Dynamical breaking of symmetries beyond the standard model and supergeometry

TL;DR

This paper explores the supersymmetric realization of the electroweak sector, demonstrating dynamical symmetry breaking and extending the model using supergeometry and supergroup structures.

Contribution

It introduces a supergeometric framework for the electroweak sector, including dynamical symmetry breaking and supercoherent states based on the supergroup SU(2|1).

Findings

Derived the super Yang-Mills Lagrangian for the model

Defined supercoherent states to extend symmetry

Discussed the physical implications of additional degrees of freedom

Abstract

Group theoretical realizations containing the electroweak sector of the Standard Model are discussed from the supersymmetry point of view. Dynamical breaking of the symmetry is performed and the corresponding quadratic (super Yang-Mills) Lagrangian is obtained. Supercoherent states of the Klauder-Perelomov type are defined to enlarge the symmetry taking into account the geometry of the coset based in the simplest supergroup SU (2 | 1) as the structural basis of the electroweak sector of the SM. The extended model is superintegrable and the superconnection in the odd part takes a dynamical character. The physical and geometrical implications of the additional degrees of freedom interpreted as a hidden sector of the representation are briefly discussed.

Equations

$$J=\left(\begin{array}[c]{cc}M&\phi\\ \overline{\phi}&N\end{array}\right).$$

$$\lambda_{1}=\left(\begin{array}[c]{ccc}0&1&0\\ 1&0&0\\ 0&0&0\end{array}\right),\quad\lambda_{2}=\left(\begin{array}[c]{ccc}0&-i&0\\ i&0&0\\ 0&0&0\end{array}\right),\quad\lambda_{3}=\left(\begin{array}[c]{ccc}1&0&0\\ 0&-1&0\\ 0&0&0\end{array}\right),\quad\lambda_{8}=\left(\begin{array}[c]{ccc}-1&0&0\\ 0&-1&0\\ 0&0&-2\end{array}\right)$$

$$\lambda_{4}=\left(\begin{array}[c]{ccc}0&0&1\\ 0&0&0\\ 1&0&0\end{array}\right),\quad\lambda_{5}=\left(\begin{array}[c]{ccc}0&0&-i\\ 0&0&0\\ i&0&0\end{array}\right),\quad\lambda_{6}=\left(\begin{array}[c]{ccc}0&0&0\\ 0&0&1\\ 0&1&0\end{array}\right),\quad\lambda_{7}=\left(\begin{array}[c]{ccc}0&0&0\\ 0&0&-i\\ 0&i&0\end{array}\right).$$

$$\lambda_{0}\equiv\sqrt{\frac{2}{3}}\left(\begin{array}[c]{ccc}1&0&0\\ 0&1&0\\ 0&0&-1\end{array}\right).$$

$$\mathcal{K}=\frac{1}{\sqrt{1+\phi^{\dagger}\phi}}\left(\begin{array}[c]{cc}\mathbb{I}_{2\times 2}&\phi\\ \phi^{\dagger}&1\end{array}\right)$$

$$\phi=\frac{\tan\sqrt{v^{\dagger}v}}{\sqrt{v^{\dagger}v}}v\text{ \ \ \ and \ \ \ \ }v=\left(\begin{array}[c]{c}\theta_{4}-i\theta_{5}\\ \theta_{6}-i\theta_{7}\end{array}\right)$$

$$\mathcal{H}=\left(\begin{array}[c]{cc}we^{-i\theta_{8}/\sqrt{3}}&\mathbf{0}_{2\times 1}\\ \mathbf{0}_{1\times 2}&e^{-2i\theta_{8}/\sqrt{3}}\end{array}\right)$$

$$\mathcal{U}=\mathcal{K}\mathcal{H},$$

$$w$$

$$\displaystyle\text{\ }\begin{array}[c]{c}\text{\ }\theta^{2}=\theta_{1}^{2}+\theta_{2}^{2}+\theta_{3}^{2},\qquad\theta_{-}=\left(\theta_{1}-i\theta_{2}\right),\qquad\theta_{+}=\left(\theta_{1}+i\theta_{2}\right).\end{array}$$

$$\mathcal{U}=\frac{1}{\sqrt{1+Z^{\dagger}Z}}\left(\begin{array}[c]{cc}we^{-i\theta_{8}/\sqrt{3}}&\phi e^{-2i\theta_{8}/\sqrt{3}}\\ \phi^{\dagger}we^{-i\theta_{8}/\sqrt{3}}&e^{-2i\theta_{8}/\sqrt{3}}\end{array}\right).$$

