# Infrared fixed point and anomalous dimensions
in a composite Higgs model

Anna Hasenfratz  Department of Physics, University of Colorado, Boulder, CO
80309, USA  Ethan T. Neil  Department of Physics, University of Colorado,
Boulder, CO 80309, USA  Yigal Shamir  Raymond and Beverly Sackler School of
Physics and Astronomy, Tel Aviv University, 69978 Tel Aviv, Israel  Benjamin
Svetitsky  Raymond and Beverly Sackler School of Physics and Astronomy, Tel
Aviv University, 69978 Tel Aviv, Israel  Oliver Witzel  Center for Particle
Physics Siegen, Theoretische Physik 1, Naturwissenschaftlich-Technische
Fakultät, Universität Siegen, 57068 Siegen, Germany

###### Abstract

We use lattice simulations and the continuous renormalization-group method,
based on the gradient flow, to study a candidate theory of composite Higgs and
a partially composite top. The model is an SU(4) gauge theory with four Dirac
fermions in each of the fundamental and two-index antisymmetric
representations. We find that the theory has an infrared fixed point at
$g^{2}\simeq 15.5$ in the gradient flow scheme. The mass anomalous dimension
of each representation is large at the fixed point. On the other hand, the
anomalous dimensions of top-partner operators do not exceed 0.5 at the fixed
point. This may not be large enough for a phenomenologically successful model
of partial compositeness.

††preprint: SI-HEP-2023-08

##  I  Introduction

###  I.1  Background

Compared to the Planck scale, the mass of the Higgs particle in the Standard
Model is unnaturally small. A popular solution is to suppose that the Higgs is
a pseudo Nambu-Goldstone boson arising from the spontaneous breaking of a
chiral symmetry [1, 2], which in turn is induced by a novel strong interaction
sometimes known as hypercolor (HC). Since the top quark has a mass on the same
scale, one might similarly suppose that the top quark is partially composite
[3], receiving its mass from the direct coupling to a hypercolor baryon called
the top partner.

Composite Higgs models have been extensively studied using effective field
theory techniques (for reviews, see Refs. [4, 5, 6]). It is important,
however, to seek out realizations of this paradigm as a concrete,
asymptotically free theory such as hypercolor. A list of candidate theories
satisfying a number of desirable properties was compiled by Ferretti and
Karateev [7, 8, 9, 10] (see also Ref. [11]). As a stand-alone theory—before
coupling to Standard Model fields—each model in the Ferretti–Karateev list is
a vector-like gauge theory with fermions in two different representations of
the gauge group. The top partner is a chimera: a three-fermion bound state
made out of fermions of both representations.

Without additional interaction terms, the hypercolor gauge interaction cannot
generate Standard Model fermion masses, nor can it induce electro-weak
symmetry breaking. The coupling of the top quark to its chimera partner must
come from four-fermion interactions, whose origin lies in a sector, known as
extended hypercolor (EHC), with a still higher energy scale. Such interactions
at the HC scale are naively suppressed by a factor
$\Lambda_{HC}^{2}/\Lambda_{EHC}^{2}$, where $\Lambda_{HC}$ is the scale of the
hypercolor theory, while $\Lambda_{EHC}\gg\Lambda_{HC}$ is the scale of the
EHC theory.

In the absence of a concrete realization of extended hypercolor in the
literature,111See, however, Ref. [12] it must be assumed that, generically,
the four-fermion interactions induced by the EHC theory can give rise to
unwanted flavor violations. In order to respect experimental bounds on flavor
violation, $\Lambda_{HC}^{2}/\Lambda_{EHC}^{2}$ must be small. In order to
obtain a realistic top mass, however, the effect of the suppression factor on
the top-partner mixing term must be reduced. To this end, two conditions must
be satisfied. First, some of the four-fermion operators responsible for
partial compositeness must have a large anomalous dimension. Schematically,
these operators have the form $\bar{q}B$, where $q$ is a Standard Model
fermion field, and $B$ is a hypercolor-singlet chimera operator. Therefore,
one requires the existence of chimera operators $B$ with large anomalous
dimensions within the hypercolor theory. The second requirement is that the
hypercolor theory itself must be nearly conformal, allowing the large chimera
anomalous dimensions to persist over many scales—ideally, all the way from the
EHC scale down to the hypercolor scale.222Since the days of walking
technicolor [13, 14], it has been expected that large anomalous dimensions
would appear near the sill of the conformal window. See also Ref. [15].

The hope, then, is that successful composite-Higgs models will be found near
the sill of the conformal window. Theories just below the sill are obvious
candidates, nearly conformal but ultimately confining and chirally broken.
Theories slightly above the sill, which feature an infrared fixed point when
all the fermions are massless, are also eligible, as one can induce
confinement by giving large masses to a small subset of the fermions [16, 17,
18, 19].

Figure 1 shows analytic estimates for the location of the conformal sill for
SU(4) gauge theory in the plane spanned by the number of Dirac fermions in the
fundamental 4 representation and the number of Majorana fermions in the two-
index antisymmetric 6 representation, which is real.333 The dashed green line
is the sill of the conformal window according to the 2-loop beta function,
where it first develops an IR fixed point; the dashed red line is from the
“all-orders” beta function; the dashed blue line reflects the “critical
condition” of Ref. [33]; the dashed black line stems from long-standing
analysis of Schwinger–Dyson equations—for all of these, see Ref. [33], from
which this figure was adapted, and references therein. Also shown in the
figure are two of the Ferretti–Karateev models that belong to this plane: the
so-called M6 and M11 models [7, 10]. The “2+2 model,” which contains two Dirac
fermions in each of the 4 and 6 representations,444 The two Dirac fermions in
the real 6 representation are equivalent to four Majorana fermions.  has been
studied extensively using lattice techniques. While QCD-like, and hence far
from the conformal sill, the 2+2 model served as a useful laboratory,
providing the first example studied of an asymptotically free gauge theory
with two representations, in particular one that produces chimera baryons [20,
21, 22, 23, 24, 25, 26, 27].555 For lattice studies of a composite Higgs model
based on the Sp(4) gauge group, see Refs. [28, 29, 30, 31, 32].

Figure 1:  Estimates for the location of the conformal window of the SU(4)
gauge theory with $N_{f}$ Dirac fermions in the fundamental representation and
$n_{f}$ Majorana fermions in the sextet representation. The uppermost line is
the limit of asymptotic freedom, where the 1-loop beta function changes sign.
The other lines are various analytical estimates of the sill of the conformal
window (see text). Blue circle: M6 model; red diamond: M11 model; black
square: 2+2 model; open circle: 4+4 model, the subject of this paper. Only the
lowest two lines are consistent with the IR fixed point that we find for the
4+4 model.

In this paper we move on to the “4+4 model,” where we increase the number of
Dirac fermions in each representation from two to four. The 4+4 model has a
number of desirable features. First, the M6 and M11 models can both be
embedded into the 4+4 model, and can be reached by giving a subset of the
fermions a (Dirac or Majorana) mass. In addition, as can be seen in the
figure, the 4+4 model is much more likely to be close to, or even inside, the
conformal window.

###  I.2  Method and summary of results

We extract the beta function and anomalous dimensions using a continuous
renormalization group (RG) method [34, 35].666for a slightly different
approach see Refs. [36, 37, 38]. The length scale for this RG is $\sqrt{t}$,
where $t$ is the parameter of a gradient flow (GF) generated by integrating a
diffusion equation for the gauge field [39]. The GF running coupling is
defined as [40]

$g^{2}=\frac{\mathcal{N}}{C(t;L,T)}t^{2}\left\langle E(t)\right\rangle\ .$ (1)

Here the energy density at scale $\sqrt{t}$ is
$E=\frac{1}{4}G_{\mu\nu}^{a}G_{\mu\nu}^{a}$, where $G_{\mu\nu}^{a}$ is the
flowed gauge field strength. $\mathcal{N}$ is a numerical factor which depends
on the gauge group, and $C(t;L,T)$ is a correction for the finite dimensions
$L,T$ of the volume being simulated (see below). Viewing the gradient flow as
an RG transformation, the beta function is

$\beta(g^{2})=-t\,\frac{\partial g^{2}}{\partial t}\ .$ (2)

The extension of the GF technique to fermions was developed in Ref. [41],
while the use of the continuous RG for obtaining anomalous dimensions of
fermion operators was introduced in Ref. [42].

> Figure 2:  The $\beta$ function obtained with five different lattice
> gradient flows. All flows are the same in the continuum limit; their regions
> of validity are different, though overlapping. For details see Sec. II.

