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Robert Shrock

Publications and source records attributed to Robert Shrock.

At least 55 records · Page 3Linked to original sources
arXiv2018

Scheme-Independent Calculations of Properties at a Conformal Infrared Fixed Point in Gauge Theories with Multiple Fermion Representations

In previous work we have presented scheme-independent calculations of physical properties of operators at a conformally invariant infrared fixed point in an asymptotically free gauge theory with gauge group $G$ and $N_f$ fermions in a representation $R$ of $G$. Here we generalize this analysis to the case of fermions in multiple representations, focusing on the case of two different representations. Our results include the calculation of the anomalous dimensions of gauge-invariant fermion bilinear operators, and the derivative of the beta function, evaluated at the infrared fixed point. We illustrate our results in an SU($N_c$) gauge theory with $N_F$ fermions in the fundamental representation and $N_{Adj}$ fermions in the adjoint representation.

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arXiv2018

Scheme-Independent Calculations of Anomalous Dimensions of Baryon Operators in Conformal Field Theories

We present the first analytic scheme-independent series calculations of anomalous dimensions of several types of baryon operators at an infrared fixed point (IRFP) in an asymptotically free SU(3) gauge theory with $N_f$ fermions. Separately, for an asymptotically free gauge theory with a gauge group $G$ and $N_f$ fermions in a representation $R$ of $G$, we consider physical quantities at an IRFP, including the anomalous dimension of gauge-invariant fermion bilinears and the derivative of the beta function. These quantities have been calculated in series expansions whose coefficients have been proved to be scheme-independent at each order. We illustrate the scheme independence using a variety of schemes, including the RI$^\prime$ scheme and several types of momentum subtraction (MOM) schemes.

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arXiv2017

Duality in a Supersymmetric Gauge Theory From a Perturbative Viewpoint

We study duality in $\mathcal{N}=1$ supersymmetric QCD in the non-Abelian Coulomb phase, order-by-order in scheme-independent series expansions. Using exact results, we show how the dimensions of various fundamental and composite chiral superfields, and the quantities $a$, $a/c$, and $b$ at superconformal fixed points of the renormalization group emerge in scheme-independent series expansions in the electric and magnetic theories. We further demonstrate that truncations of these series expansions to modest order yield very accurate approximations to these quantities.

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arXiv2017

Physics of the Non-Abelian Coulomb Phase: Insights from Padé Approximants

We consider a vectorial, asymptotically free SU($N_c$) gauge theory with $N_f$ fermions in a representation $R$ having an infrared (IR) fixed point. We calculate and analyze Padé approximants to scheme-independent series expansions for physical quantities at this IR fixed point, including the anomalous dimension, $γ_{\barψψ,IR}$, to $O(Δ_f^4)$, and the derivative of the beta function, $β'_{IR}$, to $O(Δ_f^5)$, where $Δ_f$ is an $N_f$-dependent expansion variable. We consider the fundamental, adjoint, and rank-2 symmetric tensor representations. The results are applied to obtain further estimates of $γ_{\barψψ,IR}$ and $β'_{IR}$ for several SU($N_c$) groups and representations $R$, and comparisons are made with lattice measurements. We apply our results to obtain new estimates of the extent of the respective non-Abelian Coulomb phases in several theories. For $R=F$, the limit $N_c \to \infty$ and $N_f \to \infty$ with $N_f/N_c$ fixed is considered. We assess the accuracy of the scheme-independent series expansion of $γ_{\barψψ,IR}$ in comparison with the exactly known expression in an ${\cal N}=1$ supersymmetric gauge theory. It is shown that an expansion of $γ_{\barψψ,IR}$ to $O(Δ_f^4)$ is quite accurate throughout the entire non-Abelian Coulomb phase of this supersymmetric theory.

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arXiv2017

$β'_{IR}$ at an Infrared Fixed Point in Chiral Gauge Theories

We present scheme-independent calculations of the derivative of the beta function, denoted $β'_{IR}$, at a conformally invariant infrared (IR) fixed point, in several asymptotically free chiral gauge theories, namely SO($4k+2$) with $2 \le k \le 4$ with respective numbers $N_f$ of fermions in the spinor representation, and E$_6$ with fermions in the fundamental representation.

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arXiv2017

Infrared Fixed Point Physics in ${\rm SO}(N_c)$ and ${\rm Sp}(N_c)$ Gauge Theories

We study properties of asymptotically free vectorial gauge theories with gauge groups $G={\rm SO}(N_c)$ and $G={\rm Sp}(N_c)$ and $N_f$ fermions in a representation $R$ of $G$, at an infrared (IR) zero of the beta function, $α_{IR}$, 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 anomalous dimensions of (gauge-invariant) fermion bilinear operators $γ_{\barψψ,IR}$ to $O(Δ_f^4)$ and of the derivative of the beta function at $α_{IR}$, denoted $β'_{IR}$, to $O(Δ_f^5)$, where $Δ_f$ is an $N_f$-dependent expansion variable. It is shown that all coefficients in the expansion of $γ_{\barψψ,IR}$ that we calculate are positive for all representations considered, so that to $O(Δ_f^4)$, $γ_{\barψψ,IR}$ 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.

