#### A Mellin space approach to the conformal bootstrap

Received: March
Mellin space approach to the conformal bootstrap
Rajesh Gopakumar 0 1 2 4
Apratim Kaviraj 0 1 2 3
Kallol Sen 0 1 2 3
Aninda Sinha 0 1 2 3
Open Access 0 1 2
c The Authors. 0 1 2
0 The University of Tokyo Institutes for Advanced Study , Kashiwa, Chiba 277-8583 , Japan
1 C.V. Raman Avenue , Bangalore 560012 , India
2 Survey No. 151, Shivakote, Hesaraghatta Hobli, Bangalore North 560 089 , India
3 Centre for High Energy Physics, Indian Institute of Science
4 International Centre for Theoretical Sciences (ICTS-TIFR)
We describe in more detail our approach to the conformal bootstrap which uses the Mellin representation of CF Td four point functions and expands them in terms of crossing symmetric combinations of AdSd+1 Witten exchange functions. We consider arbitrary external scalar operators and set up the conditions for consistency with the operator product expansion. Namely, we demand cancellation of spurious powers (of the cross ratios, in position space) which translate into spurious poles in Mellin space. We discuss two contexts in which we can immediately apply this method by imposing the simplest set of constraint equations. The rst is the epsilon expansion. We mostly focus on the Wilson-Fisher xed point as studied in an epsilon expansion about d = 4. We reproduce Feynman diagram results for operator dimensions to O( 3) rather straightforwardly. This approach also yields new analytic predictions for OPE coe cients to the same order which t nicely with recent numerical estimates for the Ising model (at mention some leading order results for scalar theories near three and six dimensions. The second context is a large spin expansion, in any dimension, where we are able to reproduce and go a bit beyond some of the results recently obtained using the (double) light cone expansion. We also have a preliminary discussion about numerical implementation of the above bootstrap scheme in the absence of a small parameter.
Conformal Field Theory; AdS-CFT Correspondence
1 Introduction
The philosophy outlined
The s-channel
The t-channel
The u-channel
The bootstrap constraints
The case of identical scalars
Double and single poles
4.2 Identity operator contribution 4.3 The bootstrap constraints 5
Scalar dimensions and OPE coe cients
Higher spin anomalous dimensions and OPE coe cients
Justi cation for truncating operator sums
A summary and comparison of results
Anomalous dimensions
OPE coe cients
Comparisons with numerics in the 3d Ising model
dimensions | a non-unitary example
-expansion in other dimensions
Large spin asymptotics
Strongly coupled theories
Weakly coupled theories
CFTs close to a free theory
Theories in 4
Witten diagrams & conformal blocks in Mellin space
The spectral function representation
Adding in the t; u channels
The bootstrap strategy implemented
A The Mack polynomial
B The continuous Hahn polynomials
A key normalisation
D t integrals in the crossed channels
E q sums in the t channel
F Simpli cations for the
F.1 s-channel
F.2 Crossed channels
Spin-0 exchange
Spin-`0 > 0 exchange
G Large ` behavior of Q ;`
H Comparision with numerical results
Introduction
Quantum Field Theory (QFT) is one of the most robust frameworks we have in theoretical
physics. Its versatility is attested by the fact that it plays a central role in many contexts
in high energy physics, condensed matter physics and statistical physics. Thanks to the
work of Wilson and others [1{8], QFT was understood beyond a perturbative Feynman
diagram expansion. The central role in this modern understanding is played by scale
xed points of the Renormalisation Group (RG)
When combined with
d dimensional Poincare invariance, these
xed points are believed to have an enhanced
SO(d; 2) conformal invariance [9]. The resulting CFTs while being dynamically nontrivial
are also strongly constrained by the conformal symmetry.
The conformal bootstrap is the philosophy that these constraints are strong enough
to largely determine the dynamical content of the CFT viz. the spectrum of operator
dimensions of primaries and their three point functions. The presence of a convergent
OPE then implies that all other correlators can be
xed in terms of this data [10{12].
2 [13], employs the associativity of the four point function, as we describe below. Recently,
making use of the progress in
nding e cient expressions for conformal blocks [14, 15],
this approach was revived for d > 2 [16] where associativity constraints, often together
with positivity on the squares of OPE coe cients, were implemented numerically through
linear programming and semi-de nite programming, together with judicious truncation of
the operator spectrum [17{31]. This has led to remarkably precise bounds on low-lying
operator dimensions in a number of nontrivial CFTs. This includes, famously, the 3d Ising
model [32{34] which is in the same universality class as the critical point of the liquid-vapour
transition of water. There are also very strong indications of such theories living at special
points (\kinks") in the numerically allowed regions of parameter space. This suggests
that these theories are special in some way and perhaps amenable to analytic treatment.
These numerical methods have also been extended to supersymmetric theories [35{37].
Furthermore, there are also certain analytic results available at large spin [38{48, 83, 84].
However, the existing approaches do not appear to be well suited for extracting analytic
results in general. Also limited progress has been made in the case where external operators
carry spin, see e.g. [49{55].
Recently, using the conformal invariance of the three point function, the leading order
) dimensions were calculated
for the Wilson-Fisher
xed point [56]. This approach was further generalized to extract
leading order anomalous dimensions for other theories in [57{60]. Results have also been
dimensions of almost conserved higher spin currents [61{63]. These results crucially rely
on the use of the equations of motion or a higher spin symmetry, that is present when
the coupling constant goes to zero. It is not immediately obvious how to systematize
these approaches to subleading orders. In [64], a dispersion relation based method of
Polyakov [12] (which had built in crossing symmetry) was re-visited and it was found that
this approach could be extended to get the subleading order anomalous dimension for the
2 operator.1 In spite of this encouraging result (though it took more than 40 years to
reach here!), it was again not clear how to extend this dispersion relation based approach
to operators with spin or to make it a starting point for a systematic algorithm. A major
stumbling block was the reliance on momentum space where the underlying conformal
symmetry is not fully manifest.
In this paper, we will describe in more detail a novel approach to the conformal
bootstrap that was recently outlined in [65]. This approach is calculationally e ective and at
the same time conceptually quite suggestive. It combines two important ingredients. The
rst goes back to an alternative approach to the above dispersion based one, also attempted
by Polyakov in his original bootstrap paper [12]. He outlined a general way in which
demanding consistency of the operator product expansion with crossing symmetry gave rise
to constraints on operator dimensions and OPE coe cients. This was then implemented in
position space which made the symmetries more manifest compared to momentum space.
The idea behind this approach was to expand the CF Td four point function not in terms of
the conventional conformal blocks but rather in terms of a new set of building blocks with
built-in crossing symmetry from the beginning. We will see, in our modern incarnation,
that these new building blocks can be chosen to be essentially tree level Witten exchange
diagrams in AdSd+1. This is very suggestive of a reorganisation of the CFT in terms of a
dual AdS description though this will not be the main thrust of the present work.
The second ingredient we introduce is to implement the above bootstrapping procedure
in Mellin space rather than position space as used in [12]. The position space approach
1It was also shown how the leading order anomalous dimension at O( 2) for large spin operators could
be extracted using large spin bootstrap arguments based on [83, 84].
made the equations in [12] quite cumbersome and not explicit, especially for exchanges
involving spin. We are familiar with this from the complicated form that Witten diagrams
take in position space. The technology of the Mellin representation has been developed
quite a bit in recent years starting from the work of Mack [66{73]. As has been amply
stressed in these works, Mellin space is very natural for a CFT and plays a role
analogous to momentum space in usual QFTs. This enables one to exploit properties such as
meromorphy and more generally, features of scattering amplitudes (to which Mellin space
amplitudes naturally transition to, in an appropriate at space limit). This, we will see,
brings us big calculational gains. We will be able to reproduce many of the analytic results
available in the literature for the conformal bootstrap in a fairly straightforward manner.
In addition, we will be able to derive new results which we subject to various cross checks.
We also give some preliminary evidence that this approach might also be workable into a
useful computational scheme, complementary to existing ones.
In the rest of this section we give a broad sketch of the new philosophy that we adopt
and state some of the new results obtained with this approach. We rst describe the ideas
in position space and only later translate them into Mellin space.
