New Selection Rules from Angular Momentum Conservation
aa r X i v : . [ h e p - ph ] J a n New Selection Rules from Angular Momentum Conservation
Minyuan Jiang, Jing Shu,
1, 2, 3, 4, 5
Ming-Lei Xiao, and Yu-Hui Zheng
1, 2 CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,Chinese Academy of Sciences, Beijing 100190, China. School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, P. R. China. CAS Center for Excellence in Particle Physics, Beijing 100049, China Center for High Energy Physics, Peking University, Beijing 100871, China School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study,University of Chinese Academy of Sciences, Hangzhou 310024, China
We derive the generalized partial wave expansion for M → N scattering amplitude in terms ofspinor helicity variables. The basis amplitudes of the expansion with definite angular momentum j consist of the Poincare Clebsch-Gordan coefficients, while j constrains the UV physics that couldgenerate the corresponding operators at tree level. Moreover, we obtain a series of selection rules thatrestrict the anomalous dimension matrix of effective operators and the way how effective operatorscontribute to some 2 → N amplitudes at the loop level. INTRODUCTION
Symmetry plays a crucial role in understanding someelegant phenomena of our nature. From Noether’s the-orem [1], continuous symmetries are always associatedwith conservation laws. At the quantum level, where thesystem takes discrete values, some possible transitions ofthe system from one quantum state to another are for-bidden due to symmetry. These phenomena are knownas the selection rules, which have been derived for transi-tions in molecules, atoms, nuclei, or even the elementaryparticle decay, such as Landau-Yang theorem [2, 3].One crucial task for the high energy physics studies isthe precision tests of the Standard Model (SM), for whichthe SM Effective Field Theory (SMEFT) is a necessarytool. Recently, some interesting progress has been madeto apply the on-shell amplitude methods to SMEFT [4–8]. Moreover, various constraints have been observed onthe relations among operators or between operators andobservables [9–15], which indicates the existence of someselection rules in particle scattering processes due to con-servation laws.In [5], we have matched the effective operator ba-sis with on-shell amplitude basis, dubbed operator-amplitude basis correspondence, which helps us writedown all dimension 6 operators in the SMEFT withoutredundancy [16]. Moreover, we can easily associate theproperties of amplitudes to effective operators. For 2 → d jνν ′ , which is pre-cisely an amplitude basis as shown later in the paper. Bythe operator-amplitude basis correspondence, d jνν ′ inducean operator basis labeled by the angular momentum j ,which is a recombination of the Warsaw-like basis [17].This basis has the privilege that by angular momentumconservation, several selection rules, which restrict theway an operator could contribute to a process, becomeapparent.In this letter, we first define the partial wave basis of multi-particle states and develop the formalism to usethe spinor helicity variables to construct the partial waveamplitude basis not only for 2 → M → N processes. As a byproduct, one can seewhether specific processes or operators can be gener-ated at the tree level. Moreover, with this technique,we can obtain two kinds of selection rules for effectiveoperators. Firstly, we find non-trivial constraints on theanomalous dimension matrix that are beyond the non-renormalization relations found in [13]. Then we provesome entirely vanishing contributions from certain oper-ators to specific amplitudes at one loop, part of which isalso presented in [15]. GENERALIZED PARTIAL WAVESIntroduction to Spinor-Helicity Variables
The spinor-helicity variables (or helicity spinors), bothfor massless and massive particles, are defined as [18] p µ σ µα ˙ α = p α ˙ α = λ Iα ˜ λ ˙ αI , (1)where I taking r = rank( p α ˙ α ) values is called little groupindex whose contraction reveals the little group invari-ance of p . The hermiticity of p α ˙ α requires that λ ∗ α = ± ˜ λ ˙ α .For massless particles, r = 1 and I can be omitted, but λ and ˜ λ are required to have opposite charges underlittle group U (1), i.e. helicities. For massive particles, r = 2 and I is the index for (anti-)fundamental represen-tation of the little group SU (2). It is convenient to setdet λ Iα = det ˜ λ ˙ αI = m to satisfy the on-shell condition p = det( p α ˙ α ) = m .The helicity spinors constitute the solutions to equa-tion of motion for particles of all spins. Therefore allPoincar´e information – momentum p , spin and its pro-jection ( s, σ ), helicity h – can be encoded by helicityspinors. Amplitudes can be directly constructed by he-licity spinors and should satisfy two conditions: 1. allspinor indices are contracted; 2. little group repre-sentations of all external particles should be respected.Hence an all-massless amplitude with helicities h i (in all-outgoing convention) must take the form A h ,...,h N ∼ K ( s I ) Y i λ r i i ˜ λ ¯ r i i , ¯ r i − r i = 2 h i , (2)where spinor contractions are omitted. s I = ( P i ∈I p i ) are Mandelstam variables, and the function K containsall the analytic information like poles and branch cuts.The unfactorizable part[19] is defined as basis amplitudes B in [5]. When there is a massive particle with spin s , theamplitude should have 2 s totally symmetric little groupindices coming from its spinor variables, A { I ...I s } = λ I α · · · λ I s α s A { α ...α s } (3)where A { α ...α s } also takes the form of eq. (2) accordingto the helicities of the other massless particles, but with2 s totally symmetric uncontracted spinor indices. Partial Wave Basis of Multi-Particle States
