# Tachyons and (non)vanishing scalar masses in six-dimensional gauge theories
with flux compactification

Victor Franken 111This report has been written during an internship at CPHT-
Ecole Polytechnique, supervised by Emilian<EMAIL_ADDRESS>
(24 August 2021)

###### ?abstractname?

In this paper, we study the possibility to obtain a massless scalar boson for
which quantum corrections to the mass vanish at all loop-order, which has been
recently understood to be due to a shift symmetry making the scalar a
Goldstone boson. We present the effective four-dimensional Lagrangian of a
six-dimensional gauge theory compactified on a torus with magnetic flux.
Because of this magnetic field, a symmetry of translation in the extra
dimensions is broken which implies the existence of a massless scalar boson.
We then explicitly check that a model with two U(1) gauge symmetries contains
a scalar boson with finite mass but protected from large quantum corrections.
Finally, we study the presence of tachyons in the model with non-abelian gauge
symmetry. In particular, we propose a way to eliminate these tachyons and we
compute the full mass spectrum of the scalars in this theory. Finally, we show
that our method preserve the chirality of fermions in the model.

######  ?contentsname?

  1. 1 Introduction
  2. 2 Interacting model in 6 dimensions with flux compactification
    1. 2.1 Flux compactification on a torus
    2. 2.2 N=1 supersymmetric gauge theory in 6 dimension with flux compactification
    3. 2.3 Non supersymmetric fermion with abelian flux background
    4. 2.4 Quantum corrections
    5. 2.5 Underlying symmetry
    6. 2.6 U(1)xU(1) gauge group
  3. 3 Six-dimensional Yang-Mills theories with flux compactification
    1. 3.1 Tachyons in 6D Yang-Mills theories with flux compactification
    2. 3.2 Eliminating the tachyons
    3. 3.3 Chiral fermions in the SU(2) model without tachyon
  4. 4 Conclusion and Outlook
  5. A Appendix: Conventions

##  1 Introduction

The Standard Model of fundamental interactions has been a successfully tested
theory for decades. Nevertheless, it has left some fundamental questions
unanswered, as the hierarchy problem which has been the main guideline for the
search of physics beyond the standard model. These unanswered questions
suggest that the Standard Model cannot be the fundamental theory of
fundamental interactions. Thus, at high energy scales, one should find signs
of new physics, i.e. deviations from the Standard Model predictions. As new
physics has not been observed yet, the scale of possible underlying physics
goes higher and higher and the fine-tuning necessary to obtain the 125 GeV of
the Higgs gets stronger with time.

This report is mainly focused on the hierarchy problem which consists of
explaining the low mass of the Higgs boson while the quantum corrections to
its mass are huge in the framework of the standard model. A desirable solution
could be that the Higgs mass is zero at tree level and only due to quantum
corrections. During this internship, we explored the promising possibility to
use flux compactification for this purpose.

Flux compactification plays an important role in string theory and field
theories. It has been studied extensively as it has many nice properties.
Indeed, it leads to a multiplicity of chiral fermions, explains the number of
lepton-quark generations [1], it can break supersymmetry [2], and stabilize
the compact dimensions [3]. In the context of compact extra dimensions, it is
known that there is a possibility to use zero-modes of higher dimensional
gauge fields as Higgs field in four dimensions [4, 5]. It appears that
identifying these zero modes with the Higgs scalar gives a finite Higgs mass
protected by the size of the extra dimensions $m^{2}_{Higgs}\propto L^{-2}$
[6]. Thus this mechanism is interesting if the extra dimensions are large.

However, if the extra dimension is small, i.e. if the UV cutoff is much higher
than the electroweak scale, one has to find an alternative. Using magnetic
compactification, it has been shown recently that quantum corrections to the
mass of the zero modes of the scalar field induced by extra components of the
gauge field vanish. This was first shown by [7] in a supersymmetric model. The
vanishing of the one-loop correction has been studied more carefully in [8],
taking particular care to the regularization of divergent momentum integrals
via dimensional regularization. In a second paper [9], the authors of [7]
studied the cancellation of one-loop corrections in more detail and showed
explicitly how this cancellation is due to an exact shift symmetry in the
higher dimensional theory.

This was done without supersymmetry as the cancellation is due to the shift
symmetry and not SUSY. The one-loop corrections to the zero-mode mass have
then been calculated in a six-dimensional Yang-Mills theory with flux
compactification [10]. As expected, the quantum corrections are canceled.
Moreover, [11] studied the possibility to produce a nonvanishing finite Wilson
line scalar mass. Indeed, the Wilson line must have a finite mass if one wants
to identify it with the Higgs field which is massive. They therefore
classified the interactions terms breaking the shift symmetry. As for
comsological applications, [12] proposed a new inflation scenario where the
massless Wilson line is identified with an inflaton.

This paper is organized as follows. First, we introduce flux compactification
by a simple example in abelian gauge theories. Then, we present the
supersymmetric model proposed by [7] and compute the 4d effective Lagrangian.
The non-supersymmetric model of [9] is then introduced. In the next two
subsections, we compute the one-loop corrections to the Wilson line scalar
mass and explain its cancellation with a shift symmetry of the 6d action [9].
We then show explicitly how a $U(1)\crossproduct U(1)$ gauge symmetry could
lead to a finite Higgs mass with vanishing one-loop corrections. This could
lead to a finite Higgs mass protected from high corrections due to the
possibly high compactification scale.

In section 3, we study a six-dimensional Yang-Mills theory compactified on a
torus with magnetic flux. [7] showed the presence of tachyons in the $SU(2)$
model, implying that the effective action corresponds to an expansion around
an extremal point which is not the true ground state. Rather than studying
tachyon condensation, we try to eliminate the tachyon by adding a scalar boson
charged under the gauge symmetry, which gets a vacuum expectation value.
Finally, we check if this model could include chiral fermions.

##  2 Interacting model in 6 dimensions with flux compactification

###  2.1 Flux compactification on a torus

Let us begin by introducing flux compactification in a simple example. Flux
compactification produces a mass spectrum in the spirit of the Landau levels
in quantum mechanics. Let us consider a scalar field of charge q under the
abelian gauge symmetry of coupling constant g carried by the gauge field
$A^{M}$. The six dimensional action for the scalar is

$S_{6}=-\int d^{6}xD_{M}\overline{\chi}D^{M}\chi$ (2.1)

with

$D_{M}=\partial_{M}+igqA_{M}$ (2.2)

First we must compactify the extra dimensions. A common way to compactify the
extra dimensions is to compactify on a torus. For a factorizable torus, the
compact dimensions are treated as circles of length $2\pi r$. This correspond
to the identification $y\leftrightarrow y+2\pi r$. In this case, $V_{n}=(2\pi
r)^{n}$. A magnetic flux in the background correspond to a constant flux
density $f$ in the internal dimensions. One can make the following gauge
choice:

$A_{5}=-\frac{1}{2}fx_{6}\qquad A_{6}=\frac{1}{2}fx_{5}\qquad F_{56}=f$ (2.3)

For a torus of finite volum $L^{2}$, the flux is quantized 333This is showed
in the appendix of [9]. For another approach, see footnote 2 of [8]..

$\frac{qg}{2\pi}\int_{T^{2}}F=\frac{qg}{2\pi}L^{2}f=N\in\mathbb{Z}$ (2.4)

We choose without loss of generality $qf<0$. By splitting the kinetic term
into a 4D kinetic term and a kinetic term on $T^{2}$, one can find :

$S_{6}=-\int\eta^{\mu\nu}D_{\mu}\overline{\chi}D_{\nu}\chi-\overline{\chi}H_{2}\chi$
(2.5)

where

$H_{2}=-2qgf(a^{\dagger}a+1/2)$ (2.6)

is written by defining a creation and an annihilation operator functions of
the internal space components.

$\displaystyle\begin{split}a=\frac{i}{\sqrt{-2qgf}}(\overline{\partial}-qgfz),\qquad
a^{\dagger}=\frac{i}{\sqrt{-2qgf}}(\partial+qgf\overline{z})\end{split}$ (2.7)

with444the common notation is to write the nth extra component with index n,
even if consistency with the 4-vector notation would ask for index n-1.

$z=\frac{1}{2}(x_{5}+ix_{6})\qquad\partial=\frac{\partial}{\partial
z}=\partial_{5}-i\partial_{6}$ (2.8)

The ladder operators $a$, $a^{\dagger}$ satisfy the relation
$[a,a^{\dagger}]=1$. Thus, based on the work of [2], we have showed that the
internal Hamiltonian can be written in the form of a harmonic oscillator
Hamiltonian, completing the analogy with the Landau levels. We denote the
internal fields $\xi_{n,j}$ where $n$ is the index for the Landau levels and
$j$ is the degeneracy index going from 0 to $|N|-1$. Starting from the lowest
mass fields,

$a\xi_{0,j}=0\qquad a^{\dagger}\overline{\xi}_{0,j}=0,$ (2.9)

one defines all the field profiles using the ladder operators.

$\xi_{n,j}=\frac{1}{\sqrt{n!}}\left(a^{\dagger}\right)^{n}\xi_{0,j}\qquad\overline{\xi}_{n,j}=\frac{1}{\sqrt{n!}}\left(a\right)^{n}\overline{\xi}_{0,j}$
(2.10)

They satisfy the same orthonormality relation as the Kaluza-Klein level.
However, the KK levels are not orthonormal to the Landau levels.

$\int_{T^{2}}d^{2}x\xi_{n^{\prime},j^{\prime}}\overline{\xi}_{n,j}=\delta_{n,n^{\prime}}\delta_{j,j^{\prime}}$
(2.11)

The function $\xi_{n,j}$ forms a complete set of functions and we can derefore
develop the charged fields with respect to the Landau levels.

$\chi(x_{M})=\sum_{n,j}\chi_{n,j}(x_{\mu})\xi_{n,j}(x_{m})\qquad\overline{\chi}(x_{M})=\sum_{n,j}\overline{\chi}_{n,j}(x_{\mu})\overline{\xi}_{n,j}(x_{m})$
(2.12)

Using the harmonic oscillator algebra, the 6D action 2.5 then becomes :

$S_{4}=\int
d^{4}x\sum_{n,j}\left(-D_{\mu}\overline{\chi}_{n,j}D^{\mu}\chi_{n,j}+2qgf(n+1/2)\overline{\chi}_{n,j}\chi_{n,j}\right)$
(2.13)

which contains a mass term for the scalars $\chi_{n,j}$.

$m_{n,j}=-2qgf(n+1/2)=\frac{2\pi|N|}{L^{2}}(n+1/2)$ (2.14)

###  2.2 N=1 supersymmetric gauge theory in 6 dimension with flux
compactification

The simplest 6D matter multiplet is the hypermultiplet which decomposes into
two 4D chiral multiplets of opposite charge. We decompose these chiral
multiplets $Q,\tilde{Q}$ with respect to the Landau levels.

$\displaystyle\begin{split}Q(x_{M},\theta,\overline{\theta})&=\sum_{n,j}Q_{n,j}(x_{\mu},\theta,\overline{\theta})\xi_{n,j}(x_{m})\\\
\tilde{Q}(x_{M},\theta,\overline{\theta})&=\sum_{n,j}\tilde{Q}_{n,j}(x_{\mu},\theta,\overline{\theta})\overline{\xi}_{n,j}(x_{m})\end{split}$
(2.15)

In this section, we will recall the models presented in [7] and compute the
effective mass spectrum for a charged matter multiplet with abelian gauge
interactions compactified on a torus with magnetic flux.

To bring an abelian gauge interaction into the model, we will need an
uncharged 6D vector multiplet. Since it is uncharged, it decomposes in the
usual KK modes as in equation (A.11).

The $N=2$ vector multiplet corresponds in $N=1$ SUSY to one vector multiplet
and one chiral multiplet, we note them $V$ and $\phi$. To see this more
explicitly, the 6D vector field is composed of one six components vector field
(4 d.o.f.) and a Dirac gaugino (4 d.o.f) which adds up to 8 d.o.f..

In four dimensions, the vector field split into one 4-vector field and one
complex scalar field and the Dirac gaugino split into two Weyl gauginos. Thus,
in four dimensions, we get one vector field, one complex scalar and two
spinors which correspond to a 4D N=2 vector multiplet or one N=1 vector
multiplet plus one chiral multiplet. Let us remind that $x_{5}$ and $x_{6}$
play the role of indices for the superfields $V$ and $\phi$, this dependence
will be implicit for the rest of this work. We will only consider the zero
modes of the uncharged sector: $\phi_{0}$ and $V_{0}$.

It is $\phi_{0}$ that contains the internal component of the gauge field and
therefore the magnetic flux. Indeed,
$\left.\phi\right|_{\theta=\overline{\theta}=0}=\frac{1}{\sqrt{2}}(A_{6}+iA_{5})$.
Let us decompose explicitly $\phi_{0}$ into its vacuum expectation value (vev)
and its perturbation. The perturbation is $\varphi$, the Wilson line. And the
vev is the magnetic flux’s contribution to the background.

$\left.\phi_{0}\right|_{\theta=\overline{\theta}=0}=\frac{f}{2\sqrt{2}}(x_{5}-ix_{6})+\varphi$
(2.16)

The supersymmetric action for a N=2 vector multiplet is[13]:

$S_{6}=\int d^{6}x\left\\{\int d^{4}\theta\left(\partial
V\overline{\partial}V+\phi\overline{\phi}+\sqrt{2}V(\overline{\partial}\phi+\partial\overline{\phi})\right)+\left(\frac{1}{4}\int
d^{2}\theta W^{\alpha}W_{\alpha}+h.c.\right)\right\\}$ (2.17)

The action for a hypermultiplet interacting with the vector multiplet
$\\{V,\phi\\}$ is the following. The hypermultiplet being chiral, one must
choose the gaugino field to be of the same chirality.

