Breaking rotational symmetry in two-flavor color superconductors

6 Pages·0.28 MB·English

by  H. Müther

-
Breaking rotational symmetry in two-flavor color superconductors
H. Mu¨ther and A. Sedrakian
Institut fu¨r Theoretische Physik, Universit¨at Tu¨bingen, D-72076 Tu¨bingen, Germany
The color superconductivity under flavor asymmetric conditions relevant to the compact star
phenomenology is studied within the Nambu-Jona-Lasinio model. We focus on the effect of the
deformation of the Fermi surfaces on the pairing properties and the energy budget of the super-
conductingstate. We find that at finiteflavor asymmetries the color superconductingBCS state is
unstable towards spontaneous quadrupole deformation of the Fermi surfaces of the d and u quarks
intoellipsoidal form. Theground stateofthephasewith deformed Fermisurfaces correspondstoa
superposition of prolate and oblate deformed Fermi ellipsoids of d and u quarks.
3
0 I. INTRODUCTION The loss of the condensation energy due to the sepa-
0
ration of the Fermi spheres of the u and d quarks, how-
2
ever, might favor a non-Fermi liquid occupation of the

n A possible outcome of the de-confinement phase tran- fermionicstates, wherethe condensationenergyis main-
a sition in hadronic matter, when it is compressed in the
tained at the cost of extra kinetic energy. One possi-
J centersofcompactstarstodensitiesseveraltimesthenu-
bility is the deformation of the Fermi spheres (DFS),
8 clear saturation density, is the formation of a two-flavor
to the lowest order, in the ellipsoidal form (hereafter -
2 (u and d) quark matter coexisting with leptons under
DFS phase) [6]. While deviations from the spherically
β stability and global charge neutrality [1]. It is be-
2 symmetric form of the Fermi surfaces cost extra kinetic
lieved that the strong interaction among quarks in the
v energy, the gain in the condensation energy due to an
7 attractive channel(s) pairs the u and d quarks of two increase in the phase-space overlap between the states
1 color (leaving one color unpaired) to form a Bardeen- that pair can stabilize the system. Note that we as-
3 Cooper-Schrieffer (BCS) type color superconductor (the
sume that the pairing interaction is isotropic in space;
2 so called 2SC phase) [2, 3]. In the asymptotic regime
such a deformation spontaneously breaks the rotational
1
ofhigh-densities rigorousperturbative resultscanbe ob-
2 O(3)symmetryoftheoriginalLagrangiandowntoO(2).
tained in the weak coupling regime [4]; at the intermedi-
0 Both LOFF and DFS superconducting phases break the
ate densities, relevant for compact stars, one has to rely
/ global space symmetries. Another possibility that main-
h upon effective-field theory models of QCD, such as the
tainsthesesymmetriesistherearrangementoftheFermi
p
Nambu-Jona-Lasinio (NJL) model.
- surface of the d-quarks [interior gap pairing (IGP)] [7].
p In compact stars the color superconductivity is likely TheIGPassumesd-quarkholeslocatedinastriparound
e
to occur at finite isospin asymmetry, where the u and theFermisurfaceoftheu-quarksareliftedtotheexcited
h
d quarks fill two distinct Fermi spheres. The separation states above the Fermi surface of the d quarks and the
:
v of their Fermi energies is of the order of the electron pairingoccursatthe shoresofthe u-quarkFermisea[7].
