Triplet Representation Research Articles

Common origin of nonzero θ13 and baryon asymmetry of the Universe in a TeV scale seesaw model with A4 flavor symmetry

We study the possibility of generating non-zero reactor mixing angle $\theta_{13}$ and baryon asymmetry of the Universe within the framework of an $A_4$ flavour symmetric model. Using the conventional type I seesaw mechanism we construct the Dirac and Majorana mass matrices which give rise to the correct light neutrino mass matrix. Keeping the right handed neutrino mass matrix structure trivial so that it gives rise to a (quasi) degenerate spectrum of heavy neutrinos suitable for resonant leptogenesis at TeV scale, we generate the non-trivial structure of Dirac neutrino mass matrix that can lead to the light neutrino mixing through type I seesaw formula. Interestingly, such a setup naturally leads to non-zero $\theta_{13}$ due to the existence of anti-symmetric contraction of the product of two triplet representations of $A_4$. Such antisymmetric part of triplet products usually vanish for right handed neutrino Majorana mass terms, leading to $\mu-\tau$ symmetric scenarios in the most economical setups. We constrain the model parameters from the requirement of producing the correct neutrino data as well as baryon asymmetry of the Universe for right handed neutrino mass scale around TeV. The $A_4$ symmetry is augmented by additional $Z_3 \times Z_2$ symmetry to make sure that the splitting between right handed neutrinos required for resonant leptogenesis is generated only by next to leading order terms, making it naturally small. We find that the inverted hierarchical light neutrino masses give more allowed parameter space consistent with neutrino and baryon asymmetry data.

The effective action of a BPS Alice string

Recently a BPS Alice string has been found in a U(1)times SO(3) gauge theory coupled with three charged complex scalar fields in the triplet representation (in JHEP 1709:046 arXiv:1703.08971 [hep-th], 2017). It is a half BPS state preserving a half of the supercharges when embedded into a supersymmetric gauge theory. In this paper, we study zero modes of a BPS Alice string. After presenting U(1) and translational zero modes, we construct the effective action of these modes. In contrast to a previous analysis of the conventional Alice string for which only large distance behaviors are known, we can exactly perform a calculation in the full space thanks to the BPS properties.

Spontaneous CP violation in multi-Higgs potentials with triplets of \u0394(3n2) and \u0394(6n2)

Motivated by discrete flavour symmetry models, we analyse Spontaneous CP Violation (SCPV) for potentials involving three or six Higgs fields (both electroweak doublets and singlets) which fall into irreducible triplet representations of discrete symmetries belonging to the Δ(3n2) and Δ(6n2) series, including A4, S4, Δ(27) and Δ(54). For each case, we give the potential and find various global minima for different regions of the parameter space of the potential. Using CP-odd basis Invariants that indicate the presence of Spontaneous CP Violation we separate the VEVs into those that do or do not violate CP. In cases where CP is preserved we reveal a CP symmetry of the potential that is preserved by those VEVs, otherwise we display a non-zero CP-odd Invariant. Finally we identify interesting cases where there is Spontaneous Geometrical CP Violation in which the VEVs have geometrical phases.

A Sommerfeld toolbox for colored dark sectors

We present analytical formulas for the Sommerfeld corrections to the annihilation of massive colored particles into quarks and gluons through the strong interaction. These corrections are essential to accurately compute the dark matter relic density for coannihilation with colored partners. Our formulas allow us to compute the Sommerfeld effect, not only for the lowest term in the angular momentum expansion of the amplitude, but for all orders in the partial wave expansion. In particular, we carefully account for the effects of the spin of the annihilating particle on the symmetry of the two-particle wave function. This work focuses on strongly interacting particles of arbitrary spin in the triplet, sextet and octet color representations. For typical velocities during freeze-out, we find that including Sommerfeld corrections on the next-to-leading order partial wave leads to modifications of up to 10 to 20 percent on the total annihilation cross section. Complementary to QCD, we generalize our results to particles charged under an arbitrary unbroken SU(N) gauge group, as encountered in dark glueball models. In connection with this paper a Mathematica notebook is provided to compute the Sommerfeld corrections for colored particles up to arbitrary order in the angular momentum expansion.