$S_{6}=\int d^{6}x\left\\{\int
d^{4}\theta\left(\overline{Q}e^{2gqV}Q+\overline{\tilde{Q}}e^{-2gqV}\tilde{Q}\right)+\left(\int
d^{2}\theta\tilde{Q}(\partial+\sqrt{2}qg\phi)Q+h.c.\right)\right\\}$ (2.18)

By replacing the appropriate parts with ladder operators and the chiral fields
by their expansions in KK and Landau modes, one can show that the effective
action is [7]:

$\displaystyle\begin{split}S_{4}&=\int d^{4}x\left[\int
d^{4}\theta\left(\overline{\varphi}\varphi+\sum_{n,j}\left(\overline{Q}_{n,j}e^{2gqV_{0}}Q_{n,j}+\overline{\tilde{Q}}_{n,j}e^{-2gqV_{0}}\tilde{Q}_{n,j}\right)+2fV_{0}\right)\right.\\\
&+\left.\int
d^{2}\theta\left(\frac{1}{4}W^{\alpha}_{0}W_{\alpha,0}+\sum_{n,j}\left(-i\sqrt{-2gq(n+1)}\tilde{Q}_{n+1,j}Q_{n,j}+\sqrt{2}qg\tilde{Q}_{n,j}\varphi
Q_{n,j}\right)+h.c.\right)\right]\end{split}$ (2.19)

This action contains a superpotential with mass terms for the charged
superfields, and an interaction term coupling them to the Wilson lines. The
kinetic terms in the action gives kinetic terms for the charged fields, the 4D
gauge fields, the Wilson line, and a Fayet-Iliopoulos term $2fV_{0}$. In order
to obtain the physical mass spectrum and interactions terms for the component
fields, one has to eliminate the auxiliary fields. We denote the component
fields as follows.

$\displaystyle\begin{split}Q&=\mathcal{Q}+\sqrt{2}\theta\chi+\theta^{2}F\\\
V_{0}&=-\theta\sigma^{\mu}\overline{\theta}A_{\mu}+i\overline{\theta}\overline{\theta}\theta\lambda-i\overline{\theta}\theta\theta\overline{\lambda}+\frac{1}{2}\overline{\theta}^{2}\theta^{2}D\end{split}$
(2.20)

This leads to the effective Lagrangian for the on-shell component fields given
in section 3.2 of [7]. We will not give the full expression which is quite
tedious. However, it has the following form:

$\mathscr{L}_{eff}=\mathscr{L}_{kin}+\mathscr{L}_{M}+\mathscr{L}_{int}-\frac{1}{2}f^{2}$
(2.21)

and gives the following masses:

$\displaystyle\begin{split}m^{2}_{\tilde{\mathcal{Q}}_{n,j}}&=m^{2}_{\mathcal{Q}_{n,j}}=\frac{2\pi|N|}{L^{2}}(2n+1)\\\
m_{\Psi_{n,j}}^{2}&=\frac{4\pi|N|}{L^{2}}(2n+1)\end{split}$ (2.22)

where $\Psi_{n,j}$ is formed by chiral fermions paired up as:

$\Psi_{n,j}=\begin{pmatrix}\tilde{\chi}_{n+1,j}\\\ \chi_{n,j}\end{pmatrix}$
(2.23)

and the $\tilde{\chi}_{0,j}$ are the only charged massless fields. Moreover,
the interacting part of the Lagrangian contains interaction terms for the
Wilson line.

$\displaystyle\begin{split}\mathscr{L}_{int}&\supset-i\sqrt{2}gq\sum_{n,j}\sqrt{\alpha(n+1)}\varphi\left(\overline{\tilde{\mathcal{Q}}}_{n+1,j}\tilde{\mathcal{Q}}_{n,j}-\overline{\mathcal{Q}}_{n,j}\mathcal{Q}_{n+1,j}\right)+h.c.\\\
&-2q^{2}g^{2}\sum_{n,j}|\varphi|^{2}\left(|\mathcal{Q}_{n,j}|^{2}+|\mathcal{\mathcal{Q}}_{n,j}|^{2}\right)-\left(\sqrt{2}qg\varphi\tilde{\chi}_{n,j}\chi_{n,j}+h.c.\right)\end{split}$
(2.24)

where we introduced $\alpha=-2qgf$. These are the interactions that will
contribute to the quantum corrections to the Wilson line mass.

###  2.3 Non supersymmetric fermion with abelian flux background

In the next section, we will compute quantum corrections to the Wilson line
scalar mass and show that it vanishes at first order. Before that, we would
like to propose a simpler model than the one introduced above. As we will see,
the cancellation is independent of supersymmetry. Therefore, we introduce a
simple model to work without supersymmetry: A 6D fermion charged under an
abelian gauge theory with a magnetic flux in the background. This model was
presented in detail in [9].

Let us write the 6D Lagrangian for a single fermion with abelian gauge
symmetry.

$S_{6}=\int
d^{6}x\left(-\frac{1}{4}F^{MN}F_{MN}+i\overline{\Psi}\Gamma^{M}D_{M}\Psi\right)$
(2.25)

where $D_{M}=\partial_{M}+iqA_{M}$. Before moving on, we introduce some
conventions for our six-dimensional fermion. The 6D Weyl spinor is splitted
into two two-components Weyl spinors $\psi$ and $\chi$ with charge $q$ and
$-q$, respectively.

$\displaystyle\begin{split}\Psi=\begin{pmatrix}\psi_{L}\\\
\psi_{R}\end{pmatrix}\quad,\qquad\psi_{L}&=\begin{pmatrix}\psi\\\
0\end{pmatrix}\quad\psi_{R}=\begin{pmatrix}0\\\
\overline{\chi}\end{pmatrix}\\\
\gamma_{5}\psi_{L}=-\psi_{L}&\qquad\gamma_{5}\psi_{R}=\psi_{R}\end{split}$
(2.26)

For the gamma matrices $\Gamma^{M}$ we choose the following basis.

$\Gamma^{\mu}=\begin{pmatrix}\gamma^{\mu}&0\\\
0&\gamma^{\mu}\end{pmatrix}\qquad\Gamma^{5}=\begin{pmatrix}0&i\gamma_{5}\\\
i\gamma_{5}&0\end{pmatrix}\qquad\Gamma^{6}=\begin{pmatrix}0&-\gamma_{5}\\\
\gamma_{5}&0\end{pmatrix}$ (2.27)

These 6D gamma matrices satisfy the algebra
$\\{\Gamma_{M},\Gamma_{N}\\}=-2\eta_{MN}$. Developing the fermionic part of
the action (2.25), one find

$S_{6f}=\int
d^{6}x\left(-i\psi\sigma^{\mu}\overline{D}_{\mu}\overline{\psi}-i\chi\sigma^{\mu}D_{\mu}\overline{\chi}-\chi(\partial+\sqrt{2}q\phi)\psi-\overline{\chi}(\overline{\partial}+\sqrt{2}q\overline{\phi})\overline{\psi}\right)$
(2.28)

Similarly, the gauge part of the action is developed as follows.

$S_{6g}=\int
d^{6}x\left(-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-\partial^{\mu}\overline{\phi}\partial_{\mu}\phi-\frac{1}{4}(\partial\overline{\phi}+\overline{\partial}\phi)^{2}-\frac{1}{2}\overline{\partial}A^{\mu}\partial
A_{\mu}-\frac{i}{\sqrt{2}}\partial_{\mu}A^{\mu}(\partial\overline{\phi}-\overline{\partial\phi})\right)$
(2.29)

Again, we compactify on a torus $T^{2}$ with a quantized magnetic flux in the
background (2.3). To obtain the 4D effective action, a method similar to the
one used in the preceding sections is used. For simplicity, let us choose
here, without loss of generality, $qf>0$. The ladder operators then take the
form

$\displaystyle\begin{split}a_{+}^{\dagger}&=\frac{i}{\sqrt{2qf}}(\overline{\partial}-qfz)\qquad
a_{+}=\frac{i}{\sqrt{2qf}}(\partial+qf\overline{z})\\\
a_{-}^{\dagger}&=\frac{i}{\sqrt{2qf}}(\partial-qf\overline{z})\qquad
a_{-}=\frac{i}{\sqrt{2qf}}(\overline{\partial}+qfz)\end{split}$ (2.30)

and satisfy the relations
$[a_{\pm},a^{\dagger}_{\pm}]=1,\quad[a_{\pm},a_{\pm}]=[a_{\pm},a^{\dagger}_{\mp}]=0$.
One defines the orthonormal set of wave functions as

$\displaystyle\begin{split}a_{+}\xi_{0,j}=0&\qquad
a_{-}\overline{\xi}_{0,j}=0\\\
\xi_{n,j}=\frac{i^{n}}{\sqrt{n\\!}}(a_{+}^{\dagger})^{n}\xi_{0,j}&\qquad\overline{\xi}_{n,j}=\frac{i^{n}}{\sqrt{n\\!}}(a_{-}^{\dagger})^{n}\overline{\xi}_{0,j}\end{split}$
(2.31)

where $j$ takes values between $0$ and $|N|-1$. One can expand the Weyl
spinors with respect to these mode functions similarly to (2.12).

$\displaystyle\begin{split}\psi&=\sum_{n,j}\psi_{n,j}\xi_{n,j}\\\
\chi&=\sum_{n,j}\chi_{n,j}\overline{\xi}_{n,j}\end{split}$ (2.32)

As for the gauge fields, they do not feel the magnetic flux and thus expand
with respect to KK modes functions. An important remark is needed: The
fermionic zero modes are chiral. Indeed, starting from the Lagrangian (2.28),
one can write the Euler-Lagrange equations for the Weyl fermions. Let us
compute as an example the equation for $\psi$.

$\displaystyle\begin{split}\mathscr{L}_{f}&\supset-i\psi\sigma^{\mu}\partial_{\mu}\overline{\psi}-\chi(\partial+qf\overline{z})\psi\\\
\frac{\partial\mathscr{L}_{f}}{\partial\psi^{\alpha}}&=\partial^{\mu}\left(\frac{\partial\mathscr{L}_{f}}{\partial(\partial_{\mu}\psi^{\alpha})}\right)\Rightarrow-i\sigma^{\mu}\partial_{\mu}\overline{\psi}-i\sqrt{2qf}a_{-}^{\dagger}\chi=0\end{split}$
(2.33)

where we used $\phi=\frac{f}{\sqrt{2}}\overline{z}+\varphi$ and partial
integration to isolate $\psi$. By performing a similar calculation for
$\chi,\overline{\chi},\overline{\psi}$ and combining the coupled equations,
one can find

$\displaystyle\begin{split}\Box\psi-\mathcal{M}_{+}^{2}\psi&=0\\\
\Box\chi-\mathcal{M}_{-}^{2}\chi&=0\\\ \end{split}$ (2.34)

where

$\displaystyle\begin{split}\mathcal{M}_{+}^{2}&=2qfa_{+}^{\dagger}a_{+}\\\
\mathcal{M}_{-}^{2}&=2qf(a_{-}^{\dagger}a_{-}+1)\end{split}$ (2.35)

where we used the property
$\sigma^{\mu}\overline{\sigma}^{\nu}\partial_{\mu}\partial_{\nu}=\Box$. Hence,
there are $|N|$ left-handed fermionic zero modes.

Using the decomposition in Landau and KK modes, one can write the 4D effective
Lagrangian of the theory555Again, we only include the zero modes of the
uncharged fields $A_{\mu},\varphi$..

$\displaystyle\begin{split}S_{4}&=\int
d^{4}x\Bigg{[}-\partial^{\mu}\overline{\varphi}_{0}\partial_{\mu}\varphi_{0}+\sum_{n,j}\Bigg{(}i\overline{\psi}_{Lj}\gamma^{\mu}D_{\mu}\psi_{Lj}+i\overline{\Psi}_{n,j}\gamma^{\mu}D_{\mu}\Psi_{n,j}\Bigg{.}\Bigg{.}\\\
&+\sqrt{2qf(n+1)}\overline{\Psi}_{n,j}\Psi_{n,j}+\sqrt{2}q\varphi_{0}\left(\overline{\Psi}_{0,j}\frac{1-\gamma_{5}}{2}\psi_{Lj}+\overline{\Psi}_{n+1,j}\frac{\gamma_{5}-1}{2}\Psi_{n,j}\right)\\\
&+\Bigg{.}\Big{.}\sqrt{2}q\overline{\varphi_{0}}\left(\overline{\psi}_{Lj}\frac{1+\gamma_{5}}{2}\Psi_{0,j}+\overline{\Psi}_{n,j}\frac{1+\gamma_{5}}{2}\Psi_{n+1,j}\right)\Bigg{)}\Bigg{]}\end{split}$
(2.36)

where $\psi_{Lj}$ are $|N|$ left-handed fermions and $\Psi_{n,j}$ is an
infinite tower of massive Dirac fermions.

$\psi_{Lj}=\begin{pmatrix}\psi_{0,j}\\\
0\end{pmatrix}\qquad\Psi_{n,j}=\begin{pmatrix}\psi_{n+1,j}\\\
\overline{\chi}_{n,j}\end{pmatrix}$ (2.37)

Therefore, the model with magnetic flux contains massless chiral fermions and
the Wilson line scalar is massless. We will see later that the vanishing of
the Wilson line scalar mass is due to a continuous shift symmetry. Moreover,
the vacuum expectation value of the WL scalar does not give mass to the chiral
fermions, which would be necessary to construct a realistic model.

###  2.4 Quantum corrections

In this section, we will compute one-loop corrections to the Wilson line
scalar mass. We will begin by the quantum corrections without flux as an
example. Then, we will show that the quantum corrections in the case with flux
vanish in both supersymmetric and non-supersymmetric cases.

####  2.4.1 Quantum corrections without flux

Let us consider the action without flux found in [7]. Expressed in terms of
the component fields, it reads

$\displaystyle\begin{split}\mathscr{L}_{4}&\supset-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-\partial_{\mu}\overline{\varphi}\partial^{\mu}\varphi+\sum_{n,m}\Bigg{(}-D_{\mu}\overline{\mathcal{Q}}_{n,m}D^{\mu}\mathcal{Q}_{n,m}-|M_{n,m}+\sqrt{2}gq\varphi|^{2}\overline{\mathcal{Q}}_{n,m}\mathcal{Q}_{n,m}\Bigg{.}\\\
&\Bigg{.}-i\chi_{n,m}\sigma^{\mu}D_{\mu}\overline{\chi}_{n,m}-i\tilde{\chi}_{n,m}\sigma^{\mu}D_{\mu}\overline{\tilde{\chi}}_{n,m}+\left(M_{n,m}+\sqrt{2}gq\varphi\right)\tilde{\chi}_{n,m}\chi_{n,m}+h.c.\Bigg{)}\end{split}$
(2.38)

From this Lagrangian, one obtains the one-loop contributions to the $\varphi$
mass. We will separate the bosonic and fermionic contributions.

