Research Article𝐷𝑆

Mesons in Asymmetric Hot and Dense Hadronic Matter

Divakar Pathak and Amruta Mishra

Department of Physics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110 016, India

Correspondence should be addressed to Amruta Mishra; amruta@physics.iitd.ac.in

Received 8 June 2015; Revised 9 August 2015; Accepted 18 August 2015

Academic Editor: Juan José Sanz-Cillero

Copyright © 2015 D. Pathak and A. Mishra. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited. The publication of this article was funded by SCOAP3.

The in-medium properties of 𝐷𝑆mesons 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, the𝐷

𝑆mesons are observed to experience net attractive interactions in a dense hadronic medium, hence reducing themasses of the𝐷+

𝑆

and 𝐷−𝑆mesons 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 massdegeneracy of the 𝐷

𝑆mesons 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 between𝐷+𝑆and𝐷−

𝑆mesons is also observed to

grow with an increase in baryonic density and strangeness content of the medium. However, all medium effects analyzed are foundto 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 upcomingCompressed Baryonic Matter (CBM) experiment at the future Facility for Antiproton and Ion Research (FAIR), GSI, where matterat high baryonic densities is planned to be produced.

1. Introduction

An effective description of hadronic matter is fairly commonin low-energyQCD [1–3]. Realizing that baryons andmesonsconstitute the effective degrees of freedom in this regime,it is quite sensible to treat QCD at low-energies as aneffective theory of these quark bound states [1].This approachhas been vigorously pursued in various incarnations overthe years, with the different adopted strategies representingmerely different manifestations of the same underlying phi-losophy. The actual manifestations range from the quark-meson coupling model [4, 5], phenomenological, relativisticmean-field theories based on the Walecka model [6], alongwith their subsequent extensions, the method of QCD sumrules [7–9], as well as the coupled channel approach [10, 11]for treating dynamically generated resonances, which hasfurther evolved into more specialized forms, namely, thelocal hidden gauge theory [12, 13], and formalisms based onincorporating heavy-quark spin symmetry (HQSS) [14, 15]into the coupled channel framework [16–20]. Additionally,

the method of chiral-invariant Lagrangians [1] (which willalso be embraced in this work) has developed over theyears into a very successful strategy. This method uses aneffective field theoretical model in which the specific formof hadronic interactions is dictated by symmetry principlesand the physics governed predominantly by the dynamicsof chiral symmetry—its spontaneous breakdown implyinga nonvanishing scalar condensate ⟨𝑞𝑞⟩ in vacuum. Onenaturally expects then that the hadrons composed of thesequarks would also be modified in accordance with thesecondensates [2, 3]. But while all hadrons would be subjecttomediummodifications from this perspective, pseudoscalarmesons have a special role in this context. In accordancewith Goldstone’s theorem [1], spontaneous breaking of chiralsymmetry leads to the occurrence of massless pseudoscalarmodes, the so-called Goldstone bosons, which are generallyidentified with the spectrum of light pseudoscalar mesons,like the pions or kaons and antikaons [2, 3]. In a strict sense,however, none of these physical mesons is a true Goldstonemode, since they are all massive, while Goldstone modes

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2015, Article ID 697514, 16 pageshttp://dx.doi.org/10.1155/2015/697514

2 Advances in High Energy Physics

are supposed to be massless [21]. The origin of the massesof these mesons is related to the nonzero masses of lightquarks, as can be easily discerned from the Gell-Mann-Oakes-Renner (GOR) relations and, hence, from explicitsymmetry breaking terms (or explicit mass terms) in thechiral effective framework [1]. In fact, if one considers thelimiting situation of vanishing quark masses (𝑚𝑞 → 0), themasses of these pseudoscalar mesons would also vanish,so that the perfect Goldstone modes are indeed recovered.For this reason, the physically observed light pseudoscalarmesons are dubbed pseudo-Goldstone bosons [21]. In purelythis sense, therefore, there is an inherent similarity betweenall these classes of pseudoscalar mesons; masses are acquiredthrough explicit quarkmass terms, and only themagnitude ofthese mass terms differs between these cases, being small forthe pions (since 𝑚

𝑢, 𝑚

𝑑< 10MeV), comparatively larger for

the kaons and antikaons (since 𝑚𝑠∼ 150MeV), and appre-

ciably larger for the charmed pseudoscalar mesons. Thus,as we advance from the pions to the strange pseudoscalarmesons, with increasing mass, these mesons depart morefrom the ideal Goldstone mode character pertaining to thetheorem and there is considerable departure of the charmedpseudoscalar mesons from Goldstone mode behaviour dueto the explicit chiral symmetry breaking arising from thelarge charm quark mass (𝑚

𝑐≃ 1.3GeV). An understanding

of the in-medium properties of the pseudoscalar mesonshas been an important topic of research, both theoreticallyand experimentally. Within the chiral effective approach,the pseudoscalar mesons are modified in the mediumdue to the modifications of the quark condensates in thehadronic medium. For pions, it is observed, however, thatmedium effects for them are weakened by the smallnessof explicit symmetry breaking terms [2, 3]. Considerablydetailed analysis of medium effects has been performedover the years, particularly in a chiral 𝑆𝑈(3) approach, forstrange pseudoscalar mesons (kaons and antikaons) [22–25].For studying the charmed mesons, one needs to generalizethe 𝑆𝑈(3) model to 𝑆𝑈(4), in order to incorporate theinteractions of the charmedmesons to the light hadrons. Sucha generalization from 𝑆𝑈(3) to 𝑆𝑈(4) was initially done in[26], where the interaction Lagrangian was constructed forthe pseudoscalar mesons for 𝑆𝑈(4) from a generalization ofthe lowest order chiral 𝑆𝑈(3) Lagrangian. Since the chiralsymmetry is explicitly broken for the 𝑆𝑈(4) case due to thelargemass of the charmquark (𝑚

𝑐≃ 1.3GeV), which ismuch

larger than the masses of the light quarks, for the study ofthe charmed (𝐷) pseudoscalar mesons [27–29], we adopt thephilosophy of generalizing the chiral 𝑆𝑈(3)model to 𝑆𝑈(4) toderive the interactions of thesemesons with the light hadronsbut use the observed masses of these heavy hadrons as wellas empirical/observed values of their decay constants [30].With all these studies proving to be informative, the mostnatural direction of extension of this approach would be toanalyze these medium effects for a strange-charmed system(the 𝐷𝑆 mesons). Apart from pure theoretical interest, anunderstanding of the medium modifications of 𝐷𝑆 mesonsis important, since these can make a considerable differenceto experimental observables in the (ongoing and future)relativistic heavy ion collision experiments, besides being

significant in questions concerning their production andtransport in such experimental situations. For instance, in arecent work, He et al. [31] have shown that the modificationsof the 𝐷

𝑆meson spectrum can serve as a useful probe for

understanding key issues regarding hadronization in heavyion collisions. It is suggested that, by comparing observablesfor𝐷 and𝐷𝑆 mesons, it is possible to constrain the hadronictransport coefficient.This comparison is useful since it allowsfor a clear distinction between hadronic and quark-gluonplasma behavior.

However, as far as the existing literature on this strange-charmed system of mesons is concerned, we observe thatonly the excited states of 𝐷𝑆 mesons have received con-siderable attention, predominantly as dynamically generatedresonances in various coupled channel frameworks [32,33]. One must bear in mind that, in certain situations, amolecular interpretation of these excited states (resonances)is more appropriate [34] for an explanation of their observed,larger than expected lifetimes. From this perspective, awhole plethora of possibilities have been entertained for theexcited𝐷

𝑆states, the standard quark-antiquark picture aside.

These include their description as molecular states, borneout of two mesons, four-quark states, or the still furtherexotic possibilities—as two-diquark states and as a mixtureof quark-antiquark and tetraquark states [32]. Prominentamong these is a treatment of𝐷∗

𝑆0(2317) as a𝐷𝐾 bound state

[35, 36], 𝐷𝑆1(2460) as a dynamically generated 𝐷∗𝐾 reso-

nance [37], and 𝐷∗𝑆2(2573) being treated within the hidden

local gauge formalism in coupled channel unitary approach[38, 39], as well as the vector 𝐷∗

𝑆states and the 𝐷+

𝑆(2632)

resonance treated in a multichannel approach [40]. Theformer three have also been covered consistently under thefour-quark picture [41]. So, while considerable attention hasbeen paid towards dynamically generating the higher excitedstates of the 𝐷𝑆 mesons, there is a conspicuous dearth [42]of available information about the medium modifications ofthe lightest pseudoscalar 𝐷𝑆 mesons, 𝐷𝑆(1968.5), the onethat we know surely is well described within the quark-antiquark picture. In fact, to the best of our knowledge, theentire existing literature about the (𝐽

𝑃= 0

−)𝐷

𝑆mesons

in a hadronic medium is limited to the assessment of theirspectral distributions andmedium effects on the dynamicallygenerated resonances borne out of the interaction of these𝐷

𝑆mesons with other hadron species, in the coupled channel

analyses of [42–45]. This situation is quite unlike their open-charm, non-strange counterparts, the𝐷mesons, which boastof a sizeable amount of literature having been extensivelyinvestigated using a multitude of approaches over the years.In stark contrast, the available literature concerning the in-medium behavior of pseudoscalar 𝐷

