KEK-TH-2358
Gauge Symmetry Restoration by Higgs Condensation
in Flux Compactifications on Coset Spaces

We consider Yang-Mills theory on a $(4+d)$-dimensional manifold $\mathbb{R}^{4}\times{\cal M}$ with the action

$$S\ =\ \int dv\,{\rm Tr}\left[-\frac{1}{4}F_{MN}F^{MN}\right],\hskip 28.45274ptdv\ :=\ \frac{1}{g_{\rm YM}^{2}}d^{4+d}X\sqrt{-G},$$

(2.1)

where $M,N=0,1,\cdots,3+d$ and $G_{MN}$ is a metric on $\mathbb{R}^{4}\times{\cal M}$. The overall normalization of the action is chosen such that each matrix component of the gauge field $A_{M}$ is canonically normalized. Our convention for the field strength is

$$F_{MN}\ :=\ \nabla_{M}A_{N}-\nabla_{N}A_{M}+i[A_{M},A_{N}],$$

(2.2)

where $\nabla_{M}$ is the covariant derivative with respect to the metric $G_{MN}$.

We investigate this theory around a background gauge field $\bar{A}_{M}$. The gauge field $A_{M}$ is then decomposed as $A_{M}=\bar{A}_{M}+a_{M}$. In the following, we often use the notation

$$\bar{D}_{M}a_{N}\ :=\ \nabla_{M}a_{N}+i[\bar{A}_{M},a_{N}].$$

(2.3)

We employ the background field gauge

$$\bar{D}^{M}a_{M}\ =\ \nabla^{M}a_{M}+i[\bar{A}^{M},a_{M}]\ =\ 0.$$

(2.4)

The corresponding gauge-fixing term is given by

$$S_{\rm gf}\ =\ \int dv\,{\rm Tr}\left[-\frac{1}{2}\left(\bar{D}^{M}a_{M}\right)^{2}\right].$$

(2.5)

By expanding $a_{M}$ into Kaluza-Klein modes on ${\cal M}$, we can obtain a four-dimensional gauge theory coupled to various matter fields.

Let $x^{\mu}$ ($\mu=0,\cdots,3$) be coordinates on $\mathbb{R}^{4}$, and let $y^{\alpha}$ ($\alpha=1,\cdots,d$) be coordinates on $\cal M$. Accordingly, the gauge field $a_{M}$ is decomposed into $a_{\mu}$ and $\phi_{\alpha}$. We assume that the background gauge field $\bar{A}_{M}$ is of the form

$$\bar{A}_{M}\ =\ (0,\bar{A}_{\alpha}),\hskip 28.45274pt\partial_{\mu}\bar{A}_{\alpha}\ =\ 0.$$

(2.6)

This means that we put an $x^{\mu}$-independent gauge flux on ${\cal M}$. Note that the extra-dimensional components of the gauge field, $\phi_{\alpha}$, provide a set of adjoint matters, transforming homogeneously under the gauge transformations as they are defined by a difference of two gauge fields $A_{\alpha}$ and $\bar{A}_{\alpha}$. We also set the background metric of $\mathbb{R}^{4}\times{\cal M}$ as

$$G_{MN}\ =\ \left[\begin{array}[]{cc}\eta_{\mu\nu}&0\\ 0&h_{\alpha\beta}(y)\end{array}\right].$$

(2.7)

The total action $S+S_{\rm gf}$ consists of the following three parts:

	$\displaystyle S_{1}$	$\displaystyle=$	$\displaystyle\int dv\,{\rm Tr}\left[-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2}\left(\partial^{\mu}a_{\mu}\right)^{2}\right]$		(2.8)
		$\displaystyle=$	$\displaystyle\int dv\,{\rm Tr}\left[-\frac{1}{2}\left(\partial_{\mu}a_{\nu}\right)^{2}-i\partial_{\mu}a_{\nu}[a^{\mu},a^{\nu}]+\frac{1}{4}[a_{\mu},a_{\nu}]^{2}\right],$

