We use the worldline formalism to derive a universal relation for the lower boundary of the conformal window in non-supersymmetric QCD-like theories. The derivation relies on the convergence of the expansion of the fermionic determinant in terms of Wilson loops. The expansion shares a similarity with the lattice strong coupling expansion and the genus expansion in string theory. Our result relates the lower boundary of the conformal window in theories with different representations and different gauge groups. Finally, we use SQCD to estimate the boundary of the conformal window in QCD-like theories and compare it with other approaches.
We develop a model for hadrons in the framework of the worldline formalism. While being based wholly in four-dimensional quantum field theory, it shares many features with holographic approaches: Already by the use of the worldline formalism the approach appears intrinsically quantum mechanical. As auxiliary fifth dimension Schwinger's proper time is grouped with the physical four spacetime dimensions into an AdS5 geometry, which is warped due to conformal-symmetry breaking effects. Hidden local symmetry is emergent. The four-dimensional sources are extended to five-dimensional fields by a Wil…
We propose a novel constraint on the gauge dynamics of strongly interacting gauge theories stemming from the a theorem. The inequality we suggest is used to provide a lower bound on the conformal window of four dimensional gauge theories.
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 study the nonperturbative dynamics of nonsupersymmetric asymptotically free gauge theories with fermionic matter in distinct representations of the SO(N) and Sp(2N) gauge groups. We use different analytic methods to unveil the associated conformal windows for the relevant matter representations. We propose a direct test for confronting and establishing the validity of the analytic methods used to constrain the conformal windows. By comparing the resulting windows for SU, Sp and SO a pleasing universal picture emerges.
We apply the worldline formalism to scalar quantum electrodynamics (QED) to find a Bern-Kosower type master formula for generalized Compton scattering, on-shell and off-shell. Moreover, we use it to study the non-perturbative gauge parameter dependence of amplitudes in scalar QED and, as our main result, find a simple non-perturbative transformation rule under changes of this parameter in x-space in terms of conformal cross ratios. This generalizes the well-known Landau-Khalatnikov-Fradkin transformation (LKFT). We also exemplify how the LKFT works in perturbation theory.
The dynamics with an infrared stable fixed point in the conformal window in QCD like theories (with a relatively large number of fermion flavors) is studied. The dependence of masses of colorless bound states on a bare fermion mass is described. In particular it is shown that in such dynamics, glueballs are much lighter than bound states composed of fermions, if the value of the infrared fixed point is not too large. This yields a clear signature for the conformal window, which is in particular can be useful for lattice computer simulations of these gauge theories.
We study the conformal window of gauge theories containing fermionic matter fields, where the gauge group is any of the exceptional groups with the fermions transforming according to the fundamental and adjoint representations and the orthogonal groups where the fermions transform according to a spinorial representation. We investigate the phase diagram using a purely perturbative four loop analysis, the all-orders beta function and the ladder approximation.
We investigate generic properties of the conformal phase transition in gauge theories featuring Higgs-like fundamental particles. These theories provide an excellent arena to properly investigate conformal dynamics and discover novel features. We show that the phase transition at the boundary of the Higgs conformal window is not smooth but a jumping one for the known perturbative examples. In addition the general conditions under which the transition is either jumping or smooth are provided. Jumping implies that the massive spectrum of the theory will jump at the phase transition. It, however,…
We investigate the properties of near-conformal dynamics in a sector of large charge when approaching the lower boundary of the conformal window from the chirally broken phase. To elucidate our approach we use the time-honored example of the phenomenologically relevant SU(2) color theory featuring $N_f$ Dirac fermions transforming in the fundamental representation of the gauge group. In the chirally broken phase we employ the effective pion Lagrangian featuring also a pseudo-dilaton to capture a possible smooth conformal-to-non-conformal phase transition. We charge the baryon symmetry of the L…