$\displaystyle\begin{split}\delta
m_{b,1}^{2}&=2g^{2}q^{2}\sum_{n,m}\int\frac{d^{4}k_{eu}}{(2\pi)^{4}}\left(\frac{1}{k_{eu}^{2}+|M_{n,m}|^{2}}-\frac{|M_{n,m}|^{2}}{(k_{eu}^{2}+|M_{n,m}|^{2})^{2}}\right)\\\
\delta
m_{f,1}^{2}&=-4g^{2}q^{2}\sum_{n,m}\int\frac{d^{4}k_{eu}}{(2\pi)^{4}}\frac{k_{eu}^{2}}{(k_{eu}^{2}+|M_{n,m}|^{2})^{2}}\\\
&{\Rightarrow\delta m_{1}^{2}=\delta m_{b,1}^{2}+\delta
m_{f,1}^{2}=0}\end{split}$ (2.39)

where $k_{eu}^{2}=\sum_{i}k^{2}$ is the euclidean norm squared of $k$.
Starting from now on, the euclidean notation will be understood in one-loop
integrals. The bosonic and fermionic contributions cancel each other, as
expected for a supersymmetric theory.

?figurename? 1: Bosonic (first two diagrams) and fermionic (last diagram)
contributions to the Wilson line mass without flux.

However, these quantum corrections are infinite sums of quadratically
divergent terms [14] and a consistent treatment asks for a regularization
scheme as well as renormalization conditions. This is discussed in [7]. It is
useful to use the Schwinger representation of propagators

$\frac{1}{p^{2}+m^{2}}=\int_{0}^{\infty}dte^{-t(p^{2}+m)}$ (2.40)

This allows us to rewrite the bosonic correction as

$\delta
m_{b,1}^{2}=\frac{g^{2}q^{2}}{\pi^{3}L^{2}}\sum_{r,s}\frac{1}{r^{2}+s^{2}}$
(2.41)

in which the full divergence is contained in the term $r=s=0$. A counterterm
is needed to eliminate this divergence. For the finite part, the latter result
has been compared to other works using different methods, as dimensional
regularization or the Wilson line effective potential.

####  2.4.2 Quantum corrections with flux in a supersymmetric model

Let us now compute the one-loop corrections for the Wilson line scalar mass
with flux compactification. We thus have to take into account the full tower
of Landau levels of bosons and fermions. Using the interaction Lagrangian
(2.24), one can compute the one loop correction to the scalar mass.

$\displaystyle\begin{split}\delta
m_{b,1}^{2}&=-4q^{2}g^{2}|N|\sum_{n}\int\frac{d^{4}k}{(2\pi)^{4}}\left(\frac{n}{k^{2}+\alpha(n+1/2)}-\frac{n+1}{k^{2}+\alpha(n+3/2)}\right)\\\
\delta
m_{f,1}^{2}&=4q^{2}g^{2}|N|\sum_{n}\int\frac{d^{4}k}{(2\pi)^{4}}\left(\frac{n}{k^{2}+\alpha
n}-\frac{n+1}{k^{2}+\alpha(n+1)}\right)\end{split}$ (2.42)

where $\alpha=-2qgf$. Using the Schwinger representation, one finds

$\displaystyle\begin{split}\delta
m_{b,1}^{2}&=-4q^{2}g^{2}|N|\sum_{n}\int_{0}^{\infty}dt\frac{1}{t^{2}}\left(ne^{-\alpha(n+1/2)t}-(n+1)e^{-\alpha(n+3/2)t}\right)=0\\\
\delta
m_{f,1}^{2}&=4q^{2}g^{2}|N|\sum_{n}\int_{0}^{\infty}dt\frac{1}{t^{2}}\left(ne^{-\alpha
nt}-(n+1)e^{-\alpha(n+1)t}\right)=0\end{split}$ (2.43)

?figurename? 2: Bosonic (first two diagrams) and fermionic (last diagram) one-
loop contributions to the Wilson line mass with flux.

where the cancellation is showed by using the Schwinger representation and
inverting the summation and integration after having integrated on momentum
space. Surprisingly, we have shown that both contributions to the correction
vanish individually. Therefore, the quantum correction vanishes as in the case
without flux but, in this case, the cancellation is not due to supersymmetry.

In [8], particular care is given to the regularization of the quantum
corrections. Indeed, the one-loop integrals are divergent and one needs to
perform a UV regularization consistent with the symmetry of the theory. The
dimensional regularization used ensures that the series of these integrals are
well defined. Furthermore, in the limit of removing the regulator, any
divergences of the series give the form of the corresponding counterterms.
Furthermore, using dimensional regularization ensures that all symmetries are
respected (including gauge symmetries). In the limit of removing the
regularization, i.e. $\epsilon=4-d\rightarrow 0$, $\delta m^{2}(q^{2})$ where
$q^{2}$ is the external momenta has a pole in $\epsilon$ dictating the form of
the counterterm.

$\delta
m_{1}^{2}(q^{2})\propto\alpha^{2}\left(-\frac{q^{2}}{\alpha}\left(\frac{1}{\epsilon}+\frac{C_{1}}{\alpha}\right)+C_{2}q^{2}+\mathcal{O}(q^{2}/\alpha^{3})\right)$
(2.44)

where $C_{1,2}$ are real constants. This counterterm implies the presence of a
ghost state of squared mass proportional to $\alpha=-2qgf$. After having
identified the counterterm, they set $q^{2}=0$ and find $\delta
m_{b,1}^{2}=\delta m_{f,1}^{2}=0$ as expected.

One would like to show that the quantum corrections to the Wilson line mass in
a non-supersymmetric model still vanish. As we will see in the next
subsection, this is precisely the case. This suggests that another symmetry,
due to the magnetic flux, should protect the scalar mass.

In [7], it is also shown that no $|\varphi|^{4}$-term is generated at one-loop
order. As predicted by [2], this suggests that the entire one-loop effective
potential vanishes.

####  2.4.3 Quantum corrections with flux in a non-supersymmetric model

Let us compute the quantum corrections in the non-supersymmetric model of
section 2.3. We will use the action (2.36) which gives two fermionic
contributions to the corrections to the Wilson line (WL) scalar mass. One
obtain from these diagrams the following correction.

$\displaystyle\begin{split}\delta
m_{1}^{2}&=-2q^{2}|N|\sum_{n}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{2k^{2}}{(k^{2}+2qfn)(k^{2}+2qf(n+1))}\\\
&=4q^{2}|N|\sum_{n}\int\frac{d^{4}k}{(2\pi)^{4}}\left(\frac{n}{k^{2}+2qfn}-\frac{n+1}{k^{2}+2qf(n+1)}\right)\\\
&=\frac{q^{2}}{4\pi^{2}}|N|\int_{0}^{\infty}dt\frac{1}{t^{2}}\sum_{n}\left(ne^{-2qnft}-(n+1)e^{-2qf(n+1)}\right)\\\
&=\frac{q^{2}}{4\pi^{2}}|N|\int_{0}^{\infty}dt\frac{1}{t^{2}}\left(\frac{e^{2qft}}{(e^{2qft}-1)^{2}}-\frac{e^{2qft}}{(e^{2qft}-1)^{2}}\right)\\\
&=0\end{split}$ (2.45)

?figurename? 3: One-loop contributions to the Wilson line mass with flux.

Hence, the cancellation of one-loop correction is indeed independent of
supersymmetry. So what is the origin of this cancellation? We will investigate
this question in the next section.

Cancellation of the one-loop corrections in a Yang-Mills SU(2) non-
supersymmetric model has been verified in [10]. They use a model similar to
the SU(2) one in [7]. Quantum corrections to the WL mass also vanish thanks to
the shift symmetry under the translation in extra space. In this extension,
the cancellation is non-trivial since they take into account the ghost
fields666Faddev-Popov ghosts are introduced into gauge quantum field theories
to maintain consistency of the path integral formulation. In Yang-Mills
theories, the ghosts are complex scalar fields but they anticommute like
fermions. Hence, we talk of non-physical particles. contributions.

####  2.4.4 Nonvanishing finite contributions to the scalar mass

The results above cannot be applied as it stands to the hierarchy problem.
Indeed, if one wants to identify the WL scalar with the Higgs boson of the
standard model, the WL scalar must be a pseudo-Nambu-Goldstone boson with
finite mass. Therefore, it is necessary to have some mechanism generating a
term breaking the translational symmetry in compactified space. In [11], the
possibility to generate finite scalar mass for the Higgs boson has been
studied. They generalize one-loop quantum corrections and derive the
conditions for the corrections to be nonvanishing and finite. Dimensional
regularization is used to treat UV divergences in loop integrals.

The model used contain a scalar $\Phi$ and a fermion field with SU(2) gauge
symmetry in six dimensions with flux compactification on a torus. A particular
attention is given to the following simple term that generate nonvanishing
correction.

$\mathscr{L}_{breaking}=\kappa(\overline{\varphi}+\varphi)\overline{\Phi}\Phi$
(2.46)

where $\kappa$ is a dimensionless coupling constant. This term gives the
following correction to the scalar mass.

$\delta m_{1}^{2}=\frac{|N|\ln{2}}{32\pi^{2}}\frac{\kappa^{2}}{L^{2}}$ (2.47)

and $\delta m_{1}^{2}=0$ is reproduced for $\kappa\rightarrow 0$. This Wilson
line scalar mass could considered as a Higgs mass. Indeed, even if the
compactification scale $1/L$ is the Planck scale, a realistic Higgs mass at
the TeV scale could be realized by the interaction term by some dynamics at
the TeV scale $\kappa/L$.

###  2.5 Underlying symmetry

In this section we will try to explain the cancellation of one-loop correction
to the WL mass. Let us begin by observing that the actions (2.29) and (2.28)
without magnetic flux are invariant under translation
$\delta_{T}=\epsilon\partial+\overline{\epsilon}\overline{\partial}$ on the
torus $T^{2}$, acting on $\chi,\psi,A^{\mu}$. By adding a magnetic flux in the
background, we explicitly break the translational symmetry because the
magnetic field depends on $x_{5},x_{6}$. This breaking of the invariance can
be compensated by a shift in the scalar fluctuation $\varphi$.

$\delta_{T}\varphi=(\epsilon\partial+\overline{\epsilon}\overline{\partial})\varphi+\frac{\overline{\epsilon}}{\sqrt{2}}f$
(2.48)

The generators of translations do not commute with the mass operators (2.35).
However, the mode functions $\psi_{n,j}$ and $\chi_{n,j}$ are eigenfunctions
of these operators. Therefore, the action of the translation on the mode
function cannot be trivial. Instead, one will see that all the tower is
reshuffled.

To construct a general transformation of the mode functions one can combine
the translation
$\delta_{T}=\epsilon\partial+\overline{\epsilon}\overline{\partial}$ with
another symmetry of the action. The action is invariant under the following
local transformation.

$\varphi_{\Lambda}=-\frac{1}{\sqrt{2}}\partial\Lambda\quad\psi_{\Lambda}=e^{q\Lambda}\psi\quad\chi_{\Lambda}=e^{-q\Lambda}\chi\quad\Lambda=f(\alpha\overline{z}-\overline{\alpha}z)$
(2.49)

where $\alpha$ is a complex parameter. These transformations where first
considered in [15]. For the WL, the transformation corresponds therefore to a
shift $\delta_{\Lambda}\varphi=\frac{\overline{\alpha}}{\sqrt{2}}f$. Combining
the two symmetries of the Lagrangian, one obtains the simple infinitesimal
transformation:

$\delta\psi=(\delta_{T}+\delta_{\Lambda})\psi=-i\sqrt{2qf}(\epsilon
a_{+}+\epsilon a_{+}^{\dagger})\psi$ (2.50)

Therefore, the transformation above reshuffle the different modes, connecting
neighbouring mode functions. Using equation (2.31), one can find explicitly
how the different mode functions are reshuffled.

$\displaystyle\begin{split}\delta\psi&=\sum_{n,j}\delta\psi_{n,j}\xi_{n,j}\\\
\delta\psi_{n,j}&:=\sqrt{2qf}(\epsilon\sqrt{n+1}\psi_{n+1,j}-\overline{\epsilon}\sqrt{n}\psi_{n-1,j})\end{split}$
(2.51)

Analogously,

$\displaystyle\begin{split}\delta\chi&=\sum_{n,j}\delta\chi_{n,j}\xi_{n,j}\\\
\delta\chi_{n,j}&:=\sqrt{2qf}(\overline{\epsilon}\sqrt{n+1}\chi_{n+1,j}-\epsilon\sqrt{n}\chi_{n-1,j})\end{split}$
(2.52)

The invariance of (2.36) under these transformation is explicitly verified in
[9]. As expected, the invariance is showed under the condition that $\varphi$
tranforms as

$\delta\varphi_{0}=\sqrt{2}\overline{\epsilon}f$ (2.53)

Indeed, the shift part of $\delta_{T}\varphi$ corresponds to the
transformation of $\varphi_{0}$ above while the translation part is
$\epsilon\partial+\overline{\epsilon}\overline{\partial}$. As explained in
[9], this part of the transformation corresponds to the massive part of the
scalar KK tower. This massive part of the uncharged field has to be considered
if one wants to write the full Lagrangian of the theory which contains new
interactions terms. Furthermore, the full Lagrangian has to be taken into
account to compute higher loop order corrections. For example, the Kaluza-
Klein modes of the vector fields appear in two-loop order correction to the WL
scalar mass. The transformation laws of the massive gauge fields mode
functions are computed in [9] and will be summarized in section 2.6. It
appears that the full 4D Lagrangian also has an exact symmetry under which the
scalar zero modes transform under the shift (2.53).

How does this shift symmetry explain the cancellation of the one-loop
correction to the scalar mass? Let us summarize what we did here.

We started by identifying two symmetries of the 6D actions, a translation, and
gauge invariance. From these, we identified a transformation law of the four-
dimensional fields which leaves the 4D effective action invariant. In
particular, the complex scalar zero-mode transforms with a shift which
prevents the generation of a mass term. Indeed, the mass of a scalar field is
generated by the presence of an extremum in the scalar potential. If the
Lagrangian is invariant under a shift of this scalar, this implies the absence
of a unique ground state and therefore the absence of mass. Therefore, the
shift symmetry only allows for derivative terms. Moreover, the fact that the
full action containing the full tower of vector and scalar preserves this
shift symmetry of $\varphi_{0}$ suggests that the scalar mass is protected at
two-loop order and higher. The calculation of two-loop order corrections has
been done in [16] and the cancellation has been shown up to this order.

Furthermore, the Wilson line scalar can be identified with the Nambu-Goldstone
boson of the translational symmetry on the torus [7, 16]. Indeed, the shift
symmetry (2.53) is precisely the definition of an NG boson. And one can see
that $\varphi_{0}$ is the NG boson corresponding to the breaking of the
translational symmetry spontaneously broken by the vacuum expectation value of
the internal components of the gauge field.

