QCD at finite isospin density

4 Pages·0.13 MB·English

by  D. T. Son

QCD at Finite Isospin Density
D.T. Son1,3 and M.A. Stephanov2,3
1 Physics Department, Columbia University, New York, New York 10027
2 Department of Physics, University of Illinois, Chicago, Illinois 60607-7059
3 RIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973
(May 2000)
QCDatfiniteisospinchemicalpotentialµI hasnofermionsignproblemandcanbestudiedonthe
lattice. Wesolvethis theory analytically in two limits: at low µI,where chiral perturbation theory
isapplicable,andatasymptoticallyhighµI,whereperturbativeQCDworks. Atlowisospindensity
the ground state is a pion condensate, whereas at high density it is a Fermi liquid with Cooper
pairing. The pairs carry the same quantum numbers as the pion. This leads us to conjecture that
thetransitionfromhadrontoquarkmatterissmooth,whichpassesseveraltests. Ourresultsimply
anontrivialphasediagraminthespaceoftemperatureandchemicalpotentialsofisospinandbaryon
1
0 number.
0
2
Introduction.—AmpleknowledgeofQCDintheregime are turned off. Once this is done, we have a nontrivial
n
a of finite temperature and baryon density is crucial for regime which, as has been emphasized recently in [4], is
J understanding a wide range of phenomena from heavy accessiblebypresentlatticeMonte Carlomethods,while

0 ion collisions to neutron stars and cosmology. First- being,asweshallsee,analyticallytractableinvariousin-
1 principles lattice numerical Monte Carlo calculations teresting limits. As a result, the system we consider has

provide a solid basis for our knowledge of the finite- a potential to improve substantially our understanding
2
temperatureregime. However,theregimeoffinitebaryon of cold dense QCD. This regime carries many attractive
v
5 chemicalpotentialµB isstillinaccessiblebyMonteCarlo traits of two-color QCD [5,6], but is realized in a physi-
2 becausepresentmethodsofevaluatingtheQCDpartition cally relevant theory — QCD with three colors.
2 function require taking a path integral with a measure Positivity and QCD inequalities.—Since the Euclidean
5
which includes a complex fermion determinant. Ignoring versionof our theory has a real and positive fermion de-
0
the determinant (as in the popular quenched approxi- terminant, some rigorous results on the low-energy be-
0
0 mation) leads to qualitatively wrong answers for finite havior can be obtained from QCD inequalities [7,6]. In
/ µ [1]. Such a contrastto the case of µ =0, where the vacuum QCD, the latter rely on the following property
h B B
quenched approximation proved useful, comes from the of the Euclidean Dirac operator =γ (∂+iA)+m:
p
D ·
- fact that the latter corresponds to an unphysical theory
p with pairs of quarks of opposite baryon charges (conju- γ5 γ5 = †. (1)
e D D
gate quarks) [2]. This is one of the main reasons why
h
which,inparticular,impliespositivitydet 0. Forthe
v: our understanding of QCD at finite baryon density is correlator of a generic meson M = ψ¯Γψ,Dw≥e can write,
still rudimentary. Many interesting phenomena, such as
Xi color superconductivity and color-flavor locking [3], oc- using (1) and the Schwartz inequality:
ar clautrtiactefitencihtenibqaureyso.n density, beyond the reach of current hM(x)M†(0)iψ,A =−hTrS(x,0)ΓS(0,x)ΓiA =
To understand the regime of finite baryon density one Tr (x,0)Γiγ5 †(x,0)iγ5Γ A Tr (x,0) †(x,0) A, (2)
h S S i ≤h S S i
would need to follow the transition from hadronic to
where −1 and Γ γ Γ†γ . The inequality is satu-
quark degrees of freedom by increasing the density of S ≡D ≡ 0 0
ratedfor mesons with Γ=iγ τ , since commutes with
a conserved charge (such as baryon number), i.e., with- 5 i D
isospinτ ,whichmeansthatthepseudoscalarcorrelators
out invoking the temperature. This is the motivation i
arelarger,point-by-point,thanallotherI =1mesoncor-
for us to turn to QCD at finite chemical potential µ of
I
relators[8]. As a consequence,one obtains an important
isospin (moreprecisely,ofthethirdcomponent,I ). Na-
3
restriction on the pattern of the symmetry breaking: for
ture provides us with nonzero µ systems in the form of
I example, it cannot be driven by a condensate of ψ¯γ ψ ,
isospin-asymmetric matter. These always contain both h 5 i
which would give 0+ Goldstones.
isospin density and baryon density. In any realistic set-
At finite isospin density, µ = 0, positivity still holds
ting µI µB. In this paper, however, we shall consider I 6
≪ [4] and certain inequalities can be derived (in contrast
an idealization in which µ is nonzero while µ = 0.
