The mass anomalous dimension is determined in SU(2) gauge theory coupled to $N_f$ fermions in the adjoint representation for $N_f = 2$, $3/2$, $1$ and $1/2$, where half-integer flavor numbers correspond to Majorana fermions. The numerical method is based on gradient flow. The results show near-conformal behavior for $N_f = 2$, $3/2$ and $1$. Particular emphasis is placed on $N_f = 2$, which is relevant for a strongly interacting extension of the Standard Model and has been studied in several previous investigations. We check whether the method is able to resolve discrepancies in earlier results for this theory. Overall, the method based on the gradient flow delivers reliable results in qualitative agreement with previously known numerical data.
In this work we consider the $SU(N)$ gauge theory with two Dirac fermions in the adjoint representation, in the limit of large $N$. In this limit the infinite-volume physics of this model can be studied by means of the corresponding twisted reduced model defined on a single site lattice. Making use of this strategy we study the reduced model for various values of $N$ up to 289. By analyzing the eigenvalue distribution of the adjoint Dirac operator we test the conformality of the theory and extract the corresponding mass anomalous dimension.
{We present the results of a numerical investigation of SU(2) gauge theory with $N_f=3/2$ flavours of fermions, corresponding to 3 Majorana fermions, which transform in the adjoint representation of the gauge group. At two values of the gauge coupling, the masses of bound states are considered as a function of the PCAC quark mass. The scaling of bound states masses indicates an infrared conformal behaviour of the theory. We obtain estimates for the fixed-point value of the mass anomalous dimension $γ^*$ from the scaling of masses and from the scaling of the mode number of the Wilson-Dirac oper…
We summarize our results concerning the spectrum and mass anomalous dimension of SU(2) gauge theories with various numbers of fermions in the adjoint representation, where each Majorana fermion corresponds effectively to half a Dirac flavour $N_f$. The most relevant examples for extensions of the standard model are supersymmetric Yang-Mills theory ($N_f=1/2$) and Minimal Walking Technicolour ($N_f=2$). In addition to these theories we will also consider the cases of $N_f=1$ and $N_f=3/2$. The results comprise the particle spectrum of glueballs, triplet and singlet mesons, and possible fraction…
In this work we present the results of our investigation of \su{2} gauge theory with two Dirac fermions in the adjoint representation (aQCD2), which belongs to the class of strongly interacting gauge theories that are of basic interest for extensions of the Standard Model. We have done numerical lattice simulations of this theory at two different values of the gauge coupling and several fermion masses. Our results include the particle spectrum and the mass anomalous dimension. The spectrum contains new exotic fermion-gluon states and flavour-singlet mesons. The mass anomalous dimension is dete…
Theories with fermions in the adjoint representation have several interesting applications in extensions of the standard model. The conformal window for these theories is of particular interest for technicolour extensions. We present here our newest results for the spectrum of $N_f=2$ adjoint QCD and compare them with the predictions for a conformal behaviour. The comparison with supersymmetric Yang-Mills theory, investigated with the same methods, will help to distinguish more clearly the conformal and the confining scenario. The spectrum includes additional fermionic states that are not pres…
We present a major update on our investigations of SU(2) gauge theory with one Dirac flavor in the adjoint representation on the lattice. In particular we consider larger volumes, as well as four different values of the gauge coupling. We provide results for the spectrum including gluonic, fermionic, and hybrid observables, Polyakov loops, and the anomalous dimension of the fermionic condensate from the Dirac mode number. These data confirm that the theory is close to the lower boundary of the conformal window for adjoint fermions. Our investigations provide important insights regarding the re…
The infrared effective theory of adjoint QCD with one Dirac flavour is still under debate. The theory could be confining, conformal, or with a massless fermion in the infrared. The study of chiral symmetry seems to be important to answer this question. While previous investigations have considered Wilson fermions, we present here the first results for this theory based on overlap fermions to avoid explicit chiral symmetry breaking. These indicate spontaneous chiral symmetry breaking by the formation of a fermion condensate. We have also investigated the running coupling of the theory, which in…
