On finite-temperature holographic QCD in the Veneziano limit

Holographic models in the T=0 universality class of QCD in the limit of large number N_c of colors and N_f massless fermion flavors, but constant ratio x_f=N_f/N_c, are analyzed at finite temperature. The models contain a 5-dimensional metric and two scalars, a dilaton sourcing TrF^2 and a tachyon dual to \bar qq. The phase structure on the T,x_f plane is computed and various 1st order, 2nd order transitions and crossovers with their chiral symmetry properties are identified. For each x_f, the temperature dependence of p/T^4 and the quark-antiquark -condensate is computed. In the simplest case, we find that for x_f up to the critical x_c\sim 4 there is a 1st order transition on which chiral symmetry is broken and the energy density jumps. In the conformal window x_c<x_f<11/2, there is only a continuous crossover between two conformal phases. When approaching x_c from below, x_f\to x_c, temperature scales approach zero as specified by Miransky scaling.


Defining V-QCD
Degrees of freedom are two scalar fields: The tachyon τ ↔qq , and the dilaton λ ↔ TrF 2 λ is identified as the 't Hooft coupling g 2 N c Need to choose V g , V f 0 , a, and κ . . .

Match with QCD behavior at qualitative level
The simplest and most reasonable choices do the job! Turning on finite chemical potential Work in progress! Standard method: add a gauge field A µ dual toqγ µ q A 0 can be integrated out ⇒ one integration constant, which can be mapped to µ 10/14 Computation of pressure

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Holographic beta functions Generalization of the holographic RG flow of IHQCD The full equations of motion boil down to two first order partial non-linear differential equations for β and γ  In this model, after rescalings, this parameter can be mapped to a parameter (τ 0 or r 1 ) that controls the diverging tachyon in the IR x has become continuous in the Veneziano limit

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Effective potential: zero tachyon Start from Banks-Zaks region, τ * = 0, chiral symmetry conserved V eff defines a β-function as in IHQCD -Fixed point guaranteed in the BZ region, moves to higher λ with decreasing x Fixed point λ * runs to ∞ either at finite x(< x c ) or as x → 0 Banks-Zaks Conformal Window x → 11/2 x > x c x < x c ?? Effective potential: what actually happens

Banks-Zaks
Conformal Window x → 11/2 x > x c x < x c Where is x c ?
How is the edge of the conformal window stabilized? Tachyon IR mass at λ = λ * ↔ quark mass dimension Why γ * = 1 at x = x c ?
No time to go into details . . . the question boils down to the linearized tachyon solution at the fixed point

Mass dependence
For m > 0 the conformal transition disappears The ratio of typical UV/IR scales Λ UV /Λ IR varies in a natural way m/Λ UV = 10 −6 , 10 −5 , . . . , 10 x = 2, 3.5, 3.9, 4.25, 4.5 The case of N = 1 SU(Nc ) superQCD with N f quark multiplets is known and provides an interesting (and more complex) example for the nonsupersymmetric case. From Seiberg we have learned that: x = 0 the theory has confinement, a mass gap and Nc distinct vacua associated with a spontaneous breaking of the leftover R symmetry Z Nc .
At 0 < x < 1, the theory has a runaway ground state.
At x = 1, the theory has a quantum moduli space with no singularity. This reflects confinement with ChSB.
At x = 1 + 1/Nc , the moduli space is classical (and singular). The theory confines, but there is no ChSB.
At 1 + 2/Nc < x < 3/2 the theory is in the non-abelian magnetic IR-free phase, with the magnetic gauge group SU(N f − Nc ) IR free.
At 3/2 < x < 3, the theory flows to a CFT in the IR. Near x = 3 this is the Banks-Zaks region where the original theory has an IR fixed point at weak coupling. Moving to lower values, the coupling of the IR SU(Nc ) gauge theory grows. However near x = 3/2 the dual magnetic SU(N f − Nc ) is in its Banks-Zaks region, and provides a weakly coupled description of the IR fixed point theory.
At x > 3, the theory is IR free.

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Saturating the BF bound (sketch) Does the nontrivial (ChSB) massless tachyon solution exist? Two possibilities: x > x c : BF bound satisfied at the fixed point ⇒ only trivial massless solution (τ ≡ 0, ChS intact, fixed point hit) x < x c : BF bound violated at the fixed point ⇒ a nontrivial massless solution exist, which drives the system away from the fixed point Conclusion: phase transition at x = x c As x → x c from below, need to approach the fixed point to satisfy the boundary conditions ⇒ nearly conformal, "walking" dynamics 56/14 x = 2, 3, 3.5, 3.9 57/14