A general multi-particle state is usually written in thetensor representation of Poincar´e group for a scatteringprocess, which we may call the tensor basis . Now thatwe can trade the Poincar´e information p i , s i , σ i for thespinor variables λ Ii , ˜ λ Ji , we express the tensor basis as | Ψ N i ⊗ = N Ni =1 | λ Ii , ˜ λ Ji , n i i . In this section we are inter-ested in getting the Poincar´e information for the wholemulti-particle state – the total momentum P , total an-gular momentum j and its projection σ , hence we needto decompose the tensor basis into an irreducible repre-sentation of Poincar´e group, which we call partial wavebasis : | Ψ N i j = | P, j, σ, a, { n }i = | χ I , ˜ χ J , a, { n }i , (4)where { P, j, σ } are replaceable by a pair of auxiliary helic-ity spinors defined by P = χ I ˜ χ I , a is a label for possibledegeneracy, and { n } is the collection of particle speciesinformation.Note that in quantum mechanics, we learned aboutClebsch-Gordan (CG) coefficients for angular momen-tum addition, which does not involve the partial wavefunction, as they are only CG coefficients of the spa-tial rotation group SU (2). However, here we are talkingabout the tensor decomposition of Poincar´e representa-tions. Hence the conversion of basis is described by CGcoefficients of the Poincar´e group instead. In the follow-ing whenever “CG coefficient” is mentioned, it means the Poincar´e CG coefficient . In this letter, we derive the gen-eralized N -particle Poincar´e CG coefficients in terms ofhelicity spinors as the following overlap function ⊗ h Ψ N | Ψ N i j = h{ p i , s i , σ i } N | P, j, σ, a i≡ C P,j,σ,a { p i ,s i ,σ i } N δ ( P − X p i ) . (5) We can convert it to a function of helicity spinors byadopting the helicity spinor representation for both statevectors C P,j,σ,a { p i ,s i ,σ i } N ≡ ¯ C j,a { s i } N ( { λ i , ˜ λ i } N , χ, ˜ χ ) . (6)The total angular momentum j is reflected by the re-quirement that ¯ C j,a should include 2 j factors of χ or ˜ χ with symmetric little group indices, whose 2 j + 1 com-ponents give the value of the σ label on the left handside. ¯ C j,a has the same form of a basis amplitude withan auxiliary particle with spin j , which can be expressedsimilar to eq. (3) as¯ C j,a { s i } N = f j,a ; { α ,...,α j }{ s i } N χ I α · · · χ I j α j , (7)where f is the multi-particle wave function in the all- χ basis. The wave functions in other basis involving ˜ χ areequivalent to it via the Dirac equation for ( χ, ˜ χ ) with themass replaced by √ P .While in general f can have extra SU (2) indices formassive external particles, we will only be focusing onthe multi-massless-particle states that are relevant formassless EFTs, where s i are replaced by helicities h i .Let us take the simplest example – a two massless particlestate, with helicity h , h . The CG coefficient in spinorrepresentation ¯ C j,a is like an amplitude for two masslessparticles and one massive spin- j particle, whose generalform is shown in [18] as¯ C jh ,h ∼ [12] j + h + h s (3 j + h + h ) / ( h χ i j − h + h h χ i j + h − h ) { I ...I j } , (8)where we adopt the usual notation that the ǫ contractionof λ i λ j is denoted by h ij i and that of ˜ λ i ˜ λ j is denoted by[ ij ]. Here the normalization by a power of s = P keepsit dimensionless. The label for degeneracy a is omittedhere since we have a unique solution.Due to angular momentum conservation, the S-matrixis block diagonal in the partial wave basis. Therefore, wecan partial wave expand a general scattering amplitudeas[20] A ( { p i , σ i , n i } N ; { p ′ i , σ ′ i , n ′ i } M ) ≡ ⊗ h Ψ M |M| Ψ N i ⊗ = X j,a,b j h Ψ M |M jab | Ψ N i j X σ C P,j,σ,b { p ′ i ,s ′ i ,σ ′ i } M ( C P,j,σ,a { p i ,s i ,σ i } N ) ∗ ≡ X j,a,b M jab ( s ) B j,a → b { s i } N →{ s ′ i } M . (9)Since the CG coefficient part P σ CC ∗ only involves thePoincar´e information of the external particles, it is com-pletely determined by symmetry and serves as a basisfor a generic amplitude, while the coefficient matrics M j carry the information of the dynamics. We shall call theCG coefficient part with appropriate normalization the partial wave amplitude basis B j , which is nothing but aspecial choice of amplitude basis that have definite an-gular momenta. The sum over σ could be translated tothe contraction of little group indices in helicity spinorrepresentation B j,a → b { s i } N →{ s ′ i } M = X σ C P,j,σ,b { p ′ i ,s ′ i ,σ ′ i } M ( C P,j,σ,a { p i ,s i ,σ i } N ) ∗ = ( ¯ C j,b { s ′ i } M ) I ,...I j ( ¯ C j,a { s i } N ) ∗ I ,...I j . (10)If we write them in the all- | χ i basis, we can further sim-plify it using the identity χ Iα χ Jβ ǫ IJ = −√ sǫ αβ and get B j,a → b { s i } N →{ s ′ i } M = ( −√ s ) j f j,b { s i } N ( f j,a { s ′ i } M ) ∗ . (11)Taking complex conjugation of wavefunctions is equiv-alent to flipping all the helicities, for instance ( f jh ,h ) ∗ =( − j + h − h f j − h , − h . Thus we can rederive the Wignerd-matrix as the partial wave amplitude basis of 2 → B j { h ,h }→{ h ,h } ∼ ( − j − ∆ s j + h/ [12] j − h − h [34] j + h + h × X i w i h i i h i ∆ − ∆ ′ + i h i j − ∆ − i h i j +∆ ′ − i ,w i = (2 j )!