Positivity of QCD at asymptotic density

6 Pages·0.11 MB·English

by  Deog Ki Hong

February1,2008 20:16 WSPC/TrimSize: 9inx6inforProceedings hong˙sewm02
3
0
0
POSITIVITY OF QCD AT ASYMPTOTIC DENSITY
2

n
a
J DEOGKI HONG

7
Department of Physics, Pusan National University,
1
Busan 609-735, Korea

E-mail: [email protected]
1
v
5
Inthistalk,ItrytoshowthatthesignproblemofdenseQCDisduetomodeswhose
2
frequencyishigherthanthechemicalpotential. Aneffectivetheoryofquasi-quarks
1
near the Fermi surface has a positive measure in the leading order. The higher-
1
order corrections make the measure complex, but they are suppressed as long as
0
3 the chemical potential is sufficiently larger than ΛQCD. As a consequence of the
positivity of the effective theory, we can show that the global vector symmetries
0
except theU(1)baryonnumberareunbrokenatasymptoticdensity.
/
h
p
-
p 1. Introduction
e
h Itisnowfirmlybelievedthatquantumchromodynamics(QCD)successfully
:
v describesthestronginteraction. Itspredictiononthehadroninteractionat
i highenergyiswellconfirmedbyexperiments. Thecouplingextractedfrom
X
various hadronic processes scales logarithmically with respect to energy as
r
a QCD predicted 1,2. Furthermore, the low energy dynamics of hadrons is
in good agreement of the chiral symmetry breaking of QCD 3. It explains
successfully why pions and kaons are much lighter than baryons and why
hadron spectra are not in parity-doublet, though the strong interaction
preserves parity.
Being the theory of strong interaction, QCD should be able to tell us
how matter behaves atextreme environments,as encounteredin heavy ion
collisions, in early universe, or in compact stars. One salient feature of
dense matter is that it undergoes phase transitions at extreme environ-
ments. QCDpredictsphasetransitionsforhotanddense matterasonein-
creases the temperature or the density of matter. On dimensional ground,
thecriticaltemperatureandthecriticalchemicalpotentialhavetobeofor-
derofΛ ,whichis theonlydimensionalparameterthatQCDdynamics
QCD
depends on; T Λ and µ Λ . In fact, some of these predic-
C QCD C QCD
∼ ∼
tions on hot matter have been confirmed by lattice calculations 4. Lattice
1
February1,2008 20:16 WSPC/TrimSize: 9inx6inforProceedings hong˙sewm02
2
calculation shows T = 175 MeV, which is close to Λ ( 213 MeV).
C QCD
≃
However,little progressinlatticeQCDhasbeenmadetoprobethedensity
phase transition in matter except when the chemical potential is small 5,
since lattice QCDatfinite density suffersa notorioussignproblemdue the
complexity of the measure 6.
Inthis talk,I willargue thatthe signproblemcanbe solvedfor certain
quantities at high density, thus allowing lattice calculation, and the QCD
measure becomes positive at asymptotic density.
2. High density effective theory
Quark matter on lattice is described by a partition function given as
Z(µ)= dA det(M)e−S(A), (1)
Z
where M = γµDµ +µγ4 is the Euclidean Dirac operator with a chemical
E E E
potential µ. In general the measure of dense QCD is complex, since there
is no matrix P that satisfies for arbitrary gauge field A
M(A)=P−1M(A)†P. (2)
However, we claim that the complexity of the measure is due to fast
modes, whosefrequency is largerthan the chemicalpotential, ω >µ. Ifwe
areinterestedinFermisurfacephenomenaorlowenergydynamicsofdense
matter, most of degrees of freedom in QCD are irrelevant. For instance,
modes in the deep Dirac sea are hard to excite at low energy due to Pauli
blocking by the states in the Fermi sea and thus decoupled to physics near
the Fermi surface. On the other hand, modes near the Fermi surface are
easy to excite, since it does not cost much energy to put in or remove the
modes near the Fermi surface.
