# On product decomposition

Ming-Deh A. Huang (USC<EMAIL_ADDRESS> Computer Science
Department,University of Southern California, U.S.A<EMAIL_ADDRESS>
###### Abstract.

Given a finite set $W$ in $\bar{k}^{n}$ where $\bar{k}$ is the algebraic
closure of a field $k$ one would like to determine if $W$ can be decomposed as
$\prod_{i=1}^{n}V_{i}$ where $V_{i}\subset\bar{k}$ under a linear
transformation, that is,
$W\stackrel{{\scriptstyle\lambda}}{{\to}}\prod_{i=1}^{n}V_{i}$ where
$\lambda\in Gl_{n}(\bar{k})$. We assume that $W$ is presented as
$W=Z(\mathcal{F})$, the zero set of a polynomial system $\mathcal{F}$ in $n$
variables over $k$. We study algebraic characterization of such product
decomposition. For decomposition into component sets of the same cardinality
we obtain a stronger characterization and show that the decomposition in this
case is essentially unique (up to permutation and scalar multiplication of
coordinates). We investigate computational problems that arise from the
decomposition problem.

###### Key words and phrases:

polynomial system, product decomposition, last fall degree

###### 2010 Mathematics Subject Classification:

13P10, 13P15

##  1\. Introduction

Suppose $W\subset\bar{k}^{n}$ is a finite set where $k$ is a field and
$\bar{k}$ is the algebraic closure of $k$. We say that $W$ has a product
decomposition if $W=\lambda(\prod_{i=1}^{n}V_{i})$ where $V_{i}\subset\bar{k}$
is finite and $\lambda\in Gl_{n}(\bar{k})$.

We study algebraic characterizations of product decomposition. In § 2 we begin
with a simple characterization in Theorem 2.1. We then provide a stronger
characterization in Theorem 2.2 for the case where $W\subset\bar{k}^{n}$ has
product decomposition
$W\stackrel{{\scriptstyle\lambda}}{{\to}}\prod_{i=1}^{n}V_{i}$ with
$\lambda\in Gl_{n}(\bar{k})$, $V_{i}\subset\bar{k}$ and $|V_{i}|=d$ for
$i=1,\ldots,n$. Moreover in Theorem 2.3 we show that in this case product
decomposition is unique up to permutation and scalar multiplication of
coordinates, provided $d$ is not divisible by the characteristic of $k$.

Suppose $W=Z({\mathcal{G}})$, the zero set of a set ${\mathcal{G}}$ consisting
of $n$ polynomials $g_{1},\ldots,g_{n}$ of degree bounded by $d$ in
$\bar{k}[x_{1},\ldots,x_{n}]$. Then by Theorem 2.2 $Z({\mathcal{G}})$ has
product decomposition into component sets of the same cardinality if and only
if there exists $\mu\in Gl_{n}(\bar{k})$ such that letting ${\mathcal{L}}$ be
the linear vector space spanned by ${\mathcal{G}}$, then the isomorphic linear
vector space $\mu^{*}({\mathcal{L}})=\\{h\circ\mu:h\in{\mathcal{L}}\\}$ has a
basis $f_{i}(x_{i})$, $i=1,\ldots,n$, where each $f_{i}(x_{i})$ is of degree
$d$ without multiple roots. Therefore such a decomposition exists for $W$ if
and only if

(1.1) $\left(\begin{array}[]{c}g_{1}\\\ .\\\ .\\\
g_{n}\end{array}\right)=\rho\circ\left(\begin{array}[]{c}f_{1}\\\ .\\\ .\\\
f_{n}\end{array}\right)\circ\lambda$

for some $f_{i}(x_{i})$, $i=1,\ldots,n$, of degree $d$ without multiple roots,
and $\lambda,\rho\in Gl_{n}(k)$.

Treating $f_{i}$, $\lambda,\rho$ as unknown, we arrive at a problem of solving
a system of polynomials of degree $d$ in $O(n^{2})$ unknown and $O(n^{d})$
conditions. In general algebraic algorithms for solving polynomial systems
take time exponential in the number of variables, which is $O(n^{2})$ in this
case (see for example [4]). Whether the system characterized by Eq. 1.1 admits
more efficient solution is an interesting open question.

In § 3 we study last fall degrees of polynomial systems associated with
algebraic sets that admit product decomposition. More generally we consider
polynomial systems specified by a set ${\mathcal{G}}=\\{g_{1},\ldots,g_{n}\\}$
with $g_{i}\in k[x_{1},\ldots,x_{n}]$ and $\deg(g_{i})\leq d$ for
$i=1,\ldots,n$, such that there exists $\mu\in Gl_{n}(k)$ where the linear
space spanned by $g_{i}\circ\mu$, $i=1,\ldots,n$, contains some $f_{i}\in
k[x_{i}]$ with $\deg f_{i}\leq d$ for $i=1,\ldots,n$. We study the last fall
degrees [2, 3] of such polynomial systems. In Theorem 3.1 we prove
$d_{{\mathcal{G}}}\leq d$, where $d_{{\mathcal{G}}}$ is the last fall degree
of ${\mathcal{G}}$. If $d=O(1)$ then $d_{{\mathcal{G}}}=O(1)$. Finding
$k$-rational solutions amounts to solving the polynomial system
${\mathcal{G}}\cup{\mathcal{E}}$, where ${\mathcal{E}}$ consists of the field
equations $x_{i}^{q}-x_{i}$, $i=1,\ldots,n$ and $k=\mathbb{F}_{q}$. In Theorem
3.5 we show that the last fall degree $d_{{\mathcal{G}}\cup{\mathcal{E}}}\leq
d+q$. In situations where the number of $k$-rational solutions is $O(1)$, it
follows from Proposition 2.11 of [3] that the problem can be solved in time
$n^{O(d+q)}$, hence $n^{O(q)}$ if $d=O(1)$. An improvement can be made in
finding $k$-rational point by reducing the problem to solving a system
${\mathcal{G}}\cup{\mathcal{E}}^{\prime}$ in $O(nm)$ variables and degree
$O(dp)$, where $q=p^{m}$ and ${\mathcal{E}}^{\prime}$ consists of
$x_{ij}^{p}-x_{i\ j+1}$ for $i=1,\ldots,n$ and $j=0,\ldots,m-1$, and $j+1$ is
taken modulo $m$. In Theorem 3.5 we show that the last fall degree
$d_{{\mathcal{G}}\cup{\mathcal{E}}^{\prime}}\leq dp$. So if the number of
$k$-rational solutions is $O(1)$, then it follows that the problem can be
solved in time $(nm)^{O(dp)}$, which is $(nm)^{O(1)}$ providing $d=O(1)$ and
$p=O(1)$.

##  2\. Algebraic characterization

We begin with a simple algebraic characterization of product decomposition.

######  Theorem 2.1.

Suppose $W\subset\bar{k}^{n}$. The following two statements are equivalent:

  1. (1)

$W\stackrel{{\scriptstyle\lambda}}{{\to}}\prod_{i=1}^{n}V_{i}$ with
$\lambda\in Gl_{n}(\bar{k})$ and $V_{i}$ is a finite subset of $\bar{k}$ for
$i=1,\ldots,n$.

  2. (2)

There exist polynomials $f_{i}(x_{i})$, $i=1,\ldots,n$, each having distinct
roots in $\bar{k}$, and $\lambda\in Gl_{n}(\bar{k})$ such that
$W=Z(\\{f_{1}\circ\lambda,\ldots,f_{n}\circ\lambda\\})$.

Proof Suppose $W\stackrel{{\scriptstyle\lambda}}{{\to}}\prod_{i=1}^{n}V_{i}$
with $\lambda\in Gl_{n}(\bar{k})$ and $V_{i}$ is a finite subset of $\bar{k}$
for $i=1,\ldots,n$. Let $f_{i}(x_{i})=\prod_{\alpha\in V_{i}}(x_{i}-\alpha)$
for $i=1,\ldots,n$. Then for $\alpha\in\bar{k}^{n}$, $\alpha\in W$ if and only
if $\lambda(\alpha)\in\prod_{i=1}^{n}V_{i}$, if and only if
$f_{i}\circ\lambda(\alpha)=0$ for $i=1,\ldots,n$. So (1) implies (2).

Now assume (2). Put $V=\prod_{i=1}^{n}V_{i}$ where $V_{i}$ is the set of roots
of $f_{i}(x_{i})$. Then $\alpha\in W$ if and only if
$f_{i}\circ\lambda(\alpha)=0$ for $i=1,\ldots,n$ if and only if
$\lambda(\alpha)\in V$. So (2) implies (1). $\Box$

Suppose $I$ is an ideal of $\bar{k}[x_{1},\ldots,x_{n}]$. For $i\geq 0$, let
$I_{i}$ denote the submodule of $I$ consisting of all $f\in I$ with $\deg
f=i$.

For $S\subset\bar{k}[x_{1},\ldots,x_{n}]$ let $Z(S)$ denote the zero set of
$S$, that is $Z(S)=\\{v\in\bar{k}^{n}:f(v)=0\ \ \forall f\in S\\}$. For
$V\subset\bar{k}^{n}$ let $I(V)=\\{f\in\bar{k}[x_{1},\ldots,x_{n}]:f(v)=0\ \
\forall v\in V\\}$.

Theorem 2.2 and Theorem 2.3 below provide a stronger characterization for the
case of product decomposition into components of the same cardinality. More
precisely we restrict our attention to product decomposition of the form
$W\stackrel{{\scriptstyle\lambda}}{{\to}}\prod_{i=1}^{n}V_{i}$ with
$\lambda\in Gl_{n}(\bar{k})$ and $V_{i}$ is a finite subset of $\bar{k}$ of
the same cardinality $d$ for $i=1,\ldots,n$.

######  Theorem 2.2.

Suppose $W\subset\bar{k}^{n}$ and $W=Z({\mathcal{G}})$ where ${\mathcal{G}}$
consists of $n$ polynomials $g_{1},\ldots,g_{n}$ of degree bounded by $d$ in
$\bar{k}[x_{1},\ldots,x_{n}]$. Then the following are equivalent:

  1. (1)

$W\stackrel{{\scriptstyle\lambda}}{{\to}}\prod_{i=1}^{n}V_{i}$ with
$\lambda\in Gl_{n}(\bar{k})$, $V_{i}\subset\bar{k}$ and $|V_{i}|=d$ for
$i=1,\ldots,n$.

  2. (2)

$g_{1},\ldots,g_{n}$ are linearly independent and they form a basis of
$I(W)_{d}$ as a $\bar{k}$-vector space. Moreover there are $f_{i}(x_{i})$,
$i=1,\ldots,n$, of degree $d$ without multiple roots such that $f_{i}(x_{i})$,
$i=1,\ldots,n$, form a basis for the vector space spanned by $g_{i}\circ\mu$,
$i=1,\ldots,n$, for some $\mu\in Gl_{n}(\bar{k})$.

######  Theorem 2.3.

Suppose $W\subset\bar{k}^{n}$ and $|W|=d^{n}$. Suppose $d$ is not divisible by
the characteristic of $k$. If
$W\stackrel{{\scriptstyle\lambda}}{{\to}}\prod_{i=1}^{n}V_{i}$ with
$\lambda\in Gl_{n}(\bar{k})$, $V_{i}\subset\bar{k}$ and $|V_{i}|=d$ for
$i=1,\ldots,n$, then such product decomposition is unique up to permutation
and coordinate-wise scalar multiplication. More explicitly if
$W\stackrel{{\scriptstyle\lambda^{\prime}}}{{\to}}\prod_{i=1}^{n}V^{\prime}_{i}$
with $V^{\prime}_{i}\subset\bar{k}$ and $|V^{\prime}_{i}|=d$ for
$i=1,\ldots,n$ then $\lambda^{\prime}=\rho\circ\lambda$ where
$\rho(x_{1},\ldots,x_{n})=(\alpha_{1}x_{\sigma_{1}},\ldots,\alpha_{n}x_{\sigma_{n}})$
for all $(x_{1},\ldots,x_{n})\in\bar{k}^{n}$, where
$\alpha_{1},\ldots,\alpha_{n}\in\bar{k}$ and $(\sigma_{1},\ldots,\sigma_{n})$
is a permutation of $(1,\ldots,n)$.

To prove the two theorems, we will need some technical preparation.

######  Lemma 2.4.

Suppose $V_{i}$ is a finite set of $\bar{k}$ with $|V_{i}|=d_{i}$ for
$i=1,\ldots,n$. For $f(x_{1},\ldots,x_{n})\in\bar{k}[x_{1},\ldots,x_{n}]$ with
$\deg_{x_{i}}f<d_{i}$ for $i=1,\ldots,n$, if
$f(\alpha_{1},\ldots,\alpha_{n})=0$ for all
$(\alpha_{1},\ldots,\alpha_{n})\in\prod_{i=1}^{n}V_{i}$, then $f=0$.

Proof The lemma can be proved by induction on $n$. The statement is true for
$n=1$ since a polynomial in $x_{1}$ of degree less than $d_{1}$ cannot have
$d_{1}$ roots unless it is 0.

Suppose the statement is true for $n-1$.

Write $f=\sum_{i=0}^{d_{1}-1}f_{i}x_{1}^{i}$ where
$f_{i}\in\bar{k}[x_{2},\ldots,x_{n}]$. For $j>1$ since $\deg_{x_{j}}f<d_{j}$
and $\deg_{x_{j}}f=\max_{i=0,\ldots,d_{1}-1}\deg_{x_{j}}f_{i}$, it follows
that $\deg_{x_{j}}f_{i}<d_{j}$ for $i=0,\ldots,d_{1}-1$.

Let $(\beta_{2},\ldots,\beta_{n})\in\prod_{i=2}^{n}V_{i}$. Then by assumption
$f(x_{1},\beta_{2},\ldots,\beta_{n})$ vanishes at every $\beta_{1}\in V_{1}$,
hence it has at least $d_{1}$ roots. Since its degree in $x_{1}$ is no greater
than $\deg_{x_{1}}f$, which is less than $d_{1}$, we conclude that it is 0. So
$f_{i}(\beta_{2},\ldots,\beta_{n})=0$ for $i=0,\ldots,d_{1}-1$.

Since $\deg_{x_{j}}f_{i}<d_{j}$ for $j=2,\ldots,n$ and
$f_{i}(\beta_{2},\ldots,\beta_{n})=0$ for all
$(\beta_{2},\ldots,\beta_{n})\in\prod_{i=2}^{n}V_{i}$, it follows by induction
that $f_{i}=0$. This is true for each $i=0,\ldots,d_{1}-1$. So $f=0$. $\Box$

######  Lemma 2.5.

Let $f_{i}(x_{i})$ be a monic polynomial in $x_{i}$ of degree $d$ without
multiple roots over $\bar{k}$, for $i=1,\ldots,n$. Let
$I\subset\bar{k}[x_{1},\ldots,x_{n}]$ be the ideal generated by
$f_{i}(x_{i})$, $i=1,\ldots,n$. Then $I_{d}$ is a $\bar{k}$-vector space with
$f_{i}$, $i=1,\ldots,n$, as a basis.

Proof Let $V_{i}$ be the set of roots of $f_{i}$ for $i=1,\ldots,n$. If $f\in
I$ then $f$ vanishes at all points of $\prod_{i=1}^{n}V_{i}$. Suppose $f\in
I_{d}$. Then $f$ as a polynomial in $\bar{k}[x_{1},\ldots,x_{n}]$ of degree
$d$ can be written in the form $f=\sum_{i=1}^{n}\alpha_{i}f_{i}(x_{i})+g$
where $\alpha_{i}\in\bar{k}$ and $g\in\bar{k}[x_{1},\ldots,x_{n}]$ where
$\deg_{x_{i}}g<d$ for $i=1,\ldots,n$. Since
$\sum_{i=1}^{n}\alpha_{i}f_{i}(x_{i})\in I$ and $f\in I$, it follows $g\in I$
hence $g$ vanishes at all points of $\prod_{i=1}^{n}V_{i}$. Since
$\deg_{x_{i}}g<d$ for $i=1,\ldots,n$, it follows from Lemma 2.4 that $g=0$.
Therefore $f=\sum_{i=1}^{n}\alpha_{i}f_{i}(x_{i})$ with
$\alpha_{i}\in\bar{k}$. $\Box$

######  Lemma 2.6.

Suppose $V,W\subset\bar{k}^{n}$ and $V=\lambda(W)$ where $\lambda\in
Gl_{n}(\bar{k})$. Then we have a $\bar{k}$-linear bijection
$I(V)\stackrel{{\scriptstyle\lambda^{*}}}{{\to}}I(W)$ where
$\lambda^{*}(f)=f\circ\lambda$ for $f\in I(V)$. Moreover
$\lambda^{*}(I(V)_{i})=I(W)_{i}$ for all $i\geq 0$.