###  2.6 U(1)xU(1) gauge group

As mentioned in 2.4.4, there is a necessity to break the translational
symmetry to generate a finite mass of the TeV order for the Higgs boson.
Rather than adding ad-hoc interactions terms as [11], we would like to
investigate the possibility to generate non-vanishing finite corrections to
the scalar mass using only the fundamental structure of the theory. It has
been mentioned in [7] that in the case of a gauge group with several U(1)
gauge factors, the WL lines are the pseudo-Nambu-Goldstone bosons of the
translational symmetry.

Let us recall that the physical reason for the cancellation of quantum
corrections to the WL scalar mass in the U(1) case is that the WL can be seen
as the Nambu-Goldstone boson of the translational symmetry on the torus. In a
case where the gauge symmetry contains more than one $U(1)$, the situation
becomes more subtle. Contrary to the discussion in [7], let us discuss this
matter in the non-supersymmetric model of [9] and section 2.3, and let us
study the simplest case of a $U(1)_{1}\crossproduct U(1)_{2}$ gauge symmetry.
There are two gauge fields and therefore two Wilson Line scalars denoted
$\varphi^{1}$ and $\varphi^{2}$. The covariant derivative reads

$D_{M}=\partial_{M}+iq_{\alpha}A_{M}^{\alpha}$ (2.54)

where $q_{\alpha}$ is the charge of $\Psi$ associated to the gauge symmetry
$U(1)_{\alpha}$ and $A^{\alpha}_{M}$ is the gauge vector field associated with
$U(1)_{\alpha}$.

$A_{5}^{\alpha}=-\frac{1}{2}f^{\alpha}x_{6},\quad
A_{6}^{\alpha}=\frac{1}{2}f^{\alpha}x_{5},\qquad F_{56}^{\alpha}=f^{\alpha}$
(2.55)

One could easily include an arbitrary number of fermions families $N_{f}$. In
this case, the charge would take the form of a matrix $q_{i\alpha}$.

Following the same procedure as in section 2.5, one can show (equations (2.62)
to (2.65)) that under the condition that the Lagrangian transforms into a
total derivative, the Wilson lines transform as

$q_{\alpha}\delta\varphi_{0}^{\alpha}=\sqrt{2}q_{\alpha}\overline{\epsilon}f^{\alpha}$
(2.56)

This indicates that a field charged under two U(1) gauge symmetries should
feel an effective flux depending on the two charges and the flux of the gauge
groups $U(1)_{\alpha}$. Conversely, a field may not feel any flux even though
there is a flux in the background on the two gauge fields. For example, let us
consider two fermions doublets $\\{\Psi^{i}\\}_{i=1,2}$ with the following
charges.

$q_{i\alpha}=\begin{pmatrix}1&1\\\ -1&1\end{pmatrix}$ (2.57)

For the flux choice $f^{1}=f^{2}=f$, one has

$q_{i\alpha}\delta\varphi_{0}^{\alpha}=\sqrt{2}f\begin{pmatrix}2\overline{\epsilon}^{1}\\\
0\end{pmatrix}$ (2.58)

and therefore, the second fermion $\Psi^{2}$ does not feel any flux. Indeed,
if, $q_{i\alpha}f^{\alpha}=0$, the ladder operators (2.30) do not satisfy the
harmonic oscillator algebra and there are no longer Landau levels in the
spectrum of the theory.

Additionally, one can see that in this example, there is one Goldstone boson
corresponding to $\varphi_{0}^{1}+\varphi_{0}^{2}$ while
$\varphi_{0}^{1}-\varphi_{0}^{2}$ does not transform nonlinearly. Hence, there
is only one Goldstone boson. This is a general feature of this theory. There
are two translational symmetries in the extra space. One along $x_{5}$ and the
other along $x_{6}$. When at least one of the gauge fields contains a flux in
its background, both of these symmetries are broken and a complex NG boson
arises. There is no other symmetry to be broken and therefore any other
symmetry would be accidental and potentially broken by interactions terms.
Therefore one can have at most one Nambu-Goldstone boson and any other NG
boson would be a "pseudo-NG boson" with a non-vanishing mass.

The reason why we are interested in these pseudo-NG bosons is that even if
their mass are not zero, they should be small compared to a standard boson
mass. Let us verify that. One can do the calculations of section 2.3 with two
$U(1)$ instead of one and verify that the effective action has exactly the
same form as (2.36) with $q\varphi_{0}$ replaced by
$q_{\alpha}\varphi_{0}^{\alpha}$. One can even take an arbitrary number of
fermion families and check that the total action is simply a sum over
$i=1,2,\dots N_{f}$ of the action (2.36) for $\Psi^{i}$. The interactions
terms are:

$\displaystyle\begin{split}\mathscr{L}_{4}&\supset\sum_{i=1}^{N_{f}}\sum_{n\geq
0}\sum_{j=0}^{N_{i}-1}\Bigg{(}\sqrt{2}q_{i\alpha}\varphi_{0}^{\alpha}\left(\overline{\Psi}_{0,j}^{i}\frac{1-\gamma_{5}}{2}\psi_{Lj}^{i}+\overline{\Psi}^{i}_{n+1,j}\frac{\gamma_{5}-1}{2}\Psi^{i}_{n,j}\right)\Bigg{.}\\\
&\Bigg{.}+\sqrt{2}q_{i\alpha}\overline{\varphi_{0}}^{\alpha}\left(\overline{\psi}^{i}_{Lj}\frac{1+\gamma_{5}}{2}\Psi^{i}_{0,j}+\overline{\Psi}^{i}_{n,j}\frac{1+\gamma_{5}}{2}\Psi^{i}_{n+1,j}\right)\Bigg{)}\end{split}$
(2.59)

where $N_{i}$ is the quantization of the flux associated to the charge $i$.

$\frac{q_{i\alpha}}{2\pi}f^{\alpha}=N_{i}\in\mathbf{Z}$ (2.60)

With the interaction Lagrangian above, one can compute the one-loop correction
to the Wilson lines masses. Here, because of the multitude of Wilson lines,
one have to consider possible mixing between the two scalars. Indeed, there
are eight one-loop diagrams to consider.

?figurename? 4: One-loop contributions to the Wilson lines masses for
$N_{f}=1$.

One has for an arbitrary number of families $N_{f}$:

$\displaystyle\begin{split}(\delta
m_{1}^{2})_{\alpha\beta}&=-2\sum_{i=1}^{N_{f}}q_{i\alpha}q_{i\beta}|N_{i}|\sum_{n}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{2k^{2}}{(k^{2}+2q_{i\gamma}f^{\gamma}n)(k^{2}+2q_{i\nu}f^{\nu}(n+1))}\\\
&=4\sum_{i=1}^{N_{f}}q_{i\alpha}q_{i\beta}|N_{i}|\sum_{n}\int\frac{d^{4}k}{(2\pi)^{4}}\left(\frac{n}{k^{2}+2q_{i\gamma}f^{\gamma}n}-\frac{n+1}{k^{2}+2q_{i\nu}f^{\nu}(n+1)}\right)\\\
&=0\end{split}$ (2.61)

if $q_{i\alpha}f^{\alpha}\neq 0$. We used the fact that the series of
integrals above vanish as showed in equations (2.45), independently of
$q_{i\alpha}$. Thus, one-loop corrections of Wilson lines scalar masses always
vanish for non-vanishing effective flux.

Nonetheless, the scalar mass should be finite and non-vanishing in some cases.
Therefore, one expects that quantum corrections to the WL masses should be
finite in some cases beyond one-loop order. However, this result is
interesting because it confirms that even the pseudo-NG bosons masses are
protected from one-loop order corrections and one could hope to produce a
realistic mass in the TeV range. Further calculations at two-loop order would
be needed to compute the precise spectrum of Wilson lines.

If finite corrections are expected, one should explicitly observe symmetry
breaking terms in the Lagrangian. The transformations of fermions Landau modes
founded in [7] can be generalized to the $U(1)_{1}\crossproduct U(1)_{2}$ case
with $N_{f}$ families.

$\displaystyle\begin{split}\delta\psi_{n,j}^{i}&=\sqrt{2q_{i\alpha}f^{\alpha}}\left(\epsilon^{i}\sqrt{n+1}\psi^{i}_{n+1,j}-\overline{\epsilon}^{i}\sqrt{n}\psi^{i}_{n-1,j}\right)\\\
\delta\chi_{n,j}^{i}&=\sqrt{2q_{i\alpha}f^{\alpha}}\left(-\epsilon^{i}\sqrt{n}\chi^{i}_{n-1,j}+\overline{\epsilon}^{i}\sqrt{n+1}\chi^{i}_{n+1,j}\right)\end{split}$
(2.62)

As explained before, we expect the symmetry to be broken by interaction terms.
One can compute the transformation of the interaction terms of the complete
Lagrangian given in [9]. The variation of the mass term

$\displaystyle\begin{split}\delta\left(-\sum_{n,j}\sqrt{2q_{i\alpha}f^{\alpha}(n+1)}\chi^{i}_{n,j}\psi^{i}_{n+1,j}\right)&=-{2q_{i\alpha}f^{\alpha}}\sum_{n,j}\sqrt{n+1}\left(-\epsilon^{i}\sqrt{n}\chi^{i}_{n-1,j}\psi^{i}_{n+1,j}+\overline{\epsilon}^{i}\sqrt{n+1}\chi^{i}_{n+1,j}\psi^{i}_{n+1,j}\right.\\\
&\left.+\epsilon^{i}\sqrt{n+2}\chi_{n,j}^{i}\psi^{i}_{n+2,j}-\overline{\epsilon}^{i}\sqrt{n+1}\chi_{n,j}^{i}\psi_{n,j}^{i}\right)=2\overline{\epsilon}^{i}q_{i\alpha}f^{\alpha}\sum_{n,j}\chi^{i}_{n,j}\psi^{i}_{n,j}\end{split}$
(2.63)

which must compensate the variation of the Yukawa term

$\displaystyle\begin{split}\delta\left(-\sum_{n,j}\sqrt{2}q_{i\alpha}\varphi_{0}^{\alpha}\chi_{n,j}^{i}\psi^{i}_{n,j}\right)&=\sum_{n,j}\left(-\sqrt{2}q_{i\alpha}\delta(\varphi_{0}^{\alpha})\chi_{n,j}^{i}\psi^{i}_{n,j}-\sqrt{2}q_{i\alpha}\varphi_{0}^{\alpha}\delta(\chi_{n,j}^{i}\psi^{i}_{n,j})\right)\\\
&=-\sqrt{2}q_{i\alpha}\delta(\varphi_{0}^{\alpha})\sum_{n,j}\chi_{n,j}^{i}\psi^{i}_{n,j}\end{split}$
(2.64)

These equations imply the transformation law of the Wilson lines.

$q_{i\alpha}\delta\varphi_{0}^{\alpha}=\sqrt{2}\overline{\epsilon}^{i}q_{i\alpha}f^{\alpha}$
(2.65)

Thus, we have found the transformation law mentioned at the beginning of this
section. As stated then, one can see that this transformation cannot leave the
full Lagrangian invariant. However, the fact that the $N_{f}$ shift symmetries
are preserved by the Yukawa interaction explains the fact that the one-loop
correction vanishes. We will now show that the interaction term involved in
two-loop corrections will break these $N_{f}$ shift symmetries to only
preserve one particular shift transformation. Following the work [9], one can
write the transformation law of massive modes of gauge fields.

$\displaystyle\begin{split}\delta\varphi_{l,m}^{\alpha}&=\left(\epsilon^{\alpha}M_{l,m}-\overline{\epsilon}^{\alpha}\overline{M}_{l,m}\right)\varphi_{l,m}^{\alpha}\\\
\delta
A_{\mu;l,m}^{\alpha}&=\left(\epsilon^{\alpha}M_{l,m}-\overline{\epsilon}^{\alpha}\overline{M}_{l,m}\right)A_{\mu;l,m}^{\alpha}\end{split}$
(2.66)

where we have introduced two different parameters $\epsilon^{\alpha}$
corresponding to the transformation of the two sets of gauge fields. Let us
consider as an example an interaction term between massive gauge fields found
in [9] and extend it to the $U(1)\crossproduct U(1)$ case.

$\mathscr{L}_{4}\supset-\sqrt{2}\sum_{i=1}^{N_{f}}\sum_{l,m;n,j;n^{\prime},j^{\prime}}C_{n,j;n^{\prime},j^{\prime}}^{l,m}q_{i\alpha}\varphi_{l,m}^{\alpha}\chi_{n,j}^{i}\psi_{n,j}^{i}$
(2.67)

where

$C_{n,j;n^{\prime},j^{\prime}}^{l,m}=\int_{T^{2}}d^{2}xe^{zM_{l,m}-\overline{z}\overline{M}_{l,m}}\overline{\xi}_{n,j}\xi_{n^{\prime},j^{\prime}}.$
(2.68)

The transformation of this term under (2.66) gives

$\displaystyle\begin{split}&-\sqrt{2}\sum_{i=1}^{N_{f}}\sum_{l,m;n,j;n^{\prime},j^{\prime}}q_{i\alpha}\Bigg{[}-(\overline{\epsilon}^{\alpha}\overline{M}_{l,m}-\epsilon^{\alpha}M_{l,m})C_{n,j;n^{\prime},j^{\prime}}^{l,m}\Bigg{.}\\\
&+\sqrt{2q_{i\alpha}f^{\alpha}}\left(-\epsilon^{i}\sqrt{n+1}C_{n+1,j;n^{\prime},j^{\prime}}^{l,m}+\overline{\epsilon}^{i}\sqrt{n}C_{n-1,j;n^{\prime},j^{\prime}}^{l,m}\right)\\\
&\Bigg{.}+\sqrt{2q_{i\alpha}f^{\alpha}}\left(\epsilon^{i}\sqrt{n^{\prime}}C_{n,j;n^{\prime}-1,j^{\prime}}^{l,m}-\overline{\epsilon}^{i}\sqrt{n^{\prime}+1}C_{n,j;n^{\prime}+1,j^{\prime}}^{l,m}\right)\Bigg{]}\varphi_{l,m}^{\alpha}\chi_{n,j}^{i}\psi_{n,j}^{i}\\\
&=\sum_{i=1}^{N_{f}}\sum_{l,m;n,j;n^{\prime},j^{\prime}}\sqrt{2}q_{i\alpha}\big{(}(\overline{\epsilon}^{\alpha}-\overline{\epsilon}^{i})\overline{M}_{l,m}-(\epsilon^{\alpha}-\epsilon^{i})M_{l,m}\big{)}C_{n,j;n^{\prime},j^{\prime}}^{l,m}\varphi_{l,m}^{\alpha}\chi_{n,j}^{i}\psi_{n,j}^{i}\end{split}$
(2.69)

which is not zero in general. Indeed, only one particular case gives a zero
variation :

$\epsilon^{i}=\epsilon^{\alpha}\equiv\epsilon\qquad\forall i,\alpha$ (2.70)

Hence, there is only one parameter for the shift symmetry of the Wilson lines
: only one shift symmetry. The only linear combination of the Wilson lines
transforming with a shift and there being the Golstone boson is

$\frac{\varphi^{1}_{0}+\varphi_{0}^{2}}{\sqrt{2}}$ (2.71)

##  3 Six-dimensional Yang-Mills theories with flux compactification

###  3.1 Tachyons in 6D Yang-Mills theories with flux compactification

To construct an effective theory interesting from the phenomenological point
of view, it is inevitable to extend the model to Yang-Mills gauge theories. It
has been shown in [7] that the effective action for non-abelian flux contains
tachyons, i.e. fields with an imaginary mass. This was predicted by [2].
Indeed, the mass shift for the component of a multiplet living in six
dimensions with flux compactification is:

$\delta\mathcal{M}_{q}^{2}=(2n+1)|qgf|+2qgf\Sigma$ (3.1)

$\Sigma$ is the internal helicity which is equal to $\pm 1/2$ for fermions,
$0$ for scalars and $A^{\mu}$ the 4D components of the scalar field, and $\pm
1$ for $A_{5,6}$ the internal components of the vector field. Let us note
however that Bachas’s mass equation does not predict all the spectrum found in
[7] because some of the fields are absorbed vie the Stückelberg
mechanism777The Stückelberg mechanism is a particular case of the Brout-
Englert-Higgs mechanism where the Higgs mass has been sent to infinity..
Indeed, the flux background breaks the gauge symmetry and the charged vector
gauge fields acquire a mass. Thus they have to absorb some scalar field to
complete their longitudinal degree of freedom, and one expects to find a
Goldstone modes among the scalars of the theory.

