Quark matter and color superconductivity · Quark matter and color superconductivity Massimo Mannarelli INFN-LNGS [email protected] Review of Modern Physics 86, 509 (2014) WG2

Quark matter and

color superconductivity

Massimo Mannarelli INFN-LNGS

[email protected]

Review of Modern Physics 86, 509 (2014) WG2 meetingCatania, November 7 - 9, 2017

mailto:[email protected]

mailto:[email protected]

Outline• Background

• Competing condensates

• Color supercondusctors

• Crystalline Color Superconductors

• Conclusions

BACKGROUND

3

What is matter made of?

JP Particles (mass in MeV)0� ⇥0 (135), ⇥± (140), � (547), �⌅ (958), K± (494), K0, K0 (498)1� ⇤±,0 (771), ⌅ (783), K⇥±, K⇥0, K⇥0 (892), ⇧ (1020)12

+p (938), n (939), ⇥ (1116), ⌅±,0 (1193), ⇤0,� (1318)

32

+ �++, �+, ��, �0 (1232), ⌅⇥±,0 (1385), ⇤⇥±,0 (1318), ⌃� (1672)

mesons

baryons

{{

Hadrons

baryons mesons

Feel the strong interaction

4

LeptonsDo not feel the strong interaction

Q Particles (mass in MeV)�1 e (0.5) µ (105) ⇤ (1777)

0 ⇥e ⇥µ ⇥⇥quarks and gluons

Quantum chromodynamics

↵,� = 1, 2, 3 ↵,iquark field:

gluon gauge fields: Aa

LQCD = (i�µDµ + µ�0 �M) � 1

4Fµ⌫a F a

µ⌫

QCDNon-Abelian gauge theory AND non-perturbative at energy scales below

Kaczmarek and ZantowPhysical Review D 71(11):114510

⇤QCD ⇠ 300MeV

F aµ⌫ = @µA

a⌫ � @⌫A

aµ + gfabcA

bµA

c⌫

i, j = 1, 2, . . . , Nf

color indexflavor index

a = 1, . . . , 8 adjoint color indexField content }

https://www.researchgate.net/profile/Olaf_Kaczmarek


https://www.researchgate.net/journal/1550-7998_Physical_Review_D





↵,� = 1, 2, 3 ↵,iquark field:


LQCD = (i�µDµ + µ�0 �M) � 1

4Fµ⌫a F a

µ⌫



⇤QCD ⇠ 300MeV

confinement

F aµ⌫ = @µA

a⌫ � @⌫A

aµ + gfabcA

bµA

c⌫

i, j = 1, 2, . . . , Nf










↵,� = 1, 2, 3 ↵,iquark field:


LQCD = (i�µDµ + µ�0 �M) � 1

4Fµ⌫a F a

µ⌫



⇤QCD ⇠ 300MeV

confinement

asymptotic freedom

F aµ⌫ = @µA

a⌫ � @⌫A

aµ + gfabcA

bµA

c⌫

i, j = 1, 2, . . . , Nf










↵,� = 1, 2, 3 ↵,iquark field:


LQCD = (i�µDµ + µ�0 �M) � 1

4Fµ⌫a F a

µ⌫



⇤QCD ⇠ 300MeV

confinement

asymptotic freedom

F aµ⌫ = @µA

a⌫ � @⌫A

aµ + gfabcA

bµA

c⌫

i, j = 1, 2, . . . , Nf



Despite of this, the majority of studies byNambu-Jona Lasinio (NJL) contact models







6

For three flavor massless quarks the QCD Lagrangian has the symmetry group

gauge groupglobal chiral symmetry

global baryonic number

SU(3)c ⇥ SU(3)L ⇥ SU(3)R| {z }� [U(1)e.m.]

⇥U(1)B

Symmetries of QCD

6

For three flavor massless quarks the QCD Lagrangian has the symmetry group

gauge groupglobal chiral symmetry

global baryonic number

SU(3)c ⇥ SU(3)L ⇥ SU(3)R| {z }� [U(1)e.m.]

⇥U(1)B

Symmetries of QCD

h C�5 i

h i

h �2�5 i

Chiral condensate: Locks chiral rotations

Diquark condensate: Breaks the gauge group and may lock chiral rotations

The ground state may have a lower symmetry because quarks form condensates

Pion condensate: Locks chiral rotations and breaks U(1)e.m.