I B
withthecaseofµ =0whenthereisnopositivity). Now
Such a system is unstable with respect to weak decays =γ (∂+iA)+B 61µ γ τ +m, and Eq. (1) is not true
which do not conserveisospin. However,since we arein- D · 2 I 0 3
anymore,sincetheoperationontheright-handsideof(1)
terested in the dynamics of strong interactionalone, one
changesthe relativesignofµ . However,providedm =
can imagine that all relatively slow electroweak effects I u
m , interchanging up and down quarks compensates for
d
1
this sign change (the u and d quarks play the role of The tilt angle α is determined by minimizing the vac-
mutually conjugate quarks [2]), i.e, uum energy. The energy is degenerate with respect to
the angle φ, corresponding to the spontaneous breaking
τ γ γ τ = †. (3)
1 5 5 1 of the U(1) symmetry generated by I in the La-
D D L+R 3
grangian(4). Thegroundstateisapionsuperfluid,with
Instead of isospin τ in (3) one can also use τ (but not
1 2
onemasslessGoldstonemode. Sincewestartfromathe-
τ ). Equation (3), in place of the now invalid Eq. (1),
3
orywiththreepions, therearetwomassivemodeswhich
ensures that det 0. Repeating the derivation of the
D ≥ canbe identified with π and a linearcombinationof π+
QCD inequalities, by using (3) we find that the lightest 0
meson,orthecondensate,mustbeinchannelsψ¯iγ5τ1,2ψ, and π−. At the condensation threshold, mπ0 = mπ and
i.e., a linear combination of π− u¯γ5d and π+ d¯γ5u the mass of the other mode is 2mπ, while for |µI|≫mπ
∼ ∼ both masses approach µ .
states. Indeed, as shown below, in the two analytically | I|
The isospin density is found by differentiating the
tractableregimesofsmallandlargeµ the lightestmode
I
ground state energy with respect to µ and is equal to:
is a massless Goldstone which is a linear combination of I
u¯γ5d and d¯γ5u. n =f2µ sin2α=f2µ 1 m4π , µ >m . (7)
Small isospin densities.—When µI is small compared I π I π I(cid:18) − µ4 (cid:19) | I| π
I
to the chiral scale (taken here to be m ), we can use
ρ
chiral perturbation theory. For zero quark mass and For µI just above the condensation threshold, µI
| | | | −
zero µI the pions are massless Goldstones of the spon- mπ ≪ mπ, Eq. (7) reproduces the equation of state of
taneously broken SU(2) SU(2) chiral symmetry. If the dilute nonrelativistic pion gas [6].
L R
the quarks have small eq×ual masses, the symmetry is It is also possible to find baryon masses, i.e., the en-
only SU(2) . The low-energy dynamics is governed ergy cost of introducing a single baryoninto the system.
L+R
by the familiar chiral Lagrangian for the pion field Σ The most interesting baryons are those with lowest en-
SU(2): = 1f2Tr[∂ Σ∂ Σ† 2m2ReΣ], which contain∈s ergy and highest isospin, i.e. neutron n and ∆− isobar.