We propose a five-dimensional framework for modeling the background geometry associated to ordinary Yang-Mills (YM) as well as to nonsupersymmetric gauge theories possessing an infrared fixed point with fermions in various representations of the underlying gauge group. The model is based on the improved holographic approach, on the string theory side, and on the conjectured all-orders beta function for the gauge theory one. We first analyze the YM gauge theory. We then investigate the effects of adding flavors and show that, in the holographic description of the conformal window, the geometry …
We analyze a beta function with the analytic form of Novikov-Shifman-Vainshtein-Zakharov result in the five dimensional gravity-dilaton environment. We show how dilaton inherits poles and fixed points of such beta function through the zeros and points of extremum in its potential. Super Yang-Mills and supersymmetric QCD are studied in detail and Seiberg's electric-magnetic duality in the dilaton potential is explicitly demonstrated. Non-supersymmetric proposals of similar functional form are tested and new insights into the conformal window as well as determinations of scheme-independent value…
We propose an ansatz for the nonperturbative beta function of a generic non-supersymmetric Yang-Mills theory with or without fermions in an arbitrary representation of the gauge group. While our construction is similar to the recently proposed Ryttov-Sannino all order beta function, the essential difference is that it allows for the existence of an unstable ultraviolet fixed point in addition to the predicted Bank-Zaks -like infrared stable fixed point. Our beta function preserves all of the tested features with respect to the non-supersymmetric Yang-Mills theories. We predict the conformal wi…
We investigate the conformal window of four-dimensional gauge theories with fermionic matter fields in multiple representations. Of particularly relevant examples are the ultra-violet complete models with fermions in two distinct representations considered in the context of composite Higgs and top partial-compositeness. We first discuss various analytical approaches to unveil the lower edge of the conformal window and their extension to the multiple matter representations. In particular, we argue that the scheme-independent series expansion for the anomalous dimension of a fermion bilinear at …
We study the conformal window of asymptotically free gauge theories containing $N_f$ flavors of fermion matter transforming to the vector and two-index representations of $SO(N),~SU(N)$ and $Sp(2N)$ gauge groups. For $SO(N)$ we also consider the spinorial representation. We determine the critical number of flavors $N_f^{\rm cr}$, corresponding to the lower end of the conformal window, by using the conjectured critical condition on the anomalous dimension of the fermion bilinear at an infrared fixed point, $γ_{\barψψ}=1$ or equivalently $γ_{\barψψ}(2-γ_{\barψψ})=1$. To compute the anomalous dim…
We identify and characterise the conformal window in gauge theories relevant for beyond the standard model building, e.g. Technicolour, using the criteria of metric confinement and causal analytic couplings, which are known to be consistent with the phase diagram of supersymmetric QCD from Seiberg duality. Using these criteria we find perturbation theory to be consistent throughout the predicted conformal window for several of these gauge theories and we discuss recent lattice results in the light of our findings.
We investigate the effects of four-fermion interactions on the phase diagram of strongly interacting theories for any representation as function of the number of colors and flavors. We show that the conformal window, for any representation, shrinks with respect to the case in which the four-fermion interactions are neglected. The anomalous dimension of the mass increases beyond the unity value at the lower boundary of the new conformal window. We plot the new phase diagram which can be used, together with the information about the anomalous dimension, to propose ideal models of walking technic…
Using lattice simulations, we study the extent of the conformal window for an $\text{SU}(3)$ gauge theory with $N_f$ Dirac fermions in the fundamental representation. We present evidence that the infrared behavior is conformal for $12 \leq N_f \leq 16$, governed by an infrared fixed point, while confinement and chiral symmetry breaking are present for $N_f \leq 8$.
Interacting fixed points in four-dimensional gauge theories coupled to matter are investigated using perturbation theory up to three loop order. It is shown how fixed points, scaling exponents, and anomalous dimensions are obtained as a systematic power series in a small parameter. The underlying ordering principle is explained and contrasted with conventional perturbation theory and Weyl consistency conditions. We then determine the conformal window with asymptotic safety from the complete next-to-next-to-leading order in perturbation theory. Limits for the conformal window arise due to fixed…
We review recent progress in the lattice investigations of near-conformal non-abelian gauge theories relevant for dynamical symmetry breaking and model building of composite Higgs models. The emphasis is placed on the mass spectrum and the running renormalized coupling. The role of a light composite scalar isosinglet particle as a composite Higgs particle is highlighted.
