# Time reversal invariant topological 1D and 2D superconductors: doubling the
Sau-Luchtin-Tewari-Sarma [1] and Oreg-Refael-von Oppen [2, 3] proposals

Garry Goldstein1<EMAIL_ADDRESS>
###### Abstract

In this work we present doubled versions of the Sau-Luchtin-Tewari-Sarma [1]
and Oreg-Refael-von Oppen [2, 3] proposals thereby obtaining time reversal
invariant p-wave superconductivity in both 1D and 2D. This construction is
much like the Kane-Mele spin Hall model [4], which is a time reversal
invariant doubling of the Haldane model [5]. We show that the low energy
effective action for these doubled versions of the Sau-Luchtin-Tewari-Sarma
[1] and Oreg-Refael-von Oppen [2, 3] models correspond to single band p-wave
time reversal invariant superconductors with pseudospin degree of freedom
instead of spin degree of freedom. There are Majorana fermions at the ends of
wires or in vortex cores of these superconductors. Furthermore these Majorana
fermions are shown to be stable to small perturbations. In the supplement we
present a physical realization of the system with cold atoms [6] and show that
a related “no-go” theorem given in Ref. [7, 8] has too restrictive assumptions
to apply to this proposal.

###  I Introduction

The scientific community has been studying topological superconductivity and
superfluidity for a relatively long time. It is now known that the ${}^{3}He$
-A and -B phases are topological superfluids. The ${}^{3}He$ -A and -B phases
have been characterized by topological bulk invariants [9]. $p+ip$
superconductors in 2D also posses two topologically invariant phases and
exhibit Majorana fermions on edges between them [10]. The smoking gun for
topological superconductors and superfluids is the existence of zero energy
Majorana modes in the vortices of their order parameters [11, 19, 8, 16, 14,
3, 2, 15, 13, 1, 17, 9, 12]. Majorana fermions are real fermions which are
their own antiparticles [20]. Sau et. al. [1] suggested creating Majorana
fermions in vortices of ferromagnetic insulator/semiconductor/s-wave
superconductor superstructures. The authors of [1] showed that Majorana
fermions exist in this setup by solving the single vortex core problem for the
effective Hamiltonian for the superstructure [1]. Alicea [11] extended the
work of Sau et al. [1] to replace the external magnetic field with a
ferromagnet. Shen presented [17] an equivalence between the model in Sau et.
al. [1] and spinless $p+ip$ superconductors. In this work for clarity in the
supplement we extend these results and show that the low energy effective
action for large magnetic fields is that of a spinless $p+ip$ superconductor
[17] (however this is not the main thrust of this work). These ideas about 2D
s-wave superconductor proximitized semiconductors were extended to 1D by [2,
3] where it was shown that such ferromagnetic insulator/semiconductor/s-wave
superconductor heterostructures but with 1D semiconductor wires have Majorana
modes at the ends of the wires much like 1D spinless p-wave superconductors.
In this work, in the supplement [21], we show that the low energy effective
action is that of a 1D spinless p-wave superconductor as well.

The main thrust of this this work is to extend the ideas of Sau et al. [1] as
well as the 1D case of Refs. [2, 3] to the case of two band s-wave
superconductor proximitized semiconductors with both Kane Mele like [4] and
Rashba [22] spin orbit coupling. Furthermore no large magnetic fields (or
ferromagnets) are involved as in the Sau-Luchtin-Tewari-Sarma [1] and Oreg-
Refael-von Oppen [2, 3] proposals \- which is highly advantageous to
superconductivity [23]. We replace the time reversal symmetry breaking effects
of the magnetic field with spin orbit coupling (which is time reversal (TR)
invariant) thereby doubling the proposals in Refs. [1, 2, 3] in a TR invariant
way. This is much like the Kane-Mele spin Hall proposal [4] doubled the
Haldane model [5] in a TR invariant way. We show that to leading order the low
energy effective action in 2D is time reversal invariant p-wave
superconductivity with two copies of $p+ip$ superconductors with opposite
chirality, the two copies are in pseudospin space rather then in spin space as
is the usual case with TR invariant superconductors [17, 19]. Furthermore in
the supplement we show that “no-go” theorems derived previously [7, 8] make
assumptions that are not general enough to cover proximity effects from multi-
band superconductors considered here and as such do not apply to the setup
presented in this work. Majorana fermions are found in vortex cores for the
superconductors in this work and their stability is proved for small
perturbations. In 1D, for the superconductors in this work, we find Majorana
fermions at the end of wires (these are also stable to small perturbations)
with the effective low energy theory of the doubled Oreg-Refael-von Oppen [2,
3] proposal is that of a single band time reversal invariant p-wave
superconductor with pseudospin rather then spin degree of freedom.

###  II General considerations about time reversal and the type of orbitals
needed for our construction

In order to obtain the doubled Sau-Luchtin-Tewari-Sarma [1] and Oreg-Refael-
von Oppen [2, 3] proposals we require two spinful orbitals with specific time
reversal properties given by Eq. (5) below. Any two orbitals with this
transform property under time reversal will do. Here we will show that these
time reversal properties are not exotic by presenting an explicit physical
realization of such orbitals with
$\left|Y_{1}^{1}\right\rangle,\,\left|Y_{-1}^{1}\right\rangle$ (which
transform under TR as in Eq. (5) ). Here $Y_{m}^{l}$ are spherical harmonics.
For the rest of the paper, for simplicity, the reader may focus on this
realization if they so please, though many others are possible.

####  II.1 Physical setup

We now present explicit orbitals with time reversal properties given by Eq.
(5) below. For concreteness we will consider the case of $p_{x}$ and $p_{y}$
orbitals which are a linear combination of
$\left|Y_{1}^{1}\right\rangle,\,\left|Y_{-1}^{1}\right\rangle$ . We will work
in the basis:

$\left|+\right\rangle=\left|Y_{1}^{1}\right\rangle\sim\exp\left(i\theta\right),\;\left|-\right\rangle=\left|Y_{-1}^{1}\right\rangle\sim-\exp\left(-i\theta\right).$
(1)

Here $\theta$ is the azimuthal angle. For each lattice site for the systems we
will consider there will be four relevant basis states:

$\left|+,\uparrow\right\rangle,\>\left|+,\downarrow\right\rangle,\>\left|-,\uparrow\right\rangle,\>\left|-,\downarrow\right\rangle$
(2)

We will use $\tau$ as the Pauli matrices within the orbital space and $\sigma$
to be the Pauli matrices within the spin space, below we shall also introduce
$\mu$ which are Pauli matrices for particle-hole space. We note that
$\left|Y_{1}^{2}\right\rangle,\;\left|Y_{-1}^{2}\right\rangle$ are just as
good for our purposes of obtaining Eq. (5) as well as many other orbitals.

####  II.2 Time reversal symmetry

Under time reversal symmetry we have that
$\vec{\sigma}\rightarrow-\vec{\sigma}$. Furthermore under time reversal
$Y_{m}^{l}\rightarrow\left(-1\right)^{m}Y_{-m}^{l}$ (as $i\rightarrow-i$, see
Eq. (1)) which means that we have that in the model we propose below the time
reversal operator is given by

$T:-\tau_{x}i\sigma_{y}K$ (3)

Where $K$ is complex conjugation. This means that we have

$T:\tau_{z}\rightarrow-\tau_{z},\,\tau_{x}\rightarrow\tau_{x},\,\tau_{y}\rightarrow\tau_{y}$
(4)

Furthermore under time reversal $\mathbf{k}\rightarrow-\mathbf{k}$ (here
$\mathbf{k}$ is the pseudo-momentum in the first Brillouin zone). As such we
have that:

$\displaystyle T:$ $\displaystyle
c_{\mathbf{k}+\uparrow}^{\dagger}\rightarrow-
c_{-\mathbf{k}-\downarrow}^{\dagger},\>c_{\mathbf{k}+\downarrow}^{\dagger}\rightarrow+c_{-\mathbf{k}-\uparrow}^{\dagger},$
$\displaystyle c_{\mathbf{k}-\uparrow}^{\dagger}\rightarrow-
c_{-\mathbf{k}+\downarrow}^{\dagger},\>c_{\mathbf{k}-\downarrow}^{\dagger}\rightarrow+c_{-\mathbf{k}+\uparrow}^{\dagger}$
(5)

###  III Doubling the Sau-Luchtin-Tewari-Sarma proposal (p-wave time reversal
preserving Hamiltonian)

####  III.1 Global analysis

We consider a two band spinful BDG Hamiltonian given by:

$\displaystyle H\left(\mathbf{k}\right)$
$\displaystyle=\mu_{z}\left[\left(\frac{\mathbf{k}^{2}}{2m}-\mu\right)+\lambda_{SO}\sigma_{z}\tau_{z}+\lambda_{R}\left[\mathbf{k}_{y}\sigma_{x}-\mathbf{k}_{x}\sigma_{y}\right]\right]$
$\displaystyle+\Delta_{s}\mu_{x}$ (6)

