Dirac sea effects on Heavy Quarkonia decay widths in magnetized matter a field theoretical model of composite hadrons Amruta Mishra

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Dirac sea eﬀects on Heavy Quarkonia decay widths in magnetized

matter – a ﬁeld theoretical model of composite hadrons

Amruta Mishra∗

Department of Physics, IIT, Delhi, New Delhi – 110 016, India

S.P. Misra†

Institute of Physics, Bhubaneswar – 751005, India

Abstract

We study the partial decay widths of charmonium (bottomonium) states to D ¯

D (B¯

B) mesons

in magnetized (nuclear) matter using a ﬁeld theoretical model of composite hadrons with quark

(and antiquark) constituents. These are computed from the mass modiﬁcations of the decaying

and produced mesons within a chiral eﬀective model, including the nucleon Dirac sea eﬀects. The

mass modiﬁcations of the open charm (bottom) mesons are calculated from their interactions with

the nucleons and the scalar mesons, whereas the mass shift of the heavy quarkonium state is

obtained from the medium change of a scalar dilaton ﬁeld, χ, which mimics the gluon condensates

of QCD. The Dirac sea contributions are observed to lead to a rise (drop) in the quark condensates

as the magnetic ﬁeld is increased, an eﬀect called the (inverse) magnetic catalysis. These eﬀects

are observed to be signiﬁcant and the anomalous magnetic moments (AMMs) of the nucleons

are observed to play an important role. For ρB=0, there is observed to be magnetic catalysis

(MC) without and with AMMs, whereas, for ρB=ρ0, the inverse magnetic catalysis (IMC) is

observed when the AMMs are taken into account, contrary to MC, when the AMMs are ignored.

In the presence of a magnetic ﬁeld, there are also mixings of spin 0 (pseudoscalar) and spin 1

(vector) states (PV mixing) which modify the masses of these mesons. The magnetic ﬁeld eﬀects

on the heavy quarkonium decay widths should have observable consequences on the production the

heavy ﬂavour mesons, which are created in the early stage of ultra-relativistic peripheral heavy ion

collisions, at RHIC and LHC, when the produced magnetic ﬁelds can still be extremely large.

∗Electronic address: amruta@physics.iitd.ac.in

†Electronic address: misrasibaprasad@gmail.com

arXiv:2210.09192v3 [hep-ph] 23 Aug 2023

I. INTRODUCTION

The study of the in-medium properties of the heavy ﬂavour mesons [1], in particular in the

presence of strong magnetic ﬁelds, has been a topic of intense research due to its relevance

in relativistic heavy ion collision experiments. The heavy ﬂavour mesons are created at the

early stage when the magnetic ﬁelds resulting from ultra-relativistic peripheral heavy ion

collisions, estimated to be huge [2], can still be extremely large. The heavy quarkonium

states and the open heavy ﬂavour mesons have been studied extensively in the literature

using the potential models [3–13], the QCD sum rule approach [14–31], the coupled channel

approach [32–38], the quark meson coupling (QMC) model [39–47], as well as using a chiral

eﬀective model [48–55]. Studies of heavy quarkonium states ( ¯

QQ bound states, Q=c, b) in

presence of a gluon ﬁeld, assuming the distance between Qand ¯

Qto be small as compared

to the scale of the gluonic ﬂuctuations, show that the mass modiﬁcations of these states arise

from the medium modiﬁcation of the scalar gluon condensate in the leading order [56–58].

A study of the mass modiﬁcations of the charmonium states due to the gluon condensates as

well as ¯

DD meson loop [59] showed that the dominant contributions are due to the medium

modiﬁcations of the gluon condensates. In a chiral eﬀective model, the in-medium masses

of the heavy quarkonium (charmonium and bottomonium) have been computed from the

medium change of a scalar dilaton ﬁeld [50, 51, 55], which simulates the gluon condensates

of QCD within the eﬀective hadronic model.

