Spin dependent Fermi Liquid parameters and properties of polarized quark matter

0.31 MB·English

by  Kausik Pal

Spin dependent Fermi Liquid parameters and properties of
polarized quark matter
Kausik Pal,∗ Subhrajyoti Biswas, and Abhee K. Dutt-Mazumder
High Energy Physics Division, Saha Institute of Nuclear Physics,
1/AF Bidhannagar, Kolkata 700064, India.
9
0 We calculate the spin dependent Fermi liquid parameters (FLPs), single particle
0
2 energies and energy densities of various spin states of polarized quark matter. The

n
a expressions for the incompressibility(K) and sound velocity (c ) in terms of the spin
1
J

5 dependent FLPs and polarization parameter (ξ) are derived. Estimated values of K

] and c reveal that the equation of state (EOS) of the polarized matter is stiffer than
h 1
p
the unpolarized one. Finally we investigate the possibility of the spin polarization
-
p
e (ferromagnetism) phase transition.
h
[

2 PACS numbers: 04.40.Dg, 12.38.Bx,12.39.-x, 14.70.Dj, 26.60.-c, 97.60.Jd
v
4 Keywords: Ferromagnetism, Quark matter, Landau parameters.
0
4
0
.
9
0
I. INTRODUCTION
8
0
:
v
Oneoftheimportantresearch areasofthecontemporaryhighenergy physics hasbeenthe
i
X
study of matter under extreme conditions. Such a matter, inthe laboratorycan be produced
r
a
by colliding heavy ions at ultra-relativistic energies. Due to asymptotic freedom of quantum
chromodynamics(QCD),itispredictedthatthehadronicmatterathightemperatureand/or
density can undergo a series of phase transitions like confinement-deconfinement and/or
chiral phase transition [1, 2]. In the high density regime QCD predicts the existence of color
superconducting state [3, 4, 5]. These apart, the possibility of spin polarized quark liquid
i.e. the existence of ferromagnetic phase in dense quark system has also been suggested
recently with which we are presently concerned [2, 6]. The properties of dense quark system
are particularly relevant for the study of various astrophysical phenomenon.
∗
Electronic address: [email protected]
2
The part of the motivation to study the ferromagnetic phase transition in dense quark
matter (DQM), as mentioned in [6] is provided by the discovery of ‘magnetars’ [7] where an
extraordinarily high magnetic field ∼ 1015G exists [6, 8]. In [6], it is argued, that the origin
of such a high magnetic field can be attributed to the existence of spin polarized quark
matter [9]. To examine the possibility of ferromagnetism in DQM in ref.[6] a variational
calculation is performed where it is observed that there exists a critical density below which
spin polarized quark matter is energetically favorable than unpolarized state. Subsequently
various other calculations were also performed to investigate this issue [2, 4, 5, 7, 8, 9, 10].
Forexample, in[5]it isshown that there is no contradiction between color superconductivity
and ferromagnetism and both of these phase can co-exist. In [10], the same problem was
studied in the large N and N limit while keeping N /N fixed where it was shown that spin
c f c f
polarizedstatecanexist, however, inpresence ofmagneticscreening, colorsuperconductivity
or dense chiral waves disappear. It might be mentioned that such screening is nowsupported
by the lattice calculation [10, 11]. In[5] it is analytically shown that, if quarks are massless,
ferromagnetismdoesnotappearwhichisconsistentwiththeconclusiondrawnin[10]. Ref.[8]
shows that ferromagnetism might appear in quark matter with Goldstone boson current
where the magnetization is shown to be related to triangle anomalies.
In the present work, we apply relativistic Fermi liquid theory (RFLT) to study the possi-
bility of para-ferro phase transition in DQM. The relativistic Fermi liquid theory was devel-
oped by Baym and Chin [14] where it has been shown how the various physical quantities
like chemical potential (µ), incompressibility (K), sound velocity (c ) etc. can be expressed
1
in terms of the Landau parameters (LPs) calculated relativistically. However, the formalism
developed in [14] is valid for unpolarized matter and LPs calculated there are spin averaged.
