EFT Asymptotics: the Growth of Operator Degeneracy
EEFT Asymptotics: the Growth of OperatorDegeneracy
Tom Melia a and Sridip Pal ∞ a Kavli Institute for the Physics and Mathematics of the Universe (WPI), UTIAS, The Universityof Tokyo, Kashiwa, Chiba 277-8583, Japan ∞ School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, U.S.A.
E-mail: [email protected] , [email protected] Abstract:
We establish formulae for the asymptotic growth (with respect to the scalingdimension) of the number of operators in effective field theory, or equivalently the number of S -matrix elements, in arbitrary spacetime dimensions and with generic field content. Thiswe achieve by generalising a theorem due to Meinardus and applying it to Hilbert series—partition functions for the degeneracy of (subsets of) operators. Although our formulaeare asymptotic, numerical experiments reveal remarkable agreement with exact results atvery low orders in the EFT expansion, including for complicated phenomenological theoriessuch as the standard model EFT. Our methods also reveal phase transition-like behaviourin Hilbert series. We discuss prospects for tightening the bounds and providing rigorouserrors to the growth of operator degeneracy, and of extending the analytic study and utilityof Hilbert series to EFT. a r X i v : . [ h e p - t h ] O c t ontents d = 2 asymptotics 93.2 Plane partitions 123.3 Relation of plane partitions to d = 3 Hilbert series 13 d = 4 d = 4 d , j B.1 Scalar field theory in d = 2 n + 1 for n ≥ d = 2 n + 1 for n ≥ j fields in d = 4 C Combinatorical geometry for SO ( d ) saddle points in arbitrary dimension 41 C.1 Take III—The SO (4) double cover way 42C.2 Bosonic theory 42C.3 Fermionic theory 44– 1 – Introduction
The principle of effective field theory (EFT) is to parameterise all possible contributionsto the S -matrix that can occur in a quantum field theory of given particle content andsymmetry. Its phenomenological utility relies on a hierarchy of experimental importance ofthese contributions, and on the possibility of probing higher order terms via an improvementin experimental precision. Both of these aspects are necessary, for example, for the searchfor new physics at the Large Hadron Collider within the framework of the standard modelEFT (SMEFT) to be a viable endeavour. The rate of growth of the number of independent S -matrix contributions that can be constrained with increasing experimental precision isan important practical issue: even at the lowest few orders in the SMEFT expansion, theirnumber is cumbersome [1–3]. Experimental practicalities aside, the results of this paper canbe used to determine that at around the 120th order in the SMEFT expansion, the numberof possible independent measurements that can be performed exceeds the number of atomsin the observable universe (around ). Our purpose, however, is not to establish wheresuch cosmologically large thresholds lie. Rather it is to develop new analytic methods bywhich such numbers can be obtained, and to explore what these techniques can reveal aboutEFT itself through its analytic study.The analytic study of EFTs/ S -matrices dates back to the discovery of the strong in-teraction and the birth of S -matrix theory ( e.g. [4]). Many of the ideas and methods havefound utility in modern amplitude calculations in perturbative QCD and other theories; forreviews see e.g. [5, 6]. In the past few years, building upon the success of the modern CFTbootstrap program [7], a number of new analytic results have been obtained in an S -matrixbootstrap approach [8–19]. Here, we focus on a much simpler mathematical object that hasrecently been introduced to study EFT/ S -matrices—Hilbert series.Hilbert series have appeared in the particle physics literature as partition functionsto enumerate gauge and flavour invariants [20–24], and were applied to EFT operatordegeneracy counting in [3, 25–30]. A study of their utility as analytic probes was initiatedin [26, 28], establishing, for example, recursion relations between Hilbert series for theoriesof different field content in d = 2 spacetime dimensions, and all-order in derivatives resultsfor degeneracy of S -matrix elements with a low fixed number of external particles. Theaim of this paper is to extend what we know analytically about Hilbert series, and thusEFTs/ S -matrices, by extracting the full asymptotic growth (with respect to the scalingdimension) of number of operators from the Hilbert series, going beyond the basic studieson 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 withscaling dimension ∆ as ∆ → ∞ .Appropriately graded Hilbert series correspond to partition functions of free quantumfield theories. There is thus a connection to many of the techniques to study partition func-tions that appear prominently in the study of CFT. The methods and formulae establishedby Cardy [31] provide the growth of operator degeneracy in d = 2 . We review the use ofmodular invariance of the partition function for d = 2 scalar EFT to obtain the asymptoticgrowth of all operators, and those captured by the Hilbert series (i.e. a projection from the– 2 –pace of all operators to the spin zero subspace). In d = 2 , we can also utilize the connec-tion of the Hilbert series to integer partitions, as established in [26], to capture asymptoticgrowth via the famous Hardy-Ramanujan formula.In higher dimensions and including more general field content—fermionic and higherspin representations, possibly with internal symmetry degrees of freedom—modular invari-ance of the partition function is lost, and much less is known. Leading behaviour of thefree field partition function (for all operators) of a scalar field in arbitrary d was presentedin [32]. We develop and apply a theorem due to Meinardus, so as to obtain exact results forthe 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. Some-what surprisingly, the asymptotic formulae we obtain are in remarkably good agreementwith exact results for operator degeneracy at low scaling dimension.In line with our above goal of analytic exploration, we also explore subtleties that arisein 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 fermionsgoes to zero.As a way of defining the techniques we use to mine asymptotics and introducing Hilbertseries themselves, consider a quantum field theory with a discrete spectrum { ∆ n } andcorresponding degeneracies { a n } , which means that there are a n number of operators withscaling dimension ∆ n . The Plethystic Exponential (PE) is defined as the sum, P E ( q ) = (cid:88) n a n q ∆ n . (1.1)We will work with the variables q = e − β . To estimate the growth of the degeneracies, wewrite P E ( q ) = (cid:88) n a n e − β ∆ n ≡ (cid:90) ∞ d ∆ ρ (∆) e − β ∆ , (1.2)where ρ (∆) ≡ (cid:88) n a n δ (∆ − ∆ n ) . (1.3)Thus formally we have ρ (∆) = 12 πı (cid:90) Γ+ ı ∞ Γ − ı ∞ dβ P E (cid:16) q = e − β (cid:17) e β ∆ . (1.4)Our objective is to estimate asymptotic growth of a n for large n (or equivalently ρ (∆) forlarge ∆ ). The idea is that the asymptotic growth of a n is encoded in the behaviour of PEin the q → limit [32]. In particular, we have [31, 32] ρ (∆) (cid:39) ∆ →∞ πı (cid:90) Γ+ ı ∞ Γ − ı ∞ dβ P E ( q → e β ∆ . (1.5) The intuition behind this is that q → limit, P E ( q ) diverges. Formally P E ( q = 1) = (cid:80) n a n divergestoo. Thus the P E ( q → encodes the growth of (cid:80) Nn =1 a n as N → ∞ , hence the growth of a n as n → ∞ . – 3 –or EFT we typically want to count operators which are singlets under the Lorentzsymmetry group and possibly some internal symmetry group. To do this, one needs to turnon variables (fugacities), w i , for these groups and project out only the singlet terms fromthe PE integrating over the group measure, schematically H ( q ) = (cid:90) dµ Lorentz (cid:90) dµ internal P ( q, w i ) P E ( q, w i ) , (1.6)where dµ are the Haar measures of the symmetry groups, and where we also included a pro-jector P ( q, w i ) − (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 classof IBP related operators for each conformal primary operator in the spectrum. ImposingIBP 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 areinterested in obtaining asymptotic form of this generating function in the β → limit, andin what follows we will show it takes the form H ( β →
0) = exp (cid:34) d − (cid:88) k =0 a k β − k + b log( β ) + c (cid:35) , a d − > . where d is the space-time dimension. Again one can write, H ( β ) = (cid:90) ∞ d ∆ ρ g (∆) e − β ∆ (1.7)where ρ g is the density of number of operators in the singlet sector. Similar considerationsto the above yield ρ g (∆) (cid:39) ∆ →∞ πı (cid:90) Γ+ ı ∞ Γ − ı ∞ dβ H ( β → e β ∆ (1.8)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. (1.5) and Eq. (1.8)is a continuous function, the left hand side is a distribution. The proper way to interpretthis is to smear ρ (∆) over a small window of [∆ − δ, ∆ + δ ] and then estimate the numberof states lying in that window as ∆ → ∞ , keeping δ fixed. The mathematical machinerythat one needs to achieve this lies in the Tauberian theorems (See [33] and appendix Cof [34] for a basic introduction) In D conformal field theories (free and/or interacting) itis possible to estimate the density of states using these techniques and even put an upperbound on the spectral gap in interacting CFTs [35–38]. We will not attempt to make ourgeneral analysis mathematically rigorous in the sense just described and leave it as a futureavenue 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,and detail the connection to the Hardy-Ramanujan formula. Moving away from d = 2 the– 4 –odular properties are lost. In Sec. 3 we introduce a new trick that can quickly obtainasymptotic formulae for the growth of operators for more general partition functions, up toan unknown order one number; we test its accuracy by comparing to the known asymptoticgrowth of plane partitions, and we show how the PE for a real scalar field in d = 3 can berelated to the partition function of plane partitions. In Sec. 4 we obtain our main results:by generalising a theorem of Meinardus, we find exact asymptotic formulae that can beapplied to the PE for EFTs in general d and including particles of spin. Sec. 5 presentsthe saddle point techniques used to make the projections of the PE to singlets of spacetimeand internal symmetries, and we provide general formulae for the resulting Hilbert series.In Sec. 6 we detail a subtlety in taking the saddle point approximation when fermions arepresent, and observe phase transition-like behaviour in the Hilbert series. We apply ourresults to the SMEFT in Sec. 7, and observe remarkable numerical agreement at low massdimension in this theory. Sec. 8 contains a discussion of refinements and generalisationsthat can be made to the analysis we present here.We highlight some of our main results: Eq. (4.40) for the leading behaviour of the freefield partition function in arbitrary d and for arbitrary spin j ; the general lessons in 5.2, inparticular, Eq. (5.13) for the asymptotic growth of operators in EFTs in four dimensions,including the exact order one multiplicative factors; and, Eq. (7.12) for the asymptoticgrowth of operators in the SMEFT. We begin by considering a single scalar field φ in dimensions. This section followsthe analysis and methods introduced in [31]. We will first estimate the growth of all theoperators as encoded in the Plethystic exponential. Using an appropriate projector, thePE can be turned into a Hilbert series, which encodes the number of scalars appearing inthe 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 SL (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 con-struct, not necessarily invariant under Lorentz group. It is given by P E = exp (cid:34) ∞ (cid:88) n =1 n