We note that $\left[H\left(\mathbf{k}\right),\tau_{z}\right]=0$ so the two
orbitals decouple and the Hamiltonian is doubled and
$\lambda_{SO}\sigma_{z}\tau_{z}$ is similar to the spin orbit coupling used by
Kane and Mele 4 [4] except $\tau_{z}$ is not in the valley space for graphene
4 [4]. We now use the basis:

$\displaystyle\Psi_{\mathbf{k}}^{\tau_{z}}=$
$\displaystyle\left(c_{\mathbf{k}+\uparrow},c_{\mathbf{k}+\downarrow},c_{-\mathbf{k}+\downarrow}^{\dagger},-c_{-\mathbf{k}+\uparrow}^{\dagger}\right.$
$\displaystyle\left.c_{\mathbf{k}-\uparrow},c_{\mathbf{k}-\downarrow},c_{-\mathbf{k}-\downarrow}^{\dagger},-c_{-\mathbf{k}-\uparrow}^{\dagger}\right)^{T}$
(7)

As such:

$H\left(\mathbf{k}\right)=\left(\begin{array}[]{cc}H_{+}\left(\mathbf{k}\right)&0\\\
0&H_{-}\left(\mathbf{k}\right)\end{array}\right)$ (8)

with

$\displaystyle H_{\pm}\left(\mathbf{k}\right)=$
$\displaystyle\mu_{z}\left[\left(\frac{\mathbf{k}^{2}}{2m}-\mu\right)\pm\lambda_{SO}\sigma_{z}+\lambda_{R}\left[\mathbf{k}_{y}\sigma_{x}-\mathbf{k}_{x}\sigma_{y}\right]\right]$
$\displaystyle+\Delta_{s}\mu_{x}$ (9)

being two time reversed copies of the Sau-Luchtin-Tewari-Sarma Hamiltonians
(the total Hamiltonian is TR invariant see Eq. (5)). Furthermore since this is
an exact doubling the phase boundaries (where the Hamiltonian become gapless)
are identical to those in the proposal by Sau et. al. [1] with the critical
boundary given by [1]:

$\mu=\sqrt{\lambda_{SO}^{2}-\Delta_{s}^{2}}$ (10)

####  III.2 Effective low energy theory

#####  III.2.1 Hamiltonian ignoring the effects of pairing

We now introduce the spinor
$\Psi_{\mathbf{k}}=\left(c_{\mathbf{k}+\uparrow},c_{\mathbf{k}+\downarrow},c_{\mathbf{k}-\uparrow},c_{\mathbf{k}-\downarrow}\right)^{T}$
then we consider the Hamiltonian given by:

$H\left(\mathbf{k}\right)=\Psi_{\mathbf{k}}^{\dagger}\mathcal{H}\left(\mathbf{k}\right)\Psi_{\mathbf{k}}$
(11)

with:

$\mathcal{H}\left(\mathbf{k}\right)=\left(\frac{\mathbf{k}^{2}}{2m}-\mu\right)+\lambda_{SO}\tau_{z}\sigma_{z}+\lambda_{R}\left(\mathbf{k}_{x}\sigma_{y}-\mathbf{k}_{y}\sigma_{x}\right)$
(12)

It is straightforward to check that
$\left[\mathcal{H}\left(\mathbf{k}\right),\tau_{z}\right]=0$ and that the
Hamiltonian in Eq. (12) is TR invariant and corresponds to the non-pairing
piece of the Hamiltonian in Eq. (6). Now we will assume that
$\lambda_{SO}\gg\lambda_{R}$ so that the low energy subspace of the
Hamiltonian has a basis given by [21]:

$\displaystyle\left|\Uparrow\right\rangle$
$\displaystyle=\left|+,\downarrow\right\rangle-\frac{\lambda_{R}}{2\lambda_{SO}}\left(i\mathbf{k}_{x}-\mathbf{k}_{y}\right)\left|+,\uparrow\right\rangle+...$
$\displaystyle\left|\Downarrow\right\rangle$
$\displaystyle=\left|-,\uparrow\right\rangle-\frac{\lambda_{R}}{2\lambda_{SO}}\left(-i\mathbf{k}_{x}-\mathbf{k}_{y}\right)\left|-,\downarrow\right\rangle+...$
(13)

this being a Kramers doublet, with the Hamiltonian is this basis being given
by [21]:

$H\left(\mathbf{k}\right)=\left[\left(\frac{\mathbf{k}^{2}}{2m}-\mu\right)-\lambda_{SO}\right]\left[c_{\mathbf{k},\Uparrow}^{\dagger}c_{\mathbf{k},\Uparrow}+c_{\mathbf{k},\Downarrow}^{\dagger}c_{\mathbf{k},\Downarrow}\right]+...$
(14)

#####  III.2.2 Adding a small pairing

We consider adding

$H_{\Delta}\left(\mathbf{k}\right)=\Delta
c_{\mathbf{k}+\uparrow}^{\dagger}c_{-\mathbf{k}+\downarrow}^{\dagger}+\Delta^{*}c_{\mathbf{k}-\uparrow}^{\dagger}c_{-\mathbf{k}-\downarrow}^{\dagger}+h.c.$
(15)

to the Hamiltonian in Eq. (11) which corresponds to the pairing piece of the
Hamiltonian in Eq. (6).. Then we have that:

$\displaystyle T\sum_{\mathbf{k}}H_{\Delta}\left(\mathbf{k}\right)T=$
$\displaystyle=\sum_{\mathbf{k}}\left[-\Delta^{*}c_{-\mathbf{k}-\downarrow}^{\dagger}c_{\mathbf{k}-\uparrow}^{\dagger}-\Delta
c_{-\mathbf{k}+\downarrow}^{\dagger}c_{\mathbf{k}-\uparrow}+h.c.\right]$
$\displaystyle=\sum_{\mathbf{k}}H_{\Delta}\left(\mathbf{k}\right),$ (16)

which means the pairing term in Eq. (15) is time reversal invariant. Now we
consider the case where $\Delta$ is the smallest energy scale to the
Hamiltonian in Eq. (11). As such it is sufficient to do zeroth order
perturbation theory in $\Delta$. Now we write [21]:

$\displaystyle c_{\mathbf{k}+\uparrow}^{\dagger}$
$\displaystyle=-\frac{\lambda_{R}}{2\lambda_{SO}}\left(-i\mathbf{k}_{x}-\mathbf{k}_{y}\right)c_{\mathbf{k}\Uparrow}^{\dagger}+...$
$\displaystyle c_{\mathbf{k}+\downarrow}^{\dagger}$
$\displaystyle=c_{\mathbf{k}\Uparrow}^{\dagger}+...$ $\displaystyle
c_{\mathbf{k}-\uparrow}^{\dagger}$
$\displaystyle=c_{\mathbf{k}\Downarrow}^{\dagger}+...$ $\displaystyle
c_{\mathbf{k}-\downarrow}^{\dagger}$
$\displaystyle=-\frac{\lambda_{R}}{2\lambda_{SO}}\left(+i\mathbf{k}_{x}-\mathbf{k}_{y}\right)c_{\mathbf{k}\Downarrow}^{\dagger}+...$
(17)

As such computing matrix elements of the Hamiltonian in Eq. (15) we see that
within the low energy subspace:

$\displaystyle H_{\Delta}\left(\mathbf{k}\right)=$
$\displaystyle\frac{\lambda_{R}\Delta}{2\lambda_{SO}}\left(i\mathbf{k}_{x}-\mathbf{k}_{y}\right)c_{\mathbf{k}\Uparrow}^{\dagger}c_{-\mathbf{k}\Uparrow}^{\dagger}+h.c.$
$\displaystyle-\frac{\lambda_{R}\Delta^{*}}{2\lambda_{SO}}\left(-i\mathbf{k}_{x}-\mathbf{k}_{y}\right)c_{\mathbf{k}\Downarrow}^{\dagger}c_{-\mathbf{k}\Downarrow}^{\dagger}+h.c.$
(18)

As such the total low energy effective Hamiltonian is given by:

$\displaystyle H\left(\mathbf{k}\right)=$
$\displaystyle\left[\left(\frac{\mathbf{k}^{2}}{2m}-\mu\right)-\lambda_{SO}\right]\left[c_{\mathbf{k},\Uparrow}^{\dagger}c_{\mathbf{k},\Uparrow}+c_{\mathbf{k},\Downarrow}^{\dagger}c_{\mathbf{k},\Downarrow}\right]$
$\displaystyle+\left[\frac{\lambda_{R}\Delta}{2\lambda_{SO}}\left(-i\mathbf{k}_{x}-\mathbf{k}_{y}\right)c_{\mathbf{k}\Uparrow}^{\dagger}c_{-\mathbf{k}\Uparrow}^{\dagger}\right.$
$\displaystyle\;\left.-\frac{\lambda_{R}\Delta^{*}}{2\lambda_{SO}}\left(i\mathbf{k}_{x}-\mathbf{k}_{y}\right)c_{\mathbf{k}\Downarrow}^{\dagger}c_{-\mathbf{k}\Downarrow}^{\dagger}+h.c.\right]$
(19)

Which is a 2D TR invariant Hamiltonian with pseudospin degree of freedom:
$\Uparrow,\,\Downarrow$.

