D_{S} Mesons in Asymmetric Hot and Dense Hadronic Matter

Mesons in Asymmetric Hot and Dense Hadronic Matter

Divakar Pathak dpdlin@gmail.com Department of Physics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110 016, India    Amruta Mishra amruta@physics.iitd.ac.in Department of Physics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110 016, India

The in-medium properties of mesons are investigated within the framework of an effective hadronic model, which is a generalization of a chiral SU(3) model, to SU(4), in order to study the interactions of the charmed hadrons. In the present work, the mesons are observed to experience net attractive interactions in a dense hadronic medium, hence reducing the masses of the and mesons from the vacuum values. While this conclusion holds in both nuclear and hyperonic media, the magnitude of the mass drop is observed to intensify with the inclusion of strangeness in the medium. Additionally, in hyperonic medium, the mass degeneracy of the mesons is observed to be broken, due to opposite signs of the Weinberg-Tomozawa interaction term in the Lagrangian density. Along with the magnitude of the mass drops, the mass splitting between and mesons is also observed to grow with an increase in baryonic density and strangeness content of the medium. However, all medium effects analyzed are found to be weakly dependent on isospin asymmetry and temperature. We discuss the possible implications emanating from this analysis, which are all expected to make a significant difference to observables in heavy ion collision experiments, especially the upcoming Compressed Baryonic Matter (CBM) experiment at the future Facility for Antiproton and Ion Research (FAIR), GSI, where matter at high baryonic densities is planned to be produced.

14.40.Lb; 21.65.Cd; 21.65.Jk; 11.30.Rd; 12.38.Lg

I Introduction

An effective description of hadronic matter is fairly common in low-energy QCD leshouches (); Rev_1997 (); RMP2010 (). Realizing that baryons and mesons constitute the effective degrees of freedom in this regime, it is quite sensible to treat QCD at low-energies as an effective theory of these quark bound states leshouches (). This approach has been vigorously pursued in various incarnations over the years, with the different adopted strategies representing merely different manifestations of the same underlying philosophy. The actual manifestations range from the quark-meson coupling model QMC1 (); QMC2 (), phenomenological, relativistic mean-field theories based on the Walecka model serotwalecka (), along with their subsequent extensions, the method of QCD sum rules QSR_original_and_Shifman_review (), as well as the coupled channel approach Oset_PRL98 (); Oset_NPA98 () for treating dynamically generated resonances, which has further evolved into more specialized forms, namely the local hidden gauge theory Hidden_local_gauge_formalism_Refs1 (); Hidden_local_gauge_formalism_Refs2 (), as well as formalisms based on incorporating heavy-quark spin symmetry (HQSS) Heavy_Quark_Spin_Symmetry_Refs1 (); Heavy_Quark_Spin_Symmetry_Refs2 () into the coupled channel framework HQSS_Example_Papers1 (); HQSS_Example_Papers2 (); HQSS_Example_Papers3 (); HQSS_Example_Papers4 (); HQSS_Example_Papers5 (). Additionally, the method of chiral invariant Lagrangians leshouches () (which shall also be embraced in this work), has developed over the years into a very successful strategy. The same constitutes an effective field theoretical model in which the specific form of hadronic interactions is dictated by symmetry principles, and the physics governed predominantly by the dynamics of chiral symmetry – its spontaneous breakdown implying a non-vanishing scalar condensate in vacuum. One naturally expects then, that the hadrons composed of these quarks would also be modified in accord with these condensates Rev_1997 (); RMP2010 (). But while all hadrons would be subject to medium modifications from this perspective, pseudoscalar mesons have a special role in this context. In accordance with the Goldstone’s theorem leshouches (), spontaneous breaking of chiral symmetry leads to the occurrence of massless pseudoscalar modes, the so-called Goldstone bosons, which are generally identified with the spectrum of light pseudoscalar mesons, like