Proof If $f\in I(V)$ then $f(V)=f(\lambda(W))=0$ hence $\lambda^{*}(f)\in
I(W)$. It is easily checked that $\lambda^{*}$ is $\bar{k}$-linear. It is
injective since $f\circ\lambda=0$ implies
$(f\circ\lambda)\circ\lambda^{-1}=f=0$. It is surjective since for $g\in
I(W)$, $g=f\circ\lambda$ where $f=g\circ\lambda^{-1}\in I(V)$. Finally the map
is degree preserving since $\lambda$ is linear. Hence
$\lambda^{*}(I(V)_{i})=I(W)_{i}$ for all $i\geq 0$. $\Box$

######  Lemma 2.7.

Suppose $g(x)$ is univariate and monic of degree $d$ where $d$ is not
divisible by the characteristic of the field $k$. Suppose
$\ell=\sum_{i=1}^{n}a_{i}x_{i}$ with $a_{i}\in\bar{k}$ for $i=1,\ldots,n$. If
$g\circ\ell=\sum_{i=1}^{n}\beta_{i}f_{i}(x_{i})$ where $\beta_{i}\in\bar{k}$
for $i=1,\ldots,n$, then $\ell=a_{i}x_{i}$ for some $i$.

Proof We will prove the lemma by induction on $n$. The case $n=1$ is trivial.
Suppose inductively the lemma is true for $n-1$. Let $\alpha_{i}$,
$i=1,\ldots,d$, be the roots of $g$. Then
$g\circ\ell=\prod_{i=1}^{d}(\ell-\alpha_{i})$. So we have

$\prod_{i=1}^{d}(\ell-\alpha_{i})=\sum_{i=1}^{n}\beta_{i}f_{i}(x_{i}).$

We see that for the left-hand-side of the equality, the coefficient for
$x_{1}^{d-1}x_{2}$ is $da_{1}^{d-1}a_{2}$, which must be 0 by virtue of the
right hand side. Since $d$ is not divisible by the characteristic of $k$, we
must have $a_{1}a_{2}=0$. Say $a_{1}=0$, then
$\ell=a_{2}x_{2}+\ldots+a_{n}x_{n}$ and $\beta_{1}=0$. We are reduced to the
case of $n-1$. By induction we conclude that $\ell=a_{i}x_{i}$ for some $i\neq
1$. Similarly if $a_{2}=0$ we conclude that $\ell=a_{i}x_{i}$ for some $i\neq
2$. $\Box$

Proof of Theorem 2.2 Assume (2). Then the space spanned by the $n$ polynomials
$g_{i}\circ\mu$, $i=1,\ldots,n$ has dimension exactly $n$, since it contains
$f_{i}(x_{i})$, $i=1,\ldots,n$, which are clearly independent. Moreover the
space is also spanned by $f_{i}(x_{i})$, $i=1,\ldots,n$. Let $V_{i}$ be the
set of roots of $f_{i}(x_{i})$ for $i=1,\ldots,n$. Let
$V=\prod_{i=1}^{n}V_{i}$. Then by Lemma 2.5, $f_{i}(x_{i})$, $i=1,\ldots,n$
generate $I(V)$ and they also form a basis $I(V)_{d}$. Let $\lambda=\mu^{-1}$.
Then $\alpha\in W$ if and only if
$g_{i}(\alpha)=(g_{i}\circ\mu)(\lambda(\alpha))=0$ if and only if
$f_{i}(\lambda(\alpha))=0$, $i=1,\ldots,n$, if and only if $\lambda(\alpha)\in
V$. So $\lambda$ maps bijectively from $W$ to $V$. So (2) implies (1).

Assume (1). We claim that $g_{1},\ldots,g_{n}$ are linearly independent.
Otherwise say $g_{n}$ is linearly dependent on $g_{1},\ldots,g_{n-1}$. Then
$Z({\mathcal{G}})=Z(\\{g_{1},\ldots,g_{n-1}\\})$, which has dimension greater
than 0, and we have a contradiction.

By Lemma 2.6, $I(W)_{d}$ has dimension equal to the dimension $I(V)_{d}$. By
Lemma 2.5, $I(V)_{d}$ has dimension $n$. Therefore $g_{1},\ldots,g_{n}$ form a
basis of $I(W)_{d}$. Let $\mu=\lambda^{-1}$. By Lemma 2.6, $g_{i}\circ\mu$,
$i=1,\ldots,n$, form a basis of $I(V)_{d}$. Let $V_{i}$ be the set of roots of
$f_{i}(x_{i})$ for $i=1,\ldots,n$. Then by Lemma 2.5 $f_{i}(x_{i})$,
$i=1,\ldots,n$, form a basis of $I(V)_{d}$, which as argued above is also
spanned by $g_{i}\circ\mu$, $i=1,\ldots,n$. So (1) implies (2). $\Box$

Proof of Theorem 2.3 Suppose
$W\stackrel{{\scriptstyle\lambda}}{{\to}}\prod_{i=1}^{n}V_{i}$ and
$W\stackrel{{\scriptstyle\lambda^{\prime}}}{{\to}}\prod_{i=1}^{n}V^{\prime}_{i}$
with $|V_{i}|=|V^{\prime}_{i}|=d$ for $i=1,\ldots,n$ and
$\lambda,\lambda^{\prime}\in Gl_{n}(\bar{k})$. Let $V=\prod_{i=1}^{n}V_{i}$
and $V^{\prime}=\prod_{i=1}^{n}V^{\prime}_{i}$. Then
$V\stackrel{{\scriptstyle\rho}}{{\to}}V^{\prime}$ with
$\rho=\lambda^{\prime}\circ\lambda^{-1}\in Gl_{n}(\bar{k})$. Let
$f_{i}(x_{i})=\prod_{\alpha\in V_{i}}(x_{i}-\alpha)$ and
$g_{i}(x_{i})=\prod_{\alpha\in V^{\prime}_{i}}(x_{i}-\alpha)$ for
$i=1,\ldots,n$. Then $g_{i}\in I(V^{\prime})_{d}$ and by Lemma 2.6
$g_{i}\circ\rho\in I(V)_{d}$ for $i=1,\ldots,n$. By Lemma 2.5,
$g_{i}\circ\rho=\sum_{j=1}^{n}\beta_{ij}f_{j}(x_{j})$ with
$\beta_{ij}\in\bar{k}$, for $i,j=1,\ldots,n$. Now the theorem follows easily
from Lemma 2.7. $\Box$

##  3\. Last fall degree and Computation

Let $\mathcal{F}$ be a finite set of polynomials in $k[x_{1},\ldots,x_{n}]$.
For $i\in\mathbb{Z}_{\geq 0}$, we let $V_{\mathcal{F},i}$ be the smallest
$k$-vector space such that

  1. (1)

$\\{f\in\mathcal{F}:\deg(f)\leq i\\}\subseteq V_{\mathcal{F},i}$;

  2. (2)

if $g\in V_{\mathcal{F},i}$ and if $h\in k[x_{1},\ldots,x_{n}]$ with
$\deg(hg)\leq i$, then $hg\in V_{\mathcal{F},i}$.

We write $f\equiv_{i}g\pmod{\mathcal{F}}$, for $f,g\in k[x_{1},\ldots,x_{n}]$,
if $f-g\in V_{\mathcal{F},i}$.

Put $R=k[x_{1},\ldots,x_{n}]$ and let $R_{\leq i}$ denote the $k$-vector space
consisting of polynomials in $R$ of degree no greater than $i$ for $i\geq 0$.
The _last fall degree_ as defined in [2] (see also [3]) is the largest $d$
such that $V_{\mathcal{F},d}\cap R_{\leq d-1}\neq V_{\mathcal{F},d-1}$. We
denote the last fall degree of $\mathcal{F}$ by $d_{\mathcal{F}}$.

As shown in [2, 3] the last fall degree is intrinsic to a polynomial system,
independent of the choice of a monomial order, always bounded by the degree of
regularity, and invariant under linear change of variables and linear change
of equations.

######  Theorem 3.1.

Suppose $W\subset\bar{k}^{n}$ and
$W\stackrel{{\scriptstyle\lambda}}{{\to}}\prod_{i=1}^{n}V_{i}$ with
$\lambda\in Gl_{n}(\bar{k})$, $V_{i}\subset\bar{k}$ and $|V_{i}|\leq d$ for
$i=1,\ldots,n$. Suppose ${\mathcal{F}}\subset\bar{k}[x_{1},\ldots,x_{n}]$,
$Z({\mathcal{F}})=W$ and the vector space spanned by
$\lambda^{*}({\mathcal{F}})$ contains $f_{i}$ for $i=1,\ldots,n$ where
$f_{i}(x_{i})=\prod_{\alpha\in V_{i}}(x_{i}-\alpha)$. Then
$d_{{\mathcal{F}}}\leq d$.

Proof The last fall degree is invariant under linear change of variables and
linear change of equations (Proposition 2.6 (part v) of [3]). Therefore we are
reduced to the case that $W=\prod_{i=1}^{n}V_{i}$ and ${\mathcal{F}}$ contains
$f_{i}(x_{i})=\prod_{\alpha\in V_{i}}(x_{i}-\alpha)$ for $i=1,\ldots,n$. We
have $f_{i}(x_{i})=x_{i}^{d}-h_{i}(x_{i})$ with $\deg h_{i}<d$. So
$x_{i}^{d_{i}}\equiv_{d_{i}}h_{i}(x_{i})\pmod{f_{i}}$. Inductively we get
$x_{i}^{j}\equiv_{j}h_{ij}(x_{i})\pmod{f_{i}}$ for some $h_{ij}(x_{i})$ of
degree less than $d_{i}$, for all $j\geq 0$. From this it is easy to see that
for all $f\in\bar{k}[x_{1}\ldots x_{n}]$, we have $f\equiv_{\max(d,\deg
f)}\bar{f}\pmod{I}$ where $\bar{f}\in\bar{k}[x_{1},\ldots,x_{n}]$ with
$\deg_{x_{i}}\bar{f}<d_{i}$ for all $i$. Finally $f\in I$ if and only if
$\bar{f}\in I$, and since $\deg_{x_{i}}\bar{f}<d_{i}$ for all $i$, we conclude
from Lemma 2.4 $\bar{f}\in I$ if and only if $\bar{f}=0$. It follows that
$I_{i}=V_{{\mathcal{F}},i}$ for $i\geq d$. Hence $d_{{\mathcal{F}}}\leq d$.
$\Box$

###  3.1. Last fall degree with field equations

Let $k$ be a finite field with $q=p^{m}$ elements where $p$ is prime. Suppose
$\lambda\in Gl_{n}(k)$ and ${\mathcal{F}}\subset k[x_{1},\ldots,x_{n}]$ is a
finite set. Let
$\lambda^{*}({\mathcal{F}})=\\{f\circ\lambda:f\in{\mathcal{F}}\\}$. Let
${\mathcal{E}}=\\{x_{i}^{q}-x_{i}:i=1,\ldots,n\\}$, the set of field equations
for $k$. Let $\vec{x}=(x_{1},\ldots,x_{n})$. Suppose $\lambda=(a_{ij})$ with
$1\leq i,j\leq n$ and $a_{ij}\in k$. Let
$\ell_{i}(\vec{x})=\sum_{j=1}^{n}a_{ij}x_{j}$ for $i=1,\ldots,n$, so that
$\lambda(\vec{x})=(\ell_{i}(\vec{x}))_{i=1}^{n}$. Let
$\lambda({\mathcal{E}})=\\{\ell_{i}(\vec{x}^{q}-\vec{x}):i=1,\ldots,n\\}$.

######  Lemma 3.2.

$d_{\lambda^{*}({\mathcal{F}})\cup{\mathcal{E}}}=d_{{\mathcal{F}}\cup{\mathcal{E}}}$.

Proof As noted before, the last fall degree is invariant under linear change
of variables and linear change of equations (Proposition 2.6 (part v) of [3]).
So
$d_{\lambda^{*}({\mathcal{F}}\cup{\mathcal{E}})}=d_{{\mathcal{F}}\cup{\mathcal{E}}}$.
Since

$(\vec{x}^{q}-\vec{x})\circ\lambda=((\ell_{i}(\vec{x}))^{q}-\ell_{i}(\vec{x}))_{i=1}^{n}=(\ell_{i}(\vec{x}^{q}-\vec{x}))_{i=1}^{n},$

it follows that $\lambda^{*}({\mathcal{E}})=\lambda({\mathcal{E}})$. Since
${\mathcal{E}}$ and $\lambda({\mathcal{E}})$ are related by a linear change of
equations, so are $\lambda^{*}({\mathcal{F}})\cup\lambda({\mathcal{E}})$ and
$\lambda^{*}({\mathcal{F}})\cup{\mathcal{E}}$, so
$d_{\lambda^{*}({\mathcal{F}})\cup{\mathcal{E}}}=d_{\lambda^{*}({\mathcal{F}})\cup\lambda({\mathcal{E}})}$.
Therefore

$d_{{\mathcal{F}}\cup{\mathcal{E}}}=d_{\lambda^{*}({\mathcal{F}}\cup{\mathcal{E}})}=d_{\lambda^{*}({\mathcal{F}})\cup\lambda^{*}({\mathcal{E}})}=d_{\lambda^{*}({\mathcal{F}})\cup\lambda({\mathcal{E}})}=d_{\lambda^{*}({\mathcal{F}})\cup{\mathcal{E}}}.$

$\Box$

We observe that $Z({\mathcal{F}}\cup{\mathcal{E}})=Z_{k}({\mathcal{F}})$ where
$Z_{k}({\mathcal{F}})=Z({\mathcal{F}})\cap k^{n}$. The degree of field
equations for $k=\mathbb{F}_{q}$ can be reduced by introducing more variables.
More precisely $k=\mathbb{F}_{q}=Z(x^{q}-x)\simeq
Z(\\{x^{p}-x_{1},x_{1}^{p}-x_{2},\ldots,x_{m-1}^{p}-x\\})$ where $x\in k$
corresponds to $(x,x_{1},\ldots,x_{m-1})$, with $x_{1}=x^{p}$,…,
$x_{m-1}=x_{m-2}^{p}$. Now consider $nm$ variables $x_{ij}$, $i=1,\ldots,n$,
$j=0,\ldots,m-1$, and identify $f(x_{1},\ldots,x_{n})\in{\mathcal{F}}$ with
$f(x_{10},\ldots,x_{n0})$. Let ${\mathcal{E}}^{\prime}$ include the
polynomials $x_{i0}^{p}-x_{i1}$,…,$x_{i\ m-1}^{p}-x_{i0}$ for $i=1,\ldots,n$.
Then $Z({\mathcal{E}}^{\prime})\simeq k^{n}$ where $(x_{ij})$ corresponds to
$(x_{i0})$, and $Z({\mathcal{F}}\cup{\mathcal{E}}^{\prime})\simeq
Z_{k}({\mathcal{F}})$.

Let $\sigma$ denotes the Frobenius $p$-th power map: $\sigma(x)=x^{p}$ for
$x\in\bar{k}$. Suppose $\lambda\in Gl_{n}(k)$. Let $\tilde{\lambda}\in
Gl_{nm}(k)$ such that $\tilde{\lambda}$ is block-diagonally decomposed as
$(\lambda_{0},\lambda_{1},\ldots,\lambda_{m-1})$ with
$\lambda_{j}=\lambda^{\sigma^{j}}$ acting on the block
$(x_{1j},\ldots,x_{nj})$. Let $\vec{x}_{j}=(x_{1j},\ldots,x_{nj})$, for
$j=0,\ldots,m-1$. Then
$x_{ij}\circ\tilde{\lambda}=(\lambda_{j}(\vec{x}_{j}))_{i}$, so
$\vec{x}_{j}\circ\tilde{\lambda}=\lambda_{j}(\vec{x}_{j})$.

