1 / N expansion for pion gas Fermi systems NR Fermi gas and - - PowerPoint PPT Presentation

1/N expansion

• T. Brauner

O(N) sigma model

Auxiliary field technique Next-to-leading order Summary – part I

Fermi systems

NR Fermi gas Dense quark matter Summary – part II

1/N expansion for pion gas and strongly interacting Fermi systems

Tom´ aˇ s Brauner

Institut f¨ ur Theoretische Physik Goethe Universit¨ at Frankfurt am Main

R+R Budapest, 3 April 2009

1/N expansion

• T. Brauner

O(N) sigma model

Auxiliary field technique Next-to-leading order Summary – part I

Fermi systems

NR Fermi gas Dense quark matter Summary – part II

Outline

1

O(N) sigma model and 1PI-1/N expansion Auxiliary field technique Next-to-leading order Summary – part I

2

Strongly interacting Fermi systems Nonrelativistic Fermi gas Dense quark matter Summary – part II

1/N expansion

• T. Brauner

O(N) sigma model

Auxiliary field technique Next-to-leading order Summary – part I

Fermi systems

NR Fermi gas Dense quark matter Summary – part II

Outline

1

O(N) sigma model and 1PI-1/N expansion Auxiliary field technique Next-to-leading order Summary – part I

2

Strongly interacting Fermi systems Nonrelativistic Fermi gas Dense quark matter Summary – part II

1/N expansion

• T. Brauner

O(N) sigma model

Auxiliary field technique Next-to-leading order Summary – part I

Fermi systems

NR Fermi gas Dense quark matter Summary – part II

Introduction

QCD at low temperatures: include only the lightest DOFs, i.e., the Goldstone bosons of the spontaneously broken SUL(Nf)×SUR(Nf) chiral symmetry. For Nf = 2 make use of the isomorphism SU(2)×SU(2) ≃ SO(4) and study the O(4) model. O(N) model studied in the 1PI-1/N expansion since long time ago.

Coleman, Jackiw, and Politzer (1974): LO at T = 0; Root (1974): NLO at T = 0 Meyers-Ortmanns, Pirner, and Schaefer (1993): LO at T = 0 Andersen, Boer, and Warringa (2004): NLO pressure at T = 0

O(N) model also studied extensively in the 2PI-1/N formalism.

Baym and Grinstein (1977); Amelino-Camelia and Pi (1993) Petropoulos (1999); Lenaghan and Rischke (2000)

Here: renormalized 1PI-1/N expansion to NLO including solution of gap equation at nonzero temperature. Related work: Jakov´

ac (2008); Fej˝

• s, Patk´
• s, and Sz´

ep (2009)

1/N expansion

• T. Brauner

O(N) sigma model

Auxiliary field technique Next-to-leading order Summary – part I

Fermi systems

NR Fermi gas Dense quark matter Summary – part II

Introduction

QCD at low temperatures: include only the lightest DOFs, i.e., the Goldstone bosons of the spontaneously broken SUL(Nf)×SUR(Nf) chiral symmetry. For Nf = 2 make use of the isomorphism SU(2)×SU(2) ≃ SO(4) and study the O(4) model. O(N) model studied in the 1PI-1/N expansion since long time ago.

Coleman, Jackiw, and Politzer (1974): LO at T = 0; Root (1974): NLO at T = 0 Meyers-Ortmanns, Pirner, and Schaefer (1993): LO at T = 0 Andersen, Boer, and Warringa (2004): NLO pressure at T = 0

O(N) model also studied extensively in the 2PI-1/N formalism.

Baym and Grinstein (1977); Amelino-Camelia and Pi (1993) Petropoulos (1999); Lenaghan and Rischke (2000)

Here: renormalized 1PI-1/N expansion to NLO including solution of gap equation at nonzero temperature. Related work: Jakov´

ac (2008); Fej˝

• s, Patk´
• s, and Sz´

ep (2009)

1/N expansion

• T. Brauner

O(N) sigma model

Auxiliary field technique Next-to-leading order Summary – part I

Fermi systems

NR Fermi gas Dense quark matter Summary – part II

O(N) model

Generalize the O(4) model to arbitrary N by a suitable redefinition of the couplings.

L = 1

2(∂µφi)2 + λb 8N(φiφi −Nf 2

π,b)2

Some leading-order, O(N), contributions to the pressure: Some next-to-leading-order, O(1), contributions to the pressure:

1/N expansion

• T. Brauner

O(N) sigma model

Auxiliary field technique Next-to-leading order Summary – part I

Fermi systems

NR Fermi gas Dense quark matter Summary – part II

Auxiliary field technique

Introduce a new auxiliary field α and add pure Gaussian integral over α.

