# Simple obstructions and cone reduction

F. Qu  Northeast Normal University, School of Mathematics and Statistics,
Changchun, Jilin, China<EMAIL_ADDRESS>
###### Abstract.

Let $X$ be a Deligne-Mumford stack locally of finite type over an
algebraically closed field $k$ of characteristic zero. We show that the
intrinsic normal cone $C_{X}$ of $X$ is supported in the subcone
$\mathbb{V}(\Omega_{X}[-1])$ ($h^{1}/h^{0}((\Omega^{1}_{X})^{\vee})$) of its
intrinsic normal sheaf $N_{X}$. This leads to an alternative proof of cone
reduction by cosections for $C_{X}$. We also discuss vanishing of simple
obstructions under the Buchweitz-Flenner semiregularity map for sheaves.

###### 1991 Mathematics Subject Classification:

14C17,14N35

##  1\. Introduction

For a closed immersion between schemes $f\colon X\to Y$, we have the normal
sheaf $N_{f}$ and normal cone $C_{f}$ defined by

$N_{f}=\mathrm{Spec}_{\mathcal{O}_{Y}}\operatorname{Sym}I/I^{2},C_{f}=\mathrm{Spec}_{\mathcal{O}_{Y}}\oplus_{d\geq
0}I^{d}/I^{d+1},$

where $I$ denotes the ideal sheaf of $X$ in $Y$. As functors of closed
immersions between schemes locally of finite type over a base field $k$, the
functors $N$ and $C$ are extended to morphisms between algebraic stacks ([Kr1,
Kr2, KKP, AP]), and are involved in the construction of virtual fundamental
classes [LT, BeFa]. See [BH] and the references therein for extensions beyond
higher stacks.

For a DM stack $X$ locally of finite type over $k$, the normal sheaf and
normal cone of the map $X\to\mathrm{Spec}k$ are the intrinsic normal sheaf and
intrinsic normal cone $N_{X}$ and $C_{X}$ introduced in [BeFa]. And it is
proved that they carry information about deformations of maps from
$\mathrm{Spec}k$ to $X$. More precisely, for any $k$-point

$\xi\colon\mathrm{Spec}k\to X,$

the coarse moduli space of the pullback of $N_{X}$ along $\xi$ is
$\operatorname{Ext}^{1}_{k}(\xi^{*}\mathbb{L}_{X},k)$, where $\mathbb{L}_{X}$
denotes the cotangent complex of $X$. The vector space
$\operatorname{Ext}^{1}_{k}(\xi^{*}\mathbb{L}_{X},k)$ is the universal
obstruction space for $\xi$ ([BeFa, Lemma 4.6]), and its subspace of
curvilinear obstructions is the coarse moduli space of the pullback of $C_{X}$
along $\xi$ ([BeFa, Lemma 4.7]). Globally, universality of $N_{X}$ builds into
the definition of an obstruction theory for $X$ and it is equivalent to a
closed embedding of $N_{X}$ into a cone stack ([BeFa, Theorem 4.5]), while
curvilinearity of $C_{X}$ lacks transparent description.

When $\operatorname{char}k=0$, curvilinear extensions are simple
extensions.(See Section 2.3.) The key observation in this paper is that,
simple obstructions, i.e., obstructions for simple extensions, are contained
in the subspace

$\operatorname{Ext}^{1}_{k}(\mathbf{L}\xi^{*}\Omega^{1}_{X},k)$

of $\operatorname{Ext}^{1}(\xi^{*}\mathbb{L}_{X},k)$, and we are led to
consider the subcone

$\mathbb{V}(\Omega_{X}^{1}[-1]),\text{or}\
h^{1}/h^{0}((\Omega^{1}_{X})^{\vee}),$

of $N_{X}$.(See Section 2.2 for notation.) If $k$ is also algebraically
closed, then we show $C_{X}$ is supported in $\mathbb{V}(\Omega_{X}^{1}[-1])$
(Theorem 3.2). It is proved using Lemma 2.4 which deduce global information
about support of closed substacks from their pointwise coarse sheaves. Here
the support of a closed substack $C$ is the reduced substack
$C_{\mathrm{red}}$ associated to it.

Cone reduction for a subcone of $N_{X}$ concerns its support. It is easy to
see cone reduction by cosections for $\mathbb{V}(\Omega_{X}^{1}[-1])$. A
cosection

$\sigma\colon N_{X}\to A^{1}_{k}$

is a map between cones and is induced by a map
$\mathcal{O}_{X}[1]\to\mathbb{L}_{X}^{\geq-1}$ from $\mathcal{O}_{X}[1]$ to
the truncated cotangent complex of $X$ in $\mathrm{D}_{\mathrm{qcoh}}(X)$.
Since there is no nonzero map from $\mathcal{O}_{X}[1]$ to
$\Omega^{1}_{X}$[0], we conclude $\mathbb{V}(\Omega_{X}^{1}[-1])$ is supported
in the kernel $\sigma^{-1}\\{0\\}$. This observation improves [KL, Corollary
4.5]. Note that we consider maps
$\mathcal{O}_{X}[1]\to\mathbb{L}_{X}^{\geq-1}$ without additional obstruction
theories for $X$. More generally, for any
$\mathcal{G}[1]\to\mathbb{L}_{X}^{\geq-1}$, where $\mathcal{G}$ is a coherent
sheaf on $X$, the obvious vanishing of $\mathcal{G}[1]\to\Omega_{X}^{1}[0]$
can be viewed as a global version of [BuFl, Proposition 6.13(1)].
Semiregularity can also be treated globally from this perspective as in [Ch,
Section 5] and [Sc, Section 3].

We also deduce a relative version. For a map $f\colon X\to Y$ between DM
stacks locally of finite type over $k$, and $Y$ smooth, we look for subcones
of $N_{f}$ that contain $C_{f}$. The cone $\mathbb{V}(\Omega_{f}[-1])$ is too
small for closed immersions. As $C_{X}$ is given by $[C_{f}/f^{*}T_{Y}]$, it
follows from Theorem 3.2 that $C_{f}$ is supported in

$\mathbb{V}(\\{f^{*}{\Omega_{Y}^{1}}\to\Omega_{X}^{1}\\}[-1])$

(Proposition 3.4) . Here $\\{f^{*}\Omega_{Y}^{1}\to\Omega_{X}^{1}\\}$ is the
two-term complex with $f^{*}\Omega_{Y}^{1}$ in degree $-1$ and
$\Omega_{X}^{1}$ degree 0.

Cone reduction by cosections can also be treated in the framework of derived
algebraic geometry and derived differential geometry. See [AKLPR, Section 6],
[BKP, Appendix A.1], and [Sa, Proposition 5.1].

Recent interests in cone reduction come from the work of Oh-Thomas ([OT]). In
their approach to construct algebraic virtual classes on moduli spaces of
sheaves on complex Calabi-Yau 4 folds, it is of interest to show a normal cone
inside a quadratic bundle is isotropic or supported in some maximal isotropic
subbundle. The isotropic condition in [OT] is established using the Darboux
theorem ([BBJ]) for shifted symplectic structures ([PTVV]). The local
counterparts to these global questions concern the subspace of curvilinear
obstructions inside the standard obstruction space with its quadratic form
induced by Serre Duality, and imply global results for the reduced normal
cone. However, these local questions are not as trivial to establish as those
in this paper and seems out of reach for us.

The paper is organized as follows. Preliminary results are collected in
Section 2, from which the the main results in Section 3 follow rather easily.
In Section 4, we globalize the semiregularity map([BuFl]) for sheaves to a map
in the derived category, and show vanishing of obstructions for simple
extensions.

### Notation and convention

We work over a base field $k$. Denote $\mathbf{Sets}$ the category of sets,
and $\mathbf{Grpd}$ the category of groupoids. Denote $\mathbf{DM}_{k}$ the
category of Deligne-Mumford stacks locally of finite type over $k$, and
$\mathbf{Sch}_{k}$ its full subcategory of schemes in $\mathbf{DM}_{k}$. For
$X\in\mathbf{DM}_{k}$ and a ring $R$ of finite type over $k$, $X(R)$ denotes
the set of morphisms $\mathrm{Spec}R\to X$ in $\mathbf{DM}_{k}$. Functors for
sheaves and complexes are derived. For sheaves, we will emphasis left or right
derived functor by adding $\mathbf{L}$ or $\mathbf{R}$.

##  2\. Preliminaries

###  2.1. Coarse sheaf and support

####  2.1.1. $\pi_{0}$

Let $\pi_{0}\colon\mathbf{Grpd}\to\mathbf{Sets}$ be the functor that sends a
groupoid to its isomorphic classes of objects. It is the left adjoint of the
inclusion $\mathbf{Sets}\to\mathbf{Grpd}$.

We view $\mathbf{Sch}_{k}$ as a site with the big étale topology. For a stack

$Y\colon\mathbf{Sch}_{k}\to\mathbf{Grpd},$

denote $\pi_{0}(Y)$ the sheafification of the composition

$\pi_{0}\circ Y\colon\mathbf{Sch}_{k}\to\mathbf{Sets}.$

Clearly, $\pi_{0}(Y)$ is functorial in $Y$ and there is a natural
transformation

$Y\to\pi_{0}(Y).$

Recall a map $M\to N$ between stacks over $\mathbf{Sch}_{k}$ is fully faithful
if the relative diagonal $M\to M\times_{N}M$ is an isomorphism. Equivalently,
for any $T\in\mathbf{Sch}_{k}$, $M(T)\to N(T)$ is fully faithful in
$\mathbf{Grpd}$.

######  Lemma 2.1.

Let $M\to N$ be a fully faithful map of stacks, then

${M}$${N}$${\pi_{0}(M)}$${\pi_{0}(N)}$

is a cartesian diagram in the $(2,1)$-category of stacks over
$\mathbf{Sch}_{k}$.

###### Proof.

If

$G\to H$

is a fully faithful map in $\mathbf{Grpd}$, then

${G}$${H}$${\pi_{0}(G)}$${\pi_{0}(H)}$

is (homotopy) cartesian in $\mathbf{Grpd}$. It follows that

${M}$${N}$${\pi_{0}\circ M}$${\pi_{0}\circ N}$

is cartesian, and we see the lemma holds by taking stackification. ∎

######  Remark 2.2.

If $g\colon Z\to Y$ is a closed immersions between algebraic stacks, then $g$
is fully faithful. From the lemma we see that $\pi_{0}(Z)\to\pi_{0}(Y)$ is
representable by closed immersions. In particular, $\pi_{0}(Z)\to\pi_{0}(Y)$
is fully faithful, and we view $\pi_{0}(Z)$ as a subsheaf of $\pi_{0}(Y)$.

######  Remark 2.3.

For the closed immersion $C_{X}\to N_{X}$, the lemma is proved during the
proof of [Be, Proposition 2.2].

####  2.1.2. Support of closed substacks

Let $N$ be an algebraic stack over $X\in\mathbf{DM}_{k}$, and $C,D$ closed
substacks of $N$. For any point $f\colon T\to X$ in $\mathbf{DM}_{k}$, we have
subsheaves $\pi_{0}(f^{*}C)$ and $\pi_{0}(f^{*}D)$ of $\pi_{0}(f^{*}N)$. Here
$f^{*}C,f^{*}D$ are the fiber products $T\times_{X}C,T\times_{X}D$.

The following lemma is a variation on [CL, Proposition 2.1].

######  Lemma 2.4.

Let $k$ be an algebraic closed field. If for any $k$\- point

$\xi\colon\mathrm{Spec}k\to X,$

the sheaf $\pi_{0}(\xi^{*}C)$ is contained in $\pi_{0}(\xi^{*}D)$ as
subsheaves of $\pi_{0}(\xi^{*}N)$, then $C$ is supported in $D$. In other
words, the inclusion map $C_{\mathrm{red}}\to N$ factors through the inclusion
$D\to N$. Here $C_{\mathrm{red}}$ denotes the reduced stack associated to $C$.

###### Proof.

By passing to smooth covers, we can assume $N$ and $X$ are in
$\mathbf{Sch}_{k}$. For schemes in $\mathbf{Sch}_{k}$, closed points are
dense, and $C$ is supported in $D$ if closed points of $C$ belongs to those of
$D$. As $k$ is algebraically closed, closed points are $k$-points, and we only
need to show $C(k)\subset D(k)$.

Any $k$-point of $C$ lies over some closed point $\xi\colon\mathrm{Spec}k\to
X$. By Lemma 2.1, we have cartesian diagrams

${\xi^{*}C}$${\xi^{*}N}$${\pi_{0}(\xi^{*}C)}$${\pi_{0}(\xi^{*}N),}$${\xi^{*}D}$${\xi^{*}N}$${\pi_{0}(\xi^{*}D)}$${\pi_{0}(\xi^{*}N).}$

From the assumption $\pi_{0}(\xi^{*}C)\subset\pi_{0}(\xi^{*}D)$, we see
$\xi^{*}C\subset\xi^{*}D$ as closed subschemes of $\xi^{*}N$, and closed
points of $C$ over $\xi$ belong to $D(k)$.

