first-style=short LP short = LP, long = Linear Programming, tag = abbrev SDP
short = SDP, long = Semi-definite Programming, tag = abbrev OPE short = OPE,
long = Operator Product Expansion, tag = abbrev GN short = GN model, long =
Gross-Neveu model – the theory of $N$ fermions with quartic interaction, tag =
abbrev GNY short = GNY model, long = Gross-Neveu-Yukawa model – a scalar field
coupled to the Gross-Neveu model via Yukawa interactions, tag = abbrev IR
short = IR , long = Infrared – used to indicate low energies, tag = abbrev UV
short = UV , long = Ultraviolet – used to indicate high energies, tag = abbrev
TQFT short = TQFT , long = Topological Quantum Field Theory, tag = abbrev MFT
short = MFT , long = Mean Field Theory, tag = abbrev QFT short = QFT , long =
Quantum Field Theory, tag = abbrev CFT short = CFT , long = Conformal Field
Theory, tag = abbrev SCFT short = SCFT , long = Supersymmetric Conformal Field
Theory, tag = abbrev CFTdp short = CFT${}_{d^{+}}$ , long = Conformal Field
Theory in $d$ dimensions with Euclidean signature, tag = abbrev CFTdm short =
CFT${}_{d^{-}}$ , long = Conformal Field Theory in $d$ dimensions with
Lorentzian signature, tag = abbrev d short = $d$ , long = Spacetime dimension:
we consider either $\mathbb{R}^{d}$ or $\mathbb{R}^{d-1,1}$, sort = a, tag =
nomencl T short = $T^{\mu\nu}$ , long = Stress energy tensor of a given field
theory, sort = a, tag = nomencl phi short = $\phi$ , long = A local operator
in the trivial representation of the rotation subgroup of conformal group,
sort = b, tag = nomencl cO short = $\mathcal{O}$ , long = A local operator in
the generic representation of the rotation subgroup of conformal group, sort =
c, tag = nomencl psi short = $\psi$ , long = A local spin$-\frac{1}{2}$
operator of conformal group in $\mathbb{R}^{2,1}$, sort = c, tag = nomencl
Delta short = $\Delta$ , long = Scaling dimension of a local operator:
$\mathcal{O}_{i}(\lambda x)=\lambda^{-\Delta_{i}}\mathcal{O}_{i}(x)$, sort =
d, tag = nomencl Delta3 short = $\Delta_{abc}$ , long = Shorthand notation for
the frequently used combination $\Delta_{a}+\Delta_{b}-\Delta_{c}$, sort = d2,
tag = nomencl spin short = $\ell$ , long = Spin of a local operator in three
spacetime dimensions, sort = e, tag = nomencl z short = $z$ , long =
Holomorphic cross ratio of four points in the standard conformal frame, sort =
f, tag = nomencl zb short = $\overline{z}$ , long = Anti-holomorphic cross
ratio of four points in the standard conformal frame, sort = g, tag = nomencl
mu short = $\mu(\mathcal{O})$ , long = Plancherel measure of the conformal
group, sort = h, tag = nomencl Dissertation Director: Professor David Poland

# Analytic Studies of Fermions in the Conformal Bootstrap

Soner Albayrak

(June 2021)

###### Abstract

In this thesis, we analyze unitary conformal field theories in three
dimensional spaces by applying analytic conformal bootstrap techniques to
correlation functions of non-scalar operators, in particular Majorana
fermions. Via the analysis of these correlation functions, we access several
sectors in the spectrum of conformal field theories that have been previously
unexplored with analytic methods, and we provide new data for several operator
families. In the first part of the thesis, we achieve this by the large spin
expansions that have been traditionally used in the conformal bootstrap
program for scalar correlators, whereas in the second part we carry out the
computations by combining several analytic tools that have been recently
developed such as weight shifting operators, harmonic analysis for the
Euclidean conformal group, and the Lorentzian inversion formula. We compare
these methods and demonstrate the superiority of the latter by computing
nonperturbative correction terms that are inaccessible in the former. A better
analytic grasp of the spectrum of fermionic conformal field theories can help
in many directions including making new precise analytic predictions for
supersymmetric models, computing the binding energies of fermions in curved
space, and describing quantum phase transitions in condensed matter systems
with emergent Lorentz symmetry.

2021

To my family

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Historical background
      1. 2.1.1 Brief account of conformal symmetry in the past centuries
      2. 2.1.2 Recent developments in the field
    2. 2.2 Primer on conformal field theories
      1. 2.2.1 Conformal transformations and conformal algebra
        1. 2.2.2 Conformal group and its linear realizations
          1. 2.3 Lightning review of the conformal bootstrap
            1. 3 Fermions in Conformal Field Theories: A Perturbative Approach
              1. 3.1 Fermions in $d=3$
                1. 3.1.1 Notations & Conventions
                2. 3.1.2 Double twist sector of $3d$ fermions
              2. 3.2 Large spin expansion
                1. 3.2.1 Lightcone bootstrap
                2. 3.2.2 Fermion conformal blocks and the crossing equations
                3. 3.2.3 Results
                  1. 4 Fermions in Conformal Field Theories: Non-Perturbative Corrections
                    1. 4.1 Perturbative vs Non-perturbative
                    2. 4.2 A detour: effect of non-perturbative terms in scalar CFTs
                    3. 4.3 Review: harmonic analysis on the conformal group
                      1. 4.3.1 Shadow transform and shadow coefficients
                      2. 4.3.2 Euclidean pairings
                      3. 4.3.3 Euclidean inversion formula
                      4. 4.3.4 Partial Waves and Conformal Blocks
                      5. 4.3.5 $6j$ Symbols
                      6. 4.3.6 Weight-Shifting Operators
                    4. 4.4 Mean Field Theory OPE Coefficients
                      1. 4.4.1 $\expectationvalue{\psi\phi\phi\psi}$
                      2. 4.4.2 $\expectationvalue{\psi_{1}\psi_{2}\psi_{2}\psi_{1}}$
                      3. 4.4.3 $\expectationvalue{\psi\psi\psi\psi}$
                      4. 4.4.4 MFT Coefficients for Higher Twist Towers
                    5. 4.5 CFT data of a general spectrum
                      1. 4.5.1 Applications in the computation of anomalous dimensions
                      2. 4.5.2 Computation of corrections to OPE Coefficients: a working example
                        1. 5 Discussion
                          1. A $6j$ Symbols
                            1. A.1 Spinning Down the $6j$ Symbol
                            2. A.2 OPE Function and its Decomposition
                            3. A.3 $\mathcal{K}$ Coefficients
                            4. A.4 Symmetries of $6j$ Symbols

###### List of Figures

  1. 4.1 Spectrum in the Ising model
  2. 4.2 OPE coefficients in the Ising model
  3. 4.3 Spectrum in the O(2) model
  4. 4.4 OPE coefficients in the O(2) model
  5. 4.5 Definition of the shadow coefficients in diagrammatic notation
  6. 4.6 Definition of the OPE function in diagrammatic notation
  7. 4.7 Partial wave expansion in diagrammatic notation
  8. 4.8 Bubble annihilation coefficients in diagrammatic notation
  9. A.1 Diagrammatic illustration of the decomposition of two fermion $6j$ symbol
  10. A.2 Diagrammatic illustration of the decomposition of four fermion $6j$ symbol
  11. A.3 $6j$ symbol written as a tetrahedron in diagrammatic notation

###### List of Tables

  1. 3.1 Double-twist families for Majorana fermions in 3d
  2. 4.1 Results for the twists and OPE coefficients in the 3d Ising model
  3. 4.2 Results for the twists and OPE coefficients in the O(2) model

[include=noSuchClass,name=Acknowledgments, heading =chapter*]

I would like to thank my supervisor, David Poland, for his invaluable support
throughout my Ph.D. years. I appreciate his mentoring, his academic guidance,
and his patience with me: he is a great collaborator and a tremendous advisor
to have, and I am extremely grateful to have been his PhD student. I sincerely
hope that one day I can be an advisor like him, with such an encouraging
approach towards people and an insightful approach towards physics.

I also would like to thank my dear friend Savan Kharel for practically being a
second mentor for me. I have learned substantially from him about a lot of
things, not least the academic world, and I am utterly thrilled that our
casual discussions have led to several collaborations and diverse
publications.

There are numerous friends and colleagues that I have crossed paths with in
Yale particle theory group and in Walter Burke Institute at Caltech; and, I
would like to acknowledge their support through my Ph.D. years (they are
simply too many to name). In particular, I thank David Meltzer who has been a
great friend and a treasured collaborator: I’m indebted to him for his
contributions to the work presented in this thesis. Additionally, I’d like to
thank Chandramouli Chowdhury: I certainly enjoyed our collaborations. Lastly,
I would like to give my special thanks to Taylan Nurlu, who help me forget a
theoretical physicist’s inevitable doom of being never understood hence never
sincerely appreciated by their friends and family, as he has always been there
to listen and try to understand what I do throughout this thesis. This is in
fact the second thesis I’m thanking him with that exact sentence!

Acknowledgment is also due to my thesis committee: I thank Witold Skiba,
Thomas Appelquist, Keith Baker, and Simon Caron-Huot for reading this thesis
and for giving their much appreciated feedback.

Finally, I owe my sincerest thanks to my family for their everlasting love. My
father Halim, my mother Münevver, and my brother $\dot{\text{I}}$lker have
always been there for me whenever I needed them. In particular, I’d like to
express my gratitude to my wife Bengisu for her endless mental support. I
could not have completed my Ph.D. without my family’s help and this thesis is
dedicated to them.

[include=abbrev,name=Abbreviations, heading =chapter*]

[include=nomencl,name=Nomenclature, heading =chapter*]

##  Chapter 1 Introduction

It would not be an overstatement to say that quantum field theories (QFTs)
with conformal symmetry are tremendously important in modern theoretical
physics. Understanding such theories may shed light on numerous areas,
including but not limited to

  1. 1.

the study of quantum gravity through the gauge gravity correspondence,

  2. 2.

beyond the standard model physics via phenomenological applications,

  3. 3.

condensed matter systems through critical phenomena,

  4. 4.

effective field theories perturbed around CFTs via renormalization flow.

Such relevance of conformal field theories (CFTs) has stimulated an intensive
amount of research into analytic and numerical tools to understand the
underlying principles and restrictions of conformal symmetries in field
theories. In two dimensional spacetimes, the restrictions of conformal
symmetry are enhanced to those of the _Virasaro_ symmetry, hence two
dimensional CFTs enjoy a greater number of constraints which in turn enable
their classification and in some cases their complete solution. In contrast,
CFTs in general dimensions are harder to solve analytically and the analysis
of these theories has been relatively idle in the $20^{\text{th}}$ century.

In the past decade, there has been a revival in the analytic study of
conformal field theories, particularly through nonperturbative tools and
general constraints such as unitarity and causality. This program, coined the
_conformal bootstrap_ in analogy to the s-matrix bootstrap program of 1960s,
has yielded novel insight into the analytic structure of CFTs, both in
Euclidean signature — convergence of the operator product expansion (OPE),
analytic expansions for the conformal blocks, the existence of the operators
that shift the conformal weights of local operators, etc. — and in Lorentzian
signature — the existence of infinite double twist families, analyticity in
spin, light-ray operators, etc.

Despite the stupendous progress of the last decade, the analytic studies in
CFTs have been mostly focused on the direct study of the _scalar_ operators,
i.e. local operators in the trivial representation of the rotation subgroup of
the conformal group.111Among some of the exceptions for the analytic bootstrap
with external spinning operators, we can name the lightcone bootstrap [1, 2,
3, 4, 5, 6, 7], Mellin space techniques [8, 9], and mean field theory
computations [10]. Conformal symmetry tells us that symmetric traceless
representations of the $\mathrm{SO}(d)$ group appear in the OPE of scalar
operators, so the analysis of scalar operators is deemed sufficient if one is
only interested in such operators. This is especially a natural choice as the
direct analysis of spinning operators through correlation functions of such
operators is far more complicated for a variety of technical reasons. However
analysis of correlation functions of scalar operators alone (or correlation
functions of any bosonic operators for that matter) does not give access to
the fermionic operators in the spectrum of CFTs.

There are various reasons as to why CFTs with fermionic operators are of great
relevance. In $d=4$, we expect non-trivial non-supersymmetric fermionic CFTs
because of the UV Lagrangian descriptions based on matter coupled to gauge
theories that flow in the IR to weakly coupled Caswell-Banks-Zaks fixed points
[11, 12]. The situation is similar in $d=3$ where we can consider IR fixed
points of QED coupled to matter or the Gross-Neveu-Yukawa (GNY) models.
Furthermore, there is a fermionic sector in any supersymmetric conformal field
theory (SCFT) in any $d$,222We note that superalgebras actually only exist for
$d\leq 6$ [13]. and a better analytic grasp of such theories is not only
important for their theoretical relevance but also for their utility in the
description of many condensed matter systems (such as a critical point on the
boundary of topological superconductors [14]).

In this thesis we address this missing link in the literature by extending the
analytic progress achieved for bosonic CFTs to $3d$ CFTs with fermionic
sectors. We will do this by considering correlation functions of Majorana
fermions with scalars and with each other; this allows us to reach various new
sectors such as the fermionic double twist operators.333The irreducible
representations of the $\mathrm{so}(2,1)$ algebra are real, so we do not lose
any generality by considering Majorana spinors of the $\mathrm{Pin}(2,1)$
group. This is in contrast to $d=4$ case where the irreducible representations
of the $\mathrm{so}(3,1)$ algebra are complex hence Majorana fermions are
special cases (Technically, in both $d=3,4$, we have pseudo-Majorana spinors
instead of Majorana spinors but we will gloss over this distinction [15]). Our
work has two main consequences: firstly we provide explicit new data that is
applicable in any unitary $3d$ CFT; secondly we detail and discuss two
different technologies to achieve this — an old style method that has been
used for scalars for some time now, and a new method that combines several
analytic tools that have been recently developed in the bootstrap literature.
We believe that our results can be utilized in various concrete CFTs such as
the GNY models and that our method for the analysis of the fermionic
correlators be extended to higher-spin operators, which would ultimately
enrich our understanding of conformal field theories even further.

#### Outline of the thesis

We start with a general introduction of conformal symmetry and the conformal
bootstrap program in Chapter 2. In particular, we review the historical
development of both, alongside with a brief mathematical description of
conformal symmetry. We then specialize to fermions in three dimensional
spacetime in Chapter 3: we set our conventions, define the spectrum of
fermionic theories, review the traditional so-called lightcone bootstrap
approach, and carry out the computations in this scheme to obtain the CFT
data.

The results obtained in Chapter 3 are _not_ analytic in the spin $\ell$ of the
relevant CFT operators; in particular, they should be regarded as asymptotic
results around $\ell\sim\infty$. At finite spin, we may get spurious poles
which ruin the analyticity: such spurious poles are actually canceled by what
we call the _non-perturbative_ corrections, i.e. terms that are exponentially
suppressed in $\ell$ — hence are invisible to the framework of Chapter 3. In
Chapter 4, we detail this difference and demonstrate its importance in
concrete scalar CFTs such as the $3d$ Ising and $O(N)$ models. In these
theories, we obtain the non-perturbative corrections through a recently
developed analytic tool (the Lorentzian inversion formula). In the rest of the
chapter, we discuss how to extend this approach to spinning operators, review
necessary ingredients, and apply the derived relations to obtain the non-
perturbative corrections for various fermionic operators. The backbone of this
approach relies on the $6j$ symbols of the conformal group; in Appendix A, we
provide further details for these objects.

#### Citations to Previous Work

Chapter 3 is essentially identical to

> S. Albayrak, D. Meltzer, and D. Poland, _“More Analytic Bootstrap:
> Nonperturbative Effects and Fermions,”_ JHEP 08 (2019) 040, arXiv:1904.00032
> [hep-th]

Chapter 4 is excerpted from

> S. Albayrak, D. Meltzer, and D. Poland, _“More Analytic Bootstrap:
> Nonperturbative Effects and Fermions,”_ JHEP 08 (2019) 040, arXiv:1904.00032
> [hep-th]
>
> S. Albayrak, D. Meltzer, and D. Poland, _“The Inversion Formula and 6j
> Symbol for 3d Fermions,”_ JHEP 09 (2020) 148, arXiv:2006.07374 [hep-th]

Appendix A is excerpted from

> S. Albayrak, D. Meltzer, and D. Poland, _“The Inversion Formula and 6j
> Symbol for 3d Fermions,”_ JHEP 09 (2020) 148, arXiv:2006.07374 [hep-th]

#### Other PhD Work of the Author

> S. Albayrak, S. Kharel and D. Meltzer, _“On duality of color and kinematics
> in (A)dS momentum space,”_ arXiv:2012.10460 [hep-th]

> S. Albayrak, C. Chowdhury and S. Kharel, _“On loop celestial amplitudes for
> gauge theory and gravity,”_ Phys.Rev.D 102 (2020)

> S. Albayrak and S. Kharel, _“Spinning loop amplitudes in Anti-de Sitter
> space,”_ Phys.Rev.D 103 (2021) 2

> S. Albayrak, C. Chowdhury and S. Kharel, _“Study of momentum space scalar
> amplitudes in AdS spacetime,”_ Phys.Rev.D 101 (2020) 12

> S. Albayrak and S. Kharel, _“Towards the higher point holographic momentum
> space amplitudes. Part II. Gravitons,”_ JHEP 12 (2019) 135

> S. Albayrak, C. Chowdhury and S. Kharel, _“New relation for Witten
> diagrams,”_ JHEP 10 (2019) 274

> S. Albayrak and S. Kharel, _“Towards the higher point holographic momentum
> space amplitudes,”_ JHEP 02 (2019) 040

##  Chapter 2 Preliminaries

###  2.1 Historical background

####  2.1.1 Brief account of conformal symmetry in the past centuries

In this section, we will review the evolution of the conformal bootstrap. A
more appropriate history should actually start around $19^{\text{th}}$
century, encompassing the story of conformal transformations themselves. Such
a review would take us from the times of Joseph Liouville and Sophus Lie, and
would include all the relevant topics along the timeline of conformal
symmetries. As it happens, there is a such a review and interested reader is
directed there [16]. Below, we will recap the developments in the last century
and set the scene for the emergence of the conformal bootstrap as we
understand it today.111See [17] for a thorough analysis along with an
extensive historical account.

The end of the $19^{\text{th}}$ century and the first half of $20^{\text{th}}$
century have witnessed exiting advancements in the area of scale invariance
among which the first observation of critical opalescence and the exact
solution of $2d$ Ising model come to mind [18, 19]. Conformal invariance also
entered into physics around this time through the Maxwell equations. Later
with the advancement of the Wilsonian renormalization group, it was realized
that CFTs have a much broader usage than their mere application in statistical
and classical physics because any QFT connects to a CFT in the UV and another
one in the IR via renormalization group flow.222Technically endpoints of
renormalization group flow are either TQFTs or _scale invariant_ theories,
which usually tend to be conformally invariant as well.

