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Title Operator bases, S-matrices, and their partition functions

Permalink https://escholarship.org/uc/item/31n0p4j4

Journal Journal of High Energy Physics, 2017(10)

ISSN 1126-6708

Authors Henning, B Lu, X Melia, T et al.

Publication Date 2017-10-01

DOI 10.1007/JHEP10(2017)199

Peer reviewed

eScholarship.org Powered by the California Digital Library University of California JHEP10(2017)199 Springer July 7, 2017 : October 6, 2017 , October 27, 2017 : c,d,e : Received Accepted Published [email protected] , -matrix kinematics and symmetry S Published for SISSA by https://doi.org/10.1007/JHEP10(2017)199 and Hitoshi Murayama c,d,e -matrix — and derive a matrix integral expression [email protected] S , Tom Melia b -matrices, and their partition = 5 scalar fields. This construction universally applies to S n . 3 Xiaochuan Lu, a The Authors. c

Relativistic quantum systems that admit scattering experiments are quan- , [email protected] In addition to counting, we discuss construction of the basis. Simple algorithms follow We introduce a partition function, termed the Hilbert series, to enumerate the op- Kashiwa 277-8583, Japan E-mail: [email protected] Berkeley, California 94720, U.S.A. Theoretical Physics Group, Lawrence BerkeleyBerkeley, National California Laboratory, 94720, U.S.A. Kavli Institute for theTodai Physics Institutes and for Mathematics Advanced Study, of University the of Universe Tokyo, (WPI), Department of Physics, Yale University, New Haven, Connecticut 06511, U.S.A. Department of Physics, UniversityDavis, of California California, 95616, U.S.A. Department of Physics, University of California, -matrix to derive the structure of the EFT operator basis, providing complementary de- -point amplitudes. b c e d a Open Access Article funded by SCOAP from the algebraic formulationoperators in involving momentum space. up to fields We with explicitly spin, compute since the then basis operator for basis for scalars encodes the momentum dependence of with equations of motion (on-shell) and integration by partserator (momentum basis conservation). — correspondingly,to the compute the Hilbertdimension, series. with arbitrary The field expression content is and general, (linearly easily realized) applied symmetries. in any spacetime considerations are encoded in theS operator spectrum of thescriptions EFT. in (i) In position this space papermentum utilizing we space the use via conformal the algebra an and algebraicimplemented cohomology formulation as and in an (ii) terms ideal. mo- of These a frameworks ring systematically of handle momenta redundancies with associated kinematics Abstract: titatively described by effective field theories, where Brian Henning, Operator bases, functions JHEP10(2017)199 - S 1706.08520 Effective Field Theories, Differential and Algebraic Geometry, Scattering Although primarily discussed in the language of EFT, some of our results — conceptual We discuss in detail the operator basis for non-linearly realized symmetries. In the ArXiv ePrint: and quantitative — may bethe AdS/CFT of correspondence. broader use in studying conformal fieldKeywords: theories as well as Amplitudes presence of massless particles,matrix there in the is form freedom ofto to soft limits. the impose operator The additional most implementation basis structure na¨ıve for for for on massless non-linear pions, scalars realizations. the leads which we confirm using the standard CCWZ formulation JHEP10(2017)199 47 48 13 9 43 47 57 3 , 64 = 2 42 d 49 24 35 = 5 basis n 54 33 55 14 14 K G n, = 4 29 30 43 d M ) 10 35 K d ( 61 23 : generators and algebraic properties 15 – i – O n 47 distinguishable scalars in 51 and n J ) to d 39 21 6 12 17 6 = 5 operator basis n 54 22 1 5.4.4 Generalizations with spin 5.6.1 The 5.4.1 The5.4.2 Cohen-Macaulay property A5.4.3 set of primary invariants Relationship in between the absence of Gram conditions 6.1 Hilbert6.2 series for a real Closed scalar form6.3 field Hilbert in series for Hilbert6.4 series for spinning particles Quantifying the effects of EOM, IBP, and Gram redundancy 5.5 Conformal primaries5.6 and elements of Algorithms for operator construction and the 5.7 Summary 5.3 Interpreting the5.4 ring in terms Constructing of the operators basis and at amplitudes higher 4.4 Reweighting and summary of main formulas 5.1 Polynomial ring5.2 in particle momenta Tour of the four point ring 3.4 Summary 4.1 Deriving a4.2 matrix integral formula Multiple for fields4.3 the and Hilbert internal series symmetries Parity invariance: from SO( 3.1 Unitary conformal3.2 representations Character formulae 3.3 Character orthogonality 2.1 Single particle2.2 modules The operator2.3 basis via cohomology The and operator2.4 conformal basis representation as theory Feynman Non-linear rules realizations 2.5 Gauge symmetries,2.6 topological terms and A discrete partition spacetime function symmetries for the operator basis 6 Applications and examples of Hilbert series 5 Constructing operators: kinematic polynomial rings 4 Counting operators: Hilbert series 3 Conformal representations and characters Contents 1 Introduction 2 The operator basis JHEP10(2017)199 86 96 84 94 of the EFT, K in its own right, we K -matrix. S r 2 so 4 82 S 75 ] 34 83 84 ) , s r 66 24 (2 , s O . By considering the set – 1 – 23 K 102 , s 68 97 14 ) is Weyl invariant , s 89 g ( 79 13 f , s 65 12 s [ H C -matrix theory and effective field theory are equivalent: the starting S 92 72 73 The purpose of this work is to formalize the rules governing the operator basis and C.1 Representations andC.2 characters of Understanding theC.3 parity odd character: Hilbert folding seriesC.4 with parity Examples B.1 A simplificationB.2 when The measures for the classical Lie groups 7.1 Linearly transforming7.2 building blocks Computing the7.3 Hilbert series Summary addressing specific dynamics. At(and a other practical internal level, symmetries), as itequations well amounts of to as motion dealing imposing (EOM) with Lorentzredundancies). and redundancies invariance At associated total a with derivatives more (alsosingle fundamental called particle level, we states integration are together by accounting with parts for Poincar´einvariance of (IBP) Poincar´ecovariance of the ments. This is dictated byfield kinematics theory and (EFT), symmetry this principles. parameterizationwhich is In is embodied the defined in context to the of be operator effective the basis set of allinvestigate operators the that structure lead they to induceare physically on aiming distinct to phenomena. get as much mileage from kinematics and selection rules as possible before 1 Introduction The basic tenets of point is to take assumed particle content and parameterize all possible scattering experi- F Counting helicity amplitudes G EOM for non-linear realizations D Computation of ∆ E Hilbert series for C Parity A Character formulae for classical LieB groups Weyl integration formula 8 Discussion 7 Non-linear realizations JHEP10(2017)199 - - ]. S S , or 32 ] for – (1.1) 8 are in ]. Our 29 1 − 32 d ] it was shown ) are the usual 10 ]). Similar types of ), whose solution is s, t, u 3 , ]. In [ 2 2.12 -dimensional scattering d 28 – where ( 26 , in eq. ( 10 stu 4 , M
], found concrete implementation in and -matrix by enumerating independent O 6 Hilbert series of the operator basis S – b tu 4 w + the -matrix, and as such can address questions ] has demonstrated underlying connections -dimensions consisting of four scalar fields X S O∈K 9 d su ] was a counting argument, that showed the H = + 10 – 2 – b w st K -point amplitudes in one higher dimension [ n = Tr H -point conformal correlator involves spinning objects, the n ]. 25 – ], spawning the modern CFT bootstrap program (see e.g. [ ], in the past 25 years has seen remarkable, useful, and beautiful 7 10 1 is the same as the number of solutions to the crossing equations. ]. The number of such tensor structures is the same as the number 4 29 K for short. The Hilbert series is a counting function, just as the usual M 1)-dimensional conformal correlators [ − d ) a freely generated ring in 5 is some weighting function. We call Hilbert series b w . In many ways, the Hilbert series can be thought of as a partition function of the 2 spacetime dimensions. Our main purpose is to introduce and develop a number of In a similar vein, when an More concretely, the AdS/CFT correspondence in the flat space limit of AdS indicates Effective field theory is a description of the One basic and very useful means of studying the operator basis is to consider the We focus on the exploration of operator bases in a wide class of relativistic EFTs in 2 CFTs a decade ago [ βE ≥ − correlator is first decomposedconformal-cross into ratios tensor [ structures whichof then such multiply structures functions for ofA corresponding the general procedure forHilbert series counting techniques the capture this tensor information structures and more: was they recently account given for in not only [ the Mandelstam variables. The initialnumber clue of in objects [ in The Hilbert series techniquescalculation developed of in determining thisalgorithms the paper we numbers allow present an of aid in easy operators their and in explicit straightforward construction. more general cases, and simple amplitudes and ( that solutions to theone-to-one crossing correspondence equations with for operatorsand scalar in derivatives. four In point our functions analysis,(see this in section set CFT is described by d > an overview). The AdS/CFT correspondencebetween [ these two areas [ that there is a (not entirely understood) correspondence between concerning the use of generalconsistency physical conditions principles with to the derive goalmatrix and of theory implement ideas, probing — e.g. physical [ bootstrap observables.results — A based revival on of unitarity older andgeneral causality considerations, (for also recent discussed reviews last see century e.g. [ [ physical partition function (pathe integral) counts states inmatrix. the Loosely Hilbert speaking, space it captures weightedobservables. the by rank Hilbert of series the will play a central role in our study of operator bases. partition function on where simply d systematic methods to dealexpansion. with EOM This andlow-order treatment terms IBP of of redundancies course an atphenomenological operator also all theories basis, provides as orders a experimental a continuing in energy topic standard the and of procedure precision EFT interest thresholds for to are various determine crossed. relativistic JHEP10(2017)199 ]. 34 , 33 , only appears once since d µ d p ]. We subsequently build ··· 37 1 – µ 1 1 p aims to establish the logic of to tabulate more detailed and d 35 7 2 – ...µ 1 3 µ  – 3 – -matrix. The operator basis follows from the construction of S using on-shell quantities, bypassing the introduction of a K , is that it elucidates analytic properties more clearly. On the . From this perspective, it is clear the EOM redundancy is K by definition This is not merely some pleasing alternative viewpoint — it often clarifies 2 : the operator basis. 2 -matrix. We introduce what we term ‘single particle modules’, which consist of all Establishing the operator basis along this line of reasoning has distinct advantages. Our presentation is somewhat lengthy — as well as containing new results and tech- EFTs are very good at clearly isolating relevant degrees of freedom — this has given At a superficial level, EFT may also offer some novelty in connecting modern ampli- S For example, something containing the epsilon tensor, like Note that we are only advocating the idea of conceptually divorcing operator content (kinematics) from 1 2 is why covariant derivatives behave as commuting objects in our analysis.two epsilon Both tensors these multiplied rules together can be decomposedWilson into coefficients products (dynamics). of The thethe first metric. can, latter, and the should, Lagrangian be remains established an with indispensable only tool. physical quantities. For across the operator basis. It derives the rulesLagrangian. governing the validity and use of certain rules. One example is the use of the free-field EOM. Another essence, these modules aremodulo an EOM abstraction ofsimply a an field along on-shell withwhich condition; a provides it the tower is of freedomoperators to its precisely perform via derivatives this field (tensor) EOM redefinitions products — [ of as these used modules, in thus accounting LSZ for reduction EOM — redundancy operator bases from physicalthe principles, main as ideas well we as present provide in the the readerSection bulk with of an this overview work. of the possible Lorentz spin operators that interpolate an asymptotic single particle state. In tions of existing resultsaims in to the both (mainly clarifyresults the mathematics) and structure literature. new of derivations/interpretations this The ofseparate paper, following known summaries and results; summary are to for included make the at theexplicit reader’s distinction the convenience, results between end and new of formulae sections from each section. Section of operators, embodied in other hand, the remarkable success ofsensible, EFT is potentially a even serious for hint that studying truncation strongly is quantitatively coupled field theoriesniques, [ it also exhibits a number of self-contained, heuristic, and physics-oriented deriva- CFT (largely on account of theobject operator-state described correspondence) by — the while the EFT actual is physical the birth to anSCET, entire SM zoo EFT, of etc.practical EFTs reasons), In in which can most high mask discussions, analytic energy properties. however, physics, One EFTs hope e.g. are in truncated chiral studying the (for lagrangians, entire very set HQET, good be applied an infinite numberas of well times, as and secondary appear invariants which as can denominators be in used thetude only Hilbert and once. series), CFT ideasfrequently simply formulated in by position the space language and typically discussed used in to terms discuss of EFTs: operators — EFTs similar are to tensor structures but also the Mandelstam invariants (which serve as generators that can JHEP10(2017)199 ]. H we 41 ; ∆ ], and B 0 40 H . It is only 0 . H d ≤ ] and we have relied 41 This section reviews results dimensions necessary for treating d A new and central result of this work are scalar conformal primaries), ] where we gave the application to the SM , and present a matrix integral for interpolates a single particle state. Moreover, H 39 – 4 – φ k O ∈ K + ∆ D are elements of certain kinematic polynomial rings). 0 H = O ∈ K H . 4; here we show how to derive the Hilbert series in arbitrary ). The use of conformal representation theory — formulation 7 are co-closed but not co-exact 0-forms), > dimensions was worked out by Dolan in [ 4.16 . d 2 ]. A question about soft limits naturally leads to non-linearly realized O ∈ K 39 = 4 in our previous work [ d ], they are presented in full detail and in greater generality here. tabulates character formulae for the classical Lie groups. In appendix : conformal representations and characters. : counting operators: Hilbert series. 39 A , 4 3 38 conformal representation theory ( an algebraic description ( cohomology ( 2 dimensions, eq. ( More precisely, we write As character theory plays an important role in our analysis, we note here that Next, we discuss the treatment of linearly realized internal symmetries, covariant We then give an overview of the new techniques we use to further impose IBP. We find ≥ (2) (3) (1) outlined) in EFT at mass dimension spacetime dimension, including relevant operators with mass dimension corresponds to a finite number of operators not properly accounted for in is a matrix integral (groupd integral) formula for the(2) Hilbert — series to of handle an IBPseries. operator redundancy This basis, allows treatment is us in valid toconformal when obtain the group a single — complete particle this modules derivationin form of encompasses four representations the a dimensions, of Hilbert the and large in class particular of the EFTs SM that EFT. This includes was gauge the theories approach used (but only derive the Weyl integration formulaWhile and these provide the results measures are foraimed textbook the to classical material, be Lie more the groups. physics-oriented treatment than of that theSection which Weyl is integral usually formula encountered. is formal characters in heavily on this reference. However,poses we that present elaborates a on largely derivations self-contained and version includes for physical our intuitions pur- appendix that supplement [ take it up in full in section Section in the representation theory ofIBP the via conformal formulation group in (2)representations mentioned of above. the conformal The group use is of of particular character importance. formulae The for treatment irreducible of con- work [ derivatives, and topological terms;also some presented in of [ thisinternal methodology symmetries; was we pioneered give in an [ overview of what this implies for the operator basis and While all three formulations have been introduced at some level in our previous this picture allows usat to the give end a of physical section derivation of the Hilbertthree series, mathematical which formulations is for outlined doing so, in terms of are obvious from the perspective that JHEP10(2017)199 . In H algebra sp ]. Counting This section 45 , . 44 C ] to deal with the problem of an explicit formula for ∆ 43 , 40 only pertains to certain marginal or for a derivation utilizing differential H 7 derive ], and for the Standard Model (SM) EFT 38 = 4 has also been addressed using topological – 5 – is valid and relatively straightforward to derive d 0 ]. In this paper we elucidate the algebraic modules H . 38 7 ) can be found, contained to a few pages, in Weyl’s original d ( O = 1 dimension [ d ]. ) invariance. Previous documentation of the character theory of the parity ]. They were adapted in the two papers [ for a physical derivation, and section 46 d ( 2 42 : constructing operators: kinematic polynomial rings. dimensions. Many of these details are contained in appendix O ]. They have since found application in non-relativistic EFTs [ ]. We expand upon this, including a self-contained discussion on the ‘folding’ of a 5 r 39 47 ) to = 2 The application of commutative algebra to the IBP problem was first explored in our After giving a precise definition of the rings relevant for an operator basis involving We also explain in detail how to impose parity on the operator basis — moving from Hilbert series have been previously used in the particle physics literature to study flavor d d a connection between the elementsa of construction the algorithm ring which we andthe use case conformal to of primaries, construct indistinguishable and scalars. the finally ring detail with five pointprevious kinematics work in in scalar fields, we usedimensionality, parity, and four particle indistinguishability point on the kinematicsand construction of we to the reflect operators, showcase on various theprove some aspects connection general to properties of constructing of the theuniversal physical properties rings effects scattering and (that amplitudes. application they of of are We these Cohen-Macaulay), scalar then and rings discuss to the cases involving spin. We provide Feyman rules. EOM andconditions IBP and redundancies momentum manifest conservationof as — momenta. kinematic they As constraints are we of implementedconstructing shall on-shell as the see, operator an the basis: Hilbert ideala series in many basis is algorithms the rely an (as ring on indispensable well input tool as that human towards comes explicitly intuition) from for the finding Hilbert series. in Section develops an algebraic descriptionoperator — basis. formulation The (3) basic — idea in is order to to construct explicitly a build ring the in momentum whose elements are SO( odd and even pieceswork of [ Dynkin diagram, which explains the — at first mysterious — appearance of the series for enumerating independent operatorsfirst with for any scalar number EFTs in ofin 0+1 derivatives were ref. dimensions studied [ in ref. [ of conformal primaries infield free theory [ CFTs in relevant operators — theclearly cohomology which picture we allows discuss us in section to understand its structureinvariants [ more finding Lorentz invariant operators involving multipleof types of the field, operator but basis only for where the no subset more than one derivative appears in an operator. Hilbert tions of the conformalthe group, general that case, we where areconformal the able group, single to particle we modules emphasize(see are that section not unitary representationsforms). of Moreover, the our observations indicate that ∆ in formulation (2), when the single particle modules correspond to unitary representa- JHEP10(2017)199 , 2 3 , (for ≥ 6.2 (2.1) = 2 σ d d ]. , exploring 38 6.3 = 4 dimensions, scalars in d . In this section we present H reductions, Gram constraints -matrix. We proceed on this S , x distinguishable · ip ) are constructed which interpolate − denote such a state with spin x e . ( ) l i p 8 σ ( polynomial , spin under the relevant little group, and σ l p 2 | p U ], and use this to give an identification of = 48 i ∼ 2 , – 6 – σ m In this section we show how to include invariance 37 p | ) x ], local fields Φ ( l 49 , quantifying how much EOM, IBP, and Gram conditions Φ | 0 h 6.4 ), thus providing a significant generalization of [ 5.15 , studying Hilbert series for real scalar field in 6.1 reduction). : non-linear realizations. : applications and examples of Hilbert series. 7 6 exponential We conclude with a short discussion in section other possible internal quantumnow, numbers. we ignore Let other possible quantumstandard numbers pathway for to simplicity of field discussion). theorythis Following [ single the particle state 2.1 Single particleAssume modules that we are givenare a particles set which of can particles beparticle that states we part are have of specified asymptotic in by access their and to mass out — states i.e. in these scattering experiments. The single 2 The operator basis Under some milddescribed physical either assumptions, by at an interactingthe low CFT latter, enough or and an energies we infraredroute a look free by relativistic theory. imposing to We the system build are consequences the is concerned of EFT with Poincar´esymmetry. to describe the Hilbert series takes a form very similarwith to conformal the irreps; case where here, the however, singleout we particle to modules appeal be) coincide to the Hodge relevant theory and to marginal determine operators (what belonging turn to ∆ Section of the operatora basis linearization under procedure, non-linearly `alathe realized single CCZW internal particle [ symmetry modules.we groups. Having appeal constructed to We this formulation module follow (1) (and and thus use dealt Hodge with theory EOM), to address the IBP redundancy. The Hilbert series with spinning particles, inparticles, particular the relation and to those how involving onlyamplitudes this scalar of spin; provides and information section “cut on down” the the operator basis tensor (EOM andgive decomposition IBP give of scattering rank conditions — termed Gramstudies conditions are: — are section particularlyand interesting), etc.. the impact The case ofderiving indistinguishability a on closed the form expression structuredimensions, for of the where the Hilbert Gram operator series of conditions basis; are section more readily understood; section dimensions, given in eq. ( Section some novel calculations using ouris developed how framework. Common the tosecondary Hilbert all invariants of series (generators), these information reveals examples on structure relations between in invariants the (spacetime operator basis, such as primary and (quotient rings in particle momenta) that form the operator bases of EFTs in JHEP10(2017)199 , i ). µ σ x p ( p l (2.3) (2.2) | . They m = 0; and σ µ... u µ p ) a Poincar´etransfor- ) is the momentum , a is an asymptotic particle ). It is worth elaborating i x 2.1 σ ( , l , the field equations can be . For now, however, we stick p φ x | · ,σ K , † p ip x a · − with U(Λ e ip } i x − 2 · σ e µ ) consistent with locality. To capture ip ) x p p · p − p ( | 1 ( ip e µ σ ) l ), which is simply the starting point for ]. Because σ l µ { − to be the set of all distinct interpolating U e p p u , a 50 2.1 l . . . φ i ∼ R can be represented by the set of wavefunctions i ∼ i ∼ i ∼ U(Λ – 7 – σ p p p l | | | . Importantly, these derivatives can only be added φ p l ) together with an infinite tower of derivatives on φ φ φ | | R ) } φ x µ 0 i → 2 ( x ∂ h l µ | ( σ l φ ∂ 0 p h 1 φ | | ) — is eq. ( µ 0 { p h ( ∂ | σ l 0 h U consists of l φ ; fields with spin will have a transverse condition R single particle module ) is linear in the creation operator 3 2 x ( m = 0. The possible interpolating fields are l φ σ , i.e. = i σ 2 from which these objects arise. This dual momentum space picture will prove ]. Specifically, every field obeys a Klein-Gordon equation due to the on-shell p p modulo EOM. Equivalently, | i 49 . The correspondence between these two representations — interpolating fields σ l which has a minimum number of Lorentz indices to specify the spin component is some unspecified Lorentz index structure. Up to some normalization (hence the } φ l l p ) | φ p ) versus wavefunctions To clear up the abstraction, let’s look at a specific example. Consider some scalar We define the The different types of interpolating fields are not mysterious. There is some basic ( Recall that it is the linearized EOM which determines the propagator and is used in LSZ reduction. x σ l 3 in the above equation), the right hand side is determined purely by properties of ( l . Beyond this, the only available object on the r.h.s. of ( U to the position space picture in terms of interpolatingparticle fields. state, Φ the familiar correspondence betweenthought operators of and as Feynman simply rules. astate Fourier In transform, practice, although this it slightly canuseful, obscures be especially the in Hilbert giving space a concrete algorithm to obtain theory Lagrangian. fields for top of { condition spinning, massless fields havetogether, further they constraints are in relativistic the wavestate, form equations these of [ look Bianchi like identities. free-fieldare Taken equations structurally of motion equivalent but to involve the the physical linearized mass equation of motion obtained from the field and assuming determined via consistency with mation [ field σ corresponding to derivatives acting on in a manner consistenton with this the point. field From equations obeyed by ∼ under Poincar´etransformations. In turn, this dictates theThere field equation(s) are obeyed many by Φ —there in is fact, an infinite entire —the set different most of types general distinct of set wavefunctions of interpolating local fields; interactions, equivalently, we must take all distinct interpolating fields. where JHEP10(2017)199 ) 0 ]; p ( 37 σ µ This – (2.4) (2.5)  k > 4 35 the leading ··· (0) ) with + p L 3 ( ].) The ability  σ µ  O 51 ν with k 1 p ) Λ n ( − − , is dimensionless only L )  dropped. An order 2 ) = 0, and (obviously) . Correspondingly, the p 3 + ] ) for creating the single p L ( F σ ν n i ( 3 1 σ ν  2 δφ Λ δ σ µ σ  ··· g 1 }  6 1 µ 4 µ p EOM) can be eliminated in favor µ [ + p | = 1, the two are equivalent. p p x + ) d n k (1) 2 x µ d ( L  l = ) = . R φ O p + n | , which changes the Lagrangian by = 0, leading to the single particle . ( k 0 1 . . . p       O Λ h (0) σ µν       1 k ν 1 µν L − } µ Λ u φ ν { F µ  = } leading order } 2 p − F 2 µ 2 L ∂ L 2 µ φ φ 2 F . . . µν . . . µ µ φ ∂ δφ δ 1 F ∂ 1 ∂ = µ → 2 1 1 µ { { µ φ 3 ∂ + { ∂ µ – 8 – 2 ∂  µ = 0, transversality       O       F 2 k 1 coincides with the conformal representation of a free = 1 p µ Λ = φ φ ∂ -matrix elements, the only thing that matters is that a unitary conformal representation (one indication of − d F R R S . The single particle module is R (proportional to the ...µ 2   1 not ) µ , and terms suppressed by more than m k (  ··· k is 1 Λ = + = F 2 ⊂ L ) giving the wavefunction p (1) R coincides with the conformal representation for a free vector. Out- x µν δφ L (0) ( above. However, in the special case of δφ δ L F F δ µ L νµ + R O ∂ F k 1 (0) − Λ δφ L δ suppressed by  ) ) = denotes the symmetric, traceless component — the trace is removed because k ( x = ( L L µν ∆ F {· · · } = 0, allowed wavefunctions take the form ] = 4). Let us briefly pause to emphasize a well known point concerning removing the EOM As a second example, consider a massless spin-one particle. The basic interpolating ν A generic EFT Lagrangian is a truncated expansion p d 4 µ [ variation terms in the ∆ redundant operator of higher order terms through the field redefinition The very first termpensation. cancels This the is redundant clearly different operator, from while plugging all in the the full other EOM, terms which amounts give to its keeping higher only order the first com- it tells us thatan in interpolating computing field the giveparticle the state. proper This wavefunction clearly ( allows for redefinitions of anorder interpolating term, field. generically appear inoperators the can Lagrangian always be as removed oneoperation by in a flows a field Lagrangian between redefinition. is performing various (Hereis a we energy well field emphasize redefinition, known scales. that not and the pluggingto has valid in Such the perform been EOM. field explicitly redefinitions emphasized is in formally the justified literature, by e.g. the [ LSZ reduction procedure [ this is that the couplingin constant in the Maxwell action, redundancy. In the fulla fledged field Lagrangian theory and — a which requires renormalization additional prescription structure — such operators as proportional to the EOM In four dimensions, side of four dimensions the polarization vector.p By on-shell field equations are module For future reference, we notescalar that field. field is of the on-shell condition where JHEP10(2017)199 if 0 l = 0. (2.8) (2.6) (2.7) µ n p ∼ O + l O is the set of is the set of ··· + J J ) means that we µ 1 d we need to apply p d  J 6 K ⊂ J , i.e. ) group relations like Fierz J d . ] , , φ, . . .
) } denote the set of all operators formed d φ 2 µ R is the set of all local operators modulo J i ∂ SO( to gauge invariant operators. 1 . To select out n  φ µ can equivalently be built by taking tensor 2 J J { J J Let ∂ h d J sym 5  . – 9 – must have the same quantum numbers as the φ, ∂ ] is a differential ring formed from implementing =0 J µ µ ∞ denote an asymptotic multiparticle state. Then, n M  α ∂ J ; i = = n φ φ, ∂ [ [ σ defined above. The superscript SO( K C J n C p ∼ ; = = J ··· -matrix. Use crossing symmetry to take all particles as in- ; S 1 σ 1 p | to be non-trivial, = . For the single scalar field we then have i i i α Φ . Implementing this equivalence relation allows us to formally identify | l α | R ) S | 00 0 O h ∂ ) invariance. Since we know that we want Lorentz scalar operators — and that denotes taking the derivative of every element of d + ( J 0 l d O It is at least conceptually clear how to get Lorentz singlets, although determining in- The operator basis is clearly a subset of In other words, thisIf is there justified is because a we gauge are symmetry, perturbing one free further fields. restricts = 5 6 l equivalence classes with relation apply SO( a scalar which is a total derivative must be the divergence of some vector — we can be O the operator basis as the differential ring where no momentum, meaning they cannot be a total derivative. dependent scalars can beidentities, extraordinarily Gram difficult determinants, due to etc.mine SO( operators What up is to conceptually a less obvious total is derivative. how to This deter- is an equivalence relation, Poincar´einvariance of the coming and let in order for vacuum. In particular, itInteractions is must a respect Lorentz scalar this and invariance as carries well: no momentum, operators are Lorentz singlets and carry where we take symmetric tensor productsparticle due modules to Bose form statistics. freeoperators In which field the participate case representations in that of the the single operator the state conformal correspondence. group, Making use of theproducts single of particle the modules, equations of motion.the Mathematically, equations of motion as an ideal. Explicitly, for a single scalar field Local operators are built byoperators taking as products simply of a the product interpolatingfied of fields. other in operators our Viewing framework is composite because a the perturbativea asymptotic statement, Fock which multi-particle space is states from justi- in the scatteringfrom single are products built particle as states. of the interpolating fields, i.e. 2.2 The operator basis via cohomology and conformal representation theory JHEP10(2017)199 = 4 (2.9) d do not i field plus at the top Φ l l R φ φ . We will see K ) fields in even l  . We will be liberal ), we have solved the , schematically, 7 O 2.5 However, our experience l, . . . , l, ), ( 8 2.4 . . ] for a related discussion concerning clearly we solve total derivative 53       i 7.2 consisting of only one Φ O O . . . 2 O R ∂ J ∂ the operator basis is spanned by scalar,       1) fields. Generically, the as some column vector with  l the operator basis consists of co-closed but O , X φ R , but the essential feature is that the derivative ∼ – 10 – 3 n N       l l l φ . . . 2 φ ∂φ ∂       for notation). Note that this covers gauge fields in 3 and decompose these into conformal representations. Only the via tensor products of the i correspond to spin (1 Φ J R followed by a tower of derivatives acting on e F 7 i . . O  F ] (see section = 52 L,R F These conclusions about the conformal group are established rigorously when the single The construction of the single particle modules is a great hint at how to handle the Some basic definitions ofIn Hodge the theory general are case, reviewed although in the section single particle modules mimic the structure of conformal representa- 7 8 tions, they are decisively notin unitary representations. unitary This conformal is representations. duetations, to with EOM It shortening ghosts is conditions accounting not possibleMaxwell for present theory that the outside these shortening four could conditions. dimensions. be See considered [ non-unitary represen- that, in general, there is no special conformal2.3 generator which makes these The words operator precise. basisThe as operator Feynman basis rules canthis also description be also described proceeds in via the the language single of particle Feynman rules. modules, but The this route time to formulated in where coincide with unitary representationsindicates of that the this largely conformal doesan group. not example matter of for this determiningwith when the terminology we majority study and of non-linear say realizations “primary” in and section “descendant”, although it is to be understood conformal primaries particle modules are unitaryfree representations scalars of and the conformal spinorsdimensions group. in [ any This dimension is the as case well for as free spin ( is a lowering operatoruse within conformal representation the theory conformaltensor to algebra. products organize of the What the totalprimary is derivative operators equivalence: obviously are suggested we not total take isators. derivatives. to Therefore, Lorentz invariance we means arrive we at take the scalar oper- conclusion that These multiplets follow preciselysentations the are structure reviewed of in detail conformal in multiplets. section Conformal repre- derivatives. In building equivalence if we canderivative decompose operator back into a sum of multiplets consisting of a non-total not co-exact 0-forms equivalence up to total derivatives.of Viewing the multiplet followed bytotal a derivative tower equivalence of problem derivatives, for e.g. eqs. operators ( in even more precise in the equivalence relation: JHEP10(2017)199 . An i (2.14) (2.15) (2.10) (2.11) (2.12) (2.13) σ  in the ideal. µ n p , -point amplitude + 9 n n S ··· × ) d + . , ) µ 1 n } . p SO( S ) µ i i i p ij i i { s ]. 2 n 2 n ( ( l I , p 54 component removed, implying , p A M 2 ) l ]; holographic arguments suggest i p ···
··· , n 29 , , p 2 1 σ i 2 1 ,   p i , p ( h with 2 µ n n . ) are given by the ring p ··· . p n µ h ] 1 , ...σ p µ n σ j 1 . + M ]  ). Focusing on the scalar particle, the wave- I σ p 2.11 , p µ p g · ; as this cumbersome notation is presently unnecessary =0 ( p 1) dimensions corresponds to the amplitude ∞ ··· i particles have arbitrary individual spin), ]. It essentially is the naive expectation: it is [ n n M σ l p I S ··· C – 11 – + − n ) = X 32 × U , = ) } = µ 1 d µ 1 d ]. A general prescription for counting the number = µ i p p K K [ is that it captures the objects an ij h p ) = φ 32 SO( n, s are Lorentz scalar polynomials linear in each { (the i C – n . , R ] l M h n } , p µ n M 29 i i  σ = ...σ , p (  1
n { I σ ( φ g n ··· R means to take polynomials symmetric under permutations , ,...,σ 1 1 µ n σ p n ,...,σ [ 1 S A σ C sym h A dimensions. To this point, the number of tensor structures — the ) are simply polynomials in ≡ . d n n 2.3 M M above — coincides [ = 0. This last condition is accounted for by placing we will label these rings as I µ n ) is the decomposition into helicity amplitudes [ 5 g p ) invariant polynomials and enforce equivalence under momentum conservation + d 2.15 ··· The basic physical content of In section + 9 µ 1 of such structures wasthe recently number given of in different helicity [ configurationsis, one eq. can ( take for the external particles. That we use the simpler Here, the little group tensors analogous decomposition is donethat for CFT the correlators conformal [ decomposition decomposition in in ( number of (tensor structure refers to littlepurely group of indices) the multiplying Mandelstam amplitudes invariants which are functions Feynman amplitude of overall spin The amplitude can be decomposed into a finite number of Lorentz singlet tensor structures can depend on. Typically we think of an amplitude as polarization tensors dotted into a Hence the independent Feynman rules in eq. ( and the operator basis (for a single scalar field) is where the superscript of the momenta.the SO( To getp the independent, Lorentz invariant Feynman rules we take functions in eq. ( from which momentum space with the wavefunctions JHEP10(2017)199 ), 0, a 2.4 (2.16) (2.17) . Hence, )–( → i , leaving universal X n p i 2.3 iπ M e which do not = φ ξ , an ansatz for the n R in eqs. ( , but instead with the M 11 i µ with φ j ),
x R 1 ( − i µ ) implies that the gener- ξ ) (with possibly different u µ ξu i 2.15 2.12 . ) to X n , M x 2.17 ·
i ) group relations could cause some = Tr ip d in eq. ( ) to a future work. i µ − , p ) multiplying the j ) is i e I i n σ l µ g u p ) is simply some low-energy vector field 2.16 M 2.3 ij I δ f 2.17 = 0. This naturally makes one guess that the I , i.e. delete all modes from 2 i ∼ M φ from our field theory knowledge. p j – 12 – p R π 0. Suppose we have some massless scalar states ∼ | )
∈ x } → . To ensure the absence of scattering as ( i appearing above need not be exactly identical to that l φ i µ p j Φ n | { 0 M h a posteriori . The two are related by n , . . . , n 7 M = 1 i ] for the very interesting question about constructing EFTs from is if the interpolating fields vanish with momentum. In a suggestive 56 ) to include spinning particles will be some decomposition into poly- , , see section ) tells us that any derivatives acting on the vector interpolating field with i µ 10 55 u i 2.12 i p 0. Concretely, renaming the field in eq. ( 2.17 π interpolate the single pion state, but differ when interpolating multi-pion states. → i µ (serving as appropriate avatars for the ), nor even the same for each term in the sum (it is also not clear we can get j µ I p i ≡ | -matrix — namely, Poincar´esymmetry — to build up the operator basis. Are i f and 2.12 S i µ u = 0 Equation ( At this stage the field appearing in eq. ( In terms of the operator basis, the decomposition in eq. ( At present, this claimThe is justified CCWZ construction does not directly work with the symmetry current σ 10 11 p but without the “top”vanish as component Maurer-Cartan form both here; we refer tosoft-limits. [ must be traceless and symmetric,appropriate because single particle module is essentially the scalar module which (1) interpolates a scalar stateFrom and these (2) two is pieces divergenceless because ofacting the knowledge, in scalar all is some we massless. unknown canfield manner conclude as on is some a that specific the scalar field function function. has requires To a additional go derivative input. further and We determine do this not pursue this avenue sufficient condition notation, the first such interpolating field in eq. ( which should be ringing bells about Goldstone’s theorem. of the there any other assumptionsIR that free we theories can is makethis the within means absence this interactions of framework? vanish long| as The range hallmark interactions. of If there are massless particles, momentum dependence, which isaim an in this ingredient work to in developthe all a precise healthy scattering understanding description processes. of of the the universal We general ingredient therefore case2.4 in eq. ( Non-linear realizations Our approach thus far has been to start with an IR free theory and use the symmetries in eq. ( a clean direct sumchanges. decomposition). The For essential example, point SO( permutation is groups that to the account modules forto identical all versus aspects distinguishable of particles) the are operator basis. This is precisely because they are analyzing possible alization of eq. ( nomials We use a tilde here because the JHEP10(2017)199 . . -
µ G (or has ), A A , so d l A + µν (2.20) (2.18) (2.19) φ 2.8 d µ F µ ∂ A ∼ = ∧· · ·∧ ] µ ν A ∧· · ·∧ . D should be d A ,D ∧     d → µ K } A µ ∧ 2 2) act trivially). D ∂ i µ A u − i µ . . . 1 u = d d µ { . The point is we work F A D SO(     ⊃ = d . Second, note that for the l = ∧· · ·∧ φ 2) F , u µ F i R − D σ . d p | G ]. l × and ⇒ φ ) d µ 48 , ∂ , then the operators in | µν SO( 0 37 x F G (where the brackets denote symmetrization · is correct via the conventional CCWZ La-  h ip u J = µν − d R e i F – 13 – ). Of course, we also promote  } x σ 2 · + J µ p ip 2.5 ) |  p l − ν 1 e φ µ = D µ { µ µ p p ( K D ij ij | D δ δ 0 h , designed to guarantee the correct little group transforma- . . . a → n i ∼ i ∼ ν d j j p p D π π ∼ µ | | } even though it is a total derivative, i µ f D 2 u d i µ | µ u 0 1 h 1 − µ d { µ , as is apparent in eq. ( µ D F | A 0 . We first note that gauge invariance is straightforward to impose because h a ··· plays no role in interpolating the single particle state (for one thing, we will show this ansatz for 2 T µ 7 µ a 1 µν , not A µ F µν F d F = µ An important question is whether we mis-identify or omit operators by working with For concreteness, let’s discuss the familiar vector case Some comments should be made about massless particles with spin, which lead to gauge ··· 1 µν µ due to topological terms. For example, in even dimensionsLikewise, in we odd will dimensions pick we will upits not the non-abelian pick theta up generalization), the term Chern-Simons whichcomments term is apply gauge to invariant other up massless to spinning a fields, total as derivative. well Similar as to topological terms such as because two annihilation operators). Thiswe can can always also replace beof justified the by indices) noting when that constructing [ operators. the field strengths instead of the gauge fields. The answer is yes, and it is unsurprisingly we work with gaugepurposes covariant of quantities constructing like partial the derivative operator basis, the covariant derivative will behave like a tion (ensuring that the non-compact directions of ISO( with In the case thaton there them, are we multiple know vector thisF single must particle correspond states to with a a non-abelian symmetry gauge imposed theory and we work with symmetries. The construction outlineding presently objects, ultimately since works it with is simplythere the transform- is field strengths a which gauge interpolate potentialtives single field of, particle states. schematically which the Of field course, strength is some number of exterior deriva- invariant. At an abstract level, we simply append an additional superscript to eq. ( Some care should be takensymmetry to is identify non-linearly the realized, appropriate as degrees mentioned of above. freedom, such as when a In section grangian description of non-linear realizations [ 2.5 Gauge symmetries,If topological the terms and EFT discrete has spacetime an symmetries internal symmetry group appropriate single particle module is JHEP10(2017)199 n /P M . We (2.21) C above, we and 2.3 ) and angular – P q . The character Φ ] is the partition . For calculations 2.1 χ Φ C χ ) group integral over the , and ). We have not, however, organized as per the r.h.s. d d T ( , J O P , to take the schematic form, That is, PE[ . H ] , it is clear that multiplying by 1 Φ ) and 12 P d χ )). PE[ ), where we can only have 1 P 3.12 d ) d SO( – 14 – ; in essence we are removing a factor of momentum dµ J Z and label the states via a character Φ ∼ R H . P ∝ . That is, we find the Hilbert series, i Φ x χ (the PE is defined below in eq. ( ) of each state in the module. The tower of states in a single particle module i x J ) by exploring the difference between SO( ). Since their characters are proportional to 2.12 ), that is 2.9 i x To now get at states in the operator basis, we need to enforce momentum conservation Since asymptotic scattering states are built as a Fock space, the multi-particle parti- Take a single particle module Let us also comment on discrete spacetime symmetries ; Crossing symmetry implies we can take all particles as incoming or outgoing, which is why we only q 12 ( particle module we associatesions a and character transformation which propertiesmodule labels corresponds under to its the a states Lorentz conformalcharacter. by representation, group. the their associated When scaling character the is dimen- a single conformal particle need to consider one set of multiparticle states. 3 Conformal representations andA characters central tool to identifying the operator spectrum is the use of characters. To each single of eq. ( will remove total derivative states from from every multiparticle module. Finally, Lorentzover invariance boosts is simply and applied by rotations,angular averaging which variables is implemented using an SO( exponential (PE), which ismechanics the (although familiar the generating name functionfunction may for on not free systems be in familiar). statistical and Lorentz invariance. Consider the multiparticle states in momentum ( arise from translations,P which is reflected in the charactertion by function is a constructed factor from the of single-particle ‘momentum’ partition function using the plethystic we might expect from aand partition useful function, we object will for findfollowing studying the sections, operator Hilbert here bases. series we a Deferring outline very a interesting a more simple precise physical description derivationis to of essentially the the a single Hilbert particle series. partition function with weights for the energy ( attempted to include charge conjugation,back nor to to Minkowski space systematically where address time-reversal the becomes Wick2.6 an rotation optional symmetry. A partitionNow function that we for have the outlined the operatorcan definition sketch basis of the the form operator that basis in the sections partition function — Hilbert series — takes on the basis. As with pions, we work with the Maurer-Cartan forms). in this work, wehave work made in some Euclidean effort spacein to SO( eq. incorporate ( parity, primarily to allow us to probe the rings theta, Hopf, or Wess-Zumino terms in non-linear realizations (where instead of working JHEP10(2017)199 ) ), 13 d C ]. , = 1, (3.1) d 57 which ). The + 2 r c d are con- 2 d , l : b µ The SO( 2. For K ··· = , 14 ≥ 1 r ). l d ) the translation C ]. , C = ( , as it is associated . For this purpose, , 38 , respectively, while we l ] R l 4 + 2 descendants and scaling dimension + 2 ∈ d [∆; ), where the minus sign l d r D , l SO( and : it has rank ' , called ··· ] l d l , 1 R 2), which leads to finite dimensional O 2) , of spin [∆; / ), in SO( , l 1 A d, d . ∆ − 15 −       l ). l C O ] uses , primary 2 l O ) structure uncovered in [ 41 µ . . . 1 O ∂ C µ spin quantum numbers 1 , + 2 ) is compact, the spin labels are quantized, ∂ µ r d d ∂ ) are constructed as follows: a highest weight       C – 15 – , called l , ∼ -dimensions SO( O ]. We work in the orthonormal basis, where the generators . Due to the non-compact nature, the translation ] d l µ + 2 41 K d [∆; R = ] in our analysis, although we differ in notation on one point. For † follow [ µ 41 i + 1 quantum numbers ( P r 2 and explains the SL(2 / N ]. We allow spinors, so we actually work with the covering group Spin( ∈ − 58 ]; a somewhat gentler treatment in four dimensions can be found in [ , respectively. ] 41 l are lowering operators while the special conformal generators ; in contrast, the scaling dimension is continuous, ∆ [∆; Z µ e χ 1 2 P ∈ and r ] l , l [∆; Heuristically, this picture is easy to understand in terms of field theory operators. The Unitary representations of SO( The representations of the conformal group and their characters can be used as a This minus sign (with ∆ positive) leads to infinite dimensional representations for We adopt the conventions of [ Our conventions on factors of χ ··· 15 13 14 , 1 of the Lie algebrathis are basis, split weights intointroduction, of the see representations Cartan e.g. are subalgebra [ although together eigenvalues we with of will raising not the be and Cartan careful lowering generators. to operators. make∆ this In For distinguishment can a throughout be the physicsrepresentations negative text. oriented when (the ∆ free scalar field has scaling dimension characters of long anduse short irreducible representations [ unitary irreps consist of some∆ operator together with an infinite tower of derivatives acting on operators may be appliedweight state, an implying infinite unitary number irreps are of infinite times dimensional. withoutphysical annihilating intuition comes some from lowest conformaldence field theories, implies where that the operators state-operator are correspon- organized into irreps of the conformal group. In essence, state is specified andIn the addition to representation the is raisinggenerators filled and out lowering operators by of applyingjugate SO( the raising lowering operators, operators. i.e. ciated with the dilatationl operator. As SO( with non-compact directions. Informal summary, group irreducible are representations labeled (irreps) by is of a the convention con- due to our working with SO( We work with the conformalsubgroup group is in the usualmeans rotation that its group representations in are labeledextra Euclidean by two space directions inanother the label conformal ∆ group for increase representations, the called rank the by scaling one, dimension thereby as giving it us is physically asso- powerful tool to addresswe IBP redundancy, review as and wecomprehensive summarize will and show the excellent in necessary treatment section ofgiven results by conformal Dolan in characters [ in conformal arbitrary representation dimensions theory. is 3.1 A Unitary conformal representations JHEP10(2017)199 ) r (3.3) (3.2) . Due d . Under . In odd r r l l the EOM for SO(2 + | = 2 φ 1 r , namely that l − r r 1 l l ≥ | . The conditions = 2 for even l 1 / r , respectively, while − N (traceless symmetric r -form field strengths, 0 we have conserved , l 1, and Λ µν 2 d ∈ N T ) − s 1 2 r ∈ 0) , n > 1 and Λ , n l ≤ ): a unitary irrep is labeled − d i ··· is annihilated: it leads to null ≥ · · · ≥ ··· r . , , ) are non-negative integers. In even ) with 1 µ ≤ 1 2 r l ≤ . s ∂ 0 | Λ i ,  0) with = (0 = ( = 0 +1 = 0. satisfying these conditions, ∆ needs ≤ l , for 1 l l p ··· = 0 for fermions, etc., the free field l , s, n n > l , | µν , with 1 +1 ··· T i Z l > ] for 1 ··· , / ∂ψ and stress-tensor µ 0 | ∈ ∂ − l s, µ p +1 59 i j 1 for all other i l n, l , l | 2 for 2 for +1 i = ( − / l 41 = − = − l l = ( ]. In such a case, these states are removed and i i 2) – 16 – l ) corresponds to the usual Young diagrams used to label p − 22 for for l d r 60 / / i ··· − l = − + Then for each 2) 1) i d d boxes in the second row, etc.. = ( n , and 2 | , . . . , l − − 16 d l + 1 and 2 ( ]. These representations are correspondingly labeled by l for the irrep to be unitary. This lower bound is hence l d d 1 representations. | ( ( l l Z = 0 for scalars, ,( = 52 i 1 2 l = l + 1). ∆            φ | r 2 ∈ ∆ 1 long = l ≥ ∂ i for odd | l l representation. Accordingly, unitary irreps that are not short 1 2 , and is given by [ ∆ , conservation dictates , ≥ µν T short = 0 denotes the position (the serial number) of the last component in satisfy ∆ boxes in first row, 0 for SO(2 l s r 1 l , some descendant obtained by applying = 0 corresponds to the free scalar, while for ≥ l + 1, the Dynkin labels are Λ ), satisfies ) that has the same absolute value as the first component , the Dynkin labels are Λ r r ≤ r n r indices), where the bounds read: r l , l l , l n = 0; for p ) with = 2 = 2 = 1 and 2 are a conserved current φ ··· unitarity bound , s d ··· d 2 n , ≤ , ∂ 1 2 l ··· ≥ · · · ≥ , l 1 1 are the familiar ones for finite dimensional irreps of SO( l s, 3 are generalized higher spin conserved currents). In each instance, some descendant = ( l Our method involves working with free fields, using them as building blocks to con- To gain intuition, let us specialize to Physically, the unitarity bound imposes representations to have scaling dimension Requiring a representation be unitary places conditions on ∆ and These are the conditions such that the Dynkin labels (Λ l l ≥ 16 = ( = ( representations are short representations. dimensions dimensions this convention, for integer valuesrepresentations of with struct the operator basis. Ineven dimensions, odd we dimensions, may the have only scalars,and chiral free spinors, higher fields (anti) spin are self-dual scalars generalizations and [ l spinors; in to the free field EOM, e.g. currents ( n is annihilated by thedictates derivative action. For example, for a free scalar field tensors with Saturation for rated, i.e. ∆ = ∆ and subsequently negative norm statesthe [ irrep is calledare a sometimes referred to as where 1 l the components of greater than or equal to that of free fields or conserved currents. When a bound is satu- on by and to satisfy a lowercalled bound the ∆ JHEP10(2017)199 ) A C , a can , r (3.5) (3.6) (3.4) ) pa- , with G +2 T i = 2 d iθ ∈ e such that d g = SO( G i , then it takes , ∈ ∈ l ) ) is the contri- x g x x ) = dim( ∆ h ( ; , any ) vector and sym- ) ) (see appendix G q d > d G d ( ( l and ) d χ q ( ∆ , iθ + 1)-dimensional torus P . In the second equality q l l e ) 2 r x R ⊗ ⊗ ( = ) χ ) ) d 1 . q ( l ) . . . l R χ , 1 ( ( χ ) 1 2 − r ∆ of scaling dimension ∆ and spin x q ] ( = l , x ) , i.e. there exists r 2 )  th symmetric product of the vector ( [∆; R G ) = n ] for unitary irreps of the conformal n , x ⊗ R x 1 ) ( 41 d ) R ( sym representation of SO( d ··· rank( ) torus, i.e. χ ( l χ , l d . . . ) generates the contributions from all the 1 ∆ χ n x ) is the trace of a group element in that rep- q ) (the derivative is a SO( − 1 , associated with the scaling dimension, and ; ∆ + 1 sym ∆ + 2 sym x d q q ( , x R ( =0 ) ∞ ) 1 scaling dim spin X n – 17 –  = U(1) d ( x ( ) we parameterize the ( . For a connected Lie group n ≡ T l P C G ) ) , d l O ( sym x 2 ∈ diag( ; l O χ µ . . . q 1 0), of SO( n + 2 O ∂ g ( µ 7→ ) 1 d ∂ d µ ) ∆+ ( r ) is ∂ q P ,..., x (2  ; 0 ) for h = =0 ∞ q , Consider a unitary irrep g X ] n ( ( l ] l ) in the vector representation has eigenvalues R ) r [∆; d = (1 ( [∆; ) = R l χ x ; q SO(2 ↔ ( . ] ) = Tr l ∈ g ) R ( d ) for the parameters of the SO( ) by the following variables: ) is the representation formed by the h ( [∆; r c R ∈ ) that any unitary matrix can be diagonalized by another unitary matrix. ). If the irrep does not saturate a unitarity bound, i.e. ∆ χ 2 d i χ ) denotes the character of the spin r b ( N θ x , l . This is simply a diagonalization theorem, generalizing the familiar statement n ( ) = T d , . . . , x ( l r ··· and 1 ∈ χ = U( , x 1 C l G gh 1 ∈ = ( = ( − q θ Long representations. l the form for Since the trace isrameters conjugation of invariant, the it torus.(recall, depends For only SO( onx the rank( In this section wegroup. reproduce The character character formulae forresentation, [ a representation be conjugated into theh maximal torus 3.2 Character formulae which can be thoughtbution of from as the the primary momentumdescendants. block, generating This while function: functiongroup can element be computed as follows. In even dimensions, for general formulae), and weabove, have we used have the defined fact a quantity where metrization comes because partial derivatives commute).acting A on group the element abovecomponents representation acting is on an the infinitematrix primary dimensional the and character matrix descendants. with Evidently, block upon diagonal tracing over such a representation, where sym JHEP10(2017)199 (3.9) (3.7) (3.8) (3.11) (3.12) (3.10) , r ¯ a
1 . r − r ¯ a x
. 1 r )] − r a g
x . ( r , V r )) x is given by  . a x ) )
( uχ 3 i r G x ··· x ( qh . 1 , ) . f q/x PE[ ¯ a i ) 1 −
# − ··· ≡ ) 1 ug th fully symmetric products of (1 1 − 1 q/x m ¯ a ) + 2 n ! 1 x  x − th symmetric product of functions,
)(1 2 ) ( − i 1 1 n x 1 m f ( (1 − 1 a det , g 1
q x f x  ( V m ) we get )(1 , ) 1 ) m i ) u V 2 1 x x − 1 under a group a ( ) = x 3.6 det − r q x
Tr 0 ( f =1 x (1 1 ∞ a V monomial formed by the above eigenvalues ( f m X m
, x − ) x u r r 1 " n ) =  =1 + + 3 ). Therefore i ( Y g (1 n 1 m n 3 2 x , x ( n q = – 18 – ) ) ( ) ) ) r = r x x =1 d V 1 a coefficient of r =1 −  i Y ( ( ( sym ( ∞ ··· a ( +¯ X n m f f χ , 1 n n r +¯   ) 1 n a r u + 1, we have

