Constructing effective field theories via their harmonics
IIPMU19-0001
Constructing effective field theories via their harmonics
Brian Henning
1, 2 and Tom Melia Department of Physics, Yale University,New Haven, Connecticut 06511, USA D´epartment de Physique Th´eorique, Universit´e de Gen`eve,24 quai Ernest-Ansermet, 1211 Gen`eve 4, Switzerland Kavli Institute for the Physics and Mathematics of the Universe (WPI),The University of Tokyo Institutes for Advanced Study,The University of Tokyo, Kashiwa, Chiba 277-8583, Japan (Dated: February 20, 2019)
Abstract
We consider the construction of operator bases for massless, relativistic quantum field theories,and show this is equivalent to obtaining the harmonic modes of a physical manifold (the kinematicGrassmannian), upon which observables have support. This enables us to recast the approachof effective field theory (EFT) through the lens of harmonic analysis. We explicitly constructharmonics corresponding to low mass dimension EFT operators. a r X i v : . [ h e p - ph ] F e b . INTRODUCTION The approach of EFT is to consider all possible contributions to a given physical ob-servable. Particle scatterings and decays only have support on a physical manifold wheremomentum conservation and on-shell conditions are satisfied. These constraints manifestas what are termed equations of motion (EOM) and integration by parts (IBP) relationsbetween operators in the EFT, and have been the subject of extensive study spanning thepast few decades [1–8].In a series of papers [6–8], it was shown that these constraints are ultimately a conse-quence of the Poincar´e symmetry of spacetime; this insight enabled a systematic enumerationof basis elements ( i.e. operator counting) in an EFT. In particular, by considering a largerspacetime symmetry—that of the conformal group—it was shown the operator basis nat-urally consists of conformal primary operators, which could then be counted using Hilbertseries techniques.In this note, we put operator construction on the same footing as operator enumeration,by detailing the systematic construction of the conformal primary operators that providea privileged choice of basis for the S -matrix of the theory (for other approaches to oper-ator basis construction, see [5, 6, 8–12]). The presentation is designed to accompany thepaper [13], which considers more generally the entire operator spectrum (not just Lorentzscalars), as is relevant for more general correlation functions. This note also proceeds moreheuristically than [13]—in particular, by including a number of worked examples—and omitsmany mathematical details. We have endeavoured to provide pointers to [13] in the rele-vant places. We would, however, like to refer the interested reader to [13] for a reinforcedconnection to ideas in conformal field theory (CFT), and modern (Hamiltonian truncation)non-perturbative methods.We consider four dimensional relativistic theories of massless particles, and allow for allparticle spins by working with spinor helicity variables, which encode both kinematic andhelicity information. In these variables a U ( N ) action on the phase space of N particles isrevealed, which generalises the U (1) N ⊂ U ( N ) particle little group scalings. This symmetryplays a crucial role, first via a duality with the conformal group SU (2 , (cid:39) SO (4 ,
2) thatin [13] we term ‘conformal-helicity duality’, and second via its symmetry breaking patternwhich, in the case of EFTs, is down to U ( N − × U (2), identifying the physical manifold2s the Grassmann manifold G ( C N ) = U ( N ) /U ( N − × U (2) (the kinematic Grassman-nian [14]).A new picture of EFT emerges—that of harmonic analysis on the Grassmann manifold.There is a tight analogy with the harmonic analysis of a sphere: functions f = f ( x, y, z ), withcoordinates subject to the constraint x + y + z = 1, can be expanded in terms of sphericalharmonics on the sphere, f = (cid:80) l,m c lm Y lm . In the EFT case, observables O ( { p i } ), subjectto the constraints p i = 0 and (cid:80) i p µi = 0 involving particles of any spin can be similarlydecomposed into harmonics of the Grassmannian, O = (cid:80) (cid:126)l c (cid:126)l (cid:101) Y (cid:126)l (with Wilson coefficients c (cid:126)l ,and with (cid:126)l a vector of quantum numbers to be specified below). For the case of the sphere,harmonic polynomials in x , y and z are annihilated by the Laplacian, ∇ ; these form a basisof spherical harmonics when restricted to the sphere. For the EFT case, we will constructharmonic polynomials in spinor variables which are annihilated by a generalised Laplacianoperator, K , that turns out to be the special conformal generator (whence (cid:101) Y (cid:126)l are primary);these form a basis for the S -matrix.The note has the following structure. In Sec. II we detail the construction of the EFTharmonics, presenting the main result from [13] and providing additional heuristic moti-vation. In Sec. III we use this result to explicitly construct low-lying harmonics therebyproviding EFT bases at low mass dimension. Sec. IV concludes. II. CONSTRUCTING EFT HARMONICS