∆L = N 2λb

• α− iλb

2N(φiφi −Nf 2

π,b)

2

L = 1

2(∂µφi)2 − i 2α(φiφi −Nf 2

π,b)+ N

2λb α2

Systematic renormalization of divergences possible order by

• rder in 1/N by redefinition of the parameters.

f 2

π,b = f 2 π +a0 + 1

Na1 +··· , 1 λb = 1 λ +b0 + 1 Nb1 +···

Introduce explicit chiral-symmetry breaking term.

L → L −

√ NHσ

1/N expansion

• T. Brauner

O(N) sigma model

Auxiliary field technique Next-to-leading order Summary – part I

Fermi systems

NR Fermi gas Dense quark matter Summary – part II

1/N expansion Andersen and TB (2008)

Introduce the chiral condensate φ0 and auxiliary field condensate M, and shift the fields.

σ → √ Nφ0 +σ, α → iM2 + α √ N

The auxiliary field trick reduces resummation of all NLO graphs to a single Gaussian integral.

SNLO

eff

βV = 1 2(N −3)∑

Z

P log(P2 +M2)−NHφ0 − NM4

2λ + 1 2NM2(φ2

0 −f 2 π)

−1 2NM2a0 − 1 2NM4b0+1 2∑

Z

P χTD−1χ∗ − 1

2M2a1 − 1 2M4b1

D−1 =

1

2Π(P,M)+ 1 λ +b0

−iφ0 −iφ0 P2 +M2

• ,

χ =

• α

σ

• Π(P,M) = ∑

Z

Q

1 Q2 +M2 1 (P+Q)2 +M2

1/N expansion

• T. Brauner

O(N) sigma model

Auxiliary field technique Next-to-leading order Summary – part I

Fermi systems

NR Fermi gas Dense quark matter Summary – part II

LO effective potential

LO thermodynamic potential includes the condensate contributions and thermal fluctuations of massless pions.

VLO = M2 2 (f 2

π −φ2 0)+ T4

64π2 J0(βM)+ M4 64π2 32π2 λ +log µ2 m2 + 1 2

• +Hφ0

where

J0(βM) = 32 3T4

Z ∞

0 dp p4

ωp n(ωp)

and

ωp =

• p2 +M2

LO spectrum at N = 4: 4 massless pions governing the low-T pressure. The correction to 3 at NLO. LO renormalization by the LO counterterms.

a0 = Λ2 16π2 , b0 = − 1 32π2 log Λ2 µ2

LO β-function: β(λ) =

λ2 16π2.

1/N expansion

• T. Brauner

O(N) sigma model

Auxiliary field technique Next-to-leading order Summary – part I

Fermi systems

NR Fermi gas Dense quark matter Summary – part II

NLO effective potential

NLO expression for the effective potential (pressure):

VNLO = −1 2∑

Z

P logJ(P,M)+ 1

2M2a1 + 1 2M4b1 J(P,M) = 1 2Π(P,M)+ 1 λ +b0 + φ2 P2 +M2

Contribution from the dynamics of σ and α; must be evaluated numerically. NLO effective action ⇒ 1/N correction to the pion mass. In chiral limit, pion exactly massless at each order of 1/N. NLO effective action ⇒ LO sigma propagator.

1/N expansion

• T. Brauner

O(N) sigma model

Auxiliary field technique Next-to-leading order Summary – part I

Fermi systems

NR Fermi gas Dense quark matter Summary – part II

Extraction of divergences

Expand the log in VNLO in inverse powers of momentum.

logJ = log

• κ+log Λ2

P2

• + 2

P2

• M2 + G−2M2

κ+log Λ2

P2

• − 2

P4   2M4 + 3M2(G− 3

2M2)

κ+log Λ2

P2

+ (G−2M2)2

• α+log Λ2

P2

2   +···

G = 16π2φ2

0 +T2J1(βM)−M2 log µ2

M2 − 32π2M2 λ , κ = 1+32π2 1 λ +b0

1/N expansion

• T. Brauner

O(N) sigma model

Auxiliary field technique Next-to-leading order Summary – part I

Fermi systems

NR Fermi gas Dense quark matter Summary – part II

Extraction of divergences

Expand the log in VNLO in inverse powers of momentum.

logJ = log

• κ+log Λ2

P2

• + 2

P2

• M2 + G−2M2

κ+log Λ2

P2

• − 2

P4   2M4 + 3M2(G− 3

2M2)

κ+log Λ2

P2

+ (G−2M2)2

• α+log Λ2

P2

2   +···

G = 16π2φ2

0 +T2J1(βM)−M2 log µ2

M2 − 32π2M2 λ , κ = 1+32π2 1 λ +b0

• Quartic UV-divergence, independent of M,φ0,ρ0.