𝑆mesons can at best be

described as scanty, and there is need for more work on thissubject. If one considers this problem from the point of viewof the (extended) chiral effective approach, this scantinessis most of all because of the lack of a proper frameworkwhere the relevant form of the interactions for the𝐷𝑆 mesonswith the light hadrons (or in more generic terms, of meson-baryon interactions with the charm sector covered), basedon arguments of symmetry and invariance, could be writtendown. Clearly, such interactions would have to be based on

Advances in High Energy Physics 3

𝑆𝑈(4) symmetry and bear all these pseudoscalar mesons andbaryons in 15-plet and 20-plet representations, respectively,withmeson-baryon interaction terms still in accordance withthe general framework for writing chiral-invariant structures,as well as bearing appropriate symmetry breaking termsobeying the requisite transformation behavior under chiraltransformations [1], which is quite a nontrivial problem. Oflate, such formalisms have been proposed in [43, 46], as anextension of the frameworks based on chiral Lagrangians,where 𝑆𝑈(4) symmetry forms the basis for writing downthe relevant interaction terms. However, since the mass ofthe charm quark is approximately 1.3 GeV [47], which isconsiderably larger than that of the up, down, and strangequarks, the 𝑆𝑈(4) symmetry is explicitly broken by this largecharm quark mass. Hence, this formalism only uses thesymmetry to derive the form of the interactions, whereasan explicit symmetry breaking term accounts for the largequark mass through the introduction of mass terms of therelevant (𝐷 or 𝐷

𝑆) mesons. Also, 𝑆𝑈(4) symmetry being

badly broken implies that any symmetry and order in themasses and decay constants, as predicted on the basis of𝑆𝑈(4) symmetry, would not hold in reality. The same isacknowledged in this approach [46] and, as has been alreadymentioned, one does not use the masses and decay constantsas expected on the basis of 𝑆𝑈(4) symmetry but rather theirobserved Particle Data Group (PDG) [47] values. Overall,therefore, 𝑆𝑈(4) symmetry is treated (appropriately) as beingbroken in this approach. Also, it is quite well-known, throughboth model-independent [48] and model-dependent [29]calculations, that the light quark condensates (⟨𝑢𝑢⟩, ⟨𝑑𝑑⟩) aremodified significantly in a hadronic medium with mediumparameters like density and temperature; the strange quarkcondensate ⟨𝑠𝑠⟩ is comparatively stolid and its variation issignificantly more subdued, while upon advancing to thecharm sector the variation in the charmed quark condensate⟨𝑐𝑐⟩ is altogether negligible in the entire hadronic phase [48].These observations form the basis for treating the charmdegrees of freedom of open-charm pseudoscalar mesons asfrozen in the medium, as was the case in the treatments of[28, 29, 46].Thus, as we advance from pions and kaons to thecharmed pseudoscalar mesons, the generalization is perfectlynatural but with the aforementioned caveats. Provided allthese aspects are taken into account, a generalization ofthis chiral effective framework to open-charm pseudoscalarmesons is quite reasonable and sane, and the predictions fromsuch an extended chiral effective approach bode very well[28] with alternative calculations based on the QCD sum ruleapproach, quark-meson coupling model, coupled channelapproach, and studies of quarkonium dissociation usingheavy-quark potentials from lattice QCD at finite tempera-tures. Additionally, it is interesting to note that this approach,followed in [28, 29, 46] for the charmedpseudoscalarmesons,has recently been extended to the bottom sector and used tostudy the medium behavior of the open bottom pseudoscalar𝐵, 𝐵, and 𝐵

𝑆mesons [49, 50]. The inherent philosophy

beneath this extension continues to be the same; the dynam-ics of the heavy quark/antiquark is treated as frozen, and theinteractions of the light quark (or antiquark) of the meson,with the particles constituting the medium, are responsible

for the medium modifications. With this subsequent gener-alization as well, the physics of the medium behavior thatfollows from this approach is in agreement [49] with worksbased on alternative, independent approaches, like the heavymeson effective theory, quark-meson coupling model, andthe QCD sum rule approach. Thus, these aforementioned,prior works based on the generalization of the original chiraleffective approach to include heavy flavored mesons, aretotally concordant with results from alternative approachesfollowed in the literature, which lends an aura of credibilityto this strategy. Given this backdrop, it is clear that theseformalisms wipe out the reason why such an investigation forthe𝐷𝑆 mesons within the effective hadronic model, obtainedby generalizing the chiral 𝑆𝑈(3)model to 𝑆𝑈(4), has not beenundertaken till date and permit this attempt to fill the void.

We organize this paper as follows. In Section 2, we outlinethe chiral 𝑆𝑈(3)

𝐿× 𝑆𝑈(3)

𝑅model (and its generalization

to the 𝑆𝑈(4) case) used in this investigation. In Section 3,the Lagrangian density for the 𝐷

𝑆mesons, within this

extended framework, is explicitly written down and is usedto derive their in-medium dispersion relations. In Section 4,we describe and discuss our results for the in-mediumproperties of 𝐷

𝑆mesons, first in the nuclear matter case and

then in the hyperonic matter situation, following which webriefly discuss the possible implications of these mediummodifications. Finally, we summarize the entire investigationin Section 5.

2. The Effective Hadronic Model

As mentioned previously, this study is based on a general-ization of the chiral 𝑆𝑈(3)𝐿 × 𝑆𝑈(3)𝑅 model [51], to 𝑆𝑈(4).We summarize briefly the rudiments of the model, whilereferring the reader to [51, 52] for the details. This is aneffective hadronic model of interacting baryons and mesons,based on a nonlinear realization of chiral symmetry [53, 54],where chiral invariance is used as a guiding principle, indeciding the form of the interactions [55–57]. Additionally,the model incorporates a scalar dilaton field, 𝜒, to mimic thebroken scale invariance of QCD [52]. Once these invariancearguments determine the form of the interaction terms, oneresorts to a phenomenological fitting of the free parametersof the model, to arrive at the desired effective Lagrangiandensity for these hadron-hadron interactions. The generalexpression for the chiral model Lagrangian density reads

L =Lkin +∑𝑊

LBW +Lvec +L0 +Lscale break

+LSB.

(1)

In (1),Lkin is the kinetic energy term, whileLBW denotes thebaryon-meson interaction term. Here, baryon-pseudoscalarmeson interactions generate the baryon masses. Lvec treatsthe dynamical mass generation of the vector mesons throughcouplings with scalar mesons. The self-interaction terms ofthese mesons are also included in this term.L

0contains the

meson-meson interaction terms, which induce spontaneousbreaking of chiral symmetry. Lscale break introduces scaleinvariance breaking, via a logarithmic potential term in the

4 Advances in High Energy Physics

scalar dilaton field, 𝜒. Finally, LSB refers to the explicitsymmetry breaking term. This approach has been employedextensively to study the in-medium properties of hadrons,particularly pseudoscalar mesons [22–25]. As was observedin Section 1 as well, this would be most naturally extended tothe charmed (nonstrange and strange) pseudoscalar mesons.However, that calls for this chiral 𝑆𝑈(3) formalism to begeneralized to 𝑆𝑈(4), which has been addressed in [43,46]. For studying the in-medium behavior of pseudoscalarmesons, the following contributions need to be analyzed[28, 29, 46]:

L =LWT +L1st Range +L𝑑1

+L𝑑2

+LSME. (2)

In (2),LWT denotes theWeinberg-Tomozawa term, given bythe expression [46]

LWT = −1

2[𝐵

𝑖𝑗𝑘𝛾

𝜇((Γ

𝜇)

𝑙

𝑘

𝐵𝑖𝑗𝑙+ 2(Γ

𝜇)

𝑙

𝑗

𝐵𝑖𝑙𝑘)] , (3)

with repeated indices summed over. Baryons are representedby the tensor 𝐵𝑖𝑗𝑘, which is antisymmetric in its first twoindices [43]. The indices 𝑖, 𝑗, and 𝑘 run from 1 to 4, and onecan directly read the quark content of a baryon state, withthe following identification: 1 ↔ 𝑢, 2 ↔ 𝑑, 3 ↔ 𝑠, 4 ↔ 𝑐.However, the heavier, charmed baryons are discounted fromthis analysis. In (3), Γ

𝜇is defined as

Γ𝜇

= −𝑖

4(𝑢

†(𝜕𝜇𝑢) − (𝜕𝜇𝑢

†) 𝑢 + 𝑢 (𝜕𝜇𝑢

†) − (𝜕𝜇𝑢) 𝑢

†) ,

(4)

where the unitary transformation operator, 𝑢 = exp(𝑖𝑀𝛾5/√2𝜎0), is defined in terms of the matrix of pseudoscalarmesons, 𝑀 = (𝑀𝑎𝜆𝑎/√2), 𝜆𝑎 representing the generalizedGell-Mann matrices. Further, LSME is the scalar mesonexchange term, which is obtained from the explicit symmetrybreaking term

LSB = −1

2Tr (𝐴

𝑝(𝑢𝑋𝑢 + 𝑢

†𝑋𝑢

†)) , (5)

where 𝐴𝑝 = (1/

√2) diag[𝑚2𝜋𝑓𝜋, 𝑚

2

𝜋𝑓𝜋, (2𝑚

2

𝐾𝑓𝐾 − 𝑚

2

𝜋𝑓𝜋),

(2𝑚2

𝐷𝑓

𝐷−𝑚

2

𝜋𝑓

𝜋)] and𝑋 refers to the scalar meson multiplet

[28]. Also, the first range term is obtained from the kineticenergy term of the pseudoscalar mesons and is given by theexpression

L1st Range = Tr (𝑢𝜇𝑋𝑢

𝜇𝑋 + 𝑋𝑢

𝜇𝑢

𝜇𝑋) , (6)

where 𝑢𝜇= −𝑖((𝑢

†(𝜕

𝜇𝑢) − (𝜕

𝜇𝑢

†)𝑢) − (𝑢(𝜕

𝜇𝑢

†) − (𝜕

𝜇𝑢)𝑢

†))/4.

