#### Alladi Ramakrishnan Hall

#### (Pre-synopsis seminar) Study of $Z_N$ Symmetry in SU(N) Gauge Theories in The Presence of Matter Fields.

#### Sabiar Shaikh

##### IMSc

*Theoretical studies in Quantum Chromodynamics (QCD) show that at high*

temperature hadrons melt into quark-gluon plasma (QGP) via a transition known as

confinement-deconfinement (CD) transition. In pure SU(N) gauge theories the CD

transition is described by the Polyakov loop and the $Z_N$ (center of SU(N))

symmetry. This symmetry is spontaneously broken in the deconfined phase. When

matter fields in the fundamental representation are included, the Euclidean action

breaks this symmetry explicitly. The non-perturbative studies have shown in

SU(2)-Higgs theory (scalar QCD) the partition function averages, in the continuum

limit (for large number of temporal sites ($N_\tau$)), exhibit the corresponding

$Z_2$ symmetry in parts of the phase diagram.

In this talk, we will discuss our work on the $Z_N$ symmetry in SU(N)-Higgs

theories. We present results of non-perturbative studies for $N=3$ which suggest

that, the $Z_3$ symmetry is effectively realized,in the Higgs symmetric phase. The

nature of the CD transition changes with $N_\tau$, becomes first order in the

continuum limit (as if in pure SU(N) theory). In the same limit, in the deconfined

phase the $Z_N$ states become degenerate.

To understand the realization of $Z_N$ symmetry, next we will discuss our study on

one dimensional gauged chain of matter fields. As in SU(N) gauge theories, the

action breaks $Z_N$ symmetry explicitly. The analytic calculation of the partition

function shows that the $Z_N$ symmetry is realized in the continuum limit for bosonic

matter fields. To further probe the reason behind the realisation of $Z_N$ symmetry

we consider a simple model of $Z_2$-Higgs theory in both 3+1 and 0+1 dimensions

where it is shown that the density of states (DoS) exhibits $Z_2$ symmetry. As DoS,

so entropy, dominate the thermodynamics (over the Boltzmann factor) in the Higgs

symmetric phase, the $Z_2$ symmetry is realized.

Done