Mellin bootstrap for scalars in generic dimension
HJE
Mellin bootstrap for scalars in generic dimension
John Golden 0 1
Daniel R. Mayerson 0 1
0 University of Michigan , 450 Church Street, Ann Arbor, MI 481091020 , U.S.A
1 Department of Physics and Leinweber Center for Theoretical Physics
We use the recently developed framework of the Mellin bootstrap to study perturbatively free scalar CFTs in arbitrary dimensions. This approach uses the crossingsymmetric Mellin space formulation of correlation functions to generate algebraic bootstrap equations by demanding that only physical operators contribute to the OPE. We nd that there are no perturbatively interacting CFTs with only fundamental scalars in d > 6 dimensions (to at least second order in the perturbation). Our results can be seen as a modest step towards understanding the space of interacting CFTs in d > 6 and are consistent with the intuition that no such CFTs exist.
Conformal Field Theory; Field Theories in Higher Dimensions

1 Introduction and summary
2
3
2.1
2.2
3.1
3.2
3.3
3.4
Bootstrap
A De nitions
B Simpli cations
Setup, assumptions, and simpli cations
The Mellin bootstrap
Simpli cations
tchannel contributions from
tChannel Contributions from
Bootstrap equations to O( 2)
Summary and extensions
2
B.1
The coe cients c O;l
B.2 S1: schannel contributions for heavier operators
B.3 S2: tchannel contributions for l0 > 0
B.4 S3: tchannel contributions for heavier scalars (l0 = 0)
B.5 Contributions from
and 2
The program of the conformal bootstrap was introduced in the 1970s [1, 2] and has recently
regained attention [3{6]. This program is largely based on the fact that the consequences
of conformal symmetry are su ciently stringent to signi cantly constrain the space of
possible CFTs. Two particularly important consequences are the operator product
expansion (OPE) and crossing symmetry. Considerable insights can be gained by studying
the constraints of crossing symmetry on theories which, by construction, satisfy the OPE
(e.g. [7]). This is done by constructing an ansatz in the form of an OPE of a fourpoint
correlation with generic coe cients, and then imposing crossing symmetry and exploring
the solution space. The recent Mellin bootstrap, pionereed by Gopakumar, Kaviraj, Sen,
Sinha in [8, 9],1 approaches the problem from the complementary perspective of assuming
crossing symmetry and then imposing the correct OPE structure. In this case, one
constructs an ansatz that explicitly satis es crossing symmetry but then the correct operator
spectrum has to be imposed as a consistency condition. Demanding that the \exchanged"
1Some earlier works important for the development of the Mellin bootstrap are [10{12].
WilsonFisher xed points studied in [8, 9, 13],2
it was shown that, to a certain order in , this in nite set of algebraic equations in the
Mellin bootstrap truncates to an analytically solvable nite set. It is interesting to consider
whether this simpli cation of the Mellin bootstrap is a feature of perturbatively free CFTs
in general. In this note, we begin by investigating theories with a single fundamental
that are perturbatively close to a free point in a general dimension d (which can
be a function of the perturbation parameter as in the WilsonFisher xed point but does
not necessarily have to be). We show that the Mellin bootstrap again reduces to a
nite
set of solvable analytic equations in this more general case. Furthermore, the solutions
to these equations constrains the dimensions of
and the lowest excited scalar operator
\ 2", as well as the OPE coe cients C
T , C
and C
2 . In particular, we nd that
the scalar theory is forced to be the free scalar theory in arbitrary dimension d > 6, to
the perturbative order we are able to calculate. This provides evidence that there are no
interacting CFTs with a single fundamental scalar in d > 6 dimensions. Our results can
also easily be generalized to the case of an arbitrary number of fundamental scalars, again
providing evidence that there are no interacting CFTs with any number of fundamental
scalars in d > 6 dimensions.
Our results are perhaps not surprising: for perturbatively free scalar theories,
Lagrangian methods give an easy argument that no nontrivial marginal operator can be
constructed in d > 6 for scalar theories  therefore no interacting scalar theory would be
expected to exist. Furthermore, recall that the superconformal algebra does not close in
d > 6, so there are no superconformal eld theories in this regime [16]. However, the Mellin
bootstrap approach allows us to approach the question of existence of CFTs from a
fundamentally di erent viewpoint not involving Lagrangians at all. This approach also begins to
clarify the criteria under which the Mellin bootstrap reduces to a nite set of equations. It
would be interesting to expand our results to include fermions, other (fundamental) elds,
and to connect these techniques with the analysis of generalized WilsonFisher xed points
in [17, 18].