EFT quantifies all possible S -matrix elements between an | in (cid:105) state in a multi-particleFock space and the vacuum, (cid:104) | S | in (cid:105) . (1)We consider massless asymptotic particle states labelled by kinematic (three momenta),helicity, and possibly some internal quantum numbers. Moreover, we consider multi-particlestates that are built from distinguishable particles, deferring a discussion on exchange sym-metry to Sec. IV.We encode the kinematic information using spinor helicity variables, p µi ( σ ˙ aa ) µ = (cid:101) λ ˙ ai λ i a , (cid:101) λ ˙ ai = ( λ i a ) ∗ , (2) Massive states can be described via two massless states (up to an SU (2) little group redundancy), see e.g. [15]; we will leave extensions in this direction to future work. a, ˙ a = 1 , i = 1 , . . . , N a particle, or flavour, index (raisedon λ and lowered on (cid:101) λ to anticipate the action of a U ( N ) symmetry), such that S -matrixelements (cid:104) | S | λ , (cid:101) λ , h ; . . . ; λ N , (cid:101) λ N , h N (cid:105) = f ( { λ i , (cid:101) λ i } ) δ (4) ( N (cid:88) i =1 (cid:101) λ ˙ ai λ i a ) , (3)where f ( { λ i , (cid:101) λ i } ) is a Lorentz scalar function of the spinor variables. In eq. (3) we labelledstates in the Fock space with spinors to encode the kinematic information, and with helicities h i . In these variables, Lorentz invariant phase space is written as, d Φ N = N (cid:89) i =1 d p i δ + ( p i ) = N (cid:89) i =1 d λ i d (cid:101) λ i Vol( U (1)) , (4)where δ + ( p i ) = δ ( p i ) θ ( p ) and the volume of the little group Vol( U (1)) = 2 π .We are interested in a basis for the functions f in eq. (3). That this is equivalent toconstructing an EFT basis, taking into account EOM and IBP, follows from the standardintroduction of local operators as products of interpolating fields—see [8] for a detaileddiscussion on this point. Note that in using spinors, we automatically take into account theEOM ( i.e. the momenta are on-shell). The fields are required to transform under Poincar´ein the way dictated by the helicity of the asymptotic state. For example, λ i a transforms inthe ( j , j ) = ( ,
0) representation (rep) of Poincar´e, and thus interpolates a negative helicityfermion ψ L ; (cid:101) λ ˙ ai (cid:101) λ ˙ bi transforms in the (0 ,
1) rep and interpolates a positive helicity masslessspin-1 state, (the field-strength operator F R = ( F + i (cid:101) F ) in spinor variables); pairs of (cid:101) λ ˙ ai λ i a imply a derivative acting on the interpolating fields in the operator. In other words, F ˙ a ˙ bR = (cid:90) d λd (cid:101) λ Vol( U (1)) (cid:16)(cid:101) λ ˙ a (cid:101) λ ˙ b e i λ a (cid:101) λ ˙ a x a ˙ a a † + h.c. (cid:17) , (5) etc. In this way, f transforms under the asymptotic particle little groups with the correcthelicity weight.The delta function in eq. (3) enforces total momentum conservation, P ˙ aa = N (cid:88) i =1 (cid:101) λ ˙ ai λ i a = 0 . (6)This equation is a constraint on the variables λ that fixes the functions f in eq. (3) to lie onsome manifold ⊂ C N . This manifold is well known in the literature to be the Grassmannian,4 ( C N ) [14]. Fixing a Lorentz frame and writing, λ λ · · · λ N λ λ · · · λ N = uv , (7)one sees that the vectors u and v define a 2-plane; under Lorentz transformations u and v rotate within the plane so that, modulo these transformations, Lorentz invariant phase spaceis described as the set of 2-planes that intersect the origin in in C N , which defines G ( C N ).A more general manifold is obtained if one does not mod out by Lorentz rotations [13]—this case is most easily analysed by considering the breaking of the U ( N ) symmetry (underwhich u and v transform as fundamentals) by eq. (6) down to U ( N − U ( N ) /U ( N −
2) defines is known as the Stiefel manifold. (In the present case,the coset construction of the Grassmannian is U ( N ) /U ( N − × U (2), with the extra U (2)being the Lorentz transformations and a complex phase that are further modded out.)Returning to the analogy with the sphere where the Laplacian ∇ in essence forms anadjoint to | r | , we construct the adjoint operator to P ˙ aa as K ˙ aa = − N (cid:88) i =1 ∂∂ (cid:101) λ ˙ ai ∂∂λ i a , (8)which is the generator of special conformal transformations in spinor variables. The harmonicmodes of the Grassmannian manifold are those annihilated by K ; they are thus identifiedwith primary conformal operators. To construct a basis for the functions f in eq. (3) wetherefore turn to constructing such harmonic polynomials. A. Harmonics from Young diagrams