In this paper we extend the formalism of RFLT and the required LPs are calculated
by retaining their explicit spin dependencies. As a result, here various combination of
parameters like f++, f+−, f−+ and f−− corresponding to scattering involving up-up, up-
0,1 0,1 0,1 0,1
down, down-up or down-down spins are appear [14]. Once determined, these parameters are
used to calculate quantities like chemical potentials for the spin up and spin down quarks
or the total energy density of the system as a function of ξ = (n+−n−)/n and n together
q q q q
with various other quantities as we shall see. Here n+ and n− correspond to densities of
q q
spin up, down quarks respectively and n = n+ +n−, denotes total quark density [6]. We,
q q q
also compare some of our results with those presented in [6] where more direct approach was
3
adopted to calculate the total energy density from the loop. In addition, the present work
is extended further to estimate incompressibility and sound velocity in dense quark system
for a given fraction of spin-up or down quarks.
Furthermore, in dealing with the massless gluons, we find that naive series expansion fails
and one has to use hard density loop (HDL) corrected gluon propagator to get the finite
result for the LPs involving scattering of like spins [12]. This however does not cause any
problem for the calculation of various physical quantities like chemical potential, exchange
energy, incompressibility etc. We shall see, even though f and f (suppressing spin indices)
0 1
individually remain divergent, what appears in our case is the particular combination of
these parameters where such divergences cancel.
The plan of the paper is as follows. In Sec.II, as mentioned before, we extend the formal-
ism of RFLT to include explicit spin dependence. In Sec.III, we derive spin dependent LPs
due to one gluon exchange (OGE) for polarized quark matter. Subsequently, we calculate
chemical potential and energy density. We find the density dependence of incompressibility
(K) and first sound velocity (c ) with arbitrary spin polarization (ξ). To compare with
1
ref.[6], we present ultra-relativistic and non-relativistic results and studied para-ferro phase
transition of quark matter. Sec.IV is devoted to summary and conclusion. In Appendix, we
calculate various LPs for unlike spin states of scatterer.
II. FORMALISM
In FLT total energy density E of an interacting system is the functional of occupation
number n of the quasi-particle states of momentum p. The excitation of the system is
p
equivalent to the change of occupation number by an amount δn . The corresponding
p
energy density of the system is given by [13, 14],
d3p 1 d3p d3p′
E = E0 + (2π)3ε0psδnps + 2 (2π)3(2π)3fps,p′s′δnpsδnp′s′, (1)
s Z ss′ Z
X X
where E0 is the ground state energy density and s is the spin index, and the quasi-particle
energy can be written as,
d3p′
εps = ε0ps + (2π)3fps,p′s′δnp′s′, (2)
s′ Z
X
4
where ε0 is the non-interacting single particle energy. The interaction between quasi-
ps
particles is given by fps,p′s′, which is defined to be the second derivative of the energy
functional with respect to occupation functions,
δ2E
fps,p′s′ = . (3)
δnps δnp′s′
Since, the quasiparticles are well defined only near the Fermi surface, one assumes
ε = µs +vs(p−ps). (4)
ps f f
In FLT, the interaction parameter, fps,p′s′, is expanded on the basis of Legendre polyno-
mials, P [13, 14]. The coefficients of this expansion are known as FLPs, which are given
l
by
dΩ
flss′ = (2l+1) 4πPl(cosθ)fps,p′s′, (5)
Z
where θ is the angle between p and p′, both taken to be on the Fermi surface, and the
integration is over all directions of p [14]. Note that unlike [13, 14], here we retain explicit
spin indices without performing spin summation. We restrict ourselves for l ≤ 1 i.e. fs and
0
fs, since higher l contribution decreases rapidly as the scattering is dominated by the small
1
angles and the series converges, here, fs = 1 fss′ [15].