r ) is formed by the )] = exp 1 2 1 6 q a − 1 ) = + sym r x (2 sym  + x X ( ··· ) = χ ( , x ; χ = = =0 ··· ∞ n x + n 1 q X n 1 X   ; + = 2 ( u uf ) ) a 1 ) q = exp , x r a d ( x x +¯ =0 ( ( 1 ∞ ) +¯ ) = (2 X n PE[ a 1 f f x a +1) P   ug ; r 2 3 + )]: it is the q 0 (2 ( diag(1 x − ) = a ) ( P x 1 d ( f ( sym sym (1 7→ [ ) P n V ) =  ( x n +1) ( ) ) r det r  (2  ( (2 sym h n χ +1) r (2 sym . Using log det = Tr log, the above can be rewritten into an object called the χ G ∈ With the plethystic exponential, one can define the From the above, we note that for both even and odd dimension cases g For example, one readily finds i.e. objects like sym for plethystic exponential, This identity is central tosymmetric many tensor of products the of calculations a in representation this work: a generating function for and Plugging this in and performing the sum in eq. ( Similarly, in odd dimensions, the vector components, each distinctshould degree- show up precisely once in Because the representation sym JHEP10(2017)199 ) is d ) is φ x } ( 2 and n (3.14) (3.15) (3.13) (3.16) ) µ / V ∂ ( 2) n ··· − 1 sym d . µ ( , etc. χ { )] ) vanishes for 3 ∂ g = 0, as a result ≡ , as it is usually ( V . ( f φ 0 V ) = n 2 in the descendants. 2 2 2 ( ∧ ∂ ) that uχ 2 φ [ ≥ , } f n n < n µ 3.11 ∂ , sym PE . . . 0 1 . This fermionic generating β ≡ ··· ) + . − x 1 ) 3 e ! ( µ ) ) { ) and ( , with ∆ = ∆ = ug  = ), a similar identity holds: ∂ m φ ( 2 2 g V u ( − ( × 3.10 0 n n V + 1 + 2 + 3 (1 + )]. With this definition, we obviously . . . 2 ) ∧ ∆ 0 0 0 x V d Tr ( sym ( ∆ ∆ ∆ m χ uf u scaling dim spin [ 0). The set of components − , f +1 ) ) ) = det x x m φ ( ( g ··· } m ) ) – 19 – ). Let us first look at some examples. Consider ( φ , 1) 3 )   } 0 )], one can define the antisymmetric product of µ ( ( 2 − V φ ∂ x 3.2 n n ( ( µ ( 2 1 n, n . . . ) ) φ ∂ respectively, with µ µ f ∧ d d 1 [ =1 ( sym ( sym ∂ ∂ 1 µ ∞ χ n 1 = ( X { χ χ m  n µ ∂ ) l symmetric components { u

     u in eq. ( ∂ l =0 ∞  coefficient of PE X n = ) = The character formulae are modified when a unitarity bound ). Frequently we omit the subscript on PE ] n by contracting two of the indices while leaving the others fully x x ;0 u ( ( 17 0 φ ) . Check familiar statements like ) = exp 0) 1 n traceless , )]. V [∆ ( µ − x ug ··· n R ∂ α , ( ∧ 0 ) V + χ d n, ··· χ ( ( α [ (1 + 1 )] as the χ n 0). The shortening condition comes from the EOM µ V x , ∂ ( ) = )] = α f the Young diagram labeling the traceless symmetric representation of SO( x [ ( det ··· ( 2 n } ). In physics language, the difference compared to the symmetric case is a re- ,  χ V ∧ V (0 χ n [ {z ··· indices, corresponding to n ≡ ∧  | n dim( For anti-symmetric (exterior) tensor products, A simple exercise to familiarize these operations is to work out some examples for SU(2). The doublet 17 = 0 l of which there are only Thus the representation looks like have clear by context what is meant. Short representations. is saturated, i.e. whenthe ∆ short = ∆ representation formed by the free scalar field Similarly to thefunctions definition of sym Note that then > sum truncatesflection of since the statistics the —partition function it given antisymmetric simply by generalizes (1 product thefunction familiar can case again of be the written fermionic/bosonic into a (fermionic) plethystic exponential the same as sym and so forth. With this definition, we see from eqs. ( character is with with obtained from symmetric. Therefore, we have JHEP10(2017)199 ] l is ; l ) d e χ ( [∆ e χ (3.18) (3.19) (3.21) (3.22) (3.17) Let us (3.20a) (3.20b) . The general , ) from the long e χ ) . x . ··· 0)] ; ] , : should be removed. q ) ;0 ( φ, d d ν ) 1)] ( [ 2 , ) d (4) [4;(0 ( χ ∂ L χ 2 P F . ). That is µ − (4) [2;(1 ) l · + ∂ 2 χ , 1 q ∂ ] ∆ 3.5 0)] , µ 0)] 0)] . , , − ) ··· > , . Thus the character is given by +2;0 x 0 φ, ∂ , from ]. For long representations, 0 0), which is the same as a con- ; (1 ) (4) [3;(1 (4) [3;(1 l 2 , 0 q d . As in the above examples, e 0). Current conservation dictates χ χ ( [∆ ∂ ( = 0 ∆ , 1;(1 0)] ) χ , µ χ q d l ) − − − for ∆ ( 1). The EOM and Bianchi identity d d , ( [ − , = (1 ··· P d (4) [3;(1 ] φ, ∂ 1)] 1)] χ , ) , , l ) = , 2 e χ ;0 0 x x 0 ∂ , ) = ( ( ) d = (1 (4) [2;(1 (4) [2;(1 ) ( [∆ d 0) ( l , x 0)] χ χ l χ , ; χ ··· = (1 q , ··· = = ∆ = , ( 0 l – 20 – ] 0 ) q ] ), with the tilde, to denote the actual conformal , l ) we can write it as d n, ) ;0 1)] ( ( x , d 0 ≡ ; ) ( [∆; χ 1;(1 is different from 3.5 d q ) n χ ( [∆ ) ( − x ] e d (4) [2;(1 + e χ d χ l ( [ ; 1 and 0 ) e χ representation q e χ d ), ∆ ( has ∆ = 3 and ) = ( [∆; l − ] q : l e x χ ) ν d : ; ), we obtain the character for the short representation in : d ) φ =0 q ( [∆; ). ∞ L ( X n µ short ] χ l j 3.6 F ) with ∆ = 2 and L µν ··· · d free ( [∆; F ) = by subtracting off another conformal character vanishes automatically by anti-symmetry. Hence, the character is ∂ φ, e χ µν x 2 ] ˜ ; µ l F µ ; ) itself is an operator with ∆ = has ∆ = q l ∂ ) ( L ] 1 d µ + µ ( [∆ µ F j ;0 j · χ 0 µ ) conserved µν d ∂ ∂ [∆ ( F φ, ∂ ( : e = 0, so that descendant states proportional to ( χ µ µ χ ), without a tilde, to denote the function in eq. ( ∂ = x field strength ; φ, ∂ L µν q d ( F ) as 4 ] L µν l µ ) F ∂ d [∆; ( = 0, where 3.15 In some instances, we must subtract off another short representation, as opposed to Similar interpretations can be given to other short representations. For example, a χ µ j µ just given by For short representations (∆ = ∆ is obtained from use On the other hand,character we of will the use conformal representation labeled by [∆; The general shorteningmake rule some clarification and about notation our for notations conformal on characters. conformal characters. We will always However, the operator ( served current, and thereforethis saturates because a unitarityobtained bound by itself. subtracting the Manifestly one can see a long representation. Ansions, example of this isimply the left-handed field strength in four dimen- conserved current ∂ rather simple to interpret. Using eq. ( which clearly reflects subtracting offmultiplet ( the states ( eq. ( where we have used a tilde to emphasize that this is a short representation. This result is Using this, as well as eq. ( JHEP10(2017)199 ): B , one (3.23) (3.24) (3.25) (3.26) e χ + 1, the l that has the , = ∆ l . Note that in d − l χ ). For any is the order of the | being subtracted is a 3.20 e χ W , . | ) in odd ) 1 2 , ). The only exceptions are 1; since ∆ α , x G − ) is a short irrep itself. This = 0 6= 0 3.2 ), and ( ] for the proof. l − ··· l l . , p , 2 41 . When the integrand is a class 1 2 (1 3.23 R 0 0 ) 3.19 = G 1 for for G R − = 0 = ( l δ ), ( rt( l p Y x > x x < = 1. We use the shorthand notation ∈ ] ] α | − − i ) = l l 3.18 j for for all other cases g x j | ( 2 +2; +1; l l πix ) ) dx R d d 2
( [∆ ( [∆ χ l 1 for 0 for 1 for = 1; in these cases the by 1, i.e. ) e e p χ χ is a linear combination of the j l g l – 21 – l − by a simple replacement: Y ( p p − − e 1 l χ is defined as the last component in ] ]        l l ∗ R I ; ; l l l sgn χ ) ) | p ≡ d d l [∆ ( [∆ ( G 1 ) − W χ χ | x l p dµ l being subtracted in eq. (      → ) reduces Z e χ sgn( ) are given in eqs. ( ), or when ) that = → ) is taken over all the roots of G 1 2 ] l l ; G p 3.24 l 3.2 l  dµ ) 3.23 , d obtained from , and sgn is the standard signum function: ( [∆ rt( 1 2 1 Z l e χ , − generically saturate the bound in eq. ( l ∈ with − l l l ··· α p , 1 2    18 ) iteratively until the character being subtracted off corresponds to a long ≡ and − − 3.23 l l and ( is the invariant measure (Haar measure) on G = 0 dµ l ]. Here we only summarize the results, and refer the reader to [ The characters for all short unitary conformal irreps are given in eqs. (3.25)-(3.27) and (3.32)-(3.34) 41 18 The product over Weyl group, and the contours are taken along of [ where function (conjugation invariant) — as clearlycan is be the restricted case to for the characters torus — and the is Haar given by measure the Weyl integration formula (see appendix For a compact Lieintegration over group, the the group: characters of the irreps are orthonormal with respect to values for ∆ when long representation and the iteration terminates. 3.3 Character orthogonality Explicit examples of eq.can ( use eq. ( representation. So itthis is iteration, clear generically that the any is because generically eq. ( same absolute value as where we recall from eq. ( with the lowered spin shortening rule is JHEP10(2017)199 ). r ) in , l C (3.29) (3.27) (3.28) , given in ··· + 2): , 1 d + 2 , +2) l d 0 d ll , δ SO( = ( ) 0 2 l ) we use in this dµ q ∆∆ ( C δ . f , ) 1 πiq ) = dq + 2 − everywhere else in the 2 x x d ), namely SO( )). ; ; + 1, with 2 q 1 C q =1 r ( 19 | , ] − . We will have to carefully q 0 3.28 | l ), it is trivial to check that q . χ ; ( → I = 2 0 ) ) + 2 d d B d q 2 are orthonormal with respect 1 ( ( [∆ d 3.21 / χ together with an infinite number P ) = Z )
q l and ( 1 x . ∈ ). In particular, if the theory under ; r f − ,l q q x ( π ∆ ; ) = 2 4 1 dθ d 3.28 O ( d − π q P 4 0 ) ] Z l x ) with ∆ ) characters). We emphasize that the modified ) ( , needs to be extended to the double cover: ) d d ) and ( ( [∆; q d ) holds. Up to a proportionality constant it is just the – 22 – for both iθ 3.21 → e SO( 3.26 3.27 +2) dµ χ = d orthonormal with respect to this measure, as can be dµ ) ). The above can be simplified to a product over only q in these formulas to obtain eq. ( q Z r SO( ( Z q ). In the characters, this appears as possible square roots not f , α dµ ) piece alone and replaces C q → ), which is quite useful for practical calculations. The Haar ) = π , ) for the SO( are linear combinations of the πiq dq x 2 dθ ··· 2 ; πiq + 2) +1 (restricted to the torus) is given by e χ , r +2 q π r 1 x ( (2 2 B.11 3.25 d ] I defined in eq. ( 0 0 α . For simplicity of discussion, let us focus on half-integer scaling dµ l ] Z ; x l 0 = = (2 ) dq/ ) of scaling dimension ∆ and spin d = ( d ( [∆ ( [∆; ,l dµ ) = dµ χ α χ ∆ and ) q ( ) of SO( ) (replace x O q Z ; f C for q , + 2): ( r ] B.16 l d ∗ α r πiq dq for short irreps are ) +2 x 2 d ( [∆; e d χ =1 ··· | ) and ( ) holds (using eq. ( q 1 2 dimensions, this implies that all values of ∆ are positive, half-integer. In this | 2 dimensions, unitary irreps are infinite dimensional. Structurally, they consist of a α 1 dµ χ I x ≥ ≥ B.14 3.27 As a technical matter, one has to keep in mind the covering group Spin( With the explicit expression of the character in eq. ( The above results do not immediately generalize to non-compact groups, such as the Z This measure is normalized so that eq. ( ≡ d d 19 α primary operator of descendants consisting of derivatives acting on measure for SO( eqs. ( integrand. 3.4 Summary Conformal representations are labeled by their scalingIn dimension ∆ and spin dimensions. Then the integral over that is, one leaves the applying the contour integrals ofconsideration has eqs. either ( half-integer scaling dimensionscover or Spin( spinors, then we are inof the double the arguments eq. ( characters seen from the factaccount that for the this non-orthonormality. where the Haar measure free field limit, the scalingIn dimension of operators coincidescase, with the the characters canonical dimension. to integration over the maximal compact subgroup of SO( the positive roots, see eq.measures ( for the classical groups are tabulated in appendix conformal group. However, forwork, the there is specific a representations notion of of character SO( orthogonality that we can make use of. As we work in a x JHEP10(2017)199 , ), A are e χ 3.19 (3.33) (3.30) (3.32) (3.31) (3.34a) (3.34b) ). Such x ), ( . ; q ( ) 3.18 d ( + 1 r r P ) +2), the maximal x ( = 2 = 2 d ) , d , d d . ( l ) )] χ )] g x , g ( ∆ ; l ( q q V ) V ( ∆ i P uχ > ) [ uχ ) = 2 q/x f x q , ; ) 0 − i q ll − ( ] δ l 0 for ∆ 1 = PE [ ) (1 )(1 q/x . In this case, some descendant state d i l 0 ( [∆; ) ) = PE ∆∆ ∆ − , χ δ ) q x q ug ug x 1 ), from which we readily see that the = − ; )(1 = − i ] q ] ) is 0 1 ( l (1 ; x (1 0 (1 + 3.23 q x P ; V ) V q +2;0 r [∆ =1 ( 0 − x i Y – 23 – χ ). ( ] l P det l q [∆ (1 χ χ 1 − ∗ [∆; ∆ 3.28 r =1 − q 1 ) = ) = det i Y ] g g ( ( ;0                dµ χ , the essential idea is to consider the enlarged operator ) ) ) 0 characters with detailed expressions given in appendix ) = V V C l 2 ( ( [∆ Z , n n χ ) = q, x ∧ x ( . Some explicit examples of this are given in eqs. ( χ ] ( = + 2 sym l ) n χ ] χ d  u ) spin- ;0 ( [∆; n d 0 n u χ =0 ∞ [∆ X n e =0 sym χ ∞ X n χ n q =0 ∞ ) are orthonormal with respect to an integration over SO( X n x ; ) are the SO( ≡ q x ); we reproduce the free scalar character here: ( ) ] ( l l x ; χ [∆; q 3.20 ( χ Finally, a comment on notation. In this section we have used a superscript to denote We frequently make use of symmetric and anti-symmetric products of representations. In the free field limit, where all scaling dimensions are positive half-integer, the charac- A short irrep saturates a unitarity bound, ∆ = ∆ A long irrep does not saturate a unitarity bound. Its character is given by P . As discussed in the section drop the superscript, as we have done in this4 summary. Counting operators: HilbertWe now series proceed to buildingK the Hilbert series as a partition function on the operator basis what spacetime dimensionality functions are in,a e.g. notation is cumbersome. Since the meaning is generally clear in context, we will typically The plethystic exponential serves as a generating function for them: with the Haar measure given by eq. ( ters compact subgroup of SO( becomes null; this state andgeneral its shortening rule descendants is are summarized removed inlinear to eq. combinations ( form of the the shortand multiplet. ( The where and the momentum generating function JHEP10(2017)199 , , i φ in J Φ R φ R is then back into K J ) to the . Some of the q, x 5 ( i Φ χ is readily constructed )) how the cohomological J 3.32 7.2 . We start with the single par- ) x ( . . 0) J ) and (         ,..., 0 φ k, 3.17 } ( φ 3 . The piece corresponding to ) for can also be addressed without recourse χ } µ 2 k J φ ∂ µ + 1 2 K . . . 0 φ ∂ µ µ 1 ∆ ∂ ∂ µ to q 1 { φ, q, x µ ), ( ∂ – 24 – { =0 ∞ J ∂ Z k X 2.4 are representations of the conformal group, then         i ) = Φ = is built from tensor products of single particle modules , emphasizing its basic ingredients. R φ is constructed by taking symmetric tensor products of q, x J R as the scalar primaries. ( 4.4 φ . J K R . For this reason we give a summary of the main expression χ H K ⊂ J dimensions. Along the way, we show how the physical partition function can d and select out ). Therefore, a generating function which labels states in J 2.7 We begin by deriving an explicit integral expression for the Hilbert series for a real We stress that the reduction from To explicitly obtain the Hilbert series, we associate characters . In the special case that the i Φ -dimensions. The modifications to include multiple fields, symmetries etc. are described Recall that the operator space eq. ( The corresponding weighted character is (see eqs. ( The character generating function ticle module for a real scalar field, eq. ( 4.1 Deriving a matrixFor integral clarity formula of for presentation, thed we Hilbert will series derivein the the following Hilbert subsection. series for a real scalar field prove useful for thederivation and study details of in thesepractical the first computations real sections of scalar arefor rather field the technical presented Hilbert and in series bear in little section section weight on scalar in be obtained rather simplyeralization from to our multiple formalism. fields and Weexplained possible next how internal present to gauge the include and straightforward parity global gen- if symmetries. one It wishes is to then consider parity invariant EFTs; this will selected using character orthogonality, suchexpression that for we the ultimately Hilbert arrive series. at a matrix integral to conformal representation theory.nature of We the will IBP show redundancydifferential allows forms. in us section to obtain the Hilbert series using the language of R itself forms a (reducible) representationconformal of irreps, the we conformal identify group; decomposing allowing us to define a generating function for space JHEP10(2017)199 φ (4.3) (4.4) (4.1) (4.2) ). , φ, q, x ] ( l Z [∆; , ] e l χ . ) ,l . n [∆; ( ∆ i )). We call this the ∆ e χ ) b p ) ) ,l φ φ ( ∆ q, x 3.11 X ( ). They can be projected ,l ( 0 . Therefore, the operator , φ φ ∆  = ( ] The key insight is that the ∆ R C 0 0 , , )), which we reproduce here: ]) C 0 ,l ∆ ;0 φ χ , 0 ∆ [∆ ) and ( X C h ∆ 0 . e 3.33 X χ ll 4 ([∆ δ φ φ n 0  3.10 + ∆∆ sym = 1 + 8 δ = 1 + e ) = PE ] χ φ l , 0 = = PE l, Z [∆; ] Now it is clear that to obtain the Hilbert q, x δ 0 ( l e ) = χ i ; ⇒ ) ∆ 0 . Keeping only the scalar primaries amounts ) ,l φ φ ) should be an integral over ∆, as the scaling dimension φ = ) p n ,l ( R ( ∆ ) [∆ n ). R 0 Hence, we can rewrite the character generating ( ∆ b ( , – 25 – 4.2 φ χ b φ 8 n ] ] ( l l ,l ( n φ χ ,l C 20 0 h ∆ φ , X ∆ [∆; sym ∗ [∆; n ∆ C 2. In this sense, ∆ is effectively quantized for our purposes and χ R =1 / φ = 0 piece for convenience, and performed the sum over n ) C n ∞ ,l ,l . Therefore, the Hilbert series is defined as Z φ ) from the character generating function n 0 dµ χ n ∆ ( ∆ X =1 l, ∞ P ∈ b X n δ =0 ∞ ) = PE Z φ, p ,l X n ∆ ≡ ( 1 + ∆ p X ) will also form a conformal representation. We just need to H φ ≡ ) = = ( φ φ, q, x ) ,l
( ) = 1 + ] R with ∆ Z ;0 ] C l 0 φ, p ( φ, q, x [∆ ( [∆; H φ, q, x e ) using character orthonormality (eq. ( R Z χ ( , so we anticipate that ), as opposed to an integral, is appropriate. Z 2 n ] into conformal irreps, and then the highest weight state in each irrep, namely 2 4.2 , and so ) ] φ, q, x 0 φ J ( sym , with the plethystic exponential (see eqs. ( µ 0 some unknown multiplicities. generated by Z ∂ φ [∆ ) ,l R n ) gives the number of operators of dimension ∆, weighted by the number of fields R J ( ∆ χ φ b ( Characters nicely keep track of the decomposition into conformal irreps: = 0 In general, the sum over ∆ on the r.h.s. of eq. ( and [( , is just the operator space formed by the scalar primaries. We now follow this strategy with the definition 20 φ ∆ 8 is allowed to be continuous inof the free conformal group. field However, representations asshow our that up starting in have point the integer is decomposition tothe or take have sum tensor half-integer ∆ in products scaling eq. dimensions, ( the representations that Computing the multiplicities series, our task isout to from compute the weighted multiplicities C in the a given operator.φ For example, in four dimensions at ∆ = 8 we have two operators, where we have separated outn the to replacing each with function as K to define the Hilbert series single particle module forms precisely theR free scalar representation of thespace conformal group: decompose the primary operator, is what will survive the IBP redundancy. Then the operator basis character generating function and denote it by IBP addressed by conformal representation theory. from JHEP10(2017)199 . ] l in ] into l [∆; is the (4.6) (4.7) . For χ (4.5c) (4.5a) e (4.5b) χ [∆; e χ χ e χ , so we first ), we have the . χ
and the 1 4.5c , e χ
− 1 Z short irreps . − ] ) and (
, , Z , all 2] 3] 0)] already discussed above, +2;0 1]
4.5b − − 0 , ] ] ] r r − for characters of long and short ,..., [ [ r ;0 ;0 [∆ ;0 0 [ e e e , χ 0 0 χ χ d e [ χ χ ) [∆ ∗ [∆ , χ − − 0 1;(1 ] e − , χ χ and ,
0 ] − ] 1] 2] − 1 r d χ ∆ ∗ [ − − +  r r +2;0 ] [ [ [ [1] − χ C ). 0 +2;0 χ χ χ χ ), but there are two exceptional cases 0 Z + [∆ − φ . . . . +2;0 ] e [∆ = = = = χ ( 0 ]
0 , 4.5b 0 ;0 0 ) allows us to absorb all the nontrivial χ 1 , , (no tilde) is defined as a function while d ). Note that this orthonormality relation ∗ [∆ ∗ [ 0)] 0)] +2 ∆ , , − ∗ [∆;0 − 0 χ +2 χ χ 4.7 1 0 0 C ] , , are linear combinations of the 1)] Z ∆ 1 1 ;0 ∆ 3.28 0)]  C , 0 e
, χ C 0 1 dµ χ dµ dµ ,..., ,..., , , [∆ 1 1 – 26 – 0 1 , , , and then make use of the above orthonormality ∗ [0;0] ), that + ( + χ Z Z Z χ ] ] χ ,..., ,..., ;0 ;0 0 1 = , + , 1);(1 2);(1 0 0 3.24 ] ) = ) = ) = − − ;(1 r r [∆ [∆ ;0 , we need to pay attention to φ φ φ )–( r ( ( 1;(1 0 ∗ [1;1] e 0 is given by eq. ( ( ( ( χ χ , − − − − χ 0 0 0 0 0 , , [∆ , , d d d d 0 ∆ [ [ [ [ , d, 0 0 3.21 e + χ ∆ e e e e = 2 and instead of the two equations ( χ χ χ χ C +2 ∆ ∆ C They stem from the overlaps between the component when written as linear combinations of the However, the +2 C 0 d is given in eq. ( 0 . . . C C ≡ ≡ ≡ ≡ ∆ ∗ [2;0] ∆ 22 ] 21 χ C . ⊃ = . r 1] 2] = 0 C [1] dµ d e  χ − − l 1 e [ χ r r dµ [ [ e + 2 = χ e e − R χ χ 0 ); outside of this context we typically use : Z ) = , d φ, p φ ( ( 0 H , ) purely in terms of the , not the + 2 + 2 : ), it is not hard to see that apart from the 2 χ 0 0 : ) gives the generic expression for C 4.3 d + 2 and ∆ = 3.23 0 4.5a 6= ∆ ) that include an ∆ = ∆ ∆ = ∆ , the recursive nature exhibited in eq. ( 4.3 To properly compute the Eq. ( χ We remind the reader, per eqs. ( In two dimensions ∆ 21 22 actual character for a conformalto representation. carefully This derive distinction isirreps, important for respectively. the next several equations single equation where we have used obviousthe shorthand notations for the subscripts. Rewriting the Therefore, the coefficient eq. ( From eq. ( all possible such short irreps are we have at ∆ = ∆ example, using is among the express eq. ( relation to project out the coefficients. Carrying out this procedure carefully, we find where the Haar measure JHEP10(2017)199 ) 2 1, 23 q < φ, p 1) | (4.8) (4.9) → ( φ (4.10) (4.11) (4.13) (4.12) , | = 1. − q H
| To com- 1 ] . Z q | . ( −
;0 ). = 1, , which are d 1 0 x [ 1) starts with | ). 1) )+∆ ; x ∆ [2] χ − | − 1 0 4.5c  − − Z Z d, q q φ, p p ( Z C . ): ( 0  0)] d φ, q, x P ) + # , . . . , χ 2 H ( 4.4 p / 0 ,..., , 1] 1 0 0)] [1] ) ∆ Z , − ) − )= e ( x r χ  [ ( ,..., ) 0 1 q 1;(1 p [1] − , p/q 1]  , χ φ, p − ( ] ( C 1  ( d r ∗ [ n − 1;(1 2 −  ∧ inside the contour + − H [ / ) − is given by eq. ( 1 1 χ ) d χ ) into eq. ( p ∗ [ ) n ) 0 dµ χ ··· ) χ √ x ) as d, d p 4.5 ; 1 Z + p/q p 1 C q φ, p, x − d ( = ( 4.9 ( ( φ, p, x 2] p + ( q − P − Z ] d + =0 [ r , Z [ 1 q 1 ;0 X n
) ) 0 e χ = 0 because the expansion of ( 1 x x ). After doing so, in the second line, the = = 2] q πi ··· ∗ [∆ ; ; − dq 2 1 1 − p p
χ
r ( ( + Z ) 1 [ ] x 3.29 +2 − I ] P P C ( 0 ;0 ) ) ) x ;0 d – 27 – , d d d results in eq. ( [ ; 0 ∆ + ) 1 p p h χ 0 ∗ [∆ , 1] − SO( SO( SO(  ··· + q − ∆ − r ] 0 [ 1 [1] C dµ dµ dµ + ) are analytic functions inside the contour, since e dµ χ χ φ, p, x 1) is regular at P ] C (  ∗ [∆;0 1] ;0 Z Z Z 2 − n Z d − Z − χ [ ). + Z r ) φ, q, x = = 0 [ ( 0 ∆ χ +2 ( x 0 0 d, p 0 C ∆ 4.1 ; Z = det 1) = ∆ d, ∆ ). 1 C p . Therefore, the coefficient  ) ] p + ( ∆ )  C q p − ] X ;0 x + P r d p/q " [ + + ;  ) 3.13 d Z ) and  1 [ d p ( χ p x ( e ; [1] [1] χ =0 dµ ) from the character generating function # ∞ ] q ] e +1 X P n χ χ r ( SO( d  Z 0 0 and [1] [ [1] = /P dµ 1) ∗ [∆;0 φ, p : ] C C C terms in the second and third lines contain ( χ [1] − [1] Z ( = ⊃ = χ ∆ e H χ ∗ [∆;0 p ··· 1 ≡ ≡ χ ) = 1 + ) ) ∆ ∆ − X p 1. The expression ( , as is obvious from eq. ( " Z φ, p φ, p φ, p 0 ( ∆ ( ( < ∆ X 0 | dµ q H H With this result, we write the Hilbert series in eq. ( p | H Note that both 1 is that for the double cover group. Practically, this means that we should send ∆ Z 23 and order which follows from eq. ( with in the integrand, asintegral explained evaluates around to eq. theFinally, ( residue in the at last the line, single we pole have used the fact that In the first line above,dµ the square root in the integrand indicates that the proper measure we get Let us massage this into a more useful form. First focusing on the sum orthogonal to Computing pute the Hilbert series, we plug the where the + overlaps into JHEP10(2017)199 .  1 1) 1) (4.17) (4.14) (4.15) . − − (4.16a) (4.16b) d − d φ ( ( As a side ) q q

