Experiment-1-1

Measurements and EFT fits on detector level

Kelci Mohrman for the ATLAS and CMS Collaborations111Copyright 2023 CERN for
the benefit of the ATLAS and CMS Collaborations. Reproduction of this article
or parts of it is allowed as specified in the CC-BY-4.0 license

Department of Physics
University of Florida

> The framework of standard model effective field theory (SMEFT) provides a
> relatively model-agnostic tool with which to parameterize the off-shell
> effects of potential heavy new physics. In the top quark sector, the ATLAS
> and CMS experiments make use of data collected by the CERN LHC to search for
> these off-shell effects by probing dimension-six SMEFT operators involving
> top quarks. The analyses employ a variety of approaches, which range from
> relatively indirect (e.g. reinterpretations of unfolded measurements) to
> relatively direct (in which the SMEFT effects are propagated all the way to
> detector-level predictions). These proceedings discuss the landscape of
> detector-level SMEFT analyses in ATLAS and CMS, describe the methodology of
> the direct detector-level approach, and summarize several recent
> experimental results.

> PRESENTED AT
>
>
>
>
> $16^{\mathrm{th}}$ International Workshop on Top Quark Physics
> (Top2023), 24–29 September, 2023

##  1 Introduction

The standard model (SM) of particle physics provides a very successful
description of the known fundamental particles and their interactions.
However, the strong evidence for phenomenon that are not explained by the SM
(such as dark matter and dark energy) imply that the SM does not represent a
complete description of nature. For this reason, one of the main goals of the
CMS [1, 2] and ATLAS [3] physics programs is to search for signs of new
physics beyond the SM. However, at this point, there is not a strong
indication of a particular direction in which to search for such evidence.
Furthermore, there is no guarantee that new physics particles are light enough
to be produced on-shell within the energy range of the CERN LHC, which will
not significantly increase throughout its remaining years of operation. The
framework of standard model effective field theory (SMEFT) provides a
relatively model-agnostic method of parametrizing the off-shell effects of
heavy new physics, and can thus provide an interesting method of probing
potential new physics at the LHC.

In the SMEFT approach, the SM Lagrangian is treated as the lowest order term
in an expansion of higher-order operators that describe new physics at a mass
scale referred to as $\Lambda$, where the strength of the new physics
interactions are determined by dimensionless parameters referred to as Wilson
coefficients (WCs). The SMEFT Lagrangian can thus be written as follows:

$\mathcal{L}_{\mathrm{EFT}}=\mathcal{L}_{\mathrm{SM}}+\sum\limits_{d,i}\frac{c^{d}_{i}}{\Lambda^{d-4}}\mathcal{O}^{d}_{i}.$
(1)

In Equation 1, the $\mathcal{O}^{d}_{i}$ are the $i$ operators at dimension
$d$, the $c^{d}_{i}$ are the corresponding WCs, and the $\Lambda$ is the mass
scale of the new physics. If all of the WCs in Equation 1 are qual to zero,
the SM Lagrangian is recovered, meaning that any non-zero WC would be a sign
of new physics. It can also be noted that each term in the expansion is scaled
by an additional power of Lambda, so the lowest order terms are expected to
contribute most significantly. Here we focus on dimension-six terms, as these
are the lowest order terms that contribute to SMEFT in the top quark sector.

In the SMEFT framework, there are thousands of possible dimension-six
operators; the precise number depends on the assumptions that are made (e.g.
what flavor symmetries are assumed) [4]. Under the flavor universality
assumption, the number of dimension-six SMEFT operators is of order 60 [5].
SMEFT analyses in the top sector often choose to focus on operators involving
at least one top quark, with this requirement, the number of operators is
reduced to about 40 (though again the precise number depends on the particular
assumptions that are made) [6]. These operators can be categorized into four
main groups based on which particles are involved in the interactions to which
they give rise: interactions involving 4 heavy quarks (where heavy refers to
top and bottom), interactions involving 2 heavy quarks and 2 light quarks,
interactions involving 2 heavy quarks with bosons, and interactions involving
2 heavy quarks and leptons.

The operators in each category can impact top quark processes; an example of a
vertex from each category is shown in Figure 1. The presence of a SMEFT vertex
in a top quark processes can impact observables, such as cross sections and
kinematic distributions. The strength of these impacts are determined by the
value of the WCs. The goal of SMEFT analyses is thus to identify the true
value of these WCs, since any non-zero WC would indicate a sign of new
physics. In the absence of a signal, the limits placed on the WCs can be
interpreted in terms of the energy scale that is probed, and the limits can
also be used to help to constrain theory models.

Figure 1: Example vertex from each operator category impacting an example
process.

##  2 Using SMEFT to search for new physics

In order to find the best fit values for SMEFT WCs, an observable must be
parametrized in terms of the WCs in order to obtain a prediction that can be
compared against an observation. Let us first consider how the cross section
depends on the WCs. To calculate a cross section, one must first write down
the matrix element. With the SMEFT framework, this involves not only the SM
diagrams, but also the diagrams involving an insertion of a SMEFT vertex (e.g.
as shown in Figure 1). Squaring the matrix element to obtain the cross section
will give rise to three different types of terms: terms in which SM diagrams
are multiplied by each other (which have no WC dependence and simply
correspond to the SM contribution), terms in which SM diagrams multiply SMEFT
diagrams (which contain one power of a WC, and represent the interference
between the SM and the SMEFT), and diagrams in which SMEFT diagrams are
multiplied by each other (which contain two powers of WCs, and represent the
pure new physics contributions). The dependence of the cross section on the
WCs is thus represented by an $n$-dimensional quadratic function, where $n$ is
the number of WCs. The quadratic dependence will hold for any cross section,
inclusive or differential; as long as the SMEFT is modeled linearly in
amplitude, a quadratic dependence is expected***It should be noted that other
contributions (e.g. interference between dimension-eight SMEFT operators and
the SM, and interference between the SM and diagrams with two insertions of
dimension-six SMEFT operators) also enter at the $\Lambda^{-4}$ power. If
these $\Lambda^{-4}$ contributions are not able to be included, the quadratic
$c^{2}$ pieces are sometimes dropped from the parameterization (i.e. only the
linear pieces are kept) for consistency..

The methods which analyses attempt to find the best fit values of WCs
generally fall along a spectrum from relatively indirect approaches to
relatively direct detector-level approaches. Direct detector-level approaches
are the focus of these proceedings. With direct approaches, the SMEFT effects
are propagated all the way to detector level, and the WC limits are extracted
by comparing the predicted yield in the observable bins against the observed
number of events in the bins. Since yields are proportional to cross sections,
the predicted yields also carry a quadratic dependence on the WCs. However,
the shape of the quadratic varies from bin to bin (e.g. a bin in the high
$p_{\rm T}$ tail of a $p_{\rm T}$ distribution would be expected to have a
much steeper quadratic dependence on the WCs than would a bin in the low
$p_{\rm T}$ range of the distribution due to the fact that SMEFT effects tend
to be larger at higher energies).

In order to obtain the parameterizations for each bin, detector-level SMEFT
analyses often employ the flexible approach of event-by-event reweighting [7].
Each simulated event carries with it a weight, which corresponds to how much
of the cross section that particular event contributes to, so the weight is
essentially a small piece of the cross section, or a differential cross
section; thus, the weight of each individual event should also carry with it
an $n$-dimensional quadratic dependence on the WCs. Technical details
describing how this per-event $n$-dimensional quadratic dependence is obtained
and propagated to detector level are described in reference [8]. Once this
per-event quadratic parameterization has been obtained for the detector-level
samples, the quadratic dependence for a given bin can be found by simply
summing the quadratics for all of the events that pass the selection criteria
for this bin. Since the sum of multiple quadratics is still quadratic, the
predicted yield for an arbitrary observable bin is thus also described by a
quadratic in terms of the WCs.

An example of a detector-level SMEFT analysis which uses the event reweighting
approach is the recent CMS multilepton analysis [9]. This analysis studies
events involving two same-sign leptons or three or more leptons in order to
probe 26 WCs (from all four categories shown in Figure 1) simultaneously. Each
of the 178 observable bins in the analysis are shown in Figure 2. The
predicted yield in each bin is parameterized by a 26-dimensional quadratic
function in terms of the 26 WCs. In Figure 2, the histograms have been
reweighted to the SM (i.e. the WCs have all been set to zero, and the value in
each bin thus corresponds to the constant piece of the quadratic function).
This SMEFT-aware histogram can then be passed to the statistical analysis
tools, which will adjust the values of the 26 WCs (which will change the shape
and normalization across all 178 of the bins) in order to extract the ranges
of the 26 WCs which give rise to predicted yields that are in agreement with
the observation in each bin.