l 2 s′ l
The Landau Fermi liquid interaction fps,p′sP′ is related to the two particle forward scatter-
ing amplitude via [13, 14],
m m
q q
fps,p′s′ = ε0 ε0 Mps,p′s′, (6)
p p′
where mq is the mass of the quark and the Lorentz invariant matrix Mps,p′s′ consists of
the usual direct and exchange amplitude, which may, therefore be evaluated by conventional
Feynman rules. The dimensionless LPs are defined as Fs = Ns(0)fs [14], where Ns(0) is
l l
the density of states at the Fermi surface is given by,
d3p
Ns(0) = δ(ε −µs)
(2π)3 ps
Z
g ps2 ∂p
deg f
=
2π2 ∂ε
(cid:18) ps(cid:19)p=ps
f
g psεs
deg f f
≃ . (7)
2π2
5
Here g is the degeneracy factor. In our case g = N N where N and N are the color
deg deg c f c f
and flavor index for quark matter. For spin up (+) and spin down (−) quark, density of
states will be change accordingly. In the above expression (∂p/∂ε ) is the inverse Fermi
ps p=ps
f
velocity (1/vs) related to the FL parameter Fs,
f 1
1
= (∂p/∂ε ) = (µs/ps)(1+Fs/3). (8)
vs ps p=psf f 1
f
With Eq.(7) and Eq.(8) one reads the general relation as [16]
1
εs = µs(1+ Fs). (9)
f 3 1
The compression modulus or incompressibility (K) of the system is defined by the second
derivative of total energy density E with respect to the number density n , is given by
q
[16, 17, 18, 19, 20]
∂2E
K = 9n . (10)
q ∂n2
q
Now we introduce a polarization parameter ξ by the equations, n+ = n (1 + ξ)/2 and
q q
n− = n (1−ξ)/2underthecondition0 ≤ ξ ≤ 1[6]. TheFermimomentainthespin-polarized
q q
quark matter then are p+ = p (1 + ξ)1/3 and p− = p (1 − ξ)1/3, where p = (π2n )1/3, is
f f f f f q
the Fermi momentum of the unpolarized matter (ξ = 0). So, there are two Fermi surfaces
corresponding to spin-up (+) and spin-down (−) states, such that E ≡ E(n+,n−). We have
q q
∂E ∂E ∂n+ ∂E ∂n−
q q
= +
∂n ∂n+ ∂n ∂n− ∂n
q q q q q
1
= (1+ξ)µ+ +(1−ξ)µ− (11)
2
(cid:2) (cid:3)
Using Eq.(11), the incompressibility becomes[20]
9n ∂µ+ ∂µ−
K = q (1+ξ)2 +(1−ξ)2
4 ∂n+ ∂n−
(cid:20) q q (cid:21)
9n 1+F+ 1+F−
= q (1+ξ)2 0 +(1−ξ)2 0 , (12)
4 N+(0) N−(0)
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
6
where [14]
∂µs 1+Fs
= 0 . (13)
∂ns Ns(0)
q
Similarly, the relativistic first sound velocity is given by the first derivative of pressure P
with respect to energy density E. Since P = µsns −E [16, 20], we have,
s q
P
∂P ∂P ∂n
c2 = = q
1 ∂E ∂n ∂E
q
(1+ξ)n+∂µ+ +(1−ξ)n−∂µ−
= q ∂n+q q ∂n−q
 (1+ξ)µ+ +(1−ξ)µ− 
 n  1+F+ 1+F−
= q (1+ξ)2 0 +(1−ξ)2 0 .(14)
2[(1+ξ)µ+ +(1−ξ)µ−] N+(0) N−(0)
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
In the above Eq.(12) and Eq.(14), N±(0) and F± corresponds to density of states at Fermi
0
surface and dimensionless LP for spin up (+) and spin down (−) quark respectively. For
unpolarized matter, ξ = 0 implying µ+ = µ−, F+ = F− and N+(0) = N−(0). From Eq.(12)
0 0
and (14) we have the well known result as K = 9n ∂µ [16] and c2 = nq ∂µ [14].
q∂nq 1 µ ∂nq
III. LANDAU PARAMETERS FOR POLARIZED QUARK MATTER
In this section we calculate LPs for quark matter with explicit spin dependencies. We
choose spin s along z axis i.e. s ≡ (0,0,±1) and represent spin-up and down states by
their signs. For a four-dimensional description of the polarization state, it is convenient to
define a 4-vector aµ which, in the rest frame of each quark, is same as the three-dimensional
vector s; since s is an axial vector, aµ is a 4-pseudovector. This 4-vector is orthogonal to
the 4-momentum in the rest frame (in which aµ = (0,s),Pµ = (m ,0)); in any frame we
q
therefore have aµP = 0 [6, 21, 22].
