We derive universal formulae for integrating out heavy degrees of freedom in scalar field theories up to one-loop level in terms of covariant quantities associated with the geometry of the field manifold. The universal matching results can be readily applied to phenomenologically interesting extensions of the Standard Model, as we demonstrate using a singlet scalar example. We also discuss the role of field redefinitions in effective field theory matching and simplifications resulting from going to a field basis where interactions are encoded in a nontrivial metric on the field manifold.

United States

Xu-Xiang Li

Xiaochuan Lu

Zhengkang Zhang

The geometric universal one-loop effective action

We derive a saturation theorem for general Effective Field Theories (EFTs) constructed using spurion analysis. Let $\textit{S}$ be a set of spurion fields introduced to organize the breaking of a global symmetry $\textit{G}$ , and $\textit{H}$ be the subgroup of $\textit{G}$ that remains unbroken under a generic vacuum expectation value ⟨$\textit{S}$⟩; we show that the EFT Lagrangian constructed from the spurion analysis $\textit{saturates}$ the EFT Lagrangian without the spurions but restricted to $\textit{H}$ invariance, provided that arbitrary powers of the spurion fields are allowed. As examples, we study several implementations of the Minimal Lepton Flavor Violation (MLFV) principle, corresponding to various origins of the neutrino masses. In each scenario, we compute the Hilbert series to obtain the numbers of independent lepton flavor covariants that appear in the corresponding EFT at mass dimension 6. These numbers agree with the number of $\textit{H}$ invariants in the EFT without the spurions, demonstrating the saturation theorem. Motivated by phenomenological connections, we provide linearly independent spurion polynomials for selected lepton flavor covariants. A supplementary material file supplies , a Mathematica notebook that provides functions for computing general Hilbert series of invariants and covariants of compact classical groups. It presents examples demonstrating the use of the code, including the Hilbert series for our MLFV scenarios.

Benjamín Grinstein

Spain

Carlos Miró

Pablo Quílez

Most general EFTs from spurion analysis Hilbert series and minimal lepton flavor violation

We elaborate on a recently proposed geometric framework for scalar effective field theories. Starting from the action, a metric can be identified that enables the construction of geometric quantities on the associated functional manifold. These objects transform covariantly under general field redefinitions that relate different operator bases, including those involving derivatives. We present a novel geometric formula for the amplitudes of the theory, where the vertices in Feynman diagrams are replaced by their geometrized counterparts. This makes the on-shell covariance of amplitudes manifest, providing the link between functional geometry and effective field theories.

CERN

Switzerland

Timothy Cohen

What is the geometry of effective field theories?

Accidental symmetries in effective field theories can be established by computing and comparing Hilbert series. This invites us to study them with the tools of invariant theory. Applying this technology, we spotlight three classes of accidental symmetries that hold to all orders for non-derivative interactions. They are broken by derivative interactions and become ordinary finite-order accidental symmetries. To systematically understand the origin and the patterns of accidental symmetries, we introduce a novel mathematical construct — a (non-transitive) binary relation between subgroups that we call . Equipped with this, we derive new criteria for all-order accidental symmetries in terms of , and criteria for finite-order accidental symmetries in terms of . They allow us to verify and identify accidental symmetries more efficiently without computing the Hilbert series. We demonstrate the success of our new criteria by applying them to a variety of sample accidental symmetries, including the custodial symmetry in the Higgs sector of the Standard Model effective field theory.

Accidental symmetries, Hilbert series, and

A correction to this paper has been published: https://doi.org/10.1007/JHEP06(2024)154

Luca Merlo

Erratum to: Hilbert series for covariants and their applications to minimal flavor violation

