QCD phase diagram at small densities from simulations with imaginary mu

5 Pages·0.13 MB·English

by  Ph. de Forcrand

February1,2008 20:19 WSPC/TrimSize: 9inx6inforProceedings myproc34
3
0
0
QCD PHASE DIAGRAM AT SMALL DENSITIES
2
FROM SIMULATIONS WITH IMAGINARY µ

n
a
J

3 PH. DE FORCRAND
2
ETH Zu¨rich, CH-8093 Zu¨rich, Switzerland

and CERN, CH-1211 Gen`eve 23, Switzerland
1
v E-mail: [email protected]
9
0
O.PHILIPSEN
2
1 Center for Theoretical Physics, Massachussets Institute of Technology,
0 Cambridge, MA 02139, USA
3 E-mail: [email protected]
0
/
h
Wereviewour resultsforthe QCDphase diagramatbaryonicchemical potential
p µB ≤πT. OursimulationsareperformedwithanimaginarychemicalpotentialµI
-
forwhichthefermiondeterminantispositive. For2flavorsofstaggeredquarks,we
p
e map out the phase diagram and identify the pseudo-critical temperature Tc(µI).
h For µI/T ≤ π/3, this is an analytic function, whose Taylor expansion is found
: to converge rapidly, with truncation errors far smaller than statistical ones. The
v truncatedseriesmaythenbecontinuedtorealµ,yieldingthecorrespondingphase
Xi diagram for µB<∼500 MeV. This approach provides control over systematics and
avoidsreweighting. Weoutlineourstrategytofindthe(2+1)-flavorcriticalpoint.
r
a
1. Introduction
Substantial progress has been accomplished over the last two years toward
the numerical determination of QCD properties at finite density. This
progress has not come through a solution of the “sign problem”, which
prohibitsstandardMonteCarlosimulationsbecausetheDiracdeterminant
is no longer real positive at non-zero chemical potential µ. The standard
approach to the sign problem, which requires statistics growing exponen-
tially with the volume and the chemical potential, is still current. Rather,
progresshascomethroughachangetoamorepragmaticattitude. Itmakes
sensetoexploretheregionofsmallchemicalpotential,withmethodswhose
failureatlargeµhasbecomeacceptable. Thisisbecauseinformationsoob-
tained is phenomenologically important, especially for heavy-ion collisions
where the net quark density remains small.
1
February1,2008 20:19 WSPC/TrimSize: 9inx6inforProceedings myproc34
2
Threeapproachesarecurrentlypursued,andreviewedintheseproceed-
ings: (i) a two-parameter reweighting, in T and µ, of µ= 0 simulations 2;
(ii)aTaylorexpansionof(i)truncatedtoitslowestnon-trivialorder3;(iii)
a study at imaginary µ, followed by analytic continuation of a truncated
Taylor series to real µ. (i) is limited to small volumes because of the sign
problem; (ii) is in addition restricted to smaller chemical potentials, and
the systematic error due to the Taylor truncation is unknown. We have
adopted (iii): it can be used for arbitrarily large volumes since it has no
signproblem,andsystematicerrorscomingfromanalyticcontinuationcan
be controlled. We review this approach here, applied to the determination
ofthe pseudo-criticallineT (µ)betweenconfinementanddeconfinement1.
c
2. QCD at real and imaginary chemical potential
The grand canonical partition function Z(V,µ,T)=Tr e−(Hˆ−µQˆ)/T can
(cid:16) (cid:17)
be considered for a complex chemical potential µ=µ +iµ . Two general
R I
properties suffice to constrain the phase structure as a function of µ :
I
- Z is an even function of µ: Z(µ¯)=Z(−µ¯), where µ¯ =µ/T.
-Anon-periodicgaugetransformation,whichrotatesthe Polyakovloopby
a center element but leaves Z unchanged, is equivalent to a shift in µ 4:
I
Z(µ¯ ,µ¯ )=Z(µ¯ ,µ¯ +2π/N). (1)
R I R I
For QCD (N =3), these two properties lead to Z(3) transitions at critical
values of the imaginary chemical potential, µ¯c = 2π n+ 1 , separating
I 3 (cid:0) 2(cid:1)
regions of parameter space where the Polyakov loop angle hϕi falls in dif-
ferent Z(3) sectors. Perturbative and strong coupling considerations led
the authors of 4 to predict at µ¯c a first-orderphase transitionat high tem-
I
perature (deconfined phase) and a crossover at low temperature (confined
phase). The resulting phase diagram in the (µ ,T) plane is periodic, and
I
depicted in Fig. 1.
Thenon-analyticityclosesttotheorigin,atµ¯c = π,limitstheprospects
I 3
of analytic continuation. Our strategy, following 5, consists of fitting by
a Taylor series in µ¯2 observables measured at (µ¯ = 0,µ¯ 6= 0), then
I R I
continuingthe truncatedTaylorseriestorealµ¯. Convergenceofthe Taylor
expansion can only be checked for µ < πT, i.e. µ <500 MeV.
I 3 B∼
3. Analyticity of the (pseudo-) critical line