( j + ∆)!( j − ∆)!( j + ∆ ′ )!( j − ∆ ′ )!( j + ∆ − i )!(∆ ′ − ∆ + i )! i !( j − ∆ ′ − i )! . (12)where we defined h = − h − h + h + h as the totalhelicities in all-outgoing convention, ∆ = h − h , ∆ ′ = h − h . To understand what this result means, we go tothe center of mass frame where we have [12] = − [34] = √ s , −h i = h i = √ s sin θ , h i = h i = √ s cos θ , θ being the scattering angle. With these substitutions, werecover the Wigner d-matrix B j { h ,h }→{ h ,h } ∼ X i ( − − ∆+ h + h − i w i × (cid:20) cos θ (cid:21) j − ∆+∆ ′ − i (cid:20) sin θ (cid:21) ∆ − ∆ ′ +2 i ∼ d j ∆ , ∆ ′ ( θ ) , (13)where omitted overall factors could be absorbed into thenormalizations of the CG coefficients. Single-Particle Resonance and Bridge-CountingMethod
One immediate application of partial wave expansionis to determine the spin of resonance in a particular chan-nel. A basis amplitude generated by an effective operatorcomes from integrating out massive degrees of freedom athigh energies. Such features are captured by poles andbranch cuts in the coefficient function M jab ( s ) defined in eq. (9), while poles indicate single-particle resonancesand branch cuts indicate multi-particle resonances. Di-agrammatically, they represent the two ways to obtainan effective operator – tree-level generation and loop-level generation, the latter suppressed by loop factors.Therefore it is phenomenologically interesting to classifyoperators by the way they are generated.If we can determine the angular momentum of a basisamplitude, we immediately fix the spin of possible single-particle resonance, which characterizes the UV physics oftree-level generation. For instance, if j ≥ B = h ih i , which is generatedby an operator F µν ψσ µν ψφ , has j = 1 in { , } → { , } channel which demands the couplings V F φ and
V ψψ where V is the vector resonance. The other topologyof the tree diagram requires a j = 1 / { , } → { , } channel, which demands the couplings F ψ
Ψ. Due to the non-existence of these couplings inrenormalizable UV theories [15, 21, 22], we can excludethe tree level generation of B and the corresponding op-erator. To classify operators more efficiently, instead ofconstructing the partial wave amplitude basis as in thelast section, we wish to compute the angular momentumof a given basis amplitude. It seems easy by observ-ing the fact that the form of eq. (11) exhibits 2 j spinorcontractions between the initial state wave function andfinal state wave function, both consisting of spinor helic-ity variables. We may call the spinor brackets contract-ing spinors from both sides of a scattering channel the bridges , and the number of them in a basis amplitudeindicates the total angular momentum j = 12 . (14)This bridge-counting method is simple and efficient foramplitudes that involve few particles and momenta.However, two preconditions for the counting should bekept in mind: 1. the two wave functions in eq. (11) shouldbe both in all- χ basis, hence the bridges should be all h·i (other forms can always be converted to this form); 2.the bridges should be symmetrized over particles on bothsides of the channel. As a quick example, in the { , } →{ , } channel, amplitude B + = h ih i + h ih i has2 bridges and thus j = 1, but B − = h ih i − h ih i shouldn’t be counted as 2 bridges because it’s not sym-metrized over the bridge contractions. Actually, the lat-ter equals h ih i by Schouten identity and has 0 bridgesand j = 0. Poincar´e Algebra in Helicity Spinor Representation
While the bridge-counting method is sometimes con-venient, it often becomes cumbersome when it comes tomore complex amplitudes, which one needs to convert tothe form that satisfies the two preconditions above. Inthis section, we propose an alternative method to obtainangular momentum and more.In quantum mechanics, we use the non-relativistic J operator to obtain angular momentum of, say, a wavefunction. But in relativistic scenario, we need to usethe Pauli-Lubanski operator W µ = ǫ µνρσ P ν M ρσ , whichinduces a Casimir invariant W for Poincar´e group witheigenvalue − P j ( j +1), where j is the covariant version oftotal angular momentum. While this operator is usuallyapplied to single-particle states to classify free particles,it can also be applied to multi-particle states.Here we propose to use the helicity spinor representa-tion so that the operator can act on the form factors f in eq. (7) or basis amplitudes B . The conformal