This paper is organized as follows: section 2 brie y introduces the Mellin bootstrap
and our CFT setup, including our assumptions and the resulting simpli cations that occur
that reduce the bootstrap equations to a
nite number of algebraic equations. Section 3
discusses solving these bootstrap equations to show that the theory we obtain is forced to
be free. The necessary de nitions of functions etc. can be found in appendix A while the
simpli cations are discussed in more detail in appendix B.
2
Setup, assumptions, and simpli cations
The objects of study for this note are unitary, perturbative CFTs containing one
fundamental scalar
as the lowestdimension operators. (We will discuss in section 3.4 how to
extend our results to an arbitrary number of fundamental scalars.)
2A similar expansion analysis for the O(N ) model was made using the Mellin bootstrap in [14]; the
saturates the unitary bound and is a free scalar for any
[19].
We have a similar expansion for the dimension of the lowest excited operator, which we
denote as \ 2" following standard convention:
is to emphasize the fact that d can be an analytic function of
, as is the case in the WilsonFisher CFT). The
= 0 limit corresponds to the free scalar
CFT in d0 dimensions. The dimension of
is parameterized as
=
d
2
2
2 + l + l(
1
) + O(
2
);
Cl = C(0) + C(
1
) + O(
2
):
l l
which is given by the schematic form3
We will also consider the lowest dimension (primary) operator Jl of spin l (for l 6= 0; 2),
and has dimension
Jl and OPE coef
cient C
Jl given by:
We make the following assumptions about the theory:
A1: there is a conserved stress tensor with
= d and spin l = 2 (i.e. the operator
Jl=2 has l(=i)2 = 0 for all i).
A2: there is only one fundamental scalar, , of dimension
given in (2.1).)
Note that we do not assume any other symmetry of the theory, such as an overall Z2, as is
often done. We will instead keep our analysis generic and then note where a Z2 symmetry
(i.e. C
= 0) allows us to make even stronger claims. We do note that we have assumed
that all relevant quantities in (2.2){(2.7) allow for an analytic expansion in ; this is an
assumption we make on the nature of the perturbative expansion.4
We do not need to
assume that
is positive; in our analysis we will see that the only possible solution for
d0 > 6 is the free theory which saturates the unitarity bound in d dimensions, regardless
of the sign of . (By contrast, note that
> 0 is necessary in the d = 6
analysis
3We have in mind the symmetric, traceless current operator of spin l that is a conserved current in the
free theory. The exact form of these operators can be found in e.g. [20].
4We thank M. Paulos for stressing this point to us.
{ 3 {
of [9]; there, when
already at O(
1
).)
review now.
can be seen to violate the unitarity bound
Our goal is to determine the values of the expansion coe cients in eqs. (2.1){(2.5)
given the assumptions A1 and A2. We will do so using the Mellin bootstrap, which we
Our analysis is an extension of the Mellin bootstrap techniques introduced in [8] and
described in detail in [9].
We are interested in the fourpoint function (with identical
i, which in position space is determined completely by the function A(u; v)
where u and v are the standard conformal cross ratios:
h (x1) (x2) (x3) (x4)i = x122
x342
A(u; v);
u =
As described in the Introduction, the Mellin bootstrap involves constructing h
crossing symmetry explicitly satis ed, but without explicit agreement with the OPE. In
other words, it is not guaranteed that the terms that appear in the expansion are primaries
and their descendants. Polyakov introduced this crossing symmetric construction in [1] and
noted that in this expansion terms proportional to u
and u
log u necessarily appear.
These terms correspond to an operator of dimension 2
appearing in the spectrum. Since
there is not generically an operator of this dimension in an interacting CFT, these terms
must cancel when summed over all channels.
Requiring that this is the case imposes
nontrivial constraints on the CFT. However, actually calculating the coe cients of the
spurious u
and u
log u terms is very di cult in position space, so this approach did
not recieve much attention until Gopakumar, Kaviraj, Sen, and Sinha realized [8] that
the spurious terms in position space become spurious poles, with calculable coe cients,
in Mellin space. We now brie y describe this calculational scheme. Appendix A contains
explicit de nitions for all of the functions that follow.