Lastly, the 𝑑1and 𝑑

2range terms are

L𝑑1

=𝑑

1

4(𝐵𝑖𝑗𝑘𝐵

𝑖𝑗𝑘(𝑢𝜇)

𝑙

𝑚

(𝑢𝜇)

𝑚

𝑙) , (7)

L𝑑2

=𝑑

2

2[𝐵𝑖𝑗𝑘(𝑢𝜇)

𝑙

𝑚

((𝑢𝜇)

𝑚

𝑘𝐵

𝑖𝑗𝑙+ 2(𝑢

𝜇)

𝑚

𝑗𝐵

𝑖𝑙𝑘)] . (8)

Adopting the mean-field approximation [6, 52], the effectiveLagrangian density for scalar and vector mesons simplifies;

the same is used subsequently to derive the equations ofmotion for the nonstrange scalar-isoscalar meson 𝜎, scalar-isovector meson 𝛿, and strange scalar meson 𝜁 and for thevector-isovector meson 𝜌, nonstrange vector meson 𝜔, andstrange vector meson 𝜙, within this model.

The 𝐷𝑆meson interaction Lagrangian density and in-

medium dispersion relations, as they follow from the abovegeneral formulation, are described next.

3. 𝐷𝑆

Mesons in Hadronic Matter

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

Ltotal =Lfree +Lint. (9)

ThisLfree is the free Lagrangian density for a complex scalarfield (which corresponds to the 𝐷

𝑆mesons in this case) and

reads

Lfree = (𝜕𝜇𝐷

+

𝑆) (𝜕𝜇𝐷

−

𝑆) − 𝑚

2

𝐷𝑆

(𝐷+

𝑆𝐷

−

𝑆) . (10)

On the other hand,Lint is determined to be

Lint = −𝑖

4𝑓2

𝐷𝑆

[(2 (Ξ0

𝛾𝜇Ξ

0+ Ξ

−

𝛾𝜇Ξ

−) + Λ

0

𝛾𝜇Λ

0

+ Σ+

𝛾𝜇Σ

++ Σ

0

𝛾𝜇Σ

0+ Σ

−

𝛾𝜇Σ

−) (𝐷

+

𝑆(𝜕

𝜇𝐷

−

𝑆)

− (𝜕𝜇𝐷

+

𝑆)𝐷

−

𝑆)] +

𝑚2

𝐷𝑆

√2𝑓𝐷𝑆

[(𝜁+ 𝜁

𝑐) (𝐷

+

𝑆𝐷

−

𝑆)]

−√2

𝑓𝐷𝑆

[(𝜁+ 𝜁

𝑐) ((𝜕

𝜇𝐷

+

𝑆) (𝜕

𝜇𝐷

−

𝑆))] +

𝑑1

2𝑓2

𝐷𝑆

[(𝑝𝑝

+ 𝑛𝑛 + Λ0

Λ0+ Σ

+

Σ++ Σ

0

Σ0+ Σ

−

Σ−+ Ξ

0

Ξ0

+ Ξ−

Ξ−) ((𝜕

𝜇𝐷

+

𝑆) (𝜕

𝜇𝐷

−

𝑆))]

+𝑑

2

2𝑓2

𝐷𝑆

[(2 (Ξ0

Ξ0+ Ξ

−

Ξ−) + Λ

0

Λ0+ Σ

+

Σ+

+ Σ0

Σ0+ Σ

−

Σ−) ((𝜕

𝜇𝐷

+

𝑆) (𝜕

𝜇𝐷

−

𝑆))] .

(11)

In this expression, the first term (with coefficient −𝑖/4𝑓2𝐷𝑆

)is the Weinberg-Tomozawa term, obtained from (3), thesecond term (with coefficient 𝑚2

𝐷𝑆

/√2𝑓𝐷𝑆

) is the scalarmeson exchange term, obtained from the explicit symmetrybreaking term of the Lagrangian (see (5)), the third term(with coefficient −√2/𝑓

𝐷𝑆

) is the first range term (see (6)),and the fourth and fifth terms (with coefficients (𝑑

1/𝑓

2

𝐷𝑆

) and(𝑑

2/𝑓2

𝐷𝑆

), resp.) are the 𝑑1 and 𝑑2 terms, calculated from (7)and (8), respectively. Also, 𝜁 = 𝜁 − 𝜁

0is the fluctuation of

the strange scalar field from its vacuum value.Themean-fieldapproximation, mentioned earlier, is a useful, simplifying

Advances in High Energy Physics 5

measure in this context, since it permits us to have the fol-lowing replacements:

𝐵𝑖𝐵

𝑗→ ⟨𝐵

𝑖𝐵

𝑗⟩ ≡ 𝛿

𝑖𝑗𝜌

𝑠

𝑖,

𝐵𝑖𝛾

𝜇𝐵

𝑗→ ⟨𝐵

𝑖𝛾

𝜇𝐵

𝑗⟩ = 𝛿

𝑖𝑗(𝛿

0

𝜇(𝐵

𝑖𝛾

𝜇𝐵

𝑗)) ≡ 𝛿

𝑖𝑗𝜌

𝑖.

(12)

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

𝜌𝑖=𝛾

𝑠

(2𝜋)3∫𝑑

3𝑘(

1

exp ((𝐸∗𝑖(𝑘) − 𝜇

∗

𝑖) /𝑇) + 1

−1

exp ((𝐸∗𝑖(𝑘) + 𝜇

∗

𝑖) /𝑇) + 1

) ,

(13)

𝜌s𝑖=𝛾

𝑠

(2𝜋)3∫𝑑

3𝑘𝑚

∗

𝑖

𝐸∗

𝑖(𝑘)(

1

exp ((𝐸∗𝑖(𝑘) − 𝜇

∗

𝑖) /𝑇) + 1

+1

exp ((𝐸∗𝑖(𝑘) + 𝜇

∗

𝑖) /𝑇) + 1

) .

(14)

In the above, 𝑚∗𝑖and 𝜇∗

𝑖are the effective mass and effective

chemical potential of the 𝑖th baryon, given as𝑚∗𝑖= −(𝑔

𝜎𝑖𝜎 +

𝑔𝜁𝑖𝜁 + 𝑔

𝛿𝑖𝛿), 𝜇∗

𝑖= 𝜇

𝑖− (𝑔

𝜌𝑖𝜏

3𝜌 + 𝑔

𝜔𝑖𝜔 + 𝑔

𝜙𝑖𝜙), 𝐸∗

𝑖(𝑘) =

(𝑘2+ 𝑚

∗

𝑖

2)

1/2, and 𝛾𝑠= 2 is the spin degeneracy factor. One

can find the equations of motion for the 𝐷+𝑆and 𝐷−

𝑆mesons,

by the use of Euler-Lagrange equations on this Lagrangiandensity. The linearity of these equations follows from (11),which allows us to assume plane wave solutions (∼𝑒𝑖(�⃗�⋅ ⃗𝑟−𝜔𝑡)),and hence, “Fourier transform” these equations, to arrive atthe in-medium dispersion relations for the𝐷𝑆 mesons.Thesehave the general form

−𝜔2+ �⃗�

2+ 𝑚

2

𝐷𝑆

− Π (𝜔,�⃗�) = 0, (15)

where𝑚𝐷𝑆

is the vacuummass of the𝐷𝑆mesons andΠ(𝜔, |�⃗�|)

is the self-energy of the𝐷𝑆mesons in the medium. Explicitly,

the latter reads

Π(𝜔,�⃗�) = [(

𝑑1

2𝑓2

𝐷𝑆

(𝜌𝑠

𝑝+ 𝜌

𝑠

𝑛+ 𝜌

𝑠

Λ+ 𝜌

𝑠

Σ+ + 𝜌

𝑠

Σ0 + 𝜌

𝑠

Σ−

+ 𝜌𝑠

Ξ0 + 𝜌

𝑠

Ξ−)) + (

𝑑2

2𝑓2

𝐷𝑆

(2 (𝜌𝑠

Ξ0 + 𝜌

𝑠

Ξ−) + 𝜌

𝑠

Λ+ 𝜌

𝑠

Σ+

+ 𝜌𝑠

Σ0 + 𝜌

𝑠

Σ−)) − (

√2

𝑓𝐷𝑆

(𝜁+ 𝜁

𝑐))] (𝜔

2− �⃗�

2)

± [1

2𝑓2

𝐷𝑆

(2 (𝜌Ξ0 + 𝜌

Ξ−) + 𝜌

Λ+ 𝜌

Σ+ + 𝜌

Σ0 + 𝜌

Σ−)]𝜔

+ [

𝑚2

𝐷𝑆

√2𝑓𝐷𝑆

(𝜁+ 𝜁

𝑐)] ,

(16)

where the + and − signs in the coefficient of 𝜔 refer to 𝐷+𝑆

and 𝐷−𝑆, respectively, and we have used (12) to simplify the

bilinears. In the rest frame of these mesons (i.e., setting |�⃗�| =0), these dispersion relations reduce to

