The operator product expansion in quantum field theory is the statement that
when two local operators are “sufficiently close” to each other, they can be
replaced by another local operator, or rather a sum of them:

$\phi_{1}(x)\phi_{2}(y)\xrightarrow{x\to y}\sum_{i}f_{i}(x-y)\phi_{i}(y),$
(5.1)

It does not matter whether the operator $\phi_{i}$ on the right-hand side is
inserted at $x$ or at $y$, or at the middle point $(x+y)/2$ since this
expansion is valid in the limit in which $x$ and $y$ coincide. In fact, the
proportionality factor $f_{i}$ might diverge in the limit $x\to y$, so this is
to be understood in the sense of an asymptotic limit, whose radius of
convergence is strictly-speaking zero.

In non-perturbative quantum field theory, the OPE can be formulated in terms
of the Hilbert space: when the product of operators in eq. (5.1) is acting on
the vacuum, then the completeness of the Hilbert space implies that we can
write

$\phi_{1}(x)\phi_{2}(y)\left|0\right\rangle=\sum_{\left|\Psi\right\rangle}\left|\Psi\right\rangle\left\langle\Psi\right|\phi_{1}(x)\phi_{2}(y)\left|0\right\rangle,$
(5.2)

where the sum is over states $\left|\Psi\right\rangle$ forming an orthonormal
basis. Among these are states obtained acting with a local operator
$\phi_{i}(y)$ on the vacuum, so that we have

$\phi_{1}(x)\phi_{2}(y)\left|0\right\rangle=\sum_{i}f_{i}(x-y)\phi_{i}(y)\left|0\right\rangle+\ldots,$
(5.3)

where the object $f_{i}$ is related to a ratio of Wightman functions

$f_{i}(x-y)\approx\lim_{z\to\infty}\frac{\left\langle
0\right|\phi_{i}(z)\phi_{1}(x)\phi_{2}(y)\left|0\right\rangle}{\left\langle
0\right|\phi_{i}(z)\phi_{i}(y)\left|0\right\rangle}.$ (5.4)

This formulation remains however imprecise, and it is not quite obvious how to
make it more rigorous using only general principles of quantum field theory.

In conformal field theory, however, the operator product expansion goes to a
completely new level of rigor, thanks to the following observations:

  * •

2-point functions of primary operators are _diagonal_ (i.e. only identical
primaries have a 2-point function), which implies that states created by
different primaries are orthogonal. The norm of primary states is also known
in terms of the operators’ normalization.

  * •

3-point functions are known (see section 4), which means that the
proportionality coefficients $f_{i}$ are in fact fixed up to an overall
multiplicative factor.232323There is more than one factor if the operators
$\phi_{1}$ and $\phi_{2}$ carry spin.

  * •

There is no other contribution to the OPE beyond those of local operators
acting on the vacuum. This property is due to the state/operator
correspondence that is discussed next.

###  5.2 The state/operator correspondence

In conformal field theory there exists a one-to-one correspondence between the
states on a given time slice and local operators defined by their scaling
dimensions and representations under the Lorentz group.

The fact that local operators define states is obvious in Minkowski space-
time, by the Wightman axioms.242424After appropriate smearing, as discussed in
section 3. But it is also true in the analytic continuation to Euclidean
space. To understand this, remember that a local operator inserted at a
generic Minkowski coordinate $x$ can be expressed using eq. (3.3) as an
operator living on the surface $x^{0}=0$ and evolved unitarily,

$\phi(x)\left|0\right\rangle=e^{ix^{0}P^{0}}e^{-i\vec{x}\cdot\vec{P}}\phi(0)\left|0\right\rangle.$
(5.5)

Since the spectrum of $P^{0}$ is non-negative, this can be analytically
continued to any value of $x^{0}$ in the upper-half complex plane; on the
contrary, if $x^{0}$ had a negative imaginary part, then there would be states
of arbitrarily high energy, with divergent norm. By going to purely imaginary
values, $x^{0}=i\tau$, we can therefore define a state of the theory through
an operator insertion in Euclidean space, provided that this operator is
inserted at Euclidean time $\tau>0$.

This has an immediate interpretation in terms of the Euclidean path integral:
a state on the Euclidean surface $\tau=0$ (which is _identical_ to the
original Minkowski time slice $x^{0}=0$) is defined by a path integral over
all field configurations restricted to the region $\tau<0$. This is valid with
any number of operators inserted at separated points of Euclidean time
$\tau<0$ (including no operators, corresponding to the vacuum state). But the
identification also works the other way around: a given state on the surface
$\tau=0$ defines a boundary condition for the path integral at $\tau<0$, which
might correspond to some number of local operators inserted in the “past” (or
a superposition of such configurations). In this way, any Euclidean
correlation function in which the operators are ordered in Euclidean time
$\tau$ can be given a Hilbert space interpretation.

Once we adopt the Euclidean path integral perspective, then it does not matter
whether we use $P^{0}$ as the Hamiltonian, or another conformal generator, as
long as the surface at $\tau=0$ is part of the foliation that it defines. For
instance, one can use the so-called conformal Hamiltonian
$\frac{1}{2}(P^{0}-K^{0})$, which foliates Euclidean space as shown in figure
2. We mentioned already in section 2 that this combination of generators is in
fact equivalent to the dilatation generator $D$, in the sense that they are
related by a rotation in the compact subgroup $\text{SO}(d+1)$ of the
conformal group $\text{SO}(d+1,1)$. This means that there exists a conformal
transformation that maps the surface $\tau=0$ to the unit sphere, and the
other surfaces related by Hamiltonian evolution to spheres of different radii
(see again figure 2). Foliating Euclidean space with spheres centered at the
origin and using the dilatation generator as a Hamiltonian is called _radial
quantization_. In comparison, the physically-equivalent choice of Hamiltonian
$\frac{1}{2}(P^{0}-K^{0})$ is often called _N-S quantization_.

The importance of these two quantizations in conformal field theory is due to
the fact that they have fixed points, unlike equal-time quantization: evolving
back in “time”, the path integral shrinks to a ball of arbitrarily small
radius surrounding one of these fixed points (the origin in radial
quantization, or the south pole in N-S quantization). This means that any
state on the unit sphere, or on the plane $\tau=0$, can be related by
Hamiltonian evolution to a state _localized_ at the fixed point. It is
therefore equivalent to a state created by a local operator inserted there.
This is in essence the argument showing that the state/operator correspondence
goes both ways: a local operator defines a state, but any state also defines a
local operator (or rather a superposition of local operators).

This correspondence provides a posteriori a justification for our choice to
organize the operators into irreducible representations of the Lorentz group
with a definite scaling dimension: the diagonalization can be performed at the
level of the Hilbert space, and then radial quantization used to argue that
each state corresponds to a local operator. Note that there are both primary
and descendant states, in the sense that the local operator does not have to
be a primary.

Many introductory courses in conformal field theory begin in fact with radial
quantization, as it provides a compelling picture in Euclidean space. The
connection with unitary quantum field theory is not easy to establish, though.
Hermitian conjugation in the unitary theory requires taking $x^{0}=i\tau$ to
$(x^{0})^{*}=-i\tau$ , i.e. it corresponds to a reflection across the surface
of $\tau=0$ in N-S quantization. In radial quantization, this becomes
inversion $r\to r^{-1}$: correlation functions compute the overlap between
states corresponding to a path integral inside the unit sphere with states
corresponding to a path integral outside the sphere. The smearing of states
required in Minkowski space-time is replaced by the limit $r\to\infty$ needed
to define conjugate states in terms of local operators. Once this limit is
taken into account, the unitarity bounds can be derived using the interplay of
generators $P_{\mu}$ and $K_{\mu}$, which are in fact conjugate to each other
under inversion. This procedure reproduces the results of our section 3, but
it might be less intuitive.

