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Robert Shrock - One of the best experts on this subject based on the ideXlab platform.
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scheme independent series expansions at an infrared zero of the Beta Function in asymptotically free gauge theories
Physical Review D, 2016Co-Authors: Thomas A Ryttov, Robert ShrockAbstract:We consider an asymptotically free vectorial gauge theory, with gauge group $G$ and ${N}_{f}$ fermions in a representation $R$ of $G$, having an infrared (IR) zero in the Beta Function at ${\ensuremath{\alpha}}_{\mathrm{IR}}$. We present general formulas for scheme-independent series expansions of quantities, evaluated at ${\ensuremath{\alpha}}_{\mathrm{IR}}$, as powers of an ${N}_{f}$-dependent expansion parameter, ${\mathrm{\ensuremath{\Delta}}}_{f}$. First, we apply these to calculate the derivative $d\ensuremath{\Beta}/d\ensuremath{\alpha}$ evaluated at ${\ensuremath{\alpha}}_{\mathrm{IR}}$, denoted ${\ensuremath{\Beta}}_{\mathrm{IR}}^{\ensuremath{'}}$, which is equivalent to the anomalous dimension of the $\mathrm{Tr}({F}_{\ensuremath{\mu}\ensuremath{\nu}}{F}^{\ensuremath{\mu}\ensuremath{\nu}})$ operator, to order ${\mathrm{\ensuremath{\Delta}}}_{f}^{4}$ for general $G$ and $R$, and to order ${\mathrm{\ensuremath{\Delta}}}_{f}^{5}$ for $G=\mathrm{SU}(3)$ and fermions in the fundamental representation. Second, we calculate the scheme-independent expansions of the anomalous dimension of the flavor-nonsinglet and flavor-singlet bilinear fermion antisymmetric Dirac tensor operators up to order ${\mathrm{\ensuremath{\Delta}}}_{f}^{3}$. The results are compared with rigorous upper bounds on anomalous dimensions of operators in conformally invariant theories. Our other results include an analysis of the limit ${N}_{c}\ensuremath{\rightarrow}\ensuremath{\infty}$, ${N}_{f}\ensuremath{\rightarrow}\ensuremath{\infty}$ with ${N}_{f}/{N}_{c}$ fixed, calculation and analysis of Pad\'e approximants, and comparison with conventional higher-loop calculations of ${\ensuremath{\Beta}}_{\mathrm{IR}}^{\ensuremath{'}}$ and anomalous dimensions as power series in $\ensuremath{\alpha}$.
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new scheme transformations and application to study scheme dependence of an infrared zero of the Beta Function in gauge theories
Physical Review D, 2014Co-Authors: Gongjun Choi, Robert ShrockAbstract:We present two new one-parameter families of scheme transformations and apply these to study the scheme dependence of the infrared zero in the Beta Function of an asymptotically free non-Abelian gauge theory up to four-loop order. Our results provide a further quantitative measure of this scheme dependence, showing that for moderate values of the gauge coupling and the parameter specifying the scheme transformation, this dependence is relatively mild. We also remark on a generalized multi-parameter family of rational scheme transformations.
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generalized scheme transformations for the elimination of higher loop terms in the Beta Function of a gauge theory
Physical Review D, 2014Co-Authors: Robert ShrockAbstract:We construct and study a generalized one-parameter class of scheme transformations, denoted $S_{R,m,k_1}$ with $m \ge 2$, with the property that an $S_{R,m,k_1}$ scheme transformation eliminates the $\ell$-loop terms in the Beta Function of a gauge theory from loop order $\ell=3$ to order $\ell=m+1$, inclusive. These scheme transformations are applied to the higher-loop calculation of the infrared zero of the Beta Function of an asymptotically free gauge theory with multiple fermions. We show that scheme transformations in this generalized class satisfy a set of criteria for physical acceptability over a larger range of numbers of fermions than previously studied scheme transformations. We also present an interesting modification of a different type of scheme transformation that removes the three-loop term in the Beta Function.