######  Lemma 3.3.

$d_{{\mathcal{F}}\cup{\mathcal{E}}^{\prime}}=d_{\lambda^{*}{{\mathcal{F}}}\cup{\mathcal{E}}^{\prime}}$

Proof Again, the last fall degree is invariant under linear change of
variables and linear change of equations (Proposition 2.6 (part v) of [3]). So
$d_{\tilde{\lambda}^{*}({\mathcal{F}}\cup{\mathcal{E}}^{\prime})}=d_{{\mathcal{F}}\cup{\mathcal{E}}^{\prime}}$.
Let $\vec{x}_{i}=(x_{1i},\ldots,x_{ni})$, for $i=0,\ldots,m-1$. Let
$\vec{y}_{i}=\lambda_{i}(\vec{x}_{i})$ for $i=0,\ldots,m-1$. Let
${\mathcal{E}}^{\prime\prime}$ consists of $y_{ij}^{p}-y_{i\ j+1}$ for
$i=1,\ldots,n$ and $j=0,\ldots,m-1$ (and where $j+1$ is taken $\mod m$). Then

$(\vec{x}_{j}^{p}-\vec{x}_{j+1})\circ\tilde{\lambda}=(\lambda_{j}(\vec{x}_{j}))^{p}-\lambda_{j+1}(\vec{x}_{j+1})=\vec{y}_{j}^{p}-\vec{y}_{j+1},$

where the index $j+1$ is taken $\mod m$. Hence
$\tilde{\lambda}^{*}({\mathcal{E}}^{\prime})={\mathcal{E}}^{\prime\prime}$.

Let $\mu=\lambda^{-1}$, and $\mu_{i}=\lambda_{i}^{-1}$ for $i=0,\ldots,m-1$.
Then for $j=0,\ldots,m-1$, $\vec{x}_{j}=\mu_{j}(\vec{y}_{j})$ and

$\vec{x}_{j}^{p}-\vec{x}_{j+1}=(\mu_{j}(\vec{y}_{j}))^{p}-\mu_{j+1}(\vec{y}_{j+1})=\mu_{j+1}(\vec{y}_{j}^{p}-\vec{y}_{j+1}),$

where the index $j+1$ is taken $\mod m$.

We conclude that
${\mathcal{E}}^{\prime}=\tilde{\mu}({\mathcal{E}}^{\prime\prime})$ where
$\tilde{\mu}\in Gl_{mn}(k)$ is block-diagonally decomposed as
$(\mu_{0},\mu_{1},\ldots,\mu_{m-1})$.

We have

$d_{{\mathcal{F}}\cup{\mathcal{E}}^{\prime}}=d_{\tilde{\lambda}^{*}({\mathcal{F}}\cup{\mathcal{E}}^{\prime})}=d_{\lambda^{*}({\mathcal{F}})\cup{\mathcal{E}}^{\prime\prime}}=d_{\lambda^{*}({\mathcal{F}})\cup\tilde{\mu}({\mathcal{E}}^{\prime\prime})}=d_{\lambda^{*}({\mathcal{F}})\cup{\mathcal{E}}^{\prime}}.$

$\Box$

###  3.2. Solving for $k$-rational points

######  Lemma 3.4.

Let $k$ be a finite field with $|k|=q=p^{m}$. Let $f\in k[x_{0}]$ with $d=\deg
f$ and ${\mathcal{F}}=\\{f,x_{0}^{p}-x_{1},\ldots,x_{m-1}^{p}-x_{0}\\}$. Then
$x_{i}\equiv_{pd}g_{i}$ with $g_{i}\in k[x_{0}]$ and $\deg g_{i}<d$ for
$i=1,\ldots,m-1$. Suppose $g=gcd(f,x_{0}^{q}-x_{0})\in k[x_{0}]$. Then
$g\equiv_{dp}0\pmod{{\mathcal{F}}}$.

Proof We have $x_{0}^{p}\equiv_{\max\\{p,d\\}}g_{1}\pmod{f}$ for some
$g_{1}\in k[x_{0}]$ with $\deg g_{1}<d$. We have
$g_{1}^{p}\equiv_{pd}g_{2}\pmod{f}$ for some $g_{2}\in k[x_{0}]$ with $\deg
g_{2}<d$. Inductively we have $g_{i}^{p}\equiv_{pd}g_{i+1}\pmod{f}$ with
$g_{i+1}\in k[x_{0}]$ and $\deg g_{i+1}<d$, for $i=1,\ldots,m-2$. In
particular it follows that $x_{0}^{q}-x_{0}\equiv h\pmod{f}$ where
$h=g_{m}-x_{0}$, so $gcd(x_{0}^{q}-x_{0},f)=gcd(f,h)$.

We have

$x_{1}\equiv_{p}x_{0}^{p}\equiv_{\max\\{p,d\\}}g_{1}\pmod{{\mathcal{F}}}$

, and inductively,

$x_{i+1}\equiv_{p}x_{i}^{p}\equiv_{pd}(g_{i})^{p}\equiv_{pd}g_{i+1}\pmod{{\mathcal{F}}},$

for $i=1,\ldots,m-2$. Finally,
$x_{0}\equiv_{p}x_{m-1}^{p}\equiv_{pd}g_{m-1}^{p}\equiv_{pd}g_{m}\pmod{{\mathcal{F}}}$.
It follows that $h\equiv_{pd}0\pmod{{\mathcal{F}}}$, consequently
$gcd(h,f)\equiv_{pd}0\pmod{{\mathcal{F}}}$. Therefore
$gcd(x_{0}^{q}-x_{0},f)\equiv_{pd}0\pmod{{\mathcal{F}}}$. $\Box$

######  Theorem 3.5.

Let $k$ be a finite field with $|k|=q=p^{m}$. Let
${\mathcal{G}}=\\{g_{1},\ldots,g_{n}\\}$ where $g_{i}\in
k[x_{1},\ldots,x_{n}]$ with $\deg g_{i}\leq d$ for $i=1,\ldots,n$. Let
$f_{i}\in k[x_{i}]$ for $i=1,\ldots,n$. Let
$\vec{g}=\left(\begin{array}[]{c}g_{1}\\\ .\\\ .\\\ g_{n}\end{array}\right)$
and $\vec{f}=\left(\begin{array}[]{c}f_{1}\\\ .\\\ .\\\
f_{n}\end{array}\right)$. Suppose $\vec{g}=\rho\circ\vec{f}\circ\lambda$ with
$\rho,\lambda\in Gl_{n}(k)$. Then (1) $d_{{\mathcal{G}}\cup{\mathcal{E}}}\leq
d+q$ where ${\mathcal{E}}=\\{x_{i}^{q}-x_{i}:i=1,\ldots,n\\}$. (2)
$d_{{\mathcal{G}}\cup{\mathcal{E}}^{\prime}}\leq dp$, where
${\mathcal{E}}^{\prime}$ consists of $x_{ij}^{p}-x_{i\ j+1}$ for
$i=1,\ldots,n$, $j=0,\ldots,m-1$ and $j+1$ is taken $\mod m$, and
$g(x_{1},\ldots,x_{n})\in k[x_{1},\ldots,x_{n}]$ is identified with
$g(x_{10},\ldots,x_{n0})\in k[x_{ij}:1=1,\ldots,n,j=0,\ldots,m-1]$.

Proof Let ${\mathcal{F}}=\\{f_{1},\ldots,f_{n}\\}$ and $\mu=\lambda^{-1}$.
Then $\vec{f}=\rho^{-1}\circ\vec{g}\circ\mu$. We see that ${\mathcal{F}}$ and
$\mu^{*}{\mathcal{G}}$ are related by $\rho^{-1}$. So by Lemma 3.2,
$d_{{\mathcal{G}}\cup{\mathcal{E}}}=d_{\mu^{*}{{\mathcal{G}}}\cup{\mathcal{E}}}=d_{{\mathcal{F}}\cup{\mathcal{E}}}$.
By Lemma 3.3,
$d_{{\mathcal{G}}\cup{\mathcal{E}}^{\prime}}=d_{\mu^{*}{{\mathcal{G}}}\cup{\mathcal{E}}^{\prime}}=d_{{\mathcal{F}}\cup{\mathcal{E}}^{\prime}}$.
So to prove (1) and (2) we are reduced to proving
$d_{{\mathcal{F}}\cup{\mathcal{E}}}\leq d+q$ and
$d_{{\mathcal{F}}\cup{\mathcal{E}}^{\prime}}\leq dp$.

To prove $d_{{\mathcal{F}}\cup{\mathcal{E}}}\leq d+q$, let $I^{\prime}$ be the
ideal generated by ${\mathcal{F}}^{\prime}:={\mathcal{F}}\cup{\mathcal{E}}$.
Let $h_{i}(x_{i})=gcd(x_{i}^{q}-x_{i},f_{i}(x_{i}))$ for $i=1,\ldots,n$. Then
$I^{\prime}$ is generated by ${\mathcal{H}}=\\{h_{i}(x_{i}):i=1,\ldots,n\\}$.
Since $h_{i}=A(x_{i}^{q}-x_{i})+Bf_{i}$ with $A,B\in k[x_{i}]$ with $\deg
A<\deg f_{i}$ and $\deg B<q$,
$h_{i}(x_{i})\equiv_{d_{i}+q}0\pmod{\\{x_{i}^{q}-x_{i},f_{i}(x_{i})\\}}$. So
$h_{i}(x_{i})\equiv_{d_{i}+q}0\pmod{{\mathcal{F}}^{\prime}}$. Let $c_{i}=\deg
h_{i}$ for $i=1,\ldots,n$. As with the proof of Theorem 3.1, it is easy to see
that for all $f\in\bar{k}[x_{1}\ldots x_{n}]$, we have
$f\equiv_{\max(c_{i},\deg f)}\bar{f}\pmod{{\mathcal{H}}}$ where
$\bar{f}\in\bar{k}[x_{1},\ldots,x_{n}]$ with $\deg_{x_{i}}\bar{f}<c_{i}$ for
all $i$. Finally $f\in I^{\prime}$ if and only if $\bar{f}\in I^{\prime}$, and
since $\deg_{x_{i}}\bar{f}<d_{i}$ for all $i$, we conclude from Lemma 2.4
$\bar{f}\in I^{\prime}$ if and only if $\bar{f}=0$. It follows that
$I_{i}=V_{{\mathcal{F}}^{\prime},i}$ for $i\geq d+q$. Hence
$d_{{\mathcal{F}}^{\prime}}\leq d+q$.

Next we prove $d_{{\mathcal{F}}\cup{\mathcal{E}}^{\prime}}\leq dp$ in a
similar fashion by using Lemma 3.4. Put
${\mathcal{F}}^{\prime\prime}={\mathcal{F}}\cup{\mathcal{E}}^{\prime}$ and let
$J$ be the ideal generated by ${\mathcal{F}}^{\prime\prime}$. Let
${\mathcal{H}}=\\{h_{1},\ldots,h_{n}\\}$ where
$h_{i}=gcd(f_{i},x_{i}^{q}-x_{i})$ for $i=1,\ldots,n$. By Lemma 3.4
${\mathcal{H}}\subset V_{{\mathcal{F}}^{\prime\prime},dp}$. Then
$J=\langle{\mathcal{H}}\cup{\mathcal{E}}^{\prime}\rangle$ and $J\cap
k[x_{i0}:i=1,\ldots,n]=\langle{\mathcal{H}}\rangle$. Moreover
$Z({\mathcal{F}}^{\prime\prime})$ maps bijectively to $Z({\mathcal{H}})\subset
k^{n}$ through the projection map sending
$(x_{ij})_{i=1,\ldots,n;j=0,\ldots,m-1}$ to $(x_{i0})_{i=1,\ldots,n}$. By
Lemma 3.4, $x_{ij}\equiv_{pd}g_{ij}\pmod{{\mathcal{F}}^{\prime\prime}}$ with
$g_{ij}\in k[x_{i0}]$ and $\deg g_{ij}<d$. We have
$x_{ij}^{2}\equiv_{pd}g_{ij}^{2}\pmod{{\mathcal{F}}^{\prime\prime}}$, and
$g_{ij}^{2}\equiv_{2d}g_{ij,2}\pmod{h_{i}}$ with $g_{ij,2}\in k[x_{i0}]$ and
$\deg g_{ij,2}<d$. Inductively we see that
$x_{ij}^{e}\equiv_{pd}g_{ij,e}\pmod{{\mathcal{F}}^{\prime\prime}}$ with
$g_{ij,e}\in k[x_{i0}]$ and $\deg g_{ij,e}<d$, for $e<p$. From this it is easy
to see that for all $f\in k[x_{ij}:i=1,\ldots,n;j=0,\ldots,m-1]$,
$f\equiv_{\max(\deg f,pd)}\bar{f}$ where $\bar{f}\in k[x_{i0}:i=1,\ldots,n]$
and $\deg_{x_{i0}}(\bar{f})<\deg h_{i}$ for all $i$. So $f\in J$ if and only
if $\bar{f}\in J\cap k[x_{i0}:i=1,\ldots,n]=\langle{\mathcal{H}}\rangle$. We
conclude from Lemma 2.4 $\bar{f}\in\langle{\mathcal{H}}\rangle$ if and only if
$\bar{f}=0$. It follows that $J_{i}=V_{{\mathcal{F}}^{\prime\prime},i}$ for
$i\geq pd$. Hence $d_{{\mathcal{F}}^{\prime}}\leq pd$. $\Box$

## References

  * [1] Bettale, L., Faugère, J.-C., and Perret, L. Cryptanalysis of HFE, multi-HFE and variants for odd and even characteristic.  Des. Codes Cryptogr. 69, 1 (2013), 1–52.
  * [2] Huang, M.-D. A., Kosters, M., and Yeo, S. L. Last fall degree, HFE, and Weil descent attacks on ECDLP  Cryptology ePrint Archive, Report 2015/573, 2015
  * [3] Huang, M.-D. A., Kosters, M., Yang, Y., and Yeo, S. L. On the last fall degree of zero-dimensional Weil descent systems  J. Symbolic Computation, Volume 87, 2018, pp. 207-222
  * [4] Y. N. Lakshman and D. Lazard. On the Complexity of Zero-dimensional Algebraic Systems. Effective Methods in Algebraic Geometry, Volume 94 of the series Progress in Mathematics, (1991), 217-225.


# Further norm and numerical radius inequalities for sum of Hilbert space
operators

D. Afraz, R. Lashkaripour and M. Bakherad  Department of Mathematics, Faculty
of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran.
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Let ${\mathbb{B}}(\mathscr{H})$ denotes the set of all bounded linear
operators on a complex Hilbert space ${\mathscr{H}}$. In this paper, we
present some norm inequalities for sums of operators which are a
generalization some recent results. Among other inequalities, it is shown that
if $S,T\in{\mathbb{B}}({\mathscr{H}})$ are normal operators, then

$\displaystyle\left\|S+T\right\|$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}(\left\|S\right\|+\left\|T\right\|)+\frac{1}{2}\min_{t>0}\sqrt{(\left\|S\right\|-\left\|T\right\|)^{2}+\left\|\frac{1}{t}f_{1}(|S|)g_{1}(|T|)+tf_{2}(|S|)g_{2}(|T|)\right\|^{2}},$

where $f_{1},f_{2},g_{1},g_{2}$ are non-negative continuous functions on
$[0,\infty)$, in which $f_{1}(x)f_{2}(x)=x$ and $g_{1}(x)g_{2}(x)=x\,\,(x\geq
0)$. Moreover, several inequalities for the numerical radius are shown.

###### Key words and phrases:

Keywords: Operator norm, Positive operator, Norm inequality, Numerical radius.

###### 2010 Mathematics Subject Classification:

Primary: 47A30, 47A12; Secondary 47A63, 47L05.