algebrain the helicity spinor representation was already givenin [23], where[24] M I ,αβ = i X i ∈I (cid:0) λ iα ∂∂λ βi + λ iβ ∂∂λ αi (cid:1) , ˜ M I , ˙ α ˙ β = i X i ∈I (cid:0) ˜ λ i ˙ α ∂∂ ˜ λ ˙ βi + ˜ λ i ˙ β ∂∂ ˜ λ ˙ αi (cid:1) . (15) M and ˜ M are induced from the Lorentz generator as M µν σ µα ˙ α σ νβ ˙ β = ǫ αβ ˜ M ˙ α ˙ β + M αβ ¯ ǫ ˙ α ˙ β . The sum is takenover particles in a group I for which we want to computeangular momentum, hence for an amplitude we sum overonly the initial or only the final state particles. It definesa scattering channel I → ¯ I that the angular momentumis associated with (¯ I is the complement of I ).From eq. (15), we note the following properties (similarfor ˜ M ) M I h ij i = 0 if i, j ∈ I or i, j ∈ ¯ I ,M I h ij i = i ( | i ih j | + | j ih i | ) , if i ∈ I , j ∈ ¯ I , (16)which inspiringly show that only bridges contribute! Us-ing eq. (15) and P I = P i ∈I λ i ˜ λ i , the Casimir invarianttakes the following form W I ( B ) = P I (cid:16) Tr ˜ M I ( B ) + Tr M I ( B ) (cid:17) −
14 Tr ( P T I M I ( B ) P I ˜ M I ( B )) (17)where M I ,αβ ( B ) ≡ M γ I ,α M I ,γβ B . It is tempting to showthe conservation of angular momentum defined by thisoperator. Recall the properties of eq. (16), we can easilyprove M I ( B ) = − M ¯ I ( B ) , Tr M I ( B ) = Tr M I ( B ) . (18) Together with P I = − P ¯ I , we find W I B = W I B , whichmeans that for any channel I → ¯ I of an amplitude B ,the angular momentum defined by the operator W I isthe same for both initial and final states.We also show that the operator W I has the correcteigenvalues. Let’s take a simplest example B = h i ,an amplitude generated by a dimension 5 operator O = ψ φ ψ φ . In the channel { , } → { , } where the an-gular distribution is d / , − , we can read out the angularmomentum j = 1 / λ s, we have ˜ M h i = 0 thus only thesecond term in eq. (17) survives( M { , } ) βα h i = iM γ { , } ,α (cid:0) | i γ h | β + | i γ h | β (cid:1) = 4 | i α h | β − h i δ βα , (19)hence W { , } h i = ( p + p ) h i − h i ] = − s h i . (20)The eigenvalue − P I j ( j +1) = − s confirms that j = 1 / W representation matrix in the spaceof basis amplitudes with the same helicity states and thesame dimension. The procedure is described in more de-tail in the appendix. SELECTION RULES
In [4, 5] it is proposed that operators subject to equa-tion of motion (EOM) and integration by part (IBP)should one-to-one correspond to the unfactorizable am-plitude it generates, dubbed “operator-amplitude basiscorrespondence”. With appropriate choice of amplitudebasis, we could easily translate selection rules for ampli-tudes to those for operators. In particular, the operatorcorresponding to the partial wave basis amplitude B j,a → b should only have matrix elements between the partialwave basis states | j, σ, a i and | j, σ ′ , b i , proportional to δ σσ ′ . It means that the operator, which we may denoteas O j,ab , would annihilate any states other than the twopartial wave basis up to crossing symmetry. Therefore,in the channel that B j,a → b is defined, the operator actingon the tensor basis state picks out only the particularpartial wave states ( j, a ) or ( j, b ). Unlike the degeneracy a, b , the angular momentum j is conserved throughoutthe whole physical scattering. With this property, wepropose the selection rules in two types of calculations,operator renormalizations and loop amplitudes, from an-gular momentum conservation. O n { n i } UV divergence O m { n i } FIG. 1. Renormalization of an effective operator O m by O n at one loop when A ′ = 0. { n i } is a collection of external legsshared by both O n and O m . Renormalization
From the on-shell perspective, an effective operator O m is renormalized by another operator O n when the loopamplitude with O n insertion contains UV divergence pro-portional to the basis amplitude corresponding to O m ,characterized by the coefficient γ mn as follows16 π A = − ( X m,n γ mn C n B m + A ′ ) 1 ǫ , (21)where A ′ is a factorizable term that may show up incertain diagrams.Suppose that there are at least two external legs ina diagram for A − loop that are shared by both O n and O m , as shown in fig. 1, the angular momenta j as wellas the degenerate label a should both be the same in thetwo basis amplitudes B n and B m . If not, this diagramshould have vanishing contribution to γ mn , up to sub-tleties caused by A ′ . We will have an example to explainthe subtlety later, but let us focus on the selection ruleassuming A ′ = 0.This non-renormalization phenomenon is independentof the criteria proposed in [13], in which operators areclassified by the (anti-)holomorphic weights ( w i , ¯ w i ) andare renormalized only by the operators of lower weights.Notably, our selection rule further picks out zero entriesin the block of the anomalous dimension matrix allowedby [13]. We also find cases where a whole type of opera-tors can not be renormalized by another type with lower(anti-)holomorphic weights. These will be more clear inthe following discussion, where we apply our selectionrule to cases at dimension 6 and 8. d = 6 To understand the patterns of