The greater appreciation of CFTs lead to their involvement in more and more
physical theories. However, the conformal bootstrap differs from other
employment of conformal symmetries in both conceptual and practical senses. To
understand that, we need to appreciate the term _bootstrap_ in physics.

The 1960’s have seen rapid proliferation of strongly interacting particles
detected in high energy collisions. The intuitive expectation that we can
explain all particles in terms of elementary ones and that local fields should
be used to calculate observables started to be doubted. In this environment,
Geoffrey Chew proposed what he called _the nuclear democracy_ [20]. It is an
approach fundamentally different in the sense that it rejects standard
reductionism in high energy physics and proposes that one should focus on
fundamental laws and how they constrain observable particles while treating
them equally instead of finding a minimal set of elementary particles to which
all others can be reduced [21, 22].

Chew’s work later turned into what we call today _the s-matrix bootstrap_
because bootstrappers were trying to obtain the scattering matrix starting
from the fundamental laws such as Lorentz invariance and unitarity. The
program was not particularly successful at that time, besides some work on
Regge trajectories such as that of Dolen, Horn, and Schmid [23, 24]; and, the
interest in s-matrix bootstrap plummeted once strong interactions were
understood in the framework of Quantum Chromodynamics. Veneziano utilized the
advancements in the Regge theories [25] however this lead to a different
direction, i.e. the rise of string theory [26].

The following year of Veneziano’s work, Wilson [27] — followed by Kadanoff
[28] — initiated the analysis of current algebras in the lines of the s-matrix
bootstrap333Like the s-matrix bootstrap, Wilson’s approach was also bypassing
conventional local fields. Nonetheless, his work with operator product
expansions were proved in the realm of QFT, initially perturbatively by
Zimmerman [29]. One can also show it nonperturbatively via radial quantization
with the path integral formalism [30]. where the authors promoted the usage of
_operator product expansion_ (OPE) associativity as a consistency condition,
which was demonstrated to be sufficient to produce an alternative solution for
$2d$ Ising model in addition to that of Onsager’s original solution [19].

We can see the appeal of these advancements: the main problem with the
s-matrix bootstrap was the lack of sufficient constraints to extract a large
set of concrete results. However, this can be circumvented if one also
includes the constraints of CFTs and the consistency of OPEs. This was first
realized by Ferrara, Grillo, and Gatto [31], and independently by Polyakov
[32]. We can safely take these papers to be the birth of the conformal
bootstrap, though it was not until 1984 that the term bootstrap was used for
this approach [33].

Despite the appreciation of its importance, progress on the conformal
bootstrap has been rather slow until the past two decades, except with $2d$
CFTs which have seen rapid development due to their mathematical simplicity
and relevance in string theories. The stagnancy in higher dimensional CFTs
changed in 2008 when Rattazzi, Rychkov, Tonni, and Vichi introduced a
numerical method into the conformal bootstrap [34]. They argued that one can
use linear programming to extract useful bounds on physical quantities such as
scaling dimensions of the operators in the CFT, _without_ analytically solving
them. Since then, the conformal bootstrap attracted much attention and created
a revival in the area. This revival also led to a nice amount of progress on
the analytical side as well, and we will discuss this in the next section.

####  2.1.2 Recent developments in the field

##### Numerics

The conformal bootstrap flourished since the introduction of _linear
programming_ (LP) into the area with the seminal paper of Rattazzi et al.
[34]. The method they introduce is a simple yet elegant application of “proof
by contradiction”: one assumes some properties about the spectrum of a CFT,
such as the scaling dimension of the lowest lying operators. One then checks
whether this spectrum is consistent with crossing symmetry and unitarity; if
it is self-consistent, we do not obtain any new information. If not, we
conclude that the assumed spectrum cannot be realized in any unitary CFT. This
way, we can put bounds on the spectrum of any unitary CFT without the need for
a full analytical solution.

Several works were published in the literature within the next three years
after [34] appeared, applying this numerical technique in numerous ways; e.g.
to bound the scaling dimension of the $\phi^{2}$ operator via the analysis of
$\expectationvalue{\phi\phi\phi\phi}$ [34, 35, 36], to bound the OPE
coefficient of two scalars with a traceless symmetric tensor $\mathcal{O}$
[37, 36], to obtain similar bounds for $\mathcal{N}=1$ supersymmetric theories
[38, 36], and to bound the central charge of generic CFTs [38, 39, 40, 36]. In
those papers, conserved currents of global symmetries and stress tensors are
also studied in (non-)supersymmetric theories.444See [41] as well.

This revival in the conformal bootstrap gained an acceleration with the
realization of an alternative to LP in [36]. In that paper, Poland, Simmons-
Duffin, and Vichi introduced _semi-definite programming_ (SDP) into the
conformal bootstrap, which they used to update the previously computed bounds
significantly by carving out various $4d$ CFTs with(out) global symmetries and
supersymmetry.

The advantages of SDP over LP are discussed in [36]; for a more detailed
review, one can refer to [42]. Here we will only mention one of these
advantages: semi-definite optimization is a very standard problem in
engineering hence there are various sophisticated implementations to address
such problems.555For a review of SDP applications, see [43].

Even after the introduction of SDP into the conformal bootstrap, there were
some analyses which kept relying on LP. Most of these papers employed the
simplex algorithm of LP implemented via the ILOG CPLEX optimizer through a
Mathematica interface [44, 45, 46, 47, 48, 49, 50]. In contrast, the bootstrap
community gradually switched to SDP: during the next 4 years after the
introductions of SDP, the conformal bootstrap analyses were mostly implemented
via the algorithm SDPA-GMP [51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62,
63]. Some papers simply used both LP and SDP in the same work [64].

In 2015 David Simmons-Duffin introduced SDPB, a SDP implementation specialized
for the conformal bootstrap applications [65]. This algorithm boosted the
numerical analyses even further and has become the de-facto optimizer used in
the conformal bootstrap community and has been actively improved thanks to
_Simons Collaboration on the Nonperturbative Bootstrap_.666See
https://github.com/davidsd/sdpb for further details. Some of the papers among
this intense research program can be grouped as follows:

  1. 1.

_Various $3d$ scalar models_: Precision islands in the Ising model [66],
$\mathcal{N}=1$ Ising and WZ models [67], $O(N)$ models [68, 66], CFTs with a
continuous global symmetry [69], stress-tensor bootstrap [70], and the random-
bond Ising model [71].

  2. 2.

$4d$ _supersymmetric theories_ : Minimial $\mathcal{N}=1$ SCFTs [72], mixed
correlators in $\mathcal{N}=1$ SCFTs [73], chiral correlators in
$\mathcal{N}=2$ SCFTs [74, 75], general $\mathcal{N}=3$ SCFTs [76], stress
tensor supermultiplets in $\mathcal{N}=4$ SCFTs [77], and $\mathcal{N}=4$
$O(N)$ Vector Models [78].

  3. 3.

_Models in $d>4$ dimensions_: Mixed correlators in the $5d$ critical $O(N)$
Models [79], and the $6d$ superconformal bootstrap [80].

  4. 4.

_Fermionic theories_ : Identical Majorana fermions in $3d$ [81], Dirac
fermions in QED3 [82], and Weyl fermions in $4d$ [83].

Of course, this is only a small fraction of the research using numerical
bootstrap techniques. In addition to these, there are a variety of different
theories where numerical bootstrap has been extensively used. These include
tetragonal CFTs [84], the half-BPS line defect [85], universality of BTZ
spectral density [86], Virasoro minimal models [87, 88], M-theory [89],
Argyres-Douglas theory [90], flavored $2d$ CFT partition functions [91],
conformal multi-flavor QCD on lattice [92], pure quantum gravity in AdS3 [93],
many-flavor gauge theories [94], long multiplets of $\mathcal{N}=2,3$ SCFTs
[95], modular constraints on CFTs with currents [96], high-precision bootstrap
of a non-unitary CFTs [97], and K3 CFT [98].

##### Analytics

The rapid progress in the numerics was accompanied by the development of
various analytical tools, allowing us to discover the rich structure of
conformal theories. Below we will briefly review this progress.

The first point that we should mention is the convergence of OPE. Being at the
heart of the conformal bootstrap program, this convergence was extensively
analyzed in [99] where the authors show the exponential suppression of
operators with higher and higher scaling dimensions. In that paper, the
authors make use of the radial quantization and the mapping between
$\mathbb{R}^{d}$ and $\mathbb{R}\times S^{d-1}$; it is also realized that
extensive usage of radial coordinates has various advantages over standard
Dolan and Osborn coordinates $\\{z,\overline{z}\\}$, especially for the
convergence of the conformal blocks written as series expansions [100, 101].

Such series expansions are particularly important since the absence of compact
and explicit expressions for the conformal blocks in odd dimensions poses a
serious practical problem.777 Even though the OPE of two scalars and a spin
$\ell$ operator was known for years, it was first in [102] that the summation
was carried out and explicit conformal blocks were derived! In that and the
following papers, Dolan and Osborn gave the closed form expressions for
conformal blocks of external scalars in even dimensions, and provided several
expressions such as integral representations for conformal blocks of external
scalars in odd dimensions [102, 103, 104].  There are some special cases where
the conformal blocks are known even in odd dimensions [105, 106], however the
generic case is yet to be derived.888 There is a new research program where
the calculations are carried out in the embedding space, including the
application of OPE [107, 108, 109, 110]. The authors provide the compact and
complete results for the necessary ingredients for any conformal block in any
dimension.  In the past decade, this problem was partially circumvented by
going to the specific kinematical limits. The most intensively studied one is
the _lightcone limit_ for which the conformal blocks in any dimension reduce
to the collinear conformal blocks whose compact form is known. In [111, 112]
the authors showed that every CFT admits a large spin expansion and the
lightcone limit probes this sector. Using the bootstrap equations, they show
the existence of infinitely many operators which organize into families of
almost the same twist.999We define twist $\tau\coloneqq\Delta-\ell$ for the
scaling dimension $\Delta$ and spin $\ell$ for the symmetric traceless
representations — all representations are such in three dimensions. For a
general mixed representation in other dimensions, $\ell$ is taken to be the
length of the first row of the Young tableau. In mean field theory, such
operators are composite objects built out of fundamental fields and they have
exactly the same twist; i.e., the operators
$\phi\partial_{\mu_{1}}\cdots\partial_{\mu_{n}}\phi$ all have the twist
$2\Delta_{\phi}$ for the scalar $\phi$ with twist $\Delta_{\phi}$.101010The
operator $\phi\partial_{\mu_{1}}\cdots\partial_{\mu_{n}}\phi$ is neither in an
irreducible representation of the rotation group nor a conformal primary. One
can ensure these properties by symmetrizing open indices and subtracting the
traces. Furthermore, one needs to replace $\partial$ with
$\frac{\overset{\rightarrow}{\partial}-\overset{\leftarrow}{\partial}}{2}$
because
$\partial=\frac{\overset{\rightarrow}{\partial}-\overset{\leftarrow}{\partial}}{2}+\frac{\overset{\rightarrow}{\partial}+\overset{\leftarrow}{\partial}}{2}$
and second term simply creates descendants. Following [112], such operators
are called _double twist operators_ in the bootstrap literature.

In a strongly interacting theory, we can no longer interpret double twist
operators as in given the schematic forms; furthermore, those operators start
to develop anomalous dimensions. However, [111, 112] proved that double twist
operators exist in any CFT, and they asymptotically have the same properties
as composite MFT operators at large spin.111111The inverse proportionality of
anomalous dimension with spin justifies viewing the large spin sector as a
perturbation around the generalized free theory. Via AdS/CFT such operators
correspond to widely separated weakly interacting particles [113, 114]

Despite being an asymptotic expansion,121212 It is shown in [115] that the
large spin expansion is asymptotic but Borel-summable. The validity of this
expansion is partially solidified in [116]. analysis of the large spin sectors
via the lightcone bootstrap received a great deal of attention [117, 118, 119,
120, 121, 122, 123, 124] where mostly the CFT data of double twist operators
are investigated. The generalization to multi-twist operators and
eikonalization are discussed in [125], and David Simmons-Duffin systematized
the lightcone bootstrap in [126]: he introduced an $\mathrm{SL}(2,\mathbb{R})$
expansion as a means to go beyond leading order in spin, discussed the
sensitivity of large spin expansion to finite spin effects, addressed the
issue of mixing, and compared the predictions of analytics with the numerics
in the case of the $3d$ Ising model.

Concurrently, Simon Caron-Huot introduced in 2017 an inversion formula which
calculates the OPE data using the correlator as an input, much like the
Froissart-Gribov formula yields the angular momentum partial wave coefficients
using the amplitude as an input in relativistic s-matrix theory [127]. More
importantly, Caron-Huot’s inversion formula establishes the analyticity in
spin in CFTs just as the Froissart-Gribov formula established the analyticity
in spin in s-matrix theory [128, 129, 130]. This is an intrinsically
Lorentzian phenomena as this inversion formula uses specific causal orderings
of the operators. This Lorentzian inversion formula was further analyzed
[131], generalized to arbitrary Lorentz representations [132], and utilized in
various works [133, 1, 134].

We stated above that the conformal blocks have been poorly understood even
though the OPE of two scalars has been known for a very long time, and that
the explicit conformal blocks were derived only less than two decades ago
[102]. However, this recently changed quite dramatically. Firstly, the
conformal blocks of traceless symmetric operator exchange for external bosonic
spinning operators were derived by a judicious use of differential operators
on scalar conformal blocks [135]. Then a procedure to calculate any conformal
block was introduced in [136], and is employed, among others, to derive
superconformal blocks [137, 138] and conformal blocks of identical external
fermions in $3d$ [81].131313The procedure is conceptually quite simple and
elegant. One uses the well known shadow formalism [139, 140, 141, 142] to
derive the _partial waves_ in the embedding space, and then obtains the
conformal blocks using a monodromy projection. Even though the process is
quite general, it does not immediately provide an explicit compact result.
There has been numerous other progress in the derivation of the conformal
blocks in a variety of dimensions as well [143, 144, 145, 146, 147, 148, 149,
150, 151]. However, the most comprehensive approach to get spinning conformal
blocks was introduced in [152]. The authors use group theoretical arguments
and manage to construct a formalism to relate different spinning three point
structures to each other by exchanging a finite dimensional representation of
the conformal group. One can use these relations inside Euclidean pairings of
three point structures and relate partial waves of different operators. Using
monodromy arguments explicitly discussed back in [136], one can then relate
spinning conformal blocks to each other, and as an important subset of that
fermion conformal blocks to bosonic ones. Alternatively, one can directly use
these _weight shifting operators_ with the Lorentzian inversion formula,
bypassing partial waves and directly working with conformal blocks
[132].141414In this thesis, we refer to the single-valued solutions of the
conformal Casimir equation as _partial waves_. They decompose into conformal
blocks and the _shadow_ conformal blocks. The main intuitive difference
between Euclidean and Lorentzian pairings is the integration range: the
integration is over all spacetime in the Euclidean inversion formula hence it
yields both conformal blocks and their shadows; in other words, partial waves
are the main objects. In contrast, the integration range is a causal diamond
in Lorentzian formula, and Lorentzian pairing of two three point function is
proportional to a conformal block. The reason for the difference in the
integration range follows from the necessity to work with the so-called
_light-ray operators_ which are continuous-spin generalization of local
operators. Being analytic in spin, Lorentzian inversion formula can be
conceptually explained only by these operators, even though it was not
originally derived through them [127, 131].

The advancement in extracting spinning conformal blocks went hand in hand with
the analysis of correlators of external spinning operators. In fact, the
authors of [135] cited above started their analysis by constructing a
formalism to analyze the spinning correlators in great generality [153].

The problem of classifying three point functions of arbitrary spin $\ell$ has
been discussed several times in the literature [154, 155, 156, 157, 158]. What
Costa, Penedones, Poland, and Rychkov did in [153] was to reintroduce the
embedding space into the conformal bootstrap and to combine it with an index-
free formalism, hence developing necessary tools to write down any bosonic
spinning three point correlator as well as the conditions on conserved
tensors.151515Usage of embedding space started with Dirac and has been on and
off since then [159, 160, 157, 161, 162, 31] — it is extensively discussed in
[153]. Likewise, derivation of the conservation conditions on three point
structures goes back to the last century [163, 156].

The extension to fermionic correlators is a little bit more complicated.
Individual cases of fermionic analysis can be found in the literature; e.g.
Weinberg used the embedding formalism to analyze $4d$ fermions back in 2010
[157], see also [81, 6, 164] for more recent studies. However, a more general
treatment only started relatively recently [165, 166, 167, 152]. In these
papers, Kravchuk and his collaborators analyzed classification of conformal
correlators, generalized Casimir recursion relations, discussed generic $4d$
bootstrap equations, and introduced the weight shifting operators.

There are a variety of other areas which benefited from and contributed to the
analytical bootstrap program. Among others, we can list some of them:

  1. 1.

_supersymmetric theories_ : analyses of $\mathcal{N}=1,2,4$ superconformal
blocks [168, 38, 169], discussion of chiral superconformal primary OPE [38,
40, 170], study of superconformal theories with global symmetries [171], and
many others [172, 173, 174, 175, 176, 177].

  2. 2.

_gauge gravity duality_ : analyses of unitarity and analyticity of holographic
s-matrix [178, 179], study of AdS3/CFT2 [180, 181, 182, 183], establishment of
a holographic connection to Witten and geodesic Witten diagrams [184, 185,
186, 187, 188].

  3. 3.

_various other applications_ : finite temperature CFTs [189, 190], Mellin
space bootstrap161616The Mellin representation of conformal correlators
actually goes all the way back to Mack [155], see also [191, 192, 193, 194].
[195, 196], multi-point conformal blocks [197, 198], defect CFTs [199, 200],
complex CFTs [201], global symmetries [202, 203], boundary and crosscap CFTs
[204], fractal Ising model [205], and so on [206, 207, 208, 209, 210, 4, 211,
212, 213].

###  2.2 Primer on conformal field theories

In this section, we will review the basics of conformal field theory. For
concreteness, we will consider the Euclidean conformal group though the
discussion can straightforwardly be extended to the Lorentzian conformal
group.