7

Quark matter phase diagram

hadronsgas

Compact Stars

Tc

T

µI

m⇡

CFL

quark-gluonplasma

coloursuperconductors

pion condensedphase

µB

7


hadronsgas

Compact Stars

Tc

T

µI

m⇡

CFL

quark-gluonplasma


h i 6= 0

h i 6= 0

h C�5 i

h �2�5 i = B sin↵

h ¯ i = B cos↵

pion condensedphase

µB

7


hadronsgas

Compact Stars

Tc

T

µI

m⇡

CFL

quark-gluonplasma


h i 6= 0

h i 6= 0

h C�5 i

h �2�5 i = B sin↵

h ¯ i = B cos↵

pion condensedphase

µB

SOME METHODS

7


pQCD

hadronsgas

Compact Stars

Tc

T

µI

m⇡

CFL

quark-gluonplasma


h i 6= 0

h i 6= 0

h C�5 i

h �2�5 i = B sin↵

h ¯ i = B cos↵

pion condensedphase

µB

SOME METHODS

7


pQCD

hadronsgas

Compact Stars

Tc

T

µI

LQCD

m⇡

CFL

quark-gluonplasma


h i 6= 0

h i 6= 0

h C�5 i

h �2�5 i = B sin↵

h ¯ i = B cos↵

pion condensedphase

µB

SOME METHODS

7


pQCD𝟀PT

hadronsgas

Compact Stars

Tc

T

µI

LQCD

m⇡

CFL

quark-gluonplasma


h i 6= 0

h i 6= 0

h C�5 i

h �2�5 i = B sin↵

h ¯ i = B cos↵

pion condensedphase

µB

SOME METHODS

7


pQCD𝟀PT

hadronsgas

Compact Stars

Tc

T

µI

LQCD

m⇡

CFL

quark-gluonplasma


h i 6= 0

h i 6= 0

h C�5 i

h �2�5 i = B sin↵

h ¯ i = B cos↵

pion condensedphase

µB

SOME METHODS

NJL-like

7


pQCD𝟀PT

hadronsgas

Compact Stars

Tc

T

µI

LQCD

m⇡

CFL

quark-gluonplasma


h i 6= 0

h i 6= 0

h C�5 i

h �2�5 i = B sin↵

h ¯ i = B cos↵

pion condensedphase

µB

SOME METHODS

NJL-like

COMPETING CONDENSATES

8

Melting the chiral condensate The chiral condensate become disvafored with increasing density.

It can melt in different ways:1) By a second order phase transition2) By a first order phase transition3) Passing through an inhomogenoeus phase

M(z) = � e2iQz

Buballa and CarignanoProg.Part.Nucl.Phys. 81 (2015) 39-96

NJL-model analysis

Chiral density wave (CDW)

color superconductor

homogeneous�SB

inhomogeneous�SB

Fight of condensates Different kind of pairings

quark-antiquark quark-hole(exciton-like)

quark-hole(CDW)

quark-quarkcolor superconductor

Not obvious which of these is energetically favored

Unfortunately it seems that depending on the model interaction any of them can be favored

T. Kojo, Y. Hidaka, L. McLerran, R. D. Pisarski, Nucl. Phys. A 843 (2010) 37

11

Inhomogeneous chiral condensate?Where does pairing occur? Why 1D modulations seem to always win?

2D projection of the Fermi spheres for . , the light region: the energy cost for exciting quasiparticle is small

µ = 335 MeV

� = 0, Q = 0 � = 0, Q = 241 MeV � = 44 MeV, Q = 241 MeV

Pairing in inhomogeneous chiral condensate is a Fermi surface phenomenon.Since the Fermi spheres are strongly modified, as far as Q is large, only 1D modulation can happen

S.Carignano, F.Anzuini, O. Benhar, MM in preparation

COLOR SUPERCONDUCTORS

The idea with a cartoonquark

point-like

baryon

~1 fm

diquark

~few fm


point-like

baryon

~1 fm

diquark

~few fm

Very high density

Liquid of deconfined quarks with correlated diquarks

} . 1fm

Focus pointEven if quark matter is deconfined, it is always self-bound by gauge fields: no need of a trap


point-like

baryon

~1 fm

diquark

~few fm

Asymptotic density: attractive perturbative interaction + bag

attractive channel

3⇥ 3 = 3A + 6S

�p

pp�

�p�

p, p0 ' pF

and

Very high density

Liquid of deconfined quarks with correlated diquarks

} . 1fm

Focus pointEven if quark matter is deconfined, it is always self-bound by gauge fields: no need of a trap

General color superconducting condensate:

Obs. the condensate has been written in a gauge variant way

General expression of the condensate

↵,� = 1, 2, 3 color indices

i, j = 1, 2, 3 flavor indices“gap parameters” for quarks whoseflavor and color is not I

14

h ↵iC�5 �ji /3X

I=1

�I"↵�I✏ijI

The condensate has a color charge, has a “flavor” charge, has a baryonic charge The corresponding symmetries can be broken or mixed. A zoo of color superconducting phases.

�I ⇠ 10� 100MeV

2 Flavor color superconductor (2SC)

�3 > 0 , �2 = �1 = 0

Suppose the strange quark mass is “large”: Strange quarks decouple

15

s u

s

s

u

u d

d

dms > µu,d


�3 > 0 , �2 = �1 = 0


15

s u

s

s

u

u d

d

dms > µu,d

Symmetry breaking

SU(3)c ⇥ SU(2)L ⇥ SU(2)R ⇥ U(1)B ⇥ U(1)S ! SU(2)c ⇥ SU(2)L ⇥ SU(2)R ⇥ U(1)B ⇥ U(1)S{� U(1)Q

{� U(1)Q


�3 > 0 , �2 = �1 = 0


• Higgs mechanism, 5 gauge bosons acquire a mass: COLOR SUPERCONDUCTOR• No chiral symmetry breaking• No global symmetry is broken: NOT A SUPERFLUID• The photon is rotated (mixed with gauge and global symmetries).

