compat=1.0.0 ††institutetext: Mathematics Department, King’s College London,
The Strand, London, WC2R 2LS, UK

#  Fermions in Boundary Conformal Field Theory : Crossing Symmetry and
$\epsilon$-Expansion

Christopher P. Herzog, Vladimir Schaub<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We use the equations of motion in combination with crossing symmetry to
constrain the properties of interacting fermionic boundary conformal field
theories. This combination is an efficient way of determining operator product
expansion coefficients and anomalous dimensions at the first few orders of the
$\epsilon$ expansion. Two necessary ingredients for this procedure are
knowledge of the boundary and bulk spinor conformal blocks. The bulk spinor
conformal blocks are derived here for the first time. We then consider a
number of examples. For $\phi$ a scalar field and $\psi$ a fermionic field, we
study the effects of a $\phi\bar{\psi}\psi$ coupling in $4-\epsilon$
dimensions, a $\phi^{2}\bar{\psi}\psi$ coupling in $3-\epsilon$ dimensions,
and a $(\bar{\psi}\psi)^{2}$ coupling in $2+\epsilon$ dimensions. We are able
to compute some new anomalous dimensions for operators in these theories.
Finally, we relate the anomalous dimension of a surface operator to the
behavior of the charge density near the surface.

##  1 Introduction

One of quantum field theory’s major successes is its ability to model the
behavior of physical systems near the point of a continuous phase transition
ZinnJustin . Following a quantum field theory to a fixed point in the
renormalization group flow, anomalous dimensions of operators at the fixed
point can be used to calculate critical exponents which in turn determine the
scaling behavior of a large number of physical quantities such as specific
heat, magnetization, and correlation length at or near the phase transition. A
key player in this story is the $O(N)$ model: a set of $N$ interacting scalar
fields with $O(N)$ symmetry. This model has been used successfully to explain
the critical behavior of the liquid-gas transition in simple gases, the
superfluid transition in liquid helium, and the Heisenberg transition in
ferromagnets. Indeed, quantum field theory can model not only these thermal
phase transitions but quantum phase transitions as well, driven by the tuning
of a coupling strength or external field at zero temperature
sachdev1999quantum . The motivation for this work is to look at a more general
class of quantum field theories that involve both relativistic fermions and
boundaries in order to broaden the set of critical systems which can be
successfully modeled.

Relativistic fermions have played an important role in the high energy
community for decades as the quarks and leptons of the standard model of
particle physics. Both in the development of the standard model and also in an
attempt to understand better certain aspects of it, such as confinement and
the existence of a Higgs particle, a number of auxiliary fermionic quantum
field theories have been developed and studied over the years, named after
their inventors – Gross, Neveu, Schwinger, Thirring, Yukawa. While the
emphasis in the condensed matter community has primarily been on non-
relativistic fermions, the attitude has evolved in the last two decades with
development of new materials such as graphene and topological insulators.
These so-called Dirac and Weyl materials Wehling:2014cla ; Vafek:2013mpa
support half-filled nodal Fermi surfaces where the dispersion relation for the
fermionic degrees of freedom is linear. Moreover, as a function of interaction
strength, some of these materials may undergo continuous phase transitions
gomes2012designer ; gutierrez2016imaging ; bao2021experimental ;
qu2022ubiquitous .

Indeed, more precise conjectures have been formulated. Take for example the
Yukawa model which has Lagrangian density

${\mathcal{L}}=-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\overline{\psi}_{a}\not{\partial}\psi_{a}-g\phi\overline{\psi}_{a}\psi_{a}-\frac{h}{4!}\phi^{4}\
.$ (1)

(See appendix A for our conventions.) Here $\phi$ is a real scalar field and
$\psi_{a}$ is a Dirac fermion in $4-\epsilon$ dimensions with
$a=1,\ldots,N_{f}$ components. Using the relevant mass term to drive the phase
transition, $\phi$ gets an expectation value and gives $\psi_{a}$ a mass,
breaking the chiral symmetry of the fermions. The case $N_{f}=2$ has been
proposed to describe a semi-metal insulator transition in graphene
Herbut:2006cs . For $N_{f}=1/4$, the model is conjectured to be supersymmetric
at the quantum critical point, and could describe the boundary of a
topological phase Grover:2013rc . Disorder can also drive phase transitions.
In the context of the replica trick, the $N_{f}\to 0$ limit may describe the
critical point between a relativistic semimetallic state and a diffusive
metallic phase in 3D Weyl semimetals Louvet:2016zbb .

While the studies above describe infinite systems, critical behavior can arise
in the presence of a boundary as well Diehl:1996kd . The critical exponents
and scaling behavior associated with the surface have been in some cases both
measured and successfully compared with theoretical predictions. For example,
Alvarado et al. alvarado1982surface  measured the surface magnetization of a
nickel ferromagnet close to the ordering transition. There is reasonable
agreement with the corresponding surface magnetization critical exponent
extracted from an epsilon expansion of the O(3) model diehl1986critical ;
diehl1980scaling ; reeve1980critical . There are also measurements of surface
critical exponents for an iron-aluminum alloy, a liquid mixture and a
molecular solid that agree with predictions from the Ising model in the
presence of a boundary mailander1990near ; burandt1993near ; sigl1986order .

To our knowledge, relatively little theoretical work has been done on critical
fermionic systems with boundaries. While a recent tour-de-force calculation
Zerf:2017zqi  computes critical exponents in the Yukawa model at fourth order
in the epsilon expansion in the infinite system, the state of the art in the
presence of a boundary is $O(\epsilon)$ Giombi:2021cnr . Refs. Giombi:2021cnr
; Carmi:2018qzm  consider also the large $N_{f}$ limit. On the experimental
side, the associated critical exponents have not yet been measured in the
infinite system nor in the presence of a boundary, yet much promising work has
been done to engineer graphene and graphene like systems which undergo this
type of chiral symmetry breaking phase transition (see e.g. gomes2012designer
; gutierrez2016imaging ; bao2021experimental ; qu2022ubiquitous ).

A major point of our work is to develop tools to make the calculation of
critical exponents for fermionic systems with boundaries more straightforward.
We are inspired by ref. Rychkov:2015naa  which pointed out the utility of the
equations of motion in reducing the number of Feynman diagrams that need to be
evaluated, and also by the revival of the conformal bootstrap program,
especially as it pertains to systems with fermions vanLoon:2017xlq ;
Giombi:2017rhm ; Iliesiu:2015qra  and boundaries Liendo:2012hy ; Dey:2020jlc ;
Kaviraj:2018tfd ; Mazac:2018biw . In the context of the fermionic CFT, the
equation of motion procedure has been implemented successfully without
boundary in refs. Ghosh:2015opa ; Raju:2015fza  to rederive anomalous
dimensions. In ref. Giombi:2021cnr  the equation of motion technique was
further extended to fermions in bCFT, where they calculated some spinorial
boundary anomalous dimensions in the $\epsilon$-expansion for the first time.
Our work can be viewed as a variation of Giombi:2021cnr , augmenting the
equation of motion technique with a conformal bootstrap approach instead of
with Feynman diagram computations.

The key idea in the conformal bootstrap is the notion of crossing symmetry. In
systems with boundaries, this symmetry can be imposed on two-point correlation
functions in the bulk. One either brings the two bulk operators close together
or close to the boundary. In the first case, the operator product decomposes
into a sum over bulk operators. As only scalar operators can have nonzero one
point functions in boundary conformal field theory, the sum reduces to a sum
over these bulk scalar one point functions. In the second case, the bulk
operators each decompose as a sum over boundary operators. The bulk two-point
function is then expressed as a sum over boundary two-point functions. The
equivalence of these two procedures is the crossing constraint.

To employ crossing symmetry, however, one must have an efficient formalism to
deal with the sums. In conformal field theory, these operators arrange
themselves into multiplets – a conformal primary along with an infinite number
of descendants obtained by acting with derivatives on the primary. In the
context of crossing symmetry, these multiplets get resummed into conformal
blocks. While the boundary conformal blocks for spinor two-point functions
were known already Nishida:2018opl ; Herzog:2019bom , here we derive the bulk
conformal blocks as well.

With this technical advance in hand, we proceed to analyze three different
interacting fermionic theories in the $\epsilon$ expansion in the presence of
a boundary. We find that the combination of crossing symmetry with the
equations of motion allows us efficiently to put nontrivial constraints on
combinations of the surface and bulk critical exponents. Similar to
Giombi:2021cnr , we consider the Yukawa model (1) in $4-\epsilon$ dimensions
and also the Gross-Neveu model in $2+\epsilon$ dimensions:

${\mathcal{L}}=-\overline{\psi}_{a}\not{\partial}\psi_{a}+\frac{g}{2}(\overline{\psi}_{a}\psi_{a})(\overline{\psi}_{b}\psi_{b})\
.$ (2)

In these two cases, we can take advantage of the known bulk critical exponents
to fix the boundary data.

It is often argued that the phase transition of these two models in the limit
$\epsilon=1$ is in the same universality class. Thus one can gain further
control over the $d=3$ dimensional theory by interpolating the results for the
two theories at $\epsilon=1$. There is however a further wrinkle in the
boundary case because of the the different choices of boundary condition for
the scalar field $\phi$. Ref. Giombi:2021cnr  argues that the Gross-Neveu
model in $2+\epsilon$ dimensions matches onto the extraordinary boundary
condition of the Yukawa model in $4-\epsilon$ dimensions. The extraordinary
phase transition corresponds to a bulk ordering phase transition at a point in
the phase diagram where the surface is already ordered,
$\langle\phi\rangle\sim r^{-\Delta_{\phi}}$. While we have some remarks to
make about the extraordinary case, the emphasis in this work is on the so-
called special and ordinary phase transitions of the Yukawa model, which in
the classical limit become Dirichlet and Neumann boundary conditions for the
scalar.111 The boundary condition for the fermion is a projection condition
$\frac{1}{2}(1+\chi\gamma_{n})\psi=0$ where $\gamma_{n}$ is the gamma matrix
perpendicular to the boundary and $\chi=\pm 1$. This boundary condition
corresponds to an “armchair” edge in graphene Biswas:2022dkg .

Our third model is the little studied

${\mathcal{L}}=-\frac{1}{2}\partial_{\mu}\phi_{a}\partial^{\mu}\phi_{a}-\overline{\psi}_{b}\not{\partial}\psi_{b}-\frac{g}{2}\phi^{2}\overline{\psi}_{b}\psi_{b}-\frac{h}{6!}(\phi^{2})^{3}$
(3)

in $d=3-\epsilon$ dimensions and the indices $a,b,\ldots$ run either from one
to $N_{s}$ or one to $N_{f}$, depending on whether they decorate a scalar or
fermion. The results we obtain for this third model are new as far as we are
aware, even in the absence of a boundary.222See however related work on
supersymmetric Chern-Simons theories in exactly $d=3$. In these theories,
kinetic terms involve gauge covariant derivatives, there is a Chern-Simons
term, and there is a flavor mixing term $(\bar{\psi}\phi)(\phi^{*}\psi)$ since
the $\phi$ and $\psi$ fields are part of the same chiral multiplet and are
assumed to transform in the same representation. We have in mind (1.3) of
Jain:2013gza .  One reason it has not been studied has to do with the behavior
of the purely bosonic version of this theory. The large $N_{s}$ limit has both
a UV and an IR fixed point which annihilate at $0<\epsilon_{c}<1$
Pisarski:1982vz ; it is thus not possible to set $\epsilon$ equal to one in
order to learn something about a 2d strongly interacting fixed point. It is
nonetheless an interesting playground in which to explore the intersection of
crossing symmetry and the equations of motion. Similar to what happens in the
purely bosonic theory with boundary diehi1987walks ; eisenriegler1988surface ,
the boundary critical exponents get nonanalytic corrections of
$O(\sqrt{\epsilon})$.

### Summary of Perturbative Results

Our application of crossing symmetry together with the equations of motion fix
a number of anomalous dimensions as well as an infinite number of OPE
coefficients to the first few orders in the $\epsilon$-expansions. Below we
tabulate the anomalous dimensions. The OPE coefficients can be found in the
text.

#### Yukawa model

At the RG fixed point, the coupling is of order $g^{2}=O(\epsilon)$. For
convenience, we express the results using a rescaled coupling
$\kappa_{s}g^{2}\equiv\lambda\epsilon$. The constant
$\kappa_{s}^{-1}=(d-2)\operatorname{Vol}(S^{d-1})$ is the usual normalization
of the free scalar two-point function in $d$ dimensions. Crossing symmetry and
the equations of motion enforce the following relations between the scaling
dimensions of $\psi$ and $\phi$, and the boundary value of $\psi$ denoted here
by $\rho$

$\displaystyle\Delta_{\psi}$ $\displaystyle=$
$\displaystyle\frac{3}{2}-\frac{\epsilon}{2}+\frac{\lambda}{8}\epsilon+O(\epsilon^{2})\
,$ (4) $\displaystyle\Delta_{\phi}$ $\displaystyle=$ $\displaystyle
1+\frac{\epsilon}{2}-\frac{3}{4}\lambda\epsilon+O(\epsilon^{2})\ ,$ (5)
$\displaystyle\Delta_{\rho}$ $\displaystyle=$
$\displaystyle\frac{3}{2}-\frac{\epsilon}{2}+\frac{(4-\chi_{s})}{8}\lambda\epsilon+O(\epsilon^{2})\
.$ (6)

The constant $\chi_{s}$ selects Neumann $\chi_{s}=1$ or Dirichlet
$\chi_{s}=-1$ boundary conditions for the scalar field $\phi$. Note the
presence or absence of a boundary cannot affect the bulk anomalous dimensions
and beta function calculation. Thus a standard beta function calculation in
the system without boundary will fix the relation between $g$ and $\epsilon$
at the conformal fixed point and thus the surface scaling dimension
$\Delta_{\rho}$. The results for the beta function can be looked up in many
places, for example Zerf:2017zqi . Our results agree with the computation of
Giombi:2021cnr  although our procedure requires less input from Feynman
diagram calculations.

#### Gross-Neveu model

This model has the advantage that with the same amount of effort, we are able
to go to one higher order in the $\epsilon$ expansion compared to the other
two models. In this case $g\sim\epsilon$ while in the other two cases
$g^{2}\sim\epsilon$. The resulting expressions are simpler in the rescaled
coupling $\lambda$ where $\kappa_{f}g\equiv\epsilon\lambda$. The constant
$\kappa_{f}^{-1}=\operatorname{Vol}(S^{d-1})$ is the usual normalization of
the free fermion two-point function in $d$ dimensions. Furthermore, crossing
symmetry and the equations of motion at $O(\epsilon^{2})$ allow us to fix

$g_{*}=\frac{\epsilon}{2(N-1)\kappa_{f}}+O(\epsilon^{2})\ ,$ (7)

without needing to invoke results from the literature for the case without
boundary.

We obtain the following scaling dimensions

$\displaystyle\Delta_{\psi}$ $\displaystyle=$
$\displaystyle\frac{1}{2}+\frac{\epsilon}{2}+\frac{N-\frac{1}{2}}{8(N-1)^{2}}\epsilon^{2}+O(\epsilon^{3})\
,$ (8) $\displaystyle\Delta_{\psi^{2}}$ $\displaystyle=$ $\displaystyle
1-\frac{1}{2(N-1)}\epsilon+\gamma^{(2)}_{\psi^{2}}\epsilon^{2}+O(\epsilon^{3})\
,$ (9) $\displaystyle\Delta_{\psi^{4}}$ $\displaystyle=$ $\displaystyle
2+O(\epsilon^{2})\ ,$ (10) $\displaystyle\Delta_{\rho}$ $\displaystyle=$
$\displaystyle\frac{1}{2}+\left(\frac{1}{2}+\frac{N-\frac{1}{2}}{2(N-1)}\right)\epsilon+\left(\frac{N-\frac{1}{2}}{4(N-1)^{2}}-\frac{\gamma^{(2)}_{\psi^{2}}}{2}\right)\epsilon^{2}+O(\epsilon^{3})\
.$ (11)

The result for $\Delta_{\rho}$ at $O(\epsilon)$ was presented already in
Giombi:2021cnr . The result $\gamma^{(2)}_{\psi^{2}}$ is known from
calculations in the literature for the case without boundary (see e.g.
Gracey:2008mf ):

$\gamma^{(2)}_{\psi^{2}}=-\frac{N-\frac{1}{2}}{4(N-1)^{2}}\ ,$ (12)

which gives the boundary anomalous dimension

$\gamma_{\rho}^{(2)}=\frac{3\left(N-\frac{1}{2}\right)}{8(N-1)^{2}}\ .$ (13)

####  $\phi^{2}\bar{\psi}_{a}\psi_{a}$ model

Similar to the 4d Yukawa model, the fixed point happens for
$g^{2}\sim\epsilon$. In contrast to the 4d Yukawa model, however, the surface
anomalous dimensions get contributions already at order
$g\sim\sqrt{\epsilon}$. The contribution occurs at leading order because the
nonzero one point function for $\phi^{2}$ in the presence of a boundary mimics
the effect of a conformal mass term for the fermion Herzog:2019bom . The one
point function $\langle\phi\rangle$ in the Yukawa model, which could play a
similar role, vanishes for Dirichlet and Neumann boundary conditions. We work
with the rescaled coupling $\lambda$ where $\kappa_{s}g\equiv
2\lambda\sqrt{\epsilon}$. The full results are

$\displaystyle\Delta_{\psi}$ $\displaystyle=$ $\displaystyle
1-\frac{\epsilon}{2}+\frac{N_{s}\lambda^{2}}{3}\epsilon+O(\epsilon^{2})\ ,$
(14) $\displaystyle\Delta_{\phi^{2}}$ $\displaystyle=$ $\displaystyle
1+O(\epsilon)\ ,$ (15) $\displaystyle\Delta_{\psi^{2}}$ $\displaystyle=$
$\displaystyle 2-\epsilon+\frac{8N_{s}}{3}\lambda^{2}\epsilon+O(\epsilon^{2})\
,$ (16) $\displaystyle\Delta_{\psi\phi}$ $\displaystyle=$
$\displaystyle\frac{3}{2}+O(\epsilon)\ ,$ (17) $\displaystyle\Delta_{\rho}$
$\displaystyle=$ $\displaystyle
1-\frac{\epsilon}{2}+\frac{\chi\chi_{s}N_{s}}{2}\lambda\sqrt{\epsilon}-\lambda^{2}N_{s}\left(\chi\chi_{s}(1+N_{f})+\frac{4}{3}\right)\epsilon+O(\epsilon^{3/2})\
.$ (18)

A standard beta function calculation in the theory without boundary, performed
in appendix D, gives the value of the coupling $\lambda$ at the fixed point

$\displaystyle\lambda_{*}^{2}$ $\displaystyle=$
$\displaystyle\frac{3}{8(2N_{f}+N_{s}-3)}+O(\epsilon)\ .$ (19)

#### Outline

In section 2, we set up the embedding formalism that will allow us to
linearize the constraints of conformal symmetry. We then use this embedding
formalism to review the form of scalar, vector and spinor correlation
functions in the presence of a boundary. Next in section 3, we use the
embedding formalism to derive the bulk and boundary conformal partial waves
(or blocks) that we will use to enforce the crossing symmetry constraint.
Section 4 is the heart of the paper. We discuss how crossing symmetry is
satisfied by free fermions, make some comments about the extraordinary phase
transition, and then embark on a solution of the crossing symmetry constraints
for our three perturbative models. Section 5 is a discussion where, among
other things, we propose that a particular surface anomalous dimension may
have observable consequences for the electron density of states near the
boundary, at the point of a quantum phase transition, for a Dirac or Weyl
material.