##  1\. Introduction

Let ${\mathbb{B}}(\mathscr{H})$ denotes the $C^{*}$-algebra of all bounded
linear operators on a complex Hilbert space ${\mathscr{H}}$ with an inner
product $\langle\cdot,\cdot\rangle$ and the corresponding norm $\|\cdot\|$. In
the case when ${\rm dim}{\mathscr{H}}=n$, we identify
${\mathbb{B}}({\mathscr{H}})$ with the matrix algebra $\mathbb{M}_{n}$ of all
$n\times n$ matrices with entries in the complex field. For
$S\in{\mathbb{B}}(\mathscr{H})$, let $S=\Re(S)+i\Im(S)$ be the Cartesian
decomposition of $S$, where the Hermitian matrices $\Re(S)=\frac{S+S^{*}}{2}$
and $\Im(S)=\frac{S-S^{*}}{2i}$ are called the real and imaginary parts of
$S$, respectively. The numerical radius of $S\in{\mathbb{B}}({\mathscr{H}})$
is defined by

$\displaystyle w(S):=\sup\\{|\langle
Sx,x\rangle|:x\in{\mathscr{H}},\|x\|=1\\}.$

It is well known that $w(\,\cdot\,)$ defines a norm on
${\mathbb{B}}({\mathscr{H}})$, which is equivalent to the usual operator norm
$\|\cdot\|$. In fact, for any $S\in{\mathbb{B}}({\mathscr{H}})$,
$\frac{1}{2}\|S\|\leq w(S)\leq\|S\|$; see. Let $r(\cdot)$ denote the spectral
radius. It is well known that for every operator
$S\in{\mathbb{B}}({\mathscr{H}})$, we have $r(S)\leq w(S)$. In, the authors
showed that
$w\left(S\right)=\sup_{\theta\in\mathbb{R}}\|\Re({{\rm{e}}^{i\theta}S})\|=\sup_{\alpha^{2}+\beta^{2}=1}\|\alpha\Re
S+\beta\Im S\|,$ which is equali to the above definition. For more facts about
the numerical radius see and references therein. Let
$S,T,X,Y\in{\mathbb{B}}({\mathscr{H}})$. The operator matrices
$\left[\begin{array}[]{cc}S&0\\\ 0&Y\end{array}\right]$ and
$\left[\begin{array}[]{cc}0&T\\\ X&0\end{array}\right]$ are called the
diagonal and off-diagonal parts of the operator matrix
$\left[\begin{array}[]{cc}S&T\\\ X&Y\end{array}\right]$, respectively.
In , it has been shown that if $S$ is an operator in
${\mathbb{B}}({\mathscr{H}})$, then

$\displaystyle w(S)\leq\frac{1}{2}\left(\|S\|+\|S^{2}\|^{\frac{1}{2}}\right).$
(1.1)

Several refinements and generalizations of inequality have been given; see
Yamazaki showed that for $S\in{\mathbb{B}}({\mathscr{H}})$ and $t\in[0,1]$ we
have

$\displaystyle w(S)\leq\frac{1}{2}\left(\|S\|+w(\tilde{S}_{t})\right).$ (1.2)

where $S=U|S|$ is the polar decomposition of $S$ and
$\tilde{S}_{t}=|S|^{t}U|S|^{1-t}$. Horn et al. proved that

$\displaystyle\|S+T\|\leq\||S|+|T|\|,$ (1.3)

where $S,T\in{\mathbb{B}}({\mathscr{H}})$ are normal. Davidson and Power
proved that if $S$ and $T$ are positive operators in
${\mathbb{B}}({\mathscr{H}})$, then

$\displaystyle\|S+T\|\leq\max\\{\|S\|,\|T\|\\}+\|ST\|^{\frac{1}{2}}.$ (1.4)

Inequality has been generalized in . In , the author extended this inequality
to the form

$\displaystyle\|S+T\|\leq\max\\{\|S\|,\|T\|\\}+\frac{1}{2}\left(\left\||S|^{t}|T|^{1-t}\right\|+\left\||S^{*}|^{1-t}|T^{*}|^{t}\right\|\right),$
(1.5)

in which $S,T\in{\mathbb{B}}({\mathscr{H}})$ and $t\in[0,1]$. In , the authors
showed that a generalization of inequality as follows:

$\displaystyle\|S+T\|\leq\max\\{\|S\|,\|T\|\\}+\frac{1}{2}\Big{(}\big{\|}f(|S|)g(|T|)\big{\|}+\big{\|}f(|S^{\ast}|)g(|T^{\ast}|)\big{\|}\Big{)},$

in which $S,T\in{\mathbb{B}}({\mathscr{H}})$ and $f,g$ be two non-negative
non-decreasing continuous functions on $[0,\infty)$ such that
$f(x)g(x)=x\,\,(x\geq 0)$. Recently, Shi et al. proved the following
inequality

$\displaystyle\left\|S+T\right\|\leq\frac{1}{2}(\left\|S\right\|+\left\|T\right\|)+\frac{1}{2}\sqrt{(\left\|S\right\|-\left\|T\right\|)^{2}+\left\|\frac{1}{t}|S|^{r}|T|^{s}+t|S|^{1-r}|T|^{1-s}\right\|^{2}}$
(1.6)

for normal operators $S,T\in{\mathbb{B}}({\mathscr{H}})$, $r,s\in[0,1]$ and
$t>0$.

In this paper, we give several further norm inequalities for sums of bounded
linear operators. These inequalities refine and generalize inequalities.
Moreover, as another application, we show a new numerical radius inequality
which is a generalization.

##  2\. main results

To present our results, we need the following lemmas.

######  Lemma 2.1.

Let $S,T,X,Y\in{\mathbb{B}}({\mathscr{H}})$. Then

$\displaystyle\left\|\left[\begin{array}[]{cc}S&T\\\
X&Y\end{array}\right]\right\|\leq\left\|\left[\begin{array}[]{cc}\|S\|&\|T\|\\\
\|X\|&\|Y\|\end{array}\right]\right\|.$

######  Lemma 2.2.

If $S,T\in{\mathbb{B}}({\mathscr{H}})$ in which $ST$ is selfadjoint, then

$\displaystyle\|ST\|\leq\|\Re(TS)\|.$

In the first result, a generalization of the inequality is obtained.

######  Theorem 2.3.

Let $S,T\in{\mathbb{B}}({\mathscr{H}})$ be normal and
$f_{1},f_{2},g_{1},g_{2}$ be non-negative continuous functions on
$[0,\infty)$, in which $f_{1}(x)f_{2}(x)=x$ and $g_{1}(x)g_{2}(x)=x\,\,(x\geq
0)$. Then

$\displaystyle\left\|S+T\right\|$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}(\left\|S\right\|+\left\|T\right\|)+\frac{1}{2}\min_{t>0}\sqrt{(\left\|S\right\|-\left\|T\right\|)^{2}+\left\|\frac{1}{t}f_{1}(|S|)g_{1}(|T|)+tf_{2}(|S|)g_{2}(|T|)\right\|^{2}}.$

###### Proof.

Let $S$ and $T$ be positive operators. Then for all $t>0$ we have

$\displaystyle\left\|S+T\right\|$ $\displaystyle=$
$\displaystyle\left\|\left[\begin{array}[]{cc}S+T&0\\\
0&0\end{array}\right]\right\|$ $\displaystyle=$
$\displaystyle\left\|\left[\begin{array}[]{cc}tf_{2}(S)&g_{1}(T)\\\
0&0\end{array}\right]\left[\begin{array}[]{cc}\frac{1}{t}f_{1}(S)&0\\\
g_{2}(T)&0\end{array}\right]\right\|$ $\displaystyle\leq$
$\displaystyle\left\|\Re\left(\left[\begin{array}[]{cc}\frac{1}{t}f_{1}(S)&0\\\
g_{2}(T)&0\end{array}\right]\left[\begin{array}[]{cc}tf_{2}(S)&g_{1}(T)\\\
0&0\end{array}\right]\right)\right\|$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad(\textrm{by Lemma })$
$\displaystyle=$
$\displaystyle\left\|\left[\begin{array}[]{cc}S&\frac{1}{2}(\frac{1}{t}f_{1}(S)g_{1}(T)+tf_{2}(S)g_{2}(T))\\\
\frac{1}{2}(tf_{2}(S)g_{2}(T)+\frac{1}{t}f_{1}(S)g_{1}(T))&T\end{array}\right]\right\|$
$\displaystyle\leq$
$\displaystyle\left\|\left[\begin{array}[]{cc}\|S\|&\frac{1}{2}\|\frac{1}{t}f_{1}(S)g_{1}(T)+tf_{2}(S)g_{2}(T)\|\\\
\frac{1}{2}\|tf_{2}(S)g_{2}(T)+\frac{1}{t}f_{1}(S)g_{1}(T)\|&\|T\|\end{array}\right]\right\|$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad(\textrm{by Lemma })$
$\displaystyle=$ $\displaystyle
r\left(\left[\begin{array}[]{cc}\|S\|&\frac{1}{2}\|\frac{1}{t}f_{1}(S)g_{1}(T)+tf_{2}(S)g_{2}(T)\|\\\
\frac{1}{2}\|\frac{1}{t}f_{2}(S)g_{2}(T)+tf_{1}(S)g_{1}(T)\|&\|T\|\end{array}\right]\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad(\textrm{since is Hermitian
matrix})$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\left\|S\right\|+\left\|T\right\|)+\frac{1}{2}\sqrt{(\left\|S\right\|-\left\|T\right\|)^{2}+\left\|\frac{1}{t}f_{1}(|S|)g_{1}(|T|)+tf_{2}(|S|)g_{2}(|T|)\right\|^{2}},$

whence

$\displaystyle\left\|S+T\right\|$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}(\left\|S\right\|+\left\|T\right\|)+\frac{1}{2}\min_{t>0}\sqrt{(\left\|S\right\|-\left\|T\right\|)^{2}+\left\|\frac{1}{t}f_{1}(|S|)g_{1}(|T|)+tf_{2}(|S|)g_{2}(|T|)\right\|^{2}}$

for positive operators $S$ and $T$. Now, if $S$ and $T$ are normal operators,
then by using inequality we have

$\displaystyle\left\|S+T\right\|$ $\displaystyle\leq$
$\displaystyle\left\||S|+|T|\right\|\qquad(\textrm{by inequality })$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}(\left\||S|\right\|+\left\||T|\right\|)+\frac{1}{2}\min_{t>0}\sqrt{(\left\||S|\right\|-\left\||T|\right\|)^{2}+\left\|\frac{1}{t}f_{1}(|S|)g_{1}(|T|)+tf_{2}(|S|)g_{2}(|T|)\right\|}$
$\displaystyle=$
$\displaystyle\frac{1}{2}(\left\|S\right\|+\left\|T\right\|)+\frac{1}{2}\min_{t>0}\sqrt{(\left\|S\right\|-\left\|T\right\|)^{2}+\left\|\frac{1}{t}f_{1}(|S|)g_{1}(|T|)+tf_{2}(|S|)g_{2}(|T|)\right\|}.$

∎

As a consequence of Theorem we have the following result.

######  Corollary 2.4.

Let $S,T\in{\mathbb{B}}({\mathscr{H}})$ be normal, $f_{1},f_{2},g_{1},g_{2}$
be non-negative continuous functions on $[0,\infty)$ such that
$f_{1}(x)f_{2}(x)=x$ and $g_{1}(x)g_{2}(x)=x\,\,(x\geq 0)$. Then

$\displaystyle\left\|S+T\right\|$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}(\left\|S\right\|+\left\|T\right\|)+\frac{1}{2}\sqrt{(\left\|S\right\|-\left\|T\right\|)^{2}+4\left\|f_{1}(|S|)g_{1}(|T|)\right\|\left\|f_{2}(|S|)g_{2}(|T|)\right\|}.$

###### Proof.

If $S$ and $T$ are normal operators, then

$\displaystyle\left\|S+T\right\|$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}(\left\|S\right\|+\left\|T\right\|)+\frac{1}{2}\min_{t>0}\sqrt{(\left\|S\right\|-\left\|T\right\|)^{2}+\left\|\frac{1}{t}f_{1}(|S|)g_{1}(|T|)+tf_{2}(|S|)g_{2}(|T|)\right\|^{2}}$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}(\left\|S\right\|+\left\|T\right\|)+\frac{1}{2}\min_{t>0}\sqrt{(\left\|S\right\|-\left\|T\right\|)^{2}+\left(\frac{1}{t}\left\|f_{1}(|S|)g_{1}(|T|)\right\|+t\left\|f_{2}(|S|)g_{2}(|T|)\right\|\right)^{2}}$
$\displaystyle=$
$\displaystyle\frac{1}{2}(\left\|S\right\|+\left\|T\right\|)+\frac{1}{2}\sqrt{(\left\|S\right\|-\left\|T\right\|)^{2}+4\left\|f_{1}(|S|)g_{1}(|T|)\right\|\left\|f_{2}(|S|)g_{2}(|T|)\right\|}.$

The last equation follows from
$\min_{t>0}h(t)=\min_{t>0}(\frac{1}{t}a+tb)=2\sqrt{ab}\,\,(a,b\geq 0)$. ∎

For invertible normal operators we get the next result.

######  Corollary 2.5.

Let $S,T\in{\mathbb{B}}({\mathscr{H}})$ be invertible normal and $f,g$ be two
positive continuous functions on $[0,\infty)$. Then

$\displaystyle\left\|S+T\right\|$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}(\left\|S\right\|+\left\|T\right\|)+\frac{1}{2}\min_{t>0}\sqrt{(\left\|S\right\|-\left\|T\right\|)^{2}+\left\|\frac{1}{t}f(|S|)g(|T|)+t|S|(f(|S|))^{-1}|T|(g(|T|))^{-1}\right\|^{2}}.$

In particular,

$\displaystyle\left\|S+T\right\|\leq\frac{1}{2}(\left\|S\right\|+\left\|T\right\|)+\frac{1}{2}\min_{t>0}\sqrt{(\left\|S\right\|-\left\|T\right\|)^{2}+\left\|\frac{1}{t}|S|^{r}|T|^{s}+t|S|^{1-r}|T|^{1-s}\right\|^{2}},$

where $r,s\in\mathbb{R}$.

###### Proof.

For the first inequality, it is enough to put $f_{1}(t)=f(t)$,
$f_{2}(t)=\frac{t}{f(t)}$, $g_{1}(t)=g(t)$ and
$g_{2}(t)=\frac{t}{g(t)}\,\,(t>0)$ in Theorem . In particular, if $f(t)=t^{r}$
and $g(t)=t^{s}\,\,(r,s\in\mathbb{R})$ we get the second inequality. ∎

######  Corollary 2.6.

Let $S,T\in{\mathbb{B}}({\mathscr{H}})$ be invertible normal and $f,g$ be two
positive continuous functions on $[0,\infty)$. Then

$\displaystyle\left\|S+T\right\|\leq\max\\{\left\|S\right\|+\left\|T\right\|\\}+\left\|f(|S|)g(|T|)\right\|^{\frac{1}{2}}\left\||S|(f(|S|))^{-1}|T|(g(|T|))^{-1}\right\|^{\frac{1}{2}}.$

In particular,

$\displaystyle\left\|S+T\right\|\leq\max\\{\left\|S\right\|,\left\|T\right\|\\}+\left\||S|^{r}|T|^{1-s}\right\|^{\frac{1}{2}}\left\||S|^{1-r}|T|^{s}\right\|^{\frac{1}{2}},$
(2.8)

where $r,s\in\mathbb{R}$.