###  5.3 The conformal OPE

The lesson we learn from radial quantization is that every state in the
Hilbert space can be written as a linear combination of states created by the
action of a single local operator on the vacuum, including primaries as well
as descendants. Applying this lesson to Wightman functions in Minkowski space-
time, we can be tempted to write the complete Hilbert space as

$\mathscr{H}\stackrel{{\scriptstyle?}}{{=}}\text{span}\big{\\{}\left|0\right\rangle,\phi(0)\left|0\right\rangle,\partial_{\mu}\phi(0)\left|0\right\rangle,\ldots\big{\\}}.$
(5.6)

The problem is that these are not normalizable states. Instead, we can
consider states created by primary operators inserted at any point $x$ in
Minkowski space-time,

$\mathscr{H}\sim\text{span}\big{\\{}\left|0\right\rangle,\phi(x)\left|0\right\rangle,\ldots\big{\\}},$
(5.7)

where we implicitly understand that the local operators should be smeared
against test functions. This gives a good description of the Hilbert space,
but it is not very practical because primary operators inserted at different
points (or smeared against distinct test functions) are not orthogonal to each
other. This leads us to consider yet another representation of the Hilbert
space, using local operators in momentum space,

$\mathscr{H}\sim\text{span}\big{\\{}\left|0\right\rangle,\widetilde{\phi}(p)\left|0\right\rangle,\ldots\big{\\}}.$
(5.8)

States carrying different momenta are orthogonal to each other, and it is
moreover sufficient to consider momenta $p$ inside the forward light cone.
This means that we can express the completeness of the Hilbert space through
the resolution of the identity

$\mathds{1}=\left|0\right\rangle\left\langle
0\right|+\sum_{i}\int\limits_{p^{0}>\left|\vec{p}\right|}\\!\\!\frac{d^{d}p}{(2\pi)^{d}}\frac{\widetilde{\phi}_{i}(p)\left|0\right\rangle\left\langle
0\right|\widetilde{\phi}_{i}(-p)}{2\pi C\,(-p^{2})^{\Delta-d/2}}+\ldots,$
(5.9)

where the sum is over all primary operators of the theory. The denominator is
simply the scalar 2-point function, eq. (3.42), and states that carry spin
have been omitted for simplicity: they could be included in the same way after
inverting the Lorentz tensor appearing in the 2-point function, which is
positive-definite by unitarity.

This identity can be applied whenever we encounter a product of two local
operators acting on the vacuum to rewrite the product as a sum of local
operators. It takes its simplest form when the operators are all expressed in
momentum space:

$\widetilde{\phi}_{1}(p_{1})\widetilde{\phi}_{2}(p_{2})\left|0\right\rangle=\sum_{i}\widetilde{f}_{i}\,\widetilde{\phi}_{i}(p_{1}+p_{2})\left|0\right\rangle.$
(5.10)

This is the conformal OPE in Minkowski space-time, which reduces higher-point
functions to lower-point ones. It is trivial when applied to 2-point
functions, as only the vacuum state appears in the sum (the identity operator
can be interpreted as a primary operator in its own right). When applied to a
3-point function, it can be used to determine the proportionality factor
$\widetilde{f}_{i}$ in terms of the scaling dimension, Lorentz representation,
and 3-point coefficients $\lambda_{12i}$. Using this OPE further, a 4-point
function can be decomposed into a sum of terms whose kinematics is completely
determined by conformal symmetry.

This form of the OPE is however rarely used, for two reasons: our
understanding of Wightman 3-point functions in momentum space is incomplete,
and the convergence of this OPE is not particularly nice. By this, we mean
that the OPE converges in the distributional sense, i.e. only after smearing
with smooth test functions. In contrast, we will see next that the Euclidean
OPE is _absolutely_ convergent!

Since we saw before that even Schwinger functions can be given a Hilbert space
interpretation, we might as well formulate the OPE directly in Euclidean
space. We write

$\phi_{1}(x_{1})\phi_{2}(x_{2})=\sum_{i}f_{i}\left(x_{1}-x_{2},\partial_{x_{2}}\right)\phi_{i}(x_{2}).$
(5.11)

Unlike the Minkowskian OPE (5.10), the convergence of this OPE is conditional:
it is only true inside a correlation function in which the points $x_{1}$ and
$x_{2}$ can be separated from all other insertions of local operators by a
quantization surface, say a sphere in radial quantization. Since the center
and radius of the sphere can be chosen at will thanks to translation and scale
symmetry, this includes many (in fact nearly all) configurations of points, as
we shall see below.

In the notation of eq. (5.11), the object $f_{i}$ does not only depend on the
distance between the two operators, but also on derivatives acting on the
primary operator $\phi_{i}$. This is a compact way of including all
descendants in the sum. Keep in mind however that there are infinitely many
such descendants, so $f_{i}$ is itself an infinite series. The only exception
is the identity operator, which enters the OPE if and only if the two primary
operators on the left-hand side of (5.11) are identical: in a 2-point
function, the OPE takes the form

$\langle\phi(x)\phi(0)\rangle=\frac{1}{(x^{2})^{\Delta_{\phi}}}\langle\mathds{1}\rangle,$
(5.12)

with $\langle\mathds{1}\rangle=1$, from which we deduce

$f_{\mathds{1}}(x,\partial)=\frac{1}{(x^{2})^{\Delta_{\phi}}}.$ (5.13)

When viewed as a primary operator, the identity does not have any descendant.
In all other cases, the value of $f_{i}(x,\partial)$ can be determined from
the 3-point function: using the OPE in the scalar 3-point function (4.23), one
can for instance deduce that

$f_{i}(x,\partial)=\frac{\lambda_{12i}}{(x^{2})^{\Delta_{12,i}}}\left[1+ax^{\mu}\partial_{\mu}+b_{1}x^{\mu}x^{\nu}\partial_{\mu}\partial_{\nu}+b_{2}x^{2}\partial^{2}+\ldots\right]$
(5.14)

where

$\displaystyle a$ $\displaystyle=\frac{\Delta_{i1,2}}{\Delta_{i}},$
$\displaystyle b_{1}$
$\displaystyle=\frac{\Delta_{i1,2}(\Delta_{i1,2}+1)}{2\Delta_{i}(\Delta_{i}+1)},$
(5.15) $\displaystyle b_{2}$
$\displaystyle=-\frac{\Delta_{i1,2}\Delta_{i2,1}}{4\Delta_{i}(\Delta_{i}+1)\left(\Delta_{i}-\frac{d-2}{2}\right)},$

and so on. This expression is valid when $\phi_{i}$ is a scalar operator, but
there are also operators with spin that enter the OPE, even though $\phi_{1}$
and $\phi_{2}$ are scalars: the operators being inserted at separated points
$x_{1}\neq x_{2}$ in space, their product carries angular momentum; for a
point-like operator on the other side of the OPE, this angular momentum is
realized as internal spin. Note however that not all Lorentz representations
appear in the OPE of two scalars, but only symmetric tensors. Moreover, when
the operators are identical, there are only tensors with even numbers of
Lorentz indices. The list of representations that appear can be determined
from the group theory associated with Lorentz symmetry. On the contrary, any
scaling dimension $\Delta$ can appear.

There are therefore still many unknowns in the OPE. We shall see in the next
section that there are in fact always infinitely many primary operators in the
sum. But when compared with the OPE in a generic quantum field theory, the
conformal OPE is extremely rigid: in a scale-invariant QFT the coefficients
$a$, $b_{1}$, $b_{2}$, $\ldots$, entering the definition of the coefficient
$f_{i}(x,\partial)$ are all theory-dependent factors. In CFT, on the contrary,
they are completely fixed by the kinematics. The only dynamical information is
contained in the OPE coefficients $\lambda_{12i}$ multiplying the contribution
of a primary and all of its descendants.

