Kaon Condensation in a Nambu--Jona-Lasinio (NJL) Model at High Density

0.44 MB·

by  Michael McNeil Forbes

MIT-CTP-3554
Kaon Condensation in a Nambu–Jona-Lasinio (NJL) Model at High Density
Michael McNeil Forbes
Center for Theoretical Physics, Department of Physics, MIT, Cambridge, Massachusetts 02139
(Dated: 30 November2005)
Wedemonstrateafully self-consistent microscopic realization ofa kaon-condensedcolour-flavour
lockedstate(CFLK0)withinthecontextofamean-fieldNJLmodelathighdensity. Theproperties
of this state are shown to be consistent with the QCD low-energy effective theory once the proper
gauge neutrality conditions are satisfied, and a simple matching procedure is used to compute the
pion decay constant, which agrees with the perturbative QCD result. The NJL model is used to
compare theenergies of theCFLK0 state totheparity even CFL state, and todetermine locations
of the metal/insulator transition to a phase with gapless fermionic excitations in the presence of a
non-zerohyperchargechemical potential and a non-zerostrange quarkmass. Thetransition points
arecomparedwithresultsderivedpreviouslyviaeffectivetheoriesandwithpartially self-consistent
6 NJLcalculations. Wefindthatthequalitativephysicsdoesnotchange,butthatthetransitionsare
0 slightly lower.
0
2
n I. INTRODUCTION A recent attempt has been made to extrapolate to large
a strange quark mass (ms) [34], but this approachhas not
J dealt with additional complications in the condensate
Recently there has been interest in the structure of
3 structure that allow different gap parameters for each
matter at extremely high densities, such as might be
2 pair of quarks.
foundinthecoresofneutronstars. Atlargeenoughden-

2 sities, the nucleons are crushed together and the quarks To deal with moderate quark masses, another ap-
v become the relevantdegreesoffreedom. The asymptotic proach has been to study Nambu–Jona-Lasinio (NJL)
1 freedom of QCD ensures that the theory is weakly cou- models [35, 36] of free quarks with contact interactions
0 pled at high enough densities. This allows one to per- that model instanton interactions or single gluon ex-
0 form weak-coupling calculations at asymptotically high change. Thesemodelsareamenabletoamean-fieldtreat-
1 densities. Such calculations have established that the mentandexhibitasimilarsymmetrybreakingpatternto
1
structure of the groundstate of quark matter is a colour QCD which results in CFL ground states [5, 6].
4
0 superconductor (see for example [1, 2, 3, 4, 5, 6, 7, 8, 9, Within these models, one can study the effects of
/ 10, 11, 12, 13, 14, 15]). In particular, at densities high moderate quark masses through self-consistent solutions
h
enough that the three lightest quarks can be treated as of the mean-field gap equations. This has led to a
p
massless, the ground state is the colour-flavour–locked plethora of phases. In particular, several analyses show
-
p (CFL) state in which all three colours and all three atransitiontoacolour-flavourlockedphasewithgapless
e flavours participate in maximally (anti)-symmetric pair- fermionic excitations (the gCFL phase). These include
h
ing [7, 10, 16, 17]. bothNJL-basedcalculations[37,38,39,40]andeffective-
:
v Determination of the QCD phase structure at moder- theory–based calculations [34, 41]. Until recently, how-
Xi atedensitiesandinthepresenceofnon-zeroquarkmasses ever, the NJL calculations have excluded the possibil-
hasproceededinseveralways. Oneapproachhasbeento ity of kaon condensation (see however [42] which consid-
r
a formulateachainofeffectivetheories,andthentomatch ers kaoncondensationin the NJL model atlow density),
coefficients acrossseveralenergy scales throughthese ef- while the effective theories do not consider the compli-
fective theories to perturbative calculations. Coefficients catedpatternsinwhichthecondensateparametersevolve
inthe low-energychiraleffective theory[18]arematched at finite quark masses.
to calculations performed in high-density effective theo- The goal of this paper is to show that one can com-
ries (HDET) [9, 19, 20] which in turn are matched to bine the analysis of the low-energy effective theories,
weakly-coupled QCD. This allows one to determine the whichexhibitkaoncondensation,withtheself-consistent
propertiesoftheGoldstonebosonsanddeterminetheef- mean-field analysis of the NJL model, which accounts
fects ofsmallquarkmasses[21,22,23,24,25,26,27,28, for the full condensate structure. In particular, we use
29, 30]. Within this framework, it has been noted that, an NJL model based on single gluon exchange to find
in the presence of a finite strange quark mass, neutral self-consistent solutions that correspond to the CFLK0
“kaons” (the lightest pseudo-Goldstone modes at high phase; we show that these phases agree with the predic-
densitywiththesamequantumnumbersastheirvacuum tions of the low-energy effective theory; and we deter-
counterpart)canBose-condenseintheCFLstatetoform mine how and where the zero temperature phase transi-
akaon-condensedCFLK0 phasewithlowercondensation tion to a gapless CFL phase occurs as one increases the
energy [31, 32, 33]. strange quark mass. In addition, unlike previous work
Unfortunately, the low-energy effective theory is only onthe NJL model, ournumericalsolutions arefully self-
reliable for small perturbations and at moderate densi- consistent: we include all condensates and self-energy
ties the strange quark mass is not a small perturbation. corrections required to close the gap equations.
