Standard Model Effective Field Theory from On-shell Amplitudes
aa r X i v : . [ h e p - ph ] M a r Standard Model Effective Field Theory from On-shell Amplitudes
Teng Ma, Jing Shu,
1, 2, 3, 4 and Ming-Lei Xiao 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
We present a general method of constructing unfactorizable on-shell amplitudes (amplitude basis),and build up their one-to-one correspondence to the independent and complete operator basis ineffective field theory (EFT). We apply our method to the Standard Model EFT, and identify theamplitude basis in dimension 5 and 6, which correspond to the Weinberg operator and operators inWarsaw basis except for some linear combinations.
I. INTRODUCTION
Effective Field theory (EFT) has wide applicationsin various aspects of physics. It serves as a powerfultool to understand the emergent coarse-graining behaviorwhere the underlying system has sophisticated patternsor is strongly coupled, such as superconductivity [1], frac-tional quantum hall effect [2], or low energy QCD [3], etc.Meanwhile, EFT provides a model-independent methodto categorize and parametrize possible unknown physicsfrom the ultraviolet (UV). In this case, our best exampleis the Standard Model (SM) EFT, which becomes a ba-sic paradigm of exploring the imprints of beyond the SMeffects.One essential step in an EFT calculation is the identifi-cation of complete operator basis [4–8]. To construct thefull set of independent operators in quantum field the-ory, one has to eliminate the redundancies from equationof motion (EOM) and integration by parts (IBP), whichyield relations between operators. Previously, people relyon symmetry [9–14] in SM to eliminate those redundan-cies and encode enumeration of operators in a Hilbertseries. Nevertheless, those methods do not give exact ex-pressions of all operators and are not naturally directedto the SMEFT calculations.In this paper, we introduce a novel way to write downall independent operators, which is based on the on-shellamplitude method [15]. Instead of using symmetries todeal with EOM and IBP, we can write down all completelocal on-shell amplitudes respecting Lorentz symmetry,SM gauge symmetry and spin-statistics theorem. Thoseon-shell amplitudes are in one to one correspondence withthe operators, which naturally form a new amplitude ba-sis . Our key observation is that for amplitude basis, theelimination of EOM and IBP redundancies are triviallyrealized by external leg on-shell conditions and momen-tum conservation, as naturally inherited from the on-shellamplitudes. This approach was used recently to infer theEFT Lagrangian for theories with a spin-0 or 1 singletcoupled to gluons [16].Our method really has its own advantages to beginwith. By using the on-shell amplitude method, the rootof our amplitude basis is nothing but the unfactoriz- able on-shell amplitudes from locality (positive power ofMadelstam variables without poles). We start directlywith the computation of unfactorizable amplitudes fromspin helicity formulism, thus automatically provide thebasic building blocks of EFT calculation, and advancedtechniques like recursion relations, unitarity cuts, etc canbe used naturally. Indeed, our method also greatly sim-plifies the calculation. The amplitude basis in d=6 forSMEFT corresponds to the Warsaw basis, except forsome linear combinations, can be derived in few pageslater in the paper.The rest of the paper is organized as follows. Wefirst discuss the general structure of the amplitude ba-sis and outline the rules of constructing the complete setof amplitude basis in a given dimension. Armed withthose tools, we explicitly construct the amplitude basisfor SMEFT in d = 5 and 6, and map them to the corre-sponding operators. Finally we conclude. II. THE STRUCTURE OF AMPLITUDE BASIS