	$\displaystyle S_{2}$	$\displaystyle=$	$\displaystyle\int dv\,{\rm Tr}\left[-\frac{1}{4}F_{\mu\alpha}F^{\mu\alpha}-\partial^{\mu}a_{\mu}\bar{D}^{\alpha}\phi_{\alpha}\right]$		(2.9)
		$\displaystyle=$	$\displaystyle\int dv\,{\rm Tr}\left[-\frac{1}{2}\left(D_{\mu}\phi_{\alpha}\right)^{2}-\frac{1}{2}\left(\bar{D}_{\alpha}a_{\mu}\right)^{2}+i[a_{\mu},\phi_{\alpha}]\bar{D}^{\alpha}a^{\mu}\right],$

where

$$D_{\mu}\phi_{\alpha}\ :=\ \partial_{\mu}\phi_{\alpha}+i[a_{\mu},\phi_{\alpha}],$$

(2.10)

and

	$\displaystyle S_{3}$	$\displaystyle=$	$\displaystyle\int dv\,{\rm Tr}\left[-\frac{1}{4}F_{\alpha\beta}F^{\alpha\beta}-\frac{1}{2}\left(\bar{D}^{\alpha}\phi_{\alpha}\right)^{2}\right]$
		$\displaystyle=$	$\displaystyle\int dv\,{\rm Tr}\left[-\frac{1}{4}\left(\bar{F}_{\alpha\beta}+\bar{D}_{\alpha}\phi_{\beta}-\bar{D}_{\beta}\phi_{\alpha}+i[\phi_{\alpha},\phi_{\beta}]\right)^{2}-\frac{1}{2}\left(\bar{D}^{\alpha}\phi_{\alpha}\right)^{2}\right],$

where $\bar{F}_{\alpha\beta}$ is the background field strength of the gauge potential $\bar{A}_{\alpha}$.

In the Kaluza-Klein reduction, the terms (LABEL:scalar_potential_original) in $S_{3}$ give the scalar potential $V(\phi)$ after an integration on $\cal M$. In particular, the mass terms of the scalars around $\phi_{\alpha}=0$ come from the following terms

$${\rm Tr}\left[\frac{1}{2}\left(\bar{D}_{\alpha}\phi_{\beta}\right)^{2}-\frac{1}{2}\phi^{\alpha}R_{\alpha\beta}\phi^{\beta}-i\phi^{\alpha}[\bar{F}_{\alpha\beta},\phi^{\beta}]\right],$$

(2.12)

where $R_{\alpha\beta}$ is the Ricci tensor for $h_{\alpha\beta}$ on $\cal{M}$. Note that we have used the equations of motion for $\bar{A}_{\alpha}$ in deriving (2.12). On the other hand, the mass terms of the vector fields are provided from the terms,

$${\rm Tr}\left[\frac{1}{2}\left(\bar{D}_{\alpha}a_{\mu}+i[\phi_{\alpha},a_{\mu}]\right)^{2}\right].$$

(2.13)

The second term gives additional contributions to mass at $\langle\phi_{\alpha}\rangle\neq 0$, whose effects we will investigate in sections 4 for ${\cal M}=S^{2}$ and section 5 for ${\cal M}=\mathbb{CP}^{2}$. We show that some of the massive Kaluza-Klein modes become massless by the second term.

In the following, we focus our attention on a compactification on a coset space. See e.g. [40, 25] for more details.

We consider a coset space ${\cal M}=G/H$ where $G$ and $H$ are Lie groups with $H\subset G$. Let us decompose generators of $G$ as $(\{t_{a}\},\{t_{m}\})$ where $\{t_{a}\}$ ($a=1,\cdots\dim H$) are a set of generators of $H$. Note that $\{t_{m}\}$ ($m=1,\cdots,d=\dim G-\dim H)$ correspond to a basis of the tangent space of $G/H$. We assume that the generators $t_{a},t_{m}$ satisfy the following commutation relations

$$[t_{a},t_{b}]\ =\ if^{c}{}_{ab}t_{c},\hskip 28.45274pt[t_{a},t_{m}]\ =\ if^{n}{}_{am}t_{n},\hskip 28.45274pt[t_{m},t_{n}]\ =\ if^{a}{}_{mn}t_{a}.$$

(2.14)

A coset space whose generators satisfy the commutation relations of this form is said to be symmetric. In the following, we use $a,b,c$ for generators of $H$ and $m,n$ for generators along $G/H$. The indices $m,n$ also represent those of coordinates of the tangent space on ${\cal M}=G/H$. In this paper, we discuss two examples of symmetric coset spaces, namely $S^{2}=SU(2)/U(1)$ and $\mathbb{CP}^{2}=SU(3)/U(2)$. In these cases, $t_{a}$ are represented in terms of block-diagonal matrices, while $t_{m}$ are given in terms of block-off-diagonal matrices, and their commutation relations are apparently of the form (2.14). Non-symmetric coset spaces are discussed in [41].

