Mesonic Excitations of QGP: Study with an Effective Model

Mesonic Excitations of QGP: Study with an Effective Model

Abstract

We study the correlations between quark-antiquark pairs in different quantum number channels in a deconfined plasma by using an effective model of QCD. Using the three flavour PNJL model, the finite temperature spectral functions for different mesonic states are studied at zero and nonzero quark chemical potentials. It is found that in the channel resonance structures survive above the chiral transition temperature , while the kaonic states seem to get washed off just above . The sensitivity of the structures to the anomaly term are carefully investigated.

pacs:
12.38.Aw, 12.38.Mh, 12.39.-x, 12.40.Yx, 11.30.Rd, 25.75.Nq

TIFR/TH/08-57

CU-PHYSICS/1-2009


I Introduction

At high temperatures and densities, strongly interacting matter is expected to undergo a transition to new phases where colour degrees of freedom are deconfined. Such conditions of high temperatures and densities can be created in the laboratory by the collision of heavy ions at large energies. A large amount of data have already been obtained from RHIC. In near future results from LHC, for low density and high temperature phase, will be available as well. On the other hand, the FAIR experiment at GSI will provide us information about extremely dense matter at low temperature. It is thus extremely relevant to study the properties of strongly interacting matter at high temperature and density.

A first principle study of hot and dense strongly interacting matter starting from QCD is not easy because the physics is nonperturbative in the temperature and density range of interest. At finite temperatures, numerical studies on lattice provide the most reliable results for the physics of the deconfined phase. The physics of excitations in real time, however, is not directly accessible from lattice and one requires an analytical continuation from Euclidean time. Studies at finite baryon number density are prohibitively difficult on the lattice, because of the well-known sign problem. Some progress has been made in recent years in extracting results for small densities (1); (2). As it stands now, the lattice predicts that the deconfinement and the chiral symmetry restoration happen roughly simultaneously (3) via a smooth crossover (4) at zero quark chemical potential, and the two transitions roughly coincide at about MeV (3). There is an indication of a critical point at a small chemical potential 300 - 500 MeV, with a first order transition for larger (1). There have also been suggestions that the first order line may not exist (2).

In view of the limitations in studying finite density and real time problems, it may be useful to get an idea of the features of the new phases of matter by studying QCD inspired models. Depending on the physics of interest, one can attempt to study a model which has features relevant to the physics issue. In particular, for issues of the transition of the hadronic matter to a chiral symmetry restored, deconfined phase the Polyakov loop extended Nambu – Jona-Lasinio (PNJL) model has been used.

In the Nambu – Jona-Lasinio (NJL) model (5); (6), as it is used currently, the interactions between quarks are taken into account by suitable four-quark terms which respect the chiral symmetry of the QCD lagrangian. Since the gluons are integrated out, with their only effect being the four-quark interaction terms, there is no confinement per se in this model. The chiral symmetry is, however, spontaneously broken by condensation: . As a result, the quarks pick up a constituent mass. Furthermore, the pseudoscalar mesons and become the Goldstone modes. It is also easy to incorporate the anomaly by introducing a suitable determinant term (6). The NJL model has been widely used to study the chiral phase transition in QCD, and also the nature of the excitations at temperatures slightly above the transition temperatures. Based on such studies, it was suggested sometime back that nontrivial strong correlations between pairs persist in the deconfined phase at moderately high temperatures (6).

Due to the lack of confinement, though, the NJL model does miss out on some important aspects of the QCD thermal transition. In particular, the chiral transition in QCD is also of deconfining nature, as the Polyakov loop, which is the confinement-deconfinement order parameter, shows a rapid change. The Polyakov loop extended NJL (PNJL) model (7) attempts to capture this feature of the QCD transition. In this model, the gluon dynamics is described by a background temporal gluon field which is coupled to the quarks by the covariant derivative. The value of the background field is determined by minimizing the corresponding potential which depends on the traced polyakov loop .

The PNJL model has been extensively used to look at the phase structure at high temperature and density. Ratti et.al. (8) have studied the two flavour version of this model in certain amount of detail. They have looked at the crossover transition around 200 MeV and also the quark number densities at different baryonic chemical potentials. Their results match nicely with the lattice, especially the fact that the chiral restoration and the deconfinement transition take place almost simultaneously. The speed of sound and the diagonal susceptibilities, calculated with the PNJL model, were also found to be in good agreement with lattice results (9). On the other hand the off-diagonal susceptibilities in two flavour PNJL model differ from the lattice results (10). Recently Fukushima has studied the phase diagram of the three flavour quark matter within the framework of the PNJL model (11).