Figure 2: Observed data and expected yields in the prefit scenario. [9]

While direct-detector level approaches are powerful in that they can provide
good sensitivity to SMEFT effects and can offer a consistent and comprehensive
modeling of final states involving complicated admixtures of processes and
SMEFT effects, there are also challenges associated with the approach. The
challenges stem mainly from the necessity of producing detector-level SMEFT
samples. These samples are computationally intensive to produce, and would
need to be reproduced if any change to the SMEFT modeling is to be
incorporated. For this reason, reinterpretations and analysis preservation may
be challenging for analyses which follow the direct detector-level approach.

##  3 Recent detector-level SMEFT analyses in CMS and ATLAS

The new detector-level SMEFT analyses (i.e. analyses completed since the
Top2022 conference in the preceding year) from ATLAS and CMS are listed below.

  * •

CMS: Search for charged lepton flavor violation (involving electrons and
muons) in trilepton final states, places limits (individually) on CLFV WCs
[10].

  * •

ATLAS: Search for charged lepton flavor violation (involving muons and taus)
in dilepton final states, places limits (individually) on CLFV WCs [11].

  * •

CMS: Search for new physics impacting associated top t($\overline{\rm t}$)X
processes in multilepton (two same-sign leptons or three or more leptons)
final states, places limits (individually and simultaneously) on 26 WCs [9].

  * •

ATLAS: Measurement of t-channel production of single top, places a limit on
one WC (from the 2 light 2 heavy category of operators) [12].

Figure 3 illustrates how these analyses map into the space of recent detector-
level SMEFT analyses. The $x$-axis shows the categories of operators (as
introduced in Section 1, though here the charged lepton flavor violating WCs
are shown separately from the two-quark-two-lepton WCs), and the $y$-axis
shows the final states studied in the analyses (categorized by the number of
leptons included in the signal regions).

Figure 3: Visual representation of some of the recent detector-level SMEFT
analyses.

##  4 Summary

Analyses in ATLAS and CMS utilize the SMEFT framework to search for the off-
shell effects of heavy new physics. A variety of approaches are employed,
ranging from indirect reinterpretations to direct approaches in which
detector-level predictions are parameterized in terms of the SMEFT WCs. While
these analyses have yet to observe any significant discrepancy with the SM,
there remain many expansions and improvements that are being pursued;
improvements in analysis techniques, improvements in EFT modeling, the
collection of more data, and combinations among analyses (both within the top
quark sector and across multiple sectors) are leading to more comprehensive
results that aim to improve the understanding of the ranges of WC values that
are consistent with the observed data.

## References

  * [1] CMS Collaboration.  The CMS experiment at the CERN LHC, JINST 3 (2008) S08004, doi:10.1088/1748-0221/3/08/S08004.
  * [2] CMS Collaboration.  Development of the CMS detector for the CERN LHC Run 3, accepted by JINST (2023), arXiv:2309.05466.
  * [3] ATLAS Collaboration.  ATLAS Collaboration, The ATLAS Experiment at the CERN LHC, JINST 3 (2008) S08003, doi: 10.1088/1748-0221/3/08/S08003.
  * [4] Brian Henning, Xiaochuan Lu, Tom Melia, and Hitoshi Murayama.  2, 84, 30, 993, 560, 15456, 11962, 261485, …: Higher dimension operators in the SM EFT.  JHEP, 08, 2017.  [Erratum: JHEP 09, 019 (2019)].
  * [5] B. Grzadkowski, M. Iskrzynski, M. Misiak, and J. Rosiek.  Dimension-Six Terms in the Standard Model Lagrangian.  JHEP, 10, 2010.
  * [6] D. Barducci et al.  Interpreting top-quark LHC measurements in the standard-model effective field theory.  2018\.
  * [7] Olivier Mattelaer.  On the maximal use of Monte Carlo samples: re-weighting events at NLO accuracy.  Eur. Phys. J. C, 76(12), 2016.
  * [8] CMS Collaboration.  Search for new physics in top quark production with additional leptons in proton-proton collisions at $\sqrt{s}={13TeV}$ using effective field theory.  JHEP, 03:095, 2021.
  * [9] CMS Collaboration.  Search for physics beyond the standard model in top quark production with additional leptons in the context of effective field theory.  2023 https://cds.cern.ch/record/2866292/files/CMS-TOP-22-006-arXov.pdf.
  * [10] CMS Collaboration.  Search for charged lepton flavor violation in the top quark sector in trilepton final states with the CMS detector at $\sqrt{s}=13~{}\mathrm{TeV}$.  Technical report, CERN, Geneva, 2023.  https://cds.cern.ch/record/2851002.
  * [11] ATLAS Collaboration.  Search for charged-lepton-flavour violating $\mu\tau qt$ interactions in top-quark production and decay with the ATLAS detector at the LHC.  Technical report, CERN, Geneva, 2023.  All figures including auxiliary figures are available at https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/CONFNOTES/ATLAS-CONF-2023-001.
  * [12] ATLAS Collaboration.  Measurement of $t$-channel production of single top quarks and antiquarks in $pp$ collisions at 13 TeV using the full ATLAS Run 2 dataset.  Technical report, CERN, Geneva, 2023.  All figures including auxiliary figures are available at https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/CONFNOTES/ATLAS-CONF-2023-026.


# Nonabelian lattice theories: Consistent measures and strata

R.Vilela Mendes
CMAFCIO, Faculdade de Ciências, Universidade de Lisboa
<EMAIL_ADDRESS><EMAIL_ADDRESS>http://label2.ist.utl.pt/vilela/

###### Abstract

The role of consistent measures in the rigorous construction of nonabelian
lattice theories is analized. General conditions that measures must fulfill to
insure consistency, positivity and a mass gap are obtained. The impact of
nongeneric strata on the nature of the Hamiltonian lattice potential is also
discussed.

##  1 Introduction

There are two ways to look at lattice theories. In one of them, to
characterize a theory in the continuum one starts from a discretized
spacetime, with a finite lattice of points and then, by successively
decreasing the lattice spacing $a$ and increasing the total volume $V$ of the
lattice, one approaches the continuum theory. At each step, taking an
Euclidean point of view, the discrete approximation of theory rests defined by
the choice of a measure. There is no need for these intermediate measures to
have a well defined physical meaning. All that is required is that in the
limit $a\rightarrow 0$ and $V\rightarrow\infty$ one obtains the desired
continuum measure. Of course, because in the limit one is dealing with the
delicate problem of measures in infinite dimensional spacetimes, the choice of
the intermediate finite dimensional measures turns out to be important to
insure the existence of the limit measure. However this is only a requirement
of mathematical convenience.

An alternative way to look at lattice theories is to consider the lattice as
an observational scaffold of some unknown theory which may or may not be
defined in the continuum, either because it possesses some intrinsic validity
cutoff or even because the spacetime manifold changes its nature at small
distances [1] [2] [3]. In this latter case the definition of the lattice and
its associate measure is much less arbitrary, because at each step one
requires an exact description of the theory at the length scale of the
lattice.

In both instances, whether as a device to approach the continuum limit or a
scaffold to define the theory at all length scales, the mathematical notion of
consistent measure (to be defined in the next subsection) is the appropriate
tool. In the first case because it insures the existence of a limit measure
and in the second because it insures probabilistic consistency in the
description of the physical system at all length scales.

Another important point in the formulation of nonabelian gauge theories is the
fact that in these theories the configuration space as well as the phase space
are not manifolds but orbifolds with singular points corresponding to states
of higher symmetry [4] [5]. Therefore the characterization of their strata is
important in particular because they might correspond to a multiplicity of
condensate backgrounds or potential minima. A multiplicity of backgrounds
might lead to a multiplicity of solutions in the (Schwinger-Dyson) gap
equation for fermion mass generation [6] [7]. The nature of the gauge strata
in the nonabelian lattice theories will be analyzed. Because quantum wave
functions naturally explore different strata, the impact of nongeneric strata
on the backgrounds spectrum is related to the nature of the lattice potential
associated to the measure.

