Unitarity Sum Research Articles

Tree-level Unitarity in SU2L×U1Y×U1Y′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ extrm{SU}{(2)}_L\ imes \ extrm{U}{(1)}_Y\ imes \ extrm{U}{(1)}_{Y^{\\prime }} $$\\end{document} Models

In models with a U(1) gauge extension beyond the Standard Model, one can derive sum rules for the couplings of the theory that are a consequence of tree-level unitarity. In this paper, we provide a comprehensive list of coupling sum rules for a general SU2L×U1Y×U1Y′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ extrm{SU}{(2)}_L\ imes \ extrm{U}{(1)}_Y\ imes \ extrm{U}{(1)}_{Y^{\\prime }} $$\\end{document} gauge theory coupled to an arbitrary set of fermion and scalar multiplets. These results are of particular interest for models of dark matter that employ an extended gauge sector mediated by a new (dark) Z′ gauge boson. For the case of a minimal extension of the Standard Model with a U1Y′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ extrm{U}{(1)}_{Y^{\\prime }} $$\\end{document} gauge boson, we clarify the definitions of the weak mixing angle and the electroweak ρ parameter. We demonstrate the utility of a generalized ρ parameter (denoted by ρ′) whose definition naturally follows from the unitarity sum rules developed in this paper.

Cosmology of complex scalar dark matter: Interplay of self-scattering and annihilation

The cosmology of a standard model (SM) gauge singlet complex scalar dark matter (DM), stabilized by a reflection symmetry, is studied including all renormalizable interactions that preserve the reflection symmetry but can break the larger global U(1) symmetry of DM number. We find an interesting interplay of the ensuing DM self-scatterings and annihilations in generating the present DM density, and possible particle-antiparticle asymmetry in the DM sector. The role of DM self-scatterings in determining its present density and composition is a novel phenomenon. The simultaneous presence of the self-scatterings and annihilations is required to obtain a non-zero asymmetry, which otherwise vanishes due to unitarity sum rules.

Unitarity, stability, and loops of unstable ghosts

We present a new understanding of the unstable ghost-like resonance which appears in theories such as quadratic gravity and Lee-Wick type theories. Quantum corrections make this resonance unstable, such that it does not appear in the asymptotic spectrum. We prove that these theories are unitary to all orders. Unitarity is satisfied by the inclusion of only cuts from stable states in the unitarity sum. This removes the need to consider this as a ghost state in the unitarity sum. However, we often use a narrow-width approximation where we do include cuts through unstable states, and ignore cuts through the stable decay products. If we do this with the unstable ghost resonance at one loop, we get the correct answer only by using a contour which was originally defined by Lee and Wick. The quantum effects also provide damping in both the Feynman and the retarded propagators, leading to stability under perturbations.

Unitarity restoring graviton radiation in the collapse regime of gravitational scattering

We investigate graviton radiation in gravitational scattering at small impact parameters $b<R\equiv 2G\sqrt{s}$ and extreme energies $s\gg M_P^2$, a regime in which classical collapse is thought to occur, and thus radiation may be suppressed also. Here however, by analyzing the soft-based representation of radiation recently proposed in the semiclassical ACV framework, we argue that gravitons can be efficiently produced in the untrapped region $|\boldsymbol{x}|\gtrsim R>b$, so as to suggest a possible completion of the unitarity sum. In fact, such energy radiation at large distances turns out to compensate and to gradually reduce to nothing the amount of energy $E'$ being trapped at small-$b$'s, by thus avoiding the quantum tunneling suppression of the elastic scattering and suggesting a unitary evolution. We finally look at the coherent radiation sample so obtained and we find that, by energy conservation, it develops an exponential frequency damping corresponding to a "quasi-temperature" of order $\hbar/R$, which is naturally related to a Hawking radiation and is suggestive of a black-hole signal at quantum level.

Classification of simple heavy vector triplet models

We investigate decay modes of spin-1 heavy vector bosons ($V'$) from the viewpoint of perturbative unitarity in a model-independent manner. Perturbative unitarity requires some relations among couplings. The relations are called unitarity sum rules. We derive the unitarity sum rules from processes that contain two fermions and two gauge bosons. We find the relations between $V'$ couplings to the SM fermions $(f)$ and $V'$ couplings to the SM gauge bosons ($V$). Using the coupling relations, we calculate partial decay widths for $V'$ decays into $VV$ and $ff$. We show that Br($W' \to WZ) \lesssim$ 2$\%$ in the system that contains $V'$ and CP-even scalars as well as the SM particles. This result is independent of the number of the CP-even scalars. We also show that contributions of CP-odd scalars help to make Br($W' \to WZ$) larger than Br($W' \to ff$) as long as the CP-odd scalars couple to both the SM fermions and the SM gauge bosons. The existence of the CP-odd scalar couplings is a useful guideline to construct models that predict Br($W' \to WZ) \gtrsim$ 2$\%$. Our analysis relies only on the perturbative unitarity of $f\bar{f} \to WW'$. Therefore our result can be applied to various models.

