IR fixed points in $SU(3)$ gauge Theories

We propose a novel RG method to specify the location of the IR fixed point in lattice gauge theories and apply it to the $SU(3)$ gauge theories with $N_f$ fundamental fermions. It is based on the scaling behavior of the propagator through the RG analysis with a finite IR cut-off, which we cannot remove in the conformal field theories in sharp contrast with the confining theories. The method also enables us to estimate the anomalous mass dimension in the continuum limit at the IR fixed point. We perform the program for $N_f=16, 12, 8 $ and $N_f=7$ and indeed identify the location of the IR fixed points in all cases.

The phase diagram for Nf \le 6 Phys. Rev. D54(1996), 7010 A new phase "conformal region" in addition to the confining region and deconfining region Phys.Rev. D87 (2013)  we intend to perform step 4 in this work Nf=2 deconfining Nf=2 conformal ropagator with different parameters as G(t ; g , m q , N ). (7) he subscript H of G(t) is suppressed here and here-. The relation between g and g and m a and m q etermined by the RG beta function B and the mass alous dimensions γ. Let us first discuss the case in h we are at the fixed point, i.e. g = g = g * and m q = 0 so that B = 0 and γ = γ * . In this case, the agator may have simplified notation as G(τ, N ) = G(t, N ). (8) The variable t takes 0, 1, 2, · · · , N t so that ≤ 1. In terms of τ , the RG relation eq.(??) reduces tly speaking, this equation is satisfied in the limit change t = t/s (see e.g. [12]) relates the propagator with different parameters as Here the UV renormalization scale µ in lattice theories is set by the inverse lattice spacing a −1 . The equation is irrelevant from µ and µ may be omitted in the relation. N = N/s. The subscript H of G(t) is suppressed here and hereafter. The relation between g and g and m a and m q are determined by the RG beta function B and the mass anomalous dimensions γ. Let us first discuss the case in which we are at the fixed point, i.e. g = g = g * and m q = m q = 0 so that B = 0 and γ = γ * . In this case, the propagator may have simplified notation as G(τ, N ) = G(t, N ).
with τ = t/N t . The variable t takes 0, 1, 2, · · · , N t so that 0 ≤ τ ≤ 1. In terms of τ , the RG relation eq.(6) reduces Here the UV renormalization scale µ in lattice theories is set by the inverse lattice spacing a −1 . The equation is irrelevant from µ and µ may be omitted in the relation. N = N/s. The subscript H of G(t) is suppressed here and hereafter. The relation between g and g and m a and m q are determined by the RG beta function B and the mass anomalous dimensions γ. Let us first discuss the case in which we are at the fixed point, i.e.
In this case, the propagator may have simplified notation asG (τ, N ) = G(t, N ).
with τ = t/N t . The variable t takes 0, 1, 2, · · · , N t so that 0 ≤ τ ≤ 1. In terms of τ , the RG relation eq.(6) reduces toG l study meson (2) points. r (PS) propa- s to (4) e mass t takes recall bation Here the UV renormalization scale µ in lattice theories is set by the inverse lattice spacing a −1 . The equation is irrelevant from µ and µ may be omitted in the relation.
is suppressed here and hereafter. The relation between g and g and m a and m q are determined by the RG beta function B and the mass anomalous dimensions γ. Let us first discuss the case in which we are at the fixed point, i.e.
In this case, the propagator may have simplified notation as The variable t takes 0, 1, 2, · · · , N t so that 0 ≤ τ ≤ 1. In terms of τ , the RG relation eq.(6) reduces tõ Strictly speaking, this equation is satisfied in the limit N, N → ∞. Scaled effective mass Strictly speaking, this equation is satisfied in the limit N, N → ∞.
To state our proposal concretely, we define the scaled effective mass m(t; N ) with respect to the reference lattice size N 0 as In the continuum limit N → ∞ Eq. (??) reduces to the form The crucial observation, which will be the core of our proposal is that, combining Eqs.(??) and (??), the scaled effective mass does not depend on N as a function of τ : at the fixed point. Suppose that we are away from the fixed point (i.e g = g * while m q = 0) in contrast. The scaled effective mass in the vicinity of the fixed point would instead show the following behavior , N → ∞.
To state our proposal concretely, we define the scaled ective mass m(t; N ) with respect to the reference lattice e N 0 as e crucial observation, which will be the core of our prosal is that, combining Eqs.(??) and (??), the scaled eftive mass does not depend on N as a function of τ : the fixed point. Suppose that we are away from the fixed point (i.e = g * while m q = 0) in contrast. The scaled effective ass in the vicinity of the fixed point would instead show e following behavior m(t, N ) = N N 0 ln G(t, N ) G(t + 1, N ) .
nuum limit N → ∞ Eq. (??) reduces to the observation, which will be the core of our pro-, combining Eqs.(??) and (??), the scaled efdoes not depend on N as a function of τ : point. that we are away from the fixed point (i.e le m q = 0) in contrast. The scaled effective vicinity of the fixed point would instead show speaking, this equation is satisfied in the limit ∞. tate our proposal concretely, we define the scaled mass m(t; N ) with respect to the reference lattice as cial observation, which will be the core of our prothat, combining Eqs. (7) and (9), the scaled effecss does not depend on N as a function of τ : xed point. ose that we are away from the fixed point (i.e while m q = 0) in contrast. The scaled effective the vicinity of the fixed point would instead show wing behavior for the global Metropolis test, "plaq" for the value of the plaquette and the fifth m q is the quark mass defined by eq.(??).  (3) 0.0047 (6) for a In is eith the co at zer the ze out a the de erator Bank tify t 6, wh fact t massl Be Wilso The q N = while estim ence. der sm is ord estim It is well-known that the convergence of the perturbation by the g one−plaquette is poor in general. The contribution of higher order terms will be large. On the other hand, the lattice coupling constant β RG is close to β MS and therefore Now, let us show the results, starting with the N f = 16 ase. As mentioned earlier, within the two-loop perturbaon, the IR fixed point is β * = 11.475, which is RG scheme dependent. On the other hand, the coupling constants different RG schemes are related to each other by a onstant as β 1 = β 2 + c 12 in the one-loop approximation. or example [11], the lattice coupling constants β RG and one−plaquette for one-plaquette action are related to that the continuum theory β MS (in the modified minimal btraction scheme) as  Nf=12, 8, 7 The location of IR fixed points The conformal window Continuum limit of propagators at IRFP continuum limit of scaled effective mass is given by the limit N --> infinity Even up to N=16, the limit is almost realized for \tau \ge 0.1. As N becomes larger, it will be realized for \tau \le 0.1 Note the limit depends on the aspect ratio and boundary conditions, but not on L= N a Note that local-local propagators are not local observables, due to the summation over the space coordinates Scaling relation for propagators G (τ, N ) = G(t, N ).
= t/N t . The variable t takes 0, 1, 2, · · · , N t so that ≤ 1. In terms of τ , the RG relation eq.(??) reduces y speaking, this equation is satisfied in the limit → ∞. state our proposal concretely, we define the scaled ve mass m(t; N ) with respect to the reference lattice 0 as continuum limit N → ∞ Eq. (??) reduces to the ucial observation, which will be the core of our prois that, combining Eqs.(??) and (??), the scaled efmass does not depend on N as a function of τ : Effective mass for Nf=16, 12, 8, 7 Local analysis of propagators parametrization using data at three points useful for seeing the characteristics