The (reduced) Mellin amplitude M(s; t) of the fourpoint function A(u; v) is de ned by
A(u; v) =
Z i1 ds dt
i1 2 i 2 i
usvt 2
( t) 2(s + t) 2(
s)M(s; t);
where the form of the measure has been chosen to highlight the pole at s =
particular, the expansion of M(s; t) around s =
has a constant and linear term in
(s
), so the overal integral has a single and double pole at this point. The residues at
these poles give rise to precisely the u
and u
log u terms that indicate the unphysical
operator described in the previous paragraph. Therefore, requiring that the coe cients of
the u
and u
log u terms be zero then corresponds to ensuring that the constant and
linear terms vanish:
(2.8)
(2.9)
i with
(2.10)
. In
M(s =
; t) = 0
and
= 0:
(2.11)
{ 4 {
This is the most general representation of `the Mellin bootstrap constraints'. While we will
not make use of it here, it is interesting to note that there are in fact an in nite class of
these constraints, as 2
(
s) has spurious poles at s =
+ n; 8 n 2 Z+.
Now we must cast M(s; t) in a form where we can impose (2.11) for all values of t.
First o , we can decompose M(s; t) into what can be thought of as exchange (Witten)
diagrams M (s;=lt=u)(s; t) where an operator O ;l is exchanged between pairs of 's; these
exchanges can happen in each of the s; t; uchannels. This gives us the explicitly crossing
symmetric representation
M(s; t) =
X c ;l M (s;)l(s; t) + M (t;)l(s; t) + M (u;)l(s; t) :
(2.12)
and O ;l (from the vertices of the exchange diagram).
Fortunately there is a convenient basis of orthogonal functions, the continuous Hahn
polynomials fQl (t)g, which allow us to partially separate the s
M (s=t=u)(s; t). In this decomposition, only a single term contributes in the schannel:
and t dependence in
M (s;)l(s; t) = q(s;)l(s)Ql2
+l(t);
where
q(s;)l(s) = q(2;;ls) + (s
)q(1;;ls) + : : : (as s !
):
q(2;;ls) and q(1;;ls) are labeled this way as they correspond to the double and single poles
in (2.10), respectively.
The contributions from the t
and u channels turn out to be identical, so we need
only include the tchannel twice. The in nite number of contributions in the t channel
which must cancel against the schannel contribution are:
(2.13)
(2.14)
(2.15)
(2.16)
(2.17)
where also:
q(t);ljl0 (s) = q(2;;ltj)l0 + (s
)q(1;;ltj)l0 + : : : (as s !
):
Note that there is a polynomial ambiguity that we have omitted from (2.13) and (2.15),
which stems from an ambiguity in the Mellin space Witten diagrams related to contact
terms [9, 12, 13, 15]. This ambiguity was shown to lead to di erent results for certain
quantities in the Mellin bootstrap approach compared to the regular bootstrap approach
at high orders in
[15]. While the presence of this ambiguity is thus certainly important,
it is also expected that it will only start contributing substantially to the relevant Mellin
bootstrap results at higher orders in
than we will be considering (speci cally, O(
4
)) [15].
Summing up the contributions from the di erent channels then gives us a more concrete
formulation of the bootstrap equations
X
c ;lq(a;;ls) + 2 X c ;l0 q
(a;t)
;ljl0
!
= 0;
the constraint M(s =
; t) = 0.
= 0 and a = 2 corresponds to
M (t;)l(s; t) =
l0
X q(t);ljl0 (s)Ql20
+l0 (t);
l0
{ 5 {
The bootstrap (2.17) represents an in nite number of equations (one for each l), and each
equation involves an in nite number of terms (from the sums over
; l0). A number of
nontrivial simpli cations occur for every l, resulting in only a
nite number of terms
contributing up to O(
2
). These terms involve the unknown parameters in eqs. (2.1){(2.5)
and form a solvable nite system of equations. These simpli cations are:
S0: the identity operator only contributes to the simple pole constraint, so we can
separate out that contribution and write the bootstrap equations as
HJEP06(218)
2q(1=;t)0;lj0 +
X
6=0
X
6=0
c ;lq(2;;ls) + 2 X c ;l0 q
c ;lq(1;;ls) + 2 X c ;l0 q
l0
l0
(2;t)
;ljl0
(1;t)
;ljl0
!
!
