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Complete Set of Dimension-Eight Operators in the Standard Model Effective Field Theory

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Complete Set of Dimension-Eight Operators in the Standard Model Effective Field Theory 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.
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