Centre for High Energy Physics (CHEP)
https://etd.iisc.ac.in/handle/2005/47
2023-06-04T04:20:48ZAnalytical Mellin-Barnes techniques with applications to two-loop SU(3) chiral perturbation theory and QED at higher loops
https://etd.iisc.ac.in/handle/2005/5432
Analytical Mellin-Barnes techniques with applications to two-loop SU(3) chiral perturbation theory and QED at higher loops
Ghosh, Shayan
The present era is one of precision in particle physics. To account for the lacunae in the otherwise
successful Standard Model, observables are calculated to high precision in various theoretical models,
which are then tested against experimental data to determine whether a given model is realised in
nature. In perturbative quantum eld theoretical models, higher order calculations require the
evaluation of multi-loop diagrams with multiple mass scales. Although an advanced technology has
been developed to evaluate these loop integrals, the majority of techniques are still numerical in
nature. In this thesis, we advance one technology that allows for the analytic evaluation of multi-loop
diagrams with several mass scales, the Mellin-Barnes (MB) technique, by studying and applying it
primarily in the context of three- avoured chiral perturbation theory (SU(3) ChPT). At two loop
order, the expressions for the pion, kaon and eta masses and decay constants depend on 'sunset'
diagrams, which appear with up to three independent masses, and the analytic evaluation of which
provides us the backdrop on which we develop our techniques.
The rst part of this work concerns itself with the development of the MB technology and its
application to the mathematics of sunset diagrams. We begin by developing an approach that allows
one to derive a minimal MB representation of a multi-loop multi-scale integral while retaining straight
line contours throughout the derivation process. After reducing the variety of vector and tensor
sunsets to a set of four scalar master integrals, this is then applied to evaluate all two mass scale
con gurations of the sunset, including (for completeness) those not arising in the ChPT context. The
same approach is used thereafter, with appropriate modi cations, to derive various MB representations
of the three mass scale integrals appearing in SU(3) ChPT. Each of these integrals is evaluated for all
accessible regions of convergence retaining their full dependence arising from dimensional
regularization, and in the ! 0 limit for the expressions that converge with physical meson mass
values. Formulae are also derived that allow one to expand these integrals to arbitrary order in .
The second part of this work focusses on physical applications of the aforementioned results in ChPT.
The sunset results are applied to obtain fully analytic expressions for m2
, m2
K, m2
, F , FK and F ,
which are subsequently truncated appropriately to obtain simpli ed representations that are
particularly suitable for tting with lattice QCD data. Such a preliminary lattice t is performed for the expression FK=F to extract values of the low energy constants (LEC) Lr
5, Cr
14 + Cr
15 and
Cr
15 + 2Cr
17. We also perform a numerical study of the meson masses and decay constants to examine
the relative contributions of their various components, and to investigate their dependence on the
values of the LEC. As another application of these analytic expressions, we nd an expansion of F
and m2
in the strange quark mass in the isospin limit, and perform the matching of the chiral SU(2)
and SU(3) low energy constants. A numerical study on this demonstrates the strong dependence of F
on the LEC in the chiral limit.
In the nal part of the thesis, we develop and demonstrate two methods of analytic continuation that
may be used to obtain results when values of the mass parameters do not allow for convergence of
Feynman integrals calculated using MB techniques. We apply the rst technique to the three mass
scale sunsets, and therefore obtain the full set of results for these integrals, i.e. we get solutions for the
sunsets for all possible values of the mass parameters. The same technique is then applied to
analytically continue the results of the most general four mass scale sunset integral to obtain results
which converge for physical values of the meson masses. We apply the second method of analytic
continuation in a non-ChPT context to demonstrate the general applicability of the methods
developed in this work. We rst calculate the complete result of a class of three-loop QED vacuum
polarisation contributions arising from non-diagonal avour charged leptons to the g 2 of each
charged lepton, and then show how one may obtain the expression for the case with an external muon
or tau leg from the results of the case of external electron leg by means of analytic continuation.
Applications of Holography
https://etd.iisc.ac.in/handle/2005/5294
Applications of Holography
Bala Subramanian, P N
This thesis consists of four parts. In the first part of the thesis, we investigate the phase
structure of Einstein-Maxwell-Scalar system with a negative cosmological constant. For
the conformally coupled scalar, an intricate phase diagram is charted out between the four
relevant solutions: global AdS, boson star, Reissner-Nordstrom black hole and the hairy
black hole. The nature of the phase diagram undergoes qualitative changes as the charge of
the scalar is changed, which we discuss. We also discuss the new features that arise in the
extremal limit.
