text:
\section{Introduction}

Physics in $2+1$ dimensions (two spacial dimensions) attracts over past decades
experimentalists and theorists. Thus, the discovery of graphene (planar structure) opens the door to new technology \cite{Geim}. Planar quantum physics differs from the physics in three spatial dimensions. This is due to the existence of the special `Chern$-$Simons' topological term in $2+1$ dimensions (($2+1$)-D). The quantum theory based on the Chern$-$Simons construction leads to fractional quantum statistics and is applied to anyon superconductivity \cite{Wilczek} and the fractional quantum Hall effect \cite{Stone}. Here we consider topologically massive theory in the framework of Maxwell$-$Chern$-$Simons gauge field description \cite{Deser}. Some aspects of Maxwell$-$Chern$-$Simons model were investigated in \cite{Banerjee}, \cite{Horvathy}, \cite{Dunne}, \cite{Colatto}. Chern$-$Simons term violates parity ($P$) and time reversal ($T$) symmetry and is responsible for the mass of the gauge field and spin-1 excitation. Within this theory the gauge invariant mass term possesses the topological nature that is different from the Higgs mechanism of the mass generation. Chern$-$Simons-like topological structures can be considered only in odd-dimensional space-time (for example in $7$-D and $11$-D supergravity). It is valuable that the perturbation quantum theory of the gauge field interaction with fermions in $(2+1$)-D do not have ultraviolet and infrared divergences \cite{Deser} and is super-renormalizable.
Gauge theories in $(2+1$)-D also attract attention because their possible connection with high temperature four-dimensional models.

In this paper, we investigate the behavior of gauge fields under the scale (dilatation) transformations to obtain the
`scale dimensionality' of gauge fields. It is also convenient to consider field equations in the form of the first-order relativistic wave equation (RWE). This allows us to apply well-known covariant methods for obtaining solutions of free gauge field which can be used in different quantum calculations.

The paper is organized as follows. In Sec.2, we obtain the canonical and the symmetrical Belinfante energy-momentum tensors and the dilatation current. The divergence of the dilatation current which is proportional to the topological mass of the gauge field is found. We make a conclusion that the `scale dimensionality' of fields is $d=1/2$.
The field equations are represented in the form of the first-order RWE in Sec.3. We obtain $6\times 6$-matrices of RWE that obey the Duffin$-$Kemmer$-$Petiau (DKP) algebra. Solutions to RWE for free fields with definite energy-momentum and spin are found in the form of projection matrices. The Schr\"{o}dinger form of equations and the quantum-mechanical Hamiltonian are obtained in Sec.4. We find projection operators extracting physical states in the Schr\"{o}dinger picture.  In Sec.5, results are discussed. In Appendix A, the dispersion relation ($p_0^2=\textbf{p}^2 +\mu^2$) is found from the characteristic equation. Appendix B is devoted to obtaining the symmetrical energy-momentum tensor by varying action on metric.

The Euclidean metric is used so that $x_\mu=(x_m,x_3)=(x_m,ix_0)$ and $x_0$ is a time, $x_0=t$, and Greek letters run $1,2,3$ and Latin letters run $1,2$. The system of units $\hbar =c=1$ is explored.

\section{Field equations, the energy-momentum tensor and dilatation current}

We start with the Lagrangian density \cite{Deser}
\begin{equation}
{\cal L} =-\frac{1}{4}F_{\mu\nu}^2+\frac{\mu}{4}\varepsilon_{\mu\nu\alpha}F_{\mu\nu}A_\alpha,
 \label{1}
\end{equation}
where $\varepsilon_{\mu\nu\alpha}$ is antisymmetric tensor, $\varepsilon_{123}=-i$, $F_{\mu\nu}=\partial_\mu A_\nu-
\partial_\nu A_\mu$ is the field strength, and $A_\mu$ is a vector-potential. Thus, there is a coupling of Maxwell and Chern$-$Simons terms in Eq.(1). Note that Chern$-$Simons term is not invariant under parity and time reversal transformations. Equations of motion follow from Lagrangian (1), and are given by
\begin{equation}
\partial_\mu F_{\mu\nu}+\frac{\mu}{2}\varepsilon_{\nu\mu\alpha}F_{\mu\alpha}=0.
 \label{2}
\end{equation}
Eq.(2) is invariant under U(1)-gauge transformations: $A_\mu\rightarrow A_\mu+\partial_\mu \Lambda$.
The dual vector $\widetilde{F}_\mu=(1/2)\varepsilon_{\mu\nu\alpha}F_{\nu\alpha}$ obeys the Klein$-$Gordon equation in $(2+1)$-D with the mass $\mu$ \cite{Deser} (see also \cite{Dunne}). In Appendix A, we demonstrate that Eq.(2) leads to the dispersion relation $p_0^2=\textbf{p}^2 +\mu^2$ which defines the mass $\mu$ of the gauge field $A_\nu$. The dual vector satisfies the equality $\partial_\mu \widetilde{F}_\mu=0$ leaving only two degrees of the vector $\widetilde{F}_\mu$. The gauge field is massive possessing the topological mass $\mu$ and spin-1 \cite{Deser}. The conserved canonical energy-momentum tensor $T_{\mu\nu}^{c}=(\partial_\nu
A_\alpha)\partial{\cal L}/\partial(\partial_\mu
A_\alpha)-\delta_{\mu\nu}{\cal L}$ reads
\begin{equation}
T_{\mu\nu}^{c}=\frac{\mu}{2}\varepsilon_{\mu\alpha\beta}A_\beta\partial_\nu A_\alpha-F_{\mu\alpha}\partial_\nu A_\alpha-\delta_{\mu\nu}{\cal L}.
\label{3}
\end{equation}
One can check using equations of motion (2), that the energy-momentum
tensor (3) is conserved tensor, $\partial_\mu T^c_{\mu\nu}=0$.
The canonical energy-momentum tensor (3) is not the
symmetric tensor, and its trace is given by
\begin{equation}
T_{\mu\mu}^{c}=-\frac{\mu}{2}\varepsilon_{\mu\nu\alpha}F_{\mu\nu}A_\alpha+\frac{1}{4}
F_{\mu\nu}^2. \label{4}
\end{equation}
We will investigate the dilatation symmetry with the help of the method
\cite{Coleman}. Thus, the canonical dilatation current is
\begin{equation}
D_\mu^c=x_\alpha T_{\mu\alpha}^{c}+\Pi_{\mu\nu} d A_\nu,
\label{5}
\end{equation}
where we took into account that the matrix $\textbf{d}$ defining the field dimension is proportional to the unit matrix.
The parameter $d$ is the scale dimensionality of the gauge field $A_\mu$ in $(2+1)$-D. Bose fields in 4-dimensional space-time have $d=1$ and Fermi fields possess $d=3/2$. Calculating
\begin{equation}
\Pi_{\mu\nu}=\frac{{\partial\cal L}}{\partial\left(\partial_\mu A_\nu \right)} =-F_{\mu\nu}+\frac{\mu}{2}\varepsilon_{\mu\nu\alpha}A_\alpha,
\label{6}
\end{equation}
we obtain from Eq.(5),(2)
\begin{equation}
\partial_\mu D_\mu^c=\frac{(d-1)\mu}{2}\varepsilon_{\mu\nu\alpha}F_{\mu\nu}A_\alpha+\frac{1-2d}{4}
F_{\mu\nu}^2.
\label{7}
\end{equation}
It is obvious that for Maxwell theory when $\mu=0$, we have the scale invariance, and the divergence of the dilatation current (7) should vanish. Therefore, we make a conclusion that the scale dimensionality $d$ of the gauge field $A_\mu$ in $(2+1)-D$ is
\begin{equation}
d=\frac{1}{2}.
\label{8}
\end{equation}
Then from Eq.(7), taking into consideration Eq.(8), we obtain
\begin{equation}
\partial_\mu D_\mu^c=-\frac{\mu}{4}\varepsilon_{\mu\nu\alpha}F_{\mu\nu}A_\alpha.
\label{9}
\end{equation}
The divergence of the dilatation current (90) is proportional to the mass parameter as it should be.
One can obtain Eq.(9) also from \cite{Coleman}
\begin{equation}
\partial_\mu D_\mu^c=\Pi_{\mu\nu}(d+1)\partial_\mu A_\nu
 +\frac{\partial{\cal L}}{\partial A_\mu}dA_\mu
 -3{\cal L}.
\label{10}
\end{equation}
The coefficient $3$ instead of $4$ in \cite{Coleman} is connected with the space-time dimension $(2+1)$ .

The non-symmetric and not gauge-invariant tensor (3) can be symmetrized
with the help of the Belinfante trick \cite{Coleman}:
\begin{equation}
T_{\mu\nu}^{B}=T_{\mu\nu}^{c}+\partial_\beta X_{\beta\mu\nu},
\label{11}
\end{equation}
where
\begin{equation}
X_{\beta\mu\nu}=\frac{1}{2}\left[\Pi_{\beta\sigma}\left(\Sigma_{\mu\nu}\right)_{\sigma\rho}
-\Pi_{\mu\sigma}\left(\Sigma_{\beta\nu}\right)_{\sigma\rho}-
\Pi_{\nu\sigma}\left(\Sigma_{\beta\mu}\right)_{\sigma\rho}\right]A_\rho.
\label{12}
\end{equation}
The symmetric Belinfante tensor (11) is also conserved, $\partial_\mu T^B_{\mu\nu}=0$ because the tensor $X_{\beta\mu\nu}$ is antisymmetric in indexes $\beta$ and $\mu$, and $\partial_\mu\partial_\beta
X_{\beta\mu\nu}=0$. The generators of the Lorentz transformations $\Sigma_{\mu\alpha}$ in Euclidian space-time have the matrix elements as follows:
\begin{equation}
\left(\Sigma_{\mu\alpha}\right)_{\sigma\rho}=\delta_{\mu\sigma}\delta_{\alpha\rho}
-\delta_{\alpha\sigma}\delta_{\mu\rho}. \label{13}
\end{equation}
Using equation (13), we find
\begin{equation}
X_{\beta\mu\nu}=\Pi_{\beta\mu}A_\nu.
\label{14}
\end{equation}
With the help of equations (14),(12), the gauge-invariant Belinfante tensor (11) becomes
\begin{equation}
T_{\mu\nu}^{B}=-F_{\mu\alpha}F_{\nu\alpha}+\frac{\mu}{4}\varepsilon_{\mu\alpha\beta}\left(
2F_{\nu\alpha}A_\beta+F_{\alpha\beta}A_\nu\right)-\delta_{\mu\nu}{\cal L}.
\label{15}
\end{equation}
Because the symmetrical energy-momentum tensor for pure Chern$-$Simons field equals zero \cite{Deser} (see also Appendix B), we obtain
\[
T_{\mu\nu}^{B}=-F_{\mu\alpha}F_{\nu\alpha}+\frac{1}{4}\delta_{\mu\nu}F^2_{\alpha\beta}.
\]
Thus, the Belinfante energy-momentum tensor coincides with Maxwell's energy-momentum tensor, and its trace is given by
\begin{equation}
T_{\mu\mu}^{B}=-\frac{1}{4}F_{\mu\nu}^2.
\label{16}
\end{equation}
It is interesting that in (3+1)-D the trace of Maxwell's energy-momentum tensor vanishes in contrast to (2+1)-D.
We define the modified dilatation current according to \cite{Coleman}:
\begin{equation}
D_{\mu}^{B}=x_\alpha T_{\mu\alpha}^{B}+V_\mu,
\label{17}
\end{equation}
where the field-virial $V_\mu$ is
\begin{equation}
V_\mu=\Pi_{\alpha\beta}\left[d\delta_{\alpha\mu}\delta_{\beta\rho}
-\left(\Sigma_{\alpha\mu}\right)_{\beta\rho}\right]A_\rho=\frac{1}{2}\Pi_{\alpha\mu}A_\alpha.
\label{18}
\end{equation}
We obtain from (18),(6)
\begin{equation}
\partial_\mu V_{\mu}=\frac{1}{4}F_{\mu\nu}^2-\frac{\mu}{4}\varepsilon_{\mu\alpha\beta}F_{\mu\alpha}A_\beta.
\label{19}
\end{equation}
As a result, it follows from Eq.(16)-(19) that the equality
\begin{equation}
\partial_\mu D_{\mu}^{B}=\partial_\mu D_{\mu}^c=-\frac{\mu}{4}\varepsilon_{\mu\nu\alpha}F_{\mu\nu}A_\alpha
\label{20}
\end{equation}
holds. The scale (dilatation ) symmetry is broken because the gauge field $A_\mu$ has the topological mass $\mu$.
The conformal symmetry is also broken as the field-virial $V_\mu$ is not the total divergence
of some local quantity $\sigma_{\alpha\beta}$ \cite{Coleman}. The same situation is realized in field theories
where the mass parameter presents (see, for example, \cite{Kruglov}). It should be noted that the scale invariance of Maxwell's field holds in (2+1)-D although the trace is not zero, Eq.(16). It was paid attention in \cite{Banerjee} on the importance of consideration of both symmetrical and canonical (Noether) angular momenta for derivation of the Chern$-$Simons vortices spin.

\section{Relativistic wave equation}

Let us introduce $6$-component wave function
\begin{equation}
\Psi (x)=\left\{ \psi _B(x)\right\} =\left(
\begin{array}{c}
\mu A_\nu(x)\\
F_{\mu\nu}(x)
\end{array}\right),
\label{21}
\end{equation}
where index $B=\nu,[\mu\nu]$, and the function $\Psi(x)$ is the direct sum of the vector $\psi_\nu (x)=\mu A_\nu(x)$, and the second rank antisymmetric tensor $\psi_{[\mu\nu]}(x)=F_{\mu\nu}(x)$.
We use the elements of the entire matrix algebra $\varepsilon
^{A,B}$ \cite{Kruglov1} with matrix elements and products
\begin{equation}
\left( \varepsilon ^{M,N}\right) _{AB}=\delta _{MA}\delta _{NB},
\hspace{0.5in}\varepsilon ^{M,A}\varepsilon ^{B,N}=\delta
_{AB}\varepsilon ^{M,N}, \label{22}
\end{equation}
where $A,B,M,N=\mu,[\mu\nu]$,
and generalized Kronecker symbol is given by
\begin{equation}
\delta_{[\mu\nu][\alpha\beta]}=\delta_{\mu\alpha}\delta_{\nu\beta}- \delta_{\mu\beta}\delta_{\nu\alpha}.
\label{23}
\end{equation}
In the $6\times6$-matrix  $\varepsilon ^{M,N}$ only one element is unity where the row $M$ and the
column $N$ cross and other elements are zero.