For a given coset space $G/H$, there is a “natural” choice for the vielbein $e^{m}_{\alpha}$ and the background gauge field $\bar{A}_{\alpha}$. Suppose we have a local embedding $g:y^{\alpha}\in G/H\rightarrow g(y)\in G$. Then the Maurer-Cartan 1-form $g^{-1}dg$ restricted on $g(G/H)$ is written as a sum

$$g^{-1}dg=ie^{a}(y)t_{a}+ie^{m}(y)t_{m},$$

(2.15)

where $e^{m}$ gives the natural choice of the vielbein on $G/H$ while $e^{a}$ provides the gauge field on the coset space. Indeed, $A=e^{a}t_{a}$ transforms under $g\rightarrow gh$ for $h\in H$, which is a gauge transformation as explained in Appendix C, as $A\rightarrow h^{-1}Ah-ih^{-1}dh$. In the following, we will consider gauge group $G_{\rm YM}$ that includes $H$ as $H\subset G_{\rm YM}$ and define the background gauge field on the coset space by

$$\bar{A}=\bar{A}_{\alpha}\ dy^{\alpha}:=\ e^{a}_{\alpha}(y)T_{a}dy^{\alpha},$$

(2.16)

where $T_{a}$ are generators of the gauge group $G_{\rm YM}$ which are the corresponding embedding of the generators $t_{a}$ of $H$ into the Lie algebra $\mathfrak{g}_{\rm YM}$ of $G_{\rm YM}$. In this paper, we consider various different embeddings of $H$ into $G_{\rm YM}$ for $G/H=S^{2}$ and $\mathbb{CP}^{2}$.

Interestingly, this background gauge field $\bar{A}_{\alpha}$ automatically satisfies the equations of motion

$$\bar{D}^{\alpha}\bar{F}_{\alpha\beta}\ =\ \nabla^{\alpha}\bar{F}_{\alpha\beta}+i[\bar{A}^{\alpha},\bar{F}_{\alpha\beta}]\ =\ 0$$

(2.17)

with respect to the vielbein $e^{m}_{\alpha}$ [42]. This can be checked as follows. First, the spin connection $\omega_{\alpha}{}^{m}{}_{n}$ defined by $de^{m}=-\omega^{m}{}_{n}\wedge e^{n}$, is obtained from the relation

$$d(g^{-1}dg)=-g^{-1}dg\wedge g^{-1}dg$$

(2.18)

or equivalently, from the relation (C.11) as

$$\omega_{\alpha}{}^{m}{}_{n}\ =\ -f^{m}{}_{an}e^{a}_{\alpha}.$$

(2.19)

Thus, the spin connection is written in terms of the component of the background gauge field $e^{a}_{\alpha}$, and the covariant derivative $\nabla^{\alpha}\bar{F}_{\alpha\beta}$ with respect to the metric on $G/H$ has the same form as the second term in (2.17). Second, by using the equation (C.10), the field strength of $\bar{A}_{\alpha}$ turns out to be

$$\bar{F}_{\alpha\beta}\ =\ e^{m}_{\alpha}e^{n}_{\beta}f^{a}{}_{mn}T_{a}.$$

(2.20)

Thus, the gauge field strength is non-vanishing in the $H$ subgroup of $G_{\rm YM}$. Inserting these expressions of (2.19) and (2.20) into (2.17), we find that it reduces to the Jacobi identity for the structure constants and the background gauge field indeed satisfies the equations of motion. See Appendix D for more details.