Standard model EFT and extended scalar sectors

One of the simplest extensions of the Standard Model is the inclusion of an additional scalar multiplet, and we consider scalars in the $SU(2)_L$ singlet, triplet, and quartet representations. We examine models with heavy neutral scalars, $m_H\sim 1-2$ TeV, and the matching of the UV complete theories to the low energy effective field theory. We demonstrate the agreement of the kinematic distributions obtained in the singlet models for the gluon fusion of a Higgs pair with the predictions of the effective field theory. The restrictions on the extended scalar sectors due to unitarity and precision electroweak measurements are summarized and lead to highly restricted regions of viable parameter space for the triplet and quartet models.

Hadronic weak decay Bb(12+)→B(12+, 32+)+V

It is shown that for the effective Lagrangian with the factorization ansatz considered here, in the two-body hadronic decay ${\mathcal{B}}_{b}({\frac{1}{2}}^{+})\ensuremath{\rightarrow}\mathcal{B}({\frac{1}{2}}^{+},{\frac{3}{2}}^{+})+V$, with ${\mathcal{B}}_{b}({\frac{1}{2}}^{+})$ belonging to the representation $\overline{3}$, the only allowed decay channel is ${\mathcal{B}}_{b}({\frac{1}{2}}^{+})\ensuremath{\rightarrow}\mathcal{B}({\frac{1}{2}}^{+})+V$, where $\mathcal{B}({\frac{1}{2}}^{+})$ belongs to the representation 8 of $SU(3)$. However, for ${\mathcal{B}}_{b}({\frac{1}{2}}^{+})$ belonging to the sextet representation 6, the allowed decay channels are ${\mathcal{B}}_{b}({\frac{1}{2}}^{+})\ensuremath{\rightarrow}\mathcal{B}({\frac{1}{2}}^{+},{\frac{3}{2}}^{+})+V$, where $\mathcal{B}({\frac{1}{2}}^{+})$ and $\mathcal{B}({\frac{3}{2}}^{+})$ belong to the octet representation ${8}^{\ensuremath{'}}$ and the decuplet 10 of $SU(3)$, respectively. The decay channel ${\mathcal{B}}_{b}({\frac{1}{2}}^{+})\ensuremath{\rightarrow}\mathcal{B}({\frac{1}{2}}^{+})+V$ is analyzed in detail. The decay rate ($\mathrm{\ensuremath{\Gamma}}$) and the asymmetry parameters $\ensuremath{\alpha},{\ensuremath{\alpha}}^{\ensuremath{'}},\ensuremath{\beta},\ensuremath{\gamma}$, and ${\ensuremath{\gamma}}^{\ensuremath{'}}$ are expressed in terms of four amplitudes. In particular, for the decay ${\mathrm{\ensuremath{\Lambda}}}_{b}\ensuremath{\rightarrow}\mathrm{\ensuremath{\Lambda}}+J/\ensuremath{\psi}$ it is shown that within the factorization framework, using heavy quark spin symmetry, the decay rate and the asymmetry parameters can be expressed in terms of two form factors ${F}_{1}$ and ${F}_{2}/{F}_{1}$, which are to be evaluated in some model. By using the values of these form factors calculated in a quark model, the branching ratio and the asymmetry parameters $\ensuremath{\alpha}$ and ${\ensuremath{\alpha}}^{\ensuremath{'}}$ are calculated numerically. For other heavy quarks belonging to the triplet and sextet representations, the results can be easily obtained by using $SU(3)$ symmetry and a phase-space factor. Finally, the decay ${\mathrm{\ensuremath{\Omega}}}_{b}^{\ensuremath{-}}\ensuremath{\rightarrow}{\mathrm{\ensuremath{\Omega}}}^{\ensuremath{-}}+J/\ensuremath{\psi}$ is analyzed within the factorization framework. It is shown that the asymmetry parameter $\ensuremath{\alpha}$ in this particular decay is zero. The branching ratio obtained in the first approximation is compared with the experimental value.