In the second part, we do a systematic study of the phases of gravity coupled to an
electromagnetic field and charged scalar in flat space, with box boundary conditions. The
scalar-less box has previously been investigated by Braden, Brown, Whiting and York (and
others) before AdS/CFT and we elaborate and extend their results in a language more
familiar from holography. The phase diagram of the system is analogous to that of AdS
black holes, but we emphasize the differences and explain their origin. Once the scalar
is added, we show that the system admits both boson stars as well as hairy black holes as
solutions, providing yet another way to evade flat space no-hair theorems. Furthermore both
these solutions can exist as stable phases in regions of the phase diagram. The final picture of
the phases that emerges is strikingly similar to that of holographic superconductors in global
AdS, discussed in part one. We also point out previously unnoticed subtleties associated to
the definition quasi-local charges for gravitating scalar fields in finite regions.
In part three, we investigate a class of tensor models which were recently outlined as potentially
calculable examples of holography, as their perturbative large-N behavior is similar
to the Sachdev-Ye-Kitaev (SYK) model, but they are fully quantum mechanical (in the sense
that there is no quenched disorder averaging). We explicitly diagonalize the simplest nontrivial
Gurau-Witten tensor model and study its spectral and late-time properties. We find
parallels to (a single sample of) SYK where some of these features were recently attributed
to random matrix behavior and quantum chaos. In particular, after a running time average,
the spectral form factor exhibits striking qualitative similarities to SYK. But we also observe
that even though the spectrum has a unique ground state, it has a huge (quasi-?)degeneracy
of intermediate energy states, not seen in SYK. If one ignores the delta function due to the
degeneracies however, there is level repulsion in the unfolded spacing distribution hinting
chaos. Furthermore, the spectrum has gaps and is not (linearly) rigid. The system also has
a spectral mirror symmetry which we trace back to the presence of a unitary operator with
which the Hamiltonian anticommutes. We use it to argue that to the extent that the model
exhibits random matrix behavior, it is controlled not by the Dyson ensembles, but by the
BDI (chiral orthogonal) class in the Altland-Zirnbauer classification.
In part four, we construct general asymptotically Klebanov-Strassler solutions of a five
dimensional SU(2) SU(2) Z2 Z2R truncation of IIB supergravity on T1;1, that break
supersymmetry. This generalizes results in the literature for the SU(2) SU(2) Z2 U(1)R
case, to a truncation that is general enough to capture the deformation of the conifold in
the IR. We observe that there are only two SUSY-breaking modes even in this generalized
set up, and by holographically computing Ward identities, we confirm that only one of them
corresponds to spontaneous breaking: this is the mode triggered by smeared anti-D3 branes
at the tip of the warped throat. Along the way, we address some aspects of the holographic
computation of one-point functions of marginal and relevant operators in the cascading gauge
theory. Our results strengthen the evidence that if the KKLT construction is meta-stable, it
is indeed a spontaneously SUSY-broken (and therefore bona fide) vacuum of string theory.
Applications of Moonshine Symmetry in String Theory
https://etd.iisc.ac.in/handle/2005/5001
Applications of Moonshine Symmetry in String Theory
Chattopadhyaya, Aradhita
In this thesis we study the applications of Mathieu moonshine symmetry to compacti cations of supersymmetric
string theories. These theories are compacti ed on a 6 dimensional manifold K3 T2. The
main ingredient in this study is a topological index called twisted elliptic genus. For a super-conformal
eld theory whose target space is a K3 there can be several automorphisms on K3 which are related
to Mathieu group M24. Under these automorphisms it was observed that the twining genera of the
twisted elliptic genus of K3 could be written in terms of the short and long representations of N = 4
super-conformal algebra and the characters of M24 [1, 2, 3]. We compute the twisted elliptic genus in
every sector for 16 of these orbifolds using the results of [2].
