The methods of lattice gauge theory may be applied to gauge theories besides QCD, in fact to any gauge group and any representation of matter fields (as long as the coupling is not chiral). Such theories are useful for model building beyond the Standard Model, for instance in technicolor models. We have carried out Monte Carlo simulations of the SU(3) gauge theory with color sextet fermions. Our result for its discrete beta function indicate an infrared fixed point that makes the theory conformal rather than confining. The lattice theory's phase diagram shows no separation of chiral and confinement scales, measured when the quark mass is nonzero.
This talk gives an overview, aimed at non-experts, of the recent progress on the studies of technicolor models on the lattice. Phenomenologically successful technicolor models require walking coupling; thus, an emphasis is put on the determination of the beta-function of various models. As a case study we consider SU(2) gauge field theory with two adjoint representation fermions, so-called minimal walking technicolor theory.
In technicolor theories using an SU($N_{TC}$) gauge group, the value of $N_{TC}$ is not, {\it a priori}, determined and is typically chosen by phenomenological criteria. Here we present a novel way to determine $N_{TC}$ from the embedding of a one-family technicolor model, with fermions in the fundamental represention of SU($N_{TC}$), in an extended technicolor theory, and use it to deduce that $N_{TC}=4$ in this framework.
We present results from high precision, large volume simulations of the lattice gauge theory corresponding to minimal walking technicolor. We find evidence that the pion decay constant vanishes in the infinite volume limit and that the dependence of the chiral condensate on quark mass m_q is inconsistent with spontaneous symmetry breaking. These findings are consistent with the all-orders beta function prediction as well as the Schroedinger functional studies that indicate the existence of a nontrivial infrared fixed point.
Technicolor theories provide an elegant mechanism for dynamical electroweak symmetry breaking. We will discuss the use of lattice simulations to study the strongly-interacting dynamics of some of the candidate theories, with matter fields in representations other than the fundamental. To be viable candidates for phenomenology, such theories need to be different from a scaled-up version of QCD, which were ruled out by LEP precision measurements, and represent a challenge for modern lattice computations.
We study the gauge dynamics of an SO(4)-gauge theory with two Dirac Wilson fermions transforming according to the vector representation of the gauge group. We determine the lattice phase diagram by locating the strong coupling bulk phase transition line and the zero PCAC mass line. We present results for the spectrum of the theory. In particular we measure the pseudoscalar, vector and axial meson masses. The data are consistent with a chiral symmetry breaking scenario rather than a conformal one. When used to break the electroweak symmetry dynamically the model leads to a natural dark matter c…
Viable candidate theories for electroweak dynamical symmetry breaking are expected to show (near) conformal behaviour in order to accommodate current phenomenological constraints. In principle, renormalisation group studies using finite volume renormalisation schemes are well-suited to verify this property in a given model. The most practical schemes are based on the Schrodinger functional (SF), but suffer from potentially large O(a) effects. Some care has to be taken to remove these effects and to set up a scheme where cutoff effects are small and under control. We here take a step in this di…
We study QCD with 2 colour-sextet quarks as a walking-Technicolor candidate. As such it provides a description of the Higgs sector of the standard model, in which the Higgs field is replaced by the Goldstone `pions' of this QCD-like theory, and the Higgs itself is the $σ$. Such a theory will need to be extended if it is to also give masses to the quarks and leptons. What we are attempting to determine is whether it is indeed QCD-like and hence walking, or if it has an infrared fixed point making it a conformal field theory. We do this by simulating its lattice version at finite temperature and…
Technicolor and other theories of dynamical electroweak symmetry breaking invoke chiral symmetry breaking triggered by strong gauge-dynamics, analogous to that found in QCD, to explain the observed W, Z, and fermion masses. In this talk we describe why a realistic theory of dynamical electroweak symmetry breaking must, relative to QCD, produce an enhanced fermion condensate. We quantify the degree to which the technicolor condensate must be enhanced in order to yield the observed quark masses, and still be consistent with phenomenological constraints on flavor-changing neutral-currents. Lattic…
We present a numerical study of spectroscopic observables in the SU(2) gauge theory with two adjoint fermions using improved source and sink operators. We compare in detail our improved results with previous determinations of masses that used point sources and sinks and we investigate possible systematic effects in both cases. Such comparison enables us to clearly assess the impact of a short temporal extent on the physical picture, and to investigate some effects due to the finite spatial box. While confirming the IR-conformal behaviour of the theory, our investigation shows that in order to …
The technicolor scenario replaces the Higgs sector of the standard model with a strongly interacting sector. One candidate for a realization of such a sector is two-technicolor Yang-Mills theory coupled to two degenerate flavors of adjoint, massless techniquarks. Using lattice gauge theory the properties of the technigluons in this scenario are investigated as a function of the techniquark mass towards the massless limit. For that purpose the minimal Landau gauge two-point and three-point correlation functions are determined, including a detailed systematic error analysis. The results are, wit…
We compare observables to the finite size scaling hypothesis in SU(2) lattice gauge theory with two Dirac fermions in the adjoint representation. The fits that we obtain yield an estimate of the anomalous mass dimension that is consistent with four loop perturbation theory: $γ= 0.51 \pm 0.16$, with the error due to systematic uncertainties in the finite size scaling analysis. The result is somewhat larger than one Schrödinger functional study (by 1.3$σ$) but consistent with another.
