Constraints on the two-flavor QCD phase diagram from imaginary chemical potential

0.52 MB·English

by  Claudio Bonati

CERN-PH-TH/2012-003
YITP-12-2
IFUP-TH/2012-01
Constraints on the two-flavor QCD phase diagram
from imaginary chemical potential
2
1 ClaudioBonati
0
DipartimentodiFisica,UniversitàdiPisaandINFN,
2
SezionediPisa,LargoPontecorvo3,56127Pisa,Italy

n
E-mail: [e-mail]
a
J
Philippede Forcrand∗†
3
1 InstituteforTheoreticalPhysics,ETHZürich,CH-8093Zürich,Switzerland
and

]
t CERN,PhysicsDepartment,THUnit,CH-1211Geneva23,Switzerland
a
E-mail: [e-mail]
l
-
p
MassimoD’Elia
e
h DipartimentodiFisica,UniversitàdiPisaandINFN,
[ LargoPontecorvo3,56127Pisa,Italy

1 E-mail: [e-mail]
v
9 Owe Philipsen
6
InstitutfürTheoretischePhysik,Goethe-UniversitätFrankfurt,60438FrankfurtamMain,
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2 Germany
. E-mail: [e-mail]
1
0
FrancescoSanfilippo
2
1 LaboratoiredePhysiqueThéorique(Bât.210)‡UniversitéParisSud,Centred’Orsay,F-91405
:
v Orsay-Cedex,France,and
i
X INFNSezionediRoma,P.leAldoMoro5,00185Roma,Italy
E-mail: [e-mail]
r
a
We reviewourknowledgeofthephasediagramofQCD asafunctionoftemperature,chemical
potential and quark masses. The presence of tricritical lines at imaginary chemical potential
m =ip T,withknownscalingbehaviourintheirvicinity,putsconstraintsonthisphasediagram,
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especially in the case of two lightflavors. We show first results in our projectto determinethe
finite-temperaturebehaviourintheN =2chirallimit.
f
TheXXIXInternationalSymposiumonLatticeFieldTheory-Lattice2011
July10-16,2011
SquawValley,LakeTahoe,California
∗Speaker.
†Temporaryaddress:YukawaInstituteforTheoreticalPhysics,KyotoUniversity,Kyoto606-8502,Japan
‡LaboratoiredePhysiqueThéoriqueestuneunitémixtederechercheduCNRS,UMR8627.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
N =2phasediagram PhilippedeForcrand
f
1. Introduction
Thephase diagram ofQCDasafunction oftemperature T and quark chemical potential m is
governed by the interplay of the chiral symmetry and the center symmetry [1]. These symmetries
are exact for zero and infinite quark masses, respectively. Therefore, varying the quark masses
away from their physical values towards these limits provides useful insight into the behaviour of
QCDatthephysical masspoint.
Similarly, making the chemical potential complex provides enhanced information on the be-
haviour of QCD at real chemical potential. Since the sign problem which plagues simulations at
non-zero m [2]isabsent when m ispureimaginary, theregimeofimaginary m isactually the only
direction in the complex m plane where complete, reliable information on the behaviour of QCD
canbeobtained. Itturnsoutthatarichphasediagramasafunctionof(T,m =im )andofthequark
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masses(m =m ,m )emerges. Thecriticalandtricriticalfeaturesofthisphasediagram,withtheir
u d s
associated scalinglaws,haveconsequences forthebehaviour ofQCDatreal m .
Here, we summarize what is known about this phase diagram and sketch (Fig. 2) a plausible
scenario, consistent with current numerical simulations augmented with reasonable assumptions
of continuity of the critical surfaces. We explore in particular the implications for the behaviour
of QCD in the two-flavor chiral limit (m =m =0,m =¥ ). In that limit, it is widely believed
u d s
thatQCDundergoesafinite-temperature, second-orderO(4)chiraltransitionatm =0,whichturns
first-order atatricritical point for some real m [3]. However, other possibilities exist. At m =0in
particular, thefinite-temperature transition mightbefirst-order. Thepresent numerical evidence is
inconclusive: using Wilsonfermions, O(4)scaling ispreferred [4],whilewithstaggered fermions
O(4) scaling has been elusive, and first-order behaviour has also been claimed [5]. Note that
behaviour consistent withO(4)hasbeenseenwithimprovedstaggeredfermions, inanN =2+1
f
setup where the strange quark mass isfixed at its physical value [6]. Approaching the chiral limit
fromtheimaginary m direction offersanovel,independent methodtohelpsettletheissue.