2
We first describe the pattern of symmetry breaking The CFL state (1) preserves parity, and is preferred
that leads to the CFL and CFLK0 states (Section II). wheninstantoneffects are considered. Excluding instan-
Then we present our numerical results, demonstrating ton effects, there is an uncountable degeneracy of phys-
some properties of these states and determining the lo- ically equivalent CFL ground states that violate parity.
cations of the zero-temperature phase transitions (Sec- These are generated from the parity even CFL by the
tion III). After a careful description of our model (Sec- broken symmetry generators.
tionIV)wederivethelow-energyeffectivetheory,paying
The symmetry breaking pattern (2) breaks 18 genera-
particularattentionto the differences betweenQCD and
tors. The quarks, however, are coupled to the eight glu-
the NJL model (Section V). Here we demonstrate that,
ons associated with the SU(3) colour symmetry and to
for small perturbations, our numerical solutions are well C
the photon of the U(1) electromagnetism (which is a
described by the effective theory, and we use our numer- EM
subgroupofthevectorflavoursymmetry). Eightofthese
icalresultstocompute the piondecayconstantf which
π gauge bosons acquire a mass through the Higgs mech-
agrees with the perturbative QCD results. Specific nu-
anism and the coloured excitations are lifted from the
merical details about our calculations and a full descrip-
low-energyspectrum. Thereremain10masslessNambu-
tion of our self-consistent parametrization are given in
Goldstoneexcitations: apseudo-scalaraxialflavouroctet
Appendix A.
of mesons, a scalar superfluid boson associated with the
We leave for future work the consideration of finite
broken U(1) baryon number generator, and a pseudo-
temperature effects, the analysis of the gapless CFLK0 scalar η′ bosBon associated with broken axial U(1) gen-
(gCFLK0), the inclusion of instanton effects, the inclu- A
erator. There remains one massless gauge boson that
sion of up and down quark mass effects, and the possi-
is a mixture of the original photon and one of the glu-
bility of other forms of meson condensation.
ons [7, 44]. With respect to this “rotated electromag-
netism” U(1) the CFL state remains neutral [45].
Q˜
Thedegeneracyofthevacuummanifoldisliftedbythe
II. COLOUR FLAVOUR LOCKING (CFL)
inclusion of a non-zero strange quark mass m . In the
s
absence of instanton effects and other quark masses, the
QCD has a continuous symmetry group of U(1) groundstateisnotneartotheparityevenCFLstate(1),
B
⊗
SU(3) SU(3) SU(3) . In addition, there is an ap- but rather, is a kaon rotated state CFLK0. As m 0
L R C s
⊗ ⊗ →
proximateU(1) axialflavoursymmetrythatisexplicitly this state approaches a state on the vacuum manifold
A
brokenby anomalies. Atsufficiently high densities,how- that is a pure kaon rotation of the parity even CFL (1).
ever, the instanton density is suppressed and this sym-
Even in the absence of quark masses, the vacuum
metry is approximately restored.
manifold degeneracy is partially lifted by the anomalous
TheCFLgroundstatespontaneouslybreaksthesecon-
breakingoftheU(1) axialsymmetrywhichwehavene-
tinuous symmetries through the formation of a diquark A
glected: Instantoneffectstendtodisfavourkaonconden-
condensate [7]
sationbyfavouringparityevenstates,andthusdelaythe
onsetoftheCFLK0untilm reachesacriticalvalue(pos-
s
hψCαaγ5ψβbi∝∆3ǫαβkǫabk+∆6(δaαδbβ +δbαδaβ). (1) siblyexcludingit). Theeffects ofanomalyandinstanton
contributionshavebeenwellstudied[6,16,26,46,47,48]
The symmetry breaking pattern (including the restored and play an important quantitative role in the phase
axial U(1) symmetry) is thus1 structure of QCD. Non-zero up and down quark masses
A
also tend to disfavour kaon condensation.
U(3) U(3) SU(3)
L⊗ R⊗ C SU(3) Z Z (2) For the purposes of this paper, we shall neglect both
L+R+C 2 2
Z3 → ⊗ ⊗ the effects of instantons, and the effects of finite up and
downquarkmasses. Thiswillensurethatkaoncondensa-
where the Z symmetries correspond to ψ ψ and
2 L → − L tionoccursforarbitrarilysmallms. Bothoftheseeffects
ψ ψ . Ithasbeennotedthatthesymmetrybreak-
R R open the possibility of a much richer phase structure,
→−
ingpatternathighdensity(2)isthesameasthatthatfor
including condensation of other mesons (see for exam-
hyper-nuclear matter at low density [43]. This leads one
ple [33, 49]). Future analyses should take these numeri-
to identify the low-energy pseudo-scalar degrees of free-
callyimportanteffectsinto account,bothinthe effective
dominboththeories. Weshallrefertothepseudo-scalar
theory and in the NJL model.
Goldstone bosons in the high-density phase as “pions”
and“kaons”etc. whenthey havethe sameflavourquan- The primary source of for kaon condensation is the fi-
tum numbers as the correspondinglow-densityparticles. nitestrangequarkmass. Tolowest-order,thisbehavesas
achemicalpotential[31,32,33](see(3)and(4)). Inthis
paper, we also consider the addition of a hypercharge
chemical potential as this removes many complications
1 The Z3 factor mods out the common centres. See (10) for the associatedwith masses andleads to a veryclean demon-
explicitrepresentation. stration of kaon condensation.
3
III. SELF-CONSISTENT SOLUTIONS to the NJL model with a finite strange-quark mass pa-
rameter m . In each of these cases, one solution corre-
s
sponds to a parity even CFL phase and the other cor-
We consider four qualitatively different phases: Two
responds to a kaon-condensed CFLK0 phase. Our nor-
areself-consistentmean-fieldsolutionstotheNJLmodel
malizations and a complete description of the model are
with a finite hypercharge chemical potential parameter
presented in Section IV. A full description of all the
µ ;theothertwoareself-consistentmean-fieldsolutions
Y
parameters required to describe these phases along with
some typical values is presented in Appendix A.