$$\Gamma$$

$$\equiv {\Gamma }_{even}+{\Gamma }_{odd}$$

$$d{\Gamma }_{even}+{\Gamma }_{even}\wedge {\Gamma }_{even}+{\Gamma }_{odd}\vee {\Gamma }_{odd}$$

$$d{\Gamma }_{odd}+{\Gamma }_{even}\wedge {\Gamma }_{odd}$$

$$\left[a\otimes X,b\otimes Y\right]=-1^{|X||b|}\left(a\wedge b\right)\otimes \left[X,Y\right].$$

$$\displaystyle\Gamma=\underset{\Omega}{\underbrace{\frac{e^{-i\theta_{8}/\sqrt{3}}}{\sqrt{1+\phi^{\dagger}\phi}}}}\left\{\underset{\widetilde{\Gamma}_{even}}{\underbrace{\left[\begin{array}[c]{cc}\overset{w}{\overbrace{\frac{i}{\sqrt{1+W^{2}}}\left[\begin{array}[c]{cc}W_{3}-i&W_{-}\\ W_{+}&-W_{3}-i\end{array}\right]}}&\mathbf{0}_{2\times 1}\\ \mathbf{0}_{1\times 2}&e^{-i\theta_{8}/\sqrt{3}}\end{array}\right]}}+\underset{\widetilde{\Gamma}_{odd}}{\underbrace{i\left[\begin{array}[c]{cc}\mathbf{0}_{2\times 2}&\phi\\ \phi^{\dagger}&0\end{array}\right]\widetilde{\Gamma}_{even}},}\right\}$$

$$\displaystyle=\frac{e^{-i\theta_{8}/\sqrt{3}}}{\sqrt{1+Z^{\dagger}Z}}\left\{\left(1+i\left[\begin{array}[c]{cc}\mathbf{0}_{2\times 2}&\phi\\ \phi^{\dagger}&0\end{array}\right]\right)\underset{\widetilde{\Gamma}_{even}}{\underbrace{\left[\begin{array}[c]{cc}w&\mathbf{0}_{2\times 1}\\ \mathbf{0}_{1\times 2}&e^{-i\theta_{8}/\sqrt{3}}\end{array}\right]}}\right\},$$

$$d\Gamma =d\left(\frac{e^{-i{\theta }_8/\sqrt{3}}}{\sqrt{1+{\varphi }^{†}\varphi }}\right)\wedge \left({\tilde{\Gamma }}_{even}+{\tilde{\Gamma }}_{odd}\right)+\frac{e^{-i{\theta }_8/\sqrt{3}}}{\sqrt{1+{\varphi }^{†}\varphi }}\left(d{\tilde{\Gamma }}_{even}+d{\tilde{\Gamma }}_{odd}\right),$$

$$\displaystyle dw=d\left(\frac{i}{\sqrt{1+W^{2}}}\right)\wedge\left(\begin{array}[c]{cc}W_{3}-i&W_{-}\\ W_{+}&-W_{3}-i\end{array}\right)+\frac{i}{\sqrt{1+W^{2}}}d\left(\begin{array}[c]{cc}W_{3}-i&W_{-}\\ W_{+}&-W_{3}-i\end{array}\right).$$

$$\displaystyle d\widetilde{\Gamma}_{even}=\left(\begin{array}[c]{cc}dw&\mathbf{0}_{2\times 1}\\ \mathbf{0}_{1\times 2}&d\left(e^{-i\theta_{8}/\sqrt{3}}\right)\end{array}\right),$$

$$\displaystyle d\widetilde{\Gamma}_{odd}=i\left(\begin{array}[c]{cc}\mathbf{0}_{2\times 2}&d\phi-\phi\wedge d\left(\ln e^{-i\theta_{8}/\sqrt{3}}\right)\\ d\phi^{\dagger}+\phi^{\dagger}\wedge d\ln w&0\end{array}\ \ \right)\widetilde{\Gamma}_{even},$$

$$\left[{\tilde{\Gamma }}_{even},{\tilde{\Gamma }}_{even}\right]$$

$$\left\{{\tilde{\Gamma }}_{odd},{\tilde{\Gamma }}_{odd}\right\}$$

$$\left[{\tilde{\Gamma }}_{even},{\tilde{\Gamma }}_{odd}\right]$$

$$d{\Gamma }_{odd}+{\Gamma }_{even}\wedge {\Gamma }_{odd}\equiv$$

$$\displaystyle\qquad-i\Omega\Gamma_{even}\left\{\left(\begin{array}[c]{cc}\mathbf{0}_{2\times 2}&d\phi-2\Omega\left(w-\mathbb{I}e^{-i\theta_{8}/\sqrt{3}}\right)\phi\\ d\phi^{\dagger}+2\Omega\left(w-\mathbb{I}e^{-i\theta_{8}/\sqrt{3}}\right)\phi^{\dagger}0&\end{array}\right)\right.$$

