introduce a cosmological constant by hand, without a good explanation. We rather ask for any model to be motivated from a deeper fundamental theory. Nowadays, a paradigm for such fundamental theory is played by Super Symmetric Field Theories, and a lot of work has been done in attempting to explain Dark Matter as well as Dark Energy as some super symmetric particle (references are given in the introduction, sec. 1). Nevertheless, any conclusive theory has been established yet to present date. We would like now to generalize the NJL potential to include a physically motivated potential from supersymmetric gauge theories. These class of models have been previously studied in Dark Energy models derived from gauge theory [

which is obtained from a non-perturbative super potential in a gauge theory, e.g. ^{8} We now add the potential in Equation (48) to our NJL model. Since the effective NJL potential in Equation (13) has a quadratic term

shown in

tent that NJL and Equation (48) are valid or useful theories. This is on the same footing than using the NJL model to study the dynamics of hadrons, without having obtained the model directly from the QCD Lagrangian. Since we are simply adding a term to the already studied NJL potential, we use the results of the previous section 2. Using eq. ii), (17), the condition to be satisfied by the minimum x is now written (remember that we wrote before

where we have defined the function

From this equation we see that, it is possible to obtain

Let us show an example. Suppose that

which has the solution

the minimum, the parameters must satisfy^{9} the energy density of our NJL fluid is

with the precise value of coefficient

The fermion model of Nambu and Jona-Lasinio (NJL) includes two different fermion states resulting from quantum effects, each one being associated with two different physical phases. For a weak coupling

Notice that in the strong coupling case

Here we studied the potential and solved the cosmological evolution for each fluid in presence of additional barotropic fluids (e.g. matter-dust or radiation).

For a weak coupling, we found a coefficient of state

On the other hand, the strong coupling case (without a cosmological constant) always causes an eventually vanishing energy density. This is due to the fact that the potential is negative when minimized, and even the additional presence of matter and/or radiation does not prevent this to happen. Since the vanishing energy (which is associated with the scale factor reaching a maximum), is followed by a contracting period, this means that a fermion condensate always makes the universe collapsed. The energy density of the field

Equation (23) has been known and well studied since long time ago. If the curvature parameter is

We also studied a variant of the strong coupling model, consisting in the addition of a cosmological constant. We found that, if the energy density

Perhaps it is worth to emphasize that, in both cases of weak and strong coupling and without considering a cosmological constant, one may induce an acceleration of the scale factor by manipulating the initial condition for the field amplitude

It is important to keep in mind that, once we settle a coupling strength (weak or strong), there is nothing in the theory to allow to switch between them, so actually a phase transition cannot be considered.

A very appealing feature of the NJL model is, in our opinion, the fact that 1) it is based on a “fundamental” symmetry (chiral symmetry), 2) the model leads to a potential which, due to quantum corrections, can adopt negative values in a natural way, and 3) it includes only one parameter: the coupling constant g (two parameters if we count the cut-off^{10} or introducing new kinds of fluids aimed to be relevant to cosmological problems, but at the expense of introducing several fields or parameters.^{11}

Finally, we saw that by considering an additional term besides the NJL potential, in the form of an inverse power (which is motivated from some supersymmetric theories), then it is possible to obtain a total potential with a positive minimum, thus allowing to explain a cosmological constant as a consequence of a field dynamics, which is a fermion particle (instead of a scalar field) governed by simple basic symmetries.

A.M. acknowledges financial support from UNAM PAPIIT Project No. IN101415 and Conacyt Fronteras Project No. 281.

Leonardo Quintanar G. and de la Macorra, A (2016) Cosmology of the Nambu-Jona-Lasinio Model. Journal of Modern Physics, 7, 1777-1800. http://dx.doi.org/10.4236/jmp.2016.713159