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study of possible ultraviolet zero of the Beta Function in gauge theories with many fermions
Physical Review D, 2014Co-Authors: Robert ShrockAbstract:We study the possibility of an ultraviolet (UV) zero in the $n$-loop Beta Function of U(1) and non-Abelian gauge theories with $N_f$ fermions for large $N_f$. The effect of scheme transformations on the coefficients of different powers of $N_f$ in the $n$-loop term in the Beta Function is calculated. A general scheme-independent criterion is given for determining whether or not the $n$-loop Beta Function has a UV zero for large $N_f$. We compare the results with exact integral representations of the leading terms in the Beta Functions for the respective Abelian and non-Abelian theories in the limit $N_f \to \infty$ limit with $N_f \alpha$ finite. As part of this study, new analytic and numerical results are presented for certain coefficients, denoted $b_{n,n-1}$, that control the large-$N_f$ behavior at $n$-loop order in the Beta Function. We also investigate various test Functions incorporating a power-law and essential UV zero in the Beta Function and determine their manifestations in series expansions in powers of coupling and in powers of $1/N_f$.
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study of scheme transformations to remove higher loop terms in the Beta Function of a gauge theory
Physical Review D, 2013Co-Authors: Robert ShrockAbstract:Since three-loop and higher-loop terms in the $\Beta$ Function of a gauge theory are scheme-dependent, one can, at least for sufficiently small coupling, carry out a scheme transformation that removes these terms. A basic question concerns the extent to which this can be done at an infrared fixed point of an asymptotically free gauge theory. This is important for quantitative analyses of the scheme dependence of such a fixed point. Here we study a scheme transformation $S_{R,m}$ with $m \ge 2$ that is constructed so as to remove the terms in the Beta Function at loop order $\ell=3$ to $\ell=m+1$, inclusive. Starting from an arbitrary initial scheme, we present general expressions for the coefficients of terms of loop order $\ell$ in the Beta Function in the transformed scheme from $\ell=m+2$ up to $\ell=8$. Extending a previous study of $S_{R,2}$, we investigate the range of applicability of the $S_{R,3}$ scheme transformation in an asymptotically free SU($N_c$) gauge theory with an infrared zero in $\Beta$ depending on the number, $N_f$, of fermions in the theory. We show that this $S_{R,3}$ scheme transformation can only be applied self-consistently in a restricted range of $N_f$ with a correspondingly small value of infrared fixed-point coupling. We also study the effect of higher-loop terms on the Beta Function of a U(1) gauge theory.
K V Stepanyantz - One of the best experts on this subject based on the ideXlab platform.
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the all loop perturbative derivation of the nsvz Beta β Function and the nsvz scheme in the non abelian case by summing singular contributions
European Physical Journal C, 2020Co-Authors: K V StepanyantzAbstract:The perturbative all-loop derivation of the NSVZ $$\Beta $$ -Function for $${{\mathcal {N}}}=1$$ supersymmetric gauge theories regularized by higher covariant derivatives is finalized by calculating the sum of singularities produced by quantum superfields. These singularities originate from integrals of double total derivatives and determine all contributions to the $$\Beta $$ -Function starting from the two-loop approximation. Their sum is expressed in terms of the anomalous dimensions of the quantum gauge superfield, of the Faddeev–Popov ghosts, and of the matter superfields. This allows obtaining the NSVZ equation in the form of a relation between the $$\Beta $$ -Function and these anomalous dimensions for the renormalization group Functions defined in terms of the bare couplings. It holds for an arbitrary renormalization prescription supplementing the higher covariant derivative regularization. For the renormalization group Functions defined in terms of the renormalized couplings we prove that in all loops one of the NSVZ schemes is given by the HD + MSL prescription.
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the all loop perturbative derivation of the nsvz Beta Function and the nsvz scheme in the non abelian case by summing singular contributions
arXiv: High Energy Physics - Theory, 2020Co-Authors: K V StepanyantzAbstract:The perturbative all-loop derivation of the NSVZ $\Beta$-Function for ${\cal N}=1$ supersymmetric gauge theories regularized by higher covariant derivatives is finalized by calculating the sum of singularities produced by quantum superfields. These singularities originate from integrals of double total derivatives and determine all contributions to the $\Beta$-Function starting from the two-loop approximation. Their sum is expressed in terms of the anomalous dimensions of the quantum gauge superfield, of the Faddeev--Popov ghosts, and of the matter superfields. This allows obtaining the NSVZ equation in the form of a relation between the $\Beta$-Function and these anomalous dimensions for the renormalization group Functions defined in terms of the bare couplings. It holds for an arbitrary renormalization prescription supplementing the higher covariant derivative regularization. For the renormalization group Functions defined in terms of the renormalized couplings we prove that in all loops one of the NSVZ schemes is given by the HD+MSL prescription.