80
ru gd bs rd gu rs bu gs bd
CFL 0 0 0 0 0 −1 +1 −1 +1
60 CFLK0 0 +1 −1 0 +1 −1 +1 −1 +1
2 2 2 2 2 2
)
V
e TABLE I: Leading order shifts in the chemical potentials
M 40 of the various quarks in the CFL and CFLK0 states in the
(
Ep presence of a hypercharge chemical potential shift µY. This
follows directly from (50).
20
0
450 500 550
|p| (MeV) µY =0 80
80
60
60
)
V
) e
V M 40
Me 40 (
( Ep
Ep 20
20
0
0 450 500 550
450 500 550 |p| (MeV) µY =µcY/2
|p| (MeV) µY =µcY/2 80
80
60
60
)
V
) e
V M 40
Me 40 (
( Ep
Ep 20
20
0
0 450 500 550
450 500 550 |p| (MeV) µY =1.20µcY
|p| (MeV) µY =µcY
FIG. 2: Lowest lying quasiparticle dispersions about the
FIG. 1: Lowest lying quasiparticle dispersions about the Fermi momentum pF = µq = 500 MeV for the CFLK0
Fermi momentum pF = µq = 500 MeV for the CFL phase phasewithdifferentvaluesofthehyperchargechemical. (The
withdifferentvaluesofthehyperchargechemical. Alldisper- µY = 0 dispersions are the same as in the top of Figure 1.)
sions haveleft-right degeneracy: we nowconsider thecolour- Again, all dispersion have a left-right degeneracy. In the top
flavour degeneracy. In the top plot µY = 0, and the lowest plotatµY =µcY/2=12.5MeV,theeight-folddegeneratelow-
dispersionhasaneight-folddegeneracyandagapof∆0=25 estbandhassplitintoeightindependentdispersions. Tolead-
MeV. The upperband contains a single quasiparticle pairing ingorderintheperturbation,thesplittingisdescribedbyTa-
(ru,gd,bs)with agap of4∆6+2∆3 =54MeV.Inthemiddle bleI, butthelack of degeneracy indicatesthat thereare also
plot, µY = µcY/2 = 12.5 MeV, and (rs,bu) and (gs,bd) pairs higher order effects. The lower plot at at µY ≈1.20µcY ≈30
areshiftingasindicatedinTableI. Inthelastplot,twopairs MeV isclose totheCFLK0/gCFLK0 transition. The gapless
havebecome gapless marking the CFL/gCFL transition. band now contains only a single mode and is charged.
4
1 1
0 0
∆ ∆
/ 0.5 / 0.5
p p
E E
p p
n n
mi mi
0 0
0 0.2 0.4 0.6 0.8 1.0 1.2 0 0.2 0.4 0.6 0.8 1.0 1.2
µY/µcY Ms2/(2µqµcY)
FIG.3: Physicalgapofthelowestlyingexcitationasafunc- FIG.4: Physicalgapofthelowestlyingexcitationasafunc-
tion of the hypercharge chemical potential. The dotted line tionofthestrangequarkmass. Thedottedlinecorrespondsto
corresponds to the CFL phase: the phase transition to the the CFL phase and the solid line corresponds to the CFLK0
gCFL occurs at µY =µcY where the gap vanishes. The solid phase. We have normalized the axes in terms of µcY = ∆0
line corresponds to the CFLK0 state. The transition to a forcomparison withthehyperchargechemicalpotentialcase.
gapless phase is delayed by a factor of 1.22. TheCFL/gCFLtransitionoccursataslightlysmallervalueof
Ms2/µq ≈45.4MeVthanthevalueof46.8MeVin[37,38,40].
Thisisduetotheeffectsoftheotherparametersonthequasi-
A. Finite Hypercharge Chemical Potential particle dispersion relations. We note that, as with µY, the
transition from the CFLK0 to a gapless phase is delayed rel-
ative to the CFL/gCFL transition, but by a slightly reduced
TheCFLphaseinthepresenceofahyperchargechem-
factor of 1.2. This is in qualitative agreement but quantita-
icalpotentialcorrespondstothefullygappedCFLphase
tivedisagreement withthefactorof4/3 foundin[34]. Theis
discussed in [37]. Here one models the effects of the most-likely theresult of ourfully self-consistent treatment of
strange quark through its shift on the Fermi surface thecondensate parameters.
p µ of the strange quarks. This can be seen by
F q
≈
expanding the free-quark dispersion
persions shift such that the physical gap in the spec-
M2
p2+M2 p + s + (3) trum becomes smaller; none of the other physical prop-
s ≈| | 2µq ··· erties change. In particular, as the hypercharge chem-
p ical potential increases, the coloured chemical potential
or, more carefully, by integrating out the antiparticles
µ = µ decreases to maintain neutrality. The values
to formulate the High-Density Effective Theory. (See for 8 − Y
of all of the gap parameters, the self-energy corrections,
example [9, 19, 20, 30].) These leading order effects are
the densities and the thermodynamic potential remain
equivalentto addinga hyperchargechemicalpotentialof
unchanged until the physical gap in the spectrum van-
magnitude
ishes. (Theapparentchangeinthemagnitudeofthegap
M2 parameters in the first figure of [37] is due to the shift
µY = s . (4) in the baryon chemical (5) which occurs if one uses the
2µ
q strange quark chemical potential shift µ rather than a
s
and a baryon chemical potential shift of hypercharge shift µY.) This is a consequence of the Q˜
neutrality of the CFL state [50]. In particular, the elec-
M2 tricchemicalpotentialremainszeroµ =0andthestate
δµ = s. (5) e
B − µ remainsaninsulatoruntiltheonsetofthegaplessmodes.
q
The same phenomena has also been noticed in the two-
We consider only the effect of the hypercharge chemi- flavour case [51, 52, 53]. As such, we can analytically
cal potential here, holding µB fixed. Note that the rel- identify the phase transition to the gCFL phase which
evant parameters are Ms and µq rather than ms and occurs for the critical chemical potential
µ = µ /3. M is the constituent quark mass that
s B s
appears in the dispersion relation whereas m is the µc =∆ (6)
s Y 0
bare quark mass parameter; likewise, µ is the corrected
q
quark chemical potential that determines the Fermi sur- where∆ =∆ ∆ is the physicalgapinthe spectrum
0 3 6
−
face whereas µ =µ /3 is the bare baryon chemical po- in the absence of any perturbations. Throughout this
s B
tential. (These distinctions are important because our paperweuseparametersarbitrarilychosensothatµc =
Y
model takes into account self-energy corrections.) ∆ = 25 MeV to correspond with the parameter values
0
The CFL phase responds in a trivial manner to a in [37, 38, 40]. We show typical quasiparticle dispersion
hypercharge chemical potential: the quasiparticle dis- relations for this state in Figure 1.