####  III.3 Stability analysis

It is known that superconductors whose effective theory is given by Eq. (19)
have a pair of Majorana Fermions inside vortex cores [17, 19], denoted by
$\gamma_{1}$ and $\gamma_{2}$. We now show that these modes are stable until a
gap closing transition. Indeed from Brillouin-Wigner perturbation theory we
know that the zero energy manifold for a vortex has an effective Hamiltonian
which converges until there is a gap closing phase transition [18]. By
requiring the Hamiltonian be Hermitian and using fermion parity conservation
we must have that the effective Hamiltonian for the zero energy subspace is
given by:

$H_{eff}=i\Gamma\gamma_{1}\gamma_{2}$ (20)

For some $\Gamma$. If we further assume the perturbation preserves time
reversal $T$ we must have that:

$Ti\Gamma\gamma_{1}\gamma_{2}T^{-1}=-i\Gamma\gamma_{1}\gamma_{2}=i\Gamma\gamma_{1}\gamma_{2}\Rightarrow
H_{eff}=0$ (21)

so the Majorana modes are stable (cannot open a gap) and hence the band
topology is stable under small TR preserving perturbations as well.

###  IV 1-D case: Doubling the Oreg-Refael-von Oppen proposal

####  IV.1 Global analysis

We consider a total (doubled) Hamiltonian given by:

$\displaystyle H\left(k\right)$
$\displaystyle=\mu_{z}\left[\left(\frac{k^{2}}{2m}-\mu\right)+\lambda_{SO}\sigma_{z}\tau_{z}+\lambda_{R}k\sigma_{x}\right]$
$\displaystyle+\Delta_{s}\mu_{x}$ (22)

We note that $\left[H\left(k\right),\tau_{z}\right]=0$ so the two bands
decouple and the Hamiltonian is doubled. We now use the basis:

$\displaystyle\Psi_{k}^{\tau_{z}}=$
$\displaystyle\left(c_{k+\uparrow},c_{k+\downarrow},c_{-k+\downarrow}^{\dagger},-c_{-k+\uparrow}^{\dagger}\right.$
$\displaystyle\left.c_{k-\uparrow},c_{k-\downarrow},c_{-k-\downarrow}^{\dagger},-c_{-k-\uparrow}^{\dagger}\right)^{T}$
(23)

So that the Hamiltonian is explicitly doubled:

$H\left(k\right)=\left(\begin{array}[]{cc}H_{+}\left(k\right)&0\\\
0&H_{-}\left(k\right)\end{array}\right)$ (24)

with

$H_{\pm}\left(k\right)=\mu_{z}\left[\left(\frac{k^{2}}{2m}-\mu\right)\pm\lambda_{SO}\sigma_{z}+\lambda_{R}k\sigma_{x}\right]+\Delta_{s}\mu_{x}$
(25)

being two time reversed copies of the Oreg-Refael-von Oppen Hamiltonians [2,
3]. Therefore the phase boundaries are identical to those in that proposal
with the critical boundary be given by the gap closing transition [2] given by
Eq. (10) above.

####  IV.2 Effective low energy theory

#####  IV.2.1 Hamiltonian no pairing

We now introduce the spinor
$\Psi_{k}=\left(c_{k+\uparrow},c_{k+\downarrow},c_{k-\uparrow},c_{k-\downarrow}\right)^{T}$
then we consider the Hamiltonian given by:

$H\left(k\right)=\Psi_{k}^{\dagger}\mathcal{H}\left(k\right)\Psi_{k}$ (26)

with:

$\mathcal{H}\left(k\right)=\left(\frac{k^{2}}{2m}-\mu\right)+\lambda_{SO}\tau_{z}\sigma_{z}+\lambda_{R}k\sigma_{x}$
(27)

It is straightforward to check that
$\left[\mathcal{H}\left(k\right),\tau_{z}\right]=0$ and that the Hamiltonian
in Eq. (27) is time reversal invariant and corresponds to the non-pairing
piece of the Hamiltonian in Eq. (22). Now we will assume that
$\lambda_{SO}\gg\lambda_{R}$ so that the low energy Hamiltonian is given by
[21]:

$\displaystyle\left|\Uparrow\right\rangle$
$\displaystyle=\left|+,\downarrow\right\rangle-\frac{\lambda_{R}}{2\lambda_{SO}}k\left|+,\uparrow\right\rangle+...$
$\displaystyle\left|\Downarrow\right\rangle$
$\displaystyle=\left|-,\uparrow\right\rangle-\frac{\lambda_{R}}{2\lambda_{SO}}k\left|-,\downarrow\right\rangle+...$
(28)

being a Kramers doublet, with

$H\left(\mathbf{k}\right)=\left[\left(\frac{k^{2}}{2m}-\mu\right)-\lambda_{SO}\right]\left[c_{k,\Uparrow}^{\dagger}c_{k,\Uparrow}+c_{k,\Downarrow}^{\dagger}c_{k,\Downarrow}\right]+...$
(29)

where we have focused just on the low energy subspace.

#####  IV.2.2 Adding pairing

We consider adding

$H_{\Delta}\left(k\right)=\Delta
c_{k+\uparrow}^{\dagger}c_{-k+\downarrow}^{\dagger}+\Delta^{*}c_{k-\uparrow}^{\dagger}c_{-k-\downarrow}^{\dagger}+h.c.$
(30)

to the Hamiltonian in Eq. (26) which corresponds to the pairing piece of the
Hamiltonian in Eq. (22) and is TR invariant see Eq. (16). Now we write [21]:

$\displaystyle c_{k+\uparrow}^{\dagger}$
$\displaystyle=-\frac{\lambda_{R}}{2\lambda_{SO}}kc_{k\Uparrow}^{\dagger}+...$
$\displaystyle c_{k+\downarrow}^{\dagger}$
$\displaystyle=c_{k\Uparrow}^{\dagger}+...$ $\displaystyle
c_{k-\uparrow}^{\dagger}$ $\displaystyle=c_{k\Downarrow}^{\dagger}+...$
$\displaystyle c_{k-\downarrow}^{\dagger}$
$\displaystyle=-\frac{\lambda_{R}}{2\lambda_{SO}}kc_{k\Downarrow}^{\dagger}+...$
(31)

As such we have that within the low energy subspace [21]:

$H_{\Delta}\left(k\right)=\frac{\lambda_{R}\Delta}{2\lambda_{SO}}kc_{k\Uparrow}^{\dagger}c_{-k\Uparrow}^{\dagger}-\frac{\lambda_{R}\Delta^{*}}{2\lambda_{SO}}kc_{k\Downarrow}^{\dagger}c_{-k\Downarrow}^{\dagger}+h.c.$
(32)

As such the total low energy effective Hamiltonian is given by [21]:

$\displaystyle H\left(k\right)$
$\displaystyle=\left[\left(\frac{k^{2}}{2m}-\mu\right)-\lambda_{SO}\right]\left[c_{k,\Uparrow}^{\dagger}c_{k,\Uparrow}+c_{k,\Downarrow}^{\dagger}c_{k,\Downarrow}\right]$
$\displaystyle+\left[\frac{\lambda_{R}\Delta}{2\lambda_{SO}}kc_{k\Uparrow}^{\dagger}c_{-k\Uparrow}^{\dagger}-\frac{\lambda_{R}\Delta^{*}}{2\lambda_{SO}}kc_{k\Downarrow}^{\dagger}c_{-k\Downarrow}^{\dagger}+h.c.\right]$
(33)

Which is a p-wave 1D time reversal invariant Hamiltonian with pseudospin
degree of freedom.

####  IV.3 Stability analysis

The stability analysis is verbatim that of Section III.3.

###  V Conclusions & outlook

In this work we have doubled in a TR preserving way the Sau-Luchtin-Tewari-
Sarma [1] and Oreg-Refael-von Oppen [2, 3] proposals much like the Kane-Mele
proposal (spin Hall insulator) [4] doubles the Haldane model [5] in a TR
preserving way. We have shown that the low energy effective action for these
models in 1D and 2D are the time reversal invariant single band p-wave
superconductors with pseudospin degree of freedom. Similarly the effective
action for the Sau-Luchtin-Tewari-Sarma [1] and Oreg-Refael-von Oppen [2, 3]
are the single band $p+ip$ and single band 1D p-wave superconductors with spin
degree of freedom. In the supplement [21] we have presented physical
realization of the system within the cold atoms setup [7, 8]. In future works
it would be of interest to study real solid state systems with spin orbit
coupling and proximity induces s-wave superconductivity for a completely
realistic realization of the models presented in this work in solid state
heterostructures which would open this proposal to many applications in
quantum computing [24].

Acknowledgements: The author would like to thank Chris Laumann and Claudio
Chamon for useful discussions.