. ) matrices,   , x 1 d 2  − 1) 1 d q ... −

φ, q −  ( ) 0)] Z Z ( 1)  ,..., 1 P − 0 0 ,  . After explaining how Z 1 φ, q, x ( 2∆ ) d ( q ) is closely related to the x 1 P ( Z ) ≤ dim[(2    0 x ) . ) , ; ) χ x 2∆ 2 ) x 1 0 +2 , D q ( d q 0 φ, q, x ; )

∆ ( ( 1  q q ) . Similarly, ∆

( 0 χ P q Z . From the above expression, it is SO( ) ) , x φ, p P q d 1 2 2∆ ( H −  dµ q H πi ) SO( (1 dq 2 φ, q Z x φ, q, x ( ( d ( 0)] dµ  p Z I 1 Z χ ,..., Z ) ) ) , ) + ∆ ) + 0 d d d d , d p 0 0)] begins at ∆ φ, p in the integrand as a consequence of using the SO( SO( SO( + q SO( ( ,...,

0
– 28 – 2 0 0 dim[(1 1 ∆ dµ dµ dµ q Z dµ ) q H n, )

d − Z Z Z → +1 Z Z ) 0 , ) d q ) = ) d SO( dim[( +2 +2 +2 ∆ p , x 1 ) 0 0 0 q 2 n ∆ ∆ ∆ dµ SO( = φ, p + − p p p ( 0 ) of the free field theory. The partition function for the free
) are evaluated in a similar way as above: φ, q Z dµ φ, φ, p, x q ∆ 1 ( = = = H ( ( is the basic skeleton for any theory one wishes to consider and q )(1 Z
Z +2 +2 0 − φ, p Z 0 Z 0 0 1 ( − ) ∆ ) expresses the Hilbert series as an integral over SO( +2 H ∆ ∆ Z q x 0 H − p p ; (1 ∆ 1 p − is specific to the case where the single particle modules are conformal Z 0)] p ( + + 4.16 ] =0 ∞ ] d d (1 Y + P H n ,..., 61 p p ;0 ) d only contains operators with mass dimension 0 0 , d p = +1 +1 ∗ [∆ H ) = d d +1 1;(1 SO( q d ( 1) 1) − d 1) dµ ∗ [ Z − − dµ χ − Z Z = ( = ( dµ χ = = ( +2 H 0 0 Z can be evaluated more explicitly, as shown in appendix H Let us summarize what we have derived. We split the Hilbert series into two parts ∆ ∆ d H p p ∆ H theory is given by [ ∆ Relation of the generating functionnote, to we the mention physical that partition the function. physical character partition generating function function is a central result; ∆ representations. Eq. ( giving a concrete computational methodmanifest for that obtaining ∆ to include multiple fields and possible symmetries (next subsection), the last two terms in The above expression for where in the second line, we have specified to the case of a single real scalarand field find that Putting them together, we find where in the firstdouble line we cover have measure, sent andfact in that going the from expansion the of first to the second line we made use of the The last two terms in ∆ JHEP10(2017)199 . ) ) ) φ ), q q } ( ( n 7.2 µ Z Z ∂ (4.18) (4.19) (4.20) φ, q, x 4.16a in even ( i ··· Z F 1 µ is generated { , eq. ( ∂ 0 ) has a simple J H ≡ q ( ] and section ) can be obtained φ Z n 39 ∂ , , , which are eigenvalues  φ, q, x 3 β ( )) − Z x 2 − e ( d h − = n n + + q 0 n 2-form field strengths . ∆  d/ φq and the formula for 0). The form of with − = 1) 1 − n  transforming under some internal sym- H + 1 } of the conformal group. This assumption 0 (1 i ,..., ): i − ∆ , q, x 0 , so that the single particle modules form 0) Φ + ∆ q Φ d , n { 0 G ) n, ,..., 3.34 = 1 + 0 H ,C . The non-linear case is more subtle and we , as well as ), we see that it is very simple to obtain i n, φ n – 29 – G ( = ( +2 ( = ψ ) continues to hold when we add more fields and d  l Z 4.17 H det 1, namely SO( 4.20 ) = R =0 ∞ under → Y 0)] = q ) only requires information on the number of generators . In the following we assume: n i ( q i x ( Φ , spinors Φ Z i Z R ) = G, φ ,..., 0 R . 1 and n, 7 φ, q, x ( → Z ) and the identity eq. ( φ ) scalars dim[( transform linearly under . These states have energy 4.1 i 3.1 = 1). The relation ( ] and related works. ,... . After removing the EOM redundancy, the operator space ). Comparing with eq. ( form representations x ) for the partition function. 2 ): send d ( i G , 63 V 1 Φ , , χ R 4.17 62 SO( φ, q, x = 0 ∈ ( ) = is easily relaxed, as thecan splitting be derived using theWe differential presently form stick technique described to in(see this [ section assumption primarily fordimensions. concreteness, where it includes The fields Φ linear representations address it in section The Z h n V The reason for this is that The determinant form of our character generating function • • 4.2 Multiple fields andFor a internal general symmetries EFT we canmetry have group multiple fields by the single particle modules arguments of the character todim( unity, as this measuresgauge the dimension symmetries. of The the representation, provides use of a plethysm simple andfound interpretation the in of character [ generating expressions function of the free-field partition function from of a given scaling dimension. In terms of characters, this is accomplished by setting the with with of the Hamiltonian inmulti-particle radial states quantization are (the simply dilatation builtarrive operator). at from As ( products the of theory the is single free, particle states, henceby combining we eq. ( is the dimension ofinterpretation: the the representation Hilbert spacestate-operator is correspondence, a the Fock single space, particle states built correspond from to single-particle states. By the where JHEP10(2017)199 . φ φ ) J 2 D (4.24) (4.22) (4.23) action (4.21c) − (4.21a) (4.21b) 2 ( † Z φ . Thus, the ) collectively = G ) 2 , G ) and adding a | # ) ) for the operator Dφ . | rank( with the associated ). . , q, x, y d i d ( q, x, y } Φ i ( even) − , i , q, x, y ≤ ) Φ G, , . . . , y (see also comments there Φ O } d { 1 i R χ ( . We emphasize that while , y i Φ ) Z 2.5 ⊗ { Φ y D Bosonic fields Fermionic fields )( ( ( = ( i φ, p, x y i i ( . Z ( Φ y  , i X  i ) Z G i G,ψ ) " C Φ , Φ ) with χ G,F χ ) χ pg i continues to hold when we allow for χ +2 i x ) d Φ , ) is a certain product of this ; − Φ  x )  d q f ; 4.1 ( ) ( y ), where SO( φ, q, x (1 q ) = PE ( )] y ( d ( i O  R ( 1 2 ( i Z -invariant scalar conformal primaries in 1)]  Φ O , = PE G,φ = G = PE  det 1 2 , χ G, ) 1
-singlets, we simply integrate over
i ) – 30 – d , q, x, y χ g g . Specifically, ( ,..., Φ i x i G ) i ,..., 1 2 ) consists of two connected components, the parity ; O ) to G x R Φ q d ;( ; (Φ ) in section d , so that ( ( , as discussed in section i 1 2 ] q dµ d − µ ( Φ O ;0 + +1;(1 i R 1 1 0 0 0 1 + Φ A φ, p Z Z Φ ( , i ) [∆ [∆ [∆ i i Φ Φ C e e e H χ χ χ , Y ) is the Haar measure of ) and the parity odd component R ) = R y d ( +2 d G det ) = ) = ) = ) = det then consists of φ, p ( ). This is worked out in appendix ), gives us the means to explicitly evaluate the last two terms dµ 0 SO(          K e χ H ) = SO( q, x, y q, x, y q, x, y 4.21 ( ( ( d ) = , q, x, y as opposed to i i i 4.16b ( ) = } φ F ψ i + ) are modified by replacing , our Hilbert series does not count their kinetic terms µν χ χ χ , is given by Φ O ψ 0 F { ( H 4.16 , q, x, y q, x, y i Z ( . To further project out the i i Φ ) of eq. ( ). Therefore, as a group, (Φ Φ i d χ Φ R , as these terms are proportional to the EOM. This is not the case for gauge fields, ) is conceptually important, computing it is often of little practical relevance: it . Letting ) in front, where φ, p Z y ( J ( / Dψ φ, p H G For a single real scalar field, when including parity invariance the main piece of the Two comments/caveats about our formalism are as follows. First, for scalar fields The operator basis We next construct the character generating function With these assumptions we have ( ). Parity acts as a reflection in ¯ d ψi H dµ ( even component Hilbert series, only contains a handful of terms, all4.3 related to operators with Parity ∆ invariance:If from the SO( EFT underO consideration is paritytimes invariant, SO( then the spacetime rotation group is as we work with on the treatment ofrepresentations, the eq. covariant ( derivative).in ∆ Second, knowing the∆ possible generating results inR eq. ( and spinor fields or where it is understood that proper care of statisticsEverything is in taken the in derivation the of multiple plethystic exponential. we have space character denotes the coordinates of the torus of JHEP10(2017)199 . ) 2 P Z = 1. 4.34 n ) (4.25) (4.28) (4.27) (4.26) ) d r ( ) factor ) and O d pg dµ − − SO( R ) denotes the (see eq. ( ), Z ). This is fairly d ∈ (1 ) P ( ) = SO(2 N (  − l r + O ρ ∈ g pg 4.26 (2 i l . O −
, + (1 ) . , with where  ag ) in eq. ( ) subgroup. Therefore, a n ), we get the determinants g P . The det I d n ) − det 1) + d 0) representations, − + ( φ, p, x g − 0 − ( 4.27 dµ O ( ∆ − = dµ − dµ Z ) = ,..., φp − ) Z 1 0 g P − is the Haar measure normalized as − ( 1 2 n, 1 0)  ) pg I, ). In this subsection, our focus is on | d (1 ρ l ( and eq. ( | + ,..., , − 0) = (  0 i 1) l P O ) n, + (1 ,..., ( 4.16a + − 0  ) on the parity even component. We recall Z g dµ , i.e. ) n, φ, p µ ( + ( v = det = – 31 – − ) = ( , = det pg − − ) and the measure det 4.16a 0 ) with measure P g 
− ( d − H dµ l − ( =0 → − ), all the determinants are over the symmetric tensor We can take the action of parity to flip the sign of ∞ ρ dµ ag Y n (1 O Z ag µ  ) + − v ), ∈ 4.26 − 2 are a consequence of further normalizing  ) = : (1 / ) = 1 g det φ, p g ( 0) ( + 1. is the identity matrix. Then the action on general tensor + . P is given by eq. ( 0). Using Z 0) + , r φ, p ,..., I dµ . In even dimensions the parity element does not commute with 0 ( ) can be taken in the form C 0 + φ, p, x 2 ,..., , d = ( − n, H 0 0 , ( ( Z Z ) is split into the parity even and odd pieces of 0 h ,..., ) factor of eq. ( Z n, = 2 − H is an infinite product of  ( 0 1 2 1 2 × x . The above is valid for tensor representations, H O ; r Z d l Z n, p 4.24 ≡ ) is segmented into the cosets of its SO( det ( ∈ ), d + ) = +1) ( d − r = ( ( /P g O l ···  φ, p ( ) that + O 0 1 l H ∈ 4.19  | ≡ l g = 1, and the factors of 1 | +1) = SO(2  For the Hilbert series in eq. ( The group r dµ (2 where below for spinors). representations representations is easy to deduce:while an a even tensor rank of tensor odd is rank invariant: (odd number of indices) flips sign, This complicates the procedure inout even dimensions; the here details we in summarize the appendix result and work Odd dimensions: all components ofrepresentation a matrix vector, and denoting the parity element.out The the basic determinants det procedure issimple to in make odd use dimensionsO of because this we relation can to takegeneral work rotations parity so to that the commute orthogonal with group is rotations a so semidirect product that into a more amenable form for explicit computation. general element where R The parity even piece bringing the parity odd piece, The integral in eq. ( reduces to the 1 from eq. ( where the integral is over all JHEP10(2017)199 , , ). ),  on ) # d for ) , so z does 2 φ P ψ − 4.26 x (4.30) (4.31) (4.29) φ, p , ; ˜ ( P 1 ] 2 function − − p + d ( g H P ). We have n 2) ) d − t z r . Their direct + 1) measure; ) + − − )) (2 R ( r , . . . , z , 1). Since 1 ψ φ P 1 φ, p z − − ( 2 ( − , d +2 C.39 p ↔ + φ 1 1 0 [1 H L − is the usual  2∆ → ψ 0) 1 1 2 p ) as a reflection about the 2) 2 , . . . , z ,..., d ,..., 1 − also needs to flip sign. In φ . 0 r z P i ( n, ) = ) (2 ϕ ( )+ ϕ P x ; ˜ , . . . , z φ, p det ϕ, p 1 ( p ( z 2), − → − ( H ( =0 ). Under this parity transformation ). We assign a single spurion ∞ 2) ) = diag(1 derivatives on top of the field Y φ ) − n d encodes how many derivatives there r − − d z P ) r r n t ( H (2 + − (2  . , O . ρ tg P 1 + ). 0), doesn’t transform under parity, neither 0 ) + −  flips sign under parity if there are an odd 1 t , . . . , z d ∆ dµ − 1 φ − r z (1 + ϕ, p , i.e. n φp , ( = ( d ,..., ∂ 0 " 0) , so that ϕ . The end result is (see eq. ( – 32 – 0 + 0 R − ∆ ). For a pseudoscalar, the procedure is essentially , , . . . , z ) ,..., ∆ H C 1 φ PE 1 t 0 dµ h , , , . . . , x z ) − ( 1 1 2 φ, p (1 x ( φ/t ( x ; ˜ ( 2  p  → = (1 ( det p → ) = H ) In even dimensions, we take ≡ l 2) + d − +1) φ th axis in − x r + d r . 1 , ϕ, p dµ ) gives (2 0 r ( + 1) is the parity transformation of the SO(2 (2 Z H r and P H , . . . , z 4.26 ) also flips sign: t 1 (2 2) = = = 2 z z − − ( r  → d ϕ O ) irreps and to work out the determinants and measure in eq. ( So far we have focused on scalar fields with trivial intrinsic parity. To r , t p forms a representation under 0 Sp(2 φ ∆ R ) only has its arguments transform, t dµ z ψ (  0) representations arise from adding , . . . , l Z φ 1 ⊕ l 2 dimensions, and ˜ L +1) ) = r − ψ − , = ( ,..., (2 0 0 r l = φ, p H ( n, ) ψ For spinors, we need to split the cases of even and odd dimensions. In even dimensions Let parity act on the spacetime coordinates (in Euclidean space) by switching the sign Let us put these together and reweight the Hilbert series in a momentum grading The measure on r − , (2 0 H other words, the spinors are chiral,sum and parity interchanges the two chiralities while a pseudoscalar already explained in thisseries section into how its to contribution includeand from parity then even proceeds for and to scalars: odd computing the one pieces, same splits except the that Hilbert on the parity odd piece, the spurion show how to addressand intrinsic spinors. parity, we discuss two typical field contents:of pseudoscalars the last coordinate, ( a scalar field where the integral is overbut the in symplectic 2 group Sp(2 Intrinsic parity. not commute with generalgeneral rotations, more effortThe is details required are to worked out track in the appendix action of Even dimensions: hyperplane orthogonal to the scheme by sending are. In this scheme, eq. ( this captures the obviousnumber statement of that derivatives. since the adjoint representation, does the measure. Therefore, we have The ( JHEP10(2017)199 , ) H ψ = 1. ψ, p → ( (4.35) (4.36) (4.32) 2 (4.33a) (4.34a) + (4.33b) (4.34b) ψ P H piece, with H ensure and the formula , ,  ) r H , the Hilbert series 2 r 2 G z iγ ) +∆ − r . 0 2 , i 1 ) , z H ) − . Because almost all of ). The full Hilbert series is − r d . , = 2 , p term. The ∆ ¯ ψ, p 1 i } ) i 0 − ) again in a similar way with we do this using Hodge theory − H C.4 r Φ 2 H { . In this section we have shown ψ, ψ, p ¯ ( + 1), namely each of them forms 7.2 ( ψ, p , . . . , z ( , . K 0 r ) ) − ), only contains operators of mass H − ), and both signs z z z ψ, ( (2 H ( H − , . . . , z D − − ψ − 0 O ( ( r z ¯ 2 ψ ψ H ) + ( ) + ∆ ) + ¯  ∓ iγ ψ . After Wick rotation, we can combine it ,..., , p +) metric, we find ¯ ¯ as the Hilbert series.  ∓ } ψ ψ, p − = = ψ, p i , ( 0 1 1 Φ = = ψ, + − − – 33 – H { ( ,..., 1 1 ( and H + P P 0 − − and internal symmetry group + h ) ) , H P P z z H ψ 1 2 } ). That is, the full Hilbert series is given by ), whose spinors are chiral. For the Hilbert series ) ) h ( ( i r r r ¯ 1 2 ψ 2 2 Φ { ) = P ψ P ) = 4.34 , z , z 1 1 , p ) = − − } ) (see also appendix i ψ, p r r ( 2 2 Φ ¯ ψ, p { H ( + 1), but they have opposite intrinsic parity. As a result, the 4.16b ψ, ( H r , . . . , z , . . . , z transform separately under H (2 0 0 z z ¯ O ψ ( ( ¯ ψ , we frequently refer to 0 P ψ P and H is odd under parity. This is expected because the little group for a massive ψ ) (see also the gauge field example in appendix is hermitian with our metric (+ ¯ ψψ rotations in all other coordinates so that parity reverses all of the spacetime r in accordance with eq. ( ◦ 2 ψ, p , transform under parity as ( ¯ ¯ ψ below can be derived by other means — in section ψ iγ − 0 H This section made use of conformal representation theory to derive the Hilbert series. In odd dimensions spinors are not chiral. In Minkowski signature, the two components H and → − ¯ dimension less than oris equal contained to in the spacetime dimensionality We stress, however, that this isfor not necessary. The splitting how this counting function can bethe explicitly form computed as a matrix integral. The final takes where the bulk ofexpression the given Hilbert in series eq. is ( contained in the 4.4 Reweighting and summaryGiven an of EFT main with formulas acounts set the of number fields of operators in the operator basis fermion in oddcalculation, dimensions one is computes SO(2 thethe parity scalar odd case. piece ψ But to account for their intrinsic parities, we send the spurions We see that a representation of mass term where Note that the signswith are 180 opposite for coordinates. After switching to the (+ ψ and again given by their average this representation and follow a procedure similar to the scalar case to compute JHEP10(2017)199 ). is c i 2 d b Φ (4.37) R ) requires is even or , . . . , x includes an . If d 1 ) expression G x d . H 4.16b # ) of the generat- ) = ( is parameterized i x G ), accounts for IBP ) in the mass grading p, x, y . . in eq. ( ( ) i , p y C 3.31 Φ } ( H i i χ . i Φ Φ { . They are a product of the Φ ). In particular, the formula B G, i ( 0 χ Φ ) character coincides with the i ). These measures are restricted H H ) d is X R d x " 0 ; p H ( i PE Φ ) , and appendix ) x ) is parameterized by C ; , 1 d p 4.3 ( . The momentum grading scheme — which +2 d p P , regardless of whether or not they are unitary ) )) x SO( – 34 – ( e and χ ) the function in eq. ( ) d x 4.21 } ; i p ) = Φ SO( ( { P ). The integral splits into two pieces — the parity even dµ d Z p, x, y and the Lorentz group SO( ( ) i y ), while that of SO( Φ G ( ) χ G G for the single particle modules dµ i Φ rank( Z χ ) character for the module times the character under for a heuristic physical derivation of d ) factor, with x ) = ) instead of SO( ; 2.6 , . . . , y d p , p 1 ( ( } y i O /P Φ { = ( ( 0 y ). The expression for the parity odd component depends on whether H is valid for all single particle modules minant constraints. products (appropriately chosen according toing the representations. statistics of each Φ The matrix integral and thetions use that of stem characters from automatically accounts finite for rank group conditions, rela- such as Fierz identities or Gram deter- As the single particle modulescies already are have accounted the for EOM in removed, these the characters. EOMThe redundan- plethystic exponential is a generating function for symmetric or anti-symmetric The characters weighted SO( a conformal representation, thenconformal character, the i.e. weighted (see SO( eq. ( to the torus usingby the Weyl integrationMeasures formula. for the The classical torus groups of are givenThe in appendix 1 redundancies. The Haar measures for 0 One may wish to choose different weights to access certain information in the Hilbert To include parity as a symmetry of the EFT, the matrix integral for The starting point for a practical computation of 4.37 • • • • • H odd. Relevant formulae can be found in section series. In this section,scheme as we a have function derived of thewe the find Hilbert weights frequently series useful and will use in subsequent sections — is to weight operators by integral over and odd components —in with ( the parity even component given by the SO( The ingredients in the above equation are for conformal representations. However, thethe assumption explicit that formula the for single ∆ particle modules are also conformal irreps. (see also section JHEP10(2017)199 , ! 5.6 µ (5.2) (5.1) ), x j (4.38) µ i p = i in terms of n =1 i X i K ] to arbitrary