A large number of appendices contain auxiliary results. Appendix A spells out
our conventions for dealing with fermions. Appendix B presents some integrals
in embedding space that were useful in deriving the conformal blocks. Appendix
C derives the conformal blocks in a different way, by explicitly summing the
OPE. Some conformal data for the 3d Yukawa model without boundary are derived
perturbatively in an $\epsilon$-expansion in appendix D. Finally, appendix E
discusses the relationship between our correlation functions in flat space
with boundary and their Weyl transformed cousins in anti-de Sitter space.

##  2 Correlators of Scalars and Spinors

Conformal invariance places strong constraints on correlation functions, even
in the presence of a boundary. Let us begin by writing down the set of
correlation functions that will be important to us in this work along with the
conformal invariance constraints imposed on them. As it requires the least
formal development, we give these correlation functions first in coordinate
space. Deriving the conformal invariance constraints, however, is more easily
done in embedding space. Moreover, embedding space will be important in the
next section when we discuss the conformal block decomposition of the
correlation functions. Thus, after listing our main cast of characters, we
embark on a discussion of embedding space.

Please note, the only new aspects of our embedding space discussion concern
spinors in the presence of a boundary. Earlier discussions of spinors in
embedding space without a boundary can be found already in refs.
Weinberg:2010ws ; Isono:2017grm . Also, our real space results for free
spinors with a boundary match the classic results McAvity:1995tm . Even in the
presence of a boundary, our ideas are implicitly contained in a treatment of
the embedding space for spinors in anti-de Sitter (AdS) and de Sitter (dS)
space Nishida:2018opl ; Pethybridge:2021rwf . The relation between AdS, dS,
and bCFT occurs because of the relation between their metrics under Weyl
rescaling and (in the case of dS) Wick rotation. We offer an embedding space
perspective on this mapping in Appendix E.

In what follows, we have occasion to discuss two point functions of bulk
spinors $\psi$, scalars $\phi$, and conserved currents $J^{\mu}$. We will also
require boundary spinors $\rho$ and occasionally boundary scalars $\tau$ and
vectors $v^{i}$. We divide the coordinates $x^{\mu}=(z,x^{i})$ into a normal
coordinate $z$ and parallel coordinates $x^{i}$.

In the case of bulk $\phi$ and boundary $\tau$ scalar fields, the tensor
structures are trivial, and one finds

$\displaystyle\langle\phi(x)\rangle$ $\displaystyle=$
$\displaystyle\frac{a_{\phi}}{z^{\Delta_{\phi}}}\ ,$ (20)
$\displaystyle\langle\phi(x)\tau(x^{\prime})\rangle$ $\displaystyle=$
$\displaystyle\frac{c_{\phi\tau}}{(2z)^{\Delta_{\phi}-\Delta_{\tau}}|x-x^{\prime}|^{2\Delta_{\tau}}}\
,$ (21) $\displaystyle\langle\phi(x)\phi^{\prime}(x^{\prime})\rangle$
$\displaystyle=$
$\displaystyle\frac{H(\xi)}{(2z)^{\Delta}(2z^{\prime})^{\Delta^{\prime}}}\ .$
(22)

The constants $\Delta_{\phi}$ and $\Delta_{\tau}$ are the scaling dimensions
of the corresponding operator insertions. In bCFT, only one point functions of
bulk scalars can get a nontrivial expectation value. All other one-point
functions will vanish because of Lorentz invariance. Note that the bulk one-
point function and bulk-boundary two-point function are fixed up to constant
coefficients $a_{\phi}$ and $c_{\phi\sigma}$ because of the absence of an
appropriate cross ratio. The bulk-two point function, however, can depend on
an undetermined function $H(\xi)$ of the conformally invariant cross-ratio

$\xi=\frac{(x-x^{\prime})^{2}}{4zz^{\prime}}\ .$ (23)

More generally, a correlation function of two bulk operators will be a sum
over some number of tensor structures, whose form is fixed up to coefficients
which are functions of the cross ratio $\xi$. Indeed, given two bulk points
$x$ and $x^{\prime}$, we can form exactly one conformally invariant cross
ratio, although sometimes it is useful to work with the related quantity
$v\equiv\xi/(1+\xi)$. The bulk coincident limit is given by $\xi\rightarrow
0$, the boundary limit by $\xi\rightarrow\infty$. The $\xi$ variable also has
an interpretation in terms of AdS geodesic distance.

For a conserved curent $J^{\mu}$, we have the correlation functions

$\displaystyle\langle J^{\mu}(x)\tau(x^{\prime})\rangle$ $\displaystyle=$
$\displaystyle\frac{c_{J\tau}}{|x-x^{\prime}|^{2(d-1)}}X^{\mu}\delta_{\Delta_{\tau},d-1}\
,$ (24) $\displaystyle\langle J^{\mu}(x)v^{i}(x^{\prime})\rangle$
$\displaystyle=$
$\displaystyle\frac{c_{Jv}}{(2z)^{d-1-\Delta_{v}}|x-x^{\prime}|^{2\Delta_{v}}}I^{\mu
i}(x-x^{\prime})\ ,$ (25) $\displaystyle\langle
J^{\mu}(x)J^{\nu}(x^{\prime})\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{|x-x^{\prime}|^{2(d-1)}}\left(Q(\xi)I_{\mu\nu}(x-x^{\prime})+(R(\xi)-Q(\xi))X_{\mu}X^{\prime}_{\nu}\right)\
.$ (26)

Here the tensor structures are

$\displaystyle I_{\mu\nu}(x)=\delta_{\mu\nu}-2\frac{x_{\mu}x_{\nu}}{x^{2}}\
,\;\;\;X_{\mu}=z\frac{v}{\xi}\partial_{\mu}\xi\
,\;\;\;X^{\prime}_{\mu}=z^{\prime}\frac{v}{\xi}\partial^{\prime}_{\mu}\xi\ .$
(27)

Conservation imposes the relation $v\partial_{v}R=(d-1)(R-Q)$. Furthermore,
$\langle J^{\mu}\tau\rangle$ must vanish unless $\tau$ has conformal dimension
$d-1$.

Finally, for spinor insertions, we have

$\displaystyle\langle\psi(x)\overline{\rho}(x^{\prime})\rangle$
$\displaystyle=$
$\displaystyle\frac{\gamma_{\mu}(x-x^{\prime})^{\mu}}{(2z)^{\Delta_{\psi}-\Delta_{\rho}}|x-x^{\prime}|^{2\Delta_{\rho}}}(c_{\psi\rho}+c^{\prime}_{\psi\rho}\gamma_{z})\
,$ (28)
$\displaystyle\langle\psi(x)\overline{\psi}^{\prime}(x^{\prime})\rangle$
$\displaystyle=$
$\displaystyle\frac{\gamma_{\mu}(x-x^{\prime})^{\mu}F(\xi)+\gamma_{z}\gamma_{\mu}(x^{\prime}-\tilde{x})^{\mu}G(\xi)}{(2z)^{\Delta}(2z^{\prime})^{\Delta^{\prime}}},$
(29)

where $\tilde{x}=(-z,x^{i})$ and the $\gamma^{\mu}$ are the usual gamma
matrices. It is natural to treat $\rho$ as a spinor with respect to the
smaller Lorentz group on the surface, and thus to take it as an eigenvector of
the normal gamma matrix $\gamma_{z}\rho=\pm\rho$. As a result, we can replace
$c_{\psi\rho}-c^{\prime}_{\psi\rho}\gamma_{z}$ in the bulk-boundary two point
function with $\frac{\mu_{\psi\rho}}{2}(1\mp\gamma_{z})$, depending on the
choice of eigenspace for $\rho$.

###  2.1 Embedding Space and Correlation Function in a Pure CFT

The projective light-cone is the natural space in which to define CFT
correlation functions because it linearizes the action of the conformal group
Weinberg:2010ws ; Costa:2011wa . It is the quotient

$\displaystyle\ifrac{\\{P^{A}\in\mathbb{R}^{1,d+1}\mid
P^{2}=0\\}}{\\{\lambda\in\mathbb{R}^{+}\mid P^{A}\sim\lambda P^{A}\\}}\ .$

Primary fields are defined to have homogeneous scaling. In particular, for a
scalar field of conformal dimension $\Delta$, one has

$\displaystyle\phi(\lambda P)=\lambda^{-\Delta}\phi(P)\ .$ (30)

The definition of primary generalizes to tensorial operators $F_{A_{1}\cdots
A_{n}}$. To furnish an irreducible representation of the conformal group,
these tensor operators must be $P$-transverse and obey a “gauge equivalence”
relation. For a current $J_{A}$, these two conditions amount to the
constraints $P^{A}J_{A}=0$ and $J_{A}\sim J_{A}+\alpha P_{A}$ for any
$\alpha$.

There exists an index-free formalism for general tensor type which simplifies
their manipulation Costa:2011wa ; Costa:2014rya . For symmetric tracelesss
tensors (STT), one contracts all free indices with a polarisation vector
$Z^{A}$ satisfying $Z^{2}=0$. We then require that $Z^{A}$ only appears in
transverse structures, i.e. in objects invariant under $Z\rightarrow Z+\alpha
P$. Because of the gauge equivalence relation, we can set $Z\cdot P=0$.
Tensorial operators are then homogeneous, $P$-transverse polynomials in the
polarisation vector $Z^{A}$.

Projection to real space is done by choosing a section of the lightcone,
specified through a parametrisation of the $P^{A}$. The projection is encoded
through a set of replacement rules for the various quantities entering the
correlation function. To be concrete, consider the Poincaré section, which
maps to a CFT living on flat space,

$\displaystyle
P^{A}=\left(P^{+},P^{-},P^{\mu}\right)=\left(1,x^{2},x^{\mu}\right)\ .$ (31)

A solution for $Z^{A}$ compatible with the constraints $Z^{2}=0$ and $Z\cdot
P=0$ is $Z^{A}=(0,x\cdot z,z^{\mu})$, provided $z^{2}=0$. The projection to
flat space can then be encoded in the replacement rules

$\displaystyle P_{ij}=-2P_{i}\cdot P_{j}$
$\displaystyle\rightarrow\absolutevalue{x_{i}-x_{j}}^{2}=\absolutevalue{x_{ij}}^{2}\
,$ (32) $\displaystyle P_{i}\cdot Z_{j}$
$\displaystyle\rightarrow(x_{i}-x_{j})\cdot z_{j}=x_{ij}\cdot z_{j}\ ,$
$\displaystyle Z_{ij}=Z_{i}\cdot Z_{j}$ $\displaystyle\rightarrow z_{i}\cdot
z_{j}\ ,$

and the resulting expressions describe the correlation function of operators
in real space.

We will write correlation functions evaluated without a boundary using the
symbol $\expectationvalue{\ldots}_{\circ}$. Using the constraints previously
considered, one can classify correlation functions of scalar and vector
operators

$\displaystyle\expectationvalue{\phi(P_{1})}_{\circ}=0\
,\;\;\;\expectationvalue{\phi(P_{1})\phi(P_{2})}_{\circ}=\frac{1}{(P_{12})^{\Delta}}\
,$ (33) $\displaystyle\expectationvalue{Z_{1}\cdot J(P_{1})Z_{2}\cdot
J(P_{2})}_{\circ}=c_{JJ}\frac{{\mathcal{S}}_{1}}{(P_{1}\cdot
P_{2})^{\Delta_{J}}}\ ,$ (34)

where we have defined the transverse tensor structure

${\mathcal{S}}_{1}\equiv Z_{1}\cdot Z_{2}-\frac{(Z_{1}\cdot P_{2})(Z_{2}\cdot
P_{1})}{P_{1}\cdot P_{2}}\ .$ (35)

If the current is furthermore conserved, then $\Delta_{J}=d-1$. Note the
transversality condition fixes the relative normalization between the two
terms in ${\mathcal{S}}_{1}$. Using the replacement rules (32), this tensor
structure becomes $I_{\mu\nu}$ in real space.

Spinors are dealt with similarly Weinberg:2010ws ; Iliesiu:2015qra ;
Isono:2017grm . A primary spinor field $\Psi$ obeys a scaling and
irreducibility relation

$\displaystyle\Psi(\lambda P)$
$\displaystyle=\lambda^{-\Delta-\frac{1}{2}}\Psi(X)\ ,$ $\displaystyle
P^{A}\Gamma_{A}\Psi(P)$ $\displaystyle=0\,.$ (36)

An extra complication is the relation between the embedding $\Psi$ and real
space $\psi$ spinor representations. Generically, we look at insertions of
Dirac spinors $\psi$ of $\mathbb{R}^{d}$. These are uplifted to Dirac spinors
$\Psi$ of $\mathbb{R}^{1,d+1}$, which are twice as big. Correspondingly, if
the Clifford algebra in real space is given through

$\displaystyle\\{\gamma_{\mu},\gamma_{\nu}\\}=2\delta_{\mu\nu}\ ,$ (37)

then one can construct the embedding space algebra through the tensor product

$\displaystyle\Gamma_{\mu}$ $\displaystyle=\gamma_{\mu}\otimes\sigma_{3}\ ,$
$\displaystyle\Gamma_{0}$ $\displaystyle=\mathbb{1}\otimes i\sigma_{2}\ ,$
$\displaystyle\Gamma_{d+1}$ $\displaystyle=\mathbb{1}\otimes\sigma_{1}\ .$
(38)

If required, a “gamma-five” $\Gamma_{\star}$ matrix can be included as well.
Multiplying a given spinorial structure by $\Gamma_{\star}$ gives a second
allowed structure, but with the same conformal block. As the examples we
consider in this work do not require the $\Gamma_{\star}$ structures, we
ignore this added complication.

It is convenient to work with scalar objects. The index-free formalism
generalises to spinors Iliesiu:2015qra ; Isono:2017grm . One contracts the
spinor field $\Psi$ with a (Grassman even) polarisation conjugate spinor
$\overline{S}$, which incorporates the constraints, hence also the projection
to real space. To be explicit, in the Poincaré section, one can take

$\displaystyle\overline{S}$
$\displaystyle=\overline{s}\begin{pmatrix}\not{x}&1\end{pmatrix}\ ,$
$\displaystyle\quad S$ $\displaystyle=\begin{pmatrix}1\\\
-\not{x}\end{pmatrix}s\ .$ (39)

Using these expressions, one can then find as before a set of replacement
rules encoding the real space limit. Allowed spinorial structures in
correlation functions are then the non-vanishing Lorentz-scalars one can build
using the elements introduced by the insertions. Structures are efficiently
written in a bra-ket notation, e.g.
$\bra{S_{1}}P_{3}\ket{S_{2}}=\bra{1}P_{3}\ket{2}=\overline{S}_{1}\not{P}_{3}S_{2}$.
We will abuse the bra-ket notation to designate both the sandwiching of the
matrices between the embedding spinor polarisations, $S$, and the real-space
polarisations, $s$. We will also write
$\bra{1}x_{12}\ket{2}=\overline{s}_{1}\gamma\cdot x_{12}s_{2}$. Which is meant
should hopefully be clear from context. Note that with the conventions spelled
out previously, this structure is real under conjugation and simultaneous
exchange of 1 and 2.

Using this notation, we can efficiently constrain correlation functions of
spinors

$\displaystyle\expectationvalue{\psi(P_{1},\overline{S}_{1})\overline{\psi}(P_{2},S_{2})}_{\circ}$
$\displaystyle=\frac{\innerproduct{1}{2}}{(P_{12})^{\Delta+\frac{1}{2}}}\ ,$
(40)
$\displaystyle\expectationvalue{\psi(P_{1},\overline{S}_{1})\overline{\psi}(P_{2},S_{2})\phi(P_{3})}_{\circ}$
$\displaystyle=\lambda^{(1)}_{\phi}\frac{\innerproduct{1}{2}}{P_{13}^{\frac{\Delta_{1}+\Delta_{3}-\Delta_{2}}{2}}P_{23}^{\frac{\Delta_{2}+\Delta_{3}-\Delta_{1}}{2}}P_{12}^{\frac{\Delta_{1}+\Delta_{2}-\Delta_{3}+1}{2}}}$
(41)
$\displaystyle+\lambda^{(2)}_{\phi}\frac{\bra{1}P_{3}\ket{2}}{P_{13}^{\frac{\Delta_{1}+\Delta_{3}+1-\Delta_{2}}{2}}P_{23}^{\frac{\Delta_{2}+\Delta_{3}+1-\Delta_{1}}{2}}P_{12}^{\frac{\Delta_{1}+\Delta_{2}-\Delta_{3}}{2}}}\
.$

These equations set our conventions for the CFT data coefficients, and will
reappear when we consider the bulk OPE of operators with a boundary. Note the
$\lambda^{(i)}$ are real numbers, provided one uses Lorentzian conjugation to
define reality. From the conventions previously given, real-space expressions
are given using replacement rules (see also Appendix A)

$\displaystyle\innerproduct{1}{2}$
$\displaystyle\rightarrow\bra{1}x_{12}\ket{2}=\overline{s}_{1}\,\gamma\cdot
x_{12}\,s_{2}\ ,$ (42) $\displaystyle\bra{1}P_{3}\ket{2}$
$\displaystyle\rightarrow\bra{1}x_{13}x_{32}\ket{2}=\overline{s}_{1}\,\gamma\cdot
x_{13}\,\gamma\cdot x_{32}\,s_{2}\ .$

These rules yield from (40) the familiar two point function of a Dirac spinor
primary

$\displaystyle\expectationvalue{\psi(x)\overline{\psi}(x^{\prime})}_{\circ}$
$\displaystyle=\frac{\bra{1}(x-x^{\prime})\ket{2}}{\absolutevalue{x-x^{\prime}}^{2\Delta+1}}\
.$ (43)

One can proceed similarly to construct correlation functions involving more
complicated correlation functions, but at this point we have what we need
before proceeding to the more elaborate case of a boundary.

###  2.2 Correlator in the Presence of a Boundary

In a boundary conformal field theory, one has to deal with an explicit
breaking of the conformal group $SO(1,d+1)\rightarrow SO(1,p+1)$, $p=d-1$. In
real space, this breaking is induced by a preferred direction specified
through a unit normal $n^{\mu}$. In the embedding picture, this breaking is
associated to a unit vector $V^{A}$, $V^{2}=1$ Liendo:2012hy . It follows that
one should distinguish two types of insertions, bulk $P\cdot V\neq 0\neq
x\cdot n$, and boundary, $P\cdot V=0=x\cdot n$.

It is useful at times to consider an invariant inner product, $A\bullet
B=A\cdot B-A\cdot VB\cdot V$. While bulk insertions are defined as in the pure
CFT case, with fields transforming as irreducible representations of
$SO(1,d+1)$, boundary ones have more structure, as bulk representations are
generically reducible with respect to the boundary symmetry group. More
precisely, one can consistently separate the normal and parallel components of
tensor operators, which then transform separately.

