Thermodynamic singularities of QCD in the complex baryo-chemical - - PowerPoint PPT Presentation
Thermodynamic singularities of QCD in the complex baryo-chemical potential plane and the extended analyticity conjecture David Mesterhzy Albert Einstein Center for Fundamental Physics, Universitt Bern In collaboration with: Misha Stephanov,
SIGN 2015, Debrecen, Hungary
Taylor series expansion of the QCD pressure p in powers of µ/T:
∞
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:
∞
Coefficients can be calculated with standard techniques at µ = 0
c2n+2
Karsch et al., Phys. Lett. B698, 256 (2011) Ejiri et al., Prog. Theor. Phys. Suppl. 153:118 (2004)
Convergence radius is limited by the nearest singularity to the real axis. Yang-Lee theory states that these are related to the zeros zn of the partition function.
For finite system partition is a entire function – zeros located off the real axis (at complex zn). Electrostatic analogy (two-dimensional Coulomb gas):
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.
Ejiri et al. Prog. Theor. Exp. Phys. 8, 083B02 (2014)
Landau-Ginzburg potential
2σ2 + b 4σ4 + c 6σ6
In the scaling regime the singular part of the potential F(t, h = 0) is proportional to a power of
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 eiϕ, s > 0:
Cuts are straight lines at an angle ϕ with negative t axis:
In the presence of a symmetry-breaking field
up to h-independent term. Critical point is shifted away from the origin in the complex t-plane:
∗
ψ t∗(h) h = 0 t plane h = 0 t∗(0) ϕ
suppressed anomaly at Tc QCD
crossover or first order
first order first order aQCD
first order
U(1)A anomaly suppressed anomaly at Tc QCD
crossover or first order
first order first order
Vicari, PoS LAT2007:023 (2007)
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) g g T T O(4) O(4)
U(2)xU(2)/U(2)
O(4) 1st order O(4)
In the chiral limit (m = 0) we identify t ∼ µ2 − µ2
c (T) in the vicinity of the phase transition
(conformal transformations µ → µ2 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 µc(T) on the real axis:
At tricritical point mean-field exponents (up to logarithmic corrections).
µc(T) µ∗(m) ϕ ψ m = 0 m = 0 µ plane Fixed T ∈ (T3, Tc) T > TE T = TE T < TE µ plane Fixed m = 0 µ∗(m) µE(m) m1/(βδ)
Stephanov, Phys. Rev. D73, 094508 (2006)
At nonvanishing (sufficiently small) quark masses and fixed T > T3, we identify h ∼ m in the vicinity of the crossover point. QCD critical end point (TE, µE) 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 − µE is magnetic-field-like – Lee-Yang zeros). Both thermal and magnetic operators with scaling dimensions yt = 1/ν and yh = βδ/ν couple linearly to the variable µ − µE. Since yh > yt, the magnetic field operator dominates near the critical point
µc(T) µ∗(m) ϕ ψ m = 0 m = 0 µ plane Fixed T ∈ (T3, Tc) T > TE T = TE T < TE µ plane Fixed m = 0 µ∗(m) µE(m) m1/(βδ)
Stephanov, Phys. Rev. D73, 094508 (2006)
T
R(T) ∼ m1/(βδ) ∼ m0.54
Below TE 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:
In the critical domain, where |h|, |τ | ≪ 1, the EOS satisfies the following scaling form
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.
and
Consider an expansion around (possibly non-vanishing) ¯
Second-order phase transition where the correlation length diverges:
while U′( ¯
In mean-field theory, τ = ¯
...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
(field-expectation value nonvanishing on both sides of the transition).
Scaling form near the LY 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 ¯
Spinodal singularity (associated with the limit of metastability) lies in the same universality class
What happens to the spinodal singularity?
Consider the cubic theory with scalar field
3
Im λ3 Re λ3 UV IR v = (λ3)2 βv
Same flow characteristics at two- and three-loop order
Two-dimensional Ising model (analytic properties of the universal scaling function)
high-T dispersion relation
8.33370(1) 8.33410
1457.55(11) 1458.21
5.03(1)×105 5.02×105
Two-dimensional Ising model (analytic properties of the universal scaling function)
low-T dispersion relation
0.0387529; 0.039(1) 0.0388954
0.18453
2.952
96.76
5.34×103
4.46×105
Extended analyticity conjecture (two-dimensional Ising model)
Fonseca and Zamalodchikov, J. Stat. Phys. 110, 527 (2003)
Two-dimensional Ising model Fonseca and Zamalodchikov, J. Stat. Phys. 110, 527 (2003)
9π/5 7π/5 5π/5 3π/5 π/5 2π
Arg( ) η LTW HTW HTW SHD SHD
Two-dimensional Ising quantum gravity Bourgine and Kostov, J. Stat. Mech. 1201, P01024 (2012)
For d = 2 the Yang-Lee singularity turns out to be located relatively far under the Langer branch cut.
“If extended analyticity assumption holds for d = 3, one should expect the effect of the Yang-Lee singularity being quite prominent at real negative H at least in the critical domain T → Tc.”
Nonperturbative functional renormalization group Scale-dependent effective action:
k (ϕ)(ϕ)2
k (ϕ)(∂µϕ)2 ϕ + Wc k (ϕ)
k
k (ϕ)
k (ϕ)
k (ϕ)
Canonical dimension
d−2 2
2
Classical potential UΛ(ϕ) (defined at the short-distance cutoff Λ) receives contributions from modes q2 ≈ k2 that are integrated out successively starting from the cutoff scale – This procedure introduces a k-dependence to the potential:
k
We obtain an infinite set of flow equations for the derivatives of the effective potential
k (ϕ) ≡ ∂nUk(ϕ) ∂ϕn
with respect to constant field ϕ:
k(ϕ) = −1
k (ϕ) = 1
k
k
k(ϕ) q2 + Wa k ′(ϕ)(q2)2 + U(3) k (ϕ)
k
k
Simplest possible truncation:
k
and Uk(ϕ) = 1
3! λ3,kϕ3
One-loop universality states that O(ǫ) should be reproduced – but deviations on the order of 10 % around d = 4.
3 4 5 6 d
η
Let’s continue:
k
and Uk(ϕ) = 1
3! λ3,kϕ3 + n≥4 1 n! λn,kϕn
One-loop O(ǫ) result is reproduced – but fixed point no longer extends to d = 4!
1 2
1 2