The dineutron as a bound chargeless nucleus of two identical nucleons has been attracting attention for many decades. In addition to the study of the formation of a bound dineutron in the outgoing channel of specific nuclear reactions, cross-section estimates have been gaining interest as research targets. We used a traditional neutron activation technique to irradiate Lu samples followed by measurements of the induced activity with HPGe spectrometer in order to detect gamma-peaks of the 175Lu ( <italic>n</italic>, 2 <italic>n</italic>) 174 gLu reaction product when this reaction channel is not open for incident neutron energies. Based on two measurements of the instrumental gamma-ray spectra of Lu samples we reliably identified the presence of the 174 gLu isotope in the outgoing channel of the 175Lu ( <italic>n</italic>, 2 <italic>n</italic>) 174 gLu nuclear reaction. Also, the cross-section for this nuclear reaction was obtained and to be equal <math altimg='si1.svg'><mrow><mn>33</mn><mo>.</mo><msubsup><mn>5</mn><mrow><mo>&#8722;</mo><mn>6.7</mn></mrow><mrow><mo>+</mo><mn>7.0</mn></mrow></msubsup></mrow></math> mb. For the first time, the dineutron as a nuclearreaction product was indirectly detected via statistically significant observation of the induced activity of 174 gLu with a corresponding cross-section estimate.

Ukraine

Ihor Kadenko

Hungary

Barna Biró

Mihály Braun

András Fenyvesi

Kateryna Okopna

Nadiia Sakhno

Loránd Zakàny

Formation of bound dineutrons in the 175Lu ( <italic>n</italic>, 2 <italic>n</italic>) 174 gLu nuclear reaction and its cross-section

We revisit the proton emitter $^{149}$Lu utilizing the recently developed triaxial relativistic Hartree-Bogoliubov theory in continuum (TRHBc). By incorporating the microscopic nuclear structure properties from the TRHBc theory into the WKB approximation, we successfully reproduce the measured proton-emission half-life of $^{149}$Lu within experimental uncertainties. A triaxial ground state characterized by (<math><mi>β</mi><mo>=</mo><mn>0.17</mn><mo>,</mo><mi>γ</mi><mo>=</mo><msup><mrow><mn>31</mn></mrow><mrow><mo>∘</mo></mrow></msup></math>) has been clarified for $^{149}$Lu. The inclusion of triaxiality significantly changes nuclear density distributions and potentials, which results in enhanced binding of both the nuclear system and the proton-emitting orbital. As a result, a slightly extended half-life for the proton emission of $^{149}$Lu is achieved after considering triaxial deformation degrees of freedom. The examination of the dependence of the decay width on triaxial deformation parameters highlights the significance of microscopic description of nuclear deformation beyond the proton dripline.

China

Qi Lu

Kai-Yuan Zhang

Shi-Sheng Zhang

Triaxial shape of the one-proton emitter Lu

We elaborate how to apply the Hilbert series method to enumerating group covariants, which transform under any given representation, including but going beyond group invariants. Mathematically, group covariants form a module over the ring of the invariants. The number of independent covariants is given by the rank of the module, which can be computed by taking a ratio of two Hilbert series. In many cases, the rank equals the dimension of the group covariant representation. When this happens, we say that there is a . We apply this technology to revisit the hypothesis of Minimal Flavor Violation in constructing Effective Field Theories beyond the Standard Model. We find that rank saturation is guaranteed in this case, leading to the important consequence that the MFV symmetry principle does not impose any restriction on the EFT, i.e. MFV SMEFT = SMEFT, in the absence of additional assumptions.

Hilbert series for covariants and their applications to minimal flavor violation

We derive an off-shell recursion relation for correlators that holds at all loop orders. This allows us to prove how generalized amplitudes transform under generic field redefinitions, starting from an assumed behavior of the one-particle-irreducible effective action. The form of the recursion relation resembles the operation of raising the rank of a tensor by acting with a covariant derivative. This inspires a geometric interpretation, whose features and flaws we investigate.

United Kingdom

Dave Sutherland

On amplitudes and field redefinitions

We present a formulation of scalar effective field theories in terms of the geometry of Lagrange spaces. The horizontal geometry of the Lagrange space generalizes the Riemannian geometry on the scalar field manifold, inducing a broad class of affine connections that can be used to covariantly express and simplify tree-level scattering amplitudes. Meanwhile, the vertical geometry of the Lagrange space characterizes the physical validity of the effective field theory, as a torsion component comprises strictly higher-point Wilson coefficients. Imposing analyticity, unitarity, and symmetry on the theory then constrains the signs and sizes of derivatives of the torsion component, implying that physical theories correspond to a special class of vertical geometry.