##  6 The conformal bootstrap

Our approach to conformal field theory so far has been an _algebraic_ one: in
section 3 we have defined primary operators as irreducible representations of
the conformal group, characterized by their transformations under Lorentz
symmetry and by a scaling dimension. In section 5 we have seen that these
operators can be combined using the operator product expansion. The OPE
coefficients play the role of the structure constants of this operator
algebra. The set of all scaling dimensions and representations under the
Lorentz group of a theory, together with the OPE coefficients,

$\big{\\{}(\Delta_{i},R_{i}),\lambda_{ijk}\big{\\}}$ (6.1)

is called the _CFT data_. A conformal field theory is completely defined by
its CFT data: any correlation function can be computed from it, by repeated
use of the OPE.

But does any CFT data define a good theory? The answer to this question is no:
the algebra of local operators must close, giving very constraining
consistency conditions.252525This is very similar in spirit to the
classification of simple Lie groups, leading to the well-known families
$\text{SU}(N)$, $\text{SO}(N)$, $Sp(2N)$ and a few exceptional groups. The
precise formulation of this closure condition depends on whether we are
talking about the Euclidean or Lorentzian conformal groups. For a conformal
field theory in Minkowski space-time, the crucial point is that the OPE must
be consistent with the micro-causality condition (3.9). But the consequences
of micro-causality are quite difficult to track down in practice.

Instead, there is a much simpler consistency condition for Schwinger functions
in Euclidean space: since these functions are symmetric under the exchange of
operators (a property that is in fact related to micro-causality), then the
Euclidean OPE must be _associative_. To illustrate this, let us focus on
4-point functions. Imposing associativity of the OPE on _all_ 4-point
functions is in fact sufficient to guarantee associativity of higher-point
functions. For simplicity let us consider as before the case of 4 identical
scalar primary operators with scaling dimension $\Delta_{\phi}$:

$\langle\phi(x_{1})\phi(x_{2})\phi(x_{3})\phi(x_{4})\rangle=\frac{1}{(x_{12}^{2}x_{34}^{2})^{\Delta_{\phi}}}g(u,v).$
(6.2)

###  6.1 Conformal blocks

The simplest way to understand how this 4-point function can be expressed in
terms of the CFT data is to use the OPE (5.11) on it twice:

$\langle\phi(x_{1})\phi(x_{2})\phi(x_{3})\phi(x_{4})\rangle=\sum_{i}f_{i}\left(x_{1}-x_{2},\partial_{x_{2}}\right)f_{i}\left(x_{3}-x_{4},\partial_{x_{4}}\right)\langle\phi_{i}(x_{2})\phi_{i}(x_{4})\rangle$
(6.3)

There is a single sum on the right-hand side because the 2-point function is
diagonal: it vanishes unless the two primary operators are identical. To put
this in a convenient form, let us factor out the OPE coefficient $\lambda_{i}$
corresponding to the 3-point function $\langle\phi\phi\phi_{i}\rangle$, as
well as a particular power of the distance, defining

$f_{i}(x,\partial)=\frac{\lambda_{i}}{(x^{2})^{\Delta_{\phi}}}f^{\prime}_{i}(x,\partial),$
(6.4)

The new function $f^{\prime}_{i}$ is then of the form

$f^{\prime}_{i}(x,\partial)=(x^{2})^{\Delta_{i}/2}\left[1+\ldots\right],$
(6.5)

where $\Delta_{i}$ is the scaling dimension of the operator entering the OPE
(the _internal_ operator), not to be confused with the scaling dimension
$\Delta_{\phi}$ of the operators in the original correlation function (the
_external_ operators). We have now

$\langle\phi(x_{1})\phi(x_{2})\phi(x_{3})\phi(x_{4})\rangle=\frac{1}{(x_{12}^{2}x_{34}^{2})^{\Delta_{\phi}}}\sum_{i}\lambda_{i}^{2}g_{i}(u,v),$
(6.6)

where we have defined

$g_{i}(u,v)=f^{\prime}_{i}\left(x_{1}-x_{2},\partial_{x_{2}}\right)f^{\prime}_{i}\left(x_{3}-x_{4},\partial_{x_{4}}\right)\frac{1}{(x_{24})^{\Delta_{i}}}.$
(6.7)

This should be compared with the representation (6.2) of the 4-point function,
in terms of which

$g(u,v)=\sum_{i}\lambda_{i}^{2}g_{i}(u,v).$ (6.8)

The $g_{i}$ are called _conformal blocks_ [21]: they represent the
contribution of a single primary and of all its descendants to the 4-point
function. They are conformally invariant,262626In comparison, the correlation
function is not invariant but _covariant_. and must therefore be functions of
the cross-ratios $u$ and $v$, even though this is not at all obvious from
their definition.

This definition is in fact not very practical: working out all the terms in
the series of $f^{\prime}_{i}$ is difficult, and there are also Lorentz
indices that need to be contracted when the intermediate operator carries
spin. But it is convenient to examine the limit $x_{1}\to x_{2}$, $x_{3}\to
x_{4}$, or equivalently $u\to 0$: in this case, the leading term in the OPE
shows that

$g_{i}(u,v)=u^{\Delta_{i}/2}\left[1+\ldots\right].$ (6.9)

Primary operators with the lowest scaling dimensions give the leading
contribution to the 4-point function in this limit. The operator with the
absolute lowest scaling dimension is the identity, for which there are no
descendants, so that272727In this special case it does not make much sense to
speak of an OPE coefficient, but it is conventional to take
$\lambda_{\mathds{1}}=1$.

$g_{\mathds{1}}(u,v)=1.$ (6.10)

Note that this statement is related to the more general _cluster decomposition
principle_ in quantum field theory: since there is no absolute scale in CFT,
the limit $x_{1}\to x_{2}$ is equivalent to a limit in which the other points
$x_{3}$ and $x_{4}$ are sent very far away, all the way to infinity, and in
this case one expects on general grounds that the correlation function
factorizes as

$\langle\phi(x_{1})\phi(x_{2})\phi(x_{3})\phi(x_{4})\rangle\xrightarrow{x_{3},x_{4}\to\infty}\langle\phi(x_{1})\phi(x_{2})\rangle\langle\phi(x_{3})\phi(x_{4})\rangle.$
(6.11)

This factorization is thus reproduced by the OPE in the limit $u\to 0$.

A better way of computing the conformal block is based on the Casimir
invariant of the conformal group [22]. From the group algebra, one can verify
that the combination of generators (2.39)

$\mathcal{C}_{2}=-\frac{1}{2}J_{MN}J^{MN}$ (6.12)

commutes with all individual generators, e.g.

$\left[\mathcal{C}_{2},P^{\mu}\right]=0.$ (6.13)

In group theory language, $\mathcal{C}_{2}$ is called the quadratic Casimir
invariant. Since it commutes with translations, its action on a state created
by a local operator is the same no matter where the operator is inserted, or
whether it is a primary or a descendant: for a scalar primary,

$\mathcal{C}_{2}\,\phi(x)\left|0\right\rangle=\Delta(\Delta-d)\,\phi(x)\left|0\right\rangle.$
(6.14)

Exercise 6.1 Compute the eigenvalue of the quadratic Casimir operator from the
commutators of the conformal generators with $\phi(x)$. For simplicity, work
at $x=0$.

