#### Analyticity in spin in conformal theories

HJE
Analyticity in spin in conformal theories
Simon Caron-Huot 0 1
0 3600 rue University , Montreal, QC H3A 2T8 , Canada
1 Department of Physics, McGill University
Conformal theory correlators are characterized by the spectrum and threepoint functions of local operators. We present a formula which extracts this data as an analytic function of spin. In analogy with a classic formula due to Froissart and Gribov, it is sensitive only to an \imaginary part" which appears after analytic continuation to Lorentzian signature, and it converges thanks to recent bounds on the high-energy Regge limit. At large spin, substituting in cross-channel data, the formula yields 1=J expansions with controlled errors. In large-N theories, the imaginary part is saturated by single-trace operators. For a sparse spectrum, it manifests the suppression of bulk higher-derivative interactions that constitutes the signature of a local gravity dual in Anti-de-Sitter space.
Conformal Field Theory; Conformal and W Symmetry; AdS-CFT Correspon-
1 Introduction 2
Review and main ingredients
1.1
Why good behavior in the Regge limit is constraining
Four-point correlator and conformal blocks
Positivity and analyticity properties of Rindler wedge correlator
Toy dispersion relation, ANEC and the bound on chaos
2.4 Invitation: theories with large N and large gap
From dispersion relation to Froissart-Gribov formula
3 Inverting the OPE: the CFT Froissart-Gribov formula
Partial waves: Euclidean case
S-matrix Froissart-Gribov formula revisited
Main derivation: conformal Froissart-Gribov formula
Final result
4
Application to operators with large spin
Generating function
Vaccum exchange
Systematics of large-J corrections
4.3.1
4.3.2
4.3.3
Subleading powers: individual block
Beyond leading-log: exact sum rule and application to 3D Ising
Quadruple cut equation: large spin in both channels
5
6
Application to AdS bulk locality and Witten diagrams
5.1
Bounding heavy operator contributions as a function of spin
Conclusion
A Conformal blocks in general dimensions
A.1 Expansions around the origin z; z ! 0
A.2 Expansions around z ! 0 and monodromy under analytic continuation
A.3 Expansions around z ! 1
B
A worked example: the 2D Ising model
{ 1 {
Introduction
The conformal bootstrap has highlighted the power of basic principles to constrain and even
solve quantum
eld theories. The program combines inequalities obtained from conformal
symmetry, unitarity and crossing symmetry to quantitatively narrow down the possible
values of observables like scaling exponents and operator product expansion (OPE) coe
cients. The modern numerical bootstrap [1] has been applied very successfully to numerous
models in various dimensions (far too many to even attempt to review here) yielding for
instance precise critical exponents for the three-dimensional Ising model [2, 3]. Of course, the
same principles constrain non-conformal theories as well [4], although it remains challenging
at the moment to use them as a quantitative solution method (see [5] for encouraging steps).
In parallel to numerical advances, an analytic approach has been developed which
implements constraints that are more readily visible in Lorentzian rather then Euclidean
signature. Namely, by focusing on a Lorentzian limit which selects the contribution from
operators with large spin, crossing symmetry predicts weighted averages over their OPE
data.
Assuming that individual contributions are su ciently regular and close to the
average, this then provides an asymptotic 1=J expansion of this data [6{9]. This assumption
has been con rmed in explicit examples, for instance in the three-dimensional Ising model
again, where the resulting expansion appears to remain accurate all the way down to spin
two [10, 11]! Gaining control over the errors in this expansion will be crucial to mesh it
with numerical approaches.
Another important application of the analytic bootstrap is to large-N theories with
a so-called sparse spectrum, which are famously conjectured to be dual to weakly
coupled theories of gravity in Anti-de-Sitter space [12]. The large-N crossing equation then
admit homogeneous solutions which are in one-to-one correspondence with possible
higherderivative bulk interactions, and whose coe cients need to be small for the bulk theory
to admit a local interpretation. It has been recognized that this smallness is tied to the
good high-energy (Regge) behavior of the theory, a feature which is particularly apparent
in Mellin space [13{16]. Physically, a peculiar feature of these higher-derivative solutions
is that they have a bounded spin, which clashes with the experience, reviewed below, that
physics should be analytic in spin.
The goal of this paper is to establish the phenomenon of analyticity in spin in conformal
eld theories, and to quantify its implications by means of an inversion formula. In the
context of the large spin bootstrap, this formula will explain why the spectrum organizes
into analytic families and provide control over individual OPE coe cients as opposed to
averages. At the same time, by upgrading the asymptotic expansion to a convergent one, it
will clarify the sense in which spin two is \large enough". In the context of large-N theories
with sparse spectrum, the same formula will bound the strength of higher-derivative bulk
interactions. In both cases, the validity of the formula will be tied to the good behavior of
correlation functions in the high-energy Regge limit.
{ 2 {
Physically, analyticity in spin re ects the fact that not any low-energy expansion can resum
into something that is sensible at high energies.
Mathematically, this can be illustrated by a simple single-variable model. Consider an
\amplitude" which admits a low-energy Taylor series:
f (E) =
1
X fJ EJ :
J=0
We suppose that we are given the following information: f (E) is analytic except for branch
cuts at real energies jEj > 1, and jf (E)=Ej is bounded at in nity. (In the physical
application below, f (E) will represent the four-point correlator and its low-energy expansion will
be provided by the Euclidean OPE; at the thresholds E =
1 some distances become
timelike.) With the stated assumptions, an elementary contour deformation argument relates
the series coe cients to the discontinuity of the amplitude, as shown in gure 1:
fJ
=
i0)) . The second line follows from the rst
using the assumed high-energy behavior to drop large arcs at in nity.
As a concrete example, one may take the function f (E) =
log(1
E2): upon inserting
its discontinuity Disc f = 2 , the integral indeed produces fJ = (1+( 1)J )=J , as expected.
Now let us focus on a single coe cient, say f2. It may seem paradoxical that it can be
recovered from the discontinuity of f (E), given that varying f2 alone in eq. (1.1) clearly
leaves Disc f (E) unchanged. The point is that given the constraint that jf (E)=Ej is bounded
at in nity, the coe cient f2 (or any nite number of coe cients) cannot be varied
independently of all the others. Rather the coe cients form a much more rigid structure, that is an
analytic function of spin, as explicited by the integral in eq. (1.3). (More precisely, there are
two analytic functions, for even and odd spins, re ecting that there are two branch cuts.)
These are the key features of the classic Froissart-Gribov formula [17{19], which is
conceptually the same but with Legendre functions instead of power laws. Historically,
the Froissart-Gribov formula established the analyticity in spin of partial amplitudes in
relativistic S-matrix theory, thus paving the way for phenomelogical applications of Regge
theory.
We will show that OPE coe cients in unitary conformal eld theories are of a similar
type: they are not independent from each other, but rather organize into rigid analytic
functions. Furthermore, they can be extracted from a \discontinuity" which would naively
seem to annihilate each individual contribution.
That this has quantitative implications can be illustrated in large-N theories with a
sparse spectrum, where we will see that the discontinuity is negligible below a gap
g2ap.
{ 3 {
1
1
The preceding formula then gives a result which decays rapidly with spin,
fJ
Z 1 dE
2gap E
E J Disc f (E)
( gap) 2J ;
(1.4)
which in the context of gauge-gravity duality will be interpreted as the expected suppression
of higher-derivative corrections, if the bulk theory is local to distances of order 1= gap times
the AdS curvature radius. Notice the essential role of the Regge limit: nothing would
be learnt from this argument for a given J if we didn't know that jf (E)=EJ j vanishes at
in nity. Physically, eq. (1.3) and the Froissart-Gribov formula can be regarded as dispersion
relations for partial waves, since their input are discontinuities of amplitudes (this is further
discussed in section 2.5).
The goal of this paper to obtain similar dispersive representations but which extract the
OPE coe cients in unitary CFTs (projecting out descendants and extracting only primary
operators). Convergence will be established for all spins higher than one, by borrowing
ideas from the recent \bound on chaos" as well as from the recent proof of the averaged
null energy condition (ANEC) [20, 21], which are reviewed in the next section.
This paper is organized as follows. In section 2 we review the main ingredients
regarding the operator product expansion, its convergence, and the ensuing positivity and
boundedness properties of discontinuities in Lorentzian signature; we also present a
simpli ed dispersion relation, valid in the Regge limit, and discuss its relationship to the just
mentioned recent work. Section 3 is purely mathematical and is devoted to deriving our
main result, the inversion formula in eq. (3.20). The starting point will be the partial wave
expansion in [22], in which scaling dimensions are continuous, and a corresponding
not-sowell-known Euclidean inverse to this formula, which exploits the orthonormality of blocks.
In section 4 we analyze the formula in the limit of large spin in a general conformal
eld theory, substituting in the convergent OPE expansion in a cross-channel to re-derive
and extend a number of results from the analytic boostrap. Section 5 discusses the
simpli cations in large-N theories with a large gap, and novel bounds on the contributions
of \heavy" operators to the crossing relation, with a brief discussion of loops in the bulk
gravity theory. Section 6 contains concluding remarks. A lengthy appendix A details
for{ 4 {
mulas for handling conformal blocks in various dimensions, while appendix B details tests
in the 2D Ising model.
Review and main ingredients
Four-point correlator and conformal blocks
We will be interested in the correlator of four conformal primary operators (which we will
take to be scalars). Up to an overall factor, it is a function of cross-ratios only:
hO4(x4)
O1(x1)i =
1
where here and below a = 12 ( 2
4), and z, z are conformal cross-ratios
zz =
x212x2324 ;
of O. They are eigenfunctions of the quadratic and quartic Casimir invariants (A.2) of the
conformal group. It will be useful to use blocks normalized so that, at small z
z:
GJ; (z; z) ! z 2 z 2
J
+J
(0
z
z
1) :
The same normalization was used in [11]. The angular dependence when z and z are both
small but of comparable magnitude can be expressed in terms of Gegenbauer
polynomials, see eq. (A.8). In even spacetime dimensions, the conformal blocks admit closed-form
expressions in terms of hypergeometric functions, for example
{ 5 {
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
wedges. In terms of lightcone coordinates ; = x1
x0, we let, as shown in gure 2:
x4 = ( ; ) =
x3 ;
x1 = (1; 1) =
x2 :
Their cross-ratio evaluates to z = (1+ )2 , or, equivalently:
=
p
1
1 + p
4
1
1
z
z
;
=
p
1
1 + p
1
1
z
z
For convergence purposes, in the -variables the blocks behave essentially like power-laws:
2 . In particular, the rst singularities being at
=
1, the OPE
converges whenever ; are both within the unit disc; for rigorous estimates we refer to [24].
Note that this holds whether or not
and
are complex conjugate of each other, which
The unit -disc covers the full complex z-plane, as can be seen from the fact that
and 1=
project onto the same value of z. They represent, however, physically distinct
Lorentzian con gurations as we now discuss.
2.2
Positivity and analyticity properties of Rindler wedge correlator
We will be interested in Lorentzian kinematics where the lightcone coordinates
and
are independent real variables. As long as both are positive, the four points lie within two
disjoint Rindler (spacelike) wedges. When both
and
are small, all points are spacelike
separated and the physics is essentially Euclidean. We will be more interested in the case
0 <
< 1 < , where, as depicted in
gure 2(a), the distances x4
x1 and x2
x3 both
become timelike. According to the standard Feynman's i0 prescription (x0
! x0(1
i0)),
should then be slightly below the cut if we are computing the time-ordered correlator,
and above the cut for its complex conjugate.
These kinematics do not lie within the radius of convergence of the s-channel sum (2.3).