= 0;
= 0:
S1: in the schannel, only the lowest dimension operator of spin l contributes to
P
c ;lq(2;;ls) up to O(
2
) (for l = 0, we will consider the two lowest operators
explicitly). For l = 2 the in nite sum reduces to the single term
X c ;l=2q ;l=2 = c =d ;l=2q(a;s)
(a;s)
=d ;l=2 + O(
2
);
6=0
as the lowest dimension operator of spin l = 2 is the stress tensor with
= d . For
l = 0 the in nite sum reduces to
X c ;l=0q ;l=0 = c = ;l=0q(a;s)
(a;s)
= ;l=0 + c = 0;l=0q(a;s)
= 0;l=0 + O(
2
)
6=0
as the two lowest dimension scalars that can contribute are
itself and 2. For other
l, the lowest dimension operator is the operator Jl, so the in nite sum reduces to:
6=0
X c ;lq(a;;ls) = c ;lq(a;s)
= Jl ;l + O(
2
):
S2: in the tchannel, for l0 > 0, we have c ;l0 q
scalars contribute to the sum Pl0 c ;l0 q
(a;;ltj)l0 = O(
2
); i.e. to this order in , only
(a;;ltj)l0 . This in nite sum over l0 thus reduces to:
X c ;l0 q
l0
(a;;ltj)l0 = c ;l0=0q ;ljl0=0 + O(
2
):
(a;t)
(2.18)
(2.19)
(2.20)
(2.21)
(2.22)
(2.23)
S3: furthermore, in the tchannel, for \heavier" scalars of
c ;0q(a;;ltj)0 = O(
2
). This reduces the in nite sum over
in the tchannel to:
>
0
, we have
X c ;l0=0q ;ljl0=0 = c = ;l0=0q(a;t)
(a;t)
=
;ljl0=0 + c = 0;l0=0q(a;t)
= 0;ljl0=0 + O(
2
): (2.24)
{ 6 {
Derivations of these simpli cations are in appendix B. With these simpli cations, the
bootstrap equations for generic l become
c Jl ;lq
and in the special cases l = 0 and l = 2 we have
2q0(1;0;tj0) + c ;0 q(1;s;)0 + 2q(1;t)
;0j0
cd ;2qd(2;;s2) + 2c ;0q(2;t)
cd ;2qd(1;;s2) + 2q0(1;2;tj0) + 2c ;0q(1;t)
c ;0 q(2;s;)0 + 2q(2;t)
;0j0
lefthand side of these equations vanish up to the given order in
will give us nontrivial
constraints on the coe cients appearing in (2.1){(2.7).
3
Bootstrap
In this section we describe the bootstrap in considerable detail in order to highlight the
di erences between the generic d case and the special d = 4
studied in [9]. Before we do so, let us describe our result in physical terms, as the preceeding
notation is a bit cumbersome.
Consider, as an example, the schematic form of (2.28):
T(schannel) + 1 tchannel + (tchannel) + (2tchannel) = O(
2
)
where by \O(channel)" we refer to an overall contribution of the form c O;lqchOan;lnel. The other
bootstrap equations take a similar form, with the schannel contribution coming from either
in the l = 0 case, T for l = 2, and generic Jl for l > 2, and the tchannel identity operator
only arises in the simplepole constraints. For d0
6, all of these terms can make
nite,
nonzero contributions through O(
1
) (though note that they do not all contribute in all
cases  for example (tchannel) = 0 in the Z2symmetric case as C
= 0). The essential
story for the d0
6 regime is that the complicated interplay between nontrivial s and
tchannel contributions produces a set of linear equations to get cancellations up to O(
2
).
This set of linear equations can then be used to determine operator dimensions, e.g. for
the WilsonFisher xed point in d = 4
and the 3 theory in d = 6
.
We will show that for d0 > 6, the
and
2 tchannel contributions necessarily begin
O(
2
), so they drop out of the bootstrap equations and all we are left with is the
tchannel identity to cancel the schannel operator.
Without these nontrivial tchannel
contributions, the only possible solution to the resulting set of equations is the free theory.
5d0 = 2 is also obviously a special case, but for the sake of brevity we will not comment on those
subtleties here.
{ 7 {
(2.25)
(2.26)
(2.27)
(2.28)
(2.29)
(2.30)
cases5
(3.1)
While we have emphasized d0 > 6 so far, the 2 tchannel contribution in fact drops out
already at d0 > 4.