∎

###  2.2. The functor $h^{1}/h^{0}((-)^{\vee})$

Let $X\in\mathbf{DM}_{k}$. Associated to $E\in\mathrm{D}_{\mathrm{coh}}^{\leq
0}(X)$ is the cone stack

$h^{1}/h^{0}(E^{\vee}).$

(See [BeFa, Proposition 2.4].) If $\tau^{[-1,0]}(E)$ is quasi-isomorphic to a
two-term complex of coherent sheaves

$\mathcal{Q}\to\mathcal{F}$

with $\mathcal{F}$ locally free, then $h^{1}/h^{0}(E^{\vee})$ is the quotient
stack

$[\mathrm{C}(\mathcal{Q})/\mathrm{C}(\mathcal{F})],$

where $\mathrm{C}(-)=\mathrm{Spec}_{\mathcal{O}_{X}}\operatorname{Sym}(-)$.
The map $\mathcal{Q}\to\mathcal{F}$ induces a map between abelian cones
$\mathrm{C}(\mathcal{F})\to\mathrm{C}(\mathcal{Q})$, and as a morphism of
abelian group schemes over $X$, we have a Picard stack
$[\mathrm{C}(\mathcal{Q})/\mathrm{C}(\mathcal{F})]$. It can also be described
as the stack which associates to a $T$-point $f\colon T\to X$ the mapping
space between $f^{*}(E[-1])$ and $\mathcal{O}_{T}$, and is denoted
$\mathbb{V}(E[-1])$ in [AP]. We will also use the $\mathbb{V}$ notation.

######  Remark 2.5.

The $\mathbb{V}$ construction using mapping spaces is more general than
$h^{1}/h^{0}$. For a 3-term complex $F$ in degree $[0,2]$ on an algebraic
stack, $\mathbb{V}(F)$ is in general a Picard 2-stack. The $\mathbb{V}$
notation depends on the convention of authors, it is either $\mathbb{V}(E)$ or
$\mathbb{V}(E^{\vee})$.

When $E$ is perfect, $\pi_{0}(h^{1}/h^{0}(E^{\vee}))$ is give by
$h^{1}(E^{\vee})$. Here $h^{1}(E^{\vee})$ denotes its extension to the big
étale site. (See e.g., [BeFa, Section 2, p. 60] for extensions from a small
site to a big site.)

From its description, it is clear the pullback of $h^{1}/h^{0}(E^{\vee})$
along $g\colon Y\to X$ is $h^{1}/h^{0}((g^{*}E)^{\vee})$. In particular, for
any point $\xi\in X(k)$, the pullback is $h^{1}/h^{0}((\xi^{*}E)^{\vee})$. As
$\tau^{[-1,0]}(\xi^{*}E)$ is quasi-isomorphic to $h^{-1}(\xi^{*}E)[1]\oplus
h^{0}(\xi^{*}E)$ in the derived category of $k$-modules, it is easy to see
$\pi_{0}(h^{1}/h^{0}((g^{*}E)^{\vee}))$ is isomorphic to
$\mathrm{C}(h^{-1}(\xi^{*}E))$, or the $k$-vector space
$\operatorname{Ext}^{1}_{k}(\xi^{*}E,k)$.

###  2.3. Simple extensions and obstructions

A closed immersion $S\to S^{\prime}$ between schemes with ideal $I$ is a
square zero extension if $I^{2}=0$, in which case the ideal sheaf $I$ has an
induced $\mathcal{O}_{S}$-module structure. In the affine case, $S\to
S^{\prime}$ corresponds to a surjection of commutative rings $A^{\prime}\to A$
with square zero kernel.

Isomorphic classes of square zero extensions of $S$ by an
$\mathcal{O}_{S}$-module $J$ can be identified with
$\operatorname{Ext}^{1}_{S}(\mathbb{L}_{S},J)$([Il, III.1.2.3]). For a square
zero extension $S\to S^{\prime}$, the element it determines in
$\operatorname{Ext}^{1}_{S}(\mathbb{L}_{S},J)$ is the map

$\mathbb{L}_{S}\to\mathbb{L}_{S/S^{\prime}}\to\tau^{\geq-1}\mathbb{L}_{S/S^{\prime}}\simeq
J[1].$

The natural map $\mathbb{L}_{S}\to\Omega_{S}^{1}$ induces an inclusion

$\operatorname{Ext}^{1}_{S}(\Omega_{S}^{1},J)\to\operatorname{Ext}^{1}_{S}(\mathbb{L}_{S},J).$

Elements of the subspace $\operatorname{Ext}^{1}_{S}(\Omega_{S}^{1},J)$
correspond to simple extensions. Alternatively, a square zero extension $S\to
S^{\prime}$ is simple, if

$I/I^{2}\to{\Omega_{S^{\prime}}^{1}}_{|_{S}}\to\Omega_{S}\to 0$

is exact on the left. (See e.g., [BuFl, Theorem 9.2 (2)].)

It is easy to see curvilinear extensions

$k\to k[t]/(t^{n+1})\to k[t]/(t^{n}),n\geq 2$

are simple when $\operatorname{char}k=0$.

Let $S\to S^{\prime}$ be a square zero extension in $\mathbf{Sch}_{k}$
classified by some map $e\colon\mathbb{L}_{S}\to J[1]$, and $f\colon S\to X$ a
map in $\mathbf{DM}_{k}$. The obstruction to lift $f$ from $S$ to $S^{\prime}$
lies in $\operatorname{Ext}^{1}_{S}(f^{*}\mathbb{L}_{X},J)$, and is given by
the map

(1) $f^{*}\mathbb{L}_{X}\to\mathbb{L}_{S}\xrightarrow{e}J[1].$

([Il, III.2.2.4])

######  Lemma 2.6.

Let $e$ be a simple extension of $S$ by $J$, then the obstruction to lift
$f\colon S\to X$ along $e$ to $S^{\prime}$ lies in the subspace

$\operatorname{Ext}^{1}_{S}(\mathbf{L}f^{*}\Omega^{1}_{X},J)$

of $\operatorname{Ext}^{1}_{S}(f^{*}\mathbb{L}_{X},J)$.

###### Proof.

The lemma holds because the map
$f^{*}\mathbb{L}_{X}\to\mathbb{L}_{S}\to\Omega_{S}^{1}[0]$ factors through
$\mathbf{L}f^{*}\Omega^{1}_{X}$. ∎

######  Remark 2.7.

Let $\xi\colon\mathrm{Spec}k\to X$ be a $k$-point of $X$. Let $A$ be an
Artinian $k$-algebra with residue field $k$. and $f\colon\mathrm{Spec}A\to X$
a deformation of $\xi$. Apply the lemma to $S=\mathrm{Spec}A$ and $J$ the
residue field of $A$, we see that the obstruction class belongs to

$\operatorname{Ext}^{1}_{A}(\mathbf{L}f^{*}\Omega^{1}_{X},k)\simeq\operatorname{Ext}^{1}_{k}(\mathbf{L}\xi^{*}\Omega^{1}_{X},k).$

In particular, curvilinear obstructions associated to $\xi$ lie in
$\operatorname{Ext}^{1}_{k}(\mathbf{L}\xi^{*}\Omega^{1}_{X},k)$.

##  3\. Cone reduction for $C_{X}$

In this section, we assume the field $k$ has $\operatorname{char}k=0$ and is
algebraically closed.

For a map $f\colon X\to Y$ in $\mathbf{DM}_{k}$ with $Y$ smooth, we have the
map between sheaves of Kähler differentials
$f^{*}{\Omega_{Y}^{1}}\to\Omega_{X}^{1}$, and its cone

$\\{f^{*}{\Omega_{Y}^{1}}\to\Omega_{X}^{1}\\}$

in $\mathrm{D}_{\mathrm{qcoh}}(X)$.

######  Lemma 3.1.

Let $f\colon X\to Y$ be a map in $\mathbf{DM}_{k}$ with $Y$ smooth. Then

$\mathbb{V}(\\{f^{*}{\Omega_{Y}^{1}}\to\Omega_{X}^{1}\\}[-1])$

is a closed subcone of $N_{f}$. In particular, for any $X\in\mathbf{DM}_{k}$,
$\mathbb{V}(\Omega_{X}^{1}[-1])$ is a closed subcone of $N_{X}$.

###### Proof.

As $Y$ is smooth, $\mathbb{L}_{Y}\simeq\Omega_{Y}^{1}$, and
$\mathbf{L}f^{*}{\Omega_{Y}^{1}}\simeq f^{*}\Omega_{Y}^{1}$.

The map $\mathbb{L}_{X}\to\Omega_{X}^{1}[0]=\tau^{\geq 0}\mathbb{L}_{X}$
induces a map between distinguished triangles

${\mathbf{L}f^{*}{\Omega_{Y}^{1}}}$${\mathbb{L}_{X}}$${\mathbb{L}_{f}}$${f^{*}{\Omega_{Y}^{1}}}$${\Omega_{X}^{1}}$${\\{f^{*}{\Omega_{Y}^{1}}\to\Omega_{X}^{1}\\},}$$\scriptstyle{\simeq}$

and we see that the cone of

$\mathbb{L}_{f}\to\\{f^{*}{\Omega_{Y}^{1}}\to\Omega_{X}^{1}\\}$

is isomorphic to $\tau^{\leq-1}\mathbb{L}_{X}[1]$ which lies in
$\mathrm{D}_{\mathrm{qcoh}}^{\leq-2}(X)$. Now we apply [BeFa, Proposition 2.6]
to finish the proof.

∎

######  Theorem 3.2.

Let $X$ be a DM stack locally of finite type over $k$, then $C_{X}$ is
supported in $\mathbb{V}(\Omega_{X}^{1}[-1])$.

###### Proof.

Consider the subcones $C_{X}$ and $\mathbb{V}(\Omega_{X}^{1}[-1])$ of $N_{X}$.
By [BeFa, Lemma 4.7] and Remark 2.7, we can apply Lemma 2.4 to conclude the
proof. ∎

######  Remark 3.3.

Let $\mathcal{X}$ be an algebraic stack locally of finite type over $k$, and
$U\to\mathcal{X}$ a smooth cover of $\mathcal{X}$ by $U\in\mathbf{Sch}_{k}$.
Then the normal cone $C_{\mathcal{X}}$ and normal sheaf
$N_{\mathcal{X}}$([AP]) are quotients of $C_{U}$ and $N_{U}$ by the group
stack $BT_{U/\mathcal{X}}$ respectively. In particular, we have a cartesian
diagram

${C_{U}}$${N_{U}}$${C_{\mathcal{X}}}$${N_{\mathcal{X}},}$

and it follows from the theorem for $U$ that $C_{\mathcal{X}}$ is supported in
$\mathbb{V}(\tau^{\geq 0}\mathbb{L}_{\mathcal{X}}[-1])$.

######  Proposition 3.4.

Let $f\colon X\to Y$ be a map in $\mathbf{DM}_{k}$ with $Y$ smooth. Then
$C_{f}$ is supported in
$\mathbb{V}(\\{f^{*}{\Omega_{Y}^{1}}\to\Omega_{X}^{1}\\}[-1])$.

###### Proof.

As $Y$ is smooth, we have an exact sequence of cones

$f^{*}T_{Y}\to C_{f}\to C_{X}$

by [BeFa, Proposition 3.1].

The distinguished triangle

$\Omega_{X}^{1}\to\\{f^{*}{\Omega_{Y}^{1}}\to\Omega_{X}^{1}\\}\to
f^{*}\Omega_{Y}^{1}[1]$

induces an exact sequence of cones

$f^{*}T_{Y}\to\mathbb{V}(\\{f^{*}{\Omega_{Y}^{1}}\to\Omega_{X}^{1}\\}[-1])\to\mathbb{V}(\Omega_{X}^{1}[-1])$

by[BeFa, Proposition 2.7].

Note that both sequences embed into the exact sequence

$f^{*}T_{Y}\to N_{f}\to N_{X}$

induced by $\mathbb{L}_{X}\to\mathbb{L}_{f}\to\mathbf{L}f^{*}\Omega_{Y}[1]$.
As $C_{X}$ is supported in $\mathbb{V}(\Omega_{X}^{1}[-1])$, we see $C_{f}$ is
supported in $\mathbb{V}(\\{f^{*}{\Omega_{Y}^{1}}\to\Omega_{X}^{1}\\}[-1])$

∎

######  Remark 3.5.

A more direct proof of Theorem 3.2 and Proposition 3.4 is to show for a closed
immersion $f\colon X\to Y$ in $\mathbf{Sch}_{k}$ with $Y$ smooth, the normal
cone $C_{f}$ is supported in the abelian cone $\mathrm{C}(K)$, where $K$ is
the kernel of the map $f^{*}\Omega_{Y}^{1}\to\Omega_{X}^{1}$. Unfortunately,
we don’t know how to prove this inside $\mathbf{Sch}_{k}$.

######  Corollary 3.6.

Let $f\colon X\to Y$ be a map in $\mathbf{DM}_{k}$ with $Y$ smooth. Let
$\mathcal{O}_{X}[1]\to\mathbb{L}^{\geq-1}_{f}$ be a cosection inducing

$\sigma\colon N_{f}\to\mathbb{A}_{k}^{1},$

If $\mathcal{O}_{X}[1]\to\mathbb{L}_{f}^{\geq-1}$ lifts to a cosection
$\mathcal{O}_{X}[1]\to\mathbb{L}_{X}^{\geq-1}$, or equivalently the cosection
$\sigma$ descends to $N_{X}$, then $C_{f}$ is supported in $\sigma^{-1}(0)$.

###### Proof.

Note that the map
$\mathcal{O}_{X}[1]\to\mathbb{L}_{f}^{\geq-1}\to\\{f^{*}{\Omega_{Y}^{1}}\to\Omega_{X}^{1}\\}$
is zero, as it factors through
$\mathcal{O}_{X}[1]\to\mathbb{L}_{X}^{\geq-1}\to\Omega_{X}^{1}$ in the diagram

${\mathcal{O}_{X}[1]}$${\mathbb{L}_{X}^{\geq-1}}$${\mathbb{L}_{f}^{\geq-1}}$${\Omega^{1}_{X}}$${\\{f^{*}{\Omega_{Y}^{1}}\to\Omega_{X}^{1}\\}.}$

∎

######  Remark 3.7.

The strategy to relate $C_{f}$ to $C_{X}$ in Proposition 3.4 and the condition
that $\sigma$ descends in the corollary appeared in [MPT, Section 3.5] for the
case of stable pairs moduli spaces.