µ
The components of the 4-vector aµ in a frame in which the particle is moving with
momentum p are found by a Lorentz transformation from the rest frame [22],
p(s·p) p·s
a = s+ ; a0 = (15)
m (ε +m ) m
q p q q
7
with ε = p2 +m2. We can define projection operator P(a) on each of spin polarization,
p q
P(a) = 1(1p+γ5a/). Accordingly the polarization density matrix ρ is given by the expression
2
1
ρ(P,s) = (P/+m )P(a), (16)
q
2m
q
which is normalized by the condition, Trρ(P,s) = 1. The mean value of the spin is then
given by the quantity [22]
1m 1m
q q
s = Tr(ργ Σ) = Tr(ργ γ)
av 0 5
2 ε 2 ε
p p
1m p(s·p)
q
= s+ , (17)
2 ε m (ε +m )
p (cid:18) q p q (cid:19)
which is reduced to s = 1s in the non-relativistic limit.
av 2
We consider the color-symmetric forward scattering amplitude of the two quarks around
the Fermi surface by the OGE interaction. The direct term does not contribute as it involves
trace of single color matrices like Trλ , which vanishes. Thus the leading contribution comes
a
from the exchange (Fock) term [6]:
1 1 −g
Mex = − U¯ (P′)g(ta) γµU (P) µν U¯ (P)g(ta) γνU (P′)
ps,p′s′ 3 3 β ji α (P −P′)2 α ij β
i j (cid:18) (cid:19)
X X(cid:2) (cid:3) (cid:2) (cid:3)
4 1
= Tr[γ ρ(P,s)γµρ(P′,s′)], (18)
9(P −P′)2 µ
where α, β is the flavor level, i,j is the quark color index, ta(= λ /2) is the color
a
matrix and g is the coupling constant. Since gluon is flavor blind, the u−channel diagrams
contribute only when α = β; i.e. scattering of quarks with same flavor[23]. This means
that the Fermi sphere of each flavor makes an independent contribution. Thus the potential
energyreceives afactorN . Ontheotherhand, thequarks withdifferent colorscantakepart
f
in the exchange process, giving rise to a factor N2. Eventually the potential energy density
c
is proportional to N N2g2. For the kinetic energy density, there arises an overall factor
f c
N N . Thus, the factor N N factorizes out of the total energy density and the competition
c f c f
between the kinetic and potential energies is not influenced by the number of flavor. The
number of flavor neither encourages or discourages ferromagnetism [10].
Without loss of generality, for the calculation of energy density and other related quan-
tities, we consider one-flavor quark matter. With the help of polarization density matrices
8
given in Eq.(16), we have from Eq.(18) the interaction amplitude as [6]
2g2 1 1
Mex = [2m2 −P.P′−(p·s)(p′ ·s′)+m2(s·s′)+
ps,p′s′ 9m2q (P −P′)2 q q (εp +mq)(εp′ +mq)
×{mq(εp +mq)(p′ ·s)(p′ ·s′)+mq(εp′ +mq)(p·s)(p·s′)+(p·p′)(p·s)(p′ ·s′)].
(19)
From Eq.(6) the quasiparticle interaction parameter is given by
m m
fex = q qMex (20)
ps,p′s′ εp εp′ ps,p′s′
Here the spin may be either parallel (s = s′) or anti-parallel (s = −s′). Thus scattering
possibilities are denoted by (+,+), (+,−), (−,−) etc. Motivated by [15], in analogy with
isospin we define spin dependent interaction parameter as f+ = 1(f++ +f+−) and f− =
pp′ 2 pp′ pp′ pp′
1(f−− +f−+). Note that, f+− = f−+.
2 pp′ pp′ pp′ pp′
For (+,+) scattering the interaction parameter is given by
g2 1
f++ = − [2m2 −p+2(1−cosθ)−p+2cosθ cosθ
pp′|p=p′=p+ 9ε+2p+2(1−cosθ) q f f 1 2
f f f
1
+ {m (ε+ +m )p+2(cos2θ +cos2θ )+p+4cosθcosθ cosθ }],
(ε+ +m )2 q f q f 1 2 f 1 2
f q
(21)
where pˆ· sˆ = cosθ ; pˆ′ · sˆ = cosθ and Fermi energy ε+ = (p+2 + m2)1/2. Since spin and
1 2 f f q
momentum have no preferred direction, we have done angular average of the spin dependent
parameter [24]:
dΩ dΩ
f++ = 1 2f++
pp′|p=p′=p+f 4π 4π pp′|p=p′=p+f
Z Z
g2 2m p+2
= − 2m2 −p+2(1−cosθ)+ q f . (22)
9ε+2p+2(1−cosθ) q f 3(ε+ +m )
f f " f q #
1 WiththehelpofEq.(5)alongwiththeEq.(22)onecanfindLPs,butitistobenotedthat
f++ or f−− are individually divergent because of the term, (1−cosθ), in the denominator of
0,1 0,1
the interaction parameter. This divergence disappear if one uses Debye screening mass for
[1] denoted hereafter fpp′ =fpp′.