the pions, or kaons and anti-kaons Rev_1997 (); RMP2010 (). In a strict sense, however, none of these physical mesons is a true Goldstone mode, since they are all massive, while Goldstone modes are supposed to be massless Rischke_QGP_Review (). The origin of the masses of these mesons is related to the non-zero masses of light quarks, as can be easily discerned from the Gell-Mann-Oakes-Renner (GOR) relations, and hence, from explicit symmetry breaking terms (or explicit mass terms) in the chiral effective framework leshouches (). In fact, if one considers the limiting situation of vanishing quark masses , the masses of these pseudoscalar mesons would also vanish, so that the perfect Goldstone modes are indeed recovered. For this reason, the physically observed light pseuodscalar mesons are dubbed pseudo-Goldstone bosons Rischke_QGP_Review (). In purely this sense, therefore, there is an inherent similarity between all these classes of pseudoscalar mesons – masses are acquired through explicit quark mass terms, and only the magnitude of these mass terms differ between these cases, being small for the pions (since MeV), comparatively larger for the kaons and antikaons (since MeV), and appreciably larger for the charmed pseudoscalar mesons. Thus, as we advance from the pions to the strange pseudoscalar mesons, with increasing mass, these mesons depart more from the ideal Goldstone mode character pertaining to the theorem and there is considerable departure of the charmed pesudoscalar mesons from Goldstone mode behaviour due to the explicit chiral symmetry breaking arising from the large charm quark mass ( 1.3 GeV). An understanding of the in-medium properties of the pseudoscalar mesons has been an important topic of research, both theoretically and experimentally. Within the chiral effective approach, the pseudoscalar mesons are modified in the medium due to the modifications of the quark condensates in the hadronic medium. For pions, it is observed however, that medium effects for them are weakened by the smallness of explicit symmetry breaking terms Rev_1997 (); RMP2010 (). Considerably detailed analysis of medium effects have been performed over the years, particularly in a chiral SU(3) approach, for strange pesudoscalar mesons (kaons and antikaons) sambuddha1 (); sambuddha2 (); mam_kaons2006 (); mam_kaons2008 (). For studying the charmed mesons, one needs to generalize the SU(3) model to SU(4), in order to incorporate the interactions of the charmed mesons to the light hadrons. Such a generalization from SU(3) to SU(4) was initially done in Ref. gamermann_oset (), where the interaction Lagrangian was constructed for the pesudoscalar mesons for SU(4) from a generalization of the lowest order chiral SU(3) Lagrangian. Since the chiral symmetry is explicitly broken for the SU(4) case due to the large mass of the charm quark (1.3 GeV), which is much larger than the masses of the light quarks, for the study of the charmed () pseudoscalar mesons mamD2004 (); arindam (); arvDprc (), we adopt the philosophy of generalizing the chiral SU(3) model to SU(4) to derive the interactions of these mesons with the light hadrons, but use the observed masses of these heavy hadrons as well as empirical/observed values of their decay constants liukolin (). With all these studies proving to be informative, the most natural direction of extension of this approach would be to analyze these medium effects for a strange-charmed system (the mesons). Apart from pure theoretical interest, an understanding of the medium modifications of mesons is important, since these can make a considerable difference to experimental observables in the (ongoing and future) relativistic heavy ion collision experiments, besides being significant in questions concerning their production and transport in such experimental situations. For instance, in a recent work, He et al. friesDs () have shown that the modifications of the meson spectrum can serve as a useful probe for understanding key issues regarding hadronization in heavy ion collisions. It is suggested that by comparing observables for and mesons, it is possible to constrain the hadronic transport coefficient. This comparison is useful since it allows a clear distinction between hadronic and quark-gluon plasma behavior.