i
X chemical potential as required by the charge neutrality Arobustfeatureofthe2SCsuperconductorsistheap-
and the stability against weak processes d → u+e+ν¯ pearance of the crystalline color superconducting (CCS)
r
a and u + e → d + ν, where e, ν and ν¯ refer to elec- state (the analog of LOFF phase) in a wide range of
tron, electron neutrino and anti-neutrino. Large sep- chemical potential asymmetries [8, 9, 10, 11, 12, 13, 14,
arations of the Fermi energies (of the order of pairing 15, 16, 17, 18]. In the CCS state the Cooper-pairs carry
gap) and corresponding Fermi momenta implies an in- finite center-of-mass momentum, their order parameter
coherence of the phase space for the u and d quarks, varies periodically in space and, hence, the CCS sponta-
which will eventually destroy the 2SC superconductiv- neouslybreakstheglobalspacesymmetriesoftheoriginal
ity. If the occupation of the single particle states of the BCSstate[8,9,10,11,12,13,14,15,16,17,18]. Oneof
non-interacting system is preserved when the pairing in- the consequences of the broken space symmetries is the
teraction is switched on, i.e. the Fermi spheres are filled existence ofnew masslessGoldstone modes (phonons)in
withquasi-particlesinanisotropicandhomogenousman- the CCS phase [15, 16, 17, 18]; another consequence is
ner,thephase-spacecoherence(atfiniteseparations)can that, the glitches in compact stars (rotational anomalies
be maintained by pairing Cooper pairs with finite cen- seen in the timing data from pulsars) could originate in
terofmassmomentum[Larkin-Ovchinnikov-Fulde-Ferrel theCCSphase,ifthecompactstarsfeaturesuchaphase
(LOFF)phase][5]. Thisisinlinewiththebasicprescrip- [8].
tionofthe LandauFermiliquidtheory whichstatesthat Thispaperstudiesnumericallythecolorsuperconduct-
instronglyinteractingFermiliquidstheoccupationprob- ing DFS phase; it extends the previous non-relativistic
abilitiesofinteractingquasi-particlesremainthesameas analysis [6] to relativistic systems and interactions spe-
for the non-interacting system when the interactions are cifictothecolorsuperconductors. Themainresultofthis
switched on adiabatically. analysis is that the spatially homogenous 2SC supercon-
2
ductor is unstable towards formation of the DFS phase SU(2) flavor space, λ is the antisymmetric Gell-Mann
f A
with quadrupole deformed Fermi surfaces. Clearly, this matrix acting in the SU(3) color space. The coupling
c
observation does not identify the true ground state of a constant G1 stands for the four-fermion contact interac-
2SC superconductor. Studies of the interplay between tion. For the sake of simplicity we neglect the effects
different non-BCS phases sketched above are needed to of the mass gap in the quark-anti-quark channel below
answer this question. For applications to compact stars the critical line for the chiral phase transition on the
thesephasesneedtobestudiedunderβ equilibriumcon- Cooper pairing. The generalization of the Lagrangian
ditions (for example see [19, 20, 21]). (1),whichleadstocoupledgapequationsinthedi-quark
The paper is organized as follows. In Sec. 2, start- and Cooper channels, is straight forward. The common
ing from the NJL Lagrangian in the chirally symmetric Ansatz for the order parameter in the 2SC phase is
phase,we derivethe thermodynamic potentialofthe fla-
vor asymmetric 2SC phase at finite temperatures. Self- ∆∝hψT(x)Cγ5τ2λ2ψ(x)i, (2)
consistentequationsforthe gapandthe partialdensities
of the u and d quarks are obtained. The numerical solu- where λ2 is the secondcomponentof the Gell-Mann ma-
tions of these equation and the resulting phase diagram trix. The Ansatz for the order parameter [Eq. (2)] im-
ofthecolorsuperconductingDFSphaseareshowninSec. pliesthatthecolorSU(3) symmetryisreducedtoSU(2)
c c
3. Sec. 4 contains our conclusions. sinceonlytwoofthequarkcolorsareinvolvedinthepair-
ing while the third color remains unpaired. More com-
plicated Ansaetze would allow for a spin 1 pairing of the
II. THERMODYNAMICS OF COLOR
quarks of the remaining color [23, 24].