Subtracted within the vacuum pressure. Quadratic UV-divergence. Absorbed in the f 2

π counterterm a1.

Logarithmic UV-divergence. Absorbed in the 1

λ counterterm b1. NLO β-function:

β(λ) = λ2 16π2

• 1+ 8

N

1/N expansion

• T. Brauner

O(N) sigma model

Auxiliary field technique Next-to-leading order Summary – part I

Fermi systems

NR Fermi gas Dense quark matter Summary – part II

NLO renormalization procedure

The quadratic divergence contains temperature-dependent terms in G! T-dependence only disappears upon using LO gap equation for M: G = 16π2f 2

π.

To obtain divergence-free gap equations, one should be able to renormalize the effective potential off the LO minimum! Way out: We only need to use the EOM for M, not for the physical condensate φ0. When this is treated as merely a constraint to eliminate M in favor of φ0, we get an effective potential of φ0 solely, which is renormalizable for any value of the classical field φ0.

Coleman, Jackiw, and Politzer (1974)

1/N expansion

• T. Brauner

O(N) sigma model

Auxiliary field technique Next-to-leading order Summary – part I

Fermi systems

NR Fermi gas Dense quark matter Summary – part II

NLO renormalization procedure

The quadratic divergence contains temperature-dependent terms in G! T-dependence only disappears upon using LO gap equation for M: G = 16π2f 2

π.

To obtain divergence-free gap equations, one should be able to renormalize the effective potential off the LO minimum! Way out: We only need to use the EOM for M, not for the physical condensate φ0. When this is treated as merely a constraint to eliminate M in favor of φ0, we get an effective potential of φ0 solely, which is renormalizable for any value of the classical field φ0.

Coleman, Jackiw, and Politzer (1974)

1/N expansion

• T. Brauner

O(N) sigma model

Auxiliary field technique Next-to-leading order Summary – part I

Fermi systems

NR Fermi gas Dense quark matter Summary – part II

Numerical results: Chiral limit

Chiral condensate increases by about 1/4, critical temperature by about 1/3 ⇒ compatible with 1/N. The parameter M acquires nonzero value even in the symmetry-broken phase. This is no contradiction with the Goldstone theorem, since at NLO M has no longer the interpretation as the pion mass.

1/N expansion

• T. Brauner

O(N) sigma model

Auxiliary field technique Next-to-leading order Summary – part I

Fermi systems

NR Fermi gas Dense quark matter Summary – part II

Numerical results: Physical point

20 40 60 80 100 200 300 φ0 [MeV] T [MeV] 120 160 200 240 100 200 300 M [MeV] T [MeV]

Besides solving the full gap equation numerically, the NLO correction to the chiral condensate can also be estimated by various means: φNLO = −

dVNLO(φLO

0 )

dφ0 d2VLO(φLO

0 )

dφ2

, φNLO = −

dVNLO(φLO

0 )

dφ0 d2VLO(φLO

0 )

dφ2

+ 1

N d2VNLO(φLO

0 )

dφ2

1/N expansion

• T. Brauner

O(N) sigma model

Auxiliary field technique Next-to-leading order Summary – part I

Fermi systems

NR Fermi gas Dense quark matter Summary – part II

NLO critical behavior

Weak coupling calculation predicts a decrease of Tc at NLO.

Tc =

• 12

1+ 2

N

fπ

Nontrivial dependence of NLO Tc on the coupling. Analytic expression for the NLO critical exponent:

νNLO = 1 2 − 4 Nπ2

Agrees with the numerical solution to the gap equation.

1/N expansion

• T. Brauner

O(N) sigma model

Auxiliary field technique Next-to-leading order Summary – part I

Fermi systems

NR Fermi gas Dense quark matter Summary – part II

Summary – part I

The auxiliary field technique allows efficient resummation

• f the thermodynamic potential up to NLO.

In order to renormalize the effective action consistently, we need to eliminate the auxiliary field first. Using this strategy, we renormalized and solved the NLO gap equation at nonzero temperature. In the chiral limit, we studied the critical behavior and coupling dependence of the critical temperature, and established connection to the analytic results.