###### Proof.

If $S,T\in{\mathbb{B}}({\mathscr{H}})$ is invertible normal and $f,g$ are two
positive continuous functions on $[0,\infty)$, then by using Theorem we have

$\displaystyle\left\|S+T\right\|$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}(\left\|S\right\|+\left\|T\right\|)+\frac{1}{2}\sqrt{(\left\|S\right\|-\left\|T\right\|)^{2}+\left\|\frac{1}{t}f(|S|)g(|T|)+t|S|(f(|S|))^{-1}|T|(g(|T|))^{-1}\right\|^{2}}$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}(\left\|S\right\|+\left\|T\right\|)+\frac{1}{2}|\left\|S\right\|-\left\|T\right\||+\frac{1}{2}\left\|\frac{1}{t}f(|S|)g(|T|)+t|S|(f(|S|))^{-1}|T|(g(|T|))^{-1}\right\|$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad(\textrm{by the
inequality}\,\,\sqrt{x^{2}+y^{2}}\leq\sqrt{x^{2}}+\sqrt{y^{2}})$
$\displaystyle=$
$\displaystyle\max\\{\|S\|,\|T\|\\}+\left\|\frac{1}{t}f(|S|)g(|T|)+t|S|(f(|S|))^{-1}|T|(g(|T|))^{-1}\right\|$
$\displaystyle\leq$
$\displaystyle\max\\{\|S\|,\|T\|\\}+\frac{1}{2}\left(\frac{1}{t}\left\|f(|S|)g(|T|)\right\|+t\left\||S|(f(|S|))^{-1}|T|(g(|T|))^{-1}\right\|\right).$

Now, we take the minimum on $t\,\,(t>0)$ and we get

$\displaystyle\left\|S+T\right\|$ $\displaystyle\leq$
$\displaystyle\max\\{\|S\|,\|T\|\\}+\frac{1}{2}\min_{t>0}\left(\frac{1}{t}\left\|f(|S|)g(|T|)\right\|+t\left\||S|(f(|S|))^{-1}|T|(g(|T|))^{-1}\right\|\right)$
$\displaystyle=$
$\displaystyle\max\\{\left\|S\right\|,\left\|T\right\|\\}+\left\|f(|S|)g(|T|)\right\|^{\frac{1}{2}}\left\||S|(f(|S|))^{-1}|T|(g(|T|))^{-1}\right\|^{\frac{1}{2}}.$

For the second inequality, put $f(t)=t^{r}$ and
$g(t)=t^{1-s}\,\,(r,s\in\mathbb{R})$ in the first inequality. ∎

######  Remark 2.7.

If $S,T\in{\mathbb{B}}({\mathscr{H}})$ are normal operator, then inequality is
a refinement and generalization of inequality for normal operators. In fact,
in this case

$\displaystyle\left\|S+T\right\|$ $\displaystyle\leq$
$\displaystyle\max\\{\left\|S\right\|,\left\|T\right\|\\}+\left\||S|^{r}|T|^{1-s}\right\|^{\frac{1}{2}}\left\||S|^{1-r}|T|^{s}\right\|^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\max\\{\left\|S\right\|,\left\|T\right\|\\}+\frac{1}{2}\left(\left\||S|^{r}|T|^{1-s}\right\|+\left\||S|^{1-r}|T|^{s}\right\|\right)$
$\displaystyle\qquad\qquad\qquad\qquad(\textrm{by the arithmetic-geometric
inequality}),$

where $r,s\in[0,1]$.

In the next result, we show a generalization of the inequality for arbitrary
operators in $S,T\in{\mathbb{B}}({\mathscr{H}})$.

######  Theorem 2.8.

Let $S,T\in{\mathbb{B}}({\mathscr{H}})$ and $f_{1},f_{2},g_{1},g_{2}$ be non-
negative continuous functions on $[0,\infty)$, in which $f_{1}(x)f_{2}(x)=x$
and $g_{1}(x)g_{2}(x)=x\,\,(x\geq 0)$. Then

$\displaystyle\left\|S+T\right\|\leq\frac{1}{2}(\left\|S\right\|+\left\|T\right\|)+\frac{1}{2}\sqrt{(\left\|S\right\|-\left\|T\right\|)^{2}+\max\\{\alpha,\beta\\}},$

where
$\alpha=\left\|\frac{1}{t}f_{1}(|S^{*}|)g_{1}(|T^{*}|)+tf_{2}(|S^{*}|)g_{2}(|T^{*}|)\right\|^{2}$
and
$\beta=\left\|\frac{1}{t}f_{1}(|S|)g_{1}(|T|)+tf_{2}(|S|)g_{2}(|T|)\right\|^{2}$.

###### Proof.

Let $\tilde{S}=\left[\begin{array}[]{cc}0&S\\\ S^{*}&0\end{array}\right]$ and
$\tilde{T}=\left[\begin{array}[]{cc}0&T\\\ T^{*}&0\end{array}\right]$. Then
$\tilde{S}$ and $\tilde{T}$ are normal operators and

$\|\tilde{S}+\tilde{T}\|=\left\|\left[\begin{array}[]{cc}0&S+T\\\
S^{*}+S^{*}&0\end{array}\right]\right\|=\|S+T\|.$

Hence, applying Theorem we get

$\displaystyle\left\|S+T\right\|=\|\tilde{S}+\tilde{T}\|$ (2.9)
$\displaystyle\leq$
$\displaystyle\frac{1}{2}(\|\tilde{S}\|+\|\tilde{T}\|)+\frac{1}{2}\sqrt{(\|\tilde{S}\|-\|\tilde{T}\|)^{2}+\left\|\frac{1}{t}f_{1}(|\tilde{S}|)g_{1}(|\tilde{T}|)+tf_{2}(|\tilde{S}|)g_{2}(|\tilde{T}|)\right\|^{2}}.$

Moreover, it follows from $\|\tilde{S}\|=\|S\|$, $\|\tilde{T}\|=\|T\|$ and

$f(|\tilde{S}|)=\left[\begin{array}[]{cc}f(|S^{*}|)&0\\\
0&f(|S|)\end{array}\right],\quad
f(|\tilde{T}|)=\left[\begin{array}[]{cc}f(|T^{*}|)&0\\\
0&f(|T|)\end{array}\right]$

that we get

$\displaystyle\left\|\frac{1}{t}f_{1}(|\tilde{S}|)g_{1}(|\tilde{T}|)+tf_{2}(|\tilde{S}|)g_{2}(|\tilde{T}|)\right\|$
$\displaystyle=$
$\displaystyle\left\|\left[\begin{array}[]{cc}\frac{1}{t}f_{1}(|{S^{*}}|)g_{1}(|{T^{*}}|)+tf_{2}(|{S^{*}}|)g_{2}(|{T^{*}}|)&0\\\
0&\frac{1}{t}f_{1}(|{S}|)g_{1}(|{T}|)+tf_{2}(|{S}|)g_{2}(|{T}|)\end{array}\right]\right\|$
$\displaystyle=$
$\displaystyle\max\left\\{\left\|\frac{1}{t}f_{1}(|{S^{*}}|)g_{1}(|{T^{*}}|)+tf_{2}(|{S^{*}}|)g_{2}(|{T^{*}}|)\right\|,\left\|\frac{1}{t}f_{1}(|{S}|)g_{1}(|{T}|)+tf_{2}(|{S}|)g_{2}(|{T}|)\right\|\right\\}.$

Using the recent equality and inequality we reach the desired result. ∎

Applying Theorem and a same argument in the proof of Corollary we get the next
result.

######  Corollary 2.9.

Let $S,T\in{\mathbb{B}}({\mathscr{H}})$. Then

$\displaystyle\left\|S+T\right\|$ $\displaystyle\leq$
$\displaystyle\max\left\\{\left\|S\right\|,\left\|T\right\|\right\\}+\max\left\\{\||S^{*}|^{r}|T^{*}|^{1-s}\|^{\frac{1}{2}}\||S^{*}|^{1-r}|T^{*}|^{s}\|^{\frac{1}{2}},\||S|^{r}|T|^{1-s}\|^{\frac{1}{2}}\||S|^{1-r}|T|^{s}\|^{\frac{1}{2}}\right\\},$

where $r,s\in[0,1]$. In particular,

$\displaystyle\left\|S+T\right\|\leq\max\left\\{\left\|S\right\|,\left\|T\right\|\right\\}+\max\left\\{\||S^{*}|^{\frac{1}{2}}|T^{*}|^{\frac{1}{2}}\|,\||S|^{\frac{1}{2}}|T|^{\frac{1}{2}}\|\right\\}.$
(2.11)

##  3\. Some results for the numerical radius

In this section, as application of norm inequalities for sums of operators, we
present some inequalities for the numerical radius.

######  Theorem 3.1.

Let $S,T\in{\mathbb{B}}({\mathscr{H}})$. Then

$\displaystyle w(S-T)$ $\displaystyle\geq$
$\displaystyle\max\\{2w(S),2w(T)\\}-\max\left\\{\left\|S\right\|,\left\|T\right\|\right\\}$
$\displaystyle-\max\left\\{\||S^{*}|^{r}|T^{*}|^{1-s}\|^{\frac{1}{2}}\||S^{*}|^{1-r}|T^{*}|^{s}\|^{\frac{1}{2}},\||S|^{r}|T|^{1-s}\|^{\frac{1}{2}}\||S|^{1-r}|T|^{s}\|^{\frac{1}{2}}\right\\},$

where $r,s\in[0,1]$.

###### Proof.

If $S,T\in{\mathbb{B}}({\mathscr{H}})$, then

$\displaystyle 2\max\\{w(S),w(T)\\}$ $\displaystyle=$ $\displaystyle
w\left(\left[\begin{array}[]{cc}S&0\\\ 0&T\end{array}\right]\right)$
$\displaystyle=$ $\displaystyle w\left(\left[\begin{array}[]{cc}S+T&0\\\
0&T+S\end{array}\right]+\left[\begin{array}[]{cc}S-T&0\\\
0&T-S\end{array}\right]\right)$ $\displaystyle\leq$ $\displaystyle
w\left(\left[\begin{array}[]{cc}S+T&0\\\
0&T+S\end{array}\right]\right)+w\left(\left[\begin{array}[]{cc}S-T&0\\\
0&T-S\end{array}\right]\right)$ $\displaystyle=$ $\displaystyle
w(S+T)+w(S-T),$

whence

$\displaystyle w(S-T)\geq 2\max\\{w(S),w(T)\\}-w(S+T).$

It follows from $w(S+T)\leq\|S+T\|$ and Corollary that

$\displaystyle w(S-T)$ $\displaystyle\geq$
$\displaystyle\max\\{2w(S),2w(T)\\}-w(S+T)$ $\displaystyle\geq$
$\displaystyle\max\\{2w(S),2w(T)\\}-\|S+T\|$ $\displaystyle\geq$
$\displaystyle\max\\{2w(S),2w(T)\\}-\max\left\\{\left\|S\right\|,\left\|T\right\|\right\\}$
$\displaystyle-\max\left\\{\||S^{*}|^{r}|T^{*}|^{1-s}\|^{\frac{1}{2}}\||S^{*}|^{1-r}|T^{*}|^{s}\|^{\frac{1}{2}},\||S|^{r}|T|^{1-s}\|^{\frac{1}{2}}\||S|^{1-r}|T|^{s}\|^{\frac{1}{2}}\right\\}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad(\textrm{by Corollary }),$

where $r,s\in[0,1]$. ∎

######  Remark 3.2.

If $S,T\in{\mathbb{B}}({\mathscr{H}})$ are normal operator, then
$|S|=|S^{*}|$, $|T|=|T^{*}|$, $w(S)=\|S\|$ and $w(T)=\|T\|$. This concludes
that Theorem appear as

$\displaystyle\|S-T\|\geq\max\left\\{\left\|S\right\|,\left\|T\right\|\right\\}-\||S|^{r}|T|^{1-s}\|^{\frac{1}{2}}\||S|^{1-r}|T|^{s}\|^{\frac{1}{2}}.$

In particular, if $S$ and $T$ are positive, then for $r=s=\frac{1}{2}$ we have

$\displaystyle\|S-T\|\geq\max\left\\{\left\|S\right\|,\left\|T\right\|\right\\}-\|S^{\frac{1}{2}}T^{\frac{1}{2}}\|.$

In the next result, we obtain an upper bound for the numerical radius.

######  Theorem 3.3.

Let $S\in{\mathbb{B}}({\mathscr{H}})$ and $f_{1},f_{2},g_{1},g_{2}$ be non-
negative continuous functions on $[0,\infty)$, in which $f_{1}(x)f_{2}(x)=x$
and $g_{1}(x)g_{2}(x)=x\,\,(x\geq 0)$. Then

$\displaystyle
w(S)\leq\frac{1}{2}\left\|S\right\|+\frac{1}{4}\max\\{\alpha,\beta\\},$

in which
$\alpha=\left\|f_{1}(|S^{*}|)g_{1}(|S|)+f_{2}(|S^{*}|)g_{2}(|S|)\right\|$ and
$\beta=\left\|f_{1}(|S|)g_{1}(|S^{*}|)+f_{2}(|S|)g_{2}(|S^{*}|)\right\|$. In
particular,

$\displaystyle
w(S)\leq\frac{1}{2}\|S\|+\frac{1}{4}\left\|f_{1}(|S^{*}|)f_{2}(|S|)+f_{2}(|S^{*}|)f_{1}(|S|)\right\|.$

###### Proof.

Using the of $w(\cdot)$ and Theorem for $t=1$ we have

$\displaystyle w(S)$ $\displaystyle=$
$\displaystyle\sup_{\theta\in\mathbb{R}}\|\Re({{\rm{e}}^{i\theta}T})\|$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sup_{\theta\in\mathbb{R}}\|{{\rm{e}}^{i\theta}S}+{{\rm{e}}^{-i\theta}S^{*}}\|$
$\displaystyle\leq$
$\displaystyle\frac{1}{4}(\left\|S\right\|+\left\|S^{*}\right\|)+\frac{1}{4}\max\\{\alpha,\beta\\}\qquad(\textrm{by
Theorem })$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left\|S\right\|+\frac{1}{4}\max\\{\alpha,\beta\\},$

in which
$\alpha=\left\|f_{1}(|S^{*}|)g_{1}(|S|)+f_{2}(|S^{*}|)g_{2}(|S|)\right\|$ and
$\beta=\left\|f_{1}(|S|)g_{1}(|S^{*}|)+f_{2}(|S|)g_{2}(|S^{*}|)\right\|$. In
the special case for $g_{1}=f_{2}$ and $g_{2}=f_{1}$ we get the second
inequality. ∎

######  Theorem 3.4.

Let $S\in{\mathbb{B}}({\mathscr{H}})$. Then

$\displaystyle w(S)\leq\max\left\\{\left\|\Re S\right\|,\left\|\Im
S\right\|\right\\}+\frac{1}{2}\||\Re S|^{\frac{1}{2}}|\Im S|^{\frac{1}{2}}\|.$

###### Proof.

Using inequality and the definition $w(\cdot)$ we have

$\displaystyle w(S)$ $\displaystyle=$
$\displaystyle\sup_{\alpha^{2}+\beta^{2}=1}\|\alpha\Re S+\beta\Im S\|$
$\displaystyle\leq$
$\displaystyle\sup_{\alpha^{2}+\beta^{2}=1}\left(\max\left\\{\left\|\alpha\Re
S\right\|,\left\|\beta\Im S\right\|\right\\}+\max\left\\{\||\alpha\Re
S|^{\frac{1}{2}}|\beta\Im S|^{\frac{1}{2}}\|,\||\alpha\Re
S|^{\frac{1}{2}}|\beta\Im S|^{\frac{1}{2}}\|\right\\}\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad(\textrm{by inequality })$
$\displaystyle\leq$ $\displaystyle\max\left\\{\left\|\Re S\right\|,\left\|\Im
S\right\|\right\\}+\sup_{\alpha^{2}+\beta^{2}=1}\left(\sqrt{|\alpha\beta|}\||\Re
S|^{\frac{1}{2}}|\Im S|^{\frac{1}{2}}\|\right)$ $\displaystyle\leq$
$\displaystyle\max\left\\{\left\|\Re S\right\|,\left\|\Im
S\right\|\right\\}+\frac{\sqrt{2}}{2}\||\Re S|^{\frac{1}{2}}|\Im
S|^{\frac{1}{2}}\|.$

∎

Conflict of Interest: The author declare that there is no conflict of
interest.

Funding: The authors declare that there is no research grants from funding
agencies, universities, or other parties.