We start from S -matrix program, which uses a set oflow point amplitudes as building blocks, and constructhigher point amplitudes by matching their residues us-ing recursion relations. Because of the renormalizability,the finite set of low point amplitudes is sufficient as thetheory input. However, in an non-renormalizable theorylike SMEFT, the irrelevant operators O are independentinteractions, which cannot be on-shell constructed by re-cursion relations without the help of symmetries. There-fore, those independent amplitudes should be viewed asthe input basis of the theory, which correspond to the in-dependent set of operators and can be classified by theirdimensions.We build the one-to-one correspondence between theamplitude basis and irrelevant operators O by enforcingthat all fields from O are on shell. The gauge symmetry,which reflects the redundancy, can be used to reduce theindependent amplitudes. Only the leading contact on-shell amplitudes which have the minimal fields from O are enough to fully construct all on-shell amplitudes fora given dimension [17]. Indeed, this is the same as the fa-mous example of Yang-Mills theory, the cubic term A ∂A captures the full information for the on-shell amplitudes,and the 4-point on-shell amplitudes are not independentfrom recursion relations (the existence of 4-point contactinteractions from A simply has no on-shell information).Since the above arguments only exploit gauge invariance,they should apply to the non-renormalizable theory.From Lorentz symmetry, the basic building blocks toconstruct the effective operators are F ± µν ≡ ( F µν ± i ˜ F µν )( ˜ F µν ≡ ǫ µνρσ F ρσ ), ψ L , ψ cL , φ and covariant derivative D µ which transforms under Lorentz group SU (2) L × SU (2) R ≡ SO (3 ,
1) as (1 , , / , , /
2) and(0 , F µν → ∂ µ A ν − ∂ ν A µ ” and “ D µ → ∂ µ ” to constructthose effective operators.There are two redundancies in the effective operatorsfrom Equation of Motion (EOM) and Integration by Part(IBP). However, in our amplitude basis, it is taken careautomatically because of the on shell condition and mo-mentum conservation. EOM for each fields will transmitthe operators involving the derivatives of such fields intoother operators. In our definition, the on shell condi-tions p = 0 suggests that operators involving (cid:3) φ , D/ψ or D µ F µν should vanish in the amplitude basis, thus thereis no such redundancy. For IBP in the amplitude basis,two operators differing by a total derivative are equiv-alent as the total derivative is the sum of all externalmomentum which equals zero.The general on-shell scattering amplitude should havethe following form M { α } = f ( λ i , ˜ λ i ) g ( s ij ) T { α } , (1)where λ i , ˜ λ i are helicity spinors of the i th leg and Mandel-stam variable s ij ≡ p i .p j (see more details about spinorformalism in App. A 1). f is the little group weight func-tion which is a function of spinor products [ ij ] = λ i ǫλ j and h ij i = ˜ λ i ǫ ˜ λ j and g is the little group invariant func-tion. T is the group factor, bearing all the internal groupindices of the external legs { α } and forming invariant ten-sors. For amplitude basis in a non-renormalizable theory,all spinor products in f and g have positive powers, whichhave no physical poles in Mandelstam variables and cannot be factorized into smaller building blocks due to lo-cality [18].According to dimension counting, it’s easy to get theoperator dimension d of an