SUPERCONDUCTING DFS PHASE
As is well known, the gap equation and the partial
densities of the up and down quarks can be found from
Ourtreatmentis basedupon the Nambu-Jona-Lasinio the fixed points of the thermodynamic potential density
(NJL) with two flavors (Nf =2) and three colors (Nc = Ω:
3) [22]. We shall assume from the outset that the color
superconductingphaseemergesinthechirallysymmetric ∂Ω ∂Ω
=0, − =ρ ; (3)
phase. TheLagrangiandensityofthemodelisthengiven ∂∆ ∂µ f
f
by
the flavor index f = u,d refers to up (u) and down (d)
Leff = ψ¯(x)(iγµ∂µ)ψ(x) quarks. In the superconducting state the self-consistent
+ G1(ψTCγ5τ2λAψ(x))†(ψTCγ5τ2λAψ(x)), solution of equations (3) corresponds to a minimum of
the thermodynamic potential of the system. For the La-
(1)
grangiandensitydefinedbyEq. (1)andthepairingchan-
where C = iγ2γ0 is the charge conjugation operator, τ2 nelAnsatzEq. (2),thefinitetemperaturethermodynam-
isthesecondcomponentofthePaulimatrixactinginthe ical potential Ω per unit volume is
1 d3p 1 S−1(iω ,p~) S−1(iω ,~p) ∆2
Ω(βµ)=− Tr ln β 11 n 12 n + , (4)
β Xωn Z (2π)32 (cid:20) (cid:18)S2−11(iωn,p~) S2−21(iωn,~p)(cid:19)(cid:21) 4G1
where β is the inverse temperature. The matrix structure of the inverse Matsubara propagators S−1(iω ,~p), (l,k =
kl n
1,2)reflectstheNambu-Gor’kovextensionoftheparticle-holespacetoaccountforthepaircorrelations. Theelements
of the Nambu-Gor’kov matrix are 2×2 matrices defined as diag[S−1] = (/p +µ γ0,/p+µ γ0), and diag[S−1] =
11 u d 22
(/p−µuγ0,/p−µdγ0); diag[S1−21] = (∆γ5τ2λ2,∆γ5τ2λ2) and diag[S2−11] = (−∆∗γ5τ2λ2,−∆∗γ5τ2λ2) with off-diagonal
elements zero. Upon carrying out the traces in the spin, flavor and color spaces and the fermionic Matsubara
summation over the frequencies ω one finds
n
d3p 1 2 ∆2
Ω=−2 2p+ log 1+e−βξij +E + log 1+e−βsijEij + , (5)
Z (2π)3( ij (cid:20)β ij β (cid:21)) 4G1
X (cid:0) (cid:1) (cid:0) (cid:1)
where the indeces i,j =(+,−) sum over the four branches of the paired and unpaired quasiparticle spectra defined,
respectively, as ξ =(p±µ)±δµ and E = (p±µ)2+|∆|2±δµ, where δµ=(µ −µ )/2 andµ=(µ +µ )/2
±± ±± u d u d
with µu and µd being the chemicalpotentials of the up anddownquarks;s+j =1 and s−j =sgn(p−µ). Expressions
p
analogous to (5) were derived in refs. [25, 26, 28, 29] for finite/zero temperature and flavor symmetric/asymmetric
3
cases. The variations of the thermodynamic potential (5) provide the gap equation
d3p ∆ βE++ βE+−
∆ = 8G1Z (2π)3(E+−+E++ (cid:20)tanh(cid:18) 2 (cid:19)+tanh(cid:18) 2 (cid:19)(cid:21)
∆ βE−+ βE−−
+ tanh +tanh , (6)
E−++E−− (cid:20) (cid:18) 2 (cid:19) (cid:18) 2 (cid:19)(cid:21))
and the partial densities of the up/down quarks
d3p ξ−−+ξ−+ βE−− ξ+−+ξ++ βE+−
ρu/d = Z (2π)3(2f(ξ−∓)−2f(ξ+±)∓(cid:20)1± E−−+E−+(cid:21)tanh(cid:18) 2 (cid:19)∓(cid:20)1∓ E+−+E++(cid:21)tanh(cid:18) 2 (cid:19)
ξ−−+ξ−+ βE−+ ξ+−+ξ++ βE++
± 1∓ tanh ± 1± tanh , (7)
(cid:20) E−−+E−+(cid:21) (cid:18) 2 (cid:19) (cid:20) E+−+E++(cid:21) (cid:18) 2 (cid:19))
wheref(ξ )aretheFermidistributionfunctionsandthe in the simple form
ij
upper/lower sign corresponds to the u/d-quarks. The
µ =µ¯ 1+(ε ±ε )(Cosθ)2 , (9)
changesinthethermodynamicpotentialduetothephase f f S A
transition to the superconducting phase at constant β
where the low(cid:2)er/upper sign corres(cid:3)ponds to up/down
and µ are equivalent to the changes in the free energy
at confstant β and ρ when the potentials are expressed quarks; εS/A = (3/2)[µ2d/µ¯0d ± µ2u/µ¯0u] and µ¯f =