Our main findings are the beta function (Fig. 2), which shows an IR fixed
point;777In comparing to the 2-loop curve, recall that it also crosses the
abscissa, giving an IR-stable fixed point, but at a strong coupling well
outside the range of the figure. the mass anomalous dimensions (Fig. 3); and
the anomalous dimensions of the chimera operators with the lowest mass
dimension, namely, three-fermion operators with no derivatives. The largest
chimera anomalous dimension is shown in Fig. 4, while the other two are shown
in Fig. 14. The IR fixed point places the model inside the conformal window.
Moreover, the anomalous dimensions of the mass operators are large, for both
representations, at the fixed point. Unfortunately, the anomalous dimensions
of all chimera operators are fairly small at the fixed point, making it
unlikely that the model can successfully account for a partially composite top
quark.

Figure 3:  Anomalous dimensions of the two mass operators. Top: fundamental
representation. Bottom: sextet representation. See Sec. III for details.
Figure 4:  The largest chimera anomalous dimension, $\gamma^{s}_{ch}$. The
other two chimera anomalous dimensions are smaller (see Fig. 14).

We obtain the flowed expectation values in Eq. (1) in the presence of a
lattice cutoff, generating ensembles of the gauge field with numerical
simulations. We present our lattice methods, including the calculation of the
beta function and its extrapolation to the continuum limit, in Sec. II. The
calculation of anomalous dimensions is the subject of Sec. III. We offer our
conclusions in Sec. IV. Further technical details of the lattice calculation
are given in the appendix.

##  II  Gauge flow and the beta function

###  II.1  Lattice strategy

The lattice action, our simulation algorithm, and the ensembles we generated
are described in the appendix. We use Wilson-clover fermions and tune the bare
masses such that the fermions of both representations are essentially
massless.

A new ingredient in the lattice action is a set of Pauli–Villars fields [43].
Without these, the presence of many fermion flavors, especially with smearing-
improved gauge connections, generates a large screening effect in the
effective action for the gauge field. In order to obtain a strong renormalized
coupling, one would be pushed towards large bare coupling
$g_{0}^{2}=N_{c}/\beta$. This, in turn, would cause large ultraviolet
fluctuations. As a consequence, these systems often encounter phase
transitions or other discontinuities, lattice artifacts that prevent the
approach to the desired renormalized coupling, especially when the fermions
are light. Our 4+4 system exhibited such a discontinuity when the original
lattice action was used. The addition of Pauli–Villars fields weakens the
induced term and allows us to reach much further into strong renormalized
coupling.

As mentioned in the introduction, we use the continuous RG technique to
determine the beta function. The 4+4 model that we simulate is a massless,
asymptotically free lattice theory. The gradient flow acts as a transformation
in coupling space: as the flow time $t$ is increased, irrelevant operators die
out and the flow converges towards the renormalized trajectory (RT) emerging
from the gaussian fixed point in the ultraviolet. Our task will be to ensure
that, given a specific flow at a specific renormalized coupling, the flow has
reached close enough to the RT that any remaining discretization effects can
be removed by a simple extrapolation.

We begin by applying a gradient flow to the gauge field of every lattice
configuration. In principle, extracting continuum results requires first
taking the infinite-volume limit and then the continuum limit of zero lattice
spacing $a$; the latter limit involves taking the dimensionless lattice flow
time $t/a^{2}\to\infty$. Eschewing a formal infinite-volume limit, we focus on
the continuum limit, which we extract from eight ensembles on a lattice volume
of $24^{3}\times 48$ sites. The bare couplings of these ensembles were
selected to range from weak to strong coupling. We have also generated
ensembles on volume $28^{3}\times 56$ for two of the bare couplings; we use
these to estimate finite-volume effects. By using lattice flow time
$t/a^{2}\leq 3.4$ we have limited the finite volume effects to no more than a
few percent.

###  II.2  Gauge flows

The continuum GF equation takes the form [39]

$\frac{\partial B_{\mu}}{\partial t}=D_{\nu}G_{\nu\mu}\ ,$ (3)

where $B_{\mu}$ is the flowed gauge field, and $G_{\nu\mu}$ the associated
field strength. The initial condition of the flow is the dynamical gauge
field, viz.,

$B_{\mu}\big{|}_{t=0}=A_{\mu}\ .$ (4)

The right-hand side of Eq. (3) is just $-\partial S_{g}/\partial B_{\mu}(x)$,
where $S_{g}=\frac{1}{4}G_{\mu\nu}^{a}G_{\mu\nu}^{a}$ is the standard
continuum gauge action in terms of the flowed field.

Flow | Sym | C43 | Wil | C23 | C13
---|---|---|---|---|---
$c_{p}$ | 5/3 | 4/3 | 1 | 2/3 | 1/3
$c_{r}$ | $-1/12$ | $-1/24$ | 0 | $1/24$ | $1/12$
Table 1:  Lattice gradient flows. The first line gives the flow’s name used in
this paper. “Sym” and “Wil” are the well-known Symanzik and Wilson flows.
$c_{p}$ and $c_{r}$ are the coefficients of the plaquette and rectangle terms
in the lattice action that generates each flow.

On the lattice, the continuum gauge field is replaced by the link variables
$U_{\mu}(x)$. Each saved lattice configuration provides initial values for the
gradient flow. We also need a discrete version of $\partial S_{g}/\partial
B_{\mu}(x)$ for the right-hand side of the flow equation (see Ref. [39]). In
this paper we apply five different gradient flows, each derived from a
different discretization of the continuum action $S_{g}$. The lattice gauge
action which generates each flow is a linear combination of the plaquette
($1\times 1$ Wilson loop) and rectangle ($1\times 2$ Wilson loop) terms, with
coefficients shown in Table 1. For proper normalization in the weak-coupling
limit, the plaquette and rectangle coefficients $c_{p}$ and $c_{r}$ are
constrained by

$c_{p}+8c_{r}=1\ .$ (5)

As discussed in detail below, we have found that increasing $c_{p}$ from 1
gives rise to flows with relatively small discretization effects at weak bare
coupling, while $c_{p}<1$ works well at strong bare coupling. In particular,
we found the range of validity of the Symanzik flow to be limited to the weak
coupling regime. We did not pursue the fully Symanzik-improved Zeuthen flow
[44], as it is unlikely that further perturbative improvement would
significantly change this conclusion.

Returning to Eq. (1) which defines the gradient flow coupling, the numerical
constant here is

$\mathcal{N}=\frac{128\pi^{2}}{3(N_{c}^{2}-1)}\simeq 28.4\ ,$ (6)

for $N_{c}=4$. The finite-volume correction term $C(t;L,T)=1+\delta$ corrects
for the zero modes of the gauge fields [40]. For volume $L^{3}\times T$ we
have888This expression is not given in Ref. [40] but was shared privately by
D. Nogradi. We thank him for his assistance.

$\delta(t;L,T)=-\frac{\pi^{2}(8t)^{2}}{L^{3}T}+\vartheta\left(t/L^{2}\right)^{3}\vartheta\left(t/T^{2}\right)\
.$ (7)

The correction $|\delta|$ is smaller than 0.005 for all our volumes in the
useful ranges of $t$.

In addition, one has to select a discretization of the “energy” $E=S_{g}$,
which is used to define the coupling in Eq. (1). We use three different
discretizations, two of which correspond to the Symanzik (S) and Wilson (W)
actions of Table 1. A third discretization is provided by the “clover” (C)
operator. As explained below, we have found that the S operator gives the
smoothest approach to the continuum limit, and so we will focus on results
obtained using this operator.

> Figure 5:  Results for the S operator measured in the Wilson flow. Top:
> Gradient flow coupling $g^{2}$ as a function of lattice flow time $t/a^{2}$.
> Bottom: $\beta(g^{2})$ vs. $g^{2}$ at flow times $2.4\leq t/a^{2}\leq 3.2$.
> Each open circle indicates the smallest flow time shown, $t/a^{2}=2.4$.
> Error bars in the top panel of this figure, as well as in Figs. 6 and 7, are
> too small to be visible.

###  II.3  Example flows and infinite volume limit

We generated each flow by numerically integrating a lattice version of the
flow equation (3) in steps of $dt=0.01$. The energy $E$ of the flowed gauge
field was recorded for the three operators S, W, and C, at intervals of
$\Delta t=0.1$. The derivative in the bare beta function (2) was estimated
with a five-point difference formula.

We show the raw flows for all eight of our $24^{3}\times 48$ ensembles in Fig.
5 for Wilson flow and in Fig. 6 for C13 flow.999 The C13 flow is equivalent to
the AFLOW introduced in Ref. [45]. The motivation there is quite different
from ours.  Both present the S operator. In each figure, the top panel shows
the gradient flow coupling $g^{2}$ as a function of the lattice flow time
$t/a^{2}$. The bottom panel shows the resulting raw beta function as a
function of $g^{2}$ for a fixed interval of flow time $2.4\leq t/a^{2}\leq
3.2$.