With the help of (21)-(23) equation (2) and equation for the field
strength $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$ may
be cast in the form of the first-order equation
\begin{equation}
\partial _\mu \left(\varepsilon ^{\nu,[\nu\mu]}+ \varepsilon ^{[\nu\mu],\nu}
\right)_{AB}\Psi _B(x)+ \mu\left(\frac{1}{2}\varepsilon ^{[\nu\mu],[\nu\mu]}+ \frac{1}{2}\varepsilon_{\nu\sigma\beta}\varepsilon^{\nu,[\sigma\beta]}\right) _{AB}\Psi_B(x)=0 ,
\label{24}
\end{equation}
and we imply a summation over all repeated indices. Equation (24) can be written in the form of the
first-order RWE:
\begin{equation}
\left( \beta _\mu \partial _\mu +\mu P\right) \Psi (x)=0 , \label{25}
\end{equation}
where we define $6\times6$-dimensional matrices
\begin{equation}
\beta_\mu=\varepsilon ^{\nu,[\nu\mu]}+ \varepsilon
^{[\nu\mu],\nu},~~~
P=\frac{1}{2}\varepsilon ^{[\nu\mu],[\nu\mu]}+
\frac{1}{2}\varepsilon_{\nu\sigma\beta}\varepsilon^{\nu,[\sigma\beta]}.
\label{26}
\end{equation}
The matrix $P$ is the projection operator, $P^2=P$.  It should be noted that it is not the
Hermitian matrix, $P^+\neq P$. The similar equation was derived for higher derivative
Podolsky's electrodynamics in $(3+1)$-D \cite{Kruglov2}. Matrices (26) can be visualized as follows:
\[
\beta_1=\left(
\begin{array}{cccccc}
0&0&0&0&0&0 \\
0&0&0&-1&0&0 \\
0&0&0&0&1&0\\
0&-1&0&0&0&0\\
0&0&1&0&0&0\\
0&0&0&0&0&0
\end{array}\right),~~~~
\beta_2=\left(
\begin{array}{cccccc}
0&0&0&1&0&0 \\
0&0&0&0&0&0 \\
0&0&0&0&0&-1\\
1&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&-1&0&0&0
\end{array}\right),
\]
\vspace{-7mm}
\begin{equation} \label{27}
\end{equation}
\vspace{-7mm}
\[
\beta_3=\left(
\begin{array}{cccccc}
0&0&0&0&-1&0 \\
0&0&0&0&0&1 \\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
-1&0&0&0&0&0\\
0&1&0&0&0&0
\end{array}\right),~~~~
P=\left(
\begin{array}{cccccc}
0&0&0&0&0&-i \\
0&0&0&0&i&0 \\
0&0&0&-i&0&0\\
0&0&0&1&0&0\\
0&0&0&0&1&0\\
0&0&0&0&0&1
\end{array}\right).
\]
One can verify that the Hermitian matrices $\beta_\mu$ obey the
DKP algebra \cite{Petiau}, \cite{Duffin}, \cite{Kemmer}:
\begin{equation}
\beta _\mu \beta _\nu \beta _\alpha +\beta _\alpha \beta _\nu
\beta _\mu =\delta _{\mu \nu }\beta _\alpha+\delta _{\alpha \nu
}\beta _\mu . \label{28}
\end{equation}
Matrices $\beta _\mu$, (27), realize the $6$-dimensional representation of DKP algebra. We notice that
there are $5$- and $10$-dimensional representations of DKP algebra which are used for the description of
scalar and vector particles in the framework of the first-order formalism in $(3+1)-D$. It easy to check that matrices
$\beta_1^2$, $\beta_2^2$, $\beta_3^2$ are projection operators.

Let us verify the U(1)-invariance of Eq.(25) using the method of \cite{Harish}. The gauge transformations under U(1)-group are given by
\begin{equation}
\Psi'(x)=\Psi(x)+\overline{P}\phi(x),
\label{29}
\end{equation}
where the matrix $\overline{P}$ is projection matrix ($\overline{P}^2=\overline{P}$), and reads
\begin{equation}
\overline{P}=\varepsilon^{\mu,\mu}.
\label{30}
\end{equation}
One may check with the aid of Eq.(22),(26) that the matrix $\overline{P}$ obeys the relations as follows:
\begin{equation}
\beta_\mu\overline{P}+\overline{P}\beta_\mu=\beta_\mu,~~~~P\overline{P}=0.
\label{31}
\end{equation}
Then using Eq.(25),(29),(31), we obtain
\begin{equation}
\left( \beta _\mu \partial _\mu + \mu P\right) \Psi'(x)=(1-\overline{P})\beta _\mu \partial _\mu\phi(x).
\label{32}
\end{equation}
The gauge transformations: $A_\mu(x)\rightarrow A_\mu (x)+\partial_\mu \Lambda (x)$ in the matrix notations lead to the function $\phi(x)$ in Eq.(29):
\[
\phi(x)=\left(
\begin{array}{c}
\mu \partial_\mu \Lambda(x)\\
0
\end{array}\right).
\]
One can verify that this function, $\phi(x)$, obeys the equation $(1-\overline{P})\beta_\mu\partial_\mu\phi(x)=0$, i.e. the gauge invariance holds in the first-order formalism considered.

\section{Hermitianizing matrix and $SO(2,1)$-covariance}

Under the $SO(2,1)$-group of coordinate transformations (the Lorentz group in $(2+1)$-D),
the wave function (21) becomes
\begin{equation}
\Psi ^{\prime }(x^{\prime })=T\Psi (x) , \label{33}
\end{equation}
where $6\times 6-$matrix $T$ gives the reducible tensor representation
of the $SO(2,1)$-group. The Lorentz covariance of Eq.(25) requires the validity of
equations
\begin{equation}
\beta_\mu TL_{\mu \nu }=T\beta _\nu,~~~~~~ PT=TP,  \label{34}
\end{equation}
where the Lorentz matrix $L=\{L_{\mu \nu }\}$ obeys the equation
$L_{\mu \alpha }L_{\nu \alpha }=\delta _{\mu \nu } $ (remember that we use Euclidian metric where
third components are imaginary).
At the infinitesimal Lorentz transformations the matrix $T$
is given by
\begin{equation}
T=1+\frac 12\varepsilon _{\mu \nu }J_{\mu \nu } , \label{35}
\end{equation}
where $J_{\mu \nu }$ are generators of the Lorentz group and
the infinitesimal Lorentz matrix is
\begin{equation}
L_{\mu \nu }=\delta _{\mu \nu }+\varepsilon _{\mu \nu } ,
\label{36}
\end{equation}
where $\varepsilon _{\mu \nu }=-\varepsilon _{\nu \mu }$ are three parameters of the group $SO(2,1)$.
From (34)-(36), we obtain equations as follows:
\begin{equation}
\beta _\mu J_{\alpha \nu }-J_{\alpha \nu }\beta_\mu =\delta
_{\alpha \mu }\beta _\nu -\delta _{\nu \mu }\beta _\alpha, ~~~~PJ_{\alpha \nu }=J_{\alpha \nu }P .
\label{37}
\end{equation}
The Lorentz group generators which obey Eq.(37) are given by
\begin{equation}
J_{\mu\nu}= \beta_\mu\beta_\nu-\beta_\nu\beta_\mu
=\varepsilon ^{\mu,\nu}-\varepsilon ^{\nu,\mu}+\varepsilon
^{[\lambda\mu],[\lambda\nu]}-\varepsilon
^{[\lambda\nu],[\lambda\mu]}
\label{38}
\end{equation}
or in the matrix form:
\[
J_{12}=\left(
\begin{array}{cccccc}
0&1&0&0&0&0 \\
-1&0&0&0&0&0 \\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&-1\\
0&0&0&0&1&0
\end{array}\right),~~~~
J_{13}=\left(
\begin{array}{cccccc}
0&0&1&0&0&0 \\
0&0&0&0&0&0 \\
-1&0&0&0&0&0\\
0&0&0&0&0&-1\\
0&0&0&0&0&0\\
0&0&0&1&0&0
\end{array}\right),
\]
\vspace{-7mm}
\begin{equation} \label{39}
\end{equation}
\vspace{-7mm}
\[
J_{23}=\left(
\begin{array}{cccccc}
0&0&0&0&0&0 \\
0&0&1&0&0&0 \\
0&-1&0&0&0&0\\
0&0&0&0&-1&0\\
0&0&0&1&0&0\\
0&0&0&0&0&0
\end{array}\right).
\]
One can find with the help of Eq.(22) the commutation relations of generators (38):
\begin{equation}
\left[ J_{\mu \nu },J_{\alpha \beta}\right] =\delta _{\nu \alpha
}J_{\mu \beta}+\delta _{\mu \beta }J_{\nu \alpha}-\delta _{\nu
\beta }J_{\mu \alpha}-\delta _{\mu \alpha }J_{\nu \beta}.
\label{40}
\end{equation}
The Hermitianizing matrix $\eta$, which enters the Lorentz-invariant $ \overline{\Psi }(x)\Psi (x)=\Psi ^{+}(x)\eta \Psi(x)$ ($\Psi ^{+}(x)$ is the Hermitian conjugated wave function), satisfies the relations \cite{Gel'fand}
\begin{equation}
\eta \beta _m=-\beta _m^{+}\eta^{+} ,\hspace{0.5in}\eta \beta
_3=\beta _3^{+}\eta^{+},\label{41}
\end{equation}
where $m=1,2$. One obtains
\begin{equation}
\eta=\varepsilon^{m,m}-\varepsilon^{3,3}+\varepsilon^{[m3],[m3]}-
\frac{1}{2}\varepsilon^{[mn],[mn]},
\label{42}
\end{equation}
or in the form
\begin{equation}
\eta=\left(
\begin{array}{cccccc}
1&0&0&0&0&0 \\
0&1&0&0&0&0 \\
0&0&-1&0&0&0\\
0&0&0&-1&0&0\\
0&0&0&0&1&0\\
0&0&0&0&0&1
\end{array}\right),
\label{43}
\end{equation}
so that $\eta^{+}=\eta$, i.e. it is the Hermitian matrix. One may verify that $\eta$
commutes with the projection operator $P$ (see Eq.(27)), $\eta P=P\eta$.
We find the ``conjugated" wave function
\begin{equation}
\overline{\Psi }(x)=\Psi ^{+}(x)\eta=
\left(\mu A_\nu,-F_{\mu\nu}\right). \label{44}
\end{equation}
From Eq.(25) one obtains the ``conjugated" equation
\begin{equation}
\overline{\Psi }(x)\left( \beta _\mu \overleftarrow{\partial} _\mu
-\mu P^+\right) =0 .
\label{45}
\end{equation}
One may check the relationship
\begin{equation}
\overline{\Psi }(x)P^+\Psi (x)=\overline{\Psi }(x)P\Psi (x), \label{46}
\end{equation}
but $P^+\neq P$. Therefore, from Eq.(25),(46), we obtain the conservation of
the charge: $\partial_\mu\left(\overline{\Psi }(x)\beta_\mu\Psi (x)\right)=0$.
For our case of neutral fields (($\textbf{A},A_0$) are real values), we have from (21), (27) and (44):
$\overline{\Psi }(x)\beta_\mu\Psi (x)=0$.

\section{Solutions to RWE}

Let us find solutions to Eq.(25) corresponding to definite energy and
momentum. Thus, in the momentum space Eq.(25) takes the form
\begin{equation}
\left(i\hat{p}+\mu P \right)\Psi(p)=0 ,
\label{47}
\end{equation}
where $\hat{p}=\beta_\mu p_\mu$, $p_\mu$ is a energy-momentum, $p_\mu=(p_1,p_2,ip_0)$, $p_\mu^2=-\mu^2$.
The matrix of the equation (47)
\begin{equation}
\Lambda= i\hat{p}+\mu P
\label{48}
\end{equation}
reads
\begin{equation}
\Lambda=\left(
\begin{array}{cccccc}
0&0&0&ip_2&ip_3&-i\mu \\
0&0&0&-ip_1&i\mu&ip_3 \\
0&0&0&-i\mu&-ip_1&-ip_2\\
ip_2&-ip_1&0&\mu&0&0\\
ip_3&0&-ip_1&0&\mu&0\\
0&ip_3&-ip_2&0&0&\mu
\end{array}\right).
\label{49}
\end{equation}
One can verify that the $6\times 6$-matrix $\Lambda$ (49) obeys the `minimal' polynomial equation
\begin{equation}
\Lambda\left(\Lambda^2-\mu^2\right)\left(\Lambda-2\mu\right)=0.
\label{50}
\end{equation}
It is obvious that the matrix
\begin{equation}
\Pi=N\left(\Lambda^2-\mu^2\right)\left(\Lambda-2\mu\right),
\label{51}
\end{equation}
where $N$ is a constant, satisfies the equation $\Lambda\Pi=0$,
so that the every column of the matrix $\Pi$ is the
solution to Eq.(25). It is convenient to require that $\Pi$ is the
projection operator \cite {Fedorov}. From the relation $\Pi^2=\Pi$, we obtain the normalization
constant $N=1/(2\mu^3)$. The projection operator (51) gives solutions to Eq.(25) for definite energy-momentum.

