Thermodynamic singularities of QCD in the complex baryo-chemical potential plane and the extended analyticity conjecture David Mesterházy Albert Einstein Center for Fundamental Physics, Universität Bern In collaboration with: Misha Stephanov, Xin An SIGN 2015, Debrecen, Hungary
Introduction: Nothing new here (sorry)
Taylor series expansion of the QCD pressure p in powers of µ/ T : � ∞ p ( µ/ T ) c n ( T )( µ/ T ) n = T 4 n = 0 ∂ n � p ( T , µ ) / T 4 � c n ( T ) = 1 � � ∂ ( µ/ T ) n � n ! � µ = 0 Coefficients can be calculated with standard techniques at µ = 0 Bernard et al. , Phys. Rev. D55, 6861 (1997); Karsch et al. , Phys. Lett. B478, 447 (2000); Ali Khan et al. , Phys. Rev. D64, 074510 (2001) Ejiri et al. , Prog. Theor. Phys. Suppl. 153:118 (2004)
Taylor series expansion of the QCD pressure p in powers of µ/ T : � ∞ p ( µ/ T ) c n ( T )( µ/ T ) n = T 4 n = 0 ∂ n � p ( T , µ ) / T 4 � c n ( T ) = 1 � � ∂ ( µ/ T ) n � n ! � µ = 0 Coefficients can be calculated with standard techniques at µ = 0 r = lim n →∞ r 2 n 1 / 2 r 2 n = � � c 2 n � � c 2 n + 2 � � Ejiri et al. , Prog. Theor. Phys. Suppl. 153:118 (2004) Karsch et al. , Phys. Lett. B698, 256 (2011)
Convergence radius is limited by the nearest singularity to the real axis. Yang-Lee theory states that these are related to the zeros z n of the partition function. � Z V = ( z − z n ) n � F ( z ) = − T V log Z N = − T log( z − z n ) V n For finite system partition is a entire function – zeros located off the real axis (at complex z n ). Electrostatic analogy (two-dimensional Coulomb gas): � φ = Re F ( z ) = − T log | z − z n | V n E = −∇ φ In the thermodynamic limit the zeros coalesce into one-dimensional curves with continuous charge density ρ ( z ) . A phase transition occurs only if these curves pinch (second order) or cross (first order) the real axis.
Complement lattice calculations at µ = 0 with low-energy effective models of QCD where zeros can be determined. A comparison of these methods can shed light on their range of applicability. � � − N Tr XX † � [det( D + m )] N f Z RM = dX exp � � 0 iX + iC D = iX † + iC 0 Stephanov, Phys. Rev. D73, 094508 (2006)
� − N Tr XX † � � [det( D + m )] N f Z RM = dX exp � � 0 iX + iC D = iX † + iC 0 Landau-Ginzburg potential Ω ( σ ) = − m σ + a 2 σ 2 + b 4 σ 4 + c 6 σ 6 Stephanov, Phys. Rev. D73, 094508 (2006) Ejiri et al. Prog. Theor. Exp. Phys. 8, 083B02 (2014)
In the scaling regime the singular part of the potential F ( t , h = 0) is proportional to a power of t ≡ ( z − z c ) / z c : � A + t 2 − α , t > 0 F sing ( t , h = 0) = A − ( − t ) 2 − α , t < 0 Off the real axis F ( t ) must be an analytic function everywhere except for discontinuities across the cuts. Their location can be determined using the electrostatic analogy, which requires φ to be continuous across the cut. With t = − s e i ϕ , s > 0 : A + cos[(2 − α )( ϕ − π )] = A − cos[(2 − α ) ϕ ] t plane Cuts are straight lines at an angle ϕ with negative t axis: h = 0 h � = 0 tan[(2 − α ) ϕ ] = cos( πα ) − A − / A + sin( πα ) t ∗ ( h ) ψ ϕ t ∗ (0) In the presence of a symmetry-breaking field F sing ( t , h ) = | h | 1 + 1 /δ Φ ( η = t / h 1 / ( βδ ) ) + . . . up to h -independent term. Critical point is shifted away from the origin in the complex t-plane: t ∗ = | η ∗ | e i ψ h 1 / ( βδ ) ∗ π ψ = 2 βδ
U (1) A anomaly suppressed anomaly at T c SU ( N f ) L ⊗ SU ( N f ) R → SU ( N f ) V U ( N f ) L ⊗ U ( N f ) R → U ( N f ) V QCD N f = 1 O (2) or first order crossover or first order N f = 2 O (4) or first order U (2) L ⊗ U (2) R / U (2) V or first order N f ≥ 3 first order first order aQCD SU (2 N f ) → SO (2 N f ) U (2 N f ) → O (2 N f ) N f = 1 O (3) or first order U (2) / O (2) or first order N f = 2 SU (4) / SO (4) or first order first order E. Vicari, PoS LAT2007:023 (2007)
U(1) A anomaly suppressed anomaly at T c QCD SU ( N f ) L ⊗ SU ( N f ) R → SU ( N f ) V U ( N f ) L ⊗ U ( N f ) R → U ( N f ) V N f = 1 crossover or first order O (2) or first order N f = 2 O (4) or first order U (2) L ⊗ U (2) R / U (2) V or first order N f ≥ 3 first order first order Vicari, PoS LAT2007:023 (2007)