> Figure 6:  Similar to Fig. 5, for the S operator measured in C13 flow.

A comparison of the two figures reveals that the C13 flow reaches further into
strong coupling than the Wilson flow: with the S operator, the former reaches
$g^{2}\simeq 22$ while the latter reaches only $g^{2}\simeq 18$. There are
similar distinctions among the three operators. For C13 flow, and within the
above $t/a^{2}$ interval, the C operator reaches $g^{2}\simeq 18$ at our
strongest bare coupling and the W operator reaches $g^{2}\simeq 20$, whereas
the S operator reaches $g^{2}\simeq 22$ as just noted. In the lower panel of
Fig. 6, we see that the beta function becomes positive for the largest
couplings. Reaching larger values of the gradient flow coupling has direct
bearing on the ability to confirm the existence of an IR fixed point.

Figures 5 and 6 show raw data—before extrapolation to the continuum. In Sec.
II.4 and Sec. II.5 we will show that cutoff effects at strong coupling remain
small for the S operator, and almost as small for the W operator. Moreover,
the strongest attainable renormalized coupling of the C13 flow remains much
larger than that of the Wilson flow in the continuum limit.

In Fig. 7 we examine the effect of increasing the lattice volume. We have
generated two $28^{3}\times 56$ ensembles—at $\beta=11$ which is a weak (bare)
coupling and at $\beta=10.5$ which is a strong coupling. In the top panel, we
show the flows at these two bare couplings for the two volumes, again using
C13 flow and the S operator. The change in $g^{2}$ and in its derivative due
to changing the lattice volume is evidently small. The bottom panel shows the
effect on the raw beta function. For both bare couplings, an increase in $L/a$
from 24 to 28 has the effect of shifting $g^{2}$ upward by about 0.1, leaving
the $\beta$ function otherwise unchanged. We expect the extrapolation to
infinite volume to be linear in $(a/L)^{4}$. It can be easily checked that the
increase $L/a=24\to 28$ takes $(a/L)^{4}$ halfway to the limit. Hence, the
expected change in $\beta(g^{2})$ is a horizontal shift by twice the amount
seen in Fig. 7, with no qualitative change in the overall shape, including the
existence of a fixed point. In the rest of this section we concentrate on our
eight $24^{3}\times 48$ ensembles.

> Figure 7:  Volume comparison for the S operator measured in C13 flow in the
> $\beta=11.0$ and 10.5 ensembles. The volumes are $24^{3}\times 48$ and
> $28^{3}\times 56$. Top: Gradient flow coupling $g^{2}$ as a function of the
> lattice flow time $t/a^{2}$. Bottom: $\beta(g^{2})$ vs. $g^{2}$ at flow
> times $2.4\leq t/a^{2}\leq 3.2$.

###  II.4  Interpolation

In order to determine the continuum limit of $\beta(g^{2})$ we will
extrapolate $t/a^{2}\to\infty$ at fixed $g^{2}$. In a theory with a rapidly
running coupling the graph of the raw data for $\beta(g^{2})$, Fig. 5 or 6,
would yield several ensembles that give different values for $\beta(g^{2})$ at
any fixed coupling $g^{2}$ [34, 36]. We would then read off the corresponding
flow time $t/a^{2}$ and $\beta(g^{2})$ for each ensemble, and take
$t/a^{2}\to\infty$.

Since our theory runs slowly, each ensemble covers only a small range of
$g^{2}$ and hence we have no overlaps between ensembles at any value of the
coupling. Therefore, we have to interpolate $\beta(g^{2})$ vs. $g^{2}$ at
fixed flow time—see Fig. 8. For a given flow time $t/a^{2}$, we identify the
$(g^{2},\beta(g^{2}))$ pair on each ensemble. This gives us eight data points
on each fixed-$t$ curve. To make vertical slices in $\beta(g^{2})$, we
interpolate between ensembles at each fixed $t$. In weak coupling,
$\beta(g^{2})\propto g^{4}$, and hence we use a polynomial interpolation even
in strong coupling, $\beta(g^{2})/g^{4}=c_{0}+c_{1}g^{2}+c_{2}g^{4}+\cdots$,
as detailed below. The curves plotted in Fig. 8 show the interpolations for
the Wilson, C23 and C13 flows, using the S operator for the coupling.

Figure 8:  Interpolation of $\beta(g^{2})/g^{4}$ vs. $g^{2}$ for selected
values of $t/a^{2}$ in Wilson flow (top left), C23 flow (top right), and C13
flow (bottom). The coupling $g^{2}$ is defined by the S operator, measured in
the eight $24^{3}\times 48$ ensembles. In each plot, moving from left to right
corresponds to stronger GF and bare couplings: the leftmost points come from
$\beta=13$ and the rightmost from $\beta=10.4$. The curves connect only the
points that were included in the interpolation.

Before performing each interpolation, we have to decide if we can use all
eight data points. As can be seen in the figure, the C13 and C23 flows show
rapid change with $t/a^{2}$ at the weakest bare couplings (the leftmost data
points). We interpret this as indication that these flows require larger flow
times in the weak coupling region to reach the vicinity of the RT.
Interpolations with cubic, or even quartic, polynomials have poor $p$-values
if we include all 8 data points. Dropping the left-most data point
$\beta=13.0$ is sufficient to raise the $p$-value above 10% if we use a cubic
interpolating polynomial for $\beta(g^{2})/g^{4}$. Even though Wilson flow
shows smaller cutoff effects at $\beta=13.0$, for consistency we include only
the seven ensembles at $\beta<13.0$ and use a cubic interpolation for both the
Wilson and C23 flows. Moreover, for the C13 flow, it can be seen in the bottom
panel of Fig. 8 that the $\beta=12.0$ point also shows large cutoff effects
for the Symanzik operator. Hence, for this flow we discard $\beta=12$ from the
interpolation as well, and use a quadratic interpolating polynomial to keep
the same number of degrees of freedom in the interpolation as for the other
flows.

The curves plotted in Fig. 8 connect only the data points that were included
in the fits. As usual in interpolating data, in later stages of the analysis
we will not use the interpolating curves outside the range of the data points
that they connect.

The Symanzik and C43 flows (not shown in the figure) follow an opposite trend,
exhibiting increasing cutoff effects in strong coupling. For those flows we
drop the data point at the strongest bare coupling, $\beta=10.4$. Again, we
use a cubic interpolating polynomial.

Figure 9:  Extrapolations $a^{2}/t\to 0$ of $\beta(g^{2})$ at $g^{2}=12.0$.
Data are (top to bottom) the S (green), W (red), and C (purple) operators
measured in the C13 flow. Data plotted with open symbols have been dropped
from the extrapolations.

###  II.5  Taking the continuum limit

We now turn to the final stage of our analysis, taking the continuum limit by
extrapolating $t/a^{2}\to\infty$. Having constructed curves for $\beta(g^{2})$
for a selection of $t$ values, we now consider the $\beta$ surface as a set of
curves $\beta(a^{2}/t)$ for a selection of $g^{2}$ values. At any given
physical coupling $g^{2}$, the beta functions extracted from our different
discretizations (S, W, C) must agree in the continuum limit; we will use this
requirement to impose cuts on the range of $g^{2}$ where each flow can be
trusted.

We show an example of the extrapolation process in Fig. 9, which shows the
extrapolation of $\beta(g^{2})$ at $g^{2}=12.0$ for the C13 flow, for all
three operators (S, W, C). The difference between the beta functions at finite
lattice flow time $t/a^{2}$ obtained for any two operators should approach
zero as $(a^{2}/t)^{\zeta}$, where $\zeta$ is the scaling exponent of the
leading irrelevant operator. At weak coupling one expects $\zeta=1+O(g^{2})$,
but in general the dependence on $g^{2}$ is not known.101010A. Hasenfratz and
C. T. Peterson, in progress.  Our data do not allow us to resolve any
statistically significant deviations from $\zeta=1$, and so we will assume
that all cutoff effects scale linearly with $a^{2}/t$.

In Fig. 9 the data points plotted with open symbols are not included in the
extrapolation. The smallest acceptable flow time depends on $g^{2}$—as well
as, more generally, on the flow—reflecting how close we are to the RT. We also
limit the maximum flow time to suppress finite volume effects. The remaining
data give adequate linear extrapolations. We denote the extrapolated values by
$\beta_{\text{S}},\beta_{\text{W}},\beta_{\text{C}}$.

We determine the error of each extrapolation with a bootstrap analysis. The
closeness of the three extrapolated values indicates that the three operators
give consistent results in this case. We will return to the precise condition
for consistency shortly.