## Supplementary Online Information

###  Appendix A Background (lightening review)

####  A.1 Topological superconductivity

#####  A.1.1 Time reversal symmetry breaking p-wave 2D superconductors

The simplest time reversal symmetry breaking p-wave 2D superconductors have
the following Hamiltonian:

$\displaystyle H\left(\mathbf{k}\right)$
$\displaystyle=\left(\frac{\mathbf{k}^{2}}{2m}-\mu\right)c_{\mathbf{k}}^{\dagger}c_{\mathbf{k}}$
$\displaystyle+\Delta\left(\mathbf{k}_{x}\pm
i\mathbf{k}_{y}\right)c_{\mathbf{k}}^{\dagger}c_{-\mathbf{k}}^{\dagger}+h.c.$
(34)

Where $\pm$ corresponds to $p_{x}+ip_{y}$ and $p_{x}-ip_{y}$ Hamiltonians. It
is known to have Majorana fermions in the vortices of its order parameter [17,
19].

#####  A.1.2 Time reversal symmetry invariant p-wave 2D Hamiltonians

The simplest time reversal symmetry preserving p-wave 2D superconductors have
the following Hamiltonian [7]:

$\displaystyle H\left(\mathbf{k}\right)=$
$\displaystyle\sum_{\sigma=\pm}\left(\frac{\mathbf{k}^{2}}{2m}-\mu\right)c_{\mathbf{k}\sigma}^{\dagger}c_{\mathbf{k}\sigma}$
$\displaystyle+\Delta\left(\mathbf{k}_{x}\pm
i\mathbf{k}_{y}\right)c_{\mathbf{k}\uparrow}^{\dagger}c_{-\mathbf{k}\uparrow}^{\dagger}+h.c.$
$\displaystyle-\Delta^{*}\left(\mathbf{k}_{x}\mp
i\mathbf{k}_{y}\right)c_{\mathbf{k}\downarrow}^{\dagger}c_{-\mathbf{k}\downarrow}^{\dagger}+h.c.$
(35)

Which is just two time reversal invariant copies of the Hamiltonian in Eq.
(34).

#####  A.1.3 Time reversal symmetry breaking p-wave 1D superconductors

The simplest time reversal symmetry breaking p-wave 1D superconductors have
the following Hamiltonian:

$H\left(k\right)=\left(\frac{k^{2}}{2m}-\mu\right)c_{k}^{\dagger}c_{k}+\left[\Delta
kc_{k}^{\dagger}c_{-k}^{\dagger}+h.c.\right]$ (36)

It is known to have Majorana fermions at the ends of 1D wires [17, 19].

#####  A.1.4 Time reversal symmetry invariant p-wave 1D Hamiltonians

The simplest time reversal symmetry preserving p-wave 2D superconductors have
the following Hamiltonian [7]:

$\displaystyle H\left(k\right)=$
$\displaystyle\sum_{\sigma=\pm}\left(\frac{k^{2}}{2m}-\mu\right)c_{\mathbf{k}\sigma}^{\dagger}c_{\mathbf{k}\sigma}$
$\displaystyle+\left[\Delta
kc_{k\uparrow}^{\dagger}c_{-k\uparrow}^{\dagger}-\Delta^{*}kc_{k\downarrow}^{\dagger}c_{-k\downarrow}^{\dagger}+h.c.\right]$
(37)

Which is just two time reversal invariant copies of the Hamiltonian in Eq.
(36).

####  A.2 Quantum mechanics

#####  A.2.1 First order perturbation theory

We note that if

$H_{0}\left|E_{0}\right\rangle=E_{0}\left|E_{0}\right\rangle$ (38)

and $H=H_{0}+V$ with $V\ll H_{0}$ then:

$\displaystyle H\left|E\right\rangle$ $\displaystyle=E\left|E\right\rangle$
$\displaystyle E$ $\displaystyle=E_{0}+\left\langle
E_{0}\right|V\left|E_{0}\right\rangle+.....$
$\displaystyle\left|E\right\rangle$
$\displaystyle=\left|E_{0}\right\rangle-\sum_{E_{0}^{\prime}\neq
E_{0}}\frac{\left\langle
E_{0}^{\prime}\right|V\left|E_{0}\right\rangle}{E_{0}^{\prime}-E_{0}}\left|E_{0}^{\prime}\right\rangle+....$
(39)

#####  A.2.2 Change of basis

Suppose you have an orthonormal basis
$\left\\{\left|U_{i}\right\rangle_{i=1,..N}\right\\}$ and a second orthonormal
basis $\left\\{\left|V_{i}\right\rangle_{i=1,...N}\right\\}$ then we have
that:

$\left|U_{j}\right\rangle=\sum_{i=1}^{N}\left|V_{i}\right\rangle\left\langle
V_{i}\mid U_{j}\right\rangle$ (40)

Now suppose there is some relevant Hamiltonian such that
$\left|V_{i}\right\rangle_{i=1,..M}$ is much lower in energy then
$\left|V_{i}\right\rangle_{i=M+1,...N}$ then in many cases we may write that:

$\left|U_{j}\right\rangle=\sum_{i=1}^{M}\left|V_{i}\right\rangle\left\langle
V_{i}\mid U_{j}\right\rangle+....$ (41)

###  Appendix B Proximity induced superconductivity

####  B.1 Sau-Luchtin-Tewari-Sarma proposal [1] (simplified treatment)

Consider the following Hamiltonian:

$H\left(\mathbf{k}\right)=\left(c_{\mathbf{k}\uparrow}^{\dagger},c_{\mathbf{k}\downarrow}^{\dagger}\right)h\left(\mathbf{k}\right)\left(\begin{array}[]{c}c_{\mathbf{k\uparrow}}\\\
c_{\mathbf{k}\downarrow}\end{array}\right)$ (42)

with

$\displaystyle h\left(\mathbf{k}\right)$
$\displaystyle=\left(\frac{\mathbf{k}^{2}}{2m}-\mu\right)+\lambda_{R}\left(\mathbf{k}_{x}\sigma_{y}-\mathbf{k}_{y}\sigma_{x}\right)+B\sigma_{z}$
$\displaystyle h_{0}\left(\mathbf{k}\right)$
$\displaystyle=\left(\frac{\mathbf{k}^{2}}{2m}-\mu\right)+B\sigma_{z}$
$\displaystyle V\left(\mathbf{k}\right)$
$\displaystyle=\lambda_{R}\left(\mathbf{k}_{x}\sigma_{y}-\mathbf{k}_{y}\sigma_{x}\right)$
(43)

Then using Eq. (39) we have that

$\displaystyle\left|\Omega\right\rangle$
$\displaystyle=\left|\downarrow\right\rangle-\frac{\lambda_{R}}{2B}\left(i\mathbf{k}_{x}-\mathbf{k}_{y}\right)\left|\uparrow\right\rangle+...$
$\displaystyle E_{\Omega}$
$\displaystyle=\left(\frac{\mathbf{k}^{2}}{2m}-\mu\right)-B+....$ (44)

Where we have assumed that $B\gg\lambda_{R}$. Now we write:

$\displaystyle c_{\mathbf{k}\downarrow}^{\dagger}$
$\displaystyle=c_{\mathbf{k},\Omega}^{\dagger}+...$ $\displaystyle
c_{\mathbf{k}\uparrow}^{\dagger}$
$\displaystyle=-\frac{\lambda_{R}}{2B}\left(-i\mathbf{k}_{x}-\mathbf{k}_{y}\right)c_{\mathbf{k},\Omega}^{\dagger}+...$
(45)

Now we add s-wave paring to the Hamiltonian in Eq. (42) with the new
Hamiltonian being given by:

$H\left(\mathbf{k}\right)=\left(c_{\mathbf{k}\uparrow}^{\dagger},c_{\mathbf{k}\downarrow}^{\dagger}\right)h\left(\mathbf{k}\right)\left(\begin{array}[]{c}c_{\mathbf{k\uparrow}}\\\
c_{\mathbf{k}\downarrow}\end{array}\right)+\left[\Delta
c_{\mathbf{k}\uparrow}^{\dagger}c_{-\mathbf{k}\downarrow}^{\dagger}+h.c.\right]$
(46)

Now within the low energy subspace spanned by $\left|\Omega\right\rangle$ we
have that:

$\displaystyle H_{\Omega}\left(\mathbf{k}\right)=$
$\displaystyle\left[\left(\frac{\mathbf{k}^{2}}{2m}-\mu\right)-B\right]c_{\mathbf{k},\Omega}^{\dagger}c_{\mathbf{k},\Omega}$
$\displaystyle+\left[\frac{\lambda_{R}\Delta}{2B}c_{\mathbf{k}\Omega}^{\dagger}c_{-\mathbf{k}\Omega}^{\dagger}\left(\mathbf{k}_{y}+i\mathbf{k}_{x}\right)+h.c.\right]$
(47)

or a 2D p-wave time reversal breaking Hamiltonian.

