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In this paper, we summarize our efforts in simulating Yang-Mills theories coupled to matter fields transforming under the fundamental and adjoint representations of the gauge group. In the context of composite Higgs scenarios, gauge theories with mixed representation fields have been suggested to describe the fundamental interactions well beyond the electroweak unification scale, and they are also closely related to supersymmetric QCD. In addition, they are studied as deformations of theories with pure adjoint matter in the context of adiabatic continuity. We provide some first results for bare parameter tuning and interdependence of the two representations. We also investigate how the chiral symmetry breaking or a conformal scenario can be realized and checked in such theories.

In the last two decades, there has been a substantial effort to extend lattice Monte Carlo simulations from quantum chromodynamics (QCD) toward the full landscape of gauge theories including different numbers of fermion fields transforming in the fundamental representation of the gauge group. Higher fermion representations have been also considered, most notably the adjoint representation of SU(2) and SU(3), and the sextet representation of SU(3). The motivations for these studies have been the search for an extension of the Standard Model or the consideration of supersymmetric gauge theories. The first studies of gauge theories coupled to fermions in two different representations have been published very recently

The first aim is an exploratory study toward the investigations of supersymmetric QCD (SQCD). SQCD is described by

A second aim is related to possible composite Higgs theories

Early simulations of SU(2) with fermions in the adjoint representation can be found in

The theory with two fundamental and one adjoint Dirac flavor, the so-called ultraminimal walking technicolor (UMWT), has been suggested as a strongly interacting completion of the Standard Model

The consideration of mixed representations has been extended and generalized in

Another relation of our investigations in the context of composite Higgs is the approach of a fine grained control of the running of the gauge coupling by different mass scales. This has been suggested and investigated for theories with a large number of flavors in the fundamental representation of SU(3)

The third line of motivation is related to a predicted analytic continuity between confinement of strongly coupled gauge theories and confinement in a semiclassical small circle regime. This provides a better analytic control in investigations of the relevance of nonperturbative semiclassical contributions in the confinement mechanism. The phenomenon is well understood in supersymmetric Yang-Mills (SYM) theory and also confirmed by numerical simulations

The current study represents the essential first step for all of these investigations, being an exploration of the parameter space spanned by the two mass parameters of the different representations and the gauge coupling. The study of the scaling of the meson mass spectrum close to the chiral limit provides a clear picture helping to distinguish the signals of a chiral symmetry breaking scenario from a conformal theory. In particular, the main target of the present investigations is the deformation of the spectrum of lowest mesonic states induced by the addition of fermions in a different representation. As a first step, we aim to identify possible unphysical bulk phases, which appear for higher representations and in particular in the context of near conformal theories. An important cross-check of our current first studies is also the connection to pure adjoint and fundamental limits.

The first step for Monte Carlo simulations is the lattice discretization of the continuum action. In our numerical simulations, the gauge part of the lattice action is represented by the Wilson gauge action built from plaquettes

Our simulation program allows flexible simulation for an arbitrary number of fermions in the adjoint and fundamental representation of an

The tuning of the RHMC algorithm has been done based on our experience with simulations of SYM. In SYM, rather precise rational approximations have been needed for simulations close enough to the chiral limit. Despite the fact that also heavier masses have been considered here, we have still kept a rather high order approximations in most of the simulations. In the HMC, different force contributions for each fermion representation and for the gauge action have to be considered. Our program allows to have multiple time scales for the integration and we have adjusted these parameters according to the relevance of the different force contributions.

In a first study of the theory, we map out the phase diagram on small lattices to identify possible unphysical bulk phases appearing in the context of IR conformal theories. In the bare parameter space of these theories, the strong coupling confining regime has to be separated from the conformal phase, which corresponds to the range of gauge couplings attracted by the IR fixed point. Such kind of behavior has been documented, for example, for MWT or theories with a large number of fundamental flavors. Since we do not know

Bulk transitions have to be considered in a more general context. Pure SU(2) Yang-Mills theory has a crossover from weak to strong couplings which becomes a bulk transition when an additional adjoint Wilson gauge action is coupled to the theory. Therefore, it is natural to expect bulk transitions for any theory with fermions in the adjoint representation. Moreover, there are a number of evidences for the bulk phases for theories with fermions in higher representation of the gauge group.