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three loop contribution of the faddeev popov ghosts to the Beta Function of cal n 1 supersymmetric gauge theories and the nsvz relation
arXiv: High Energy Physics - Theory, 2019Co-Authors: M D Kuzmichev, N P Meshcheriakov, S V Novgorodtsev, I E Shirokov, K V StepanyantzAbstract:We find the three-loop contribution to the $\Beta$-Function of ${\cal N}=1$ supersymmetric gauge theories regularized by higher covariant derivatives produced by the supergraphs containing loops of the Faddeev--Popov ghosts. This is done using a recently proposed algorithm, which essentially simplifies such multiloop calculations. The result is presented in the form of an integral of double total derivatives in the momentum space. The considered contribution to the $\Beta$-Function is compared with the two-loop anomalous dimension of the Faddeev--Popov ghosts. This allows verifying the validity of the NSVZ equation written as a relation between the $\Beta$-Function and the anomalous dimensions of the quantum superfields. It is demonstrated that in the considered approximation the NSVZ equation is satisfied for the renormalization group Functions defined in terms of the bare couplings. The necessity of the nonlinear renormalization for the quantum gauge superfield is also confirmed.
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three loop contribution of the faddeev popov ghosts to the Beta β Function of mathcal n 1 n 1 supersymmetric gauge theories and the nsvz relation
European Physical Journal C, 2019Co-Authors: M D Kuzmichev, N P Meshcheriakov, S V Novgorodtsev, I E Shirokov, K V StepanyantzAbstract:We find the three-loop contribution to the $$\Beta $$-Function of $$\mathcal{N}=1$$ supersymmetric gauge theories regularized by higher covariant derivatives produced by the supergraphs containing loops of the Faddeev–Popov ghosts. This is done using a recently proposed algorithm, which essentially simplifies such multiloop calculations. The result is presented in the form of an integral of double total derivatives in the momentum space. The considered contribution to the $$\Beta $$-Function is compared with the two-loop anomalous dimension of the Faddeev–Popov ghosts. This allows verifying the validity of the NSVZ equation written as a relation between the $$\Beta $$-Function and the anomalous dimensions of the quantum superfields. It is demonstrated that in the considered approximation the NSVZ equation is satisfied for the renormalization group Functions defined in terms of the bare couplings. The necessity of the nonlinear renormalization for the quantum gauge superfield is also confirmed.
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factorization of integrals defining the two loop Beta Function for the general renormalizable n 1 sym theory regularized by the higher covariant derivatives into integrals of double total derivatives
arXiv: High Energy Physics - Theory, 2011Co-Authors: K V StepanyantzAbstract:The integrals defining the two-loop Beta-Function for the general renormalizable N=1 supersymmetric Yang--Mills theory, regularized by higher covariant derivatives, are investigated. It is shown that they are given by integrals of double total derivatives. These integrals are not equal to zero due to appearing of delta-Functions. These delta-Functions allow to reduce the two-loop integrals to one-loop integrals, which can be easily calculated. The result agrees with the exact NSVZ Beta-Function and calculations made by different methods.
Chik Him Wong - One of the best experts on this subject based on the ideXlab platform.
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fate of a recent conformal fixed point and Beta Function in the su 3 bsm gauge theory with ten massless flavors
Proceedings of The 36th Annual International Symposium on Lattice Field Theory — PoS(LATTICE2018), 2019Co-Authors: Zoltan Fodor, Kieran Holland, Julius Kuti, Daniel Nogradi, Chik Him WongAbstract:$SU(3)$ gauge theory with $N_f$ fermions in the fundamental representation serves as a theoretical testing ground for possible infrared conformal behavior, which could play a role in BSM composite Higgs models. We use lattice simulations to study the 10-flavor model, for which it has been claimed there is an infrared fixed point in the gauge coupling Beta-Function. Our results suggest the opposite conclusion, namely we find no Beta-Function fixed point in the explored range, with qualitative agreement with the 5-loop MSbar prediction. We comment on the inconsistency between our findings and other studies.