They lie, however, within the radius of convergence of the t-channel sum, which is based
{ 6 {
around the limit x2!x3 corresponding to ;
= 1. This expansion is obtained by
interchanging z and 1
z, which using (2.9) gives a somewhat nontrivial transformation for :
G( ; ) =
will not use the full conformal blocks but rather just power laws, that is we include both
primaries and descendants independently; the notation f~ reminds us of that. The sum
converges provided that the parentheses are within the unit disc, which is the case for
and
within the complex plane minus the negative axis: ; 2 C n (
Let us rst consider the case where operators 1 and 2 are identical, and also 3 and
4. The OPE coe cients f~23Of14O, as squares of real numbers, are then positive. What
~
happens in the Lorentzian region 0< <1< is that the scaling blocks acquire a phase:
G( ; ) =
J0; 0
X f23Of14O 1 + p
~ ~
1
p
0+J0 1
1=p
1 + 1=p
where the dots stand for the prefactor in (2.10). Since the absolute value of each term is
the same as for the positive sum corresponding to the Euclidean point
! 1= , one thus
nd the following inequality:
G( ; )
G( ; 1= )
GEucl( ; )
(0 <
< 1 < ) :
(2.12)
Intuitively, this states simply that the amplitude for a projectile crossing a target is smaller
than the amplitude for them to propagate independently. This is analogous to at space
scattering, where S-matrix elements between normalized states satisfy jSj
1. In this
context, it is conventional to subtract o the free propagation by writing S = 1 + iM, and
the imaginary part of the amplitude then satis es Im M
that we can similarly de ne a CFT \amplitude" with a positive imaginary part:
0. The above inequality means
iM
G( ; )
GEucl( ; )
)
Im M
0 :
This imaginary part is equal to a double discontinuity of the correlator:
Im M
We will nd below that Im M is the argument of the CFT Froissart-Gribov formula.
It may seem unfamiliar that the imaginary part is equal to a double discontinuity (as
opposed to a single discontinuity for the usual S-matrix) but intuitively the role of the
extra discontinuity is to subtract the \1" part of the S-matrix from the correlator. This
fact will be crucial below when we discuss large-N theories.
{ 7 {
E ! Ee i , which interchanges points 3 and 4.
Since the i0 prescription on the time argument encodes the operator ordering, the
double discontinuity can also be written as a commutator squared:
dDisc G( ; ) =
h0j[O2( 1); O3(
)][O1(1); O4( )]j0i
0 :
(2.15)
2
xjk !
2
jxjkj
Indeed one can check that for the two terms with \non-scattering" operator ordering, like
hO2( 1)O3(
)O4( )O1(1)i, the continuation path for the cross-ratios z; z is equivalent to
staying within the Euclidean region, thereby reproducing the GEucl( ; ) term in eq. (2.14).
Positivity of the commutator squared (2.15) has appeared in several recent works. It
holds, in any QFT (not necessarily conformal), due to the Cauchy-Schwartz inequality
together with the property of so-called Rindler positivity (see refs. [20, 21] and section 3
of [25]). The argument here (similar to [26, 27]), valid in conformal eld theories, relied
only on the usual positivity of Euclidean (t-channel) OPE coe cients.
For unequal operators, the more precise de nition of the double discontinuity is that
it should be taken with respect to the two time-like invariants x214 and x223 successively, at
the level of the unstripped correlator on the left-hand-side of (2.1). That is, one should
take the di erence between the two di erent ways of making these invariants timelike,
i0. This gives, when translated to the stripped correlator,
dDisc G( ; )
cos( (a + b))GEucl( ; )
2
1 ei (a+b)G( ;
1
2
i0)
e i (a+b)G( ; + i0) ;
(2.16)
again in the range 0< <1< <1. As a function of the four operators, dDisc G1234 de nes a
positive-de nite matrix with respect to the two pairs 12 and 34 (e.g., it is a positive number
whenever 1 and 2 stand for the same linear combination of operators, and similarly for 3
and 4).
Convergence of the t-channel OPE (2.10) within the mentioned cut plane also
implies an important analyticity property, representing crossing symmetry. Namely, starting
from the timelike region 0< <1< , one can rotate
and
by opposite phases as shown
{ 8 {
and
and
!
in gure 3(b) and reach the region
< 1< <0. This represents crossing, since, in the
parametrization (2.9),
is the same as interchanging operators 3 and 4. This
crossing path is reminiscent of the Epstein-Glaser-Bros path in axiomatic S-matrix theory [28],
and, just like in that context, it lands us on the \wrong" side of the cut, e.g. on the
complex conjugate (anti-time-ordered) amplitude. Note that for this crossing path
must be rotated in opposite directions, because the Euclidean correlator near ;
and
= 0 is
single-valued only when
are complex conjugate of each other.
If one were to rotate
in the same direction, one would land in a physically
completely di erent kinematics, where the four points are inside timelike Milne wedges instead
of the spacelike Rindler wedges. This region is relevant to bulk high-energy scattering in
AdS/CFT and contains the so-called \bulk point" limit in two dimensions (see [12, 27]).
Although this region is interesting, we will not directly use it in this paper.
2.3
Toy dispersion relation, ANEC and the bound on chaos
The above analyticity and boundedness properties immediately imply a simple-minded
dispersion relation, whose integrand is positive de nite at least in the Regge limit (large
boost acting on 1 and 2). In this subsection we will parametrize this limit as E ! 1 with
=
=E;
= E :
(2.17)
With no loss of generality (because of the symmetry between z and z) we will assume that
< 1.
A key fact is that the correlator is bounded and approaches a limit as E ! 1 (with
xed): this is because the t-channel OPE (2.10) is dominated by its Euclidean counterpart
bounded since the individual OPE coe cients satisfy f23Of14O
~
~
which converges to G(0; 0) = 1. For operators that are not identical, the correlator is also
< max(f~223O; f~124O).
In fact, the correlator approaches a
nite constant as jEj ! 1 along any complex
direction in the lower-half plane. This can be shown by combining the t-channel and
u1= cos((arg E)=2) 1+ 2 since only the real part of p
channel OPE. By itself, the t-channel OPE only gives a numerically weaker bound jGj <
damps the exponent for each term.
This bound becomes poor near arg E =
, but in this case one can use the u-channel
OPE instead, which converges nicely. Being bounded like this, analyticity then implies that
it approaches the same constant not only along the real axis, but also along any complex
direction
arg E
0:
lim
jEj!1
M( ; E) = C:
This can be proved easily by taking a derivative of the contour integral (2.19) below,
dropping the arc at in nity, and integrating back. Under crossing t $ u, C goes to
To obtain a dispersion relation for M( ; E), we simply write down the contour integral
M( ; E) =
1 I dE0 M( ; E0)
2 i C
E
E0
;
1The statement that Im C is crossing symmetric is essentially Pomeranchuk's theorem, which states that
proton-proton and proton-antiproton total cross-sections are asymptotically equal if they grow with energy.
{ 9 {
(2.18)
C .1
(2.19)
is equal to M
the real part of C:
where E is assumed to be in the lower-half plane and the contour encircles the lower-half
plane clockwise. In fact it doesn't hurt to add \0" in the form of a similar integral but
encircling the upper-half plane, with the integrand replaced by the analytic function which
just above the real axis. The half-circles at in nity then simply add up to
M( ; E) = Re C +
1
1 Z 1 dE0 Im M( ; E0)
:
E
E0
(2.20)
(Note that the integral diverges logarithmically if Im C 6= 0, however with opposite signs at
E0 !
1 and such that it converges to the correct result if symmetrized under E0 !
E0
under the integration sign.)
The \toy" dispersion relation (2.20) converges to the four-point correlator but a caveat
is that for \low" energies jE0j < 1 one leaves the Lorentzian region and the integrand
switches sign, since going below
= 1 interchanges M and M
according to the de
nition (2.13). Furthermore, when jE0j < , one loses control over the sign and even reality of
the integrand. However, for jEj
1, these region contribute only a subdominant amount.
This is the sense in which the above is only a toy dispersion relation, but for this reason the
subsequent discussion should be understood to hold only for high enough energies jEj
1.
The toy dispersion relation still contains interesting physics. Separating explicitly the
real and imaginary parts of E: E = x
iy, the following inequalities follow from
straightforward di erentiation, assuming only that Im M > 0 on the real axis as shown above:
Im M(x
iy; ) =
y Z 1 dE0 2Im M(E0; )
1
(E0
x)2 + y2
1) log Im M(x
iy; ) =
y2
R 1 dE0 2Im M( ; E0) ((E0 x)2+y2)2
1
R 1 dE0 2Im M( ; E0) (E0 x)2+y2
1
1
(2.21)
The rst of these inequality, which essentially states that jSj
1 holds throughout the
complex lower half-plane and not only the real axis, played an important role in a recent
proof of the averaged null energy condition (ANEC) [21] (see also [29]). There one showed
using the lightcone OPE that there is a regime where the left-hand-side is dominated by
exchange of the stress tensor integrated over a null line, thereby proving positivity of its
matrix elements, and also generalized this argument to the operator of lowest twist for
each even spin
2.
The second inequality states that, locally, the amplitude cannot grow faster than
linearly (along the imaginary axis). When expressed in terms of Rindler time t = log E and
temperature T = 1=(2 ), this is equivalent to the \bound on chaos" on the Lyapunov
exponent,
2 T , proved in ref. [20]. (As discussed in appendix A of [20], the present
context of high-energy CFT scattering can be viewed as a special case of their general
nitetemperature results applied to Rindler space. Speci c properties of CFTs, tied to OPE
2
0;
0:
convergence, apparently allowed us here to simplify some steps of the proof, for example
the Phragmen-Lindelof argument used there.)
Although the bound on chaos will not be used directly in this paper, the closely related
dispersion relation (2.20) is conceptually central and our main goal will be to extend it to
OPE coe cients.
Invitation: theories with large N and large gap
The toy dispersive representation (2.20) tells us more. For example, to leading order in a
large-N theory, the double discontinuity Im M has the neat property that it is insensitive
to double-trace operators exchanged in the cross channels. This can be seen again from
the t-channel OPE (2.11), because double-trace operators have dimensions
2 +
3 or
1 +
4 plus integers, and the resulting continuation phases are then killed by the double
discontinuity (2.16) in both cases.
In the case of identical operators, the double-trace OPE coe cients are enhanced
by a factor N 2 compared with the connected contribution since they contribute at the
disconnected level. However, the connected contribution, associated with the leading 1=N 2
corrections to coe cients and anomalous dimensions, contains at most a single logarithm
of (1
z) (see for example [12]). These double-trace contributions are thus again killed by
the double discontinuity.
The fact that the imaginary part (2.16) is sensitive only to single-trace operators
is consistent with intuition from gauge-gravity correspondence: the imaginary part arises
from \on-shell" exchanged states, and in tree-level AdS/CFT these are the elementary bulk
elds dual to single-trace operators. Thus the toy dispersive representation (2.20) tells us
physically that the double-trace information is fully controlled by what the single-traces
do (strictly speaking, at this point, up to the neglected jE0j <
contributions).
Consider speci cally the case where the single-trace spectrum is sparse, in the sense
that the lowest twist operator of spin higher than 2 has a large twist
gap
spectral density will then be suppressed, as the contribution of heavy operators to the
t-channel OPE has the form
dDisc G(z; z)
X
gap
(1
p )(1
1=p )
(1 + p )(1 + 1=p )
/ e 2 gap(pz+pz)
e
gap=pE
: (2.22)
Thus the \energy" variable E introduced above is indeed essentially equivalent to CFT
energies, e.g. scaling dimensions (squared), and heavy operators produce an imaginary
part only at high energies, again in line with AdS/CFT intuition.
