Int J Theor PhysDOI 10.1007/s10773-014-2365-5

In Medium Properties of Charmed Strange Mesonsin Dense Hadronic Matter

Sushil Kumar

Received: 18 July 2014 / Accepted: 4 October 2014© Springer Science+Business Media New York 2014

Abstract The medium modifications of the charmed strange mesons in the dense hadronicmatter are investigated within chiral SU(4) model. The charmed strange meson proper-ties modifies due to their interactions with the nucleons, hyperons and the scalar mesons(scalar-isoscalar mesons (σ , ζ ), scalar isovector meson (δ)) in the dense hadronic medium.The various parameters used in the chiral model are obtained by fitting the vacuum baryonmasses and saturation properties of nuclear matter. The non-linear coupled equations ofthe scalar fields are solved to obtain their baryon density, isospin and strangeness depen-dent values. Furthermore, the dispersion relations are derived for charmed strange mesons.Effects of isospin asymmetry and strangeness on the energies of charmed strange mesonsare investigated. The in medium properties of charmed strange mesons can be particularlyrelevant to the experiments with neutron rich beams at the Facility for Antiproton and IonResearch (FAIR) at GSI, Germany, as well as to experiments at the Rare Isotope Accelerator(RIA) laboratory, USA. The present study of the in medium properties of charmed strangemesons will be of direct relevance for the observables from the compressed baryonic matter,resulting from the heavy ion collision experiments.

Keywords Charmed strange · Dense hadronic matter

1 Introduction

One of the challenging current problems in nuclear physics is to elucidate the behaviorof hadronic matter under extremely high-density and/or high-temperature environments.At zero density and temperature, the QCD vacuum has a non-trivial structure, whichmanifests itself by non-vanishing quark and gluon condensates. Chiral symmetry is sponta-neously broken, and the quark condensate (qq) is the corresponding order parameter. It is

S. Kumar (�)Department of Physics and Electronics, Hans Raj College, University of Delhi,Delhi, 110007, Indiae-mail: sushil8207@gmail.com

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theoretically expected that, at very high baryon densities (even at low temperatures), chiralsymmetry is likely to be restored, and that baryon matter can be converted into quark matter.Hadronic properties, at least those related to the light hadrons, are closely linked to the vac-uum structure through the chiral condensate. One, therefore, expects characteristic changesof those properties with increasing baryon density and temperature.

In the present investigation, we shall use a chiral SU(4) model for the description ofhadrons in the medium. To study the in medium properties of charmed strange meson, chiralSU(3)model is generalised to SU(4) [1]. However, since the chiral symmetry is broken forthe SU(4) case by the large charm mass, we use the SU(4) symmetry only to derive theinteractions of the Ds

+ and Ds− mesons in the dense hadronic medium, but use observed

values of the charmed hadron masses and empirical values of the decay constants [2]. Weinvestigate the effect of strangeness and isospin asymmetry on the charmed strange mesonsmasses at zero temperature and finite baryonic density. This study will be of direct relevancefor the upcoming experiment at the Facility for Antiproton and Ion Research (FAIR), at GSI,Germany. The charm and strange mesons produced in these experiments are expected to bemodified at high densities and/or high temperatures, which should reflect in experimentalobservables. Production of matter at high densities and the in medium properties of strangeand charm mesons are planned to be investigated extensively in CBM(Compressed BaryonicMatter) experiment at GSI FAIR. The CBM experiment at FAIR intends to explore the phasediagram of strongly interacting matter at high baryon densities and moderate temperaturesin heavy ion collisions and the medium modification of the charm and strange mesons areplanned to be investigated in these experiments. The PANDA experiment of FAIR project,(in the experiment with the annihilation of antiprotons on the nuclei) also intends to studycharm and strange mesons and charmonium spectroscopy [3].

The paper is organized as follows: the formalism is presented in Sections 2 and 3, wherewe first review the hadronic chiral SU(4) model and, next, we describe the calculationof the interaction Lagrangian and self energies of charmed strange mesons in the densehadronic matter. In Section 4 we show the results obtained for the energies of charmedstrange mesons. Section 5, summarizes the results.