We present the conformal windows of SU(N) supersymmetric and nonsupersymmetric gauge theories with vector-like matter transforming according to higher irreducible representations of the gauge group. We determine the fraction of asymptotically free theories expected to develop an infrared fixed point and find that it does not depend on the specific choice of the representation. This result is exact in supersymmetric theories while it is an approximate one in the nonsupersymmetric case. The analysis allows us to size the unparticle world related to the existence of underlying gauge theories deve…
We show that for four dimensional gauge theories in the conformal window, the Euler anomaly, known as the $a$-function, can be computed from a $2$-point function of the trace of the energy momentum tensor making it more amenable to lattice simulations. Concretely, we derive an expression for the $a$-function as an integral over the renormalisation scale of quantities related to $2$- and $3$-point functions of the trace of the energy momentum tensor.The crucial ingredients are that the square of the field strength tensor is an exactly marginal operator at the Gaussian fixed point and that the r…
A novel strong interaction beyond the standard model could provide a dynamical explanation of electroweak symmetry breaking. Experimental results strongly constrain properties of models that realise this mechanism. Whether these constraints are obeyed by any strongly interacting quantum field theory is a non-perturbative problem that needs to be addressed by first-principle calculations. Monte Carlo simulations of lattice regularised gauge theories is a powerful tool that enables us to address this question. Recently various lattice investigations have appeared that have studied candidate mode…
We discuss holographic QCD in the Veneziano limit (the V-QCD models), concentrating on phenomena near the "conformal" phase transition taking place at a critical value of the ratio x = N_f/N_c. In particular, we review the results for the S-parameter, the technidilaton, and the masses of the mesons.
In both QCD and supersymmetric QCD (SQCD) with N_f flavors there are conformal windows where the theory is asymptotically free in the ultraviolet while the infrared physics is governed by a non-trivial fixed-point. In SQCD, the lower N_f boundary of the conformal window, below which the theory is confining is well understood thanks to duality. In QCD there is just a sufficient condition for confinement based on superconvergence. Studying the Banks-Zaks expansion and analyzing the conditions for the perturbative coupling to have a causal analyticity structure, it is shown that the infrared fixe…
We use the exact expression for the S parameter in the perturbative region of the conformal window to establish its dependence on the explicit introduction of fermion masses. We demonstrate that the relative ordering with which one sends to zero either the fermion mass or the external momentum leads to drastically different limiting values of S. Our results apply to any fermion matter representation and can be used as benchmark for the determination of certain relevant properties of the conformal window of any generic vector like gauge theory with fermionic matter. We finally suggest the exist…
We extend the phase diagram of SU(N) gauge-fermion theories as function of number of flavours and colours to the region in which asymptotic freedom is lost. We argue, using large $N_f$ results, for the existence of an ultraviolet interacting fixed point at sufficiently large number of flavours opening up to a second ultraviolet conformal window in the number of flavours vs colours phase diagram. We first review the state-of-the-art for the large $N_f$ beta function and then estimate the lower boundary of the ultraviolet window. The theories belonging to this new region are examples of safe non…
The dynamics with an infrared stable fixed point in the conformal window in QCD like theories with a relatively large number of fermion flavors is reviewed. The emphasis is on the description of a clear signature for the conformal window, which in particular can be useful for lattice computer simulations of these gauge theories.
We study the SU(3) gauge theory with twelve flavours of fermions in the fundamental representation as a prototype of non-Abelian gauge theories inside the conformal window. Guided by the pattern of underlying symmetries, chiral and conformal, we analyze the two-point functions theoretically and on the lattice, and determine the finite size scaling and the infinite volume fermion mass dependence of the would-be hadron masses. We show that the spectrum in the Coulomb phase of the system can be described in the context of a universal scaling analysis and we provide the nonperturbative determinati…
A physical characterization of Landau singularities is emphasized, which should trace the lower boundary N_f^* of the conformal window in QCD and supersymmetric QCD. A natural way to disentangle ``perturbative'' from ``non-perturbative'' contributions to amplitudes below N_f^* is suggested. Assuming an infrared fixed point persists in the perturbative part of the QCD coupling even below N_f^* leads to the condition γ(N_f^*)=1, where γis the critical exponent. Using the Banks-Zaks expansion, one gets 4<N_f^*<6. This result is incompatible with the existence of an analogue of Seiberg duality in …
We investigate the theoretical predictions for thrust distribution in the electron positron annihilation to three-jets process at NNLO for different values of the number of flavors, $N_f$. To determine the distribution along the entire renormalization group flow from the highest energies to zero energy we consider the number of flavors near the upper boundary of the conformal window. In this regime of number of flavors the theory develops a perturbative infrared interacting fixed point. We then consider also the QED thrust obtained as the limit $N_c \rightarrow 0$ of the number of colors. In t…
We study the SU(2) gauge theory on the lattice with different numbers of fermions in the fundamental representation of the gauge group to explore the gauge theory phase diagram. We find evidence for an infrared fixed point for ten flavors. The theory with six flavors shows behaviors compatible with the existence of a (quasi) stable fixed point, but the large errors in the present data do not allow for decisive confirmation of this.
