SciPost Phys. 10, 104 (2021)

EFT asymptotics: The growth of operator degeneracy
Tom Melia1? and Sridip Pal2†
1 Kavli Institute for the Physics and Mathematics of the Universe (WPI),UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan
2 School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, U.S.A.
? tom.melia@ipmu.jp, † sridip@ias.edu

Abstract
We establish formulae for the asymptotic growth (with respect to the scaling dimension) of the number of operators in effective field theory, or equivalently the numberof S-matrix elements, in arbitrary spacetime dimensions and with generic field content. This we achieve by generalising a theorem due to Meinardus and applying it toHilbert series—partition functions for the degeneracy of (subsets of) operators. Although our formulae are asymptotic, numerical experiments reveal remarkable agreement with exact results at very low orders in the EFT expansion, including for complicated phenomenological theories such as the standard model EFT. Our methods also
reveal phase transition-like behaviour in Hilbert series. We discuss prospects for tightening the bounds and providing rigorous errors to the growth of operator degeneracy,
and of extending the analytic study and utility of Hilbert series to EFT.
Copyright T. Melia and S. Pal.
This work is licensed under the Creative Commons
Attribution 4.0 International License.
Published by the SciPost Foundation.

Received 27-10-2020
Accepted 03-05-2021
Check forupdatesPublished 14-05-2021
doi:10.21468/SciPostPhys.10.5.104

Contents
1 Introduction

2

2 Rigour in two dimensions
2.1 Analysis via modular properties of the Dedekind eta function
2.2 Analysis via integer partitions and the Hardy-Ramanujan formula

5
6
8

3 A new trick
3.1 Reproducing d = 2 asymptotics
3.2 Plane partitions
3.3 Relation of plane partitions to d = 3 Hilbert series

9
9
12
13

4 Asymptotic growth of operator degeneracy from Meinardus’ theorem
4.1 Bosonic Meinardus theorems
4.2 Fermionic Meinardus theorems
4.3 Leading behaviour in arbitrary dimension

14
15
18
21

5 Hilbert Series Projections: Spin, internal symmetry, and IBP

221

SciPost Phys. 10, 104 (2021)

5.1
5.2

A worked example in d = 4
General lessons

2225

6 Saddle subtleties with and without fermions
6.1 Double cover for bosons in d = 4
6.2 A phase transition in Hilbert series
6.3 Saddles of internal groups

26
27
29
29

7 The fate of the SMEFT
7.1 Putting it all together

29
31

8 Discussion

33

A The effect of equations of motion on asymptotics

35

B Meinardus theorem in arbitrary d, j
B.1 Scalar field theory in d = 2n + 1 for n ≥ 1
B.2 Fermionic field theory in d = 2n + 1 for n ≥ 1
B.3 Asymptotics of spin j fields in d = 4

36
36
38
38

C Combinatorical geometry for SO(d) saddle points in arbitrary dimension
C.1 Take III—The SO(4) double cover way
C.2 Bosonic theory
C.3 Fermionic theory

40
40
41
42

References

431

Introduction

The principle of effective field theory (EFT) is to parameterise all possible contributions to theS-matrix that can occur in a quantum field theory of given particle content and symmetry. Its
phenomenological utility relies on a hierarchy of experimental importance of these contributions, and on the possibility of probing higher order terms via an improvement in experimental
precision. Both of these aspects are necessary, for example, for the search for new physics at
the Large Hadron Collider within the framework of the standard model EFT (SMEFT) to bea viable endeavour. The rate of growth of the number of independent S-matrix contributions
that can be constrained with increasing experimental precision is an important practical issue:
even at the lowest few orders in the SMEFT expansion, their number is cumbersome [1–3].
Experimental practicalities aside, the results of this paper can be used to determine that at
around the 120th order in the SMEFT expansion, the number of possible independent measurements that can be performed exceeds the number of atoms in the observable universe(around 1080 ). Our purpose, however, is not to establish where such cosmologically large
thresholds lie. Rather it is to develop new analytic methods by which such numbers can beobtained, and to explore what these techniques can reveal about EFT itself through its analytic
study.
The analytic study of EFTs/S-matrices dates back to the discovery of the strong interaction
and the birth of S-matrix theory (e.g. [4]). Many of the ideas and methods have found utility
in modern amplitude calculations in perturbative QCD and other theories; for reviews see
2

SciPost Phys. 10, 104 (2021)

e.g. [5, 6]. In the past few years, building upon the success of the modern CFT bootstrap
program [7], a number of new analytic results have been obtained in an S-matrix bootstrap
approach [8–19]. Here, we focus on a much simpler mathematical object that has recently
been introduced to study EFT/S-matrices—Hilbert series.
Hilbert series have appeared in the particle physics literature as partition functions to enumerate gauge and flavour invariants [20–24], and were applied to EFT operator degeneracy
counting in [3, 25–30]. A study of their utility as analytic probes was initiated in [26, 28],
establishing, for example, recursion relations between Hilbert series for theories of different
field content in d = 2 spacetime dimensions, and all-order in derivatives results for degeneracy of S-matrix elements with a low fixed number of external particles. The aim of this paper
is to extend what we know analytically about Hilbert series, and thus EFTs/S-matrices, byextracting the full asymptotic growth (with respect to the scaling dimension) of number of operators from the Hilbert series, going beyond the basic studies on asymptotic growth presented
in [28]. From now on, by “asymptotic growth of operators” or “asymptotic operator growth”, we always mean the growth of number of operators with scaling dimension ∆ as ∆ → ∞.
Appropriately graded Hilbert series correspond to partition functions of free quantum field
theories. There is thus a connection to many of the techniques to study partition functions thatappear prominently in the study of CFT. The methods and formulae established by Cardy [31]provide the growth of operator degeneracy in d = 2. We review the use of modular invarianceof the partition function for d = 2 scalar EFT to obtain the asymptotic growth of all operators,
and those captured by the Hilbert series (i.e. a projection from the space of all operators to
the spin zero subspace). In d = 2, we can also utilize the connection of the Hilbert series tointeger partitions, as established in [26], to capture asymptotic growth via the famous HardyRamanujan formula.
In higher dimensions and including more general field content—fermionic and higher spin
representations, possibly with internal symmetry degrees of freedom—modular invariance ofthe partition function is lost, and much less is known. Leading behaviour of the free fieldpartition function (for all operators) of a scalar field in arbitrary d was presented in [32]. We
develop and apply a theorem due to Meinardus, so as to obtain exact results for the asymptotic growth of partition functions of the more general EFTs mentioned above. We apply our
results to obtain the asymptotic growth of operators in the SMEFT. Somewhat surprisingly,
the asymptotic formulae we obtain are in remarkably good agreement with exact results for
operator degeneracy at low scaling dimension.
In line with our above goal of analytic exploration, we also explore subtleties that arise
in evaluating the saddle points when projecting onto a singlet sector—phase transition-likebehaviour is observed in Hilbert series, e.g. in taking the limit where the number of fermions
goes to zero.
The analytic techniques and results developed here could further have application to thestudy of large particle number and high temperature limits of gases of non-interacting particles,
e.g. in determining level densities [33], and multiplicity distributions in models of hadrons invery high-energy particle collisions [34, 35].
As a way of defining the techniques we use to mine asymptotics and introducing Hilbert
series themselves, consider a quantum field theory with a discrete spectrum {∆n } and corresponding degeneracies {an }, which means that there are an number of operators with scaling
dimension ∆n . The Plethystic Exponential (PE) is defined as the sum,X
a n q ∆n .
P E(q) =(1)n

The P E(q) can also be though of as generating function of number of operators. We will work

3

SciPost Phys. 10, 104 (2021)

with the variables q = e−β . To estimate the growth of the degeneracies, we writeZ∞X
an e−β∆n ≡P E(q) =d∆ ρ(∆)e−β∆ ,

(2)

0

n

where
ρ(∆) ≡Xan δ(∆ − ∆n ) .

(3)nThus formally we have1ρ(∆) =2πı

Z Γ +ı∞dβ P E q = e−β eβ∆ .

(4)

Γ −ı∞

Our objective is to estimate asymptotic growth of an for large n (or equivalently ρ(∆) for large∆). The idea is that the asymptotic growth of an is encoded in the behaviour of PE in the q → 1
limit1 [32]. In particular, we have [31, 32]1ρ(∆) '∆→∞ 2πıZ Γ +ı∞dβ P E (q → 1) eβ∆ .

(5)

Γ −ı∞

For EFT we typically want to count operators which are singlets under the Lorentz symmetry group and possibly some internal symmetry group. To do this, one needs to turn on
variables (fugacities), w i , for these groups and project out only the singlet terms from the PE
integrating over the group measure, schematicallyZ
Z
1
H(q) = dµLorentz
dµinternal
P E(q, w i ) ,(6)
P(q, w i )where dµ are the Haar measures of the symmetry groups, and where we also included a
projector P(q, w i )−1 (the inverse of the momentum generating function) to count only classesof operators equivalent up to a total derivative, see [28] for further details. Operators withineach class are said to be related by integration by parts (IBP); there is an equivalence class of
IBP related operators for each conformal primary operator in the spectrum. Imposing IBP in a
Hilbert series means throwing out the descendant operators from the counting [28].
The above produces a Hilbert series H(q), which is a function of q = e−β . We are interested
in obtaining asymptotic form of this generating function in the β → 0 limit, and in what follows
we will show it takes the formd−1XH(β → 0) = exp
ak β −k + b log(β) + c , ad−1 > 0 ,k=0where d is the space-time dimension. Again one can write,Z∞d∆ ρ g (∆)e−β∆ ,H(β) =(7)0

where ρ g is the density of number of operators in the singlet sector. Similar considerations to
the above yieldZ Γ +ı∞1
ρ g (∆) 'dβ H (β → 0) eβ∆ .
(8)∆→∞ 2πıΓ −ı∞P
The intuition behind this is that q → 1 limit, P E(q) diverges. Formally P E(q = 1) = n an diverges too. Thus
PN
the P E(q → 1) encodes the growth of n=1 an as N → ∞ , hence the growth of an as n → ∞.
1

4

SciPost Phys. 10, 104 (2021)

The inverse Laplace transformation can be performed using saddle point approximation. For
large enough ∆, the saddle will be ∆∗ ∝ β −d .
A careful reader will notice that while the right hand side of Eq. (5) and Eq. (8) is acontinuous function, the left hand side is a distribution. The proper way to interpret this is tosmear ρ(∆) over a small window of [∆ − δ, ∆ + δ] and then estimate the number of stateslying in that window as ∆ → ∞, keeping δ fixed. The mathematical machinery that one needs
to achieve this lies in the Tauberian theorems (See [36] and appendix C of [37] for a basic
introduction) In 2D conformal field theories (free and/or interacting) it is possible to estimate
the density of states using these techniques and even put an upper bound on the spectral gap
in interacting CFTs [38–41]. We will not attempt to make our general analysis mathematicallyrigorous in the sense just described and leave it as a future avenue to explore.
The remainder of this paper is organised as follows. In Sec. 2 we review how to obtainrigorous results on the growth of operators in d = 2 using modular properties of the PE, anddetail the connection to the Hardy-Ramanujan formula. Moving away from d = 2 the modular properties are lost. In Sec. 3 we introduce a new trick that can quickly obtain asymptotic
formulae for the growth of operators for more general partition functions, up to an unknown
order one number; we test its accuracy by comparing to the known asymptotic growth of planepartitions, and we show how the PE for a real scalar field in d = 3 can be related to the partition
function of plane partitions. In Sec. 4 we obtain our main results: by generalising a theorem ofMeinardus, we find exact asymptotic formulae that can be applied to the PE for EFTs in generald and including particles of spin. Sec. 5 presents the saddle point techniques used to make theprojections of the PE to singlets of spacetime and internal symmetries, and we provide generalformulae for the resulting Hilbert series. In Sec. 6 we detail a subtlety in taking the saddle
point approximation when fermions are present, and observe phase transition-like behaviourin the Hilbert series. We apply our results to the SMEFT in Sec. 7, and observe remarkable
numerical agreement at low mass dimension in this theory. Sec. 8 contains a discussion ofrefinements and generalisations that can be made to the analysis we present here.
We highlight some of our main results: Eq. (102) for the leading behaviour of the free fieldpartition function in arbitrary d and for arbitrary spin j; the general lessons in 5.2, in particular,
Eq. (116) for the asymptotic growth of operators in EFTs in four dimensions, including theexact order one multiplicative factors; and, Eq. (145) for the asymptotic growth of operatorsin the SMEFT.