Let us comment briefly on the presence of tachyons. The abelian flux
background of section 2.2 was perturbatively stable. Indeed, the perturbation
theory is done around the stable ground state. Therefore, all fields have a
non-negative mass. For a non-abelian flux background, some of the extra-
dimensional gauge fields become tachyonic. Hence, the effective action is due
to an expansion around a local maximal point rather than a ground state.
Therefore, one would be interested in investigating tachyon condensation to
find the ground state of the theory.

In this section, we will try to compute the scalar spectrum of a model with an
SU(2) gauge symmetry with flux compactification on a torus. As supersymmetry
does not play any role in our model, we will work in a non-supersymmetric
model. A priori, the presence of tachyons should not be due to supersymmetry.
Indeed, the Bachas formula (3.1) predicts the presence of tachyon without
considering supersymmetry. We consider the kinetic Lagrangian for a 6D vector
field.

$\mathscr{L}_{6}=-\frac{1}{2}\Tr{F_{MN}F^{MN}}$ (3.2)

where the fields are in the adjoint representation of $SU(2)$ and decomposed
along the basis of the SU(2) algebra which are the Pauli matrices
$\\{T_{1},T_{2},T_{3}\\}$ normalized such that $\Tr
T_{a}T_{b}=\frac{1}{2}\delta_{ab}$. For non abelian gauge theories,

$F_{MN}=\partial_{M}A_{N}-\partial_{N}A_{M}-ig[A_{M},A_{N}]$ (3.3)

To find the 4D effective action, we split the six dimensions into the four
classical dimensions and two extra dimensions compactified on a torus $T^{2}$.

$\displaystyle\begin{split}F_{56}&=\frac{1}{\sqrt{2}}(\partial\overline{\phi}+\overline{\partial}\phi-\sqrt{2}g[\phi,\overline{\phi}])\\\
F_{\mu 5}^{2}+F_{\mu 6}^{2}&=(F_{\mu 6}+iF_{\mu 5})(F^{\mu 6}-iF^{\mu 5})\\\
F_{\mu 6}+iF_{\mu 5}&=\sqrt{2}\partial_{\mu}\phi-i\partial
A_{\mu}-i\sqrt{2}g[A_{\mu},\phi]\end{split}$ (3.4)

One can rewrite (3.2) as

$\displaystyle\begin{split}\mathscr{L}_{6}&=-\frac{1}{2}\Tr{F_{\mu\nu}F^{\mu\nu}}\\\
&-2\Tr{\left(\partial_{\mu}\overline{\phi}-\frac{i}{\sqrt{2}}\overline{\partial}A_{\mu}-i\sqrt{2}g[A_{\mu},\overline{\phi}]\right)\left(\partial_{\mu}\phi-\frac{i}{\sqrt{2}}\partial
A_{\mu}-ig[A_{\mu},\phi]\right)}\\\
&-\frac{1}{2}\Tr{\left(\partial\overline{\phi}+\overline{\partial}\phi-\sqrt{2}g[\phi,\overline{\phi}]\right)\left(\partial\overline{\phi}+\overline{\partial}\phi-\sqrt{2}g[\phi,\overline{\phi}]\right)}\end{split}$
(3.5)

The mass term of the scalar is contained in the third line of the Lagrangian.

To reformulate the theory in terms of charged components of the gauge fields,
one defines a new basis of generators ${T_{+},T_{-},T_{3}}$ where

$T_{\pm}=T_{1}\pm iT_{2}.$ (3.6)

The basis elements satisfy:

$\displaystyle\begin{split}\Tr{T_{3}^{2}}=\frac{1}{2},\qquad\Tr{T_{+}T_{-}}=1,&\qquad\Tr{T_{\pm}T_{3}}=0,\qquad\Tr{T_{\pm}^{2}}=0,\\\
[T_{+},T_{-}]=2T_{3},&\qquad[T_{3},T_{\pm}]=\pm T_{\pm}\end{split}$ (3.7)

An arbitrary field $\phi$ in the adjoint representation is decomposed as

$\displaystyle\phi$
$\displaystyle=\phi_{3}T_{3}+\phi_{+}\frac{T_{-}}{\sqrt{2}}+\phi_{-}\frac{T_{+}}{\sqrt{2}}$
(3.8) $\displaystyle\overline{\phi}$
$\displaystyle=\overline{\phi}_{3}T_{3}+\overline{\phi}_{+}\frac{T_{+}}{\sqrt{2}}+\overline{\phi}_{-}\frac{T_{-}}{\sqrt{2}}$
(3.9)

And the vector fields are decomposed in the same way. The vector field being
real, one has $\overline{A}_{\mu}3=A_{\mu,3}$ and
$\overline{A}_{\mu\pm}=A_{\mu\mp}$. We choose the basis such that the flux
background is only encode in $\phi_{3}$.

$<\phi_{\pm}>=0\qquad<\phi_{3}>=\frac{f}{2\sqrt{2}}(x_{5}-ix_{6})$ (3.10)

The relation $[T_{3},T_{\pm}]=\pm T_{\pm}$ shows that the components
$\phi_{+},A_{+}^{\mu}$ and $\phi_{-},A^{\mu}_{-}$ have opposite charges with
respect to $\phi_{3},A_{3}^{\mu}$. Therefore, their field profile will be
similar to the field profile in the abelian case. Indeed, charge conservation
forbid the existence of quadratic terms (harmonic oscillator terms) mixing
fields of different charges. Therefore, the difference between the abelian and
non-abelian cases will lie only in the interacting part of the Lagrangian.

Let us compute the mass spectrum of the scalar fields.

$\displaystyle\begin{split}\mathscr{L}_{6}&\supset-\frac{1}{4}\left(\partial\overline{\phi}_{3}+\overline{\partial}\phi_{3}+\sqrt{2}g(|\phi_{+}|^{2}-|\phi_{-}|^{2})^{2}\right)^{2}\\\
&-\frac{1}{2}\left(\partial\overline{\phi}_{+}+\overline{\partial}\phi_{-}-\sqrt{2}g(\phi_{3}\overline{\phi}_{+}-\overline{\phi}_{3}\phi_{-})\right)\left(\partial\overline{\phi}_{-}+\overline{\partial}\phi_{+}+\sqrt{2}g(\phi_{3}\overline{\phi}_{-}-\overline{\phi}_{3}\phi_{+})\right)\\\
&=-\frac{1}{4}\left(\sqrt{2}f+\partial\overline{\varphi}_{3}+\overline{\partial}\varphi_{3}+\sqrt{2}g(|\phi_{+}|^{2}-|\phi_{-}|^{2})^{2}\right)^{2}\\\
&-\frac{1}{2}\left((-i\sqrt{2qf}a_{-}^{\dagger}-\sqrt{2}g\varphi_{3})\overline{\phi}_{+}+(-\sqrt{2qf}a_{-}+\sqrt{2}g\overline{\phi}_{3})\phi_{-}\right)\\\
&\crossproduct\left((-i\sqrt{2qf}a_{+}^{\dagger}-\sqrt{2}g\overline{\varphi}_{3}){\phi}_{+}+(-\sqrt{2qf}a_{+}+\sqrt{2}g{\phi}_{3})\overline{\phi}_{-}\right)\\\
&\supset-
gf\left(|\phi_{+}|^{2}-|\phi_{-}|^{2}\right)+gf\left(a_{-}^{\dagger}\overline{\phi}_{+}+a_{-}\phi_{-}\right)\left(a_{+}^{\dagger}\phi_{+}+a_{+}\overline{\phi}_{-}\right)\end{split}$
(3.11)

Integrating over $T^{2}$ and using (2.30-2.31), one can get the four
dimensional effective Lagrangian.

$\displaystyle\begin{split}\mathscr{L}_{4}^{gaugemass}&=-gf\sum_{n,j}\left(|\phi_{+;n,j}|^{2}-|\phi_{-;n,j}|^{2}\right)\\\
&-gf\sum_{n,j;n^{\prime},j^{\prime}}\int
dz^{2}\left(\sqrt{n+1}\overline{\phi}_{+;n,j}\overline{\xi}_{n+1,j}-\sqrt{n}\phi_{-;n,j}\overline{\xi}_{n-1,j}\right)\times\\\
&\times\left(\sqrt{n^{\prime}+1}\phi_{+;n^{\prime},j^{\prime}}\xi_{n^{\prime}+1,j^{\prime}}-\sqrt{n^{\prime}}\overline{\phi}_{-;n^{\prime},j^{\prime}}\xi_{n^{\prime}-1,j^{\prime}}\right)\end{split}$
(3.12)

Using the orthonormility relation (2.11), one can rewrite this Lagrangian as

$\mathscr{L}_{4}^{mass}=gf|\phi_{-;0,j}|^{2}-gf\sum_{n,j}\begin{pmatrix}\overline{\phi}_{+;n,j}&\phi_{-;n+2,j}\end{pmatrix}\begin{pmatrix}n+2&-\sqrt{(n+1)(n+2)}\\\
-\sqrt{(n+1)(n+2)}&n+1\end{pmatrix}\begin{pmatrix}\phi_{+;n,j}\\\
\overline{\phi}_{-;n+2,j}\end{pmatrix}$ (3.13)

which gives the scalar mass spectrum. Let us summarise the different scalar
states we found.

  1. 1.

$\phi_{-;0,j}$ is a tachyon with negative mass squared.

$m_{\phi^{-}_{0,j}}^{2}=-gf$ (3.14)

This confirms the result of [7] and the formula (3.1).

  2. 2.

$\phi_{-;1,j}$ is a massless scalar.

  3. 3.

One of the eigenstates of the scalar mass matrix is the tower of Goldstone
bosons

$\phi_{n,j}^{G}=\sqrt{\frac{n+1}{2n+3}}\phi_{+;n,j}+\sqrt{\frac{n+2}{2n+3}}\overline{\phi}_{-;n+2,j}$
(3.15)

  4. 4.

The other eigenstate corresponds to the following massive scalars.

$\phi_{n,j}^{M}=-\sqrt{\frac{n+2}{2n+3}}\phi_{+;n,j}+\sqrt{\frac{n+1}{2n+3}}\overline{\phi}_{-;n+2,j}$
(3.16)

and its mass is

$m_{\phi^{M}_{n,j}}^{2}=gf(2n+3)$ (3.17)

###  3.2 Eliminating the tachyons

As an alternative to tachyon condensation, we will propose in this section
another approach which is to eliminate the tachyons. Nonetheless, studying
tachyon condensations without eliminating them could lead to interesting
results, in particular in string theory. How could we eliminate the tachyon?
Here, we propose to bring a scalar boson $\Phi$ in the adjoint representation
of the gauge group SU(2) taking a non-zero vacuum expectation value. We expect
this vev to give a positive contribution to the masses of all gauge scalar
fields. Then, for sufficient values of the vev, we could eliminate the
tachyons by increasing their mass up to a positive value.

$\Phi=\Phi_{3}T_{3}+\Phi_{-}\frac{T_{+}}{\sqrt{2}}+\Phi_{+}\frac{T_{-}}{\sqrt{2}}$
(3.18)

The Lagrangian describing the scalar boson and the gauge fields is

$\mathscr{L}_{6}=-\frac{1}{2}\Tr
F_{MN}F^{MN}-(D_{M}\Phi)^{\dagger}D^{M}\Phi-V(\Phi)$ (3.19)

where $V(\Phi)$ is the scalar potential

$V(\Phi)=-m^{2}\Phi^{\dagger}\Phi+\lambda(\Phi^{\dagger}\Phi)^{2}$ (3.20)

and $D_{M}$ is the covariant derivative.