We present the conformal behavior of the shear viscosity-to-entropy density ratio and the fermion-number diffusion coefficient within the perturbative regime of the conformal window for gauge-fermion theories.
Direct gauge mediation models using the Intriligator-Seiberg-Shih (ISS) metastable vacua suffer from the Landau pole problem of the standard model gauge couplings and the existence of R symmetry forbidding gaugino masses. These problems may be solved by using the recently proposed SUSY breaking models in a conformal window of the vector-like $SU(N_C)$ gauge theory with gauge singlets. In this paper we propose a model of gauge mediation based on the SUSY-breaking model in the conformal window, and study the dynamics for the SUSY breaking. In the model, there are massive vector-like bifundamenta…
I discuss recent progress in the uncovering of the phase diagram of non-supersymmetric gauge theories. The nature of the conformal window for higher dimensional representations suggests a possible way to construct realistic technicolor models. I then explicitly provide two such theories. One of these models also has a natural cold Dark Matter candidate.
Quantum Chromodynamics with a relatively large number of fundamentally charged quark flavours in the chiral limit is considered. A self-consistent solution of the quark, gluon and ghost propagator Dyson-Schwinger equations in Landau gauge exhibits a phase transition. Above the critical number of fermion flavours the non-perturbative running coupling develops a plateau over a wide momentum range, and the propagators follow a power law behaviour for these momenta. Hereby, the critical number of quark flavours depends crucially on the beyond-tree-level tensor structures of the quark-gluon vertex.
A physical characterization of Landau singularities is emphasized, which should trace the lower boundary N_f^* of the conformal window in QCD and supersymmetric QCD. A natural way to disentangle ``perturbative'' from ``non-perturbative'' contributions below N_f^* is suggested. Assuming an infrared fixed point is present in the perturbative part of the QCD coupling even in some range below N_f^* leads to the condition gamma(N_f^*)=1, where gamma is the critical exponent. This result is incompatible with the existence of an analogue of Seiberg free dual magnetic phase in QCD. Using the Banks-Zak…
We study four-dimensional gauge theories coupled to fermions in the fundamental and meson-like scalars. All requisite beta functions are provided for general gauge group and fermion representation. In the regime where asymptotic freedom is absent, we determine all interacting fixed points using perturbation theory up to three loop in the gauge and two loop in the Yukawa and quartic couplings. We find that the conformal window of ultraviolet fixed points is narrowed-down by finite-$N$ corrections beyond the Veneziano limit. We also find a new infrared fixed point whose main features such as sca…
We report new results for a frequently discussed gauge theory with twelve fermion flavors in the fundamental representation of the SU(3) color gauge group. The model, controversial with respect to its conformality, is important in non-perturbative studies searching for a viable composite Higgs mechanism Beyond the Standard Model (BSM). To resolve the controversy, we subject the model to opposite hypotheses inside and outside of the conformal window. In the first hypothesis we test chiral symmetry breaking ($χ{\rm SB}$) with its Goldstone spectrum, $F_π$, the $χ{\rm SB}$ condensate, and several…
We describe a lattice simulation of the masses and decay constants of the lowest-lying vector and axial resonances, and the electroweak S parameter, in an SU(3) gauge theory with $N_f = 2$ and 6 fermions in the fundamental representation. The spectrum becomes more parity doubled and the S parameter per electroweak doublet decreases when $N_f$ is increased from 2 to 6, motivating study of these trends as $N_f$ is increased further, toward the critical value for transition from confinement to infrared conformality.
Recently the Lattice Strong Dynamics Collaboration reported a reduction of the electroweak S parameter for SU(3) gauge theory with Nf=6 fermions in the fundamental representation, compared to scaled-up QCD (arXiv:1009.5967). Here I provide additional details of our calculation. I discuss our use of conserved lattice currents; the relation to vector--axial parity doubling; finite-volume effects; and the sensitivity of our results to the number of fermion doublets with chiral electroweak couplings. Results presented here include additional data, and do not affect our previously-published conclus…
We propose an alternative paradigm to the conjectured Miransky scaling potentially underlying the physics describing the transition from the conformally broken to the conformally restored phase when tuning certain parameters such as the number of flavors in gauge theories. According to the new paradigm the physical scale and henceforth also the massive spectrum of the theory jump at the lower boundary of the conformal window. In particular we propose that a theory can suddenly jump from a Quantum Chromodynamics type spectrum, at the lower boundary of the conformal window, to a conformal one wi…
We study the holographic Dynamic AdS/QCD description of a SU($N_c$) non-abelian gauge theory with $N_f$ fermions in the fundamental representation which also have Nambu-Jona-Lasinio interactions included using Witten's multi-trace prescription. In particular here we study aspects of the dynamics in and near the conformal window of the gauge theory as described by the two loop running of the gauge theory. If the number of flavours is such that the IR fixed point lies with the anomalous dimension, $γ$, of the quark bilinear above one then chiral symmetry breaking occurs. Here we display a spiral…
We discuss finite temperature phase diagrams of SU(N) gauge theory with massless fermions as a function of the number of fermion flavors. Inside the conformal window we find a phase boundary separating two different conformal phases. Below the conformal window we find different phase structures depending on if the beta function of the theory has a first or higher order zero at the lower boundary of the conformal window. We also outline how the associated behaviors will help in distinguishing different types of theories using lattice simulations.