####  2.2.1 Conformal transformations and conformal algebra

We define the conformal transformations as the transformations that leave the
metric invariant upto a local rescaling, that is

$\displaystyle g^{\prime}_{ij}(x^{\prime})=\Lambda(x)g_{ij}(x)\;.$ (2.2.1)

As a second rank covariant tensor, the metric transforms under an
infinitesimal transformation $x^{i}\rightarrow x^{i}+\epsilon^{i}(x)$ as

$\displaystyle g^{\prime}_{ij}=\frac{\partial x^{k}}{\partial x^{\prime
i}}\frac{\partial x^{\ell}}{\partial x^{\prime
j}}g_{k\ell}=g_{ij}-(\partial_{i}\epsilon_{j}+\partial_{j}\epsilon_{i})+O(\epsilon^{2})\;,$
(2.2.2)

which in turn restricts conformal transformations to satisfy the condition

$\displaystyle\partial_{i}\epsilon_{j}+\partial_{j}\epsilon_{i}=\lambda(x)g_{ij}(x)\;.$
(2.2.3)

Contracting both sides with $g^{ij}(x)$, we can solve for $\lambda(x)$ and
insert it back to obtain

$\displaystyle\partial_{i}\epsilon_{j}+\partial_{j}\epsilon_{i}=\frac{2}{\ac{d}}(\partial\cdot\epsilon)g_{ij}(x)\;.$
(2.2.4)

By acting on both sides with $\partial^{i}$ and $\partial^{i}\partial^{j}$, we
further obtain the conditions

$\displaystyle\frac{d}{2}\mathop{}\\!\mathbin{\Box}\epsilon_{j}=$
$\displaystyle\frac{2-d}{2}\partial_{j}(\partial\cdot\epsilon)+\delta
g_{j}\;,$ (2.2.5a)
$\displaystyle(d-1)\mathop{}\\!\mathbin{\Box}(\partial\cdot\epsilon)=$
$\displaystyle\partial\cdot\delta g$ (2.2.5b)

for $\delta g_{j}\coloneqq(\partial^{i}g_{ij})(\partial\cdot\epsilon)$. In
flat spacetimes, which will be the focus of this thesis, we have $\delta
g_{i}=0$, hence we can solve the equations above for the most general form of
$\epsilon_{i}$ in $d>2$:

$\displaystyle\epsilon_{i}=a_{i}+cx_{i}+\theta_{ij}x^{j}+2(b\cdot
x)x_{i}-x^{2}b_{i}$ (2.2.6)

for the antisymmetric matrix $\theta_{ij}=-\theta_{ij}$.

We see that conformal transformations in $d$-dimensional flat space are
parametrized by the scalar $c$, vectors $a_{i}$ & $b_{i}$, and the
antisymmetrix matrix $\theta_{ij}$, leading to the conclusion that the
conformal group has dimension $\frac{(d+2)(d+1)}{2}$.

To get further insight into the generators of the conformal group, let us
first consider the translation group: in QFT, this group is generated by the
the momentum operator $P^{\mu}$, which by Noether’s theorem should be the
conserved charge of a current. The relevant current here is the stress tensor
T, and we have the relation

$\displaystyle P^{\mu}(t)=-\int d^{d-1}xT^{0\mu}(x)\;,$ (2.2.7)

where the integration is over a constant time slice of the spacetime. In fact,
$P^{\mu}$ is a topologically conserved charge and we can rewrite the
integration over any hypersurface $\Sigma$ as
$\int_{\Sigma}dS_{\mu}T^{\mu\nu}(x)$; in a similar fashion, we can define a
new set of conserved charges $Q_{\epsilon}$ as

$\displaystyle
Q_{\epsilon}(\Sigma)\coloneqq-\int_{\Sigma}dS_{\mu}\epsilon_{\nu}(x)T^{\mu\nu}(x)$
(2.2.8)

if $\partial_{\mu}\left(\epsilon_{\nu}(x)T^{\mu\nu}(x)\right)=0$. Conservation
of stress tensor, $\partial_{\mu}T^{\mu\nu}=0$, then implies

$\displaystyle\left(\partial_{\mu}\epsilon_{\nu}+\partial_{\nu}\epsilon_{\mu}\right)T^{\mu\nu}=0\;.$
(2.2.9)

For a generic stress tensor this implies

$\displaystyle\partial_{\mu}\epsilon_{\nu}+\partial_{\nu}\epsilon_{\mu}=0\;.$
(2.2.10)

This is the _Killing equation_ , and it has the solutions
$\epsilon=\epsilon^{\mu}\partial_{\mu}$ as

$\displaystyle p_{\mu}\coloneqq$ $\displaystyle\partial_{\mu}\;,$ (2.2.11a)
$\displaystyle m_{\mu\nu}\coloneqq$ $\displaystyle
x_{\mu}\partial_{\nu}-x_{\nu}\partial_{\mu}\;,$ (2.2.11b)

which generate the usual Poincaré algebra with $p$ and $m$ generating
translations and rotations respectively.

If we impose that the stress tensor is _traceless_ , i.e.
$g_{\mu\nu}T^{\mu\nu}=0$, we instead have the _Conformal Killing equation_

$\displaystyle\partial_{\mu}\epsilon_{\nu}+\partial_{\nu}\epsilon_{\mu}=\lambda(x)g_{\mu\nu}(x)$
(2.2.12)

which is precisely eqn. (2.2.3). Therefore, in the field theory context,
conformal field theory is a local field theory with a _traceless_ stress
tensor. For this equation, we obtain two new set of vector fields for
$\epsilon=\epsilon^{\mu}\partial_{\mu}$:

$\displaystyle d\coloneqq$ $\displaystyle x^{\mu}\partial_{\mu}\;,$ (2.2.13a)
$\displaystyle k_{\mu}\coloneqq$ $\displaystyle
2x_{\mu}(x\cdot\partial)-x^{2}\partial_{\mu}\;,$ (2.2.13b)

where $d$ generates dilations and $k$ generates so-called special conformal
transformations. By comparing the generators in eqn. (2.2.11) and eqn.
(2.2.13) with the most general form of the conformal transformation in eqn.
(2.2.6), we conclude that the parameters $a,\;b,\;c,\;\theta$ parameterize
translations, special conformal transformations, dilations, and rotations
respectively.

It is illustrative to investigate the algebras constructed with these
generators, which we summarize in Table. (2.2.1). We can also collect the full
set of commutation relations as

$\displaystyle\commutator{m_{ij}}{m_{k\ell}}=$
$\displaystyle\left(\delta_{i\ell}\delta_{kr}\delta_{jp}+\delta_{jk}\delta_{ir}\delta_{\ell
p}+\delta_{ik}\delta_{\ell
p}\delta_{jr}+\delta_{j\ell}\delta_{ip}\delta_{kr}\right)m_{pr}\;,$ (2.2.14a)
$\displaystyle\commutator{p^{\pm}_{i}}{m_{jk}}=$
$\displaystyle\left(\delta_{ij}\delta_{k\ell}-\delta_{ik}\delta_{j\ell}\right)p^{\pm}_{\ell}\;,$
(2.2.14b) $\displaystyle\commutator{p^{\pm}_{i}}{d}=$ $\displaystyle\pm
p^{\pm}_{i}\;,$ (2.2.14c) $\displaystyle\commutator{p^{+}_{i}}{p^{-}_{j}}=$
$\displaystyle 2\delta_{ij}d-2m_{ij}\;,$ (2.2.14d)

for $p_{i}^{+}\coloneqq p_{i}$ and $p_{i}^{-}\coloneqq k_{i}$.

Table 2.1: Conformal algebra and its subalgebras.171717In the table, the
symbol $\oplus_{s}$ denotes a semi-direct sum. We remind the reader that an
algebra $g$ can be written as a semi-direct sum of its subalgebras $g_{1}$ and
$g_{2}$ as $g=g_{1}\oplus_{s}g_{2}$ if the conditions
$\commutator{\bar{g}_{1}}{\bar{g}_{1}}\in g_{1}$,
$\commutator{\bar{g}_{2}}{\bar{g}_{2}}\in g_{2}$, and
$\commutator{\bar{g}_{1}}{\bar{g}_{2}}\in g_{1}$ are satisfied for all
elements $\bar{g}_{i}\in g_{i}$. We also would like to note that one can
construct further subalgebras by replacing $t(N)$ with $c(N)$ in the table;
for example, we can define a second Euclidean subalgebra of the conformal
algebra as $\mathrm{c}(N)\oplus_{s}\mathrm{so}(N)$.

|  |  |
---|---|---|---
Name of the algebra |  Denotation |  Decomposition |  Commutation relations
Translation |  $\mathrm{t}(N)$ |  – |  $[p^{i},p^{j}]=0$
Rotation |  $\mathrm{so}(N)$ |  – |  $\begin{aligned} =m^{i\ell}g^{jk}-m^{jk}g^{i\ell}\\\ +m^{ik}g^{j\ell}-m^{j\ell}g^{ik}\end{aligned}$
Euclidean |  $\mathrm{iso}(N)$ |  $\mathrm{t}(N)\oplus_{s}\mathrm{so}(N)$ |  $[p^{k},m^{ij}]=g^{ki}p^{j}-g^{kj}p^{i}$
Dilation |  $\mathrm{d}(1)$ |  – |  –
Translation & Dilation |  – |  $\mathrm{t}(N)\oplus_{s}\mathrm{d}(1)$ |  $[p^{i},d]=p^{i}$
Euclidean & Dilation |  – |  $\mathrm{iso}(N)\oplus_{s}\mathrm{d}(1)$ |  $[m^{ij},d]=0$
Special conformal |  $\mathrm{c}(N)$ |  – |  $[k^{i},k^{j}]=0$
General conformal |  $\mathrm{gc}(N)$ |  – |  $\begin{aligned} {}[k^{k},m^{ij}]=&g^{ki}k^{j}-g^{kj}k^{i}\\\ [k^{i},d]=&-k^{i}\\\ [p^{i},k^{j}]=&2g^{ij}d-2m^{ij}\end{aligned}$
|  |  |

The interpretation of these relations is straightforward. The first equation
simply states that $m$ generates a rotation algebra. The second equation and
the absence of nonzero $\commutator{d}{m_{ab}}$ indicate that $p^{\pm}$ and
$d$ transform as a vector and a scalar under rotations respectively. The third
equation shows that $p^{\pm}$ behave like ladder operators with respect to the
dilation $d$; and finally, the last equation gives their non-commutative
nature.

####  2.2.2 Conformal group and its linear realizations

In section 2.2.1 we reviewed the conformal symmetry and the differential
representation of the conformal algebra. By an exponential map, one can extend
the analysis from the conformal algebra to the conformal group — or rather its
connected component around the identity element. Indeed, this way, we can
write the action of the conformal group under a _finite_ transformation by
starting with the infinitesimal transformation in eqn. (2.2.6); here, we
simply present the final results:

$\displaystyle P:$ $\displaystyle\;x^{i}\rightarrow x^{i}+a^{i}\;,$ (2.2.15a)
$\displaystyle M:$ $\displaystyle\;x^{i}\rightarrow e^{\theta_{ij}}x^{j}\;,$
(2.2.15b) $\displaystyle D:$ $\displaystyle\;x^{i}\rightarrow e^{c}x^{i}\;,$
(2.2.15c) $\displaystyle K:$
$\displaystyle\;x^{i}\rightarrow\frac{x^{i}-b^{i}x^{2}}{1-2b\cdot
x+b^{2}x^{2}}\;.$ (2.2.15d)

The transformation under the first three generators is fairly intuitive: we
translate, rigidly rotate, or scale the coordinates. The last transformation
on the other hand can be explained intuitively only through an _inversion_
operation:

$\displaystyle I:x^{i}\rightarrow\frac{x^{i}}{x^{2}}\;.$ (2.2.16)

Even though this operator is not an element of the conformal group,
conjugation by an inversion is actually the outer authomorphism of the
conformal group and one can show that

$\displaystyle K=-I\cdot P\cdot I\;.$ (2.2.17)

In other words, the special conformal transformation in eqn. (2.2.15d) is
simply an inversion followed by a translation followed by another inversion!

The nonlinear nature of the transformation in eqn. (2.2.15d) complicates the
construction of the representations of the conformal group in
$\mathbb{R}^{d}$. However, there actually exists another space in which the
realization of the conformal group is linear. To see that, we define the
generators $M_{AB}$

$\displaystyle M_{ij}=$ $\displaystyle m_{ij}\;,$ (2.2.18) $\displaystyle
M_{0,-1}=$ $\displaystyle d\;,$ $\displaystyle M_{i,-1}=$
$\displaystyle\frac{1}{2}\left(p_{i}^{+}+p_{i}^{-}\right)\;,$ $\displaystyle
M_{i,0}=$ $\displaystyle\frac{1}{2}\left(p_{i}^{-}-p_{i}^{+}\right)\;,$

where $A=-1,0,1,\dots,d$ and $i=1,\dots,d$. In terms of $M_{AB}$, we can
rewrite the comutation relations in eqn. (2.2.14) as

$\displaystyle\commutator{M_{AB}}{M_{CD}}=\left(g_{AD}M_{BC}+g_{BC}M_{DA}+g_{AC}M_{DB}+g_{BD}M_{AC}\right)$
(2.2.19)

for the metric

$\displaystyle g_{AB}\doteq\mathrm{diag}\left(-1,1,\dots,1\right)\;.$ (2.2.20)

We realize that eqn. (2.2.19) is the commutation relations for the rotation
algebra which indicates that the Euclidean group $\mathrm{gc}(d)$ is
isomorphic to an orthogonal algebra in $d+2$ dimensions, i.e.

$\displaystyle\mathrm{gc}(d)\simeq\mathrm{so}(d+1,1)\;.$ (2.2.21a) Similarly,
we can identify the algebra of the Lorentzian conformal algebra with another
orthogonal algebra: $\displaystyle\mathrm{gc}(d-1,1)\simeq\mathrm{so}(d,2)\;.$
(2.2.21b)

This isomorphism of the algebras suggests that we can work in
$\mathbb{R}^{d+1,1}$ ($\mathbb{R}^{d,2}$) instead of $\mathbb{R}^{d}$
($\mathbb{R}^{d-1,1}$) where the action of the conformal group is realized
linearly. This _embedding space_ approach was first suggested by Dirac, and
has been extensively used ever since [159, 157, 153]. Below, we will quickly
review this approach and illustrate how to construct invariant/covariant
objects of the conformal group (such as correlation functions) via this
approach.

The nonlinear action of the conformal group in eqn. (2.2.15) can be mapped to
the linear action of the special orthogonal group in the embedding space, once
the extra degrees of freedom are taken care of: this is achieved by mapping
_null rays_ in $\mathbb{R}^{d+1,1}$ to points in $\mathbb{R}^{d}$. The
intuitive explanation for this is that null rays are the objects in
$\mathbb{R}^{d+1,1}$ that are mapped to themselves under the action of
$\mathrm{so}(d+1,1)$ — similar to $\mathbb{R}^{d}$ being mapped to itself
under $\mathrm{gc}(d)$ — and that they have the correct degrees of freedom.

A straightforward procedure to map from the embedding space back to the
physical space is by _fixing the gauge_ of the freedom $X_{A}\sim\lambda
X_{A}$; a common choice is the _Poincaré_ section where we scale $X_{A}$ to
$P_{a}$ defined as

$\displaystyle P_{A}\coloneqq(1,x^{2},x_{i})$ (2.2.22)

in the lightcone coordinate $X_{A}=\left(X_{+},X_{-},X_{i}\right)$ for
$X_{\pm}=X_{0}\pm X_{-1}$. With this choice, we see that $\mathrm{so}(d+1,1)$
invariants map as

$\displaystyle
X_{A}Y^{A}\quad\rightarrow\quad-\frac{1}{2}(x_{i}-y_{i})^{2}\;.$ (2.2.23)

Similarly, one can show that $\mathrm{so}(d+1,1)$ tensors map as

$\displaystyle T_{A_{1}\cdots A_{\ell}}(X^{A}_{1},\dots)\quad\rightarrow\quad
t_{a_{1}\cdots a_{\ell}}(P^{A}_{1},\dots)\equiv\frac{\partial
P^{A_{1}}}{\partial x^{a_{1}}}\cdots\frac{\partial P^{A_{\ell}}}{\partial
x^{a_{\ell}}}T_{A_{1}\cdots A_{\ell}}(X^{A}_{1},\dots)\;.$ (2.2.24)

We can also write down a prescription to map fermionic representations of
$\mathrm{so}(d+1,1)$ but it is actually far easier to work in an _index-free_
fashion where auxiliary spinors and auxiliary vectors are used in the
embedding space to construct $\mathrm{so}(d+1,1)$ invariants. However, before
illustrating that, we will briefly discuss the stabilizer subgroup of the
conformal group to review the labeling of the operators in the irreducible
representations.

The stabilizer subalgebra $h$ of the conformal algebra consists of the
generators $m_{ij},\;p_{i}^{-},\;d$ if we choose it to fix the point $x=0$.
From eqn. (2.2.14) we then see that

$\displaystyle
h=\left(\mathrm{c}(N)\oplus_{s}\mathrm{d}(1)\right)\oplus_{s}\mathrm{so}(N)\;.$
(2.2.25)

We realize from eqn. (2.2.14c) that $p_{i}^{-}$ acts a lowering operator for
$d$, hence we can define _primary operator_ which satisfies

$\displaystyle\commutator{p_{i}^{-}}{\mathcal{O}(0)}=0\;,$ (2.2.26)

which is in the highest weight representation of $\mathrm{d}(1)$. As these
operators are invariant under the action of $c(N)$,181818Under the action of
$c(N)$, $\mathcal{O}(0)$ transforms as
$e^{-b^{i}p^{-}_{i}}\mathcal{O}(0)e^{b^{i}p^{-}_{i}}=\mathcal{O}(0)+b^{i}[\mathcal{O}(0),p^{-}_{i}]+\dots=\mathcal{O}(0)$
by the Hausdorff formula. the stabilizer algebra for primary operators is
effectively $\mathrm{d}(1)\oplus_{s}\mathrm{so}(N)$ for which the operators
are labeled by a continuous parameter $\Delta\in\mathbb{R}$ for
$\mathrm{d}(1)$ and a representation $\rho\in\mathrm{so}(d)$ for the conformal
algebra in $\mathbb{R}^{d}$. When we map from algebra to the group, we need to
take the fermionic representations into account as well, hence we denote

$\displaystyle\text{primary
operators:}\quad\mathcal{O}_{\Delta,\rho}\quad\text{with}\quad\Delta\in\mathbb{R}\;,\quad\rho\in\mathrm{Spin}(d)\;.$
(2.2.27a) As eqn. (2.2.14c) indicates that $p^{+}_{i}$ is a raising operator
for $d$, operators $p^{+}_{i}\cdot\mathcal{O}$, $p^{+}_{i}\cdot
p^{+}_{i}\cdot\mathcal{O}$, and so on are not primary operators. Such
operators are called _descendant operators_ and are given by eqn. (2.2.11a) as
$\displaystyle\text{descendant
operators:}\quad\frac{\partial^{n}\mathcal{O}_{\Delta,\rho}}{\partial
x^{\mu_{1}}\dots\partial x^{\mu_{n}}}\quad\text{with}\quad
n\in\mathbb{Z}^{+}\text{ and }\mathcal{O}=\text{primary}\;.$ (2.2.27b)

With the little detour of the labeling of operators under the conformal group
finished, we can get back to their construction in the embedding space, and in
particular, the index-free construction of conformal correlators which bypass
technical computations in eqn. (2.2.24). Without dwelling on the details, we
will simply review the procedure

  * •

For the bosonic representations in any $d$, introduce an auxiliary vector
$Z_{A}$ for each row of the Young diagram of the representation $\rho$ for the
operator $\mathcal{O}_{\Delta,\rho}$. In $d=3$, we can instead introduce a
single auxiliary spinor $S_{I}$ for both bosonic and fermionic
representations. In $d=4$, we instead introduce two auxiliary spinors $S_{I}$
and $\bar{S}_{I}$ for all representations.

