Research Articleπ·π
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; [email protected]
Received 8 June 2015; Revised 9 August 2015; Accepted 18 August 2015
Academic Editor: Juan JoseΜ Sanz-Cillero
Copyright Β© 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 π·πmesons are investigated within the framework of an effective hadronic model, which is a
generalization of a chiral ππ(3)model, to ππ(4), in order to study the interactions of the charmed hadrons. In the present work, theπ·
πmesons are observed to experience net attractive interactions in a dense hadronic medium, hence reducing themasses of theπ·+
π
and π·βπmesons 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 π·
πmesons 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 betweenπ·+πandπ·β
πmesons 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 [1β3]. 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 [7β9], 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 [16β20]. 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 symmetryβits spontaneous breakdown implyinga nonvanishing scalar condensate β¨ππβ© 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 Goldstoneβs 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 (ππ β 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 π
π’, π
π< 10MeV), comparatively larger for
the kaons and antikaons (since ππ βΌ 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 (π
πβ 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 ππ(3) approach, forstrange pseudoscalar mesons (kaons and antikaons) [22β25].For studying the charmed mesons, one needs to generalizethe ππ(3) model to ππ(4), in order to incorporate theinteractions of the charmedmesons to the light hadrons. Sucha generalization from ππ(3) to ππ(4) was initially done in[26], where the interaction Lagrangian was constructed forthe pseudoscalar mesons for ππ(4) from a generalization ofthe lowest order chiral ππ(3) Lagrangian. Since the chiralsymmetry is explicitly broken for the ππ(4) case due to thelargemass of the charmquark (π
πβ 1.3GeV), which ismuch
larger than the masses of the light quarks, for the study ofthe charmed (π·) pseudoscalar mesons [27β29], we adopt thephilosophy of generalizing the chiral ππ(3)model to ππ(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 π·π mesons). Apart from pure theoretical interest, anunderstanding of the medium modifications of π·π 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 π·
πmeson spectrum can serve as a useful probe for
understanding key issues regarding hadronization in heavyion collisions. It is suggested that, by comparing observablesforπ· andπ·π 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 π·π 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 theexcitedπ·
πstates, the standard quark-antiquark picture aside.
These include their description as molecular states, borneout of two mesons, four-quark states, or the still furtherexotic possibilitiesβas two-diquark states and as a mixtureof quark-antiquark and tetraquark states [32]. Prominentamong these is a treatment ofπ·β
π0(2317) as aπ·πΎ bound state
[35, 36], π·π1(2460) as a dynamically generated π·βπΎ reso-
nance [37], and π·βπ2(2573) being treated within the hidden
local gauge formalism in coupled channel unitary approach[38, 39], as well as the vector π·β
πstates and the π·+
π(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 π·π mesons, there is a conspicuous dearth [42]of available information about the medium modifications ofthe lightest pseudoscalar π·π mesons, π·π(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 (π½
π= 0
β)π·
πmesons
in a hadronic medium is limited to the assessment of theirspectral distributions andmedium effects on the dynamicallygenerated resonances borne out of the interaction of theseπ·
πmesons with other hadron species, in the coupled channel
analyses of [42β45]. This situation is quite unlike their open-charm, non-strange counterparts, theπ·mesons, 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 π·
πmesons 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 theπ·π 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
ππ(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 ππ(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 ππ(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 (π· or π·
π) mesons. Also, ππ(4) symmetry being