In Fig. 10 we show a compilation of continuum extrapolations. The three rows
of the figure show extrapolations at $g^{2}=10.0,$ 14.0, and 17.0, while the
three columns correspond, from left to right, to Wilson, C23, and C13 flows.
Each panel depicts the extrapolation process for all three operators. A
general feature revealed in the figure is that the slope of the extrapolation
is always smallest for the S operator, with the W operator coming next. The C
operator is last, having the largest slopes. This signals that discretization
effects are smallest for the S operator. Combined with the other advantages of
the S operator we have already discussed in Sec. II.3, this naturally leads to
the choice of the S operator for our main result.

Figure 10:  Extrapolations $a^{2}/t\to 0$ of $\beta(g^{2})$ for $g^{2}=10.0$
(top), 14.0 (middle), and 17.0 (bottom), which sample weak, intermediate, and
strong couplings. At each value of $g^{2}$ we plot extrapolations for data
from Wilson flow (left), C23 flow (center), and C13 flow (right). All data
points come from flow times that belong to the interval $2.4\leq t/a^{2}\leq
3.2$. Color codes and symbols are the same as in Fig. 9.

It is evident in Fig. 10 that there are cases where the extrapolated values of
the three operators are consistent with each other, whereas for other cases
they are not (for example, the panel in lower left). While these plots give a
general idea of the agreement of the extrapolations, they cannot directly be
used to determine the consistency of the extrapolations. The reason is that
data for the three operators are highly correlated. In order to correctly
assess the level of agreement between any two extrapolations, their
correlations must be taken into account.

We therefore base the criteria for consistency on the plots in Fig. 11, which
include the effects of correlations. Since we will be basing our main result
on the S operator, we plot red curves in each frame to represent
$\pm\sqrt{2}\sigma_{\text{S}}$, where $\sigma_{\text{S}}$ is the error in
$\beta_{\text{S}}$. The two bands in each frame are the $\pm 1\sigma$ bands
for the correlated differences $\beta_{\text{S}}-\beta_{\text{W}}$ and
$\beta_{\text{S}}-\beta_{\text{C}}$, which we also determine in the bootstrap
analysis. We comment that these two bands are much narrower than the span
between the red curves; this reflects the strong correlations among the three
operators.

> Figure 11:  Comparing the differences $\beta_{\text{S}}-\beta_{\text{W}}$
> (green) and $\beta_{\text{S}}-\beta_{\text{C}}$ (blue) with the standard
> deviation $\sigma_{\text{S}}$ of the S operator, at all physical couplings
> $g^{2}$ for each of the five flows. We require that the differences lie
> within the curves $\pm\sqrt{2}\sigma_{\text{S}}$ (red). The bounds of
> validity $g^{2}_{\text{min2}}$ and $g^{2}_{\text{max2}}$ in Tables 2 and 3
> have been set accordingly.

The range of $g^{2}$ values where a given flow can be trusted is determined by
the severity of discretization effects. As we have seen (e.g., in Figs. 9 and
10), the S operator shows the smallest discretization effects, with the W
operator coming next. Hence we will determine this range by requiring
consistency between $\beta_{\text{S}}$ and $\beta_{\text{W}}$. Table 2 shows
the range of $g^{2}$ in which each set of flow data satisfies two constraints:
(1) $g^{2}$ lies within the range of interpolation in Fig. 8, resulting in an
interval $[g^{2}_{\text{min1}},g^{2}_{\text{max1}}]$; (2) the operators S and
W give consistent values for the extrapolation to $a^{2}/t=0$, resulting in
the smaller interval $[g^{2}_{\text{min2}},g^{2}_{\text{max2}}]$. The precise
condition for consistency is that the mean value of
$\beta_{\text{S}}-\beta_{\text{W}}$, represented by the green curves in Fig.
11, lies between the $\pm\sqrt{2}\sigma_{\text{S}}$ curves.

The beta function resulting from this analysis is the final result of this
section. It is displayed in Fig. 2, and replicated in the top panel of Fig.
12. There is a nice agreement between the predictions of the different flows
in their overlapping ranges. On the weak-coupling side, our result connects
smoothly to perturbation theory. At strong coupling, we find an IR fixed point
at $g^{2}$ somewhere between 15.5 and 16.

For completeness, we also examine the consequences of requiring consistency of
all three operators. In this case, both of the differences
$\beta_{\text{S}}-\beta_{\text{W}}$ and $\beta_{\text{S}}-\beta_{\text{C}}$
are required to lie between the $\pm\sqrt{2}\sigma_{\text{S}}$ curves, see
Table 3. This results in a smaller range of validity for each flow, shown in
the bottom panel of Fig. 12. It can be seen that now there is a gap between
the coverage of the C43 and Wilson flows; a set of flows with $c_{p}$ and
$c_{r}$ chosen between those of C43 and Wilson is needed to fill that gap. We
note that for C13 flow, the differences $\beta_{\text{S}}-\beta_{\text{W}}$
and $\beta_{\text{S}}-\beta_{\text{C}}$ cross zero in the vicinity of the IR
fixed point (bottom right panel of Fig. 11). In other words, the differences
among the beta functions extracted from the extrapolations of the three
operators are minimal near the fixed point. This lends further credibility to
our determination of its existence and location.

Flow | $g^{2}_{\text{min1}}$ | $g^{2}_{\text{max1}}$ | $g^{2}_{\text{min2}}$ | $g^{2}_{\text{max2}}$
---|---|---|---|---
Sym | 3.7 | 13.6 | 3.7 | 4.0
C43 | 3.7 | 14.4 | 3.7 | 11.0
Wil | 6.6 | 17.3 | 6.6 | 12.5
C23 | 6.8 | 18.9 | 6.8 | 14.0
C13 | 9.8 | 21.3 | 9.8 | 17.5
Table 2:  Ranges of $g^{2}$ in which each flow is included in the final result for $\beta(g^{2})$. $g^{2}_{\text{min1}}$ and $g^{2}_{\text{max1}}$ result from limiting the ensembles included in the interpolations, while $g^{2}_{\text{min2}}$ and $g^{2}_{\text{max2}}$ come from further demanding consistency between the continuum extrapolations $\beta_{\text{S}}$ and $\beta_{\text{W}}$ (see Fig. 11). Since the second requirement does not constrain the $g^{2}$ range any further on the weak-coupling side, the $g^{2}_{\text{min1}}$ and $g^{2}_{\text{min2}}$ columns are identical. We quote $g^{2}_{\text{max2}}$ with a resolution of 0.5. The last two columns are the ranges reflected in the upper panel of Fig. 12 (and in Fig. 2).  Flow | $g^{2}_{\text{min1}}$ | $g^{2}_{\text{max1}}$ | $g^{2}_{\text{min2}}$ | $g^{2}_{\text{max2}}$
---|---|---|---|---
Sym | 3.7 | 13.6 | (none) | (none)
C43 | 3.7 | 14.4 | 3.7 | 4.5
Wil | 6.6 | 17.3 | 6.6 | 11.5
C23 | 6.8 | 18.9 | 6.8 | 13.5
C13 | 9.8 | 21.3 | 11.5 | 17.5
Table 3:  Same as Table 2, but here the determination of $g^{2}_{\text{min2}}$
and $g^{2}_{\text{max2}}$ is based on consistency of $\beta_{\text{S}}$,
$\beta_{\text{W}}$, and $\beta_{\text{C}}$, with results reflected in the
lower panel of Fig. 12.

> Figure 12:  The $\beta$ function obtained with five different GF
> transformations. Both panels show the beta function derived from the S
> operator, but the ranges in $g^{2}$ allowed for each flow differ between
> them. For the top panel (identical to Fig. 2), only consistency between the
> S and W operators is required (Table 2). For the bottom panel, all three
> operators are required to be consistent (Table 3). In the bottom panel,
> there is a large gap between the $g^{2}$ regions of the C43 and Wilson
> flows—to bridge this gap a finer sequence of flows is needed. Also, there is
> no region of validity for the Symanzik flow, and thus it is not shown.

##  III  Anomalous dimensions

In this section we compute the mass anomalous dimensions for the fundamental
and sextet representations, as well as the anomalous dimensions of top-partner
chimera operators, as functions of the renormalized coupling $g^{2}$. The
anomalous dimensions in the infrared are then given by their values at the
fixed point at $g^{2}\simeq 15.5$.

The gradient flow equation for a fermion field in the continuum is [41]

$\frac{\partial\chi}{\partial t}=\Delta(t)\chi\ ,$ (8)

where $\chi$ is the flowed fermion field. $\Delta$ is the covariant laplacian,
constructed from the flowed gauge field at the same $t$. Similarly to the
gauge field, the initial condition is

$\chi(0)\big{|}_{t=0}=\psi\ ,$ (9)

where $\psi$ is the dynamical fermion field.