The philosophy outlined
Consider a four point function (of four identical scalars, for de niteness - we will consider
the general case in section2). In essence, we expand this amplitude in a new basis of
building blocks as follows
A(u; v) = hO(1)O(2)O(3)O(4)i =
X c ;` W (s;)`(u; v) + W (t;)`(u; v) + W (u;`)(u; v) : (1.1)
Here (u; v) are the usual conformally invariant cross ratios, whose dependence captures the
nontrivial dynamical information of the four point function (we have suppressed a trivial
additional dependence on positions which is predetermined). In the second line we sum
over the entire physical spectrum of primary operators generically characterised by the
operator dimensions ( ) together with the spin (`) quantum numbers. The building block
W (s;)` can, for the moment, be viewed as the Witten exchange function in AdSd+1 | it
will be de ned more precisely later. This is diagrammatically represented in
gure 1. It
involves the four identical scalars with an exchange in the s channel of a eld of spin ` and
corresponding to a dimension
. Similarly, for the t and u-channels. The to-be determined
coe cients c ;` will turn out to be proportional to the (square) of the three point OPE
The idea behind this expansion, which we will contrast below to the usual conformal
block expansion, is that we are expanding in a basis which
1. Is conformally invariant, as Witten exchange diagrams are;
2. Is consistent with factorisation, in that the individual blocks factorise on the physical
operators with the right residues corresponding to three point functions;
3. Is crossing symmetric by construction since we are summing over all three channels.
to an operator of dimension
and spin `.
The Witten exchange diagrams satisfy the second criterion since they arise from a local
eld theory in AdS. This will be much more explicitly seen in the Mellin representation.
The last criterion ensures that we don't need to check channel duality since that is built in.
But what is not obvious now is that expanding the resulting amplitude in any one channel,
say the s-channel, is consistent with the operator product expansion. In other words, if
we expand A(u; v) in powers of u, it is not guaranteed that all the powers that appear are
those of the physical primary operators together with their descendants.
In fact, generically, such an expansion will have spurious power law dependence. For
instance, with identical external scalars (of dimension
), we will see that there are pieces
which go like u
dimension 2
ln (u). The u
would indicate the presence of an operator with
, which generically does not exist in the theory.2 These are often called
\double-trace operator" (\O2") contributions in the AdS/CFT literature since these are
there interpreted as contributions from two particle states whose energy is almost the same
(in a large N limit) as the two external (single) particle states.3
We will then obtain
constraints on operator dimensions as well as the coe cients c ;` (and thus the OPE
coe cients) from requiring that such spurious powers vanish. Note that these are strong
constraints implying an in nite number of relations since there is a full function (of v)
multiplying these powers. Though we will not make use of them in this work, there are
additional spurious powers (and logs) of the form u
+n ln (u). These can
viewed as contributions from descendants as well as other double-trace primaries (what
would have been \O@2nO" in a weakly coupled theory). One would obtain additional
constraints from requiring their vanishing but we will not explore the consequences of this
in this paper (see [87]).
We should stress that the Witten exchange diagrams are being employed as a
convenient kinematical basis for this expansion, for an arbitrary CF Td. We are not assuming
(and it does not have to be) that the theory has an AdSd+1 gravity dual. We could have
2There could be special operators in interacting superconformal theories | \chiral primaries" | for
which there indeed are physical operators with dimension 2
. Such cases would have to be treated
specially, perhaps using mixed correlators or by focussing on other spurious powers.
3The logarithmic dependence is a consequence of having identical scalars. If we had generic dimensions
i for the external operators, the spurious powers would take the form u 1+2 2 and u 3+2 4 corresponding
to the two sets of double trace operators associated with the external states in the s-channel. The logarithm
arises in the coincident limit
. We also emphasise that the logarithmic dependence has nothing to
do with anomalous dimensions since we are not making any expansion in a small parameter (yet).
alternatively used conformal blocks as a basis of expansion. But as will become clearer in
Mellin space these are not very well behaved at in nity.4 In contrast, Witten exchange
diagrams will be polynomially bounded and thus a better basis for expansion. We note
that since each Witten exchange diagram contains the conformal block contribution of the
exchanged operator and since we are summing over the full primary operator spectrum we
are not undercounting in this basis. In particular, what would have been double trace
operators are included separately in the sum | this is di erent from what we do in AdS/CFT
where we only sum over single trace primaries. In this context note also that contact four
point Witten diagrams make no appearance in our approach. We do not have to include
them since it is known that they are decomposable into the double trace conformal blocks
and thus, in our context, have purely spurious power law contributions.
Another important point to note is that we are implicitly assuming that the sums over
( ; `) in the spurious pole cancellation conditions are convergent or can be analytically
continued. In the examples we have considered in this paper, the spurious poles have
gotten contributions from only a small set of operators and hence we did not have to worry
about convergence. In the usual case with conformal blocks, convergence is demonstrated
in [100]. A preliminary discussion on convergence in Witten diagram expansion can be
found in [65]. It would be good to investigate the issue more generally. As for the physical
contributions, once the spurious pole cancellation has been achieved, the remaining sum is
just the usual sum over the physical conformal blocks which is believed to be convergent
in a nite domain.
Let us contrast this approach to the more \conventional" bootstrap approach to
CFTs [10, 11, 13] where we expand the four point function
A(u; v) =
X C ;`G(t);`(u; v) =
In this expansion, the function G ;`(u; v) represents the conformal block due to a primary
operator of dimension
and spin `. It satis es the usual quadratic Casimir di erential
equation [85]. The corresponding OPE coe cient is given by C ;`. We choose to expand
in terms of the conformal block in a particular channel, say, s-channel, as in the rst line.
The conformal blocks are
1. Are conformally invariant by construction;
2. Are consistent with factorisation, in that the individual blocks give the factorised
contribution on physical operators with the right residues;
3. Are consistent with the OPE by construction since we are summing over all physical
operators in any given channel.
4Polyakov made a similar observation in terms of the behaviour of these blocks in the spectral parameter
The last criterion now ensures compatibility with the operator product expansion in
terms of the powers that appear, but it is now not guaranteed that the resulting
amplitude is crossing symmetric. In other words, the equality of the rst line with the second
line does not automatically follow. Demanding this associativity of the OPE is the
nontrivial requirement which constrains operator dimensions and the OPE coe cients C ;`.
Recent progress has come from e cient ways to translate the constraints of associativity
and positivity (which follows from unitarity) into inequalities which can be numerically
Coming back to our approach, to convert Polyakov's scheme into a calculationally
e ective tool we mix in our second ingredient which is the Mellin representation of CFT
amplitudes. The position space amplitude A(u; v) has the Mellin representation
A(u; v) =
s)M(s; t)
The additional kinematic
factors in the measure are de ned for convenience [66]. M(s; t)
is the (reduced) Mellin amplitude for the original conformal amplitude A(u; v).
Mellin amplitudes are ideally suited to our present purpose since they share many
of the features of momentum space for standard S-matrix amplitudes. In particular, the
contributions of di erent operators show up as poles with a factorisation of the residues
into lower point amplitudes. Moreover, our building blocks, the Witten exchange diagrams,
are complicated in position space but are analytically easier to deal with in the Mellin
representation. In fact, they can be viewed as the meromorphic piece of the conformal
blocks in Mellin space, which are also known explicitly since the work of Mack. They
therefore have the same poles as the corresponding conformal blocks together with the
same residues thus exhibiting the needed factorisation. In fact, as mentioned above, the
Witten blocks are better behaved in Mellin space compared to conformal blocks: the latter
have exponential dependence on the Mellin variables compared to the former which are
polynomially bounded.