Now, we define pseudovector generators of $SO(2,1)$-group as follows \cite{Dunne}:
\begin{equation}
J_\mu=\frac{1}{2}\varepsilon_{\mu\nu\alpha}J_{\nu\alpha},
\label{52}
\end{equation}
where $J_{\nu\alpha}$ are given by (39). It is easy to check that generators (52) obey the
commutation relations:
\begin{equation}
\left[J_\mu,J_\nu\right]=\varepsilon_{\nu\mu\alpha}J_{\alpha},~~~~
\left[J_\mu,\hat{p}\right]=\varepsilon_{\mu\nu\alpha}p_{\nu}\beta_\alpha .
\label{53}
\end{equation}
The Pauli-Lubanski pseudoscalar is defined by (see \cite{Dunne})
\begin{equation}
W=p_\mu J_\mu,
\label{54}
\end{equation}
and reads
\begin{equation}
W=\left(
\begin{array}{cccccc}
0&-ip_3&ip_2&0&0&0\\
ip_3&0&-ip_1&0&0&0 \\
-ip_2&ip_1&0&0&0&0\\
0&0&0&0&ip_1&-ip_2\\
0&0&0&-ip_1&0&ip_3\\
0&0&0&ip_2&-ip_3&0
\end{array}\right).
\label{55}
\end{equation}
The $W$ ia a Hermitian matrix. One can verify that $W$ commutates with operators $\widehat{p}$, $P$ and $\Lambda$: $[W,\widehat{p}]=0$, $[W,P]=0$, $[W,\Lambda]=0$.
Operator (55) obeys the `minimal' polynomial equation:
\begin{equation}
W\left( W^2-p^2\right) =0 ,
\label{56}
\end{equation}
and we use the notation $p^2\equiv p^2_\mu$. According to the general method \cite{Fedorov}, we find
the projection operators extracting physical states with spins $\pm 1$ and non-physical states with spin $0$:
\begin{equation}
S_{(\pm 1)}=\frac 12\Sigma\left( \Sigma\pm 1\right)
,\hspace{0.5in}S_{(0)}=1-\Sigma^2,
\label{57}
\end{equation}
where
\[
\Sigma=\frac{W}{\sqrt{p^2}}=-i\frac{W}{\mu}.
\]
Projection operators (57) obey the relations: $S_{(\pm 1)}^2=S_{(\pm 1)}$, $S _{(\pm
1)}S_{(0)}=0$, $S_{(0)}^2=S_{(0)}$, $[\Pi,S_{(\pm 1)}]$=0, $[\Pi,S_{(0)}]$=0, $S_{(+1)}+S_{(- 1)}+S_{(0)}=1$.
From Eq.(55)(57), we obtain
\begin{equation}
S_{(\pm 1)}=\frac{1}{2\mu^2}\left(
\begin{array}{cccccc}
-p_2^2-p_3^2&p_1p_2\mp\mu p_3&p_1p_3\pm\mu p_2&0&0&0\\
p_1p_2\pm\mu p_3&-p_1^2-p_3^2&p_2p_3\mp\mu p_1&0&0&0 \\
p_1p_3\mp\mu p_2&p_2p_3\pm\mu p_1&-p_1^2-p_2^2&0&0&0\\
0&0&0&-p_1^2-p_2^2&p_2p_3\pm\mu p_1&p_1p_3\mp\mu p_2\\
0&0&0&p_2p_3\mp\mu p_1&-p_1^2-p_3^2&p_1p_2\pm\mu p_3\\
0&0&0&p_1p_3\pm\mu p_2&p_1p_2\mp\mu p_3&-p_2^2-p_3^2
\end{array}\right).
\label{58}
\end{equation}
It follows from Eq.(58) that spin operators $S_{(+ 1)}$ and $S_{(- 1)}$ differ only by the sign in the mass $\mu$ according to Poincar\'{e} algebra representation \cite{Deser}, \cite{Dunne}.
One finds projection operators
\begin{equation}
\Delta_{\pm 1} =\Pi S_{(\pm 1)} ,~~~~\Delta _0=\Pi S_{(0)}
\label{59}
\end{equation}
extracting solutions to Eq.(25) for definite energy-momentum and spin. The projection operator $\Delta_{\pm 1}$ extracts physical solutions and also defines the density matrix which can be used for covariant quantum calculations \cite{Fedorov}.

\section{Schr\"{o}dinger form of equations}

To obtain the Schr\"{o}dinger form of equations and quantum-mechanical Hamiltonian,
we exclude the non-dynamical components from the wave function (21).
For this purpose, we represent Eq.(2) and equation $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$ as follows:
\[
i\partial_t A_1 =F_{13}-\partial_1 A_3,~~~~~~i\partial_t A_2=F_{23}-\partial_2 A_3,
\]
\vspace{-8mm}
\begin{equation}
\label{60}
\end{equation}
\vspace{-8mm}
\[
i\partial_tF_{31}=-\partial_2F_{12}+i\mu F_{23},~~~~i\partial_tF_{23}=-\partial_1F_{12}-i\mu F_{31},
\]
\begin{equation}
F_{12}=\partial_1A_2-\partial_2A_1.
\label{61}
\end{equation}
Eq.(60) show the evolution of the dynamical components in time, but Eq.(61) does not contain derivative in time, and therefore, $F_{12}$ is non-dynamical component of fields. In addition, there is not evolution in time of the component of the potential $A_3$ ($A_3=iA_0$). It is possible to use the gauge condition $A_3=0$ \cite{Dunne} but it violates the Lorentz-covariance. Instead of this gauge, we use the Lorentz condition $\partial_\mu A_\mu=0$ which defines the evolution in time of the $A_3$-component:
 \begin{equation}
i\partial_t A_3 =\partial_1 A_1+\partial_2 A_2.
\label{62}
\end{equation}
One can exclude the auxiliary (non-dynamical) component $F_{12}$ from Eq.(60).
Replacing the non-dynamical components $F_{12}$ from Eq.(61) into Eq.(60), and taking into account Eq.(62), we obtain
\begin{equation}
i\partial _t \left(
\begin{array}{c}
 \mu A_1\\
\mu A_2\\
\mu A_3\\
F_{31}\\
F_{23}
\end{array}\right)=
\left(
\begin{array}{c}
 -\mu\left(F_{31}+\partial_1 A_3\right)\\
\mu\left(F_{23}-\partial_2 A_3\right)\\
\mu\left(\partial_1 A_1+\partial_2 A_2\right)\\
\frac{i}{\mu}\left(\partial_1\partial_2F_{31}-\partial_2^2F_{23}\right)+i\mu F_{23}\\
\frac{i}{\mu}\left(\partial_1^2F_{31}-\partial_1\partial_2F_{23}\right)-i\mu F_{31}
\end{array}\right).
 \label{63}
\end{equation}
Introducing 5-component wave function
\begin{equation}
\Phi (x)=\left(
\begin{array}{c}
\mu A_\nu (x)\\
F_{31}(x)\\
F_{23}(x)
\end{array}
\right),
\label{64}
\end{equation}
Eq.(63), with the help of the elements of the entire matrix algebra Eq.(22), can be
written in the Schr\"{o}dinger form
\begin{equation}
i\partial_t\Phi(x)={\cal H}\Phi(x),
\label{65}
\end{equation}
where the Hamiltonian is given by
\[
{\cal H}=\mu\left[\varepsilon^{n,[n3]}+i\left(\varepsilon^{[31],[23]}-
\varepsilon^{[23],[31]} \right)\right]+\partial_1\left(\varepsilon^{3,1}-
\varepsilon^{1,3}\right)+\partial_2\left(\varepsilon^{3,2}-
\varepsilon^{2,3}\right)
\]
\vspace{-8mm}
\begin{equation} \label{66}
\end{equation}
\vspace{-8mm}
\[
+\frac{i}{\mu}\left[\left(\varepsilon^{[31],[31]}-
\varepsilon^{[32],[32]}\right)\partial_1\partial_2-\varepsilon^{[31],[23]}\partial_2^2
+\varepsilon^{[23],[31]}\partial_1^2
\right],
\]
where we imply the summation on index $n$.
The wave function (64) contains only dynamical components of gauge fields, and therefore the Schr\"{o}dinger picture has some advantages by considering field interactions.
To obtain solutions to Eq.(65) for free fields, we consider the momentum space, where the Hamiltonian (66) becomes
\begin{equation}
{\cal H}=\left(\begin{array}{ccccc}
0&0&-ip_1&-\mu&0\\
0&0&-ip_2&0&\mu \\
ip_1&ip_2&0&0&0\\
0&0&0&-\frac{i}{\mu}p_1p_2&\frac{i}{\mu}\left(p_2^2+\mu^2\right)\\
0&0&0&-\frac{i}{\mu}\left(p_1^2+\mu^2\right)&\frac{i}{\mu}p_1p_2
\end{array}\right).\label{67}
\end{equation}
One may verify that the matrix (67) satisfies the `minimal' polynomial equation
\begin{equation}
{\cal H}^2\left({\cal H}^2-p_n^2\right)\left[{\cal H}^2-\left(p_n^2+\mu^2\right)\right]=0,
\label{68}
\end{equation}
and $p_n^2=p_1^2+p_2^2$. It follows from Eq.(68) that the Hamiltonian ${\cal H}$ possesses the eigenvalues:
$0$, $\pm\sqrt{p_n^2}$, $\pm \sqrt{p_n^2+\mu^2}$.
From equation (68), we obtain projection operators extracting physical states with positive
and negative energies ($p_0=\pm \sqrt{p_n^2+\mu^2}$):
\begin{equation}
\Sigma_{\pm}=\frac{\left({\cal H}\pm
\sqrt{p_n^2+\mu^2}\right){\cal H}^2\left({\cal H}^2-p_n^2\right)}{2\mu^2\left(p_n^2+\mu^2\right)^{3/2}},
\label{69}
\end{equation}
so that $\left(\Sigma_{\pm}\right)^2=\Sigma_{\pm}$, and
\begin{equation}
{\cal H}\Sigma_{\pm}=\pm \sqrt{p_n^2+\mu^2}\Sigma_{\pm}.
\label{70}
\end{equation}
The projection operator (69) can be used for constructing physical states in the space of wave functions
(64). Indeed, the wave function $\Phi_{\pm}=\Sigma_{\pm}\Phi_0$, where  $\Phi_0$ is a constant non-zero 5-column,
is an eigenfunction of the Hamiltonian ${\cal H}$ with physical eigenvalues $\pm \sqrt{p_n^2+\mu^2}$:
\begin{equation}
{\cal H}\Phi_{\pm}=\pm \sqrt{p_n^2+\mu^2}\Phi_{\pm}.
\label{71}
\end{equation}
Thus, the wave function $\Phi_{\pm}$ defines physical states in the Schr\"{o}dinger picture.

\section{Conclusion}

We have obtained the canonical and Belinfante energy momentum tensors and non-conserved dilatation current which is proportional to the topological mass of the gauge field. It was demonstrated that the `scale dimensionality' of gauge fields is $d=1/2$ in $(2+1)$-D. This is different from fields in $(3+1)$-D.
The field equations were represented in the form of the first-order RWE, and we obtained $6\times 6$-matrices of RWE which obey DKP algebra. Solutions to RWE for a free fields with definite energy-momentum and spin were obtained in the form of projection matrices. These solutions can be used for different covariant quantum-mechanical calculations.
The Schr\"{o}dinger form of equations and the quantum-mechanical Hamiltonian obtained can be explored in the first-order formalism. Thus, projection operators extracting physical states of free fields can be used in 5-dimensional space of dynamical components of gauge fields.
It should be noted that DKP form of equations was also considered in non-Abelian theory (QCD) \cite{Gribov}.

\vspace{7mm}
\textbf{\textbf{Acknowledgments}}
\vspace{7mm}

I wish to thank Prof. S. Deser for correspondence.

\vspace{7mm}
\textbf{Appendix A}
\vspace{7mm}

To demonstrate that the gauge field possesses the mass $\mu$, we represent equation of motion (2) in the momentum space.
Replacing the strength tensor $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$ in Eq.(2), one obtains the equation for vector-potential in the momentum space:
\[
\left[i\left(p^2\delta_{\mu\nu}-p_\mu p_\nu\right)+\mu\varepsilon_{\mu\alpha\nu}p_\alpha\right]A_\nu=0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(A.1)
\]
where $p^2=p_1^2+p_2^2-p_0^2$. Introducing the antisymmetric tensor $\epsilon_{\mu\nu\alpha}=i\varepsilon_{\mu\nu\alpha}$ ($\epsilon_{123}=1$), we rewrite Eq.(A.1) in the form
\[
M_{\mu\nu}A_\nu=0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(A.2)
\]
where
\[
M_{\mu\nu}=p^2\delta_{\mu\nu}-p_\mu p_\nu-\mu\epsilon_{\mu\alpha\nu}p_\alpha.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(A.3)
\]
Equation (A.2) possesses nontrivial solutions if determinant of the matrix $M=\left\{M_{\mu\nu}\right\}$ equals zero.
We find the eigenvalues of the matrix $M$ by solving the characteristic equation $\det\left(M-\lambda\right)=0$:
\[
\det\left[M-\lambda\right)=-\lambda\left(\lambda^2-2\lambda p^2+p^2\left(p^2+\mu^2\right)\right]=0.~~~~~~~~~~~~~~(A.4)
\]
From Eq.(A.4), we obtain eigenvalues of the matrix $M$:
\[
\lambda_1=0,~~~~\lambda_2=p^2+\sqrt{-p^2\mu^2},~~~~\lambda_3=p^2-\sqrt{-p^2\mu^2}.~~~~~~~~~~~~(A.5)
\]
The requirement $\lambda_2=0$ leads to the dispersion relation $p^2=-\mu^2$ defining the mass $\mu$ of the gauge field.
We notice that the matrix $M$ obeys the equation as follows:
\[
M\left[M^2-2p^2 M +p^2\left(p^2+\mu^2\right)\right]=0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(A.6)
\]
in accordance with the Hamilton$-$Cayley theorem.