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fate of a recent conformal fixed point and β Function in the su 3 bsm gauge theory with ten massless flavors
Proceedings of The 36th Annual International Symposium on Lattice Field Theory — PoS(LATTICE2018), 2019Co-Authors: Zoltan Fodor, Kieran Holland, Julius Kuti, Daniel Nogradi, Chik Him WongAbstract:$SU(3)$ gauge theory with $N_f$ fermions in the fundamental representation serves as a theoretical testing ground for possible infrared conformal behavior, which could play a role in BSM composite Higgs models. We use lattice simulations to study the 10-flavor model, for which it has been claimed there is an infrared fixed point in the gauge coupling Beta-Function. Our results suggest the opposite conclusion, namely we find no Beta-Function fixed point in the explored range, with qualitative agreement with the 5-loop MSbar prediction. We comment on the inconsistency between our findings and other studies.
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fate of a recent conformal fixed point and Beta Function in the su 3 bsm gauge theory with ten massless flavors
arXiv: High Energy Physics - Lattice, 2018Co-Authors: Zoltan Fodor, Kieran Holland, Julius Kuti, Daniel Nogradi, Chik Him WongAbstract:SU(3) gauge theory with $N_f$ fermions in the fundamental representation serves as a theoretical testing ground for possible infrared conformal behavior, which could play a role in BSM composite Higgs models. We use lattice simulations to study the 10-flavor model, for which it has been claimed there is an infrared fixed point in the gauge coupling $\Beta$-Function. Our results suggest the opposite conclusion, namely we find no $\Beta$-Function fixed point in the explored range, with qualitative agreement with the 5-loop $\overline{MS}$ prediction. We comment on the inconsistency between our findings and other studies.
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a new method for the Beta Function in the chiral symmetry broken phase
Epj Web of Conferences, 2018Co-Authors: Zoltan Fodor, Kieran Holland, Julius Kuti, Daniel Nogradi, Chik Him WongAbstract:We describe a new method to determine non-perturbatively the Beta Function of a gauge theory using lattice simulations in the p-regime of the theory. This complements alternative measurements of the Beta Function working directly at zero fermion mass and bridges the gap between the weak coupling perturbative regime and the strong coupling regime relevant to the mass spectrum of the theory. We apply this method to ${\mathrm {SU(3)} }$ gauge theory with two fermion flavors in the 2-index symmetric (sextet) representation. We find that the Beta Function is small but non-zero at the renormalized coupling value $g^2 = 6.7$, consistent with our previous independent investigation using simulations directly at zero fermion mass. The model continues to be a very interesting explicit realization of the near-conformal composite Higgs paradigm which could be relevant for Beyond Standard Model phenomenology.
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the twelve flavor boldsymbol Beta Function and dilaton tests of the sextet scalar
arXiv: High Energy Physics - Lattice, 2017Co-Authors: Zoltan Fodor, Kieran Holland, Julius Kuti, Daniel Nogradi, Chik Him WongAbstract:We discuss near-conformal gauge theories beyond the standard model (BSM) where interesting results on the twelve-flavor $\Beta$-Function of massless fermions in the fundamental representation of the SU(3) color gauge group and dilaton tests of the light scalar with two massless fermions in the two-index symmetric tensor (sextet) representation can be viewed as parts of the same BSM paradigm under investigation. We report results from high precision analysis of the twelve-flavor $\Beta$-Function \cite{Fodor:2016zil} refuting its published IRFP \cite{Cheng:2014jba,Hasenfratz:2016dou}. We present our objections to recent claims \cite{Hasenfratz:2017mdh,Hasenfratz:2017qyr} for non-universal behavior of staggered fermions used in our analysis. We also report our first analysis of dilaton tests of the light $0^{++}$ scalar in the sextet model and comment on related post-conference developments. The dilaton test is the main thrust of this conference contribution including presentation #405 on the $n_f=12$ $\Beta$-Function and presentation #260 on dilaton tests of the sextet model. They are both selected from the near-conformal BSM paradigm.