5
80 80
60 60
) )
V V
e e
M 40 M 40
( (
p p
E E
20 20
0 0
450 500 550 450 500 550
|p| (MeV) Ms2/(2µq)=0.50µcY |p| (MeV) Ms2/(2µq)=0.50µcY
80 80
60 60
) )
V V
e e
M 40 M 40
( (
p p
E E
20 20
0 0
450 500 550 450 500 550
|p| (MeV) Ms2/(2µq)=0.83µcY |p| (MeV) Ms2/(2µq)=0.84µcY
FIG. 5: Lowest lying quasiparticle dispersion relationships FIG. 6: Lowest lying quasiparticle dispersion relationships
abouttheFermimomentumpF =µq =500MeVfortheCFL about the Fermi momentum pF = µq = 500 MeV for the
phase with two different values of the strange quark mass. CFLK0 phase with two different values of the strange quark
(The Ms = 0 dispersions are the same as in the top of Fig- mass. (The Ms = 0 dispersions are the same as in the top
ure 1.) Qualitatively this has the same structure as Figure 1 of Figure 1.) Qualitatively this has the same structure as
except that middle dispersion is now split by higher order Figure 2.
mass effects.
In the CFL/gCFL transition, two modes become gap-
The splitting of the dispersions can also be easily un- lesssimultaneously: thelowerbranchesofthe(rs,bu)and
derstood from the charge neutrality condition (50) and (gs,bd) pairs. One of these modes is electrically neutral
the leading order effects are summarized in Table I. Af- (gs,bd) and it crosses the zero-energy axis giving rise to
ter setting µ8 = µY, the chemical potentials for the rs a “breach” in the spectrum. The other mode is electri-
−
and gs quarks shift by µY whereas for the bu and bd callycharged: assoonasin crosses,the electric chemical
−
quarks it shifts by +µY. Thus, the (gs,bd) and (rs,bu) potential must rise to enforce neutrality. The state now
pairs are the first to become gapless. contains gapless chargedexcitations and becomes a con-
The kaon-condensed hypercharge state is more com- ductor. The result is that the the neutral gapless mode
plicated. One can again use the appropriate charge neu- hastwolineardispersionswhilethechargedgaplessmode
tralityconditions(50)toestimatehowthequarkswillbe has a virtually quadratic dispersion when electric neu-
affectedby µY,butthe na¨ıveresultsholdonlytolowest- trality is enforced. (This was discovered in [37] and is
order. Inparticular,thecondensatesoftheCFLK0 state explained in detail in [38].)
also vary as µY increases (see Table III). These higher In the CFLK0/gCFLK0 transition, a single charged
order effects break all the degeneracybetween the quark mode becomes gapless.2 Thus, immediately beyond the
species and Figure 2 has nine independent dispersions.
We shall compare the thermodynamic potentials of
these two states later (see Figures 7 and 8), but we
point out here that the transition to a gapless colour- 2 Thismodepairsrs,gu,andbuquarksinquiteanon-trivialman-
flavour–locked state with kaon condensation (gCFLK0) ner. In the CFL, the quasiparticles form a nice block-diagonal
occurs at a larger hypercharge chemical potential than structure in which the quarks exhibit definite pairing between
two species. In the CFLK0, the block structure is more com-
the CFL/gCFL transition. This can be most easily seen
plicatedandthepairingcannotbesimplydescribed: thelowest
in Figure 3. This is in qualitative agreement with [34]
lying quasiparticle is a linear combination of the three rs, gu,
and [41], but in quantitative disagreement. andbuquark.
6
transition,thecorrespondinggCFLK0statewillalsobea The Gell-Mann matricesact onthe colourspaceand the
conductorbuttherewillbeasinglechargedgaplessmode flavourstructureisdiagonal. Wepointoutthatthisform
with almostquadratic dispersion. Additional modes will of NJL interaction has the desirable feature of explic-
continue to lower until either more modes become gap- itly breaking the independent colour SU(3) left and
CL
less,orafirstorderphasetransitiontoacompetingphase SU(3) right symmetries that some NJL models pre-
CR
occurs. serve. This is important because the condensation pat-
tern (1) does not explicitly link left and right particles:
Our model thus has the same continuous symmetries as
B. Finite Strange Quark Mass QCD,andtheonlycomplicationtodealwithisthegaug-
ing of the single colour SU(3) symmetry.
C
The second pair of CFL/CFLK0 states that we con- Ourgoalhereistoprovideanon-perturbativemodelto
sider are self-consistent solutions to the gap equation in discussthequalitativefeaturesofQCDatfinitedensities.
thepresenceofafinitestrangequarkmass. Qualitatively Wemodelthefinitedensitybyworkinginthegrandther-
weexpecttoseesimilarfeaturestothestatesatfinitehy- modynamicensembleandintroducingabaryonchemical
perchargechemicalpotential and indeed we do as shown potential for all of the quarks:
in Figures 5 and 6.
