Taylor expansion in chemical potential: p.1

Transcription

1 Taylor expansion in chemical potential: Mass dependence of the Wroblewski parameter Taylor expansion Condensates Wroblewski Parameter Ward Identities Results ourendu Gupta, Raarshi Ray (T.I.F.R., Mumbai Taylor expansion in chemical potential: p.1

2 Introduction QC matter at non-zero baryon density (baryon chemical potential has attracted much attention both theoretically and experimentally. Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.

3 Introduction QC matter at non-zero baryon density (baryon chemical potential has attracted much attention both theoretically and experimentally. Canonical Monte Carlo studies of QC on the Lattice at has been plagued by the occurance of complex Fermion determinants. Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.

4 Introduction QC matter at non-zero baryon density (baryon chemical potential has attracted much attention both theoretically and experimentally. Canonical Monte Carlo studies of QC on the Lattice at has 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. Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.

5 Introduction QC matter at non-zero baryon density (baryon chemical potential has attracted much attention both theoretically and experimentally. Canonical Monte Carlo studies of QC on the Lattice at has 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 chiral condensate and related quantities at finite chemical potential about Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.

6 Taylor Expansion First results for Taylor expansion for free energy was done by Gottlieb et.al. and for the meson correlators by Miyamura et.al.. Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.3

7 ! " Taylor Expansion First results for Taylor expansion for free energy was done by Gottlieb et.al. and for the meson correlators by Miyamura et.al.. In Taylor expansion, one can do continuum extrapolation of each term. This removes prescription dependence of putting chemical potential on the lattice Gavai Gupta, & % $ $ #. Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.3

8 ! " ' Taylor Expansion First results for Taylor expansion for free energy was done by Gottlieb et.al. and for the meson correlators by Miyamura et.al.. In Taylor expansion, one can do continuum extrapolation of each term. This removes prescription dependence of putting chemical potential on the lattice Gavai Gupta, & % $ $ # Taylor expansion is well behaved when the coefficients of the higher order terms tend to decrease. Otherwise the expansion breaks down approaching a phase transition.. Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.3

9 /.- +, * ( 7 3 General Formalism For the light u,d quarks the QC partition function is :9 < 8 1 0, 65 Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.

10 >- +, * ( General Formalism For the light u,d quarks the QC partition function is Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.

11 >- +, * ( (? A General Formalism For the light u,d quarks the QC partition function is At, the equation of state be expanded in a BC, can Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.

12 >- +, * ( (? A C 7 General Formalism For the light u,d quarks the QC partition function is At, the equation of state be expanded in a BC, can G I G F E HG Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.

13 >- ( (? A 7 ( (?? Q Q 3 3 General Formalism For the light u,d quarks the QC partition function is +, *, can BC At, the equation of state be expanded in a E G I G C F HG KLKMKMKONP BC J KLKMKMKONP JB C IHG R C J G J J 3 NUT Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.

14 ( ( [ Z >- ( J Z Z >- G G ` ZZ Z Z G G G G G G Z erivatives are obtained from those of B C erivatives of ( J V XW J XW V XW J G J ]_^ XW HG XW c b b aq, where Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.5

15 ( ( ( J >- ( J & d! >- erivatives erivatives of B C are obtained from those of d J d J J G Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.5

16 ( ( >- ( J & d Z d erivatives are obtained from those of B C erivatives of ( J d J d! >- J G J XW d d d d efhg e NP Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.5

17 >- i erivatives Expansion of arbitrary fermionic operators :9 Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.6

18 >- i V d >- i J V 0 G d d ^ d >- erivatives Expansion of arbitrary fermionic operators :9 i J [ J [ 1 0 G HG J J G Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.6

19 >- i V d >- i J V 0 G d d ^ d >- k erivatives Expansion of arbitrary fermionic operators :9 i J [ J [ 1 0 G HG J J G mk ( k l k i k i Q k i Q n n n F Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.6

20 7 Condensate Consider the operators for quark condensates and. crq q, cpo o Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.7

21 1 0 R 1 0 Condensate cpo o, Consider the operators for quark condensates and. crq q Z ZZ Z W Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.7

22 1 0 R 1 0 s q Ct o b q Condensate cpo o, Consider the operators for quark condensates and. crq q Z ZZ Z W We construct isoscalar and isovector combinations kb cvbxwy k c b b s u / crq cpo c b and where, Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.7

23 s s / / k d ZZk & Z n n n d F Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.8 Condensate For isovector chemical potential mk d l! Zk n n n k N szu

24 s s / / k d l! s s / / k d Condensate For isovector chemical potential ZZk & Z d mk n n n d l! F Zk n n n k N szu s u, 7 For iso-scalar chemical potential vanishes to all orders and, mk k d d Zk d ZZk & Z d mk n n n d l F Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.8

25 ( s s J I ' J Condensate and usceptibility We relate Taylor coefficients of condensate and susceptibility JBC J 9? J J 9 Q 3 KMKMKLK{N s Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.9