## References

  * [1] A. Abu-Omar and F. Kittaneh, A numerical radius inequality involving the generalized Aluthge transform, Studia Math. 216 (2013), 69-75.
  * [2] A. Abu-Omar and F. Kittaneh, Generalized spectral radius and norm inequalities for Hilbert space operators, International Journal of Mathematics Vol. 26, No. 11 (2015) 1550097 (9 pages).
  * [3] M. Bakherad and K. Shebrawi, Generalizations of the Aluthge transform of operators, Filomat 32 (18), 6465-6474.
  * [4] F. Kittaneh, A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix, Studia Math. 158 (2003), 11-17.
  * [5] F. Kittaneh, M.S. Moslehian and T. Yamazaki, Cartesian decomposition and numerical radius inequalities, Linear Algebra Appl. 471 (2015), 46-53.
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  * [7] K. Shebrawi and H. Albadawi, Numerical radius and operator norm inequalities, J. Math. Inequal. (2009) Article ID 492154, 11 pages.


compat=1.0.0 aainstitutetext: Instituto de Física, Pontificia Universidad
Católica de Valparaíso, Casilla 4950, Valparaíso, Chile b,cb,cinstitutetext:
Departamento de Física, Universidad Técnica Federico Santa María, y Centro
Científico-Tecnológico de Valparaíso, Casilla 110-V, Valparaíso, Chile

# Functional renormalization group flows of $\mathcal{N}=1$ supersymmetric
abelian gauge model with one quiral and one vector superfield

Jeremy Echeverria b Maximiliano Binder c Iván Schmidt
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We apply the functional renormalization group approach to a $\mathcal{N}=1$
supersymmetric gauge model with one quiral superfield coupled to a vector
$U(1)$ superfield. We find that the nonrenormalization theorem still works at
leading order in the supercovariant derivative expansion of the fields. We
also find the beta functions and we study the behavior of its fixed points at
local potential approximation. Regulators are also discussed.

##  1 Introduction

Supersymmetry is probably the most important formalism for achieving a
unification of the fundamental interactions, in particular since it is the
only known one in which fermions and bosons are related by its
transformations. Although so far there has been no experimental confirmation
of the Standard Model (SM) particles superpartners in nature, there is till
hope that indications will appear at higher energies, such as at the Large
Hadron Collider. The present situation in particle physics is that the
Standard model (SM) has proven to work extremely well; nevertheless, it is not
a unified theory and contains several unexplained parameters, so this makes us
think that there must exist a most general theory that can explain these
shortcomings drees1995implications . Supersymmetry has been shown to be a
possible solution, for example, to the hierarchy problem gildener1976gauge ,
the smallness of the cosmological constant froggatt2006smallness , the
renormalizability of supergravity grisaru1976one , among others, in addition
to being a key ingredient of string theory.
Inside supersymmetric models, supersymmetric gauge theories have an special
importance, being the main ingredient to build the minimal supersymmetric
model (MSSM) bagger1996weak , and being a powerful tool for approaching the
quantum theory of gravity through the gauge-gravity duality maldacena1999large
; witten1998anti ; gubser1998gauge . The non-perturbative analysis of
supersymmetric gauge theories is important in order to investigate the
mechanism of strong-coupling phenomena, some properties of condensation
phenomena, aspects of the symmetry breaking or the approach to quantum gravity
through it. Because of this it is clear that we need non-perturbative methods
tho study all aspects of explicit supersymmetry.
The functional renormalization group (FRG) has been an important non-
perturbative tool to apply to a wide range of physical problems
dupuis2021nonperturbative . It consist in applying the functional approach of
the Wilsonian renormalization group wilson1971renormalization ;
wilson1971renormalization2 ; wilson1974renormalization ;
fisher1998renormalization  and is based in an exact flow equation for a scale
dependent effective average action (EAA) wetterich1991average ;
wetterich1993average ; wetterich1993improvement ; wetterich1993exact . The FRG
flow of supersymmetric models has been studied in supersymmetric quantum
mechanics synatschke2009flow  and different types of Wess-Zumino models
synatschke2010n ; feldmann2016functional ; feldmann2018critical .
Abelian supersymmetric gauge theory is the simplest version of a
supersymmetric gauge theory and it has been shown that these types of models
can emerges from quantum phase transitions on topological insulators
jian2017emergence . In this work we study the beta functions and fixed points
of an abelian $\mathcal{N}=1$ supersymmetric gauge theory with one chiral
superfield in the FRG approach, using two types of regulators in the local
potential approximation (LPA). The main goal is to provide a useful way to
apply the FRG in supersymmetric gauge theories, being also the first
application of it to supersymmetric theories involve vector superfields
coupled with chiral superfields. Also is the first step to the study of more
elaborated models such as the MSSM.

##  2 Chiral superfield coupled to $U(1)$ vector superfield

We start with the massless, non interacting Wess-Zumino model
wess1974lagrangian  in $d=4$

$\displaystyle\mathcal{L}_{\textrm{WZ-
free}}=\partial_{\mu}{\phi}^{\dagger}\partial^{\mu}\phi+i\bar{\psi}\bar{\sigma}^{\mu}\partial_{\mu}\psi+{F}^{\dagger}F.$
(2.1)

Te most general interaction for this model has the form

$\displaystyle\mathcal{L}_{\textrm{int}}=W^{\prime}(\phi)F-\frac{1}{2}W^{\prime\prime}(\phi)\psi\cdot\psi+\textrm{h.c},$
(2.2)

where

$\displaystyle W=\frac{1}{2}m\phi^{2}+\frac{1}{6}y\phi^{3}$ (2.3)

is the superpotential, an holomorphic function of the scalar field. Here we
only consider massive and yukawa-type terms. The full lagrangian for the
chiral part of our model is

$\displaystyle\mathcal{L}_{\textrm{chiral}}=\partial_{\mu}{\phi}^{\dagger}\partial^{\mu}\phi+i\bar{\psi}\bar{\sigma}^{\mu}\partial_{\mu}\psi+{F}^{\dagger}F+\left\\{m\phi
F+\frac{1}{2}y\phi^{2}F-\frac{1}{2}m\psi\psi-\frac{1}{2}y\phi\psi\psi+\textrm{h.c}\right\\}.$
(2.4)

On the other hand, we need the gauge part of the lagrangian. We start with a
massless abelian supersymmetric gauge theory with a gauge-fixing term

$\displaystyle\mathcal{L}_{\textrm{abelian-
free}}=-\frac{1}{4}{F}_{\mu\nu}{F}^{\mu\nu}+i\bar{\lambda}\bar{\sigma}^{\mu}\partial_{\mu}\lambda+\frac{1}{2}D^{2}+\frac{1}{2\alpha}{A}_{\mu}{A}^{\mu},$
(2.5)

which we are going to couple it to 2.4 through the covariant derivative
$D_{\mu}=\partial_{\mu}+ig{A}_{\mu}$, where $g$ is the $U(1)$ coupling
constant. There is also necessary to add all possible renormalizable
interactions between the fields that respect Lorentz, gauge and supersymmetric
invariance. The lagrangian of the model is

$\displaystyle\mathcal{L}=$ $\displaystyle
D_{\mu}{\phi}^{\dagger}D^{\mu}\phi+i\bar{\psi}\bar{\sigma}^{\mu}D_{\mu}\psi+{F}^{\dagger}F-\frac{1}{4}{F}_{\mu\nu}{F}^{\mu\nu}+i\bar{\lambda}\bar{\sigma}^{\mu}\partial_{\mu}\lambda+\frac{1}{2}D^{2}+\frac{1}{2\alpha}{A}_{\mu}{A}^{\mu}$
$\displaystyle+\left\\{m\phi
F+\frac{1}{2}y\phi^{2}F-\frac{1}{2}m\psi\psi-\frac{1}{2}y\phi\psi\psi+\textrm{h.c}\right\\}+i\sqrt{2}g\left[{\phi}^{\dagger}\psi\lambda-\phi\bar{\psi}\bar{\lambda}\right]+g{\phi}^{\dagger}\phi
D.$ (2.6)

Is useful rewrite it in terms of an "effective potential":

$\displaystyle\mathcal{L}=$
$\displaystyle\partial_{\mu}{\phi}^{\dagger}\partial^{\mu}\phi+i\bar{\psi}\bar{\sigma}^{\mu}\partial_{\mu}\psi+{F}^{\dagger}F-\frac{1}{2}\partial_{\mu}{A}_{\nu}\partial^{\mu}{A}^{\nu}+\frac{1}{2}\partial_{\mu}{A}_{\mu}\partial^{\nu}{A}^{\nu}+i\bar{\lambda}\bar{\sigma}^{\mu}\partial_{\mu}\lambda+\frac{1}{2}D^{2}+\frac{1}{2\alpha}{A}_{\mu}{A}^{\mu}$
$\displaystyle-\mathcal{U}_{\textrm{eff}},$ (2.7)

where

$\displaystyle\mathcal{U}_{\textrm{eff}}$
$\displaystyle=ig\partial_{\mu}{\phi}^{\dagger}{A}^{\mu}\phi-
ig{A}_{\mu}{\phi}^{\dagger}\partial^{\mu}\phi-g^{2}{A}_{\mu}{A}^{\mu}{\phi}^{\dagger}\phi+g\bar{\psi}\bar{\sigma}^{\mu}{A}_{\mu}\psi$
$\displaystyle-\left\\{m\phi
F+\frac{1}{2}y\phi^{2}F-\frac{1}{2}m\psi\psi-\frac{1}{2}y\phi\psi\psi+\textrm{h.c}\right\\}$
$\displaystyle-i\sqrt{2}g\left[{\phi}^{\dagger}\psi\lambda-\phi\bar{\psi}\bar{\lambda}\right]-g{\phi}^{\dagger}\phi
D.$ (2.8)

The model can also be constructed from a lagrangian defined over the
superspace ferrara1974supergauge ; wess1974supergauge

$\displaystyle\mathcal{L}=$ $\displaystyle\int d^{2}\theta
d^{2}\bar{\theta}\Phi^{\dagger}e^{2gV}\Phi+\frac{1}{32}\int
d^{2}\theta{W}^{\alpha}{W}_{\alpha}+\frac{1}{\alpha}\int d^{2}\theta
d^{2}\bar{\theta}V^{2}+\int d^{2}\theta\left\\{W(\Phi)+\textrm{h.c}\right\\}$
(2.9)

where $\Phi(x,\theta,{\theta}^{\dagger})$ is defined as a chiral superfield
and satisfies the conditions

$\displaystyle{\bar{D}}_{\dot{\alpha}}\Phi=0$
$\displaystyle{D}_{\alpha}{\Phi}^{\dagger}=0$ (2.10)

where

$\displaystyle{D}_{\alpha}=\dfrac{\partial}{\partial\theta^{\alpha}}+i(\sigma^{\mu})_{\alpha\dot{\alpha}}\partial_{\mu}$
$\displaystyle{\bar{D}}_{\dot{\alpha}}=-\dfrac{\partial}{\partial{\bar{\theta}}^{\dot{\alpha}}}-i{\theta}^{\alpha}(\sigma^{\mu})_{\alpha\dot{\alpha}}\partial_{\mu}.$
(2.11)

The expansion in grassmann variables of the chiral superfield is

$\displaystyle\Phi(x,\theta,\bar{\theta})=$
$\displaystyle\phi(x)+\sqrt{2}\theta\psi(x)+\theta\theta
F(x)+i\partial_{\mu}\phi(x)\theta\sigma^{\mu}\bar{\theta}-\frac{i}{\sqrt{2}}\theta\theta\partial_{\mu}\psi(x)\sigma^{\mu}\bar{\theta}-\frac{1}{4}\partial_{\mu}\partial^{\mu}\phi(x)\theta\theta\bar{\theta}\bar{\theta}.$
(2.12)

Moreover, $W(\Phi)$ is a superfunction of the chiral superfields that give us
the superpotential

$\displaystyle W(\Phi)=\frac{1}{2}m\Phi^{2}+\frac{1}{6}y\Phi^{3}.$ (2.13)

On the other hand, $V(x,\theta,{\theta}^{\dagger})$ is defined as a vector
superfield and satisfies the condition

$\displaystyle V=V^{\dagger}.$ (2.14)

This superfield can be transform through

$\displaystyle V\to V+i\left(\Omega-{\Omega}^{\dagger}\right),$ (2.15)

where $\Omega$ is a chiral superfield. The transformation 2.15 is called a
$U(1)$ supergauge transformation and we can semi-fix it such that it only
holds for the usual gauge transformations. This is known as Wess-Zumino gauge
ferrara1974supergauge  and its expansion in grassmann variables is

$\displaystyle V_{\textrm{W-Z
gauge}}=\theta\sigma^{\mu}\bar{\theta}{A}_{\mu}+i\theta\theta\bar{\theta}\bar{\lambda}-i\bar{\theta}\bar{\theta}\theta\lambda+\frac{1}{2}\theta\theta\bar{\theta}\bar{\theta}D.$
(2.16)

Finally, $W_{\alpha}$ is the gauge invariant abelian field strength superfield

$\displaystyle{W}_{\alpha}\equiv\bar{D}^{2}{D}_{\alpha}V.$ (2.17)

Using 2.16, we can expand $e^{2gV}$ until $V^{2}$ terms and rewrite 2.9 in a
more useful form for our proposes

$\displaystyle\mathcal{L}=$ $\displaystyle\int d^{2}\theta
d^{2}\bar{\theta}\Phi^{\dagger}\Phi+2g\int d^{2}\theta
d^{2}\bar{\theta}\Phi^{\dagger}V\Phi+4g^{2}\int d^{2}\theta
d^{2}\bar{\theta}\Phi^{\dagger}V^{2}\Phi$ $\displaystyle+\frac{1}{32}\int
d^{2}\theta{W}^{\alpha}{W}_{\alpha}-\frac{1}{\alpha}\int d^{2}\theta
d^{2}\bar{\theta}V^{2}+\int d^{2}\theta\left\\{W(\Phi)+\textrm{h.c}\right\\}$
$\displaystyle=$ $\displaystyle\int d^{2}\theta
d^{2}\bar{\theta}\Phi^{\dagger}\Phi+\frac{1}{32}\int
d^{2}\theta{W}^{\alpha}{W}_{\alpha}-\frac{1}{\alpha}\int d^{2}\theta
d^{2}\bar{\theta}V^{2}-U(\Phi,V),$ (2.18)

with

$\displaystyle U(\Phi,V)$ $\displaystyle=-2g\int d^{2}\theta
d^{2}\bar{\theta}\Phi^{\dagger}V\Phi-4g^{2}\int d^{2}\theta
d^{2}\bar{\theta}\Phi^{\dagger}V^{2}\Phi-\int
d^{2}\theta\left\\{W(\Phi)+\textrm{h.c}\right\\}$ (2.19)

the effective potential.

