Scalar Extension Research Articles

Quasilinear quadratic forms and function fields of quadrics

Let p and q be anisotropic quadratic forms of dimension $$\ge 2$$ over a field F. In a recent article, we formulated a conjecture describing a general constraint which the dimensions of p and q impose on the isotropy index of q after scalar extension to the function field of p. This can be viewed as a generalization of Hoffmann’s Separation Theorem which simultaneously incorporates and refines some well-known classical results on the Witt kernels of function fields of quadrics. Using algebro-geometric methods, it was shown that large parts of this conjecture hold in the case where the characteristic of F is not 2. In the present article, we prove similar (in fact, somewhat sharper) results in the case where F has characteristic 2 and q is a so-called quasilinear form. In contrast to the situation where $$\mathrm {char}(F) \ne 2$$, the methods used to treat this case are purely algebraic.

Electroweak phase transition, gravitational waves and dark matter in two scalar singlet extension of the standard model

In this paper, the electroweak phase transition, the gravitational waves and the dark matter issues are investigated in two scalar singlet extension of the standard model. The detectability of the gravitational wave signals are discussed by comparing the results with the sensitivity curves of mathbf {eLISA}, mathbf {ALIA}, mathbf {DECIGO} and mathbf {BBO} detectors. It is shown that the results support the recent reports on the dark matter relic density by mathbf {Planck}mathbf {2018} collaboration and the direct detection experiment by mathbf {XENON1T} mathbf {2018} collaboration.

Graded change of ring

We investigate scalar restriction, scalar extension, and scalar coexten- sion functors for graded modules, including their interplay with coarsening functors, graded tensor products, and graded Hom functors. This leads to several character- isations of epimorphisms of graded rings.

B\u2013L Model with mathbf{S}_{3} symmetry

We make a scalar extension of the B–L gauge model where the mathbf{S}_{3} non-abelian discrete group drives mainly the Yukawa sector. Motived by the large and small hierarchies among the quark and active neutrino masses, respectively, the quark and lepton families are not treated on the same footing under the assignment of the discrete group. As a consequence, the nearest neighbor interaction (NNI) textures appear in the quark sector, leading to the CKM mixing matrix, whereas in the lepton sector, a soft breaking of the mu leftrightarrow tau symmetry in the effective neutrino mass, which comes from type I see-saw mechanism, provides a non-maximal atmospheric angle and a non-zero reactor angle.

Gravitational wave and collider probes of a triplet Higgs sector with a low cutoff

We study the scalar triplet extension of the standard model with a low cutoff, preventing large corrections to the quadratic masses that would otherwise worsen the hierarchy problem. We explore the reach of LISA to test the parameter space region of the scalar potential (not yet excluded by Higgs to diphoton measurements) in which the electroweak phase transition is strongly first-order and produces sizeable gravitational waves. We also demonstrate that the collider phenomenology of the model is drastically different from its renormalizable counterpart. We study the reach of the LHC in ongoing searches and project bounds for the HL-LHC. Likewise, we develop a dedicated analysis to test the key but still unexplored signature of pair-production of charged scalars decaying to third-generation quarks: pprightarrow toverline{b} (overline{t}b), boverline{b}. These results apply straightforwardly to other extensions of the Higgs sector such as the 2HDM/MSSM.

Complete one-loop matching for a singlet scalar in the Standard Model EFT

We present the results of the first complete one-loop matching calculation between the real singlet scalar extension of the Standard Model and the Standard Model effective field theory (SMEFT) at dimension six. Beyond their immediate relevance to the precision calculation of observables in singlet extensions of the Standard Model, our results illustrate a variety of general features of one-loop matching. We explore the interplay between non-supersymmetric non-renormalization theorems, the logarithmic dependence of Wilson coefficients, and the relevance of mixed diagrams in theories with large scale separation. In addition, we highlight some of the subtleties involved in computing observables at next-to-leading order in SMEFT by mapping our results to the T parameter at one loop.

