New results in QCD at ﬁnite µ

New results in QCD at finite µ

Rajiv Gavai and Sourendu Gupta

ILGTI: TIFR

XQCD 2008, Duke UniversityJuly 23, 2008

sg (ILGTI: TIFR) New results at finite µ XQCD 08 1 / 37

Outline

1 The finite temperature transition

2 Quark Number Susceptibilities

3 Linkage

4 The Critical End Point

5 Series sums and Pade resummations

6 Summary


The finite temperature transition

Outline



3 Linkage



6 Summary



Crawling towards the continuum

Before this year: state of the art lattice computations of physics atfinite chemical potential used lattice cutoff Λ = 4T ≃ 800 MeV nearTc .

Our earlier computation used mπ ≃ 230 MeV and spatial sizes withLT = 2, 3, 4 and 6. This enabled extrapolation to thethermodynamic limit, i.e., L → ∞.

Now: new computations with Λ = 6T ≃ 1200 MeV near Tc .

mπ remains unchanged (230 MeV), but spatial volumes are somewhatsmaller (LT = 2, 3 and 4). No extrapolation to L → ∞ yet.

20000–50000 configurations at each coupling; stochasticdetermination of traces with 500 random vectors on eachconfiguration. (Gavai and SG, Phys. Rev. D 68, 2003, 034506.)



Locating the finite temperature cross over

Cross-over coupling monitored using Polyakov Loop susceptibility: χL,an operator which enters fourth-order QNS: (T/V )〈O22〉c , and anoperator which enters eighth-order QNS: (T/V )〈O44〉c . Measuresconsistent with each other within the precision of this work.

For m/Tc = 0.1, we find βc = 5.425(5). Previous results bracketedthis: for m/Tc = 0.15 one had βc = 5.438(40) (Gottlieb et al, PRL59, 1987, 1513) and for m/Tc = 0.075 it was found thatβc = 5.41–5.43 ( Bernard et al, PR D 45, 1992, 3854).

Transition is not first order. Computations at larger volumes arerequired to distinguish cross over from second order transition.



Polyakov loop susceptibility

0

0.1

0.2

0.3

0.4

0.5

5.35 5.4 5.45 5.5 5.55

Lχ

β

Ns=12Ns=18Ns=24



Not first order

0

0.0001

0.0002

0.0003

10 12 14 16 18 20 22 24 26

/N3 s

max

Lχ

Ns


Quark Number Susceptibilities

Outline



3 Linkage



6 Summary



What is a QNS?

Taylor coefficient of the pressure in Nf = 2 QCD is

P(T , µu, µd) =∑

nu ,nd

1

nu!nd !χnu ,nd

(T )µnuu µnd

d ,

and, since the two quark flavours are degenerate, χnu ,nd= χnd ,nu .

Diagonal QNS have either nu = 0 or nd = 0.In two flavour QCD trade µu,d for µB,Q . Then

χB =2

9(χ20 + χ11) = 2χBQ

χQ =1

9(5χ20 − 4χ11) .

Transforming to µB,I3 , one has

χBI3 = 0, χI3 =1

2(χ20 − χ11) .



Off-diagonal QNS

-0.10

-0.08

-0.06

-0.04

-0.02

0

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

χ

/Τ11

2

T/Tc

Nt=

6

Nt=

4

Sees only 〈O11〉. No evidence for lattice spacing effects.



Diagonal QNS

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

χ

/Τ20

2

T/Tc

Nt=6

Nt=4

Sees 〈O11〉 and 〈O2〉. Second expectation value is cutoff dependent. Also,has a cross over. We look at its susceptibility 〈O22〉c to identify Tc .



“Susceptibility” of QNS: 〈O22〉c— 4th order QNS

0

0.4

0.8

1.2

1.6

2.0

0.5 1 1.5 2 2.5

(T/V

)<O

>

22

T/Tc

Nt=6

Nt=4

Peak at the same coupling as peak of χL. Within the 1% precision ofT/Tc , the two quantities peak at the same coupling. See Gavai andGupta, PR D72 (2005) 054006.