The system is an “electrical” conductor15

s u

s

s

u

u d

d

dms > µu,d

Symmetry breaking

SU(3)c ⇥ SU(2)L ⇥ SU(2)R ⇥ U(1)B ⇥ U(1)S ! SU(2)c ⇥ SU(2)L ⇥ SU(2)R ⇥ U(1)B ⇥ U(1)S{� U(1)Q

{� U(1)Q

Color Flavor Locked phaseCondensate Pairing of quarks of all flavors and colors

Alford, Rajagopal, Wilczek Nucl.Phys. B537 (1999) 443

Symmetry breaking

SU(3)c ⇥ SU(3)L ⇥ SU(3)R ⇥ U(1)B ! SU(3)c+L+R ⇥ Z2

� U(1)Q

{ { {

� U(1)Q

�1 = �2 = �3 = �CFL

s u

s

s

u

u d

d

d

Color Flavor Locked phaseCondensate

F C C F

< R R>< L L> SU(3)L rotation SU(3)c rotation SU(3)R rotation

Pairing of quarks of all flavors and colorsAlford, Rajagopal, Wilczek Nucl.Phys. B537 (1999) 443

Symmetry breaking


� U(1)Q

{ { {

� U(1)Q

�1 = �2 = �3 = �CFL

s u

s

s

u

u d

d

d

Color Flavor Locked phaseCondensate

• Higgs mechanism, 8 gauge bosons acquire a mass: COLOR SUPERCONDUCTOR

• χSB: 8 (pseudo) Nambu-Goldstone bosons (NGBs)

• U(1)B breaking, 1 NGB: SUPERFLUID

• “Rotated” electromagnetism, mixing angle (analog of the Weinberg angle) cos ✓ =

gpg

2 + 4e2/3

F C C F

< R R>< L L> SU(3)L rotation SU(3)c rotation SU(3)R rotation

Pairing of quarks of all flavors and colorsAlford, Rajagopal, Wilczek Nucl.Phys. B537 (1999) 443

Symmetry breaking


� U(1)Q

{ { {

� U(1)Q

�1 = �2 = �3 = �CFL

s u

s

s

u

u d

d

d

Gap equation

� / g2Z

d✓ d⇠�p

⇠2 +�2� / exp(�const/g

2)

=

Contact interaction (NJL-like model)quark

p · k = pk cos ✓

p

k

p p p

Gap equation

� / g2Z

d✓ d⇠�p


2)

=


gluon

p · k = pk cos ✓

p

k

p p p

Gap equation

� / g2Z

d✓ d⇠�p


2)

=


gluon

p · k = pk cos ✓

p

k

p p p

� / exp(�const/g)� / g2Z

d✓ d⇠�p

⇠2 +�2

µ2

✓µ2 + �2

Barrois, B. C., 1979, Ph.D. thesis,Son, D. T., 1999, Phys. Rev. D59, 094019.

=

QCD

unscreened “magnetic” gluon

p p

k

pp

� is the scale of Landau damping

collinear divergency

Focus points

The quark pairing is driven by the color interaction

As a consequence of the interaction • The SU(3)c gauge group is broken • The gap parameter is large

In a system of electrons the attractive interaction is mediated by some “external field” say phonons

As a consequence of the interaction • The U(1)em gauge group is broken• The gap parameter is “small”

Color Superconductors

Standard BCS superconductors

CRYSTALLINECOLOR

SUPERCONDUCTORS

Fixed mismatch, increasing coupling

d

u

s

PF

red green blue

UNPAIRED

no color interaction “strong” color interaction

20


d

u

s

PF

red green blue

UNPAIRED

PF

d

u

s

red green blue

2SC


20


d

u

s

PF

red green blue

UNPAIRED

PF

d

u

s

red green blue

2SC

d

u

s

PF

red green blue

CFL


20


d

u

s

PF

red green blue

UNPAIRED

PF

d

u

s

red green blue

2SC

d

u

s

PF

red green blue

CFL

quasiparticles with mixed color and flavor


20


d

u

s

PF

red green blue

UNPAIRED

PF

d

u

s

red green blue

2SC

d

u

s

PF

red green blue

CFL

Whenever there is BCS pairing, the Fermi surfaces have to match.



20


d

u

s

PF

red green blue

UNPAIRED

PF

d

u

s

red green blue

2SC

d

u

s

PF

red green blue

CFL

Whenever there is BCS pairing, the Fermi surfaces have to match.

If the mismatch is too large, pairing cannot occur. The largest chemical potential mismatch which allows paring is named the Chandrasekhar-Clogston limit



20

Quark matter in compact stars

For simplicity, no strong interaction

electric neutrality

weak interactions

2

3Nu � 1

3Nd �

1

3Ns �Ne = 0

µu = µd � µe

µd = µsu+ d $ u+ su ! s+ e+ ⌫eu ! d+ e+ ⌫e

Fermi momenta kFu = µu kFd = µd kFs =p

µ2s �m2

s

sizable strange quark mass +

weak equilibrium+

electric neutrality

mismatch of Fermi momenta



electric neutrality

weak interactions

2

3Nu � 1

3Nd �

1

3Ns �Ne = 0

µu = µd � µe



µ2s �m2

s

ud

s

Fermi spheres

kFs

kFu

kFd


weak equilibrium+

electric neutrality




electric neutrality

weak interactions

2

3Nu � 1

3Nd �

1

3Ns �Ne = 0

µu = µd � µe



µ2s �m2

s

The system can never phase separate. The color and the electromagnetic interactions forbid it.