There is a technical issue in the application of the continuous RG method to
operators made out of fermion fields, described in Ref. [42]. Consider a two-
point function of a flowed mesonic density $X^{\prime}$ with an unflowed
source $X$,

$\left\langle X(0)\,X^{\prime}(t)\right\rangle\sim t^{-(d+\eta+\gamma)/2}\ .$
(10)

The scaling formula follows from the fact that here $X(0)$ is kept at $t=0$,
while $X^{\prime}(t)$ is constructed from the flowed fermion fields at time
$t$. The exponent reflects the classical dimension of the fermion bilinear,
$d=3$; the anomalous dimension of the elementary fermion field, $\eta/2$; and
the anomalous dimension of the meson operator, $\gamma$. The reason for the
appearance of $\eta$ is that the gradient flow, unlike a blocking RG
transformation, does not preserve the normalization of the fermion kinetic
term.

One way to eliminate $\eta$ is to divide $\left\langle
X(0)\,X^{\prime}(t)\right\rangle$ by the two-point function of a conserved
current, whose anomalous dimension vanishes. We use the vector current, which
scales according to (suppressing the vector index)

$\left\langle V(0)\,V^{\prime}(t)\right\rangle\sim t^{-(d+\eta)/2}\ .$ (11)

Defining the ratio

$R(t)=\frac{\left\langle X(0)\,X^{\prime}(t)\right\rangle}{\left\langle
V(0)\,V^{\prime}(t)\right\rangle}\ ,$ (12)

we have

$R(t)\sim t^{-\gamma/2}\ .$ (13)

Now $\gamma$ can be extracted from the logarithmic derivative,

$\gamma=-2\frac{t}{R}\,\frac{\partial R}{\partial t}\ .$ (14)

We choose the operators $X$ and $X^{\prime}$ to lie in definite 3-volumes,
separated by euclidean time $x_{4}$. We require $\sqrt{8t}\ll x_{4}$, meaning
that the euclidean time separation must be large compared to the smearing of
the operators by the flow. Once this condition is satisfied, the ratio $R(t)$
and the corresponding exponent $\gamma$ are expected to be independent of
$x_{4}$.

Figure 13:  Examples of $x_{4}$ plateaus. Both panels show raw results for the
mass anomalous dimension at $\beta=10.8$. Top: C13 flow, fundamental
representation. Bottom: Wilson flow, sextet representation.

###  III.1  Mass anomalous dimensions

In the lattice calculation, for each representation we construct zero-momentum
meson correlation functions,

$C_{X}(x_{4},t)=\left\langle
X(x_{4}^{0})\,X^{\prime}(x_{4}^{0}+x_{4},t)\right\rangle\ ,$ (15)

where $X=S$ (scalar) or $P$ (pseudoscalar).111111We verified that the scalar
and pseudoscalar correlators agree within error, as expected when the chiral
limit is taken in finite volume. The source $X$ is placed at euclidean time
$x_{4}^{0}$, while the sink $X^{\prime}$ is separated by time $x_{4}$ from the
source. To shorten autocorrelation times, we shift the source plane
$x_{4}^{0}$ between successive lattice configurations of each ensemble. After
fixing to Coulomb gauge, we construct the source $X$ using gaussian-smeared
fermion fields with smearing radius $R_{0}=6$. We solve the Dirac equation
using the conjugate gradient algorithm, and then flow the solution by
integrating the fermion flow equation following Ref. [41]. We use point sinks,
projected to $\vec{p}=0$. As for the gauge field, flowed correlation functions
are recorded at interval $\Delta t=0.1$.

Having similarly computed the flowed two-point function of the vector current,
we use Eq. (14) to determine the mass anomalous dimension $\gamma_{m}$ as a
function of the euclidean time separation $x_{4}$ and the flow time $t$.
Examples of raw results for $\gamma_{m}$ are shown in Fig. 13 for both
representations. A new ingredient, clearly visible in the figure, is the
$x_{4}$ plateaus. This provides a practical criterion for satisfying the
condition $\sqrt{8t}\ll x_{4}$, and hence we extract the anomalous dimensions
using values of $x_{4}$ inside the plateau, averaging over the range $11\leq
x_{4}\leq 14$.

The rest of the calculation follows the same steps as for the beta function.
We interpolate the raw results to obtain $\gamma_{m}$ as a function of
$g^{2}$, and then use the interpolations $\gamma_{m}(t,g^{2})$ to take the
continuum limit $a^{2}/t\to 0$ at fixed $g^{2}$.

Final results for the mass anomalous dimensions are shown in Fig. 3 above. In
the weak-coupling region, the mass anomalous dimensions agree with one-loop
perturbation theory,

$\gamma_{m}=\frac{6g^{2}C_{2}}{16\pi^{2}}\ ,$ (16)

where $C_{2}$ is the quadratic Casimir operator: $C_{2}=15/8$ for the
fundamental representation and $C_{2}=5/2$ for the sextet representation. At
larger couplings, the calculated mass anomalous dimensions move below the one-
loop result. At the IR fixed point we have $\gamma_{m}^{(4)}\simeq 0.75$ for
the fundamental representation and $\gamma_{m}^{(6)}\simeq 1.0$ for the sextet
representation. Both are quite large, suggesting that while the 4+4 system is
IR conformal, it is not too far from the conformal sill.

These results, like the beta function presented above, are based on data
obtained on lattices of size $24^{3}\times 48$. We have carried out
comparisons to the two $28^{3}\times 56$ ensembles, parallel to the analysis
of $\beta(g^{2})$ in Sec. II.3. In a plot along the lines of Fig. 7, we find
of course the same shift in $g^{2}$, but, again, no change in the data beyond
the statistical error bars. Hence we expect $\gamma_{m}(g^{2})$ to move
horizontally with the beta function as we approach the infinite-volume limit,
with no other change. We obtain the same behavior for the chimera anomalous
dimensions, presented below.

###  III.2  Chimera anomalous dimensions

Many different chimera operators can be used to create a top-partner state
[46, 47]. We consider operators of the lowest possible mass dimension, which
are three-fermion operators with no derivatives. To write them, we introduce
Dirac fermions for the sextet representation, $\chi_{ABi}$, $i=1,\ldots,4$,
where $i$ is a flavor index while $A,B=1,\ldots,4$, are the hypercolor
indices, which are antisymmetrized. The fundamental representation fermions
are denoted $\psi_{Aa}$, where $a=1,\ldots,4$ is also a flavor index. The
three-fermion chimera operators are

$\displaystyle B_{iab}^{IJ}$ $\displaystyle=$
$\displaystyle\epsilon_{ABCD}\,P_{I}\,\chi_{ABi}\,\left(\psi_{Ca}^{T}\,CP_{J}\,\psi_{Db}\right),$
(17a) $\displaystyle B^{I}_{iab\mu}$ $\displaystyle=$
$\displaystyle\epsilon_{ABCD}\,P_{I}\,\gamma_{\mu}\chi_{ABi}\left(\psi^{T}_{Ca}C\gamma_{\mu}\gamma_{5}\psi_{Db}\right),\hskip
21.52771pt$ (17b)

where the labels $I,J$ take the values $R,L$. Here $C$ is the charge-
conjugation matrix and $P_{R,L}=(1\pm\gamma_{5})/2$. In Eq. (17b) there is no
summation over $\mu$.

Projecting these operators onto the quantum numbers of the top quark depends
on the details of the embedding of the Standard Model symmetries in the
hypercolor (M6 or M11) model. For the M6 model, see Refs. [46, 47]. This will
not concern us here.

We construct chimera two-point functions following closely Ref. [23]. All the
chimera correlators we consider have the general form

$C_{\pm}(x_{4},t)={\rm Tr}\left\langle
B(x_{4}^{0}+x_{4},t)\,\bar{\Lambda}(x_{4}^{0})P_{\pm}\right\rangle\ ,$ (18)

where $P_{\pm}={1\over 2}(1\pm\gamma_{4})$ are parity projectors. The source
operator $\bar{\Lambda}$ is a quark-model creation operator for the top
partner, which is kept at $t=0$ (for details, see Ref. [23]). The sink
operator $B(x_{4}^{0}+x_{4},t)$ is one of the operators in Eq. (17). We use
the same flowed fermion fields as for the mesons. As before, the sink is
projected onto zero spatial momentum.