\vspace{7mm}
\textbf{Appendix B}
\vspace{7mm}

In order to obtain Schwinger's \cite{Schwinger} symmetrical energy-momentum tensor, we rewrite Lagrangian (1) in the
curve coordinate system:
\[
{\cal L} =-\frac{1}{4}g^{\mu\alpha}g^{\beta\sigma}F_{\alpha\beta}F_{\mu\sigma}+
\frac{\mu}{4}\varepsilon_{\rho\sigma\beta}g^{\rho\mu}g^{\sigma\nu}g^{\beta\alpha}F_{\mu\nu}A_\alpha.~~~~~~~~~~~~~~~~~~(B.1)
\]
Then the symmetrical energy-momentum tensor obtained from varying action by the metric tensor $T^{sym}_{\mu\nu}=\delta S/\delta g^{\mu\nu}$ can be find from the equation
\[
T^{sym}_{\mu\nu}=2\left[\frac{\partial {\cal L}}{\partial g^{\mu\nu}}-
\partial_\alpha \frac{\partial {\cal L}}{\partial(\partial_\alpha g^{\mu\nu})}\right]-g_{\mu\nu}{\cal L}.~~~~~~~~~~~~~~~~~~~~~~~~~~~(B.2)
\]
Replacing Eq.(B.1) into (B.2), we obtain
\[
T_{\mu\nu}^{sym}=-F_\mu^{~\alpha}F_{\nu\alpha}+\frac{\mu}{4}\varepsilon_\mu^{~\alpha\beta}\left(
2F_{\nu\alpha}A_\beta+F_{\alpha\beta}A_\nu\right)-g_{\mu\nu}{\cal L}.~~~~~~~~~~~(B.3)
\]
Returning to Euclidian metric, one can verify that according to \cite{Deser} the symmetrical Chern$-$Simons energy-momentum tensor vanishes:
\[
T_{\mu\nu}^{C-S}=\frac{\mu}{4}\varepsilon_{\mu\alpha\beta}\left(
2F_{\nu\alpha}A_\beta+F_{\alpha\beta}A_\nu\right)-\delta_{\mu\nu}\left( \frac{\mu}{4}\varepsilon_{\rho\sigma\alpha}F_{\rho\sigma}A_\alpha\right)=0.~~~(B.4)
\]
Equation (B.4) is valid because of the identity:
\[
\varepsilon_{\mu\alpha\beta}\left(
2F_{\nu\alpha}A_\beta+F_{\alpha\beta}A_\nu\right)=
\delta_{\mu\nu} \varepsilon_{\rho\sigma\alpha}F_{\rho\sigma}A_\alpha,~~~~~~~~~~~~~~~~~~~~~~~~~(B.5)
\]
which can be checked by evaluating the components in left and right sides of the equation.


text:
\section{ Introduction}

Soft Collinear Effective Theory (SCET)  is an effective field
theory describing the dynamics of highly energetic particles
moving close to the light-cone interacting with a background field
of soft quanta\cite{Bauer:2000ew,Bauer:2000yr,Bauer:2002aj,
Chay:2003zp,Chay:2003ju,Bauer:2004tj,Jain:2007dy,Wang:2008rk,Fleming:2009fe,Faisel:2011kq}.
It provides a systematic and rigorous way to deal with the decays
of the heavy hadrons that involve different energy scales.
Moreover, the power counting in SCET helps to reduce the
complexity of the calculations and
 the factorization formula provided by SCET is
perturbative to all powers in $\alpha_s$ expansion.

 We can classify two different effective theories: SCET$_{I}$ and
SCET$_{II}$ according to the momenta modes in the process under
consideration. SCET$_{I}$  is applicable in the processes in which
the  momenta modes are the  collinear and the  ultra soft  as in
the inclusive decays of a heavy meson
 such as $B\rightarrow X^*_s\gamma$ at the end point region
 and $e^-p\rightarrow e^-X$  at the threshold region in which there
 are only collinear and ultra soft  momenta modes.
  SCET$_{II}$  is applicable to the semi-inclusive or exclusive decays of a heavy
 meson such as $B\rightarrow D\pi$, $B\rightarrow K\pi$, $B\rightarrow \pi\nu e$,....etc
 in which there  are only collinear and soft momenta modes.

The amplitude of $ B\to {M_1M_2}$ where $M_1$ and $M_2$ are light
mesons  in SCET can be written as follows \begin{eqnarray}{\cal A}_{B\to
M_1M_2}^{SCET}
 &=& {\cal A}_ {B\to M_1M_2}^{LO} +  {\cal A}_{B\to M_1M_2}^ {\chi} +
 {\cal A}_{B\to M_1M_2}^{ann} +  {\cal A}_{B\to M_1M_2}^{c.c} \end{eqnarray}

Here ${\cal A}_ {B\to M_1M_2}^{LO}$ denotes the leading order
amplitude in the expansion $1/m_b$, $ {\cal A}_{B\to M_1M_2}^
{\chi}$ denotes the chirally enhanced penguin amplitude,  $ {\cal
A}_{B\to M_1M_2}^{ann}$ denotes the annihilation amplitude and $
{\cal A}_{B\to M_1M_2}^{c.c}$ denotes the long distance  charm
penguin contributions. In the following we give a brief account
for each amplitude.

At leading power in $(1/m_b)$ expansion, the full QCD effective
weak Hamiltonian of the $\Delta_B=1$ decays is matched into the
corresponding weak Hamiltonian in $SCET_I$ by integrating out  the
hard scale $ m_b $. Then, the $SCET_I$ weak Hamiltonian is
matched into the weak Hamiltonian $SCET_{II}$ by integrating out
the hard collinear modes with $p^2\sim \Lambda m_b$ and the
amplitude of the $\Delta_B=1$ decays at leading order in $\alpha_s$ expansion
can be obtained via~\cite{Bauer:2002aj}:

 \begin{eqnarray}
 {\cal A}^{LO}_{B\rightarrow M_1M_2}&=& - i\big\langle
M_1M_2 \big| H^{SCET_{II}}_W \big|
\bar{B}\big\rangle\nonumber\\&=&\frac{G_F
m_B^2}{\sqrt{2}}\Big(f_{M_1}[\int^1_0
 dudzT_{M_1J}(u,z)\zeta^{B M_2}_J(z)\phi_{M_1}(u)\nonumber\\
  &+& \zeta^{B M_2}\int^1_0
 du T_{M_1\zeta}(u)\phi_{M_1}(u)]+(M_1 \leftrightarrow
 M_2)\Big).\label{amp1}
 \end{eqnarray}

At leading order in $\alpha_s$ expansion, the parameters $\zeta^{B
(M_1,M_2)}$, $\zeta_J^{B (M_1,M_2)}$ are treated
 as hadronic parameters and can be determined through the $\chi^2$ fit method using  the
 non leptonic decay  experimental data of  the branching fractions and CP
 asymmetries. The hard kernels $T_{(M_1,M_2)\zeta}$ and $T_{(M_1,M_2) J}$ can be expressed in terms of
 $c_i^{(f)}$ and $b_i^{(f)}$ which are functions of the
 Wilson coefficients  as follows \cite{Jain:2007dy}
 \begin{eqnarray} \label{Tz}
 T_{1\zeta}(u) &=
  {\cal C}_{u_L}^{BM_2} \, {\cal C}_{f_Lu}^{M_1} \, c_1^{(f)}(u)
  +   {\cal C}_{f_L}^{BM_2} \, {\cal C}_{u_Lu}^{M_1} \, c_2^{(f)}(u)
  \nonumber \\
 & +   {\cal C}_{f_L}^{BM_2} \, {\cal C}_{u_Ru}^{M_1} \, c_3^{(f)}(u)
  +  {\cal C}_{q_L}^{BM_2} \, {\cal C}_{f_Lq}^{M_1} \, c_4^{(f)}(u)
  ,\nonumber\\
 T_{1 J}(u,z) &=
   {\cal C}_{u_L}^{BM_2} \, {\cal C}_{f_Lu}^{M_1} \, b_1^{(f)}(u,z)
  +   {\cal C}_{f_L}^{BM_2} \, {\cal C}_{u_Lu}^{M_1} \, b_2^{(f)}(u,z)
   \nonumber\\
 &+   {\cal C}_{f_L}^{BM_2} \, {\cal C}_{u_Ru}^{M_1} \, b_3^{(f)}(u,z)
  +  {\cal C}_{q_L}^{BM_2} \, {\cal C}_{f_Lq}^{M_1} \, b_4^{(f)}(u,z).
\end {eqnarray}
here $f$ stands for $d$ or $s$ and  ${\cal C}_i^{BM}$ and ${\cal
C}_{i}^M$ are Clebsch-Gordan coefficients that depend on the
flavor content of the final state mesons.  $c_i^{(f)}$ and
$b_i^{(f)}$ are given by \cite{Bauer:2004tj}
\begin{eqnarray}\label{cmatch}
c_{1,2}^{(f)}&=&\lambda_u^{(f)}\Big[C_{1,2}+\frac{1}{N}C_{2,1}\Big]
-\lambda_t^{(f)}\frac{3}{2}\Big[\frac{1}{N}C_{9,10}
+C_{10,9}\Big]+ \Delta c_{1,2}^{(f)},\nonumber\\
c_3^{(f)}&=&-\frac{3}{2}\lambda_t^{(f)}\Big[C_7+\frac{1}{N}C_8\Big]
+ \Delta c_3^{(f)},\nonumber\\
c_4{(f)}&=&-\lambda_t^{(f)}\Big[\frac{1}{N}C_3+C_4
-\frac{1}{2N}C_9-\frac{1}{2}C_10\Big]+ \Delta c_4^{(f)},
\end{eqnarray}
and
\begin{eqnarray}
b_{1,2}^{(f)}&=&\lambda_u^{(f)}\Big[C_{1,2}+\frac{1}{N}
\Big(1-\frac{m_b}{\omega_3}\Big)C_{2,1}\Big]-
\lambda_t^{(f)}\frac{3}{2}\Big[C_{10,9}+\frac{1}{N}
\Big(1-\frac{m_b}{\omega_3}\Big)C_{9,10}\Big]+ \Delta b_{1,2}^{(f)},\nonumber\\
b_3^{(f)}&=&-\lambda_t^{(f)}\frac{3}{2}\Big[C_7
+\Big(1-\frac{m_b}{\omega_2}\Big)\frac{1}{N}C_8\Big]+ \Delta b_3^{(f)},\nonumber\\
b_4^{(f)}&=&-\lambda_t^{(f)}\Big[C_4+\frac{1}{N}
\Big(1-\frac{m_b}{\omega_3}\Big)C_3\Big]+\lambda_t^{(f)}\frac{1}{2}
\Big[C_{10}+\frac{1}{N}\Big(1-\frac{m_b}{\omega_3}\Big)C_9\Big]+
\Delta b_4^{(f)},
\end{eqnarray}
where $\omega_2=m_bu$ and $\omega_3=-m_b\bar{u}$. $u$ and
$\bar{u}=1-u $ are momentum fractions for the quark and antiquark
$\bar{n}$ collinear fields. The $\Delta c_i^{(f)}$ and $\Delta
b_i^{(f)}$ denote terms depending on $\alpha_s$ generated by
matching from $H_W$. The ${\cal O}(\alpha_s)$ contribution to
$\Delta c_i^{(f)}$ has been calculated
in Refs.(\cite{Beneke:1999br,Beneke:2000ry,Chay:2003ju}) and later in Ref.
(\cite{Jain:2007dy}) while the ${\cal O}(\alpha_s)$contribution to
$\Delta b_i^{(f)}$ has been calculated
in Refs.(\cite{Beneke:2005vv,Beneke:2006mk,Jain:2007dy}).

Corrections of order $\alpha_s(\mu_h)(\mu_M\Lambda/m_b^2)$ where
$\mu_M$ is the chiral scale parameter generate the so called
Chirally enhanced penguins amplitude $ {\cal A}_{B\to M_1M_2}^
{\chi}$\cite{Jain:2007dy}. $\mu_M$ for kaons and pions can be of
order $(2GeV)$ and therefore chirally enhanced terms can compete
with the order $\alpha_s(\mu_h)(\Lambda/m_b)$ terms.   The
chirally enhanced amplitude for  $B\to M_1M_2$ decays is given
by\cite{Jain:2007dy}

\begin{eqnarray}\label{chienhanced} A^\chi(\bar B\to M_1 M_2) &=& \frac{G_F
m_B^2}{\sqrt2}  \bigg\{- \frac{\mu_{M_1}  f_{M_1}}{3m_B}
\zeta^{BM_2}  \int_0^1 du R_{1}(u)\phi_{pp}^{M_1}(u)
   + (1\leftrightarrow 2)\nonumber\\
   &-&  \frac{\mu_{M_1} f_{M_1}}{3m_B} \int_0^1 du dz
   R_{1}^{J}(u,z) \zeta_J^{BM_2}(z) \phi_{pp}^{M_1}(u)
 + (1\leftrightarrow 2) \nonumber\\ & -& \frac{\mu_{M_2}f_{M_1}}{6 m_B}\int_0^1 dudz
   R_{1}^{\chi}(u,z) \zeta_{\chi}^{BM_2}(z) \phi^{M_1}(u)
 +(1\leftrightarrow 2) \bigg\}
 \end{eqnarray}

The factors $\mu_M$ are generated by pseudoscalars and so they
vanish for vector mesons~\cite{Jain:2007dy}. The pseudoscalar
light cone amplitude $\phi_{pp}^M(u)$ is defined as
\cite{Hardmeier:2003ig,Arnesen:2006vb}
\begin{equation}
\phi_{pp}^P(u) =3u[\phi_p^P(u)+\phi_\sigma^{P\prime}(u)/6+ 2
f_{3P}/(f_P \mu_P) \int dy'/y' \phi_{3P}(y-y',y)].
 \end{equation}

 The hard kernels $R_{K},R_{\pi},R_{K}^J,R_{\pi}^J,R_{K}^{\chi}$
and $ R_{\pi}^{\chi}$ can be expressed in terms of Clebsch-Gordan
coefficients for the different final states as\cite{Jain:2007dy}
\begin{eqnarray}
\label{chihard} R_{1}(u) &=  {\cal C}_{q_R}^{BM_{2}} {\cal
C}_{f_Lq}^{M_{1}}
  \Big[ c^{\chi}_{1(qfq)}+\frac{3}{2}e_q \,c^{\chi}_{2(qfq)} \Big]
\,, \\
R_{1}^J(u,z) &=  {\cal C}_{q_R}^{BM_{2}} {\cal C}_{f_Lq}^{M_{1}}
  \Big[ b^{\chi}_{3(qfq)}+\frac{3}{2}e_q \,b^{\chi}_{4(qfq)} \Big]
\,, \nonumber \\
R_{1}^{\chi}(u,z) &=
  {\cal C}_{q_L}^{BM_{2}} {\cal C}_{f_Lq}^{M_{1}} \, b^{\chi}_{1(qfq)}
 + {\cal C}_{u_L}^{BM_{2}} {\cal C}_{f_Lu}^{M_{1}}\,  b^{\chi}_{1(ufu)} \nonumber \\
& +{\cal C}_{f_L}^{BM_{2}} {\cal C}_{u_L u}^{M_{1}}\,
b^{\chi}_{1(fuu)}
  +{\cal C}_{f_L}^{BM_{2}} {\cal C}_{u_R u}^{M_{1}}\, b^{\chi}_{2(fuu)} \, .\nonumber
\end{eqnarray}
Summation over $q=u,d,s$ is implicit and  $c_{i}^{\chi}$ and
$b_{i}^{\chi}$ are expressed in terms of the short-distance Wilson
coefficients and can be found in Ref.\cite{Jain:2007dy}.

