Effect of thermal fluctuation of baryons on vector mesons and low mass dileptons

ρ

ω

eee

e

Sanyasachi Ghosh (VECC, Kolkata, India)

gluon ofpart kinetic )(,

QCD

qmDiqLduq

q

RL

55

q q

)1(2

1 )1(

2

1

qqq

)}( ) {(,

QCD RqLRqLRRduq

LL qmqqmqqDiqqDiqL Mass term 0

)e( 2/i,,,,

RL, RLRLRLRL UqUq

remain invariant under global symmetry &associated conserved Noether currents

RL SU(2)SU(2)

RLa

RLaRL qqJ ,,, )2/(

)(1 )2/(

)(1 )2/( 5

-

LRaa

A

LRaa

V

JJqqJ

JJqqJ Parity doublet

Vector current

Axial vector current

Real world

Spontaneous Breaking

of Chiral Symmetry

(S

BCS)

Broken Chiral Symmetry Restored Chiral Symmetry

Vacuum Stronglyinteracting matter

The spectral properties of vector mesons and axial Vector mesons in strongly-interacting matter may indicate about Chiral Symmetry Restoration (CSR).

Spectral function of ρ and a_1 are measured via hadronic decays of τ lepton at LEP by ALEPH and OPAL

experimental data from heavy ion collisionseems to prefer melting scenario

Rapp et.al. arXiv:0901.3289

Vector mesons are very special to study its in-medium property. Because they are directly couplewith electromagnetic (e.m.) channels.

e

e

matter) ofdimension (D

D 1

path freeMean

e.m.

Low mass enhancement due to in-medium modificationof light vector mesons (specially ρ).

Being motivated by this in-medium signature,we have evaluated spectral functions of lightVector mesons (ρ and ω) in hot and dense hadronic matter using Real-time formalism ofThermal Field theory (RTF).

Fourier transform

t

0 1 2 3 4 5

N(t)/N(0)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Probabilistic amplitude of Unstable particle

time space (t) energy space

)2/exp(~)(

)exp()0(

)(

)0(

)(2

ttimt

tt

N

tN

H

)(~ Im)( ,2/)(

1~)(~

)()2/()(~)(~

)(

)(

000

0

0

2

00

qqimq

q

qm

q

mN

qN

H

HH

5

)(~)( 000

Lim Hmqq

QFT definition of spectral function

),( xtx

),( 0 qqq

imq

mqq

H

H

22

22

0

1 Im~

)(~)(Lim

0|)()(|0 21 xx Propagator

Spectral function

= + + + ……………∞

=0 + 00 + 000 + ………….(an infinite g.p. series)

=0

0

1

= 22

1

baremq

where is called self energy term

220

1

baremq

) Im() Re(q

1

vacvac22

imbare

Vacuum interacting propagator

)q(q

1)q(

2222

mD

imq

mqq

H

H

22

22

0

1 Im~

)(~)(LimVacuum free propagator

) Im()(q

1

vac22

imphy Im = m

thvactotal

Mass shift

Width enhancement

m)(

ReReRe2

2vac

2

phy

thphythbare

m

mm

Spectral function = Im [propagator]0

e e

]})(q{

1[Im)(

22vdecayphy iqm

qA

]}q{

1[Im)(

tot22tot

barem

qA ]}Im)Re(q{

1[Im)(

tottot22tot

barem

qA ])}()(q{

1[Im)(

2phy

2tot

decayiqmmqA

VacuumMedium

-∞ +∞t-axis - ib

0-∞

(+∞,-ib/2)

+∞

t-plane

i t

) H ( exp) t H i ( exp

Time evolution operator

~ density matrix

Essence of RTF

Field Theory of vacuum Field Theory at finite temperature

dt contour

d

thermal propagator

Thermal self-energy

abD

ab),,q(q

1),,q(

0220

TqmTqD

Diagonalization

H

H

e

eTr

}{ AA

Thermal averaged

0 0 A

Vacuum expectation

)()()()(or )()( 332211 yxyxyx

Self Energy of ρ for mesonic loops :

T)k,-q(pD T)(k,D k)(q,N)2(

),( 11114

4

11

kd

Tq

(k)(q) (q)

(p=q – k)

Thermal vacuum11 D

)((..)),(Im 0,

311 pk

ba

baqkdTq

,, , ,ba,

0Lq

0Lq 0

Uq0Uq 0

Landau cuts Unitary cutsUnitary cuts

220 )( kpL mmqq 220 )( kpU mmqq

})nn(1-)n(1{nvac~Im

pkpk

m

ML

)1170(h 1ρ p p ρ

Mesonic collision rate

k k

)( MC

}nn-)n)(1n{(1vac~Im

pkpk

m

MU

ρ ρk k

pp

Bose enhancement of decay rate )( ..M

eB

)1260(a 1

)782(

N

B

N

B+

2B )m(m qMN

o

Unitary cut

)m(mB N

Self Energy of ρ for baryonic loops :

] }nn-)n)(1n{(1 }nn-)n)(1n{(1 [vac~Im

BNBN-BN

-BN

m

BU

ρ ρ

),( NN ),( NN

),( BB ),( BB

Pauli blocking of decay rate )( ..B

bP

Landau cut

] )}n-(1n-)nn{(1 )}n-(1n-)nn{(1 [vac~Im

BNBNBN-BN

m

BL

ρ

),( BB ),( BB

ρ),( NN ),( NNBaryonic collision rate )( B

C

12 2qM

o

Unitary cutLandau cut

N

BN*(1520)

N*(1650)

N*(1720)

)1232(

)1620(*

)1700(*

----------------------------vacuum part of loop (unitary cut )