Global fits in the Georgi-Machacek model

Off the beaten track of scalar singlet and doublet extensions of the Standard Model, triplets combine an interesting LHC phenomenology with an explanation for neutrino masses. The Georgi-Machacek model falls into this category, but it has never been fully explored in a global fit. We use the {\texttt{HEPfit}} package to combine recent experimental Higgs data with theoretical constraints and obtain strong limits on the mixing angles and mass differences between the heavy new scalars as well as their decay widths. We also find that the current signal strength measurements allow for a Higgs to vector boson coupling with an opposite sign to the Standard Model, but this possibility can be ruled out by the lack of direct evidence for heavy Higgs states. For these hypothetical particles, we identify the dominant decay channels and extract bounds on their branching ratios from the global fit, which can be used to single out the decay patterns relevant for the experimental searches.

Geometry of Interaction for MALL via Hughes--Van Glabbeek Proof-Nets

This article presents, for the first time, a Geometry of Interaction (GoI) interpretation inspired from Hughes--Van Glabbeek (HvG) proof-nets for multiplicative additive linear logic (MALL). Our GoI dynamically captures HvG’s geometric correctness criterion—the toggling cycle condition—in terms of algebraic operators. Our new ingredient is a scalar extension of the *-algebra in Girard’s *-ring of partial isometries over a Boolean polynomial ring with literals of eigenweights as indeterminates. To capture feedback arising from cuts, we construct a finer-grained execution formula. The expansion of this execution formula is longer than that for collections of slices for multiplicative GoI, hence it is harder to prove termination. Our GoI gives a dynamical, semantical account of Boolean valuations (in particular, pruning sub-proofs), conversion of weights (in particular, α-conversion), and additive (co)contraction, peculiar to additive proof-theory. Termination of our execution formula is shown to correspond to HvG’s toggling criterion. The slice-wise restriction of our execution formula (by collapsing the Boolean structure) yields the well-known correspondence, explicit or implicit in previous works on multiplicative GoI, between the convergence of execution formulas and acyclicity of proof-nets. Feedback arising from the execution formula by restricting to the Boolean polynomial structure yields autonomous definability of eigenweights among cuts from the rest of the eigenweights.

Loop-induced neutrino non-standard interactions

Non-Standard Interactions (NSI) of neutrinos may originate from models in which new particles interact with neutrinos. In scalar extensions of the SM, the typical approach to obtain NSI requires Fierz transformations and charged Higgses, which suffer from strong constraints from collider searches or charged lepton flavor violation processes. We propose here an alternative approach to generate NSI, namely via loop processes. We show that such loop-induced NSI from secret neutrino interactions can reach sizes of mathcal{O} (0.1 ∼ 1) compared to standard Fermi interaction. This approach can also give rise to neutrino-quark NSI.

Trace of the energy-momentum tensor and macroscopic properties of neutron stars

A generic feature of scalar extensions of general relativity is the coupling of the scalar degrees of freedom to the trace $T$ of the energy-momentum tensor of matter fields. Interesting phenomenology arises when the trace becomes positive---when pressure exceeds one third of the energy density---a condition that may be satisfied in the core of neutron stars. In this work, we study how the positiveness of the trace of the energy-momentum tensor correlates with macroscopic properties of neutron stars. We first show that the compactness for which $T=0$ at the stellar center is approximately equation-of-state independent, and given by $C = 0.262_{-0.017}^{+0.011}$ (90% confidence interval). Next, we exploit Bayesian inference to derive a probability distribution function for the value of $T$ at the stellar center given a putative measurement of the compactness of a neutron star. This investigation is a necessary step in order to use present and future observations of neutron star properties to constrain scalar-tensor theories based on effects that depend on the sign of $T$.

Regulator constants of integral representations of finite groups

AbstractLet G be a finite group and p be a prime. We investigate isomorphism invariants of $\mathbb{Z}_p$[G]-lattices whose extension of scalars to $\mathbb{Q}_p$ is self-dual, called regulator constants. These were originally introduced by Dokchitser–Dokchitser in the context of elliptic curves. Regulator constants canonically yield a pairing between the space of Brauer relations for G and the subspace of the representation ring for which regulator constants are defined. For all G, we show that this pairing is never identically zero. For formal reasons, this pairing will, in general, have non-trivial kernel. But, if G has cyclic Sylow p-subgroups and we restrict to considering permutation lattices, then we show that the pairing is non-degenerate modulo the formal kernel. Using this we can show that, for certain groups, including dihedral groups of order 2p for p odd, the isomorphism class of any $\mathbb{Z}_p$[G]-lattice whose extension of scalars to $\mathbb{Q}_p$ is self-dual, is determined by its regulator constants, its extension of scalars to $\mathbb{Q}_p$, and a cohomological invariant of Yakovlev.