badly broken implies that any symmetry and order in themasses and decay constants, as predicted on the basis ofππ(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 ππ(4) symmetry but rather theirobserved Particle Data Group (PDG) [47] values. Overall,therefore, ππ(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 (β¨π’π’β©, β¨ππβ©) aremodified significantly in a hadronic medium with mediumparameters like density and temperature; the strange quarkcondensate β¨π π β© is comparatively stolid and its variation issignificantly more subdued, while upon advancing to thecharm sector the variation in the charmed quark condensateβ¨ππβ© 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 pseudoscalarπ΅, π΅, and π΅
πmesons [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 fortheπ·π mesons within the effective hadronic model, obtainedby generalizing the chiral ππ(3)model to ππ(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 ππ(3)
πΏΓ ππ(3)
π model (and its generalization
to the ππ(4) case) used in this investigation. In Section 3,the Lagrangian density for the π·
πmesons, 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 π·
πmesons, 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 ππ(3)πΏ Γ ππ(3)π model [51], to ππ(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 [55β57]. Additionally,the model incorporates a scalar dilaton field, π, 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 +βπ
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, π. Finally, LSB refers to the explicitsymmetry breaking term. This approach has been employedextensively to study the in-medium properties of hadrons,particularly pseudoscalar mesons [22β25]. 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 ππ(3) formalism to begeneralized to ππ(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 +Lπ1
+Lπ2
+LSME. (2)
In (2),LWT denotes theWeinberg-Tomozawa term, given bythe expression [46]
LWT = β1
2[π΅
ππππΎ
π((Ξ
π)
π
π
π΅πππ+ 2(Ξ
π)
π
π
π΅πππ)] , (3)
with repeated indices summed over. Baryons are representedby the tensor π΅πππ, which is antisymmetric in its first twoindices [43]. The indices π, π, and π run from 1 to 4, and onecan directly read the quark content of a baryon state, withthe following identification: 1 β π’, 2 β π, 3 β π , 4 β π.However, the heavier, charmed baryons are discounted fromthis analysis. In (3), Ξ
πis defined as
Ξπ
= βπ
4(π’
β (πππ’) β (πππ’
β ) π’ + π’ (πππ’
β ) β (πππ’) π’
β ) ,
(4)
where the unitary transformation operator, π’ = exp(πππΎ5/β2π0), is defined in terms of the matrix of pseudoscalarmesons, π = (ππππ/β2), ππ representing the generalizedGell-Mann matrices. Further, LSME is the scalar mesonexchange term, which is obtained from the explicit symmetrybreaking term
LSB = β1
2Tr (π΄
π(π’ππ’ + π’
β ππ’
β )) , (5)
where π΄π = (1/
β2) diag[π2πππ, π
2
πππ, (2π
2
πΎππΎ β π
2
πππ),
(2π2
π·π
π·βπ
2
ππ
π)] andπ 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 (π’πππ’
ππ + ππ’
ππ’
ππ) , (6)
where π’π= βπ((π’
β (π
ππ’) β (π
ππ’
β )π’) β (π’(π
ππ’
β ) β (π
ππ’)π’
β ))/4.
Lastly, the π1and π
2range terms are
Lπ1
=π
1
4(π΅ππππ΅
πππ(π’π)
π
π
(π’π)
π
π) , (7)
Lπ2
=π
2
2[π΅πππ(π’π)
π
π
((π’π)
π
ππ΅
πππ+ 2(π’
π)
π
ππ΅
πππ)] . (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 π, scalar-isovector meson πΏ, and strange scalar meson π and for thevector-isovector meson π, nonstrange vector meson π, andstrange vector meson π, within this model.
The π·πmeson interaction Lagrangian density and in-
medium dispersion relations, as they follow from the abovegeneral formulation, are described next.
3. π·π
Mesons in Hadronic Matter
The Lagrangian density for the π·π 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 π·
πmesons in this case) and
reads
Lfree = (πππ·
+
π) (πππ·
β
π) β π
2
π·π
(π·+
ππ·
β
π) . (10)
On the other hand,Lint is determined to be
Lint = βπ
4π2
π·π
[(2 (Ξ0
πΎπΞ
0+ Ξ
β
πΎπΞ
β) + Ξ
0
πΎπΞ
0
+ Ξ£+
πΎπΞ£
++ Ξ£
0
πΎπΞ£
0+ Ξ£
β
πΎπΞ£
β) (π·
+
π(π
ππ·
β
π)
β (πππ·
+
π)π·
β
π)] +
π2
π·π
β2ππ·π
[(π+ π
π) (π·
+
ππ·
β
π)]
ββ2
ππ·π
[(π+ π
π) ((π
ππ·
+
π) (π
ππ·
β
π))] +
π1
2π2
π·π
[(ππ
+ ππ + Ξ0
Ξ0+ Ξ£
+
Ξ£++ Ξ£
0
Ξ£0+ Ξ£
β
Ξ£β+ Ξ
0
Ξ0
+ Ξβ
Ξβ) ((π
ππ·
+
π) (π
ππ·
β
π))]
+π
2
2π2
π·π
[(2 (Ξ0
Ξ0+ Ξ
β
Ξβ) + Ξ
0
Ξ0+ Ξ£
+
Ξ£+
+ Ξ£0
Ξ£0+ Ξ£
β
Ξ£β) ((π
ππ·
+
π) (π
ππ·
β
π))] .
(11)
In this expression, the first term (with coefficient βπ/4π2π·π
)is the Weinberg-Tomozawa term, obtained from (3), thesecond term (with coefficient π2
π·π
/β2ππ·π
) is the scalarmeson exchange term, obtained from the explicit symmetrybreaking term of the Lagrangian (see (5)), the third term(with coefficient ββ2/π
π·π
) is the first range term (see (6)),and the fourth and fifth terms (with coefficients (π
1/π
2
π·π
) and(π
2/π2
π·π
), resp.) are the π1 and π2 terms, calculated from (7)and (8), respectively. Also, π = π β π
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:
π΅ππ΅
πβ β¨π΅
ππ΅
πβ© β‘ πΏ
πππ
π
π,
π΅ππΎ
ππ΅
πβ β¨π΅
ππΎ
ππ΅
πβ© = πΏ
ππ(πΏ
0
π(π΅
ππΎ
ππ΅
π)) β‘ πΏ
πππ
π.