thepionLdeca4yπconstµantfµπ an−dvacπuumpionmassmπ as There are two effects of µI on the baryon masses. The
phenomenological parameters. The isospin chemical po- first comes from the isospin of the baryons, which effec-
eteffnetcitalcfaunrtbheerinbcreluadkesdSUto(2l)eLa+dRingdoowrndetroinU(µ1I)Lw+Rit.hoIutst tmivaeslsybryed23u|cµeIs|.thAelonneeu,ttrhoins emffaescst bwyou21ld|µlIe|adantdo tthhee f∆or−-
additional phenomenological parameters by promoting mation of baryon/antibaryonFermi surfaces, manifested
SU(2) SU(2) to a local gauge symmetry and view- in nonvanishing zero-temperature baryon susceptibility
L R
ing µI a×s the zeroth component of a gauge potential [6]. χB ≡ ∂nB/∂µB when µI > 23m∆. However, long before
Gauge invariance thus fixes the way µ enters the chiral that, anothereffect turns on: theπ−’s inthe condensate
I
Lagrangian: tend to repel the baryons,lifting up their masses. These
effects can be treated in the framework of the baryon
= fπ2Tr Σ Σ† m2πfπ2ReTrΣ, (4) chiral perturbation theory [9], giving
eff ν ν
L 4 ∇ ∇ − 2
µ 3µ
I I
where the covariant derivative is defined as mn =mN | |cosα, m∆− =m∆ | |cosα (8)
− 2 − 2
µ
Σ=∂ Σ I(τ Σ Στ ). (5) in the approximation of nonrelativistic baryons. Equa-
0 0 3 3
∇ − 2 −
tion (8) can be interpreted as follows: as a result of the
Byusing(4),itisstraightforwardtodeterminevacuum rotation (6) of the chiral condensate, the nucleon mass
alignment of Σ as a function of µ and the spectrum eigenstate becomes a superposition of vacuum n and p
I
of excitations around the vacuum. We are interested in states. The expectation value of the isospin in this state
negative µ , which favors neutrons over protons, as in is proportional to cosα appearing in (8). With cosα
I
neutron stars. The results are very similar to two-color given in Eq.(6), we see that the two mentioned effects
QCD at finite baryon density [6]: cancel each other when mπ µI mρ. Thus the
≪ | | ≪
(i) For µI < mπ, the system is in the same ground baryon mass never drops to zero, and χB = 0 at zero
| |
state as at µ = 0: Σ = 1. This is because the lowest temperature in the region of applicability of the chiral
I
lyingpionstatecostsapositiveenergym µ toexcite, Lagrangian.
π I
−| |
which is impossible at zero temperature. Asoneforcesmorepionsintothecondensate,thepions
(ii) When µ exceeds m it is favorable to excite π− arepackedcloserandtheirinteractionbecomesstronger.
I π
| |
quanta,whichformaBosecondensate. Inthelanguageof When µI mρ, the chiral perturbation theory breaks
∼
the effective theory, such a pion condensate is described down. To find the equation of state in this regime, full
by a tilt of the chiral condensate Σ, QCD has to be employed. As we have seen, this can be
done using present lattice techniques since the fermion
Σ=cosα+i(τ cosφ+τ sinφ)sinα,
1 2 sign problem is not present at finite µ , similar to the
I
cosα=m2/µ2. (6) two-color QCD [5].
π I
2
Asymptotically high isospin densities.—Intheopposite chemicalpotentialcannotberigorouslyruledout(though
limitofverylargeisospindensities,or µ m ,thede- it is unlikely, see below), this conjecture needs to be ver-
I ρ
| |≫
scription in terms of quark degrees of freedom applies ified by lattice calculations.
since the latter are weakly interacting due to asymp- A number of nontrivial arguments support the conti-
totic freedom. In our case of large negative µ , or n , nuity hypothesis. One notices that all fermions have a
I I
the ground state consists of d quarks and u¯ antiquarks gap at large µ , which implies that χ = 0 at T = 0.
I B
| |
which, neglecting the interaction, fill two Fermi spheres This is alsotrue atsmallµ . It is thus naturalto expect
I
with equal radii µ /2. Turning on the interaction be- that χ remains zero at T = 0 for all µ , which also
I B I
| |
tween the fermions leads to the instability with respect suggests one way to check the continuity on the lattice.