We can see from a simple series expansion that the normalization
coe cient c 0;0 already exhibits a special value at d0 = 4:
The fact that these poles behave di erently in d0 > 4 is not particularly surprising. We
know from [9] that at least in d0 = 6, the 2 operator indeed does only contribute at higher
order (otherwise it would compete with the C
exchange in the nonZ2 invariant theory);
however, what we nd here is that the 2 operator cannot contribute to the relevant orders
for any d0 > 4 (in particular d0 = 4n for n > 1 is not special in any way).
3.2
tChannel Contributions from
Now, we turn to the contributions from
in the bootstrap equations. Once again, we rst
note that a simple series expansion shows that the normalization coe cient c ;0 exhibits
a special value at d0 = 6:
c ;0 /
>
<
8
However, for d0 > 6, q(a0;t;l)j0 has an in nite sum of contributions from poles of the integrand
of that go as O( 0). In particular, this in nite sum is not convergent. This means that in
d0 > 6, we have (schematically):
where we have made explicit the factor of C(0) that is included in the normalization c ;0.
There are no other in nities that this could cancel with until the in nite tower of heavier
operators enters at O(
2
) in the bootstrap expressions; thus, we are forced to set C
= O(
3
)
in order for the bootstrap equations to make any sense to O(
2
). Note that this suppresses
the contribution of
from all the bootstrap equations (2.25){(2.30).
{ 8 {
It is interesting to compare this with the 2 case, as the
tchannel contributions take
an apparently similar form:
However, let us emphasize that the reason
must drop out of the bootstrap for d0 > 6
2
is di erent than
: we saw that the contribution of the
suppressed to O(
2
) for d0 > 4; here, the contribution of
2 operator is automatically
in d0 > 6 is actually in nite,
forcing us to set the OPE coe cient C
= O(
3
). This could not have happened for the
2 operator, as the 2 OPE is already nonzero in the free theory.
3.3
Bootstrap equations to O(
2
)
Above, we have discussed the tchannel contributions from
and 2 and found that they
all vanish. Now, we can use this to solve the bootstrap equations, which at this point
consist solely of operators in the schannel either cancelling o tchannel identity operator
contributions (in the simplepole constraints) or being set directly to O(
2
) (in the
doublepole case).
Let us consider the l = 2 bootstrap equations (2.27){(2.28) rst. The lowest order term
of (2.27) is at O(
1
), while (2.28) has terms at O( 0) and O(
1
). Demanding that (2.27)
and (2.28) hold up to O(
1
) thus gives us three equations to solve. The solution is
(
1
) = 0;
C(0) = C2;free(0);
2
C(
1
) = C20;free(0);
2
where C2;free( ) is the square of the (appropriately normalized) OPE coe cient in the free
C2;free( ) =
d (d
These results match those found in [9] for d = 4
the l = 2 case. The results of the bootstrap give:
The l 6= 0; 2 bootstrap equations (2.25){(2.26) for Jl proceed entirely analogously to
theory
where:6
(3.7)
(3.8)
(3.9)
(3.10)
(3.11)
Next, we can consider the l = 0 bootstrap equations (2.29){(2.30). Equation (2.30)
has terms at O( 0) and O(
1
), which can be set to zero to obtain:
0
C(0) = 2;
C(
1
) = 2 0(
1
)
0
2Hd0=2 3
Hd0 4 +
d
2
0
3d0
8
7d0 + 12
;
(3.12)
6Of course, (3.11) agrees with (3.9) for l = 2.
{ 9 {
where Hn is the nth harmonic number. This again matches [9] when d = 4
Finally, we turn to equation (2.29), where the crucial distinction between d0 = 4 and
2 contributions plays a critical role. Using (3.4), we nd that the O(
1
)
The d0 = 4 value again matches that of [9]. Because c 0;0q
extra order of in d0 > 4, we see that 0(
1
) is forced to vanish in this case, giving the free
(2;t)
0;0j0 is suppressed by an
To summarize, we have found that the quantities in question satisfy (for d0 > 6):
(3.13)
(3.14)
(3.15)
(3.16)
(3.17)
(3.18)
=
;free + O(
2
);
C
= 0 + O(
3
);
f0;2;lg =
f0;2;lg;free + O(
2
);
Cf0;2;lg = Cf0;2;lg;free + O(
2
):
Therefore we conclude that any CFT in d0 > 6 with a conserved stress tensor and a
single fundamental scalar must be perturbatively free up to O(
2
). Furthermore, for
Z2symmetric theories the range of validity for these results extends to d0 > 4, since the 2
operator drops out already at that dimension. We stress that these results only depend on
the value of d0, and in particular not on the sign of . These results can also be generalized
to higher orders in , as well as to include multiple scalars.