We can now translate the presence of spurious power law (as well as log) dependence
in position space into Mellin space. The u
ln u) behaviour arise from spurious
poles (and double poles) at s =
where s is the Mellin variable conjugate to the cross
ratio u. Therefore, we now demand that these residues vanish identically, i.e. as a function
of the other Mellin variable, t. This gives an in nite set of constraints on operator
dimensions and OPE coe cients.5 Here another advantage of the Mellin representation makes
its appearance. In analogy with partial wave expansions for momentum space scattering
amplitudes, there is a natural decomposition of the residues into a sum over a basis of
orthogonal polynomials in the t-variable. These polynomials (known as the continuous
Hahn polynomials in the mathematics literature) go over to the generalised Legendre (or
Gegenbauer) polynomials in an appropriate
at space limit. This decomposition makes
the imposition of our in nite set of conditions more tractable analytically. One big
simpli cation is that operators of spin-` contribute (in the s-channel) only to the orthogonal
5As mentioned above, there are additional spurious powers which lead to subsidiary spurious poles
(double as well as single) at s =
+ m (with m = 1; 2; : : :).
polynomial of degree `, as one might expect in analogy with at space scattering. In the
tchannel an in nite number of spins do contribute but this happens in a relatively controlled
way. This makes it natural to impose the vanishing residue conditions independently for
each partial wave `.6 This feature of the Mellin space approach to the conformal
bootstrap makes it very close in spirit to the at space S-matrix bootstrap (see [74, 75] for one
recently proposed way of connecting the two).
It turns out to be simplest to implement this schema when there is a small parameter to
expand in. We will focus here on two such examples. The rst is the canonical expansion
in d = (4
) dimensions for a single real scalar at the Wilson-Fisher
xed point. The
second is the large spin limit (in any dimension) in scalar theories with a twist gap.
In the former case, we will see that, in the Mellin partial wave decomposition, there are
some signi cant simpli cations when we impose the vanishing constraints. By assuming
contributions we nd that the lowest couple of orders in
get contribution only from the 2
primary exchange, in addition to the identity operator. This enables us to recover known
results for the anomalous scaling dimensions (
0, respectively) of both
The anomalous dimensions of these operators are known upto O( 5) [76{78].
= 1
0 = 2
We also nd the OPE coe cient C
2 = C0 with a new result for the O( 2) piece
C0 = 2
In fact, if we take the input from Feynman diagram calculations of the O( 3) contribution
0 then we can make a new prediction (5.35) for the corresponding O( 3) contribution
to C0 as well. By moving onto the partial wave ` we again
nd that, in the s-channel, it is
@` ) that contribute
to the rst two non-vanishing orders in . Once again, to this same order in the t-channel
it is only the 2 (and identity) contribution that is needed. This enables one to obtain, in
a fairly easy way, the nontrivial results for the anomalous dimensions of these operators
` = d
(3) being given in (5.33). The O( 3) term matches with the nontrivial Feynman
diagram computations of [79]. Our approach gives the OPE coe cients too with equal
ease unlike other methods. We thus obtain C
J` = C` to O( 3) as given in (5.36), (5.40)
6In the discussion section we will mention an alternative procedure for imposing the constraints which
setting each of the terms to zero. This set of conditions is linearly related to the set of conditions from the
the central charge cT which is related to C2 (by (5.42)). Our result
= 1
agrees with previous calculations to O( 2) [80{82] and gives a new prediction at O( 3).
As we describe in section 5.4.3 these results, after setting
= 1, compare very well
with some of the numerical results obtained for the 3d Ising model.
The second context is of the large spin asymptotics (in a general dimension d, for
Wilson-Fisher like
xed points) we consider the two regimes that have been analysed in
the literature through the (double) lightcone expansion. Our techniques here allow us to
reproduce results in both the large and small twist gap regimes. Thus we reproduce the
results of [83, 84] for the anomalous dimensions of the operators J ` in (6.15) and the leading
corrections to the OPE coe cients in (6.17). In an opposite \weakly coupled" regime [43]
we reproduce again the anomalous dimensions of the operators J ` in (6.23) together with
a new determination of the coe cients in (6.24).
The plan of the paper is as follows. In section 2 we discuss both Witten diagrams and
the usual conformal blocks in Mellin space. We also discuss the spectral function
representation of Witten diagrams that we employ in the rest of the paper. In section 3 we explain
how to implement the bootstrap in Mellin space. In section 4 we turn to the identical
scalar case which sets up the explicit -expansion calculation in section 5. Section 6 deals
with large spin asymptotics both for strongly coupled theories and weakly coupled theories.
We conclude in section 7 with a preliminary discussion on numerics and future directions.
There are several appendices containing useful identities and intermediate results.
Witten diagrams & conformal blocks in Mellin space
In this section we begin the process of migrating to Mellin space by carrying over the
familiar conformal blocks and the associated Witten exchange diagrams from position space.
We will consider the somewhat more general case of arbitrary scalar external operators
and de ne our amplitudes, setting notation in the process. Let A(x1; x2; x3; x4) denote the
four point function of four scalar operators in a CFT (the scalar Oi has dimension
Here we have pulled out overall factors in the four point function appropriate for an
schannel decomposition and de ned
The cross ratios (u; v) are de ned in the standard way
as =
u =
2) ; bs =
x212x2324 ; v =
i =
, we recover the previous expression eq. (1.3). Note that we are making
a particular choice here so that even when we consider t; u-channel exchange diagrams, we
will still be using the convention (2.4) with the overall factors as in (2.1).
Though we will not be utilising them very much, let us discuss how the conformal
blocks look in Mellin space [66, 86]. Under the transform of eq. (2.4), the conformal blocks
G(s;)`(u; v) ! B(s;)`(s; t):
The Mellin space conformal blocks B(s;)`(s; t) take the form [86]
t) (s + t + as) (s + t + bs)M(s; t)
The corresponding Mellin amplitude reads as7
A(u; v) =
f ig(s; t)M(s; t):
= ei (h
= ei 2 (2s
2s = 2h
h;`(s) is de ned, for later use, by the equality between the rst and second
lines. The Gamma functions in the numerator of
h;`(s) exhibit poles at both 2s =
d the spacetime dimension). Since we would like to project out the contribution of the
shadow poles the prefactor in brackets was introduced in [86] so that it has zeroes precisely
at these unphysical values. This cancellation of poles is made manifest in the third line.
The projection, however, leads to an exponential dependence on s at large values of this
Mellin variable.
The crucial piece of the conformal blocks in Mellin space are the P (s)
h;`(s; t) | the
so-called Mack Polynomials which are of degree ` in the Mellin variables (s; t). In addition
to the dependence on
, they also depend on the external scalars through as; bs, but we
suppress this dependence, so as not to clutter notation.8 We merely signal this dependence
through the superscript which indicates that we are considering parameters relevant to an
s-channel. The explicit form of these polynomials is given in appendix A.9
12; t =
4). See also [85].
8Both the Mack Polynomials as well as
h) and this is re ected in their subscript.
9Our normalisation of the Mack Polynomials agrees with that of Mack and di ers from that of [70] by
a factor of (
have a parametric dependence which is naturally in the
` + 2m giving residues
which are kinematic polynomials in the variable t determined by the spin ` of the
intermediate state and the level m of the conformal descendants [70].
B(s;)`(s; t) =
The dots refer to the entire function piece of the block in eq. (2.6). That is the part which
has an exponential behaviour at in nity. The Q`;m(t) polynomials are single variable
specialisations of the Mack Polynomials.
following i.e.
h;` s =
Q`;0(t) =
h;` s =
The Q`;0(t) turn out to be a family of orthogonal polynomials (continuous Hahn
Polynomials) whose properties are given in appendix B. These can be viewed as the generalisations
of the Legendre/Gegenbauer polynomials that accompany the partial wave decomposition
for scattering amplitudes.
Just as for the conformal blocks, we can consider the Mellin version of the contribution
from Witten exchange diagrams under the transform (2.4)
W (s;)`(u; v) ! M (s;)`(s; t):
Witten exchange diagrams in Mellin space have been investigated in the literature [67{70].
