Assuming fundamental fermions possess a new Abelian gauge charge that depends on flavors of both quark and lepton, we obtain a simple extension of the Standard Model, which reveals some new physics insights. The new gauge charge anomaly cancellation not only explains the existence of just three fermion generations as observed but also requires the presence of a unique right-handed neutrino $$\nu _R$$ with a non-zero new gauge charge. Further, the new gauge charge breaking supplies a residual matter parity, under which the fundamental fermions and $$\nu _R$$ are even, whereas a right-handed neutrino $$N_R$$ without the new charge is odd. Consequently, light neutrino masses in our model are generated from the tree-level type-I seesaw mechanism induced by $$\nu _R$$ and from the one-loop scotogenic contribution accommodated by potential dark matter candidates, $$N_R$$ and dark scalars, odd under the matter parity. We examine new physics phenomena related to the additional gauge boson, which could be observed at colliders. We analyze the constraints imposed on our model by current experimental limits on neutrino masses, neutral meson oscillations, -meson decays, and charged lepton flavor violating processes. We also investigate the potential dark matter candidates by considering relic density and direct detection.

Vietnam

Duong Loi

N. Duy

Cao Nam

Phung Dong

Scoto-seesaw model implied by flavor-dependent Abelian gauge charge

There is no reason why the gauge symmetry extension is family universal as in the standard model and the most well-motivated models, e.g. left-right symmetry and grand unification. Hence, we propose a simplest extension of the standard model – a flavor-dependent U(1) gauge symmetry – and find the new physics insight. For this aim, the U(1) charge, called X, is expressed as  $$X=x B+y L$$  <math> <mrow> <mi>X</mi> <mo>=</mo> <mi>x</mi> <mi>B</mi> <mo>+</mo> <mi>y</mi> <mi>L</mi> </mrow> </math>   in which x and y are free parameters as functions of flavor index, e.g. for a flavor i they take  $$x_i$$  <math> <msub> <mi>x</mi> <mi>i</mi> </msub> </math>   and  $$y_i$$  <math> <msub> <mi>y</mi> <mi>i</mi> </msub> </math>   respectively, where B and L denote normal baryon and lepton numbers. Imposing a relation involved by the color number 3, i.e.  $$-x_{1,2,\ldots ,n}=x_{n+1,n+2,\ldots ,n+m}=3y_{1,2,\ldots ,n+m}\equiv 3z$$  <math> <mrow> <mo>-</mo> <msub> <mi>x</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>+</mo> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mn>3</mn> <msub> <mi>y</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>+</mo> <mi>m</mi> </mrow> </msub> <mo>≡</mo> <mn>3</mn> <mi>z</mi> </mrow> </math>  , for arbitrarily nonzero z, we achieve a novel U(1) theory with implied X-charge. This theory not only explains the origin of the number of observed fermion families but also offers a possible solution for both neutrino mass and dark matter, which differs from  $$B-L$$  <math> <mrow> <mi>B</mi> <mo>-</mo> <mi>L</mi> </mrow> </math>   extension. Two typical models based on this idea are examined, yielding interesting results for flavor-changing neutral currents and particle colliders, besides those of neutrino mass and dark matter.