Since the PNJL model successfully reproduces a large number of quantities calculable directly from QCD one may be interested to use this model to study quantities of experimental interest which cannot be directly studied from lattice. In particular, a knowledge of the nature of excitations of the plasma is phenomenologically very important. The complete in-medium behavior of the mesonic excitations require a study of the spectral functions in these channels. As we discussed earlier, it is not easy to calculate them on lattice. A detailed study of such spectral functions in the NJL model has been carried out sometime back (6). It is clearly of interest to find out how the introduction of the Polyakov loop changes the nature of the correlations. Hansen et.al. (12) have studied the pion and sigma correlation functions in the 2-flavour PNJL model. Costa et.al. (13) have calculated the masses of the pseudoscalar mesons at finite temperature and zero density. It is, of course, interesting and phenomenologically important to study a larger set of mesonic correlations for the realistic case of 2+1 flavour PNJL model and also at finite density.

Here we aim to carry out such a study. We look at the spectral functions and pole masses of the pseudoscalar channels with different flavour contents. We also carefully investigate the effect of the anomaly term in the spectral functions. The spectral functions have also been studied in presence of a small chemical potential.

This paper is organised as follows: in the next section we briefly describe the 2+1 flavour PNJL model, as used by us. In Sec. III the basic in-medium calculations and the chiral transition have been discussed. The main results of the paper, about mesonic excitations, are in Sec. IV. A summary of our results and conclusions are available in the last section.


Ii Three Flavour PNJL Model

In this section, we briefly describe the PNJL model, and specify the parameters of the model that we have used. More details can be found in the literature (7); (8); (14).

In the NJL model the gluon dynamics is reduced to the chiral point couplings between quarks. The PNJL model also introduces the temporal gauge field, since the Polyakov loop,

(1)

is the order parameter for the confinement-deconfinement transition. Here is the temporal component of Euclidean gauge field, , and denotes path ordering. Our effective lagrangian is (14)

(2)

In the above Lagrangian is the kinetic term with gauge field interactions, . The gauge coupling is absorbed in the definition of . is the mass term which breaks the chiral symmetry explicitly. The mass of a particular flavour is denoted by and the corresponding field is . The light quark mass is considered to be MeV and the strange quark mass is considered to be MeV. The term is responsible for the four-fermi interaction among the quarks with coupling . Here we take this coupling to be positive. The next term which is a six-fermi interaction is invariant under but breaks symmetry. This term represents the axial anomaly of QCD (6). Here ’det’ stands for the determinant with respect to the flavour indices, and the anomalous coupling is represented as . This anomaly term is responsible for the flavour mixing of and mesons in the pseudoscalar channel, giving rise to the and mesons. The potential is expressed in terms of the Polyakov loop and its conjugate, , as (8).

(3)

where

(4)

and being constants and , which is a parameter here, is chosen to be MeV.

Both and are treated as classical field variables. When the quark number density is zero one has . This quantity can be considered as the order parameter for the phase transition. Furthermore has a center symmetry which encompasses the phase transition in QCD. At low temperature has a single minimum at . At high temperature has three degenerate minima at .

To study the chiral transition, we study the system of eqn.(2) in the Mean Field Approximation (MFA) to get the field equations for , , . Due to the breaking of symmetry to the quark condensate acquires a non-zero vacuum expectation value given by

(5)

where trace is over colour and spin states. Also, eight goldstone bosons appear for model. The self-consistent gap equation for the constituent masses are

(6)

Here denotes the chiral condensate of a quark with flavour where . The expression for at and can be written as (14)

(7)

being the three momentum cut off.

The parameters of the NJL part of the Lagrangian are fixed at . In the present work we have used the parameter set obtained in ref. (6),

This parameter set reproduces the physical values of the pion mass, MeV, pion decay constant, MeV, and masses of the and mesons, MeV and MeV. The parameters of are(8); (14),

so as to reproduce the lattice measurements of the Polyakov loop expectation value.