2

Rigour in two dimensions

We begin by considering a single scalar field φ in 1 + 1 dimensions. This section follows theanalysis and methods introduced in [31]. We will first estimate the growth of all the operators
as encoded in the Plethystic exponential. Using an appropriate projector, the PE can be turned
into a Hilbert series, which encodes the number of scalars appearing in the theory; we will
move on to estimating the growth of scalars using this Hilbert series. Finally, we will impose
the IBP constraints, which amounts to throwing out the S L(2, R) descendants; again our aim
is to be find an estimate of the growth of operators.
The Plethystic exponential has the information of all the operators that one can construct,
not necessarily invariant under Lorentz group. It is given by
∞

∞ 

X1
X1

(1 − t 2n )nn
P E = expχφ (t , x ) − 1 = exp
−1
,
(9)
n
n [1 − (t x)n ][1 − (t/x)n ]n=1
n=1
where χφ (t, x) is the character for the spin-0 conformal representation, see e.g. [42], with
5

SciPost Phys. 10, 104 (2021)

fugacities t for scaling dimension (which effectively counts derivatives) and x for angularmomentum.2
We emphasise the presence of the −1 in the above equation is needed to define the Plethystic exponential for the case d = 2. Without theP−1, the PE is not well-defined i.e. it has a1divergence when setting t = 0 of the form exp
n n . From a physics point of view, thisdivergence has a meaning. Its origin lies in the fact that in d = 2, φ is dimensionless and one
can construct an arbitrarily large number of operators using powers of φ without changing
the scaling dimension. To be precise, given an operator O, one can construct arbitrary number of operators by considering f (φ)O, without changing the scaling dimension. We form anequivalence class by counting two operators only once if they differ only by their number ofφs, and thus f (φ)O and O are not counted separately. The implementation
this equivaPof∞ 1
lence relation boils down to defining the PE by throwing away the term exp
n=1 n , hence
introducing a −1 in the definition of P E. In higher dimension, we do not need the factor of−1 to define the P E; this feature is unique to d = 2 where φ is dimensionless.
We will be using the variables β and ω,t = e−β , x = e2πıω ,(10)

and for use in the below we define the variables q and q̄,q = e−β−2πıω , q̄ = e−β+2πıω .
We rewrite the PE in the following fashion
∞ 
∞ 


X1
X1
1
1
P E = exp
− 1 exp
−1 .
n
n
n
1
−
q
n
1
−
q̄
n=1
n=1

2.1

(11)

(12)

Analysis via modular properties of the Dedekind eta function

The regularized PE can be related to Dedekind eta function as
PE =q1/24 q̄1/24.
η(q)η(q̄)(13)

We remark that the regularized PE is related to the torus partition function of c = 1 free bosonic
CFT modulo the zero mode contribution (see e.g. Chapter 10 of [43]).

All the operators
To study all the operators, we can turn the angular momentum fugacity off, setting q = q̄. In
order to extract the asymptotics, we must know η(q) in the q → 1 limit. The nice feature isthat the η function has a modular propertyv
t 2π 
4π2−βη(q = e ) =(14)η q̃ = e− β ,
β
and

π2
η(q̃ → 0) = exp −
6β



.

(15)

The presence of the (1 − t 2 ) in the conformal character enacts the null state condition—this is the conditionof imposing equations of motion (EOM) on operators in an EFT (see [28] for a detailed discussion). All the fields
of various spin that we will consider contain such null states (they saturate a unitarity bound in the free theory);
we present a short discussion on the effect that EOM have on operator growth in App. A.
2

6

SciPost Phys. 10, 104 (2021)

Thus in β → 0 i.e. q → 1 limit, we have−2

lim P E = lim η(q)

q→1

q→1

 2
β
β
π
−2
=
lim η(q̃) =exp.
2π q̃→0
2π
3β

(16)

Taking the inverse Laplace transform of the above, as per Eq. (5), we obtain the asymptotic
growth of all the operators, ρ(∆), as
p


2πp ∆
π2 ∆
π3
3−3/4 e 3.
ρ(∆) ' Inverse Laplace lim P E ='0 F̃1 3;
∆→∞
q→1
18
3
4 ∆5/4

(17)

Scalar operators
The analysis for counting the scalar operators is done by projecting the P E(q, q̄) onto spin 0.
This is done by contour integral of the PE over the maximal compact subgroup of S L(2, R),
namely U(1), with Haar measureIIdxdµU(1) =.
(18)|x|=1 2πı xHere we will work with the real variable ω such that
Z 1/2
H=

dω P E(q, q̄) .

(19)

−1/2

The asymptotic analysis of H can be done following the fixed spin analysis of [41]. In thiscase, we will not set q = q̄, instead keeping them complex conjugate to each other. Now we
have
v
v
β
t β + 2πıω t β − 2πıω e− 12βe− 12P E(q, q̄) ==η(q)η(q̄)2π2πη(q̃)η(q̄˜)
(20)
p
p
π2 β
β 2 + 4π2 ω2 4π2 241 + 4π2 241
β 2 + 4π2 ω2β−2πıω
'e β+2πıω=e 3(β 2 +4π2 ω2 ) .
β→0
2π
2π
Thus we have

Z 1/2
H(β) '

β→0

p
dω

−1/2

π2 β
β 2 + 4π2 ω2
e 3(β 2 +4π2 ω2 ) .
2π

(21)

Now note that in β → 0 limit, the integral is dominated by ω = 0 saddle. Thus we can do asaddle point approximation around ω = 0 and obtainZ∞ p 5/2π2β π3β23β
4π4 ω23β .
dω exp −(22)
H(β) 'e=e
3
5/2
β→0 2π3β4π−∞Thus we can see that while the leading exponential growth stays the same, the projection
onto scalar operators changes the polynomial dependence on β. There is an extra factor ofÇ

3β 3. The growth of scalar operators is be suppressed by 3−1/4 ∆−3/4 compared to the growth
4π3of all the operators. This comesÆ c from noting that the inverse Laplace transformation is dominated by the saddle β = π 3∆ . We can also verify the scaling by doing the inverse Laplacetransformation of Eq. (22) explicitly: q q1/4 3−3/4 exp 2π ∆
2π ∆
3
3
e
1

.
ρ scalar (∆) '
=
(23)
3
5/4
∆→∞ 12∆2
3∆
4∆
7

SciPost Phys. 10, 104 (2021)

IBP constraints
To impose the IBP constraint, we include a projector that is the inverse of the momentum
generating function
P(q, q̄)−1 = (1 − q)(1 − q̄) .

(24)

The regularized PE becomes
∞
X1


∞

X 1 q̄ nqn
P EIBP = (1 − q)(1 − q̄) exp
exp.
n 1 − qn
n 1 − q̄ n
n=1
n=1

(25)

First we look at the asymptotic growth of all operators, setting q = q̄; the β → 0 limit of P EIBP
is given by
 2
β3
π
P EIBP 'exp.
(26)
β→0 2π3βThus with the IBP constraint the growth of all operators are given by
 p 
p
p !
 2
2πp ∆
2πp ∆
π3 I4 2πp3∆
π2 3−3/4 e 3π
3−3/4 e 3'=.
ρIBP (∆) '
∆→∞
18∆2
12
3∆
∆9/4
4∆5/4

(27)

Compared to Eq. (17), we have suppression by β i.e π2 /3∆. We will see that in d dimensions,the role of IBP is to introduce an extra factor of β d in the β → 0 limit of the PE of Hilbert series.
This results in a generic suppression of ρ(∆) by a factor of ∆−1 (since the saddle point of theinverse Laplace transform is given by β∗d ∝ ∆−1 ) when comparing the growth of operatorswith IBP imposed to the growth of all operators.

Scalars and IBP
The introduction of IBP amounts to modification of Eq. (22) into following:
scalar
(β) '
HIBP

π2
β
(1 − e−β )(1 − e−β )e 3β2πZ∞4π4 ω2dω exp −
3β 3
−∞




p π2 9/2
3e 3β β
'
,
β→0
4π5/2

(28)

where we have pulled out the factor accounting for the IBP constraint out of the ω integral
evaluated at the saddle point ω = 0. The asymptotic growth of scalar operators with IBP
constrained imposed is then given by
p

p


2πp ∆
2πp ∆

 
π2 e 3
1 1/4 π23−3/4 e 3
scalarρIBP (∆) '=∆→∞ 36 ∆3
3∆3
3∆
4∆5/4

2.2

!
.

(29)

Analysis via integer partitions and the Hardy-Ramanujan formula

We conclude this section by noting that the q1/24 η−1 is the generating function for the numberof partitions of an integer3 i.e we haveXX
P E(q, q̄) =P(n)q nP(m)q̄ m ,
(30)
n

m

3

TM acknowledges Brian Henning in communicating formulas and for helpful discussions on the results of thissection.

8

SciPost Phys. 10, 104 (2021)

where P(n) is the number of partitions of an integer n4 . The generating function for scalars(without IBP) is obtained by picking out terms from the above that have the same powers ofq and q̄:X
X  ∆ 2
H(q) =
P(n)2 q2n =
P
q∆ .
(31)
2
n
∆
If we impose IBP, then we have
P EIBP (q, q̄) = (1 − q)(1 − q̄)P E(q, q̄) =X(P(n) − P(n − 1)) q nnX(P(m) − P(m − 1)) q̄ m ,mand the generating function for scalars with IBP imposed is given byX
HIBP (q) =(P(n) − P(n − 1))2 q2n .

(32)

(33)

n

Thus one can relate the asymptotics with asymptotic growth of number of partition P(n) ofinteger n. The asymptotic n → ∞ limit of P(n) is given by the famous Hardy-Ramanujan
formula [44]pne2π 6P(n) '.
(34)
p
n→∞ 4 3nIn particular, we have  2∆ρ(∆) = P
,
2
  


2

∆
d P(n) 2∆
scalar(∆) = P
ρIBP
,
−P
−1
'
∆→∞
2
2
dn
n=∆/2
scalar(35)

where it is to be understood that we are taking a derivative of the function appearing on the
R.H.S of Eq. (34). It is easily verified that Eq. (23) and Eq. (29) are reproduced.

3

A new trick

Although the above calculations are clean, they are not directly applicable in higher dimensions
where we do not have the luxury of having a modular covariant function in the PE. In thissection we will step down on rigour a bit and invent a new way to obtain asymptotics up toO(1) terms. In the following section we will show how to regain these O(1) terms, but wechoose to present the following trick first, as (i) it is a fast and transparent way to obtain theform of the asymptotic growth, and (ii) we will later use this trick—again for its simplicity ofpresentation—to analyse projections of the PE to obtain Hilbert series, appealing at the end to
the rigorous treatment to fix the O(1) terms.

3.1

Reproducing d = 2 asymptotics

Let us first demonstrate this trick for a single scalar field in d = 2 so as to make direct comparison with what we obtained in the previous section. We start with the regularized PE of
Eq. (12),∞∞
X 1 qnX 1 q̄ n
P E = exp
exp
.
(36)
n 1 − qn
n 1 − q̄ n
n=1
n=1
4

This is to be distinguished from the plane partition that we discuss later in §3.2

9

SciPost Phys. 10, 104 (2021)

All operators
We set q = q̄ = e−β for the analysis of all operators. Then we have∞
X e−β n2
P E = exp.
n 1 − e−nβ
n=1

(37)

Now we do the following series expansion for small β,β β 3 n3β 5 n5
2
2
=
+
1
+
−
+
+ ··· ,
βn
6
360
15120
1 − e−nβtake the right hand side of the above and do the summation



(38)



∞ −nβ 3 3
5 5 
Xe
 2 + 1 + β n − β n + β n  ,exp n  βn636015120 n=1
| {z }(39)

singular pieces

and take the β → 0 limit. The underbracket labelled “singular pieces” refers to the fact thatthese will eventually produce the singular pieces in the log P E in the β → 0 limit.
The singular pieces indicated in (39), after summing over n, give an exponentπ2π2
3β + log(β) − 2. This is very close to the actual value 3β + log(β) − log(2π) as in Eq. (16).
The non-rigorous part of this analysis is we are not proving that the non-singular terms wethrew away in Eq. (38) changes the order one multiplicative prefactor in the β → 0 asymptotics of the PE (not to be confused with the leading term in log P E, which we know exactly).
Nonetheless, we do reproduce Eq. (16) up to an order one multiplicative correction,
 2
π
−2
P E(q = q̄ → 1) ' β e exp
.
(40)
3β
We can compare the ratio of actual asymptotics and the one we just obtained:
actual asymptotics : asymptotics via new trick =
(16)

(40)

1
: e−2 = 1 : 0.85 .
2π

(41)

The non-singular pieces in Eq. (38) produce 0 if we take the β → 0 limit first and then sum
over n; on the other hand, if we first sum over n and then take the β → 0 limit, they produce
β independent order one corrections. If we keep up to β 5 term, sum over n and then take theβ → 0 limit, we findactual asymptotics : asymptotics via new trick =(16)

modified (40)4631: e− 252 = 1 : 1.000575645 .
2π

(42)

Unfortunately, we can not improve the order one number by keeping more and more higher
order terms—at some point the non-rigourous nature of our analysis ensures that the ratio will
become far off from 1. However, we will use this trick on the principle that the singular pieces(and perhaps a small number of higher order terms) reproduce the correct asymptotics up to
an order one number that is reasonably close to unity. Note that once we have the asymptotics
of P E(β), we can again use the inverse Laplace transformation to re-obtain Eq. (17), again up
to order one multiplicative terms.