####  B.2  Oreg-Refael-von Oppen proposal [2, 3] (simplified treatment)

Consider the following Hamiltonian:

$H\left(k\right)=\left(c_{k\uparrow}^{\dagger},c_{k\downarrow}^{\dagger}\right)h\left(k\right)\left(\begin{array}[]{c}c_{k\uparrow}\\\
c_{k\downarrow}\end{array}\right)$ (48)

with

$\displaystyle h\left(k\right)$
$\displaystyle=\left(\frac{k^{2}}{2m}-\mu\right)+\lambda_{R}k\sigma_{x}+B\sigma_{z}$
$\displaystyle h_{0}\left(k\right)$
$\displaystyle=\left(\frac{k^{2}}{2m}-\mu\right)+B\sigma_{z}$ $\displaystyle
V\left(k\right)$ $\displaystyle=\lambda_{R}k\sigma_{x}$ (49)

Then using Eq. (39) we have that

$\displaystyle\left|\Omega\right\rangle$
$\displaystyle=\left|\downarrow\right\rangle-\frac{\lambda_{R}}{2B}k\left|\uparrow\right\rangle+...$
$\displaystyle E_{\Omega}$
$\displaystyle=\left(\frac{k^{2}}{2m}-\mu\right)-B+....$ (50)

Where we have assumed that $B\gg\lambda_{R}$. Now we write:

$\displaystyle c_{k\downarrow}^{\dagger}$
$\displaystyle=c_{k\Omega}^{\dagger}+...$ $\displaystyle
c_{k\uparrow}^{\dagger}$
$\displaystyle=-\frac{\lambda_{R}}{2B}kc_{k,\Omega}^{\dagger}+...$ (51)

Now we add s-wave paring to the Hamiltonian in Eq. (48) with the new
Hamiltonian being given by:

$H\left(k\right)=\left(c_{k\uparrow}^{\dagger},c_{k\downarrow}^{\dagger}\right)h\left(k\right)\left(\begin{array}[]{c}c_{k\uparrow}\\\
c_{k\downarrow}\end{array}\right)+\left[\Delta
c_{k\uparrow}^{\dagger}c_{-k\downarrow}^{\dagger}+h.c.\right]$ (52)

Now within the low energy subspace spanned by $\left|\Omega\right\rangle$ we
have that:

$\displaystyle H_{\Omega}\left(k\right)=$
$\displaystyle\left[\left(\frac{k^{2}}{2m}-\mu\right)-B\right]c_{k,\Omega}^{\dagger}c_{k,\Omega}$
$\displaystyle+\frac{\lambda_{R}\Delta}{2B}c_{k\Omega}^{\dagger}c_{-k\Omega}^{\dagger}k+h.c.$
(53)

or a 1D p-wave time reversal breaking Hamiltonian.

###  Appendix C Cold atoms realizations of the proposals in the main text

We would like to emulate the Hamiltonian given in Ref. [16] which is similar
to ours but has different time reversal properties using cold atoms. The
Hamiltonian we would like to emulate is given by:

$\displaystyle H\left(k\right)$
$\displaystyle=\mu_{z}\left[\left(\frac{k^{2}}{2m}-\mu\right)+\lambda_{SO}\sigma_{x}+\lambda_{R}k\sigma_{z}\tau_{z}\right]$
$\displaystyle+\Delta_{s}\mu_{x}\tau_{z}$ (54)

We will closely be following [6]. Now we consider the setup with four lasers
see Fig. (1). We have that the Hamiltonian is given by:

$\displaystyle H$
$\displaystyle=\sum_{k}\left(\frac{k^{2}}{2m}-V\right)n_{k}+$ $\displaystyle
B\sum_{k}\left(c_{k+p+\uparrow}^{\dagger}c_{k-p+\downarrow}+c_{k+p-\uparrow}^{\dagger}c_{k-p-\downarrow}+h.c.\right)$
(55)

Where

$\displaystyle n_{k}$
$\displaystyle=c_{k+\uparrow}^{\dagger}c_{k+\uparrow}+c_{k+\downarrow}^{\dagger}c_{k+\downarrow}+c_{k-\uparrow}^{\dagger}c_{k-\uparrow}+c_{k-\downarrow}^{\dagger}c_{k-\downarrow}$
$\displaystyle B$ $\displaystyle=\frac{\Omega_{1}\Omega_{2}^{*}}{\Delta}$ (56)

and $p$ is the momentum of the lasers. We will now assume two species of
molecules with the Hamiltonian:

$\displaystyle H_{Fes}$ $\displaystyle=g\int
dxb_{+}\left(x\right)c_{+\uparrow}^{\dagger}\left(x\right)c_{+\downarrow}^{\dagger}\left(x\right)+h.c.$
$\displaystyle+g\int
dxb_{-}\left(x\right)c_{-\uparrow}^{\dagger}\left(x\right)c_{-\downarrow}^{\dagger}\left(x\right)+h.c.$
(57)

Now we assume that due to molecule condensation and an order from disorder
effect

$g\left\langle b_{+}\left(x\right)\right\rangle=-g\left\langle
b_{-}\left(x\right)\right\rangle=\Xi$ (58)

This means that:

$\displaystyle H$ $\displaystyle=\sum_{k}\left(\frac{k^{2}}{2m}-V\right)n_{k}$
$\displaystyle+\sum_{k}B\left(c_{k+p+\uparrow}^{\dagger}c_{k-p+\downarrow}+c_{k+p-\uparrow}^{\dagger}c_{k-p-\downarrow}+h.c.\right)$
$\displaystyle+\sum_{k}\Xi\left(c_{k+\uparrow}^{\dagger}c_{-k+\downarrow}^{\dagger}-c_{k-\uparrow}^{\dagger}c_{-k-\downarrow}^{\dagger}+h.c.\right)$
(59)

We now perform the transform

$\exp\left(ip\int
x\left(n_{+\uparrow}\left(x\right)-n_{+\downarrow}\left(x\right)-n_{-\uparrow}\left(x\right)+n_{-\downarrow}\left(x\right)\right)\right)$
(60)

This transforms the Hamiltonian to the form:

$\displaystyle H$
$\displaystyle=\sum_{k}\sum_{\sigma_{z}=\pm;\tau_{z}=\pm}\left(\frac{\left(k-p\sigma_{z}\tau_{z}\right)^{2}}{2m}-V\right)n_{k\sigma_{z}\tau_{z}}$
$\displaystyle+\sum_{k}B\left(c_{k+\uparrow}^{\dagger}c_{k+\downarrow}+c_{k-\uparrow}^{\dagger}c_{k-\downarrow}+h.c.\right)$
$\displaystyle+\sum_{k}\Xi\left(c_{k-p+\uparrow}^{\dagger}c_{-k+p+\downarrow}^{\dagger}-c_{k-p-\uparrow}^{\dagger}c_{-k+p-\downarrow}^{\dagger}+h.c.\right)$
$\displaystyle=\sum_{k}\left(\frac{k^{2}}{2m}-V+\frac{p^{2}}{2m}\right)n_{k}$
$\displaystyle-\sum_{k}\sum_{\sigma_{z}=\pm;\tau_{z}=\pm}\left(\frac{kp\sigma_{z}\tau_{z}}{m}\right)n_{k\sigma_{z}\tau_{z}}$
$\displaystyle+\sum_{k}B\left(c_{k+\uparrow}^{\dagger}c_{k+\downarrow}+c_{k-\uparrow}^{\dagger}c_{k-\downarrow}+h.c.\right)$
$\displaystyle+\sum_{k}\Xi\left(c_{k-p+\uparrow}^{\dagger}c_{-k+p+\downarrow}^{\dagger}-c_{k-p-\uparrow}^{\dagger}c_{-k+p-\downarrow}^{\dagger}+h.c.\right)$
(61)

Figure 1:  Thelase setup for the could atom realization.

###  Appendix D Comparison with Ref. [7]

In Appendix A of Ref. [7] the authors present results that the constructions
presented in the main text are impossible. We make no quarrel with the math
presented in Ref. [7] however their main assumption in Appendix A is that the
paring Hamiltonian multiplied by the time reversal matrix is positive
semidefinite (in which case, to summarize their results, the order parameter
cannot wind in $\mathbf{k}$ space, see e.g. Eq. (34) so the superconductor is
trivial) is too restrictive. However if we consider the form of Eq. (5) we see
their main assumption about positive semi definiteness does not apply to our
setup. Indeed we see that:

$\displaystyle
T\Delta_{\mathbf{k}}c_{\mathbf{k}+\uparrow}^{\dagger}\rightarrow-\Delta
Tc_{-\mathbf{k}+\downarrow}^{\dagger}\rightarrow-\Delta
c_{\mathbf{k}-\uparrow}^{\dagger},$ $\displaystyle
T\Delta_{\mathbf{k}}c_{\mathbf{k}+\downarrow}^{\dagger}\rightarrow\Delta
Tc_{-\mathbf{k}+\downarrow}^{\dagger}\rightarrow-\Delta
c_{\mathbf{k}-\downarrow}^{\dagger},$ $\displaystyle
T\Delta_{\mathbf{k}}c_{\mathbf{k}-\uparrow}^{\dagger}\rightarrow-\Delta^{*}Tc_{-\mathbf{k}-\downarrow}^{\dagger}\rightarrow-\Delta^{*}c_{\mathbf{k}+\uparrow}^{\dagger},$
$\displaystyle
T\Delta_{\mathbf{k}}c_{\mathbf{k}-\downarrow}^{\dagger}\rightarrow\Delta^{*}Tc_{-\mathbf{k}-\downarrow}^{\dagger}\rightarrow-\Delta^{*}c_{\mathbf{k}+\downarrow}^{\dagger}$
(62)

Where $\Delta_{\mathbf{k}}$ is the restriction of the pairing matrix to the
wavevector $\mathbf{k}$. As such we have that:

$T\Delta_{\mathbf{k}}=-\left(\begin{array}[]{cccc}0&0&\Delta^{*}&0\\\
0&0&0&\Delta^{*}\\\ \Delta&0&0&0\\\ 0&\Delta&0&0\end{array}\right)$ (63)

is not positive or negative semidefinite as it has eigenvalues
$\pm\left|\Delta\right|$, as such the main assumption of Appendix A of Ref.
[7] does not apply to our setup.