38 . In section ) exp = 1 [ n 5.5 d , p ··· , 1 p ( ) . are indistinguishable if n,k µ ( x j µ F t, φ ) i p n e derivatives then takes the general form → p ) ( and k p n ( i e i φ , p φ e φ counts powers of derivatives. p i t ··· d i Φ ) d – 35 – 1 Φ ∆ t p ( and Z fields and 1 i e → i φ i φ n ) = p Φ x d ( i d φ ··· 1 distinct p by passing to momentum space where operators are in one-to- d n d returns to the rings to tackle the general case: we establish that 5.1 Z 5.4 ∼ to reflect on the physical connection to operators, contact terms, and is a flavor index, such that k i µ 5.3 ∂ — and use this to solve the basis for five scalar fields. exhaustively parades through the basis involving four scalar fields, exploring K ··· is the mass dimension of Φ 1 µ 5.2 i ∂ , (where Φ i n φ φ We start in section ··· 1 . Section φ An operator consisting of 5.1 Polynomial ringTo in begin, particle we momenta passfields, to momentum space and consider the Fourier transform of the scalar of spin, emphasizing thatthey the dictate scalar the operators kinematicselements of studied momenta. presently of appear Some the comments universallywe ring about since explain the and a relationship conformal between simpleoperators primaries algorithm in are for constructing made elements in of section the ring — i.e. constructing symmetry. With an understandingpause of in the section ring formalismamplitudes. from the Section four pointthe example, rings we are Cohen-Macaulay, which meanswhich we allow can find us a to finite construct set any of other generators element of the uniquely. ring Here we also touch upon the case one correspondence with polynomialsequivalence relations in between polynomials, momenta. leading to EOMpolynomial a and natural rings formulation with IBP of ideals. redundanciesd This become generalizes our previous workthe consequences in of EOM and IBP redundancy, spacetime dimensionality, and permutation the physical observables ofmomentum a space picture theory is — particularly(i.e. scattering powerful for the amplitudes the forms — construction of of wedeveloped the the emphasize in basis operators), that elements the especially previous this when sections. used in conjunction with the machinery 5 Constructing operators: kinematicThe polynomial question rings ofof counting counting and and constructing constructingamplitudes the operator in QFTs. kinematic bases quantities In this can whichstructure section, form be we that map the translated out underlies basis this into operator translation of and that scattering bases. explore the kinematic As well as providing a concrete connection to obtained from the mass grading scheme by sending where ∆ their field content as well as the number of derivatives in them. This scheme is readily JHEP10(2017)199 ) or (5.6) (5.7) (5.3) (5.4) (5.5) d refers ) = 0 is n ) become ... ( 5.2 is empty we µ . , whose form . ∂ ) n I } , n acts by permu- . Here, i S , p I × n ) ) { G n, d d ··· ( ). S in eq. ( indicates EOM and , µ d 3 i M , 2 is the set of operators e SO( SO( F φ , ) K   is the Feynman rule in } ) generalizes to J i i ! i ) ) means either SO( 2 n 2 n µ i p = 1 , is equally valid. We refer the d = p { 5.6 i , where ( n,k ( R , p , p ( 3 n I , i O } F S F ) X ··· ··· 2 i µ i , S , , p p ) 2 1 2 1 { d 24 , p , p ) + ) + µ n µ n SO( } } ] p p i i µ n p p = 0. These define equivalence relations + + { { denotes the symmetry group we impose , p ( ( µ i from the ideals above); if 2 2 Σ where ( p ... ··· G momenta F F µ n ··· × p =1 , + + n n i ∼ ∼ ) ). µ 1 µ 1 µ 1 + d ) ) p ] for a primer that is relevant to our current application. P ( p p [ ) symmetry acting on the Lorentz indices of the } } denotes the operator space under consideration: – 36 – 5.3 h h i i d C O 38 / / p p ··· I ) ] ] { { µ n µ n = and the vanishing of total derivatives S ( ( + 1 1 ) , p , p d µ 1 F F 25 p = ( ··· ··· SO( ⇒ ⇒ n , , G = = = 0, µ 1 µ 1 M p p 2 i 3 [ [ p = 0). µ 3 C C O polynomial in the µ   O φ p We denote the rings under consideration by k 2 µ = = ∂ ∂ , although any characteristic zero field, such as ) n , namely polynomials invariant under d C + + S K , n × 2 2 ) is a permutation group acting on the momenta. We typically consider SO( n, d n momenta. In this case, the result given in eq. ( K ··· M S ∼ O ∼ O , SO( n, n 2 1 1 is a degree , ⊆ M is the standard notation for an ideal, and the subscript O O ) = 0 would force = 0 translates to i 2 n,k p φ ( 2 ... is the operator basis with both EOM and IBP imposed; h F ∂ . That is, we are interested in the polynomial ring, K For the case of indistinguishable scalars, the Fourier coefficients The effects of EOM and IBP lead to equivalence relations between polynomials. The ) and Σ We work in the field We continue to consider Euclidean spacetime and assume complex momentum (note that for real } d µ i = 24 25 ( p invariance under. Generically, O reader unfamiliar with commutative algebra to [ momenta, A word on notation. to the number of momentaI in the ring. with only EOM imposedimpose (remove neither EOM nor IBP, as in eq. ( the indices 1 tations of the where IBP equivalences are accounted for. identical, and the symmetry of the integral leads us to consider symmetric polynomials in Such polynomial relations between elementsThe of equivalence a class ring of are polynomials embodied under in these an two ideal redundancies of lie the in ring. the quotient ring the statement of momentum conservation, where the superscript indicates we are imposing invariance underEOM SO( depends on how thebetween derivatives operators act and in suchmomentum polynomials the space). (up operator; Because to we this a are iswith interested constant, in polynomials a Lorentz invariant scalar one-to-one under operators, an correspondence we{ SO( are concerned where JHEP10(2017)199 . ) t } − ≡ ( 2 I / d (5.9) (5.8) i G n, p (5.10) (5.11) 1) ··· . After H , i < j − 2 i ij ij n p s s ( 1 i { n p  are degree one µ i , such that there p d ≤ n 2 invariants . , called rank conditions / ij + 1) minors of the Gram }i s = 0. Contracting this with d ∆ ( + 1) µ i h{ p × n / ( ] + 1) sub-matrix has vanishing } n =1 n i d d . ij ( , + 1) s P ≤ A natural way to take Lorentz in- { 2 Mandelstam invariants d × [ .       / C n n . . . 1) 1 2 nn + 1. = s s = 0 s , n − + 1) d ) ] . d ij n } ) — which eliminates all the antisymmetric . d ( s ( . ≥ ··· ··· ··· d ij O i n ] s n n 6= µ n { 12 22 2 j . . . [ X – 37 – s s s , p C ≡ , and the antisymmetric invariants n i µ j = 26 . . . 11 12 1 ··· p s s ) s X , as the set of all ( d µ 1 ( i µ J . The Hilbert series of the rings are denoted as }       p p t O n, [ ∆ C M { ≡ = ij ) momenta can be linearly independent, which leads to nontrivial d s ( tensor. d O n  . In particular, the Gram matrix (all indistinguishable fields) or Σ being trivial (all distinguishable equations of the form M . Then any element in this ring is automatically Lorentz invariant. ij d n s n d µ i = 0, and we are left with S p . This dictates that any ( ii d s ··· gives 2. Alternatively, this reduction can be seen as the following: imposing IBP 2 2 ) invariance, as opposed to SO( is nontrivial only for the case µ i / i µ ). d , unless explicitly stated otherwise. Now our ring becomes p p ( } j 3) 1 , t 1 µ i O I ∆ 6= dimensions only p − { equations reduce the number of independent Mandelstam invariants to G n, i d d n n µ ( M ( ··· n IBP is the statement of momentum conservation: To study the effects of EOM and IBP, we begin with the case In 2 In the math literature, this result is known as the first and second fundamental theorems of invariant has µ H = 1 26 ij µ These n theory. each of the are no Gram conditionsimposing to EOM, impose. OurAs initial we will ring always has imposes EOM, in the rest of this paper it is understood that the notation matrix, we have the ring isomorphism Clearly, is at mostdeterminant, rank which gives a setor of Gram nontrivial relations conditions. among Denoting the relations among the the symmetric invariants  However, this feature doesparity not — come withoutinvariants a involving the cost. To simplify discussion, we impose fields). Finally, we workand in assigned the a momentum weight parameter gradingor scheme, where the Translation to avariance ring into of account is Lorentz to invariants. work with a ring generated by the invariants, which include the extremes Σ = JHEP10(2017)199 2 = irre- / is the (5.16) (5.15) (5.12) (5.13) n (5.14a) 3) (5.14b) acts on S n n − 2 Lorentz 1) S / n in the ideal − ( 1) i n , d X − , d ( d 1 + 1 − + 2. , decompose into , d d d d d Σ 2) , ij , such that we can consider  ≤ ≥ ≤ ≥ 2 ≥ s 1). µ n − }i p n n n n n n − ( ∆ V { , n , (distinguishable) (indistinguishable) + 1, the Gram condition that the 2. In this formula ( 2 ⊕ n d / 1) − 1) for , obtained by similar considerations. , = 1 J n + 1) for − n n − + 1) G n, S n d d (   d ,...,X ( ( 6= ( 3) for 1) for M i i ( 1 V momenta after requiring them to be on- d d + 1.) translations (IBP) and ) n n 2. d 1 2 1 2 X − − / d ⊕ d n h − ) i, j n n / − − 3) ] = ( ( n n ( } n n – 38 – n n − n V ij 1) 1 2 1 2 1, ( 1) 1) ,...,X ,...,X s n 1 1 ( { − − ↔ [ − − n from the ring. X X and d j n C h h d d 2 Mandelstam invariants. For example, we can choose ( ( / /  ij d / ] ] 6= ≤ s } } 1 = ( ( 3) = ) form the part of an ideal which accounts for IBP, = ij ij , i − s s Σ = 0 — such that the number of generators of the ring after − + 1. However, when n ji { { . For the distinguishable case, this module can be written as × 2 s ) = ) = [ [ . As before, for the distinguishable case, the d 2 n n ) i < j 5.11 ) / j ( n 2 is the number of generators of the Poincar´egroup, each one d p K of the J C C ( ( and consider the Gram conditions. Nominally we should expect K = / = n S ≤ G   σ 2) G n, n, O ) n, n i ij n ( − M = = M s M σ + 1) s n n K ) S n > d d 2 at both d, d ( 2 Mandelstam invariants. Having done this, one relation between these × ( K / → dim( 1)( dim( ≤ ) + 2, EOM, IBP and Gram conditions together imply the number of d / O d, ] with 1 d n ( + 1) determinant vanishes follows from momentum conservation. Thus, ij d − } ≤ 2) s O M n d ij + 1) n : ( − s ≥ M d { n n ( [ × n S equations in eq. ( d C 1)( ∈ n − = − σ + 1) 1) K n ) d d, d For the indistinguishable case, a useful result is that the Phrased in terms of our initial ring and ideal, the final quotient ring we are interested When Let us now take The − ( via ≤ O d n ( ij where Σ is acan subgroup be of used to eliminate ducible representations as where we have also included the dimensions for in is, causing a relation betweentransformations. the components: In summary,n the dimensions for the rings are (note that when considering IBP, Gram conditions only affect the case independent generators is givennumber by ( of independentshell components (EOM), of and the s a ring freely generated by M to see their effects when full ( where in the indistinguishable case, permutation invariance is imposed where only ( invariants remains — from IBP becomes ( conditions means we can entirely eliminate one momenta, say JHEP10(2017)199 , i } X 2 = 2 / (5.17a) (5.18a) s, t, u (5.17b) (5.18b) 3) − , }≡{ n ). Thus, (up 6 ( = 4, where it 14 ) ; this is, unfor- s 2 n t n n ; after imposing 1 5.15 = j S − p 23 · (1 i , s ) invariance. We pay p d → 24 s ] and references within). ) . = 34 . That is, 64 2 s generators, and in the final u ) 13 q ), and where the four scalars 2 t ij d − , s 1 ( s − 34 4 such that Gram conditions do O ) and SO( )(1 s . Before imposing IBP, the ring is d (1 24 ( = ≥ ij s s q 2) representation of O d 12 → , s , with a basis given by the variables − 2 , ) { 2 = 10 invariants n t / − S q )(1      n s 0 , counting powers of derivatives as per the 23 − t s = 0 to eliminate + 1) q 0 u t 1 u = 2, we need to account for Gram conditions. )(1 n − s t u 0 , s ( – 39 – , ] q d + n t u s 0 u t s 1 ] )(1 t − s, t      In 14 34 [ + s We begin with (1 q C , s s After imposing IBP conditions we have − 24 = = 4. , s 3) 3) )(1 3. = 2. 23 ≥ ≥ ≥ 13 d d s ≥ ( ( K K , s q , , O O 4 4 , d 14 − 3 sub-determinant of the Gram matrix, ) , d H M , d ) 2 = 6 Mandelstam invariants , s d ) × / ( d )(1 13 d ( 12 ( O 1) s , s O q ). IBP completely removes these components, by eq. ( O − with and without IBP, in all space-time dimensions, for distinguishable 12 = − s n = I [ , = ( (1 5.11 C n =4 G n ,G = = 1) form the natural representation of ,G ,G , are gradings introduced to count powers of the K M 4) 4) 1 K J ≥ ≥ ij − d d = s ( ( J J = q , = , n O O 4 4 ( I I I H ⊕ M ) n Requiring that every 3 (iii) previous section. (ii) Mandelstam invariants. We use familiarand notations then use the final IBP relation where expression for the Hilbert series we grade by are distinguishable. Our initialEOM ring we has have freely generated by these invariants, and the Hilbert series reflects this and indistinguishable scalar particles,particular and attention imposing to the generatorsand of how the this leads polynomial to ring, the the Hilbert relations series. between(i) them, not play a role. Consider first the case we impose parity, 5.2 Tour of theA four lot point of ring theis subtleties easy in to the identify above thepoint can ring generators already and be relations seen between in them. the Thus, ring we when consider the four ( defined in eq. ( to dealing with the additional Gramquires conditions), an understanding understanding the of structure the of this invariants oftunately, ring the an re- ( open and difficult mathematical problem (see e.g. [ where the notation refers to the partition associated with a Young diagram. Together the JHEP10(2017)199 , ) n = } θ E 2 2 η e (5.20) 5.21a (5.21a) (5.22a) (5.19a) s, t, u (5.21b) (5.22b) (5.19b) { , we say , and , n 5.4 ] . Momentum 3 ηθ , , e 2 stu n e θ [ ≡ C . 3 . Hence, we construct e . = ) } ] 8 i t θ 1 [ . e − 18 C h , 4 of the momenta indices be- t
s, t, u t / , and 4 ] 2 { 4 (1 S 2 3 + η ] tu + t 2 S ) , e 2 34 12 ⊕ 6 + − t 2 t t , s η 1 . , e + − su 1 24 ) . Such a ring is freely generated by ⊕ 1 + 6 e invariance of the ring in eq. ( [ 10 t + 1 , s (1 , t 4 C 2 i implies we can always eliminate powers → 23 ], with a basis for the module given by − st ) S s + θ s, t, u 4 . In the language of section i = ) = ) = 0. That is, [ 2 , s t 8 t 2 t t ≡ i ) 1 Still imposing parity, we now turn to the t q C )(1 14 s − 4 2 . That is, ηθ − + 2 t 5.4 t + − e 1 t , s q t s s,t,u e 6 − 2 , 2 s ( 4. − . In this way, the monomials t 13 3 + − Interestingly, further imposing IBP greatly sim- s q ) (1 – 40 – u 3 4 ( )(1 h S 2 t η , s s t (1 /  2 − h ≥ + ] st ηθ q i s 1 + 12 = 4, permutations in t 1 − h 3. u s are gradings introduced to count powers of the → − = [ / s, t → n ] [ + + (1 3 ) C ≥ 2 , d e (1 3 C s t θ e 4 q s, t stu = = 3 [ q permutations of of the Mandelstam invariants S + = = ≡ η 4 4 C , d 3 3 s − S S 1 4 h S × S and e (2) (2) × × K K = → / , S , ] ) can be explicitly “solved” as follows. Change variables to O O ) 4 4 4) 4) 2 1 5 )(1 generators. e d 2 ≥ ≥ η (2) × H q K e M d d ( , ( ( O J q J 4 ) , then the ideal , , O O O t 4 stu 4 d s, t, u 5.19a [ − M ( H + = 0. Special to M = C = O (1  s 3 u e = = = ≡ + ].) 4 4 t η ,G S S 38 and + is a freely generated module over × × ,G J s 3) 3) tu and K ≥ ≥ (2) . K d d = , t } ( ( + O K K 4 , , 2 = O O 4 4 − I M su cubic or higher, e.g. I s The ring in eq. ( H M η + , η, η ≡ 1 where, similar to above, st polynomials invariant under the symmetric polynomials, conservation simply removes the generator (obtaining just the Hilbert series is, however,(v) relatively simple). plifies the generator problem forsubject indistinguishable to particles. Wecome again simple work permutations with in We give the details(which on is not the straightforward) consequences and of how it the leads to the above Hilbert series in appendix { (iv) indistinguishable cases. Imposing only EOM, the ring is The existence of such aMacaulay direct rings, sum decomposition which is wethat a discuss special in property of section so-called Cohen- appendix in [ θ of freely span the ring, giving the direct sum decomposition (A choice of ordering scheme appears in the first equality for the Hilbert series, see e.g. the vanishes results in the condition JHEP10(2017)199 3. ≤ . ) and j (5.24) (5.25) d p as well (5.27a) (5.23a) (5.26b) (5.27b) (5.28b) (5.23b) i p u  s, t, u (2) ring in = 0 simply O ), rather than d stu = 0). ] (5.28a) 4 t p 3 p − 2 s p [ 1 p C   , 2 ] ) . d t 3 . d µ i ) = 4 indistinguishable particles ] (5.26a)

, e + 4 t d d 2 n s ν ν e s, t ( 1 [ 1 . . . d will be identically zero as we only [ . . . p − . µ µ C 2 C ⊕ µ i 2 ) µ i 3 (1 6 p 3 p . p t p 2 . requires the additional invariant to be sym 1 1 2 (2) for p . . . . g . . . g  µ i → p − 1 4 3. Then, analogous to the 9 ) invariance are only different for p  p O . 1 1 t , S ) = 2, the Gram constraint ⊕ d  ν ν d 2 2 2 ( ] 1 d . . . ⊕ )(1 e ) d , 3 µ µ 3 ⊕ , 4 q O 2 3 ...µ t g g t ] ] 1 + t p 2 1 ), and hence , e 2

1 In µ − 2 e p − 1 = 1 − 23 [ e – 41 –  s, t µ [ = [ s 1 + i (1  C (1 (1 C d ⊕ C ) and − ≡ d = = 2 ...ν = = p = = 13 1 4 4 d 1 = 2. i 4 4 ν s Finally, we omit parity and impose SO( with p S S p  S S  × × = 3 there is only one independent epsilon invariant, say d )( 4 ··· × × K K p 2 , , , ⊕ . Momentum conservation allows us to eliminate i i , d d 23 SO(3) SO(3) (2) (2) 4 K 4 K ...µ p , p , ) 4 s 4 Σ. O O 1 4  4 1 t t K K H i , M , µ S SO(3) SO(3) p 4 − 4  H × M 2 +  + i < j = 4, momentum conservation implies t symmetric invariant can be formed with one power of H × ) M s 2 12 , t ( d d s j 4 − p S i )( 1 ⊕ (2) p  13 1 O , say the s 1 + 4  j p − i = SO( = = = 4, any epsilon contraction with the p  12 ≥ s ( K K d ,G , , 3 ,G SO(2) SO(2) 4 4 p K 2 K H p M 1 ) we have = p = = 4 momenta (in  = 0. That is, , hence I 3 3 n I ≡ 5.20 e p In two dimensions, for distinguishable particles, the SO(2) invariants are ( For the four point ring then, SO( ), invariance. This means invariants involving the epsilon tensor are now available, 2 d p 1 ( sym p Finally, there is no difference betweensince SO(2) no and non-vanishing six epsilon invariants as three of the eq. (  For indistinguishable particles invariance under Note that in have For distinguishable particles in  because of the relation (vii) O However, only one power of the epsilon tensor can appear in any element of the ring, sets (vi) JHEP10(2017)199 (5.29) (5.30) = 4 point , by which , as n n . 4 A 5.2 φ derivatives gives 3 k φ = 4 distinguishable m -point tree-level am- ν n n ... and .... φ vertex), the . . . ∂ 1 + 3 ) + ν contribute as contact inter- φ n ∂ ) = 4 in section 2 φ n,k ) n ( α in the Lagrangian. For example, n labels the independent operators µ ) O powers of , . . . , p α n,k 1 n,k n ( ) essentially correspond to the same α p ( α ( . . . ∂ ) O O 1 -point scattering amplitude from Feynman µ . n n,k ( . For example, for 5.22a ∂ α point, tree-level amplitude, ) . 1 F n . φ n,k α ( α c m involving ν F ) . – 42 – α ) X n,k ] of eq. ( (which eliminates the ( α n,k . . . ∂ 3 ( α k 1 O φ in the basis (here X O ν , e 2 ∂ ) α e n ,k [ µ ) = → − (4 C α n k, X . Further contributions involving propagators are indicated F φ ∈ . . . ∂ 1 form the Feynman rules. Consider the ) corresponds to operators as 1 m ) . In this case, the operator may look simpler, but it is masking µ ) φ , . . . , p ∂ n,k 1 = ( α 5.18a p = stu ( ( F i ). Before proceeding to the more general case — where things get ⇔ n n φ k ) n ] A symmetry tu 2 s, t A [ Z and + C n in the figure. su ∈ Contact interaction contributions to the + m ... t n st are the Wilson coefficients of the operators s α c The polynomials More explicitly, we can write the general The association of a polynomial equation in the momenta of the particles with a given = 4 case to reflect on what this means. The if we impose a plitude in the EFT ofactions, a as real depicted scalar in field. figure by the The + operators space picture is advantageous.ments Another ( advantage istype in of the operator permutation as symmetry: aboveindices (with dropped, ele- right all number ofthe derivatives, fact of that course) one but has to withcontractions) go the for through flavor any the quantity exercise computed. of symmetrization (taking all possible Wick scalars the ring in eq. ( Actually, to be precise, thetion above of should have operators the is derivatives traceless generally — cumbersome, such a and construc- this is one way in which the momentum n operator is exactly what onefor obtains an when operator. deriving That the is,rise Feynman an rule to operator in a momentum momentum space space Feynman rule we mean that we foundlack a of set relations). of This generators,the enables and independent us understood operators to the — easily relationsat construct among the the them given independent (or polynomialssignificantly — more i.e. involved — let us use the relatively simple and concrete formulas of the Figure 1. diagrams associated with the operators 5.3 Interpreting theWe ring were in able terms of to operators completely and solve amplitudes the operator basis at JHEP10(2017)199 . ) n G 1; R . ≥ be a 5.32 ∈ under (5.31) K 5.4.2 n and let R f in terms G G and J ]. Let 0 for all in section 68 n > uniquely – S 0 × 66 ) n, d c ( I . O n, m , are also important for ) I M . For the groups G n, stu R ( M n suggests ) tu stu + can be expressed = I su 3 G n, e + M is finitely generated, meaning all 0. The structure of the operator basis in st ∈ ( and G : generators and algebraic properties g → n R tu n,m c – 43 – is a Cohen-Macaulay integral domain. A reader + ]. This result is obtained in the forward scattering stu -invariant elements of I 65 n,m X su G n, G 0 [ + and M ), the secondary invariants exhibit a pairing as a result ) = 2 > st d 4 s 0 , = ] on a set of primary invariants for 1 ]. such that any c 2 64 e 68 I → − , . . . , p = SO( G n, 1 . It turns out that we can always find a particularly nice set of p tu G M ( we briefly consider the generalization to the case that operators 4 ]. + A 5.2 65 we prove that su 5.4.4 generators + to be positive, we touch upon the relationship between the operator spaces st , where a necessary and sufficient condition is that the Hilbert series exhibits 5.4.1 4 I ) G . However, the Cohen-Macaulay property generally breaks down when spin is n, I ]. ∂φ 5.4.3 M G n, being not only CM, but even Gorenstein. Useful in this regard are the Hilbert 68 – M I are so-called Cohen-Macaulay (CM) rings. G n, 66 We discuss a conjecture [ In subsection In section Intuitively, we would like to find generators for the operator basis, i.e. some finite set denote the subring consisting of I M G n, = 4 ring in section G 5.4.1 The Cohen-MacaulayWe property begin by reviewinggraded some ring relevant whose results coordinatesR from form invariant a theory representation [ ofconsideration, some a symmetry fundamental group result is that are composed of fieldsAn with important spin, point as here wellunderstanding is as the that the spin case the case,in that rings due the the to for operator the scalar itself universalinvolved, fields, feature as carries a of spin. result kinematics of which polarization is tensors. encoded series for a certain palindromic form [ In section less interested in mathematicaltogether details with needs the only associated to definitionsinstances, note of such the “primary” as decomposition and when in “secondary”of invariants. eq. In ( some of operators (Feynman rules) fromn which all others cangenerators be for obtained, just the as rings weof did the for generators the (theM choice of generators is not unique, however). This is because the case. Although thecomplexity. basis becomes A notably unifying morethe theme involved, Hilbert in a series our rich ofare discussion structure the [ underlies is ring. this how these Useful properties references are for reflected background in and further information terms of the two indeed, this is true [ 5.4 Constructing theWith basis a at healthy higher understanding of the four-point operator basis, we turn to the general A well-known result is thatciated unitarity to and ( analyticity constrain theregime Wilson where coefficient asso- amplitude only involves contact interactions at tree-level, JHEP10(2017)199 i ), η G R ] task obeys (5.33) (5.32) m k )). This t t is said to k ( a G =dim( N : A theorem R , . . . , θ =0 m in 1 r k , θ t [ } P can be expressed C m primary invariants f ) = )) — upon which we t ( Gorenstein , . . . , θ 1 N θ { 5.27a may be polynomials of some smaller . In such a case, homogeneous system of param- module over are called i } η . s as a linear combination of the . ) ) ] ) is called a Hironaka decomposi- m i m in eq. ( d m free j t e t , . . . , η 5.32 has the decomposition or a parity violating term such as − 1 η sym is a G is the maximal degree of , . . . , θ =1  { n (1 , . . . , θ s j 1 , . . . , θ is called a R G r uniquely 1 φ 1 θ as a HSOP). θ =1 } θ In terms of operators, the secondary in- P R m i ( [ } can therefore be taken as a finitely generated i m C f Q m i = 3 ( G η – 44 – 27 R =1 . d s i ) the )). If we compute some Hilbert series and are able ) where . s ) = =1 i , . . . , θ M G P /t , t in 1 , . . . , θ θ R 5.32 2 G = 1. Moreover, = (1 5.33 = { φ ]. Broadly speaking, this entails (1) finding the rest ), it is a good indication (although not sufficient) that 1 R -point Mandelstam variables (the primary invariants). , θ 3 g G N η ( m 2 1 α n r 2 θ R t ]: H α { 5.33 1 can be expressed m α ), we have ) = G j , . . . , θ t η 1 ( R φ ∂ θ 3 ]. The set [ N µ ∈ , . . . , θ m C 1 g deg( θ [ φ ∂ 3 secondary invariants ≡ , and the decomposition in eq. ( C α , sometimes called irreducible secondary invariants. j 2 s s , . . . , θ e µ ] forms a subring of 1 ≤ θ m [ can always be taken as φ ∂ r 2 1 , C η α } , . . . , η ]. This decomposition shows that CM rings very much resemble vector spaces. 0 1 r ) and 1 (equivalently, i α η θ 1 , . . . , θ k 67 µ 1 , − θ ∂ r [ 3 , . . . , η (HSOP). Note that a choice of HSOP is by no means unique (a simple example: we 66 a 0 deg( 1 µ C as a module over In some cases one notices a remarkable property about the numerator of the Hilbert In the decomposition in eq. ( Given a Hironaka decomposition, it is trivial to write down the Hilbert series. Defining The basic problem is to identify some HSOP and then determine the structure of Within the generators, it is always possible to identify a set η 2 Note that Cohen-Macaulay { = µ ≡ 1 27 G k i µ series in a CMa ring: it ispairing, palindromic. indicative of That a duality, is, is the enough numerator to tell usset that the ring is syzygies can lead to negativeform terms to in reflect the the numerator HSOP,to as (with in bring the eq. it denominator ( chosen to inthe the the ring form is in CM. eq. ( d Importantly, the numerator is a sum of strictly positive terms. For a ring which is not CM, and the variants behave like seed operatorscan add — momenta in like the form of In other words, every with coefficients in be tion [ relations among relations, called(2) syzygies). is If greatly simplified, because it implies that module over eters could equally well have chosen R of the generators and (2) classifying the algebraic relations between generators (including relations among the generators,uniquely from so the that generators. we are not guaranteedof that homogeneous elements whichring are algebraically independent and their freely generated can be expressed as polynomials in some finite set of generators. In general there may be JHEP10(2017)199 ) d /I ] ( 6= 0. ; an s with [ O g there ]. (2) G · (5.35) (5.36) is CM C Σ 0 (5.34c) m (5.34a) f d (5.34b) R ] ] tells us G ]-module ] V ≤ [ µ n m 68 is a non-zero ]. then C n , i , . . . , θ R f Σ are algebraically 0 1 67 i , θ ∈ n [ , . . . , p , . . . , θ g }i 66 and C 1 µ 1 θ p ∆ [ , [ R . When { 1 , . . . , f C , C ii 29 1 R } s f i = , X is CM. We first need two { Σ , i }i Σ } ,  : corollary 3.2 of [ is particularly well behaved when ∆ ii
0 }i is a free s { /I R , t R ] , ∆ is a regular sequence in ]-module if G i , if for all non-zero ) s h{ { n [ i n , nn R } θ . C } ] m  6= 0. s ii [ deg s t regular , . . . , f C xy is a free module over 1 h{ h , . . . , θ − , . . . , s f is a non-zero divisor in is non-trivial. We will show that is a regular sequence. For the sequence [ 1 G , . , . . . , θ θ 1 11 ] C = (1 } R h f Σ 1 s s then }i [ n h i θ Σ 28 . if =1 { ∆ C R n i × 0 . }i ) ] h , we end up with ∈ d h{ R s Q ( then [ – 45 – ∆ ), although this is non-essential. Applying )). In this case it is clear that the resulting rings = is a free O d C } , . . . , θ h{ ( i 5.1 x, y H 1 Σ R 0 m i we have the following inequality of Hilbert series, θ . O = × 5.16 µ i ] ) { 0 ≤ s p d including diagonal components i [ ( i ) , and all the above rings are isomorphic to R is a regular sequence in , where C O nn ij ∅ , t } i P h s , . . . , θ in 0 is a non-zero divisor, or , 0 1 = nn regular sequence }i } R } θ = R ( 2 i 2 i ], says that a Cohen-Macaulay integral domain is Gorenstein { n } n > d p p Σ ∈ H , . . . , s ∆ 68 × h{ h{ f ) { 11 d ( . . , . . . , s s ] ] h O , . . . , θ µ n µ n 11 . From these definitions, it is clear that ] ]-module, this implies HSOP 1 i . s Σ is reductive. We need to check that the CM property holds when µ n is called a θ 1 0 m { { − } i × R n ) d , . . . , p , . . . , p every , an element ( µ 1 µ 1 , . . . , p we have , . . . , θ R p p µ 1 , . . . , f O [ [ 1 ]. Omitting EOM and IBP constraints, it is clear that , . . . , f 1 ) p } θ 1 [ f C C [ S h f 69 is a direct product, we can apply invariance under individual groups one-by- h h C { nn C R/ collectively denotes the = ( ≡ ≡ ≡ G s are Cohen-Macaulay. This essentially follows from the fundamental theorem of Σ Σ Σ 2 0 1 G is an integral domain if for all non-zero R R R First, note that We proceed to the case As With the above preparatory remarks we now come to the main point: the rings , . . . , s I R Given a ring some representation of Σ (e.g. eq. ( G n, 11 28 29 s A sequence divisor in quotiented by regular elements/sequences, i.e. independent. with equality holding if{ and only if independent elements and length equal to the dimension ofthat the for ring. a sequence is CM, which willother subsequently be characterizations of used CM toimplies show rings: that that for (1) TheAs fact that a free equivalent definition of a CM ring is that it possesses a regular sequence of algebraically where are no Gram constraints, V are CM as they are just invariant rings of a finite group, which are CM [ since we add back in EOM and IBP. one and see thewe will CM include property parity at andinvariance, per each work the with step discussion along in the section way. For simplicity of discussion, if and only if its Hilbert series is palindromic. M Hochster and Roberts which assertsMacaulay that [ the invariant ring of a reductive group is Cohen- of Stanley, theorem 4.4 in [ JHEP10(2017)199 ] n n ), ), is is S be 1 S ] Σ 64 i ) is a i d } R J 2) ( R 1 , -even U (5.38) (5.37) Σ 2 i ). For O n, N n − . But has the R /t i n M ( 1 i ]-module. V nn R m [ = dim( . Let (if = dim( i → , . . . , θ C 1 ], ] theorem 3.2 1 t R , K . We further m N m . 0  , θ 67 ) , , . . . , s or R , . . . , θ Σ i , t , t nn ) for some integer 1 ) is CM. This means 11 0 R d θ Gorenstein, and we J s [ i /t R h I , . . . , θ be such a sequence, 1 ( C a representation of R / = 1 as CM. , SO( n, 0 +1; in such cases, after } ∈ θ I H } not [ 2 V , . . . , s I d even, while R 1 n M G m n, C ) R 11 N  n is Gorenstein if its Hilbert . 2 ≤ s M t d ( { I , explicit calculations of the H n with G k − n, , we have the following. There H is a free t is Gorenstein for either n , . . . , θ n p ]-modules implies that Σ, we do not have a completely i S 1 M t , . . . , θ  ) S ] m θ 1 d ). It is straightforward to see, [ R is also a HSOP for m × ( I { V n > d when × = (1 = [ 2) ) , θ O n, } ) , 1) is CM via a straightforward modifi- d n n 2 C K d m (  nn −  M ( − 2 ) , ) n O , . . . , θ , Σ 1 O x ( with ) 1 2 ) ; tg V R ) θ + 1 and whether S Z [ n S 2 -regular sequence of length /t ) = ( r t − is Gorenstein for S , . . . , θ 1 is a regular sequence in = C , . . . , s n 1 , t (1 = ( n R ) (finite groups do not have enough symmetry − = ( ) θ ) 11 (1 × } I V S d = 2 as { s ] Σ i d  ( G . As a module over ) G , n, ) (no permutation group), as can easily be seen G { (1 i nn – 46 – i d 2) d R d , P and none in K ( is a M R 2  R det . Now, as established above, ( ) d sufficiently larger than O − } i  ) or ) x 1 n H is a regular sequence on U ( ( d ) n ≤ S N ) ; for ( g , . . . , s V ). By definition, since d ) = SO( ⊕ ( = SO( versus n } HSOP; in particular, 1 2) O ) d , 11 ⊕ 1 ). Let Σ i d ( 2 r = ( ) = dim( s R ( G 0 N SO( i 1) = , indicate that the ring is usually − { R , O O R n 1 R G , . . . , θ any ( 6 dµ ] and hence CM. G = 2 1 = − when = V dµ θ n m i d ( Z { ⊕ J G , . . . , θ R V is as well since momentum conservation effectively removes one Z = dim( 1 ∝ 1) θ ⊕ , K n 1 , ) ) { , so  d n ) = i − , . . . , θ ( ( − are CM, we can show that +1 1 n /t are also integral domains. Therefore, odd. For , t O ( V ) n nn θ 1 [ 2 1 [ 0 V , , I . Since dim( n 1 C ) M R C R G Σ i n, d ⊕ ( ) invariance, we are left with the ring is CM by Hochster-Roberts’ theorem, and hence we can find a reg- R (some care needs to be taken with the contours as we send ) ) is saturated, d R I M n H ( is Cohen-Macaulay. Similar reasoning establishes 0 ( SO( Z , . . . , s n, 1 O V R 11 ) 5.35 ∈ M R s = dim( S =  h k / 1 is a free module over -odd, depending on V 0 H i N , n R ] section 2.3). The basic sketch is as follows. Let -sequence, then R 0 J The rings Given that 66 = R 1 . This is always the case when (for theorem 6.2, that is Gorenstein for Hilbert series, e.g. seesuspect section this is generally the case for or for Gorenstein, then momentum). Including full permutations, are no Gram conditions in applying ( for some disconnected groups, such as general statement. When series exhibits the palindromicp form from the matrix integral formula the set of Σdirect sum non-invariant decomposition elements of that Then the direct sumfree decomposition module over of cation of the proofor that [ the invariant ringbe a of HSOP a for finiteto group remove is continuous CM degrees of (e.g. freedom), [ ular sequence ofwhere length dim( a R and hence Since eq. ( know that and by the matrix integral formula for the Hilbert series, JHEP10(2017)199 ] ). = n 64 S n ] S (5.40) (5.41) 2) × , J has an (5.39a) (5.39b) ) 2 d, d ( − . ≤ n G O n (  i . V i [ M , t G 2 ) ) C i d as the defining µ i ( i O ∈ K we are left with p n O = ij ) p J s S n · +1 K =1 ( , n, , n i S µ n 1 p × M − +1 ) P 2+1 d n d  ( ( / have been eliminated by , ≤ = 0 plus the additional O i 3) n } H n 2 i 2 i ) − X 1 . . . p invariant: by construction, p p 2 M n t ( − + n n h{ n s − 2 S . ] p µ n · ··· 1 , = (1 p + n h n : ] a similar relationship and were able to which permutes the +1 . X n , . . . , p corresponds to an edge between vertices -dimensional case studied here. n 2+1 2 S J / µ 1 38  . Eliminating, say, d ] / + ) p ij n, 3) K ij [ 1) , x s s − ; − C M [ ··· , determining the elements of n t +1 n ( h h ( d ( C G n n 12 and we work with + n ) are obviously P is equivalent to a graph isomorphism problem, ∼ ∼ – 47 – S ) K n M 1 ≤ K 2 n G t S n i ] = i − 5.39b 2) µ i and , I , . . . , s 2 p (1 n = 0, −  J 1 n ,...,X ) 2 +1 , and ( =1 − . Some evidence for this conjecture is that the Hilbert ) n x n i vertices so that n 2 n V n ; ij s 1 X S t are closely related since momentum conservation effectively p s n P ( ⊕ i , + + × subject to the on-shell conditions K + P } 6= , 1) j 2 i , ) µ n 1 p ) could also serve as a HSOP when the d +1 ··· ··· dµ ··· G P ( − n h{ n + + O + ( Z M . ≡ 1 V ] 1 2 12 5.39b p = = , . . . , p i X s ⊕ +1 1 µ . Additionally, in the absence of Gram constraints, the above can serve  X µ n and ) p G = ( n , t ( ) , J 5.6 d V ( G [ +1 n, K J , 2 n O C ) , . . . , p p M +1 S µ 1 = ( n = p [ . Using intuition from the fundamental theorem of symmetric polynomials, [ ]. As a graph, we take I M n momenta j C S  ] h 64 In one-dimensional field theories we observed [ Having knowledge of a HSOP helps to algorithmically construct the operator basis, n ij H s and [ 5.4.4 Generalizations withHere spin we makefields, a what few algebraic comments propertiesparticles when do with spin we expect spin? enters if the we Physically, problem. have each fermions, field gauge Moving fields, that beyond or composes scalar other an operator use this to arrive atto consistency explore conditions this and relationship recursion in relations. more It depth would in be the interesting To wit, in the absence ofHilbert permutation series symmetry, this relationship is clearly reflected in the The rings allows us to eliminatethe one momentum in constraint see subsection as a HSOP for some generalizations discussed5.4.3 below. Relationship between polynomials in eq. ( momentum conservation (i.e. when The power sum polynomialsthey are in invariant eq. under the ( representation. larger group where we recall series suggests a HSOP of minimal degree has degrees coinciding with the above. The When C e.g. [ i conjectured the following to be a HSOP for 5.4.2 A set of primary invariants in the absence of Gram conditions JHEP10(2017)199 . ? ]- K is ,l K G I n, m I G n, G is in n, G M n, ). We M K M I , which M G n, G n, K , . . . , θ 1 M ) represen- M in momen- θ c [ ∈ 2 d µ C b f K for examples. denote the set of 6 ,l , is an irrep of the , . . . , l ) = dim( I 1 φ is spanned by scalar ,l l G I n, R G n, K will still appear and the M be free as = ( (but, in general, not free, I M l G I n, . As a quotient ring, G n, (equivalently, in the OPE of not M f µ φ M will also be a HSOP for R K I G n, K M , but will also include a finite number G n, ij s M – 48 – which transform as the correspond to operators which are not total deriva- i K µ i describe kinematics of the momenta — universal to all p G n, i I M G n, P can serve as a HSOP in this case as well. However, the rings , M } } 2 i m p may not transform as a conformal primary when acted upon h{ / ) invariant polynomial. Moreover, dim( K number of polarization states. Hence, the appropriate algebraic ] are finitely generated modules over d µ n G ]. In particular, any HSOP for n, ( ,l , . . . , θ that the single particle module for scalars, I 1 O 70 M ) θ G 2 n, { S finite ∈ M ). Note that the spin does not change when we multiply a polynomial , . . . , p d µ 1 f to form Lorentz invariants. ( p [ µ i O ) C p by an ( S ,l ) is trivially generalized to count primary operators in the free theory of a given I G n, 4.37 M If one wants explicit conformal primaries, one option is to construct By construction, elements of While scalar fields may seem like a special case in field theory, the fact is that the Although this paper is focused on operators belonging to the operator basis The above indicates that, for fixed field content and arbitrary numbers of derivatives, )). How does this connect with the momentum space picture in terms of the rings x ∈ ( by the generator forconsists special of equivalence conformal classes, transformations, andan equivalence in class this containing way a any conformal representative primary. element tum space and then enforce the correct transformation behavior. An equivalent option is conformal group. Thisconformal leads primaries to appearing the inφ fact the that tensor the products of operator basis tives. However, this doesis, not mean an they element directly correspond to conformal primaries; that broad reason, and moreessential for pointed understanding ones operator hinted spectra in at more above,5.5 general understanding cases. the Conformal rings primariesRecall and from elements of section f thus see that the and hence not CM) [ operator bases for scalar fields is actually encoding the kinematics of momenta. For this content of free CFTs)eq. is ( to lookspin. at What operators about carryingelements explicit spin. in construction The of these Hilberttation operators? series of formula ( Let modules, i.e. they willnow not involve be the Cohen-Macaulay. polarizationtheme, these tensors This properties happens in are because addition reflected Gram in to the conditions the Hilbert momenta. seriesare necessarily — Lorentz see In scalars, section keeping a natural with generalization our (for example, in studying operator the relevant rings involvingformulated) will fields be with finitely generated spin andscalar of (whose fields. the In precise same particular, dimension description the asoperators the we — rings so have involving that only not theprimary primary yet invariants and secondary invariants of for operators composed of spinning particles in general will parameterizing a formulation will still consist ofof polynomials terms in accounting the for how theor polarization with tensors the can be contracted amongst themselves associated momentum (continuous degree of freedom) and a polarization tensor (or spinor) JHEP10(2017)199 2) 30 , 2 − . As ] and n (5.42) (5.43) n ( S V × 75 ) , d ( K . There is 74 O [ n, K 2, for G n, M / M 3) with the basis for . − 4 dimensions, i.e. for 2) n ,          denote the change of ( 2 Singular ]), ≥ n 34 12 13 14 23 24 − S ]. s s s s s s . Note that elements of n d ( j 77 = . 72          V kl +∆ m = 5 basis s          5 we are unaware of an exist- i ⊕ , ∆ and work with n | + m 1) , y ] appear to be the most straight- ], which addresses kinematics (in the n 1 n > jk 1 1 0 1 0 1 S − 71 s − n 66 − − ( , x × − | V 1 1 , . . . , x ) and letting ) 64 / il 1 n d − − ⊕ s ( x ij 1 . We can take this basis to coincide with ) δ − O − n ij n ( s 0 1 1 1 11 0 0 00 0 1 0 1 00 1 0 0 1 11 0 0 1 0 1 1 ]. Beyond since the free scalar field theory is conformal ij V = s          i ∼ K 73 ) G G n, y = = 5 identical scalars in – 49 – ( , . . . , s j M = 4 → n          23 O 1 2 3 4 and 1 2 and a basis ) n , s x x x X X X X n ij ( K ] for related discussions. 1 i s          i O 34 = h 6= ··· j → I . The algorithm we outline is lower-level, although it benefits was solved in [ decompose into P , . . . , s
l j 5 13 ij S~s = — regardless of whether they are actually primary or not — S s i k i , s × = K X 4) 12 G n, ! s the ≥ d Singular x ( M X n O = ( S )