10

SciPost Phys. 10, 104 (2021)

Projecting onto scalars
To apply the above trick to the asymptotic growth of scalar operators, we reintroduce thefugacity for spin and write∞
X e−nβP E(β, ω) = exp
f (β, ω) ,
(43)
n
n=1
where we havee−2πınωe2πınω
+.
1 − e−n(β+2πıω) 1 − e−n(β−2πıω)
There is a saddle at ω=0, so we first expand f around this point, keeping terms up to order
ω2 ,





2β n
βn
e
e +1
2
f (β, ω) '− 4π2 n2 ω2 
(44)
3  + · · · .
−β
n
1−e
β
n
e −1
f (β, ω) ≡We take the β → 0 limit of the terms appearing in the square brackets of the above expression




2
2
2
1 n
2 2 2
f (β, ω) ' 1 +− 4π n ω
+
+ +
.
(45)
βn
β 3 n2 nβ 2 β 3The fluctuation around the saddle is controlled by the term proportional to ω2 . The most
singular term in β → 0 limit that is present in this term is proportional to β −3 ; we keep thisterm only and plug the truncated f (β, ω) back into the expression for P E(β, ω) in Eq. (43).
Subsequently, we perform the sum over n and again take β → 0 limit,
∞
X




2
2
1
2 8π
P E(β, ω) =e+−ω
β n2 n
β 3 n2n=1

4 π4 ω2π2− 2 + log(β) −.
=β→0 3β3β 3−nβFinally we do the projection by integrating over the Haar measure, 2  Z 1/2
4 2
4
π
ω
π
H(β) ' β e−2 exp
dω exp −
β→0
3β
3β 3
−1/2
 p

 2 Z ∞4 24πω3β 5/2 π3β2 −2πdω exp −=,' β e−2 exp
e
3β
3β 3
2π3/2
−∞

(46)

(47)

where we have used the saddle point approximation and enlarged the integration region from−∞ to +∞ to integrate over the Gaussian fluctuation around the saddle. This precisely
reproduces Eq. (22) up to order one multiplicative correction.

Imposing IBP
Imposing IBP requires us to multiply the PE in Eq. (40) or H in Eq. (47) by β 2 . They match
onto Eq. (26) and Eq. (28) respecctively upto order one multiplicative correction.

11

SciPost Phys. 10, 104 (2021)

3.2

Plane partitions

In this subsection, we proceed to test our new trick on the plane partitions of an integer andrelate the generating function for plane partitions to the d = 3 Hilbert series for scalar field
theory.
A plane partition is defined to be a two dimensional array of non-negative integers{ai, j }i, j∈Z≥0 such thatai, j ≥ ai+1, j
ai, j ≥ ai, j+1 .
(48)
The sum of a plane partition is defined asn=Xai, j .

(49)

i, j

Let us define P L(n) to be the number of plane partitions of integer n. For example, for n = 3
we have 6 partitions,






1

 



 1 , 1 1 , 1 1 1 , 2 , 2 1 , 3 .
(50)
1
1
1
The generating function for P L(n) is given by [45]
∞

∞
Y
X qn
1
P L(n)q n =P E pl (q) ≡(1 − q n )−n = exp.
n (1 − q n )2
n
n=1
n=1
X

(51)

Again, in the following we will use the variable q = e−β . The asymptotic growth of P L(n) is
given by the inverse Laplace transformation of the β → 0 limit of∞
X e−nβ1
P E pl (β) = exp.
(52)
n (1 − e−nβ )2
n=1
We apply our trick (keeping the singular pieces before summing over n i.e.
5(1 − e−nβ )−2 ' β 21n2 + β1n + 12). This produces
P E pl (β) ' e−1/4 β 1/12 exp
β→0




ζ(3)
,
β2

(53)

which upon inverse Laplace transformation gives
ζ(3)7/36
exp 3 × 2−2/3 n2/3 (ζ(3))1/3 − 1/4 .
pn→∞ 211/36 3πn25/36

P L(n) '(54)

The asymptotic growth of P L(n) is known in the mathematics literature: it was first figuredout in an old paper by Wright [45], and a typographical error in Wright’s paper was pointed
out later by Mutafchiev and Kamenov [46]. The growth is given by
ζ(3)7/36
exp 3 × 2−2/3 n2/3 (ζ(3))1/3 + ζ0 (−1) .
pn→∞ 211/36 3πn25/36

P L(n) '(55)

Comparing Eq. (55) and Eq. (54) we have0

actual asymptotics : asymptotics via new trick = eζ (−1) : e−1/4 = 1 : 0.92 .
(55)

(54)

12

(56)

SciPost Phys. 10, 104 (2021)

The actual/obtained via trick asymptotics of P L(n) can also be turned into actual/obtained
via trick asymptotics of P E pl (β → 0),

ζ(3)
0
+
ζ
(1)
β→0
β2


ζ(3)
trick
1/12
P E pl (β) ' β
exp− 1/4 .
β→0
β2

Maths
P E pl(β) ' β 1/12 exp



(57)

The superscript in P E signifies how we obtain the result. Again, if we keep up to the first ordernβ5piece before summing over n (i.e (1 − e−nβ )−2 ' β 21n2 + β1n + 12
+ 12 ), we find that we obtaina better result
ζ(3)trick, first orderP E pl(β) ' β 1/12 exp
−
1/6
β→0
β2
0

actual asymptotics : asymptotics via new trick = eζ (−1) : e−1/6 = 1 : 0.99875 .
(55)

3.3

(58)

(54) incl. first order

Relation of plane partitions to d = 3 Hilbert series

The PE for a scalar field theory in d = 3 can be massaged into a form that is directly related
to the PE for plane partitions5 . The steps to do so (that we spell out in detail at this point) are
as follows where we start by considering the explicit form of PE from [28] by setting all the
fugacities to 1:∞
X e−nβ/2 1 − e−2nβ
P E(β) = exp
n (1 − e−nβ )3
n=1
∞

∞
X e−nβ/2 X
−knβ
= exp
(2k + 1)e
n k=0
n=1
∞

∞ −nβ(k+1/2)
X
X
e
= exp
(2k + 1)
n
n=1
k=0
Q∞
(59)
p n −n
∞
∞
Y
Y


1
−
q
−(2n+1)−(2k+1)
p
n=1
2n+1
k+1/2
=
=Q
1−q
=
1− q
p 2n −(2n)
∞
n=0
k=0
n=1 1 − qQ∞
p n −n
q
1
−
n=1
= Q
2
∞
n )−n
(1
−
q
n=1
P E pl (β/2)=
2 .
P E pl (β)
In the last line, we relate it to (51) introduced previously as generating function of planepartition of integer. Now using the asymptotics for plane partitions of integer, we find that


2ζ(3)
0
P E Maths (β) ' 2−1/12 β −1/12 exp
−
ζ
(−1)
.
(60)
β→0
β2
The superscript in P E signifies that we obtained the formula using Wright’s result [45]. In
comparison, our trick produces (keeping up to the first order term)


2ζ(3)
trick
1/12 −1/12+ 1/24 .
(61)PE(β) ' 2
β
exp
β→0
β2
5

The plane partion also appears in the context of higher spin CFT in 2D [47, 48].

13

SciPost Phys. 10, 104 (2021)

That is we find,0

actual asymptotics : asymptotics via new trick = 2−1/12 e−ζ (−1) : 21/12 e1/24 = 1 : 0.99184 .
(60)

(61)

(62)
At this point, we expect the readers to be convinced that our trick produces the singularβ dependence correctly. Furthermore, we observe that can produce the order one number
almost correctly, but not quite. In the following section we proceed to capture this order one
number exactly. The hint for how to do so is in the proof of the asymptotics of plane partitions,which uses a theorem named after Meinardus.

4

Asymptotic growth of operator degeneracy from Meinardus’ theorem

Our aim is to rigorously obtain the asymptotic growth of number of operators for EFTs ingeneral spacetime dimension d, encompassing particles of general spin j (that saturate theunitarity bounds). We show this can be achieved using the theory of partitions (the term‘partitions’ here is not in the sense of a direct higher dimensional generalization of the partitionof integers, as with the plane partitions above). In particular we appeal to a theorem due to
Meinardus [49]. A mathematical exposition of this theorem can be found in the Chapter 6
of [50] (see Theorem 6.2).
Meinardus’ theorem provides us with a technique to perform the two step process laid out
in the Introduction: first, we need to know the β → 0 behavior of the generating function;
second, we do an inverse Laplace transform to deduce the asymptotic growth. The theorem
itself pertains to the size of the error terms generated by these two steps. In what follows, wewill employ the techniques used in the proof of the theorem, without keeping careful track of
the error terms. This is because, while in some scenarios, the error terms could follow directly
as per Meinardus, in most of the scenarios we consider (including any projection to particularsinglet sectors), keeping track of the error is much more involved and beyond the scope of
this paper. Thus, we refer the reader to the original reference for an explicit statement of
the theorem, and proceed below by explaining the techniques we use through the example ofcounting of operators in Bosonic scalar field theory.
We will in fact need generalize some of the techniques needed for the proof for application to the general EFTs we are interested in; we will highlight these tricks we employ as theycome up in the following. In fact, all of the necessary tricks to obtain asymptotic formulae
for arbitrary d and spin j EFTs can be showcased by presenting the cases of scalars (bosonic
Meinardus theorems) and spin half fermions (fermionic Meinardus theorems) in even dimensions. We will thus work through these cases here; explicit results in odd dimensions and withparticles of higher spin j are collected in App. B. However, the leading behaviour in arbitraryd and j is universal and compact, and we present this in Sec. 4.3 below.
Before proceeding, let us make a brief comment regarding Cardy’s approach in [32]. The
technique involved in that paper is related to Meinardus’ theorem although Meinardus’ theorem was not stated explicitly, nor leveraged to find results beyond leading order. With thetechniques borrowed from Meinardus’ theorem it is possible to go beyond leading order. Furthermore, in the following we apply the technique to study the growth of subsets of operators
e.g. sitting in the singlet sector of some global symmetry group and/or Lorentz group.

14

SciPost Phys. 10, 104 (2021)

4.1

Bosonic Meinardus theorems

The purpose of this subsection is to spell out the techniques involved in Meinardus theorem
for the purpose of counting operators in Bosonic scalar field theory.
We begin by manipulating the PE for a scalar field theory in arbitrary even d ≥ 4 dimension
into the following form∞
X e−n d−22 β 1 − e −2nβ
P E(d, β) = expn(1 − e−nβ )d
n=1
∞

∞
X e−n d−2
2 β X
−knβ
(63)
= exp
f (k, d)e
n
n=1
k=0
=

∞ 
Y

d−2

1 − q k+ 2

− f (k,d)

k=0

=

∞
Y

1 − qk− f (k−d/2+1,d),k=1

where f (k, d) is given by the symmetric spin k respresentation of SO(d) i.e.
f (k, d) = dim [k, 0, 0, · · · ] .

(64)

For example
f (k, 4) = (k + 1)2 ,f (k, 6) =1
(k + 1)(k + 2)2 (k + 3) ,
12

etc.

(65)

Note the shift in the variable k in the final equality of Eq. (63) is valid becausef (k − d/2 + 1, d) = 0 for 1 ≤ k < (d − 2)/2. (For d = 4, the shift is trivially valid.)
The rational behind recasting the PE in a product form starting from k = 1 is that the
Meinardus theorem deals with such infinite product. We proceed to sketch the proof of thetheorem, and show how the leading behaviour of the PE, and hence the asymptotic growth ofoperators, can be obtained. The proof proceeds via two steps: we first figure out the P E(d, β)
in β → 0 limit, and then translate the result into asymptotics of growth of q expansion coefficients.
We start from


∞
Y
X1X

−
f
(k−d/2+1,d)
P E(d, β) =1 − qk= exp(66)
f (k − d/2 + 1, d)e−knβ .
n
n=1
k=1
k=1
We replace e−knβ by the Mellin transform and interchage the sum and the Mellin integral using
absolute convergence. Thus we haveZ
1
log P E(d, β) =ds β −s Γ (s)ζ(s + 1)D(d, s) ,(67)
2πıwhere the function D(d, s) is defined asD(d, s) ≡∞X

k−s f (k − d/2 + 1, d) .

(68)

k=1

Note the Mellin transform integral goes along a vertical line where the above sum converges.

15

SciPost Phys. 10, 104 (2021)

Four spacetime dimensions
Now let us focus on d = 4. In d = 4 the function evaluates toD(4, s) = ζ(s − 2) .

(69)

D(4, s) converges for Re(s) > 3, and it admits an analytic continuation for Re(s) > −C with
0 < C < 1 except for a simple first order pole at s = 3 with residue 1. In general, we denotethe pole of D(d, s) as α and the residue as A. Here we have
α = 3,

A = 1.