The correlation functions (18) are related by discrete (lattice) symmetries.
First, the $B^{I}_{iab\mu}$ operators are related by hypercubic rotations;
since we calculate correlation functions with $x_{4}$ dependence, we separate
the $\mu=4$ operator from $\mu=1,2,3$ and lump the latter into a “space”
component, viz.,

$\displaystyle B^{sI}_{iab}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\sum_{k=1}^{3}B^{I}_{iabk}\ ,$ (19a) $\displaystyle
B^{tI}_{iab}$ $\displaystyle=$ $\displaystyle B^{I}_{iab4}\ .$ (19b)

In addition, the correlation functions (18) are related by the usual continuum
discrete symmetries. Parity flips the chiral projectors, and so it takes, for
example, $B^{RR}(\vec{p}=0,x_{4})\leftrightarrow B^{LL}(\vec{p}=0,x_{4})$.
Euclidean time reversal likewise flips the parity projectors, and transforms
the time component according to $x_{4}\leftrightarrow T-x_{4}$, where $T$ is
the temporal extent of the lattice.121212 The product of euclidean time
reversal and parity acts on a generic fermion field as
$\psi(x)\to\gamma_{5}\psi(-x)$ in infinite volume. We improve statistics by
averaging over quartets of correlators related by parity and time reversal.

The outcome is that there are only three independent anomalous
dimensions.131313Naturally, the anomalous dimensions are independent of flavor
indices.  We denote them $\gamma^{RR}_{ch}$ for $B^{RR}$ and $B^{LL}$;
$\gamma^{RL}_{ch}$ for $B^{RL}$ and $B^{LR}$; and $\gamma^{s}_{ch}$ for
$B^{sI}$ and $B^{tI}$.

Since each chimera operator consists of two fundamental and one sextet
fermion, we normalize the flowed chimera two-point functions by constructing
ratios [cf. Eq. (13)]

$R_{\pm}(x_{4},t)=\frac{C_{\pm}(x_{4},t)}{C_{V}^{(4)}(x_{4},t)\sqrt{C_{V}^{(6)}(x_{4},t)}}\
.$ (20)

Here $C_{V}^{(4)}(x_{4},t)$ and $C_{V}^{(6)}(x_{4},t)$ are the two-point
functions of the fundamental and sextet vector currents. We then use Eq. (14)
as before to extract the anomalous dimension.

Figure 14:  Anomalous dimensions of chimera operators. Top left:
$\gamma^{RR}_{ch}$. Top right: $\gamma^{RL}_{ch}$. Bottom: $\gamma^{s}_{ch}$
(same plot as in Fig. 4).

Figure 14 shows our results for the three independent anomalous dimensions
$\gamma^{RR}_{ch}$, $\gamma^{RL}_{ch}$ and $\gamma^{s}_{ch}$. The general
behavior of each of them closely resembles that of the mass anomalous
dimensions. At our weakest couplings, the anomalous dimensions agree with one-
loop perturbation theory, whereas for larger couplings, the calculated
anomalous dimensions fall below the one-loop curve. In addition,
$\gamma^{RR}_{ch}$ and $\gamma^{RL}_{ch}$ agree within error, which is not
required by symmetry.

The one-loop predictions are given by $\gamma=cg^{2}/(16\pi^{2})$, where
$c=15/4$ for both $\gamma^{RR}_{ch}$ and $\gamma^{RL}_{ch}$, and $c=15/2$ for
$\gamma^{s}_{ch}$ [48] (see also [49]). These anomalous dimensions happen to
satisfy

$\gamma^{RR}_{ch}:\gamma^{RL}_{ch}:\gamma^{s}_{ch}:\gamma_{m}^{(4)}:\gamma_{m}^{(6)}=1:1:2:3:4\
.$ (21)

Thus, while at one loop $\gamma^{s}_{ch}$ is twice $\gamma^{RR}_{ch}$ or
$\gamma^{RL}_{ch}$, it is still smaller than both of the mass anomalous
dimensions. This pattern persists in our non-perturbative results. Indeed, the
ratios (21) are roughly preserved near the fixed point as well, where
$\gamma^{RR}_{ch}\approx\gamma^{RL}_{ch}\simeq 0.25$ and
$\gamma^{s}_{ch}\simeq 0.5$.

##  IV  Conclusions

As reported in this paper, we have employed lattice techniques to study the
4+4 model, an SU(4) gauge theory with four Dirac fermions in the fundamental
representation and with eight Majorana fermions in the sextet representation.
We used the continuous RG method, based on a gradient flow, to calculate the
beta function and a number of important anomalous dimensions. Our lattice
action includes a set of Pauli–Villars fields, which enable us to reach
otherwise inaccessible strong couplings. We find that the 4+4 model has an
infrared-stable fixed point at $g^{2}\simeq 15.5$.

We have dealt carefully with the continuum limit. Moreover, we have argued
that finite-volume effects on both the beta function $\beta(g^{2})$ and the
anomalous dimensions are limited to a common horizontal shift of a few
percent, which leaves the anomalous dimensions at the infrared fixed point
unchanged.

The M6 and M11 models of the Ferretti–Karateev list [7, 10] can be reached
from the 4+4 model by giving large masses to a suitable subset of the
fermions, while the fermion fields needed for the Ferretti–Karateev model
itself are kept massless or given much smaller masses. Assuming that the
Ferretti–Karateev model is below the conformal sill, it follows that the
decoupling of the heavy fermions will trigger chiral symmetry breaking and
confinement. The hypercolor scale $\Lambda_{HC}$ thus follows the heavy mass
scale [16, 17, 18, 19].

We also find that the mass anomalous dimensions of both representations are
quite large at the fixed point, even reaching $\gamma_{m}^{(6)}\simeq 1$ for
the sextet representation. The anomalous dimensions of all top-partner chimera
operators are smaller, reaching 0.5 or less. If one chooses masses for the
fermions of the 4+4 model, as just discussed, to reach the M6 or the M11
model, the anomalous dimensions that govern the running of operators from the
EHC scale down to the HC scale will still be controlled by the fixed point.
The critical value of the chimera anomalous dimension, where suppression by a
power of $\Lambda_{HC}/\Lambda_{EHC}$ is eliminated, is $\gamma=2$ (see for
example Ref. [6]). Our results, then, indicate that these anomalous dimensions
may not be large enough for a phenomenologically successful composite Higgs
model.

###### Acknowledgements.

Computations for this work were carried out on facilities of the USQCD
Collaboration, which are funded by the Office of Science of the U.S.
Department of Energy. A.H. and E.N. acknowledge support by DOE grant DE-
SC0010005. The work of B.S. and Y.S. was supported by the Israel Science
Foundation under grant No. 1429/21.

*

##  Appendix A Lattice matters

###  A.1  Lattice action and simulation code

As in our previous work on the 2+2 model, we use a Wilson-clover fermion
action, with a gauge-invariant kinetic term for each fermion species. We use
normalized hypercubic (nHYP) smeared gauge links for the fundamental
representation [50, 51]; gauge links for the sextet representation are
constructed from these smeared links. The clover coefficient is set equal to
unity for both fermion species [52, 53]. The gauge field action has the form
$\beta S_{\text{plaq}}+\gamma S_{\text{NDS}}$, where $S_{\text{plaq}}$ is the
usual plaquette action. $S_{\text{NDS}}$ is the nHYP dislocation suppressing
action, a smeared action designed to reduce gauge-field roughness that would
create large fermion forces in the molecular dynamics evolution [54]. We hold
the ratio $\gamma/\beta=1/125$ fixed [20].

As discussed in Sec. II, in this work we have added a set of Pauli–Villars
(PV) fields, which allow us to reach much further into strong renormalized
coupling [43]. The PV action is the same Wilson-clover action as for the
fermions, but the PV fields have opposite statistics. The ratio of PV to
fermion fields is $3:1$, that is, we have 12 PV fields in each representation.
The bare mass of the PV fields is kept at $am_{0}=1$ ($\kappa=1/10$) in order
that they decouple in the continuum limit. This can be gauged by the mass of
the ghost pion made out of two PV fields, which we find to be roughly equal to
2 in lattice units. This means that the effective gauge action induced by
integrating over the PV fields is essentially local. For comparison, we note
that the physical pions in both representations come out to have masses
dictated by the volume, $m_{\pi}\simeq\pi/L$, where the lattice size is
$L=24a$ or $28a$.

Our multi-representation code is a derivative of the MILC code [55]. We use
three nested update levels with one Hasenbush preconditioning mass. The
innermost level (`level=0`) contains the gauge update. Pseudofermion actions
are simulated at `level=1`, which takes care of the upper end of the fermion
spectrum, and at `level=2`, the outermost level, which takes care of the lower
end of this spectrum. The PV fields are simulated at `level=1`, as is
$S_{\text{NDS}}$.