(12)
Thus, the interaction Lagrangian density can be recast interms of the baryonic number densities and scalar densities,given by the following expressions:
ππ=πΎ
π
(2π)3β«π
3π(
1
exp ((πΈβπ(π) β π
β
π) /π) + 1
β1
exp ((πΈβπ(π) + π
β
π) /π) + 1
) ,
(13)
πsπ=πΎ
π
(2π)3β«π
3ππ
β
π
πΈβ
π(π)(
1
exp ((πΈβπ(π) β π
β
π) /π) + 1
+1
exp ((πΈβπ(π) + π
β
π) /π) + 1
) .
(14)
In the above, πβπand πβ
πare the effective mass and effective
chemical potential of the πth baryon, given asπβπ= β(π
πππ +
ππππ + π
πΏππΏ), πβ
π= π
πβ (π
πππ
3π + π
πππ + π
πππ), πΈβ
π(π) =
(π2+ π
β
π
2)
1/2, and πΎπ = 2 is the spin degeneracy factor. One
can find the equations of motion for the π·+πand π·β
πmesons,
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 (βΌππ(οΏ½βοΏ½β βπβππ‘)),and hence, βFourier transformβ these equations, to arrive atthe in-medium dispersion relations for theπ·π mesons.Thesehave the general form
βπ2+ οΏ½βοΏ½
2+ π
2
π·π
β Ξ (π,οΏ½βοΏ½) = 0, (15)
whereππ·π
is the vacuummass of theπ·πmesons andΞ (π, |οΏ½βοΏ½|)
is the self-energy of theπ·πmesons in the medium. Explicitly,
the latter reads
Ξ (π,οΏ½βοΏ½) = [(
π1
2π2
π·π
(ππ
π+ π
π
π+ π
π
Ξ+ π
π
Ξ£+ + π
π
Ξ£0 + π
π
Ξ£β
+ ππ
Ξ0 + π
π
Ξβ)) + (
π2
2π2
π·π
(2 (ππ
Ξ0 + π
π
Ξβ) + π
π
Ξ+ π
π
Ξ£+
+ ππ
Ξ£0 + π
π
Ξ£β)) β (
β2
ππ·π
(π+ π
π))] (π
2β οΏ½βοΏ½
2)
Β± [1
2π2
π·π
(2 (πΞ0 + π
Ξβ) + π
Ξ+ π
Ξ£+ + π
Ξ£0 + π
Ξ£β)]π
+ [
π2
π·π
β2ππ·π
(π+ π
π)] ,
(16)
where the + and β signs in the coefficient of π refer to π·+π
and π·βπ, respectively, and we have used (12) to simplify the
bilinears. In the rest frame of these mesons (i.e., setting |οΏ½βοΏ½| =0), these dispersion relations reduce to
βπ2+ π
2
π·π
β Ξ (π, 0) = 0, (17)
which is a quadratic equation in π, that is, of the form π΄π2 +π΅π + πΆ = 0, where the coefficients π΄, π΅, and πΆ depend onvarious interaction terms in (15) and read
π΄ = [1 + (π
1
2π2
π·π
β
(π+π»)
ππ
π)
+ (π
2
2π2
π·π
(2β
Ξ
ππ
π+ β
(π»βΞ)
ππ
π))
β (β2
ππ·π
(π+ π
π))] ,
(18)
π΅ = Β±[1
2π2
π·π
(2β
Ξ
ππ + β
(π»βΞ)
ππ)] , (19)
πΆ = [βπ2
π·π
+
π2
π·π
β2ππ·π
(π+ π
π)] . (20)
As before, the β+β and βββ signs in π΅ correspond to π·+πand
π·β
π, respectively. In writing the above summations, we have
used the following notation:π»denotes the set of all hyperons,π is the set of nucleons, Ξ represents the Xi hyperons (Ξβ,0),and (π»βΞ) denotes all hyperons other than Xi hyperons, thatis, the set of baryons (Ξ, Ξ£+,β,0) which all carry one strangequark. This form is particularly convenient for later analysis.Also, the optical potential ofπ·π mesons is defined as
π (π) = π (π) β βπ2 + π2
π·π
, (21)
where π (=|οΏ½βοΏ½|) refers to the momentum of the respective π·π
meson, and π(π) represents its momentum-dependent in-medium energy.