renormalizationat dimension 6 from angular momentum, we list all typesof dimension 6 operators[25] (except for F and suppress-ing possible flavor and Lorentz structures) in table I. The“channels” means the group of two particles shared by O m and O n for which we find the angular momentum j using the technique developed previously. We list alldimension 6 operators that could generate the two parti-cles in the row and place them in the column accordingto j . With this arrangement, a diagram specified by theshared particles exists for the renormalization betweenany two operators in a row, but only operators appear- Channels j = 0 j = 1 / j = 1 F + F + F φ (2 , F + ψ + F ψ φ (2 , F + φ F ψ φ (2 , F φ (2 , ψ + ψ + ψ (2 , ψ ¯ ψ (4 , ψ φ (4 , ψ (2 , F ψ φ (2 , ψ + ψ − ψ ¯ ψφ D (4 , ψ + φ ψ φ (4 , F ψ φ (2 , ψ ¯ ψφ D (4 , φφ φ D (4 , ψ φ (4 , φ (6 , ψ ¯ ψφ D (4 , φ D (4 , w, ¯ w ), so that one canfurther obtain non-renormalization relations for operators inthe same entry by [13]. ing in the same entry could renormalize each other dueto the selection rule.This table is special as a classification of operatorssince they can appear multiple times in it. First, fora type of operators, there may be terms with differentLorentz structures that have different angular momen-tum in the same channel. Hence the type may appearin multiple columnes in a row, like ψ in Table I, whose j = 0 and j = 1 basis could renormalize ψ φ and F ψ φ respectively but not interchangeable. In SMEFT, it as-serts that among O lequ and O lequ (the definitions aregiven in appendix ), O eH can only be renormalized bythe former while O eW only by the latter [10, 26, 27].Second, even for a single operator, it appears in severalrows for the different channels we can examine. There-fore, two operators not appearing in the same entry atone row may be in the same entry at another row, whichis why we say the selection rule is at the diagram level.Our j selection rule sometimes mixes with the selectionrule of gauge charges, like isospin I of SU (2) L . Oneexample in SMEFT is the H D with both j = 0 , I = 0 , { H † , H } channel. We denote the couplingsin the amplitude A ( H , H † , H , H † ) with definite ( j, I )quantum numbers in { , } → { , } channel as C j,I .Due to spin statistics, there are only two independentoperators, namely O HD and O H (cid:3) in Warsaw basis [17].In terms their Wilson coefficients, we derive C , = 3 C H (cid:3) , C , = C HD − C H (cid:3) ,C , = − C H (cid:3) − C HD , C , = − C H (cid:3) . (22)From Table I, the ψ ¯ ψ φ D type operators are onlyrenormalized at j = 1, while in SMEFT we have O Hl , O He , O Hq , O Hu , O Hd of this type with I = 0 thatare only renormalized by the combination C , , and also O Hl , O Hq of this type with I = 1 that are only renormal-ized by the combination C , . These are verified by theresult in [10, 26, 27]. d = 8 The above analysis can be straightforwardlygeneralized to the renormalization at higher dimensions.At dimension 8, we have many more types of opera-tors that have multiple Lorentz structures; hence oneshould diagonalize the W operator in the space of ba-sis amplitudes generated by them to get partial waveamplitude basis B j,a . In case there are degeneracieswhen we look at a channel with more than two par-ticles, we can further find eigenfunctions for subsets ofthe channel. In appendix , we present the partial waveamplitude basis in the channel { , , } and { , } forthe operator types ψ φ φ ψ φ D and ψ φ φ ¯ ψ F D ,labelled by superscript ( j , j ). In this basis, only( B / , / ψ φ D , B / , / F ψ ¯ ψφ D ) and ( B / , / ψ φ D , B / , / F ψ ¯ ψφ D ) can mixwith each other through RG running.Moreover, we find new non-renormalization relationsfor whole types of operators that were not predictedby [13]. For example, the F ¯ F ψ ¯ ψD operators have j = 2in the ( ψ, ¯ ψ ) channel and ψ ¯ ψφ D have j = 1. Althoughallowed by the ( ω, ¯ ω ) criteria, all F ¯ F ψ ¯ ψD type operatorscan not renormalize the ψ ¯ ψφ D type. A ′ subtlety: Now we briefly look at an examplethat has subtlety related to A ′ in eq. (21). Table I showsthat the operator ψ ¯ ψ φ D and φ could not renormalizeeach other. However, in the SMEFT, as shown in [27],˙ C H ⊃ λg C Hl has non-vanishing coefficient. The secrethides in the operator basis. Our selection rule only provesthat 1PI diagrams with no A ′ vanishes, thus the non-vanishing diagram for A − loop ( H ) must be non-1PI asshown in fig. 2. From angular momentum and isospinof O Hl , we assert that the subamplitude at the LHSof the P propagator should be proportional to the ba-sis amplitude B j =1 ,I =1 H D which renormalizes the operator( H † iτ i ←→ D µ H ) . Because it is not Warsaw basis opera-tor, we need to decompose B j =1 ,I =1 H D into those generatedby Warsaw basis O H (cid:3) and O HD , while we keep in mindthat one of the external legs is not on shell P = 0. As a O (3) Hl − λP UV divergence H D − λP FIG. 2. Renormalization of O H by O (3) Hl at one loop, with A ′ = 0. result, we have A − loopUV ( H ) ∼ ( g C Hl B j =1 ,I =1 H D ) 1 ǫ × P × ( − λ )= − λP (cid:18) g C Hl B H (cid:3) − g C Hl P (cid:19) ǫ = (cid:18) λg C Hl − λg C Hl B H (cid:3) P (cid:19) ǫ . (23)Compared to eq. (21), we see that when A ′ = 0, it isambiguous to define the “local” part of the UV divergencethat is supposed to be the anomalous dimension γ . Fromangular momentum point of view, B j =1 ,I =1 H D is preferredin the residue of the P pole, so that γ = 0 as shown inthe second line; but from operator basis point of view, B H (cid:3) is the basis contribution to the residue, hence weare left with a local piece for ˙ C H proportional to λg asshown in the third line. This computation also gives thecorrect proportionality γ O Hl →O H γ O Hl →O H (cid:3) = 2 λg g = 83 λ, (24)without any actual loop calculations, which agrees withthe result in [27]. Given the powerful predictivity, a moresystematic way to examine the A ′ = 0 situation needs tobe explored. Vanishing Loops
In the previous section, we are considering renormal-ization, so we have a “target operator” to be renormal-ized, which has definite j at the specific channel. We canalso consider the full amplitude, which in general doesnot have a definite j . However, some of them have con-strained j at specific channels, which select the operatorsthat contribute to these amplitudes at the loop level.We consider two ways that j can be constrained fortwo-particle states: • In the Center of Mass (COM) frame we can as-sure that the orbital angular momentum r × p hasvanishing projection along ˆ p , hence along this di-rection we have σ ≡ J · ˆ p = S · ˆ p , which for masslessparticles is ∆ h , the difference of helicities. Thuswe must have j ≥ | ∆ h | . After covariantizing, j d = 4FIG. 3. One loop diagram for 2 → N scattering in EFT. determined by the eigenvalue of W satisfies theconstraint in any frame. This is a generalization ofthe Weinberg-Witten theorem as noted in [18]. • From eq. (8), we find that if h = h = h thepermutation symmetry of the two particles in theCG coefficient is determined by the exponent in[12] j +2 h . Thus by spin-statistics, once the two par-ticles are identical, the permutation symmetry isallowed only if j is even. It does not apply if thereare other group factors that contribute to the per-mutation symmetry, which is discussed in detail in[28], but for simplicity, we only discuss the exam-ple when no group factors exist (also see [18]), likefor photons, so that all odd j s are forbidden. TheLandau-Yang theorem [2, 3], which states that thetwo-photon state with j = 1 is forbidden, is noth-ing but a combination of the above two criteria, thefirst forbidding the opposite helicity case and thesecond forbidding the same helicity state.Both of the constraints may be used to prove a diagram-level selection rule for the 2 → N scattering with aneffective operator contributing in the way as shown infig. 3. We should emphasize that the selection rule isonly at diagram-level; it is possible for an amplitude tohave both vanishing diagrams by the selection rule andnon-vanishing ones. An example is the contribution from F φ to A ( F + F − φφ ) as shown in fig. 4. Selection Rule A:
If the two external legs on theLHS of this diagram have helicities differ by ∆ h , it thenselects the j ≥ ∆ h partial waves also for the RHS state,thus forbidding the contribution from an operator withlower j in the specific channel. Such a selection rule maybe non-trivial when ∆ h ≥ d = 6, it can be verified that no effective operatorscan excite two-particle states with j >
1. So all the dia-grams like fig. 3 with ( F + , F − ) or ( F ± , ψ ∓ ) at the LHSand dimension 6 operators inserted at the RHS must van-ish. At higher dimensions, non-vanishing contributionswill show up, mostly because adding more derivatives toan operator makes it possible to generate higher angularmomentum partial waves.In Table II, we list all the 2 → F φ d = 4+ − F φ d = 4 + − FIG. 4. Contribution from F φ to A ( F + F − φφ ). The leftdiagram vanishes because F φ excites j = 0 partial wave forthe two external scalars. However, due to the existence of theright diagram, which does not vanish, the full amplitude isnon-zero.LHS(∆ h ) fields inloop RHS EFT operators at RHS F + F − (2) φφ φφ φ (0), φ D (0, 1), φ D (0, 1, 2) ψ + ψ − ψ ¯ ψφ D (1), ψ ¯ ψφ D (1, 2) F + F + F φ (0), F φ D (0, 1) ψ ¯ ψ φφ ψ ¯ ψφ D (1), ψ ¯ ψφ D (1,2) ψ + ψ − ¯ ψ ψ (1), ¯ ψ ψ D (1, 2) F + F + F ψ ¯ ψD (1) F + ψ − (3/2) ¯ ψφ ψ − φ ψ ¯ ψφ D (1/2), ψ ¯ ψφ D (1/2, 3/2) F + ψ + F ψ φ (1/2), F ψ φD (1/2, 3/2) F + φ (1) ψψ ψ ± ψ ± ¯ ψ ψ (0), ψ (0,1),¯ ψ ψ D (0,1), ψ D (0,1, 2) ψ + ψ − (1) φφ F ± F ± F φ (0), F φ D (0, 1) F F φφ F φ (0), F φ D (0, 1) F ± F ± F ¯ F (0) , F (0, 1, 2)TABLE II. Vanishing one loop amplitudes from contributionof specific operators. In the first column we list the two par-ticle states with a minimum angular momentum j ≥ ∆ h , andare at LHS of the diagram in fig. 3. In the third column we listthe two particle state excited by effective operators as RHSof diagram in fig. 3. The combinations of these two columngives the 2 → h . ψ ¯ ψφ Dd = 4 + − ++ ψ ¯ ψ d = 4 + − ++FIG. 5. Vanishing loops from j = 1 for two identical particlesof same helicity. [15], while the