$D_{M}\Phi=\partial_{M}-ig[A_{M},\Phi]$ (3.21)

The minima of this potential is not 0 and give a finite vacuum expectation
value to the scalar boson. A rotation of the basis can be performed such that

$<\Phi_{3}>=v=\sqrt{\frac{m^{2}}{2\lambda}}$ (3.22)

In its matrix form, $\Phi$ reads

$\Phi=\frac{1}{2}\begin{pmatrix}\Phi_{3}&\sqrt{2}\Phi_{-}\\\
\sqrt{2}\Phi_{+}&-\Phi_{3}\end{pmatrix}$ (3.23)

The adjoint representation of $SU(2)$ is real, thus $\Phi^{\dagger}=\Phi$ and

$\Tr\Phi\Phi^{\dagger}=\frac{1}{2}\big{(}\Phi_{3}^{2}+2\Phi_{+}\Phi_{-}\big{)}$
(3.24)

Moreover,

$\vec{\Phi}\cdot\vec{\Phi}=\Phi_{1}^{2}+\Phi_{2}^{2}+\Phi_{3}^{2}=\Phi_{3}^{2}+2\Phi_{+}\Phi_{-}=2\Tr\Phi\Phi^{\dagger}$
(3.25)

such that

$\mathscr{L}_{6}=-\frac{1}{2}\Tr F_{MN}F^{MN}-2\Tr D_{M}\Phi
D^{M}\Phi\qquad\text{with}\quad<\Phi_{3}>=v\neq 0$ (3.26)

Let us compute the mass spectrum of the model. One can write the covariant
derivative of $\Phi$ in components in the basis $\\{T_{3},T_{+},T_{-}\\}$.
First, the commutator reads

$[A_{M},\Phi]=(A_{M}^{-}\Phi_{+}-A_{M}^{+}\Phi_{-})T_{3}+(A_{M}^{3}\Phi_{-}-A_{M}^{-}\Phi_{3})\frac{T_{+}}{\sqrt{2}}+(A_{M}^{+}\Phi_{3}-A_{M}^{3}\Phi_{+})\frac{T_{-}}{\sqrt{2}}$
(3.27)

which gives for the covariant derivative:

$\displaystyle\begin{split}D_{M}\Phi=&\left(\partial_{M}\Phi_{3}-ig(A_{M}^{-}\Phi_{+}-A_{M}^{+}\Phi_{-})\right)T_{3}+\left(\partial_{M}\Phi_{-}-ig(A_{M}^{3}\Phi_{-}-A_{M}^{-}\Phi_{3})\right)\frac{T_{+}}{\sqrt{2}}\\\
&+\left(\partial_{M}\Phi_{+}-ig(A_{M}^{+}\Phi_{3}-A_{M}^{3}\Phi_{+})\right)\frac{T_{-}}{\sqrt{2}}\end{split}$
(3.28)

From this decomposition, one can see that the vev of $\Phi_{3}$ gives indeed a
mass to the tachyons. Using the trace properties of $T_{3,+,-}$, one can
compute the traces in the Lagrangian (3.26) and find that

$\Tr D_{M}\Phi
D^{M}\Phi\supset\frac{1}{2}g^{2}\left(v^{2}A_{M}^{+}A^{M}_{-}+v^{2}A_{M}^{+}A^{M}_{-}\right)=g^{2}v^{2}A_{M}^{+}A^{M}_{-}$
(3.29)

Moreover,

$\displaystyle\begin{split}\varphi^{i}=\frac{f}{\sqrt{2}}(A_{6}^{i}+iA_{5}^{i})&\qquad\phi_{\pm}=\frac{1}{\sqrt{2}}(\phi^{1}\pm
i\phi^{2})\\\
\Rightarrow\phi_{\pm}=\frac{1}{\sqrt{2}}(A_{6}^{\pm}+iA_{5}^{\pm})&\qquad\overline{\phi}_{\pm}=\frac{1}{\sqrt{2}}(A_{6}^{\mp}-iA_{5}^{\mp})\end{split}$
(3.30)

which allows to express the extra components of the vector field decomposed in
the basis $T_{3,+,-}$ in terms of the higgs-gauge scalar field $\phi$
decomposed in the same basis.

$\displaystyle\begin{split}A_{6}^{+}=\frac{1}{\sqrt{2}}(\phi_{+}+\overline{\phi}_{-})&\qquad
A_{5}^{+}=\frac{1}{\sqrt{2}i}(\phi_{+}-\overline{\phi}_{-})\\\
A_{6}^{-}=\frac{1}{\sqrt{2}i}(\overline{\phi}_{+}+{\phi}_{-})&\qquad
A_{5}^{-}=\frac{1}{\sqrt{2}i}(\phi_{-}-\overline{\phi}_{+})\end{split}$ (3.31)

The trace (3.29) then reads

$\Tr D_{M}\Phi D^{M}\Phi\supset
g^{2}v^{2}(\overline{\phi}_{+}\phi_{+}+\overline{\phi}_{-}\phi_{-})$ (3.32)

We must be cautious with this result. Indeed, the term above is not the only
contribution to the scalar masses. Indeed, mixing terms f the type
$\Phi_{+}\phi_{-},\Phi_{-}\phi_{+}$ should appear because of the definition of
the covariant derivative. We expect to find a mass matrix mixing the states
$\phi_{+}$, $\phi_{-}$, and $\Phi_{\pm}$. Therefore, one should have in the
Lagrangian a 3 by 3 mass matrix including the 2 by 2 matrix of (3.13). We
expect the size of the matrix to be 3 and not 4 because the scalar field
$\Phi$ is real and thus

$\overline{\Phi}_{\pm}=\Phi_{\mp}$ (3.33)

Now that we have clarified this issue, let us compute the relevant terms in
the Lagrangian by including the mass terms of $\Phi$ and $\phi$.

$\displaystyle\begin{split}\Tr D_{M}\Phi
D^{M}\Phi&\supset\frac{1}{2}\left(\partial_{M}\Phi_{-}-ig(<A_{M}^{3}>\Phi_{-}-A_{M}^{-}<\Phi_{3}>)\right)\left(\partial^{M}\Phi_{+}-ig(A_{M}^{+}<\Phi_{3}>-<A_{M}^{3}>\Phi_{+})\right)\\\
&+\frac{1}{2}\left(\partial^{M}\Phi_{+}-ig(A_{M}^{+}<\Phi_{3}>-<A_{M}^{3}>\Phi_{+})\right)\left(\partial_{M}\Phi_{-}-ig(<A_{M}^{3}>\Phi_{-}-A_{M}^{-}<\Phi_{3}>)\right)\\\
&\supset\left(\partial_{5}\Phi_{-}-ig(<A_{5}^{3}>\Phi_{-}-A_{5}^{-}v)\right)\left(\partial_{5}\Phi_{+}-ig(A_{5}^{+}v-<A_{5}^{3}>\Phi_{+})\right)\\\
&+\left(\partial_{6}\Phi_{-}-ig(<A_{6}^{3}>\Phi_{-}-A_{6}^{-}v)\right)\left(\partial_{6}\Phi_{+}-ig(A_{6}^{+}v-<A_{6}^{3}>\Phi_{+})\right)\end{split}$
(3.34)

The expression above depends on $\partial_{5}$ and $\partial_{6}$. One would
like to introduce $\partial$ and $\overline{\partial}$ to obtain the ladder
operators that will act on the Landau levels of the charged fields. From
(2.8), one can find

$\partial_{5}=\frac{\partial+\overline{\partial}}{2}\qquad\partial_{6}=\frac{\overline{\partial}-\partial}{2i}$
(3.35)

Additionally, we will use (2.3,3.10) to write the third component of $A_{5}$
and $A_{6}$ as

$A_{5}=-\frac{f}{2i}(z-\overline{z})\qquad A_{6}=\frac{f}{2}(z+\overline{z})$
(3.36)

The mass terms in the Lagrangian (3.26) then read

$\displaystyle\begin{split}\mathscr{L}_{6}^{mass}=&-\frac{1}{2}\left((\partial+\overline{\partial})\Phi_{-}-ig\left(-\frac{f}{i}(z-\overline{z})\Phi_{-}+\frac{\sqrt{2}}{i}(\overline{\phi}_{+}-\phi_{-})v\right)\right)\times\\\
&\times\left((\partial+\overline{\partial})\Phi_{+}-ig(\frac{\sqrt{2}}{i}(\partial_{+}-\overline{\phi}_{-})v+\frac{f}{i}(z-\overline{z})\Phi_{+})\right)\\\
&-\frac{1}{2}\left(-i(\overline{\partial}-\partial)\Phi_{-}-ig\left(f(z+\overline{z})\Phi_{-}-\sqrt{2}(\overline{\phi}_{+}+\phi_{-})v\right)\right)\times\\\
&\times\left(-i(\overline{\partial}-\partial)\Phi_{+}-ig\left(\sqrt{2}v(\phi_{+}+\overline{\phi}_{-})-f(z+\overline{z})\Phi_{+}\right)\right)\end{split}$
(3.37)

We can reorganize the terms as follows.

$\displaystyle\begin{split}\mathscr{L}_{6}^{mass}=&-\frac{1}{2}\left((\partial-
gf\overline{z})\Phi_{-}+(\overline{\partial}+gfz)\Phi_{-}-\sqrt{2}gv(\overline{\phi}_{+}-\phi_{-})\right)\times\\\
&\times\left((\partial+gf\overline{z})\Phi_{+}+(\overline{\partial}-gfz)\Phi_{+}-\sqrt{2}gv({\phi}_{+}-\overline{\phi}_{-})\right)\\\
&+\frac{1}{2}\left((\partial-
gf\overline{z})\Phi_{-}-(\overline{\partial}+gfz)\Phi_{-}+\sqrt{2}gv(\overline{\phi}_{+}+\phi_{-})\right)\times\\\
&\times\left((\partial+gf\overline{z})\Phi_{+}-(\overline{\partial}-gfz)\Phi_{+}-\sqrt{2}gv({\phi}_{+}+\overline{\phi}_{-})\right)\end{split}$
(3.38)

We have explicitly highlighted the ladder operators (2.30) in the Lagrangian.
This allows us to write the mass terms for $\Phi$ in the Lagrangian in the
following compact form.

$\displaystyle\begin{split}\mathscr{L}_{6}^{scalarmass}=&-\frac{1}{2}\left(-i\sqrt{2gf}(a_{-}^{\dagger}+a_{-})\Phi_{-}-\sqrt{2}gv(\overline{\phi}_{+}-\phi_{-})\right)\left(-i\sqrt{2gf}(a_{+}+a_{+}^{\dagger})\Phi_{+}-\sqrt{2}gv({\phi}_{+}-\overline{\phi}_{-})\right)\\\
&+\frac{1}{2}\left(-i\sqrt{2gf}(a_{-}^{\dagger}-a_{-})\Phi_{-}+\sqrt{2}gv(\overline{\phi}_{+}+\phi_{-})\right)\left(-i\sqrt{2gf}(a_{+}-a_{+}^{\dagger})\Phi_{+}-\sqrt{2}gv({\phi}_{+}+\overline{\phi}_{-})\right)\\\
&=2gf\left(a_{-}^{\dagger}\Phi_{-}a_{+}^{\dagger}\Phi_{+}+a_{-}\Phi_{-}a_{+}\Phi_{+}\right)-2g^{2}v^{2}(\overline{\phi}_{+}\phi_{+}+\phi_{-}\overline{\phi}_{-})\\\
&+2i\sqrt{gf}gv\left(a_{-}^{\dagger}\Phi_{-}\overline{\phi}_{-}-a_{-}\Phi_{-}\phi_{+}+\phi_{-}a_{+}^{\dagger}\Phi_{+}-\overline{\phi}_{+}a_{+}\Phi_{+}\right)\end{split}$
(3.39)

As we expected, the Lagrangian contains mass terms (3.32) for the charged
gauge scalar fields, mass terms for the charged scalar fields $\Phi_{+}$ and
$\Phi_{-}$, and also mass terms mixing the two types of scalar fields. To this
contribution, one must add the gauge scalar mass terms (3.11). The complete
scalar masses Lagrangian is

$\displaystyle\begin{split}\mathscr{L}_{6}^{mass}&=2gf\left(a_{-}^{\dagger}\Phi_{-}a_{+}^{\dagger}\Phi_{+}+a_{-}\Phi_{-}a_{+}\Phi_{+}\right)+2i\sqrt{gf}gv\left(a_{-}^{\dagger}\Phi_{-}\overline{\phi}_{-}-a_{-}\Phi_{-}\phi_{+}+\phi_{-}a_{+}^{\dagger}\Phi_{+}-\overline{\phi}_{+}a_{+}\Phi_{+}\right)\\\
&-(2g^{2}v^{2}+gf)|\phi_{+}|^{2}-(2g^{2}v^{2}-gf)|\phi_{-}|^{2}+gf\left(a_{-}^{\dagger}\overline{\phi}_{+}+a_{-}\phi_{-}\right)\left(a_{+}^{\dagger}\phi_{+}+a_{+}\overline{\phi}_{-}\right)\end{split}$
(3.40)

Starting from this Lagrangian, one can compute the effective 4D Lagrangian by
decomposing the charged scalar fields $\Phi_{\pm}$ with respects to the Landau
mode functions (2.31).

$\displaystyle\begin{split}\Phi_{+}&=\sum_{n,j}\Phi_{+;n,j}\xi_{n,j}\\\
\Phi_{-}&=\sum_{n,j}\Phi_{-;n,j}\overline{\xi}_{n,j}\end{split}$ (3.41)

Integrating these definitions into the Lagrangian (3.40) and using the
definition of Landau mode functions (2.31) allows to make the ladder operators
act on mode function.