Let us build basis polynomials out of n λ s and (cid:101) n (cid:101) λ s, at fixed N ≥
2. Because thepolynomials are Lorentz scalars, n and (cid:101) n must be even, with the spinors contracted as[ j j ] . . . [ j (cid:101) n − j (cid:101) n ] (cid:104) i i (cid:105) . . . (cid:104) i n − i n (cid:105) , (9)where we use bracket notation (cid:104) ij (cid:105) = λ i a λ ja , [ ij ] = (cid:101) λ i ˙ a (cid:101) λ ˙ aj , and where the indices i ..i n , j ..j (cid:101) n are (unspecified as yet) particle indices.We consider raised particle number indices on λ as U ( N ) indices, such that λ i a transformsunder SL (2 , C ) × U ( N ) as spinor × fundamental. Similarly, (cid:101) λ ˙ ai transforms as (the complex5 IG. 1. Young diagram for the harmonic modes of the Grassmannian. conjugate representation) spinor × anti-fundamental. That is, the indices i to i n in eq. (9)can be interpreted as (raised) U ( N ) indices, and the indices j to j (cid:101) n can be interpreted as(lowered) conjugate U ( N ) indices. The latter can be raised using the epsilon tensor,[ j j ] (cid:15) j j k ..k N − . . . [ j (cid:101) n − j (cid:101) n ] (cid:15) j (cid:101) n − j (cid:101) n l ..l N − (cid:104) i i (cid:105) . . . (cid:104) i n − i n (cid:105) , (10)with summation over all j indices.The key result of [13] is that the basis polynomials furnish a particular representation of U ( N ), labelled by eigenvalues n and (cid:101) n . Equivalently one can label by mass dimension ∆,and helicity h , ∆ = 12 ( n + (cid:101) n ) + N , (11) h = 12 ( n − (cid:101) n ) . (12)Finite dimensional representations of U ( N ) are in one-to-one correspondence with Youngdiagrams—see e.g. [16]. That is, the Young diagrams encode the symmetrisation patternto be applied to the indices in eq. (10), to form a U ( N ) irreducible representation. Theparticular Young diagram that renders eq. (10) a harmonic mode of the Grassmannian isgiven in Fig. 1. The indices k . . . k N − in eq. (10) are associated with the first column whichis shaded blue (to indicate it corresponds to (cid:101) λ indices raised with an epsilon tensor); theindices l . . . l N − in eq. (10) are associate with the right-most blue column; the indices i , i To provide a translation to the notation used in [13], here n = l + l and (cid:101) n = ˜ l + ˜ l in the Lorentzscalar case where l = l and ˜ l = ˜ l . We note that more general non-Lorentz-scalar operators are furtherlabelled by spin eigenvalues, j and j . i n − , i n with the final column: . A basis for the U ( N ) rep is supplied by semi-standard Young tableaux, as discussed inthe next subsection. For now, we want to reflect upon why it is that this representation isprimary.To begin to understand this result, let us start by considering holomorphic operators—that is, functions consisting purely of λ s. These are obviously primary (annihilated by K ).We consider basis functions that are polynomials in a fixed number n of λ s. These λ carrytwo indices, λ i a . A simple but important observation is that if a symmeterisation patternis applied to one index, the other index automatically inherits this pattern. For example, λ i a λ j b + ( i ↔ j ) = λ i a λ j b + λ j a λ i b , (13)is a symmeterisation in particle indices i and j , but the resulting expression is also symmetricin a and b . Similarly, λ i a λ j b − ( i ↔ j ) = λ i a λ j b − λ j a λ i b , (14)anti-symmeterises in i and j ; the anti-symmetery is inherited by a and b as well. This worksfor general symmeterisation patterns that are encoded by the Young diagrams. So, whena polynomial in n λ s is organised into a singlet representation of SL (2 , C )—correspondingto a Young diagram with n/ n/ U ( N ) indices inherit the exact same symmeterisation pattern, f ( n )hol ( { λ } ) = g SL (2 , C ) ⊗ g U ( N ) = , (15)Note that this implies that U ( N ) representations corresponding to Young diagrams withmore than two rows— i.e. that are anti-symmetrised on more than two indices—can neverbe constructed, e.g. λ i a λ j b λ k c + (anti-sym in i, j, k ) = 0, for all a , b , c .7he above considerations apply to anti-holomorphic basis functions in (cid:101) n (cid:101) λ s: again, the U ( N ) representation is dictated by the symmeterisation pattern on the Lorentz indices suchthat the functions are Lorentz scalars, f ( (cid:101) n )anti-hol ( { (cid:101) λ } ) = g (cid:63)SL (2 , C ) ⊗ g U ( N ) == , (16)where we used a barred Young diagram to denote the conjugate U ( N ) representation; in thelast equality we redrew this as the (cid:15) tensor conjugated diagram.Now we turn to the non-holomorphic case, concerning n λ s and (cid:101) n (cid:101) λ s. Such operators onlyappear for N ≥
