PHYSICAL REVIEW D 104, 015026 (2021)
Complete set of dimension-eight operators in the standard model effective field theory
† ‡ Hao-Lin Li,1,* Zhe Ren ,1,2 Jing Shu,1,2,3,4,5, Ming-Lei Xiao ,1, Jiang-Hao Yu ,1,2,4,5,6,§ and Yu-Hui Zheng1,2 1CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China 2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China 3CAS Center for Excellence in Particle Physics, Beijing 100049, People’s Republic of China 4Center for High Energy Physics, Peking University, Beijing 100871, People’s Republic of China 5School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, University of Chinese Academy of Sciences, Hangzhou 310024, People’s Republic of China 6International Centre for Theoretical Physics Asia-Pacific, Beijing/Hangzhou, People’s Republic of China
(Received 5 January 2021; accepted 11 June 2021; published 23 July 2021)
We present a complete list of the dimension-eight operator basis in the standard model effective field theory using group theoretic techniques in a systematic and automated way. We adopt a new form of operators in terms of the irreducible representations of the Lorentz group and identify the Lorentz structures as states in a SUðNÞ group. In this way, redundancy from equations of motion is absent and that from integration by part is treated using the fact that the independent Lorentz basis forms an invariant subspace of the SUðNÞ group. We also decompose operators into the ones with definite permutation symmetries among flavor indices to deal with subtlety from repeated fields. For the first time to our knowledge, we provide the explicit form of independent flavor-specified operators in a systematic way. Our algorithm can easily be applied to higher-dimensional standard model effective field theory and other effective field theories, making these studies more approachable.
DOI: 10.1103/PhysRevD.104.015026
I. INTRODUCTION highly motivated to study NP phenomena involving only the SM particles within the framework of effective theories. The standard model (SM) of particle physics is a great Effective field theory (EFT) provides a systematical triumph of modern physics. It has successfully explained framework for parametrizing various NP based on only the almost all experimental results and predicted a wide variety field content, the Lorentz invariance, and the gauge sym- of phenomena with unprecedented accuracy. Despite its metries in the SM. The Lagrangian of such an EFT contains great success, however, the SM fails to account for some not only the renormalized SM Lagrangian but also all the basic properties of our universe, e.g., neutrino masses, higher-dimensional invariant operators, which parametrize matter-antimatter asymmetry, and the existence of dark all the possible deviations from the SM. Assuming that NP matter. This has motivated both the theorists and the appears at the scale Λ above the electroweak scale,1 the experimentalists to make a dedicated effort to search for general Lagrangian can be parametrized as pieces of evidence of new physics (NP) beyond the SM. Until now, direct searches have not yielded anything of X −4X 4 1 d significance, which already pushed the NP scale to be L Lð Þ CðdÞOðdÞ; : SMEFT ¼ SM þ Λ i i ð1 1Þ above the tera-electron-volt (TeV) scale. Therefore, it is d>4 i
which describes the standard model effective field theory *[email protected] † (SMEFT), with C identified as the Wilson coefficients. The [email protected] i ‡ 5 [email protected] only possible dimension-five (d ¼ ) operator is the famous §[email protected] Weinberg operator [1], with lepton number violation encoded. The dimension-six operators were first listed in Published by the American Physical Society under the terms of Ref. [2], and a subtle problem arises due to redundancies the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, 1New physics could also exist below the electroweak scale, but and DOI. Funded by SCOAP3. such a scenario is not considered here.
2470-0010=2021=104(1)=015026(60) 015026-1 Published by the American Physical Society LI, REN, SHU, XIAO, YU, and ZHENG PHYS. REV. D 104, 015026 (2021) among the operators. It is often convenient to obtain a The main difficulty of listing the independent operators complete set of independent operators, namely the non- arises from how to effectively eliminate the redundancies redundant operator basis. This task is highly nontrivial among operators with derivatives. Operators with deriva- because different structured operators may be related by tives often involve two types of redundancies: (1) operators theequationofmotion(EOM),integrationbyparts(IBP),and differing by the classical EOM are related to each other Fierz identities. These redundancies could be avoided by through field redefinitions; (2) operators differing by a total imposing the EOMs and IBPs explicitly, the independent derivative are equivalent in perturbative calculations, the dimension-six operators in the Warsaw basis [3] were con- so-called IBP. At lower dimensions where limited operators structed based on this principle, and the complete renorm- with derivatives are present, the EOM and IBP relations alization group equations are written in Refs. [4–6].In could be imposed explicitly to eliminate all the redundan- Ref. [7,8] the complete set of dimension-seven operators cies, as was done when the dim-6 and dim-7 operators were has been obtained. Recently the Hilbert series method [9–11] written down [3,7]. The on-shell amplitude method [28–32] has been applied to enumerate the SMEFT operators up to has been applied to the dimension-six SMEFT [29–31] dimension 15 [12–15], but it is only designed to count which solves the EOM redundancy but still needs to impose the number of independent operators in each dimension. conditions to treat the IBP redundancy. Nevertheless, at Besides, a few other papers [16–19] also developed programs dimension eight or higher, the number of such operators to count the number of operators in alternative ways. Although increases tremendously, which makes the task very tedious partial lists of the dimension-8 (dim-8) and higher-dimensional and prone to error. The Hilbert series method, applied to the operators have been obtained [20–24], writing down a SMEFT, deals with these redundancies via decomposing complete set of the nonredundant operators explicitly at the field derivatives into irreducible representations of the dimension eight and higher is still a challenging task. Lorentz group and removing all the descendants while Our goal in this paper is to find a complete set of keeping the primaries in each irreducible representation of dimension-eight operators in the SMEFT framework. For a the conform group. In spite of its efficiency at counting, this physical process, if the leading NP contribution directly method does not help us write down the operator basis comes from the dimension-eight operators, or if the explicitly. One step forward along this line is Ref. [33],in contribution from the dimension-six operators is subdomi- which independent Lorentz structures were constructed as nant or highly constrained, the dimension-eight operators “harmonics” on the sphere of momentum conservation that should be seriously considered, even though their Wilson is exempt from the IBP redundancy, but the issue of coefficients are suppressed by a higher inverse power of the identical particles, namely the repeated field problem for NP scale. The first example is the neutral triple gauge boson operators, was not taken into account. Reference [16] couplings (nTGC) ZZV and ZγV, for which no dimension- generates an overcomplete list of operators at first, and six operator contributes and thus dimension-eight operators then reduces it to an independent basis by putting all the dominate [22,25]. Furthermore, in the dimension-six oper- redundant relations into a matrix, which has also been ator basis, the Wilson coefficient for the quartic gauge applied [24] to write down the partial list of the dimension- boson coupling (QGC) is related to the one in the triple eight operators involving only the bosonic fields. gauge boson coupling (TGC), while at dimension eight Another difficulty is how to obtain independent flavor there is no such correlation in the Wilson coefficients [26]. structures when repeated fields are present. In the literature Similar is true for various Higgs gauge boson couplings. [3,18,34], the concept of Lagrangian terms is ambiguous, For the four-fermion interactions, let us take the nonstand- and it is usually subtle to talk about their flavor structures. ard neutrino interaction (NSI) as an example. At the In particular, the Q3L type operators were pointed out to dimension-six level, new physics which induces the neutral have only one independent term [3,35] instead of two terms current NSI also induces an operator involving the charged shown in the older literature [1], while for both of them, current NSI, which has been tightly constrained by experi- extra efforts are needed to provide independent flavor- ments. Thus we expect the dimension-eight operators could specified operators. It is especially confusing when more dominate the neutral current NSI processes [27]. The than one term has to be written down, when the dependence presence of the electric dipole moment (EDM) which among their flavor-specified operators is even more can also be generated by the dimension-eight operators obscure. Later when the dimension-seven operators were directly indicates the existence of the CP violation in the listed, this issue of flavor structure was completely ignored UV theory, and in some cases dimension-eight operators in Ref. [7] but later addressed in Ref. [36] by imposing can be the leading order contribution as its counterpart at several flavor relations explicitly with a tedious procedure. the dimension-six operator vanishes. The dimension-eight Reference [18] provides a systematic way to deal with operators also generate new kinds of four-fermion inter- flavor structures with repeated fields, in which permutation actions with quite different Lorentz structures. Overall, the symmetries of all the factor structures [Lorentz, SUð3Þ, dimension-eight operator basis deserves a detailed study SUð2Þ] in the operator are combined via inner product with all the nonredundant operators written explicitly. decomposition into irreducible representations of the flavor
015026-2 COMPLETE SET OF DIMENSION-EIGHT OPERATORS IN THE … PHYS. REV. D 104, 015026 (2021) tensors. Again, this was only for counting, and thus did definite Lorentz and gauge group permutation symmetries not work out any of the symmetrized factor structures of the same set of repeated fields. To obtain the sym- explicitly. metrized Lorentz and gauge group structures, we introduce In this work, we provide a new and systematic method to the basis symmetrizer in the minimal left ideal of the list all the independent operators by using group theoretic symmetric group algebra, which, applied to the factor techniques, which solves the two main difficulties men- structures, generates a basis transforming as irrep of the tioned above. Inspired by the correspondence between symmetric group. Using these bases, we obtain the flavor- operators and on-shell amplitudes [28–32] and the independent operators at dimension-eight that constitute SUðNÞ transformation of on-shell amplitudes [33,37], our main result. we start by adopting a new form of operators constructed The paper is organized as follows. In Sec. II,we from building blocks, fields with or without derivatives, in introduce notations for fields and operators used in our the irreducible representations (irreps) of the Lorentz paper and define the terminologies for operators at different group, for which the EOM redundancy is absent. The levels. In Sec. III, we first discuss the problem of repeated Lorentz structure of operators is then identified as states fields and show in Sec. III A that solving this problem that transform linearly under the SUðNÞ group and form a leads to the demand of finding symmetrized Lorentz and large interclass space, in which total derivatives form an gauge group bases. Then we explain our algorithm to invariant subspace of the SUðNÞ. Group theory indicates obtain these symmetrized bases in details in Secs. III B and that the nonredundant Lorentz structures with respect to the III C for the Lorentz and gauge groups, respectively. In IBP should also form an invariant space consisting of irrep Sec. III D we show how to obtain the operators with spaces, and a basis for them is easily found by translating definite permutation symmetries of flavor indices from the semistandard Young tableau (SSYT) of these irreps. In ingredients discussed above via inner product decom- addition, we develop a procedure to list all classes of position of the symmetric group. In Sec. IV, we exhibit Lorentz structures at a given dimension, and for each of a table showing numbers of operators for each subclass in them we can directly obtain the corresponding irrep of the terms of the fermion flavor number, and we list the SUðNÞ and the labels to be filled in, which is sufficient for complete set of dim-8 SMEFT operators organized by enumerating all the SSYTs and Lorentz structures. Since the number of fermions. Our conclusion is presented in we directly obtain an independent basis of Lorentz struc- Sec. V. In Appendix A, we list useful identities and tures in the process, we never need to actually use the EOM examples of format conversions between Lorentz repre- and IBP relations as in other literature, and the correctness sentations, both for fermions and gauge bosons, which are of our result is theoretically guaranteed. used in presenting our results. In Appendix B, we introduce For the sake of generality, we treat the gauge group some basics of symmetric groups Sm and a few group structures systematically which, as far as we know, was not theory tools we used in the paper, including the basis λ presented yet before. Gauge symmetry demands singlet symmetrizer b and the projection operator involved in the combinations of fields with various representations, inner product decomposition. described by tensor product decomposition. To construct these singlets explicitly, we turn all the constituting fields II. STANDARD MODEL EFFECTIVE into forms with only fundamental indices, and adopt the FIELD THEORY Littlewood-Richardson rules to merge their Young dia- A. SM fields: Building block, and notation grams into a singlet Young diagram, during which we keep their fundamental indices in the diagram as labels. What we The Lagrangian of SMEFT should be invariant under the finally get is a singlet Young tableau, each column Lorentz group and the SM gauge group. We start by representing a Levi-Civita tensor in the group structure defining the building blocks of the effective operators: that contracts with the indices inside the column. In this fields and their covariant derivatives. The building blocks way, we get an independent basis of group structures are characterized by their representations under the Lorentz 2 C 2 2 consisting of Levi-Civita’s. group SLð ; Þ¼SUð Þl × SUð Þr and the SM gauge 3 2 1 Having obtained the complete basis of Lorentz and groups SUð ÞC × SUð ÞW × Uð ÞY. The representation gauge group structures, it is easy to combine them into a under Lorentz symmetry is given by ðjl;jrÞ, the quantum 2 2 basis of operators with only free flavor indices. However, as numbers of the SUð Þl and SUð Þr components of the mentioned above, permutation symmetries among repeated Lorentz group SLð2; CÞ. We adopt the following notations fields induce symmetries of the flavor indices, which we on the field constituents: shall resolve by constructing the factor structures with (i) Since all the SM fermions are chiral-like, we use the definite permutation symmetries and combining them via two-component Weyl spinor notation, which trans- inner product decomposition. According to the plethysm forms as irreps of the Lorentz group technique, operators with definite permutation symmetry of ψ ∈ 1 2 0 ψ † ∈ 0 1 2 flavor indices can be systematically addressed by obtaining α ð = ; Þ; α_ ð ; = Þ; ð2:1Þ
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TABLE I. The field content of the standard model, along with (iii) We use the chiral basis of the gauge boson FL=R ¼ 1 ˜ their representations under the Lorentz and gauge symmetries. 2 ðF iFÞ because they transform under irreps of The representation under the Lorentz group is denoted by ðjl;jrÞ, − the Lorentz group, which is important for us to study while the helicity of the field is given by h ¼ jr jl. The number the constraints on the Lorentz structures. They of fermion flavors is denoted as n , which is three in the standard f transform to the normal gauge field strength as model. All of the fields are accompanied with their Hermitian † conjugates that are omitted: ðFLαβÞ ¼ FRα_ β_ for gauge bosons, † † † i μν ðψ αÞ ¼ðψ Þ_ for fermions, and H for the Higgs, which are σ ∈ 1 0 α FLαβ ¼ 2 Fμν αβ ð ; Þ; under the conjugate representations of all the groups. i μν 2 2 3 2 1 F α_ β_ ¼ − Fμνσ¯ _ ∈ ð0; 1Þ: ð2:3Þ Fields SUð Þl × SUð Þr hSUð ÞC SUð ÞW Uð ÞY Flavor R 2 α_ β A −1 81 GLαβ (1, 0) 01 I −1 13 The spinor indices of FL=R are symmetric in order to WLαβ (1, 0) 01 −1 11 form (1, 0) or (0, 1) representations, as can be proved BLαβ (1, 0) 01 by the property of σμν, defined in Appendix A1. 1 −1 2 12−1 2 Lαi ð2 ; 0Þ = = nf 1 The field constituents without derivatives are given in α 0 −1 2 11 eC ð2 ; Þ = 1 nf Table I. As shown in Table I, the indices for the (anti) 1 0 −1 2 321 6 Qαai ð2 ; Þ = = nf fundamental representation of SUð2Þ , SUð2Þ , SUð3Þ , a 1 ¯ l r C u α 0 −1=2 3 1 −2=3 n C ð2 ; Þ f and SU 2 are denoted by α; β; γ; δ , α_; β_; γ_; δ_ , a 1 ¯ ð ÞW f g f g d α 0 −1=2 3 1 1=3 n C ð2 ; Þ f fa; b; c; dg, and fi; j; k; lg, respectively. We use subscripts
Hi (0,0) 0 121=2 1 to indicate the fundamental representation and superscripts to indicate the antifundamental representation. The indices 3 2 for the adjoint representation of SUð ÞC and SUð ÞW are where the indices α and α_ denote the fundamental denoted by fA; B; C; Dg and fI;J;K;Lg, respectively. In 2 2 representation of SUð Þl, and SUð Þr, respectively. case flavor indices are needed, we use fp; r; s; tg. In the We further adopt the all-left chirality convention for final result, the spinor indices are left implicit. the fermions: Q and L are the left-handed compo- Not only the SM fields but also the fields with covariant nents of the quark and lepton doublet fields, and uC, derivatives are the building blocks, although the cova- dC, and eC are the left-handed components of the riant derivative itself is not. In our notation, the covariant anti-up, anti-down, and anti-electron fields. The derivatives only act on the nearest field on the right, and the transformation to the four-component Dirac spinor gauge group indices on that field should be understood as
notation is given in Appendix A1. the indices of the whole, for example, DQpai ¼ðDQpÞai. (ii) We use the following notation for the SM Higgs Regarding the Lorentz index on D, we also adopt the doublet: SLð2; CÞ notation for convenience,
†i Hi ∈ ð0; 0Þ;H∈ ð0; 0Þ; ð2:2Þ μ Dαα_ ¼ Dμσαα_ ∈ ð1=2; 1=2Þ; ð2:4Þ where the index i denotes the (anti)fundamental 2 with the ordinary covariant derivative Dμ defined by representation of SUð ÞW. We avoid using the notation H˜ ¼ ϵH†, as it is essentially the same as † 2 ∂ − A A − aτa − 0 H but with the original SUð ÞW antifundamental Dμ ¼ μ igsGμ T igWμ ig QYBμ; ð2:5Þ indices lowered to the fundamental one by the ϵ tensor. In our final results all the gauge group indices with the SUð3Þ and SUð2Þ generators TA and τa as well as ˜ 1 are left explicit; consequently whenever there is a H the Uð ÞY charge QY determined by the fields it acts on. †j n present in other literature, it translates into ϵijH in Thus covariant derivatives of a field D D Φ, in which Φ our notation. denotes the SM field, could be expressed in general as
D ð1Þ ð1Þ D ðrþhÞ ðrþhÞ Φ ðrþhþ1Þ ðr−hÞ ;h<0; r−jhj α α_ α α_ α α ðD ΦÞ ð1Þ ðr−hÞ ð1Þ ðrþhÞ ≡ ð2:6Þ α α α_ α_ Φ 0 Dαð1Þα_ ð1Þ Dαðr−hÞα_ ðr−hÞ α_ ðr−hþ1Þ α_ ðrþhÞ ;h>; where h is the helicity of the massless particle annihilated number of Lorentz indices is correct for any Φ, for example, by the field and r ¼jhjþnD is half the total number of the r ¼ 5=2 building block of field Q, which has helicity spinor indices in this building block. One could verify the h ¼ −1=2, is given by
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2 ðD QaiÞαð1Þαð2Þαð3Þα_ ð1Þα_ ð2Þ ¼ Dαð1Þα_ ð1Þ Dαð2Þα_ ð2Þ Qαð3Þai: ð2:7Þ operator are fixed, the indices of the Lorentz and gauge groups are also fixed, and then the simplest way to With these building blocks, operators are simply con- assemble these constituents into a singlet is to contract structed as combinations of building blocks that form the all the indices with invariant tensors for each group. In this singlet representation, under all of the Lorentz group and way we obtain a basis of operators that are direct products gauge groups. In general, when the constituents of the of three-factor structures:
YN ffg fgg fhg fαg;fα_g − Θffg ri jhijΦ ¼ TSU3TSU2 TLorentz ðD iÞαð1Þ αðr−hÞα_ ð1Þ α_ ðrþhÞ i i i i i¼1 fgg;fhg fgg fhg Mffg ¼ TSU3TSU2 fgg;fhg; ð2:8Þ
where the invariant tensors TSU3, TSU2, and TLorentz form practical purposes, we group the effective operators into polynomial rings generated by the following corresponding several levels of clusters defined as below: ingredients: (i) The biggest cluster is called a class, which involves operators with the same kinds of fields in terms of 3 ∶ ABC ABC δAB λA b ϵ ϵabc spin, and the same number of derivatives, denoted SUð Þ f ;d ; ; ð Þa; abc; ; as FnF ψ nψ ϕnϕ DnD . One could be more accurate by 2 ∶ ϵIJK δIJ τI j ϵ ϵij SUð Þ ; ; ð Þi ; ij; ; setting the definite number of left/right-handed μν μν μ μαα_ αβ α_ β_ fermions and gauge bosons, so that we get sub- Lorentz∶ σαβ; σ¯ _ ; σαα_ ; σ¯ ; ϵ ; ϵ˜ : ð2:9Þ α_ β classes such as
fαg;fα_g n−1 † n1 In the second line, we collapse T with the building ψ n−1=2 ϕn0 ψ n1=2 nD Lorentz FL FR D : Mffg blocks to a formal Lorentz singlet fgg;fhg, which we will often refer to as a Lorentz structure, with free flavor and One could come up with any combinations of ni at gauge group indices that are specified after fixing the this level and rule out the ones that are not able constituting fields.2 to form Lorentz invariants later, but we propose The dimension of Θ can be derived as criteria for selecting the Lorentz invariant sub- classes, which makes the program more effective XN at higher dimensions. dimðΘÞ¼ ðri − jhijþdimðΦiÞÞ ¼ N þ r; ð2:10Þ (ii) In a (sub)class, we further group together operators i¼1 with the same constituting fields, selected by the P requirement of conservation laws: the combination where N is the particle number, r ¼ i ri turns out to be of fields should be able to form a singlet of any the mass dimension of the on-shell amplitude generated symmetric groups (besides the Lorentz group) the by Θ. theory has. This level of cluster is called a type, In Sec. III we will put additional constraints on the form denoted by a sequence of fields and derivatives. An of building blocks, Eq. (2.8), and our master formula of example of the type is operator basis Eq. (3.80) will be constructed as particular nonredundant combinations of them. 2 † † Q uCLH D; ð2:11Þ
B. Invariants at different levels: Class, type, term, which corresponds to n−1=2 ¼ 3;n0 ¼ 1;n1=2 ¼ 1; and operator nD ¼ 1. Note that Lorentz structures may not be The above subsection defines the building blocks that we fixed at this level, especially at higher dimensions, use in constructing invariants, i.e., operators; for the sake of when a type could contain quite a number of clarity, we specify the terminologies used in the rest of the independent ones. At this level, though, we could paper that describe the invariants at different levels. For identify the groups of repeated fields in a type, which put constraints on the form of independent operators within a type. 2We would like to point out here that formally each field in our notation has a flavor index, even for gauge bosons and the Higgs (iii) At this level, we define the (Lagrangian) terms that which can take only one possible value. The reason will be clear have different interpretations from the other liter- in Sec. III A. ature, e.g., [18]. We define a term as an operator with
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free flavor indices that transform as an irreducible complicates the game. To demonstrate the problem let us tensor of the auxiliary flavor symmetry group take a look at the dim-5 Weinberg operator in the SMEFT: SUðnfÞ for each set of repeated fields with nf number of flavors. We occasionally refer to the 11 α α f1 f2 Θff1f2; g ≡ ϵi1j1 ϵi2j2 ϵ 1 2 L L H H ; ð3:1Þ irreducible tensor nature of terms as the flavor α1;i1 α2;i2 j1 j2 symmetry of the operator. An example of a term where α1 2 are spinor indices, i1 2 and j1 2 are SUð2Þ for the type in Eq. (2.11) is ; ; ; W indices, and Θff1f2;11g has the same notation defined in prst abc jk μ † Eq. (2.8) with Φi’s specified to LLHH.Wehavealso Θ ¼ iϵ ϵ ððL Q ðQ σ u tc Þ pi sbk raj C grouped the flavor indices of each set of repeated fields σμ † †k þðLpiQrajðQsbk uCtc ÞÞDμH : ð2:12Þ together where the flavor indices for H have already been set to 1 as we only have one Higgs in the SM. In the One can verify that Θprst is symmetric under the following, we shall drop the flavor indices of H when their exchange of r, s flavor indices of two Q’s, which absences do not obscure our explanation. One can verify that Θprst indicates is an irreducible tensor represented Θf1f2 ¼ Θf2f1 , which we will prove later, is a result of the ’ by under SUðnfÞ group for Q s. This definition antisymmetric nature of the Lorentz and gauge structures; of the term renders enumerating independent flavor- therefore the number of independent operators are nfðnf þ specified operators defined below trivial by finding 1Þ=2 if we have nf flavors of L. Generally, to count and all the SSYTs of the corresponding Young diagram enumerate independent operators with flavor indices speci- of irrep of SUðn Þ. We shall explicitly illustrate this f fied, one can view Θffkg as a tensor of the SUðn Þ group and point and describe the algorithm to obtain a com- f decompose it into different irreps of the SUðn Þ group; then plete set of terms up to any dimensions in detail in f Sec. III A. the independent operators are given by setting the flavor (iv) Finally, the flavor-specified operators are defined as indices according to all the SSYTs of the corresponding irrep independent flavor assignments in a term. The with numbers in the tableaux weakly increased in each row corresponding example gives and strictly increased in each column. Following the example above, Θf1f2 in Eq. (3.1) is symmetric under the exchange of Θ1111 ϵabcϵjk σμ † f1 and f2, and hence it is represented by the Young tableau: ¼ i ððL1iQ1bkðQ1aj uC1c Þ .Ifnf ¼ 2, then there are three semistandard Young σμ † †k þðL1iQ1ajðQ1bk uC1c ÞÞDμH : ð2:13Þ tableaux: , ,and , which correspond to indepen- 11 12 22 dent operators Θ , Θ ,andΘ . All the other choices of f1 and f2 can be expressed by the linear combination of these III. OPERATOR BASIS FOR LORENTZ AND three using Fock’s conditions [38], and in this example, we GAUGE SYMMETRIES simply have Θ21 ¼ Θ12. A. Motivation and mathematical preparation The LLHH example above seems to be too trivial as it is easy to find out the symmetric properties among the flavor In this subsection, we first explain why we need definite indices f1 and f2. However, the situation becomes com- permutation symmetries of the Lorentz and gauge group plicated when the number of repeated fields in a set goes structures for the term we defined in Sec. II B as well as 3 how they are related to the permutation symmetry of the up. The simplest nontrivial example is Q L in the dim-6 flavor indices, and then we give a gentle introduction to SMEFT. In the Warsaw basis [3] it is expressed as mathematical tools used in generating symmetrized ϵabcϵ ϵ aj T bk cm T i Lorentz and gauge group structures. ji km½ðqr Þ Cqs ½ðqt Þ Clp ; ð3:2Þ 2 1. Why permutation symmetry where q and l are four component SUð ÞW quark and lepton doublets of which the relation to our two-component Given a type of the operator, one can enumerate all the 3 notations are shown in Appendix A1; a, b, c are SUð ÞC independent ways to construct a singlet under both Lorentz 2 indices; i, j, k, m are SUð ÞW indices; and p, r, s, t are and gauge symmetries with flavor indices unspecified. flavor indices of fermions, respectively. From Eq. (3.2),itis Fixing the flavor indices of such a Lorentz and gauge very hard to tell the independent components of the flavor- singlet completely determines the form of a flavor-specified specified operators. However, we separate this operator operator. If there are no repeated fields, then different into three terms in our notation, and each term is an irrep choices of flavor for each field correspond to different of SUðn Þ, operators. However, the presence of repeated fields f
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ð3:3Þ
ð3:4Þ
ð3:5Þ where the subscripts of Θ are partitions of the integer 3 that have a one-to-one correspondence with the Young diagrams (YD) on the left of each equation. From the above notation, we can immediately write down the independent combinatorial choices of r, s, t by enumerating the SSYTs of each irrep:
and then each Young tableau can pair with three choices of formula Sðλ;nfÞ [18]. λ is a partition of the integer m that lepton flavor p ¼ 1, 2, 3 resulting in totally 3 × ð1 þ 8 þ can be written in the form of the subscripts in Eqs. (3.3)– 10 Þ independent operators. In addition to the benefits (3.5) serving as a character of irreps of Sm. x goes from 1 to discussed above, we would like to point out that our ffkg dλ labeling basis vectors in the irrep λ. Θ λ by definition definition of term, combined with the algorithm obtaining ð ;xÞ satisfies the following relation: a complete set of independent terms described below, will automatically avoid the redundancy that needs to be ffπ g π ∘ Θffkg ≡ Θ ðkÞ resolved by obscure relations between different terms with ðλ;xÞ ðλ;xÞ 3 X flavor indices permuted. For example, Q L was initially ΘffkgDλ π ¼ ðλ;yÞ ð Þyx; ð3:6Þ written as two independent terms in Ref. [3] and was y corrected to only one in the form of Eq. (3.2) later. We will not proceed with this example further in this subsection as it for π ∈ Sm, where the action π ∘ on tensors is defined involves technical details discussed in Sec. III D, which Dλ π according to the first line and ð Þyx is the matrix may blur the big picture we would like to convey. Therefore representation of the S group for irrep λ. In the presence we shall continue with the LLHH example below and show m of multiple sets of repeated fields, one can generalize Θffkg the roadmap constructing these SUðnfÞ irreps. … into Θffk;pm; g as irreps of the groups: The above discussion demonstrates the desire to obtain Θffkg as an irrep of SUðnfÞ, where ffkg are the flavor SU ¼ SUðn1Þ ⊗ SUðn2Þ ⊗ ; indices of a set of repeated fields. Because of the Schur- f f ¯ ⊗ ⊗ Weyl duality and also pointed out in Ref. [18], if one can S ¼ Sm1 Sm2 ; ð3:7Þ Θffkg λ construct a set of tensor ðλ;xÞ that transform as an irrep of 1 2 where n ;n ; …, and m1;m2; …, are numbers of flavor and the symmetric group S (m is the number of the repeated f f m fields for different sets of repeated fields, respectively. An fields) in terms of permuting subscripts k, then any one of immediate conclusion that can be drawn from the above Θffkg the ðλ;xÞ is the irrep of SUðnfÞ with the number of discussion is that the operators involving repeated fields independent components obtained by the hook content with only one flavor such as gauge bosons or Higgs must be
015026-7 LI, REN, SHU, XIAO, YU, and ZHENG PHYS. REV. D 104, 015026 (2021) in a totally symmetric representation, which is simply a result of ð3:11Þ Sðλ; 1Þ¼0; ∀ λ ≠ ½m ; ð3:8Þ where [m] is the totally symmetric irrep. where we have used the Grassmann nature of the lep- Now we are going to show that permuting flavor indices ton field. of a given set of repeated fields in a term is equivalent to So far, it is obvious from Eq. (3.10) that those of gauge permuting the corresponding indices related to the gauge and Lorentz indices determine the permutation property of and Lorentz structures. In general, a term of a given type Mλ1 λ2 λ3 can be formally expressed as a linear combination of a set flavor indices. Hence, if a set of j , TSU3, and TSU2 λ of factorizable Θ’s according to Eq. (2.8) as transform according to irreps 1;2;3 of Sm for the certain set of repeated fields, then the direct product space spanned by … … … ff ;…g fgk; g fhk; g ffk; g Θ k ¼ T 3 T 2 M … … ; ð3:9Þ λ λ λ SU SU fgk; g;fhk; g Θ M 1 2 3 ∈ 1 … f ðλ1;x1Þ;ðλ2;x2Þ;ðλ3;x3Þ ¼ x1 TSU3;x2 TSU2;x3 jxi ; ;dλi g; … … fgk; g fhk; g ð3:12Þ where TSU3 ;TSU2 are the same TSU3;TSU2 in Eq. (2.8) with the indices of each set of repeated fields grouped with dλ the dimension of irreps λ , can be decomposed into together; the same argument applies for the correspondence i i … λ Mffk; g Mffg invariant subspaces that form different irrepsP of . The inner between fg ;…g;fh ;…g here and fgg;fhg in Eq. (2.8). k k product decomposition λ1 ⊙ λ2 ⊙ λ3 ¼ λ⊢ rλλ tells that Concretely, in the LLHH example we have m the multiplicity of the invariant subspace of irrep λ is rλ.For the LLHH example we discussed above, one can find that T 3 ¼ 1; ffπ ;11g 11 2 SU M ðkÞ Mffk; g M½ forms a total symmetric fiπ g;fj g ¼ fi g;fj g ¼ 1 i1i2;j1j2 ðkÞ m k m f g ϵi1j1 ϵi2j2 TSU2 ¼ ; representation of S2 of two lepton fields, the same is for the 2 ff1f2;11g α α f1 f2 ½ M ϵ 1 2 L L H H : SUð3Þ gauge group factor T 3 1 ¼ 1. The permuted fi1i2;j1j2g ¼ α1;i1 α2;i2 j1 j2 C SU ; fiπ ;j g 2 ðkÞ m ϵi2j1 ϵi1j2 SUð ÞW gauge group factor TSU2 ¼ seems For a given type, we call in the rest of our paper a complete not equal to the unpermuted one. However, this is just a … … … fgk; g fhk; g ffk; g ½2 set of independent T 3 ;T 2 , and M … … as SU SU fgk; g;fhk; g result of the simplification of the symmetric one TSU2;1 ¼ ϵi2j1 ϵi1j2 ϵi1j1 ϵi2j2 2 3 the Lorentz and gauge basis from which a term is ð þ Þ= when contracting with Hj1 Hj2 , so constructed. It is enough to show the permutation relations it also forms a totally symmetric representation of S2.One between flavor and combined gauge and Lorentz structures can find from the inner product decomposition ½2 ⊙ ½2 ⊙ hold for these factorizable bases, and the same is true for a ½2 ¼½2 that the only resulting irrep is the symmetric one general term. A permutation of the flavor indices of the first ½2 , and indeed the Θf1f2 is totally symmetric under … set of repeated fields in Θffk; g can be expressed as permutation of the flavor indices. In general, this decom- position is more complicated and contains the irreps with … … fg ;…g fh ;…g ffπðkÞ; g ffk; g k k dimensions larger than one. We will present a nontrivial π ∘Θ ¼ T 3 T 2 M … … |fflfflfflfflfflffl{zfflfflfflfflfflffl} SU SU fgk; g;fhk; g 3 permute flavor example Q LWL in Sec. III D and refer to Appendix B2for … … … more general cases. Although we only consider the fgπðkÞ; g fhπðkÞ; g ffπðkÞ; g ¼ T 3 T 2 M … … permutation symmetry for a single set of repeated fields SU SU fgπðkÞ; g;fhπðkÞ; g in Eqs. (3.10) and (3.12), it is straightforward to generalize fgk;…g fhk;…g ffk;…g ¼ðπ ∘T 3 Þðπ ∘T 2 Þðπ ∘M … … Þ; |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}SU SU fgk; g;fhk; g it to multiple sets of repeated fields under the product group |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ¯ permute gauge permute Lorentz S as the permutations acting on different sets of repeated fields simply commute with each other. ð3:10Þ Up to now, we have changed the problem of finding a … ff ;…g Θffk; g Θ k π ∘ term as irreps of SU into finding a series of ðλ;iÞ;… where T again only permutes the gauge indices of the ¯ first set of the repeated fields; ðπ ∘ MÞ is equal to the as irreps of S, and then further into finding the correspond- λ2;… λ3;… λ1;… ¯ M … permutation of all the subscripts of the Lorentz indices and ing TSU3;x2;…;TSU3;x3;…, and x1; as irreps of S. Before the associated derivatives of the first set of repeated fields while leaving the gauge and flavor indices unchanged. We 3This is also an example discussed in Eq. (3.8) that the demonstrate that the second to the third line in Eq. (3.10) repeated fields of one flavor must form a total symmetric does hold for M and ðπ ∘ MÞ defined above with the representation under permutation. As the Lorentz structure of π 12 2 fikg;fjmg LLHH example in Eq. (3.1) with ¼ð Þ, H is trivial, therefore TSU2 must be symmetric under h1, h2.