totheformationandcondensationofCooperpairs,simi- Anotherargumentcomesfromconsideringthe limitof
lar,tosomeextent,tothediquarkpairingathighbaryon alargenumberofcolorsN . Infinite-temperatureQCD,
c
density [3]. In our case,µ <0, the Cooperpair consists the fact that the number of gluon degrees of freedom is
I
of a u¯ and a d in the color singlet channel. The order (N2)whilethatofhadronsis (N0)hintsatafirstor-
O c O c
parameter has the same quantum numbers as the pion derconfinement-deconfinementphasetransition. Atvery
condensate at lower densities, largeµ thermodynamicquantitiessuchasthedensityof
I
isospin n are proportional to N . On the other hand,
I c
u¯γ5d =0. (9) in the large N limit the pion decay constant scales as
h i6 c
f2 = (N ),andaccordingtoEq.(7)theisospindensity
Because of Cooper pairing, the fermion spectrum ac- π O c
inthepiongasisalsoproportionaltoN . Physically,the
quires a gap ∆ at the Fermi surface, where c
repulsion between pions becomes weaker as one goes to
∆=bµI g−5e−c/g, c=3π2/2 (10) large Nc, thus more pions are stacked at a given chem-
| | ical potential. As a result, the N dependence of ther-
c
where g should be evaluated at the scale µ . This be- modynamic quantities is the same in the quark and the
I
| |
havior comes from the long-range magnetic interaction, hadronic regimes.
as in the superconducting gap at large µ [10]. The Confinement.— At large µ , gluons softer than ∆ are
B I
constant c is smaller by a factor of √2 compared to the notscreenedby the Meissner orby the Debye effect [15]:
latter case due to the stronger one-gluon attraction in the condensate does not break gauge symmetry (in con-
the singlet qq¯channel compared to the 3¯ diquark chan- trast to the color superconducting condensate [3]) and
nel. Consequently, the gap (10) is exponentially larger therearenolow-lyingcolorexcitationstoscreentheelec-
thanthediquarkgapatcomparablebaryonchemicalpo- tricfield. Thus,thegluonsectorbelowthe ∆scaleisde-
tentials. By using the methods of [11], one can estimate scribed by pure gluodynamics, which is confining. This
b 104. means there are no quark excitations above the ground
≈
The perturbative one-gluon exchange responsible for state: allparticlesandholesmustbeconfinedincolorless
pairing at large µ does not distinguish u¯d and u¯γ d objects, mesons and baryons, just like in vacuum QCD.
I 5
channels: the attraction is the same in both. The u¯γ d Ifthereisnotransitionalongtheµ axis,weexpectcon-
5 I
channelis favoredby the instanton-inducedinteractions, finementatallvaluesofµ . Sincetherunningstrongcou-
I
which explains the fact that the condensate is a pseu- pling α at the scale of ∆ is small, the confinement scale
s
doscalar and breaks parity. This is consistent with our Λ′ (whichis,ingeneral,differentfromΛ )ismuch
QCD QCD
observation that QCD inequalities also constrain the less than ∆. At large µ , we thus predict a temperature
I
I =1 condensate to be a pseudoscalar at any µ . driven deconfinement phase transition at a temperature
I
Quark-hadron continuity.—Since the order parameter T′ of order Λ′ , which is expected to be of first order
c QCD
(9)has the same quantumnumbers andbreaksthe same as in pure gluodynamics. Since Λ′ ∆ the hadronic
QCD ≪
symmetry as the pion condensate in the low-density spectrum is similar to that of a heavy quarkonium, with
regime, it is plausible that there is no phase transition ∆ playing the role of the heavy quark mass.
along the µ axis. In this case the Bose condensate of The (T,µ ) phase diagram.—By considering nonzero
I I
weakly interacting pions smoothly transforms into the µ , we make the phase diagram of QCD three dimen-
I
superfluidstateofu¯dCooperpairs. Thesituationisvery sional: (T,µ ,µ ). Two planes in this three-dimensonal
B I
similar to that of stronglycoupled superconductors with space are of a special interest: the µ =0 (T,µ ) plane,
B I
a“pseudogap”[12],andpossiblyofhigh-temperaturesu- which is completely accessible by present lattice tech-
perconductors[13]. This alsoparallelsthe continuitybe- niques, and the T = 0 (µ ,µ ) plane, where the neu-
I B
tween nuclear and quark matter in three-flavor QCD as tron star matter belongs. Two phenomena determine
conjectured by Scha¨fer and Wilczek [14]. We hence con- the (T,µ ) phase plane (Fig.1): pion condensation and
I
jecture that, in two-flavor QCD, one can move continu- confinement.
ously from the hadron phase to the quark phase with-
out encountering a phase transition. Since a first order
deconfinement phase transition at intermediate isospin
3
T
<u γ d>=0
5
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I
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I B
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4