Inhomogeneous condensation in effective models for QCD using the - - PowerPoint PPT Presentation

Inhomogeneous condensation in effective models for QCD using the finite mode approach

Achim Heinz Chiral group, April 27th 2015

The phase diagram of Quantum Chromodynamics

LQCD = ¯ q

• ı /

D − m

• q − 1

4G a

µνG a µν

/ D = γµ ∂µ − ıgQCDAa

µta

, G a

µν = ∂µAa ν − ∂νAa µ − gQCDf abcAb µAc ν

Parity doublet model

Chiral limit mq → 0 (bare masses) QCD Lagrangian is then invariant under global chiral U(Nf )L × U(Nf )R transformations → chiral condensate φ order parameter of QCD. Parity doublet model for nucleons Besides the nucleon also a chiral partner of the nucleon is

• introduced. Chiral symmetry is preserved at all densities, this is in

contrast to other models e.g. Walecka model.

The chiral density wave

Inhomogeneous condensation: < σ >= φ(x) Recent studies of quark models show the relevance of inhomogeneous condensation at moderate densities. Are spatial modulations relevant for a state-of-the-art hadron model? Ansatz for chiral density wave: σ ∼ φ cos(2 f x) , π ∼ φ sin(2 f x)

Dynamical quantities as a function of µ

900 950 1000 1050 1100Μ MeV 50 100 150

Φ, Χ MeV CDW

900 950 1000 1050 1100Μ MeV 50 100 150

Φ, Χ MeV homogeneous phase

900 950 1000 1050 1100Μ MeV 50 100 150

Φ, Χ MeV homogeneous phase

900 950 1000 1050 1100Μ MeV 50 100 150

Φ, Χ MeV homogeneous phase

900 950 1000 1050 1100Μ MeV 50 100 150

Φ, Χ MeV homogeneous phase

900 950 1000 1050 1100Μ MeV 50 100 150

Φ, Χ MeV CDW

900 950 1000 1050 1100Μ MeV 50 100 150

Φ, Χ MeV CDW

solid line: φ, dashed line: ¯ χ

900 950 1000 1050 1100Μ MeV 100 200 300 400

f MeV

f is of the size of 1.5 fm

µ = 923 MeV transition to nuclear matter, µ = 975 MeV transition to the CDW. Homogeneous nuclear matter at ρ0, mixed phase between 2.49 ρ0 to 10.75 ρ0, purely inhomogeneous phase above 10.75 ρ0.

Finite-mode approach

“Wish list” for the calculation of arbitrary modulations

arbitrary modulations: σ(x) is not an input several inhomogeneous fields: σ(x), π(x), ω(x), ... no limitations on the dimension of the model: 1 + 1, 1 + 2, ... higher dimensional modulations: σ(x, y), σ(x, y, z), ...

New numerical tool: so-called “finite-mode approach” Lattice inspired method, which allows to access all points of the “wish list”.

Not so fast, there are a few limitations: numerically complex finite size corrections

• ptimization and tests

Gross-Neveu model in the finite-mode approach

LGN =

Nc

• i=1

¯ ψi (γµ∂µ + γ0µ) ψi + 1 2g 2 Nc

• i=1

¯ ψiψi 2

in 1 + 1 dimensions renormalizable asymptotic freedom discrete chiral symmetry in the large Nc limit chiral symmetry is broken

GN model in the finite-mode approach

Effective action of the GN model: S = Nc 1 2λ

• d2x σ(x)2 − ln (detQ)
• ,

Q = γµ∂µ + γ0µ + σ(x)

finite spacetime volume ˆ L0ˆ L1 fermion fields as superposition of plane waves: ˆ ψj(ˆ x0, ˆ x1) =

• n0,n1

ηj,n0,n1 e−ı(ˆ

k0ˆ x0+ˆ k1ˆ x1)

ˆ L0ˆ L1 with the momenta: ˆ k0 = 2π ˆ L0 (n0 − 1/2) , ˆ k1 = 2π ˆ L1 n1 , n0, n1 ∈ N finite number temporal modes N0 and space modes N1

GN model in the finite-mode approach

Discretized effective action for a homogeneous condensate ˆ σ(x) ≡ ˆ σ:

S Nc = ˆ L1ˆ L0 ˆ σ2 2λ −

N0

• n0=1

N1

• n1=−N1

ln 2π ˆ L0 1 2 − n0 2 + 2π ˆ L1 n1 2 + ˆ σ2 − ˆ µ2 2 +

• 2ˆ

µ 2π ˆ L0 1 2 + n0 2

finite number of modes N0 and N1 introduces momentum cutoffs: ˆ kcut = 2π ˆ L0 N0 , ˆ kcut

1

= 2π ˆ L1 (N1 + 1/2) Effective action as a function of: λ, N0, N1, ˆ kcut and ˆ kcut