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[†]Mario Fernández Navarro

# Flavour hierarchies and $B$-anomalies in a twin Pati-Salam theory of flavour

###### Abstract

In this proceedings, based on arXiv:2209.00276, I present a model which can
simultaneously explain and connect the flavour hierarchies of the Standard
Model with flavour anomalies in $B$-physics. I will briefly introduce the
model and highlight the main features, including a common origin of Yukawa
couplings and vector leptoquark $U_{1}$ couplings to Standard Model fermions.
A GIM-like mechanism allows for large leptoquark couplings which can explain
the $B$-anomalies, while protecting from the most dangerous FCNCs that could
be mediated by a heavy coloron and $Z^{\prime}$. Finally, I will highlight
some of the most promising signals at low energy processes which can test the
model in the upcoming future. The analysis has been updated with the late 2022
measurements of $R_{K^{(*)}}$ by LHCb.

##  1 Introduction

Fundamental fermions in the Standard Model (SM) come in three copies, denoted
as “flavours”, which share universal gauge interactions but have different
masses and mixings, also known as flavour parameters. The origin of flavour in
the SM remains as a complete mystery, as it lacks of any dynamical explanation
to the high number of flavour parameters and their hierarchical patterns. A
theory of flavour beyond the SM should provide a solution to the long-lasting
“flavour puzzle”.

Simultaneously, the non-universal structure of such a theory of flavour could
leave its imprints in flavour physics observables, which are becoming
accessible up to a high precision level in the current generation of colliders
and meson factories. In this direction, a conspicuous series of anomalies in
flavour observables emerged in the last years. Of particular interest were the
anomalies in $B$-meson decays, namely the $R_{K^{(*)}}$ [1, 2] and
$R_{D^{(*)}}$ [3] ratios hinting for the breaking of the SM lepton flavour
universality (LFU), a feature which might well be a low energy signal of the
theory addressing the origin of flavour hierarchies. The origin of flavour was
connected to the $B$-anomalies in models featuring $Z^{\prime}$ bosons [4, 5]
or scalar leptoquarks [6].

Interestingly, the vector leptoquark
$U_{1}(\boldsymbol{3},\boldsymbol{1},2/3)$ was the only leptoquark capable of
addressing both “$B$-anomalies” simultaneously [7] (and further improve the
global fit to $b\rightarrow s\ell\ell$ data via loop effects [8]). Moreover,
hierarchical couplings of $U_{1}$ to SM fermions were required to
simultaneously address both anomalies, which might well be related to the
flavour hierarchies of the SM in the UV theory. The model presented here was
built to address this 2021 picture of $B$-anomalies via $U_{1}$ exchange, and
connect their origin to the origin of Yukawa couplings in the SM. However, in
late 2022 the LHCb collaboration updated their measurements of $R_{K^{(*)}}$
[9], which are now in good agreement with the SM predictions. We consider
these new measurements in this analysis, and show that the model is still
compatible with the new data.

The massive $U_{1}$ leptoquark is well known to arise from spontaneous
breaking of the traditional Pati-Salam gauge group (PS). However, this
breaking happening at the TeV scale is at odds with current bounds from
$K_{L}\rightarrow\mu e$ and other processes, unless further model building is
performed (see e.g. [10, 11] for alternative PS embeddings). In this
direction, a 4321 gauge symmetry,

$G_{4321}\equiv SU(4)\times SU(3)_{c}^{{}^{\prime}}\times SU(2)_{L}\times
U(1)_{Y^{\prime}}\,,$ (1)

was proposed as a possible origin of $U_{1}$ at the TeV scale [12, 13, 14,
15]. UV completions of the 4321 group had been proposed in recent years to
connect the flavour hierarchies of the SM with the $B$-anomalies via $U_{1}$
[16, 17], however all of them consider that the third family of fermions
transforms under the TeV scale $SU(4)$ (the others are singlets), which leads
to strong constraints via high-$p_{T}$ production of heavy gauge bosons
arising after 4321 breaking [18, 19].

Instead, we proposed a twin Pati-Salam theory of flavour [20, 21] which
features a TeV scale 4321 group under which all SM fermions are treated in the
same way, they are singlets under the TeV scale $SU(4)$. Three families of
vector-like fermions are charged under $SU(4)$ and couple to $U_{1}$,
generating effective couplings to SM fermions via mixing. Simultaneously, the
choice of the scalar sector and the twin PS symmetry forbid SM-like Yukawa
couplings, which are generated via the same mixing that led to the leptoquark
couplings. This provides a link between $B$-physics and the theory of flavour.
In this manuscript, we will briefly introduce the model and summarise the main
phenomenological consequences.

##  2 The model

The theory is based on two copies of the traditional PS group,

Field | $SU(4)_{PS}^{I}$ | $SU(2)_{L}^{I}$ | $SU(2)_{R}^{I}$ | $SU(4)_{PS}^{II}$ | $SU(2)_{L}^{II}$ | $SU(2)_{R}^{II}$ | $Z_{4}$
---|---|---|---|---|---|---|---
$\psi_{1,2,3}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{4}$ | $\mathbf{2}$ | $\mathbf{1}$ |  $\alpha$, 1, 1
$\psi_{1,2,3}^{c}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{\overline{4}}$ | $\mathbf{1}$ | $\mathbf{\overline{2}}$ |  $\alpha$, $\alpha^{2}$, 1
$\psi_{4,5,6}$ | $\mathbf{4}$ | $\mathbf{2}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | 1, 1, $\alpha$
$\overline{\psi}_{4,5,6}$ | $\mathbf{\overline{4}}$ | $\mathbf{\overline{2}}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | 1, 1, $\alpha^{3}$
$\psi_{4,5,6}^{c}$ | $\mathbf{\overline{4}}$ | $\mathbf{1}$ | $\mathbf{\overline{2}}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | 1, 1, $\alpha$
$\overline{\psi^{c}}_{4,5,6}$ | $\mathbf{4}$ | $\mathbf{1}$ | $\mathbf{2}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | 1, 1, $\alpha^{3}$
$\phi$ | $\mathbf{4}$ | $\mathbf{2}$ | $\mathbf{1}$ | $\mathbf{\overline{4}}$ | $\mathbf{\overline{2}}$ | $\mathbf{1}$ | 1
$\overline{\phi}$,$\,\overline{\phi^{\prime}}$ | $\mathbf{\overline{4}}$ | $\mathbf{1}$ | $\mathbf{\overline{2}}$ | $\mathbf{4}$ | $\mathbf{1}$ | $\mathbf{2}$ | 1, $\alpha^{2}$
$H$ | $\mathbf{\overline{4}}$ | $\mathbf{\overline{2}}$ | $\mathbf{1}$ | $\mathbf{4}$ | $\mathbf{1}$ | $\mathbf{2}$ | 1
$\overline{H}$ | $\mathbf{4}$ | $\mathbf{1}$ | $\mathbf{2}$ | $\mathbf{\overline{4}}$ | $\mathbf{\overline{2}}$ | $\mathbf{1}$ | 1
$\Omega_{15}$ | $\mathbf{15}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | 1
Table 1: The field content under $G_{422}^{I}\times G_{422}^{II}\times Z_{4}$.
The model consists of three left-handed chiral fermion families
$\psi_{1,2,3}$, $\psi_{1,2,3}^{c}$ under the second PS group, plus three VL
fermion families $\psi_{4,5,6}$, $\psi_{4,5,6}^{c}$ and their conjugates under
the first PS group. Personal Higgs doublets are contained in $H$, $\bar{H}$,
one for each second and third family charged fermion. The Higgs singlets in
$\phi$, $\bar{\phi}$ are called Yukons. In addition to the fields shown here,
we require further high energy Higgs fields (not shown) whose VEVs will break
the second PS group at a high scale, leaving the first unbroken (see [20,
21]). We also need further Higgs fields, which break the two left-right gauge
groups into their diagonal subgroup.

$G_{422}=SU(4)\times SU(2)_{L}\times SU(2)_{R}\,,$ (2)

plus a shaping discrete symmetry $Z_{4}$, i.e.