4, which reflects the familiar fact that Mandelstam invariants are trivial for N ≤ λ s and (cid:101) λ s separately have their SL (2 , C ) indices symmeterised into theLorentz scalar patterns as in the holomorphic and anti-holomorphic cases above; again the U ( N ) indices and conjugate U ( N ) indices will inherit the same pattern. What is differentthis time, is that now the resulting U ( N ) representation is reducible, f ( n, (cid:101) n ) ( { λ, (cid:101) λ } ) = g SL (2 , C ) ⊗ g (cid:63)SL (2 , C ) ⊗ ( g U ( N ) ⊗ g U ( N ) )= (17)= . (18)8n the last equality, the U ( N ) tensor decomposition is indicated, displaying only the leadingterm; this term coincides with the Young diagram in Fig. 1 and renders the polynomialharmonic, which we prove at the end of this section. This term is leading in the sense that itis the only U ( N ) representation in the decomposition that does not contain an overall factorof momentum, P , and thus the only primary operator/ harmonic mode in the decomposion.We now turn to proving this.The familiar diagrammatic ‘box placing’ rules for carrying out tensor decompositionswith Young diagrams (Littlewood-Richardson rules, again, see e.g. [16]) can be applied tothe product in eq. (17). The leading term appearing in eq. (18) is in fact the simplestrepresentation obtained using these rules—no white boxes have been shifted around, andthe Young tableaux have been simply stuck together.What of the other ‘ . . . ’ terms in eq. (18)? The box placing rules specify that we end upwith a Young diagram that has either one or two white boxes at the bottom of a blue boxcolumn of length N −
2. For the case of one white box under a column of N − j j ] (cid:15) j j k ..k N − (cid:104) i | + (anti-sym in k , . . . , k N − , i ) . By the antisymmetry, the indices k , . . . , k N − , i must be distinct choices of 1 . . . N (other-wise the anti-symmetrisation sets this factor to zero); without loss of generality, we considerthe choice 1 , . . . , N −
1. Each cyclicly related set of terms in the above anti-sym is propor-tional (by a sign) to N − (cid:88) k =1 [ N k ] (cid:104) k | = [ N | P − [ N N ] (cid:104) N | = [ N | P , (19)using P = (cid:80) Nk =1 | k ] (cid:104) k | and [ N N ] = 0. Eq. (19), as promised, contains a factor of totalmomentum, P , and thus the operator is a descendent.For the case of two white boxes under a column of N − j j ] (cid:15) j j k ..k N − (cid:104) i |(cid:104) i | + (anti-sym in k , .., k N − , i , i ) . (The spinors (cid:104) i | and (cid:104) i | could be contracted, (cid:104) i i (cid:105) ; the below arguments are valid in thiscase too.) The indices k , . . . , k N − , i , i are anti-symmeterised permutations of the set 1 ..N .Evidently, for any fixed value of i , one can factor out P as per eq. (19); in fact, one can9asily show that in summing over the other values of i , a factor of P can be pulled outoverall.This shows that the additional U ( N ) representations are descendents, because they havethe overall factor of P . We will return to a proof that the leading Young diagram eq. (18)is annihilated by K very shortly, showing that it is primary, after the introduction of semi-standard Young tableaux. B. States from semi-standard Young tableaux
For a given Young diagram, one can construct the states of the corresponding U ( N )representation using semi-standard Young tableau (SSYT), which we will see provides thelabelling of the little group scaling. We recall that a SSYT is a filling of the boxes of a Youngdiagram with the numbers 1 through N (repeated use of a number is allowed) subject tothe following rules: • The numbers along the rows must weakly increase ( i.e. reading from left to right eachsubsequent number must be greater than or equal to the previous one) • The numbers down the columns must strongly increase ( i.e. reading from top tobottom each subsequent number must be greater than the previous one)The number of valid SSYT is equal to the dimension of the U ( N ) representation. Forexample, for the eight-dimensional adjoint representation of U (3) we find eight SSYT fillings:1 12 1 13 1 22 1 23 1 32 1 33 2 23 2 33 . For a given SSYT of the Young diagram in Fig. 1, one easily constructs the basis polynomialin λ and (cid:101) λ using the diagram symmeterisation rules (sym on rows, anti-sym on columns).It is then straightforward