  * •

We construct the most general function of the auxiliary vectors/spinors and
position vectors which satisfy following constraints:

    1. 1.

Invariant under the action of $\mathrm{SO}(d+1,1)$.

    2. 2.

Homogeneous in both the auxiliary vectors/spinors and position vectors with
degrees of homogeneity fixed for a given representation.

    3. 3.

Restricted to the null cone $X^{2}=0$.

    4. 4.

Transverse, i.e. position and auxilary vector/spinor are orthogonal. In $3d$,
this means $S_{I}X^{I}_{\;J}=0$.191919Here $X^{I}_{\;J}$ is the position
vector written in spinor indices.

  * •

The resultant expression is projected to the Poincaré section by
$X_{A}\rightarrow P_{A}$ for $P_{A}$ in eqn. (2.2.22) and by projecting
auxiliary vectors/spinors accordingly. For example, in $3d$ we project the
$5d$ spinor $S_{I}$ as

$\displaystyle S_{I}\rightarrow\begin{pmatrix}s_{\alpha}\\\
x^{\alpha}_{\;\beta}s^{\beta}\end{pmatrix}$ (2.2.28)

in terms of the position $x^{\alpha}_{\;\beta}\equiv
x_{\mu}(\gamma^{\mu})^{\alpha}_{\;\beta}$ and the $3d$ spinor $s_{\alpha}$.

We can illustrate this procedure for the construction of the two point
function $\expectationvalue{\mathcal{O}(x_{1},s_{1})\mathcal{O}(x_{2},s_{2})}$
in $3d$, where we use the shorthand notation

$\displaystyle\ac{cO}(x,s)\coloneqq s_{\alpha_{1}}\dots
s_{\alpha_{2\ell}}\mathcal{O}^{\alpha_{1}\dots\alpha_{2\ell}}(x)$ (2.2.29)

for the spinor indices $\alpha_{i}$. In embedding space, we are then trying to
construct
$\expectationvalue{\mathcal{O}(X_{1},S_{1})\mathcal{O}(X_{2},S_{2})}$ which is
homogeneous in $X_{i}$ and $S_{i}$ with degrees $-\Delta-\ell$ and 2spin. We
then write down the only such $\mathrm{SO}(d+1,1)$ invariant:

$\displaystyle\expectationvalue{\mathcal{O}(X_{1},S_{1})\mathcal{O}(X_{2},S_{2})}=c\frac{\left((S_{1})_{I}(S_{2})^{I}\right)^{2\ell}}{\left((X_{1})_{A}(X_{2})^{A}\right)^{\Delta+\ell}}\;,$
(2.2.30)

where we note that
$(S_{1})_{I}(X_{1})^{I}_{\;J}(S_{2})^{J}=(S_{1})_{I}(X_{2})^{I}_{\;J}(S_{2})^{J}=0$
by the transverness condition. With eqn. (2.2.23) and eqn. (2.2.28), we then
obtain

$\displaystyle\expectationvalue{\mathcal{O}(x_{1},s_{2})\mathcal{O}(x_{2},s_{2})}\propto\frac{\left[(s_{1})_{\alpha}(x_{12})^{\alpha}_{\;\beta}(s_{2})^{\beta}\right]^{2\ell}}{x_{12}^{2\Delta+2\ell}}\;.$
(2.2.31)

###  2.3 Lightning review of the conformal bootstrap

In this section, we will briefly review the historical philosophy of the
conformal bootstrap and its modern applications in practice.

As we noted in section 2.1.1, the bootstrap approach historically bypasses
_constructionism_ : one starts with some fundamental conditions and derive the
theory with a bottom-up approach, using these conditions as constraints on the
landscape of the theory. Conformal symmetry by itself is sufficient to fix the
form of two and three point correlation functions, and one aims to use other
conditions such as unitarity and operator product expansion associativity to
entirely determine the form of all correlation functions, importantly, without
resorting to any input from a microscopic theory.

This method is in stark contrast to the traditional top-down approach where
one usually starts a theory by writing down a Lagrangian. This is an old habit
of physicists and it served them extremely well for the past few centuries.
However there is an important philosophical difference between the Lagrangian
approach and that of the bootstrap: starting in the UV by writing a
Lagrangian, one inherently assumes that physics is _constructive_ , i.e. one
can not only reduce all physics to some fundamental laws and particles but
also obtain useful information regarding phenomena observed in the IR by
starting with fundamental laws and particles in the UV and by _constructing
their way back_. In contrast, the main objects of the conformal bootstrap are
correlation functions, hence there is no need for constructionism as one
already works with the theoretically closest objects to the observables.

Despite this philosophical difference between Lagrangian and bootstrap
approaches, most physicists including the bootstrappers already accept both a
reductionist and a constructionist approach towards nature — and hence combine
the bootstrap approach with the Lagrangian theories — and we will assume in
the rest of the thesis that both top-down and bottom-up approaches are
equivalent in this sense.202020Interested reader can refer to the infamous
paper of Anderson [214] and a more recent relevant talk, $27^{\text{th}}$
Occam lecture of Slava Rychkov — see https://www.merton.ox.ac.uk/event/27th-
ockham-lecture-reductionism-vs-bootstrap-are-things-big-always-made-things-
elementary.

Let us switch gears and review the modern employment of the conformal
bootstrap; for simplicity, we will consider the analytic bootstrap for scalar
4-point functions $\expectationvalue{\phi_{1}\phi_{2}\phi_{3}\phi_{4}}$, which
take the general form

$\displaystyle\expectationvalue{\ac{phi}_{1}(x_{1})\phi_{2}(x_{2})\phi_{3}(x_{3})\phi_{4}(x_{4})}=\left(\frac{x_{23}}{x_{13}}\right)^{\Delta_{12}}\left(\frac{x_{13}}{x_{14}}\right)^{\Delta_{43}}\frac{G(z,\bar{z})}{x_{12}^{\Delta_{1}+\Delta_{2}}x_{34}^{\Delta_{3}+\Delta_{4}}}\;,$
(2.3.1)

where $x_{ij}=x_{i}-x_{j}$ and $\Delta_{ij}=\Delta_{i}-\Delta_{j}$, with
Deltai being the scaling dimension of $\phi_{i}$. The conformal cross-ratios
(z,zb) are given by

$z\overline{z}=\frac{x_{12}^{2}x_{34}^{2}}{x_{13}^{2}x_{24}^{2}}\quad,\quad(1-z)(1-\overline{z})=\frac{x_{14}^{2}x_{23}^{2}}{x_{13}^{2}x_{24}^{2}}\;.$
(2.3.2)

The function $G(z,\bar{z})$ can be expanded in conformal blocks for the
$\phi_{1}\phi_{2}\rightarrow\phi_{3}\phi_{4}$ OPE as

$\displaystyle
G(z,\bar{z})=\sum_{\mathcal{O}}f_{12\mathcal{O}}f_{43\mathcal{O}}g_{h,\bar{h}}^{r,s}(z,\bar{z})\;,$
(2.3.3)

where we have introduced the variables

$\displaystyle r$ $\displaystyle=\frac{\Delta_{1}-\Delta_{2}}{2}\;,\qquad$
$\displaystyle s=\frac{\Delta_{3}-\Delta_{4}}{2}\;,$ (2.3.4a) $\displaystyle
h$ $\displaystyle=\frac{\Delta-\ell}{2}\;,$
$\displaystyle\bar{h}=\frac{\Delta+\ell}{2}\;,$ (2.3.4b)

where $\Delta$ and $\ell$ are the scaling dimension and spin of the exchanged
operator. This parameterization is convenient for the lightcone expansion,
where $\overline{h}$ is the natural expansion parameter.

For a spacelike configuration of the operators, we have
$\expectationvalue{\phi_{1}\phi_{2}\phi_{3}\phi_{4}}=\expectationvalue{\phi_{3}\phi_{2}\phi_{1}\phi_{4}}$
which with eqn. (2.3.1) and eqn. (2.3.3) indicate

$\displaystyle\sum_{\mathcal{O}\in\phi_{1}\times\phi_{2}}f_{12\mathcal{O}}f_{43\mathcal{O}}\frac{g_{h,\bar{h}}^{r,s}(z,\bar{z})}{\left((1-z)(1-\overline{z})\right)^{(\Delta_{1}+\Delta_{3})/2}}=\sum_{\mathcal{O}\in\phi_{3}\times\phi_{2}}f_{32\mathcal{O}}f_{41\mathcal{O}}\frac{g_{h,\bar{h}}^{r,s}(1-z,1-\bar{z})}{(z\overline{z})^{(\Delta_{1}+\Delta_{3})/2}}\;.$
(2.3.5)

This is the crossing equation: it constraints the dynamical data — OPE
coefficient $f$’s — in $\mathbb{C}$ spanned by $z,\overline{z}$ where the
conformal blocks, $g(z,\overline{z})$, are theory-independent kinematic
objects.

In general dimensions the conformal blocks are not known in a simple closed
form,212121See [17] for a review of various methods to calculate the blocks.
but in the limit where two operators become light-like separated they display
a universal behavior which makes analytic results possible. In terms of the
conformal cross-ratios, if we take the limit $z\rightarrow 0$, then the
leading order behavior of conformal blocks in any dimension is

$g_{h,\overline{h}}^{r,s}(z,\overline{z})\underset{z\rightarrow
0}{\simeq}z^{h}\overline{z}^{\overline{h}}{}_{2}F_{1}\biggl{[}\genfrac{.}{.}{0.0pt}{}{\overline{h}-r\mskip
8.0mu\overline{h}+s}{2\overline{h}};\overline{z}\biggr{]}\;,$ (2.3.6)

where ${}_{2}F_{1}$ is the standard hypergeometric function. This
approximation is sufficient to compute the leading large-$\ell$ corrections to
the spectrum of double-twist operators using analytic bootstrap techniques, as
was first demonstrated in [112, 111].

In this thesis we will be interested in higher order corrections in the
large-$\ell$ expansion [115, 119, 126], hence we will need subleading terms in
the above expansion. In three dimensions, one can use dimensional reduction to
expand the conformal block in terms of the 2d conformal blocks [215], or
equivalently use an $\mathrm{SL}(2,\mathbb{R})$ expansion [126]. These two
methods are equivalent since the 2d blocks are a simple combination of 1d, or
$\mathrm{SL}(2,\mathbb{R})$, blocks.

The conformal blocks in any dimension can be expanded as

$\displaystyle g_{h,\overline{h}}^{r,s}(z,\overline{z})={}$
$\displaystyle{}\sum\limits_{n=0}^{\infty}\sum\limits_{j=-n}^{n}A_{n,j}^{r,s}(h,\overline{h})z^{h+n}k^{r,s}_{2(\bar{h}+j)}(\bar{z})\;,$
(2.3.7a) $\displaystyle k^{r,s}_{2\bar{h}}(\bar{z})\coloneqq{}$
$\displaystyle{}\overline{z}^{\overline{h}}{}_{2}F_{1}\biggl{[}\genfrac{.}{.}{0.0pt}{}{\overline{h}-r,\overline{h}+s}{2\overline{h}};\overline{z}\biggr{]}\;,$
(2.3.7b)

where $k^{r,s}_{2\bar{h}}(z)$ is the $\mathrm{SL}(2,\mathbb{R})$ block. Using
the decomposition mentioned above, or by solving the Casimir differential
equation, the first two levels are straightforward to work out and are given
by

$\displaystyle A^{r,s}_{0,0}(h,\overline{h})$ $\displaystyle=1\;,$ (2.3.8a)
$\displaystyle A^{r,s}_{1,-1}(h,\overline{h})$
$\displaystyle=\frac{h-\bar{h}}{2h-2\bar{h}+1}\;,$ (2.3.8b) $\displaystyle
A^{r,s}_{1,0}(h,\overline{h})$
$\displaystyle=\frac{1}{2}\left(\frac{rs\left(h-2\bar{h}^{2}+2\bar{h}-1\right)}{(2h-1)(\bar{h}-1)\bar{h}}-h-r+s\right)\;,$
(2.3.8c) $\displaystyle A^{r,s}_{1,1}(h,\overline{h})$
$\displaystyle=\frac{(h+\bar{h}-1)(\bar{h}-r)(\bar{h}+r)(\bar{h}-s)(\bar{h}+s)}{4\bar{h}^{2}(2\bar{h}-1)(2\bar{h}+1)(2h+2\bar{h}-1)}\;,$
(2.3.8d)

for $d=3$. By using such an expansion one can solve eqn. (2.3.5) order by
order in $z$.

##  Chapter 3 Fermions in Conformal Field Theories: A Perturbative Approach

###  3.1 Fermions in $d=3$

There are important and relevant $3d$ CFTs with fermionic operators such as
the $3d$ supersymmetric Ising and GNYs, and they are invoked to explain
several phenomena such as the phase transitions in graphene and the time-
reversal symmetry breaking in d-wave superconductors. We refer to [17] for
details; we’ll only briefly review GNY below as an example.

The GNY can be described by the action [216, 217, 218]

$\displaystyle S=\frac{1}{2}\int
d^{d}x\left(\bar{\psi}_{i}(\not{\partial}+g\phi)\ac{psi}^{i}+\partial_{\mu}\phi\partial^{\mu}\phi+m^{2}\phi^{2}+\lambda\phi^{4}\right)$
(3.1.1)

for the flavor index $i=1,\dots,N$. In $d=4-\epsilon$, one can show by working
perturbatively at large $N$ that this theory has an IR fixed point which
coincides with the UV fixed point of the GN [219, 220]. This is similar to the
familiar relation between the Wilson- Fisher fixed points of the $\phi^{4}$
theory and the UV fixed points of the non-linear sigma model in $d=2+\epsilon$
[221]. The critical point is believed to survive up to $\epsilon=1$ (hence
$d=3$), where large $N$ studies have been carried out [222, 223, 224].
Reference [225] furthermore studies the conformal OPE expansion alongside the
large $N$ expansion of this model.

The numerical bootstrap has addressed fermions in $d=3$ — along with the
contact to GNY models — in [81] and [226]: we refer the reader to those papers
for further details.

####  3.1.1 Notations & Conventions

In this thesis, we follow the same conventions with [1] for $3d$ fermions, so
we will be brief here. We use the Minkowski metric in mostly plus signature
$\eta_{\mu\nu}=\text{diag}(-1,1,1)$, and as our focus are fermions we will be
considering the double cover of the $3d$ conformal group $\mathrm{SO}(3,2)$,
which is isomorphic to $\mathrm{Sp}(2,\mathbb{R})$. We will hence use
representations of the symplectic group; in particular, the smallest
fundamental representation will act on a real two dimensional vector space,
describing Majorana fermions.

The conformal group $\mathrm{SO}(3,2)$ can be realized linearly in two higher
dimensions as we reviewed in section 2.2.2, therefore we embed
$\mathrm{Sp}(2,\mathbb{R})$ representations as the projective null
representations of $\mathrm{Sp}(4,\mathbb{R})$ in this 5d embedding space, use
the fact that conformal symmetry acts as linear isometries in this formalism,
and project back to physical structures by fixing the extra degrees of freedom
in the embedding space.

In practice, one removes the extra degrees of freedom of the embedding space
by going to the so-called Poincaré section,

$\displaystyle X^{A}\rightarrow(x^{\mu},1,x^{2})\;,$ (3.1.2)

where we are working in the lightcone coordinates
$X^{A}=(X^{\mu},X^{+},X^{-})$ and $X^{\pm}$ are related to the Cartesian
coordinates as $X^{\pm}=X^{4}\pm X^{3}$. One can reverse this projection and
lift any point to the embedding space in this Poincaré section.111The only
exception is the point $\infty$: one cannot use the same Poincaré section
$(x^{\mu},1,x^{2})$ used for finite points for the point at infinity as well.
A simple way to see this is as follows. We start with
$X^{a}=(x^{\mu},X^{+},X^{-})$ and impose nullness to obtain
$X^{A}=(x^{\mu},X^{+},x^{2}/X^{+})$. If we now consider an inversion as
$x^{\mu}_{R}=x^{\mu}/x^{2}$, we see that
$X^{A}_{R}=(x^{\mu}/x^{2},X^{+},(x^{2}X^{+})^{-1})\sim(x^{\mu},x^{2}X^{+},1/X^{+})$
where we use projectiveness of the representation. For $x^{2}\neq 0$, we can
choose $X^{+}=1/x^{2}$ for $X_{R}^{A}$ and $X^{+}=1$ for $X^{A}$, which means
we can write both $x^{\mu}$ and $x_{R}^{\mu}$ using the same Poincaré section
$(x^{\mu},1,x^{2})$. For $x^{2}=0$, this is no longer possible, and the
reflected point $x_{R}^{\mu}=x^{\mu}/x^{2}=\infty$ should be instead in the
Poincaré section $(x^{\mu},x^{2},1)$.

As developed in [153] and generalized to 3d spinors in [81], one can encode
spinors by polynomials with auxiliary spinor fields:

$\displaystyle\mathcal{O}(x,s)={}$ $\displaystyle
s_{\alpha_{1}}s_{\alpha_{2}}\dots
s_{\alpha_{2\ell}}\mathcal{O}^{\alpha_{1}\alpha_{2}\dots\alpha_{2\ell}}(x)\;,$
(3.1.3)
$\displaystyle\mathcal{O}^{\alpha_{1}\alpha_{2}\dots\alpha_{2\ell}}(x)={}$
$\displaystyle\frac{1}{(2\ell)!}\frac{\partial^{2\ell}}{\partial_{s_{a_{1}}}\partial_{s_{a_{2}}}\dots\partial_{s_{a_{2\ell}}}}\mathcal{O}(x,s)\;.$

In this formalism, we can relate the spinor field $\Psi(X,S)\coloneqq
S_{I}\Psi^{I}(X)$ in the embedding space and the spinor field
$\psi(x,s)\coloneqq s_{\alpha}\psi^{\alpha}(x)$ in the physical space as

$\Psi(X,S)=\frac{1}{(X^{+})^{\Delta_{\psi}}}\psi(x,s)\;,$ (3.1.4)

where $S_{I}$ is taken to the Poincaré section via setting

$S_{I}=\sqrt{X^{+}}\begin{pmatrix}s_{\alpha}\\\
x^{\alpha}_{\;\;\beta}s^{\beta}\end{pmatrix}\;.$ (3.1.5)

Here we use the matrices $\gamma$ and $\Gamma$ to convert the indices,

$X^{I}_{\;\;J}\equiv X^{A}(\Gamma_{A})^{I}_{\;\;J}\;,\qquad
x^{\alpha}_{\;\;\beta}\equiv x^{\mu}(\gamma_{\mu})^{\alpha}_{\;\;\beta}\;.$
(3.1.6)

The $3d$ gamma matrices and $\mathrm{Sp}(2,\mathbb{R})$ invariant $\epsilon$
are defined as

$\displaystyle(\gamma_{0})^{\alpha}_{\;\;\beta}=i(\sigma_{2})_{\alpha\beta}\;,\quad(\gamma_{1})^{\alpha}_{\;\;\beta}=(\sigma_{1})_{\alpha\beta}\;,\quad(\gamma_{2})^{\alpha}_{\;\;\beta}=(\sigma_{3})_{\alpha\beta}\;,\quad\epsilon_{\alpha\beta}=\epsilon^{\alpha\beta}=i(\sigma_{2})_{\alpha\beta}\;,$
(3.1.7)

where $\sigma_{i}$ are the standard Pauli matrices:

$\displaystyle\sigma_{1}=\left(\begin{array}[]{cc}0&1\\\ 1&0\\\
\end{array}\right)\quad,\quad\sigma_{2}=\left(\begin{array}[]{cc}0&-i\\\
i&0\\\ \end{array}\right)\quad,\quad\sigma_{3}=\left(\begin{array}[]{cc}1&0\\\
0&-1\\\ \end{array}\right).$ (3.1.8g)

The embedding space gamma matrices and $\mathrm{Sp}(4,\mathbb{R})$ invariant
$\Omega$ are then defined as:222We note that gamma matrices are defined such
that the generators of $\mathrm{Sp}(2,\mathbb{R})$ and
$\mathrm{Sp}(4,\mathbb{R})$ are given as
$-\frac{i}{4}[\gamma^{\mu},\gamma^{\nu}]$ and
$-\frac{i}{4}[\Gamma^{A},\Gamma^{B}]$, respectively.