2. Three-dimensional Columbiaplot
ThethermalbehaviourofQCDatm =0,asafunctionofthequarkmassesm =m ≡m and
u d u,d
m issummarized inthewell-known Columbia plotFig.1(left). TheN =3chiral symmetry and
s f
the Z center symmetry are achieved in the lower left and upper right corners, respectively. This
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gives rise tofirst-order transitions. Forintermediate quark masses, numerical simulations indicate
a smooth crossover as a function of temperature. Hence, the first-order regions must be bounded
bysecond-ordercriticallines: thechiralcriticallineinthelowerleftcorner,andthedeconfinement
critical line in the upper right corner. In the absence of further symmetry, the universality class is
expected (and has been numerically verified) to be that of the 3d Ising model. The chiral critical
linejoins withthem =0axisatatricritical point, forastrange quarkmassmtric whichislarger
u,d s
thanthephysicalstrangequarkmassoncoarselattices[7]orsmallerwhenusingimprovedactions
[6]. TheN =2chirallimitisobtained intheupperleftcorner.
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When the chemical potential is turned on, the two critical lines sweep critical surfaces as a
functionofm . Forbothlines,ithasbeenobservedthatthefirst-orderregionshrinks,asrepresented
Fig. 1 (right) [7, 8, 9]. Here, wewant to show real and imaginary m in asingle figure. Therefore,
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N =2phasediagram PhilippedeForcrand
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Real world
N f = 2 Pure m Heavy quarks
¥ Gauge
1st
2On(4d) o?rder d2necdo onrfdineerd
Z(2)
mtric QCD critical point DISAPPEARED
s N f = 3
crossover
phys. N f = 1
point
m s chiral X crossover 1rst ¥
2nd order 0
Z(2)
1st m
m s
u,d
0
0 m u , m d ¥
Figure 1: (Left) “Columbia plot”: schematic phase transition behaviourof N =2+1QCD for different
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choicesofquarkmasses(m ,m )atm =0. Twocriticallinesseparatetheregionsoffirst-ordertransitions
u,d s
(lightorheavyquarks)fromthecrossoverregioninthemiddle,whichincludesthephysicalpoint. (Right)
Critical surfaces swept by the critical lines as m is turned on. For light quarks [7, 8, 10] as well as for
heavyquarks[9],numericalsimulationsindicatethatthefirst-orderregionshrinksasthechemicalpotential
isturnedon.
we adopt (m /T)2 for the z coordinate: real and imaginary m appear above and below the m =0
plane, respectively. Ofparticular interestistheRoberge-Weiss plane(m /T)2=−(p /3)2. Wenow
arguethatthe3-dimensional phasediagramofN =2+1QCDislikelytobedescribed byFig.2.
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3. Phasediagram intheRoberge-Weiss plane
Thetwosymmetriesofthepartition function
m m 2p n
Z(m )=Z(−m ), Z =Z +i (3.1)
T (cid:18)T 3 (cid:19)
(cid:16) (cid:17)
imply reflection symmetry in the imaginary m direction about the “Roberge-Weiss” values m =
ip T/3(2n+1) which separate different sectors of the center symmetry [11]. Transitions between
neighbouring sectorsareoffirstorderforhighT andanalyticcrossovers forlowT [11,12,13],as
indicated Fig. 3 (left). The corresponding first-order transition lines may end with a second-order
critical point, or with a triple point, branching off into two first-order lines. Which of these two
possibilities occursdependsonthenumberofflavorsandthequarkmasses.
Recent numerical studies have shown that a triple point is found for heavy and light quark
masses, while for intermediate masses one finds a second-order endpoint. As a function of the
quarkmass,thephasediagramatm /T =ip /3isassketchedFig.3(middle). Thishappensforboth
N =2[14]andN =3[15].
f f
IfoneassumesthattheN =2andN =3tricritical pointsareconnected toeachotherinthe
f f
(m ,m )quarkmassplane,theresultingphasediagramisdepictedFig.3(right),withtwotricriti-
u,d s
callinesseparatingregionsoffirst-orderandofsecond-ordertransitions. Thisphasediagramisthe
equivalent of the Columbia plot, now at imaginary chemical potential m /T =ip /3. Note that the
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N =2phasediagram PhilippedeForcrand
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(((mmm ///TTT)))222
mm
ss mm
uu,,dd
❶❶
000
mm ❂❂ ¥¥¥
uu,,dd
❷❷
❷❷
---------(((((((((ppppppppp /////////333333333)))))))))222222222 ❸❸
❸❸
Figure 2: 3d phase diagram. The verticalaxis is (m /T)2, so that real and imaginary chemicalpotentials
areaboveandbelowthe m =0plane,respectively. The“bottomplane”correspondstotheRoberge-Weiss
transitionvaluem /T =ip /3.Thethickerredlinesaretricritical. Tricriticalpointsmarked“2”and“3”have
beenidentifiedfortheN =2[14]andN =3[15]theories,respectively.Theobjectofthepresentstudyis
f f
thebluelineinthe“backplane”m =¥ (N =2)joiningtwotricriticalpoints.
s f
assumption of continuity of the tricritical lines can be checked directly by numerical simulations,
sincethereisnosignproblemforimaginary m .