Diagonal fourth-order QNS

0

1

2

3

4

5

6

7

8

0.5 1 1.5 2 2.5

χ 40

T/Tc

Nt=6

Nt=4

Non-zero for T > Tc . Has contribution from 〈O4〉, which hasnon-vanishing value for the ideal gas.



The operator O4

0

0.4

0.8

1.2

1.6

2.0

0.5 1 1.5 2 2.5

(T/V

)<O

> 4

T/Tc

Nt=6

Nt=4

Rapid cross over from a small value in the hadronic phase to anon-vanishing value for the ideal gas.



“Susceptibility” of O4: 〈O44〉c— 8th order QNS

0

100

200

300

400

500

600

700

800

0.5 1 1.5 2 2.5

(T/V

) <

O

>44

T/Tc

This quantity peaks at the same coupling as χL and 〈O22〉c . Within theprecision of our measurement there is no dependence of the cross overcoupling on these observables.



The operator O6— 6th order QNS

0

0.4

0.8

1.2

1.6

2.0

0.5 1 1.5 2 2.5

(T/V

)<O

> 6

T/Tc

Nt=6

Nt=4

The operator expectation value 〈O6〉 has structure below Tc and hence its“susceptibility” cannot be used to probe the cross over coupling. Similarobservation for 〈O8〉.


Linkage

Outline



3 Linkage



6 Summary


Linkage

Linkage between quantum numbers

Linkage between quantum numbers F and G is

C(FG)|F =χFG

χF

= C(GF )|F .

Measures amount of G excited per unit fluctuation in F . Thequantities C(FG)|F and C(FG)|G not necessarily equal.

C(ud)|u = −2/3 at low temperature, since the pion is the lightestexcitation, and at high temperature it vanishes. For Nf = 2 one hasC(ud)|u = C(ud)|d .

C(BQ)|Q = 0 at low temperatures (pion is the lightest particle) and1/5 at high temperature.

C(BQ)|B = 1/2 at all temperatures by definition.


Linkage

Results

-1.0

-0.8

-0.6

-0.4

-0.2

0

0.5 1 1.5 2 2.5

(ud

)|u

C

T/Tc

0

0.05

0.10

0.15

0.20

0.5 1 1.5 2 2.5

C (B

Q)|

Q

T/Tc

Quick cross over from hadron gas behaviour to quark gas behaviour.Rounding seen close to Tc . Finite-size effects need to be investigated:LT = 4. Earlier computation with Nt = 4 saw less rounding with LT = 6.


The Critical End Point

Outline



3 Linkage



6 Summary



Finite size effects

At critical point correlation length becomes infinite, appropriatesusceptibilities diverge and free energy becomes singular ... in theinfinite volume limit (van Hove’s theorem).

No numerical computation ever performed on infinite volumes.

Deduce the existence of a critical point through extrapolations: finitesize scaling (FSS) well developed for direct simulations.

Example: peak of susceptibility scales as power of volume. Smallereffect: position of peak shifts from its infinite volume position by adifferent power of volume—

χmax(L) ∝ Lp, Tc(L) = Tc − a/Lq, (p, q > 0).

FSS not well developed for series expansions; some aspects are known.



Series expansions

n

ln c

n

n (L )

*

*

*

n (L )

n (L )3

1

2

L > L > L1 2 3

For a divergent quantity: χ(T , µB) =∑

n cn(T )µnB , the leading finite

volume effects in the series coefficients.



Nt = 4

0

1

2

3

4

5

6

5 10 15 20 25

2 4 6 8 10/T

Bµ

mπNs

Ns

At fixed T/Tc ≃ 0.95. Circles: ratio of order 0 and 2; boxes: ratio oforder 2 and 4. Gavai and SG, Phys. Rev. D 71, 2005, 114014.



Nt = 6: Radius of convergence

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 1 2 3 4 5 6 7 8

µ/(3

Τ)

n

LT=4; T/Tc=0.99

Filled symbols: (χ(0)/χ(n))1/n. Open symbols:√

χ(n−1)/χ(n+1).




0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 1 2 3 4 5 6 7 8

µ/(3

Τ)

n

LT=4; T/Tc=0.97


χ(n−1)/χ(n+1).