1/N expansion

• T. Brauner

O(N) sigma model

Auxiliary field technique Next-to-leading order Summary – part I

Fermi systems

NR Fermi gas Dense quark matter Summary – part II

Outline

1

O(N) sigma model and 1PI-1/N expansion Auxiliary field technique Next-to-leading order Summary – part I

2

Strongly interacting Fermi systems Nonrelativistic Fermi gas Dense quark matter Summary – part II

1/N expansion

• T. Brauner

O(N) sigma model

Auxiliary field technique Next-to-leading order Summary – part I

Fermi systems

NR Fermi gas Dense quark matter Summary – part II

A few preliminary words

Keywords

Condensed-matter physics: Bardeen–Cooper–Schrieffer (BCS) theory

• f superconductivity.

Bose–Einstein condensation (BEC). High-energy physics: Strongly interacting nuclear/quark matter. Quantum chromodynamics (QCD).

Physical realization

Atomic gases: Clear experimental realization of BEC and later, BCS–BEC crossover. QCD phase diagram: Attempts to apply the well-developed theory of BCS–BEC crossover to dense quark matter.

1/N expansion

• T. Brauner

O(N) sigma model

Auxiliary field technique Next-to-leading order Summary – part I

Fermi systems

NR Fermi gas Dense quark matter Summary – part II

BCS and BEC

= ⇒ Smooth crossover induced by changing ratio

• f Cooper pair size and interparticle distance. Eagles, Leggett

Atomic gases: shrink Cooper pairs by tuning interaction. QCD & friends: increase interparticle distance.

1/N expansion

• T. Brauner

O(N) sigma model

Auxiliary field technique Next-to-leading order Summary – part I

Fermi systems

NR Fermi gas Dense quark matter Summary – part II

Attractive nonrelativistic Fermi gas

Euclidean Lagrangian for an attractive, balanced, two-component Fermi gas with large scattering length:

L = ∑

σ=↑,↓

ψ†

σ

• ∂τ − ∇2

2m −µ

• ψσ −Gψ†

↑ψ† ↓ψ↓ψ↑

Decouple the interaction by introducing a pairing field, φ ∼ Gψ↓ψ↑, and the Nambu spinor Ψ = (ψ↑,ψ†

↓)T:

L = |φ|2

G −Ψ†D−1Ψ,

D−1 =

• −∂τ + ∇2

2m +µ

φ φ∗ −∂τ − ∇2

2m −µ

• Nikoli´

c and Sachdev (2007); Veillette, Sheehy and Radzihovsky (2007)

1/N expansion

• T. Brauner

O(N) sigma model

Auxiliary field technique Next-to-leading order Summary – part I

Fermi systems

NR Fermi gas Dense quark matter Summary – part II

What is N?

Make N copies of the spin- 1

2 fermion.

(ψ↑,ψ↓) → (ψ1↑,ψ1↓,...,ψN↑,ψN↓) The Euclidean Lagrangian generalizes to:

L = ψ†

iσ

• ∂τ − ∇2

2m −µ

• ψiσ − G

Nψ†

i↑ψ† i↓ψj↓ψj↑

The spin SU(2) symmetry extends to Sp(2N). Remains unbroken by the pairing field condensate. φ(x) ∼ G

N

∑

i=1

ψi↓(x)ψi↑(x) No unwanted NG bosons.

1/N expansion

• T. Brauner

O(N) sigma model

Auxiliary field technique Next-to-leading order Summary – part I

Fermi systems

NR Fermi gas Dense quark matter Summary – part II

Counting factors of 1/N

Bosonized action:

S = N

Z β

0 dτ

Z

d3x|φ(x,τ)|2 G −NTrlogD−1[φ(x,τ)] Make formal expansion in 1/N and at the end set N = 1. Each boson propagator contributes 1/N and each fermion loop in the effective boson self-interaction gives N ⇒ 1/N expansion equivalent to expansion in bosonic loops. LO in 1/N ⇔ MFA. NLO in 1/N ⇔ one-boson loop corrections. 1/1 is not really a small expansion parameter, but at least gives a systematic ordering to corrections beyond MFA.