##  3 The functional renormalization group

The dependence on the scale of the model constants and operators will be
threaded within the frame of functional renormalization group. This is
constructed through the flow equation of the effective average action
$\Gamma_{k}$, an action functional that is scale dependent and interpolates
between the classical action at microscopic scales and the quantum effective
action wetterich1991average . On a certain scale of energy, where the system
does not receive contributions of energy fluctuations, we introduce the energy
cutoff scale $\Lambda$, and when the effective average action approximates
$\Lambda$ we shall obtain the classical action, i.e the fluctuations may be
frozen

$\displaystyle\Gamma_{k\to\Lambda}=S.$ (3.1)

On the other hand, if the effective average action approximates to $0$, all
fluctuations should be consider, i.e it will be

$\displaystyle\Gamma_{k\to 0}=\Gamma.$ (3.2)

The flow on the energy scale of the effective average action is determined
from the Wetterich equation wetterich1993exact

$\displaystyle\partial_{t}\Gamma_{k}=\frac{1}{2}\textrm{STr}\left\\{\left[\Gamma^{(2)}_{k}+R_{k}\right]^{-1}\partial_{t}R_{k}\right\\},$
(3.3)

where $t=\ln{\frac{k}{\Lambda}}$ and the supertrace denotes sum over all types
of indices and momentum integration. $\Gamma^{(2)}_{k}$ is the second
functional derivative of the effective average action with respect to the
fields111in momentum space.:

$\displaystyle\left[\Gamma^{(2)}_{k}\right]_{ij}(p,q)=\frac{\vec{\delta}}{\delta\Psi_{i}(p)}\Gamma_{k}\frac{\reflectbox{$\vec{\reflectbox{$\delta$}}$}}{\delta\Psi_{j}(q)}$
(3.4)

where $\Psi$ is a collection of fields, and for this case, we use the
arrangement

$\displaystyle\Psi^{\dagger}=({\phi}^{\dagger}(q),\phi(-q),{F}^{\dagger}(q),F(-q),D(q),{A}_{\mu}(q),\bar{\psi}(q),\psi^{T}(-q),\bar{\lambda}(q),\lambda^{T}(-q))$
$\displaystyle\Psi=(\phi(q),{\phi}^{\dagger}(-q),F(q),{F}^{\dagger}(-q),D(-q),{A}_{\nu}(-q),\psi(q),\bar{\psi}^{T}(-q),\lambda(q),\bar{\lambda}^{T}(-q)).$
(3.5)

On the other hand, $R_{k}=\Delta S^{(2)}_{k}$ is a matrix containing all
regulator functions and it has of the form

$\displaystyle\Delta S_{k}=\int d^{4}q\Psi^{\dagger}(q)R_{k}\Psi(q).$ (3.6)

##  4 Supersymmetric regulators

The goal of using the functional renormalization group is that it preserve
supersymmetry in all steps of its implementation. To guarantee this we need to
add a supersymmetric regulating term $\Delta S_{k}$ in the formulation of EAA.
The types of regulators that can be used in supersymmetric theories in the
local potential approximation(LPA) has been studied in, for example
feldmann2016functional ; feldmann2018critical , and we are going to generalize
them for vector superfields. Following feldmann2016functional  we use the
regulator term

$\displaystyle\Delta S_{k}=$ $\displaystyle\int d^{2}\theta
d^{2}\bar{\theta}\Phi^{\dagger}\rho_{2}(D,\bar{D})\Phi+2g\int d^{2}\theta
d^{2}\bar{\theta}\Phi^{\dagger}\rho_{G}(D,\bar{D})V\Phi+4g^{2}\int d^{2}\theta
d^{2}\bar{\theta}\Phi^{\dagger}\rho_{S}(D,\bar{D})V^{2}\Phi$
$\displaystyle+\frac{1}{32}\int
d^{2}\theta{W}^{\alpha}\tau_{2}(D,\bar{D}){W}_{\alpha}+\left\\{\int
d^{4}xd^{2}\theta\Phi\rho_{1}(D,\bar{D})\Phi+\textrm{h.c}\right\\}+\int
d^{4}xd^{2}\theta d^{2}\bar{\theta}V\tau_{1}(D,\bar{D})V.$ (4.1)

That can be simplified to the form

$\displaystyle\Delta S_{k}=$ $\displaystyle\int d^{2}\theta
d^{2}\bar{\theta}\Phi^{\dagger}r_{2}(\square)\Phi+2g\int d^{2}\theta
d^{2}\bar{\theta}\Phi^{\dagger}r_{G}(\square)V\Phi+4g^{2}\int d^{2}\theta
d^{2}\bar{\theta}\Phi^{\dagger}r_{S}(\square)V^{2}\Phi$
$\displaystyle+\frac{1}{32}\int
d^{2}\theta{W}^{\alpha}t_{2}(\square){W}_{\alpha}+\left\\{\int
d^{4}xd^{2}\theta\Phi r_{1}(\square)\Phi+\textrm{h.c}\right\\}+\int
d^{4}xd^{2}\theta d^{2}\bar{\theta}Vt_{1}(\square)V.$ (4.2)

The reason for using distinct names for the regulator functions in each term
is due to the dimensionless process, as each function is accompanied by
different types of fields which will contain different wave function
renormalizations. The explicit form of the $R_{k}$ operator is shown in
Appendix Appendix B: Operators LPA’.
According to feldmann2016functional , we use two types of regulators:

  * •

Callan Symanzik regulator(CS): $r_{1}=t_{1}=1$ and
$r_{2}=t_{2}=r_{G}=r_{S}=0$.

  * •

Litim-type regulator(LT): $r_{1}=t_{1}=0$ and
$r_{2}=t_{2}=r_{G}=r_{S}=\left(\dfrac{1}{q}-1\right)\theta(1-q^{2})$

for dimensionless and renormalized regulators.

##  5 Supersymmetric and gauge invariant LPA’ truncation

We study the FRG flows of the LPA’ truncation for this model, that is,
consider the effective potential 2.7 as scale dependent. The truncation used
for $\Gamma_{k}$ is

$\displaystyle\Gamma_{k}=$ $\displaystyle\int
d^{4}x\left\\{Z_{\Phi}\partial_{\mu}{\phi}^{\dagger}\partial^{\mu}\phi+iZ_{\Phi}\bar{\psi}\bar{\sigma}^{\mu}\partial_{\mu}\psi+Z_{\Phi}{F}^{\dagger}F\right.-iZ_{G}g_{k}\partial_{\mu}{\phi}^{\dagger}{A}^{\mu}\phi+iZ_{G}g_{k}{A}_{\mu}{\phi}^{\dagger}\partial^{\mu}\phi$
$\displaystyle+Z_{S}g_{k}^{2}{A}_{\mu}{A}^{\mu}{\phi}^{\dagger}\phi-
Z_{G}g_{k}\bar{\psi}\bar{\sigma}^{\mu}{A}_{\mu}\psi+i\sqrt{2}Z_{G}g_{k}\left[{\phi}^{\dagger}\psi\lambda-\phi\bar{\psi}\bar{\lambda}\right]+Z_{G}g_{k}{\phi}^{\dagger}\phi
D$
$\displaystyle+Z_{V}\left(-\frac{1}{2}\partial_{\mu}{A}_{\nu}\partial^{\mu}{A}^{\nu}+\frac{1}{2}\partial_{\mu}{A}^{\mu}\partial_{\nu}{A}^{\nu}\right.+\frac{1}{2\alpha}{A}_{\mu}{A}^{\mu}\left.+i\bar{\lambda}\bar{\sigma}^{\mu}\partial_{\mu}\lambda+\frac{1}{2}D^{2}\right)$
$\displaystyle+\left.\left[\dfrac{\partial
W_{k}}{\partial\phi}F-\frac{1}{2}\dfrac{\partial^{2}W_{k}}{\partial\phi^{2}}\psi\psi+\textrm{h.c}\right]\right\\},$
(5.1)

or in superspace formulation

$\displaystyle\Gamma_{k}=$ $\displaystyle Z_{\Phi}\int d^{2}\theta
d^{2}\bar{\theta}\Phi^{\dagger}\Phi+\frac{Z_{V}}{32}\int
d^{2}\theta{W}^{\alpha}{W}_{\alpha}-\frac{Z_{V}}{\alpha}\int d^{2}\theta
d^{2}\bar{\theta}V^{2}-U_{k}(\Phi,V)$ $\displaystyle=$ $\displaystyle
Z_{\Phi}\int d^{2}\theta d^{2}\bar{\theta}\Phi^{\dagger}\Phi+2g_{k}Z_{G}\int
d^{2}\theta d^{2}\bar{\theta}\Phi^{\dagger}V\Phi+4g_{k}^{2}Z_{S}\int
d^{2}\theta d^{2}\bar{\theta}\Phi^{\dagger}V^{2}\Phi$
$\displaystyle+\frac{Z_{V}}{32}\int
d^{2}\theta{W}^{\alpha}{W}_{\alpha}-\frac{Z_{V}}{\alpha}\int d^{2}\theta
d^{2}\bar{\theta}V^{2}+\int
d^{2}\theta\left\\{W_{k}(\Phi)+\textrm{h.c}\right\\}.$ (5.2)

where ($Z_{\Phi}$, $Z_{V}$, $Z_{g}$) are the wave function renormalizations
and does not depend on the energy scale. Besides we have

$\displaystyle W_{k}=$
$\displaystyle\frac{1}{2}m_{k}\phi^{2}+\frac{1}{6}y_{k}\phi^{3},$
$\displaystyle Z_{G}=$ $\displaystyle
Z_{\Phi}Z_{g}\sqrt{Z_{V}}\quad\textrm{and}$ $\displaystyle Z_{S}=$
$\displaystyle Z_{\Phi}Z^{2}_{g}Z_{V}.$ (5.3)

The explicit form of the operator $\Gamma^{(2)}$ is shown in Appendix Appendix
B: Operators LPA’.

##  6 Flow equations

Now we are going to present the results for the non-renormalization theorem
and the beta functions for the dimensionless gauge ($g_{k}$) and Yukawa
($y_{k}$) couplings, in addition to the dimensionless mass ($m_{k}$), in the
LPA’ truncation. First, we write 5.1 and 4.2 in momentum space; second,
because the wave function renormalizations do not depend on the energy scale,
we can restrict all fields to real constants
$\Psi_{i}(q)=(\sqrt{2\pi})^{4}\Psi_{i}\delta(q)$. Finally, in Appendix
Appendix A: Conventions and dimensionless-renormalized quantities we show the
dimensionless and renormalized form of the functions, which are used to write
the beta functions in a dimensionless form.

###  6.1 Non-renormalization theorem

If we replace 5.1 in the left side of 3.3 and we set
$\psi=\bar{\psi}=\lambda=\bar{\lambda}=D=A=0$ we obtain

$\displaystyle\partial_{t}\Gamma_{k}=(\sqrt{2\pi})^{4}\delta(0)\partial_{t}\left[Z_{\Phi}{F}^{\dagger}F+\dfrac{\partial
W_{k}}{\partial\phi}F+\dfrac{\partial{W}^{\dagger}_{k}}{\partial{\phi}^{\dagger}}{F}^{\dagger}\right],$
(6.1)

thus, we can obtain the flow of the superpotential taking the $F$ or
${F}^{\dagger}$ derivative and setting $F={F}^{\dagger}=0$ respectively. Doing
this we obtain

$\displaystyle\partial_{t}W=0,$ (6.2)

showing that the non-renormalization theorem works in the LPA’ approximation.

###  6.2 Beta Functions and Fixed Points

In $d$ dimensions the beta functions are

$\displaystyle\beta_{g_{k}}$
$\displaystyle=\left(\eta_{g}+\frac{d-4}{2}\right)g_{k}$
$\displaystyle\beta_{y_{k}}$
$\displaystyle=\left(-\frac{3}{2}\eta_{\Phi}-\frac{d-4}{2}\right)y_{k},$
$\displaystyle\beta_{m_{k}}$ $\displaystyle=\left(\eta_{\Phi}-1\right)m_{k},$
(6.3)

where $\eta_{\mathcal{O}}=-\partial_{t}\ln{Z_{\mathcal{O}}}$ is the LPA’
anomalous dimension of the operator $\mathcal{O}$ and
$\beta_{\mathcal{O}}=\partial_{t}\mathcal{O}$ is the beta function. We compute
the anomalous dimensions in the same way as before but setting
$\psi={\psi}^{\dagger}=F={F}^{\dagger}=A=\lambda={\lambda}^{\dagger}=0$ in 3.3

$\displaystyle\partial_{t}\Gamma_{k}=(\sqrt{2\pi})^{4}\delta(0)\partial_{t}\left[g_{k}Z_{G}{\phi}^{\dagger}\phi
D+\frac{Z_{V}}{2}D^{2}\right].$ (6.4)

We compute the anomalous dimensions using

  * •

$\eta_{\Phi}$ taking the $\phi,{\phi}^{\dagger}$ derivatives of 6.1 at
$\phi={\phi}^{\dagger}=0$,

  * •

$\eta_{V}$ taking the $D$ second derivative of 6.4 at $D=0$,

  * •

$\eta_{G}$ taking the $\phi,{\phi}^{\dagger},D$ derivatives of 6.4 at
$\phi={\phi}^{\dagger}=D=0$

and using 5.3 to write

$\displaystyle\eta_{g}=\eta_{G}-\eta_{\Phi}-\frac{1}{2}\eta_{V}.$ (6.5)

####  6.2.1 Callan-Symanzik regulator

Using the CS regulator the beta functions in $d=4$ are

$\displaystyle\beta^{\textrm{CS}}_{y_{k}}$
$\displaystyle=\frac{24\sqrt{2}\pi^{2}y_{k}^{3}}{96+192\pi+48\pi^{3}+6\pi^{4}+3(2+\pi)^{4}m_{k}-16\pi^{2}(\sqrt{2}y_{k}^{2}-9)},$
$\displaystyle\beta^{\textrm{CS}}_{m_{k}}$
$\displaystyle=-\frac{3(2+\pi)^{4}m_{k}(m_{k}+2)}{96+192\pi+48\pi^{3}+6\pi^{4}+3(2+\pi)^{4}m_{k}-16\pi^{2}(\sqrt{2}y_{k}^{2}-9)},$
$\displaystyle\beta^{\textrm{CS}}_{g_{k}}$
$\displaystyle=\frac{48\sqrt{2}\pi^{2}g_{k}(g_{k}^{2}+y_{k}^{2})}{48+96\pi+24\pi^{3}+3\pi^{4}+3(2+\pi)^{4}m_{k}-8\pi^{2}(\sqrt{2}y^{2}-9)}.$
(6.6)

Explicit forms of the anomalous dimensions in $d$ dimensions are shown in
Appendix Appendix C: Anomalous Dimensions LPA’.
In this case we find that the system only exhibit gaussian fixed points
($g_{k}=y_{k}=m_{k}=0$).

####  6.2.2 Litim-Type regulator

Using the LT regulator the beta functions in $d=4$ are

$\displaystyle\beta^{\textrm{LT}}_{y_{k}}$
$\displaystyle=-\frac{36\sqrt{2}\pi^{\frac{5}{2}}(m_{k}^{2}+1)y_{k}^{3}}{9\sqrt{\pi}(m_{k}^{2}-1)^{3}+8\sqrt{2}\pi^{\frac{5}{2}}(m_{k}^{2}+1)y_{k}^{2}}$
$\displaystyle\beta^{\textrm{LT}}_{m_{k}}$
$\displaystyle=\frac{\sqrt{\pi}m_{k}\left(16\sqrt{2}\pi^{2}y_{k}^{2}(m_{k}^{2}+1)-9(2+\pi)^{4}(m_{k}^{2}-1)^{3}\right)}{9\sqrt{\pi}(m_{k}^{2}-1)^{3}+8\sqrt{2}\pi^{\frac{5}{2}}(m_{k}^{2}+1)y_{k}^{2}}$
$\displaystyle\beta^{\textrm{LT}}_{g_{k}}$
$\displaystyle=\frac{g_{k}\left(m_{k}^{2}(m_{k}^{2}-1)^{3}\left(144+\pi+72\pi^{3}+9\pi^{4}+216\pi^{2}\right)-4\pi^{2}\sqrt{2}(7m_{k}^{4}+4m_{k}^{2}-3)\right)}{(m_{k}^{2}-1)^{4}\left(144+\pi+72\pi^{3}+9\pi^{4}+216\pi^{2}\right)+8\sqrt{2}\pi^{2}(m_{k}^{4}-1)y_{k}^{2}}.$
(6.7)

Explicit form of the anomalous dimensions are shown in Appendix Appendix C:
Anomalous Dimensions LPA’. In this case we find non-gaussian fixed points for
$g_{k}=0$, $y_{k}\neq 0$ and $m_{k}\neq 0$. We can see in fig 1 that there
exists an unstable fixed point at $m_{k}=1$ and a stable one, which changes
continuously its position when $y_{k}$ grows. We can see in fig 2 the
continuous change in the fixed point value for different values of $y_{k}$.

(a) $y_{k}=10$.

(b) $y_{k}=50$.

Figure 1: Flow Diagrams in the $g_{k}-m_{k}$ projection, for different values
of $y_{k}$. Here $m(g)$ is the $y(x)$ axis. Figure 2: Fixed point, for
different values of $m_{k}$ and $y_{k}$.

##  7 Comments and conclusions

We have obtained the beta functions for an abelian supersymmetric gauge theory
in LPA’ with a supersymmetric regularization for two different regulator
functions. We found that the non-renormalization theorem still works when
considering coupling with a vector superfield in LPA’. For the case of Callan-
Symanzik-type (CS) regulator we found that there are no non-gaussian fixed
points, i.e, it does not exist a point
$\left\\{g_{*},y_{*},m_{*}\right\\}\neq\left\\{0,0,0\right\\}$ for wich
$\beta_{g}=\beta_{y}=\beta_{m}=0$. On the other hand, for the Litim-type (LT)
regulator we found that there exists stable non-gaussian fixed points of the
form $\left\\{g_{*},m_{*}\right\\}=\left\\{0,m\right\\}$ and unestable non-
gaussian fixed points $\left\\{g_{*},m_{*}\right\\}=\left\\{0,1\right\\}$ for
any value of $y_{*}$. In the case $y=0$ only exists the unestable fixed point.
Clearly both types of regularizations give completely different results. In
the study of the Wess-Zumino models the CS regulator has been shown to be a
non-optimized regulatorgies2009supersymmetry  in comparison with the optimized
LT regulatorlitim2001optimized . In fact, a mass-type regulator for the vector
superfield in the Wess-Zumino gauge as CS regulator seems not be a good option
because it does not work to regularized $\lambda$ and $D$ field, because the
Wess-Zumino gauge allows that quadratic terms in vector superfield be
proportional to $A^{2}$.
The most important contribution of this work has been to present a useful way
to study beta functions of supersymmetric gauge theories in the non-
perturbative regime, expanding previous works in Wess-Zumino models to include
vector superfields, using convenient regulator functions. We have studied an
abelian gauge theory which involves only one chiral superfield in a LPA’,
focusing in the application of the FRG in this type of models, which in future
work can be generalized to models with more field content and next order
truncations in a superderivative expansion.