(70)

Using the analytic structure of D(d, s) we rewrite Eq. (67) specifying the contour:
Z 1+α+ı∞1
log P E(d, β) =ds β −s Γ (s)ζ(s + 1)D(d, s) .
2πı 1+α−ı∞(71)

Now the idea is to move the contour to the left , crossing past the pole to a vertical lineRe(s) = −C. There are two obstacles to such a movement. Firstly, the integrand has a simple
first order pole at s = α = 3 coming from D(4, s) with residue AΓ (α)ζ(α + 1)β −α . Secondly,
there is a second order pole at s = 0. We pick up a contribution from both poles, resulting in
log P E(d, β) = AΓ (α)ζ(α + 1)β −α − D(4, 0) log(β) + D0 (4, 0)
Z −C+ı∞1+dsβ −s Γ (s)ζ(s + 1)D(d, s) .
2πı −C−ı∞
|
{z
}

(72)

Error term

R −C+ı∞1

After showing that 2πı −C−ı∞ dsβ −s Γ (s)ζ(s + 1)D(d, s) is indeed an error term, one arrivesat (Lemma 6.1, Eq. 6.2.1 of [50] ),log P E(4, β) ' AΓ (α)ζ(α + 1)β −α − D(4, 0) log(β) + D0 (4, 0) = 2ζ(4)β −3 + ζ0 (−2) . (73)β→0The second part of the Meinardus theorem involves translating the above result in asymptoticsof ρ(∆); this is essentially done using the saddle point method and carefully estimating theerror. Evaluating the inverse Laplace transform in the saddle approximation we find for ascalar field in d = 4,
1
4π∆3/40(74)ρ(∆) 'exp+ ζ (−2) .
pp p4∆→∞ 2 2 8 15∆5/8
3 15
In Fig. 1 we show the exact ρ(∆) obtained from a Taylor series expansion of the PE, comparedwith the asymptotic formula Eq. (74). In the upper panel of the plot, the asymptotic curve is
indistinguishable from the data. In the lower panel we show the error of the asymptotic result:even at lowest mass dimensions this error is less than ten percent, and it decreases below theper mille level at dimension ∆ ' 500.
Before moving on, we can compare the expression for the growth of P E(d, β) in β → 0
limit derived above to that obtained via the new trick of Sec. 3. The latter, (keeping up to firstorder term in β before summing over n) yields
log P E(4, β) ' = 2ζ(4)β −3 −
β→0

13
.
360

(75)

Now we have0

actual asymptotics : asymptotics via new trick = e−ζ (−2) : e−13/360 = 1 : 0.935 .
exp[(73)]

exp[(75)]

16

(76)

SciPost Phys. 10, 104 (2021)<latexit sha1_base64="UwSPYPb8ReH17e6mFlvvD/5Nc/E=">AAACYnicbVFNTxsxFPQu5SsUCHBsD1ajqkEI2A1B5YjaHnqkUgNIcRp5nbeJhfcD+y1qavlP9saJCz+kTrJSW+BJlsYzb57tcVIqaTCK7oNw6dXyyuraemPj9ebWdnNn99IUlRbQE4Uq9HXCDSiZQw8lKrguNfAsUXCV3Hye6Vd3oI0s8u84LWGQ8XEuUyk4emrYnLJUc2FjZzuUmVuNtuMWoH82sPGp33wBhZz+sKfHZ84x+FkyBSm27cLZpayU9G/XyXHXOXtSz+jOZ7gD9gu8+qF92Nl3TMvxBPeHzVZ0FM2LPgdxDVqkroth8zcbFaLKIEehuDH9OCpxYLlGKRS4BqsMlFzc8DH0Pcx5BmZg5xE5+t4zI5oW2q8c6Zz912F5Zsw0S3xnxnFinmoz8iWtX2Hqg5J5WSHkYnFQWimKBZ3lTUdSg0A19YALLf1dqZhwnxz6X2n4EOKnT34OLjtHscffuq3zT3Uca+QNeUfaJCYfyTn5Si5IjwjyECwHW8F28Bg2wp1wb9EaBrVnj/xX4ds/Zk20ng==</latexit>

1p p
2 2 8 15

5/8

exp

✓

4⇡ 3/4p+ ⇣ 0 ( 2)
3 4 15◆Figure 1: The growth of operators for a single real scalar in d = 4. Upper panel: The
thick dashed line corresponds to exact data, obtained by a direct expansion of the
PE. The thin grey curve that is indistinguishable from the data in the upper panel isthe asymptotic result. Lower panel: Relative error between data and the asymptotic!
rrr
result.
28 1
28 2⇡
15 ⇡
8

15 8

exp
5/8

3/4

4

15 3

233563

9912

1/4

4

28 12

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All even spacetime dimensions
We now proceed to analyse scalar field theory in d = 2n dimensions for n ≥ 3. Here we needa modified version of the theorem, which allows multiple but finite number of poles for thefunction D(d, s) for Re(s) > −C with 0 < C < 1. Let us illustrate the concept in d = 6 andd = 8 dimensions. Starting from the general Hilbert series, we find, for example, in d = 6, 8,
using Eq. (68)1
D(6, s) =[ζ(s − 4) − ζ(s − 2)]
12
(77)
1
[ζ(s − 6) − 5ζ(s − 4) + 4ζ(s − 2)] .
D(8, s) =360
Now these functions converge for Re(s) > 5, 7 and their analytic continuation have poles at
αi (d), given by
α1 (6) = 5 , α2 (6) = 3(78)
α1 (8) = 7 , α2 (8) = 5 α3 (8) = 3 .
The corresponding residues are given by
A1 (6) = 1/12 , A2 (6) = −1/12
A1 (8) = 1/360 , A2 (8) = −1/72 A3 (8) = 1/90 .
The modified Lemma 6.1 reads now
X
log P E(β) 'Ai Γ (αi )ζ(αi + 1)β −αi − D(d, 0) log(β) + D0 (d, 0) ,
β→0

i

17

(79)

(80)

SciPost Phys. 10, 104 (2021)

where the sum over poles come from shifting the defining contour from Re(s) = 1 + α1 toRe(s) = −C. For d = 6, 8 we have

2π6
π4
1  0
0
−
+
ζ
(−4)
−
ζ
(−2)
β→0 945β 5540β 3 12
(81)
8

π
π6
π4
1  0
d=8
0
0
log P E(β) '−++ζ (−6) − 5ζ (−4) + 4ζ (−2) .
β→0 4725β 72835β 5 4050β 3 360

log P E d=6 (β) 'The results obtained from our trick (keeping up to first order term before summing over
n) is
2π6
π4
379
−
+
log P E d=6 (β) '53
β→0 945β
540β
302400
(82)
8
6
π
π
π4
52729
d=8
log P E
(β) '
−
+
−
.
β→0 4725β 72835β 5 4050β 3 38102400
In comparison, we have for d = 6
actual asymptotics : asymptotics via new trick = e 12 [ζ (−4)−ζ (−2)] : e 302400 = 1 : 0.998 , (83)1exp[(81)]and for d = 83790exp[(82)]

actual asymptotics : asymptotics via new trick
exp[(81)]

=e

4.2

0

1
360

exp[Eq. (82)][ζ0 (−6)−5ζ0 (−4)+4ζ0 (−2)]

:e

52729
− 38102400(84)= 1 : 0.999 .

Fermionic Meinardus theorems

For the fermionic Meinardus theorem, we follow similar steps as in the scalar case to massage
the PE to the following form:
PE

(f )

(β) =

∞ 
Y

d−1

1 + q n+ 2

 g(n,d)

,

(85)

n=0

where g(n, d) is given by the dimension of the following representation of SO(d):g(n, d) = dim [n + 1/2, 1/2, · · · ] .

(86)

We focus here on even spacetime dimensions and we haveg(n, 4) = (n + 1)(n + 2) , g(n, 6) =1
(1 + n)(2 + n)(3 + n)(4 + n) etc.
6

In even dimensions, the canonical dimension of a spin 1/2 fermion is half-integer. Thus
we can not directly apply the Meinardus theorem in its original form. Another modification isneeded.
We recast the PE in the following way
P E ( f ) (β) =

∞ 
Y

1+q

k=0

where we have definedn+ d−12

 g(n,d)

=

∞ 
Y

1+q

k=1

k− 21

 g(k−d/2,d)p  g̃(k,d)
1 + ( q)k=Q,
p 2k  g̃(2k,d)
∞
1
+
(
q)
k=1
(87)



d
g̃(k, d) = g k/2 + 1/2 − , d .
2

18

Q∞

k=1

SciPost Phys. 10, 104 (2021)

Again, in the first equality the shift in the variable k is valid using the fact g(k − d/2, d) = 0
for 1 ≤ k < d/2 and k ∈ Z+ . The function g̃ is constructed in a way such thatg̃ (2k − 1, d) = g k − 1/2 − d2 , d and the last equality follows. Thus the PE can be written
as a ratio of two auxiliary PEs that are more amenable to the application of Meinardus’ theorem:
P E aux.1 (β/2),(88)
P E ( f ) (β) =P E aux.2 (β)
where the auxiliary PEs are given by
P E aux.1 (β) =
PE

aux.2

(β) =

∞
Y
k=1
∞
Y

1 + e−kβ g(k/2+1/2− d2 ,d)

,
(89)

1+e

d

−kβ g(k+1/2− 2 ,d).

k=1

We again define D functions for each of these PEs,
D1 (d, s) =∞X

k−s g(k/2 + 1/2 −

k=1
∞
X

d
, d) ,2(90)d
D2 (d, s) =k g(k + 1/2 − , d) .
2
k=1
−s

Let us denote the poles of these functions as αi,1 and α j,2 with residues Ai,1 and A j,2 , where
the second index 1, 2 refers to D1 and D2 .
The analogue of Eq. (71) will readlog P Eaux.

1(d, β) =2πı

Z 1+α+ı∞dsβ −s Γ (s)(1 − 2−s )ζ(s + 1)D(d, s) ,(91)

1+α−ı∞

where the extra factor (compared to the Bosonic case) comes due to Fermionic nature of thePE; this factor changes the behaviour of the integrand around s = 0—In particular, we don’t
obtain any log β piece. This is the final modification of Meinardus’ theorem we will need. We
findlog P E ( f ) (β → 0) = log P E aux.1 (β/2 → 0) − log P E aux.2 (β → 0)


X

−αi,1
−αi,1
=
ζ(αi,1 + 1)(β/2)
Ai,1 Γ (αi,1 ) 1 − 2
i

(92)

!
−

X


A j,2 Γ (α j,2 ) 1 − 2−α j,2 ζ(α j,2 + 1)β −α j,2+ [D1 (d, 0) − D2 (d, 0)] log(2) .

j

For example, in d = 4 we haved1
D1 (4, s) =k g k/2 + 1/2 − , d = [ζ(s − 2) − ζ(s)]
2
4


X
d
1
D2 (4, s) =k−s g k + 1/2 − , d = ζ(s − 2) − ζ(s) .
2
4
X

−s



(93)

We have poles at α1 = 3 and α2 = 1 for both of the PE. The residues are different
A1,1 = −A2,1 =

1
,
4

1
A1,2 = 1 , A2,2 = − .
4
19

(94)

SciPost Phys. 10, 104 (2021)

233563991<latexit sha1_base64="ZYo2tHvf4TxvgzYKVh3So0j1WaY=">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</latexit>

r
8

14 1
exp
15 8 5/8<latexit sha1_base64="XDoNxOxPiUQUCrjZ739Rt3VJtDM=">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</latexit>

r
4

14 2⇡
15 3

3/4

r
4

15 ⇡
14 24

1/4

!

Figure 2: The growth of operators for a fermion in d = 4. Upper panel: The thickdashed line corresponds to exact data, obtained by a direct expansion of the PE.
The thin grey curve that is indistinguishable from the data in the upper panel isthe asymptotic result. Lower panel: Relative error between data and the asymptotic
result.
Hence, we have using Eq. (92)log P E(β → 0) =71
7π4 −3 π2 −1
ζ(4)β −3 − ζ(2)β −1 =
β −
β .
4
8
360
48

(95)

It is readily checked that this matches exactly with the result obtained from our trick (in fact,independent of the number of higher order terms one keeps before summing over n).
Performing the inverse Laplace transform, one obtains the asymptotic growth for the all
operators in the fermionic PE in d = 4,
v
v

v
t
t
t
1
15
π
8 14
4 14 2π4ρ(∆) '(96)
exp∆3/4 −∆1/4 .
∆→∞
15 4 ∆5/8
15 3
14 24
In Fig. 2, the asymptotic formula above is compared with the exact values (data) from expanding the fermionic PE in d = 4. The asymptotic result in fact ‘over counts’ the numberof operators by a factor of two at a fixed mass dimension, and the plotted asymptotic curvein Fig. 2 includes a factor of a half. We can understand this using intuition from plethora ofexamples in 2D CFT which says that the asymptotic formula is a count of the number operators
lying in a window of δ = 1/2, centred at ∆, barring one of the end points if both the end points
have operators. Similar results appeared in the context of the number of partitions of integers
in [51], and are used in appendix B of [41]. We return to this point in the discussion section
below. For now we simply reflect on the very good agreement between data and asymptotic
expression shown in Fig. 2: as for the case of the scalar field, is at the level of ten percent atvery low mass dimension, and decreases below the per mille level at dimension ∆ ' 500.

20

SciPost Phys. 10, 104 (2021)

4.3

Leading behaviour in arbitrary dimension

The hilbert series in β → 0 limit has the following form:d−1XH(β → 0) = exp
ak β −k + b log(β) + c , ad−1 > 0 .
k=0

The leading singularity ad−1 β −(d−1) comes from the residue of the rightmost pole of the function D(d, s). In even dimension, the residue of the rightmost pole is determined by the leading
term in D(s). We consider massless fields of spin j that satisfy the unitarity bound. We recall
the definition of D(s):X
D(s) =
k−s g(k − k∗ ) , g(k) = dim k + j, j, j · · · , j  ,(97)| {z }
d/2−1 times

where k∗ is some constant shift depending on the canonical dimension of the field. Now theright most pole in D(s) comes from the large k piece of g(k). We use the identity
2dim [ j, j, · · · , j] ,g(k) = dim k + j, j, j, · · · , j  '| {z }| {z } k→∞ Γ (d − 1)(98)(d/2−1) times

d/2−1 times

to extract the leading pieceD(d, s) 3∞X

k−s+d−2

k=1

2 f (r, j)2dim [ j, j, · · · , j] =ζ(s − d + 2) ,| {z } Γ (d − 1)
Γ (d − 1)(99)

f (r, j) ≡ dim [ j, j, · · · , j] .
| {z }(100)(d/2−1) times

where we have defined f (r, j) as(d/2−1) times

One can immediately identify f (r, j) with the dimension of spin [ j, j, · · · , j] representationof the little group SO(d − 2) with r being the rank of the group SO(d). A few explicit forms
are given below for low numerical values of r as a function of s:f (2, j) = 1
f (3, j) = (2 j + 1)
1
f (4, j) = (2 j + 3)(2 j + 2)(2 j + 1)
61
f (5, j) =(2 j + 1)(2 j + 2)(2 j + 3)2 (2 j + 4)(2 j + 5) .
360

(101)

The leading singularity of D(d, s) appears at s = d −1. Using the leading behavior of f (r, j)
2 f (r, j)
we find that the residue at the leading singularity of D(d, s) is given by Γ (d−1) . As a result, in
β → 0 limit, we havelog H(β) ' 2 f (r, j)χ(statistics)ζ(d)β d−1 + O(β d−2 ) .
β→0

(102)

Here, χ(statistic) is given by¨
χ(statistic) =1,

for Bosonic

1−2

−d+1

21

,

for Fermionic

.

