###### Contents

  1. 1 Introduction
  2. 2 Brief review of perturbative T$\bar{\text{T}}$, J$\bar{\text{T}}$ and J$\bar{\text{J}}$ deformations to 2D CFT
  3. 3 Krylov complexity in deformed CFTs
    1. 3.1 Review of Lanczos coefficients and Krylov complexity
    2. 3.2 T$\bar{\text{T}}$
      1. 3.2.1 Autocorrelation function and Lanczos coefficients
      2. 3.2.2  Krylov complexity
    3. 3.3 J$\bar{\text{T}}$
    4. 3.4 J$\bar{\text{J}}$
  4. 4 Discussion and Conclusion
  5. A Important integrals
    1. A.1 Integrals for T$\bar{\text{T}}$
    2. A.2 Integrals for J$\bar{\text{T}}$
      1. A.2.1 $\mathcal{I}^{1}(z_{1},\bar{z}_{1})$
      2. A.2.2 $\mathcal{I}^{2}(z_{1},\bar{z}_{1},z_{2},\bar{z}_{2})$
      3. A.2.3 $\mathcal{I}^{3}(z_{1},\bar{z}_{1})$
      4. A.2.4 $\mathcal{I}^{4}(z_{1},\bar{z}_{1},z_{2},\bar{z}_{2})$
    3. A.3 Integrals for J$\bar{\text{J}}$
      1. A.3.1 $\mathcal{I}^{5}(z_{1},\bar{z}_{1})$
      2. A.3.2 $\mathcal{I}^{6}(z_{1},\bar{z}_{1},z_{2},\bar{z}_{2}$)
  6. B $|\phi_{n}(t)|^{2}$ for T$\bar{\text{T}}$

##  1 Introduction

The concept of quantum complexity originally emerged as a measure of
algorithmic efficiency in quantum computation. It involved assigning distinct
costs to various configurations of quantum gates for transitioning from a
specified reference state to a desired target state. While Nielsen complexity
of a unitary operator is defined as the minimal distance of it from the
identity in the group manifold [1, 2, 3, 4], the computational complexity
counts the minimum number of elementary quantum gates necessary to achieve a
unitary transformation to reach the target state from the reference state. In
recent years, there has been significant interests in assigning cost for
preparing a state within holography, spurred by Susskind’s conjecture linking
the notion of cost assignment to black hole physics [5]. Due to several
universal features exhibited by computational complexity in chaotic systems of
finite size, [5] conjectured its relations with geometric aspects of anti-de
Sitter (AdS) black hole geometries, particularly concerning the expansion of
black hole interiors over time. Among the most extensively examined proposals
concerning holographic complexity are the “complexity=volume” [5, 6],
“complexity=action” [7, 8] and “complexity=volume 2.0” [9] proposals111Several
works have explored these propositions [10, 11, 12, 13, 14, 15, 16, 17, 18,
19, 20] and also examined the violation of universal bounds within this
framework [21, 22], for instance, the Lloyd’s bound [23].. Recently, a broad
array of gravitational observables has been introduced as
“complexity=anything” [24, 25, 26]. The endeavor to comprehend holographic
complexity through a dual microscopic description has motivated advancements
in defining and examining complexity within quantum field theories. However
the relationship to holographic complexity has largely been constrained to
qualitative comparisons. For example, the time evolution of holographic
complexity exhibits a linear growth up to saturating to almost a constant
value, similar to the computational complexity. In [27], a mapping between
computational complexity and a geometric entity within the gravity theory was
derived from first principles and can be naturally applied to AdS black hole
spacetimes and their dual thermofield double (TFD) states.
Gradually complexity has evolved into a crucial tool to probe quantum chaos
[28, 29, 30, 31, 32, 33], motivated by the expectation that chaotic systems
exhibit maximum complexity. In parallel to this developments, there exist
several other approaches to diagnose quantum chaos including level spacing
statistics [34, 35], exponential growth of out-of-time-order correlators
(OTOCs) [36], eigenstate thermalization hypothesis [37, 38, 39]. One of the
significant challenges in this regard is to precisely examine if there exists
a notion which unifies these varied approaches. Recently, a new measure of
quantum state/operator complexity known as the Krylov complexity has been
introduced in [40] to analyze chaotic behaviour of quantum systems. Krylov
complexity essentially characterizes the spread of an operator or equivalently
a state evolving with time in a distinct Hilbert space spanned by what is
known as the Krylov basis222Krylov state complexity is also known as spread
complexity in the literature.. This measure of complexity has more universal
feature compared to existing ones, since for a maximally entangled state the
Krylov complexity solely relies on the Hamiltonian’s spectrum and does not
depend on the choice of fundamental gates or any local information unlike both
the Nielsen and computational complexity. Time evolution of an unitary
operator in a space of operators in the Heisenberg picture can be described as
the evolution under the operation of a super operator known as the Liouvillian
whose repeated action leads to nested commutators involving the Hamiltonian
and the operator itself. This in turn leads to the definition of an invariant
subspace known as the Krylov subspace spanned by orthonormalized operators
constructed from iterative action of the Liouvillian [41, 40].

It was argued in [40], that for maximally chaotic systems the Lanczos
coefficients exhibit a linear growth (maximally allowed growth) with the
number of basis vectors (up to a log correction) satisfying an universal
operator growth hypothesis. While the OTOC typically exhibits an exponential
growth over time, indicating a rapid scrambling of the information and
displaying an extreme sensitivity to the initial conditions, the Krylov
complexity also grows exponentially fast as $e^{\lambda_{K}t}$, with
$\lambda_{K}$ being the Krylov exponent and $t$ is time. Furthermore it was
conjectured in [40], that twice the rate of linear growth of Lanczos ( which
is same as the Krylov exponent ) serves as an upper bound on the quantum
Lyapunov exponent $\lambda_{L}$ derived from OTOCs
$\lambda_{L}\leq\lambda_{K}$333The relationship that twice the linear growth
rate of the Lanczos coefficients equals the Krylov exponent can be
demonstrated using a continuum version of the recursion equation for operator
amplitudes [42]. . Since quantum Lyapunov exponent $\lambda_{L}$ itself
satisfies the Maldacena-Shenker-Stanford (MSS) bound
$\lambda_{L}\leq\frac{2\pi}{\beta}$ [36], combining the two bounds a modified
bound $\lambda_{L}\leq\lambda_{K}\leq\frac{2\pi}{\beta}$ was conjectured to
hold for any quantum system in [43] (from now on we will refer to this as the
extended MSS bound). Although the operator growth hypothesis has turned out to
be very efficient in distinguishing chaotic and non-chaotic systems in lattice
models, it was later shown to be far too universal in quantum systems with
infinite degrees of freedom such as QFTs. Infact even in very simple
integrable field theories such as free scalar field theories the Lanczos
coefficients exhibit linear growth and the Krylov complexity also grows
exponentially [44, 45]. Furthermore, in the same article it was also shown a
primary operator in any two dimensional CFT also exhibits a linear growth of
Lanczos coefficients leading to exponential growth of Krylov complexity with
Krylov exponent saturating the conjectured bound. This indicates that the
notion of the Krylov complexity is more subtle and its behaviour might in turn
be state dependent through the choice of the inner product in the Krylov space
as shown in[46].

Due to many intriguing features such as the ones mentioned above, the
behaviour of Krylov complexity has been extensively investigated in a vast
number of different settings such as free and interacting Quantum Field
Theories(QFTs), random matrix theories, open quantum systems etc [44, 45, 47,
48, 49, 50, 33, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60]. Another interesting
behavior exhibited by the Krylov complexity of quantum chaotic systems,
encompassing notable models such as the SYK model and matrix models reveals a
fascinating pattern wherein complexity exhibits four distinct dynamical
phases: a steadily rising ramp culminating in a peak, succeeded by a declining
slope leading to a plateau [33]. Notably, the duration of these phases, as
well as the magnitudes of the peak and plateau, exhibit exponential dependence
on entropy. This observation draws parallel to the characteristic slope-dip-
ramp-plateau structure observed in the spectral form factor [61, 62, 63],
offering valuable insights into the intricate dynamics of chaotic systems.
These phases were also observed in integrable systems [51, 64] which exhibits
saddle-dominated scrambling. Apart from the above described properties, the
notion of Krylov complexity also seems to possess interesting geometrical
aspects. Remarkably for low rank algebras, Krylov operator demonstrates a
connection to the vector fields responsible for generating isometries within
the classical phase space geometry which in turn is also related to Nielsen
complexity [65, 66, 67]. Its expectation value corresponds to the volume
delineated by the classical motion unfolding within the associated geometry
[68, 65].
In this work, we delve into the behaviour of Krylov complexity for two
dimensional conformal field theories in the presence of perturbative
deformations. In particular, we will consider two irrelevant deformations
induced by composite operators constructed from abelian current and energy
momentum tensor namely T$\bar{\text{T}}$ and J$\bar{\text{T}}$ along with one
exactly marginal deformation J$\bar{\text{J}}$ constructed only from abelian
currents [69, 70, 71]. We first compute the thermal autocorrelation functions
from the first order correction of the correlation functions due to the
deformations. Following this, we determine the first order correction to the
Lanczos coefficients and operator wavefunctions due to the deformations for
both the positive and negative values of the deformation parameter.
Furthermore, finally we examine the growth of Krylov complexity to understand
the implications of the above mentioned deformations. The key insight of our
study requires to understand the regime of validity of perturbative techniques
while computing the Lanczos coefficients and Krylov complexity growth. It is
reasonable to expect that perturbation techniques will remain valid as long as
the probability of the operator wavefunction remains conserved over time [52].
In [72], stability of autocorrelation function was examined under a specific
perturbations to Lanczos coefficients. Our results demonstrate a correction to
the Lanczos coefficients, operator wavefunctions and the growth of Krylov
complexity for the T$\bar{\text{T}}$ deformations treated perturbatively,
while J$\bar{\text{J}}$ and J$\bar{\text{T}}$ deformation do not affect the
CFT results. This may establish an intriguing behavior of complexity of field
theories with the presence of irrelevant deformations, since Lyapunov exponent
computed from OTOC do not receive any correction for perturbative
T$\bar{\text{T}}$ and J$\bar{\text{T}}$ deformations [73].
This paper is organized as the following. We begin by reviewing some
properties of T$\bar{\text{T}}$, J$\bar{\text{T}}$ and J$\bar{\text{J}}$
deformations to 2D CFT in section 2, focusing mainly on the two point
correlation functions within the perturbative framework. In section 3, we
provide the main results containing the corrections to the autocorrelation
functions, Lanczos coefficients, operator wavefunctions and growth of Krylov
complexity due to the perturbative deformations of 2D CFTs, following a brief
review on the computation of Krylov complexity. We conclude in section 4, with
possible outlooks and future directions. We provide all the details concerning
the integrals required for the derivations of correlation functions in the
appendix A.

##  2 Brief review of perturbative T$\bar{\text{T}}$, J$\bar{\text{T}}$ and
J$\bar{\text{J}}$ deformations to 2D CFT

The space of QFTs can be navigated through the renormalization group (RG)
flows, wherein conformal field theories are the fixed points. Investigating
flows diverging from fixed points proves equally intriguing. These
deformations of QFTs, contingent upon the operators driving the flow, fall
into three broad categories: relevant, marginal, and irrelevant. Although,
irrelevant deformations of QFTs inherently entail an infinite array of counter
terms in the Lagrangian, however, recent revelations indicate that for certain
specialized classes of irrelevant deformations within two-dimensional
spacetime, this process is more tractable and even solvable. In [69, 74], an
irrelevant deformation induced by T$\bar{\text{T}}$ composite operator444In
our case with CFT,
T$\bar{\text{T}}=4\pi^{2}(T_{zz}T_{\bar{z}\bar{z}}-T^{2}_{z\bar{z}})=T\bar{T}$
where $T_{zz}$ and $T_{\bar{z}\bar{z}}$ are the holomorphic and
antiholomorphic component of the CFT stress tensor and $T_{z\bar{z}}=0$.
constructed from the components of the conserved energy momentum tensor was
introduced. Given a one parameter family of 2D CFTs, the action of deformed
theory along the flow satisfies,

$\frac{\partial S}{\partial\lambda}=\int\sqrt{g}\leavevmode\nobreak\
d^{2}x(\text{T}\bar{\text{T}})_{\lambda}$ (2.1)

where $\lambda$ is the coupling parameter. Despite the fact that this
deformation flows toward the ultraviolet (UV) regime, numerous physically
significant quantities like deformed S matrix [75], finite volume spectrum
using integrability methods, torus partition function [76, 77, 78, 79] could
be computed exactly. This deformation is integrable and one can show that for
integrable field theories, infinite number of conserved charges continue to
remain conserved along the flow [69]. Since most QFTs are not integrable, but
the surprising solvability of T$\bar{\text{T}}$ even in that case leads to the
random geometry picture prescribed in [76]. Another interesting property of
this deformation to a QFT is that it is equivalent to coupling the theory to
two dimensional Jackiw-Teitelboim (JT) gravity [75]. The holographic dual of
this deformed CFT is a cut-off AdS gravity theory [80, 81, 82].

For our purpose of computation of Krylov complexity, we are interested in the
two point correlation function defined within the perturbative framework. Now
we may express the deformed action at leading order as,

$\displaystyle S(\lambda)=S_{0}+\lambda\int\sqrt{g}\leavevmode\nobreak\
d^{2}x(\text{T}\bar{\text{T}})_{0}+\mathcal{O}(\lambda^{2})$ (2.2)

where $S_{0}$ is the undeformed action and $(\text{T}\bar{\text{T}})_{0}$ is
the perturbing operator of a CFT. For a two dimensional Euclidean CFT on a
complex plane whose coordinates are denoted by ($z,\bar{z}$), the first order
correction to a two point function involving two primary operators can be
computed following the OPE of energy momentum tensor and subsequently
utilizing the Ward identity associated with the conserved energy momentum
tensor as follows[83, 73, 84],

$\displaystyle\langle
O(z_{1},\bar{z}_{1})O(z_{2},\bar{z}_{2})\rangle_{\lambda}=$
$\displaystyle\lambda\int
dzd\bar{z}\bigg{[}\sum_{i}\left(\frac{h_{i}}{(z-z_{i})^{2}}+\frac{\partial_{z_{i}}}{z-z_{i}}\right)\left(\frac{\bar{h}_{i}}{(\bar{z}-\bar{z}_{i})^{2}}+\frac{\partial_{\bar{z}_{i}}}{\bar{z}-\bar{z}_{i}}\right)$
$\displaystyle\qquad\qquad\qquad\qquad\hskip 99.58464pt\times\langle
O(z_{1},\bar{z}_{1})O(z_{2},\bar{z}_{2})\rangle_{0}\bigg{]}$ (2.3)

In the above equation the two point correlation function involving two primary
operators for the undeformed CFT is given by,

$\displaystyle\langle
O_{1}(z_{1},\bar{z}_{1})O_{2}(z_{2},\bar{z}_{2})\rangle_{0}=\frac{1}{(z_{1}-z_{2})^{2h}(\bar{z}_{1}-\bar{z}_{2})^{2\bar{h}}}=\frac{1}{z_{12}^{2h}\bar{z}_{12}^{2\bar{h}}}$
(2.4)

with $h$ and $\bar{h}$ being the scaling dimensions of the operators. It is
worth emphasizing that within the perturbation framework, the operators are
assumed not to be deformed. Rather one can think that the vacuum state of free
CFT got corrections due to the perturbations, resulting in a corrected vacuum
expectation value. To examine the Lanczos coefficients and the Krylov
complexity we require the thermal two point function. This can be obtained by
the usual conformal map from CFT on a Euclidean plane to a cylinder.