$\displaystyle\Gamma_{0}=\gamma_{2}\otimes\gamma_{0}\;,\quad\Gamma_{1}=\mathbbm{I}\otimes\gamma_{1}\;,\quad\Gamma_{2}=\mathbbm{I}\otimes\gamma_{2}\;,\quad\Gamma_{3}=\gamma_{0}\otimes\gamma_{0}\;,\quad\Gamma_{4}=\gamma_{1}\otimes\gamma_{0}\;,\quad\Omega=\epsilon\otimes\mathbbm{I}$
(3.1.9)

with the embedding space metric $g_{IJ}=\text{diag}\left(-,+,+,+,-\right)$. In
lightcone coordinates, the gamma matrices take the form

$\displaystyle(\Gamma^{\mu})^{I}_{\;\;J}=\begin{pmatrix}\left(\gamma^{\mu}\right)^{\alpha}_{\;\;\beta}&0\\\
0&\left(\gamma^{\mu}\right)_{\alpha}^{\;\;\beta}\end{pmatrix}\;,\quad(\Gamma^{+})^{I}_{\;\;J}=\begin{pmatrix}0&2\epsilon^{\alpha\beta}\\\
0&0\end{pmatrix}\;,\quad(\Gamma^{-})^{I}_{\;\;J}=\begin{pmatrix}0&0\\\
2\epsilon_{\alpha\beta}&0\end{pmatrix}.$ (3.1.10)

Finally, we raise (lower) the spinor indices with $\mathrm{SP}$ invariants by
acting from right (left), e.g. $x_{\alpha}=\epsilon_{\alpha\beta}x^{\beta}$
and $x^{\alpha}=x_{\beta}\epsilon^{\beta\alpha}$.

To describe the embedding space spinor structures, we define the shorthand
notation

$\displaystyle\expectationvalue{S_{1}X_{2}X_{3}\dots X_{n-1}S_{n}}\equiv$
$\displaystyle-S_{1}\cdot X_{2}\cdot X_{3}\cdots X_{n-1}\cdot S_{n}$ (3.1.11)
$\displaystyle=$
$\displaystyle-(S_{1})_{I}(X_{2})^{I}_{\;\;J}(X_{3})^{J}_{\;\;K}\cdots(X_{n-1})^{L}_{\;\;M}(S_{n})^{M}.$

In this convention, we diagonalize the spectrum and normalize the operators
$\mathcal{O}^{\Delta,\ell}$ such that they satisfy

$\displaystyle\expectationvalue{\mathcal{O}^{\Delta,\ell}(X_{1},S_{1})\mathcal{O}^{\Delta,\ell}(X_{2},S_{2})}=i^{2\ell}\frac{\expectationvalue{S_{1}S_{2}}^{2\ell}}{X_{12}^{2\Delta+2\ell}}.$
(3.1.12)

For example, two-point functions of fermions in embedding and physical spaces
read as

$\expectationvalue{\Psi(X_{1},S_{1})\Psi(X_{2},S_{2})}=i\frac{\expectationvalue{S_{1}S_{2}}}{X_{12}^{\Delta_{\psi}+\frac{1}{2}}}\;,\qquad\expectationvalue{\psi^{\alpha}(x_{1})\psi_{\beta}(x_{2})}=i\frac{(x_{12})^{\alpha}_{\;\;\beta}}{x_{12}^{2\Delta_{\psi}+1}}\;.$
(3.1.13)

Conformal symmetry also fixes the form of three point correlators albeit non-
uniquely unless two of the operators are scalars. Therefore, in general, one
writes down there point correlator
$\expectationvalue{\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}}$ in terms of
three point structures
$\expectationvalue{\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}}^{(i)}$ as

$\displaystyle\expectationvalue{\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}}=\sum\limits_{i}\lambda^{i}_{\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}}\expectationvalue{\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}}^{(i)}\;,$
(3.1.14)

where $\lambda$ are the OPE coefficients. We stress that
$\expectationvalue{\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}}^{(i)}$ are
_not_ physical three point correlator; in particular, they are formal
expressions and do not necessarily satisfy Fermi-statistics if operators are
interchanged. Throughout this paper, we will refer to these objects as three-
point structures, for which we will follows the conventions of [81, 148] and
choose three point structures as:

$\displaystyle\expectationvalue{\phi_{1}\phi_{2}\mathcal{O}_{3}}^{(1)}$
$\displaystyle=\frac{\expectationvalue{S_{3}X_{1}X_{2}S_{3}}^{\ell}}{X_{12}^{(\Delta_{123}+\ell)/2}X_{23}^{(\Delta_{231}+\ell)/2}X_{31}^{(\Delta_{312}+\ell)/2}},$
(3.1.15a)
$\displaystyle\expectationvalue{\psi_{1}\phi_{2}\mathcal{O}_{3}}^{(1)}$
$\displaystyle=\frac{\expectationvalue{S_{1}S_{3}}\expectationvalue{S_{3}X_{1}X_{2}S_{3}}^{\ell-\frac{1}{2}}}{X_{12}^{(\Delta_{123}+\ell-\frac{1}{2})/2}X_{23}^{(\Delta_{231}+\ell-\frac{1}{2})/2}X_{31}^{(\Delta_{312}+\ell+\frac{1}{2})/2}},$
(3.1.15b)
$\displaystyle\expectationvalue{\psi_{1}\phi_{2}\mathcal{O}_{3}}^{(2)}$
$\displaystyle=\frac{\expectationvalue{S_{1}X_{2}S_{3}}\expectationvalue{S_{3}X_{1}X_{2}S_{3}}^{\ell-\frac{1}{2}}}{X_{12}^{(\Delta_{123}+\ell+\frac{1}{2})/2}X_{23}^{(\Delta_{231}+\ell+\frac{1}{2})/2}X_{31}^{(\Delta_{312}+\ell-\frac{1}{2})/2}},$
(3.1.15c)
$\displaystyle\expectationvalue{\psi_{1}\psi_{2}\mathcal{O}_{3}}^{(1)}$
$\displaystyle=\frac{\expectationvalue{S_{1}S_{2}}\expectationvalue{S_{3}X_{1}X_{2}S_{3}}^{\ell}}{X_{12}^{(\Delta_{123}+\ell+1)/2}X_{23}^{(\Delta_{231}+\ell)/2}X_{31}^{(\Delta_{312}+\ell)/2}},$
(3.1.15d)
$\displaystyle\expectationvalue{\psi_{1}\psi_{2}\mathcal{O}_{3}}^{(2)}$
$\displaystyle=\frac{\expectationvalue{S_{1}S_{3}}\expectationvalue{S_{2}S_{3}}\expectationvalue{S_{3}X_{1}X_{2}S_{3}}^{\ell-1}}{X_{12}^{(\Delta_{123}+\ell-1)/2}X_{23}^{(\Delta_{231}+\ell)/2}X_{31}^{(\Delta_{312}+\ell)/2}},$
(3.1.15e)
$\displaystyle\expectationvalue{\psi_{1}\psi_{2}\mathcal{O}_{3}}^{(3)}$
$\displaystyle=\frac{\left(X_{23}\expectationvalue{S_{1}S_{3}}\expectationvalue{S_{2}X_{1}S_{3}}+X_{13}\expectationvalue{S_{2}S_{3}}\expectationvalue{S_{1}X_{2}S_{3}}\right)\expectationvalue{S_{3}X_{1}X_{2}S_{3}}^{\ell-1}}{X_{12}^{(\Delta_{123}+\ell-1)/2}X_{23}^{(\Delta_{231}+\ell)/2}X_{31}^{(\Delta_{312}+\ell)/2}},$
(3.1.15f)
$\displaystyle\expectationvalue{\psi_{1}\psi_{2}\mathcal{O}_{3}}^{(4)}$
$\displaystyle=\frac{\left(X_{23}\expectationvalue{S_{1}S_{3}}\expectationvalue{S_{2}X_{1}S_{3}}-X_{13}\expectationvalue{S_{2}S_{3}}\expectationvalue{S_{1}X_{2}S_{3}}\right)\expectationvalue{S_{3}X_{1}X_{2}S_{3}}^{\ell-1}}{X_{12}^{(\Delta_{123}+\ell-1)/2}X_{23}^{(\Delta_{231}+\ell)/2}X_{31}^{(\Delta_{312}+\ell)/2}},$
(3.1.15g)

where we define for brevity

$\displaystyle X_{ab}\coloneqq-2X_{a}\cdot X_{b}\;,$ (3.1.16a) and
$\displaystyle\ac{De\ell ta3}\coloneqq\Delta_{a}+\Delta_{b}-\Delta_{c}\;.$
(3.1.16b)

For integer spin we can also convert to vector notation,

$\displaystyle\mathcal{O}^{\alpha_{1}\dots\alpha_{2J}}=$
$\displaystyle\mathcal{O}^{\mu_{1}\dots\mu_{J}}\gamma_{\mu_{1}}^{\alpha_{1}\alpha_{2}}\cdots\gamma_{\mu_{J}}^{\alpha_{2J-1}\alpha_{2J}},$
(3.1.17) $\displaystyle\mathcal{O}^{\mu_{1}\dots\mu_{J}}=$
$\displaystyle\left(-\frac{1}{2}\right)^{J}\gamma^{\mu_{1}}_{\alpha_{1}\alpha_{2}}\cdots\gamma^{\mu_{J}}_{\alpha_{2J-1}\alpha_{2J}}\mathcal{O}^{\alpha_{1}\dots\alpha_{2J}},$

where $\mathcal{O}^{\mu_{1}\dots\mu_{J}}$ is a symmetric traceless tensor. If
we introduce the auxiliary polarization vectors $z_{\mu}$, and insist that

$\displaystyle z_{\mu_{1}}\dots
z_{\mu_{\ell}}\mathcal{O}^{\mu_{1}\dots\mu_{\ell}}=s_{\alpha_{1}}\dots
s_{\alpha_{2\ell}}\mathcal{O}^{\alpha_{1}\dots\alpha_{2\ell}}$ (3.1.18)

holds, we find the following relation:

$\displaystyle z_{\mu}=s_{\alpha}s_{\beta}\gamma^{\alpha\beta}.$ (3.1.19)

We can now use these relations to convert two and three-point functions to
vector notation for integer spin. In particular, eqn. (3.1.12) and eqn.
(3.1.15a) become

$\displaystyle\expectationvalue{\mathcal{O}^{\Delta,\ell}(X_{1},Z_{1})\mathcal{O}^{\Delta,\ell}(X_{2},Z_{2})}=$
$\displaystyle\frac{1}{2^{\ell}}\frac{H_{12}^{\ell}}{X_{12}^{\Delta_{\ell}}},$
(3.1.20a)
$\displaystyle\expectationvalue{\phi(X_{1},Z_{1})\phi(X_{2},Z_{2})\mathcal{O}(X_{3},Z_{3})}=$
$\displaystyle\frac{V_{3}^{\ell}}{X_{12}^{(\Delta_{123}-\ell)/2}X_{23}^{(\Delta_{231}+\ell)/2}X_{31}^{(\Delta_{312}+\ell)/2}},$
(3.1.20b)

where we define

$\displaystyle H_{12}\equiv{}$ $\displaystyle{}-2\left[(Z_{1}\cdot
Z_{2})(X_{1}\cdot X_{2})-(X_{1}\cdot Z_{2})(Z_{1}\cdot X_{2})\right],$
(3.1.21) $\displaystyle V_{3}\equiv{}$ $\displaystyle{}\frac{(Z_{3}\cdot
X_{1})(X_{2}\cdot X_{3})-(Z_{3}\cdot X_{2})(X_{1}\cdot X_{3})}{X_{1}\cdot
X_{2}},$

in the conventions of [153].

Associated to every representation $\mathcal{O}$ is a shadow representation
$\widetilde{\mathcal{O}}$ which has dimension $\widetilde{\Delta}$ and the
same $\mathrm{SO}(d)$ representation $\rho$333In even dimensions we actually
need the reflected representation $\rho^{R}$, but in odd dimensions the two
are equivalent so to simplify the discussion we will ignore the distinction.
where we define

$\displaystyle\widetilde{\Delta}=d-\Delta$ (3.1.22)

for brevity. Then there exists a conformally-invariant pairing:

$\displaystyle\left(\widetilde{\mathcal{O}}^{\dagger},\mathcal{O}\right)=\int
d^{d}x\widetilde{\mathcal{O}}^{\dagger}_{\alpha_{1}\cdots\alpha_{2J}}(x)\mathcal{O}^{\alpha_{1}\cdots\alpha_{2J}}(x),$
(3.1.23)

where $\widetilde{\mathcal{O}}^{\dagger}$ has dimension $\widetilde{\Delta}$.
We will work with real operators, so we can drop the $\dagger$ in the
expressions which follow.

In the rest of the thesis we will implicitly contract indices when an operator
and its shadow are being integrated over, we will always use spinor indices,
and suppressed spinor indices always go from southwest to northeast. In
particular, this means that order of correlators in an expression matters in
our conventions, for instance

$\displaystyle\int
dx\expectationvalue{\mathcal{O}_{1}\mathcal{O}(x)\mathcal{O}_{2}}\expectationvalue{\mathcal{O}_{3}\widetilde{\mathcal{O}}(x)\mathcal{O}_{4}}=(-1)^{2\ell_{\mathcal{O}}}\int
dx\expectationvalue{\mathcal{O}_{3}\widetilde{\mathcal{O}}(x)\mathcal{O}_{4}}\expectationvalue{\mathcal{O}_{1}\mathcal{O}(x)\mathcal{O}_{2}}\
.$ (3.1.24)

By using the conformally-invariant pairing in eqn. (3.1.23), we can define
_shadow transformation_ which maps a representation $\mathcal{O}$ to
$\widetilde{\mathcal{O}}$:

$\displaystyle\mathbf{S}[\mathcal{O}](x)\equiv\int
dy\mathcal{O}(y)\expectationvalue{\widetilde{\mathcal{O}}(y)\widetilde{\mathcal{O}}(x)}.$
(3.1.25)

We will adopt the convention that the shadow transform always acts by
multiplying the two-point function from the left. The distinction is
immaterial for bosonic correlators, but when we introduce fermions the
ordering does matter.

####  3.1.2 Double twist sector of $3d$ fermions

It is shown in [111] that every CFT has a double twist sector which
asymptottes to MFT operators at large spin. This spectrum is important for
various reasons; for example, it is an open question whether several relevant
operators such as stress tensor lie in these trajectories. Likewise, these
operators correspond to two particle states in the bulk via the AdS/CFT
correspondence. Finally, double twist operators are the relevant spectrum in a
channel to reproduce low lying operators such as exchange of the identity
operator and stress tensor in the crossed channel in the light-cone limit.

In the free theory limit, double twist operators can be written down as
composite operators of the form

$\displaystyle:A_{\alpha_{1}\dots\alpha_{p}}\partial_{\beta_{11}\beta_{12}}\cdots\partial_{\beta_{q1}\beta_{q2}}(\partial^{2})^{r}B_{\rho_{1}\dots\rho_{s}}:-\text{
total derivatives}\;,$ (3.1.26)

where total derivative parts are extracted from the normal-ordered product to
ensure that the resultant operator is a primary. In addition, the open indices
above are symmetrized or antisymmetrized (contracted) dependent on the
representation.

We group such operators into towers which has the same twist
$\tau=\Delta-\ell$. In particular, we observe that inclusion of a partial
derivative with open indices does not change the twist hence we define and
denote the double twist towers as

$\displaystyle{}[AB]_{n}\xrightarrow{\text{free theory
limit}}\\{A(\partial^{2})^{n}B,\;A\partial_{\alpha_{1}\beta_{1}}(\partial^{2})^{n}B,\;A\partial_{\alpha_{1}\beta_{1}}\partial_{\alpha_{2}\beta_{2}}(\partial^{2})^{n}B,\;\cdots\\}\;,$
(3.1.27)

where $A$ and $B$ may have open or contracted indices. In a free theory, the
double twist tower $[AB]_{n}$ is a collection of composite operators
schematically given in the equation above444We emphasize that the expression
$A(\partial^{2})^{n}B$ and so on are schematic as the correct form should be
normal-ordered with total derivative pieces removed as defined in eqn.
(3.1.26). and all these operators have the same twist
$\tau_{[AB]_{n}}=\tau_{A}+\tau_{B}+2n$. Once we move away from the free
theory, we can no longer describe double twist operators with such simple
composite structures; nevertheless, they are still in the spectrum and they
satisfy

$\displaystyle\lim\limits_{\ell\rightarrow\infty}\tau_{[AB]_{n}}=\tau_{A}+\tau_{B}+2n\;,$
(3.1.28)

hence we can define a double twist tower as a collection of operators with the
same twist accumulation point as spin goes to infinity. At finite spin, we
have

$\displaystyle\tau_{[AB]_{n}}=\tau_{A}+\tau_{B}+2n+\gamma_{[AB]_{n}}(n,\ell)\;,$
(3.1.29)

where $\gamma$ is the _anomalous dimension_. One of the successes of the
lightcone bootstrap was to show that for any CFT555We restrict ourselves to
local CFTs with twist gap. In $d=3$, any unitary CFT does have the required
twist gap.