###  A.2  Ensembles

For each value of the gauge coupling $\beta$, we determine the critical point
$K_{c}=(\kappa_{4}^{c},\kappa_{6}^{c})$ by tuning the fermion masses to zero
for both representations, as calculated from the axial Ward identities (AWI).
We require the actual AWI masses to be $|m_{q}|\lesssim 0.002$, as we have
seen that these masses are small enough to have negligible effect on the flow.
The ensembles used in this work are shown in Table 4. In practice, we tuned to
$K_{c}$ for each of our $24^{3}\times 48$ ensembles. In the $28^{3}\times 56$
ensembles we kept the same $(\kappa_{4},\kappa_{6})$ as for the smaller volume
and verified that the AWI masses are practically unchanged.

$L/a$ | $\beta$ | $\phantom{X}\kappa_{4}$ | $\phantom{X}\kappa_{6}$ | lattices | $\phantom{XX}m_{q4}$ | $\phantom{XX}m_{q6}$
---|---|---|---|---|---|---
24 | 10.4 | 0.12805 | 0.12905 | 35 | $\phantom{-}0.0017(1)$ | $-0.0003(1)$
24 | 10.5 | 0.12787 | 0.12876 | 25 | $\phantom{-}0.0013(1)$ | $\phantom{-}0.0011(1)$
24 | 10.6 | 0.12775 | 0.12855 | 35 | $-0.0007(1)$ | $\phantom{-}0.0005(1)$
24 | 10.8 | 0.12750 | 0.12804 | 25 | $-0.0028(1)$ | $\phantom{-}0.0040(1)$
24 | 11.0 | 0.12717 | 0.12777 | 25 | $-0.0010(1)$ | $\phantom{-}0.0015(1)$
24 | 11.5 | 0.12670 | 0.12724 | 25 | $-0.0012(1)$ | $-0.0009(1)$
24 | 12.0 | 0.12633 | 0.12680 | 45 | $\phantom{-}0.00154(3)$ | $\phantom{-}0.00139(3)$
24 | 13.0 | 0.12600 | 0.12640 | 45 | $\phantom{-}0.00141(1)$ | $\phantom{-}0.00010(1)$
28 | 10.5 | 0.12787 | 0.12876 | 25 | $\phantom{-}0.0018(1)$ | $\phantom{-}0.0015(1)$
28 | 11.0 | 0.12717 | 0.12777 | 25 | $-0.00096(4)$ | $\phantom{-}0.0016(1)$
Table 4:  Parameters and fermion masses for the ensembles. The volume is
$L^{3}\times T$, with $T=2L$. The fifth column gives the number of
configurations that were used in the flow analysis. We saved a configuration
every 5 trajectories (after equilibration). $m_{q4}$ and $m_{q6}$ are the AWI
masses of the fundamental and sextet representations, respectively.