It is known that they have the same poles and residues as the corresponding conformal
blocks. However, they are polynomially bounded for large (s; t), in contrast to the
exponential dependence of conformal blocks B(s;)`(s; t). They therefore take the form
M (s;)`(s; t) =
+ R` 1(s; t)
where R` 1(s; t) is a polynomial10 of degree at most (`
1) in (s; t). Note that the rst
term is identical to that in (2.7). The second term is an additional polynomial ambiguity
coming from freedom in the choice of three point vertices in the bulk AdS in de ning the
exchange diagram. The meromorphic piece is however xed to be the same as that of the
conformal blocks (2.7). Since our interest is to use an appropriate basis, we will choose the
ambiguity to our convenience. A particularly simple choice of basis would, for instance, be
to only use the meromorphic piece of the conformal block i.e. just the rst term in (2.11).
10Even though we will not need the form of R` explicitly, we can x it by writing the Mellin amplitude
as a meromorphic piece plus a regular piece. The Mellin amplitude (2.23), which is in terms of a spectral
function, hence xes R` implicitly.
We can write this sum (for any `) in terms of a nite sum of hypergeometric functions. Our
choice will actually involve the additional polynomial piece R` 1(s; t) as well. In the case
of a scalar exchange, however, such terms don't enter and the answer for the corresponding
sum in (2.11) is particularly simple [67{69]
M (s;)`=0(s; t) =
; 1 : (2.12)
In forthcoming work [87] we will employ this direct method to explicitly write down
the Witten exchange function. In the current paper, we use an alternative approach to
writing the exchange diagram in terms of a spectral function representation. While this
introduces some additional terminology, it will have some advantages for implementing our
bootstrap philosophy.
The spectral function representation
Our starting point will be the spectral representation of the Witten exchange function in
position space (in, say, the s-channel). Following (2.1) we de ne
W (s;)`(xi) =
The spectral representation is then a decomposition in terms of conformal partial waves
(see for e.g. [88], section 6). This follows from a \split" representation of the bulk-to-bulk
propagator in terms of two bulk-to-boundary propagators with a spectral parameter
is integrated over. The latter can be expressed in terms of conformal partial waves
W (s;)`(u; v) =
The conformal partial waves F (;s`)(u; v) are closely related [70, 85] to the conformal blocks
being just linear combinations of a block of ctitious dimension
= h +
and its shadow
with dimension d
= h
(s;)`( ) =
F (;s`)(u; v) =
n( ; `) =
+ `)=2, 1 = (h
+ `)=2 and
x;y = (x + y) (x
The spectral function
(s;)`( ) itself is the dynamical piece that contains information
about the exchanged operator with dimension
. We can further break it as
conformal group, see for e.g. appendix B of [89]))
`( ) =
(h + `) (h +
and a piece which is dynamical (i.e. (s;)`( )) and thus knows about
. The explicit
ex(s;)`( ) =
) ( 3+ 4 h+`+ ) ( 3+ 4 h+`
: (2.19)
contribute.11
The interpretation as a spectral function comes from the fact that we can evaluate the
integral along the imaginary axis by closing the contour (when the integral is well behaved
at in nity) on, say, the right half plane and picking up the residues at the simple poles.
These then correspond to the primary operators which are exchanged whose contribution,
along with their conformal descendants, is captured by the conformal block in (2.15). The
contour is chosen to enclose either an operator or its shadow, but not both. The superscript
in (s) (and (s)) signi es the channel and is re ected in the dependence on the
i in (2.19).
Let us see which primaries contribute. The spectral function in (2.19) has simple
h corresponding to the operator ( ; `) (and its shadow). But it also
has simple poles at h
2 + ` + 2n and h
4 + ` + 2m where
n and m are non-negative integers. In a generic theory there are no operators of this
literature. This is because in a weakly coupled (\generalised free eld") theory, like in the
large N limit, these would be the dimensions of double trace operators of the schematic
form O1@`(@2)nO2 (and similarly with O3 and O4). It is known [90, 91] that precisely
these double trace primary operators (of spin `) do contribute to the Witten diagram (in
the s-channel) and the spectral function merely reproduces this fact. Note that when we
contour on the right half plane only the poles with the plus sign will (typically)
The full spectral function (with the Plancherel measure) thus takes the form
(s;)`( ) =
) ( 3+ 4 h+`+ ) ( 3+ 4 h+`
We have seen that this has the right behaviour to reproduce the known properties of the
Witten exchange diagram. There is one subtlety though that we should mention in this
form of the spectral function. There are a nite number of extra poles in the integral from
the denominator factors of (h
1)` whose contributions need to be cancelled by adding
11Note that this would imply that there are no poles coming from
1;bs in the integrand. For
general as, there may be a
nite set of poles on the right but an in nite set of poles on the left. Our choice
of contour is such that the entire in nite chain is on one side of the contour.
lower order spin terms (`0 < `). These have been explicitly studied in [88]. However, these
additional terms will not contribute to the terms of interest to us which are the residues
at the double trace poles which come purely from the above piece.
agrees with what was constructed by Polyakov for what he called the `unitary' amplitude.
Polyakov, of course, did not come to this from Witten exchange diagrams. He rst
constructed a spectral function for what he called the `algebraic amplitude' which is nothing
other than the conformal block itself. This turns out to have the form
Al;g`( ) =
This algebraic spectral function
Al;g`( ) is designed to reproduce the conformal block on the
l.h.s. if we insert it in the r.h.s. of (2.14) instead of (s;)`( ). We see that now the
integral over the right half plane only gets contributions from the single pole at
after using (2.15), (2.16). Note that
Al;g`( ) in (2.21) di ers from
(s) ( ) given in (2.20)
by the four numerator -functions in the latter which were the double trace contributions.
Al;g`( ) su ers from the problem that it diverges as one goes to
i1. As we will
see later, this is related to the poor behaviour of the conformal block in Mellin space, at
in nity along the imaginary axis. To cure this problem, Polyakov prescribes adding certain
additional factors of -functions to the numerator. These turn out to be precisely (in his
case, for identical scalars) the double trace ones which appear in the numerator of (2.20)
so that we indeed get the spectral function appropriate to the Witten exchange diagram!
We can now translate this spectral representation to Mellin space. We use the fact
that the partial wave appearing in (2.14) has the Mellin representation [85]
F (;s`)(u; v) =
in terms of the Mack Polynomials as well as the
Mellin measure f ig(s; t) in (2.4).
channel Witten exchange diagram in Mellin space to be
Then, combining (2.22) with (2.14) we have the spectral representation for the
s(s;`)(s) de ned in (2.6) and the standard
M (s;)`(s; t) =
Adding in the t; u channels
Our discussion was for the s-channel contribution W (s;)`(u; v) to (1.1) and the corresponding
M (s;)`(s; t) in Mellin space. It is not di cult to extend the discussion to the other two
channels. The main point to keep in mind is that our conventions are chosen, for de niteness,
for an s-channel expansion. Thus we pull out the same external factor as in (2.1) when we
are considering the reduced amplitude in the t and u-channels also. This is even though
the natural de nition for the Witten diagram in the t-channel would involve an interchange
of subscripts (2; 4) (and similarly (2; 3) for the u-channel) of the s-channel which gives, for
instance, answer (2.13)
W (t;)`(xi) =
where at =
4) and bt = 12
2). If we recast this in the form (2.1) by pulling
out the same external factor as in that equation, then this corresponds to multiplying
W (t;)`(u; v) by an extra factor of u 2 ( 3+ 4)v 21 ( 2+ 3). In a similar way, an extra factor of
1
1
u 2 ( 1+ 4) multiplies W (u;`)(u; v). Here, both W (t;;`u)(u; v) are obtained from W (s;)`(u; v) by
the interchange of labels (2; 4) (and (2; 3), respectively).
We can translate this to Mellin space in a straightforward manner. Thus the t-channel
partial wave (the analogue of (2.22) reads, with the above prefactor, as
1 1
= u 2 ( 3+ 4)v 2 ( 2+ 3)
(at + s + t) (bt + s + t)
Here the integrand on the r.h.s. is obtained from the corresponding one of the
schannel (2.22), with the interchange (s $ t) i.e. of labels (2; 4). The superscript t on the
Here `0 denotes the spin in the t-channel.
But now observe that by shifting variables
we can make the r.h.s. of (2.25) now in the same form as (2.22) i.e.