K S , ~s ( 5 given by ] have dedicated packages for these problems; see the end of this subsection , determining the relevant symmetrized Mandelstam variables appropriate M 5 S 1) 76 , = [ × 1 ] beyond 4-point correlation functions. In the math literature, the invariant ring − 4) 5 n ≥ S 10 ( ] d V ( 2) , O ) ⊕ For concreteness, we take We are unaware of an explicit construction of these variables in the physics literature, A related point is the question of endowing an inner product on (3 K After posting a preprint of this paper we became aware of [ S , ) ( V 5 n 30 [ ( V given by the orthogonal linearthe combinations usual of one built from tabloids of the standard tableaux (e.g. [ Macaulay2 for an example using from its conceptual simplicity and flexibility to handle therepresentations varying of cases of interest to us. and note that such a settion could of be of [ technical use, forC example, to investigate the generaliza- ing solution. In these cases,forward algorithmic line approaches [ of attack. Freely available computer programs, like operator basis with the goalthen of apply determining this a algorithm set toM of the primary case and of secondary invariants.for We 5-point amplitudes. different degrees in are automatically orthogonal underdilatation the operator with inner different product eigenvalues. since they5.6 are eigenvectors of Algorithms the forIn operator this construction section and we the explain a simple algorithm for explicitly constructing elements of the eigenstates of this operator. See [ a natural inner productthis inherited we from can orthogonalize field the theory, elements namely of so the that two-point primary function. operators With obey to construct the conformal Casimir in momentum space and find polynomials which are absence of Gram conditions) with similar tools as presented here. See also [ Fixing an order basis matrix we have, for example, at JHEP10(2017)199 . k n c S . If × . To n with ) S n (5.47) (5.48) (5.46) (5.44) (5.45) d S ( K × m ] ) O α m n, d m ( K M O n, elements of ) = 1. First, . . . x M a 1 k x α 1 ) to reflect the , . . . , x + 1 we account 1 x x = 0. This in turn [ 5.33 ≡ C i µ elements in the ring. p α and then evaluate the n > d i k + 1), one instead picks . x . ) = deg( µ n ) as

random points. If there ij , P elements of ! σ . k . s ) . If ( n k
k k σ c S t ρ m x = ( to be in the set of partitions k i n > d ) | and c R i , . . . , p 2) σ . By working directly with the , α ) α µ 1 ( 2 i j 2 | a is trivial) then these can be ∈ elements in p =0 p p − 2) x , k X , · ) n k 2 0 i ( }i m − p = ρ ∆( h n ∆ ≥ ( ) 0 ) = i . ρ h{ σ m d ] ( t ij α m f j s 1) , . . . , x is just a linear algebra problem. A simple e , − n 1 independent degree t 1 ) is non-trivial ( subject to S x s j k − 0 k ∈ ( (1 } X n c n µ σ ( P A ! ∆ =1 , . . . , x ) and ⊕ m i 1 ) – 50 – { 1 n the ideal ≥ · · · ≥ n x ( Q [ 1 ρ we have 5.42 , . . . , p α ⇔ C ). ) =

1 µ | h m x p ) = ( ∗ t = = ] is spanned by the monomials ) ( ∗ denote the set of polynomials obtained by symmetrizing 5.42 + 1 . When 1 n α f m n ) S d m − x S k ( ( × S , . . . , x × R 7→ ) ) via eq ( ) )  1 A ≤ d d ) ( K . m σ ∈ x ( K ( a = x O n n, O ) , . . . , x n, ( x ij ) per eq. ( 1 s H m f k x M ( . Let µ i [ , we obtain the representation matrices p k A Sρ C , . . . , x n 1 , this is done by first symmetrizing over the degree parts, , S = x k k , . . . , x ∈ m is an overcomplete span of the degree m 1 x α ) σ k ( + have been removed by momentum conservation and the Gram conditions are ]. This produces an overcomplete basis for degree A i points ( m ··· X denote the Reynolds operator which symmetrizes a polynomial: k c + ∗ 1 + 1, one can find the linearly independent elements by numerically evaluating the α d into at most , . . . , x The set Let Now let’s explain in more detail. For simplicity, we set deg( We first outline how to determine a basis for elements of fixed degree in 1 = as functions of the random values for the momenta k x ≤ | [ a k α are no Gram constraintsrandom ( points ( c provides momenta, finite rank conditions are automatically accounted for. Following this procedure, find a set of linearmethod independent is elements in to evaluate the elements numerically on (at least) At a fixed degree | the monomials. Since weof are symmetrizing, we can restrict The Taylor expansion tells us that there are one computes the Hilbertdegrees series of a and set brings of it primary to and secondary the invariants, form as in eq. ( C n elements on enough randomlyfor chosen Gram points conditions ( by insteadx picking random values for where the expressed in terms of the At a fixed degree In terms of the coordinates For an element JHEP10(2017)199 . 5 be S × (5.49) 4) min ≥ (5.50b) k d ( O ) K S , ( 5 are a basis for M ..., i x + which are linearly ). ) to obtain our con- . ) = 1) (5.50a) 10 k = 5 identical scalars t ( ij 2                   s t 5.46 B n 35 45 12 13 14 15 23 24 25 34 s s s s s s s s s s + 13 →                   9 (deg( t t                   where the contains new primary and/or and six secondary invariants 1 5 ) − + 9 } k S ( ] 6 1 8 5 t ) B − , θ 6 5 given in eq. ( t 1 + 8 k − , θ − c 4 7 1 t , . . . , x 1 , θ − 15 ) = 1 (send )(1 3 x t a 5 1 1 [ + 4 t , θ x − C 6 + 2 t − θ 11 1 1 11 1 1 9 t { − − − − with the accordingly. We then increment the degree and – 51 – + 5 + )(1 1 1 1 k ) 1 1 1 1 4 5 c 8 − − − t t t 31 min 1 1 = k ) = deg( ). − 1 1 1 1 + ( − −

+ 2 , linearly independent elements which span the degree ij 111 ) 7 B s 1 k k 4 t 1 1 ( )(1 t is non-empty. The Hilbert series will indicate how many 5.42 − 3 1 1 11 + t B ) = 1. We note in passing the palindromic nature of the 1

k 6 1 1 1 1 1 1 1 1 + 2 ( t − − 0 3 η B t                   , eq. ( polynomials built from already identified primary and secondary )(1 1 + = 2) + ) = 2). The Hilbert series indicates that a minimal set of gen- 2 , k t 2 . Note ij (3                   t s n 2 3 4 5 1 1 2 3 4 5 V − where the subscript indicates the degree. The degree-zero secondary S x x x x x X X X X X × } ) should be considered primary invariants and how many secondary, and (1 , the Hilbert series may indicate that                   d k 15 ) ( K = 1 + = O n, , η = 5 operator basis min 5 9 k S M ( n , η × B denote a set of degree 8 4) ) ≥ , η k d 7 ( ( representation 4 dimensions, i.e. find a set of primary and secondary invariants for O B , η ) 6 K 5 , S ≥ The Hilbert series is readily calculated as To determine primary and secondary invariants, we go degree-by-degree. Let ( 5 S Explicitly, the change of basis is , η d 0 H elements in 31 η ventional grading deg( erators consists of five{ primary invariants is just thenumerator, trivial which element, indicates the ring is Gorenstein. where, for simplicity, we set deg( We now use thein above algorithm to obtainThis the ring operator is basis isomorphicthe for to the invariant ring computing the Jacobian determinant, which isare non-vanishing algebraically if independent. and This only algorithm ifno the terminates further polynomials when primary the or Hilbert secondary series invariants indicates remain. 5.6.1 The repeat. At degree secondary invariants: one identifies theseindependent by from finding degree the elements of invariants (see below forwhere an to example). look for Thea primary Hilbert candidate and series secondary set provides invariants. of Finally, much primary one information invariants needs is on to algebraically make independent. sure This that is easily done by k the lowest degree for which elements in one then splits up the elements of we let JHEP10(2017)199 = 5 ∗ =0. x = 0. ) ; we 2, to i 4 2 1 b i , (5.53) (5.51) (5.52) x x X x 2 4 1 X a x x 2 3 x = 1 1 x i = 0. x , 2 . In particu- . , 3 + 2 x 1 2 5 . By direct 5 θ − . 5 ... x } x ) 4 R − x  ∗ + ) 2 3 x σ ), setting 2 3 . 4 ∈ ( 2 3 2 1 4 x 2 + 4 x 2 . Therefore there x x x 2 ) x 1 2) θ 2 2 5 3 , ) 1 i x θ 5.49 x x 1 0 3 0 1 2 ( 5 + − (3 x = )) and setting 4 2 = 3 2 ρ 4 5 −  x ∗ x 5 ∗ x x ) 3 1 4 . The first two elements ) − S 3 3 4 2 1 x = } x 5.49 ∈ 2 3 x x x ( 2 2 x ∗ − σ 2 2 3 , . . . , x ) , x 2 5 x 5 ) 3 ∗ x x x i x !# ) x ( 1 P 2 x 4 − 3 3 2 x − . We only need to symmetrize + 3 x 1 x 5! 3 x x ( (1) (2) 5 + 3 1 x 4 3 2 4 x x x S 1 x 5 ] x

x 2 x ≡ ( = x , 5 3 2 2 2 2 2 + 2 x 2 1 and using eq. ( − x 2 ( ) } 2 ∗ θ θ x x i 1 , x 2 5 a ∗ 5 ) ( 1 x ( 2 ∗ 3 45 ) x x 1 x 1 ) , ) (using eq. ( x x 2 s 2 + 1 x 2 (1) (2) ∗ a x − x x ) x x ∗ x + x − 3 1 . | | + 4 3 , . . . , x 2 2 is trivial to solve and we may simply take ) factor and take ( 2 1 , x ∗ ∗ ∗ 1 4 x 4 1 − x i x 3 ) ∗ ) ) x − ( ··· 1 x 2 1 x x ) 2 ( 2 1 + 4 5 4 4 x [ , 15 X ( 2 2 5 ] are spanned by the monomials 2 x , x 2 1 + ∗ x x x x ( 2 4 1 5 ( C ) is ∗ x x 2 ) x 2 4 3 3 x x a ) , 2 1 3 12 2 1 1 ), is precisely what is obtained by taking x x x 3 1 ∗ s − x − → x x = ( 2 2 − ) 2 – 52 – x ( 2 4 2 2 x x 5.48 ( 3 2 2 + 3 { ij 1 1 x − x x { 5.51 θ x s + 2 4 3 1 2 , . . . , x x x 5 = x ( ( 1 3 4 x x = and ( + x 3 ( 1 x x + 3 4 [ 2 2 ∗ , x x (2) 2 ) x have the smallest possible integer coefficients, e.g. if ( ∗ (1) (2) x (3) C + 2 1 ) 3 x ) we drop the x x 1 A 4 1 ∗

2 4 A 1 x x + 2 ) should be understood under this convention. x + 2 32 ∗ ∗ f x 2 x x ... ) ) 4 ( 4 3 2 4 1 4 1 − { x − x x + x x (although the relationship is not hard to find by hand in this 2 2 5.6.1 2 1 − . The Hilbert series indicates there is a new primary invari- ( ( 5 2 1 x x + x } x = 2 1 x 2 1 2 ∗ (4) 4 2 4 x ) x (3 − x x " 4 ) or ( − B 2 2 3 3 2 15 (4) − x x x x = 3 3 3 1 A and converting the = 5.51 x ∗ x + + − 2 = 2, we need two random points ) 2 , as given in eq. ( 2 45 2 x )

2 = , so that the set in eq. ( s 1 1 x θ 2 x 1 ∗ x (4) ) x + . The Hilbert series tells us this element is a primary invariant, so we set ] vanish under symmetrization, e.g. ( ) ) indicates there are no degree one elements. Indeed, all degree one elements ( x σ 1 5 3 } , B (

RowReduce ∗ ∗ x so that all terms in x ) ) 2) ··· 2 , = ( 2 1 4 1 ∗ x (3 x x ∗ f + 5.50b 1 ρ ( ( ) and ( x 2 1 { { , . . . , x 5 2 13 2 1 1 x S s ( ∈ x x = = = ( At degree three we consider The degree two elements of Eq. ( Let us use the numerical evaluation described above to find the relationship between At degree four [ σ Throughout this section, we take the symmetrization operation up to a constant. We choose to + and the elements of 3 C P (2) (4) 32 θ 2 2 2 12 1 5! normalize lar, equalities like eqs. ( s vanish; the last one does not and we can take it to be the primary invariant The linear dependence between ( B We note that where the symmetrization gives of symmetrize over these to obtainover elements of θ case). Since evaluate the polynomials on: calculation, the firstB four elementsant are at degree equal four,must to while be some the each linear other other; combination degree such four that therefore, element we comes from can take This is precisely what is obtained by taking JHEP10(2017)199 (5.54) . 2 2 A, B, C, D . It turns θ } − 15 . ∗ 2 , η )
9 4 into a polynomial 2 5 x x , η 3 2 4 45 8 x s x 2 , η x + 7 − 1 5 x , η ··· 6 x 6. This will always be true 4 + , η (45)) x , e.g. ), the primary invariants can + 2( , 2 4 4 12 x ∗ θ s ,..., . ) , so that its square is invariant results in an expression over half = 1 ∗ 4 1 + 5 ) (34) 5 x 4 , 5 0 5.39b S = 2 η x commands.) We then choose some x i 3 { 3 , x x (23) 2 2 , i 45 0 = 3( , . . . , x x x s 1 1 contains algorithms devoted to describing x x + + ⇒ 5 2( ). x a ··· , so a set of irreducible secondary invariants
2 − into 8 x x – 53 – 1 + η ∗ 1 ) 7 − 6 45 Mathematica x 4 1 i 12 η but odd under s ring (A,B,C,D); s x Singular − 5 . The expressions for the secondary invariants are, + or 3 2 5( } 4 A 9 x 9 ··· ≡ 3 (input matrices which generate the group, e.g. η , η x = 4 6 8 + 2 θ η # x , η 6 12 One needs to ensure that the candidate primary invariants are 7  s ]. With a few lines of input, it returns a set of primary and = − , η 4 There are six secondaries 6 x 75 15 η The software , 3 η Per the discussion around eq. ( { 1 0 3 0 1 2 x 74 1  x " .) Expanding 5 S corresponding to the permutations (12) .) NullSpace a >matrix A[5][5] = ... >matrix P,S,IS = invariant >LIB ‘‘finvar.lib’’; >ring R=0,(x1,x2,x3,x4,x5),dp; x The output matrices P,secondary S, invariants, and respectively. IS give a set of primary, secondary, and irreducible invariant rings [ secondary invariants: out we cancan take be given by unfortunately, too long to include in this paper. a page long, so the above is quite a simplification. algebraically independent, e.g. by checking theoption rank is of to the run Jacobian an matrix. elimination Another routine using, for example, Gr¨obnerbases. (We found this byunder noting the the alternating degree group under three polynomial in parentheses is invariant be chosen to be thewhen polynomials there are no Gramalways conditions be to used. consider.more Of For compact example, course, form the we (in above found terms algorithm a of can the different degree six invariant that has a This algorithm proceeds in a straightforward manner. We make a few final comments Secondary invariants. Computer packages. Algebraic independence. Primary invariants. (This is the linear combination thatin comes the from converting (We have cast theother above linear equations combination to as serve as the primary invariant The relationship lies in the kernel of this matrix, JHEP10(2017)199 : . 5.4 j = 0) θ (5.56) (5.55) . This µ i µ n p become i I G n, P to show up, M , . . . , p in section I µ 1 G n, I p G n, M M = 4, and tabulate the , d , this quotient ring is Σ are the Lorentz invariant j i p ij · }i s i i = 8. ∆ p as a linear combination of the 6= { n j . in the momenta , ] = ) n P m ij s n,k derivatives have a one-to-one corre- = ( i up to F k X uniquely 5 , . . . , θ ,...,X 1 1 are the set of Gram conditions (vanishing θ is a possible permutation group in the case [ X imply that they are fundamental to under- } h n C i ∆ I . S , where the structure of the rings η ] { G n, n } – 54 – ⊆ s ij =1 of section M i M s polynomials = 0) and IBP (momentum conservation, n { [ S = k 2 i . The case studies we examine are as follows. C × p I h 4 G scalar fields and n, (4) ), and Σ = O can be expressed primary and secondary invariants of the n ) M K ij S Σ s I ( n, × G ) n, d M ( K M same O n, whose coefficients are polynomials in the primary invariants ∈ M i η f ). = 0 are removed by EOM, the composed of we detail the case of a single real scalar field in 2 i . 5.56 ) p 5 + 1) minors in S = 4 case, we obtained explicit solutions to these rings with relative ease 6.1 = n,k , showing that the rings are generated by the familiar Mandelstam variables × d ( ( n ii 4) O , and then used this to find a set of primary and secondary invariants for the s ≥ × 5.2 d ( K (or symmetric combinations thereof). The insight gained here provides the founda- , O 5 5.6 } + 1) M In section The description of the operator basis in terms of quotient rings immediately lends itself The kinematic nature of the rings We showed that there is a precise notion of generators for the rings In the d s, t, u 6 Applications and examplesIn of this Hilbert section series weintegral compute formula the derived Hilbert in series section in several cases ofHilbert interest series using for the the matrix rings to algorithmic approaches forsection constructing the basis.ring We outlined a simple algorithm in standing general operator spectra,spin. including Here when we expect operatorssupplemented the are by composed some of additional generators fieldsIn physically the with connected case to of polarizationlike spin, that tensors. however, in we eq. generally ( will not have a clean direct sum decomposition In other words, any secondary invariants tion for understanding the ringssignificantly at more higher involved. we can always findsecondary a invariants, direct sum decomposition of the rings into primary invariants and of ( the particles are indistinguishable. in section { where the consequences of momentum conservation, Operators spondence with homogeneous degree leads to a formulation ofideal the to operator account basis for in EOM termsredundancies. (on-shell, of Phrased a in polynomial terms ring of quotiented by Lorentz an invariants 5.7 Summary JHEP10(2017)199 ] in dis- and , we (6.1) 32 n n 5 φ , . Although  5 2 1 x x ) takes the form ], which were ob- − 1 H, 32 4.37 , )   2 + ∆ 1 2  x x ) t/x ) is encoded in the Hilbert x − ; − t 1 ( ). The basic idea to enumerate 2.15 P discussed in section )(1 ) 2   2 n φ, t t 2 S ( tx suppression. x × 1 , − 1 H = 4 − 2 x (4) (1 d φt O ) 1 φ − )(1 K  S 1 + ( 1 n, ). In this scheme, eq. ( 4  t M PE t/x
− ) terms in 2 4.38 exponential − x – 55 – x 1 ; = 1 1 t φ x ( )(1 H 1 P − ∆ )) and tx 1 and arbitrary numbers of derivatives. This corresponds 0 − 2 is to expand the plethystic exponential to order φ SO(4) φ 2 B.16 φ (1 3 dimensions. Both the above examples highlight how the dµ πix dx , 2 Z 1 ) = 1 = 2 x ; πix dx t ) = d ( 2 4 1 P ), φ, t ( we compute a closed form expression for the Hilbert series for = we examine the number of operators as a function of mass dimension we examine the Hilbert series when the particles carry spin. The number H 4.14 = 4 dimensions with the EFT of a single real scalar field and compute the 6.2 6.4 6.3 scalars in SO(4) d .) dµ F only contributes to the H We first discuss the SO(4) case, and subsequently include parity. As in section In section In section In section i.e. ∆ all terms for athen fixed evaluate the power SO(4) of contour integrals using the residue theorem. the measure given by (see eq. ( and, as per eq. ( with leads to more complicated contour integrals (compare to theuse next the subsection). momentum weighting scheme eq. ( We work in Hilbert series for fixed powersto of computing the Hilbert seriesthe for computation is the straightforward, rings the permutation symmetryoperationally, makes the it plethystic somewhat difficult; exponential accounts for this symmetry in the integrand, but as we turn onthat various EOM constraints. and The IBPoperators, main redundancies while (and, Gram give perhaps conditions a give at polynomial an first, suppression surprising) in result the is 6.1 total number of Hilbert series for a real scalar field in of tensor structures inseries. the amplitude This decomposition reproducestained eq. the using ( recent a determination differentappendix of technique. these (For numbers completeness, [ we re-derive the results of [ tinguishable Hilbert series reflects propertiesand of secondary the invariants, the operator role of basesin gram — difficulty conditions, such when etc. — as particles as are dimensionality, well primary identical. as the marked increase JHEP10(2017)199 , , ) ) = ). ). 4 . tu φ φ 2 | 6.4 C ) + (6.4) (6.5) (6.6) (6.2) (6.3) 4.30 = 4). φt ) gives 5.22a φ, t n su 4 + ( , φ + 4 H ( − t st , O H , = )  ) to . − 4 ) ... + ∆  z H 4 ) ( t # + x ∆ f ) and appendix ; − 2 t , 12 ( 1. x 4 2 t ; x  4.3 P 2 ) + 6 < ) 2 t )( 3 1 2 | ( + 2 x 4 2 takes the form (eq. ( z t t | , ( x (2) 10 4 − f − − t 4 2 t ) )) P 4. 1 x H z ) term corresponds to 2 (1 + 4 1 ( − t  4 φ x ≤ 6 , φ, t f ). 2 4, so that in the average eq. ( z
 ) t ( t ( − ) 2 ) 8 n )(1 t t − t f ≤ x (4) give the same result for + 6.1 1 4 ( ( + 8 t ) ) 2 H 4 O n − 2 n n − t for φ ( ( ) − 1 2 n (1 4 1 D N ) + for z φ ( | x n n ) + (1 + f φ φ )( 4 φ, t x | πix ) and the contour integrals, taken around dx 4 1 ( ; φ 2 + ); e.g. the = x t + SO(4) ( 4 x 1 2 ) 6.1 H = t ) + 3 ; H n Expanding PE H – 56 – 2 t ( (2) φ ) = ( = − ( z 6 2 1 ( = t 33 O P n

f φP ) (1 n φ ) 2 2 | − " 4 φ ) t | Sp(2) − ) = φ z = 4. ( φ, t − 1 (4) PE H )(1 ( n , which is related to the fact that the 1-, 2-, and 3-point amplitudes f 4 ) O φ, t 3 t , dµ ) = (1 ( . They have momentum space Feynman rules 1, x H φ 2 4 2 H ; φ t ) + 6 − ) t x (4) = ( ) (recall that SO(4) and ; φ 4 is scalar in the strict sense, namely not a pseudo-scalar. The − O 3 4 ) (1 σ t/x φ t (2) 1 | z φ 4 ∂ H ( ) ( ( − P φ 2 f P terms respectively correspond to the representative operators 5.22b ) )  φ, t = 0) — i.e. they are the first three elements of the ring in eq. ( = 1 8 ( 6 φ ) we use 1 )(1 t t 4! } u 4 H z 4 Sp(2) ν φ ( tx − + | φ ∂ ) t f dµ µ − { (1 )] = + and 3 cases are, of course, easier to compute. The results end up being trivial: , and 4 ∂ z s Z φ, t (4) case we also need to include parity (see section 2 , (1 ( φ ( 2 φ 4 the Hilbert series becomes increasingly more intricate. We write O f , and , [ H 4 = 4 is the parity even contribution given in eq. ( ) = t ) = 2 4 = 1 φ , and ( + n > | φ (up to 2 ) φ, t n ] sym t, x H ( , 2 ( 4 ) − φ, t For For the Carrying out the above procedure, we obtain As a concrete example, take stu φ ( = 1, are evaluated using the residue theorem recalling that φ The (2) H µ | H i 33 P ∂ , x φ are trivial in massless theories. It is straightforward to check that we obtain the expected result with Hilbert series is where in the momentum weighting scheme) the [( and Under parity, we assume in agreement with eq. ( In the expansion above we can begin to read off the operators in the basis. For example, These factors are| then inserted into eq. ( the fourth symmetric product of the argument of the plethystic exponential, where in place of JHEP10(2017)199 ) 5 ), i 10 (4) d t t (5) O ≥ D.3 (6.7) − N ) cases · n ( (1 O 3 =1 ) and ( m i , . A useful t Q D.2 ) and · = 2 36 = 2 with primary ) Hilbert series / 4) terms. In the d 4 . We will look at D φ 3) n ( ≥ . )]; to get the Hilbert − O n n x i n ) ; terms, eqs. ( 35 , namely ( t i x j (3) are not. ( n 10 ( ; φ . φ O t i ); the Hilbert series sheds P ) is tractable). We leave a ( φ ) − 2 P t n ) = 5 both the SO( 6.5 i

t 2 + g )) linear combinations of   1 (
) m 2 . + − i P − i . 8 − 2 8 det 2) ,n t k n t t 5.14 2 / t/x − f 2 . +1 so − n i + ( 2 i t n  −
k,l 6 6 ) . Then, by the residue theorem, f − 2 = 1 t t 1 =0 2 + 25  x n i t 1 t f − i 1 6 − )(1 4 , P + ) t t 1 l P 2)! − 4 k n ) − tx + 16 t 2 1 + −  t tx 4 − − m k 2 2 t 3 (see eq. ( l n t i x + 9 − + (1 ( − =0 + 100 + (1 i X n − 2 4 t  + ( k t (1 3 = (2) case. While this follows from the machin- n + 36 K − 1 − 2 O . Define n +1 SO(2) ) is ≡ n, t t l − ) ,l πix dx – 59 – ) 2 n 1 H 2 2 = 1 t ) = 6.8 + 100 h − t /  1 k 2 x, t i x − 1 2 I t ( − is conjugate to n k f − k (1 = x ) k,l + 1) g ( 2 f = t 1 at m d 1 = ( (2) K −  − d − O k,l n, ,n f 2 n , it is easiest to recognize − m K H =0 i X − SO(2) ∂ n, n ∂x C 1) = (1 th derivative with respect to f = H ) m − πi = 7 : 1 + 25 = 3 := 4 := 5 := 6 : 1 + 1 16 1 1 + + + 4 9 dx m K 2 d ( k,l ( n n n n n SO(2) n, , note that the first derivative is given by f n ) ! I H 1 m m ( k,l f denote the ) m ( denoting a binomial coefficient. k,l f
serve as primary invariants. The reflection property of the binomial coefficient tells b a ij We can include parity and work out the With this, the Hilbert series in eq. ( To evaluate In two dimensions the above takes the form s ery worked out in appendix that on the parity odd component The total number of secondary invariants is given by that some choice of the us that the numerator is palindromic; in fact, we see Pascal’s triangle: The denominator reflects the dimensionality of the ring and indicates the anticipated result Making use of this recursively, one readily finds with and let the above integral is equal to namely it has a pole of order JHEP10(2017)199 , . 2
ij / s +1 3) term ) 2 2 (6.19) (6.18) (6.17) (6.15) (6.16) 2 − 2 − t n n ( 2)( . i . 2 − t  3 minors of 1 n # . − i  × n 2 3 ) t t 1 i  − 1 = 3 case. Using the i − − 3, i.e. ( n d 1) n x (  − − ) 1  8 n − 1 term is equal to t we have ) + ( ,n 8 i t/x 4 n − 1 t + 1 ) t  2 − i − n 2 to t 2 + + 15 n ) — gives a straightforward way /  f − 6 6 i − , t t t ( 4 3) )(1
t 6 − t 4 n 2 , i.e. half of the secondary invariants − ) t 6.12 − − 1 1 i t
− + 6 − n 2 ( − n n 2 + + 6 + 45 2) 4 2 = 1, as does its first derivative; this indi- ( ) n  t 2 2 4 − ) 2   2 − n t t t t )(1 t ), one readily finds t n K 1 2 n =0 i SO(3) i 2( − n, tx − − i − + 21 n N 1 2 P x 6.12 − + 55 2 n (1 (  ; moreover, the t 2 1  =
– 60 – " t (1 2 , which corresponds to the dimension of the ring, = − 1 2 2 2  2 − ,n 6) − 2 − =0 ) n 2 i X n n − − x =0 − ) i K n X n 3 2 n SO(3) n, − 2 (2 t − 3 f ) n − −  t H ) term out of this sum. After some manipulation we obtain, ) − (1 n t 2 ) 2 − t πi 2 ) dx 1 t t 2 πix dx − (1 2 − − (1 + = 3 := 4 := 5 := 6 := 7 : 1 + 1 1 6 + + 10 1 1 + 3 ), the Hilbert series is given by I ) in the numerator, making it more difficult to unravel what is 6 6 n n n n n 1 (1 I t -tensor). − n ( ) is equal to −  n n ) B.17 n 2 ) = 2 O ) ) t 2 t ), these primary invariants are not completely removed from the ring 2 t 1 t 6.15 ) in higher dimensions. Let us work out the (2) K − 3 ij − − s O n, − ( 6.7 (1 (1 H O (1 = (1 = = K K ). The polynomial in the sum vanishes at SO(3) n, SO(3) n, H H 5.14 The same basic technique — making use of eq. ( In two dimensions, the Gram constraints are the vanishing of the 3 We stripped off the factoreq. (1 ( cates that we can factor a (1 Making use of the residue theorem and eq. ( constraints kick in at going on. to evaluate eq. ( SO(3) measure in eq. ( primary invariants are removedconstraints due are to the— Gram they conditions. should still However, bein since present the the in sum some Gram in waysas as eq. we secondary can ( invariants. use Indeed, each the of these terms twice without hitting a Gram constraint. The Gram The total number of secondaryin invariants the is SO(2) ringare are proportional parity to invariant, while the the other half areThey parity reduce odd the (implying dimensionality they of the ring from The first few numerators are (they are palindromic for even Using the binomial expansion (1 JHEP10(2017)199 is 1) — − k c . d 2 4 (6.23) (6.20) (6.21) (6.22) − 37 n ] obtain of fields , 3 C 3 n 32 − 12 − where n t n C k . C . + t 3 i k 2 − c t 10 n t k # C 3  P + 6 3 − n 9 9 i t − t ) = n = 4 one finds t . + = 2 ( d  7 + 20 . The authors of [ ) t ]); to aid in accessibility, in  t 8  G 3 for a fixed number spin t − 32 3) ( + 3 H , 2 − + 1 K 6 5.4.4 6 i − t n 28 − t + 21 K ] there is a tensor structure for each – ). That is, , while in  n n 7 SO(3) 2 n, + t t 54 − 2( 4 H + 10 26 n -odd. t − 4.3 ,  5 C n 2 t 3 + 10 − + 36 1) t ) + [  3 t 6 3 − t (  t d + 6 n i − 4 3) 3 t + 4 K n = ( − – 61 – ] from the scattering viewpoint. SO(3) + 56 2 n, −  t
+ 6 n 5 n H " 32 2) t 2 3 3  − 2( t − n − 2 1 n =0 and QFT i X n 2(  38 + 36 1 3 3 − + 10 4 − ) = − d t n . The appearance of the Catalan numbers appears common in any 2 t n ) t
( t +1 m (3) K 2 + 21 m O 2) for massless particles. n, 3 t (1 + 1) = 2 H − − d
≡ → m m ) 2 (3), the parity odd piece of the Hilbert series is obtained simply by + 20 t t ( ( 2 O t ) be the Hilbert series for this set, i.e. = = 2 we found above t K K (
in SO(3) Hilbert series (see section d SO(3) SO(3) m n, n, m t -point amplitude involving particles with spin. As discussed in section 2 G spin N N ] developed a general procedure for counting these tensor structures. We will n ) — the amplitude can be decomposed into a finite number of tensor structures (possibly with spin, possibly identical, etc.) and arbitrary numbers of deriva- we re-derive the results of [ H +1 1 32 → − m n F t 2.15 Φ = = 4 := 5 := 6 := 7 : 1 + 6 1 + 3 1 + 1 + = 3 : 1 m We ask what operators belong to the operator basis In this section we explore how the Hilbert series reflects this amplitude decomposition. In the case of The first few terms for the numerator are C For our purposes, the word “helicity” refers to the spin states under the relevant little group, SO( n n n n n ,..., 37 38 1 tives. Let dimensionality: in for massive particles or SO( about the underlying algebra, intheir results connection primarily with in section thecorrespondence language of is decomposing between conformal CFT correlationappendix functions (the Φ invariants. In this helicity amplitudeindependent decomposition [ helicity configuration. Recently, [ show how the Hilbert series also computes this number, as well as provides information 6.3 Hilbert seriesConsider for an spinning particles see eq. ( (which depend on the polarization tensors) multiplying scalar functions of the Mandelstam sending We note that the numerators are palindromic for We see that thetotal numerators number are of palindromic, secondary indicating invariants is these proportional rings to are the Gorenstein. Catalan number The where the polynomial in the numerator is JHEP10(2017)199 3 )], 6 and t 2 − (6.24) (6.25) ) 2 t )(1 4 t − 39 − . As usual, k [(1 ∂ ). / n ) 6 Φ t 6.24 − ] (the corresponding ··· 2 ). Hence, the number t 5 1 32 6 , t — that the appropriate permutations). The second − 5 dimensions, while table = 4 7 5 7 7 4 19 10 , = (1 + S ) d . )(1 )]. ) 4 t R 4 4 t 6 5.4.4 ( # tensor t F t t ( ) = 4 2 6 H scalar fields. − − t d G = spin G n − G scalar = 7 (1 − )(1 4 4 L H 2 = 4. The Hilbert series is / 1 t H t F 6 t d 1 6 → H )(1 5 dimensions. The top half of the table is − t + 2 t 4 ,