FOCUS POINT

ud

s

Fermi spheres

kFs

kFu

kFd


weak equilibrium+

electric neutrality


FFLO-phaseTwo flavor quark matter

For the superconducting FFLO phase is favored with Cooper pairs of non-zero total momentum

In weak coupling

�µ1 ' �0p2 �µ2 ' 0.75�0

�µ1 < �µ < �µ2

• In coordinate space • In momentum space

u

d

6/

P

e P d

u2q

2bµ

2q

6

2eq

q

q

< (pu) (pd) > ⇠ � �(pu + pd � 2q) < (x) (x) > ⇠ � ei2q·x

Crystalline structures: CCSC phase


• Complicated structures can be obtained combining more plane waves









• “no-overlap” condition between ribbons



• “no-overlap” condition between ribbons

Rajagopal and Sharma Phys.Rev. D74 (2006) 094019

X

Y

Z

X

Y

Z

CX 2cube45z

• Three flavors

< ↵iC�5 �j >⇠X

I=2,3

�I

X

qmI 2{qm

I }

e2iqmI ·r✏I↵�✏Iij

qmI = q nm

I

simplifications

0 50 100 150 200 250

M2

S/µ [MeV]

-50

-40

-30

-20

-10

0

En

erg

y D

iffe

ren

ce [1

06 M

eV4]

unpaired 2PW SqX 2Tet

gCFL

CFL

Cub

eX

2Cube45z

2Octa90xy

Free energy estimate

Rajagopal and Sharma Phys.Rev. D74 (2006) 094019

NJL + GL expansion!!

Fermionic dispersion lawsQuark quasiparticles have an anisotropic gapless dispersion law:

Velocity of fermions in two different structures

E = c(✓,�) ⇠

BCC FCC

direction dependent velocity

MM et al. Review of Modern Physics 86, 509 (2014)

Displacement of the crystal structure

myy

myx

myzxy

z

stress tensor

deformation tensor

�ij = Kukk�ij + 2⌫

✓uij �

1

3ukk�ij

◆

ui = x

0i � xidisplacement vector

uij =ui,j + uj,i

2

shear moduluscompressibility

Elastic deformation of a stressed crystal (Landau Lifsits, vol. 7)


myy

myx

myzxy

z

stress tensor

deformation tensor


✓uij �

1

3ukk�ij

◆

ui = x


uij =ui,j + uj,i

2



• Crystalline structure given by the spatial modulation of the gap parameter

• It is this pattern of modulation that is rigid (and can oscillate)


myy

myx

myzxy

z

stress tensor

deformation tensor


✓uij �

1

3ukk�ij

◆

ui = x


uij =ui,j + uj,i

2



• Crystalline structure given by the spatial modulation of the gap parameter

• It is this pattern of modulation that is rigid (and can oscillate)

MM, Rajagopal and Sharma Phys.Rev. D76 (2007) 074026

20 to 1000 times more rigid than the crust of neutron stars⌫CCSC ⇠ 2.47MeV/fm3

Rotating and/or oscillating strange/hybrid stars with a CCSC crust/core, which are somehow deformed to nonaxisymmetric configurations, can efficiently emit gravitational waves

Lin, Phys.Rev. D76 (2007) 081502, Phys.Rev. D88 (2013)

Haskell et al. Phys.Rev. Lett.99. 231101 (2007)

Knippel et al. Phys.Rev. D79 (2009) 083007

Rupak and Jaikumar Phys.Rev. C88 (2013) 065801

........

Gravitational waves

http://dx.doi.org/10.1103/PhysRevLett.99.231101


http://inspirehep.net/author/profile/Knippel%2C%20Bettina?recid=812007&ln=it

http://inspirehep.net/author/profile/Knippel%2C%20Bettina?recid=812007&ln=it

Gravitational waves from “mountains”

If the star has a non-axial symmetric deformation (mountain) it can emit gravitational waves

ellipticity

z

x

yGW amplitude

✏ =Ixx

� Iyy

Izz

h =16⇡2G

c4✏Izz⌫2

r



ellipticity

z

x

y

• The deformation can arise in the crust or in the core

• Deformation depends on the breaking strain and the shear stress

GW amplitude

✏ =Ixx

� Iyy

Izz

h =16⇡2G

c4✏Izz⌫2

r



ellipticity

z

x

y

• The deformation can arise in the crust or in the core

• Deformation depends on the breaking strain and the shear stress

GW amplitude

✏ =Ixx

� Iyy

Izz

h =16⇡2G

c4✏Izz⌫2

r

• Large shear modulus

• Large breaking strain To have a “large” GW amplitude

Gravitational waves from mountains

3

0 200 400 600 800

µ (MeV)

0

5

10

15

20

25

30

Δ (

MeV

)

σmax

=10-3

σmax

=10-2

FIG. 1: The areas above the solid lines are excluded bythe direct upper limit for the Crab pulsar obtained from theS3/S4 runs for the cases σmax = 10−3 and 10−2. The dotted(dashed) line is the constraint set by the spin-down limit forσmax = 10−3 (10−2). The rectangular box is the theoreticallyallowed region of µ and ∆.