In the next section, we study the sensitivity of the π·π
meson effective mass on various characteristic parameters ofhadronic matter, namely, baryonic density (π
π΅), temperature
(π), isospin asymmetry parameter π = ββππΌ
3ππ
π/π
π΅, and the
strangeness fractionππ = β
π|π
π|π
π/π
π΅, where π
πand πΌ
3πdenote
the strangeness quantum number and the third componentof isospin of the πth baryon, respectively.
4. Results and Discussion
Before describing the results of our analysis of π·π 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 (π·) mesons within this effective hadronicmodel [46]. In particular, we use the same values of theparameters π
1and π
2of the range terms (π
1= 2.56/π
πΎand
π2= 0.73/π
πΎ), fitted to empirical values of kaon-nucleon
scattering lengths for the πΌ = 0 and πΌ = 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 π·π meson decay constant, ππ·
π
,which is treated as follows. By extrapolating the results of [57],we arrive at the following expression for ππ·
π
in terms of thevacuum values of the strange and charmed scalar fields:
ππ·π
=β (π
0+ π
π0)
β2. (22)
For fitting the value of ππ·π
, we retain the same values of π0and π
0as in the earlier treatments of this chiral effectivemodel
[23, 51] and determine ππ0from the expression forπ
π·in terms
of π0and π
π0[28], using the Particle Data Group (PDG) value
ofππ·= 206MeV. Substituting these in (22), our fitted value of
ππ·π
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 π
π·π
= 235MeV in this investigation.Also, we treat the charmed scalar field (π
π) as being arrested at
its vacuum value (ππ0) 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 (ππ βΌ 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 [58β60]. As our last approxima-tion, we point out that, in the current investigation, we workwithin the βfrozen glueball limitβ [52], where the scalar dilatonfield (π) is regarded as being frozen at its vacuum value (π
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-mediumbehaviorofπ·
πmesons, beginning first with the nuclearmatter (π
π = 0)
situation and including the hyperonic degrees of freedomonly later.This approach has the advantage thatmany featuresof the π·
πmeson 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 theπ
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 π·+
πand π·β
π; all other interaction terms are
absolutely identical for them. Thus, a direct consequence ofthe vanishing of Weinberg-Tomozawa contribution is thatπ·
+
πand π·β
πare degenerate in nuclear matter. As mentioned
earlier, theπ·πmeson dispersion relations, with |οΏ½βοΏ½| = 0, takes
the quadratic equation form (π΄π2 + π΅π + πΆ = 0), where
the coefficients (given by (18)β(20)) reduce to the followingexpressions in nuclear matter:
π΄(π)
= [1 + (π
1
2π2
π·π
(ππ
π+ π
π
π)) β (
β2
ππ·π
(π+ π
π))] ,
(23)
π΅(π)= 0, (24)
πΆ(π)= [βπ
2
π·π
+
π2
π·π
β2ππ·π
(π+ π
π)] , (25)
where the subscript (π) emphasizes the nuclear mattercontext. For solving this quadratic equation, we require thevalues of π, 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 π
π΅and π. 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 π·
πmesons, given by (23) to (25). The
variation of the π·π meson in-medium mass, π(οΏ½βοΏ½ = 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 (π = 100MeV). It isobserved that the in-medium mass of π·
πmesons 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 theπ·
πmeson effective mass in nuclear matter:
π
= ππ·π
[
[
1 β (π+ π
π) /β2π
π·π
1 + (π1/2π2
π·π
) (ππ π+ ππ
π) β (β2/ππ·
π
) (π + ππ)
]
]
1/2
.
(26)
From this exact closed-form solution, several general con-clusions regarding the in-medium behavior of π·
πmesons in
nuclear matter can be drawn. On the basis of this expression,the π
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 π (we assume the value of ππ to be frozen atits vacuum value and hence π
π= 0). However, the density
dependence of π is observed to be quite subdued in nuclearmatter. The total change in the value of π from its vacuumvalue is observed; that is, (π)max. β 15MeV. This value isacquired at a baryonic density of ππ΅ β 3π0 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
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TotalScalar
Range
(πB/π0)
Ener
gyπ(k=0)
(MeV
)
(a) π = 0
1800
1850
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0 1 2 3 4 5 6
TotalScalar
Range
(πB/π0)
Ener
gyπ(k=0)
(MeV
)
(b) π = 100MeV
Figure 1:The various contributions to theπ·πmeson energies, π(οΏ½βοΏ½ = 0), in nuclear matter, plotted as functions of baryonic density in units of
the nuclear saturation density (ππ΅/π
0). In each case, the isospin asymmetric situation (π = 0.5), as described in the legend, is also compared
against the symmetric situation (π = 0), represented by dotted lines.