Unitarity sum rules, three-site moose model, and the ATLAS 2 TeV diboson anomalies

We investigate $W'$ interpretations for the ATLAS 2 TeV diboson anomalies. The roles of the unitarity sum rules, which ensure the perturbativity of the longitudinal vector boson scattering amplitudes, are emphasized. We find the unitarity sum rules and the custodial symmetry are powerful enough to predict various nontrivial relations among $WWZ'$, $WZW'$, $WWh$, $WW'h$ and $ZZ'h$ coupling strengths in a model independent manner. We also perform surveys in the general parameter space of $W'$ models and find the ATLAS 2 TeV diboson anomalies may be interpreted as a $W'$ particle of the three site moose model, i.e., a Kaluza-Klein like particle in a deconstructed extra dimension model. It is also shown that the non standard-model-like Higgs boson is favored by the present data to interpret the ATLAS diboson anomalies as the consequences of the $W'$ and $Z'$ bosons.

Unitarity-controlled resonances after the Higgs boson discovery

If the recently discovered Higgs boson's couplings deviate from the Standard Model expectation, we may anticipate new resonant physics in the weak boson fusion channels resulting from high scale unitarity sum rules of longitudinal gauge boson scattering. Motivated by excesses in analyses of multi-leptons $+$ missing energy $+$ jets final states during run 1, we perform a phenomenological investigation of these channels at the LHC bounded by current Higgs coupling constraints. Such an approach constrains the prospects to observe such new physics at the LHC as a function of very few and generic parameters and allows the investigation of the strong requirement of probability conservation in the electroweak sector to high energies.

Does unitarity imply finiteness of electroweak oblique corrections at one loop? Constraining extra neutral Higgs bosons

Introducing an arbitrary number of neutral Higgs bosons in the electroweak symmetry breaking sector, we derive a set of conditions among Higgs couplings which need to be satisfied to maintain the unitarity of the high energy scattering amplitudes of weak gauge bosons at the tree level (unitarity sum rules). It is shown that the unitarity sum rules require the tree level $\ensuremath{\rho}$ parameter to be 1, without explicitly invoking the custodial symmetry arguments. The one loop finiteness of the electroweak oblique corrections is automatically guaranteed once these unitarity sum rules are imposed among Higgs couplings. Severe constraints on the lightest Higgs coupling (125 GeV Higgs coupling) and the mass of the second lightest Higgs boson are obtained from the unitarity and the results of the electroweak precision tests (oblique parameter measurements). These results are compared with the effective theory of the light Higgs boson, and we find simple relationships between the mass of the second lightest Higgs boson in our framework and the ultraviolet cutoff in the effective theory framework.

Composite Higgs sketch

The coupling of a composite Higgs to the standard model fields can deviate substantially from the standard model values. In this case perturbative unitarity might break down before the scale of compositeness is reached, which would suggest that additional composites should lie well below this scale. In this paper we account for the presence of an additional spin 1 custodial triplet of rhos. We examine the implications of requiring perturbative unitarity up to the compositeness scale and find that one has to be close to saturating certain unitarity sum rules involving the Higgs and the rho couplings. Given these restrictions on the parameter space we investigate the main phenomenological consequences of the spin 1 triplet. We find that they can substantially enhance the Higgs di-photon rate at the LHC even with a reduced Higgs coupling to gauge bosons. The main existing LHC bounds arise from di-boson searches, especially in the experimentally clean channel where the charged rhos decay to a W-boson and a Z, which then decay leptonically. We find that a large range of interesting parameter space with 700 GeV < m(rho) < 2 TeV is currently experimentally viable.

Superallowed nuclear β decay: symmetry breaking, CVC and CKM unitarity

Currently, the most restrictive test of the unitarity of the Cabibbo-Kobayashi-Maskawa (CKM) matrix is anchored by nuclear beta decay. Precise measurements of the ft-values for superallowed beta transitions between analog 0 + states are used to determine G V , the vector coupling constant; this, in turn, yields V u d , the up-down element of the Cabibbo-Kobayashi-Maskawa (CKM) matrix. When the value for V u d obtained from the latest survey of world data is combined with V u s and V u b , the other top-row elements, it leads to the most demanding test available of the unitarity of that matrix. Important ingredients required in the extraction of V u d from experiment are calculated radiative corrections as well as corrections to account for the isospin symmetry breaking that occurs between the analog parent and daughter states of each superallowed transition. The latter are well supported by comparison with experiment. The current value for the unitarity sum is 0.99995(61), a result that is in remarkable agreement with Standard Model expectations.