However, as far as the existing literature on this strange-charmed system of mesons is concerned, we observe that only the excited states of mesons have received considerable attention, predominantly as dynamically generated resonances in various coupled channel frameworks excited_ds_PPNP_review (); excited_ds_review (). One must bear in mind that in certain situations, a molecular interpretation of these excited states (resonances), is more appropriate close_nature_2003 () for an explanation of their observed, larger than expected lifetimes. From this perspective, a whole plethora of possibilities have been entertained for the excited states, the standard quark-antiquark picture aside. These include their description as molecular states borne out of two mesons, four-quark states, or the still further exotic possibilities - as two-diquark states and as a mixture of quark-antiquark and tetra-quark states excited_ds_PPNP_review (). Prominent among these is a treatment of as a bound state simonov (); wang_and_wang (), that of as a dynamically generated resonance ds2460_plb2007 (), being treated within the hidden local gauge formalism in coupled channel unitary approach oset_ds2573 (); Oset_conf_ds_molecular (), as well as the vector states and the resonance treated in a multichannel approach prl_ds2632_2004 (). The former three have also been covered consistently under the four-quark picture ds_4quark_prl2004 (). So, while considerable attention has been paid to dynamically generating the higher excited states of the mesons, there is a conspicuous dearth lutzkorpa () of available information about the medium modifications of the lightest pseudoscalar mesons, , the one that we know surely, is well described within the quark-antiquark picture. In fact, to the best of our knowledge, the entire existing literature about the mesons in a hadronic medium is limited to the assessment of their spectral distributions and medium effects on the dynamically generated resonances borne out of the interaction of these mesons with other hadron species, in the coupled channel analyses of Refs. lutzkorpa (); hofmannlutz (); JimenezTejero_2009vq (); JimenezTejero_2011fc (). Compared with their open-charm, non-strange counterparts, the mesons, which have been extensively investigated using a multitude of approaches over the years and consequently boast of an extensive literature, the literature concerning the medium behavior of mesons can at best be described as scanty, and there is need for more work on this subject. If one considers this problem from the point of view of the (extended) chiral effective approach, this scantiness is most of all, because of the lack of a proper framework where the relevant form of the interactions for the mesons with the light hadrons (or in more generic terms, of meson-baryon interactions with the charm sector covered), based on arguments of symmetry and invariance, could be written down. Clearly, such interactions would have to be based on symmetry and bear all these pseudoscalar mesons and baryons in 15-plet and 20-plet representations respectively, with meson-baryon interaction terms still in accordance with the general framework for writing chiral-invariant structures, as well as bearing appropriate symmetry breaking terms obeying the requisite transformation behavior under chiral transformations leshouches (), which is quite a non-trivial problem. Of late, such formalisms have been proposed in Refs. hofmannlutz (); arvDepja (), as an extension of the frameworks based on chiral Lagrangians, where symmetry forms the basis for writing down the relevant interaction terms. However, since the mass of the charm quark is approximately GeV PDG2012 (), which is considerably larger than that of the up, down and strange quarks, the symmetry is explicitly broken by this large charm quark mass. Hence, this formalism only uses the symmetry to derive the form of the interactions, whereas an explicit symmetry breaking term accounts for the large quark mass through the introduction of mass terms of the relevant ( or ) mesons. Also, symmetry being badly broken implies that any symmetry and order in the masses and decay constants, as predicted on the basis of symmetry, would not hold in reality. The same is acknowledged in this approach arvDepja () and as has been already mentioned, one does not use the masses and decay constants as expected on the basis of symmetry, but rather, their observed, Particle Data Group (PDG) PDG2012 () values. Overall, therefore, symmetry is treated (appropriately) as being broken in this approach. Also, it is quite well known, both through model-independent roder () and model-dependent arvDprc () calculations that the light quark condensates are modified significantly in a hadronic medium with medium parameters like density and temperature, the strange quark condensate is comparatively stolid and its variation is significantly more subdued, while upon advancing to the charm sector, the variation in the charmed quark condensate is altogether negligible in the entire hadronic phase roder (). These observations form the basis for treating the charm degrees of freedom of open charm pseudoscalar mesons as frozen in the medium, as was the case in the treatments of Refs. arindam (); arvDprc (); arvDepja (). Thus, as we advance from pions and kaons to the charmed pseudoscalar mesons, the generalization is perfectly natural but with the aforementioned caveats. Provided all these aspects are taken into account, a generalization of this chiral effective framework to open charm pseudoscalar mesons is quite reasonable and sane, and the predictions from such an extended chiral effective approach bodes very well arindam () with alternative calculations based on the QCD sum rule approach, quark meson coupling model, coupled channel approach, as well as studies of quarkonium dissociation using heavy-quark potentials from lattice QCD at finite temperatures. Additionally, it is interesting to note that this approach, followed in Refs. arindam (); arvDprc (); arvDepja () for the charmed pseudoscalar mesons, has recently been extended to the bottom sector and used to study the medium behavior of the open bottom pseudoscalar , and mesons DP_Bm_PRC2015 (); DP_Bsm_IJMPE (). The inherent philosophy beneath this extension continues to be the same – the dynamics of the heavy quark/anti-quark is treated as frozen, and the interactions of the light quark (or anti-quark) of the meson, with the particles constituting the medium, are responsible for the medium modifications. With this subsequent generalization as well, the physics of the medium behavior that follows from this approach, is in agreement DP_Bm_PRC2015 () with works based on alternative, independent approaches, like the heavy meson effective theory, quark-meson coupling model, as well as the QCD sum rule approach. Thus, these aforementioned, prior works based on the generalization of the original chiral effective approach to include heavy flavored mesons, are totally concordant with results from alternative approaches followed in the literature, which lends an aura of credibility to this strategy. Given this backdrop, it is clear that these formalisms wipe out the reason why such an investigation for the mesons within the effective hadronic model, obtained by generalizing the chiral SU(3) model to SU(4), has not been undertaken till date, and permit this attempt to fill the void.