 Annihilation amplitudes ${\cal A}_{B\to M_1M_2}^{ann}$ have been studied
 in PQCD and QCD factorization in Refs.(\cite{Keum:2000ph,Lu:2000em,Beneke:2001ev,Kagan:2004uw}).
Within SCET, the annihilation contribution becomes factorizable
and  real at leading order,  $ {\cal O}(\alpha_s(m_b)\Lambda/m_b)$
\cite{Manohar:2006nz}.  Their size are small and contains large
uncertainty compared to the other contributions\cite{Arnesen:2006vb,Jain:2007dy}.

  In SCET, charm penguins are  treated as non perturbative and its amplitude is parameterized as
 \begin{equation} {\cal A}_{B\to M_1 M_2}^{c.c}=|{\cal A}_{B\to M_1
M_2}^{c.c}|e^{i \delta_{cc}} \end{equation} where $\delta_{cc}$ is the strong
phase of the charm penguin. The modulus and the phase of the charm penguin
are fixed through the fitting with non leptonic decays in a
similar way to the hadronic parameters $\zeta^{B (M_1,M_2)}$,
$\zeta_J^{B (M_1,M_2)}$.

\section{SM contribution to the CP asymmetries of $B\to \pi K^*$ and $B\to
\rho K$ decays}\label{sec:SM}

Withen SCET, the predicted  branching ratios of the decay modes
$B\to \pi K^*$ and $B\to \rho K$  are in agreements with their
corresponding experimental values in most of the decay
modes\cite{Wang:2008rk,Faisel:2011qt}. On the other hand, the SM
predictions for the CP asymmetries of $B^+\to \pi^0 K^{*\,+}$ has
different sign in comparison with the experimental measurement and
the predicted CP asymmetries in many of the decay modes are in
agreement with the experimental measurements due to the large
errors in these measurements\cite{Faisel:2011qt}. Moreover, the
predicted CP asymmetry of $\bar{B}\to \pi^0 \bar{K}^{*\,0}$ and
$B^+\to\rho^0 K^+$ disagree with the experimental results within
$1\sigma.$ error of the experimental data.  Note, SCET provides
large strong  phases and thus with new sources of weak CP
violation one can expect enhancement of these asymmetries. This
possibility will be discussed in the next section considering
supersymmetry (SUSY) as a possible candidate of physics beyond SM
that has new sources of weak phases.

\section{SUSY contributions to the CP asymmetries of$ B\to \rho K$ and $ B\to \pi K^*$}\label{sec:SUSY}

Supersymmetry  has new sources for CP violation which can account
for the baryon number asymmetry and affect other CP violating
observables in the B and K decays. The effects of these phases on
the CP asymmetries in semi-leptonic $\tau$ decays has been studied
in Refs.(\cite{Delepine:2006fv,Delepine:2007qg,Delepine:2008zzb}).
In SUSY, Flavor Changing Neutral Current(FCNC) and CP quantities
are sensitive to particular entries in the mass matrices of the
scalar fermions. Thus it is useful to adopt a model independent-
parametrization, the so-called Mass Insertion Approximation (MIA)
where all the couplings of fermions and sfermions to neutral
gauginos are flavor diagonal~\cite{Hall:1985dx}.  The Flavor
Changing structure of the $A-B$ sfermion propagator  is exhibited
by its non-diagonality and it can be expanded as
\begin{equation}
\langle \tilde{f}_{A}^{a}\tilde{f}_{B}^{b\ast }\rangle \simeq \frac{i\delta _{ab}}{k^{2}-\tilde{m}%
^{2}}+\frac{i(\Delta
_{AB}^{f})_{ab}}{(k^{2}-\tilde{m}^{2})^{2}}+O(\Delta ^{2}),
\end{equation}%
where $a,b=(1,2,3)$ are flavor indices, $\Delta$ are the
off-diagonal terms in the $(M^2_{\tilde{f}})_{AB}$ and $I$ is the
unit matrix. It is convenient to define a dimensionless quantity
$(\delta
_{AB}^{f})_{ab}\equiv (\Delta _{AB}^{f})_{ab}/\tilde{m}^{2}.$ As long as $%
(\Delta _{AB}^{f})_{ab}$ is smaller than $\tilde{m}^{2}$ we can
consider only the first order term in $(\delta _{AB}^{f})_{ab}$ of
the sfermion propagator expansion.

 The parameters $(\delta _{AB}^{f})_{ab}$ can be constrained
through vacuum stability argument~\cite{Casas:1996de},
experimental measurements concerning FCNC and CP violating
phenomena~\cite{Gabbiani:1996hi}. Recent studies about other
possible constraints can be found in
Refs.(~\cite{Crivellin:2008mq,Crivellin:2009ar,Crivellin:2010gw}).

The mass insertions $(\delta^u_{RL})_{32}$ and
$(\delta^u_{LR})_{32}$ are not constrained by $b \to s \gamma$
 and so we can set them as $(\delta^u_{RL})_{32}=(\delta^u_{LR})_{32}= e^{i\delta_u}$
 where $\delta_u$ is the phase that can vary from $-\pi$ to $\pi$.  It should be noted that in order
 to have a well defined Mass Insertion Approximation scheme, it is necessary to have
 $|(\delta^f_{AB})_{ab} | < 1$  but here in order
 to maximize the SUSY CP-violating contributions we take it of order one.
 Applying $b \to s \gamma$ constraints leads to the following parametrization~\cite{Huitu:2009st}

 \begin{equation}
 (\delta^d_{LL})_{23}= e^{i\delta_d}~~~~~~~~~~~ (\delta^d_{LR})_{23}=(\delta^d_{RL})_{23}= 0.01 e^{i\delta}\label{masinscons}
 \end{equation}

 We consider two scenarios, the first one with a
single mass insertion where we keep only one mass insertion per
time and take the other mass insertions to be zero and the second
scenario with two mass insertions will be considered only in the
cases when one single mass insertion is not sufficient to
accommodate the experimental measurement. After setting the
different  mass insertions as mentioned above, we find that, the
terms that contain the mass insertions $(\delta^{u}_{RL})_{32}$
and $(\delta^{u}_{LR})_{32}$ will be small in comparison  with the
other terms and thus we expect that their contributions to the
asymmetries will be small. These terms are obtained from diagrams
mediated by the chargino exchange and thus we see that gluino
contributions give the dominant contributions. The results  are
presented in Figures(\ref{singlemas},\ref{singlemas2},\ref{singlemas3},
\ref{singlemas4})

\begin{figure}[tbhp]
\includegraphics[width=6.5cm,height=7cm]{Graph11}
\hspace{1.cm}
\includegraphics*[width=6.5cm,height=7cm]{Graph22}
\medskip
\caption{ CP asymmetries  versus the phase of the
$(\delta^{d}_{AB})_{23}$ where A and B denote the chirality i.e.
L, R. for 3 different mass insertions.
 The left diagram corresponds to  $A_{CP}(B^+\to \pi^+\bar{K}^{*\,0})$ while the right diagram corresponds to
 $A_{CP}(B^+\to \pi^0 K^{*\,+})$. In both diagrams we take only one mass insertion
per time and vary the phase of  from $-\pi$ to $\pi$. The
horizontal lines in both diagrams represent the experimental
measurement to $1\sigma$\cite{Faisel:2011qt}.} \label{singlemas}
\end{figure}

 In Figure\ref{singlemas},we show the  CP asymmetries,
$A_{CP}(B^+\to \pi^+\bar{K}^{*\,0})$ and  $A_{CP}(B^+\to \pi^0
K^{*\,+})$  versus the phase of the $(\delta^{d}_{AB})_{32}$
 where A and B denote the chirality i.e. L and R. for 3 different mass insertions. The horizontal lines
in both diagrams represent the experimental measurements to
$1\sigma$.  As can be seen from  Figure\ref{singlemas}~left, for
all gluino mass insertions, the value of the CP asymmetry
$A_{CP}(B^+\to \pi^+\bar{K}^{*\,0})$ is enhanced to accommodate
the experimental measurement of the asymmetry within $1\sigma$ for
many values of the phase of the mass insertions. On the other
hand,Figure \ref{singlemas}right shows that  the CP asymmetry
$A_{CP}(B^+\to \pi^0 K^{*\,+})$ is enhanced  to accommodate  the
experimental measurement within $1\sigma$ for all values of the
phase of the mass insertions.

 \begin{figure}[tbhp]
\includegraphics[width=6.5cm,height=7cm]{Graph33}
\hspace{1.cm}
\includegraphics*[width=6.5cm,height=7cm]{Graph44}
\medskip
\caption{CP asymmetries  versus the phase of the
$(\delta^{d}_{AB})_{23}$ where A and B denote the chirality i.e.
L, R. for 3 different mass insertions.
 The left diagram corresponds to  $A_{CP}(B^0\to \pi^0\bar{K}^{*\,0})$ while the right diagram corresponds to
$A_{CP}(B^0\to \pi^-K^{*\,+})$. In both diagrams we take only one
mass insertion per time and vary the phase of  from $-\pi$ to
$\pi$. The horizontal lines in both diagrams represent the
experimental measurement to $1\sigma$\cite{Faisel:2011qt}.}
\label{singlemas2}
\end{figure}

In Fig.\ref{singlemas2}  we show  the two asymmetries,
$A_{CP}(B^0\to \pi^0\bar{K}^{*\,0})$ and  $A_{CP}(B^0\to
\pi^-K^{*\,+})$ versus the phase of the $(\delta^{d}_{AB})_{32}$
as before.  Clearly from  Fig.\ref{singlemas2} left,
$A_{CP}(B^0\to \pi^0\bar{K}^{*\,0})$  lies within $1\sigma$ range
of its experimental value for  many values of the phase of the
mass insertion $(\delta^{d}_{LR})_{23}$ only. The reason is that
the two mass insertions $(\delta^d_{LL})_{23}$ and
$(\delta^d_{RL})_{23}$ have equal contributions to the CP
asymmetries which  will be smaller than the case of using
$(\delta^d_{LR})_{23}$.  On the other hand, Fig.\ref{singlemas2}
right, one sees that  $A_{CP}(B^0\to \pi^-K^{*\,+})$ can be
accommodated within $1\sigma$ for many values of the phase of the
three gluino mass insertions.

Finally we present  the CP asymmetries of the decay modes $B^+\to
\rho^+ K^{0}$ and $B^+\to \rho^0 K^{+}$ in Fig.(\ref{singlemas3}).
In Fig.(\ref{singlemas3}) we do not show the horizontal lines
representing the $1\sigma$ range of the experimental measurement
as the three curves of the $A_{CP}(B^+\to \rho^+ K^{0})$
corresponding to the three gluino mass insertions totally lie in
this $1\sigma$ range for all values of the phase of the mass
insertions.  On the other hand, Fig.\ref{singlemas3} right, we see
that $A_{CP}(B^+\to \rho^0 K^{+})$  can not be accommodated within
$1\sigma$ for any value of the phase of all gluino mass
insertions. This motivates us to consider the second scenario with
two mass insertions which is shown in Fig.\ref{singlemas4}.

\begin{figure}[tbhp]
\includegraphics[width=6.5cm,height=7cm]{Graph55}
\hspace{1.cm}
\includegraphics*[width=6.5cm,height=7cm]{Graph66}
\medskip
\caption{CP asymmetries  versus the phase of the
$(\delta^{d}_{AB})_{23}$ where A and B denote the chirality i.e.
L, R. for 3 different mass insertions.
 The left diagram corresponds to  $A_{CP}(B^+\to \rho^+ K^{0})$ while the right diagram corresponds to
$A_{CP}(B^+\to \rho^0 K^{+})$. In both diagrams we take only one
mass insertion per time and vary the phase of  from $-\pi$ to
$\pi$\cite{Faisel:2011qt}.} \label{singlemas3}
\end{figure}

The left diagram correspond to gluino contributions where we keep
the two mass insertions $(\delta^{d}_{LR})_{23}$ and
$(\delta^{d}_{RL})_{23}$  and set the other mass insertions to
zero.  The right diagram correspond to both gluino and chargino
contributions where we keep  the two mass insertions
$(\delta^{d}_{LR})_{23}$ and $(\delta^{u}_{RL})_{32}$ and set the
other mass insertions to zero. In both diagrams we assume that the
two mass insertion have equal phases and we vary the phase  from
$-\pi$ to $\pi$.  As can be seen from Fig.\ref{singlemas4} left,
two gluino mass insertions can not accommodate the experimental
measurement for any value of the phase of the mass insertion. On
the other hand from Fig.\ref{singlemas4} right, two mass
insertions one corresponding to chargino contribution and the
other corresponding to  gluino contribution can not  accommodate
the experimental measurements. We find that in order to
accommodate the CP symmetry in this case the Wilson coefficient
$C^{\tilde{g}}_{9}$ should be increased at least by a factor
$-6\pi/\alpha$ without violating any constraints on the SUSY
parameter space which is shown in Fig.\ref{singlemas5}.

\begin{figure}[tbhp]
\includegraphics[width=6.5cm,height=7cm]{Graph77}
\hspace{1.cm}
\includegraphics*[width=6.5cm,height=7cm]{Graph88}
\medskip
\caption{CP asymmetry of $A_{CP}(B^+\to \rho^0 K^{+})$ versus the
phase of the mass insertion  for 2 different mass insertions.
 The left diagram correspond to gluino contributions where we keep
 the two mass insertions $(\delta^{d}_{LR})_{23}$ and $(\delta^{d}_{RL})_{23}$
 and set the other mass insertions to zero.   The right diagram correspond to
both gluino and chargino contributions where we keep
 the two mass insertions $(\delta^{d}_{LR})_{23}$ and $(\delta^{u}_{RL})_{32}$
 and set the other mass insertions to zero.  The horizontal lines in both diagrams represent the experimental
measurements to $1\sigma$\cite{Faisel:2011qt}.} \label{singlemas4}
\end{figure}

\begin{figure}[tbhp]
\includegraphics[width=6.5cm,height=7cm]{Graph7}
\medskip
\caption{CP asymmetry of $A_{CP}(B^+\to \rho^0 K^{+})$ versus the
phase of the mass insertion  for 2 different mass insertions
correspond to gluino contributions where we keep the two mass
insertions $(\delta^{d}_{LR})_{23}$ and $(\delta^{d}_{LL})_{23}$
and set the other mass insertions to zero.  We assume that the two
mass insertion have equal phases and we vary the phase  from
$-\pi$ to $\pi$. The horizontal lines in the diagram represent the
experimental measurements to $1\sigma$\cite{Faisel:2011qt}.}
\label{singlemas5}
\end{figure}

\section{Conclusion}\label{sec:conclusion}

 In this talk we discussed SUSY contributions to the direct CP asymmetries of  $B \to \pi K^*$ and
$B\to\rho K$ decays within Soft Collinear Effective Theory. We
considered non minimal flavor  SUSY models and applied  the mass
insertion approximation to analyze SUSY contributions to the CP
asymmetries of $B \to \pi K^*$ and $B\to\rho K$ decays.  We  show
that in most decay channels, direct CP asymmetries can be
significantly enhanced by the SUSY contributions mediated by
gluino exchange and thus accommodate the experimental results. For
the decay mode $B^+\to \rho^+ K^{0}$, we find that the enhancement
is not enough to accommodate the CP asymmetry.  To accommodate the
CP asymmetry of $B^+\to \rho^+ K^{0}$, the Wilson coefficient
$C^{\tilde{g}}_{9}$ should be increased at least by a factor
$-6\pi/\alpha$.