Superallowed0+→0+nuclearβdecays: A new survey with precision tests of the conserved vector current hypothesis and the standard model

A new critical survey is presented of all half-life, decay-energy, and branching-ratio measurements related to 20 superallowed ${0}^{+}\ensuremath{\rightarrow}{0}^{+}\ensuremath{\beta}$ decays. Compared with our last review, there are numerous improvements: First, we have added 27 recently published measurements and eliminated 9 references, either because they have been superseded by much more precise modern results or because there are now reasons to consider them fatally flawed; of particular importance, the new data include a number of high-precision Penning-trap measurements of decay energies. Second, we have used the recently improved isospin symmetry-breaking corrections, which were motivated by these new Penning-trap results. Third, our calculation of the statistical rate function $f$ now accounts for possible excitation in the daughter atom, a small effect but one that merits inclusion at the present level of experimental precision. Finally, we have re-examined the systematic uncertainty associated with the isospin symmetry-breaking corrections by evaluating the radial-overlap correction using Hartree-Fock radial wave functions and comparing the results with our earlier calculations, which used Saxon-Woods wave functions; the provision for systematic uncertainty has been changed as a consequence. The new ``corrected'' $\mathcal{F}t$ values are impressively constant and their average, when combined with the muon lifetime, yields the up-down quark-mixing element of the Cabibbo-Kobayashi-Maskawa (CKM) matrix, ${V}_{\mathit{ud}}=0.97425\ifmmode\pm\else\textpm\fi{}0.00022$. The unitarity test on the top row of the matrix becomes $|{V}_{\mathit{ud}}|{}^{2}+|{V}_{\mathit{us}}|{}^{2}+|{V}_{\mathit{ub}}|{}^{2}=0.99995\ifmmode\pm\else\textpm\fi{}0.00061$. Both ${V}_{\mathit{ud}}$ and the unitarity sum have significantly reduced uncertainties compared with our previous survey, although the new value of ${V}_{\mathit{ud}}$ is statistically consistent with the old one. From these data we also set limits on the possible existence of scalar interactions, right-hand currents, and extra $Z$ bosons. Finally, we discuss the priorities for future theoretical and experimental work with the goal of making the CKM unitarity test even more definitive.

Probing the Standard Model with Superallowed Nuclear Beta Decay

Central to the Electroweak Standard Model is the Cabibbo-Kobayashi-Maskawa (CKM) matrix, a unitary 3 × 3 matrix that relates the weak- to the mass-eigenstates of the three generations of quarks. The Standard Model itself does not prescribe the values of the individual matrix elements—they must each be determined experimentally—but it does insist that the full matrix must satisfy all conditions of unitarity. Thus, any conclusive experimental demonstration that the unitarity conditions are not met would prove the need for “new physics” extending beyond the current Standard Model. Conversely, if the experiment were instead to confirm unitarity, this result would circumscribe the scope of any possible new physics to an extent limited only by the quoted experimental uncertainties on the unitarity sum. Either way, there is compelling motivation here for precision measurements of the CKM matrix elements aimed at the most demanding tests possible of the matrix's unitarity.

Dynamical properties of the Anderson impurity model within a diagrammatic pseudoparticle approach

The Anderson model of a twofold spin degenerate impurity level in the limit of infinite Coulomb repulsion, $U\ensuremath{\rightarrow}\ensuremath{\infty}$, coupled to one and two degenerate conduction bands or channels, is considered in pseudo-particle representation. We extend the conserving T-matrix approximation (CTMA), a general diagrammatic approximation scheme based on a fully renormalized computation of two-particle vertex functions in the spin and in the charge channel, to the calculation of thermodynamic and spectral properties. In the single-channel case, the CTMA yields in the Kondo regime a temperature independent Pauli spin susceptibility for temperatures below the Kondo temperature ${T}_{K}$ and down to the lowest temperatures considered, reproducing the exact spin screening in the Fermi liquid state. The impurity spectral density appears to remain non-singular down to the lowest temperatures, in agreement with Fermi liquid behavior. However, the unitarity sum rule, which is crucial for an impurity solver like the CTMA to be applicable within dynamical mean field theories for strongly correlated lattice models, is overestimated at the lowest temperatures. We argue that this shortcoming may be due to numerical imprecision and discuss an appropriate scheme for its correction. In the two-channel case, the spectral density calculated within CTMA exhibits qualitatively the correct non-Fermi liquid behavior at low temperatures, i.e., a power law singularity.