Plugging this into the second inequality (2.21) one immediately sees that any theory
with a sparse spectrum will nearly saturate the bound on chaos within at least the energy
range 1
E
2
gap (provided only that the light operator contribution to the dispersive
relation is not enhanced by a power of
g2ap, which would preclude the heavy operators
from dominating it.) Conversely, the bound on chaos can only be saturated, locally for
some value of energy, if most of the spectral density is concentrated at a much higher
energy, which is a weak statement of a \gap." Also, since Im M is locally bounded, the
HJEP09(217)8
gap certainly can't be larger than the inverse coe cient of the linear growth, which is
essentially the stress tensor two-point function cT ; this re ects the familiar observation
that in weakly coupled gravity theories, the string scale never exceeds the Planck scale.
These are all moral implications of the dispersion relation, but to make them (and
others) fully quantitative one would like to have a dispersion relation whose integrand
remains physically sensible and positive even away from the Regge limit. We were not
able to derive one. However, in the next section we will short-circuit this technical issue
by switching our focus to the OPE data instead of the correlator itself, and deriving a
Froissart-Gribov formula.
From dispersion relation to Froissart-Gribov formula
The importance of the concept of analyticity in spin was emphasized already in the
introduction, for example in the two contexts of large spin expansions and also bulk locality.
Being related to analyticity in energy, itself tied to causality, in a sense it is a physical
re ection of causality. The Froissart-Gribov formula is an integral representation for
partial wave coe cients which makes analyticity manifest and quantitative. Here we brie y
review its connection to dispersion relation in the context of the at-space S-matrix. One
considers 2 ! 2 scattering, and de ne as usual the projection of the amplitude into the
partial wave of angular momentum J :
d(cos )(sin )d 4 CJ (cos )M(s; t( )); t( ) =
(1
cos ); (2.23)
s
4m2
2
where CJ (cos ) are Gegenbauer polynomials (Legendre polynomials PJ in four spacetime
dimensions). Here s; t; u are the usual Mandelstam variables for 2 ! 2 scattering. The
Froissart-Gribov formula follows from using a xed-s dispersive representation of the
amplitude, in terms of t- and u-channel cuts:
M(s; t) =
Disct M(s; t0) + (t $ u) :
(2.24)
HJEP09(217)8
ing
one thus nd:
aJ (s) =
dx0(1
x02) d 2 4 CJ (x0)
1) this gives:
Z 1 d(cosh )
0 cosh
x0
The integration threshold t0 will not be important, and convergence and subtractions will
be discussed shortly. To see analyticity in spin, one simply plugs eq. (2.24) into (2.23);
changing variable to t0 = s 4m2 (cosh
2
Interchanging the order of integrations, the x0-integral can be done once and for all. De
nQJ (x)
Z 1 dx0 Cj (x0)
1
x
x0
1
1
x02
x2
d 4
2
;
aJ (s) = atJ (s) + ( 1)J aJu (s);
Disct M(s; t0) + (t $ u) :
(2.25)
(2.26)
(2.27)
where
d(cosh ) (sinh )d 4QJ (cosh ) Disct M(s; t);
t =
s
4m2
2
(cosh
eq. (2.28) is known as the Froissart-Gribov formula. It shows that, while aJ was a-priori
de ned only for integer J , its t- and u-channel contributions atj;u(s) are separately analytic
in spin. They are power-behaved at large imaginary J , as opposed to the direct
evaluation of (2.23) which would grow like e i J , and general arguments show that an analytic
continuation with this property is unique.2
The regime of validity of the Froissart-Gribov formula (2.28) is the same as that of
the dispersion relation (2.24) from which is originates.
Depending on the high-energy
behavior, this dispersion relations may receive ambiguities, known as subtractions, that
are polynomials in t. These polynomials a ect a
nite number of the aJ 's in eq. (2.23).
Thus partial wave coe cients are analytic in spin except for a
nite number of low spins,
which cannot be obtained from the Froissart-Gribov integral. The minimum spin starting
from which the formula works is equal to the exponent controlling the power behavior of
the amplitude in the Regge limit (large jsj at xed t).
In the next section, we will derive using group-theoretic arguments a Froissart-Gribov
formula for OPE data in conformal eld theories. The derivation will by-pass the dispersive
representation which we do not have, but to which the formula is morally equivalent. The
good high-energy behavior of CFT correlators discussed above, will imply that the formula
works for all spins except possibly J = 0; 1.
3
Inverting the OPE: the CFT Froissart-Gribov formula
A prerequisite rst step to derive integral representations is to get rid of the discreteness
of . This rst step was already carried out in ref. [22], where the sum over dimensions
was replaced by an integral over continuous dimensions:
G(z; z) = 112134 + X1 Z d=2+i1 d
J=0 d=2 i1
2 i
c(J; ) FJ; (z; z):
(3.1)
The rst term is the identity contribution. The goal of this section is to derive a Lorentzian
inverse to this representation, which will express the coe cients c(J; ) analytically in
J and in terms of positive Lorentzian data, in analogy with the Froissart-Gribov
formula (2.28).
This section is exclusively mathematical. After brie y reviewing the expansion (3.1),
we derive an analog to the Euclidean inverse (2.23) and then perform its analytic
continuation to Lorentzian signature, obtaining our main result (3.20). Many technical details are
moved to appendix A. Physical implications are discussed in the next sections.
2Because of the ( 1)J multiplying the u-channel cut contribution, the even and odd spin partial waves
constitute two independent well-behaved analytic functions.
HJEP09(217)8
The idea behind eq. (3.1) is to expand correlators over an orthogonal basis of eigenfunctions
FJ;
of the Casimir invariants of the conformal group [22]. These are the quadratic and
quartic di erential operators in eqs. (A.2).
These invariants are self-adjoint only when acting on single-valued complex functions
(that is, which do not have branch cuts in Euclidean kinematics z = z ), otherwise,
integration-by-parts would receive extra boundary terms from the branch cuts. Thus such
an expansion only has a chance to work in the space of single-valued functions.
Fortunately, the physical correlator G(z; z) is of this type. The individual conformal blocks GJ;
entering the OPE sum are not, and to make a basis of eigenfunctions one must use instead
the single-valued combinations [22], also known as harmonic functions:
whose coe cients can be expressed using a frequently-recurring products of -function:
KJ; =
(
1)
d
2
J+ ;
=
2
a
2 + a
Single-valuedness of the harmonic functions FJ; (z; z) can be understood from an integral
representation [30], which involves three-point functions of the exchanged operator and of
its \shadow" related by
! d
symmetrical:
in eq. (3.2). With no loss of generality we can thus assume that shadow coe cients are
. This explains also why the shadow block appears
c(J; )
KJ;
=
c(J; d
KJ;d
)
:
(3.2)
(3.3)
(3.4)
(3.5)
(3.6)
For our purposes, single-valuedness of eq. (3.2) can also be veri ed explicitly using the
analytic continuation formulas in appendix A.2. It requires the spin to be an integer.
Under the assumption that the harmonic functions FJ; 's are orthogonal, the
expansion (3.1) can be immediately inverted to give the c(J; )'s, by simply integrating against
FJ; :
3
The integration runs over the full complex plane (with z = z ). The measure is xed by
self-adjointness of the Casimir di erential operators in eq. (A.2), which works out to give
(z; z) =
z
zz
z d 2 (1
z)(1
z)
(zz)2
a+b
:
The orthogonality assumption is actually a simple consequence of the Casimir equations
(modulo minor convergence issues to be discussed shortly).
The normalization factor
N (J; ), given in eq. (A.14), can be calculated using only the behavior of the blocks near
3I thank V. Goncalves and J. Penedones for initial collaboration on this Euclidean inversion formula. I
have also been made aware of related work by B. van Rees and M. Hogervorst on this subject.
the origin, where the radial integral boils down to a Mellin transform (which also explains
the choice of contour in eq. (3.1)). Both of these issues are discussed in detail in A.1.
We will not dwell too much on eq. (3.5), since it is only an intermediate step toward
the Lorentzian formula that we are after. Let us only describe the connection between the
partial wave expansion (3.1) and the usual OPE sum (2.3). In Euclidean kinematics where
j j < 1, the blocks GJ; (z; z) vanish exponentially at large real
, like power-laws j j .
Therefore it is natural to close the contour to the right in eq. (3.1) on the rst term of
eq. (3.2), and to the left in the shadow block (which will produce an identical result due to
the mentioned shadow symmetry). The OPE sum is therefore reproduced provided that
the partial wave coe cients c(J; ) have poles with appropriate residues:
cJ; =
Res 0= c(J; 0)
(
generic) :
(3.7)
The function c(J; ) thus encodes the usual OPE data through the position and residue of
its poles. The fact that this data is encoded in an analytic function of
will be important
physically, since the discreteness of the original OPE data would otherwise preclude our
next step. The quali er \generic" was added in the preceding equation due to some special
cases that we now discuss.
Subtleties, convergence, contour, etc.
This subsubsection contains technical
comments that the reader may skip on rst reading. The preceding equations are too hasty for
two reasons: the blocks GJ; themselves have poles, and the integral (3.5) doesn't always
converge. These two issues are xed here.
Let us rst discuss convergence of the inverse transform (3.5) near z = 0. The integral
in this region can be de ned easily by analytic continuation. Indeed one can make the
integral convergent by subtracting
nitely many terms in the small-z expansion of the
amplitude times block, which can then be integrated back using the analytic formula:
Z 1 djzj z p
z j j
0 j j
p
1
:
(3.8)
Being analytic in
, this prescription automatically preserves self-adjointness of the Casimir
operators. This formula also shows explicitly how the blocks in the usual OPE sum turn
into poles of the function c(J; ).
There are a few special cases at low dimensions, which can all be understood by
thinking of the representation (3.1) as an inverse Mellin transform, which it is near the origin.
Due to the unitarity bound
J
d 2 for J
1, these only a ect the J = 0 contribution.
For operators with dimension less than d=2, one should deform the contour so it picks
only the positive-residue pole on the left of d=2 instead of its re ection on the right.
For operators of dimension precisely d=2, one should use a principal-value contour so
that only half of the residue contributes. (In two dimensions, this case also includes
conserved currents with J = 1.)
The unit block is orthogonal to all FJ; . This is why it appears separately in eq. (3.1).
HJEP09(217)8
The blocks GJ; have poles, which were not included in eq. (3.7). These poles are all
below the unitarity bound
= J +d 2, but are nonetheless to the right of the contour
= d=2, and so they should be included. Poles of conformal blocks were studied e.g.
in [31, 32]. Their possible location is heavily constrained by the Casimir equations, which
imply that the residue solves the same equations and thus must be related by one of
the symmetries (A.5). The only poles with
> d=2 are at
= j + d
2
m with
m = 0; 1; 2; : : : and with residue proportional to Gj 1 m;j+d 1, which is then a physical
block (above the unitarity bound and with integer spin). The proportionality factor rJ;
is given in eq. (A.19). Therefore all the residues produce physical blocks. Collecting their
coe cient, we nd that the correct formula for the OPE coe cients is not (3.7) but rather
This is consistent with the inversion formula (3.5) since its integrand (through the factor
N (J; )) diverges at
J d = 0; 1; 2, producing poles unrelated to z ! 0 divergences.
However these poles of the integrand cancels in the combination (3.9).
Finally let us analyze the region z ! 1 (z ! 1 is similar). Convergence there depends
on the dimension of the lightest operator exchanged in the t-channel, compared to that of
the external operators. Thus for generic external operators the integral will not converge.
