One-loop Matching of the Type-II Seesaw Model onto the Standard Model Effective Field Theory
Xu Li, Di Zhang, Shun Zhou
备注：
37 pages, 3 figures, the version accepted for publication in JHEP; A Mathematica Notebook containing all results for Wilson coefficients is attached in the ancillary files
In this paper, we continue to construct the low-energy effective field theories (EFTs) of the canonical seesaw models, which are natural extensions of the Standard Model (SM) to accommodate tiny but nonzero neutrino masses. Different from three right-handed neutrino singlets in the type-I seesaw model, the Higgs triplet in the type-II seesaw model participates directly in the electroweak gauge interactions, rendering the EFT construction more challenging. By integrating out the heavy Higgs triplet in the functional-integral formalism, we carry out a complete one-loop matching of the type-II seesaw model onto the so-called Standard Model Effective Field Theory (SMEFT). It turns out that 41 dimension-six operators (barring flavor structures and Hermitian conjugates) in the Warsaw basis of the SMEFT can be obtained, covering all those 31 dimension-six operators in the case of type-I seesaw model. The Wilson coefficients for 41 dimension-six operators are computed up to $\mathcal{O}\left( M^{-2}_Δ\right)$ with $M^{}_Δ$ being the mass scale of the Higgs triplet. Moreover, the branching ratios of rare radiative decays of charged leptons $l^-_α\to l^-_β+ γ$ are calculated in the EFT and compared with that in the full theory in order to demonstrate the practical application and the correctness of our EFT construction.
Using the functional method, one-loop matching of the type-I, -II and -III seesaw models are investigated and the results are presented in both the Green's and the Warsaw bases. Although these models generate the same dimension-5 Weinberg operator, they could induce quite different types of dimension-6 effective operators that can be utilized for model discrimination. We also find the threshold effects from one-loop matching could be significant, which turn out to allow triggering electroweak symmetry breaking radiatively in type-II seesaw while forbid that in type-I/-III models. An analytical…
The canonical type-I seesaw model with three heavy Majorana neutrinos is one of the most natural extensions of the standard model (SM) to accommodate tiny Majorana masses of three ordinary neutrinos. At low-energy scales, Majorana neutrino masses and unitarity violation of lepton flavor mixing have been extensively discussed in the literature, which are respectively generated by the unique dimension-five Weinberg operator and one dimension-six operator in the seesaw effective field theory (SEFT) with the tree-level matching. In this work, we clarify that a self-consistent calculation of radiat…
In this paper, we accomplish the complete one-loop matching of the type-I seesaw model onto the Standard Model Effective Field Theory (SMEFT), by integrating out three heavy Majorana neutrinos with the functional approach. It turns out that only 31 dimension-six operators (barring flavor structures and Hermitian conjugates) in the Warsaw basis of the SMEFT can be obtained, and most of them appear at the one-loop level. The Wilson coefficients of these 31 dimension-six operators are computed up to $\mathcal{O}\left( M^{-2}\right)$ with $M$ being the mass scale of heavy Majorana neutrinos. As th…
The renormalization group (RG) running of the neutrino mass operator is required for comparing the predictions of neutrino models at high energy with the experimental data at low energies. In the type I seesaw scenario with n_G right-handed neutrinos, the RG running is also performed in the effective theories above and between the thresholds given by the masses of the right-handed neutrinos. At these thresholds, the effective theories are matched. When calculating the two-loop RG running, the matching has to be performed at the one-loop level. In this work, we calculate the one-loop matching f…
In this talk, we present the results of the complete one-loop matching of the seesaw model onto its low-energy effective theory by integrating out three right-handed neutrinos at the one-loop level. We find that there are 31 independent dimension-6 (dim-6) operators (barring flavor structures and Hermitian conjugates) in the Warsaw basis, and the standard-model couplings and the Wilson coefficient of the Weinberg operator acquire threshold corrections at the one-loop level. The complete Lagrangian up to dim-6 at the one-loop level is derived, which is indispensable for consistent explorations …
In terms of flavor invariants, we establish the intimate connection between leptonic CP violation in the canonical seesaw model for neutrino masses and that in the seesaw effective field theory (SEFT). For the first time, we calculate the Hilbert series and explicitly construct the primary flavor invariants in the SEFT by considering both the dimension-five Weinberg operator ${\cal O}^{αβ}_5 = \overline{\ell^{}_{α\rm L}} \widetilde{H} \widetilde{H}^{\rm T} \ell^{\rm C}_{β\rm L}$ and the dimension-six operator ${\cal O}^{αβ}_6 = \overline{\ell^{}_{α\rm L}} \widetilde{H} {\rm i} \not\!\partial \…
The quantity 1/lambda_B, the inverse moment of the B-meson light-cone wave function, plays an important role in exclusive B-decays. I calculate the matching coefficient of lambda_B defined in QCD onto lambda_B defined in HQET. This is useful for comparing results that have been obtained in QCD to results obtained in HQET.