We report new results on nearly conformal gauge theories with fermions in the fundamental representation of the SU(3) color gauge group as the number of fermion flavors is varied in the Nf = 4-16 range. To unambiguously identify the chirally broken phase below the conformal window we apply a comprehensive lattice tool set in finite volumes which includes the test of Goldstone pion dynamics, the spectrum of the fermion Dirac operator, and eigenvalue distributions of random matrix theory. We also discuss the theory inside the conformal window and present our first results on the running of the r…
We discuss various aspects of Conformal Field Theories on the Lattice. We mainly investigate the SU(3) gauge theory with N_f degenerate fermions in the fundamental representation, employing the one-plaquette gauge action and the Wilson fermion action. First we make a brief review of our previous works and thereby clarify the reason why we conjecture that the conformal window is 7 le N_f le 16. Secondly, we introduce a new concept, "conformal theories with an IR cutoff" and point out that any numerical simulation on a lattice is bounded by an IR cutoff Lambda_{IR}. Then we make predictions that…
We derive universal scaling laws for physical observables such as the critical temperature, the chiral condensate, and the pion decay constant as a function of the flavor number near the conformal window of many-flavor QCD in the chiral limit. We argue on general grounds that the associated critical exponents are all interrelated and can be determined from the critical exponent of the running gauge coupling at the Caswell-Banks-Zaks infrared fixed point. We illustrate our findings with the aid of nonperturbative functional Renormalization Group (RG) calculations and low-energy QCD models.
We review the hot QCD transition with varying number of flavors, from two till the onset of the conformal window. We discuss the universality class for $N_f=2$, along the critical line for two massless light flavors, and a third flavor whose mass serves as an interpolator between $N_f = 2$ and $N_f=3$. We identify a possible scaling window for the 3D $O(4)$ universality class transition, and its crossover to a mean field behaviour. We follow the transition from $N_f=3$ to larger $N_f$, when it remains of first order, with an increasing coupling strength; we summarize its known properties, incl…
Inspired by the model of Jarvinen and Kiritsis, we present a simple holographic model for the on set of chiral symmetry breaking at the edge of the conformal window in QCD in the Veneziano limit. Our most naive model enforces the QCD two loop running coupling on a D3/D7 holographic brane system. The mass of the holographic field, describing the chiral condensate in the model, is driven below the BF bound when the running is sufficiently strong, triggering chiral symmetry breaking for N_f/N_c<2.9. This model though contains too great a remnant of supersymmetry and does not correctly encode the …
We present our investigation of SU(2) gauge theory with 8 flavours, and SU(3) gauge theory with 12 flavours. For the SU(2) case, at strong bare coupling, $β\lesssim 1.45$, the distribution of the lowest eigenvalue of the Dirac operator can be described by chiral random matrix theory for the Gaussian symplectic ensemble. Our preliminary result indicates that the chiral phase transition in this theory is of bulk nature. For the SU(3) theory, we use high-precision lattice data to perform the step-scaling study of the coupling, $g_{\rm GF}$, in the Gradient Flow scheme. We carefully examine the re…
Recently, we developed an effective theory of pions and a light dilatonic meson for gauge theories with spontaneously broken chiral symmetry that are close to the conformal window. The pion mass term in this effective theory depends on an exponent $y$. We derive the transformation properties under dilatations of the renormalized fermion mass, and use this to rederive $y=3-γ_m^*$, where $γ_m^*$ is fixed-point value of the mass anomalous dimension at the sill of the conformal window. This value for $y$ is consistent with the trace anomaly of the underlying near-conformal gauge theory.
