Infrared fixed point physics in SO(Nc) and Sp(

Abstract

We study properties of asymptotically free vectorial gauge theories with gauge groups G = SO(Nc) and G = Sp(Nc) and N f fermions in a representation R of G, at an infrared (IR) zero of the beta function, αIR, in the non-Abelian Coulomb phase. The fundamental, adjoint, and rank-2 symmetric and antisymmetric tensor fermion representations are considered. We present scheme-independent calculations of the anomalous dimensions of (gauge-invariant) fermion bilinear operators γψ ψ,IR to O(∆ 4 f ) and of the derivative of the beta function at αIR, denoted β ′ IR , to O(∆ 5 f ), where ∆ f is an N f -dependent expansion variable. It is shown that all coefficients in the expansion of γψ ψ,IR that we calculate are positive for all representations considered, so that to O(∆ 4 f ), γψ ψ,IR increases monotonically with decreasing N f in the non-Abelian Coulomb phase. Using this property, we give a new estimate of the lower end of this phase for some specific realizations of these theories.

Physics of the non-Abelian Coulomb phase: Insights from Padé approximants

“…Moreover, as reviewed below, in a gauge theory with N ¼ 1 supersymmetry, an exact expression is known for the anomalous dimension of the (gauge-invariant) fermion bilinear operator, and the Taylorseries expansion of this exact expression in powers of Δ f yields κ j coefficients that are all positive. These results led to our conjecture in [13], elaborated upon in our later works, that, in addition to the manifestly positive κ 1 and κ 2 , and our findings in [14][15][16]18] that κ 3 and κ 4 are positive for all of the groups and representations for which we calculated them, (i) the higher-order κ j coefficients with j ≥ 5 are also positive in (vectorial, asymptotically free) nonsupersymmetric gauge theories. In turn, this conjecture led to several monotonicity conjectures, namely that (ii) for fixed s, γψ ψ;IR;Δ s f increases monotonically as N f decreases in the non-Abelian Coulomb phase, and (iii) for fixed N f in the NACP, γψ ψ;IR;Δ s f is a monotonically increasing function of s, so that (iv) for fixed N f in the NACP and for finite s, γψ ψ;IR;Δ s f is a lower bound on the actual anomalous dimension γψ ψ;IR , as defined by the infinite series (2.10).…”

“…Although κ 3 and κ 4 contain terms with negative as well as positive signs, one of the important results of our explicit calculations of κ 3 and κ 4 for SUðN c Þ, SOðN c Þ, and SpðN c Þ gauge groups and a variety of representations, including fundamental, adjoint, and rank-2 symmetric and antisymmetric tensors, was that for all of these theories, κ 3 and κ 4 are also positive [14][15][16]18]. Moreover, as reviewed below, in a gauge theory with N ¼ 1 supersymmetry, an exact expression is known for the anomalous dimension of the (gauge-invariant) fermion bilinear operator, and the Taylorseries expansion of this exact expression in powers of Δ f yields κ j coefficients that are all positive.…”

“…We refer the reader to our previous papers [12][13][14][15][16][17][18] for details. The requirement of asymptotic freedom implies that N f must be less than an upper (u) bound N u , where [20] …”

“…The calculation of κ 4 was given for SU(3) and R ¼ F in [13] and for SUðN c Þ in [15], Our calculation of κ 4 for general G and R used the b l coefficients up to b 5 from [27] and c l up to c 4 from [28]. Specific expressions for κ j and plots of γψ ψ;IR;Δ s f for G ¼ SUðN c Þ and the fundamental, adjoint, and rank-2 symmetric and antisymmetric representations were given in [12][13][14][15][16][17] and for G ¼ SOðN c Þ and SpðN c Þ in [18]. In [14] we discussed operational criteria for the accuracy of the Δ f expansion, and we briefly review some points here.…”

“…In earlier work we had presented values of γψ ψ;IR [2] (see also [3]) and β 0 IR [4] calculated as conventional n-loop series expansions in powers of the n-loop (nl) IR coupling α IR;nl . In [12][13][14][15][16][17][18] we calculated values of γψ ψ;IR and β 0 IR via our schemeindependent expansions.…”