# Collider Tests of Flavored Resonant Leptogenesis in the $U(1)_{X}$ Model

Garv Chauhan

###### Abstract

We study the generation of baryon asymmetry through the flavored resonant
leptogenesis in the $U(1)_{X}$ extension of the Standard Model. Being a
generalization of the $U(1)_{B\text{-}L}$, $U(1)_{X}$ is an ultraviolet-
complete model of the right-handed neutrinos (RHNs), whose CP violating out-
of-equilibrium decays lead to the generation of baryon asymmetry via
leptogenesis. We can also explain the neutrino masses via the seesaw mechanism
in this model. We consider three different cases for different $U(1)_{X}$
charges of the scalar particle responsible for $U(1)_{X}$ breaking at TeV-
scale. These include the popular $U(1)_{B\text{-}L}$ and $U(1)_{R}$ models, as
well as a $U(1)_{C}$ model which maximizes the collider signal. We numerically
solve the flavored Boltzmann transport equations to calculate the total baryon
asymmetry. We show that all three cases considered here can naturally explain
the observed baryon asymmetry of the Universe in a large portion of the
available parameter space, while satisfying the neutrino oscillation data. We
find that the $U(1)_{C}$ case offers successful leptogenesis in a larger
portion of the parameter space as compared to $U(1)_{B\text{-}L}$ and
$U(1)_{R}$. We also perform a comparative study between the flavored and
unflavored leptogenesis parameter space. Finally, we also study the collider
prospects for all these scenarios using the lepton number violating signal of
$pp\to\ell^{\pm}\ell^{\pm}+$jets mediated by the $Z^{\prime}$ boson associated
with $U(1)_{X}$. We find that HL-LHC may be able to probe a small portion of
the relevant parameter space having successful leptogenesis, if neutrinos have
normal mass ordering, while a $\sqrt{s}=100$ TeV future collider can access a
much larger region of the parameter space, thereby offering an opportunity to
test resonant leptogenesis in the $U(1)_{X}$ model.

##  1 Introduction

Although the Standard Model (SM) of particle physics has been highly
successful in describing the microscopic physics to the smallest length scales
probed so far, there is strong empirical evidence for physics beyond the SM
(BSM). In particular, the observation of neutrino oscillations [1] necessarily
implies that at least two of the three neutrino mass eigenvalues must be
nonzero, which immediately demands some BSM explanation, as neutrinos are
exactly massless in the SM. Perhaps the simplest way to generate neutrino mass
is by the so-called seesaw mechanism [2, 3, 4, 5, 6] where one adds SM-singlet
Majorana fermions, also known as right-handed neutrinos (RHNs), which give
rise to a small Majorana mass for the $SU(2)_{L}$-doublet neutrinos after
electroweak symmetry breaking.

It is interesting that the same RHNs responsible for neutrino mass could also
explain another important evidence for BSM physics, namely, the observed
matter-antimatter asymmetry of the Universe [7], via the mechanism of
leptogenesis [8]. The basic idea is that the out-of-equilibrium decays of the
RHNs to the SM lepton and Higgs doublets can produce a nonzero lepton
asymmetry in the early Universe, which is reprocessed into a baryon asymmetry
by the $(B+L)$-violating electroweak sphaleron transitions [9]. However, the
vanilla leptogenesis scenario imposes a lower bound on the mass of the RHNs,
$M_{N}\gtrsim 10^{9}$ GeV [10, 11],111Including flavor effects can in
principle lower this value to $\sim 10^{6}$ GeV [12]. thus precluding the
possibility of testing it in laboratory experiments.

A low-energy alternative is the resonant leptogenesis mechanism [13], which
relies on the resonant enhancement of the CP asymmetry from RHN decays via
self-energy contributions when the masses of two RHNs are quasi-degenerate
[14]. This can potentially bring the scale of $M_{N}$ down to the electroweak
scale222RHN scale can even be lighter below GeV-scale in the parametric regime
of ARS leptogenesis [15]. [16, 17, 18, 19], thus offering the hope to test
this mechanism at the Large Hadron Collider (LHC) and future colliders.

