Pion condensation in the twoflavor chiral quark model at finite - - PowerPoint PPT Presentation

Pion condensation in the two–flavor chiral quark model at finite baryochemical potential

P´ eter Kov´ acs KFKI Research Institute of Particle and Nuclear Physics of HAS, Theoretical Department

2009.04.03

• Motivation
• The constituent quark model and its renormalization
• Equations at one–loop level, the OPT
• Diagonalized propagators
• Parametrization
• Results at lowest order in ρ

(Phys.Rev.D78:116008,2008.)

• Conclusion

Collaborator: T. Herpay

1

Where can pion condensation occur in nature?

• Quark matter can exist in neutron stars −

→ at very large bariochemical potential (µB ≈ 1 GeV) If the isospin chemical potential is also different from zero − → possibility of pion condensation

• In heavy ion collisions µI is tunable with different isotopes of an element

Neutrino emission from pion condensed quark matter → direct Urca processes: d → u + e− + ¯ ν u + e− → d + ν = ⇒ It might will be investigated experimentaly

2

In 2 flavoured NJL model (L. He et al Phys. Rev. D74, 036005 (2004)):

• if µI < 140 MeV(= mπ) → no pion condensation
• if 140 MeV< µI < 230 MeV → BEC phase
• if µI > 230 MeV → BCS phase

Interesting feature of pion condensation found in SU(2) PNJL model: At sufficiently high temperature the condesate evaporates above a certain µI. (e.g. Z. Zhang, Y. Liu: hep-ph/0610221v3) Up to now:

• Investigations in SU(2) NJL and PNJL model (BEC, BCS and CFL phases)
• Investigation in O(4) model in the large N limit (leading order) (BEC phase)

3

The model and its renormalization

Our starting point is the renormalized SU(2)L × SU(2)R symmetric Lagrangian with explicit symmetry breaking term L = 1 2

• ∂µφ∂µφ − m2φ2

− λ 4φ4 + hφ0 + i ¯ ψγµ∂µψ − gF 2 ¯ ψTiφiψ + 1 2

• δZ∂µφ∂µφ − δm2φ2

− δλ 4 φ4 ψ = (u, d)T − → doublet quark fileds φ = (φ0, φ1, φ2, φ3) ≡ (σ, π1, π2, π3) − → sigma and pion scalar fields h − → symmetry breaking external field Ti = (τ0, iτiγ5) − → quark–boson coupling matrix The renormalized (finite) parameters of the Lagrangian: m2, λ, gF δz, δm2, δλ are the usual (infinite) counterterms (Fermions are treated at tree level → no wavefunction renormalization)

4

The genarating functional: Z =

• DφDΠD ¯

ψDψ exp

• i

−iβ dt

• d3x(Π ˙

φ + i ¯ ψγ0 ˙ ψ − H + µBQB + µIQI)

• ,

where the Hamiltonian is H = 1 2

• Π2 + (∇φ)2 + m2φ2

+ λ 4φ4 − hφ0 + i ¯ ψγi∂iψ + gF 2 ¯ ψTαφαψ − 1 2δZΠ2 + 1 2δZ(∇φ)2 + δλ 4 φ4 + 1 2δm2φ2,

(i = 1, 2, 3)

and the canonical momenta of the scalar fields Π = δL δ ˙ φ = (1 + δZ) ˙ φ. QB, QI are the conserved barion and isospin charges QB =

• d3x1

3(u†u + d†d) QI =

• d3x
• (1 + δZ)(π2 ˙

π1 − π1 ˙ π2) + 1 2

• u†u − d†d
• .

5

Symmetry breaking: At small T when either h = 0 or h = 0 and m2 < 0 ⇒ φ0 ≡ σ ≡ v = 0 At large µI ⇒ φ1 ≡ π1 ≡ ρ = 0 and φi ≡ πi = 0 for i = 2, 3 Shifting the corresponding fields

Z = Z DφD ¯ ψDψ » e−

R −iβ dt R d3x ¯ ψGf −1ψe−i R −iβ dt R d3x ˜ LI

Z DΠei

R −iβ dt R d3x(Π ˙ φ− ˜ HB)

– .

where the Π dependent part of Π ˙ φ − ˜ HB: Π ˙ φ − ˜ HB = − 1 2(1 − δZ)

• Π2

0 − 2Π0(1 + δZ) ˙

φ0

• −

1 2(1 − δZ)

• Π2

3 − 2Π3(1 + δZ) ˙

φ3

• −

1 2(1 − δZ)

• Π2

1 − 2Π1(1 + δZ)( ˙

φ1 − µIφ2)

• −

1 2(1 − δZ)

• Π2

2 − 2Π2(1 + δZ)( ˙

φ2 + µI(φ1 + ρ))

• .