Thomas A Ryttov - One of the best experts on this subject based on the ideXlab platform.
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scheme independent series expansions at an infrared zero of the Beta Function in asymptotically free gauge theories
Physical Review D, 2016Co-Authors: Thomas A Ryttov, Robert ShrockAbstract:We consider an asymptotically free vectorial gauge theory, with gauge group $G$ and ${N}_{f}$ fermions in a representation $R$ of $G$, having an infrared (IR) zero in the Beta Function at ${\ensuremath{\alpha}}_{\mathrm{IR}}$. We present general formulas for scheme-independent series expansions of quantities, evaluated at ${\ensuremath{\alpha}}_{\mathrm{IR}}$, as powers of an ${N}_{f}$-dependent expansion parameter, ${\mathrm{\ensuremath{\Delta}}}_{f}$. First, we apply these to calculate the derivative $d\ensuremath{\Beta}/d\ensuremath{\alpha}$ evaluated at ${\ensuremath{\alpha}}_{\mathrm{IR}}$, denoted ${\ensuremath{\Beta}}_{\mathrm{IR}}^{\ensuremath{'}}$, which is equivalent to the anomalous dimension of the $\mathrm{Tr}({F}_{\ensuremath{\mu}\ensuremath{\nu}}{F}^{\ensuremath{\mu}\ensuremath{\nu}})$ operator, to order ${\mathrm{\ensuremath{\Delta}}}_{f}^{4}$ for general $G$ and $R$, and to order ${\mathrm{\ensuremath{\Delta}}}_{f}^{5}$ for $G=\mathrm{SU}(3)$ and fermions in the fundamental representation. Second, we calculate the scheme-independent expansions of the anomalous dimension of the flavor-nonsinglet and flavor-singlet bilinear fermion antisymmetric Dirac tensor operators up to order ${\mathrm{\ensuremath{\Delta}}}_{f}^{3}$. The results are compared with rigorous upper bounds on anomalous dimensions of operators in conformally invariant theories. Our other results include an analysis of the limit ${N}_{c}\ensuremath{\rightarrow}\ensuremath{\infty}$, ${N}_{f}\ensuremath{\rightarrow}\ensuremath{\infty}$ with ${N}_{f}/{N}_{c}$ fixed, calculation and analysis of Pad\'e approximants, and comparison with conventional higher-loop calculations of ${\ensuremath{\Beta}}_{\mathrm{IR}}^{\ensuremath{'}}$ and anomalous dimensions as power series in $\ensuremath{\alpha}$.
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supersymmetry inspired qcd Beta Function
Physical Review D, 2008Co-Authors: Thomas A Ryttov, Francesco SanninoAbstract:We propose an all orders Beta Function for ordinary Yang-Mills theories with or without fermions inspired by the Novikov-Shifman-Vainshtein-Zakharov Beta Function of N=1 supersymmetric gauge theories. The Beta Function allows us to bound the conformal window. When restricting to one adjoint Weyl fermion we show how the proposed Beta Function matches the one of supersymmetric Yang-Mills theory. The running of the pure Yang-Mills coupling is computed and the deviation from the two loop result is presented. We then compare the deviation with the one obtained from lattice data also with respect to the two loop running.
Daniel Nogradi - One of the best experts on this subject based on the ideXlab platform.
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fate of a recent conformal fixed point and Beta Function in the su 3 bsm gauge theory with ten massless flavors
Proceedings of The 36th Annual International Symposium on Lattice Field Theory — PoS(LATTICE2018), 2019Co-Authors: Zoltan Fodor, Kieran Holland, Julius Kuti, Daniel Nogradi, Chik Him WongAbstract:$SU(3)$ gauge theory with $N_f$ fermions in the fundamental representation serves as a theoretical testing ground for possible infrared conformal behavior, which could play a role in BSM composite Higgs models. We use lattice simulations to study the 10-flavor model, for which it has been claimed there is an infrared fixed point in the gauge coupling Beta-Function. Our results suggest the opposite conclusion, namely we find no Beta-Function fixed point in the explored range, with qualitative agreement with the 5-loop MSbar prediction. We comment on the inconsistency between our findings and other studies.