##  4\. semiregularity map for sheaves

### 4.1.

Let $M$ be a DM stack in $\mathbf{DM}_{k}$ with an obstruction theory
$F\to\mathbb{L}_{M}$, i.e., a map in $\mathrm{D}_{\mathrm{coh}}(M)$ whose cone
lies in $\mathrm{D}_{\mathrm{coh}}^{\leq-2}(M)$. Let $\rho\colon G\to F$ be a
map in $\mathrm{D}_{\mathrm{coh}}(M)$, it functions as a global semiregularity
map. For any map $f\colon S\to M$ and square zero extension $\mathbb{L}_{S}\to
J[1]$ we obtain the map

$\operatorname{Ext}^{1}_{S}(f^{*}F,J)\xrightarrow{f^{*}\rho\circ}\operatorname{Ext}^{1}_{S}(f^{*}G,J),$

which we view as a semiregularity type map.

### 4.2.

Now we move on to the semiregularity map for sheaves in [BuFl], we will
globalize the map as above. For a discussion of works related to semiregular
maps, see [Le, Chapter 3].

Let $\pi\colon X\to B$ be a flat family of smooth projective varieties over an
algebraically closed field $k$ with $\operatorname{char}k=0$, and $\mu\colon
M\to B$ a moduli scheme of stable sheaves on the fibers of $\pi$. (See e.g.,
[HL, Chapter 4].) Consider the cartesian digram,

${M\times_{B}X}$${X}$${M}$${B,}$$\scriptstyle{q}$$\scriptstyle{p}$$\scriptstyle{\pi}$$\scriptstyle{\mu}$

where we used $p,q$ to denote projection maps.

We have a map

(2) $(p_{*}\mathop{\mathcal{H}om}\nolimits(E,E)[1])^{\vee}\to\mathbb{L}_{\mu}$

obtained from the Atiyah class of a twisted universal sheaf $E$ on
$M\times_{B}X$ relative to $q$. (See e.g., [Ku, Section 1.4].) And an
obstruction theory can be constructed from this map ([HT, Section 4]).

Next, we will construct the map in $\mathrm{D}_{\mathrm{coh}}(M)$ with target
$(p_{*}\mathop{\mathcal{H}om}\nolimits(E,E)[1])^{\vee}$, or dually, a map with
source $p_{*}\mathop{\mathcal{H}om}\nolimits(E,E)[1]$, that induces the local
semiregularity map up to $\pm$ signs. For each integer $n\geq 0$, consider the
composition

$E\otimes\wedge^{n}\mathbb{L}_{p}[n]\xrightarrow{\mathrm{At}(E)\otimes\mathrm{id}}E\otimes\mathbb{L}_{p}[1]\otimes\wedge^{n}\mathbb{L}_{p}[n]\xrightarrow{\mathrm{id}\otimes\wedge}E\otimes\wedge^{n+1}\mathbb{L}_{p}[n+1],$

where $\mathrm{At}(E)\colon E\to E\otimes\mathbb{L}_{p}[1]$ denotes the Atiyah
class of $E$ relative to $p$. It induces a map

$\mathrm{At}(E)\circ\colon\mathop{\mathcal{H}om}\nolimits(E,E\otimes\wedge^{n}\mathbb{L}_{p}[n])\to\mathop{\mathcal{H}om}\nolimits(E,E\otimes\wedge^{n+1}\mathbb{L}_{p}[n+1]).$

These maps determine

$(\mathrm{At}(E)\circ)^{n}[1]\colon\mathop{\mathcal{H}om}\nolimits(E,E)[1]\to\mathop{\mathcal{H}om}\nolimits(E,E\otimes\wedge^{n}\mathbb{L}_{p}[n])[1].$

Composing with the trace map

$\mathop{\rm
tr}\nolimits\colon\mathop{\mathcal{H}om}\nolimits(E,E\otimes\wedge^{n}\mathbb{L}_{p}[n])[1]\simeq\mathop{\mathcal{H}om}\nolimits(E,E)\otimes\wedge^{n}\mathbb{L}_{p}[n+1]\xrightarrow{\mathop{\rm
tr}\nolimits_{E}\otimes\mathrm{id}}\wedge^{n}\mathbb{L}_{p}[n+1]$

and pushforward alone p, we obtain the map

(3) $p_{*}\mathop{\mathcal{H}om}\nolimits(E,E)[1]\to
p_{*}\wedge^{n}\mathbb{L}_{p}[n+1].$

The product of these maps over $n$

$p_{*}\mathop{\mathcal{H}om}\nolimits(E,E)[1]\to\prod_{n\geq
1}p_{*}\wedge^{n}\mathbb{L}_{p}[n+1]$

in $\mathrm{D}_{\mathrm{coh}}(M)$ is dual to the desired global semiregularity
map up to $\pm$ signs on each factor indexed by $n$.

As $\mathbb{L}_{p}\simeq q^{*}\mathbb{L}_{\pi}\simeq q^{*}\Omega_{\pi}^{1}$,
we see

$p_{*}\wedge^{n}\mathbb{L}_{p}[n+1]\simeq
p_{*}q^{*}\Omega^{n}_{\pi}[n+1]\simeq\mu^{*}\mathbf{R}\pi_{*}\Omega_{\pi}^{n}[n+1].$

Note that

$\mathbf{R}\pi_{*}\Omega_{\pi}^{n}\simeq\bigoplus_{j\geq
0}\mathbf{R}^{j}\pi_{*}\Omega_{\pi}^{n}[-j]$

is a direct sum of shifted locally free coherent sheaves by [De, Theorem 6.1,
Proposition 5.5(i)].

Vanishing of simple extensions under the semiregularity map was proved in
[BuFl], an alternative proof for smooth projective families follows from the
lemma below.

######  Lemma 4.1.

Let $M$ be a DM stack, $m$ an integer, $\mathcal{G}$ a locally free coherent
sheaf on $M$ with a map $\phi\colon\mathcal{G}[m]\to\mathbb{L}_{M}$. Let
$f\colon\mathrm{Spec}A\to M$ be a map. If $m\neq 0$, then the composition

$f^{*}\mathcal{G}[m]\xrightarrow{f^{*}\phi}f^{*}\mathbb{L}_{M}\to\mathbb{L}_{A}\to\Omega_{A}^{1}$

is zero. It follows that for any $A$-module $J$ and map
$e\colon\Omega_{A}^{1}\to J[1]$ in
$\mathrm{D}_{\mathrm{qcoh}}(\mathrm{Spec}A)$ the composition

$f^{*}\mathcal{G}[m]\xrightarrow{f^{*}\phi}f^{*}\mathbb{L}_{M}\to\mathbb{L}_{A}\to\Omega_{A}^{1}\xrightarrow{e}J[1]$

is zero.

###### Proof.

As $f^{*}\mathcal{G}$ is a locally free coherent sheaf on $\mathrm{Spec}A$,

$\hom_{\mathrm{D}_{\mathrm{qcoh}}(\mathrm{Spec}A)}(f^{*}\mathcal{G}[m],\Omega_{A}^{1})=\operatorname{Ext}^{-m}_{A}(f^{*}\mathcal{G},\Omega_{A}^{1})=0$

if $m\neq 0$. For $m=0$,

$\operatorname{Ext}^{-m}_{A}(f^{*}\mathcal{G},J[1])=\operatorname{Ext}^{1}_{A}(f^{*}\mathcal{G},J)=0.$

∎

## References

  * [AKLPR] D. Aranha, A. A. Khan, A. Latyntsev, H. Park, C. Ravi, Localization theorems for algebraic stacks, arXiv:2207.01652v1.
  * [AP] D. Aranha, P. Pstrągowski, The intrinsic normal cone for Artin stacks, Ann. Inst. Fourier, 74 (2024), no. 1, 71–120.
  * [BKP] Y. Bae, M. Kool, H. Park, Counting surfaces on Calabi-Yau 4-folds I: foundations, arXiv:2208.09474v1.
  * [Be] K. Behrend, Donaldson-Thomas type invariants via microlocal geometry, Ann. of Math. (2) 170 (2009), no. 3, 1307–1338.
  * [BeFa] K. Behrend, B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), 45–88.
  * [BH] O. Ben-Bassat, J. Hekking, Blow-ups and normal bundles in connective and nonconnective derived geometries, arXiv:2303.11990v2.
  * [BBJ] C. Brav, V. Bussi, D. Joyce, A Darboux theorem for derived schemes with shifted symplectic structure, J. Amer. Math. Soc. 32 (2019), no. 2, 399–443.
  * [BuFl] R.-O. Buchweitz, H. Flenner, A semiregularity map for modules and applications to deformations, Compos. Math. 137 (2003) 13–210.
  * [Ch] H.-L. Chang, Derived Kodaira Spencer map, cosection lemma, and semiregularity, arXiv:0808.0988v2.
  * [CL] H.-L. Chang, J. Li, Semi-perfect obstruction theory and Donaldson-Thomas invariants of derived objects, Comm. Anal. Geom. 19 (2011), no. 4, 807–830.
  * [De] P. Deligne, Théorème de Lefschetz et critères de dégénérescence de suites spectrales, Publ. Math. Inst. Hautes Études Sci. 35 (1968),107-126.
  * [HL] D. Huybrechts, M. Lehn, The geometry of moduli spaces of sheaves, 2nd ed. Cambridge Mathematical Library, Cambridge University Press, Cambridge (2010).
  * [HT] D. Huybrechts, R. P. Thomas, Deformation-obstruction theory for complexes via Atiyah and Kodaira-Spencer classes, Math. Ann. 346 (2010), no. 3, 545–569.
  * [Il] L. Illusie, Complexe cotangent et déformations I, Lecture Notes in Mathematics. 239, Springer-Verlag, 1971.
  * [KL] Y.-H. Kiem, J. Li, Localizing virtual cycles by cosections, J. Amer. Math. Soc. 26 (2013), no. 4, 1025–1050.
  * [KKP] B. Kim, A. Kresch, T. Pantev, Functoriality in intersection theory and a conjecture of Cox, Katz, and Lee, J. Pure Appl. Algebra 179 (2003), no. 1-2, 127–136.
  * [Kr1] A. Kresch, Canonical rational equivalence of intersections of divisors, Invent. Math. 136 (1999), no. 3, 483–496.
  * [Kr2] A. Kresch, Cycle groups for Artin stacks, Invent. Math. 138 (1999), no. 3, 495–536.
  * [Ku] N. Kuhn, The Atiyah class on algebraic stacks, arXiv:2306.13934v1.
  * [Le] E. Lepri, $L_{\infty}$ morphisms and semiregularity, PhD Thesis, Università di Roma La Sapienza, 2023.
  * [LT] J. Li, G. Tian, Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998), no. 1,119–174.
  * [MPT] D. Maulik, R. Pandharipande, R. P. Thomas, Curves on K3 surfaces and modular forms, with an appendix by A. Pixton, J. Topol. 3 (2010), no. 4, 937–996.
  * [OT] J. Oh, R. P. Thomas, Counting sheaves on Calabi-Yau 4-folds I, Duke Math. J. 172 (2023), no. 7, 1333–1409.
  * [PTVV] T. Pantev, B. Toën, M. Vaquié, G.Vezzosi, Shifted symplectic structures, Publ. Math. Inst. Hautes Études Sci. 117 (2013), 271–328.
  * [Sa] M. Savvas, Cosection localization and vanishing for virtual fundamental classes of d-manifolds, Adv. Math. 398, (2022), 108232.
  * [Sc] T. Schürg, More reduced obstruction theory, Math. Res. Lett. 22 (2015), no. 1, 203–217.


# Resonant leptogenesis in minimal $U(1)_{X}$ extensions of the Standard Model

Arindam Das<EMAIL_ADDRESS>Institute for the Advancement of
Higher Education, Hokkaido University, Sapporo 060-0817, Japan  Department of
Physics, Hokkaido University, Sapporo 060-0810, Japan  Yuta Orikasa
<EMAIL_ADDRESS>Institute of Experimental and Applied Physics, Czech
Technical University in Prague, Husova 240/5, 110 00 Prague 1, Czech Republic

###### Abstract

We investigate a general $U(1)_{X}$ scenario where we introduce three
generations of Standard Model (SM) singlet Right Handed Neutrinos (RHNs) to
generate the light neutrino mass through the seesaw mechanism after the
breaking of $U(1)_{X}$ and electroweak symmetries. In addition to that, a
general $U(1)_{X}$ scenario involves an SM-singlet scalar field and due to the
$U(1)_{X}$ symmetry breaking the mass of a neutral beyond the SM (BSM) gauge
boson $Z^{\prime}$ is evolved. The RHNs, being charged under $U(1)_{X}$
scenario, can explain the origin of observed baryon asymmetry through the
resonant leptogenesis process. Applying observed neutrino oscillation data we
study $Z^{\prime}$ and BSM scalar induced processes to reproduce the observed
baryon asymmetry. Hence we estimate bounds on the $U(1)_{X}$ gauge coupling
$(g_{X})$ and the mass of the $Z^{\prime}$ $(M_{Z^{\prime}})$ for different
$U(1)_{X}$ charges and benchmark masses of RHN and SM-singlet scalar. Finally
we compare our results with limits obtained from the existing limits from LEP-
II and LHC. We find that depending on the $U(1)_{X}$ charges, the masses of
RHNs and SM-singlet scalar resonant leptogenesis could provide stronger limit
on $g_{X}$ for $M_{Z^{\prime}}>5.8$ TeV which could be probed by high energy
scattering experiment in future.