Flavor-dependent U(1) extension inspired by lepton, baryon and color numbers

The  $$SU(3)_L\otimes U(1)_X$$  <math> <mrow> <mi>S</mi> <mi>U</mi> <msub> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> <mi>L</mi> </msub> <mo>⊗</mo> <mi>U</mi> <msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>X</mi> </msub> </mrow> </math>   symmetry actually studied is directly broken to the electroweak symmetry  $$SU(2)_L\otimes U(1)_Y$$  <math> <mrow> <mi>S</mi> <mi>U</mi> <msub> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> <mi>L</mi> </msub> <mo>⊗</mo> <mi>U</mi> <msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>Y</mi> </msub> </mrow> </math>   by a Higgs triplet, predicting a relevant new physics at TeV scale. This work argues, by contrast, that the higher weak isospin  $$SU(3)_L$$  <math> <mrow> <mi>S</mi> <mi>U</mi> <msub> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> <mi>L</mi> </msub> </mrow> </math>   might be broken at a high energy scale, much beyond 1 TeV, by a Higgs octet to an intermediate symmetry  $$SU(2)_L\otimes U(1)_{T_8}$$  <math> <mrow> <mi>S</mi> <mi>U</mi> <msub> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> <mi>L</mi> </msub> <mo>⊗</mo> <mi>U</mi> <msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mi>T</mi> <mn>8</mn> </msub> </msub> </mrow> </math>   at TeV, before the latter  $$U(1)_{T_8}$$  <math> <mrow> <mi>U</mi> <msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mi>T</mi> <mn>8</mn> </msub> </msub> </mrow> </math>   recombined with  $$U(1)_X$$  <math> <mrow> <mi>U</mi> <msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>X</mi> </msub> </mrow> </math>   defines (i.e., broken to)  $$U(1)_Y$$  <math> <mrow> <mi>U</mi> <msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>Y</mi> </msub> </mrow> </math>   by a Higgs singlet. The new physics coupled to  $$SU(3)_L$$  <math> <mrow> <mi>S</mi> <mi>U</mi> <msub> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> <mi>L</mi> </msub> </mrow> </math>   breaking phase is decoupled, whereas what remains is a novel family-nonuniversal abelian model,  $$U(1)_{T_8}\otimes U(1)_X$$  <math> <mrow> <mi>U</mi> <msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mi>T</mi> <mn>8</mn> </msub> </msub> <mo>⊗</mo> <mi>U</mi> <msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>X</mi> </msub> </mrow> </math>  , significantly overhauling the standard model as well as yielding consistent results for neutrino mass, dark matter, W-mass anomaly, and FCNC, differently from the usual 3-3-1 model.

Physics implication from higher weak isospin decomposition

Quark has an electric charge either  $$-1/3$$  <math> <mrow> <mo>-</mo> <mn>1</mn> <mo>/</mo> <mn>3</mn> </mrow> </math>   or 2/3 and a baryon number 1/3, where the divisions 3’s match the color number. Although the electric charge and the baryon number have a nature distinct from the color charge, the matching is necessary for the standard model or a relevant  $$B-L$$  <math> <mrow> <mi>B</mi> <mo>-</mo> <mi>L</mi> </mrow> </math>   extension consistent at quantum level, since the relevant anomaly  $$[SU(2)_L]^2U(1)_A$$  <math> <mrow> <msup> <mrow> <mo>[</mo> <mi>S</mi> <mi>U</mi> <msub> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> <mi>L</mi> </msub> <mo>]</mo> </mrow> <mn>2</mn> </msup> <mi>U</mi> <msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>A</mi> </msub> </mrow> </math>   for  $$A=Y$$  <math> <mrow> <mi>A</mi> <mo>=</mo> <mi>Y</mi> </mrow> </math>   or  $$B-L$$  <math> <mrow> <mi>B</mi> <mo>-</mo> <mi>L</mi> </mrow> </math>   must vanish. If elementary particles have a new U(1) charge differently from A, such anomaly is not cancelled for each family. However, if we demand that the anomaly is cancelled over all families, this relates the color number to the family number instead of the electric charge and baryon number, and interestingly the family number guides us to a novel U(1) theory. We will discuss the implication of this theory for neutrino mass, recent W-boson mass anomaly, FCNC, and particle colliders.

Tran Hung

Abelian charge inspired by family number

The recent precision measurement of the W-boson mass reveals an exciting hint for the new physics as of the 3-3-1 model. We indicate that the 3-3-1 model contains distinct sources by itself that may cause the W-mass deviation, as measured, such as the tree-level Z- $$Z'$$  <math> <msup> <mi>Z</mi> <mo>′</mo> </msup> </math>   mixing, the tree-level W-Y and Z- $$Z'$$  <math> <msup> <mi>Z</mi> <mo>′</mo> </msup> </math>  -X mixings, as well as the non-degenerate gauge vector (X, Y) and new Higgs doublets. We point out that the gauge vector doublet negligibly contributes to this mass shift, whereas the rest of the effects with tree-level mixings governed by Z- $$Z'$$  <math> <msup> <mi>Z</mi> <mo>′</mo> </msup> </math>   and new Higgs doublets are significant. A discussion of scalar sextet contributions is also given.

