On Yang--Mills Theories with Chiral Matter at Strong Coupling Page: 3 of 46
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1 Introduction
Recent advances in strongly coupled Yang-Mills theories, both analytic and
numerical, are quite spectacular. At the same time, our knowledge of (ano-
maly free) Yang-Mills theories with chiral matter remains at a rudimentary
level. All existing methods of exploration fail in the chiral case. Lattice tech-
niques at the moment do not lead to a manifestly gauge invariant formulation
of the non-Abelian chiral gauge theories. (For progress in this direction, see
[1, 2, 34, 4] and references therein.) Even if a satisfactory gauge invariant lat-
tice formulation was constructed, the well known problem of complex fermion
determinants would render numerical simulations impractical. Analytic ar-
senals of theorists dealing with chiral matter at strong coupling are poor, to
put it mildly. The 't Hooft matching [5] is a useful tool, generally speaking.
However, since we will mostly focus on ZK-orbifold theories with no contin-
uous global axial symmetries, the 't Hooft matching [5] is not applicable in
this case. AdS/QCD modeling and string theory techniques (so far) do not
provide insights into the strong coupling chiral dynamics either.
In essence, the only fact which can be considered established is the pertur-
bative equivalence of the ZK orbifold theories at large N to supersymmetric
SU(KN) Yang-Mills theory, with an appropriate resealing of the gauge cou-
plings [6]. This equivalence does not extend beyond perturbation theory in
the chiral case (K 3) due to spontaneous breaking of the chiral symme-
try in the parent theory [7] (which is used in the projection).2 Thus, the
above planar equivalence tells us nothing about such basic features of the
theory as the vacuum degeneracy/nondegeneracy, patterns of the discrete
chiral symmetry breaking and so on, let alone the spectrum of composite
colorless hadrons.
In this paper we discuss dynamics of non-Abelian gauge theories with
chiral fermion sectors at strong coupling applying and developing ideas sug-
gested in [10]. Our primary target is the so-called ZK orbifold theories at
K > 3. These theories are obtained from supersymmetric SU(KN) Yang-
Mills theory by ZK orbifold projection. The gauge group is [SU(N)]K, i.e
they contain K gluon sectors which are connected to each other only through
bifundamental Weyl fermions of the type <j - (1, ... , Ni, Ny1, ... 1) which
2Discussion of the non-perturbative fate of planar equivalence for ZK orbifolds (K > 3)
was initiated in Refs. [8, 9].2
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Shifman, M.; /Minnesota U., Theor. Phys. Inst. /Saclay, SPhT; Unsal, Mithat & /SLAC /Stanford U., Phys. Dept. On Yang--Mills Theories with Chiral Matter at Strong Coupling, article, August 20, 2008; [Menlo Park, California]. (https://digital.library.unt.edu/ark:/67531/metadc902228/m1/3/: accessed April 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.