###  1.1 Projective limits and consistent measures as tools to approach the
continuum or to describe physical systems at all scales

The definition of consistent measures on a sequence of spaces (lattices in
this case) requires that the sequence be defined in an appropriate way. This
leads to the notion of:

Projective limit

Let $(I,\preceq)$ be a directed111Partially ordered and for any $i_{1},i_{2}$
in $I$, there is an $i_{d}$ in $I$ such that $i_{1}\preceq i_{d}$ and
$i_{2}\preceq i_{d}$ set and $\left\\{X_{i};i\in I\right\\}$ a family of
topological separable spaces indexed by $I$. For every pair $i,j\in I$ with
$i\preceq j$ there is a continuous map $\pi_{ij}:X_{j}\rightarrow X_{i}$ such
that

$\pi_{ij}\circ\pi_{jk}=\pi_{ik}\hskip 28.45274pt\text{for\hskip
28.45274pt}i\preceq j\preceq k$ (1)

This is called a projective family of spaces.

The projective limit of the family is the subset $X$ of elements of the
Cartesian product $\Pi_{i\in I}X_{i}$ such that its elements
$x=\left(x_{i}\right)$ satisfy

$x_{i}=\pi_{ij}x_{j}\hskip 28.45274pt\text{for\hskip 28.45274pt}i\preceq j$
(2)

Furthermore there are mappings $\pi_{i}:X\rightarrow X_{i}$ such that

$\pi_{i}=\pi_{ij}\circ\pi_{j}\hskip 28.45274pt\text{for\hskip
28.45274pt}i\preceq j$ (3)

The projective limit $X$ is denoted

$X=\underleftarrow{\lim}X_{i}$ (4)

Consistent measures

Of special interest is the case where each space $X_{i}$ is equipped with a
probability measure $\mu_{i}$. The family of measures
$\left\\{\mu_{i}\right\\}$ is called a consistent family if for every
measurable set $A$ in $X_{i}$

$\mu_{i}\left(A\right)=\mu_{j}\left(\pi_{ij}^{-1}\left(A\right)\right)\hskip
28.45274pt\text{for\hskip 28.45274pt}i\preceq j$ (5)

The measure on the projective limit

The next question is to know when, given a consistent family of measures,
there is also a measure $\mu$ on the projective limit $X$. The most general
requirement is probably the following [8] [9]:

For every $\varepsilon>0$ there is a compact subset $K\subset X$ such that

$\mu_{i}\left(X_{i}\setminus\pi_{i}\left(K\right)\right)\leq\varepsilon\hskip
28.45274pt\text{for every }i\in I$ (6)

and $\mu$ is denoted

$\mu=\underleftarrow{\lim}\mu_{i}$ (7)

The existence of an infinite-dimensional measure as a limit of consistent
finite-dimensional measures is a powerful concept in the sense that the limit
measure might even be of different nature from the finite-dimensional ones,
for example not being absolutely continuous in relation to the reference
measure of the finite-dimensional ones.

A classical illustration of these concepts is Kolmogorov’s construction of a
stochastic process $\left\\{X_{t}:t\in\mathbb{R}^{+}\right\\}$ from its
finite-dimensional distributions [10]:

Here $I=\mathbb{R}^{+}$, the $X_{i}$ spaces are discrete subsets
$\left\\{t_{1},t_{2},\cdots,t_{n}\right\\}$ of $\mathbb{R}^{+}$ ordered by
inclusion with measures $\mu_{t_{1},t_{2},\cdots,t_{n}}$ being the finite-
dimensional probability distributions. The mappings $\pi_{ij}$ are simply the
coordinate projections from the sets
$\left\\{t_{1},t_{2},\cdots,t_{n}\right\\}$ to its subsets. One important
point to notice in this construction is that the projective limit (a subset of
the Cartesian product) is not simply the continuum limit of the stochastic
process, instead it also contains, in a consistent manner, the description of
the process at all levels of observation.

In this paper this framework will be applied to the description of physical
systems extended in space-time, that is fields. Therefore it is natural to
consider the directed set to be a space-time lattice that, by subdivision of
its length elements, is successively refined and ordered by inclusion. Notice
that there is no need to identify the lattice with the physical system, the
lattice might simply be a observational scaffold of the physical system at
successively smaller scales. There is also no need to assume that the physical
systems are defined up to arbitrarily smaller scales. The directed set $I$ may
very well stop at some nonzero scale. The definition of the sequence of spaces
and their mappings is not particularly difficult. The important point is of
course to find the consistent measures relevant to each physical system.

###  1.2 Heat kernels

Heat kernels are wonderful mathematical objects and, as has been pointed out
before [11] [12], appear in all manner of places and disguises. But where does
the wonderfulness of heat kernels come from?

Given the operator $-O_{x}+\frac{\partial}{\partial t}$ where the operator $O$
is an elliptic operator, in particular a Laplacian in $\mathbb{R}^{n}$, on a
manifold or on a group manifold, the heat kernel is the solution of the
equation

$\left(-O_{x}+\frac{\partial}{\partial t}\right)K_{t}\left(x,y\right)=0$ (8)

What makes the heat kernel a powerful gadget is the convolution222Convolution
defined as $\int K_{t}\left(x-y\right)f\left(y\right)d\mu\left(y\right)$ or
$\int K_{t}\left(xy^{-1}\right)f\left(y\right)d\mu_{H}\left(y\right)$ for a
group approximating property

$\lim_{n\rightarrow\infty}K_{\frac{1}{n}}\ast f=f$ (9)

$f$ being a continuous and bounded function, as well as its representation as
a theta series

$K\left(t,x,y\right)=\sum_{k}\phi_{k}\left(x\right)\phi_{k}\left(y\right)e^{-\lambda_{k}t}$
(10)

for a manifold, with the $\phi_{k}^{\prime}s$ being a complete orthonormal set
of eigenvectors of the Laplacian and $\left\\{\lambda_{k}\right\\}$ the set of
their eigenvalues or a similar expression for a group manifold with $x$ and
$y$ group elements and the $\phi_{k}^{\prime}s$ representation characters.
Another important property, which naturally follows from the evolution
equation (8) is the convolution semigroup property

$K_{t_{1}}\ast K_{t_{2}}=K_{t_{1}+t_{2}}$ (11)

This property turns out to be of critical importance for the construction of
consistent measures. Notice also that a large family of convolution semigroups
of the heat kernel type do exist not only associated to Laplacians but also to
other more general Lévy kernels [13].

###  1.3 A projective lattice family for spacetime fields

In $n$-dimensional spacetime consider a set $\left(\mathcal{L},\preceq\right)$
of successively finer hypercubic lattices ordered by inclusion. Starting from
some initial hypercubic lattice $L_{0}$ with lattice spacing $a_{0}$, the
successive elements $L_{i}\in\mathcal{L}$ in this ordered set are obtained
both by regular subdivision of already existent plaquettes as well as by the
addition of new square or rectangular plaquettes. For the purpose of
definition of the mappings $\pi_{ij}:L_{j}\rightarrow L_{i}$ with $i\preceq j$
the elementary sets are the plaquettes, the plaquettes in $L_{j}$ being mapped
on the corresponding plaquettes on $L_{i}$ or on the empty set when they are
new plaquettes which do not correspond to a subdivision of the plaquettes in
$L_{i}$. (this is analogous to the coordinate projections in the Kolmogorov
construction). The property $\pi_{ij}\circ\pi_{jk}=\pi_{ik}$ for $i\preceq
j\preceq k$ being verified, a projective family of lattices is obtained, with
projective limit defined as the subset $L$ of elements of the Cartesian
product $\Pi_{i\in I}L_{i}$ such that its elements $L=\left(L_{i}\right)$
satisfy

$L_{i}=\pi_{ij}L_{j}\hskip 28.45274pt\text{for\hskip 28.45274pt}i\preceq j$
(12)

It should be pointed out that the family of lattices may either be infinite if
the lattice spacing $a_{n}\rightarrow 0$ and (or) the total volume
$V\rightarrow\infty$ or finite for finite volume and finite cutoff $a_{c}$.