Iii Chiral transition in PNJL model

In the mean field approximation, the thermodynamic potential of the PNJL model can be written as (14)

(8)

where is the inverse quark propagator,

(9)

and are the Matsubara frequencies of fermions. Using the identity , and performing the colour trace, we get

(10)

where is the single quasiparticle energy. In the above integrals, the vacuum integral has a cutoff whereas the medium dependent integrals have been extended to infinity.

To look at thermal behavior of different observables, we need to study the variation with temperature of the minimum of the thermodynamic potential. In fig. 1 we show the thermal behavior of the chiral condensates for both the light and the strange quarks, in both NJL and PNJL models. The NJL results can be obtained from eq. (10) by setting , and . Fig. 1 shows a rapid drop in in both the models in the temperature range 150-200 MeV, without any singularity, indicating a chiral symmetry restoring crossover. This is consistent with what lattice studies find for QCD (4). From the points of inflexion of one estimates 170 MeV for NJL and 190 MeV for PNJL models. Figure 1 also shows the behavior of the Polyakov loop, (15). The figure shows that deconfinement temperature, , as expected from lattice studies of QCD (3).


Figure 1: (Color online) (a) (red, solid) and (green, dashed), as a function of at for the NJL model. (b) The same for the PNJL model; also shown is the Polyakov loop expectation value, (blue, dash-dotted). (c) Same as (b) but for MeV. Here (blue, dash-dotted) and (pink, dotted) are different.

At large quark densities, the chiral symmetry restoring transition in PNJL model becomes first order. Figure 1 also shows the behavior of the chiral condensate and the Polyakov loop in the presence of a chemical potential = 320 MeV, which is just above the critical point for this model. The transition temperature at this is 85 MeV. Note the sharp fall of , indicating a first order transition. All the quantities shown in the figure show a jump at the transition point, consistent with a discontinuous transition, though the drop in and the rise in are only a few %, and both these quantities have rather complicated temperature dependence above the transition point. In the presence of a chemical potential, there is no charge conjugation symmetry so , as seen in fig. 1.

One problem of the mean-field study of PNJL model is the absence of color neutrality at finite density (17); (18). This arises because of the fact that the Polyakov loop, which couples to the dynamical quarks, serves as an external coloured field, leading effectively to separate chemical potentials for quarks of different color. As a remedy to this problem, it has been suggested to introduce a colour chemical potential which will enforce color neutrality(17). While this affects the individual number densities of the quarks of different colors, the total number densities of quarks and antiquarks, and , do not change significantly (18). We will discuss in Sec. IV.3 the implication of this for our study.

In fig.2 we have shown the variation of and with temperature at different chemical potentials. Here and are the constituent quark masses, eq.(6). At = 0, The change in the mass is smooth. But the change becomes sharper on introducing a chemical potential. While the introduction of the Polyakov loop changes the constituent quark masses considerably around the phase boundary, at very high temperature both the models give similar results, since there . A similar variation of the strange quark mass is obtained in the two models.


Figure 2: (Color online) (a) Variation of (a) with temperature, at and MeV for both NJL and PNJL model. Solid (red) and dashed (green) lines are for NJL model with = 0 and 320 MeV, respectively, while dashed-dot (blue) lines and dotted (pink) lines show the corresponding figures for PNJL model. (b) Same as (a) but for .

The phase diagram we obtain is shown in fig. 3. The critical point in our calculation was found to be at = (314 MeV, 92 MeV). While lattice estimates of the critical chemical potential vary considerably between different groups, they tend to be much lower than the PNJL value, 150 MeV. This probably indicates that the naive PNJL (or NJL) model may not be a good model for the QCD transition at large 300 MeV, since nuclear excitations become important in this regime.


Figure 3: (Color online) Phase diagram in the PNJL model. The solid line at large ends in a critical point at = (314 MeV, 92 MeV). At smaller one gets a crossover, denoted by the dotted line. Putting in an explicit temperature dependence in the anomaly term (denoted PNJL1; see text) washes off the first order line completely.

Another point to mention here is that the anomaly term is introduced explicitly in the NJL and PNJL models by a six fermion interaction term. In QCD, of course, this term is introduced due to instantons, which are topological objects in the gauge sector (19). It is not known how well the anomalous symmetry is restored in the deconfined phase. In Ref. (6) it was speculated that the symmetry is restored well below . While there is no evidence of that from lattice studies, it is an open question whether the symmetry is restored (or at least, the effect of the anomaly term substantially reduced) at above . If we include an explicit temperature dependence of the coefficient of the anomaly term, , where is the transition temperature at quark chemical potential , then the transition becomes considerably softer and the first order line completely disappears, while the transition temperature moves to a somewhat lower value (see fig. 3). We use PNJL1 to denote the model with such a temperature dependence of in eq. (2). One might be tempted to make a conclusion that the existence of critical point is strongly related to the strength of the instanton-induced anomaly term at the transition point (20).