−𝜔2+ 𝑚

2

𝐷𝑆

− Π (𝜔, 0) = 0, (17)

which is a quadratic equation in 𝜔, that is, of the form 𝐴𝜔2 +𝐵𝜔 + 𝐶 = 0, where the coefficients 𝐴, 𝐵, and 𝐶 depend onvarious interaction terms in (15) and read

𝐴 = [1 + (𝑑

1

2𝑓2

𝐷𝑆

∑

(𝑁+𝐻)

𝜌𝑠

𝑖)

+ (𝑑

2

2𝑓2

𝐷𝑆

(2∑

Ξ

𝜌𝑠

𝑖+ ∑

(𝐻−Ξ)

𝜌𝑠

𝑖))

− (√2

𝑓𝐷𝑆

(𝜁+ 𝜁

𝑐))] ,

(18)

𝐵 = ±[1

2𝑓2

𝐷𝑆

(2∑

Ξ

𝜌𝑖 + ∑

(𝐻−Ξ)

𝜌𝑖)] , (19)

𝐶 = [−𝑚2

𝐷𝑆

+

𝑚2

𝐷𝑆

√2𝑓𝐷𝑆

(𝜁+ 𝜁

𝑐)] . (20)

As before, the “+” and “−” signs in 𝐵 correspond to 𝐷+𝑆and

𝐷−

𝑆, respectively. In writing the above summations, we have

used the following notation:𝐻denotes the set of all hyperons,𝑁 is the set of nucleons, Ξ represents the Xi hyperons (Ξ−,0),and (𝐻−Ξ) denotes all hyperons other than Xi hyperons, thatis, the set of baryons (Λ, Σ+,−,0) which all carry one strangequark. This form is particularly convenient for later analysis.Also, the optical potential of𝐷𝑆 mesons is defined as

𝑈 (𝑘) = 𝜔 (𝑘) − √𝑘2 + 𝑚2

𝐷𝑆

, (21)

where 𝑘 (=|�⃗�|) refers to the momentum of the respective 𝐷𝑆

meson, and 𝜔(𝑘) represents its momentum-dependent in-medium energy.

In the next section, we study the sensitivity of the 𝐷𝑆

meson effective mass on various characteristic parameters ofhadronic matter, namely, baryonic density (𝜌

𝐵), temperature

(𝑇), isospin asymmetry parameter 𝜂 = −∑𝑖𝐼

3𝑖𝜌

𝑖/𝜌

𝐵, and the

strangeness fraction𝑓𝑠= ∑

𝑖|𝑆

𝑖|𝜌

𝑖/𝜌

𝐵, where 𝑆

𝑖and 𝐼

3𝑖denote

the strangeness quantum number and the third componentof isospin of the 𝑖th baryon, respectively.

4. Results and Discussion

Before describing the results of our analysis of 𝐷𝑆 mesonsin a hadronic medium, we first discuss our parameter choiceand the various simplifying approximations employed in thisinvestigation.The parameters of the effective hadronic modelare fitted to the vacuummasses of baryons, nuclear saturationproperties, and other vacuum characteristics in the mean-field approximation [51, 52]. In this investigation, we haveused the same parameter set that has earlier been used to

6 Advances in High Energy Physics

study charmed (𝐷) mesons within this effective hadronicmodel [46]. In particular, we use the same values of theparameters 𝑑

1and 𝑑

2of the range terms (𝑑

1= 2.56/𝑚

𝐾and

𝑑2= 0.73/𝑚

𝐾), fitted to empirical values of kaon-nucleon

scattering lengths for the 𝐼 = 0 and 𝐼 = 1 channels, asemployed in earlier treatments [22, 23, 28, 46]. For an exten-sion to the strange-charmed system, the only extra parameterthat needs to be fitted is the 𝐷𝑆 meson decay constant, 𝑓𝐷

𝑆

,which is treated as follows. By extrapolating the results of [57],we arrive at the following expression for 𝑓𝐷

𝑆

in terms of thevacuum values of the strange and charmed scalar fields:

𝑓𝐷𝑆

=− (𝜁

0+ 𝜁

𝑐0)

√2. (22)

For fitting the value of 𝑓𝐷𝑆

, we retain the same values of 𝜎0and 𝜁

0as in the earlier treatments of this chiral effectivemodel

[23, 51] and determine 𝜁𝑐0from the expression for𝑓

𝐷in terms

of 𝜎0and 𝜁

𝑐0[28], using the Particle Data Group (PDG) value

of𝑓𝐷= 206MeV. Substituting these in (22), our fitted value of

𝑓𝐷𝑆

comes out to be 235MeV, which is close to its PDG valueof 260MeV and, particularly, is of the same order as typicallattice QCD calculations for the same [47]. We therefore per-sist with this fitted value 𝑓

𝐷𝑆

= 235MeV in this investigation.Also, we treat the charmed scalar field (𝜁

𝑐) as being arrested at

its vacuum value (𝜁𝑐0) in this investigation, as has been con-

sidered in other treatments of charmedmesons in a hadroniccontext [28, 46]. This neglect of charm dynamics appearsnatural from a physical viewpoint, owing to the large massof a charm quark (𝑚𝑐 ∼ 1.3GeV) [47]. The same was verifiedin an explicit calculation in [48], where the charm condensatewas observed to vary very weakly in the temperature range ofinterest to us in this regime [58–60]. As our last approxima-tion, we point out that, in the current investigation, we workwithin the “frozen glueball limit” [52], where the scalar dilatonfield (𝜒) is regarded as being frozen at its vacuum value (𝜒

0).

This approximation was relaxed in a preceding work [29]within this effective hadronic model, where the in-mediummodifications of this dilaton field were found to be quitemeager. We conclude, therefore, that this weak dependenceonly serves to justify the validity of this assumption.

Wenext describe our analysis for the in-mediumbehaviorof𝐷

𝑆mesons, beginning first with the nuclearmatter (𝑓

𝑠= 0)

situation and including the hyperonic degrees of freedomonly later.This approach has the advantage thatmany featuresof the 𝐷

𝑆meson in-medium behavior, common between

these regimes, are discussed in detail in a more simplifiedcontext, and the effect of strangeness becomes a lot clearer.

In nuclear matter, the Weinberg-Tomozawa term and the𝑑

2range term vanish, since they depend on the number

densities and scalar densities of the hyperons, and have nocontribution from the nucleons. It follows from the self-energy expression that only the Weinberg-Tomozawa termdiffers between 𝐷+

𝑆and 𝐷−

𝑆; all other interaction terms are

absolutely identical for them. Thus, a direct consequence ofthe vanishing of Weinberg-Tomozawa contribution is that𝐷

+

𝑆and 𝐷−

𝑆are degenerate in nuclear matter. As mentioned

earlier, the𝐷𝑆meson dispersion relations, with |�⃗�| = 0, takes

the quadratic equation form (𝐴𝜔2 + 𝐵𝜔 + 𝐶 = 0), where

the coefficients (given by (18)–(20)) reduce to the followingexpressions in nuclear matter:

𝐴(𝑁)

= [1 + (𝑑

1

2𝑓2

𝐷𝑆

(𝜌𝑠

𝑝+ 𝜌

𝑠

𝑛)) − (

√2

𝑓𝐷𝑆

(𝜁+ 𝜁

𝑐))] ,

(23)

𝐵(𝑁)= 0, (24)

𝐶(𝑁)= [−𝑚

2

𝐷𝑆

+

𝑚2

𝐷𝑆

√2𝑓𝐷𝑆

(𝜁+ 𝜁

𝑐)] , (25)

where the subscript (𝑁) emphasizes the nuclear mattercontext. For solving this quadratic equation, we require thevalues of 𝜁, as well as the scalar densities of protons andneutrons, which are obtained from a simultaneous solutionof coupled equations of motion for the scalar fields, subjectto constraints of fixed values of 𝜌

𝐵and 𝜂. The behavior

of the scalar fields, thus obtained, has been discussed indetail in [29]. Here, we build upon these scalar fields andproceed to discuss the behavior of solutions of the in-mediumdispersion relations of 𝐷

𝑆mesons, given by (23) to (25). The

variation of the 𝐷𝑆 meson in-medium mass, 𝜔(�⃗� = 0), with

baryonic density in nuclear matter, along with the individualcontributions to this net variation, is shown in Figure 1 forboth zero and finite temperature value (𝑇 = 100MeV). It isobserved that the in-medium mass of 𝐷

𝑆mesons decreases

with density, while being weakly dependent on temperatureand isospin asymmetry parameter. We can understand theobserved behavior through the following analysis. From (23)to (25), we arrive at the following closed-form solution for the𝐷

𝑆meson effective mass in nuclear matter:

𝜔

= 𝑚𝐷𝑆

[

[

1 − (𝜁+ 𝜁

𝑐) /√2𝑓

𝐷𝑆

1 + (𝑑1/2𝑓2

𝐷𝑆

) (𝜌𝑠𝑝+ 𝜌𝑠

𝑛) − (√2/𝑓𝐷

𝑆

) (𝜁 + 𝜁𝑐)

]

]

1/2

.