→ t 2 t
− − t [(1 4 4 6 t t 2 / 4 4 t − = 4 t t t 2 d )) 16 8 (1 + + + 2 + 2 = 1 2 2 − + 4 3 + 5 t t 2 2 2 2 2 2 R 4 t t t t t t F — we have t – 62 – = 4 gauge fields in 2 L d F ) = + n = 4 gauge fields in t H 2 Numerator 3 + 5 2 + 3 2 + 3 2 + 3 ( t 4 n S × = 4 identical gauge fields is equal to 7, 3 + 5 )], and 4 4 4 4 6 4 t n S S S S = 4 identical scalars is 1 SO(4) F (4) 2(3 + (5) 6 + 4 − — special to 4 n × × × × O O H = 4 gauge fields in = 4 identical photons in G # tensor structures = lim , is the symmetry group we impose invariance under. Further, SO(5) 6 + 9 SO(4) 4(3 + n )(1 G n n 4 L,R (4) (5) S t F 1: O O − SO(4) SO(5) ⊆ is for conserved currents). Hilbert series for → = 5 gauge fields. In these tables we arranged the denominators of the [(1 t 1 / n 2 − t d Σ, Σ = × 3 ) for the distinguishable and identical cases, respectively. The last column lists the R ) 6 F t ) denote the Hilbert series corresponding to lists the Hilbert series for d L t ( F 2 ( Hilbert series for − H O ) = = 4 Hilbert series for )(1 S G 4 scalar R d t As an example, take Table The number of tensor structures is read off by taking the ratio of spin to scalar Hilbert F In terms of chiral fields H 3 L = ( − 39 F The of SO(4) tensor structures for H problem in CFT algebra when spin is involved is finitely generated overdoes the the algebra same for for scalars. Hilbert series to beIn equal every case, to the the number denominators of tensor of structures the agrees corresponding precisely with scalar [ Hilbert series. series and sending This formula follows from the notion — discussed in section number of tensor structures in the 4-point amplitude for gaugethe fields, see number eq. of ( independentG operators schematically oflet the form Φ Table 2. for distinguishable gauge fields;column the lists bottom the identical (accounts(1 numerators for of the Hilbert series; the denominators are given by (1 JHEP10(2017)199 : , 5 5 S × 32 32 243 122 243 122 (4) # tensor O ) S G 11 t 26 ) for identical. The t ; in particular, the 12 18 18 t t (5) : 1; ( t 1 5 + 73 18 O − , + 13 t ) 18 10
S t t 25 ( + 108 t )(1 + 108 34 34 , = 4 t t + 28 10 = 5 identical gauge fields is 17 16 32 d t t t (4) t + 28 + 63 16 = 192 + 27 2 t + 2 + 2 n 9 − O 1 t 16 ; t − 24 32 32 → 4 t t t 6 t )(1 q t + 103 + 117 + 39

30 8 t − −
t + 51 10 3 + 39 16 14 14 8 t t + 32 t 30 30 − t ) with a numerator of strictly positive − − t t 14 33 t 7 7 t 23 5 Cohen-Macaulay. t t 28 )(1 t − − − + 42 6 5.33 + 36 + 117 + 110 t 39 7 + 42 28 28 32 12 t not + 2 t t + 49 t t 15 − − 12 t t 7 7 2 12 4 t 22 26 t t t − − − )(1 Hilbert series for – 63 – + 30 is the number of possible helicity states for the + 35 4 = 5 gauge fields in 35 t 6 + 89 i 26 30 26 5 t t t + 103 t + 35 10 n − h S 6 8 + 59 3 + 13 8 t − 10 . t 3 t 14 10 5 t t t − − − × 24 21 10 + 16 t t = 5 gauge fields. The denominators of the Hilbert series are + 28 5 29 24 24 − = 5 gauge numerator t t t t + 63 (4) 8 + 65 n 7 7 4 + 28 t ) 32 ) 32 8 t 2 + 32 + 110 + 71 t O 4 4 8 = 5 identical scalars is given in table , where t t t + 3 − − t i , n 35 + 16 4 4 22 13 20 h n t t t ) to a form like eq. ( 28 22 22 + 17 4 2 (see text). − t t t − − t + 36 (4) 6 6 = 32 = 2 / 2 + 17 =1 t 2 2 6 + 145 / n i t t t O t 6 , but for 6.25 t + 2 + 59 + 4 + 4 + 92 + 88 + 4 t Q + 7 3 + 1) 2 27 20 20 12 19 20 t 4 t t t t t t 5 + 16 + 7 t ≡ Hilbert series for have numerators given by SO(4) : 1 + : exercise for the reader. For the tensor structures, we point out that 32 = 2 2 4 4 + 5 t t 5 +12 4 +88 t +89 +91 2 +18 2 +18 Numerator S ) . The number of terms in the numerator is max n G scalar × ( h, 3 H N (5) 5 5 5 5 O S S S S Same as table (4) 16(1 + 5 (5) 22 + 145 ) for distinguishable fields and (1 S × × × × O O , and 122 = (3 G 5 5 SO(4) 32(1 + 5 SO(5) 22 + 95 (4) Hilbert series for ) ;( G 2 (5) (4) O 1 t An explicit counting of the independent helicity amplitudes is straightforward to obtain The number of tensor structures is simply the number of independent helicity am- As a second example, the O O − SO(5) SO(4) th particle. Symmetry may relate configurations and reduce this number, as long as the bounded by i symmetry operation preserves the kinematics of the scattering configuration. by using momentum conservation and Lorentz transformations to fix a scattering config- The numerator indicates there arestructures is 6 equal secondary to invariants. 192 Therefore, theplitudes. number of Intuitively, when tensor for setting each up external a particle; scattering therefore, experiment the one number of picks independent a helicity helicity configurations state is shown in table 243 = 3 Note that the Hilbertimpossible series to contains bring a eq. negative ( terms. sign in This implies the the numerator; underlying in algebra particular, is it is Table 3. (1 corresponding table JHEP10(2017)199 4, ]. 2 or / ) give (6.27) (6.26) d > d 78 , ( ) ] = 1 for max = ). These n O ∂ 64 ( h, d n ( = 2 case we N O d ) and d . (2)), blue (SO(2))) i ] = 1 and [ h O ]. If our interest is in φ n =1 32 i Y = suppression: not accounting .
) max /d case, both SO( n 1 ( h, − 1 N = 2 (green ( . ∆ d F d → ∞ operators of mass dimension ∆ grows β d in the above equation. exponential all we apply Molien’s formula as in [ , and exhibit the expected exponential sup- → ∞ 2 = 2 case we evaluate the formulas we presented d (∆) exp – 64 – d d → ∞ a d ≈ (∆) . For the 2 and grade the Hilbert series with [ (red) results and the where we re-derive the results of [ / ) 4 : #tensor structures = d 2) F ( all ops even or odd, independent tensor structures in ], ρ → ∞ − 61 d d and evaluate up to grading dimension 80. For the → ∞ ) max n ( h, d n > d > N -dimensions the number of ] = ( d φ If → ∞ = 2 and d d 2, for / ] — see appendix 32 + 1) = 2 and d ) factor in the integrand. For the ) The results of this are plotted in figure In order to make a reasonable comparison, we choose to work away from the canonical We can already illustrate the comparison between the polynomial suppression of EOM Asymptotic behavior of the full partition function of scalar field theory is known — max n ( h, /P N equivalent results. pression between the results. This is incurves contrast enforcing with neither the EOM polynomial or suppression IBP that (dotted), is apparent EOM between only the (dashed), and EOM and IBP (1 scaling dimension [ both do this with and without parity invariance; for the and IBP with the exponential suppressionHilbert series of up Gram to constraints moderately by high explicitextreme orders. evaluation cases: of To the illustrate the Gramabove, constraints, we including take two or omittingthe EOM shortening of and the IBP character of redundancies the through scalar field inclusion/omission and of inclusion/omission of the momentum exponential growth but differthat in the enforcing prefactor Lorentz (which(primaries) invariance is leads (scalar a to power operators) power lawnote law and that in suppression enforcing then ∆). of finite momentum rank the Thisfor conditions overall implies conservation linear leads number dependencies to of amounts operators. to taking However, exponentially at large ∆ [ One can show that the density of scalar operators, as well as scalar primaries, have the same 6.4 Quantifying the effectsIt of is interesting EOM, to IBP, explore andredundancy. some Gram For basic concreteness, quantification redundancy we of do the effects this of in EOM, the IBP, context andfor of Gram a the free EFT of scalar a in real scalar field. all of the symmetryparity by can rotating the still momenta act into( on some the configuration. polarization When results tensors, and and the this other cases will are leave explained us in with appendix low-dimensional field theories, then the most applicable result is the intuitive one: The reason every helicity configuration is independent in this case is because we exhaust uration [ JHEP10(2017)199 . , G G and ] = 1 using d φ a u ] = 0 gives R φ 80 was based on the 70 4 = 2 dimensions, with and d DL 60 1 @ . ¶= G , D ] = 0 leads to an infinity of operators at 1 we give a derivation of φ 50 @ Φ= 7.1 ], in which the building blocks of the H 84 – 40 79 – 65 – zero mode in the Hilbert series and then using [ 30 1 ). In section − ) φ 2.18 , for theories of pions, scattering in the soft limit places − Grading dimension transform linearly under the internal symmetry group 20 i 2.4 in eq. ( u 10 R (2) and SO(2)) imposed on the operator basis, respectively. The solid curves are 8 6 4 2 1 O 16 14 12 10 22 20 18 )) to future work. 1; green and blue curves include these rank conditions in = 2 dotted curve looks exponentially bigger than the dashed and solid curves is an artifact

10 10 10 10

10 10 10 10 10 10 10 d o foperators of No.  Growth of the number of independent scalar operators with grading dimension ([ 6.27 We leave a more detailed analysis of the precise scaling (i.e. coefficients ] = 1 in our comparison. The canonical dimension [ d φ 40 ] = 1) in the EFT of a real scalar field, up to dimension 80. Red curves are when spacetime ∂ As mentioned in section That the in eq. ( 40 = 2 curves with the same exponential behavior and relative polynomial suppression. d they vanish with momentum.gle We particle expect these module considerations to lead directly toof the taking sin- [ any mass dimension; removingd the (1 However, many phenomenological theoriesmost are famously built chiral fromLagrangian perturbation transform a theory in non-linear a [ realization more complicated of way under an additional requirement on the fields interpolating the single particle states such that 7 Non-linear realizations The formalism for computingassumption the that Hilbert the series fields described Φ in section after imposing both EOMthe and dotted IBP curves redundancy; are the without dashed EOM curves and are IBP imposed. after only imposing(solid). EOM; β Figure 2. and [ rank conditions (Gram conditions)dimension are not enforced, whichwithout is parity ( equivalent to considering spacetime JHEP10(2017)199 . . g V are ) µ (7.1) (7.2) (7.3) (7.4) N v (7.5a) (7.5b) presents and , however, ]. It turns , and their µ 1 in the coset = SU( is valued in ). In a non- K u 7.2 84 − denoting the x h µ ( H to / i h w g π ξ, ξ / J g ∈ transformations , so that µ ∈ ]. We will report the µ u i iv is stipulated to trans- 84 , X − broken to ) ξ global µ , , R µ iu ) g, ξ G v ( − N 1 + ∈ , with = − µ . g µ G . u ) . We first construct the enlarged SU( ). To better see what polynomials ) w 1 gξh 1 x 4 ⊂ × ( − = − , ξ h = L a h π ) µ 0 H µ T /f ∂ ξ Note in particular that ∂ i a N µ ( ( -invariant polynomial of v X h → h ) → 41 G ), it is helpful to define the Maurer-Cartan x + + ( ξ + G i , i 1 1 1 7.2 iπ = SU( − must transform as X − − e i µ h – 66 – h h in the definition, µ G µ u µ µ i.e. i w ) = = hu hv hw x ( ξ , ) operators compared with the results in [ , as it depends on ξ µ ) → → → 6 ∂ p µ µ 1 µ ( v g, ξ u − w local ] as an alternative approach to compute the Hilbert series. ( O ξ belonging to the unbroken algebra and h . In this section, we only consider 0 39 ≡ h ξ ) is H µ ∈ = ∈ w ), it is easy to see that the Maurer-Cartan form transforms as g, ξ µ ( ) v gξ h 7.2 g, ξ ( , the components of ] description of non-linearly realized symmetries, providing a concrete h h by taking tensor products of the modules. In going from , with ∈ 48 g , J ) ] 1 denoting the unbroken generators. 37 − h 48 h , µ ∈ does not correspond to a conformal representation. Instead we use the differential ∂ 37 ( a u h T R As a non-linear realization, under a transformation We have applied our non-linear realization counting formalism to the chiral Lagrangian This module forms the starting point for building the Hilbert series; section The typical convention includes a factor of 41 Hermitian. Formulas that follow are easily modified to adhere to the standard convention. the Lie algebra space. Following eq. ( Since However, observe that are invariant under the transformationform eq. ( with form as where we require linear realization, the buildingmatrix block [ of the EFT Lagrangian is the unitary representation and its derivatives; namely, the Lagrangianderivatives. is a details elsewhere. 7.1 Linearly transformingConsider building blocks a spontaneous symmetrybroken generators. breaking We are interested in the EFT of the Goldstone bosons case form technique described in [ under the symmetry breaking scenarioWe of found a smaller numberout of that there is a redundancy in the list of operators presented in [ justification for the form of this single particle module. the derivation mirroring theoperator approach space taken in section we can no longer straightforwardly appeal to conformal representation theory, as in this the CCWZ [ JHEP10(2017)199 , ) ξ µ µ µν D u , F (7.8) (7.7) ξ (7.6c) (7.6a) (7.6d) (7.6b) under a , and global and G H for products , we have to G ∈ ξ G as g -invariance. H G is inhomogeneous, this reads µ × , with v µν , namely if we define with a “matter field” H G . To see this, consider F µ µ ∂ × u H G , . , and 1 µ − ξ , D = 0 µ , u µ (and hence the field strength µν u − F ] = 0 µ . Because these building blocks only , ν . v . , µ = 1 , , . However, a crucial feature is that — µν 1 1 ] + 1 1 drops out of the Lagrangian. Therefore, µ − , w , − − ξu F ν v µν − 1 − h µ ξ h h ξ h F − w = µ µ , u + µν µ µ µ v µ µ u + [ ∂ hF hD gξh hu hD , and = 1, + ξv µ µ – 67 – ξ + [ 1 ≡ → → → → → w 1 − D − µ ν µ is the gauge field for the local group − ξ ξ ξ µ µ µ , u ∂ ξ µν µ µ µ u D ν together with the derivative µ D D ∂ ∂ F u − v ) = 0. In terms of D µ ξ ν v = = 1 − ) implies we can always trade derivatives of w 1 ξ − -invariance is equivalent to the original µ ν µ ξ − is homogeneous: ∂ u ( 7.3 ξ H D µ µ µ µ ∂ , and field strength D D D . But since µ ξ 1 D − ξ , imposing H an independent degree of freedom from the field transforms homogeneously while the transformation of ] which transforms homogeneously. into a linearly realized theory of the local group µ ν is the only component that retains explicit dependence on , with their transformation properties under the group u , not ξ G µν ,D ) as a bi-linear transformation under a larger group µ µ u F 7.2 D [ only enters without derivatives. Then to make invariant terms under and It seems that we have converted the original non-linear realization theory of the global Now we see that the linearly transforming building blocks of the Lagrangian are ≡ ξ local. From this point of view, , and ξ , covariant derivative µ µν µ which trivially follows from symmetry u contrary to a usualis gauge actually theory — thethe gauge identity field of the Maurer-Cartan form i.e. form the combination we are left withtransform the under building blocks transformation. However, eq. ( of Note that F D This can be understoodin from eq. a ( different perspectiveH by regarding the transformationthe of covariant derivative. As usual, the covariant derivative also leads to a field strength similar to a gauge field. If we put then the transformation of We see that JHEP10(2017)199 µ u 2 and D µ (7.10) (7.9a) (7.9b) . This u µ ). Given . These u µ G D , etc. ) 3 µ . µ u 2 u ) imply that ) implies a similar µ , all local operators D 7.9 G 1 7.9b µ ( -invariant as well. (see appendix G ξ ,D ) 2 ). Therefore, the gauge field . ] could have both components: µ x h . ν ( / u h g ξ , 1 / ] , u . By forming operators invariant = 0, and this further simplifies to h g µ ν , are constrained, we can expect its ] ] µ ( ] h ν ν H ν / u µ , u g . Moreover, ( ] u , u µ , u , u ν µ , ,D µ µ µ u h µ u u , u u u [ , , / is fully determined by the field u . [ [ and the covariant derivative µ ). However, this also means that while the h g − + [ u . But [ − − x µ h µν ( h ∈ ∈ = 0 = = 0 ] u ξ / F ν = = ] ] µ µ i g b µν u µ u , u ν F µν – 68 – ∈ µ u 42 , then [ µ ,T ,X D h ν F a u µ a D − ∈ D u T T )) of the field [ [ ν ν , i.e. out of T − u µ ] = [ D is µ 7.9 u ∼ ν ) reads ν ] D − is considered as a local group because in the transforma- u µ ), antisymmetric combinations of the covariant derivatives ν µ , u u 7.8 u µ H D µ u X,X [ , we are guaranteed that these are 7.9a building blocks, D , as it depends on H two and . So, up to equation of motion, all operators are built from local µ h u ∈ ) is µν g, ξ F ( h ) shows that the field strength ), . By virtue of eq. ( is a symmetric space, [ is considered as “gauged”, its local transformation parameter is not introduced 7.9a µ 7.2 D H G/H The equation of motion for To sum up, in a non-linear realization of the global symmetry Eq. ( If and 42 µ Note that the usual chiral symmetry breaking patterns are symmetric spaces. which can obtained from variationthat of the the action divergence with and respectharmonic curl to behavior (eq. to ( be determined. Concretely, the EOM and eq. ( u can always be eliminatedresult for for the polynomials curl in of symmetrized the derivatives acting on under the unbroken group 7.2 Computing the HilbertWe identified series above the building blocks of operators to be the linearly transforming objects group as a free parameter,should but not be instead expected fully as fixed an by independent the field field fromcan the be “matter” built field fromobjects just transform linearly under the unbroken group result is expected. Recalltion that eq. ( Splitting into components, eq. ( we clearly have Recalling that JHEP10(2017)199 ) d ) at that (7.11) (7.14) (7.15) (7.12) (7.13) + 1. If can be J k D ∧ u, q, x, y ( D Z , ) , y Below we will use ( )] ) character for the d 43 H,u . , and (3) independent because } χ ν  q, x, y H ∂ , u ( 1 The above considerations ) µ u y { ) to denote the character of − ( . y . : D uχ ) ( consists of operators in k x  J H,u ; µ K χ q H,u . ) ( , u and χ -weighted SO( ν x P q       ( µ u )  } u ) for 2 ) = PE [ 0) ν , 3 q u µ }  ··· 2 , u . − 0 µ = , µ 2 u µ µ u q, x, y . . . µ (1 1 +1 ( u  u µ D ) k 2 { ( 1 u u, q, x, y µ ) character generating function D χ R D { ( ( d ) = . Then, the – 69 – n +1 D Z , we have used H k The operator basis 4 q ] = 1, this block possesses mass dimension       sym µ χ q, x, y u = n to the field ( to keep track of the mass dimension, the appropriately ) = u u u q u,k instead of the usual derivatives R =0 χ ∞ X n . As [ D ). Therefore, the generating function for the operator space q, x, y µ =0 ∞ symmetric coset: ( u k X } ) = k u,k 3.17 can be obtained by summing over µ χ ) = u D R ) is ··· u and therefore effectively be treated as zero in our analysis. u, q, x, y 1 q, x, y ( R µ ( µ ( ) character of this block should be { u Z u n d . This method has its footing in Hodge theory. χ D J ≡ sym µ u =0 k ∞ n ). Now we explain how to address IBP redundancy by counting differential form D L H The general expression is straightforward to obtain, although the final result is not very enlightening. under the internal symmetry group = 43 µ -weighted SO( operators in However, for a symmetric space it takes a simple form: IBP addressed by Hodge theory. are (1) Lorentz scalars,under (2) integration invariant by under parts. thehand, With unbroken the the group first SO( twoand conditions are straightforward to impose (by integrating over SO( where we have assigned a weight where we have usedJ eq. ( where following the notation inu section full generating tower The corresponding weighted characterblock can be constructedwe as introduce follows. the weight First variable q consider the cohomology to address IBP redundancy;with we point the out covariant here derivative thatrewritten we can using discuss cohomology The character generating function lead us to the following single particle module can always be eliminated in favor of polynomials of JHEP10(2017)199 . - k ) 1)- ,
− , J (7.18) (7.20) (7.16) (7.17) -form. d k d µ . x -forms d k . The Hodge k the operator µ , i.e. 1-forms x δ -forms d , in which case ∧ · · · ∧ d +1 +1 k k -form to a ( . µ ∧· · ·∧ -form, i.e. it is given k δβ
x 1 d d µ d = x . Now we can formulate c d 1 k k ...µ O ω +1 δ exact ...µ k 1 , µ , + -forms are taken equivalent if µ 1 k k c 1 b 0 d . If ω − ! 1 c d k ...µ ...µ O ·O 1 1 e O µ µ = =  ∂ co-closed but not co-exact k k The dual of any 0-form is a a 0 . co-exact 0-forms ω + + d # . Namely, for counting we have ...µ O b 0 b d 1 co-closed 44 +1 µ # if e . O O k ω b 0 − )! 1) = = k
− a 1)-form. In the dual picture, the equivalence 0 a d # (co-closed 1-forms)# (co-closed but not co-exact 1-forms) (7.19) ( − 1 ∼ O e O O − d d =1 − − a 0 – 70 – X k !( co-exact d k O ( if if 0-forms = + is said to be
b 0 b d ) k k as taking a divergence, similar to the intuition that d e
µ O = 0 (the double divergence of some form vanishes by ω x = # δ ) reads d # (co-exact 1-forms) ∼ O 2 ∼
δ a 0 a − d -forms e 7.16 O is said to be O = # (1-forms) k operators in the basis are closed but not exact k ∧ · · · ∧ ω # 1 k µ -form. So the precise statement about IBP redundancy (formulated = 0, then x d 1) operators d k is essentially obtained by contraction with the epsilon tensor. Minus signs, tensor − ( ∗ k -dimensional manifold in coordinates is # denotes the Hodge dual. In words: two δω d ω k d =0 ∗ k )-form, X k ...µ k ω ( 1 ∗ on a µ + 1)-form. Hodge duality gives a natural adjoint to d: the codifferential − ) reads = ω k d = ! k ω )-form
1 k k k 7.16 (we are ignoring minus signs in definitions), which takes a − = − consists of all 0-forms that are not co-exact d is a ( k d ∗ # (co-exact 0-forms) = # (1-forms) e ω -form k ω d K k ω ∗ ∗ ∗ With this language, eq. ( To proceed, we need a little more terminology. The exterior derivative takes a IBP redundancy imposes an equivalence relation among the scalar operators in = 0 follows trivially, A operators k = 44 # dual i.e. the ( densities, raised/lowered indices, etc. are not important for our discussion, so we will ignore these. We can iterate the logic for # (co-exact 1-forms), ultimately arriving at the sequence: that are not co-closed: the IBP redundancy in thebasis original picture (as opposed to the dual picture): Co-exact 0-forms come from 1-forms that do not vanish when acting with form. Intuitively, one thinkstakes of a curl.antisymmetry). Unsurprisingly, If δω they differ by an exact in the dual picture) is that form to aδ ( The dual of a scalarby the which is exterior the derivative d divergencerelation acting of ( on a a 1-form ( is an where where, anticipating our language,equivalent up the to subscript thebecomes denotes more divergence that familiar of scalars by are a taking 0-forms, 1-form. the Hodge taken dual. The wording of this equivalence relation JHEP10(2017)199 , ) . H = 0. . The = 0 (7.21) (7.22) µ 4 k u ν µ u, q, x, y u in front of ( D k Z p ··· ) +1 i ν ) H,u u , ) ) χ ) ) ( i H,u as counting co-closed ν 1 H,u H,u H χ u − ( 1 χ χ H d ( ( µ are listed in table k ∧ 2 3 − d D is given by 1 d H ( − − p u, p, x, y d d − 0 ∧ 1 ( ) by the factor d , ∧ ∧ − +1 k H i Z , u ...... ) d 2 3 ν ) d ) − x d u − − p 1) x ; d d u ; 1 p − 1 u, p d u u , for non-linear realizations and their p ( -forms which appear in ··· ( d d ( p k P 1 p p u, q, x, y and P H ν ( 3 4 +1 ) , are co-closed essentially because the u k Z
1) 1) x k , . . . , d H,u ( ν ) = u 1) ) − − k x ··· ( ( d − ( ) + ∆ 1 − uχ ( ) = 1 ν d 2 k  SO( p ∧ ( k − u, p k d ). So in the end, ( ∧ dµ µ – 71 – ). ∗ 0 y ( χ ···
H 1 k Z u H µ 1 ) 4.37  1) -forms, y − dµ


( d k ) = − k u u u ( =1 ∧ H i 3 2 k X k p − − − 1 ( u, p ...... d d dµ d ∗ ( ∗ ∧ ∧ d ∧ ) = =0 H Z k X k ). To further project out invariants of the unbroken group ν ∗ ∗ ∗ u and piece, we enumerate all the co-closed but not co-exact forms. For ) = 4.11 u ··· H 1 u, p ν ). We additionally gain the interpretation of ∆ ( u 0 k . -form operators. ν H k ...... H 4.36 ··· 1 -forms to properly weight the mass dimension. Therefore, while integrating piece, it is straightforward to count all rank operators contributions to ∆ ν -forms -forms k k ) Haar measure, we should multiply 0 2-forms 3-forms 1-forms d k − d H d , the type of these forms and their contributions to ∆ µ ) character orthogonality. Note that we also need to include a factor vanishes: Co-closed but not co-exact k d ··· is co-closed (i.e. its divergence is zero) due to the equation of motion 1 µ µ u u  ( 1 To compute the ∆ For the µ D 1-form The Hodge dual ofcurl the of wedge products, which is structurally identical to eq. ( each rank where we have used eq.we ( integrate over the Haar measure but not co-exact using SO( the number of over the SO( counts all possible formscases (with misidentified appropriate in signs) the and first set. another which This corrects leads to exceptional splitting the Hilbert series intoexactly as two in pieces: eq. ( Table 4. contributions to ∆ As indicated in the equation, the counting is naturally divided into two sets: one which JHEP10(2017)199 is u , the (7.24) (7.25) (7.26) (7.27) (7.23) H (7.28a) (7.28b) is given by . . p # , but without # ) for the oper-  φ  ) ) R y y ( ( H,u H,u ). The power of χ u, q, x, y χ  (  , . k H,u ) Z k − y H − R d ( is the covariant derivative. d ( . ∧ k d ∧ µ broken to a subgroup k H,u ∧ , k , ) ≤ − D χ ) − d G d  , u 1 u u, p (       +1 and , − +1 k } k H ) )] 3 1) x H µ } 1) u, p, x, y ; 2 u − ( q − µ ( 2 ( ( Z u µ µ ) + ∆ . . . 1 q, x, y ) d P =1 u d =1 ( ) µ D k X x k X { 2 u 1 ; d u, p q µ 1 d p p D ( { ( p – 72 – uχ 0 − D P H ) ) + (1 ) +       x y  y ( under the unbroken group ( and the generating function ( ) ) = = d µ u u H,u u ) = ) = PE [ H,u R column, the antisymmetric product of the character func- u, p SO( R ( uχ uχ 2 . This structure follows from the CCWZ formalism and is dµ H 2 H p φ p q, x, y ) for " R ( Z " ) in the corresponding operator, while the power of u ) ) u, q, x, y y ( ∈ χ y y ( µ ( ( Z u H φ q, x, y H H ( u dµ dµ dµ χ gives the character of the representation Z Z Z  ) y are ) = ) is the character of only contains operators with mass dimension ( ) = ) = y ( J H transforms linearly under the subgroup u, p H,u 1 is co-closed as it is a constant whose divergence must be zero. Clearly, none of u, p u, p ( χ ( ( ∗ µ H,u  0 H u χ k H H ∆ ∧ ∆ Making use of differential forms, the IBP redundancy is recast as an equivalence of The character In the contributions to ∆ is structurally identical to the single particle module of a scalar field, u where scalar (0-form) operators up to aclasses co-exact leads 0-form. to Counting a the number natural of splitting such equivalence of the Hilbert series as ator space where where R the “top” component physically related to soft limits of pion amplitudes. single particle module takes the form Clearly, ∆ 7.3 Summary For a non-linearly realized theory of a global symmetry tion given by the powerthe of mass dimension ofneeded the to operator bring plus it the into rank a of 0-form). the Putting form all (the the number listed of contributions divergences together, we get the operators in the listthere can is be written no as derivative a involvedco-exact divergence in forms. of any another We of operator, do these simply notconjecture operators. because have this So a is they proof the are that case. all this co-closed list but exhausts not all the possibilities, but we Finally, JHEP10(2017)199 H -form k -matrix imply? S requirements of shift sym- ]. This suggests that it should ) is specific to a theory consisting 56 , are universally valid, arising from -matrix to construct the operator ], and general non-renormalization ), and we conjecture these are the dimensions, with scalars, fermions, a priori S 0 55 87 d 7.28b H 7.28b counts co-closed by not co-exact H in eq. ( . We explicitly identified the operators in ∆ ]. H – 73 – G 86 , 85 and the formula for H ] and e.g. [ 65 + ∆ . The formula for ∆ 0 H kinematic selection rules arising from the full Poincar´esymmetry of the = , . . . , d H ] in our operator basis language. = 1 . This would be in the vein of imposing k 88 2 a priori 2-form fields, and detailed the inclusion of parity invariance on the theory. Our 6 dimensions has also been recently obtained [ d/ Of course, a whole wealth of new structure appears in going from considering the Through requiring locality, and by further considering the behavior of scattering am- After making use of Poincar´einvariance of the We presented quite general results for EFTs in d < -matrix. construction, but it would be interesting to make thisoperator connection. basis to thethe full, phenomena dynamical of EFT. the Ittheorems holomorphy would, [ of for the instance, SM be EFT interesting [ to study in be possible to impose suchin behavior section at the levelmetries of on the single theories. particlebuilding module, We as blocks did discussed that not form follow the this single line particle of modules thought for non-linear here, theories instead via identifying a the CCWZ basis, the natural questionThis is: transitions us what beyond dothe pure unitarity constraint kinematics and of causality/analyticity into analyticity has theWilson of been coefficients, realm the shown see of to [ dynamics. enforce positivity In constraints this on context, plitudes in the soft limit, a systematic classification of Lorentz invariant single scalar EFTs discussion of how topaper deal we with made use non-linearly of realized character theory;representation internal we theory symmetries. found in a direct Throughout the and case the usefulof connection of these to linearly conformal ideas realized for symmetries, the and non-linear a story. slight modification use of all S and results easily include invariances under linear internal/gauge groups, and we included a and eliminate redundancies inwe the have operator shown bases how ofin to relativistic this treat EFTs. way the Most we redundanciesindependence particularly, detailed associated of the with construction all EOM ofquantum (in-principle) and theory. operator physical IBP This bases measurements can identities; that one be properly can seen account as make for a in the final a squeeze of relativistic the EFT approach — making full only such operators. 8 Discussion In this paper we have introduced a number of systematic techniques that allowed us to study a natural inclusion/exclusionderivation accounting of of these IBP resultsoperators, also redundancy. tells us In thatsolely the ∆ of general pions which case,associated non-linearly the realize to each term in the integrand of eq. ( The splitting JHEP10(2017)199 ] appeared, -dimensions. d 90 , = 1 dimensions 89 d . 6 – 74 – ]. As with the other facets of the operator basis evidenced in this 38 During the finalization of this manuscript, the preprints [ 1 dimensions? What can we learn from asymptotics? One motivation to explore Finally, as far as the enumeration of operators in bases is concerned, much remains to The formulation in terms of a polynomial ring of kinematic variables that we developed d > by the U.S. DOEPHY-1316783 and under PHY-1638509. Contract DE-AC02-05CH11231,Scientific HM and Research was (C) also by (No. supported theResearch 26400241 by NSF on and the Innovative under 17K05409), Areas JSPS MEXT grants (No. Grant-in-Aid Grant-in-Aid 15H05887, for for 15K21733), Scientific and by WPI, MEXT, Japan. and Witek Skiba for conversations.ingenuity And we we thank sadly Francis Dolan, never whoseXL got off the to is beaten know path supported personally,DE-AC02-05CH11231, for by and lifting DOE by us WPI, grant MEXT, fromresources DE-SC0009999; Japan; one provided TM to TM through also acknowledges is ERC computational grant supported number by 291377: U.S. LHCtheory. DOE HM grant was supported Acknowledgments We would likeManohar, to Adam Martin, thank DavidJunpu Poland, Hsin-Chia Wang Kyoji for Cheng, useful Saito, discussions. Andy Ben BH Stergiou, is Jed Gripaios, grateful Thompson, to Zuhair Walter and Goldberger, Siddarth Khandker, Prabhu, Aneesh paper, here tooquantum one field expects theory. a richer structureNote as added. we move fromwith some quantum overlap mechanics with to the ideas of this paper. study about the Hilbertin series. Can we obtainalong a these fully lines closed is formEFT for to expansion, similar a capture to relativistic analytic the theory in recursion properties relations our that and previous properties are work found [ not in evident in any truncated for scalar fields waslently, very constructing the useful independent for Feynman explicitly rules).ring constructing We picture believe the for that operator developingstructure spinning a basis of particles similar (or these would amplitudes; equiva- thisand be would Mandelstam very go structures beyond useful that the was to more presented basic understand in counting section more of the about tensor the JHEP10(2017)199 (A.3) (A.4) (A.5) (A.6) (A.1) (A.2) , λ . , ) ) α α x − x − ). In particular, the prod- − (1 . ) A.1 ) (1 G j ) . λ i G rt( ( x Y acting on the vector be the half-sum of positive roots, ∈ W + α ρ , eq. ( , Y . We denote the coordinates of the rt ) ∈ W ρ r ∈ λ ( α w ) = w ρ , = det( α. α l x x ) ) − ρ r + w G ( r ρ x ( A λ σ 1) + A ) = ]. − ) be the highest weight vector — in the or- rt 2 X − r ( ∈ 91 α/ α . . . x )(1 W ) = – 75 – − 1 2 ∈ α x x X (1) 1 ( w x l λ σ , . . . , l = − χ x 1 − = l 2 ρ σ λ (1 α/ 1) ) A = ( x G − ( ( l ( ) r + G S ( Y rt ∈ + ∈ X σ Y α rt ∈ α = ). Let r ρ = ) is the sign of group element A ρ ∗ ρ w is equal to the Jacobian factor that shows up in the Weyl integration ( A A ) in terms of various determinants. For further details on how to obtain ) can be shown to be equal to a certain product over the positive roots ρ , . . . , x A 1 ) characters. ]; here, we only review the bare bones of the formula and then specify sgn ρ A.3 d x A.2 − = 92 A , = ( w be a compact, connected Lie group of rank is an antisymmetric sum of the Weyl group = 91 x 1) ρ λ G in eq. ( in eq. ( − A A ρ ∗ ρ W The WCF for the classical groups can be cast into a more “user-friendly” expression A Let A Since the Weyl groupover contains the permutation group,these, this as allows well as us other to useful rewrite formulas, the see sum [ involving determinants. The basic identitygives is a that the determinant, anti-symmetric sum of permutations where the second equality follows fromuct the definition of formula, where ( (this is sometimes referred to as Weyl’s denominator formula): where The WCF then gives the character for the irreducible representation to be torus by thonormal basis — of an irreducible representation. Let The Weyl character formula (WCF)of provides irreducible an representations. explicit formulaexample The to WCF [ obtain is the covered characters directly in to most SO( group theory textbooks, for A Character formulae for classical Lie groups JHEP10(2017)199 r 2 Z i ) r r λ (A.8) (A.9) (A.7) ( − σ (A.11) (A.10) x 0) in the − , ) r ( r ,   λ σ ) ··· ) 1 , x r , ( r ) where the − r 
λ σ 2 ). Note that the j x x x ··· /Z = (0 1 + r 2  , ··· A.6 l − i r 1 Z

 x λ ( (1)  (1) . . . 1 1 1 − σ j ) and the modding out λ σ − λ n r x 1 1 x 1 − i r − 1 − r − , . . . , x x   ) x x S
1 1 . . . 1 (1) 1 − 1 . − + + − r − j ) λ σ = x j j , . . . , λ x ) by first summing over the sign r 1 x x ) λ i 1 y X 1 x x r +  x λ + − i odd flips  1 − A.3 + r x ]: x i − ··· ···  ( SO(2 y ) = ( r − ( 91 r λ V r ( 1) 1) W , the Weyl group consists of permuta- 1 λ − σ + det r ≤ − − = r r x  r , . . . , x ( ( X j