However one can de ne the integral by cutting o a small circle around 1, for example, and
dropping the singular terms as its radius goes to zero. As long as the same cuto is used
for all spins (so that the spurious pole cancelation just mentioned continues to operate),
the resulting expression will only have the correct physical poles, and the OPE coe cients
extracted from eq. (3.9) will correctly reproduce the correlator.
We conclude that the
integral (3.5) may or may not converge near z ! 1 but this seems devoid of consequences
for generic z; z. This is akin to the Fourier transform of a singular distribution, which is
ambiguous up to contact terms in coordinate space and polynomials in momentum space.
All of these subtleties can be tested explicitly in simple examples, like the 2D Ising
model, as discussed in appendix B.
3.2
S-matrix Froissart-Gribov formula revisited
In section 2.5 we saw how the angular momentum projection (2.23) and dispersion
relation (2.24) naturally combine into the (S-matrix) Froissart-Gribov formula (2.28), thus
relating partial waves coe cients to discontinuities of amplitudes. In CFT we have
constructed analogous ingredients: the Euclidean inversion formula (3.5) and the Regge-limit
dispersion relation (2.20), and one could imagine again substituting the later into the
former. Unfortunately, being restricted to the Regge limit, that dispersion relation is not
good enough for our purposes.
Fortunately, there exists a second derivation of the S-matrix Froissart-Gribov formula,
which does not require a dispersion relation. It is essentially the contour deformation
argument from the introduction; we reformulate it here in terms of variables that will
be more convenient shortly, focusing for simplicity on d = 3 scattering where the SO(2)
1=w0
channel appear on the right, related to each other by w ! 1=w, and two copies of the u-channel
cut appear on the left.
Gegenbauer polynomials are just cosines. The trick is to rewrite eq. (2.23) as a contour
integral in the variable w = ei :
The main features in the w-plane are shown in gure 4. On the positive real axis there are
two branch points, where t(w0) = t0 > 0. They are related by w ! 1=w and both represent
the same physical point, the t-channel threshold. Similarly on the negative axis there are
two copies of the u-channel cut. To proceed, assuming J large enough that wJ times the
amplitude vanishes near the origin of the unit disc, we simply close the contour toward the
interior. The integral becomes a sum of contributions from the t- and u-channel cuts as in
eq. (2.27), with
atJ (s) =
Z w0 dw
0
w
wJ Disct M(s; t(w)) ;
which is equivalent to eq. (2.28) via the change of variable 2 cosh
= w + 1=w. Whereas the
Euclidean contour integral (3.10) only made physical sense for integer spin J , the
discontinuity integral eq. (3.11) is fundamentally Lorentzian and makes sense for continuous spin.
3.3
Main derivation: conformal Froissart-Gribov formula
We now adapt this contour deformation argument to CFTs. The geometry is easier to
visualize in terms of the -coordinates (2.9), where the \radial" and \angular" variables
are naturally identi ed with the magnitude and phase of . The Euclidean inversion
formula (3.5) then runs over the unit disc:
( ; ) =
(1
2)(1
I
2 (1
jwj=1 iw
dw
4
)(
( ; ) g( ; ) FJ; ( ; ) = w
= w 1
) d 2
(1
)(1
)
a+b
2
(1 + )(1 + )
:
:
Here, the measure in -coordinates, including the Jacobian from the change of variable, is
(3.11)
(3.12)
(3.13)
We also recall that, at this stage, J is still restricted to be an integer.
1 1=
in the CFT case. The singularities at w =
integrable and come only from the measure factor jz
j
z d 2
. The branch points at w =
1 are
and
w =
1=
pose no problems, but the arcs at w = 0 and w ! 1 correspond to the Regge limit
where the integrand should vanish at large positive spin.
HJEP09(217)8
At xed
< 1, the complex-w plane contains the same features as in the preceding
subsection. There are two copies of the t-channel cut starting at w =
and w = 1=
(corresponding to
starting at w =
and w =
= 1, respectively), and two copies of the u-channel cut,
Roughly speaking, the idea is to split the block FJ; into a part which vanishes like wJ
near the origin and another which vanishes like w J at in nity, so that we can close the
w-contour to the interior or exterior in each case. This was the trick in the S-matrix case.
The behavior of the blocks FJ; is however more complicated. As explained in appendix A,
there are 8 solutions to the Casimir equations corresponding to a given value of J and
which can be conveniently labelled (for generic J and
z
z
1:
gjp;ure(z; z) = z 2 j z 2
+j
The 8 solutions are then related by the symmetries (A.5). The limit w ! 0 is governed
by these asymptotics but after analytic continuation around the point z = 1. The block
FJ; (z; z) then becomes a complicated combination of all 8 basic solutions. The possible
exponents are such that two solutions vanish like wJ , two diverge like w J , and the four
remaining ones have exponents controlled by
. Ideally we would like to get rid of the 6
nondecreasing solutions, which is not obviously possible. A systematic strategy is to simply
close the contour in eq. (3.12) to the interior of the unit disc, and then try to remove the
nonvanishing terms near the origin by adding zero in the form of an integral over blocks
eJ; ( ; ) along the closed contours C , circling the upper- and lower-half planes, as shown
in gure 5. That is, omitting the ; arguments, we write
c(J; ) =
I dw
C iw
g N (J; ) FJ; + X I
dw
C iw
g eJ;
!
:
(3.15)
Naturally, the extra functions eJ; should be eigenfunctions of the same Casimir equations.
These should also vanish at large w, in order to not re-introduce the problem there. Since
large w corresponds to 0
z
z
1, this singles out two of the eigenfunctions:
eJ; (z; z) = eJ;;1 gpu+re1 d;j+d 1(z; z) + eJ;;2 gpure
1
;j+d 1(z; z)
(w ! 1) ;
(3.16)
where eJ;;1 are unknown coe cients. This ansatz vanishes like w J at large w as can be
seen from (3.14). Equation (3.15), with the arcs at in nity dropped, thus holds for any
choice of these coe cients. We would like to
nd a choice such that the arcs at the origin
also cancel, at least for su ciently large spin, that is
N (J; )FJ; + eJ+;;1 gpu+re1 d;j+d 1 + eJ+;;2 gpure
1
;j+d 1 / w
J
(w ! 0 + i0) :
(3.17)
Taking the coe cient of each 6 unwanted solution gives 6 constraints on two parameters.
To compute these constraints we performed the analytic continuation of the above
eigenfunctions to the origin w ! 0 using a sequence of elementary monodromies:
lowers w below 1= , which takes z counter-clockwise around 1 and changes the blocks gpure
rst, one
according to eq. (A.22). Then one takes w below 1 and interchange the arguments z and
z using eq. (A.23). Finally, one takes w below , which takes z counter-clockwise around
1 and is done using eq. (A.22) again. Although the individual steps are simple, composing
these three steps produces a complicated linear combination of all 8 solutions, so we do
not reproduce it here. It needs to be added to the continuation of FJ; to w < , obtained
using eq. (A.22) once.
Thus we obtain 6 constraints on two free parameters eJ+;;1=2. This system is
overconstrained so it is not obvious a priori that a solution exists. It is not too surprising that
the system is overconstrained, since a solution is at best expected when the spin is an
integer. It is still not obvious whether it should have any solution in that case, but the
physical analogy between the correlator and the S-matrix suggests that a solution should
exist. Indeed, a solution does exist! It is given as (3.16) with
eJ+;;1 =
1
4 J+ e i (a+b) ;
eJ+;;2 =
1
4 J+
d
2
d
2
(
(d
1)
1)
and the complex conjugate expressions for e below the axis. Pleasantly, this is precisely
such that eq. (3.16) organizes into a regular block G +1 d;J+d 1 (note the interchange of
dimension and spin compared to the usual blocks.) Also, most of the complicated factors
from N (J; ) have canceled out. Furthermore, we
nd that, after adding e
below the
axis, the contributions from the four regions (w 2 (0; r) [ (r; 1) [ (1; 1=r) [ (1=r; 1)), which
all project onto the interval 0 < z; z < 1, are all proportional to the same block! The
di erent regions are related to each other by the same phases as in the positive-de nite
double discontinuity (2.16), times absolute value of the measure, exactly like we physically
hoped for! Finally, the contribution from the cuts with w < 0 produce a similar result but
with an extra factor ( 1)J .
It would be nice to understand more deeply why the above solution exists at all,
perhaps by obtaining it through some integral representation. In the rest of this paper we
concentrate on its implications.
We note also that the process of extracting the coe cient of a given power of z in eq. (4.2)
commutes with doing the z integral (for J > 1), whose boundary can then be taken to be 0.6
Finally, let us record the precise relation between the generating function (4.2) and
OPE coe cients. According to the inversion formula (3.19), operators with even and odd
spins constitute two independent analytic families, and the t and u channel contributions
should be added with a relative sign depending on whether the desired operator has even
or odd spin,
X
m
C
( )(z; ) =
Cm( )( ) z 21 m( )( ) ;
C
( )
C
t
Cu :
(4.4)
The expansion coe cients Cm( )( ) and m( )( ) then encode the OPE coe cients and twists
of the operators. Both are analytic in
and to obtain physical operators one should restrict
to those values of
for which the spin 12 (
) happens to be an integer. Actually, C is not
quite the OPE coe cient, because, according to eq. (3.9), the
-residue of 1=(
J
( ))
should be taken with J
xed. This produces a extra Jacobian factor:
f12Of43O =
1
d m( )( )
d
! 1
Cm( )( )
2
m( )( ) =J
(4.5)
(4.6)
(4.7)
where the sign ( ) depends on the parity of J . Precisely this formula, and the existence of a
function C(z; ), has been deduced empirically before from large-J expansions, see [11, 33].
For operators of subleading twist one should replace C by c to account for the subtraction
of descendants according to eq. (4.3).
4.2
Vaccum exchange
A simple but important case where the integral (4.2) can be done analytically is the unit
operator in the t-channel. One then expects s-channel OPE coe cients to be products of
-functions. In fact, since one can vary the dimensions of the external operators, this gives
an in nite set of integrals which can be done analytically. Taking the pair of operators (4,3)
to be the same as (1,2), the vacuum contribution to the generating function, is, according
to eqs. (2.10) and (4.2),
C+(z; )
z 1 +2 2 I(a;a)
1
2
( )
(vacuum exchange) ;
where as before a =
2
2 1 , and using a standard Euler-type representation for the hyper
geometric function k , we could evaluate the following general integral:
I(a0;b)( )
(z) b
2
2
1
a
z
z
20 b
2 + b
1
3
5
2
0
2
2 + 20 + 1
1
:
6This follows from the fact that the correlator dDisc G(z; z) is bounded as z = z ! 0, so the
lowerendpoint of the range of integration can only produce powers of z higher than z( +J)=2 1, independently
of m and k in eq. (4.3). This is never of the form z
(
J)=2 for any J > 1.
This formula will be used extensively in this section.
Some comments are in order. First, the integral vanishes when the exponent 0 di ers
from 2b or
2a by a nonnegative integer, in agreement with the discussion in subsection 2.4:
physically these exponents represent \double trace" t-channel operators of twist
2 +
3
or
1 +
4 plus even integers. Note also that the integral naively diverges for su ciently
negative exponent 0, but the result is actually
nite. This can be understood from the
derivation in the preceding section, where the double-discontinuity arose from the contour
integrals in
gure 5: the contours smoothly avoid the branch point z = 1 and so can't
diverge there, and thus the integral is explicitly analytic in 0 (except for the last factor
which comes from divergences around 0).
