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Pion condensate versus chiral density wave at zero temperature Patrick Kneschke (University of Stavanger) in collaboration with Jens O. Andersen (NTNU Trondheim) June 25, 2018 Strong and ElectroWeak Matter, Barcelona 2018 J. O. Andersen, P.
Patrick Kneschke (University of Stavanger)
in collaboration with Jens O. Andersen (NTNU Trondheim)
June 25, 2018 Strong and ElectroWeak Matter, Barcelona 2018
“Chiral density wave versus pion condensation at finite density and zero temperature”, Phys. Rev. D 97, 076005
− → inhomogeneous phases predicted around µB = 1 GeV possible
Fukushima et al. Rept.Prog.Phys. 74 (2011) 014001
The task is to find phase boundaries and critical points! Perturbative QCD breaks down at low energies Lattice QCD can be used to find the phase transition at µ = 0, but problematic at non zero chemical potential ⇒ sign problem (complex fermion determinant) Effective theories of QCD, that model chiral symmetry and/or confinement are used to map
(Nambu-Jona-Lasinio model, quark meson model, chiral perturbation theory)
Patrick Kneschke (University of Stavanger) in collaboration with Jens O. Andersen (NTNU Trondheim) () Pion condensate versus chiral density wave at zero temperature June 25, 2018 2
The isospin chemical potential µI introduces an imbalance between up- and down quarks. µu = µ + µI , µd = µ − µI At large enough densities bose condensation of charged pions is possible.
TΧ TΡ TBCS
50 100 150 50 100 150 200 ΜI MeV T MeV
I = mπ 2
No sign problem for isospin chemical potential ⇒ Lattice QCD works well and can be used for comparison with model calculations
Patrick Kneschke (University of Stavanger) in collaboration with Jens O. Andersen (NTNU Trondheim) () Pion condensate versus chiral density wave at zero temperature June 25, 2018 3
Inhomogeneous chiral condensate: Cooper pair of quarks with non-zero net momentum q ⇒ complex order parameter ∆ei
q· x
For simplicity we consider a one-dimensional modulation ∆eiqz At finite temperature the order is destroyed by thermal fluctuations of phonons ⇒ map out the µ − µI phase diagram at zero temperature previous study in 1+1 dimensional NJL model showed pion and inhomogeneous condensate don’t coexist → We extend this to a 3+1 dimensional QM model
Patrick Kneschke (University of Stavanger) in collaboration with Jens O. Andersen (NTNU Trondheim) () Pion condensate versus chiral density wave at zero temperature June 25, 2018 4
Lagrangian of the QM model in Minkowski space is:
1 √ 2 (π1 ± iπ2)
2
+ (∂µ + 2iµI δµ,0) π− ∂µ − 2iµI δµ,0 π+ + 1 2 m2(σ2 + π2
3 + 2π+π−) + λ
24 (σ2 + π2
3 + 2π+π−)2 − hσ
+ ψf
∂ + γ0(µ + µI τ3) − g(σ + iγ5 τ · π)
Replace the meson fields with constant background fields in mean field approximation. Chiral symmetry is broken by chiral condensate, here in the form of a chiral density wave (CDW): < σ >= φ0 cos(qz) , < π3 >= φ0 sin(qz) Allow for charged pion condensate: < π1 >= π0 , < π2 >= 0 Introducing the new variables ∆ = gφ0 and ρ = gπ0 we get the meson potential: V0 = 1 2 q2 ∆2 g2 + 1 2 m2 g2 ∆2 + 1 2 m2 − 4µ2
I
g2 ρ2 + 1 24 λ g4 (∆2 + ρ2)2 − h g ∆δq,0
Patrick Kneschke (University of Stavanger) in collaboration with Jens O. Andersen (NTNU Trondheim) () Pion condensate versus chiral density wave at zero temperature June 25, 2018 5
In the large Nc limit we consider only quark corrections to the potential: V1 = − 1 βV log
D
/ D = i / ∂ + γ0(µ + µI τ3) − ∆ + 1
2 qτ3γ5γ3 − iργ5τ1
V1 = −Nc
1 2 E ±
f
+ 1 2 E ±
¯ f
+ T log
f
−µ)
+ T log
¯ f
+µ)
with the quark energies: E ±
u = E(±q, −µI ) ,
E ±
d = E(±q, µI ) ,
E ±
¯ u = E(±q, µI ) ,
E ±
¯ d = E(±q, −µI )
E(q, µI ) =
⊥ +
+ ∆2 + q
2 2 + µI 2 + ρ2 1 2
Patrick Kneschke (University of Stavanger) in collaboration with Jens O. Andersen (NTNU Trondheim) () Pion condensate versus chiral density wave at zero temperature June 25, 2018 6
1 ǫ + log Λ2 M2
2
0 = m2 ren + δm2 etc.