(103)

SciPost Phys. 10, 104 (2021)

The half integer j in even dimension can be handled in a similar way. Furthermore, the function
f (r, j)χ(statistic) is additive if we have multiple fields.
For odd dimension, the only free fields that saturate the unitarity bounds are scalars andspin 1/2 fermions [52]. Now f (r, j) becomes fodd (r, j) defined as
fodd (r, 0) = 1 & fodd (r, 1/2) = 2 r−1 ,where r = d−1
2 .

5Hilbert Series Projections: Spin, internal symmetry, and IBP

In d = 2, we have seen that the effect of projecting onto the spin zero sector suppresses
operator growth by a factor of β 3/2 . The introduction of IBP further suppresses this growth
by a factor of β 2 . Here we will see how things work out in arbitrary dimension. We will also
introduce internal symmetry and project onto singlets of these groups too. We find it more
useful to do a concrete example in d = 4, rather than making everything very general andabstract. The idea is to watch out for the patterns in the simple d = 4 calculation and deduce
the results for arbitrary d dimension.

5.1

A worked example in d = 4

For our example, we consider complex scalars φ and φ † (charges q b and −q b ) and spin 1/2
fermions ψ and ψ† (charges q f and −q f ) There are some subtleties that lead to a factor of
two if the theory contains only bosonic degree of freedom that we will discuss in the following
section.
We will be using the SU(2) L × SU(2)R language for the fugacities α and γ of the Lorentz
group as in e.g. [3]. The Haar measure isZ
Z
ZdµLorentz =dµSU(2) L (α)

dµSU(2)R (γ) ,(104)

where e.g.
ZdµSU(2) L =I

dα(1 − α2 ) .
α|α|=1

The PE for this example EFT, is given by∞
X 1 q n (1 − q2n )
P E = exp
χU(1) (nq b ω) + χU(1) (−nq b ω)n P(q n , αn , γn )
n=1



3
∞
X
2 n α n + α−n − q n (γ n + γ−n )q1
× exp  (−1)n+1
χU(1) (nq f ω)
n , αn , γn )
n
P(q
n=1



3
∞
n
n
−n
n
n
−n
X
1 q 2 γ + γ − q (α + α )× exp  (−1)n+1
χU(1) (−nq f ω) ,n
P(q n , αn , γn )
n=1

(105)

(106)

where the character of the U(1) internal symmetry is, in terms of the angular fugacity w,
χU(1) (ω) = e2πıω ,22

SciPost Phys. 10, 104 (2021)

and where the momentum generating function isP(q, α, γ) = (1 − q αγ) 1 − q αγ−1 1 − q α−1 γ 1 − q γ−1 α−1 .

(107)

In the following we will work with fugacities β and ω1,2 defined asq = e−β , α = e2πıω1 , γ = e2πıω2 .
Let us analyse the PE using our by now familiar trick from Sec. 3. First we expand in theβ → 0 limit aslog(P E) =222 X e−nβ  243
7 16π (ω1 + ω2 )+ 2 2+− −χU(1) (nq b ω) + χU(1) (−nq b ω)
3
3
5
3
n
β n
β n
βn 6
β n
n

3
2
2
2 
X (−1)n+1 e− 2 nβ32
3 16π (ω1 + ω2 )
2
+
+
+ −
+
n
β 3 n3 β 2 n2 β n 4
β 5 n3n
× χU(1) (nq f ω) + χU(1) (−nq f ω) ,(108)
where we kept only the singular terms in β.
The saddle
will be determined by the leading term proportional to β −3 for the ω integral−3
i.e on 2β
g b (ω) + g f (ω) where g b/ f are given by
4 2 2
4π4 q3b |ω|3 2π4 q4b ω4π4 2π q b ω
g b (ω) =
=
−
+
−
n4
45
3
3
3
n

4
4
4
4
2
2
X
2π q f ω
2 cos 2πnq f ω
7π4 π q f ω−+.
g f (ω) =(−1)n+1
=
n4
360
3
3
n

X 2 cos(2πnq ω)
b

(109)

Note this term is independent of the spin, depending only on statistics. We keep up to thequadratic piece and do the integral over ω. We remark that the quadratic piece can be obtainedby expanding χU(1) (nq b/ f ω) + χU(1) (−nq b/ f ω) first and then summing over n, and often thisis a more practical approach to extract the asymptotics rather doing the sum explicitly as in
Eq. (109). However, the way we proceeded above reveals the full periodic nature of the PE in
ω. We will assume that U(1) charge is quantized in units of some base charge, with some field
having base charge, and hence we can simple rescale all charges such that the base charge isunity6 —in this case, Eq. (109) shows that the saddle is indeed ω = 0.
If one worked with base charge of integer k > 1, then there would be k saddles to consider. The result should
P bemultiplied by k because of this, but would be suppressed by a factor of k coming from a 1/Q (where Q2 = i qi2 )after the projection to U(1) singlets, see Eq. (115) below. If on the other hand the base charge was 1/k, one
should use the k-th cover of U(1), receiving a suppression by a factor of 1/k in the measure; this would similarly
be cancelled by the 1/Q after projection.
623

SciPost Phys. 10, 104 (2021)

Expanding, and performing the sum on n in the terms with the angular fugacities,log(P E) =
X e−nβ  2
4
3
7
+
+
−
(dim b )n
β 3 n3 β 2 n2 β n 6n 
8π2 (ω21 + ω22 ) π4
(dim b )−β545X (−1)n+1 e− 23 nβ  2
3
2
3
+
+
+
+
(dim f )n
β 3 n3 β 2 n2 β n 4n
8π2 (ω21 + ω22 ) 7π4(dim f )
−
β5
360


4 2 2
π4 q2f ω2πqω
1b+ 2β −3 −(dim b )− (dim f )
,
3
2
3

(110)

where we have instated factors of dim b and dim f so as to keep track of the contribution ofbosonic and fermionic degrees of freedom in this example (where dim b = dim f = 2).
The fugacity independent pieces will be
log P E(β) 3 Aβ −3 + Bβ −1 + Cζ0 (−2) ,(111)
where A, B have the universal form already seen above for scalars and spin-half fermioins,namely
 4

π
7π4
A=
dim b +
dim f
45
360
 2

(112)
π
dim f .
B=−
48
The order one piece is not correctly obtained through the above, but it is the result derived
from using the Meinardus theorem,
C = dim b .

(113)

Projecting onto Lorentz scalars is done by integrating the above about the saddle with theHaar measure for SU(2) × SU(2),Z∞−∞dω1 µSU(2) (ω1 )Z∞dω2 µSU(2) (ω2 ) exp −−∞8π2 (ω21 + ω22 )
β5



π4
7π4
dim b +
dim f
45
360




−3
91125β 15
7
=
dim
.
dim
+
b
f
32π13
8

(114)
If we had multiple scalars and fermions, with charges qi , the ω dependent piece always contains a β 32n4 n2 ω2 term multiplied by a sum over charges,
P E(β) 3 exp −

2π ω
3β 3
4

2

dim
b /2
X
i

!
X
1
q2bi +
q2f i  ,2 i
dim f /2in the above example we have q b1 = −q b2 = q b and q f 1 = −q f 2 = q f and this matches with
Eq. (110).
24

SciPost Phys. 10, 104 (2021)

Projecting onto U(1) singlets is done in following way:


 v
Z∞
t 3 β 3/2
2π4 ω2 X 2
1X 2
dω exp −
q
+
q
=
bi
3β 3
2 i fi
2π3 Q
−∞
i


X
1X 2
q
.
Q2 ≡
q2bi +
2 i fi
i

5.2

(115)

General lessons

General Lessons in d = 4
We start by presenting the master formula for the asymptotic behaviour of an EFT in d = 4,
including particles up to spin j = 1 and including IBP relations. It follows from results thatappeared above (and in App. B.3 for the spin 1 terms), and application of the saddle point
approximations described in the previous section to project out singlets of general internal
symmetry groups, G. The formula reads,−3   
 91125β 15 
3
7
dim g
2
dimB + dim fH(β) ' Kβ
β4
β→0
32π13
8
(116)


 
β
−3
−1
0
+ higher spin>1 ,× exp Aβ + Bβ + Cζ (−2) + D log
2π
where


π4
7π4
A=
dimB +
dim f ,
45
360

 2
π2
π
dim1/2 +dim1 ,
B=−
48
6


(117)

C = dimB ,1
D = − dim1 .
2
Here dimB counts bosonic degrees of freedom (dof), dim f counts fermionic dof, dim1/2 counts
spin-1/2 dof, dim1 counts spin-1 dof, and dim g counts the dimension of the internal symmetrygroup G. The exact value of K can be determined case by case; it depends on the symmetrygroup at hand, and also on the representations of the fields transforming under the symmetry.
If dim f = 0, the result should be multiplied by a factor of two (see following section)
Let us discuss each of the terms in Eq. (116), working backwards, from right to left.
• The leading piece (exponential term) comes from the counting of all degrees of freedom
in the EFT, turning off fugacities characterizing spin and internal symmetry.
• Imposing IBP provides a suppression of β 4 .
• Projecting onto Lorentz scalars provides a suppression of β 15 , and an order one multiplicative piece that depends on the number of bosonic and fermionic dof.
• Projecting onto singlets of an internal symmetry group G provides a suppression in β thatis dependent on the dimension of G, and an order one multiplicative piece K, describedabove.

25

SciPost Phys. 10, 104 (2021)

General Lessons in arbitrary d
The below results also follow from a direct application of the above techniques• The leading piece (exponential term) again comes from the counting of all degrees of
freedom in the EFT, turning off fugacities characterizing spin and internal symmetry (seealso Sec. 4.3).
• Imposing IBP provides a suppression of β d .
• Projecting onto singlet under any Lie group induces suppresion by(β) g(d,G)×dim g ,where g is the dimension of the Lie algebra corresponding to Lie group G. The functiong(d, G) is given by¨g(d, G) = d+1
where G = Lorentz ,
2 ,
(118)
d−1
g(d, G) = 2 ,where G = Global symmetry .
That is, we find2

– Projecting onto Lorentz scalars implies a suppression by β d(d −1)/4 (irrespective of
the spin of the fields).
– Projecting on to gauge singlet implies suppression by (β (d−1)/2 )` , where the quantity ` can be extracted from the volume measure of the gauge group in the fugacity
going to 1 limit. For a continuous Lie group G, this identifies ` = dim g as thedimension of the Lie algebra.
The suppression by IBP projection and onto singlets of the Lorentz and internal symmetrygroups is less important than sub-leading (larger than logarithmic in β) corrections in the
exponent. The (unquantified) polynomial suppression of IBP projection and of projection onto
singlets of the Lorentz was pointed out in [28].

6Saddle subtleties with and without fermions

For a purely Bosonic theory, one needs to multiply the result presented Eq. (116) by a factorof 2. To be precise, if we view H(β → 0) as a function of dim f , we haveH(β → 0, dim f = 0) = 2 lim H(β → 0, dim f.
(119)
dim f 7→0

The above signals phase transition-like behaviour, and arises because of the presence of
extra saddles that appear in the limit. The symmetry reason behind such phenomenon isthat the presence of fermions break the symmetry (ω1 , ω2 ) 7→ (ω1 + 21 , ω2 + 12 ), i.e. when
dim f = 0 , there is a symmetry enhancement. In terms of α = e2πıω1 and γ = e2πıω2 , thesymmetry is implemented by (α, γ) 7→ (−α, −γ). While the bosonic PE is symmetric under thistransformation, the fermionic (or the ones with both bosons and fermions) ones are not.
In the α, γ variables there are two saddles to keep track of: when α = γ = ±1. The
apparent discontinuity in Eq. (119) comes about because for dim f = 0, we have degeneratesaddles but when dim f 6= 0, the symmetry is broken, and so is the degeneracy. Consequently,
one of the saddles gets suppressed. This phenomenon happens universally in all dimensions,
see App. C for the details. Keeping both the saddles restore the continuity of H(β) as a functionof number of fermions. We explicitly show this in Sec. 6.2 for d = 4.
26

SciPost Phys. 10, 104 (2021)

Figure 3: Calculation using SU(2) × SU(2). The figure depicts saddles on(ω1 , ω2 ) plane. The center of the square is at (0, 0). The corners are at(1/2, 1/2), (1/2, −1/2), (−1/2, 1/2), (−1/2, 1/2). For the bosonic case, all the saddles contribute. When we add fermions in the mix, the green colored saddles at(0, ±1/2), (±1/2, 0) don’t contribute anymore, leading to an overall factor of 1/2.
We have drawn circular regions arounds the saddle to denote how much the fluctuation around the saddle contributes. For example, each of the corner ones contributes
one quarter of the center one.