The first order correction to the two point correlation function of two
dimensional CFT on a cylinder at a finite temperature $1/\beta$, due to the
$\text{T}\bar{\text{T}}$ deformation is expressed as follows,

$\displaystyle\langle\mathcal{O}\left(w_{1},\bar{w}_{1}\right)\mathcal{O}\left(w_{2},\bar{w}_{2}\right)\rangle_{\beta,\lambda}=$
$\displaystyle\langle\mathcal{O}\left(w_{1},\bar{w}_{1}\right)\mathcal{O}\left(w_{2},\bar{w}_{2}\right)\rangle_{\beta,0}$
$\displaystyle\qquad+\lambda\int d^{2}w\left\langle
T(w)\bar{T}(\bar{w})\mathcal{O}\left(w_{1},\bar{w}_{1}\right)\mathcal{O}\left(w_{2},\bar{w}_{2}\right)\right\rangle_{\beta,0}$
(2.5)

where the coordinates on cylinder are denoted by $(w,\bar{w})$ and these are
related to the coordinates on the complex plane $(z,\bar{z})$ through the
following conformal transformation

$\begin{split}&z_{i}=e^{\frac{2\pi}{\beta}w_{i}},\quad\bar{z}_{i}=e^{\frac{2\pi}{\beta}\bar{w}_{i}};\\\
&w=x+it,\quad\bar{w}=x-it\end{split}$ (2.6)

Under the above conformal transformation the following correlator involving
the energy momentum tensors transforms as follows

$\displaystyle\langle\text{T}(w)\bar{\text{T}}(\bar{w})\mathcal{O}(w_{1},\bar{w}_{1})\mathcal{O}(w_{2},\bar{w}_{2})\rangle_{\beta}$
$\displaystyle=$ $\displaystyle\prod_{i=1,2}\Big{(}\frac{2\pi
z_{i}}{\beta}\Big{)}^{h_{i}}\Big{(}\frac{2\pi\bar{z}_{i}}{\beta}\Big{)}^{h_{i}}\Big{(}\frac{2\pi
z}{\beta}\Big{)}^{2}\Big{(}\frac{2\pi\bar{z}}{\beta}\Big{)}^{2}\Big{\langle}\Big{(}T(z)-\frac{c}{24z^{2}}\Big{)}\Big{(}\bar{T}(\bar{z})-\frac{c}{24\bar{z}^{2}}\Big{)}\mathcal{O}(z_{1},\bar{z}_{1})\mathcal{O}(z_{2},\bar{z}_{2})\Big{\rangle}_{0}$
(2.7)

Utilizing the above expression in section 2 it is easy to obtain,

$\displaystyle\frac{\langle\mathcal{O}(w_{1},\bar{w}_{1})\mathcal{O}(w_{2},\bar{w}_{2})\rangle_{\beta,\lambda}}{\langle\mathcal{O}(w_{1},\bar{w}_{1})\mathcal{O}(w_{2},\bar{w}_{2})\rangle_{\beta,0}}=1+\lambda\Big{(}\frac{2\pi}{\beta}\Big{)}^{4}\int
d^{2}w|z|^{4}\frac{\langle(T(z)-\frac{c}{24z^{2}})(\bar{T}(\bar{z})-\frac{c}{24\bar{z}^{2}})\mathcal{O}(z_{1},\bar{z}_{1})\mathcal{O}(z_{2},\bar{z}_{2})\rangle_{0}}{\langle\mathcal{O}(z_{1},\bar{z}_{1})\mathcal{O}(z_{2},\bar{z}_{2})\rangle_{0}}$

The above equation will be split into four different integrals for convenience

$\displaystyle\frac{\langle\mathcal{O}(w_{1},\bar{w}_{1})\mathcal{O}(w_{2},\bar{w}_{2})\rangle_{\beta,\lambda}}{\langle\mathcal{O}(w_{1},\bar{w}_{1})\mathcal{O}(w_{2},\bar{w}_{2})\rangle_{\beta,0}}=1+\lambda\Big{(}\frac{2\pi}{\beta}\Big{)}^{2}\Big{(}I_{1}+I_{2}+I_{3}+I_{4}\Big{)}$
(2.9)

where we have set $h=\bar{h}$ for the operator $\mathcal{O}$ and accounted the
change in measure of the integral. The four integrals
$I_{1},I_{2},I_{3},I_{4}$ are given as

$\displaystyle I_{1}$ $\displaystyle=|z_{12}|^{4h}\int
dzd\bar{z}\,z\bar{z}\langle
T(z)\bar{T}(\bar{z})\mathcal{O}(z_{1},\bar{z}_{1})\mathcal{O}(z_{2},\bar{z}_{2})\rangle_{0}$
(2.10) $\displaystyle I_{2}$ $\displaystyle=-\frac{c|z_{12}|^{4h}}{24}\int
dzd\bar{z}\,\frac{z}{\bar{z}}\langle
T(z)\mathcal{O}(z_{1},\bar{z}_{1})\mathcal{O}(z_{2},\bar{z}_{2})\rangle_{0}$
(2.11) $\displaystyle I_{3}$ $\displaystyle=-\frac{c|z_{12}|^{4h}}{24}\int
dzd\bar{z}\,\frac{\bar{z}}{z}\langle\bar{T}(\bar{z})\mathcal{O}(z_{1},\bar{z}_{1})\mathcal{O}(z_{2},\bar{z}_{2})\rangle_{0}$
(2.12) $\displaystyle I_{4}$ $\displaystyle=\left(\frac{c}{24}\right)^{2}\int
dzd\bar{z}\,\frac{1}{z\bar{z}}$ (2.13)

The integral $I_{4}$ is divergent and can be shown to depend only on the
cutoff. Therefore it will not contribute to any dynamics. For the other three
integrals, we will report the final results upto $\mathcal{O}(\epsilon)$ terms
below and provide the details of the computation in appendix A

$\displaystyle I_{1}$ $\displaystyle=2\pi
h^{2}\biggr{[}{|z_{1}|^{2}+|z_{2}|^{2}\over|z_{12}|^{2}}\left({4\over\epsilon}+2\gamma-5+2\ln\pi+2\ln|z_{12}|^{2}+\ln\mu^{2}\right)$
$\displaystyle\qquad\qquad-\left({2\over\epsilon}+\gamma-3+\ln\pi+\ln|z_{12}|^{2}+\ln\mu\right)\biggr{]},$
$\displaystyle I_{2}$ $\displaystyle={c\,h\over
24}\left[2\pi+\pi\ln|z_{1}z_{2}|^{2}-{2\pi\over
z_{12}}\left(z_{1}\ln|z_{1}|^{2}-z_{2}\ln|z_{2}|^{2}\right)\right],$
$\displaystyle I_{3}$ $\displaystyle={c\,\bar{h}\over
24}\left[2\pi+\pi\ln|z_{1}z_{2}|^{2}-{2\pi\over\bar{z}_{12}}\left(\bar{z}_{1}\ln|z_{1}|^{2}-\bar{z}_{2}\ln|z_{2}|^{2}\right)\right].$
(2.14)

Finally, utilizing the coordinate transformation given in eq. 2.6 in section
2, we can obtain the two point correlation function on the cylinder. We will
compute the autocorrelation function from these results in the next section.
There exist additional families of solvable QFT deformations. For instance, in
theories featuring a conserved holomorphic $U(1)$ current, one can define
another irrelevant deformation known as the J$\bar{\text{T}}$ deformation [85,
86, 87]. Analogous to the T$\bar{\text{T}}$ deformation, the deformed action
can be defined as given by the following expression

$\frac{\partial S}{\partial\lambda}=\int\sqrt{g}\leavevmode\nobreak\
d^{2}x\leavevmode\nobreak\ (\text{J}\bar{\text{T}})_{\lambda}$ (2.15)

Within the framework of perturbative analysis, the first order correction to
the two point function on the plane from OPEs of the operators are determined
as follows [73, 87],

$\displaystyle\langle
O(z_{1},\bar{z}_{1})O(z_{2},\bar{z}_{2})\rangle_{\lambda}$ $\displaystyle=$
$\displaystyle\lambda\int
dzd\bar{z}\left(\frac{q_{1}}{z-z_{1}}+\frac{q_{2}}{z-z_{2}}\right)\frac{\bar{h}\bar{z}^{2}_{12}}{(\bar{z}-\bar{z}_{1})^{2}(\bar{z}-\bar{z}_{2})^{2}}\langle
O(z_{1},\bar{z}_{1})O(z_{2},\bar{z}_{2})\rangle_{0}$ (2.16)

where $q$’s are the charges of the primary operators $O$’s under the operation
of the $U(1)$ current. We will write the thermal two point function from
section 2 by applying the coordinate transformation given in eq. 2.6,

$\displaystyle\langle\mathcal{O}\left(w_{1},\bar{w}_{1}\right)\mathcal{O}\left(w_{2},\bar{w}_{2}\right)\rangle_{\beta,\lambda}$
$\displaystyle=\langle\mathcal{O}\left(w_{1},\bar{w}_{1}\right)\mathcal{O}\left(w_{2},\bar{w}_{2}\right)\rangle_{\beta,0}$
$\displaystyle\quad+\lambda\int d^{2}w\left\langle
J(w)\bar{T}(\bar{w})\mathcal{O}\left(w_{1},\bar{w}_{1}\right)\mathcal{O}\left(w_{2},\bar{w}_{2}\right)\right\rangle_{\beta}$
(2.17)

Considering the transformation properties of J and $\bar{\text{T}}$ operators
under eq. 2.6,

$\displaystyle\langle
J(w)\bar{T}(\bar{w})\mathcal{O}(w_{1},\bar{w}_{1})\mathcal{O}(w_{2},\bar{w}_{2})\rangle_{\beta}$
$\displaystyle=$ $\displaystyle\prod_{i=1,2}\Big{(}\frac{2\pi
z_{i}}{\beta}\Big{)}^{h_{i}}\Big{(}\frac{2\pi\bar{z}_{i}}{\beta}\Big{)}^{h_{i}}\Big{(}\frac{2\pi\bar{z}}{\beta}\Big{)}^{2}\Big{(}\frac{2\pi{z}}{\beta}\Big{)}\langle
J(z)(\bar{T}(\bar{z})-\frac{c}{24\bar{z}^{2}})\mathcal{O}(z_{1},\bar{z}_{1})\mathcal{O}(z_{2},\bar{z}_{2})\rangle_{0}$
(2.18)

we can manipulate section 2 to yield

$\displaystyle\frac{\langle\mathcal{O}(w_{1},\bar{w}_{1})\mathcal{O}(w_{2},\bar{w}_{2})\rangle_{\beta,\lambda}}{\langle\mathcal{O}(w_{1},\bar{w}_{1})\mathcal{O}(w_{2},\bar{w}_{2})\rangle_{\beta,0}}=$
$\displaystyle 1+\Big{(}{{2\pi}\over\beta}\Big{)}\lambda\int
d^{2}z\,{\bar{z}}\frac{\langle
J(z)(\bar{T}(\bar{z})-\frac{c}{24\bar{z}^{2}})\mathcal{O}(z_{1},\bar{z}_{1})\mathcal{O}(z_{2},\bar{z}_{2})\rangle_{0}}{\langle\mathcal{O}(z_{1},\bar{z}_{1})\mathcal{O}(z_{2},\bar{z}_{2})\rangle_{0}}$
$\displaystyle=1+\Big{(}{{2\pi}\over\beta}\Big{)}\lambda(\mathbb{I}_{1}+\mathbb{I}_{2})$
(2.19)

For convenience, we will split the second term on the RHS involving
integration into two different integrals with $h=\bar{h}$

$\displaystyle\mathbb{I}_{1}(z_{1},\bar{z}_{1};z_{2},\bar{z}_{2})$
$\displaystyle=|z_{12}|^{4h}\int dzd\bar{z}\,{\bar{z}}\langle
J(z)\bar{T}(\bar{z})\mathcal{O}(z_{1},\bar{z}_{1})\mathcal{O}(z_{2},\bar{z}_{2})\rangle_{0}$
(2.20) $\displaystyle\mathbb{I}_{2}(z_{1},\bar{z}_{1};z_{2},\bar{z}_{2})$
$\displaystyle=-\frac{c|z_{12}|^{4h}}{24}\int
dzd\bar{z}\,{1\over\bar{z}}\langle
J(z)\mathcal{O}(z_{1},\bar{z}_{1})\mathcal{O}(z_{2},\bar{z}_{2})\rangle_{0}$
(2.21)

We provide the detailed computation of these integrals in the appendix A.2 and
the final contributions discarding the ${\cal{O}(\epsilon)}$ terms are,

$\begin{split}\mathbb{I}_{1}&={\pi\lambda\bar{h}\over
2\bar{z}_{12}}(\bar{z}_{1}+\bar{z}_{2})(q_{2}-q_{1})\left({4\over\epsilon}-5+2\gamma+2\ln\pi+2\ln\mu|z_{12}|^{2}\right)+{\pi\over\bar{z}_{12}}\lambda\,\bar{h}\,(q_{2}\bar{z}_{2}-q_{1}\bar{z}_{1})\\\
\mathbb{I}_{2}&={\pi c\over
24}\left[\left(q_{1}+q_{2}\right)\left({2\over\epsilon}+\gamma+\ln\pi\right)+q_{1}\ln|z_{1}|^{2}+q_{2}\ln|z_{2}|^{2}\right]\end{split}$
(2.22)

Since the autocorrelation function is related to the unequal time thermal two
point function of the same operator $\mathcal{O}$, the charges $q_{1}$ and
$q_{2}$ are equal i.e $q_{1}=q_{2}=q$. Hence, the results of the above
integrals simplify remarkably to,