in terms of appropriafte thermodynamicalvariables. The µf0−µf2/2.The parameters εS and εA describe the de-
formationsoftheFermisurfaces;thesymmetricpart(ε )
free energy F is related to the thermodynamic poten- S
isthemeasureoftheconformalexpansion/contractionof
tial Ω by the relation F = Ω + µ ρ + µ ρ . Below,
u u d d theFermisurfaces;theantisymmetricpart(ε )describes
we choose to minimize the free energy of the system A
the relative deformations of the Fermi spheres of u and
at constant temperature and density of the matter as
dquarks. Note thatthe expansion(8) need notconserve
a function of the flavor asymmetry parameter defined as
the volume of a Fermi sphere; the isotropic expansion
α ≡ (ρ −ρ )/(ρ +ρ ). Thus, fixing the net density
d u d u parameters are self-consistently found form the normal-
of the quark matter and the flavor asymmetry, fixes the
ization to the same matter density as in the case for the
chemicalpotentialsµ bymeansofEqs. (6)and(7). The
f undeformed state.
chemical potentials are isotropic if the Fermi surfaces of
thespeciesarespherical. Toaccommodatethepossibility
of their deformation we expand the chemical potentials
III. THE PHASE DIAGRAM
in Legendre polynomials with respect to an (arbitrary)
axis of spontaneous symmetry breaking of the spherical
As the NJL model is non-renormalizable,the momen-
symmetry of the Fermi surfaces:
tumintegralsinthe gapequation(6)needto be regular-
∞ izedbyacut-off;weemployathree-dimensionalmomen-
µ = µ P(Cosθ). (8) tum space cut-off |p| < Λ. The phenomenological value
f f,l l
l=0 of the coupling constant G1 in the hqqi Cooper channel
X
is related to the coupling constant in the hqq¯i di-quark
The l = 0 terms are constants which merely renormal- channel by the relation G1 = Nc/(2Nc−2)G; the latter
ize the chemical potentials from their values for the case coupling constant and the cut-off are fixed by adjust-
of vanishing pairing interactions. Since the interactions ing the model to the vacuum properties of the system
are translationally invariant, the l = 1 terms can not [25, 26, 27]. We employ the parameter set G1 =3.10861
contribute, since they manifestly break the translational GeV−2 andG1Λ2 =1.31fromRef. [26]. Forthis param-
symmetry. (The breaking of translationalsymmetry can eter set the value of the gap is 40 MeV at the baryonic
be caused by the kinetic energy contribution, as is the density ρ =0.31fm−3 which correspondsto the chemi-
B
case for the LOFF/CCS phases.) We shall keep below calpotential value µ=320MeV in the flavorsymmetric
the leading order l=2 terms which break the rotational stateatT =0. Figure1summarizesthemainfeaturesof
symmetryofthesystembydeformingtheFermisurfaces the DFSphase inthe physicallyrelevantregimeofflavor
into an ellipsoidal form. The corresponding expansion asymmetries 0.1≤α≤0.3 likely to occur in the charge-
coefficients are treated below as variational parameters neutralmatterunderβ equilibrium. Thecolorsupercon-
to minimize the free energy of the color superconductor. ductinggap(leftpanel)andthefreeenergydifferencebe-
With the leading order in the deformation terms kept, tweenthe superconducting state and normalstate (right
thechemicalpotentialsoftheuanddquarkscanbecast panel)areshownasafunctionofdeformationparameter
4
ε for several flavor asymmetries at the baryonic den-
A
sity ρ = 0.31 fm−3 and temperature T = 2 MeV. The
B
conformal deformation has been constrained to ε = 0.