Since the spin connection $\omega_{\alpha}{}^{m}{}_{n}$ and the background gauge field $\bar{A}_{\alpha}$ are given in terms of the same quantity $e^{a}_{\alpha}$, the covariant derivative of $\phi_{m}:=e^{\alpha}_{m}\phi_{\alpha}$ can be written as

$$\bar{D}_{\alpha}\phi_{m}\ :={\partial_{\alpha}\phi_{m}-\omega_{\alpha}{}^{n}{}_{m}\phi_{n}+i[\bar{A}_{\alpha},\phi_{m}]}=\ \partial_{\alpha}\phi_{m}+ie^{a}_{\alpha}\left(-if^{n}{}_{am}\phi_{n}+[T_{a},\phi_{m}]\right).$$

(2.21)

This shows that the field $\phi_{m}$ can be regarded as a field on the flat $\mathbb{R}^{d}$ which couples to a gauge field $e^{a}_{\alpha}$ as a tensor product of two representations. Actually, the second commutation relation in (2.14) implies that $t_{m}$ form a representation $R_{t}$ of $H$ on which the generators are given by $if^{n}{}_{am}$. Therefore, $\phi_{m}$ belongs to the tensor product representation of $R_{t}$ and the adjoint representation of $G_{\rm YM}$ and can be decomposed into various irreducible representations of $H$. This property plays an important role in the investigations of mass spectrum of various fields with different spins and charges on $G/H$.

Besides the beautiful properties we have seen above, there are further advantages in choosing a symmetric coset space $G/H$ as the internal manifold $\cal M$. Most importantly, many properties of the complete set of functions on $G/H$ are well-known and we can explicitly perform the Kaluza-Klein reduction of any field on $G/H$ [40]. For the coset space $S^{2}$, these functions are given by the monopole harmonics [43]. For a general coset space $G/H$, the Peter-Weyl theorem tells us that each mode functions in the complete set on $G/H$ is labeled by a representation of $G$. As mentioned above, the field $\phi_{m}$ on $G/H$ can be regarded as belonging to a particular representation of $H$. This information can be incorporated by taking into account the irreducible decomposition of the representation of $G$ with respect to $H$. See Appendix F for more details.

By using the mode functions, the mass of each Kaluza-Klein mode in the four-dimensional sense can be obtained explicitly [36, 31]. As the mode functions are labeled by the representation of $G$ and its decomposition with respect to $H$, the mass is given in terms of group-theoretic quantities. Namely, it is given in terms the second Casimir invariants of certain representations. We review it in Appendix G, which will be used in the proof of the tachyonic behavior of the symmetric Higgs field observed in the following sections.

Our first choice of the embedding of the $H=\rm U(1)$ generator $T$ in $G_{\rm YM}=\rm SU(3)$ is

$$T\ =\ \frac{1}{2}\left[\begin{array}[]{ccc}1&0&0\\ 0&-1&0\\ 0&0&0\end{array}\right].$$

(4.3)

The background flux $\bar{F}_{+-}=-2iT$ breaks the gauge group ${\rm SU}(3)$ to its Cartan subgroup ${\rm U}(1)\times{\rm U}(1)$.

Let us now find the symmetric Higgs field satisfying the relation (3.5) for the U(1) generator $T$. We first define $T$-charges $q_{ij}$ of $\phi_{\pm,ij}$ fields by

$$[T,\phi_{\pm}]_{ij}\ =\ q_{ij}\,\phi_{\pm,ij},$$

(4.4)

where $i,j=1,2,3$ are indices of $3\times 3$ matrices, and no summation is taken. For the choice (4.3) of $T$, the $T$-charges are given in the matrix notation as

$$q\ =\ \left[\begin{array}[]{ccc}0&1&\frac{1}{2}\\ -1&0&-\frac{1}{2}\\ -\frac{1}{2}&\frac{1}{2}&0\end{array}\right].$$

(4.5)

Then, the condition (3.5) for the symmetric Higgs field and the commutation relation (4.1) tell us that the $(i,j)=(1,2)$ and $(2,1)$ components of $\phi_{\pm,ij}$ with $T$-charge $\pm 1$ provide us with the symmetric Higgs fields. Thus there is only one symmetric Higgs field (and its complex conjugate) given by

$$\phi_{+}(x)\ =\ \left[\begin{array}[]{ccc}0&\varphi(x)&0\\ 0&0&0\\ 0&0&0\end{array}\right],\hskip 28.45274pt\phi_{-}(x)\ =\ \left[\begin{array}[]{ccc}0&0&0\\ \varphi^{\dagger}(x)&0&0\\ 0&0&0\end{array}\right],$$

(4.6)

where we have used $\phi_{-}=(\phi_{+})^{\dagger}$.