$\begin{split}\mathbb{I}_{1}&={\pi\over\bar{z}_{12}}\lambda\,\bar{h}\,q(\bar{z}_{2}-\bar{z}_{1})\\\
\mathbb{I}_{2}&={\pi c\over
24}\left[2q\left({2\over\epsilon}+\gamma+\ln\pi+\ln\mu\right)+q\ln(|z_{1}z_{2}|^{2})\right]\end{split}$
(2.23)

Analogous to the T$\bar{\text{T}}$ case, we may compute the thermal two point
function with deformations expressing eq. 2.23 in cylindrical coordinates and
obtain the corresponding autocorrelation function. Computation of correlation
function of the deformed theory non-perturbatively was first proposed in [76]
and was also investigated in [88, 87]. These approaches promise to yield
highly nontrivial insights into the renormalization flow structure of
correlation functions.
Finally, we consider another marginal deformation to Euclidean CFTs induced by
a bilinear operator J$\bar{\text{J}}$ constructed from both the conserved
holomorphic and antiholomorphic $U(1)$ currents [76, 88, 87]. From the OPE and
the Ward identity, we may obtain the deformed two point function on the
complex plane as follows

$\displaystyle\langle
J(z)\bar{J}(\bar{z})O(z_{1},\bar{z}_{1})O(z_{2},\bar{z}_{2})\rangle_{\lambda}=$
$\displaystyle\lambda\int
dzd\bar{z}\left(\frac{q_{1}}{z-z_{1}}+\frac{q_{2}}{z-z_{2}}\right)\left(\frac{{\bar{q}}_{1}}{\bar{z}-\bar{z}_{1}}+\frac{{\bar{q}}_{2}}{\bar{z}-\bar{z}_{2}}\right)$
$\displaystyle\qquad\langle
O(z_{1},\bar{z}_{1})O(z_{2},\bar{z}_{2})\rangle_{0}$ (2.24)

The deformed thermal correlation function is therefore given by

$\displaystyle\frac{\langle\mathcal{O}\left(w_{1},\bar{w}_{1}\right)\mathcal{O}\left(w_{2},\bar{w}_{2}\right)\rangle_{\beta,\lambda}}{\langle\mathcal{O}\left(w_{1},\bar{w}_{1}\right)\mathcal{O}\left(w_{2},\bar{w}_{2}\right)\rangle_{\beta,0}}=1+\lambda\int
d^{2}w\frac{\left\langle
J(w)\bar{J}(\bar{w})\mathcal{O}\left(w_{1},\bar{w}_{1}\right)\mathcal{O}\left(w_{2},\bar{w}_{2}\right)\right\rangle_{\beta}}{\langle\mathcal{O}\left(w_{1},\bar{w}_{1}\right)\mathcal{O}\left(w_{2},\bar{w}_{2}\right)\rangle_{\beta,0}}$
(2.25)

Utilizing the conformal transformation given in eq. 2.6 the required
correlator on the cylinder may be related to that on the complex plane as
follows

$\displaystyle\langle
J(w)\bar{J}(\bar{w})\mathcal{O}(w_{1},\bar{w}_{1})\mathcal{O}(w_{2},\bar{w}_{2})\rangle_{\beta}$
(2.26) $\displaystyle=$ $\displaystyle\prod_{i=1,2}\Big{(}\frac{2\pi
z_{i}}{\beta}\Big{)}^{h_{i}}\Big{(}\frac{2\pi\bar{z}_{i}}{\beta}\Big{)}^{h_{i}}\Big{(}\frac{2\pi\bar{z}}{\beta}\Big{)}\Big{(}\frac{2\pi{z}}{\beta}\Big{)}\langle
J(z)\bar{J}(z)\mathcal{O}(z_{1},\bar{z}_{1})\mathcal{O}(z_{2},\bar{z}_{2})\rangle_{0}$
(2.27)

Utilizing the above expression and section 2 in eq. 2.25 reduces the
computation of the required thermal two point correlation into evaluation of
the following integrals

$\displaystyle{\cal Y}_{1}=\int dzd\bar{z}\leavevmode\nobreak\
\frac{q_{1}\bar{q}_{1}}{|z-z_{1}|^{2}},\quad{\cal Y}_{2}=\int
dzd\bar{z}\leavevmode\nobreak\ \frac{q_{2}\bar{q}_{2}}{|z-z_{2}|^{2}}$
$\displaystyle{\cal Y}_{3}=\int dzd\bar{z}\leavevmode\nobreak\
\frac{q_{2}}{z-z_{2}}\frac{\bar{q}_{1}}{\bar{z}-\bar{z}_{1}},\quad{\cal
Y}_{4}=\int dzd\bar{z}\leavevmode\nobreak\
\frac{q_{1}}{z-z_{1}}\frac{\bar{q}_{2}}{\bar{z}-\bar{z}_{2}}$ (2.28)

The details of the computation of the above integrals are provided in appendix
A.3. While ${\cal Y}_{1}$ and ${\cal Y}_{2}$ only contribute at ${\cal
O}(\epsilon)$, ${\cal Y}_{3}$ and ${\cal Y}_{4}$ contribute at the leading
order which leads us to the following result

$\displaystyle\frac{\langle\mathcal{O}\left(w_{1},\bar{w}_{1}\right)\mathcal{O}\left(w_{2},\bar{w}_{2}\right)\rangle_{\beta,\lambda}}{\langle\mathcal{O}\left(w_{1},\bar{w}_{1}\right)\mathcal{O}\left(w_{2},\bar{w}_{2}\right)\rangle_{\beta,0}}=1-\pi\lambda\left(q_{1}\bar{q}_{2}+\bar{q}_{1}{q}_{2}\right)\left({2\over\epsilon}+\gamma+\ln\pi+\ln\mu|z_{12}|^{2}\right)+\mathcal{O}(\epsilon)$
(2.29)

Once again since we are finally interested in computing the autocorrelation
function, we may take $q_{1}=q_{2}=q$ to obtain

$\displaystyle\frac{\langle\mathcal{O}\left(w_{1},\bar{w}_{1}\right)\mathcal{O}\left(w_{2},\bar{w}_{2}\right)\rangle_{\beta,\lambda}}{\langle\mathcal{O}\left(w_{1},\bar{w}_{1}\right)\mathcal{O}\left(w_{2},\bar{w}_{2}\right)\rangle_{\beta,0}}=1-2\pi\lambda
q^{2}\left({2\over\epsilon}+\gamma+\ln\pi+\ln\mu|z_{12}|^{2}\right)+\mathcal{O}(\epsilon)$
(2.30)

One can also construct a general class of integrable deformations of CFTs in
particular of the WZW type with non abelian current-current operators. These
are known as $\lambda$-deformations, first constructed in [89] which reduce to
J$\bar{\text{J}}$ deformation at the perturbative limit. In particular the
$\lambda$-deformed action reads

$S_{k,\lambda}(g)=S_{\text{WZW}}(g)+\frac{k}{2\pi}\int
d^{2}z\,J^{a}(z)\,(\lambda^{-1}-D^{T})_{ab}\bar{J}^{b}(\bar{z})$ (2.31)

where $g\in G$ a semi-simple group element, $k$ is the integer level of the
undeformed WZW model and $\lambda$ is the coupling constant. Rescaling the
current as $J^{a}\to(1/\sqrt{k})J^{a}$ the perturbative limit reads

$S_{k,\lambda}(g)=S_{\text{WZW}}(g)+\frac{\lambda}{2\pi}\int
d^{2}z\,J^{a}(z)\bar{J}^{a}(\bar{z})$ (2.32)

The deformed two point functions have been computed non-perturbatively in [90,
91] and reads

$\langle{\mathcal{O}}(z_{1},\bar{z}_{1})\mathcal{O}(z_{2},\bar{z}_{2})\rangle=\frac{\delta_{IJ}}{z_{12}^{\gamma^{(I)}_{R,R^{\prime}}(k,\lambda)}\bar{z}_{12}^{\gamma^{(I)}_{R,^{\prime}R}(k,\lambda)}}$
(2.33)

Compared to CFTs, here in eq. 2.33 the scaling dimension of operators in the
two point function got replaced by anomalous dimensions
$\gamma^{(I)}(k,\lambda)$.

In the next section, we proceed to evaluate the Lanczos coefficients and the
Krylov complexity for perturbative T$\bar{\text{T}}$, J$\bar{\text{T}}$ and
J$\bar{\text{J}}$ deformations. For the case of T$\bar{\text{T}}$, the Lanczos
coefficients and Krylov exponent extracted from the large time behaviour of
Krylov complexity receive correction at the first order in $\lambda$. We
demonstrate that the corrected Lanczos coefficients do not exhibit linear
growth with $n$ in the valid perturbation regime. Moreover, through numerical
analysis, we illustrate that the Krylov complexity for positive value of
deformation parameter seemingly contravenes the conjectured bound for the
Krylov exponent within the finite dimensional Krylov space, proposed in [40].
In contrast to this for the J$\bar{\text{T}}$ case, the Lanczos coefficients
and the Krylov complexity undergo no corrections at the first order in
$\lambda$. On the other hand, we will also demonstrate that although the
Lanczos coefficients and the Krylov complexity receive corrections for the
J$\bar{\text{J}}$ deformation at first order in $\lambda$, both the large-$n$
behaviour of $b_{n}$ and the Krylov exponent are exactly same as that of the
undeformed CFT. We do not expect any corrections for both the Lanczos
coefficients and Krylov complexity due to the deformation in eq. 2.33, since
there is no nontrivial time dependent corrections involved in the two point
function compared to the undeformed CFT case.

##  3 Krylov complexity in deformed CFTs

In this section, we very briefly review the notion of Krylov complexity and
Lanczos coefficients555For detailed reviews we refer the reader to [40, 92,
44, 93, 94, 95, 32, 33, 96, 45, 97, 98] and then proceed to compute these
quantities for perturbative T$\bar{\text{T}}$, J$\bar{\text{T}}$ and
J$\bar{\text{J}}$ deformations of a two dimensional Euclidean conformal field
theory which were described in the previous section.

###  3.1 Review of Lanczos coefficients and Krylov complexity

The evolution of an operator in the Heisenberg picture is given by

$\mathcal{O}(t)=e^{iHt}\mathcal{O}(0)e^{-iHt}$ (3.1)

Upon expanding the above expression order by order in $t$ leads to nested
commutators of the operator of interest at $t=0$, whose span forms an
invariant subspace known as the Krylov space,
$\operatorname{span}\left\\{\mathcal{L}^{n}\mathcal{O}\right\\}_{n=0}^{+\infty}=\operatorname{span}\\{\mathcal{O},[H,\mathcal{O}],[H,[H,\mathcal{O}]],\ldots\\}$.
The ${\mathcal{L}}$ itself is known as the Liouvillian whose action on
operators is defined by the commutator $[H,.]$. An orthonormal basis for the
Krylov space mentioned above can be obtained utilizing the Gram-Schmidt
procedure, which in this particular case is known as the Lanczos algorithm
[41, 40]. This algorithm begins with the initial operator as the zeroth
element i.e $\mid\mathcal{O}(t=0)):=\mid\mathcal{O}_{0})$ and leads to the
entire basis given as follows

$\displaystyle\mid\tilde{\mathcal{O}}_{n})$
$\displaystyle=\mathcal{L}\mid\mathcal{O}_{n-1})-b_{n-1}\mid\mathcal{O}_{n-2})-a_{n-1}|{\cal
O}_{n-1})$ (3.2) $\displaystyle\text{with}\leavevmode\nobreak\
\leavevmode\nobreak\ \mid\mathcal{O}_{n})$
$\displaystyle=b_{n}^{-1}\mid\tilde{\mathcal{O}}_{n}),\quad
b_{n}=(\tilde{\mathcal{O}}_{n}\mid\tilde{\mathcal{O}}_{n})^{1/2},\quad
a_{n}=(\mathcal{O}_{n}\mid\mathcal{L}\mid\mathcal{O}_{n})$ (3.3)

The coefficients $b_{n}$ and $a_{n}$ are called the Lanczos coefficients. The
construction of the Krylov space, requires a positive semi-definite inner
product for instance like Wightman inner product

$\displaystyle(\mathcal{O}_{1}\mid\mathcal{O}_{2})=\operatorname{Tr}\left(\rho^{\frac{1}{2}}\mathcal{O}_{1}^{\dagger}\rho^{\frac{1}{2}}\mathcal{O}_{2}\right)$
(3.4)

where $\rho=e^{-\beta H}$ is the thermal density matrix. Therefore, the
operator at any time can be expanded in the Krylov basis

$\displaystyle|\mathcal{O}(t))=\sum_{n}i^{n}\varphi_{n}(t)|\mathcal{O}_{n})$
(3.5)

In the above expression $\varphi_{n}(t)$ are treated as the probability
amplitude for the operator to be in the basis state $|\mathcal{O}_{n})$ and
the Heisenberg equation for the operator reduces to a one dimensional
recursive equation resembling that of an electron hopping on a 1d chain which
is as follows

$\displaystyle\partial_{t}\varphi_{n}(t)=b_{n}\varphi_{n-1}(t)-b_{n+1}\varphi_{n+1}(t)+ia_{n}\varphi_{n}(t)$
(3.6)

This recursion relation has to be solved with the boundary conditions
$\varphi_{-1}(t)=0$, along with $b_{0}=0$, and $\varphi_{n}(0)=\delta_{n,0}$.
The Krylov complexity characterizes the average position of $\phi_{n}$’s in
the 1d chain and therefore can be defined as

$\displaystyle
K_{\mathcal{O}}(t)=\left(\mathcal{O}(t)|n|\mathcal{O}(t)\right)=\sum_{n=0}^{\infty}n\left|\varphi_{n}(t)\right|^{2}$
(3.7)

Having reviewed the notion of the Lanczos coefficients and the Krylov
complexity of an operator of a quantum system, we now proceed towards their
computation in a two dimensional CFT whose Hamiltonian is deformed by
T$\bar{\text{T}}$, J$\bar{\text{T}}$ and J$\bar{\text{J}}$ composite
operators. We will assume that the primary operators of the seed CFT are
undeformed and the time evolution of these operators depend on the Hamiltonian
of the theory which is changing along the flow. Thus finite time operators
will change due to their dependence on deformed Hamiltonian [88],

$\displaystyle\mathcal{O}(t,0)=e^{itH_{\lambda}}\mathcal{O}(0,0)e^{-itH_{\lambda}}$
(3.8)

where $H_{\lambda}=\int dxT_{00}^{(\lambda)}(0,x)$. Note that the zeroth order
wave function $\varphi_{0}(t)$ which is the primary quantity crucial for all
the computations is known as the autocorrelation function. For a Hermitian
operator $\mathcal{O}$, this autocorrelation function is given by

$\displaystyle C_{0}(t)\equiv\varphi_{0}(t)$
$\displaystyle=(\mathcal{O}(t)|\mathcal{O}(0))$
$\displaystyle=\langle\mathcal{O}(t-i\beta/2)\mathcal{O}(0)\rangle_{\beta}$
(3.9)

where the last line which relates the autocorrelation function to the thermal
two point function is obtained by utilizing the definition of Wightman inner
product in eq. 3.4, which incorporates a shift in Euclidean time by $\beta/2$
to avoid the divergence. From the above expression one can observe that the
autocorrelation function is directly related to the thermal two point
function. One can compute moments from the Taylor series expansion
coefficients of $C_{0}(t)$ around $t=0$. These moments satisfy a recursion
relation which is ultimately utilized to compute the Lanczos coefficients
$b_{n}$. In [40] it was conjectured that $b_{n}$ satisfy a linear growth for
large $n$, for maximal chaotic systems evolving under a Hamiltonian involving
local interactions. This aspect can also be understood from the high frequency
behaviour of the power spectrum obtained from the Fourier transform of the
autocorrelation function,

$f^{2}(\omega)={1\over 2\pi}\int_{-\infty}^{\infty}\leavevmode\nobreak\
dte^{i\omega t}C_{0}(t)$ (3.10)

If $C_{0}(t)$ has a pole at $t=i/\omega_{0}$, then $f^{2}(\omega)$ has an
exponential asymptotic behaviour

$f^{2}(\omega)|_{\omega\rightarrow\infty}\sim e^{-\omega/\omega_{0}}$ (3.11)

It is important to note that asymptotic linear growth of $b_{n}$ implies
exponential decay of high frequency tail of the power spectrum but not vice-
versa [43].