Of course, PNJL1 implies a substantial reduction of the anomaly term already before . A more realistic temperature dependence of could be a sharp drop around . To investigate the effect of such a temperature dependence, we also consider eqn. (2) with , where . We call this model PNJL2 (21). PNJL2 has a phase diagram similar to PNJL i.e. the critical point has the same position at = (314 MeV, 92 MeV). However, as we will see later, the nature of the mesonic excitations around can be substantially different between the two.


Iv Mesonic Excitations and Anomaly

In order to understand the properties of the medium beyond the bulk thermodynamic properties, we need to look at the low-lying excitations. The simplest gauge invariant excitations in QCD are the mesonic excitations. Masses and decay widths of the mesonic resonances are calculated from the correlations of -type operators in QCD vacuum. In the medium, we are interested in spectral changes due to interactions with the medium; in particular, we would be interested in possible resonance-like structures in the deconfined phase. We need therefore to look at the spectral functions.

Here we study spectral functions of pseudoscalar and scalar mesonic states. They are of interest because of their close connection with the chiral symmetry breaking and its restoration. The temperature dependence of the spectral function has been studied in the Nambu-Jona-Lasinio model in ref. (6). There it was found that narrow structures persist in the symmetry restored phase, at moderately high temperatures. Here we intend to see how the coupling with the Polyakov loop affects these results. Also we study the sensitivity of these structures on the coefficients of the anomaly term. We also extend the studies to nonzero quark chemical potentials.

iv.1 Formalism

The spectral function for a given mesonic channel in a system at temperature T can be defined through the Fourier transform of the real time two point functions and (22),

(11)
(12)

Here is the suitable hadronic operator and the subscript denotes the connected part of the correlation function. Using the Kubo-Martin-Schwinger(KMS) condition (22)

(13)

one can connect to the retarded correlation function,

(14)

Inserting a complete set of states in eq. (11) and using eq. (13), one gets the expansion

(15)

where Z is the partition function, and refers to the four-momenta of the state .

A stable mesonic state contributes a function-like peak to the spectral function:

(16)

where is the mass of the state. For an unstable particle one gets a peak with a finite width, e.g., the Breit-Wigner form for states with narrow decay width. We want to study how the spectral function changes due to collisions with the thermal medium.

At the level of approximation we are working, the collective excitations, the fluctuation of the mean field around the vacuum, can be handled within the Random Phase Approximation (RPA) (23). In this approximation, which is equivalent to summing over the ring diagrams, the retarded correlation function is given by

(17)

Here G is the suitable coupling constant and is the one-loop polarisation function for the mesonic channel under consideration,

(18)

is the quark propagator and is the effective vertex factor. Eqn. (18) has been evaluated in the literature; for the NJL model, the effective formulae were calculated in ref. (6). The whole effect of the introduction of the background Polyakov loop can be absorbed into a modification of the Fermi-Dirac distribution functions (12)

(19)
(20)

For NJL model, and the distribution functions become the usual Fermi-Dirac distribution function.

From the correlation function one can calculate the spectral function using eq.(14). The position of the pole controls the decay of the correlator and is called the pole mass. It can be obtained by solving

(21)




Table 1: Pseudoscalar coupling strengths in NJL model, where , , (6); (5).

The coupling constants for the different flavour combinations are given in table 1. The cases of and need a special mention. These mesons arise from the mixing of the flavour singlet and octet states, i.e. and mesons. The mixing arises because of an interplay between the anomaly term and the nondegeneracy between and . The masses of and may be obtained from the roots of the equation

(22)

where

(23)

The polarisation functions are obtained from eq.(18). On the other hand pole masses of and can be obtained directly from , and eqn. (21).

iv.2 Results at finite temperatures

The pion and sigma channels provide the most interesting observables with mesonic states, since they are directly associated with the chiral symmetry. In the chiral symmetry restored phase, pion and sigma should be degenerate. Below , on the other hand, pion is the goldstone mode of the spontaneously broken chiral symmetry and is therefore much lighter than the sigma.