The knowledge of the phase diagram of strongly coupled theories as function of the number of colors, flavors and matter representation plays a fundamental role when constructing viable extensions of the standard model (SM) featuring dynamical electroweak symmetry breaking. Here I summarize the state-of-the-art of the phase diagram for SU(N) gauge theories with fermionic matter transforming according to arbitrary representations of the underlying gauge group. I critically report on the latest results from first principle lattice simulations and then review the principal models of (near) conform…
We construct a lattice gauge theory using reduced staggered fermions and gauge fields which provides a non-perturbative realization of a {\it complete} technicolor model; one which treats both strong and weakly coupled gauge sectors on an equal footing. We show that the model is capable of developing a Higgs phase at non zero lattice spacing via the formation of fermion condensates. We further show that while the broken symmetry associated with this phase has a vector character in the lattice theory it is realized as an axial symmetry in the continuum limit in agreement with the Vafa Witten th…
QCD with 2 flavours of massless colour-sextet quarks is studied as a theory which might exhibit a range of scales over which the running coupling constant evolves very slowly (walks). We simulate lattice QCD with 2 flavours of sextet staggered quarks to determine whether walks, or if it has an infrared fixed point, making it a conformal field theory. Our initial simulations are performed at finite temperatures $T=1/N_ta$ ($N_t=4$ and $N_t=6$), which allows us to identify the scales of confinement and chiral-symmetry breaking from the deconfinement and chiral-symmetry restoring transitions. Unl…
The SU(2) gauge theory with two massless Dirac flavors constitutes the building block of several models of Technicolor. Furthermore it has also been used as a template for the construction of a natural light asymmetric, or mixed type, dark matter candidate. We use explicit lattice simulations to confirm the pattern of chiral symmetry breaking by determining the Goldstone spectrum and therefore show that the dark matter candidate can, de facto, be constituted by a complex Goldstone boson. We also determine the phenomenologically relevant spin-one and spin-zero isovector spectrum and demonstrate…
The research of strongly coupled beyond-the-standard-model theories has generated significant interest in non-abelian gauge field theories with different number of fermions in different representations. Motivated by the increased interest to various technicolor scenarios, we study the non-perturbative improvement of the Wilson-clover action with SU(2) gauge fields and 2 flavors of fermions in the fundamental and adjoint representations. The Sheikholeslami-Wohlert coefficients are fixed using Schroedinger functional boundary conditions. The adjoint representation theory is a candidate for a "mi…
QCD with two flavours of massless colour-sextet quarks is considered as a model for conformal/walking Technicolor. If this theory possess an infrared fixed point, as indicated by 2-loop perturbation theory, it is a conformal(unparticle) field theory. If, on the other hand, a chiral condensate forms on the weak-coupling side of this would-be fixed point, the theory remains confining. The only difference between such a theory and regular QCD is that there is a range of momentum scales over which the coupling constant runs very slowly (walks). In this first analysis, we simulate the lattice versi…
We suggest the model with the gauge group $ ... \otimes SU(6) \otimes SU(5)\otimes SU(4) \otimes SU(3) \otimes SU(2) \otimes U(1)$. This group is the infinite continuation of the gauge group $SU(4) \otimes SU(3) \otimes SU(2) \otimes U(1)$ of Farhi - Susskind model. The constructed model contains fermions from the fundamental representations of any SU(N) subgroups of the gauge group. In the construction of the model we use essentially the requirement that it posseses an additional discrete symmetry $\cal Z$ that is the continuation of the $Z_6$ symmetry of the Standard Model. It has been found…
We present a Monte Carlo renormalisation group study of the SU(2) gauge theory with two Dirac fermions in the adjoint representation. Using the two lattice matching technique recently advocated and exploited in [arXiv:0907.0919], we measure the running of the coupling and the anomalous mass dimension.