$$\displaystyle\qquad+\left.i\left(\begin{array}[c]{cc}\mathbf{0}_{2\times 2}&-\phi\wedge d\ln\left(\Omega e^{-i\theta_{8}/\sqrt{3}}\right)\\ \phi^{\dagger}\wedge d\ln\left(\Omega w\right)&0\end{array}\right)\right\}.$$

$$\displaystyle\mathcal{W\rightarrow}\widetilde{\mathcal{W}}\mathcal{\equiv}\widetilde{W}\cdot\sigma\rightarrow\frac{e^{-i\theta_{8}/\sqrt{3}}}{\sqrt{1+\phi^{\dagger}\phi}}\frac{i}{\sqrt{1+W^{2}}}\underset{W\cdot\sigma}{\underbrace{\left(\begin{array}[c]{cc}W_{3}&W_{-}\\ W_{+}&-W_{3}\end{array}\right)}},$$

$$-\frac{1}{\sqrt{3}}B\cdot {\mathbb{I}}_{2\times 2}\to -\frac{1}{\sqrt{3}}\tilde{B}\cdot {\mathbb{I}}_{2\times 2}\underset{B-d\theta }{\underbrace{\to }}-\frac{e^{-i{\theta }_8/\sqrt{3}}}{\sqrt{1+{\varphi }^{†}\varphi }}\frac{i}{\sqrt{1+W^2}}{\mathbb{I}}_{2\times 2}.$$

$$B\to B-d\theta .$$

$$d{\Gamma }_{odd}+{\Gamma }_{even}\wedge {\Gamma }_{odd}$$

$$\displaystyle\equiv-i\left(\begin{array}[c]{cc}\mathbf{0}_{2\times 2}&d\widetilde{\phi}-2i\left[\widetilde{W}\cdot\sigma-\frac{g^{\prime}}{\sqrt{3}}\widetilde{B}\cdot\mathbb{I}_{2\times 2}\right]\widetilde{\phi}\\ d\widetilde{\phi}^{\dagger}+2i\widetilde{\phi}^{\dagger}\left[\widetilde{W}\cdot\sigma-\frac{g^{\prime}}{\sqrt{3}}\widetilde{B}\cdot\mathbb{I}_{2\times 2}\right]&0\end{array}\right),$$

$$D_{\mu }\tilde{\varphi }={\partial }_{\mu }\tilde{\varphi }-2i\left[\sigma \cdot {\tilde{W}}_{\mu }-\frac{1}{\sqrt{3}}{\mathbb{I}}_{2\times 2}{\tilde{B}}_{\mu }\left(1-2\sqrt{1-{\tilde{W}}^2}\right)\right]\tilde{\varphi }.$$

$$D_{\mu }\varphi ={\partial }_{\mu }\tilde{\varphi }-2i\left(g{\tilde{W}}_{\mu }\cdot \sigma -g^{\prime }{\mathbb{I}}_{2\times 2}{\tilde{B}}_{\mu }\right)\tilde{\varphi },$$

$${\sin }^2{\theta }_W=\frac{g{\prime }^2}{g^2+g{\prime }^2}\to \frac{{\left(1-2\sqrt{1-{\tilde{W}}^2}\right)}^2}{3+{\left(1-2\sqrt{1-{\tilde{W}}^2}\right)}^2}.$$

Full text

††preprint:

identifier

1. I Introduction

2. II Gauge structure and superconnections: $SU(2/1)$

3. II.1 Supergroup structure

4. II.2 Element of the supergroup: $\mathcal{K\in B}_{1},\mathcal{H\in}$ $\mathcal{B}_{0}:$

5. III Supercurvature and Lagrangian

6. III.1 Odd supercurvature and Weinberg angle

7. III.2 Even supercurvature

8. IV Superconnections and supergeometry

9. V Extended supercurvature

10. VI Superlagrangian

11. VI.1 Case 1: $\widetilde{\psi},\widetilde{\varphi}=const.$

12. VI.2 Case 2: $\widetilde{\psi}=\widetilde{\psi}\left(z\right),\widetilde{\varphi}=\widetilde{\varphi}\left(z\right)$

13. VII Symmetry breaking mechanism

14. VIII Concluding remarks and outlook

15. IX Appendix I. Coset coherent states

16. X Appendix II. Superconnection and algebra generating the group

17. XI Appendix III. The extended superconnection: $\kappa^{AB}$

18. XII Appendix IV. Determination of the ground state vectors $V_{0}$: the meaning of the physical vacuum

I Introduction

As it is well known, the Standard Model (SM) of particle physics and General Relativity are extremely successful phenomenologically based theories. In the case of the SM, the apparent last building block associated with the Higgs sector has been detected experimentally in 2012. However, the discovery of the Higgs boson with the mass of about $125$ GeV doesn’t solve the hierarchy problem of the SM. There are also several questions coming from both astrophysical and cosmological scenarios where the necessity of new exotic states (possibly from supersymmetrical or composite) is quite obvious. This incompleteness of the SM, which is the case to be particularly treated here, is a clear indication that new models with a richer geometric and algebraic structure should be considered. With this aim and motivation in this paper we will apply the model described in our previous works to solve the problem of the symmetry rupture as the content of fields, both those corresponding to the SM, plus other additional ones emerging from the group structure.

In the language of superbundles in our previous work Arbuzov:2017frk , the fiber was defined as $E=E(U_{4}$, $F=E^{2}(V_{4}^{\prime},S)$, $G=SO(4,2))$ where $P=P(U_{4}$, $F=G=SO(4,2))$ was the associated bundle and $G$ can be viewed as the bundle $P(G/H,H)$. The geometrical meaning in this statement, is that the soldering of $E$ is obtained by identifying the tangent space of $V_{4}^{\prime}$ of $F$ at the point, namely, $\psi=\psi^{0}$ with the local tangent space $T_{x}$ of $U_{4}$ at $x$ through an isomorphism. Consequently, the point $\psi^{0}$ $\left(\text{the origin of }V_{4}^{\prime}\right)$ is geometrically the contact point between the base space (spacetime) and the fiber at $x\in U_{4}$.

Now we treat the simplest supergroup as the principal bundle $SU\left(2|1\right)$ that have several very interesting features. In particular, the matrix representation is performed exactly as the $SU\left(3\right)$ Gell-Mann matrices with the only change in the structure of the diagonal matrix $\Lambda_{8}$ that is of non-zero trace. This important characteristic of the supergroup $SU\left(2|1\right)$ facilitates calculations (which are usually rather complex, in particular for the construction of supercoherent states). Moreover, the interpretation of the results can be performed from the point of view of a grand unification theory beyond the SM Cirilo-Lombardo:2015fua .

II Gauge structure and superconnections: $SU(2/1)$

The superconnection was introduced by Quillen in mathematics, it is a supermatrix belonging to a given supergroup $S$ valued over elements of a Grassmann algebra of forms. The even part of the supermatrix is valued over the gauge potentials of the even subgroup $G\subset S$, one-forms $Bdx$ on the base $M$ manifold of the bundle, realizing the “gauging” of $G$. The odd part of the supermatrix, representing the quotient $S/G=H\subset S$, is valued over zero-forms in that Grassmann algebra, physically interpreted as the Higgs multiplet in a spontaneously broken $G$ gauge theory. In some quantum treatments which are set to reproduce geometrically the ghost fields and BRST equations, the Grassmann algebra is taken over the complete bundle variable.

The first physical example of a superconnection preceded Quillen’s theory. This was the Neeman’s et al. supergroup proposal Neeman:2005dbs ; Fairlie:1979at for an algebraically irreducible electroweak unification. Lacking Quillen’s generalized formulation, the model appeared to suffer from spin-statistics interpretative complications for physical fields. The structural $Z$ grading of Lie superalgebras, as previously used in physics (e.g. in SUGRA etc.) corresponds to the grading inherent in quantum statistics, i.e. to Bose–Fermi transitions, so that the invariance under the supergroup represents symmetry between bosons and fermions. In the proposal Neeman:2005dbs , however, though the superconnection itself does fit the quantum statistics ansatz. This is realized through the order of the forms in the geometrical space of the Grassmann algebra, rather than through the quantum statistics of the particle Hilbert space.

II.1 Supergroup structure

The generator structure for exponential representation of the group under considerations reads Neeman:2005dbs ; Bars:1982ep ; Arbuzov:2017frk

[TABLE]

The even generators in this structure are

[TABLE]

and the odd ones are

[TABLE]

The diagonal matrices are related to the following important operators: $I_{L}^{3}\equiv\frac{1}{2}\lambda_{3}$, the gradation operator $U_{Q}\equiv\sqrt{3}\lambda_{8}$, and the new operator that is fixed by the orthogonality in $U(2/1)$:

[TABLE]

In previous studies the latter was related with the chirality of the model defined by $\sqrt{\frac{3}{2}}\lambda_{0}$.