µ
Quantitatively, we notice a few differences with previ- µ= B1. (9)
ousanalysesconcerningthelocationsofthephasetransi- 3
tionstogaplessstates. Ourparametershavebeenchosen
With only this chemical potential and no quark masses,
to match the parameters in [37, 38, 40]. They find that
the gCFL/CFL transition occurs at M2/µ= 46.8 MeV, ourmodelhasanU(3)L U(3)R SU(3)C/Z3 continuous
s ⊗ ⊗
global symmetry in which the left-handed quarks trans-
but the CFL/gCFL transition happens noticeably ear-
lier with our model at M2/µ = 43.9 MeV. This is due form as (3¯,1,3) and the right handed quarks transform
s as (1,3¯,3). In the chiral basis we have explicitly4
to a corresponding six-percent reduction in the conden-
sate parameters and represents the effects of performing
a fully self-consistent calculation. ψL e−iθLF∗L⊗C 0 ψL (10)
moAdneostihnerthdeiffCerFeLnKce0csotnacteern(sseethFeiagp.p4e).arTanhcisetorfagnaspitlieosns ψR!→ 0 e−iθRF∗R⊗C! ψR!
occurs at M2/µ = 52.5 MeV in our model—a factor of
s where F and C are SU(3) matrices. For an attractive
1.2 larger than the CFL/gCFL transition. This is some
interaction, this NJL model exhibits the same symme-
tenpercentsmallerthanthefactorof4/3derivedin[34].
try breaking pattern as QCD (2) with a restored ax-
This is likely due to the more complicated condensate
ial symmetry. The difference between this NJL model
structureweconsiderandtheinclusionofself-energycor-
and QCD is that the NJL model contains no gauge
rections.
bosons. Thus, there are 18 broken generatorswhich cor-
respondtomasslessGoldstonebosons,andnoneofthese
is eaten. To effectively model QCD, we must remove
IV. NJL MODEL
the extra coloured Goldstone bosons. At the mean-field
level, this is done by imposing gauge neutrality condi-
We base our analysis onthe followingHamiltonian for
tions [45, 54, 55]. Once the appropriate chemical po-
the NJL model
tentials are introduced, the dependence on the vacuum
d3~p expectation values of the coloured Goldstone modes is
H = ψ† (α~ ~p µ+γ M)ψ +H . (7)
(2π)3 ~p · − 0 ~p int cancelled and the low-energy physics of the NJL model
Z
matches that of QCD.
Here we consider 9 species of quarks = 3 colours 3
× The usual NJL model has a local interaction, but this
flavours: Includingtherelativisticstructure,thereare36
is not renormalizable and needs regulation. For the pur-
quark operators in the vector ψ. The matrices µ and
poses of this paper, we introduce a hard cutoff on each
M are the quark chemical potentials and masses respec-
tively.
We take the interaction to be a four-fermion contact
interaction with the quantum numbers of single gluon
Ournormalizationsandconventions are
exchange:3
Tr[λAλB]= 1δAB,
2
Hint =g (ψ¯γµλAψ)(ψ¯γµλAψ). (8) γ5=iγ0γ1γ2γ3= −1 0 ,
Z 0 1!
γC =iγ2γ0.
Wealsousenaturalunitswherec=~=kB =1.
3 HerethematricesλAaretheeight3×3Gell-Mannmatricesand 4 From this explicit representation we can see how the centres of
theγµ aretheDiracmatriceswhichwetakeinthechiralbasis. thecolourandflavoursoverlapgivingrisetotheZ3 factor.
7
of the momenta Λ =θ Λ ~p to mimic the effects of To impose a charge neutrality condition, we instead
~p
−k k
asymptotic freedom at large momenta: varyµ (alongwithwiththeotherparameters)toobtain
R
(cid:0) (cid:1)
a neutral solution (again we note that the total charge
Hint = (2πg)9 d3~pd3~p′d3~qd3~q′ ΛpΛp′ΛqΛq′× aconrdreoctthederpcaorarrmeleatteirosnµs o)f.tOhnecsettahteisdsoelpuetniodnoisnlfyouonnd,tδhµe
R
Z
×δ(3)(~p−~p′+~q−~q′)(ψ¯~pγµλAψ~p′)(ψ¯~qγµλAψ~q′). iµscomδpµudteetderamndintehde.rDeqeuspirietdetbhaerefacchtetmhiactalthpeotseenlft-ieanleµrg=y
R
−
Tostudythismodelweperformavariationalcalculation correctionsdependonlyonthecorrectedparameters(µR
by introducing the quadratic Hamiltonian etc.), the thermodynamic potential depends on both the
corrected and the bare parameters and so this last step
H = d3~p ψ†E(~p)ψ +ψTBψ +ψ†B†ψ∗ (11) is important.
0 (2π)3 ~p ~p ~p ~p ~p ~p One must alsobe carefulabout which thermodynamic
Z (cid:16) (cid:17)
potential is used to compare states when neutrality con-
where
ditions are enforced as we are no longer in the grand
E(~p)=α~ ~p µ+γ M A (12) ensemble. The differences between the potentials of the
· − 0 − relevantensembles are proportionalto terms of the form
and then computing the following upper bound [56] on Qµ,however,soforneutralityconditions,Q=0,andthe
the thermodynamic potential Ω of the full system: thermodynamic potential may still be used to compare
states.
Ω Ω + H H . (13)
≤ 0 h − 0i0
Ω is the thermodynamic potential of the quadratic
0
model and the expectation value is the thermal av- A. Numerical Techniques
hi0
erage with respect to the quadratic ensemble defined by
H0. Inprinciple,thequadraticmodelisexactlysolvable, We sketchhere the method used to calculate the ther-
thusthe upperboundcanbe computed. Onethenvaries modynamic potential and perform the variational min-
the parameters A and B to minimize this upper bound, imization. First, we express the Hamiltonian H in the
obtaining a variational approximation for the true en- following simplified form
semble. In the zero-temperature limit, this is equivalent
tosimplyminimizingtheexpectationvalueoftheHamil- H =Ψ†HΨ+gΨ†Γ†ΨΨ†ΓΨ (14)
tonian over the set of all Gaussian states.