Now,as(m /T)2 isvariedbetween zeroandtheRoberge-Weiss value−(p /3)2,theColumbia
plot must change from Fig. 1 (left) to Fig. 3 (right). Assuming continuity of the critical surfaces
atimaginary m ,whichagaincanbecheckedbynumericalsimulations, theresulting3-dimensional
phase diagram is that of Fig. 2. The two red surfaces (“chiral” and “deconfinement”) are critical.
They are bounded by lines, among which the following are tricritical: (i) the two lines in the
(m /T)2 =−(p /3)2 Roberge-Weiss plane; (ii) the line in the m =0 chiral plane. Note that the
u,d
N =2(i.e. m =¥ )“backplane” contains two tricritical points on the chiral critical surface: one
f s
intheRoberge-Weissplane,theotheronthem =0verticalaxis(seeFig.2). Thelocationofthe
u,d
latterisrelatedtothevalueofthetricritical strange quarkmass.
4. Tricritical scaling
Inthevicinity ofatricritical point, scalinglawsapply. Thephasediagram issimilartothatof
a metamagnet, with two external fields: H, which respects the symmetry, and H† which breaks it
(likeastaggeredandanordinarymagneticfield),depictedFig.4(left). ThethreesurfacesS ,S ,S
0 + −
indicatefirst-ordertransitions. TheymeetatalineoftriplepointsLt ,depictedbyasolidline. They
are bounded by second-order transition lines, depicted by dotted lines. All four lines meet at the
tricritical point(T,H).
t t
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N =2phasediagram PhilippedeForcrand
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1st order
triple
2nd order
T Tricritical
T m 3d Ising
s
1st tricritical
order
Tricritical
triple
?
0 0.5 1 1.5 2 2.5 3 3.5 4
µTi/(π3) m mu,d
Figure3:(Left)Genericphasediagramasafunctionofimaginarychemicalpotentialandtemperature.Solid
linesarefirst-orderRoberge-Weisstransitions. Thebehaviouralongdottedlinesdependsonthenumberof
flavorsandthequarkmasses. (Middle)ForN =2 andN =3, theendpointoftheRoberge-Weissline is
f f
atriplepoint(where3first-orderlinesmeet)forlightorheavyquarkmasses,andanIsingcriticalpointfor
intermediatequarkmasses. Thus, two tricriticalmassesexist. (Right)Thesimplestassumptionis thatthe
N =2andN =3tricriticalpointsarejoinedbytricriticallines[15]. Thisassumptioncanbecheckedwith
f f
N =2+1imaginary-m simulations.
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Inourcase, thescaling exponents governing thebehaviour nearthetricritical pointaremean-
field,becauseQCDbecomes3-dimensionalasthecorrelationlengthdivergeswhilethetemperature
isfixedtothatofthetricritical point, andd=3istheuppercriticaldimension fortricriticality. Of
course, thisimpliesthepresenceofpotentially largelogarithmic corrections toscaling.
Here, we are interested in the second-order lines S , corresponding to a departure from the
±
symmetry plane H† = 0. Along these lines, the scaling law is H† (cid:181) |t|5/2, where the reduced
temperature t is measured along the tangent to Ll . For Nf =2 QCD, tricritical scaling should be
satisfiednearthetricritical points:
(i)(m /T)2=−(p /3)2: thenH†∼ (m /T)2+(p /3)2 , t ∼(m −m ),sothat
u,d tric
(cid:2) (cid:3)
(m /T)2+(p /3)2 (cid:181) (m −m )5/2 (4.1)
u,d tric
(cid:2) (cid:3)
(ii)m =0: thenH†∼m , t ∼ (m /T)2−(m /T)2| ,sothat
u,d u,d tric
(cid:2) (cid:3)
m (cid:181) (m /T)2−(m /T)2| 5/2 (4.2)
u,d tric
(cid:2) (cid:3)
It is not clear how broad the scaling window is around each of these two tricritical points. The
two scaling windows might overlap, leading to a very constrained system of equations, with 3
unknowns (the two constants of proportionality in eqs.(4.1,4.2) and (m /T)2| — m having
tric tric
beendeterminedalreadyin[14])andoneconstraint(continuity ofthederivativeattheintersection
ofthetwoscalingcurves),leadingtoaphasediagramasinFig.5(left),whichcouldbedetermined
from onlytwopointsmeasured byMonteCarlo. Reasonforsuchoptimism canbefound inFig.4
(right), wherethescalingwindowaroundthethirdN =2tricritical point,corresponding toheavy
f
(u,d)quarks,isshowntoextendfarintotheregionofrealchemicalpotential[15,16].
