# Rees algebra of Maximal Order Pfaffians and its diagonal subalgebras

Neeraj Kumar  Department of Mathematics, IIT Hyderabad, Kandi, Sangareddy -
502285<EMAIL_ADDRESS>and  Chitra Venugopal  Department of
Mathematics, IIT Hyderabad, Kandi, Sangareddy - 502285
<EMAIL_ADDRESS>
###### Abstract.

Given a skew-symmetric matrix $X$, the Pfaffian of $X$ is defined as the
square root of the determinant of $X$. In this article, we give the explicit
defining equations of Rees algebra of Pfaffian ideal $I$ generated by maximal
order Pfaffians of generic skew-symmetric matrices. We further prove that all
diagonal subalgebras of the corresponding Rees algebra of $I$ are Koszul. We
also look at the Rees algebra of Pfaffian ideals of linear type associated to
certain sparse skew-symmetric matrices. In particular, we consider the
tridiagonal matrices and identify the corresponding Pfaffian ideal to be of
Gröbner linear type and as the vertex cover ideal of an unmixed bipartite
graph. As an application of our results, we conclude that all their powers
have linear resolutions.

###### Key words and phrases:

Pfaffians, Koszul algebra, Cohen-Macaulay, Diagonal Subalgebra

###### 2020 Mathematics Subject Classification:

Primary 13C40, 13D02; Secondary 13H10

## Introduction

Let $A$ be a graded Noetherian ring and $I$ be a homogeneous ideal of $A$.
Then Rees algebra of $I$ denoted by $\mathcal{R}(I)$ is a bigraded algebra
defined as $\oplus_{i\geq 0}I^{i}$. Rees algebras form a principal class of
bigraded algebras and contain a great deal of information about the powers of
the ideal $I$. Moreover geometrically, it corresponds to the blowup of
Spec$(A)$ along the variety of $I$. In general, corresponding to a homogeneous
ideal $I$ of a ring $A$, finding the explicit defining equations of Rees
algebra is not easy. Some study has been done for certain classes of ideals
like perfect ideals of grade $2$ [37, 38, 7, 23], perfect Gorenstein ideals of
grade $3$ [33, 37], determinantal ideals [8, 9] etc. An ideal $I$ of a ring
$A$ is said to be of linear type if the Rees algebra of $I$ is isomorphic to
it’s symmetric algebra. We are interested in the study of Rees algebra of
ideals generated by $d$-sequence (a notion introduced by Huneke in [24, 25])
as they form a class of ideals of linear type. The motivation to explore Rees
algebra corresponding to $d$-sequence comes from the analogous study in [15,
41, 12, 31] for the ideals generated by regular sequence.

In this article, we look at a particular class of ideals called the Pfaffian
ideals, which come corresponding to skew-symmetric matrices. Let $X$ be a
skew-symmetric matrix and let det $X$ denote its determinant. Then Pfaffian of
$X$ denoted by $\text{Pf}(X)$ is defined as the square root of det $X$ i.e,
$\text{Pf}(X)^{2}=\text{det }X$ (cf. [3]). The Pfaffian ideal of X denoted by
$\text{Pf}_{n-1}(X)$ is the ideal obtained by considering Pfaffians of
submatrices of order $n-1$ obtained by deleting a row and the corresponding
column of the matrix $X$ (cf. [11]). In [11], Buchsbaum and Eisenbud proved
that every Gorenstein ideal of codimension $3$ in a commutative Noetherian
ring $A$ can be identified as the ideal of Pfaffians of order $(n-1)$ of some
$n\times n$ alternating matrix of rank $n-1$. Under some assumptions on the
entries of the skew-symmetric matrix, Pfaffian ideals are found to be of
linear type (cf. [5], [18]). We attempt to study the diagonal subalgebras of
the corresponding Rees algebra.

The notion of diagonal subalgebras was introduced by Simis, Trung and Valla in
[41] generalizing the concept of Segre product of graded algebras. The
diagonal subalgebras of certain classes of equigenerated homogeneous ideals of
a standard graded polynomial ring can be viewed as the homogeneous coordinate
ring of some rational varieties embedded in projective space (cf. [41]). It is
also known that for $c,e\in\mathbb{N}$ and a homogeneous ideal $I$ of
$B=K[x_{1},\ldots,x_{n}]$, if $I^{e}$ is generated by forms of degree $d\leq
c-1$, then $K[(I^{e})_{c}]$ can be identified as a diagonal subalgebra of Rees
algebra in a natural way [41, 15]. One of the key challenges in the study of
diagonal subalgebras is to find suitable conditions on a bigraded algebra $R$
such that certain algebraic properties of $R$ are inherited by $R_{\Delta}$
(Definition 1.2).

We are interested in looking at the Koszulness and Cohen-Macaulay property of
the diagonals of Rees algebras of equigenerated homogeneous ideals. There is a
lot of literature on these properties of diagonals of bigraded algebras (cf.
[41, 15, 34, 31, 1]).

A standard graded $K$-algebra $A$ is _Koszul_ if the non-zero entries of
matrices representing the maps in the minimal free resolution of $K$ are
homogeneous of degree $1$. There are several articles that have discussed the
Koszul property of diagonal subalgebras of the Rees algebras of ideals (cf.
[15, 12, 21, 6, 31, 1]. Explicit lower bounds are known for residual
intersections ([1]) and when the ideals are complete intersections [15, 31].
More generally, in [15] it has been proved that for any bigraded $K$-algebra
$R$, $R_{\Delta}$ is Koszul for $c,e\gg 0$.

In [41] for $\Delta=(1,1)$, the authors discuss some classes of ideals for
which the corresponding diagonal subalgebra of the Rees algebra is Cohen-
Macaulay. Complete intersections and certain classes of straightening closed
ideals in algebras with straightening law, like determinantal ideals generated
by the maximal minors of a generic matrix, are some of the ideals looked into.
In [15], Conca, Herzog Trung and Valla solve an open problem posed in [41]
regarding the conditions on $(c,e)$ which guarantees Cohen-Macaualay property
of $K[(I^{e})_{c}]$ when $I\subset B$ is a homogeneous complete intersection
minimally generated by $r$ forms of degree $d_{1},\ldots,d_{r}$. In case of
some classes of perfect ideals of height two as well, certain bounds on $c,e$
are known for which the diagonals of the corresponding Rees algebras are
Cohen-Macaulay (cf. [1]). In general, it is known that if a standard bigraded
ring $R$ is Cohen-Macaulay, then $R_{\Delta}$ is Cohen-Macaulay for large
integers $c\gg e>0$ (cf. [34]).

In this article, we primarily look at equigenerated Pfaffian ideals so that
the associated Rees algebra is standard bigraded, thus forming a standard
graded K-algebra with respect to total degree. In particular, it makes sense
to look at the Koszul property of such graded K-algebras.

Some of the important results discussed in this article are the following.

  1. (i)

Let $I=$Pf${}_{n-1}(X)$ where $X$ is a generic skew-symmetric matrix of odd
order $n=2r+1$, $r\in\mathbb{N}$ (defined in Section 1). Then we prove the
following.

    1. (a)

The $c$-th Veronese subalgebra of the corresponding Pfaffian ring is Koszul
for $c\geq n/4.$

    2. (b)

$\mathcal{R}(I)$ is generated by quadrics but need not be Koszul always.

    3. (c)

All diagonals of $\mathcal{R}(I)$ are Koszul.

  2. (ii)

For the Pfaffian ideal $I$ coming from a tridiagonal matrix of the form
mentioned in Theorem 3.2, $\mathcal{R}(I)_{\Delta}$ is Koszul and Cohen-
Macaulay for all $\Delta$. In this case, Pf${}_{n-1}(X)$ is an equigenerated
monomial ideal of Gröbner linear type (defined in Section 3), which can be
identified as the vertex cover ideal of an unmixed bipartite graph, thereby
giving information about the linear resolution of powers of the vertex cover
ideal.

  3. (iii)

For $I=$Pf${}_{n-1}(X)$ where $X$ has the form given in Proposition 3.10,
$\mathcal{R}(I)_{\Delta}$ is Koszul and Cohen-Macaulay for all $\Delta$.

We focus mainly on the maximal order Pfaffians, as in the non-maximal case, it
is not always true that the defining ideal of Rees algebra is generated by
quadrics, a property which is essential for the study of Koszulness of Rees
algebra.

The reader may be familiar with some of the observations made in this article.
However, in order to maintain the study self-contained, we reproduce some
arguments and independently establish the results. All the computations in
this article are done using Macaulay2 ([22]).

##  1\. Preliminaries

We consider $K$ to be a field of characteristic zero throughout the article.
Consider the skew-symmetric matrix
$X=\begin{bmatrix}0&x_{12}&\ldots&x_{1\;n}\\\ -x_{12}&0&\ldots&x_{2\;n}\\\
\vdots&\vdots&&\vdots\\\ -x_{1\;n}&-x_{2\;n}&\ldots&0\\\ \end{bmatrix}$ of odd
order $n=2r+1$, $r\in\mathbb{N}\cup\\{0\\}$, where the entries $x_{ij}$ for
$i=1,\ldots,n-1$, $j=2,\ldots,n$ are indeterminates. This is the form of a
generic skew-symmetric matrix of odd order. Let $B=K[X]$ where $K[X]$ is the
polynomial ring with indeterminates being the non-zero entries of $X$ and
$I=\text{Pf}_{n-1}(X)=\text{Pf}_{2r}(X).$ Then $B/I$ is said to be a Pfaffian
ring.

######  Note 1.1.

Let $X$ be a skew-symmetric matrix of odd order. Then by Pf${}_{\bar{l}}(X)$
we mean the Pfaffian of the submatrix of $X$ obtained by removing $l^{th}$ row
and corresponding column.

For a field $K$, $A$ is said to be a standard graded $K$-algebra if
$A=\oplus_{i\geq 0}A_{i}$ such that $A_{0}=K$, $A_{1}$ is a finite dimensional
$K$-vector space and $A_{i}A_{j}=A_{i+j}$ for every
$i,j\in\mathbb{N}\cup\\{0\\}$. Any standard graded $K$-algebra $A$ can be
identified as $B/I$, where $B$ is a standard graded polynomial ring over $K$
($K$-algebra) and $I$ is it’s homogeneous ideal. For an equigenerated ideal
$\mathfrak{I}$ of $A$, the Rees algebra of $\mathfrak{I}$ in $A$ is a standard
bigraded $K$-algebra defined as $\mathcal{R}(\mathfrak{I})=\bigoplus_{n\geq
0}{\mathfrak{I}}^{n}$, where standard bigraded means
$\mathcal{R}(\mathfrak{I})_{1}=\oplus_{i\geq
0}\mathcal{R}(\mathfrak{I})_{(i,0)}$ and
$\mathcal{R}(\mathfrak{I})_{2}=\oplus_{i\geq
0}\mathcal{R}(\mathfrak{I})_{(0,i)}$ are standard graded $K$-algebras. Similar
to the graded case, $\mathcal{R}(\mathfrak{I})$ has a presentation $S/J$ as
quotient of the standard bigraded polynomial ring $S=K[X,Y]$ over a field $K$,
by a bihomogeneous ideal $J$. An ideal $\mathfrak{I}$ of a ring $A$ is of
linear type if the defining relations of $\mathcal{R}(\mathfrak{I})$ are
linear in the indeterminates $Y$.

######  Definition 1.2.

For two integers $c,e\geq 0$ with $(c,e)\neq(0,0)$ the $(c,e)$-diagonal is the
$\Delta=\\{(cs,es):s\in\mathbb{Z}\\}$ of $\mathbb{Z}^{2}$. The diagonal
subalgebra of a bigraded algebra $R$ along $\Delta$ is defined as the graded
algebra $R_{\Delta}:=\bigoplus_{i\geq 0}R_{(ci,ei)}$ (cf. [41]).

It is analogous to the notion of Veronese subalgebras of graded algebras. For
$c\in\mathbb{N}$ and a standard graded $K$-algebra $A$,
$A^{(c)}=\oplus_{j\in\mathbb{N}}A_{cj}$ is defined as the $c$-th Veronese
subalgebra of $A$.

