Taylor expansion in chemical potential

Taylor expansion in chemicalpotential:

Mass dependence of the Wroblewski parameter

Sourendu Gupta, Rajarshi Ray

(T.I.F.R., Mumbai)

Taylor expansion

Condensates

Wroblewski Parameter

Ward Identities

Results

Taylor expansion in chemical potential: – p.1

IntroductionQCD matter at non-zero baryon density (baryonchemical potential �� ) has attracted much attentionboth theoretically and experimentally.

Canonical Monte Carlo studies of QCD on the Lattice athas been plagued by the occurance of complex

Fermion determinants.

Possible way out includes the reweighting techniques,analytic continuation from computations at imaginary

, and Taylor expansions.

We present Taylor series expansion of the chiralcondensate and related quantities at finite chemicalpotential about

Taylor expansion in chemical potential, NEFT, Dec 2003, Mumbai. – p.2


Canonical Monte Carlo studies of QCD on the Lattice at��

has been plagued by the occurance of complexFermion determinants.

Possible way out includes the reweighting techniques,analytic continuation from computations at imaginary

, and Taylor expansions.






Possible way out includes the reweighting techniques,analytic continuation from computations at imaginary�� , and Taylor expansions.






Possible way out includes the reweighting techniques,analytic continuation from computations at imaginary�� , and Taylor expansions.

We present Taylor series expansion of the chiralcondensate and related quantities at finite chemicalpotential about ��


Taylor ExpansionFirst results for Taylor expansion for free energy wasdone by Gottlieb et.al.

� � � � �� and for the meson correlators byMiyamura et.al.

� � �� .

In Taylor expansion, one can do continuumextrapolation of each term. This removes prescriptiondependence of putting chemical potential on the latticeGavai Gupta, .

Taylor expansion is well behaved when the coefficientsof the higher order terms tend to decrease. Otherwisethe expansion breaks down approaching a phasetransition.




� � �� .

In Taylor expansion, one can do continuumextrapolation of each term. This removes prescriptiondependence of putting chemical potential on the lattice!

Gavai

"

Gupta,

� � �� # �� $ � �$ � �% &

.

Taylor expansion is well behaved when the coefficientsof the higher order terms tend to decrease. Otherwisethe expansion breaks down approaching a phasetransition.




� � �� .

In Taylor expansion, one can do continuumextrapolation of each term. This removes prescriptiondependence of putting chemical potential on the lattice!

Gavai

"

Gupta,

� � �� # �� $ � �$ � �% &

.

Taylor expansion is well behaved when the coefficientsof the higher order terms tend to decrease. Otherwisethe expansion breaks down ' approaching a phasetransition.


General FormalismFor the light u,d quarks the QCD partition function is

( � � ) * +, � -. � ) / 0, 12 3 465 7

8 � � �:9 2<; =; �2 �



( � � ) * +, � ->



( � � ) * +, � ->

At �2 ��

, the equation of state

�? �@ A �@ = BC (

, canbe expanded in a Taylor series



( � � ) * +, � ->

At �2 ��


�? �@ A �@ = BC (


� � =; � 4; � 7 � � � � =; �; � �ED 2 C 2 �2 D �� F 2HG

I2 G �2 �G D � � �



( � � ) * +, � ->

At �2 ��


�? �@ A �@ = BC (


� � =; � 4; � 7 � � � � =; �; � �ED 2 C 2 �2 D �� F 2HG

I2 G �2 �G D � � �

C 2 � =?

J B C (J �2

KLKMKMKONP 3 Q R I2HG � =?

J S BC (J �2 J �G

KLKMKMKONP 3NUT 3 Q


DerivativesDerivatives of

B C (

are obtained from those of(

J (J �2 � -> V �XW ) Y2 Z2 [

J S (J �2 J �G � -> V �XW ) YG ZG �XW ) Y2 Z2

D \@ �XW ) YG ZG ) YG ZG D �XW ) YG Z ZG ]_^ 2HG `

where

Z2 � a Q, ) Y2 � b cb



B C (


J (J �2 � -> d Y

J S (J �2 J �G � -> ! d Y YD d S &



B C (


J (J �2 � -> d Y

J S (J �2 J �G � -> ! d Y YD d S &

d Y � �XW ) Y2 Z2d Y Y � d Y d Yd S � efhg eNPTaylor expansion in chemical potential, NEFT, Dec 2003, Mumbai. – p.5

DerivativesExpansion of arbitrary fermionic operators

i �:9 ; =; � � � ->



i �:9 ; =; � � � ->

J iJ �2 � -> V d 02 1Y D 02 1 [

J S iJ �G J �2 � -> V d 02 1Y Y D ^ 2HG d 02 1S D d 02 1Y 0G 1 D 02 G 1 [



i �:9 ; =; � � � ->

J iJ �2 � -> V d 02 1Y D 02 1 [

J S iJ �G J �2 � -> V d 02 1Y Y D ^ 2HG d 02 1S D d 02 1Y 0G 1 D 02 G 1 [

j k � j i Q k D j i Y k �D l j i S k@ j i Q k j ( S km � S� F D n n n


CondensateConsider the operators for quark condensates 4 � cpo o,and 7 � crq q

.

We construct isoscalar and isovector combinations

where, and



.

0 4 14 �@ �XW ) Y4 Z 4 ) Y4 R 0 7 14 � �

0 4 4 14 � �W �� ) Y4 Z 4 ) Y4 ) Y4 Z 4@ ) Y4 Z Z4 ) Y4 �


where, and



.