The chiral eﬀective model, in the original version with three ﬂavours of quarks (SU(3)

model) [60–63], has been used extensively in the literature, for the study of ﬁnite nuclei [61],

strange hadronic matter [62], light vector mesons [63], strange pseudoscalar mesons, e.g. the

kaons and antikaons [64–67] in isospin asymmetric hadronic matter, as well as for the study

of bulk matter of neutron stars [68]. Within the QCD sum rule framework, the light vector

mesons [69, 70], as well as, the heavy quarkonium states [16–18], in (magnetized) hadronic

matter have been studied, using the medium changes of the light quark condensates and

gluon condensates calculated within the chiral SU(3) model. Using the in-medium masses of

the heavy ﬂavour mesons in the (magnetized) hadronic matter, calculated within the chiral

eﬀective model, the partial decay widths of the heavy quarkonium states to the open heavy

ﬂavour mesons have been studied in (magnetized) hadronic medium [51, 71], using a light

quark-antiquark pair creation model [72], namely the 3P0model [73–76] as well as using a

ﬁeld theoretical model for composite hadrons with quark (and antiquark) constituents [77–

81]. The eﬀects of magnetic ﬁeld on the masses of the heavy ﬂavour mesons have been studied

in Refs. [82–89], and, it is observed that the spin-magnetic ﬁeld interaction leads to mixing

between the pseudoscalar meson and the longitudinal component of the vector meson (PV

mixing). This results in a dominant rise (drop) in the mass of the longitudinal component of

the vector meson (pseudoscalar) meson for the heavy quarkonia (charmonia and bottomonia)

states as well as for open charm (bottom) mesons [84–89]. In the presence of a magnetic

ﬁeld, the studies of the eﬀects of Dirac sea (DS) in the quark matter sector [90–93] within

the Nambu-Jona-Lasinio model [94–96], are observed to lead to enhancement of the light

quark condensates with increase in the magnetic ﬁeld, an eﬀect called the magnetic catalysis

(MC). The opposite trend of the light quark condensates with magentic ﬁeld, namely the

inverse magnetic catalysis (IMC) is observed in some lattice QCD calculations [97], where

the crittical temperature, Tcis seen to decrease with increase in the magnetic ﬁeld. For the

nuclear matter, the eﬀects of Dirac sea (DS) have been studied using the Walecka model as

well as an extended linear sigma model in Ref. [98]. These are observed to lead to magnetic

catalysis (MC) eﬀect for zero temperature and zero density, which is observed as a rise in

the eﬀective nucleon mass with the increase in magnetic ﬁeld. In Ref. [99], the contributions

of Dirac sea of the nucleons to the self-energies of the nucleons have been studied in the

Walecka model by summing over the scalar (σ) and vector (ω) tadpole diagrams, in a weak

magnetic ﬁeld approximation of the fermion propagator. At zero density, the eﬀects of the

Dirac sea are seen to lead to magnetic catalysis (MC) eﬀect at zero temperature [99]. When

the anomalous magnetic moments (AMMs) of the nucleons are taken into account, at a ﬁnite

density and zero temperature, there is observed to be a drop in the eﬀective nucleon mass

with increase in the magnetic ﬁeld. This behaviour with the magnetic ﬁeld is observed when

the temperature is raised from zero to non-zero values, upto the critical temperature, Tc,

when the nucleon mass has a sudden drop, corresponding to the vacuum to nuclear matter

phase transition. The decrease in Tcwith increase in value of Bis identiﬁed with the inverse

magnetic catalysis (IMC) [99].

In the present work, the partial decay widths of the charmonium (bottomonium) states

to open heavy ﬂavour mesons, D¯

D(B¯

B) are studied in magnetized (nuclear) matter using

a ﬁeld theoretical model of composite hadrons. As the matter created in ultra-relativiistic

peripheral heavy ion collisions is dilute, we study the partial decay widths of the lowest

quarkonium states in the charm and bottom sectors, ψ(3770) and Υ(4S) (which decay to

D¯

Dand B¯

Bin vacuum). These are investigated for ρB= 0 as well as for ρB=ρ0, the

nuclear matter saturation density, for symmetric as well as asymmetric nuclear matter in

the presence of an external magnetic ﬁeld. The study of eﬀects of temperature on the open

charm and charmonium masses (and hence on the charmonium decay widths) [50, 51] have

been observed to be marginal for small densities (upto ρ0). Within the chiral eﬀective model,

the mass shift of the heavy quarkonium states and the open heavy ﬂavour mesons arise from

the medium modiﬁcations of the dilaton ﬁeld and the scalar ﬁelds, which have marginal

modiﬁcations due to temperature, and, hence the temperature eﬀects on the quarkonium

decay widths (due to mass modiﬁcation of these mesons) are not taken into account in the

present study. The magnetic eﬀects are the most dominant eﬀects for the (dilute) matter

resulting from ultra-relativistic peripheral collisons, which include the contributions from

the magnetized Dirac sea of nucleons as well as PV mixing, in additon to the Landau

level contributions for the charged hadrons. In the chiral eﬀective model, the eﬀects of the

Dirac sea are incorporated to the nucleon propagator, through summation of scalar (σ,ζ

and δ) and vector (ωand ρ) tadpole diagrams. When the anomalous magnetic moments

(AMMs) of the nucleons are not taken into account, for zero density as well as for ρB=ρ0,

magnetic catalysis (MC) is observed. However, when the AMMs of nucleons are considered,

for ρB=ρ0(both for symmetric and asymmetric nuclear matter), inverse magnetic catalysis

(IMC) is observed, i.e., the quark condensate is observed to be reduced with rise in the

magnetic ﬁeld.