In [41], the authors have given the presentation of diagonal subalgebra of a
bigraded algebra $R$ in the following way. Consider $R\cong S/J$, where
$S=K[x_{1},\ldots,x_{n},y_{1},\ldots,y_{m}]$ is a bigraded polynomial ring and
$J$ a bihomogeneous ideal of $S$. Then

$R_{\Delta}=S_{\Delta}/J_{\Delta},$ (1)

where $S_{\Delta}$ is the Segre product of $K[x_{1},\ldots,x_{n}]^{(c)}$ and
$K[y_{1},\ldots,y_{m}]^{(e)}$ and $J_{\Delta}=\oplus_{i\geq 0}J_{(ci,ei)}$.

Now let $\Delta=(1,1)$. Then $S_{\Delta}=K[x_{i}y_{j}|\,1\leq i\leq n,\,1\leq
j\leq m]$ and a presentation of $S_{\Delta}$ can be seen as $S_{\Delta}\cong
K[T]/I_{2}(T)$ where $T=(t_{ij})$ is an $n\times m$ matrix of indeterminates
for $1\leq i\leq n$, $1\leq j\leq m$ and $I_{2}(T)$ is the ideal generated by
the $2$-minors of $T$. This presentation is obtained by mapping $t_{ij}$ in
$K[T]$ to $x_{i}y_{j}$ in $S_{\Delta}$. The following lemma gives the form for
the generators of $J_{\Delta}$ in $K[T]$ when $\Delta=(1,1)$.

######  Lemma 1.3.

([41, Lemma 2.1]) Let $R\cong S/J$ be a standard bigraded algebra and $J$ a
bihomogeneous ideal of $S$ generated by $g_{1},\ldots,g_{r}$ with deg
$g_{i}=(a_{i},b_{i})$. Then for $c_{i}=\text{max}\\{a_{i},b_{i}\\}$, the
generators of $J_{\Delta}$ has the form $g_{i}m$ where $m$ is a monomial of
degree $(c_{i}-a_{i},c_{i}-b_{i})$, $i=1,\ldots,r$.

For an equigenerated ideal of a standard graded polynomial ring, if $c\geq
ed+1$, then the dimension of the corresponding diagonal subalgebra of Rees
algebra is found to be independent of the diagonal.

######  Lemma 1.4.

([15, Lemma 1.3(ii)]) Let $I$ be an equigenerated ideal of a standard graded
polynomial ring in $n$ indeterminates with the degree of the generators being
denoted by $d$. If $c\geq ed+1$, then $\dim\,\mathcal{R}(I)_{\Delta}=n$ where
$\Delta=(c,e)$.

Let $A$ be a standard graded $K$-algebra where $K$ is a field and $M$ be a
finitely generated $A$-module. Let
$t_{i}^{A}(M)=\sup\\{j:\operatorname{Tor}_{i}^{A}(M,K)_{j}\neq 0\\}$ with
$t_{i}^{A}(M)=-\infty$ if $\operatorname{Tor}_{i}^{A}(M,K)_{j}=0$ for all
$j\geq 0$. Then regularity of $M$ denoted by reg${}_{A}(M)$ is defined as
$\operatorname{reg}_{A}(M)=\sup\\{t_{i}^{A}(M)-i,\,i\geq 0\\}$. Similarly,
corresponding to a standard bigraded $K$-algebra, there is an analogous notion
of $x$-regularity and $y$-regularity (refer [2, Section 2] for definitions).

Let $M$ be an $A$-module generated by elements of same degree, say $d$. Then
$M$ is said to have a $d$-linear resolution over $A$ if reg${}_{A}(M)=d$. If
the degree $d$ of the generators of an $A$-module $M$ is clear from the
context, just the terminology ’linear resolution’ is used. A standard graded
$K$-algebra $A$ is Koszul if the minimal $A$-free resolution of the residue
field is linear, that is reg${}_{A}(K)=0$.

######  Remark 1.5.

We recall the following results related to Koszulness of bigraded algebras and
linear resolutions of ideals for later use.

  * (i)

Let $R$ be a bigraded $K$-algebra with the free modules in the minimal
bigraded free resolution being denoted by
$F_{i}=\oplus_{(a,b)\in\mathbb{N}^{2}}S(-a,-b)^{\beta_{i,a,b}}$. Then for
$\Delta=(c,e)$, $R_{\Delta}$ is Koszul if
max$\\{\dfrac{a}{c},\dfrac{b}{e}:\beta_{i,a,b}\neq 0\\}\leq i+1$ for every $i$
([15, Theorem 6.2]). This shows how information about the shifts in the
$S$-free resolution of $\mathcal{R}(I)$ helps is obtaining lower bounds for
$(c,e)$ for which $\mathcal{R}(I)_{\Delta}$ is Koszul.

  * (ii)

If $R$ is a Koszul bigraded $K$-algebra then $R_{\Delta}$ is Koszul for all
$\Delta$ ([6, Theorem 2.1]).

  * (iii)

Let $I$ be an equigenerated ideal of a standard graded ring $A$. If
$\mathcal{R}(I)$ is Koszul then $I^{n}$ has linear resolution for all $n\geq
0$ ([6, Corollary 3.6]).

Similarly, there are results which help in getting bounds for $\Delta$ from
the $S$-free resolution of $\mathcal{R}(I)$ such that
$\mathcal{R}(I)_{\Delta}$ is Cohen-Macaulay.

######  Lemma 1.6.

[15, Lemma 3.10] Let $S=K[x_{1},\ldots x_{n},y_{1},\ldots,y_{m}]$ be a
standard bigraded polynomial ring. Then for $\Delta=(c,e)$ and
$a,b\in\mathbb{Z}$,

  * (i)

$\dim$ $S(-a,-b)_{\Delta}=n+m-1$.

  * (ii)

If $0<b<m$ or $0<a<n$, then $S(-a,-b)_{\Delta}$ is Cohen-Macaulay if
$c>\max\\{-a,-n+a\\}$ and $e>\max\\{-b,-m+b\\}$.

The following are some other results which will be used in this article
repeatedly.

######  Lemma 1.7.

[40, Lemma 2.2] Let $f_{1},\ldots,f_{r}$ be a sequence of elements in $A$ such
that with respect to some monomial order on $A$, $(LT(f_{i}),LT(f_{j}))=1$ for
all $i,j\in\mathbb{N}$ with $i\neq j$. Then $f_{1},\ldots,f_{r}$ is a regular
sequence in $A$.

######  Lemma 1.8 (Depth Lemma).

Let $A\cong B/I$ where $B$ is the standard graded polynomial ring and $I$ a
homogeneous ideal of $B$, with the free modules in the minimal graded free
resolution being denoted by $F_{i}$,
$i=1,\ldots,\operatorname{proj\,dim}_{B}A$. Then the following inequality
holds,

$\operatorname{depth}\,(A)\geq\min\\{\operatorname{depth}\,(F_{i})-i;\,i\geq
0\\}.$

###### Proof.

It is obtained by applying depth lemma iteratively on the short exact sequence
of modules. ∎

Let $B=K[x_{1},\ldots,x_{n}]$ and $I=\langle u_{1},\ldots,u_{s}\rangle$ be a
monomial ideal in $B$. Let $I_{r}$ denote the set of all sequences
$\alpha=(i_{1},\ldots,i_{r})$ in $[s]$ of length $r$ such that $i_{1}\leq
i_{2}\leq\ldots\leq i_{r}$. For any $\alpha\in I_{r}$, let
$u_{\alpha}=u_{i_{1}}u_{i_{2}}\ldots u_{i_{r}}$ and
$t_{\alpha}=t_{i_{1}}t_{i_{2}}\ldots t_{i_{r}}$ and for any $\alpha,\beta\in
I_{r}$, define
$t_{\alpha,\beta}=\dfrac{\operatorname{lcm}[u_{\alpha},u_{\beta}]}{u_{\beta}}t_{\beta}-\dfrac{\operatorname{lcm}[u_{\alpha},u_{\beta}]}{u_{\alpha}}t_{\alpha}$.
Then,

$\mathcal{R}(I)\cong B[t_{1},\ldots,t_{s}]/J$ (2)

where $J=SJ_{1}+S(\cup_{r=2}^{\infty}J_{r})$ with
$J_{r}=\\{t_{\alpha,\beta}:\alpha,\beta\in I_{r}\\}$ (cf. [42]).

######  Definition 1.9.

Let $A$ be a commutative ring and $\\{\textbf{a}\\}=\\{a_{1},\ldots,a_{n}\\}$
a sequence of elements in $A$. Then for an $A$-module $M$, $\\{\textbf{a}\\}$
forms a $d$-sequence on $M$ if the following holds:

  1. (1)

$a_{i}M\notin\langle a_{1},\ldots,\hat{a_{i}},\ldots,a_{n}\rangle M$ for
$i=1,\ldots,n$.

  2. (2)

$(\langle a_{1},\ldots,a_{i}\rangle M:a_{i+1}a_{k}M)=(\langle
a_{1},\ldots,a_{i}\rangle M:a_{k}M)$ for all $k\geq i+1$ and $i=0,\ldots,n-1$.

##  2\. Pfaffian ideals of generic skew-symmetric matrices

Assume $X$ to be a generic skew-symmetric matrix of odd order $n=2r+1$,
$r\in\mathbb{N}$ and $I=$Pf${}_{n-1}(X)$. Then the ideal Pf${}_{n-1}(X)$
generated by the maximal order Pfaffians are found to be of linear type (cf.
[5]). Huneke proved that the Pfaffian sequence forms a weak $d$-sequence [26,
1.20] and further remarked that it seems to form a $d$-sequence [25].
Recently, we gave a proof to show that it indeed forms a $d$-sequence [32]. In
fact from the proof it is not difficult to see that it forms an unconditioned
$d$-sequence.

Structure theorem of ideals of codimension $3$ [11] gives the minimal free
resolution of Pfaffian ring in generic case corresponding to an alternating
map. In the following lemma, we mention the differentials (explicitly) in the
resolution which helps in studying the related Rees algebra.

######  Lemma 2.1.

Let $B=K[X]$ and $I=$Pf${}_{n-1}(X)$. Then the minimal graded free resolution
of $B/I$ has the following form.

$0\longrightarrow B(-2r-1)\stackrel{{\scriptstyle
d_{3}}}{{\longrightarrow}}B^{n}(-r-1)\stackrel{{\scriptstyle
d_{2}}}{{\longrightarrow}}B^{n}(-r)\stackrel{{\scriptstyle
d_{1}}}{{\longrightarrow}}B\longrightarrow 0.$ (3)

where

$d_{1}=\begin{bmatrix}Pf_{\bar{1}}(X)&Pf_{\bar{2}}(X)&\cdots&Pf_{\bar{n}}(X)\end{bmatrix},$

$d_{2}=(a_{ij})=\left\\{\begin{array}[]{ccc}(-1)^{i+j}x_{n+1-i\;\;n+1-j}&if&i>j\\\
0&if&i=j\\\ (-1)^{i+j+1}x_{n+1-j\;\;n+1-i}&if&i<j.\\\ \end{array}\right.$

and

$d_{3}=\begin{bmatrix}Pf_{\bar{1}}(X)\\\ Pf_{\bar{2}}(X)\\\ \vdots\\\
Pf_{\bar{n}}(X)\\\ \end{bmatrix}$

###### Proof.

In $(\ref{min_free_res})$ we have $d_{1}\circ d_{2}=0$ and $d_{2}\circ
d_{3}=0$. Let $d_{2}(B^{n})=\langle g_{1},\ldots,g_{n}\rangle$ where
$g_{i}^{\prime}s$ form a minimal generating set of the image. Since the
cardinality of the minimal generating set of $d_{2}(B^{n})$ is $n$ and
considering the appropriate shifts, complex $(\ref{min_free_res})$ can be seen
as a graded free resolution of $B/I$. Finally since
$d_{2}(B^{n})\subseteq\mathfrak{m}B^{n}$, we get the required minimal graded
free resolution of $B/I$. ∎

######  Remark 2.2.

As a consequence of the above result, for a generic skew-symmetric matrix $X$
and $I=$Pf${}_{n-1}(X)\subseteq B=K[X]$, we observe the following.