Inserting these expressions into the scalar potential (4.2), we obtain the scalar potential for the symmetric Higgs field $\varphi$

$$V(\phi)\ =\ \frac{1}{4}\left(1-|\varphi|^{2}\right)^{2}.$$

(4.7)

Thus $\varphi$ will acquire vev at $|\varphi|=1$. At the origin $\varphi=0$, as mentioned before, the gauge symmetry $\rm SU(3)$ is broken to ${\rm U}(1)\times{\rm U}(1)\subset{\rm SU}(3)$ by the background gauge flux. When the symmetric Higgs field acquires vev at $|\varphi|=1$, the gauge symmetry is expected to be further broken to ${\rm U}(1)$ by the Higgs mechanism. Thus the expected symmetry breaking pattern is as follows:

$$G_{\rm YM}={\rm SU}(3)\xrightarrow{{\rm background}\ {\rm flux}}{\rm U}(1)\times{\rm U}(1)\xrightarrow{{\rm Higgs}\ {\rm vev}}{\rm U}(1)\ ?$$

(4.8)

This is the usual argument for the gauge symmetry breaking in the context of the coset space dimensional reduction in which only the low lying states are taken into considerations. However, the conclusion of the gauge symmetry breaking is suspicious in view of the higher dimensional gauge theory with the Kaluza-Klein reduction. The reason is the following. Note that we have vanishing scalar potential $V({|\varphi|=1})=0$ at the the global minimum of $V(\phi)$. Since the scalar potential originally comes from the terms (LABEL:scalar_potential_original), the vanishing scalar potential implies that the gauge field $A_{\alpha}$ at the symmetric Higgs vev $|\varphi|=1$ must be a pure gauge, and we must conclude that the full gauge symmetry ${\rm SU}(3)$ is recovered at the symmetric Higgs vacuum, instead of being broken to U(1).

In the rest of this section, in order to show the restoration of the gauge symmetry, we will explicitly see that some of the originally massive Kaluza-Klein vector fields become massless at vev $|\varphi|=1$, and eight massless vector fields emerge at the symmetric Higgs vacuum. These massless vector fields are the gauge fields due to the general argument by Weinberg [44].

Mass term of the vector field $a_{\mu}$ comes from the term (2.13), and a vector field is massless in the presence of the symmetric Higgs vev if and only if

$$\bar{D}_{+}a_{\mu}(x,y)+i[T_{+},a_{\mu}(x,y)]\ =\ 0,\hskip 28.45274ptT_{+}\ :=\ \left[\begin{array}[]{ccc}0&1&0\\ 0&0&0\\ 0&0&0\end{array}\right]$$

(4.9)

is satisfied. Note that $T$, $T_{+}$ and $T_{-}:=(T_{+})^{\dagger}$ form an $su(2)$ subalgebra of $su(3)$, and $a_{\mu}$ is in the adjoint representation $\bf 8$ of $su(3)$. Thus, by the irreducible decomposition of $\bf 8$ of $su(3)$ into $\bf 3\oplus 2\oplus 2^{\prime}\oplus 1$ of $su(2)$, the condition (4.9) can be decomposed into the following four conditions. First, for the representation $\bf 3$, we have

$$[1]\ {\rm Massless\ Cond.\ for\ {\bf 3}}\hskip 28.45274pt\bar{D}_{+}\left[\begin{array}[]{c}a_{\mu,12}\\[2.84526pt] a_{\mu,11}-a_{\mu,22}\\[2.84526pt] a_{\mu,21}\end{array}\right]\ =\ -i\left[\begin{array}[]{c}a_{\mu,22}-a_{\mu,11}\\[2.84526pt] 2a_{\mu,21}\\[2.84526pt] 0\end{array}\right].$$

(4.10)