##  I Introduction

The origin of baryon asymmetry of the universe is an unsolved puzzle of modern
cosmology. The WIMP satellite WMAP:2010qai  observed that the ratio of the
baryon minus anti-baryon density $(n_{B-\bar{B}})$ over the entropy density of
the universe $(s)$ to be $Y_{B}=8.7\times 10^{-11}$ at an accuracy of $10\%$
precision level. A strong first order phase transition could be required to
explain the origin of baryon (B) asymmetry induced by the electroweak
baryogenesis Cohen:1993nk ; Funakubo:1996dw ; Trodden:1998ym  within the
Standard Model (SM) framework, however, the observation of the SM Higgs mass
around 125 GeV CMS:2012qbp ; ATLAS:2012yve  does not indicate such a
phenomenon. As a result electroweak baryogenesis is possibly ruled out under
the SM scenario.

In addition to the anomaly associated with the observed baryon asymmetry there
are interesting developments in the lepton sector of the SM which are observed
in the form of tiny neutrino mass, neutrino flavor oscillation and mixing
ParticleDataGroup:2022pth  and they can not be explained within the SM
framework. As a result an approach of stepping beyond the SM (BSM) was
important which evolved in the form of seesaw mechanism Minkowski:1977sc ;
Yanagida:1979as ; Gell-Mann:1979vob ; Mohapatra:1979ia ; Glashow:1979nm  where
the origin of tiny neutrino mass could be explained through the suppression of
a heavy Majorana mass scale. In this case the SM is extended by SM-singlet
heavy Majorana type Right Handed Neutrinos (RHNs) at the intermediate scale
which create lepton number violation by unit 2 through a dimension five
Weinberg operator Weinberg:1979sa .

An attempt to connect the baryon asymmetry of the universe and possibility of
tiny neutrino mass could be established by the seesaw scenario by an
attractive possibility called leptogenesis Fukugita:1986hr . The out of
equilibrium decay of the Majorana type RHNs can generate lepton (L) asymmetry
in the universe which can be converted into baryon asymmetry through the
sphaleron transition violating $(B+L)$ quantum number Manton:1983nd ;
Klinkhamer:1984di . Hence the ratio between baryon and lepton conversion rate
can be $-\frac{28}{79}$ following Khlebnikov:1988sr ; Iso:2010mv ;
Chauhan:2021xus  for three (one) generations of SM fermions (Higgs). In this
context we mention that thermal leptogenesis imparts a lower bound on the
Majorana neutrinos $(M_{N})$ to generate sufficient amount of baryon asymmetry
as $M_{N}>10^{9}$ GeV Plumacher:1996kc ; Buchmuller:2002rq ; Buchmuller:2004nz
. Such heavy Majorana neutrinos can not be observed at the hadron and lepton
colliders.

In order to test heavy Majorana neutrinos at high energy colliders we need to
think possible alternative scenarios which may explain the origin of small
neutrino mass and observed baryon asymmetry simultaneously. General U$(1)$
extensions of the SM are such scenarios where these aspects can be addressed
nicely. In this case, the SM is extended by a general $U(1)_{X}$ gauge group
where three generations of the RHNs are introduced with an SM-singlet BSM
scalar. These RHNs not only help generating the tiny neutrino mass though the
seesaw mechanism after the general $U(1)$ symmetry breaking, but also cancel
the gauge gravity anomalies Oda:2015gna ; Das:2016zue . If general $U(1)$
symmetry breaking scale lies around the TeV scale, the corresponding neutral
gauge boson commonly known as $Z^{\prime}$ and the RHNs can be produced at
high energy colliders Das:2022rbl . In this scenario there is another
interesting aspect which affects the $Z^{\prime}$ searches. After the anomaly
cancellation we find that the left and right handed fermions of the model
interact differently with the $Z^{\prime}$ boson. The general $U(1)$ scenarios
considered in this article are at the TeV scale prohibiting thermal
leptogenesis to occur because in this case the RHNs are lighter than the bound
of $10^{10}$ GeV. The CP asymmetry parameter in this case is proportional to
the modulus squared of the Dirac Yukawa coupling $(Y_{D})$ between the RHNs
and the SM lepton doublet. Due to the smallness of $Y_{D}$, the right amount
of baryon asymmetry of the universe can not be generated. It has been also
found that if two RHNs are almost degenerate in masses, an enhancement
Flanz:1996fb  of the CP asymmetry parameter can take place making the
leptogenesis scenario viable for the TeV scale RHNs. This is called resonant
leptogenesis Pilaftsis:1997jf ; Pilaftsis:2003gt . In this case maximum
attainable enhancement could be achieved if mass difference between the two
generations of the RHNs is $(\mathcal{O}(\Gamma_{N_{i}}))$, the total decay
width $(\Gamma_{N_{i}})$ of either of the generations of the RHNs $(N_{i})$.
Hence in principle by tuning the mass difference between any two generations
of the RHNs, the CP asymmetry could be attained around $\mathcal{O}(1)$,
however, in the presence of general $U(1)$ extensions the lepton asymmetry
through RHN decay is suppressed due to the interaction with the TeV scale
$Z^{\prime}$ gauge boson. As a result such a scenario makes the generation of
the CP asymmetry of the universe non-trivial Blanchet:2009bu .

In this paper we study resonant leptogenesis in general $U(1)$ extensions of
the SM where $U(1)_{X}$ and $U(1)_{q+xu}$ symmetry breaks at the TeV scale.
Solving Boltzmann equations and considering different model parameters we show
the dependance of baryon asymmetry via leptogenesis on neutrino Dirac Yukawa
couplings and TeV scale nearly degenerate heavy Majorana neutrino masses. In
this analysis we involve neutrino oscillation parameters in order to relate
the baryon asymmetry via leptogenesis with realistic data. In this analysis we
also study the role of the Dirac CP phase in the $U_{\rm{PMNS}}$ matrix in
order to explain baryon asymmetry of the universe.

We organize the paper in the following way. In Sec. II we describe the model.
In Sec. III we analyse the resonant leptogenesis scenario. In Sec. IV we apply
neutrino oscillation data to investigate the role of Dirac CP phase to
generate right amount of baryon asymmetry of the universe. We discuss our
results in Sec. IV and conclude it in Sec. V.

##  II General $U(1)$ extensions of Standard Model

General $U(1)$ extensions of the SM involve an SM-singlet scalar $\Phi$ and
three generations of the SM-singlet RHNs. Apart from participating in the
neutrino mass generation mechanism, the RHNs also help to cancel gauge and
mixed gauge-gravity anomalies. The particle content is given in Tab. 1 where
generation independent $U(1)_{X}$ charges $(x_{f},f=\\{q,\leavevmode\nobreak\
u,\leavevmode\nobreak\ d,\leavevmode\nobreak\ \ell,\leavevmode\nobreak\
e,\leavevmode\nobreak\ \nu\\})$ of the particles are given.

| SU(3)c | SU(2)L | U(1)Y | U(1)X | U(1)q+xu
---|---|---|---|---|---
$q_{L}^{i}$ | 3 | 2 | $\frac{1}{6}$ | $x_{q}=\frac{1}{6}x_{H}+\frac{1}{3}x_{\Phi}$ | $\frac{1}{3}$
$u_{R}^{i}$ | 3 | 1 | $\frac{2}{3}$ | $x_{u}=\frac{2}{3}x_{H}+\frac{1}{3}x_{\Phi}$ | $\frac{x}{3}$
$d_{R}^{i}$ | 3 | 1 | $-\frac{1}{3}$ | $x_{d}=-\frac{1}{3}x_{H}+\frac{1}{3}x_{\Phi}$ | $\frac{2-x}{3}$
$\ell_{L}^{i}$ | 1 | 2 | $-\frac{1}{2}$ | $x_{\ell}=-\frac{1}{2}x_{H}-x_{\Phi}$ | $-1$
$e_{R}^{i}$ | 1 | 1 | $-1$ | $x_{e}=-x_{H}-x_{\Phi}$ | $-(\frac{2+x}{3})$
$N_{R}$ | 1 | 1 | $0$ | $x_{\nu}=-x_{\Phi}$ | $\frac{-4+x}{3}$
$H$ | 1 | 2 | $-\frac{1}{2}$ | $\frac{x_{H}}{2}$ | $\frac{1-x}{3}$
$\Phi$ | 1 | 1 | $0$ | $2x_{\Phi}$ | $2(\frac{-4+x}{3})$
Table 1:  Particle content of the general $U(1)$ extensions of the SM where
$i(=1,2,3)$ represents the family index. The quantities $x_{H}$, $x_{\Phi}$
and $x$ are the real parameters.

We write the Yukawa interactions following $\mathcal{G}_{\rm SM}\otimes$
$U(1)_{X}$ gauge symmetry as

${\cal L}^{\rm
Yukawa}=-Y_{u}^{\alpha\beta}\overline{q_{L}^{\alpha}}Hu_{R}^{\beta}-Y_{d}^{\alpha\beta}\overline{q_{L}^{\alpha}}\tilde{H}d_{R}^{\beta}-Y_{e}^{\alpha\beta}\overline{\ell_{L}^{\alpha}}\tilde{H}e_{R}^{\beta}-Y_{D}^{\alpha\beta}\overline{\ell_{L}^{\alpha}}HN_{R}^{\beta}-Y_{N}^{\alpha}\Phi\overline{(N_{R}^{\alpha})^{c}}N_{R}^{\alpha}+{\rm
H.c.}\leavevmode\nobreak\ ,$ (1)

where $\alpha,\beta$ are the generation indices indicating three generations
of the fermions involved in the theory. Here $H$ is the SM Higgs doublet, and
$\tilde{H}=i\tau^{2}H^{*}$ with $\tau^{2}$ being the second Pauli matrix.
Hence using $U(1)_{X}$ neutrality we can write the following conditions as

$\displaystyle-\frac{1}{2}x_{H}$ $\displaystyle=$ $\displaystyle-x_{q}+x_{u}\
=\ x_{q}-x_{d}\ =\ x_{\ell}-x_{e}=\ -x_{\ell}+x_{\nu}\leavevmode\nobreak\
;\leavevmode\nobreak\ 2x_{\Phi}=-2x_{\nu}\leavevmode\nobreak\ .$ (2)

Using the gauge and mixed gauge-gravity anomaly cancellation conditions we can
relate general $U(1)_{X}$ charges of the fermions in the following way

$\displaystyle{\rm U}(1)_{X}\otimes\left[{\rm SU}(3)_{C}\right]^{2}$
$\displaystyle\ :$ $\displaystyle 2x_{q}-x_{u}-x_{d}$ $\displaystyle\ =\ 0,$
$\displaystyle{\rm U}(1)_{X}\otimes\left[{\rm SU}(2)_{L}\right]^{2}$
$\displaystyle\ :$ $\displaystyle 3x_{q}+x_{\ell}$ $\displaystyle\ =\ 0,$
$\displaystyle{\rm U}(1)_{X}\otimes\left[{\rm U}(1)_{Y}\right]^{2}$
$\displaystyle\ :$ $\displaystyle x_{q}-8x_{u}-2x_{d}+3x_{\ell}-6x_{e}$
$\displaystyle\ =\ 0,$ $\displaystyle\left[{\rm
U}(1)_{X}\right]^{2}\otimes{\rm U}(1)_{Y}$ $\displaystyle\ :$
$\displaystyle{x_{q}}^{2}-{2x_{u}}^{2}+{x_{d}}^{2}-{x_{\ell}}^{2}+{x_{e}}^{2}$
$\displaystyle\ =\ 0,$ $\displaystyle\left[{\rm U}(1)_{X}\right]^{3}$
$\displaystyle\ :$
$\displaystyle{6x_{q}}^{3}-{3x_{u}}^{3}-{3x_{d}}^{3}+{2x_{\ell}}^{3}-{x_{\nu}}^{3}-{x_{e}}^{3}$
$\displaystyle\ =\ 0,$ $\displaystyle{\rm U}(1)_{X}\otimes\left[{\rm
grav.}\right]^{2}$ $\displaystyle\ :$ $\displaystyle
6x_{q}-3x_{u}-3x_{d}+2x_{\ell}-x_{\nu}-x_{e}$ $\displaystyle\ =\ 0,$ (3)

respectively. Solving Eqs. (2) and (3) we express the fermionic $U(1)_{X}$
charges in terms of $x_{H}$ and $x_{\Phi}$ and the $U(1)_{X}$ charges of the
SM fermions can be written as linear combination of the $U(1)_{Y}$ and B$-$L
charges. Hence we find that left and right handed fermions are differently
charged under the $U(1)_{X}$ gauge group and as a result they interact
differently with the associated neutral BSM gauge boson $Z^{\prime}$
manifesting a chiral nature of the model. Such a chiral behavior affect any
interaction process mediated by the $Z^{\prime}$ associating the fermions of
this model. In this scenario, if we fix $x_{\Phi}=1$ and $x_{H}=0$ we can
reproduce the B$-$L scenario and with $x_{H}=-2$ we find that the $U(1)_{X}$
charges of the left handed fermions become zero, however, the right handed
fermions have nonzero $U(1)_{X}$ charges. This is called $U(1)_{\rm R}$
scenario. In addition to that there are other properties of this model if we
change $x_{H}$ keeping $x_{\Phi}=1$ fixed as 1. For example, in case of
$x_{H}=-1$, the $U(1)_{X}$ charge of $e_{R}$ vanishes, in case of
$x_{H}=-0.5$, the U$(1)_{X}$ charge of $u_{R}$ vanishes and in case of
$x_{H}=1$, the $U(1)_{X}$ charge of $d_{R}$ vanishes, respectively. However,
for these respective $x_{H}$ values remaining fermions interact with the
$Z^{\prime}$. The $U(1)_{X}$ gauge coupling is considered as $g_{X}$ for this
model. This is a free parameter and it appears as either $g^{\prime}x_{H}$ or
$g^{\prime}x_{\Phi}$ in the kinetic part of the Lagrangian. The renormalizable
scalar potential in this scenario can be written as