  1. (i)

For a standard graded $K$-algebra $A$, it is known that $A^{(c)}$ is Koszul
for $c\gg 0$ (cf. [4]). Using Lemma 2.1 and Remark 1.5(i) we have
$(B/I)^{(c)}$ is Koszul for $c\geq n/4.$

  2. (ii)

Since $I$ is of linear type, the explicit defining relations of
$\mathcal{R}(I)$ in $S=K[X,Y]$ will have the form
$\begin{bmatrix}y_{1}&y_{2}&\ldots&y_{n}\end{bmatrix}.\;\;d_{2}$ where
$d_{2}=(a_{ij})=\left\\{\begin{array}[]{ccc}(-1)^{i+j}x_{n+1-i\;\;n+1-j}&if&i>j\\\
0&if&i=j\\\ (-1)^{i+j+1}x_{n+1-j\;\;n+1-i}&if&i<j.\\\ \end{array}\right.$

Some interesting colon conditions are satisfied by the defining relations of
Rees algebra which is discussed in the following Lemma.

######  Lemma 2.3.

Consider the setup as in Remark $\ref{generic_rees}(ii)$. Let the defining
relations of $\mathcal{R}(I)=S/J$ obtained above be denoted by
$g_{1},\ldots,g_{n}$ where $S=K[X,Y]$ and $J=\langle
g_{1},\ldots,g_{n}\rangle$, a homogeneous ideal of $S$. Then the following
holds.

  1. (1)

$g_{1},\ldots,g_{n-1}$ forms an $S$-regular sequence.

  2. (2)

$(\langle g_{1},\ldots,g_{n-1}\rangle:_{S}g_{n})=\langle
g_{1},\ldots,g_{n-1},y_{n},\text{Pf}_{\bar{1}}(X)\rangle$.

  3. (3)

$(\langle g_{1},\ldots,g_{n-1}\rangle:_{S}y_{n})=J.$

  4. (4)

$(\langle
y_{n},g_{1},\ldots,g_{n-1}\rangle:_{S}\text{Pf}_{\bar{1}}(X))=\langle
y_{1},\ldots,y_{n}\rangle.$

###### Proof.

Let $J^{\prime}=\langle g_{1},\ldots,g_{n-1}\rangle$.

  1. (1)

Consider graded lexicographic term order on $S$ induced by
$x_{12}>x_{23}>x_{34}>x_{45}>\cdots>x_{n-1n}$ followed by the remaining
indeterminates. Then from Lemma 1.7, we get that $g_{1},\ldots,g_{n-1}$ forms
an $S$-regular sequence. Infact any $n-1$ relations among $g_{1},\ldots,g_{n}$
can be proved to form a regular sequence by a corresponding change in the
order on indeterminates.

  2. (2)

It is not difficult to see that $g_{n}y_{n}=-(\sum_{i=1}^{n-1}g_{i}y_{i})$ and
$g_{n}\text{Pf}_{\bar{1}}(X)=-(\sum_{i=1}^{n-1}g_{i}\text{Pf}_{\overline{n-i+1}}(X))$.
Hence the inclusion $\supseteq$ follows. Since $\langle
g_{1},\ldots,g_{n-1},y_{n},\text{Pf}_{\bar{1}}(X)\rangle$ contains
$J^{\prime}$, to prove the equality it suffices to show that $\langle
g_{1},\ldots,g_{n-1},y_{n},\text{Pf}_{\bar{1}}(X)\rangle$ is a prime ideal.
This is equivalent to proving $\overline{\text{Pf}_{\bar{1}}(X)}$ is a prime
element in $S/\langle g_{1},\ldots,g_{n-1},y_{n}\rangle$ where - denotes an
element in the respective quotient ring. First we claim that
Pf${}_{\bar{1}}(X)$ is an irreducible element in $S$. To prove this, consider
$f=\text{Pf}_{\bar{1}}(X)$ as a polynomial in one variable $x_{23}$ that is,
$f\in S^{\prime}[x_{23}]$ where $S^{\prime}=K[X^{\prime},Y]$,
$X^{\prime}=X\backslash\\{x_{23}\\}$. Let $S^{\prime\prime}$ be the field of
fractions of $S^{\prime}$. Then we can write
$f=g(x^{\prime},y)x_{23}+h(x^{\prime},y)$ where
$g(x^{\prime},y),h(x^{\prime},y)\in S^{\prime\prime}$ and
$\text{gcd}(g(x^{\prime},y),h(x^{\prime},y))=1$. Then clearly $f$ is an
irreducible polynomial in $S^{\prime\prime}[x_{23}]$. Now assume that
$f=p(x,y)q(x,y)$ where $p(x,y),q(x,y)\in S$. Then since $f$ is irreducible in
$S^{\prime\prime}[x_{23}]$, without loss of generality let $p(x,y)$ be a unit
in $S^{\prime\prime}[x_{23}]$. This implies $p(x,y)\in S^{\prime\prime}$, but
since $p(x,y)\in S$, we have $p(x,y)\in S^{\prime}$. This in turn gives
$p(x,y)|\text{gcd}(g(x^{\prime},y),h(x^{\prime},y))$. Thus $p(x,y)$ must be a
unit in $S$. Hence $\text{Pf}_{\bar{1}}(X)$ is an irreducible element in $S$.
Hence we have from the relations $J^{\prime}$, if
$\overline{\text{Pf}_{\bar{1}}(X)}|\bar{x}\bar{y}$ in $S/J^{\prime}$ with
$\overline{\text{Pf}_{\bar{1}}(X)}\not|\,\bar{x}$ and
$\overline{\text{Pf}_{\bar{1}}(X)}\not|\,\bar{y}$, then
$\overline{y_{n}}|\bar{x}\bar{y}$. This implies if $\bar{x}\bar{y}\neq 0$ in
$S/\langle g_{1},\ldots,g_{n-1},y_{n}\rangle$ with
$\overline{Pf_{\bar{1}}(X)}|\bar{x}\bar{y}$, then either
$\overline{\text{Pf}_{\bar{1}}(X)}|\bar{x}$ or
$\overline{\text{Pf}_{\bar{1}}(X)}|\bar{y}$. Thus
$\overline{\text{Pf}_{\bar{1}}(X)}$ is a prime element in $S/\langle
g_{1},\ldots,g_{n-1},y_{n}\rangle$.

  3. (3)

From $(2)$ of this lemma we have the inclusion $\supseteq$. And since $J$ is a
prime ideal containing $J^{\prime}$, we get the other inclusion.

  4. (4)

We have coefficient of $y_{i}$ in $g_{j}$ is
$\left\\{\begin{array}[]{ccc}a_{ij}&for&i\neq j\\\ 0&for&i=j\\\
\end{array}\right.$ where $a_{ij}$ are the entries of the matrix $d_{2}$ in
Remark $\ref{generic_rees}(ii)$. Then for each $i=1,\ldots,n-1$, we can
represent Pf${}_{\bar{1}}(X)=\sum_{j=1}^{n-1}\alpha_{ij}a_{ij}$ where
$\alpha_{ij}\in B$ which in turn gives the relation,
Pf${}_{\bar{1}}(X)y_{i}=\sum_{j=1}^{n-1}\alpha_{ij}g_{j}$mod$(y_{n})$ for each
$i=1,\ldots,n$. Thus the inclusion $\supseteq$ follows. The equality can now
be seen as a consequence of the primality of $\langle
y_{1},\ldots,y_{n}\rangle$ and the inclusion $\langle
y_{n},g_{1},\ldots,g_{n-1}\rangle\subseteq\langle y_{1},\ldots,y_{n}\rangle.$

∎

The properties mentioned in Lemma 2.3 are found to be surprisingly similar to
those satisfied by the defining relations of the Rees algebra of an ideal
generated by regular sequence ( [31, Lemma 3.2]).

######  Proposition 2.4.

Let $B=K[X]$ and $I=\text{Pf}_{n-1}(X)=\text{Pf}_{2r}(X).$ Then the defining
relations of $\mathcal{R}(I)$ given in Remark $\ref{generic_rees}(ii)$ forms a
$d$-sequence. Moreover it generates an almost complete intersection ideal.

###### Proof.

It is known that Rees algebra of $I$ is Cohen-Macaulay [20, Proposition 2.8].
This implies height of the defining ideal of Rees algebra is $n-1$. From Lemma
2.3$(1)$ we have $\\{g_{1},\ldots,g_{n-1}\\}$ forms a regular sequence. To
prove that $\\{g_{1},\ldots,g_{n}\\}$ forms a $d$-sequence, it suffices to
show $(J^{\prime}:g_{n})=(J^{\prime}:{g_{n}}^{2})$ where $J^{\prime}=\langle
g_{1},\ldots,g_{n-1}\rangle$. Clearly
$(J^{\prime}:g_{n})\subseteq(J^{\prime}:{g_{n}}^{2})$. To see the other
inclusion let $\alpha\in(J^{\prime}:{g_{n}}^{2}).$ Then $\alpha\cdot
g_{n}\in(J^{\prime}:g_{n})$. Since $(J^{\prime}:g_{n})$ is a prime ideal (from
the proof of Lemma 2.3$(2)$) and $g_{n}\notin(J^{\prime}:g_{n})$, this implies
$\alpha\in(J^{\prime}:g_{n})$. Hence the equality follows.

Since the defining ideal is minimally generated by $n$ elements, one more than
the height of the ideal and the sequence forms a $d$-sequence, by [27, Lemma
4.2], it generates an almost complete intersection.

∎

It is evident from Remark $\ref{generic_rees}(ii)$ that the defining relations
of Rees algebra of maximal order Pfaffians of generic skew-symmetric matrix
are generated by quadrics. So, it makes sense to ask whether something
stronger holds. Therefore, we pose the following question.

######  Question 2.5.

Is it true that the Rees algebra of maximal order Pfaffians in generic case is
always Koszul?

The study that follows indicate that it is not true in general.

Let $X_{1}$ be generic skew-symmetric matrix of order $3$ and consider
$B_{X_{1}}=K[X_{1}]$ with $I_{1}=\text{Pf}_{2}(X_{1})=\langle
x_{12},x_{13},x_{23}\rangle$, the graded maximal ideal of $B_{X_{1}}$. Then
the defining ideal of $\mathcal{R}(I_{1})$ is given by $2$-minors of the
matrix $\begin{pmatrix}x_{12}&x_{13}&x_{23}\\\ y_{1}&y_{2}&y_{3}\end{pmatrix}$
and the minimal bigraded $S$-free resolution of $\mathcal{R}(I_{1})$ has the
form,

$0\longrightarrow\begin{matrix}S(-1,-2)\\\ \oplus\\\
S(-2,-1)\end{matrix}\longrightarrow\begin{matrix}S(-1,-1)^{3}\\\
\end{matrix}\longrightarrow S\longrightarrow 0.$

Since $\operatorname{reg}_{S}(\mathcal{R}(I_{1}))=1$ and $S$ seen as a graded
ring with respect to total degree is Koszul, by transfer of Koszulness ([14,
Theorem $2$]), $\mathcal{R}(I_{1})$ is Koszul. In fact the defining ideal of
$\mathcal{R}(I_{1})$ is observed to be generated by a
Gr$\ddot{\textrm{o}}$bner basis of quadrics with respect to graded reverse
lexicographic order induced by $x_{12}>x_{13}>x_{23}>y_{1}>y_{2}>y_{3}$.

Now, let $X_{2}$ be a generic skew-symmetric matrix of order $5$. Consider
$B_{X_{2}}=K[X_{2}]$ with $I_{2}=\text{Pf}_{4}(X_{2})=\langle
x_{14}x_{23}-x_{13}x_{24}+x_{12}x_{34},\;x_{15}x_{23}-x_{13}x_{25}+x_{12}x_{35},\;x_{15}x_{24}-x_{14}x_{25}+x_{12}x_{45},\;x_{15}x_{34}-x_{14}x_{35}+x_{13}x_{45},\;x_{25}x_{34}-x_{24}x_{35}+x_{23}x_{45}\rangle$.
From the following Betti table of the Pfaffian ring $B_{X_{2}}/I_{2}$, we see
that $I_{2}$ does not have a linear resolution.