tree-level: m2 = − 1 2
σ − 3m2 π
σ − m2 π
π
, h = m2
πfπ , g = mq
fπ
2
σ − 3δm2 π
σ − δm2 π
π
− λ δf 2
π
f 2
π
δh = δmπfπ + h δfπ fπ , − δg2 g2 = δf 2
π
f 2
π
= δZπ , δmq = 0
Σσ,π
σ,π
σ,π = 0
⇒ m2
σ,π,ren = m2 σ,π is pole of the propagator
∂ ˆ Σσ,π ∂p2
σ,π
⇒ residue at the pole is one
Patrick Kneschke (University of Stavanger) in collaboration with Jens O. Andersen (NTNU Trondheim) () Pion condensate versus chiral density wave at zero temperature June 25, 2018 7
m2
MS = m2 + 4g2Nc
(4π)2
m2
q
− 2m2
q − 1
2
σ − 4m2 q
σ) + 3
2 m2
πF(m2 π)
λMS = λ + 12g2Nc (4π)2f 2
π
q
m2
q
+ F(m2
σ)
σ
m2
q
+ F(m2
π) + m2 πF ′(m2 π)
π
m2
q
+ 2F(m2
π) + m2 πF ′(m2 π)
g2
MS = g2
(4π)2
m2
q
+ F(m2
π) + m2 πF ′(m2 π)
hMS = h
(4π)2
m2
q
+ F(m2
π) − m2 πF ′(m2 π)
2 q2 ∆2 g2
MS
+ 1 2 m2
MS
∆2 g2
MS
+ 1 2
MS − 4µ2 I
ρ2 g2
MS
+ λMS 24
g4
MS
− hMS ∆ gMS δ0,q + Vfin
Patrick Kneschke (University of Stavanger) in collaboration with Jens O. Andersen (NTNU Trondheim) () Pion condensate versus chiral density wave at zero temperature June 25, 2018 8
We set q = 0 and choose mπ = 140 MeV , mσ = 600 MeV , mq = 300 MeV , fπ = 93 MeV
mq ΧSB 2nd order Ρ 1st order Ρ
100 200 300 400 50 100 150 200 Μ MeV ΜI MeV 50 100 150 200 50 100 150 200 250 300 ΜI MeV ,Ρ MeV 100 200 300 400 50 100 150 200 250 300 Μ MeV MeV 50 100 150 200 50 100 150 200 250 300 ΜI MeV ,Ρ MeV
Patrick Kneschke (University of Stavanger) in collaboration with Jens O. Andersen (NTNU Trondheim) () Pion condensate versus chiral density wave at zero temperature June 25, 2018 9
We determine the position of the phase transition with a Ginzburg-Landau Analysis: V = α0 + α2ρ2 + α4ρ4; V1 ≈ −2Nc
2 ρ2
− 1 8 ρ4 (
Vfin,2 = −2Ncρ2
p2 + ∆2 − µI 2 − 1
µ2
I
(p2 + ∆2)
3 2
(4π)2 ρ2µ2
I
1 s
I
(4π)2 F(4µ2
I ),
Vfin,4 = 1 4 Ncρ4
1
3 − 2 (p2 + ∆2)
3 2
= 2Nc (4π)2 ρ4
s2 + 1 s3
µ2
I
− 2
1 s
Pion condensate versus chiral density wave at zero temperature June 25, 2018 10
Evaluated at the vacuum value ∆ = mq : α2 = 1 2 f 2
π
m2
q
π
4m2
qNc
(4π)2f 2
π
m2
πF ′(m2 π)
I
4m2
qNc
(4π)2f 2
π
π) + m2 πF ′(m2 π) − F(4µ2 I )
α4 = 1 8 m2
σf 2 π
m4
q
4m2
qNc
(4π)2f 2
π
4m2
q
m2
σ
σ) + F(m2 π) + m2 πF ′(m2 π)
8 m2
πf 2 π
m4
q
4m2
qNc
(4π)2f 2
π
m2
πF ′(m2 π)
(4π)2
s2 + 1 s3
µ2
I
− 2
1 s
2 we get α2 = 0 and α4 > 0 ⇒ second order phase transition. This also applies for all µ < mq − mπ 2 (vacuum phase)