###  1.4 Strata [14], [15], [16]

Let $G$ be a compact Lie group acting on a manifold $\mathcal{M}$. The action
of $G$ on $\mathcal{M}$ leads to a stratification of $\mathcal{M}$
corresponding to the classes of equivalent orbits $\left\\{g\mathcal{M};g\in
G\right\\}$. Let $S_{x}$ denote the isotropy (or stabilizer) group of
$x\in\mathcal{M}$

$S_{x}=\left\\{\gamma\in G:\gamma x=x\right\\}$ (13)

The stratum $\Sigma\left(x\right)$ of $x$ is the set of points having isotropy
groups $G-$conjugated to that of $x$

$\Sigma\left(x\right)=\left\\{y\in\mathcal{M}:\exists\gamma\in G:S_{y}=\gamma
S_{x}\gamma^{-1}\right\\}$ (14)

If $G$ is a symmetry group for a physical system with states in $\mathcal{M}$,
the configuration space of the system is the quotient space $\mathcal{M}/G$
and a stratum is the set of points in $\mathcal{M}/G$ that corresponds to
orbits with conjugated isotropy groups. The map that, to each orbit, assigns
the conjugacy class of its isotropy group is called the type. The set of
strata carries a partial ordering of types,
$\Sigma_{x}\subseteq\Sigma_{x^{\prime}}$ if there are representatives $S_{x}$
and $S_{x^{\prime}}$ of the isotropy groups such that $S_{x}\supseteq
S_{x^{\prime}}$. The maximal element in the ordering of types is the class of
the center $Z(G)$ of $G$ and the minimal one is the class of $G$ itself.

In gauge theories, one deals with the strata of the connections, the strata
being in one-to-one correspondence with the Howe subgroups of $G$, that is,
the subgroups that are centralizers of some subset in $G$. Given an holonomy
group $H_{\tau}$ associated to a connection $A$ of type $\tau$, the stratum of
$A$ is classified by the conjugacy class of the isotropy group $S_{\tau}$,
that is, the centralizer of $H_{\tau}$,

$S_{\tau}=Z\left(H_{\tau}\right)$ (15)

an important role being also played by the centralizer of the centralizer

$H_{\tau}^{\prime}=Z\left(Z\left(H_{\tau}\right)\right)$ (16)

that contains $H_{\tau}$ itself. If $H_{\tau}^{\prime}$ is a proper subgroup
of $G$, the connection $A$ reduces locally to the subbundle
$P_{\tau}=\left(\mathcal{M},H_{\tau}^{\prime}\right)$. Whether or not all the
strata types exist for the action of $G$ on $\mathcal{M}$ depends on the
structure of $\mathcal{M}$ itself. Global reduction depends on the topology of
$\mathcal{M}$, but it is always possible if $P=\left(\mathcal{M},G\right)$ is
a trivial bundle. $H_{\tau}^{\prime}$ is the structure group of the maximal
subbundle associated to type $\tau$. Therefore the types of strata are also in
correspondence with types of reductions of the connections to subbundles. If
$S_{\tau}$ is the center of $G$ the connection is called irreducible, all
others are called reducible. The stratum of the irreducible connections is
called the generic stratum. It is open and dense.

###  1.5 Summary of results

In Refs.[17] and [18], by using measures that satisfy a semigroup law, it was
possible to check the consistency of the measures at the one-plaquette level.
Here the construction is extended for plaquettes sharing common edges, which
by induction implies the possibility to construct pure gauge consistent
measures in finite or infinite lattices. This is the done in Section 2. The
positivity of the transfer matrix and the existence of a mass gap is also
established. Then Section 3 explores the construction of consistent measures
when there are also fermion matter fields in the lattice. Consistency of a
particular measure is checked, here however only at the one-plaquette level.
Finally in Section 4 one discusses the role of non-generic strata in lattice
theories.

##  2 Nonabelian pure gauge lattice theory: A review and some developments

Here I will draw and extend the results already obtained in Refs. [17] and
[18] concerning the construction of a consistent measure, with also some new
results concerning the positivity of the measures as well as an alternative
discussion of the mass gap.

###  2.1 Consistent interacting measures

A state of the nonabelian theory corresponds to the assignment of an element
$g$ of a nonabelian group $G$ to each edge of the lattice. As a reference
measure, the Haar measure of the group $G$ is also associated to each edge.
The set of independent Haar measures in the edges clearly establish a
consistent family of measures in the lattice and therefore there also exists a
reference Haar measure in the projective limit. In addition the theory is
invariant under a direct product group

$\mathcal{G}=\underset{vertices}{\Pi}G_{i}$ (17)

that is, an independent copy of $G$ at each vertex which acts on the $ij$ edge
group variable $g$ as

$g\rightarrow g_{i}gg_{j}^{-1}$ (18)

The next step is to find nontrivial measures, that is measures that couple the
group elements in different edges, but that satisfy the consistency
requirement along the projective lattice family. In particular one looks for
densities that multiply the reference Haar measure. In [17] and [18] this was
achieved by considering the construction of the projective family in such a
way that at each step only one plaquette is subdivided, together with the
adding of new plaquettes. However the procedure is much more general, with the
same consistency condition being obtained for the measure densities. If the
subdivided plaquettes have no edge in common the proof of the condition for
consistency of the measure applies without any modification. The other
possibility is when the plaquettes that are subdivided have a common edge.
Here I will show that the measure consistency condition is unaltered. This is
done by explicit calculation. Consider the two subdivided plaquettes in Fig.1

Figure 1: Subdivision of two contiguous plaquettes

The measure density associated to the plaquettes is a central function of the
product of the group elements around the plaquettes and denote by
$p\left(G_{1}G_{2}G_{3}^{-1}G_{4}^{-1}\right)$ and
$p\left(H_{1}H_{2}H_{3}^{-1}G_{2}^{-1}\right)$ the densities associated to the
large plaquettes and by $p^{\prime}$ the corresponding densities of the small
plaquettes. Furthermore assume that the densities, in addition to being
central functions, satisfy the following semigroup properties

$\displaystyle\int
p^{\prime}\left(G_{i}X\right)p^{\prime}\left(X^{-1}G_{j}\right)d\mu_{H}\left(X\right)$
$\displaystyle=$ $\displaystyle p^{\prime\prime}\left(G_{i}G_{j}\right)$
$\displaystyle\int
p^{\prime\prime}\left(G_{i}X\right)p^{\prime\prime}\left(X^{-1}G_{j}\right)d\mu_{H}\left(X\right)$
$\displaystyle=$ $\displaystyle p\left(G_{i}G_{j}\right)$ (19)