$\beta_{(i,i+j)}=\;\begin{tabular}[]{c| c c c
c}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}1}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}2}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}3}\\\
\hline\cr{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0}&1&0&0&0\\\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}1}&0&5&5&0\\\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}2}&0&0&0&1\\\
\end{tabular}$

As a consequence, $\mathcal{R}(I_{2})$ as a graded ring with respect to total
degree is not Koszul (Remark 1.5 (iii)).

Let $I_{1}$ and $I_{2}$ be as defined before.

######  Proposition 2.6.

Let $X$ be a generic skew-symmetric matrix of odd order $n$ and $B=K[X]$ with
$I=\text{Pf}_{n-1}(X)$ for $n=3,5$. Then,

  1. a)

$\mathcal{R}(I_{1})$ is Koszul, whereas $\mathcal{R}(I_{2})$ is not Koszul.

  2. b)

$I^{j}$ have linear resolution for all $j\geq 2$.

  3. c)

reg${}_{x}^{S}(\mathcal{R}(I^{j}))=0$ for $j\geq 2$.

###### Proof.

  1. a)

Koszulness of $\mathcal{R}(I_{1})$ and non-Koszulness of $\mathcal{R}(I_{2})$,
both follows from previous discussion.

  2. b)

For $n=3$, since $\mathcal{R}(I_{1})$ is Koszul, we have in fact
reg${}_{B}(I_{1})^{j}=2j$ for $j\geq 1$. For $n=5$, the observation follows
from the result by Cutkosky, Herzog and Trung which says that if an ideal $I$
of a ring $A$ is generated by $d$-sequence of $n$ forms and is equigenerated
of degree $r$ then for all $j\geq n+1$,
reg${}_{A}(I^{j})=(j-n-1)r+$reg${}_{A}(I^{n+1})$ [16, Corollary 3.8] and the
fact that in our case reg${}_{A}(I^{n+1})=(n+1)r$.

  3. c)

Follows from b) and [10, Theorem 5.2].

∎

######  Remark 2.7.

The following is a realization of diagonal subalgebra of Rees algebra
$\mathcal{R}(I_{1})$, as a quotient of standard graded $K$-algebra for
$\Delta=(1,1)$.

For $n=3$, let $T=\begin{pmatrix}t_{11}&t_{12}&t_{13}\\\
t_{21}&t_{22}&t_{23}\\\ t_{31}&t_{32}&t_{33}\\\ \end{pmatrix}$ where for
$1\leq j\leq 3$, $t_{1j}=x_{12}y_{j}$, $t_{2j}=x_{13}y_{j}$ and
$t_{3j}=x_{23}y_{j}$. Then $\mathcal{R}(I_{1})_{\Delta}$ can be identified
with $K[T]/I_{2}(T)+\langle
t_{12}-t_{23},\;t_{21}-t_{32},\;t_{11}-t_{33}\rangle$ where $I_{2}(T)$ denotes
the ideal generated by $2$-minors of $T$ (From equation (1) and Lemma 1.3).
From the above defining relations we get, $\mathcal{R}(I_{1})_{\Delta}\cong
K[T]/\langle-
t_{12}t_{21}+t_{11}t_{22},\;t_{11}t_{21}-t_{12}t_{31},\;t_{21}^{2}-t_{22}t_{31},\;t_{11}t_{12}-t_{13}t_{21},\;t_{11}^{2}-t_{13}t_{31},\;t_{11}t_{21}-t_{12}t_{31},\;t_{12}^{2}-t_{13}t_{22},\;t_{11}t_{12}-t_{13}t_{21},\;-t_{12}t_{21}+t_{11}t_{22}\rangle.$

Similarly we can write an expression for $\mathcal{R}(I_{2})$ for
$\Delta=(1,1)$.

It is evident from Proposition 2.6 that $\mathcal{R}(I)$ is not always Koszul
for a maximal order Pfaffian ideal $I$ of a general skew symmetric matrix $X$.
However, the following result shows that, regardless of this, all of its
diagonals are always Koszul.

######  Theorem 2.8.

Let $B=K[X]$ and $I=$Pf${}_{n-1}(X)$. Then $\mathcal{R}(I)_{\Delta}$ is Koszul
for all $\Delta=(c,e)$, $c>0$, $e>0$.

###### Proof.

We have the defining relations of the Rees algebra in the given case to be of
bidegree $(1,1)$. Since the defining relations satisfy certain nice properties
from Lemma 2.3, we can consider the following complex.

$\mathbb{F.}:\cdots\stackrel{{\scriptstyle
y_{n}}}{{\longrightarrow}}T(-2d,-3)\stackrel{{\scriptstyle
g_{n}}}{{\longrightarrow}}T(-d,-2)\stackrel{{\scriptstyle
y_{n}}}{{\longrightarrow}}T(-d,-1)\stackrel{{\scriptstyle
g_{n}}}{{\longrightarrow}}T\longrightarrow 0.$ (4)

Then using an idea similar to the one in [31, Theorem 3.1] and [12, Theorem
3.2], the required result is obtained. ∎

It is known that Rees algebra of maximal generic Pfaffians is Cohen-Macaulay.
Following are some observations that can be made regarding the Cohen-
Macaulayness of diagonals of the same.

######  Remark 2.9.

Let $X$ be a generic skew-symmetric matrix of odd order $n$ and $B=K[X]$ with
$I=\text{Pf}_{n-1}(X)$.

  1. (1)

Then $\mathcal{R}(I)_{\Delta}$ is Cohen-Macaulay for $c\gg 0$ and $e>0$. This
can be seen as a consequence of $I$ being generated by $d$-sequence which
implies vanishing of y-regularity of $\mathcal{R}(I)$ [39, Corollary 3.2] and
[15, Corollary 3.14].

  2. (2)

In particular, for the previous cases of $n=3,5$, $\mathcal{R}(I)_{\Delta}$ is
Cohen-Macaulay if $c\geq 2e+1$, $c,e>0$ which is a consequence of the repeated
application of depth lemma, Lemma 1.6 and Lemma 1.4.

##  3\. Pfaffian ideals of sparse skew-symmetric matrices

For a generic skew-symmetric matrix $X$ of odd order $n$, computations suggest
that as $n$ increases, the number of generators for $I=\text{Pf}_{n-1}(X)$
increases largely. Moreover, the expression for the generators of $I$ becomes
too complex thereby making the study of the minimal graded free resolutions of
the corresponding Rees algebra hard. Thus for the generators of a Pfaffian
ideal and the defining ideal of the Rees algebra to satisfy some properties,
it makes sense to focus on sparse form of skew-symmetric matrices.

Note that we primarily focus on maximal order Pfaffians since an ideal $I$
generated by non-maximal order Pfaffians most often lead to the defining
relations of Rees algebra being of total degree greater than $2$. For example,

$X=\begin{bmatrix}0&x_{12}&0&x_{14}&0&0&0\\\ -x_{12}&0&x_{23}&0&x_{25}&0&0\\\
0&-x_{23}&0&x_{34}&0&x_{36}&0\\\ -x_{14}&0&-x_{34}&0&x_{45}&0&x_{47}\\\
0&-x_{25}&0&-x_{45}&0&x_{56}&0\\\ 0&0&-x_{36}&0&-x_{56}&0&x_{67}\\\
0&0&0&-x_{47}&0&-x_{67}&0\\\ \end{bmatrix}_{7\times 7}$

In the above case, for $I=$Pf${}_{4}(X)$ we get $\mathcal{R}(I)$ to have $52$
generators of bidegree $(1,1)$, $14$ generators of bidegree $(0,2)$ and $3$
generators of bidegree $(0,3)$. Hence it is clearly not generated by quadrics
(with respect to total degree).

In an attempt to study sparse skew-symmetric matrices, we first focus on
Pfaffians of tridiagonal matrices of the form given in Theorem 3.2. Following
are some of the terms which will be used for the same.

A homogeneous ideal $I$ of a standard graded ring $A$ is of Gröbner linear
type if the ideal is of linear type with the linear relations of the defining
ideal of $\mathcal{R}(I)\cong K[X,Y]/J$ forming a Gröbner basis with respect
to some monomial order on $K[X,Y]$. A sequence of monomials
$m_{1},\ldots,m_{s}$ in a set of indeterminates $X$ is said to be an
$M$-sequence if for all $1\leq i\leq s$, there exists a total order on the set
of indeterminates, say $x_{1}<\ldots<x_{n}$ with $m_{i}=x_{1}^{a_{1}}\cdots
x_{n}^{a_{n}}$ and $a_{1}>0,\ldots,a_{n}>0$, such that whenever $x_{k}/m_{j}$
with $1\leq k\leq n$ and $i<j$, then $x_{k}^{a_{k}}\cdots x_{n}^{a_{n}}/m_{j}$
(cf. [13]). It is proved that ideals generated by $M$-sequence are of Gröbner
linear type [13, Theorem 2.4 (i)]. For a monomial $m$ and an indeterminate
$x$, let $O_{x}(m)$ denote the exponent of $x$ in $m$. Then a sequence of
monomials $m_{1},\ldots,m_{s}$ in the set of indeterminates $X$ is said to be
of interval type if for all $1\leq i<j\leq s$ and $x/\text{gcd}(m_{i},m_{j})$,
one has $O_{x}(m_{i})\leq O_{x}(m_{k})$ for all $i\leq k\leq j$ (cf. [13]). It
is known that sequence of interval type implies $M$-sequence [13, Proposition
3.2].

######  Lemma 3.1.

Let $\footnotesize{X=\begin{bmatrix}0&x_{1}&0&0&\ldots&0&0\\\
-x_{1}&0&x_{2}&0&\ldots&0&0\\\ 0&-x_{2}&0&x_{3}&\ldots&0&0\\\
0&0&-x_{3}&0&\ldots&0&0\\\ \vdots&\vdots&\vdots&\vdots&&\vdots&\vdots\\\
0&0&0&0&\ldots&0&x_{n-1}\\\ 0&0&0&0&\ldots&-x_{n-1}&0\\\
\end{bmatrix}_{n\times n}}$, where $n\in\mathbb{N}$.

Then,

  1. (1)

det $X=0$, if $n$ is odd.

  2. (2)

det $X=\prod_{i=2k+1}{x_{i}}^{2}$, $k=0,\ldots,\dfrac{n-2}{2}$, if $n$ is
even.

###### Proof.

Let $X$ be a matrix of the above form.

  1. (1)

Follows from the property of the matrix being skew-symmetric.

  2. (2)

The proof follows by induction on n. Clearly for $n=2,4$ the determinant is
given by $x_{1}^{2}$ and $x_{1}^{2}x_{3}^{2}$ respectively. Thus the statement
holds in these cases. Let $n$ be an even integer such that $n>4$, then the
Laplace expansion of the determinant along the last column and then the last
row gives det $X=(-x_{n-1})(-x_{n-1})$det $X^{\prime}$ where $X^{\prime}$ is a
matrix of order $n-2$ whose determinant is given by the induction hypothesis.
Thus the required result is obtained.

∎

######  Theorem 3.2.

Let $\footnotesize{X_{3}=\begin{bmatrix}0&x_{12}&0&0&\ldots&0&0\\\
-x_{12}&0&x_{23}&0&\ldots&0&0\\\ 0&-x_{23}&0&x_{34}&\ldots&0&0\\\
0&0&-x_{34}&0&\ldots&0&0\\\ \vdots&\vdots&\vdots&\vdots&&\vdots&\vdots\\\
0&0&0&0&\ldots&0&x_{n-1\;n}\\\ 0&0&0&0&\ldots&-x_{n-1\;n}&0\\\
\end{bmatrix}_{n\times n}}$
where $n=2r+1,\,r\in\mathbb{N}\cup\\{0\\}$. Then $I=Pf_{n-1}(X_{3})$ is a
monomial ideal of Gröbner linear type and $\mathcal{R}(I)$ is Koszul and
Cohen-Macaulay.

######  Note 3.3.

In case of a generic skew-symmeric matrix, it is known from [5, Theorem 2.2]
that the Pfaffian ideal $I=$Pf${}_{n-1}(X)$ is of linear type. But results in
that direction cannot be directly applied in the sparse cases like the one
above. Hence we will separately show that the Pfaffian ideal in this case is
of linear type.

