Study of chemical potential effects on hadron mass by lattice QCD

Pushkina Irina*

Hadron Physics & Lattice QCD, Japan 2004

Three main points• What do we know from first principles?• Why is QCD at finite density difficult?• What can we do in practice?

*) School of Biosphere Science, Hiroshima University, Japan

Experimental data regarding in-medium hadrons

CERES: Observation of the large low mass e+e- pair enhancement in CERN SPS in Pb+Au collisions at 158 AGeV/c and 40 AGeV/c (nucl-ex/0212015). The data may only be reproduced if the properties of the intermediate in the hot and dense medium are modified.

KEK: at KEK, invariant mass spectra of electron-positron pairs were measured in the region below the meson mass for the p+C and p+Cu collisions (nucl-ex/0011013). The possible signature of the modification at a normal nuclear-matter density.

STAR: The invariant mass of meson decays in Star experiment at Au+Au collisions at RHIC shows 60-70 MeV downward shift of the peak from the vacuum value (nucl-ex/0211001). The modification of the spectral function at finite T and .

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Introduction of chemical potential

€

Z = Tre−

1

κTH −μN( )

= DUDψ Dψ∫ e−

1

κTSG −ψ Δψ

= DU det Δ∫ e−

1

κTSG

A thermodynamical system is described by the partition function

For staggered fermions, the fermion determinant

€

Δ x,y( ) = amqδ x,y +1

2η i x( ) U i x( )δ x+ˆ i ,y

−U i+ y( )δ x−ˆ i ,y{ }

i=1

3

∑

€

+1

2η 4 x( ) e+aμU 4 x( )δ x+ˆ 4 ,y

− e−aμU 4+ y( )δ x−ˆ 4 ,y{ }

The chemical potential is introduced as,

Gavai considered more general form than (Phys.Rev.D32 (1985) 519)Creutz discussed how the chemical potential appears in lattice fermion formulation (hep-lat/9905024)Hasenfratz and Karsch have shown that such formula avoids nonphysical divergence of the free energy of quarks (Phys.Lett.125B (1983) 308)€

e±aμ

€

U t x( ) → eaμU t x( )

U t+ x( ) → e−aμU t

+ x( )

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Introduction of chemical potential

€

Γ5 x,y( ) = −1( )x1+x2 +x3 +x4 δ x,y

€

Δ ( )+ =Γ5Δ −μ( )Γ5

Due to the complex nature of the fermion determinant, the standard Monte Carlo simulation is very difficult to obtain physical quantities.Quench approximation of lattice QCD is not appropriate for finite density system.

The property of fermion determinant

€

Uν+ x + ˆ μ ( )

€

U μ x + ˆ ν ( )

€

Uν x( )€

U μ+ x( )

€

x + ˆ μ + ˆ ν

€

x + ˆ ν

€

x + ˆ μ

€

x€

ˆ μ

€

ˆ ν

€

a

€

=0,

μ / = 0,

the fermion determinant is real

Δ becomes complex!

⎧ ⎨ ⎩

€

U μ x( ) = exp iaAμ x( )( )

€

x + ˆ μ

€

x

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Response of observables with respect to

The singlet and non-singlet quark number susceptibilities (S.Gottlieb et al., Phys.Rev.D55 (1997) 6852) The susceptibilities for quenched and 2 flavors QCD (Phys.Rev.D65 (2002) 054506). The susceptibilities for 3 flavors with improved staggered fermions (MILC, hep-lat/0209079).The responses of meson screening masses and the quark condensation with respect to the chemical potential at =0 (QCD-TARO Collaboration, Phys.Rev.D65 (2002) 054501).

€

/ = 0

Direct simulationsare very hard!

Many challengingefforts…

€

∂ O

∂μ,∂ 2 O

∂μ 2,...

QCD-TARO Collaboration:

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A. Nakamura Ph. de Forcrand M. Garcia Perez H. Matsufuru I.-O. Stamatescu T. Takaishi T. Umeda

Response of hadron masses with respect to

range of our work

Tc

T

c

QGP phase

3CSC

2CSChadron phase

€

+1

2

μ

T

⎛

⎝ ⎜

⎞

⎠ ⎟2

T

2

∂ 2M

∂μ 2

μ =0

+Oμ

T

⎛

⎝ ⎜

⎞

⎠ ⎟3 ⎡

⎣ ⎢

⎤

⎦ ⎥

A schematic representation ofpossible QCD phase diagram Our strategy is to expand the ha

dronic quantities in the vicinity of zero at finite temperature, and explore their changes through the response to the chemical potential at = 0.

€

M μ( )T

=M

T μ = 0

+μ

T

⎛

⎝ ⎜

⎞

⎠ ⎟∂M

∂μμ = 0

+

The hadron correlator

€

C z( ) = H x, y,z, t( )H 0,0,0,0( )+

x,y,t

∑ = Ci

i

∑ e− ˆ m i ˆ z

Here, and

€

ˆ m 1 = am1, ˆ m 2 = am2

€

ˆ z = z /a

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Lattice simulations

Problem:

the meson propagator part

€

G = Tr g ˆ μ u( )n:0Γg ˆ μ d( )0:n

Γ+[ ]

the quark propagator

€

g ˆ μ ( ) = D U; ˆ μ ( )−1

€

H n( )H 0( )+

= G =dU[ ]GΔe−SG∫dU[ ]Δe−SG∫

the Dirac matrix

€

d

€

u+-meson

€

Δ =det D U; ˆ μ u( )( )N f / 4

det D U; ˆ μ d( )( )N f / 4

how to get the derivative of the correlator from lattice simulations?