In fig. 4 we show the temperature dependence of the pole masses of the pion and the sigma in the NJL and the PNJL model. We see that close to there is substantial difference between the two models. While the two states come close to each other, they are not degenerate just above . As mentioned in Sec. III, results of the two models converge at higher temperatures. This, combined with the fact that is lower for NJL model, leads to a smoother change in NJL model above than in PNJL. The nondegeneracy between the pion and the sigma can also be seen in the fig.4(b) , which show the spectral functions in the pion and sigma channels at a temperature of 1.1 .

There can be two sources for the nondegeneracy: the finite (though small) current quark masses, and the remnant symmetry. To investigate the effect of the symmetry, we look at PNJL2, which introduces a cutoff in , as explained in sec. III. As can be seen in fig. 4, this significantly alters the spectral functions, and removes most of the nondegeneracy of the and above . Also, while there is a clear structure in both of these channels above , the structure is considerably broader in the absence of the anomaly term. When studying remnant bound state structures from QCD-inspired models, it therefore is important to keep in mind the effect of the anomaly term, since unlike in QCD, the anomaly terms are put in here by hand and therefore their contributions across phase transition may be different from that in QCD.

Figure 4: (Color online) (a) Pion and sigma mass versus temperature, at MeV for both NJL (red, solid and blue, dash-dotted) and PNJL (green,dashed and pink, dotted) model. (b) Spectral functions for the pion and sigma in the PNJL model, just above . Also shown is the effect of suppression of the anomaly term at (PNJL2; see Sec. III) on the spectral functions .

In figure 5 we have plotted the masses of , , and mesons respectively. We see features similar to those in fig. 4: significant differences around the transition region due to the introduction of the Polyakov loop. It is also seen that very close to the transition region, the masses of the singlet and octet states start approaching each other, indicating that the effect of the anomaly term starts getting reduced. However, without an explicit cutoff like PNJL2, the nondegeneracy between these two states is clearly visible at 1.1 .

Figure 5: (Color online) Masses of the different states in NJL(green, dashed) and PNJL (red, solid) model.

The spectral functions of the and states, calculated in the PNJL model, are shown in fig. 6. Here we also investigate the effect of the anomaly term by comparing the calculations in PNJL2 model. We find here that the effect of the anomaly term is considerable in but not in . This can be understood by looking at the two relevant couplings, and , in table 1. Because of the different numerical factors, and also the different combinations of the quark condensates, the anomalous coupling of is substantially larger than that of . Interestingly, the broadening due to anomaly suppression is much less even for than was seen for the pion. In these channels, a clear (though broad) resonance structure is seen just above .

Figure 6: (Color online) (a) Spectral functions of at temperatures of 0.9 and . Note the function peak for below . (b) The same above . Also shown are the effects of cutting off the anomaly term above at (PNJL2; see Sec. III).

The behavior of the pole mass for the kaon state, fig. 7, is qualitatively similar to the pion. However, the spectral function above , shown in the same figure, shows a very different behavior. The sharp peak at is completely washed off already by 1.1 : there is no indication of a strong correlation between quarks in this channel.

Figure 7: (Color online) (a) Pole mass of the kaon state, in PNJL (solid, red) and NJL (dashed, green) models, at different temperatures. (b) Spectral function, from the PNJL model. The sharp peak at seems to be completely washed off already by 1.1 .

iv.3 Results at finite densities

One of the advantages of model studies is that one can study regions in parameter space which are not easy to study directly from QCD in a controlled way. The agreement of PNJL model results with QCD at finite temperatures encourages us to investigate effects of nonzero baryon densities, by introducing a quark chemical potential. As mentioned in Sec. III, however, we do not expect the PNJL model to be a good model for QCD transition at large . We therefore investigate the mesonic excitations in this model at = 150 MeV. This is around the region where lattice studies suggest the beginning of the first order line, while the PNJL model still finds a crossover transition at these densities. From fig. 3 we also get 170 MeV at = 150 MeV, which is not too far from what one gets from lattice.