For a symmetric tensor with $\ell$ Lorentz indices, the eigenvalue contains an
additional term that is simply the quadratic Casimir of the Lorentz/rotation
group in $d$ dimensions,

$\mathcal{C}_{2}\,\phi^{\mu_{1}\ldots\mu_{\ell}}(x)\left|0\right\rangle=\big{[}\Delta(\Delta-d)+\ell(\ell+d-2)\big{]}\phi^{\mu_{1}\ldots\mu_{\ell}}(x)\left|0\right\rangle.$
(6.15)

The key idea is now to evaluate the value of this Casimir operator between
states that are each created by two local operators, i.e. compute the
correlation function

$\langle\phi(x_{1})\phi(x_{2})\mathcal{C}_{2}\phi(x_{3})\phi(x_{4})\rangle$
(6.16)

Applying the OPE once, this can be written

$\langle\phi(x_{1})\phi(x_{2})\mathcal{C}_{2}\phi(x_{3})\phi(x_{4})\rangle=\sum_{i}f_{i}\left(x_{3}-x_{4},\partial_{x_{4}}\right)\langle\phi(x_{1})\phi(x_{2})\mathcal{C}_{2}\phi_{i}(x_{4})\rangle,$
(6.17)

where there are now two possible ways of computing each term on the right-hand
side: either $\mathcal{C}_{2}$ acts to the right, and then the eigenvalue
equation (6.15) can be used, or it acts to the left. In the second case, the
individual generators forming $\mathcal{C}_{2}$ must be commuted with
$\phi(x_{1})$ and $\phi(x_{2})$ successively. This is most easily done in a
convenient reference frame, for instance in the $z$-frame described in section
4 and shown in figure 5. In this case, one finds that the action of
$\mathcal{C}_{2}$ on each individual conformal block is a second-order
differential operator in $z$ and $\bar{z}$, which we denote by
$D_{z,\bar{z}}$. This means that every conformal block satisfies a
differential equation of the form

$D_{z,\bar{z}}\,g_{i}(z,\bar{z})=\big{[}\Delta_{i}(\Delta_{i}-d)+\ell_{i}(\ell_{i}+d-2)\big{]}g_{i}(z,\bar{z}).$
(6.18)

Together with a boundary condition provided by eq. (6.9), this is sufficient
to determine the conformal blocks entirely. In even space-time dimensions, the
solution is in the form of products of hypergeometric functions. In odd
dimensions, there is no known closed-form solution, but the Casimir equation
can be conveniently solved term-by-term in a series expansion.

Note that in our specific example where the external operators are all
identical, the conformal blocks do not depend on the scaling dimension
$\Delta_{\phi}$, but only on the scaling dimension $\Delta_{i}$ and spin
$\ell_{i}$ of the internal operator. This property is not true in general.

Figure 5: Two convenient conformal frames that describe the 4-point function.
On the left, 3 points are mapped to $0$, $1$ and $\infty$ using a conformal
transformation, and the complex coordinate $z$ describes the position of the
fourth point. Applying a conformal transformation in the plane, one can reach
the configuration on the right, with operators placed at antipodal points on a
pair of concentric circles, with the larger circle having unit radius; this
configuration is parameterized by the complex coordinate $\rho$, satisfying
$\left|\rho\right|\leq 1$.

###  6.2 OPE convergence

So far we have been using the OPE in the 4-point function without worrying
about its convergence. This is fine as long as the points $x_{1}$ and $x_{2}$
can be separated from $x_{3}$ and $x_{4}$ by a sphere. In terms of the
coordinate $z$, this is obviously true for all $\left|z\right|<1$ (the unit
disk delimited by a dashed line in figure 5): then the product of operators
$\phi(x_{1})\phi(x_{2})$ can be replaced by a sum of local operators at
$x_{1}=0$ by the state-operator correspondence, and therefore the OPE
converges since it is a Hilbert-space sum. But the domain of convergence is in
fact much larger: radial quantization can be used around any point in space,
not just $x_{1}$, and it is not hard to see that in most configurations one
can draw a circle surrounding $x_{1}$ and $x_{2}$, but excluding $x_{3}$ and
$x_{4}$. This is actually possible in all cases except when $z$ is real with
$z>1$.282828In a generic frame, the only case in which the OPE
$\phi(x_{1})\times\phi(x_{2})$ does not converge is when all four points are
on a circle (this includes a line) and the points $x_{1}$ and $x_{2}$ are not
consecutive, i.e. they are separated by $x_{3}$ and $x_{4}$.

This is even more easily seen in a different reference frame: using a
conformal transformation in the plane of $z$, one can map the 4 points to a
configuration depicted on the right-hand side of figure 5, with the pairs of
points $(x_{1},x_{2})$ and $(x_{3},x_{4})$ placed at antipodal points on two
circles centered at the origin [23]. The map is given by

$\rho=\frac{z}{\left(1+\sqrt{1-z}\right)^{2}}\qquad\Leftrightarrow\qquad
z=\frac{4\rho}{(1+\rho)^{2}}.$ (6.19)

It takes all of the complex $z$ plane to the unit disk in $\rho$, with the
half-line $z>1$ being mapped to the unit circle. In this frame, it is now
obvious that the OPE converges in radial quantization unless
$\left|\rho\right|=1$.

Figure 6: The $\rho$-coordinate configuration of figure 5 as seen on the
radial quantization cylinder. The two segment $x_{1}-x_{2}$ and $x_{3}-x_{4}$
form an angle $\theta$, and they are separated in cylinder time by
$\tau=\log(r)$. $\theta$ and $\tau$ are related to the flat-space
configuration by $\rho=e^{\tau+i\theta}=re^{i\theta}$.

This conformal frame is also very useful for understanding the convergence
properties of the OPE. It can be seen as a configuration on the radial
quantization cylinder, as in figure 6, with coordinates

$x_{1}=(\tau,\vec{n}),\qquad x_{2}=(\tau,-\vec{n}),\qquad
x_{3}=(0,\vec{n}^{\prime}),\qquad x_{4}=(0,-\vec{n}^{\prime}),$ (6.20)

where $\tau=\log(r)$ is the “time” component, and the “space” components are
unit vectors $\vec{n}$ and $\vec{n}^{\prime}$ that parameterize a direction on
the sphere $S^{d-1}$, and form an angle $\theta$ between them. The connection
with the flat-space configuration is through

$\rho=r\,e^{i\theta}.$ (6.21)

A conformal block corresponds to the projection of this configuration onto
intermediate primary and descendant states with scaling dimensions
$\Delta_{i}+n$ ($n$ being an integer) and spin $j$ (whose range is determined
by the spin $\ell$ of the primary). On general grounds, one can therefore
expect the conformal block to take the form

$g_{i}(r,\theta)=\sum_{n,j}B_{n,j}r^{\Delta_{i}+n}\mathcal{C}_{j}^{(d-2)/2}(\cos\theta),$
(6.22)

where $\mathcal{C}_{j}^{(d-2)/2}(\cos\theta)$ is a Gegenbauer polynomial
obtained from the contraction of the symmetric and traceless tensor
$\vec{n}^{\mu_{1}}\cdots\vec{n}^{\mu_{j}}$ with
$\vec{n}^{\prime\mu_{1}}\cdots\vec{n}^{\prime\mu_{j}}$. The coefficients
$B_{n,j}$ are positive by unitarity because they are the norm of eigenstates
of definite spin (labeled by $n$ and $j$). They are in fact rational functions
of $\Delta_{i}$. This representation is very useful: on the one hand, it gives
an efficient way of evaluating the conformal block at any given level of
precision by truncating the series in $n$, since by assumption $r<1$; on the
other hand, it shows that the $g_{i}$ are nice (i.e. analytic) functions of
scaling dimension $\Delta_{i}$ for all values of $\Delta_{i}$ above the
unitarity bound.

Beyond individual conformal blocks, the $\rho$ coordinates also show how the
OPE converges as a function of $r$: in a configuration with $r\ll 1$, the
series is dominated by operators of low $\Delta$ (both primary and
descendants), and a good approximation of the 4-point function can be obtained
from a truncated OPE. For instance, a configuration in which the operators are
placed on the corners of a square corresponds to $z=\frac{1}{2}$, or
$\left|\rho\right|=(1+\sqrt{2})^{-2}\approx 0.17$. Even though the point
$x_{1}$ and $x_{2}$ are not obviously close to each other in this
configuration, $r=\left|\rho\right|$ is a small number and the convergence of
the OPE is very fast.