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fate of a recent conformal fixed point and β Function in the su 3 bsm gauge theory with ten massless flavors
Proceedings of The 36th Annual International Symposium on Lattice Field Theory — PoS(LATTICE2018), 2019Co-Authors: Zoltan Fodor, Kieran Holland, Julius Kuti, Daniel Nogradi, Chik Him WongAbstract:$SU(3)$ gauge theory with $N_f$ fermions in the fundamental representation serves as a theoretical testing ground for possible infrared conformal behavior, which could play a role in BSM composite Higgs models. We use lattice simulations to study the 10-flavor model, for which it has been claimed there is an infrared fixed point in the gauge coupling Beta-Function. Our results suggest the opposite conclusion, namely we find no Beta-Function fixed point in the explored range, with qualitative agreement with the 5-loop MSbar prediction. We comment on the inconsistency between our findings and other studies.
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fate of a recent conformal fixed point and Beta Function in the su 3 bsm gauge theory with ten massless flavors
arXiv: High Energy Physics - Lattice, 2018Co-Authors: Zoltan Fodor, Kieran Holland, Julius Kuti, Daniel Nogradi, Chik Him WongAbstract:SU(3) gauge theory with $N_f$ fermions in the fundamental representation serves as a theoretical testing ground for possible infrared conformal behavior, which could play a role in BSM composite Higgs models. We use lattice simulations to study the 10-flavor model, for which it has been claimed there is an infrared fixed point in the gauge coupling $\Beta$-Function. Our results suggest the opposite conclusion, namely we find no $\Beta$-Function fixed point in the explored range, with qualitative agreement with the 5-loop $\overline{MS}$ prediction. We comment on the inconsistency between our findings and other studies.
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a new method for the Beta Function in the chiral symmetry broken phase
Epj Web of Conferences, 2018Co-Authors: Zoltan Fodor, Kieran Holland, Julius Kuti, Daniel Nogradi, Chik Him WongAbstract:We describe a new method to determine non-perturbatively the Beta Function of a gauge theory using lattice simulations in the p-regime of the theory. This complements alternative measurements of the Beta Function working directly at zero fermion mass and bridges the gap between the weak coupling perturbative regime and the strong coupling regime relevant to the mass spectrum of the theory. We apply this method to ${\mathrm {SU(3)} }$ gauge theory with two fermion flavors in the 2-index symmetric (sextet) representation. We find that the Beta Function is small but non-zero at the renormalized coupling value $g^2 = 6.7$, consistent with our previous independent investigation using simulations directly at zero fermion mass. The model continues to be a very interesting explicit realization of the near-conformal composite Higgs paradigm which could be relevant for Beyond Standard Model phenomenology.
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the twelve flavor boldsymbol Beta Function and dilaton tests of the sextet scalar
arXiv: High Energy Physics - Lattice, 2017Co-Authors: Zoltan Fodor, Kieran Holland, Julius Kuti, Daniel Nogradi, Chik Him WongAbstract:We discuss near-conformal gauge theories beyond the standard model (BSM) where interesting results on the twelve-flavor $\Beta$-Function of massless fermions in the fundamental representation of the SU(3) color gauge group and dilaton tests of the light scalar with two massless fermions in the two-index symmetric tensor (sextet) representation can be viewed as parts of the same BSM paradigm under investigation. We report results from high precision analysis of the twelve-flavor $\Beta$-Function \cite{Fodor:2016zil} refuting its published IRFP \cite{Cheng:2014jba,Hasenfratz:2016dou}. We present our objections to recent claims \cite{Hasenfratz:2017mdh,Hasenfratz:2017qyr} for non-universal behavior of staggered fermions used in our analysis. We also report our first analysis of dilaton tests of the light $0^{++}$ scalar in the sextet model and comment on related post-conference developments. The dilaton test is the main thrust of this conference contribution including presentation #405 on the $n_f=12$ $\Beta$-Function and presentation #260 on dilaton tests of the sextet model. They are both selected from the near-conformal BSM paradigm.