As is well known, the smallest neutrino mass turns out to be vanishing in the minimal seesaw model, since the effective neutrino mass matrix $M^{}_ν$ is of rank two due to the fact that only two heavy right-handed neutrinos are introduced. In this paper, we point out that the one-loop matching condition for the effective dimension-five neutrino mass operator can make an important contribution to the smallest neutrino mass. By using the available one-loop matching condition and two-loop renormalization group equations in the supersymmetric version of the minimal seesaw model, we explicitly calc…
The neutrino option is a scenario where the electroweak scale, and thereby the Higgs mass, is generated simultaneously with neutrino masses in the seesaw model. This occurs via the leading one loop and tree level diagrams matching the seesaw model onto the Standard Model Effective Field Theory. We advance the study of this scenario by determining one loop corrections to the leading order matching results systematically, performing a detailed numerical analysis of the consistency of this approach with Neutrino data and the Standard Model particle masses, and by examining the embedding of this s…
The canonical seesaw models are one of the simplest and most natural scenarios that can account simultaneously for neutrino masses and matter-antimatter asymmetry in our universe. Below the seesaw scale, one can integrate out the heavy degrees of freedom to construct the seesaw effective field theory (SEFT). In this talk, we investigate the connection between the full seesaw model and the low-energy SEFT from a brand-new perspective: the invariant theory. Using the powerful tool of Hilbert series, we demonstrate the intimate relation between the flavor space of the SEFT and that of its ultravi…
We investigate matching conditions and threshold corrections between full and effective theories based on the type I seesaw mechanism. In general, using an intuitive Feynman diagrammatical approach, we compute the amplitudes before and after integrating out heavy right-handed neutrinos at the matching scale. In particular, we derive the one-loop matching conditions between the full and the effective theories. The matching conditions of the parameters are influenced by one-loop corrections to the corresponding vertices as well as wave function corrections for the Higgs and the lepton fields. Ou…
This letter presents results for the two-loop matching coefficients for the dimension eight operators that contribute to Higgs production via gluon fusion. The coefficients can be used to calculate the first correction to the infinite top mass limit to Higgs production with large transverse momentum at two loops. To date such processes have been studied at two loop order only in the leading term in the top mass expansion. These corrections become enhanced in processes with large final state invariant mass, typical of multijet processes.
A HEFT Perspective on the Type-II Seesaw Model and the Complete Basis of Lepton-Number-Violating Operators
School of Physics and Information Technology, Shaanxi Normal University（陕西师范大学物理与信息技术学院）；Center for Theoretical Physics of the Universe, Institute for Basic Science (IBS)（基础科学研究院宇宙理论物理中心）
The scotogenic neutrino seesaw model is a minimal extension of the standard model with three $\mathbb{Z}_2$-odd right-handed singlet fermions $N$ and one $\mathbb{Z}_2$-odd Higgs doublet $η$ that can accommodate the tiny neutrino mass and provide a dark matter candidate in a unified picture. Due to lack of experimental signatures for electroweak scale new physics, it is appealing to assume these new particles are well above the electroweak scale and take the effective field theory approach to study their effects on low energy observables. In this work we apply the recently developed functional…
We match the seesaw model for generating neutrino masses onto the Standard Model Effective Field Theory (SMEFT). We perform this matching at tree level up to dimension seven in the operator expansion. We explain how some of the perturbations of the neutrino mass matrix due to operators of mass dimension greater than five are tied to integrating out the heavy Majorana mass eigenstates in sequence. We demonstrate that the low energy limit of seesaw models are well described by the SMEFT, particularly when constructed using a flavour space expansion. Flavour space expansions of seesaw models are …
The operator definition of generalised transverse-momentum-dependent (GTMD) distributions is exploited to compute for the first time the full set of one-loop corrections to the off-forward matching functions. These functions allow one to obtain GTMDs in the perturbative regime in terms of generalised parton distributions (GPDs). In the unpolarised case, non-perturbative corrections can be incorporated using recent determinations of transverse-momentum-dependent (TMD) distributions. Evolution effects for GTMDs closely follow those for TMDs and can thus be easily accounted for up to next-to-next…