to read off the field content by the little group scaling for eachparticle; equivalently these are the eigenvalues of the U (1) N ⊂ U ( N ) generators. Notethat the little group scaling of pairs of λ i , (cid:101) λ i cancel; for each such pair one should count aderivative to the field content of the harmonic/operator i.e. λ ia (cid:101) λ i ˙ a = p ia ˙ a is the momentumof the i th particle (a derivative acting on the field for the i th particle). While each term inthe polynomial must scale the same way under the little group overall, the pairs of λ i , (cid:101) λ i could appear (and do appear) for different particle numbers i in different terms.10e point out that the SSYT fillings will separately construct harmonics for all possiblespins of each external state. For example, harmonics corresponding to each of the operators F L F L φ , F L φ F L , and φ F L F L will be included separately. However, it is clear thatthese operators are of exactly the same form and can be related to each other with a simpleparticle index permutation. We emphasise we are dealing with all-distinguishable particles,and that such a permutation is between particle species; it is not the (anti)-symmeterisationnecessary when to describe indistinguishable particles. We can define a set of reduced SSYT which mods out such permutations between particle species with a simple ordering rule:order on SSYT filling: ≥ ≥ . . . ≥ N s . That this is true is proven in the appendix.As promised, we now return to the proof that all states of the representation shown inFig. 1 are annihilated by K = − (cid:80) ∂ (cid:101) ∂ . Consider the highest weight state, correspondingto the filling of all the boxes in the first row with 1s, all those in the second row with 2s, etc. . Such a state is trivially annihilated by K : it consists only of polynomials in thefour variables λ , λ , (cid:101) λ N − and (cid:101) λ N . The rest of the proof follows by group theory: since K is a U ( N ) singlet, its action commutes with the action of the U ( N ) raising and loweringoperators, and as such annihilates all the states in the representation.We conclude this section with a discussion on the orthogonality of the harmonics con-structed via the Young tableaux of Fig. 1, under the phase space measure of eq. (4). First,operators at different N are orthogonal due to the Fock space structure of the Hilbert space.Given the U ( N ) symmetry of phase space, it is also clear that U ( N ) representations withdifferent n, (cid:101) n are automatically orthogonal. What of the states within each representation?The integral over the little group for each individual particle ensures that states with differ-ent eigenvalues of the torus U (1) N are automatically orthogonal as well. In general, however,there exist degenerate subspaces where more than one operator has equal little group eigen-values (the SSYT are permutations of each other). In such cases, state orthogonality is notguaranteed; we postpone discussion of this point (and details of normalisation with respectto the phase space volume) to a future detailed, systematic study of the harmonics. For an explicit formulation of phase space in terms of Grassmannian variables, see [17]. II. EFT SPECTRA AT LOW MASS DIMENSION
It is instructive to work through the construction of harmonics/operators at low valuesof n and (cid:101) n i.e. at low mass dimension, ∆. In the following, we work through examples thatsuffice to construct an EFT basis up to mass dimension six.The formalism above provides a recipe to perform the construction:1. Write down the Young diagram corresponding to the choice of n and (cid:101) n , as shown inFig. 1.2. Write down all semi-standard Young tableau (SSYT) fillings to construct the U ( N )states. The operators we construct are summarised in Tables I, II, III.We will highlight the special features of this conformal basis as we come across them. Ofparticular importance are the structure of the harmonics when annihilation by K is non-trivial. Such a case happens when the corresponding operator involves derivatives, which isalso where IBP relations come into play; these operators are necessarily non-holomorphic.Another feature is the grouping of harmonics/operators with differing field content as statesof the same U ( N ) representation.Below we normalise the Young tableaux permutations with a factor 1 /k , k = (cid:89) i ∈ rows (cid:89) j ∈ columns p i ! q j ! , (20)where p i is the number of boxes in the i th row, and q j is the number of boxes in the j thcolumn of the tableaux. A. Harmonics of type ( n, (cid:101) n ) = (2 , , (0 , We begin with harmonics for which ( n, (cid:101) n ) = (2 , , N ≥