Boundary operators in embedding space can be introduced as before, but now
with some extra constraints. They cannot depend on the normal direction, and
hence $P\cdot V=0$. Furthermore, if we contract a symmetric traceless boundary
tensor with a polarization vector $W_{A}$ such that $W^{2}=0$, then we should
choose $W$ such that $W\cdot V=0$.

Correlation functions can then be constructed as in the pure CFT case: by
building the most general Lorentz invariant object in the embedding given the
objects and constraints at hand, and then projecting them to real space. The
conformally invariant cross ratio can be written in embedding space:

$\displaystyle\xi=\frac{-2P_{1}\cdot P_{2}}{(-2P_{1}\cdot V)(-2P_{2}\cdot V)}\
.$ (44)

For the bulk scalar and vector, we have almost immediately

$\displaystyle\expectationvalue{\phi(P)}$
$\displaystyle=\frac{a_{\phi}}{(2P\cdot V)^{\Delta}}\ ,$ (45)
$\displaystyle\expectationvalue{\phi(P_{1})\phi(P_{2})}$
$\displaystyle=\frac{H(\xi)}{(2P_{1}\cdot V)^{\Delta_{1}}(2P_{2}\cdot
V)^{\Delta_{2}}}\ ,$ (46) $\displaystyle\expectationvalue{Z_{1}\cdot
J(P_{1})Z_{2}\cdot J(P_{2})}$ $\displaystyle=\frac{1}{(-2P_{1}\cdot
P_{2})^{\Delta_{J}}}(Q(\xi){\mathcal{S}}_{1}+(R(\xi)-Q(\xi)){\mathcal{S}}_{2})$
(47)

where ${\mathcal{S}}_{1}$ is the same as in the case without boundary, but now
there is a new structure:

${\mathcal{S}}_{2}\equiv\left(Z_{1}\cdot V-\frac{(Z_{1}\cdot P_{2})(P_{1}\cdot
V)}{P_{1}\cdot P_{2}}\right)\left(Z_{2}\cdot V-\frac{(Z_{2}\cdot
P_{1})(P_{2}\cdot V)}{P_{1}\cdot P_{2}}\right)\ .$ (48)

These expressions project down to the real space results (20), (22), and (26).
As they are anyway well known, we will leave the embedding space form of (21),
(24), and (25) as exercises.

In this work, we are concerned with insertions of fields transforming as
spinors under the rotation group, both as bulk and boundary insertions. The
initial question is how to generalise the analysis of boundary tensors to that
of spinors.

On the boundary, the preserved generators of the conformal group will commute
with the matrices $\not{V}$ in the embedding, respectively $\not{n}$ in real
space. Hence, when we consider generic boundary spinor insertions $\rho(x)$,
$x\cdot n=0$, we can impose irreducibility by requiring

$\displaystyle\not{n}\rho(x)$ $\displaystyle=\pm\rho(x)\ ,$
$\displaystyle\not{V}\rho(P)$ $\displaystyle=\pm\rho(P)\ .$ (49)

The boundary primaries are then “chiral spinors” $\rho_{\pm}$. In odd
dimensions $d=2k+1$, a Dirac spinor of $Spin(d)$ is also a Dirac spinor of
$Spin(p)$. Then, the $\rho_{\pm}$ can be thought of as Weyl fermions under the
stabiliser group of the boundary, i.e. intrinsically to the boundary theory,
they are simply Weyl spinors. When $d=2k+2$, the analysis is more intricate;
it turns out that $\rho_{\pm}$ transform under the two unitarily inequivalent
Dirac spinor representations of $Spin(p)$. Intrinsically, we then have a
doublet of Dirac spinors, which are exchanged under “parity”. These are
however technical details, useful when working intrinsically, with only
boundary insertions.

When we consider a mix of bulk and boundary insertions, it is most convenient,
as for tensors, to encode the boundary objects using constrained bulk
representations, as we will do from now on. This is efficiently done by using
the projectors $\Pi_{\pm}=\frac{1}{2}\left(\mathds{1}\pm\not{V}\right)$, and
$\mathbb{P}_{\pm}=\frac{1}{2}\left(\mathds{1}\pm\not{n}\right)$. As before,
one can work in an index-free formalism, by contracting all objects with
polarisation spinors, but now the polarisations for boundary insertions have a
definite helicity.

One should however be cautious when comparing real space and embedding
expressions though, as the change in signature makes it such that
$\overline{\rho}_{\pm}(x)\not{n}=\mp\overline{\rho}_{\pm}(x)$, while
$\overline{\rho}_{\pm}(P)\not{V}=\pm\overline{\rho}_{\pm}(P)$, i.e. Dirac
conjugation exchanges helicity eigenvalue in real space, but not in the
embedding. For this reason, embedding space becomes useful to derive
conformally invariant structures, but those are simpler to analyse in real
space. For completeness, one can define polarisation spinors

$\displaystyle\overline{S}^{(\pm)}\not{V}$
$\displaystyle=\pm\overline{S}^{(\pm)}\ ,$ $\displaystyle\not{V}S^{\pm}$
$\displaystyle=\pm S^{(\pm)}\ ,$ (50) $\displaystyle\overline{s}_{\pm}\not{n}$
$\displaystyle=\pm\overline{s}_{\pm}\ ,$ $\displaystyle\not{n}s_{\pm}$
$\displaystyle=\mp s_{\pm}\ .$ (51)

These subtleties can create confusion. For example, if we consider the bra-ket
notation, with an added label for helicity $\bra{1,\alpha}$, this leads to
projections from embedding space to real space where we have schematically

$\displaystyle\bra{1,\alpha}$ $\displaystyle\rightarrow\bra{1,\alpha}(\ldots)\
,$ (52) $\displaystyle\ket{2,\alpha}$
$\displaystyle\rightarrow(\ldots)\ket{2,-\alpha}\ .$ (53)

Although the rules may appear intricate, they are simple to deploy. Consider
for example the two point function of the boundary spinor

$\displaystyle\expectationvalue{\rho_{\alpha}(x)\overline{\rho}_{\beta}(y)}=\frac{\bra{1,\alpha}(x-y)\ket{2,-\beta}}{\absolutevalue{x-y}^{2\Delta+1}}=\overline{s}_{1}\mathbb{P}_{\alpha}\frac{\gamma\cdot(x-y)}{\absolutevalue{x-y}^{2\Delta+1}}\mathbb{P}_{-\beta}s_{2}\
.$ (54)

Clearly $\alpha=\beta$ for this correlator not to vanish. Note again that
under complex conjugation as well as exchange of $(1,x)\leftrightarrow(2,y)$,
this correlator is mapped to itself, since the kets exchange helicity. The
embedding space spinor structure in a pure CFT is $\bra{1}\ket{2}$, which is
diagonal; hence we need to project $\ket{2,+}$ in the embedding into something
proportional to $\ket{2,-}$ in real space,

$\displaystyle\expectationvalue{\rho_{\alpha}(P_{1})\overline{\rho}_{\beta}(P_{2})}=\frac{\bra{1,\alpha}\ket{2,\beta}}{(P_{12})^{2\Delta+1}}\
.$ (55)

Being even more explicit, we can view the real-space and embedding space
expressions as the same ones, under the following set of supplementary
replacement rules

$\displaystyle P_{i}\cdot V$ $\displaystyle\rightarrow x_{i}\cdot n\ ,$ (56)
$\displaystyle\bra{i}V\ket{j}$
$\displaystyle\rightarrow\bra{1}n(-\tilde{x}_{i}+x_{j})\ket{2}\ ,$
$\displaystyle\bra{i}\ket{j,\alpha}$
$\displaystyle\rightarrow\bra{i}(x_{i}-x_{j})\ket{j,-\alpha}\ ,$
$\displaystyle\bra{i,\alpha}\ket{j}$
$\displaystyle\rightarrow\bra{i,\alpha}(x_{i}-x_{j})\ket{j}\ ,$
$\displaystyle\bra{i,\alpha}\ket{j,\beta}$
$\displaystyle\rightarrow\bra{i,\alpha}(x_{i}-x_{j})\ket{j,-\beta}\ ,$

and we define $\tilde{x}^{\mu}=x^{\mu}-2n^{\mu}x\cdot n$, the boundary
reflection of $x^{\mu}$.

With these details spelled out, we can turn to the classification of
correlation functions. The simplest spinorial correlator is the bulk-boundary
correlator of $\psi$ and $\rho$

$\displaystyle\expectationvalue{\psi(P_{1})\overline{\rho}_{\alpha}(P_{2})}$
$\displaystyle=\mu_{\Delta_{2}}\frac{\innerproduct{1}{2,\alpha}}{(2P_{1}\cdot
V)^{\Delta_{1}-\Delta_{2}}(-2P_{1}\cdot P_{2})^{\Delta_{2}+\frac{1}{2}}}\ .$
(57)

The real-space result for the bulk-boundary two-point function of spinors (28)
follows straightforwardly by projection. It is useful to spell out the
$\langle\rho\bar{\psi}\rangle$ correlation function as well:

$\displaystyle\expectationvalue{\rho_{\alpha}(x_{1},s_{1})\overline{\psi}(x_{2},s_{2})}$
$\displaystyle=\mu_{\Delta_{1}}^{\dagger}\frac{\bra{1,\alpha}x_{12}\ket{2}}{(2z_{2})^{\Delta_{2}-\Delta_{1}}\absolutevalue{x_{12}}^{2\Delta_{1}+1}}\
,$ (58)

which follows by taking the complex conjugate of (28). One can in general
perform a phase redefinition of $\rho_{\alpha}$ to make $\mu_{\Delta}$ real
valued, as we now assume.

The natural next step in complexity is to consider spinor bulk-bulk
correlators. There are two allowed spinorial structures. For spinors in
embedding space, we have

$\displaystyle\expectationvalue{\psi(P_{1},\overline{S}_{1})\overline{\psi}(P_{2},S_{2})}$
$\displaystyle=\frac{F(\xi)\innerproduct{1}{2}+G(\xi)\bra{1}V\ket{2}}{(2P_{1}\cdot
V)^{\Delta_{1}+\frac{1}{2}}(2P_{2}\cdot V)^{\Delta_{2}+\frac{1}{2}}}\ .$ (59)

The spinor two point function and its two structures are the central object of
study of this paper. More intuition for these structures can be found from the
real space expression (29), where the second structure is associated to the
reflection of the first one by the boundary. These structures match the
structures that appear for free fermions McAvity:1993ul ; McAvity:1995tm , but
we see from this analysis that no additional structures can appear and be
consistent with conformal invariance.333We are ignoring the possibility of
$\Gamma_{\star}$ structures here.

In the rest of this paper, we investigate in detail these functions $F(\xi)$
and $G(\xi)$. We decompose them in a distinguished basis, the conformal
blocks, and constrain them using crossing symmetry, in a few perturbative
models. The derivation of the bulk conformal blocks and perturbative analysis
of our three models form the core of our results.

The kinematic classification for more complicated insertions, for example
tensors and tensor-spinors, is done more straightforwardly in the real space
picture. The tactic is to use the different tensor and spinorial structures
already found, and to multiply them together before taking out all
$\gamma$-traces.

##  3 The Operator Product Expansion and Conformal Blocks

We determined that the correlator of interest,
$\langle\psi(x)\overline{\psi}(x^{\prime})\rangle$, is fixed up to two
functions of a cross-ratio, $F(\xi)$ and $G(\xi)$. These functions, however,
are further constrained by the existence of two different operator product
expansions (OPEs). The first one is the boundary OPE, where we take a bulk
field and Taylor-expand it near the boundary, yielding a sum over boundary
primary operators and their descendants. Meanwhile, one can bring two bulk
insertions close together, and merge them into a sum of bulk operators; this
is the bulk OPE. Grouping the sums into conformal multiplets, each of which
consists of a primary operator and all its descendants, we find the conformal
block decomposition of the correlation function.

In this section, we derive the conformal blocks. The boundary conformal blocks
were first derived in Nishida:2018opl ; Herzog:2019bom  from AdS; while our
methods are different, the results are the same. Our results for the bulk
conformal blocks, however, are new. We compute the blocks by using a
representation of them as conformal integrals. We are then able to check the
computation by explicitly resumming the OPE, the details of which are
presented in appendix C.

###  3.1 Conformal Blocks

The conformal blocks depend on the type (bulk, boundary), and the properties
of the exchanged operator:

$\displaystyle\expectationvalue{\psi(x)\overline{\psi}(y)}$
$\displaystyle=\sum_{\Delta,i=1,2}a_{\Delta}\lambda^{(i)}_{\Delta}\mathcal{W}_{\Delta,i}(x,y)$
(60)
$\displaystyle=\sum_{\widehat{\Delta},\alpha=\pm}\mu_{\widehat{\Delta},\alpha}^{2}\widehat{\mathcal{W}}_{\widehat{\Delta},\alpha}(x,y)\
,$ (61)

where the hat is used to indicate the boundary. Equivalently, this is an
expansion for the invariant functions $F(\xi)$ and $G(\xi)$:

$\displaystyle F(\xi)$
$\displaystyle=\sum_{\Delta,i=1,2}a_{\Delta}\lambda^{(i)}_{\Delta}f_{\Delta,i}(\xi)$
$\displaystyle G(\xi)$
$\displaystyle=\sum_{\Delta,i=1,2}a_{\Delta}\lambda^{(i)}_{\Delta}g_{\Delta,i}(\xi)$
(62)
$\displaystyle=\sum_{\widehat{\Delta},\alpha=\pm}\mu_{\widehat{\Delta},\alpha}^{2}\widehat{f}_{\widehat{\Delta},\alpha}(\xi)\
,$
$\displaystyle=\sum_{\widehat{\Delta},\alpha=\pm}\mu_{\widehat{\Delta},\alpha}^{2}\widehat{g}_{\widehat{\Delta},\alpha}(\xi)\
.$ (63)

This basis of blocks is easy to work with. With the reality structure we
picked, the blocks and coefficients multiplying them are manifestly real. It
turns out that $\widehat{f}_{\widehat{\Delta},\alpha}$ is $\alpha$
independent, while $\widehat{g}_{\widehat{\Delta},\alpha}$ is proportional to
$\alpha$. Moreover, $f_{\Delta,2}=0=g_{\Delta,1}$.

We distinguish between the $\mathcal{W}_{\Delta,i}(x,y)$ and
$\widehat{\mathcal{W}}_{\widehat{\Delta},\alpha}(x,y)$, which we call
conformal partial waves (CPWs) and the $f_{\Delta,i}(\xi)$,
$g_{\Delta,i}(\xi)$, $\widehat{f}_{\Delta,\alpha}(\xi)$, and
$\widehat{g}_{\Delta,\alpha}(\xi)$ we call conformal blocks Dolan:2004up .
There are two distinct methods for computing these objects: solving a
differential equation and computing a sum. The differential equation route,
involves identifying the CPWs as eigenfunctions for the bulk or boundary
Casimir operators. The sum route involves resumming the OPE Dolan:2001wg ;
Billo:2016vm . The OPE relations are best given in real space. For the bulk
OPE one has

$\displaystyle\psi_{1}(x)\overline{\psi}_{2}(y)$
$\displaystyle=\sum_{\Delta,i=1,2}\frac{\lambda_{\Delta}^{(i)}\mathfrak{C}^{(i)}_{(\Delta_{1},\Delta_{2},\Delta)}[x-y,\partial_{y}]\phi_{\Delta}(y)}{\absolutevalue{x-y}^{\Delta_{1}+\Delta_{2}+1-\Delta}}+(\ell>0)$
(64)
$\displaystyle=\sum_{\Delta}\frac{\left(\lambda_{\Delta}^{(1)}\gamma\cdot(x-y)+\lambda_{\Delta}^{(2)}\mathbb{1}\absolutevalue{x-y}\right)}{\absolutevalue{x-y}^{\Delta_{1}+\Delta_{2}+1-\Delta}}\phi_{\Delta}\left(y\right)+(\ell>0)+desc.$

While for the boundary OPE we find :

$\displaystyle\psi_{\Delta}(x,z)$
$\displaystyle=\sum_{\widehat{\Delta},\alpha}\frac{\mu_{\widehat{\Delta},\alpha}}{(2z)^{\Delta_{1}-\Delta}}\mathfrak{D}_{(\Delta,\widehat{\Delta})}^{(\alpha)}(z,x,\partial_{x})\rho_{\widehat{\Delta},\alpha}\left(x\right)$
(65)
$\displaystyle=\sum_{\widehat{\Delta},\alpha}\frac{\mu_{\widehat{\Delta},\alpha}}{(2z)^{\Delta_{1}-\Delta}}\rho_{\widehat{\Delta},\alpha}\left(x\right)+desc.$

In appendix C, we systematically determine the form of the sum over
descendants, and then perform the resummation to obtain the conformal blocks.
The resummation is technical, and we present the differential equation route
here instead, computing the blocks through the shadow formalism as conformal
integrals Ferrara:1972uy ; Ferrara:1973yt .

The CPWs are uniquely fixed by their transformation properties under the
conformal group and matching with the OPE leading terms. More specifically,
they inherit the transformation properties of the whole correlator, but form
an eigenfunction of the conformal Casimir Dolan:2012wt . As such, any good
ansatz satisfying these conditions will give a representation of the CPW. We
represent the CPWs as conformal integrals – as projections of the conformally-
invariant sewing of lower point functions. These integrals are most easily
evaluated in the embedding space Simmons-Duffin:2014wb . The conformal
integral is given symbolically by

$\displaystyle\int D^{d}X$ (66)

and can integrate weight $-d$ functions in the variable $X^{A}$. Its defining
property is that it forms a conformally invariant integration measure over the
point $X^{A}$. We also encounter the reduced integral with $d\rightarrow
p=d-1$, which is invariant under the conformal group preserved by the
boundary. Any such conformal integral can generically be reduced to a
fundamental case, which is fixed from dimensional analysis

$\int D^{d}X(-2Y\cdot X)^{-d}\propto(-Y\cdot Y)^{-\frac{d}{2}}\ .$

The partial waves then follow from sewing points of simpler correlators using
this integral. One must further take into account a projection $\mathcal{P}$,
and a normalisation $\mathcal{N}$. These are fixed by matching with the bulk
or boundary OPE. The conformal integral yields a mixture of the block
corresponding to the exchange of operator with dimension $\Delta$, as well as
its shadow, which corresponds to the exchange of the unphysical operator with
dimension $d-\Delta$. This second contribution must be projected out. The
normalisation is fixed by matching to the OPE leading term.

One encounters integrals which are a product of power laws with typical
insertions in the numerator of the form $(Z\cdot X)^{l}$. These can be reduced
to the fundamental case given earlier using standard methods. In appendix B,
we explicitly perform this analysis in full generality to compute all possible
conformal integrals for the partial wave type considered in this work.

###  3.2 Conformal Integral for Boundary Block

The boundary blocks are eigenvectors of the boundary conformal Casimir acting
on either of the spinor insertions. We construct an ansatz by sewing together
two bulk to boundary spinor correlators

$\displaystyle\widehat{\mathcal{W}}_{\Delta,\alpha}(P_{1},P_{2},\overline{S}_{1},S_{2})=\frac{\mathcal{P}}{\mathcal{N}}\int
D^{p}X\expectationvalue{\psi(P_{1},\overline{S}_{1})\overline{\rho}_{\Delta}(X)}\not{X}\Pi_{\alpha}\expectationvalue{\rho_{p-\Delta}(X)\overline{\psi}(P_{2},S_{2})}\
.$ (67)

By writing $\rho_{\Delta}(X)$ in place of $\rho_{\Delta,\alpha}(X,S)$, we have
implicitly removed the $S$ dependence from the correlation function. We claim
this object has the correct properties to be a CPW. It is an eigenvector of
the boundary Casimir operator. The extra power of $X_{A}$ is required by
dimensional analysis, and the $\Pi_{\alpha}$ picks out a conformal block of a
particular helicity.