In fig. 8, we show the pion and sigma spectral functions above for = 150 MeV. We see that the spectral functions have a sharp structure even substantially above , as in the case for = 0. The nondegeneracy of the two channels and the effect of the anomaly terms are also similar to the = 0 case. In the figures 8(b) and 8(c) we show the variation of the pion pole mass with , for two different temperatures. Interestingly, the quantitative difference in the masses calculated in PNJL and NJL models increases with density. Of course, as we have discussed above, we do not expect these models to be a good approximation to QCD at values of 150 MeV.

Figure 8: (Color online) (a) Spectral functions of pion near the chiral transition point, for = 150 MeV. Also shown is the effect of suppression of anomaly term at (PNJL2; see Sec. III). (b) Pion mass versus , at MeV. Solid red line is for PNJL model and dashed green line is for NJL model. (c) Same as (b) but at MeV.

The spectral functions of and are shown in fig. 9, for = 150 MeV. This figure has some interesting features. First, the interplay of the temperature and chemical potential gives rise to a nonmonotonous behavior in the peak of the spectral function in the deconfined phase. The peak position drops sharply at , and then rises again. Second, we note that the peak, which is below the peak below or at , becomes very broad already at 1.05 , with the peak position slightly above that of the peak. If this feature is true for full QCD and is not an artefact of the model, it will have interesting phenomenological implications.

Figure 9: (Color online) (a) The spectral functions of the channel around the chiral transition temperatures, for = 150 MeV. (b) Same as (a) but for .

Let us now discuss the sensitivity of our results on enforcement of colour neutrality (Sec. III). Our calculations at finite density have been carried out at MeV and . At such T and , if we want to enforce color neutrality by the introduction of a color chemical potential, we need a color chemical potential (in the notation of Ref. (17)) with a small , while is negligible. Here and are Gell-mann matrices in color space. While this changes the individual colour densities, the total quark number density () and the total scalar density () do not have any appreciable change. Similar results about the total quark density has also been reported in ref.(18). In fact the change in the scalar density is less than under such conditions. Here we have studied spectral functions of colorless, meson-like objects, which are sensitive to the scalar density. Hence we do not expect our results at finite chemical potential to be affected appreciably by the issue of colour neutrality.


V Conclusion

To conclude, we have studied the meson-like excitations in the deconfined plasma at both zero and nonzero quark chemical potential, using the Polyakov loop extended Nambu – Jona-Lasinio (PNJL) model.

The phase diagram of the PNJL model shows a first order line of transitions at large , ending in a critical point at (314 MeV, 92 MeV). We note, however, that the phase diagram can be quite sensitive to the temperature dependence of the coefficient of the anomaly term. In particular, a suppression of the anomaly term at large temperatures may lead to a complete wash off of the first order line.

The in-medium behavior of the mesonic correlations was studied by looking at the spectral functions. In the pion and sigma channels, reasonably sharp structure was found still at 1.1 ; also the two states are not degenerate at these temperatures. While both these features can be phenomenologically interesting, it was also found that these features are quite sensitive to the anomaly term. In particular, suppressing the anomaly term above leads to much broader and degenerate structures in these two channels. Since the structure of the resonance above is of importance for RHIC phenomenology, this sensitivity implores one to look for the restoration or otherwise of the anomaly at by looking at suitable observables directly on lattice.

The and the , on the other hand, show much less sensitivity to the anomaly term. Both these channels show a clear resonance at 1.1 . shows very little sensitivity to a suppression of the anomaly term. is more sensitive, but the resonance-like structure remains even when the anomaly term is suppressed. In the PNJL model, the kaon channel does not show any sharp structure above . All these behaviors will be of interest for RHIC phenomenology.

We also looked at the mesonic spectral functions just above for a nonzero . At = 150MeV, which should be in the regime covered by the RHIC energy scan, PNJL shows a crossover transition. The features of the spectral functions in the pion and sigma channels are found to be quite similar to those at = 0. A more interesting behavior was found in the channels. The interplay of the finite and finite lead to a nonmonotonous behavior of the peak in the channel. Also it was found that the relative positions of the and peaks get reversed above : the peak comes at a lower than the peak at 1.05 . It will be interesting to explore the phenomenological implications of these behaviours.


Acknowledgements.
P.D. thanks CSIR for financial support. A.B. thanks CSIR for support through the project 03(1074)/06/EMR-II and acknowledges UGC UPE grant (computational support). SKG thanks DST, Govt. of India for financial support under IRHPA scheme.

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This is a comment super asjknd jkasnjk adsnkj
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