As first investigation, we monitor the expectation value of plaquette on small lattices as a function of

The strong coupling phase transitions shown by discontinuity of the average plaquette as a function of the bare mass parameter. Figure

The SU(2) gauge theory with two fermions in the fundamental representation has been recently investigated in a series of publications; see

In the present studies, we use the parameter

With the common scale setting, the physical value of the vector meson mass in the chiral limit can be compared to previous results for the SU(2)

The vector meson mass of the SU(2)

Overall, there is a reasonable agreement between our study and earlier results in the pure fundamental case despite the absence of a continuum extrapolation. From the perspective of the pure fundamental theory, there seems to be no reason not to consider coarser lattices. The main limitation is due to the bulk phase induced by the adjoint fermions.

The SU(2) pure adjoint limit corresponds to SYM theory. The simulations with one Majorana fermion require in this limit the RHMC algorithm and have a significantly higher computational cost than the pure fundamental limit. The validation of the results in this limit against our previous data is required since the simulation setup and the lattice action have been changed.

There are no simple physical mesonic states without disconnected contributions in this theory. The adjoint pion mass, corresponding to

We have generated only a small number of runs, and consequently the fit range is insufficient for a precise chiral extrapolation. The rough estimates based on a linear fit are

The mass of the gluino-glue particle in the pure adjoint limit (supersymmetric Yang-Mills theory). The plot shows a linear chiral extrapolation in units of

In the previous sections, we have confirmed the reliability of our simulations in the limiting pure fundamental and pure adjoint cases. The current first study of the mixed representation theory is organized in such a way that both limiting cases can be reached with the same bare coupling. This means that the range of

A possible scenario for the theory is a spontaneous chiral symmetry breaking in the limit where both masses tend to zero (chiral limit). Like in QCD, Goldstone bosons are expected to interact accordingly to chiral perturbation theory in the small mass regime. Alternatively, this theory could be close to an infrared conformal fixed point (IRFP). Near an IRFP, the masses of all particles scale to zero with an exponent provided by the mass anomalous dimension. In a walking or near conformal regime, the behavior could be quite similar, even though it is difficult to quantify the distance to the conformal case. Hence, it could be that the theory shows already signs of conformality even if a chiral symmetry breaking scenario is obtained in the deep infrared.

In the following, we investigate to what extends these two scenarios are reflected in the numerical data. The fact that two different fermion representations are considered leads to some complications, for instance, related to the parameter tuning and scale setting, that will be explained in the following.

The tuning of the bare mass in one representation shows a clear dependence on the mass parameter of the other, as illustrated in Fig.

Dependence of the pseudoscalar mass in one representation on the bare mass parameter of the other. In Fig.

Range of masses obtained from the partially conserved axial current (PCAC) relation for both of the representations. The adjoint PCAC mass

Another observation is a significant increase of the flow scale. At

Chiral perturbation theory with two different representations has been worked out in

We consider first a fit of

Dependence of the pseudoscalar mass on the PCAC mass of the same representation. (a) shows the fundamental and (b) the adjoint case. The leading order of chiral perturbation theory predicts a linear behavior of

In order to include the corrections from the mutual interactions of the fields in the other representation, the fit must include two dimensions. Due to our currently limited data, we consider only the leading contributions in

The fitted parameters are summarized in Table

Summary of the fit results for the pion masses. The fit range has been restricted to

Apart from

Dependence of the pseudoscalar decay constant on the PCAC mass of the same representation. (a) shows the fundamental and (b) the adjoint case. Compared to the functions

Summary of the fit results for the pseudoscalar decay constants. The fit ranges are the same as in Table

Besides the expected massless states in the scenario of chiral perturbation theory, we can also look at the vector states

Dependence of the vector meson mass on the PCAC mass of the same representation. (a) shows the fundamental and (b) the adjoint case. The fit is done with a constant and up to quadratic corrections ignoring the dependence on the mass of the other representation.

A generic fit ansatz for

Summary of the fit results for the pion masses. The last two columns specify the fit ranges and reduced chi-square. Note that the reduced chi-square without the second representation (

The range of

The figures have been presented in lattice units to show the general functional dependence and avoid additional errors from the scale setting. In Fig.

In addition, we want to comment further on the relevance of higher order corrections in view of a larger reduced chi-square for

We have also done a number of simulations at a larger value of

Chiral extrapolations of the gluino-glue mass (a) and the scale

Dependence of the vector meson mass on the PCAC mass of the same representation as in Fig.