(26)

From this exact closed-form solution, several general con-clusions regarding the in-medium behavior of 𝐷

𝑆mesons in

nuclear matter can be drawn. On the basis of this expression,the 𝑑

1range term appearing in the denominator will lead to a

decrease in themediummass with increase in density.The in-medium mass is, additionally, also modified by the mediumdependence of 𝜁 (we assume the value of 𝜁𝑐 to be frozen atits vacuum value and hence 𝜁

𝑐= 0). However, the density

dependence of 𝜁 is observed to be quite subdued in nuclearmatter. The total change in the value of 𝜁 from its vacuumvalue is observed; that is, (𝜁)max. ≈ 15MeV. This value isacquired at a baryonic density of 𝜌𝐵 ≈ 3𝜌0 and thereafter itappears to saturate. This behavior of the strange scalar field,though ubiquitous in this chiral model [29, 51], is not justlimited to it. Even in calculations employing the relativisticHartree approximation to determine the variation of thesescalar fields [61, 62], a behavior similar to this mean-fieldtreatment is observed. We also point out that the saturationof the scalar meson exchange and first range term follows

Advances in High Energy Physics 7

1800

1850

1900

1950

2000

0 1 2 3 4 5 6

TotalScalar

Range

(𝜌B/𝜌0)

Ener

gy𝜔(k=0)

(MeV

)

(a) 𝑇 = 0

1800

1850

1900

1950

2000

0 1 2 3 4 5 6

TotalScalar

Range

(𝜌B/𝜌0)

Ener

gy𝜔(k=0)

(MeV

)

(b) 𝑇 = 100MeV

Figure 1:The various contributions to the𝐷𝑆meson energies, 𝜔(�⃗� = 0), in nuclear matter, plotted as functions of baryonic density in units of

the nuclear saturation density (𝜌𝐵/𝜌

0). In each case, the isospin asymmetric situation (𝜂 = 0.5), as described in the legend, is also compared

against the symmetric situation (𝜂 = 0), represented by dotted lines.

as a direct consequence of the saturation of 𝜁, as is impliedby their proportionality in (11) and (25). At larger densities,while the terms having 𝜁 saturate, the contribution from the𝑑

1range term continues to rise. With 𝜁 saturating at a value

of around 15MeV and the value of 𝑓𝐷𝑆

chosen to be 235MeVin the present investigation, the last term in the numeratoras well as the denominator in (26) turns out to be muchsmaller as compared to unity. Moreover, the 𝑑

1term in the

denominator in the expression for the in-mediummass of𝐷𝑆

given by (26) also turns out to be much smaller as comparedto unity, with (𝑑

1(𝜌

𝑠

𝑝+ 𝜌

𝑠

𝑛)/2𝑓

2

𝐷𝑆

) ≃ 0.05 at a density of𝜌

𝐵= 6𝜌

0. Hence the smallness of this term as compared

to unity is justified for the entire range of densities we areconcerned with, in the present investigation. In order to readmore into our solution, we expand the argument of the squareroot, in a binomial series (assuming (𝜁 +𝜁

𝑐)/√2𝑓𝐷

𝑆

≪ 1 and(𝑑

1(𝜌

𝑠

𝑝+𝜌

𝑠

𝑛)/2𝑓

2

𝐷𝑆

) ≪ 1), and retain up to only the first-orderterms. This gives, as the approximate solution,

𝜔 ≈ 𝑚𝐷𝑆

[1 +

(𝜁+ 𝜁

𝑐)

2√2𝑓𝐷𝑆

−𝑑

1

4𝑓2

𝐷𝑆

(𝜌𝑠

𝑝+ 𝜌

𝑠

𝑛)] . (27)

Moreover, since 𝜌𝑠𝑖≈ 𝜌𝑖 at small densities, the 𝑑1 range term

contribution in the above equation approximately equals(𝑑

1/4𝑓

2

𝐷𝑆

)𝜌𝐵

at low densities. The first of the terms inthe approximate expression given by (27) is an increasingfunction while the second one is a decreasing function ofdensity. Their interplay generates the observed curve shape,the repulsive contribution being responsible for the smallhump in an otherwise linear fall, at small densities (0 < 𝜌

𝐵<

0.5𝜌0). While the attractive 𝑑1 term would have produced alinear decrease right away, the role of repulsive contributionis to impede this decrease, hence producing the hump. Atmoderately higher densities, however, the contribution fromthe second term outweighs the first, which is why we see alinear drop with density. This is observed to be the case, tillaround 𝜌

𝐵≈ 2𝜌

0. At still larger densities, the approximation

(𝜌𝑠𝑖≈ 𝜌

𝑖) breaks down, though our binomial expansion

is still valid. Since scalar density falls slower than numberdensity, the term [−(𝑑

1/4𝑓

2

𝐷𝑆

)(𝜌𝑠

𝑝+ 𝜌

𝑠

𝑛)] will fall faster than

[−(𝑑1/4𝑓

2

𝐷𝑆

)𝜌𝐵]. This is responsible for the change in slope

of the curve at intermediate densities, where a linear fallwith density is no longer obeyed. So, at intermediate andlarge densities, the manner of the variation is dictated bythe scalar densities, in the 𝑑1 range term. From (18) to (20),one would expect the mass modifications of 𝐷𝑆 mesons tobe insensitive to isospin asymmetry, since, for example, inthis nuclear matter case, the dispersion relation bears isospinsymmetric terms like (𝜌𝑠

𝑝+ 𝜌

𝑠

𝑛). (This is in stark contrast with

earlier treatments of kaons and antikaons [22, 23] as wellas the 𝐷 mesons [28] within this effective model, where thedispersion relations had terms like (𝜌𝑠

𝑝−𝜌

𝑠

𝑛) or (𝜌

𝑝−𝜌

𝑛), which

contributed to asymmetry.) However, the𝐷𝑆meson effective

mass is observed to depend on asymmetry in Figure 1, thoughthe dependence is weak. For example, the values of 𝜔(𝑘 =0), for the isospin symmetric situation (𝜂 = 0), are 1913,1865, and 1834MeV, respectively, at 𝜌

𝐵= 2𝜌

0, 4𝜌

0, and

6𝜌0, while the corresponding numbers for the (completely)

asymmetric (𝜂 = 0.5) situation are 1917, 1875, and 1849MeV,respectively, at 𝑇 = 0. Since the 𝜂-dependence of LSME and

8 Advances in High Energy Physics

L1st Range contributions must be identical, one can reason

from Figure 1 that this isospin dependence of 𝐷𝑆meson

effective mass is almost entirely due to the 𝑑1range term

(∼(𝜌𝑠𝑝+ 𝜌

𝑠

𝑛)), which was expected to be isospin symmetric.

This apparently counterintuitive behavior has been observedearlier in [29], in the context of𝐷mesons.This is because thevalue of (𝜌𝑠

𝑝+ 𝜌

𝑠

𝑛) turns out to be different for symmetric and

asymmetric situations, contrary to naive expectations. Sincethe scalar-isovector 𝛿 meson (𝛿 ∼ ⟨𝑢𝑢 − 𝑑𝑑⟩) is responsiblefor introducing isospin asymmetry in this effective hadronicmodel [51], owing to the equations of motion of the scalarfields being coupled, the values of the other scalar fields turnout to be different in the symmetric (𝛿 = 0) and asymmetric(𝛿 ̸= 0) cases. The same is also reflected in the values ofthe scalar densities calculated from these scalar fields, whichleads to the observed behavior.

Additionally, we observe from a comparison of Figures1(a) and 1(b) that the magnitude of the 𝐷

𝑆meson mass drop

decreases with an increase in temperature from 𝑇 = 0 to𝑇 = 100MeV. For example, in the symmetric (𝜂 = 0)situation, at 𝑇 = 0, the 𝐷

𝑆meson mass values, at 𝜌

𝐵=

2𝜌0, 4𝜌

0, and 6𝜌

0, are 1913, 1865, and 1834MeV, respectively,

which grow to 1925, 1879, and 1848MeV, respectively, at 𝑇 =100MeV. Likewise, with 𝜂 = 0.5, the corresponding valuesread 1917, 1875, and 1849MeV at 𝑇 = 0, while the samenumbers, at 𝑇 = 100MeV, are 1925, 1883, and 1855MeV.Thus, though small, there is a definite reduction in themagnitude of the mass drops, as we go from 𝑇 = 0 to𝑇 = 100MeV, in each of these cases. This behavior canbe understood, from the point of view of the temperaturevariation of scalar condensates, in the following manner. Itis observed that the scalar fields decrease with an increasein temperature from 𝑇 = 0 to 100MeV [27, 29]. In [27],the same effect was understood as an increase in the nucleonmass with temperature. The equity of the two argumentscan be seen by invoking the expression relating the baryonmass to the scalar fields’ magnitude, 𝑚∗

𝑖= −(𝑔

𝜎𝑖𝜎 + 𝑔

𝜁𝑖𝜁 +

𝑔𝛿𝑖𝛿) [51]. The temperature dependence of the scalar fields

(𝜎, 𝜁, 𝛿) has been studied within the model in [29], whichare observed to be different for the zero and finite baryondensities. At zero baryon density, themagnitudes of the scalarfields 𝜎 and 𝜁 are observed to remain almost constant upto a temperature of about 125MeV, above which these areobserved to decrease with temperature. This behaviour canbe understood from the expression of 𝜌𝑖

𝑠given by (14) for the

situation of zero density, that is, for 𝜇∗𝑖= 0, which decreases

with increase in temperature. The temperature dependenceof the scalar density in turn determines the behaviour of thescalar fields. The scalar fields which are solved from theirequations of motion behave in a similar manner as the scalardensity. At finite densities, that is, for nonzero values of theeffective chemical potential, 𝜇∗