\section*{Acknowledgement}

Gaber Faisel would like to thank David Delepine and Mostafa
Shalaby for their collaborations in the realization of this work.
Gaber Faisel's  work is supported by the National Science Council
of R.O.C. under grants NSC 99-2112-M-008-003-MY3 and NSC
99-2811-M-008-085.

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}\label{sec:1}
Already two of the magnetic moments of the baryon decuplet have been
measured. The most precise experiment for the $\Delta^{++}$ is a pion
bremsstrahlung analysis \cite{boss91}. The magnetic moment of the
$\Omega^-$ has been measured by the E756 collaboration \cite{dieh91}
where $\Omega^-$ hyperons are produced by a polarized neutral spin
transfer reaction. The final result from the succeeding experiment
called E800 is expected within the year \cite{rame93}. Theoretical
predictions for the $\Delta^{++}$ and $\Omega^-$ magnetic moments
have been given in many models. In particular, the simple, additive
quark model predicts the ratio of the magnetic moment of $\Delta^{++}$
and proton to be 2, and it predicts the ratio of the magnetic moment of
$\Omega^-$ and $\Lambda^0$ to be 3.

The analysis of Ref.~\cite{boss91} finds the ratio $\mu(\Delta^{++})/\mu(p)$
to be $1.62\pm 0.18$, which is significantly smaller than the additive
quark model. Most models however predict a ratio 2 or higher, a value
that is only compatible with older experiments.

We recently investigated the predictive power of a relativistic
constituent quark model formulated on the light-front \cite{schl93}. It
provides a simple model wherein we have overall an excellent and
consistent picture of the magnetic moments and the semileptonic
decays of the baryon octet. The parameters of this model
are the constituent quark mass $m$ and the scale parameter $\beta$, which
is a measure for the size of the baryon. All parameters except $\beta$
for $\Omega^-$ have been determined and fixed in Ref.~\cite{schl93}.
We predict the magnetic moments for the baryon decuplet and find
that the ratio $\mu(\Delta^{++})/\mu(p)$ is in excellent agreement with
the experiment, while our ratio $\mu(\Omega^-)/\mu(\Lambda^0)$ is
slightly higher than the experimental ratio.

In Sec.~\ref{sec:2} we give a brief summary of our model as described in
Ref.~\cite{schl93}. Section~\ref{sec:3} contains the explicit expressions
for the magnetic moments of the baryon decuplet. The numerical
results are presented in Sec.~\ref{sec:4}, and are compared with experiment
and other calculations. We summarize our investigation in a
concluding Sec.~\ref{sec:5}.

\section{Quark-model wave function}\label{sec:2}
The constituent quark model that we are going to use is described in a
previous paper \cite{schl93}. In order to fix the notation we repeat
here the essential formalism.

We shall formulate our model on the light-front which is specified by the
invariant hypersurface $x^+=x^0+x^3=0$.
The following notation is used: The
four-vector is given by $x = (x^+,x^-,x_\perp)$, where $x^\pm = x^0
\pm x^3$ and $x_\perp=(x^1,x^2)$.
Light-front vectors are denoted by boldface ${\bf x} =
(x^+,x_\perp)$, and they are covariant under kinematic Lorentz
transformations \cite{chun88}. The three momenta ${\bf p}_i$ of the quarks
can be transformed to the total and relative momenta to facilitate
the separation of the center of mass motion
\cite{bakk79}:

\begin{eqnarray}
{\bf P}&=& {\bf p}_1+ {\bf p}_2 +{\bf p}_3,
\quad \xi={p_1^+\over p_1^++p_2^+}\;,
\quad
\eta={p_1^++p_2^+\over P^+}\;,\nonumber\\
&&\\
q_\perp&=&(1-\xi)p_{1\perp}-\xi p_{2\perp}\;, \quad
Q_\perp =(1-\eta)(p_{1\perp}+p_{2\perp})-\eta p_{3\perp}\;.\nonumber
\end{eqnarray}
Note that the four-vectors are not conserved, i.e., $p_1+p_2+p_3\not= P$.
In the light-front dynamics the Hamiltonian takes the form
\begin{equation}
H={P^2_\perp + M^2 \over P^+}\;,
\end{equation}
where $M$ is the mass operator with the interaction term $W$
\begin{eqnarray}
M &=&M_0+W\;, \nonumber\\
M_0^2&=&{Q_\perp^2\over \eta(1-\eta)}+{M_3^2\over \eta}+{m_3^2\over 1-\eta},
\label{eq:2.3} \\
M_3^2&=&{q_\perp^2\over \xi (1-\xi)}+{m_1^2 \over \xi}+{m_2^2\over 1-\xi}\;,
\nonumber
\end{eqnarray}
with $m_i$ being the masses of the constituent quarks. To get a clearer
picture of $M_0$ we transform to $q_3$ and $Q_3$ by
\begin{eqnarray}
\xi&=&{E_1+q_3\over E_1+E_2}\;, \quad \eta={E_{12}+Q_3\over E_{12}+E_3}\;,
\nonumber\\
&&\\
E_{1/2}&=&({\bf q}^2+m_{1/2}^2)^{1/2}\;,\quad
E_{3}=({\bf Q}^2+m_{3}^2)^{1/2}\;,\quad
E_{12}=({\bf Q}^2+M_{3}^2)^{1/2}\;,\nonumber
\end{eqnarray}
where ${\bf q}=(q_1,q_2,q_3)$, and ${\bf Q}=(Q_1,Q_2,Q_3)$.
The expression for the mass operator is now simply
\begin{equation}
M_0=E_{12}+E_3\;, \quad M_3=E_1+E_2\;.
\end{equation}

All relevant matrix elements we investigate are
related to
\begin{equation}
\left< {\bf p}\,'\left|\bar q \gamma^+ q\right| {\bf p}\right>
\sqrt{p^{'+}p^+} \equiv {\cal M}^+ ,
\end{equation}
where the state $| {\bf p}\,\rangle\equiv|p\rangle/\sqrt{p^+}$ is
normalized according to
\begin{equation}
\left< {\bf p}\,' | {\bf p} \,\right>=\delta ({\bf p}\,' -{\bf p}\,) .
\end{equation}
The plus component of the matrix element is \cite{schl93}:
\begin{equation}
{\cal M}^+=
3{N_c\over (2\pi)^6}\int d^3qd^3Q\left({E'_3E'_{12}M\over E_3E_{12}M'}
\right)^{1/2}\Psi^\dagger({\bf q}',{\bf Q}',
\lambda')\Psi({\bf q},{\bf Q},\lambda)\;.
\label{eq:me}
\end{equation}

We can write the $\Delta^{++}$ for instance as
\begin{equation}
|\Delta^{++}\rangle = (uuu)\chi\phi .
\label{eq:wf}
\end{equation}
with $\chi$ being the spin wave function and $\phi$ being the momentum
distribution. For the latter we choose a function of $M^2_0$, in
particular we choose the same harmonic oscillator and pole type wave
function as in Ref.~\cite{schl93}:
\begin{eqnarray}
\phi_H & = & N_H \exp{(-M^2_0/2\beta^2)} ,\nonumber\\
\phi_P & = & N_P (1+M^2_0/\beta^2)^{-3.5} .
\label{eq:phi}
\end{eqnarray}
The normalization constants $N_H$ and $N_P$ are given by the constraint:
\begin{equation}
\frac{N_c}{(2\pi)^6}\int d^3q d^3Q |\phi|^2 = 1 .

\end{equation}
The spin wave function $\chi$ for the spin $\frac{3}{2}$ and
$\frac{1}{2}$ are given by
\begin{eqnarray}
\chi_\frac{3}{2} & = & \uparrow\uparrow\uparrow ,\nonumber\\
\chi_\frac{1}{2} & = & (\uparrow\uparrow\downarrow+
\uparrow\downarrow\uparrow+
\downarrow\uparrow\uparrow)/\sqrt{3} .

\end{eqnarray}
In order to get the baryon to be an eigenfunction of the spin operator we
still have to rotate the quark spins by the Melosh transformation
\cite{melo74} as follows:
\begin{equation}
\uparrow={\cal R}_i\pmatrix{1\cr 0} \hbox{  and  }
\downarrow={\cal R}_i\pmatrix{0\cr 1}\;.
\end{equation}
The Melosh rotation for three particles can be written as
\begin{eqnarray}
{\cal R}_1&=&{1\over \sqrt{a^2+Q_\perp^2}\sqrt{c^2+q_\perp^2}}
\pmatrix{ac-q_RQ_L&-aq_L-cQ_L\cr
         cQ_R+aq_R&ac-q_LQ_R}\;,\nonumber\\
{\cal R}_2&=&{1\over \sqrt{a^2+Q_\perp^2}\sqrt{d^2+q_\perp^2}}
\pmatrix{ad+q_RQ_L&aq_L-dQ_L\cr
         dQ_R-aq_R&ad+q_LQ_R}\;,\label{eq:melosh}\\
{\cal R}_3&=&{1\over \sqrt{b^2+Q_\perp^2}}\pmatrix{b&Q_L\cr
         -Q_R&b}\;,\nonumber
\end{eqnarray}
where we have defined the following quantities:
\begin{eqnarray}
a&=&M_3+\eta M\;,\quad b=m_3+(1-\eta)M\;,\nonumber\\
c&=&m_1+\xi M_3\;, \quad d=m_2+(1-\xi)M_3\;,\nonumber\\
q_R&=&q_1+iq_2\;,\quad q_L=q_1-iq_2\;,\\
Q_R&=&Q_1+iQ_2\;,\quad Q_L=Q_1-iQ_2\;.\nonumber
\end{eqnarray}

\section{Magnetic moments for the baryon decuplet}\label{sec:3}
The electromagnetic current matrix element for spin $\frac{3}{2}$
particles can be written as
\begin{equation}
\langle p',s'|J^\mu(0)|p,s\rangle=\bar u_\alpha(p',s'){\cal
O}^{\alpha\mu\beta} u_\beta(p,s) ,

\end{equation}
where $u_\alpha(p,s)$ is a Rarita-Schwinger spin-vector with momentum $p$
and spin $s$. The Lorentz covariant form for the tensor ${\cal
O}^{\alpha\mu\beta}$ may be written as
\begin{equation}
{\cal
O}^{\alpha\mu\beta}=g^{\alpha\beta}\left(F_1\gamma^\mu+\frac{F_2}{2M_B}
i\sigma^{\mu\nu}K_\nu\right)+\frac{K^\alpha K^\beta}{2M_B^2}
\left(F_3\gamma^\mu+\frac{F_4}{2M_B} i\sigma^{\mu\nu}K_\nu\right)

\end{equation}
with momentum transfer $K=p'-p$ and baryon mass $M_B$. For $K^2=0$ the
form factors $F_1$ and $F_2$ are respectively equal to the charge and the
anomalous magnetic moment, and the magnetic
moment is $\mu=F_1(0)+F_2(0)$. For this analysis we are therefore not
interested in the form factors $F_3$ and $F_4$. In order to use
Eq.~(\ref{eq:me}) we
express the form factors $F_1$ and $F_2$ in terms of the plus component
of the current:
\begin{eqnarray}
F_1(0) & = & \bigg\langle p,\frac{3}{2}\bigg|J^+\bigg|
p,\frac{3}{2}\bigg\rangle ,\nonumber\\
K_\perp F_2(0) & = & 2\left[ \sqrt{3} M_B \bigg\langle p',\frac{1}{2}
\bigg|J^+\bigg|p,\frac{3}{2}\bigg\rangle +
K_\perp \bigg\langle p,\frac{3}{2}\bigg|J^+\bigg|p,\frac{3}{2}\bigg\rangle
\right].
\label{eq:plus}
\end{eqnarray}
By inserting our wave function from Eq.~(\ref{eq:wf}) into the spin conserving
matrix element $\langle p,\frac{3}{2}|J^+|p,\frac{3}{2}\rangle$ we get
the charge of the baryon. For the spin flipping matrix element we get
\begin{equation}
\bigg\langle p',\frac{1}{2}\bigg|J^+\bigg|p,\frac{3}{2}\bigg\rangle=K_\perp f ,