Once this integral is computed, we extract the putative blocks, and from
there, fix the normalisations. These follow from the matching with the OPE in
some limit of the cross-ratio. In the case of the boundary blocks, we have the
normalisation conditions

$\displaystyle\lim_{\xi\rightarrow\infty}\xi^{\Delta+\frac{1}{2}}\hat{f}_{\Delta,\alpha}(\xi)$
$\displaystyle=\frac{1}{2}\ ,$
$\displaystyle\lim_{\xi\rightarrow\infty}\xi^{\Delta+\frac{1}{2}}\hat{g}_{\Delta,\alpha}(\xi)$
$\displaystyle=\frac{-\alpha}{2}\ .$ (68)

The integral is given by

$\displaystyle=\frac{\overline{S}_{1}\Gamma_{A}\Pi_{\alpha}S_{2}}{(2P_{1}\cdot
V)^{\Delta_{1}-\Delta}(2P_{2}\cdot
V)^{\Delta_{2}-\tilde{\Delta}}}\frac{\mathcal{P}}{N}\int
D^{p}X\frac{X^{A}}{(-2P_{1}\bullet X)^{\Delta+\frac{1}{2}}(-2P_{2}\bullet
X)^{\tilde{\Delta}+\frac{1}{2}}}\ ,$ (69)

where $\tilde{\Delta}=p-\Delta$. It is now a straightforward exercise to use
the formulas given in appendix B, plus the normalization conditions, to obtain
a closed form expansion for the boundary blocks. After the dust settles, one
finds the remarkably simple expression for the partial wave

$\displaystyle\widehat{\mathcal{W}}_{\Delta,\alpha}$
$\displaystyle=\frac{1}{2\xi^{\Delta+\frac{1}{2}}}\frac{1}{(2P_{1}\cdot
V)^{\Delta_{1}+\frac{1}{2}}(2P_{2}\cdot V)^{\Delta_{2}+\frac{1}{2}}}$ (70)
$\displaystyle\times\overline{S}_{1}\Bigg{(}\,_{2}F_{1}\left(\Delta+\frac{1}{2},\Delta-\frac{p}{2};1-p+2\Delta;-\frac{1}{\xi}\right)$
$\displaystyle-\alpha\not{V}\,_{2}F_{1}\left(\Delta+\frac{1}{2},1+\Delta-\frac{p}{2};1-p+2\Delta;-\frac{1}{\xi}\right)\Bigg{)}S_{2}\
,$

and the expression for the blocks follows:

$\displaystyle\widehat{f}_{\widehat{\Delta},\alpha}(u)$
$\displaystyle=\frac{1}{2}\xi^{-\widehat{\Delta}-\frac{1}{2}}\,_{2}F_{1}\left(\widehat{\Delta}+\frac{1}{2},\widehat{\Delta}-\frac{p}{2};2\widehat{\Delta}-p+1;-\frac{1}{\xi}\right)\
,$ (71) $\displaystyle\widehat{g}_{\widehat{\Delta},\alpha}(u)$
$\displaystyle=\frac{-\alpha}{2}\xi^{-\widehat{\Delta}-\frac{1}{2}}\,_{2}F_{1}\left(\widehat{\Delta}+\frac{1}{2},1+\widehat{\Delta}-\frac{p}{2};2\widehat{\Delta}-p+1;-\frac{1}{\xi}\right)\
.$ (72)

Closely related expressions appear in Nishida:2018opl ; Herzog:2019bom ;
Kawano:1999au .

Because of the simple $\alpha$ dependance of these blocks, it is convenient to
write the block decomposition using the linear combination that appears in the
first and second structure:

$\displaystyle\sum_{\alpha}\mu_{\widehat{\Delta},\alpha}^{2}\widehat{f}_{\widehat{\Delta},\alpha}(\xi)$
$\displaystyle=\nu_{\widehat{\Delta}}^{(1)}\widehat{f}_{\widehat{\Delta}}(\xi)\
,$ $\displaystyle\nu_{\widehat{\Delta}}^{(1)}$
$\displaystyle=\frac{\mu_{\widehat{\Delta},+}^{2}+\mu_{\widehat{\Delta},-}^{2}}{2}\
,$ (73)
$\displaystyle\sum_{\alpha}\mu_{\widehat{\Delta},\alpha}^{2}\widehat{g}_{\widehat{\Delta},\alpha}(\xi)$
$\displaystyle=\nu_{\widehat{\Delta}}^{(2)}\widehat{g}_{\widehat{\Delta}}(\xi)\
,$ $\displaystyle\nu_{\widehat{\Delta}}^{(2)}$
$\displaystyle=\frac{\mu_{\widehat{\Delta},-}^{2}-\mu_{\widehat{\Delta},+}^{2}}{2}\
.$

###  3.3 Bulk Block From Conformal Integrals

Consider now the bulk CPW. It is by definition an eigenvector of the bulk
conformal Casimir acting on the two spinor insertions. It then follows that it
can be computed through the ansatz

$\displaystyle\mathcal{W}_{\Delta,i}(P_{1},P_{2},\overline{S}_{1},S_{2})=\frac{\mathcal{P}}{\mathcal{N}}\int
D^{d}X\expectationvalue{\psi(P_{1},\overline{S}_{1})\overline{\psi}(P_{2},S_{2})\phi_{\Delta}(X)}_{\circ}^{(i)}\expectationvalue{\phi_{d-\Delta}(X)}\
,$ (74)

with $\expectationvalue{\ldots}_{\circ}^{(i)}$ enumerating the independent
structures in the correlator. Indeed, provided the integration is invariant
under the whole conformal group, as well as the projection, then this object
manifestly satisfies the properties to be a CPW, since the pure CFT three
point functions are eigenvectors of the bulk Casimir operator. One must
further fix the normalization. The CPW asymptotics are constrained by the bulk
OPE, which translate into the conditions

$\displaystyle\lim_{\xi\rightarrow
0}\xi^{\frac{\Delta_{1}+\Delta_{2}+1-\Delta}{2}}f_{\Delta,1}(\xi)$
$\displaystyle=1\ ,$ $\displaystyle f_{\Delta,2}(\xi)$ $\displaystyle=0\ ,$
(75) $\displaystyle g_{\Delta,1}(\xi)$ $\displaystyle=0\ ,$
$\displaystyle\lim_{\xi\rightarrow
0}\xi^{\frac{\Delta_{1}+\Delta_{2}-\Delta}{2}}g_{\Delta,2}(\xi)$
$\displaystyle=1\ .$

With these elements fixed, one can proceed with the evaluation of the
integrals to find the blocks. We now have two different cases to consider. The
first one is $\mathcal{W}_{\Delta,1}$, which is just a straightforward
conformal integral with no insertion

$\displaystyle\mathcal{W}_{\Delta,1}(P_{1},P_{2},\overline{S}_{1},S_{2})$
$\displaystyle=\frac{\overline{S}_{1}S_{2}}{(2P_{1}\cdot
V)^{\Delta_{1}+\frac{1}{2}}(2P_{2}\cdot
V)^{\Delta_{2}+\frac{1}{2}}}\xi^{\frac{\Delta-\Delta_{1}-\Delta_{2}-1}{2}}$
(76)
$\displaystyle\times\,_{2}F_{1}\left(\frac{\Delta+\Delta_{1}-\Delta_{2}}{2},\frac{\Delta-\Delta_{1}+\Delta_{2}}{2};\Delta-\frac{d}{2}+1;-\xi\right)\
.$

For the computation of the second partial wave, $\mathcal{W}_{\Delta,2}$, we
encounter an integral with one insertion in the numerator, whose explicit form
simplifies to give

$\displaystyle\mathcal{W}_{\Delta,2}(P_{1},P_{2},\overline{S}_{1},S_{2})$
$\displaystyle=\frac{\overline{S}_{1}\not{V}S_{2}}{(2P_{1}\cdot
V)^{\Delta_{1}+\frac{1}{2}}(2P_{2}\cdot
V)^{\Delta_{2}+\frac{1}{2}}}\xi^{\frac{\Delta-\Delta_{1}-\Delta_{2}}{2}}$ (77)
$\displaystyle\times\,_{2}F_{1}\left(\frac{\Delta+1+\Delta_{1}-\Delta_{2}}{2},\frac{\Delta+1-\Delta_{1}+\Delta_{2}}{2};\Delta-\frac{d}{2}+1;-\xi\right)\
.$

##  4 Solutions to Crossing Symmetry

We have derived the block decomposition of the two point function of bulk
spinor fields. The constraint of crossing symmetry is the equality

$\displaystyle\sum_{\Delta}\begin{pmatrix}[2]\lambda a_{\Delta}^{(1)}\
f_{\Delta}^{(\Delta_{1},\Delta_{2})}(\xi)\\\ \lambda a_{\Delta}^{(2)}\
g_{\Delta}^{(\Delta_{1},\Delta_{2})}(\xi)\end{pmatrix}=\sum_{\widehat{\Delta}}\begin{pmatrix}[2]\nu^{(1)}_{\widehat{\Delta}}\
\widehat{f}_{\widehat{\Delta}}(\xi)\\\ \nu^{(2)}_{\widehat{\Delta}}\
\widehat{g}_{\widehat{\Delta}}(\xi)\end{pmatrix}$ (78)

with functions given by

$\displaystyle f_{\Delta}^{(\Delta_{1},\Delta_{2})}(\xi)$
$\displaystyle=\sqrt{\xi}^{\Delta-\Delta_{1}-\Delta_{2}-1}\,_{2}F_{1}\left(\frac{\Delta+\Delta_{1}-\Delta_{2}}{2},\frac{\Delta-\Delta_{1}+\Delta_{2}}{2};\Delta+\frac{1-p}{2};-\xi\right)\
,$ (79) $\displaystyle g_{\Delta}^{(\Delta_{1},\Delta_{2})}(u)$
$\displaystyle=\sqrt{\xi}^{\Delta-\Delta_{1}-\Delta_{2}}\,_{2}F_{1}\left(\frac{\Delta+\Delta_{1}-\Delta_{2}+1}{2},\frac{\Delta-\Delta_{1}+\Delta_{2}+1}{2};\Delta+\frac{1-p}{2};-\xi\right)\
,$ $\displaystyle\widehat{f}_{\widehat{\Delta}}(\xi)$
$\displaystyle=\frac{1}{\sqrt{\xi}^{2\widehat{\Delta}+1}}\,_{2}F_{1}\left(\widehat{\Delta}+\frac{1}{2},\widehat{\Delta}-\frac{p}{2};2\widehat{\Delta}-p+1;-\frac{1}{\xi}\right)\
,$ $\displaystyle\widehat{g}_{\widehat{\Delta}}(\xi)$
$\displaystyle=\frac{1}{\sqrt{\xi}^{2\widehat{\Delta}+1}}\,_{2}F_{1}\left(\widehat{\Delta}+\frac{1}{2},1+\widehat{\Delta}-\frac{p}{2};2\widehat{\Delta}-p+1;-\frac{1}{\xi}\right)\
.$

In this section, we explore this constraint in the context of the
$\epsilon$-expansion of interacting QFT. We consider situations with
$\Delta_{1}=\Delta_{2}=\Delta_{\psi}$. To simplify the notation, we drop
redundant information, abbreviating
$f^{(\Delta_{\psi},\Delta_{\psi})}_{\Delta}(\xi)\equiv f_{\Delta}$ and
$g^{(\Delta_{\psi},\Delta_{\psi})}_{\Delta}(\xi)\equiv g_{\Delta}$. Moreover,
in this special case $\Delta_{1}=\Delta_{2}$, we can use (73) to write the
$\nu_{\widehat{\Delta}}^{(i)}$ in terms of the manifestly non-negative
quantities $\mu_{\widehat{\Delta},\alpha}^{2}$.

The starting point of our analysis is a free spinor. We follow with the
generalized free field (GFF) version of the spinor, i.e. a spinorial field
with a generalized dimension but whose correlation functions still follow from
Wick’s Theorem. We then consider three different interacting models generated
by classically marginal deformations near dimensions $4$, $3$ and $2$:

  * •

The Yukawa model, $\mathcal{L}_{\rm int}=-g\phi\overline{\psi}\psi$, which
describes the interaction of a fermion with a scalar field, which we study in
$d=4-\epsilon$. At first order in $\epsilon$, the addition of the field $\phi$
to the bulk OPE induces a reorganisation of the field content through a fake-
primary effect.

  * •

The Gross-Neveu model, $\mathcal{L}_{\rm
int}=\frac{g}{2}(\overline{\psi}_{a}\psi^{a})^{2}$, of an interacting $O(N)$
vector of fermions in $d=2+\epsilon$. In this model at the fixed point one has
$g\sim\epsilon$, which allows us to fix the correlation function to order
$\epsilon^{2}$. Surprisingly, this decomposition also fixes the value of the
zero of the beta function, although it leaves unfixed one bulk anomalous
dimension.

  * •

A Yukawa-type model, $\mathcal{L}_{\rm
int}=-\frac{g}{2}\phi_{a}\phi^{a}\overline{\psi}\psi$, which describes a
fermion interacting with an $O(N_{s})$-vector of scalar fields. In
$d=3-\epsilon$, this model remarkably generates an expansion in
$\sqrt{\epsilon}$ rather than $\epsilon$ for boundary data. Similar behavior
was remarked for the $\phi^{6}$ scalar field theory in $3-\epsilon$ dimensions
diehi1987walks ; eisenriegler1988surface . More surprisingly still, crossing
symmetry of $\expectationvalue{\psi\overline{\psi}}$ does not fully fix the
boundary anomalous dimension at the next order. This ambiguity is resolved by
bootstrapping the leading order deformation of the scalar-scalar bulk
correlator.

In each case, we reproduce data concerning the bulk critical point from this
boundary bootstrap. We employ the same methodology for each interacting model.
Starting from the equation of motion applied to the correlation function, we
find differential equations for $F(\xi)$, $G(\xi)$. We solve them
perturbatively in $\epsilon$, up to integration constants. This ansatz is then
decomposed on the bulk and boundary block basis. Compatibility fixes the
integration constants and allows us to read off the anomalous dimensions and
field content of the theory.

###  4.1 (Gaussian) Free Theory

Let us first consider the free theory

$\displaystyle\expectationvalue{\psi(x)\overline{\psi}(x^{\prime})}$
$\displaystyle=\frac{\bra{1}(x-x^{\prime})\ket{2}}{\absolutevalue{x-x^{\prime}}^{\frac{p+1}{2}}}+\chi\frac{\bra{1}n(x^{\prime}-\tilde{x})\ket{2}}{\absolutevalue{\tilde{x}-x^{\prime}}^{\frac{p+1}{2}}}$
(80)
$\displaystyle=\mathcal{W}_{0}^{(0)}+\chi\mathcal{W}_{p}^{(1)}=(1-\chi)\widehat{\mathcal{W}}_{\frac{p}{2},+}+(1+\chi)\widehat{\mathcal{W}}_{\frac{p}{2},-}\
,$

where the boundary condition on the spinor is imposed through a choice of
$\chi$. The second line solves the crossing symmetry constraint. In terms of
the functions $F(\xi)$ and $G(\xi)$, we have more precisely

$\displaystyle\begin{pmatrix}[2]F(\xi)\\\ G(\xi)\end{pmatrix}$
$\displaystyle=\begin{pmatrix}[2]\xi^{-\frac{p+1}{2}}\\\
\chi(1+\xi)^{-\frac{p+1}{2}}\end{pmatrix}=\begin{pmatrix}[2]f_{0}\\\
\chi\,g_{p}\end{pmatrix}=\begin{pmatrix}[2]\widehat{f}_{\frac{p}{2}}\\\
\chi\,\widehat{g}_{\frac{p}{2}}\end{pmatrix}\ .$ (81)

The positivity of the boundary blocks constrains $\chi$,

$\displaystyle\absolutevalue{\mu_{\frac{p}{2},\alpha}}^{2}\geq
0\Rightarrow-1\leq\chi\leq 1\ .$ (82)

Indeed, a very similar constraint appears when looking at the boundary blocks
for a free scalar. In both cases, the choices $\chi=\pm 1$ play a special
role. In the scalar case, they select for either Dirichlet or Neumann boundary
conditions. In the spinor case, decomposing the boundary value of $\psi$ into
boundary spinors of plus or minus chirality with respect to $\gamma_{z}$, they
set one of the boundary spinors $\rho_{\pm}$ to zero.

We can use Wick contraction to explain the operators that show up in the bulk
decomposition:

$\displaystyle\psi(x)\overline{\psi}(x^{\prime})$
$\displaystyle=\wick{\c{\psi}(x)\overline{\c{\psi}}(x^{\prime})}+{:}\psi(x)\overline{\psi}(x^{\prime})$
$\displaystyle=\frac{\gamma\cdot(x-x^{\prime})}{\absolutevalue{x-x^{\prime}}^{p+1}}-\frac{\mathbb{1}}{\Tr(\mathbb{1})}{:}\overline{\psi}(x)\psi(x){:}-\frac{\gamma^{\mu}}{p+1}{:}\overline{\psi}(x)\gamma_{\mu}\psi(x){:}+\ldots$

where we used the generalized Fierz identity on the spinor bilinear and Taylor
expanded the second insertion to bring it to a coincident point. The ellipsis
denotes terms depending on derivatives and possibly higher powers of the gamma
matrices. Because we are in a bCFT, the one point functions of all operators
with nonzero spin vanish. Indeed only the first two terms survive after taking
an expectation value. As there is a unique dimension zero operator, the
identity, in a CFT, we already knew the origin of $\mathcal{W}_{0}^{(0)}$.
However, now we see very clearly that $\mathcal{W}_{p}^{(1)}$ corresponds to
the mass operator ${:}\overline{\psi}\psi{:}$.

The richer case of a GFF spinor follows straightforwardly. The two point
function is taken to be instead

$\displaystyle\expectationvalue{\psi(x)\overline{\psi}(x^{\prime})}$
$\displaystyle=\frac{\bra{1}(x-x^{\prime})\ket{2}}{\absolutevalue{x-x^{\prime}}^{2\Delta+1}}+\chi\frac{\bra{1}n(x^{\prime}-\tilde{x})\ket{2}}{\absolutevalue{\tilde{x}-x^{\prime}}^{2\Delta+1}}\
,$ (83)

corresponding to the bulk decomposition

$\displaystyle F(\xi)$ $\displaystyle=f_{0}\ ,$ (84) $\displaystyle G(\xi)$
$\displaystyle=\chi\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!}\frac{\left(\Delta+\frac{1}{2}\right)_{k}\left(\Delta-\frac{p}{2}\right)_{k}}{\left(2\Delta-\frac{p+1}{2}+k\right)_{k}}\,g_{2\Delta+2k}\
.$ (85)

The bulk blocks in the second structure correspond naturally to the exchange
of double trace operators of the schematic form
${:}\overline{\psi}\Box^{k}\psi{:}$. In the limit
$\Delta\rightarrow\frac{p}{2}$, we retrieve the free theory result of a single
contribution from the spinor bilinear, while the remaining operators become
null. This example clearly show the non-positivity of the coefficients
entering the bulk block expansion.