$\displaystyle\lim\limits_{\ell\rightarrow\infty}\ell^{\tau_{0}}\gamma_{[AB]_{n}}(n,\ell)=\text{
constant}\;,$ (3.1.30)

where $\tau_{0}$ is the twist of lowest twist non-identy operator in the
spectrum of the crossed channel, which is usually the stress tensor. For
example, we expect a double twist family

$\displaystyle{}[\psi_{\alpha}\psi^{\alpha}]_{0}\xrightarrow{\text{free theory
limit}}\\{\psi_{\alpha}\psi^{\alpha},\;\psi_{\alpha}\partial_{\beta_{11}\beta_{12}}\psi^{\alpha},\;\psi_{\alpha}\partial_{\beta_{11}\beta_{12}}\partial_{\beta_{21}\beta_{22}}\psi^{\alpha},\;\cdots\\}$
(3.1.31)

in the spectrum of fermionic CFTs: one aim of the conformal bootstrap is to
find an analytic form for the anomalous dimension of this tower.

Let’s now step back and briefly discuss what kind of representations we expect
for double twist operators for $3d$ CFTs. We know that the irreducible
representations of the $\mathrm{so}(3)$ algebra are labeled by one index
$j\in\frac{1}{2}\mathbb{Z}_{\geq 0}$, where tensor products of two irreps have
the well known reduction

$\displaystyle j_{1}\otimes
j_{2}=(j_{1}+j_{2})\oplus(j_{1}+j_{2}-1)\oplus(j_{1}+j_{2}-2)\oplus\cdots\oplus\absolutevalue{j_{1}-j_{2}}\;.$
(3.1.32)

For example, $\frac{1}{2}\otimes\frac{1}{2}=1\oplus 0$ indicates that we have
both spin-1 and spin-0 operators constructed out of two fermions, i.e.
$\psi_{\\{\alpha}\psi_{\beta\\}}$ and
$\psi_{[\alpha}\psi_{\beta]}=-\frac{1}{2}\epsilon_{\alpha\beta}(\psi_{\sigma}\psi^{\sigma})$.
In what follows, we will assume that all open $\mathrm{Sp}(2,\mathbb{R})$
indices are symmetrized as any antisymmetric pair can be rewritten in terms of
contracted indices and $\epsilon_{\alpha\beta}$. So, we say that
$\frac{1}{2}\otimes\frac{1}{2}=1\oplus 0$ indicates that
$\psi_{\alpha}\times\psi_{\beta}$ can be decomposed into
$\psi_{\alpha}\psi_{\beta}$ and $\psi_{\alpha}\psi^{\alpha}$.

In some cases, we are interested in CFTs with parity symmetry where the
relevant algebra is enlarged from $\mathrm{so}(3)$ to $\mathrm{o}(3)$ which
are labeled by the previous index $j$ and a sign $p=\pm 1$. We will denote the
representation as $j^{p}$, where $p$ indicates if the corresponding irrep is
parity even or parity odd. The decomposition of products of two irreps now
becomes

$\displaystyle j_{1}^{+}\otimes
j_{2}^{+}=(j_{1}+j_{2})^{+}\oplus(j_{1}+j_{2}-1)^{-}\oplus(j_{1}+j_{2}-2)^{+}\oplus\cdots\oplus\absolutevalue{j_{1}-j_{2}}^{+|-}\;,$
(3.1.33)

where parity alternates between different irreps. Likewise, if some of the
parities are flipped on left hand side, so are those in the right hand side,
e.g. $j_{1}^{+}\otimes j_{2}^{-}=(j_{1}+j_{2})^{-}\oplus\cdots$.

We now turn to the bispinors we can construct out of $\psi^{\alpha}$ and
$\psi^{\beta}$. The representation of the Majorana fermion can be taken as
either $\frac{1}{2}^{+}$ or $\frac{1}{2}^{-}$: this ambiguity reflects the
fact that it is not the $\mathrm{O}(2,1)$ but the $\mathrm{Pin}(2,1)$ group
which acts on the spinors. Nevertheless, as we will always be interested in
composite operators or correlators with even number of fermions, our results
will be ambiguity-free. Let us choose that $\psi_{\alpha}$ transforms under
the $\mathrm{Pin}(2,1)$ with the representation $\frac{1}{2}^{+}$. We then
have the decomposition

$\displaystyle\frac{1}{2}^{+}\otimes\frac{1}{2}^{+}=1^{+}\oplus 0^{-}\;,$
(3.1.34)

which means we can construct a parity even vector and a parity odd scalar out
of two $\psi$.666One can also check this explicitly. If we choose parity
transformation as a reflection $x^{1}\rightarrow-x^{1}$ and
$x^{0,2}\rightarrow x^{0,2}$, the Majorana fermion transforms as
$\psi^{\alpha}\rightarrow\pm(\gamma^{1}\psi)^{\alpha}$ where the ambiguity is
the same one discussed above. By using
$\psi_{\alpha}=\epsilon_{\alpha\beta}\psi^{\beta}$, we can show that
$\psi_{\alpha}\rightarrow\mp(\psi\gamma^{1})_{\alpha}$, hence
$\psi_{\alpha}\psi^{\alpha}\rightarrow-(\psi\gamma^{1})_{\alpha}(\gamma^{1}\psi)^{\alpha}=-\psi_{\alpha}\psi^{\alpha}$
indicating that $\psi_{\alpha}\psi^{\alpha}$ is a parity-odd scalar. One can
similarly check the parities of other operators. These are exactly
$\psi_{\alpha}\psi_{\beta}$ and $\psi_{\alpha}\psi^{\alpha}$, which should
then appear in the double twist towers $[\psi_{\alpha}\psi_{\beta}]_{n}$ and
$[\psi_{\alpha}\psi^{\alpha}]_{n}$ .

We can now include a partial derivative, which is parity even spin$-1$ object.
That means

$\displaystyle\frac{1}{2}^{+}\otimes\frac{1}{2}^{+}\otimes 1^{+}=2^{+}\oplus
1_{2}^{-}\oplus 0^{+}\;,$ (3.1.35)

where $a$ denotes the multiplicity of the representation $j$ in the notation
$j_{a}$. These objects are

$\displaystyle\psi_{\alpha}\partial_{\gamma\lambda}\psi_{\beta}\;,\quad\psi_{\alpha}\partial_{\beta\gamma}\psi^{\alpha}\;,\quad\psi_{\alpha}\partial_{\beta\gamma}\psi^{\gamma}\;,\quad\psi_{\alpha}\partial^{\alpha}_{\;\;\beta}\psi^{\beta}\;.$
(3.1.36)

First two objects already appear in $[\psi_{\alpha}\psi_{\beta}]_{n}$ and
$[\psi_{\alpha}\psi^{\alpha}]_{n}$ double twist towers. We are left to
conclude that inclusion of a derivative yields only two new towers: a parity
odd tower $[\psi_{a}(\partial\psi)_{\beta}]_{n}$ and a parity even tower
$[\psi_{a}(\partial\psi)^{\alpha}]_{n}$.

Considering more derivatives will not produce new towers as we expect a total
of four different double twist towers. To see that, we first note that out of
the irreducibly representations in the decomposition of p-derivatives, i.e.
$1^{+}\otimes 1^{+}\cdots\otimes 1^{+}$, only the fully symmetric
representation is relevant as contracted derivatives can be rewritten in terms
of $(\partial^{2})$ and fewer number of derivatives with open indices due to
the identity
$\partial^{\alpha}_{\;\gamma}\partial^{\gamma}_{\;\beta}=(\partial^{2})\delta^{\alpha}_{\beta}$.
Hence, the independent towers in the decomposition of two fermions with
$p-$derivatives appear in $\frac{1}{2}^{+}\otimes\frac{1}{2}^{+}\otimes
p^{+}\in\frac{1}{2}^{+}\otimes\frac{1}{2}^{+}\otimes 1^{+}\otimes
1^{+}\cdots\otimes 1^{+}$ which is decomposed as

$\displaystyle\frac{1}{2}^{+}\otimes\frac{1}{2}^{+}\otimes
p^{+}=(p+1)^{+}\oplus p_{2}^{-}\oplus(p-1)^{+}\;,$ (3.1.37)

which are the towers

$\displaystyle{}[\psi_{\alpha}\psi_{\beta}]_{n}\;,\quad[\psi_{\alpha}\psi^{\alpha}]_{n}\;,\quad[\psi_{\alpha}(\partial\psi)_{\beta}]_{n}\;,\quad[\psi_{\alpha}(\partial\psi)^{\alpha}]_{n}\;.$
(3.1.38)

With the double twist towers constructed as above, we finally turn to the
_selection rules_ on spin imposed by the fact that double twist operators
satisfy certain symmetries as they are defined modulo total derivatives. For
example, via integration by parts we see that

$\displaystyle\psi_{\alpha}\partial_{\mu_{1}}\dots\partial_{\mu_{\ell}}\partial^{2n}\psi_{b}=(-1)^{\ell+1}\psi_{\beta}\partial_{\mu_{1}}\dots\partial_{\mu_{\ell}}\partial^{2n}\psi_{\alpha}+\text{
total derivatives}\;,$ (3.1.39)

hence
$\psi_{(\alpha}\partial_{\mu_{1}}\dots\partial_{\mu_{p}}\partial^{2n}\psi_{b)}$
is nonzero only if $p+1\in 2\mathbb{Z}^{+}$. This then indicates that the
tower $[\psi_{\alpha}\psi_{\beta}]_{n}$ has operators of even spin only. By a
similar analysis, we show that only odd/even spin operators contribute to each
double twist tower: we summarize them in Table. (3.1).

Table 3.1: We list the different double-twist families for Majorana fermions in 3d, with their twist accumulation points, parities, and spins. Family |  Twist |  Parity |  Spin
---|---|---|---
$[\psi_{(\alpha}\psi_{\beta)}]_{n,\ell}$ |  $2\Delta_{\psi}+2n-1$ |  Even |  $\ell\geq 2$, Even
$[\psi_{\alpha}\psi^{\alpha}]_{n,\ell}$ |  $2\Delta_{\psi}+2n$ |  Odd |  $\ell\geq 0$, Even
$[\psi_{(\rho}\partial^{\rho}_{\;\alpha}\psi_{\beta)}]_{n,\ell}$ |  $2\Delta_{\psi}+2n$ |  Odd |  $\ell\geq 1$, Odd
$[\psi_{\alpha}(\partial\psi)^{\alpha}]_{n,\ell}$ |  $2\Delta_{\psi}+2n+1$ |  Even |  $\ell\geq 0$, Even

###  3.2 Large spin expansion

####  3.2.1 Lightcone bootstrap

##### Short review

In the following calculations, we review the work of [126] which solves the
lightcone bootstrap in a perturbative expansion. We start by considering the
4-point function of identical scalars, $\expectationvalue{\phi\phi\phi\phi}$.
The 4-point function is invariant under $1\leftrightarrow 3$ (or
$s\leftrightarrow t$) crossing, which implies

$\left(\frac{(1-z)(1-\bar{z})}{z\overline{z}}\right)^{2h_{\phi}}\sum_{\mathcal{O}}P_{\phi\phi\mathcal{O}}g_{h_{\mathcal{O}},\overline{h}_{\mathcal{O}}}(z,\overline{z})=\sum_{\mathcal{O}}P_{\phi\phi\mathcal{O}}g_{h_{\mathcal{O}},\overline{h}_{\mathcal{O}}}(1-\overline{z},1-z)\;,$
(3.2.1)

where $P_{\phi\phi\mathcal{O}}=f_{\phi\phi\Omega}^{2}$.

We can now consider this equation in the lightcone limit $z\ll 1-\bar{z}\ll
1$. In the limit $z\ll 1$, the identity operator dominates on the left-hand
side, while taking $1-\bar{z}\ll 1$ allows us to use the
$\mathrm{SL}(2,\mathbb{R})$ expansion on the right-hand side. In this limit,
the crossing equation becomes

$\left(\frac{1-\bar{z}}{z}\right)^{2h_{\phi}}\approx\sum_{\mathcal{O}}P_{\phi\phi\mathcal{O}}(1-\bar{z})^{h_{\mathcal{O}}}k_{2\overline{h}_{\mathcal{O}}}(1-z)\;.$
(3.2.2)

By the arguments of [111, 112], in order to match the $z\rightarrow 0$
divergence on the left-hand side, which is not present in any individual
$t$-channel block, we need to sum over operators with unbounded spin on the
right-hand side. Specifically, we need a tower of “double-twist” operators in
the $t$-channel, $[\phi\phi]_{0,\ell}$, such that $h_{0,\ell}\rightarrow
2h_{\phi}$ as $\ell\rightarrow\infty$.

At this point we could use a Bessel function approximation of the blocks to
derive the large-$\ell$ asymptotics of the OPE coefficients, but it will be
more useful to use our knowledge of $1d$ generalized free field theories to
write down the exact sum [126]

$\displaystyle\sum_{\begin{subarray}{c}\bar{h}=-a+\ell\\\
\ell=0,1,...\end{subarray}}S_{a}(\bar{h})k_{2\bar{h}}(1-z)=\left(\frac{z}{1-z}\right)^{a}\;,$
(3.2.3a) where $\displaystyle
S_{a}(\bar{h})=\frac{\Gamma(\bar{h})^{2}\Gamma(\bar{h}-a-1)}{\Gamma(-a)^{2}\Gamma(2\bar{h}-1)\Gamma(\bar{h}+a+1)}\;.$
(3.2.3b)

From (3.2.3), we now have at large $\ell$, where $\bar{h}\approx
2h_{\phi}+\ell$, the following result for the OPE coefficients:

$P_{\phi\phi\mathcal{O}}(\bar{h})\sim S_{-2h_{\phi}}(\bar{h})\;.$ (3.2.4)

The “$\sim$” is because with this approach we can only find the asymptotic
expansion for the OPE coefficients at large $\bar{h}$. Note that by expanding
to higher orders in $1-\bar{z}$ one can also prove the existence of operators
$[\phi\phi]_{n,\ell}$ which have $h_{n,\ell}\rightarrow 2h_{\phi}+n$ as
$\ell\rightarrow\infty$.

To extend these calculations to higher orders in the large-$\bar{h}$
expansion, we can use the $\mathrm{SL}(2,\mathbb{R})$ expansion on both sides
of the crossing equation (3.2.1), expand in $z\ll 1-\bar{z}\ll 1$, and then
use (3.2.3) to unambiguously match generic powers of $z$ in the $s$-channel to
the large-spin asymptotics of double-twist operators in the $t$-channel.

Of course, there are subtleties in this procedure which we have glossed over.
First, for the arguments of [111, 112] to work when matching a power of $z$ in
the $s$-channel to an infinite sum of $t$-channel blocks, we need the
$s$-channel term to be more divergent than any individual $t$-channel block.
From (3.2.1), we see this is only true if $h_{\mathcal{O}}<2h_{\phi}$.
However, as noted in [115], we can make any generic, individual power of
$z^{a}$ on the left-hand side of (3.2.1) as divergent as we like by repeatedly
acting with a $\mathrm{SL}(2,\mathbb{R})$ Casimir differential operator:

$\displaystyle\mathcal{C}\equiv(1-z)^{2}z\partial_{z}^{2}+(1-z)^{2}\partial_{z}\;.$
(3.2.5)

Since the $t$-channel $\mathrm{SL}(2,\mathbb{R})$ blocks are eigenfunctions of
this Casimir, these differential operators leave the form of the $t$-channel
expansion unchanged. Therefore, by acting with this differential operator
sufficiently many times we can make the $s$-channel more divergent than the
crossed channels.

In the terminology of [126], generic powers $z^{a}$ are “Casimir-singular”. On
the other hand, terms like $z^{n}$ and $z^{n}\log(z)$, with $n$ a non-negative
integer, are called “Casimir-regular”. If we repeatedly act with $\mathcal{C}$
on these terms, we eventually get $0$. Therefore, we cannot use the arguments
of [111, 112] to match these terms with large-spin asymptotics of double-twist
operators and they are more sensitive to finite-spin effects.

In the study of the lightcone bootstrap, there are a few places where Casimir-
regular terms can appear. The first is if we start the
$\mathrm{SL}(2,\mathbb{R})$ sum (3.2.3) at a generic point $\bar{h}_{0}$:

$\sum_{\begin{subarray}{c}\bar{h}=\bar{h}_{0}+\ell\\\
\ell=0,1,...\end{subarray}}S_{a}(\bar{h})k_{2\bar{h}}(1-z)=\left(\frac{z}{1-z}\right)^{a}+\mathcal{A}(\bar{h}_{0})\;,$
(3.2.6)

where $\mathcal{A}(\bar{h}_{0})$ is defined in [126] (we will not need its
explicit form). Since the choice of starting point only affects a finite
number of blocks, changing the lower limit of the sum will not affect
predictions for large-spin asymptotics.

Another important issue is that our sums over blocks are not actually integer
spaced. In general, the double-twist operators will get anomalous dimensions
which also depend on the spin. Therefore, for a tower of double-twist
operators $\mathcal{O}_{\ell}$ parametrized by the spin $\ell$ we have

$\bar{h}_{\mathcal{O}_{\ell}}=2h_{\phi}+\ell+\delta h_{\mathcal{O}_{\ell}}\;,$
(3.2.7)

where $\delta h_{\mathcal{O}_{\ell}}$ is half the anomalous dimension with
respect to the generalized free field value.

To account for this effect, we can reparametrize our sum by inserting a
Jacobian:

$\sum\limits_{\ell=0}^{\infty}\frac{\partial\bar{h}_{\mathcal{O}_{\ell}}}{\partial\ell}P(\bar{h}_{\mathcal{O}_{\ell}})k_{2\bar{h}_{\mathcal{O}_{\ell}}}(1-z)=\sum\limits_{\ell=0}^{\infty}P(2h_{\phi}+\ell)k_{4h_{\phi}+2\ell}(1-z)+\ldots\;,$
(3.2.8)

where the dropped terms are Casimir-regular in $z$. In the current discussion
we will not be concerned with matching Casimir-regular terms, but will rather
focus on how we can use the $\mathrm{SL}(2,\mathbb{R})$ expansion to match
individual conformal blocks in the $s$-channel.

Thus, let us now consider the effect of single generic conformal block
$g_{h_{i},\bar{h}_{i}}(z,\bar{z})$ in the s-channel. In the limit $z\ll
1-\bar{z}\ll 1$ it has the general form:

$\displaystyle
g_{h_{i},\bar{h}_{i}}(z,\bar{z})=\left(\frac{z}{1-z}\right)^{h_{i}}\left(A_{i}\log(1-\bar{z})+B_{i}+\Omega(1-\bar{z})\right)+\ldots\;,$
(3.2.9)

where we have only written the leading order terms. It is straightforward to
include $\mathrm{SL}(2,\mathbb{R})$ descendants using the results of [126,
215], and to expand to higher orders in $(1-\bar{z})$ using the explicit form
of the hypergeometric functions, where such terms are needed to fix the
corrections for higher twist towers, i.e. $[\phi\phi]_{n>0}$.777We will drop
the label $\ell$ when referring to a given twist tower. Here we will be
primarily interested in the form of the correction for the leading twist
$[\phi\phi]_{0}$ operators.