Then

$\displaystyle\int
p^{\prime}\left(g_{1}y_{1}y_{4}^{-1}g_{4}G_{4}^{-1}\right)p^{\prime}\left(g_{1}^{-1}G_{1}G_{2}g_{2}^{-1}y_{2}^{-1}y_{1}^{-1}\right)p^{\prime}\left(y_{4}y_{3}g_{3}G_{3}^{-1}g_{4}^{-1}\right)p^{\prime}\left(y_{2}g_{2}g_{3}^{-1}y_{3}^{-1}\right)$
(20) $\displaystyle\times
p^{\prime}\left(h_{1}z_{1}z_{4}^{-1}g_{2}G_{2}^{-1}\right)p^{\prime}\left(h_{1}^{-1}H_{1}H_{2}h_{2}^{-1}z_{2}^{-1}z_{1}^{-1}\right)p^{\prime}\left(z_{4}z_{3}h_{3}H_{3}^{-1}g_{2}^{-1}\right)p^{\prime}\left(z_{2}h_{2}h_{3}^{-1}z_{3}^{-1}\right)$
$\displaystyle\times\prod_{i=1,\cdots,4}d\mu_{H}\left(g_{i}\right)d\mu_{H}\left(y_{i}\right)d\mu_{H}\left(z_{i}\right)d\mu_{H}\left(G_{i}\right)\prod_{j=1,\cdots,3}d\mu_{H}\left(h_{j}\right)d\mu_{H}\left(H_{j}\right)$
$\displaystyle=$ $\displaystyle\int
p^{\prime}\left(X_{1}X_{4}^{-1}G_{4}^{-1}\right)p^{\prime}\left(X_{1}^{-1}G_{1}G_{2}X_{2}\right)p^{\prime}\left(X_{3}G_{3}^{-1}X_{4}\right)p^{\prime}\left(X_{2}^{-1}X_{3}^{-1}\right)$
$\displaystyle\times
p^{\prime}\left(Y_{1}Y_{4}^{-1}G_{2}^{-1}\right)p^{\prime}\left(H_{1}H_{2}Y_{2}Y_{1}^{-1}\right)p^{\prime}\left(Y_{3}H_{3}^{-1}Y_{4}\right)p^{\prime}\left(Y_{2}^{-1}Y_{3}^{-1}\right)$
$\displaystyle\times\prod_{i=1,\cdots,4}d\mu_{H}\left(X_{i}\right)d\mu_{H}\left(Y_{i}\right)d\mu_{H}\left(G_{i}\right)\prod_{j=1,\cdots,3}d\mu_{H}\left(H_{j}\right)$
$\displaystyle=$ $\displaystyle\int
p^{\prime\prime}\left(X_{4}^{-1}G_{4}^{-1}G_{1}G_{2}X_{2}\right)p^{\prime\prime}\left(X_{2}^{-1}G_{3}^{-1}X_{4}\right)p^{\prime\prime}\left(H_{1}H_{2}Y_{2}Y_{4}^{-1}G_{2}^{-1}\right)p^{\prime\prime}\left(Y_{2}^{-1}H_{3}^{-1}Y_{4}\right)$
$\displaystyle\times
d\mu_{H}\left(X_{1}\right)d\mu_{H}\left(X_{2}\right)d\mu_{H}\left(X_{4}\right)d\mu_{H}\left(Y_{2}\right)d\mu_{H}\left(Y_{4}\right)\prod_{i=1,\cdots,4}d\mu_{H}\left(G_{i}\right)\prod_{j=1,\cdots,3}d\mu_{H}\left(H_{j}\right)$
$\displaystyle=$ $\displaystyle\int
p\left(G_{3}^{-1}G_{4}^{-1}G_{1}G_{2}\right)p\left(H_{3}^{-1}G_{2}^{-1}H_{1}H_{2}\right)\prod_{i=1,\cdots,4}d\mu_{H}\left(G_{i}\right)\prod_{j=1,\cdots,3}d\mu_{H}\left(H_{j}\right)$

The first step uses centrality, invariance of the Haar measure and the change
of variables

$\displaystyle X_{1}$ $\displaystyle=$ $\displaystyle
g_{1}y_{1};X_{2}=g_{2}^{-1}y_{2}^{-1};X_{3}=y_{3}g_{3};X_{4}=g_{4}^{-1}y_{4}$
$\displaystyle Y_{1}$ $\displaystyle=$ $\displaystyle
h_{1}z_{1};Y_{2}=h_{2}^{-1}z_{2}^{-1};Y_{3}=z_{3}h_{3};Y_{4}=g_{2}^{-1}z_{4}$
(21)

and the second and the third the semigroup properties (19). This an explicit
check of the consistency condition (5).

A quite similar construction holds if the new plaquette that is subdivided
shares other edges with other already subdivided plaquettes. By induction,
with the reasoning here and in Ref.[17] it is established that:

###### Proposition 1

Let $\mathcal{L}$ be a (finite or infinite)projective lattice family of
compact nonabelian gauge theory with the product Haar measure as reference
measure. Then a sufficient condition for the existence of a consistent measure
in $\mathcal{L}$ is that the (plaquette) densities be central functions
satisfying the semigroup conditions (19).

Notice that in (19) the equality sign might be simply replaced by
”proportional to”, with the scaling factor being absorbed by the measure
normalization.

The choice of the semigroup defines the particular physical theory that is
implemented (or observed) in the lattice. In [17] and [18] it has been checked
that the heat kernel associated to the group $G$ , having the semigroup
properties (11) it also approximates at small lattice spacing the formal
measure associated to the Yang-Mills Lagrangian. It might therefore be used as
a rigorous definition of this as yet undefined theory. In this case then, the
measure density associated to each plaquette is the heat kernel
$K\left(U_{\square},\beta\right)$.

From the consistency condition one sees that as the plaquettes are subdivided
along the consistent family of lattices, one should replace the $\beta$
parameter in the heat kernel associated to each particular plaquette in the
following way:

$\displaystyle\beta$ $\displaystyle\rightarrow$
$\displaystyle\beta^{\prime}=\frac{\beta}{4}\hskip 28.45274pt\text{for
subdivision of one }a-\text{plaquette into }4\text{ }a/2-\text{plaquettes}$
$\displaystyle\beta$ $\displaystyle\rightarrow$
$\displaystyle\beta^{\prime\prime}=\frac{\beta}{2}\hskip 28.45274pt\text{for
subdivision of one }a-\text{plaquette into }2\text{ rectangular plaquettes}$

Hence one has $\beta\sim a^{2}$. To obtain the relation of the $\beta$
parameter to the usual coupling constant $g$ in lattice theories one should
compare the small $\beta$ limit of the heat kernel with, for example, the
Wilson action e${}^{-S_{W}}$

$S_{W}=-\frac{2}{g^{2}}\sum_{\square}\mathnormal{Re}Tr\left(U_{{}_{\square}}\right)$
(23)

This comparison was performed in [17] for $SU\left(2\right)$ and
$SU\left(3\right)$, the result being that heat kernel coefficient $\beta$
corresponds to the square of the coupling constant

$\beta\sim g^{2}$ (24)

Consistency of the measure is important not only to insure a correct matching
of the description of the physical system at all length scales, but also to
establish the existence of a continuous limit when $a\rightarrow 0$. Notice
however than in the $\beta\rightarrow 0$ limit the heat kernel (the plaquette
density) ceases to be a continuous function, meaning that the limit measure
exists but is not absolutely continuous in relation to the product Haar
measure. It is however easy to give a precise meaning to this limiting density
in the framework of a gauge projective triplet (see [17] Sect.III).

Here I have been assuming uniformity of the lattice spacing $a$ at each length
scale. However, sometimes it is useful, for example for the Hamiltonian
formulation, to have a different size for one of the axis, which one may
identify as time. Then one would have $\beta_{t}$ and $\beta_{s}$
corresponding respectively to the time and and space directions. When a
plaquette is subdivided only in time direction with the space direction kept
fixed, it is the second replacement in (LABEL:CM5a) that applies.

In addition to the Wilson measure, several modified lattice measures have been
proposed in the past, either to avoid lattice artifacts or to improve the
speed of convergence in numerical calculations. Most of these improved actions
do not implement measures that are consistent in the sense considered here. An
exception are the papers by Drouffe [19] and Menotti and Onofri [20] who also
propose the use of the heat kernel measure, although they mostly emphasize a
better convergence of the strong coupling expansion rather than its role as a
consistent measure in a projective family. The heat kernel measure has also
been used by Klimek and Kondracki in their construction of two-dimensional QCD
[21]. Another advantage of the heat kernel measure is the positivity of the
transfer matrix, as has already been pointed out in the past [22]. Because the
explicit form of the transfer matrix is important for the Hamiltonian
formulation, the transfer matrix and the proof of positivity will be briefly
sketched in the next subsection.

###  2.2 Positivity of the transfer matrix

Given a lattice Euclidean measure, a condition for this measure to correspond
to a physical theory, with an operator representation in Hilbert space, is the
positivity of the transfer matrix. The transfer matrix propagates the system
from one time to the next. In the Hilbert space formulation time translations
are generated by the Hamiltonian. Therefore once the positivity is proved, the
Hamiltonian may be obtained by taking the logarithm of the transfer matrix and
identifying the negative of the term linear in the lattice time-spacing as the
Hamiltonian.