underlying mechanism–angular momen-tum conservation–is more manifest here. For dimension8 cases, which are not studied in [15], we also find twovanishing contributions, namely the contribution from F φ D to A ( φφF + F − ) and F ¯ F to A ( ψ + ψ − F ± F ± ).The other dimension 8 operators in this table can excitepartial waves that reach the “threshold” of j = ∆ h andthus have non-vanishing contributions to the correspond-ing processes. For these cases, the amplitude selects spe-cific combinations of the operators that may contribute,which is a strong constraint for EFT phenomenology. Forexample, the H D type operators (defined in the ap-pendix) contribute to the amplitude A ( B + B − H α H † ˙ β )at one loop level only in the form of the j = 2 partialwave amplitude basis, with coefficients proportional tothe combinations C , = 16 ( C H D + 13 C H D + C H D ) ,C , = 16 ( C H D − C H D + C H D ) , (25)while the isospin further selects the first one C , . Selection Rule B:
If the LHS state in fig. 3 con-sists of two identical particles (same gauge charges andsame helicities), the j = 1 partial wave is forbidden onthe RHS, selecting the operator that contributes in thisspecific way. As an example, consider the one-loop am-plitude A ( ψ + ψ − F + F + ) from ψ ¯ ψφ D and ψ ¯ ψ , as infig. 5, with two gauge bosons being identical, like B inthe SM. Because the other two-fermion state created bythese two effective operators have exactly j = 1, thisamplitude must vanish. CONCLUSION
In this letter, we have derived the CG coefficients ofPoincar´e group in terms of spinor helicity variables. Weshowed that they are the normalized amplitude basis be-tween the multi-particle state and an auxiliary massiveparticle state, the spin of which gives the total angu-lar momentum. Then we obtain the partial wave am-plitude basis B j with definite angular momentum, wherethe 2 → W in thespinor helicity representation.More importantly, these techniques allow us to findnew selection rules based on angular momentum conser-vation. By using the operator-amplitude basis correspon-dence, we assign an angular momentum to the operatorcorresponding to a partial wave amplitude basis in a cer-tain channel. When inserted into one-loop diagrams, ei-ther in calculating the renormalization of effective oper-ators or in the loop calculation of full amplitudes, onlythe operators with correct angular momentum would beselected. First, we show how such selection rules areapplied to predict new zeros and non-trivial proportion-alities in the anomalous dimension matrix of effectiveoperators. Second, we prove two kinds of constraintson the total angular momentum for two-massless-particlestates as the analog of the famous Weinberg-Witten andLandau-Yang theorems, both of which prevent the oper-ators with wrong angular momentum from contributingto the 2 → N amplitudes by the conservation law. ACKNOWLEDGEMENTS
J.S. is supported by the National Natural Sci-ence Foundation of China (NSFC) under grantNo.11947302, No.11690022, No.11851302, No.11675243and No.11761141011 and also supported by theStrategic Priority Research Program of the ChineseAcademy of Sciences under grant No.XDB21010200 andNo.XDB23000000. M.L.X. is supported by the Na-tional Natural Science Foundation of China (NSFC) un-der grant No.2019M650856 and the 2019 InternationalPostdoctoral Exchange Fellowship Program. [1] E. Noether, Gott. Nachr. , 235 (1918),[Transp. Theory Statist. Phys.1,186(1971)],arXiv:physics/0503066 [physics].[2] C. N. Yang, Phys. Rev. , 242 (1950).[3] L. D. Landau, Dokl. Akad. Nauk SSSR. , 207 (1948).[4] Y. Shadmi and Y. Weiss, JHEP , 165 (2019),arXiv:1809.09644 [hep-ph].[5] T. Ma, J. Shu, and M.-L. Xiao, (2019),arXiv:1902.06752 [hep-ph].[6] B. Henning and T. Melia,Phys. Rev. D100 , 016015 (2019),arXiv:1902.06754 [hep-ph].[7] G. Durieux and C. S. Machado, (2019),arXiv:1912.08827 [hep-ph].[8] A. Falkowski, (2019), arXiv:1912.07865 [hep-ph].[9] J. Elias-Miro, J. R. Espinosa, andA. Pomarol, Phys. Lett.
B747 , 272 (2015),arXiv:1412.7151 [hep-ph]. [10] E. E. Jenkins, A. V. Manohar, and M. Trott,JHEP , 087 (2013), arXiv:1308.2627 [hep-ph].[11] R. Alonso, E. E. Jenkins, and A. V. Manohar,Phys. Lett. B739 , 95 (2014), arXiv:1409.0868 [hep-ph].[12] Z. Bern, E. Sawyer, and J. Parra-Martinez, (2019),arXiv:1910.05831 [hep-ph].[13] C. Cheung and C.-H. Shen,Phys. Rev. Lett. , 071601 (2015),arXiv:1505.01844 [hep-ph].[14] A. Azatov, R. Contino, C. S. Machado,and F. Riva, Phys. Rev.