as a direct consequence of the saturation of π, as is impliedby their proportionality in (11) and (25). At larger densities,while the terms having π saturate, the contribution from theπ
1range term continues to rise. With π saturating at a value
of around 15MeV and the value of ππ·π
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 π
1term in the
denominator in the expression for the in-mediummass ofπ·π
given by (26) also turns out to be much smaller as comparedto unity, with (π
1(π
π
π+ π
π
π)/2π
2
π·π
) β 0.05 at a density ofπ
π΅= 6π
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 (π +π
π)/β2ππ·
π
βͺ 1 and(π
1(π
π
π+π
π
π)/2π
2
π·π
) βͺ 1), and retain up to only the first-orderterms. This gives, as the approximate solution,
π β ππ·π
[1 +
(π+ π
π)
2β2ππ·π
βπ
1
4π2
π·π
(ππ
π+ π
π
π)] . (27)
Moreover, since ππ πβ ππ at small densities, the π1 range term
contribution in the above equation approximately equals(π
1/4π
2
π·π
)ππ΅
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 < π
π΅<
0.5π0). While the attractive π1 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 π
π΅β 2π
0. At still larger densities, the approximation
(ππ πβ π
π) breaks down, though our binomial expansion
is still valid. Since scalar density falls slower than numberdensity, the term [β(π
1/4π
2
π·π
)(ππ
π+ π
π
π)] will fall faster than
[β(π1/4π
2
π·π
)ππ΅]. 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 π1 range term. From (18) to (20),one would expect the mass modifications of π·π mesons tobe insensitive to isospin asymmetry, since, for example, inthis nuclear matter case, the dispersion relation bears isospinsymmetric terms like (ππ
π+ π
π
π). (This is in stark contrast with
earlier treatments of kaons and antikaons [22, 23] as wellas the π· mesons [28] within this effective model, where thedispersion relations had terms like (ππ
πβπ
π
π) or (π
πβπ
π), which
contributed to asymmetry.) However, theπ·πmeson effective
mass is observed to depend on asymmetry in Figure 1, thoughthe dependence is weak. For example, the values of π(π =0), for the isospin symmetric situation (π = 0), are 1913,1865, and 1834MeV, respectively, at π
π΅= 2π
0, 4π
0, and
6π0, while the corresponding numbers for the (completely)
asymmetric (π = 0.5) situation are 1917, 1875, and 1849MeV,respectively, at π = 0. Since the π-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 π·πmeson
effective mass is almost entirely due to the π1range term
(βΌ(ππ π+ π
π
π)), which was expected to be isospin symmetric.
This apparently counterintuitive behavior has been observedearlier in [29], in the context ofπ·mesons.This is because thevalue of (ππ
π+ π
π
π) turns out to be different for symmetric and
asymmetric situations, contrary to naive expectations. Sincethe scalar-isovector πΏ meson (πΏ βΌ β¨π’π’ β ππβ©) 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 (πΏ = 0) and asymmetric(πΏ ΜΈ= 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 π·
πmeson mass drop
decreases with an increase in temperature from π = 0 toπ = 100MeV. For example, in the symmetric (π = 0)situation, at π = 0, the π·
πmeson mass values, at π
π΅=
2π0, 4π
0, and 6π
0, are 1913, 1865, and 1834MeV, respectively,
which grow to 1925, 1879, and 1848MeV, respectively, at π =100MeV. Likewise, with π = 0.5, the corresponding valuesread 1917, 1875, and 1849MeV at π = 0, while the samenumbers, at π = 100MeV, are 1925, 1883, and 1855MeV.Thus, though small, there is a definite reduction in themagnitude of the mass drops, as we go from π = 0 toπ = 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 π = 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 fieldsβ magnitude, πβ
π= β(π
πππ + π
πππ +
ππΏππΏ) [51]. The temperature dependence of the scalar fields
(π, π, πΏ) 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 π and π 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 ππ
π given by (14) for the
situation of zero density, that is, for πβπ= 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, πβ
π, 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 π and π fields with temperature. These scalar fieldsare observed to have an initial increase in their magnitudesup to a temperature of around 125β150MeV, followed by adecrease as the temperature is further raised [29].This kind ofbehavior of the scalar π 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
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TotalWT
ScalarRange
(πB/π0)
Ener
gyπ(k=0)
(MeV
)
(a) π·+π
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TotalWT
ScalarRange
(πB/π0)
Ener
gyπ(k=0)
(MeV
)
(b) π·βπ
Figure 2: The various contributions to the effective mass of π·+πand π·β
πmesons, as a function of baryonic density, for typical values of
temperature (π = 100MeV) and strangeness fraction (ππ = 0.5). In each case, the asymmetric situation (π = 0.5), as described in the legend,
is also compared against the symmetric situation (π = 0), represented by dotted lines.