26 ( s s J I ' J I " Condensate and usceptibility We relate Taylor coefficients of condensate and susceptibility JBC J 9? J J 9 Q 3 KMKMKLK{N s For small quark masses the [Gavai Gupta, Phys.Rev. 67 ( ] are mass independent Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.9

27 ( s s J I ' J I } " " Condensate and usceptibility We relate Taylor coefficients of condensate and susceptibility JBC J 9? J J 9 Q 3 KMKMKLK{N s For small quark masses the [Gavai Gupta, Phys.Rev. 67 ( ] Also the order susceptibilities have been found to be mass independent [Gavai are mass independent Gupta, Phys.Rev. 68 ( ] Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.9

28 s 9 Chiral Ward Identity The pseudo-scalar correlator Ward Identity / I ~ satisfies the Chiral / I~ Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.10

29 s 9 J I / ZZ I ~ 9 Chiral Ward Identity The pseudo-scalar correlator Ward Identity / I ~ satisfies the Chiral / I~ J 9 s 9 Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.10

30 s 9 J I / ZZ I ~ 9 ƒ 9 ' Chiral Ward Identity The pseudo-scalar correlator Ward Identity / I ~ satisfies the Chiral / I~ J 9 s 9 For and independent of m. s, meson screening mass is Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.10

31 k k 7 C I " Wroblewski Parameter trangeness enhancement in heavy-ion collisions can be expressed in terms of the Wroblewski parameter. C I ˆ C Gavai Gupta, Phys.Rev. 65 ( Lattice computation at the very small physical quark mass is costly. Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.11

32 k k 7 C I " Wroblewski Parameter trangeness enhancement in heavy-ion collisions can be expressed in terms of the Wroblewski parameter. C I ˆ C Gavai Gupta, Phys.Rev. 65 ( Lattice computation at the very small physical quark mass is costly. One can however simulate at a higher mass and do an expansion in mass. Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.11

33 k k 7 C I ", Wroblewski Parameter trangeness enhancement in heavy-ion collisions can be expressed in terms of the Wroblewski parameter. C I ˆ C Gavai Gupta, Phys.Rev. 65 ( Lattice computation at the very small physical quark mass is costly. One can however simulate at a higher mass and do an expansion in mass. The first order term will be. ŠŒ UŽ Ž Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.11

34 k k 7 C I ", Wroblewski Parameter trangeness enhancement in heavy-ion collisions can be expressed in terms of the Wroblewski parameter. C I ˆ C Gavai Gupta, Phys.Rev. 65 ( Lattice computation at the very small physical quark mass is costly. One can however simulate at a higher mass and do an expansion in mass. The first order term will be. ŠŒ UŽ Ž Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.11

35 $ Computations We used degenerate flavors of staggered quarks in quenched QC. Measurements were done at, and Continuum extrapolations were obtained from simulations done at and lattices.. Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.1

36 Numerical Results First order derivatives are related to the mass derivative of the number density, which is zero at consistent with our computations. In the continuum, both the condensate and the second derivative have divergences which are to be cured by renormalization. ince the divergence is in the ultraviolet limit, ratios of finite temperature quantities with those at zero temperature should be well behaved. Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.13

37 0. m u m d 0.1 Tc T 1.5Tc 0. C r /N τ Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.1

38 0. m u m d 0.1 Tc T Tc 0. C r /N τ Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.1

39 0. m u m d 0.1 Tc T 3Tc 0. C r /N τ Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.1

40 m u m d 0.1 Tc C r Tc Tc 1.5Tc /N τ Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.15

41 ummary Lattice QC computation at finite baryon chemical potential was explored via Taylor expansion. Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.16

42 ummary Lattice QC computation at finite baryon chemical potential was explored via Taylor expansion. Taylor coefficients can be computed by usual lattice techniques and continuuum extrapolations can be obtained. Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.16

43 ummary Lattice QC computation at finite baryon chemical potential was explored via Taylor expansion. Taylor coefficients can be computed by usual lattice techniques and continuuum extrapolations can be obtained. We computed the Taylor coefficients of the chiral condensate upto second order, above the critical temperature. They satisfy various Ward identities and Maxwell relations. Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.16

44 ummary Lattice QC computation at finite baryon chemical potential was explored via Taylor expansion. Taylor coefficients can be computed by usual lattice techniques and continuuum extrapolations can be obtained. We computed the Taylor coefficients of the chiral condensate upto second order, above the critical temperature. They satisfy various Ward identities and Maxwell relations. Ward identities and Maxwell relations relate Taylor coefficients of various quantities. Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.16

45 ummary Lattice QC computation at finite baryon chemical potential was explored via Taylor expansion. Taylor coefficients can be computed by usual lattice techniques and continuuum extrapolations can be obtained. We computed the Taylor coefficients of the chiral condensate upto second order, above the critical temperature. They satisfy various Ward identities and Maxwell relations. Ward identities and Maxwell relations relate Taylor coefficients of various quantities. Through a Maxwell relation we were able to connect the Taylor coefficients of the chiral condensate with the mass dependence of the Wroblewski parameter. Taylor expansion in chemical potential, NEFT, ec 003, Mumbai. p.16