Based on the extrapolation of terrestrial materials, val-ues as high as 10−2 has been suggested for neutron starcrusts [36, 37], which may also be favored by the enor-mous energy (∼ 1046 erg) liberated in the 2004 December27 giant flare of SGR 1806-20 according to the magnetarmodel [38, 39]. In this work, we consider the effects ofσmax in the range 10−3 to 10−2, values that have beenused in the study of compact stars by other authors (e.g.,[18, 19, 27, 38]).

Using the shear modulus given in Eq. (5), the max-imum quadrupole moment for a solid quark star in thecrystalline color superconducting phase can be estimatedby Eq. (4). However, the moment of inertia Izz is neededto calculate the corresponding maximum equatorial ellip-ticity. Using the empirical formula for strange stars givenby Bejger and Haensel (see Eq. (10) of [40]), Eq. (2) givesthe maximum equatorial ellipticity as

ϵmax = 2.6 × 10−4

!

ν

MeV/fm3

"

#σmax

10−3

$

!

1.4M⊙

M

"2

×

!

R

10 km

"4 %

1 + 0.14

!

M

1.4M⊙

" !

10 km

R

"&−1

. (6)

We use the values M = 1.4M⊙, R = 10 km, andσmax = 10−2 in order to compare with [19], in whichOwen obtained ϵmax ∼ 2 × 10−4 for solid strange stars(with the quarks clustered in groups of about 18). Forthe estimated range of the shear modulus given above,we find that ϵmax could be as large as ∼ 5 × 10−2 forsolid quark stars in a crystalline color superconductingphase. This relatively large value of ϵmax is about fourorders of magnitude larger than the tightest upper limitobtained by the combined S3/S4 result for the pulsarPSR J2124-3358 [30].

Constraints set by the Crab pulsar. Eq. (1) suggeststhat the observational upper limits on h0 obtained fromknown isolated pulsars can be used to set a limit on ϵ

assuming a value of Izz. However, the moment of inertiais very sensitive to the poorly known dense matter equa-tion of state (EOS). It can change by a factor of sevendepending on the stiffness of the EOS [40]. Alternatively,with Eq. (2), one can use Eq. (1) to set a limit on thepulsar’s quadrupole moment without assuming a valueof Izz [41]. The limit can in turn set a constraint on theshear modulus of crystalline color superconducting quarkmatter by Eq. (4). In particular, with the expression (5)for ν, we can define an exclusion region in the ∆−µ planeby the the following constraint:

∆µ <∼ 7.3 × 104 MeV2

!

10 km

R

"3 !

1 Hz

f

"

×

'(

h0

10−24

)

!

M

1.4M⊙

" !

10−3

σmax

" !

r

1 kpc

"

*1/2

,(7)

where h0 is the observational upper limit on h0 for agiven pulsar.

Under the assumption that the pulsar is an isolatedrigid body and that the observed spin-down of the pulsaris due to the loss of rotational kinetic energy as gravita-tional radiation, one can also obtain the so-called spin-down limit on the gravitational-wave amplitude hsd =(5GIzz |f |/2c3r2f)1/2, where f is the time derivative ofthe pulsar’s spin frequency [30]. As it is expected thatthe strain amplitude satisfies h0

<∼ hsd in general, we

can derive a constraint on the product ∆µ based on thespin-down limit:

(∆µ)sd

<∼ 2.1 × 105 MeV2

!

10−3

σmax

"1/2 !

M

1.4M⊙

"3/4

×

!

10 km

R

"5/2(

|f |

10−10 Hz s−1

)1/4!

f

1 Hz

"−5/4

×

%

1 + 0.14

!

M

1.4M⊙

" !

10 km

R

"&1/4

, (8)

where we have used Eq. (10) of [40] for the moment ofinertia for strange stars.

In [30] it is reported that the gravitational-wave strainupper limit for the Crab pulsar (h0 = 3.1× 10−24) is theclosest to the spin-down limit (at a ratio of 2.2). For theother pulsars, the direct observational upper limits aretypically at least one hundred times larger than the spin-down limits. For pulsars in globular clusters, in whichcases the spin-down measurement is obscured by the clus-ter’s dynamics, the gravitational-wave observations pro-vide the only direct upper limits. In the following, weshall focus on the constraints set by the observationaldata for the Crab pulsar.

The S3/S4 results for the Crab pulsar are plotted inFig. 1. In the figure, the solid lines represent Eq. (7) whenthe equality holds for the breaking strain σmax = 10−3

and 10−2, assuming that M = 1.4M⊙ and R = 10 km.The pulsar’s spin frequency is f = 29.8 Hz and the dis-tance is r = 2 kpc. For comparison, the dotted and