S
The gap is normalized to its value in the flavor symmet-
ric and undeformed state ∆00 ≡ ∆(α = 0,εA = 0) = 40
MeV.Thefree-energydifferenceδF =F −F islikewise
S N
normalized to its value in the flavor symmetric and un-
deformed state δF(α=0,ε =0)=37 MeV fm−3. The
A
β equilibrated quark matter requires an excess of the d
over u quarks, therefore the range of the flavorasymme-
try is restricted to the positive values. The deformation
parameterε assumesbothpositiveandnegativevalues.
A
ThemainfeaturesseeninFig. 1are: (i)forafixedα6=0
and ε > 0, the gap is larger in the DFS state than in
A
theordinaryBCSstate(ε =0);(ii)theminimumofthe FIG. 2: The deformed Fermi surfaces of the d and u quarks
A
freeenergycorrespondstotheDFSstatewithε ≃0.25, fortheflavorasymmetryα=0.3anddeformation parameter
A
and its position weakly depends on the value of α. εA=0.25. TherightpartofthisfigurereferstotheFermisur-
face of the d-quarks, exhibiting a prolate deformation, while
theoblate shapeontheleft corresponds totheFermisurface
of theu-quarks.
0
The behavior of the gap as a function of α and
1 ε is best understood in terms of the symmetric and
A
anti-symmetric combinations of the quasiparticle spec-
tra ES = (E−+ +E−−)/2 and EA = (E−+ −E−−)/2.
-0.5 Since we assume that εS = 0 the symmetric combina-
∆ /∆00 F/F00 t[∼ionOo(fεthδeµ)s]p.eIcntrathies wcaesaekEly aff=ec0teodnebyretchoevedresfotrhmeaBtiCoSn
0.5 A A
result with perfectly overlappingFermi surfaces. The ef-
fect of finite E atε =0 is to induce a phase-spacede-
A A
coherenceinthekernelofthegapequation;thisblocking
-1
effect reduces the magnitude of the gap.
Asthechemicalpotentialshiftδµandε contributeto
A
0
-0.2 0 0.2 -0.2 0 0.2 E with different signs, switching on finite ε > 0 acts
ε ε A A
Α Α
to reduce E and the magnitude of the gap increases
A
due to the restoration of the phase-space overlap. Once
the condition ε >δµ is satisfied further increase in the
FIG. 1: The color superconducting gap (left panel) and the A
freeenergy(rightpanel)asafunctionofrelativedeformation deformation acts to increase the de-coherence and, thus,
parameter εA for the values of the flavor asymmetry α = decrease the magnitude of the gap.
0 (solid lines), 0.1 (dashed lines), 0.2 (short dashed lines) Fornegativevaluesofε thedeformationandtheshift
A
and 0.3 (dashed dotted lines) at density ρB =0.31 fm−3 and inthechemicalpotentialscontributetoE withthesame
A
temperature T =2 MeV. sign,thereforethedeformationactstoincreasethephase-
space de-coherence which reduces the magnitude of the
Positive values of ε imply a prolate deformation of gap. ThesefeaturesareseeninFig. 3,whereweshowthe
A
the Fermi surface of the d-quarks and an oblate defor- pairing gap as a function of the flavor asymmetry α and
mation for the Fermi surface of the u-quark. Figure 2 the relative deformation of the Fermi surfaces ε for the
A
illustrates the Fermi surfaces for the parameter choice same density andtemperature as in Fig. 1. [We assume,
α = 0.3 and ε = 0.25, which corresponds to the de- as for Fig. 1, that the conformal expansion/contraction
A
formation at which the minimum in the free energy is of the Fermi surfaces is absent, ε =0].