2. The relevant reduced SSYT aredisplayed in Tab. I. They correspond to operators of field content φ N − ψ L and φ N − ψ R ,respectively. We re-emphasise that we consider distinguishable particles at this point; the Or any other method of constructing the states, e.g. start with the the highest weight state and applylowering operators. ·· N − (2 ,
0) (0 , φ N − ψ L φ N − ψ R TABLE I. Reduced SSYT for Lorentz scalar operators of the form ( n, (cid:101) n ) = (2 , , (0 ,
2) for all N ≥ particle index is suppressed in the Table, but we indicate it explicitly in the following con-struction: φ . . . φ N ψ L ψ L : 12 = 12! ( λ a λ a − λ a λ a ) = (cid:104) (cid:105) . (21) φ . . . φ N − ψ R N − ψ R N :12 ·· N − = 1( N − (cid:101) λ j ˙ a (cid:101) λ ˙ aj ( (cid:15) j j ..N − + anti-sym. in 1 ..N − , (22a)= [ N − N ] , (22b)where in eq.(22a) summation over j and j is implied. Putting back in the flavour permuta-tions, there are N ( N − / i.e. L ↔ R in all fields, and are thusrelated by switching λ ↔ (cid:101) λ , or (cid:104) (cid:105) ↔ [ ]. B. Low ‘frequency’ harmonics for N = 3 Next, we fix the number of particles in the harmonic to be N = 3, and consider harmonicsof low n and (cid:101) n . The case N = 3 is special, as the construction given in Fig. 1 does not13
12 2 1 12 3 1 1 1 2(4 ,
0) (4 ,
0) (0 ,
4) (0 , φF L F L ψ L φF R F R ψ R ,
0) (6 ,
0) (6 , φ ξ L ξ L F L ψ L F L ,
6) (0 ,
6) (0 , φ ξ R ξ R F R ψ R F R TABLE II. Reduced SSYT for Lorentz scalar operators with N = 3, for low values of n , (cid:101) n . ξ denotes a spin 3 / produce a valid Young tableau when both n and (cid:101) n are non-zero. This reflects the fact that allof the Lorentz scalar harmonics/operators for N = 3 are holomorphic (or anti-holomorphic).In Table II we consider the cases ( n, (cid:101) n ) = (4 , , (0 , , (6 , , (0 , U ( N )representation; for example the harmonics φF L and F L ψ L both appear as states in the (4 , N = 3 operators in Tab. II are constructed as follows. φ F L F L :1 12 2 = 1(2!) ( λ a λ a λ b λ b + tab. perms ) , = 2 (2!) ( λ a λ a λ b λ b − λ a λ a λ b λ b − λ a λ a λ b λ b + λ a λ a λ b λ b ) , = (cid:104) (cid:105) . (23) F L ψ L ψ L :1 12 3 = 2 (2!) ( λ a λ a λ b λ b − λ a λ a λ b λ b − λ a λ a λ b λ b + λ a λ a λ b λ b ) , = (cid:104) (cid:105)(cid:104) (cid:105) . (24)14 ξ L ξ L : 1 1 12 2 2 = 1(3!) (2!) ( λ a λ a λ b λ b λ c λ c + tab. perms ) , = (cid:104) (cid:105) . (25) ξ L F L ψ L : 1 1 12 2 3 = 1(3!) (2!) ( λ a λ a λ b λ b λ c λ c + tab. perms ) , = (cid:104) (cid:105) (cid:104) (cid:105) . (26) F L F L F L : 1 1 22 3 3 = 1(3!) (2!) ( λ a λ a λ b λ b λ c λ c + tab. perms ) , = (cid:104) (cid:105)(cid:104) (cid:105)(cid:104) (cid:105) . (27)In the above we made use of the formula λ i a λ ja = − λ ia λ j a . The remaining right-handedholomorphic operators in Tab. II can be obtained (up to flavour permutations) by exchanging λ → (cid:101) λ i.e. (cid:104) ij (cid:105) → [ ij ]. However, we work out one case from the tableaux explicitly, forillustrative purposes, ψ R ψ R F R : 1 2 = 12! (cid:101) λ j ˙ a (cid:101) λ ˙ aj (cid:101) λ j ˙ b (cid:101) λ ˙ bj ( (cid:15) j j (cid:15) j j + (cid:15) j j (cid:15) j j ) , = [13][32] , (28)where in the above summation over j , j , j , j is implied. This is indeed the conjugate of(a flavour permutation of) eq. (24). C. Low ‘frequency’ harmonics for N = 4 As a last class of examples, we consider harmonics involving N = 4 fields; reduced SSYTfor ( n, (cid:101) n ) = (4 , , (2 , , (0 ,
4) are shown in Tab. (III). There are two new features evident inthe Table that were not present for the cases considered above. The first feature is that nownon-holomorphic harmonics appear (the middle column). We will comment on the detailedform of these operators below. The second feature is that there are distinct harmonics withthe same field content: two copies of the harmonic ψ L /ψ R appear in the (4 , / (0 , U ( N )15 ,
0) (2 ,
2) (0 , φ F L ψ L ψ R φ F R φF L ψ L φ ψ L ψ R D φF R ψ R ψ L φ D ψ R ψ L φ D ψ R TABLE III. Reduced SSYT for Lorentz scalar operators with with N = 4, for low values of n , (cid:101) n . representation, and two copies of φ D appear in the (2 ,