The type-I seesaw model is probably the most straightforward and best studied extension of the Standard Model that can account for the tiny active neutrino masses determined from neutrino oscillation data. In this article, we calculate the complete set of one-loop corrections to charged lepton flavour violating processes within this model. We give the results both using exact diagonalisation of the neutrino mass matrix, and at at leading order in the seesaw expansion (i.e. $\mathcal{O}(v^2/M_R^2)$). Furthermore, we perform the matching onto the $SU(2)_L$ invariant Standard Model Effective Fiel…
Besides the three canonical seesaw mechanisms, the hybrid scenario, i.e., the so-called type-(I+II) seesaw mechanism containing both the right-handed neutrinos $N^{}_{\rm R}$ and the triplet Higgs $Φ$ is also an appealing extension of the Standard Model (SM) to account for tiny neutrino masses. Recently, the seesaw effective field theories (SEFTs) of the three canonical seesaw mechanisms have already been completely constructed up to one-loop level. In this work, we carry out the one-loop matching of the type-(I+II) seesaw mechanism onto the corresponding type-(I+II) SEFT, which is by no means…
In this paper, we systematically study the algebraic structure of the ring of the flavor invariants and the sources of CP violation in the seesaw effective field theory (SEFT), which is obtained by integrating out heavy Majorana neutrinos in the type-I seesaw model at the tree level and thus includes the dimension-five Weinberg operator and one dimension-six operator. For the first time, we calculate the Hilbert series and explicitly construct all the primary flavor invariants in the SEFT. We show that all the physical parameters can be extracted using the primary invariants and any CP-violati…
One-loop Renormalization of the Type-I Seesaw Model in the On-shell Scheme
Institute of High Energy Physics, Chinese Academy of Sciences（中国科学院高能物理研究所）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）；Center for High Energy Physics, Henan Academy of Sciences（河南省科学院高能物理中心）
We derive the conditions for matching high-energy renormalizable Quantum Field Theories onto low-energy nonrenormalizable ones by means of the FDR approach described in Fortsch.Phys. 63 (2015) 132-141. Our procedure works order-by-order in the loop expansion and avoids the addition of higher dimensional interactions into the nonrenormalizable Lagrangian. To illustrate our strategy, we match the high-energy fermion-loop corrections computed in the complete electroweak theory onto the nonrenormalizable four-fermion Fermi model. As a result, the Fermi Lagrangian can be used without modifications …
The resummed differential thrust rate in e+e- annihilation is calculated using Soft-Collinear Effective Theory (SCET). The resulting distribution in the two-jet region T~1 is found to agree with the corresponding expression derived by the standard approach. A matching procedure to account for finite corrections at T < 1 is then described. There are two important advantages of the SCET approach. First, SCET manifests a dynamical seesaw scale q = p^2/Q in addition to the center-of-mass energy Q and the jet mass scale p ~ Q ~ sqrt(1 - T). Thus, the resummation of logs of p/q can be cleanly distin…
Dismissing traditional naturalness concerns while embracing the Higgs boson mass measurement and unification motivates careful analysis of trans-TeV supersymmetric theories. We take an effective field theory (EFT) approach, matching the Minimal Supersymmetric Standard Model (MSSM) onto the Standard Model (SM) EFT by integrating out heavy superpartners, and evolving MSSM and SMEFT parameters according to renormalization group equations in each regime. Our matching calculation is facilitated by the recent covariant diagrams formulation of functional matching techniques, with the full one-loop SU…
Mapping UV theories onto low energy effective descriptions is a procedure known as matching. The last decade has seen tremendous progress in the development of new tools for efficiently performing matching calculations, by relying on so-called functional methods. This white paper summarizes the status of functional matching. Specifically, matching for relativistic theories is a fully solved problem up to one-loop order in perturbation theory, and to arbitrary order in the effective field theory expansion. A streamlined prescription that has been partially automated facilitates the application …
We calculate the subleading corrections to the thrust rate using Soft-Collinear Effective Theory to factorize the rate and match onto jet and soft operators that describe the degrees of freedom of the relevant scales. We work in the perturbative regime where all the scales are well above Λ_QCD. The thrust rate involves an incomplete sum over final states that is enforced by a measurement operator. Subleading corrections require matching onto not only the higher dimensional dijet operators, but also matching onto subleading measurement operators in the effective theory. We explicitly show how t…
SUSY meets SMEFT: Complete one-loop matching of the general MSSM
Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-Alpes, CNRS/IN2P3（亚原子与宇宙物理学实验室，格勒诺布尔阿尔卑斯大学，法国国家科学研究中心/国家核物理与粒子物理研究所）；Centro de Ciências Naturais e Humanas, Universidade Federal do ABC（自然与人文科学中心，ABC联邦大学）；Departament de Física Teòrica, IFIC (Universitat de València - CSIC)（理论物理系，瓦伦西亚大学-西班牙国家科学研究委员会联合研究所）；Institute for Theoretical Particle Physics and Cosmology, RWTH Aachen University（理论粒子物理学与宇宙学研究所，亚琛工业大学）