In practice, it is difficult to vary with respect to all where H is a hermitian matrix. In order to do this and
possible quadratic models since the space is of uncount- include the “anomalous” correlations ψψ , we must use
able dimensionality. In this paper we restrict ourselves an augmented “Nambu-Gorkov”spinohr i
to minimizing over homogeneous and isotropic systems.
This is equivalent to performing a fully self-consistent
ψ
mean-field analysis. The condition for the right hand Ψ= , (15)
side of (13) to be stationary with respect to the varia- ψC!
tional parameters is equivalent to the self-consistent gap
equation. whereψC isthechargeconjugatedspinor. Thisdoubling
The microscopic analysis presented in this paper con- of the degrees of freedom requires careful attention to
sists of choosing reasonable parameterizations of A avoid double counting.
(which includes the chemical potentials, masses and re- Tosimplifythe presentationofourmethodinthis sec-
latedcorrections)andB(which includes the gapparam- tion, we shall ignore this complication and assume that
eters ∆) that are closed under the self-consistency con- Ψ contains a single set of operators with no duplicate
dition, and numerically finding stationary points of this degrees of freedom. We also consider only a single inter-
system of equations. (As A and B are arbitrary 36 36 action term, and subsume the momentum structure into
×
matrices subject only to A = A† and B = BT, a full the matrix structure. Explicitly dealing with these com-
parametrizationconsistsof2556parameters−andwastoo plicationsisstraightforwardandthedetailsarepresented
costly for the present author to consider. However, the in [57].5
parametrizationchosenisquitenaturalandfully closed.)
Once the parameters A and B are found, the properties
of the ensemble can be computed by diagonalizing the
quadratic Hamiltonian. 5 Toderivethefullequations,oneintroducestheaugmentedstruc-
As discussed in Section VE and [45], we must impose ture whileimposingconstraints on thematrices throughout the
variation. For example, one must ensure that H = −CHTC
theappropriategaugechargeneutralityconditions. This
where ΨC = CΨ. To derive the proper momentum structure,
is done by introducing bare gauge chemical potentials
onesimplyattachesmomentumindices: forhomogeneous states
into the model and choosing these to ensure the final allcorrelationshavetheformhΨ~†pΨ~qi∝δ3(~p−~q)andthemo-
solution is neutral. mentum structurefollowstrivially.
8
We now express the variational Hamiltonian as a secant-like update to the Jacobian requiring far fewer
function evaluations per step of the algorithm.
H0 =Ψ†(H+Σ)Ψ=Ψ†H0Ψ. (16) Inmanycases,theSchwinger-Dysonequation(18)con-
verges through simple iteration. With charge neutrality
The matrix of variationalparameters Σ may be thought constraints,thisisoftennolongerthecase,buttheBroy-
of as the self-energy corrections. All of the two-point den update is sufficient to restore convergence.
correlations are determined from the corrected Hamilto-
nianmatrixH ,withthe“anomalous”correlationsbeing
0
found off the diagonal:
B. CFL at ms=0.
1
hΨΨ†i=F−, hΨ∗ΨTi=[F+]T, F± = 1+e±βH0. As an example, consider the parity even CFL state.
Theself-consistencyconditionsarefully closedwhenone
At finite temperatures,there is a one-to-onerelationship includes four variational parameters. There are two gap
between E and the matrix F− = 1 F+. Armed with parameters∆3 and∆6correspondingtothediquarkcon-
−
thisresult,thevariationalboundtakestheexplicitform6 densate(1),onechemicalpotentialcorrectionδµB tothe
baryon chemical potential and an induced off-diagonal
Ω 1 Trln[F−] Tr[ΣF+]+ cchanemthicuaslpboeteexnptiraelssµeodct. ThequadraticHamiltonian(11)
≤ β −
+g Tr[Γ†F+]Tr[ΓF+]+Tr[Γ†F−ΓF+] . (17) H =ψ† α~ ~p 1µ δµ + 1ψTγ γ ∆ψ + h.c.
0 · − 3 B − 2 C 5
(cid:16) (cid:17)
(cid:0) (cid:1)
From this, one may find the stationary points by by dif- where
ferentiating with respect to F+. At finite temperature
thisisformallyequivalenttofindingthestationarypoints [∆]αaβb =∆3ǫαβkǫabk+∆6(δaαδbβ +δbαδaβ), (19)
by varying with respect to Σ. Differentiating with re- δµ= 1δµ +µ , (20)
spect to Σ is complicated by the functional dependence 3 B oct
andtheresultisnotexpressibleasasimplematrixequa-
and
tion. The conditions ∂Ω/∂F+ = 0 yield the fully self-
ij
consistent Schwinger-Dysonequations which may be ex- 8
pressed as: [µoct]αaβb =µoct [λA]αa[λA]βb−8[λ0]αa[λ0]βb!.
A=1
X
Σ=g Γ†Tr[ΓF+]+ΓTr[Γ†F+]+Γ†F−Γ ΓF+Γ† .