$\displaystyle V\ =\
m_{H}^{2}(H^{\dagger}H)+\lambda_{H}(H^{\dagger}H)^{2}+m_{\Phi}^{2}(\Phi^{\dagger}\Phi)+\lambda_{\Phi}(\Phi^{\dagger}\Phi)^{2}+\lambda_{\rm
mix}(H^{\dagger}H)(\Phi^{\dagger}\Phi)\leavevmode\nobreak\ ,$ (4)

where the fields $\Phi$ and $H$ are SM-singlet scalar and SM Higgs doublet,
respectively. These scalar fields can be approximated separately in the
analysis of scalar potential where $\lambda_{\rm mix}$ is very small
Oda:2015gna ; Das:2016zue . After the breaking of $U(1)_{X}$ gauge symmetry
and electroweak symmetry, the scalar vacuum expectation values (VEVs) for the
fields $H$ and $\Phi$ are developed as

$<H>\ =\ \frac{1}{\sqrt{2}}\begin{pmatrix}v+h\\\ 0\end{pmatrix},\quad{\rm
and}\quad<\Phi>\ =\ \frac{v_{\Phi}+\phi}{\sqrt{2}},$ (5)

where $v=246$ GeV is marked as electroweak scale at the potential minimum and
$v_{\Phi}$ is considered as a free parameter. After the $U(1)_{X}$ gauge
symmetry is broken with a limit $v_{\Phi}\gg v$, the mass of $Z^{\prime}$ is
evolved and can be defined as

$M_{Z^{\prime}}=2g_{X}v_{\Phi}x_{\Phi}$ (6)

which is a free parameter in this model. From Eq. (1), we find that the RHNs
interact with $\Phi$ which can generate the Majorana mass for heavy neutrinos
after $U(1)_{X}$ symmetry is broken. After the electroweak symmetry breaking,
the Dirac mass term is generated from the Yukawa coupling involving left
handed lepton and SM Higgs doublets respectively. These mass terms finally
allow seesaw mechanism to explain the origin of tiny neutrino mass and flavor
mixing. We write the Majorana and Dirac mass terms from Eq. 1 as

$m_{N^{\alpha}}\ =\
\frac{Y^{\alpha}_{N}}{\sqrt{2}}v_{\Phi},\,\,\,\,\,m_{D}^{\alpha\beta}\ =\
\frac{Y_{D}^{\alpha\beta}}{\sqrt{2}}v,$ (7)

respectively.

###  $Z^{\prime}$ interactions with the fermions

General $U(1)$ extensions of the SM allow a neutral BSM gauge boson which can
interact with the SM fermions $(f)$. The interaction Lagrangian involves
$U(1)_{X}$ charges manifesting chiral behavior as

$\displaystyle\mathcal{L}^{\rm{int}}=-g_{X}(\overline{f}\gamma_{\mu}q_{f_{L}}P_{L}f+\overline{f}\gamma_{\mu}q_{f_{R}}P_{R}f)Z_{\mu}^{\prime}\leavevmode\nobreak\
,$ (8)

where $P_{L(R)}=(1\pm\gamma_{5})/2$ is the projection operator, $q_{L(R)}$ is
the corresponding general $U(1)$ charge of the left (right) handed fermions
which could be found from Tab. 1. These quantities depend on $x_{H}$ and
$x_{\Phi}$ respectively. Hence we calculate partial decay widths of
$Z^{\prime}$ into charged fermions as

$\displaystyle\Gamma(Z^{\prime}\to\bar{f}f)$
$\displaystyle=N_{C}\frac{g_{X}^{2}M_{Z^{\prime}}}{24\pi}\left[\left(q_{f_{L}}^{2}+q_{f_{R}}^{2}\right)\left(1-\frac{m_{f}^{2}}{M_{Z^{\prime}}^{2}}\right)+6q_{f_{L}}q_{f_{R}}\frac{m_{f}^{2}}{M_{Z^{\prime}}^{2}}\right]\sqrt{1-4\frac{m_{f}^{2}}{M_{Z^{\prime}}^{2}}}\leavevmode\nobreak\
,$ (9)

where $m_{f}$ is the SM fermion mass and $N_{C}=1\leavevmode\nobreak\ (3)$ for
the SM leptons (quarks). The partial decay width of $Z^{\prime}$ into a pair
of light neutrinos for three generations neglecting neutrino masses is given
by

$\displaystyle\Gamma(Z^{\prime}\to\nu\nu)=3\frac{M_{Z^{\prime}}g_{X}^{2}}{24\pi}q_{f_{L}}^{2}\leavevmode\nobreak\
,$ (10)

where $q_{f_{L}}$ is the $U(1)_{X}$ charge of the SM lepton doublet,
$Z^{\prime}$ gauge boson can decay into a pair of heavy Majorana neutrinos and
the partical decay width is given by

$\displaystyle\Gamma(Z^{\prime}\to
N_{R}^{\alpha}N_{R}^{\alpha})=\frac{g_{X}^{2}M_{Z^{\prime}}}{24\pi}x_{\nu}^{2}\left(1-\frac{m_{N_{i}}^{2}}{M_{Z^{\prime}}^{2}}\right)^{\frac{3}{2}}\leavevmode\nobreak\
,$ (11)

where $x_{\nu}$ is the general $U(1)$ charge of the RHNs and $m_{N_{i}}$ is
its mass. The general $U(1)$ charge of the RHNs could be obtained from Tab. 1.

### Heavy neutrino interactions

The neutrino mass in the model is generated by the seesaw mechanism after
$U(1)_{X}$ and electroweak symmetries are broken. Hence the neutrino mass
matrix in this scenario can be written as

$\displaystyle m_{\nu}\ =\ \begin{pmatrix}0&&m_{D}\\\
m_{D}^{T}&&m_{N}\end{pmatrix}.$ (12)

where we consider $M_{N}$ as a diagonal matrix without the loss of generality.
Diagonalizing this mass matrix we obtain the light neutrino mass eigenvalue as

$\displaystyle m_{\nu}\simeq-m_{D}m_{N}^{-1}m_{D}^{T}.$ (13)

As a result we write down light neutrino flavor eigenstate $(\nu_{\alpha})$ in
terms of light $(\nu_{i})$ and heavy $(N_{i})$ mass eigenstates

$\displaystyle\nu_{\alpha}\simeq U_{\alpha i}\nu_{i}+V_{\alpha i}N_{i},$ (14)

where $\alpha$ and $i$ are the generation indices. In this expression
$U_{\alpha i}$ is a $3\times 3$ light neutrino mixing matrix which can be
expressed as $(1-\frac{\epsilon}{2})U_{\rm PMNS}$ with
$\epsilon=V^{\ast}V^{T}$, a non-unitary parameter. We define

$\displaystyle V_{\alpha i}\simeq m_{D}m_{N}^{-1}$ (15)

as the mixing between light and heavy neutrinos. Through the light-heavy
mixing, the SM gauge singlet heavy neutrinos interact with the SM gauge
bosons. Here $U_{\rm PMNS}$ is a $3\times 3$ matrix which diagonalizes the
light neutrino mass matrix as

$\displaystyle U_{\rm PMNS}^{T}\leavevmode\nobreak\
m_{\nu}\leavevmode\nobreak\ U_{\rm PMNS}=diag(m_{1},m_{2},m_{3}).$ (16)

In presence of $\epsilon$ the mixing matrix $U$ becomes non-unitary. Finally
under the influence of light-heavy neutrino mixing the charged-current
interactions can be expressed in terms of neutrino mass eigenstates as

$\displaystyle{\mathcal{L}_{CC}\supset-\frac{g}{\sqrt{2}}W_{\mu}\bar{e}\gamma^{\mu}P_{L}V_{\alpha
i}N_{i}+\rm{h.c}.},$ (17)

where $e$ represents the three generations of the charged leptons. Similarly,
in terms of mass eigenstates, the neutral-current interactions can be written
as

$\displaystyle{\mathcal{L}_{NC}\supset-\frac{g}{2c_{w}}Z_{\mu}\left[\overline{N}_{m}\gamma^{\mu}P_{L}(V^{\dagger}V)_{mi}N_{i}+\left\\{\overline{\nu}_{m}\gamma^{\mu}P_{L}(U^{\dagger}V)_{mi}N_{i}+\rm{h.c.}\right\\}\right],}$
(18)

where $c_{w}\equiv\cos\theta_{w}$ with $\theta_{w}$ being the weak mixing
angle. Finally we observe that heavy neutrinos $(N)$ decay into $\ell W$, $\nu
Z$ and $\nu h$ respectively where $h$ is the SM Higgs boson. For sterile
neutrinos heavier than $W$, $Z$ and $h$, the decays are on-shell, i.e., 2-body
followed by the further decays of the SM bosons. The corresponding partial
decay widths are

$\displaystyle\Gamma(N_{i}\rightarrow\ell_{\alpha}W)$ $\displaystyle=$
$\displaystyle\frac{|V_{\alpha
i}|^{2}}{16\pi}\frac{(m_{N_{i}}^{2}-M_{W}^{2})^{2}(m_{N_{i}}^{2}+2M_{W}^{2})}{m_{N_{i}}^{3}v_{h}^{2}},$
$\displaystyle\Gamma(N_{i}\rightarrow\nu_{\alpha}Z)$ $\displaystyle=$
$\displaystyle\frac{|V_{\alpha
i}|^{2}}{32\pi}\frac{(m_{N_{i}}^{2}-M_{Z}^{2})^{2}(m_{N_{i}}^{2}+2M_{Z}^{2})}{m_{N_{i}}^{3}v_{h}^{2}},$
$\displaystyle\Gamma(N_{i}\rightarrow\nu_{\alpha}h)$ $\displaystyle=$
$\displaystyle\frac{|V_{\alpha
i}|^{2}}{32\pi}\frac{(m_{N_{i}}^{2}-M_{h}^{2})^{2}}{m_{N_{i}}v_{h}^{2}}.$ (19)

##  III Resonant Leptogenesis

Figure 1:  The ratios of the B-L asymmetry with scalar contributions
($Y_{B-L}^{w}$) and without the contribution ($Y_{B-L}^{wo}$) as functions of
the scalar mass. The red solid line represents $m_{N_{1}}=2$ TeV and the blue
dashed line represents $m_{N_{1}}=4$ TeV. The U(1)X gauge coupling and
$Z^{\prime}$ boson mass for different $x_{H}$ fixing $g_{X}=0.1$ and
$M_{Z^{\prime}}=6$ TeV.

The Boltzmann equations that govern the yields of the singlet neutrinos, the
B$-$L charge and a scalar are written as

$\displaystyle\frac{dY_{N_{i}}}{dz}$ $\displaystyle=$
$\displaystyle-\frac{z}{sH(m_{N_{1}})}\left[\left(\frac{Y_{N_{i}}}{Y_{N_{i}}^{eq}}-1\right)\left(\gamma_{D_{i}}+2\gamma_{h,s}+4\gamma_{h,t}\right)+\left(\left[\frac{Y_{N_{i}}}{Y_{N_{i}}^{eq}}\right]^{2}-1\right)\gamma_{Z^{\prime}}\right.$
(20)
$\displaystyle\left.+\left(\left[\frac{Y_{N_{i}}}{Y_{N_{1}}^{eq}}\right]^{2}-\left[\frac{Y_{\Phi}}{Y_{N_{1}}^{eq}}\right]^{2}\right)\left(\gamma_{Z^{\prime},\Phi}+\gamma_{N,\Phi}\right)\right],$
$\displaystyle\frac{dY_{\Phi}}{dz}$ $\displaystyle=$
$\displaystyle-\frac{z}{sH(m_{N_{1}})}\left[\sum_{i=1}^{2}\left(\left[\frac{Y_{\Phi}}{Y_{N_{1}}^{eq}}\right]^{2}-\left[\frac{Y_{N_{i}}}{Y_{N_{1}}^{eq}}\right]^{2}\right)\left(\gamma_{Z^{\prime},\Phi}+\gamma_{N,\Phi}\right)+\left(\left[\frac{Y_{\Phi}}{Y_{N_{1}}^{eq}}\right]^{2}-1\right)\gamma_{Z^{\prime},h}\right],$
(21) $\displaystyle\frac{dY_{B-L}}{dz}$ $\displaystyle=$
$\displaystyle-\frac{z}{sH(m_{N_{1}})}\left[\sum_{i=1}^{2}\left(\frac{1}{2}\frac{Y_{B-L}}{Y_{\ell}^{eq}}+\epsilon_{i}\left(\frac{Y_{N_{i}}}{Y_{N_{i}}^{eq}}-1\right)\right)\gamma_{D_{i}}+\frac{Y_{B-L}}{Y_{\ell}^{eq}}\left(2\left(\gamma_{N}+\gamma_{N,t}+\gamma_{h,t}\right)+\sum_{i=1}^{2}\frac{Y_{N_{i}}}{Y_{N_{i}}^{eq}}\gamma_{h,s}\right)\right],$
(22) $\displaystyle\epsilon_{i}$ $\displaystyle=$ $\displaystyle-\sum_{j\neq
i}\frac{m_{N_{i}}}{m_{N_{j}}}\frac{\Gamma_{j}}{m_{N_{j}}}\left(\frac{V_{j}}{2}+S_{j}\right)\frac{{\rm
Im}\left[\left(y_{D}y_{D}^{\dagger}\right)_{ij}^{2}\right]}{\left(y_{D}y_{D}^{\dagger}\right)_{ii}\left(y_{D}y_{D}^{\dagger}\right)_{jj}},$
(23) $\displaystyle V_{j}$ $\displaystyle=$ $\displaystyle
2\frac{m_{N_{j}}^{2}}{m_{N_{i}}^{2}}\left[\left(1+\frac{m_{N_{j}}^{2}}{m_{N_{i}}^{2}}\right)\log\left(1+\frac{m_{N_{i}}^{2}}{m_{N_{j}}^{2}}\right)-1\right]$
(24) $\displaystyle S_{j}$ $\displaystyle=$
$\displaystyle\frac{m_{N_{j}}^{2}\Delta M_{ij}^{2}}{\left(\Delta
M_{ij}^{2}\right)^{2}+m_{N_{i}}^{2}\Gamma_{j}^{2}},\ \Delta M_{ij}^{2}\equiv
m_{N_{j}}^{2}-m_{N_{i}}^{2},$ (25)

and $\gamma$’s are given by

$\displaystyle\gamma_{a}=\frac{m_{N}}{64\pi^{4}z}\int
ds\hat{\sigma}_{a}(s)\sqrt{s}K_{1}\left(\frac{z\sqrt{s}}{m_{N}}\right).$ (26)

$\hat{\sigma}_{a}$’s are reduced cross sections listed in Appendix. A.