015026-8 COMPLETE SET OF DIMENSION-EIGHT OPERATORS IN THE … PHYS. REV. D 104, 015026 (2021) we delve into the details of obtaining all the independent one by permuting the arguments. It can be shown in λ symmetrized group factors and Lorentz structures, we first Appendix B1 that a series of new functions FxðfpkgÞ λ digress to introduce the mathematical tools used in the rest generated by applying bx to a function FðfpkgÞ defined by of the section. λ λ FxðfpkgÞ ¼ bx ∘ FðfpkgÞ X 2. Group algebra and left ideal λ ¼ c ;iπ ∘ Fðfp gÞ As mentioned above, in this section, we explain the key x i k i mathematical tools to obtain the Lorentz and gauge X λ;i ≡ c F pπ 3:18 structure in different irreps λ of the symmetric group x ðf iðkÞgÞ ð Þ associated with the repeated fields. We introduce the idea i of group algebra and the method using them to generate a transform as an irrep of λ under the permutation series of symmetrized functions transforming as an irrep of the Sm group under permutations defined in (3.6) from an λ λ π ∘ F ðfp gÞ ¼ F ðfpπ gÞ asymmetrized one. The first concept is the group algebra i x k Xx iðkÞ ˜ λ λ space Sm of Sm, which is defined as a set consisting of ¼ FyðfpkgÞDyxðπiÞ: ð3:19Þ formal linear combinations of the group elements in the y group Sm [39], X The function here has general meanings: in the LLHH ˜ i Sm∶frjr ¼ r πi for ri ∈ C; πi ∈ Smg: ð3:13Þ example above, the function can be referred to the gauge i i1i2;j1j2 group tensor TSU2 with arguments i1;2;j1;2 or the Lorentz structure M ¼ Mðα1; α2Þ with the arguments α1 2. The addition and multiplication rules of the elements in the ; In addition, we would like to mention that our con- group algebra are λ vention for bx follows Chapter 6 of the textbook in [38], X λ i i ˜ where b is proportional to the Young symmetrizer of the c1r þ c2q ¼ π ðc1r þ c2q Þ for r;q ∈ S ;c1;c2 ∈ C; 1 i m λ i normal Young tableau of , i.e., the Young tableau with the numbers 1 to n appearing in order from left to right and ð3:14Þ ½2;1 X X from the top row to the bottom row. For example, b1 is i j π π π Δk i j proportional to the Young symmetrizer of , which is r · q ¼ r q ð i · jÞ¼ k ijr r ; ð3:15Þ i;j i;j;k equal to the multiplication of sλ, the sum over all possible horizontal permutations, and aλ, the sum over all possible Δk where the matrix Pij with only one nonzero element vertical permutations weighted by their signatures, 1 for π π π Δk defined by i · j ¼ k k ij is the regular representation even and odd permutations, respectively. For ,wehave of the group. Obviously, a linear vector space structure is contained in the group algebra. In this sense, the group s½2;1 ¼ E þð12Þ; ð3:20Þ algebra elements have a dual role of vectors and linear operators. ˜ a½2;1 ¼ E − ð13Þ; ð3:21Þ It is well known that the Sm [38,39] can be decomposed λ into invariant subspaces transforming as irrep ofP the Sm λ λ;iπ expanded by a set of group algebra elements bx ¼ i cx i such that ð3:22Þ X λ λ λ πi · bx ¼ byDyxðπiÞ; ð3:16Þ y ¼ E þð12Þ − ð13Þ − ð132Þ: ð3:23Þ where the indices x, y go from 1 to dλ and the dimension of λ ½2;1 rst irrep λ, DyxðπiÞ is the same one in Eq. (3.6), the matrix When we apply b1 on a tensor T , we will associate the representation of λ [39]. The invariant subspaces expanded tensor indices r, s, t to the numbers 1,2,3 (not to confuse λ ˜ with flavors), and then formally we have by bx are actually a minimal left ideal Lλ of Sm such that ˜ r · b ∈ Lλ; ∀ r ∈ Sm;b∈ Lλ: ð3:17Þ ð3:24Þ Alternatively, one can view the group algebra elements as symmetrizers that act on a function generating another
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The resulting symmetrized tensor is symmetric for the operator basis we find has nonvanishing on-shell permutation of labels r, s as they appear in the same row in amplitudes, which also form an amplitude basis. the Young tableau, which is a general property of the Young (d) Integration by part. In perturbative quantum field symmetrizers. theory, we have
XDμY ∼−DμXY: ð3:28Þ B. Lorentz basis: SUðNÞ × S¯ irreps To obtain an independent set of Lorentz structures, in In other words, operators are equivalent modulo total literature such as the Warsaw basis [3], one usually writes derivatives. From the on-shell point of view, it is down all the possible Lorentz invariant combinations equivalent to the momentum conservation law. This of the building blocks, and then removes all the redun- may be the most subtle one, because it is the way people dancies by imposing the following relations among oper- eliminate this redundancy while counting that prevents ators repeatedly: the listing of the independent operator basis. In this (a) Fierz identity. As explained in Appendix A1, for Weyl section, we develop a new method to deal with IBP. spinors, the Fierz identities can be expressed as We aim at a systematic treatment of all of these redun- dancies before we write down operators, so that we do not σμ σν 2ϵ ϵ gμν αα_ ββ_ ¼ αβ α_ β_ ; ð3:25Þ need to examine them in an overcomplete list. Section III B 1 tackles the redundancies (b), (c), and the first half of (a), while Sec. III B 2 deals with the Schouten identities and the αβ γ βγ α γα β ϵ δκ þ ϵ δκ þ ϵ δκ ¼ 0; IBP. Finally in Sec. III B 3, we symmetrize the Lorentz κ_ κ_ κ_ structures over repeated fields for a specific type and obtain ϵ˜α_ β_ δγ_ þ ϵ˜β_ γ_δα_ þ ϵ˜γ_ α_ δ_ ¼ 0: ð3:26Þ β the Lorentz basis.
For the first identity, we choose to replace the left-hand 1. Lorentz invariance: Enumerating the classes side whenever it appears in the operator by the right- hand side. As our building blocks do not contain any We start by further analyzing the building block defined Lorentz indices μ, ν, etc., there would be no chance to in Eq. (2.6) and reduce them to irreps of the Lorentz group. σ use the matrices in TLorentz, Thus we are left with By applying the following relations: only the ϵ tensors for both dotted and undotted spinor indices in the Lorentz invariant tensor. The other two σμ σν − 2ϵ ϵ i ϵ σ¯ μν D½αα_ Dβ β_ ¼ DμDν αα_ _ ¼ D αβ α_ β_ þ ½Dμ;Dν αβ _ ; identities, also known as the Schouten identities, will ½ β β 2 α_ β ψ σμ ψ −ϵ ψ be tackled later. D½αα_ β ¼ Dμ αα_ β ¼ αβð=D Þα_ ; − ½ (b) ½Dμ;Dν ¼ iFμν. We also choose to replace the left- μ νρ μ ν D αα_ F β γ DμFνρσ σ 2D Fμνϵαβσ ; : hand side whenever it appears in the operator by the ½ L ¼ ½αα_ β γ ¼ γα_ ð3 29Þ right-hand side. Note that the replacement changes the 2 μ type of operator, and thus it should not be counted in where D ¼ D Dμ and the original type as an independent operator. Effec- μ 0 Dμσαα_ tively, we treat ½Dμ;Dν as zero while counting γμ =D ¼ Dμ ¼ μββ_ ; operators of a given type. Dμσ¯ 0 (c) Equation of motion. Classically there are the EOM relations for each kind of fields we note that any pair of antisymmetric spinor indices in a building block would lead to factors that vanish according 2 to the redundancies (b) or (c). As a consequence, we are left D ϕ þ Jϕ ¼ 0;i=Dψ þ Jψ ¼ 0; with building blocks in which all spinor indices, dotted or μν ν 0 DμF þ JA ¼ : ð3:27Þ undotted, are totally symmetric, respectively. After raising the dotted indices, we could express the remaining building For quantum fields, these are not rigorous operator blocks as equations. Nevertheless, operators differing by EOM 1 2 α_ ð Þα_ ð Þ…α_ ðrþhÞ r h r − h r h are related with each other by field redefinitions and r−jhjΦ ð Þ ≡ r−jhjΦ α_ þ ∈ þ ðD Þ ð1Þ ð2Þ ðr−hÞ ðD Þαr−h ; ; are hence physically equivalent. To remove this ðα α …α Þ 2 2 redundancy of field redefinition, we choose to replace ð3:30Þ the first term (the kinetic term) whenever it appears in the operator by the source term JΦ. Again, because the where, without ambiguity, we abbreviate the totally sym- type is changed during the replacement, we effectively metric indices (indicated by the parentheses) by an index treat the kinetic terms as zero while counting operators with a power. Now the remaining building block trans- of a given type. This choice guarantees that the forms as irreps under the Lorentz group as shown above.
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We would like to emphasize here that, with the above should be understood as a product of epsilons with possibly construction we throw away all the possible kinetic terms different spinor indices, which are only distinguished by that may be replaced with EOMs in our building blocks, the building block Eq. (3.30) they come from. Such and our algorithm enumerates the complete and indepen- operators with certain N and the numbers of epsilons ˜ M dent operators basis type by type at a fixed dimension, once ðn; nÞ form a basis of the linear space ½ N;n;n˜ , which still operator bases for all the types for a fixed dimension are have redundancy from the Schouten identity and the IBP. found, our goal is completed; thus the interchange between To solve the IBP problem, we decompose the space as types and different dimensions due to EOMs never plays a M A ⊕ B B ½ N;n;n˜ ¼½ N;n;n˜ ½ N;n;n˜ , where the subspace ½ N;n;n˜ role in our method. contains all the Lorentz structures with total derivatives. With this notation, together with our treatment of the For any Lorentz structure M ∈ ½M we have redundancy (a) using Eq. (3.25), we arrive at a general form of Lorentz structure modulo (b) and (c) redundancies as M ¼ MA þ MB; MA ∈ ½A ; MB ∈ ½B : ð3:32Þ YN α_ riþhi αiαj ⊗n ⊗n˜ ri−jhij i M ϵ ϵ˜α_ α_ D Φ − ∈ M ˜ ; A ¼ð Þ ð i j Þ ð iÞαri hi ½ N;n;n If this decomposition is possible, the subspace ½ would i¼1 i be the space of the nonredundant Lorentz structures, ð3:31Þ because for any two Lorentz structures in ½A , their difference is also in ½A and cannot be a total derivative. where N is the number of building blocks in the operator, We will achieve this decomposition in the next subsection. corresponding to the number of particles in the on-shell Before that, we would like to show how Lorentz amplitude it generates. Here we recognize the epsilon invariance constrains the classes appearing at a certain tensors introduced is the Lorentz invariant tensor TLorentz dimension. We derive some nontrivial constraints among in Eq. (2.6), in which Eq. (3.26) guarantees that only the the parameters in Eq. (3.31). The contractions of spinor epsilon tensors appear. The power of the epsilon tensors indices lead to the following relations:
X X n˜ þ n ¼ ri ¼ r; n˜ − n ¼ hi ≡ h; ð3:33Þ X i X i X nD ¼ ðri − jhijÞ ¼ 2n þ h − jhij¼2n˜ − h − jhij ≤ minð2n; 2n˜Þ: ð3:34Þ i i i Here we find another interpretation of r as the total number of ϵ’s. The second line gives one constraintP on the number of − derivatives necessary for an operator with given helicity combination hi, that the number must equal h i jhij mod 2 and is bounded by twice the minimum of n and n˜. Another constraint is already shown in [40] and comes from the following fact indicated by r ≥ jhj: 1 X ≥ − ≥−2 ∀ ⇒ − ≥−2 n ri hi hi; i 2 ðri hiÞ¼n min hi; i 1 X ˜ ≥ ≥ 2 ∀ ⇒ ˜ ≥ 2 n ri þ hi hi; i 2 ðri þ hiÞ¼n max hi; ð3:35Þ i from which we deduce X X X nD ¼ ðri − hiÞ − 2jhij ≥−4 min hi − 2jhij; 0 0 Xi hXi< Xhi< nD ¼ ðri þ hiÞ − 2jhij ≥ 4 max hi − 2jhij: ð3:36Þ i hi>0 hi>0
In sum, we arrive at the complete constraint on nD: 0 P 1 h − jhij; mod 2 B i C B P C B −4 − 2 C 2 2˜ ≥ ≥ min hi jhij minð n; nÞ nD maxB 0 C: ð3:37Þ B hi< C @ P A 4 max hi − 2jhij hi>0
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TABLE II. All the subclasses of Lorentz structures at dimension eight.