2) representation. These operatorsare independent, so it is important that they are both included; the rules for constructingthe reduced SSYT ensure this happens.The left-handed holomorphic ones are constructed as follows (the first two are identicalto the operators in eq. (23) and eq. (24), respectively, differing only by the addition of anextra φ field). φ φ F L F L : 1 12 2 = (cid:104) (cid:105) . (29) φ F L ψ L ψ L : 1 12 3 = (cid:104) (cid:105)(cid:104) (cid:105) . (30) ψ L ψ L ψ L ψ L : 1 23 4 = 1(2!) ( λ a λ a λ b λ b + tab. perms ) , = 12 ( (cid:104) (cid:105)(cid:104) (cid:105) + (cid:104) (cid:105)(cid:104) (cid:105) ) . (31) ψ L ψ L ψ L ψ L : 1 32 4 = 1(2!) ( λ a λ a λ b λ b + tab. perms ) , = 12 ( −(cid:104) (cid:105)(cid:104) (cid:105) + (cid:104) (cid:105)(cid:104) (cid:105) ) . (32)16he right-handed holomorphic harmonics in Tab. III are obtained via conjugation ofeqs. (29)-(32), and so we do not present their construction explicitly.Turning finally to the non-holomorphic harmonics, we have, ψ L ψ L ψ R ψ R :1 12 2 = 1(2!) (cid:101) λ j ˙ a (cid:101) λ ˙ aj ( (cid:15) j j λ a λ a + tab. perms ) , = 2 (2!) (cid:101) λ j ˙ a (cid:101) λ ˙ aj ( (cid:15) j j λ a λ a − (cid:15) j j λ a λ a − (cid:15) j j λ a λ a + (cid:15) j j λ a λ a ) , = (cid:104) (cid:105) [34] . (33) φ φ ψ L ψ R D :1 12 3 = 1(2!) (cid:101) λ j ˙ a (cid:101) λ ˙ aj ( (cid:15) j j λ a λ a + tab. perms ) , = 2(2!) (cid:101) λ j ˙ a (cid:101) λ ˙ aj ( (cid:15) j j λ a λ a + (cid:15) j j λ a λ a + tab. anti-syms ) , = 12 ( (cid:104) (cid:105) [34] − (cid:104) (cid:105) [24]) . (34) φ φ φ φ D : 1 23 4 = 1(2!) (cid:101) λ j ˙ a (cid:101) λ ˙ aj ( (cid:15) j j λ a λ a + tab. perms ) , = 1(2!) (cid:101) λ j ˙ a (cid:101) λ ˙ aj ( (cid:15) j j λ a λ a + (cid:15) j j λ a λ a + (cid:15) j j λ a λ a + (cid:15) j j λ a λ a + tab. anti-syms ) , = 14 ( (cid:104) (cid:105) [42] − (cid:104) (cid:105) [41] − (cid:104) (cid:105) [32] + (cid:104) (cid:105) [31]) . (35) φ φ φ φ D : 1 32 4 = 1(2!) (cid:101) λ j ˙ a (cid:101) λ ˙ aj ( (cid:15) j j λ a λ a + tab. perms ) , = 14 ( (cid:104) (cid:105) [43] − (cid:104) (cid:105) [41] − (cid:104) (cid:105) [32] + (cid:104) (cid:105) [21]) . (36)The last three of these have non-trivial annihilation by K . For example, the harmonic ineq. (34) with operator content φ ψ L ψ R D , N (cid:88) i =1 ∂∂ (cid:101) λ ˙ ai ∂∂λ i a (cid:18)
12 ( (cid:104) (cid:105) [34] − (cid:104) (cid:105) [24]) (cid:19) = 12 (cid:18) λ a (cid:101) λ ˙ a − λ a (cid:101) λ ˙ a (cid:19) = 0 . (37)17sing momentum conservation, one could rewrite the operator eq. (34) as another equallyvalid operator basis element, e.g. simply (cid:104) (cid:105) [34] or (cid:104) (cid:105) [24], but it is only the combination ∝ ( (cid:104) (cid:105) [34] − (cid:104) (cid:105) [24]) that that is a conformal primary and is annihilated by K as ineq. (37); it is in this sense that the harmonics form a privileged basis. IV. DISCUSSION
The general construction above applies to the distinguishable particles case. To takeinto account exchange symmetry one must (anti-)symmeterise over the identical (fermionic)bosonic fields in an operator. The particle index can also be interpreted as a gauge or othersymmetry index; further bookkeeping is required here too. The kinematic constructiondetailed here is a necessary first step (and the above considerations can be easily applied byhand, if not entirely systematically at present).To the EFTer, the systematic nature of the construction is clearly appealing. The auto-matic orthogonality of (the majority of operators) at different N and with different U (1) N eigenvalue of basis elements also has utility: converting from a UV Lagrangian/other EFTparameterisation is then simple, via a projection (cid:82) d Φ N Y ∗ L other . It will be useful to furtherstudy orthogonality in the degenerate eigenvalue case. It would also be interesting to explorehow this ‘mathematically singled out’ basis fares in phenomenological applications.There is deep structure in the operator basis which should be explored further. One of theinteresting features is the mixing of different particle species within the same harmonic ( e.g. the columns in Tab. III)—does this imply any relation between different phenomenologicalobservables? We note that these harmonic blocks are the same grouping as the classes in thenon-renormalisation theorems [18–20], and may shed further light on the structure of EFTanomalous dimension matrices/amplitude non-interference [21] results. Of further interest iswhether the harmonic picture presented here sheds further light or provides tools for studingpositivity-type constraints on Wilson coefficients [22–26]; it would also be interesting tounderstand the connection between this natural basis and natural bases for amplidutes e.g. partial waves. 18 CKNOWLEDGEMENTS
We thank Peter Cox, Marc Riembau, and Francesco Riva for conversations, and RodrigoAlonso and Peter Cox for comments on the draft. BH is funded by the Swiss NationalScience Foundation under grant no. PP002-170578. TM is supported by the World PremierInternational Research Center Initiative (WPI), MEXT, Japan, and by JSPS KAKENHIGrant Number JP18K13533.