Novel effects of the W-boson mass shift in the 3-3-1 model

We present the features of the fully flipped 3-3-1-1 model and show that this model leads to dark matter candidates naturally. We study two dark matter scenarios corresponding to the triplet fermion and singlet scalar candidates, and we determine the viable parameter regimes constrained from the observed relic density and direct detection experiments.

Dark matter in the fully flipped 3-3-1-1 model

The present matter content of our universe may be governed by a  $$U(1)_{B-L}$$  <math> <mrow> <mi>U</mi> <msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mrow> <mi>B</mi> <mo>-</mo> <mi>L</mi> </mrow> </msub> </mrow> </math>   symmetry – the simplest gauge completion of the seesaw mechanism which produces small neutrino masses. The matter parity results as a residual gauge symmetry, implying dark matter stability. The Higgs field that breaks the  $$B-L$$  <math> <mrow> <mi>B</mi> <mo>-</mo> <mi>L</mi> </mrow> </math>   charge inflates the early universe successfully and then decays to right-handed neutrinos, which reheats the universe and generates both normal matter and dark matter manifestly.

Asymmetric matter from  $$B-L$$  <math> <mrow> <mi>B</mi> <mo>-</mo> <mi>L</mi> </mrow> </math>   symmetry breaking

It is shown that for a higher weak isospin symmetry, SU(P) L  with P ≥ 3, the baryon minus lepton charge B − L neither commutes nor closes algebraically with SU(P) L  similar to the electric charge Q, which all lead to a SU(3) C  ⊗ SU(P) L  ⊗ U(1) X  ⊗ U(1) N  gauge completion, where X and N determine Q and B − L, respectively. As a direct result, the neutrinos obtain appropriate masses via a canonical seesaw. While the version with P = 3 supplies the schemes of single-component dark matter well established in the literature, we prove in this work that the models with P ≥ 4 provide the novel scenarios of multicomponent dark matter, which contain simultaneously at least P−2 stable candidates, respectively. In this setup, the multicomponet dark matter is nontrivially unified with normal matter by gauge multiplets, and their stability is ensured by a residual gauge symmetry which is a remnant of the gauge symmetry after spontaneous symmetry breaking. The three versions with P = 4 according to the new lepton electric charges are detailedly investigated. The mass spectrum of the scalar sector is diagonalized when the scale of the U(1) N  breaking is much higher than that of the usual 3-4-1 symmetry breaking. All the interactions of gauge bosons with fermions and scalars are obtained. We figure out viable parameter regimes given that the multicomponent dark matter satisfies the Planck and (in)direct detection experiments.

Le Thuy

Multicomponent dark matter in noncommutative B − L gauge theory

We argue that neutrino mass and dark matter can arise from an approximate symmetry. This idea can be realized in a minimal setup of the flipped 3-3-1 model, which discriminates lepton families while keeping universal quark families and uses only two scalar triplets in order for symmetry breaking and mass generation. This proposal contains naturally an approximate non-Abelian symmetry which consequently leads to an approximate matter parity. The approximate symmetries produce small neutrino masses in terms of type II and III seesaws and may make dark matter long lived. Additionally, dark matter candidate is either unified with the Higgs doublet by gauge symmetry or acted as an inert multiplet. The Peccei-Quinn symmetry is discussed. The gauge and scalar sectors are exactly diagonalized. The signals of the new physics at colliders are examined.

Dang Soa

Neutrino mass and dark matter from an approximate B

It is well established that the SU(P)L gauge symmetry for P ≥ 3 can address the question of fermion generation number due to the anomaly cancellation, but it neither commutes nor closes algebraically with electric and baryon-minus-lepton charges. Hence, two U(1) factors that determine such charges are required, yielding a complete gauge symmetry, SU(P)L ⊗ U(1)X ⊗ U(1)N, apart from the color group. The resulting theory manifestly provides neutrino mass, dark matter, inflation, and baryon asymmetry of the universe. Furthermore, this gauge structure may present kinetic mixing effects associated to the U(1) gauge fields, which affect the electroweak precision test such as the ρ parameter and Z couplings as well as the new physics processes. We will construct the model, examine the interplay between the kinetic mixing and those due to the symmetry breaking, and obtain the physical results in detail.