6

Making whole squares in the above brakets Then performing the Π integration → produce the inverse bosonic propagators and the tree-level EoS (linear terms). Propagator matirices:

iGf

ij −1 =

„(−iωn + 1

3µB + 1 2µI)γ0 − γipi − gF 2 v

−i

gF 2 γ5ρ

−i

gF 2 γ5ρ

(−iωn + 1

3µB − 1 2µI)γ0 − γipi − gF 2 v

« , iGb

44 −1= (−iωn)2 − E2 π3,

iGb

kl −1=

B @ (−iωn − µI)2 − E2

π3 − λρ2

−λρ2 − √ 2λvρ −λρ2 (−iωn + µI)2 − E2

π3 − λρ2

− √ 2λvρ − √ 2λvρ − √ 2λvρ (−iωn)2 − E2

π3 − 2λv2

1 C A

Tree-level EoS: EoStree

σ

= v(m2 + λ(v2 + ρ2)) − h = 0, EoStree

π1

= ρ(m2 + λ(v2 + ρ2) − µ2

I (1 + δZ)) = 0.

7

Renormalizaiton conditions → finiteness of the perturbative N–point functions (the propagator and the four point boson vertex) Practically it is easier to obtain the counterterms from the one–loop EoS Note: there is some arbitrariness in choosing the finite parts With cut-off regularization: δm2 = −6λ(Λ2 − m2 ln Λ2 l2

b

) + g2

F

4π2NcΛ2, δλ = 12λ2 ln Λ2 l2

b

− g2

F

32π2Nc ln Λ2 el2

f

, δZ = −Nc g2

F

16π2 ln Λ2 e2l2

f

, lb, lf → bosonic and fermionic renormalization scales

8

Equations at one–loop level, the OPT

At finite temperature tree level mass squares can become negative → some sort

• f resummation is needed

Using the optimized perturbation theory (OPT):

• a temperature dependent mass term introduced in the Lagrangian
• the difference is treated as a higher order counterterm
• the new mass parameter determined by the FAC criterion (m1–loop = mtree) →

can be transformed to an equation for a resummed particle mass

• conserves Ward–identities

9

Equation for the resummed π3 mass: m2

π3(T, µ) = m2 + δm2 + (λ + δλ)(v2 + ρ2) + Σπ3(ω = p = 0, m2 π3, T, µ)

One–loop level EoS for σ:

v „ m2 + δm2 + (λ + δλ)(v2 + ρ2) + λ X

p

Z Tr{HbGb(ωn, p, µI)} + gF X

p

Z Tr{HfGf(ωn, p, µI, µB)} « = h

comparing the two equations → a Ward–identity is recognized vm2

π3 = h

One–loop level EoS for π1:

ρ „ m2 + δm2 + (λ + δλ)(v2 + ρ2) − µ2

I (1 + δZ) + λ

X

p

Z Tr{RbGb(ωn, p, µI)} +gF X

p

Z Tr{RfGf(ωn, p, µI, µB)} « = 0,

10

Calculation of the loop integrals requires the diagonalization of the propagators → Straightforward but the eigenvalues are non–rational functions of ωn → diagonalization for small ρ → Landau–type analysis Up to O(ρ3) m2

π3 = m2 + λv2 + t(0)(m2 π3, v, T, µI,B) + (λ + t(2)(m2 π3, v, T, µI,B))ρ2

Due to the Ward identity the EoS for σ remains vm2

π3 = h

ρ

• µ2

I − m2 − λv2 − r(0)(m2 π3, v, T, µI,B) − (λ + r(2)(m2 π3, v, T, µI,B))ρ2 + O(ρ4)

• = 0

⇒ Pion condensate non–zero only if the roots are real If λ + r(2) > 0 and µ2

I − m2 − λv2 − r(0) > 0

ρ =

• µ2

I − m2 − λv2 − r(0)(m2 π3, v, T, µI,B)

λ + r(2)(m2

π3, v, T, µI,B)

⇒ The transition is of second order

11

Diagonalized propagators

OB =    1 − |a|2 ρ2 b(1 − 2av)ρ2 − √ 2aρ b∗(1 − 2a∗v)ρ2 1 − |a|2 ρ2 − √ 2a∗ρ √ 2aρ √ 2a∗ρ 1 − 2 |a|2 ρ2    + O(ρ3), a = a(ωn, µ) = λv/(µ2 + 2λv2 + 2iωnµ) and b = b(ωn, µ) = iλ/(4µωn) i ˜ Gπ+ = 1 (ωn + iµI)2 + E2

π

− ρ2 λ(2µ2

I + 2λv2 − 4iµIωn)

((ωn + iµI)2 + E2

π)2(µ2 I + 2λv2 − 2iµIωn) + O(ρ4),

i ˜ Gπ− = 1 (ωn − iµI)2 + E2

π

− ρ2 λ(2µ2

I + 2λv2 + 4iµIωn)

((ωn − iµI)2 + E2

π)2(µ2 I + 2λv2 + 2iµIωn) + O(ρ4),

i ˜ Gσ = 1 ω2

n + E2 σ

− ρ2 λ(µ2

I + 2λv2)(µ2 I + 6λv2 + 4µ2 I ω2 n)

(ω2

n + E2 σ)2((µ2 I + 2λv2)2 + 4µ2 I ω2 n) + O(ρ4),

while the π3 propagator is iGπ3 = 1 ω2

n + E2 π

− ρ2 λ (ω2

n + E2 π)2 + O(ρ4).