###### Proof.

Using Lemma 3.1, we get the generators of the Pfaffian ideal
$I=\text{Pf}_{n-1}(X_{3})$ to have the following form,

$p_{1}=\prod_{k=1}^{r}x_{2k\;2k+1},\text{
}p_{i}=x_{12}\prod_{j<i}x_{j\;j+1}\prod_{i<k\leq 2r}x_{k\;k+1},$ (5)

where $i=2p+1,\,1\leq p\leq r;\;j=1+2l,\,l\geq 1;\;k=i+1+2m,\,m\geq 0$. That
is, we have $\\{p_{i}:i=2p+1,\,0\leq p\leq r\\}$ to be the generating set of
the Pfaffian ideal $I=$Pf${}_{n-1}(X_{3}).$

Now for $1\leq k<m\leq r+1$, assume that $x_{j\;j+1}|\text{gcd}(p_{k},p_{m})$
for $1\leq j\leq n-1$. Then there are two possibilities.

  1. (1)

$j<k.$
This implies that $j$ is odd and $j<l$ for all $k\leq l\leq m$. Thus
$x_{j\;j+1}|p_{l}$.

  2. (2)

$j>k.$
Then we have $j$ to be even. Hence $j>m$ and so $j>l$ for all $k\leq l\leq m$.
Thus $x_{j\;j+1}|p_{l}$.

Since the generators are squarefree monomials, this implies that
$\\{p_{i}:i=2p+1,\,1\leq p\leq r\\}$ forms a sequence of interval type. In
particular, it forms an $M$-sequence with respect to some total order and thus
is of Gröbner linear type.

Let $\mathcal{R}(I)\cong S/J$ where
$S=K[x_{12},\ldots,x_{n-1\;n},y_{1},\ldots,y_{r+1}]$ and $J$ is the defining
ideal of $\mathcal{R}(I)$. From the relation (2) in Section 1, we get
$J=\langle x_{i\;i+1}y_{j}-x_{i+1\;i+2}y_{j+1};\,i=2k+1,\,0\leq k\leq
r-1,\,j=\dfrac{i+1}{2}\rangle$. Then from Lemma 1.7, with respect to graded
reverse lexicographic term order induced by
$x_{12}>x_{13}>\ldots>x_{n-1\,n}>y_{1}>y_{2}>\ldots>y_{r+1}$, it can be
concluded that $J$ is generated by a regular sequence and so is a complete
intersection of quadrics. This implies $\mathcal{R}(I)$ is Cohen-Macaulay.
Koszulness of $\mathcal{R}(I)$ follows from observing that the defining
relations of Rees algebra is generated by a Gröbner basis of quadrics.

∎

######  Note 3.4.

For $n,m\in\mathbb{N}$, let $(n\;m)$ be the pair of integers which denotes
indices of the indeterminates as entries in the matrix $X_{3}$.

######  Corollary 3.5.

The ideal of Pfaffians Pf${}_{n-1}(X_{3})$ can be seen as the vertex cover
ideal of an unmixed bipartite graph.

###### Proof.

Let $G$ be the following bipartite graph.

(12)(23)(45)(67)(n-1 n)(n-3 n-2)(34)(56)(n-4 n-3)(n-2 n-1)

Then it suffices to show that the set of all minimal vertex covers of $G$ are
given by $\\{(2k\;2k+1)\\}$, $1\leq k\leq r$ and
$\\{(1\,2),(j\;j+1),(k\;k+1)\\}$, where $i=2p+1,\,1\leq p\leq
r;\;j=1+2l,\,l\geq 1;\;k=i+1+2m,\,m\geq 0$. We prove this by induction on $r$
where the order of the matrix is $n=2r+1$. For $r=1$ and $r=2$, the minimal
vertex covers are given by $\\{(1\,2)\\},\\{(2\,3)\\}$ and
$\\{(1\,2),(3\,4)\\}$, $\\{(1\,2),(4\,5)\\}$, $\\{(2\,3),(4\,5)\\}$
respectively. Then the result is true for the base cases. Now let $r>2$. The
minimal vertex covers of $G$ on $2(r-1)$ vertices by induction hypothesis will
have the form $\\{(2k\;2k+1)\\}$, $1\leq k\leq r-1$ and $i=2p+1,\,1\leq p\leq
r-1;\;j=1+2l,\,l\geq 1;\;k=i+1+2m,\,m\geq 0$. Observe that only one vertex
cover say $u=\\{(l\;l+1)\\}$, $l=2m+1$, $0\leq m\leq r-2$ contains
$(2r-3\;2r-2)$ and the rest of them contains $(2r-2\;2r-1)$. Then $u$ can be
extended by adjoining either $(2r-1\;2r)$ or $(2r\;2r+1)$ whereas the other
vertex covers containing $(2r-2\;2r-1)$ can be extended only by adding
$(2r\;2r+1)$. This is because if instead $(2r-1\;2r)$ is added, the edge
between $(2r-3\;2r-2)$ and $(2r\;2r+1)$ would have empty intersection with the
set. Thus the minimal vertex covers of $G$ on $n=2r+1$ vertices is given by
$\\{(2k\;2k+1)\\}$, $1\leq k\leq r$ and $\\{(1\,2),(j\;j+1),(k\;k+1)\\}$,
where $j=1+2l,\,l\geq 1;\;i=2p+1,\,1\leq p\leq r;\;k=i+1+2m,\,m\geq 0$.

Thus the ideal of Pfaffians Pf${}_{n-1}(X_{3})$ in Theorem 3.2 can be seen as
the vertex cover ideal of the above unmixed bipartite graph $G$.

∎

As a consequence of the above results, the following can be said.

######  Corollary 3.6.

For $j\in\mathbb{N}$, $I^{j}$ has linear resolutions where $I$ is the vertex
cover ideal of the graph $G$.

###### Proof.

Follows from Remark 1.5 (iii) and Theorem 3.2. ∎

In [30], the authors gave some bounds on $(c,e)$ for which the diagonal
subalgebras of hypersurfaces become Cohen-Macaulay. In the following
proposition we try to extend it to the diagonals of algebras defined by
complete intersections.

######  Proposition 3.7.

Let $R=S/J$ be a standard bigraded $K$-algebra where
$S=K[x_{1},\ldots,x_{n},y_{1},\ldots,y_{m}]$ and $J$ a bihomogeneous ideal of
$S$ generated by regular sequence $\\{f_{1},\ldots,f_{l}\\}$ of bidegree
$(a_{i},b_{i})$, $i=1,\ldots,l$. Let $d_{1}=max\\{a_{i},\,1\leq i\leq l\\}$
and $d_{2}=max\\{b_{i},\,1\leq i\leq l\\}$. Then for $c\geq-n+ld_{1}$ and
$e\geq-m+ld_{2}$, $R_{\Delta}$ is Cohen-Macaulay for the following cases.

  1. (1)

For $\Delta=(c,e)$, if $R$ satisfies the property that all its associated
primes do not contain $S_{(c,0)}$ and $S_{(0,e)}$. In particular, this is true
if $R$ is a domain.

  2. (2)

If $R=R_{1}\otimes_{K}R_{2}$ with
$\text{dim}R=\text{dim}R_{1}+\text{dim}R_{2}$ where dim denotes Krull
dimension.

###### Proof.

We have that $R$ is defined by a complete intersection. Hence the Koszul
complex on $\\{f_{1},\ldots,f_{l}\\}$ resolves $R$. Applying $\Delta$ to the
resolution and then using Lemma 1.8 and Lemma 1.6 gives depth($R_{\Delta})\geq
m+n-l-1$.

  1. (1)

Assume for $\Delta=(c,e)$, $R$ satisfies the property that all its associated
primes do not contain $S_{(c,0)}$ and $S_{(0,e)}$. Then relative dimension of
$R$ coincides with it’s Krull dimension [41, Section 2.2]. Hence from the idea
similar to [41, Proposition 2.3], it can be concluded that
dim($R_{\Delta})=m+n-l-1$.

  2. (2)

Let $R=R_{1}\otimes_{K}R_{2}$ with
$\text{dim}R=\text{dim}R_{1}+\text{dim}R_{2}$. Then since from [15, Lemma
2.7], dim$(R_{\Delta})\leq m+n-l-1$, from the lower bound on depth we obtain
dim$(R_{\Delta})=m+n-l-1$.

∎

As an application of the previous results, we have the following observations.

######  Theorem 3.8.

Let $I=Pf_{n-1}(X_{3})$. Let $c,e$ be positive integers. Then,

  1. (a)

$\mathcal{R}(I)_{\Delta}$ is Koszul for all $\Delta=(c,e)$.

  2. (b)

$\mathcal{R}(I)_{\Delta}$ is Cohen-Macaulay for all $\Delta=(c,e)$.

###### Proof.

Since $\mathcal{R}(I)$ is Koszul, Koszulness of $\mathcal{R}(I)_{\Delta}$ can
be seen as a consequence of Remark 1.5 (ii).

We have that $\mathcal{R}(I)$ is a complete intersection domain. Hence the
Cohen-Macaulayness of all diagonals follow from Proposition 3.7

∎

######  Remark 3.9.

  1. (1)

Let $I=$Pf${}_{n-1}(X_{3})$, $\Delta=(1,1)$ and $T=(t_{ij}),\,1\leq i\leq
n-1,\,1\leq j\leq r+1$. Then

$\mathcal{R}(I)_{\Delta}\cong K[T]/I_{2}(T)+\langle
t_{ij}-t_{i+1j+1}:\,i=2k+1,\,0\leq k\leq r-1,\,j=\dfrac{i+1}{2}\rangle.$

  2. (2)

In general, Pfaffian ideals need not be generated by monomials or binomials.
Hence it is not always possible to associate a graph to a Pfaffian ideal. For
instance for generic skew-symmetric matrices of order greater than $3$, the
generators are seen to be neither monomials nor binomials.

  3. (3)

But if the Pfaffian ideal is a squarefree monomial ideal then it corresponds
to a simplicial complex $\Delta^{\prime}$, where the Pfaffian ideal can be
identified as the Stanley Reisner ideal of $\Delta^{\prime}$ ([36]). In fact
Theorem 3.2 is a special case where the Pfaffian ideal can be viewed as the
vertex cover ideal of an unmixed bipartite graph.

  4. (4)

Corresponding to a Pfaffian ideal, even if a graph is associated to it, the
correspondence need not be unique. For example,

Let $\footnotesize{X=\begin{bmatrix}0&x_{12}&0&0&0\\\
-x_{12}&0&x_{23}&x_{24}&0\\\ 0&-x_{23}&0&x_{34}&0\\\
0&-x_{24}&-x_{34}&0&x_{45}\\\ 0&0&0&-x_{45}&0\\\ \end{bmatrix}_{5\times 5}}$

This matrix also corresponds to the same Pfaffian ideal Pf${}_{4}(X_{3})$, the
one in Theorem 3.2. Hence the correspondence between the skew-symmetric
matrices and the graph is not unique.

Another form of sparse skew-symmetric matrix that we have considered is the
following.

######  Proposition 3.10.

Let $n=2r+1,\,r\in\mathbb{N}\cup\\{0\\}$ and let $X_{4}=\begin{bmatrix}O&A\\\
-A^{T}&B\\\ \end{bmatrix}$, $O$ is a zero block matrix,

$A=\begin{bmatrix}x_{1\;r+2}&x_{1\;r+3}&\ldots&x_{1\;n-1}&x_{1\;n}\\\
x_{2\;r+2}&x_{2\;r+3}&\ldots&x_{2\;n-1}&x_{2\;n}\\\
\vdots&\vdots&\vdots&\vdots&\vdots\\\
x_{r+1\;r+2}&x_{r+1\;r+3}&\ldots&x_{r+1\;n-1}&x_{r+1\;n}\\\
\end{bmatrix},\;\text{ and
}\;B=\begin{bmatrix}0&x_{r+2\;r+3}&\ldots&x_{r+2\;n-1}&x_{r+2\;n}\\\
-x_{r+2\;r+3}&0&\ldots&x_{r+3\;n-1}&x_{r+3\;n}\\\
\vdots&\vdots&\vdots&\vdots&\vdots\\\
-x_{r+2\;n}&-x_{r+3\;n}&\ldots&-x_{n-1\;n}&0\\\ \end{bmatrix},$

where $A$ is an $(r+1)\times r$ matrix and $B$ is a skew-symmetric matrix of
order $r\times r$. Then,

  1. (1)

$I=Pf_{n-1}(X_{4})$ is generated by an unconditioned $d$-sequence.