N ðn; n˜Þ5 Subclasses 4 4 (4,0) FL þ H:c: 2 ψψ† ψ 4 2 ψ 2ϕ 2 2 ϕ2 2 (3,1) FL D þ H:c: D þ H:c:FL D þ H:c:FL D þ H:c: 2 2 ψψ† ψ 2ψ †2 2 ψ 2ϕ 2 (2,2) FLFR FLFR D D FR D þ H:c: 2 2 † 2 3 4 4 FLFRϕ D ψψ ϕ D ϕ D ψ 4 2 ψ 2ϕ 3 ϕ2 5 (3,0) FL þ H:c:FL þ H:c: FL þ H:c: ψ 2ψ †2 2 ψ †2ϕ ψ 3ψ †ϕ ψψ†ϕ2 (2,1) FL þ H:c:FL þ H:c: D þ H:c:FL D þ H:c: ψ 2ϕ3 2 ϕ4 2 D þ H:c:FL D þ H:c: ψ 4ϕ2 ψ 2ϕ3 2 ϕ4 6 (2,0) þ H:c:FL þ H:c:FL þ H:c: (1,1) ψ 2ψ †2ϕ2 ψψ†ϕ4D ϕ6D2 7 (1, 0) ψ 2ϕ5 þ H:c: 8 (0,0) ϕ8
The minimum is a correction to the constraint shown 2. Lorentz structures as SUðNÞ states in [40]. Back to the problem of finding the subspace ½A from M In light of the above relations, we can enumerate the as proposed after Eq. (3.32), we first claim property of its classes of Lorentz structures for a given dimension after the elements in the format of Eq. (3.31) that such a Lorentz following steps: structure is completely determined by the epsilon tensors, (i) From Eqs. (2.10) and (3.33), we get d ¼ n þ n˜ þ N. α α_ 4 because the numbers of i and i on these tensors fix all the We start by iterating N from 3 to d, while for each N ri−jhij parameters of the building blocks D Φi. For example, we could iterate n; n˜ under the constraint n þ n˜ ¼ α α α α given ϵ 1 3 ϵ 2 3 ϵα_ α_ , we obtain d − N. 3 4 (ii) Given the tuple ðN;n;n˜Þ, we iterate n from 0 to D ϵα1α3 ϵα2α3 ϵ˜ minð2n; 2n˜Þ according to Eq. (3.37). Provided the α_ 3α_ 4 α_ † number of derivatives n , we have the following ⇒ M ϵα1α3 ϵα2α3 ϵ˜ ψ ψ ψ 3 ψ α_ 4 D ¼ α_ 3α_ 4 ð 1Þα1 ð 2Þα2 ðD 3Þα2 ð 4Þ : relations implied by Eq. (3.34): 3 ð3:39Þ X 2n−1 þ n−1=2 ¼ jhij − h ¼ 2n − nD; Those who are familiar with spinor helicity variables i should recognize that these epsilons are nothing but the X α α 6 ˜ spinor brackets ϵ i j ∼ hiji, ϵα_ α_ ∼ ½ij . Therefore our 2n1 þ n1=2 ¼ jhijþh ¼ 2n − nD: ð3:38Þ i j i claim here is exactly the amplitude-operator correspon- dence [29,41]. In this subsection, unless stated otherwise, we claim that a product of epsilons refers to the Lorentz (iii) Then we find all tuples ni ¼ðPn−1;n−1=2;n0;n1=2;n1Þ structure determined by it, and a linear combination of that satisfy Eq. (3.38) and i ni ¼ N, making sure them refers to the linear combination of the corresponding that nD satisfies the minimum given in Eq. (3.37) at Lorentz structures. It gives us a hint on how to identify ½B , the meantime. In this way, we find all the combi- the subspace of Lorentz structures with total derivatives. nations of ðni;nDÞ that could form Lorentz invariant _ First, a derivative on field Φi has a pair of indices ðαi; αiÞ, structures, each of which determines a subclass of which must also be found in the epsilons. Hence there has operators. α α to be a factor of ϵ i j ϵ˜α_ α_ in the epsilons. Therefore, a total 5 i k P At dimension eight, we list all the subclasses in Table II. α α ϵ i j ϵ˜ derivative is thus represented by a factor of i α_ iα_ k , which is the character of Lorentz structures in B . 4 ½ N ¼ 3 is a special case when there is the so-called special To identify the complement space ½A , we use a trick: by kinematics that renders n ¼ 0 or n˜ ¼ 0. Particularly it implies that μ introducing an SUðNÞ group for which Lorentz structures nD ¼ 0 when N ¼ 3. For example, we have Dμϕ1D ϕ2ϕ3 ¼ 1 2 2 2 2 ðϕ1ϕ2D ϕ3 − ϕ1D ϕ2ϕ3 − D ϕ1ϕ2ϕ3Þ which is redundant due 6 to EOM of ϕi in our treatment. Recall the r is the number of ϵ’s, which corresponds to the 5We only list classes with n ≥ n˜, while all the classes with number of spinor brackets in the on-shell amplitude, or in other n 015026-12 COMPLETE SET OF DIMENSION-EIGHT OPERATORS IN THE … PHYS. REV. D 104, 015026 (2021) M ∈ ½M transform linearly, and both ½M and ½B are into ½A and ½B . Specifically, because of Eq. (3.40),it invariant spaces, ½A must also be an invariant space that amounts to the decomposition of the tensor representations consists of whole representation spaces. This group is formed by products of the epsilons. defined by the following transformations of the epsilons: In terms of SUðNÞ YD in which a box represents X X fundamental representation, ϵ and ϵ˜ form irreducible α α i j α α † k † l ϵ i j → U U ϵ k l ; ϵ˜α_ α_ → U U ϵ˜α_ α_ : k l i j i j k l representations and , respectively, k;l k;l due to the antisymmetry of their indices. Given specific ð3:40Þ labels i, j for the epsilons, they are states in these In other words, the undotted spinor index αi with subscript i representation spaces, indicated by the Young tableau. running from 1 to N transforms as 2 × N of the SLð2; CÞ × For example, when N ¼ 5,wehave SUðNÞ group, while the dotted index α_ i transforms as 2¯ × N¯ . Obviously, the transformation does not change the tuple ðN;n;n˜Þ, which means that ½M is invariant. It is also ð3:42Þ easy to prove the invariance of ½B , X X X α α j † n α α ϵ i j ϵ˜α_ α_ → UmU ϵ i m ϵ˜α_ α_ : ð3:41Þ i k k i n E i m;n i where is the Levi-Civita tensor of SUðNÞ. Then we use the Littlewood-Richardson (LR) rules [38] Now the task is converted to finding irreducible represen- to decompose their products. First, we examine the tensor M tation spaces of SUðNÞ in ½ N;n;n˜ and classifying them power of each type of the epsilons. Since ð3:43Þ We can use the Schouten identity to eliminate representa- increasing down the columns and nondecreasing along the tions with more than two rows, either dotted or undotted, rows. Fock’s conditions [38] for such YDs are nothing but the Schouten identities. Therefore, choosing the SSYT basis automatically eliminates the redundancy from the Schouten identity. For example,7 ð3:44Þ and hence we are left with only the first term in the decom- position Eq. (3.43). Similarly if we multiply more epsilons of the same kind, we should be left with only the following YDs: ð3:46Þ 7Note that it seems as if we did not perform the row symmetrization for the Young tableau, which was done in Ref. [33]. It is due to our different treatments of the action of permutations on the SUðNÞ tensors: in Ref. [33] the action of permutation, say (12), means permuting the specific labels 1 and 2, while we treat the action of (12) as permuting the first and second indices in the tensor. While the two treatments give ð3:45Þ different sets of Lorentz structures, both of them are the independent basis of the same space. In our treatment, the Young This reflects the fact that the spinor indices take only two tableau would have indicated the column antisymmetry rather than the row symmetry of the resulting tensor. That is why we values, forbidding antisymmetry over more than two of only need to translate each column to an ϵ tensor to generate this them. The independent basis of the representation space is desired feature. The same translation of the Young tableau is used given by the SSYTs, where labels filled in the YD are in the next subsection regarding gauge group tensors. 015026-13 LI, REN, SHU, XIAO, YU, and ZHENG PHYS. REV. D 104, 015026 (2021) The two terms on the right-hand side corresponding to the SSYTs are our standard basis, and the left-hand side corresponding to the non-SSYT can be expressed by the standard basis via the Schouten identity. Finally we use the LR rule to obtain the tensor product of these two YDs. Using the LR rules we place boxes from ½n2 into the YD ½n˜ N−2 ð3:47Þ where the first term describes the case when all the boxes are put to the right of the original ½n˜ N−2 boxes and the terms in are cases when boxes are put under the original YD. Following the correspondence Eq. (3.42), whenever a box from ½n2 is placed under a column in ½n˜ N−2 , the resulting column with N − 1 rows represent a tensor, X X k1;…;k −2;l1;l2 α α k1;…;k −2;i;x α α E N ϵ˜α α ϵ i j þðantisym over k1; …;k −2;iÞ¼ E N ϵ˜α α ϵ y j ; ð3:48Þ l1 l2 N x y l1;l2 y which contains a factor of total derivative as discussed previously. According to Eq. (3.41), the whole representation space B M is contained in the subspace of ½ N;n;n˜ . It rules out all but the first term in Eq. (3.47); thus we conclude that the remaining YD, which we define as the primary YD, is the only irrep contained in ½A , namely ð3:49Þ A The primary YD ½ N;n;n˜ is exactly the space of Lorentz structures without any of the redundancies listed at the beginning of this section. Our next task is to obtain a complete basis of this space, which is again the SSYTs. Fock’s conditions between non-SSYT and the SSYT basis are equivalent to the Schouten identity or the IBP. An example of Fock’s condition reflecting the IBP is ð3:50Þ 2 where D ϕ5 is understood to be eliminated by the EOM. according to Eq. (3.38), we can easily find the ½A that As shown by Table II, ½M usually contains more than the class is included, in which the labels in it come from α α one class of operators, and so does A . The full set of its ϵ i j Ei jkϵ˜ ϵ ½ either or α_ jα_ k. The number of the former, with SSYT basis includes a lot of nonphysical fillings that index αi, equals the number of α indices in the building involve fields with large helicities (gravitino, graviton, and block i, which is ri − hi according to Eq. (3.31), while the even higher). Thus it would be wise to single out a subset of number of the latter, the same as the number of ϵ˜ without them as the Lorentz structures for a given class. By index α_ , equals n˜ − ðr þ h Þ. Together with Eq. (3.38), ˜ i i i obtaining the tuple ðN;n;nÞ from the helicities and nD we get 015026-14 COMPLETE SET OF DIMENSION-EIGHT OPERATORS IN THE … PHYS. REV. D 104, 015026 (2021) 1 X the identity relations (a)–(d) shown at the beginning of this #i ¼ n˜ − 2hi ¼ nD þ jhij − 2hi; ð3:51Þ 2 section to prove the redundancies. With our strategy, the hi>0 redundancyrelations,suchasEqs.(35)–(37) in [42],are ’ which surprisingly does not depend on ri and are hence nothing but Fock s conditions between the Young tableau, completely and uniquely determined by the class informa- which is automatically tackled by choosing the SSYT. tion ðfhig;nDÞ. Another example where a class contains several indepen- ψ 4 #1 2 #2 Our strategy is now clear: for each subclass, we find the YD dent Lorentz structures is FL , which has ¼ ; ¼ of Eq. (3.49) determined by Eq. (3.38),anduseEq.(3.51) to #3 ¼ #4 ¼ #5 ¼ 1 and n ¼ 3; n˜ ¼ 0.TheYDhasthesame deduce the tuple of labels f1#1; …;N#Ng to fill in the YD; shape as the above example, but they indicate different the SSYTs obtained this way8 correspond to the complete representation spaces due to their different ðN;n;n˜Þ.The and independent basis of Lorentz structures. As an example, SSYT basis of Lorentz structures for this class is given by consider the class of operators ψψψψ†D at dimension seven, with hi ¼f−1=2; −1=2; −1=2; 1=2g and nD ¼ 1. With Eqs. (3.38) and (3.51) we have #1 ¼ #2 ¼ #3 ¼ 2; #4 ¼ 0 and n ¼ 2; n˜ ¼ 1. The only SSYT is given by ð3:52Þ ð3:53Þ which leads to the Lorentz structure Eq. (3.39). It means that To count the number of the basis for a given class, we can Eq. (3.39) is the only independent Lorentz structure of this treat the YD of Eq. (3.49) as a product of YDs with the same class, which sounds counterintuitive. Indeed, in [42] the labels, since the latter is determined by the class information authors pointed out several redundancies of the dim-7 #i: they have to be totally symmetric one-row YD [#i]. For operators listed in [43] and found the correct independent the case of Eq. (3.52),wehave½#1 ¼½#2 ¼½#3 ¼ operator basis. One of the redundancies was about this (label 4 does not contribute), and hence we examine the particular class of operators, for which they explicitly apply decomposition of their product ð3:54Þ and find only one target YD in it, which means only one SSYT with a certain filling exists. Similarly, for the case of Eq. (3.53) we have ð3:55Þ where the multiplicity of the target YD precisely we quickly count the number of independent basis but also we can write them down by a translation from the reproduces the number of SSYT we listed in Eq. (3.53). SSYTs. This makes our approach superior to the com- In summary, by identifying Lorentz structures as states petitors and allows us to achieve a systematic way to list in the SUðNÞ representation space ½A ˜ , not only can N;n;n the operators in generic effective field theories. 8Note that in [33] the complete basis is given by the so-called 3. Permutation: Counting and listing reduced SSYTs, which eliminates the overcounting of classes the Lorentz basis while enumerating the SSYTs. But since we start from a certain class, we do not suffer from the overcounting of classes. Thus the The Lorentz structures we obtained as SSYTs in the condition of SSYT is sufficient. above subsection did not take into account the permutation 015026-15 LI, REN, SHU, XIAO, YU, and ZHENG PHYS. REV. D 104, 015026 (2021) symmetries of possible repeated fields when we specify the slightly different from that of the Lorentz structure M type. For the purpose of counting, we adopt the technique itself, which we defined in Sec. III A as λ1. There are two of plethysm. Since the repeated fields with the same sources of differences: helicities must have an equal number of labels to be filled (i) λ characterizes the permutation symmetry of labels into the YD, instead of taking a direct product of the [#i]as filled in the YD, which are indices of the combina- in Eq. (3.54), we take the plethysm with particular tion of the Lorentz structure and n˜ factors of E from permutation symmetry. In particular, for any YD Y we have the Hodge duals of the ϵ˜’s. As E is totally anti- X symmetric for any subset of labels, each E contrib- ⊗m m Y ¼ dλY ⓟ λ; ð3:56Þ utes total antisymmetry ½1 i to the ith repeated λ⊢m fields. (ii) The SSYT does not know about spin statistics; λ where dλ is the dimension of the Sm irrep . In the example hence the permutation symmetry of fermionic re- of Eqs. (3.52) and (3.54), suppose the three ψ’s are repeated peated fields has not taken into account their Grass- 3 † fields as in the type of operator Q HeCD, where the three mann feature, which should have contributed an 3 Q’s could have permutation symmetries [3], [2, 1], and ½1 , extra ½1mi . for which we derive the plethysm The property of inner product λ ⊙ ½1m ¼λT; ∀ λ⊢m then suggests that the final permutation symmetry of the Lorentz structure is given by Y Y T ð3:57Þ λ1 ¼ λΦ × λΦ; n˜ is even; fermionY bosonY T λ1 ¼ λΦ × λΦ; n˜ is odd: ð3:58Þ fermion boson These are nothing but classifications of the result in Take the example in Eq. (3.52) where the only Lorentz 2 3 Eq. (3.54). Note that d½2;1 ¼ , so the YDs in the second structure has the permutation symmetry λ ¼½1 as shown 3 † line should be counted twice while matching with in Eq. (3.57), the type of operators Q HeCD has n˜ ¼ 1, Eq. (3.54). Among the results, we find the target YD, and the repeated field Q is a fermion, which means 13 3 namely , which only appears in ½ symmetry. The λM ¼ λ ¼½1 . As for the case in Eq. (3.53), we take the λ 3 permutation symmetry obtained here, whichQ in general type WLQ L as an example which has repeated field Q, and should include all sets of repeated fields λ ¼ Φ λΦ,is we compute the plethysm ð3:59Þ It indicates that the three SSYTs obtained in Eq. (3.53) are grouped into ½3 T ¼½13 and ½2; 1 T ¼½2; 1 representation spaces. In order to construct the bases of these representation spaces as combinations of the original SSYT states Mξ, we apply λ the projectors bx introduced in Sec. III A 2 to all of the SSYTs, Mλ1 ≡ λM 1 … ξ;x bx ξ;x¼ ; ;dλ; ð3:60Þ where the difference between the symmetries λ → λ1 should be noticed. Each of the projections either forms a representation space of symmetry ½λ1 according to Eq. (3.19), 015026-16 COMPLETE SET OF DIMENSION-EIGHT OPERATORS IN THE … PHYS. REV. D 104, 015026 (2021) X π ∘ Mλ1 Mλ1 D π π ∈ ¯ ξ;x ¼ ξ;y ð Þyx; S; ð3:61Þ y or vanishes by the projection. For the example (3.53) where we got three independent Lorentz structures for the type 3 M WLQ L, which we denote as ξ¼1;2;3, respectively, we obtain 3 3 3 3 1 M½1 ≡ M½1 M½1 M½1 M M M 1 1;1 ¼ 2;1 ¼ 3;1 ¼ 3 ð 1 þ 2 þ 3Þ; 1 1 M½2;1 ≡ M½2;1 −M½2;1 M M − 2M M − 2M M x 1;x ¼ 3;x ¼ 3 ð 1 þ 2 3Þ; 3 ð 1 2 þ 3Þ ; x M½2;1 0 0 ; 2;x ¼f ; g ð3:62Þ ½13 ½2;1 ½2;1 hence we get the symmetrized Lorentz structures as M1 and Mx ;x¼ 1, 2. Note that bx acting on M1 and M3 produce the same representation space. In general, when there are multiple numbers of the same representation space ½λ1 , λ picking out linearly independent spaces from the nonvanishing projections of bx is nontrivial, which is why we use Fock’s conditions to convert the symmetrized Lorentz structures to combinations of the original basis Mξ. Generically we obtain X Mλ1 Kλ1;xM ξ;x ¼ ξζ ζ; ð3:63Þ ζ λ1;x 9 λ1 λ1 λ1;x ¯ where the coefficient matrix K has rank N . Now we can select N number of rows from K as ξ ¼ ξ1; …; ξN λ1 λ Mλ1 which provide an independent set of the ½ 1 -symmetry Lorentz basis as ξ¯;x. In the above example, we have 0 1 0 1 0 1 1 1 1 1 1 − 2 1 − 2 1 B 3 3 3 C B 3 3 3 C B 3 3 3 C ½13 ;1 B 1 1 1 C ½2;1 ;1 B 000C ½2;1 ;2 B 000C K ¼ @ 3 3 3 A; K ¼ @ A; K ¼ @ A; ð3:64Þ 1 1 1 − 1 − 1 2 − 1 2 − 1 3 3 3 3 3 3 3 3 3 which have ranks N ½3 ¼ N ½2;1 ¼ 1. In that there are no invariants using Levi-Civita tensors. Here we postulate a way multiplicities of the representation spaces, we are allowed to express all Tξ in terms of Levi-Civita tensors of the SUðNÞ to omit the subscript ξ¯ as in Eq. (3.62). group provided that each field is expressed in a tensor of fundamental indices only. The algorithm is to use the LR rule repeatedly but with indices associated with the correspond- C. Gauge basis: Littlewood-Richardson rule ing irreps filled in during the construction of a singlet YD. Mλ After obtaining the symmetrized Lorentz structures x, From this procedure, one can obtain different singlet Young we are now ready to find a set of symmetrized gauge group tableaux with N rows as different ways to construct a SUðNÞ λ λ factors TSU3;x and TSU2;x in Eq. (2.8): the procedure is similar singlet. Each Young tableau then translates into a Tξ as a to finding the symmetrized Lorentz structures discussed product of ϵ tensors with the indices setting to the corre- above. We shall find all the independent group factors Tξ sponding indices in each column in a consistent manner. We λ 2 first, and then symmetrize them by applying b ’s discussed in illustrate the procedure by constructing the SUð ÞW group x 3 2 Sec. III A 2 to the gauge group indices of the repeated fields: factor of the operator Q LWL. Suppose the SUð ÞW indices for three Q’sandL are j, k, l, and i, respectively, while that λ λ ∘ Tξ;x ¼ bx Tξ: ð3:65Þ for WL is I. The first step is to convert all the nonfundamental indices into fundamental ones. The only field that needs this In principle, one can obtain all the independent Tξ by I preprocessing in our case is WL, and we convert it by recursively using Clebsch-Gordan coefficients (CGCs) of the τI x ϵ contracting with ð Þm1 xm2 , which leads to corresponding gauge group; however, this method cannot I τI x ϵ give nice forms of group factors expressed in terms of WL;m1m2 ¼ WLð Þm1 xm2 ; ð3:66Þ where the summation over the repeated indices is implied. 