For the representations $\bf 2$ and $\bf 2^{\prime}$, we have

$$[2]\ {\rm Massless\ Cond.\ for}\ {\bf 2}\hskip 28.45274pt\bar{D}_{+}\left[\begin{array}[]{c}a_{\mu,13}\\[2.84526pt] a_{\mu,23}\end{array}\right]\ =\ -i\left[\begin{array}[]{c}a_{\mu,23}\\[2.84526pt] 0\end{array}\right]$$

(4.11)

with the condition for their conjugate components $a_{\mu,31},a_{\mu,32}$ which is equivalent to

$$[2^{\prime}]\ {\rm Massless\ Cond.\ for}\ {\bf 2^{\prime}}\hskip 28.45274pt\bar{D}_{-}\left[\begin{array}[]{c}a_{\mu,13}\\[2.84526pt] a_{\mu,23}\end{array}\right]\ =\ -i\left[\begin{array}[]{c}0\\[2.84526pt] a_{\mu,13}\end{array}\right],$$

(4.12)

and finally,

$$[3]\ {\rm Massless\ Cond.\ for\ {\bf 1}}\hskip 28.45274pt\bar{D}_{+}a_{\mu,33}\ =\ 0$$

(4.13)

for the singlet representation $\bf 1$. A vector field satisfying one of these conditions become massless at the global minimum of the Higgs potential $V(\varphi)$ at $|\varphi|=1$. These are a set of first-order partial differential equations which can be written explicitly by using the formulas in Appendix E, and the number of massless vector fields can be found by solving the above differential equations. In the following, instead of solving them explicitly, we solve these conditions by reducing to a group-theoretic problem.

For this purpose, we need to understand the action of the covariant derivative $\bar{D}_{\pm}$ on $a_{\mu}$ [40]. The action can be simplified by choosing a suitable complete set of functions on $S^{2}$ which can be used to expand $a_{\mu}(x,y)$. Generally speaking, as explained in Appendix F, due to the Peter-Weyl theorem, a complete set of functions on a group manifold $G$ is given by the representation matrices $\rho^{R}(g)_{IJ}$ for all the representation $R$ of $G$ and their components $I,J=1,\cdots,\dim R$. Then a complete set of functions on $G/H$ is obtained by imposing particular transformation laws under $H$, corresponding to the $T$-charge of functions on $G/H$. Collecting all the representations of $H$, the complete set on $G$ is recovered.

In the case of $S^{2}={\rm SU}(2)/{\rm U}(1)$, a complete set on $S^{2}$, collecting all the charges of $H={\rm U}(1)$, is given by

$$f^{j}_{mm^{\prime}}(y)\hskip 14.22636pt{\rm where}\ \ j\ =\ 0,\ \frac{1}{2},\ 1,\ \frac{3}{2},\ \cdots,\hskip 14.22636pt-j\ \leq\ m,m^{\prime}\,\leq\ j.$$

(4.14)

Each $j$ corresponds to the spin $j$ representation of ${\rm SU}(2)$. As explained in Appendix F, the function $f^{j}_{mm^{\prime}}$ has $T$-charge $m$ of $H={\rm U}(1)$. Thus, $m=0$ gives the usual spherical harmonics, while $m\neq 0$ modes are the monopole spherical harmonics with $T$-charge $m$, which are relevant in the monopole background.

A field $\chi(y)$ on $S^{2}$ with the $T$-charge $q$ is then expanded in terms of $f^{j}_{qm^{\prime}}(y)$ as

$$\chi(y)\ =\ \sum_{j}\sum_{m^{\prime}=-j}^{j}c^{j}_{m^{\prime}}f^{j}_{qm^{\prime}}(y),$$

(4.15)

where the sum of $j$ is taken over all values of the spin $j$ whose magnetic quantum number $m^{\prime}$ can take $q$. Explicitly, $j$ in the sum must satisfy

$$-j\ \leq\ q\ \leq\ j,\hskip 28.45274ptj-q\ \in\ \mathbb{Z}.$$

(4.16)

From this expansion, we obtain $2j+1$ complex-valued fields, labeled by $m^{\prime}$, with the $T$-charge $q$ from each $j$.