€

Γ=γ5,

γ μ ,

⎧ ⎨ ⎩

for pseudoscalar mesonfor vector meson

the fermion determinant

€

ˆ μ = aμ =μ

NzT

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Lattice simulations

Crucial fact:

€

d

d ˆ μ det D = Tr ˙ D D−1

[ ]det D

the derivatives are taken before doing the integration numerically !

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€

d2

d ˆ μ S2

Re G = 4Re Tr P ˙ D P ˙ D P( )n:0

Γγ 5Pn:0+ γ 5Γ

+[ ] − 2Re Tr P ˙ ̇ D P( )

n:0Γγ 5Pn:0

+ γ 5Γ+

[ ]

€

−2Re Tr P ˙ D P( )n:0

Γγ 5 P ˙ D P( )n:0

+γ 5Γ

+[ ]

€

+1

2Re Tr Pn:0Γγ 5Pn:0

+ γ 5Γ+

[ ] Tr ˙ ̇ D P[ ] − Tr ˙ D P ˙ D P[ ] +1

2Tr ˙ D P[ ]

2 ⎛

⎝ ⎜

⎞

⎠ ⎟

⎧ ⎨ ⎩

€

−Tr Pn:0Γγ 5Pn:0+ γ 5Γ

+[ ] Tr[ ˙ ̇ D P] − Tr ˙ D P ˙ D P[ ] +

1

2Tr ˙ D P[ ]

2 ⎫ ⎬ ⎭

€

+2 ImTr P ˙ D P( )n:0

Γγ 5Pn:0+ γ 5Γ

+[ ]ImTr ˙ D P[ ]

Lattice size: 122×24×6 Polyakov line susceptibility

Lattice simulations

Quark mass: amq = 0.100The R-algorithm is used to generate 1000 configurations. The mesonic correlator and its derivatives were calculated every 25 sweeps with molecular dynamics step = 0.2 and trajectory length of 50 steps in this run. The fitting range is z = 1-23 for pseudoscalar meson and z = 4-20 for vector meson.

€

Tr1

D

∂D

∂μ

⎡

⎣ ⎢

⎤

⎦ ⎥= ξ α

+ 1

D

∂D

∂μξ α

ξ

€

ξα+ 1

D

∂D

∂μξ αξ β

+ 1

D

∂D

∂μξ β

ξ

= Tr1

D

∂D

∂μ

⎡

⎣ ⎢

⎤

⎦ ⎥

2

+ δα ,β Tr1

D

∂D

∂μ

⎡

⎣ ⎢

⎤

⎦ ⎥

2

Z2 noise method with 200 noise vectors is used for the calculation of fermionic operators.

ξα are L vectors of complex Gaussian random numbers,

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α = 1, … , L

€

a = 0.1 ~ 0.27 fm

m = 74 ~ 192 MeV

⎧ ⎨ ⎩

Mesonic correlator

The mesonic correlator:

€

z( ) = C1 e− ˆ m 1 ˆ z + e− ˆ m 1 N z − ˆ z ( )( ) + C2 e− ˆ m 2 ˆ z + e− ˆ m 2 N z − ˆ z ( )

( )

€

z( ) = ′ C 1 e− ˆ ′ m 1 ˆ z + e− ˆ ′ m 1 N z − ˆ z ( )( ) +

€

+ ′ C 2 −1( )ˆ z

e− ˆ ′ m 2 ˆ z + e− ˆ ′ m 2 N z − ˆ z ( )( )

€

Tr D−1( )

n:0Γ D−1

( )n:0

+Γ+

[ ]

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First order responses HP&LQCD Japan 2004

Isoscalar chemical potential

€

ˆ μ V = ˆ μ u = − ˆ μ d

Isovector chemical potential

€

ˆ μ S = ˆ μ u = ˆ μ d

The first order responses are equal to 0!

€

d

d ˆ μ VG = −2Re Tr D−1 ˙ D D−1

( )n:0

Γγ 5 D−1( )

n:0

+γ 5Γ[ ]

€

ˆ μ u

€

ˆ μ d

€

ˆ μ u

€

ˆ μ d

Second order responses

Second order response of -meson Second order response of meson

In the confinement phase the second order response is not changed much with increasing temperatureIn the deconfinement phase the behavior of the second order response of both pseudoscalar and vector mesons is quite similar

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Conclusion & Outlook

The behavior of hadrons can be investigated at finite

The extension of investigation to the chiral limit The influence of the vector chemical potential on the screenin

g mass of pseudoscalar meson in the deconfinement phase The possibility to check the various scenarios concerning the

nature of vector mesons (Harada & Sasaki, hep-ph/0109034)

The investigation of baryons at finite quark number chemical potential is in progress!

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