2 Chiral SU(4) Hadronic Model

In the present investigation, we use an chiral SU(4) [1] model for the study of charmedstrange meson. This is generalization of SU(3) chiral model [4], which is a relativisticfield theoretical model of baryons and mesons build on chiral symmetry [5–7] and bro-ken scale invariance [4, 8]. A non linear realization of chiral symmetry is adopted thathas been successful in a simultaneous description of finite nuclei and hyperons potentials[5–7]. The same model with linear realization of chiral symmetry has been used to studythe hadronic matter in vacuum as well as in the medium [9]. Chiral SU(3) model have beenquite successful in modeling hadron interactions. The general form of the Lagrangian is asfollows:

L = Lkin +∑

W=X,Y,V ,A,u

LBW + Lvec + L0 + LSB. (1)

The Lagrangian contains the baryon octet and the spin-0 and spin-1 meson multiplets aselementary degrees of freedom. In (1), Lkin is the kinetic energy term, LBW includes theinteraction term of the baryons with the spin-0 and spin-1 mesons, the former generatingthe baryon masses, Lvec generate the masses of the spin-1 mesons through interactions with

Int J Theor Phys

the spin-0 fields and contain quartic self interactions of vector fields. L0 gives the meson-meson interaction terms which induce the spontaneous breaking of chiral symmetry as wellas scale invariance breaking logarithmic potential. Finally, LSB introduces an explicit sym-metry breaking of the chiral symmetry. In the present investigation, we have used the meanfield approximation, where all the meson fields are treated as classical fields. In this approx-imation, only the scalar and the vector fields contribute to the baryon-meson interaction,since for all the other mesons, the expectation values are zero [9, 10]. The interactions ofthe scalar mesons and vector mesons with the baryons are given as,

LBscal + LBvec = −∑

i

ψi

[gωiγ0ω + gφiγ0φ + gρiγ0τ3ρ + m∗

i

]ψi. (2)

The interaction of the vector mesons, of the scalar fields and the interaction correspondingto the explicitly symmetry breaking in the mean field approximation are given as,

Lvec = 1

2

(m2

ωω2 + m2ρρ2 + m2

φφ2) χ2

χ20

+ g4(ω4 + 6ω2ρ2 + ρ4 + 2φ4), (3)

L0 = −1

2k0χ

2(σ 2 + ζ 2 + δ2) + k1(σ2 + ζ 2 + δ2)2

+k2

(σ 4

2+ δ4

2+ 3σ 2δ2 + ζ 4

)+ k3χ(σ 2 − δ2)ζ

−k4χ4 − 1

4χ4ln

χ4

χ40

+ d

3χ4ln

((σ 2 − δ2)ζ

σ 20 ζ0

(χ

χ0

)3)

(4)

and

LSB = −(

χ

χ0

)2 [m2

πfπσ +(√

2m2kfk − 1√

2m2

πfπ

)ζ

]. (5)

The effective mass of the baryon of species i is given as,

m∗i = −gσiσ − gζ iζ − gδiτ3δ, (6)

where m∗i is the effective mass of the baryon of type, i(i = p, n, , �±,0,�−,0), gσ i, gζ i

and gδi , are the coupling constants for the interaction of baryon species i with the σ, ζ andδ fields respectively and τ3 is the third component of isospin.

The baryon-scalar meson interactions generate the baryon masses through the couplingof baryons to the nonstrange scalar meson σ , strange scalar mesons ζ , and also to scalarisovector meson δ. The coupled equations of motion for the non-strange scalar field σ ,strange scalar field ζ , scalar isovector field δ and the dilaton field χ , are given as,

k0χ2σ − 4k1(σ

2 + ζ 2 + δ2)σ − 2k2(σ3 + 3σδ2) − 2k3χσζ

− d

3χ4 2σ

σ 2 − δ2+

(χ

χ0

)2

m2πfπ −

∑gσiρ

si = 0, (7)

k0χ2ζ − 4k1(σ

2 + ζ 2 + δ2)ζ − 4k2ζ3 − k3χ(σ 2 − δ2)

− d

3

χ4

ζ+

(χ

χ0

)2 [√2m2

kfk − 1√2m2

πfπ

]−

∑gζ iρ

si = 0, (8)