In the previous section, we have reviewed the thermal two point functions
involving primary operators of a two dimensional CFT with the first order
perturbative correction induced by T$\bar{\text{T}}$, J$\bar{\text{T}}$ and
J$\bar{\text{J}}$ deformations following [73, 81]. We will utilize these
results and translate them in Euclidean time by $\beta/2$ to obtain the series
expansion

$\displaystyle C_{0}^{\lambda}(t)$
$\displaystyle=\langle\mathcal{O}(t-i\beta/2)\mathcal{O}(0)\rangle_{\beta,\lambda}$
$\displaystyle=\langle\mathcal{O}(t-i\beta/2)\mathcal{O}(0)\rangle_{\beta,0}+\lambda\langle\mathcal{O}(t-i\beta/2)\mathcal{O}(0)\rangle^{\lambda}_{\beta}+O(\lambda^{2})$
(3.12)

where $\langle\mathcal{O}(t-i\beta/2)\mathcal{O}(0)\rangle^{\lambda}_{\beta}$
denotes the first order correction to the autocorrelation function. One should
note that this autocorrelation function needs to be properly normalised with a
factor of $C_{0}^{\lambda}(0)$ to obtain the autocorrelation function

$C_{0}(t)={C_{0}^{\lambda}(t)\over C_{0}^{\lambda}(0)},$

before proceeding with the computations for Lanczos coefficients. Determining
the required correlators involve evaluating numerous integrals whose
computational details are in appendix A. Once the autocorrelation function is
determined, the Lanczos coefficients are computed utilizing either the moment
method or the Toda chain technique. Since these two methods are explained in
quite extensive details in several recent literature, we do not discuss them
here, however we refer the interested readers to detailed discussions in [40,
41, 44, 46]. From the autocorrelation function and the Lanczos coefficients
the operator wave functions are then obtained by solving the recursion
relation in eq. 3.6. Finally, the Krylov complexity is determined using eq.
3.7.

###  3.2 T$\bar{\text{T}}$

In this section we will focus on the final results and discussion of the
plausible physical interpretations of the behaviour of Krylov complexity under
the above described deformation.

####  3.2.1 Autocorrelation function and Lanczos coefficients

We compute the autocorrelation function for a T$\bar{\rm T}$ deformed 2D CFT
from section 2, which yields

$\displaystyle C_{0}^{\lambda}(t)=$ $\displaystyle{1\over\left(\cosh{\pi
t\over\beta}\right)^{2\Delta}}+\tilde{\lambda}\bigg{\\{}f_{1}(\nu_{1},\beta,t)-f_{2}(\nu_{2},\beta,t)\bigg{\\}}$
(3.13)

with

$\displaystyle f_{1}(\nu_{1},\beta,t)$
$\displaystyle={\pi\Delta^{2}\over\left(\cosh{\pi
t\over\beta}\right)^{2\Delta+2}}\left[\nu_{1}+\frac{1}{2}\ln\left(\cosh{\pi
t\over\beta}\right)^{2}\right]$ (3.14) $\displaystyle f_{2}(\nu_{2},\beta,t)$
$\displaystyle={\pi\Delta^{2}\over\left(\cosh{\pi
t\over\beta}\right)^{2\Delta}}\left[\nu_{2}+\frac{1}{2}\ln\left(\cosh{\pi
t\over\beta}\right)^{2}\right]$ (3.15)

where we have denoted

$\displaystyle\tilde{\lambda}=\Big{(}\frac{2\pi}{\beta}\Big{)}^{2}\lambda,\qquad
h=\bar{h}=\Delta/2$ $\displaystyle\nu_{1}={1\over
4}\bigg{[}{4\over\epsilon}+2\gamma-5+2\ln\pi\mu+4\ln 4\bigg{]},$
$\displaystyle\nu_{2}={1\over
2}\bigg{[}{2\over\epsilon}+\gamma-3+\ln\pi\mu+2\ln
4\bigg{]}=\nu_{1}-\frac{1}{4}.$ (3.16)

Here $\mu$ is an arbitrary mass scale appearing due to the dimensional
regularization procedure. We then calculate the first order corrections to the
Lanczos coefficients, given by

$\displaystyle
b_{n}=b_{0}(n)+\tilde{\lambda}b_{1}(n)\bigg{\\{}\nu_{1}-b_{2}(n)\bigg{\\}}+{\cal
O}(\lambda^{2}),$ (3.17)

where,

$\displaystyle b_{0}(n)$
$\displaystyle=\frac{\pi}{\beta}\sqrt{n(2\Delta+n-1)}$ (3.18) $\displaystyle
b_{1}(n)$
$\displaystyle=\frac{\pi^{2}\Delta\sqrt{n(2\Delta+n-1)}(2\Delta+2n-1)}{2\beta(2\Delta+1)}=b_{0}(n){\pi\Delta(2\Delta+2n-1)\over(2\Delta+1)}$
(3.19) $\displaystyle b_{2}(n)$
$\displaystyle=\frac{4\Delta(n-1)}{(2\Delta+1)(2\Delta+2n-1)}-\frac{1}{2\Delta}+H_{n+2\Delta-1}-H_{2\Delta-1}$
(3.20)

where $H_{n}=\sum_{k=1}^{n}{1\over k}$ and $b_{0}(n)$ represents the
undeformed case [45]. Since the operator growth hypothesis is concerned with
the asymptotic large-n behaviour of Lanczos coefficients, let us now examine
the same utilizing the expressions above. One can easily check that the
$b_{0}(n)$ grow linearly whereas the $O(\lambda)$ contributions involving
$b_{1}(n)$ and $b_{1}(n)b_{2}(n)$ both grows quadratically. However since they
are multiplied by a perturbative parameter $\lambda$, their contributions are
small for low values of $n$. Furthermore, these contributions turn out to be
negative for $n>4$ with positive value of $\lambda$, resulting in a quick
dampening of the growth as depicted in the plots given in Fig. 1.

(a) $\lambda=0.07$

(b) $\lambda=0.08$

(c) $\lambda=0.10$

(d) $\lambda=0.12$

Figure 1: Lanczos coefficients of a two dimensional CFT contrasted with the
same post T$\bar{\text{T}}$ deformation until the perturbation breaks down.
$b_{n}$ for the deformed theory will be greater than the undeformed one until
$n=4$. A quick numerical calculation will show from eq. 3.17 that
$b_{n}=b_{0}(n)$, when $n=4.265$.

####  3.2.2  Krylov complexity

Utilizing the Lanczos coefficients determined above, the operator wave
functions are obtained by solving the recursive one dimensional equation given
by eq. 3.6 perturbatively. We also set $\Delta=2$ for computational
convenience (the behaviour of complexity is similar for other values of
$\Delta$ as well). The probabilities corresponding to the operator wave
functions are as follows

$\displaystyle|\phi_{n}(t)|^{2}=P_{0}(n,t)-\tilde{\lambda}\\{P_{1}(n,t)+\nu_{1}P_{2}(n,t)\\}+{\cal
O}(\lambda^{2})$ (3.21)

where $P_{0}(n,t)$ is the unperturbed probability and $P_{1},P_{2}$ are the
corrections due to the deformation. The functional form of $P_{0},P_{1},P_{2}$
are given as follows

$\displaystyle P_{0}(n,t)$
$\displaystyle=\frac{1}{6}(n+1)(n+2)(n+3)\leavevmode\nobreak\
\text{sech}^{8}\left(\frac{t}{2}\right)\tanh^{2n}\left(\frac{t}{2}\right)$
$\displaystyle P_{1}(n,t)$ $\displaystyle=\pi
f_{0}^{2}(n,t)\sum_{i=1}^{12}g_{i}(n,t)$ $\displaystyle P_{2}(n,t)$
$\displaystyle=\pi f_{0}^{2}(n,t)\sum_{i=1}^{5}h_{i}(n,t)$ (3.22)

We have provided all the functions $f_{0},g_{i},h_{i}$ in appendix B.

The Krylov complexity itself is not analytically tractable and hence we resort
to numerical computations. Also note that since $b_{n}$’s go to zero after a
reasonably large-$n$, we should not be summing the above probabilities all the
way upto infinity to obtain the Krylov complexity. In addition, it is
pertinent to inquire about the reliability of the Krylov complexity derived
from a perturbative autocorrelation function, questioning until which point in
time it remains valid.

(a) $\lambda=0.07$

(b) $\lambda=0.08$

(c) $\lambda=0.10$

(d) $\lambda=0.12$

Figure 2: Sum of absolute squares of Probability post $T\bar{T}$ deformation
until the perturbation for the autocorrelation function breaks down for
various values of the deformation parameter $\lambda$.

This question was also addressed in [52] where the behaviour of Krylov
complexity was examined numerically from a perturbative autocorrelation
function. As described in [52], there are two time scales involved in such an
analysis. There is a $t_{\rm max}$ which characterizes the time at which the
perturbative series corresponding to the autocorrelation function breaks down
i.e terms of $O(\lambda^{0})$ and $O(\lambda)$ in the autocorrelation function
start competing with each other. However, there is another time scale $t_{c}$
which emerges as a consequence of truncating the infinite series in eq. 3.7 to
a finite sum, beyond which the probabilities’ sum $\sum_{i}|\phi_{n}(t)|^{2}$
begins to decay to values less than 1. The numerics will be only valid if
$t_{c}>t_{\rm max}$ otherwise the finite sum effects start distorting the
behaviour of the Krylov complexity before the perturbation breaks down. Taking
all these into account by choosing appropriate $\lambda$ such that
$t_{c}>t_{\rm max}$ the behaviour of Krylov complexity is depicted in the plot
given in Fig 3, while the probability conservation is shown in Fig. 2.

Note that the plot in Fig 3 seems to indicate that the Krylov complexity is
growing faster compared to the undeformed CFT one with an exponent
$\lambda_{K}>\frac{2\pi}{\beta}$. Several comments are in place here as this
seems to be violating the bound conjectured in [40]. Note that this is a
perturbative result and it is possible that the apparent violation might go
away once higher order corrections are taken into account [43]. One analogous
example is the case of higher spin symmetry where a violation of the MSS bound
for the Lyapunov exponent obtained from OTOCs was observed if one takes into
account a finite number of higher spin conformal blocks [99]. However, it was
argued that if one takes into account all the higher spin blocks (an infinite
number of them) they may arrange into a geometric series which can be summed
and the resultant Lyapunov exponent shows no violation of the bound anymore.

(a) $\lambda=0.07$

(b) $\lambda=0.08$

(c) $\lambda=0.10$

(d) $\lambda=0.12$

Figure 3: Krylov complexity of a two dimensional CFT contrasted with the the
same post T$\bar{\rm T}$ deformation for various values of the deformation
parameter $\lambda$.

For a comprehensive understanding we will also consider negative value of
deformation parameter. So far we were following the convention of [80, 81]
with the deformation parameter $\lambda>0$, which is canonically known as the
"bad sign". The origin of the name came from the fact that at sufficiently
large energy of the undeformed theory the deformed spectrum become complex.
The advantage of this sign is the observed holographic duality between the
T$\bar{\rm T}$ deformed theory and putting Dirichlet boundary condition at
some finite radius of AdS space. On the other hand one can also consider
deformations with $\lambda<0$ corresponding to the "good sign", where the
undeformed energy can be taken to be infinitely large. Interestingly in this
case the T$\bar{\rm T}$ deformed CFT shares intriguing similarities with a
class of non-local theories called little string theory which is in turn dual
to gravity theories on linear dilaton background [100].

(a) $\lambda=-0.07$

(b) $\lambda=-0.08$

(c) $\lambda=-0.10$

(d) $\lambda=-0.12$

Figure 4: Lanczos coefficients of a two dimensional CFT contrasted with the
same post T$\bar{\text{T}}$ deformation until the perturbation breaks down for
negative values of the deformation parameter.

(a) $\lambda=-0.07$

(b) $\lambda=-0.08$

(c) $\lambda=-0.10$

(d) $\lambda=-0.12$

Figure 5: Sum of absolute squares of Probability post $T\bar{T}$ deformation
until the perturbation for the autocorrelation function breaks down for
negative values of the deformation parameter $\lambda$.

(a) $\lambda=-0.07$

(b) $\lambda=-0.08$

(c) $\lambda=-0.10$

(d) $\lambda=-0.12$

Figure 6: Krylov complexity of a two dimensional CFT contrasted with the the
same post T$\bar{\rm T}$ deformation for various negative values of the
deformation parameter $\lambda$.