χ φ ( t n , x n ) (cid:35) = exp (cid:34) ∞ (cid:88) n =1 n (1 − t n )[1 − ( tx ) n ][1 − ( t/x ) n ] (cid:35) , (2.1)where χ φ ( t, x ) is the character for the spin-0 conformal representation, see e.g. [39], withfugacities t for scaling dimension (which effectively counts derivatives) and x for angularmomentum. We will be using the variables β and ω , t = e − β , x = e πıω , (2.2) The presence of the (1 − t ) in the conformal character enacts the null state condition—this is thecondition of 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 boundin the free theory); we present a short discussion on the effect that EOM have on operator growth in App. A. – 5 –nd for use in the below we define the variables q and ¯ q , q = e − β − πıω , ¯ q = e − β +2 πıω . (2.3)We rewrite the PE in the following fashion P E = exp (cid:34) ∞ (cid:88) n =1 n (cid:18) − ( q ¯ q ) n (1 − q n )(1 − ¯ q n ) (cid:19)(cid:35) = exp (cid:34) ∞ (cid:88) n =1 n (cid:18) − q n − (cid:19)(cid:35) exp (cid:34) ∞ (cid:88) n =1 n (cid:18) − ¯ q n − (cid:19)(cid:35) exp (cid:34) ∞ (cid:88) n =1 n (cid:35)(cid:124) (cid:123)(cid:122) (cid:125) Throw away . (2.4)There is a singularity in the above sum since in d = 2 , φ is dimensionless and one canconstruct an arbitrarily large number of operators using powers of φ without changing thescaling dimension. To be precise given an operator O , one can construct arbitrary numberof operators by considering f ( φ ) O , which has the same dimension. We form an equivalenceclass by counting two operators only once if they differ by their number of φ s only, and thus f ( φ ) O and O are not counted separately. The implementation of this equivalence relationboils down to regularizing the PE by throwing away the term exp (cid:2)(cid:80) ∞ n =1 1 n (cid:3) . The regularized PE can be related to Dedekind eta function as
P E = q / ¯ q / η ( q ) η (¯ q ) . (2.5)We remark that the regularized PE is related to the torus partition function of c = 1 freebosonic CFT modulo the zero mode contribution (see e.g. Chapter 10 of [40]). All the operators
To study all the operators, we can turn the angular momentum fugacity off, setting q = ¯ q .In order to extract the asymptotics, we must know η ( q ) in the q → limit. The nice featureis that the η function has a modular property η ( q = e − β ) = (cid:114) πβ η (cid:18) ˜ q = e − π β (cid:19) . (2.6)and η (˜ q →
0) = exp (cid:20) − π β (cid:21) (2.7)Thus in β → i.e. q → limit, we have lim q → P E = lim q → η ( q ) − = β π lim ˜ q → η (˜ q ) − = β π exp (cid:20) π β (cid:21) (2.8)Taking the inverse Laplace transform of the above, as per Eq. (1.5), we obtain the asymp-totic growth of all the operators, ρ (∆) , as ρ (∆) (cid:39) ∆ →∞ Inverse Laplace (cid:20) lim q → P E (cid:21) = π ˜ F (cid:18) π ∆3 (cid:19) (cid:39) − / e π √ ∆ √ ∆ / (2.9)– 6 – calar operators The analysis for counting the scalar operators is done by projecting the
P E ( q, ¯ q ) onto spin . This is done by contour integral of the PE over the maximal compact subgroup of SL (2 , R ) , namely U (1) , with Haar measure (cid:73) dµ U (1) = (cid:73) | x | =1 dx πıx . (2.10)Here we will work with the real variable ω such that H = (cid:90) / − / dω P E ( q, ¯ q ) (2.11)The asymptotic analysis of H can be done following the fixed spin analysis of [38]. Inthis case, we will not set q = ¯ q , instead keeping them complex conjugate to each other.Now we have P E ( q, ¯ q ) = e − β η ( q ) η (¯ q ) = (cid:114) β + 2 πıω π (cid:114) β − πıω π e − β η (˜ q ) η (˜¯ q ) (cid:39) β → (cid:112) β + 4 π ω π e π β +2 πıω + π β − πıω = (cid:112) β + 4 π ω π e π β β π ω . (2.12)Thus we have H ( β ) (cid:39) β → (cid:90) / − / dω (cid:112) β + 4 π ω π e π β β π ω . (2.13)Now note that in β → limit, the integral is dominated by ω = 0 saddle. Thus we can doa saddle point approximation around ω = 0 and obtain H ( β ) (cid:39) β → β π e π β (cid:90) ∞−∞ dω exp (cid:18) − π ω β (cid:19) = √ β / π / e π β . (2.14)Thus we can see that while the leading exponential growth stays the same, the pro-jection onto scalar operators changes the polynomial dependence on β . There is an extrafactor of (cid:113) β π . The growth of scalar operators is be suppressed by − / ∆ − / comparedto the growth of all the operators. This comes from noting that the inverse Laplace trans-formation is dominated by the saddle β = π (cid:112) c . We can also verify the scaling by doingthe inverse Laplace transformation of Eq. (2.14) explicitly: ρ scalar (∆) (cid:39) ∆ →∞ e π (cid:113) ∆3 = (cid:18) (cid:19) / − / exp (cid:18) π (cid:113) ∆3 (cid:19) / . (2.15) IBP constraints
To impose the IBP constraint, we include a projector that is the inverse of the momentumgenerating function P ( q, ¯ q ) − = (1 − q )(1 − ¯ q ) . (2.16)– 7 –he regularized PE becomes P E
IBP = (1 − q )(1 − ¯ q ) exp (cid:34) ∞ (cid:88) n =1 n q n − q n (cid:35) exp (cid:34) ∞ (cid:88) n =1 n ¯ q n − ¯ q n (cid:35) . (2.17)First we look at the asymptotic growth of all operators, setting q = ¯ q ; the β → limit of P E
IBP is given by
P E
IBP (cid:39) β → β π exp (cid:20) π β (cid:21) . (2.18)Thus with the IBP constraint the growth of all operators are given by ρ IBP (∆) (cid:39) ∆ →∞ π I (cid:16) π √ ∆ √ (cid:17) (cid:39) π
12 3 − / e π √ ∆ √ ∆ / = (cid:18) π (cid:19) − / e π √ ∆ √ / . (2.19)Compared to Eq. (2.9), we have suppression by β i.e π / . We will see that in d dimen-sions, the role of IBP is to introduce an extra factor of β d in the β → limit of the PE ofHilbert series. This results in a generic suppression of ρ (∆) by a factor of ∆ − (since thesaddle point of the inverse Laplace transform is given by β d ∗ ∝ ∆ − ) when comparing thegrowth of operators with IBP imposed to the growth of all operators. Scalars and IBP
The introduction of IBP amounts to modification of Eq. (2.14) into following: H scalarIBP ( β ) (cid:39) β π (1 − e − β )(1 − e − β ) e π β (cid:90) ∞−∞ dω exp (cid:18) − π ω β (cid:19) (cid:39) β → √ e π β β / π / (2.20)where we have pulled out the factor accounting for the IBP constraint out of the ω integralevaluated at the saddle point ω = 0 . The asymptotic growth of scalar operators with IBPconstrained imposed is then given by ρ scalarIBP (∆) (cid:39) ∆ →∞ π e π √ ∆ √ ∆ = (cid:18) (cid:19) / (cid:18) π (cid:19) − / e π √ ∆ √ / (2.21) We conclude this section by noting that the q / η − is the generating function for thenumber of partitions of an integer i.e we have P E ( q, ¯ q ) = (cid:88) n P ( n ) q n (cid:88) m P ( m )¯ q m , (2.22)where P ( n ) is the number of partitions of an integer n . The generating function for scalars(without IBP) is obtained by picking out terms from the above that have the same powersof q and ¯ q : H ( q ) = (cid:88) n P ( n ) q n = (cid:88) ∆ P (cid:18) ∆2 (cid:19) q ∆ . (2.23) TM acknowledges Brian Henning in communicating formulas and for helpful discussions on the resultsof this section. – 8 –f we impose IBP, then we have
P E
IBP ( q, ¯ q ) = (1 − q )(1 − ¯ q ) P E ( q, ¯ q ) = (cid:88) n ( P ( n ) − P ( n − q n (cid:88) m ( P ( m ) − P ( m − q m , (2.24)and the generating function for scalars with IBP imposed is given by H IBP ( q ) = (cid:88) n ( P ( n ) − P ( n − q n . (2.25)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-Ramanujanformula [41] P ( n ) (cid:39) n →∞ e π √ n √ n . (2.26)In particular, we have ρ scalar (∆) = (cid:20) P (cid:18) ∆2 (cid:19)(cid:21) ,ρ scalarIBP (∆) = (cid:20) P (cid:18) ∆2 (cid:19) − P (cid:18) ∆2 − (cid:19)(cid:21) (cid:39) ∆ →∞ (cid:18) dP ( n ) dn (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) n =∆ / , (2.27)where it is to be understood that we are taking a derivative of the function appearing onthe R.H.S of Eq. (2.26). It is easily verified that Eq. (2.15) and Eq. (2.21) are reproduced. Although the above calculations are clean, they are not directly applicable in higher dimen-sions where we do not have the luxury of having a modular covariant function in the PE. Inthis section we will step down on rigour a bit and invent a new way to obtain asymptoticsup to O (1) terms. In the following section we will show how to regain these O (1) terms,but we choose to present the following trick first, as (i) it is a fast and transparent wayto obtain the form of the asymptotic growth, and (ii) we will later use this trick—againfor its simplicity of presentation—to analyse projections of the PE to obtain Hilbert series,appealing at the end to the rigorous treatment to fix the O (1) terms. d = 2 asymptotics Let us first demonstrate this trick for a single scalar field in d = 2 so as to make directcomparison with what we obtained in the previous section. We start with the regularizedPE of Eq. (2.4), P E = exp (cid:34) ∞ (cid:88) n =1 n q n − q n (cid:35) exp (cid:34) ∞ (cid:88) n =1 n ¯ q n − ¯ q n (cid:35) . (3.1)– 9 – ll operators We set q = ¯ q = e − β for the analysis of all operators. Then we have P E = exp (cid:34) ∞ (cid:88) n =1 e − βn n − e − nβ (cid:35) . (3.2)Now we do the following series expansion for small β , − e − nβ = 2 βn + 1 + β − β n
360 + β n · · · , (3.3)take the right hand side of the above and do the summation exp ∞ (cid:88) n =1 e − nβ n βn + 1 (cid:124) (cid:123)(cid:122) (cid:125) singular pieces + βn − β n
360 + β n , (3.4)and take the β → limit. The underbracket labelled “singular pieces” refers to the factthat these will eventually produce the singular pieces in the log P E in the β → limit.The singular pieces indicated in (3.4), after summing over n , give an exponent π β +log( β ) − . This is very close to the actual value π β + log( β ) − log(2 π ) as in Eq. (2.8). Thenon-rigorous part of this analysis is we are not proving that the non-singular terms we threwaway in Eq. (3.3) changes the order one multiplicative prefactor in the β → asymptoticsof the PE (not to be confused with the leading term in log P E , which we know exactly).Nonetheless, we do reproduce Eq. (2.8) up to an order one multiplicative correction,
P E ( q = ¯ q → (cid:39) βe − exp (cid:20) π β (cid:21) . (3.5)We can compare the ratio of actual asymptotics and the one we just obtained:actual asymptotics (2.8) : asymptotics via new trick (3.5) = 12 π : e − = 1 : 0 . . (3.6)The non-singular pieces in Eq. (3.3) produce if we take the β → limit first and thensum over n ; on the other hand, if we first sum over n and then take the β → limit, theyproduce β independent order one corrections. If we keep up to β term, sum over n andthen take the β → limit, we findactual asymptotics (2.8) : asymptotics via new trick modified (3.5) = 12 π : e − = 1 : 1 . . (3.7)Unfortunately, we can not improve the order one number by keeping more and more higherorder terms—at some point the non-rigourous nature of our analysis ensures that the ratiowill become far off from . However, we will use this trick on the principle that the singularpieces (and perhaps a small number of higher order terms) reproduce the correct asymptoticsup to an order one number that is reasonably close to unity. Note that once we have theasymptotics of P E ( β ) , we can again use the inverse Laplace transformation to re-obtainEq. (2.9), again up to order one multiplicative terms.– 10 – rojecting onto scalars To apply the above trick to the asymptotic growth of scalar operators, we reintroduce thefugacity for spin and write