Similarly, in the u-channel, we have
(t;)`0 (t) =
(u;`)0 (s + t) =
`0) +s+t
(bs + s + t) (as + s + t)
3) and bu = 12 ( 2
: (2.28)
This was for the partial waves in the t; u-channels. We can now employ the
corresponding versions of (2.14) to write the expressions for the corresponding Witten exchange
diagrams in the spectral representation in Mellin space i.e. the counterparts of (2.23).
Combining (2.25) with the analogue of (2.14), we nd
M (t;)`0 (s; t) =
M (u;)`0 (s; t) =
And similarly in the u-channel with (2.29)
Here the spectral weights, (t);`0 ( ); (u)
subscripts (2 $ 4) and (2 $ 3) respectively.
The bootstrap strategy implemented
;`0 ( ) are given by (2.20) with the exchange of
With all this machinery in place, we are now ready to come to the crux of our strategy.
As mentioned in the introduction, we write the four point function as a sum over a set of
crossing symmetric Witten exchange diagrams as in (1.1). In position space this can be
written, using the spectral representation (2.14), as
A(u; v) = X Z i1
1 1
+ c(t);` (t);`( )u 2 ( 3+ 4)v 2 ( 2+ 3)F (;t`)(u; v)
1
+ c(u;)` (u;)`( )u 2 ( 1+ 4)F (;u`)(u; v) :
Here the sum over ; ` is over the entire physical (primary) operator spectrum of the CFT.
Note that we have, in general, to-be-determined coe cients c(s;;`t;u) which are mutually
related by exchanges of the labels (e.g. (2 $ 4) or (2 $ 3)). This ensures that the full
amplitude is crossing symmetric.
Since we are not making an expansion of the amplitude in terms of conformal blocks
in a xed channel, we are not guaranteed that this expansion will have the right power
law dependences on the positions (or equivalently, cross-ratios) that is consistent with the
OPE. For instance, in the case of identical scalars we see from (4.3) that the spectral
func(s) ( ) has double poles (at h +
is the dimension of the common
external scalar). When we perform the
integral, this double pole gives rise to u
terms in the sum, as well as u
terms. Both of these dependences would imply the
presence of an operator with dimension 2
in the spectrum which is generically not the case.
More generally, we will have spurious power laws of the form u 1+2 2 and u 3+2 4 when
we expand (3.1) in the s-channel. There are generically no operators corresponding to
dimensions ( 1 +
4). Thus we have to demand that these terms
identically vanish after including the contributions from the other channels and on summation
over ( ; `).
As discussed, it will be easier to implement this in Mellin space. In other words, we
look at the total Mellin space amplitude corresponding to (3.1) which we obtain by putting
together (2.23), (2.31) and (2.32)
M(s; t) =
c(s;)`M (s;)`(s; t) + c(t);`M (t;)`(s; t) + c(u;)`M (u;)`(s; t)
c(s;)` (s;)`( ) (s;`)(s)P (;s`)(s; t)
+ c(u;)` (u;)`( ) (u;`)(s + t)P (;u`) s
and s = 12
The de nition of the Mellin transform in (2.4) imply that the spurious powers in
4). When the external scalars are identical, these two sets of spurious
. It is important to note that
these are statements about the full Mellin space amplitude and not just the reduced one,
M(s; t). In other words, recalling the notation of (2.4) we need to examine the spurious
f ig(s; t)M(s; t). In particular, for identical scalars, the
. So we will need to look at the
s) piece of
constant as well as terms linear in (s
) of M(s; t) to isolate the poles of interest to us.
In either case, the residues at these spurious poles will be a function of t and we will
obtain an in nite number of constraints on our CFT by setting these identically to zero.
Below, we will individually look at the Mellin amplitudes in each channel, for non-identical
scalars, and isolate the residues. We can then add them all up and
nd the conditions for
consistency with the OPE. In the following section we will examine the special features
that arise for identical scalars.
The s-channel
We start with the unitary block in the s-channel (i.e. the Mellin transform of the Witten
exchange diagram) given in eq. (2.23).
where, as in (2.20), we have the spectral function
M (s;)`(s; t) =
(s;)`( ) =
(s;`)(s) =
2 = (h +
`)=2; 2 = (h
Here we have introduced, for compactness, the notation [85]:
We are to carry out the
integral by closing the contour on the right half plane.
The \physical pole" in the spectral function is the one at
= (
h).12 In this case
the factors of
cancel out with the corresponding factors in
s) in the denominator of
f ig(s; t) of eq. (2.4).13 The residue at this
physical pole in
has factors of -functions from the numerator of (3.5) which give rise to
the physical pole in the s-variable (as well as for the shadow) i.e.
2s =
2s = (2h
` + 2n; (n = 0; 1; 2 : : :) :
When we do the s integral (again closing the contour appropriately) of the Mellin
amplitude, this gives rise to the physical contribution with a u
`)=2+n dependence but does
not pick up the shadow.
spectral function (3.4)
However, there are other poles in
which give rise to the spurious poles in s that
we described earlier. For instance, when we consider the poles from the numerator of the
the residue will get a contribution from the numerator of (3.5) /
The denominator of (3.5) then cancels with the Mellin measure but the above piece gives
at s = 12
spurious poles at s = 21
2) + n. By a similar argument there are spurious poles also
4) + n. Instead, if we had taken the poles from the numerator of (3.5) i.e.
= 2s
then the residue contribution from the numerator of (3.4) would be again /
s). Thus the residues of the Mellin amplitude evaluated on these second set
of poles gives a factor of two to the previous contribution.15 We have already discussed
(see around (2.20) and footnote 10) the absence of any role from all the other poles of the
spectral function.
2. The two variable
. We use the
general relation (2.9) to obtain
= 4 `( 1 +
12We are assuming
0. If not, we have to deform the contour so that we include this pole but not
that of the shadow operator which would now lie on the right half plane [66].
13For the scalar, this can be explicitly seen in the denominator
factors in (2.11) which cancel against
f14iAg(ss;wte)llaatsthaeshpahdyoswicaplipecoele i(nhs.
` 1
2
s) which will always be understood to be present
but which we will ignore since we will choose to close the Mellin contour so as to exclude this set of poles.
15The poles at s = 12
2) + n with (n > 0) actually come with a multiplicity. But since we will be
As a result the nett residue of the Mellin amplitude at the unphysical pole is
M (s)(s; t)
s= 12 ( 1+ 2)
Here we have de ned
s) s= 12 ( 1+ 2)
q(s;)` =
denotes contribution from the physical pole
as well as the other spurious
pole at s = 12 ( 3 +
4) which gets an identical contribution to above with (1; 2) replaced
by (3; 4). The case of identical scalars will be discussed separately in the next section.
In the next subsections we look at the t; u-channels and similar pole contributions
1
that lead to anomalous u 2 ( 1+ 2) behaviour in the amplitude. Demanding that these
cancel against the above s-channel contribution will give us constraints. Note that the
cancellation conditions involve a whole function of t. To facilitate the comparison, we
have expanded the functional dependence on t in terms of the orthogonal polynomials
Q`;01+ 2+`(t). We will expand the amplitudes in the other channels in terms of the same
polynomials and use the orthogonality to set the nett coe cients of each Q`;01+ 2+`(t) to
zero.16 We also note another important utility of this particular orthogonal decomposition
| a spin ` Witten diagram contributes only to the orthogonal polynomial labelled by the
same `. This property makes this decomposition the analogue of the usual partial wave
decompositions for
at space scattering amplitudes. The contributions from a speci c
spin is, however, a special feature of the s-channel decomposition and will not hold in the
t-channel (see (3.16)).
The t-channel
The unitary block in the t-channel is given by (2.31)
M (t;)`0 (s; t) =
polynomials Q`;03+ 4+`(t).
(t);`( ) =
dimension. This is mostly negative except for
comes positive.
1:733) where this ips sign and
bepn
iS4
i42
-pn
in2
pS
function of the dimension of the exchanged operator dimensions ( ). All of these plots are mostly
positive except for near the unitarity bound given by
d 2 + `. The sign ip near the unitarity
bound is given by the inset plots.
This will demonstrate how one may hope to see a bound arising from these numerics.