N f = 2 QCD effective theory � � Tr( ∂ µ Φ † )( ∂ µ Φ ) + τ Tr Φ † Φ + 1 4 u (Tr Φ † Φ ) 2 + 1 d 3 x 4 v Tr( Φ † Φ ) 2 + S = (det Φ † ) 2 + (det Φ ) 2 � � + w ( g )(det Φ † + det Φ ) + 1 4 x ( g )(Tr Φ † Φ )(det Φ † + det Φ ) + 1 � 4 y ( g ) g g O(4) w , x , y = O ( g ) O(4) T T U(2)xU(2)/U(2) O(4) 1st order O(4) Taking into account the U (1) A anomaly, we expect that the chiral-symmetry restoration transition is in the three-dimensional O (4) universality class If that assumption holds, expect an analytic crossover for nonzero values of the quark masses (first order transition is quite stable with respect to the presence of non-vanishing quark masses)
In the chiral limit ( m = 0 ) we identify t ∼ µ 2 − µ 2 c ( T ) in the vicinity of the phase transition (conformal transformations µ → µ 2 do not affect the angles φ and ψ away from µ = 0 ). At a given temperature T , we are therefore inquiring about the location of the singularities in the complex µ -plane (Fisher zeros, relevant perturbation temperature-like). The (two) cuts originate at the branching point located at µ c ( T ) on the real axis: α ≈ − 0 . 25 , A + / A − ≈ 1 . 6 O (4) universality class : ϕ ≈ 77 ◦ , ϕ ≈ 77 ◦ At tricritical point mean-field exponents (up to logarithmic corrections). m = 0 m � = 0 T = T E T > T E T < T E µ plane µ plane Fixed T ∈ ( T 3 , T c ) Fixed m � = 0 µ ∗ ( m ) µ ∗ ( m ) m 1 / ( βδ ) ϕ ψ µ E ( m ) µ c ( T ) Stephanov, Phys. Rev. D73, 094508 (2006)
At nonvanishing (sufficiently small) quark masses and fixed T > T 3 , we identify h ∼ m in the vicinity of the crossover point. QCD critical end point ( T E , µ E ) lies in the three-dimensional Ising universality class. The initial direction of the cuts near this point is perpendicular to the real axis (relevant perturbation µ − µ E is magnetic-field-like – Lee-Yang zeros). Both thermal and magnetic operators with scaling dimensions y t = 1 /ν and y h = βδ/ν couple linearly to the variable µ − µ E . Since y h > y t , the magnetic field operator dominates near the critical point m = 0 m � = 0 T = T E T > T E T < T E µ plane µ plane Fixed T ∈ ( T 3 , T c ) Fixed m � = 0 µ ∗ ( m ) µ ∗ ( m ) m 1 / ( βδ ) ϕ ψ µ E ( m ) µ c ( T ) Stephanov, Phys. Rev. D73, 094508 (2006) R ( T ) ∼ m 1 / ( βδ ) ∼ m 0 . 54 µ 2 min T Below T E the singularity continues to move away from the origin, and the radius is now determined by the spinodal point of the first order phase transition.
Equation of state (EOS) for Ising field theory: h ( τ, ϕ ) = U ′ ( ϕ ) h = c h H (1 + O ( ∆ T , H )) τ = c τ ∆ T (1 + O ( ∆ T , H )) In the critical domain, where | h | , | τ | ≪ 1 , the EOS satisfies the following scaling form h ( τ, ϕ ) = ϕ | ϕ | δ − 1 f � τ | ϕ | − 1 /β � where f ( z = τ | ϕ | − 1 /β ) is a universal, dimensionless scaling function, uniquely defined up to normalization; The exponents β and δ characterize the asymptotic scaling behavior of the magnetization ϕ for vanishing h and τ , respectively. | h | = f (0) | ϕ | δ and τ = z 0 | ϕ | 1 /β
Consider an expansion around (possibly non-vanishing) ¯ ϕ : � � � � ϕ + 1 δϕ + 1 τ + 1 δϕ 2 + 1 ϕδϕ 3 + 1 ϕ 3 ϕ 2 4! λ 4 δϕ 4 + const U ( ϕ ) = τ ¯ 3! λ 4 ¯ 2 λ 4 ¯ 3! λ 4 ¯ 2 Second-order phase transition where the correlation length diverges: U ′′ ( ¯ ϕ ) = 0 while U ′ ( ¯ ϕ ) = ¯ h ϕ = 0 and small δϕ = ϕ − ¯ ¯ ϕ : Ising universality class (associated by the breaking of the Z 2 -symmetry of the order parameter ϕ 2 ). In mean-field theory, τ = ¯ h = 0 , λ 4 > 0 , and critical exponents given by: β = 1 Ising critical point : 2 , δ = 3 ...and from scaling relations: α = 0 , γ = 1 , η = 0 , and ν = 1 2 .
Additional singularities in the high-temperature phase ( τ > 0 ) at non-vanishing (complex) values of the external field h – cannot be associated with a characteristic symmetry-breaking pattern (field-expectation value nonvanishing on both sides of the transition). � 2 τ � 3 / 2 h = 2 ϕ = ± i λ 4 ¯ τ > 0 : 3 τ ¯ 3 λ 4 ϕ ) ≡ δ U ′ � ϕ � δ h = U ′ ( ϕ ) − U ′ ( ¯ EOS: Scaling form near the LY critical point: δ h = δϕ | δϕ | δ − 1 g � µ 2 | δϕ | − 1 /β � µ 2 = τ + 1 ϕ 2 ( Λ ) 2 λ 4 ¯ β = 1 , δ = 2 Lee-Yang critical point : ... and using scaling relations: α = − 1 , γ = 1 , η = 0 , and ν = 1 2
Same argument goes through in the low-temperature phase ( τ < 0 ) where we identify a set of critical points with values ¯ h that lie on the real axis Re h : � − 2 τ � 3 / 2 h = ± λ 4 ¯ τ < 0 : , 3 λ 4 Spinodal singularity (associated with the limit of metastability) lies in the same universality class of Lee-Yang theory What happens to the spinodal singularity?
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