II.2 Element of the supergroup: $\mathcal{K\in B}_{1},\mathcal{H\in}$

$\mathcal{B}_{0}:$

We have seen that in the ordinary parameterization, an element of the supergroup can be written as Bars:1982ep ; Cirilo-Lombardo:2015fua ; Cirilo-Lombardo:2016ogj

[TABLE]

with

[TABLE]

and

[TABLE]

[TABLE]

with

[TABLE]

[TABLE]

Consequently, the element of the supergroup can be explicitly written as

[TABLE]

Note that

i) in $\mathcal{U}$, the even part of the group structure (corresponding with the electroweak sector, namely $SU(2)\otimes U(1))$ is exactly preserved;

ii) the non diagonal blocks (the odd part) in $\mathcal{U}$ can be interpreted as new fermion-boson interaction;

iii) $\mathcal{H=}\exp\left(\underset{k=1,2,3,8}{\sum\theta_{k}\sigma_{k}}\right)$ and $w\mathcal{=}\exp\left(\underset{k=1,2,3}{\sum\theta_{k}\sigma_{k}}\right)\in SU(2)_{L}$;

iv) there are other parameterization symmetrical (Borel type) ones involving ladder operators that we will use in the construction of the corresponding supercoherent states.

III Supercurvature and Lagrangian

Due to the identification of the superconnection and $\mathcal{U}$, we can see that from

[TABLE]

one can obtain the super-Riemannian curvature in the language of superforms:

[TABLE]

To compute the above equations that define the supercurvature, it is useful to have in mind that the tensor product of a commutative superalgebra of differential forms and a Lie superalgebra is again a Lie superalgebra with the product

[TABLE]

The next step is to compute the above quantities explicitly. To this end we rewrite the connection in the original form as

[TABLE]

Consequently,

[TABLE]

[TABLE]

The supercurvatures with the above definitions can be computed in the usual way.

III.1 Odd supercurvature and Weinberg angle

The explicit computation of the odd supercurvature gives

[TABLE]

Now we use the detailed association (at the end in the Appendix III), to identify the physical fields, namely

[TABLE]

For consistency, it should be supplemented with the transformation law of the gauge of the 1-form associated with $\lambda_{8}$:

[TABLE]

Then, rewriting accordingly we arrive to

[TABLE]

where we defined $g^{\prime}\equiv\left(1-2\sqrt{1-\widetilde{W}^{2}}\right)$. The covariant derivative induced via pullback (it is necessary to make the correspondence with the electroweak sector of the SM) can be immediately identified Fairlie:1979at ; Weinberg:1967tq

[TABLE]

The electroweak mixing angle $\theta_{W}$ is determined via the relation between the gauge couplings $g$ and $g^{\prime}$ in the above equation:

[TABLE]

then

[TABLE]

If $W^{2}\sim 0$ then

[TABLE]

III.2 Even supercurvature

In the same way as before for the odd part of the supercurvature, we get the even part in the form

[TABLE]

[TABLE]

Consequently, the full supercurvature takes the form

[TABLE]

Note that a tilde indicates here that the respective quantities are affected by the induced supercurvature due the pullback from the algebra (vector space) to the group representation.

IV Superconnections and supergeometry

The strategy to extend the symmetry without breaking the group theoretical features of the model is realized as follows.

i) If we have two diffeomorphic (or gauge) non-equivalent $SU(2|1)$ valuated superconnections, namely $\Gamma^{AB}$ and $\widetilde{\Gamma}^{AB}$. Their difference transforms as a second rank three-supertensor under the action of $SU(2|1)$:

[TABLE]

ii) If we calculate now the curvature from $\widetilde{\Gamma}^{AB}$, we obtain

[TABLE]

where the $SU(2|1)$ supercovariant derivative is defined in the usual way (see the previous Section)

[TABLE]

iii) Redefining the $SU(2|1)$ three vectors as $V_{2}^{A}\equiv\psi^{A}$ and $V_{1}^{B}\equiv\varphi^{B}$ (in order to put all in the standard notation), the 2-form $\kappa^{AB}$ can be constructed as

[TABLE]

where $U$ is a super-scalar function. Then we introduce all into the $\widetilde{R}^{AB}$ and get

[TABLE]

Note that the supercurvature $\widetilde{R}^{AB}$ splits into even and odd parts as indicated in the previous Section, being the capital letters the multi-index $A,B,C$ etc. corresponding to the supercoordinates of the $su(2|1)$ superspace.

iv) Let us define

[TABLE]

with the extended superconnection $\widetilde{\Gamma}^{AB}=\Gamma^{AB}+\kappa^{AB}$, then

[TABLE]

where $\left(\varphi^{B}\right)^{2}=\left(\varphi_{\text{ }B}\varphi^{B}\right)$ and $\left(\psi\cdot\varphi\right)=\psi_{B}\varphi^{B}$ etc.

