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Infrared Fixed Point Physics in ${\Rm SO}(N C)$ and ${\Rm Sp}(N C)$ Gauge Theories

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Infrared Fixed Point Physics in ${\Rm SO}(N C)$ and ${\Rm Sp}(N C)$ Gauge Theories University of Southern Denmark Infrared Fixed Point Physics in ${\rm SO}(N_c)$ and ${\rm Sp}(N_c)$ Gauge Theories Ryttov, Thomas A.; Shrock, Robert Published in: Physical Review D DOI: 10.1103/PhysRevD.96.105015 Publication date: 2017 Document version: Final published version Citation for pulished version (APA): Ryttov, T. A., & Shrock, R. (2017). Infrared Fixed Point Physics in ${\rm SO}(N_c)$ and ${\rm Sp}(N_c)$ Gauge Theories. Physical Review D, 96(10), [105015]. https://doi.org/10.1103/PhysRevD.96.105015 Go to publication entry in University of Southern Denmark's Research Portal Terms of use This work is brought to you by the University of Southern Denmark. Unless otherwise specified it has been shared according to the terms for self-archiving. If no other license is stated, these terms apply: • You may download this work for personal use only. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying this open access version If you believe that this document breaches copyright please contact us providing details and we will investigate your claim. Please direct all enquiries to [email protected] Download date: 29. Sep. 2021 PHYSICAL REVIEW D 96, 105015 (2017) Infrared fixed point physics in SOðNcÞ and SpðNcÞ gauge theories Thomas A. Ryttov1 and Robert Shrock2 1CP3-Origins and Danish Institute for Advanced Study University of Southern Denmark, Campusvej 55, Odense, Denmark 2C. N. Yang Institute for Theoretical Physics Stony Brook University, Stony Brook, New York 11794, USA (Received 15 September 2017; published 22 November 2017) We study properties of asymptotically free vectorial gauge theories with gauge groups G ¼ SOðNcÞ and ¼ ð Þ α G Sp Nc and Nf fermions in a representation R of G, at an infrared (IR) zero of the beta function, IR, 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 γ ðΔ4 Þ anomalous dimensions of (gauge-invariant) fermion bilinear operators ψψ¯ ;IR to O f and of the α β0 ðΔ5 Þ Δ derivative of the beta function at IR, denoted IR,toO f , where f is an Nf-dependent expansion γ variable. It is shown that all coefficients in the expansion of ψψ¯ ;IR that we calculate are positive for all ðΔ4Þ γ representations considered, so that to O f , ψψ¯ ;IR increases monotonically with decreasing Nf 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. DOI: 10.1103/PhysRevD.96.105015 I. INTRODUCTION in a chirally symmetric (deconfined) non-Abelian Coulomb phase (NACP) [4]. Here the value α ¼ α is an exact IR The evolution of an asymptotically free gauge theory IR fixed point of the renormalization group, and the corre- from the ultraviolet (UV) to the infrared is of fundamental sponding theory in this IR limit is scale-invariant and importance. The evolution of the running gauge coupling generically also conformal invariant [5]. g ¼ gðμÞ, as a function of the Euclidean momentum scale, μ The physical properties of the conformal field theory at , is described by the renormalization-group (RG) beta α IR are of considerable interest. These properties clearly function, β ¼ dg=dt, or equivalently, βα ¼ dα=dt, where g cannot depend on the scheme used for the regularization αðμÞ¼gðμÞ2=ð4πÞ and dt ¼ d ln μ (the argument μ will and renormalization of the theory. (For technical conven- often be suppressed in the notation). The asymptotic ience, we restrict our discussion here to mass-independent freedom (AF) property means that the gauge coupling schemes.) In usual perturbative calculations, one computes approaches zero in the deep UV, which enables one to a given quantity as a series expansion in powers of α to perform reliable perturbative calculations in this regime. some finite n-loop order. With this procedure, the result is Here we consider a vectorial, asymptotically free gauge scheme-dependent beyond the leading term(s). For exam- theory (in four spacetime dimensions) with two types of ple, the beta function is scheme-dependent at loop order gauge groups, namely the orthogonal group, G ¼ SOðN Þ, c l ≥ 3 and the terms in an anomalous dimension are and the symplectic group (with even N ), G ¼ SpðN Þ, c c scheme-dependent at loop order l ≥ 2 [6]. This applies, and N copies (“flavors”) of Dirac fermions transforming f in particular, to the evaluation at an IR fixed point. A key according to the respective (irreducible) representations R fact is that as N (considered to be