9
gluons or equivalently use HDL corrected gluon propagator while evaluating the scattering
amplitudes [12, 24]. Note that the combination f++(−−) − 1f++(−−) is, however, finite as
0 3 1
in this case the divergences cancel and we do n(cid:16)ot calculate the LPs(cid:17)separately. It would,
however, beinteresting tosee howdotheresults modifyifHDLcalculationsareperformedto
evaluate f++(−−), f+− and the corresponding physical quantities. The numerical estimates
0,1 0,1
suggest that for the results what we present here, the effect of HDL corrections are expected
to be small.
From Eq.(5),
1 g2 +1 2m p+2
f++ − f++ = − 2m2 −p+2(1−cosθ)+ q f d(cosθ)
0 3 1 18ε+2p+2 q f 3(ε+ +m )
f f Z−1 " f q #
g2 2m p+2
= − 2m2 −p+2 + q f . (23)
9ε+2p+2 q f 3(ε+ +m )
f f " f q #
The above combination will appear in the calculation of the chemical potential and other
relevant quantities. For (+,−) scattering, the angular averaged interaction parameter yields
g2 m p+2 m p−2 1
f+− = 1− q f + q f × .
p=p+f,p′=p−f 9ε+f ε−f " (3(ε+f +mq) 3(ε−f +mq)) (m2q −ε+f ε−f +p+f p−f cosθ)#
(cid:12)
(cid:12) (24)
(cid:12)
It is to be noted that, individual LPs for scattering of unlike spin states are finite i.e. free
of divergences, in contrast to the case involving scattering of like spin states (For details see
Appendix).
A. Chemical potential
Now we proceed to calculate chemical potential, which, in principle, will be different for
spin-up and spin-down quarks, denoted by µs with s (or s′) = +,− for matter containing
unequaldensities ofupanddownquarks. Todetermine thechemicalpotentialwitharbitrary
polarization ξ, we take the distribution function with explicit spin index (s or s′), so that
variation of distribution function gives [13, 20, 25]
δns = −Ns(0) fss′δns′ −δµs , (25)
q 0 q
" s′ #
X
10
where Ns(0) is given by the Eq.(7). The Eq.(25) yields
∂µs 1 ∂ns′
= + fss′ q . (26)
∂ns Ns(0) 0 ∂ns
q s′ q
X
Separately for spin-up and spin-down states we have
∂µ+ 1 +f++ f+− ∂n+
= N+(0) 0 0 q ,
∂µ−   f−+ 1 +f−− ∂n− 
0 N−(0) 0 q
     (27)
where the superscripts ++ and +− denote scattering of quasiparticle with up-up and up-
down spin states. For unpolarized matter the upper and lower component become equal
which gives rise to the well known result [14]
g µp2 1
deg f
µdµ = p + (f − f ) dp . (28)
f 2π2 0 3 1 f
(cid:20) (cid:21)
In general the chemical potential (both for spin-up and spin-down) is the combination of like
and unlike spin states. By adjusting the constant of integration [14], the chemical potential
of spin-up quark turns out to be
g2 11 p+ +ε+ 2 p+ε+
µ+ = ε+ − m2ln f f + p+m − f f
f 6π2ε+ 6 q m 3 f q 2
f " q ! #
g2 2m3 p+ +p− 4m2ε+ p+ +p− p+ε− +p−ε+
q f f q f f f f f f f
+ − ln + ln +ln
72π2ε+ p+ p+ −p− p+ p+ −p− p+ε− −p−ε+
f " f f f ! f ( f f ! f f f f !)
p− +ε− p+ +p−
−14m2ln f f +2m p− −3m p−ln f f
q m q f q f p+ −p−
q ! f f !
m p+ε− +p−ε+ p− +ε−
− q(2m2 +3p+2)ln f f f f +6m ε+ln f f
p+ q f p+ε− −p−ε+ q f m
f f f f f ! q !
m ε−ε+ −m2 −p−p+
+ q{2ε−(2m −ε+)−p−2}ln f f q f f +6p−ε− . (29)
p+ f q f f ε−ε+ −m2 +p−p+ f f
f f f q f f ! #
In the above equation the term in the first square bracket arises due to the scattering of
like spin states (++), while the latter comes from the scattering of unlike spin states (+−).
Similarly, for spin-down quark, one may determine µ− by replacing p± with p∓ and ε±
f f f
with ε∓ in Eq.(29).
f