P E ( β, ω ) = exp (cid:34) ∞ (cid:88) n =1 e − nβ n f ( β, ω ) (cid:35) , (3.8)where we have f ( β, ω ) ≡ e − πınω − e − n ( β +2 πıω ) + e πınω − e − n ( β − πıω ) . There is a saddle at ω =0, so we first expand f around this point, keeping terms up to order ω , f ( β, ω ) (cid:39) (cid:20) − e − βn (cid:21) − π n ω (cid:34) e βn (cid:0) e βn + 1 (cid:1) ( e βn − (cid:35) + · · · . (3.9)We take the β → limit of the terms appearing in the square brackets of the aboveexpression f ( β, ω ) (cid:39) (cid:20) βn (cid:21) − π n ω (cid:20) β n + 2 nβ + 1 β + n (cid:21) (3.10)The fluctuation around the saddle is controlled by the term proportional to ω . The mostsingular term in β → limit that is present in this term is proportional to β − ; we keepthis term only and plug the truncated f ( β, ω ) back into the expression for P E ( β, ω ) inEq. (3.8). Subsequently, we perform the sum over n and again take β → limit, P E ( β, ω ) = ∞ (cid:88) n =1 e − nβ (cid:20) βn + 1 n (cid:21) − ω (cid:20) π β n (cid:21) = β → π β − β ) − (cid:0) π ω (cid:1) β . (3.11)Finally we do the projection by integrating over the Haar measure, H ( β ) (cid:39) β → βe − exp (cid:20) π β (cid:21) (cid:90) / − / dω exp (cid:34) − (cid:0) π ω (cid:1) β (cid:35) (cid:39) βe − exp (cid:20) π β (cid:21) (cid:90) ∞−∞ dω exp (cid:34) − (cid:0) π ω (cid:1) β (cid:35) = √ β / π / e π β − , (3.12)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 preciselyreproduces Eq. (2.14) up to order one multiplicative correction. Imposing IBP
Imposing IBP requires us to multiply the PE in Eq. (3.5) or H in Eq. (3.12) by β . Theymatch onto Eq. (2.18) and Eq. (2.20) respecctively upto order one multiplicative correction.– 11 – .2 Plane partitions In this subsection, we proceed to test our new trick on the plane partitions of an integerand relate the generating function for plane partitions to the d = 3 Hilbert series for scalarfield theory.A plane partition is defined to be a two dimensional array of non-negative integers { a i,j } i,j ∈ Z ≥ such that a i,j ≥ a i +1 ,j a i,j ≥ a i,j +1 (3.13)The sum of a plane partition is defined as n = (cid:88) i,j a i,j (3.14)Let us define P L ( n ) to be the number of plane partitions of integer n . For example, for n = 3 we have partitions, , (cid:34) (cid:35) , (cid:104) (cid:105) , (cid:34) (cid:35) , (cid:104) (cid:105) , (cid:104) (cid:105) . (3.15)The generating function for P L ( n ) is given by [42] P E pl ( q ) ≡ (cid:88) n P L ( n ) q n = ∞ (cid:89) n =1 (1 − q n ) − n = exp (cid:34) ∞ (cid:88) n =1 q n n − q n ) (cid:35) . (3.16)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 → limit of P E pl ( β ) = exp (cid:34) ∞ (cid:88) n =1 e − nβ n − e − nβ ) (cid:35) . (3.17)We apply our trick (keeping the singular pieces before summing over n i.e. (1 − e − nβ ) − (cid:39) β n + βn + ). This produces P E pl ( β ) (cid:39) β → e − / β / exp (cid:20) ζ (3) β (cid:21) , (3.18)which upon inverse Laplace transformation gives P L ( n ) (cid:39) n →∞ ζ (3) / / √ πn / exp (cid:104) × − / n / ( ζ (3)) / − / (cid:105) (3.19)The asymptotic growth of P L ( n ) is known in the mathematics literature: it was firstfigured out in an old paper by Wright [42], and a typographical error in Wright’s paper waspointed out later by Mutafchiev and Kamenov [43]. The growth is given by P L ( n ) (cid:39) n →∞ ζ (3) / / √ πn / exp (cid:104) × − / n / ( ζ (3)) / + ζ (cid:48) ( − (cid:105) (3.20)– 12 –omparing Eq. (3.20) and Eq. (3.19) we haveactual asymptotics (3.20) : asymptotics via new trick (3.19) = e ζ (cid:48) ( − : e − / = 1 : 0 . . (3.21)The actual/obtained via trick asymptotics of P L ( n ) can also be turned into actual/obtainedvia trick asymptotics of P E pl ( β → , P E
Maths pl ( β ) (cid:39) β → β / exp (cid:20) ζ (3) β + ζ (cid:48) (1) (cid:21) P E trick pl ( β ) (cid:39) β → β / exp (cid:20) ζ (3) β − / (cid:21) (3.22)The superscript in P E signifies how we obtain the result. Again, if we keep up to the firstorder piece before summing over n (i.e (1 − e − nβ ) − (cid:39) β n + βn + + nβ ), we find thatwe obtain a better result P E trick, first order pl ( β ) (cid:39) β → β / exp (cid:20) ζ (3) β − / (cid:21) actual asymptotics (3.20) : asymptotics via new trick (3.19) incl. first order = e ζ (cid:48) ( − : e − / = 1 : 0 . . (3.23) d = 3 Hilbert series
The PE for a scalar field theory in d = 3 can be massaged into a form that is directlyrelated to the PE for plane partitions. The steps to do so (that we spell out in detail atthis point) are as follows where we start by considering the explicit form of PE from [28]by setting all the fugacities to : P E ( β ) = exp (cid:34) ∞ (cid:88) n =1 e − nβ/ n − e − nβ (1 − e − nβ ) (cid:35) = exp (cid:34) ∞ (cid:88) n =1 e − nβ/ n ∞ (cid:88) k =0 (2 k + 1) e − knβ (cid:35) = exp (cid:34) ∞ (cid:88) k =0 (2 k + 1) ∞ (cid:88) n =1 e − nβ ( k +1 / n (cid:35) = ∞ (cid:89) k =0 (cid:16) − q k +1 / (cid:17) − (2 k +1) = ∞ (cid:89) n =0 (cid:16) − √ q n +1 (cid:17) − (2 n +1) = (cid:81) ∞ n =1 (cid:0) − √ q n (cid:1) − n (cid:81) ∞ n =1 (cid:0) − √ q n (cid:1) − (2 n ) = (cid:81) ∞ n =1 (cid:0) − √ q n (cid:1) − n (cid:2)(cid:81) ∞ n =1 (1 − q n ) − n (cid:3) = P E pl ( β/ P E pl ( β )] (3.24)In the last line, we relate it to (3.16) introduced previously as generating function of planepartition of integer. Now using the asymptotics for plane partitions of integer, we find that P E
Maths ( β ) (cid:39) β → − / β − / exp (cid:20) ζ (3) β − ζ (cid:48) ( − (cid:21) (3.25)– 13 –he superscript in P E signifies that we obtained the formula using Wright’s result [42]. Incomparison, our trick produces (keeping up to the first order term)
P E trick ( β ) (cid:39) β → / β − / exp (cid:20) ζ (3) β + 1 / (cid:21) (3.26)That is we find,actual asymptotics (3.25) : asymptotics via new trick (3.26) = 2 − / e − ζ (cid:48) ( − : 2 / e / = 1 : 0 . . (3.27)At this point, we expect the readers to be convinced that our trick produces the singular β dependence correctly. Furthermore, we observe that can produce the order one numberalmost correctly, but not quite. In the following section we proceed to capture this orderone number exactly. The hint for how to do so is in the proof of the asymptotics of planepartitions, which uses a theorem named after Meinardus. Our aim is to rigorously obtain the asymptotic growth of number of operators for EFTsin general spacetime dimension d , encompassing particles of general spin j (that saturatethe unitarity bounds). We show this can be achieved using the theory of partitions (theterm ‘partitions’ here is not in the sense of a direct higher dimensional generalization ofthe partition of integers, as with the plane partitions above). In particular we appeal to atheorem due to Meinardus [44]. A mathematical exposition of this theorem can be foundin the Chapter 6 of [45] (see Theorem . ).We will in fact need to generalize the theorem for application to the general EFTs weare interested in; we will highlight the tricks we employ as they come 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 thuswork through these cases here; explicit results in odd dimensions and with particles ofhigher spin j are collected in App. B. However, the leading behaviour in arbitrary d and j is universal and compact, and we present this in Sec. 4.3 below. We begin by manipulating the PE for a scalar field theory in arbitrary even d ≥ dimensioninto the following form P E ( d, β ) = exp (cid:34) ∞ (cid:88) n =1 e − n d − β n − e − nβ (1 − e − nβ ) d (cid:35) = exp (cid:34) ∞ (cid:88) n =1 e − n d − β n ∞ (cid:88) k =0 f ( k, d ) e − knβ (cid:35) = ∞ (cid:89) k =0 (cid:16) − q k + d − (cid:17) − f ( k,d ) = ∞ (cid:89) k =1 (cid:16) − q k (cid:17) − f ( k − d/ ,d ) (4.1)– 14 –here f ( k, d ) is given by the symmetric spin k respresentation of SO ( d ) i.e. f ( k, d ) = dim [ k, , , · · · ] . (4.2)For example f ( k,
4) = ( k + 1) , f ( k,
6) = 112 ( k + 1)( k + 2) ( k + 3) , etc. (4.3)Note the shift in the variable k in the final equality of Eq. (4.1) is valid because f ( k − d/ , d ) = 0 for ≤ k < ( d − / . (For d = 4 , the shift is trivially valid.)The rational behind recasting the PE in a product form starting from k = 1 is thatthe Meinardus theorem deals with such infinite product. We proceed to sketch the proofof the theorem, and show how the leading behaviour of the PE, and hence the asymptoticgrowth of operators, can be obtained. The proof proceeds via two steps: we first figure outthe P E ( d, β ) in β → limit, and then translate the result into asymptotics of growth of q expansion coefficients.We start from P E ( d, β ) = ∞ (cid:89) k =1 (cid:16) − q k (cid:17) − f ( k − d/ ,d ) = exp (cid:34)(cid:88) n =1 n (cid:88) k =1 f ( k − d/ , d ) e − knβ (cid:35) . (4.4)We replace e − knβ by the Mellin transform and interchage the sum and the Mellin integralusing absolute convergence. Thus we have log P E ( d, β ) = 12 πı (cid:90) ds β − s Γ( s ) ζ ( s + 1) D ( d, s ) , (4.5)where the function D ( d, s ) is defined as D ( d, s ) ≡ ∞ (cid:88) k =1 k − s f ( k − d/ , d ) . (4.6)Note the Mellin transform integral goes along a vertical line where the above sum converges. Four spacetime dimensions
Now let us focus on d = 4 . In d = 4 the function evaluates to D (4 , s ) = ζ ( s − . (4.7) D (4 , s ) converges for Re ( s ) > , and it admits an analytic continuation for Re ( s ) > − C with < C < except for a simple first order pole at s = 3 with residue . In general, wedenote the pole of D ( d, s ) as α and the residue as A . Here we have α = 3 , A = 1 (4.8)Using the analytic structure of D ( d, s ) we rewrite Eq. (4.5) specifying the contour: log P E ( d, β ) = 12 πı (cid:90) α + ı ∞ α − ı ∞ ds β − s Γ( s ) ζ ( s + 1) D ( d, s ) (4.9)– 15 –ow the idea is to move the contour to the left , crossing past the pole to a verticalline Re ( s ) = − C . There are two obstacles to such a movement. Firstly, the integrand hasa simple first order pole at s = α = 3 coming from D (4 , s ) with residue A Γ( 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 ) = A Γ( α ) ζ ( α + 1) β − α − D (4 ,
0) log( β ) + D (cid:48) (4 , πı (cid:90) − C + ı ∞− C − ı ∞ dsβ − s Γ( s ) ζ ( s + 1) D ( d, s ) (cid:124) (cid:123)(cid:122) (cid:125) Error term (4.10)After showing that πı (cid:82) − C + ı ∞− C − ı ∞ dsβ − s Γ( s ) ζ ( s + 1) D ( d, s ) is indeed an error term, onearrives at (Lemma . , Eq. . . of [45] ), log P E (4 , β ) (cid:39) β → A Γ( α ) ζ ( α +1) β − α − D (4 ,
0) log( β )+ D (cid:48) (4 ,
0) = 2 ζ (4) β − + ζ (cid:48) ( − . (4.11)The second part of the Meinardus theorem involves translating the above result in asymp-totics of ρ (∆) ; this is essentially done using the saddle point method and carefully estimatingthe error. Evaluating the inverse Laplace transform in the saddle approximation we findfor a scalar field in d = 4 , ρ (∆) (cid:39) ∆ →∞ √ √ / exp (cid:32) π ∆ / √
15 + ζ (cid:48) ( − (cid:33) . (4.12)In Fig. 1 we show the exact ρ (∆) obtained from a Taylor series expansion of the PE,compared with the asymptotic formula Eq. (4.12). In the upper panel of the plot, theasymptotic curve is indistinguishable from the data. In the lower panel we show the errorof the asymptotic result: even at lowest mass dimensions this error is less than ten percent,and it decreases below the per mille level at dimension ∆ (cid:39) .Before moving on, we can compare the expression for the growth of P E ( d, β ) in β → limit derived above to that obtained via the new trick of Sec. 3. The latter, (keeping up tofirst order term in β before summing over n ) yields log P E (4 , β ) (cid:39) β → = 2 ζ (4) β − − (4.13)Now we haveactual asymptotics exp[ (4.11) ] : asymptotics via new trick exp[ (4.13) ] = e − ζ (cid:48) ( − : e − / = 1 : 0 . . (4.14) All even spacetime dimensions
We now proceed to analyse scalar field theory in d = 2 n dimensions for n ≥ . Here weneed a modified version of the theorem, which allows multiple but finite number of polesfor the function D ( d, s ) for Re ( s ) > − C with < C < . Let us illustrate the concept in– 16 – p p / e ⇡ 3 /
43 4 p + ⇣ (
15 + ⇣ ( ◆
The thick dashedline corresponds to exact data, obtained by a direct expansion of the PE. The thin grey curve that isindistinguishable from the data in the upper panel is the asymptotic result.