Singlet scalar and 2HDM extensions of the Standard Model: CP-violation and constraints from (g \u2212 2)\u03bc and eEDM

We study popular scalar extensions of the Standard Model, namely the singlet extension, the 2-Higgs doublet model (2HDM) and its extension by a singlet scalar. We focus on the contributions of the added scalars to the anomalous magnetic moment of the muon, (g − 2)μ in the presence of CP-violation, and the electric dipole moment of the electron (eEDM) in these models. In the absence of CP-violation, CP-even and CP-odd scalars contribute with an opposite sign to the anomalous magnetic moment of the muon and as a result these models generally require very light scalars to explain the observed discrepancy in (g − 2)μ. We study the effect of CP-violation on the anomalous magnetic moment of the muon and its compatibility with the eEDM constraints. We show that given the current status of the global set of constraints applied on all values of cot β, in the CP-violating scalar extensions, there exist no viable parameter space in agreement with both aμ and eEDM bounds.

Thermal resummation and phase transitions

The consequences of phase transitions in the early universe are becoming testable in a variety of manners, from colliders physics to gravitational wave astronomy. In particular one phase transition we know of, the electroweak phase transition (EWPT), could potentially be first order in BSM scenarios and testable in the near future. If confirmed this could provide a mechanism for baryogenesis, which is one of the most important outstanding questions in physics. To reliably make predictions it is necessary to have full control of the finite temperature scalar potentials. However, as we show the standard methods used in BSM physics to improve phase transition calculations, resumming hard thermal loops, introduces significant errors into the scalar potential. In addition, the standard methods make it impossible to match theories to an EFT description reliably. In this paper we define a thermal resummation procedure based on partial dressing (PD) for general BSM calculations of phase transitions beyond the high-temperature approximation. Additionally, we introduce the modified optimized partial dressing (OPD) procedure, which is numerically nearly as efficient as old incorrect methods, while yielding identical results to the full PD calculation. This can be easily applied to future BSM studies of phase transitions in the early universe. As an example, we show that in unmixed singlet scalar extensions of the SM, the (O)PD calculations make new phenomenological predictions compared to previous analyses. An important future application is the study of EFTs at finite temperature.

Picard groups for tropical toric schemes

From any monoid scheme $X$ (also known as an $\mathbb{F}_1$-scheme) one can pass to a semiring scheme (a generalization of a tropical scheme) $X_S$ by scalar extension to an idempotent semifield $S$. We prove that for a given irreducible monoid scheme $X$ (satisfying some mild conditions) and an idempotent semifield $S$, the Picard group $Pic(X)$ of $X$ is stable under scalar extension to $S$ (and in fact to any field $K$). In other words, we show that the groups $Pic(X)$ and $Pic(X_S)$ (and $Pic(X_K)$) are isomorphic. In particular, if $X_\mathbb{C}$ is a toric variety, then $Pic(X)$ is the same as the Picard group of the associated tropical scheme. The Picard groups can be computed by considering the correct sheaf cohomology groups. We also define the group $CaCl(X_S)$ of Cartier divisors modulo principal Cartier divisors for a cancellative semiring scheme $X_S$ and prove that $CaCl(X_S)$ is isomorphic to $Pic(X_S)$.

Light scalar dark matter extension of the type-II two-Higgs-doublet model

We examine the type-II two-Higgs-doublet model with a light scalar dark matter ($S$) after imposing the constraints from the Higgs searches at the LHC and dark matter experiments. We first assume that both two CP-even Higgses ($h$ and $H$) are portals between the DM and SM sectors, and the CP-odd Higgs ($A$) and $H$ are heavier than 130 GeV. We find that the DM with a mass of $10\sim 50$ GeV is disfavored by the joint constraints of the 125 GeV Higgs signal data, the relic density, XENON1T (2017), PandaX-II (2017) and the Fermi-LAT. Next, we consider a special scenario in which the heavy CP-even Higgs is taken as the 125 GeV Higgs. The light CP-even Higgs is the only portal between the DM and SM sectors, and the DM mass is slightly below Higgs resonance. We find that the signal data of the 125 GeV Higgs restrict $\tan\beta$ to be in the range of $1\sim 1.5$ for $m_h<$ 62 GeV. The $gg\to A\to hZ$ and $b\bar{b}\to h \to \tau^+\tau^-$ channels at the LHC can impose lower limits and upper limits on $\tan\beta$, respectively. For appropriate values of $\tan\beta$, $\lambda_h$ and $m_h$, the DM with a mass of $10\sim 50$ GeV is allowed by the constraints of the Higgs searches at the LHC and dark matter experiments. For example, $\tan\beta$ is restricted to be in the range of $1.0\sim1.5$ for 10 GeV $<m_s<$ 26 GeV, and $\frac{m_h}{2m_S}>$ 1.125 is excluded for 30 GeV $<m_S<$ 50 GeV.