\end{equation}
where $f$ is given for the different baryons as follows:
\begin{eqnarray}
f(\Delta^{++}) & = & 2 I_\Delta , \nonumber\\
f(\Delta^{+}) & = & I_\Delta , \nonumber\\
f(\Delta^{0}) & = & 0 , \nonumber\\
f(\Delta^{-}) & = & -I_\Delta , \nonumber\\
f(\Sigma^{*+}) & = & (4I_{\Sigma^*}^{(2)}-I_{\Sigma^*}^{(3)})/3 ,\nonumber\\
f(\Sigma^{*0}) & = & (I_{\Sigma^*}^{(2)}-I_{\Sigma^*}^{(3)})/3 , \nonumber\\
f(\Sigma^{*-}) & =& (-2I_{\Sigma^*}^{(2)}-I_{\Sigma^*}^{(3)})/3 ,\nonumber\\
f(\Xi^{*0}) & = & (2I_{\Xi^*}^{(3)}-2I_{\Xi^*}^{(2)})/3 , \nonumber\\
f(\Xi^{*-}) & = & (-I_{\Xi^*}^{(3)}-2I_{\Xi^*}^{(2)})/3 ,\nonumber\\
f(\Omega^-) & = & -I_\Omega .
\end{eqnarray}
The integral $I$ is given by
\begin{equation}
I=\frac{N_c}{(2\pi)^6}\int d^3q d^3Q |\phi|^2(A_1+A_2+A_3)/\sqrt{3}
\label{eq:i}
\end{equation}
where the quantities $A_i$ are
\begin{eqnarray}
A_1&=&{\eta\left( a-{Q_\perp^2\over 2(1-\eta)M}\right)\over a^2+Q_
\perp^2}{c^2\over c^2+q^2_\perp}\;,\nonumber\\
A_2&=&{\eta\left( a-{Q_\perp^2\over 2(1-\eta)M}\right)\over a^2+Q_
\perp^2}{d^2\over d^2+q^2_\perp}\;,\nonumber\\
A_3&=&{{Q_\perp^2\over 2M}-\eta b\over b^2+Q_\perp^2}\;.\nonumber
\end{eqnarray}
Note that for equal $u$ and $d$ quark masses there is an equality $A_1=A_2$
under the integral. The masses $m_i$ in our equations are set as follows
$(m=m_u=m_d)$:
\begin{eqnarray}
I_\Delta\quad :&&\quad m_1=m_2=m_3=m\;,\nonumber\\
I_{\Sigma^*}^{(2)}\quad :&&\quad m_1=m_3=m,\quad m_2=m_s\;,\nonumber\\
I_{\Sigma^*}^{(3)}\quad :&&\quad m_1=m_2=m,\quad m_3=m_s\;,\nonumber\\
I_{\Xi^*}^{(2)}\quad :&&\quad m_1=m_3=m_s,\quad m_2=m\;,\nonumber\\
I_{\Xi^*}^{(3)}\quad :&&\quad m_1=m_2=m_s,\quad m_3=m\;,\nonumber\\
I_\Omega\quad :&&\quad m_1=m_2=m_3=m_s\;.\nonumber
\end{eqnarray}
In the nonrelativistic limit, $\beta/m\to 0$, and for equal quark masses
the integral $I$ does vanish.

\section{Results and discussions}\label{sec:4}

We have calculated the magnetic moments of the decuplet baryons using
Eqs.~(\ref{eq:plus})-(\ref{eq:i}). The parameters of the model, the
constituent quark mass $m$ and the scale parameter $\beta$, have been
determined and fixed by a successful fit to the electroweak
properties of the baryon octet ($\beta_\Delta=\beta_N,
\beta_{\Sigma^*}=\beta_\Sigma, \beta_{\Xi^*}=\beta_\Xi$)
\cite{schl93}. The only new
parameter $\beta_\Omega$ is chosen to fit nicely into the raising pattern
of the $\beta$s. The parameters for both wave functions $\phi_H$ and
$\phi_P$ in Eq.~(\ref{eq:phi}) are summarized in Table~\ref{table:para}.
The corresponding results for the magnetic moments for both wave functions
(H) and (P) are given in Table~\ref{table:res} and \ref{table:compare},
together with other calculations.

In the simple nonrelativistic quark model (NQM) \cite{pdg}
the magnetic moment of a decuplet baryon is the sum of the magnetic
moments of each quark composing the baryon. This is quite different to
the fact that the magnetic moment is derived from the elastic electron
scattering at non-zero momentum transfer. The lattice result (Latt) is
taken from a recent lattice simulation of quenched QCD \cite{lein93}. The
other model calculations include results from a cloudy bag model (CB)
\cite{kriv87}, the Skyrme model (Skyr) \cite{kim89}, a Bethe-Salpeter
formalism (BS) \cite{mitr84}, an additive quark model \cite{chao90} with
effective quark masses (EM) and a calculation in which relativistic
corrections (RC) to the baryon magnetic moments are considered
\cite{geor83}. The experimental value (Expt) for the $\Delta^{++}$
is taken
from a recent pion bremsstrahlung analysis \cite{boss91}, and the
experimental value for the $\Omega^-$ is
taken from a recent investigation
where $\Omega^-$ hyperons are produced by a polarized neutral spin
transfer reaction \cite{dieh91}.

It is instructive to compare the different results of the ratios of the
magnetic moments $\mu(\Delta^{++})/\mu(p)$ and
$\mu(\Omega^-)/\mu(\Lambda^0)$. In the NQM these ratios are parameter
free and given to be 2 and 3 respectively.

The experimental value for $\mu(\Delta^{++})/\mu(p)$ of
Ref.~\cite{boss91} is lower than 2 even if we include the uncertainty of
the model dependence of the measurement $(\pm 0.16)$. Only the BS
calculation and our result are in excellent agreement with experiment. The
results from Latt and CB are even larger than 2, a result only compatible
with older experimental values \cite{nefk78,witt88}. In this sense the
magnetic moment of the $\Delta^{++}$ is a good mean to distinguish between
the different models.

The experimental value for $\mu(\Omega^{-})/\mu(\Lambda^0)$ of
Ref.~\cite{dieh91} is slightly higher than 3. It is interesting
to note that every model beyond the NQM gives also a value larger than 3,
except for EM. Our value 3.49 is closest to the central value of the
experiment, although most of the calculations are within the statistical
$(\pm 0.28)$ and systematic $(\pm 0.23)$ error of this experiment. We
could even reduce our value for the ratio $\mu(\Omega^-)/\mu(\Lambda^0)$
by using a larger value of $\beta_\Omega$ or by choosing an appropriate
anomalous magnetic moment of the strange quark.

\section{Summary}\label{sec:5}
The magnetic moments of the baryon decuplet are calculated in a
relativistic constituent quark model using the light-front
formalism. The parameters of the model are fixed by fitting the baryon
octet physics, except for $\beta_\Omega$ which is chosen in a natural
way. It is a challenge for every hadronic model to get consistent values
for the magnetic moments for both the $\Delta^{++}$ and $\Omega^-$.
Our calculation for the magnetic
moment ratio $\mu(\Delta^{++})/\mu(p)$ is in excellent agreement
with the experimental ratio, while our ratio
$\mu(\Omega^-)/\mu(\Lambda^0)$ is slightly higher than the experimental
ratio.

\acknowledgments

It is a pleasure to thank W.~Jaus for helpful discussions. I am grateful
to X.~Ji for his hospitality at M.I.T. where the draft of this paper was
written.
This work was supported in part by the Schweizerischer Nationalfonds and
in part by the Department of Energy, contract DE-AC03-76SF00515.


text:
\section{Introduction}
\label{sec:intro}

Graphs are appealing mathematical tools for modelling pairwise relationships between data points, without being limited by the rigid structure of Euclidean spaces. In a graph, the data points are represented as vertices, whereas the pairwise relationships between vertices are described by weighted edges. Examples of graph-based representations can be found in numerous application domains, such as biology, social networks, financial and banking, mobility and traffic patterns.

The main focus of this paper is spectral embedding, which aims at representing the nodes of a graph into a low-dimensional space that preserves the connectivity patterns described by the graph edges. In this regard, one of most popular approaches consists of embedding the graph vertices into a subspace spanned by the eigenvectors of the graph Laplacian matrix corresponding to the $K$ smallest eigenvalues \cite{Schaeffer2007, Fortunato2010}. This leads to an optimization problem that can be solved by performing the eigendecomposition of the Laplacian matrix  \cite{Belkin2003}. However, this technique may not scale well when the number of graph vertices grows big.

A possible way to reduce the computational requirements of eigendecomposition involves the use of Nystr\"om method, a random sampling technique to approximate the eigenvectors of a large matrix \cite{Fowlkes2004}. Similarly, one can constrain the original eigendecomposition problem to satisfy the Nystr\"om formula, so as to obtain a reduced-size problem that achieves a lower approximation error \cite{vladymyrov16}.

An alternative approach to speed up the computation of eigenvectors is to use the power method from the numerical linear algebra literature \cite{Boutsidis2015}. From an optimization viewpoint, the power method simultaneously updates all the coordinates of the iterate, but they converge to the optimal value at different speeds. Hence, it is possible to derive a coordinate-wise version of the power method for a further speedup \cite{Lei2016}.

Yet another direction to efficiently deal with spectral embedding involves the use of stochastic gradient optimization. The idea is to cast the algebraic problem of identifying the principal eigenvectors of a matrix as the optimization of a trace function subject to an orthogonality constraint \cite{Xu2000, Wang2010}. Therefore, a natural way to tackle this problem consists of using the notion of Riemannian gradients \cite{Han17}.

\paragraph*{Contribution}
In this work, we propose a stochastic optimization algorithm to deal with spectral embedding. Our main contribution consists of reformulating the latter so that the orthogonality constraint is replaced with an orthogonalization matrix in the criterion. As the gradient can be computed through a Cholesky factorization, our reformulation enables us to develop an efficient algorithm based on mini-batch gradient descent \cite{Reddi2018, Dudar2017}. The peculiarity of our approach is that the graph nodes are processed by splitting them in chunks of manageable size. This allows us to deal with much larger graphs than those tackled by similar methods \cite{Belkin2003, Han17}.

\paragraph*{Outline}
The paper is organized as follows. In Section~\ref{sec:aggregation}, we formulate the spectral embedding problem. In Section~\ref{sec:embedding}, we describe the proposed reformulation and the stochastic gradient optimization algorithm to solve it. In Section~\ref{sec:results}, we compare our approach to similar spectral embedding methods. Finally, the conclusion is drawn in Section~\ref{sec:conclusion}.

\section{Problem formulation}
\label{sec:aggregation}

In this paper, we aim at clustering a graph $\mathcal{G}(V,E,W)$ which consists of a set $V$ of vertices, and a set $E$ of edges with non-negative weights $w_{i,j}$, where $W = [w_{i,j}]_{i,j} \in \mathbb{R}^{N\times N}$ is the weighted adjacency matrix, with $w_{i,i} = 0$ for all $i$.

The degree of a vertex $i$ in the graph $\mathcal{G}$, denoted as $d(i)$, is defined as the sum of weights of all the edges incident to $i$ in the graph $\mathcal{G}$. The degree matrix $D$ is then defined as
\begin{equation}
D_{i,j} = \begin{cases} d(i) \quad & \textup{if $i=j$} \\
0 \quad & \textup{otherwise.}
\end{cases}
\end{equation}
Based on $W$ and $D$, the Laplacian matrix of $\mathcal{G}$ is
\begin{equation}
L = D - W.
\end{equation}
In the literature, $L$ is called the unormalized Laplacian matrix. There are also normalized versions of this matrix, such as
\begin{equation}
L_{\rm{sym}} = D^{-\frac{1}{2}}(D-W)D^{-\frac{1}{2}}.
\end{equation}

The Laplacian matrix is of broad interest in the studies of spectral graph theory \cite{Chung1997}. Although closely related to each other, the different definitions of the Laplacian matrix don't enjoy the same properties. For example, $L$ and $L_{\rm{sym}}$ are both real and symmetric matrices, but they  have a different set of eigenvalues and orthonormal eigenvectors. Moreover, each definition of the Laplacian matrix comes with its own algorithm to perform spectral embedding and clustering \cite{Luxburg2007}. In this paper, we use the unnormalized graph Laplacian $L$ defined above, since its eigenvectors are orthonormal and its eigenvalues are non-negative, with the smallest equal to zero.

\subsection{Spectral Clustering}
\label{subsec:single_layer_clustering}

Spectral clustering is a standard application of spectral embedding \cite{Malik2000, Ng2001,Luxburg2007}. The idea is to represent the data points to be clustered as the vertices of a graph, whose edges describe the pairwise relationships between those points (e.g., computed with k-nearest neighbours).
The core operation is to embed the graph vertices in a low-dimensional space, where the projected points can be trivially clustered. More specifically, for a $K$-way clustering, one aims at finding the semi-orthogonal matrix $U = \left[ u_1 \,|\dots|\, u_N \right]^\top$ such that its rows $u_n \in \mathbb{R}^K$ minimize the pairwise distances on the graph edges, namely
\begin{align}\label{eq:spectral_embbedding}
\operatorname*{minimize}_{U\in\mathbb{R}^{N\times K}}\;\;  \operatorname{Tr}(U^\top LU) := \sum_{i=1}^N\sum_{j=1}^N w_{i,j} \|u_i - u_j\|^2 \nonumber  \\
\quad{\rm s.t.}\quad U^\top U = {\bf I_{K\times K}}.
\end{align}
Each row of the matrix $U$ is a point in the low-dimensional space $\mathbb{R}^K$ representing a graph vertex. Strongly-connected vertices are thus mapped to close vectors in $\mathbb{R}^K$. One can apply a $K$-mean algorithm on the rows of the solution $\bar{U}$, so that $K$ clusters are formed by grouping together the vertices that are the most strongly connected by the graph.

\section{Proposed algorithm}
\label{sec:embedding}

It is well known that the solution to Problem \eqref{eq:spectral_embbedding}  is the matrix formed by the eigenvectors associated to the  $K$ smallest eigenvalues of the Laplacian matrix $L$. While there exist many efficient algorithms for spectral embedding \cite{Belkin2003, Fowlkes2004, Boutsidis2015, vladymyrov16, Lei2016, Han17}, they may not always scale well as the graph size $N$ grows big. Therefore, we propose an alternative approach to solve Problem \eqref{eq:spectral_embbedding}, which consists of reformulating the latter so that it can be dealt with a stochastic optimization algorithm, such as mini-batch gradient descent \cite{Reddi2018, Dudar2017}.