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k0χ2δ − 4k1(σ

2 + ζ 2 + δ2)δ − 2k2(δ3 + 3σ 2δ) + 2k3χδζ

+ 2d

3χ4

(δ

σ 2 − δ2

)−

∑τ3gδiρ

si = 0, (9)

k0χ(σ 2 + ζ 2 + δ2) − k3(σ2 − δ2)ζ + χ3

[1 + ln

(χ4

χ40

)]+ (4k4 − d)χ3

−4

3dχ3ln

(((σ 2 − δ2

)ζ

σ 20 ζ0

)(χ

χ0

)3)+ 2χ

χ20

[m2

πfπσ +(√

2m2kfk− 1√

2m2

πfπ

)ζ

]=0.

(10)

where, ρi and ρsi are the vector density and scalar density of the baryon of type,

i (i = p, n, ,�±,0,�−,0) are given as,

ρi = γi

∫d3k

(2π)3

m∗i

E∗i (k)

(1

e(Ei∗(k)−μi

∗)/T + 1− 1

e(Ei∗(k)+μi

∗)/T + 1

), (11)

The above coupled equations of motion are solved at zero temperature, to obtain thedensity dependent values of the scalar fields (σ, ζ and δ) and dilaton field (χ ) in the densehadronic medium. The isospin asymmetry of the hadronic medium is defined through theparameter η, given by,

η =(ρn − ρp

)

(2ρB)+ (ρ�− − ρ�+ )

(ρB)+ (ρ�− − ρ�+)

(2ρB), (12)

where ρi is the number density of the baryon of type, i (i = p, n, ,�±,�−,0) and ρB isthe baryon density. The strangeness fraction, fs of the medium is defined by

fs = �iρiSi

(ρB)(13)

where Si is the number of strange quarks of baryon’ i.

3 Charmed Strange Mesons in Dense Hadronic Matter

In this section we study the Ds+ and Ds

− mesons properties in dense hadronic matter. Wederive the interaction lagrangian and dispersion relations for the charmed strange mesonsin the dense hadronic matter. The modification in the charmed strange mesons (Ds

+ andDs

−) energies arise due to their interactions with the nucleons, hyperons and the scalarmesons. The interaction of the pseudoscalar mesons to the vector mesons, in addition tothe pseudoscalar meson-nucleon vectorial interaction, leads to a double counting in thelinear realization of the chiral effective theory. Within the nonlinear realization of the chiraleffective theories, such an interaction does not arise in the leading or subleading order, butonly as a higher order contribution [11]. The interaction Lagrangian density of the charmedstrange mesons (D+

s and D−s ) is given as,

L = − i

8f 2D

[(4

3 0γ μ 0 + 2�−γ μ�− + 2�+γ μ�+ + 3�−γ μ�− + �0γ μ�0

)

× (D−

s (∂μD+s ) − (∂μD−

s )D+s

)

+ − 1√2

[m2

πfπσ +(

2m2kfk − m2

πfπ

)ζ +

(2m2

DsfDs − m2

πfπ

)ζc

]

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− 1

f 2D

(D+

s D−s

) [(m2

kfk + m2Ds

fDs − m2πfπ

)(ζ + ζc)

]

+[

1√2f 2

Ds

(−2

√2

(fDs + fK − fπ

)(ζ + ζc) − 2

(fDs + fK − fπ

)2)

+ d1

4f 2Ds

(pp + nn + 0 0 + �+�+ + �0�0 + �−�− + �−�− + �0�0)

+ d2

2f 2Ds

(2

3 0 0 + �−�− + �0�0

)]((∂μD−

s )(∂μD+s )

). (14)

In (14), the first term is the vectorial Weinberg Tomozawa interaction term, obtainedfrom the baryon-pseudo-scalar meson interaction Lagrangian for the SU(4) case is asfollows.