In this work, we study the resonant leptogenesis mechanism in an ultraviolet-
complete model of the RHNs in terms of the $U(1)_{X}$ extension of the SM [20,
21, 22, 23, 24]. The $U(1)_{X}$ symmetry can be identified as the linear
combination of the $U(1)_{Y}$ in SM and the $U(1)_{B-L}$ gauge group, and
hence, can be regarded as a generalization of the $U(1)_{B-L}$ extension of
the SM [25, 26, 27], where the RHN fields are an essential ingredient required
for anomaly cancellation. The presence of the extra neutral gauge boson
$Z^{\prime}$ associated with the $U(1)_{X}$ breaking affects the lepton
asymmetry calculation in a nontrivial way [28]. We take this into account and
also include the flavor effects from both RHNs and charged leptons, which are
known to be important in resonant leptogenesis [18, 29]. A crucial input for
leptogenesis is the complex Dirac Yukawa coupling matrix, which we parametrize
using the Casas-Ibarra parametrization [30] to satisfy the neutrino
oscillation data, while also highlighting the role of the Dirac CP phase in
the generation of the lepton asymmetry. We then perform a numerical scan over
the masses of the heavy gauge boson and RHNs to carve out the parameter space
consistent with successful leptogenesis for three different benchmark charges
of the $U(1)_{X}$ scalars.

We also calculate the collider prospects of the allowed parameter space for
leptogenesis using the lepton number violating (LNV) signal $pp\to
Z^{\prime}\to NN\to\ell^{\pm}\ell^{\pm}+{\rm jets}$ [27, 31, 32, 33, 34, 35].
We find that the high-luminosity LHC (HL-LHC) will be sensitive to a small
portion of the allowed parameter space, especially for the normal hierarchy of
neutrino masses, while a future 100 TeV collider can probe a much wider range
of parameter space, thus making resonant leptogenesis in the $U(1)_{X}$ model
truly testable at the Energy Frontier.

##  2 General $U(1)_{X}$ Scenario

The general $U(1)_{X}$ extension of the SM is based on the $SU(2)_{C}\times
SU(2)_{L}\times U(1)_{Y}\times U(1)_{X}$ gauge group. The particle content
involves adding three generations of the SM-singlet RHNs and a SM-singlet
scalar $\Phi$, all charged under $U(1)_{X}$. The RHNs in addition to
contributing to the neutrino mass generation via seesaw mechanism, also play a
crucial role in canceling the gauge and mixed gauge-gravity anomalies [24].
The particle content of the model along with the $U(1)_{X}$ charges is given
in Table 1. Note that the fermion charges are generation-independent. The
scalar charges $x_{H}$, $x_{\Phi}$ are real parameters. For $x_{H}=0$ and
$x_{\Phi}=1$, we recover the $U(1)_{B-L}$ model [25, 26]. Without loss of
generality, we fix $x_{\Phi}=1$ in this paper. As a result, the $x_{H}$ charge
simply acts as an angle between the $U(1)_{Y}$ and $U(1)_{B-L}$ directions.
$x_{H}=-2$ corresponds to the $U(1)_{R}$ model [36], whereas $x_{H}=-1.2$
corresponds to the $U(1)_{C}$ model with maximum enhancement in the
$Z^{\prime}\to NN$ branching ratio [37]. We will consider these three
benchmark values of $x_{H}$ in the following numerical analysis. The
$U(1)_{X}$ gauge coupling $g_{X}$ is another free parameter in this model and
we will also fix some benchmark values for it for a given $Z^{\prime}$ mass to
be consistent with the current LHC constraints [38].