Lower panel:
Relativeerror between data and the asymptotic result. d = 6 and d = 8 dimensions. Starting from the general Hilbert series, we find, for example,in d = 6 , , using Eq. (4.6) D (6 , s ) = 112 [ ζ ( s − − ζ ( s − D (8 , s ) = 1360 [ ζ ( s − − ζ ( s −
4) + 4 ζ ( s − (4.15)Now these functions converge for Re ( s ) > , and their analytic continuation have polesat α i ( d ) , given by α (6) = 5 , α (6) = 3 α (8) = 7 , α (8) = 5 α (8) = 3 (4.16)The corresponding residues are given by A (6) = 1 / , A (6) = − / A (8) = 1 / , A (8) = − / A (8) = 1 / (4.17)The modified Lemma . reads now log P E ( β ) (cid:39) β → (cid:88) i A i Γ( α i ) ζ ( α i + 1) β − α i − D ( d,
0) log( β ) + D (cid:48) ( d, , (4.18)– 17 –here the sum over poles come from shifting the defining contour from Re ( s ) = 1 + α to Re ( s ) = − C . For d = 6 , we have log P E d =6 ( β ) (cid:39) β → π β − π β + 112 (cid:2) ζ (cid:48) ( − − ζ (cid:48) ( − (cid:3) log P E d =8 ( β ) (cid:39) β → π β − π β + π β + 1360 (cid:2) ζ (cid:48) ( − − ζ (cid:48) ( −
4) + 4 ζ (cid:48) ( − (cid:3) (4.19)The results obtained from our trick (keeping up to first order term before summingover n ) is log P E d =6 ( β ) (cid:39) β → π β − π β + 379302400log P E d =8 ( β ) (cid:39) β → π β − π β + π β − (4.20)In comparison, we have for d = 6 actual asymptotics exp[ (4.19) ] : asymptotics via new trick exp[ (4.20) ] = e [ ζ (cid:48) ( − − ζ (cid:48) ( − : e = 1 : 0 . , (4.21)and for d = 8 actual asymptotics exp[ (4.19) ] : asymptotics via new trick exp[ Eq. (4.20) ] = e [ ζ (cid:48) ( − − ζ (cid:48) ( − ζ (cid:48) ( − : e − = 1 : 0 . . (4.22) For the fermionic Meinardus theorem, we follow similar steps as in the scalar case to massagethe PE to the following form:
P E ( f ) ( β ) = ∞ (cid:89) n =0 (cid:16) q n + d − (cid:17) g ( n,d ) (4.23)where g ( n, d ) is given by the dimension of the following representation of SO ( d ) : g ( n, d ) = dim [ n + 1 / , / , · · · ] (4.24)We focus here on even spacetime dimensions and we have g ( n,
4) = ( n + 1)( n + 2) , g ( n,
6) = 16 (1 + n )(2 + n )(3 + n )(4 + n ) etc. In even dimensions, the canonical dimension of a spin / fermion is half-integer. Thuswe can not directly apply the Meinardus theorem in its original form. Another modificationis needed.We recast the PE in the following way P E ( f ) ( β ) = ∞ (cid:89) k =0 (cid:16) q n + d − (cid:17) g ( n,d ) = ∞ (cid:89) k =1 (cid:16) q k − (cid:17) g ( k − d/ ,d ) = (cid:81) ∞ k =1 (cid:0) √ q ) k (cid:1) ˜ g ( k,d ) (cid:81) ∞ k =1 (cid:0) √ q ) k (cid:1) ˜ g (2 k,d ) , (4.25)– 18 –here we have defined ˜ g ( k, d ) = g (cid:18) k/ / − d , d (cid:19) . Again, in the first equality the shift in the variable k is valid using the fact g ( k − d/ , d ) = 0 for ≤ k < d/ and k ∈ Z + . The function ˜ g is constructed in a way such that ˜ g (2 k − , d ) = g (cid:0) k − / − d , d (cid:1) and the last equality follows. Thus the PE can be writtenas a ratio of two auxiliary PEs that are more amenable to the application of Meinardus’theorem: P E ( f ) ( β ) = P E aux.1 ( β/ P E aux.2 ( β ) , (4.26)where the auxiliary PEs are given by P E aux.1 ( β ) = ∞ (cid:89) k =1 (cid:16) e − kβ (cid:17) g ( k/ / − d ,d ) ,P E aux.2 ( β ) = ∞ (cid:89) k =1 (cid:16) e − kβ (cid:17) g ( k +1 / − d ,d ) . (4.27)We again define D functions for each of these PEs, D ( d, s ) = ∞ (cid:88) k =1 k − s g ( k/ / − d , d ) .D ( d, s ) = ∞ (cid:88) k =1 k − s g ( k + 1 / − d , d ) . (4.28)Let us denote the poles of these functions as α i, and α j, with residues A i, and A j, , wherethe second index , refers to D and D .The analogue of Eq. (4.9) will read log P E aux. ( d, β ) = 12 πı (cid:90) α + ı ∞ α − ı ∞ dsβ − s Γ( s )(1 − − s ) ζ ( s + 1) D ( d, s ) (4.29)where the extra factor (compared to the Bosonic case) comes due to Fermionic nature ofthe PE; this factor changes the behaviour of the integrand around s = 0 —In particular, wedon’t obtain any log β piece. This is the final modification of Meinardus’ theorem we willneed. We find log P E ( f ) ( β →
0) = log
P E aux.1 ( β/ → − log P E aux.2 ( β → (cid:32)(cid:88) i A i, Γ( α i, ) (cid:0) − − α i, (cid:1) ζ ( α i, + 1)( β/ − α i, (cid:33) − (cid:88) j A j, Γ( α j, ) (cid:0) − − α j, (cid:1) ζ ( α j, + 1) β − α j, + [ D ( d, − D ( d, (4.30)For example, in d = 4 we have – 19 – p p / e ⇡ 3 /
43 4 p + ⇣ (
15 + ⇣ ( ◆
The thick dashed linecorresponds to exact data, obtained by a direct expansion of the PE. The thin grey curve that isindistinguishable from the data in the upper panel is the asymptotic result.
Lower panel:
Relativeerror between data and the asymptotic result. D (4 , s ) = (cid:88) k − s g (cid:18) k/ / − d , d (cid:19) = 14 [ ζ ( s − − ζ ( s )] D (4 , s ) = (cid:88) k − s g (cid:18) k + 1 / − d , d (cid:19) = ζ ( s − − ζ ( s ) (4.31)We have poles at α = 3 and α = 1 for both of the PE. The residues are different A , = − A , = 14 , A , = 1 , A , = − (4.32)Hence, we have using Eq. (4.30) log P E ( β →
0) = 74 ζ (4) β − − ζ (2) β − = 7 π β − − π β − (4.33)It is readily checked that this matches exactly with the result obtained from our trick (infact, 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 theall operators in the fermionic PE in d = 4 , ρ (∆) (cid:39) ∆ →∞ (cid:114) / exp (cid:32) (cid:114) π / − (cid:114) π
24 ∆ / (cid:33) (4.34)In Fig. 2, the asymptotic formula above is compared with the exact values (data) fromexpanding 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 curve– 20 –n Fig. 2 includes a factor of a half. We can understand this using intuition from plethoraof examples in D CFT which says that the asymptotic formula is a count of the numberoperators lying in a window of δ = 1 / , centred at ∆ , barring one of the end points if boththe end points have operators. Similar results appeared in the context of the number ofpartitions of integers in [46], and are used in appendix B of [38]. We return to this pointin the discussion section below. For now we simply reflect on the very good agreementbetween data and asymptotic expression shown in Fig. 2: as for the case of the scalar field,is at the level of ten percent at very low mass dimension, and decreases below the per millelevel at dimension ∆ (cid:39) . The hilbert series in β → limit has the following form: H ( β →
0) = exp (cid:34) d − (cid:88) k =0 a k β − k + b log( β ) + c (cid:35) , a d − > . The leading singularity a d − β − ( d − comes from the residue of the rightmost pole of thefunction D ( d, s ) . In even dimension, the residue of the rightmost pole is determined by theleading term in D ( s ) . We consider massless fields of spin j that satisfy the unitarity bound.We recall the definition of D ( s ) : D ( s ) = (cid:88) k − s g ( k − k ∗ ) , g ( k ) = dim k + j, j, j · · · , j (cid:124) (cid:123)(cid:122) (cid:125) d/ − times (4.35)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 g ( k ) = dim k + j, j, j, · · · , j (cid:124) (cid:123)(cid:122) (cid:125) d/ − times (cid:39) k →∞ d − dim [ j, j, · · · , j ] (cid:124) (cid:123)(cid:122) (cid:125) ( d/ − times (4.36)to extract the leading piece D ( d, s ) (cid:51) ∞ (cid:88) k =1 k − s + d − d − dim [ j, j, · · · , j ] (cid:124) (cid:123)(cid:122) (cid:125) ( d/ − times = 2 f ( r, j )Γ( d − ζ ( s − d + 2) , (4.37)where we have defined f ( r, j ) as f ( r, j ) ≡ dim [ j, j, · · · , j ] (cid:124) (cid:123)(cid:122) (cid:125) ( d/ − times . (4.38)One can immediately identify f ( r, j ) with the dimension of spin [ j, j, · · · , j ] representa-tion of the little group SO ( d − with r being the rank of the group SO ( d ) . A few explicit– 21 –orms are given below for low numerical values of r as a function of s : f (2 , j ) = 1 f (3 , j ) = (2 j + 1) f (4 , j ) = 16 (2 j + 3)(2 j + 2)(2 j + 1) f (5 , j ) = 1360 (2 j + 1)(2 j + 2)(2 j + 3) (2 j + 4)(2 j + 5) (4.39)The leading singularity of D ( d, s ) appears at s = d − . Using the leading behavior of f ( r, j ) we find that the residue at the leading singularity of D ( d, s ) is given by f ( r,j )Γ( d − . Asa result, in β → limit, we have log H ( β ) (cid:39) β → f ( r, j ) χ ( statistics ) ζ ( d ) β d − + O ( β d − ) . (4.40)Here, χ ( statistic ) is given by χ ( statistic ) = (cid:40) , for Bosonic − − d +1 , for Fermionic (4.41)The half integer j in even dimension can be handled in a similar way. Furthermore, thefunction 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 scalarsand spin / fermions [47]. Now f ( r, j ) becomes f odd ( r, j ) defined as f odd ( r,
0) = 1 & f odd ( r, /
2) = 2 r − where r = d − . In d = 2 , we have seen that the effect of projecting onto the spin zero sector suppressesoperator growth by a factor of β / . The introduction of IBP further suppresses this growthby a factor of β . Here we will see how things work out in arbitrary dimension. We willalso introduce internal symmetry and project onto singlets of these groups too. We find itmore useful to do a concrete example in d = 4 , rather than making everything very generaland abstract. The idea is to watch out for the patterns in the simple d = 4 calculation anddeduce the results for arbitrary d dimension. d = 4 For our example, we consider complex scalars φ and φ † (charges q b and − q b ) and spin / fermions ψ and ψ † (charges q f and − q f ) There are some subtleties that lead to a factorof two if the theory contains only bosonic degree of freedom that we will discuss in thefollowing section. – 22 –e will be using the SU (2) L × SU (2) R language for the fugacities α and γ of theLorentz group as in e.g. [3]. The Haar measure is (cid:90) dµ Lorentz = (cid:90) dµ SU (2) L ( α ) (cid:90) dµ SU (2) R ( γ ) , (5.1)where e.g. (cid:90) dµ SU (2) L = (cid:73) | α | =1 dαα (1 − α ) . (5.2)The PE for this example EFT, is given by P E = exp (cid:34) ∞ (cid:88) n =1 n q n (1 − q n ) P ( q n , α n , γ n ) (cid:0) χ U (1) ( nq b ω ) + χ U (1) ( − nq b ω ) (cid:1)(cid:35) × exp (cid:34) ∞ (cid:88) n =1 ( − n +1 n q n ( α n + α − n − q n ( γ n + γ − n )) P ( q n , α n , γ n ) χ U (1) ( nq f ω ) (cid:35) × exp (cid:34) ∞ (cid:88) n =1 ( − n +1 n q n ( γ n + γ − n − q n ( α n + α − n )) P ( q n , α n , γ n ) χ U (1) ( − nq f ω ) (cid:35) , (5.3)where the character of the U (1) internal symmetry is, in terms of the angular fugacity w , χ U (1) ( ω ) = e πıω , and where the momentum generating function is P ( q, α, γ ) = (1 − q αγ ) (cid:0) − q αγ − (cid:1) (cid:0) − q α − γ (cid:1) (cid:0) − q γ − α − (cid:1) . (5.4)In the following we will work with fugacities β and ω , defined as q = e − β , α = e πıω , γ = e πıω . Let us