In the $t=0$ hyperplane, the spatial group elements at each edge
$U_{i}\left(0,\overrightarrow{x}\right)$ are the wave function coordinates for
the Schrödinger picture, scalar products being defined with the Haar measure.
It is also useful to restrict (or project) the Hilbert space to gauge-
invariant functions. The transfer matrix $T$ is an operator defined from the
partition function $Z$ by

$Z=\lim_{N\rightarrow\infty}Tr\left[T^{N}\right]$ (25)

$N$ being the number of lattice spacings along the time direction. Denoting by
$U_{i}\left(na,\overrightarrow{x}\right)$ and
$U_{0}\left(na,\overrightarrow{x}\right)$ the space-like and time-like group
elements at time $na$, the partition function may be written

$Z=\prod_{n}\int\prod_{i,\overrightarrow{x}}d\mu_{H}\left(U_{i}\left(na,\overrightarrow{x}\right)\right)d\mu_{H}\left(U_{0}\left(na,\overrightarrow{x}\right)\right)\prod_{j<l}K_{jl}\left(na,\overrightarrow{x},\beta_{s}\right)\prod_{l^{\prime}}K_{0l^{\prime}}\left(na,\overrightarrow{x},\beta_{t}\right)$
(26)

$K_{\mu\nu}\left(na,\overrightarrow{x},\beta_{\bullet}\right)$ being the heat
kernel associated to the $\mu\nu-$plaquette at $\overrightarrow{x}$, where
different $\beta$ coefficients are associated to time and space directions.
From (26) it follows that, denoting by $U^{(n)}$ a generic space configuration
at time $na$, the matrix elements of the transfer matrix are

$\left\langle
U^{(n)}\left|T\right|U^{(n+1)}\right\rangle=\prod_{\overrightarrow{x}}\prod_{j<l}K_{jl}^{1/2}\left(na,\overrightarrow{x},\beta_{s}\right)\prod_{s}K_{0s}\left(na,\overrightarrow{x},\beta_{t}\right)\prod_{j^{\prime}<l^{\prime}}K_{j^{\prime}l^{\prime}}^{1/2}\left(\left(n+1\right)a,\overrightarrow{x},\beta_{s}\right)$
(27)

The next step is to show the positivity of this operator

$\left\langle\Psi\left(U\right)\left|T\right|\Psi\left(U\right)\right\rangle\geq
0$ (28)

$\Psi\left(U\right)$ being gauge invariant states. From (27) it is seen that
$T$ is the product of three operators

$ABA$

of which $A$ only involves elements at a fixed time and only $B$ connects
different time hyperplanes. Therefore with
$\Phi\left(U\right)=A\Psi\left(U\right)$ it suffices to prove positivity of
the $B$ operator,

$B=\prod_{\overrightarrow{x}}\prod_{s}K_{0s}\left(na,\overrightarrow{x},\beta_{t}\right)$
(29)

which follows from the positivity of the heat kernel of compact Lie groups.
Therefore, for a lattice theory associated to a compact Lie group G, the
transfer matrix obtained from the heat kernel measure is a positive operator.

An alternative proof of the positivity of the transfer matrix might involve
time-reflection positivity as in [23], by choosing the $t=0$ hyperplane at mid
distance between two lattice space hyperplanes and a gauge where all the edges
along the time direction are set to the group identity. Then one sees that the
time-positive and time-negative parts of the $A$ operator are symmetric and
that in the $B$ operator the only component that involves time-positive and
time-negative edges does so in a symmetric way. This latter proof would
however be more general, because it also applies to any positive linear
combination of traces of plaquette operators, not only to the heat kernel.

From the logarithm of the positive transfer matrix, an Hamiltonian may be
obtained as the negative of the term linear in the lattice spacing. In
particular the potential term is

$V=-\frac{1}{2\beta_{t}}\sum_{j<l}\ln
K_{jl}\left(\overrightarrow{x},\beta_{s}\right)$ (30)

$K_{jl}\left(\overrightarrow{x},\beta_{s}\right)$ denoting the heat kernel
associated to the $jl-$spatial plaquette at $\overrightarrow{x}$.

##  3 Hamiltonian and the mass gap

Here one considers an Hamiltonian formulation of the lattice theory, letting
the lattice size along the time direction tend to zero, $a_{t}\rightarrow 0$,
and the one along the space directions $a_{s}$ kept fixed. Therefore also
$\beta_{t}\rightarrow 0$ in the consistent measure. From the previous analysis
[17] one already knows that for small $\beta$ one obtains the same limit as
for the Wilson action. Therefore one may use for the kinetic term the same
term as in Kogut-Susskind Hamiltonian [24] and for the potential term the
function $V$ in (30)

$H=c\sum\pi_{ij}^{\alpha}\pi_{ij}^{\alpha}+V$ (31)

where $c$ is a positive constant related to the coupling constants or, in the
case of an Hamiltonian constructed from the consistent measure, a function of
$\beta_{t}$ and $\beta_{s}$. The operators $\pi_{ij}^{\alpha}$ act on the
group element $U_{ij}$ of the $ij$ spatial edge by

$\left[\pi_{ij}^{\alpha},U_{ij}\right]=-\xi^{\alpha}U_{ij}$ (32)

$\xi^{\alpha}$ being an element of the Lie algebra and $\pi_{ij}^{\alpha}$ may
be written as $i\frac{\partial}{\partial\theta_{ij}^{\alpha}}$ if
$U_{ij}=e^{i\theta_{ij}^{\alpha}\xi^{\alpha}}$

To estimate a bound on the lowest nontrivial eigenvalue $\lambda_{1}$ one uses
the bound on the Rayleigh quotient [25]

$\lambda_{1}=\inf_{\psi\left(U\right)\neq
0}\frac{\int\left(\left|\pi\psi\left(U\right)\right|^{2}+V\left|\psi\left(U\right)\right|^{2}\right)d\mu\left(U\right)}{\int\left|\psi\left(U\right)\right|^{2}d\mu\left(U\right)}$
(33)

Let $\psi\left(U\right)$ be normalized,
$\left(\psi\left(U\right),\psi\left(U\right)\right)=1$. Then with $K$ an heat
kernel

$\left(\psi\left(U\right),K\psi\left(U\right)\right)\leq\left(\psi\left(U\right),\psi\left(U\right)\right)^{1/2}\left(K\psi\left(U\right),K\psi\left(U\right)\right)^{1/2}\leq
1$ (34)

the second inequality following from the contractive semigroup property of the
heat kernel. Then

$\left(\psi\left(U\right),\ln K\psi\left(U\right)\right)\leq 0$

and the potential $V$ in equation (30) is $\geq 0$. Therefore to prove that
$\lambda_{1}>0$ it suffices to analyze the $V=0$ case. For the kinetic part of
the Hamiltonian the lowest gauge invariant state corresponds to one excited
plaquette. In the Hamiltonian formulation one chooses a gauge where all the
edges along the time direction are set to the identity of the group. This is
not a complete gauge, yet remaining to fix the gauge in the space slices.
There one chooses a vertex $x_{0}$ of the plaquette to be excited and uses
this point to establish a maximal tree gauge on the space-like edges. In this
gauge the plaquettes nearest to the $x_{0}$ vertex have only one link that is
not set to the identity. Therefore to excite the plaquette is the same as to
excite this link. From the parametrization of the group elements

$U=e^{i\theta^{\alpha}\xi^{\alpha}}$ (35)

where $\theta^{\alpha}\in\left[0,2\pi\right)$ or $\left[0,\pi\right)$ for a
compact group, it then follows that regularity of the compact boundary
conditions implies

$\lambda_{1}>0$

In conclusion:

###### Proposition 2

At any spatial lattice spacing, the nonabelian lattice theory with heat kernel
measure has a positive mass gap.

Compactness of the group and the contractive nature of the heat kernel
semigroup are the main ingredients leading to this result.

In [18] a similar conclusion was reached using Wentzell-Freitlin estimates
associated to the ground state stochastic process. However, because they rely
on some hypotheses on the construction of the ground state, I think that the
derivation above is simpler and more satisfactory.

##  4 Nonabelian lattice gauge theory with matter fields

In addition to the nonabelian gauge fields, physical theories also contain
matter fields which conventionally are defined to live on the vertices of the
lattice. For pure gauge theories the natural gauge invariant quantity is the
plaquette product of group elements. With fermions however the basic element
is
$\overline{\psi}\gamma_{\mu}\left(\nabla_{\mu}+\nabla_{\mu}^{\ast}\right)\psi$
with $\nabla_{\mu}$ and $\nabla_{\mu}^{\ast}$ denoting the forward and
backward covariant difference operators along the $\mu$ coordinate,

$\displaystyle\nabla_{\mu}\psi\left(x\right)$ $\displaystyle=$
$\displaystyle\frac{1}{a}\left(U_{\mu}\left(x\right)\psi\left(x+a\widehat{\mu}\right)-\psi\left(x\right)\right)$
$\displaystyle\nabla_{\mu}^{\ast}\psi\left(x\right)$ $\displaystyle=$
$\displaystyle\frac{1}{a}\left(\psi\left(x\right)-U_{\mu}\left(x-a\widehat{\mu}\right)^{{\dagger}}\psi\left(x-a\widehat{\mu}\right)\right)$
(36)

Therefore a fermion gauge measure density
$\nu\left(U,\overline{\psi},\psi\right)$ might be a function

$\nu\left(U,\overline{\psi},\psi\right)=f\left(\left\\{\overline{\psi}\gamma_{\mu}\left(\nabla_{\mu}+\nabla_{\mu}^{\ast}\right)\psi\right\\}\right)$
(37)

$\left\\{\overline{\psi}\gamma_{\mu}\left(\nabla_{\mu}+\nabla_{\mu}^{\ast}\right)\psi\right\\}$
denoting the set of all fermion edge strings.

