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 1 / N expansion for pion gas Fermi systems NR Fermi gas and strongly interacting Fermi systems Dense quark matter Summary – part II Tom´ aˇ s Brauner Institut f¨ ur Theoretische Physik Goethe Universit¨ at Frankfurt am Main R + R Budapest, 3 April 2009

Outline 1 / N expansion T. Brauner O ( N ) sigma model Auxiliary field technique Next-to-leading order O ( N ) sigma model and 1PI-1 / N expansion 1 Summary – part I Auxiliary field technique Fermi systems NR Fermi gas Next-to-leading order Dense quark matter Summary – part II Summary – part I Strongly interacting Fermi systems 2 Nonrelativistic Fermi gas Dense quark matter Summary – part II

Outline 1 / N expansion T. Brauner O ( N ) sigma model Auxiliary field technique Next-to-leading order O ( N ) sigma model and 1PI-1 / N expansion 1 Summary – part I Auxiliary field technique Fermi systems NR Fermi gas Next-to-leading order Dense quark matter Summary – part II Summary – part I Strongly interacting Fermi systems 2 Nonrelativistic Fermi gas Dense quark matter Summary – part II

Introduction 1 / N expansion QCD at low temperatures: include only the lightest DOFs, T. Brauner i.e., the Goldstone bosons of the spontaneously broken O ( N ) sigma model SU L ( N f ) × SU R ( N f ) chiral symmetry. Auxiliary field technique For N f = 2 make use of the isomorphism Next-to-leading order Summary – part I SU ( 2 ) × SU ( 2 ) ≃ SO ( 4 ) and study the O ( 4 ) model. Fermi systems NR Fermi gas O ( N ) model studied in the 1PI-1 / N expansion since long Dense quark matter Summary – part II 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˝ os, Patk´ os, and Sz´ ep (2009)

Introduction 1 / N expansion QCD at low temperatures: include only the lightest DOFs, T. Brauner i.e., the Goldstone bosons of the spontaneously broken O ( N ) sigma model SU L ( N f ) × SU R ( N f ) chiral symmetry. Auxiliary field technique For N f = 2 make use of the isomorphism Next-to-leading order Summary – part I SU ( 2 ) × SU ( 2 ) ≃ SO ( 4 ) and study the O ( 4 ) model. Fermi systems NR Fermi gas O ( N ) model studied in the 1PI-1 / N expansion since long Dense quark matter Summary – part II 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˝ os, Patk´ os, and Sz´ ep (2009)

O ( N ) model 1 / N expansion Generalize the O ( 4 ) model to arbitrary N by a suitable T. Brauner redefinition of the couplings. O ( N ) sigma model Auxiliary field technique L = 1 2 ( ∂ µ φ i ) 2 + λ b 8 N ( φ i φ i − Nf 2 π , b ) 2 Next-to-leading order Summary – part I Fermi systems Some leading-order, O ( N ) , contributions to the pressure: NR Fermi gas Dense quark matter Summary – part II Some next-to-leading-order, O ( 1 ) , contributions to the pressure:

Auxiliary field technique 1 / N expansion T. Brauner Introduce a new auxiliary field α and add pure Gaussian O ( N ) sigma model integral over α . Auxiliary field technique Next-to-leading order Summary – part I � 2 ∆ L = N � α − i λ b 2 N ( φ i φ i − Nf 2 Fermi systems π , b ) 2 λ b NR Fermi gas Dense quark matter Summary – part II L = 1 2 ( ∂ µ φ i ) 2 − i π , b )+ N 2 α ( φ i φ i − Nf 2 α 2 2 λ b Systematic renormalization of divergences possible order by order in 1 / N by redefinition of the parameters. π + a 0 + 1 1 = 1 λ + b 0 + 1 f 2 π , b = f 2 Na 1 + ··· , Nb 1 + ··· λ b Introduce explicit chiral-symmetry breaking term. √ L → L − NH σ

1 / N expansion Andersen and TB (2008) 1 / N expansion T. Brauner Introduce the chiral condensate φ 0 and auxiliary field O ( N ) sigma model condensate M , and shift the fields. Auxiliary field technique Next-to-leading order √ α → iM 2 + α Summary – part I σ → N φ 0 + σ , √ N Fermi systems NR Fermi gas Dense quark matter The auxiliary field trick reduces resummation of all NLO Summary – part II graphs to a single Gaussian integral. S NLO P log ( P 2 + M 2 ) − NH φ 0 − NM 4 = 1 + 1 Z eff 2 NM 2 ( φ 2 0 − f 2 2 ( N − 3 ) ∑ π ) β V 2 λ − 1 2 NM 2 a 0 − 1 2 NM 4 b 0 + 1 P χ T D − 1 χ ∗ − 1 2 M 2 a 1 − 1 Z 2 M 4 b 1 2 ∑ � 1 2 Π ( P , M )+ 1 − i φ 0 � � α � λ + b 0 D − 1 = , χ = P 2 + M 2 − i φ 0 σ 1 1 Z Π ( P , M ) = ∑ Q 2 + M 2 ( P + Q ) 2 + M 2 Q