In a conformal scenario, the behavior of the theory is influenced by an infrared fixed point (IRFP) of the massless theory. If the scalar matter fields and their interactions were included in the Lagrangian of our theory, the corresponding SQCD theory has

In the conformal case, the masses are relevant directions of the renormalization group transformation in the vicinity of the IRFP, while the gauge coupling is irrelevant. If one considers

The situation becomes more complicated for a theory with two different representations, since the anomalous dimensions of the two relevant mass parameters are not the same. We therefore recall some of the arguments presented in

In our case, we can consider the ratio of vector and pseudoscalar mesons, for example, in order to check whether it can be represented by a functional dependence

For this reason, let us first investigate the leading dependence

Double log representation of the dominant dependence of the bound state masses on the two mass parameters. (a) shows the fundamental and (b) the adjoint case. The linear fit assumes an approximate scaling near an IRFP.

Summary of the approximate leading scaling dimensions obtained with the fits in logarithmic representation, Fig.

The leading scaling can be used to investigate finally the subleading dependence

The ratio of vector over pseudoscalar mass in the fundamental representation as a function of

We have presented the first results of the simulations with a gauge theory coupled to fermions in adjoint and fundamental representation. The present work provides a preparatory study for further simulations of SQCD, UMWT, or compactified SYM with fundamental matter. In the current study, we have considered SU(2) gauge theory with one Majorana fermion in the adjoint representation and two fundamental flavors. This corresponds to two flavor SU(2) SQCD without scalar fields.

We have presented cross-checks with existing results in the pure adjoint and fundamental limit. In a first investigations we have also determined the reliable range of parameters by an investigation of the bulk transition.

The main objective of this work has been the determination of signals for a chiral symmetry breaking or a conformal scenario since the realization of these distinct infrared scenarios will affect any further investigation of the theory. We have derived scaling relations for theories with two different fermion representation according to an IR conformal scenario. We have compared our numerical data to the expected functional dependence in the conformal and the chiral symmetry breaking case. In the conformal case, we have found no consistent scaling determined by scaling dimensions

Our investigation shows that the theory is consistent with a chiral symmetry breaking scenario, but still close to the conformal window. The running of the gauge coupling is quite small and mass ratios are nearly constant. This situation is similar to what is expected for SU(2) SQCD with two flavors, since the conformal window is predicted to start close by at

The bare parameter tuning with Wilson fermions of one representation seems to be considerably affected by the other. The dependence on physical mass parameters is provided by the two PCAC masses. We observe a rather mild dependence of the states in one representation on the mass of the other. The most significant effect has the adjoint fermion mass since the fundamental meson masses clearly depend on it. Furthermore, the adjoint mesons and the gluino-glue can be extrapolated to the chiral limit considering as a good approximation only the dependence on the adjoint mass.

Regarding the simulations of SU(2) two flavor SQCD, the following strategy could be derived from the data. Since the system is close to the conformal window, one might neglect the running of marginal couplings like the gauge coupling and set them to their tree level value. The important extrapolation of the pure SYM part to the chiral limit can be done as a first approximation just from the adjoint PCAC mass, which might capture the leading dependence. Note, however, that due to Yukawa couplings the chiral transformations of fundamental and adjoint fermions are not independent in SQCD. The Yukawa terms are only invariant under a combined transformation of the two fermion fields and the scalar field.

In order to investigate in more detail, the conformal scenario with two different representations, a larger number of adjoint fields should be considered. One interesting investigation is the simulation of UMWT, which means one Dirac instead of one Majorana fermion in the adjoint representation.

The current data correspond to a first investigation of the theory. A larger statistic, a more complete scan of the relevant parameter range, and a more careful considerations of methods such as the mode number measurement for two fermion representations would be required to provide more precise data. Note, however, that the parameter scan with two independent fermion representations will require a considerably larger amount of computational resources.

The choice of the parameters provides particular challenges in the current close to conformal theory with two representations. In principle, one would like to avoid bulk phases for the complete range of mass parameters, which is achieved by a sufficiently large

The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V.

An important issue for the simulation is a freezing of the topological charge leading to simulations at an effectively fixed topology. It is known that this effect sets in for QCD simulations at fine lattices and large

Subset of the Monte Carlo history for the topological charge determined from the gradient flow. Runs for different mass parameters on a

The method used to measure the topological charge and the effect of topological freezing on

The same data as in Figs.

This table is a summary of the data of the pure adjoint (

This table summarizes the runs with two dynamical representations. The clover coefficient was again set to the one-loop values of