1 , ≤ i

V  x x  x λ i x 1 x ··· A.4 = λ . . . x (1) −  det , . . . , y ···  − σ ··· ··· 2 1  1 (1) 1 j ) labeled by x y − ρ λ σ 1 1  1 2 ( (1) r 1 r 2 x − i + − − λ σ
. x r +det r V j j x x 1 l can be computed from eq. (

 x (1) 1 j 1 1 − ) = ) 1 + λ σ r 1 λ + r 1 − − r x X − i x j . . . x − i ( x x j all flips l ρ i x ) x X r x +   h = + + −  SO(2 ρ For even dimensions, 0). − σ + (2 l ) even flips 1 2 r 1 j r r A 1 χ σ σ 1) λ i x x ) is the Vandermonde determinant det x r = i ). − 1) 1) 1 2 

( , and we have applied the determinant formula eq. ( ,..., x j d SO(2 ρ l r − − l 2 ( ( S = = A = r r det ∈ + + ) − X S S  σ j r j ρ , . . . , y ∈ ∈ y X X 1 2 1 2 ρ 1 different sign flips we can perform on σ σ , r y 1 = − ( r = = = = ρ SO(2 = i is requiring only even numbers of sign flips. The half-sum on positive roots is ) λ − V j y A r λ r 2 The numerator Z SO(2 λ = ( A ducible representation of SO(2 with Using which manifestly agrees with eq. ( where where second determinant above vanishesabove, if we obtain the denominator: flips and then the permutations in the Weyl group [ tions together with anare even the number of signby flips: ρ WCF for SO( JHEP10(2017)199 (A.16) (A.12) (A.13) (A.14) (A.15a) (A.15b) ] = 2 mod 8 91 d 0), we get , , 4  , , x 1 2 3 2  ··· − r x x 1 2 , x 1 + c.c. − i x 4 3 − x + c.c. x x 1 2 = (0 r r 1 2 − 3 l x x x 1 1 2 x + i  2 x x 0), in even and odd dimen- 4 r x .  x ... 2 3 +  r , x x i  =1   3 4 1 1 r i , j ,..., x j + 1 2 x x λ x λ 0 1 2 For reference, here we record the  Q − 1 3 , − i − i + i x x ) 2 x . The half-sum on positive roots is x r 1 x x i x 1 r 1 2 x x r − r − x + − Z − . In even dimensions, spinors are chiral. + x = (1  1 2 1 j 1 3 j + i , r n λ i l λ i − i x x + x r 2 4 x =1 2 4 x r x i  + X r   x x  x x S ) 1 3 2 1 = 1 mod 8. In odd dimensions, spinors are either x x r 3 x x =1 det ∗ i . When specifying to − r = + c.c. i 1 X x d j – 77 – x x l r x 2 characters are r = det 1 2 / , . . . , x + + 1 + +1) x x + r ) = ) = 1 + r r , − 1 1 2 4 x x 4 +1) + r ( ( x x r x 3 1 + 3 3 0) 0) l SO(2 + + + 1, the Weyl group consists of permutations together x x x x j + 2 2 2 2 1 ,..., ,..., SO(2 ρ W r +1) ) + c.c. , . . . , x x 0 0 x x x x − r , r , 1 ( A 1 1 1 r (2 (1 (2 (1 − 1 x x x x r V = 2 x χ χ √ + √ √ √ = ) irreps that are either utilized frequently in this work or may d d + j = = = = ) = λ 1 . ). Following a similar procedure to the above, we get [ j x ) ) ) ) x l ( 1 2 ( 1 2 1 2 1 2 1 2 , , , (3) ( + 1 2 1 2 1 2 V χ , , (5) ( +1) 1 2 1 2 1 2 r χ = 3, 5, 7, and 9 spin 1 = , ( (7) (2 l ,..., 1 2 so that χ + d (9) ( 3 2 χ l j χ +1) r − + − ρ r , r 1 2 SO(2 ρ = 5 : = 7 : = 9 : = 3 : = = A d d d d − λ j λ r The properties of spinors depends on The character of the vector representation, For odd dimensions, = ( where + c.c. means + complexThese conjugate, spinors may be real, pseudo-real, or complex. They are complex in real or pseudo-real; therefore,conjugate. the characters The of spinors in odd dimensions are always self- specific characters of SO( be useful for Hilbert series calculations. sions is with Explicit formulas for certain representations. Combining it all together, the character is given by with any number ofρ sign flips: where JHEP10(2017)199 ] , 91 3.3 (A.18) (A.17) -forms have 2 d etc. They cor- + c.c. , 3 , the 0) d x 4 2 + 1) matrices, + c.c. x x r 1 ( ,..., 4 3 independent, this is the x 0 x x × 2 , 1 2 = 2, 4, 6, and 8: 2 r 3 1 x x x 3 +1 x , d 1 x r 1 x + 1 x x , x r , + 1) 4 i r r x i (1 + r i 3 2 i − 3 4 + , x x + + 1) with highest weight vector − x x 1 +1 3 1 2 0) 3 r x We reproduce results found in e.g. [ r +1 2 x x x 2 x r j + . One readily sees these properties in 1 x i 1 x r l j x ∗ x x r
h x ). . Leaving ) = 4 mod 8 spinors are pseudo-real, while ,..., + h 1 r r + 0 r d +2 4 − i , det k 2 4 + x 1 x 2 + det l x x 2 3 , 3 1 3 3  x x 1 x =1 – 78 – (1 x x 1 1 x . In r i 2 ). Computations in this paper frequently involve x +4) 2 ∗ x ,..., x ) = k , x d 1
Q x r l x ) (4 ( ( r 0) r + χ r +1 = k + 4 2 + +1) + x r 4 3 3 = 3 +1 ,..., + c.c. x +1) and Sp(2 +2) r ,...,l x x x 0 1 3 ) of SO( k ) + c.c. r 1 SU( l 2 2 x 2 l , 2 d x x (4 ( χ x x x +2 1 2 x 1 2 1 1 k χ 1 , so that all the square roots in the above characters disappear. 1 x 2 x x x x 2 i l x x x x + 1) and is a Schur polynomial. 1 x  = (1 √ √ r √ √ √ r = 2 mod 4); in this case the characters for the chiral spinors are = √ r l +4) ,..., ) → d k = = = = 1 = = 0), and = = l i (4 ( ) ) +1 , ) ) ) ) ) ) x k r 1 2 1 2 1 2 1 2 χ 1 2 1 2 2 1 2 1 2 , , l , = 8 mod 8; in both cases, the characters for the chiral representations − (2) ( 1 2 − 1 2 − , 1 2 − , , − , , 1). (2) ( χ 1 2 (4) ( 1 2 d 1 2 1 2 1 2 , , , χ +2) (4) ( χ ( (6)  1 2 ,..., 2 1 1 2 k , , . . . , l 1 χ (6) ( χ (8) ( l 1 2 1 ) can be written as a ratio determinants of ( (4 ( l χ (8) ( χ r χ χ ,..., 1 = ( , l 1 = 6 mod 8 (i.e. , . . . , l , 1 = 8 : = 6 : = 2 : = 4 : l d The character for the representation of SU( Finally, we mention that the fundamental weights in the orthonormal basis have d d d d = ( = (1 where character formula for U( Character formulae for SU( for character formulae in the orthogonal basis for thel remaining classical groups. highest weight vectors respond to the vectorl and anti-symmetric representations. For even As a side comment we remindwith the reader the that covering if group we include Spin( integrating fermions over we are the actually group working we using need contour to integrals; make as sureachieved to discussed by do sending at this the properly end for of the covering section group. In practice, this is simply conjugate: they are real in are self-conjugate: the equation below, which gives the characters for spinors in and JHEP10(2017)199 , ) ), r N (B.1) G/T (A.19) ∈ , . . . , l . It is a elements 1 r commute). l times; we h | ), SO( | T N . W = ( | ) = which carries W !. T | l and G ). For example, N G ∈ T , W = ) ∈ 1 gh | . As is obvious by the 1 − t N − /T S ) h | . hth T and rank( ( ( i ) , for f i N 1 n i − − is conjugate to with ) i = +1 g − N r ) = = 1. We will elaborate on this hth W S + +1 i , G l r , } ( ( can be conjugated into a maximal = dµ T − j − j g ] gives a very thorough explanation. Thus, and switching domains of integration G x x G ∈ . The different elements are basically R ) (Jacobian) 1 G/T 92 1 , ∈ − − 45 T h − ) − . matrices, ( ) ) g g i i r ( W − − and ) with highest weight vector dµ hth gT g f . The discrete subgroup in r × | g ) +1 +1 T = r r g r ) appears. The Weyl integration formula is ( j G ( , + G/T g x i G l – 79 – ∈ Z h ( j dµ ) g x 3.26 in t { G h ( det Z T . normalized as dµ ) = can be written as det G T T ( G/T Z G N ) = ∝ ∈ since the torus acts trivially on itself (as all elements of ) in eq. ( x ) ( α g T ) g x r ( f belong to − ) l Sp(2 g W ( χ (1 ) . The proportionality constant is fixed by noting that the mapping dµ G can be diagonalized by another group element: ) contains T ) be the normalizer of G T rt( g T ( Z ∈ ( × is the order of the Weyl group (the number of elements in α N | N ) the Weyl group is the permutation group is conjugate to multiple elements in Q , i.e. W g N 1 | G/T ). be a compact, connected Lie group with dim( − G be a function on the group. We consider averaging over this function, is the Haar measure on | N G f to ⊂ W ) consists of all group elements which, acting by conjugation, leave elements of the torus within the | dµ = SU( T G T ( Let Let In fact, The character for a representation of Sp(2 , where G Precisely, let N T 45 i.e. torus. Obviously, The elements which actdefinition, non-trivially the make elements of up the Weyl group, i.e. where there is a Jacobian factorfrom from using where measure below. Since every we can rewrite the above integral to be over out these permutations is called thein Weyl group for fact, which we will nottorus prove, that every element related by permutations of the eigenvalues of easily read by physicists. Afterus, doing the this, usual we Weyl then integrationcan explain formula remove how, essentially this for calculates redundancy, the the which cases can samethe that significantly thing Haar ease interest measures, computations. when Finally,and restricted we Sp(2 record to the torus, for the classical groups SU( B Weyl integration formula In this appendix wefactor explain the Weyl integration formula,covered in focusing many especially group on theoryWe how texts; hope the chapter to IV summarize of and [ pseudo-derive some of the main features in a way that is more can be written as a ratio of determinants of JHEP10(2017)199 . . 1 = g − t t g / (B.5) (B.2) (B.3) (B.4) t g | ). ) t )= . First, it g 1 ( Ad 3.26 t . − , so that the . We rewrite ) 1 − 1 is the Cartan δg − − hth . t −
. by Ad hth = h hth = n 1 ( g . A g 1 = ) − f t , det(1 ω − ) ( g it is straightforward to dg G h f δg 1 are the generators of . ( t i − δg is a class function, so that / i A g dµ t | f ) by πix dx ∧ · · · ∧ t . Just as the invariant volume , 2 1 dgh h G/T G B A 1 Ad Z r AB =1 ω ω t − i Y δ / subspace, where A ) we have n − h g g | ω , we have the invariant volume on , , where =1 is trivial and we have t ) 1 | , while it is conjugated under right multipli-
...A t g x i / ) = − 1 1 x n AB g | δg B g δg . A − d δ times. Below we will explicitly compute t ( I Ad |  = Tr d ! | A → η ) det(1 ≡ G/H
t 1 = t − = n 1 ( W
i δg δg δg | − dg det = : 1 | dµ dθ g 0 G − n T – 80 – g p g r ω δg δg =1 Z i Y ) det(1 dg dg = Tr | → t ≡ is π ( when we change variables to From 0 = 2 g 2 1 Tr W ν 0 | iω T dµ covers − ds Z 47 dx = Tr T ∧ · · · ∧ × = G µ Z
1 ) = 2 1 | ) = ∈ g ω t dx − ( ( 1 ds 1 W G/T f | µν = − ) η dµ ω g . The trace ensures invariance under both left and right multiplication. 0 T ( δg δg n hth = ) = Z d . It can be constructed from the Maurer-Cartan forms g dµ is valued in the Lie algebra δgg ( 2 1 G 7→ = G f A 0− , is invariant under left and right multiplication and therefore provides ) ds Z ) = Tr t G g g g A 1 2 ( h, t dµ → − ω ds ). In this case, the integral over dµ ), so that the above coincides with the expression given in eq. ( ,( t δg dg , the measure of the torus is simply α = ( G : G r x 0 f Z = 46 ω − gg 1 → − (1 ) = → ) T 1 δg g G is invariant under left multiplication, − = U(1) × With the explicit parameterization of the measure on Let us now compute the Jacobian from the change of variables The Weyl integration formula is particularly useful when rt( In plainer language, thisδg is the determinant in the adjoint representation with zero’s omitted. T ∈ hth 46 47 α ( the measure as cation, on a metricthe space group find the induced measure on an invariant measure on themetric group. is where we have normalized the generators as Tr( Note that The metric where Π is helpful to recallinvariant some measure facts on about the Haar measure, which is the unique left and right As The determinant is equal to the product over the roots of subalgebra. f the Jacobian. Upon doing so, we arrive at the Weyl integration formula, In the Jacobian Ad isNote the that adjoint map, the which acts determinant on the is Lie restricted algebra to the G/T JHEP10(2017)199 , 1 − G (B.7) (B.8) (B.9) (B.6) hth 7→ ) . ,
) ] h, t 1 1 − − :( G , hth i ( δh, t 1 [ → f − ) h dt T
h 1 ( × − δt δt h dµ , + Tr ) . This means the third trace
t G/T α = 0. , 1 , / G/T h t dh , + Tr x 1 ) −
t g Z , we have achieved a parameteri- / − i t + , t − g ) t . dt dt δh / | / ] ] 1 1
g dt ) ! ]( | 1 g 1 t (1 − − t ) in the last line. Similarly, 1 ) − t − ∈ G ] + 2 δh δh, t Ad . In matrix form the metric reads, ] Ad 1 [ ) δh t t ]( Ad rt( − Y t / δh, t − ∈ − ), we arrive at the formula quoted in the h dt h − dt [ δh t δh t g − α Ad subspace. ]( Ad + + = t − t ⊕ is valued in B.3 δh, t = + Tr − 1 1 / 1 t − = [1 t 1
g − − − / Ad . Upon inserting this for the Weyl integration δh [1 – 81 – 1 − 1 = [ g = dt, | − G

− = det(1 ) det(1 δh − ) ) [1 g h t t t δh | ( 1 = η ] + ] = δh dh t h t δh t − + Ad dt dt ]( dµ t δh t 1 t − δg AB det T − − | η δh, t δh, t − 1 subspace. This means it is picking up the roots of Z h δh t [ . Similarly, Tr Ad − t while [1 δh | p t factor arises because the map h and used + Tr δh = [ = = / t − 1 ∈ W 1 g
| det(1 ] 1 ∈ − [1 1 is the adjoint action on the Lie algebra. Then the metric is dh . The Jacobian factor is the determinant in the adjoint repre- | − h 1 − dg h G 1 δt − W ) = 1 | − h t dt g − ( = Tr δh, t h ≡ f we have ][ 2 ) t δh t 1 g ) is a class function, eq. ( δh ds ( − . Since g ( δh, t [ dt dµ f 1 ) = hth , the subtraction G − g t Z δh = ( ∈ t g = Tr ≡ -fold covering of 2 | ) vanishes since δh δt ds W | With this Jacobian factor, we arrive at the Weyl integration formula We write B.6 where the product isformula over when the rootsmain of text. where we recall that the is a sentation, restricted toso the that and hence where the determinant is restricted to the where zation of the algebra as the direct sum where Ad given by Consider the commutator While in ( so that where we defined Inserting JHEP10(2017)199 ) ρ A.1 ) for g (B.11) ( ) factor: f α − x ) = − ). , which grows gh (1 | α ) 1 x G − ( W
i | )
h + − ) α ( α . rt ( α ) 1 2 f ∈ w (1 ) x α x ) 1 2 α . times. To only pick up ( − x G Q −
( w | α − + α (1 x ) 1 2 ) x rt ) (B.10) W G α | − ( ∈ ( − α − x + α w . x ) Y 1 2 rt 1 ) (1 Q ∈ + ) ) for the − α − ( α rt ) x G ) w ( , with each covering related by a A ∈ Y G ρ (1 ) x α ( + ( ) G G w + Y rt w ( G − x ( Y ∈ rt + 1) w + α ∈ Y (1 rt ) − Y α rt ∈ 1) ( G copies of W α ∈ ( h # − ) ) ∈ | α + ( i ρ ρ X ρ ( ( w Y ), and show that it is equal to a sum over rt i − w w ∈ W α → | x x x α πix dx − into the product; the third line follows from the action ) 48 2 ) = x W W W W α ρ ), α ∈ ∈ ∈ ∈ – 82 – x X X X X − α r − w w w w − =1 x i Y x x − = = = " -fold covering of | ) = )(1 − − (1 ) is Weyl invariant α α ) =1 | − g W x (1 i G | ) x ( x )(1 | G − f rt( α − ( I Y ∈ x + α (1 = rt ) )(1 | − ∈ α G G ( x α 1 W (1 + | ) − map is a Q dµ rt G ( ∈ (1 out of the product. G ) α + Z ) G Y rt ρ ( ( Q ∈ → + w α Y rt − ∈ T x α ) = . In this case the Weyl integration formula is, in some sense, redundant: α × x − G/T G/T (1 ⊂ ) G is the half-sum of positive roots to bring ) is Weyl invariant, then for each term in the sum above we can perform the inverse W g rt( Note that all class-functions are Weyl invariant; in particular, characters obey this The key to proving the above replacement is to take the Jacobian factor, ρ ( To show this, use the Weyl denominator formula (see appendix ∈ f ∈ α 48 In the first linethat we used the Weyl denominatorof formula; the Weyl in group the on second theroots root line and system we it (algebraically, used maps the the Weylto the group definition bring root is eq. a generated system ( by factor into reflections of itself); of to the simple arrive at the fourth line we again used the definition of (on a computer, itfactorially essentially with reduces the computation rank time of by the group). a factor of Haar measures to be This simplification can significantly ease computations, especially for large rank groups If Weyl transformation so that we end up with property. Therefore, forplacement. all Explicitly, computations if described the in function the we integrate main over text is we Weyl invariant, can we do can take this the re- Q Weyl transformations of in the Weyl integration formula.the Note positive that roots. the product on the right hand side is over only Let us assume thath the function we integrate oversince is the Weyl invariant: Weyl transformation, we are pickingthe up contribution the once, same we contribution can replace B.1 A simplification when JHEP10(2017)199 (B.16) (B.14) (B.15) (B.12) (B.13)  ); we omit the i j    x i i x j j i j x x B.4 x x x x − − − 1 − 1 1
1 α  x j    i x j j 1 i − j x x x 1 x i x 1 1 i x  −  x i − ) , is as in eq. ( 1 2 i 1 x − G 1 1 T x −  1 rt( − 1 Y dµ − ∈  +1 1 α i r j  1   x i x ≤ j i  i  j x x i i Y  x x − i

= ( 1 r − O vanishes as its representation l r ) − intrinsic parity ! r , − l ,...,l 1 1 1 ,...,l  χ l − 1 − ( +1 l r r SO(2 ( , ) can be induced from the irreps of c s R 1 r R 1 . . . . 1 00 0 0 0 − ) with 1 − r (2 − ⊕ 1 r − and ) O − r , x s labeled by r r ) c 0) 1 ,l , r l ), with parity exchanging chiral subspaces − 1 1 − ,l r r R 1 − − .

r r coincides with a character of Sp(2 ) ) − (2 r r r r . 2 ) for any parity odd element . O − l , . . . , x ,...,l ,...,l . ··· ··· ··· ··· − 1 so χ – 85 – 1 1 ,...,l l l g x 1 SO(2 SO(2 ( ( , . . . , x ( l ( ! ( 1  R R 2 2 − 2 R ρ ≡ c s The irreps of = = ˜ . . . , x x 2 0 ) 0, 2 2 s r 0) ). In this case, the parity even character , c ,l r 1 − ≥ , x 1 − 1 − r (2 r

l r − 1 ) O ) ]). Instead of following this standard technical procedure, let r r ,...,l ! , x ,...,l (2 1 1 92 (2 1 l 1 1 l O ( x c s O ( R ) representation space . . . R 1 0 00 0 0 r 1 s ≥ · · · ≥ c itself forms an irrep of − 1 l l ) (e.g. [ 0 : actually induces two inequivalent irreps of

r R l > = 0 :                    R r r ) characters, while the parity odd character l l 1) arguments, which we collectively denote by . An immediate consequence is that the character = 0, r ) with i r − r iθ l e r . We can assign an intrinsic parity to these representations — as an SO(2 l = R ]. This coincidence might be a bit mysterious; in the next subsection, we explain i , . . . , l 1 x 47 l In summary, the general irreducible representations of When Consider the SO(2 = ( l scalar, vector vs pseudo-vector, etc.character, In this while case, the the parity parity odd evento character character an overall signago reflecting [ the intrinsic parityhow it assignment), arises a from result folding worked the out Lie a algebra Weyl the two SO(2 matrix is off block diagonal. within representation, General irreps and characters. its subgroup SO(2 us understand it in a more heuristic way. parity exchanges the two chiral spaces sum of the two is an irrep of with on these ( by an orthogonal transformation, i.e.The by corresponding a group eigenvalues are conjugation, where defining/vector representation matrix to the form The defining representation and the torus of the parity odd component. JHEP10(2017)199 ) k ), r µ x ( (2 + l (C.6) (C.7) (C.8) (C.9) O (C.5a) (C.5b) χ . Since i µ | : l ): . x ) R (˜ x (˜ − l ) χ 1 − 2) r , − r µ ). ) irrep with positive x ,...,l r . 1 l r l i , Sp(2 ( 2 R (2 4.3 µ µ ∈ χ X , O x , the parity odd character µ so  P | P l P = 0 is more complicated and = k R + r i H ∈ g X l 2) is not a subgroup of ) = 0 ) = k µ µ is labeled by its eigenvalues θ | . In the basis with the Cartan x x l = (˜ (˜ − k i = , which we explain below. =1 R in the representation − l − l r H r µ r k − i | 2 k g . µ µ θ P | . i so P i k , χ , χ e =1 | r k µ is a direct sum of two chiral spaces (see H ) | to denote the k µ Consider the parity even character x l P θ k h ( ), namely i l transform under parity. This is achieved by ) l R µ r e r contribute to the character =1 l R l | r k 0. The case of ∈ = − R µ X µ P P h i > – 86 – i l ,..., µ e 1 r R | l l | , . . . , µ ( ∈ ) = k 2 X µ ) representation µ χ h P r H as usual. Using , µ P l + 1 (2 k g R ) = ) + µ ( iθ ∈ O l X x + e µ ( g ) ) diagonal, each state ( = ( r l ), we therefore need to identify all the states invariant under = ,l x µ ) = k (˜ ) ··· , x ) = Tr x ) irreps with x − l 1 in the (˜ r l ( x χ l ( , . . . , r (˜ − l ) = Tr (2 + 2 χ χ − l χ , g x and O χ ( weight r + l ), especially considering that Sp(2 ) = ) = µ r r χ = 1 x x x ( ( 0 this is fairly easy: because (2 k l + l + ( − ··· 2. In this subsection, we provide an understanding of this. χ χ > , only states invariant under 1 O 0 k r ) = 0 for µ 1 ≤ l H x x µµ r denotes the set of states invariant under (˜ δ 0 : , called the − l ≡ action, parity either acts trivially on a state, or exchanges two different k P l )) that get exchanged under parity, there is simply no state invariant under parity. = χ > = 0 : 2 R µ H i r r 0 x Z ) is l l C.4 In order to compute µ x (˜ | l − µ parity. For eq. ( Hence, requires a closer check onstudying how the the action states in of parity on the root system of where As a h with χ generators under The character is given by a sum over all the weights formulae for except for Computing characters from the weights. which is the trace of intrinsic parity assignment.negative intrinsic It parity is assignment (see straightforward the to discussion modify in section formulaeC.2 for the Understanding case the of parityAt odd a first character: glance it folding is rather surprising that the symplectic group shows up in the character Throughout this appendix, we use the label with corresponding characters JHEP10(2017)199 : r r by 2 Lie D r r so 2 2 1 and (C.13) (C.12) (C.10) (C.11) so so = 0, the − of r l . Starting from , . . . , r i l co-roots | . Parity exchanges = 1 = (corresponding to Lie j i We adopt the follow- i 1 l 6= ) irrep with − r r i P | for B (2 , j O e +1 i e ∓ , i . − e i 0) 0) 0) 1) , this amounts to studying if the i In an . l i , , ,  | e 1 l − +1) 0 0 1 | r − , , , , . − 1 1 m = 0 0 r . r P , , , = r 2 r and ··· ··· E i 1 0 0 ··· α ) obtained by folding the i m r by the outer automorphism. (c) The Dynkin j α l : 2 so i e E r ··· − ,..., ,..., ↔ 2 r Dynkin diagram possesses a reflection symmetry α P | 2 0 1 2  ,..., ,..., ,..., 1 r , are so ··· m 2 i − 0 0 0 1 sp − (c) (a) (b) D 2 e , , , , r r E of – 87 – − 1 0 0 0 2 1 m ), an outer automorphism of the Lie algebra 2  α , , , , , : m 1 so E 1 E . . . P . Since m 0) is invariant under parity, 1 roots 3(a) i = (1 = (0 = (0 = (0 = (0 , = E µ 1 r 2 1 1 2 | i − P − − α α α µ r r r | = α α i 1 1 µ − − r r r , . . . , l D 1 P | C B l and the parity outer automorphism. r = ( 2 l so (corresponding to Lie algebra , i.e. 1 r associated with the simple roots ) obtained by folding the − e 1 r i − C Relevant Dynkin diagrams: (a) The + E r 2 1 − so r e = r We wish to determinestring if of lowering operator is invariant under the conjugation: Strings of lowering operatorshighest and weight state folding this highest weight state, alloperators the other states are obtained by applying a string of lowering with the complete setthe of last roots two given simplearound by roots, the which horizontal axis corresponds (see to figure the reflection of the Dynkin diagram Root system of ing convention. Theα simple roots for about the horizontal axis.algebra This under symmetry the correspondsalgebra to parity the transformation. outer automorphism (b)diagram of of The the Dynkinthe diagram outer automorphism. Figure 3. JHEP10(2017)199 . r ]. 4 51 E 94 ]. D 93 ) and . This (C.17) (C.15) (C.16) except 1 and (C.14c) relative ). (C.14a) (C.14b) ). They i − r x 1 r E (˜ associated E 3(a) ) ˜ − α = 3 for 1 1 r , 1 long − 2) − 2 n E r − r − − r of the orbit of is r r E ˜ E ,...,l 1 (figure 1 l − Sp(2 ( r r ), and 2 χ α 6 , . . . , α so E 1 average . , ) = . Namely α 2 x . However, it is probably ) as well as a set of notes [ (˜ 6=4 − , ··· n r 0) 2 , . 2 E D 1 two , − 2 0) − , − t r r , n (note that ˜ E 2 one A , 2 ,...,l s 0 1 ) l − ( r 3(c) simple roots , . . . , r χ E , 2 1 0 , ,..., i − in between r α Wikipedia page r = 1 E ( n =1 = (0 E 0 can transform into under the outer automor- . X i r 1 i ··· 1 n α , − , i α – 88 – i r r 2 and + − 1 = E E E t r only ever enter in the combination ( 1 1 i shown in figure r E = = = ˜ − − α = 2 for most cases ( r r 2 1 E 1 1 1 s − α n E ) − − − r r 2 parity symmetrized The folding of a Dynkin diagram is a procedure to use P P P = i E r 1 and sp 1 1 − E E 1 − . This is why we have − r r r − 2 P P r E ˜ α − E r E ( P ] generalized Weyl’s character formula to disconnected groups. 2 95 sp ··· commute with each other, and they both commute with all r E and is the total number of them. Note that root vectors that are transformed 1 − n r denote all the simple roots that E 0 i α , as is obvious from the Dynkin diagram. It then follows that a parity invariant string The version in Wikipedia is to find a Lie subalgebra that is invariant under the outer 2 We additionally found some math.stackexchange posts helpful ( − 51 r phism, and among each other by the outerThe automorphism nontrivial new are simple supposed root to has be orthogonal to each other. We point out that Kostant [ the roots under the outer automorphism, where taking orbits of the automorphism.on the The roots two procedures is are equivalent to dual doing to the each other other:automorphism. on doing the one Therefore co-roots. weautomorphism. are For looking the for root a system, smaller it group corresponds to invariant under taking the the outer Two definitions ofan folding. outer automorphism ofroot a system. Dynkin It diagram is andless discussed, hence known for its in example, root the inOperationally, system the the community two to versions that correspond obtain to there a averaging over are new an outer two automorphism kinds versus of folding one can define [ effectively work with a newis set of precisely the root systemto for the other roots).obtain the Pictorially, diagram we for “fold” the Dynkin diagram for The crucial feature is that therefore act together, effectivelyto giving the the new combined simple lowering root operator So in the procedure of finding the parity invariant states from the highest weight state, we Note that E must contain an equalpair number up of From the action of parity on the simple roots, we have JHEP10(2017)199 2 − r and 2 r . The sp (C.21) (C.20) (C.18) (C.19) 1 D leads to − , r r r 2 . They are D A . Note that n so , hence short 2 − /n r 2 sp root system, not a new , ) , . ) , namely 0) g , p, x , n 1 + ) and we find a new simple root (Φ , ). Φ , is given by 0 3(c) Z 0 x ∆ ) form the basis for the root lattice (˜ p ) H C.10 1 2 pg are subset of the original root lattice, Φ − 2) − 1 ,..., r r − α − − . r 0 i ,...,l (1 (1 1 α ) l  in eq. ( Sp(2 ( n ) = (0 ( l χ r r n =1 , . . . , α . Note that this is a 0 det X i 1 α α ) α det – 89 – d = + shown in figure ) = ( i x 1 O 3(b) 1 and =0 (˜ ∞ ˜ α − Y n − r 1 0) dµ r , − α 1 C r ( − Z ) = r α 2 1 . Yet the characters are given by the weights of the r . = ,...,l 2 ) = 1 1 , p, x l 1 − − ( so , p − r together with χ r (Φ folds to C ˜ (Φ 1 α Z r 0 − shown in figure r , the subalgebra invariant under parity is obviously D H r 1 α 2 − under the outer automorphism. This is what we need if we want to r so B root, and invariant , namely 1 long is not a subalgebra of − r 2 , while the second to 2 It is easy to see that two definitions of folding lead to dual lattices. Using the convention This second procedure picks up all the weights in a given representation that are There is an alternative way to define the folded Dynkin diagram and the corresponding In our case of 1 − r so , the first definition of folding leads to a new root with squared length 2 − 2 r r with the character generating function B C.3 Hilbert seriesWe with now show parity how to computea the general Hilbert field series Φ, when the including main parity as piece a of symmetry. the For Hilbert series, that every roots areE normalized to lengthroots, two while for the simply-laced second rootrelated definition by lattices leads roots to and a co-roots. long Therefore, root it with is squared natural that length the 2 first folding of invariant under the outer automorphism andthis hence is contribute the to the correctthen character. version a Therefore, of foldingsp relevant to our discussions.weight lattice. The This new is simple why root is Obviously they are on the originalall root vectors lattice generated because by the the coefficients newand are set these integers. of vectors simple Namely roots areweights ˜ invariant in under the the new sublattice outer are automorphism. invariant. Correspondingly, all the by the outer automorphism,characters. and For this we purpose, want weaveraging: to introduce the identify new the set weights of simple that roots contribute by to the sum, the without of sublattice of the original root lattice because of theroot non-integer (weight) coefficients. lattice. Weof take vectors the sublatticeinclude of the the root outer and automorphism weight as lattices a that part consist of the group, namely the group is extended outer automorphism flips theaveraging sign affects of the the simple roots last component of the root vectors. Indeed, the This is a short root. ˜ JHEP10(2017)199 µ 1) = , 1 µ − r x (C.24) (C.25) (C.23) (C.22a) (C.22b) are , . . . , x 1 . + x  g ) is segmented d = ( ( dµ x O .  , which corresponds ) with those depending ) r 1 x − r (˜ x − l , implements a restriction, i.e. i , . . . , x ) aχ x ↔ 1  , x does not have to be nonzero. , p r  x , and the ones paired up under . Therefore, all the weights P r Z (Φ r I = ( ) µ − ). To see this, let us take a close µ , µ  6= PE Φ. As the group n x 0 , n x # pg is the Haar measure normalized as H , ( ∂ ) 1) ) + l ) n , − − n ↔ − d 1 g χ ). Note that ( x ) + ( − r (˜ (1 l  r = 0, while the paired up weights have the µ , p − l  O variables Tr C.28 ) = χ I r n (Φ r n dµ µ + n det a 6= g + ,  ( ) 0 l , . . . , x = – 90 – ), this integral can be split accordingly: 1 =1 n − ∞ H dµ d X n x g ( h  ( ( " l 1 2  Z ) as well as the integration measure dµ ≡ ), e.g. eq. ( O − Tr 1 52 ). However, note that x . − ), ) = ) = ag P r r  = exp P µ , p , p The evaluation of this determinant is a bit subtle, because g − + ) (  g , − (Φ (Φ Z (1 1 l 0 , . . . , x  ). = , 1 ag − ) representation formed by 0 H P r x = − d − − ( H 1 g  ag = ( O ). (1 Z l + x − g , . . . , µ = 1. The key ingredients we need to discuss are how to evaluate a 1) ), det P 1 ) − d (1 r r l  ( µ  dµ O SO(2 SO(2 = ( R P  ∈ 1 variables ˜ . The parity action exchanges the coordinate µ . The invariant weights must have ) denotes the = r − P  µ r  n ) ( g x µ r d l ( O ) = Tr(Res For notational convenience, let As explained before, in a general representation, the eigenvalues of ··· Although cumbersome, we are trying to be careful with our notation. Introducing x ( 1 dµ 52 µ 1 + l allows us toon compare the equations dependingχ on the parity structure look at the structure of x to flipping the lastfall into component two of categories, the the ones weights invariant under parity as opposed to the usual paradigm, e.g. Tr The reason is that the trace evaluation for the parity odd element is nontrivial: Evaluating det the usual plethystic exponential identity does not apply: where R determinant of the form det into the parity even and odd pieces where JHEP10(2017)199 # ) k                2 . x . (C.30) (C.26) (C.27) (C.28) (C.29) (˜ # . l ) k 2 2Γ ! x k k (˜ 2 l 2 a 0 1 1 0 2Γ ,