In contrast to its counterpart discussed before, $b_{n}$’s do not go to zero
for $\lambda<0$ in Fig. 4. One should still be cautious and observe that there
is a finite $n_{\text{max}}$ after which $O(\lambda)$ correction starts being
of the same order as $O(\lambda^{0})$. Furthermore, as discussed earlier
because of the truncation of infinite sum corresponding to Krylov complexity,
there is a time $t_{c}$ only up to which the probabilities sum to unity (Fig.
5) and there is a time $t_{\rm max}$ upto which the perturbative series for
autocorrelation remains valid. Taking all these into account and sticking to
the regime where $t_{c}>t_{\text{max}}$, Krylov complexity for this case is
given in fig. 6. Interestingly in this case $\lambda_{K}<{2\pi\over\beta}$
obeying the MSS bound. But as can be seen from fig. 4, the growth of $b_{n}$
in this case is superlinear for $n>4$. Therefore, although with $\lambda<0$ we
recover the MSS bound the price one had to pay is the superlinear growth of
$b_{n}$. In fact, this observation is quite intriguing because of two
interesting facts. The first one is that the growth of $b_{n}$ should be
linearly bounded for local theories. On the other hand it has been observed
that the asymptotic density of states for T$\bar{\rm T}$ deformed CFT with the
"good" sign interpolates between Cardy behavior and Hagedorn behavior. The
Hagedorn behavior indicates non-locality of T$\bar{\rm T}$ deformed CFTs. This
exact observation fuelled the series of works in [100, 101, 102], showing
T$\bar{\rm T}$ deformed CFT as the boundary description of two dimensional
vacua of little string theory. Therefore the superlinear behavior of the
Lanczos coefficients of T$\bar{\rm T}$ deformed CFT for $\lambda<0$ is quite
interesting but not so unexpected.

(a) $\lambda=\pm 0.07$

(b) $\lambda=\pm 0.08$

(c) $\lambda=\pm 0.10$

(d) $\lambda=\pm 0.12$

Figure 7: Power spectrum of a two dimensional CFT contrasted with the the same
post T$\bar{\rm T}$ deformation for both positive and negative values of the
deformation parameter $\lambda$.

Again we should emphasize that it can be a artifact of perturbative approach
and to confirm this apparent superlinear growth one needs to compute higher
order correction to the autocorrelation function666We thank Anatoly Dymarsky
for drawing our attention to this point.. In this regard we also studied the
power spectrum by using (3.10). Note that $C_{0}^{\lambda}(t)$ given in (3.13)
has both higher order poles and logarithmic branch points at
$t=(2m+1)i\beta/2$ where $m=0,\pm 1\cdots$, compared to the undeformed case.
Therefore one can expect corrections to the the asymptotic exponential tail of
power spectrum discussed in (3.11) originating from higher order poles and
branch points. The analytical computations seems quite challenging and are
beyond the scope of our current work. We have considered the power spectrum
numerically for both the positive and negative values of the deformation
parameter given in Fig. 7. One can observe that the power spectrum tail for
positive deformation parameter is surpassing the undeformed CFT power spectrum
tail which may support the apparent violation of universal operator growth
hypothesis.

###  3.3 J$\bar{\text{T}}$

Following the same suit as before, we will now proceed with J$\bar{\rm T}$
deformation, picking up our computations for thermal two point function from
eq. 2.23.

#### Autocorrelation function and Lanczos coefficients

The autocorrelation function for this case can be obtained from eq. 2.23 to
give,

$\displaystyle C_{0}^{\lambda}(t)=\frac{1}{\cosh(\frac{\pi
t}{\beta})^{2\Delta}}\bigg{[}1+\Big{(}\frac{\pi\lambda\Delta}{\beta}\Big{)}q\left(-1+\frac{\pi
c}{6\Delta}\nu_{2}\right)\bigg{]}$ (3.23)

Since the autocorrelation function is proportional to that of the undeformed
CFT at the leading order in $\lambda$, the Lanczos coefficients $b_{n}$
corresponding to the above autocorrelation function receives no correction at
this order and are given by

$\displaystyle b(n)$
$\displaystyle=\frac{\pi\sqrt{n(2\Delta+n-1)}}{\beta}+{\cal O}(\lambda^{2})$

#### Krylov complexity

Since $b_{n}$s receive no correction at the leading order in $\lambda$ and the
autocorrelation function is proportional to that of the undeformed CFT case,
the wave functions are well known and the Krylov exponent is
$\lambda_{K}=\frac{2\pi}{\beta}$, same as the undeformed CFT.

###  3.4 J$\bar{\text{J}}$

In this subsection, we determine the Lanczos coefficients and the Krylov
complexity of an operator of a 2D CFT with J$\bar{\text{J}}$ deformation.

#### Autocorrelation function and Lanczos coefficients

The autocorrelation function can be computed likewise,

$\displaystyle C_{0}^{\lambda}(t)=\frac{1}{\Big{(}\cosh\frac{\pi
t}{\beta}\Big{)}^{2\Delta}}\Big{[}1-{2\pi\lambda
q^{2}}\bigg{(}\nu_{1}+2\log\bigg{(}\cosh\frac{\pi
t}{\beta}\bigg{)}\bigg{)}\bigg{]}+{\cal O}(\lambda^{2})$

where $\nu_{1}=\left({2\over\epsilon}+\gamma+\ln\pi+2\ln 4\right)$. Which is
computed through the result quoted in eq. 2.30. The corresponding Lanczos
Coefficients are as follows

$\displaystyle b(n)$
$\displaystyle=\frac{\pi\sqrt{n(2\Delta+n-1)}}{\beta}+\lambda\bigg{(}\frac{2\pi^{2}q^{2}\Delta}{\beta^{2}}\bigg{)}\sqrt{\frac{n}{2\Delta+n-1}}+{\cal
O}(\lambda^{2})$

As we can see for large $n$,

$\displaystyle b_{n}\approx\alpha n+\delta$ (3.27)

where the slope is $\alpha={\pi\over\beta}$ and hence unaffected by the
deformation.

#### Krylov Complexity

The necessary probability amplitudes are given by

$\displaystyle|\phi(n,t)|^{2}=|\phi_{\rm
undef}(n,t)|^{2}\bigg{[}1+\lambda\frac{4\pi q^{2}}{\beta}\left\\{\Delta
H_{n+2\Delta-1}-\gamma\Delta-\Delta\psi^{(0)}(2\Delta)-2\Delta\log\left[\cosh\left(\frac{\pi
t}{\beta}\right)\right]\right\\}\bigg{]}$ (3.28)

where $|\phi_{\rm undef}(n,t)|$ correspond to the probability amplitudes for
undeformed CFT and is given by

$\displaystyle|\phi_{\rm
undef}(n,t)|^{2}=\frac{\Gamma(n+2\Delta)}{\Gamma(2\Delta)\Gamma(n+1)}\tanh^{2n}\left(\frac{\pi
t}{\beta}\right)\text{sech}^{4\Delta}\left(\frac{\pi t}{\beta}\right)$ (3.29)

The Krylov Complexity can be computed analytically in this case and is given
by

$\displaystyle K(t)=2\Delta\Big{(}1+\lambda\frac{2\pi
q^{2}}{\beta}\Big{)}\sinh^{2}\left(\frac{\pi t}{\beta}\right)$ (3.30)

which grows exponentially at long times with the undeformed Krylov exponent
with $\lambda_{K}=\frac{2\pi}{\beta}$. Hence unlike the T$\bar{\text{T}}$
case, here the Krylov complexity for a 2d CFT under J$\bar{\text{J}}$
deformation saturates the bound for Krylov exponent conjectured in [40].

##  4 Discussion and Conclusion

In this work, we have explored the behaviour of the Lanczos coefficients and
the Krylov complexity for T$\bar{\text{T}}$, J$\bar{\text{T}}$ and
J$\bar{\text{J}}$ deformed two-dimensional CFTs perturbatively upto first
order of the deformation parameter. To this end, we began by first computing
the thermal two point functions in these deformed theories by mapping the CFT
on a Euclidean plane to a cylinder and then utilizing the dimensional
regularization scheme to get a handle on the spacetime integrals.
Subsequently, we determine the autocorrelation function which is related to
the thermal two-point function through the Wightman inner product. Our final
results show some intriguing features of these deformations. Before discussing
the implication of these results, let us summarise the main observations

  * •

T$\bar{\text{T}}$ deformation: In this case, we obtain the Lanczos
coefficients, operator wave functions analytically. However we resort to
numerical analysis for the Krylov complexity. Lanczos coefficients pick up
$\mathcal{O}(\lambda)$ corrections and the Krylov exponent seemingly violates
the universal bound proposed in [43] i.e $\lambda_{K}>{2\pi\over\beta}$. For
$\lambda<0$ we find that $\lambda_{K}<{2\pi\over\beta}$ but the Lanczos
coefficients exhibit a superlinear growth violating the operator growth
hypothesis [40].

  * •

J$\bar{\text{T}}$ deformation: Lanczos coefficients remain unaltered at
$\mathcal{O}(\lambda)$ and the Krylov exponent also remains same as that of
the undeformed case saturating the conjectured bound.

  * •

J$\bar{\text{J}}$ deformation: Although for small-$n$, the Lanczos
coefficients receive correction at $\mathcal{O}(\lambda)$, their large-$n$
behaviour is same as that of the undeformed CFT. The Krylov complexity grows
exponentially at long times while the Krylov exponent remains to be
${2\pi\over\beta}$, which is exactly same as that of the undeformed CFT.

As we demonstrated, our observations are a result of perturbative expansion of
autocorrelation functions as well as a careful truncation of the infinite
probability sum to a finite sum such that it still remains unity. However, to
confirm our observation beyond the perturbative domain, one should either
compute all higher order corrections to the Lanczos coefficients and resum or
pursue a non-perturbative approach towards computing Krylov complexity for
deformed CFTs. While J$\bar{\rm J}$ deformation can be analyzed non-
perturbatively, it is quite challenging to get the exact non-perturbative
thermal two-point function for T$\bar{\rm T}$ deformed theories [71, 88].
Towards the first avenue, before computing the Lanczos coefficients one can
naively expand the autocorrelation function to further higher orders of the
deformation parameter $\lambda$ and subsequently observe their dependency on
the central charge thus breaking universality. Furthermore, our computation of
dimensionally regularized thermal two point function is evaluated up to
$\mathcal{O}(\lambda)$ and therefore any $\mathcal{O}(\lambda^{2})$ correction
in the Lanczos coefficients cannot be presumably _trusted_ in the domain of
this formalism. For a comprehensive understanding one has to modify the
recursion relations involving the Lanczos coefficients accordingly. We would
like to point out that even though our observed violation of the extended MSS
bound for T$\bar{\text{T}}$ deformed CFTs might apparently be seen as an
artifact of our perturbation scheme, but our other observations with the same
scheme of perturbation suggests otherwise. For the "good" sign of
T$\bar{\text{T}}$ deformed CFT we observe a superlinear growth of $b_{n}$
reinforcing the fact of it being a non-local QFT, observed earlier in the
literature [100, 101, 102]. Also we have seen no violation of the extended MSS
bound for J$\bar{\text{T}}$ as well as J$\bar{\text{J}}$ deformed CFTs.
Therefore the observed violation of MSS bound for T$\bar{\text{T}}$ deformed
CFT with $\lambda>0$ deserves a more thorough investigation in the near
future.

It is worth mentioning that, in [73] the authors proposed that
T$\bar{\text{T}}$ deformed CFTs with large central charge, will not alter the
maximal chaos bound ($\lambda_{L}=\frac{2\pi}{\beta}$) identified by the
exponential growth of OTOCs when treated perturbatively up to the first order
in the deformation parameter. Furthermore, since the Lyapunov exponent derived
from the OTOC remains unaffected, it was expected that the gravity dual of
T$\bar{\text{T}}$ deformed CFTs will correspond to a theory which saturates
the bound on chaos. One key assumption in [73] is that the T$\bar{\text{T}}$
deformed theory retains its integrability up to the first order of
deformation. Since our result apparently violates the extended MSS bound in
certain region at this order, it may have interesting implications for its
gravity dual [81, 103, 104].

Going beyond the first order correction in the deformation parameter for the
Lanczos coefficients is quite challenging although it is much needed since we
do not get any $\mathcal{O}(\lambda)$ corrections specially for
J$\bar{\text{T}}$ as well as J$\bar{\text{J}}$ deformed theories. Therefore at
present we are trying to develop a non-perturbative approach to compute Krylov
complexity for field theories with deformations to confirm if the apparent
violation of the conjectured bound for the Krylov exponent in the
T$\bar{\text{T}}$ deformations case, is just an artifact of the limitations of
our perturbative approach or it has some deeper implications. We hope to
address this exciting issue in near future.

## Acknowledgements

The work of AC is supported by the European Union Horizon 2020 research and
innovation programme under the Marie Skłodowska Curie grant agreement number
101034383. The work of VM was supported by the Brain Pool program funded by
the Ministry of Science and ICT through the National Research Foundation of
Korea (RS-2023-00261799). The work of AM is supported by POSTECH BK21
postdoctoral fellowship. VM and AM acknowledge the support by the National
Research Foundation of Korea (NRF) grant funded by the Korean government
(MSIT) (No. 2022R1A2C1003182). AC acknowledges the hospitality of Örebro
University during the final course of this work. We would like to thank Hugo
A. Camargo, Viktor Jahnke, Mitsuhiro Nishida and Pantelis Panopoulos for
useful discussions. We thank Anatoly Dymarsky for insightful comments on the
draft.

##  Appendix A Important integrals

In this section, we provide a comprehensive overview of the computational
details employed to determine the integrals of thermal two point functions on
cylinder for all three deformations.