9Actually, the linear dependence among the rows of the matrix λ1;x λ Next, we are going to form the Young tableaux with indices K should not depend on x, just as the projector bx either projects out a full representation space, nonvanishing for all x,or j; k; l; i; m1;m2 according to the LR rule. There are three ’ annihilate a Lorentz structure, vanishing for all x. Therefore, the different TSU2;ξ s which correspond to three different paths to λ rank N 1 is also independent of x. construct 3 × 2 YDs. We illustrate them in the following: 015026-17 LI, REN, SHU, XIAO, YU, and ZHENG PHYS. REV. D 104, 015026 (2021) ð3:67Þ ð3:70Þ where the first line tells the order of the fields in forming the ð3:68Þ singlet YDs. We follow the above paths to fill each box with the corresponding indices of the field and translate them into products of ϵ’s: ð3:69Þ ð3:71Þ ð3:72Þ ð3:73Þ ð3:74Þ With this set of TSU2;ξ, we can project out the corres- λ λ ð3:75Þ ponding TSU2;x by using the symmetrizers bx. To find out which λ the three Q’s can take, we first need to enumerate 2 ’ all the SUð ÞW irreps constructed by Q s that can form a singlet with the rest of the fields L and W. In this example ð3:76Þ both the quadruplet and the doublet are capable. Next, one can pick out the λ that after taking plethysm with the 2 ’ SUð ÞW irrep of Q s are able to produce the quadruplet and From the above equation, we find that [3] and [2,1] are the doublet: possible choices, and we have 3 3 1 ½ ½ ji km1 lm2 T b ∘ T 2 1 ϵ ϵ ϵ i; j; k SU2;1 ¼ 1 SU ; ¼ 6 ½ þðperm Þ 1 − ¼ TSU2;1 3 ðTSU2;2 þ TSU2;3Þ; ð3:77Þ 2 1 2 1 1 ½ ; ½ ; jm1 ki lm2 ji km1 lm2 jm2 km1 li jm1 km2 li T b ∘ T 2 1 ϵ ϵ ϵ ϵ ϵ ϵ − ϵ ϵ ϵ − ϵ ϵ ϵ SU2;1 ¼ 1 SU ; ¼ 3 ½ þ 2 1 − ¼ 3 TSU2;2 3 TSU2;3; ð3:78Þ 015026-18 COMPLETE SET OF DIMENSION-EIGHT OPERATORS IN THE … PHYS. REV. D 104, 015026 (2021) 2 1 2 1 1 ½ ; ½ ; jm1 km2 li ji km2 lm1 jm2 ki lm1 jm1 ki lm2 T b ∘ T 2 1 ϵ ϵ ϵ ϵ ϵ ϵ − ϵ ϵ ϵ − ϵ ϵ ϵ SU2;2 ¼ 2 SU ; ¼ 3 ½ þ 1 2 − ¼ 3 TSU2;2 þ 3 TSU2;3: ð3:79Þ From the first to the second lines in the above equations, an example to demonstrate the procedure of the inner we have used the Schouten identity and the fact that any product decomposition of a single symmetric group S3, and ϵm1m2 terms proportional to can be dropped as WL;m1m2 is a the generalization to arbitrary sets of repeated fields will be symmetric tensor. In addition, one can verify that the manifest. ½3 projection of b1 on T 2 2 or T 2 3 gives a null space We use the projection operator defined in Theorem 4.2 in SU ; SU ; λ λ λ ½2;1 ð 1;x1Þ;ð 2;x2Þ;ð 3;x3Þ while that of bx ’s generates the same space as the one we Ref. [39] to obtain the generalized CGCs Cðλ;xÞ;j generate above from Eq. (3.78) to Eq. (3.79). of the symmetric group with the definition 3 Readers can follow this method to derive the SUð ÞC group factor for this type of operator, which is quite trivial X ðλ1;x1Þ;ðλ2;x2Þ;ðλ3;x3Þ λ λ λ3 3 Θ M 1 ⊗ 2 ⊗ 1 λ ¼ C x1 T T ; yielding T½ ¼ ϵabc given that the indices of three Q’s ð ;xÞ;j ðλ;xÞ;j SU3;x2 SU2;x3 SU3;1 x1;x2;x3 are a, b, c. It is obvious that this group factor is in the ½13 ð3:80Þ representation of S3. The above construction can be generalized to operator Θ types with more than one set of repeated fields. The where ðλ;xÞ;j is the xth basis vector in the jth (label of projection operations for different sets of repeated fields multiplicity) irrep λ from the decomposition, which is simply commute with each other. Therefore one can essentially a linear combination of various factorizable obtain a set of symmetrized group factors transforming terms defined in Eq. (2.8). The details of using a projection ¯ as irreps of the direct product symmetric group S defined in operator to extract CGCs are given in Appendix B2, and Sec. III A. here we directly provide the relevant CGCs of S3 for our 3 example Q LWL. As we have obtained in the above two D. Flavor basis: Inner product decomposition sections, the permutation symmetries of the Lorentz struc- 13 2 The above two subsections describe the systematic ways ture can be ½ or [2, 1], those of the SUð ÞW group factor 3 3 to generate the Lorentz structures and the group factors as can be ½ or [2, 1], while the SUð ÞC group factor only ¯ 3 irreps of S. Now we are at the stage to show how to use takes ½1 . Therefore there are four possibilities to form these ingredients to construct operators with certain flavor direct product representations, of which the inner product 3 permutation symmetry. Still, we shall take the Q LWL as decompositions are ð3:81Þ One can observe that the first three combinations of the detail for the last one in the following. The relevant CGCs permutation symmetries are trivial as the decomposition for the last decomposition are summarized in Table III. only results in a single irrep of S3, so we only show the Since in our case, the multiplicities of each irrep is 1, the 015026-19 LI, REN, SHU, XIAO, YU, and ZHENG PHYS. REV. D 104, 015026 (2021) last indices of the subscripts of C are all 1. Also as the first one in our convention is that it is equal to the discussed in Ref. [18], for irreps with dimension larger Young symmetrizer of the normal Young tableau of the than 1, we only need to choose one of the basis vectors corresponding YD discussed in Sec. III A 2, which helps us from the decomposed invariant space as others generate simplify our forms of operators in Sec. IV. We shall come the same flavor space. Here we always select the first back to this point later in Sec. IVA. vector in our basis, and this is why the third subscript Therefore we obtain three terms from the last line indices of C is always 1. In principle it is equivalent to of Eq. (3.81): select any one of the basis vectors; the reason we choose 2 13 2 1 2 1 2 1 2 1 1 13 2 1 2 1 2 1 2 1 Θprst ¼ T½ ðT½ ; M½ ; þ T½ ; M½ ; Þþ T½ ðT½ ; M½ ; þ T½ ; M½ ; Þ; ð3:82Þ ð½3 ;1Þ;1 3 SU3;1 SU2;1 1 SU2;2 2 3 SU3;1 SU2;1 2 SU2;2 1 1 13 2 1 2 1 2 1 2 1 2 1 2 1 Θprst ¼ T½ ðT½ ; M½ ; þ T½ ; M½ ; þ T½ ; M½ ; Þ; ð3:83Þ ð½2;1 ;1Þ;1 3 SU3;1 SU2;1 1 SU2;2 2 SU2;2 1 prst 1 ½13 ½2;1 ½2;1 ½2;1 ½2;1 Θ 3 ¼ T ðT M − T M Þ; ð3:84Þ ð½1 ;1Þ;1 2 SU3;1 SU2;2 1 SU2;1 2 where r, s, t, and p are the flavor indices of Q’s and L, respectively. As each factor is rather lengthy, we only show the full prst expression of Θ 3 here: ð½1 ;1Þ;1 i prst abc I i I jk lm1 jl km1 μν μν Θ 3 ¼ ϵ ðτ Þ W fð2ϵ ϵ − ϵ ϵ Þ½ðL σ Q ÞðQ Q Þ − ðL σ Q ÞðQ Q Þ ð½1 ;1Þ;1 12 m1 Lμν pi sbk raj tcl pi raj skb tcl jl km1 jk lm1 μν μν −ð2ϵ ϵ − ϵ ϵ Þ½ðLpiσ QsbkÞðQrajQtclÞþ2ðLpiσ QrajÞðQskbQtclÞ g i ¼ − ϵabcðτIÞi WI ½ϵjm1 ϵklðL σμνQ ÞðQ Q Þþϵjlϵkm1 ðL σμνQ ÞðQ Q Þ ; ð3:85Þ 4 m1 Lμν pi sbk raj tcl pi raj skb tcl where the Schouten identity has been used in the last line. So far, we have demonstrated the whole process to obtain prst 3 Θ 3 a term with a concrete example Q LW . We summarize our One can verify that ð½1 ;1Þ;1 is indeed totally antisymmetric L about indices r, s, t as it should be. algorithm to find a complete set of independent terms for a given dimension in a flow chart in Fig. 1 and realize automated treatment in a Mathematica code. TABLE III. The relevant CGCs of S3 inner product decom- position. Given a dimension, one can enumerate the operator classes that determine the number of fields of each spin and the Flavor sym Relevant CGCs number of the derivative. Further, by finding the correspond- ð½2;1 ;1Þ;ð½13 ;1Þ;ð½2;1 ;1Þ 2 ing SSYTs, one can obtain the Lorentz structure candidates C ¼ ð½3 ;1;1Þ 3 without EOM and IBP redundancy. All of these above the ð½2;1 ;1Þ;ð½13 ;1Þ;ð½2;1 ;2Þ 1 Cð½3 ;1Þ;1 ¼ 3 first dash-dotted line in the figure are model independent, ð½2;1 ;2Þ;ð½13 ;1Þ;ð½2;1 ;1Þ 1 which can be applied to any Lorentz invariant EFTs. Cð½3 ;1Þ;1 ¼ 3 2 1 2 13 1 2 1 2 After specifying the UV model, one can determine the Cð½ ; ; Þ;ð½ ; Þ;ð½ ; ; Þ ¼ 2 ð½3 ;1Þ;1 3 types of operators for each class and, indeed, determine the 3 ð½2;1 ;1Þ;ð½1 ;1Þ;ð½2;1 ;1Þ 1 independent Lorentz and gauge structures Mξ’s and Tξ’s. Cð½2;1 ;1Þ ¼ 3 2 1 1 13 1 2 1 2 Afterward, taking into account the information of repeated Cð½ ; ; Þ;ð½ ; Þ;ð½ ; ; Þ ¼ 1 ð½2;1 ;1Þ;1 3 fields from the specific type, one can symmetrize the Mξ’s ð½2;1 ;2Þ;ð½13 ;1Þ;ð½2;1 ;1Þ 1 ’ Cð½2;1 ;1Þ;1 ¼ 3 and Tξ s to obtain a set of Lorentz and gauge group basis 2 1 2 13 1 2 1 2 ¯ ð½ ; ; Þ;ð½ ; Þ;ð½ ; ; Þ 0 that transform as irreps of S. Finally, by putting these Cð½2;1 ;1Þ;1 ¼ ingredients together to form the Lorentz and gauge singlets ð½2;1 ;1Þ;ð½13 ;1Þ;ð½2;1 ;1Þ ¯ C 3 0 ð½1 ;1Þ ¼ that transform as direct product representations of S, and ð½2;1 ;1Þ;ð½13 ;1Þ;ð½2;1 ;2Þ 1 using inner product decomposition to decompose them C 3 ¼ ð½1 ;1Þ;1 2 ¯ ð½2;1 ;2Þ;ð½13 ;1Þ;ð½2;1 ;1Þ 1 back into the irreps of S, one obtains several irrep spaces, C 3 ¼ − ð½1 ;1Þ;1 2 each corresponding to an independent term with a definite ð½2;1 ;2Þ;ð½13 ;1Þ;ð½2;1 ;2Þ C 3 0 flavor permutation symmetry. The symmetrization and the ð½1 ;1Þ;1 ¼ inner product decomposition below the second dash-dotted 015026-20 COMPLETE SET OF DIMENSION-EIGHT OPERATORS IN THE … PHYS. REV. D 104, 015026 (2021) FIG. 1. Flow chart for finding all the independent terms at a given dimension. The content above the first dash-dotted line is model independent and can be applied to any EFT. The content below the second dash-dotted line is our main contribution to this work. We automatize the whole procedure in a Mathematica code. line are our unique contributions that are not present in the reduced to products of gμν, σμν, and ϵμνρη. Relevant literature yet. formulas are listed in Appendix A1. (ii) We are using two-component spinors for all the IV. LISTS OF OPERATOR BASIS fermion fields as they are the most natural way to deal with chiral fermions. Conversion rules to four- A. Preview of the result component spinor notation are provided in Appen- In this section, we list all the dimension-eight terms of dix A1. Because of the way we deal with the Fierz operators, grouped by the classes of Lorentz structures. In identity Eq. (3.25), the Lorentz structures we exhibit Table IV we show the statistics of all the SMEFT do not contain any vector, axial, or tensor couplings dimension-eight results organized by subclasses, with links for four-fermion interactions. Readers could use referring to the corresponding lists of operators in the Fierz identities also presented in Appendix A1 to following subsections. The subclasses with nontrivial convert the operators to any forms they like. Exam- ples are also provided beside the lists in Sec. IV D. polynomials of nf, the fermion flavor number, as the total (iii) We also convert the chiral basis of gauge bosons number of operators are those for which we need to take ˜ care of the repeated field issues. The statistics for the B FL=R to the Hermitian fields F; F by using the violation (ΔB ¼ 1) are listed with underlines, while the formula in Appendix A2. After this is done, some lepton number violation is not shown because B − L is of the types, even from different subclasses, merge into one. conserved at dimension eight. (iv) The following common notations are adopted to For readers’ convenience, we further perform several reduce some of our terms: notation changes and simplifications on the basis of the terms directly produced by our algorithm: ↔μ μ − μ ≡ μ ≡ □ (i) In the Lorentz structures, we convert the derivatives XD Y D XY XD Y; DμD ; and the gauge bosons to the form with Lorentz †i † †i I j † I H Hi ≡ ðH HÞ;Hðτ Þ Hj ≡ ðH τ HÞ; indices μ; ν; ρ; …. This is done by grouping the i μν F1¼F2¼F 2 spinor contractions into chains that start and end at F1μνF2 ≡ ðF1F2Þ ≡ F : ð4:1Þ fermions, and traces that start and end at the same F or D. On the one hand, we reduce the σ products in (v) The most subtle simplification is trying to super- the chains to the three basic bilinear forms ψχ, ficially reduce the length of terms10 in order to ψσμχ†, and ψσμνχ and their conjugates, where all ψ χ spinor indices are suppressed and , are both left- 10In this paragraph, term without quotes only indicates a handed Weyl spinors as in our convention for the monomial in a polynomial expression rather than a level of fermion fields. On the other hand, all the traces are operators in our construction. 015026-21 LI, REN, SHU, XIAO, YU, and ZHENG PHYS. REV. D 104, 015026 (2021) TABLE IV. Complete statistics of dimension-eight operators in the SMEFT, while the numbers with underlines are for the B-violating operators. N in the leftmost column shows the number of particles. ðn; n˜Þ are the numbers of ϵ and ϵ˜ in the Lorentz structure, which determines the primary YD ½A the subclasses belong to. Note that our definition of “term” is different from the other literature, and the numbers are larger than those in, for instance, [18] because they did an extra step of merging before the counting. However, the number of operators is exactly the same as in [15,18]. The links in the rightmost column refer to the list(s) of the terms in given subclasses. N N N type; term, and operator show the number of types, terms, and Hermitian operators, respectively (independent conjugates are counted). ˜ N ðn; nÞ Subclasses N type N term N operator Equations 4 4 (4,0) FL þ H:c: 14 26 26 (4.19) 2 ψψ† 22 2 (3,1) FL D þ H:c: 22 22 nf (4.50) ψ 4 2 4 41814 12 4 3 5 − 1 D þ H:c: þ þ nfþnfð nf Þ (4.74), (4.77), (4.79) ψ 2ϕ 2 32 2 FL D þ H:c: 16 32 nf (4.43) 2 ϕ2 2 FL D þ H:c: 812 12 (4.14) 2 2 (2,2) FLFR 14 17 17 (4.19) ψψ† 35 2 FLFR D 27 35 nf (4.49), (4.50) ψ 2ψ †2 2 17 4548 1 2 75 2 11 6 4 – D þ þ 2 nfð nf þ Þþ nf (4.73), (4.78) (4.80) ψ 2ϕ 2 16 2 FR D þ H:c: 16 16 nf (4.43) ϕ2 2 FLFR D 56 6 (4.14) ψψ†ϕ2 3 16 2 D 716 nf (4.31), (4.32) ϕ4D4 13 3 (4.8) ψ 4 12 10 66 54 42 4 2 3 9 1 5 (3,0) FL þ H:c: þ þ nfþ nfð nf þ Þ (4.85), (4.87), (4.88), (4.90) 2 ψ 2ϕ 60 2 FL þ H:c: 32 60 nf (4.46), (4.47) 3 ϕ2 FL þ H:c: 66 6 (4.16) ψ 2ψ †2 84 24 172 32 2 2 59 2 − 2 24 4 – – (2,1) FL þ H:c: þ þ nfð nf Þþ nf (4.83) (4.84), (4.87) (4.91) 2 ψ 2ϕ 36 2 FR þ H:c: 32 36 nf (4.46), (4.47) ψ 3ψ †ϕ 32 14 180 56 3 135 − 1 3 29 3 – D þ H:c: þ þ nfð nf Þþnfð nf þ Þ (4.65), (4.68) (4.71) ψψ†ϕ2 92 2 FL D þ H:c: 38 92 nf (4.39), (4.40) ψ 2ϕ3 2 36 2 D þ H:c: 636 nf (4.28) ϕ4 2 FL D þ H:c: 46 6 (4.10) ψ 4ϕ2 12 44818 5 5 4 2 2 8 4 2 6 (2,0) þ H:c: þ þ ð nf þ nfÞþ 3 ð nf þ nfÞ (4.54), (4.58), (4.61), (4.63) ψ 2ϕ3 22 2 FL þ H:c: 16 22 nf (4.36) 2 ϕ4 FL þ H:c: 810 10 (4.12) ψ 2ψ †2ϕ2 23 10 57 14 2 42 2 2 3 3 3 − 1 – (1,1) þ þ nfð nf þ nf þ Þþ nfð nf Þ (4.53), (4.54), (4.58) (4.62) ψψ†ϕ4 13 2 D 713 nf (4.24), (4.25) ϕ6D2 12 2 (4.8) ψ 2ϕ5 6 2 7 (1, 0) þ H:c: 66 nf (4.21) 8 (0,0) ϕ8 11 1 (4.8) Total 48 471þ70 1070þ196 993ðnf ¼ 1Þ; 44807ðnf ¼ 3Þ present them in the paper better. Take Eq. (3.85) as guess that by performing total antisymmetrization an example, where two terms exist after expansion. on one of the terms over r, s, t should reproduce Θprst The two terms together guarantee the total antisym- 13 1 1, such as prst ð½ ; Þ; metry of the Q flavors r, s, t in Θ 3 . It is fair to ð½1 ;1Þ;1 ð4:2Þ 015026-22 COMPLETE SET OF DIMENSION-EIGHT OPERATORS IN THE … PHYS. REV. D 104, 015026 (2021) ½13 where the Young symmetrizer specified by the Young tableaux , equivalent to the projector b1 as explained in Sec. III A 2, acts on the flavor indices r, s, t, so that the permutation symmetry over these indices are guaranteed by the property of the projector Eq. (3.16) to Eq. (3.19). An example of the nontrivial symmetrizer for the mixed symmetry is ð4:3Þ where … represents the possible presence of flavor indices of other sets of repeated fields. For terms involving more than one set of repeated fields, the symmetrizer is specified by several Young tableaux, for example, ð4:4Þ Back to the example of Eq. (3.85), where only one ½13 of different types as a complex type, and a self- 3 operator exists for the type WLQ L, there is no doubt that conjugate type as a real type. Since we do not Eq. (4.2) can reproduce it up to an overall constant factor. In present conjugates of the complex types, operators this way, we reduce the length of our terms with the definite of these types should be counted twice in the sense flavor symmetry and hide the complexity into the corre- of Hermitian degrees of freedom. For the real types, sponding Young symmetrizer leaving rather simple forms although the operators presented may not be Her- exhibited in the following sections. The case becomes more mitian on their own, their conjugates must be complicated when a certain type containing more than one combinations of operators in the same type and copy of irrep of the same flavor symmetry λ. In these cases a should not be counted separately, so these operators dedicated “desymmetrization” procedure [44] is performed are only counted as 1 Hermitian degree of freedom. to find out the monomials such that after acting on the The numbers presented in Table IVare all counted in corresponding Young symmetrizer they are symmetrized to this manner. We have also listed the B-violating independent terms. Moreover, if several irreducible flavor operators separately in Sec. IV D. tensors can be written as different Young symmetrizers acting on the same term, these tensors can be merged into B. Classes involving bosons only one tensor with reducible flavor symmetry, which is how term was used in [18]. The Q3L operator at dimension six is In the following sections, we list our operators in terms one of such examples, which is why there is only one term of subclasses, ordered by the number of fermions and for it in the Warsaw basis [3]. However, the principle for the gauge bosons. The subclasses shown here are summarized merging does not exist so far, the number of such a term is in Table II, while those not showing up are redundant an ambiguous quantity as discussed in [18]. We emphasize according to our treatments of various redundancy relations listed at the beginning of Sec. III B. that our term defined in Sec. II B does not have such 8− ϕ nD nD ambiguity, and we prefer not to do the merging but instead Class D : Operators with only scalars. The subclasses available in Table II are nD ¼ 0, 2, 4. The shorten our notation with the trick of the Young symmetr- 8 izer mentioned above. Lorentz structure of the all-scalar subclass ϕ is trivial, (vi) Finally, instead of listing subclasses sorted by the shown as tuple ðN;n;n˜Þ, we list chirality-blind classes sorted by the number of fermions to fit the needs of ϕ1ϕ2ϕ3ϕ4ϕ5ϕ6ϕ7ϕ8: ð4:5Þ phenomenologists. Within a class, operators are listed as either “complex” types or “real” types. For the subclass ϕ6D2, all the Lorentz structures are given We refer to a type of operators whose conjugates are by the algorithm in Sec. III B 2 as follows: α α_ α α_ α α_ ϕ1ðDϕ2Þα_ ϕ3ϕ5ðDϕ4Þαϕ6; ϕ1ðDϕ2Þα_ ϕ3ϕ4ðDϕ5Þαϕ6; ϕ1ðDϕ2Þα_ ϕ3ϕ4ϕ5ðDϕ6Þα; α α_ α α_ α α_ ϕ1ϕ2ðDϕ3Þα_ ðDϕ4Þαϕ5ϕ6; ϕ1ϕ2ðDϕ3Þα_ ϕ4ðDϕ5Þαϕ6; ϕ1ϕ2ðDϕ3Þα_ ϕ4ϕ5ðDϕ6Þα; α α_ α α_ α α_ ϕ1ϕ2ϕ3ðDϕ4Þα_ ðDϕ5Þαϕ6; ϕ1ϕ2ϕ3ðDϕ4Þα_ ϕ5ðDϕ6Þα; ϕ1ϕ2ϕ3ϕ4ðDϕ5Þα_ ðDϕ6Þα; ð4:6Þ while those for the subclass ϕ4D4 are given as 015026-23 LI, REN, SHU, XIAO, YU, and ZHENG PHYS. REV. D 104, 015026 (2021) _ _ _ ϕ 2ϕ αβ ϕ 2ϕ α_ β ϕ ϕ α ϕ β 2ϕ α_ β ϕ ϕ 2ϕ αβ 2ϕ α_ β 1ðD 2Þα_ β_ 3ðD 4Þαβ ; 1ðD 2Þα_ ðD 3Þβ_ ðD 4Þαβ ; 1 2ðD 3Þα_ β_ ðD 4Þαβ : ð4:7Þ Plugging fields from Table I to these Lorentz structures, making sure that the total hypercharge is zero, we get the following three types of operators, and by going through our algorithm, we obtain six terms in all as a nonredundant basis: † 4 O 4 †4 H H ðH HÞ ; Oð1;2Þ † 2□ † † † 2 H3H†3D2 ðH HÞ ðH HÞ; ðH HÞjH DμHj ; Oð1∼3Þ † □2 † † 2 † □ † μ H2H†2D4 ðH HÞ ðH HÞ; jH DμDνHj ; ðH DμHÞ ðH D HÞ: ð4:8Þ The superscripts of the O’s label the terms in the particular type, in the order of left to right and top to bottom. The first operator modifies the shape of the Higgs potential, and the rest could renormalize the Higgs field and thus modify the Higgs couplings uniformly. 