Orthogonal involutions on central simple algebras and function fields of Severi–Brauer varieties

An orthogonal involution σ on a central simple algebra A, after scalar extension to the function field F(A) of the Severi–Brauer variety of A, is adjoint to a quadratic form qσ over F(A), which is uniquely defined up to a scalar factor. Some properties of the involution, such as hyperbolicity, and isotropy up to an odd-degree extension of the base field, are encoded in this quadratic form, meaning that they hold for the involution σ if and only if they hold for qσ. As opposed to this, we prove that there exists non-totally decomposable orthogonal involutions that become totally decomposable over F(A), so that the associated form qσ is a Pfister form. We also provide examples of nonisomorphic involutions on an index 2 algebra that yield similar quadratic forms, thus proving that the form qσ does not determine the isomorphism class of σ, even when the underlying algebra has index 2. As a consequence, we show that the e3 invariant for orthogonal involutions is not classifying in degree 12, and does not detect totally decomposable involutions in degree 16, as opposed to what happens for quadratic forms.

Towards a viable scalar interpretation of RD(*)

Recent measurements of semileptonic B-meson decays seemingly imply violations of lepton flavor universality beyond the Standard Model predictions. With three-level explanations based on extended Higgs sectors being strongly challenged by the measurements of the $B_c^-$ lifetime, new theories invoking leptoquark or vector fields appear as the only feasible answer. However, in this work we show that simple scalar extensions of the Standard Model still offer a possible solution to the $B$ physics puzzle, owing to sizeable loop-level corrections which mimic the effects of new vector contributions. Considering a simplified model characterised by a charged and a neutral scalar particle, we verify the compatibility of the observed $R_{D^{(*)}}$ signal with the relevant collider bounds. We also study an embedding of the simplified model into a three-Higgs-doublet framework, and investigate its main phenomenological constraints.

Solitary wave propagation in a two-dimensional lattice

We construct a family of exact planar solitary wave solutions in a two-dimensional lattice. The system under consideration is a scalar two-dimensional extension of a nonintegrable Fermi–Pasta–Ulam problem with a piecewise quadratic potential. The constructed solutions exhibit an anisotropic dependence on the angle of propagation. Through a detailed analysis of explicit solutions, we show that conventional quasicontinuum models fail to fully describe this dependence. However, a truncated series approximation of the constructed solution that includes sufficiently short wavelengths captures this effect quite well.

Hyperbolicity and near hyperbolicity of quadratic forms over function fields of quadrics

Let p and q be anisotropic quadratic forms over a field F of characteristic $$\ne 2$$ , let s be the unique non-negative integer such that $$2^s < {\mathrm {dim}( p)} \le 2^{s+1}$$ , and let k denote the dimension of the anisotropic part of q after scalar extension to the function field F(p) of p. We conjecture that the dimension of q must lie within k of an integer multiple of $$2^{s+1}$$ . This can be viewed as a direct generalization of Hoffmann’s Separation theorem. Among other cases, we prove that the conjecture is true if $$k<2^{s-1}$$ . When $$k=0$$ , this shows that any anisotropic form representing an element of the kernel of the natural restriction homomorphism $$W(F)\rightarrow W(F(p))$$ has dimension divisible by $$2^{s+1}$$ .

Motifs et adjoints

We show in many cases the existence of adjoints to extension of scalars on categories of motivic nature, in the framework of field extensions. This is to be contrastedwith themore classical situationwhere one dealswith a finite typemorphism of schemes. Among various applications, one is a construction of a “Tate–Šafarevič motive” attached to an abelian variety over a function field. We also deduce a possible approach to Bloch’s conjecture on surfaces, by reduction to curves.