This is a fantastic property of the _Euclidean_ OPE. The situation is not as
nice in Minkowski space-time. The power-law dependence in $r$ seen in eq.
(6.22) is the consequence of exponential damping in Euclidean “time” $\tau$
($r^{\Delta_{i}}=e^{\tau\Delta_{i}}$ with negative $\tau$), following from
Hamiltonian evolution in radial quantization. In Minkowski space-time,
independently of the choice of quantization, the evolution is always _unitary_
, e.g. $e^{iEt}$: there is in principle no reason why operators of higher
scaling dimension should contribute less than those of lower dimension in a
generic configuration. This does not mean that the Minkowskian OPE is
uninteresting, but it is more difficult to harness than the Euclidean one.

###  6.3 The crossing equation and simple solutions

We have so far only discussed one particular OPE, in which the points $x_{1}$
and $x_{2}$ were assumed to be closed to each other in a conformal sense. But
since Schwinger functions are symmetric under the exchange of operators, there
is no reason not to consider different OPEs, e.g. between the operators
$\phi(x_{1})$ and $\phi(x_{3})$, or $\phi(x_{1})$ and $\phi(x_{4})$. In the
vast majority of configurations of 4 points, the space can be cut in half by a
sphere in three inequivalent ways. Even in the special case in which the 4
points are on a circle, there are still two inequivalent ways of surrounding
pairs of points with a sphere. This means that we expect different OPEs to
converge in all cases.292929An equivalent statement is that Schwinger function
can be obtained by analytic continuation from multiple Wightman functions with
distinct orderings of operators.

$\sum\limits_{i}$ $=\leavevmode\nobreak\ \sum\limits_{j}$

Figure 7: Diagrammatic representation of the crossing equation at the core of
the conformal bootstrap, relating two distinct OPEs.

This implies that besides the expansion

$g(u,v)=\sum_{i}\lambda_{i}^{2}g_{i}(u,v),$ (6.23)

one can write a similar expansion for the 4-point function in a configuration
related by crossing symmetry, e.g.

$\left(\frac{u}{v}\right)^{\Delta_{\phi}}g(v,u)=\left(\frac{u}{v}\right)^{\Delta_{\phi}}\sum_{i}\lambda_{i}^{2}g_{i}(v,u).$
(6.24)

Since the two are equal, we must have

$\sum_{i}\lambda_{i}^{2}g_{i}(u,v)=\left(\frac{u}{v}\right)^{\Delta_{\phi}}\sum_{i}\lambda_{i}^{2}g_{i}(v,u).$
(6.25)

This equation is represented pictorially in figure 7. Note that in the case of
identical external operators the two sums are over the same set of primary
operators, but in general these could be different sums.

This is a relatively simple equation, but it is quite difficult to solve. We
saw above that the OPE is dominated by intermediate operators of low
$\Delta_{i}$ in the limit $u\to 0$, but this only applies to the left-hand
side: taking the same limit on the right-hand side, one reaches the boundary
of the domain of convergence of the OPE. Using the known expression for the
conformal blocks in terms of hypergeometric functions, it can be shown that
the limit $z\to 1$ approaches a branch cut around which

$g_{i}(z,\bar{z})\propto\log(1-z).$ (6.26)

This means that the right-hand side of the crossing equation is dominated by
terms of the form

$\left(\frac{u}{v}\right)^{\Delta_{\phi}}g_{i}(v,u)\propto
z^{2\Delta_{\phi}}\log(z),$ (6.27)

in the limit $z\to 0$. A finite sum of terms of this form cannot reproduce the
leading constant contribution from the identity operator on the left-hand
side. This is the first wisdom contained in the crossing equation (6.25): it
can never be satisfied block-by-block, but only by an infinite sum of
conformal blocks, and there must therefore be an infinite number of primary
operators in the OPE.

Before looking into the clever way in which the conformal bootstrap deals with
this problem, let us examine the simplest solution to crossing symmetry that
we know of. A possible function that transforms covariantly under the
conformal group and is crossing symmetric is the following combination of
2-point functions:

$\displaystyle\langle\phi(x_{1})\phi(x_{2})\rangle\langle\phi(x_{3})\phi(x_{4})\rangle+\langle\phi(x_{1})\phi(x_{3})\rangle$
$\displaystyle\langle\phi(x_{2})\phi(x_{4})\rangle$ $\displaystyle+$
$\displaystyle\langle\phi(x_{1})\phi(x_{4})\rangle\langle\phi(x_{2})\phi(x_{3})\rangle,$
(6.28)

corresponding to

$g(u,v)=1+u^{\Delta_{\phi}}+\left(\frac{u}{v}\right)^{\Delta_{\phi}}.$ (6.29)

Performing an expansion at small $u$ and matching each term with a
hypothetical primary operator, one finds an infinite spectrum of operators
characterized by their spin $\ell$ and scaling dimension

$\Delta_{i}=2\Delta_{\phi}+2n+\ell$ (6.30)

where $n=0,1,2,\ldots$ This defines a valid 4-point correlation function in a
theory called _generalized free field theory_ (or sometimes mean free theory,
or Gaussian theory). It is similar to a free theory in the sense that the OPE
$\phi\times\phi$ contains operators that have the same scaling dimension and
spin as _composites_ of the field $\phi$, of the schematic form

$\left[\phi(\partial^{2})^{n}\partial^{\mu_{1}}\ldots\partial^{\mu_{\ell}}\phi\right].$
(6.31)

The free scalar theory is a special realization of this in which only
operators with $n=0$ are present in the OPE (the other vanish by the equation
of motion $\partial^{2}\phi=0$, or they are descendants), and these operators
are higher-spin conserved current. Generalized free field theory is usually
considered non-local: its definition through the above 4-point function does
not include an energy-momentum tensor.303030Also it can be defined through the
non-local Lagrangian
$\mathscr{L}=\frac{1}{2}\phi(\partial^{2})^{d/2-\Delta_{\phi}}\phi$. It is
nevertheless a physically interesting case because the large-$N$ limit of
gauge theory takes precisely this form: if one considers the 4-point function
of a gauge-invariant composite operator such as the fermion bilinear
$\bar{\psi}_{a}\psi_{a}$, then its decomposition into conformal blocks is
given at leading order in $1/N$ by a sum of “double-trace” operators with
scaling dimensions $2\Delta_{\phi}+2n+\ell$, whereas “single-trace” operators
(among which the energy-momentum tensor) only enter at sub-leading order in
$1/N$.

###  6.4 The numerical bootstrap

The revolutionary idea that gave birth to the modern conformal bootstrap
appeared in 2008 [24].313131The “original” conformal bootstrap was developed
in two dimensions [25], but it relies on very different techniques than the
“modern” conformal bootstrap valid in any $d$. It leverages in particular the
infinite Virasoro algebra that is specific to $d=2$. The starting point is to
rewrite the crossing equation (6.25) as

$\sum_{i}\lambda_{i}^{2}\left[v^{\Delta_{\phi}}g_{i}(u,v)-u^{\Delta_{\phi}}g_{i}(v,u)\right]=0,$
(6.32)

and to realize that it can be viewed as an equation in an infinite vector
space

$\sum_{i}\lambda_{i}^{2}\,\vec{F}_{i}(\Delta_{\phi},\Delta_{i},\ell_{i})=0,$
(6.33)

where the $\lambda_{i}^{2}$ are positive coefficients (remember that
$\lambda_{i}$ must be real in a unitary quantum field theory). The components
of the vector $\vec{F}_{i}$ correspond to
$v^{\Delta_{\phi}}g_{i}(u,v)-u^{\Delta_{\phi}}g_{i}(v,u)$ evaluated at an
infinite set of points $(u,v)$, or equivalently as the coefficients of a
Taylor expansion around some preferred point.323232Traditionally the numerical
bootstrap uses a Taylor expansion around the point $u=v=\frac{1}{4}$, which
corresponds to $z=\frac{1}{2}$, i.e. the symmetric point that has the fastest
converging OPEs. $\vec{F}_{i}$ has infinitely many components, but the
equation is also true if we truncate it to any finite subset. Some interesting
results can already be obtained from considerations about a 2-dimensional
subset, but typically the more stringent bounds that are shown below require
scanning over spaces with a large number of dimensions.