We organize this article as follows: in section II, we outline the Chiral  Model (and its generalization to the case) used in this investigation. In section III, the Lagrangian density for the mesons, within this extended framework, is explicitly written down, and is used to derive their in-medium dispersion relations. In section IV, we describe and discuss our results for the in-medium properties of mesons, first in the nuclear matter case, and then in the hyperonic matter situation, following which, we briefly discuss the possible implications of these medium modifications. Finally, we summarize the entire investigation in section V.

Ii The Effective Hadronic Model

As mentioned previously, this study is based on a generalization of the chiral  model Pap_prc99 (), to . We summarize briefly the rudiments of the model, while referring the reader to Refs. Pap_prc99 (); Zsch () for the details. This is an effective hadronic model of interacting baryons and mesons, based on a non-linear realization of chiral symmetry weinberg67 (); weinberg68 (), where chiral invariance is used as a guiding principle, in deciding the form of the interactions coleman1 (); coleman2 (); bardeenlee (). Additionally, the model incorporates a scalar dilaton field, , to mimic the broken scale invariance of QCD Zsch (). Once these invariance arguments determine the form of the interaction terms, one resorts to a phenomenological fitting of the free parameters of the model, to arrive at the desired effective Lagrangian density for these hadron-hadron interactions. The general expression for the chiral model Lagrangian density reads:


In eqn.(1), is the kinetic energy term, while denotes the baryon-meson interaction term. Here, baryon-pseudoscalar meson interactions generate the baryon masses. treats the dynamical mass generation of the vector mesons through couplings with scalar mesons. The self-interaction terms of these mesons are also included in this term. contains the meson-meson interaction terms, which induce spontaneous breaking of chiral symmetry. introduces scale invariance breaking, via a logarithmic potential term in the scalar dilaton field, . Finally, refers to the explicit symmetry breaking term. This approach has been employed extensively to study the in-medium properties of hadrons, particularly pseudoscalar mesons mam_kaons2006 (); mam_kaons2008 (); sambuddha1 (); sambuddha2 (). As was observed in section I as well, this would be most naturally extended to the charmed (non-strange and strange) pseudoscalar mesons. However, that calls for this chiral formalism to be generalized to , which has been addressed in arvDepja (); hofmannlutz (). For studying the in-medium behavior of pseudoscalar mesons, the following contributions need to be analyzed arvDepja (); arindam (); arvDprc ():


In eqn.(2), denotes the Weinberg-Tomozawa term, given by the expression arvDepja () -


with repeated indices summed over. Baryons are represented by the tensor , which is antisymmetric in its first two indices hofmannlutz (). The indices and run from 1 to 4, and one can directly read the quark content of a baryon state, with the identification: . However, the heavier, charmed baryons are discounted from this analysis. In eqn.(3), is defined as -


where the unitary transformation operator, ), is defined in terms of the matrix of pseudoscalar mesons, , representing the generalized Gell-Mann matrices. Further, is the scalar meson exchange term, which is obtained from the explicit symmetry breaking term -


where , and refers to the scalar meson multiplet arindam (). Also, the first range term is obtained from the kinetic energy term of the pseudoscalar mesons, and is given by the expression:


where . Lastly, the and range terms, are:


Adopting the mean field approximation serotwalecka (); Zsch (), the effective Lagrangian density for scalar and vector mesons simplifies; the same is used subsequently, to derive the equations of motion for the non-strange scalar-isoscalar meson , scalar-isovector meson , strange scalar meson , as well as for the vector-isovector meson , non-strange vector meson  and the strange vector meson , within this model.

The meson interaction Lagrangian density and in-medium dispersion relations, as they follow from the above general formulation, are described next.

Iii Mesons in Hadronic Matter

The Lagrangian density for the mesons in isospin-asymmetric, strange, hadronic medium is given as -


This is the free Lagrangian density for a complex scalar field (which corresponds to the mesons in this case), and reads:


On the other hand, is determined to be:


In this expression, the first term (with coefficient ) is the Weinberg-Tomozawa term, obtained from eqn.(3), the second term (with coefficient ) is the scalar meson exchange term, obtained from the explicit symmetry breaking term of the Lagrangian (eqn.(5)), third term (with coefficient ) is the first range term (eqn.(6)) and the fourth and fifth terms (with coefficients and , respectively) are the and terms, calculated from eqns.(7) and (8), respectively. Also, , is the fluctuation of the strange scalar field from its vacuum value. The mean-field approximation, mentioned earlier, is a useful, simplifying measure in this context, since it permits us the following replacements:


Thus, the interaction Lagrangian density can be recast in terms of the baryonic number densities and scalar densities, given by the following expressions:


In the above, and are the effective mass and effective chemical potential of the baryon, given as, , and , and is the spin degeneracy factor. One can find the equations of motion for the  and  mesons, by the use of Euler-Lagrange equations on this Lagrangian density. The linearity of these equations follows from eqn.(11), which allows us to assume plane wave solutions , and hence, ‘Fourier transform’ these equations, to arrive at the in-medium dispersion relations for the mesons. These have the general form:


where,  is the vacuum mass of the mesons and is the self-energy of the mesons in the medium. Explicitly, the latter reads:


where the and signs in the coefficient of , refer to  and  respectively, and we have used equations (12) and (13) to simplify the bilinears. In the rest frame of these mesons (i.e. setting ), these dispersion relations reduce to:


which is a quadratic equation in , i.e. of the form , where the coefficients and depend on various interaction terms in Eq. (16), and read:


As before, the and signs in , correspond to  and , respectively. In writing the above summations, we have used the following notation: denotes the set of all hyperons, is the set of nucleons, represents the Xi hyperons and denotes all hyperons other than Xi hyperons, i.e. the set of baryons which all carry one strange quark. This form is particularly convenient for later analysis. Also, the optical potential of mesons, is defined as:


where , refers to the momentum of the respective meson, and represents its momentum-dependent in-medium energy.