$G_{422}^{I}\times G_{422}^{II}\times Z_{4}\,.$ (3)

The content of the model is depicted in Table 1. We assume that three chiral
fermion families transform under the second PS group, $G_{422}^{II}$, which is
broken at a high scale $M_{\mathrm{High}}\gtrsim 1\,\mathrm{PeV}$ down to
$G_{4321}$, the latter bound imposed by $K_{L}\rightarrow\mu e$. On the other
hand, three vector-like (VL) fermion families transform under the first PS
group, $G_{422}^{I}$, in such a way that they couple to the TeV scale
leptoquark $U_{1}$ arising after the breaking of $G_{4321}$ down to the SM.
They further couple to a TeV scale coloron
$g^{\prime}(\mathbf{8},\mathbf{1},0)$ and
$Z^{\prime}(\mathbf{1},\mathbf{1},0)$ that also arise after the breaking of
$G_{4321}$. The renormalizable mass Lagrangian is given by

$\displaystyle\begin{aligned}
\mathcal{L}_{\mathrm{mass}}^{ren}&=y_{ia}^{\psi}\overline{H}\psi_{i}\psi_{a}^{c}+y_{a3}^{\psi}H\psi_{a}\psi_{3}^{c}+x_{ia}^{\psi}\phi\psi_{i}\overline{\psi}_{a}+x_{a2}^{\psi^{c}}\overline{\psi_{a}^{c}\phi^{\prime}}\psi_{2}^{c}+x_{a3}^{\psi^{c}}\overline{\psi_{a}^{c}\phi}\psi_{3}^{c}+x_{16}^{\psi}\phi\psi_{1}\overline{\psi}_{6}+x_{61}^{\psi^{c}}\overline{\psi_{6}^{c}\phi}\psi_{1}^{c}&{}\\\
{}&+M_{ab}^{\psi}\psi_{a}\overline{\psi_{b}}+M_{ab}^{\psi^{c}}\psi_{a}^{c}\overline{\psi_{b}^{c}}+M_{66}^{\psi}\psi_{6}\overline{\psi_{6}}+M_{66}^{\psi^{c}}\psi_{6}^{c}\overline{\psi_{6}^{c}}&{}\\\
{}&+\lambda_{15}^{aa}\Omega_{15}\psi_{a}\overline{\psi_{a}}+\lambda_{15}^{66}\Omega_{15}\psi_{6}\overline{\psi_{6}}+\bar{\lambda}_{15}^{aa}\Omega_{15}\psi_{a}^{c}\overline{\psi_{a}^{c}}+\bar{\lambda}_{15}^{66}\Omega_{15}\psi_{6}^{c}\overline{\psi_{6}^{c}}+\mathrm{h.c.}\,,&{}\end{aligned}$
(4)

where $i,j=2,3$ and $a,b=4,5$. It can be seen that due to the choice of the
scalar sector and the twin PS symmetry, SM-like Yukawa couplings are not
present. The Lagrangian above can be written as a mass matrix, and its
diagonalisation will lead to effective Yukawa couplings for SM fermions.

We shall refer to the Higgs doublets contained in $H$, $\bar{H}$ as personal
Higgs doublets, since under the SM decomposition there will be a separate
Higgs for each fermion mass as we shall see shortly. The Higgs singlet fields
in $\phi$ , $\bar{\phi}$ are called Yukons, since they are necessary to
generate the effective Yukawa couplings. They develop different VEVs in their
quark and lepton components ($\phi_{3}$ and $\phi_{1}$), breaking $G_{4321}$
at the TeV scale down to the SM.

###  2.1 Effective Yukawa couplings and leptoquark couplings

The diagonalisation of the full mass matrix leads to mixing between VL and
chiral fermions. Each VL family mixes with one chiral family, with mixing
angles given by

$s_{34}^{Q,L}=\frac{x_{34}^{\psi}\left\langle\phi_{3,1}\right\rangle}{\sqrt{\left(x_{34}^{\psi}\left\langle\phi_{3,1}\right\rangle\right)^{2}+\left(M_{4}^{Q,L}\right)^{2}}}\,,\>s_{25}^{Q,L}=\frac{x_{25}^{\psi}\left\langle\phi_{3,1}\right\rangle}{\sqrt{\left(x_{25}^{\psi}\left\langle\phi_{3,1}\right\rangle\right)^{2}+\left(M_{5}^{Q,L}\right)^{2}}}\,,\>s_{16}^{Q,L}=\frac{x_{16}^{\psi}\left\langle\phi_{3,1}\right\rangle}{\sqrt{\left(x_{16}^{\psi}\left\langle\phi_{3,1}\right\rangle\right)^{2}+\left(M_{6}^{Q,L}\right)^{2}}}\,,$
(5)

which further lead to effective Yukawa couplings and $U_{1}$ couplings for SM
fermions, as depicted in the diagrams of Fig. 1. It can be seen that both the
Yukawa couplings and $U_{1}$ couplings originate via the same physics, and are
connected via the same mixing angles.

$Q_{3}$$M^{Q}_{4}$$u^{c}_{3}$$\phi_{3}$$H_{t}$$\overline{Q_{4}}$$Q_{4}$

(a)

$Q_{3}$$M^{Q}_{4}$$M^{L}_{4}$$L_{3}$$\phi_{3}$$U_{1}$$\phi_{1}$$\overline{Q}_{4}$$Q_{4}$$L_{4}$$\overline{L}_{4}$

(b)

$Q_{i}$$M^{\psi^{c}}_{4}$$u^{c}_{j}$$H_{c}$$\overline{\phi}_{3}$$\overline{u^{c}_{4}}$$u^{c}_{4}$

(c)

$u^{c}_{3}$$M^{\psi^{c}}_{4}$$M^{\psi^{c}}_{4}$$e^{c}_{3}$$\overline{\phi}_{3}$$U_{1}$$\overline{\phi}_{1}$$\overline{u^{c}_{4}}$$u^{c}_{4}$$e^{c}_{4}$$\overline{e^{c}_{4}}$

(d)

Figure 1: (Top) The diagram on the left leads to the mass of the top quark,
and similar diagrams can be written for all third family charged fermions. The
diagram on the right leads to effective couplings of the third family left-
handed fermions to the $U_{1}$ vector leptoquark. (Bottom) The diagram on the
left leads to the mass of second family charged fermions, $i,j=2,3$, and their
mixing with the third family. The diagram on the right leads to effective
couplings of the third family right-handed fermions to the $U_{1}$ vector
leptoquark, which are suppressed as they are connected to the origin of second
family fermion masses.

The large top mass requires $\left\langle\phi_{3}\right\rangle/M_{4}^{Q}\sim
1$, hence the 3-4 mixing angle is expected to be very large. This is in good
agreement with the expectation of $R_{D^{(*)}}$, which requires large
couplings of $U_{1}$ to third family fermions, featuring a nice connection
between the flavour puzzle and $B$-anomalies only present in our model.
Moreover, the origin of second family masses requires heavy masses for EW
singlets VL fermions, leading to the hierarchy of VL masses,

$M_{a}^{\psi}\ll M_{a}^{\psi^{c}}$ (6)

for $a=4,5,6$. This way, leptoquark currents involving right-handed (EW
singlets $\psi^{c}$) chiral fermions are suppressed by small mixing angles,
connected to the origin of second family fermion masses, leading to a model
which dominantly predicts left-handed leptoquark currents as preferred by the
global fits of $B$-anomalies (see e.g. [22, 7, 23]). This way, we obtain
another nice connection between the flavour puzzle and $B$-anomalies, see the
bottom diagrams in Fig. 1.

###  2.2 GIM-like mechanism and FCNCs

The explanation of $R_{D^{(*)}}$ requires large flavour-violating $U_{1}$
couplings that compete with the tree-level SM charged current. Obtaining such
large couplings while suppressing possible FCNCs mediated by the $Z^{\prime}$
and $g^{\prime}$ is a challenge for 4321 models, which is usually addressed
via adopting an ad-hoc flavour structure, calling for an understanding in
terms of a UV theory which the twin PS model can provide.

$Q_{i}$$M^{Q}_{a}$$M^{L}_{b}$$L_{j}$$\phi_{3}$$U_{1}$$\phi_{1}$$\overline{Q}_{a}$$Q_{a}$$L_{b}$$\overline{L}_{b}$

(a)

$Q_{i}$$M^{Q}_{a}$$M^{Q}_{a}$$Q_{i}$$\phi_{3}$$g^{\prime},Z^{\prime}$$\phi_{3}$$\overline{Q}_{a}$$Q_{a}$$Q_{a}$$\overline{Q}_{a}$

(b)

Figure 2: (Left) Flavour off-diagonal effective couplings of the $U_{1}$
vector leptoquark to SM fermions. (Right) Flavour diagonal effective couplings
of the heavy coloron and $Z^{\prime}$ to SM quarks. Similar diagrams are
obtained for leptons with the $Z^{\prime}$ boson.