We perform a lattice study of the topological susceptibility and instanton size distribution of the $\su{2}$ gauge theory with two adjoint Dirac fermions (also known as Minimal Walking Technicolor), which is known to be in the conformal window. In the theory deformed with a small mass term, by drawing a comparison with the pure gauge theory, we find that topological observables are decoupled from the fermion dynamics. This provides further evidence for the infrared conformality of the theory. A study of the instanton size distribution shows that this quantity can be used to detect the onset of…
One exciting possibility of new physics beyond the Standard Model is that the fundamental Higgs sector is replaced by a strongly-interacting gauge theory, known as technicolor. A viable theory must break chiral symmetry dynamically, like in QCD, to generate Goldstone bosons which become the longitudinal components of the W and Z. By measuring the eigenvalues of the Dirac operator, one can determine if chiral symmetry is in fact spontaneously broken. We simulate SU(3) gauge theory with n_s=2 and 3 staggered flavors in the fundamental representation, corresponding to N_f=8 and 12 flavors in the …
We summarize basic features associated to dynamical breaking of the electroweak symmetry. The knowledge of the phase diagram of strongly coupled theories as function of the number of colors, flavors and matter representation plays a fundamental role when trying to construct viable extensions of the standard model (SM). Therefore we will report on the status of the phase diagram for SU(N) gauge theories with fermionic matter transforming according to arbitrary representations of the underlying gauge group. We will discuss how the phase diagram can be used to construct unparticle models. We will…
We map the phase diagram of gauge theories of fundamental interactions in the flavor-temperature plane using chiral perturbation theory to estimate the relation between the pion decaying constant and the critical temperature above which chiral symmetry is restored. We then investigate the impact of our results on models of dynamical electroweak symmetry breaking and therefore on the electroweak early universe phase transition.
We construct the ultraviolet completion of the Standard Model that contains an infinite sequence of Hypercolor gauge groups. So, the whole gauge group of the theory is $... \otimes SU(5)\otimes SU(4) \otimes SU(3) \otimes SU(2) \otimes U(1)$. Here SU(4) is the Technicolor group of Farhi - Susskind model. The breakdown of chiral symmetry due to the the Technicolor gives rise to finite $W$ and $Z$ boson masses in a usual way. The other Hypercolor groups are not confining. We suggest the hypothesis that the fermion masses are not related in any way to technicolor gauge group. We suppose that the …
Since the pioneering work of Eichten and Lane it has been known that the scale of the interactions responsible for the generation of the strange-quark mass in extended technicolor theories must, absent any "GIM-like" mechanism for suppressing flavor-changing neutral currents, be greater than of order 1000 TeV. In this note we point out that the constraint from the neutral D-meson system is now equally strong, implying that the charm quark mass must also arise from flavor dynamics at a scale this high. We then quantify the degree to which the technicolor condensate must be enhanced in order to …
Conformal technicolor is a paradigm for new physics at LHC that may solve the problems of strong electroweak symmetry breaking for quark masses and precision electroweak data. We give explicit examples of conformal technicolor theories based on a QCD-like sector. We suggest a practical method to test the conformal dynamics of these theories on the lattice.
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…
Here I summarize some of the salient features of technicolor theories with technifermions in higher dimensional representations of the technicolor gauge group. The expected phase diagram as function of number of flavors and colors for the two index (anti)symmetric representation of the gauge group is reviewed. After having constructed the simplest walking technicolor theory one can show that it is not at odds with the precision measurements. The simplest theory also requires, for consistency, a fourth family of heavy leptons. The latter may result in an interesting signature at LHC. In the cas…
We investigate theories in which the technifermions in higher dimensional representations of the technicolor gauge group dynamically break the electroweak symmetry of the standard model. For the two-index symmetric representation of the gauge group the lowest number of techniflavors needed to render the underlying gauge theory quasi conformal is two. We confront the models with the recent electroweak precision measurements and demonstrate that the two technicolor theory is a valid candidate for a dynamical breaking of the electroweak symmetry. The electroweak precision measurements provide use…
We calculate the vacuum-expectation-value, the meson mass and the meson decay constant to next-to-next-to-leading-order in the chiral expansion for QCD-like theories with general $N_F$ degenerate flavours for the cases with a complex representation, a real and a pseudoreal representation, i.e. with global symmetry and breaking patters $SU(N_F)_L\times SU(N_F)_R\to SU(N_F)_V$, $SU(2N_F)\to SO(2N)$ and $SU(2N_F)\to Sp(2N_F)$. These calculations should be useful for lattice calculations for dynamical electroweak symmetry breaking and related cases.