In the next section, we study the sensitivity of the meson effective mass on various characteristic parameters of hadronic matter, viz. baryonic density (), temperature (), isospin asymmetry parameter , and the strangeness fraction , where and denote the strangeness quantum number, and the third component of isospin of the baryon, respectively.

Iv Results and Discussion

Before describing the results of our analysis of mesons in a hadronic medium, we first discuss our parameter choice and the various simplifying approximations employed in this investigation. The parameters of the effective hadronic model are fitted to the vacuum masses of baryons, nuclear saturation properties and other vacuum characteristics in the mean field approximation Pap_prc99 (); Zsch (). In this investigation, we have used the same parameter set that has earlier been used to study charmed () mesons within this effective hadronic model arvDepja (). In particular, we use the same values of the parameters and of the range terms ( and ), fitted to empirical values of kaon-nucleon scattering lengths for the and channels, as were employed in earlier treatments arvDepja (); arindam (); sambuddha1 (); sambuddha2 (). For an extension to the strange-charmed system, the only extra parameter that needs to be fitted is the meson decay constant, , which is treated as follows. By extrapolating the results of Ref.bardeenlee (), we arrive at the following expression for  in terms of the vacuum values of the strange and charmed scalar fields:


For fitting the value of , we retain the same values of and as in the earlier treatments of this chiral effective model Pap_prc99 (); sambuddha2 (), and determine from the expression from the expression for  in terms of and arindam (), using the Particle Data Group (PDG) value of MeV. Substituting these in eqn.(23), our fitted value of  comes out to be 235 MeV, which is close to its PDG value of 260 MeV, and particularly, is of the same order as typical lattice QCD calculations for the same PDG2012 (). We therefore persist with this fitted value MeV in this investigation. Also, we treat the charmed scalar field () as being arrested at its vacuum value () in this investigation, as has been considered in other treatments of charmed mesons in a hadronic context arindam (); arvDepja (). This neglect of charm dynamics appears natural from a physical viewpoint, owing to the large mass of a charm quark ( GeV) PDG2012 (). The same was verified in an explicit calculation in Ref.roder (), where the charm condensate was observed to vary very weakly in the temperature range of interest to us in this regime LQCD1 (); LQCD2 (); LQCD3 (). As our last approximation, we point out that in the current investigation, we work within the ‘frozen glueball limit’ Zsch (), where the scalar dilaton field () is regarded as being frozen at its vacuum value (). This approximation was relaxed in a preceding work arvDprc () within this effective hadronic model, where the in-medium modifications of this dilaton field were found to be quite meager. We conclude, therefore, that this weak dependence only serves to justify the validity of this assumption.

We next describe our analysis for the in-medium behavior of mesons, beginning first with the nuclear matter situation, and including the hyperonic degrees of freedom only later. This approach has the advantage that many features of the meson in-medium behavior, common between these regimes, are discussed in detail in a more simplified context, and the effect of strangeness becomes a lot clearer.

In nuclear matter, the Weinberg-Tomozawa term and the range term vanish, since they depend on the number densities and scalar densities of the hyperons, and have no contribution from the nucleons. It follows from the self-energy expression that only the Weinberg-Tomozawa term differs between  and ; all other interaction terms are absolutely identical for them. Thus, a direct consequence of the vanishing of Weinberg-Tomozawa contribution is that,  and  are degenerate in nuclear matter. As mentioned earlier, the meson dispersion relations, with , eqns.(19-21), reduce to the quadratic equation form , with the following coefficients (in nuclear matter):


where the subscript emphasizes on the nuclear matter context. For solving this quadratic equation, we require the values of , as well as the scalar densities of protons and neutrons, which are obtained from a simultaneous solution of coupled equations of motion for the scalar fields, subject to constraints of fixed values of  and . The behavior of the scalar fields, so obtained, has been discussed in detail in Ref.arvDprc (). Here, we build upon these scalar fields and proceed to discuss the behavior of solutions of the in-medium dispersion relations of mesons, given by equations (24) to (26).