1 S Nc = 2π2N0(N1 + 1/2)ˆ σ2 λˆ kcut ˆ kcut

1

−

+N0

• n0=1

N1

• n1=−N1

ln

• ˆ

kcut n0 − 1/2 N0 2 +

• ˆ

kcut

1

n1 N1 + 1/2 2 + ˆ σ2 − ˆ µ2 2 +

• 2ˆ

µˆ kcut n0 − 1/2 N0 2

Choosing and determining suitable parameters

The temperature ˆ T is proportional to 1/N0: low temperature ˆ T ≈ 0, where ˆ σ( ˆ T) ≈ ˆ σ0: N0 → N00 critical temperature ˆ T = ˆ Tc, where ˆ σ( ˆ T) = 0: N0 → N0c ≪ N00 ˆ Tc = ˆ Tc(λ, N00, N0c, N1, ˆ kcut , ˆ kcut

1

) N00 and N1 are limited by computer resources λ and ˆ kcut follow from the gap equation 0 =

d d ˆ σ S N for ˆ

T ≈ 0 and ˆ T = ˆ Tc N0c and ˆ kcut

1

are fixed due to

∂ ∂N0c ˆ

Tc = 0 and

∂ ∂ˆ kcut

1

ˆ Tc = 0 Only N00 and N1 are free parameters

Tc as a function of of ˆ kcut

1

and N1

50 100 150 200 250 300k

• 1

cut 0.560 0.562 0.564 0.566 0.568 0.570

T

• c

for small ˆ kcut

1

no improvement with increasing N1

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

k

• 1

cut

N1 0.5

0.560 0.562 0.564 0.566 0.568 0.570

T

• c

for large ˆ kcut

1 /(N1 + 0.5) on

improvement with increasing N1

N1 = 64 (green curve) N1 = 128 (red curve) N1 = 256 (blue curve)

• ptimal choice for ˆ

kcut

1

at

∂ ∂ˆ kcut

1

ˆ Tc = 0 N00 = 256 and N0c = 36, λ and ˆ kcut are fixed, black lines indicate the infinite volume continuum result ˆ Tc = eC /π

Performance of the finite mode approach

• N00 480

N00 120 N00 96 N00 72 N00 48

100 200 300 400

N1

0.56680 0.56685 0.56690 0.56695 0.56700 0.56705 0.56710

T

• c

ˆ Tc as a function of N1 and N00, the black line the infinite volume continuum result ˆ Tc = eC /π. Fast convergence for symmetric choices N00 = N1 (filled black circles)

GN model phase diagram

• II

III I

0.0 0.5 1.0 1.5

Μ

• 0.0

0.1 0.2 0.3 0.4 0.5 0.6

T

• 0.2

0.4 0.6 0.8 1.0

x

• 1.0

0.5 0.5 1.0

Σ

• The shape of the condensate

ˆ σ(x) close to I / III phase boundary. I: homogeneous broken phase II: restored phase III: inhomogeneous phase ˆ σ(ˆ x1) =

+M

• m=−M

cme−iˆ

pˆ x1

ˆ p = 2π ˆ L1 m , c+m = (c−m)∗

ˆ σ := σ/σ0 , ˆ µ := µ/σ0 , ˆ t := T/σ0

χGross-Neveu model in the finite-mode approach

LχGN =

N

• i=1

¯ ψi (γµ∂µ) ψi + g2 2   N

• i=1

¯ ψiψi 2 + N

• i=1

¯ ψiıγ5ψi 2  discrete chiral symmetry → continuous chiral symmetry ˆ σ(x) =

N

• i=1

¯ ψiψi

• =

10

• n=−10

cneınx ˆ η(x) =

N

• i=1

¯ ψiıγ5ψi

• =

10

• n=−10

dneınx

• nly two phases are present:

restored phase with ˆ σ(x) = ˆ η(x) = 0 chiral density wave (CDW) with ˆ σ(x) = ˆ σ cos(2 f x) , and ˆ η(x) = ˆ σ sin(2 f x)

CDW restored phase 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Μ

• 0.1

0.2 0.3 0.4 0.5 0.6 0.7

t

χGross-Neveu model in the finite-mode approach

0.2 0.4 0.6 0.8 1.0

x

• 1.0

0.5 0.5 1.0

Σ , Η

• ˆ

µ = 0.295 at ˆ t = 0.0945, first coefficient

0.2 0.4 0.6 0.8 1.0

x

• 1.0

0.5 0.5 1.0

Σ , Η

• ˆ

µ = 0.585 at ˆ t = 0.0945, second coefficient

0.2 0.4 0.6 0.8 1.0

x

• 1.0

0.5 0.5 1.0

Σ , Η

• ˆ

µ = 0.875 at ˆ t = 0.0945, third coefficient

CDW is not an input! It follows from a dynamical minimization of the σ- and the η-condensate.