amplitude basis d = n + m = n + [ f ] + [ g ] , (2)where n is the number of legs and [ f ] and [ g ] are thedimensions of f and g where [ g ] is always an even inte-ger. The scattering processes are classified in terms offermion number n ψ and gauge boson number n A . Eachfermion contributes one helicity spinor, and each gauge boson contributes two, thus the number of spinor prod-ucts is at least m ≥ n ψ + n A . Using Eq(2) we have32 n ψ + 2 n A ≤ d (3)which gives a finite possibilities below a certain dimen-sion. For instance, to get amplitude basis below d = 6,what we need are scattering amplitudes with ( n ψ , n A )satisfying n ψ + 2 n A ≤
6. We list all possible ampli-tude basis in ( n ψ , n A , h ) where h is the total helicity with h ≥ h <
0. We leave scalar number unspecified to shortenthe list, because adding scalar does not change the formof Lorentz factor. For each ( n ψ , n A , h ), we examine allpossible helicity combinations (up to the conjugation).For a given helicity assignment, we can write down thenet powers of helicity spinors of all legs. For instance f ( ψ + ψ + φ ) ∼ λ λ ,f ( ψ + ψ − φ ) ∼ λ ˜ λ . (4)To contract the spinor indices, we use the complete set ofClifford algebra { , σ µ , σ µν , σ µνρ , iǫ µνρξ } to construct bi-linears. Specifically, { , σ µν } can be used to contract two λ s or two ˜ λ s, while { σ µ , σ µνρ } can be used to contract a λ and a ˜ λ . The more spacetime indices ( µ, ν, . . . ) a bilinearhas, the more momenta we need to add to contract withthem, which increases m . For instance to contract a λ i and a ˜ λ j , the lowest dimension combination we can writedown is ( λ i σ µ ˜ λ j ) p kµ ≡ [ i | p k | j i = [ ik ] h kj i . Following thisrule, the lowest dimension amplitudes for the cases (4)are f ( ψ + ψ + φ ) ∼ ( λ ǫλ ) ≡ [12] ,f ( ψ + ψ − φ ) ∼ ( λ ǫσ µ ˜ λ ) p µ ≡ [1 | p | i . (5)Moreover, Mandelstam variables can be added freely to g , as it is helicity blind. For each helicity assignment,there is a kinematic factor with minimum m , and thus aminimum dimension d , which we call primary amplitude .It is the leading amplitude for a given scattering states.We work out all primary amplitudes in the TABLE Ifor d ≤
6. For the (4 , ,
2) case, there are different waysof spinor contraction combinations, and we can define f ± ( ψ + ψ + ψ + ψ + ) = ([13][24] ± [14][23]) after applyingSchouten identity [12][34] + [13][42] + [14][23] = 0. III. COUNTING EFFECTIVE OPERATORS INSM EFT
The results in the previous section are simple conse-quences of Lorentz invariance, gauge invariance and lo-cality. When applying to SM matter fields in TABLE II,one has to take into account the SM gauge quantum num-bers and respect Fermi or Boson statistics for identicalfields. In this section, we explicitly construct the com-plete amplitude basis at dim-5 and dim-6 for SM EFT. ( n ψ , n A , h ) Primary amplitude m min n s d min (0,0,0) f ( φ n s ) = 1 0 n s > f ( A + A + φ n s ) = [12] f ( A + A + A + ) = [12][23][31] 3 6(2,0,1) f ( ψ + ψ + φ n s ) = [12] 1 4(2,0,0) f ( ψ + ψ − φ ) = [1 | p | i n s > f ( A + ψ + ψ + φ n s ) = [12][13] 2 5(4,0,2) f ( ψ + ψ + ψ + ψ + ) = [12][34] ∗ f ( ψ + ψ + ψ − ψ − ) = [12] h i d ≤
6. The ∗ for the (4 , ,
2) case stands for multiple ways of spinor con-traction.