In the boundary channel, we find a sum over an infinite tower of single trace
operators, with schematic form ${:}\partial_{n}^{k}\rho_{\pm}{:}$

$\displaystyle F(\xi)$
$\displaystyle=\sum_{k=0}^{\infty}\frac{1}{k!}\frac{\left(\Delta+\frac{1}{2}\right)_{k}(2\Delta-p)_{k}}{4^{k}\left(\Delta-\frac{p-1}{2}\right)_{k}}\,\widehat{f}_{\Delta+k}\
,$ (86) $\displaystyle G(\xi)$
$\displaystyle=\chi\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!}\frac{\left(\Delta+\frac{1}{2}\right)_{k}(2\Delta-p)_{k}}{4^{k}\left(\Delta-\frac{p-1}{2}\right)_{k}}\,\widehat{g}_{\Delta+k}\
,$ (87)

where now, in the free field limit, the Dirac equation imposes
$\partial_{n}\rho_{+}=-\gamma^{i}\partial_{i}\rho_{-}$, making all fields,
except the leading ones, descendant. As previously, reality of the expansion
coefficients imposes $\chi^{2}\leq 1$.

Starting from free theory, one can try to perform a slight perturbation to the
operator spectrum of the theory and its CFT data, to explore a conformal
manifold. Generically, it is not so easy to find such manifolds. In bCFT, the
parameter $\chi$ seems to give us an example of a marginal parameter. However,
one expects that $\chi$ is fixed to $\pm 1$ in a free theory from a stress-
tensor Ward Identity Herzog:2021spv . To find continuous deformations away
from mean-field theory which satisfy crossing symmetry, it is then convenient
to perturb more than the field content. One path to a non-trivial deformation
is the $\epsilon$-expansion, where one analytically continues the dimension
$d$, or $p=d-1$, to non-integer values. This deformation gives us a parameter
with which to explore points away from these Gaussian systems. Our goal for
the remainder of this section is to explore some examples of solutions to
crossing symmetry in the $\epsilon$-expansion.

###  4.2 The Yukawa Model at $1$-loop

We now turn to a more interesting system, the Yukawa Model with a boundary.
Our starting point is to assume there exists a perturbative fixed point,
$g^{2}\sim\epsilon$, and draw the consequences from the equations of motions
at leading order in $\epsilon$. This is analogous to the computation of
Giombi:2021cnr . The generic form of the two point function is as before

$\displaystyle\expectationvalue{\psi(x)\overline{\psi}(x^{\prime})}$
$\displaystyle=\frac{F(\xi)\bra{1}(x-x^{\prime})\ket{2}+G(\xi)\bra{1}n(x^{\prime}-\tilde{x})\ket{2}}{(4x\cdot
n\,x^{\prime}\cdot n)^{\Delta_{\psi}+\frac{1}{2}}}\ ,$ (88)
$\displaystyle\not{\partial}_{x}\expectationvalue{\psi(x)\overline{\psi}(x^{\prime})}\overset{\leftarrow}{\not{\partial}}_{x^{\prime}}$
$\displaystyle=\frac{\mathcal{D}_{F}\bra{1}(x-x^{\prime})\ket{2}+\mathcal{D}_{G}\bra{1}n(x^{\prime}-\tilde{x})\ket{2}}{(4x\cdot
n\,x^{\prime}\cdot n)^{\Delta_{\psi}+\frac{1}{2}}}\ ,$

where the differentiated blocks are given by

$\displaystyle\mathcal{D}_{G}$
$\displaystyle=-4\left(\xi(1+\xi)F^{\prime\prime}(\xi)+\left(\frac{p+3}{2}+\left(p+2\right)\xi\right)F^{\prime}(\xi)+\left(\Delta_{\psi}+\frac{1}{2}\right)\left(p-\Delta_{\psi}+\frac{1}{2}\right)F(\xi)\right)\
,$ (89) $\displaystyle\mathcal{D}_{G}$
$\displaystyle=-4\left(\xi(1+\xi)G^{\prime\prime}(\xi)+\left(\frac{p+1}{2}+(p+2)\xi\right)G^{\prime}(\xi)+\left(\Delta_{\psi}+\frac{1}{2}\right)\left(p-\Delta_{\psi}+\frac{1}{2}\right)G(\xi)\right)\
.$

Replacing the explicit form of the Yukawa interaction inside the correlator,
and evaluating the resulting four point function using Wick contractions, we
find

$\displaystyle\not{\partial}_{x}\expectationvalue{\psi(x)\overline{\psi}(x^{\prime})}\overset{\leftarrow}{\not{\partial}}_{x^{\prime}}$
$\displaystyle=-g^{2}\expectationvalue{\phi(x)\phi(x^{\prime})}\expectationvalue{\psi(x)\overline{\psi}(x^{\prime})}$
(90)
$\displaystyle=-\frac{g^{2}H(\xi)F(\xi)\bra{1}(x-x^{\prime})\ket{2}+g^{2}H(\xi)G(\xi)\bra{1}n(x^{\prime}-\tilde{x})\ket{2}}{(4x\cdot
n\,x^{\prime}\cdot n)^{\Delta_{\psi}+\frac{1}{2}+\Delta_{\phi}}}\ .$

We will take $\kappa_{s}g^{2}\equiv\lambda\epsilon$, and use the free field
propagator for $H(\xi)$. The anomalous dimension one finds for the spinor will
enter only through the explicit factors of $\Delta_{\psi}$ in
$\mathcal{D}_{F/G}$. The scalar field two point function depends on a
reflection coefficient $\chi_{s}$,

$H(\xi)=\frac{1}{\xi}+\frac{\chi_{s}}{1+\xi}\ .$ (91)

The equations of motion simplify to two ordinary differential equations:

$\displaystyle\mathcal{D}_{F}$ $\displaystyle=-\epsilon\lambda H(\xi)F(\xi)\
,$ (92) $\displaystyle\mathcal{D}_{G}$ $\displaystyle=-\epsilon\lambda
H(\xi)G(\xi)\ ,$

which we solve perturbatively in $\epsilon$, setting
$\Delta_{\psi}=\frac{p}{2}+\gamma_{\psi}^{(1)}\epsilon+O(\epsilon^{2})$. At
each order in $\epsilon$, we need to specify two boundary conditions for $F$
and two for $G$. At zeroth order, the boundary conditions mean setting the
correlation function to be the free theory result. To $O(\epsilon^{2})$, the
solution is

$\displaystyle F(\xi)$
$\displaystyle=\frac{1}{\xi^{2}}+\frac{\epsilon}{\xi^{2}}\bigg{(}\delta\kappa-\frac{1}{2}\left(\frac{1}{1+\xi}+\log(1+\frac{1}{\xi})\right)+c_{1}\left(\frac{1}{1+\xi}+\log(1+\xi)\right)$
(93)
$\displaystyle\quad\quad\quad\quad\quad\quad+\frac{\lambda}{8}\left(\frac{1+2\chi_{s}}{1+\xi}+\log(1+\frac{1}{\xi})\right)\bigg{)}+O(\epsilon^{2})\
,$ $\displaystyle G(\xi)$
$\displaystyle=\frac{\chi}{(1+\xi)^{2}}+\frac{\epsilon\chi}{(1+\xi)^{2}}\bigg{(}\delta\chi+\frac{1}{2}\left(\frac{1}{\xi}+\log(1+\frac{1}{\xi})\right)-c_{2}\left(\frac{1}{\xi}-\log(\xi)\right)+$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+\frac{\lambda}{8}\left(\frac{2+\chi_{s}}{\xi}+\chi_{s}\log(1+\frac{1}{\xi})\right)\bigg{)}+O(\epsilon^{2})\
.$

with integration constants $\delta\kappa$, $\delta\chi$, $c_{1}$, and $c_{2}$.
The constant $\delta\kappa$ renormalizes the contribution from the identity
block, while the difference between $\delta\chi$ and $\delta\kappa$ controls
shifts of the boundary condition.

The task at hand is now to fix these integration constants by comparison with
the boundary and bulk OPE’s of the spinor correlation function. These OPEs
depend on a number of undetermined coefficients as well: the anomalous
dimensions and OPE coefficients. Nevertheless, we will see that the comparison
fixes all but a few of these unknown quantities. The remaining quantities in
the cases we study can then be fixed by studying the corresponding CFT without
boundary. For example, here we are unable to determine $\lambda$ through the
crossing constraint. The value of $\lambda$ is the same whether or not there
is a boundary however, and hence this value is easily looked up in the
literature Fei:2016sgs ; Zerf:2017zqi .

We start with the boundary channel and the function $F(\xi)$. The interaction
gives an anomalous dimension to the fields present in the free theory.
Moreover the anomalous dimension of the bulk field converts it essentially to
a GFF at this order in the $\epsilon$-expansion, which means we expect to find
at order $\epsilon$ a tower (86) of spinors on the boundary. One can
explicitly verify the matching

$\displaystyle
F(\xi)=(1+\delta\kappa\,\epsilon)\,\widehat{f}_{\frac{p}{2}+\gamma_{\rho}}+\frac{\epsilon\lambda}{4}\left(\chi_{s}\,\widehat{f}_{\frac{3}{2}+1}+\sum_{k=1}^{\infty}\frac{\Gamma(k)\Gamma(2+k)}{\Gamma(1+2k)}\,\widehat{f}_{\frac{3}{2}+k}\right)\
,$ (94)

which allows us to fix $c_{1}=\frac{1}{2}-\gamma_{\rho}^{(1)}$. At a technical
level, we are fixing the anomalous dimension for $\rho$ by matching the
coefficient of a logarithm that appears in (93) with a logarithm that appears
in the boundary OPE decomposition. The matching requires expanding a
hypergeometric function in its indices. A simple approach which works here is
to expand the hypergeometric first in a near boundary limit and then to expand
the Taylor series coefficients as functions of the indices of the
hypergeometric. (Later for the bulk blocks, one can replace a near boundary
limit with a coincident limit.)

We choose to write the decomposition (94) this way to bring some attention to
the final sum. As in the GFF model, the single trace operators are of the form
$\partial_{n}^{k}\rho$. At level $k=1$, however, we also have the operator
${:}\phi\rho{:}$. The primary field exchanged is made from a mixing of the
two. This is how we understand this decomposition of the $k=1$ single-trace
operator data into two contributions, one of which is coming from the $k=1$
term in the sum.

To compare the tower with the GFF model (86) we considered above, take
$\Delta=\frac{p}{2}+\gamma_{\psi}^{(1)}\epsilon$ in the decomposition. At
leading order in epsilon we find

$\displaystyle\lim_{\Delta\to\frac{p}{2}+\gamma_{\psi}^{(1)}\epsilon}F_{GFF}(\xi)$
$\displaystyle=\widehat{f}_{\Delta_{\psi}}+2\gamma_{\psi}^{(1)}\epsilon\sum_{k=1}^{\infty}\frac{\Gamma(k)\Gamma(2+k)}{\Gamma(1+2k)}\times\widehat{f}_{\frac{p}{2}+k}+\mathcal{O}(\epsilon^{2})\
,$ (95)

from which it is clear that this tower of operators is that expected
perturbatively from the GFF result.

As we obtained on the boundary the infinite sum (86) of a GFF, we expect to
find something like (84) and (85) in the bulk as well. For $F$, the GFF
decomposition in the bulk is extremely simple, as there is only the identity,
whose dimension is protected. We fix its normalisation to $1$, which in turn
fixes
$\delta\kappa=\gamma_{\rho}^{(1)}-\frac{\lambda\left(1+2\chi_{\phi}\right)}{8}$.
We are able to use a finite number of manually adjusted blocks. One has

$\displaystyle F(\xi)$ $\displaystyle=1\times
f_{0}+\epsilon\bigg{(}\underbrace{-\frac{\lambda\chi_{s}}{4}f_{2}}_{:\phi^{2}:}+\underbrace{\frac{\lambda+8\gamma_{\rho}^{(1)}}{16}f_{4}}_{:\phi\bar{\psi}\psi:}\bigg{)}\
.$ (96)

The matching of the logarithm from the ansatz and the block imposes the
condition

$\displaystyle\gamma_{\psi}^{(1)}=\frac{\lambda}{8}\ ,$ (97)

our first anomalous dimension! This result matches the literature Fei:2016sgs
; Zerf:2017zqi , but we have determined it in a new way.

We can now turn to the decomposition on the boundary of the $G(\xi)$ function.
The procedure is analogous to the previous one

$\displaystyle G(\xi)$
$\displaystyle=\chi(1+\delta\chi\,\epsilon)\,\widehat{g}_{\frac{p}{2}+\gamma_{\rho}^{(1)}\epsilon}+\frac{\epsilon\chi\lambda}{4}\left(\widehat{g}_{\frac{3}{2}+1}-\chi_{s}\sum_{k=1}^{\infty}(-1)^{k}\frac{\Gamma(k)\Gamma(2+k)}{\Gamma(1+2k)}\,\widehat{g}_{\frac{3}{2}+k}\right)\
,$ (98)

which fixes an integration constant $c_{2}=\frac{1}{2}-\gamma_{\rho}^{(1)}$.
Requiring that the boundary condition of the fermion is preserved as we crank
up the interaction fixes $\delta\kappa=\delta\chi$ by comparing the
coefficients multiplying the leading boundary spinor blocks in each structure.
As previously, the coefficients of the tower are easily understood by
comparing with the GFF limit

$\displaystyle\lim_{\Delta_{\psi}\rightarrow\frac{p}{2}+\gamma_{\psi}^{(1)}\epsilon}G_{GFF}(\xi)=\chi\,\widehat{g}_{\Delta_{\psi}}+\chi
2\gamma_{\psi}^{(1)}\epsilon\sum_{k=1}^{\infty}(-1)^{k}\frac{\Gamma(k)\Gamma(2+k)}{\Gamma(1+2k)}\,\widehat{g}_{\frac{3}{2}+k}\
.$ (99)

Finally, we can consider the bulk decomposition of $G(\xi)$. This last
matching is more delicate than the others. Taylor expanding $G(\xi)$ around
$\xi=0$, one encounters a term of order $\xi^{-1}$, which would correspond to
a block with $\Delta=1$, the free scalar $\phi$. However, such a block naively
diverges, because it has a simple pole associated to the exchange of the null
descendant $\Box\phi$ Penedones:2015aga . More explicitly, by considering the
form of the block and using a hypergeometric identity we find

$\displaystyle g_{\Delta}(\xi)$
$\displaystyle=\frac{(\Delta+1)^{2}}{4\left(\frac{p-3}{2}-\Delta\right)}\times\frac{g_{\Delta+2}(\xi)}{\left(\Delta-\frac{p-1}{2}\right)}$
(100)
$\displaystyle+\xi^{\frac{\Delta}{2}-\Delta_{\psi}}\,{}_{2}F_{1}\left(\frac{\Delta+1}{2},\frac{\Delta+1}{2};1+\Delta-\frac{p-1}{2};-\xi\right)\
.$

In the free-field limit, $\Delta=\frac{p-1}{2}$, the first term gives a
singular contribution, while the second term is finite. From the equations of
motions however, we expect $\Box\phi\sim\overline{\psi}\psi$; hence $\phi$
will gain an anomalous dimension,
$\Delta_{\phi}=\frac{p-1}{2}+\gamma_{\phi}^{(1)}\epsilon$, and
$\bar{\psi}\psi$ becomes a descendant Fei:2016sgs . The wrong thing to do
would be to assume $\overline{\psi}\psi$ remains a primary of dimension nearly
three. This situation is often called the “fake primary effect”, where a
primary field at threshold signals itself through a block which corresponds to
the exchange of its null-descendant. This effect was noticed and explained in
applications of the numerical bootstrap Karateev:2019pvw .

The diverging first term of (100), in the limit $\epsilon\rightarrow 0$,
reproduces the contribution from a primary with dimension $3$ instead of $1$.
The logarithmic contributions of (93) can now be matched to the bulk OPE
expansion, and one finds

$\displaystyle G(\xi)$
$\displaystyle=\chi\left(-\gamma_{\phi}^{(1)}\epsilon+\nu_{\phi}\epsilon^{2}\right)\,g_{\Delta_{\phi}}-\frac{\epsilon\lambda\chi\chi_{s}}{8}\sum_{k=1}^{\infty}(-1)^{k}\frac{\Gamma(k)\Gamma(k+2)}{\Gamma(1+2k)}\,g_{3+2k}$
(101)

with the identifications

$\displaystyle\gamma_{\rho}^{(1)}$
$\displaystyle=-\frac{1}{2}+\frac{\lambda(4-\chi_{\phi})}{8}\
,\;\;\;\gamma_{\phi}^{(1)}=\frac{1}{2}-\frac{3}{4}\lambda\ ,$ (102)
$\displaystyle\nu_{\phi}$
$\displaystyle=-\gamma_{\phi}^{(2)}+\frac{1}{2}-\frac{1}{32}(3\lambda-2)(3\lambda(1+\chi_{s})-4)\
.$ (103)

We have computed two more anomalous dimensions as a function of the coupling
$\lambda$! The result for $\gamma_{\phi}^{(1)}$ matches the literature
Fei:2016sgs ; Zerf:2017zqi , while the result for $\gamma_{\rho}^{(1)}$
matches the more recent Giombi:2021cnr , but our methods are different. In
their work, the bulk anomalous coefficients are used to extrapolate the
boundary data, while we effectively derive both at the same time.

Note that the tower of double trace operators in (101) furthermore has a
structure that matches (85) in an $\epsilon$-expansion with
$\Delta_{\psi}=\frac{p}{2}+\epsilon\gamma_{\psi}^{(1)}$.

###  4.3 Coupling to a scalar undergoing an extraordinary phase transition

We have studied the coupling of fermions to a scalar field. Notoriously,
scalar fields in the presence of a boundary admits multiple interesting
phases, one of which is the extraordinary transition. In this configuration,
the scalar field develops an expectation value Diehl:1996kd , contrasting
starkly with the Dirichlet and Neumann boundary conditions that we just
considered. Let us investigate the effect of the extraordinary transition for
$\phi$ in the Yukawa model. Treating the expectation value of the scalar as a
background field for the fermion, the spinor equation of motion is effectively
modified to

$\left(\not{\partial}\psi=-g\phi\psi\right)\rightarrow\left(\not{\partial}\psi=-\frac{\mu}{x\cdot
n}\psi\right)\ ,$ (104)

where $\mu=\frac{ga_{\phi}}{2}$ is related to the expectation value of the
scalar $\phi$. We obtain an effective position dependent mass term for the
fermion. If we were to try to write an action for such a fermion with a
position dependent mass (in absence of the scalar), we would find that the
fermion is sensitive to the breaking of the bulk conformal invariance
everywhere and not just through its boundary condition at $z=0$. By an abuse
of language, we might call this sensitivity spontaneous breaking of local bulk
conformal invariance. We call this breaking spontaneous because the full
theory, including the scalar, remains sensitive to the breaking of bulk
conformal invariance only through the boundary condition. We call the
invariance local to emphasize that the boundary condition anyway breaks it. In
the AdS picture, Weyl rescaling changes the position dependent mass to a
constant mass, and the situation is reminiscent of the Higgs mechanism
Carmi:2018qzm ; Herzog:2019bom . Indeed, generically for QFT in AdS, there is
no bulk conformal invariance.