6− Class Fϕ nD DnD : Operators with one gauge boson and arbitrary scalars. According to Table II, only one subclass ϕ4 2 FL D survives our criteria, which contains the following three independent Lorentz structures: αβϕ ϕ ϕ α_ ϕ αβϕ ϕ ϕ ϕ α_ αβϕ ϕ ϕ ϕ α_ FL1 2ðD 3Þαα_ ðD 4Þβ 5;FL1 2ðD 3Þαα_ 4ðD 5Þβ;FL1 2 3ðD 4Þαα_ ðD 5Þβ: ð4:9Þ Together with their Hermitian conjugates, they combine into the form with F; F˜ , that become real in this notation. In the SMEFT, we have the following real types: Iμν † †τI ˜ Iμν † †τI ð1∼4Þ W ðH HÞðDμH DνHÞ; W ðH HÞðDμH DνHÞ; O 2 †2 2 WH H D Iμν † † I ˜ Iμν † † I W ðDμH DνHÞðH τ HÞ; W ðDμH DνHÞðH τ HÞ; Oð1;2Þ μν † † ˜ μν † † BH2H†2D2 jB ðH HÞðDμH DνHÞ; B ðH HÞðDμH DνHÞ: ð4:10Þ 2 4− Class F ϕ nD DnD : Operators with two gauge bosons and arbitrary scalars. Table II contains two subclasses of this form, 2 4 with nD ¼ 0, 2. The only Lorentz structure in the subclass F ϕ is αβ ϕ ϕ ϕ ϕ FL1 FL2αβ 3 4 5 6: ð4:11Þ In the SMEFT we get the following types under this subclass: ð1;2Þ 2 † 2 A ˜ Aμν † 2 O 2 2 †2 jG ðH HÞ ; ðGμνG ÞðH HÞ ; G H H I Jμν †τI †τJ 2 † 2 ð1∼4Þ W μνW ðH HÞðH HÞ;WðH HÞ ; O 2 2 †2 W H H I ˜ Jμν † I † J I ˜ I † 2 W μνW ðH τ HÞðH τ HÞ; ðW W ÞðH HÞ ; Oð1;2Þ 2 † 2 ˜ † 2 B2H2H†2 jB ðH HÞ ; ðBBÞðH HÞ ; Oð1;2Þ Iμν †τI † ˜ Iμν †τI † BWH2H†2 jBμνW ðH HÞðH HÞ;BμνW ðH HÞðH HÞ: ð4:12Þ When all the Higgs bosons are put to their vacuum expectation values (VEVs), the operators normalize the kinetic terms of gauge bosons and thus modify the corresponding gauge couplings uniformly. On the other hand, there are three independent Lorentz structures in the subclass F2ϕ2D2: _ αβ γ ϕ ϕ α_ αβ ϕ γ ϕ α_ αβϕ 2ϕ α_ β FL1 FL2α ðD 3Þβα_ ðD 4Þγ ;FL1 FL2αβðD 3Þα_ ðD 4Þγ ;FL1 2ðD 3Þαβα_ β_ FR4 : ð4:13Þ Again, combined with their Hermitian conjugates, we obtain the following real types: 015026-24 COMPLETE SET OF DIMENSION-EIGHT OPERATORS IN THE … PHYS. REV. D 104, 015026 (2021) ð1∼3Þ 2 μ † A ˜ A μ † Aμ Aνλ † O 2 † 2 jG ðD H DμHÞ; ðG G ÞðD H DμHÞ;GλG ðDμH DνHÞ; G HH D 2 μ † IJK I μ Jνλ † K W ðD H DμHÞ;iϵ W λ W ðDμH τ DνHÞ; ð1∼6Þ I ˜ I μ † IJK I ½μ ˜ Jν λ † K O 2 † 2 ðW W ÞðD H DμHÞ;iϵ W λW ðDμH τ DνHÞ; W HH D Iμ Iνλ † IJK I ðμ ˜ JνÞλ † K W λW ðDμH DνHÞ;iϵ W λW ðDμH τ DνHÞ; ð1∼3Þ 2 μ † ˜ μ † μ νλ † O 2 † 2 jB ðD H DμHÞ; ðBBÞðD H DμHÞ;Bλ B ðDμH DνHÞ; B HH D I μ † I ˜ I μ † I ðBW ÞðD H τ DμHÞ; ðBW ÞðD H τ DμHÞ; ð1∼6Þ ½μ Iν λ † I ½μ ˜ Iν λ † I O † 2 iB λW ðDνH τ DμHÞ;iBλW ðDνH τ DμHÞ; ð4:14Þ BWHH D ðμ IνÞλ † I ðμ ˜ IνÞλ † I B λW ðDνH τ DμHÞ;BλW ðDνH τ DμHÞ; ½μ ν λ μ νλ ν μλ where brackets for the indices are shorthand notations for (anti)symmetrization F1 λF2 ≡ F1λF2 − F1λF2 and ðμ νÞλ μ νλ ν μλ F1 λF2 ≡ F1λF2 þ F1λF2 . The operators of these types contribute to the neutral triple gauge boson couplings, which do not appear at lower dimensions [22]. Class F3ϕ2: Operators with triple gauge bosons. Note that the operators of class F3D2 are absent due to our treatment about EOM. The only Lorentz structure in the subclass F3ϕ2 is αβ γ ϕ ϕ FL1 FL2α FL3βγ 4 5: ð4:15Þ Note that the types B3HH†;BG2HH†;B2WHH†, and G2WHH† cannot exist, even though they are able to form Lorentz invariant gauge singlets. The reason is that the only Lorentz structure shown above is totally antisymmetric for the three gauge bosons. In case no antisymmetric structures from the gauge group sectors, such as the structure constants, are available, the operators must vanish due to the commuting nature of any repeated gauge bosons in it. The nonvanishing types, which all involve totally antisymmetric structure constants, are shown below Oð1;2Þ ABC A Bμ Cνλ † ABC A B μ ˜ Cνλ † G3HH† jf G μνG λG H H; f G μνG λG H H; Oð1;2Þ ϵIJK I J μ Kνλ † ϵIJK I Jμ ˜ Kνλ † W3HH† j W μνW λW H H; W μνW λW H H; Oð1;2Þ ϵIJK Iμ Jνλ †τK ϵIJK Iμ ˜ Jνλ †τK BW2HH† j BμνW λ W H H; BμνW λW H H: ð4:16Þ These operators contribute to the anomalous triple gauge boson couplings. 4 2 2 Class F : Operators with four gauge bosons. There is one Lorentz structure of subclass FLFR and three Lorentz 4 structures of subclass FL, αβ α_ β_ FL1 FL2αβFR3α_ β_ FR4 ; ð4:17Þ αβ γδ αβ γ δ αβ γδ FL1 FL2 FL3αβFL4γδ;FL1 FL2α FL3β FL4γδ;FL1 FL2αβFL3 FL4γδ: ð4:18Þ After symmetrization described in Sec. III B 3, we find no Lorentz structure that is antisymmetric over the gauge bosons, 3 2 which implies that the type BW whose SUð ÞW structure has to be totally antisymmetric must vanish. The nonvanishing types are given below: GAGB GAGB ;dACEdBDE GAGB GCGD ;fACEfBDE GAGB GCGD ; ð Þð Þ ð Þð Þ ð Þð Þ ð1∼9Þ A B A ˜ B ACE BDE A B C ˜ D ACE BDE A B C ˜ D O 4 ðG G ÞðG G Þ;d d ðG G ÞðG G Þ;f f ðG G ÞðG G Þ; G ðGAG˜ BÞðGAG˜ BÞ;dACEdBDEðGAG˜ BÞðGCG˜ DÞ;fACEfBDEðGAG˜ BÞðGCG˜ DÞ; I I J J I I J ˜ J I ˜ I J ˜ J ð1∼6Þ ðW W ÞðW W Þ; ðW W ÞðW W Þ; ðW W ÞðW W Þ; O 4 W ðWIWJÞðWIWJÞ; ðWIWJÞðWIW˜ JÞ; ðWIW˜ JÞðWIW˜ JÞ; 015026-25 LI, REN, SHU, XIAO, YU, and ZHENG PHYS. REV. D 104, 015026 (2021) ð1∼3Þ 2 2 2 ˜ ˜ ˜ O 4 jðB ÞðB Þ; ðB ÞðBBÞ; ðBBÞðBBÞ; B 2 2 2 I ˜ I A ˜ A 2 A ˜ A I ˜ I ð1∼7Þ G W ;GðW W Þ; ðG G ÞW ; ðG G ÞðW W Þ; O 2 2 G W ðGAWIÞðGAWIÞ; ðGAWIÞðGAW˜ IÞ; ðGAW˜ IÞðGAW˜ IÞ; 2 2 2 ˜ A ˜ A 2 A ˜ A ˜ ð1∼7Þ G B ;GðBBÞ; ðG G ÞB ; ðG G ÞðBBÞ; O 2 2 G B ðGABÞðGABÞ; ðGABÞðGAB˜ Þ; ðGAB˜ ÞðGAB˜ Þ; 2 2 2 ˜ I ˜ I 2 I ˜ I ˜ ð1∼7Þ B W ;WðBBÞ; ðW W ÞB ; ðW W ÞðBBÞ; O 2 2 W B ðWIBÞðWIBÞ; ðWIBÞðWIB˜ Þ; ðWIB˜ ÞðWIB˜ Þ; ABC A B C ABC A B ˜ C ð1∼4Þ d ðBG ÞðG G Þ;d ðBG ÞðG G Þ; O 3 ð4:19Þ BG dABCðBG˜ AÞðGBGCÞ;dABCðBG˜ AÞðGBG˜ CÞ: C. Classes involving two fermions 1. No gauge boson involved 2 5−nD nD In this subsection we deal with the classes ψ ϕ D . Note from Eq. (2) that for odd nD we have fermions of opposite helicities, or chirality conserving, and for even nD we have them with the same helicities or chirality violating. Class ψ 2ϕ5: The only Lorentz structure of this subclass is α ψ 1 ψ 2αϕ3ϕ4ϕ5ϕ6ϕ7: ð4:20Þ In the SMEFT, these are Yukawa terms with additional Higgses, which are all complex types: † 2 O 3 †2 ϵil a QuCH H j ðQpaiuC r ÞHlðH HÞ ; † † 2 O 2 †3 a i QdCH H jðdC pQraiÞH ðH HÞ ; † † 2 O 2 †3 i eCLH H jðeCpLriÞH ðH HÞ : ð4:21Þ After taking the Higgs VEV, they give rise to additional contributions to the SM fermion Yukawa couplings. According to Appendix A1, the relevant bilinear of two-component spinors can be converted to the four-component notation as a ¯ a a ¯ a ¯ I A μ ðQpaiΓuCr Þ¼ður ΓqpaiÞ; ðdCpΓQraiÞ¼ðdpΓqraiÞ; ðeCpΓLriÞ¼ðepΓlriÞ; Γ ¼ 1; τ ; λ ;D : ð4:22Þ Class ψ 2ϕ4D: The subclass has to be ψψ†ϕ4D, which has the following Lorentz structures: α † α_ α † α_ α † α_ ψ 1 ϕ2ðDϕ3Þαα_ ϕ4ϕ5ψ 6 ; ψ 1 ϕ2ϕ3ðDϕ4Þαα_ ϕ5ψ 6 ; ψ 1 ϕ2ϕ3ϕ4ðDϕ5Þ αα_ ψ 6 : ð4:23Þ In the SMEFT, all but one of the types are real: ↔ ↔ μ † ai † † μ † aj †i † iðQ σ Q ÞðH DμHÞðH HÞ;iðQ σ Q r ÞH H ðH DμHÞ; Oð1∼4Þ pai r pai j QQ†H2H†2D ↔ μ † aj †i † μ † aj †i † ðQpaiσ Q r ÞH HjDμðH HÞ;iðQpaiσ Q r ÞH DμHjðH HÞ; ↔ a μ † † † O † 2 †2 jiðuC σ uCra ÞðH DμHÞðH HÞ; uCuCH H D p ↔ μ † † † O † a σ 2 †2 jiðdC dC ra ÞðH DμHÞðH HÞ; dCdCH H D p ↔ ↔ μ † i † † μ † j †i † iðL σ L ÞðH DμHÞðH HÞ;iðL σ L rÞH H ðH DμHÞ; Oð1∼4Þ pi r pi j LL†H2H†2D ↔ μ † j †i † μ † j †i † ðLpiσ L rÞH HjDμðH HÞ;iðLpiσ L rÞH DμHjðH HÞ; ↔ μ † † † O † 2 †2 jiðeC σ eCr ÞðH DμHÞðH HÞ: ð4:24Þ eCeCH H D p 015026-26 COMPLETE SET OF DIMENSION-EIGHT OPERATORS IN THE … PHYS. REV. D 104, 015026 (2021) The only complex type is jk a μ † † O † 3 † jϵ ðuC σ dCra ÞðH HÞH DμH : ð4:25Þ uCdCH H D p j k After taking VEV for two of the Higgses, these are the five neutral fermion currents and one charged fermion current coupled with the neutral and charged Higgs current, which are already present at dimension six, but with additional v2=Λ2 2 suppression. Note that for the left-handed fermions Qi and Li, new terms exist due to the richness of the SUð ÞW structures. The conversion of these fermion currents to four-component spinor notation is shown by the following examples: μ † μ μ † i ¯i μ I A μ ðeCpσ ΓeCr Þ¼ðe¯pγ ΓerÞ; ðLpiσ ΓL rÞ¼−ðlrγ ΓlpiÞ; Γ ¼ 1; τ ; λ ;D : ð4:26Þ Class ψ 2ϕ3D2: The subclass ψ 2ϕ3D2 contains six independent Lorentz structures: α β α_ α β α_ α β α_ ψ 1 ψ 2 ðDϕ3Þαα_ ðDϕ4Þβϕ5; ψ 1 ψ 2 ðDϕ3Þαα_ ϕ4ðDϕ5Þβ; ψ 1 ψ 2 ϕ3ðDϕ4Þαα_ ðDϕ5Þβ; α β α_ α β α_ α β α_ ψ 1 ψ 2αðDϕ3Þα_ ðDϕ4Þβϕ5; ψ 1 ψ 2αðDϕ3Þα_ ϕ4ðDϕ5Þβ; ψ 1 ψ 2αϕ3ðDϕ4Þα_ ðDϕ5Þβ: ð4:27Þ Types of this subclass in the SMEFT are similar to the Yukawa terms, which are all complex, with additional Higgs and derivatives: ↔ ↔ ik a † μ ik μν a † iϵ ðQpaiuC r ÞDμHkðH D HÞ;iϵ ðQpaiσ uC r ÞDμHkðH DνHÞ; 1∼6 Oð Þ ik a † μ ik μν a † 2 † 2 ϵ ðQ uC ÞH ðDμH D HÞ; ϵ ðQ σ uC ÞH ðDμH DνHÞ; QuCH H D pai r k pai r k ik a μ † ik μν a † ϵ ðQ uC ÞDμH D ðH HÞ; ϵ ðQ σ uC ÞDμH DνðH HÞ; pai r k pai r k ↔ ↔ a μ †i † a μν †i † ið dCpQraiÞD H ðH DμHÞ;ið dCpσ QraiÞDμH ðH DνHÞ; 1∼6 Oð Þ a †i μ † a μν †i † †2 2 ð dC Q ÞH ðD H DμHÞ; ð dC σ Q ÞH ðDμH DνHÞ; QdCHH D p rai p rai a μ †i † a μν †i † ð dC Q ÞD H DμðH HÞ; ð dC σ Q ÞDμH DνðH HÞ; p rai p rai ↔ ↔ μ †i † μν †i † iðeCpLriÞD H ðH DμHÞ;iðeCpσ LriÞDμH ðH DνHÞ; 1∼6 Oð Þ †i μ † μν †i † †2 2 ðeC L ÞH ðD H DμHÞ; ðeC σ L ÞH ðDμH DνHÞ; ð4:28Þ eCLHH D p ri p ri μ †i † μν †i † ðeCpLriÞD H DμðH HÞ; ðeCpσ LriÞDμH DνðH HÞ; but due to the derivatives, these are new Lorentz structures at dimension eight. In some of the terms, the dipole moment bilinear appears, which is converted to four-component notation as σμνΓ ¯ σμνΓ σμνΓ a ¯ aσμνΓ Γ 1 τI λA μ ðeCp LriÞ¼ðep lriÞ; ðQpai uCr Þ¼ður qpaiÞ; ¼ ; ; ;D : ð4:29Þ Class ψ 2ϕ2D3: With three derivatives, we have only two independent Lorentz structures as follows: _ _ ψ α ϕ β ϕ ψ † α_ β ψ αϕ 2ϕ β ψ † α_ β 1 ðD 2Þα_ ðD 3Þαβ_ ðD 4Þβ ; 1 2ðD 3Þαα_ β_ ðD 4Þβ ; ð4:30Þ which can easily be checked by enumerating the SSYT of shape and labels f1; 1; 1; 2; 2; 3; 3; 4g (cf. Sec. III B 2). The types in the SMEFT are very similar to those of the subclass ψ 2ϕ4D, with five real types: 015026-27 LI, REN, SHU, XIAO, YU, and ZHENG PHYS. REV. D 104, 015026 (2021) ↔ ↔ † ai † μ † ai μ † ν iðQ σμQ Þ□ðH D HÞ;iðQ σμDνQ ÞðD H D HÞ; Oð1∼4Þ pai r pai r QQ†HH†D3 ↔ ↔ † aj †i μ † aj μ †i ν iðQ σμQ Þ□ðH D H Þ;iðQ σμDνQ ÞðD H D H Þ; pai r j pai r j 1;2 ↔ ↔ Oð Þ a † † μ a † μ † ν † † 3 σ ra □ σ ra uCu HH D iðuC p μuC Þ ðH D HÞ;iðuC p μDνuC ÞðD H D HÞ; C 1;2 ↔ ↔ Oð Þ a † † μ a † μ † ν † † 3 iðdC σμd ra Þ□ðH D HÞ;iðdC σμDνd ra ÞðD H D HÞ; dCdCHH D p C p C ↔ ↔ † i † μ † i μ † ν iðL σμL Þ□ðH D HÞ;iðL σμDνL ÞðD H D HÞ; Oð1∼4Þ pi r pi r LL†HH†D3 ↔ ↔ † j †i μ † j μ †i ν iðL σμL Þ□ðH D H Þ;iðL σμDνL ÞðD H D H Þ; pi r j pi r j 1 2 ↔ ↔ Oð ; Þ † † μ † μ † ν † † 3 iðeC σμe Þ□ðH D HÞ;iðeC σμDνe ÞðD H D HÞ; ð4:31Þ eCeCHH D p C r p C r and one complex type: ij a ν μ † O † 2 3 jiϵ ðuC σ D dCra ÞDμH DνH : ð4:32Þ uCdCH D p i j If we use the Fierz identity of SUðNÞ group Eq. (A26), we can perform the following transformation: ↔ 1 ↔ ↔ σ † aj □ †i μ ¯ γ □ † μ ¯ γ τI □ †τI μ iðQpai μQ r Þ ðH D HjÞ¼2 iðqr μqpÞ ðH D HÞþiðqr μ qpÞ ðH D HÞ: ð4:33Þ It could help convert our terms in Eq. (4.31) to more common forms, such as 8 8 ↔ ↔ > † ai † μ > † μ > iðQ σμQ Þ□ðH D HÞ > iðq¯ γμq Þ□ðH D HÞ > pai r > r p > ↔ > ↔ < † aj †i μ < I † I μ iðQ σμQ Þ□ðH D H Þ iðq¯ γμτ q Þ□ðH τ D HÞ pai r j ⇒ r p > ↔ > ↔ : ð4:34Þ > † ai μ † ν > μ † ν > iðQ σμDνQ ÞðD H D HÞ > iðq¯ γμDνq ÞðD H D HÞ > pai r > r p > ↔ > ↔ : σ † aj μ †i ν : ¯ γ τI μ †iτI ν iðQpai μDνQ r ÞðD H D HjÞ iðqr μ DνqpÞðD H D HjÞ 2. One gauge boson involved Class Fψ 2ϕ3: The only independent Lorentz structure of this subclass is αβψ ψ ϕ ϕ ϕ FL1 2α 3β 4 5 6: ð4:35Þ The operators with these Lorentz structures in the dimension eight SMEFT are ik A μν A a b † O 2 † ϵ GμνðQ σ ðλ Þ uC ÞH ðH HÞ; GQuCH H pai b r k 1 2 ð ; Þ km I j I μν a †i km I i I μν a † O 2 † ϵ τ Wμν Q σ uC H H H ; ϵ τ Wμν Q σ uC H H H ; WQuCH H ð Þm ð pai r Þ j k ð Þm ð pai r Þ kð Þ μν † O 2 † ϵik σ a BQuCH H BμνðQpai uC r ÞHkðH HÞ; μν † † O †2 A a σ λA b i GQdCHH GμνðdC p ð ÞaQrbiÞH ðH HÞ; 015026-28 COMPLETE SET OF DIMENSION-EIGHT OPERATORS IN THE … PHYS. REV. D 104, 015026 (2021) ð1;2Þ I a μν †i † I I i I a μν †l † O †2 WμνðdC σ Q ÞH ðH τ HÞ; ðτ Þ WμνðdC σ Q ÞH ðH HÞ; WQdCHH p rai l p rai μν † † O †2 a σ i BQdCHH BμνðdC p QraiÞH ðH HÞ; ð1;2Þ I μν †i † I I i I μν †l † O †2 WμνðeC σ L ÞH ðH τ HÞ; ðτ Þ WμνðeCpσ L ÞH ðH HÞ; WeCLHH p ri l ri μν † † O †2 σ i BeCLHH BμνðeCp LriÞH ðH HÞ: ð4:36Þ Note that these are all complex types, whose real part and imaginary part contribute to the electric and magnetic dipole moments of the fermions, respectively, after the Higgses take their VEV. One may refer to Eq. (4.29) for the conversion to four-component spinor notation. Class OðFψ 2ϕ2DÞ: In this class, the two spinors have opposite helicities and form a fermion current, while the gauge boson couples with both the fermion current and the Higgs current. Two independent Lorentz structures are present: αβψ ϕ ϕ ψ †α_ αβψ ϕ ϕ ψ †α_ FL1 2αðD 3Þβα_ 4 5 ;FL1 2α 3ðD 4Þβα_ 5 : ð4:37Þ In the SMEFT, real types with neutral fermion currents are as follows: ↔ A ν A a † bi μ † A ν A a † bi † μ GμνðQ σ ðλ Þ Q ÞD ðH HÞ;iGμνðQ σ ðλ Þ Q ÞðH D HÞ; pai b r pai b r ↔ ˜ A ν A a † bi μ † ˜ A ν A a † bi † μ GμνðQ σ ðλ Þ Q ÞD ðH HÞ;iGμνðQ σ ðλ Þ Q ÞðH D HÞ; Oð1∼8Þ pai b r pai b r GQQ†HH†D ↔ A ν A a † bj μ †i A ν A a † bj †i μ GμνðQ σ ðλ Þ Q r ÞD ðH H Þ;iGμνðQ σ ðλ Þ Q r ÞðH D H Þ; pai b j pai b j ↔ ˜ A σν λA a † bj μ †i ˜ A σν λA a † bj †i μ GμνðQpai ð ÞbQ r ÞD ðH HjÞ;iGμνðQpai ð ÞbQ r ÞðH D HjÞ; ↔ I σν † ai †τI I σν † ai †τI μ WμνðQpai Q r ÞDμðH HÞ;iWμνðQpai Q r ÞðH D HÞ; ↔ ˜ I σν † ai †τI ˜ I σν † ai †τI μ WμνðQpai Q r ÞDμðH HÞ;iWμνðQpai Q r ÞðH D HÞ; ↔ I k I ν † aj †i I k I ν † aj †i μ τ μν σ μ τ μν σ ð1∼12Þ ð Þj W ðQpai Q r ÞD ðH HkÞ;ið Þj W ðQpai Q r ÞðH D HkÞ; O † † 4:38 WQQ HH D ↔ ð Þ τI k ˜ I σν † aj †i τI k ˜ I σν † aj †i μ ð Þj WμνðQpai Q r ÞDμðH HkÞ;ið Þj WμνðQpai Q r ÞðH D HkÞ; ↔ I i I ν † aj †k I i I ν † aj †k μ ðτ Þ WμνðQ σ Q ÞDμðH H Þ;iðτ Þ WμνðQ σ Q ÞðH D H Þ; k pai r j k pai r j ↔ τI i ˜ I σν † aj †k τI i ˜ I σν † aj †k μ ð ÞkWμνðQpai Q r ÞDμðH HjÞ;ið ÞkWμνðQpai Q r ÞðH D HjÞ; ↔μ ν † ai μ † ν † ai † BμνðQ σ Q ÞD ðH HÞ;iBμνðQ σ Q ÞðH D HÞ; pai r pai r ↔μ ˜ ν † ai μ † ˜ ν † ai † BμνðQ σ Q ÞD ðH HÞ;iBμνðQ σ Q ÞðH D HÞ; Oð1∼8Þ pai r pai r BQQ†HH†D ↔μ ν † aj μ †i ν † aj †i BμνðQ σ Q ÞD ðH H Þ;iBμνðQ σ Q ÞðH D H Þ; pai r j pai r j ↔μ ˜ ν † aj μ †i ˜ ν † aj †i BμνðQpaiσ Q r ÞD ðH HjÞ;iBμνðQpaiσ Q r ÞðH D HjÞ; ↔μ A a σν λA b † μ † A a σν λA b † † ð1∼4Þ GμνðuC p ð ÞauC rbÞD ðH HÞ;iGμνðuC p ð ÞauC rbÞðH D HÞ; O † † GuCu HH D ↔μ C ˜ A a σν λA b † μ † ˜ A a σν λA b † † GμνðuC p ð ÞauC rbÞD ðH HÞ;iGμνðuC p ð ÞauC rbÞðH D HÞ; ↔μ I a σν † μ †τI I a σν † †τI ð1∼4Þ WμνðuC p uC raÞD ðH HÞ;iWμνðuC p uC raÞðH D HÞ; O † † WuCu HH D ↔μ C ˜ I a ν † μ † I ˜ I a ν † † I WμνðuC pσ uC raÞD ðH τ HÞ;iWμνðuC pσ uC raÞðH τ D HÞ 015026-29 LI, REN, SHU, XIAO, YU, and ZHENG PHYS. REV. D 104, 015026 (2021) ↔ a ν † μ † a ν † † μ BμνðuC σ u ÞD ðH HÞ;iBμνðuC σ u ÞðH D HÞ; Oð1∼4Þ p C ra p C ra † † BuCu HH D ↔ C ˜ a ν † μ † ˜ a ν † † μ BμνðuC pσ uC raÞD ðH HÞ;iBμνðuC pσ uC raÞðH D HÞ; ↔ A a ν A b † μ † A a ν A b † † μ GμνðdC σ ðλ Þ d ÞD ðH HÞ;iGμνðdC σ ðλ Þ d ÞðH D HÞ; Oð1∼4Þ p a C rb p a C rb † † GdCd HH D ↔ C ˜ A a σν λA b † μ † ˜ A a σν λA b † † μ GμνðdC p ð ÞadC rbÞD ðH HÞ;iGμνðdC p ð ÞadC rbÞðH D HÞ; ↔ I a ν † μ † I I a ν † † I μ WμνðdC σ d ra ÞD ðH τ HÞ;iWμνðdC σ d ra ÞðH τ D HÞ; Oð1∼4Þ p C p C † † WdCd HH D ↔ C ˜ I a ν † μ † I ˜ I a ν † † I μ WμνðdC pσ dCra ÞD ðH τ HÞ;iWμνðdC pσ dCra ÞðH τ D HÞ; ↔ a ν † μ † a ν † † μ BμνðdC σ d ra ÞD ðH HÞ;iBμνðdC σ d ra ÞðH D HÞ; Oð1∼4Þ p C p C † † BdCd HH D ↔ C ˜ a ν † μ † ˜ a ν † † μ BμνðdC pσ dCra ÞD ðH HÞ;iBμνðdC pσ dCra ÞðH D HÞ; ↔ I σν † i †τI I σν † i †τI μ WμνðLpi L rÞDμðH HÞ;iWμνðLpi L rÞðH D HÞ; ↔ ˜ I σν † i †τI ˜ I σν † i †τI μ WμνðLpi L rÞDμðH HÞ;iWμνðLpi L rÞðH D HÞ; ↔ I k I ν † j †i I k I ν † j †i μ ðτ Þ WμνðL σ L rÞDμðH H Þ;iðτ Þ WμνðL σ L rÞðH D H Þ; Oð1∼12Þ j pi k j pi k WLL†HH†D ↔ τI k ˜ I σν † j †i τI k ˜ I σν † j †i μ ð Þj WμνðLpi L rÞDμðH HkÞ;ið Þj WμνðLpi L rÞðH D HkÞ; ↔ I i I ν † j †k I i I ν † j †k μ ðτ Þ WμνðL σ L ÞDμðH H Þ;iðτ Þ WμνðL σ L ÞðH D H Þ; k pi r j k pi r j ↔ τI i ˜ I σν † j †k τI i ˜ I σν † j †k μ ð ÞkWμνðLpi L rÞDμðH HjÞ;ið ÞkWμνðLpi L rÞðH D HjÞ; ↔ ν † i μ † ν † i † μ BμνðL σ L ÞD ðH HÞ;iBμνðL σ L ÞðH D HÞ; pi r pi r ↔ ˜ ν † i μ † ˜ ν † i † μ BμνðL σ L ÞD ðH HÞ;iBμνðL σ L ÞðH D HÞ; Oð1∼8Þ pi r pi r BLL†HH†D ↔ ν † j μ †i ν † j †i μ BμνðL σ L rÞD ðH H Þ;iBμνðL σ L rÞðH D H Þ; pi j pi j ↔ ˜ ν † j μ †i ˜ ν † j †i μ BμνðLpiσ L rÞD ðH HjÞ;iBμνðLpiσ L rÞðH D HjÞ; ↔ I ν † μ † I I ν † † I μ WμνðeC σ e ÞD ðH τ HÞ;iWμνðeC σ e ÞðH τ D HÞ; Oð1∼4Þ p C r p C r † † WeCe HH D ↔ C ˜ I ν † μ † I ˜ I ν † † I μ WμνðeCpσ eC rÞD ðH τ HÞ;iWμνðeCpσ eC rÞðH τ D HÞ; ↔ ν † μ † I ν † † I μ BμνðeC σ e ÞD ðH τ HÞ;iBμνðeC σ e ÞðH τ D HÞ; Oð1∼4Þ p C r p C r † † ð4:39Þ BeCe HH D ↔ C ˜ ν † μ † I ˜ ν † † I μ BμνðeCpσ eC rÞD ðH τ HÞ;iBμνðeCpσ eC rÞðH τ D HÞ; while complex types with charged currents also exist: Oð1;2Þ ϵij A a σν λA b † μ ϵij ˜ A a σν λA b † μ † 2 j GμνðuC p ð ÞadCrb ÞHiD Hj; GμνðuC p ð ÞadCrb ÞHiD Hj; GuCdCH D Oð1;2Þ ϵjk τI i I a σν † μ ϵjk τI i ˜ I a σν † μ † 2 j ð Þ WμνðuC p dCra ÞHiD Hj; ð Þ WμνðuC p dCra ÞHiD Hj; WuCdCH D k k 1 2 Oð ; Þ ϵij a σν † μ ϵij ˜ a σν † μ † 2 j BμνðuC p dCra ÞHiD Hj; BμνðuC p dCra ÞHiD Hj: ð4:40Þ BuCdCH D These operators involve new Lorentz structures that were absent at lower dimensions. The conversion to the four spinor notation for the fermion currents can be found in Eqs. (4.26) and (4.33). 