0:518 which is the value for the 3d-Ising model.
As is clear from the spin block plots, close to the unitarity bound (
the spin blocks are negative but positive elsewhere. The scalar block on the other
hand is positive only for a small region of
1:733) (see
In the rest of the range of
0, the scalar block is negative. If we assume that
the non-zero spin operators, which contribute most to the constraints, are close to
the unitarity bound (as happens for the 3d Ising model), then their contribution to
the constraint equation will be negative. So the only way to satisfy the constraint
equation would be if the scalar block were to give a positive contribution. This gives
1:733) which is indeed the case for the
2 operator. Note that we
used just one constraint for illustrative purposes to demonstrate that investigating
numerics along these directions ought to be a promising future endeavour. Of course,
one needs to demonstrate, since there is an in nite sum over the spectrum
the resulting numerics converge. Our preliminary investigations, of theories living at
the border of the known allowed regions, using presently available numerical methods,
does suggest that the approach above will lead to convergent numerics.27 A more
thorough investigation of this issue should be carried out.
Higher orders in : we have had striking success in using our approach to obtain
results to O( 3) | therefore it is natural to ask how to go to the next order in
the -expansion. Indeed one would like to know if there is a systematic approach
that allows one to go to any arbitrary order in the expansion, if one so desired.28
Once we set up the formalism obtaining the O( 3) results was conceptually and
mathematically straightforward (though, perhaps a bit tedious) and needed very few
and rather mild assumptions. The two main inputs were the existence of a conserved
stress tensor and the leading behaviour of OPE coe cients for higher order operators
which we know from the perturbation expansion | for instance C
2 begins at O( 2)
since it is the square of the OPE coe cient which is assumed to be O( ). We had
pointed out in section 5, that at O( 4), the constraint equations at the spurious pole
(s =
) involved an in nite number of operators. However, it is plausible that,
by appropriately combining the enormous amount of information in the additional
constraints at s =
+ n, one can give an algorithm to continue to higher orders
in a controlled manner. Furthermore, we have only investigated the case of
identical scalars. Thus we may need to combine the information from other spurious
poles with correlators of other scalar operators. At present, our approach yielded
information about operators which were bilinear in the elementary scalar . It is
possible to extend our results to operators with higher powers of
[87], for which
some information is known in the -expansion [101{103]. At some stage we expect the
non-unitary behaviour in 4
to show up for some large dimension operator [104].
Our approach, however, did not crucially rely on unitarity as the non-unitary example
showed and should be able to capture this behaviour in 4
dimensions. It
will also be interesting to use our approach to study the theories in higher dimensions
considered in [105{107].
Other small expansion parameters: we have studied our equations with two small
expansion parameters, namely
and 1` , and found quite remarkable simpli cations.
It will therefore be interesting to investigate our equations when we have other small
expansion parameters. These could be, for example, large dimensions for external
operators, large spacetime dimension d limit, strong or weak coupling limits and of
course, large N . In these cases one might hope to have similar simpli cations which
organise the bootstrap conditions so that there is a controlled way of incorporating
the contributions from di erent families of operators. Recently in [108], a systematic
procedure has been outlined to solve the conventional boostrap equations in the
large spin limit using \twist blocks". These twist blocks resum the contribution of
27We thank Slava Rychkov for suggesting this check.
28Note that the -expansion, like any perturbative QFT expansion, is only an asymptotic one and needs to
be Pade-resummed to obtain something useful. The question is whether there is an in-principle systematic
method to obtain the n-th term in this expansion using our Mellin space approach.
all operators of degenerate twist and di erent spins and appear to be a useful way
to compute the anomalous dimension of large spin operators with arbitrary twists.
One could attempt a similar procedure to approximate the Mack polynomials in this
limit and set up the analogous equations in Mellin space.
Technical hurdles: in order to use our constraint equations systematically, one
bottleneck is the integration over the spectral parameter in the crossed channel and
another is the lack of a compact expression for the Mack polynomials. In the way we
have currently set-up the equations there were coincidences which led to remarkable
(almost) cancellations between various -poles. This fact enabled us to go to higher
than what one may have naively expected from [12, 64]. In fact, as we saw
in our calculation, in the crossed channel only the
2 operator contributed to yield
the nontrivial results at O( 3). In [87] we will show how to get rid of the integration
over the spectral parameter leading to an enormous simpli cation in the form of the
equations. This should enable a systematic investigation of a plethora of questions,
some of which have been indicated above. It will also be desirable to have a better
understanding of the Mack polynomials to see if more compact representations for
them exist, compared to the present one. One could also perhaps try to see if there
is a geometric way (in AdSd+1) of understanding our consistency conditions.29 This
could lead to a new way of doing quantum
eld theory for critical phenomena which
uses a di erent set of diagrams rather than Feynman diagrams to systematize general
perturbative expansions.
Connection with AdS/CFT: our building blocks are Witten diagrams in Mellin space.
This suggests the tantalizing possibility of AdS/CFT playing an important role to
understand the Wilson-Fisher xed point. Of course, any such dual string theory is
likely to be in the quantum regime. In a companion paper [93], the present method
is extended to O(N ) both in the -expansion as well as in the large-N limit showing
that it works analogously, yielding the rst few subleading orders. A systematic study
of our constraints in the large-N limit will be an important question to investigate
in the future in order to explicate the connection with a weakly coupled string
theory/Vasiliev theory. We do not have further insights to o er at this stage, but clearly
it will be fascinating to unearth a direct connection between string theory and the
3d Ising model.
Acknowledgments
Special thanks to J. Penedones for collaboration during the initial stages of this work and
for discussions. We acknowledge useful discussions with S. Giombi, T. Hartman, J. Kaplan,
I. Klebanov, G. Mandal, S. Minwalla, D. Poland, S. Pufu, Z. Komargodski, J. Maldacena,
H. Osborn, E. Perlmutter, L. Rastelli, M. Serone, S. Wadia and especially S. Rychkov.
We also thank all our other colleagues at IISc, ICTS and TIFR-Mumbai for numerous
discussions and encouragement during various stages of this work. R.G. acknowledges the
29There is a geometric way of understanding the conventional conformal blocks in terms of geodesic
Witten diagrams [109].
support of the J. C. Bose fellowship of the DST. A.S. acknowledges support from a DST
Swarnajayanti Fellowship Award DST/SJF/PSA-01/2013-14. This work would not have
been possible without the unstinting support for the basic sciences by the people of India.
The Mack polynomial
The Mack polynomials P (;s`)(s; t) are explicitly known [66, 70, 85], albeit in terms of a
multiple sum
Here we employ the notation in [85]
1 =
1 =
2 =
2 =
1;bs are de ned in (2.17) with as; bs de ned in (2.2) and the li-s are given by,
l1 = 2
l3 = 2 + bs + k + m ;
l2 = 2 + as + k + m
l4 = 2
The continuous Hahn polynomials
In (2.9) we specialised the two variable Mack polynomials de ned in appendix A to obtain
polynomials of degree ` in one variable.
Q`;+0`(t) =
P (+s)` h;` s =
Note that we have suppressed the dependence on the parameters as; bs These
polynomials have a number of remarkable properties which enable us to simplify the
bootstrap conditions.
The rst is that the multiple sum that de nes the Mack polynomials in (A.1) collapses
into a simple single sum which is, in fact, a familiar 3F2 hypergeometric function.
Q`;+0`(t) =
Here and in the rest of this appendix, for generality, we replace the s-channel parameters
(as; bs) by arbitrary parameters (a; b).
The second remarkable fact is that the Q`;0 polynomials are orthogonal and known
in the mathematics literature as the continuous Hahn polynomials [110]. They obey the
orthonormality condition,
b)Q`;+0`(t)Q`0+;0`0(t) = `( =2) `;`0 ;
`( =2) =
`+ =2;a `+ =2;b
We note another useful property of the Q`;0 polynomials (in the special case of
identical scalars) which follows from properties of the hypergeometric function 3F2.
