Renormalization group evolution of dimension-seven operators in standard model effective field theory and relevant phenomenology

Abstract

We showed in a previous publication that there are six independent dimension-seven operators violating both lepton and baryon numbers (L = −B = 1) and twelve ones violating lepton but preserving baryon number (L = 2, B = 0) in standard model effective field theory, and we calculated one-loop renormalization for the former six operators. In this work we continue our efforts on renormalization of the operators. It turns out this could become subtle because the operators are connected by nontrivial relations when fermion flavors are counted. This kind of relations does not appear in lower dimensional operators.We show how we can extract anomalous dimension matrix for a flavor-specified basis of operators from counterterms computed for the above flavor-blind operators without introducing singular inverse Yukawa coupling matrices. As a phenomenological application, we investigate renormalization group effects on nuclear neutrinoless double β decay. We also discuss very briefly its analog in the meson sector, K ± → π ∓ µ ± µ ± , and indicate potential difficulties to compute its decay width. 1

Long-distance contributions to neutrinoless double beta decay π−→π+ee

“…While the vanishing phase space does not allow the π − → π + ee decay happen in nature (This problem does not exist for the K − → π + ee decay, which is proposed by Ref. [24]), the hadronic matrix element is well defined within the Standard Model and is equivalent to the one from π − e + → π + e − scattering, where π ± and e ± carry zero spatial momentum. As the crossed-channel analog to the π − π − → ee decay, the process of π − → π + ee can be combined together with π − π − → ee and serves as a cross-check for the prediction from χPT.…”

Effective field theory approach to lepton number violating decays $$ {K}^{\pm}\to {\pi}^{\mp }{l}_{\alpha}^{\pm }{l}_{\beta}^{\pm }: $$ long-distance contribution

“…The above upper bounds on branching ratios then translate into the lower bounds on their inverse cubic roots as displayed in table 1. The bounds are rather weak, especially when compared with those from nuclear 0νββ decays [32,38]. This relative weakness originates from much smaller data samples accumulated in kaon experiments than the number of nuclei available in a ton-level experiment of 0νββ decays as we estimated roughly in ref.…”

“…[31] so that flavor symmetries are apparently realized as advocated in ref. [32]. This only concerns the subset of operators that violate lepton number but conserve baryon number.…”

“…Then in section 4 we match LEFT upwards the scale to the standard model effective field theory (SMEFT), assuming that there are no new particles with a mass of order Λ EW or lower. The relevant leading LNV operators in SMEFT are the dim-5 and -7 ones known in the literature [29][30][31][32] and reproduced in appendix A. We express the branching ratios in terms of the Wilson coefficients in SMEFT and the LECs of strong dynamics through the above two-step matching, whose complete expressions can be found in appendix B, and make some numerical estimates.…”

Enumerating the operators of an effective field theory

“…As mentioned earlier, the notation µ m means that µ = {µ 1 , µ 2 , · · · } is a partition of the integer m (so the sum of the µ i add up to m). Therefore µ m represents a sum over all partitions µ of m 11. Those µ's having a Young diagram with more rows than the available flavors (n) should not be considered in this last expression; they are too anti-symmetric and as a consequence they are not associated to any operator (S (µ, n) = 0).…”