The nonzero result for negative integer exponent, where the double-discontinuity
naively vanishes, can be interpreted as a
-function terms at z = 1: physically, it is
not too surprising to learn that the discontinuity of a propagator 1=x223 is a -function.
Mathematically the \double discontinuity" of 1=(1
z)
1=(x223x124) is however ambiguous
and to directly compute the integral in this case requires to go back to the contour integral
in
gure 5. In practice, a good way to deal with a general, singular double-discontinuity
in eq. (4.2) is to use eq. (4.7) to integrate its singular part analytically, thus leaving an
unambiguous remainder.
Incidentally, a converse to formula (4.7) appeared recently in [11] while this work was
being completed, where the following sum of SL2(R) blocks was derived:
1
m=0
X I(a0;b)( 0 + 2m)k 0+2m(z) =
1
z
z
20 b
(z) b
R 0; 0
(a;b) (z);
(4.8)
where R is an explicitly known remainder, with the property that is has a vanishing
doublediscontinuity around z = 1.7 This property was termed \Casimir-regular" in [11]. This
is a nice consistency check which con rms that the SL2(R) inversion integral (4.2) indeed
\inverts" the coe cient of k .
At large spin or equivalently large , the collinear block k (z) behave like ( ) =2 and
so the inversion integral is dominated by z ! 1. There the t-channel blocks vanish with
an exponent governed by their twist. One thus expect the large- limit of the s-channel
OPE coe cients to decay with the twist 0 of t-channel exchanged operators, and, indeed,
the above integral decays like 1= ( 0+a b+3=2).
It follows that, at large
, the t-channel unit contribution to the generating
functional (4.6) can't be cancelled by any other operator, if the theory has a twist gap above
the unit operator (which is always the case for a unitary theory in d > 2). This is
precisely the conclusion reached in [6, 7]: operators with twist arbitrarily close to
1 +
2
must exist at large spin. The same argument holds for subleading powers of z, leading to
primaries of dimension
1 +
2 + 2m + J . Following [11], we refer to these as double-twist
7For identical operators, we record for reference the expression from [11]:
R(0;;00)(1 z) =
1
2
2
0
2
0+
2
20 + k
k) (k + 1)2
0
2
k 1
z
1 z
k
:
(4.9)
operators [12]m(J ). A central focus of the analytic bootstrap program is to understand the
1= corrections to these multi-twist families, to which we now turn.
Systematics of large-J corrections
The t-channel unit contribution (4.6) gets corrected, at subleading orders in 1= , by a
convergent sum over t-channel primaries of subleading twist, each decaying like 1= 0 . First
we focus on an individual primary, and then we discuss the e ects of the in nite summation.
Our starting point will be the generating functional (4.2) with the correlator
represented by its t-channel OPE. Accounting for the prefactor in the crossing relation (2.10),
C(z; ) =
X f14O0 f23O0
J0; 0
Z 1 dz(1
z)a+b
z2
k (z) dDisc4
2
The sum converges (at xed z) since all sampled values of z lie within the convergence
radius of the OPE. We now rst tentatively take the z ! 0 limit term by term | this will
not be completely correct, but almost!
As discussed in appendix A.3, for each block there are two towers of terms in the z ! 0
limit, starting from two exponents 1 =
1 +
2 and 2 =
3 +
4
:
z; 1
z)
The computation of the functions H in general dimension is detailed in appendix. There
is a slight issue when the initial and nal operator pairs are identical, as is needed to get
matrix elements between identical double-twist operators: then the two exponents i are
identical and logarithms of z appear. The z ! 0 expansion must then be rewritten slightly:
z 1+2 2
GJ0; 0 (1 z; 1 z) =
1
X z 1+ 22+2n
n=0
1
2
log z HJlo0g;;n0 (1
z) + HJre0g;;n0 (1
z) :
The 12 log z term is interpreted, to rst order in the anomalous dimension, as a shift to the
dimension of the double-twist operators. At large spin, one only needs the limit z ! 1 of the
H functions, which can be determined simply from SL2(R) blocks (see eq. (A.21)) and reads
HJ(l0o;g)0;0(1
z)
2
( 0 + J 0)
0+J0 2 (1
2
0 J0
z) 2 (1 + O(1
z)) :
Plugging into (4.7) and dividing by the unit operator contribution in eq. (4.6), we thus
obtain the rst-order correction to double-twist anomalous dimensions from exchange of
a single t-channel operator of spin J 0 and dimension
0:
[12]0 (J )
( 1 +
2 + J )
( 0 + J 0) I(a0;a)J0
0+J0 2
2
I(a;a)
1
1
2
( )
2
( )
2f11O0 f22O0
(4.13)
(4.14)
where
=
[12]0 (J ) + J is the conformal spin, and the approximation signs are up to
subleading orders in 1= (coming both from the truncation of the (1
z) series and of the
integral (4.7)).
Eq. (4.14) is in perfect agreement with formulas from sections 3 and 4 of [6]. This is
a nice check of all the factors in our inversion formula (3.20).
A comment is in order regarding the u-channel cut contribution, neglected in the
above, but which is to be added with an extra sign ( 1)J in eq. (3.19). The u-channel
term would be absent in certain situations where the operators are distinct, but in general
one should sum up the t- and u-channel contributions prior to dividing by the vacuum
exchange. Thus even and odd spins generally describe two totally independent analytic
families of operators. In the case where the operators 1; 2 are identical, the above formula
remains valid but should be restricted to even spins.
In the case of stress tensor exchange, the OPE coe cient is xed by symmetries. With
our blocks normalized as in (2.4) and the T T normalization CT de ned as in [2], the OPE
coe cients reads fiiT = 2pCT (d 1)
id
The main di erence so far compared with these works, is that the inversion formula
applies to each individual operator in the even/odd spin families, as opposed to giving
only their average large-spin properties. In particular, this establishes the existence of
each individual double-twist operator (for su ciently large spin that we can ignore the
possibility of coe cients summing up to zero or very large anomalous dimensions), and it
explains why they organize into analytic families in the rst place.
. Plugging it into (4.14) it then also agrees with [6, 7].
Subleading powers: individual block
To generate subleading terms in 1=J from a given block is now straightforward: one simply
expands the function HJlo0g;;00 (1
z) to higher orders in (1
z)=z and use the analytic
integral (4.7). This can be done to any desired order using the quartic equation satis ed
by H, see appendix A.3. This however only produces an asymptotic expansion in 1=J ,
because the series in (1
z)=z does not converge for z < 12
. The inversion formula makes
it possible to do better, since in principle one can just do the z integral numerically and
conceptually there is no need to expand in 1=J .
To illustrate this in a simple concrete example, consider the exchange of a t-channel
primary of spin J 0 = 0. The z ! 0 limit of the t-channel block, HJ(l0o;g)0;0(1
evaluated analytically in terms of a hypergeometric function given in eq. (A.36). Thus the
z), can be
contribution of an individual J 0 = 0 primary to the double-twist anomalous dimension is,
restricting eq. (4.10) to lowest twist:
[12]0 (J ) ( 1 + 2 +J )
dDisc" 2F1 2
0 ; 20 ; 0 d 2 2 ;1 z
0=2
#
The approximation sign is only because we have used the coe cient of 12 log z in the
generating function as a proxy for the anomalous dimension, as above, but this formula exactly
gives the coe cient of 12 log z. (This proxy will be relaxed shortly.)
Expanding the integral at large spin
we reproduce, for example, the rst few terms
of the large spin expansion given for four scalars in the 3D Ising model in eq. (5.6) and
footnote 14 of [10] (with jt2here = 14 (
1=J expansion to all orders! It converges for
> d
of spin J
2 due to unitarity bounds.
2)here). Thus the above integral indeed resums the
1
2, which includes all operators
Beyond leading-log: exact sum rule and application to 3D Ising
If one could sum up all t-channel primaries, the convergent expansion (4.10) would
reproduce exactly all s-channel OPE coe cients. However, in practice, one must address the
fact that the in nite sum over primaries does not commute with taking the z ! 0 limit
of each term. This is because the true z ! 0 limit of the generating function involves
power-laws z =2, in contrast with individual t-channel blocks, which have at most single
logarithms times double-twist powers.
A simple solution is to subtract a known sum, such as the SL2(R) sum in eq. (4.8).
Let us restrict, for notational simplicity, to the case where all four operators are the same
scalar , and to the lowest twist; generalization will be straightforward. From the t-channel
sum (4.10) we know that
C0( )z
+ 12 0( ) + subleading in z =
X f
2
J0; 0
O0 I~J0; 0 (z; );
where C0( ) and 0( ) are the lowest-twist OPE coe cicent and anomalous dimension
appearing in eq. (4.4), and I~ stands for the universal (theory-independent) integral of a
block in eq. (4.10). Subtracting eq. (4.8) with z ! 1 z (and an arbitrary 0) and dividing
by z
, this replaces the exponent on the left by a constant and logarithm:
1
2
1 "
J0=0
A( )+B( ) logz= X
C0( )I(00;(0))( 0+2J 0)k 0+2J0 (1 z)+
z
1
Xf
2
0
O0 I~J0; 0 (z; )
(4.16)
#
z!0
(4.17)
:
(4.18)
The point is that we can now take the z ! 0 limit termwise and obtain two equations giving
the coe cients A; B as convergent sums over t-channel primaries. The coe cients A and
B are then directly related to the s-channel OPE coe cients and anomalous dimension
through the expression for the remainder R given in eq. (4.9):
A( ) + B( ) log z
1
2
C0( )R(00;(0)); 0 (1
log z + O( 02) :
corrections of order 02
solved iteratively.
Thus, when the anomalous dimension is small, the sum rule in eq. (4.17) reduces to the
naive procedure of extracting OPE coe cients and anomalous dimensions from regular and
logarithmic terms, as was done above, but in general the k = 0 term in R incorporates
. Note also that even though C and 0 appear on both sides of the
equation, their e ect is much more important on the left-hand side, so the equation can be
At subleading twists, the same logic gives rise to two sum rules for each exponent
= 2
+ 2m where m = 0; 1; 2; : : :. The number of subtraction needed for convergence is
equal to the number of s-channel operators with twist less than 2
+2m at that particular
value of , which we generally expect to be nite.
Somewhat reminiscent sum rules have been used recently in Mellin space [34, 35].
These authors expand the correlator in terms of Witten diagrams instead of conformal
blocks and then require that spurious \double trace" operators of twist 2
+ 2m cancel
out. It would be interesting to better understand the connection.
Using the numerical data for the 3D Ising model provided graciously in [11], we could
numerically check the above sum rule (4.18); including the operators from the [
]
and [ 0] families tabulated in the appendices of that paper, we checked the sum rule for
the s-channel stress tensor to 10 3 accuracy for its dimension and OPE coe cient. This is
impressively accurate although not signi cantly di erent than the quality of the asymptotic
expansions already considered in [11]. We leave it as an open question to identify which
operators must be included to increase the precision beyond this point.
An interesting possibility is to use the convergent sums to control the errors. For
example, schematically, an alternative way to extract the lowest twist at a given
is from
( ) = lim
z!0
(2
C(z; )
:
(4.19)
At nite z, both the numerator and denominator are convergent t-channel sums, and
evaluating the sum at nite but small z (say 10 3) the error will be proportional to z. Both the
numerator and denominator are generically sums of positive terms. Since the denominator
starts with 1, the e ect of any given t-channel primary is stronger on the numerator than
denominator. For the right-hand-side to not exceed 2
1
0:036 for the stress tensor
( = 5), which the -exchange already nearly comes close to saturating, then gives an upper
bound on the remaining operator contributions. When we change the values of , these
get smaller, with the higher-dimension contributions decaying more rapidly with
, thus
allowing to bound the uncertainties on other operators using the error on the stress tensor.