Therefore, we need to know the energy spectra of QCD near the Fermi
surfacetofindouttherelevantmodesforthelowenergydynamicsofdense
QCD. This is in general very difficult since it amounts to solving QCD.
However,theproblembecomeseasieratextremedensitybecausethetypical
momentum transfer in the scattering of quarks near the Fermi surface is
quite large compared to Λ .
QCD
Due to asymptotic freedom, the QCD interaction of modes near the
Fermisurfacecanbetreatedperturbativelyandthespectrumisdetermined
approximately by the energy eigenvalue equation of free Dirac particles;
(α~ p~ µ)ψ =E ψ , (3)
± ± ±
· −
February1,2008 20:16 WSPC/TrimSize: 9inx6inforProceedings hong˙sewm02
3
where ψ are the eigenstates of α~ ~p with eigenvalues p~. At low energy
±
· ±| |
(E < µ), the states ψ near the Fermi surface, p~ µ, are easily excited,
+
| |∼
while ψ , corresponding to the states in the Dirac sea, are completely
−
decoupled. Therefore, the relevant modes for the low-energy QCD at high
density (µ Λ ) are ψ modes and the soft gluons.
QCD +
≫
Consider a quark near the Fermi surface and decompose the quark mo-
mentum into the Fermi momentum and a residual momentum as
p =µv +l , l <µ,
µ µ µ µ
| |
where v = (0,~v ) and ~v = p~ /µ is the Fermi velocity, neglecting the
µ F F F
quark masses. In the leading approximation in 1/µ expansion, the energy
ofthequarkneartheFermisurfacedependsonlyontheresidualmomentum
paralleltotheFermivelocity,whiletheperpendicularcomponent,~l ,labels
⊥
the degeneracy on the Fermi surface. Therefore, the integration over the
perpendicular component should give the area of the Fermi surface,
d2l =4πp2. (4)
Z ⊥ F
Now, at low energy E < µ, the Fermi velocity of the quark near the
Fermi surface does not change under any scattering, since any change in
the Fermi velocity can be absorbed into the redefinition of the residual
momentum. So, it is convenient to define a Fermi-velocity dependent field
which carries the residual momentum only,
1+α~ ~v
ψ (~v ,x)= · Fe−iµ~vF·~xψ(x).
+ F
2
Integrating out the irrelevant modes, ψ and hard gluons, we get the high
−
density effective theory (HDET) of QCD for dense quark matter at low
energy 7.
3. Positivity at asymptotic density
The effective theory is described by
c
L =ψ¯ iγµD ψ (~v ,x) 1ψ¯ γ0(D/ )2ψ + , (5)
eff + k µ + F − 2µ + ⊥ + ···
where c is a dimensionless constant due to loop effects of the irrelevant
1
modes and the ellipsis denotes the higher order terms in 1/µ expansion.
We note that the Dirac operator of the effective theory in Euclidean
space is related to its hermitian conjugate by a similarity transformation,
M = γE D(A) = γ M† γ . (6)
eft k · 5 eft 5
February1,2008 20:16 WSPC/TrimSize: 9inx6inforProceedings hong˙sewm02
4
Therefore, HDET has a positive measure in the leading order. Since the
next-to-leadingtermishermitian,whilethe leadingtermis anti-hermitian,
the sign problem comes in at the next-leading order. However, the sign
problem is suppressed by 1/µ.
To implement HDET on lattice, it is useful to introduce an operator
formalism in which the velocity is realized as an operator,
i ∂
~v = − , (7)
√ 2 ∂~x
−∇
since one needs to know the Fermi velocity for a given configuration of
quarks. Then, the quasi-quarks near the Fermi surface are described by
ψ =exp( iµx v α v)ψ. (8)
+
− · ·
Now, the effective Lagrangiandensity becomes
L =ψ¯ γµ ∂µ+iAµ ψ , (9)
+ + k + +
(cid:0) (cid:1)
where Aµ =e−iX Aµ e+iX and X =µx v α v. Note that γµ∂µ =γµ∂µ,
+ · · k
since v ∂v γ =∂ γ .
· · ·
The partition function of dense QCD can be rewritten as
Z(µ)= dA detM (A )e−Seff(A+), (10)
+ eff +
Z
and the effective action is given as
1 M2
Seff(A)=Z d4xE4FµaνFµaν + 16π Aa⊥µAa⊥µ+···, (11)
X~vF
 