Fingerprints of heavy scales in electroweak effective Lagrangians

The couplings of the electroweak effective theory contain information on the heavy-mass scales which are no-longer present in the low-energy Lagrangian. We build a general effective Lagrangian, implementing the electroweak chiral symmetry breaking SU(2)L ⊗ SU(2)R → SU(2)L+R, which couples the known particle fields to heavier states with bosonic quantum numbers JP = 0± and 1±. We consider colour-singlet heavy fields that are in singlet or triplet representations of the electroweak group. Integrating out these heavy scales, we analyze the pattern of low-energy couplings among the light fields which are generated by the massive states. We adopt a generic non-linear realization of the electroweak symmetry breaking with a singlet Higgs, without making any assumption about its possible doublet structure. Special attention is given to the different possible descriptions of massive spin-1 fields and the differences arising from naive implementations of these formalisms, showing their full equivalence once a proper short-distance behaviour is required.

General analysis of the charged Higgs sector of theY=0triplet-singlet extension of the MSSM at the LHC

We investigate the extended Higgs sectors, specially the charged Higgs sector in a supersymmetric $Y=0$ $SU(2)$ triplet and a Standard Model (SM) gauge singlet extension of SM. We show that in this model the allowed data for the Higgs boson interaction eigenstates tend to group into separate blocks for $SU(2)$ triplet, doublet and singlet. The triplet sector has two degenerate paired states, each pair composed of a mostly-triplet charged Higgs and of a mostly-triplet scalar or pseudoscalar state. The mostly-doublet sector involves a Standard Model like Higgs of 125 GeV and extra mass-degenerate states, composed of a charged, a scalar and a pseudoscalar. The CP-odd component of the singlet scalar, after supersymmetry breaking, takes the role of a pseudo-Nambu-Goldstone mode in the $Z_3$ symmetric case, while the CP-even one becomes decoupled. In a second part of our study we investigate the different decay processes allowed to a charged Higgs boson of this model. Specifically, we search for general signatures of the TNMSSM in order to distinguish among Higgs fields belonging to $SU(2)$ doublet, triplet and singlet representations at the LHC. We also propose few golden plated final state modes carrying the distinctive signatures of this model which could be investigated in collider searches. We also show how in the decoupling limit of the triplet ($|\lambda_T| \simeq 0$) affects the decays as well as the production channels at the LHC.

CP-odd invariants for multi-Higgs models: Applications with discrete symmetry

CP-odd invariants provide a basis independent way of studying the CP properties of Lagrangians. We propose powerful methods for constructing basis invariants and determining whether they are CP-odd or CP-even, then systematically construct all of the simplest CP-odd invariants up to a given order, finding many new ones. The CP-odd invariants are valid for general potentials when expressed in a standard form. We then apply our results to scalar potentials involving three (or six) Higgs fields which form irreducible triplets under a discrete symmetry, including invariants for both explicit as well as spontaneous CP violation. The considered cases include one triplet of Standard Model (SM) gauge singlet scalars, one triplet of SM Higgs doublets, two triplets of SM singlets, and two triplets of SM Higgs doublets. For each case we study the potential symmetric under one of the simplest discrete symmetries with irreducible triplet representations, namely $A_4$, $S_4$, $\Delta(27)$ or $\Delta(54)$, as well as the infinite classes of discrete symmetries $\Delta(3n^2)$ or $\Delta(6n^2)$.