6.1

Double cover for bosons in d = 4

We begin by considering a single scalar field theory. The PE can either be written in thelanguage of SU(2) × SU(2) or in the language of SO(4). The purpose of this section is toclarify that the both will provide us with the same answer. If we include fermions in thetheory, then we need to work with the double cover of SO(4). This is is taken up in App. C,
and generalized so as to studyng this phenomena in arbitrary d.

Take I—The SU(2) × SU(2) way
H(β) is given byH(β) =

I

I
dα


∞

X e−nβ11 − e−2nβ22dγ(1 − α )(1 − γ ) exp,
αγn P(e−nβ , αn , γn )
n=1


(120)

with P defined in Eq. (107)
Now note that in the β → 0 limit, the singularity appears when αγ = α/γ = 1. This admits
two solutions
α = γ = ±1 .
(121)
The angular fugacities ωi are defined as
α = e2πıω1 , γ = e2πıω2 .

(122)

As α, γ traverse the circle once, we have a square region swept out in the (ω1 , ω2 ) plane withcenter at (0, 0) and vertices at (±1/2, ±1/2), (±1/2, ∓1/2). From Eq. (121) it follows that inthe (ω1 , ω2 ) plane the saddles exist at the points (see fig. 3)(ω1 , ω2 ) = {(0, 0), (1/2, 1/2), (1/2, −1/2), (−1/2, 1/2), (−1/2, −1/2)} .

(123)

We have already considered the (0, 0) saddle in the previous section. Now there are four more
saddle points, which are the corners of square on (ω1 , ω2 ) plane over which we are doing the
ωi integrals. The fluctuation around this each corner provides 1/4 of the contribution comingfrom the fluctuation around saddle (0, 0). This is self evident because a full circle around (0, 0)27

SciPost Phys. 10, 104 (2021)

contributes to fluctuation integral whereas, only a quarter chunk of the circle contributes forthe corner points. The leading value of the integral from all of the saddles is the same. So wehave following expression for H(β)X
H(β) =P E(β, ωi = saddle value) × fluctuation contribution
saddles


π4 −3
0
= exp
β + ζ (−2) (1 + 4 × 1/4)
45
Z ∞
Z∞



8π6
2
2
dω1 µ(ω1 )(ω + ω2 )dω2 µ(ω2 ) exp −45β 5 1−∞−∞
 4
91125β 15
π −3
0
exp
β + ζ (−2) .
=
16π13
45


(124)

When we add fermions in the mix, the saddle α = γ = −1 produces a suppressed contribution
at leading order (these are depicted by the dashed green lines in fig. 3). This is evident fromthe character of fermion being (α + α−1 ) − q(γ + γ−1 ). Thus we recover the result presented
in Eq. (116).

Take II—The SO(4) Way
H(β) is given byH(β) =

II
d x1

d x 2 µSO(4) (x i ) exp∞
X e−nβ 1 − e−2nβn=1n P(nβ, x in )

,

(125)

where
µSO(4) (x i ) =1(1 − x 1 x 2 )(1 − x 1 /x 2 ) ,
x1 x2

(126)

and
P(β, x i ) = 1 − e−β x 1

1 − e−β x 1−1

1 − e−β x 2
1 − e−β x 2−1 .

(127)

Now note that in the β → 0 limit, the singularity appears when x 1 = x 2 = 1. In terms of
the angular variable ω̃i (where x i = e2πıω̃i ) we have the following saddle (the only one)(ω̃1 , ω̃2 ) = (0, 0)(128)

H(β) = P E(β, ω̃i = saddle value) × fluctuation contribution
 4 Z ∞
Z∞


4π6
π −3
0
2
2
= exp
d ω̃2 µSO(4) (ωi ) exp −β + ζ (−2)
d ω̃1(ω + ω2 )45
45β 5 1−∞−∞
 4
91125β 15
π −3
0
=
exp
β + ζ (−2) .
16π13
45
(129)
Thus Eq. (124) matches with Eq. (129). Note that we can not add fermions in the SO(4)language—first we need to go to the double cover, see App. C for the details.

28

SciPost Phys. 10, 104 (2021)

6.2

A phase transition in Hilbert series

In this section, we take a closer look at the β → 0 behaviour of Hilbert series as a function ofdim f . In particular, we want to inspect how the following comes about




.
(130)
H(β → 0, dim f = 0) = 2 lim H(β → 0, dim f
dim f 7→0

Let us consider the Hilbert series for b scalars and f spin-1/2 fermions,
I
I


1
2
2
H(β, f ) = dα dγ(1 − α )(1 − γ )
αγ


∞
X 1 be−nβ (1 − e−2nβ ) + (−1)n+1 f e−3nβ/2 (αn + α−n ) − e−nβ (γn + γ−n )
.
× exp
n
P(e−nβ , αn , γn )
n=1
(131)
In the β → 0 limit, the singularity appears when α = γ = ±1. When f 6= 0, the leading
singularity is α = γ = 1, and the subleading one is α = γ = −1. Let us keep them both. Now
we have (we denote the Hilbert series as H 0 to distinguish it from the one where we take only
the leading saddle)
 4
91125β 15
π (b + 7/8 f ) f π200
H (β → 0, f ) =exp−+ bζ (−2)
32π13 (b + 7/8 f )3
45β 3
48β
 4

(132)
91125β 15
π (b − f ) π2 f
0
+
exp
+
+ bζ (−2) .
32π13 (b − f )3
45β 3
24β
Now one can see that for f = 0 the two saddles coincide and instead of Eq. (130) we have
H 0 (β → 0, f → 0) = H 0 (β → 0, f = 0) .

(133)

So the apparent discontinuity/phase transition in Eq. (130) gets resolved by adding contribution from the second saddle; the second saddle becoming equally important when f = 0 We
also remark that for b ≤ f , the other saddle is unstable one, and one should not include it.

6.3

Saddles of internal groups

When taking projecting to singlets of some internal symmetry group, similar saddle point
subtleties can arise. We already mentioned one in the case of a U(1) symmetry in footnote 6.
Another example: consider the Hilbert series for a single field transforming in the adjoint
representation of SU(N ). In this case there are N saddles that contribute equally. However,
the inclusion of another field transforming in e.g. the fundamental representation suppresses
all but one saddle (at the centre of the hypercube swept out by the angular fugacities), and asimilar discontinuity to the one above occurs. We leave a detailed study of such discontinuities
in Hilbert series to future work.

7

The fate of the SMEFT

We finally turn to applying our results to a a complicated phenomenological EFT, namely the
SMEFT. This has field content shown in Table 1 (conjugates of all fields must also be included).
We will consider this theory with Ng copies of fermionic generations (the SM has Ng = 3).
Hilbert series methodology was applied to this theory in [3] to systematically enumerate operators at mass dimension eight and above (results up to mass dimension 15 were presented).
Let us assemble the components that form the leading behaviour of the PE for the SMEFT
in the β → 0 limit.
29

SciPost Phys. 10, 104 (2021)

Table 1: The field content of the SMEFT wiith representations under the Lorentz
group SU(2) L × SU(2)R , and the gauge group SU(3)c × SU(2)W × U(1)Y of the SM.
Field
Q
L
uc
dc
ec
GL
WL
BL
H

SU(2) L
2
2
2
2
2
3
3
3
1

SU(2)R
1
1
1
1
1
1
1
1
1

SU(3)c
3
1
3
3
1
8
1
1
1

SU(2)W
2
2
1
1
1
1
3
1
2

U(1)Y
1/6
−1/2
−2/3
1/3
1
0
0
0
1/2

Leading exponential
The leading exponential piece can be read straight from Eq. (116),
log P E(β) 3 Aβ −3 + Bβ −1 + Cζ0 (−2) + D log(β/2π) ,

(134)

with
π4
7π4
dimB +
dim f
45
360


π2
π2
1
A=
, B=−
dim1/2 +dim1 , C = dimB , D = − dim1 ,
48
6
2
(135)
where from Table 1 (remembering to count the dof in the gauge group representations, andto include the conjugate fields) we have,dimB = 28, dim f = 30Ng , dim1 = 24, dim1/2 = 30Ng .

(136)

Projection onto Lorentz scalars and IBP
These pieces can also be read straight from Eq. (116), and we find a suppression by a factorof
−391125β 157dimB + dim f
,
(137)
32π13
8
for the projection to scalars, and factor of β 4 for from the IBP projector.

Projection onto U(1) singlets
We have the suppression found in Eq. (115), by a factor ofvXt 3 β 3/2
1X 2
2
2
, where Q ≡q bi +
qf i .
2π3 Q
2
i
i

(138)

For the SMEFT, scaling the charges that appear in Table 1 by a factor of 6 (see footnote 6), wehave
Q2 = 36 + 120Ng .
(139)

30

SciPost Phys. 10, 104 (2021)

Projection onto SU(2) singlets
There is a the single saddle point at ω = 0, and the integral over fluctuation is found to be
Z∞



1
4π4 ω2
adj
fund
fund
dimB + dim f + 4dim bdµSU(2) (ω) exp −
3β 3
2
−∞
(140)
v


−3/2
t 3
1
adj
=3
dimfund+ dimfund+ 4dimB
β 9/2 .
Bf
4π72
Here dimfund
counts the number of bosonic dof in the fundamental representation (rep) ofB
adj

SU(2); dimfund
counts the number of fermionic dof in the fundamental rep of SU(2); dimBfcounts the number of bosonic dof in the adjoint rep of SU(2). For the SMEFT we haveadj

dimfund= 2 , dimfund= 8Ng , dimB = 2 .
B
f

(141)

Projection onto SU(3) singlets
There is a the single saddle point at k1 = k2 = 0, and the integral over fluctuation is found to
be
Z∞


4π4 (k12 − k1 k2 + k22 ) 
adj
fund
fund
dµSU(3) (k1 , k2 ) exp −dim f + dim ¯ + 12dim b
f
6β 3
−∞
(142)
p 12 
−4
432 3βadj=+ 12dimB,
dimfund+ dimfundff¯
π9where dimfund
now counts the number of dof in the fundamental representation of SU(3), etc..
B
For the SMEFT, we haveadj

dimfund= dimfund= 4Ng , dimB = 2 .
¯
f
f

7.1

(143)

Putting it all together

Assembling the above components and the IBP suppression, we can construct the Hilbert series
for the SMEFT in the β → 0 limit:
 4

π2
7π
0
44286750β 25 exp 180β
3 (15N g + 16) − 8β (5N g + 32) + 28ζ (−2)H(β) ∼
.
(144)
Æ
β→0
343π15 (Ng + 3)4 (2Ng + 5)3/2 10Ng + 3(15Ng + 16)3
Performing the inverse Laplace transform, one obtains the asymptotic growth of operatorsin the SMEFT!
p v
π 5Ng + 32
2π 2 t112 3/4
4
ρ(∆) ∼ N exp∆ − p q7Ng +∆1/4 + 28ζ0 (−2) , (145)
4
112
∆→∞
3
15
4 2 7Ng + 15
where N is given by3/8 5/8 10
27783 75
3 π 15Ng + 16 27/8

 Æ
N=
.
p
4
1024000 2∆55/8 Ng + 3 4 2Ng + 5 3/2 10Ng + 3(146)

We note that in performing the inverse Laplace transform, there is a simple way to probe
the effect of higher order terms in the asymptotic series we present above. The transform isZ
dβ H(β)eβ∆ ,
31

(147)

SciPost Phys. 10, 104 (2021)

Figure 4: The effect of a linear shift in ∆ to capture sub-leading corrections to theasymptotic formula for the growth of operators in the SMEFT, with one generation offermions, Ng = 1. The thick dashed line is the exact results for the number of operators in the SMEFT as a function of ∆. The thin dashed curve indicates the asymptotic
result given in Eq. (145). The solid grey curve indicates Eq. (145) evaluated with thee given by Eq. (149).
shifted ∆where H(β) contains the exponential termH(β) 3 eAβ−3+Bβ −1 +Cβ 0 +D log(β/2π)+...

.