To count dim-5 amplitude basis in SM EFT, we needto combine the amplitudes in TABLE I and appropri-ate group factors. Group factors are not always uniquefor a given set of group indices, and when there aremultiple choices, we use superscripts to label them. Inparticular, superscripts ± indicate permutation symme-try among the same type of indices, such as T ± αβ ˙ α ˙ β = ( δ α ˙ α δ β ˙ β ± δ α ˙ β δ β ˙ α ). Among the kinematic factors in TA-BLE I, we find that only the f ( ψ + ψ + φ ) combination isthe SM gauge singelt, which is M ( L α L β H γ H δ ) = [12]( ǫ αγ ǫ βδ + ǫ αδ ǫ βγ ) , (6)where the group factor is chosen to satisfy the spinstatistics. Together with its conjugate f ( ψ − ψ − φ ) withopposite helicity, we find the only 2 dim-5 amplitudebasis in SM EFT, which are the Weinberg operators O (5) = ( HL ) + h.c. .Dim-6 amplitude basis in SM EFT are obtained in thesame way. It is interesting that the classes of our am-plitude basis in TABLE I already reproduce the classesof operator summarized in [5] as the Warsaw basis. Be-cause we choose the group factor basis and Mandelstamvariables according to the permutation symmetry, the re-sultant amplitude basis for the same scattering states weget might be the linear combination of the operators de-fined in Warsaw basis. We list the correspondence below:1. Class M ( φ n s ) ( O ∼ ϕ and ϕ D ): Operator Amplitude Basis O H M ( H αβγ H † α ˙ β ˙ γ ) = T + αβγ ˙ α ˙ β ˙ γ O HD − O H (cid:3) M + ( H αβ H † α ˙ β ) = s T + αβ ˙ α ˙ β O HD + O H (cid:3) M − ( H αβ H † α ˙ β ) = ( s − s ) T − αβ ˙ α ˙ β where T + αβγ ˙ α ˙ β ˙ γ ≡ δ α ˙ α δ β ˙ β δ γ ˙ γ + δ β ˙ α δ α ˙ β δ γ ˙ γ + δ γ ˙ α δ β ˙ β δ α ˙ γ + δ β ˙ α δ γ ˙ β δ α ˙ γ + δ α ˙ α δ γ ˙ β δ β ˙ γ + δ γ ˙ α δ α ˙ β δ β ˙ γ isfully symmetric for SU (2) L indices αβγ and ˙ α ˙ β ˙ γ . T ± αβ ˙ α ˙ β ≡ δ α ˙ α δ β ˙ β ± δ β ˙ α δ α ˙ β is the (ant-)symmetricgroup structure for indices αβ and ˙ α ˙ β . s and s − s are the symmetric and antisymmetricMandelstam variables at the order s .2. Class M ( A + A + φ ) and M ( A − A − φ ) ( O ∼ X ϕ ):Instead of operators with definite CP, the ampli-tude basis are more naturally written down for def-inite chirality. They have easy linear relations.Warsaw Amplitude Basis O HB + O H ˜ B M ( B + B + H α H † ˙ α ) = [12] δ α ˙ α O HB − O H ˜ B M ( B − B − H α H † ˙ α ) = h i δ α ˙ α O HW B + O H ˜ W B M ( B + W i + H α H † ˙ β ) = [12] τ iα ˙ β O HW B − O H ˜ W B M ( B − W i − H α H † ˙ β ) = h i τ iα ˙ β O HW + O H ˜ W M ( W i + W j + H α H † ˙ β ) = [12] T ij + α ˙ β O HW − O H ˜ W M ( W i − W j − H α H † ˙ β ) = h i T ij + α ˙ β O HG + O H ˜ G M ( G A + G B + H α H † ˙ β ) = [12] T AB + α ˙ β O HG − O H ˜ G M ( G A − G B − H α H † ˙ β ) = h i T AB + α ˙ β where τ i is the Pauli matrix, T ij + α ˙ β ≡ δ ij δ α ˙ β and T AB + α ˙ β ≡ δ AB δ α ˙ β .3. Class M ( A + A + A + ) and M ( A − A − A − ) ( O ∼ X ):Warsaw Amplitude Basis O W + O ˜ W M ( W i + W j + W k + ) = [12][23][31] ǫ ijk O W − O ˜ W M ( W i − W j − W k − ) = h ih ih i ǫ ijk O G + O ˜ G M ( G A + G B + G C + ) = [12][23][31] f ABC O G − O ˜ G M ( G A − G B − G C − ) = h ih ih i f ABC where ǫ ijk and f ABC are SU (2) L and SU (3) c struc-ture constant.4. Class M ( ψ + ψ + φ ) ( O ∼ ψ ϕ ) + h.c.:Warsaw Amplitude Basis O eH M ( L α eH β H † α ˙ β ) = [12] T + αβ ˙ α ˙ β O dH M ( Q aα d ˙ a H β H † α ˙ β ) = [12] T + αβ ˙ α ˙ β δ a ˙ a O uH M ( Q aα u ˙ a H βγ H † ˙ α ) = [12] T + α ( βγ ) ˙ α δ a ˙ a where the group structure T + α ( βγ ) ˙ α ≡ ǫ αβ δ γ ˙ α + ǫ αγ δ β ˙ α symmetric for indices β and γ . If we flip thehelicity of fermions, we get another three indepen-dent amplitude basis in the class of M ( ψ − ψ − φ ),which is the conjugation of above operator basis.The expressions of these new amplitude basis areobtained by replacing the helicity factor [12] with h i in above expressions.5. Class M ( ψ + ψ − φ ) ( O ∼ ψ ϕ D ):Note that momentum conservation implies that[1 | p | i = [1 | p − p | i for n = 4, and is anti-symmetric