Indeed, bulk conformal invariance is an extra assumption. The natural form of
conformal invariance in the presence of a boundary is $SO(1,p+1)$ invariance.
That the bulk itself is invariant under the full $SO(1,d+1)$ far away from the
boundary is a supplementary assumption. This is obvious in the AdS picture,
where a generic QFT in AdS generates correlation functions satisfying boundary
conformal invariance, although the bulk itself will not be a CFT.

Given this modified equation of motion, we would like to determine the
boundary OPE of the spinor field $\psi$ through analyzing its correlation
function with boundary operators

$\displaystyle\left(\gamma^{\nu}\partialderivative{x^{\nu}}+\frac{\mu}{z}\right)\expectationvalue{\psi(x,z)\overline{\rho}_{\alpha}(y)}=0\Rightarrow\begin{cases}\Delta_{\psi}&=\frac{p}{2}\\\
\Delta_{\rho,\alpha}&=\frac{p}{2}-\alpha\mu\end{cases}\ ,$ (105)

The bulk field has a free-field dimension, as needed for its descendant to be
a (null) primary field. The boundary spinors $\rho_{\pm}$ now have different
dimensions, but they are still shadow dual in the sense that their dimensions
sum to $p$. (The scalar field case is very similar Herzog:2019bom ;
Behan:2020vy .) Noting that there is a unitarity bound at
$\widehat{\Delta}\geq\frac{p-1}{2}$, we find the usual bound of
$\absolutevalue{\mu}\leq\frac{1}{2}$, where alternate quantisation of the
spinor in AdS is allowed. For $\absolutevalue{\mu}>\frac{1}{2}$, there will be
a single quantisation scheme. One can then use the block decomposition to find
the two point function this field content induces

$\displaystyle\expectationvalue{\psi(x)\overline{\psi}(x^{\prime})}=\nu_{+}\widehat{\mathcal{W}}_{\frac{p}{2}-\mu,+}+\nu_{-}\widehat{\mathcal{W}}_{\frac{p}{2}+\mu,-}\
.$ (106)

Up to a power law prefactor, the same expression appears in computing the
propagator of a free massive spinor field in AdS (see appendix E). If we
further impose boundary conditions on the spinor field, we can restrict to
only one helicity. Normalizing the coincident limit, one fixes

$\displaystyle\expectationvalue{\psi(x)\overline{\psi}(y)}=\frac{2\Gamma(\frac{p+1}{2}-\mu\alpha)\Gamma(1-\mu\alpha)}{\Gamma(\frac{p+1}{2})\Gamma(1-2\mu\alpha)}\widehat{\mathcal{W}}_{\frac{p}{2}-\alpha\mu,\alpha}\
.$ (107)

Unitarity places constraints on when this boundary conformal partial wave can
be expressed as a sum over bulk conformal partial waves. If we look at the
boundary $\hat{g}_{\frac{p}{2}-\alpha\mu}(\xi)$ conformal block in the
coincident limit, we find a sum of two power series in $\xi$, the first with
leading term $\xi^{\frac{1}{2}-\frac{p}{2}}$ and the second with constant
leading term. The more singular power series with the
$\xi^{\frac{1}{2}-\frac{p}{2}}$ behaviour is absent precisely in the case
$\mu=0$. Furthermore, the more singular power series requires the existence of
a bulk operator of dimension one in order to be expressed as a sum over bulk
conformal blocks. However, this dimension is only consistent with the bulk
unitarity bound when $d\leq 4$. In fact, in the case $d=4$, this bulk scalar
must be free and thus it cannot appear in the bulk OPE of the two spinor
operators. By this argument, crossing symmetry for such a spinor with a
position dependent mass term can only be consistent with unitarity in the case
$d<4$. (A similar argument looking at $\hat{f}_{\frac{p}{2}-\alpha\mu}(\xi)$
gives a weaker bound, that $d<5$.)

###  4.4 The Gross-Neveu Model at 2-loops

We now focus the Gross-Neveu Model in the $\epsilon$ expansion $d=2+\epsilon$,
up to order $\epsilon^{2}$. We will consider a model of $N$ fermions, with
flavour index $a,b$.

The starting point of our analysis is the equation of motion

$\displaystyle\not{\partial}\psi_{a}(x)=g\left(\sum_{b}\bar{\psi}_{b}\psi_{b}\right)\psi_{a}(x)\
.$ (108)

At the fixed point, we anticipate
$g=\frac{\lambda\epsilon}{\kappa_{f}}+\mathcal{O}(\epsilon^{2})$. Acting on
the two point function at leading order, we obtain the system of equations

$\displaystyle\not{\partial}\expectationvalue{\psi_{a}(x)\overline{\psi}_{b}(x^{\prime})}$
$\displaystyle=-\epsilon\left(2N-1\right)\frac{\lambda\chi}{2x\cdot
n}\expectationvalue{\psi_{a}(x)\overline{\psi}_{b}(x^{\prime})}\ .$ (109)

In terms of the functions $F(\xi)$ and $G(\xi)$, the equation of motion
translates into the differential equations

$\displaystyle 2\xi
F^{\prime}(\xi)+\left(1+p\right)F(\xi)+\epsilon\lambda\chi\left(2N-1\right)G(\xi)$
$\displaystyle=0\ ,$ (110) $\displaystyle
2(1+\xi)G^{\prime}(\xi)+\left(1+p\right)G(\xi)+\epsilon\lambda\chi\left(2N-1\right)F(\xi)$
$\displaystyle=0\ .$ (111)

In order to be able to write the two-point function purely in terms of
$F(\xi)$ and $G(\xi)$ (and hence preserve conformal invariance), we find at
this stage that we must set $\Delta_{\psi}=\frac{p}{2}+O(\epsilon^{2})$. We
solve this system in an $\epsilon$-expansion, and find

$\displaystyle F(\xi)$
$\displaystyle=\frac{1}{\xi}+\frac{\epsilon}{\xi}\left(\delta\kappa+\frac{\lambda(1-2n)}{2}\log(1+\xi)-\frac{1}{2}\log(\xi)\right)\
,$ (112) $\displaystyle G(\xi)$
$\displaystyle=\frac{\chi}{(1+\xi)}+\frac{\epsilon}{(1+\xi)}\left(\delta\chi+\chi\left(\frac{\lambda(1-2n)}{2}\log(\xi)-\frac{1}{2}\log(1+\xi)\right)\right)\
.$ (113)

There are two integration constants, $\delta\kappa$ associated with the
normalization of the coincident limit, and $\delta\chi$ associated with the
choice of boundary condition for the spinor.

These functions admit an expansion in terms of bulk and boundary blocks. In
fact, one only needs a finite number of blocks in each channel to reproduce
$F(\xi)$ and $G(\xi)$ at $O(\epsilon)$

$\displaystyle\begin{pmatrix}F(\xi)\\\
G(\xi)\end{pmatrix}=\begin{pmatrix}(1+\delta\kappa\,\epsilon)f_{0}+\epsilon\lambda\frac{(1-2n)}{2}f_{2}\\\
(\chi+\delta\chi\epsilon)g_{p+\gamma_{\sigma}^{(1)}\epsilon}\end{pmatrix}=\begin{pmatrix}(1+\delta\kappa\,\epsilon)\widehat{f}_{\frac{p}{2}+\gamma_{\rho}^{(1)}\epsilon}\\\
(\chi+\delta\chi\epsilon)\widehat{g}_{\frac{p}{2}+\gamma_{\rho}^{(1)}\epsilon}\end{pmatrix}\
.$ (114)

We denote by $\sigma\equiv\overline{\psi}\psi$ the mass operator in this
theory, which has dimension nearly one. The bulk field of dimension two is
then naturally interpreted as $\sigma^{2}\equiv(\overline{\psi}\psi)^{2}$. The
matching works provided one takes

$\displaystyle\chi^{2}$ $\displaystyle=1\ ,$
$\displaystyle\gamma_{\psi}^{(1)}$ $\displaystyle=0\ ,$
$\displaystyle\gamma_{\rho}^{(1)}$ $\displaystyle=\frac{(2n-1)\lambda}{2}\ ,$
$\displaystyle\gamma_{\sigma}^{(1)}$ $\displaystyle=-(2n-1)\lambda\ .$ (115)

We fix $\delta\kappa=0$ by normalising the identity block, and we absorb
$\delta\chi$ into $\chi$. Crossing symmetry imposes at leading order $\chi=\pm
1$, and we will now anticipate and require that this condition is preserved by
the interaction.

To push to the next order in $\epsilon$, we use the equation of motion twice
to constrain
$\langle\not{\partial}\psi(x)\overline{\psi}(x^{\prime})\overset{\leftarrow}{\not{\partial}}\rangle$,
similar to what we did for the Yukawa model above. The equation of motion
produces a factor of $g^{2}\sim\epsilon^{2}$, allowing us to evaluate the
right hand side using free fields and Wick contractions.

One can then use as an input our formulas for $F$ and $G$ to order $\epsilon$
in order to find an equation for the $\epsilon^{2}$ components. Proceeding as
described, we find

$\displaystyle\not{\partial}_{x}\expectationvalue{\psi_{a}(x)\overline{\psi}_{b}(y)}\overset{\leftarrow}{\not{\partial}}_{y}=$
$\displaystyle\
-g^{2}\sum_{m,n}\expectationvalue{\bar{\psi}_{m}(x)\psi_{m}(x)\psi_{a}(x)\overline{\psi}_{b}(y)\bar{\psi}_{n}(y)\psi_{n}(y)}\
.$ (116)

Note the equations of motion for $\overline{\psi}_{a}$ and $\psi_{a}$ have
opposite sign. To evaluate these products of two point functions, it is
convenient to specialise to points $x^{\mu}=zn^{\mu}$, $y^{\mu}=wn^{\mu}$.
When evaluating objects such as
$\expectationvalue{\psi(x)\overline{\psi}(x)}$, one must consider the regular
part of the expression, as usual in a normal-ordered correlator. The final
expression simplifies greatly and can be encapsulated as in the Yukawa case by
a set of eigenvalue equations

$\displaystyle\mathcal{D}_{F}$
$\displaystyle=-\epsilon^{2}\lambda^{2}H(\xi)F(\xi)\ ,$ (117)
$\displaystyle\mathcal{D}_{G}$
$\displaystyle=-\epsilon^{2}\lambda^{2}H(\xi)G(\xi)\ ,$ (118)

with a function $H(\xi)$ encoding the Wick contractions given by

$\displaystyle
H(\xi)=\left(2N-1\right)\left(2N-1+\frac{1}{\xi}-\frac{1}{(1+\xi)}\right)\ ,$
(119)

where we have used that $\chi^{2}=1$. The two point functions at order
$\epsilon^{2}$ can then be solved for

$\displaystyle F(\xi)=\ldots$
$\displaystyle+\frac{\epsilon^{2}}{\xi}\bigg{(}\frac{\log(\xi)^{2}}{8}+c_{1}\log(1+\xi)+\lambda^{2}\frac{(2N-1)}{2}\big{(}\text{coth}^{-1}\left(1+2\xi\right)+(N-1)\log(\xi)\log(1+\xi)\big{)}$
$\displaystyle+\frac{\lambda(\lambda+1)(2N-1)}{8}\log(1+\xi)^{2}-\lambda(1-2(N-1)\lambda)\frac{(2N-1)}{4}\text{Li}_{2}(-\xi)\bigg{)}\
,$ (120) $\displaystyle G(\xi)=\ldots$
$\displaystyle+\frac{\epsilon^{2}\chi}{(1+\xi)}\bigg{(}\frac{\log(1+\xi)^{2}}{8}+c_{2}\log(\xi)+\lambda\frac{2N-1}{4}\left(\log(\xi)\log(1+\xi)-2\lambda\text{coth}^{-1}\left(1+2\xi\right)\right)$
$\displaystyle+\frac{\lambda(\lambda+1)(2N-1)}{8}\log(\xi)^{2}+\lambda(1-2(N-1)\lambda)\frac{(2N-1)}{4}\text{Li}_{2}(-\xi)\bigg{)}\
.$ (121)

The variables $c_{1}$ and $c_{2}$ are new integration constants. We have fixed
two integration constants – the $O(\epsilon^{2})$ analogs of $\delta\kappa$
and $\delta\chi$ – from the field normalisation as well as the definition of
$\chi$.

One can then expand the relevant pieces in blocks. The bulk expansion of the
$F(\xi)$ function is

$\displaystyle F(\xi)=$ $\displaystyle
f_{0}+\left(-\frac{\lambda(2N-1)}{2}\epsilon+\left(c_{1}-\lambda(\lambda(2N-3)-1)\frac{(2N-1)}{4}\right)\epsilon^{2}\right)f_{\Delta_{\sigma^{2}}}$
$\displaystyle+\epsilon^{2}\frac{\lambda^{2}(2N-1)}{4}\sum_{k=1}^{\infty}\frac{(2N-1+(-1)^{k+1})\sqrt{\pi}\Gamma(k)}{4^{k}(1+k)\Gamma(\frac{1}{2}+k)}f_{2+2k}\
.$ (122)

From the matching of the logarithms, one fixes the anomalous dimension of the
leading operators $\psi$ and $\sigma^{2}$

$\displaystyle\Delta_{\psi}$
$\displaystyle=\frac{1+\epsilon}{2}+\lambda^{2}\frac{(2N-1)}{4}\epsilon^{2}+\ldots\
,$ (123) $\displaystyle\Delta_{\sigma^{2}}$
$\displaystyle=2+\epsilon-2\lambda(N-1)\epsilon+\ldots\ .$ (124)

These agree with results computed for the theory without boundary ZinnJustin ;
Fei:2016sgs .

The boundary decomposition for $F(\xi)$ in contrast is given by

$\displaystyle F(\xi)=$
$\displaystyle\left(1+\lambda\pi^{2}(1-2(N-1)\lambda)\frac{(2N-1)}{24}\epsilon^{2}\right)\widehat{f}_{\Delta_{\rho}}$
$\displaystyle+\epsilon^{2}\frac{\lambda^{2}(2N-1)}{8}\sum_{k=1}^{\infty}\frac{(k-(-1)^{k}+1)\sqrt{\pi}\Gamma(1+k)}{4^{k}(k+1)\Gamma(\frac{3}{2}+k)}\widehat{f}_{\frac{3}{2}+k}\
.$ (125)

Matching logarithms constrains anomalous dimensions. Expanding the dimension
of the leading boundary spinor as

$\Delta_{\rho}=\frac{p}{2}+\gamma_{\rho}^{(1)}\epsilon+\gamma_{\rho}^{(2)}\epsilon^{2}\
,$

we find $c_{1}=-\gamma_{\rho}^{(2)}$.

We can now turn to the second function $G(\xi)$ and its bulk expansion. From
the matching of the logarithms, one encounters a quadratic constraint which
fixes the value of $\lambda$ to that needed for the vanishing of the
$\beta$-function, $\lambda=\frac{1}{2(N-1)}$ or $\lambda=0$. Using the non-
trivial choice, one finds

$\displaystyle G(\xi)$ $\displaystyle=\chi\times
g_{\Delta_{\sigma}}+\epsilon^{2}\frac{\lambda^{2}(2N-1)}{2}\sum_{k=1}^{\infty}\frac{(2N-1+(-1)^{k}k)\sqrt{\pi}\Gamma(k)}{4^{k}k\Gamma(\frac{1}{2}+k)}g_{1+2k}\
.$ (126)

This matching fixes the integration constant

$\displaystyle c_{2}$
$\displaystyle=\frac{\gamma_{\sigma}^{(2)}}{2}-\frac{(2N-1)}{8(N-1)^{2}}\ .$
(127)

We have further expanded the scaling dimension of the mass operator as

$\displaystyle\Delta_{\sigma}$
$\displaystyle=p+\gamma_{\sigma}\epsilon+\gamma_{\sigma}^{(2)}\epsilon^{2}+\ldots$
$\displaystyle=p-(2N-1)\lambda\epsilon+\gamma_{\sigma}^{(2)}\epsilon^{2}+\ldots\
.$ (128)

Finally, one can expand $G(\xi)$ in the basis of boundary blocks, yielding the
decomposition

$\displaystyle G(\xi)$
$\displaystyle=\chi\left(1-\epsilon^{2}\lambda\pi^{2}(1-2(N-1)\lambda)\frac{(2N-1)}{24}\right)\widehat{g}_{\Delta_{\rho}}$
$\displaystyle-\epsilon^{2}\frac{\lambda^{2}(2N-1)}{8}\sum_{k=1}^{\infty}\frac{(-1)^{k}(k-(-1)^{k}+1)\sqrt{\pi}\Gamma(1+k)}{4^{k}(k+1)\Gamma(\frac{3}{2}+k)}\widehat{g}_{\frac{3}{2}+k}\
,$ (129)

which further requires $c_{1}=-\gamma_{\rho}^{(2)}$. Comparing the
coefficients of the boundary blocks $\widehat{f}_{\Delta_{\rho}}$ and
$\widehat{g}_{\Delta_{\rho}}$ in (4.4) and (4.4), we see that the boundary
condition is preserved at the fixed point

At this order in the $\epsilon$ expansion, we lack a constraint to fully fix
our system based on crossing symmetry alone. From the
$\expectationvalue{\sigma\sigma}$ correlator at $O(\epsilon^{2})$ for the
theory without boundary, one can derive or look up $\gamma_{\sigma}^{(2)}$
Fei:2016sgs ,

$\displaystyle\gamma_{\sigma}^{(2)}$
$\displaystyle=-\frac{(2N-1)}{2}\lambda^{2}=-\frac{(2N-1)}{8(N-1)^{2}}\ ,$
(130)

which further fixes

$\displaystyle\gamma_{\rho}^{(2)}$
$\displaystyle=3\frac{(2N-1)}{4}\lambda^{2}\ .$ (131)

Thus the simple input of $\gamma_{\sigma}^{(2)}$ into the constraint of
crossing symmetry fully specifies the 2-loop correction to this system.

###  4.5 The 3D Yukawa Model

It is natural to look for a model of interacting fermions in $d=3-\epsilon$
dimensions. From power counting, one can construct a classically marginal
interaction term of the form $\phi^{2}\overline{\psi}\psi$. This model is
quite similar to the Yukawa model studied before. One has an equation of
motion

$\displaystyle\not{\partial}\psi=-\frac{g}{2}\phi^{2}\psi\ ,$ (132)

with $\kappa_{s}g\equiv 2\lambda\sqrt{\epsilon}$ at the fixed point, for some
order one $\lambda$ determined by the zero of the beta function. The main
difference is that the CFT data admits an expansion in powers of
$\sqrt{\epsilon}$ rather than $\epsilon$. From a perturbative perspective, the
change comes about because the operator ${:}\phi(x)^{2}{:}$ has a nonzero
expectation value in the theory with a boundary even when $g=0$;
$\langle\phi^{2}\rangle=N_{s}\chi_{s}/(2x\cdot n)$ where $\chi_{s}=\pm 1$
selects the boundary conditions, Dirichlet or Neumann, for the scalar field,
and the $N_{s}$ is the number of scalar fields. This makes the leading order
result analogous to the conformally massive fermion previously
discussed.444This “conformal mass” argument can also be invoked to show that a
purely $\phi^{6}$ bosonic model will involve an expansion in
$\sqrt{\epsilon}$: the equation of motion for the scalar will take the
schematic form $\Box\phi\sim\langle\phi^{4}\rangle\phi$. Interestingly, refs.
diehi1987walks ; eisenriegler1988surface  derive this $\sqrt{\epsilon}$
behavior in a different way, through mixing in the RG flow of a bulk
$\phi^{6}$ coupling and an induced boundary $\phi^{4}$ coupling.