Patrick Kneschke (University of Stavanger) in collaboration with Jens O. Andersen (NTNU Trondheim) () Pion condensate versus chiral density wave at zero temperature June 25, 2018 11
pion mass dependence
The explicit symmetry breaking term: −hMS ∆ gMS δ0,q ∼ −m2
πfπ
∆ g δ0,q If the pion mass is to large, the homogeneous phase is always preferred. The width of the inhomogeneous phase decreases with increasing pion mass:
260 280 300 320 340 100 200 300 400 μ (MeV) Δ(MeV), q (MeV) 10 20 30 40 300 310 320 330 340 350 mΠMeV ΜMeV
A more general ansatz, like a chiral soliton lattice can save the inhomogeneous phase even at the physical point, but here we stay in the chiral limit.
Patrick Kneschke (University of Stavanger) in collaboration with Jens O. Andersen (NTNU Trondheim) () Pion condensate versus chiral density wave at zero temperature June 25, 2018 12
chiral limit
ΡΡ0 homogeneous inhomogeneous
260 280 300 320 340 50 100 150 ΜMeV ΜIMeV
50 100 150 200 100 200 300 400 ΜI MeV Ρ MeV 280 290 300 310 320 330 340 350 100 200 300 400 Μ MeV ,Ρ,q MeV 280 290 300 310 320 330 340 350 100 200 300 400 Μ MeV Ρ MeV
Patrick Kneschke (University of Stavanger) in collaboration with Jens O. Andersen (NTNU Trondheim) () Pion condensate versus chiral density wave at zero temperature June 25, 2018 13
We find the onset of pion condensation at exactly µI = mπ 2 for µ < mq − mπ 2 CDW only exists for small pion masses (and at zero temperature). No overlap between pion condensate and inhomogeneous condensate is found in the chiral limit. More sophisticated models, like a chiral soliton lattice should be studied. Extension to an inhomogeneous pion condensate possible in the future.
Patrick Kneschke (University of Stavanger) in collaboration with Jens O. Andersen (NTNU Trondheim) () Pion condensate versus chiral density wave at zero temperature June 25, 2018 14
We find the onset of pion condensation at exactly µI = mπ 2 for µ < mq − mπ 2 CDW only exists for small pion masses (and at zero temperature). No overlap between pion condensate and inhomogeneous condensate is found in the chiral limit. More sophisticated models, like a chiral soliton lattice should be studied. Extension to an inhomogeneous pion condensate possible in the future.
Patrick Kneschke (University of Stavanger) in collaboration with Jens O. Andersen (NTNU Trondheim) () Pion condensate versus chiral density wave at zero temperature June 25, 2018 14