analyse the PE using our by now familiar trick from Sec. 3. First we expand inthe β → limit as log( P E ) = (cid:88) n e − nβ n (cid:18) β n + 4 β n + 3 βn − − π ( ω + ω ) β n (cid:19) (cid:0) χ U (1) ( nq b ω ) + χ U (1) ( − nq b ω ) (cid:1) + (cid:88) n ( − n +1 e − nβ n (cid:18) β n + 3 β n + 2 βn + 34 − π ( ω + ω ) β n (cid:19) × (cid:0) χ U (1) ( nq f ω ) + χ U (1) ( − nq f ω ) (cid:1) , (5.5)where we kept only the singular terms in β .The saddle for the ω integral will be determined by the leading term proportional to β − i.e on β − [ g b ( ω ) + g f ( ω )] where g b/f are given by g b ( ω ) = (cid:88) n πnq b ω ) n = π − π q b ω π q b | ω | − π q b ω g f ( ω ) = (cid:88) n ( − n +1 πnq f ω ) n = 7 π − π q f ω π q f ω (5.6)– 23 –ote this term is independent of the spin, depending only on statistics. We keep up tothe quadratic piece and do the integral over ω . We remark that the quadratic piece canbe obtained by expanding χ U (1) ( nq b/f ω ) + χ U (1) ( − nq b/f ω ) first and then summing over n ,and often this is a more practical approach to extract the asymptotics rather doing the sumexplicitly as in Eq. (5.6). However, the way we proceeded above reveals the full periodicnature of the PE in ω . We will assume that U (1) charge is quantized in units of some basecharge, with some field having base charge, and hence we can simple rescale all chargessuch that the base charge is unity —in this case, Eq. (5.6) shows that the saddle is indeed ω = 0 .Expanding, and performing the sum on n in the terms with the angular fugacities, log( P E ) = (cid:88) n e − nβ n (cid:18) β n + 4 β n + 3 βn − (cid:19) ( dim b ) − π ( ω + ω ) β (cid:20) π (cid:21) ( dim b )+ (cid:88) n ( − n +1 e − nβ n (cid:18) β n + 3 β n + 2 βn + 34 (cid:19) ( dim f ) − π ( ω + ω ) β (cid:20) π (cid:21) ( dim f )+ 2 β − (cid:34) − ( dim b ) π q b ω − ( dim f ) 12 π q f ω (cid:35) , (5.7)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 ) (cid:51) (cid:2) Aβ − + Bβ − (cid:3) + Cζ (cid:48) ( − (5.8)where A, B have the universal form already seen above for scalars and spin-half fermioins,namely A = (cid:18) π dim b + 7 π dim f (cid:19) B = − (cid:18) π dim f (cid:19) (5.9)The order one piece is not correctly obtained through the above, but it is the result derivedfrom using the Meinardus theorem, C = dim b . (5.10) If one worked with base charge of integer k > , then there would be k saddles to consider. The resultshould be multiplied by k because of this, but would be suppressed by a factor of k coming from a /Q (where Q = (cid:80) i q i ) after the projection to U (1) singlets, see Eq. (5.12) below. If on the other hand thebase charge was /k , one should use the k -th cover of U (1) , receiving a suppression by a factor of /k inthe measure; this would similarly be cancelled by the /Q after projection. – 24 –rojecting onto Lorentz scalars is done by integrating the above about the saddle withthe Haar measure for SU (2) × SU (2) , (cid:90) ∞−∞ dω µ SU (2) ( ω ) (cid:90) ∞−∞ dω µ SU (2) ( ω ) exp (cid:20) − π ( ω + ω ) β (cid:18) π dim b + 7 π dim f (cid:19)(cid:21) = 91125 β π (cid:18) dim b + 78 dim f (cid:19) − . (5.11)If we had multiple scalars and fermions, with charges q i , the ω dependent piece alwayscontains a β n n ω term multiplied by a sum over charges, P E ( β ) (cid:51) exp − π ω β dim b / (cid:88) i q bi + 12 dim f / (cid:88) i q fi in the above example we have q b = − q b = q b and q f = − q f = q f and this matches withEq. (5.7).Projecting onto U (1) singlets is done in following way: (cid:90) ∞−∞ dω exp (cid:34) − π ω β (cid:32)(cid:88) i q bi + 12 (cid:88) i q fi (cid:33)(cid:35) = (cid:114) π β / QQ ≡ (cid:32)(cid:88) i q bi + 12 (cid:88) i q fi (cid:33) (5.12) 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 pointapproximations described in the previous section to project out singlets of general internalsymmetry groups, G . The formula reads, H ( β ) (cid:39) β → (cid:104) Kβ dim g (cid:105) (cid:34) β π (cid:18) dim B + 78 dim f (cid:19) − (cid:35) (cid:2) β (cid:3) × exp (cid:20) Aβ − + Bβ − + Cζ (cid:48) ( −
2) + D log (cid:18) β π (cid:19) + higher spin>1 (cid:21) , (5.13)where A = (cid:18) π dim B + 7 π dim f (cid:19) ,B = − (cid:18) π dim / + π dim (cid:19) ,C = dim B ,D = − dim . (5.14)– 25 –ere dim B counts bosonic degrees of freedom (dof), dim f counts fermionic dof, dim / counts spin-1/2 dof, dim counts spin-1 dof, and dim g counts the dimension of the internalsymmetry group G . The exact value of K can be determined case by case; it depends onthe symmetry group at hand, and also on the representations of the fields transformingunder the symmetry. If dim f = 0 , the result should be multiplied by a factor of two (seefollowing section)Let us discuss each of the terms in Eq. (5.13), working backwards, from right to left.• The leading piece (exponential term) comes from the counting of all degrees of freedomin the EFT, turning off fugacities characterizing spin and internal symmetry.• Imposing IBP provides a suppression of β .• Projecting onto Lorentz scalars provides a suppression of β , and an order one mul-tiplicative 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 that is dependent on the dimension of G , and an order one multiplicative piece K ,described above. 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 offreedom in the EFT, turning off fugacities characterizing spin and internal symmetry(see also 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 . Thefunction g ( d, G ) is given by (cid:40) g ( d, G ) = d +12 , where G = Lorentz ,g ( d, G ) = d − , where G = Global symmetry . (5.15)That is, we find – Projecting onto Lorentz scalars implies a suppression by β d ( d − / (irrespectiveof the spin of the fields). – Projecting on to gauge singlet implies suppression by ( β ( d − / ) (cid:96) , where thequantity (cid:96) can be extracted from the volume measure of the gauge group in thefugacity going to limit. For a continuous Lie group G , this identifies (cid:96) = dim g as the dimension of the Lie algebra.– 26 –he 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 theexponent. The (unquantified) polynomial suppression of IBP projection and of projectiononto singlets of the Lorentz was pointed out in [28]. For a purely Bosonic theory, one needs to multiply the result presented Eq. (5.13) by afactor of 2. To be precise, if we view H ( β → as a function of dim f , we have H ( β → , dim f = 0) = 2 (cid:18) lim dim f (cid:55)→ [ H ( β → , dim f ] (cid:19) (6.1)The above signals phase transition-like behaviour, and arises because of the presenceof extra saddles that appear in the limit. The symmetry reason behind such phenomenonis that the presence of fermions break the symmetry ( ω , ω ) (cid:55)→ ( ω + , ω + ) , i.e. whendim f = 0 , there is a symmetry enhancement. In terms of α = e πıω and γ = e πıω , thesymmetry is implemented by ( α, γ ) (cid:55)→ ( − α, − γ ) . While the bosonic PE is symmetric underthis transformation, the fermionic (or the ones with both bosons and fermions) ones arenot. In the α, γ variables there are two saddles to keep track of: when α = γ = ± . Theapparent discontinuity in Eq. (6.1) comes about because for dim f = 0 , we have degeneratesaddles but when dim f (cid:54) = 0 , the symmetry is broken, and so is the degeneracy. Conse-quently, one of the saddles gets suppressed. This phenomenon happens universally in alldimensions, see App. C for the details. Keeping both the saddles restore the continuity of H ( β ) as a function of number of fermions. We explicitly show this in Sec. 6.2 for d = 4 . 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 isto clarify that the both will provide us with the same answer. If we include fermions inthe theory, then we need to work with the double cover of SO (4) . This is is taken up inApp. C, and generalized so as to studyng this phenomena in arbitrary d . Take I—The SU (2) × SU (2) way H ( β ) is given by H ( β ) = (cid:73) dα (cid:73) dγ (cid:20) αγ (1 − α )(1 − γ ) (cid:21) exp (cid:34) ∞ (cid:88) n =1 e − nβ n − e − nβ P ( e − nβ , α n , γ n ) (cid:35) , (6.2)with P defined in Eq. (5.4)Now note that in the β → limit, the singularity appears when αγ = α/γ = 1 . Thisadmits two solutions α = γ = ± . (6.3)– 27 – igure 3 . Calculation using SU (2) × SU (2) . The figure depicts saddles on ( ω , ω ) plane. Thecenter of the square is at (0 , . The corners are at (1 / , / , (1 / , − / , ( − / , / , ( − / , / .For the bosonic case, all the saddles contribute. When we add fermions in the mix, the greencolored saddles at (0 , ± / , ( ± / , don’t contribute anymore, leading to an overall factor of / .We have drawn circular regions arounds the saddle to denote how much the fluctuation around thesaddle contributes. For example, each of the corner ones contributes one quarter of the center one. The angular fugacities ω i are defined as α = e πıω , γ = e πıω . (6.4)As α, γ traverse the circle once, we have a square region swept out in the ( ω , ω ) planewith center at (0 , and vertices at ( ± / , ± / , ( ± / , ∓ / . From Eq. (6.3) it followsthat in the ( ω , ω ) plane the saddles exist at the points (see fig. 3) ( ω , ω ) = { (0 , , (1 / , / , (1 / , − / , ( − / , / , ( − / , − / } . (6.5)We have already considered the (0 , saddle in the previous section. Now there are fourmore saddle points, which are the corners of square on ( ω , ω ) plane over which we are doingthe ω i integrals. The fluctuation around this each corner provides / of the contributioncoming from the fluctuation around saddle (0 , . This is self evident because a full circlearound (0 , contributes to fluctuation integral whereas, only a quarter chunk of the circlecontributes for the corner points. The leading value of the integral from all of the saddlesis the same. So we have following expression for H ( β ) H ( β ) = (cid:88) saddles P E ( β, ω i = saddle value ) × fluctuation contribution = exp (cid:20) π β − + ζ (cid:48) ( − (cid:21) (1 + 4 × / (cid:18)(cid:90) ∞−∞ dω µ ( ω ) (cid:90) ∞−∞ dω µ ( ω ) exp (cid:20) − π β ( ω + ω ) (cid:21)(cid:19) = 91125 β π exp (cid:20) π β − + ζ (cid:48) ( − (cid:21) . (6.6)When we add fermions in the mix, the saddle α = γ = − produces a suppressed contribu-tion at leading order (these are depicted by the dashed green lines in fig. 3). This is evidentfrom the character of fermion being ( α + α − ) − q ( γ + γ − ) . Thus we recover the resultpresented in Eq. (5.13). – 28 – ake II—The SO (4) Way H ( β ) is given by H ( β ) = (cid:73) dx (cid:73) dx µ SO (4) ( x i ) exp (cid:34) ∞ (cid:88) n =1 e − nβ n − e − nβ P ( nβ, x ni ) (cid:35) (6.7)where µ SO (4) ( x i ) = 1 x x (1 − x x )(1 − x /x ) , (6.8)and P ( β, x i ) = (cid:16) − e − β x (cid:17) (cid:16) − e − β x − (cid:17) (cid:16) − e − β x (cid:17) (cid:16) − e − β x − (cid:17) . (6.9)Now note that in the β → limit, the singularity appears when x = x = 1 . In termsof the angular variable ˜ ω i (where x i = e πı ˜ ω i ) we have the following saddle (the only one) (˜ ω , ˜ ω ) = (0 , (6.10) H ( β ) = P E ( β, ˜ ω i = saddle value ) × fluctuation contribution = exp (cid:20) π β − + ζ (cid:48) ( − (cid:21) (cid:18)(cid:90) ∞−∞ d ˜ ω (cid:90) ∞−∞ d ˜ ω µ SO (4) ( ω i ) exp (cid:20) − π β ( ω + ω ) (cid:21)(cid:19) = 91125 β π exp (cid:20) π β − + ζ (cid:48) ( − (cid:21) (6.11)Thus Eq. (6.6) matches with Eq. (6.11). 