  2. (2)

$\mathcal{R}(I)$ is Koszul and Cohen-Macaulay.

  3. (3)

$I^{j}$ have linear resolution for all $j\geq 1$.

###### Proof.

Let $B=K[X_{4}]$. Then the generators of the Pfaffian ideal
$I=$Pf${}_{n-1}(X_{4})$ is given by the $r$-minors of the first $r+1$ rows and
the last $(n-r-1)=r$ columns of $X_{4}$ which is $r+1\choose r$ in number.

  1. (1)

From [25, Proposition 1.1], the Pfaffians form an unconditioned $d$-sequence.

  2. (2)

From example $1.2$ in [25], it follows that $\mathcal{R}(I)\cong S/J$ where
$S=K[X_{4},Y]$, $Y=\begin{bmatrix}y_{1},\ldots,y_{k}\\\ \end{bmatrix}$ for
$k=1,\ldots,r+1$ and the defining ideal $J$ has the following form,

$J=\langle g_{i},i=1,\ldots,r\rangle\text{ where
}g_{i}=\sum_{k=1}^{r+1}(-1)^{k+1}x_{k\;n-(i-1)}y_{k},\;i=1,\dots r.$

Consider the following order on the indeterminates of $S$,
$x_{1\;n}>x_{2\;n-1}>\ldots>x_{r\;n-(r-1)}$ followed by the other
indeterminates, with term order being graded lexicographic. Then from Lemma
1.7, $J$ is a complete intersection of quadrics. Thus being complete
intersection rings defined by quadrics, $\mathcal{R}(I)$ is Koszul and Cohen-
Macaulay.

  3. (3)

Since $\mathcal{R}(I)$ is Koszul, reg${}_{B}(I^{j})=rj$ for $j\geq 1$ follows
from Remark 1.5 (iii).

∎

######  Theorem 3.11.

Let $I=Pf_{n-1}(X_{4})$. Then $\mathcal{R}(I)_{\Delta}$ is Koszul and Cohen-
Macaulay for all $\Delta=(c,e)$, $c,e>0$.

###### Proof.

Koszulness of $\mathcal{R}(I)_{\Delta}$ for all $\Delta$ follows from
Proposition 3.10 and Remark 1.5 (ii). Since $\mathcal{R}(I)$ is a complete
intersection domain, $\mathcal{R}(I)_{\Delta}$ is Cohen-Macaulay for all
$\Delta$ from Proposition 3.7. ∎

Acknowledgement. The first author is partially supported by the MATRICS grant,
SERB India. The second author is financially supported by the INSPIRE
fellowship, DST, India.

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# Combined constraints for scalars and pseudoscalars from
$\left(g-2\right)_{l}$ and $e^{+}e^{-}$ colliders

Aleksandr Pustyntsev  Institut für Kernphysik and $\text{PRISMA}^{+}$ Cluster
of Excellence, Johannes Gutenberg Universität, D-55099 Mainz, Germany  Marc
Vanderhaeghen  Institut für Kernphysik and $\text{PRISMA}^{+}$ Cluster of
Excellence, Johannes Gutenberg Universität, D-55099 Mainz, Germany

(August 28, 2024)

###### Abstract

Scalars and pseudoscalars with masses in the MeV to GeV range are of interest
in different extensions of the Standard Model. Such particles are often
associated with dark matter, the strong CP problem and the
$\left(g-2\right)_{\mu}$ anomaly. In this work we investigate limits for
masses of such particles and their couplings to photons and leptons which can
be derived from present and future $e^{+}e^{-}$ collider experiments and
current $\left(g-2\right)_{l}$ measurements. Our work expands upon previous
studies in several ways, demonstrating that the interplay of both couplings is
a decisive factor in this type of analyses.

Axions and ALPs and $e^{+}e^{-}$ experiments

##  I Introduction

After being first proposed more than fourty years ago, the Peccei-Quinn
mechanism so far remains the most persuasive solution to the strong CP
problem, that is, the experimentally observed absence of CP violations in
quantum chromodynamics Peccei and Quinn (1977a, b); Weinberg (1978); Wilczek
(1978). The Peccei-Quinn theory comes down to the postulate of a new global
symmetry $U\left(1\right)$, the spontaneous breaking of which gives rise to a
new pseudoscalar particle called axion, dynamically driving the CP-violating
phase to zero.

Axion-like particles (or ALPs) are a natural generalization of this idea, but
are not constrained by a linear mass-coupling relation and are not necessarily
associated with the strong CP problem. ALPs arise naturally in various
extensions of the Standard Model Jaeckel and Ringwald (2010); Baker  _et al._
(2013); Daido  _et al._ (2017); Bagger  _et al._ (1994); Arvanitaki  _et al._
(2010) and, in addition, both the QCD axion and ALPs are considered to be
promising dark matter candidates as being both very long-lived and weakly-
interacting Daido  _et al._ (2017); Adams  _et al._ (2022); Ringwald (2012);
Arias  _et al._ (2012); Giannotti  _et al._ (2016).

Over the past decades, extensive efforts have been made to search for axions
and ALPs, encompassing both laboratory searches and astronomical observations
Graham  _et al._ (2015); Irastorza and Redondo (2018); Bauer  _et al._ (2017);
Mimasu and Sanz (2015); Safronova  _et al._ (2018); Izaguirre  _et al._
(2017); Regis  _et al._ (2021); Lees  _et al._ (2022). Recently, there has
been a surge of interest in the MeV to GeV range Dolan  _et al._ (2017); Alves
and Weiner (2018); Millea  _et al._ (2015); Jaeckel and Spannowsky (2016);
Beacham  _et al._ (2020); Gavela  _et al._ (2020); Mimasu and Sanz (2015);
Bauer  _et al._ (2017); Agrawal  _et al._ (2021); Antel  _et al._ (2023). This
area remains relatively unexplored by current experiments, particularly when
contrasted with the sub-MeV mass region where astrophysical constraints can be
applied. Additionally, most of the existing studies assume a single ALP-photon
coupling scenario, while the recent works Liu  _et al._ (2023a, b); Pustyntsev
and Vanderhaeghen (2024) indicate that accounting for both ALP-photon and ALP-
lepton couplings is important in deducing empirical constraints from processes
involving leptons.

Meanwhile, the measurement of the muon’s anomalous magnetic moment
$\left(g-2\right)_{\mu}$ is a powerful tool to test potential effects beyond
the Standard Model due to its enhanced sensitivity to New Physics Aoyama  _et
al._ (2020); Athron  _et al._ (2021). With the current Standard Model
calculations of $\left(g-2\right)_{\mu}$ Aoyama  _et al._ (2020), there is an
unexplained up to $5\sigma$ discrepancy between theory and experiment, namely
$a_{\text{exp.}}-\,a_{\text{th.}}=249\left(48\right)\times 10^{-11}$ Aguillard
_et al._ (2023) (although recent lattice calculations show smaller discrepancy
Borsanyi  _et al._ (2021)). In this context, scalars and pseudoscalars
motivated by the Standard Model extensions could serve as a “two birds with
one stone” solution. Nevertheless, it is one of the very few viable one-field
explanations of the observed anomaly Marciano  _et al._ (2016); Buen-Abad  _et
al._ (2021); Liu  _et al._ (2023b); Chen  _et al._ (2016); Liu  _et al._
(2019).

In the present work both analyses are extended. Focusing on $3\gamma$ final
states, we rigorously examine constraints which can be derived from existing
$e^{+}e^{-}$ collider data for scalar and pseudoscalar particles when lepton
and photon couplings are simultaneously taken into consideration.
Additionally, we show projections for the upcoming Belle II data collection,
which by 2030 is expected to result in a record 50 $\text{ab}^{-1}$ of
integrated luminosity, with the prospect of significantly narrowing down the
parameter space for potential ALPs in the MeV to GeV mass range, where current
constraints remain relatively loose. We combine these bounds with ones which
can be derived from the present $\left(g-2\right)_{e}$ data Fan  _et al._
(2023); Aoyama  _et al._ (2015) and test the available parameter space for the
possible $\left(g-2\right)_{\mu}$ resolution. Our results show that the
interplay of lepton and gauge boson ALP couplings plays a crucial role in both
studies.

The paper is organized as follows. Section II provides the overview of the ALP
formalism with the discussion of couplings, potential contribution to
$\left(g-2\right)_{l}$ and $e^{+}e^{-}$ annihilation with the corresponding
constraints derived. In the section III we repeat our calculations for a
potential scalar New Physics particle. Section IV summarizes our work.

##  II Studying ALPs through $\left(g-2\right)_{l}$ and $e^{+}e^{-}$ colliders

In this study, we assume that ALPs in the MeV to GeV range interact with
electroweak gauge bosons and leptons, i.e. decay exclusively into visible
states and are isolated from the QCD sector. The latter statement is justified
by the fact that such couplings typically lead to tightly constrained flavour-
changing processes Dolan  _et al._ (2015); Lees  _et al._ (2011).

The following section provides a short review of the relevant ALP
interactions. Lepton universality implies that the parameter space includes
only three variables: the ALP mass $m_{a}$ and its couplings to photons and
electrons, which are denoted by $g_{a\gamma\gamma}$ and $g_{aee}$,
respectively. We do not focus on any specific UV-complete ALP model, thus the
exact origin of $g_{a\gamma\gamma}$ and $g_{aee}$ is unspecified and these
quantities must be treated as effective couplings, with both tree-level values
and possible loop corrections to them.

Figure 1: Yukawa-like correction to the lepton dipole moment.

In this formalism we provide the calculations of the 1-loop corrections to the
$\left(g-2\right)_{l}$ and of the $e^{+}e^{-}\to 3\gamma$ cross section with
an intermediate ALP. The signal over background ratio is estimated and
constraints are derived for the scenario involving an interplay of both lepton
and photon ALP-couplings.

###  II.1 Lepton coupling and shift symmetry

The shift symmetry for an ALP implies that only derivative couplings with
leptons are allowed. The lowest order operator of such an effective
interaction with the shift symmetry has the form

$\mathcal{L}_{all}=-\frac{g_{all}}{2m_{l}}\partial_{\mu}a\,\bar{l}\gamma^{5}\gamma^{\mu}l,$
(1)

where $l$ stands for the lepton field, $m_{l}$ denotes its mass and $g_{all}$
is the dimensionless coupling constant. At the level of tree diagram
calculations $\mathcal{L}_{all}$ can be equivalently reduced to a pseudoscalar
Yukawa interaction

$\mathcal{L}_{all}=-ig_{all}a\,\bar{l}\gamma^{5}l.$ (2)

It is clear that lepton universality requires a large enhancement of the ALP
coupling to the muon, as compared to the electron, namely

$g_{a\mu\mu}\approx\frac{m_{\mu}}{m_{e}}g_{aee},$ (3)

and even more enhanced coupling - to the tau-lepton.

Such an enhanced coupling of ALPs to muons, compared to the one to electrons,
opens the possibility of the $\left(g-2\right)_{\mu}$ resolution without the
violation of $\left(g-2\right)_{e}$. Conversely, without this enhanced
coupling, resolving the muon anomaly would inevitably conflict with
constraints on the electron’s anomalous magnetic moment.

Figure 2: Barr-Zee correction to the lepton dipole moment. Figure 3:
$e^{+}e^{-}$ annihilation into three photons involving the $g_{all}$ and
$g_{a\gamma\gamma}$ couplings. Graphs obtained from these by crossing are not
shown, but are evaluated too.

The latter already exhibits a relatively strong agreement between theoretical
predictions and experimental results. Recent measurements of
$\left(g-2\right)_{e}$ have improved the precision of previous results by a
factor of 2.2, achieving an impressive agreement up to 12 digits Fan  _et al._
(2023).