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arXiv2017

Scheme-Independent Calculations of Physical Quantities in an ${\cal N}=1$ Supersymmetric Gauge Theory

We consider an asymptotically free, vectorial, ${\cal N}=1$ supersymmetric gauge theory with gauge group $G$ and $N_f$ pairs of chiral superfields in the respective representations ${\cal R}$ and $\bar {\cal R}$ of $G$, having an infrared fixed point (IRFP) of the renormalization group at $α_{IR}$. We present exact results for the anomalous dimensions of various (gauge-invariant) composite chiral superfields $γ_{Φ_{\rm prod}}$ at the IRFP and prove that these increase monotonically with decreasing $N_f$ in the non-Abelian Coulomb phase of the theory and that scheme-independent expansions for these anomalous dimensions as powers of an $N_f$-dependent variable, $Δ_f$, exhibit monotonic and rapid convergence to the exact $γ_{Φ_{\rm prod}}$ throughout this phase. We also present a scheme-independent calculation of the derivative of the beta function, $dβ/dα|_{α=α_{IR}}$, denoted $β'_{IR}$, up to $O(Δ_f^3)$ for general $G$ and ${\cal R}$, and, for the case $G={\rm SU}(N_c)$, ${\cal R}=F$, we give an analysis of the properties of $β'_{IR}$ calculated to $O(Δ_f^4)$.

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arXiv2017

Higher-Order Scheme-Independent Series Expansions of $γ_{\barψψ,IR}$ and $β'_{IR}$ in Conformal Field Theories

We study a vectorial asymptotically free gauge theory, with gauge group $G$ and $N_f$ massless fermions in a representation $R$ of this group, that exhibits an infrared (IR) zero in its beta function, $β$, at the coupling $α=α_{IR}$ in the non-Abelian Coulomb phase. For general $G$ and $R$, we calculate the scheme-independent series expansions of (i) the anomalous dimension of the fermion bilinear, $γ_{\barψψ,IR}$, to $O(Δ_f^4)$ and (ii) the derivative $β' = dβ/dα$, to $O(Δ_f^5)$, both evaluated at $α_{IR}$, where $Δ_f$ is an $N_f$-dependent expansion variable. These are the highest orders to which these expansions have been calculated. We apply these general results to theories with $G={\rm SU}(N_c)$ and $R$ equal to the fundamental, adjoint, and symmetric and antisymmetric rank-2 tensor representations. It is shown that for all of these representations, $γ_{\barψψ,IR}$, calculated to the order $Δ_f^p$, with $1 \le p \le 4$, increases monotonically with decreasing $N_f$ and, for fixed $N_f$, is a monotonically increasing function of $p$. Comparisons of our scheme-independent calculations of $γ_{\barψψ,IR}$ and $β'_{IR}$ are made with our earlier higher $n$-loop values of these quantities, and with lattice measurements. For $R=F$, we present results for the limit $N_c \to \infty$ and $N_f \to \infty$ with $N_f/N_c$ fixed. We also present expansions for $α_{IR}$ calculated to $O(Δ_f^4)$.

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arXiv2017

Higher-Order Scheme-Independent Calculations of Physical Quantities in the Conformal Phase of a Gauge Theory

We consider an asymptotically free vectorial SU($N_c$) gauge theory with $N_f$ massless fermions in a representation $R$, having an infrared fixed point (IRFP) of the renormalization group at $α_{IR}$ in the conformal non-Abelian Coulomb phase. The cases with $R$ equal to the fundamental, adjoint, and symmetric rank-2 tensor representation are considered. We present scheme-independent calculations of the anomalous dimension $γ_{\barψψ,IR}$ to $O(Δ_f^4)$ and $β'_{IR}$ to $O(Δ_f^5)$ at this IRFP, where $Δ_f$ is an $N_f$-dependent expansion parameter. Comparisons are made with conventional $n$-loop calculations and lattice measurements. As a test of the accuracy of the $Δ_f$ expansion, we calculate $γ_{\barψψ,IR}$ to $O(Δ_f^3)$ in ${\cal N}=1$ SU($N_c$) supersymmetric quantum chromodynamics and find complete agreement, to this order, with the exactly known expression. The $Δ_f$ expansion also avoids a problem in which an IRFP may not be manifest as an IR zero of a higher $n$-loop beta function.

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arXiv2016

On the Question of a Possible Infrared Zero in the Beta Function of the Finite-$N$ Gross-Neveu Model

We investigate whether the beta function of the finite-$N$ Gross-Neveu model, as calculated up to the four-loop level, exhibits evidence for an infrared zero. As part of our analysis, we calculate and analyze Padé approximants to this beta function and evaluate effects of scheme dependence. From our study, we find that in the range of coupling where the perturbative calculation of the four-loop beta function is reliable, it does not exhibit robust evidence for an infrared zero.