In these strings the fermions are entities defined in a product space

$\displaystyle\overline{\psi}$ $\displaystyle=$
$\displaystyle\overline{\chi}\otimes\overline{\phi}\in
V_{\overline{\chi}}\otimes V_{\overline{\phi}}$ $\displaystyle\psi$
$\displaystyle=$ $\displaystyle\chi\otimes\phi\in V_{\chi}\otimes V_{\phi}$
(38)

$V_{\overline{\chi}}$ and $V_{\chi}$ being Grassman spaces and
$V_{\overline{\phi}}$ and $V_{\phi}$ representation spaces of the gauge group.
The density in (37) is to be multiplied by

$\prod_{i\in\text{all sites}}d\overline{\psi}_{i}d\psi_{i}\prod_{j\in\text{all
edges}}dU_{j}$ (39)

Formally expanding (37) and by Berezin integration over the Grassman
variables$\phi$

$\prod_{j\in\text{all edges}}dU_{j}\sum_{P}(-1)^{P}\prod_{i=1\cdots
N}\left.\partial_{P(\overline{\psi}_{i}}\partial_{\psi_{i})}f\left(\left\\{\overline{\psi}\gamma_{\mu}\left(\nabla_{\mu}+\nabla_{\mu}^{\ast}\right)\psi\right\\}\right)\right|_{\overline{\psi}=\psi=0}$
(40)

$P$ being a permutation over all sites. The argument

$\mu=\sum_{P}(-1)^{P}\prod_{i=1\cdots
N}\left.\partial_{P(\overline{\psi}_{i}}\partial_{\psi_{i})}f\left(\left\\{\overline{\psi}\gamma_{\mu}\left(\nabla_{\mu}+\nabla_{\mu}^{\ast}\right)\psi\right\\}\right)\right|_{\overline{\psi}=\psi=0}$
(41)

in (38) is a function only of the group elements in the sites and the edges.
To obtain a consistent measure try the following ansatz

$\mu=\mu\left(\prod_{plaquettes}\overline{\phi}_{i}U_{\mu}\phi_{i+\mu}\overline{\phi}_{i+\mu}U_{\nu}\phi_{i+\mu+\nu}\left(\overline{\phi}_{i+\nu}U_{\mu}\phi_{i+\nu+\mu}\right)^{{\dagger}}\left(\overline{\phi}_{i}U_{\nu}\phi_{i+\nu}\right)^{{\dagger}}\right)$
(42)

that is, a product of plaquette-strings. Now the consistency and the choice of
the function $\mu,$may be verified as before, here only at the single
plaquette level. Consider the four group strings around a plaquette in Fig.2A.
Subdividing this plaquette as in Fig.2B, and denoting by $\mu^{\prime}$ the
new measure associated to the subdivided plaquette, the functional integral
becomes

Figure 2: Subdivision of a plaquette with fermion fields

$\displaystyle I$ $\displaystyle=$
$\displaystyle\int\mu^{\prime}\left(\overline{\phi}_{1}g_{1}\theta_{1}\overline{\theta}_{1}y_{1}\theta_{5}\left(\overline{\theta}_{4}y_{4}\theta_{5}\right)^{-1}\left(\overline{\phi}_{1}G_{4}g_{4}^{-1}\theta_{4}\right)^{-1}\right)\mu^{\prime}\left(\overline{\theta}_{1}g_{1}^{-1}G_{1}\phi_{2}\overline{\phi}_{2}g_{2}\theta_{2}\left(\overline{\theta}_{5}y_{2}\theta_{2}\right)^{-1}\left(\overline{\theta}_{1}y_{1}\theta_{5}\right)^{-1}\right)$
(43)
$\displaystyle\times\mu^{\prime}\left(\overline{\theta}_{4}y_{4}\theta_{5}\overline{\theta}_{5}y_{3}\theta_{3}\left(\overline{\phi}_{4}G_{3}g_{3}^{-1}\theta_{3}\right)^{-1}\left(\overline{\theta}_{4}g_{4}\phi_{4}\right)^{-1}\right)\mu^{\prime}\left(\overline{\theta}_{5}y_{2}\theta_{2}\overline{\theta}_{2}g_{2}^{-1}G_{2}\phi_{3}\left(\overline{\theta}_{3}g_{3}\phi_{3}\right)^{-1}\left(\overline{\theta}_{5}y_{3}\theta_{3}\right)^{-1}\right)$
$\displaystyle\times\prod_{i=1}^{5}d\mu_{H}\left(\overline{\theta}_{i}\right)d\mu_{H}\left(\theta_{i}\right)\prod_{j=1}^{4}d\mu_{H}\left(g_{j}\right)d\mu_{H}\left(y_{j}\right)$

Let

$g_{1}\theta_{1}\overline{\theta}_{1}y_{1}\theta_{5}=Y_{1};\;g_{2}\theta_{2}\overline{\theta}_{2}y_{2}^{-1}\theta_{5}=Y_{2};\;\overline{\theta}_{5}y_{3}\theta_{3}\overline{\theta}_{3}g_{3}=Y_{3}^{-1};\;\overline{\theta}_{5}y_{4}^{-1}\theta_{4}\overline{\theta}_{4}g_{4}=Y_{4}^{-1}$
(44)

Then using centrality, invariance of the Haar measure and integrating over the
remaining variables

$\displaystyle I$ $\displaystyle=$
$\displaystyle\int\mu^{\prime}\left(\overline{\phi}_{1}Y_{1}Y_{4}^{-1}G_{4}^{-1}\phi_{1}\right)\mu^{\prime}\left(Y_{1}^{-1}G_{1}\phi_{2}\overline{\phi}_{2}Y_{2}\right)\mu^{\prime}\left(Y_{4}Y_{3}^{-1}G_{3}^{-1}\phi_{4}\overline{\phi}_{4}\right)\mu^{\prime}\left(Y_{2}^{-1}G_{2}\phi_{3}\overline{\phi}_{3}Y_{3}\right)$
$\displaystyle\times\prod_{i}^{4}d\mu_{H}\left(Y_{i}\right)$

If there is a family of measures $\mu^{\prime}$, $\mu^{\prime\prime}$ and
$\mu^{\prime\prime\prime}$ satisfying the semigroup property as discussed
before (Eq.(19))

$\displaystyle I$ $\displaystyle=$
$\displaystyle\int\mu^{\prime\prime}\left(Y_{4}^{-1}G_{4}^{-1}\phi_{1}\overline{\phi}_{1}G_{1}\phi_{2}\overline{\phi}_{2}Y_{2}\right)\mu^{\prime\prime}\left(G_{3}^{-1}\phi_{4}\overline{\phi}_{4}Y_{4}Y_{2}^{-1}G_{2}\phi_{3}\overline{\phi}_{3}\right)d\mu_{H}\left(Y_{2}\right)d\mu_{H}\left(Y_{4}\right)$
(45) $\displaystyle=$
$\displaystyle\int\mu^{\prime\prime}\left(G_{4}^{-1}\phi_{1}\overline{\phi}_{1}G_{1}\phi_{2}\overline{\phi}_{2}Y_{2}Y_{4}^{-1}\right)\mu^{\prime\prime}\left(Y_{4}Y_{2}^{-1}G_{2}\phi_{3}\overline{\phi}_{3}G_{3}^{-1}\phi_{4}\overline{\phi}_{4}\right)d\mu_{H}\left(Y_{2}\right)d\mu_{H}\left(Y_{4}\right)$
$\displaystyle=$
$\displaystyle\mu\left(G_{4}^{-1}\phi_{1}\overline{\phi}_{1}G_{1}\phi_{2}\overline{\phi}_{2}G_{2}\phi_{3}\overline{\phi}_{3}G_{3}^{-1}\phi_{4}\overline{\phi}_{4}\right)$

Hence the consistency condition on the measures may also be fulfilled with
fermions on the lattice, by using central measures with the semigroup
property. The consistency condition was verified after integration over the
Grassman variables. The full measure in (41) is implicitly defined as the
measure that by integration over the Grassman part of the fermion variables
leads to (42).