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 study the gauge sector of Minimal Walking Technicolor, which is an SU(2) gauge theory with nf=2 flavors of Wilson fermions in the adjoint representation. Numerical simulations are performed on lattices Nt x Ns^3, with Ns ranging from 8 to 16 and Nt=2Ns, at fixed β=2.25, and varying the fermion bare mass m0, so that our numerical results cover the full range of fermion masses from the quenched region to the chiral limit. We present results for the string tension and the glueball spectrum. A comparison of mesonic and gluonic observables leads to the conclusion that the infrared dynamics is gi…
We investigate non perturbatively scattering properties of Goldstone Bosons in an SU(2) gauge theory with two Wilson fermions in the fundamental representation. Such a theory can be used to build extensions of the Standard Model that unifies Technicolor and pseudo Goldstone composite Higgs models. The leading order contribution to the scattering amplitude of Goldstone bosons at low energy is given by the scattering lengths. In the context of technicolor extensions of the Standard Model the scattering lengths are constrained by WW scattering measurements. We first describe our setup and in part…
The electroweak gauge symmetry is allowed to be spontaneously broken by the strongly interacting vector-like gauge dynamics. When the gauge coupling of a theory runs slowly in a wide range of energy scale, the theory is a candidate for walking technicolor. This may open up the possibility that the origin of all masses may be traced back to the gauge theory. We use the \SF method to see whether the gauge coupling of 10-flavor QCD "walks" or not. Preliminary result is reported.
We present a lattice calculation of $L_{10}$, one of the low energy constants in Chiral Perturbation Theory, and the charged-neutral pion squared mass splitting, using dynamical overlap fermion. Exact chiral symmetry of the overlap fermion allows us to reliably extract these quantities from the difference of the vacuum polarization functions for vector and axial-vector currents. In the context of the technicolor models, these two quantities are read as the $S$-parameter and the pseudo-Nambu-Goldstone boson mass respectively, and play an important role in discriminating the models from others. …
SU(3) gauge theory coupled to N_f = 2 fermions in the sextet representation is a promising candidate for a technicolor inspired Standard Model extension. In this note the progress in the past few years aimed at understanding the non-perturbative properties of the model is reviewed. The main difficulties lying ahead in order to make robust conclusions from lattice simulations are outlined.
We present the report of the LatKMI collaboration on the lattice QCD simulation for the cases of 4 and 8 flavors. The Nf=8 in particular is interesting from the model-building point of view: The typical walking technicolor model with the large anomalous dimension is the so-called one-family model (Farhi-Susskind model). Thus we explore the walking behavior in LQCD with 8 HISQ quarks by comparing with the 4-flavor case (in which the chiral symmetry is spontaneously broken). We report preliminary results on the spectrum, analyzed through the chiral perturbation theory and the finite-size hypersc…
We complete a program of study of SU(N) gauge theories coupled to two flavors of fermions in the two-index symmetric representation by performing numerical simulations in SU(4). The beta function, defined and calculated via the Schrödinger functional, runs more slowly than the two-loop perturbative result. The mass anomalous dimension levels off in strong coupling at a value of about 0.45, rendering this theory unsuitable for walking technicolor. A large-N comparison of this data with results from SU(2) and SU(3) reveals striking regularities.
In realistic technicolor models containing many fermions, the electroweak baryogenesis offers a natural scenario for generating baryon number asymmetry. One of the key ingredients is the occurrence of the first order phase transition at finite temperature. As a first step toward the exploration of this possibility on the lattice, we develop an agile method to identify the critical mass for a given Nf, separating the first order and the crossover transition. We explain the outline of our method and demonstrate it by determining the critical mass of Nf-flavors in the presence of light two-flavor…
Slowly walking technicolor models provide a mechanism for electroweak symmetry breaking whose nonperturbative lattice investigation is rather challenging. Here we demonstrate walking near a conformal fixed point considering the 2-d lattice O(3) model at vacuum angle $θ\approx π$. The essential features of walking technicolor models are shared by this toy model and can be accurately investigated by numerical simulations. We show results for the running coupling and the beta-function and we perform a finite size scaling analysis of the massgap close to the conformal point.