We present some results for SU(2) with one adjoint Dirac flavour from lattice studies. Data for the spectroscopy, the static potential, topological charge, and the anomalous dimension of the fermionic condensate are included. Our findings are found to be in- consistent with conventional confining behaviour, instead pointing tentatively towards a theory lying within or very near the onset of the conformal window, with an anomalous dimension of the fermionic condensate of almost 1. Implications of these findings on the building of models of strongly-interacting dynamics beyond the standard model…
Motivated by recent scenarios of exotic infrared behaviour and by earlier lattice findings, we present results for the SU(2) gauge theory with one Dirac flavor in the adjoint representation. This provides a major update on our previous investigation of this theory, including data for four values of the gauge coupling $β$, and for smaller masses and larger volumes than previously considered. Results for the particle spectrum, topological observables, and the anomalous dimension from both hyperscaling and the Dirac mode number are presented. At the finest coupling, we observe a mass anomalous di…
Via large and small $N_c$ relations we derive nonperturbative results about the conformal window of two-index theories. Using Schwinger-Dyson methods as well as four-loops results we estimate subleading corrections and show that naive large number of colors extrapolations are unreliable when $N_c$ is less than about six. Nevertheless useful nonperturbative inequalities for the size of the conformal windows, for any number of colors, can be derived. By further observing that the adjoint conformal window is independent of the number of colors we argue, among other things, that: The large $N_c$ t…
We determine the anomalous dimensions of baryon operators for the three color theory as function of the number of massless flavours within the conformal window to the maximum known order in perturbation theory. We show that the anomalous dimension of the baryon is controllably small, within the $δ$-expansion, for a wide range of number of flavours. We also find that this is always smaller than the anomalous dimension of the fermion mass operator. These findings challenge the partial compositeness paradigm.
We study SU(2) gauge theory with two fermion flavors in the adjoint representation. Using a clover improved HEX smeared action and the gradient flow running coupling allows us to simulate with larger lattice size than before. We find an infrared fixed point after a continuum extrapolation in the range $4.5 \lesssim g^{*2} \lesssim 5$. We also measure the mass anomalous dimension and find the value $ 0.25 \lesssim γ^* \lesssim 0.28 $ at the fixed point.
Non-Abelian gauge theories with fermions transforming in the adjoint representation of the gauge group (AdjQCD) are a fundamental ingredient of many models that describe the physics beyond the Standard Model. Two relevant examples are N=1 Supersymmetric Yang-Mills (SYM) theory and Minimal Walking Technicolor, which are gauge theories coupled to one adjoint Majorana and two adjoint Dirac fermions, respectively. While confinement is a property of N=1 SYM, Minimal Walking Technicolor is expected to be infrared conformal. We study the propagators of ghost and gluon fields in the Landau gauge to co…
We review methods and results for extracting the anomalous dimensions of operators from lattice field theory calculations. The most important application is the anomalous mass dimension in conformal or nearly conformal gauge field theories which might be related to dynamical electroweak symmetry breaking. Some discussion of the underlying theory of renormalization and mixing of operators is also included.
Four fermion interactions appear in many models of Beyond Standard Model physics. In Technicolour and composite Higgs models Standard Model fermion masses can be generated by four fermion terms. They are also expected to modify the dynamics of the new strongly interacting sector. In particular in technicolour models it has been suggested that they can be used to break infrared conformality and produce a walking theory with a large mass anomalous dimension. We study the SU(2) gauge theory with 2 adjoint fermions and a chirally symmetric four fermion term. We demonstrate chiral symmetry breaking…
SU(2) gauge theory with one and two adjoint fermions towards the continuum limit
The Cyprus Institute（塞浦路斯研究所）；Swansea University（斯旺西大学）；University of Jena（耶拿大学）；Universidad de Zaragoza（萨拉戈萨大学）；Universidad Autónoma de Madrid（马德里自治大学）；University of Southern Denmark（南丹麦大学）；Helmholtz Zentrum Dresden Rossendorf（德累斯顿罗森多夫亥姆霍兹中心）；Queen Mary University of London（伦敦玛丽女王大学）
SU(2) gauge theory with one Dirac flavour in the adjoint representation is investigated on a lattice. Initial results for the gluonic and mesonic spectrum, static potential from Wilson and Polyakov loops, and the anomalous dimension of the fermionic condensate from the Dirac mode number are presented. The results found are not consistent with conventional confining behaviour, instead tentatively pointing towards a theory lying within or very near the onset of the conformal window, with the anomalous dimension of the fermionic condensate in the range $0.9 \lesssim γ_* \lesssim 0.95$. The implic…