where A = A A , M = N /(2π2)g µ is the Debye mass, and the
⊥ k f s
−
ellipsis denotes terms supprespsed by 1/µ. Therefore, we see that at the
leading order HDET has a positive measure and the lattice calculation is
possible 8.
To estimate the size of the higher-order contributions, we calculate the
correction to the vacuum energy by the naive dimensional analysis 9. We
found
δE α Λ
vac s
, (12)
E ∼ 2π µ
vac
where Λ Λ is the energy scale that we are interested in. Therefore,
QCD
≃
the positivity of HDET is good, as long as the chemical potential is much
larger than Λ .
QCD
February1,2008 20:16 WSPC/TrimSize: 9inx6inforProceedings hong˙sewm02
5
As an application of the positivity of QCD at asymptotic density, one
canestablisharigorousinequalityliketheVafa-Wittentheorem10 toshow
that the color-flavor locked (CFL) phase 11 is in fact exact at asymptotic
density.
Consider the correlator of the SU(3) flavor currents
V
JA(~v ,x)JB(~v ,y) A = Trγ TASA(x,y;∆)γ TBSA(y,x;∆), (13)
µ F ν F − µ ν
(cid:10) (cid:11)
where JA(~v ,x) = ψ¯ (~v ,x)γ TAψ (~v ,x) and we have introduced an
µ F + F µ + F
infraredcut-off∆,whichbreakstheU(1)baryonnumbersymmetrya. The
anomalous propagator can be rewritten as
1 ∞
SA(x,y;∆)= x y = dτ x e−iτ(−iM) y (14)
h |M | i Z h | | i
0
where D =∂+iA and
D V ∆
M =γ · , (15)
0(cid:18) ∆ D V¯ (cid:19)
·
with V = (1,~v ), V¯ = (1, ~v ). Since the eigenvalues of M are bound
F F
−
from the below by ∆, we have the following inequality:
1 ∞ e−∆R
x y dτe−∆τ xx y y = xx y y . (16)
(cid:12)h |M | i(cid:12)≤Z h | i h | i ∆ h | i h | i
(cid:12) (cid:12) R p p p p
(cid:12) (cid:12)
(cid:12) (cid:12)
Since the measure of HDET is positive, the vector current correlator falls
off exponentially even after integrating over the gauge fields. Therefore,
there is no Nambu-Goldstone mode along the vector channel. Combining
this with the Cooper theorem, we prove that the CFL phase is exact.
Inconclusion,wehaveshownthatdense QCDis positiveatasymptotic
density. Furthermore,alatticecalculationshouldbe possibleusingHDET,
an effective theory for quasi-quarksnear the Fermi surface, as long as µ
≫
Λ . As a consequence of the positivity, we were able to show that the
QCD
(global) vector symmetries except the U(1) baryonnumber are not broken
in QCD at asymptotic density.
aNote that any infraredregulator has to break the U(1) baryon number to open agap
attheFermisurface
February1,2008 20:16 WSPC/TrimSize: 9inx6inforProceedings hong˙sewm02
6
Acknowledgments
I would like to thank Steve Hsu for the collaboration on which this talk is
based. ThisworkissupportedinpartbytheKOSEFgrantnumber1999-2-
111-005-5andalsobytheacademicresearchfundofMinistryofEducation,
Republic of Korea,Project No. BSRI-99-015-DI0114.
References
1. D.J. Gross and F. Wilczek, Phys. Rev.Lett. 30 (1973) 1343.
2. H.D. Politzer, Phys.Rev. Lett.30 (1973) 1346.
3. J. Gasser and H. Leutwyler, AnnalsPhys. 158, 142 (1984).
4. F. Karsch, E. Laermann and A. Peikert, Nucl. Phys. B 605, 579 (2001)
[arXiv:hep-lat/0012023].
5. Z.FodorandS.D.Katz,Phys.Lett.B534,87(2002)[arXiv:hep-lat/0104001];
JHEP 0203, 014 (2002) [arXiv:hep-lat/0106002].
6. See, for instance, S. Hands, Nucl. Phys. Proc. Suppl. 106, 142 (2002)
[arXiv:hep-lat/0109034]; I. M. Barbour, S. E. Morrison, E. G. Klepfish,
J. B. Kogut and M. P. Lombardo, Nucl. Phys. Proc. Suppl. 60A, 220 (1998)
[arXiv:hep-lat/9705042]; M. G. Alford, Nucl. Phys. Proc. Suppl. 73, 161
(1999) [arXiv:hep-lat/9809166]; S. Chandrasekharan and U. J. Wiese, Phys.
Rev.Lett. 83, 3116 (1999) [arXiv:cond-mat/9902128].
7. D. K. Hong, Phys. Lett. B 473, 118 (2000) [hep-ph/9812510]; Nucl. Phys. B
582, 451 (2000) [hep-ph/9905523].
8. D. K. Hong and S. D. H. Hsu, Phys. Rev. D 66, 071501 (2002) [arXiv:hep-
ph/0202236].
9. A. Manohar and H. Georgi, Nucl. Phys. B 234, 189 (1984); H. Georgi and
L.Randall,Nucl.Phys.B276,241(1986); H.Georgi,Phys.Lett.B298,187
(1993).
10. C.VafaandE.Witten,Phys.Rev.Lett.53,535(1984); Nucl.Phys.B234,
173 (1984).
11. M. G. Alford, K. Rajagopal and F. Wilczek, Nucl. Phys. B 537, 443 (1999)
[arXiv:hep-ph/9804403].