(148)

Here the + . . . denotes terms which vanish as β → 0. The term linear in β, i.e. +Kβ in theabove exponent, generates an effective shift in ∆ as can be seen from Eq. (147). That is,e = ρ(∆ + K) that captures its effect.
including the linear term one can define an effective ρ(∆)
Note that this implements sub-leading corrections to the asymptotic result, which will change
the result at low, finite ∆, but not as ∆ → ∞.
When using the (non-rigorous) new trick of Sec. 3 we often kept such higher order terms in
β in the exponential at intermediate stages. By applying the trick to the SMEFT, and retaining
terms up to linear order in β, we find an effective shift in ∆e =∆+∆67 17Ng
+
.
60
64

(149)

We have observed experimentally that when the number of degrees of freedom in the EFT islarge, such as it is for the SMEFT, this linear shift obtained by the trick brings the asymptotic
result in better agreement with low scaling dimension data. (When the dof are small, it has
little effect.) We do not speculate on why this is so here; our errors are not under control,
precluding us from quantitative analysis. Nevertheless, it is interesting that this shift implements subleading corrections to Eq. (145) that bring it into remarkable agreement with low
mass dimension data. This is illustrated in Fig. 4. The thick dashed line corresponds to theexact number of operators as a function of mass dimension ∆ in the SMEFT, for the case of
Ng = 1. In obtaining the exact results for the SMEFT beyond mass dimension 15 that are usedin Fig. 4, as well as Fig. 5 below, we used the computer code accompanying the paper [53].
The thin dashed red curve indicates the result of Eq. (145), and the solid grey curve indicatese given by Eq. (149).
Eq. (145) evaluated with the shifted ∆Fig. 5 compares our results to exact data in the SMEFT for Ng = 1 and Ng = 3 (the fulle in Eq. (149). Given that
SMEFT), in both cases using Eq. (145) evaluated with the shifted ∆our formulae are only asymptotic, the agreement to exact data at low mass dimensions isstriking. The agreement for Ng = 3 appears even better, in line with our observations that the32

SciPost Phys. 10, 104 (2021)

0

r
p 10
427 3/8
3⇡
2 p 4 427 e 3/4
15
exp @
2
⇡
p
4
55/8
e
3
15
991232000 2233563449<latexit sha1_base64="ZYo2tHvf4TxvgzYKVh3So0j1WaY=">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</latexit>q1
15
47 4 427⇡
e 1/4 + 28⇣ 0 ( 2)A
p
4 2

0
r
p 10
217 3/8p
7⇡
15@ 2 2 4 217 ⇡ e 3/4
exp
p
p
3
15
131072000 4 2 13 e 55/8

152023473<latexit sha1_base64="D/NJq8eKQXK/rZDWeJmB2X09H0s=">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</latexit>q1
15
37 4 217⇡
e 1/4 + 28⇣ 0 ( 2)Ap
4 2

Figure 5: The growth of operators in the SMEFT, comparing exact data with theasymptotic formulae. The thick dashed curve is data for one generation of fermions,
Ng = 1; the thick dotted curve is data for three generations of fermions Ng = 3. The
solid grey curves indicate the asymptotic result Eq. (145) evaluated with the shifted
e given by Eq. (149).
∆
shift works well for a large number of dof. However, such statements are only qualitative; weleave a quantitative assessment of error terms to future work.

8

Discussion

We have developed new techniques for studying EFTs/S-matrices, probing the asymptotic
growth of operators through a study of the analytic behaviour of Hilbert series. Our methods also revealed phase transition-like discontinuities in evaluating projections onto singletsof Lorentz and internal symmetry groups. These results are a further step in the broader direction of using Hilbert series to study EFTs analytically. We end with a discussion of a few
directions we can see to pursue so as to generalise and improve upon the above techniques.
Firstly, it would be interesting to study the asymptotics of finer-grained Hilbert series—
the behaviour of the growth of operators projected onto different spin representations, or
as a function of a fixed particular number of fields in an operator (see Sec 5 of [28]). The
latter is particularly interesting from the point of view of studying S-matrix elements, as iscorresponds to fixing the number external legs. If it is in any way possible to study of theconvergence properties of the S-matrix of a dynamic theory by methods that mirror the known
results about convergence of the OPE in CFT [54–57], our results on the growth of operator
degeneracy must enter at some level.
The Hilbert series appears in disguise in the calculation of superconformal index [58] andpartition functions in large N quantum field theories [59, 60]. One often looks at large N
and/or high temperature behavior of the free energy of these theories, that can be obtained
33

SciPost Phys. 10, 104 (2021)

from the partition function. From this vantage point, the Hilbert series is in fact closely related
to the physical partition function on S 1 × S d−1 via radial quantization, where S 1 is the thermal
circle and the β → 0 behavior of the Hilbert series is nothing but the high temperature behaviorof the partition function. Thus our result can be thought of as a study of growth of number ofstates in free CFT on a cylinder. In particular, we have performed a refined study of growthof all the states, the states sitting in various singlet sectors (singlet under the Lorentz group
and/or internal symmetry group) of the theory.
Results on the leading term in the high temperature partition function for free CFTs (ourSec. 4.3) have appeared many times in the literature as a function of the number of bosonicor grassmannian variables of the theory under consideration. This began with the work of
Cardy [32], and more recently with explorations in a holographic context in [61], and wasagain revisited in [62–64]. These papers find the same leading scaling of log H(β) in theβ → 0 limit as us. Our contribution is to pinpoint the the order one coefficients accompanying the leading term in log H(β) as a function of spin and statistics of the field content inany dimension. Furthermore, the refined analysis where we project onto the singlet sectorsof various symmetry group required us to evaluate the subleading corrections with precise
coefficients as well. To add to list of what have been explored in the literature, we note theapplication of Meinardus theorem in context of counting of BPS operators in [65], the apperance of Cardy formula in supersymmetric theories in [66] and a recent revival of Cardy likeformulas in [67–70]. It would be nice to investigate whether some of the results appearing
here would be useful in that context.
Our results can be quite generally applied to partition functions; for instance it would beinteresting to apply them to Hilbert series of non-relativistic EFTs [29], and applications totheories of e.g. superfluidity [71] and more general condensed matter systems [72], whichmight exhibit a different high temperature behaviour. It would also be interesting to work out
Meinardus theorems when other fugacities are present in the partition.
How can the accuracy and robustness of our formulas be improved? This question is particularly motivated given that (i) we already see remarkably encouraging agreement even at low
mass dimension e.g. in the SMEFT such that one might wonder how good they can get, and(ii) results in the mathematics literature improve upon the Meinardus formula for the planepartitions P L(n) considered in Sec. 3 to obtain integer results within error up to partitions on
the order of n ® 6400, (a number of around 300 digits) [73]. (Such precision would evenbe useful to turn the asymptotic formulas into a calculational tool in lieu of a Taylor series
expansion of a Hilbert series to obtain the exact projection onto invariants.)
To elaborate further, we note that the actual density of states/operators is a distributionthat has support on a set of discrete values. It is more appropriate to smear this distribution
over a small window and estimate the number of operators with dimension lying in that window by establishing a rigorous upper and lower bound. In 2D CFT, this has been achieved ina series of papers [38–41], with a result of the following form,(2δ − 1)ρ0 (∆) ≤Z ∆+δ

d∆0 ρ(∆0 ) ≤ (2δ + 1)ρ0 (∆) ,

(150)

∆−δ

where we have smeared the density of operators ρ(∆0 ) over a window centred at ∆ with widthδ and taken the limit ∆ → ∞ with δ fixed. Here ρ0 (∆) is the smooth approximation of ρ(∆)
which can be naively obtained by doing inverse Laplace transformation; the above equation
immediately tells us that it is accurate up to order one multiplicative error.
One might wonder whether one needs to smear over a window for the asymptotic analysis
that we have done, thus losinig control over the order one number. It is helpful to get some
intuition from scenarios in 2D where such smearing is not actually needed. Famous examplesof this include the growth of bosonic excitaions in 2D free boson CFT (which is related to the34

SciPost Phys. 10, 104 (2021)

integer partitions [44] and the 2D Hilbert series in Sec. 2), and the growth of operators in
extremal CFTs (holomorphic CFTs with c = 24k where k is a positive integer; for k = 1 thepartition function is given by j function [74]). In such cases, one can obtain a convergent
Rademacher sum [75, 76] for the exact degeneracy of operators. The technical reason behindsuch an improvement over a generic CFT is the fact that we know the operators are regularly
spaced (gapped by an integer). (This goes in other direction as well. In generic 2D CFTs, one
can show the Cardy inequality is saturated iff we have integer spaced spectra asymptotically,
which in turn implies all the operators are gathered together at some integer spaced points
with huge degeneracy [41].)
The examples in higher dimension that we consider here are of this kind: the operator
spectrum is regularly gapped. Thus one can hope to write down a formula without smearing
and hope to get the order one number correct. The only way to rigorously estimate this is tofigure out correction terms and show that they are suppressed e.g. by following the proof of
Meinardus theorem (see chapter 6 in [50]) keeping track of the error terms. We remark that wehave adapted and generalized only a part of the Meinardas theorem for our purposes, omitting
the part where one estimates the error rigorously in a step analogous to doing inverse Laplacetransformation. A more ambitious goal would be to write down something like a convergent
Rademacher sum in higher dimension. We have not attempted either of the above approaches
in this paper, but we expect that one can improve upon our result.

Acknowledgements
T.M. and S.P. thank UCSD for support and hospitality where this work was initiated. We thankAneesh Manohar and Masahito Yamazaki for comments on the manuscript. T.M. is grateful to
Brian Henning, Xiaochuan Lu, and Hitoshi Murayama for many important and fruitful discussions about Hilbert series. In obtaining the exact results for the SMEFT beyond mass dimension 15, we acknowledge use of the form computer code accompanying the paper [53]. T.M. is
supported by the World Premier International Research Center Initiative (WPI) MEXT, Japan,
and by JSPS KAKENHI grants JP18K13533, JP19H05810, JP20H01896 and JP20H00153. S.P.
acknowledges the support from Ambrose Monell Foundation and DOE grant DE-SC0009988.

A

The effect of equations of motion on asymptotics

We illustrate the effect of EOM on the asymptotic growth of operators with the example of asingle real scalar field in d = 4. Without EOM imposed, the PE is∞
X e−β n1
P ENo EOM = exp.
(151)
n (1 − e−β n )4
n=1
Applying the trick introduced in Sec. 3, keeping only singular terms in β, we find


ζ(5)
π4
ζ(3) 19 log(β)
43lim P ENo EOM (β) = exp
+
+
+
−
.
β→0
β4
90β 3 3β 2
720
288

(152)

Next consider a more general partition
P E(K, β) = exp∞
X e−β n 1 − e−Kβ nn=1n (1 − e−β n )4

35

,

(153)

SciPost Phys. 10, 104 (2021)

with the case K = 2 corresponding to the PE for a real scalar field in d = 4 with EOM imposed.
Again keeping only singular terms in β in applying the trick of Sec. 3, the leading behaviouras β → 0 is,
 4
π K
1
1 2
lim P E(K, β) = exp
π (K − 2)(K − 1)K−(K − 2)Kζ(3) +
3
2
β→0
90β
2β
36β
(154)

1
1
2 2
3
2
+ (K − 2) K log(β) +−12K + 45K − 46K .
24
72
The behaviour of this general partition function is exponentially suppressed for any finite valueof K compared to the PE for the scalar field without EOM imposed. Note that all sub-leading
terms with β dependence vanish when setting K = 2 to recover the physical case of imposing
EOM
 4

π
1
lim P E(K = 2, β) = exp
.
(155)
−
β→0
45β 3 9B

Meinardus theorem in arbitrary d, j

In this appendix we collect results not presented in the main text that are obtained by generalizing Meinardus’ theorem and applying it to EFTs in spacetime dimensions d and with fieldsof spin j.

B.1

Scalar field theory in d = 2n + 1 for n ≥ 1

In d = 2n + 1 dimension, the canonical dimension of the scalar field is half integer. Thus we
can not directly apply Meinardus theorem (or Lemma 6.1 or its modified form Eq. (80)). A
different kind of modification is needed. This is somewhat analogous to the case of fermionsin even dimensions.
We recast the PE in following way:
P E(β) =

∞ 
Y

d−2

1 − q n+ 2

− f (n,d)

=

n=0

∞
Y

1 − q k−1/2

− f (k−(d−1)/2,d)k=1

p − f˜(k,d)
1 − ( q)k,=Q
p 2k − f˜(2k,d)
∞
1
−
(
q)
k=1
Q∞

(156)

k=1

where f (n, d) is given by the symmetric spin n respresentation of SO(d) i.e.
f (n, d) = dim [n, 0, 0, · · · ] .

(157)

In the second step,
we performed a change of variable and defined˜This is constructed in a way such thatf (k, d) = f (k/2 + 1/2 − d−1
2 , d).
f˜(2k − 1, d) = f (k − (d − 1)/2, d). Note, in the second equality, shift in k is valid sincef (k − (d − 1)/2, d) = 0 for 1 ≤ k < (d − 1)/2 and k being integer. Thus the PE can be written
as a ratio of two auxiliary PE, on which one can apply the modified Meinardus theorem:
P E(β) =P E aux1 (β/2),
P E aux2 (β)

36

(158)

SciPost Phys. 10, 104 (2021)

P E aux1 (β) =∞Y

1 − e−kβ− f (k/2+1/2− d−1
2 ,d)

,

k=1

PE

aux2

(β) =

(159)

∞
Y

1−e

d−1

−kβ − f (k+1/2− 2 ,d).

k=1

The D functions corresponding to the auxiliary PEs are found to be


X
d −1
D1 (d, s) =k−s f k/2 + 1/2 −
,d ,
2
k


X
d −1
−s
D2 (d, s) =k f k + 1/2 −
,d ,
2
k

(160)

and let us denote the poles as αi,1 , α j,2 with residues Ai,1 and A j,2 where the index 1 and 2
refer to the P E aux1 and P E aux2 . In terms of these variables we havelog P E(β → 0) = log P E aux1 (β/2 → 0) − log P E aux2 (β → 0)
!


X
X
=
Ai,1 Γ (αi,1 )ζ(αi,1 + 1)(β/2)−αi,1 −A j,2 Γ (α j,2 )ζ(α j,2 + 1)β −α j,2ij0− [D1 (d, 0) log(β/2) − D2 (d, 0) log(β)] + D1 (d, 0) − D20 (d, 0) .