The Yukawa-like 1-loop contribution to the lepton dipole moment is shown in
Fig. 1. It is insensitive to whether (1) or (2) are used and results in

$\Delta
a_{l}^{Y}=-\frac{g_{all}^{2}}{8\pi^{2}}\frac{1}{r_{a}^{2}}\int_{0}^{1}\frac{\left(1-z\right)^{3}}{\frac{\left(1-z\right)^{2}}{r_{a}^{2}}+z}dz,$
(4)

where we denoted

$a_{l}=\frac{\left(g-2\right)_{l}}{2},\quad\quad\quad
r_{a}=\frac{m_{a}}{m_{l}}.$ (5)

As this contribution is negative, while the $\left(g-2\right)_{\mu}$
discrepancy is positive, it is evident that such a photophobic ALP scenario is
not capable of resolving the mentioned anomaly on its own. However, accounting
for the additional correction from the ALP-photon interaction, shown in Fig.
2, changes the situation, since this Barr-Zee contribution can be of an
opposite sign. Consequently, the sum of the diagrams shown in Figs. 1 and 2
can yield a positive overall contribution to $a_{l}$.

###  II.2 Low energy interactions between ALPs and gauge bosons

The generic gauge-invariant Lagrangian of ALPs interaction with electroweak
vector bosons is given by Dolan  _et al._ (2017)

$\mathcal{L}_{aEW}=-\frac{g_{aBB}}{4}aB^{\mu\nu}\tilde{B}_{\mu\nu}-\frac{g_{aWW}}{4}a\textbf{W}^{\mu\nu}\tilde{\textbf{W}}_{\mu\nu},$
(6)

$B^{\mu\nu}$ and $\textbf{W}^{\mu\nu}$ refer to the $U\left(1\right)$ and
$SU\left(2\right)$ field tensors, respectively. The corresponding dual
pseudotensors are defined in the standard way,
$\tilde{B}_{\mu\nu}=\frac{1}{2}\varepsilon_{\mu\nu\lambda\sigma}{B}^{\lambda\sigma}$
and similarly for $\textbf{W}^{\mu\nu}$, while $g_{aBB}$ and $g_{aWW}$ are the
coupling constants of $\mbox{GeV}^{-1}$ dimension.

After the symmetry breaking the effective interaction of ALPs and photons is
given by

$\mathcal{L}_{a\gamma\gamma}=-\frac{g_{a\gamma\gamma}}{4}aF^{\mu\nu}\tilde{F}_{\mu\nu}-\frac{g_{a\gamma
Z}}{2}aF^{\mu\nu}\tilde{Z}_{\mu\nu},$ (7)

where the new coupling constants were defined

$\displaystyle
g_{a\gamma\gamma}=g_{aBB}\cos^{2}{\theta_{w}}+g_{aWW}\sin^{2}{\theta_{w}},$
(8) $\displaystyle g_{a\gamma
Z}=\frac{g_{aWW}-g_{aBB}}{2}\sin{\left(2\theta_{w}\right)},$ (9)

with $\theta_{w}$ being the weak mixing angle. The ratio of $g_{aBB}$ and
$g_{aWW}$ is of key importance for ALPs studies at $e^{+}e^{-}$ colliders. In
particular, $a\gamma Z$ interaction potentially leads to the anomalous decay
$Z\to 3\gamma$, which was a subject of LEP studies Acciarri  _et al._ (1995a,
b). Such a process would be suppressed if the condition $g_{aWW}\approx
g_{aBB}$ is fulfilled.

Searches for flavor-changing processes, however, impose stringent constraints
on the coupling $g_{aWW}$ Izaguirre  _et al._ (2017); Lees  _et al._ (2022),
while the limits on $g_{aBB}$ are more relaxed. It is therefore reasonable to
assume that the coupling between ALPs and W is, at best, subdominant, i.e.
$g_{aBB}\gg g_{aWW}$. In this case one can express

$g_{a\gamma\gamma}\approx-\cot{\theta_{w}}g_{a\gamma Z}\approx-1.9\,g_{a\gamma
Z}.$ (10)

Of course, the Lagrangian (6) also leads to a non-zero $aZZ$ and $aWW$
interactions. However, they do not contribute to $\left(g-2\right)_{l}$ at
1-loop level and are irrelevant in the domain of interest corresponding to
$e^{+}e^{-}$ colliders with center-of-momentum energy below 10 GeV, since the
direct production of $Z$ and $W$ bosons is not possible.

The corresponding correction to $\left(g-2\right)_{l}$ is shown in Fig. 2 with
the internal boson line standing for either photon or $Z$. The leading (log-
enhanced) contribution is

$\begin{split}&\Delta a_{l}^{BZ}\simeq\\\
&\frac{g_{all}m_{l}}{8\pi^{2}}\ln{\Lambda^{2}}\left(g_{a\gamma\gamma}-\frac{4\sin^{2}{\theta_{w}}-1}{4\sin{\theta_{w}}\cos{\theta_{w}}}g_{a\gamma
Z}\right),\end{split}$ (11)

where $\Lambda$ denotes the high-energy cut-off, referring to the new physics
scale, which we set to $1$ TeV.

Clearly, the effect of $g_{a\gamma Z}$ interaction is suppressed and provides
only a small correction to $g_{a\gamma\gamma}$ contribution. The expressions
for finite parts are summarized in Appendix A.

It is useful to note that the overall sign of (11) depends on the relative
sign between $g_{a\gamma\gamma}$ and $g_{all}$. We assume them to be of the
same sign - in this case $\Delta a_{l}^{BZ}$ is positive, which is consistent
with $\left(g-2\right)_{\mu}$ anomaly.

###  II.3 ALP searches at $e^{+}e^{-}$ colliders

One of the most promising methods for detecting ALPs is by probing their
coupling to photons. Numerous experiments have been dedicated to this search
over the past years, targeting ALPs in cosmic rays Graham  _et al._ (2015);
Irastorza and Redondo (2018), atomic and molecular effects Safronova  _et al._
(2018), beam dumps and collider experiments Bauer  _et al._ (2017); Mimasu and
Sanz (2015); Ablikim  _et al._ (2023); Abudinén  _et al._ (2020); Sirunyan
_et al._ (2019); Blumlein  _et al._ (1991); Banerjee  _et al._ (2020); Riordan
_et al._ (1987); Bergsma  _et al._ (1985); Jaeckel and Spannowsky (2016); Aad
_et al._ (2021); Knapen  _et al._ (2017). A summary of current limits on ALP
couplings can be found in O’Hare (2020).

It this work we focus on the $e^{+}e^{-}$ collider kinematics. Currently,
there are results from a Belle II 2018 test run with the 445 $\text{pb}^{-1}$
of integrated luminosity. The data allowed to establish an upper limit on the
cross section $\sigma_{a}$ for the process $e^{+}e^{-}\to 3\gamma$ involving
an intermediate in the mass range $0.2<m_{a}<9.7$ GeV Abudinén  _et al._
(2020). The corresponding coupling limit was derived under the assumption that
$a\to 2\gamma$ is the only allowed decay channel, making use of the formula

$\left.\sigma_{a}\right|_{g_{all}=0}=\frac{\alpha
g^{2}_{a\gamma\gamma}}{24}\left(1-\frac{m_{a}^{2}}{s}\right)^{3},$ (12)

where $\alpha\equiv e^{2}/4\pi$. The integration over the full solid angle
$4\pi$ is implied, but due to the absence of forward-backward enhancement in
this process, it leads only to a minor correction. The outcomes of this test
run were already on par with the results from the LEP run at the $Z$-pole,
which we discuss below.

Projections for the upcoming Belle II data collection were also studied Dolan
_et al._ (2017) and later generalized to the scenario when ALP-electron
interaction is taken into consideration, leading to a notable modification
Pustyntsev and Vanderhaeghen (2024); Liu  _et al._ (2023a). Fig. 3 shows the
relevant diagrams for such search. The total decay width of ALPs is assumed to
be a sum of the following contributions

Figure 4: Exclusion limits on the $\left(m_{a},g_{a\gamma\gamma}\right)$
parameter space for the fixed value of $g_{a\mu\mu}=10^{-3}$. The colored
regions represent the existing bounds, the blue dashed line is the projection
for the upcoming data collection at Belle II. The grey band refers to the
parameter space that could potentially resolve the $\left(g-2\right)_{\mu}$
discrepancy.

$\Gamma_{a}=\Gamma_{a\gamma\gamma}+\sum_{e,\mu,\tau}\Gamma_{all},$ (13)

where we denoted

1

1

Figure 5: Exclusion limits on the $\left(g_{a\mu\mu},g_{a\gamma\gamma}\right)$
parameter space for masses $m_{a}=0.4$, $1.5$ and $5$ GeV. The colored regions
represent the existing bounds, the blue dashed line is the projection for the
upcoming data collection at Belle II. The grey band refers to the parameter
space that could potentially resolve the $\left(g-2\right)_{\mu}$ discrepancy.

$\displaystyle\Gamma_{a\gamma\gamma}=\frac{g^{2}_{a\gamma\gamma}m_{a}^{3}}{64\pi},$
(14)
$\displaystyle\Gamma_{all}=\frac{g_{all}^{2}}{4\pi}\sqrt{\frac{m^{2}_{a}}{4}-m_{l}^{2}}\,\theta\left[m_{a}-2m_{l}\right].$
(15)

For comparable values of $m_{a}g_{a\gamma\gamma}$ and $g_{a\mu\mu}$ it is
typically dominated by leptonic decays.

For the upcoming Belle II data, the cross section over the Belle II geometry
and require three resolved photons within the
$37.3^{\circ}<\theta<123.7^{\circ}$ angular region in the lab frame (same as
in the 2018 analysis). The event selection threshold is set to $1$ GeV in the
center-of-momentum frame for the 2018 data and $0.25$ GeV for the projection.
For a gaussian distributed signal the aimed sensitivity can be estimated as

$\frac{\sigma_{a}}{\sigma_{b}}=\frac{N}{\sqrt{L\cdot\sigma_{b}}},$ (16)

where $L$ denotes the integrated luminosity, $N$ is the number of standard
deviations that determines whether or not a fluctuation is considered as a
signal and $\sigma_{b}$ is the background cross section (which is primarily
dominated by the QED $e^{+}e^{-}\to 3\gamma$ annihilation). For our analysis,
we set $N=2$, corresponding to a 95% confidence level.

In addition to the Belle II exclusion limits, we also incorporate the adapted
boundaries from the results of LEP run at $Z$-pole Acciarri  _et al._ (1995b,
a), which imposed stringent constraints on possible decays $Z\to 2\gamma$ and
$Z\to 3\gamma$, significantly narrowing down potential New Physics parameter
space. Furthermore, we include results from BESIII Ablikim  _et al._ (2023)
analyses of $J/\psi\to 3\gamma$ decays (notably, its results remain
independent of $g_{all}$).

Coupling of potential new particles to leptons, primarly muons, was also a
subject of studies at $e^{+}e^{-}$ colliders Lees  _et al._ (2020, 2016).
BaBar analysis of $\tau^{+}\tau^{-}e^{+}e^{-}$ and
$\tau^{+}\tau^{-}\mu^{+}\mu^{-}$ final states established a boundary
$g_{a\mu\mu}\lesssim 10^{-3}$ for ALPs lighter than $4$ GeV (assuming 100$\%$
branching ratio to leptons). The resolution of $\left(g-2\right)_{\mu}$
problem is achieved if the product of couplings satisfy
$g_{a\gamma\gamma}g_{a\mu\mu}\approx 10^{-7}$, which then implies
$g_{a\gamma\gamma}\approx 10^{-4}\,\text{GeV}^{-1}$.

###  II.4 Results for pseudoscalars

For illustrative purposes, we choose the value $g_{a\mu\mu}=10^{-3}$ to
project onto the $\left(m_{a},g_{a\gamma\gamma}\right)$ plane in Fig. 4. It is
evident that in the coming years, Belle II will be capable to cover a
considerable part of the parameter space for potential New Physics, especially
in the lower mass region, where it has a much enhanced sensitivity compared to
LEP.