The fermion mass terms and flavor mixing are introduced by the Yukawa
interaction terms written as

${\cal L}_{\rm
Yukawa}=-Y_{e}^{\alpha\beta}\overline{\ell_{L}^{\alpha}}\tilde{H}e_{R}^{\beta}-Y_{\nu}^{\alpha\beta}\overline{\ell_{L}^{\alpha}}HN_{R}^{\beta}-Y_{N}^{\alpha}\Phi\overline{(N_{R}^{\alpha})^{c}}N_{R}^{\alpha}-Y_{u}^{\alpha\beta}\overline{q_{L}^{\alpha}}Hu_{R}^{\beta}-Y_{d}^{\alpha\beta}\overline{q_{L}^{\alpha}}\tilde{H}d_{R}^{\beta}+{\rm
H.c.}~{},$ (2.1)

where $\tilde{H}=i\sigma^{2}H^{*}$ and $\alpha,\beta=1,2,3$ are the generation
indices. Note here $H$ is the SM Higgs doublet.

| $SU(3)_{c}$ | $SU(2)_{L}$ | $U(1)_{Y}$ | $U(1)_{X}$
---|---|---|---|---
$q_{L}^{\alpha}$ | 3 | 2 | $\frac{1}{6}$ | $q_{q}=\frac{1}{6}x_{H}+\frac{1}{3}x_{\Phi}$
$u_{R}^{\alpha}$ | 3 | 1 | $\frac{2}{3}$ | $q_{u}=\frac{2}{3}x_{H}+\frac{1}{3}x_{\Phi}$
$d_{R}^{\alpha}$ | 3 | 1 | $-\frac{1}{3}$ | $q_{d}=-\frac{1}{3}x_{H}+\frac{1}{3}x_{\Phi}$
$\ell_{L}^{\alpha}$ | 1 | 2 | $-\frac{1}{2}$ | $q_{\ell}=-\frac{1}{2}x_{H}-x_{\Phi}$
$e_{R}^{\alpha}$ | 1 | 1 | $-1$ | $q_{e}=-x_{H}-x_{\Phi}$
$N_{\alpha}$ | 1 | 1 | $0$ | $q_{N}=-x_{\Phi}$
$H$ | 1 | 2 | $-\frac{1}{2}$ | $\frac{x_{H}}{2}$
$\Phi$ | 1 | 1 | $0$ | $2x_{\Phi}$
Table 1:  The particle content of the general $U(1)_{X}$ model. Here
$\alpha=1,2,3$ represents the family index.

The renormalizable scalar potential in this model is given by

$\displaystyle V\ =\
m_{H}^{2}(H^{\dagger}H)+\lambda_{H}(H^{\dagger}H)^{2}+m_{\Phi}^{2}(\Phi^{\dagger}\Phi)+\lambda_{\Phi}(\Phi^{\dagger}\Phi)^{2}+\lambda_{\rm
mix}(H^{\dagger}H)(\Phi^{\dagger}\Phi)~{}.$ (2.2)

In the limit of small $\lambda_{\rm mix}$, the scalar fields $H$ and $\Phi$
can be analyzed separately [23, 24]. The $U(1)_{X}$ gauge symmetry and
electroweak symmetry are respectively broken by the vacuum expectation values
(VEVs) of $\Phi$ and $H$, given by

$\langle H\rangle\ =\ \frac{1}{\sqrt{2}}\begin{pmatrix}v+h\\\
0\end{pmatrix},\quad{\rm and}\quad\langle\Phi\rangle\ =\
\frac{v_{\Phi}+\phi}{\sqrt{2}},$ (2.3)

where $v=246$ GeV is the electroweak scale and $v_{\Phi}$ is a free parameter,
which can be traded for the $Z^{\prime}$ mass. In other words, after the
$U(1)_{X}$ symmetry is broken and assuming $v_{\Phi}\gg v$, the $Z^{\prime}$
mass can be written as

$m_{Z^{\prime}}\simeq 2g_{X}v_{\Phi}x_{\Phi}\,.$ (2.4)