𝑖, however, the temperature

dependence of the scalar density is quite different fromthe zero density situation. For finite baryon density, withincrease in temperature, there are contributions also fromhigher momenta, thereby increasing the denominator of theintegrand on the right-hand side of the baryon scalar densitygiven by (14). This leads to a decrease in the scalar density. Atfinite baryon densities, the competing effects of the thermal

distribution functions and the contributions from the highermomentum states give rise to the temperature dependenceof the scalar density, which in turn determine the behaviourof the 𝜎 and 𝜁 fields with temperature. These scalar fieldsare observed to have an initial increase in their magnitudesup to a temperature of around 125–150MeV, followed by adecrease as the temperature is further raised [29].This kind ofbehavior of the scalar 𝜎 field on temperature at finite densitieshas also been observed in the Walecka model in [63]. In fact,we point out that a decrease in the scalar condensates with anincrease in temperature, though small in the hadronic regime(

Advances in High Energy Physics 9

0 1 2 3 4 5 61800

1850

1900

1950

2000

TotalWT

ScalarRange

(𝜌B/𝜌0)

Ener

gy𝜔(k=0)

(MeV

)

(a) 𝐷+𝑆

0 1 2 3 4 5 61800

1850

1900

1950

2000

TotalWT

ScalarRange

(𝜌B/𝜌0)

Ener

gy𝜔(k=0)

(MeV

)

(b) 𝐷−𝑆

Figure 2: The various contributions to the effective mass of 𝐷+𝑆and 𝐷−

𝑆mesons, as a function of baryonic density, for typical values of

temperature (𝑇 = 100MeV) and strangeness fraction (𝑓𝑠= 0.5). In each case, the asymmetric situation (𝜂 = 0.5), as described in the legend,

is also compared against the symmetric situation (𝜂 = 0), represented by dotted lines.

degeneracy of 𝐷+𝑆and 𝐷−

𝑆now stands broken. For example,

the values of (𝑚𝐷+

𝑆

, 𝑚𝐷−

𝑆

) at 𝜌𝐵= 𝜌

0, 2𝜌

0, 4𝜌

0, and 6𝜌

0

are observed to be (1948, 1953), (1918, 1928), (1859, 1879),and (1808, 1838)MeV, respectively, as can be seen from thefigure. Thus, except at vacuum (𝜌

𝐵= 0), mass difference of

𝐷+

𝑆and 𝐷−

𝑆is nonzero at finite 𝜌

𝐵, growing in magnitude

with density. This mass degeneracy breaking is a directconsequence of nonzero contributions from the Weinberg-Tomozawa term, which follows from (11), (16), and (19). Thesame may be reconciled with Figure 2, from which it followsthat, except for this Weinberg-Tomozawa term acquiringequal and opposite values for these mesons, all other termsare absolutely identical for them. Once again, on the basis ofthe following analysis of the 𝐷

𝑆meson dispersion relations

at zero momentum, we insist that this observed behavior isperfectly consistent with expectations.

The general solution of the equivalent quadratic equation(17) is

𝜔 =

(−𝐵 + √𝐵2 − 4𝐴𝐶)

2𝐴

≈ −𝐵

2𝐴+ √𝐶

1

𝐴+

𝐵2

8𝐴√𝐴𝐶1

+ ⋅ ⋅ ⋅ ,

(28)

where we have disregarded the negative root and haveperformed a binomial expansion of (1 + 𝐵2/4𝐴𝐶

1)

1/2, with𝐶

1= −𝐶. Upon feeding numerical values, we observe that the

expansion parameter (𝐵2/4𝐴𝐶1) is much smaller than unity

for our entire density variation, which justifies the validity ofthis expansion for our analysis. The same exercise also allowsus to safely disregard higher-order terms and simply write

𝜔hyp ≈ √𝐶

1

𝐴−𝐵

2𝐴. (29)

Further, with the same justification as in the nuclear mattercase, both (𝐶1/𝐴)

1/2 and (1/𝐴) can also be expanded bino-mially. For example, for the second term, this gives

|𝐵|

2𝐴=|𝐵|

2+ O(

𝜌𝑖𝜌𝑠

𝑖

𝑓4

𝐷𝑆

) , (30)

where the contribution from these higher-order terms issmaller owing to the large denominator, prompting us toretain only the first-order terms. Here, we point out that since𝐵 = ±|𝐵| (+ sign for 𝐷+

𝑆and − sign for 𝐷−

𝑆), this term,

which represents the Weinberg-Tomozawa contribution tothe dispersion relations, breaks the degeneracy of the 𝐷

𝑆

mesons. On the other hand, the first term in (29) is commonbetween 𝐷+

𝑆and 𝐷−

𝑆mesons. Thus, the general solution of

the 𝐷𝑆meson dispersion relations in the hyperonic matter

context can be written as

𝜔hyp = 𝜔com ∓ 𝜔brk, (31)

10 Advances in High Energy Physics

where

𝜔com = √𝐶1

𝐴≈ 𝑚

𝐷𝑆

[1 − (𝑑1

4𝑓2

𝐷𝑆

∑

(𝑁+𝐻)

𝜌𝑠

𝑖)

− (𝑑

2

4𝑓2

𝐷𝑆

(2∑

Ξ

𝜌𝑠

𝑖+ ∑

(𝐻−Ξ)

𝜌𝑠

𝑖)) + (

(𝜁+ 𝜁

𝑐)

2√2𝑓𝐷𝑆

)] ,

(32)

𝜔brk =|𝐵|

2𝐴≈|𝐵|

2= [

1

4𝑓2

𝐷𝑆

(2∑

Ξ

𝜌𝑖 + ∑

(𝐻−Ξ)

𝜌𝑖)] . (33)

Thus, 𝜔𝐷+

𝑆

= 𝜔com − 𝜔brk and 𝜔𝐷−𝑆

= 𝜔com + 𝜔brk(where 𝜔brk is necessarily positive), which readily explainswhy the mass of𝐷+

𝑆drops more than that of𝐷−

𝑆, with density.

Also, this formulation readily accounts for symmetric fallof the medium masses of 𝐷+

𝑆and 𝐷−

𝑆mesons, about 𝜔com

in Figure 3. Trivially, it may also be deduced that 𝜔brk isextinguished in nuclear matter, so that the mass degeneracyof 𝐷+

𝑆and 𝐷−

𝑆is recovered from these equations. However,

we observe that though the curve corresponding to 𝜔com isexactly bisecting the masses of 𝐷+

𝑆and 𝐷−

𝑆at small densities,

this bisection is no longer perfect at high densities. This canbe understood in the following manner. In essence, we arecomparing the density dependence of a function 𝑓(𝜌𝑠

𝑖), with

the functions 𝑓(𝜌𝑠𝑖) ± 𝑔(𝜌

𝑖). Since scalar densities fall sublin-

early with the number density, it follows that the fall can notbe absolutely symmetric at any arbitrary density. In fact, sincea decreasing function of scalar density will fall faster than thatof a number density, one expects 𝜔com to lean away from thecurve for𝐷−

𝑆meson and towards the curve corresponding to

𝐷+

𝑆, which is exactly what is observed in Figure 3.Since this disparity between 𝐷+

𝑆and 𝐷−

𝑆originates from

the Weinberg-Tomozawa term, whose magnitude is directlyproportional to the hyperonic number densities, one expectsthis disparity to grow with an increase in the strangenesscontent of the medium.The same also follows from the aboveformulation, since the mass splitting between the two, Δ𝑚 =(𝑚𝐷−

𝑆

− 𝑚𝐷+𝑆

) ≡ (𝜔𝐷−𝑆

− 𝜔𝐷+𝑆

) = −2𝜔brk, should grow withboth 𝑓

𝑠at fixed 𝜌

𝐵(i.e., a larger proportion of hyperons) and

𝜌𝐵at fixed (nonzero) 𝑓

𝑠(i.e., a larger hyperonic density), in

accordancewith (33). Naively, one expects this𝑓𝑠dependence

to be shared by the two 𝐷𝑆mesons; however, closer analysis

reveals that this is not the case, as shown in Figure 4 where weconsider the 𝑓

𝑠dependence of the 𝐷

𝑆meson medium mass.

It is observed that while the 𝑓𝑠dependence for 𝐷+

𝑆meson is

quite pronounced, the same for the𝐷−𝑆is conspicuously sub-

dued. Counterintuitive as it may apparently be, this observedbehavior follows from the above formulation. Since 𝜔𝐷−

𝑆

=

𝜔com +𝜔brk, at any fixed density, the first of these is a decreas-ing function of 𝑓

𝑠, while the second increases with 𝑓

𝑠. These

opposite tendencies are responsible for the weak 𝑓𝑠depen-

dence of𝐷−𝑆meson.On the other hand, because𝜔

𝐷+

𝑆

= 𝜔com−

𝜔brk, the two effects add up to produce a heightened overalldecreasewith𝑓

𝑠for the𝐷+

𝑆meson, aswe observe in the figure.

A comparison of these hyperonic matter (𝑓𝑠̸= 0) solu-

tions with the nuclear matter (𝑓𝑠= 0) solutions is shown in

Figure 5 for typical values of the parameters. It is observed

0 1 2 3 4 5 61800

1850

1900

1950

2000

(𝜌B/𝜌0)

D−SD+S

𝜔com

Ener

gy𝜔(k=0)

(MeV

)Figure 3:𝐷

𝑆mesonmass degeneracy breaking in hyperonic matter,

as a function of baryonic density, for typical values of the otherparameters (𝑇 = 100MeV, 𝑓

𝑠= 0.5, and 𝜂 = 0.5). The medium

masses of 𝐷+𝑆and 𝐷−

𝑆are observed to fall symmetrically about 𝜔com

(see text).

from Figure 5(a) that, at small densities, the nuclear mattercurve, which represents both 𝐷+

𝑆and 𝐷−

𝑆, bisects the mass

degeneracy breaking curve, akin to 𝜔com in Figure 3. How-ever, at larger densities, both of these 𝑓

𝑠̸= 0 curves drop

further than the 𝑓𝑠= 0 curve. From a casual comparison of

expressions, it appears that the nuclear matter solution, givenby (27), also enters the hyperonic matter solution, (32), albeitas a subset of 𝜔com. It is tempting to rearrange the latter then,such that this nuclear matter part is separated out, generatingan expression of the type 𝜔hyp = 𝜔nucl + 𝑓(𝜌𝑖, 𝜌

𝑠

𝑖)|

𝑖=𝐻, which

would also entail the segregation of the entire 𝑓𝑠dependence.