for the two scalars ([1 | p + p || i = 0).Hence there is only one independent term.Warsaw Amplitude Basis O He M ( ee † H α H † ˙ α ) = [1 | p | i δ α ˙ α O Hu M ( u ˙ a u † a H α H † ˙ α ) = [1 | p | i δ α ˙ α δ a ˙ a O Hd M ( d ˙ a d † a H α H † ˙ α ) = [1 | p | i δ α ˙ α δ a ˙ a O Hud M ( d ˙ a u † a H αβ ) = [1 | p − p | i ǫ αβ δ a ˙ a O † Hud M ( u ˙ a d † a H † α ˙ β ) = [1 | p − p | i ǫ ˙ α ˙ β δ a ˙ a O (3) HL + O (1) HL M + ( L α L † ˙ α H β H † ˙ β ) = [1 | p | i T + αβ ˙ α ˙ β O (3) HL − O (1) HL M − ( L α L † ˙ α H β H † ˙ β ) = [1 | p | i T − αβ ˙ α ˙ β O (3) HQ + O (1) HQ M + ( Q aα Q † ˙ a ˙ α H β H † ˙ β ) = [1 | p | i T + αβ ˙ α ˙ β δ a ˙ a O (3) HQ − O (1) HQ M − ( Q aα Q † ˙ a ˙ α H β H † ˙ β ) = [1 | p | i T − αβ ˙ α ˙ β δ a ˙ a
6. Class M ( A + ψ + ψ + φ ) ( O ∼ ψ Xϕ ) +h.c.: Warsaw Amplitude Basis O eB M ( B + eL α H † ˙ α ) = [12][13] δ α ˙ α O dB M ( B + d ˙ a Q aα H † ˙ α ) = [12][13] δ α ˙ α δ a ˙ a O dG M ( G A + d ˙ b Q aα H † ˙ α ) = [12][13] δ α ˙ α λ Aa ˙ b O eW M ( W i + eL α H † ˙ β ) = [12][13] τ iα ˙ β O dW M ( W i + d ˙ a Q aα H † ˙ β ) = [12][13] τ iα ˙ β δ a ˙ a O uB M ( B + u ˙ a Q aα H β ) = [12][13] ǫ αβ δ a ˙ a O uW M ( W i + u ˙ a Q aα H β ) = [12][13] τ iβα δ a ˙ a O uG M ( G A + u ˙ b Q aα H β ) = [12][13] ǫ αβ λ Aa ˙ b where λ Aa ˙ b is the generator matrix of SU (3) c and τ iβα ≡ τ iαγ ǫ γβ . We can get the independent ampli-tude basis in the class of M ( A − ψ − ψ − φ ) whose ex-pressions can also be obtained by replacing squareproduct [12][13] with angle product h ih i .7. Class M ( ψ + ψ + ψ − ψ − ) ( O ∼ ¯ LL ¯ LL , ¯ RR ¯ RR ,¯ LL ¯ RR , ¯ LR ¯ RL , LLRR ) :The operators with all left and right handedfermions, and with group factor containing ǫ abc , vi-olate Baryon number conservation.Warsaw Amplitude Basis O (3) qq + O (1) qq M + ( Q aα Q bβ Q † ˙ a ˙ α Q † ˙ b ˙ β ) = [12] h i T + αβ ˙ α ˙ β T + ab ˙ a ˙ b O (3) qq − O (1) qq M − ( Q aα Q bβ Q † ˙ a ˙ α Q † ˙ b ˙ β ) = [12] h i T − αβ ˙ α ˙ β T − ab ˙ a ˙ b O (3) lq + O (1) lq M ± ( Q aα L β Q † ˙ a ˙ α L † ˙ β ) = [12] h i T + αβ ˙ α ˙ β δ a ˙ a O (3) lq − O (1) lq M ± ( Q aα L β Q † ˙ a ˙ α L † ˙ β ) = [12] h i T − αβ ˙ α ˙ β δ a ˙ a O ll M ( L α L β L † ˙ α L † ˙ β ) = [12] h i T + αβ ˙ α ˙ β O (8) qu + O (1) qu M ± ( Q aα u ˙ b Q † ˙ a ˙ α u † b ) = [12] h i δ α ˙ α T + ab ˙ a ˙ b O (8) qu − O (1) qu M ± ( Q aα u ˙ b Q † ˙ a ˙ α u † b ) = [12] h i δ α ˙ α T − ab ˙ a ˙ b O (8) qd + O (1) qd M ± ( Q aα d ˙ b Q † ˙ a ˙ α d † b ) = [12] h i δ α ˙ α T + ab ˙ a ˙ b O (8) qd − O (1) qd M ± ( Q aα d ˙ b Q † ˙ a ˙ α d † b ) = [12] h i δ α ˙ α T − ab ˙ a ˙ b O (8) ud + O (1) ud M ± ( u ˙ a d ˙ b u † a d † b ) = [12] h i T + ab ˙ a ˙ b O (8) ud − O (1) ud M ± ( u ˙ a d ˙ b u † a d † b ) = [12] h i T − ab ˙ a ˙ b O uu M ( u ˙ a u ˙ b u † a u † b ) = [12] h i T + ab ˙ a ˙ b O dd M ( d ˙ a d ˙ b d † a d † b ) = [12] h i T + ab ˙ a ˙ b O lu M ( L α u ˙ a L † ˙ α u † a ) = [12] h i δ α ˙ α δ a ˙ a O ld M ( L α d ˙ a L † ˙ α d † a ) = [12] h i δ α ˙ α δ a ˙ a O qe M ( Q aα eQ † ˙ a ˙ α e † ) = [12] h i δ α ˙ α δ a ˙ a O ledq M ( Q aα d ˙ a L † ˙ α e † ) = [12] h i δ α ˙ α δ a ˙ a O † ledq M ( L α eQ † ˙ a ˙ α d † a ) = [12] h i δ α ˙ α δ a ˙ a O le M ( L α eL † ˙ α e † ) = [12] h i δ α ˙ α O eu M ( eu ˙ a e † u † a ) = [12] h i δ a ˙ a O ed M ( ed ˙ a e † d † a ) = [12] h i δ a ˙ a O ee M ( e e † ) = [12] h iO duq M ( Q aα L β u † b d † c ) = [12] h i ǫ αβ ǫ abc O † duq M ( u ˙ b d ˙ c Q † ˙ a ˙ α L † ˙ β ) = [12] h i ǫ ˙ α ˙ β ǫ ˙ a ˙ b ˙ c O qqu M ( Q aα Q bβ u † c e † ) = [12] h i ǫ αβ ǫ abc O † qqu M ( u ˙ c eQ † ˙ a ˙ α Q † ˙ b ˙ β ) = [12] h i ǫ ˙ α ˙ β ǫ ˙ a ˙ b ˙ c