Lin, Phys.Rev. D76 (2007) 081502

4

3 4 5 6 7 8

ρ / ρ

1e-14

1e-13

1e-12

1e-11

1e-10

1e-09

1e-08

1e-07

ε

Full star

Naked core

Core

n

FIG. 2: In the left panel, the ellipticities obtained from our analysis are shown as a function of the core transitiondensity ρc/ρn, where ρn is the nuclear saturation density. We consider two breaking strains, a maximum valueσbr = 10−2 and a minimum of σbr = 10−5. For each of these values we examine the region between the maximumand the minimum shear modulus µ, obtained for the range of parameters in [10], cf. (2). The shaded regionindicates the region permitted by LIGO observations and the horizontal line is the current LIGO upper limit ofϵ = 7.1×10−7 for PSR J2124-3358 [5]. If the breaking strain is close to the maximum the observations are alreadyconstraining the theory. If it is close to the minimum, however, there are no significant constraints yet. In theright panel we compare the ellipticity obtained by considering the full star, only the core of the full star and justthe naked core, with no fluid around it. We take the breaking strain to be σbr = 10−5. As one can see the resultsfor the two cores do not differ significantly, while the ellipticity of the core plus fluid star is smaller, more so asthe core size decreases at higher transition densities.

der to make real improvements, one would (again) wantto use a realistic equation of state. This would requirethe calculation to be carried out within General Relativ-ity. Then the determination of the background solutionis straightforward, but the calculation of the mountainsize would require implementing the General Relativistictheory of elasticity, see for example [15]. To date, therehave been no such calculations. Work in this directionshould clearly be encouraged.

Finally, we need to improve our understanding of thebreaking strain. The range of values that we have used,10−5 ≤ σbr ≤ 10−2, is relevant for a crust consisting ofnormal matter. The upper limit represents the limit onCoulomb force dominated micro crystals while the lowerlimit is a pessimistic estimate of large scale breaking.However, the physics of the core is very different fromthat of the crust. There is no reason to believe that theestimates on the breaking strain for the crust should beapplicable to the core. We used these estimates simplybecause no data relevant for our study is available. Theresponse of the crystalline quark matter to large stressesis also uncertain. Normal matter will be predominantlybrittle and break into pieces when the temperature is suf-ficiently far below the melting temperature and will re-spond by plastic flow (up to some limit) otherwise. Howan elastic quark core will respond is completely unknown.Yet, for our purposes it may not matter which scenariois realised as long as the timescale for plastic flow at agiven strain is longer than the observation time.

We thank Krishna Rajagopal for useful discussions.This work was supported by PPARC/STFC via grant

numbers PP/E001025/1 and PP/C505791/1.

[1] G. Ushomirsky, C. Cutler, L. Bildsten, MNRAS 319 902(2000)

[2] B. Haskell, D.I. Jones, N. Andersson, MNRAS 373 1423(2006)

[3] B. Haskell, L. Samuelsson, K. Glampedakis, N. An-dersson, Modelling magnetically deformed neutron stars,preprint arXiv:0705.1780

[4] T. Akgun, I. Wasserman, Toroidal Magnetic Fieldsin Type II Superconducting Neutron Stars, preprintarXiv:0705.2195

[5] B. Abbott et al, Upper limits on gravitational wave emis-sion from 78 radio pulsars, preprint arXiv:gr-qc/0702039

[6] B.J. Owen, Phys. Rev. Lett. 95 211101 (2005)[7] P. Haensel, pp 129-149 in Relativistic Gravitation and

Gravitational Radiation, Ed. J-A. Marck and J-P. Lasota(Cambridge Univ Press, Cambridge 1997)

[8] K. Rajagopal, R. Sharma, Phys. Rev. D. 74 094019(2006)

[9] K. Rajagopal, R. Sharma, J. Phys. G. 32 S483 (2006)[10] M. Mannarelli, K. Rajagopal, R. Sharma, The rigidity of

crystalline color superconducting quark matter, preprintarXiv:hep-ph/0702021

[11] E. Kim, M.H.W. Chan, Science 305 1951 (2004)[12] A.T. Dorsey, P.M. Goldbart, J. Toner, Phys. Rev. Lett.

96 055301 (2006)[13] C. Josserand, T. Pomeau, S. Rica, Phys. Rev. Lett. 98

195301 (2007)[14] M. Alford, S. Reddy, Compact stars with color supercon-

ducting quark matter, preprint nucl-th/0211046[15] M. Karlovini, L. Samuelsson, Class. Quantum Grav. 20

3613 (2003)

Andersson et al. Phys.Rev. Lett.99. 231101 (2007)

Using the non-observation of GW from the Crab by the LIGO experiment

allowed regions





Gravitational waves from mountains

3

0 200 400 600 800

µ (MeV)

0

5

10

15

20

25

30

Δ (

MeV

)

σmax

=10-3

σmax

=10-2

FIG. 1: The areas above the solid lines are excluded bythe direct upper limit for the Crab pulsar obtained from theS3/S4 runs for the cases σmax = 10−3 and 10−2. The dotted(dashed) line is the constraint set by the spin-down limit forσmax = 10−3 (10−2). The rectangular box is the theoreticallyallowed region of µ and ∆.

Based on the extrapolation of terrestrial materials, val-ues as high as 10−2 has been suggested for neutron starcrusts [36, 37], which may also be favored by the enor-mous energy (∼ 1046 erg) liberated in the 2004 December27 giant flare of SGR 1806-20 according to the magnetarmodel [38, 39]. In this work, we consider the effects ofσmax in the range 10−3 to 10−2, values that have beenused in the study of compact stars by other authors (e.g.,[18, 19, 27, 38]).