\subsection{Forward-backward splitting}
The main difficulty in solving Problem \eqref{eq:spectral_embbedding} via gradient descent arises from the orthogonality constraint. A possible way to circumvent this issue is to make use of forward-backward splitting \cite{Combettes2005}, which boils down to the following iterations:
\begin{equation}
(\forall t\in\mathbb{N})\qquad U_{t+1} = {\sf QR}\Big( U_t - \gamma_t LU_t \Big).
\end{equation}
Hereabove, $LU$ is the gradient of $\frac{1}{2}{\rm Tr}(U^\top LU)$, whereas the operator ${\sf QR}(\cdot)$ extracts the semi-orthogonal matrix $Q$ obtained with the QR decomposition of $U_t - \gamma_t LU_t$. Indeed, any rectangular matrix $U\in\mathbb{R}^{N\times K}$ with $N\ge K$ can be factorized as $U=Q R$, where $Q\in\mathbb{R}^{N\times K}$ is semi-orthogonal, and $R\in\mathbb{R}^{K\times K}$ is upper triangular. This was the approach followed in \cite{Han17}, where a Riemannian gradient was used instead of the regular gradient.

\subsection{Reformulation with implicit constraint}
We propose an alternative approach to deal with the orthogonality constraint in Problem \eqref{eq:spectral_embbedding}. Indeed, the latter can be enforced implicitly by using the upper triangular matrix $R$ of the QR decomposition of $U$, which is defined as
\begin{equation}
U = Q R.
\end{equation}
When $U$ is positive definite, the QR decomposition is unique, and $R$ is equal to the upper triangular factor of the Cholesky decomposition  $U^\top U =  R^\top R$. Then, it is possible to extract the Q-factor of $U$ as
\begin{equation}
Q = U R^{-1}.
\end{equation}
This equality allows us to reformulate Problem \eqref{eq:spectral_embbedding} as
\begin{equation}
\label{eq:embedding_reformulated}
\operatorname*{minimize}_{U\in\mathbb{R}^{N\times K}}\;\; J(U):={\rm Tr}\Big((U R^{-1})^\top L (U R^{-1})\Big).
\end{equation}
Note that the matrix $U$ is not semi-orthogonal, but $U R^{-1}$ is. Also, $R$ is a function of $U$, for which we can compute the gradient w.r.t.\ $U$ \cite{Murray2016}. Therefore, the reformulated problem can be solved using standard gradient descent, yielding
\begin{equation}\label{eq:gradient_descent}
(\forall t\in\mathbb{N})\qquad U_{t+1} = U_t - \gamma_t \nabla J(U_t).
\end{equation}

\subsection{Stochastic gradient descent}\label{sec:sgd}
The main advantage in solving Problem \eqref{eq:embedding_reformulated} is that we can modify the update in Eq. \eqref{eq:gradient_descent} by replacing the function $J$ with a stochastic approximation $J_t$, yielding
\begin{equation}\label{eq:stochastic_gradient_descent}
(\forall t\in\mathbb{N})\qquad U_{t+1} = U_t - \gamma_t \nabla J_t(U_t).
\end{equation}

Before we can define the above function $J_t$, let us denote by $\mathcal{I}$ the set of indexes associated to the nonzero elements in the upper triangular part of Laplacian matrix $L$, namel
\begin{equation}
\mathcal{I} =\Big\{(i,j)\in\{1,\dots,N\}^2 \;|\; w_{i,j} \neq 0 \;{\rm and}\; j > i\Big\}.
\end{equation}
The Laplacian matrix can be thus decomposed as follows
\begin{equation}
L = \sum_{(i,j)\in\mathcal{I}} \underbrace{ w_{i,j} \big(e_i e_i^\top + e_j e_j^\top - e_i e_j^\top - e_j e_i^\top\big) }_{\mathsf{L}_{i,j} \in \mathbb{R}^{N\times N}},
\end{equation}
where $e_n$ is the $n$-th column of the $N\times N$ identity matrix. Thus, the original criterion in Problem \eqref{eq:spectral_embbedding} can be rewritten as
\begin{equation}
\operatorname{Tr}(U^\top LU)  = \sum_{(i,j)\in\mathcal{I}} \operatorname{Tr}(U^\top \mathsf{L}_{i,j} U) = \sum_{(i,j)\in\mathcal{I}} w_{i,j} \|u_i - u_j\|^2.
\end{equation}

To build the function $J_t$ appearing in Eq. \eqref{eq:stochastic_gradient_descent}, at every iteration $t$ we select a subset of indexes
\begin{equation}
\mathcal{S}_t \subset \mathcal{I},
\end{equation}
we use those indexes to build a partial Laplacian matrix
\begin{equation}
\mathbf{L}_t = \sum_{(i,j)\in\mathcal{S}_t} \mathsf{L}_{i,j},
\end{equation}
and we approximate the criterion in Eq. \eqref{eq:embedding_reformulated} as
\begin{equation}
J_t(U) := {\rm Tr}\Big((U R^{-1})^\top \mathbf{L}_t (U R^{-1})\Big).
\end{equation}
Here is where the proposed reformulation comes into play. As the term $\mathbf{L}_t$ contains many rows/columns filled with zeros, we never have to work with the full-size matrix $U$. Specifically, the function $J_t(U)$ boils down to
\begin{equation}
J_t(U) := \sum_{(i,j)\in\mathcal{S}_t} w_{i,j} \|(u_i-u_j)^\top R^{-1} \|^2.
\end{equation}
As for the Cholesky decomposition $U^\top U =  R^\top R$, we compute the full-size matrix product at the algorithm initialization
\begin{equation}
M_0 = U_0^\top U_0.
\end{equation}
Since the update in Eq.  \eqref{eq:stochastic_gradient_descent} only modifies the terms $u_i$ and $u_j$ such that $(i,j)\in\mathcal{S}_t$, we can efficiently update it as
\begin{align}
M_{t+1} = M_t &+ \sum_{(i,j)\in\mathcal{S}_t} \big(u_{i,t+1} u_{i,t+1}^\top - u_{i,t} u_{i,t}^\top \big)\nonumber\\
&+ \sum_{(i,j)\in\mathcal{S}_t} \big(u_{j,t+1} u_{j,t+1}^\top - u_{j,t} u_{j,t}^\top\big).
\end{align}
Putting all together, we arrive at the proposed optimization algorithm, which can be readily implemented with modern tools for automatic differentiation \cite{paszke2017}.

\section{Numerical results}
\label{sec:results}

We evaluate the proposed algorithm in the context of multilayer graph clustering. An overview is given in the following.

\subsection{Multilayer Graph Clustering}
With the increasing richness of datasets, applications even rely often on multiple sources of information to characterize the relationships between data points. This leads to multi-layer graph representations, where data points are modelled as nodes shared across the layers, each of which describes a set of relationships (graph edges) on their set of observations (shared graph nodes). For example, a social network can be represented by a multilayer graph, where each layer corresponds to a different type of relationship among the same group of persons, such as friendships, music interests, etc.

More precisely, a multilayer graph consists of $S$ edge layers sharing the same vertices. Each layer $s$ is represented by a Laplacian matrix $L^s$, along with a spectral embedding matrix $U^s$ obtained as the solution to Problem \eqref{eq:spectral_embbedding}. These matrices $(L^s,U^s)_{1\le s\le S}$ can be merged into a representative embedding matrix $U$ via the minimization of the projection distance in a Grassman manifold,\footnote{By definition, a Grassmann manifold is the set of $K$-dimensional linear subspaces in $\mathbb{R}^N$. Each point in this manifold can be represented by a semi-orthogonal matrix $U \in \mathbb{R}^{N\times K}$. The distance between two points $U_1$ and $U_2$ is then defined based on a set of principal angles $\{\theta_k\}_{1\le k\le K}$ between the corresponding subspaces. These angles are the fundamental measure used to define various distances on the Grassmann manifold, such as the squared projection distance \cite{Hamm2008}, defined as
	\begin{equation*}
	d^2_{\sf proj}(U_1,U_2)
	= K - \operatorname{Tr}(U_1U_1^\top U_2U_2^\top) \\

	\end{equation*}
}
 defined as
\begin{equation}
d_{\sf proj}(U,\{U^s\}_{1\le s\le S}) = KS - \sum_{s=1}^S \operatorname{Tr}\big(UU^\top U^s{U^s}^\top\big).
\end{equation}
Consequently, multilayer graph clustering boils down to \cite{Dong_TSP_2014}
\begin{align}\label{eq:multilayer_embedding}
\operatorname*{minimize}_{U\in\mathbb{R}^{N\times K}}\;  \operatorname{Tr}\Big( \sum_{s=1}^{S}U^\top\big( L^s - & \alpha \, U^s {U^s}^\top\big) U\Big) \nonumber  \\
& \quad{\rm s.t.}\quad U^\top U = {\bf I_{K\times K}}.
\end{align}
This is equivalent to perform spectral embedding on the aggregated Laplacian matrix defined as
\begin{equation}\label{eq:aggregated_laplacian}
L_{\sf MLG} = \sum_{s=1}^{S} (L^s - \alpha \, U^s {U^s}^\top),
\end{equation}
where $\alpha>0$ is a regularization parameter.

For the sake of completeness, note that there exist other approaches to deal with a multilayer graph, such as averaging the Laplacian matrices of individual layers \cite{argyriou_2005_nips,Tang_20012, Chen_Hero_2017}, or treating them as points in a Grassmann manifold \cite{Sindhwani2005, Kumar2011, Wang_TIP_2013, Dong_TSP_2014}.

\subsection{Datasets}
In our experiments, we consider a synthetic dataset, as well as a real dataset coming from the Yelp challenge. Both have a multilayer graph representation of data.

As for the synthetic data, we consider a multilayer graph with $S = 3$ layers and $K=5$ clusters. Data points are drawn from a Gaussian mixture model with five components, each representing a cluster. The goal with this dataset is to recover the five clusters of the graph vertices, using the layers constructed from the point clouds.

Yelp is a popular website for reviewing and rating local businesses. In our experiments, we only extract star ratings, text reviews, and review evaluations (users can mark reviews as ``cool'', ``useful'', and ``funny''), ignoring the other information in the dataset. An example is shown in Fig. \ref{fig:yelp}. Our goal is to cluster the businesses by predicting their ratings ($K=3$).

To build a multilayer graph on Yelp data, we proceed as follows. We preprocess the text reviews using sentiment analysis. This yields a polarity score within the range $[-1, 1]$ on which we build a 5-nearest neighbor (NN) graph. We also build a 5-NN graph on the other features (``cool'', ``useful'', and ``funny'' evaluations), leading to $S=4$ graph layers.

\begin{figure}[t]
	\centering
	\includegraphics[width=\linewidth]{yelp}
	\caption{Yelp data before the preprocessing of text reviews.}
	\label{fig:yelp}
\end{figure}

\subsection{Comparisons}
We compare spectral embedding algorithms on Problem \eqref{eq:multilayer_embedding}, assuming that the aggregated Laplacian matrix defined in Eq. \eqref{eq:aggregated_laplacian} is given as input. This means that a spectral embedding problem is solved to compute each matrix $U^s$. For brevity, these problems are not considered into our  analysis.

\paragraph*{Problem setting}
We evaluate the performance of three spectral embedding algorithms: matrix eigendecomposition \cite{Belkin2003}, stochastic forward-backward splitting \cite{Han17}, and our method described in Section \ref{sec:sgd}. Note that ``matrix eigendecomposition`` uses the normalized version of the aggregated Laplacian matrix. The normalization is important to obtain a good clustering performance. However, this causes the method to slow down considerably, which gives rise to the interest of considering alternative approaches. The other methods use the unnormalized Laplacian matrix.

\paragraph*{Implementation details}
All the experiments were conducted in Python/Numpy on a 40-core Intel Xeon CPU at 2.5 GHz with 128GB of RAM. As for the proposed method, we used a step-size $\gamma=10^{-3}$, and mini-batches of size $4'000$. The latter is a critical parameter, as mini-batches should be big enough to capture the structure of the Laplacian matrix. For this reason, it is also highly important that mini-batches are sampled at random from the entire dataset at each step, and not be fixed across iterations.

\paragraph*{Result analysis}
We use three criteria to measure the clustering performance: Purity, Normalized Mutual Information (NMI), and Rand Index (RI). The results reported in Tables \ref{tab:synt} and \ref{tab:real} for both synthetic and Yelp data show that the proposed method achieves almost the same clustering performance as the matrix eigendecomposition, while being faster than the other compared methods. The small differences in the performance are related to the stochastic nature of our optimization algorithm. Running it for less iterations would allow us to get even faster results, yet less accurate. There is thus a tradeoff between speed and accuracy, as usual in optimization.

\begin{table}[t]
\centering
\caption{Clustering on synthetic data ($N=10'000$). \label{tab:synt}}
\begin{tabular}{ l   c  cc   c   c c}
\toprule
Method & Time & Iter.& Purity & NMI & RI \\
\midrule
Eigendecomp. & 36.3 s & - & 0.93 & 0.83 & 0.85 \\
Stochastic FB & 22.8 s & 3000 & 0.94 & 0.84 & 0.86 \\
Proposed & 5.2 s & 500 & 0.92 & 0.79 & 0.82 \\
\bottomrule
\end{tabular}
\end{table}

\begin{table}[t]
\caption{Clustering on Yelp data ($N=11'160$). \label{tab:real}}
\centering
\begin{tabular}{ l   cc   c   c   c c}
\toprule
Method & Time & Iter. & Purity & NMI & RI \\
\midrule
Eigendecomp. & 69.7 s & - & 0.87 & 0.65 & 0.56\\
Stochastic FB & 50.6 s & 3000 & 0.88 & 0.79 & 0.88 \\
Proposed & 11.4 s & 500 & 0.86 & 0.71 & 0.84 \\
\bottomrule
\end{tabular}
\end{table}

\section{Conclusion}
\label{sec:conclusion}

In this paper, we proposed a reformulation of spectral embedding that allowed us to devise a new optimization algorithm based on stochastic gradient descent. The experimental results showed that the proposed approach is more efficient than matrix eigendecomposition and stochastic Riemannian optimization on medium-size datasets. This confirms that our approach is capable of scaling to larger datasets than what similar spectral embedding methods can handle.

There are a number of extensions that we are currently investigating, such as the possibility to extend our approach to the Riemann manifold of semi-positive definite matrices \cite{ElGheche2018_arxiv}, and to simultaneously compute the embedding matrices of each layer and the aggregated graph. Moreover, a GPU implementation of the proposed algorithm is under development.

\bibliographystyle{IEEEbib}