degeneracy of π·+πand π·β
πnow stands broken. For example,
the values of (ππ·+
π
, ππ·β
π
) at ππ΅= π
0, 2π
0, 4π
0, and 6π
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 (π
π΅= 0), mass difference of
π·+
πand π·β
πis nonzero at finite π
π΅, 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 π·
πmeson 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
π =
(βπ΅ + βπ΅2 β 4π΄πΆ)
2π΄
β βπ΅
2π΄+ βπΆ
1
π΄+
π΅2
8π΄βπ΄πΆ1
+ β β β ,
(28)
where we have disregarded the negative root and haveperformed a binomial expansion of (1 + π΅2/4π΄πΆ
1)
1/2, withπΆ
1= βπΆ. Upon feeding numerical values, we observe that the
expansion parameter (π΅2/4π΄πΆ1) 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
πhyp β βπΆ
1
π΄βπ΅
2π΄. (29)
Further, with the same justification as in the nuclear mattercase, both (πΆ1/π΄)
1/2 and (1/π΄) can also be expanded bino-mially. For example, for the second term, this gives
|π΅|
2π΄=|π΅|
2+ O(
ππππ
π
π4
π·π
) , (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 sinceπ΅ = Β±|π΅| (+ sign for π·+
πand β sign for π·β
π), this term,
which represents the Weinberg-Tomozawa contribution tothe dispersion relations, breaks the degeneracy of the π·
π
mesons. On the other hand, the first term in (29) is commonbetween π·+
πand π·β
πmesons. Thus, the general solution of
the π·πmeson dispersion relations in the hyperonic matter
context can be written as
πhyp = πcom β πbrk, (31)
10 Advances in High Energy Physics
where
πcom = βπΆ1
π΄β π
π·π
[1 β (π1
4π2
π·π
β
(π+π»)
ππ
π)
β (π
2
4π2
π·π
(2β
Ξ
ππ
π+ β
(π»βΞ)
ππ
π)) + (
(π+ π
π)
2β2ππ·π
)] ,
(32)
πbrk =|π΅|
2π΄β|π΅|
2= [
1
4π2
π·π
(2β
Ξ
ππ + β
(π»βΞ)
ππ)] . (33)
Thus, ππ·+
π
= πcom β πbrk and ππ·βπ
= πcom + πbrk(where πbrk is necessarily positive), which readily explainswhy the mass ofπ·+
πdrops more than that ofπ·β
π, with density.
Also, this formulation readily accounts for symmetric fallof the medium masses of π·+
πand π·β
πmesons, about πcom
in Figure 3. Trivially, it may also be deduced that πbrk isextinguished in nuclear matter, so that the mass degeneracyof π·+
πand π·β
πis recovered from these equations. However,
we observe that though the curve corresponding to πcom isexactly bisecting the masses of π·+
πand π·β
πat 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 π(ππ
π), with
the functions π(ππ π) Β± π(π
π). 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 πcom to lean away from thecurve forπ·β
πmeson and towards the curve corresponding to
π·+
π, which is exactly what is observed in Figure 3.Since this disparity between π·+
πand π·β
πoriginates 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, Ξπ =(ππ·β
π
β ππ·+π
) β‘ (ππ·βπ
β ππ·+π
) = β2πbrk, should grow withboth π
π at fixed π
π΅(i.e., a larger proportion of hyperons) and
ππ΅at fixed (nonzero) π
π (i.e., a larger hyperonic density), in
accordancewith (33). Naively, one expects thisππ dependence
to be shared by the two π·πmesons; however, closer analysis
reveals that this is not the case, as shown in Figure 4 where weconsider the π
π dependence of the π·
πmeson medium mass.
It is observed that while the ππ dependence for π·+
πmeson is
quite pronounced, the same for theπ·βπis conspicuously sub-
dued. Counterintuitive as it may apparently be, this observedbehavior follows from the above formulation. Since ππ·β
π
=
πcom +πbrk, at any fixed density, the first of these is a decreas-ing function of π
π , while the second increases with π
π . These
opposite tendencies are responsible for the weak ππ depen-
dence ofπ·βπmeson.On the other hand, becauseπ
π·+
π
= πcomβ
πbrk, the two effects add up to produce a heightened overalldecreasewithπ
π for theπ·+
πmeson, aswe observe in the figure.