extended from positive of the gauge group, where R is the fundamental (F), adjoint f integers to positive real numbers) approaches the upper (A), or rank-2 symmetric (S2) or antisymmetric (A2) tensor. limit allowed by the requirement of asymptotic freedom, It may be recalled that for SOðNcÞ, the adjoint and A2 ð Þ denoted Nu [given in Eq. (2.3) below], it follows that representations are equivalent, while for Sp Nc , the α → 0 IR . Consequently, one can express a physical quantity adjoint and S2 representations are equivalent. For technical α convenience, we take the fermions to be massless [1].In evaluated at IR in a manifestly scheme-independent way as a series expansion in powers of the variable the case of SOðNcÞ, we do not consider Nc ¼ 2, since SOð2Þ ≅ Uð1Þ, and a U(1) gauge theory is not asymptoti- Δ ¼ − ð Þ cally free (but instead is infrared-free). f Nu Nf: 1:1 If Nf is sufficiently large (but less than the upper limit implied by asymptotic freedom), then the beta function has For values of Nf in the non-Abelian Coulomb phase such α Δ an IR zero, at a coupling denoted IR, that controls the UV that f is not too large, one may expect this expansion to IR evolution [2,3]. Given that this is the case, as the to yield reasonably accurate perturbative calculations of α Euclidean scale μ decreases from the UV to the IR, αðμÞ physical quantities at IR [7]. Some early work on this type α – increases toward the limiting value IR, and the IR theory is of expansion was reported in [7,8].In[9 13] we have 2470-0010=2017=96(10)=105015(17) 105015-1 © 2017 American Physical Society THOMAS A. RYTTOV and ROBERT SHROCK PHYSICAL REVIEW D 96, 105015 (2017) presented scheme-independent calculations of a number of In this paper we report our completion of this task for physical quantities at an IR fixed point in an asymptotically the gauge groups SOðNcÞ and SpðNcÞ, with fermions free vectorial gauge theory with a general (simple) gauge transforming according to the (irreducible) representations group G and Nf massless fermions in a representation R listed above, namely F, A, S2, and A2. In the Cartan R of G, including the anomalous dimension of a (gauge- classification of Lie algebras, An ¼ SUðn þ 1Þ, Bn ¼ ðΔ4Þ ð2 þ 1Þ ¼ ð2 Þ ¼ ð2 Þ ð Þ invariant) bilinear fermion operator up to O f and the SO n , Cn Sp n , and Dn SO n . For SO Nc dβ 0 ≥ 6 derivative of the beta function at α , jα¼α ≡ β ,upto with even Nc, we restrict to Nc since the algebra Dn is IR dα IR IR ≥ 3 ð Þ OðΔ5Þ. These results for general G and R were evaluated simple if n , and for Sp Nc , we restrict to even Nc, f owing to the D ¼ Spð2nÞ correspondence of Lie algebras. for G ¼ SUðN Þ with several fermion representations. n c Henceforth, these restrictions on N will be implicit. Since the global chiral symmetry is realized exactly in c We calculate the coefficients κ to OðΔ4Þ in the Δ series the non-Abelian Coulomb phase, the bilinear fermion j f f γ operators can be classified according to their representation expansion of the anomalous dimension ψψ¯ ;IR of the ψψ¯ properties under this symmetry, including flavor-singlet (gauge-invariant) fermion bilinear . Again, this is the and flavor-nonsinglet. Let γψψ¯ denote the anomalous same for the flavor-singlet and flavor-nonsinglet bilinears dimension of the (gauge-invariant) fermion bilinear, ψψ¯ [14], so we use the same notation for both. Stating our γ results at the outset, we find that (in addition to the and let ψψ¯ ;IR denote its value at the IR fixed point. The γ manifestly positive κ1 and κ2) κ3 and κ4 are positive for scheme-independent expansion of ψψ¯ ;IR can be written as both the SOðNcÞ and SpðNcÞ theories and for all of the X∞ representations that we consider. Some earlier work on the γ ¼ κ Δj ð Þ ð Þ ð Þ ψψ¯ ;IR j f: 1:2 conformal window in SO Nc and Sp Nc gauge theories, j¼1 including estimates of the lower end of this conformal window from perturbative four-loop results and Schwinger- We denote the truncation of the right-hand side of Eq. (1.2) Dyson methods, was reported in [16,17]. γ so the upper limit on the sum over j is the maximal power p We will also use our calculation of ψψ¯ ;IR to estimate the rather than ∞ as γψψ¯ Δp . The anomalous dimension γψψ¯ ;IR; f ;IR value of Nf, denoted Nf;cr, that defines the lower end of the is the same for the flavor-singlet and flavor-nonsinglet non-Abelian Coulomb phase. We do this by combining γ p fermion bilinears [14], and hence we use the simple the monotonic behavior that we find for ψψ¯ ;IR;Δ for all p γ κ f notations ψψ¯ ;IR and j for both.
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