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arXiv2016

Study of the Six-Loop Beta Function of the $λϕ^4_4$ Theory

We investigate whether the six-loop beta function of the $λϕ^4_4$ theory exhibits evidence for an ultraviolet zero. As part of our analysis, we calculate and analyze Padé approximants to this beta function. Extending our earlier results at the five-loop level, we find that in the range of $λ$ where the perturbative calculation of the six-loop beta function is reliable, the theory does not exhibit robust evidence for an ultraviolet zero.

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arXiv2016

Scheme-Independent Calculation of $γ_{\barψψ,IR}$ for an SU(3) Gauge Theory

We present a scheme-independent calculation of the infrared value of the anomalous dimension of the fermion bilinear, $γ_{\barψψ,IR}$ in an SU(3) gauge theory as a function of the number of fermions, $N_f$, via a series expansion in powers of $Δ_f$, where $Δ_f=(16.5-N_f)$, to order $Δ_f^4$. We perform an extrapolation to obtain the first determination of the exact $γ_{\barψψ,IR}$ from continuum field theory. The results are compared with calculations of the $n$-loop values of this anomalous dimension from series in powers of the coupling and from lattice measurements.

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arXiv2016

Infrared Zero of $β$ and Value of $γ_m$ for an SU(3) Gauge Theory at the Five-Loop Level

We calculate the value of the coupling at the infrared zero of the beta function of an asymptotically free SU(3) gauge theory at the five-loop level as a function of the number of fermions. Both a direct analysis of the beta function and analyses of Padé approximants are used for this purpose. We then calculate the value of the five-loop anomalous dimension, $γ_m$, of the fermion bilinear at this IR zero of the beta function.

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arXiv2016

An Integral Formalism for the Construction of Scheme Transformations in Quantum Field Theory

We present an integral formalism for constructing scheme transformations in a quantum field theory. We apply this to generate several new useful scheme transformations. A comparative analysis is given of these scheme transformations in terms of their series expansion coefficients and their resultant effect on the interaction coupling, in particular at a zero of the beta function away from the origin in coupling-constant space.

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arXiv2016

Dynamical Symmetry Breaking in Chiral Gauge Theories with Direct-Product Gauge Groups

We analyze patterns of dynamical symmetry breaking in strongly coupled chiral gauge theories with direct-product gauge groups $G$. If the gauge coupling for a factor group $G_i \subset G$ becomes sufficiently strong, it can produce bilinear fermion condensates that break the $G_i$ symmetry itself and/or break other gauge symmetries $G_j \subset G$. Our comparative study of a number of strongly coupled direct-product chiral gauge theories elucidates how the patterns of symmetry breaking depend on the structure of $G$ and on the relative sizes of the gauge couplings corresponding to factor groups in the direct product.

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arXiv2016

Study of the Renormalization-Group Evolution of ${\cal N}=1$ Supersymmetric Gauge Theories Using Padé Approximants

We study asymptotically free SU($N_c$) gauge theories with ${\cal N}=1$ supersymmetry, including the purely gluonic theory and theories with $N_f$ copies of a pair of massless chiral superfields in the respective representations $R$ and $\bar R$ of SU($N_c$). The cases in which $R$ is the fundamental representation and the symmetric and antisymmetric rank-2 tensor representation are considered. We calculate Padé approximants to the beta functions for these theories in the $\overline{\rm DR}$ scheme up to four-loop order for the gluonic theory and up to three-loop order for the theories with matter superfields and compare results for IR zeros and poles with results from the NSVZ beta function. Our calculations provide a quantitative measure, for these theories, of how well finite-order perturbative results calculated in one scheme reproduce properties of a known beta function calculated in a different scheme.

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arXiv2015

$A_k \bar F$ Chiral Gauge Theories

We study asymptotically free chiral gauge theories with an SU($N$) gauge group and chiral fermions transforming according to the antisymmetric rank-$k$ tensor representation, $A_k \equiv [k]_N$, and the requisite number, $n_{\bar F}$, of copies of fermions in the conjugate fundamental representation, $\bar F \equiv \overline{[1]}_N$, to render the theories anomaly-free. We denote these as $A_k \, \bar F$ theories. We take $N \ge 2k+1$ so that $n_{\bar F} \ge 1$. The $A_2 \, \bar F$ theories form an infinite family with $N \ge 5$, but we show that the $A_3 \, \bar F$ and $A_4 \,\bar F$ theories are only asymptotically free for $N$ in the respective ranges $7 \le N \le 17$ and $9 \le N \le 11$, and that there are no asymptotically free $A_k \, \bar F$ theories with $k \ge 5$. We investigate the types of ultraviolet to infrared evolution for these $A_k \, \bar F$ theories and find that, depending on $k$ and $N$, they may lead to a non-Abelian Coulomb phase, or may involve confinement with massless gauge-singlet composite fermions, bilinear fermion condensation with dynamical gauge and global symmetry breaking, or formation of multifermion condensates that preserve the gauge symmetry. We also show that there are no asymptotically free, anomaly-free SU($N$) $S_k \, \bar F$ chiral gauge theories with $k \ge 3$, where $S_k$ denotes the rank-$k$ symmetric representation.

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