In search for a composite Higgs boson (techni-dilaton) in the walking technicolor, we present our preliminary results on the first observation of a light flavor-singlet scalar in a candidate theory for the walking technicolor, the Nf=8 QCD, which was found in our previous paper to have spontaneous chiral symmetry breaking together with remnants of the conformality. Based on simulations with the HISQ-type action on several lattice sizes with various fermion masses, we find evidence of a flavor-singlet scalar meson with mass comparable to that of the Nambu-Goldstone pion in both the small fermio…
We study the meson spectrum of the SU(2) gauge theory with two Wilson fermions in the fundamental representation. The theory unifies both Technicolor and composite Goldstone Boson Higgs models of electroweak symmetry breaking. We have calculated the masses of the lightest spin one vector and axial vector mesons. In addition, we have also obtained preliminary results for the mass of the lightest scalar (singlet) meson state. The simulations have been done with multiple masses and two different lattice spacings for chiral and continuum extrapolations. The spin one meson masses set lower limits f…
Quarks in the adjoint representation have been a subject of study for both conceptual and practical purposes. Conceptually, their differences when it comes to confining and chiral symmetry properties has long been suspected to hold important information on the relation of these two distinguished properties of QCD-like gauge theories. Practically, they have been studied as both a possibility to access finite density quark systems as well as candidate theories for technicolor in beyond-the-standard-model settings. The most elementary object describing such particles is their propagator, though i…
We construct two-dimensional ${\cal N} = (2, 2)$ supersymmetric gauge theories on a Euclidean spacetime lattice with matter in the two-index symmetric and anti-symmetric representations of SU($N_c$) color group. These lattice theories preserve a subset of the supercharges exact at finite lattice spacing. The method of topological twisting is used to construct such theories in the continuum and then the geometric discretization scheme is used to formulate them on the lattice. The lattice theories obtained this way are gauge-invariant, free from fermion doubling problem and exact supersymmetric …
SU(2) gauge theory with two fermions transforming under the adjoint representation may appear conformal or almost conformal in the infrared, and is one of the candidate theories for building models for technicolor. Early lattice Monte Carlo studies of this model have used unimproved Wilson fermion formulation, which can be expected to have large lattice cutoff effects. In this paper we present the calculation of the O(a) improved lattice Wilson-clover action of the theory. The Sheikholeslami-Wohlert coefficient has been determined non-perturbatively, and various boundary improvement terms, nee…
We consider the spin-zero spectrum of a strongly coupled gauge theory. In particular, we focus on the dynamical mass of the isosinglet scalar resonance in the presence of a four-fermion interaction external to the gauge dynamics. This is motivated by the extended technicolor framework for dynamical electroweak symmetry breaking. Applying the large-$N$ limit, we sum all the leading-order contributions, and find that the corrections to the mass of the isosinglet scalar resonance can be large, potentially reducing its value from ${\cal O}(1)$ TeV to the observed value of 125 GeV.
We investigate chiral and conformal properties of the lattice QCD with eight flavors (Nf=8) through meson spectrum using the Highly Improved Staggered Quark (HISQ) action. We also compare our results with those of Nf=12 and Nf=4 which we study on the same systematics. We find that the decay constant F_pi of the pseudoscalar meson "pion" is non-zero, with its mass M_pi consistent with zero, both in the chiral limit extrapolation of the chiral perturbation theory (ChPT). We also measure other quantities which we find are in accord with the pi data results: The rho meson mass is consistent with n…
Walking technicolor theory attempts to realize electroweak symmetry breaking as the spontaneous chiral symmetry breakdown caused by the gauge dynamics with slowly varying gauge coupling constant and large mass anomalous dimension. Many-flavor QCD is one of the candidates owning these features. We focus on the SU(3) gauge theory with ten flavors of massless fermions in the fundamental representation, and compute the gauge coupling constant in the Schrödinger functional scheme. Numerical simulation is performed with $O(a)$-unimproved lattice action, and the continuum limit is taken in linear in …
We present preliminary results for simulations of SO(4)-gauge theory with two Dirac Wilson fermions transforming according to the vector representation. We map out the phase diagram including the strong coupling bulk phase transition line as well as the zero PCAC-mass line. In addition, we measure the pseudo scalar and vector meson masses, and investigate whether the theory features chiral symmetry breaking. If the theory is used for breaking the electroweak symmetry dynamically it is the orthogonal group equivalent of the Minimal Walking Technicolor model but with the following distinctive fe…