=1 ∞ k k k X . 2 2 . a . − + 1 ) =1 . ∞ n k k k X . x . (˜ 1 − l = 2 = 2 ) + χ . n n n . n x . ) (˜ a − l n                . χ − i ( n )                n a x ) for , . (˜ =1 ∞ . n X n − l . =1 x ∞                X n χ (˜ − l . " ! " . − . P r ) µ . 0 x . i − r ! ( i ) x ) + 2Γ ) 1 + l 2 = exp = exp 2 n x χ − – 91 – P r x x # (˜ h # 0 µ r (˜ l (˜ ) ) l 1 0 0 1 2 x Γ − l takes the form n − n Γ − 2 g

χ 2 g

( a ≡ − ( a l P + l g ) P + µ − µ x Tr ) = ) for Tr x (˜ h x ) + l n n n n . n . x a x x ) Γ . . (˜ (˜ ( . a PE . − l i − l + l n =1 − ∞ ) . χ χ X n ( . x . aχ . " diagonal, (˜ . − h . ( − l I + I µ =0 g ∞ µ aχ X n x ) = x h " n . − = exp = PE f . . g . . ) ( . l − ) is the central result for handling determinants on the parity odd compo-                Tr                ag = PE ). It is easily generalized to the fermionic case: is completely washed out, as expected. It is now clear that the mismatch in ) = exp = 7→ − 1 r − C.29 − P (2 (1 g ) happens only for even powers: l ag + O g in det = r C.24 The determinant then follows: (1 + x l − g det then in the basis with where we have defined eq. ( Equation ( nent of where in the secondon line we made a further diagonalization. Note that the dependence JHEP10(2017)199 ) r 2), = 4 (2 − + d (C.36) (C.34) (C.35) (C.31) (C.32) r O (C.33a) (C.33b) . # . ) ) 2 , x x ) . ; ˜ function, we use , whose derivative (˜ 2 p − ) φ ( p , 0) g x − ( ) 2) ; ˜ Z − f ,..., 2 p x − 0 ( p r ; 1 2) p n, 2) (2 − ( ( − 1 r ) − Γ P r r 1 ) n (2 (2 and the gauge field in − 2 Sp(2 = 1. The measure on g p P P r ( ) = ) can be easily obtained from i  dµ f . , ) and hence is also a representa- =0 ∞ ) ) − r X n ): = 2 dµ x x ) = ) = Z 0 p/x (˜ ( + − dµ d R ) can be computed using the Weyl 1 2 2) 0) 0) C.2 C.22b 2∆ r − ). From above, we know that − pg pg p = Z r 2 (2 ,..., ,..., ) + ) splits into two separate integrals over 0 0 1 2 − − φ )(1 − r i dµ + n, n, ) O ( Sp(2 ( C.22 g (1 (1 (2 ). Focusing on the R ( − 0) of SO(2 px χ χ ) + ) +   g f O x 0 + ) − 4.1 (˜ r g det det ( ) = ) = – 92 – 0) +∆ . (1 for some given field content. In the following, we ,..., f x x 1 . As the folded root system is that of Sp(2 n 0 ( (˜ 2) + SO(2 ,..., −  =1 0 i − Y 0) 0) r B φp n, Z r ) n, , , dµ dµ ), the characters are − ( 2 ,..., ,..., ) − 1 0 0 p χ x Z Z Sp(2 ) n (˜ n, n, (1 C.5 ) = ( + ( − ( − p x 1 2 1 2 Our first example is the real scalar field 2) ( n dµ 0) χ χ ) ( − r =0 l = r ∞ . = ,..., X n ) = 0 r 0 ). The measure of g − The integral over n, ( r ) = (1 SO(2 Sp(2 ∆ ( f − dµ ) = 2 dµ dµ φp r 2 are from normalizing ), det pg " r / (2 d = = O − (2 =0 ∞ + − Y n  dµ (1 O dµ dµ = PE  ) function is given in eq. ( Z ) = ˜ x det ). According to eq. ( r (2 ): ), or even more straightforwardly from eq. ( φ, p, x φ, p, ( ( O + − Z Z forms the representation C.29 C.29 The vector representation determinant in eq. ( φ n The eq. ( dimensions. Real scalar field in ∂ tion of Our task is to computeshow the two functions explicit examples — the real scalar field in C.4 Examples To compute the Hilbert series, one follows eq. ( where the factors of 1 is obviously that ofintegration formula SO(2 given in appendix the end result is simply the components eq. ( Integration measure. JHEP10(2017)199 . # i ) , ) 2 x ), the x (˜ (C.39) (C.40) (C.41) (C.38) 0). In i ; ˜ as well (C.37a) (C.37b) 2 2) 0) 1)] C.5 p − 0) ( r − . , ,..., 2) ) 0 ,..., ,..., , 0 0 − x , r Sp(2 (1 , ; ˜ [2;(1 which are the (1 1 p (2 e χ n , ( pχ . P h µν 2) # + 2 ) sym F − , 2 +2 p r ) 1) representation of 0 1)] 1). By eq. ( R = (1 2 x , (2 ,  ; ˜ − l , 2∆ 2 P , x = 1 p p 1 [2;(1 p ) = PE # + 1 ( 2 x e 0) ) χ x p + 1 ( − φ 2) n (˜ x  1 , (˜ − 1) ,..., n 0) r 0 ) F − )+ 0) , = (2 x h n, x ,..., ( ; ˜ ) = ( ; ˜ 0 ,..., i +1 P 2) , . In the end we obtain p n p 0 ) R n ( − 2 ( x ( ( (1 x l n, r +2 p n . 2) − ( ; ˜ χ 2) 0 e in the ( = PE p − F χ − − i ( r r n 2∆ Sp(2 sym # 1 p (2 2) ) + p )  (2 χ 2 2 − L,R 2 n P P F r =0 φ F p ∞ 0 1) representations of SO(4). The conformal , x ) corresponding to , x X n (2 n = = 1 1 ∆ r  ∂ =0 P x x ∞ ) − , ) + ( X n i ( φp ) − L,R x 1) 1) " ) x , F ; ˜ – 93 – , ( x , p p/x = (1 ), evaluated on ) = i ( ; +1 +1 0) PE l x p n n 2) − ) ( (˜ ( ( + . ) ,..., − x (4) representation χ χ 2) 3.17 r 0 0) r ; ˜ 1 n − O 2 )(1 (2 p n, (2 r i + ( ( ,..., p into chiral components by defining the chiral components 2+ P ). In the second equation, the first sum in the brackets is 0 P ) = 0 ) = χ 2) ) 1 0 2 px − x n n, 2 − Sp(2 ( ( − i 2) representation relation µν ∆ p F p r p 1 − , x x χ F 1) 1 , (2 − − n φp =0 =0 ∞ ∞ x + (1 p P As a second example, we consider gauge fields in four dimensions. X n " X n r ( +1 (1 i contain derivatives 1 " " n h 2) 1) x − ( =0 , − ∞ =1 ( 1 2 1 2 i − Y X r n χ 1 r +1 L,R − =1 = = n r i ( + F = 4. ) = PE ) = ) = Sp(2 χ ˜ x P x x ) = PE d (˜ (˜ 2 dµ which transform as the 0) 0) , x ) = Z 53 φ, p, 1 , ,..., ,..., x ( 0 0 e (˜ F − ) uses the Sp(2 ) = n, n, 2) 0) ( − (  Z − Γ χ r F 6= 0. ,..., F, p, x n n φ, p 0 ( r ( p , p = l C.37a ) + Gauge fields are handled by working with their field strengths r Sp(2 (1 − In Minkowski signature there is a factor of , Z =0 =0 ∞ ∞ χ (2 0 X X n n 53 L,R H Using these, we obtain for the parity even piece characters are rank 2 antisymmetric representationfour dimensions of we SO(2 can split F representations for SO(4), whose direct sum gives the Gauge fields in In addition to its physical interest,with this case is an example of how to address representations Gathering it all together yields the final result for the parity odd piece of the Hilbert series Eq. ( as the scalar conformal character, eq. ( The two sums are evaluated as follows: JHEP10(2017)199 , i ) , 2 ) 2 (D.1c) (C.44) (C.42) (C.43) (D.1a) (D.1b) , x ) to the 1 . , x looks like ) 1 x , . 2 ; x d 1 i p ; − , x ( p 4.16b d 1 1) ( q , 1)] x

, 2 ; −  (4) , x p 1)  ( P ; 1 2 x,  [2;(1 p (4) ) ( 2 − ( e i χ p 1) P ) ) , from eq. ( 1)] 2 F  , + x 2 +1 ( p H n  ( i ) + 1) [2;(1 2 , , ) 2 χ + 1 e χ x , q, x, y 0 2 +1 }  , x ∆ = i n q + 1 F, p F 2 (

i 1 ( . h Φ x 2 x ) i Γ ; { − ) i ) x , x ( p n 0 ( + ( 1) Z + 2 1) , H ,  1)] 2 1 x , = PE ) for general field content, i.e. based on ) 2(2+ 1 x x +1 p x ; # + , q, x, y ) + n , p 2 ; 2 − ( } [2;(1 } + 1 1 q 1) p i i F χ . 1 e ( χ ( , 1 x ). Generalizing ∆ 2 Φ Φ 2 F, p x P F − { ( { 1)] x H , + h ( (  ) +  ( + 0)] ). Noting that , , 1 1) 4.21 x Z ) p – 94 – 0 0)] 1) H ( , , x h PE x [2;(1 ), the Hilbert series for gauge fields in 4 x, H ( − 1) ) )  e [4;(0 , χ ( +1 h 2 C.29  x p n [4;(0 and ∆ 2 χ  1 2 1) ( ( χ +1 , ) χ , x , 2 1 F n ) χ C.33 − d + 1 2 − ( x ) h +1 x 1 H + x 1 n χ ( n  H ) = 1 ; + ( ) 0)] x SO( n , 2 1 d p PE χ 0)] ( , x x ) h 2+ 2(2+ dµ . F, p x + ∆ p 2 [3;(1 SO( ( (4) 1 2 ; H p 1 0 [3;(1 χ F p Z p P 4.2 ( dµ =0 h + 1 + ) χ ∞ H H − ) X n − y 2 + 1 + ) = 2 ( (2) 2 1 − Z =0 x ∞ x 1 2 X n G 1)] 1 ) ( P F , x + ∆ x x y x − 1)] 1 " ( ) , , 1) ( d , dµ x x results with eq. ( p ( G (   +1 Z [2;(1 [2;(1  2 2 n +1 dµ ( χ p χ p d Z = PE Γ +2 SO(4) Sp(2) 1) 0 Z = = = = ) = PE ∆ d − dµ dµ p p 1)] 1)] , − Z Z factor is computed using eq. ( , = = ( = F, p, x 2 1 [2;(1 = = − ( H e [2;(1 χ H H − Z 54 + − ∆ e , , χ This is somewhat schematic; for example, if gauge field is non-abelian there are other pieces pertaining Z ∆ ∆ 0 0 54 H H where our focus is obviously on ∆ to the gauge group. See section general field content, we have D Computation of ∆ In this appendix, we explicitlygeneral evaluate ∆ generating representations of eq. ( with Combining the we have The with the conformal characters JHEP10(2017)199 . , Z # ··· a in (D.3) (D.4) (D.2) is in a 0 1+ G,F mod 4. ∆ χ j j φ ⊃ ) q ¯ ¯ r ψ ψ j a -form field ¯ , G, G, F 2 d # : if G,φ . In this case χ χ +
i i † . χ i ¯ ¯ G a ψ ψ mod 4, due to , and hence the φ j ¯ # ψ j F is the minimum
G, G, ( r φ i G, . ∗ G,φ ¯ χ χ 0 i ψ χ j j a χ ¯ ¯ X G, ψ ψ 2 i i G,ψ = 1 , χ . ¯ ¯ ) j ψ ψ χ − , and the 1 2 r i G 2 sym φ + G,φ is even or odd — so does ∧ 1) 2 i i d i 0 n 1 n d n . . . ) (= number of h p p p           2.15 , -point configurations , + 2, to rotate ] n 4 6 g 1) for massive particles. ··· . i > 3 3 − t , 4. Because − + i 4 , 2 d e 2 3 , . . . , y           t M 2 independent variables, eq. ( 1 0 d 1 d d d / . . . , = y p p p [ 5 ] for determining the number of 4 e C           i + 1) , 32 f n > d > 4 d + ). t ( , t d , to get to the COM frame ( 6 3 =1           3 i F.9 M 1 1 1 1 − h t µ 0 − − − − . . . 0 1) = 0 d 1 d d d − M p p p we can obtain explicit formulas for , where we consider 4 3 − be the number of allowed helicity states for S g d d           ] ( i − n 34 6.3 h + ≥ 3 – 97 – being 1, , s n e i ··· into (12) plane; (4) use 24 f = µ 3 + , s p to some configuration, and then examine the helicity 3 23 µ n           , we let . Otherwise, this number may be reduced. 2 3 0 3 1 3 i n . . . , and 0 0 , s p p p ); (2) use boosts , t h 4 Φ µ n 14 g 2 We begin with the case           p g , =1 to denote the number of independent tensor structures. i n , s 3 + ], we first fix a scattering frame by using the Poincar´esymme- , . . . , p = ) are Lorentz generators): (1) use translation generators to set momentum − 2) for massless particles and SO( µ 1 h 13 there is no general expression in terms of the Q n ,..., 1, to rotate 32 p 2 ( , h ··· 1           µν ) is, − e , s 3 ≡ p 0 2 N g + . . . 0 0 M d 12 p i > = − , ]). µ 3 s E.1 , th elementary symmetric polynomial in the variables . We recall that polarization tensors are transverse and are specified 2 [ p 2 i ) i t           n < d max n 1 t 54 n C , ( = h, Φ M + 2 µ 2 g N p           , 0 1 1 . For = . . . + 0 0 p i e p ,..., ) h µ 1 1 n p           ( h 4: no symmetry. N denotes the 2 scattering, we can rotate to the following scattering configuration 0)); (3) use being i , − e i y th particle and use n ··· We go case-by-case, beginning with the situation where there is no remaining symmetry Following the logic of [ The basic setup is described in section i , In words the steps are ( 0 55 , → 0 2 conservation ( p plane; etc. Note that this procedure leaves us with terms of the be easily computed from a group integral,n see > eq. d ( > all the Poincar´egenerators to adjust the2 momenta of particles. For example, if we consider the momenta, i.e. stabilizes thefigurations. scattering If frame, no then such this symmetry exists,is may then relate simply the various number helicity of independent con- helicity amplitudes and work our way up in complexity. For For a given set ofthe fields Φ try to bring the momenta configurations in this frame. If there is some remaining symmetry which does not change independent tensor structures inhelicity amplitudes the [ amplitude decomposition eq. ( from fields Φ by the little group SO( with F Counting helicity amplitudes In this appendix we re-derive the results of reference [ where decomposition of eq. ( and setting JHEP10(2017)199 n S does ⊆ (F.3) (F.2) ; this is odd 3 (when n 1) acts 2) , S 2 − ) , − is covered max n n , there can 1 = Σ ( ( h, n V . Recall that n > d S N n kin n , are permuted Σ ⊆ ,..., 2 i n < d n ⊆ m 4 , contractions with i ) that stabilizes the d S kin n m P only for ( ≤ . d } ∈ = O d i n µ i representation of h , where parity is the only p u d 1) , =1 + n i ··· = (2 t 1 ) invariant tensor structures is V 1 Q µ i n d (14)(23) + ) vanishes by momentum conser- i p ( = d h , s O F.1 n ...µ =1 ), i Y ) 1 max n µ (  h, = ) N s, t, u n (13)(24) in eq. ( ( ) invariant tensor structures is essentially h ). The various cases to consider are: , d µ d N ( p O action, these orbits involve at most two different 5.16 – 98 – 2 4 : stabilize the configuration. Hence, Σ Z is isomorphic to the n If we have a permutation group Σ (12)(34) is even or odd, respectively. 4, the number of 4 S , eq. ( , ) symmetry, the parity element diag(1 S n ) ∈ d ) e S max ( ( d > n σ and therefore can be used to relate different polarization ( h, n > d > O = N , when subject to on-shell and momentum conservation, fill 56 ≡ { ij n s . Here we explain the case for kin 4 2 if d When parity is a symmetry of the theory it can relate helicity / S 3), so all ≤ . In particular, the permutations 3 ≤ n + 1) S parity is part of a larger spacetime symmetry 4, the last component of n representation of ) max representation of n ( 2) h, d > 3. , 2) N , (2 n < d = ≤ 2 V 4: parity. − n odd). Hence, for n n ( i V 2 or ( 2. h according to not exist for for is just a fancy way of saying the usual ( / / these vanish by momentum conservation). Therefore, for The On-shell and momentum conservation fix all Mandelstam invariants ( d > 4: permutation symmetry. -tensor necessarily involve polarization tensors. d 3: More precisely, parity acts trivially only if the configuration is all spin-0 helicity states; To determine independent amplitudes, we take orbits of the polarization configurations When  ) ) We are working back in Euclidean space, as we have done throughout this paper. In Minkowski space, max max = n n ≤ = ( ( ≤ = 4: 56 h, h, the following results hold for applying both parity and time-reversal symmetry. n n be permutations that stabilizerelate polarization the configurations. momenta Call configurationthe this and Mandelstam permutation therefore invariants subgroup can Σ out be the used to moreover, there is at most(i.e. one all such configuration, and itN exists if and only if n the under the action of parity. Ashelicity parity is configurations, a soN the number of vation. If the theorytrivially possesses on the configuration, states. In then invariant description, we can form n configurations when spacetime symmetry which stabilizesbelow the (when scattering frame.scattering The frame). case of no further operations thatHence, can the be number done of to helicity relate amplitudes the is various just external polarization tensors. Because we have used all of the Poincar´esymmetry to achieve this configuration, there are JHEP10(2017)199 . kin n ) is (F.4) 57 −− = 4 the ) where x (1). n + 1. ( i i ζ group, e.g. ζ n 2 = − , and therefore i Z . Below we give d 3) singlets 2) h , 2 2 ≡ − − which stabilize this d n is a polarizations ( m living here are n V SO( S kin 4 its representation under ∈ S i V σ              ). Note that 1 2 ) symmetry acts non-trivially 0 4 k k . . . 3, e.g. from fixing a scattering 0 0 lit i p m , we can bring all the momenta − − G = 4, the orbit of (+ + ≥ ) symmetry,              d m n + n < d th particle, = 1 it is the little group for the particle. i = rank(              n act non-trivially on s 1 2 0 3 . . . ). 0 0 n p k k , ++); in this way one finds seven distinct or- When S lit i kin n              ∈ . We will typically denote this as G – 99 – i matters only if all four particles are identical, − − σ = V kin 4              p 0 2 ) + 2( . . . 0 0 0 (4) when we include parity, as in table p − 2) symmetry; for 4. O −− −              d n > + particles tensored together. n ) rotations — as visualized in the above equation — and impose ) (and a possible Σ              0 1 m . . . m 0 0 0 p p ) the character of . Therefore, Σ = 4 identical gauge bosons in ) = 2(+ + g }              + 1)-dimensional submanifold for ). Hence, there is a remaining SO( ( i n 3) acts as: , or if we have two pairs of identical particles. n V −− F.1 − kin 4 − Tr ) parameterize the torus of d S s d be the relevant little group for the ≡ (13)(24) (+ + = 1 is trivial for = , i ◦ lit i ) ζ 3) acts e n kin 4 kin = 2 it is a remaining SO( G σ ( − , . . . , x Σ { Σ configuration. In other words, all stabilize the kinematics. Any nontrivial subgroup of n : leftover rotational symmetry. 1 d After fixing a momentum configuration, there are no ) invariance for the kin 4 x living here S 4: Let The number of such invariants is counted with our favorite routine: character orthog- As polarizations are transverse to momenta, this SO( m , and ∈ For SO( on polarizations σ = ( 57 lit i G x be invariant under SO( onality. The idea isto to the take subgroup the of representation SO( SO( of a particle under its little group, restrict In order for the amplitude to be Lorentz invariant, the polarization configurations need to see this is aframe ( as in eq. ( on some of theremaining polarization SO( states. For example, picking a scattering frame for bits for SO(4) and fivea orbits concrete for formula for counting these orbits in general. n < d into some hyperplane, leaving rotations in the orthogonal submanifold. It is not hard to We determine the independent helicityFor configurations example, fromP the for distinct orbits under Σ n > JHEP10(2017)199 58 A = 1 (F.9) (F.6) (F.7) (F.8a) (F.5a) s (F.8b) (F.5b) x ) repre- ). m , , m = 2 ( 2 − 1 ) is trivial. − 1 ). O x . x ··· m 1 + − 2 ) to SO(2) gives + 4.3 1 x = 1 ), breaks Poincar´e= − 1 and the integral in − 1 F.5 are coordinates for + x ) “higgses” or “breaks” , i x F.4 +1 2 i r ): r h . x h F.1 1 x m − 1 + ) characters, appendix denote the SO( n =1 x i Y n 1 + 1 + i + 1 + . − 1 1 , . . . , x (1) = ) 1 V and section ) by setting the appropriate 1 i x x x x s ζ x ( . C . + + n + + 2 + (1) = d ζ 1 1) 2 1 n 2 1 1 x x ζ x x = ). Let ··· , . . . , x n . ) is trivial, then we simply multiply all 1 ··· ,..., ) = i x x so that F.4 ) = ) = 2 1 ( ( V ). Introducing the shorthand notation ) become , i x x 1 ) kin n ) = (1) ( ( r ζ ζ lit i , x 1 1 m ζ ζ 1 ζ x G F.7 lit i ( x +1) ( G SO( n → , . . . , s (3) (2) 0) − , the scattering frame, e.g. eq ( ) χ , . . . , x d (4) (1 x Res 1 – 100 – , e.g. eq. ( ( + 1 i χ x ) into n ( SO( ≡ V r i ζ m i ζ n < d dµ ) = ) = V = ··· 2 1 ≡ ) Z x i ) ) is our usual notation for SO( ), together and average over SO( ( , x x x , x ( 1 ζ = i ( ( 1 i x ) ) iθ ζ F.7 ( ζ n ) n e ζ ( l ( h = 4 then restricting the characters in eq. ( m χ N = n + 1). SO( there is no remaining rotational symmetry, i.e. SO( i n = 5 the characters for a graviton and a massive vector — little x d − dµ 1), the relevant character is given by ) is readily obtained from d d ) characters are explained in appendix g ≥ ( Z ), then the restriction is obtained by setting n i n ( ].) If parity is a symmetry, then the group integral is over SO( = 5 and V m ,..., O 1 32 d = 0 in → , = 5 graviton to SO(2) : = 5 massive vector to SO(2) : d r i θ T d d n ) only acts on a subspace of . When 1) m ) becomes trivial, , , . . . , x 1 1 x − = 5 graviton: = 5 massive vector: F.9 n < d ( In fact, as is hopefully clear, the present discussion actually applies to all cases, not We are now in a position to write down the number of independent polarization con- SO( For example, in Restrict Restrict In a very particle physics language, the momentum configuration in eq. ( d = 1. Specifically, embedding SO( d d ≡ 58 i up to a possible average over the parity action when the entire Poincar´esymmetry, whileSO( when just In particular, the restrictedeq. characters ( in eq. ( (See eq. 5.8 in [ figurations. If the kinematic permutationthe group restricted Σ characters, eq. ( x For example, if The character Tr x the torus of(i.e. SO( setting We remind the reader that (also, if needed, sentation obtained by restricting the group action to this subspace, groups SO(3) and SO(4), respectively — are JHEP10(2017)199 ) + ).) , is 2 = 4 ) G F.12 6.2 g n ) hold (F.12) (F.13) (F.14) (F.10) (F.11) ( ∈ V contains g . n Tr F.14 ),  ) Φ g ) and ( 1 2 2 ( ρ x . , (  2  F.11 ,..., 2 ζ ) = . ) and ( 2 ) ) 1 g )  2 2 ( ) 2 gives the character ρ ). Taking the trace x 2 V x ( 1 2 ( x j F.13 1 ( 2 kin ζ 4 ζ i 2 S ρ sym ζ + 2 ) j + 3 2 + 3Tr( 2 1 i ) if all particles are identical ) 4 x 4 ) . ( ρ ) x 1 ( x  ρ ζ ( 2 F.3 2 + ζ ) ζ 2 2 2 +  Tr( j ) x 2 2 1 4  i x = 3 one has to be more careful. We ( ) ( 1 4 ρ ζ , then the appropriate representation x 3) 1 n 1 j ( in eq. ( j ζ − v 2 1 = d )  i ζ )], we have for the characters + 3 g 2 ρ 1 2 x kin 4 4 ( ) ( = 4. We assume that Φ (4) ( SO( ) S σ 1 2 3) x ij f j d x ( [ ρ − 4 ( dµ 1 i d ζ = ζ ρ kin 4  →  Z S SO( 1 4 i 1 2 – 101 – (2) ··· v = σ j dµ = = ) 2 (1) i as h  σ  ( 4 ρ Z j ) ) 1 V x i The little group for massive particles is SO(3), so that x N (1) ( = ( ρ σ ζ 2 j )  1 59 h i kin 4 . , ζ ( 4 ρ ) S
kin ). That is, 4 , we obtain the familiar formula Tr ) if two pairs of particles are identical (here we picked the pairs X ∈ N x 2 2 2 S ( σ S j } 1 x 2 ∈ F.9 i 1 4 ( ζ σ δ  4 χ j 1 j 4 P i , is a recipe for symmetrizing indices. For example, if 1 + : i δ all identical: 1 2! δ 2 V ) , n (13)(24) x ··· is kin,pairs 4 , ( ) 1 Σ = 4 this turns out to okay and the results in eqs. ( j χ = 4 e Let us enumerate the cases in 1 V kin,pairs 4 ( i 2 n n 1 2 δ 3), as in eq. ( ≡ { is non-trivial we do the following. Recall that taking the symmetric product of a ] for details. − )] = two pairs identical: , which has usually been phrased in this work as an action on the character d 32 kin n x ,
( ) 2 χ [ g kin,pairs 4 = 4 We have been a bit cavalier in the above by ignoring the underlying statistics of the To obtain the appropriate counting for these cases, we average eqs. ( The point of reviewing the symmetric products is that it is the exact same procedure If Σ 2 ( Really, it is the covering group U(1), but CPT ensures that all one-dimensional representations come have two polarization states. V n = 4 cases. 59 i at least one particlestructures with is spin just (if one). they TheΦ are little all group for scalars, massless then particles trivially is the SO(2), number so that ofin all tensor pairs. massless particles. For whether the particles are bosonsrefer or to fermions, [ but for d and over SO( (One may wish toSimilarly, compare for this Σ to formula for the fourth symmetric product, eq. ( Extending this to an action on functions, sym to take tensor productsthe with possible any non-trivial type permutation of groupsor are permutation Σ symmetry. In particular,to for be 1,3 and 2,4). Taking the tensor product with, for example, representation, sym the representation matrix acting on acting on sym by contracting with Tr JHEP10(2017)199 . + 2 are 2 h 2 or ( (G.2) (G.1) / (F.15) (F.16) 2 8 P} 1)). It h , ) − 1 max h , , ) = 1 and = { ( 4 1 1), which is ) P , N h ( − ( 4 4 , ζ . N = 7, as in table ! ··· ) ) ) . Here, h P ( 4 P ( ( P} 4 is non-trivial: N 1 ζ , ζ 1 (1) is diag(1 . { kin 4 ··· ) O ) µ )). Therefore we obtain u P 2 µ ( g 1 u ( ζ (1) = i ) is non-zero only if all particles V O + tr ( (3) we can take parity to represent 2 i π O . is even or odd. We can also derive f h F.16 (2) represents as diag(1 i . ) = 1, while for any half-integer spin − h ) simply becomes a sum over these two 4 O + 2). i =1 i Y 4 we simply have P . ) h 2 = max ( =1 h , + 1). i

h 4 i F.9 ( + 1) = 4
ζ n 1 . =1 2 ξ 1 2 = 0 i Y Q h d N n > h (1). For µ 2 ( 1 µ 2 = = O D − ) means Tr h u h ) with the integral being trivial. We have 1 ( 2 ) = 4 µ ) – 102 – 1 2 n g x − h ( h ( ( D ) h ( ξ n> max i 4 1 4 1 h , µ O 1) (whose restriction to ζ ( + 1. If ζ 4 F.13 N h i − D 1 2 = l N , ··· ) 1 h = ) = tr ( 4 − = 2 g 2 ) ) π , ( h i h N f ) with As discussed above, the answer is simply 1 ( 1 ( 4 4 h ζ . − = N N F.9 P} . , K Analogous to cases (2) and (3) above, we obtain has 1 = 1. L . i X (recall that l ∈{ We use eq. ( (3). The integral in eq. ( g = 1. . kin O kin,pairs 4 kin 4 4 h 1 2 S 2 depending on whether + 3). If the four particles are massless this gives kin,pairs 4 kin 4 kin / 4 2 ) = S h 2 ( (2) or 2 4 (1 + 1) h with spin O ζ i = ) ) = 0. Therefore, the second term in eq. ( max ) h , ( h 4 P ( 4 ( (1) = i N N 6), which reproduces the entry in table with spin are massivewe and recover have the integer stated spin, result. in which case ζ For all massless particles withremains spin, the parity in same upon restrictingon to the vector asis diag( easy to seeζ that for any integer spin the identity and paritygroups elements restricted fromelements, their representations in the little ( this result from eq. ( = 4, we need to distinguish the cases when parity and/or Σ n representation is At the leading order, the non-linear Lagrangian is (6) With parity, Σ G EOM for non-linearIn realizations this appendix, we show that the EOM (at the leading order of the EFT) in non-linear (4) With Parity, (5) No parity, Σ (3) No parity, (1) No parity, Σ (2) With parity, Σ If a massive Φ JHEP10(2017)199 ]. δξ ]. 1 − ]. (G.3) (G.4) (G.5) ξ Int. J. fields): , SPIRE SPIRE , π IN IN [ SPIRE . IN ].  ) [ SpringerBriefs Zh. Eksp. Teor. , , δξ δξ ( 1  − = 0. Since µ SPIRE ξ δξ D IN ) K [ 1 1 as (1973) 643] [ µ L − − u ξ (1973) 161 δ ξ ξ µ µ ) Dimensions + µ D D  76 3 B 53 ) u 1 arXiv:1310.5353 ξ µ , µ − The analytic S-matrix ≥ + ( δξ ξ ]. D ( D D ]. 1 µ δξ = 1 − ]. + (  D µ ξ , we have − Holography from Conformal Field 1 ) u ξ SPIRE δξ − δξ δξ )
ξ SPIRE IN δξ
: ξ arXiv:1308.1697 1 Bounding scalar operator dimensions in ( 1 µ SPIRE IN Annals Phys. ][ , K + − 1 − Erratum ibid. , . IN ξ D ][ L ξ ξ − Covariant expansion of the conformal [ ( µ ξ µ µ ][ 1 u − D = 0 D − 1 ξ µ µ − µ µ u D ξ ) = u u µ – 103 – . ) + (1972) 77 δξ  D − Tensor representations of conformal algebra and ]. ( δξ µ ) + ) ( δξ µ u 1 1 δξ δξ ) D − ( ( − 1 B 49 arXiv:0807.0004 ) ξ (which is of course induced by variations in ξ 1 1 δξ limit of superconformal field theories and supergravity − [ µ ) arXiv:0907.0151 ( − − − hep-th/9711200 ξ ), which permits any use, distribution and reproduction in ξ µ [  1 ξ ξ [ Scattering Amplitudes N δu u µ µ − µ + µ ]. µ ξ u u u ξ u µ µ µ D  µ u u u (     D tr ( tr 2 2 π π
SPIRE (2008) 031 tr tr tr tr Nucl. Phys. f f 1 2 2 2 2 π π π π , IN (2009) 079 arXiv:1601.05000 2 2 , we conclude that the EOM is − CC-BY 4.0 f f f f The Large- [ 12 h (1999) 1113 − − Nonhamiltonian approach to conformal quantum field theory / δξ 10 This article is distributed under the terms of the Creative Commons g = = = 2 = 2 = 2 = 2 38 EPFL Lectures on Conformal Field Theory in A brief introduction to modern amplitude methods = K JHEP µ L , JHEP δ (1974) 23 [ δu , . This variation will result in a variation in 66 δξ CFT + D in Physics (2016) Theor. Phys. Theory Fiz. 4 four-point function conformally covariant operator product expansion Cambridge University Press, Cambridge (1966). ξ I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, J.M. Maldacena, A.M. Polyakov, R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, S. Rychkov, H. Elvang and Y.-t. Huang, S. Ferrara, A.F. Grillo, G. Parisi and R. Gatto, S. Ferrara, A.F. Grillo and R. Gatto, R.J. Eden, P.V. Landshoff, D.I. 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