− Most of this structure is all well-known and discussed
(cid:16) (1(cid:17)8) many times in the literature, however, there has been
Inprinciple,onemayderiveanalyticexpressionsforthese no mentionofthe parameterµ because mostanalyses
oct
equations, but, in practise, the matrices are 72 72 and neglect the self-energy corrections.
computing F± analytically—even when many×approxi-
Neglecting the correction to the baryon chemical po-
mations are made—is quite tedious. Instead, we simply
tential is reasonable since it has little physical signifi-
use these expressions numerically. The required diag-
cance: itsimplyentersasaLagrangemultipliertoestab-
onalization is then efficiently performed using standard
lishafinitedensity. Assuch,theeffectivecommonquark
numerical linear algebra tools. The traces involved in-
chemical potential
clude momentum integrals, but for homogeneous states
these are one-dimensional and thus also quite efficient. µ = 1µeff = 1(µ +δµ ) (21)
Thebiggestchallengeistosolvesimultaneouslytheequa- q 3 B 3 B B
tions presentin(18). This isdone byfirstprojectingout
istherelevantphysicalparameterdefiningtheFermisur-
the the limited subspaces describe in Appendix A and
face. To comparestates in the grandensemble,however,
then employing a multi-dimensional root-finder.
one must fix the bare rather than the effective chemical
Sincethesearchspaceislarge( 45parametersforthe potentials. Thisiswhatwehavedoneinourcalculations.
CFLK0 states), traditional root-∼finders are prohibitive Numerically, we find that the corrections δµ cause µ
B q
because they recompute the Jacobianateachstep. Here tovarybyonlyafewpercentaswevarytheperturbation
we use a modified Broyden algorithm [58, 59] to provide parameters µ and m .
Y s
There is no bare parameter corresponding to µ .
oct
Thus, it is spontaneously induced and should be treated
on the same footing as ∆. To see that such a parameter
6 For a fully self-consistent analysis, we must include the term must exist, consider changing to the “octet” basis using
−ghΨ†Ψ†ihΨΨi. These correlations vanish in this simplified
the augmented Gell-Mann matrices
analysis,butareincludedwhenthefullaugmentedstructure(15)
is considered as discussed in [57]. Note also that momentum
integrationisimplicitinthematrixmultiplicationandtraces. ψ˜A =2[λA]αaψ(αa) (22)
9
where λ0 = 1/√6. In this basis, the off-diagonal con- C. CFL at µY,ms6=0
densate becomes diagonal with one singlet parameter
4∆ +2∆ and eight octet parameters ∆ ∆ :
6 3 6 3
− Once one introduces a strange quark mass, one must
introduce additionalparameters. A simple wayto deter-
4∆ +2∆
6 3 mine which parameters are required is to add the mass,
∆˜ = ∆6−∆3  (23) then compute the gap equation and see which entries in
... the self-energy matrix are non-zero. By doing this for a
 
  variety of random values of the parameters, one can de-
 ∆6−∆3 termine the dimension of the subspace required to close
 
the gapequationandintroducethe requiredparameters.
It is clear thatin the CFL, the singletchanneldecouples
from the octet channel: there is no symmetry relating InthecaseoftheCFLstatewithnon-zerohypercharge
these and the two gap parameters are related by the nu- chemical potential, one only needs to introduce the pa-
merical value of the coupling g. This decoupling is also rameters µY and µ8 to ensure gauge neutrality: As dis-
present in the chemical potential corrections. One linear cussedinSec.IIIA,noneoftheotherparameterschange.
combinationcorrespondstotheidentity: thiscorrectsthe To go beyond the transition into the gCFL phase, how-
baryonchemicalpotentialδµ . Theotheristheinduced ever, or to extend the results to non-zero temperature,
B
µ . one must introduce additional parameters. These in-
oct
cludetheperturbationµ ,thegaugechemicalpotentials
Numerically,wecalibrateourmodelwiththisCFLso- Y
µ , µ , and µ required to enforce neutrality, as well as
lution. In particular, we chose our parameters to repro- 3 8 e
ninegapparameters∆ ,∆ ,∆ ,∆ ,∆ ,∆ ,∆ ,
duce the results of [40]. We use a hardcutoff atΛ=800 12 23 13 45 67 89 11
∆ , and ∆ that fully parametrize the triplet and sex-
MeV, and a coupling constant chosen so that, with an 22 33
tet diquark condensates. (These latter nine parameters
effective quark chemical potential of µ = 500 MeV one
q
correspond to the parameters φ , ϕ and σ defined in
hasaphysicalgapinthespectrumof∆ =∆ ∆ =25 i i i
0 3 6
− reference [39].) The additional parameters are chemical
MeV.Thisfixesthe followingparametervalueswhichwe
potentials similar to µ which are induced by the gap
hold fixed for all of our calculations: oct
equations. The fullsetofparametersindiscussedinAp-
pendix A.
Λ=800 MeV, (24a)
Adding a strange quark mass is more complicated.
gΛ2 =1.385, (24b)
First of all, we need to introduce additional Lorentz
µB/3=549.93 MeV. (24c) structure. Forhomogeneousandisotropicsystems,there
are eight possible relativistic structures:
With these parameters fixed, the fully self-consistent
mean-fieldCFLsolutionhasthefollowingvariationalpa-
rameters: A=1 δµ+γ δµ γ δm γ γ δm ,
⊗ 5⊗ 5− 0⊗ − 0 5⊗ 5
B=γ γ ∆+γ ∆ +γ γ γ κ+γ γ κ .
C 5⊗ C ⊗ 5 0 C 5⊗ 0 C ⊗ 5
∆ =25.6571 MeV,
3
∆ =0.6571 MeV,
6
Introducing quark masses requires one to introduce the
δµ /3= 49.93 MeV,
B − additionalLorentzstructureκ[60]toclosethegapequa-
µoct = 0.03133 MeV. tions, but these are found to be small. In total, one re-
−
quiresabout20parameterstofullyparametrizetheCFL
As firstnotedin[7],anddiscussedin[60],the parameter in the presence of a strange quark mass (see Table IV).