Figure 1: (Color Online) The various contributions to the meson energies, , in nuclear matter, plotted as functions of baryonic density in units of the nuclear saturation density (/). In each case, the isospin asymmetric situation (), as described in the legend, is also compared against the symmetric situation (), represented by dotted lines.

The variation of the meson in-medium mass, , with baryonic density in nuclear matter, along with the individual contributions to this net variation, is shown in Fig.1 for both zero and a finite temperature value ( MeV). It is observed that the in-medium mass of mesons, decreases with density, while being weakly dependent on temperature and isospin asymmetry parameter. We can understand the observed behavior through the following analysis. From equations (24) to (26), we arrive at the following closed form solution for the meson effective mass in nuclear matter:


From this exact closed-form solution, several general conclusions regarding the in-medium behavior of mesons in nuclear matter, can be drawn. On the basis of this expression, the range term appearing in the denominator will lead to a decrease in the medium mass with increase in density. The in-medium mass, in addition, also is modified by the medium dependence of (we assume the value of to be frozen at its vacuum value and hence ). However, the density dependence of is observed to be quite subdued in nuclear matter. It is observed that the total change in the value of from its vacuum value, i.e. . This value is acquired at a baryonic density of , and thereafter it appears to saturate. This behavior of the strange scalar field, though ubiquitous in this chiral model Pap_prc99 (); arvDprc (), is not just limited to it. Even in calculations employing the relativistic Hartree approximation to determine the variation of these scalar fields mam_rha1_2004 (); mam_rha_2004 (), a behavior similar to this mean field treatment is observed. We also point out that the saturation of the scalar meson exchange and first range term follows as a direct consequence of the saturation of , as is implied by their proportionality in equations (11) and (26). At larger densities, while the terms having saturate, the contribution from the range term continues to rise. With saturating at a value of around 15 MeV and the value of chosen to be 235 MeV in the present investigation, the last term in the numerator as well as the denominator, in equation (27) turn out to be much smaller as compared to unity. Moreover, the term in the denominator in the expression for the in-medium mass of given by equation (27) also turns out to be much smaller as compared to unity, with at a density of . Hence the smallness of this term as compared to unity is justified for the entire range of densities we are concerned with, in the present investigation. In order to read more into our solution, we expand the argument of the square root, in a binomial series (assuming ), and retain up to only the first order terms. This gives, as the approximate solution:


Moreover, since  at small densities, the range term contribution in the above equation approximately equals  at low densities. The first of the terms in the approximate expression given by Eq. (28) is an increasing function while the second one is a decreasing function of density. Their interplay generates the observed curve shape - the repulsive contribution being responsible for the small hump in an otherwise linear fall, at small densities (). While the attractive term would have produced a linear decrease right away, the role of repulsive contribution is to impede this decrease, hence producing the hump. At moderately higher densities, however, the contribution from the second term outweighs the first, which is why we see a linear drop with density. This is observed to be the case, till around . At still larger densities, the approximation ( ) breaks down, though our binomial expansion is still valid. Since scalar density falls slower than number density, the term will fall faster than . This is responsible for the change in slope of the curve at intermediate densities, where a linear fall with density is no longer obeyed. So, at intermediate and large densities, the manner of the variation is dictated by the scalar densities, in the range term. From equations (19) to (21), one would expect the mass modifications of mesons to be insensitive to isospin asymmetry, since, e.g. in this nuclear matter case, the dispersion relation bears isospin-symmetric terms like (+). (This is in stark contrast with earlier treatments of kaons and antikaons sambuddha1 (); sambuddha2 () as well as the mesons arindam () within this effective model, where the dispersion relations had terms like () or (), which contributed to asymmetry.) However, the meson effective mass is observed to depend on asymmetry in Fig.1, though the dependence is weak. For example, the values of , for the isospin symmetric situation , are , and MeV, respectively, at while the corresponding numbers for the (completely) asymmetric situation are , and MeV, respectively, at . Since the -dependence of and contributions must be identical, one can reason from Fig.1 that this isospin dependence of meson effective mass is almost entirely due to the range term ( ( + )), which was expected to be isospin symmetric. This apparently counterintuitive behavior has been observed earlier in arvDprc (), in the context of mesons. This is because, the value of ( + ) turns out to be different for symmetric and asymmetric situations, contrary to naive expectations. Since the scalar-isovector  meson () is responsible for introducing isospin asymmetry in this effective hadronic model Pap_prc99 (), owing to the equations of motion of the scalar fields being coupled, the values of the other scalar fields turn out to be different in the symmetric and asymmetric cases. The same is also reflected in the values of the scalar densities calculated from these scalar fields, which leads to the observed behavior.