In the same manner we also define

[TABLE]

V Extended supercurvature

From the original superconnection extended in the way described in the previous Section, the extended supercurvature is computed in the straightforward manner as

[TABLE]

where $2\left(\widetilde{\psi}^{2}\widetilde{\varphi}^{2}+\left(\widetilde{\psi}\cdot\widetilde{\varphi}\right)^{2}\right)=\widetilde{\psi}^{2}\widetilde{\varphi}^{2}$ due to the $N=1$ Fierz identities111for $N=1$ we use $\left(\psi\varphi\right)\left(\psi\varphi\right)=-\frac{1}{2}\psi^{2}\varphi^{2}$ we get $D\widetilde{W}_{k}=\left(d\widetilde{W}_{k}+\widetilde{W}_{i}\wedge\widetilde{W}_{j}\epsilon_{i}jk\right)\sigma_{k}$ and $D\widetilde{B}=d\widetilde{B}$. Note that, at the supercurvature level, a Dirac-type term, namely $\left(\widetilde{\psi}d\widetilde{\varphi}-\widetilde{\varphi}d\widetilde{\psi}\right)+2\widetilde{\psi}^{2}\widetilde{\varphi}^{2}$ plus the couplings with $\widetilde{\phi^{\dagger}}$ and $\widetilde{\phi}$ are geometrically induced by the extension of the original superalgebra.

VI Superlagrangian

VI.1 Case 1: $\widetilde{\psi},\widetilde{\varphi}=const.$

From Eq. (92) in the constant case for the fiducial vectors, namely for $d\widetilde{\varphi}=d\widetilde{\psi}=0$, we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where we defined

[TABLE]

Gauge couplings $g$ and $g^{\prime}$ are not modified in the covariant derivative. Consequently, assuming similar conditions as in the SM (see that the even sector is just $SU(2)\otimes U(1)$), so the masses of the $\widetilde{W}$ and $\widetilde{Z}=\frac{\sqrt{3}\widetilde{W}^{3}-\widetilde{B}}{2}$ gauge bosons become

[TABLE]

For $g=1$ the expressions are simplified even more.

VI.2 Case 2: $\widetilde{\psi}=\widetilde{\psi}\left(z\right),\widetilde{\varphi}=\widetilde{\varphi}\left(z\right)$

In this particular case we propose

[TABLE]

Consequently, multiplying (99) by $\left(\widetilde{\psi}\widetilde{\varphi}\right)^{-1}$ we obtain the differential equation for the contraction $\left(\widetilde{\psi}\widetilde{\varphi}\right)$ and get the following result

[TABLE]

Note that the equation is on the third components of the respective fiducial vectors $\psi^{A}$ and $\varphi^{B}$and $z$ is a super coordinate of the manifold $SU(2\mid 1)$ -valuated. Here $z\equiv z^{A}E_{A}$ is a general coordinate of the supergeometry (induced via the pullback with $\left\{E_{A}\right\}$ the corresponding vector basis) from $su(2\mid 1))$. The above result is new and is very important due to the possibility to change the character of the $su(2\mid 1)$ valuated supercovariant derivative into a dynamical one.

Then, the extended supercurvature becomes

[TABLE]

As in the standard case, we can take $\phi$ adimensionalized as appear in the exponential representation of the $SU(2\mid 1)$ (from the group to the physical scenario). To this end we construct the $\phi$ field as

[TABLE]

including the values $v$, $h$ for the Higgs field and $\pi$ and some bare quantity $\phi_{0}$ to be determined. Extracting from the geometrically induced extended superpotential $V\left(\widetilde{\phi^{\dagger},}\widetilde{\phi},\widetilde{\psi},\widetilde{\varphi}\right)$, see Eq. (96), the (adimensionalized) mass term for $h$

[TABLE]

Then, at the tree level we obtain the adimensionalized Higgs mass as

[TABLE]

Consequently the physical mass is given by

[TABLE]

The normalizing field $\phi_{0}$ can be determined from the expression (100) by the constant of integration that would indicate that $\rho^{-1}\sim\phi_{0}.$

VII Symmetry breaking mechanism

In spite of great theoretical achievements on can’t say that symmetry breaking (spontaneous or not) in gauge field theories and nonlinear phenomenological Lagrangians is completely well understood. A rather close relationship of spontaneous symmetry breaking with nonlinearly realized symmetries was stressed long time ago by many people ColWess ; VA ; BO ; SamWess ; IK . In the formalism of nonlinear realizations, we are dealing with the invariance of a Lagrangian under a given group $G$ of field transformations. Generally these invariances, however, are partially linear and partially nonlinear. It happens that there is a subgroup $H$ of $G$, called the stability group, which in well studied spontaneously broken theories corresponds to the symmetry group of the vacuum state and acts on all fields and the covariant derivatives linearly. The remaining transformations of the larger group of invariance, namely, those belonging to the coset $G/H$, change the vacuum state and produce nonlinear transformations on the physical fields and covariant derivatives.