We would also like to emphasize that the projection curve is quite
conservative. The setup and selection procedures are likely to be further
optimized compared to the 2018 test run, so we expect the final curve to be
even more improved. In particular, upgrading the spatial resolution of photons
would allow to access the currently unexplored region $m_{a}<0.2\,\text{GeV}$
Abudinén  _et al._ (2020).

On the other hand, it is evident that more data probing the ALP-lepton
coupling is necessary to comprehensively explore the parameter space relevant
to $\left(g-2\right)_{\mu}$. In this work we are focused on $3\gamma$ final
states, but the detailed study of leptonic final states at BESIII or Belle II
has the potential to improve upon existing constraints, in the latter case -
even with the relatively limited data available from its test run.

Next, fig. 5 shows our results for three different ALP masses, $m_{a}=0.4$,
$1.5$ and $5$ GeV as a projection onto the
$\left(g_{a\mu\mu},g_{a\gamma\gamma}\right)$ plane. In the limit $g_{all}\to
0$ these bounds are in agreement with the single $g_{a\gamma\gamma}$ coupling
scenario.

Figure 6: Exclusion limits on the $\left(m_{s},g_{s\gamma\gamma}\right)$
parameter space for the fixed value of $g_{s\mu\mu}=10^{-3}$. The colored
regions represent the existing bounds, the blue dashed line is the projection
for the upcoming data collection at Belle II. The magenta and grey bands refer
to the parameter space that could potentially resolve the
$\left(g-2\right)_{\mu}$ discrepancy under the scenarios where
$g_{s\gamma\gamma}$ and $g_{s\mu\mu}$ have the same and opposite signs,
respectively.

We see that in fact a large part of the parameter space for
$\left(g-2\right)_{\mu}$-solving ALPs can be ruled out with experimental data
which are already available. A sizeable further part of the parameter space
corresponding to the solution for $\left(g-2\right)_{\mu}$ can be tested after
Belle II finishes the data collection and BESIII collects more data on
$J/\psi$ decays (at the time of writing this article BESIII collaboration
released a pre-print on updated ALP limits Ablikim  _et al._ (2024), with a
four-fold increased statistics of $J/\psi$ decays compared to the previous
result Ablikim  _et al._ (2023)).

1

1

Figure 7: Exclusion limits on the $\left(g_{s\mu\mu},g_{s\gamma\gamma}\right)$
parameter space for masses $m_{s}=0.4$, $1.5$ and $5$ GeV. The colored regions
represent the existing bounds, the blue dashed line is the projection for the
upcoming data collection at Belle II. The magenta and grey bands refer to the
parameter space that could potentially resolve the $\left(g-2\right)_{\mu}$
discrepancy under the scenarios where $g_{s\gamma\gamma}$ and $g_{s\mu\mu}$
have the same and opposite signs, respectively.

We also observe that the sensitivity decreases for relatively large ALP-lepton
couplings, as the branching ratio $\Gamma_{a\gamma\gamma}/\Gamma_{a}$ becomes
suppressed. Nevertheless, this regime could be potentially explored with
$e^{+}e^{-}\to\tau^{+}\tau^{-}\mu^{+}\mu^{-}$ and other similar reactions,
which would further constrain the parameter space and challenge the
pseudoscalar explanation of $\left(g-2\right)_{\mu}$ already at the current
generation of colliders and with existing data.

On the other hand, if the discrepancy in $\left(g-2\right)_{\mu}$ is
eventually resolved within the Standard Model framework, the region above the
grey band will be excluded, thereby utterly reducing further the parameter
space available for potential ALPs.

##  III Studying scalars through $\left(g-2\right)_{l}$ and electron-positron
colliders

Scalars in the few hundred MeV to GeV mass range represent another scenario
frequently discussed in the context of New Physics Beacham  _et al._ (2020);
Batell  _et al._ (2017); Marciano  _et al._ (2016); Abdallah  _et al._ (2020);
Winkler (2019); Chen  _et al._ (2016). Despite the current measurements of
Higgs boson properties fitting well within the Standard Model, the structure
of the Higgs sector could potentially be more complicated, possibly including
an additional scalar doublet Athron  _et al._ (2021); Berlin  _et al._ (2019);
Arcadi  _et al._ (2020). Furthermore, particles of this nature typically have
couplings to fermions that are proportional to $m_{f}$. Possible dark scalars
are also widely discussed in different contexts and can be favored for many
reasons Mondino  _et al._ (2021); Chen  _et al._ (2016); Davoudiasl  _et al._
(2012); Fabbrichesi  _et al._ (2020); Athron  _et al._ (2017).

In general, the limits for scalars are less extensively studied compared to
those for pseudoscalars. The impact of scalars on the anomalous magnetic
moment of the muon was discussed in Chen  _et al._ (2016); Marciano  _et al._
(2016); Liu  _et al._ (2019). In the following, we discuss the differences
between scalar and pseudoscalar scenarios, and present bounds derived from
both magnetic moment measurements and $e^{+}e^{-}$ collider data.

The Lagrangian of scalar-lepton and scalar-photon interactions has some
notable modifications, compared to the case of pseudoscalars. It is
effectively given by

$\mathcal{L}=-g_{sll}s\,\bar{l}l-\frac{g_{s\gamma\gamma}}{4}sF^{\mu\nu}F_{\mu\nu}-\frac{g_{s\gamma
Z}}{2}sF^{\mu\nu}Z_{\mu\nu}.$ (17)

The relation (10) between $\gamma\gamma$ and $\gamma Z$ couplings still holds
in the scalar case. One-loop corrections to $\left(g-2\right)_{l}$ are given
by

$\displaystyle\Delta
a^{Y}_{l}=\frac{g_{sll}^{2}}{8\pi^{2}}\frac{1}{r_{s}^{2}}\int_{0}^{1}\frac{\left(1-z\right)^{2}\left(1+z\right)}{z+\frac{\left(1-z\right)^{2}}{r_{s}^{2}}}\,dz,$
(18) $\displaystyle\begin{split}&\Delta a_{l}^{BZ}\simeq\\\
&\frac{g_{sll}m_{l}}{8\pi^{2}}\ln{\Lambda^{2}}\left(g_{s\gamma\gamma}-\frac{4\sin^{2}{\theta_{w}}-1}{4\sin{\theta_{w}}\cos{\theta_{w}}}g_{s\gamma
Z}\right).\end{split}$ (19)

The Yukawa contribution is positive and is capable of resolving the
$\left(g-2\right)_{\mu}$ discrepancy on it’s own, as explored in Chen  _et
al._ (2016). When the scalar-photon coupling is incorporated, two scenarios
emerge. Either $g_{see}$ and $g_{s\gamma\gamma}$ have opposite signs, leading
to partial cancellation between Fig. 1 and Fig. 2, or they share the same sign
and sum up. Both possibilities can accommodate the muon anomaly resolution, in
contrast to the pseudoscalar case, where only one scenario aligns. For small
values of $g_{s\gamma\gamma}$ the difference between them vanishes.

Implementing these adjustments, we can apply the same search strategy which
was used for ALPs in the previous section of this work. The partial widths of
$s\to\gamma\gamma$ and $s\to l^{+}l^{-}$ decays are

$\displaystyle\Gamma_{s\gamma\gamma}=\frac{g^{2}_{s\gamma\gamma}m_{s}^{3}}{64\pi},$
(20) $\displaystyle\Gamma_{sll}=\frac{g_{sll}^{2}}{\pi
m_{s}^{2}}\left(\frac{m_{s}^{2}}{4}-m_{l}^{2}\right)^{3/2}\theta\left[m_{s}-2m_{l}\right].$
(21)

Fig. 6 shows the projection onto $\left(m_{s},g_{s\gamma\gamma}\right)$ plane
for the fixed value $g_{s\mu\mu}=10^{-3}$. Fig. 7 illustrates the constraints
on the $\left(g_{s\mu\mu},g_{s\gamma\gamma}\right)$ parameter space for three
different masses, $m_{s}=0.4$, $1.5$ and $5$ GeV. Unlike the ALP case, the
parameter space aligning within the $\left(g-2\right)_{\mu}$ resolution is not
restricted and extends down to $g_{s\gamma\gamma}\to 0$. The rest of our
conclusions regarding the pseudoscalar case also apply to the scalar case.

##  IV Conclusion

In this paper we thoroughly examined ALPs and scalars coupled to leptons and
photons in a minimal way. Constraints from $e^{+}e^{-}\to 3\gamma$
annihilation and electron magnetic moment from both existing and upcoming
experiments were derived, and the available parameter space for the possible
$\left(g-2\right)_{\mu}$ resolution was tested.

Our analysis has emphasized the significance of simultaneous consideration of
couplings of ALPs and scalars to both leptons and photons, revealing the
intricate interplay between these interactions and their implications for
experimental constraints. We have demonstrated the potential for narrowing
down the parameter space for ALPs and scalars in the MeV to GeV mass range and
conclude that while existing experimental results already impose stringent
bounds on the parameter space relevant to $\left(g-2\right)_{\mu}$, a
substantial portion of the parameter space remains unexplored.

There is still room for further improvement in this work. A detailed study of
processes such as $e^{+}e^{-}\to\text{leptons}$ at Belle II and BESIII
collider experiments could provide a direct probe of $g_{all}$ couplings and
yield more stringent bounds on them. This could potentially narrow down the
parameter space further or even provide the complete coverage of the parameter
space relevant to $\left(g-2\right)_{\mu}$.

In addition to that, we assumed that studied scalars and pseudoscalars have
100$\%$ branching ratio into visible states. However, in many theoretical
models such particles play the role of mediators between Standard Model
particles and the dark sector. Clearly, the contribution of light dark matter
particles with sub-GeV masses could modify the established constraints if the
pair production threshold is exceeded. On the other hand, incorporating such
particles produces a lot of uncertainties, which greatly complicates the
analysis of experimental bounds.

## Acknowledgments

This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation), in part through the Research Unit [Photon-photon
interactions in the Standard Model and beyond, Projektnummer 458854507 - FOR
5327], and in part through the Cluster of Excellence [Precision Physics,
Fundamental Interactions, and Structure of Matter] (PRISMA+ EXC 2118/1) within
the German Excellence Strategy (Project ID 39083149).

##  Appendix A Finite parts of Barr-Zee loop contributions

The finite part of the Barr-Zee diagram shown in Fig. 2 can be represented in
terms of the function

$\Delta=m_{l}^{2}y^{2}+m_{a}^{2}\left(1-x\right)-i\varepsilon.$ (22)

ALP contribution calculated with the Lagrangian (1) in case of the internal
photon line is given by

$\begin{split}&\text{Finite}\left[\Delta
a^{BZ}_{l}\right]=-\frac{m_{l}g_{all}g_{a\gamma\gamma}}{8\pi^{2}}\\\
&\times\int_{0}^{1}\int_{0}^{x}\biggl{\\{}\left(1+3y\right)\ln{\Delta}+\frac{m_{l}^{2}y^{3}}{\Delta}\biggr{\\}}dy\,dx.\end{split}$
(23)

If the internal line represents a $Z$-boson instead, we need to replace

$\displaystyle
g_{a\gamma\gamma}\to-\frac{4\sin^{2}{\theta_{w}}-1}{4\cos{\theta_{w}}\sin{\theta_{w}}}g_{a\gamma
Z},$ (24) $\displaystyle\Delta\to\Delta+m_{Z}^{2}\left(x-y\right).$ (25)

We also note that the difference between this result and the one which can be
obtained with the reduced Lagrangian (2) is only a small constant term in the
finite part of the Barr-Zee diagram. This term, in principle, can be absorbed
by the redefinition of $\Lambda$.

In case of a scalar particle and the internal photon line we get

$\text{Finite}\left[\Delta
a^{BZ}_{l}\right]=-\frac{m_{l}g_{sll}g_{s\gamma\gamma}}{4\pi^{2}}\int_{0}^{1}\int_{0}^{x}\ln{\Delta}\,dy\,dx.$
(26)

The contribution of a $Z$-boson can be obtained by performing in this formula
substitutions similar to (24) and (25).

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