The transformation
n; k1; k2; 1 =
factor (k3
obtain the relation
If we take n = `, k1 =
+ ` 1, k2 = b + 2 + t, k3 = 2
a and k4 = 2 + b, we see that this
Q`;+0`(t) = ( 1)`Q`;+0`
Now let us use this to show that for identical scalars, and for an even spin ` exchange in
the s-channel, we have the t-channel expansion coe cient in (3.18) equal to that in the
u-channel (3.23). In the t-channel, q`(t)(s) for identical scalars is given by (4.7),
q(t);`j`0( =2) = `( =2) 1
while in the u-channel q(u;)`j`0(s) is given by,
q(u;)`j`0( =2) = `( =2) 1
2( =2 + t) ( 2
(`t0)( )P (;t`)0( =2
( t) ( =2 + t + 2
) ( =2 + t + 2
(`u0)( )P (;u`)0( =2
Now if we use the identity (B.6), the above two expressions become equal under the
exchange t $
t. Hence one can conclude in general for identical scalars and even spin
exchange in s-channel, that the t-channel is equal to the u-channel.
Finally the more general polynomials Q`;+m`(t) ,
Q`;+m`(t) =
P +` h;` s =
appear at the descendant poles. However, analogues of the above nice properties of Q`;0(t)
are not known for the polynomials with m > 0.
We have been expanding the amplitude in terms of Witten diagram blocks as in (1.1) with
to-be determined coe cients c ;`. These c ;` are proportional to the (square of the) OPE
coe cients C ;` which appear in the conventional conformal block expansion of (1.2). We
need to x the relative normalisation between the two if we are to be able to compute the
OPE coe cients C ;`. We will do so in this appendix.
This is simplest to do in position space. The conformal blocks G ;`(u; v) are normalised
Thus we need to ensure that as (u ! 0; v ! 1)
= C ;`u 2 (1
By explicitly evaluating the l.h.s. we will be able to obtain the relative normalisations. We
will actually carry this out for non-identical scalars for generality.
The Witten diagram in position space has the Mellin representation is obtained from
the general expression (2.4) by plugging in M (s)(s; t) as the reduced mellin amplitude. The
latter is given in the spectral representation by (2.23). In the integral over the spectral
parameter, we now focus on the contribution from the physical pole at
h. This gives
c ;`W (s;)`(u; v) = c ;`
bs) (s + t + as) (s + t + bs)
) ( 1 + 2h + `
( 2h + ` +
( 2h + `+
Thus we will multiply the above expression by a factor of two.
To proceed we expand vt in 1
v. This will give,
contribution to the spectral integral from a
= 2h
` s. Evaluating the residue
2 ` . Actually, there there is another
` which gives an identical contribution.
( 2h + ` +
( 2h+`+
Here we have used the relation
+ t + bs Q`;0(t; as; bs) :
h;` s =
= 2 2`(
1)`Q`;0(t) :
The binomial expansion coe cient ( 1)m t
= ( 1)m
O(tm 1) is a polynomial in t of degree m. Therefore we can rewrite it as a sum over
the orthogonal polynomials Q`0;0(t) for 0
Q`;0(t) = 2`t` + O(t` 1) we must have ( 1)m t
m
m. Since Q`;0 is normalized such that
Qm;0(t) +
. Using the
orthonormality of the Q`;0 polynomials as given in appendix B, we can evaluate the
contribution to (C.4) which goes as (1
expansion. Doing the t integral gives the contribution
c ;`W (s;)`(u; v) = u 2 ` (1
( 2h + ` +
The omitted terms denote higher powers of u and 1
N 1;` =
( 2h + ` +
t integrals in the crossed channels
This section will deal with performing the t integrals in (4.7). We will carry out the integrals
for generic `; `0. The technique can be generalised to the non-identical scalar cases of (3.18)
and (3.23) but we will not do so here since we only need the identical scalar results. Let
Q`;+0` =
2` 2( =2 + `) ( + `
2( =2) ( + 2`
1; =2 + t
Now the hypergeometric function can be written as a sum as below,
1; =2 + t
; 1 =
q(t);`j`0( =2) = `( =2) 1
(t);`0( )P (;t`)( =2
2`(( =2)`)2 ( `)q( + `
(( =2)q)2 q!
In the second equality we have used the explicit forms of the Mack Polynomials in (A.1)
that, we will use the integral representation of the 2F1 function, which is given by,
With this let us evaluate the t-channel q(t);` as given in (3.18),
Shifting the variables t ! t s
and using the mapping,
a = 2 +
+k ; b = 2 +
+k ; c = 2k + + +
+ 2 + 2 ; (D.7)
denotes that we have shifted the contour along the real axis so that all the poles
of the same sign lie on the same side of the contour. We can further use,
z) =2 t = X
above, by picking up the power of zq. So we have,
The idea is to use the above as a generating funtion for the part 2 + t q in the t-integral
z) =2 t :
2F1(a; b; c; z) =
2 i (a) (b) (c
ds (s) (c
b + s) (a
s) (b s)(1
= (1 z)
2k + + + 2 + 2 + 2
Collecting the powers of zq, we nally have,
q + + 2k + + + 2 + 2 2
+ k)(1 z) =2 t
Using this in (D.3) we get,
q(t);`j`0( =2)
2`(( =2)`)2 ( `)q( + ` 1)q
(( =2)q)2 q!
The above expression also gives q(u;)`j`0 in the u-channel.
= 0. Also
q + + 2k + + + 2 + 2 2
The hypergeometric 3F2 simply reduces to 1. Also note that
q(t);`j`0=0( =2) =
2`(( =2)`)2
`( =2)( + `
(( =2)q)2 q!
= (h + )=2 and 1 = 2 =
2`(( =2)`)2 ( `)q( +` 1)q
(( =2)q)2 q!
`
2 X ( 1)q2 ` (2` + 1) (` + 1 + q)
q=0 q! (` + 1
q) (` + 1)3 (2 + q)
1;0 1;0=Qi (li) is 1. So
q sums in the t channel
In this section, we will demonstrate, how the expression in (4.8) can lead to simple
expressions, in an
expansion. Here we will show this for the leading term in (4.9). Let us write
down the full expression, given by,
h)2)`!q! (` +
( ) ( ) ( + q) ( + q)
= h+
= h
2 . The expression can be expanded in . The
integral can
be carried out by evaluating residues at only the poles
= 2
discussed in section 5.3 and appendix F, the other poles are subleading. The leading term
is given by,
10 + 16` + (1+2`) 45 E +
18H2` + 18Hq
18H`+q + 36 (` + 1)
where, note that the problematic terms are Hq and H`+q. Using the identity,
we can pull out the k sum and rst perform the q sum over these term. After the q sum,
we can perform the sum over k to get,
q=0 q! (` + 1
q) (` + 1)3 (2 + q)
(1 + 2`)(18Hq
18H`+q)
= 9
22+` (3=2 + `)
The remaining terms can be handled with the usual sum over q to obtain,
q=0 q! (` + 1
q) (` + 1)3 (2 + q)
10 + 16` + (1 + 2`) 45 +
18H2` + 36 (` + 1)
2 `(1 + 2`)(2`)!
`(1 + `)(`!)3
Adding these two separate contributions, we nd that,
q(2;;`tj)0 =
2 X ( 1)q2 ` (2` + 1) (` + 1 + q)
q=0 q! (` + 1
q) (` + 1)3 (2 + q)
10+16`+(1+2`) 45 E +
18H2` +18Hq
18H`+q + 36 (` + 1)
indicate subleading terms in . One must be careful while handling the above
expression, since with the normalization inside c ;` that multiplies this, the whole thing
starts from O( 2).
Simpli cations for the
dimensions rested on several
simplications, which occur when we Taylor expand our equations in . In this appendix we will
address all of them.