LO effective potential 1 / N expansion T. Brauner LO thermodynamic potential includes the condensate O ( N ) sigma model contributions and thermal fluctuations of massless pions. Auxiliary field technique Next-to-leading order Summary – part I V LO = M 2 0 )+ T 4 64 π 2 J 0 ( β M )+ M 4 � 32 π 2 + log µ 2 m 2 + 1 � Fermi systems 2 ( f 2 π − φ 2 + H φ 0 64 π 2 NR Fermi gas λ 2 Dense quark matter Z ∞ 0 dp p 4 J 0 ( β M ) = 32 Summary – part II � p 2 + M 2 where and n ( ω p ) ω p = 3 T 4 ω p 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. a 0 = Λ 2 32 π 2 log Λ 2 1 16 π 2 , b 0 = − µ 2 λ 2 LO β -function: β ( λ ) = 16 π 2 .

NLO effective potential 1 / N expansion T. Brauner NLO expression for the effective potential (pressure): O ( N ) sigma model Auxiliary field technique Next-to-leading order Summary – part I V NLO = − 1 P log J ( P , M )+ 1 2 M 2 a 1 + 1 Z 2 M 4 b 1 Fermi systems 2 ∑ NR Fermi gas Dense quark matter φ 2 J ( P , M ) = 1 2 Π ( P , M )+ 1 Summary – part II 0 λ + b 0 + P 2 + M 2 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.

Extraction of divergences 1 / N expansion Expand the log in V NLO in inverse powers of momentum. T. Brauner O ( N ) sigma model   � �  2 M 4 + 3 M 2 ( G − 3 2 M 2 ) κ + log Λ 2 M 2 + G − 2 M 2 + ( G − 2 M 2 ) 2 � � + 2 − 2 Auxiliary field technique log J = log  + ···   P 2 P 2 κ + log Λ 2 P 4 κ + log Λ 2 � 2 Next-to-leading order � α + log Λ 2 P 2 P 2 P 2 Summary – part I Fermi systems 0 + T 2 J 1 ( β M ) − M 2 log µ 2 M 2 − 32 π 2 M 2 � 1 � NR Fermi gas G = 16 π 2 φ 2 κ = 1 + 32 π 2 , λ + b 0 Dense quark matter λ Summary – part II

Extraction of divergences 1 / N expansion Expand the log in V NLO in inverse powers of momentum. T. Brauner O ( N ) sigma model   � �  2 M 4 + 3 M 2 ( G − 3 2 M 2 ) κ + log Λ 2 M 2 + G − 2 M 2 + ( G − 2 M 2 ) 2 � � + 2 − 2 Auxiliary field technique log J = log  + ···   P 2 P 2 κ + log Λ 2 P 4 κ + log Λ 2 � 2 Next-to-leading order � α + log Λ 2 P 2 P 2 P 2 Summary – part I Fermi systems 0 + T 2 J 1 ( β M ) − M 2 log µ 2 M 2 − 32 π 2 M 2 � 1 � NR Fermi gas G = 16 π 2 φ 2 κ = 1 + 32 π 2 , λ + b 0 Dense quark matter λ Summary – part II Quartic UV-divergence, independent of M , φ 0 , ρ 0 . Subtracted within the vacuum pressure. Quadratic UV-divergence. Absorbed in the f 2 π counterterm a 1 . Logarithmic UV-divergence. Absorbed in the 1 λ counterterm b 1 . NLO β -function: λ 2 � 1 + 8 � β ( λ ) = 16 π 2 N

NLO renormalization procedure 1 / N expansion T. Brauner The quadratic divergence contains temperature-dependent O ( N ) sigma model Auxiliary field technique terms in G ! Next-to-leading order Summary – part I T -dependence only disappears upon using LO gap equation Fermi systems for M : G = 16 π 2 f 2 π . NR Fermi gas Dense quark matter To obtain divergence-free gap equations, one should be able Summary – part II 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 φ 0 . When this is treated as merely a constraint to eliminate M in favor of φ 0 , we get an effective potential of φ 0 solely, which is renormalizable for any value of the classical field φ 0 . Coleman, Jackiw, and Politzer (1974)