The crossing equation then becomes

$\left(\frac{z}{1-z}\right)^{-2h_{\phi}}+\sum_{i}\left(\frac{z}{1-z}\right)^{h_{i}-2h_{\phi}}(A_{i}\log(1-\bar{z})+B_{i}+\order{1-\bar{z}})\\\
=\sum\limits_{\mathcal{O}\in[\phi\phi]_{0}}P_{\phi\phi\mathcal{O}}(1-\bar{z})^{h_{\mathcal{O}}-2h_{\phi}}k_{2\bar{h}_{\mathcal{O}}}(1-z)+\ldots\;.$
(3.2.10)

To match the $\log$ terms we have to expand in the anomalous dimension,
$h_{\mathcal{O}}=2h_{\phi}+\delta h_{\mathcal{O}}$, and we find

$\displaystyle P_{\phi\phi\mathcal{O}}\sim{}$ $\displaystyle
2\frac{\partial\bar{h}_{\mathcal{O}}}{\partial\ell}\left[S_{-2h_{\phi}}(\bar{h}_{\mathcal{O}})+\sum_{i}B_{i}S_{h_{i}-2h_{\phi}}(\bar{h}_{\mathcal{O}})\right]\;,$
(3.2.11a) $\displaystyle\delta h_{\mathcal{O}}P_{\phi\phi\mathcal{O}}\sim{}$
$\displaystyle
2\frac{\partial\bar{h}_{\mathcal{O}}}{\partial\ell}\left[\sum_{i}A_{i}S_{h_{i}-2h_{\phi}}(\bar{h}_{\mathcal{O}})\right]\;.$
(3.2.11b)

The factors of $2$ are because we only sum over double-twist operators of even
spin in the $t$-channel. To find the asymptotic, large-spin behavior of the
anomalous dimensions we then just need the ratio of these terms:

$\delta
h_{\mathcal{O}}\sim\frac{\sum_{i}A_{i}S_{h_{i}-2h_{\phi}}(\bar{h}_{\mathcal{O}})}{S_{-2h_{\phi}}(\bar{h}_{\mathcal{O}})+\sum_{i}B_{i}S_{h_{i}-2h_{\phi}}(\bar{h}_{\mathcal{O}})}\;.$
(3.2.12)

In order to compute $P_{\phi\phi\mathcal{O}}$ we can then plug this into the
relation
$\frac{\partial\bar{h}_{\mathcal{O}}}{\partial\ell}=\left(1-\frac{\partial\delta
h_{\mathcal{O}}}{\partial\bar{h}_{\mathcal{O}}}\right)^{-1}$.

##### Coefficient Expansions

In eqn. (3.2.3b), we provided the MFT coefficient $S_{a}(\overline{h})$ for
identical operators. Its generalization $S_{a}^{r,s}(\overline{h})$ are
defined as

$\displaystyle
S_{a}^{r,s}(\overline{h})\coloneqq\frac{1}{\Gamma(-a-r)\Gamma(-a-s)}\frac{\Gamma(\overline{h}-r)\Gamma(\overline{h}-s)}{\Gamma(2\overline{h}-1)}\frac{\Gamma(\overline{h}-a-1)}{\Gamma(\overline{h}+a+1)}\;,$
(3.2.13)

which satisfy the one dimensional Mean Field Theory sum [126]:

$\displaystyle\sum\limits_{\begin{subarray}{c}\overline{h}=\ell-a\\\
\ell=0,1,\dots\end{subarray}}S_{a}^{r,s}(\overline{h})(1-z)^{\overline{h}}{}_{2}F_{1}\biggl{[}\genfrac{.}{.}{0.0pt}{}{\overline{h}-r,\overline{h}+s}{2\overline{h}};1-z\biggr{]}=\frac{z^{r+a}}{(1-z)^{a}}\;.$
(3.2.14)

$S_{a}^{r,s}(\overline{h})$ scales like
$4^{-\overline{h}}\overline{h}^{-\frac{3}{2}-2a-r-s}$ at large $\overline{h}$,
meaning that we can expand $S_{a}^{r,s}(\overline{h})$ as

$\displaystyle
S_{a}^{r,s}(\overline{h})=\sum\limits_{k=0}^{\infty}c_{k,a}^{r,s,m,n}S_{a-\frac{m+n-k}{2}}^{r+m,s+n}(\overline{h})\;,$
(3.2.15)

with $\overline{h}$-independent coefficients $c_{k,a}^{r,s,m,n}$. Since we are
working at next-to-next-to-leading order in $1/\overline{h}$, we can truncate
this expansion as

$\displaystyle S_{a}^{r,s}(\overline{h})\approxeq
c_{0,a}^{r,s,m,n}S_{a-\frac{m+n}{2}}^{r+m,s+n}(\overline{h})+c_{1,a}^{r,s,m,n}S_{a-\frac{m+n-1}{2}}^{r+m,s+n}(\overline{h})+c_{2,a}^{r,s,m,n}S_{a-\frac{m+n-2}{2}}^{r+m,s+n}(\overline{h})\;.$
(3.2.16)

Armed with this, let us consider the following summation which appears
repeatedly in eqn. (3.2.36) after insertion of the ansatz in eqn. (3.2.38):

$\displaystyle\sum_{\ell=0}^{\infty}\frac{\partial\overline{h}}{\partial\ell}S_{a}^{0,0}(\overline{h})(1-z)^{\overline{h}}{}_{2}F_{1}\biggl{[}\genfrac{.}{.}{0.0pt}{}{\overline{h}-m,\overline{h}+n}{2\overline{h}};1-z\biggr{]}\;.$
(3.2.17)

For parity-even structures, $m,n=0$, hence we can immediately use eqn.
(3.2.37). For parity-odd structures, we can expand $S^{0,0}$ in terms of
$S^{m,n}$ and obtain

$\sum_{\ell=0}^{\infty}\frac{\partial\overline{h}}{\partial\ell}S_{a}^{0,0}(\overline{h})(1-z)^{\overline{h}}{}_{2}F_{1}\biggl{[}\genfrac{.}{.}{0.0pt}{}{\overline{h}-m\mskip
8.0mu\overline{h}+n}{2\overline{h}};1-z\biggr{]}\\\
=z^{a+\frac{m-n}{2}}\left(c_{0,a}^{r,s}+c_{1,a}^{r,s}{\sqrt{z}}+\left(\frac{2a-m-n}{2}c_{0,a}^{r,s}+c_{2,a}^{r,s}\right)z+\order{z^{3/2}}\right)\;,$
(3.2.18)

where $c_{k,a}^{m,n}\equiv c_{k,a}^{0,0,m,n}$. This equation simply means that
we replace factors of $\left(\frac{z}{1-z}\right)^{a}$ in eqn. (3.2.39) with
the expression above for parity-odd structures.

We list below the coefficients $c_{0}$, $c_{1}$, and $c_{2}$ for the reader’s
convenience:

$\displaystyle
c_{k,a}^{r,s,m,n}=\kappa_{k}\frac{\Gamma\left(-a-\frac{m}{2}+\frac{n}{2}-r-\frac{k}{2}\right)\Gamma\left(-a+\frac{m}{2}-\frac{n}{2}-s-\frac{k}{2}\right)}{\Gamma(-a-r)\Gamma(-a-s)}\;,\quad
k\in\\{0,1,2\\}\;,$ (3.2.19)

with

$\displaystyle\kappa_{0}=$ $\displaystyle 1\;,$ (3.2.20a)
$\displaystyle\kappa_{1}=$ $\displaystyle-\frac{m(m+2r)+n(n+2s)}{2}\;,$
(3.2.20b) $\displaystyle\kappa_{2}=$
$\displaystyle\frac{1}{8}m^{4}+\frac{4r-1}{8}m^{3}+\frac{n(1+2n+4s)+4r(r-1)-2(a+1)}{8}m^{2}$
$\displaystyle+\frac{n^{2}(1+4r)+n(8rs-4)-4a(n-2)+4(a^{2}-r^{2}+1)}{8}m$
$\displaystyle+\frac{n}{8}\left((n+2s)^{2}(n-1)-2n-2a(n-4)+4(a^{2}+1)\right)\;.$
(3.2.20c)

####  3.2.2 Fermion conformal blocks and the crossing equations

Having reviewed lightcone bootstrap for scalars, we will now move on to
4-point functions of fermions $\expectationvalue{\psi\psi\psi\psi}$ in 3d. Our
focus for now will be to compute the first several perturbative $1/\bar{h}$
corrections to the $[\psi\psi]_{0}$ coefficients and anomalous dimensions,
generalizing the method of section 3.2.1 to fermions. As illustrated in [126],
such perturbative calculations are in many cases numerically sufficient and
also quite useful in providing consistency checks on formulas obtained from
the inversion formula approach.

We will be using the method of [81] to generate the fermion conformal blocks
by hitting the four-scalar conformal block with differential operators in the
embedding space.888A review of the embedding space formalism can be found in
section 3.1.1. Specifically, a contribution to
$\expectationvalue{\psi\psi\psi\psi}$ takes the form:

$\left(\frac{X_{24}}{X_{14}}\right)^{\frac{\Delta_{12}}{2}}\left(\frac{X_{14}}{X_{13}}\right)^{\frac{\Delta_{34}}{2}}\frac{t_{I}g^{I;a,b}_{h,\bar{h}}(z,\bar{z})}{X_{12}^{\frac{\Delta_{1}+\Delta_{2}+1}{2}}X_{34}^{\frac{\Delta_{3}+\Delta_{4}+1}{2}}}\\\
=\mathrm{D}_{a}\widetilde{\mathrm{D}}_{b}\left[\left(\frac{X_{24}}{X_{14}}\right)^{\frac{\Delta_{12}}{2}}\left(\frac{X_{14}}{X_{13}}\right)^{\frac{\Delta_{34}}{2}}\frac{g_{h,\bar{h}}(z,\bar{z})}{X_{12}^{\frac{\Delta_{1}+\Delta_{2}}{2}}X_{34}^{\frac{\Delta_{3}+\Delta_{4}}{2}}}\right]\;,$
(3.2.21)

where $t_{I}$ are different 4-point structures in embedding space, $a,b$ are
indices denoting possible 3-point structures, and we have suppressed the
dependence of the blocks on the external dimensions. The operators
$\mathrm{D}_{a}$ are defined as follows:

$\displaystyle\mathrm{D}_{1}=\expectationvalue{S_{1}S_{2}}\Pi_{1^{+}2^{+}}\;,$
(3.2.22a)
$\displaystyle\mathrm{D}_{2}=-\expectationvalue{S_{1}\frac{\delta}{\delta
X_{1}}\frac{\delta}{\delta X_{2}}S_{2}}\Pi_{1^{-}2^{-}}\;,$ (3.2.22b)
$\displaystyle\mathrm{D}_{3}=\expectationvalue{S_{1}\frac{\delta}{\delta
X_{1}}S_{2}}\Pi_{1^{-}2^{+}}-\expectationvalue{S_{2}\frac{\delta}{\delta
X_{2}}S_{1}}\Pi_{1^{+}2^{-}}\;,$ (3.2.22c)
$\displaystyle\mathrm{D}_{4}=\expectationvalue{S_{1}\frac{\delta}{\delta
X_{1}}S_{2}}\Pi_{1^{-}2^{+}}+\expectationvalue{S_{2}\frac{\delta}{\delta
X_{2}}S_{1}}\Pi_{1^{+}2^{-}}\;,$ (3.2.22d)

where $\Pi_{a^{i}b^{j}}$ are operators which shift external dimensions such
that

$\displaystyle\Pi_{a^{i}b^{j}}:\\{\Delta_{a},\Delta_{b}\\}\rightarrow\\{\Delta_{a}+\frac{i}{2},\Delta_{b}+\frac{j}{2}\\}\;.$
(3.2.23)

The corresponding operators acting on the points $(X_{3},X_{4})$ are found via
the replacement

$\displaystyle\widetilde{\mathrm{D}}_{a}\coloneqq\mathrm{D}_{a}\evaluated{}_{(1,2)\rightarrow(3,4)}\;.$
(3.2.24)

An alternative basis denoted by $\mathcal{D}_{i}$ was used in [81], given
by:999There is also an additional term in $\mathcal{D}_{4}$ which vanishes for
identical external operators.

$\displaystyle\mathcal{D}_{1}=$ $\displaystyle\;\mathrm{D}_{1}\;,$ (3.2.25a)
$\displaystyle\mathcal{D}_{2}=$
$\displaystyle\;\frac{1}{4(\overline{h}-h)(\overline{h}+h-1)}\left(\mathrm{D}_{2}-(2h+2\Delta_{\psi}-4)(2h-2\Delta_{\psi}+1)\mathrm{D}_{1}\right)\;,$
(3.2.25b) $\displaystyle\mathcal{D}_{3}=$
$\displaystyle\;\frac{1}{2(\overline{h}+h-1)}\mathrm{D}_{3}\;,$ (3.2.25c)
$\displaystyle\mathcal{D}_{4}=$
$\displaystyle\;\frac{1}{2(\overline{h}-h)}\mathrm{D}_{4}\;.$ (3.2.25d)

These two bases have different merits. One nice feature of the
$\mathrm{D}_{i}$ basis is that the operators are independent of $(h,\bar{h})$,
so there is a clean separation between the calculation of double-twist data
and the differential operators, i.e. we do not want these operators to also
depend on the anomalous dimensions. By comparison $\mathcal{D}_{i}$ generates
the most natural basis of embedding space, 3-point tensor structures. We find
that it is most convenient to use the $\mathrm{D}_{i}$ basis when performing
the calculations and presenting the results.

The operators $\mathrm{D}_{1}$ and $\mathrm{D}_{2}$ generate parity-even
structures, whereas $\mathrm{D}_{3}$ and $\mathrm{D}_{4}$ generate parity-odd
ones. When the external fermions are identical, there are also selection rules
on the spins of the exchanged operators: $\mathrm{D}_{1}$, $\mathrm{D}_{2}$,
and $\mathrm{D}_{3}$ are associated to operators of even spin while odd spins
are associated with $\mathrm{D}_{4}$.

Now let us write down the condition from crossing symmetry. For convenience we
will define the prefactor $\mathsf{p}$ as

$\displaystyle{\rm Phys.\leavevmode\nobreak\ Rev.\leavevmode\nobreak\
E}\left(\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}\right)\equiv\left(\frac{X_{24}}{X_{14}}\right)^{\frac{\Delta_{12}}{2}}\left(\frac{X_{14}}{X_{13}}\right)^{\frac{\Delta_{34}}{2}}X_{12}^{-\frac{\Delta_{1}+\Delta_{2}}{2}}X_{34}^{-\frac{\Delta_{3}+\Delta_{4}}{2}}\;,$
(3.2.26)

and introduce the shorthand notations

$\displaystyle{\rm Phys.\leavevmode\nobreak\ Rev.\leavevmode\nobreak\ E}_{i}$
$\displaystyle\equiv{\rm Phys.\leavevmode\nobreak\ Rev.\leavevmode\nobreak\
E}\left(\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}\right)\;,$ (3.2.27a)
$\displaystyle{\rm Phys.\leavevmode\nobreak\ Rev.\leavevmode\nobreak\
E}_{\psi}$ $\displaystyle\equiv{\rm Phys.\leavevmode\nobreak\
Rev.\leavevmode\nobreak\
E}\left(\Delta_{\psi}+\frac{1}{2},\Delta_{\psi}+\frac{1}{2},\Delta_{\psi}+\frac{1}{2},\Delta_{\psi}+\frac{1}{2}\right)\;.$
(3.2.27b)

Then we can write a more compact form of the conformal block for identical
fermions:

$\displaystyle
t_{I}g^{I;a,b}_{h,\bar{h}}(z,\bar{z})=\frac{\left(\mathrm{D}_{a}\widetilde{\mathrm{D}}_{b}\left[{\rm
Phys.\leavevmode\nobreak\ Rev.\leavevmode\nobreak\
E}_{i}\;g_{h,\bar{h}}(z,\bar{z})\right]\right)_{\Delta_{i}\rightarrow\Delta_{\psi}}}{{\rm
Phys.\leavevmode\nobreak\ Rev.\leavevmode\nobreak\ E}_{\psi}}\;,$ (3.2.28)

and crossing symmetry implies

$\displaystyle{\rm Phys.\leavevmode\nobreak\ Rev.\leavevmode\nobreak\
E}_{\psi}\sum\limits_{\mathcal{O}}t_{I}P_{\mathcal{O}}^{a,b}g^{I;ab}_{h,\overline{h}}(z,\overline{z})=-\left({\rm
Phys.\leavevmode\nobreak\ Rev.\leavevmode\nobreak\
E}_{\psi}t_{I}\right)\evaluated{}_{1\leftrightarrow
3}\sum\limits_{\mathcal{O}}P_{\mathcal{O}}^{a,b}g^{I;ab}_{h,\overline{h}}(1-\overline{z},1-z)\;,$
(3.2.29)

where we use $P$ for the coefficients in the $\mathrm{D}_{a}$ differential
basis.

To simplify some expressions, we will first consider the contributions of
double-twist operators in the $(12)\rightarrow(34)$ OPE. Then later we will
take $1\leftrightarrow 3$ so that they appear in the $t$-channel, and match
their sum to the contributions of individual $s$-channel blocks, as in section
3.2.1.

The 3-point structures will be unimportant in the following discussion so we
will simplify the notation of the left-hand side of (3.2.29) to

$\displaystyle\sum\limits_{\begin{subarray}{c}\text{different}\\\ \text{
structures}\end{subarray}}\sum\limits_{\mathcal{O}}\left(\mathrm{D}\widetilde{\mathrm{D}}\left[{\rm
Phys.\leavevmode\nobreak\ Rev.\leavevmode\nobreak\
E}_{i}P_{\mathcal{O}}g_{h,\overline{h}}(z,\overline{z})\right]\right)\evaluated{}_{\Delta_{i}\rightarrow\Delta_{\psi}}\;.$
(3.2.30)

We can also interchange the order of differential operators and summation over
relevant operator families. Since ${\rm Phys.\leavevmode\nobreak\
Rev.\leavevmode\nobreak\ E}$ is independent of the exchanged operator, the
double-twist sum reduces to

$\displaystyle\sum\limits_{\begin{subarray}{c}\text{different}\\\ \text{
structures}\end{subarray}}\left(\mathrm{D}\widetilde{\mathrm{D}}\left[{\rm
Phys.\leavevmode\nobreak\ Rev.\leavevmode\nobreak\
E}_{i}\sum\limits_{\mathcal{O}}P_{\mathcal{O}}g_{h,\overline{h}}(z,\overline{z})\right]\right)_{\Delta_{i}\rightarrow\Delta_{\psi}}\;.$
(3.2.31)

To go any further, we need to specify which operators $\mathcal{O}$ must
appear to reproduce the lightcone limit of the crossed channel. In particular,
we know that an infinite sum of double-twist operators is needed to reproduce
the identity operator in the crossed channel [126, 112, 111]. The required
operators and their quantum numbers are schematically shown in Table 3.1.