Recall that the s-channel has P
c ;`q(s;)`, which is a sum over all operators of spin `. The
rst simpli cation here is that only the lowest dimension operator of spin ` contributes to
the sum to the order we consider. This is the operator with dimension
` = 2+`
For the 4 theory, we are considering, there are higher dimension operators with
` + 2 + 2m + m + O( 2). These operators have the generic form O2m;`
(@2)m@` . Using
2m;` =
the equation of motion, O2m;`
b @c , where among the a + b + c (= 2m
derivatives, 2m
2 derivatives are contracted and ` derivatives carry indices. We will show
that these operators are suppressed in an
expansion. We will demonstrate this only for
the q(2;;`s) term. The q(1;;`s) follows a similar logic. Using
from (4.6) for
4(1 + m + `)
Here C2m;` is the OPE coe cient of this operator. Now these operators do not exist in
the free theory. Since Om;` has four -s it is easy to guess that the 3-point function starts
hence C2m;`
). The ope coe cient C2m;` goes as square of this quantity and
O( 2). Accounting for this, we must have,
2m;`;` = O( 4) :
There can be other \heavier" operators too, with spin `, also contributing to the sum
(s) . However, such operators are composites with a higher number of -s, for
. Now such operators will have the OPE coe cient C ;`
which are even further suppressed in . So the corresponding q(s;)` will begin from O( 6) or
beyond. Because of this, when we considered only up to the O( 3) term, keeping only the
` operator had su ced.
2m;0;0 =
( 1)2m+1 ( m + 1) 2 2(2 + 2m)
2 4(1 + m)
Then, again since C2m;0
only go to O( 2) order, by keeping just one operator.
O( 2), we have c2m;0q(s)
Crossed channels
In the crossed channels, we have P
we had considered only a certain set of poles and neglected others under the assumption
that the other set of poles would contribute at a higher order in the
expansion. Here
we will put these assumptions on solid grounds. We will rst go through the case when
the exchanged operator is a spin 0 particle (scalar) and then move on to substantiate the
same arguments for the case of spin `0 exchanged operators. In the subsection below, we
will rst demonstrate the case of the scalar exchange (for identical external scalars, which
;`0 c ;`q(t);`j`0 , which is a sum over all operators in the
O( 3). Thus, for ` = 0 we could
is the case of interest).
We begin by explicitly writing down the spin 0 contribution for the t channel (since
the crossed channels give identical contributions for the identical scalar case, we can just
consider either one of them) in (D.10),
2 `c ;0 ( ` + q) (2` + 2s) (2s + ` + q
h)2)`!q! ( `) (` + s)2 (q + s)2 (2s + ` 1) ( ) ( )
+ q + s) (
We will show that the leading
dependence comes from only the 2 operator and also
considering only two poles in the spectral integral is su cient. This happens because of
cancellations. Let us introduce the notation,
a nontrivial cancellation among residues of poles for every operator. We will demonstrate
this with the log term, which is q(2;;`tj)0. The power law term which is q(1;;`tj)0 will have similar
f (q; s) =
1)q (q + s
`!q! (` + s)2 (q + s)2 (h + q + 2(s
The poles of the above integral, lying on the positive real part of the - contour,
c ;0q(t);`j0(s) =
Note that for n2
q the poles III become double poles.
= 2
1. Lowest dimension scalar: consider the operator 2 with dimension
0 = 2 +
(1) + O( 2). This is the only operator that contributes in the crossed channels up
to the O( 3) order. In our computation we used only the poles
h. Let us see why the other poles do not contribute. Let us assume
n1 < q and n2 < q + 1. Then residues at the poles II and II give (where we have used
h = 2
=2 and the familiar dimension of ,
= 1
=2 + (2) 2),
Res =2
h+2n1 =
Res =h 2
+2s+2n2 =
C2;0( 1) n1+1( 0(1))2(1 + 0(1))2f2n1(q; 1) (q
C2;0( 1) n2( 0(1))2(1 + 0(1))2f2+2n2(q; 1) (q
Here C2;0 is the OPE coe cient of 2
. The subleading terms in
are cumbresome
quite nicely this cancellation is till the O( 3) order. Hence we have,
Res =2
h+2n + Res =h 2
+2s+2n 2 = O( 4) ;
with n being a positive integer. This cancellation is true for any function f (q; s).
When the poles III become a double pole for n2
q, the individual residues start
from a di rent order, viz Res =2
Res =h 2
the cancellation (F.8) till O( 3) would still hold.
O( 1). However
2. Heavier scalars: now let us look at heavier scalars with dimensions of the form
2m;0 = 2 + 2m +
+ O( ). Here m is a positive integer. It was argued in
section F.1 that OPE coe cients of such operators begin from C2m;0
With this taken into account one makes the following observations:
O( 2) at least.
Res =
h + Res =2
h+2m + Res =h 2
Res =2
h+n + Res =h 2
Res =2
h = O( 5)
+2s+2m 2 = O( 4)
+2s+2n 2 = O( 4)
n 6= m :
Hence we see, none of the higher dimensional scalars can contribute to the crossed
Spin-`0 > 0 exchange
law term c ;`0 q
(1;;`sj)`0 we can simply put s =
For spin `0 operators too, there are analogous cancellations just like what we have seen
above. Let us take the general expression (D.10). To avoid tedious expressions, we will not
give the explicit forms of the individual residues. We will just indicate the pairs of poles
that cancel each other under -expansion. For both the log term c ;`0 q
, which will give us the following poles,
h) | from the denominator of the spectral weight (t);`0 ( )
h + 2n) | from the numerator
factors of the spectral weight
= (h + `0 + 2n) | from the other
factors in numerator
= (h
2 + `0) | from the denominator Pochhammer terms
in the spectral weight .
operators, with dimensions
following cancellations,
1. Lowest dimension spin `0: the lowest dimension operators with spin `0 are the J`0
`0 = `0 + 2
+ O( 2) . For these operators we nd the
Res =
h + Res =2
+`0 h + Res =h 2+`0 = O( 4)
Res =2
h+`0+2n+2 + Res =h+`0+2n = O( 4) where
n = 0; 1; 2;
Res =h 1
Res =h 2
Res =h+`0 3 = O( 4) :
2. Higher dimensional spin `0 operators: there are heavier operators with spin `0 as
we discussed in F.1. These operators, labelled O2m;`0 have the dimensions
2m;` =
` + 2 + 2m + m + O( 2) . We had argued that since these operators must have
the composition
coe cients go like C2m;`0
operators, we have the cancellations,
c , or a higher number of -s, their OPE
O( 2) or higher. Taking this into account for these
Res =2 +`0 h + Res =h 2+`0 = O( 4)
Res =
h + Res =2
h+`0+2m + Res =h+`0+2m 2 = O( 4)
Res =2
Res =h 1
Res =h 2
Res =h+`0 3 = O( 4) :
Thus we conclude none of the operators in the crossed channels contribute except the 2
operator for which only two poles are su cient up to the O( 3) order.
Large ` behavior of Q ;`
In this appendix we will derive the large ` approximation for Q`;0. Let us start by deriving
an approximation for the 3F2 hypergeometric function. It has the integral representation,
n; k1; k2; 1 =
Now as n ! 1, we have [111],
2F1( n; k2; k3; z)
z) k1+k4 12F1( n; k2; k3; z)dz : (G.1)
(k2)(nz) k2 + (k3 k2)( nz)k2 k3(1 z)k3 k2+n :
For identical external scalars, the Q`;0 has a 3F2 which is equal to the above under the map
So we get,
n; k1; k2; 1 =
(k1) ( k2 + k3) ( k2 + k4)
Finally using this in (B.2) and putting the values of k1;2;3;4, we get the large `
approxima1, k2 = s + t, k3 = 2 and k4 = 2
. Since n
second term in the parentheses above. Then (G.1) becomes,
n; k1; k2; 1 =
(k3) (k4)
1 Z 1
Carrying out the z integral it gives,
(k3) (k4)
k3 k2 we can neglect the
(k3) (k4) (k1
Comparision with numerical results
In this appendix we compare the results obtained in section 5.2 with numerical data found
in [112]. To compare, we have taken the OPE coe cients (5.36) and obtained their square
roots. Then we re-expand in
to O( 3) to obtain f J` , and put
= 1. As expected the
match worsens as we go higher in spin. The reason for this is discussed in section 5.4.3.
` = 2
` = 4
` = 6
` = 8
` = 10
` = 12
` = 14
` = 16
f J` j =1
Percentage Deviation
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