$\displaystyle\begin{split}\mathscr{L}_{6}^{mass}&=\sum_{n,j;n^{\prime},j^{\prime}}2gf\left(-i\sqrt{n+1}\Phi_{-;n,j}\overline{\xi}_{n+1,j}\times(-i\sqrt{n^{\prime}+1})\Phi_{+;n^{\prime},j^{\prime}}\xi_{n^{\prime}+1,j^{\prime}}\right.\\\
&\left.+i\sqrt{n}\Phi_{-;n,j}\overline{\xi}_{n-1,j}\times
i\sqrt{n^{\prime}}\Phi_{+;n^{\prime},j^{\prime}}\overline{\xi}_{n^{\prime}-1,j^{\prime}}\right)\\\
&+2i\sqrt{gf}gv\sum_{n,j;n^{\prime},j^{\prime}}\Big{(}-i\sqrt{n+1}\Phi_{-;n,j}\overline{\xi}_{n+1,j}\overline{\phi}_{-;n^{\prime},j^{\prime}}\xi_{n^{\prime},j^{\prime}}-i\sqrt{n}\Phi_{-;n,j}\overline{\xi}_{n-1,j}{\phi}_{+;n^{\prime},j^{\prime}}\xi_{n^{\prime},j^{\prime}}\Big{.}\\\
&\Big{.}-i\sqrt{n+1}{\phi}_{-;n^{\prime},j^{\prime}}\overline{\xi}_{n^{\prime},j^{\prime}}{\Phi}_{+;n,j}\xi_{n+1,j}-i\sqrt{n}\overline{\phi}_{+;n^{\prime},j^{\prime}}\overline{\xi}_{n^{\prime},j^{\prime}}{\phi}_{+;n,j}\xi_{n-1,j}\Big{)}\\\
&-\sum_{n,j;n^{\prime},j^{\prime}}\Big{(}(2g^{2}v^{2}+gf)\overline{\phi}_{+;n,j}\overline{\xi}_{n,j}\phi_{+;n^{\prime},j^{\prime}}\xi_{n^{\prime},j^{\prime}}+(2g^{2}v^{2}-gf){\phi}_{-;n,j}\overline{\xi}_{n,j}\overline{\phi}_{+;n^{\prime},j^{\prime}}\xi_{n^{\prime},j^{\prime}}\Big{)}\\\
&-gf\sum_{n,j;n^{\prime},j^{\prime}}\Big{(}\sqrt{n+1}\overline{\phi}_{+;n,j}\overline{\xi}_{n+1,j}-\sqrt{n}\phi_{-;n,j}\overline{\xi}_{n-1,j}\Big{)}\times\\\
&\times\left(\sqrt{n^{\prime}+1}\phi_{+;n^{\prime},j^{\prime}}\xi_{n^{\prime}+1,j^{\prime}}-\sqrt{n^{\prime}}\overline{\phi}_{-;n^{\prime},j^{\prime}}\xi_{n^{\prime}-1,j^{\prime}}\right)\end{split}$
(3.42)

The orthonormality relation (2.11) can be applied and one can get after
integration on $T^{2}$:

$\displaystyle\begin{split}\mathscr{L}_{4}^{mass}&=-2gf\sum_{n,j}\left(n\Phi_{-;n-1,j}\Phi_{+;n-1,j}+(n+1)\Phi_{-;n+1,j}\Phi_{+;n+1,j}\right)\\\
&+2\sqrt{gf}gv\sum_{n,j}\left(\sqrt{n}\Phi_{-;n-1,j}\overline{\phi}_{-;n,j}+\sqrt{n+1}\Phi_{-;n+1,j}\phi_{+;n,j}+\sqrt{n}\phi_{-;n,j}\Phi_{+;n-1,j}+\sqrt{n+1}\overline{\phi}_{+;n,j}\Phi_{+;n+1,j}\right)\\\
&-(2g^{2}v^{2}+gf)|\phi_{+;n,j}|^{2}-(2g^{2}v^{2}-gf)|\phi_{-;n,j}|^{2}-gf\sum_{n,j}\left(\sqrt{n}\overline{\phi}_{+;n-1,j}-\sqrt{n+1}\phi_{-;n+1,j}\right)\times\\\
&\times\left(\sqrt{n}\phi_{+;n-1,j}-\sqrt{n+1}\overline{\phi}_{-;n+1,j}\right)\end{split}$
(3.43)

Let us perform some translations in the Landau levels.

$\displaystyle\begin{split}\mathscr{L}_{4}^{mass}&=-2gf\sum_{n,j}\left((2n+3)\Phi_{-;n+1,j}\Phi_{+;n+1,j}+\Phi_{-;0,j}\Phi_{+;0,j}\right)+2\sqrt{gf}gv\sum_{n,j}\left(\Phi_{-;0,j}\overline{\phi}_{-;1,j}+\phi_{-;1,j}\Phi_{+;0,j}\right.\\\
&\left.+\sqrt{n+2}\Phi_{-;n+1,j}\overline{\phi}_{-;n+2,j}+\sqrt{n+1}\Phi_{-;n+1,j}\phi_{+;n,j}+\sqrt{n+2}\phi_{-;n+2,j}\Phi_{+;n+1,j}+\sqrt{n+1}\overline{\phi}_{+;n,j}\Phi_{+;n+1,j}\right)\\\
&-(2g^{2}v^{2}+gf)|\phi_{+;n,j}|^{2}-(2g^{2}v^{2}-gf)(|\phi_{-;n+2,j}|^{2}+|\phi_{-;0,j}|^{2}+|\phi_{-;1,j}|^{2})\\\
&-gf\sum_{n,j}\left(\sqrt{n+1}\overline{\phi}_{+;n,j}-\sqrt{n+2}\phi_{-;n+2,j}\right)\left(\sqrt{n+1}\phi_{+;n,j}-\sqrt{n+2}\overline{\phi}_{-;n+2,j}\right)-gf\sum_{j}|\phi_{-;0,j}|^{2}\end{split}$
(3.44)

We can now rewrite this Lagrangian in a more practical way. The matrix form of
the Lagrangian will allow us to compute the mass spectrum and the expression
of the different eigenstates.

$\displaystyle\begin{split}\mathscr{L}_{4}^{mass}&=-\sum_{n,j}\begin{pmatrix}\overline{\phi}_{+;n,j}&\phi_{-;n+2,j}&\Phi_{-;n+1,j}\end{pmatrix}\times\\\
&\times\begin{pmatrix}gf(n+2)+2g^{2}v^{2}&-gf\sqrt{(n+1)(n+2)}&-2gv\sqrt{gf(n+1)}\\\
-gf\sqrt{(n+1)(n+2)}&gf(n+1)+2g^{2}v^{2}&-2gv\sqrt{gf(n+2)}\\\
-2gv\sqrt{gf(n+1)}&-2gv\sqrt{gf(n+2)}&2gf(2n+3)\end{pmatrix}\begin{pmatrix}\phi_{+;n,j}\\\
\overline{\phi}_{-;n+2,j}\\\ \Phi_{+;n+1,j}\end{pmatrix}\\\
&-2\sum_{j}\begin{pmatrix}\phi_{-;1,j}&\Phi_{-;0,j}\end{pmatrix}\begin{pmatrix}g^{2}v^{2}&-gv\sqrt{gf}\\\
-gv\sqrt{gf}&gf\end{pmatrix}\begin{pmatrix}\overline{\phi}_{-;1,j}\\\
\Phi_{+;0,j}\end{pmatrix}-(2g^{2}v^{2}-gf)|\phi_{-;0,j}|^{2}\end{split}$
(3.45)

At first sight, one can see that some appropriate value of the vev could
eliminate the tachyon (3.14). Moreover the Lagrangian contains two matrices
that need to be studied in more depth. To simplify the calculation, one can
write the 3x3 matrix as follows.

$\mathcal{M}_{3x3}^{2}=\begin{pmatrix}b^{2}+2c^{2}&-ab&-2ac\\\
-ab&a^{2}+2c^{2}&-2bc\\\ -2ac&-2bc&2(a^{2}+b^{2})\end{pmatrix}$ (3.46)

whose determinant is zero, as expected. Here,

$a=\sqrt{gf(n+2)}\qquad b=\sqrt{gf(n+1)}\qquad c=gv$ (3.47)

Let us summarize the resulting scalar mass spectrum.

  1. 1.

The 3x3 matrix and the 2x2 matrices both have a massless eigenstate.

$\displaystyle\begin{split}\Phi^{G,1}_{n,j}&=\sqrt{\frac{n+1}{2n+3+v/f}}\phi_{+;n,j}+\sqrt{\frac{n+2}{2n+3+v/f}}\overline{\phi}_{-;n+2,j}+\sqrt{\frac{v/f}{2n+3+v/f}}\Phi_{+;n+1,j}\\\
\Phi^{G,2}_{j}&=\sqrt{\frac{1}{1+gv^{2}/f}}\overline{\phi}_{-;1,j}+\sqrt{\frac{1}{1+f/gv^{2}}}\Phi_{+;0,j}\end{split}$
(3.48)

  2. 2.

The 3x3 matrix have two other eigenstates which are massive and the 2x2 matrix
have a second eigenstate which is also massive.

$\displaystyle\begin{split}\Phi^{M,1}_{n,j}&=-\sqrt{\frac{v}{f}\frac{n+1}{(2n+3+v/f)(2n+3)}}\phi_{+;n,j}\\\
&-\sqrt{\frac{v}{f}\frac{n+2}{(2n+3+v/f)(2n+3)}}\overline{\phi}_{-;n+2,j}+\sqrt{\frac{2n+3}{2n+3+v/f}}\Phi_{+;n+1,j}\\\
\Phi^{M,2}_{n,j}&=-\sqrt{\frac{n+2}{2n+3}}\phi_{+;n,j}+\sqrt{\frac{n+1}{2n+3}}\overline{\phi}_{-;n+2,j}\\\
\Phi^{M,3}_{j}&=-\sqrt{\frac{1}{1+f/gv^{2}}}\overline{\phi}_{-;1,j}+\sqrt{\frac{1}{1+gv^{2}/f}}\Phi_{+;0,j}\end{split}$
(3.49)

and their masses are

$\displaystyle\begin{split}(m_{\Phi^{M,1}_{n,j}})^{2}&=2(gf(2n+3)+g^{2}v^{2})\\\
(m_{\Phi^{M,2}_{n,j}})^{2}&=gf(2n+3)+2g^{2}v^{2}\\\
(m_{\Phi^{M,3}_{j}})^{2}&=2(gf+g^{2}v^{2})\end{split}$ (3.50)

  3. 3.

The last scalar is the one that was a tachyon before the addition of the
scalar $\Phi$. As we hoped, the addition of $\Phi$ cures the problem by adding
a positive value to the mass. Indeed, $\phi_{-;0,j}$ is not a tachyon if

$2gv^{2}\geq f$ (3.51)

We have shown that by adding a scalar in the adjoint representation, the
tachyons can be eliminated. Furthermore, for sufficiently high values of the
vev of this scalar, the non-abelian flux background is perturbatively stable.
In other words, none of the fields is tachyonic and the effective action is
due to an expansion around the true ground state. In addition, let us note
that the elimination of the tachyon depends on the physical parameters of the
model. For fixed magnetic flux and vev, we can see that the presence of
tachyon depends on the gauge coupling. In particular, one may wonder what
happens when one renormalize the theory to energy scale below the limit
$g=\frac{f}{2v^{2}}$. Let us note in parallel that the presence of the scalar
field does not change the fact that the Wilson line $\varphi_{3}$ is still
massless in this model. The Lagrangian terms for $\Phi$ do not give rise to a
$\phi_{3}$ mass term.

###  3.3 Chiral fermions in the SU(2) model without tachyon

In this last section, we would like to add fermions into the model. Indeed, if
we wants to use the Wilson line as a Higgs field, it is important to verify
that fermions in this model can be chiral. Let us add a $SU(2)$ doublet in the
model. The covariant derivative for this doublet is

$D_{M}=\partial_{M}+igqA_{M}$ (3.52)

and the complete Lagrangian must contains a covariant derivative term for the
gauge field, the fermion doublet and the scalar triplet, a scalar potential,
and an interaction term between the scalar field and the fermion doublet
(Yukawa term).

$\mathscr{L}_{6}=-\frac{1}{2}\Tr
F_{MN}F^{MN}+i\overline{\Psi}\Gamma^{M}\left(\partial_{M}+igqT^{a}A^{a}_{M}\right)\Psi-(D_{M}\Phi)^{\dagger}D^{M}\Phi-V(\Phi)+\overline{\Psi}T^{a}\Psi\Phi^{a}$
(3.53)

where

$\Psi=\begin{pmatrix}\Psi^{1}\\\
\Psi^{2}\end{pmatrix}\qquad\Psi^{i}=\begin{pmatrix}\psi^{i}\\\ 0\\\ 0\\\
\overline{\chi}^{i}\end{pmatrix}$ (3.54)

and

$A_{M}=T^{a}A^{a}_{M}=\frac{1}{2}\begin{pmatrix}A_{M}^{3}&\sqrt{2}A_{M}^{-}\\\
\sqrt{2}A_{M}^{+}&-A_{M}^{3}\end{pmatrix}$ (3.55)

Let us assign a charge $q_{1}$ and $q_{2}$ to the fermions of the doublet. We
have already calculated in section 2.3 that the fermionic part of the
Lagrangian reads (2.28) which combine classical 4D fermionic terms and the
following internal components terms.

$\mathscr{L}_{6f}\supset-\chi(\partial+\sqrt{2}gq\phi)\psi-\overline{\chi}(\overline{\partial}+\sqrt{2}gq\overline{\phi})\overline{\psi}$
(3.56)

where $\phi=T^{a}\phi^{a}$. Using the matrix form (3.55), one finds

$\mathscr{L}_{6f}\supset-\begin{pmatrix}\chi^{1}&\chi^{2}\end{pmatrix}\begin{pmatrix}\partial+\sqrt{2}gq\phi_{3}&2gq\phi_{i}\\\
2gq\phi_{+}&\partial-\sqrt{2}gq\phi_{3}\end{pmatrix}\begin{pmatrix}\psi^{1}\\\
\psi^{2}\end{pmatrix}-\begin{pmatrix}\overline{\chi}^{1}&\overline{\chi}^{2}\end{pmatrix}\begin{pmatrix}\overline{\partial}+\sqrt{2}gq\overline{\phi}_{3}&2gq\overline{\phi}_{+}\\\
2gq\overline{\phi}_{-}&\overline{\partial}-\sqrt{2}gq\overline{\phi}_{3}\end{pmatrix}\begin{pmatrix}\overline{\psi}^{1}\\\
\overline{\psi}^{2}\end{pmatrix}$ (3.57)

Let us develop these matrices multiplications in components.

$\displaystyle\begin{split}\mathscr{L}_{6f}\supset&-\chi^{1}(\partial+\sqrt{2}gq_{1}\phi_{3})\psi^{1}-\chi^{2}(\partial-\sqrt{2}gq_{2}\phi_{3})\psi^{2}-\overline{\chi}^{1}(\overline{\partial}+\sqrt{2}gq_{1}\overline{\phi}_{3})\overline{\psi}^{1}-\overline{\chi}^{2}(\overline{\partial}-\sqrt{2}gq_{2}\overline{\phi}_{3})\overline{\psi}^{2}\\\
&-2g\left(q_{1}\chi^{2}\phi^{+}\psi^{1}+q_{2}\chi^{1}\phi_{-}\psi^{2}+q_{1}\overline{\chi}^{2}\overline{\phi}_{-}\overline{\psi}^{1}+q_{2}\overline{\chi}^{1}\overline{\phi}_{+}\overline{\psi}^{2}\right)\end{split}$
(3.58)

This Lagrangian contains interactions terms between the charged fields and
mass terms for the fermion fields. Using the annihilation and creation
operators, we obtain the mass-squared operators for fermions. For
$q_{1}=-q_{2}=1/2$, one has

$\displaystyle\begin{split}\mathcal{M}^{2}_{\psi^{1}}&=gfa_{+}^{\dagger}a_{+}\\\
\mathcal{M}^{2}_{\psi^{2}}&=gf(a_{-}^{\dagger}a_{-}+1)\\\
\mathcal{M}^{2}_{\chi^{1}}&=gfa_{+}^{\dagger}a_{+}\\\
\mathcal{M}^{2}_{\chi^{2}}&=gf(a_{-}^{\dagger}a_{-}+1)\\\ \end{split}$ (3.59)

One has to take into account the Yukawa terms. We denote the Yukawa coupling
constant $h$. The Yukawa terms takes the form

$\displaystyle\begin{split}\mathscr{L}_{Yuk}&=h\overline{\Psi}\Phi\Psi=\frac{1}{2}\begin{pmatrix}\overline{\Psi}^{1}&\overline{\Psi}^{2}\end{pmatrix}\begin{pmatrix}\Phi_{3}&\sqrt{2}\Phi_{-}\\\
\sqrt{2}\Phi_{+}&-\Phi_{3}\end{pmatrix}\begin{pmatrix}\Psi^{1}\\\
\Psi^{2}\end{pmatrix}\\\ &=0\end{split}$ (3.60)

Indeed, for chiral 6D fermions, the Lorentz invariant $\overline{\Psi}\Psi$ is
zero. Therefore, there are no Yukawa terms in our simple model containing only
one doublet.