###  A.1 Integrals for T$\bar{\text{T}}$

In the following we will explicitly evaluate $I_{1}$,

$\displaystyle I_{1}=|z_{12}|^{4h}\int dzd\bar{z}\,z\bar{z}\langle
T(z)\bar{T}(\bar{z})\mathcal{O}(z_{1},\bar{z}_{1})\mathcal{O}(z_{2},\bar{z}_{2})\rangle_{0}$
(A.1)

We will state only the important steps of derivation for the rest. With some
manipulations and disregarding contact terms, we have the relation [73]

$\langle
T(z)\bar{T}(\bar{z})\mathcal{O}(z_{1},\bar{z}_{1})\mathcal{O}(z_{1},\bar{z}_{2})\rangle_{0}=h^{2}{|z_{12}|^{4}\over|z-z_{1}|^{4}|z-z_{2}|^{4}|z_{12}|^{4h}}.$
(A.2)

where we have assumed $h=\bar{h}$. In the following, we are going to evaluate
the integrals utilizing the following Feynman parametrization,

$\displaystyle{1\over|z-z_{1}|^{4}|z-z_{2}|^{4}}=6\int_{0}^{1}du{u(1-u)\over\left(|\tilde{z}|^{2}+u(1-u)|z_{12}|^{2}\right)^{4}}$
(A.3)

where

$\displaystyle z=\tilde{z}+uz_{1}+(1-u)z_{2}.$ (A.4)

Therefore we have the relation

$\displaystyle I_{1}$ $\displaystyle=$ $\displaystyle h^{2}|z_{12}|^{4}\int
d^{2}z\frac{z\bar{z}}{|z-z_{1}|^{4}|z-z_{2}|^{4}}$ (A.5) $\displaystyle=$
$\displaystyle
6h^{2}|z_{12}|^{4}\int_{0}^{1}du\,u(1-u)\int{d^{2}\tilde{z}A(\tilde{z},\bar{\tilde{z}})\over\left(|\tilde{z}|^{2}+u(1-u)|z_{12}|^{2}\right)^{4}},$

with

$A(\tilde{z},\bar{\tilde{z}})=|\tilde{z}|^{2}+\tilde{z}(u\bar{z}_{12}+\bar{z}_{2})+\bar{\tilde{z}}(uz_{12}+z_{2})+u|z_{1}|^{2}+(1-u)|z_{2}|^{2}-u(1-u)|z_{12}|^{2}.$

One should note that the integration over the second and third term of
$A(\tilde{z},\bar{\tilde{z}})$ will contribute to zero once we parametrize
$\tilde{z}$ in polar coordinates as $\tilde{z}=\rho e^{i\theta}$. Therefore
finally we have the relation

$\displaystyle I_{1}=$
$\displaystyle\overbrace{6h^{2}|z_{12}|^{4}\int_{0}^{1}du\,u(1-u)F_{1}^{2}(u)}^{T_{1}}+\overbrace{6h^{2}|z_{12}|^{4}|z_{1}|^{2}\int_{0}^{1}du\,u^{2}(1-u)F_{2}^{2}(U)}^{T_{2}}$
$\displaystyle+\underbrace{6h^{2}|z_{12}|^{4}|z_{2}|^{2}\int_{0}^{1}du\,u(1-u)^{2}F_{2}^{2}(U)}_{T_{3}}-\underbrace{6h^{2}|z_{12}|^{6}\int_{0}^{1}du\,u^{2}(1-u)^{2}F_{2}^{2}(U)}_{T_{4}},$

where $F_{1}^{2}(u),F_{2}^{2}(u)$ integrals are divergent. Therefore we
compute the integrals by dimensional regularization scheme according to

$F_{1}^{d}(u)=2V_{s^{d-1}}\int_{0}^{\infty}{\rho^{d-1}\rho^{2}d\rho\over\left(\rho^{2}+u(1-u)|z_{12}|^{2}\right)^{4}};\quad
F_{2}^{d}(u)=2V_{s^{d-1}}\int_{0}^{\infty}{\rho^{d-1}d\rho\over\left(\rho^{2}+u(1-u)|z_{12}|^{2}\right)^{4}}.$
(A.7)

with $V_{s^{d-1}}={\pi^{d\over 2}\over\Gamma({d\over 2})}$.

Using the following relation

$\int_{0}^{\infty}dt\,{t^{m-1}\over\left(t^{2}+a^{2}\right)^{n}}={1\over(a^{2})^{n-{m\over
2}}}{\Gamma({m\over 2})\Gamma({n-{m\over 2})}\over 2\Gamma({n})},$ (A.8)

we compute

$\displaystyle F_{1}^{d}(u)={d\pi^{d/2}\over
12\left(|z_{12}|^{2}\right)^{3-d/2}}{\Gamma(3-{d\over
2})\over\left(u(1-u)\right)^{3-d/2}},\quad F_{2}^{d}(u)={\pi^{d/2}\over
6\left(|z_{12}|^{2}\right)^{4-d/2}}{\Gamma(4-{d\over
2})\over\left(u(1-u)\right)^{4-d/2}}.$ (A.9)

Finally we can carry out the integration over $u$ in $T_{1}$ using the
following integral,

$\int_{0}^{1}{du\over\left(u(1-u)\right)^{2-d/2}}={2^{3-d}\sqrt{\pi}\Gamma\left({d\over
2}-1\right)\over\Gamma({d\over 2}-{1\over 2})},$ (A.10)

yielding

$T_{1}={d\over 2}h^{2}\pi^{d/2}\left(|z_{12}|^{2}\right)^{{d\over
2}-1}\Gamma\left(3-{d\over 2}\right){2^{3-d}\sqrt{\pi}\Gamma\left({d\over
2}-1\right)\over\Gamma({d\over 2}-{1\over 2})}$ (A.11)

This converges for $\text{Re}(d)>2$ but can be analytically continued to any
complex value of $d$. Therefore we can now take the limiting case
$d=2+\epsilon$ in eq. A.11, with $\epsilon$ being the regularization
parameter,

$T_{1}=\mu^{\epsilon\over 2}\left(1+{\epsilon\over
2}\right)\,h^{2}\,\pi\pi^{\epsilon/2}\left(|z_{12}|^{2}\right)^{\epsilon/2}\left(1-{\epsilon\over
2}\right){\Gamma(1-{\epsilon\over 2})\Gamma({\epsilon\over
2})\over\Gamma({\epsilon\over 2})\Gamma({1\over 2}+{\epsilon\over
2})}2^{1-\epsilon}\sqrt{\pi}\,\Gamma({\epsilon\over 2}).$ (A.12)

where $\mu$ is introduced as an arbitrary mass scale in order to keep the
correct units for the integral under dimensional regularisation. Furthermore
one can use the properties of Gamma functions

$\begin{matrix}\Gamma(1-z)\Gamma(z)=\displaystyle{\pi\over\sin\pi
z}\quad\text{Euler's reflection formula}\\\ \\\ \Gamma(z)\Gamma(z+{1\over
2})=2^{1-2z}\sqrt{\pi}\,\Gamma(2z)\quad\text{Legendre duplicate
formula}\end{matrix}$ $\lim_{x\rightarrow 0}\Gamma(x)={1\over
x}-\gamma+{6\gamma^{2}+\pi^{2}\over 12}x+\mathcal{O}(x^{2})$

in eq. A.12 to reduce

$\displaystyle T_{1}=2\pi
h^{2}\left({2\over\epsilon}+\gamma+\ln\pi+\ln|z_{12}|^{2}+\ln\mu\right)+\mathcal{O}(\epsilon).$
(A.13)

To compute $T_{2}$ and $T_{3}$ we require the following two integrals which
give same results,

$\int_{0}^{1}{u^{2}(1-u)\over\left(u(1-u)\right)^{4-{d\over
2}}}du=\int_{0}^{1}{u(1-u)^{2}\over\left(u(1-u)\right)^{4-{d\over
2}}}du={2^{4-d}\sqrt{\pi}\,\Gamma({d\over 2}-2)\over\Gamma({d\over 2}-{3\over
2})}$ (A.14)

Using the same approach as before and analytically continue to any complex
value for $d$ and using $d=2+\epsilon$ as before we have

$\displaystyle T_{2}$ $\displaystyle=$
$\displaystyle|z_{1}|^{2}h^{2}{2\pi\over|z_{12}|^{2}}\left({4\over\epsilon}+2\gamma-5+2\ln\pi+2\ln|z_{12}|^{2}+\ln\mu^{2}\right)+\mathcal{O}(\epsilon),$
(A.15) $\displaystyle T_{3}$ $\displaystyle=$
$\displaystyle|z_{2}|^{2}h^{2}{2\pi\over|z_{12}|^{2}}\left({4\over\epsilon}+2\gamma-5+2\ln\pi+2\ln|z_{12}|^{2}+\ln\mu^{2}\right)+\mathcal{O}(\epsilon).$
(A.16)

In a similar manner, the following integral

$\displaystyle\int_{0}^{1}\frac{u^{2}(1-u)^{2}}{(u(1-u))^{4-\frac{d}{2}}}=\frac{\Gamma\left(\frac{d}{2}-1\right)^{2}}{\Gamma(d-2)}$
(A.17)

yields

$T_{4}=2\pi\,h^{2}\left({4\over\epsilon}+2\gamma-3+2\ln\pi+2\ln|z_{12}|^{2}+2\ln\mu\right)+\mathcal{O}(\epsilon).$
(A.18)

Hence we finally have the final result for $I_{1}$ by adding all contributions
from $T_{i}$’s with (i=1 to 4),

$\displaystyle I_{1}$ $\displaystyle=$ $\displaystyle 2\pi
h^{2}{|z_{1}|^{2}+|z_{2}|^{2}\over|z_{12}|^{2}}\left({4\over\epsilon}+2\gamma-5+2\ln\pi+2\ln|z_{12}|^{2}+\ln\mu^{2}\right)$
(A.19) $\displaystyle-2\pi
h^{2}\left({2\over\epsilon}+\gamma-3+\ln\pi+\ln|z_{12}|^{2}+\ln\mu\right)+\mathcal{O}(\epsilon).$

Next we will evaluate

$\displaystyle I_{2}$ $\displaystyle=-\frac{c|z_{12}|^{4h}}{24}\int
dzd\bar{z}\,\frac{z}{\bar{z}}\langle
T(z)\mathcal{O}(z_{1},\bar{z}_{1})\mathcal{O}(z_{2},\bar{z}_{2})\rangle_{0}$
$\displaystyle={-ch}\int
dzd\bar{z}\,\frac{z}{\bar{z}}\left[\frac{1}{(z-z_{1})^{2}}+\frac{1}{(z-z_{2})^{2}}-{2\over{(z-z_{1})z_{12}}}+{2\over{(z-z_{2})z_{12}}}\right]$
(A.20)

Likewise

$\displaystyle I_{3}$ $\displaystyle={-ch}\int
dzd\bar{z}\,\frac{\bar{z}}{z}\left[\frac{1}{(\bar{z}-\bar{z}_{1})^{2}}+\frac{1}{(\bar{z}-\bar{z}_{2})^{2}}-{2\over{(\bar{z}-\bar{z}_{1})\bar{z}_{12}}}+{2\over{(\bar{z}-\bar{z}_{2})\bar{z}_{12}}}\right]$
(A.21)

Before tackling these integrals, let us focus on the following,

$\displaystyle\mathcal{I}^{0}(z_{i})=\int
dzd\bar{z}\,{z\over\bar{z}(z-z_{i})}$ $\displaystyle=$ $\displaystyle\int
dzd\bar{z}\,{z^{2}(\bar{z}-\bar{z}_{i})\over|z|^{2}|z-z_{i}|^{2}}$ (A.22)
$\displaystyle=$ $\displaystyle\int_{0}^{1}du\int
d^{2}\hat{z}{B(\hat{z})\over\left(|\hat{z}|^{2}+u(1-u)|z_{i}|^{2}\right)^{2}},$

where $\hat{z}=z-(1-u)z_{i}$ and

$\displaystyle B(\hat{z})=z^{2}(\bar{z}-\bar{z}_{i})=$ $\displaystyle
2z_{i}(1-u)|\hat{z}|^{2}-u(1-u)^{2}|z_{i}|^{2}z_{i}+\bar{\hat{z}}z_{i}^{2}(1-u)^{2}+|\hat{z}|^{2}\hat{z}$
(A.23) $\displaystyle-\hat{z}^{2}u\bar{z}_{i}-2|z_{i}|^{2}\hat{z}u(1-u).$

One should note that only the first two terms will contribute to the integral
with the polar coordinate parametrization, therefore we have

$\displaystyle\mathcal{I}^{0}(z_{i})=$ $\displaystyle
4z_{i}V_{s^{2-1}}\int_{0}^{1}du\,(1-u)\int_{0}^{\infty}{\rho^{2-1}\rho^{2}d\rho\over\left(\rho^{2}+u(1-u)|z_{i}|^{2}\right)^{2}}$
$\displaystyle-2|z_{i}|^{2}z_{i}V_{s^{2-1}}\int_{0}^{1}du\,u(1-u)^{2}\int_{0}^{\infty}{\rho^{2-1}d\rho\over\left(\rho^{2}+u(1-u)|z_{i}|^{2}\right)^{2}}$
(A.24)

This can be regularized through the same procedure as before to have

$\displaystyle\mathcal{I}^{0}(z_{i})={\pi z_{i}}\left({1\over
2}-{2\over\epsilon}-\gamma-\ln\pi-\ln|z_{i}|^{2}-\ln\mu\right)+\mathcal{O}(\epsilon)$
(A.25)

One can easily find that $I_{2}$ and $I_{3}$ can be written as the derivatives
of $\mathcal{I}^{0}$

$\displaystyle I_{2}$ $\displaystyle=-{c\,h\over
24}\left[\partial_{z_{1}}\mathcal{I}^{0}(z_{1})+\partial_{z_{2}}\mathcal{I}^{0}(z_{2})-{2\over
z_{12}}\mathcal{I}^{0}(z_{1})+{2\over z_{12}}\mathcal{I}^{0}(z_{2})\right]$
$\displaystyle I_{3}$ $\displaystyle=-{c\,h\over
24}\left[\partial_{\bar{z}_{1}}\mathcal{I}^{0}(\bar{z}_{1})+\partial_{\bar{z}_{2}}\mathcal{I}^{0}(\bar{z}_{2})-{2\over\bar{z}_{12}}\mathcal{I}^{0}(\bar{z}_{1})+{2\over\bar{z}_{12}}\mathcal{I}^{0}(\bar{z}_{2})\right].$
(A.26)

Hence the final results are,

$\displaystyle I_{2}={c\,h\over
24}\left[2\pi+\pi\ln|z_{1}z_{2}|^{2}-{2\pi\over
z_{12}}\left(z_{1}\ln|z_{1}|^{2}-z_{2}\ln|z_{2}|^{2}\right)\right]=\bar{I}_{3}$
(A.27)

###  A.2 Integrals for J$\bar{\text{T}}$

Now we will consider the integrals required for the computation of thermal two
point function of J$\bar{\text{T}}$ deformed CFT.