Appendix A: Reduced tableaux
When operators are related by simple index permutations between particle species thatdo not change the form of the operator, e.g. F L F L φ , F L φ F L , and φ F L F L , we wish todefine a rule so as to only consider one of them. A canonical choice is to keep only operatorsin which the fields are helicity ordered, such that those of lower helicity are assigned lowerparticle indices. In the example above, this would be the operator F L F L φ . (Right handedfields have positive helicity, so if we replace all instances of L → R in the above example,the canonical choice would be φ F R F R .)More precisely, an operator is not of this canonical form if the following is true: thereexists a pair of fields in the operator that have particle index i and j with i < j , but havehelicities satisfying h i > h j . After removing such operators, we call the remaining set reducedoperators. We will show that the SSYT corresponding to a reduced operator satisfiesorder on SSYT filling: ≥ ≥ . . . ≥ N s . (A1)Before turning to the proof, note that if h i = h j there is no notion of a canonical orderon i or j in defining a reduced operator. That is, the set of reduced operators includesoperators related by non-trivial permutations of the indices between fields of equal helicity.For illustratation, we take two examples from the text. First, consider the SSYT for N = 3,1 12 3 : F L ψ L ψ L . (A2)The corresponding operator has a single field of helicity − − , andit is of reduced form. There are no non-trivial index permutations between the two fermions.(The Young tableaux corresponding to this permutation is not semi-standard—it would be19he filling ((1 , , (3 , F L ψ L ψ L and F L ψ L ψ L , which correspond to SSYT fillings((1 , , (2 , , , (3 , N = 4,1 23 4 and 1 32 4 , (A3)both of which are operators with field content ψ L ψ L ψ L ψ L and are related by non-trivialparticle index permutations between particles of equal helicity. Both are included in thereduced set of operators. We now turn to proving that the statement on SSYT in eq. (A1) follows for an operatorthat is of reduced form. First, consider the holomorphic case. Here, each field of helicity h i necessitates 2 | h i | copies of λ i in the operator, which in turn necessitates 2 | h i | copies of thebox i in the SSYT filling. Since for a reduced operator | h | ≥ | h | ≥ . . . ≥ | h N | , and allhelicities h i ≤ h i ≥
0. A field ofhelicity h i necessitates 2 h i copies of (cid:101) λ i in the operator. Each Lorentz contracted pair of (cid:101) λ i ˙ a (cid:101) λ ˙ aj necessitates a column in the SSYT of N − N , excluding i and j . Since h N ≥ h N − ≥ . . . ≥ h , the number N will be excluded in theSSYT more (or equal) times than the number N −
1, which in turn will be excluded more(or equal) times than N − etc. , and again the condition eq. (A1) follows.For the non-holomorphic case, first let us assume that no derivatives are present in theoperator. In this case, we split the particles into negatice helicity (to which we apply thesame reasoning in the holomorphic case) and into positive helicity (to which we apply thenon-holomorphic reasoning), and conclude again that the condition eq. (A1) holds.Finally we need to show that derivatives do not change the counting. A derivative impliesa pair (cid:101) λ i λ i (no sum on i ) in the operator. It is useful to consider the (cid:101) λ i as contributing This highlights an issue with how the particle index permutations are implemented across the setof reduced operatorss. To illustrate this, denote the two operators from the example (A3) as( ψ L ψ L ψ L ψ L ) A and ( ψ L ψ L ψ L ψ L ) B . Now consider two reduced operators that exist in the N = 7ring: ( ψ L ψ L ψ L ψ L ) A φ φ φ and ( ψ L ψ L ψ L ψ L ) B φ φ φ . These are related by index permuta-tions to the non-reduced operators ( ψ L ψ L ψ L ψ L ) A φ φ φ and ( ψ L ψ L ψ L ψ L ) B φ φ φ . It wouldbe incorrect to perform the permutation to the two operators differently, such that one could obtain e.g. ( ψ L ψ L ψ L ψ L ) A φ φ φ and ( ψ L ψ L ψ L ψ L ) B φ φ φ , which are in fact identical operators. − N , excluding i ; when it is contractedwith a (cid:101) λ j , a box j is further removed. The λ i in the pair contributes a (white) box i .Thus we see that the contribution of (cid:101) λ i λ i to the SSYT filling is to add a set of N boxesthat contains one copy each of the numbers 1 to N . As such, it does not affect the conditioneq. (A1). [1] W. Buchmuller and D. Wyler, “Effective Lagrangian Analysis of New Interactions andFlavor Conservation,” Nucl. Phys.
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