A comparison of these hyperonic matter (ππ ΜΈ= 0) solu-
tions with the nuclear matter (ππ = 0) solutions is shown in
Figure 5 for typical values of the parameters. It is observed
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(πB/π0)
DβSD+S
πcom
Ener
gyπ(k=0)
(MeV
)Figure 3:π·
πmesonmass degeneracy breaking in hyperonic matter,
as a function of baryonic density, for typical values of the otherparameters (π = 100MeV, π
π = 0.5, and π = 0.5). The medium
masses of π·+πand π·β
πare observed to fall symmetrically about πcom
(see text).
from Figure 5(a) that, at small densities, the nuclear mattercurve, which represents both π·+
πand π·β
π, bisects the mass
degeneracy breaking curve, akin to πcom in Figure 3. How-ever, at larger densities, both of these π
π ΜΈ= 0 curves drop
further than the ππ = 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 πcom. It is tempting to rearrange the latter then,such that this nuclear matter part is separated out, generatingan expression of the type πhyp = πnucl + π(ππ, π
π
π)|
π=π», which
would also entail the segregation of the entire ππ dependence.
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 ππ = 0 situation, which are radicallydifferent from the solutions obtained in the ππ ΜΈ= 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 ππ ΜΈ= 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 πcom reduces to πnucl at small densities.This expectation is fueled further by their comparison in
Advances in High Energy Physics 11
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(πB/π0)
fs = 0
fs = 0.3fs = 0.5
Ener
gyπ(k=0)
(MeV
)
(a) π·+π
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(πB/π0)
fs = 0
fs = 0.3fs = 0.5
Ener
gyπ(k=0)
(MeV
)
(b) π·βπ
Figure 4: The sensitivity of the medium mass of (a) π·+πand (b) π·β
πmesons, to the strangeness fraction π
π , at typical values of temperature
(π = 100MeV) and isospin asymmetry parameter (π = 0.5).
0 1 2 3 4 5 61800
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2000
(πB/π0)
DβS (fs = 0.5)
D+S (fs = 0.5)
Both (fs = 0)
Ener
gyπ(k=0)
(MeV
)
(a)
0 1 2 3 4 5 61800
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(πB/π0)
πcom
πNucl
Ener
gyπ(k=0)
(MeV
)
(b)
Figure 5: (a) A comparison of medium mass of π·πmesons, in nuclear and hyperonic matter, maintaining the same values of parameters as
before. (b) Comparison of the nuclear matter solution, πnucl, with the πcom in hyperonic matter, as observed in Figure 3.
12 Advances in High Energy Physics
0
0 250 500 750 1000
πB = π0πB = 4π0
U(π,k)
(MeV
)
β50
β100
β150
β200
Momentum, k (MeV)
(a) π·+π
0 250 500 750 1000
πB = π0πB = 4π0
U(π,k)
(MeV
)
0
β50
β100
β150
β200
Momentum, k (MeV)
(b) π·βπ
Figure 6:The optical potentials of the (a)π·+πand (b)π·β
πmesons, as a function of momentum π (β‘|οΏ½βοΏ½|), in cold (π = 0), asymmetric (π = 0.5)
matter, at different values of ππ΅. In each case, the hyperonic matter situation (π
π = 0.5), as described in the legend, is also compared against
the nuclear matter (ππ = 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 ππ
πβ ππ in
this regime, using which (32) reduces to
πcom β ππ·π
(1 +
(π+ π
π)
2β2ππ·π
βπ 1
4π2
π·π
ππ΅)
β ππ·π
(π 2Ξ
4π2
π·π
) ,
(34)
with π 1 = (π1 + π2) β 1.28π1, with our choice of parameters,π 2 (β‘π2) = 0.22π 1, and with the difference term, Ξ = (π
π
Ξ0 +
ππ
Ξβ βπ
π
πβπ
π
π).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 ππ΅, as weconcluded from (27). The marginal increase of π above π1is compensated by a marginal increase in π for the π
π ΜΈ= 0
case, as compared to the ππ = 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 π·
πmesonsβ in-medium interactions, especially at
large ππ΅. From the point of view of the π·π meson dispersionrelations, this is conclusively because of the contributionsfrom the extra, hyperonic terms, which result in an overallincrease of the coefficient π΄hyp significantly above π΄nucl,particularly at large ππ .