Using the shear modulus given in Eq. (5), the max-imum quadrupole moment for a solid quark star in thecrystalline color superconducting phase can be estimatedby Eq. (4). However, the moment of inertia Izz is neededto calculate the corresponding maximum equatorial ellip-ticity. Using the empirical formula for strange stars givenby Bejger and Haensel (see Eq. (10) of [40]), Eq. (2) givesthe maximum equatorial ellipticity as

ϵmax = 2.6 × 10−4

!

ν

MeV/fm3

"

#σmax

10−3

$

!

1.4M⊙

M

"2

×

!

R

10 km

"4 %

1 + 0.14

!

M

1.4M⊙

" !

10 km

R

"&−1

. (6)

We use the values M = 1.4M⊙, R = 10 km, andσmax = 10−2 in order to compare with [19], in whichOwen obtained ϵmax ∼ 2 × 10−4 for solid strange stars(with the quarks clustered in groups of about 18). Forthe estimated range of the shear modulus given above,we find that ϵmax could be as large as ∼ 5 × 10−2 forsolid quark stars in a crystalline color superconductingphase. This relatively large value of ϵmax is about fourorders of magnitude larger than the tightest upper limitobtained by the combined S3/S4 result for the pulsarPSR J2124-3358 [30].

Constraints set by the Crab pulsar. Eq. (1) suggeststhat the observational upper limits on h0 obtained fromknown isolated pulsars can be used to set a limit on ϵ

assuming a value of Izz. However, the moment of inertiais very sensitive to the poorly known dense matter equa-tion of state (EOS). It can change by a factor of sevendepending on the stiffness of the EOS [40]. Alternatively,with Eq. (2), one can use Eq. (1) to set a limit on thepulsar’s quadrupole moment without assuming a valueof Izz [41]. The limit can in turn set a constraint on theshear modulus of crystalline color superconducting quarkmatter by Eq. (4). In particular, with the expression (5)for ν, we can define an exclusion region in the ∆−µ planeby the the following constraint:

∆µ <∼ 7.3 × 104 MeV2

!

10 km

R

"3 !

1 Hz

f

"

×

'(

h0

10−24

)

!

M

1.4M⊙

" !

10−3

σmax

" !

r

1 kpc

"

*1/2

,(7)

where h0 is the observational upper limit on h0 for agiven pulsar.

Under the assumption that the pulsar is an isolatedrigid body and that the observed spin-down of the pulsaris due to the loss of rotational kinetic energy as gravita-tional radiation, one can also obtain the so-called spin-down limit on the gravitational-wave amplitude hsd =(5GIzz |f |/2c3r2f)1/2, where f is the time derivative ofthe pulsar’s spin frequency [30]. As it is expected thatthe strain amplitude satisfies h0

<∼ hsd in general, we

can derive a constraint on the product ∆µ based on thespin-down limit:

(∆µ)sd

<∼ 2.1 × 105 MeV2

!

10−3

σmax

"1/2 !

M

1.4M⊙

"3/4

×

!

10 km

R

"5/2(

|f |

10−10 Hz s−1

)1/4!

f

1 Hz

"−5/4

×

%

1 + 0.14

!

M

1.4M⊙

" !

10 km

R

"&1/4

, (8)

where we have used Eq. (10) of [40] for the moment ofinertia for strange stars.

In [30] it is reported that the gravitational-wave strainupper limit for the Crab pulsar (h0 = 3.1× 10−24) is theclosest to the spin-down limit (at a ratio of 2.2). For theother pulsars, the direct observational upper limits aretypically at least one hundred times larger than the spin-down limits. For pulsars in globular clusters, in whichcases the spin-down measurement is obscured by the clus-ter’s dynamics, the gravitational-wave observations pro-vide the only direct upper limits. In the following, weshall focus on the constraints set by the observationaldata for the Crab pulsar.

The S3/S4 results for the Crab pulsar are plotted inFig. 1. In the figure, the solid lines represent Eq. (7) whenthe equality holds for the breaking strain σmax = 10−3

and 10−2, assuming that M = 1.4M⊙ and R = 10 km.The pulsar’s spin frequency is f = 29.8 Hz and the dis-tance is r = 2 kpc. For comparison, the dotted and

Lin, Phys.Rev. D76 (2007) 081502

4

3 4 5 6 7 8

ρ / ρ

1e-14

1e-13

1e-12

1e-11

1e-10

1e-09

1e-08

1e-07

ε

Full star

Naked core

Core

n

FIG. 2: In the left panel, the ellipticities obtained from our analysis are shown as a function of the core transitiondensity ρc/ρn, where ρn is the nuclear saturation density. We consider two breaking strains, a maximum valueσbr = 10−2 and a minimum of σbr = 10−5. For each of these values we examine the region between the maximumand the minimum shear modulus µ, obtained for the range of parameters in [10], cf. (2). The shaded regionindicates the region permitted by LIGO observations and the horizontal line is the current LIGO upper limit ofϵ = 7.1×10−7 for PSR J2124-3358 [5]. If the breaking strain is close to the maximum the observations are alreadyconstraining the theory. If it is close to the minimum, however, there are no significant constraints yet. In theright panel we compare the ellipticity obtained by considering the full star, only the core of the full star and justthe naked core, with no fluid around it. We take the breaking strain to be σbr = 10−5. As one can see the resultsfor the two cores do not differ significantly, while the ellipticity of the core plus fluid star is smaller, more so asthe core size decreases at higher transition densities.

der to make real improvements, one would (again) wantto use a realistic equation of state. This would requirethe calculation to be carried out within General Relativ-ity. Then the determination of the background solutionis straightforward, but the calculation of the mountainsize would require implementing the General Relativistictheory of elasticity, see for example [15]. To date, therehave been no such calculations. Work in this directionshould clearly be encouraged.