However, careful analysis reveals that it is impossible toachieve this kind of a relation, since, in spite of an identicalexpression, in hyperonic matter, the RHS of (27) does notgive the nuclear matter solution. This is because the nuclearmatter solution involves the scalar fields (and scalar densitiescomputed using these scalar fields) obtained as a solution ofcoupled equations in the 𝑓𝑠 = 0 situation, which are radicallydifferent from the solutions obtained in the 𝑓𝑠 ̸= 0 case. (Thishas been observed to be the case, in almost every treatmentof hyperonic matter within this effective model but perhapsmost explicitly in [46].)Thus, one can not retrieve the nuclearmatter solution from the scalar fields obtained as solutionsin the 𝑓𝑠 ̸= 0 case; moreover, due to the complexity of theconcerned equations, it is impossible to achieve a closed-formrelation between the values of the scalar fields (and hence alsothe scalar densities) obtained in the two cases. In fact, thelow-density similarity between Figures 3 and 5(a) might leadone to presume that 𝜔com reduces to 𝜔nucl at small densities.This expectation is fueled further by their comparison in

Advances in High Energy Physics 11

0 1 2 3 4 5 61800

1850

1900

1950

2000

(𝜌B/𝜌0)

fs = 0

fs = 0.3fs = 0.5

Ener

gy𝜔(k=0)

(MeV

)

(a) 𝐷+𝑆

0 1 2 3 4 5 61800

1850

1900

1950

2000

(𝜌B/𝜌0)

fs = 0

fs = 0.3fs = 0.5

Ener

gy𝜔(k=0)

(MeV

)

(b) 𝐷−𝑆

Figure 4: The sensitivity of the medium mass of (a) 𝐷+𝑆and (b) 𝐷−

𝑆mesons, to the strangeness fraction 𝑓

𝑠, at typical values of temperature

(𝑇 = 100MeV) and isospin asymmetry parameter (𝜂 = 0.5).

0 1 2 3 4 5 61800

1850

1900

1950

2000

(𝜌B/𝜌0)

D−S (fs = 0.5)

D+S (fs = 0.5)

Both (fs = 0)

Ener

gy𝜔(k=0)

(MeV

)

(a)

0 1 2 3 4 5 61800

1850

1900

1950

2000

(𝜌B/𝜌0)

𝜔com

𝜔Nucl

Ener

gy𝜔(k=0)

(MeV

)

(b)

Figure 5: (a) A comparison of medium mass of 𝐷𝑆mesons, in nuclear and hyperonic matter, maintaining the same values of parameters as

before. (b) Comparison of the nuclear matter solution, 𝜔nucl, with the 𝜔com in hyperonic matter, as observed in Figure 3.

12 Advances in High Energy Physics

0

0 250 500 750 1000

𝜌B = 𝜌0𝜌B = 4𝜌0

U(𝜔,k)

(MeV

)

−50

−100

−150

−200

Momentum, k (MeV)

(a) 𝐷+𝑆

0 250 500 750 1000

𝜌B = 𝜌0𝜌B = 4𝜌0

U(𝜔,k)

(MeV

)

0

−50

−100

−150

−200

Momentum, k (MeV)

(b) 𝐷−𝑆

Figure 6:The optical potentials of the (a)𝐷+𝑆and (b)𝐷−

𝑆mesons, as a function of momentum 𝑘 (≡|�⃗�|), in cold (𝑇 = 0), asymmetric (𝜂 = 0.5)

matter, at different values of 𝜌𝐵. In each case, the hyperonic matter situation (𝑓

𝑠= 0.5), as described in the legend, is also compared against

the nuclear matter (𝑓𝑠= 0) situation, represented by dotted lines.

Figure 5(b), which shows clearly that they coincide at smalldensities. However, in light of the above argument, it followsthat they are absolutely unrelated. Their coincidence at smalldensities can be explained by invoking the relation 𝜌𝑠

𝑖≈ 𝜌𝑖 in

this regime, using which (32) reduces to

𝜔com ≈ 𝑚𝐷𝑆

(1 +

(𝜁+ 𝜁

𝑐)

2√2𝑓𝐷𝑆

−𝜅1

4𝑓2

𝐷𝑆

𝜌𝐵)

− 𝑚𝐷𝑆

(𝜅2Δ

4𝑓2

𝐷𝑆

) ,

(34)

with 𝜅1 = (𝑑1 + 𝑑2) ≈ 1.28𝑑1, with our choice of parameters,𝜅2 (≡𝑑2) = 0.22𝜅1, and with the difference term, Δ = (𝜌

𝑠

Ξ0 +

𝜌𝑠

Ξ− −𝜌

𝑠

𝑝−𝜌

𝑠

𝑛).The contribution from this second term in (34)

is significantly smaller than the first term at small densities,owing to the smaller coefficient, as well as the fact that thisdepends on the difference of densities rather than on theirsum (like the first term). This first part of (34) can be likenedto the approximate nuclear matter solution at small 𝜌𝐵, as weconcluded from (27). The marginal increase of 𝜅 above 𝑑1is compensated by a marginal increase in 𝜁 for the 𝑓

𝑠̸= 0

case, as compared to the 𝑓𝑠 = 0 situation, and hence thetwo curves look approximately the same at small densities inFigure 5(b). At larger densities, however, this simple picturebreaks down, and the fact that they are unrelated becomesevident. Nevertheless, we may conclude in general fromthis comparison that the inclusion of hyperonic degrees offreedom in the hadronic medium makes it more attractive,regarding 𝐷

𝑆mesons’ in-medium interactions, especially at

large 𝜌𝐵. From the point of view of the 𝐷𝑆 meson dispersionrelations, this is conclusively because of the contributionsfrom the extra, hyperonic terms, which result in an overallincrease of the coefficient 𝐴hyp significantly above 𝐴nucl,particularly at large 𝑓𝑠.

Finally, we consider momentum-dependent effects byinvestigating the behavior of the in-medium optical poten-tials for the 𝐷

𝑆mesons. These are shown in Figure 6, where

we consider them in the context of both asymmetric nuclear(𝑓

𝑠= 0) and hyperonic matter (𝑓

𝑠= 0.5), as a function of

momentum 𝑘 (=|�⃗�|) at fixed values of the parameters 𝜌𝐵, 𝜂,

and 𝑇. As with the rest of the investigation, the dependenceof the in-medium optical potentials on the parameters 𝑇and 𝜂 is quite weak; hence, we neglect their variation in thiscontext. In order to appreciate the observed behavior of theseoptical potentials, we observe that as per its definition, (21),at zero momentum, optical potential is just the negative ofthe mass drop of the respective meson (i.e., 𝑈(𝑘 = 0) =−Δ𝑚(𝑘 = 0) ≡ Δ𝑚(𝜌

𝐵, 𝑇, 𝜂, 𝑓

𝑠)). It follows then that, at

𝑘 = 0, the two 𝐷𝑆mesons are degenerate in nuclear matter;

moreover, it is observed that this degeneracy extends to thefinite momentum regime.This is because, from (15) and (16),the finite momentum contribution is also common for 𝐷+

𝑆

and 𝐷−𝑆in nuclear matter. In hyperonic matter, however,

nonzero contribution from these terms breaks the degeneracyin the 𝑘 = 0 limit (as was already discussed before), and finitemomentum preserves this nondegeneracy. Consequently, atany fixed density, the curves for different values of 𝑓

𝑠differ

in terms of their 𝑦-intercept but otherwise appear to runparallel. Further, as we had reasoned earlier for the 𝑘 = 0

Advances in High Energy Physics 13

case, the effect of increasing 𝑓𝑠on the mass drops of 𝐷+

𝑆and

𝐷−

𝑆is equal and opposite about the nuclear matter situation at

small densities, which readily explains the behavior of opticalpotentials for these mesons at 𝜌

𝐵= 𝜌

0. At larger densities,

for example, the 𝜌𝐵= 4𝜌

0case shown in Figure 6, the lower

values of optical potentials for both𝐷+𝑆and𝐷−

𝑆, as compared

to the 𝑓𝑠 = 0 case, can be immediately reconciled with thebehavior observed in Figure 5. In fact, the large differencebetween the𝐷+

𝑆optical potentials for the 𝑓𝑠 = 0 and 𝑓𝑠 = 0.5

cases, as compared to that for 𝐷−𝑆, is a by-product of their

zero-momentum behavior, preserved at nonzeromomenta asa consequence of (15) and (16).