We present an update of our ongoing study of the SU(2) gauge theory with one flavor of Dirac fermion in the adjoint representation. Compared to our previous results we now have data at larger lattice volumes, smaller values of the fermion mass, and also larger values of $β$. We present data for the spectrum of mesons, baryons, glueballs, and the hybrid fermion-glue state, as well as new estimates of the mass anomalous dimension from both finite-size hyperscaling and the Dirac mode number, and discuss the implications of these data for the presence or otherwise of chiral symmetry breaking in th…
We investigate relevant properties of composite operators emerging in nonsupersymmetric, four-dimensional gauge-Yukawa theories with interacting conformal fixed points within a precise framework. The theories investigated in this work are structurally similar to the standard model of particle interactions, but differ by developing perturbative interacting fixed points. We investigate the physical properties of the singlet and the adjoint composite operators quadratic in the Higgs field, and discover that the singlet anomalous dimension is substantially larger than the adjoint one. The numerica…
We compute, to the first non-trivial order in the $ε$-expansion of a perturbed scalar field theory, the anomalous dimensions of an infinite class of primary operators with arbitrary spin $\ell=0,1,..$, including as a particular case the weakly broken higher-spin currents, using only constraints from conformal symmetry. Following the bootstrap philosophy, no reference is made to any Lagrangian, equations of motion or coupling constants. Even the space dimensions d are left free. The interaction is implicitly turned on through the local operators by letting them acquire anomalous dimensions. Whe…
This talk is an overview of our recent investigations of supersymmetric and near conformal gauge theories. We have studied extensively $\mathcal{N}=1$ super Yang-Mills theory, most recently with the gauge group SU(3). In addition we have investigated theories that show indications for a conformal behavior at an infrared fixed point. We have included a mixed fundamental and adjoint fermion action setup in our studies. I will explain how this is related to the investigation of supersymmetric QCD on the lattice and present some first studies of the main obstacles that need to be addressed in the …
We study SU(2) gauge theory with two Dirac fermions in the adjoint representation of the gauge group on the lattice. Using clover improved Wilson fermion action with hypercubic truncated stout smearing we perform simulations at larger coupling than earlier. We measure the evolution of the coupling constant using the step scaling method with Schrödinger functional and study the remaining discretization effects. At weak coupling we observe significant discretization effects, which make it difficult to obtain a fully controlled continuum limit. Nevertheless, the data remains consistent with the e…
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 …
We study four-dimensional conformal field theories with an $SU(N)$ global symmetry by employing the numerical conformal bootstrap. We consider the crossing relation associated with a four-point function of a spin~$0$ operator~$ϕ_i^{\Bar{k}}$ which belongs to the adjoint representation of $SU(N)$. For~$N=12$ for example, we found that the theory contains a spin~$0$ $SU(12)$-breaking relevant operator when the scaling dimension of~$ϕ_i^{\Bar{k}}$, $Δ_{ϕ_i^{\Bar{k}}}$, is smaller than~$1.71$. Considering the lattice simulation of many-flavor quantum chromodynamics with $12$~flavors on the basis o…
We propose a continuous real space renormalization group transformation based on gradient flow, allowing for a numerical study of renormalization without the need for costly ensemble matching. We apply our technique in a pilot study of SU$(3)$ gauge theory with $N_f = 12$ fermions in the fundamental representation, finding the mass anomalous dimension to be $γ_m = 0.23(6)$, consistent with other perturbative and lattice estimates. We also present the first lattice calculation of the nucleon anomalous dimension in this theory, finding $γ_N = 0.05(5)$.
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 fixed point of the renormalization group. We calculate scheme-independent series expansions for the anomalous dimensions of higher-spin bilinear fermion operators at this infrared fixed point up to $O(Δ_f^3)$, where $Δ_f$ is an $N_f$-dependent expansion variable. Our general results are evaluated for several special cases, including the case $G={\rm SU}(N_c)$ with $R$ equal to the fundamental and adjoint representations.