Higher orders in .
While our results are certainly sugggestive, they are at relatively
low orders in . To what extent can we push this to deeper orders? We have found two
regimes in which we can go further:
d = 4n + 2 + O(
1
); or
d = 2n + 2 + O(
1
) and the CFT is Z2symmetric;
for n > 1; n 2 Z+. As described in appendix B (see especially B.1), when either of these
is true, the simpli cations S1S3 actually hold to O(
4
) (note this also requires l(
1
) = 0,
which is guaranteed by (3.14)). This means that the right hand side of (2.25){(2.30) is
O(
4
) and we can then perform the bootstrap analysis above at orders O(
2
) and O(
3
) as
well. The analysis of the equations at these orders proceeds entirely analogously as above,
and results in determining the quantities in question to higher order in :
7
=
;free + O(
4
);
C
= 0 + O(
5
);
f0;2;lg =
f0;2;lg;free + O(
4
);
Cf0;2;lg = Cf0;2;lg;free + O(
4
):
7Note that we expect the polynomial ambiguities in the Mellin space Witten diagrams in (2.13) and (2.15)
to start contributing at O(
4
) [15], which would complicate the analysis beyond this order in
even further.
Bootstrap for multiple scalars.
Our analysis was done assuming only one fundamental
that saturates the unitary bound when the theory is free at
= 0. However, it
is fairly straightforward to see that we can relax this assumption to include an arbitrary
number N of fundamental scalars (i), with dimensions:
(i) =
d
2
2
We can perform N versions of the four identical scalar bootstrap that we have performed
above in the case of one scalar. The simpli cations S1S3 still hold with arbitrary index
structures on the heavy operators Ok;l;m. The rest of the bootstrap proceeds analogously
 for instance, the l = 2 bootstrap equations will give e.g. (1(i)) = 0 for every i. The only
possible extra nontriviality that one might worry about is the existence of a conserved
spin1 current (such as in the O(N ) model). However, since our results force the scalar
theory into a free theory, the relevant spin 1 current is automatically conserved, anyway,
without needing any further input.8
Our analysis is thus insensitive to any demand of
global symmetries the theory could have, as the resulting theory is always free.
We thus conclude that our analysis can be extended to the case of N fundamental
scalars (i) with arbitrary indices placed on the relevant operators. It is surprising that
with very little input or assumptions, we are able to see clear evidence that no perturbative
interacting CFTs with any number of fundamental scalars exist in d0 > 6, without needing
any notion at all of the Lagrangian. Perhaps by considering more of the in nite class of
spurious poles in the Mellin bootstrap one could further relax our assumptions to fully
explore the world of CFTs in d0 > 6.
Acknowledgments
We thank Marco Baggio, Anthony Charles, Ben Safdi, and Miguel Paulos for useful
discussions, and especially Henriette Elvang for many helpful suggestions and guidance during
this project. This work was supported by the U.S. Department of Energy under grant
DEFG0295ER40899 and JG was supported in part by a Van Loo Postdoctoral Fellowship.
A
De nitions
In this appendix, we collect a number of necessary and useful de nitions and expressions
we need; for more details and the derivations of these expressions, we refer the reader to [9]
where they originally appeared. We will often use the notation:
and the de nition of the Pochhammer symbol:
8Note that the spin2 conserved current, the stressenergy current, is not automatically conserved. We
need to demand its conservation in order to nd the condition (1()i) = 0.
h =
d
2
;
(a)b =
(a + b)
(a)
:
(A.1)
(A.2)
The coe cient c ;l is given by:
c ;l = C ;lN ;l;
where C ;l is the square of the
OPE coe cient C normalization coe cient:
O ;l
, and
N ;l is the
The function q(s;)l(s) is given by:
from which the coe cients q(2;;ls) and q(1;;ls) can be extracted using:
q(2;;ls) = q(s;)l(s =
);
q(1;;ls) =
These are given by:
q(2;;ls) =
q(1;;ls) =
(l
(l
+ l
h)
+ 2
The function q(t);ljl0 (s) is given by:
q(t);ljl0 (s) = l(s) 1 X
X
g
Here, we used the subexpressions:
2
h)
+ 2
2
)
+ 2)
+ 2
q
s
;
2
2 +
(A.3)
(A.4)
(A.5)
(A.6)
(A.7)
(A.8)
#
; 1 :
(A.9)
(A.10)
(A.11)
(
s)m
(
s)l0 2k m
2l(s)l2
(
1
)ql!(2s + l
1)q
Qi4=1 (li) (2s + l
1)l
(k + q + s +
+ 2
(s)q2q!(l
q)!