Improvements to the method of dispersion relations forBnonleptonic decays

We bring some clarifications and improvements to the method of dispersion relations in the external masses variables that we proposed recently for investigating the final state interactions in the B nonleptonic decays. We first present arguments for the existence of an additional term in the dispersion representation, which arises from an equal-time commutator in the Lehmann-Symanzik-Zimmerman formalism and can be approximated by the conventional factorized amplitude. The reality properties of the spectral function and the Goldberger-Treiman procedure to perform the hadronic unitarity sum are analyzed in more detail. We also improve the treatment of the strong interaction part by including the contributions of both t- and u-channel trajectories in the Regge amplitudes. Applications to the ${B}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ and ${B}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{K}^{+}$ decays are presented.

Dispersion relations and rescattering effects inBnonleptonic decays

Recently, the final state strong interactions in nonleptonic B decays were investigated in a formalism based on hadronic unitarity and dispersion relations in terms of the off-shell mass squared of the $B$ meson. We consider an heuristic derivation of the dispersion relations in the mass variables using the reduction LSZ formalism and find a discrepancy between the spectral function and the dispersive variable used in the recent works. The part of the unitarity sum which describes final state interactions is shown to appear as spectral function in a dispersion relation based on the analytic continuation in the mass squared of one final particles. As an application, by combining this formalism with Regge theory and SU(3) flavour symmetry we obtain constraints on the tree and the penguin amplitudes of the decay $B^0\to \pi^+\pi^-$.

The quantum field theory of kaon oscillations

We consider the covariant formulation of the kaon mixing in the context of the propagator method. It results important to check the possibility of a sizable effect in the vacuum regeneration of kaons. We discuss all those terms which may give relevant contributions to modify the exponential decay law of the Wigner-Weisskopf narrow width approximation. Moreover, we examine the characteristic structure of the complex singularities of the matrix propagator and we provide a generalized form of the Bell-Steinberger unitarity sum rule.

Unitarity constraints on the B and B* form factors from QCD analyticity and heavy quark spin symmetry.

A method of deriving bounds on the weak meson form factors, based on perturbative QCD, analyticity and unitarity, is generalized in order to fully exploit heavy quark spin symmetry in the ground state $(L=0)$ doublet of pseudoscalar $(B)$ and vector $(B^*)$ mesons. All the relevant form factors of these mesons are taken into account in the unitarity sum. They are treated as independent functions along the timelike axis, being related by spin symmetry only near the zero recoil point. Heavy quark vacuum polarisation up to three loops in perturbative QCD and the experimental cross sections $\sigma(e^+e^- \rightarrow \Upsilon)$ are used as input. We obtain bounds on the charge radius of the elastic form factor of the $B$ meson, which considerably improve previous results derived in the same framework.

The pomeron as an expanding grey disk

In the crossing symmetric Reggeon calculus based on the 1N expansion, s-channel unitarity sum rules are written down explicitly to each order in 1N. Some of the lowest order sum rules are shown to imply a non-vanishing bare triple-Pomeron coupling at t = 0, irrespectively of the bare Pomeron intercept.

A new model for elastic deuteron-deuteron scattering

Straightforward application of the Glauber multiple scattering theory is dramatically challenged by data on elastic detueron-deuteron (d-d) scattering. The challenge has been argued to be met by an improved representation of the ground-state wave function of the deuteron as an admixture ofS-andD-waves. In the light of the failure of the Glauber and geometrical picture models in general, to explain proton-proton (pp) and proton-antiproton $$(p\bar p)$$ scattering data up to and including collider energies and for all-momentum trasfers, this argument becomes less and less compelling and more and more unconvincing. A model inspired by unitarity and which produces substantial elastic scattering through a unitarity sum over a specific class of intermediate states is presented. The model fits not only d-d, but also pp, $$p\bar p$$ and αN→αN (N=α, d,3He) data for all energies and momentum transfers. No detailed knowledge of ground-state wave functions is required.

Charged Particle Emission and Detection Sources and the One-Body and the Many-Body Coulomb Scattering

A detailed study is carried out of the fundamental roles of the emission and detection sources in the one-body and the many-body Coulomb scattering. It is shown that the emission and detection sources, as amplitudes of emission and detection of charged particles, are to have given time-dependent structures for a correct formulation of the problem. The method incorporates the long range effect of the Coulomb field which persists even at large distance separations. Most importantly, composite sources are necessarily introduced for the many-body Coulomb scattering which act as emission and detection sources of “clusters” of charged particles and reflect the physically important fact that charged particles feel the presence of each other even before or after “collisions”. A complete concellation of the so-called Dalitz phase factor naturally follows leading to infra-red finite transition amplitudes (as opposed only of transition probabilities) by a systematic use, in the process, of a unitarity sum with respect to the external sources.