We discuss the contribution of a scalar boson. We assume that the scalar
mixing is very small and neglect its effects. The scalar boson couples to the
$Z^{\prime}$ boson and right-handed neutrinos. For the $Z^{\prime}$ exchange
process, the scalar boson is in thermal equilibrium and the decoupling
temperature is similar to that of the right-handed neutrino. Therefore the
deviation of the scalar density from thermal equilibrium changes the baryon
asymmetry in our universe. Fig.1 shows the ratio of the B$-$L asymmetry with a
scalar contribution and without a scalar contribution. When the scalar boson
mass is less than 1 TeV, the scalar contribution is very large. We discuss the
CP-asymmetry parameter. We consider resonant leptogenesis and two neutrino
masses are degenerated:

$\displaystyle\Delta M_{12}^{2}=m_{N_{1}}\Gamma_{2}.$ (27)

We assume that the third generation of the right-handed neutrino is a DM
candidate and the rank of the neutrino mass matrix is two. In the case, the
dirac neutrino mass matrix can be written the following using the Casas-Ibarra
parametrization Casas:2001sr ,

$\displaystyle
m_{D}=U_{MNS}^{\ast}\left(\begin{array}[]{ccc}\sqrt{m_{1}}&0&0\\\
0&\sqrt{m_{2}}&0\\\ 0&0&\sqrt{m_{3}}\\\
\end{array}\right)O\left(\begin{array}[]{ccc}\sqrt{m_{N_{1}}}&0&0\\\
0&\sqrt{m_{N_{2}}}&0\\\ 0&0&\sqrt{m_{N_{3}}}\\\ \end{array}\right)$ (34)

where

$\displaystyle m_{1}=0,m_{2}=m_{2NH},m_{3}=m_{3NH},$ (35) $\displaystyle
O=\left(\begin{array}[]{ccc}0&0&1\\\ \cos\alpha&\sin\alpha&0\\\
-\sin\alpha&\cos\alpha&0\\\ \end{array}\right),$ (39)

for the NH case and

$\displaystyle m_{1}=m_{1IH},m_{2}=m_{2IH},m_{3}=0,$ (40) $\displaystyle
O=\left(\begin{array}[]{ccc}\cos\alpha&\sin\alpha&0\\\
-\sin\alpha&\cos\alpha&0\\\ 0&0&1\\\ \end{array}\right),$ (44)

for the IH case. $\alpha$ is a complex phase.

In the NH case, the CP-asymmetry parameter is given by observed values and a
complex phase $\alpha=\alpha_{r}+i\alpha_{i}$:

$\displaystyle|\epsilon_{1}|$ $\displaystyle=$
$\displaystyle\left|\frac{1}{2}\frac{{\rm
Im}\left[\left(m_{D}m_{D}^{\dagger}\right)_{ij}^{2}\right]}{\left(m_{D}m_{D}^{\dagger}\right)_{ii}\left(m_{D}m_{D}^{\dagger}\right)_{jj}}\right|$
(45) $\displaystyle=$
$\displaystyle\left|\frac{(m_{2NH}^{2}-m_{3NH}^{2})\sin(2\alpha_{r})\sinh(2\alpha_{i})}{(m_{2NH}-m_{3NH})^{2}\cos(2\alpha_{r})^{2}-(m_{2NH}+m_{3NH})^{2}\cosh(2\alpha_{i})^{2}}\right|$
(46) $\displaystyle\leq$
$\displaystyle\frac{m_{3NH}-m_{2NH}}{2(m_{3NH}+m_{2NH})}$ (47)
$\displaystyle=$ $\displaystyle 0.353,$ (48)

where $m_{2NH}=0.00861\ {\rm eV},m_{3NH}=0.0502\ {\rm eV}$ Esteban:2020cvm .
The equality of Eq. (47) holds when
$\alpha=\pm\frac{\pi}{4}+\frac{i}{2}\log(1+\sqrt{2})$. We use the maximum CP-
asymmetry parameter in this paper.

Next, we discuss the CP-asymmetry parameter for the IH case. The maximum CP-
asymmetry parameter is given by

$\displaystyle|\epsilon_{1}|$ $\displaystyle=$
$\displaystyle\left|\frac{(m_{1IH}^{2}-m_{2IH}^{2})\sin(2\alpha_{r})\sinh(2\alpha_{i})}{(m_{1IH}-m_{2IH})^{2}\cos(2\alpha_{r})^{2}-(m_{1IH}+m_{2IH})^{2}\cosh(2\alpha_{i})^{2}}\right|$
(49) $\displaystyle\leq$
$\displaystyle\frac{m_{2IH}-m_{1IH}}{2(m_{2IH}+m_{1IH})}=0.00377,$ (50)

where $m_{1IH}=0.0492\ {\rm eV},m_{2IH}=0.0500\ {\rm eV}$ Esteban:2020cvm .
The maximum value is very smaller than NH case and the other contributions are
very small, therefore, the generated baryon asymmetry becomes very small. We
don’t consider the IH case in this paper.

##  IV Results and discussions

We numerically solved the Boltzman equations given in Eqs. (20), (21) and (22)
from $z_{min}=0.001$ to $z_{max}=m_{N_{1}}/T_{sph}$ with initial conditions:

$\displaystyle Y_{N_{i}}(z_{min})$ $\displaystyle=$ $\displaystyle
Y_{N_{i}}^{eq}(z_{min}),$ (51) $\displaystyle Y_{\Phi}(z_{min})$
$\displaystyle=$ $\displaystyle Y_{\Phi}^{eq}(z_{min}),$ (52) $\displaystyle
Y_{B-L}(z_{min})$ $\displaystyle=$ $\displaystyle 0,$ (53)

where $i$ runs 1 and 2, $T_{sph}=150$ GeV is the sphaleron decoupling
temperature Burnier:2005hp . We fixed the complex phase at
$\alpha=\pi/4+i/2\log(1+\sqrt{2})$ and the mass of the second right-handed
neutrino at $m_{N_{2}}^{2}=m_{N_{1}}^{2}+m_{N_{1}}\Gamma_{2}$ as discussed in
Sec. III. The B$-$L density is converted to the baryon number density
($Y_{B}$) through the sphaleron process Kuzmin:1985mm  by the following
formula:

$\displaystyle Y_{B}=\frac{28}{79}Y_{B-L}(z_{max}).$ (54)

The green and cyan lines in the Fig. 2 represent the observed baryon number
density lines, $Y_{B}=8.7\times 10^{-11}$, with different Majorana mass and
scalar mass. The regions above the lines don’t create a sufficient baryon
asymmetry because we use the maximum CP-asymmetry parameter. Limits obtained
from the resonant leptogenesis (RL) scenario are shown in Fig. 2 in darker
green and cyan limes for fixed scalar mass $m_{s}=1$ TeV with varying RHN
masses $m_{N}=0.5$ TeV (solid green), $m_{N}=1$ TeV (green dashed), $m_{N}=2$
TeV (green dotted) and $m_{N}=3$ TeV (green dot-dashed), respectively. We also
show the limits for the RL scenario for fixed RHN mass at $m_{N}=1$ TeV with
different scalar masses $m_{s}=10$ TeV (cyan dot-dashed) and $m_{s}=0.1$ TeV
(cyan dotted), respectively.

Figure 2: Limits from dilepton (dotted), diject (dot-dashed) searches of CMS
(magenta), ATLAS (blue), LEP-II (red, solid) and prospective ILC experiments
for $\sqrt{s}=250$ GeV (red, dotted), 500 GeV (red, dashed) and 1 TeV (red,
dot-dashed). We show the limits from resonant leptogenesis (RL) scenario
(green and cyan) for different benchmarks of RHNs $(m_{N})$ and scalar
$(m_{s})$ masses varying $x_{H}$. We represent RL limes for fixed scalar mass
$m_{s}=1$ TeV with varying RHN mass $m_{N}=0.5$ TeV (solid green), $m_{N}=1$
TeV (green dashed), $m_{N}=2$ TeV (green dotted) and $m_{N}=3$ TeV (green dot-
dashed), respectively. RL limits for fixed RHN mass at $m_{N}=1$ TeV with
different scalar mass $m_{s}=10$ TeV (cyan dot-dashed) and $m_{s}=0.1$ TeV
(cyan dotted), respectively

Depending on the choice of $x_{H}$, the constraints obtained for different
choices of $m_{N}$ and $m_{s}$ vary for a range of the $Z^{\prime}$ mass $3$
TeV $M_{Z^{\prime}}\leq 10$ TeV.

Now we estimate bounds on $g_{X}$ for different $M_{Z^{\prime}}$ from the LEP-
II searches Eichten:1983hw ; Electroweak:2003ram ; ALEPH:2013dgf  for
different $x_{H}$ considering $M_{Z^{\prime}}\gg\sqrt{s}$ utilizing contact
interaction for $e^{-}e^{+}\to f\bar{f}$ process as

$\displaystyle{\cal L}_{\rm eff}\ =\
\frac{g_{X}^{2}}{(1+\delta_{ef})(\Lambda_{AB}^{f\pm})^{2}}\sum_{A,B=L,R}\eta_{AB}(\overline{e}\gamma^{\mu}P_{A}e)(\overline{f}\gamma_{\mu}P_{B}f)\,,$
(55)

where by convention we take $g_{X}^{2}/4\pi$ as 1 by where $\delta_{ef}=1\
(0)$ for $f=e$ ($f\neq e$). Here $\eta_{AB}=\pm 1$ or 0, and
$\Lambda_{AB}^{f\pm}$ is assumed to be the scale of contact interaction where
constructive (destructive) interference with SM processes $e^{+}e^{-}\to
f\bar{f}$ Kroha:1991mn ; Carena:2004xs  are represented by plus (minus) sign.
The $Z^{\prime}$ exchange matrix element under $U(1)_{X}$ scenario can be
written as

$\displaystyle\frac{g^{\prime
2}}{{M_{Z^{\prime}}}^{2}-s}[\overline{e}\gamma^{\mu}(\tilde{x}_{\ell}P_{L}+\tilde{x}_{e}P_{R})e][\overline{f}\gamma_{\mu}(\tilde{x}_{f_{L}}P_{L}+\tilde{x}_{f_{R}}P_{R})f]\,,$
(56)

where $\tilde{x}_{f_{L}}$ and $\tilde{x}_{f_{R}}$ are the $U(1)_{X}$ charges
of left handed and right handed fermions $(f_{L},\leavevmode\nobreak\ f_{R})$,
respectively from Table 1. Matching Eqs. (55) and (56) we find

$\displaystyle M_{Z^{\prime}}^{2}\ \gtrsim\
\frac{{g^{\prime}}^{2}}{4\pi}|{x_{e_{A}}}x_{f_{B}}|(\Lambda_{AB}^{f\pm})^{2}\,,$
(57)

taking for LEP-II scenario $M_{Z^{\prime}}^{2}\gg s$ where $\sqrt{s}=209$ GeV.
Hence we estimate bounds on $M_{Z^{\prime}}/g_{X}$ for different values of
$\Lambda_{AB}^{f\pm}$ at $95\%$ for different $x_{H}$ from ALEPH:2013dgf
assuming universality in the contact interactions where $AB=LL,\ RR,\ LR,\
RL,\ VV$ and $AA$. The upper limits on $M_{Z^{\prime}}/g_{X}$ are $5$ TeV,
$2.2$ TeV, $7.0$ TeV $11.1$ TeV and $18$ TeV for $x_{H}=-2$, $-1$, $0$, $1$
and $2$, respectively which are ruled out by LEP-II data at $\sqrt{s}=209$
GeV. In the same line we estimate prospective limits on $M_{Z^{\prime}}/g_{X}$
for different $x_{H}$ at the ILC with different center of mass energies
$\sqrt{s}=250$ GeV, $500$ GeV and $1$ TeV considering limits on
$\Lambda_{AB}^{f\pm}$ from LCCPhysicsWorkingGroup:2019fvj  at $95\%$ C. L. The
prospective upper bounds on $M_{Z^{\prime}}/g_{X}$ at $\sqrt{s}=250$ GeV for
$x_{H}=-2$, $-1$, $0$, $1$ and $2$ are $31.6$ TeV, $16.3$ TeV, $48.2$ TeV,
$79.0$ TeV and $113.7$ TeV, respectively. The prospective upper bounds on
$M_{Z^{\prime}}/g_{X}$ at $\sqrt{s}=500$ GeV for $x_{H}=-2$, $-1$, $0$, $1$
and $2$ are estimated as $54.4$ TeV, $26.3$ TeV, $81.6$ TeV, $139.1$ TeV and
$199.7$ TeV, respectively. Finally we estimate prospective upper limits on
$M_{Z^{\prime}}/g_{X}$ for $x_{H}=-2$, $-1$, $0$, $1$ and $2$ as $88.6$ TeV,
$47.7$ TeV, $137.2$ TeV, $238.2$ TeV and $339.2$ TeV, respectively at
$\sqrt{s}=1$ TeV. Hence the estimated lines are shown in Fig. 2 for LEPII
(red, solid), ILC250 (red, dotted), ILC500 (red, dashed) and ILC1000 (red,
dot-dashed), respectively.