) (k + q + s +
(q + 2s + 2k +
+
+ 2 + 2
+ 2) (k + s +
(k + s +
"
3F2
q;
1
k
q
;l0 ( ) =
l(s) =
2 i((
(2s + l
(
1
)l4ll!
h)2
2) ( ) (
)(h +
1)l0 (h
1)l2 (2s + 2l
; 1
s
2
(
2k
q
2 +
2s
; 1
k
2) 2(
4(l + s)
1) (2s + l
2)
1)
;
and the de nition of the complicated summation:
X
g
and nally the shorthand notations:
l0!
l0
The expression (A.9) simpli es considerably for l0 = 0 (with
The coe cients q(2;;ltj)l0 and q(1;;ltj)l0 can be extracted from q(t);ljl0 (s) in a similar way as in (A.6).
For completeness, we also give the expression for the continuous Hahn polynomials Ql (t),
which we give in a convenient form:
Q
2
l
+l(t) =
(2
2l(
)
2
l
+ l
1)l
3F2
"
l; 2
+ l
;
1;
+ t
#
; 1 :
These satisfy the orthonormality condition:
1 Z +i1
2 i
i1
B
Simpli cations
dt 2
(
+ t) 2( t)Ql2
+l(t)Ql20
+l0 (t) = l
(
) l;l0 :
Here we give more information about the simpli cations S1{S3 (the simpli cation S0 relies
only on the fact that we are working with identical scalars, therefore the derivation given
in [9] follows through). These simpli cations deal with the contributions of generic
operators Ok;2m;l that contains k 's, l uncontracted derivatives, and 2m contracted derivatives.
Such an operator will have spin l and dimension:
Ok;2m;l = k
d
2
2
+ l + 2m + k(1;2)m;l + k(2;2)m;l
2 + O(
3
):
(B.1)
the coe cients k(i;)2m;l.
B.1
The coe cients c
O;l
For a given k; m; l, one possible such operator is (schematically) k 1@a1
there could possibly be other operators (by distributing the derivatives di erently among
the 's).9 However, any such operators Ok;2m;l and Ok;2m;l will only di er by the value of
0
First of all, let us consider the coe cients c O;l for the heavy operators Ok;2m;l with
dimension (B.1). These coe cients are given by (A.3); we discuss the factor C
Ok;2m;l and
N ;l separately.
e.g. the three point function
k vanishes for k > 2); this implies that:
In the free theory, the three point function of
Ok;2m;l for k > 2 vanishes (because
HJEP06(218)
Next, we note that the normalization factor N ;l of every heavy operator with k = 2
(or a higher power) as the C coe cient is the square of the
Ok;2m;l OPE coe cient (which
is reasonable to assume goes as O(
1
)). For k = 2, we can consider only (primary) operators
where the derivatives hit only one of the 's, i.e. of the schematic form
For m > 0, these operators identically vanish in the free theory so that we again have (B.2).
Only the operators with k = 2; m = 0 (i.e. the Jl operators) may have:
C
Ok;2m;l = O(
2
);
C
O2;0;l = O( 0):
goes as:10
while when k > 2, we have:
N ;l =
(
1
)
k=2;2m;l
2
g(h0)O(
2
) + O(
4
);
1
2
N ;l =
(k
2)(h0
1)
m
2
g~(h0)O( 0);
N ;l = O(
2
):
c O;l = O(
2
):
(B.2)
(B.3)
(B.4)
(B.5)
(B.6)
(B.7)
where g(h0) and g~(h0) are unimportant additional factors involving h0. Note that the
inverse gamma function has a zero when its argument is a negative integer; the only way to
achieve this here is if k = 2n1; h0 = n2 with n1; n2 integers, or if h0 = 2n + 1 with n integer
(since k is always an integer). Remembering that d = 2h, this is precisely the condition
presented in (3.16). So, when (3.16) holds, we have:
9Note that, in [9], only operators with k = 2 were possible; this is because they explicitly consider the
4 theory where
3 is a descendant operator. Note that we could possibly be \overcounting" the number of
operators when considering generic Ok;2m;l (i.e. also considering some descendants), but this is irrelevant
since we prove we can ignore their contribution to the bootstrap equations anyway.