The phasetransitionorcrossoverbetweenconfinementanddeconfinement,
as β (or T) is varied while keeping µ fixed, is characterized by a peak of
February1,2008 20:19 WSPC/TrimSize: 9inx6inforProceedings myproc34
3
T
0 0.5 1
3µ /(2πT)
Ι
Figure 1. Schematic phase diagram inthe (µI,T)plane. The solidlinemarks afirst-
orderZ3 transition.
thesusceptibilityχ=VN (O−hOi)2 forobservableO. Thecriticalline
t
(cid:10) (cid:11)
β (µ) is thus defined implicitly via
c
∂χ ∂2χ
(cid:12) =0, (cid:12) <0. (2)
∂β(cid:12) ∂β2(cid:12)
(cid:12)µ,βc (cid:12)µ,βc
(cid:12) (cid:12)
In a finite volume V, χ(β,µ) is analytic, and from the implicit function
theorem β (µ) also is. Moreover, it is an even function of µ just like χ, so
c
that it can be expanded in powers of µ2:
β (µ)=β (µ=0)+ c (aµ)2n. (3)
c c X n
n=1
Analytic continuation between real and imaginary µ (for |µ| < πT) is
3
accomplished by flipping the sign of µ2.
4. Numerical results for two light flavors
We have studied QCD with 2 flavors of staggered fermions (83×4 lattice,
quark mass am =0.025, R-algorithm). The lattice spacing is a ∼0.3 fm,
q
and sizeable corrections can be expected in the continuum extrapolation.
The predicted nature of the Z(3) transition at (aµ )c = π/12 is
I
confirmed, and the phase diagram is that of Fig. 1. The confinement-
deconfinement transition line is obtained from the peaks of the plaquette,
quark condensate and Polyakovloop susceptibilities, which give consistent
values β (aµ ) (see Fig. 2). These values are then fitted by a Taylor series
c I
in(aµ )2 overtheintervalaµ ∈[0,π/12]. Thedataarewelldescribedbya
I I
linearfit, andagreementhardlyimproveswithaquadraticfit(seeFig.3a).
February1,2008 20:19 WSPC/TrimSize: 9inx6inforProceedings myproc34
4
0.0001 0.003
aµ=0.0 aµ=0.0
8e-05 aµI=0.15 0.0025 aµI=0.15
aµI=0.23 aµI=0.23
I 0.002 I
χplaquette46ee--0055 χ|P|0.0015
0.001
2e-05
0.0005
50.24 5.26 5.28 5.3 5.32 5.34 5.36 50.24 5.26 5.28 5.3 5.32 5.34 5.36
β β
Figure2. Susceptibilities of theplaquette (left)andof the magnitude ofthe Polyakov
loop(right)forvariousµI’s,asafunctionofβ.
The quadratic coefficient ∝(aµ )4 is zero within errors. Therefore,we can
I
safely approximate the critical line as
β (aµ )=5.2865(18)+0.596(40)(aµ )2 . (4)
c I I
Wetranslateintophysicalunitsusingtheperturbativetwo-loopβ-function,
which suffices for our present accuracy. This yields, as a function of the
baryonic chemical potential µ
B
T (µ ) µ 2
c B B
=1−0.00563(38) , (5)
T (µ =0) (cid:16) T (cid:17)
c B
using T (µ = 0) = 173(8) MeV, while the next-order term O((µ /T)4) is
c B
statisticallyinsignificantuptoµ ∼500MeV(seeFig.3b). Similarresults
B
have been obtained for 4 flavors 6.
5. How to find the (2+1)-flavor critical endpoint
In the case of3 flavors(degenerate ornot), the deconfinement “transition”
changes from crossover to first-order at a critical point (T ,µ ), which can
c c
be identified by a study of cumulant ratios or Lee-Yang zeroes. As the
quark mass m is varied, the critical point describes an analytic curve
q
T (µ) (Fig. 4), which can be Taylor expanded. On the lattice:
c
β (µ)=β (µ=0)+ b (aµ)2n . (6)
c c X n
n=1
As in Section 4, data can be obtained at imaginary µ, then analytically
continued to real µ by flipping the sign of µ2.
February1,2008 20:19 WSPC/TrimSize: 9inx6inforProceedings myproc34
5
5.34
5.33 180
fit O(aµ)2
Ι
5.32 fit O(aµ)4 175
Ι
V
β5.31 e
M
170
5.3 T/
5.29 165
5.28 0 0.02 0.04(aµ)20.06 0.08 0.1 1600 100 200 µ /3M00eV 400 500 600
Ι B
Figure3. (left)Phasediagraminthe((aµI)2,β)plane. Therightmostpointliesbeyond
theZ3 transition,andisthereforeexcludedfromthefit. (right)Analyticallycontinued
phasediagraminthe(µB,T)plane.
crossover increasing mq →
T
first order
Mu
Figure 4. Critical lines in the (T,µ) plane for different quark masses mq. The bold
upward parabolic curve characterizes second-order transition points, separating the
crossoverandthefirst-orderregimes. Itcanbeanalyticallycontinued tonegativeµ2.
References
1. Ph. deForcrand and O.Philipsen, Nucl. Phys. B 642 (2002) 290 [arXiv:hep-
lat/0205016]; arXiv:hep-lat/0209084.
2. Z.Fodor and S.D. Katz, JHEP 0203 (2002) 014 [arXiv:hep-lat/0106002].
3. C. R.Allton et al., Phys.Rev.D 66 (2002) 074507 [arXiv:hep-lat/0204010].
4. A.Roberge and N. Weiss, Nucl. Phys. B 275 (1986) 734.
5. M. P. Lombardo, Nucl. Phys. Proc. Suppl. 83 (2000) 375 [arXiv:hep-
lat/9908006]; A. Hart, M. Laine and O. Philipsen, Phys. Lett. B 505 (2001)
141 [arXiv:hep-lat/0010008].
6. M. D’Elia and M. P. Lombardo, arXiv:hep-lat/0209146.