0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 1 2 3 4 5 6 7 8

µ/(3

Τ)

n

LT=4; T/Tc=0.94


χ(n−1)/χ(n+1).



Radius of convergence

0

1

2

3

4

5

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2 2.2 2.4

/T

*(2)

µ

T/Tc

Nt=6

Nt=4

Lattice spacing dependence quantifies possible systematic errors.



Nt = 6: Finite size scaling

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 1 2 3 4 5 6 7 8

µ/(3

Τ)

n

LT=2; T/Tc=0.94


χ(n−1)/χ(n+1).



Nt = 6: Finite size scaling

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 1 2 3 4 5 6 7 8

µ/(3

Τ)

n

LT=4; T/Tc=0.94


χ(n−1)/χ(n+1).



Critical end point

Multiple criteria agree:

Stability of radius of convergence with order and estimatorPinching of the radius of convergence with T .Smallest T where all the coefficients are positive.Finite size effects roughly correct; more planned for the future.

This gives

TE

Tc= 0.94 ± 0.01 and

µEB

TE= 1.8 ± 0.1

with Nf = 2 when mπ/mρ ≃ 0.3 at a finite volume with LT = 4 andlattice cutoff of a = 1/6TE .

For a lattice cutoff of a = 1/4TE at the same renormalized quarkmass and on the same volume we had found a similar value forTE/Tc and µE

B/TE = 1.3 ± 0.3. Extrapolation to L → ∞ reducedthis to 1.1 ± 0.1.


Series sums and Pade resummations

Outline



3 Linkage



6 Summary



Fluctuations of Baryon number

Suggestion by Stephanov, Rajagopal, Shuryak; Asakawa, Heinz, Muller;Jeon, Koch

P(∆B) = exp

(

−(∆B)2

2VTχB

)

.

Extrapolate χB to finite chemical potential: peak at Tc?

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0.8 1 1.2 1.4 1.6 1.8 2

Ser

ies

coef

ficie

nts

for

χ B

T/Tcsg (ILGTI: TIFR) New results at finite µ XQCD 08 32 / 37


Sum the series

0

1

2

3

4

5

0 0.5 1 1.5 2 2.5 3

χ

/ΤΒ

2

µ/Τ

n=2

n=4

Summing the series never shows critical behaviour: sum is a polynomialand smoothly behaved. The sum peaks at Tc : incorrect (see SG, SEWM2006).



Critical fluctuations

0

1

2

3

4

5

0 0.5 1 1.5 2 2.5 3

χ

/ΤΒ

2

µ/Τ

P01

P11

Use Pade approximants for the extrapolations: divergence at the criticalend point (see Lombardo, Mumbai 2005). Error propagation requires care:see arXiv:0806.2233 [hep-lat].


Summary

Outline



3 Linkage



6 Summary


Summary

Summary 1: finite temperature

1 Simulations of Nf = 2 QCD (staggered quarks, Wilson action) withrenormalized quark mass mπ ≃ 230 MeV with 1/a ≃ 1200 MeV andLT = 2, 3 and 4.

2 Finite temperature cross over located at βc = 5.425(5), consistentwith previous computations at neighbouring masses. Consistentmeasurements obtained with χL, (T/V )〈O22〉c and (T/V )〈O44〉cwithin precision of this computation.

3 Cutoff artifacts seen in many QNS. Surprisingly, measurements aremore well-behaved at smaller lattice spacing (see χ60 and χ80, forexample).


Summary

Summary 2: finite chemical potential

1 Very stable estimate of the critical end point: three criteria agree.

TE

Tc

= 0.94 ± 0.01 andµE

B

TE= 1.8 ± 0.1

with lattice cutoff of a = 1/6TE ≃ 1100 MeV, compared toµE

B/TE = 1.3 ± 0.3 at a = 1/4TE ≃ 750 MeV on the same spatialvolume.

2 Series extrapolation needs resummation: Pade approximants are onepossible resummation.

3 Linkage between u and d quantum numbers disappears at T ≃ Tc

when µB = 0. How abrupt? Requires finite size scaling.


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New results in QCD at ﬁnite µ