Finally, we need to improve our understanding of thebreaking strain. The range of values that we have used,10−5 ≤ σbr ≤ 10−2, is relevant for a crust consisting ofnormal matter. The upper limit represents the limit onCoulomb force dominated micro crystals while the lowerlimit is a pessimistic estimate of large scale breaking.However, the physics of the core is very different fromthat of the crust. There is no reason to believe that theestimates on the breaking strain for the crust should beapplicable to the core. We used these estimates simplybecause no data relevant for our study is available. Theresponse of the crystalline quark matter to large stressesis also uncertain. Normal matter will be predominantlybrittle and break into pieces when the temperature is suf-ficiently far below the melting temperature and will re-spond by plastic flow (up to some limit) otherwise. Howan elastic quark core will respond is completely unknown.Yet, for our purposes it may not matter which scenariois realised as long as the timescale for plastic flow at agiven strain is longer than the observation time.

We thank Krishna Rajagopal for useful discussions.This work was supported by PPARC/STFC via grant

numbers PP/E001025/1 and PP/C505791/1.

[1] G. Ushomirsky, C. Cutler, L. Bildsten, MNRAS 319 902(2000)

[2] B. Haskell, D.I. Jones, N. Andersson, MNRAS 373 1423(2006)

[3] B. Haskell, L. Samuelsson, K. Glampedakis, N. An-dersson, Modelling magnetically deformed neutron stars,preprint arXiv:0705.1780

[4] T. Akgun, I. Wasserman, Toroidal Magnetic Fieldsin Type II Superconducting Neutron Stars, preprintarXiv:0705.2195

[5] B. Abbott et al, Upper limits on gravitational wave emis-sion from 78 radio pulsars, preprint arXiv:gr-qc/0702039

[6] B.J. Owen, Phys. Rev. Lett. 95 211101 (2005)[7] P. Haensel, pp 129-149 in Relativistic Gravitation and

Gravitational Radiation, Ed. J-A. Marck and J-P. Lasota(Cambridge Univ Press, Cambridge 1997)

[8] K. Rajagopal, R. Sharma, Phys. Rev. D. 74 094019(2006)

[9] K. Rajagopal, R. Sharma, J. Phys. G. 32 S483 (2006)[10] M. Mannarelli, K. Rajagopal, R. Sharma, The rigidity of

crystalline color superconducting quark matter, preprintarXiv:hep-ph/0702021

[11] E. Kim, M.H.W. Chan, Science 305 1951 (2004)[12] A.T. Dorsey, P.M. Goldbart, J. Toner, Phys. Rev. Lett.

96 055301 (2006)[13] C. Josserand, T. Pomeau, S. Rica, Phys. Rev. Lett. 98

195301 (2007)[14] M. Alford, S. Reddy, Compact stars with color supercon-

ducting quark matter, preprint nucl-th/0211046[15] M. Karlovini, L. Samuelsson, Class. Quantum Grav. 20

3613 (2003)

Andersson et al. Phys.Rev. Lett.99. 231101 (2007)

...we can restrict the parameter space!

Using the non-observation of GW from the Crab by the LIGO experiment

allowed regions





Conclusions

•Hadronic matter at very high densities is made of deconfined quarks

•The color interaction may produce a color superconducting phase

•The crystalline color superconducting phase maybe realized in “realistic” conditions

•Oscillations of hybrid or strange stars may lead to observable signals

BACKUP

extreme

10 g cm-3

Increasing baryonic densityDensity

H

He

.....

Fe

neutron drip

neutrons and protons

Cooper pairs of quarksNGBs

Degrees of freedom

light nuclei

heavy nuclei

quarks and gluonsCooper pairs of quarks?

.....

10 11 g cm-3

......

weakcoupling

confining

strong coupling

↵s ⌘ ↵s(µ)

very largequarkdrip

quarksoup

neutron protonsoup

⇢ 32

10 14 g cm-3

CSO part

atmosphere

outer crust

inner crust

core

}}

Quark model

33

Quarks and gluons are the building blocks of hadrons

The theory describing quarks and gluons is Quantum Chromodynamics (QCD): a nonabelian SU(3) gauge theory. Quarks form a triplet in the fundamental representaion Gluons are the vector gauge bosons associated to the octet adjoint representation

neutronproton

ud

u dd

u....

BARYONS MESONS

....u d

pions

Mn ⇠ 1GeV � mu,d M⇡ ⇠ 135 MeV � mu,d

Q quark flavor (mass in MeV)

+2/3 u (3) c (1300) t (170000)

�1/3 d (5) s (130) b (4000)

Documents

Quark matter and color superconductivity · Quark matter and color superconductivity Massimo Mannarelli INFN-LNGS [email protected] Review of Modern Physics 86, 509 (2014) WG2