M ( ψ + ψ + ψ + ψ + ) ( O ∼ ¯ LR ¯ LR , LLLL , RRRR ) + h.c.: Notice that f ( ψ + ψ + ψ + ψ + ) has two choices. Wedefine combinations f ± ≡ [13][24] ± [23][14] withspecific permutation symmetries. O (3) lequ is not theweak current interaction, but defined as differentspinor contractions σ µν σ µν . The failure of a unifiednotation in Warsaw basis proves the advantage ofthe amplitude basis as a systematic classification.Warsaw Amplitude Basis O (8) quqd + O (1) quqd M + ( Q aα Q bβ u ˙ a d ˙ b ) = f − ǫ αβ T − ab ˙ a ˙ b O (8) quqd − O (1) quqd M − ( Q aα Q bβ u ˙ a d ˙ b ) = f + ǫ αβ T + ab ˙ a ˙ b − O (3) lequ + O (1) lequ M + ( L α Q aβ u ˙ a e ) = f + ǫ αβ δ a ˙ a − O (3) lequ − O (1) lequ M − ( L α Q aβ u ˙ a e ) = f − ǫ αβ δ a ˙ a O qqq M ( Q aα Q bβ Q cγ L δ ) = f − T − αβγδ ǫ abc O duu M ( u a ˙ b d ˙ c e ) = f + ǫ ˙ a ˙ b ˙ c Following the same procedure, we can obtain theamplitude basis in the class of M ( ψ − ψ − ψ − ψ − ) byreplacing square product with angle product.Summing up all 8 classes of amplitude basis, we get3 + 8 + 4 + 6 + 9 + 16 + 12 + 26 = 84 basis (hermitianconjugates are counted separately), recovering the wellknown result. IV. CONCLUSION AND OUTLOOK
In this letter, we propose a novel way of writing downall independent effective operators from the unfactoriz-able on-shell amplitudes. This particular basis is referredas the amplitude basis since all operators are in one to onecorrespondence with the on-shell amplitudes. We providethe general rules to construct those primary amplitudesand classify them by the external legs and helicity as-signments so that all operators in the amplitude basiscan be enumerated systematically for a given dimension.Then we further demonstrate how to use our method togenerate all independent dim-5 and dim-6 operators inSMEFT while respecting the SM gauge symmetry andspin-statistics constrains. Interestingly, we find that op-erators in our amplitude basis for d = 6 SMEFT is thewell known Warsaw basis, except for some linear combi-nations. Our method starts from the on-shell amplitudesthus it is naturally convenient for EFT calculation andfree from redundancies connected by EOM and IBP.Our result here is only a small tip of the iceberg forthe on shell effective field theory. There are various in-teresting aspects that can be done or will be finishedvery soon (some related applications are discussed inRef [19, 20]). The procedure can be applied to moresophisticated cases together with tools to deal with ten-sor structures in d = 7, 8 SMEFT [21]. Applicationsto specific processes can be demonstrated as Ref. [16].The SMEFT is a massless case, applications to the EFTwith massive particles [22] are under investigation. Thecurrent setup can be encoded into the computation ofWilson coefficients of the amplitude basis if we know theunderlying theory. Applications to other types of EFTand related concepts may also spark off intriguing results. Acknowledgements
We thank Song He and Hui Luo for useful discussionsand comments. J.S. thanks the hospitality of HKUSTJockey Club Institute for Advanced Study while workingon this project. T.M. is supported in part by projectY6Y2581B11 supported by 2016 National PostdoctoralProgram for Innovative Talents. J.S. is supported by theNational Natural Science Foundation of China (NSFC)under grant No.11847612, No.11690022, No.11851302,No.11675243 and No.11761141011 and also supported bythe Strategic Priority Research Program of the ChineseAcademy of Sciences under grant No.XDB21010200 andNo.XDB23000000. M.L.X. is supported by 2019 the In-ternational Postdoctoral Exchange Fellowship Program.