To analyse the system, we proceed as in the Gross-Neveu model, and first act
once with the equation of motion for $\psi$,

$\displaystyle\not{\partial}\expectationvalue{\psi_{a}(x)\overline{\psi}_{b}(x^{\prime})}$
$\displaystyle=-\lambda\sqrt{\epsilon}\frac{N_{s}\chi_{s}}{2x\cdot
n}\expectationvalue{\psi_{a}(x)\overline{\psi}_{b}(x^{\prime})}\ .$ (133)

The equation of motion gives an expansion for the two point function

$\displaystyle F(\xi)$
$\displaystyle=\frac{1}{\xi^{3/2}}+\frac{\lambda\sqrt{\epsilon}}{\xi^{3/2}}N_{s}\chi\chi_{\phi}\left(\sqrt{\frac{\xi}{1+\xi}}-\text{sinh}^{-1}\sqrt{\xi}\right)\
,$ (134) $\displaystyle G(\xi)$
$\displaystyle=\frac{\chi}{(1+\xi)^{3/2}}+\frac{\lambda\sqrt{\epsilon}}{(1+\xi)^{3/2}}N_{s}\chi_{\phi}\left(\sqrt{\frac{1+\xi}{\xi}}-\text{sinh}^{-1}\sqrt{\xi}\right)\
.$ (135)

We absorbed two integrations constants into the normalisation of the
coincident limit and the definition of $\chi$. The matching to the blocks
imposes $\chi^{2}=1+\mathcal{O}(\epsilon)$. In fact, we will from now require
that this is preserved also at higher order. These functions can be written as

$\displaystyle\begin{pmatrix}F(\xi)\\\
G(\xi)\end{pmatrix}=\begin{pmatrix}f_{0}-\lambda\sqrt{\epsilon}\frac{N_{s}\chi\chi_{s}}{3}\,f_{3}\\\
\chi\,g_{p}+\lambda\sqrt{\epsilon}N_{s}\chi_{s}\,g_{1}\end{pmatrix}=\begin{pmatrix}(1+N_{s}\chi\chi_{\phi}(1-\log(2))\lambda\sqrt{\epsilon})\,\widehat{f}_{\frac{p}{2}+\gamma_{\rho}\sqrt{\epsilon}}\\\
\chi(1+N_{s}\chi\chi_{\phi}(1-\log(2))\lambda\sqrt{\epsilon})\,\widehat{g}_{\frac{p}{2}+\gamma_{\rho}\sqrt{\epsilon}}\end{pmatrix}\
.$ (136)

This matching is contingent on the modification at $O(\sqrt{\epsilon})$ of the
dimension of the boundary field

$\displaystyle\Delta_{\rho}$ $\displaystyle=\frac{p}{2}+\frac{\lambda
N_{s}\chi\chi_{\phi}}{2}\sqrt{\epsilon}+O(\epsilon)\ .$ (137)

The deformation of the conformal block decomposition at $O(\sqrt{\epsilon})$
is particularly simple. That is because the anomalous dimensions of the bulk
fields are untouched at this order in the expansion, while the boundary
spectrum is highly constrained by the conformal mass structure.

To go to the next order, we mimic the Gross-Neveu discussion above and act
with the equations of motions on both $\psi$ and $\overline{\psi}$ in the
spinor two-point function. We find the same differential operators studied in
the Yukawa model. The source for the system of differential equations is given
by the Wick contraction of four scalar insertions. We find for $F(\xi)$ and
$G(\xi)$ at $O(\epsilon)$ that

$\displaystyle F(\xi)$
$\displaystyle=\ldots+\frac{\epsilon}{\xi^{3/2}}\bigg{(}\frac{\log(\xi)}{2}+2c_{1}\left(\text{sinh}^{-1}\left(\sqrt{\xi}\right)-\sqrt{\frac{\xi}{1+\xi}}\right)-2\lambda^{2}N_{s}\chi_{\phi}\sqrt{\frac{\xi}{1+\xi}}$
(138)
$\displaystyle-\lambda^{2}N_{s}\left(\frac{\log(\xi)}{3}+\log(1+\xi)\right)+\frac{\lambda^{2}N_{s}^{2}}{2}\left(\text{sinh}^{-1}\left(\sqrt{\xi}\right)^{2}-2\sqrt{\frac{\xi}{\xi+1}}\text{sinh}^{-1}\left(\sqrt{\xi}\right)\right)\bigg{)}\
,$ $\displaystyle G(\xi)$
$\displaystyle=\ldots+\frac{\epsilon}{(1+\xi)^{3/2}}\bigg{(}\delta\chi^{(2)}+\chi\frac{\log(1+\xi)}{2}+2c_{2}\left(\text{sinh}^{-1}\left(\sqrt{\xi}\right)-\sqrt{\frac{1+\xi}{\xi}}\right)$
(139)
$\displaystyle+\lambda^{2}N_{s}\chi\left(\frac{\log(1+\xi)}{3}+\log(\xi)\right)+\frac{\lambda^{2}\chi
N_{s}^{2}}{2}\left(\text{sinh}^{-1}\left(\sqrt{\xi}\right)^{2}-2\sqrt{\frac{1+\xi}{\xi}}\text{sinh}^{-1}\left(\sqrt{\xi}\right)\right)$
$\displaystyle+\lambda^{2}\chi
N_{s}^{2}+2\lambda^{2}N_{s}\chi\chi_{\psi}\sqrt{\frac{1+\xi}{\xi}}\bigg{)}\ .$

We expand these functions using the boundary and bulk conformal blocks. For
$F(\xi)$, we find in the bulk

$\displaystyle F(\xi)$
$\displaystyle=f_{0}+\left(-\frac{\lambda\sqrt{\epsilon}N_{s}\chi\chi_{s}}{3}+\epsilon\frac{2c_{1}}{3}\right)\,f_{3}-2N_{s}\chi_{s}\epsilon\lambda^{2}\,f_{1}$
$\displaystyle+\epsilon\frac{\lambda^{2}N_{s}}{2}\sum_{k=0}^{\infty}\frac{(-1)^{k}\Gamma(k+\frac{1}{2})\Gamma(k-\frac{1}{2})+2^{-2k+1}kN_{s}\pi\Gamma(2k-1)}{(1+k)\sqrt{\pi}\Gamma(\frac{1}{2}+2k)}\,f_{2+2k}\
.$ (140)

Matching the logarithms, one finds further the bulk anomalous dimension of the
fermion

$\displaystyle\Delta_{\psi}$
$\displaystyle=\frac{p}{2}+\frac{\lambda^{2}N_{s}}{3}\epsilon+O(\epsilon^{2})\
.$ (141)

The boundary expansion of $F(\xi)$ is instead

$\displaystyle F(\xi)$
$\displaystyle=(1+N_{s}\chi\chi_{s}(1-\log(2))\lambda\sqrt{\epsilon}+\delta\nu_{F}\lambda^{2}\epsilon)\,\widehat{f}_{\Delta_{\rho}}$
(142)
$\displaystyle+\frac{2N_{s}\lambda^{2}}{3}\epsilon\sum_{k>0}\frac{(4k^{2}+6k\chi_{s}-3(-1)^{k}-1)}{4^{k}(2k-1)k}\,\widehat{f}_{1+k}\
.$

Matching the logarithms partially constrains the anomalous dimension of
$\rho$,

$c_{1}=\frac{4N_{s}\lambda^{2}}{3}-\gamma_{\rho}^{(2)}\ ,$ (143)

where we write
$\Delta_{\rho}=\frac{p}{2}+\gamma_{\rho}^{(1)}\sqrt{\epsilon}+\gamma_{\rho}^{(2)}\epsilon+\ldots$.
The parameter $\delta\nu_{F}$ is complicated

$\displaystyle\delta\nu_{F}\lambda^{2}$ $\displaystyle=2c_{1}(\log
2-1)+\frac{N_{s}\lambda^{2}}{2}(-4\chi_{s}+N_{s}(\log 2-2)\log 2)\ .$ (144)

We turn now to $G(\xi)$. The bulk expansion is

$\displaystyle G(\xi)$
$\displaystyle=(\chi+\epsilon\delta\chi^{(2)})g_{\Delta_{\sigma}}+(N_{s}\chi_{s}\lambda\sqrt{\epsilon}+2(N_{s}\chi\chi_{s}\lambda^{2}-c_{2})\epsilon)\,g_{1}$
(145)
$\displaystyle+\chi\frac{\lambda^{2}N_{s}}{2}\epsilon\sum_{k=1}^{\infty}\frac{4^{-k}N_{s}\pi\Gamma(2k)+\frac{1}{3}(-1)^{k+1}\Gamma(k+\frac{1}{2})\Gamma(k+\frac{3}{2})}{k\sqrt{\pi}\Gamma(2k+\frac{1}{2})}\,g_{2k+2}$
$\displaystyle+\chi\chi_{s}\lambda^{2}N_{s}\epsilon\sum_{k=1}^{\infty}\frac{\sqrt{\pi}\Gamma\left(k-\frac{1}{2}\right)^{2}}{\Gamma\left(2k-\frac{1}{2}\right)}g_{2k+1}\
.$

Matching the logarithms, the dimension of the bulk mass term is

$\displaystyle\Delta_{\sigma}=p+\frac{8N_{s}}{3}\lambda^{2}\epsilon+O(\epsilon^{2})\
.$ (146)

Finally, one can expand $G(\xi)$ in boundary blocks, giving

$\displaystyle G(\xi)$
$\displaystyle=\chi(1+N_{s}\chi\chi_{s}(1-\log(2))\lambda\sqrt{\epsilon}+\delta\nu_{G}\lambda^{2}\epsilon)\widehat{g}_{\Delta_{\rho}}$
(147)
$\displaystyle+\chi\frac{2N_{s}\lambda^{2}\epsilon}{3}\sum_{k>0}\frac{3+(-1)^{k}(1-4k^{2}-6k\chi_{s})}{4^{k}(2k-1)k}\widehat{g}_{1+k}\
.$

One finds an additional constraint on the anomalous dimension of the boundary
spinor, $c_{2}=-\frac{4}{3}N_{s}\lambda^{2}-\gamma_{\rho}^{(2)}$, which allows
us to relate $c_{1}$ and $c_{2}$, using a similar constraint (143) from the
boundary expansion of $F(\xi)$. The $O(\epsilon)$ contribution to the boundary
$\rho$ block takes the form

$\displaystyle\chi\delta\nu_{G}\lambda^{2}$ $\displaystyle=2c_{2}(\log
2-1)+\frac{N_{s}\lambda^{2}}{2\chi}(4\chi_{s}+N_{s}((\log 2-2)\log
2+2)+\delta\chi^{(2)}\ .$

Insisting on the boundary conditions $\delta\nu_{F}=\delta\nu_{G}$ then fixes
$\delta\chi^{(2)}$ in terms of $\lambda$.

Unfortunately, there is a degeneracy in the system of equations which does not
allow us to specify both $c_{1}$ and $c_{2}$, and so we cannot determine
$\gamma_{\rho}^{(2)}$ either. This situation is unsatisfying but can be
remediated. Crossing symmetry of
$\expectationvalue{\psi(x)\bar{\psi}(x^{\prime})}$ is not enough to fix
$\gamma_{\rho}^{(2)}$, but we can look at
$\expectationvalue{\phi(x)\phi(x^{\prime})}$ as well. To see how this would
help, consider the bulk expansion we found for the $G$ function

$\displaystyle G(\xi)$ $\displaystyle=\ldots+\bigg{(}\left[\lambda
a_{\phi^{2}}^{(2)}\right]_{(1)}\sqrt{\epsilon}+\left[\lambda
a_{\phi^{2}}^{(2)}\right]_{(2)}\epsilon\bigg{)}g_{1}+\ldots\,.$ (148)

The coefficient $\left[\lambda a_{\phi^{2}}^{(2)}\right]_{(2)}$ contains
$c_{2}$, or equivalently $\gamma_{\rho}$, and is the only unknown left. This
term contains a contribution from the expectation value $a_{\phi^{2}}$ at
order $\sqrt{\epsilon}$, and from the coupling $\lambda_{\phi^{2}}^{(2)}$ at
order $\epsilon$. Notice that the latter can be computed using perturbation
theory in the theory without a boundary. In such a setup, each interaction
vertex $g$ contributes one fermion propagator, while
$\lambda_{\phi^{2}}^{(1)}$ and $\lambda_{\phi^{2}}^{(2)}$ multiply
respectively structures with odd and even numbers of gamma matrices. From this
diagrammatic argument, we gather that $\lambda_{\phi^{2}}^{(2)}$ receives
contributions only for odd powers of the coupling,
$g^{2n+1}\sim\epsilon^{n}\sqrt{\epsilon}$; hence the ratio of the two
coefficients multiplying the block measures precisely the correction to
$a_{\phi^{2}}$ at order $g$,

$\displaystyle\frac{\left[\lambda
a_{\phi^{2}}^{(2)}\right]_{(2)}}{\left[\lambda
a_{\phi^{2}}^{(2)}\right]_{(1)}}$
$\displaystyle=\frac{[a_{\phi^{2}}]_{(1)}}{[a_{\phi^{2}}]_{(0)}}\ .$ (149)

Crucially, this ratio can be determined from an analysis of
$\expectationvalue{\phi(x)\phi(x^{\prime})}$ at order $\sqrt{\epsilon}$. The
equation of motions for the scalar is

$\displaystyle\Box\phi_{a}=g\overline{\psi}_{b}\psi_{b}\phi_{a}+\frac{h}{5!}(\phi^{2})^{2}\phi_{a}\
.$ (150)

At the fixed point, at order $\sqrt{\epsilon}$, we can neglect the second
term. Applying this equation once to the scalar correlator allows us to solve
for $H(\xi)$,

$\displaystyle H(\xi)$
$\displaystyle=\frac{1+\delta\kappa\sqrt{\epsilon}}{\sqrt{\xi}}+\frac{\chi_{s}+c_{3}\sqrt{\epsilon}}{\sqrt{1+\xi}}+2\lambda\sqrt{\epsilon}N_{f}\chi\left(\frac{\chi_{s}\sinh[-1](\sqrt{\xi})}{\sqrt{\xi}}+\frac{\coth[-1](\sqrt{\frac{1+\xi}{\xi}})-\chi_{s}}{\sqrt{1+\xi}}\right)\,.$
(151)

We still need to fix the two unknowns $c_{3}$ and $\delta\kappa$. From the
normalisation of the coincident limit, we fix $\delta\kappa=0$. To fix
$c_{3}$, we pick a boundary condition through some $\chi_{s}=\pm$, and adjust
$c_{3}$ to maintain compatibility of $H(\xi)$ with either Dirichlet or Neumann
boundary conditions. This gives $c_{3}=-2\lambda N_{f}\chi\chi_{s}$. With
$\expectationvalue{\phi(x)\phi(x^{\prime})}$ fully fixed, one can look at its
bulk decomposition. The bulk conformal block decomposition of the correlator
has the generic form

$\displaystyle H(\xi)$ $\displaystyle=\sum_{\Delta}\lambda
a_{\Delta}h_{\Delta}\,,$ (152) $\displaystyle=\sum_{\Delta}\lambda
a_{\Delta}\xi^{\frac{\Delta-2\Delta_{\phi}}{2}}\,_{2}F_{1}\left(\frac{\Delta}{2},\frac{\Delta}{2};\Delta-\frac{p-1}{2};-\xi\right)\,,$

where we have used the explicit expressions for the bulk conformal blocks
McAvity:1995tm . The equations of motion are compatible with conformal
invariance only for $\Delta_{\phi}=\frac{1}{2}+\mathcal{O}(\epsilon)$. For
$\epsilon=0$, the only two exchanged bulk operators are the identity and
$\phi^{2}$. The identity operator contribution receives no correction beyond
the field rescaling which we have fixed. One can systematically match
expressions (151) and (152). Focusing on the block of dimension $1$, we find

$\displaystyle H(\xi)$
$\displaystyle=\ldots+(\chi_{s}+c_{3}\sqrt{\epsilon})h_{1}+\ldots\,,$
$\displaystyle=\ldots+([\lambda a_{\phi^{2}}]_{(0)}+[\lambda
a_{\phi^{2}}]_{(1)}\sqrt{\epsilon})g_{1}+\ldots$ (153)

We can now repeat the argument made previously for the fermion. $\lambda$
measures the coupling of
$\expectationvalue{\phi(x)\phi(y):\phi^{2}(z):}_{\circ}$, whose first
correction must come from one $h$ or two $g$ vertices, at order $\epsilon$.
Hence, the ratio of the two coefficients must measure precisely the relative
shift in the expectation value of $\phi^{2}$. This now fixes the fermion
correlator, and gives a prediction for the boundary anomalous dimension of the
leading spinor

$\displaystyle\gamma_{\rho}^{(2)}$
$\displaystyle=-\lambda^{2}N_{s}\left(\chi\chi_{s}(1+N_{f})+\frac{4}{3}\right)\
.$ (154)

In other dimensions, we could compare our results with explicit perturbative
results in the literature. Since we lacked those in this specific case, we
went through their derivation to be able to showcase the agreement. The
renormalisation group analysis of this system, given in Appendix D, indeed
reproduces the bulk scaling dimensions $\Delta_{\sigma}$ and $\Delta_{\psi}$
we derived here.

##  5 Discussion

A central accomplishment of this work is the derivation of the bulk and
boundary conformal blocks for spinor two point functions in boundary CFT.
While the boundary blocks were known in some form previously Nishida:2018opl ;
Herzog:2019bom , the bulk blocks are completely new, and we have put both to
use for the first time in imposing crossing symmetry constraints on the spinor
two point function. In combination with the equations of motion, we used these
constraints to derive a number of bulk and surface anomalous dimensions as
well as towers of OPE coefficients in three different interacting models, in
the $\epsilon$-expansion.

Each of our models presented an interesting feature. The Yukawa model in
$4-\epsilon$ dimensions exhibited the fake primary effect Karateev:2019pvw ,
as we were cleanly able to demonstrate using our bulk conformal blocks.
Naively there should be a primary of $\Delta\approx 3$ corresponding to the
operator $\overline{\psi}\psi$ when in fact, because of the equations of
motion, this operator is a descendant of $\phi$ and there is no $\Delta\approx
3$ primary in the bulk OPE of the spinor two-point function.

For the Gross-Neveu model in $2+\epsilon$ dimensions, for the same amount of
work, we were able to get results to $O(\epsilon^{2})$, in contrast to the
other two models where our expansions terminate at $O(\epsilon)$. As a result,
we were able to compute the anomalous dimension for the leading fermion
surface operator $\rho$ to one higher order (18) than had been computed
heretofore Giombi:2021cnr .