The scalar $\Omega_{15}$ transforming in the adjoint of $SU(4)^{I}$ splits the
bare masses of VL quarks and leptons $M_{aa}^{Q,L}$, which otherwise would be
identical due to the twin PS symmetry. The mass matrix of VL fermions is
diagonalised via different transformations for quark and leptons, leading to
large flavour-violating $U_{1}$ couplings,

$\left.\begin{array}[]{c}\left(\begin{array}[]{ccc}M_{4}^{Q}&0&0\\\
0&M_{5}^{Q}&0\\\
0&0&M_{6}^{Q}\end{array}\right)=V_{45}^{Q}\left(\begin{array}[]{ccc}M_{44}^{Q}&M_{45}^{\psi}&0\\\
M_{54}^{\psi}&M_{55}^{Q}&0\\\
0&0&M_{66}^{Q}\end{array}\right)V_{45}^{\bar{Q}\dagger}\\\
\left(\begin{array}[]{ccc}M_{4}^{L}&0&0\\\ 0&M_{5}^{L}&0\\\
0&0&M_{6}^{L}\end{array}\right)=V_{45}^{L}\left(\begin{array}[]{ccc}M_{44}^{L}&M_{45}^{\psi}&0\\\
M_{54}^{\psi}&M_{55}^{L}&0\\\
0&0&M_{66}^{L}\end{array}\right)V_{45}^{\bar{L}\dagger}\end{array}\right\\}\Rightarrow\frac{g_{4}}{\sqrt{2}}Q_{a}^{\dagger}\gamma_{\mu}\left(\begin{array}[]{ccc}c_{\theta_{LQ}}&-s_{\theta_{LQ}}&0\\\
s_{\theta_{LQ}}&c_{\theta_{LQ}}&0\\\
0&0&1\end{array}\right)L{}_{b}U_{1}^{\mu}+\mathrm{h.c.}$ (7)

where the zeroes are enforced by the $Z_{4}$ symmetry. This way, we obtain a
CKM-like matrix $W_{LQ}\equiv V_{45}^{Q}V_{45}^{L\dagger}$ in $SU(4)^{I}$
flavour space. Remarkably, this way we obtain large flavour-violating
leptoquark currents in VL quark-lepton flavour space, while neutral currents
mediated by $g^{\prime}$ and $Z^{\prime}$ remain flavour universal.
Afterwards, 3-4, 2-5 and 1-6 mixing is performed (see Eq. (5)), inducing the
required structure of $U_{1}$ couplings that contribute to LFU ratios. On the
other hand, neutral currents become flavour diagonal, controlled by the mixing
angles $s_{34}^{Q,L}$, $s_{25}^{Q,L}$ and $s_{16}^{Q,L}$, see Fig. 2.

  * •

If $s_{25}^{Q,L}\approx s_{16}^{Q,L}$ then there is no 1-2 tree-level FCNCs,
featuring a GIM-like mechanism which allows for large flavour-violating
leptoquark couplings to explain the anomalies but protects from the most
dangerous FCNCs. A down-aligned flavour structure is achieved in the 2-3
sector (see [21]), protecting from tree-level contributions to
$B_{s}-\bar{B}_{s}$ meson mixing.

  * •

$s_{34}^{Q}\approx 1$ is motivated by the top mass and $R_{D^{(*)}}$, as
anticipated before. $s_{16}^{Q}\approx 0.2$ is compatible with bounds from
coloron production at LHC.

  * •

We choose a suitable benchmark $\left\langle\phi_{3,1}\right\rangle\approx
0.6\,\mathrm{TeV},\,0.3\,\mathrm{TeV}$ and
$M_{44}^{Q,L}<M_{55}^{Q,L},M_{66}^{Q,L}\approx
1.2\,\mathrm{TeV}-\,0.8\,\mathrm{TeV}$ and we explore the parameter space of
$x_{34}^{\psi}$ and $x_{25}^{\psi}$ (see [21] for further details about the
benchmark).

##  3 Phenomenology

(a)

(b)

Figure 3: (Left) Parameter space in the plane ($x_{34}^{\psi}$,
$x_{25}^{\psi}$) compatible with the LFU ratios. The rest of parameters are
fixed as in the benchmark of [21] for both panels. The blue region is excluded
by the $\Delta M_{s}$ bound, the region excluded only due to the contribution
via the 5th lepton is also shown in lighter blue for comparison. The blue and
red stars show benchmark points for 2021 and 2022 data. (Right) $\delta(\Delta
M_{s})=\Delta M_{s}^{\mathrm{NP}}/\Delta M_{s}^{\mathrm{SM}}$ as a function of
the 5th family lepton vector-like mass term. $x_{25}^{\psi}$ is varied in the
range $x_{25}^{\psi}=[0.3,\,0.35]$ ($[0.4,\,0.45]$) preferred by
$R_{K^{(*)}}^{2022}$ ($R_{K^{(*)}}^{2021}$ ), obtaining the yellow (blue)
band. The gray region is excluded by the bound $\delta(\Delta M_{s})<0.11$
[21]. The stars denote benchmark points.

Fig. 3a shows that the model is compatible with both LFU ratios at 1$\sigma$,
although the good parameter space is narrow with 2022 data on $R_{K^{(*)}}$. A
large contribution to $B_{s}-\bar{B}_{s}$ meson mixing, namely to $\Delta
M_{s}$, arises at 1-loop mediated by $U_{1}$ and constrains part of the
parameter space. This loop is dominated by the 5th VL charged lepton, which is
required to be light in order to pass the stringent test of $\Delta M_{s}$,
see Fig. 3b. This way, a light vector-like lepton is predicted with a mass
around 1 TeV, accessible to direct searches at LHC.

Signals in $\tau$ LFV decays, as per Figs. 4a and 4b, are directly related to
the nature of the model as a theory of flavour. The model allows for large
leptonic 2-3 FCNCs mediated by the $Z^{\prime}$ boson, and proportional to
$\tau-\mu$ mixing arising through a diagram similar to that of Fig. 1c but for
charged leptons. This $Z^{\prime}$ contribution is intrinsic to the twin PS
model, and dominates over a further 1-loop contribution mediated by $U_{1}$
which is common to all 4321 models, hence allowing to disentangle the twin PS
theory from other proposals. A wide range of the parameter space will be
tested in the future by Belle II.

(a)

(b)  (c)

(d)

Figure 4: (Top) Branching fractions of LFV $\tau$ decays as a function of the
2-3 charged lepton mixing sine $s_{23}^{e}$ (for $\tau\rightarrow 3\mu$) and
$x_{34}^{\psi}$ (for $\tau\rightarrow\mu\gamma$). The purple region denotes
the $Z^{\prime}$ contribution while the blue region denotes the $U_{1}$
contribution. For $\tau\rightarrow 3\mu$ we have varied
$x_{25}^{\psi}=[0.3,0.35]$ which is compatible with the 2022 LFU ratios, for
$\tau\rightarrow\mu\gamma$ we have varied $s_{23}^{e}=[V_{cb},5V_{cb}]$.
(Bottom) On the left panel, the branching fraction of $B\rightarrow K\tau\tau$
as a function of $x_{34}^{\psi}$. On the right, $\delta(B\rightarrow
K\nu\nu)=\mathcal{B}(B\rightarrow K\nu\nu)/\mathcal{B}(B\rightarrow
K\nu\nu)_{\mathrm{SM}}-1$ as a function of the 5th family lepton vector-like
mass term. In both panels, $x_{25}^{\psi}$ is varied in the range
$x_{25}^{\psi}=[0.3,\,0.35]$ ($[0.4,\,0.45]$) preferred by
$R_{K^{(*)}}^{2022}$ ($R_{K^{(*)}}^{2021}$ ), obtaining the yellow (blue)
band. The hatched region is excluded by the $\Delta M_{s}$ bound, and the
stars denote benchmark points.

On the other hand, the model predicts large contributions to $b\rightarrow
s\tau\tau$ and $b\rightarrow s\nu_{\tau}\nu_{\tau}$ connected to the
explanation of $R_{D^{(*)}}$, as depicted in Figs. 4c and 4d . In particular,
the full parameter space of the model will be tested by Belle II via
$B\rightarrow K^{(*)}\nu\nu$.

##  4 Conclusions

We have presented a model where the flavour hierarchies are related to flavour
anomalies in $B$-physics. The model features two Pati-Salam gauge groups
broken at different scales, leading to a TeV scale 4321 gauge group. SM-like
fermions are all singlets under the TeV scale $SU(4)$, and hence originally do
not couple to the $U_{1}$ vector leptoquark arising after the breaking of the
4321 symmetry. SM-like Yukawa couplings are forbidden due to the choice of the
scalar sector along with the twin PS symmetry. Both the effective Yukawa
couplings and the $U_{1}$ couplings for SM fermions arise through mixing with
vector-like fermions transforming under the TeV scale $SU(4)$. This way,
flavour hierarchies and contributions to LFU ratios via $U_{1}$ find a common
origin, and are connected via the same physics.

The model predicts a rich phenomenology at low energies, and will be tested by
Belle II via signals in LFV $\tau$ decays and $B\rightarrow K^{(*)}\nu\nu$. A
plethora of new states are predicted at the TeV scale, which could be directly
detected at the LHC, including the heavy $U_{1}$, coloron and $Z^{\prime}$.
Remarkably, a vector-like lepton is preferred to be around 1 TeV to pass the
stringent bounds of $B_{s}-\bar{B}_{s}$ meson mixing. The model is still
compatible with 2022 data on $R_{K^{(*)}}$, although the window to explain
$R_{D^{(*)}}$ at $1\sigma$ is now very narrow.

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