Appendix A: Notation
In this section we list the conventions throughout thiswork.
1. Conventions for spinor helicity formalism
Since the Lorentz group SO (3 ,
1) is isomorphism with SU (2) L × SU (2) R , the four-vector momentum p µ can bemapped into a two-by-two matrix via p α ˙ α = p µ σ µα ˙ α (A1)where σ µ = (1 , σ i ) is a four-vector of Pauli matrices andthe undotted and dotted indices transform under theusual spinor representations of the Lorentz group. Wecan find the determinant of p α ˙ α is Lorentz scalardet[ p ] = p µ p µ , (A2)which vanishes for massless on-shell particle. So the van-ishing determinant of massless on-shell particle momen-tum indicates that p α ˙ α is a two-by-two matrix of at mostrank one, which can be written as the outer product oftwo two-component objects which are called spinors p α ˙ α = −| p ] α h p | ˙ α . (A3)Given two massless particles i and j , we can defineLorentz invariant building blocks of spinor helicity for-malism h ij i = ǫ αβ | i i α | j i β [ ij ] = ǫ ˙ α ˙ β | i ] ˙ α | j ] ˙ β (A4) where we use the short-hand notation | i i ≡ | p i i ( | i ] ≡| p i ]) and ǫ αβ is 2-index Levi-Civita symbols. The Man-delstam invariants can be written in terms of these ob-jects s ij = ( p i + p j ) = 2 p i .p j = h ij i [ ij ] . (A5)Because spinors are two dimension objects, one can al-ways write a spinor as a linear combination of two linearlyindependent spinors and thus have the identity[ ij ][ kl ] + [ ik ][ lj ] + [ il ][ jk ] = 0 , (A6)which is known as the Schouten identity.The lightlike momentum decomposition in Eq. A3 isinvariant under the scaling | p i → t | p i [ p | → t − [ p | . (A7)For real momentum, the scaling factor t is just a purephase. So the transformation in Eq. (A7) correspondsto the SO (2) little group transformation of the lightlikemomentum, which is called little group scaling.For an on-shell amplitude, the i th external leg withhelicity h i scales as t − h i and neither propagators or ver-tices can scale under little group. So the on-shell ampli-tude transform homogeneously under little group scaling A n (1 h , h , ..., n h n ) → Y i t h i A n (1 h , h , ..., n h n ) . (A8)The transformation of the on-shell scattering amplitudesunder little group scaling can help determine the littlegroup weight function f (see the review [23, 24]).
2. Conventions for SM fields and gauge symmetry
In this section, we list the notations of SM fields andtheir gauge symmetry indices in Tab. (II), where allfermions are listed as left-handed Weyl fermions. We re-quire that the anti-fundamental representation of SU (3) c are denoted by dotted letters ˙ a, ˙ b, ... and the indices of theconjugate of SU (2) L doublets of SM left-handed fermionsand Higgs doublet with hypercharge 1 / α, ˙ β, ... . Appendix B: Warsaw Basis