Predictions of the Higgs Mass and the Weak Mixing Angle in the 6D Gauge-Higgs Unification

In the gauge-Higgs unification with multiple extra spaces, the Higgs self-coupling is of the order of $g^2$ and Higgs is predicted to be light, being consistent with the LHC results. When the gauge group is simple, the weak mixing angle is also predictable. We address a question whether there exists a model of gauge-Higgs unification in 6-dimensional space-time, which successfully predicts the mass ratios of the Higgs boson and weak gauge bosons. First, by use of a useful formula we give a general argument on the condition to get a realistic prediction of the weak mixing angle $sin^{2}theta_{W} = 1/4$, and find that triplet and sextet representations of the minimal SU(3) gauge group lead to the realistic prediction. Concerning the Higgs mass, we notice that in the models with one Higgs doublet, the predicted Higgs mass is always the same: $M_H = 2 M_W$. However, by extending our discussion to the models with two Higgs doublets, the situation changes: we obtain an interesting prediction $M_{H} leq 2M_{W}$ at the leading order of the perturbation. Thus it is possible to recover the observed Higgs mass, 125 GeV, for a suitable choice of the parameter. The situation is in clear contrast to the case of the minimal supersymmetric standard model, where $M_{H} leq M_{Z}$ at the classical level and the predicted Higgs mass cannot recover the observed value.

Lepton flavor violating decays of μ and τ leptons in a gauge group SU(2)L × SU(2)R × SU(2)Y

The electroweak unification group [Formula: see text] is proposed for the charged lepton flavor violating decays of the muon [Formula: see text] and tau [Formula: see text] leptons. The group [Formula: see text] is in the lepton space. The left-handed leptons and antileptons are assigned to the fundamental representation [Formula: see text] of the semi-simple group. The gauge group [Formula: see text] is spontaneously broken to [Formula: see text], where [Formula: see text] is the hypercharge, by introducing a scalar multiplet [Formula: see text] which belongs to the triplet representation 3 of the [Formula: see text] and is singlet under [Formula: see text]. At this stage charged vector bosons [Formula: see text] of [Formula: see text] which mediate the lepton flavor violating decays acquire masses and are decoupled with one Higgs scalar [Formula: see text]. The residual group [Formula: see text] has all the features of the left–right electroweak unification group extensively studied in the literature. The probability for lepton flavor violating decays is [Formula: see text].

Highly accurate doubling algorithms for M-matrix algebraic Riccati equations

The doubling algorithms are very efficient iterative methods for computing the unique minimal nonnegative solution to an M-matrix algebraic Riccati equation (MARE). They are globally and quadratically convergent, except for MARE in the critical case where convergence is linear with the linear rate 1 / 2. However, the initialization phase and the doubling iteration kernel of any doubling algorithm involve inverting nonsingular M-matrices. In particular, for MARE in the critical case, the M-matrices in the doubling iteration kernel, although nonsingular, move towards singular M-matrices at convergence. These inversions are causes of concerns on entrywise relative accuracy of the eventually computed minimal nonnegative solution. Fortunately, a nonsingular M-matrix can be inverted by the GTH-like algorithm of Alfa et al. (Math Comp 71:217---236, 2002) to almost full entrywise relative accuracy, provided a triplet representation of the matrix is known. Recently, Nguyen and Poloni (Numer Math 130(4):763---792, 2015) discovered a way to construct triplet representations in a cancellation-free manner for all involved M-matrices in the doubling iteration kernel, for a special class of MAREs arising from Markov-modulated fluid queues. In this paper, we extend Nguyen's and Poloni's work to all MAREs by also devising a way to construct the triplet representations cancellation-free. Our construction, however, is not a straightforward extension of theirs. It is made possible by an introduction of novel recursively computable auxiliary nonnegative vectors. As the second contribution, we propose an entrywise relative residual for an approximate solution. The residual has an appealing feature of being able to reveal the entrywise relative accuracies of all entries, large and small, of the approximation. This is in marked contrast to the usual legacy normalized residual which reflects relative accuracies of large entries well but not so much those of very tiny entries. Numerical examples are presented to demonstrate and confirm our claims.