The main novelty of the inversion formula (3.20), compared with formulas from
refs. [10] and [11] which include similar physics, is that conceptually it produces convergent
sums that are valid for any individual spin J > 1, as opposed to inverting a crossing
equation as a series in 1=J . Although this does not seem to make a big numerical di erence for
the 3D Ising model, this does explain conceptually why the 1=J expansions of [10, 11]
appear to work all the way down to the stress tensor. It will be interesting to see how this helps
make error estimates that can be used in practice by the numerical bootstrap program.
4.3.3
Quadruple cut equation: large spin in both channels
We conclude this discussion with a brief analysis of the interplay between operators of
large spin in both channels. Just like the double discontinuity around z = 1, which kills
individual s-channel blocks and allows to focus on the analytic-in-spin part, by taking a
further double-discontinuity at z = 0 one can focus on the part which is analytic-in-spin
in both channels.
We do this by de ning a generating function C( 0; ) that depends on two conformal
spins. Speci cally, we integrate C(z; ) over z using a measure similar to z in eq. (4.2).
HJEP09(217)8
Restricting, for simplicity, to identical external operators of dimension
, we thus de ne:
0 (1
dz
Z 1 dz
z z2
1
z G(z; z) :
C( 0; )
0 k 0 (1
k (z) qDisc
(4.20)
Here qDisc is the quadruple discontinuity: the double-discontinuity around z = 1 followed
by the double-discontinuity around z = 0. At large spins 0, , the integral is dominated
by the corner (z; z) ! (0; 1), which is the usual double-lightlike limit.
Plugging in the expansion (4.4), this can be interpreted as a sum over families of
operators in the s-channel:
C( 0; ) =
X Cm( ) I(0m;0() )( 0) + subleading ;
(4.21)
HJEP09(217)8
where m = m
2
are anomalous dimension (de ned relative to double-twist operators)
and the (known) omitted terms originate only from the di erence between z =2 and (z=(1
z)) =2 in the small z expansion and are subleading at large 0. On the other hand, the
factors to the power
in eq. (4.20) have been chosen so that the formula is crossing
symmetrical under interchange of
and 0, and so it can also be interpreted as a sum over
What can be learnt from equating (4.21) and (4.22)?
As an example, in the 3D Ising model, considering the
correlator and taking
0
1 to project onto the lowest s-channel trajectory [ ]0, the equality reduces to:
C~0( )
1
( 12 0( ))2 ( 0=2) 0( )
1 + subleading =
X
m
C~m0( 0)
1
( 12 m0( 0))2 ( =2) m0( 0)
(4.23)
where C~0( )
( =22) 23 C0( ), and again we omit computable 1= corrections. The
doubletwist anomalous dimension [ ]0 vanishes at large
The left-hand side can thus be expanded into powers 1= (m
) where m = 2; 3; : : :
Comparing with the right-hand-side, where the power of
correspond to the twist of operators,
one concludes that multi-twist families f[ ]; [
]; : : :g of twist m
must exist, and one
also predicts their (averaged) OPE coe cients. For example, from the m = 2 case, the [ ]
OPE coe cient must approach a constant asymptotically. Physically, this ensures that the
t-channel OPE reproduces the correct term 18 2 log2 z predicted by exponentiation of the
leading anomalous dimension (this was also discussed recently in [11]). For the [ ] family,
(see eq. (4.14)): 0( ) /
f 2 =
the formula predicts OPE coe cients which grow like log( 0), etc.
Finally, let us mention another interesting situation where the quadruple discontinuity
seems particularly apt at capturing the physics: the interplay between isolated lowest-twist
trajectories in both channels, which gives, in a general CFT:
C~0( )
( 12 0( ))2 ( 0=2) 0( ) =
1
( 12 00( 0))2 ( =2) 00( 0)
Taking the logarithm on both sides, one sees that the right-hand-side can grow at most
linearly with log( ). Imposing this on the left-hand-side, one concludes that, for such
isolated trajectories, there must exist constants such that:
lim
!1
lim
0!1
This gives a simple proof that the well-known logarithmic scaling behavior of gauge theories
is the most general possibility consistent with crossing symmetry, as originally proved
in [36]. The limit for the OPE coe cient, when cusp 6= 0, also agrees with the result there.
(The extension remains relatively simple when there is only one operator in one channel
but many in the other, see [37].) It would be nice to understand how subleading twist
trajectories in both channels interact with each other, perhaps combining the quadruple
discontinuity with the methods of [38, 39]; we leave this for the future.
e
gap(pz+pz) :
5
Application to AdS bulk locality and Witten diagrams
The inversion formula (3.20), involving a bounded and positive de nite integrand
dominated by single-trace operators, seems an ideal tool to analyze CFTs with gravity duals.
As an application we derive here upper bounds on higher-derivative interactions.
Bounding heavy operator contributions as a function of spin
Consider a theory with a sparse spectrum of single-trace operators, characterized by a large
gap
gap, which for simplicity we de ne here as the lowest twist of single-trace primaries
with spin J > 2. The contributions to the t-channel OPE can be separated into light and
heavy operators according to their twist. Double-trace primaries contribute to the double
discontinuity only at subleading order in 1=N , as discussed in section 2.4, and the heavy
contribution to the double discontinuity can be estimated as in eq. (2.22):
dDisc G heavy = (prefactor)
X
J> gap
cJ0; 0
p
1
1 + p
0+J0
0 J0
p
1
1 + p
convergence of the Euclidean OPE, and 11+pp
This inequality is in fact mathematically rigorous, since (prefactor) P cJ0; 0
1 due to
e 2p
e p
z
. We stress the importance
of focusing on the double discontinuity, otherwise double-trace operators (which exist below
the gap) would also contribute.
Changing variables to (z; z) = e t with
small, the heavy contribution to the
inverphases
(5.1)
sion formula (3.20) becomes, using (A.9),
ct(J; ) heavy = C
Z 1 d
0
j 1
1
dt sinh(t) d 2C~ +1 d(cosh t) dDisc G heavy ;
(5.2)
where C =
p (
( d 2 1 ) (
d ) is some constant. For
of integration, the t integration is nonsingular but the
integral is strongly suppressed
e 2
p cosh(t) gap , one gets bounds of the type
away from its lower endpoint; plugging in the upper bound from eq. (5.1), dDisc G
= d2 + i along the physical contour
d
2
c(J;
+ i )heavy
#
( 2gap)J 1
where # is a universal (theory-independent) constant (that may depend on ).
To be clear, eq. (5.3) does not assume that any large-N factorization or even distinction
between single- and multi- traces exists above the gap, only that a full, unitary, theory exists
in the UV limit
! 0 at nite N , and that the double traces below the gap are numerically
suppressed (for the double-discontinuity) due to the parameter 1=N . Also no statement is
needed about the UV behavior of the correlator order by order in 1=N .
To put this bound into perspective we can look at the case J = 2, where c(2; ) must
contain the stress tensor pole. We recall that the inversion integral (5.2) is justi ed for
J > 1, which includes this case. The contribution to c(2; ) of a
nite number of light
t-channel operators, as studied in the preceding section, only produce poles at double-trace
twists, which are only near the stress tensor pole if
1 +
2
2)
0, that is if the
external scalar operators nearly saturate the unitary bound, which we do not expect to
happen in strongly coupled theories. Therefore the stress tensor residue
1=cT
1=N must
be saturated by heavy t-channel operators (a similar conclusion is obtained on the gravity
side [40]), so (5.3) gives a bound of the type cT
2
gap with a calculable coe cient.
From the gravity perspective, this is the statement (expected from unitarity) that new
states must appear below the Planck scale. (Similar parametric bounds were obtained
in Mellin space [15]; the improvement here stems from dDisc G being locally bounded, in
contrast to the Mellin amplitude, which makes the argument nonperturbative in 1=N .)
Since the double discontinuity is positive de nite and locally bounded, one can use the
stress tensor contribution to control its overall normalization, thus rewriting eq. (5.3) as
(5.3)
HJEP09(217)8
d
2
c(J;
+ i )heavy
1
cT ( 2gap)J 2
(5.4)
again with some computable coe cient. Again this is valid for J > 1.
This bound can be compared to expectations from e ective eld theory in AdS. In this
setup, \heavy" operators represent elds of large mass in AdS units, which can be integrated
out when computing correlators of light
elds, as depicted in
gure 6. This produces
a series of higher-derivative corrections suppressed by inverse power of the heavy mass,
of the schematic form (@2= gap)m 4 with various contractions. As explained originally
in [12], the Witten diagrams associated with these interactions give rise to solutions to the
crossing equation which are supported by double trace primaries with
nitely many spins,
J = 0; : : : m.
We see that the bound (5.4) coincides with the expected optimal one in the case
where derivatives are organized to produce the maximal angular momentum (in the channel
under consideration): each extra factor of @2 is then suppressed by
g2ap. In general, the
3
tions, suppressed by powers of the heavy mass. (b) The double-discontinuity of a one-loop correction
is equal to a product of trees.
bound (5.4) is however weaker because it detects the angular momentum of an interaction
rather than its mass dimension.
This is still highly constraining due to crossing symmetry, because it is not possible
for a four-scalar interaction to have very many derivatives without having large spin in at
least one of the s-, t- or u-channel. Therefore, except for nitely many exceptions (such
as the six-derivative interaction represented by the
at space amplitude stu, which has
spin 2 in all channels), the Regge limit constraint (5.4) proves that the coe cients of all
higher-derivative interactions contributing to a four-point correlator must be small and
decay with increasing dimension, as conjectured in [12]. We also expect the bounds to
be more constraining for external operators in spin, due to the restrictions on their local
self-interactions.
We thus believe that the bound (5.4) goes a long way toward establishing that all
largeN theories with a large gap admit a local gravity dual, although it will be important to
improve it toward the expected optimal bounds, which will presumably require information
from limits other then the Regge limit. Also it will be important to gain better control
over the low spins | for example the spin 0 interaction 4 | which are generally expected
to have also small coe cients in AdS/CFT but over which we provide no control here.
Another important question is whether a theory dual to a CFT with a large gap can have
a light spin-two particle beyond the graviton, which is not expected on the gravity side
but not ruled out by the present arguments. (Of course, with supersymmetry, there can be
more restrictions; for instance the constraint implemented in [16], that the correlator cannot
grow faster in the Regge limit than spin-two exchange, fully determined the correlators in
the N = 4 theory and is rigorously justi ed by the above bounds.)
Finally, let us brie y comment on loop corrections at large N . Since the double
discontinuity extracts, roughly speaking, the coe cient of log2(1
z), double-trace operators
start to contribute to it at the one-loop order, but only in a way proportional to the leading
1=N anomalous dimension of t-channel operators, which can be extracted from tree-level
amplitudes. (A similar conclusion was reached recently in Mellin space [41].) Note that,
because of mixing among double traces, one has to sum over all the intermediate primaries
O, O~. All we would like to add here is that, since the amplitude M already extracts
one discontinuity (see eq. (2.13)), the double-discontinuity at one-loop can be written in a
very suggestive form: Im M = M
M (the tensor product sign representing multiplica
tion in (J 0; 0) space, divided by free theory OPE coe cient of the t-channel intermediate
operators, f O2O~[OO~]).
6
The aim of this paper was to present a mathematical formula. The formula is in eq. (3.20).