We list all the standard Warsaw basis operators be-low. We mostly keep the notations used in this paper,and for consistency we label the right handed fermionsas u R = Cu ∗ , d R = Cd ∗ , e R = Ce ∗ , where C = iσ forWeyl spinors and C = iγ γ for Dirac spinors. We usefour-component Dirac spinors here as the original War-saw paper did. σ µν = [ γ µ , γ ν ] and T A = λ A / SU (3) c SU (2) L U (1) Y G ± A W ± i B ± Q aα u ˙ a ¯3 1 -2/3 d ˙ a ¯3 1 L α -1/2 e H α SU (3) c × SU (2) L × U (1) Y . All fermions are in the form of left hand. X X ϕ O G f ABC G Aµν G Bνρ G Cρµ O H ( ∼ ) B H † H ( ∼ ) B µν B µν O ˜ G f ABC ˜ G Aµν G Bνρ G Cρµ O H ( ∼ ) W B H † τ i H ( ∼ ) W iµν B µν O W ǫ ijk W iµν W jνρ W kρµ O H ( ∼ ) W H † H ( ∼ ) W iµν W iµν O ˜ W ǫ ijk ˜ W iµν W jνρ W kρµ O H ( ∼ ) G H † H ( ∼ ) G Aµν G Aµν ϕ and ϕ D ψ ϕ O H ( H † H ) O eH ( H † H )( ¯ Le R H ) O H (cid:3) ( H † H ) (cid:3) ( H † H ) O uH ( H † H )( ¯ Qu R ˜ H ) O HD | H † D µ H | O eH ( H † H )( ¯ Qd R H ) ψ Xϕ ψ ϕ D O eB ( ¯ Lσ µν e R ) HB µν O He ( H † i ←→ D µ H )(¯ e R γ µ e R ) O dB ( ¯ Qσ µν d R ) HB µν O Hu ( H † i ←→ D µ H )(¯ u R γ µ u R ) O dG ( ¯ Q λ A σ µν d R ) HG Aµν O Hd ( H † i ←→ D µ H )( ¯ d R γ µ d R ) O eW ( ¯ Lσ µν e R ) τ i HW iµν O Hud ( ˜ H † iD µ H )(¯ u R γ µ d R ) O dW ( ¯ Qσ µν d R ) τ i HW iµν O (1) Hl ( ˜ H † iD µ H )( ¯ Lγ µ L ) O uB ( ¯ Qσ µν u R ) ˜ HB µν O (3) Hl ( ˜ H † iD iµ H )( ¯ Lτ i γ µ L ) O uW ( ¯ Qσ µν u R ) τ i ˜ HW iµν O (1) Hq ( ˜ H † iD µ H )( ¯ Qγ µ Q ) O uG ( ¯ Q λ A σ µν u R ) ˜ HG Aµν O (3) Hq ( ˜ H † iD iµ H )( ¯ Qτ i γ µ Q )( ¯ LR )( ¯ LR ) LLLL ( B/ ) O (1) quqd ( ¯ Qu R ) ǫ ( ¯ Qd R ) O qqq ǫ abc ( Q a ǫCQ b )( Q c ǫCL ) O (8) quqd ( ¯ QT A u R ) ǫ ( ¯ QT A d R ) RRRR ( B/ ) O (1) lequ ( ¯ Le R ) ǫ ( ¯ Qd R ) O duu ǫ abc ( d aR Cu bR )( u cR Ce R ) O (3) lequ ( ¯ Lσ µν e R ) ǫ ( ¯ Qσ µν u R )( ¯ LL )( ¯ LL ) ( ¯ LL )( ¯ RR ) O ll ( ¯ Lγ µ L )( ¯ Lγ µ L ) O le ( ¯ Lγ µ L )(¯ e R γ µ e R ) O (1) qq ( ¯ Qγ µ Q )( ¯ Qγ µ Q ) O lu ( ¯ Lγ µ L )(¯ u R γ µ u R ) O (3) qq ( ¯ Qγ µ τ i Q )( ¯ Qγ µ τ i Q ) O ld ( ¯ Lγ µ L )( ¯ d R γ µ d R ) O (1) lq ( ¯ Lγ µ L )( ¯ Qγ µ Q ) O qe ( ¯ Qγ µ Q )(¯ e R γ µ e R ) O (3) lq ( ¯ Lγ µ τ i L )( ¯ Qγ µ τ i Q ) O (1) qu ( ¯ Qγ µ Q )(¯ u R γ µ u R )( ¯ RR )( ¯ RR ) O (8) qu ( ¯ Qγ µ λ A Q )(¯ u R γ µ λ A u R ) O ee (¯ eγ µ e )(¯ eγ µ e ) O (1) qd ( ¯ Qγ µ Q )( ¯ d R γ µ d R ) O uu (¯ uγ µ u )(¯ uγ µ u ) O (8) qd ( ¯ Qγ µ λ A Q )( ¯ d R γ µ λ A d R ) O dd ( ¯ d R γ µ d R )( ¯ d R γ µ d R ) ( ¯ LR )( ¯ RL ) O eu (¯ e R γ µ e R )(¯ u R γ µ u R ) O ledq ( ¯ Le R )( ¯ d R Q ) O ed (¯ e R γ µ e R )( ¯ d R γ µ d R ) LLRR ( B/ ) O (1) ud (¯ u R γ µ u R )( ¯ d R γ µ d R ) O duq ǫ abc ( d aR Cu bR )( Q c ǫCL ) O (8) ud (¯ u R γ µ λ A u R ) O qqu ǫ abc ( Q a ǫCQ b )( u cR Ce R )( ¯ d R γ µ λ A d R ) Appendix C: Example
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