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. In this section, we take a closer look at the β → behaviour of Hilbert series as a functionof dim f . In particular, we want to inspect how the following comes about H ( β → , dim f = 0) = 2 (cid:18) lim dim f (cid:55)→ [ H ( β → , dim f ] (cid:19) (6.12)Let us consider the Hilbert series for b scalars and f spin-1/2 fermions, H ( β, f ) = (cid:73) dα (cid:73) dγ (cid:20) αγ (1 − α )(1 − γ ) (cid:21) × exp (cid:34) ∞ (cid:88) n =1 n be − nβ (1 − e − nβ ) + ( − n +1 f e − nβ/ (cid:2) ( α n + α − n ) − e − nβ ( γ n + γ − n ) (cid:3) P ( e − nβ , α n , γ n ) (cid:35) (6.13)In the β → limit, the singularity appears when α = γ = ± . When f (cid:54) = 0 , the leadingsingularity is α = γ = 1 , and the subleading one is α = γ = − . Let us keep them both.– 29 –ield SU (2) L SU (2) R SU (3) c SU (2) W U (1) Y Q / L − / u c − / d c / e c G L W L B L H / 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. Now we have (we denote the Hilbert series as H (cid:48) to distinguish it from the one where wetake only the leading saddle) H (cid:48) ( β → , f ) = 91125 β π ( b + 7 / f ) exp (cid:20) π ( b + 7 / f )45 β − f π β + bζ (cid:48) ( − (cid:21) + 91125 β π ( b − f ) exp (cid:20) π ( b − f )45 β + π f β + bζ (cid:48) ( − (cid:21) (6.14)Now one can see that for f = 0 the two saddles coincide and instead of Eq. (6.12) we have H (cid:48) ( β → , f →
0) = H (cid:48) ( β → , f = 0) (6.15)So the apparent discontinuity/phase transition in Eq. (6.12) gets resolved by adding contri-bution 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 includeit. When taking projecting to singlets of some internal symmetry group, similar saddle pointsubtleties can arise. We already mentioned one in the case of a U (1) symmetry in footnote 4.Another example: consider the Hilbert series for a single field transforming in the adjointrepresentation 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 suppressesall but one saddle (at the centre of the hypercube swept out by the angular fugacities),and a similar discontinuity to the one above occurs. We leave a detailed study of suchdiscontinuities in Hilbert series to future work. We finally turn to applying our results to a a complicated phenomenological EFT, namelythe SMEFT. This has field content shown in Table 1 (conjugates of all fields must also be– 30 –ncluded). We will consider this theory with N g copies of fermionic generations (the SMhas N g = 3 ). Hilbert series methodology was applied to this theory in [3] to systematicallyenumerate operators at mass dimension eight and above (results up to mass dimension 15were presented).Let us assemble the components that form the leading behaviour of the PE for theSMEFT in the β → limit. Leading exponential
The leading exponential piece can be read straight from Eq. (5.13), log
P E ( β ) (cid:51) (cid:2) Aβ − + Bβ − (cid:3) + Cζ (cid:48) ( −
2) + D log( β/ π ) (7.1)with A = (cid:18) π dim B + 7 π dim f (cid:19) , B = − (cid:18) π dim / + π dim (cid:19) , C = dim B , D = − dim . (7.2)where from Table 1 (remembering to count the dof in the gauge group representations, andto include the conjugate fields) we have,dim B = 28 , dim f = 30 N g , dim = 24 , dim / = 30 N g . (7.3) Projection onto Lorentz scalars and IBP
These pieces can also be read straight from Eq. (5.13), and we find a suppression by a factorof β π (cid:18) dim B + 78 dim f (cid:19) − (7.4)for the projection to scalars, and factor of β for from the IBP projector. Projection onto U (1) singlets We have the suppression found in Eq. (5.12), by a factor of (cid:114) π β / Q , where Q ≡ (cid:32)(cid:88) i q bi + 12 (cid:88) i q fi (cid:33) (7.5)For the SMEFT, scaling the charges that appear in Table 1 by a factor of 6 (see footnote 4),we have Q = 36 + 120 N g . (7.6) Projection onto SU (2) singlets There is a the single saddle point at ω = 0 , and the integral over fluctuation is found to be (cid:90) ∞−∞ dµ SU (2) ( ω ) exp (cid:20) − π ω β (cid:18) dim fund B + 12 dim fund f + 4 dim adj b (cid:19)(cid:21) = 3 (cid:114) π (cid:18) dim fund B + 12 dim fund f + 4 dim adj B (cid:19) − / β / (7.7)– 31 –ere dim fund B counts the number of bosonic dof in the fundamental representation (rep) of SU (2) ; dim fund f counts the number of fermionic dof in the fundamental rep of SU (2) ; dim adj B counts the number of bosonic dof in the adjoint rep of SU (2) . For the SMEFT we havedim fund B = 2 , dim fund f = 8 N g , dim adj B = 2 . (7.8) Projection onto SU (3) singlets There is a the single saddle point at k = k = 0 , and the integral over fluctuation is foundto be (cid:90) ∞−∞ dµ SU (3) ( k , k ) exp (cid:20) − π ( k − k k + k )6 β (cid:16) dim fund f + dim fund ¯ f + 12 dim adj b (cid:17)(cid:21) = 432 √ β π (cid:16) dim fund f + dim fund ¯ f + 12 dim adj B (cid:17) − , (7.9)where dim fund B now counts the number of dof in the fundamental representation of SU (3) ,etc.. For the SMEFT, we havedim fund f = dim fund ¯ f = 4 N g , dim adj B = 2 . (7.10) Assembling the above components and the IBP suppression, we can construct the Hilbertseries for the SMEFT in the β → limit: H ( β ) ∼ β → β exp (cid:16) π β (15 N g + 16) − π β (5 N g + 32) + 28 ζ (cid:48) ( − (cid:17) π ( N g + 3) (2 N g + 5) / (cid:112) N g + 3(15 N g + 16) (7.11)Performing the inverse Laplace transform, one obtains the asymptotic growth of oper-ators in the SMEFT ρ (∆) ∼ ∆ →∞ N exp π √ (cid:114) N g + 11215 ∆ / − π (5 N g + 32)4 √ (cid:113) N g + ∆ / + 28 ζ (cid:48) ( − , (7.12)where N is given by N = 27783 (cid:0) (cid:1) / / π (15 N g + 16) / √ / ( N g + 3) (2 N g + 5) / (cid:112) N g + 3 (7.13)We note that in performing the inverse Laplace transform, there is a simple way toprobe the effect of higher order terms in the asymptotic series we present above. Thetransform is (cid:90) dβ H ( β ) e β ∆ (7.14)– 32 – p p / e ⇡ 3 /
43 4 p + ⇣ (
15 + ⇣ ( ◆
24 / !
43 4 p + ⇣ (
15 + ⇣ ( ◆
T.M. and S.P. thank UCSD for support and hospitality where this work was initiated. Wethank Aneesh Manohar and Masahito Yamazaki for comments on the manuscript. T.M.is grateful to Brian Henning, Xiaochuan Lu, and Hitoshi Murayama for many importantand fruitful discussions about Hilbert series. In obtaining the exact results for the SMEFTbeyond mass dimension 15, we acknowledge use of the form computer code accompanying– 36 –he paper [71]. T.M. is supported by the World Premier International Research Center Ini-tiative (WPI) MEXT, Japan, and by JSPS KAKENHI grants JP18K13533, JP19H05810,JP20H01896 and JP20H00153. S.P. acknowledges the support from Ambrose Monell Foun-dation 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 exampleof a single real scalar field in d = 4 . Without EOM imposed, the PE is P E
No EOM = exp (cid:34) ∞ (cid:88) n =1 e − βn n − e − βn ) (cid:35) (A.1)Applying the trick introduced in Sec. 3, keeping only singular terms in β , we find lim β → P E
No EOM ( β ) = exp (cid:20) ζ (5) β + π β + ζ (3)3 β + 19 log( β )720 − (cid:21) . (A.2)Next consider a more general partition P E ( K, β ) = exp (cid:34) ∞ (cid:88) n =1 e − βn n − e − Kβn (1 − e − βn ) (cid:35) (A.3)with the case K = 2 corresponding to the PE for a real scalar field in d = 4 with EOMimposed. Again keeping only singular terms in β in applying the trick of Sec. 3, the leadingbehaviour as β → is, lim β → P E ( K, β ) = exp (cid:20) π K β − β ( K − Kζ (3) + 136 β π ( K − K − K + 124 ( K − K log( β ) + 172 (cid:0) − K + 45 K − K (cid:1)(cid:21) . (A.4)The behaviour of this general partition function is exponentially suppressed for any finitevalue of K compared to the PE for the scalar field without EOM imposed. Note that allsub-leading terms with β dependence vanish when setting K = 2 to recover the physicalcase of imposing EOM lim β → P E ( K = 2 , β ) = exp (cid:20) π β − (cid:21) . (A.5) B Meinardus theorem in arbitrary d , j In this appendix we collect results not presented in the main text that are obtained bygeneralizing Meinardus’ theorem and applying it to EFTs in spacetime dimensions d andwith fields of spin j . – 37 – .1 Scalar field theory in d = 2 n + 1 for n ≥ In d = 2 n + 1 dimension, the canonical dimension of the scalar field is half integer. Thus wecan not directly apply Meinardus theorem (or Lemma . or its modified form Eq. (4.18)).A different kind of modification is needed. This is somewhat analogous to the case offermions in even dimensions.We recast the PE in following way: P E ( β ) = ∞ (cid:89) n =0 (cid:16) − q n + d − (cid:17) − f ( n,d ) = ∞ (cid:89) k =1 (cid:16) − q k − / (cid:17) − f ( k − ( d − / ,d ) = (cid:81) ∞ k =1 (cid:0) − ( √ q ) k (cid:1) − ˜ f ( k,d ) (cid:81) ∞ k =1 (cid:0) − ( √ q ) k (cid:1) − ˜ f (2 k,d ) (B.1)where f ( n, d ) is given by the symmetric spin n respresentation of SO ( d ) i.e. f ( n, d ) = dim [ n, , , · · · ] . (B.2)In the second step, we performed a change of variable and defined ˜ f ( k, d ) = f ( k/ / − d − , d ) . This is constructed in a way such that ˜ f (2 k − , d ) = f ( k − ( d − / , d ) . Note, inthe second equality, shift in k is valid since f ( k − ( d − / , d ) = 0 for ≤ k < ( d − / and k being integer. Thus the PE can be written as a ratio of two auxiliary PE, on whichone can apply the modified Meinardus theorem: P E ( β ) = P E aux ( β/ P E aux ( β ) (B.3) P E aux ( β ) = ∞ (cid:89) k =1 (cid:16) − e − kβ (cid:17) − f ( k/ / − d − ,d ) ,P E aux ( β ) = ∞ (cid:89) k =1 (cid:16) − e − kβ (cid:17) − f ( k +1 / − d − ,d ) . (B.4)The D functions corresponding to the auxiliary PEs are found to be D ( d, s ) = (cid:88) k k − s f (cid:18) k/ / − d − , d (cid:19) ,D ( d, s ) = (cid:88) k k − s f (cid:18) k + 1 / − d − , d (cid:19) , (B.5)and let us denote the poles as α i, , α j, with residues A i, and A j, where the index and refer to the P E aux and P E aux . In terms of these variables we have log P E ( β →