1/N expansion

• T. Brauner

O(N) sigma model

Auxiliary field technique Next-to-leading order Summary – part I

Fermi systems

NR Fermi gas Dense quark matter Summary – part II

NLO: Tc Abuki and TB (2008)

• =

Tc reduced by a constant factor in the BCS limit! Chemical potential in the BCS limit governed by PT. Reproduces second-order analytic formula. µ εF = 1+ 4 3πkFa+ 4(11−2log2) 15π2 (kFa)2

1/N expansion

• T. Brauner

O(N) sigma model

Auxiliary field technique Next-to-leading order Summary – part I

Fermi systems

NR Fermi gas Dense quark matter Summary – part II

NLO: T = 0

• =

• =

1/N corrections moderate even around unitarity. Gap reduced by a constant factor in the BCS limit. Different factor than for Tc ⇒ departure from BCS ratio π

eγ !

1/N expansion

• T. Brauner

O(N) sigma model

Auxiliary field technique Next-to-leading order Summary – part I

Fermi systems

NR Fermi gas Dense quark matter Summary – part II

Quark matter: What is N? Abuki and TB, PRD78 (2008) 125010

We already have three colors, how about SU(N)c? 1/3 might be a reasonable expansion parameter. However:

• No trace over SU(N) indices!

Full RPA series not resummed at any finite order in 1/N. Would coupling ∼ O(1) help? No, 1/N expansion then even impossible.

1/N expansion

• T. Brauner

O(N) sigma model

Auxiliary field technique Next-to-leading order Summary – part I

Fermi systems

NR Fermi gas Dense quark matter Summary – part II

N is not color!

1/N expansion based on extension of color SU(3) will not lead to Cooper pairing. Resums different class of diagrams than needed. Solution: Introduce a new quantum number. φa ∼ G

N

∑

i=1

ψiCγ5Qaψi Global symmetry now SU(3)c ×SO(N)×flavor group. SO(N) again unbroken by Cooper pairs. Perform 1/N expansion and set N = 1 at the end. We lose the color expansion parameter 1/3. This construction can be applied to any pattern

• f relativistic fermion pairing.

1/N expansion

• T. Brauner

O(N) sigma model

Auxiliary field technique Next-to-leading order Summary – part I

Fermi systems

NR Fermi gas Dense quark matter Summary – part II

1/N expansion

Inverse boson propagator at zero momentum (Thouless criterion): Universal nonrelativistic expression with necessary relativistic modifications and a pairing-dependent algebraic prefactor.

G−1(0) = G−1

(0)(0)+ NB

NF ∑

Z

dQG(0)(Q) ∑

e,f=±

Z

d3k (2π)3

• 1+ef m2 +k·(k+q)

εkεk+q

• I(eξe

k,fξf k+q;iΩN)

I(a,b;iΩN) = 1 8a2 tanh βa

2 +tanh βb 2 −βacosh−2 βa 2

iΩN +b+a + 1 8a2 tanh βa

2 −tanh βb 2

iΩN +b−a + 1 4a tanh βa

2 +tanh βb 2

(iΩN +b+a)2

Fluctuations distinguish otherwise MF-identical patterns: pairing NB NF NB/NF “BCS” 1 N 1/N 2SC 3 6N 1/2N CFL 9 9N 1/N

1/N expansion

• T. Brauner

O(N) sigma model

Auxiliary field technique Next-to-leading order Summary – part I

Fermi systems

NR Fermi gas Dense quark matter Summary – part II

Numerical results in HDET

Correction to critical temperature given by a universal function with a model-dependent algebraic prefactor. log Tc T(0)

c

= −NB NF X T(0)

c

µ

• NR limit

UR limit

1/N expansion

• T. Brauner

O(N) sigma model

Auxiliary field technique Next-to-leading order Summary – part I

Fermi systems

NR Fermi gas Dense quark matter Summary – part II

Impact on the phase diagram

In the three-flavor chiral limit, Cooper instability in the 2SC and CFL channels appears at the same temperature. Fluctuation effects distinguish between the patterns. CFL pairing disfavored by fluctuations ⇒ narrow window of 2SC at arbitrarily high density. Effect similar to finite strange quark mass.

1/N expansion

• T. Brauner

O(N) sigma model

Auxiliary field technique Next-to-leading order Summary – part I

Fermi systems

NR Fermi gas Dense quark matter Summary – part II

Summary – part II

General remarks on 1/N expansion for Fermi gases

Perturbative extrapolation based on MF values of ∆,T,µ,... Avoids problems with self-consistency, technically easy. Only reliable when the NLO corrections are small.

Color-superconducting quark matter

BCS more likely, though BEC not ruled out. Fluctuation corrections non-negligible, may affect competition of various pairing patterns. Improvements necessary: Fermi surface mismatch (mass & chemical potential), color neutrality etc. Generalization below the critical temperature.