D95 , 065014 (2017),arXiv:1607.05236 [hep-ph].[15] N. Craig, M. Jiang, Y.-Y. Li, and D. Sutherland, (2019),arXiv:2001.00017 [hep-ph].[16] For more complicated cases, one needs to use the reducedYoung Tableau [6] or momentum twistors [8] to deal withmomentum conservation.[17] B. Grzadkowski, M. Iskrzynski, M. Misiak,and J. Rosiek, JHEP , 085 (2010),arXiv:1008.4884 [hep-ph].[18] N. Arkani-Hamed, T.-C. Huang, and Y.-t. Huang,(2017), arXiv:1709.04891 [hep-th].[19] Here “unfactorizable” means that the amplitude does nothave any kinematic poles or branch cuts on which it couldbe factorized by unitarity.[20] The partial wave expansion for long-range scattering istricky, which involves zero-poles in other channels andan infinite tower of j in the summation. In this letter, wetemporarily ignore such amplitudes, and mainly focus onthe basis amplitudes.[21] C. Arzt, M. B. Einhorn, andJ. Wudka, Nucl. Phys. B433 , 41 (1995),arXiv:hep-ph/9405214 [hep-ph].[22] J. de Blas, J. C. Criado, M. Perez-Victoria, and J. San-tiago, JHEP , 109 (2018), arXiv:1711.10391 [hep-ph].[23] E. Witten, Commun. Math. Phys. , 189 (2004),arXiv:hep-th/0312171 [hep-th].[24] We are using a slightly different normalization than thatin [23].[25] We use ( φ, ψ α , ¯ ψ ˙ α , F αβ , ¯ F ˙ α ˙ β ) to denote generic fieldstransforming under Lorentz group SU (2) L × SU (2) R ≡ SO (3 ,
1) as (0,0), (1/2,0), (0,1/2), (1,0) and (0,1). And D is the covariant derivative. We also use ( φ, ψ ± , F ± ) todenote on-shell scalar, fermion, and vector particles with ± helicities in scattering amplitudes.[26] E. E. Jenkins, A. V. Manohar, and M. Trott,JHEP , 035 (2014), arXiv:1310.4838 [hep-ph].[27] R. Alonso, E. E. Jenkins, A. V. Manohar, and M. Trott,JHEP , 159 (2014), arXiv:1312.2014 [hep-ph].[28] H. Li, Z. Ren, J. Shu, M.-L. Xiao, J.-H. Yu, and Y.-H.Zheng, work in progress. Relevant effective operators in SMEFT
We list the the dimension 6 operators in Warsaw basis[17] relevant in the main text in table III.For the dimension 8 φ D type operators, we write the ψ H ψ O eH H † H (¯ leH ) O lequ (¯ le ) ǫ jk (¯ qu ) ψ H D O lequ (¯ lσ µν e ) ǫ jk (¯ qσ µν u ) O Hl ( H † i ←→ D µ H )(¯ lγ µ l ) ψ XH O Hl ( H † i ←→ D µ τ a H )(¯ lγ µ τ a l ) O eW (¯ lσ µν e ) τ a HW aµν O He ( H † i ←→ D µ H )(¯ eγ µ e ) O eB (¯ lσ µν e ) HB µν O Hq ( H † i ←→ D µ H )(¯ qγ µ q ) H D O Hq ( H † i ←→ D µ τ a H )(¯ qγ µ τ a q ) O H (cid:3) ( H † H ) (cid:3) ( H † H ) O Hu ( H † i ←→ D µ H )(¯ uγ µ u ) O HD ( H † D µ H ) ∗ ( H † D µ H ) O Hd ( H † i ←→ D µ H )( ¯ dγ µ d ) H O H ( H † H ) TABLE III. Relevant dimension 6 operators in SMEFT. corresponding amplitude basis as B ( H α H β H † ˙ α H † ˙ β ) = ( δ α ˙ α δ β ˙ β + δ β ˙ α δ α ˙ β )( s − s ) , B ( H α H β H † ˙ α H † ˙ β ) = ( δ α ˙ α δ β ˙ β − δ β ˙ α δ α ˙ β ) s ( s − s ) , B ( H α H β H † ˙ α H † ˙ β ) = ( δ α ˙ α δ β ˙ β + δ β ˙ α δ α ˙ β ) s . (26)while the Wilson coefficients are denoted as C H D i . Diagonalization of W matrix Since W commutes with all Poincar´e generators anddilatation, it has a matrix representation in the spaceof basis amplitudes with the same particle states anddimension, whose basis can be described by the reducedSemi-Standard Young Tableau (rSSYT) [6]. Thereforewe can simply diagonalize it and obtain a partial waveamplitude basis. We have realized the W operator in Mathematica , which quickly does the diagonalization.To present a non-trivial result, we take the operator ψ φ D as an example, which has 6 terms regardless ofgroup factor. We choose the channel ψ φ φ → ψ φ ,where we first classify the basis amplitudes in terms of j . The algorithm goes as follows • Find an initial amplitude basis B i , i = 1 , . . . , i B i O i h ih i [25] − ψ σ µ ¯ σ ν ψ ( D µ φ ) φ ( D ν φ )2 h ih i [23] ψ σ µ ¯ σ ν ψ ( D µ φ )( D ν φ ) φ h ih i [35] − ψ σ µ ¯ σ ν ψ φ ( D ν φ )( D µ φ )4 h ih i [25] − ψ ψ ( D µ φ ) ψ ( D µ φ )5 h ih i [23] − ψ ψ ( D µ φ )( D µ φ ) φ h ih i [35] − ψ ψ φ ( D µ φ )( D µ φ ) (27) • Applying W to them according to eq. (17), andget the coefficient matrix W B i = P j W ij s B j W = −
34 0 0 2 0 00 −
34 1 − −
114 0 0 −
20 0 0 −
154 0 00 0 0 − −
34 00 0 − − . (28) • Diagonalizing W to obtain the eigenvalues − j i ( j i +1) and the corresponding eigenvectors B j i .After the diagonalization, we get a dimension 2 anddimension 4 degenerate eigenspaces for j = 1 / j = 3 / a to label. Sometimes it is conve-nient to label the degeneracy by angular momenta ofsubgroup of particles, say j . Using the quantum num-bers ( j , j ), the 6 amplitudes can be further classi-fied into 4 eigenspaces (3 / , / / , / / , / / , / B / , / ψ φ D = − B + 2 B + 2 B + 9 B + 3 B + B , B / , / ψ φ D = −B + 2 B + B , B / , / ψ φ D = 2 B − B + 3 B + B . (29)In the j = 1 / j , where j = 3 / / B / , / ψ φ D = −B + 2 B − B , (30) which is independent of B / , / . Finally, the remainingdimension 2 linear space with j = j = j = 1 / j andobtain the eigenstates B / , / , ψ φ D = B − B + B , B / , / , ψ φ D = −B − B + B . (31)The reason to obtain one-dimensional eigenspaces is toshow the possibility of labelling all the degenerate statesby angular momenta of subsets of particles. It also pre-vents mixing between the degenerate operators via renor-malization, since they generate different partial wavestates in the shared channel. For example, when con-sider the mixing between operator types ψ φ φ ψ φ D and ψ φ φ ¯ ψ F D , we also diagonalize W for the lat-ter and obtain the partial wave amplitude basis in thechannels { , , } and { , } the as following: B / , / F ψ ¯ ψφ D = 3 h ih i [24] + h ih i [34] , B / , / F ψ ¯ ψφ D = h ih i [35] . (32)After this diagonalization, we conclude that only thepairs (cid:16) B / , / ψ φ D , B / , / F ψ ¯ ψφ D (cid:17) and (cid:16) B / , / ψ φ D , B / , / F ψ ¯ ψφ D (cid:17)(cid:17)