In this context, following our previous work Arbuzov:2017frk ; dc , the natural interpretation in the language of coherent states is extremely clear and concise, namely the group structure is chosen in the form

[TABLE]

The subgroup $H$ gives the transformations defining the vacuum states and consequently the set of fiducial vectors spanning the vacuum sub-space.

The coset $G/H$ gives the remaining transformations changing the vacuum states and therefore producing nonlinear (Bogoliubov) transformations of physical fields and induced covariant derivatives. As we can see from the previous Sections, the breaking of the supersymmetries in this case is driven by the supercoherent states (constructed by the action of the supergroup $SU(2\mid 1)$ on supervectors as described in Appendixes I and III) in the same way as in Ref. Arbuzov:2017frk . However, in this case, the original representation is still preserved because everything is an internal space of symmetry: it only extends in contrast to our previous work where the representation also included the external space-time.

VIII Concluding remarks and outlook

In this paper we developed a possible description of the electroweak sector of the SM using the methods described in our paper Arbuzov:2017frk . As we see, using naturally a coherent superstate based on the simplest supergroup $SU(2\mid 1)$ which is the group of dynamic symmetry of the supersphere. This is naturally isomorphic to the group $OSp(2\mid 2)$ from the point of view of algebra, keeping invariant in the natural factorization, the even part $SU(2)\otimes U(1)$. Our interpretation of the odd sector is that physically it might be a hidden counterpart of the Standard Model.

In one of the cases discussed here that is particularly interesting, the diagonal part corresponding to the even sector that defines a geometrically induced differential equation for a component of a super-field. This differential equation can be considered non-homogeneous, specifically equal to the constant of the mean value of the condensate $v^{2}$ or superscalar product between such coherent superfields that extend the original superalgebra. The resolution of the differential equation results in a solution 101 as indicated in figures 1—5. They describes the compact solutions (1,2) and noncompact ones (3,4) respectively. Fig. 5 represents the 3-dimensional picture and Fig. 6 the superpotential. This fact, namely getting the supersymmetric solitonic solutions Mshif shows the superintegrability of the super-extended model.

It is good to note that the superpotential is an exact difference of squares in contrast to the standard case where the quadratic part appears. With respect to the statistics and other issues corresponding to the structure of the group, here the field corresponding to the Higgs appears as odd in the representation but as $SU(2)$ scalar (doublet) from the Lagrangian point of view. In this same line we see that the field associated with the Higgs must be a scalar of $SU(2)$ because it is extended by means of the ”fermionic condensate” given by the fiducial vectors (vacuum, $\psi\varphi$) of the supercoherent states (see Figs. (96,97)). Consequently, there is no problem with respect to the bosonic character of $\phi$. However, the very geometrical structure of the simplest supergroup must define the correctness of the sign of the dynamical terms of the additional superfields.

This fact (under investigation now) is in part connected with the observation made in Ref. Fairlie:1979at : changing the value of $\lambda_{8}$ to have trace equal to $-2$, the structure of the representation goes from $SU(3)$ to $SU(2\mid 1)$, i.e. from a Lie structure to a graded Lie one. In our case, the final structure is of a dynamic character.

Regarding the relationship with the approach of nonlinear realizations, the link is direct considering that the field $\widetilde{\phi}$ which plays the role of Higgs could be clearly eliminated at the expense of the fields of the hidden super-sector $\widetilde{\psi},\widetilde{\varphi}$ and the constant $v$(playing the final role of expectation value). This observation is consistent considering that the antidiagonal part of the supercurvature (101) it is precisely the Maurer-Cartan superform associated with the breaking of some (super) symmetry in a standard way: canceling precisely that superform (e.g. $D\widetilde{\phi}=d\widetilde{\phi})$

[TABLE]

with $X\equiv(z-z_{0})\sqrt{2}v$, the result is easily obtained as

[TABLE]

being $z\equiv z^{A}E_{A}$ the supercoordinate $SU(2\mid 1)$ valuated (with the superbasis carrying the superalgebra symmetry). If we define again $\tanh\left(X/2\right)=\Lambda$ then (107) can be written as

[TABLE]

The above results are very important:

1. the ”Higgs” doublet $\widetilde{\phi}$ as we see is soliton type superintegrable for determined values of $\rho<0$ (our ”fermion condensate”) and for the other case it shape turns to be divergent as we see in figure in this Section.