Figure 8: A toy example of the behavior of vectors $\vec{F}_{i}$ in a
3-dimensional space. For each spin $\ell_{i}=0,2,4,6,\ldots$, $\vec{F}_{i}$
draws a smooth curve on the unit sphere as $\Delta_{i}$ varies between the
unitarity bound $\Delta_{\text{min}}$ and infinity. All curves are contained
in the upper hemisphere, except the scalar curve $\ell=0$ that enters the
lower hemisphere when $\Delta<\Delta_{*}$. This means that a putative theory
with no scalar primary operator satisfying $\Delta<\Delta_{*}$ is excluded, as
the crossing equation (6.33) cannot be satisfied: all vectors $\vec{F}_{i}$
point on the same side of the horizontal _separating plane_ , so they cannot
add up to zero. Note that the curves shown here are made up and do not
correspond to actual functionals acting on conformal blocks.

The norm of the vector $\vec{F}_{i}$ is irrelevant in this equation, as it
gets multiplied by a positive factor $\lambda_{i}^{2}$ that we do not know.
But the direction in which the vector $\vec{F}_{i}$ points is crucial: if all
vector $\vec{F}_{i}$ point in the same general direction, for all spins
$\ell_{i}$ and all $\Delta_{i}$ above the corresponding unitarity bound, then
the equation does not have a solution, because no non-trivial linear
combinations of $\vec{F}_{i}$ can ever sum to zero. A toy example of this
mechanism is shown in figure 8. This observation is at the core of all
numerical bootstrap algorithms, which go along the following lines: we begin
with making a hypothesis about the scaling dimension of $\phi$, and of the
operator with the lowest scaling dimension appearing in the OPE
$\phi\times\phi$;333333We know that there exists a crossing symmetric solution
for all values of $\Delta_{\phi}$, namely the generalized free field theory
one. Therefore it is not sufficient to make a hypothesis on $\Delta_{\phi}$
alone. then we study the behavior of the vector $\vec{F}_{i}$ under this
assumption and try to find a separating plane so that all vectors point on the
same side of that plane; if such a plane is found, the equation cannot be
satisfied, which implies that the hypothesis is wrong. Iterating over this
strategy allows to exclude entire regions in the space of possible parameters.
Figure 9 shows the result of this procedure in $d=3$ dimensions.

Figure 9: Figure taken from ref. [26], showing with a dashed line an upper
bound on the scaling dimension $\Delta_{\epsilon}$ of the first operator in
the OPE of two identical scalar operators with scaling dimension
$\Delta_{\sigma}$, obtained from the conformal bootstrap procedure. Without
making any theory-specific assumptions, this bound displays a “kink” close to
the Ising CFT discussed in section 6.5, and marked with a cross. The blue
region is the remaining allowed region after an additional assumption is made,
namely that the OPE takes the form of eq. (6.40). Note that this figure is
outdated: by now the size of the island has shrunk to something smaller than
the uncertainty of the best Ising model Monte-Carlo simulations.

###  6.5 Example: the Ising model in 3 dimensions

Very often, a lot of mileage is gained when one can input additional
assumptions about some target theory. Let us for instance consider the theory
given by the action

$S=\int
d^{3}x\left[-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m^{2}\phi^{2}-\frac{g}{4!}\phi^{4}\right].$
(6.34)

in which $g>0$ so that the potential is bounded below. Note that the free
scalar field has mass dimension $\left[\phi\right]=\frac{1}{2}$ in $d=3$, and
therefore

$\left[m^{2}\right]=2,\qquad\qquad\left[g\right]=1.$ (6.35)

Since both $m^{2}$ and $g$ have positive mass dimensions, they are _relevant_
operators: they determine the dynamics in the low-energy limit (IR), but their
importance decreases at high energy (UV): at energies $E\gg g,|m|$, this
theory approaches the free, massless scalar field. On the other hand, at low
energies the physics depends obviously on $g$ and $m$: when $m^{2}\gg g^{2}$,
this is a theory of a massive scalar of mass $m$; when $m^{2}\ll-g^{2}$, then
the potential has two minima at

$\langle\phi\rangle=\pm\sqrt{-\frac{6m^{2}}{g}},$ (6.36)

with excitations of mass $\sqrt{2}|m|$ around it. Clearly, this theory has two
phases, and there is therefore an intermediate value of $m^{2}$ where a phase
transition must happen (or working in units set by $g$, a critical value of
the dimensionless ratio $m^{2}/g^{2}$). Note that this theory has a
$\mathbb{Z}_{2}$ symmetry corresponding to $\phi\to-\phi$, which is
spontaneously broken in one phase and not in the other.

It turns out that the theory describing the IR physics exactly at the phase
transition is a conformal field theory. Unlike the two phases surrounding it,
it admits excitations of arbitrarily small energy (but they are not
particles). How do we know that? Feynman diagram computations cannot be
trusted in the IR: the approximation given by the (asymptotic) perturbative
series is valid in the UV, but it breaks down in the IR, as in QCD. One way of
understanding the phase transition is to examine the theory in $d\neq 3$
dimensions: the same theory in $d=4-\varepsilon$ has a perturbative fixed
point, called the Wilson-Fisher fixed point. The position of the fixed point
depends on the renormalization scheme, but there are other quantities that are
scheme-independent. This is for instance the case of the anomalous dimension
of the operator $\phi$. In the limit $\varepsilon\to 1$, the state-of-the-art,
many-loop computation gives343434See ref. [27] for a recent, comprehensive
review of perturbative computations in the $\varepsilon$ expansion.

$\gamma_{\phi}\approx
0.0182\qquad\Leftrightarrow\qquad\Delta_{\phi}=\frac{d-2}{2}+\gamma_{\phi}\approx
0.5182.$ (6.37)

This estimate of the scaling dimension coincides with that of a statistical
physics model: the _Ising model_ is a theory of classical spins
$\sigma_{i}=\pm 1$ on a lattice with nearest-neighbor interactions,
characterized by the Hamiltonian

$H=-J\sum_{\langle ij\rangle}\sigma_{i}\sigma_{j}.$ (6.38)

This model has a critical value of $J$ at which the continuum limit is
described by a CFT. At this value, Monte-Carlo simulations indicate that the
correlation between two spins decreases with a power of the distance given by

$\Delta_{\sigma}\approx 0.5181.$ (6.39)

The two theories have completely different microscopic descriptions: one is a
quantum field theory describing particles in Minkowski space-time, and the
other is a simple theory on a Euclidean lattice. Even more surprisingly, the
same critical exponent is found in experiments, such as the critical point of
water and other liquids. This is an example of _universality_.

The explanation for this coincidence is that there are not many candidate
conformal field theories that can describe the phase transition of the
$\phi^{4}$ theory and of the Ising model. These two theories have in common:

  * •

A global $\mathbb{Z}_{2}$ symmetry (respectively $\phi\to-\phi$ and
$\sigma_{i}\to-\sigma_{i}$) that is broken in one phase and unbroken in the
other;

  * •

Exactly two relevant operators, both of them scalars, which we will denote
$\sigma$ (odd) and $\epsilon$ (even under $\mathbb{Z}_{2}$).