We next discuss the possible implications of thesemedium effects. In the present investigation, we haveobserved a reduction in the mass of the 𝐷

𝑆mesons, with

an increase in baryonic density. This decrease in 𝐷𝑆meson

mass can result in the opening up of extra reaction channelsof the type 𝐴 → 𝐷+

𝑆𝐷

−

𝑆in the hadronic medium. In fact,

a comparison with the spectrum of known charmoniumstates [47, 65] sheds light on the possible decay channels,as shown in Figure 7. As a starting approximation, we haveneglected the variation of energy levels of these charmoniawith density, since we intuitively expect medium effects tobe larger for 𝑠𝑐 (or 𝑐𝑠) system, as compared to 𝑐𝑐 system.However, this approximation can be conveniently relaxed,for example, as was done in [46, 66] for charmonia andbottomonia states, respectively. It follows from Figure 7that above certain threshold density, which is given by theintersection of the 𝐷

𝑆meson curve with the respective

energy level, the mass of the 𝐷+𝑆𝐷

−

𝑆pair is smaller than

that of the excited charmonium state. Hence, this decaychannel opens up above this threshold density. The mostimmediate experimental consequence of this effect wouldbe a decrease in the production yields for these excitedcharmonia in collision experiments. Additionally, since anextra decay mode will lower the lifetime of the state, oneexpects the decay widths of these excited charmonia to bemodified as a result of these extra channels. Though it can bereasoned right away that reduction in lifetime would causea broadening of the resonance curve, more exotic effects canalso result from these level crossings. For example, it has beensuggested in [67] that if the internal structure of the hadronsis also accounted for, there is even a possibility for thesein-medium widths to become narrow. This counterintuitiveresult originates from the node structure of the charmoniumwavefunction, which can even create a possibility for thedecay widths to vanish completely at certain momenta of thedaughter particles. Also, if the medium modifications of thecharmonium states are accounted for, the decay widths willbe modified accordingly (as was observed in the context ofthe𝐷mesons, in [46]).

Apart from this, one expects signatures of these mediumeffects to be reflected in the particle ratio (𝐷+

𝑆/𝐷

−

𝑆). A produc-

tion asymmetry between𝐷+𝑆and𝐷−

𝑆mesons might be antici-

pated owing to their unequal mediummasses, as observed inthe current investigation. Also, due to the semileptonic andleptonic decaymodes of the𝐷𝑆 mesons [47], one also expectsthat, accompanying an increased production of 𝐷𝑆 mesonsdue to a lowered mass, there would be an enhancement

0 1 2 3 4 5 63600

3650

3700

3750

3800

3850

3900

3950

fs = 0

fs = 0.2

fs = 0.3

fs = 0.5

X(3940)

𝜒c2(2P) (3927)

𝜓(3770)

X(3915)

X(3872)

𝜂C(2S) (3639)

(𝜌B/𝜌0)

𝜓(2S) (3686)

Ener

gy𝜔(k=0)

forD

+ SD− S

pair

(MeV

)

Figure 7: Mass of the 𝐷+𝑆𝐷

−

𝑆pair, compared against the (vacuum)

masses of the excited charmonia states, at typical values of temper-ature (𝑇 = 100MeV) and asymmetry (𝜂 = 0.5), for both nuclearmatter (𝑓

𝑠= 0) and hyperonic matter (𝑓

𝑠= 0.2, 0.3, and 0.5)

situations. The respective threshold density (see text) is given by thepoint of intersection of the two concerned curves.

in dilepton production as well. Further, we expect thesemedium modifications of the 𝐷

𝑆mesons to be reflected in

the observed dilepton spectra, due to the well-known factthat dileptons, with their small interaction cross section inhadronic matter, can serve as probes of medium effects incollision experiments [68, 69]. The attractive interaction ofthe𝐷

𝑆mesons with the hadronic medium might also lead to

the possibility of formation of𝐷𝑆-nucleus bound state, which

can be explored at the CBM experiment at FAIR in the futurefacility at GSI [70].

We now discuss how the results of the present investi-gation compare with the available literature on the mediumeffects for pseudoscalar 𝐷

𝑆(1968.5) mesons. As has already

been mentioned, [42–45] have treated the medium behaviorof 𝐷𝑆 mesons, using the coupled channel framework. Thebroad perspective that emerges from all these analyses is thatof an attractive interaction in the medium [44, 45] between𝐷𝑆 mesons and baryon species like the nucleons or theΛ, Σ, Ξhyperons [42–45], which is consistent with what we haveobserved in this work. Additionally, one observes in theseapproaches an attractive medium interaction even with thecharmed baryons [43]. Some of these interaction channelsalso feature resonances, which is reflected in the relevantscattering amplitudes picking up imaginary parts as well, forexample, the 𝑁

𝑐𝑠(2892) state in the 𝐷+

𝑆𝑁 channel [42, 43].

Treating such resonances comes naturally to the coupled

14 Advances in High Energy Physics

channel framework, since, by default, the strategy is aimedat considering the scattering of various hadron species offone another and assessing scattering amplitudes and crosssections, and so forth [10, 11]. In the present work, we haveinvestigated the in-medium masses of the𝐷

𝑆mesons arising

due to their interactions with the baryons and scalar mesonsin the nuclear (hyperonic) medium and have not studied thedecays of these mesons. As has already been mentioned themass modifications of the 𝐷𝑆 mesons can lead to openingup of the new channels for the charmonium states decayingto 𝐷+

𝑆𝐷

−

𝑆at certain densities as can be seen from Figure 7.

In fact, even if one contemplates working along the linesof [46] for calculating these decay widths, the situation is abit more complicated for these 𝐷𝑆 mesons, since the states𝑋(3872),𝑋(3915), and𝑋(3940), which are highly relevant inthis context (as one can see from Figure 7), are not classifiedas having clear, definite quantum numbers in the spectrum ofexcited charmonium states [47]. This makes the assessmentof medium effects for these states more difficult in com-parison, since the identification of the relevant charmoniumstates with definite quantum numbers in the charmoniumspectrum, such as 1𝑆 for 𝐽/𝜓, 2𝑆 for 𝜓(3686), and 1𝐷 for𝜓(3770), was a crucial requirement for evaluating their massshifts in the medium in the QCD second-order Stark effectstudy in [46]. For some of these unconventional states, forexample,𝑋(3872), this absence of definite identification withstates in the spectrum has led to alternative possibilitiesbeing explored for these states. A prominent example is thepossibility of amolecular structure for this𝑋(3872) state [65],borne out of contributions from the (𝐷+𝐷∗− + 𝐷−𝐷∗+) and(𝐷

0𝐷

∗0

+ 𝐷0

𝐷∗0) components in the 𝑠-wave.

Lastly, we point out that the medium effects describedin this paper, and the possible experimental consequencesentailed by these, are especially interesting in wake of theupcoming CBM [70] experiment at FAIR, GSI, where highbaryonic densities are expected to be reached. Due to thestrong density dependence of thesemediumeffects, we expectthat each of these mentioned experimental consequenceswould intensify at higher densities and should be palpable inthe aforementioned future experiment.

5. Summary

To summarize, we have explored the properties of𝐷𝑆mesons

in a hot and dense hadronic environment, within the frame-work of the chiral 𝑆𝑈(3)

𝐿× 𝑆𝑈(3)

𝑅model, generalized to

𝑆𝑈(4). The generalization of chiral 𝑆𝑈(3) model to 𝑆𝑈(4)is done in order to derive the interactions of the charmedmesons with the light hadrons, needed for the study of thein-medium properties of the𝐷𝑆 mesons in the present work.However, realizing that the chiral symmetry is badly brokenfor the 𝑆𝑈(4) case due to the large mass of the charm quark,we use the interactions derived from 𝑆𝑈(4) for the𝐷𝑆 mesonbut use the observed masses of these heavy pseudoscalarmesons as well as empirical/observed values of their decayconstants [30]. The𝐷

𝑆mesons have been considered in both

(symmetric and asymmetric) nuclear and hyperonic matter,at finite densities that extend slightly beyond what can beachieved with the existing and known future facilities and

at both zero and finite temperatures. Due to net attractiveinteractions in the medium, 𝐷

𝑆mesons are observed to

undergo a drop in their effective mass. These mass drops arefound to intensify with an increase in the baryonic densityof the medium, while being largely insensitive to changesin temperature as well as the isospin asymmetry parameter.However, upon adding hyperonic degrees of freedom, themass degeneracy of 𝐷+

𝑆and 𝐷−

𝑆is observed to be broken.

The mass splitting between 𝐷+𝑆and 𝐷−

𝑆is found to grow

significantly with an increase in baryonic density as well asthe strangeness content of the medium. Through a detailedanalysis of the in-medium dispersion relations for the 𝐷𝑆mesons, we have shown that the observed behavior followsprecisely from the interplay of contributions from variousinteraction terms in the Lagrangian density. We have brieflydiscussed the possible experimental consequences of thesemedium effects, for example, in the 𝐷+

𝑆/𝐷

−

𝑆ratio, dilepton

spectra, possibility of formation of exotic 𝐷𝑆-nucleus bound

states, and modifications of the decays of charmonium states𝐷

+

𝑆𝐷

−

𝑆pair in the hadronic medium. The medium modifi-

cations of the 𝐷𝑆mesons are expected to be considerably

enhanced at large densities and hence the experimentalconsequences may be accessible in the upcoming CBMexperiment, at the future facilities of FAIR, GSI [70].

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

Divakar Pathak acknowledges financial support from Uni-versity Grants Commission, India (Sr. no. 2121051124, Ref.no. 19-12/2010(i)EU-IV). Amruta Mishra would like to thankDepartment of Science andTechnology,Government of India(Project no. SR/S2/HEP-031/2010), for financial support.

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