2. When $\rho$ is zero, the cut branch appears, passing the field $\widetilde{\phi}$from a compact configuration (soliton type) to a divergent one.

3. We can see from (108) that $\widetilde{\phi}$ depends on the fields $\widetilde{\psi},\widetilde{\varphi}$ of the hidden supersector of the model.

4)From the dynamic point of view the difference with the solutions in the similar cases obtained in other approaches (e.g. nonlinear realizations) is the exponential dependence with respect to $\rho$ that gives the soliton shape (location / confinement) of the Higgs doublet.

In the next step, explicit form of the $\psi,\varphi$ supercoherent states will be performed and the full supergroup $SU(2,2\mid 1)$ will be considered in the same way as a basis of a model containing the extension of GR and SM in a unified way.

Acknowledgments

D.J.C.-L. is grateful to the Bogoliubov Laboratory of Theoretical Physics JINR for hospitality and CONICET-Argentina for financial support.

IX Appendix I. Coset coherent states

Let us remind the definition of coset coherent states

[TABLE]

Consequently the orbit is isomorphic to the coset, i.e.

[TABLE]

Analogously, if we remit to the operators

[TABLE]

Then the orbit

[TABLE]

with

[TABLE]

The orbits are identified with cosets spaces of $G$ with respect to the corresponding stability subgroups $H_{0}$ and $H$ being the vectors $V_{0}$ in the second case defined within a phase. From the quantum viewpoint $\left|V_{0}\right\rangle\in\mathcal{H}$ (the Hilbert space) and $\rho_{0}\in\mathcal{F}$ (the Fock space) are $V_{0}$ normalized fiducial vectors (embedded unit sphere in $\mathcal{H}$).

X Appendix II. Superconnection and algebra generating the group

i) The superconnection as defined in the case of Fairlie:1979at ; Neeman:2005dbs involves the vector space via gauging over the superalgebra where the even part is, phenomenologically speaking, as in the electroweak sector

[TABLE]

ii) In our case, we are working over the group structure (pullback) where the physics lives

[TABLE]

The difference with respect to the previous case is evident.

3. From the algebra generating the group (e.g. $\mathcal{J\rightarrow U)}$ via the pullback we obtain the nonlinear structure of fields as follows

[TABLE]

[TABLE]

[TABLE]

Consequently, the explicit form of the even supercurvature is

[TABLE]

The explicit odd part reads

[TABLE]

We have $g^{\prime}=\left(1-2\sqrt{1-W^{2}}\right)$, therefore this expression takes the form give in Section IV.

Remark 1

Note that in the above redefined ”tilde” quantities, the geometry of the group manifold is included.

Remark 2

Note that in the case of the $B$ field, it is induced by the phase $\theta_{8}$, by $\left|\theta\right|$ (e.g. $W_{3},W_{\pm})$ and by the supersymmetric sector of the model.

XI Appendix III. The extended superconnection: $\kappa^{AB}$

First at all we denote here the supergeometric product for two supervectors as usual: $\left(V_{1}V_{2}\right)\equiv V_{1}\cdot V_{2}+V_{1}\wedge V_{2}$, in particular for the spinor parameters $\left(\phi_{1}\phi_{2}\right)=\phi_{1\alpha}\phi_{2}^{\alpha}+\phi_{1\alpha}\wedge\phi_{2\beta}$. Now we construct $\kappa^{AB}$ as follows:

i) we select two fiducial spinors $\psi^{0},\varphi^{0}\in$ $SU(2|1)$ ground state, that mean that are annihilated by the lower (upper) group operators as defined in appendix

ii) $\psi^{0},\varphi^{0}$ under the action of the supergroup in the Borel symmetrical representation take the following form

[TABLE]

where $\Omega=\left(1-\overline{\rho}\rho+\omega^{\ast}\omega-\overline{\Phi}\Phi\right)$ and primed parameters indicate that the fiducial vectors are, in principle, under the action of different elements of $SU(2|1)$:

[TABLE]

If the elements of the group are the same then

[TABLE]

and when the symmetry is broken changing the vacuum (inverse Bogoliubov transformation) we obtain as expected

[TABLE]

XII Appendix IV. Determination of the ground state vectors $V_{0}$: the meaning of the physical vacuum

i) In the pure boson $V_{0}$ case

[TABLE]

and the coherent vector reads

[TABLE]

ii) In the pure fermion $V_{0}$ case

[TABLE]

and the coherent vector reads

[TABLE]

iii) In the boson-fermion (symmetric) $V_{0}$ case

[TABLE]

with the coherent vector

[TABLE]

The specific choice depends on the structure of the Fock space.