Additionally, we observe from a comparison of figures 1a and 1b that the magnitude of the meson mass drop decreases with an increase in temperature from T=0 to T=100 MeV. For example, in the symmetric () situation, at , the meson mass values, at , are MeV respectively, which grow to MeV respectively, at MeV. Likewise, with , the corresponding values read MeV at , while the same numbers, at MeV, are MeV. Thus, though small, there is a definite reduction in the magnitude of the mass drops, as we go from to MeV, in each of these cases. This behavior can be understood, from the point of view of the temperature variation of scalar condensates, in the following manner. It is observed that the scalar fields decrease with an increase in temperature from to MeV arvDprc (); mamD2004 (). In Ref.mamD2004 (), the same effect was understood as an increase in the nucleon mass with temperature. The equity of the two arguments can be seen by invoking the expression relating the baryon mass to the scalar fields’ magnitude, Pap_prc99 (). The temperature dependence of the scalar fields (, , ) have been studied within the model in arvDprc (), which are observed to be different for the zero and finite baryon densities. At zero baryon density, the magnitudes of the scalar fields and are observed to remain almost constant upto a temperature of about 125 MeV, above which these are observed to decrease with temperature. This behaviour can be understood from the expression of given by equation (15) for the situation of zero density, i.e. for =0, which decreases with increase with temperature. The temperature dependence of the scalar density in turn determines the behaviour of the scalar fields. The scalar fields which are solved from their equations of motion, behave in a similar manner as the scalar density. At finite densities, i.e., for nonzero values of the effective chemical potential, , however, the temperature dependence of the scalar density is quite different from the zero density situation. For finite baryon density, with increase in temperature, there are contributions also from higher momenta, thereby increasing the denominator of the integrand on the right hand side of the baryon scalar density given by Eq. (15). This leads to a decrease in the scalar density. At finite baryon densities, the competing effects of the thermal distribution functions and the contributions from the higher momentum states give rise to the temperature dependence of the scalar density, which in turn determine the behaviour of the and fields with temperature. These scalar fields are observed to have an initial increase in their magnitudes upto a temperature of around 125-150 MeV, followed by a decrease as the temperature is further raised arvDprc (). This kind of behavior of the scalar field on temperature at finite densities has also been observed in the Walecka model in Ref. LiKoBrown (). In fact, we point out that a decrease in the scalar condensates with an increase in temperature, though small in the hadronic regime ( MeV), is well-known in general model-independent terms Gerber_Leutwyler_NPB_1989 (), and was also observed to be a consistent feature of all linear sigma models in the model-independent work of Röder et al. roder (). Since these scalar fields serve as an input in calculating the scalar densities Pap_prc99 (), a decrease in the magnitude of the latter accompanies a decrease in the former, at larger temperatures. From the point of view of the dispersion relations, this results in a decrease in the coefficient , and owing to the inverse dependence of on , increases the value of in the finite temperature case, as compared to the situation. Or stating it differently, the difference of this , from the vacuum value, i.e the mass drop, decreases. Thus, from a physical viewpoint, since the origin of these mass drops is the attractive in-medium interactions, one can say that the reduction in the mass drop magnitudes is due to a weakening of the attractive strength of these in-medium interactions, represented in these models by a reduction in the quark condensates with increasing temperatures. Additionally, it is also observed from Fig.1 that isospin dependence of the meson mass, feeble anyways, weakens further with temperature. For example, as mentioned earlier, mass of the mesons at , for the and cases, are and MeV respectively (a difference of MeV), when , which changes to and MeV at MeV (a MeV difference). This is, once again, due to a decrease in the magnitude of , with temperature, at any fixed value of the parameters and . In particular, the difference between the value of  for the and cases is observed to decrease with temperature (as was shown explicitly in Ref. arvDprc ()). Since asymmetry is introduced through , a decrease in the difference of the values of , between symmetric and asymmetric situations, with temperature, follows naturally.

Next, we generalize our analysis by including hyperonic degrees of freedom as well, in the medium. However, as mentioned previously, in the ensuing discussion of mesons in hyperonic matter, we focus predominantly on the new physics arising via the introduction of strangeness in the medium, since, e.g. the weak dependence on isospin asymmetry, or a weak reduction in the mass drop magnitudes at higher temperatures, has, in principle, the same explanation that stood in the nuclear matter context.

Figure 2: (Color Online) The various contributions to the effective mass of  and  mesons, as a function of baryonic density, for typical values of temperature ( MeV) and strangeness fraction (). In each case, the asymmetric situation (), as described in the legend, is also compared against the symmetric situation (), represented by dotted lines.

