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@MASTERSTHESIS{Bradley:1037848,
author = {Bradley, Grant},
title = {{S}tudy of {E}xotic {H}adrons with {L}attice {QCD} and
{D}istillation},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
type = {Masterarbeit},
reportid = {FZJ-2025-00991},
pages = {51 p.},
year = {2024},
note = {Masterarbeit, Rheinische Friedrich-Wilhelms-Universität
Bonn, 2024},
abstract = {Quantum Chromodynamics is the SU(3) gauge field theory
governing the strong force, dictating the dynamics of quarks
and gluons. One can have two or three quark states, mesons
and baryons, respectively, bound together by gluons. QCD
phenomena are informed by two main quark correlations:
confinement, whereby color forces allow $q,\bar{q}$ to be
correlated only into color singlets, and chiral symmetry
breaking. The regime in which the coupling constant of the
theory $\alpha_s$ is still small, perturbation theory can
reliably be used. However, to probe the hadronic world ($\mu
\leq 1 GeV$), perturbation theory ceases to be a useful
tool. By discretizing Minkowski spacetime into Euclidean
spacetime, we can calculate the hadron spectrum and other
quantities such as matrix elements. This non-perturbative
approach is called Lattice QCD (henceforth LQCD), the only
systematically improvable and regularization scheme
independent method to probe the strong force. The property
of quark confinement can be realized in the lattice version
of QCD. For more information on the underlying theory,
consult~\cite{10.5555/3029317},\cite{gupta1998introductionlatticeqcd},\cite{Gattringer2009QuantumCO},\cite{Griffiths:1987tj},\cite{Cheng1984GaugeTO}.The
quark model was proposed by Gell-Mann and
Zweig~\cite{Gell-Mann:1964ewy}\cite{Zweig:1964ruk} as a
refined classification scheme for hadrons in which all
hadrons are built out of spin-$\frac{1}{2}$ quarks that
transform according to the fundamental representation of
SU(3). The basic tenants are: There are three flavors of
quarks ($u,d,s$) in the fundamental representation
\textbf{3}, with their antipartners in the \textbf{3*},
mesons are $q\bar{q}$ bound states which reside in SU(3)
singlets and octets, and that baryons are $qqq$ bound
states. This original ``simple" quark model contained
paradoxes that were resolved by Greenberg, Nambu and
others~\cite{Han:1965pf} with the introduction of a hidden
quark degree of freedom, color; Each quark comes in three
differnt colors (red,green,blue) which are packaged in a
triplet under a color SU(3) group. This gives rise to the
postulate that only color singlet states are physically
observable states. Thus, the theory of QCD hinges upon the
exchange of gluons between colored quarks.Hadronic states
that do not fit into the traditional quark model have been
coined XYZ states \cite{Brambilla:2019esw} or exotics, such
as tetraquark and pentaquark states\cite{Cheung_2017}.
Multiquark states are posited to be combinations of
conventional mesons or diquark-antidiquark pairs, which
imply colored building blocks. QCD should predict whether
tetraquark states exist, thus, we can leverage the lattice
to construct color-flavor-spatial-spin structures resembling
that of compact tetraquarks, such as the $T_{cc}^+(3875)$,
which is the focus of this project. The lack of a consensus
as to how the experimental data is to be interpreted, namely
the make-up of the aforementioned multiquark building
blocks, has spawned a flurry of research efforts in the
Lattice QCD community~\cite{Cheung_2017}. The inner workings
of QCD and the nature of unstable hadronic resonances, which
comprise most of the observed spectrum to date, will be
further illuminated once the grand question of
interpretation is resolved. Very few exotic hadrons have
been studied on the lattice and thus lack a rigorous
theoretical basis.The class of exotics that we aim to
explore are doubly charmed tetraquarks in isospin channels
$I=0,1$. Namely, the tetraquarks with flavor content $\bar
c\bar s ud$, $c\bar s u\bar d$, $cc\bar u\bar d$ and $c\bar
c u\bar d$; The third flavor profile is known as
$T_{cc}^+(3875)$ \cite{LHCb:2021vvq}. The flavor content is
based on the decay channel $D^0D^0\pi^+$ and has isospin 0.
The experimental data shows that this is the longest-lived
exotic hadron. This exotic has mass of roughly 3875 MeV and
manifests as a peak in the mass spectrum of $D^0D^0\pi^+$
mesons. We will use meson-meson interpolators to explore
these isospin quantum numbers as opposed to
diquark-antidiquark operators; The use of the latter was
previously the gold standard in the study of exotic hadrons,
but as of late, dimeson operators are primarily used when
the heavy quark mass is \textbf{not} close to the bottom
quark mass. For instance, it is posited that $T_{bb}$ is
likely a diquark-antidiquark state, thus, employing
diquark-antidiquark operators is important in this case.The
aim of this study is to \textbf{establish a pole in the
corresponding scattering amplitude $t(E_{cm})$} using
distillation smearing on the lattice with coupled dimeson
interpolating operators. Assuming we have a suitably large
basis of interpolators in the relevant channels of interest,
we can compute the spectrum and energy shifts with respect
to the $DD^*$ threshold for a heavy quark mass close to the
charm quark mass. Moreover, we will investigate whether this
species of tetraquark exists within said threshold. We can
extract the scattering amplitude from a L\"{u}scher analysis
of the lattice data, thereby obtaining the finite volume
energies. We endeavor to show that lattice calculations are
in agreement with phenomenology, namely, that the $DD^*$
interaction is repulsive in the $I=1$ channel and attractive
in the $I=0$ channel, which logically follows from the $I=0$
assignment for the $T_{cc}^+$ state.In Chapter 1 we give a
brief background of continuum QCD and the path integral
formalism to describe contemporary hadron spectroscopy
methods that we employ on the lattice, namely quark field
smearing with distillation, and two point correlators. In
chapter 2 we describe the computational workflow, tools
involved, and the HPC cost associated with the calculation
of the eigenbasis, meson elementals, perambulators. Chapter
3 details interpolating operator construction for meson and
di-mesons within the distillation framework, the use of
derivative (extended) operators, and how to account for
mesons at zero and non-zero momentum. The relevant group
theory is introduced, to be expanded on in the appendix. In
chapter 4 we expand on the end-point analysis of meson
correlators and present results for our study of mesonic
signal saturation with distillation, specifically the
dependence on the size of the distillation basis and number
of source insertions. This study dictates the ideal rank of
the distillation basis to use when calculating the
perambulators and elementals for each ensemble; We can
proceed to compute the spectrum and energy shifts with
respect to the $DD^*$ threshold for a heavy quark mass close
to the charm quark mass. In chapter 5 we describe remaining
work, notably a rigorous lattice determination of the quark
mass dependence of the $T_{cc}(3875)$ using a large basis of
$DD^*$ interpolating operators in various irreducible
representations and total momenta. At the end of the day, we
must search for poles using finite volume energies from the
scattering amplitude, which indicate where an attractive
potential is not deep enough to hold a bound state, thus
permitting us to make a phenomenological interpretation from
the lattice analysis.},
cin = {JSC},
cid = {I:(DE-Juel1)JSC-20090406},
pnm = {5111 - Domain-Specific Simulation $\&$ Data Life Cycle Labs
(SDLs) and Research Groups (POF4-511) / NRW-FAIR
(NW21-024-A)},
pid = {G:(DE-HGF)POF4-5111 / G:(NRW)NW21-024-A},
typ = {PUB:(DE-HGF)19},
doi = {10.34734/FZJ-2025-00991},
url = {https://juser.fz-juelich.de/record/1037848},
}
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