S
obtained. Figure 4 shows the difference between the free ener-
As can be seen from Fig. 2, the deformation of the gies of the superconducting and normal states δF nor-
Fermi surfaces leads to an area in which the Fermi sur- malized to its value in the flavor-symmetric BCS state
faces of u- and d-quarks touch each other, which means δF00 = δF(α = 0,εA = 0). Numerically, the con-
that for this part of the phase space there is no blocking tribution from the entropy difference between the nor-
effectreducingthepairingcorrelation. Asaconsequence, mal and superconducting phases is negligible. Small
the superconducting gapis largerthanforthe configura- ε > 0,α 6= 0 perturbations from the flavor symmet-
A
tion with spherical Fermi surfaces leading also to more ric/undeformedstateincreaseδF signalinganinstability
attractive free energy. of the BCS state towardsdeformations of the Fermi sur-
5
gap as a function of the parameters α and ε , which is
A
due to the dominant contributionfromthe condensation
energy (cf. Fig. 1).
IV. CONCLUDING REMARKS
∆/∆
00
1.2 Color superconducting 2SC phase becomes unstable
1
0.8
0.6 towards formation of a superconducting phase with de-
0.4
0.2 formed Fermi surfaces of up and down quarks for finite
00
separations of their Fermi levels. Small asymmetries in
0.1
the populations of the of the up/down quarks are suffi-
0.2 cient to achieve the bifurcation point where the form of
0.4
0.3 0.3 the Fermi surfaces changes from the spherical to the el-
α 0.2
0.4 0.1 lipsoidal form. Although the departures from the spher-
0
0.5 -0.2 -0.1 δ ε ical form of the Fermi surfaces costs extra kinetic en-
ergy, there is net energy gain because the deformation
increasesthephasespacecoherencebetweenthefermions
forming Cooper pairs and, therefore, the gain in the po-
FIG.3: Thecolorsuperconductinggapasafunctionofflavor tential energy. As mentioned in the introduction, the
asymmetry α and the relative deformation parameter εA at presentstudy does notestablishthe true groundstate of
density ρB = 0.31 fm−3 and temperature T = 2 MeV. The the 2SC superconductor under flavor asymmetry. This
pairing gap is in units ∆(α=0,εA=0)=40 MeV. can be done by a careful comparison of the condensa-
tion energies of the DFS and CCS phases. With the
form of the lattice of the CCS phase established in the
Ginzburg-Landauregime[13]itbecomesfeasibletocarry
out a combined study of the DFS and CCS phases [30] .
For applications to the compact stars, our treatment
needs an extention which will include the β equilibrium
andchargeneutralityamongthequarksandleptons. The
F / F 00
0 onset of the DFS phase (or its combinations with non-
-0.2
-0.4 BCSsuperconductingstates)willaffectthe propertiesof
-0.6
-0-.18 the compact stars in a number of ways. One straightfor-
0.4 ward implication is the modification of the specific heat
0.3
0.2 0.5 of the superconducting phase which affects the thermal
0.1 0.4 cooling of compact stars: we anticipate that the spe-
0 0.3 cificheatwillbesuppressedlinearlyinthe DFSphaseas
δ ε -0.1 0.2 α comparedto the exponentialsuppressioninthe ordinary
-0.2 0.1
BCS-state. The spontaneous breaking of the rotational
0
symmetry implies emergence of massless bosonic modes
(Goldstone’s theorem). The deformation of the Fermi
surfaces opens new channels for neutrino radiation via
FIG.4: Thedifferencebetweenthefreeenergiesofthesuper-
the bremsstrahlung process of the type u → u+ν +ν¯
conducting and normal phases as a function of flavor asym-
metryαandtherelativedeformationparameterεA. Thefree and d →d+ν+ν¯ since the phase-space probability for
energyscaleisnormalizedtoitsvalueδF(α=0,εA=0)=37 a transitionof a quasi-particlefrom one point onthe de-
MeV fm−3. formed Fermi surface to another (say, from the equator
to the pole) is non-zero.
faces.
For flavor asymmetries α > 0.3 and positive values
of the deformation parameter the superconducting state
can exists only in the DFS phase. For negative values of
ε there is no energy gain related to the deformation of WethankMichaelBuballaandMicaelaOertelforhelp-
A
the Fermi surfaces (except a marginal effect in the limit fulcommentsonthefirstversionofthispaper. Thiswork
α → 0 and ε → 0). Quite generally the shape of the has been supported by the Sonderforschungsbereich 382
A
freeenergysurfacereflectsthefunctionalbehaviorofthe of DFG.
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