Fig.2 shows the variation of the in-medium mass of the mesons, along with the various individual contributions, in hyperonic medium as a function of baryonic density, for typical values of temperature, isospin asymmetry parameter and strangeness fraction. The most drastic consequence of the inclusion of hyperons in the medium is that, the mass degeneracy of  and  now stands broken. For example, the values of at , are observed to be , , and MeV, respectively, as can be seen from the figure. Thus, except at vacuum (), mass difference of  and  is non-zero at finite , growing in magnitude with density. This mass degeneracy breaking is a direct consequence of non-zero contributions from the Weinberg-Tomozawa term, which follows from equations (11), (17) and (20). The same may be reconciled with Fig.2, from which it follows that except for this Weinberg-Tomozawa term acquiring equal and opposite values for these mesons, all other terms are absolutely identical for them. Once again, on the basis of the following analysis of the meson dispersion relations at zero momentum, we insist that this observed behavior is perfectly consistent with expectations.

Figure 3: (Color Online) meson mass degeneracy breaking in hyperonic matter, as a function of baryonic density, for typical values of the other parameters ( MeV, and ). The medium masses of  and  are observed to fall symmetrically about (see text).

The general solution of the equivalent quadratic equation, eqn.(18), is:


where we have disregarded the negative root, and have performed a binomial expansion of , with . Upon feeding numerical values, we observe that the expansion parameter is much smaller than unity for our entire density variation, which justifies the validity of this expansion for our analysis. The same exercise also allows us to safely disregard higher-order terms, and simply write:


Further, with the same justification as in the nuclear matter case, both and can also be expanded binomially. For example, for the second term, this gives


where the contribution from these higher order terms is smaller owing to the large denominator, prompting us to retain only the first order terms. Here, we point out that since , ( sign for  and sign for ), this term, which represents the Weinberg-Tomozawa contribution to the dispersion relations, breaks the degeneracy of the mesons. On the other hand, the first term in eqn.(30) is common between  and  mesons. Thus, the general solution of the meson dispersion relations in the hyperonic matter context, can be written as:




Thus, , and , (where is necessarily positive), which readily explains why the mass of  drops more than that of , with density. Also, this formulation readily accounts for symmetric fall of the medium masses of  and  mesons, about in Fig.3. Trivially, it may also be deduced that is extinguished in nuclear matter, so that the mass degeneracy of  and  is recovered from these equations. However, we observe that though the curve corresponding to is exactly bisecting the masses of  and  at small densities, this bisection is no longer perfect at high densities. This can be understood in the following manner. In essence, we are comparing the density dependence of a function , with the functions . Since scalar densities fall sub-linearly with the number density, it follows that the fall can not be absolutely symmetric at any arbitrary density. In fact, since a decreasing function of scalar density will fall faster than that of a number density, one expects to lean away from the curve for  meson and towards the curve corresponding to , which is exactly what is observed in Fig.3.

Figure 4: (Color Online) The sensitivity of the medium mass of (a) , and (b)  mesons, to the strangeness fraction , at typical values of temperature ( MeV) and isospin asymmetry parameter ().

Since this disparity between  and  originates from the Weinberg-Tomozawa term, whose magnitude is directly proportional to the hyperonic number densities, one expects this disparity to grow with an increase in the strangeness content of the medium. The same also follows from the above formulation, since the mass splitting between the two, , should grow with both, at fixed  (i.e. a larger proportion of hyperons), as well as with  at fixed (non-zero) (i.e. a larger hyperonic density), in accordance with eqn.(34). Naively, one expects this dependence to be shared by the two mesons; however, closer analysis reveals that this is not the case, as shown in Fig.4 where we consider the dependence of the meson medium mass. It is observed that while the dependence for  meson is quite pronounced, the same for the  is conspicuously subdued. Counterintuitive as it may apparently be, this observed behavior follows from the above formulation. Since , at any fixed density, the first of these is a decreasing function of , while the second increases with . These opposite tendencies are responsible for the weak dependence of  meson. On the other hand, because , the two effects add up to produce a heightened overall decrease with for the  meson, as we observe in the figure.

Figure 5: (Color Online) (a) A comparison of medium mass of mesons, in nuclear and hyperonic matter, maintaining the same values of parameters as before. (b) Comparison of the nuclear matter solution, , with the in hyperonic matter, as observed in Fig.3.

A comparison of these hyperonic matter solutions with the nuclear matter solutions is shown in Fig.5 for typical values of the parameters. It is observed from Fig.5a that at small densities, the nuclear matter curve, which represents both  and , bisects the mass degeneracy breaking curve, akin to in Fig.3. However, at larger densities, both these curves drop further than the curve. From a casual comparison of expressions, it appears that the nuclear matter solution, given by eqn.(28), also enters the hyperonic matter solution, eqn.(33), albeit as a subset of . It is tempting to rearrange the latter then, such that this nuclear matter part is separated out, generating an expression of the type