UV primary | $\Delta$ | IR primary | $\Delta$ | $\mathbb{Z}_{2}$
---|---|---|---|---
$\phi$ | 0.5 | $\sigma$ | 0.51815 | odd
$\phi^{2}$ | 1 | $\epsilon$ | 1.14126 | even
$\phi^{3}$ | 1.5 | – | – | –
$\phi^{4}$ | 2 | $\epsilon^{\prime}$ | $>3$ | even
$T^{\mu\nu}\sim\partial^{\mu}\phi\partial^{\nu}\phi$ | 3 | $T^{\mu\nu}$ | 3 | even
Table 1: List of primary operators in the free UV limit of the theory
described by the Lagrangian (6.34), and corresponding primaries in the IR CFT.

This second point requires some clarifications. In the lattice model, in
addition to $J$, the phase diagram is characterized by an interaction with an
external magnetic field, corresponding to the additional term $\delta
H=-\mu\sum\sigma_{i}$ in the Hamiltonian. In the quantum field theory, this
claim is supported by the list of relevant primary operators in the UV, given
in the first column of table 1. Two of these primaries can reasonably be
expected to still be primary operators at the interacting fixed point: $\phi$
and $\phi^{2}$ have sufficiently low scaling dimensions to start with. On the
contrary, $\phi^{4}$ is the relevant operator that triggers the
renormalization group flow in the UV, so we expect it to become irrelevant in
the IR (otherwise the flow would continue). $\phi^{3}$ is special: as soon as
the coupling $g$ is turned on, the equation of motion
$(-\partial^{2}+m^{2})\phi=\frac{1}{3!}\phi^{3}$ implies that it is not an
independent primary operator anymore, but rather a descendant of $\phi$. Among
operators that carry spin, the only relevant one is the energy-momentum tensor
$T^{\mu\nu}$: this operator exists in any QFT, and therefore it can be
expected to be there in the IR as well, with an unchanged scaling dimension.

The conformal bootstrap philosophy is quite orthogonal to this discussion, in
the sense that it does not care about the microscopic details (the lattice
model or the Lagrangian theory). It relies instead purely on symmetry
arguments. Besides conformal symmetry that is built in the method, the
$\mathbb{Z}_{2}$ transformation properties of $\sigma$ and $\epsilon$ imply
that their OPEs have the following schematic form,

$\displaystyle\sigma\times\sigma$
$\displaystyle=\mathds{1}+\epsilon+T^{\mu\nu}+\ldots$
$\displaystyle\sigma\times\epsilon$ $\displaystyle=\sigma+\ldots$ (6.40)
$\displaystyle\epsilon\times\epsilon$
$\displaystyle=\mathds{1}+\epsilon+T^{\mu\nu}+\ldots$

where the dots indicate contributions from all (infinitely many) irrelevant
primary operators. Using these properties as an input, and studying all
4-point functions involving $\sigma$ and $\epsilon$, one is able to make the
allowed region of parameters ($\Delta_{\sigma},\Delta_{\epsilon}$) shrink to
an island surrounding the value known experimentally, see figure 9. With the
more recent numerical analysis, the size of this allowed island has shrunk to
become much smaller than the uncertainty of Monte-Carlo simulations, so that
the current best theoretical prediction for the critical exponents of the
Ising model stems from the conformal bootstrap ($\Delta_{\sigma}\approx
0.5181489$).

This is very impressive, and it illustrates admirably how the concept of
_universality_ in statistical physics works: given some very general
assumptions (conformal symmetry, only two relevant operators, and a
$\mathbb{Z}_{2}$ symmetry), the conformal bootstrap essentially establishes
that there is a unique theory and provides excellent numerical results with
rigorous error bars. Moreover, at the intersection between the allowed and
disallowed regions (i.e. on the boundary of the island), the crossing equation
(6.33) must be satisfied in a peculiar way: many vectors must lie precisely on
the separating plane, and only they can have non-zero OPE coefficients for the
equation to be satisfied. Reading out the scaling dimensions $\Delta_{i}$ and
spins $\ell_{i}$ associated with these vectors, one gets a numerical estimate
of the spectrum of primary operators entering the OPE. Figure 10 shows such an
estimate for the Ising model. The spectrum of operators is shown there as a
function of the spin $\ell_{i}$ and of the difference $\Delta_{i}-\ell_{i}$
(called the “twist”) so that an underlying structure clearly appears: the
operators are organized into _conformal Regge trajectories_.

Figure 10: Spectrum of operators entering the $\sigma\times\sigma$ OPE in the
3-dimensional Ising model, for spins up to 40 and scaling dimensions of the
order of the spin, based on data taken from ref. [28].

This observation matches the state-of-the-art analytical understanding of the
crossing equation [29, 30]: every operator with a low scaling dimension
$\Delta_{i}$ (hence low spin by the unitarity bound) is related by crossing
symmetry to a family of operators with scaling dimensions
$2\Delta_{i}+2n+\ell$. This is what we saw in generalized free field theory,
but it works reasonably well in the Ising model, as seen in the figure 10: for
$\Delta-\ell<4$, there are 3 families of operators that are aligned with the
values $\Delta_{i}-\ell_{i}=2\Delta_{\sigma}$, $2\Delta_{\epsilon}$, and
$2\Delta_{\sigma}+2$. The equivalence between these families and operators of
low scaling dimension has been established rigorously in the limit of infinite
spin $\ell\to\infty$, and the leading correction to their twist, proportional
to a power of the spin $\ell$, is well understood [31, 32]. In theories that
are sufficiently close to generalized free fields, the equivalence goes even
further [33, 28, 34].

###  6.6 Other conformal bootstrap results

The reach of the conformal bootstrap is not limited to the Ising model. Our
understanding of many conformal field theories has been significantly improved
by the conformal bootstrap. This is particularly true of theories in 2 and 3
dimensions. In 3 dimensions, there are generalizations of the Lagrangian
(6.34) to $N$ scalar fields $\phi_{a}$, with interactions of the form
$\left(\sum\phi_{a}^{2}\right)^{2}$ and an $\text{O}(N)$ symmetry, for which
the bootstrap has made compelling physical predictions:

  * •

The $\text{O}(2)$ model describes the superfluid transition of liquid helium.
There is an unresolved discrepancy between the best measurement of this
transition (an experiment performed in the space shuttle) and the best Monte-
Carlo simulations. In this case, the theoretical bounds set by the conformal
bootstrap are consistent with the Monte-Carlo result [35], calling for a new
experiment.

  * •

The $\text{O}(3)$ model describes the critical behavior of Heisenberg magnets.
These are statistical physics models in which the magnetization is isotropic:
it can point in any direction in space, without a preferred direction. In
practice, however, it seems very difficult to achieve the required level of
isotropy on a lattice or in a solid: the $\text{O}(3)$ symmetry tends to be
spontaneously broken to its cubic subgroup (the finite group of symmetries of
the cube). The crucial question here is whether the 4-index operator
$\mathcal{O}_{ijkl}$ that can trigger a renormalization group flow from the
$\text{O}(3)$ model to a CFT with cubic symmetry is relevant or not. Since its
scaling dimension is accidentally very close to 3, this is hard to determine
from lattice simulations. The conformal bootstrap has now established on
rigorous grounds that the scaling dimension of this operator is below 3, and
hence that the $\text{O}(3)$ symmetry naturally tends to be broken [36].

There are many more interesting results and exciting open questions related to
the conformal bootstrap, and this is not the place to list them exhaustively
(the list would not stay up-to-date for long). For a recent summary, see for
instance the _Snowmass 2022 White Paper_ on the numerical conformal bootstrap
[37].