We calculate limits on $g_{X}-M_{Z^{\prime}}$ in the $U(1)_{X}$ scenario for
different $x_{H}$ from the dilepton and dijet searches in ATLAS and CMS
ATLAS:2019erb ; CMS:2019tbu  experiments of the the LHC. The corresponding
cross sections for these scenarios are estimated as $\sigma_{\rm Model}$
involving the $Z^{\prime}$ contribution for different $x_{H}$ from $U(1)_{X}$
model considering a trial value of the general $U(1)_{X}$ gauge coupling
$g_{\rm{Model}}$ at $\sqrt{s}=14$ TeV varying $M_{Z^{\prime}}$. Comparing
these estimated cross sections with the observed cross sections from the LHC
$(\sigma_{\rm{Obs.}})$ to estimate limits on the gauge coupling for different
$M_{Z^{\prime}}$ and $x_{H}$ following

$\displaystyle g^{\prime}\ =\
\sqrt{g_{\rm{Model}}^{2}\left(\frac{\sigma_{\rm{Obs.}}}{\sigma_{\rm{Model}}}\right)}.$
(58)

to estimate $95\%$ constraints on the $g_{X}-M_{Z^{\prime}}$. The LHC limits
for the dilepton (dotted) and dijet (dot-dashed) events from CMS (magenta) and
ATLAS (blue) are shown in Fig. 2. The dilepton bounds provide strong
constrains on $g_{X}-M_{Z^{\prime}}$ plane for different $x_{H}$.

Finally from Fig. 2 we compare the limits obtained from resonant leptogenesis
and collider searches for different $x_{H}$. We find that limits obtained from
resonant leptogenesis become stronger than the limits obtained from LEP-II,
LHC for $M_{Z^{\prime}}>5.8$ TeV for $x_{H}=-2$ and $-1$. The constraints
obtained from LEP-II become stronger than the CMS limits for $x_{H}\leq 0$,
ruling out some parts of the resonant leptogenesis parameters for $x_{H}=0$
and $1$, respectively. We find that For $x_{H}=2$, LEP-II limits rule out the
limits obtained from the resonant leptogenesis. We find that prospective
limits in case of 250 GeV ILC can be comparable with the limits obtained by
the resonant lespogenesis scenario. Hence these limits could be probed at 250
GeV ILC for $x_{H}=-1$. Other prospective limits from ILC at different center
of mass energies more than 250 GeV and different $x_{H}$ could provide
stronger constraints than the limits obtained from resonant leptogenesis.

##  V Conclusions

We consider a general $U(1)_{X}$ scenario where we have three generations of
SM-singlet RHNs whichc are charged under general $U(1)_{X}$ scenario. After
cancelling gauge and mixed gauge-gravity anomalies we find that left and right
handed charged leptons interact differently with $Z^{\prime}$. These scenarios
affect the generation of CP asymmetry mediated by $Z^{\prime}$ and scalars
while induced by SM and BSM fermions and scalars applying resonant
leptogenesis. Reproducing the CP asymmetry applying different benchmark
scenarios of the RHN and SM-singlet BSM scalar masses we estimate the bounds
on $g_{X}-M_{Z^{\prime}}$ plane for different general $U(1)_{X}$ charges. We
estimate limits on general $U(1)_{X}$ coupling for different $M_{Z^{\prime}}$
using $M_{Z^{\prime}}>\sqrt{s}$ for LEP-II and prospective ILC using electron-
positron scattering. Comparing with the estimated dilepton and dijet cross
sections at the proton-proton collider with the LHC searches we estimate
limits on $g_{X}-M_{Z^{\prime}}$ plane for different $U(1)_{X}$ charges. We
find that limits obtained from resonant leptogenesis provide stronger bounds
compared to those obtained from LHC for $M_{Z^{\prime}}>5.8$ TeV. Existing
LEP-II limits from electron-positron collisions will provide stronger limits
when $x_{H}\geq 0,$ 1 and partially rule out a part of the parameter regions
found using resonant leptogenesis scenario when $M_{Z^{\prime}}>5.8$ TeV.
Limits obtained from LEP-II for $x_{H}=2$ provide stronger bounds compared to
resonant leptogenesis scenario. We find that for $x_{H}=-1$ and $-2$ limits
obtained from resonant leptogenesis are stronger than any collider limits for
$M_{Z^{\prime}}>5.8$ TeV. These parameter regions involving TeV scale RHNs,
BSM scalar and $Z^{\prime}$ gauge boson depending on different general
$U(1)_{X}$ charges could be probed at high energy colliders experiments in
future.

###### Acknowledgements.

##  Appendix A Reduced cross sections

The reduced cross sections involved in this analysis can be given by
$\hat{\sigma}=\sigma\\{\frac{8}{s}[(p_{f}.p_{i})^{2}-m_{f}^{4}]\\}$ where
$\sigma$ is the total cross section in CM frame for different $2\to 2$
processes participating in leptogenesis, $p_{i}$ and $p_{f}$ are momenta of
initial and final state particles. We write the reduced cross sections below:

  * (i)

RHN pair production from different initial states involving SM charged
fermions:

$\displaystyle\hat{\sigma}_{Z^{\prime}}\equiv\hat{\sigma}(f\bar{f}\leftrightarrow
N_{i}N_{i})=\frac{g_{X}^{4}x_{\nu}^{2}s^{2}}{12\pi}\Bigg{[}\frac{(1-\frac{4m_{N_{i}}^{2}}{s})^{\frac{3}{2}}}{(s-{m_{Z^{\prime}}}^{2})^{2}+{\Gamma_{Z}^{\prime}}^{2}{m_{Z^{\prime}}}^{2}}\Bigg{]}(C_{A}^{2}+C_{V}^{2})$
(59)

where the vector $(C_{V}=\frac{q_{f_{L}}+q_{f_{R}}}{2})$ and axial-vector
$(C_{A}=\frac{q_{f_{L}}-q_{f_{R}}}{2})$ couplings can be derived using the
general $U(1)$ charges of the left and right handed fermions from Tab. 1. The
center of mass cross section will be averaged over color factor for quarks
$3^{2}$ in initial states. We calculate the reduced cross sections for the
scalar $(h)$ initialed process $hh\to f\bar{f}$ mediated by $Z^{\prime}$ in
the following as

$\displaystyle\hat{\sigma}_{Z^{\prime},h}\equiv\hat{\sigma}(hh\leftrightarrow
f\bar{f})=\frac{g_{X}^{4}x_{h}^{2}s^{2}}{12\pi}\Big{[}\frac{(1-\frac{4m_{h}^{2}}{s})^{\frac{3}{2}}}{{(s-{m_{Z^{\prime}}}^{2})^{2}+{\Gamma_{Z}^{\prime}}^{2}{m_{Z^{\prime}}}^{2}}}\Big{]}(C_{A}^{2}+C_{V}^{2}).$
(60)

where $x_{h}$ and $x_{\nu}$ are the general $U(1)_{X}$ charges of the scalars
and RHNs. In addition to that, we also calculate the reduced cross section for
the scalar initiated process $hh\to h_{\rm SM}h_{\rm SM}$ into a pair of SM
Higgs $(h_{\rm SM})$ in the following as

$\displaystyle\hat{\sigma}(hh\leftrightarrow h_{\rm SM}h_{\rm
SM})=\frac{g_{X}^{4}x_{h}^{2}x_{h_{\rm
SM}}^{2}s^{2}}{12\pi}\frac{(1-\frac{4m_{h}^{2}}{s})^{\frac{3}{2}}(1-\frac{4m_{h_{\rm
SM}}^{2}}{s})^{\frac{3}{2}}}{{(s-{m_{Z^{\prime}}}^{2})^{2}+{\Gamma_{Z}^{\prime}}^{2}{m_{Z^{\prime}}}^{2}}}.$
(61)

where $m_{h_{\rm SM}}$ and $x_{h_{\rm SM}}$ are the mass and general $U(1)$
charge of $h_{\rm SM}$, respectively.

  * (ii)

RHN pair production from different generation of RHNs:

$\displaystyle\hat{\sigma}(N_{i}N_{i}\leftrightarrow
N_{j}N_{j})=\frac{g_{X}^{4}x_{\nu}^{4}\sqrt{(s-4m_{N_{j}}^{2})(s-4m_{N_{i}}^{2})}}{72\pi
s\\{(s-m_{Z^{\prime}}^{2})^{2}+\Gamma_{Z^{\prime}}^{2}m_{Z^{\prime}}^{2}\\}}\Bigg{[}(s-4m_{N_{j}}^{2})(s-4m_{N_{i}}^{2})+12\frac{m_{N_{i}}^{2}m_{N_{j}}^{2}}{m_{Z^{\prime}}^{4}}(s-m_{Z^{\prime}}^{2})^{2}\Bigg{]}$
(62)

where $i\neq j$. RHN pair production form same generation:

$\displaystyle\hat{\sigma}(N_{i}N_{i}\leftrightarrow N_{i}N_{i})$
$\displaystyle=$
$\displaystyle\frac{g_{x}^{4}x_{\nu}^{4}}{128\pi}\Bigg{[}\frac{(s-4m_{N_{i}}^{2})^{3}}{3s((s-m_{Z^{\prime}}^{2})^{2}+m_{Z^{\prime}}^{2}\Gamma_{Z^{\prime}}^{2})}+\frac{(s-4m_{N_{i}}^{2})}{sm_{Z^{\prime}}^{4}(s-4m_{N_{i}}^{2}+m_{Z^{\prime}}^{2})}\Big{\\{}m_{Z^{\prime}}^{2}(s-4m_{N_{i}}^{2})^{2}+$
(63) $\displaystyle
2(m_{Z^{\prime}}^{2}-2m_{N_{i}}^{2})^{3}+s(8m_{N_{i}}^{4}+3m_{Z^{\prime}}^{4})-4m_{N_{i}}^{2}(m_{Z^{\prime}}^{4}+4m_{N_{i}}^{4}+m_{Z^{\prime}}^{2}m_{N_{i}}^{2})\Big{\\}}+$
$\displaystyle+\Big{\\{}\frac{(3m_{Z^{\prime}}^{2}-4m_{N_{i}}^{2})(s-4m_{N_{i}}^{2})^{2}+m_{Z^{\prime}}^{4}(3s-20m_{N_{i}}^{2})+2m_{Z^{\prime}}^{2}(m_{Z^{\prime}}^{4}+8m_{N_{i}}^{4})}{sm_{Z^{\prime}}^{2}(s-4m_{N_{i}}^{2}+2m_{Z^{\prime}}^{2})}\Big{\\}}$
$\displaystyle\log\Big{(}\frac{m_{Z^{\prime}}^{2}}{s-4m_{N_{i}}^{2}+m_{Z^{\prime}}^{2}}\Big{)}\Bigg{]}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $

  * (iii)

RHN pair production from $h$ in $s-$channel:

$\displaystyle\hat{\sigma}_{Z^{\prime},\Phi}\equiv\hat{\sigma}(hh\leftrightarrow
N_{i}N_{i})=\frac{{g_{X}^{4}}x_{h}^{2}x_{\nu}^{2}s^{2}}{12\pi}\frac{(1-\frac{4m_{N_{i}}^{2}}{s})^{\frac{3}{2}}(1-\frac{4{m_{h}}^{2}}{s})^{\frac{3}{2}}}{m_{Z^{\prime}}^{2}\Gamma_{Z^{\prime}}^{2}+(s-m_{Z^{\prime}}^{2})^{2}}.$
(64)

Reduced cross section for $h$ pair production from RHN in $t-$channel and
$u-$channel processes:

$\displaystyle\hat{\sigma}_{N,\Phi}\equiv\hat{\sigma}(N_{i}N_{i}\leftrightarrow
hh)=\frac{Y_{N}^{4}}{8\pi}\Bigg{(}1-\frac{4m_{N_{i}}^{2}}{s}\Bigg{)}\Bigg{[}\frac{s-4m_{N_{i}}^{2}}{2m_{N_{i}}^{2}}+\Bigg{(}1-\frac{4m_{N_{i}}^{2}}{s}\Bigg{)}^{\frac{3}{2}}\log\Bigg{(}\frac{s-\sqrt{s(s-4m_{N_{i}}^{2})}}{s+\sqrt{s(s-4m_{N_{i}}^{2})}}\Bigg{)}\Bigg{]}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $ (65)

  * (iv)

Scalar mediated $\ell N^{i}\to tq$ process in $s-$channel:

$\displaystyle\hat{\sigma}_{h,s}\equiv\hat{\sigma}(N^{i}\ell\leftrightarrow
tq)=\frac{3\pi\alpha^{2}m_{t}^{2}}{M_{W}^{4}\sin\theta_{W}^{4}}(m_{D}^{\dagger}m_{D})_{ij}\Bigg{(}1-\frac{m_{N_{i}}^{2}}{s}\Bigg{)}^{2}$
(66)

Scalar mediated $N^{i}t\to\ell q$ process in $t-$channel:

$\displaystyle\hat{\sigma}_{h,t}\equiv\hat{\sigma}(N^{j}t\to\ell
q)=\frac{3\pi\alpha^{2}m_{t}^{2}}{M_{W}^{4}\sin\theta_{W}^{4}}(m_{D}^{\dagger}m_{D})_{ij}\Bigg{[}1-\frac{m_{N_{i}}^{2}}{s}+\frac{m_{N_{i}}^{2}}{s}\log\Bigg{(}1+\frac{s-m_{N_{i}}^{2}}{m_{h}^{2}}\Bigg{)}\Bigg{]}$
(67)

where $\alpha=\frac{e^{2}}{4\pi}$.