Firstly we study the heterotic compacti cations of N = 2 super-symmetric strings compacti ed on
orbifolds of K3 T2 and E8 E8 where g0 is an action on K3 corresponding to [M24] along with a 1=N
shift on one of the circles of T2. We compute the gauge and gravitational threshold corrections in these
theories. Here we need a topological index called the new supersymmetric index. The un-orbifolded
result for K3 was known for gauge couplings in [4] and the gravitational ones were computed in [5]. We
observe that the di erences in gauge couplings can be written in terms of the twisted elliptic genus of
K3 for standard embeddings. For non-standard embeddings we studied two orbifold realizations of K3
as T4=Z2 and T4=Z4 and computed the threshold di erences. The result could be written in terms the
twisted elliptic genus of K3 and the elliptic genus of K3. From the gravitational corrections we predict
the Gopakumar Vafa invariants and the Euler character for the dual Calabi Yau geometries. We also
observe that the conifold singularities of these manifolds are manifested in twisted sectors only and only
the genus zero Gopakumar-Vafa invariants at those points are non-zero.
Secondly we study the properties of 1/4 BPS dyons in type II string compacti ed on K3 T2 orbifolded
with an action of g0 which corresponds to automorphisms of K3 corresponding to the conjugacy classes of
Mathieu group M24 and a 1=N shift in one of the circles of T2. For these compacti cations the counting
function for these dyons can be computed from Siegel modular forms given by the lift of the twisted
elliptic genus. These give the correct sign as predicted from black hole physics as conjectured by Sen [6].
We also study the properties of 1/4th BPS dyons in type II string theory compacti ed on Z2 and Z3
orbifolds on T6 with 1=N shift in one of the S1 and encountered some violations to this conjecture which
points to the existence of non-trivial hair modes. We associate mock modular forms corresponding to
single centred black holes and extend the work of Dabholkar-Murthy-Zagier [7] to these orbifolds of K3
and also for the toroidal orbifolds.
In computing the twisted elliptic genus and new super-symmetric index in various twisted sectors we
encounter several identities between some 0(N) modular forms. With a bit more analysis we determine
the exact location of the zeros of some weight 2 Eisenstein series of 0(N) in the fundamental domain
of 0(N) where N = 2; 3; 5; 7. The location of their zeros were controlled by those of Eisenstein series of
weight 4 and 6.
Aspects of conformal field theories at finite temperature
https://etd.iisc.ac.in/handle/2005/5238
Aspects of conformal field theories at finite temperature
Dutta Chowdhury, Subham
In this thesis we have studied broadly two aspects of thermal field theory. We began by examining how
the macroscopic system (described by relativistic hydrodynamics) behalves in presence of microscopic
anomalies. We are able to relate macroscopic transport coefficients to the anomalous conservation equations
of the microscopic theory. It is to be noted that, using the perturbative methods that we develop,
we are able to relate both the mixed and pure gravitational anomalies to their respective transport coe
fficients. Our results agree with other methods used to study this relationship. Using our perturbative
approach, we are also able to understand the breakdown of the replacement rule for gravitino systems.
Global anomalies instead of perturbative anomalies can also be used to x the macroscopic transport
coefficients. By computing the global anomalies associated with particular systems, we were able to
write down thermal effective actions which reproduce the anomalies. We show that such effective actions
can be used to compute the transport coefficients and obtain a match with our perturbative results. We
also provide a topological understanding of the replacement rule. As a further check of our formalism,
we compute perturbatively using the formalism developed in [11], the anomalous transport coefficient
(corresponding to pure gravitational anomaly) for self dual tensors in d = 6 and obtain a match with
the global anomaly result.
In the second part of the thesis we look at constraints that can be placed on spectral densities in
a conformal field theory at fi nite temperature. Sum rules provide important constraints on spectral
densities of any quantum field theory. We relate the weighted integral of spectral densities over frequency
to the energy density of the theory. We show that the proportionality constant can be written down
in terms of Hofman-Maldacena variables t2 and t4, which determine the three point function of stress
tensors of a parity preserving CFT. For CFTs dual to two derivative Einstein gravity, we nd agreement
of our sum rule derived from general conformal invariance with holographic methods. We also obtain
correction to the holographic shear sum rule for theories with quadratic curvature corrections to the
Einstein gravity.
We extend the conformal collider physics formalism developed by Maldacena et al to study three point
functions involving a stress tensor T, a U(1) current j, in 2 + 1 dimensional parity violating conformal
field theories. We show that large N Chern Simons theories coupled to fundamental fermions/ bosons
saturate our derived bounds. This is consistent with the observations that the scaling dimensions of spin
operators in these theories saturate the unitarity bound ( s s + 1) and hence perhaps the conformal
collider bounds as well.