10We already set (
1
) = 0 here for simplicity; this does not change the leading power of in N ;l.
Putting everything together, we can conclude that for any heavy operator Ok;2m;l,
Since (B.7) holds for these operators, heavier operators in the schannel begin contributing
For the simpli cation S2, we want to prove that operators Ok;2m;l0 for l0 > 0 (with k
2
We rst note that the integrand in the integral expression for q(a;t)
O;ljl0 (for a = 1; 2) has
When (3.16) holds and when 2(1;0);l = l(
1
) = 0, (B.6) and (B.4) imply that we have:
B.2
S1: schannel contributions for heavier operators
= 0 for heavier operators Ok;2m;l with k > 2 and/or m > 0,
Expanding (A.5) around
gives
at O(
2
), i.e. we have:
and m
0) satisfy:
for l = 0 and l = 2.11
the following poles:
I.
II.
III.
IV.
=
= 2
O
h
+ l0
= h + l0 + 2n3
= h
1 + n4;
h + 2n2
(n4 < l0)
In poles II and III, n2; n3; can be any nonnegative integer; these are in nite families of
poles; for the poles IV, n4 is a nonnegative integer in the range 0; 1;
; l0 1. The integral
expression for q(a;t)
but the residues can be seen to combine to give vanishing contributions up to O( 0) (for
O;ljl0 will then have contributions from the residues at each of these poles,
both q(1;t) and q(2;t) separately). The cancellations that occur for any Ok;2m;l0 are:
ResII(n2=n3+1) + ResIII(n3) = O( 0);
ResIV (n4) = O( 0);
n3 6= m
n4 = 0; 1;
; l0
1;
2:
(B.12)
(B.13)
11In checking these simpli cations for l 6= 2, it is computationally favorable to already set (
1
) = 0 (as
brie y mentioned in [9]). The only place where we need to keep (
1
) explicitly nonzero is in checking the
simpli cations for q(2;t) when l = 2.
The rest of the poles cancel in slightly di erent ways for m = 0 and m > 0:
Thus, we can conclude that q(a;t)
holds.
to O(
4
).
When (3.16) holds and l
O;ljl0 = O( 0). Together with (B.7), this proves that (B.11)
(
1
) = 0, (B.7) implies that (B.11) becomes suppressed
HJEP06(218)
B.4
S3: tchannel contributions for heavier scalars (l0 = 0)
For the simpli cation S3, we want to show that for operators Ok;2m;l0 with l0 = 0 (i.e.
scalar operators) that are heavy, i.e. k > 2 and/or m > 0, we have:
c O;l0=0q(a;t)
O;ljl0=0 = O(
2
):
The analysis proceeds in a similar way to S2. The poles I, II, III (with l0 = 0) are present
in the integrand for q(a;t), while the poles IV are now absent. The pole cancellations (B.12)
still holds. For m > 0, the cancellations (B.15) still hold. The nal poles present satisfy:
(B.17)
(B.18)
(B.19)
ResI + ResII(n2=0) = O( 0)
ResII(n2=0) = O( 0)
(m = 0);
(m > 0):
We again have (B.7), so we can conclude that (B.17) indeed holds. When (3.16) holds and
l
(
1
) = 0, (B.7) implies that (B.17) becomes suppressed to O(
4
).
B.5
Contributions from and
2
in (B.18), so that all of the poles of q(a;t)
For the operators and
2 in the tchannels, the cancellations (B.12) again holds for
q(a;t)
f ; 2g;lj0. The only other possible contributions can come from ResI and ResII(n2=0).
For d0 larger than the special dimension 4 (for 2) or 6 (for ), these are also suppressed as
f ; 2g;lj0 go as O( 0). However, there is a discontinuity
for q(2;t)
f ; 2g;0j0 in this cancellation for d0 = 4 (for 2), as discussed in section 3.1. For
there are in fact discontinuities at both d0 = 4 (for ResII(n2=0)) and d0 = 6 (for ResI ).
However these
discontinuities end up playing no signi cant role as the
operator does not
contribute for the Z2 case at d0 = 4 and then, as discussed in section 3.2, the normalization
coe cient c ;0 = O(
1
) in d0 = 6.
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings (TASI
2015), June 1{26, Boulder, U.S.A. (2015), arXiv:1602.07982 [INSPIRE].
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(2017) 019 [arXiv:1612.05032] [INSPIRE].
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