Finally, we consider momentum-dependent effects byinvestigating the behavior of the in-medium optical poten-tials for the π·
πmesons. These are shown in Figure 6, where
we consider them in the context of both asymmetric nuclear(π
π = 0) and hyperonic matter (π
π = 0.5), as a function of
momentum π (=|οΏ½βοΏ½|) at fixed values of the parameters ππ΅, π,
and π. As with the rest of the investigation, the dependenceof the in-medium optical potentials on the parameters πand π 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., π(π = 0) =βΞπ(π = 0) β‘ Ξπ(π
π΅, π, π, π
π )). It follows then that, at
π = 0, the two π·πmesons 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 π·+
π
and π·βπin nuclear matter. In hyperonic matter, however,
nonzero contribution from these terms breaks the degeneracyin the π = 0 limit (as was already discussed before), and finitemomentum preserves this nondegeneracy. Consequently, atany fixed density, the curves for different values of π
π differ
in terms of their π¦-intercept but otherwise appear to runparallel. Further, as we had reasoned earlier for the π = 0
Advances in High Energy Physics 13
case, the effect of increasing ππ on the mass drops of π·+
πand
π·β
πis equal and opposite about the nuclear matter situation at
small densities, which readily explains the behavior of opticalpotentials for these mesons at π
π΅= π
0. At larger densities,
for example, the ππ΅= 4π
0case shown in Figure 6, the lower
values of optical potentials for bothπ·+πandπ·β
π, as compared
to the ππ = 0 case, can be immediately reconciled with thebehavior observed in Figure 5. In fact, the large differencebetween theπ·+
πoptical potentials for the ππ = 0 and ππ = 0.5
cases, as compared to that for π·βπ, 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 π·
πmesons, with
an increase in baryonic density. This decrease in π·πmeson
mass can result in the opening up of extra reaction channelsof the type π΄ β π·+
ππ·
β
πin 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 π π (or ππ ) system, as compared to ππ 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 π·
πmeson curve with the respective
energy level, the mass of the π·+ππ·
β
πpair 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 oftheπ·mesons, in [46]).
Apart from this, one expects signatures of these mediumeffects to be reflected in the particle ratio (π·+
π/π·
β
π). A produc-
tion asymmetry betweenπ·+πandπ·β
πmesons might be antici-
pated owing to their unequal mediummasses, as observed inthe current investigation. Also, due to the semileptonic andleptonic decaymodes of theπ·π mesons [47], one also expectsthat, accompanying an increased production of π·π 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)
πc2(2P) (3927)
π(3770)
X(3915)
X(3872)
πC(2S) (3639)
(πB/π0)
π(2S) (3686)
Ener
gyπ(k=0)
forD
+ SDβ S
pair
(MeV
)
Figure 7: Mass of the π·+ππ·
β
πpair, compared against the (vacuum)
masses of the excited charmonia states, at typical values of temper-ature (π = 100MeV) and asymmetry (π = 0.5), for both nuclearmatter (π
π = 0) and hyperonic matter (π
π = 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 π·
πmesons 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 oftheπ·
πmesons with the hadronic medium might also lead to
the possibility of formation ofπ·π-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 π·
π(1968.5) mesons. As has already
been mentioned, [42β45] have treated the medium behaviorof π·π mesons, using the coupled channel framework. Thebroad perspective that emerges from all these analyses is thatof an attractive interaction in the medium [44, 45] betweenπ·π mesons and baryon species like the nucleons or theΞ, Ξ£, Ξhyperons [42β45], 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 π
ππ (2892) state in the π·+
ππ 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 theπ·
πmesons 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 π·π mesons can lead to openingup of the new channels for the charmonium states decayingto π·+
ππ·
β
πat 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 π·π mesons, since the statesπ(3872),π(3915), andπ(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 1π for π½/π, 2π for π(3686), and 1π· forπ(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,π(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 thisπ(3872) state [65],borne out of contributions from the (π·+π·ββ + π·βπ·β+) and(π·
0π·
β0
+ π·0
π·β0) components in the π -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 ofπ·πmesons
in a hot and dense hadronic environment, within the frame-work of the chiral ππ(3)
πΏΓ ππ(3)
π model, generalized to
ππ(4). The generalization of chiral ππ(3) model to ππ(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 theπ·π mesons in the present work.However, realizing that the chiral symmetry is badly brokenfor the ππ(4) case due to the large mass of the charm quark,we use the interactions derived from ππ(4) for theπ·π mesonbut use the observed masses of these heavy pseudoscalarmesons as well as empirical/observed values of their decayconstants [30]. Theπ·
πmesons 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, π·
πmesons 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 π·+
πand π·β
πis observed to be broken.
The mass splitting between π·+πand π·β
πis 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 π·πmesons, 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 π·+
π/π·
β
πratio, dilepton
spectra, possibility of formation of exotic π·π-nucleus bound
states, and modifications of the decays of charmonium statesπ·
+
ππ·
β
πpair in the hadronic medium. The medium modifi-
cations of the π·πmesons 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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