-------------------------thermal part of loops (unitary cut)

---------------------------- loops (Landau cut )

----------------------------

loops (Landau cut)

MeV 150decay MeV 25.. M

eB MeV 55MC MeV 20B

C

MeV 55MC

MeV 20BC

MeV 150decay

MeV 25. MeB

) , ,

(

11 ah

H

),,,

(**

NNNNNN

NB

MeV 250 & MeV 150 B T

Effect of baryonic chemical potential on ρ spectral function in low mass region:

),,,( 0BTqqA

Effect of temperature on ρ spectral function in low mass region:

),,,( 0BTqqA

Effect of momentum of ρ in off mass shell on its spectral function in low mass region:

),,,( 0BTqqA

N

B

S. Sarkar & S. GhosharXiv:1204.0893

ρ meson spectral function

ω meson spectral function

S. Ghosh and S. Sarkar arXiv:1207.2251 [nucl-th]

I )(J)( )()(J dFie)(2nd

)exp(

lh442

4int

yyAxAxyxd

xdLiSFI

)( )( )( )( int xAxJexAxJeL lh

Strong interacting current

Leptonic current

Z

)Eexp(

)2()2( N

2

, ,

I3

23

31

3

21

pp FI

FISVpdVpd

Final phase space Thermal average

Dilepton multiplicity

WL

qxdd

dN (...)

44

Formalism ofdilepton :

WL

qd

dR (...)

4

)0(J)(J e ),(W hhiq40 xxdiqq x

ssdduudduuem

3

1 2/)(

3

1 2/)(J

HM ),,,( (...) 0

,,4 B

ii TqqAL

qd

dR

J 3

1 J

3

1 J J

em

ssdduuem

3

1

3

1

3

2J

QM

4qd

dRq

q

l

l

2

Contribution of ω is down by a factor ~ 10

Effect of mesonic as well as baryonic medium modification of ρ on dilepton rate in low mass region :

),,,(A (...) 0422 B

T

TqqLqd

dR

dyqddM

dR

qxdd

d44

0 }T,q,N{q

23

},{ 0 qq

)(xu

)(xT

Dilepton production ininvariant mass after Space time evolution :

xdqxdd

xxud 444

})T(,q),(N{q

])}T(,q),(N{q

[)()(

4424

qxdd

xxudqMdydxd

dM

MdNT

Invariant mass spectra

Fluid element

Low mass enhancement at SPS :

Phys.Rev. C85 (2012) 064906J K Nayak, J Alam, T Hirano, S Sarkar and B Sinha  

S. Sarkar & S. GhosharXiv:1204.0893

Meson (S.Ghosh, S.Mallik. S.Sarkar Eur. Phys. C 70, (2010) 251) + Bayon (S.Ghosh, S.Sarkar Nucl. Phys. A 870, (2011) 94) -loop self-energies

Meson loop self-energies (S.Ghosh, S.Mallik. S.Sarkar Eur. Phys. C 70, (2010) 251) + Baryon part from Eltesky et. al. (Phys. Rev. C 64, (2001) 035202)

S. Sarkar & S. GhosharXiv:1204.0893

Analytic structure of ρ meson propagator at finite temperatureEur. Phys. J. C 70, 251 (2010).S.Ghosh, Sourav Sarkar, (VECC) , S. Mallik, (SINP)

ρ self-energy at finite temperature and density in thereal-time formalismNucl. Phys. A 870, 94 (2011) S. Ghosh, Sourav Sarkar, (VECC)

Observing many-body effects on lepton pair production from lowmass enhancement and flow at RHIC and LHC energiesEur. Phys. J. C 71, 1760 (2011).S. Ghosh, Sourav Sarkar, Jan-e Alam, (VECC)

The $\rho$ meson in hot hadron matter and low mass dilepton spectra.Nucl. Phys. A 862, 294 (2011) S. Ghosh, Sourav Sarkar, Jan-e Alam, (VECC)

In-medium vector mesons and low mass lepton pairs from heavy ionCollisionsarXiv:1204.0893 [nucl-th] Sourav Sarkar, S. Ghosh

Elliptic flow of thermal dileptons as a probe of QCD matter Phys.Rev. C (R) 85 (2012) 031903

Payal Mohanty, Victor Roy, S. Ghosh, Santosh K. Das, Bedangadas Mohanty, Sourav Sarkar, Jane Alam, Asis K. Chaudhuri, (VECC)

Theoretical work on dilepton by VECC group (India)

Understanding low mass enhancement of SPS datain the language of Thermal Field Theory :

Mesonic collision rate

Baryonic collision rate

Bose enhancement of decay rate

Vacuum decay rate+

+

+

Hdronic Matter

Quark Matter

Total

+

(with Feynman boundary condition)

])()([2

),,( )( )( 0

ttitti ettetti

qtt

(with K.M.S.boundary condition)

])}()({

)}()([{2

),,(

)(

)(0

i

i

ef

efi

qD

Temporal Fourier transform

2200

1),(

mqqq

Temporal Fourier transform

220

210

120

110

00 ),(DD

DDqqDab

)(),,(][ 02

2

2

ttqttt

spatial Fourier transform spatial Fourier transform

)(),,(][ 02

2

2

cqD

)(),,(])[( 40

2 xxxxttmx

Free scalar propagator in vacuum satisfies

)(),,(])[( 40

2 xxxxDm cc

Free scalar propagator in medium satisfies

0,for

0,for ,),(),(

),(),(00

tDeCe

tBeAeDtiti

titi

Extra slides

Real part of self-energy

Baryonic loops

Mesonic loops