The outline of the paper is as follows. In section II, we describe brieﬂy the chiral eﬀective

model used to calculate the masses of the charmonium (bottomonium) and open charm

(bottom) mesons, accounting for the eﬀects of the Dirac sea for the nucleons. The PV mixing

eﬀects are also taken into account which modify the masses of the heavy quarkonium states

as well as open heavy ﬂavour mesons. In section III, the computations of the decay widths of

ψ(3770) →D¯

Dand Υ(4S)→B¯

Busing the ﬁeld theoretical model of composite hadrons are

brieﬂy described, and, the salient features of the model are presented in Appendix A. The

results of the partial decay widths in magnetized (nuclear) matter are discussed in section

IV and the summary of the present work are given in section V.

II. MASS MODIFICATIONS OF CHARM AND BOTTOM MESONS

We describe breiﬂy the chiral eﬀective model used to study the open charm (bottom)

mesons and the charmonium (bottomonium) states in magnetized nuclear matter. The

model is a generalization of a chiral SU(3) model, based on a nonlinear realization of chiral

symmetry, and, the breaking of scale invariance of QCD. The scale symmetry breaking

is incorporated through a scalar dilaton ﬁeld (which mimics the scalar gluon condensate)

and the mass modiﬁcations of the heavy quarkonium states are obtained from medium

modiﬁcations of the dilaton ﬁeld. The in-medium masses of the open heavy (charm and

bottom) ﬂavour mesons are obtained by generalizing the chiral SU(3) model to include the

interactions of the open charm and bottom mesons with the light hadrons.

In the presence of a magnetic ﬁeld, the Lagrangian for SU(3) model has the form [100]

L=Lkin +X

WLBW +Lvec +L0+Lscalebreak +LSB +LBγ

mag,(1)

where Lkin refers to the kinetic energy terms of the baryons and the mesons, LBW is the

baryon-meson interaction term, Lvec describes the dynamical mass generation of the vector

mesons via couplings to the scalar mesons and contain additionally quartic self-interactions

of the vector ﬁelds, L0contains the meson-meson interaction terms, Lscalebreak is the scale

invariance breaking term and LSB describes the explicit chiral symmetry breaking. The

kinetic energy terms are given as

Lkin =iTrBγµDµB+1

2TrDµXDµX+ Tr(uµXuµX+XuµuµX) + 1

2TrDµY DµY

2DµχDµχ−1

4Tr ˜

Vµν ˜

Vµν −1

4Tr (Aµν Aµν )−1

4Tr (Fµν Fµν ),(2)

where, Bis the baryon octet, Xis the scalar meson multiplet, Yis the pseudoscalar chiral

singlet, χis the scalar dilaton ﬁeld, Vµν =∂µVν−∂νVµ,Aµν =∂µAν−∂νAµ, and Fµν =

∂µAν−∂νAµ, are the ﬁeld strength tensors of the vector meson multiplet, Vµ, the axial

vector meson multiplet Aµand the photon ﬁeld, Aµ. In Eq. (2),

uµ=−i

4[(u†(∂µu)−(∂µu†)u)−(u(∂µu†)−(∂µu)u†)],(3)

and the covariant derivative of a ﬁeld Φ(≡B, X, Y, χ) reads DµΦ = ∂µΦ + [Γµ,Φ], with

Γµ=−i

4[(u†(∂µu)−(∂µu†)u)+(u(∂µu†)−(∂µu)u†)],(4)

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DiracseaeffectsonHeavyQuarkoniadecaywidthsinmagnetizedmatter–afieldtheoreticalmodelofcompositehadronsAmrutaMishra∗DepartmentofPhysics,IIT,Delhi,NewDelhi–110016,IndiaS.P.Misra†InstituteofPhysics,Bhubaneswar–751005,IndiaAbstractWestudythepartialdecaywidthsofcharmonium(bottomonium)statestoD¯D(B¯B)meson...

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Dirac sea effects on Heavy Quarkonia decay widths in magnetized matter a field theoretical model of composite hadrons Amruta Mishra

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