LWT = −1

2

∑

i,j,k,l

Bijk γ μ((�μ)l

k Bij l + 2 (�μ)lj Bilk

), (15)

In the above, the baryons belong to the 20-plet representation and mesons belong to the 16-plet representation. The baryons are represented by the tensor Bijk, which are antisymmetricin the first two indices [1]. The indices i, j, k run from one to four, where one can read offthe quark content of a baryon state by the identifications 1 ↔ u, 2 ↔ d, 3 ↔ s, 4 ↔ c.The baryon states are given as [1],

B121 = p, B122 = n, B213 = 2√6 0,

B132 = 1√2�0 − 1√

6 0, B231 = 1√

2�0 + 1√

6 0,

B232 = �−, B311 = �+, B233 = �−, B313 = �0 (16)

where we have written down only the baryons containing the three light quarks,u, d and squarks. The second term and third term in (14) is the scalar meson exchange term, which isobtained from the explicit symmetry breaking term

LSB = −1

2TrAp(uXu + u†Xu†) (17)

where, Ap given as,

Ap = 1/√

2diag(m2

πfπ , m2πfπ , 2m2

KfK − m2πfπ , 2m2

DsfDs − m2

πfπ

), (18)

X is the scalar meson multiplet [1]. In the above, u is given as,

u = exp

[i√2σ0

Mγ5

], (19)

with M = 1√2πaλa as the pseudoscalar meson multiplet [1]. The next three terms of (14)

are the range terms. The first range term is obtained from the kinetic energy term of thepseudoscalar mesons which is defined as,

LRT−1 = T r(uμXuμX + XuμuμX

). (20)

Int J Theor Phys

where, uμ is given as,

uμ = − i

2

[u†(∂μu) − u

(∂μu†

)]. (21)

The range terms d1 and d2 of equation (14) are obtained from the expressions

Ld1 = d1

4

4∑

i,j,k,l=1

BijkBijk(uμ)m

l(uμ)

lm (22)

and

Ld2 = d2

2

4∑

i,j,k,l=1

Bijk(uμ)lm(uμ)m

kBij l (23)

respectively.The dispersion relations for the D+

s and D−s mesons are obtained by the Fourier

transformations of equations of motion. These are given as,

− ω2 + �k2 + m2Ds

− �(ω, |�k|) = 0. (24)

where, mD±s

is the vacuum mass of the D±s meson and �D±

s(ω, |�k|) is the self-energy of the

D±s mesons in the medium. The self-energy �(ω, |�k|, ) for the D+

s meson is given as

�D+s(ω, |�k|) = 1

4f 2Ds

[2ρ�− + 2ρ�0 + 3ρ�− + ρ�0 + 4

3ρ 0

]ω

+ 1√2f 2

Ds

(ζ ′ + ζ ′c)

(m2

kfk + m2Ds

fDs − m2πfπ

)

−[

1

f 2Ds

(fDs + fk − fπ )(m2

kfk − m2πfπ

)− 1

fDs

(fk − fπ )(m2

Ds

)]

+[ √

2

f 2Ds

(fDs + fK − fπ)(ζ ′ + ζ ′c) + 1

f 2Ds

(fK − fπ )2 + 2

fDs

(fK − fπ )

+ d1

4f 2Ds

(ρs

p + ρsn + ρs

0 + ρs�+ + ρs

�0 + ρs�− + ρs

�0 + ρs�−

)

+ d2

2f 2Ds

(2

3ρs

0 + ρs�− + ρs

�0

)](ω2 − �k2). (25)

where ζ ′ = (ζ − ζ0) and ζ ′c = (

ζc − ζc0

), are the fluctuations of the fields ζ and ζc from

their vacuum expectation values in the dense hadronic medium. A non-zero value of thescalar isovector field, δ means the medium has isospin asymmetry and the vacuum expec-tation value of scalar isovector field δ will be zero. In the present investigation, isospinasymmetry is originating from the scalar isovector field δ, Weinberg Tomozava Term (vec-torial interaction) as well as isospin dependent range term (d2 term). Also, the fluctuation,ζ ′c in the heavy charm quark condensate (ζc = cc) from the vacuum value has been observed

to be negligible [12] and its contribution to the in-medium masses of D+s and D−

s mesons

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will be neglected in the present investigation. For the D−s , the expression for self-energy is

given as,

�D−s(ω, |�k|) = − 1

4f 2Ds

[2ρ�− + 2ρ�0 + 3ρ�− + ρ�0 + 4

3ρ 0

]ω

+ 1√2f 2

Ds

(ζ ′ + ζ ′c)

(m2

kfk + m2Ds

fDs − m2πfπ

)