Finally, to our knowledge, our Yukawa model in $3-\epsilon$ dimensions with
boundary has not been studied before, and thus our results are new. A quirk of
this model is that it exhibits an expansion in $\sqrt{\epsilon}$ rather than
$\epsilon$, similar to what happens for the scalar $\phi^{6}$ model in
$3-\epsilon$ dimensions in the presence of a boundary diehi1987walks ;
eisenriegler1988surface . To perform the analysis on this little studied 3d
model with boundary, we had to back up first and perform a standard
perturbative analysis on the theory without a boundary, in order to locate the
zero of the beta functions. Interestingly, we found that the beta functions
vanish in exactly $d=3$ for $N_{s}=N_{f}=1$, which may indicate this case is
supersymmetric. It would be interesting to look at this case more closely, and
also to see if the zero of the beta function can be extended to larger
$N_{f}=N_{s}\neq 1$ by allowing for interactions between fermions and bosons
that mix flavor. Another natural question is whether, in the large-$N$ limit,
one finds that the two fixed points collide for an intermediate value of
$\epsilon<1$, as is the case for the purely bosonic system Pisarski:1982vz .
On a tangential note, the tools we outlined allow for a generalisation to
spinor-currents, from which one could discuss quantitatively supersymmetry in
the presence of a boundary, at the level of correlators and blocks
Erdmenger:2002ex ; Drukker:2017dgn ; Herzog:2018lqz .

One clear direction for further study is to explore how many anomalous
dimensions can be calculated in this fashion and to what order in $\epsilon$.
While we got away with using free two-point functions on the right hand side
of the equations of motion, to proceed further than we did requires
considering interacting correlation functions on the right hand side. Thus
technically, the problem is more difficult but not obviously insuperable. It
would be interesting to have more analytical knowledge and control of this
expansion, making contact with the works on analytical bootstrap in bCFT
Bissi:2022mrs ; Kaviraj:2018tfd ; Mazac:2018biw ; Bissi:2018mcq ; Dey:2020jlc
. We have found that the crossing equation for the spinor is extremely similar
to the one of the scalar. We hope to present a more in depth analysis of the
spinor crossing equations in the future.

The surface anomalous dimension $\Delta_{\rho}$ is probably the most
interesting quantity to come out this work. While we demonstrated the bulk
anomalous dimensions can be computed efficiently using crossing symmetry in
boundary CFT, these bulk dimensions are unaffected by the presence of a
boundary and were known previously in the 4d Yukawa model and 2d Gross-Neveu
model.

It would be interesting to find an experimentally measurable observable
associated with $\Delta_{\rho}$. One possibility is angle resolved photo-
emission spectroscopy (ARPES). In such an experiment, a high energy photon is
used to eject electrons from the surface of the material. The energy and
momentum of the ejected electrons can then be used to reconstruct the electron
spectral function and by extension the fermionic Green’s function. In
principle, then, it should be possible to extract the surface anomalous
dimension $\Delta_{\rho}$ from the fermionic Green’s function near the edge of
a Dirac or Weyl material. Whether such a measurement is possible in practice,
we do not know.

A possibility for the measurement of a related surface anomalous dimension is
to use electron transport. Consider $J^{\mu}=\bar{\psi}\gamma^{\mu}\psi$,
which we will assume to be conserved $\partial_{\mu}J^{\mu}=0$. In the
presence of a boundary at $z=0$, conservation requires also that the flux of
the current through the boundary vanish, $J^{z}|_{z=0}=0$.

Let us try to understand how this condition affects the structure of the
boundary OPE for the current. Conformal invariance, bulk conservation
$\partial_{\mu}J^{\mu}=0$, and unitarity together imply that $J^{\mu}$ can be
decomposed into a boundary scalar $\tau$ of dimension $\Delta=d-1$ (see (24))
and a tower of boundary vectors $v^{i}$ of dimension $\Delta>d-2$ (see (25))
Liendo:2012hy .555A boundary conserved current, which saturates the unitarity
bound, with $\Delta=d-2$, must decouple from the bulk current, precisely
because it is conserved.

Consider the current-current two point function $\langle
J_{\mu}(x)J_{\nu}(x^{\prime})\rangle$ (26) for the particular configuration
where $x$ and $x^{\prime}$ are on a line perpendicular to the boundary. In
this case, we can write

$\displaystyle\langle J_{z}(x)J_{z}(x^{\prime})\rangle$ $\displaystyle=$
$\displaystyle\frac{R(v)}{|x-x^{\prime}|^{2(d-1)}}\ ,\;\;\;\langle
J_{i}(x)J_{j}(x^{\prime})\rangle=\frac{Q(v)\delta_{ij}}{|x-x^{\prime}|^{2(d-1)}}\
.$ (155)

The scalar of dimension $d-1$ that is allowed to contribute to this two point
function has a $R(\xi)\sim v^{d-1}(v^{-1}+v)$ which is finite in the boundary
limit $v\to 1$. Thus, in order for $J^{\mu}$ to be conserved, this scalar must
be absent from the bOPE of $J^{\mu}$. In contrast, the tower of boundary
vectors have a $R(\xi)\sim(v^{-1}-v)^{\Delta-d+2}$ close to the boundary.
Thus, the unitarity bound $\Delta>d-2$ is sufficient to guarantee current
conservation for these operators.

Given this tower of boundary vectors and the unitarity bound, there will be
one vector with smallest conformal dimension which determines the near
boundary behavior of $J^{\mu}$ through the bOPE. In a free fermion theory,
this boundary vector is just the boundary value of $\bar{\psi}\gamma^{i}\psi$,
and has dimension $d-1$. More generally, we assume that this operator has
dimension $\Delta_{1}=d-1+\gamma_{1}$ and that there is a gap in conformal
dimension before the next boundary vector. Note that $\gamma_{1}$ can be
positive or negative but boundary unitarity enforces $\gamma_{1}>-1$.

What are the physical implications of this smallest boundary vector operator?
By dimensional analysis of the bOPE, it follows that $J^{\mu}$ has near
boundary behavior $J^{i}\sim z^{\gamma_{1}}$. Indeed, the $z^{\gamma_{1}}$
scaling of $J^{t}$ should be a visible feature of the Fermi sea near the
boundary for a material close to a phase transition. The density of states
should satisfy this power law behavior, either vanishing (for $\gamma_{1}>0$)
or diverging (for $\gamma_{1}<0$) close to the boundary. Moreover, this
scaling should be reflected in the behavior of spatial currents $J^{i}$ near
and parallel to the surface.

To be more precise, we can isolate the contribution of this smallest boundary
vector to the current-current two point function by looking at the near
boundary limit:

$\displaystyle\langle J^{z}({x})J^{z}({x}^{\prime})\rangle$
$\displaystyle\sim$ $\displaystyle\frac{(zz^{\prime})^{\gamma_{1}+1}}{|{\bf
x}-{\bf x}^{\prime}|^{2\Delta_{1}+2}}\ ,$ (156) $\displaystyle\langle
J^{i}({x})J^{j}({x}^{\prime})\rangle$ $\displaystyle\sim$
$\displaystyle\frac{(zz^{\prime})^{\gamma_{1}}}{|{\bf x}-{\bf
x^{\prime}}|^{2\Delta_{1}}}\frac{\Delta_{1}-d+2}{8(d-1)}\left(\delta^{ij}-\frac{2{\bf
x}^{i}{\bf x}^{j}}{|{\bf x}|^{2}}\right)\ .$ (157)

Indeed $\langle J^{z}J^{z}\rangle$ will vanish by unitarity, consistent with
conservation. The $\langle J^{i}J^{j}\rangle$ contribution on the other hand
looks like the correlation function of a boundary vector with dimension
$\Delta_{1}$, however with a power law dependent profile as one moves away
from the boundary

For our fermionic theories, we determined $\gamma_{\rho}$, the anomalous
dimension of the boundary value of the $\psi$ fields. The anomalous dimension
$\gamma_{1}$ will have a contribution from $2\gamma_{\rho}$. In general,
however, it may also get contributions from connected loop diagrams that
involve both $\rho$ fields. It is an interesting project for the future to try
to compute these anomalous dimensions precisely, especially as they may have
directly observable consequences.

## Acknowledgments

We would like to thank D. Anninos, A. Chalabi, P. Kravchuk, E. Lauria, S.
Parameswaran, S. Simon, A. Stergiou, and B. van Rees for discussion. We would
also like to thank H. W. Diehl for his comments on the draft. C. H. would like
to acknowledge the hospitality of the Oxford University Physics Department
where part of this work was developed. C. H. was supported in part by a
Wolfson Fellowship from the Royal Society. This work was supported by the U.K.
Science & Technology Facilities Council Grant ST/P000258/1.

##  Appendix A Conventions

The theories we consider are reflection-positive Euclidean CFTs, which under
analytic continuation should map to unitary Lorentzian CFTs. We use Greek
indices ranging from $1$ to $d$, contracted with a metric $\delta_{\mu\nu}$.
Intrinsic coordinates on the boundary are written using Latin indices
$i,j,k=0,\ldots,p-1$ with $p=d-1$. Embedding coordinates are written
$A,B,C\ldots=0,\ldots,d+1$, with one time-dimension. The (inward pointing)
unit-normal to the boundary is taken to be $n^{\mu}=\delta^{d,\mu}$, without
loss of generality. The induced metric on the boundary is
$h_{\mu\nu}=\eta_{\mu\nu}-n_{\mu}n_{\nu}$, and intrinsically $\delta_{ij}$.
Lightcone coordinates are given in the embedding using $P^{\pm}=P^{0}\pm
P^{d+1}$, i.e. $\eta^{+-}=-2$, $\eta_{+-}=-\frac{1}{2}$.

Our convention for the Clifford algebra is that
$\\{\gamma_{\mu},\gamma_{\nu}\\}=2\delta_{\mu\nu}$. We generically consider
Dirac spinors unless otherwise specified. In real space, since we work in
Euclidean signature, we take $\overline{\psi}$ and $\psi$ to be independent
fields, upon which we impose a reality condition coming from Lorentzian
signature. In the embedding, conjugation follows through in the same way, and
so one must consider how Dirac conjugation works with two time dimensions. The
index-free spinors are made up from spinor bilinears; hence their complex-
conjugation properties are straightforward.

The embedding gamma-matrices are given by

$\displaystyle\Gamma_{\mu}$ $\displaystyle=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\otimes\gamma_{\mu}\ ,$ $\displaystyle\Gamma_{+}$
$\displaystyle=\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}\otimes\mathbb{1}\ ,$
$\displaystyle\Gamma_{-}$ $\displaystyle=\begin{pmatrix}0&0\\\
-1&0\end{pmatrix}\otimes\mathbb{1}\ .$ (158)

For a generic point on the lightcone $P^{A}$ in the Poincaré patch, with real-
space coordinate $x^{\mu}$, we have

$\displaystyle\not{P}=\begin{pmatrix}\not{x}&1\\\
-x^{2}&-\not{x}\end{pmatrix}=\begin{pmatrix}1\\\
-\not{x}\end{pmatrix}\begin{pmatrix}\not{x}&1\end{pmatrix}\ .$ (159)

Because of the broken rotational symmetry, it is convenient to work with
projectors of definite helicity on the boundary. In real space, we use the
normal $n^{\mu}$, and in the embedding we use $V^{A}=\delta^{A,d}$. The analog
of the polarization tensor $Z$ is the polarization spinor $S$, which must
satisfy a similar transversality condition $\not{P}S=0$. This null eigenvalue
condition is solved by

$\displaystyle S$ $\displaystyle=\begin{pmatrix}1\\\ -\not{x}\end{pmatrix}s\
,$ $\displaystyle\overline{S}$
$\displaystyle=\overline{s}\begin{pmatrix}\not{x}&1\end{pmatrix}\ .$ (160)

We use Dirac bra-ket notation to efficiently write the spinor-bilinear. The
convention is that $\bra{i}\equiv\overline{S}_{i}$, $\ket{i}\equiv S_{i}$, and
that vectors inside (unless otherwise specified) are contracted with
$\Gamma$-matrices. Projection to real-space is performed by evaluating the
bilinear using its expression in the tensor-product space, to reduce it to a
real-space bilinear. The twistor form of $\not{P}$ (159) is particularly
convenient for this purpose. For example, the structures entering the spinor-
spinor-scalar three point function are

$\displaystyle\bra{1}\ket{2}\rightarrow\bra{1}x_{12}\ket{2}\
,\;\;\;\bra{1}P_{3}\ket{2}\rightarrow\bra{1}x_{13}x_{32}\ket{2}\ .$ (161)

And these are manifestly real, as they map to themselves under complex-
conjugation and exchange $1\leftrightarrow 2$.

To discuss conjugate spinors, we settle on a reality structure to impose on
our correlators. This translates into reality conditions on the different
spinorial structures and OPE coefficients. Although the space is Euclidean and
so are the spinors, we impose the reality structure suitable for spinors of
$\mathbb{R}^{2,d}$ in the embedding, $\mathbb{R}^{1,d-1}$ in real space. This
means that the correlators we write will not be Hermitian as seen from the
Euclidean viewpoint.

The breaking of rotational symmetry induced by the planar boundary is
efficiently understood by the analogy with Weyl-spinors. A Dirac spinor in the
bulk can be decomposed on the boundary into its two helicity parts. We encode
boundary spinors as constrained Dirac spinors of the same type as the bulk
ones. They are written $\rho_{\pm}$, and satisfy $\gamma\cdot
n\rho_{\pm}=\pm\rho_{\pm}$. Their polarisations are written with a
supplementary label, $s_{i}^{(\pm)}\equiv\ket{i,\mp}$ ,
$\overline{s}_{i}^{(\pm)}\equiv\bra{i,\pm}$. Naturally, one has
$\bra{i,\pm}=\bra{i}\mathbb{P}_{\pm}$ and likewise for the kets. The perhaps
counterintuitive notation is such that $\overline{s}_{+}\rho_{+}$ and
$\overline{\rho}_{+}s_{+}$ are scalars, complex-conjugate to each other, and
compatible with $\not{n}\rho_{+}=\rho_{+}$ and
$\overline{\rho}_{+}\not{n}=-\overline{\rho}_{+}$.

##  Appendix B Spinning Conformal Integral

In this appendix we collect useful results for the explicit evaluation of
generic CPWs using the embedding space form of the shadow formalism. The idea
is that partial waves can be computed through an integral representation using
a conformally invariant sewing of simpler correlation functions. The resulting
formulae take particularly simple form in the embedding space.

###  B.1 Reduction to Standard Integral

We first note the following useful formulae:

$\displaystyle\int\frac{D^{d}X}{(-2Y\cdot X)^{d}}$
$\displaystyle=\frac{\sqrt{\pi}^{d}\Gamma(\frac{d}{2})}{\Gamma(d)}\frac{1}{(-Y\cdot
Y)^{\frac{d}{2}}}\ ,$ (162)
$\displaystyle\prod_{i=1}^{n}\frac{1}{A_{i}^{a_{i}}}$
$\displaystyle=\frac{\Gamma(\sum_{i=1}^{n}a_{i})}{\prod_{i=1}^{n}\Gamma(a_{i})}\int\left(\prod_{i=2}^{n}\frac{d\alpha_{i}}{\alpha_{i}}\alpha_{i}^{a_{i}}\right)\frac{1}{(A_{1}+\alpha_{2}A_{2}+\ldots\alpha_{n}A_{n})^{\sum_{i=1}^{n}a_{i}}}\
,$ (163) $\displaystyle\left(\sum_{i=1}^{n}A_{i}\right)^{l}$
$\displaystyle=\sum_{r_{1}+r_{2}+\ldots r_{n}=l}\binom{l}{r_{1}\ldots
r_{n}}(A_{1})^{r_{1}}\ldots(A_{n})^{r_{n}}\ .$ (164)

We can now evaluate the most general thing we will encounter

$\displaystyle\mathcal{I}^{(l)}_{(a_{i})}(P_{i},Z)$
$\displaystyle=\mathcal{P}\int D^{d}X(Z\cdot
X)^{l}\prod_{i=1}^{n}\frac{1}{(-2P_{i}\cdot X)^{a_{i}}}$ (165)
$\displaystyle=\mathcal{P}\frac{\Gamma(d+l)}{\prod_{i=1}^{n}\Gamma(a_{i})}\int\left(\prod_{i=2}^{n}\frac{d\alpha_{i}}{\alpha_{i}}\alpha_{i}^{a_{i}}\right)\frac{(Z\cdot
X)^{l}}{\Big{(}-2\underbrace{(P_{1}+\alpha_{2}P_{2}+\ldots\alpha_{n}P_{n})}_{Y}\cdot
X)\Big{)}^{d+l}}$
$\displaystyle=\mathcal{P}\frac{\Gamma(d+l)}{\prod_{i=1}^{n}\Gamma(a_{i})}\frac{1}{2^{l}(d)_{l}}\int\left(\prod_{i=2}^{n}\frac{d\alpha_{i}}{\alpha_{i}}\alpha_{i}^{a_{i}}\right)\left(Z\cdot\partialderivative{Y}\right)^{l}\int
D^{d}X\frac{1}{\left(-2Y\cdot X\right)^{d}}$
$\displaystyle=\mathcal{P}\frac{\sqrt{\pi}^{d}\Gamma(\frac{d}{2}+l)}{\prod_{i=1}^{n}\Gamma(a_{i})}\int\left(\prod_{i=2}^{n}\frac{d\alpha_{i}}{\alpha_{i}}\alpha_{i}^{a_{i}}\right)\frac{(Y\cdot
Z)^{l}}{(-Y\cdot Y)^{\frac{d}{2}+l}}$
$\displaystyle=\frac{\sqrt{\pi}^{d}\Gamma(\frac{d}{2}+l)}{\prod_{i=1}^{n}\Gamma(a_{i})}\sum_{r_{1}+\ldots+r_{n}=l}\binom{l}{r_{1}\ldots
r_{n}}(Z\cdot P_{1})^{r_{1}}\ldots(Z\cdot
P_{n})^{r_{n}}\mathcal{F}_{(a_{i}+r_{i})}(P_{i})\ ,$

where in the last line we defined the object which we refer to as the
“standard integral”:

$\displaystyle\mathcal{F}_{(a_{i})}(P_{i})=\mathcal{P}\int\left(\prod_{i=2}^{n}\frac{d\alpha_{i}}{\alpha_{i}}\alpha_{i}^{a_{i}}\right)\frac{1}{\Big{(}-(P_{1}+\alpha_{2}P_{2}+\ldots)\cdot(P_{1}+\alpha_{2}P_{2}+\ldots)\Big{)}^{\frac{\sum
a_{i}}{2}}}\ .$ (166)

The task of evaluating the most general correlator is then reduced to fixing
$\mathcal{F}$. Note the integral still must be projected, via ${\mathcal{P}}$,
on the right hypergeometric subspace.

###  B.2 Evaluation of the bulk standard integral

We consider 3 points, which are contracted with a full dot product $\cdot$,
and one of them not on the lightcone, $P_{3}=-V$. The standard integral
becomes

$\displaystyle\mathcal{F}_{(a,b,c)}$
$\displaystyle=\mathcal{P}\int\frac{d\beta
d\gamma}{\beta\gamma}\frac{\beta^{b}\gamma^{c}}{\Big{(}-(P_{1}+\beta