Wilson is a Python library for matching and running Wilson coefficients of higher-dimensional operators beyond the Standard Model. Provided with the numerical values of the Wilson coefficients at a high new physics scale, it automatically performs the renormalization group evolution within the Standard Model effective field theory (SMEFT), matching onto the weak effective theory (WET) at the electroweak scale, and QCD/QED renormalization group evolution below the electroweak scale down to hadronic scales relevant for low-energy precision tests. The matching and running encompasses the complete…
In this study we present a universal effective action for one-loop matching of all scalar leptoquarks. We use both the Universal One-Loop Effective Action (UOLEA) and covariant diagrams to evaluate the Wilson coefficients directly in the Green basis for up-to dimension-6 operators. On the technical side, we use the newly developed method of evaluating supertraces, to further validate the results stemming from the use of covariant diagrams. As an application, we perform a fully functional matching onto Standard Model Effective Field Theory (SMEFT) of a model with two scalar leptoquark fields: a…
The Standard Model electroweak vacuum has been found to be metastable, with the true stable vacuum given by a large, phenomenologically unacceptable vacuum expectation value $\approx M_{P}$. Moreover, it may be unstable in an inflationary universe. Motivated by the necessity of physics beyond the Standard Model and to accommodate non-zero neutrino masses, we investigate vacuum stability within type-II seesaw and left-right symmetric models. Our analysis is performed by solving the renormalisation group equations, carefully taking into account the relevant threshold corrections. We demonstrate …
This paper presents STrEAM (SuperTrace Evaluation Automated for Matching), a Mathematica package that calculates all functional supertraces which arise when matching a generic UV model onto a relativistic Effective Field Theory (EFT) at one loop and to arbitrary order in the heavy mass expansion. STrEAM implements the covariant derivative expansion to automate the most tedious step of the streamlined functional matching prescription presented in arXiv:2011.02484 . The code and an example notebook are available at https://www.github.com/EFTMatching/STrEAM .
We simplify the one-loop functional matching formalism to develop a streamlined prescription. The functional approach is conceptually appealing: all calculations are performed within the UV theory at the matching scale, and no prior determination of an Effective Field Theory (EFT) operator basis is required. Our prescription accommodates any relativistic UV theory that contains generic interactions (including derivative couplings) among scalar, fermion, and vector fields. As an example application, we match the singlet scalar extended Standard Model (SM) onto SMEFT.
Several experiments search for μ- e flavour change, for instance in μ->e conversion, μ-> e γ, and μ-> 3e. This paper studies how to translate these experimental constraints from low energy to a New Physics scale M >> mW. A basis of QCD and QED-invariant operators (as appropriate below mW) is constructed, then run to mW with one-loop RGEs of QCD and QED. At mW, these operators are matched onto SU(2)-invariant dimension-six operators, which can continue to run up with electroweak RGEs. As an example, the μ-> e γbound is translated to the scale M, where it constrains two sums of operators. The co…
We discuss different choices that can be made when matching a general high-energy theory -- with the restriction that it should not contain heavy gauge bosons -- onto a general renormalisable effective field theory at one loop, with particular attention to the quartic scalar couplings and Yukawa couplings. This includes a generalisation of the counterterm scheme that was found to be useful in the case of high-scale/split supersymmetry, but we show the important differences when there are new heavy scalar fields in singlet or triplet representations of $SU(2)$. We also analytically compare our …
We explain how nonperturbative effects can be systematically included when constructing an effective field theory. We concentrate in particular on the matching calculation, by which a low energy theory without a heavy particle degree of freedom is matched onto the full theory. The matching calculation requires some new formalism, because for a large class of theories, instantons occur in a continuous range of scales from far below the mass of the heavy particle to far above. We show how the complete IR finiteness of the usual matching formalism arises in this new context. As an example, we for…
The $x$-dependence of light-cone distribution amplitude (LCDA) can be directly calculated from a quasi distribution amplitude (DA) in lattice QCD within the framework of large-momentum effective theory (LaMET). In this paper, we study the one-loop renormalization of the quasi-DA in the regularization-independent momentum subtraction (RI/MOM) scheme. The renormalization factor for the quasi parton distribution function can be used to renormalize the quasi-DA provided that they are implemented on lattice and in perturbation theory in the same manner. We derive the one-loop matching coefficient t…