Given a four-point correlator in a conformal eld theory, the formula returns the
operator dimensions and OPE coe cients which lead to it. This data is an analytic function
of the spin of the exchanged operator, and the formula, similarly to a dispersive
representation, quanti es the consequences of this fact. The input is that the Lorentzian correlator
admits a sensible high-energy Regge limit, which physically is a consequence of crossing
symmetry and of the positivity of Euclidean OPE data. A simpli ed formula, which
exploits only the SL2(R) conformal symmetries of a null line, is given in eq. (4.2).
We have illustrated the formula in a number of applications and tests. In section 4
we showed how it concisely encodes a body of existing results on operators with large
spin. Most importantly, it explains conceptually why these operators organize into analytic
families in the rst place. It provides convergent sums, instead of asymptotic series in 1=J .
We hope that this will be of great help to control its errors.
In theories with a large-N factorization, the formula is saturated by single-trace
operators. In the case where the spectrum is sparse this enables to bound OPE coe cients
of double-trace operators of spin larger than 2, see eq. (5.4). These bounds match the
expectations from a dual AdS theory that would be local down to distances of order 1= gap
in AdS units. We believe that this goes a long way toward proving that any large-N CFT
with a large gap has a local gravity dual, as originally conjectured in [12]. They do not
saturate the expectations yet however, as was expected since they use only the Regge limit
and not the more general high-energy
xed-angle limit.
We feel that a lot remains to understand, and that the tool could be useful for other
questions. Analyticity in spin is only guaranteed to apply to OPE data for spin J
2, and
it would be important to better understand spins 0,1 (which in this paper are only covered
by the Euclidean inversion formula (3.5)). One might ask for instance if strong scalar
selfinteractions are possible in theories with AdS gravity dual. Also it would be nice to better
understand why our main formula eq. (3.20) exists at all, due to the overconstrained nature
of the system we solved. This could also help generalize the formula to external operators
with spin; such a generalization could shed new light on the relation between the a and c
central charges in theories with AdS gravity dual [42, 43]. In two dimensions, extending
the formula to Virasoro blocks would also be interesting. Our nding that correlators are
analytic in spin for J
2 squares well with the recent conjecture of [44], that the only
theories with Virasoro primaries of bounded spin have maximal spin 0 (and are Liouville
theory speci cally), although from the present perspective it is not immediately clear why
spin 1 should not be possible. In the context of the 1=N expansion, it could be fruitful
to pursue the analogy with S-matrix unitarity and higher-point correlators sketched at
the end of the preceding section | this could help organize multi-twist operators in more
general theories. In general theories, it would be exciting to implement the error control
on the large-spin expansion suggested in section 4, so as to be enable using these analytic
predictions within the numerical bootstrap.
Finally, it is worth remembering that the Froissart-Gribov formula originated over 50
years ago in the context of the S-matrix bootstrap. Have all its applications been found?
Acknowledgments
I would like to thank David Simmons-Du n and the participants of the \S-matrix
bootstrap workshop" held in Lausanne, January 2017, for comments and stimulating
discussions.
A
Conformal blocks in general dimensions
Conformal blocks are characterized by the dimensions and spin of the exchanged primary
operator. This data is encoded in the quadratic and quartic Casimir invariants of the
conformal algebra [45, 46]:
1
2
c2 =
[J (J + d
2) +
c4 = J (J + d
2)(
d)] ;
d + 1) ;
which are the eigenvalues of the following di erential operators:
Here
C2 = Dz + Dz + (d
C4 =
zz
z
z
d 2
2)
z
zz
z
[(1
2 d
(1
abz
(A.1)
(A.2)
(A.3)
HJEP09(217)8
( 3
4) as in the main text.
In even spacetime dimensions, explicit closed-form solutions can be obtained in terms
of hypergeometric functions, as given in eqs. (2.5). In the general case, it is necessary to
rely on other methods such as the various series expansions discussed in this appendix.
Since we will be interested in analytic continuations, it will be useful to consider the
most general solution to these equations. These are most concisely described when J and
are generic (such that J and J
are non-integer) | non-generic cases can then be
obtained as limits. One can then choose solutions that are pure power laws in the limit
0
z
z
1, labelled by their exponents:
gJp;ure(z; z) = z 2 J z 2
+J
(1 + integer powers of z=z; z) :
(A.4)
There are in fact eight independent solutions, which are obtained from the above by acting
with the three independent Z2 symmetries of the Casimir eigenvalues (A.1):
J
d
J ;
! d
;
! 1
J :
A.1
Expansions around the origin z; z ! 0
In the limit that z; z ! 0, the dependence on their ratio is controlled by the Gegenbauer
di erential equation with x = cos
= 12 (pz=z + pz=z):
gJp;ure = (zz) 2 (fJ (x) + O(zz));
(1
HJEP09(217)8
This is the d-dimension generalization of spherical harmonics, e.g. Legendre polynomials.
This is physically expected since in Euclidean kinematics x is the cosine of an angle. The
pure power solutions corresponding to (A.4) can be written as
(A.5)
(A.6)
(A.7)
fJ (x) = (2x)J 2F1
J 1
2
2
J
; 2
J
d 1
2 ; x2
On the other hand, there is always a solution analytic around x = 1, known as Gegenbauer
polynomial (normalized here to C~j (1) = 1)
C~J (x)
(J + 1) (d
(J + d
2)
2) Cd=2 1(x) = 2F1
J
J; J + d
2;
d
1 1
;
2
x
:
(A.8)
It is, of course, only a polynomial when J is an integer. Comparing its large-x asymptotics
with the normalization (2.4), we conclude that the blocks behave near the origin like
:
2
GJ; (z; z) =
23 dp
d 1
2
(J + d
2)
C~J (x) + O(zz) :
(A.9)
More generally, the Gegenbauer polynomials can be written as a sum of two pure power
solutions. This gives an exact decomposition of the regular solution G used in the main text:
GJ; (z; z) = gJp;ure(z; z) +
(J + d
2)
d 2
2
gpuJre d+2; (z; z):
(A.10)
This will be useful since the pure power solutions gJp;ure have simpler asymptotic expansions
and analytic continuations. (This formula is valid for generic dimension. The limit to
even spacetime dimension can be singular for integer spins, but this does not appear to
a ect our nal formulas.)
Orthonormality of conformal blocks. Interchanging the order of integrations in the
Euclidean inversion formula (3.5), with
= d2 + i , one nds the following integral:
IJ; ;J0; 0
Z
d2z (z; z) g(z; z) FJ; d2 +i (z; z)FJ0; d2 +i 0 (z; z) :
(A.11)
It is easy to verify that, at least when a; b are small enough, the functions F are su ciently
regular near 0, 1, 1 that the Casimir operators are self-adjoint. For example, when
a = b = 0, the F 's have at most logarithmic singularities near z = 1, and the
doublederivatives in Dz come with a factor (1
z)@z2 which ensures that all boundary terms near
z = 1 arising from integration-by-parts vanish. When a; b 6= 0, the integral may become
divergent near z = 1, but as discussed in section 3.1 any contribution from the region
z; z
1 has a speci c J; dependence which removes any possible physical consequence.
Self-adjointness of the quadratic and quartic Casimirs implies that the above integral
vanishes unless both blocks have the same eigenvalues. For real
this forces J = J 0 and
0. The latter is a distributional term which can only appear if the integral develops
a singularity, which in turn can only come from near the origin where the behavior of the
functions is similar to that of a Mellin transform:
0) + non-singular
The angular integral, thanks to the appropriate factor for spherical harmonics in d
dimensions which arises from the measure (3.6), j(z
z)jd 2
/ (sin =2)d 2, takes the form of the
of the orthogonality relation of Gegenbauer polynomials and forces the spins to be equal:
Z 1
1
x2) d 2 3 C~J (x)C~J0 (x) = JJ0 (J + d
2)
(J; J 0 = integers) :
By combining this with the normalization of the single-valued functions F in eq. (3.2), we
obtained the normalization factor in the Euclidean inversion formula (3.5):
N (J; ) =
Subleading terms and poles of conformal blocks. Subleading terms in zz can be
worked out systematically using the methods of [23]: one postulates an expansion in terms
of powers times Gegenbauer polynomials,
gJp;ure(z; z) =
1
X (zz) +2m
m=0
m
X
k= m
Ak;mC~J+k(x) :
The quadratic Casimir equation then gives a rst order recurrence relation for the coe
cients ak;m; the spin changes only by one unit at each step due to the addition properties
of angular momentum. We reproduce this recursion relation here in the more general case
where a; b 6= 0. It is seeded by Ak;0 = k;0 times the prefactor in eq. (A.9) and reads:
2 c2(J + k;
+ m) c2(J; ) Ak;m =
+m 1;J+k 1AJ+k 1;m 1 +
+m 1;J+kAJ+k+1;m 1
(A.12)
(A.13)
(A.14)
(A.15)
(A.16)
:
(A.17)
where
E;J =
(J +d 2)(E+J +2a)(E+J +2b)
2J +d 2
E;J =
J (E J d+2+2a)(E j d+2+2b)
2J +d 2
This recursion relation is valid whether or not J is integer and holds equally for GJ; and
gJp;ure, one simply has to replace C~J (x) by
(J+ d 2 2 )
(J+d 2) fJ (x) in the latter case.
Using this recursion relation, it is possible to determine the poles in the conformal
blocks GJ;
as a function of
. Since the denominators come from the Casimir, it is easy
to check that the only poles are at
= J + d
2
m, where m = 0; 1; 2; : : :. This has a
simple physical interpretation since
J + d
2 is the unitarity bound (for generic spin
J ): the poles appear when the unitarity bound is crossed. The residue must be one of the
solutions related by the symmetries (A.5). Working out the proportionality factor we nd
that the following combination is pole-free for
> d=2:
r +1 d;J+d 1G +1 d;J+d 1(z; z);
(A.18)
where rJ; is a messy-looking product of -functions (with x =
j
d + 2):
1) (
+ 2
J + d2
(2
x)
a + x2
(J + 1) (J + d
2) (x)
a + 2 x
2
2
b +b +2 x2x : (A.19)
A.2
Expansions around z ! 0 and monodromy under analytic continuation
Many applications involve the collinear limit z ! 0. The z dependence is then controlled by
the conformal symmetry SL2(R) of a one-dimensional null line; in particular the quadratic
Casimir (A.2) reduces to a hypergeometric equation whose solution appeared already in
the two- and four-dimensional blocks in eq. (2.5):
gJp;ure(z; z) z!!0 z 2 J k +J (z);
k (z) = z =2 2F1( =2 + a; =2 + b; ; z):
(A.20)
collinear limit of g1pure
;1 J (z; z).
In the limit z ! 1, the hypergeometric function admits the standard expansion
lim k (z) =
(a + b) ( )
(1
z) a b +
( a
b) ( )
( =2
+ : : :
(A.21)
where the dots stand for in nite towers of integer power corrections to the two terms. This
formula will be used to seed the double null limit z ! 0; z ! 1 in the next subsection.
The second solution to the quadratic equation involves k2
(z) and controls the
When we analytically continue, in particular going to
the Regge sheet by taking z counter-clockwise around 1 (while retaining z small), these
two solutions mix. The continuation, which can be worked out from (A.21), reads
gjp;ure(z;z) = gjp;ure(z;z) 1 2i
"
sin( (J + ))
sin
J +
2
sin
J +
2
;1 j (z;z)
J+
(A.22)
with
de ned in eq. (3.3). In the text we also need the continuation as z goes
counterclockwise around 1, with z xed. The trick is to do this in multiple steps, rst interchanging
z and z in the pure power solutions. By analyzing the hypergeometric function (A.7) near
x = 1, we obtain the following connection formula, if z is analytically continued to the
right of z in a counter-clockwise fashion (so that x goes counter-clockwise around 1):
gJp;ure(z;z) = gpuJre d+2; (z;z)
e i d 2 2 ( J
d 2
2 ) (1 J
d 2
2 )
( J ) (3 J d)
ei J sin( d 2
2 )
sin
(J + d 2 2 )
:
(A.23)
The continuation of gjp;ure(z; z) counter-clockwise around z = 1 can then be obtained simply
by applying (A.23) followed by (A.22) and then (A.23) again. (We caution the reader that
the limit of integer spin should be approached with care in the preceding formula, since
the large solution fJ (x) diverges in this limit and both terms contribute.)