## References

  * Georgi and Kaplan [1984] H. Georgi and D. B. Kaplan, Composite Higgs and Custodial SU(2), Phys. Lett. 145B, 216 (1984).
  * Dugan  _et al._ [1985] M. J. Dugan, H. Georgi, and D. B. Kaplan, Anatomy of a Composite Higgs Model, Nucl. Phys. B254, 299 (1985).
  * Kaplan [1991] D. B. Kaplan, Flavor at SSC energies: A New mechanism for dynamically generated fermion masses, Nucl. Phys. B365, 259 (1991).
  * Contino [2011] R. Contino, The Higgs as a Composite Nambu-Goldstone Boson, in  _Physics of the large and the small, TASI 09, proceedings of the Theoretical Advanced Study Institute in Elementary Particle Physics, Boulder, Colorado, USA, 1-26 June 2009_ (2011) pp. 235–306, arXiv:1005.4269 [hep-ph] .
  * Bellazzini  _et al._ [2014] B. Bellazzini, C. Csáki, and J. Serra, Composite Higgses, Eur. Phys. J. C74, 2766 (2014), arXiv:1401.2457 [hep-ph] .
  * Panico and Wulzer [2016] G. Panico and A. Wulzer, The Composite Nambu-Goldstone Higgs, Lect. Notes Phys. 913 (2016), arXiv:1506.01961 [hep-ph] .
  * Ferretti and Karateev [2014] G. Ferretti and D. Karateev, Fermionic UV completions of Composite Higgs models, JHEP 03, 077, arXiv:1312.5330 [hep-ph] .
  * Ferretti [2014] G. Ferretti, UV Completions of Partial Compositeness: The Case for a SU(4) Gauge Group, JHEP 06, 142, arXiv:1404.7137 [hep-ph] .
  * Ferretti [2016] G. Ferretti, Gauge theories of Partial Compositeness: Scenarios for Run-II of the LHC, JHEP 06, 107, arXiv:1604.06467 [hep-ph] .
  * Belyaev  _et al._ [2017] A. Belyaev, G. Cacciapaglia, H. Cai, G. Ferretti, T. Flacke, A. Parolini, and H. Serodio, Di-boson signatures as Standard Candles for Partial Compositeness, JHEP 01, 094, arXiv:1610.06591 [hep-ph] .
  * Vecchi [2017] L. Vecchi, A dangerous irrelevant UV-completion of the composite Higgs, JHEP 02, 094, arXiv:1506.00623 [hep-ph] .
  * Cacciapaglia  _et al._ [2021] G. Cacciapaglia, S. Vatani, and C. Zhang, The Techni-Pati-Salam Composite Higgs, Phys. Rev. D 103, 055001 (2021), arXiv:2005.12302 [hep-ph] .
  * Bando  _et al._ [1988] M. Bando, T. Kugo, and K. Yamawaki, Nonlinear Realization and Hidden Local Symmetries, Phys. Rept. 164, 217 (1988).
  * Hill and Simmons [2003] C. T. Hill and E. H. Simmons, Strong Dynamics and Electroweak Symmetry Breaking, Phys. Rept. 381, 235 (2003), [Erratum: Phys.Rept. 390, 553–554 (2004)], arXiv:hep-ph/0203079 .
  * Kaplan  _et al._ [2009] D. B. Kaplan, J.-W. Lee, D. T. Son, and M. A. Stephanov, Conformality Lost, Phys. Rev. D 80, 125005 (2009), arXiv:0905.4752 [hep-th] .
  * Brower  _et al._ [2016] R. C. Brower, A. Hasenfratz, C. Rebbi, E. Weinberg, and O. Witzel, Composite Higgs model at a conformal fixed point, Phys. Rev. D 93, 075028 (2016), arXiv:1512.02576 [hep-ph] .
  * Hasenfratz  _et al._ [2017] A. Hasenfratz, C. Rebbi, and O. Witzel, Large scale separation and resonances within LHC range from a prototype BSM model, Phys. Lett. B 773, 86 (2017), arXiv:1609.01401 [hep-ph] .
  * Witzel and Hasenfratz [2019] O. Witzel and A. Hasenfratz (Lattice Strong Dynamics), Constructing a composite Higgs model with built-in large separation of scales, PoS LATTICE2019, 115 (2019), arXiv:1912.12255 [hep-lat] .
  * Appelquist  _et al._ [2021] T. Appelquist _et al._ (Lattice Strong Dynamics), Near-conformal dynamics in a chirally broken system, Phys. Rev. D 103, 014504 (2021), arXiv:2007.01810 [hep-ph] .
  * Ayyar  _et al._ [2018a] V. Ayyar, T. DeGrand, M. Golterman, D. C. Hackett, W. I. Jay, E. T. Neil, Y. Shamir, and B. Svetitsky, Spectroscopy of SU(4) composite Higgs theory with two distinct fermion representations, Phys. Rev. D97, 074505 (2018a), arXiv:1710.00806 [hep-lat] .
  * Ayyar  _et al._ [2018b] V. Ayyar, T. Degrand, D. C. Hackett, W. I. Jay, E. T. Neil, Y. Shamir, and B. Svetitsky, Baryon spectrum of SU(4) composite Higgs theory with two distinct fermion representations, Phys. Rev. D97, 114505 (2018b), arXiv:1801.05809 [hep-ph] .
  * Ayyar  _et al._ [2018c] V. Ayyar, T. DeGrand, D. C. Hackett, W. I. Jay, E. T. Neil, Y. Shamir, and B. Svetitsky, Finite-temperature phase structure of SU(4) gauge theory with multiple fermion representations, Phys. Rev. D97, 114502 (2018c), arXiv:1802.09644 [hep-lat] .
  * Ayyar  _et al._ [2018d] V. Ayyar, T. DeGrand, D. C. Hackett, W. I. Jay, E. T. Neil, Y. Shamir, and B. Svetitsky, Partial compositeness from baryon matrix elements on the lattice, (2018d), arXiv:1812.02727 [hep-ph] .
  * Ayyar  _et al._ [2019] V. Ayyar, M. F. Golterman, D. C. Hackett, W. Jay, E. T. Neil, Y. Shamir, and B. Svetitsky, Radiative Contribution to the Composite-Higgs Potential in a Two-Representation Lattice Model, Phys. Rev. D 99, 094504 (2019), arXiv:1903.02535 [hep-lat] .
  * Golterman  _et al._ [2021] M. Golterman, W. I. Jay, E. T. Neil, Y. Shamir, and B. Svetitsky, Low-energy constant $L_{10}$ in a two-representation lattice theory, Phys. Rev. D 103, 074509 (2021), arXiv:2010.01920 [hep-lat] .
  * Cossu  _et al._ [2019] G. Cossu, L. Del Debbio, M. Panero, and D. Preti, Strong dynamics with matter in multiple representations: $\mathrm{SU}(4)$ gauge theory with fundamental and sextet fermions, Eur. Phys. J. C 79, 638 (2019), arXiv:1904.08885 [hep-lat] .
  * Del Debbio  _et al._ [2023] L. Del Debbio, A. Lupo, M. Panero, and N. Tantalo, Multi-representation dynamics of SU(4) composite Higgs models: chiral limit and spectral reconstructions, Eur. Phys. J. C 83, 220 (2023), arXiv:2211.09581 [hep-lat] .
  * Bennett  _et al._ [2018] E. Bennett, D. K. Hong, J.-W. Lee, C. J. D. Lin, B. Lucini, M. Piai, and D. Vadacchino, Sp(4) gauge theory on the lattice: towards SU(4)/Sp(4) composite Higgs (and beyond), JHEP 03, 185, arXiv:1712.04220 [hep-lat] .
  * Bennett  _et al._ [2019] E. Bennett, D. K. Hong, J.-W. Lee, C. J. D. Lin, B. Lucini, M. Piai, and D. Vadacchino, Sp(4) gauge theories on the lattice: $N_{f}=2$ dynamical fundamental fermions, JHEP 12, 053, arXiv:1909.12662 [hep-lat] .
  * Bennett  _et al._ [2022] E. Bennett, D. K. Hong, H. Hsiao, J.-W. Lee, C. J. D. Lin, B. Lucini, M. Mesiti, M. Piai, and D. Vadacchino, Lattice studies of the Sp(4) gauge theory with two fundamental and three antisymmetric Dirac fermions, Phys. Rev. D 106, 014501 (2022), arXiv:2202.05516 [hep-lat] .
  * Hsiao  _et al._ [2022] H. Hsiao, E. Bennett, D. K. Hong, J.-W. Lee, C. J. D. Lin, B. Lucini, M. Piai, and D. Vadacchino, Spectroscopy of chimera baryons in a Sp(4) lattice gauge theory, (2022), arXiv:2211.03955 [hep-lat] .
  * Bennett  _et al._ [2023] E. Bennett, J. Holligan, D. K. Hong, H. Hsiao, J.-W. Lee, C. J. D. Lin, B. Lucini, M. Mesiti, M. Piai, and D. Vadacchino, Sp($2N$) Lattice Gauge Theories and Extensions of the Standard Model of Particle Physics (2023) arXiv:2304.01070 [hep-lat] .
  * Kim  _et al._ [2020] B. S. Kim, D. K. Hong, and J.-W. Lee, Into the conformal window: Multirepresentation gauge theories, Phys. Rev. D 101, 056008 (2020), arXiv:2001.02690 [hep-ph] .
  * Hasenfratz and Witzel [2020] A. Hasenfratz and O. Witzel, Continuous renormalization group $\beta$ function from lattice simulations, Phys. Rev. D 101, 034514 (2020), arXiv:1910.06408 [hep-lat] .
  * Hasenfratz and Witzel [2019] A. Hasenfratz and O. Witzel, Continuous $\beta$ function for the SU(3) gauge systems with two and twelve fundamental flavors, PoS LATTICE2019, 094 (2019), arXiv:1911.11531 [hep-lat] .
  * Peterson  _et al._ [2022] C. T. Peterson, A. Hasenfratz, J. van Sickle, and O. Witzel, Determination of the continuous $\beta$ function of SU(3) Yang-Mills theory, PoS LATTICE2021, 174 (2022), arXiv:2109.09720 [hep-lat] .
  * Hasenfratz  _et al._ [2023] A. Hasenfratz, C. T. Peterson, J. van Sickle, and O. Witzel, $\Lambda$ parameter of the SU(3) Yang-Mills theory from the continuous $\beta$ function, (2023), arXiv:2303.00704 [hep-lat] .
  * Fodor  _et al._ [2018] Z. Fodor, K. Holland, J. Kuti, D. Nogradi, and C. H. Wong, A new method for the beta function in the chiral symmetry broken phase, EPJ Web Conf. 175, 08027 (2018), arXiv:1711.04833 [hep-lat] .
  * Lüscher [2010] M. Lüscher, Properties and uses of the Wilson flow in lattice QCD, JHEP 08, 071, [Erratum: JHEP 03, 092 (2014)], arXiv:1006.4518 [hep-lat] .
  * Fodor  _et al._ [2012] Z. Fodor, K. Holland, J. Kuti, D. Nogradi, and C. H. Wong, The Yang-Mills gradient flow in finite volume, JHEP 11, 007, arXiv:1208.1051 [hep-lat] .
  * Lüscher [2013] M. Lüscher, Chiral symmetry and the Yang–Mills gradient flow, JHEP 04, 123, arXiv:1302.5246 [hep-lat] .
  * Carosso  _et al._ [2018] A. Carosso, A. Hasenfratz, and E. T. Neil, Nonperturbative Renormalization of Operators in Near-Conformal Systems Using Gradient Flows, Phys. Rev. Lett. 121, 201601 (2018), arXiv:1806.01385 [hep-lat] .
  * Hasenfratz  _et al._ [2021] A. Hasenfratz, Y. Shamir, and B. Svetitsky, Taming lattice artifacts with Pauli-Villars fields, Phys. Rev. D 104, 074509 (2021), arXiv:2109.02790 [hep-lat] .
  * Ramos and Sint [2016] A. Ramos and S. Sint, Symanzik improvement of the gradient flow in lattice gauge theories, Eur. Phys. J. C 76, 15 (2016), arXiv:1508.05552 [hep-lat] .
  * Hasenfratz and Witzel [2021] A. Hasenfratz and O. Witzel, Dislocations under gradient flow and their effect on the renormalized coupling, Phys. Rev. D 103, 034505 (2021), arXiv:2004.00758 [hep-lat] .
  * Golterman and Shamir [2015] M. Golterman and Y. Shamir, Top quark induced effective potential in a composite Higgs model, Phys. Rev. D91, 094506 (2015), arXiv:1502.00390 [hep-ph] .
  * Golterman and Shamir [2018] M. Golterman and Y. Shamir, Effective potential in ultraviolet completions for composite Higgs models, Phys. Rev. D97, 095005 (2018), arXiv:1707.06033 [hep-ph] .
  * DeGrand and Shamir [2015] T. DeGrand and Y. Shamir, One-loop anomalous dimension of top-partner hyperbaryons in a family of composite Higgs models, Phys. Rev. D92, 075039 (2015), arXiv:1508.02581 [hep-ph] .
  * Buarque Franzosi and Ferretti [2019] D. Buarque Franzosi and G. Ferretti, Anomalous dimensions of potential top-partners, SciPost Phys. 7, 027 (2019), arXiv:1905.08273 [hep-ph] .
  * Hasenfratz and Knechtli [2001] A. Hasenfratz and F. Knechtli, Flavor symmetry and the static potential with hypercubic blocking, Phys. Rev. D64, 034504 (2001), arXiv:hep-lat/0103029 [hep-lat] .
  * Hasenfratz  _et al._ [2007] A. Hasenfratz, R. Hoffmann, and S. Schaefer, Hypercubic smeared links for dynamical fermions, JHEP 05, 029, arXiv:hep-lat/0702028 [hep-lat] .
  * Bernard and DeGrand [2000] C. W. Bernard and T. A. DeGrand, Perturbation theory for fat link fermion actions,  _Lattice field theory. Proceedings, 17th International Symposium, Lattice’99, Pisa, Italy, June 29-July 3, 1999_ , Nucl. Phys. Proc. Suppl. 83, 845 (2000), arXiv:hep-lat/9909083 [hep-lat] .
  * Shamir  _et al._ [2011] Y. Shamir, B. Svetitsky, and E. Yurkovsky, Improvement via hypercubic smearing in triplet and sextet QCD, Phys. Rev. D83, 097502 (2011), arXiv:1012.2819 [hep-lat] .
  * DeGrand  _et al._ [2014] T. DeGrand, Y. Shamir, and B. Svetitsky, Suppressing dislocations in normalized hypercubic smearing, Phys. Rev. D90, 054501 (2014), arXiv:1407.4201 [hep-lat] .
  * [55] MILC Collaboration, http://www.physics.utah.edu/~detar/milc/.