To remove clutter, we will denote different double-twist families generically
as $[\psi\psi]_{n}$ below. Their contribution then reads as

$\displaystyle\sum\limits_{\begin{subarray}{c}\text{different}\\\ \text{
structures}\end{subarray}}\Bigg{(}\mathrm{D}\widetilde{\mathrm{D}}\bigg{[}{\rm
Phys.\leavevmode\nobreak\ Rev.\leavevmode\nobreak\
E}_{i}\sum\limits_{n=0}^{\infty}\sum\limits_{\ell=0}^{\infty}P_{[\psi\psi]_{n}}g_{h_{[\psi\psi]_{n}},\,\bar{h}_{[\psi\psi]_{n}}}(z,\overline{z})\bigg{]}\Bigg{)}_{\Delta_{i}\rightarrow\Delta_{\psi}}\;,$
(3.2.32)

where the summation over all relevant families $[\psi\psi]_{n}$ appearing in
Table 3.1 is implicit.

Restoring possible dependence on the external dimension differences $r,s$
(which arise from the shift operators $\Pi$), we will now use Eq. (2.3.7) in a
slightly modified form after applying $g(z,\overline{z})=g(\overline{z},z)$
symmetry:

$\displaystyle
g_{h,\overline{h}}^{r,s}(z,\overline{z})=\sum\limits_{n=0}^{\infty}\sum\limits_{j=-n}^{n}A_{n,j}^{r,s}(h,\overline{h})\bar{z}^{h+n}k^{r,s}_{2(\bar{h}+j)}(z)\;.$
(3.2.33)

By expanding

$\displaystyle h_{[\psi\psi]_{n}}=h_{[\psi\psi]_{0}}+n+\delta
h_{[\psi\psi]_{n}}\;,$ (3.2.34)

the leading part of the double-twist sum at small $\bar{z}$ can then be
rewritten as

$\displaystyle\sum\limits_{\begin{subarray}{c}\text{different}\\\ \text{
structures}\end{subarray}}\Bigg{(}\mathrm{D}\widetilde{\mathrm{D}}\bigg{[}{\rm
Phys.\leavevmode\nobreak\ Rev.\leavevmode\nobreak\
E}_{i}\sum\limits_{\ell=0}^{\infty}P_{[\psi\psi]_{0}}\overline{z}^{h_{[\psi\psi]_{0}}+\delta
h_{[\psi\psi]_{0}}}k^{r,s}_{2\overline{h}_{[\psi\psi]_{0}}}(z)\bigg{]}\Bigg{)}_{\Delta_{i}\rightarrow\Delta_{\psi}}\left(1+\order{{\overline{z}}}\right)\;.$
(3.2.35)

To reproduce $\log$ terms in the crossed channel, we will expand to linear
order in the anomalous dimension:101010Higher order terms in $\delta h$ are
matched with multi-twist operators in the $s$-channel.

$\displaystyle\sum\limits_{\begin{subarray}{c}\text{different}\\\ \text{
structures}\end{subarray}}\Bigg{(}\mathrm{D}\widetilde{\mathrm{D}}\bigg{[}{\rm
Phys.\leavevmode\nobreak\ Rev.\leavevmode\nobreak\
E}_{i}\;\overline{z}^{h_{[\psi\psi]_{0}}}\bigg{\\{}\sum\limits_{\ell=0}^{\infty}P_{[\psi\psi]_{0}}k^{r,s}_{2\overline{h}_{[\psi\psi]_{0}}}(z)+\log(\overline{z})\sum\limits_{\ell=0}^{\infty}P_{[\psi\psi]_{0}}\delta
h_{[\psi\psi]_{0}}k^{r,s}_{2\overline{h}_{[\psi\psi]_{0}}}(z)\bigg{\\}}\bigg{]}\Bigg{)}_{\Delta_{i}\rightarrow\Delta_{\psi}}.$
(3.2.36)

In [126] it was shown how to sum over ${}_{2}F_{1}$ hypergeometric functions
to reproduce terms Casimir-singular in $z$:

$\displaystyle\sum_{\ell=0}^{\infty}\frac{\partial\overline{h}}{\partial\ell}S_{a}^{r,s}(\overline{h})k^{r,s}_{2\overline{h}}(1-z)=\left(\frac{z}{1-z}\right)^{a}+[\cdots]_{z}\;,$
(3.2.37)

which is the generalization of eqn. (3.2.3) to non-identical external scaling
dimensions. The explicit form of $S_{a}^{r,s}(h)$ is given in eqn. (3.2.13),
though it will not be necessary for the following calculations.

As explained in section 3.2.1, we make the ansatz

$\displaystyle P_{[\psi\psi]_{0}}^{a,b}(\bar{h})=$
$\displaystyle\left(\frac{\partial\overline{h}_{[\psi\psi]_{0}}}{\partial\ell}\right)\sum\limits_{\\{i\\}}A_{a,b,i}S_{i}^{0,0}(\overline{h}_{[\psi\psi]_{0}})\;,$
(3.2.38a) $\displaystyle(\delta hP)^{a,b}_{[\psi\psi]_{0}}(\bar{h})=$
$\displaystyle\left(\frac{\partial\overline{h}_{[\psi\psi]_{0}}}{\partial\ell}\right)\sum\limits_{\\{j\\}}B_{a,b,j}S_{j}^{0,0}(\overline{h}_{[\psi\psi]_{0}})\;,$
(3.2.38b)

and insert this into eqn. (3.2.36) to obtain

$\displaystyle\frac{1}{2}\sum\limits_{\begin{subarray}{c}\text{different}\\\
\text{
structures}\end{subarray}}\left(\mathrm{D}\widetilde{\mathrm{D}}\left[{\rm
Phys.\leavevmode\nobreak\ Rev.\leavevmode\nobreak\
E}_{i}\overline{z}^{h_{[\psi\psi]_{0}}}\left(\sum\limits_{\\{i\\}}A_{a,b,i}\left(\frac{1-z}{z}\right)^{i}+\log(\overline{z})\sum\limits_{\\{j\\}}B_{a,b,j}\left(\frac{1-z}{z}\right)^{j}\right)\right]\right)_{\Delta_{i}\rightarrow\Delta_{\psi}}\;.$
(3.2.39)

The $1/2$ in front accounts for the fact that we are summing over operator
families with even-integer spacing [126], since we are dealing with identical
external fermions. In particular, $P^{4,4}=0$ for even $\ell$, and
$P^{1,1}=P^{1,2}=P^{2,1}=P^{2,2}=P^{3,3}=0$ for odd $\ell$ [81].

After acting with the differential operators we take $1\leftrightarrow 3$ and
$(z,\bar{z})\rightarrow(1-z,1-\bar{z})$ and match to individual $s$-channel
blocks. We will find the sets $\\{i\\}$ and $\\{j\\}$, and the coefficients
$A$ and $B$ by matching the $s$-channel.

There is actually a subtle point we skipped while going from eqn. (3.2.37) to
eqn. (3.2.39). As evident from eqn. (3.2.22), we need
$S^{\pm\frac{1}{2},\pm\frac{1}{2}}$ to be able to use eqn. (3.2.37) for
parity-odd structures even though we used $S^{0,0}$ in our ansatz above. We
resolve this in appendix 3.2.1 by expanding $S_{a}^{r+n,r+m}$ in terms of
$S_{a+k}^{r,s}$.

####  3.2.3 Results

In this section, we first discuss identity matching and find the MFT
solutions. Then we consider the exchange of parity-even and parity-odd
operators of arbitrary dimension and spin, and calculate their contribution to
the OPE coefficients and anomalous dimensions of the double-twist families
$[\psi_{(\alpha}\psi_{\beta)}]_{0,\ell}$,
$[\psi_{\alpha}\psi^{\alpha}]_{0,\ell}$, and
$[\psi_{(\rho}\partial^{\rho}_{\;\alpha}\psi_{\beta)}]_{0,\ell}$ at leading
and sub-leading order in the small $z$ expansion. As special cases, we will
present the contributions due to stress tensor exchange and scalar exchanges.

The reader is reminded that the contributions of all double-twist families are
present, but we simply match the subset of terms relevant for the above
families. For example, we match the
$\order{z^{1-\Delta_{\psi}}(1-\bar{z})^{0}}$ contribution of the stress tensor
without matching the $\order{z^{-\Delta_{\psi}}(1-\bar{z})^{1}}$ contribution
from identity exchange, even though the latter is more dominant in the
lightcone limit. However, these contributions come from different twist
families in the crossed channel, and there is no mixing for the terms leading
order in $(1-\bar{z})$. For instance, both
$[\psi_{(\alpha}\psi_{\beta)}]_{0,\ell}$ and
$[\psi_{(\alpha}\psi_{\beta)}]_{1,\ell}$ bring contributions of order
$\order{z^{-\Delta_{\psi}}(1-\bar{z})^{1}}$, however only
$[\psi_{(\alpha}\psi_{\beta)}]_{0,\ell}$ brings
$\order{z^{-\Delta_{\psi}}(1-\bar{z})^{0}}$ terms. So by requiring
$[\psi_{(\alpha}\psi_{\beta)}]_{0,\ell}$ to reproduce these in the crossed
channel, we can extract its OPE coefficients and anomalous dimensions.

Below, we will suppress the label for the double twist families whenever there
is no ambiguity. For example, we will simply write the OPE coefficient
$f^{1}_{[\psi_{(\alpha}\psi_{\beta)}]_{0,\ell}}$ as $f^{1}$. We can extract
the relevant family due to the conditions listed in Table 3.1 and the fact
that the four types of 3-point functions $f^{a}$ are associated with (parity,
spin) as: ($+$, even), ($+$, even), ($-$, even), and ($-$, odd), respectively.

##### Identity matching

Let us first focus on the identity contribution alone. In the $s$-channel the
relevant terms are $\order{z^{-\frac{1}{2}-\Delta_{\psi}}(1-\bar{z})^{0}}$ and
$\order{z^{\frac{1}{2}-\Delta_{\psi}}(1-\bar{z})^{0}}$. We reproduce these
terms in the $t$-channel by tuning $A_{a,b,i}$ in eqn. (3.2.38a); for example,
we need

$\displaystyle\left\\{A_{2,2,\frac{3}{2}-\Delta_{\psi}},A_{2,2,\frac{5}{2}-\Delta_{\psi}}\right\\}$
(3.2.40)

for the double-twist family $[\psi_{(\alpha}\psi_{\beta)}]_{0,\ell}$. The fact
that $A_{1,1,i}$, $A_{1,2,i}$, and $A_{2,1,i}$ are zero reflects the vanishing
of the 3-point coefficient $f^{1}$ at all orders.

As there is no anomalous dimension for the identity exchange alone we have
$\frac{\partial\overline{h}}{\partial\ell}=1$, hence we can immediately get
$P^{a,b}$ with eqn. (3.2.38), then solve for the physical 3-point
coefficients111111The $(-1)^{\ell}$ term here would be absent in the notation
of [81], however we need it as our conformal block normalization in eqn.
(2.3.6) differs from that paper by a factor of $(-1)^{\ell}$.

$\displaystyle
P_{\mathcal{O}}^{a,b}=(-1)^{\ell}f_{\psi_{1}\psi_{2}\mathcal{O}}^{a}f_{\psi_{3}\psi_{4}\mathcal{O}}^{b}\;,$
(3.2.41)

where the identity itself has the OPE coefficients
$f^{1}_{\psi\psi\mathbb{1}}=i$ and $f^{2}_{\psi\psi\mathbb{1}}=0$.

Using the steps described above, we compute the OPE coefficients

$\displaystyle f^{1}=\,$ $\displaystyle 0\;,$ (3.2.42a) $\displaystyle
f^{2}=\,$
$\displaystyle\frac{f_{0}}{4\overline{h}^{2}}\left(1-\frac{8\Delta_{\psi}-17}{16\bar{h}}-\frac{256\Delta_{\psi}^{3}-2112\Delta_{\psi}^{2}+4208\Delta_{\psi}-2787}{1536\overline{h}^{2}}+\order{\frac{1}{\overline{h}}}^{3}\right)\;,$
(3.2.42b) $\displaystyle f^{3}=\,$
$\displaystyle\frac{f_{0}}{2\overline{h}}\sqrt{\frac{\Delta_{\psi}-1}{2\overline{h}}}\left(1-\frac{24\Delta_{\psi}-31}{16\bar{h}}+\order{\frac{1}{\overline{h}}}^{2}\right)\;,$
(3.2.42c) $\displaystyle f^{4}=\,$
$\displaystyle\frac{f_{0}}{2\overline{h}}\sqrt{\frac{\Delta_{\psi}-1}{2\overline{h}}}\left(1+\frac{8\Delta_{\psi}-1}{16\bar{h}}+\order{\frac{1}{\overline{h}}}^{2}\right)\;,$
(3.2.42d)

where for convenience we have defined the prefactor $f_{0}$ as

$\displaystyle f_{0}\equiv
i\sqrt[4]{\pi}\frac{2^{\frac{3}{2}-\bar{h}}\bar{h}^{\Delta_{\psi}-\frac{1}{4}}}{\Gamma\left(\Delta_{\psi}+\frac{1}{2}\right)}\;.$
(3.2.43)

Note that the results take a slightly simpler form in the $\mathcal{D}$ basis.
E.g., at leading order

$\displaystyle f^{2}\evaluated{}_{\mathcal{D}}=f_{0}\;,\qquad
f^{3,4}\evaluated{}_{\mathcal{D}}=f_{0}\sqrt{\frac{\Delta_{\psi}-1}{2\overline{h}}}\;,$
(3.2.44)

which follows from eqn. (3.2.25).

##### Matching the exchange of a generic parity-even operator

Let us turn to the contribution of the exchange of a generic parity-even
operator $\mathcal{O}^{+}_{\tau,\ell}$ of twist $\tau$ and spin $\ell$ in the
$s$-channel to the double-twist families $[\psi_{(\alpha}\psi_{\beta)}]_{0}$,
$[\psi_{\alpha}\psi^{\alpha}]_{0}$, and
$[\psi_{(\rho}\partial^{\rho}_{\alpha}\psi_{\beta)}]_{0}$ in the $t$-channel.

When calculating corrections to the anomalous dimensions of double-twist
families we need to recall that the contributions of multiple operators are
not additive. Additionally, in general there can be multiple double-twist
families that mix with each other. The full formula for their anomalous
dimension matrix is121212The anomalous dimension matrix (3.2.45) is diagonal
if there is no mixing between the double-twist families.

$\displaystyle\frac{\gamma_{[\psi\psi]}}{2}=\delta
h_{[\psi\psi]}=\frac{\sum\limits_{\mathcal{O}}\left(\delta
h_{[\psi\psi]}P_{[\psi\psi]}^{a,b}J^{-1}_{[\psi\psi]}\right)_{\mathcal{O}}}{\sum\limits_{\mathcal{O}}\left(P_{[\psi\psi]}^{a,b}J^{-1}_{[\psi\psi]}\right)_{\mathcal{O}}}\;,$
(3.2.45)

where $J$ is the Jacobian

$\displaystyle J_{[\psi\psi]}\equiv\frac{\partial\delta
h_{[\psi\psi]}}{\partial\overline{h}}\;.$ (3.2.46)

Here $\mathcal{O}$ runs over all exchanged operators in the $s$-channel.
Likewise, the OPE coefficients $f^{a}_{[\psi\psi]}$ are given as

$\displaystyle(-1)^{\ell}f_{[\psi\psi]}^{a}f_{[\psi\psi]}^{b}=J_{[\psi\psi]}\sum\limits_{\mathcal{O}}\left(P^{a,b}_{[\psi\psi]}J^{-1}_{[\psi\psi]}\right)_{\mathcal{O}}\;.$
(3.2.47)

In the large $\bar{h}$ expansion, we can of course truncate the summation over
operators in twist to extract the large $\bar{h}$ behavior. For example, in
the first few orders, we see that

$\displaystyle\frac{\gamma_{[\psi\psi]}}{2}=\;\delta h_{[\psi\psi]}$
$\displaystyle=\frac{\left(\delta
h_{[\psi\psi]}P_{[\psi\psi]}^{a,b}\right)_{\mathbbm{O}}}{\left(P_{[\psi\psi]}^{a,b}\right)_{\mathbb{1}}+\left(P_{[\psi\psi]}^{a,b}\right)_{\mathbbm{O}}}\;,$
(3.2.48a) $\displaystyle(-1)^{\ell}f_{[\psi\psi]}^{a}f_{[\psi\psi]}^{b}$
$\displaystyle=\left(P^{a,b}_{[\psi\psi]}\right)_{\mathbb{1}}+\left(P^{a,b}_{[\psi\psi]}\right)_{\mathbbm{O}}\;,$
(3.2.48b)

for the identity operator $\mathbb{1}$ along with the operator with minimum
twist $\mathbbm{O}$, which is usually either the stress tensor or a scalar of
low dimension.

At leading order in $1/\bar{h}$ one can easily isolate the contribution of any
operator. Only the identity operator contributes in the denominator in eqn.
(3.2.45), allowing one to write an isolated contribution to the anomalous
dimension. Likewise, at leading order, one can immediately calculate an
individual contribution to $f^{a}$.

Once we go beyond leading order, we can work with
$\left(P_{[\psi\psi]}^{a,b}J^{-1}_{[\psi\psi]}\right)_{\mathcal{O}}$
and$\left(\delta
h^{a,b}_{[\psi\psi]}P_{[\psi\psi]}^{a,b}J^{-1}_{[\psi\psi]}\right)_{\mathcal{O}}$,
make an ansatz for their large $\bar{h}$ behavior, and then calculate the
corrections to the 3-point coefficients and anomalous dimensions. We find

$\displaystyle\left(P^{1,1}J^{-1}\right)_{\mathcal{O}_{\tau,\ell}^{+}}=$
$\displaystyle-\frac{f_{+}^{2}}{16\bar{h}^{4}(f_{\mathcal{O}}^{\prime
1})^{2}(H_{\ell+\frac{\tau}{2}-1})^{2}}$ (3.2.49a)
$\displaystyle\times\left[\left\\{\left(f_{\mathcal{O}}^{\prime
1}\right)^{2}(5-4\Delta_{\psi})-4\left(f_{\mathcal{O}}^{2}\right)^{2}\ell(\ell+\tau-1)(-2\Delta_{\psi}+\tau+1)^{2}\right\\}^{2}+\order{\frac{1}{\overline{h}}}\right]\;,$
$\displaystyle\left(P^{2,2}J^{-1}\right)_{\mathcal{O}_{\tau,\ell}^{+}}=$
$\displaystyle-\frac{f_{+}^{2}\left(f_{\mathcal{O}}^{\prime
1}\right)^{2}}{16\bar{h}^{4}}\Bigg{[}1+\frac{-8\Delta_{\psi}+4\tau+17}{8\bar{h}}+\order{\frac{1}{\overline{h}}}^{2}\Bigg{]}\;,$
(3.2.49b) $\displaystyle\hskip 5.05942pt\left((\delta
hP)^{2,2}J^{-1}\right)_{\mathcal{O}_{\tau,\ell}^{+}}=$