  * (v)

RHN mediated $\ell h\to\ell h$ process in $s-$channel and $t-$ channel:

$\displaystyle\hat{\sigma}_{N}\equiv\hat{\sigma}(\ell h\leftrightarrow\ell h)$
$\displaystyle=$
$\displaystyle\frac{2\alpha^{2}\pi}{\sin^{4}\theta_{W}M_{W}^{4}s}\Bigg{\\{}\sum_{i=1}^{2}m_{N_{i}}^{2}(m_{D}^{\dagger}m_{D})_{ii}^{2}\Bigg{[}\frac{s}{m_{N_{i}}^{2}}+\frac{2s}{\mathcal{D}_{i}}+\frac{s^{2}}{2\mathcal{D}_{i}^{2}}-\Bigg{(}1+2\frac{s+m_{N_{i}}^{2}}{\mathcal{D}_{i}}\Bigg{)}\log\Bigg{(}1+\frac{s}{m_{N_{i}}^{2}}\Bigg{)}\Bigg{]}$
(68)
$\displaystyle+2m_{N_{1}}m_{N_{2}}\mathcal{R}e\Bigg{[}(m_{D}^{\dagger}m_{D})_{12}^{2}\Bigg{]}\Bigg{[}\frac{s}{\mathcal{D}_{1}}+\frac{s}{\mathcal{D}_{2}}+\frac{s^{2}}{2\mathcal{D}_{1}\mathcal{D}_{2}}-\frac{(s+m_{N_{1}}^{2})(s+m_{N_{1}}^{2}-2m_{N_{2}}^{2})}{(m_{N_{1}}^{2}-m_{N_{2}}^{2})\mathcal{D}_{2}}\log\Bigg{(}1+\frac{s}{m_{N_{1}}^{2}}\Bigg{)}$
$\displaystyle-\frac{(s+m_{N_{2}}^{2})(s+m_{N_{2}}^{2}-2m_{N_{1}}^{2})}{(m_{N_{2}}^{2}-m_{N_{1}}^{2})\mathcal{D}_{1}}\log\Bigg{(}1+\frac{s}{m_{N_{2}}^{2}}\Bigg{)}\Bigg{]}\Bigg{\\}}$

where
$\mathcal{D}_{i}=\frac{(s-m_{N_{i}}^{2})^{2}+m_{N_{i}}^{2}\Gamma_{i}^{2}}{(s-m_{N_{i}}^{2})}$
is the off-shell part of the propagator and it is a dimensionless quantity.

  * (vi)

RHN mediated $\ell\ell\leftrightarrow hh$ process in $t-$ channel:

$\displaystyle\hat{\sigma}_{N,t}\equiv\hat{\sigma}(\ell\ell\leftrightarrow
hh)$ $\displaystyle=$
$\displaystyle\frac{2\pi\alpha^{2}}{M_{W}^{4}\sin\theta_{W}^{4}}\Bigg{\\{}\sum_{i=1}^{2}(m_{D}^{\dagger}m_{D})_{ii}^{2}\Bigg{[}\frac{s}{2(s+m_{N_{i}}^{2})}+\frac{m_{N_{i}}^{2}}{s+2m_{N_{i}}^{2}}\log\Bigg{(}1+\frac{s}{m_{N_{i}}^{2}}\Bigg{)}\Bigg{]}$
(69) $\displaystyle+$
$\displaystyle\mathcal{R}e\Bigg{[}(m_{D}^{\dagger}m_{D})_{12}^{2}\Bigg{]}\frac{m_{N_{1}}m_{N_{2}}}{(m_{N_{1}}^{2}-m_{N_{2}}^{2})(s+m_{N_{1}}^{2}+m_{N_{2}}^{2})}\Bigg{[}(s+2m_{N_{1}}^{2})\log\Bigg{(}1+\frac{s}{m_{N_{2}}^{2}}\Bigg{)}$
$\displaystyle-$
$\displaystyle(s+2m_{N_{2}}^{2})\log\Bigg{(}1+\frac{s}{m_{N_{1}}^{2}}\Bigg{)}\Bigg{]}\Bigg{\\}}.$

## References

  * (1) WMAP Collaboration, E. Komatsu et al., “Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation,” Astrophys. J. Suppl. 192 (2011) 18, arXiv:1001.4538 [astro-ph.CO].
  * (2) A. G. Cohen, D. B. Kaplan, and A. E. Nelson, “Progress in electroweak baryogenesis,” Ann. Rev. Nucl. Part. Sci. 43 (1993) 27–70, arXiv:hep-ph/9302210.
  * (3) K. Funakubo, “CP violation and baryogenesis at the electroweak phase transition,” Prog. Theor. Phys. 96 (1996) 475–520, arXiv:hep-ph/9608358.
  * (4) M. Trodden, “Electroweak baryogenesis,” Rev. Mod. Phys. 71 (1999) 1463–1500, arXiv:hep-ph/9803479.
  * (5) CMS Collaboration, S. Chatrchyan et al., “Observation of a New Boson at a Mass of 125 GeV with the CMS Experiment at the LHC,” Phys. Lett. B 716 (2012) 30–61, arXiv:1207.7235 [hep-ex].
  * (6) ATLAS Collaboration, G. Aad et al., “Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC,” Phys. Lett. B 716 (2012) 1–29, arXiv:1207.7214 [hep-ex].
  * (7) Particle Data Group Collaboration, R. L. Workman et al., “Review of Particle Physics,” PTEP 2022 (2022) 083C01.
  * (8) P. Minkowski, “$\mu\to e\gamma$ at a Rate of One Out of $10^{9}$ Muon Decays?,” Phys. Lett. B 67 (1977) 421–428.
  * (9) T. Yanagida, “Horizontal gauge symmetry and masses of neutrinos,” Conf. Proc. C 7902131 (1979) 95–99.
  * (10) M. Gell-Mann, P. Ramond, and R. Slansky, “Complex Spinors and Unified Theories,” Conf. Proc. C 790927 (1979) 315–321, arXiv:1306.4669 [hep-th].
  * (11) R. N. Mohapatra and G. Senjanovic, “Neutrino Mass and Spontaneous Parity Nonconservation,” Phys. Rev. Lett. 44 (1980) 912.
  * (12) S. L. Glashow, “The Future of Elementary Particle Physics,” NATO Sci. Ser. B 61 (1980) 687.
  * (13) S. Weinberg, “Baryon and Lepton Nonconserving Processes,” Phys. Rev. Lett. 43 (1979) 1566–1570.
  * (14) M. Fukugita and T. Yanagida, “Baryogenesis Without Grand Unification,” Phys. Lett. B 174 (1986) 45–47.
  * (15) N. S. Manton, “Topology in the Weinberg-Salam Theory,” Phys. Rev. D 28 (1983) 2019.
  * (16) F. R. Klinkhamer and N. S. Manton, “A Saddle Point Solution in the Weinberg-Salam Theory,” Phys. Rev. D 30 (1984) 2212.
  * (17) S. Y. Khlebnikov and M. E. Shaposhnikov, “The Statistical Theory of Anomalous Fermion Number Nonconservation,” Nucl. Phys. B 308 (1988) 885–912.
  * (18) S. Iso, N. Okada, and Y. Orikasa, “Resonant Leptogenesis in the Minimal B-L Extended Standard Model at TeV,” Phys. Rev. D 83 (2011) 093011, arXiv:1011.4769 [hep-ph].
  * (19) G. Chauhan and P. S. B. Dev, “Interplay between resonant leptogenesis, neutrinoless double beta decay and collider signals in a model with flavor and CP symmetries,” Nucl. Phys. B 986 (2023) 116058, arXiv:2112.09710 [hep-ph].
  * (20) M. Plumacher, “Baryogenesis and lepton number violation,” Z. Phys. C 74 (1997) 549–559, arXiv:hep-ph/9604229.
  * (21) W. Buchmuller, P. Di Bari, and M. Plumacher, “Cosmic microwave background, matter - antimatter asymmetry and neutrino masses,” Nucl. Phys. B 643 (2002) 367–390, arXiv:hep-ph/0205349. [Erratum: Nucl.Phys.B 793, 362 (2008)].
  * (22) W. Buchmuller, P. Di Bari, and M. Plumacher, “Leptogenesis for pedestrians,” Annals Phys. 315 (2005) 305–351, arXiv:hep-ph/0401240.
  * (23) S. Oda, N. Okada, and D.-s. Takahashi, “Classically conformal U(1)’ extended standard model and Higgs vacuum stability,” Phys. Rev. D 92 no. 1, (2015) 015026, arXiv:1504.06291 [hep-ph].
  * (24) A. Das, S. Oda, N. Okada, and D.-s. Takahashi, “Classically conformal U(1)’ extended standard model, electroweak vacuum stability, and LHC Run-2 bounds,” Phys. Rev. D 93 no. 11, (2016) 115038, arXiv:1605.01157 [hep-ph].
  * (25) A. Das, S. Mandal, T. Nomura, and S. Shil, “Heavy Majorana neutrino pair production from Z’ at hadron and lepton colliders,” Phys. Rev. D 105 no. 9, (2022) 095031, arXiv:2202.13358 [hep-ph].
  * (26) M. Flanz, E. A. Paschos, U. Sarkar, and J. Weiss, “Baryogenesis through mixing of heavy Majorana neutrinos,” Phys. Lett. B 389 (1996) 693–699, arXiv:hep-ph/9607310.
  * (27) A. Pilaftsis, “CP violation and baryogenesis due to heavy Majorana neutrinos,” Phys. Rev. D 56 (1997) 5431–5451, arXiv:hep-ph/9707235.
  * (28) A. Pilaftsis and T. E. J. Underwood, “Resonant leptogenesis,” Nucl. Phys. B 692 (2004) 303–345, arXiv:hep-ph/0309342.
  * (29) S. Blanchet, Z. Chacko, S. S. Granor, and R. N. Mohapatra, “Probing Resonant Leptogenesis at the LHC,” Phys. Rev. D 82 (2010) 076008, arXiv:0904.2174 [hep-ph].
  * (30) J. A. Casas and A. Ibarra, “Oscillating neutrinos and $\mu\to e,\gamma$,” Nucl. Phys. B 618 (2001) 171–204, arXiv:hep-ph/0103065.
  * (31) I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, T. Schwetz, and A. Zhou, “The fate of hints: updated global analysis of three-flavor neutrino oscillations,” JHEP 09 (2020) 178, arXiv:2007.14792 [hep-ph].
  * (32) Y. Burnier, M. Laine, and M. Shaposhnikov, “Baryon and lepton number violation rates across the electroweak crossover,” JCAP 02 (2006) 007, arXiv:hep-ph/0511246.
  * (33) V. A. Kuzmin, V. A. Rubakov, and M. E. Shaposhnikov, “On the Anomalous Electroweak Baryon Number Nonconservation in the Early Universe,” Phys. Lett. B 155 (1985) 36.
  * (34) E. Eichten, K. D. Lane, and M. E. Peskin, “New Tests for Quark and Lepton Substructure,” Phys. Rev. Lett. 50 (1983) 811–814.
  * (35) LEP, ALEPH, DELPHI, L3, OPAL, LEP Electroweak Working Group, SLD Electroweak Group, SLD Heavy Flavor Group Collaboration, t. S. Electroweak, “A Combination of preliminary electroweak measurements and constraints on the standard model,” arXiv:hep-ex/0312023.
  * (36) ALEPH, DELPHI, L3, OPAL, LEP Electroweak Collaboration, S. Schael et al., “Electroweak Measurements in Electron-Positron Collisions at W-Boson-Pair Energies at LEP,” Phys. Rept. 532 (2013) 119–244, arXiv:1302.3415 [hep-ex].
  * (37) H. Kroha, “Compositeness limits from E+ E- annihilation revisited,” Phys. Rev. D 46 (1992) 58–69.
  * (38) M. Carena, A. Daleo, B. A. Dobrescu, and T. M. P. Tait, “$Z^{\prime}$ gauge bosons at the Tevatron,” Phys. Rev. D 70 (2004) 093009, arXiv:hep-ph/0408098.
  * (39) LCC Physics Working Group Collaboration, K. Fujii et al., “Tests of the Standard Model at the International Linear Collider,” arXiv:1908.11299 [hep-ex].
  * (40) ATLAS Collaboration, G. Aad et al., “Search for high-mass dilepton resonances using 139 fb-1 of $pp$ collision data collected at $\sqrt{s}=$13 TeV with the ATLAS detector,” Phys. Lett. B 796 (2019) 68–87, arXiv:1903.06248 [hep-ex].
  * (41) CMS Collaboration, “Search for a narrow resonance in high-mass dilepton final states in proton-proton collisions using 140$\leavevmode\nobreak\ \mathrm{fb}^{-1}$ of data at $\sqrt{s}=13\leavevmode\nobreak\ \mathrm{TeV}$,”.