Symmetry breaking patterns of the 3-3-1 model at finite temperature

We consider the minimal version of an extension of the standard electroweak model based on the $SU(3)_c \times SU(3)_L \times U(1)_X$ gauge symmetry (the 3-3-1 model). We analyze the most general potential constructed from three scalars in the triplet representation of $SU(3)_L$, whose neutral components develop nonzero vacuum expectation values, giving mass for all the model's massive particles. {}For different choices of parameters, we obtain the particle spectrum for the two symmetry breaking scales: one where the $SU(3)_L \times U(1)_X$ group is broken down to $SU(2)_L\times U(1)_Y$ and a lower scale similar to the standard model one. Within the considerations used, we show that the model encodes two first-order phase transitions, respecting the pattern of symmetry restoration. The last transition, corresponding to the standard electroweak one, is found to be very weak first-order, most likely turning second-order or a crossover in practice. However, the first transition in this model can be strongly first-order, which might happen at a temperature not too high above the second one. We determine the respective critical temperatures for symmetry restoration for the model.

Coupling between Majorana fermions and Nambu-Goldstone bosons inside a non-Abelian vortex in dense QCD

Quark matter at high density may exhibit color superconductivity. As magnetic flux tubes in metallic superconductors, color magnetic flux tubes as non-Abelian vortices appear in the color-flavor locked phase of high density QCD. ${\mathbb C}P^2$ Nambu-Goldstone bosons and Majorana fermions belonging to the triplet representation are known to be localized around a non-Abelian vortex. In this paper, we determine the coupling of these bosonic and fermionic modes by using the nonlinear realization method.

Study of lepton flavor violation in flavor symmetric models for lepton sector

Flavor symmetric model is one of the attractive Beyond Standard Models (BSMs) to reveal the flavor structure of the Standard Model (SM). A lot of efforts have been put into the model building and we find many kinds of flavor symmetries and setups are able to explain the observed fermion mass matrices. In this paper, we look for common predictions of physical observables among the ones in flavor symmetric models, and try to understand how to test flavor symmetry in experiments. Especially, we focus on the BSMs for leptons with extra Higgs $SU(2)_L$ doublets charged under flavor symmetry. In many flavor models for leptons, remnant symmetry is partially respected after the flavor symmetry breaking, and it controls well the Flavor Changing Neutral Currents (FCNCs) and suggests some crucial predictions against the flavor changing process, although the remnant symmetry is not respected in the full lagrangian. In fact, we see that $\tau^- \to e^+ \mu^- \mu^-$ $( \mu^+ e^- e^-)$ and $e^+ e^- \to \tau^+\tau^-$ $(\mu^-\mu^+)$ processes are the most important in the flavor models that the extra Higgs doublets belong to triplet representation of flavor symmetry. For instance, the stringent constraint from the $\mu \to e \gamma$ process could be evaded according to the partial remnant symmetry. We also investigate the breaking effect of the remnant symmetry mediated by the Higgs scalars, and investigate the constraints from the flavor physics: the flavor violating $\tau$ and $\mu$ decays, the electric dipole moments, and the muon anomalous magnetic moment. We also discuss the correlation between FCNCs and nonzero $\theta_{13}$, and point out the physical observables in the charged lepton sector to test the BSMs for the neutrino mixing.

Δ(27) family symmetry and neutrino mixing

The observed neutrino mixing, having a near maximal atmospheric neutrino mixing angle and a large solar mixing angle, is close to tri-bi-maximal. This structure may be related to the existence of a discrete non-Abelian family symmetry. In this paper the family symmetry is the non-Abelian discrete group Δ(27), a subgroup of SU(3) with triplet and anti-triplet representations. Different frameworks are constructed in which the mixing follows from combining fermion mass terms with the vacuum structure enforced by the discrete symmetry. Mass terms for the fermions originate from familon triplets, anti-triplets or both. Vacuum alignment for the family symmetry breaking familons follows from simple invariants.