0.2 0.4 0.6 0.8 1.0

Μ

• 4

3 2 1

S

• eff

effective action at ˆ t = 0.189. The different colours correspond to different local minima

The NJL model in the finite-mode approach

Lagrangian of the NJL model LNJL =

Nc

• i=0

¯ ψi / ∂ψi − G Nc   Nc

• i=0

¯ ψiψi 2 + Nc

• i=0

¯ ψiıγ5ψi 2  S Nc = λ 2

• d4x m∗2(x) − ln (det Q)
• ,

Q = γµ∂µ + γ0µ + m∗2(x) Looks similar but: 1 + 3 dimensional model not renormalizable continuous chiral symmetry regularization scheme effects phase diagram computational effort increases strongly

Regularization with the Pauli-Villars scheme

Discretized effective action of the NJL model, ˆ m∗2(x) ≡ ˆ m∗2, ˆ µ = 0:

S Nc = ˆ L1ˆ L0 ˆ m∗2 2λ − 2

N0

• n0=1

N1

• n1,2,3=−N1

ln   2π ˆ L0 1 2 − n0 2 +

3

• i=1
• 2π

ˆ Li ni 2 + ˆ m∗2  

Adding further heavy fermions will regularize the sum:

ln   2π ˆ L0 1 2 − n0 2 +

3

• i=1

2π ˆ L1 ni 2 + ˆ m∗2   →

m

• a=0

Caln   2π ˆ L0 1 2 − n0 2 +

3

• i=1

2π ˆ L1 ni 2 + ˆ m∗2 + αaˆ Λ2

PV

  , with C0 = 1 and α0 = 0

How to chose Ca and αa?

Regularization with the Pauli-Villars scheme

n1 → ∞, spherical coordinates

m

• a=0

Ca 2π ˆ L1 n1 2 ln 2π ˆ L0 1 2 − n0 2 + 2π ˆ L1 n1 2 + ˆ m∗2 + αaˆ Λ2

PV

• ,

with C0 = 1 and α0 = 0 → 2π ˆ L1 n1 2 ln 2π ˆ L1 n1

• m
• a=0

Ca + 2π ˆ L0 1 2 − n0 2 + ˆ m∗2

• m
• a=0

Ca + ˆ Λ2

PV m

• a=0

Caαa + O(1/n2

1)

High modes are suppressed when: m

a=0 Caαa = 0 , and m a=0 Ca = 0 2π ˆ L1 n1 ≪ ˆ

ΛPV , spherical coordinates

m

• a=0

Ca 2π ˆ L1 n1 2 ln 2π ˆ L0 1 2 − n0 2 + 2π ˆ L1 n1 2 + ˆ m∗2 + αaˆ Λ2

PV

• ,

with C0 = 1 and α0 = 0 → 2π ˆ L1 n1 2 ln 2π ˆ L0 1 2 − n0 2 + 2π ˆ L1 n1 2 + ˆ m∗2

• + const. + O(1/ˆ

Λ2

PV )

NJL model Phase diagram

II III I

0.25 0.30 0.35 0.40 0.45Μ GeV 0.00 0.02 0.04 0.06 0.08 0.10 0.12

T GeV

II III I

1 2 3 4

Μ GeV

0.00 0.05 0.10 0.15 0.20

T GeV

I: homogeneous broken phase II: restored phase III: inhomogeneous region

m0 = 350 MeV, Ni = 120, and N00 = 120

NJL model Phase diagram

1 2 3 4

Μ GeV

0.00 0.05 0.10 0.15 0.20

T GeV

1 2 3 4

Μ GeV

0.00 0.05 0.10 0.15 0.20

T GeV

m0 = 350 MeV m0 = 300 MeV

black line: two regulating masses blue line: three regulating masses Ni = 120, and N00 = 120

Summary and Outlook

relevance of inhomogeneous phases at moderate densities extension to three flavours further parity doublets ∆ resonances spatial ω0 modulations . . . solid approach to calculate inhomogeneous phases higher dimensional modulations interweaving chiral spirals further effective models diquark condensation . . .

• D. Parganlija, F. Giacosa, D. H. Rischke, arXiv:1003.4934 [hep-ph]
• D. Parganlija, P. Kovacs, G. Wolf, F. Giacosa and D. H. Rischke,

arXiv:1208.2054 [hep-ph]

• S. Janowski and F. Giacosa, arXiv:1312.1605 [hep-ph]
• S. Gallas, F. Giacosa, G. Pagliara, arXiv:1105.5003v1 [hep-ph]
• A. H., [arXiv:1301.3430]
• A. H., F. Giacosa, D. H. Rischke, arXiv:1312.3244 [nucl-th]
• M. Wagner, arXiv:hep-ph/0704.3023v1 [hep-lat]
• M. Thies, K. Urlichs, [hep-th/0302092]
• G. Basar, G. V. Dunne and M. Thies, arXiv:0903.1868 [hep-th]
• D. Nickel, arXiv:0906.5295 [hep-ph]
• S. Carignano, M. Buballa, arXiv:1111.4400 [hep-ph] and arXiv:1203.5343

[hep-ph]