$\displaystyle\mathbb{I}_{1}(z_{1},\bar{z}_{1};z_{2},\bar{z}_{2})$
$\displaystyle=|z_{12}|^{4h}\int dzd\bar{z}\,{\bar{z}}\langle
J(z)\bar{T}(\bar{z})\mathcal{O}(z_{1},\bar{z}_{1})\mathcal{O}(z_{2},\bar{z}_{2})\rangle_{0}$
(A.28) $\displaystyle\mathbb{I}_{2}(z_{1},\bar{z}_{1};z_{2},\bar{z}_{2})$
$\displaystyle=-\frac{c|z_{12}|^{4h}}{24}\int
dzd\bar{z}\,{1\over\bar{z}}\langle
J(z)\mathcal{O}(z_{1},\bar{z}_{1})\mathcal{O}(z_{2},\bar{z}_{2})\rangle_{0}$
(A.29)

We split $\mathbb{I}_{1}$ in two parts,

$\displaystyle\mathbb{I}_{1}=\mathbb{K}_{1}+\mathbb{K}_{2}$ (A.30)

where

$\displaystyle\mathbb{K}_{1}$ $\displaystyle=\lambda
q_{1}\bar{h}\bar{z}^{2}_{12}\int
dzd\bar{z}\,\frac{\bar{z}}{(z-z_{1})(\bar{z}-\bar{z}_{1})^{2}(\bar{z}-\bar{z}_{2})^{2}}$
(A.31) $\displaystyle\mathbb{K}_{2}$ $\displaystyle=\lambda
q_{2}\bar{h}\bar{z}^{2}_{12}\int
dzd\bar{z}\,\frac{\bar{z}}{(z-z_{2})(\bar{z}-\bar{z}_{1})^{2}(\bar{z}-\bar{z}_{2})^{2}}$
(A.32)

Plugging the OPE of J and ${\cal O}$, $\mathbb{I}_{2}$ boils down to

$\displaystyle\mathbb{I}_{2}=-\frac{c}{24}\int
dzd\bar{z}\,\frac{1}{\bar{z}}\left(\frac{q_{1}}{z-z_{1}}+\frac{q_{2}}{z-z_{2}}\right)$
(A.33)

To solve (A.31), (A.32) and (A.33), we would first focus on the following four
integrals

$\displaystyle\mathcal{I}^{1}(z_{1},\bar{z}_{1})$ $\displaystyle=$
$\displaystyle\int dzd\bar{z}\,{1\over\bar{z}(z-z_{1})}$ (A.34)
$\displaystyle\mathcal{I}^{2}(z_{1},\bar{z}_{1},z_{2},\bar{z}_{2})$
$\displaystyle=$ $\displaystyle\int
dzd\bar{z}\,{\bar{z}\over(z-z_{1})(\bar{z}-\bar{z}_{2})}$ (A.35)
$\displaystyle\mathcal{I}^{3}(z_{1},\bar{z}_{1})$ $\displaystyle=$
$\displaystyle\int dzd\bar{z}{\bar{z}\over|z-z_{1}|^{2}}$ (A.36)
$\displaystyle\mathcal{I}^{4}(z_{1},\bar{z}_{1},z_{2},\bar{z}_{2})$
$\displaystyle=$ $\displaystyle\int
dzd\bar{z}\,{\bar{z}\over(z-z_{1})(\bar{z}-\bar{z}_{1})(\bar{z}-\bar{z}_{2})}$
(A.37)

####  A.2.1 $\mathcal{I}^{1}(z_{1},\bar{z}_{1})$

We will start by rewriting the integral as

$\displaystyle\int dzd\bar{z}{1\over\bar{z}(z-z_{1})}$ $\displaystyle=$
$\displaystyle\int
dzd\bar{z}{z(\bar{z}-\bar{z}_{1})\over|z|^{2}|z-z_{1}|^{2}}$ (A.38)
$\displaystyle=$ $\displaystyle\int_{0}^{1}du\int
d\hat{z}{C(\hat{z})\over\left(|\hat{z}|^{2}+u(1-u)|z_{1}|^{2})\right)^{2}}$

where $\hat{z}=z-(1-u)z_{1}$ and
$C(\hat{z})=|\hat{z}|^{2}-u(1-u)|z_{1}|^{2}-uz_{1}\hat{z}+(1-u)z_{1}\bar{\hat{z}}$
Only the first two terms in $C(\hat{z})$ will contribute to the integral with
the polar coordinate parametrization. Therefore this integral can be broken up
in the following two terms

$\displaystyle\mathcal{I}^{1}(z_{1},\bar{z}_{1})$ $\displaystyle=$
$\displaystyle\overbrace{2V_{S^{2}-1}\int_{0}^{1}du\,\int_{0}^{\infty}{\rho^{2-1}\rho^{2}d\rho\over\left(\rho^{2}+u(1-u)|z_{1}|^{2}\right)^{2}}}^{U_{1}}$
(A.39)
$\displaystyle-\overbrace{2V_{S^{2}-1}|z_{1}|^{2}\int_{0}^{1}du\,u(1-u)\int_{0}^{\infty}{\rho^{2-1}d\rho\over\left(\rho^{2}+u(1-u)|z_{1}|^{2}\right)^{2}}}^{U_{2}}$

Using the dimensional regularization techniques we have the simple relation
upto $\mathcal{O}(\epsilon)$ that

$U_{1}=-\pi\left({2\over\epsilon}-1+\gamma+\ln\pi+\ln|z_{1}|^{2}\right),\quad
U_{2}=\pi.$

which finally results in,

$\mathcal{I}^{1}(z_{1},\bar{z}_{1})=-\pi\left({2\over\epsilon}+\gamma+\ln\pi+\ln|z_{1}|^{2}+\ln\mu\right)+\mathcal{O}(\epsilon)$
(A.40)

####  A.2.2 $\mathcal{I}^{2}(z_{1},\bar{z}_{1},z_{2},\bar{z}_{2})$

In this case, we can first introduce the Feynman parametrization as the
following

$\displaystyle\int dzd\bar{z}\,{\bar{z}\over(z-z_{1})(\bar{z}-\bar{z_{2}})}$
$\displaystyle=$ $\displaystyle\int
dzd\bar{z}\,{\bar{z}(\bar{z}-\bar{z}_{1})(z-z_{2})\over|z-z_{1}|^{2}|z-z_{2}|^{2}}$
(A.41) $\displaystyle=$ $\displaystyle\int_{0}^{1}du\int
d^{2}\tilde{z}{D(\tilde{z})\over\left(|\tilde{z}|^{2}+u(1-u)|z_{12}|^{2}\right)^{2}}$

where

$\displaystyle D(\tilde{z})$ $\displaystyle=$ $\displaystyle
u^{2}(u-1)|z_{12}|^{2}\bar{z}_{12}+u(u-1)|z_{12}|^{2}\bar{z}_{2}+(2u-1)\bar{z}_{12}|\tilde{z}|^{2}+\bar{z}_{2}|\tilde{z}|^{2}+\cdots$
(A.42)

with the definition $z=\tilde{z}+uz_{1}+(1-u)z_{2}$. One should note that all
the rest of the terms in $\cdots$ in the expression of
$\bar{z}(\bar{z}-\bar{z}_{1})(z-z_{2})$ will integrate to zero. Therefore one
can break the integral for convenience as per the following,

$\displaystyle\mathcal{I}^{2}(z_{1},\bar{z}_{1},z_{2},\bar{z}_{2})$
$\displaystyle=$
$\displaystyle\overbrace{2V_{S^{2}-1}|z_{12}|^{2}\bar{z}_{12}\int_{0}^{1}du\,u^{2}(u-1)\int_{0}^{\infty}{\rho^{2-1}d\rho\over\left(\rho^{2}+u(1-u)|z_{12}|^{2}\right)^{2}}}^{\mathbb{T}_{1}}$
(A.43)
$\displaystyle+\overbrace{2V_{S^{2}-1}|z_{12}|^{2}\bar{z}_{2}\int_{0}^{1}du\,u(u-1)\int_{0}^{\infty}{\rho^{2-1}d\rho\over\left(\rho^{2}+u(1-u)|z_{12}|^{2}\right)^{2}}}^{\mathbb{T}_{2}}$
$\displaystyle+\overbrace{2V_{S^{2}-1}\bar{z}_{12}\int_{0}^{1}du\,(2u-1)\int_{0}^{\infty}{\rho^{2-1}\rho^{2}d\rho\over\left(\rho^{2}+u(1-u)|z_{12}|^{2}\right)^{2}}}^{\mathbb{T}_{3}}$
$\displaystyle+\overbrace{2V_{S^{2}-1}\bar{z}_{2}\int_{0}^{1}du\int_{0}^{\infty}{\rho^{2-1}\rho^{2}d\rho\over\left(\rho^{2}+u(1-u)|z_{12}|^{2}\right)^{2}}}^{\mathbb{T}_{4}}.$

Following the same steps as before for dimensional regularization we have the
following result upto $\mathcal{O}(\epsilon)$

$\begin{split}\mathbb{T}_{1}&=-{\pi\over
2}\bar{z}_{12},\quad\mathbb{T}_{2}=-\pi\bar{z}_{2},\quad\mathbb{T}_{3}=0\\\
\mathbb{T}_{4}&=-\pi\bar{z}_{2}\left({2\over\epsilon}-1+\gamma+\ln\pi+\ln|z_{12}|^{2}\right).\end{split}$
(A.44)

Collecting all these together we have

$\mathcal{I}^{2}(z_{1},\bar{z}_{1},z_{2},\bar{z}_{2})=-{\pi\over
2}\bar{z}_{12}-\pi\bar{z}_{2}\left({2\over\epsilon}+\gamma+\ln\pi+\ln|z_{12}|^{2}+\ln\mu\right)+\mathcal{O}(\epsilon).$
(A.45)

####  A.2.3 $\mathcal{I}^{3}(z_{1},\bar{z}_{1})$

To get the results of this integral in our preferred format we will rewrite
the integral as

$\displaystyle\int dzd\bar{z}{\bar{z}\over|z-z_{1}|^{2}}$ $\displaystyle=$
$\displaystyle\int dzd\bar{z}{\bar{z}|z|^{2}\over|z|^{2}|z-z_{1}|^{2}}$ (A.46)
$\displaystyle=$ $\displaystyle\int_{0}^{1}du\int
d^{2}\hat{z}{E(\hat{z})\over\left(|\hat{z}|^{2}+u(1-u)|z_{1}|^{2}\right)^{2}}$

where $\hat{z}=z-(1-u)z_{1}$ and
$E(\hat{z})=(1-u)^{3}|z_{1}|^{2}\bar{z}_{1}+2(1-u)\bar{z}_{1}|\hat{z}|^{2}$
with all other terms finally integrate to zero. Hence we have the relation

$\displaystyle\int dzd\bar{z}{\bar{z}\over|z-z_{1}|^{2}}$ $\displaystyle=$
$\displaystyle\overbrace{2V_{S^{2-1}}|z_{1}|^{2}\bar{z}_{1}\int_{0}^{1}du\,(1-u)^{3}\int_{0}^{\infty}d\rho{\rho^{2-1}d\rho\over\left(\rho^{2}+u(1-u)|z_{1}|^{2}\right)^{2}}}^{\mathbb{T}_{5}}$
(A.47)
$\displaystyle+\overbrace{4V_{S^{2-1}}\bar{z}_{1}\int_{0}^{1}du\,(1-u)\int_{0}^{\infty}d\rho{\rho^{2-1}\rho^{2}d\rho\over\left(\rho^{2}+u(1-u)|z_{1}|^{2}\right)^{2}}}^{\mathbb{T}_{6}}$

Again utilizing the dimensional regularization process we obtain,

$\mathbb{T}_{5}=\pi\bar{z}_{1}\left({2\over\epsilon}-{3\over
2}+\gamma+\ln\pi+\ln|z_{1}|^{2}\right),\quad\mathbb{T}_{6}=-{\pi}\bar{z}_{1}\left({2\over\epsilon}-1+\gamma+\ln\pi+\ln|z_{1}|^{2}\right).$

Therefore we have the final result

$\mathcal{I}^{3}(z_{1},\bar{z}_{1})=-{1\over
2}{\pi}\bar{z}_{1}+\mathcal{O}(\epsilon)$ (A.48)

####  A.2.4 $\mathcal{I}^{4}(z_{1},\bar{z}_{1},z_{2},\bar{z}_{2})$

We can rewrite $\mathcal{I}^{4}$ according to,

$\displaystyle\mathcal{I}^{4}(z_{1},\bar{z}_{1},z_{2},\bar{z}_{2})$
$\displaystyle=$ $\displaystyle{1\over\bar{z}_{12}}\int
dzd\bar{z}\left({\bar{z}\over|z-z_{1}|^{2}}-{\bar{z}\over{(z-z_{1})(\bar{z}-\bar{z}_{2})}}\right)$
(A.49) $\displaystyle=$
$\displaystyle{1\over\bar{z}_{12}}\left[\mathcal{I}^{3}(z_{1},\bar{z}_{1})-\mathcal{I}^{2}(z_{1},\bar{z}_{1},z_{2},\bar{z}_{2})\right]$

by plugging

$\displaystyle{1\over(\bar{z}-\bar{z}_{1})(\bar{z}-\bar{z}_{2})}$
$\displaystyle=$ $\displaystyle{1\over
z_{12}}\left({1\over\bar{z}-\bar{z}_{1}}-{1\over\bar{z}-\bar{z}_{2}}\right).$
(A.50)

Therefore, we finally have

$\displaystyle\mathcal{I}^{4}(z_{1},\bar{z}_{1},z_{2},\bar{z}_{2})={\pi\over\bar{z}_{12}}\left[-{\bar{z}_{1}\over
2}+{\bar{z}_{12}\over
2}+{\bar{z}_{2}}\left({2\over\epsilon}+\gamma+\ln\pi+\ln|z_{12}|^{2}+\ln\mu\right)\right]+\mathcal{O}(\epsilon)$

With all these calculations we now have the simple relation

$\displaystyle\mathbb{K}_{1}$ $\displaystyle=$ $\displaystyle\lambda
q_{1}\bar{h}\bar{z}_{12}^{2}\int
dzd\bar{z}{\bar{z}\over(z-z_{1})(\bar{z}-\bar{z}_{1})^{2}(\bar{z}-\bar{z}_{2})^{2}}=\lambda
q_{1}\bar{h}\bar{z}_{12}^{2}\,\partial_{\bar{z}_{1}}\partial_{\bar{z_{2}}}\mathcal{I}^{4}(z_{1},\bar{z}_{1},z_{2},\bar{z}_{2})$
$\displaystyle=$ $\displaystyle-{\pi\over
2\bar{z}_{12}}(\bar{z}_{1}+\bar{z}_{2})\lambda
q_{1}\bar{h}\left({4\over\epsilon}-5+2\gamma+2\ln\pi+2\ln|z_{12}|^{2}+\ln\mu^{2}\right)-{\pi\over\bar{z}_{12}}\lambda
q_{1}\bar{h}\bar{z}_{1}$

A similar analysis will yield

$\displaystyle\mathbb{K}_{2}$ $\displaystyle=$ $\displaystyle{\pi\over
2\bar{z}_{12}}(\bar{z}_{1}+\bar{z}_{2})\lambda
q_{2}\bar{h}\left({4\over\epsilon}-5+2\gamma+2\ln\pi+2\ln|z_{12}|^{2}+\ln\mu^{2}\right)+{\pi\over\bar{z}_{12}}\lambda
q_{2}\bar{h}\bar{z}_{2}$

A simple addition of (A.2.4) and (A.2.4) will generate the integral $I_{1}$ in
(A.30) upto $\mathcal{O}(\epsilon)$.