0) = log
P E aux ( β/ → − log P E aux ( β → (cid:32)(cid:88) i A i, Γ( α i, ) ζ ( α i, + 1)( β/ − α i, (cid:33) − (cid:88) j A j, Γ( α j, ) ζ ( α j, + 1) β − α j, − [ D ( d,
0) log( β/ − D ( d,
0) log( β )] + D (cid:48) ( d, − D (cid:48) ( d, (B.6)– 38 – =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 D (3 , s ) = ζ ( s − , D (3 , s ) = 2 ζ ( s − ,α , (3) = 2 , α , (3) = 2 A , (3) = 1 , A , (3) = 2 (B.7)leading to Eq. (3.25). d=5 For d = 5 we have D (5 , s ) = 124 [ ζ ( s − − ζ ( s − α , (3) = 4 , α , (3) = 2 A , (3) = 1 / , A , (3) = − / (B.8)and D (5 , s ) = 112 [4 ζ ( s − − ζ ( s − ,α , (5) = 4 , α , (5) = 2 A , (5) = 1 / , A , (3 , s ) = − / (B.9)Applying Eq. (B.6) we find log P E d =5 ( β →
0) = 2 ζ (5) β − ζ (3)12 β + 17 log( β )2880 + 112880 log(2) − ζ (cid:48) ( −
3) + 124 ζ (cid:48) ( − (B.10)Using the trick, one can find that log P E trick d =5 ( β →
0) = 2 ζ (5) β − ζ (3)12 β + 17 log( β )2880 + 68 log (cid:0) (cid:1) − (B.11)Now we have actual asymptotics exp[ (B.10) ] : asymptotics via new trick exp[ (B.11) ] = 1 : 0 . . (B.12) B.2 Fermionic field theory in d = 2 n + 1 for n ≥ The canonical dimension of the fermionic field is an integer. This mimics the case of scalarsin even dimension. Thus we have
P E ( β ) = ∞ (cid:89) n =0 (cid:16) q n + d − (cid:17) g ( n,d ) = ∞ (cid:89) k =1 (cid:16) q k (cid:17) g ( k − ( d − / ,d ) (B.13) g ( n, d ) = dim [ n + 1 / , / , · · · ] , (B.14)– 39 –or example g ( n,
3) = 2( n + 1) , g ( n,
5) = 23 ( n + 1)( n + 2)( n + 3) . The limit in k can be shifted to k = 1 in the above because g ( k − ( d − / , d ) = 0 for k < ( d − / and k ∈ Z + .Let us apply this on spin / fermions in d = 3 and d = 5 . Explicitly we have D (3 , s ) = 2 ζ ( s − , α (3) = 2 , A (3) = 2 D (5 , s ) = 23 [ ζ ( s − − ζ ( s − , α (5) = 4 , α (5) = 2 , A (5) = − A (5) = 23 (B.15)Using the results following from Eq. (4.29), we obtain log P E ( f ) d =3 ( β ) (cid:39) β → ζ (3) β − − log 26log P E ( f ) d =5 ( β ) (cid:39) β → ζ (5)4 β − ζ (3)2 β + 11180 log(2) (B.16)Here we have put in the supersprcipt f explicitly to denote that these are fermionic PE. Itcan be 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 theunitarity bound. Fields of spin j have dimension j + 1 . So all the bosonic fields haveinteger dimension while the fermionic field have half-integer dimension. Bosonic fields j ∈ Z The PE is given by
P E ( j, β ) = exp (cid:34) ∞ (cid:88) n =1 e − ( j +1) nβ n (cid:18) j + 1 − je − nβ + (2 j − e − nβ (1 − e − nβ ) (cid:19)(cid:35) = ∞ (cid:89) n =0 (cid:16) − e − ( n + j +1) β (cid:17) − ( n +1)( n +2 j +1) = (cid:81) ∞ k =1 (cid:0) − e − kβ (cid:1) − ( k − j ) (cid:81) jk =1 (1 − e − kβ ) − ( k − j ) , (B.17)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 → limit: j (cid:89) k =1 (cid:16) − e − kβ (cid:17) − ( k − j ) (cid:39) β → j (cid:89) k =1 ( kβ ) j − k = β ( j − j (4 j +1) j (cid:89) k =1 ( k ) j − k . – 40 –he numerator can be handled using Meinardus approach. The relevant function D isgiven by D ( j, s ) = ∞ (cid:88) k =1 k − s ( k − j ) = ζ ( s − − j ζ ( s ) (B.18)Hence we have log P E ( j, β ) (cid:39) β → Γ(3) ζ (4) β − − j Γ(1) ζ (2) β − − j β + ζ (cid:48) ( −
2) + 12 j log(2 π ) −
16 ( j − j (4 j + 1) log β − j (cid:88) k =1 ( j − k ) log k = π β − π j β − j (2 j − j + 1) log( β ) + ζ (cid:48) ( −
2) + 12 j log(2 π ) − j (cid:88) k =1 ( j − k ) log k (B.19) Fermionic fields j ∈ Z / Z The PE is given by
P E ( j, β ) = exp (cid:34) ∞ (cid:88) n =1 ( − n +1 e − ( j +1) nβ n (cid:18) j + 1 − je − nβ + (2 j − e − nβ (1 − e − nβ ) (cid:19)(cid:35) = ∞ (cid:89) n =0 (cid:16) e − ( n + j +1) β (cid:17) ( n +1)( n +2 j +1) = (cid:81) ∞ k =1 (cid:0) e − ( k − / β (cid:1) ( k − / − j (cid:81) j +1 / k =1 (cid:0) e − ( k − / β (cid:1) ( k − / − j = j +1 / (cid:89) k =1 (cid:16) e − ( k − / β (cid:17) j − ( k − / (cid:81) ∞ k =1 (cid:0) e − kβ/ (cid:1) k / − j (cid:81) ∞ k =1 (1 + e − kβ ) k − j (B.20)Thus we have log P E ( j, β ) (cid:39) β → j (2 j − j + 1) log 2 + lim β → log P E aux ( β/ P E aux ( β ) (B.21)We apply Meinardus theorem on P E aux and P E aux . Now we have D ( j, s ) = 14 (cid:2) ζ ( s − − j ζ ( s ) (cid:3) , D ( j, s ) = (cid:2) ζ ( s − − j ζ ( s ) (cid:3) (B.22)Thus we have log P E ( j, β ) (cid:39) β → π β − π j β + 16 j (2 j − j + 1) log 2 (B.23) C Combinatorical geometry for SO ( d ) saddle points in arbitrary dimen-sion 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.– 41 – igure 6 . Calculation using the double cover of SO (4) . The figure depicts saddles on (˜ ω , ˜ ω ) plane. The center of the square is at (0 , . The corners are at (1 , , (1 , − , ( − , , ( − , . Forthe bosonic case, all the saddles contribute. When we add fermions in the mix, the green colouredsaddles at (0 , ± , ( ± , don’t contribute anymore, leading to an overall factor of / . We havedrawn circular regions arounds the saddle to denote how much the fluctuation around the saddlecontributes. For example, each of the corner ones contributes one quarter of the center one. 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 H ( β ) = 14 (cid:73) twice dx (cid:73) twice dx µ SO (4) ( x i ) exp (cid:34) ∞ (cid:88) n =1 e − nβ n − e − nβ P ( nβ, x ni ) (cid:35) , (C.1)where the factor of in front of the measure normalizes it to unity. As x , x traverses thecircle twice, we have a square region swept out on (˜ ω , ˜ ω ) plane with center at (0 , andveritces at ( ± , ± , ( ± , ∓ ), and the positions of the saddles are at ( ω , ω ) = { (0 , , (0 , , (0 , − , (1 , , ( − , , (1 , , (1 , − , ( − , , ( − , − } (C.2)Again we have to sum over the saddles along with the fluctuations around it. Compared tothe contribution coming from the fluctuation around the middle saddle (0 , , the cornersones i.e ( ± , , (1 , ± produces / of the contribution, while each of the (0 , ± , ( ± , produces / , see Fig. 6. So we have (1 + 4 × / × /
2) = 4 coming from all the saddlesand this kills the / normalization factor appearing in front of the measure as a result ofgoing to the double cover. In this way it matches with Eq. (6.6) and Eq. (6.11).Adding fermions kills off the saddle at (0 , ± , ( ± , , so now we have the center onesand the corner ones giving a contribution of × / ; we must include again theoverall factor of / in the defining integral. Thus we land up with an overall factor of 1/2and reproduce the result in Eq. (5.13). C.2 Bosonic theory
There is only one saddle at ˜ ω i = 0 for the following integral: H ( β ) = (cid:73) (cid:89) i dx i µ SO ( d ) ( x i ) exp (cid:34) ∞ (cid:88) n =1 e − nβ n − e − nβ P ( nβ, x ni ) (cid:35) (C.3)– 42 – igure 7 . Calculation using double cover of SO (6) : the figure depicts saddles on (˜ ω , ˜ ω , ˜ ω )3 − space. The center of the cube is at (0 , , marked red and circled. The corners are at ( a , a , a ) where a i = ± . 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 dimensional object edge)and green/black colored saddles at the corners (corresponding to dimensional object cube) don’tcontribute anymore, leading to a factor of / . The saddles corresponding to the zero dimensionalcenter and midpoints of the faces ( dimensional object) always contribute. Fluctuation around thesaddles corresponding to k dimensional object contributes k − compared to that of the center one. If we use the double cover then we have following expression in even d dimensions H ( β ) = 12 d/ (cid:73) twice (cid:89) i dx i µ SO ( d ) ( x i ) exp (cid:34) ∞ (cid:88) n =1 e − nβ n − e − nβ P ( nβ, x ni ) (cid:35) (C.4)We are going to show the extra d/ factor gets killed by presence of other saddles. Now thesaddle points are distributed over a hypercube of length centered at (0 , , · · · ) , aligned tothe axes. There are d/ saddle points, they are given bySaddles = { ( a , a , · · · a d/ ) : a i ∈ { , ± }} (C.5)For d = 4 , we have / = 9 points, they comes in three kinds (see fig Fig. 6 andFig. 7). One point is at the center, four of them are at the midpoints of edges, and fourof them are at the corners. The leading contribution from these saddles are the same,but the sub-leading contribution coming from the fluctuation around these three kinds ofsaddles is different. Geometrically, one can map these three kind of saddles with existenceof geometrical objects, which are embedded in the dimensional square (the integrationregion). We have exactly three such kinds: zero, one or two dimensional objects. The zerodimensional objects are the corners, the one dimensional objects are the edges, and thetwo dimension object is the full square. Thus the saddles are mapped to two dimensionalobject, one dimensional objects and zero dimensional objects. This pattern survives– 43 –n higher dimension. There are d/ different type of saddles, which can be mapped to d/ different type of dimensional objects which can embedded.The number of k dimensional objects inside a d/ dimensional object is given by theexpression d/ − k Binomial [ d/ , k ] . This follows because the k dimensional object can befound by setting k entries of { a , a , · · · a d/ } to and setting rest of them to ± . The k entries then scan be chosen in Binomial [ d/ , k ] ways and rest of them can be filled in d/ − k ways, leading to the above expression. Also, note that the compared to the contributioncoming from the fluctuation around the saddle at the center (which is mapped to the d/ dimensional object) saddles corresponding to the k dimensional object contribute a factor of k − d/ . This can be obtained if we think coordinate-wise, the coordinates set at contributecompletely to the fluctuation, while the ones at ± contributes / of the full contribution,thus leading to the k − d/ suppression. We have a total contribution compared to the singlecover d/ d/ (cid:88) k =0 d/ − k Binomial [ d/ , k ] (cid:124) (cid:123)(cid:122) (cid:125) of saddles k − d/ = 1 (C.6)Now if we add fermions to the mix, some of the saddles will get suppressed at leadingorder 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 / , · · · / case in even dimension. The Hilbertseries is given by H ( β ) = 12 d/ (cid:73) twice (cid:89) i dx i µ SO ( d ) ( x i ) P E ( x i , β ) P E ≡ exp (cid:34) ∞ (cid:88) n =1 ( − n +1 e − nβ ( d − / n χ [1 / , ··· / , +1 / − χ [1 / , ··· / , − / e − nβ P ( nβ, x ni ) (cid:35) (C.7)The fermionic characters have branch-cuts. Thus the saddles that contribute to theleading order necessarily have the following form { a , a , · · · a d/ : a i ∈ { , ± } & of ± entries is even } (C.8)Thus compared to Eq. (C.6), we only sum over even k i.e. we have total contribution d/ (cid:98) d/ (cid:99) (cid:88) k =0 d/ − k Binomial [ d/ , k ] (cid:124) (cid:123)(cid:122) (cid:125) of saddles k − d/ = 12 (C.9)Thus we arrive at H ( β → , dim f = 0) = 2 (cid:18) lim dim f (cid:55)→ [ H ( β → , dim f ] (cid:19) . (C.10)– 44 – eferences [1] B. Grzadkowski, M. Iskrzynski, M. Misiak, and J. Rosiek, Dimension-Six Terms in theStandard Model Lagrangian , JHEP (2010) 085, [ arXiv:1008.4884 ].[2] R. Alonso, E. E. Jenkins, A. V. Manohar, and M. Trott, Renormalization Group Evolution ofthe Standard Model Dimension Six Operators III: Gauge Coupling Dependence andPhenomenology , JHEP (2014) 159, [ arXiv:1312.2014 ].[3] B. Henning, X. Lu, T. Melia, and H. Murayama,