For single-valued combinations, the two ways of reaching the Regge sheet by rotating
either z or z counter-clockwise around 1 should give the same result. The preceding two
formulas can therefore be used to con rm single-valuedness of the combination FJ; in
eq. (3.2).
Subleading terms.
To expand in subleading powers of z one can proceed following
methods similar to [47]. In fact in the main text what we really need is the expansion of
the block times measure:
z)a+b gpu+re1 d;J+d 1(z; z) =
1
X z J 2 +m
m=0
m
X
k= m
BJ(m;;k)k +J+2k (z):
The coe cients BJ(m;;k) (which are those entering eq. (4.3)) can then be obtained
recursively using the quadratic Casimir equation. To simplify the expansion it turns out to be
convenient to consider a slightly di erent prefactor:
z1 d z
z
z
d 2
2
1
m=0
X z j 2 +m h(Jm;)(z):
We expand each term as a sum of SL2 blocks, h(Jm;)(z) = Pkm=
m h(Jm;;k)k +2k(z) with
+ J and
X
k
k(k +
1) + m(m +
+ 1
d) h(Jm;;k)k +2k(z)
1
2
1
4
d + m + a
+ (d
2)(d
m
4) X
m0=1
1
2
2m0
zm0
d + m + b h(Jm; 1)(z)
2m0 1
zm0 1
h(Jm; m0)(z) :
The right-hand side can be expressed as a sum over SL2 blocks using the recursion relation:
1
z
k (z) = k 2(z) +
1
2
2ab
2)
k (z) +
1 2)(b2
2( 2
1)
1 2)
k +2(z) :
(A.27)
In this way the coe cients h(Jm;;k) can be obtained recursively. Multiplying by (1
d 2
z=z) 2 (1
z)a+b and exchanging J $ 1
in z, then gives the desired BJ(m;;k) coe cients. As simple examples we nd BJ(0;;0) = 1 and
to go from (A.25) to (A.24) and re-expanding
BJ(1;;1) =
(d 2)(J+2) .
2J+d
(A.24)
(A.25)
(A.26)
Operators with low twist and high spin in the s-channel are controlled by the behavior of
t-channel blocks G(1
z; 1
z) in the limit z ! 0; z ! 1. The argument of the block itself
thus appraochs (0; 1). An expansion in powers of z around this limit, that is, in powers
of the second argument of the block, was de ned in (4.11) and corresponds to the twist
expansion in the s-channel. Reverting to (z; z) arguments it can be written equivalently as
X
1
X (1
i=1;2 m=0
z)pi+mHJ(i;);m(z) :
(A.28)
those entering eq. (4.11).)
The quadratic Casimir equation is singular in this limit z ! 1, since the operator Dz
reduces m by one unit. One concludes that there can be only two towers of terms, beginning
with the two exponents corresponding to the zero-modes of Dz: p1 = 0 and p2 =
a
b,
which represent double-twist s-channel operators. (When these blocks are used in t-channel,
which is related by 1 $ 3 to the s-channel, one should use that ajt channel =
2
2 3 and
1
2 4 . The HJ(i;);m functions de ned by eq. (A.28) then become precisely
The solutions are more conveniently described by decomposing the blocks into pure
power solutions (A.10). Normalizing the contributions so that they have simple behavior
gJp;ure(z; z) =
X
1
X (1
i=1;2 m=0
z)pi+mc(Ji+)
H~J(i;);m(z)
where in the z ! 0 limit, H~J(i;);m(z) = z 2
coe cients, coming from the collinear expansion (A.21), are
J times a tower of integer powers, and the
c(1) =
( ) ( a
a) ( =2
c(2) =
( ) (a + b)
( =2 + a) ( =2 + b)
:
In the case where a + b = 0, which occurs for identical operators, logarithms of z appear
and this need to be rewritten, following eq. (4.12), as
(A.29)
(A.30)
1
m=0
1
2
gJp;ure(z; z) = X (1
z) + cJn+
H~ Jlo;g;m(z) + H~Jre;g;m(z)
; (A.31)
where H~ Jlo;g=reg;0=z 2 J respectively tend to 1 or 0 as z ! 0. The coe cients are now
clog =
2 ( )
( =2
a) ( =2 + a)
c n =
1
2
1
2
( =2
a) +
( =2 + a)
(1) ; (A.32)
where (x) = 0(x)= (x) is the polygamma function.
When this expansion is used in the t-channel, the expansion in powers of z is dual to
1=J corrections in the s-channel. The coe cients are determined by the Casimir equations,
but this is actually subtle, since in this limit both the quadratic and quartic Casimirs mix
the m = 0 and m = 1 terms. To resolve this, we start from the equation for C4 and
substitute in the equation for C2 to remove the o ending lowering operators Dz; after
2)z(@z + 1piz ). That the equation is quartic was to be expected since
there needs to be 8 solutions, and there are only two exponents p. In the logarithmic case
a + b = 0, one simply replaces pi by d=d log(1
We don't know whether this quartic di erential equation can be solved in terms of more
elementary functions. In any case, it can be solved numerically, or also straightforwardly
as a power series in z: the above di erential equation directly translates into a fourth-order
recurrence relation. For example, focusing on the case where d = 3 and a = b = 0 as used
in the main text, and expanding in the variable y = 1 z z , we nd that
2Dz
Y
c2 + 2
d
2Dz + Y
c2 GJ; (z; z) ;
(1
z)@z] is the operator entering (A.2). The limit
2Dz
c2 + 2
d
2Dz + Y~
c2 H~J(i;);0(z) ;
some algebra we nd:
c4GJ; (z; z) =
c4H~J(i;);0(z) =
where now Y~ =
zz
z
z
2) zzzz [(1
d 2
d 2
1
z
z
z ! 0 is now regular and gives a closed equation for m = 0:
zz
z
z
z
1
z
2 d
2 d
(A.33)
(A.34)
y2 +:::
(A.35)
(A.36)
H~ Jlo;g;0(y)=y 2 J = 1+
J 1)( (1 2J )+J )
2(2
1)(2J 1)
y + (
J )2 1
3(4J 2 8J +3)
2(4J 3 12J 2 +13J 6)+
J 2 +J 2(J 2)
8(2
1)(2
J )(2J 3)(2J 1)
H~Jre;g;0(y)=y 2 J =
y
2(2
1)(2J 1)
2(3 2J )+2 (3 2J +2J 2)+3 5J +J 2
4(2
1)(2
+1)(2J 3)(2J 1)
y2 +:::
In the special case J = 0, the equation factorizes, and correspondingly we nd that the
series simpli es and can be summed exactly:
H~0(1;);0(z) = 2F1
H~0(2;);0(z) = (1
2
+ a;
+ b;
2
z) a b2F1
2
2
; z ;
2
d
2
2
; z :
In the limit a + b ! 0, either of these solution converges to H~ 0lo;g;0(y) as quoted in the main
text.
Finally, we note that once the quartic equation for the z-dependence of the leading
z ! 1 term is solved, it does not need to be used anymore since the quadratic equation
expresses the subleading (1
z) terms in terms of z-derivatives of the leading solution.
In this section we test the Lorentzian inversion formula on explicit correlators of the
twodimensional critical Ising model. This model contains two scalar Virasoro primaries:
and
, which have mass dimension
= 18 and
= 1 and are odd and even under a Z2 symmetry,
respectively. Their four-point correlators, with the conventions from subsection 2.1, are [48]
(expressions from [49]):
G
G
A
C
G
G
1
2
1=16(1 + 6 + 2
)2(1 + )1=8
p
2)1=4
2
; G
G
B
D
G
G
1 + 2 2
1
2
;
1 + 14 2 + 4 2
(1
2)2
:
Note that G
B and G
Let us consider G
C represent di erent channels of the same correlator.
A in detail, leaving the others to the reader (the others are also
interesting as they manifest the divergences at z ! 1 discussed below eq. (3.9). To compute
its double discontinuity, we need to treat ;
as independent variable and take
either above or below the axis (see eq. (2.14)), which gives
dDisc GA( ; ) =
1
p1 (p
2
+ p ) + p
2)1=4(1
2)1=4
which here is to be evaluated in the range 0 < ; < 1 (thus di ering from the formulas in
the main text by
! 1= ) Note that it is positive, as required. To get the OPE coe cients,
according to our main inversion formula (3.20), we just need to integrate this against the
two-dimensional (global) conformal blocks given in eq. (2.5). Since both the global blocks
and Ising correlator take on a factorized form, we get a sum of factorized integrals:
A
cJ;
= 1 + ( 1)J
I0 1=4(J + ) I0 1=4(J +2
) + I1=12=4(J + ) I1=12=4(J +2
)
p I1=12=4(J + ) I0 1=4(J +2
p I0 1=4(J + ) I1=12=4(J +2
where, in terms of the -variables, using the measure given in (3.13), the basic integral is
1
2
J+
2
Ipp10 ( )
Z 1 d (1
This looks very di cult because k is an hypergeometric function with argument , but
in fact in the case that a = b = 0 it can be written in terms of a hypergeometric with
argument 2
:
k (z) a=b=0 = (4 ) =2 2F1
1
; ;
2 2
+ 1
2
; 2
k ( ):
Changing variable to u = 2 the integral can then be computed as a generalized
hypergeometric function,
Ipp10 ( ) =
2
3 (p1 + 2)
+2p0 2
4
+2p0+4p1+6
4
2
2
k ( ) p0 (1
2)p1 :
! 1= ,
(B.2)
) ; (B.3)
(B.4)
(B.5)
; 1 :
(B.6)
3F2
1
; ;
2 2
+ 2p0
4
2
+ 1
2
+ 2p0 + 4p1 + 6
4
The usual, discrete OPE coe cients are then obtained from these partial wave coe cients
by taking residues, according to (3.9):
A
cJ;
Res 0=
8 cA(J; 0)
: cA(J; 0)
( 0 J) 2( J+2
2
0) 2( 02 J0 ) cA( 0 1; J +1)
For the rst few OPE coe cients (up to dimension 7), for example, this formula gives:8
A
c0;1 =
A
c4;5 =
1
2
1
65536
1
2
1
4
A
; c2;2 =
A
c6;6 =
1
A
c4;4 =
A
c2;6 =
A
c0;4 =
A
; c6;7 =
1
;
1
The extra 12 in the rst case is due to the dimension being equal to d=2, as explained below
eq. (3.8). We have checked that, upon substituting into the OPE sum (2.3), these numbers
reproduce the series expansion of the Euclidean correlator GA! (Using computer algebra
we have checked the match up to dimension 15.) This con rms the extraction of OPE
data (with respect to the rigid, not Virasoro, conformal symmetry) from the Lorentzian
inversion formula, as an analytic function of dimension and spin.
We have also veri ed that the analytic result (B.3) agrees numerically with the
Euclidean inversion integral (3.5):
(
generic)
J = 3; 5; : : :)
(B.8)
HJEP09(217)8
Z
j j<1
A
cJ;
= N (J; )
d
2
( ; ) GA( ; ) FJ; ( ; ) :
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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