−[

1

f 2Ds

(fDs + fk − fπ )(m2

kfk − m2πfπ

)− 1

fDs

(fk − fπ )(m2

Ds

)]

+[ √

2

f 2Ds

(fDs +fK −fπ

)(ζ ′ + ζ ′

c) + 1

f 2Ds

(fK − fπ)2 + 2

fDs

(fK − fπ )

+ d1

4f 2Ds

(ρs

p + ρsn + ρs

0 + ρs�+ + ρs

�0 + ρs�− + ρs

�0 + ρs�−

)

+ d2

2f 2Ds

(2

3ρs

0 + ρs�− + ρs

�0

)](ω2 − �k2). (26)

4 Results And Discussions

In the present work, we study the effect of strangeness on the energies of charmedstrange meson. Here we have used the following model parameters. The values, gσN =10.56684,gσλ = 7.0423, gσ� = 5.6355, gσ� = 3.5426, gζN = −0.4670, gζλ =3.4875, gζ� = 5.4771, gζ� = 8.1297, gδN = −2.0929,gδλ = 0, gδ� = −6.2960 andgδ� = −4.2031, are determined by fitting vacuum baryon masses. The other parametersas fitted to the asymmetric nuclear-matter saturation properties in the mean-field approx-imation are, gωN = 13.3265, gωλ = 8.8844, gω� = 8.8844, gω� = 4.4422, gφN =13.3265, gφλ = 8.8843, gφ� = 8.8843, gφ� = 4.4422, gρN = −5.4885,gρλ = 0, gρ� =8.8844 and gρ� = 4.4422, [13]. The values of the hadron masses used are, mp = 938 MeV,mn = 939 MeV, mλ = 1115 MeV, m� = 1193 MeV, m� = 1315 MeV, mω = 783 MeV,mφ = 1020 MeV and mρ = 770 MeV. The other parameters used in the calculations are,k0 = 2.536559, k1 = 1.3544, k2 = −4.775199, k3 = −2.77257, k4 = −0.21887,χ0 =409.76060972,σ0 = −93.3029, ζ0 = −106.7663,mk = 498 MeV, mπ = 4139 MeV,fk = 93.3 MeV, fπ = 122.143 MeV [14]. The values of the coefficients d1 and d2 are cal-culated from kaon nucleon scattering length in isospin I=0 and I=1 channels and are takenas 2.56/mk , and 0.73/mk respectively. Unrealistic hyperons potentials arise in the linear σ

model due to the different types of coupling of the spin-0 and spin-1 mesons to baryonsand a direct coupling of the σ with the strange condensate. Both effects can be switched offin the nonlinear realization. However, the experimentally extracted value for the centralpotential of U = 28 ± 1 MeV [15] cannot be reproduced. Fortunately, explicit symmetrybreaking can be introduced in the nonlinear realization without affecting, Partially Con-served Axial Current (PCAC) relations. This allows for a parametrization of the hyperonpotentials and the inclusion of additional terms for the hyperons scalar meson coupling [8]given as,

Lhyp = m3Tr(BB + B[B, S]) Tr(X − X0), (27)

Int J Theor Phys

where X, represents a scalar and Sab = − 1

3

[√3(λ8)

ab − δa

b

]. In the mean field approxima-

tion this leads to following additional mass term [14], given as

m∗i = m∗

i + am3[√

2(σ − σ0) + (ζ − ζ0)]. (28)

The explicit symmetry breaking term contributes only for hyperons at finite baryon densitiesalong the hypercharge direction. With the parameter m3 adjusted to the potential of −28MeV. The additional mass term is only for and � hyperons. The values of m3 used are0.642 MeV and −0.001 MeV for and � hyperon respectively. The modifications of scalarfields, modify the masses of nucleons and hyperons in dense hadronic medium. Interactionsof modified nucleons, hyperons and the scalar fields with charmed strange mesons modifytheir properties.

Figure 1, show the variation of the energies of D+s and D−

s at zero momentum andzero temperature, with baryon density (ρB ) in the units of nuclear-matter saturation density(ρ0), for isospin asymmetry parameter η as 0.0,0.3 and 0.5 respectively. At each value of η,energy is shown for three strangeness fractions (fs = 0.0,0.3,0.5).