In order to discuss the massless condition (4.9) for vector fields, it is sufficient to know the action of $\bar{D}_{+}$ on the mode functions $f^{j}_{mm^{\prime}}(y)$. From (G.3) in Appendix G, this action turns out to be given by

$$\bar{D}_{+}f^{j}_{mm^{\prime}}(y)\ =\ -i\sum_{n=-j}^{j}\left(T^{(j)}_{+}\right)_{mn}f^{j}_{nm^{\prime}}(y),$$

(4.17)

where $T^{(j)}_{+}$ is the spin-$j$ representation of $t_{+}$. Note that it is valid irrespective of the value of the symmetric Higgs field. Thus, the condition (4.9) for massless vector fields is reduced to algebraic relations of the coefficients $c^{j}_{\mu,i_{1}i_{2},m^{\prime}}$ in the mode expansion

$$a_{\mu,i_{1}i_{2}}(y)\ =\ \sum_{j}\sum_{m^{\prime}=-j}^{j}c^{j}_{\mu,i_{1}i_{2},m^{\prime}}f^{j}_{q(i_{1},i_{2}),m^{\prime}}(y)$$

(4.18)

between the first and the second terms in (4.9). Here $q(i_{1},i_{2})$ is the $T$-charge of $(i_{1},i_{2})$-component of $a_{\mu}$. The first term in (4.9) is a multiplication of $T^{(j)}_{+}$ on the complete set $f^{j}_{mm^{\prime}}(y)$ due to the covariant derivative $\bar{D}_{+}$ while the second term is the adjoint action of $T_{+}$ due to the symmetric Higgs vev. If we can choose the expansion coefficients of $a_{\mu}$ such that these two actions have the same effect, then we obtain a massless vector field.

Let us check whether this condition can be satisfied. First, we consider (4.11) of the massless condition for representation $\bf 2$, i.e., $q=\pm 1/2$. Thus, the representations of $\rm SU(2)$ are restricted to be $j=k+1/2$ for non-negative integers $k$. This can be written as

$\displaystyle\sum_{k=0}^{\infty}\sum_{m^{\prime}=-k-\frac{1}{2}}^{k+\frac{1}{2}}c^{k}_{m^{\prime}}\left(T^{(k+\frac{1}{2})}_{+}\right)_{\frac{1}{2},n}f^{k+\frac{1}{2}}_{nm^{\prime}}(y)$

$\displaystyle=$

$\displaystyle\sum_{k=0}^{\infty}\sum_{m^{\prime}=-k-\frac{1}{2}}^{k+\frac{1}{2}}\tilde{c}^{k}_{m^{\prime}}f^{k+\frac{1}{2}}_{-\frac{1}{2},m^{\prime}}(y)$

(4.19)

and

$\displaystyle\sum_{k=0}^{\infty}\sum_{m^{\prime}=-k-\frac{1}{2}}^{k+\frac{1}{2}}\tilde{c}^{k}_{m^{\prime}}\left(T^{(k+\frac{1}{2})}_{+}\right)_{-\frac{1}{2},n}f^{k+\frac{1}{2}}_{nm^{\prime}}(y)$

$\displaystyle=$

$\displaystyle 0,$

(4.20)

where we renamed the coefficients $c^{j}_{\mu,13,m^{\prime}}$ and $c^{j}_{\mu,23,m^{\prime}}$ as $c^{k}_{m^{\prime}}$ and $\tilde{c}^{k}_{m^{\prime}}$, respectively. These equations are satisfied if and only if $c^{0}_{m^{\prime}}=\tilde{c}^{0}_{m^{\prime}}$ are the only non-zero coefficients. Note that in our normalization and notation

$$\left(T^{(\frac{1}{2})}_{+}\right)_{\frac{1}{2},-\frac{1}{2}}\ =(\sigma_{+})_{12}=\ 1\,,$$

(4.21)

where $\sigma_{+}$ is the Pauli matrix. The same coefficients also solve (4.12). Consequently, we have shown that

$$a_{\mu,13}(x,y)\ =\ \sum_{m^{\prime}=\pm\frac{1}{2}}c^{0}_{m^{\prime}}(x)f^{\frac{1}{2}}_{\frac{1}{2},m^{\prime}}(y),\hskip 28.45274pta_{\mu,23}(x,y)\ =\ \sum_{m^{\prime}=\pm\frac{1}{2}}c^{0}_{m^{\prime}}(x)f^{\frac{1}{2}}_{-\frac{1}{2},m^{\prime}}(y)$$

(4.22)

and their conjugates give us four massless vector fields.