###### Acknowledgements.

This research was partially supported by project Proyecto ANID PIA/APOYO
AFB180002 (Chile). J.E.P has been funded by Beca doctorado nacional Chile ANID
N°21212381 and project FONDECYT N°1180232.

## Appendix A: Conventions and dimensionless-renormalized quantities

For momentum space we use the convention

$\displaystyle F(q)=\dfrac{1}{\left(\sqrt{2\pi}\right)^{d}}\displaystyle\int
d^{d}xF(x)e^{i{q}_{\mu}{x}^{\mu}}.$ (.1)

In order to write anomalous dimensions and beta functions in term of
dimensionless and renormalized quantities we defined

$\displaystyle m_{k}=$ $\displaystyle Z_{\Phi}km_{k,R}$ $\displaystyle y_{k}=$
$\displaystyle\sqrt{(Z_{\Phi})^{3}}k^{\frac{d-4}{2}}y_{k,R}$ $\displaystyle
g_{k}=$ $\displaystyle Z_{g}k^{\frac{d-4}{2}}g_{k,R},$ (.2)

also we need the regulator functions in a dimensionless form, for that we
redefine

$\displaystyle r_{1}=$ $\displaystyle Z_{\Phi}kr_{1,R}$ $\displaystyle t_{1}=$
$\displaystyle Z_{V}kt_{1,R}$ $\displaystyle r_{2}=$ $\displaystyle
Z_{\Phi}r_{2,R}$ $\displaystyle t_{2}=$ $\displaystyle Z_{V}t_{2,R}$
$\displaystyle r_{G}=$ $\displaystyle Z_{g}Z_{\Phi}\sqrt{Z_{V}}r_{G,R}$
$\displaystyle r_{S}=$ $\displaystyle Z_{\Phi}Z_{V}Z_{g}^{2}r_{S,R}.$ (.3)

It is convenient to solve the integrals in Appendix Appendix C: Anomalous
Dimensions LPA’ change the integration variable by $q_{R}\to\dfrac{q}{k}$ and
defining $r^{\prime}_{i,R}(q_{R}^{2})=r_{i,R}(k^{2}q_{R}^{2})$. Finally, we
omit all $R$ subscripts and primes.

## Appendix B: Operators LPA’

The way to found the explicit form of the operators $R_{k}$ and
$\Gamma^{(2)}_{k}$ is, in first place, write the regulator 4.2 and the action
5.1 in momentum space, then restrict all fields to real constants
$\Psi_{i}(q)=(\sqrt{2\pi})^{4}\Psi_{i}\delta(q)$ and compute the matrix
$R_{k}$ and the second derivatives of EAA.
In our case, the explicit form of operators are

$\displaystyle R_{k}=\begin{pmatrix}R_{kBB}&R_{kBF}\\\
R_{kFB}&R_{kFF}\end{pmatrix},$ (.4)

where

$\displaystyle R_{kBB}=$
$\displaystyle\begin{pmatrix}r_{2}q^{2}+g_{k}r_{G}D+g_{k}r_{S}A^{2}&0&0&2r_{1}&g_{k}r_{G}\phi&2g_{k}^{2}r_{S}\phi
A\\\
0&r_{2}q^{2}+g_{k}r_{G}D+g_{k}r_{S}A^{2}&2r_{1}&0&g_{k}r_{G}{\phi}^{\dagger}&2g_{k}^{2}r_{S}{\phi}^{\dagger}A\\\
0&2r_{1}&r_{2}&0&0&0\\\ 2r_{1}&0&0&r_{2}&0&0\\\
g_{k}r_{G}{\phi}^{\dagger}&g_{k}r_{G}\phi&0&0&t_{2}&0\\\
2g_{k}^{2}r_{S}{\phi}^{\dagger}A&2g_{k}^{2}r_{S}\phi
A&0&0&0&t_{1}+\frac{q^{2}}{\alpha}t_{2}+2g_{k}^{2}r_{S}\phi{\phi}^{\dagger}\end{pmatrix}$
$\displaystyle R_{kFB}=$
$\displaystyle\begin{pmatrix}0&0&\sqrt{2}g_{k}r_{G}\psi^{T}\sigma^{2}&0\\\
0&\sqrt{2}g_{k}r_{G}\bar{\lambda}\sigma^{2}&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\\\
g_{k}r_{G}\bar{\sigma}^{\mu}\bar{\psi}&g_{k}r_{G}\bar{\sigma}^{\mu}\psi^{T}&0&0\end{pmatrix}$
$\displaystyle R_{kBF}=$
$\displaystyle\begin{pmatrix}0&0&0&0&0&g_{k}r_{G}\bar{\sigma}^{\mu}\psi\\\
0&\sqrt{2}g_{k}r_{G}\lambda\sigma^{2}&0&0&0&g_{k}r_{G}\bar{\sigma}^{\mu}\bar{\psi}^{T}\\\
\sqrt{2}g_{k}r_{G}\bar{\psi}^{T}\sigma^{2}&0&0&0&0&0\\\
0&0&0&0&0&0\end{pmatrix}$ $\displaystyle R_{kFF}=$
$\displaystyle\begin{pmatrix}r_{2}{q}_{\mu}{\bar{\sigma}}^{\mu}+g_{k}r_{G}{A}_{\mu}{\bar{\sigma}}^{\mu}&ir_{1}\sigma^{2}&0&0\\\
-ir_{1}\sigma^{2}&r_{2}{q}_{\mu}{\bar{\sigma}}^{T\mu}+g_{k}r_{G}{A}_{\mu}{\bar{\sigma}}^{T\mu}&\sqrt{2}g_{k}r_{G}{\phi}^{\dagger}\sigma^{2}&0\\\
0&\sqrt{2}g_{k}r_{G}\phi\sigma^{2}&t_{2}{q}_{\mu}{\bar{\sigma}}^{\mu}&0\\\
0&0&0&t_{2}{q}_{\mu}{\bar{\sigma}}^{\mu T}\end{pmatrix}.$

In the same way

$\displaystyle\Gamma_{k}=\begin{pmatrix}\Gamma_{kBB}&\Gamma_{kBF}\\\
\Gamma_{kFB}&\Gamma_{kFF}\end{pmatrix},$ (.5)

with

$\displaystyle\Gamma_{kBB}=$
$\displaystyle\begin{pmatrix}\gamma&y_{k}{F}^{\dagger}&0&m_{k}+y_{k}{\phi}^{\dagger}&g_{k}Z_{G}\phi&2g_{k}^{2}Z_{S}\phi
A\\\
y_{k}F&\gamma&m_{k}+y_{k}\phi&0&g_{k}Z_{G}{\phi}^{\dagger}&2g_{k}^{2}Z_{S}{\phi}^{\dagger}A\\\
0&m_{k}+y_{k}{\phi}^{\dagger}&Z_{\Phi}&0&0&0\\\
m_{k}+y_{k}\phi&0&0&Z_{\Phi}&0&0\\\
g_{k}Z_{G}{\phi}^{\dagger}&g_{k}Z_{G}\phi&0&0&Z_{V}&0\\\
2g_{k}^{2}Z_{S}{\phi}^{\dagger}A&2g_{k}^{2}Z_{S}\phi
A&0&0&0&\frac{q^{2}}{\alpha}Z_{V}+2g_{k}^{2}Z_{S}\phi{\phi}^{\dagger}\end{pmatrix}$
$\displaystyle\Gamma_{kFB}=$
$\displaystyle\begin{pmatrix}0&-\frac{iy_{k}\bar{\psi}}{8\pi^{2}}\sigma^{2}&\sqrt{2}g_{k}Z_{G}\psi^{T}\sigma^{2}&0\\\
\frac{iy_{k}\psi^{T}}{8\pi^{2}}\sigma^{2}&\sqrt{2}g_{k}Z_{G}\bar{\lambda}\sigma^{2}&0&0\\\
0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
g_{k}Z_{G}\bar{\psi}\bar{\sigma}^{\mu}&g_{k}Z_{G}\psi^{T}\bar{\sigma}^{\mu}&0&0\end{pmatrix}$
$\displaystyle\Gamma_{kBF}=$
$\displaystyle\begin{pmatrix}0&-\frac{iy_{k}\bar{\psi}^{T}}{8\pi^{2}}\sigma^{2}&0&0&0&g_{k}Z_{G}\psi\bar{\sigma}^{\mu}\\\
\frac{iy_{k}\psi}{8\pi^{2}}\sigma^{2}&\sqrt{2}g_{k}Z_{G}\lambda\sigma^{2}&0&0&0&g_{k}Z_{G}\bar{\psi}^{T}\bar{\sigma}^{\mu}\\\
\sqrt{2}g_{k}Z_{G}\bar{\psi}^{T}\sigma^{2}&0&0&0&0&0\\\
0&0&0&0&0&0\end{pmatrix}$ $\displaystyle\Gamma_{kFF}=$
$\displaystyle\begin{pmatrix}Z_{\phi}{q}_{\mu}{\bar{\sigma}}^{\mu}+g_{k}Z_{G}{A}_{\mu}{\bar{\sigma}}^{\mu}&-\dfrac{i(m_{k}+y_{k}{\phi}^{\dagger})}{4\pi^{2}}\sigma^{2}&0&0\\\
\dfrac{i(m_{k}+y_{k}\phi)}{4\pi^{2}}\sigma^{2}&Z_{\phi}{q}_{\mu}{\bar{\sigma}}^{T\mu}+g_{k}Z_{G}{A}_{\mu}{\bar{\sigma}}^{T\mu}&\sqrt{2}g_{k}Z_{G}{\phi}^{\dagger}\sigma^{2}&0\\\
0&\sqrt{2}g_{k}Z_{G}\phi\sigma^{2}&Z_{V}q_{\mu}{\bar{\sigma}}^{\mu}&0\\\
0&0&0&Z_{V}q_{\mu}{\bar{\sigma}}^{\mu T}\end{pmatrix}.$ (.6)

Where

$\displaystyle\gamma=$ $\displaystyle
Z_{\Phi}q^{2}+Z_{S}g_{k}^{2}A^{2}+Z_{G}g_{k}D.$ (.7)

## Appendix C: Anomalous Dimensions LPA’

In $d$ dimensions, the explicit form of the dimensionless anomalous dimensions
in both regulators are

$\displaystyle\eta_{\Phi}^{\textrm{CS}}=$
$\displaystyle-\int^{\infty}_{0}dq\frac{\Omega_{d}}{(2\pi)^{d}}q^{d-1}\frac{4y_{k}^{2}\alpha_{1}(1-\eta^{\textrm{CS}}_{\Phi}+\partial_{t}-q\partial_{q})r_{1}}{(\alpha_{1}^{2}-q^{2})^{3}}$
$\displaystyle\eta_{V}^{\textrm{CS}}=$
$\displaystyle-\int^{\infty}_{0}dq\frac{\Omega_{d}}{(2\pi)^{d}}q^{d-1}\frac{8g_{k}^{2}\alpha_{1}(1-\eta^{\textrm{CS}}_{\Phi}+\partial_{t}-q\partial_{q})r_{1}}{(\alpha_{1}^{2}-q^{2})^{3}}$
$\displaystyle\eta_{g}^{\textrm{CS}}=$
$\displaystyle\int^{\infty}_{0}dq\frac{\Omega_{d}}{(2\pi)^{d}}q^{d-1}\frac{4\alpha_{1}\left[3g_{k}^{2}\left(\alpha_{1}^{2}-q^{2}\right)-y_{k}^{2}\left(\alpha_{1}^{2}+5q^{2}\right)\right](1-\eta^{\textrm{CS}}_{\Phi}+\partial_{t}-q\partial_{q})r_{1}}{(\alpha_{1}^{2}-q^{2})^{4}},$
(.8)

where

$\displaystyle\alpha_{1}=m+2r_{1}.$ $\displaystyle\eta_{\Phi}^{\textrm{LT}}=$
$\displaystyle-\int^{1}_{0}dq\frac{\Omega_{d}}{(2\pi)^{d}}q^{d-1}\frac{y_{k}^{2}\alpha_{2}\left(\alpha_{2}^{2}q^{2}+m_{k}^{2}\right)(-\eta_{\Phi}^{\textrm{LT}}+\partial_{t}-q\partial_{q})r_{2}}{(\alpha_{2}^{2}q^{2}-m_{k}^{2})^{3}}$
$\displaystyle\eta_{V}^{\textrm{LT}}=$
$\displaystyle-\int^{1}_{0}dq\frac{\Omega_{d}}{(2\pi)^{d}}q^{d-1}\frac{1}{(\alpha_{2}^{2}q^{2}-m_{k}^{2})^{3}}\left\\{2g_{k}^{2}\alpha_{2}\alpha_{G}\left[\alpha_{G}\left(\alpha_{2}^{2}q^{2}+m_{k}^{2}\right)(-\eta_{\Phi}^{\textrm{LT}}+\partial_{t}-q\partial_{q})r_{2}\right.\right.$
$\displaystyle\left.\left.-\alpha_{2}\left(\alpha_{2}^{2}q^{2}-m_{k}^{2}\right)(-\eta_{g}^{\textrm{LT}}-\eta_{\Phi}^{\textrm{LT}}-\tfrac{1}{2}\eta_{V}^{\textrm{LT}}+\partial_{t}-q\partial_{q})r_{G}\right]\right\\}$
$\displaystyle\eta_{g}^{\textrm{LT}}=$
$\displaystyle-\int^{1}_{0}dq\frac{\Omega_{d}}{(2\pi)^{d}}q^{d-1}\frac{1}{(\alpha_{2}^{2}q^{2}-m_{k}^{2})^{4}\alpha_{t}^{2}}\left\\{\alpha_{t}\left[y_{k}^{2}\alpha_{t}\left(m_{k}^{4}\left(r_{2}-r_{G}\right)-8m_{k}^{2}q^{2}\alpha_{2}^{2}\alpha_{G}-q^{4}\alpha_{2}^{4}\left(4+r_{2}+3r_{G}\right)\right)\right.\right.$
$\displaystyle-\left.g_{k}^{2}\alpha_{2}\left(\alpha_{2}^{4}q^{4}-m_{k}^{4}\right)\alpha_{G}^{2}\left(3+2r_{G}+t_{2}\right)\right](-\eta_{\Phi}^{\textrm{LT}}+\partial_{t}-q\partial_{q})r_{2}$
$\displaystyle-\alpha_{2}\left(\alpha_{2}^{2}-m_{k}^{2}\right)\left[g_{k}^{2}\alpha_{2}\left(\alpha_{2}^{2}q^{2}-m_{k}^{2}\right)\alpha_{G}^{3}(-\eta_{V}^{\textrm{LT}}+\partial_{t}-q\partial_{q})t_{2}\right.$
$\displaystyle-\alpha_{t}\left(g_{k}^{2}\alpha_{2}\left(\alpha_{2}^{2}q^{2}-m_{k}^{2}\right)\alpha_{G}\left(4+3r_{G}+t_{2}\right)-y_{k}^{2}\alpha_{t}\left(\alpha_{2}^{2}q^{2}+m_{k}^{2}\right)\right)\times$
$\displaystyle\times\left.\left.(-\eta_{g}^{\textrm{LT}}-\eta_{\Phi}^{\textrm{LT}}-\tfrac{1}{2}\eta_{V}^{\textrm{LT}}+\partial_{t}-q\partial_{q})r_{G}\right]\right\\},$
(.9)

where $\alpha_{2}=r_{2}+1$, $\alpha_{G}=r_{G}+1$, $\alpha_{t}=t_{2}+1$ and
$\Omega_{d}=\dfrac{(2\pi)^{\frac{d+1}{2}}}{\Gamma\left[\frac{d+1}{2}\right]}$
is the surface of a $d-$dimensional sphere.

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