0 4 14 �@ �XW ) Y4 Z 4 ) Y4 R 0 7 14 � �

0 4 4 14 � �W �� ) Y4 Z 4 ) Y4 ) Y4 Z 4@ ) Y4 Z Z4 ) Y4 �


s / � �� j cb b k tC q s�u � �� j cvbxwy b k

where,

b � o q and

cb � cpo crq


CondensateFor isovector chemical potential � 4 �@ � 7 � �

s / � � � � s / � � �

D !� l j d S k@ j d S k j km D j d Y ZD Z Z k & � S� F D n n n

j szu kN � j Z k �D n n n

For iso-scalar chemical potential ,vanishes to all orders and,


CondensateFor isovector chemical potential � 4 �@ � 7 � �

s / � � � � s / � � �

D !� l j d S k@ j d S k j km D j d Y ZD Z Z k & � S� F D n n n

j szu kN � j Z k �D n n n

For iso-scalar chemical potential � 4 � � 7 � �, s�u

vanishes to all orders and,

s / � � � � s / � � �D j d Y D Z k �D ! � l j d Y Y k@ j d Y Y k j km

D � l j d S k@ j d S k j km D j d Y ZD Z Z k & � S� F D n n n


Condensate and SusceptibilityWe relate Taylor coefficients of condensate andsusceptibility

s � =; � � � �?

J BC (J9

' s S � =; � � � J S s � =; � �J � SKMKMKLK{N 3 Q � �

=J I 4 4J9

For small quark masses the are mass independent[Gavai Gupta, Phys.Rev. D 67 (2003) 034501] Also the

order susceptibilities have been found to be massindependent [Gavai Gupta, Phys.Rev. D 68 (2003) 034506]



s � =; � � � �?

J BC (J9


=J I 4 4J9

For small quark masses the I 4 4 are mass independent[Gavai

"

Gupta, Phys.Rev. D 67 (2003) 034501]

Also theorder susceptibilities have been found to be mass

independent [Gavai Gupta, Phys.Rev. D 68 (2003) 034506]



s � =; � � � �?

J BC (J9


=J I 4 4J9

For small quark masses the I 4 4 are mass independent[Gavai

"

Gupta, Phys.Rev. D 67 (2003) 034501] Also the

� | }

order susceptibilities have been found to be massindependent [Gavai

"Gupta, Phys.Rev. D 68 (2003) 034506]


Chiral Ward IdentityThe pseudo-scalar correlator I�~ / satisfies the ChiralWard Identity

s � =; � � � 9 I ~ /

For and , meson screening mass isindependent of m .



s � =; � � � 9 I ~ /

I Z Z~ / � = � � �9 s S � = � � �9 =

J I 4 4J9

For and , meson screening mass isindependent of m .



s � =; � � � 9 I ~ /

I Z Z~ / � = � � �9 s S � = � � �9 =

J I 4 4J9

For

=�� =�� and 9 � � � � � = �, meson screening mass is

independent of m ' s S � �.


Wroblewski ParameterStrangeness enhancement in heavy-ion collisions canbe expressed in terms of the Wroblewski parameter.

�� = � � j C � kj C 4D C 7 k � I�� = �I 4 4 � = �

Gavai

"

Gupta, Phys.Rev. D65 (2002) 094515.

Lattice computation at the very small physical quarkmass is costly.

One can however simulate at a higher mass and do anexpansion in mass.

The first order term will be .


Wroblewski ParameterStrangeness enhancement in heavy-ion collisions canbe expressed in terms of the Wroblewski parameter.

�� = � � j C � kj C 4D C 7 k � I�� = �I 4 4 � = �

Gavai

"

Gupta, Phys.Rev. D65 (2002) 094515.

Lattice computation at the very small physical quarkmass is costly.

One can however simulate at a higher mass and do anexpansion in mass.

The first order term will be

) , ��U� �.


ComputationsWe used 2 degenerate flavors of staggered quarks inquenched QCD.

Measurements were done at

� � =�� , � � � =�� and$ � � =�� .

Continuum extrapolations were obtained fromsimulations done at

�� ; �; � �; � �and

� �lattices.


Numerical ResultsFirst order derivatives are related to the mass derivativeof the number density, which is zero at � � � �consistent with our computations.

In the continuum, both the condensate and the secondderivative have divergences which are to be cured byrenormalization.

Since the divergence is in the ultraviolet limit, ratios offinite temperature quantities with those at zerotemperature should be well behaved.


-0.4

-0.2

0

0.2

0.4

0 0.005 0.01 0.015 0.02

Cr 1

1/Nτ2

mu = md = 0.1 TcT = 1.5Tc


-0.4

-0.2

0

0.2

0.4

0 0.005 0.01 0.015 0.02

Cr 1

1/Nτ2

mu = md = 0.1 TcT = 2Tc


-0.4

-0.2

0

0.2

0.4

0 0.005 0.01 0.015 0.02

Cr 1

1/Nτ2

mu = md = 0.1 TcT = 3Tc


-0.05

-0.045

-0.04

-0.035

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0 0.005 0.01 0.015 0.02

Cr 2

1/Nτ2

mu = md = 0.1 Tc

1.5Tc

2Tc

3Tc


SummaryLattice QCD computation at finite baryon chemicalpotential was explored via Taylor expansion.

Taylor coefficients can be computed by usual latticetechniques and continuuum extrapolations can beobtained.

We computed the Taylor coefficients of the chiralcondensate upto second order, above the criticaltemperature. They satisfy various Ward identities andMaxwell relations.

Ward identities and Maxwell relations relate Taylorcoefficients of various quantities.

Through a Maxwell relation we were able to connect theTaylor coefficients of the chiral condensate with themass dependence of the Wroblewski parameter.


Taylor expansion in chemical potential

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