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N =2phasediagram PhilippedeForcrand
f
H L 9.5
+
L
− S
+ 9
SS
−−
L H† 8.5 Imaginary m Real m
τ
T
S (Tt,Ht) m /c 8
0 7.5
L
λ
7
6.6+1.6((p /3)2+(m /T)2)2/5
•
6.5
T -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
N
T (m /T)2
Figure 4: (Left) Schematic phase diagram of a metamagnet. The external field H† breaks the symme-
try, while H does not. (Right) For heavy quarks, tricritical scaling in the vicinity of the Roberge-Weiss
imaginary-m valueextendsfarintotheregionofrealm [15].
5. Preliminary N =2results
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Following the above discussion, we performed simulations of N =2 QCD, with staggered
f
quarks of masses am =0.01 and 0.005 on N =4 lattices, scanning in (m /T)2 to determine the
q t
value of imaginary m corresponding to a second-order transition. Our observable is the Binder
cumulant of the quark condensate. Consistent results are obtained from the finite-size scaling of
the plaquette distribution. The two critical points are shown Fig. 5 (right). Disappointingly, it
seems impossible to smoothly match two tricritical scaling curves passing through these points.
Additional masses are needed to determine the critical curve. Nevertheless, assuming convexity
of the critical curve already constrains the m =0 tricritical point to lie at (m /T)2 &−0.3. The
u,d
figureillustrates thecasewherethispointliesatm =0. Itmightalsolieat(m /T)2>0,sothatthe
m =0 chiral transition would be first-order. Additional small-mass measurements are underway
andwillsettlethisissue. Notethatwearesimulatingtwo-flavourQCDbytakingthesquarerootof
the staggered determinant, and approaching the chiral limit at fixed, rather coarse lattice spacing.
Thisisthewrongorderoflimits,andisthemostlikelyapproach toexposeafailureofrooting.
Finally, our phase diagram Fig. 2 makes it clear that, if the transition in the massless N =2
f
theoryisO(4),turningonarealchemicalpotential (i.e. goinguptheverticalaxisintheback)will
not make it become first-order. Obtaining such behaviour requires that the chiral critical surface
bendawayfromtheN =3chiralpoint,orthatanother,non-chiralcriticalsurfaceappearatlargem .
f
Acknowledgements:
This work is partially supported by the German BMBF, project Hot Nuclear Matter from
Heavy Ion Collisions and its Understanding from QCD, No. 06MS254. We thank the Minnesota
Supercomputer Institute and HLRSStuttgartfor providing computer resources. Large scale simu-
lations havebeenperformed ontwoGPUclustersinPisaandGenoaprovided byINFN.
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N =2phasediagram PhilippedeForcrand
f
❂
MMMMMaaaaatttttccccchhhhhiiiiinnnnnggggg tttttrrrrriiiiicccccrrrrriiiiitttttiiiiicccccaaaaalllll ssssscccccaaaaallllliiiiinnnnnggggg aaaaarrrrrooooouuuuunnnnnddddd mmmmm RRRRRWWWWW aaaaannnnnddddd aaaaarrrrrooooouuuuunnnnnddddd mmmmm=====00000 0 mc1
mc2
00000
-0.2
OOOOO(((((44444)))))
-----00000.....22222
❂❂❂❂❂
-0.4
-----00000.....44444
22222/T)/T)/T)/T)/T) -----00000.....66666 CCCCCrrrrrooooossssssssssooooovvvvveeeeerrrrr 2(mu/T) -0.6
mmmmm(((((
-----00000.....88888 -0.8
FFFFFiiiiirrrrrsssssttttt ooooorrrrrdddddeeeeerrrrr
-----11111
❷❷❷❷❷ -1
RRRRRWWWWW
00000 00000.....0000011111 00000.....0000022222 00000.....0000033333 00000.....0000044444 00000.....0000055555 ❷
0 0.01 0.02 0.03 0.04 0.05
mmmmm
qqqqq mq
Figure5: (Left)Strategyused: todeterminethecriticalline,wefixthequarkmassandmeasurethecorre-
spondingcriticalimaginarychemicalpotential. Thetwotricriticalscalinglinesmaymatchsmoothlyifthe
scalingwindowsoverlap.(Right)Preliminaryresults:thescalingwindowsdonotseemtooverlap;convexity
placesthem =0tricriticalpointat(m /T)2&−0.3.Forillustration,itisplacedatm =0inthefigure.
u,d
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