(161)

d=3
Let us do the case d = 3 explicitly (we have done this before using Wright’s result for planepartitions in Sec. 3; here we employ a method generalizable to arbitrary odd dimension). We
have the following data
D1 (3, s) = ζ(s − 1) ,D2 (3, s) = 2ζ(s − 1) ,α1,1 (3) = 2 , α2,1 (3) = 2(162)

A1,1 (3) = 1 , A2,1 (3) = 2 ,leading to Eq. (60).

d=5For d = 5 we have1[ζ(s − 3) − ζ(s − 1)]24
α1,1 (3) = 4 , α2,1 (3) = 2
D1 (5, s) =(163)

A1,1 (3) = 1/24 , A2,1 (3) = −1/24 ,and1[4ζ(s − 3) − ζ(s − 1)] ,12
α2,2 (5) = 2

D2 (5, s) =α1,2 (5) = 4 ,A1,2 (5) = 1/3 ,(164)

A2,2 (3, s) = −1/12 .

Applying Eq. (161) we findlog P Ed=5 (β → 0) =17 log(β)
2ζ(5)
ζ(3)
11
7 0
1 0
−
+
+
log(2) −ζ (−3) +ζ (−1) .
β4
12β 2
2880
2880
24
24
(165)
37

SciPost Phys. 10, 104 (2021)

Using the trick, one can find that

3
68
log
− 131
17
log(β)
2ζ(5)
ζ(3)
2
trick
log P Ed=5(β → 0) =
−
+
+
.
4
2
β
12β
2880
11520

(166)

Now we haveactual asymptotics : asymptotics via new trick = 1 : 0.996 .
exp[(165)]

B.2

(167)

exp[(166)]

Fermionic field theory in d = 2n + 1 for n ≥ 1

The canonical dimension of the fermionic field is an integer. This mimics the case of scalars ineven dimension. Thus we have
P E(β) =

∞ 
Y

d−1

1 + q n+ 2

 g(n,d)

=

n=0

∞
Y

1 + qk g(k−(d−1)/2,d)

(168)

k=1

g(n, d) = dim [n + 1/2, 1/2, · · · ] ,(169)

for example
2(n + 1)(n + 2)(n + 3) .
3
The limit in k can be shifted to k = 1 in the above because g(k − (d − 1)/2, d) = 0 for
k < (d − 1)/2 and k ∈ Z+ .
g(n, 3) = 2(n + 1) ,g(n, 5) =Let us apply this on spin 1/2 fermions in d = 3 and d = 5. Explicitly we haveD(3, s) = 2ζ(s − 1) , α(3) = 2 , A(3) = 2
22
D(5, s) = [ζ(s − 3) − ζ(s − 1)] , α1 (5) = 4 , α2 (5) = 2 , A1 (5) = −A2 (5) = .
3
3

(170)

Using the results following from Eq. (91), we obtainlog 23
ζ(3)β −2 −β→0 2
6
15ζ(5) ζ(3)11(f )log P Ed=5 (β) '−+log(2) .
4
2
β→0 4β2β180(f )log P Ed=3 (β) '(171)

Here we have put in the supersprcipt f explicitly to denote that these are fermionic PE. It canbe verified that our trick reproduces these asymptotics exactly.

B.3

Asymptotics of spin j fields in d = 4

We focus on d = 4 dimensional field theories with arbitraty spin j field, saturating the unitarity
bound. Fields of spin j have dimension j + 1. So all the bosonic fields have integer dimension
while the fermionic field have half-integer dimension.

38

SciPost Phys. 10, 104 (2021)

Bosonic fields j ∈ Z
The PE is given by
P E( j, β) = exp∞
X e−( j+1)nβ  2 j + 1 − 4 je−nβ + (2 j − 1)e−2nβ =∞Y(1 − e−nβ )4nn=11 − e−(n+ j+1)β−(n+1)(n+2 j+1)

(172)

n=0

Q∞
=Q

−kβ
k=1 1 − ejk=1

1 − e−kβ−(k2 − j 2 )−(k2 − j 2 ) ,where the factor (n + 1)(n + 2 j + 1) comes from the dimension of the SO(4) representation[n + j, j].
The quantity in the denominator will produce a polynomial factor in β in β → 0 limit:
j
Y

1−e

 2 2
−kβ −(k − j )

j
j
Y
Y
1
2
2
j 2 −k2( j−1) j(4 j+1)
6
'
(kβ)
=β
(k) j −k .

β→0

k=1

k=1

k=1

The numerator can be handled using Meinardus approach. The relevant function D isgiven by∞X
D( j, s) =k−s (k2 − j 2 ) = ζ(s − 2) − j 2 ζ(s) .
(173)
k=1

Hence we havelog P E( j, β) ' Γ (3)ζ(4)β −3 − j 2 Γ (1)ζ(2)β −1 −
β→0

j2
1
log β + ζ0 (−2) + j 2 log(2π)
2
2

j
X
1
− ( j − 1) j(4 j + 1) log β −( j 2 − k2 ) log k6k=1

π2 j 2 1
π41=−− j(2 j − 1)(2 j + 1) log(β) + ζ0 (−2) + j 2 log(2π)
45β 3
6β
6
2
j
X
−
( j 2 − k2 ) log k .

(174)

k=1

Fermionic fields 2 j ∈ Z/2Z
The PE is given by
P E( j, β) = exp

∞
X

(−1)

n+1 e

−( j+1)nβ

n=1

=

∞
Y

1 + e−(n+ j+1)βn

2 j + 1 − 4 je−nβ + (2 j − 1)e−2nβ(1 − e−nβ )4

(n+1)(n+2 j+1)

n=0

Q∞
=Q
=

(k−1/2)2 − j 2

k=1

1 + e−(k−1/2)β

j+1/2
k=1

1 + e−(k−1/2)β

j+1/2
Y

1+e

(175)

(k−1/2)2 − j 222−(k−1/2)β j −(k−1/2)

Q∞
k=1

Q∞

1 + e−kβ/2−kβ
k=1 1 + e

k=139k2 /4− j 2k2 − j 2.

SciPost Phys. 10, 104 (2021)

Figure 6: Calculation using the double cover of SO(4). The figure depicts saddles on (ω̃1 , ω̃2 ) plane. The center of the square is at (0, 0). The corners are at(1, 1), (1, −1), (−1, 1), (−1, 1). For the bosonic case, all the saddles contribute. When
we add fermions in the mix, the green coloured saddles at (0, ±1), (±1, 0) don’t contribute anymore, leading to an overall factor of 1/2. We have drawn circular regions arounds the saddle to denote how much the fluctuation around the saddle
contributes. For example, each of the corner ones contributes one quarter of thecenter one.
Thus we have
P E1aux (β/2)
1log P E( j, β) 'j(2 j − 1)(2 j + 1) log 2 + lim log
.
β→0 6
β→0
P E2aux (β)(176)

We apply Meinardus theorem on P E1aux and P E2aux . Now we have
D1 ( j, s) =1
ζ(s − 2) − 4 j 2 ζ(s) ,4D2 ( j, s) = ζ(s − 2) − j 2 ζ(s) .

(177)

Thus we have
π2 j 2 1
7π4−+ j(2 j − 1)(2 j + 1) log 2 .
β→0 360β 3
12β6log P E( j, β) '

C

(178)

Combinatorical geometry for SO(d) saddle points in arbitrary
dimension

We begin with presenting a third take on the calculation of the saddle points of a real scalarin Sec. 6—doing the calculation in the SO(4) double cover. We then show how the phasetransition behaviour discussed in Sec. 6 occurs in arbitrary spacetime dimension.

C.1

Take III—The SO(4) double cover way

In this case, the contours must traverse the unit circle in the complex plane twiice. H(β) is
given by
∞II
X e−nβ 1 − e−2nβ1H(β) =d x1
d x 2 µSO(4) (x i ) exp,(179)
4 twicen P(nβ, x in )
twice
n=1
where the factor of 14 in front of the measure normalizes it to unity. As x 1 , x 2 traverses thecircle twice, we have a square region swept out on (ω̃1 , ω̃2 ) plane with center at (0, 0) andveritces at (±1, ±1), (±1, ∓1)), and the positions of the saddles are at(ω1 , ω2 ) = {(0, 0), (0, 1), (0, −1), (1, 0), (−1, 0), (1, 1), (1, −1), (−1, 1), (−1, −1)} .
40

(180)

SciPost Phys. 10, 104 (2021)

Again we have to sum over the saddles along with the fluctuations around it. Compared to thecontribution coming from the fluctuation around the middle saddle (0, 0), the corners ones i.e(±1, 1), (1, ±1) produces 1/4 of the contribution, while each of the (0, ±1), (±1, 0) produces
1/2, see Fig. 6. So we have (1 + 4 × 1/4 + 4 × 1/2) = 4 coming from all the saddles and thiskills the 1/4 normalization factor appearing in front of the measure as a result of going to the
double cover. In this way it matches with Eq. (124) and Eq. (129).
Adding fermions kills off the saddle at (0, ±1), (±1, 0), so now we have the center ones
and the corner ones giving a contribution of 1 + 4 × 1/4 = 2; we must include again theoverall factor of 1/4 in the defining integral. Thus we land up with an overall factor of 1/2
and reproduce the result in Eq. (116).

C.2

Bosonic theory

There is only one saddle at ω̃i = 0 for the following integral:∞I Y
X e−nβ 1 − e−2nβH(β) =
d x i µSO(d) (x i ) exp.
n P(nβ, x in )
n=1
i

(181)

If we use the double cover then we have following expression in even d dimensions
∞IY
X e−nβ 1 − e−2nβ1d x i µSO(d) (x i ) exp.
H(β) = d/2(182)
n P(nβ, x in )2
twice in=1
We are going to show the extra 2d/2 factor gets killed by presence of other saddles. Now thesaddle points are distributed over a hypercube of length 2 centered at (0, 0, · · · ), aligned to theaxes. There are 3d/2 saddle points, they are given by
Saddles = {(a1 , a2 , · · · ad/2 ) : ai ∈ {0, ±1}} .

(183)

For d = 4, we have 34/2 = 9 points, they comes in three kinds (see fig Fig. 6 and Fig. 7).
One point is at the center, four of them are at the midpoints of edges, and four of them areat the corners. The leading contribution from these saddles are the same, but the sub-leading
contribution coming from the fluctuation around these three kinds of saddles is different. Geometrically, one can map these three kind of saddles with existence of geometrical objects,
which are embedded in the 2 dimensional square (the integration region). We have exactly
three such kinds: zero, one or two dimensional objects. The zero dimensional objects are the
4 corners, the one dimensional objects are the 4 edges, and the two dimension object is the fullsquare. Thus the saddles are mapped to 1 two dimensional object, 4 one dimensional objects
and 4 zero dimensional objects. This pattern survives in higher dimension. There are d/2
different type of saddles, which can be mapped to d/2 different type of dimensional objects
which can embedded.
The number of k dimensional objects inside a d/2 dimensional object is given by the expression 2d/2−k Binomial [d/2, k]. This follows because the k dimensional object can be foundby setting k entries of {a1 , a2 , · · · ad/2 } to 0 and setting rest of them to ±1. The k entries thenscan be chosen in Binomial [d/2, k] ways and rest of them can be filled in 2d/2−k ways, leading
to the above expression. Also, note that the compared to the contribution coming from thefluctuation around the saddle at the center (which is mapped to the d/2 dimensional object)saddles corresponding to the k dimensional object contribute a factor of 2k−d/2 . This can beobtained if we think coordinate-wise, the coordinates set at 0 contribute completely to thefluctuation, while the ones at ±1 contributes 1/2 of the full contribution, thus leading to the41

SciPost Phys. 10, 104 (2021)

Figure 7: Calculation using double cover of SO(6): the figure depicts saddles on(ω̃1 , ω̃2 , ω̃3 ) 3−space. The center of the cube is at (0, 0), marked red and circled.
The 8 corners are at (a1 , a2 , a3 ) where ai = ±1. For the bosonic case, all the saddles
contribute. When we add fermions in the mix, the green colored saddles at the
middle point of edges (corresponding to 1 dimensional object edge) and green/blackcolored saddles at the corners (corresponding to 3 dimensional object cube) don’t
contribute anymore, leading to a factor of 1/2. The saddles corresponding to thezero dimensional center and midpoints of the faces (2 dimensional object) always
contribute. Fluctuation around the saddles corresponding to k dimensional object
contributes 2k−3 compared to that of the center one.
2k−d/2 suppression. We have a total contribution compared to the single coverd/2
1 X

2d/2−k Binomial [d/2, k] 2k−d/2 = 1 .
{z}
2d/2 k=0 |(184)#of saddles

Now if we add fermions to the mix, some of the saddles will get suppressed at leading
order compared to the one at the center. We will take up this problem in next subsection.

C.3

Fermionic theory

For simplicity let us consider the spin [1/2, · · · 1/2] case in even dimension. The Hilbert series
is given byIY
1
H(β) = d/2
d x i µSO(d) (x i )P E(x i , β)2twice i∞(185)
X (−1)n+1 e−nβ(d−1)/2 χ[1/2,···1/2,+1/2] − χ[1/2,···1/2,−1/2] e−nβ
P E ≡ exp
.
n
n
P(nβ,
x
)
i
n=1
The fermionic characters have branch-cuts. Thus the saddles that contribute to the leading
order necessarily have the following form
{a1 , a2 , · · · ad/2 : ai ∈ {0, ±1} & # of ± 1 entries is even} .

42

(186)

SciPost Phys. 10, 104 (2021)

Thus compared to Eq. (184), we only sum over even k i.e. we have total contribution
1

bd/4c
X

2d/2 k=012d/2−2k Binomial [d/2, 2k] 22k−d/2 = .
{z
}
|
2

(187)

#of saddles

Thus we arrive at
H(β → 0, dim f = 0) = 2lim

dim f 7→0

H(β → 0, dim f




.

(188)

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