2, 84, 30, 993, 560, 15456, 11962, 261485,...: Higher dimension operators in the SM EFT , JHEP (2017) 016, [ arXiv:1512.03433 ].[Erratum: JHEP 09, 019 (2019)].[4] R. J. Eden, P. V. Landshoff, D. I. Olive, and J. C. Polkinghorne, The analytic S-matrix .Cambridge Univ. Press, Cambridge, 1966.[5] R. Ellis, Z. Kunszt, K. Melnikov, and G. Zanderighi,
One-loop calculations in quantum fieldtheory: from Feynman diagrams to unitarity cuts , Phys. Rept. (2012) 141–250,[ arXiv:1105.4319 ].[6] H. Elvang and Y.-t. Huang,
Scattering Amplitudes , arXiv:1308.1697 .[7] R. Rattazzi, V. S. Rychkov, E. Tonni, and A. Vichi, Bounding scalar operator dimensions in4D CFT , JHEP (2008) 031, [ arXiv:0807.0004 ].[8] M. F. Paulos, J. Penedones, J. Toledo, B. C. van Rees, and P. Vieira, The S-matrixbootstrap. Part I: QFT in AdS , JHEP (2017) 133, [ arXiv:1607.06109 ].[9] M. F. Paulos, J. Penedones, J. Toledo, B. C. van Rees, and P. Vieira, The S-matrix bootstrapII: two dimensional amplitudes , JHEP (2017) 143, [ arXiv:1607.06110 ].[10] M. F. Paulos, J. Penedones, J. Toledo, B. C. van Rees, and P. Vieira, The S-matrix bootstrap.Part III: higher dimensional amplitudes , JHEP (2019) 040, [ arXiv:1708.06765 ].[11] L. Córdova and P. Vieira, Adding flavour to the S-matrix bootstrap , JHEP (2018) 063,[ arXiv:1805.11143 ].[12] D. Mazac and M. F. Paulos, The analytic functional bootstrap. Part I: 1D CFTs and 2DS-matrices , JHEP (2019) 162, [ arXiv:1803.10233 ].[13] D. Mazac and M. F. Paulos, The analytic functional bootstrap. Part II. Natural bases for thecrossing equation , JHEP (2019) 163, [ arXiv:1811.10646 ].[14] L. Córdova, Y. He, M. Kruczenski, and P. Vieira, The O(N) S-matrix Monolith , JHEP (2020) 142, [ arXiv:1909.06495 ].[15] D. Karateev, S. Kuhn, and J. a. Penedones, Bootstrapping Massive Quantum Field Theories , JHEP (2020) 035, [ arXiv:1912.08940 ].[16] M. Correia, A. Sever, and A. Zhiboedov, An Analytical Toolkit for the S-matrix Bootstrap , arXiv:2006.08221 .[17] A. Homrich, J. a. Penedones, J. Toledo, B. C. van Rees, and P. Vieira, The S-matrixBootstrap IV: Multiple Amplitudes , JHEP (2019) 076, [ arXiv:1905.06905 ].[18] S. Komatsu, M. F. Paulos, B. C. Van Rees, and X. Zhao, Landau diagrams in AdS andS-matrices from conformal correlators , arXiv:2007.13745 .[19] S. Caron-Huot, D. Mazac, L. Rastelli, and D. Simmons-Duffin, Dispersive CFT Sum Rules , arXiv:2008.04931 . – 45 –
20] S. Benvenuti, B. Feng, A. Hanany, and Y.-H. He,
Counting BPS Operators in GaugeTheories: Quivers, Syzygies and Plethystics , JHEP (2007) 050, [ hep-th/0608050 ].[21] B. Feng, A. Hanany, and Y.-H. He, Counting gauge invariants: The Plethystic program , JHEP (2007) 090, [ hep-th/0701063 ].[22] J. Gray, A. Hanany, Y.-H. He, V. Jejjala, and N. Mekareeya, SQCD: A Geometric Apercu , JHEP (2008) 099, [ arXiv:0803.4257 ].[23] E. E. Jenkins and A. V. Manohar, Algebraic Structure of Lepton and Quark FlavorInvariants and CP Violation , JHEP (2009) 094, [ arXiv:0907.4763 ].[24] A. Hanany, E. E. Jenkins, A. V. Manohar, and G. Torri, Hilbert Series for Flavor Invariantsof the Standard Model , JHEP (2011) 096, [ arXiv:1010.3161 ].[25] L. Lehman and A. Martin, Hilbert Series for Constructing Lagrangians: expanding thephenomenologist’s toolbox , Phys. Rev. D (2015) 105014, [ arXiv:1503.07537 ].[26] B. Henning, X. Lu, T. Melia, and H. Murayama, Hilbert series and operator bases withderivatives in effective field theories , Commun. Math. Phys. (2016), no. 2 363–388,[ arXiv:1507.07240 ].[27] L. Lehman and A. Martin,
Low-derivative operators of the Standard Model effective fieldtheory via Hilbert series methods , JHEP (2016) 081, [ arXiv:1510.00372 ].[28] B. Henning, X. Lu, T. Melia, and H. Murayama, Operator bases, S -matrices, and theirpartition functions , JHEP (2017) 199, [ arXiv:1706.08520 ].[29] A. Kobach and S. Pal, Hilbert Series and Operator Basis for NRQED and NRQCD/HQET , Phys. Lett. B (2017) 225–231, [ arXiv:1704.00008 ].[30] M. Ruhdorfer, J. Serra, and A. Weiler,
Effective Field Theory of Gravity to All Orders , JHEP (2020) 083, [ arXiv:1908.08050 ].[31] J. L. Cardy, Operator content of two-dimensional conformally invariant theories , NuclearPhysics B (1986) 186 – 204.[32] J. L. Cardy,
Operator content and modular properties of higher-dimensional conformal fieldtheories , Nuclear Physics B (1991), no. 3 403 – 419.[33] J. Qiao and S. Rychkov,
A tauberian theorem for the conformal bootstrap , JHEP (2017)119, [ arXiv:1709.00008 ].[34] D. Das, S. Datta, and S. Pal, Charged structure constants from modularity , JHEP (2017)183, [ arXiv:1706.04612 ].[35] B. Mukhametzhanov and A. Zhiboedov, Modular invariance, tauberian theorems andmicrocanonical entropy , JHEP (2019) 261, [ arXiv:1904.06359 ].[36] S. Ganguly and S. Pal, Bounds on the density of states and the spectral gap in CFT , Phys.Rev. D (2020), no. 10 106022, [ arXiv:1905.12636 ].[37] S. Pal and Z. Sun,
Tauberian-Cardy formula with spin , JHEP (2020) 135,[ arXiv:1910.07727 ].[38] B. Mukhametzhanov and S. Pal, Beurling-Selberg Extremization and Modular Bootstrap atHigh Energies , SciPost Phys. (2020), no. 6 088, [ arXiv:2003.14316 ].[39] F. Dolan, Character formulae and partition functions in higher dimensional conformal fieldtheory , J. Math. Phys. (2006) 062303, [ hep-th/0508031 ]. – 46 –
40] P. Di Francesco, P. Mathieu, and D. Senechal,
Conformal Field Theory . Graduate Texts inContemporary Physics. Springer-Verlag, New York, 1997.[41] G. H. Hardy and S. Ramanujan,
Asymptotic formulaæ in combinatory analysis , Proceedingsof the London Mathematical Society s2-17 (1918), no. 1 75–115,[ https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/plms/s2-17.1.75 ].[42] E. Wright,
Asymptotic partition formulaei. plane partitions , The Quarterly Journal ofMathematics (1931), no. 1 177–189.[43] L. Mutafchiev and E. Kamenov,
On the asymptotic formula for the number of planepartitions of positive integers , arXiv preprint math/0601253 (2006).[44] G. Meinardus, Asymptotische aussagen über partitionen , Mathematische Zeitschrift (1953), no. 1 388–398.[45] G. E. Andrews, The theory of partitions . No. 2. Cambridge university press, 1998.[46] A. Ingham,
A Tauberian theorem for partitions , Annals of Mathematics (1941) 1075–1090.[47] W. Siegel,
All Free Conformal Representations in All Dimensions , Int. J. Mod. Phys. A (1989) 2015.[48] G. Mack, Convergence of operator product expansions on the vacuum in conformal invariantquantum field theory , Comm. Math. Phys. (1977), no. 2 155–184.[49] D. Pappadopulo, S. Rychkov, J. Espin, and R. Rattazzi, OPE Convergence in ConformalField Theory , Phys. Rev. D (2012) 105043, [ arXiv:1208.6449 ].[50] P. Kravchuk, J. Qiao, and S. Rychkov, Distributions in CFT. Part I. Cross-ratio space , JHEP (2020) 137, [ arXiv:2001.08778 ].[51] J. Qiao, Classification of Convergent OPE Channels for Lorentzian CFT Four-PointFunctions , arXiv:2005.09105 .[52] J. Kinney, J. M. Maldacena, S. Minwalla, and S. Raju, An Index for 4 dimensional superconformal theories , Commun. Math. Phys. (2007) 209–254, [ hep-th/0510251 ].[53] B. Sundborg,
The Hagedorn transition, deconfinement and N=4 SYM theory , Nucl. Phys. B (2000) 349–363, [ hep-th/9908001 ].[54] O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas, and M. Van Raamsdonk,
A Firstorder deconfinement transition in large N Yang-Mills theory on a small S**3 , Phys. Rev. D (2005) 125018, [ hep-th/0502149 ].[55] D. Kutasov and F. Larsen, Partition sums and entropy bounds in weakly coupled CFT , JHEP (2001) 001, [ hep-th/0009244 ].[56] C. Behan, Density of states in a free CFT and finite volume corrections , Phys. Rev. D (2013), no. 2 026015, [ arXiv:1210.5655 ].[57] E. Shaghoulian, Modular forms and a generalized Cardy formula in higher dimensions , Phys.Rev. D (2016), no. 12 126005, [ arXiv:1508.02728 ].[58] E. Shaghoulian, Modular Invariance of Conformal Field Theory on S × S and CircleFibrations , Phys. Rev. Lett. (2017), no. 13 131601, [ arXiv:1612.05257 ].[59] J. Lucietti and M. Rangamani,
Asymptotic counting of BPS operators in superconformal fieldtheories , J. Math. Phys. (2008) 082301, [ arXiv:0802.3015 ]. – 47 –
60] L. Di Pietro and Z. Komargodski,
Cardy formulae for SUSY theories in d = d = , JHEP (2014) 031, [ arXiv:1407.6061 ].[61] S. Choi, J. Kim, S. Kim, and J. Nahmgoong, Large AdS black holes from QFT , arXiv:1810.12067 .[62] S. Choi, J. Kim, S. Kim, and J. Nahmgoong, Comments on deconfinement in AdS/CFT , arXiv:1811.08646 .[63] J. Kim, S. Kim, and J. Song, A 4d N=1 Cardy Formula , arXiv:1904.03455 .[64] S. M. Hosseini, K. Hristov, Y. Tachikawa, and A. Zaffaroni, Anomalies, Black strings and thecharged Cardy formula , JHEP (2020) 167, [ arXiv:2006.08629 ].[65] S. Hellerman, D. Orlando, S. Reffert, and M. Watanabe, On the CFT Operator Spectrum atLarge Global Charge , JHEP (2015) 071, [ arXiv:1505.01537 ].[66] A. Nicolis, R. Penco, F. Piazza, and R. Rattazzi, Zoology of condensed matter: Framids,ordinary stuff, extra-ordinary stuff , JHEP (2015) 155, [ arXiv:1501.03845 ].[67] S. Govindarajan and N. S. Prabhakar, A superasymptotic formula for the number of planepartitions , arXiv:1311.7227 .[68] E. Witten, Three-Dimensional Gravity Revisited , arXiv:0706.3359 .[69] H. Rademacher and H. S. Zuckerman, On the fourier coefficients of certain modular forms ofpositive dimension , Annals of Mathematics (1938), no. 2 433–462.[70] H. Rademacher, The fourier coefficients of the modular invariant j( τ ) , American Journal ofMathematics (1938), no. 2 501–512.[71] C. B. Marinissen, R. Rahn, and W. J. Waalewijn, ..., 83106786, 114382724, 1509048322,2343463290, 27410087742, ... efficient Hilbert series for effective theories , Phys. Lett. B (2020) 135632, [ arXiv:2004.09521 ].].