015026-30 COMPLETE SET OF DIMENSION-EIGHT OPERATORS IN THE … PHYS. REV. D 104, 015026 (2021) ψ 2ϕ 2 ψ 2ϕ 2 ψ 2ϕ Class F D : There are two subclasses of this form. One is FL D , a dimension-six class FL with two additional derivatives, which has two independent Lorentz structures: αβψ γ ψ ϕ α_ αβψ ψ γ ϕ α_ FL1 2 ðD 3Þαβα_ ðD 4Þγ ;FL1 2αðD 3Þβα_ ðD 4Þγ : ð4:41Þ ψ 2ϕ 2 The other subclass is FR D , where the flip of helicity for the gauge boson is made possible by the presence of the two additional derivatives. The Lorentz structure of this subclass is unique: ψ αψ β 2ϕ α_ β_ 1 2 ðD 3Þα_ βαβ_ FR4 : ð4:42Þ Converting to the F; F˜ basis, these two subclasses mix together. Below we present the operators of this class in the SMEFT, which are all complex types: ij A νλ A a b μ ϵ G ðQ σ ðλ Þ uC ÞD DνH ; ð1∼3Þ μλ pai b r j O 2 GQuCHD ϵij ˜ A σνλ λA a b μ ϵij A λA a μ b ν GμλðQpai ð ÞbuC r ÞD DνHj; GμνðQpaið ÞbD uC r ÞD Hj; ik I j I νλ a μ ϵ ðτ Þ W ðQ σ uC ÞD DνH ; ð1∼3Þ k μλ pai r j O 2 WQuCHD ϵik τI j ˜ I σνλ a μ ϵjk τI i I μ a ν ð ÞkWμλðQpai uC r ÞD DνHj; ð ÞkWμνðQpaiD uC r ÞD Hj; ϵij σνλ a μ ð1∼3Þ BμλðQpai uC r ÞD DνHj; O 2 BQuCHD ij ˜ νλ a μ ij μ a ν ϵ BμλðQpaiσ uC r ÞD DνHj; ϵ BμνðQpaiD uC r ÞD Hj; A a νλ A b μ †i G ðdC σ ðλ Þ Q ÞD DνH ; ð1∼3Þ μλ p a rbi O † 2 GQdCH D ˜ A a σνλ λA b μ †i A a λA b μ ν †i GμλðdC p ð ÞaQrbiÞD DνH ;GμνðdC pð ÞaD QrbiÞD H ; I i I a νλ μ †j ðτ Þ W ðdC σ Q ÞD DνH ; ð1∼3Þ j μλ p rai O † 2 WQdCH D τI i ˜ I a σνλ μ †j τI i I a μ ν †j ð ÞjWμλðdC p QraiÞD DνH ; ð ÞjWμνðdC pD QraiÞD H ; a σνλ μ †i ð1∼3Þ BμλðdC p QraiÞD DνH ; O † 2 BQdCH D ˜ a νλ μ †i a μ ν †i BμλðdC pσ QraiÞD DνH ;BμνðdC pD QraiÞD H ; I i I νλ μ †j ðτ Þ W ðeCpσ L ÞD DνH ; ð1∼3Þ j μλ ri O † 2 WeCLH D τI i ˜ I σνλ μ †j τI i I μ ν †j ð ÞjWμλðeCp LriÞD DνH ; ð ÞjWμνðeCpD LriÞD H ; σνλ μ †i ð1∼3Þ BμλðeCp LriÞD DνH ; O † 2 ð4:43Þ BeCLH D ˜ νλ μ †i μ ν †i BμλðeCpσ LriÞD DνH ;BμνðeCpD LriÞD H : 2 ψ 2ϕ The Lorentz structures here are also new at dimension while for FR we have only one independent Lorentz eight. To convert to four-component spinor notation, one structure, may refer to Eqs. (4.29) and (4.22). ψ αψ ϕ α_ β_ 1 2α 3FR4α_ β_ FR5 : ð4:45Þ 3. Two gauge boson involved 2ψ 2ϕ Class F : Two subclasses are involved, with the ˜ same opposite helicities for the gauge bosons and fermions. After converting to the F; F basis, the second in Eq. (4.44) 2 2 and the one in Eq. (4.45) combine to the form as products For the subclass F ψ ϕ, we obtained two independent L of a Yukawa coupling and a gauge kinetic term, while the Lorentz structures, first in Eq. (4.44) is a distinct one. The types of this class in the SMEFT can be found by adding two gauge bosons to αβ γψ ψ ϕ αβ ψ γψ ϕ FL1 FL2α 3β 4γ 5;FL1 FL2αβ 3 4γ 5; ð4:44Þ the Yukawa terms, which are all complex: 015026-31 LI, REN, SHU, XIAO, YU, and ZHENG PHYS. REV. D 104, 015026 (2021) ij ABC A B C a b ij 2 a ϵ d ðG G ÞðQ ðλ Þ uC ÞH ; ϵ G ðQ uC ÞH ; pai b r j pai r j Oð1∼5Þ ij ABC A ˜ B C a b ij A ˜ A a 2 ϵ d ðG G ÞðQpaiðλ Þ uC r ÞHj; ϵ ðG G ÞðQpaiuC r ÞHj; G QuCH b ij ABC A Bμ νλ C a b ϵ f GμνG λðQpaiσ ðλ Þ uC r ÞHj; b ϵij 2 a ϵij I ˜ I a ð1∼3Þ W ðQpaiuC r ÞHj; ðW W ÞðQpaiuC r ÞHj; O 2 W QuCH K i IJK jk I Jμ νλ a ðτ Þ ϵ ϵ WμνW λðQpaiσ uC r ÞHj; k jk I i A I A a b ϵ ðτ Þ ðG W ÞðQ ðλ Þ uC ÞH ; Oð1∼3Þ k pai b r j GWQuCH jk I i A ˜ I A a b jk I i Iμ A νλ A a b ϵ ðτ Þ ðG W ÞðQ ðλ Þ uC ÞH ; ϵ ðτ Þ W λGμνðQ σ ðλ Þ uC ÞH ; k pai b r j k pai b r j ð1;2Þ ij 2 a ij ˜ a O 2 ϵ B ðQ uC ÞH ; ϵ ðBBÞðQ uC ÞH ; B QuCH pai r j pai r j ij A A a b ϵ ðBG ÞðQ ðλ Þ uC ÞH ; Oð1∼3Þ pai b r j BGQuCH ij ˜ A A a b ij Aμ νλ A a b ϵ ðBG ÞðQpaiðλ Þ uC r ÞHj; ϵ BμνG λðQpaiσ ðλ Þ uC r ÞHj; b b ϵjk τI i I a ð1∼3Þ ð ÞkðBW ÞðQpaiuC r ÞHj; O BWQuCH ϵjk τI i ˜ I a ϵjk τI i Iμ σνλ a ð ÞkðBW ÞðQpaiuC r ÞHj; ð ÞkW λBμνðQpai uC r ÞHj; ABC A B a C b †i 2 a †i d ðG G ÞðdC ðλ Þ Q ÞH ;GðdC Q ÞH ; p a rbi p rai Oð1∼5Þ ABC A ˜ B a C b †i A ˜ A a †i 2 † d ðG G ÞðdC pðλ Þ QrbiÞH ; ðG G ÞðdC pQraiÞH ; G QdCH a ABC A Bμ a σνλ λC b †i f GμνG λðdC p ð ÞaQrbiÞH ; 2 a †i W dC Q H ; ð1∼3Þ ð p raiÞ O 2 † W QdCH I ˜ I a †i τK i ϵIJK I Jμ a σνλ †j ðW W ÞðdC pQraiÞH ; ð Þj WμνW λðdC p QraiÞH ; I i I A a A b †j ðτ Þ ðW G ÞðdC ðλ Þ Q ÞH ; ð1∼3Þ j p a rbi O † ð4:46Þ GWQdCH τI i I ˜ A a λA b †j τI i Iμ A a σνλ λA b †j ð ÞjðW G ÞðdC pð ÞaQrbiÞH ; ð ÞjW λGμνðdC p ð ÞaQrbiÞH ; ð1;2ÞÞ 2 a †i ˜ a †i O 2 † B ðdC Q ÞH ; ðBBÞðdC Q ÞH ; B QdCH p rai p rai A a λA b †i ð1∼3Þ ðBG ÞðdC pð ÞaQrbiÞH ; O † BGQdCH ˜ A a λA b †i Aμ a σνλ λA b †i ðBG ÞðdC pð ÞaQrbiÞH ;BμνG λðdC p ð ÞaQrbiÞH ; I i I a †j ðτ Þ ðBW ÞðdC Q ÞH ; ð1∼3Þ j p rai O † BWQdCH I i ˜ I a †j I i Iμ a νλ †j ðτ Þ ðBW ÞðdC Q ÞH ; ðτ Þ W λBμνðdC σ Q ÞH ; j p rai j p rai 1 2 ð ; Þ 2 †i A ˜ A †i O 2 † G eCpL H ; G G eCpL H ; G eCLH ð riÞ ð Þð riÞ 2 †i ð1∼3Þ W ðeCpLriÞH ; O 2 † W eCLH I ˜ I †i K i IJK I Jμ νλ †j ðW W ÞðeCpL ÞH ; ðτ Þ ϵ WμνW λðeCpσ L ÞH ; ri j ri 1 2 ð ; Þ 2 †i ˜ †i O 2 † B eCpL H ; BB eCpL H ; B eCLH ð riÞ ð Þð riÞ I i I †j ðτ Þ ðBW ÞðeCpL ÞH ; ð1∼3Þ j ri O † ð4:47Þ BWeCLH τI i ˜ I †j τI i Iμ σνλ †j ð ÞjðBW ÞðeCpLriÞH ; ð ÞjW λBμνðeCp LriÞH : Conversion to four-component spinor notation in this class can be found in Eqs. (4.22) and (4.29). 2ψ 2 ψψ† Class F D: The gauge bosons can have the same or opposite helicities, leading to two subclasses FLFL=R D, each of which contains only one independent Lorentz structure, αβ γ ψ ψ †α_ αβψ ψ † α_ β_ FL1 FL2α ðD 3Þβγα_ 4 ;FL1 2αðD 3Þβα_ β_ FR4 : ð4:48Þ 015026-32 COMPLETE SET OF DIMENSION-EIGHT OPERATORS IN THE … PHYS. REV. D 104, 015026 (2021) Without other fields carrying hypercharges, the fermions in this class have to form a neutral current, which demands the types in the SMEFT to be all real: ↔ ↔ ABC Aμ Bν λ C a † bi ABC Aμ Bν λ C a † bi if G νG λ Q σ λ DμQ ;idG νG λ Q σ λ DμQ ; ð pai ð Þb r Þ ð pai ð Þb r Þ 1∼5 ↔ ↔ ð Þ μ ν λ † μ ν λ † O 2 † ABC ˜ A B σ λC a bi A A σ ai G QQ D if G νG λðQpai ð Þ DμQ r Þ;iGνG λðQpai DμQ r Þ; b ↔ ABC Aμ ˜ Bν σλ λC a † bi if G νG λðQpai ð ÞbDμQ r Þ; ↔ ↔ IJK Iμ Jν λ K i † aj Iμ Iν λ † ai iϵ W νW λðQ σ ðτ Þ DμQ Þ;iWνW λðQ σ DμQ Þ; Oð1∼4Þ pai j r pai r W2QQ†D ↔ ↔ IJK ˜ Iμ Jν λ K i † aj IJK Iμ ˜ Jν λ K i † aj iϵ W νW λðQ σ ðτ Þ DμQ Þ;iϵ W νW λðQ σ ðτ Þ DμQ Þ; pai j r pai j r ↔ O 2 † μ ν λ † ai B QQ D iB νB λðQpaiσ DμQ r Þ; ↔ ↔ ABC Aμ Bν a σλ λC b † ABC Aμ Bν a σλ λC b † if G νG λðuC p ð ÞaDμuCrbÞ;idG νG λðuC p ð ÞaDμuCrbÞ; 1∼5 ↔ ↔ Oð Þ ABC ˜ Aμ Bν a λ C b † Aμ Aν a λ † 2 † if G νG λðuC σ ðλ Þ Dμu Þ;iGνG λðuC σ Dμu Þ; G uCuCD p a C rb p C ra ↔ ABC Aμ ˜ Bν a λ C b † if G νG λðuC σ ðλ Þ Dμu Þ; p a C rb ↔ μ ν λ † O 2 † I I a σ ð4:49Þ W uCuCD iW νW λðuC p DμuC raÞ; ↔ μ ν λ † O 2 † a σ B uCuCD iB νB λðuC p DμuC raÞ; ↔ ↔ ABC Aμ Bν a λ C b † ABC Aμ Bν a λ C b † if G νG λ dC σ λ Dμd ;idG νG λ dC σ λ Dμd ; ð p ð Þa C rbÞ ð p ð Þa C rbÞ 1∼5 ↔ ↔ Oð Þ ABC ˜ Aμ Bν a λ C b † Aμ Aν a λ † 2 † if G νG λðdC σ ðλ Þ Dμd Þ;iGνG λðdC σ Dμd Þ; G dCdCD p a C rb p C ra ↔ ABC Aμ ˜ Bν a λ C b † if G νG λðdC σ ðλ Þ Dμd Þ; p a C rb ↔ μ ν λ † O 2 † I I a σ W dCdCD iW νW λðdC p DμdC raÞ; ↔ μ ν λ † O 2 † a σ B dCdCD iB νB λðdC p DμdC raÞ; ↔ O 2 † Aμ Aν λ † i G LL D iG νG λðLpiσ DμL rÞ; ↔ ↔ IJK Iμ Jν λ K i † j Iμ Iν λ † i iϵ W νW λðL σ ðτ Þ DμL Þ;iWνW λðL σ DμL Þ; Oð1∼4Þ pi j r pi r W2LL†D ↔ ↔ IJK ˜ Iμ Jν λ K i † j IJK Iμ ˜ Jν λ K i † j iϵ W νW λðL σ ðτ Þ DμL Þ;iϵ W νW λðL σ ðτ Þ DμL Þ; pi j r pi j r ↔ O 2 † μ ν λ † i B LL D iB νB λðLpiσ DμL rÞ; ↔ μ ν λ † O 2 † A A σ G eCeCD iG νG λðeCp DμeC rÞ; ↔ μ ν λ † O 2 † I I σ W eCeCD iW νW λðeCp DμeC rÞ; ↔ μ ν λ † O 2 † σ B eCeCD iB νB λðeCp DμeC rÞ; 015026-33 LI, REN, SHU, XIAO, YU, and ZHENG PHYS. REV. D 104, 015026 (2021) ↔ ↔ Aμ Iνλ A a I i † bj Aμ ˜ Iνλ A a I i † bj iG λW ðQ σνðλ Þ ðτ Þ DμQ Þ;iGλW ðQ σνðλ Þ ðτ Þ DμQ Þ; Oð1∼4Þ pai b j r pai b j r GWQQ†D ↔ ↔ Aν Iμλ σ λA a τI i † bj Aν ˜ Iμλ σ λA a τI i † bj iG λW ðQpai νð Þbð ÞjDμQ r Þ;iGλW ðQpai νð Þbð ÞjDμQ r Þ; ↔ ↔ Aμ νλ A a † bi Aμ ˜ νλ A a † bi iG λB ðQ σνðλ Þ DμQ Þ;iGλB ðQ σνðλ Þ DμQ Þ; Oð1∼4Þ pai b r pai b r GBQQ†D ↔ ↔ Aν μλ σ λA a † bi Aν ˜ μλ σ λA a † bi iG λB ðQpai νð ÞbDμQ r Þ;iGλB ðQpai νð ÞbDμQ r Þ; ↔ ↔ Iμ νλ I i † aj Iμ ˜ νλ I i † aj iW λB ðQ σνðτ Þ DμQ Þ;iWλB ðQ σνðτ Þ DμQ Þ; Oð1∼4Þ pai j r pai j r WBQQ†D ↔ ↔ Iν μλ σ τI i † aj Iν ˜ μλ σ τI i † aj iW λB ðQpai νð ÞjDμQ r Þ;iWλB ðQpai νð ÞjDμQ r Þ; ↔ ↔ Aμ νλ a σ λA b † Aμ ˜ νλ a σ λA b † ð1∼4Þ iG λB ðuC p νð ÞaDμuC rbÞ;iGλB ðuC p νð ÞaDμuC rbÞ; O † GBuCu D ↔ ↔ C Aν μλ a σ λA b † Aν ˜ μλ a σ λA b † iG λB ðuC p νð ÞaDμuC rbÞ;iGλB ðuC p νð ÞaDμuC rbÞ; ↔ ↔ Aμ νλ a σ λA b † Aμ ˜ νλ a σ λA b † ð1∼4Þ iG λB ðdC p νð ÞaDμdC rbÞ;iGλB ðdC p νð ÞaDμdC rbÞ; O † GBdCd D ↔ ↔ C Aν μλ a σ λA b † Aν ˜ μλ a σ λA b † iG λB ðdC p νð ÞaDμdC rbÞ;iGλB ðdC p νð ÞaDμdC rbÞ; ↔ ↔ Iμ νλ I i † j Iμ ˜ νλ I i † j iW λB ðL σνðτ Þ DμL Þ;iWλB ðL σνðτ Þ DμL Þ; Oð1∼4Þ pi j r pi j r WBLL†D ↔ ↔ ð4:50Þ Iν μλ σ τI i † j Iν ˜ μλ σ τI i † j iW λB ðLpi νð ÞjDμL rÞ;iWλB ðLpi νð ÞjDμL rÞ: Conversion of the relevant fermion currents to four- consequently B − L is conserved for all the dimension- component notation can be found in Eqs. (4.26) and (4.33). eight operators. Note that repeated fermions start to appear in this D. Classes involving four fermions section, for which Young symmetrizers are applied to The classes of Lorentz structures with four fermions are the terms to retain particular flavor symmetries, as the most populated in the dimension-eight SMEFT; thus to explained in Secs. III A and IVA. present in a less dense way, we separate the types in 1. Two scalars involved different lists by the number of quarks involved. Those with three quarks and one lepton violate both the baryon number Class ψ 4ϕ2: There are two subclasses in this class: and lepton number ΔB ¼ ΔL ¼ 1, which is the only ψ 2ψ †2ϕ2 and ψ 4ϕ2 þ H:c:, and the independent Lorentz source of these violations at dimension eight, and structures are α † †α_ α β α β ψ 1ψ 2αϕ3ϕ4ψ 5 α_ ψ 6 ; ψ 1ψ 2ψ 3αψ 4βϕ5ϕ6; ψ 1ψ 2αψ 3ψ 4βϕ5ϕ6: ð4:51Þ With the two scalars taken to be ðH†HÞ, we get the same types as the four-fermion operators at dimension six with the additional Higgses. There are new types at dimension eight with the two scalars taken to be the Higgses with the same 2 †2 2 hypercharges H or H , whose SUð ÞW indices must be symmetric to avoid the repeated field constraint. This demands at 2 least another pair of SUð ÞW doublets in the four fermions, which excludes the following types that are also Lorentz 2 invariant gauge singlets, but with all four fermions as SUð ÞW singlets: †2 2 † † 2 † 2 † 2 2 †2 3 2 dCdC uCH ;dCeCeCuCH ;dCuC uCH ;dC eCuCH ;eCuC H : ð4:52Þ Operators of this class contribute to the four-fermion interactions if the Higgs fields take their VEV, and operators involving two or four L’s are relevant to the neutrino nonstandard interactions. 015026-34 COMPLETE SET OF DIMENSION-EIGHT OPERATORS IN THE … PHYS. REV. D 104, 015026 (2021) 1. Operators involving only quarks: There are six real types from all combinations of the three quark currents: ð4:53Þ An addition of four complex types exist: ð4:54Þ Y Oð1∼10Þ Recall the definition of Young symmetrizer in Sec. III A 2, we can obtain the following relations for type Q2Q†2HH† : Oð1Þ † aj † bk † aj † bk †i Q2Q†2HH† ¼ðQpaiQrbj þ QraiQpbjÞðQ s Q t þ Q t Q s ÞH Hk; Oð4Þ † aj † bk − † aj † bk †i Q2Q†2HH† ¼ðQpaiQrbj þ QraiQpbjÞðQ s Q t Q t Q s ÞH Hk; Oð6Þ − † aj † bk † aj † bk †i Q2Q†2HH† ¼ðQpaiQrbj QraiQpbjÞðQ s Q t þ Q t Q s ÞH Hk; Oð8Þ − † aj † bk − † aj † bk †i Q2Q†2HH† ¼ðQpaiQrbj QraiQpbjÞðQ s Q t Q t Q s ÞH Hk; ð4:55Þ as an example of how Y’s act on the terms. The conversion from the two-component spinors to the four-component spinors, with extra transformation via Fierz identity, are shown by the following examples: 015026-35 LI, REN, SHU, XIAO, YU, and ZHENG PHYS. REV. D 104, 015026 (2021) 1 † aj † bk ¯ aj ¯ bk ¯ ajγμ ¯ bkγ ðQpaiQrbjÞðQ s Q t Þ¼ðqpaiCqrbjÞðq s Cq t Þ¼2 ðq s qpaiÞðq t μqrbjÞ; 1 † † a b ¯ a ¯ b ¯ a γμ ¯ bγ ðuC sauC tbÞðuC puC r Þ¼ðusaCutbÞðu pCu r Þ¼2 ðu p usaÞðu r μutbÞ; 1 a † cj † ¯ a ¯ cj − ¯ aγμ ¯ cjγ ðQpaiuC r ÞðQ s uC tcÞ¼ðu r qpaiÞðq s uutcÞ¼ 2 ðu r utcÞðq s μqpaiÞ: ð4:56Þ The Hermitian conjugate of a non-Hermitian operator of this class is, for example, ik jm b a † † † ai † † bj †k †m ½ϵ ϵ ðQpaiuC s ÞðQrbjuC t ÞHkHm ¼ ϵikϵjmðuC sbQ p ÞðuC taQ r ÞH H : ð4:57Þ Operators involving d quark and leptons of this class can be converted similarly. 2. Operators involving three quarks with ΔB ¼ ΔL ¼ 1: All the types with three quarks are complex, and there are seven of them in this class: ð4:58Þ Oð1∼7Þ Y’ Here are examples in the type Q3LHH† about how the Young symmetrizers s act on the operators: Oð1Þ ϵabcϵimϵjn †k Q3LHH† ¼ ½ðQrajQtcmÞðLpiQsbkÞþðQsajQtcmÞðLpiQrbkÞ H Hn abc im jn †k − ϵ ϵ ϵ ½ðQtajQrcmÞðLpiQsbkÞþðQtajQscmÞðLpiQrbkÞ H Hn; Oð4Þ ϵabcϵimϵjn †k Q3LHH† ¼ ½ðQrajQtcmÞðLpiQsbkÞþðQsajQtcmÞðLpiQrbkÞ H Hn abc im jn †k þ ϵ ϵ ϵ ½ðQtajQrcmÞðLpiQsbkÞþðQrajQscmÞðLpiQtbkÞ H Hn abc im jn †k þ ϵ ϵ ϵ ½ðQtajQscmÞðLpiQrbkÞþðQsajQrcmÞðLpiQtbkÞ H Hn; Oð6Þ ϵabcϵimϵjn − †k Q3LHH† ¼ ½ðQrajQtcmÞðLpiQsbkÞ ðQsajQtcmÞðLpiQrbkÞ H Hn abc im jn †k − ϵ ϵ ϵ ½ðQtajQrcmÞðLpiQsbkÞþðQrajQscmÞðLpiQtbkÞ H Hn abc im jn †k þ ϵ ϵ ϵ ½ðQtajQscmÞðLpiQrbkÞþðQsajQrcmÞðLpiQtbkÞ H Hn: ð4:59Þ 3. Operators involving two leptons and two quarks: There are six real types as combinations of the three quark currents and the two lepton currents: 015026-36 COMPLETE SET OF DIMENSION-EIGHT OPERATORS IN THE … PHYS. REV. D 104, 015026 (2021) † † † † † † ðL Q ÞðL i Q akÞH jH ; ðL Q ÞðL kQ aiÞH jH ; pi raj s t k pi raj s t k ð1∼5Þ † j † ak †i † j † ai † O † † † ðL Q ÞðL sQ ÞH H ; ðL Q ÞðL sQ ÞðH HÞ; QQ LL HH pi raj t k pi raj t ðL Q ÞðL† kQ† ajÞH†iH ; pi raj s t k 1 2 Oð ; Þ † † aj †i † † ai † † † † ðeCpQ ÞðeC Q ÞH H ; ðeCpQ ÞðeC Q ÞðH HÞ; QQ eCe HH rai s t j rai s t C 1 2 Oð ; Þ a † j † †i a † i † † † † † ðL uC ÞðL suC ÞH H ; ðL uC ÞðL uC ÞðH HÞ; uCu LL HH pi r ta j pi r s ta C † † a † O † † † e u eCpuC H H ; uCuCeCeCHH ð C s C taÞð r Þð Þ 1 2 Oð ; Þ a † † j †i a † † i † † † † ðdC L ÞðdC L ÞH H ; ðdC L ÞðdC L ÞðH HÞ; dCd LL HH p ri sa t j p ri sa t C † † a † O † † † ðd e ÞðdC eCrÞðH HÞ: ð4:60Þ dCdCeCeCHH C sa C t p There are also five complex types, in which three involve repeated Higgses: ϵij a † ϵij a † ð1∼4Þ ðeCpQsajÞðLriuC t ÞðH HÞ; ðeCpLriÞðQsajuC t ÞðH HÞ; O † QuCLeCHH ik a †j ik a †j ϵ ðeCpQ ÞðL uC ÞH H ; ϵ ðeCpL ÞðQ uC ÞH H ; saj ri t k ri saj t k ik † † j a O † † 2 ϵ ðe L ÞðQ uC ÞH H ; QuCL eCH C s t pai r j k ð1;2Þ a †i †j a †i †j O †2 ðdC eCrÞðL Q ÞH H ; ðeCrQ ÞðdC L ÞH H ; QdCLeCH p si taj taj p si 1 2 Oð ; Þ † † j a †i † † i a † † † † ðeC L ÞðdC Q ÞH H ; ðeC L ÞðdC Q ÞðH HÞ; QdCL e HH s t p rai j s t p rai C ik † † j a O † † 2 ϵ ðd L ÞðL uC ÞH H : ð4:61Þ uCdCLL H C sa t pi r j k 4. Operators involving only leptons: The combinations of the two kinds of lepton currents give three real types of operators in this class: ð4:62Þ There is one more complex type with repeated Higgses: ð4:63Þ 2. One derivative involved Class ψ 4ϕD: The subclass of this form must contain three spinors of the same helicities and one spinor of the opposite helicity, namely ψ 3ψ †ϕD. A total of three independent Lorentz structures exist in this subclass, 015026-37 LI, REN, SHU, XIAO, YU, and ZHENG PHYS. REV. D 104, 015026 (2021) α β †α_ α β †α_ α β †α_ ψ 1ψ 2ðDψ 3Þαβα_ ϕ4ψ 5 ; ψ 1ψ 2ψ 3αðDϕ4Þβα_ ψ 5 ; ψ 1ψ 2αψ 3ðDϕ4Þβα_ ψ 5 : ð4:64Þ All the types in this subclass must be complex. 1. Operators involving only quarks: The six types are all of the combinations of the two quark Yukawa terms and the three quark kinetic terms: ð4:65Þ To see how Y’s act on operators one can refer to Eqs. (4.55) and (4.59). The conversion from the two-component spinors to the four-component spinors are shown by the following examples: 015026-38 COMPLETE SET OF DIMENSION-EIGHT OPERATORS IN THE … PHYS. REV. D 104, 015026 (2021) a σμ † bj − ¯ a ¯ bjγμ ðQpaiuC s ÞðQrbj Q t Þ¼ ðu s qpaiÞðq t qrbjÞ; a b μ † ¯ a ¯ b μ ðQpaiuC s ÞðuC r σ uC tbÞ¼ðu s qpaiÞðu r γ utbÞ; a σμ † bj ¯ a γμ ¯ bj ðQrbiDμQsajÞðdC p Q t Þ¼ðqrbiCDμqsajÞðd p Cqt Þ 1 ¯ a γμγν ¯ bjγ ¼ 2 ðd p DμqsajÞðqt νqrbiÞ; a c μ † ¯ a ¯ c μ ðuC r DμuC sÞðQpaiσ uC tcÞ¼ðu r CDμu sÞðqpaiCγ utcÞ ¯ a ¯ c μ ¯ a μ ¯ c ¼ðu r qpaiÞðDμu sγ utcÞ − ðu r γ utcÞðDμu sqpaiÞ: ð4:66Þ The Hermitian conjugate of a non-Hermitian operator of this class is, for example, ϵik σμ † bj † ϵ † † ai σμ † bj †k ½ ðQpaiuCsaÞðQrbj Q t ÞDμHk ¼ ikðuC saQ p ÞðQtbj Q r ÞDμH : ð4:67Þ 2. Operators involving three quarks with ΔB ¼ ΔL ¼ 1: There are seven B-violating types in this class: ð4:68Þ 3. Operators involving two leptons and two quarks: The combinations of two quark Yukawa terms and two lepton kinetic terms, and the combinations of one lepton Yukawa term and three quark kinetic terms, constitute seven types here, while † one more type uCdCLeCHD not as such a combination is present: 015026-39 LI, REN, SHU, XIAO, YU, and ZHENG PHYS. REV. D 104, 015026 (2021) † aj μ †i † ai μ †j ðeCpσμQ t ÞðLriD QsajÞH ; ðeCpσμQ t ÞðLriD QsajÞH ; 1∼6 ð Þ † aj μ †i † ai μ †j O † † ðeCpL ÞðQ σμQ ÞD H ; ðeCpL ÞðQ σμQ ÞD H ; QQ LeCH D ri saj t ri saj t † aj μ †i † ai μ †j ðeCpQ ÞðL σμQ ÞD H ; ðeCpQ ÞðL σμQ ÞD H ; saj ri t saj ri t ik μ a † j ij μ a † k ϵ ðQrajD uC s ÞðLpiσμL t ÞHk; ϵ ðQrajD uC s ÞðLpiσμL t ÞHk; 1∼6 ð Þ ik a † j μ ij a † k μ O † ϵ ðL Q ÞðuC σμL ÞD H ; ϵ ðL Q ÞðuC σμL ÞD H ; QuCLL HD pi raj s t k pi raj s t k ik a † j μ ij a † k μ ϵ ðLpiuC s ÞðQrajσμL t ÞD Hk; ϵ ðLpiuC s ÞðQrajσμL t ÞD Hk; ij a μ † ij a μ † ϵ ðeCpQ ÞðuC σ e ÞDμH ; ϵ ðeCpuC ÞðQ σ e ÞDμH ; Oð1∼3Þ rai s C t j s rai C t j † † ð4:69Þ QuCeCeCHD ij μ a ϵ ðeCpσ eC tÞðQraiDμuC s ÞHj; a † j μ †i a † i μ †j ðdC pσμL t ÞðLriD QsajÞH ; ðdC pσμL tÞðLriD QsajÞH ; 1∼6 ð Þ a † j μ †i a † i μ †j O † † ðdC L ÞðQ σμL ÞD H ; ðdC L ÞðQ σμL ÞD H ; QdCLL H D p ri saj t p ri saj t a † j μ †i a † i μ †j ðdC pQsajÞðLriσμL t ÞD H ; ðdC pQsajÞðLriσμL tÞD H ; a μ † †i μ † a †i ðdC eCrÞðQ σ e ÞDμH ; ðeCrσ e ÞðdC Q ÞDμH ; Oð1∼3Þ p sai C t C t p sai † † † QdCeCeCH D a μ †i ðeCrDμQsaiÞðdC pσ eC tÞH ; a μ † †i a μ † †i ðeCpL ÞðuC σ u ÞDμH ; ðeCpuC ÞðL σ u ÞDμH ; Oð1∼3Þ ri s C ta s ri C ta † † † uCuCLeCH D μ a †i ðeCpσ uC taÞðLriDμuC s ÞH ; ij a μ † ij a μ † ϵ ðeCpL ÞðuC σ d ÞDμH ; ϵ ðeCpuC ÞðL σ d ÞDμH ; Oð1∼3Þ ri s C ta j s ri C ta j † † uCdCLeCHD ij μ a ϵ ðeCpσ dC taÞðLriDμuC s ÞHj; a μ † †i a μ † †i ðdC eCrÞðL σ d ÞDμH ; ðdC L ÞðeCrσ d ÞDμH ; Oð1∼3Þ p si C ta p si C ta † † ð4:70Þ dCdCLeCH D a μ † †i ðeCrDμLsiÞðdC pσ dC taÞH : 4. Operators involving only leptons: The two following types are simply the lepton Yukawa term combined with one of the lepton kinetic terms: ð4:71Þ 3. Two derivatives involved Class ψ 4D2: There are two subclasses of this form: ψ 2ψ †2D2 and ψ 4D2 þ H:c:, and five independent Lorentz structures are involved: _ _ ψ αψ β ψ † ψ † α_ β ψ αψ ψ † β ψ † α_ β 1 2ðD 3Þαα_ β_ ðD 4Þβ ; 1 2αðD 3Þα_ β_ ðD 4Þβ ; α βγ α_ α β γ α_ α βγ α_ ψ 1ðDψ 2Þα_ ψ 3αðDψ 4Þβγ; ψ 1ψ 2ðDψ 3Þαα_ ðDψ 4Þβγ; ψ 1ψ 2αðDψ 3Þα_ ðDψ 4Þβγ: ð4:72Þ μν Note that in converting to the conventional form of Lorentz structures, we avoid having parts such as σ Dμψ because they are related to Dνψ by the EOM redundancy. The types in this class are exactly the dimension-six four-fermion types plus two extra derivatives, which include 15 real types and seven complex types, among which are four B-violating types. 015026-40 COMPLETE SET OF DIMENSION-EIGHT OPERATORS IN THE … PHYS. REV. D 104, 015026 (2021) 1. Operators involving only quarks: There are six all-quark real types as follows: ð4:73Þ and one complex type: ð4:74Þ To see how Y’s act on operators one can refer to Eqs. (4.55) and (4.59). The conversion from the two-component spinors to the four-component spinors follows Eq. (4.56) as † ai † bj μν ¯ ai ¯ bj μν ðDμQ s DνQ t ÞðQpaiσ QrbjÞ¼ðDμq s CDνq t ÞðqpaiCσ qrbjÞ 1 ¯ aiγρσμν ¯ bjγ ¼ 2 ðDμq s qrbjÞðDνq t ρqpaiÞ; † † a μν b a μν b ðDμuC saDνuC tbÞðuC pσ uC r Þ¼ðDμusaCDνutbÞðu¯ pσ Cu¯ r Þ 1 ¯ a σμνγρ ¯ bγ ¼ 2 ðu p DνutbÞðu r ρDμusaÞ; † ai † μν b ¯ ai ¯ b μν ðDμQ s DνuC tbÞðQpaiσ uC r Þ¼ðDμq s DνutbÞðu r σ qpaiÞ: ð4:75Þ The Hermitian conjugate of a non-Hermitian operator of this class is, for example, ij b a μν † † † aj † bi μν † ½ϵ ðDμQsajDνuC t ÞðdC pσ QrbiÞ ¼ ϵijðDνuC tbDμQ s ÞðQ r σ¯ dC paÞ: ð4:76Þ Operators involving leptons can be converted similarly. 