##  7 Conclusions

There are many more known CFTs that we have not mentioned so far:

  * •

Free massless theories are conformal. This is the case of the free boson and
free fermion in any number of dimensions, but also of the $n$-form gauge
theory in $d=2n+2$ dimensions (e.g. the free vector theory in $d=4$). There
are also theories with any number of free fields. We are used to describe the
Hilbert space of free theories as a Fock space (with states of a given
“particle number”), but there is also a conformal basis for them in terms of
primary and descendants that turns out to be useful in the effective field
theory approach, or in Hamiltonian truncation.353535This is a numerical
technique that studies renormalization group flows using the CFT data as
input: the Hilbert space of a UV CFT is truncated by some cutoff before a
relevant deformation is introduced; the new Hamiltonian can then be
diagonalized numerically in the truncated basis, and IR observables measured;
iterating the process while varying the cutoff gives a sense of how the
observables depend on the scale [38].

  * •

There are theories in which the $\beta$-function has a perturbative fixed
point. Typical examples are deformations of the free scalar theory with a
potential of the type $\phi^{n}$, such as $\phi^{4}$ in $d=4-\varepsilon$
dimensions (it is possible to make sense of non-integer dimensions in
perturbation theory): given the action

$S=\int
d^{d}x\left[-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{g}{4!}\phi^{4}\right],$
(7.1)

the $\beta$-function for $g$ can be computed to be

$\beta_{g}=\mu\frac{dg}{d\mu}=-\varepsilon+\frac{3g}{(4\pi)^{2}}+\mathcal{O}(g^{2}),$
(7.2)

and it vanishes when

$\frac{g_{*}}{(4\pi)^{2}}=\frac{\varepsilon}{3}.$ (7.3)

Higher-order corrections are negligible in the limit $\varepsilon\ll 1$. There
are similar fixed points with a $\phi^{3}$ interaction in $d=6+\varepsilon$
dimensions, or with a $\phi^{6}$ interaction around $d=3$.

  * •

A similar type of fixed point can be found in the $\beta$-function of
$SU(N_{c})$ gauge theories with $N_{f}$ fermions (in the fundamental
representation), which is given at leading order in the gauge coupling
$\alpha=g^{2}/(4\pi)^{2}$ by

$\beta_{\alpha}=\mu\frac{d\alpha}{d\mu}=-\frac{2}{3}\alpha^{2}\left(11N_{c}-2N_{f}\right)+\mathcal{O}(\alpha^{3}).$
(7.4)

When $N_{f}=\frac{11}{2}N_{c}$, the leading order term in this
$\beta$-function vanishes, so the next-to-leading term becomes important.
Around that value, the first two terms are of similar importance, and one
finds a perturbative fixed point when $N_{f}\lesssim\frac{11}{2}N_{c}$. This
is called the (Caswell-)Banks-Zaks fixed point. A theory in this situation is
asymptotically free like QCD, but it approaches an interacting conformal field
theory at low energy. For an $SU(3)$ gauge theory like QCD, this critical
value is at $N_{f}=16.5$. There is strong evidence from lattice simulation
that a theory with $N_{f}=16$ is conformal. On the other hand, a theory with
low $N_{f}$ (QCD has 3 light quarks) is clearly confining, meaning that its
low-energy limit is a theory of massless Goldstone bosons (if the quarks are
massless), or of massive pions. There is a critical value $N_{f}^{*}\approx
12$ above which we expect an interacting conformal field theory in the low-
energy limit. The domain $N_{f}^{*}\leq N_{f}\leq\frac{11}{2}N_{c}$ is called
the _conformal window_. Note that gauge theories with different gauge groups
and/or fermions coupling differently to the gauge fields (i.e. transforming in
different representations) can also have a conformal window. It is even
possible to engineer gauge theories with perturbative UV fixed points using
not only fermions but also scalars.

  * •

There are also theories with extended supersymmetry in which the
$\beta$-function is exactly zero at all orders in perturbation theory, for
instance $\mathcal{N}=4$ supersymmetric Yang-Mills. More generally, in
theories with sufficiently many supersymmetries, it is often sufficient to
engineer the matter content to make the $\beta$-function zero at leading
order, and then non-renormalization theorems ensure their vanishing at all
orders. When combined with supersymmetry, the conformal algebra gets extended
to a bigger and more rigid structure. All superconformal algebras have been
classified and are restricted to dimensions $d\leq 6$ (see refs. [39, 40] for
recent reviews).

  * •

In two dimensions, we mentioned in section 2 that there are many more Killing
vectors and hence conformal generators. In theories that have an energy-
momentum tensor, this gives rise to the infinite-dimensional Virasoro algebra,
which has been used to solve completely a class of _minimal models_. One can
also place a 2-dimensional CFT on a torus and study the modular properties of
its partition function in a bootstrap setup. This is a vast topic that has its
own literature [41, 42, 43, 44, 45, 46].

  * •

Finally, another example of CFT is the family of theories obtained by putting
a quantum field theory in $(d+1)$-dimensional anti-de-Sitter (AdS) space-time.
AdS admits a compactification with a spherical boundary, and correlation
functions on that boundary are isomorphic to that of a $d$-dimensional
conformal field theory.363636Another type of correspondence applies to late-
time correlators in de Sitter space-time [47]. There are two quite distinct
situations: On the one hand, it is possible to consider theories of quantum
gravity that approach AdS space-time asymptotically; these are dual to a true
CFT with an energy-momentum tensor (and some peculiar properties). On the
other hand, one can also place a local quantum field theory in a fixed AdS
background, and study the CFT correlators on its boundary; this CFT does not
necessarily have an energy-momentum tensor, and it is in fact very similar to
the generalized free field theory (or a large-N theory). Studying the limit of
infinite AdS radius in this latter approach reveals interesting connections
with flat-space scattering amplitudes. This observation is at the heart of the
recently-revived S-matrix bootstrap techniques [48].

This non-exhaustive list shows how important conformal field theory has become
in theoretical physics. But as particle physicists know, nature is certainly
not scale-invariant, so why should we care so much about CFT?

Sometimes the best motivation to study a subject is its beauty, and we hope
that this modest introduction to the topic is reflecting at least part of its
appeal. But there are also very practical reasons for particle physicists to
get interested in CFT. For one, some of the conformal bootstrap results
presented here provide a unique glimpse into the dynamics of strongly-coupled
quantum field theory, one that is impossible to reach with perturbation
theory. Moreover, the constraints on possible CFTs in Minkowski space in $3+1$
space-time dimensions that will be derived in the future will certainly teach
us important lessons about gauge theories such as QCD, grand unified theories,
or various beyond-the-standard-model scenarios. So far, generic bootstrap
bounds have remained relatively weak and featureless in $3+1$ dimensions. The
CFTs that are known to exist are gauge theories, and this provides an
additional difficulty since the information on the gauge group is encoded in
the CFT data in a complicated way. This should not however be taken as a
failure of the conformal bootstrap, but rather as a challenge for the next
generation of theoretical physicists!

## Acknowledgments

I am indebted to the Institute for Theoretical Physics at the University of
Bern (Mikko Laine and Thomas Becher in particular) for giving me the
opportunity to teach a graduate class on my favorite topic. I would like to
thank the students and postdocs who took part in the class for their
involvement and their critical questions, as well as Johan Henriksson, Slava
Rychkov, Flip Tanedo, and Luca Vecchi for their feedback on the notes. I would
like finally to say a big thanks to all members of the conformal bootstrap
community for several enjoyable years of research together, which I wished
could have continued.

[Courses] are fantastically good for learning physics. The lecturer learns a
lot of physics. After my first few studies, just about everything I learned
about physics came from teaching it. I don’t know if the students learned a
lot, but I certainly did. So I consider teaching physics very important.

Leonard Susskind [49]

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