Important to note − → the transformation matrix depends on ρ, ωn, µ

12

OF =   1 +

g2

F

32k2

0ρ2

−i gF

4k0γ0γ5ρ

−i gF

4k0γ0γ5ρ

1 +

g2

F

32k2

0ρ2

  , where k0 = (−iωn + 1

3µB)γ0 and the matrix is hermitian

i ˜ Gu/d = − 1 / pu/d − mf − ρ2 g2

F

8k0 1 / pu/d − mf γ0 1 / pu/d − mf where / pu/d = (−iωn + µu/d)γ0 − γipi and µu/d = µB/3 ± µI/2. Calculation of one–loop contributions (bosonic case): Tr{BbGb} = Tr{BbO−1

B OBGbO−1 B OB} = Tr{OBBbO−1 B ˜

Gb} = Tr{ ˜ Bb ˜ Gb} where ˜ Gb = diag( ˜ G−1

π+, ˜

G−1

π−, ˜

G−1

σ )

13

Parameterization

For finite temperature calculations → necessary to fix the parameters of the model, namely: m2, λ, gF, h and v Four physical quantity:

• one–loop pion mass: Mπ = 138 MeV
• one–loop sigma mass: Mσ = 500 MeV
• tree level u–d fermon mass: mf = 938/3 MeV
• pion decay constant: fπ = 93 MeV

and the EoS for σ is used.

M 2

π = m2 + λv2 + 3λ(T b 0(Mπ, lb) + T b 0(mσ, lb)) + 2g2 FNcT f,π

(mf, lf) M 2

σ

= m2 + 3λv2 + 3λ(T b

0(Mπ, lb) + T b 0(mσ, lb)) + 18λ2v2Bb 0(mσ, lb)

+ 6λ2v2Bb

0(Mπ, lb) + 6g2 FT f,σ

(mf, lf),

v = fπ, gF = 2mf fπ , h = fπM 2

π,

14

note: fixing the finite part of δZ → fixes the fermionic renormalization scale lf The bosonic scale dependence of the parameters:

510 520 530 540 550 560 570 580 590 300 400 500 600 700 800 900 [MeV] lb lf sqrt(-m2) 400 600 800 1000 1200 1400 1600 300 400 500 600 700 800 900 [MeV] lb Mσ=500 MeV fπλ mσ

Left panel m2 and lf Right panel the tree level σ mass and λ At lb ≈ 600 MeV the tree level σ mass equals its one – loop level value → σ mass becomes selfconsistent In addition m and lf moderately depend on the renormalization scale Choosing the following scale range: lb ∈ [400 MeV, 800 MeV]

15

Results

The second order boundary for the occurrence of the pion condensation in the µI − µB − T space: µ2

I − m2 π3(T, µI, µB) − R1–loop(T, µI, µB) = 0

100 200 300 400 500 600 700 800 100 200 300 400 500 600 700 20 40 60 80 100 120 140 160 T[MeV] µB [MeV] µI [MeV] T[MeV]

16

20 40 60 80 100 120 140 160 100 200 300 400 500 600 T[MeV] phase boundary: µB= 0 MeV µB=400 MeV µI=mπ: µB= 0 MeV µB=400 MeV

condensation starts at µI = 131 MeV

17

The one–loop pole masses of the charged pions detrmined at ρ = 0:

• M pole

π±

2 =

• mtree

π±

2 + Σπ±(ω = M pole

π± , p = 0, T, µI,B),

The known solutions of v(T, µI,B) and mπ3(T, µI,B) are used

100 200 300 400 500 600 700 100 200 300 400 500 600 [MeV] µI [MeV] mπ+ : T=150 MeV T=140 MeV T= 90 MeV mπ- : T=150 MeV T=140 MeV T= 90 MeV

18

Conclusion

• Bosonic scale dependence was investigated and a moderate dependence

was found

• The 2nd order boundary of pion condensation was determined in the

µI − µB − T space at one–loop level using a Landau-type of analysis.

• The surface starts steeply with increasing µI at fixed µB and towards large

values of µB the pion condensed region shrinks and even disappears at around µB = 830 MeV. (However, at such a high energy one should take into account the effects of the strange quark.)

• Investigating different sections of the surface at one–loop level the pion

condensation curve slightly differ from the µI = mπ3 curve at small µI and this deviation increases with increasing µI.

• µI dependence of charged pion pole masses were obtained.
• Possible continuation: Next to leading order in ρ: scaling, BEC-BCS transition;

Three flavor; Polyakov–loop