So far, in the derivation leading to the consistency condition in Eq.(45),
$G_{i},g_{i},y_{i},\phi_{i},\overline{\phi}_{i},\theta_{i},\overline{\theta}_{i}$
have been considered arbitrary group elements. However specification of a
physical theory requires the choice of particular representations for these
group elements. In $SU\left(n\right)$ non Abelian gauge theories is usual to
choose the defining $n^{2}-1$ dimensional representation for the group
elements associated to the edges of the lattice. Then to the group elements
$\phi_{i},\overline{\phi}_{i},\theta_{i},\overline{\theta}_{i}$ at the
vertices one may associate the fundamental $n$ and $\overline{n}$
representations. These elements always appear in the measure in the
$\phi_{i}\overline{\phi}_{i}$ and $\theta_{i}\overline{\theta}_{i}$
combinations which will decompose into a scalar and a $n^{2}-1$ dimensional
representation. Hence the same measure may contain both the pure gauge part
and the matter fields.

##  5 Strata and the lattice potential

The measures, that have been discussed before, provide the probability of each
particular group configuration in the lattice. In particular they provide the
integration measure that controls the fluctuations around the ground state.
Let us parametrize the group elements in the usual way

$U_{\mu}\left(x\right)=e^{\int_{x}^{x+\widehat{\mu}}ds\tau_{a}A_{a}^{\mu}\left(s\right)}$
(46)

Of particular interest are ground state configurations corresponding to
condensates. Invariance of the measure implies that the fluctuations are
around zero mean,

$\left\langle A_{a}^{\mu}\right\rangle=\left\langle
F_{a}^{\mu\nu}\right\rangle=0$ (47)

$F_{a}^{\mu\nu}$ being the plaquette field, while quantities like
$\left\langle A_{a}^{\mu}A_{a\mu}\right\rangle$ and $\left\langle
F_{a}^{\mu\nu}F_{a\mu\nu}\right\rangle$ may be different from zero.

The multiplicity of backgrounds may be related to the strata of the gauge
group operating in the lattice. The configuration space of a pure gauge
lattice $L_{i}$ (associated to the group $G$) in the projective family
$\mathcal{L}$ is

$\mathcal{M=}G^{\otimes N}$ (48)

$N$ being the number of edges in the lattice. On $\mathcal{M}$ acts the gauge
group

$\mathcal{G=}G^{\otimes N_{0}}$ (49)

$N_{0}$ being the number of vertices in the lattice. The action of the group
may be reduced from $G^{\otimes N_{0}}$ to $G$ by choosing a maximal tree
gauge [27], connecting a particular vertex $x_{0}$ to all vertices in the
lattice and assigning to the identity all group elements in the edges along
the tree. The remaining edges not in the tree have both vertices group-
identified (more precisely, parallel transported) to the point $x_{0}$.
Therefore the reduced system becomes

$\mathcal{M}^{\prime}\mathcal{=}G^{\otimes M}$ (50)

where in general $M<<N$ and in $\mathcal{M}^{\prime}$ acts the $G$ group by
the conjugate action

$\left(g_{1},g_{2},\cdots,g_{M}\right)\rightarrow\left(gg_{1}g^{-1},gg_{2}g^{-1},\cdots,gg_{M}g^{-1}\right)$
(51)

$g\in G$. $\mathcal{M}^{\prime}$ is effectively a set of $M$ $\ x_{0}-$ based
loops acted upon by $G$

The strata of the lattice configuration space are the strata of the action of
$G$ on $G^{M}$. They are at most as many as the number of Howe subgroups of
$G$ and for $G=SU\left(n\right)$ they were fully characterized by the authors
of refs. [27] [28].

Let, for example, $G=SU\left(3\right)$. Let each orbit be characterized by the
pair $\left(\lambda_{1},\lambda_{2}\right)$ of eigenvalues of the group
elements (the two independent diagonal elements of the maximal torus) and
$M\geq 2$. If the set $\left(g_{1},g_{2},\cdots,g_{M}\right)$ has no common
eigenspace the stabilizer subgroup is the center of $SU\left(3\right)$ and
this is the generic stratum. If there is one common eigenspace the stabilizer
is $U\left(1\right)$. If there are three different common one-dimensional
eigenspaces the stabilizer is $U\left(1\right)\times U\left(1\right)$ and if
there is one common two-dimensional eigenspace the stabilizer is
$U\left(2\right)$. Finally if there is a common three-dimensional eigenspace,
meaning that all elements in $\left(g_{1},g_{2},\cdots,g_{M}\right)$ are the
identity the stabilizer is $SU\left(3\right)$. Hence there are $5$ different
strata.

Classical dynamics in the lattice takes place in the phase space, the
cotangent bundle $T\mathcal{M}^{\prime\ast}=TG^{\otimes M\ast}$. If the
initial condition lies in an orbit of a particular stratum, the classical
system remains there for all its undisturbed classical evolution. Therefore
for classical dynamics it makes sense to consider and classify dynamics in the
different strata although, for random initial conditions, the generic strata
will be almost surely chosen.

However for quantum mechanics, the situation is different because the wave
function will surely explore different strata and, the generic stratum having
full measure, it would seem that it is only the generic stratum that matters.
Nevertheless, some authors [4] [5] have argued that in systems with gauge
symmetry, where the configuration space is a orbifold with singularities
corresponding to points of non-generic higher symmetry, one may find
concentrations of the wave functions near the non-generic strata. This may
depend on the form of the Hamiltonian used in the Schrödinger equation.

The role of the non-generic strata in the Hamiltonian formulation of the
lattice theory may be analyzed with the Hamiltonian (31) and the potential
(30). Let for definiteness $G=SU\left(3\right)$. The heat kernel that enters
the potential (30) is a function of the two angles
$\left(\theta_{1},\theta_{2}\right)$ in the maximal torus
$diag\left\\{\exp\left(i\theta_{1}\right),\exp\left(i\theta_{2}\right),\exp\left(-i\left(\theta_{1}+\theta_{2}\right)\right)\right\\}$
of the group element associated to each plaquette. When the lattice is refined
to small lattice spacing the heat kernel becomes [17]

$K\left(\beta_{s}\right)\underset{\beta_{s}\rightarrow
0}{\longrightarrow}\exp\left\\{-\frac{1}{2\beta_{s}}\left(\theta_{1}^{2}+\theta_{2}^{2}+\theta_{1}\theta_{2}\right)\right\\}$
(52)

that is, it contributes to the potential an harmonic term
$\frac{1}{4\beta_{t}\beta_{s}}\left(\theta_{1}^{2}+\theta_{2}^{2}+\theta_{1}\theta_{2}\right)$.

Consider now a spatial lattice of dimension $N\times N\times N$. A maximal
tree gauge starting for an upper corner is essentially equivalent to an axial
gauge $A_{3}=0$ $\left(U_{3}=1\right)$. There are then $2\times N^{3}$
independent edge group elements, and $3\times N^{3}$ plaquette group elements,
not all independent. Of these there are $2\times N^{3}$ plaquettes along the
$3$ direction with two non-trivial edges each and $N^{3}$ plaquettes along the
$1,2$ planes with four nontrivial edges.

After the gauge fixing each edge group element is still acted upon by the
conjugate action as in (51) and the classification of the strata is as
described before. Consider now the non-generic stratum with stabilizer
$U\left(1\right)$. With a common eigenvalue in all independent links one
obtains $\theta_{1}=0$ for all the plaquettes. Therefore for each fixed
$\left\\{\theta_{2}^{(ij)}\right\\}$ configuration one obtains a minimum of
the potential on this stratum. Likewise, in the strata with $SU\left(2\right)$
stabilizer one obtains $\theta_{1}=\theta_{2}=0$ for the plaquettes, another
potential minimum. Hence, in the potential of this high dimensional
Schrödinger problem there are multiple different local minima associated to
the non-generic strata. It is intriguing to realize that this multiple minima
situation is the one that might lead to a fast growing point spectrum for the
gauge backgrounds, as shown in [29] using an inverse scattering argument.

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