Invariant approach to [formula omitted] in unbroken Δ(27)

The invariant approach is a powerful method for studying CP violation for specific Lagrangians. The method is particularly useful for dealing with discrete family symmetries. We focus on the CP properties of unbroken Δ(27) invariant Lagrangians with Yukawa-like terms, which proves to be a rich framework, with distinct aspects of CP, making it an ideal group to investigate with the invariant approach. We classify Lagrangians depending on the number of fields transforming as irreducible triplet representations of Δ(27). For each case, we construct CP-odd weak basis invariants and use them to discuss the respective CP properties. We find that CP violation is sensitive to the number and type of Δ(27) representations.

Left-right supersymmetry after the Higgs boson discovery

We perform a thorough analysis of the parameter space of the minimal left-right supersymmetric model in agreement with the LHC data. The model contains left- and right-handed fermionic doublets, two Higgs bidoublets, two Higgs triplet representations, and one singlet, insuring a charge-conserving vacuum. We impose the condition that the model complies with the experimental constraints on supersymmetric particles masses and on the doubly-charged Higgs bosons, and require that the parameter space of the model satisfy the LHC data on neutral Higgs signal strengths at $2\sigma$. We choose benchmark scenarios by fixing some basic parameters and scanning over the rest. The LSP in our scenarios is always the lightest neutralino. We find that the signals for $H\to \gamma \gamma$ and $H \to VV^\star$ are correlated, while $H \to b \bar b$ is anti-correlated with all the other decay modes, and also that the contribution from singly-charged scalars dominate that of the doubly-charged scalars in $H\to \gamma \gamma$ and $H \to Z\gamma$ loops, contrary to Type-II seesaw models. We also illustrate the range for mass spectrum of the LRSUSY model in light of planned measurements of the branching ratio of $H\to \gamma \gamma$ to 10% level.

Componentwise accurate fluid queue computations using doubling algorithms

Markov-modulated fluid queues are popular stochastic processes frequently used for modelling real-life applications. An important performance measure to evaluate in these applications is their steady-state behaviour, which is determined by the stationary density. Computing it requires solving a (nonsymmetric) M-matrix algebraic Riccati equation, and indeed computing the stationary density is the most important application of this class of equations. Xue, Xu and Li [Numer. Math., 2012] provided a componentwise first-order perturbation analysis of this equation, proving that the solution can be computed to high relative accuracy even in the smallest entries, and suggested several algorithms for computing it. An important step in all proposed algorithms is using so-called triplet representations, which are special representations for M-matrices that allow for a high-accuracy variant of Gaussian elimination, the GTH-like algorithm. However, triplet representations for all the M-matrices needed in the algorithm were not found explicitly. This can lead to an accuracy loss that prevents the algorithms to converge in the componentwise sense. In this paper, we focus on the structured doubling algorithm, the most efficient among the proposed methods in Xue et al., and build upon their results, providing (i) explicit and cancellation-free expressions for the needed triplet representations, allowing the algorithm to be performed in a really cancellation-free fashion; (ii) an algorithm to evaluate the final part of the computation to obtain the stationary density; and (iii) a componentwise error analysis for the resulting algorithm, the first explicit one for this class of algorithms. We also present numerical results to illustrate the accuracy advantage of our method over standard (normwise-accurate) algorithms.

Scalar representations in the light of electroweak phase transition and cold dark matter phenomenology

The extension of the standard model's minimal Higgs sector with aninert SU(2)L scalar doublet can provide light dark matter candidateand simultaneously induce a strong phase transition for explainingBaryogenesis. There is however no symmetry reasons to prevent theextension using scalars with higher SU(2)L representations. Bymaking random scans over the models' parameters, we showthat in the light of electroweak physics constraints, strong firstorder electroweak phase transition and the possibility of havingsub-TeV cold dark matter candidate thehigher representations are rather disfavored compared to the inertdoublet. This is done by computing generic perturbativitybehavior and impact on electroweak phase transitions of higherrepresentations in comparison with the inert doublet model. Explicitphase transition and cold dark matter phenomenology within the contextof the inert triplet and quartet representations are used for detailedillustrations.