Next, we consider (4.10) for the massless condition for representation $\bf 3$, i.e., $q=0,\pm 1$. Thus, the representations of $\rm SU(2)$ are restricted to be $j=l$ for non-negative integers $l$. The range of $l$ depends on the $T$-charges. The linear combinations

$$-a_{\mu,12},\hskip 28.45274pt\frac{1}{\sqrt{2}}(a_{\mu,11}-a_{\mu,22}),\hskip 28.45274pta_{\mu,21}$$

(4.23)

as the independent fields are convenient for our purpose. Then, the condition (4.10) can be written as

	$\displaystyle\sum_{l=1}^{\infty}\sum_{m^{\prime}=-l}^{l}c^{l}_{m^{\prime}}\left(T^{(l)}_{+}\right)_{1,n}f^{l}_{nm^{\prime}}(y)$	$\displaystyle=$	$\displaystyle\sqrt{2}\sum_{l=0}^{\infty}\sum_{m^{\prime}=-l}^{l}\widetilde{c}^{l}_{m^{\prime}}f^{l}_{0,m^{\prime}}(y),$		(4.24)
	$\displaystyle\sum_{l=0}^{\infty}\sum_{m^{\prime}=-l}^{l}\widetilde{c}^{l}_{m^{\prime}}\left(T^{(l)}_{+}\right)_{0,n}f^{l}_{nm^{\prime}}(y)$	$\displaystyle=$	$\displaystyle\sqrt{2}\sum_{l=1}^{\infty}\sum_{m^{\prime}=-l}^{l}\widehat{c}^{l}_{m^{\prime}}f^{l}_{-1,m^{\prime}}(y),$		(4.25)
	$\displaystyle\sum_{l=1}^{\infty}\sum_{m^{\prime}=-l}^{l}\widehat{c}^{l}_{m^{\prime}}\left(T^{(l)}_{+}\right)_{-1,n}f^{l}_{nm^{\prime}}(y)$	$\displaystyle=$	$\displaystyle 0.$		(4.26)

These equations are satisfied if and only if $c^{1}_{m^{\prime}}=\widetilde{c}^{1}_{m^{\prime}}=\widehat{c}^{1}_{m^{\prime}}$ are the only non-zero coefficients. Then, the following three combinations

	$\displaystyle-a_{\mu,12}$	$\displaystyle=$	$\displaystyle{\sum_{m^{\prime}=-1,0,1}c^{1}_{m^{\prime}}f^{1}_{1,m^{\prime}}(y),}$		(4.27)
	$\displaystyle\frac{1}{\sqrt{2}}(a_{\mu,11}-a_{\mu,22})$	$\displaystyle=$	$\displaystyle\sum_{m^{\prime}=-1,0,1}c^{1}_{m^{\prime}}f^{1}_{0,m^{\prime}}(y),$		(4.28)
	$\displaystyle a_{\mu,21}$	$\displaystyle=$	$\displaystyle\sum_{m^{\prime}=-1,0,1}c^{1}_{m^{\prime}}f^{1}_{-1,m^{\prime}}(y),$		(4.29)

with the condition $a_{\mu}^{\dagger}=a_{\mu}$, give us 3 massless vector fields. In fact, this can be easily anticipated by rewriting (4.10) as

$$\bar{D}_{+}\left[\begin{array}[]{c}-a_{\mu,12}\\[2.84526pt] \frac{1}{\sqrt{2}}(a_{\mu,11}-a_{\mu,22})\\[2.84526pt] a_{\mu,21}\end{array}\right]\ =\ -iT^{(1)}_{+}\left[\begin{array}[]{c}-a_{\mu,12}\\[2.84526pt] \frac{1}{\sqrt{2}}(a_{\mu,11}-a_{\mu,22})\\[2.84526pt] a_{\mu,21}\end{array}\right].$$

(4.30)

Namely, these three components form the triplet of the $su(2)$ subalgebra, as mentioned before. Note that $a_{\mu,11}-a_{\mu,22}$ has also a contribution from $j=0$ which was massless before the Higgs condensation. This becomes massive due to the Higgs mechanism.