015026-41 LI, REN, SHU, XIAO, YU, and ZHENG PHYS. REV. D 104, 015026 (2021) 2. Operators involving one lepton and three quarks with ΔB ¼ ΔL ¼ 1: The four B-violating types are ð4:77Þ 3. Operators involving two leptons and two quarks: Combinations of three kinds of quark currents and two kinds of lepton currents provide six real types: † j μ † ai † j † ai σμν ð1∼4Þ ðLpiQrajÞðDμL sD Q t Þ; ðDμL sDνQ t ÞðLpi QrajÞ; O † † 2 QQ LL D † i μ † aj † i † aj μν ðL Q ÞðDμL D Q Þ; ðDμL DνQ ÞðL σ Q Þ; pi raj s t s t pi raj Oð1;2Þ † μ † ai † † ai σμν † † 2 ðeCpQraiÞðDμeC sD Q t Þ; ðDμeC sDνQ t ÞðeCp QraiÞ; QQ eCeCD 1 2 Oð ; Þ a † i μ † † i † σμν a † † 2 ðLpiuC r ÞðDμL sD uC taÞ; ðDμL sDνuC taÞðLpi uC r Þ; uCuCLL D Oð1;2Þ a † μ † † † σμν a † † 2 ðeCpuC r ÞðDμeC sD uC taÞ; ðDμeC sDνuC taÞðeCp uC r Þ; uCuCeCeCD 1 2 Oð ; Þ a † μ † i † † i a σμν † † 2 ðdC pLriÞðDμdC saD L tÞ; ðDμdC saDνL tÞðdC p LriÞ; dCdCLL D Oð1;2Þ a † μ † † † a σμν † † 2 ðdC peCrÞðDμdC saD eC tÞ; ðDμdC saDνeC tÞðdC p eCrÞ: ð4:78Þ dCdCeCeCD Two additional complex types are present: ϵij μ a ϵij σμν a ð1∼3Þ ðeCpLriÞðDμQsajD uC t Þ; ðeCp LriÞðDμQsajDνuC t Þ; O 2 QuCLeCD ij μ a ϵ ðeCpQ ÞðDμL D uC Þ; saj ri t 1 2 Oð ; Þ a † μ † i † † i a σμν † † 2 ðdC pQraiÞðDμeC sD L tÞ; ðDμeC sDνL tÞðdC p QraiÞ: ð4:79Þ QdCL eCD 4. Operators involving only leptons: Two kinds of lepton currents form three real types with all leptons: ð4:80Þ 4. One gauge boson involved ψ 4 ψ 2ψ †2 Class F : There are two subclasses in this class: FL þ H:c: with only one Lorentz structure, αβψ ψ ψ † ψ †α_ FL1 2α 3β 4 α_ 5 ; ð4:81Þ 015026-42 COMPLETE SET OF DIMENSION-EIGHT OPERATORS IN THE … PHYS. REV. D 104, 015026 (2021) ψ 4 and FL þ H:c: with three independent Lorentz structures, αβψ γψ ψ ψ αβψ ψ γψ ψ αβψ ψ ψ γψ FL1 2 3α 4β 5γ;FL1 2α 3 4β 5γ;FL1 2α 3β 4 5γ: ð4:82Þ In converting to the conventional form, the gauge boson always contracts with the σμν (one may convert to other forms ˜ μν β μν β ˜ via Fierz identities, which we choose not to do), and due to the identity Fμνðσ Þα ¼ iFμνðσ Þα , the F and F are equivalent; hence we only use F instead of F˜ in our operators. The types in this class are simply the dimension-six four- fermion types with an additional gauge boson, depending on the gauge charges of the fermions: B is always available as all 1 the fermions are charged under Uð ÞY; G is available whenever quarks are present; W is available whenever Q or L is present. 1. Operators involving only quarks: Based on the six real types with four quarks, B and G can be added to all of them, while W can be added to the three types with Q. Overall, 6 þ 6 þ 3 ¼ 15 real types exist: ð4:83Þ 015026-43 LI, REN, SHU, XIAO, YU, and ZHENG PHYS. REV. D 104, 015026 (2021) ð4:84Þ The only complex four-quark type with additional G, W,orB constitute the three complex types: 015026-44 COMPLETE SET OF DIMENSION-EIGHT OPERATORS IN THE … PHYS. REV. D 104, 015026 (2021) ð4:85Þ To see how Y’s act on operators one can refer to Eqs. (4.55) and (4.59). The conversion from the two-component spinors to the four-component spinors are similar to Eq. (4.75). The Hermitian conjugate of a non-Hermitian operator of this class is, for example, ij μν c a † μν † † ai † cj † ½ϵ B ðQraiuC t ÞðdC pσμνQscjÞ ¼ ϵijB ðuC tcQ r ÞðQ s σ¯ μνdC paÞ: ð4:86Þ Other operators of this class can be converted similarly. 2. Operators involving one lepton and three quarks with ΔB ¼ ΔL ¼ 1: In the four B-violating four-fermion couplings 2 2 at dimension six, uC dCeC consists of only the SUð ÞW singlet, which cannot couple to W in this class. Therefore we have 4 þ 4 þ 3 ¼ 11 types in all: ð4:87Þ 015026-45 LI, REN, SHU, XIAO, YU, and ZHENG PHYS. REV. D 104, 015026 (2021) ð4:88Þ 3. Operators involving two leptons and two quarks: Among the six real types with two leptons and two quarks, two 2 6 6 4 16 involve only SUð ÞW singlets. Hence we have þ þ ¼ real types: λA a A † j † bi σμν λA b A † ajσ¯ μν † i ð1∼4Þ ð ÞbGμνðL sQ t ÞðLpi QrajÞ; ð ÞaGμνðQrbiLpjÞðQ t L sÞ; O † † GQQ LL A a A † i † bj μν A b A † aj μν † i ðλ Þ GμνðL Q ÞðL σ Q Þ; ðλ Þ GμνðQ L ÞðQ σ¯ L Þ; b s t pi raj a rbj pi t s τI j I † i † ak σμν τI k I † ajσ¯ μν † i ð ÞkWμνðL sQ t ÞðLpi QrajÞ; ð Þj WμνðQrakLpiÞðQ t L sÞ; 1∼6 ð Þ I j I † k † ai μν I k I † aj μν † i O † † ðτ Þ WμνðL Q ÞðL σ Q Þ; ðτ Þ WμνðQ L ÞðQ σ¯ L Þ; WQQ LL k s t pi raj j rai pk t s τI i I † k † aj σμν τI k I † ajσ¯ μν † i ð ÞkWμνðL sQ t ÞðLpi QrajÞ; ð Þi WμνðQrajLpkÞðQ t L sÞ; † j † ai σμν † ajσ¯ μν † i ð1∼4Þ BμνðL sQ t ÞðLpi QrajÞ;BμνðQraiLpjÞðQ t L sÞ; O † † BQQ LL † i † aj μν † aj μν † i BμνðL sQ t ÞðLpiσ QrajÞ;BμνðQrajLpiÞðQ t σ¯ L sÞ; 015026-46 COMPLETE SET OF DIMENSION-EIGHT OPERATORS IN THE … PHYS. REV. D 104, 015026 (2021) Oð1;2Þ A a A † † bi μν A b A † ai μν † † † ðλ Þ GμνðeC Q ÞðeCpσ Q Þ; ðλ Þ GμνðQ eC ÞðQ σ¯ eC Þ; GQQ eCe b s t rai a rbi p t s C Oð1;2Þ τI i I † † aj σμν τI j I † aiσ¯ μν † † † ð Þ WμνðeC Q t ÞðeCp Q Þ; ð Þ WμνðQ eC ÞðQ eC Þ; WQQ eCe j s rai i raj p t s C Oð1;2Þ † † ai μν † ai μν † † † BμνðeC Q ÞðeCpσ Q Þ;BμνðQ eC ÞðQ σ¯ eC Þ; BQQ eCe s t rai rai p t s C Oð1;2Þ A b A † i † μν a A a A b † μν † i † † ðλ Þ GμνðL uC ÞðL σ uC Þ; ðλ Þ GμνðuC L ÞðuC σ¯ L Þ; GuCu LL a s tb pi r b r pi ta s C Oð1;2Þ τI i I † j † σμν a τI j I a † σ¯ μν † i † † ð Þ WμνðL suC ÞðL uC Þ; ð Þ WμνðuC L ÞðuC L Þ; WuCu LL j ta pi r i r pj ta s C Oð1;2Þ † i † μν a a † μν † i † † BμνðL uC ÞðL σ uC Þ;BμνðuC L ÞðuC σ¯ L Þ; BuCu LL s ta pi r r pi ta s C ð1;2Þ A b A † † μν a A a A b † μν † O † † ðλ Þ GμνðeC uC ÞðeCpσ uC Þ; ðλ Þ GμνðuC eC ÞðuC σ¯ eC Þ; GuCu eCe a s tb r b r p ta s C C ð1;2Þ † † μν a a † μν † O † † BμνðeC uC ÞðeCpσ uC Þ;BμνðuC eC ÞðuC σ¯ eC Þ; BuCu eCe s ta r r p ta s C C Oð1;2Þ A b A † † i a μν A a A b † i μν † † † ðλ Þ GμνðdC L ÞðdC σ L Þ; ðλ Þ GμνðL dC ÞðL σ¯ dC Þ; GdCd LL a sb t p ri b ri p t sa C Oð1;2Þ τI i I † † j a σμν τI j I a † iσ¯ μν † † † ð Þ WμνðdC L t ÞðdC L Þ; ð Þ WμνðL dC ÞðL dC Þ; WdCd LL j sa p ri i rj p t sa C Oð1;2Þ † † i a μν a † i μν † † † BμνðdC L ÞðdC σ L Þ;BμνðL dC ÞðL σ¯ dC Þ; BdCd LL sa t p ri ri p t sa C 1 2 ð ; Þ A b A † † a μν A a A b † μν † O † † ðλ Þ GμνðdC eC ÞðdC σ eCrÞ; ðλ Þ GμνðeC dC ÞðeC σ¯ dC Þ; GdCd eCe a sb t p b r p t sa C C Oð1;2Þ † † a μν a † μν † † † BμνðdC saeC tÞðdC pσ eCrÞ;BμνðeCrdC pÞðeC tσ¯ dC saÞ: ð4:89Þ BdCdCeCeC Both of the two complex four-fermion types with two leptons and two quarks can couple to all of the three gauge bosons; hence we have 2 þ 2 þ 2 ¼ 6 complex types: ij A a Aμν b ij A a Aμν b ϵ ðλ Þ G ðeCpuC ÞðL σμνQ Þ; ϵ ðλ Þ G ðL uC ÞðeCpσμνQ Þ; Oð1∼3Þ b t ri saj b ri t saj GQuCLeC ij A a Aμν b ϵ ðλ Þ G ðQsajuC t ÞðeCpσμνLriÞ; b jk I i Iμν a jk I i Iμν a ϵ ðτ Þ W ðeCpuC ÞðL σμνQ Þ; ϵ ðτ Þ W ðL uC ÞðeCpσμνQ Þ; Oð1∼3Þ k t ri saj k ri t saj WQuCLeC jk I i Iμν a ϵ ðτ Þ W ðQsajuC t ÞðeCpσμνLriÞ; k ij μν a ij μν a ϵ B ðeCpuC ÞðL σμνQ Þ; ϵ B ðL uC ÞðeCpσμνQ Þ; Oð1∼3Þ t ri saj ri t saj BQuCLeC ij μν a ϵ B ðQ uC ÞðeCpσμνL Þ; saj t ri Oð1;2Þ A b A † † i a μν A b A a † i μν † † † ðλ Þ GμνðeC L ÞðdC σ Q Þ; ðλ Þ GμνðQ dC ÞðL σ¯ eC Þ; GQdCL e a s t p rbi a rbi p t s C Oð1;2Þ τI i I † † j a σμν τI j I a † iσ¯ μν † † † ð Þ WμνðeC L t ÞðdC Q Þ; ð Þ WμνðQ dC ÞðL eC Þ; WQdCL e j s p rai i raj p t s C Oð1;2Þ † † i a σμν a † iσ¯ μν † † † BμνðeC sL tÞðdC p QraiÞ;BμνðQraidC pÞðL t eC sÞ: ð4:90Þ BQdCL eC 4. Operators involving only leptons: There should be no G coupled to the four-lepton types; hence we have 0 þ 3 þ 2 ¼ 5 real types as follows: 015026-47 LI, REN, SHU, XIAO, YU, and ZHENG PHYS. REV. D 104, 015026 (2021) ð4:91Þ further imply that the on-shell language may be much V. CONCLUSION closer to the essence of effective field theory than the In this paper, we provided the full result of the inde- traditional field theory language. pendent dimension-eight operator basis in the standard The second obstacle is to get a form with definite model effective field theory. Although the number of the permutation symmetries among the flavor indices. In liter- dimension-eight operators was already counted [12–15,18], ature, although the technique of plethysm is already widely and part of the list, only gauge bosons and the Higgs boson used [12–15,18] to perform a systematic counting of oper- involved, was also given in Refs. [22,24], it is the first time atorswithrepeatedfields,itisnotenoughforwritingdownthe that the two-fermion and four-fermion operators are listed explicit form of the operators. We propose a systematic in full form that constitute over half of the complete list. method to solve this issue. To obtain the basis particularly for What is more important is that the form of the operators we an irreducible representation space of the permutation sym- provide here has definite symmetry over the flavor indices, metry S¯, which permutes fields only within the group of making it possible to identify independent flavor-specified repeated fields, we apply the left ideal projector of the group operators. These flavor-independent operators were never algebra to an already-found independent basis, either for the obtained in the past, nor a systematic approach, for various Lorentz structure or the gauge group tensors. Then by use of higher-dimensional operators, including the Warsaw basis the Clebsch-Gordan coefficients of the inner product decom- in dimension six [3]. position, we combine all the symmetrized factors to get a To achieve the goal, we need to overcome two main flavor tensor with definite permutation symmetry. The obstacles. The first is to list all the independent Lorentz independent flavor-specified operators are thus given by, structures. The methods used in literature, such as the again, the semistandard Youngtableau.This essential feature Hilbert series, are usually good for counting the number of on the flavor structure makes our result more practically independent Lorentz structures, but not suitable for writing useful than the other papers on listing higher-dimensional down the explicit form of the operators. Inspired by operators. With the flavor structure addressed, we list the [33,37], we introduce a SUðNÞ transformation of the complete and independent set of the flavor-specified dimen- operators, which divides the space of Lorentz structures sion-9operatorsoftheSMEFTintheforthcomingpaper[44]. into complementary invariant subspaces, one of which After the complete list of operators is written, it is worth- consists of those with factors of total derivatives. The other while to investigate various phenomenological applications invariant subspace, which turns out to be a single irreduc- of these operators. As mentioned in the Introduction, if the ible representation space, is hence a linear space of contribution from dimension-six operators is subdominant independent operators regarding the integration by parts. or highly constrained, the dimension-eight operators should Group theory allows us to use the semistandard Young be seriously considered, even though their Wilson coef- tableau to enumerate a basis for this irreducible represen- ficients are suppressed by a higher inverse power of the new tation space, which is the basis of Lorentz structures we are physics scale. We notice there are several new Lorentz looking for. It is worth mentioning that the notation of structures that only appear at the dimension-eight level, and operators used in this derivation is largely inspired by the there are several dimension-eight operators dominant over on-shell amplitudes, which is made possible by a corre- the dimension-six operators. These phenomenological spondence proposed in Refs. [28–32]. This work may applications deserve a closer look in the future. 015026-48 COMPLETE SET OF DIMENSION-EIGHT OPERATORS IN THE … PHYS. REV. D 104, 015026 (2021) The whole procedure is implemented and automized by Lorentz indices, two-component Weyl spinor vs four- Mathematica, and our code can easily be applied to higher component Dirac spinor, various forms of four-fermion dimensions of SMEFT and other EFTs beyond the SM. In couplings related by Fierz identities, and the chiral basis ˜ terms of efficiency, listing the dimension-eight operators FL=R vs Hermitian basis F; F of the gauge bosons. only cost less than 2 min on a personal laptop. 1. Identities for spinors ACKNOWLEDGMENTS a. 1.σ techniques We are grateful to Christopher W. Murphy for cor- respondence on their work on the same topic, and we This part is devoted to conversions between Lorentz also thank Adam Martin and Veronica Sanz for valuable structures written with all spinor indices, while all factors 2 C discussions. H. L. L. thanks Zhou Xu for helpful dis- are in irreducible representations of SLð ; Þ, and the form cussion on programming problems. J. H. Y. thanks Jordy with the usual Lorentz indices μ, ν, etc., running over de Vries for discussion that initializes this project and 0,1,2,3, on derivatives and the gauge bosons. The key of the appreciates the hospitality of Amherst Center for conversion is at the reduction of σ products. We adopt the Fundamental Interactions, University of Massachusetts at following definitions: the metric is “mostly minus” gμν ¼ Amherst,whileworkingonthisproject.J. S.,M. L. X.,andJ. diagðþ1; −1; −1; −1Þ; the Levi-Civita tensors are ϵ0123 ¼ 12 H. Y thank Tom Melia for useful comments. H. L. L., Z. R., −ϵ0123 ¼þ1 and ϵ ¼ ϵ21 ¼þ1; the sigma matrices are μ i μ μαα_ αα_ iαα_ μ and J. H. Y. are supported by the National Natural Science defined as σαα_ ¼ð1αα_ ; ταα_ Þ , σ¯ ¼ð1 ; −τ Þ , with Foundation of China (NSFC) under Grants No. 12022514 identity 1 and Pauli matrices τi;i¼ 1, 2, 3. The two and No. 11875003. J. H. Y.is also supported by the National sigmas are related by raising and lowering indices by Natural Science Foundation of China (NSFC) under Grant the ϵ tensor No. 12047503. J. S. is supported by the National Natural Science Foun- dation of China under Grants No. 12025507, _ μ σ¯ μαα_ ϵαβϵα_ βσ : No. 11690022, No. 11947302, and is supported by the ¼ ββ_ Strategic Priority Research Program and Key Research Program of Frontier Science of the Chinese Academy of We also define Sciences under Grants No. XDB21010200, No. XDB23010000, and No. ZDBS-LY-7003. M. L. X. is μν β i μ ν ν μ β ðσ Þα ¼ ðσ σ¯ − σ σ¯ Þα ; ðA1Þ supported by the National Natural Science Foundation of 2 China (NSFC) under Grant No. 2019M650856 and the 2019 International Postdoctoral Exchange Fellowship Program. σ¯ μν α_ i σ¯ μσν − σ¯ νσμ α_ ð Þ β_ ¼ 2 ð Þ β_ ; ðA2Þ Note added.—Reference [34] also presents a list of the σ dimension-eight operators in the standard model effective which directly induce the decomposition of two products: field theory. There are two main differences between our σμσ¯ ν β μνδβ − σμν β works. First, we provide a systematic and automated ð Þα ¼ g α ið Þα ; ðA3Þ method in which we obtain an independent basis directly σ¯ μσν α_ μνδα_ − σ¯ μν α_ in which the EOM and IBP redundancies are entirely ð Þ β_ ¼ g β_ ið Þ β_ : ðA4Þ absent. This would help our method apply to more com- plicated cases where the correctness of our result is For more than two σ’s multiplying as a chain, we may use guaranteed from the first principle. Second, in contrast the following three σ decompositions: to [34], the form of the operators we provide has definite symmetry over the flavor indices, and thus the independent μ ν ρ μν ρ μρ ν νρ μ μνρλ ðσ σ¯ σ Þ _ ¼ g σ −g σ þg σ þiϵ σ _ ; ðA5Þ flavor-specified operators could be obtained easily as a αβ αβ_ αβ_ αβ_ λαβ semistandard Young tableau. μ ν ρ αβ_ μν ναβ_ μρ ναβ_ νρ μαβ_ μνρλ αβ_ ðσ¯ σ σ¯ Þ ¼ g σ¯ −g σ¯ þg σ¯ −iϵ σ¯ λ ; ðA6Þ APPENDIX A: CONVERSION BETWEEN NOTATIONS to recursively reduce it toward a linear combination of 1; σμ; σ¯ μ; σμν, and σ¯ μν. The Hermitian conjugates of these Various people have various conventions for how bilinears are given by operators are written, while our result is presented only in one of them. In this Appendix, we provide a complete set ψ ψ † ψ †ψ † of identities for conversions of Lorentz structures between ð 1 2Þ ¼ 2 1; μ † † μ † different conventions, together with a bunch of examples, ðψ 1σ ψ 2Þ ¼ ψ 2σ ψ 1; in order to make it easier for different readers to use our ψ σμνψ † ψ †σ¯ μνψ † result. Relevant conventions are SLð2; CÞ vs SOð3; 1Þ ð 1 2Þ ¼ 2 1: ðA7Þ 015026-49 LI, REN, SHU, XIAO, YU, and ZHENG PHYS. REV. D 104, 015026 (2021) To compute the trace of a σ’s chain, one simply reduces the Q 0 0 chain to the above basic forms and takes the trace as qL ¼ ;uR ¼ † ;dR ¼ † ; 0 uC dC follows: L 0 μ μ μν μν lL ¼ ;eR ¼ † ; ðA12Þ Tr1 ¼ 2; Trσ ¼ Trσ¯ ¼ Trσ ¼ Trσ¯ ¼ 0: ðA8Þ 0 eC σ ¯ 0 † ¯ 0 ¯ 0 The frequently used example of four chain and trace is qL ¼ð ;Q Þ; uR ¼ðuC; Þ; dR ¼ðdC; Þ; given as follows: ¯ 0 † ¯ 0 lL ¼ð ;L Þ; eR ¼ðeC; Þ: ðA13Þ σμσ¯ νσρσ¯ κ gμνgρκ − gμρgνκ gνρgμκ iϵμνρκ 1 ¼ð þ þ Þ The conversion rules of the fermion bilinears in the SM to μν ρκ μρ νκ νρ μκ μνρλ κ − iðg σ − g σ þ g σ þ iϵ σλ Þ; the four-component notation are obtained by substituting μ ν ρ κ μν ρκ μρ νκ νρ μκ μνρκ these fields into the relations in Eq. (A11), such as Trðσ σ¯ σ σ¯ Þ¼2g g − 2g g þ 2g g þ 2iϵ ; μ ν ρ κ μν ρκ μρ νκ νρ μκ μνρκ μ † μ † † T Trðσ¯ σ σ¯ σ Þ¼2g g − 2g g þ 2g g − 2iϵ : uCσ uC ¼ u¯γ u; eCL ¼ el;¯ uCdC ¼ u Cd: ðA9Þ ðA14Þ c. A brief introduction to Fierz identities b. Converting two-component to four-component spinor The following 16 bilinear forms constitute a complete basis of the 4 × 4 Hermitian matrices In this part, we use Ψ; Ψ¯ to denote four-component ξ χ S spinors and , to denote two-component left-handed Γ1 ¼ 1; ðA15Þ spinors, while their Hermitian conjugates ξ†, χ† are right- V V μ handed spinors. Generally, we may combine a left-handed Γ1 through Γ4 ¼ γ ; ðA16Þ Weyl spinor ξα and an independent right-handed Weyl χ†α_ T T μν spinor into a four-component Dirac spinor Γ1 through Γ6 ¼ σ ; ðA17Þ ξ μ α ¯ † 0 α † ΓA ΓA γ γ Ψ ¼ ; Ψ ¼ Ψ γ ¼ðχ ; ξ Þ: ðA10Þ 1 through 4 ¼ 5; ðA18Þ χ†α_ α_ P Γ1 ¼ γ5: ðA19Þ We can then write down the spinor bilinears that are commonly used Labels S, V, T, A, P denote scalar, vector, tensor, axial- μν i μ ν vector, and pseudoscalar, respectively, while σ ¼ 2½γ ;γ ; ¯ α † †α_ Ψ1Ψ2 ¼ χ ξ2α þ ξ χ ; 0 1 2 3 −10 1 1α_ 2 γ5 ¼ iγ γ γ γ ¼ð 0 1Þ. The inner product between them is ¯ μ α μ †α_ † μαα_ Ψ1γ Ψ2 ¼ χ1σαα_ χ2 þ ξ1α_ σ¯ ξ2α; defined as _ ¯ μν α μν β † μν α_ β A B AB Ψ σ Ψ χ σ ξ ξ σ¯ _ χ Γ Γ δ A B 1 2 ¼ 1ð Þα 2β þ 1α_ ð Þ β 2; Tr½ a b ¼ gab; ; ¼ S; V; T; A; P; T α † †α_ 1 … A 1 … B Ψ1 CΨ2 ¼ ξ1ξ2α þ χ1α_ χ2 ; a ¼ ; ; dim ;b¼ ; ; dim : ðA20Þ ΨT γμΨ ξασμ χ†α_ χ† σ¯ μαα_ ξ 1 C 2 ¼ 1 αα_ 2 þ 1α_ 2α; Regarding g as the metric depending on our choice of _ T μν α μν β † μν α_ †β coordinates in each subspace, and using it to raise and Ψ Cσ Ψ2 ¼ ξ ðσ Þ ξ2β þ χ ðσ¯ Þ _ χ ; 1 1 α 1α_ β 2 lower indices, the inner product induces an orthogonality ¯ ¯ T † †α_ α 4 4 Ψ1CΨ2 ¼ ξ1_αξ2 þ χ1χ2α; relation, which allowsP any × matrix M to be expanded aΓ a ¯ μ ¯ T α μ †α_ † μαα_ in this basis as M ¼ a M a, with coordinates M ¼ Ψ1γ CΨ χ σ ξ ξ σ¯ χ2α; 2 ¼ 1 αα_ 2 þ 1α_ TrðMΓaÞ. _ ¯ μν ¯ T † μν α_ †β α μν β Fierz transformations of four-fermion couplings are the Ψ1σ CΨ2 ¼ ξ1α_ ðσ¯ Þ β_ ξ2 þ χ1ðσ Þα χ2β; ðA11Þ linear transformations: μ ϵ αβ σ X X X γ0γ2 αβ 0 −ϵ 0 γμ 0 αβ_ A A B B where C ¼ i ¼ð0 α_ β_ Þ¼ð 0 −ϵ Þ, ¼ðσ¯ μαβ_ 0 Þ Γ Γ a Γ Γ b ϵ α_ β_ ð a Þijð Þkl ¼ CAB ð b Þilð Þkj: ðA21Þ μν i μ ν a B b in the chiral representation, and σ ¼ 2 ½γ ; γ ¼ μν β ðσ Þα 0 ð 0 σ¯ μν α_ Þ. In the SM, the four-component chiral fer- ð Þ β_ According to the orthogonalityP Eq. (A20), we infer ΓAΓBΓAaΓBb mions are related to our notations of two-component immediately CAB ¼ a Trð a b Þ. Calculating all fermions as the CAB to get the following formula: 015026-50 COMPLETE SET OF DIMENSION-EIGHT OPERATORS IN THE … PHYS. REV. D 104, 015026 (2021) 0 δ δ 1 0 10 δ δ 1 ij kl 1=41=41=4 −1=41=4 il kj B γμ γ C B CB γμ γ C B ð Þijð μÞkl C B 1 −1=20−1=2 −1 CB ð Þilð μÞkj C B C B CB C B 1 σμν σ C B CB 1 σμν σ C B 2 ð Þijð μνÞkl C ¼ B 3=20−1=203=2 CB 2 ð Þilð μνÞkj C: ðA22Þ B C B CB C @ γμγ γ γ A @ −1 −1 20−1 21A@ γμγ γ γ A ð 5Þijð μ 5Þkl = = ð 5Þilð μ 5Þkj γ γ 1 4 −1 4141414 γ γ ð 5Þijð 5Þkl = = = = = ð 5Þilð 5Þkj ¯ ¯ Both sides of Eq. (A22) contract with Ψ1iΨ2jΨ3kΨ4l, 0 1 0 10 δ δ 1 CijCkl −1=41=41=41=4 −1=4 il jk B γμ γ C B CB γμ γ C B ð CÞijðC μÞkl C B −1 −1=20 1=21CB ð Þilð μÞjk C B C B CB C B 1 σμν σ C B CB 1 σμν σ C B 2 ð CÞijðC μνÞkl C ¼ B −3=20−1=20−3=2 CB 2 ð Þilð μνÞjk C: ðA23Þ B C B CB C @ γμγ γ γ A @ 1 −1 20 12 −1 A@ γμγ γ γ A ð 5CÞijðC μ 5Þkl = = ð 5Þilð μ 5Þjk γ γ −1 4 −1 414 −1 4 −1 4 γ γ ð 5CÞijðC 5Þkl = = = = = ð 5Þilð 5Þjk ¯ ¯ Both sides of Eq. (A23) contract with Ψ1iΨ2jΨ3kΨ4l. With these Fierz identities, some four-fermion interactions can be transformed to couplings of neutral fermion currents, 1 1 1 1 1 ¯ ¯ − ¯ ¯ − ¯γμ ¯γ − ¯σμν ¯σ ¯γμγ ¯γ γ − ¯γ ¯γ ðdlÞðldÞ¼ 4 ðddÞðllÞ 4 ðd dÞðl μlÞ 8 ðd dÞðl μνlÞþ4 ðd 5dÞðl μ 5lÞ 4 ðd 5dÞðl 5lÞ 1 − ¯γμ ¯γ ¼ 2 ðd dÞðl μlÞ; ðA24Þ 1 1 1 1 1 ¯ ¯ − ¯ ¯ ¯γμ ¯γ ¯σμν ¯σ ¯γμγ ¯γ γ − ¯γ ¯γ ðlCqÞðlCqÞ¼ 4 ðllÞðqqÞþ4 ðl lÞðq μqÞþ8 ðl lÞðq μνqÞþ4 ðl 5lÞðq μ 5qÞ 4 ðl 5lÞðq 5qÞ 1 ¯γμ ¯γ ¼ 2 ðl lÞðq μqÞ: ðA25Þ It is worth mentioning that Γa may also be generators of fundamental SUðNÞ, denoted by Ta. Since f1;Tag is a complete A set of N × N Hermitian matrices, substituting Γa ¼ Ta; ΓI ¼ 1; TrðTaTbÞ¼δab into Eq. (A21), we get the Fierz identity for SUðNÞ group as X 1 δ δ − δ δ ðTaÞijðTaÞkl ¼ il kj ij kl: ðA26Þ a N d. Examples Under Fierz identities, some terms can be transformed into bilinear form which readers may be more familiar with. Here are some examples. Oð1∼4Þ (i) Example 1, type QQ†HH†D3 , ↔ ↔ ↔ ν † ai μ † ¯ ai ν μ † ¯ ν μ † iðQpaiσμD Q r ÞðD H DνHÞ¼iðqr γμD qpaiÞðD H DνHÞ¼iðqrγμD qpÞðD H DνHÞ; ðA27Þ ↔ ↔ ν † aj μ †i ¯ aj ν μ †i iðQpaiσμD Q r ÞDνHjD H ¼ iðqr γμD qpaiÞDνHjD H 1 ↔ ¯ aiγ ν μ †j ¼ 2 iðqr μD qpaiÞDνHjD H ↔ 1 ↔ ¯ ajγ ν μ †i − ¯ aiγ ν μ †j þ i ðqr μD qpaiÞDνHjD H 2 ðqr μD qpaiÞDνHjD H 1 ↔ ↔ ¯ γ ν μ † ¯ γ τI ν μ †τI ¼ 2 iðqr μD qpÞðD H DνHÞþiðqr μ D qpÞðD H DνHÞ: ðA28Þ 015026-51 LI, REN, SHU, XIAO, YU, and ZHENG PHYS. REV. D 104, 015026 (2021) Hence, the basis can be transformed into 8 8 ↔ ↔ > † ai † μ > † μ > iðQ σμQ Þ□ðH D HÞ > iðq¯ γμq Þ□ðH D HÞ > pai r > r p > ↔ > ↔ < † aj †i μ < I † I μ iðQ σμQ Þ□ðH D H Þ iðq¯ γμτ q Þ□ðH τ D HÞ pai r j ⇒ r p > ↔ > ↔ : ðA29Þ > † ai μ † ν > μ † ν > iðQ σμDνQ ÞðD H D HÞ > iðq¯ γμDνq ÞðD H D HÞ > pai r > r p > ↔ > ↔ : σ † aj μ †i ν : ¯ I μ †i I ν iðQpai μDνQ r ÞðD H D HjÞ iðqrγμτ DνqpÞðD H τ D HjÞ ð1∼4Þ (ii) Example 2, type O † † † , QQ uCuCHH 1 b † ai † † ¯ b ¯ ai † − ¯ aiγμ ¯ bγ † ðQpaiuC r ÞðQ s uC tbÞðH HÞ¼ður qpaiÞðqs utbÞðH HÞ¼ 2 ðqs qpaiÞður μutbÞðH HÞ 1 − ¯ γμ ¯ γ † ¼ 2 ðqs qpÞður μutÞðH HÞ; ðA30Þ 1 a † ci † † ¯ a ¯ ci † − ¯ ciγμ ¯ aγ † ðQpaiuC r ÞðQ s uC tcÞðH HÞ¼ður qpaiÞðqs utcÞðH HÞ¼ 2 ðqs qpaiÞður μutcÞðH HÞ 1 1 − ¯ ciγμ ¯ aγ † − ¯ aiγμ ¯ cγ † ¼ 2 ðqs qpaiÞður μutcÞðH HÞ 3 ðqs qpaiÞður μutcÞðH HÞ 1 − ¯ aiγμ ¯ cγ † 6 ðqs qpaiÞður μutcÞðH HÞ 1 1 − ¯ γμλA ¯ γ λA † − ¯ γμ ¯ γ † ¼ 2 ðqs qpÞður μ utÞðH HÞ 6 ðqs qpÞður μutÞðH HÞ; ðA31Þ 1 b † aj † †i ¯ b ¯ aj †i − ¯ ajγμ ¯ bγ †i ðQpaiuC r ÞðQ s uC tbÞH Hj ¼ður qpaiÞðqs utbÞH Hj ¼ 2 ðqs qpaiÞður μutbÞH Hj 1 1 − ¯ ajγμ ¯ bγ †i − ¯ aiγμ ¯ bγ †j ¼ 2 ðqs qpaiÞður μutbÞH Hj 2 ðqs qpaiÞður μutbÞH Hj 1 − ¯ aiγμ ¯ bγ †j 4 ðqs qpaiÞður μutbÞH Hj 1 1 − ¯ γμτI ¯ γ †τI − ¯ γμ ¯ γ † ¼ 2 ðqs qpÞður μutÞðH HÞ 4 ðqs qpÞður μutÞðH HÞ; ðA32Þ 1 1 a † cj † †i − ¯ γμτIλA ¯ γ λA †τI − ¯ γμτI ¯ γ †τI ðQpaiuC r ÞðQ s uC tcÞH Hj ¼ 2 ðqs qpÞður μ utÞðH HÞ 6 ðqs qpÞður μutÞðH HÞ 1 1 − ¯ γμλA ¯ γ λA † − ¯ γμ ¯ γ † 4 ðqs qpÞður μ utÞðH HÞ 12 ðqs qpÞður μutÞðH HÞ: ðA33Þ Hence, the basis can be transformed into 8 8 > a † cj † †i > ¯ γμ ¯ γ † > ðQpaiuC r ÞðQ s uC tcÞH Hj > ðqs qpÞður μutÞðH HÞ > > < a † ci † † < ¯ γμλA ¯ γ λA † ðQpaiuC r ÞðQ s uC tcÞðH HÞ ðqs qpÞður μ utÞðH HÞ ⇒ : ðA34Þ > † aj † † > μ I † I > b i > q¯ γ τ q u¯ γμu H τ H > ðQpaiuC r ÞðQ s uC tbÞH Hj > ð s pÞð r tÞð Þ :> :> μ † b † ai † † ¯ γ τIλA ¯ γ λA τI ðQpaiuC r ÞðQ s uC tbÞðH HÞ ðqs qpÞður μ utÞðH HÞ 015026-52 COMPLETE SET OF DIMENSION-EIGHT OPERATORS IN THE … PHYS. REV. D 104, 015026 (2021) Oð1∼5Þ (iii) Example 3, type QQ†LL†HH† , 1 † k † aj †i ¯kγμ ¯ ajγ †i ðLpiQrajÞðL sQ t ÞH Hk ¼ 2 ðls lpiÞðqt μqrajÞH Hk 1 1 ¯ γμτI ¯ γ †τI ¯ γμ ¯ γ † ¼ 2 ðls lpÞðqt μqrÞðH HÞþ4 ðls lpÞðqt μqrÞðH HÞ; ðA35Þ 1 1 † j † ai † ¯ γμτI ¯ γ τI † ¯ γμ ¯ γ † ðLpiQrajÞðL sQ t ÞðH HÞ¼2 ðls lpÞðqt μ qrÞðH HÞþ4 ðls lpÞðqt μqrÞðH HÞ; ðA36Þ 1 1 † i † ak †j ¯ γμ ¯ γ τI †τI ¯ γμ ¯ γ † ðLpiQrajÞðL sQ t ÞH Hk ¼ 2 ðls lpÞðqt μ qrÞðH HÞþ4 ðls lpÞðqt μqrÞðH HÞ; ðA37Þ 1 † j † ak †i ¯jγμ ¯ akγ †i ðLpiQrajÞðL sQ t ÞH Hk ¼ 2 ðls lpiÞðqt μqrajÞH Hk 1 1 ¯ γμτI ¯ akγ τI l †m ¯ γμ ¯ akγ †l ¼ 2 ðls lpÞðqt μqralÞð ÞmH Hk þ 4 ðls lpÞðqt μqralÞH Hk 1 1 ¯ γμτI ¯ γ τJ †τIτJ ¯ γμτI ¯ γ †τI ¼ 2 ðls lpÞðqt μ qrÞðH HÞþ4 ðls lpÞðqt μqrÞðH HÞ 1 1 ¯ γμ ¯ γ τI †τI ¯ γμ ¯ γ † þ 4 ðls lpÞðqt μ qrÞðH HÞþ8 ðls lpÞðqt μqrÞðH HÞ 1 1 ϵIJK ¯ γμτI ¯ γ τJ †τK ¯ γμτI ¯ γ τI † ¼ 2 i ðls lpÞðqt μ qrÞðH HÞþ4 ðls lpÞðqt μ qrÞðH HÞ 1 1 ¯ γμτI ¯ γ †τI ¯ γμ ¯ γ τI †τI þ 4 ðls lpÞðqt μqrÞðH HÞþ4 ðls lpÞðqt μ qrÞðH HÞ 1 ¯ γμ ¯ γ † þ 8 ðls lpÞðqt μqrÞðH HÞ; ðA38Þ 1 1 † k † ai †j ϵIJK ¯ γμτI ¯ γ τK †τJ ¯ γμτI ¯ γ τI † ðLpiQrajÞðL sQ t ÞH Hk ¼ 2 i ðls lpÞðqt μ qrÞðH HÞþ4 ðls lpÞðqt μ qrÞðH HÞ 1 1 ¯ γμτI ¯ γ †τI ¯ γμ ¯ γ τI †τI þ 4 ðls lpÞðqt μqrÞðH HÞþ4 ðls lpÞðqt μ qrÞðH HÞ 1 ¯ γμ ¯ γ † þ 8 ðls lpÞðqt μqrÞðH HÞ; ðA39Þ Hence, the basis can be transformed into 8 8 > † i † ak †j > ¯ γμ ¯ γ † > ðLpiQrajÞðL sQ t ÞH Hk > ðls lpÞðqt μqrÞðH HÞ > > > L Q L† kQ† ai H†jH > ¯ γμ ¯ γ τI †τI <> ð pi rajÞð s t Þ k <> ðls lpÞðqt μ qrÞðH HÞ L Q L† jQ† ak H†iH ⇒ ¯ γμτI ¯ γ τI † > ð pi rajÞð s t Þ k > ðls lpÞðqt μ qrÞðH HÞ : ðA40Þ > > > † j † ai † > ¯ γμτI ¯ γ †τI > ðLpiQrajÞðL sQ t ÞðH HÞ > ðls lpÞðqt μqrÞðH HÞ > > : † k † aj †i : ϵIJK ¯ γμτI ¯ γ τJ †τK ðLpiQrajÞðL sQ t ÞH Hk ðls lpÞðqt μ qrÞðH HÞ Oð1∼5Þ (iv) Example 4, type L2L†2HH† , 1 † j † i † ¯ γμ ¯ γ † ðLpiLrjÞðL sL tÞðH HÞ¼2 ðls lrÞðlt μlpÞðH HÞ; ðA41Þ 1 † j † k †i ¯ γμ ¯kγ †i ðLpiLrjÞðL sL t ÞH Hk ¼ 2 ðls lrÞðlt μlpiÞH Hk 1 1 ¯ γμ ¯ γ τI †τI ¯ γμ ¯ γ † ¼ 2 ðls lrÞðlt μ lpÞðH HÞþ4 ðls lrÞðlt μlpÞðH HÞ: ðA42Þ 015026-53 LI, REN, SHU, XIAO, YU, and ZHENG PHYS. REV. D 104, 015026 (2021) Since different Y’s indicate different flavor symmetries, operators with different Y’s should not be mixed if one does not want to confuse the flavor symmetry. means we will get a minus sign if we exchange p,r. The basis can be transformed into ðA43Þ ðA44Þ Oð1∼4Þ (v) Example 5, type L2L†2D2 , 1 † i μ † j ¯ γ μ¯ γν ðLpiLrjÞðDμL sD L t Þ¼2 ðDμls νlpÞðD lt lrÞ: ðA45Þ † i † j μν μ † i † j Operator iðDμL sDνL t ÞðLpiσ LrjÞ is equivalent to ðLpiD LrjÞðL sDμL t Þ up to IBP and EOM, and 1 μ † i † j ¯ γ μ¯ γν ðLpiD LrjÞðL sDμL t Þ¼2 ðls νlpÞðD lt DμlrÞ: ðA46Þ Hence, the basis can be transformed into ðA47Þ Oð1∼4Þ (vi) Example 6, type BQ3L , ðA48Þ Oð1∼4Þ It should be noted that BQ3L is a complex type, which means the Hermitian conjugate of independent operators of this type are still independent operators. 015026-54 COMPLETE SET OF DIMENSION-EIGHT OPERATORS IN THE … PHYS. REV. D 104, 015026 (2021) At last we give some examples of how Y’s act on operators, ðA49Þ with Oð1Þ † aj † bk † aj † bk †i Q2Q†2HH† ¼ðQpaiQrbj þ QraiQpbjÞðQ s Q t þ Q t Q s ÞH Hk; Oð4Þ † aj † bk − † aj † bk †i Q2Q†2HH† ¼ðQpaiQrbj þ QraiQpbjÞðQ s Q t Q t Q s ÞH Hk; Oð6Þ − † aj † bk † aj † bk †i Q2Q†2HH† ¼ðQpaiQrbj QraiQpbjÞðQ s Q t þ Q t Q s ÞH Hk; Oð8Þ − † aj † bk − † aj † bk †i Q2Q†2HH† ¼ðQpaiQrbj QraiQpbjÞðQ s Q t Q t Q s ÞH Hk; ðA50Þ and ðA51Þ with Oð1Þ ϵabcϵimϵjn †k Q3LHH† ¼ ½ðQrajQtcmÞðLpiQsbkÞþðQsajQtcmÞðLpiQrbkÞ H Hn abc im jn †k − ϵ ϵ ϵ ½ðQtajQrcmÞðLpiQsbkÞþðQtajQscmÞðLpiQrbkÞ H Hn; Oð4Þ ϵabcϵimϵjn †k Q3LHH† ¼ ½ðQrajQtcmÞðLpiQsbkÞþðQsajQtcmÞðLpiQrbkÞ H Hn abc im jn †k þ ϵ ϵ ϵ ½ðQtajQrcmÞðLpiQsbkÞþðQrajQscmÞðLpiQtbkÞ H Hn abc im jn †k þ ϵ ϵ ϵ ½ðQtajQscmÞðLpiQrbkÞþðQsajQrcmÞðLpiQtbkÞ H Hn; Oð6Þ ϵabcϵimϵjn − †k Q3LHH† ¼ ½ðQrajQtcmÞðLpiQsbkÞ ðQsajQtcmÞðLpiQrbkÞ H Hn abc im jn †k − ϵ ϵ ϵ ½ðQtajQrcmÞðLpiQsbkÞþðQrajQscmÞðLpiQtbkÞ H Hn abc im jn †k þ ϵ ϵ ϵ ½ðQtajQscmÞðLpiQrbkÞþðQsajQrcmÞðLpiQtbkÞ H Hn: ðA52Þ ˜ 2. Conversion between FL=R and F;F Feynman rule calculations. In this subsection we summa- From Sec. III B 1 it is clear that we are strongly inclined rize the conversion rules between the two bases. We start by writing down the definitions: to use the chiral basis of the gauge bosons FL=R, which massively simplifies our derivations. Physically, it may be 1 1 due to their direct correspondence with on-shell particles ˜ μν μνρη ˜ F ¼ ϵ Fρη;F¼ ðF ∓ iFÞ; ðA53Þ with definite helicities. However, the other basis that is 2 L=R 2 more commonly used, F; F˜ , also has many privileges such as its Hermiticity and definite CP. Moreover, a lot of from which we can easily deduce the following useful applications are also based on the F; F˜ basis, such as the identities: 015026-55 LI, REN, SHU, XIAO, YU, and ZHENG PHYS. REV. D 104, 015026 (2021) 1 ˜ ρν − νρ ˜ − ˜ δν 1 F1μρF2 ¼ F1 F2ρμ 2 ðF1F2Þ μ; ðA54Þ ρν δν 2 − ˜ FRμρFR ¼ 8 μðF iFFÞ; ðA63Þ 1 ˜ ˜ ρν νρ ν F1μρF2 ¼ F1 F2ρμ þ ðF1F2Þδμ: ðA55Þ 1 1 2 ρν ρν 2δν FLμρFR ¼ 2 FμρF þ 8 F μ: ðA64Þ In the following, we present various situations where we ˜ explicitly do the conversions as examples: When F1, F2 are antisymmetric, thus ðF1F2Þ¼ðF1F2Þ¼ 0 (for instance, they have antisymmetric group indices), we a. Operators involving one gauge bosons can deduce When the gauge boson contracts with a two form Oμν † 1 with property Oμν ¼ Oμν, we have ρν 2 ρν νρ ˜ − ˜ ρν F1LμρF2L ¼ 4 ð F1μρF2 þ iF1 F2ρμ iF1μρF2 Þ; μνO O μν O ˜ μν CFL μν þ H:c: ¼ðReCÞ μνF þðImCÞ μνF ; ðA65Þ A56 ð Þ 1 ρν ρν νρ ˜ ˜ ρν F1 μρF2 ¼ ð2F1μρF2 − iF1 F2ρμ þ iF1μρF2 Þ; † R R 4 while for Oμν ¼ Oνμ we get instead ðA66Þ μνO O μν O ˜ μν CFL μν þ H:c: ¼ðImCÞ μνF þðReCÞ μνF : 1 ðA57Þ ρν νρ ˜ ˜ ρν F1LμρF2R ¼ 4 ðiF1 F2ρμ þ iF1μρF2 Þ: ðA67Þ In particular, when F ;F contract with σμν, it is further L R For examples to get operators in Eq. (4.49) we performed simplified to the following conversions: μν σ β μν σ β − ˜ μν σ β FL ð μνÞα ¼ F ð μνÞα ¼ iF ð μνÞα ; ↔ ABC A μ B ν σλ λC a † bi μν σ¯ α_ 0 id GL νGR λðQpai ð ÞbDμQ r Þ FL ð μνÞ β_ ¼ ; ðA58Þ ↔ i ABC Aμ Bν λ C a † bi ¼ d G νG λðQ σ ðλ Þ DμQ Þ; ðA68Þ μν σ β 0 2 pai b r FR ð μνÞα ¼ ; μν σ¯ α_ μν σ¯ α_ ˜ μν σ¯ α_ FR ð μνÞ β_ ¼ F ð μνÞ β_ ¼ iF ð μνÞ β_ : ðA59Þ ↔ ABC A μ B ν σλ λC a † bi f GL νGR λðQpai ð ÞbDμQ r Þ ˜ ↔ Note that the basis F; F are degenerate when contracting i ABC Aμν ˜ B A ˜ Bνμ λ C a † bi ¼ f ðG Gνλ þ GλνG ÞðQ σ ðλ Þ DμQ Þ; with σμν. In our result, for instance, Eq. (4.36), we adopt F 4 pai b r ˜ instead of F in the operators. ðA69Þ b. Operators involving two gauge boson while for the complex type with complex Wilson coef- μν For the X Xμν contractions, we have ficient C we get 1 ↔ μν ˜ ABC A μ B ν λ C a † bi F1 F2 μν 0; F1 F2 F1F2 − iF1F2 ; σ λ L R ¼ ð L LÞ¼2 ð Þ Cf GL νGL λðQpai ð ÞbiDμQ r ÞþH:c: ↔ 1 ABC Aμν B λ C a † bi ˜ ¼ðReCÞf G GνλðQpaiσ ðλ Þ iDμQ r Þ ðF1RF2RÞ¼ ðF1F2 þ iF1F2Þ: ðA60Þ b 2 1 ABC Aμν ˜ B A ˜ Bνμ þ ðImCÞf ðG Gνλ − GλνG Þ Thus in the operator they are recombined as 2 ↔ σλ λC a † bi O O ˜ O × ðQpai ð ÞbiDμQ r Þ: ðA70Þ CðF1LF2LÞ þ H:c: ¼ðReCÞðF1F2Þ þðImCÞðF1F2Þ ; σ ðA61Þ When two FL or FR contract with μν, we write the conversion rules similar with Eqs. (A58) and (A59), where O† ¼ O is Hermitian. Contractions of the form ν XμνX ρ are converted as μλ ν σ β μλ ν σ β F1L F2Lλ ð μνÞα ¼ F1 F2λ ð μνÞα ; ðA71Þ 1 ρν ν 2 ˜ F μρF ¼ δμðF þ iFFÞ; ðA62Þ μλ ν σ¯ α_ μλ ν σ¯ α_ L L 8 F1R F2Rλ ð μνÞ β_ ¼ F1 F2λ ð μνÞ β_ : ðA72Þ 015026-56 COMPLETE SET OF DIMENSION-EIGHT OPERATORS IN THE … PHYS. REV. D 104, 015026 (2021) c. Operators involving more gauge bosons With the above point of view, which treats the group If all gauge bosons contract with each other, they vanish elements as a function, the group multiplication rule is for any mixed helicity configurations inherent by the composition rule of the function such that ν ρ μ 0 ν ρ μ 0 π ðπ ðkÞÞ¼ðπ · π ÞðkÞ; ðB3Þ F1Lμ F2Lν F3Rρ ¼ ;F1Lμ F2Rν F3Rρ ¼ ; ðA73Þ i j i j but survive when all the helicities are the same, where · is the ordinary group multiplication and i and j are labels of the group elements. Further, we can define the 1 group elements as an operation that permutes the order of ν ρ μ ν ρ μ − ν ρ ˜ μ F1Lμ F2Lν F3Lρ ¼ 2 ðF1μ F2ν F3ρ iF1μ F2ν F3ρ Þ; arguments of a function such that it becomes another function of the same set of arguments with the original ðA74Þ order, 1 ν ρ μ ν ρ μ ν ρ ˜ μ F1Rμ F2Rν F3Rρ ¼ ðF1μ F2ν F3ρ þ iF1μ F2ν F3ρ Þ: π ∘ F p1;p2; …;p F pπ 1 ;pπ 2 ; …;pπ 2 i ð mÞ¼ ð ið Þ ið Þ iðmÞÞ ≡ … ðA75Þ Fπi ðp1;p2; ;pmÞ; ðB4Þ Similar features hold for more gauge bosons contracting and without loss of clarity, we shorten the above notation π ∘ ≡ together. Some nonvanishing examples of four gauge boson as i FðfpkgÞ ¼ FðfpπiðkÞgÞ Fπi ðfpkgÞ. More specifi- contractions are cally, the above operation changes the kth argument of the function F to the argument that originally seats at the 1 π CB2 G2 þ H:c: ¼ ðReCÞðB2G2 þðBB˜ ÞðGG˜ ÞÞ iðkÞth slot in F, or equivalently, moves the ith argument to R L 2 π−1 π ∘ the i ðkÞ slot. The operation i is essentially a map 1 2 2 that converts a function to another function; hence the þ ðImCÞðB ðGG˜ Þ − ðBB˜ ÞG Þ; ðA76Þ 2 composition rule of this map can be defined. It is easy to show that such a composition rule naturally preserves the μ λρ ν CBLμνBL λBL BL ρ þ H:c: group multiplication rule: μ λρ ν μ λρ ˜ ν ¼ðReCÞBμνB λB B ρ þðImCÞBμνB λB B ρ: ðπi ∘ πjÞ ∘ FðfpkgÞ ≡ πi ∘ ðπj ∘ FðfpkgÞÞ ðA77Þ π ∘ π ∘ ¼ i Fπj ðfpkgÞ ¼ i FðfpπiðkÞgÞ ¼ FðfpπiðπjðkÞÞgÞ ¼ Fðfpðπi·πjÞðkÞgÞ APPENDIX B: MATHEMATICAL TOOLS ¼ðπi · πjÞ ∘ FðfpkgÞ; ðB5Þ 1. Convention in permutation operation The elements of symmetric groups S are permutations which means the correspondence between the group m π π ∘ of m objects. Two most popular ways to represent the element i and the operation i on functions is a homomorphism. elements of the Sm are cycles notation and matrix notation. In Sec. III A, we mention that to generate a set of bases of For example, a typical element in Sm that permutes the first three objects and exchanges the last two objects can be the Lorentz structures and the group factors transforming expressed in the following form: under a certain irrep of the symmetric group, one only needs to act on an unsymmetrized Lorentz structure or λ 12345 group factor a set of group algebra elements bx that form a π ¼ð123Þð45Þ¼ : ðB1Þ 23154 basis of the same irrep in the group algebra space. Therefore, we need to generalize the concept of group In the matrix notation, the numbers in the first row can be elements as operations on functions to the group algebra viewed as the labels or the positions of the objects and the space. We define a group algebra element as an operation corresponding numbers in the second row are the labels or on a function basedP on the definition in Eq. (B4). For a iπ ˜ the positions of those objects after permutation. In this generic element r ¼ i r i in the Sm, the corresponding sense, the permutation can also be viewed as a function that operation r ∘ on functions is defined as maps the numbers in the first row to the numbers in the X i second row; i.e., in the above example we have r ∘ ¼ r πi ∘ : ðB6Þ i πð1Þ¼2; πð2Þ¼3; πð3Þ¼1; This operation still changes a function to another func- π 4 5 π 5 4 ð Þ¼ ; ð Þ¼ : ðB2Þ tion with the same set of arguments, while this resulting 015026-57 LI, REN, SHU, XIAO, YU, and ZHENG PHYS. REV. D 104, 015026 (2021) ½2;1 function is a linear combination of the original one with T1 ðp1;p2;p3;p4Þ arguments permuted: ½2;1 X ¼ b1 ∘ Tðp1;p2;p3;p4Þ i r ∘ FðfpkgÞ ¼ r πi ∘ FðfpkgÞ 1 ϵp1p2 ϵp3p4 ϵp2p1 ϵp3p4 − ϵp2p4 ϵp3p1 − ϵp1p4 ϵp3p2 Xi ¼ 3 ð þ Þ i ≡ ¼ r Fðfpπ ðkÞgÞ FrðfpkgÞ: ðB7Þ 1 i ϵp1p4 ϵp2p3 ϵp1p3 ϵp2p4 i ¼ 3 ð þ Þ; ðB14Þ One can verify that the generalization preserves the ½2;1 multiplication rule in the group algebra: T2 ðp1;p2;p3;p4Þ ½2;1 ¼ b ∘ Tðp1;p2;p3;p4Þ π ∘ F ðfp gÞ ¼ π ∘ r ∘ Fðfp gÞ 1 i r k i k 1 ¼ F ðfπ ðkÞgÞ ¼ ð−ϵp2p1 ϵp3p4 þ ϵp1p3 ϵp2p4 − ϵp2p3 ϵp1p4 þ ϵp3p1 ϵp2p4 Þ Xr i 3 j ¼ r Fðfp π π gÞ 1 ð i· jÞðkÞ ¼ ðϵp1p2 ϵp3p4 − ϵp1p4 ϵp2p3 Þ; ðB15Þ Xi 3 ¼ rjðπ · π ÞFðfp gÞ i j k and again readers can verify with the Schouten identity that i they transform according to Eq. (B12). ¼ðπi · rÞ ∘ FðfpkgÞ: ðB8Þ λ In this case, one can obtain a set of functions FxðfpkgÞ by 2. Projection operator and CGCs λ exerting bx defined in Eq. (3.16) such that We define the projection operator in the direct product ⊗ space V ¼ Vλi of the Sm group, Fλðfp gÞ ≡ bλ ∘ Fðfp gÞ; ðB9Þ x k x k X X dλ Pj D−1 π U π ; π ∘ λ λ λ π λi ¼ ! λ ð Þji ð Þ ðB16Þ i FxðfpkgÞ ¼ FyðfpkgÞDyxð iÞ: ðB10Þ m π y As an example we show in the following how to where dλ is the dimension of the λ representation, m! is the D π generate the basis of [2, 1] representation of S3 with order of the Sm group, and λð Þji is the matrix repre- p1p2p3p4 ϵp1p2 ϵp3p4 T ¼ Tðp1;p2;p3;p4Þ¼ . First one can sentation of π in irrep λ:UðπÞ is the representation of Sm on ½2;1 ½2;1 verify that the group algebra elements b1 and b2 below V defined by do form a basis of [2, 1] irrep in the group algebra space, X Y k j U π ⊗v i ⊗v i Dλ π ; 1 ð Þ λ ¼ λ i ð Þj k ðB17Þ ½2;1 12 − 13 − 123 i i i i i i b1 ¼ 3 ½e þð Þ ð Þ ð Þ ; ðB11Þ ji i vji λ 2;1 1 where λ is the jith basis vector of i irrep. b½ ¼ ½−ð12Þþð23Þ − ð123Þþð132Þ ; ðB12Þ i 2 3 The Theorem 4.2 in Ref. [39] states that for any v ∈ V, j fPλ v;i¼ 1; …;dλg transform as irrep λ if they are not null such that any permutation π in S3 acting on either of them i such that will result in a linear combination of them. This set of bases ½2;1 X generates a matrix representation D ðπÞ of S3 with the j j UðπÞðPλ vÞ¼ ðPλ vÞDλ ðπÞ : ðB18Þ two generators (12) and (123) given by i k i ki k 1 −1 −11 D½2;1 ðð12ÞÞ ¼ ; D½2;1 ðð123ÞÞ ¼ : In practice, we chose j ¼ 1 and generated invariant sub- 0 −1 −10 spaces of irrep λ by iterating v for different basis vector ⊗ vki until we get the number of the linear independent ðB13Þ λi subspaces equal to the number of multiplicity of irrep λ in Readers can verify that the relation in Eq. (3.16) does hold the inner product decomposition ⊙ λi. The CGCs can be with the definitions in Eqs. (B12) and (B13). 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