For fs = 0 and zero momentum, charmed strange meson energies decreases withincrease in density, after a slight initial rise (irrespective of the value of the isospin asym-metry parameter). At isospin asymmetry parameter η = 0.0, the decrease in the energyof D+

s (D−s ) at zero momentum, is 29.33(29.33) MeV with increase in the baryon density

from ρ0 to 4ρ0. This decrease in the D+s (D−

s ) energies, at zero momentum modifies to29.70(29.70) MeV at η = 0.5. So one can conclude that for non strange matter, decreasein the energy of charmed strange meson with density is almost independent of the isospinasymmetry of the medium.

For a given value of the strangeness fraction , there is increase in the energy of D+s (D−

s )

with increase in the isospin asymmetry of the medium. At fs = 0.0 and ρB = ρ0, theincrease in the energy of D+

s (D−s ) as we move from η = 0.0 to η = 0.5, is 3.35(3.35) MeV.

This increase in the D+s (D−

s ) energies with isospin asymmetry, at zero momentum becomes2.93(2.93) MeV at ρB = 4ρ0. At fs = 0.5 and ρB = ρ0, the increase in the energy ofD+

s (D−s ) as we move from η = 0.0 to η = 0.5, is 3.49(1.86) MeV. At a higher density of

ρB = 4ρ0, the increase in the energy of the charmed strange meson D+s (D−

s ), as we movefrom η = 0 to η = 0.5 is 6.41(15.47) MeV. So one can conclude that isospin asymmetry isparticularly significant for higher strangeness fraction at high densities.

For isospin symmetric matter (η = 0.0), energy of charmed strange mesons increaseswith increase in the strangeness. At baryon density of ρB = ρ0, the increase in the energy ofD+

s (D+s ) at zero momentum, is 3.946(7.27) MeV, as we move from fs = 0.0 to fs = 0.5.

At higher density of ρB = 4ρ0, the increase in the energy of D+s (D+

s ) at zero momentummodifies to 13.62(27.36) MeV, as we move from fs = 0.0 to fs = 0.5. For isospin asym-metric matter (η = 0.5) energies of D+

s and D+s meson increase, as the strangeness fraction

is increased from fs = 0.0 to fs = 0.5. At baryon density of ρB = ρ0, the increase inthe energy of D+

s (D+s ) at zero momentum, is 4.09(5.78) MeV, as we move from fs=0.0 to

fs = 0.5. At higher density of ρB = 4ρ0, the increase in the energy of D+s (D+

s ) at zeromomentum modifies to 17.1(39.9) MeV, as we move from fs = 0.0 to fs = 0.5.

Thus it can be seen that for a given value of η, the energy of charmed strange mesonincreases, as we move to higher strangeness i.e from nuclear medium to hadronic medium,while for a fixed isospin asymmetry, the energies of charmed strange mesons decreases withincrease in the baryon density. The decrease in the energy is more pronounced at higher den-sities. The reason for this behavior is dominance of the d1 term (attractive) with increasingcontributions from the hyperons, in addition to nucleons.

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Fig. 1 (Color online) Variation of the Ds±) a,c,e for D+

s and b,d,f for D−s ) energy at zero momentum and

zero temperature, plotted as functions of the baryon density (ρB ) in units of nuclear-matter saturation density(ρ0), for the isospin asymmetry parameter, η as 0,0.3 and 0.5. For each value of η, the energies are shown forfs = 0.0,0.3,0.5 as the solid, dotted and short dashed lines respectively

Int J Theor Phys

5 Summary

In summary, we present the baryon density and strangeness dependence of the energies ofthe charmed strange mesons in the dense hyperonic matter. The nucleons, hyperons and thescalar mesons get modified in dense hadronic medium and then they modify the proper-ties of the charmed strange mesons. We have also investigated the isospin and strangenesseffects on the energies of charmed strange mesons. Dependence of in medium energies ofcharmed strange mesons on the isospin asymmetry of the medium seems to be even moredominant for larger values of the strangeness fractions and high densities. In the CBM exper-iment at FAIR at GSI, Germany and experiments at the proposed Rare Isotope Accelerator(RIA) laboratory, USA matter may be produced at high density and /or high temperature.

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