∆ is requiredto closethe gapequation,but is smallbe-
6
With the inclusion of a bare quark mass m one in-
cause the sextet channel is repulsive. In weakly-coupled s
ducesachiralcondensate ψ¯ψ whichinturngeneratesa
QCD, ∆6 is suppressed by an extra factor of the cou- h i
correctiontothequarkmass. Theresultingparameterin
pling. This effect is numerically captured in the NJL
H (11)istheconstituentquarkmassM whichappears
model. The parameter µ is also required to close the 0 s
oct
in the dispersion relationships for the quarks. It is this
gapequation when the Hartree-Fockterms are included.
constituentquarkmassthatmustbeusedwhencalculat-
Itis alsonumericallysuppressed. Recentcalculationsof-
ing the effective chemical potential shift (4). Generally
ten omit ∆ and µ : we see that this is numerically
6 oct
the constituent quark mass is quite a bit larger than the
justified.
bare quark mass parameter m . For example, close to
s
The physicalgapinthe spectrumalsodefines the crit-
the phase transition, we have m 83 MeV while the
s
ical hypercharge chemical potential for the CFL/gCFL ≈
constituentquarkmassisM 150MeV(seeTableIV).
s
transition (6): ≈
We havecheckedthatourcalculationsarequantitatively
consistent with the calculations presented in [61] in this
µcY =∆0 =25.00 MeV. (26) regard.
10
D. CFLK0 parity
X Y, A A†, Σ Σ†. (30a)
Applying a kaon rotation to the CFL state breaks the ↔ ↔ ↔
parityofthestate,andmixestheparityevenparameters The field content of the effective theories is thus:
µ, m, ∆ and κ with their parity odd counterparts µ5, H, η′: Two singlet fields corresponding to the U(1)
m , ∆ and κ . The full set of parameters and typical
5 5 5 phasesofAandV. ThefieldassociatedwithV isa
numerical values is presented in Appendix A.
scalar boson associated with the superfluid baryon
numbercondensation. WeshalldenotethisfieldH.
ThefieldassociatedwithAisapseudo-scalarboson
V. LOW-ENERGY EFFECTIVE THEORY
associatedwiththeaxialbaryonnumbersymmetry
and shall be identified with the η′ particle. As dis-
To describe the low-energy physics of these models,
cussedinSectionII,the axialsymmetry symmetry
we follow a well established procedure: identify the low-
is anomalously broken in QCD and the η′ is not
energydegreesoffreedomandtheirtransformationprop-
strictly masslessdue to instantoneffects, but these
erties,identifytheexpansionparameters(powercounting
are suppressedat highdensity. We ignorethese ef-
scheme), write down the most general action consistent
fects. Our NJL model thus contains no instanton
with the symmetries andpower counting,anddetermine
vertex and our low-energy theory will contain no
the arbitrary coefficients by matching to experiment or
Wess-Zumino-Witten terms [63, 64]. It would be
anothertheory. Inourcase,wewillmatchontothemean-
interesting to include both of these terms and re-
fieldapproximationoftheNJLmodel. Theresultinglow-
peat this calculation as these effects are likely not
energyeffective theoryhasbeenwellstudied[18,62]: we
small [48].
use this presentation to establish our conventions, and
to contrast the effective theory of QCD with that of the πa: Eight pseudo-scalar mesons πa corresponding to the
microscopic NJL model. broken axial flavour generators. As colour singlets
these remain as propagating degrees of freedom in
both QCD andNJL models. These have the quan-
A. Degrees of Freedom tumnumbersofpions,kaonsandtheetaandtrans-
form as an octet under the unbroken symmetry.
The coset space in the NJL model is isomorphic to
φa: Eight scalar bosons φa corresponding to the bro-
U(3) U(3). This can be fully parametrized with two
⊗ ken coloured generators. These are eaten by the
SU(3) matrices X andY and twophysicalphases A and
gauge bosons in QCD and are removed from the
V whichonecanphysicallyidentifywiththecondensates:
low-energy theory. This gives masses to eight of
the gauge bosons and decouples them from low-
√V†A[X] ǫ ǫ ψaαψbβ , (27a) energy physics. In the NJL model these bosons
cγ ∝ abc αβγh L L i still remain as low-energy degrees of freedom, but
√V†A†[Y]cγ ∝ǫabcǫαβγhψRaαψRbβi. (27b) decouple from the colour singlet physics when one
properly enforces colour neutrality.
These thus transform as follows:
Thereareadditionalfieldsandeffectsthatshouldbecon-
X F XC†, (28a) sideredaspartofacompletelow-energytheory,butthat
L
→ we neglect:
Y F YC†, (28b)
R
→ 1. The appropriately “rotated electromagnetic field”
A e2i(θR−θL)A, (28c)
associated with the unbroken U(1) symmetry re-
→ Q˜
V e2i(θR+θL)V. (28d) mains massless. Both the CFL and CFLK0 states
→
remain neutral with respect to this field, however,
Note that the condensation pattern X = Y = 1, A = and we do not explicitly include it in our formula-
V =1isunbrokenbytheresidualsymmetrywhereFL = tion.
F = C and also by the Z symmetries where θ ,θ =
R 2 L R
π. This is the reason for the extra factor of two in 2. The leptons are not strictly massless, but the elec-
±the phases. In QCD the degrees of freedom are similar, tron and muon are light enough to consider in the
but one must consider only colour singlet objects. Thus, low-energy physics. In particular, they contribute
the low-energy theory for QCD should include only the to the charge density in the presence of an electric
colour singlet combination chemical potential and at finite temperature. In
this paper, leptonic excitations play no role since
Σ=XY† F ΣF† (29) we consider only T =0 and both CFL andCFLK0
→ L R quarkmatteriselectricallyneutralforµ =0. The
e
and the colour singlet phases A and V. Note also that leptons play an implicit role in fixing µ such that
Q˜
thesehavethefollowingtransformationpropertiesunder µ =0 in both insulating phases.
e