Let us us compute the spectrum to verify that it satisfy the Bachas formula
(3.1).

$\displaystyle\begin{split}\mathscr{L}_{6}^{fermion-
mass}&=-\chi^{1}(\partial+gqf\overline{z})\psi^{1}-\chi^{2}(\partial+gqf\overline{z})\psi^{2}-\overline{\chi}^{1}(\overline{\partial}+gfz)\overline{\psi}^{1}-\overline{\chi}^{2}(\overline{\partial}+gqfz)\overline{\psi}^{2}\\\
&=i\sqrt{2gqf}\sum_{n,j;n^{\prime},j^{\prime}}\Big{[}\left(\chi^{1}_{n,j}a_{+}\psi^{1}_{n^{\prime},j^{\prime}}+\overline{\chi}_{n,j}^{2}a_{-}\overline{\psi}_{n^{\prime},j^{\prime}}^{2}\right)\overline{\xi}_{n,j}\xi_{n^{\prime},j^{\prime}}+\left(\chi^{2}_{n,j}a_{+}\psi_{n^{\prime},j^{\prime}}^{2}+\overline{\chi}_{n,j}^{1}a_{-}\overline{\psi}_{n^{\prime},j^{\prime}}^{1}\right)\xi_{n,j}\overline{\xi}_{n^{\prime},j^{\prime}}\Big{]}\\\
&=-\sqrt{2gqf(n+1)}\sum_{n,j}\Big{[}\chi^{1}_{n,j}\psi^{1}_{n+1,j}+\overline{\chi}^{2}_{n,j}\overline{\psi}^{2}_{n+1,j}+\chi^{2}_{n,j}\psi^{2}_{n+1,j}+\overline{\chi}^{1}_{n,j}\overline{\psi}^{1}_{n+1,j}\Big{]}\end{split}$
(3.61)

This confirm the Bachas formula (3.1) in the case of fermions which predicted
$\delta\mathcal{M}^{2}_{fermions}=gf(n+1/2\pm 1/2)$. Indeed, the Lagrangian
below implies

$\displaystyle\begin{split}m^{2}_{\psi_{n,j}}&=gfn\\\
m^{2}_{\chi_{n,j}}&=gf(n+1)\end{split}$ (3.62)

And therefore the fermions are chiral, as one may have already see with
equations (LABEL:fermions_mass_SU2). In particular, the left-handed zero modes
are massless while the right-handed zero modes are massive. This is an
important feature that the scalar $\Phi$ preserve the chirality of fermions
since chiral fermions are necessary to phenomenology.

##  4 Conclusion and Outlook

In this work, we aimed to continue the work undertaken by [7, 9] to use the
gauge vector field of a six-dimensional model with flux compactification on a
torus to obtain a massless scalar boson.

First, we have explained that the extra component of the gauge fields with a
magnetic flux in the background forms a complex scalar field - the Wilson line
- which is massless independently of supersymmetry. Indeed, contrary to the
case without flux where the Wilson line receives quantum corrections depending
on the volume of the torus, the Wilson line scalar is protected from one-loop
corrections in U(1) and SU(2) gauge symmetries. However, the SU(2) model
contains a finite number of tachyons which encourages investigating the nature
of the true ground state by tachyon condensation or to eliminate these
tachyons.

Furthermore, we recalled that it has been shown that the Wilson line scalar
mass is expected to vanish at all-loop order. Indeed, the Lagrangian of the
model is invariant under a shift symmetry of the Wilson line. Therefore, the
massless scalar is the Goldstone boson of a translational symmetry in the
internal space.

Since one would like to construct a realistic model of the Higgs boson, we
investigated the possibility to obtain a non-vanishing mass protected from
large corrections. For this purpose, we added a gauge symmetry to the original
one and proved that there can be at most only one Nambu-Goldstone boson. The
two gauge fields produce two Wilson lines which have a shift symmetry leaving
the Yukawa term of the Lagrangian invariant. However, only one particular
combination of these two shift symmetries leaves the other interactions terms
invariant. Therefore, when the effective fluxes felt by the fermions do not
vanish, both of the Wilson lines are pseudo-Nambu-Goldstone bosons with non-
vanishing two-loop mass correction. This opens the possibility to interpret
the Pseudo scalars as Higgs bosons with finite mass protected from large
quantum corrections. However, we must make an important comment. The Wilson
lines we are talking about here are uncharged under the gauge symmetry groups
while the Higgs boson of the Standard Model is a charged SU(2) doublet. Thus,
one would need a Wilson line produced by the U(1) gauge group that would be a
doublet of SU(2) (the one of the SM) while preserving the shift symmetry of
$U(1)$.

Our second aim was to investigate the possibility to eliminate the tachyons
present in the model with an SU(2) gauge symmetry. We first showed that the
presence of tachyons was independent of supersymmetry. Then, we added a scalar
boson in the adjoint representation of SU(2) and computed the complete charged
scalar mass spectrum. By diagonalizing the mass matrices obtained, we
concluded that the model contains two charged Goldstone bosons that are
absorbed via the Stückelberg mechanism by the charged vector fields
$A^{\mu}_{\pm;n,j}$, three massive scalar modes of masses (3.50), and one
possible tachyonic mode. This latter tachyon is shown to be eliminated for
appropriates values of the vacuum expectation value of the uncharged component
of SU(2) scalar triplet. In particular, the vev must be greater than
$\sqrt{\frac{f}{2g}}$. Finally, it was verified that the scalar field $\Phi$
of our model preserve the chirality of the fermions.

The present paper suggests several extensions.

  1. 1.

As explained above, one could incorporate the SU(2) symmetry of the SM such
that that the Higgs boson is the Wilson line of the U(1) gauge fields with
magnetic flux in the background and is a doublet under SU(2). Verification of
the fact that the shift symmetry is preserved would be needed.

  2. 2.

We have shown that the Wilson lines in a $U(1)\crossproduct U(1)$ case have a
non-vanishing mass protected from one-loop corrections. One would like to
precise this result by computing the two-loop quantum corrections to find the
compute the Higgs mass and check that a TeV value is possible. In particular,
one can envisage large extra dimensions where $1/R$ would be of the order of
$\leavevmode\nobreak\ 10TeV$ such that a Higgs boson protected from 1-loop or
even 2-loop corrections would have a realistic mass.

  3. 3.

One could investigate the inequality (3.51) and the possibility that tachyons
appear depending on the gauge coupling value.

  4. 4.

We have mainly focused our attention on the hierarchy problem. However, an
easier application of the models introduced in this work would be for
inflation models. Indeed, contrary to the Higgs boson, the inflaton is
uncharged, as the scalar field of this work. As an example, [12] is a recent
model of inflation using the massless Wilson line as an inflaton.

## Acknowledgement

This work is dedicated to Charles Franken, my grandfather.

Finally, I would like to thank Emilian Dudas for his great guidance and
supervision throughout my internship at CPHT.

##  ?appendixname? A Appendix: Conventions

In this work, we follow the conventions of [17]. For example, we work with the
Minkowski metric $g_{mn}=diag(-1,1,1,1)$. We will use Weyl spinors. For a
Dirac spinor $\Psi$ with four components, we write :

$\Psi=\begin{pmatrix}\psi_{\alpha}\\\
\overline{\chi}^{\dot{\alpha}}\end{pmatrix}$ (A.1)

where the two components of $\Psi$ are Weyl spinors and have two components.
The Dirac spinor is a representation (1/2,1/2) of the Lorentz group and the
two Weyl components are the two irreducible representations (1/2,0) and
(0,1/2) respectively. They are chirality eigenstates. Indeed,

$\displaystyle\Psi_{L}$
$\displaystyle=\frac{1-\gamma_{5}}{2}\psi=\begin{pmatrix}\psi_{\alpha}\\\
0\end{pmatrix}$ (A.2) $\displaystyle\Psi_{R}$
$\displaystyle=\frac{1+\gamma_{5}}{2}\psi=\begin{pmatrix}0\\\
\overline{\chi}^{\dot{\alpha}}\end{pmatrix}$ (A.3)

where we use the Weyl basis for the gamma matrices and
$\alpha,\dot{\alpha}\in\\{1,2\\}$ are the Weyl spinors components.

$\gamma^{\mu}=\begin{pmatrix}0&\sigma_{\alpha\dot{\beta}}^{m}\\\
\overline{\sigma}^{m\dot{\alpha}\beta}&0\end{pmatrix}\quad,\quad\gamma_{5}=\gamma^{5}=-i\gamma_{0}\gamma_{1}\gamma_{2}\gamma_{3}=\begin{pmatrix}-\mathbb{1}_{2\crossproduct
2}&0\\\ 0&\mathbb{1}_{2\crossproduct 2}\end{pmatrix}$ (A.4)

where $\sigma^{i}$ are the Pauli matrices,
$\sigma^{0}=\overline{\sigma}^{0}=diag(-1,-1)$ and
$\overline{\sigma}^{1,2,3}=-\sigma^{1,2,3}$. The gamma matrices satisfy the
Clifford algebra.

$\\{\gamma_{m},\gamma_{n}\\}=-2\eta_{mn}$ (A.5)

The Dirac adjoint is defined by $\overline{\Psi}=\Psi^{\dagger}\gamma_{0}$.
Therefore,

$\overline{\Psi}=\begin{pmatrix}\chi^{\alpha}&\overline{\psi}_{\dot{\alpha}}\end{pmatrix}.$
(A.6)

Let us make some precision on spinor notations. We define Weyl spinors as the
objets carrying the fundamental representation of $SL(2,\mathbb{C})$ (the set
of nxn matrices of determinant 1). A Weyl spinor transforms under an element
of Sl($2,\mathbb{C}$) as

$\psi_{\alpha}\rightarrow\mathcal{M}_{\alpha}^{\beta}\psi_{\beta}$ (A.7)

whith $\alpha,\beta=1,2$ labelling the components. The complex conjugate of a
representation transforms as

$\overline{\psi}_{\dot{\alpha}}\rightarrow\mathcal{M}^{*\dot{\beta}}_{\dot{\alpha}}\overline{\psi}_{\dot{\beta}}$
(A.8)

where we used dotted indices to make the difference explicit. One can see that
$\overline{\psi}_{\dot{\alpha}}=(\psi_{\alpha})^{*}$. Finally, it is useful to
introduce the antisymmetric tensor

$\displaystyle\begin{split}-\epsilon_{\alpha\beta}=-\epsilon_{\dot{\alpha}\dot{\beta}}=\epsilon^{\alpha\beta}=\epsilon^{\dot{\alpha}\dot{\beta}}=\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\end{split}$ (A.9)

with the following properties.

$\displaystyle\begin{split}\psi\chi=\chi\psi=\psi^{\alpha}\chi_{\alpha}&\qquad\overline{\psi}\overline{\chi}=\overline{\chi}\overline{\psi}=\overline{\psi}_{\dot{\alpha}}\overline{\chi}^{\dot{\alpha}}\\\
\psi^{\alpha}=\epsilon^{\alpha\dot{\beta}}\psi_{\dot{\beta}}&\qquad\chi^{\alpha}\sigma^{m}_{\alpha\beta}\overline{\psi}^{\beta}=-\overline{\psi}_{\dot{\alpha}}\overline{\sigma}^{m\dot{\alpha}\beta}\chi_{\beta}\\\
\overline{\sigma}^{m\dot{\alpha}\alpha}&=\epsilon^{\dot{\alpha}\dot{\beta}}\epsilon^{\alpha\beta}\sigma^{m}_{\beta\dot{\beta}}\end{split}$
(A.10)

We refer to [17] or [18] for further explanations on spinors and their
notations.

Let us introduce our notations and conventions for Kaluza-Klein expensions of
higher dimensional fields. Considering a N=1 vector multiplet $V$ as an
example, we develop the superfield in its KK modes.

$\displaystyle\begin{split}V(x_{M},\theta,\overline{\theta})&=\sum_{n,m}V_{n,m}(x_{\mu},\theta,\overline{\theta})\psi_{n,m}(x_{m})\end{split}$
(A.11)

where $V_{n,m}$ are the KK modes and $\psi_{n,m}$ are a complete set of mode
functions. Following the convention of [7], we will choose them to be:

$\psi_{n,m}(x_{m})=\frac{1}{2\pi r}\exp{\frac{i}{r}(nx_{5}+mx_{6})}$ (A.12)

where $L^{2}=(2\pi r)^{2}$ is the area of the torus $T^{2}$ and
$n,m\in\mathbb{Z}$. They satisfy the following relations:

$\displaystyle\begin{split}\int_{T^{2}}d^{2}x\psi_{n,m}\overline{\psi}_{k,l}&=\delta_{n,k}\delta_{m,l}\\\
\int_{T^{2}}d^{2}x\overline{\psi}_{n,m}\psi_{k,l}\psi_{r,s}&=\frac{1}{L}\delta_{n,k+r}\delta_{m,l+s}\\\
\int_{T^{2}}d^{2}x\overline{\psi}_{n,m}\psi_{k,l}\psi_{r,s}\psi_{u,v}&=\frac{1}{L^{2}}\delta_{n,k+r+u}\delta_{m,l+s+v}\end{split}$
(A.13)

The reality of $V$ implies $\overline{V}_{n,m}=V_{-n,-m}$. We often note

$M_{mn}=\frac{2\pi}{r}(m+in)$ (A.14)

which are called KK masses

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