$\displaystyle\mathbb{I}_{1}$ $\displaystyle={\pi\lambda\bar{h}\over
2\bar{z}_{12}}(\bar{z}_{1}+\bar{z}_{2})(q_{2}-q_{1})\left({4\over\epsilon}-5+2\gamma+2\ln\pi+2\ln|z_{12}|^{2}+\ln\mu^{2}\right)$
$\displaystyle\qquad\qquad\qquad+{\pi\over\bar{z}_{12}}\lambda\,\bar{h}\,(q_{2}\bar{z}_{2}-q_{1}\bar{z}_{1})$
(A.54)

We can evaluate $\mathbb{I}_{2}$ as follows,

$\displaystyle\mathbb{I}_{2}$ $\displaystyle=$ $\displaystyle-\frac{c}{24}\int
dzd\bar{z}\,\frac{1}{\bar{z}}\left(\frac{q_{1}}{z-z_{1}}+\frac{q_{2}}{z-z_{2}}\right)$
(A.55) $\displaystyle=$ $\displaystyle-{c\over
24}\left(q_{1}\mathcal{I}^{1}(z_{1},\bar{z}_{1})+q_{2}\mathcal{I}^{1}(z_{2},\bar{z}_{2})\right)$
$\displaystyle=$ $\displaystyle{\pi c\over
24}\left[\left(q_{1}+q_{2}\right)\left({2\over\epsilon}+\gamma+\ln\pi+\ln\mu\right)+q_{1}\ln|z_{1}|^{2}+q_{2}\ln|z_{2}|^{2}\right]$

###  A.3 Integrals for J$\bar{\text{J}}$

In this case, we need to evaluate the following two integrals

$\displaystyle\mathcal{I}^{5}(z_{1},\bar{z}_{1})$ $\displaystyle=$
$\displaystyle\int{dzd\bar{z}\over|z-z_{1}|^{2}}$ (A.56)
$\displaystyle\mathcal{I}^{6}(z_{1},\bar{z}_{1},z_{2},\bar{z}_{2})$
$\displaystyle=$
$\displaystyle\int{dzd\bar{z}\over(z-z_{1})(\bar{z}-\bar{z}_{2})}$ (A.57)

####  A.3.1 $\mathcal{I}^{5}(z_{1},\bar{z}_{1})$

We rewrite $\mathcal{I}^{5}$ in the following way,

$\displaystyle\int{dzd\bar{z}\over|z-z_{1}|^{2}}$ $\displaystyle=$
$\displaystyle\int dzd\bar{z}{|z|^{2}\over|z|^{2}|z-z_{1}|^{2}}$ (A.58)
$\displaystyle=$ $\displaystyle\int_{0}^{1}du\int
d^{2}\hat{z}{E(\hat{z})\over\left(|\hat{z}|^{2}+u(1-u)|z_{1}|^{2}\right)^{2}}$

with $\hat{z}=z-(1-u)z_{1}$, and
$E(\hat{z})=z\bar{z}=|\hat{z}|^{2}+(1-u)^{2}|z_{1}|^{2}+(1-u)(\bar{z}_{1}\hat{z}+z_{1}\bar{\hat{z}}).$
Only the first two terms in $E(\hat{z})$ will contribute in the integral,
therefore we have

$\displaystyle\int{dzd\bar{z}\over|z-z_{1}|^{2}}$ $\displaystyle=$
$\displaystyle\overbrace{2V_{S^{2-1}}\int_{0}^{1}du\int_{0}^{\infty}{\rho^{2-1}\rho^{2}d\rho\over\left(\rho^{2}+u(1-u)|z_{1}|^{2}\right)^{2}}}^{{\cal{T}}_{1}}$
(A.59)
$\displaystyle+\overbrace{2V_{S^{2-1}}|z_{1}|^{2}\int_{0}^{1}du(1-u)^{2}\int_{0}^{\infty}{\rho^{2-1}d\rho\over\left(\rho^{2}+u(1-u)|z_{1}|^{2}\right)^{2}}}^{\mathcal{T}_{2}}$

These integrals leads to,

$\begin{split}\mathcal{T}_{1}=-\pi\left({2\over\epsilon}-1+\gamma+\ln\pi+\ln|z_{1}|^{2}\right)+\mathcal{O}(\epsilon)\\\
\mathcal{T}_{2}=\pi\left({2\over\epsilon}-1+\gamma+\ln\pi+\ln|z_{1}|^{2}\right)+\mathcal{O}(\epsilon),\end{split}$
(A.60)

finally contributing to

$\mathcal{I}^{5}(z_{1},\bar{z}_{1})=\mathcal{O}(\epsilon)$ (A.61)

####  A.3.2 $\mathcal{I}^{6}(z_{1},\bar{z}_{1},z_{2},\bar{z}_{2}$)

Similar to the earlier integrals, in this case we will rewrite the integral
likewise,

$\displaystyle\int{dzd\bar{z}\over(z-z_{1})(\bar{z}-\bar{z}_{2})}$
$\displaystyle=$ $\displaystyle\int
dzd\bar{z}{(\bar{z}-\bar{z}_{1})(z-z_{2})\over|z-z_{1}|^{2}|z-z_{2}|^{2}}$
(A.62) $\displaystyle=$ $\displaystyle\int_{0}^{1}du\int
d^{2}\tilde{z}{F(\tilde{z})\over\left(|\tilde{z}|^{2}+u(1-u)|z_{12}|^{2}\right)^{2}}$

where as usual $z=\tilde{z}+uz_{1}+(1-u)z_{2}$ and

$F(\tilde{z})=|\tilde{z}|^{2}-u(1-u)|z_{12}|^{2}+\cdots$

where "$\cdots$" denotes other terms which will not contribute to this
integral. Hence

$\displaystyle\int{dzd\bar{z}\over(z-z_{1})(\bar{z}-\bar{z}_{2})}$
$\displaystyle=$
$\displaystyle\overbrace{2V_{S^{2-1}}\int_{0}^{1}du\int_{0}^{\infty}{\rho^{2-1}\rho^{2}d\rho\over\left(\rho^{2}+u(1-u)|z_{12}|^{2}\right)^{2}}}^{\mathscr{T}_{3}}$
$\displaystyle-\overbrace{2V_{S^{2-1}}|z_{12}|^{2}\int_{0}^{\infty}du\,u(1-u)\int_{0}^{\infty}{\rho^{2-1}d\rho\over\left(\rho^{2}+u(1-u)|z_{12}|^{2}\right)^{2}}}^{\mathscr{T}_{4}}$

Carrying out the dimensional regularization, we obtain

$\mathscr{T}_{3}=-\pi\left({2\over\epsilon}-1+\gamma+\ln\pi+\ln|z_{12}|^{2}\right),\quad\mathscr{T}_{4}=\pi,$

which furthermore gives,

$\mathcal{I}^{6}(z_{1},\bar{z}_{1},z_{2},\bar{z}_{2})=-\pi\left({2\over\epsilon}+\gamma+\ln\pi+\ln|z_{12}|^{2}+\ln\mu\right)+\mathcal{O}(\epsilon)$
(A.64)

Collecting all these calculations together we finally have

$\displaystyle{\cal Y}_{1}+{\cal Y}_{2}+{\cal Y}_{3}+{\cal Y}_{4}$ (A.65)
$\displaystyle=$
$\displaystyle|q_{1}|^{2}\,\mathcal{I}^{5}(z_{1},\bar{z}_{1})+|q_{2}|^{2}\,\mathcal{I}^{5}(z_{2},\bar{z}_{2})+q_{1}\bar{q}_{2}\,\mathcal{I}^{6}(z_{1},\bar{z}_{1},z_{2},\bar{z}_{2})+\bar{q}_{1}q_{2}\,\mathcal{I}^{6}(z_{2},\bar{z}_{2},z_{1},\bar{z}_{1})$
$\displaystyle=$
$\displaystyle-\pi\left(q_{1}\bar{q}_{2}+\bar{q}_{1}q_{2}\right)\left({2\over\epsilon}+\gamma+\ln\pi+\ln|z_{12}|^{2}+\ln\mu\right)+\mathcal{O}(\epsilon).$

##  Appendix B $|\phi_{n}(t)|^{2}$ for T$\bar{\text{T}}$

The probabilities corresponding to the operator wave functions are as follows

$\displaystyle|\phi_{n}(t)|^{2}=P_{0}(n,t)-\tilde{\lambda}\\{P_{1}(n,t)+\nu_{1}P_{2}(n,t)\\}+{\cal
O}(\lambda^{2})$ (B.1)

The functions $P_{0},P_{1},P_{2}$ in the above expression are given as follows

$\displaystyle P_{0}(n,t)$
$\displaystyle=\frac{1}{6}(n+1)(n+2)(n+3)\text{sech}^{8}\left(\frac{t}{2}\right)\tanh^{2n}\left(\frac{t}{2}\right)$
$\displaystyle P_{1}(n,t)$ $\displaystyle=\pi
f_{0}^{2}(n,t)\bigg{[}g_{1}(n,t)+g_{2}(n,t)+g_{3}(n,t)+g_{4}(n,t)+g_{5}(n,t)+g_{6}(n,t)+g_{7}(n,t)$
$\displaystyle\qquad+g_{8}(n,t)+g_{9}(n,t)+g_{10}(n,t)+g_{11}(n,t)+g_{12}(n,t)\bigg{]}$
$\displaystyle P_{2}(n,t)$ $\displaystyle=\pi
f_{0}^{2}(n,t)\bigg{[}h_{1}(n,t)+h_{2}(n,t)+h_{3}(n,t)+h_{4}(n,t)+h_{5}(n,t)\bigg{]}$

where $f_{0},g_{i}$ and $h_{i}$ are given by

$\displaystyle f_{0}(n,t)$ $\displaystyle=\text{sech}^{9}\left(\frac{\pi
t}{\beta}\right)\tanh^{n-2}\left(\frac{\pi t}{\beta}\right)$ $\displaystyle
h_{1}(n,t)$
$\displaystyle=\frac{1}{768}(n+1)(n+2)(n+3)(n+4)\cosh\left(\frac{2\pi
t}{\beta}\right)$ $\displaystyle h_{2}(n,t)$
$\displaystyle=\frac{1}{960}(n+1)(n+2)(n+3)(n+4)(2n+5)\cosh\left(\frac{4\pi
t}{\beta}\right)$ $\displaystyle h_{3}(n,t)$
$\displaystyle=-\frac{1}{512}(n+1)(n+2)(n+3)(n+4)\cosh\left(\frac{6\pi
t}{\beta}\right)$ $\displaystyle h_{4}(n,t)$
$\displaystyle=-\frac{1}{3840}(n+1)(n+2)(n+3)(n+4)(2n+5)\cosh\left(\frac{8\pi
t}{\beta}\right)$ $\displaystyle h_{5}(n,t)$
$\displaystyle=\frac{1}{1536}(n+1)(n+2)(n+3)(n+4)\cosh\left(\frac{10\pi
t}{\beta}\right)$ $\displaystyle g_{1}(n,t)$
$\displaystyle=-\frac{(n+1)(n+2)(n+3)(2(n-1)n-5)}{1920}\cosh\left(\frac{2\pi
t}{\beta}\right)\log\left[\cosh^{2}\left(\frac{\pi t}{\beta}\right)\right]$
$\displaystyle g_{2}(n,t)$
$\displaystyle=-\frac{(n+1)(n+2)(n+3)(n+4)\left(12H_{n+4}-25\right)}{9216}\cosh\left(\frac{2\pi
t}{\beta}\right)$ $\displaystyle g_{3}(n,t)$
$\displaystyle=\frac{(n+1)(n+2)(n+3)(n+4)\left(-60(2n+5)H_{n+4}+274n+625\right)}{57600}\cosh\left(\frac{4\pi
t}{\beta}\right)$ $\displaystyle g_{4}(n,t)$
$\displaystyle=\frac{1}{480}(n+1)(n+2)(n+3)(n(n+4)+5)\cosh\left(\frac{4\pi
t}{\beta}\right)\log\left[\cosh^{2}\left(\frac{\pi t}{\beta}\right)\right]$
$\displaystyle g_{5}(n,t)$
$\displaystyle=\frac{\left((n+1)(n+2)(n+3)(n+4)\left(12H_{n+4}-25\right)\right)}{6144}\cosh\left(\frac{6\pi
t}{\beta}\right)$ $\displaystyle g_{6}(n,t)$
$\displaystyle=\frac{(n+1)(n+2)(n+3)(2n-5)(2n+3)}{3840}\cosh\left(\frac{6\pi
t}{\beta}\right)\log\left[\cosh^{2}\left(\frac{\pi t}{\beta}\right)\right]$
$\displaystyle g_{7}(n,t)$
$\displaystyle=-\frac{(n+1)(n+2)(n+3)(n(4n+11)+15)}{1920}\log\left[\cosh^{2}\left(\frac{\pi
t}{\beta}\right)\right]$ $\displaystyle g_{8}(n,t)$
$\displaystyle=\frac{(n+1)(n+2)(n+3)(n+4)\left(60(2n+5)H_{n+4}-274n-625\right)}{230400}\cosh\left(\frac{8\pi
t}{\beta}\right)$ $\displaystyle g_{9}(n,t)$
$\displaystyle=-\frac{1}{384}(n+1)^{2}(n+2)(n+3)\cosh\left(\frac{8\pi
t}{\beta}\right)\log\left[\cosh^{2}\left(\frac{\pi t}{\beta}\right)\right]$
$\displaystyle g_{10}(n,t)$
$\displaystyle=\frac{(n+1)(n+2)(n+3)(n+4)\left(25-12H_{n+4}\right)}{18432}\cosh\left(\frac{10\pi
t}{\beta}\right)$ $\displaystyle g_{11}(n,t)$
$\displaystyle=\frac{1}{768}(n+1)(n+2)(n+3)\cosh\left(\frac{10\pi
t}{\beta}\right)\log\left[\cosh^{2}\left(\frac{\pi t}{\beta}\right)\right]$
$\displaystyle g_{12}(n,t)$
$\displaystyle=\frac{(n+1)(n+2)(n+3)(n+4)\left(60(2n+5)H_{n+4}-274n-625\right)}{76800}$

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