It can be clearly seen after the breaking of $U(1)_{X}$ and $\Phi$ VEV is
developed, the third term in the Eq. (2.1) leads to the generation of Majorana
mass of the RHNs. The second term after the $H$ develops VEV generates the
Dirac mass term from the Yukawa coupling. The combination of these Dirac and
Majorana mass terms leads to Type-I seesaw formula that can explain the
neutrino masses and mixing. The Dirac and Majorana mass terms arising from Eq.
(2.1) can be written as

$m_{D}^{\alpha\beta}\ =\
\frac{Y_{\nu}^{\alpha\beta}}{\sqrt{2}}v,\,\,\,\,\,m_{N^{\alpha}}\ =\
\frac{Y^{\alpha}_{N}}{\sqrt{2}}v_{\Phi}.$ (2.5)

respectively.

###  2.1 Fermions and $Z^{\prime}$ interactions

The new gauge boson $Z^{\prime}$ interacts with the SM fermions via $U(1)_{X}$
gauge coupling. The chiral interaction terms in the Lagrangian for fermions
interacting with $Z^{\prime}$ is given by

$\displaystyle\mathcal{L}_{\rm{int}}=-g_{X}(\overline{f}\gamma_{\mu}q_{f_{L}}P_{L}f+\overline{f}\gamma_{\mu}q_{f_{R}}P_{R}f)Z_{\mu}^{\prime}~{},$
(2.6)

where $q_{f_{L(R)}}$ is the corresponding $U(1)_{X}$ charge of the left
(right) handed fermions [cf. Table 1] and $P_{L(R)}=(1\pm\gamma_{5})/2$ are
the usual projection operators. Using this, we can calculate the partial decay
widths of $Z^{\prime}$ into charged fermions as follows

$\displaystyle\Gamma(Z^{\prime}\to\bar{f}f)$
$\displaystyle=N_{C}\frac{M_{Z^{\prime}}g_{X}^{2}}{24\pi}\sqrt{1-4\frac{m_{f}^{2}}{M_{Z^{\prime}}^{2}}}\left[\left(q_{f_{L}}^{2}+q_{f_{R}}^{2}\right)\left(1-\frac{m_{f}^{2}}{M_{Z^{\prime}}^{2}}\right)+6q_{f_{L}}q_{f_{R}}\frac{m_{f}^{2}}{M_{Z^{\prime}}^{2}}\right]~{},$
(2.7)

For our collider study, the relevant decay mode is $Z^{\prime}\to NN$ whose
partical decay width is given by

$\displaystyle\Gamma(Z^{\prime}\to
N_{\alpha}N_{\alpha})=\frac{M_{Z^{\prime}}g_{X}^{2}}{24\pi}q_{N}^{2}\left(1-\frac{M_{N}^{2}}{M_{Z^{\prime}}^{2}}\right)^{\frac{3}{2}}~{},$
(2.8)

where $q_{N}$ is the $U(1)_{X}$ charge of RHNs and $M_{N}$ is the RHN mass.

###  2.2 RHN interactions

As discussed earlier after breaking of $U(1)_{X}$ and EW symmetry i.e. after
$\Phi$ and $H$ develop VEVs, it leads to the generation of RHN Majorana mass
$M_{N}$ and Dirac mass term respectively. This leads to the generation of
neutrino masses in this model. The full neutrino mass matrix takes the
standard seesaw form as

$\displaystyle M_{\nu}\ =\ \begin{pmatrix}0&&M_{D}\\\
M_{D}^{T}&&M_{N}\end{pmatrix},$ (2.9)

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

$\displaystyle m_{\nu}\simeq-M_{D}M_{N}^{-1}M_{D}^{T},$ (2.10)

in the seesaw limit $m_{D}\ll M_{N}$.

The light neutrino flavor eigenstate $(\nu_{\alpha})$ can be approximately
decomposed in terms of light $(\nu_{i})$ and heavy $(N_{i})$ mass eigenstates

$\displaystyle\nu_{\alpha}\simeq U_{\alpha i}\nu_{i}+V_{\alpha i}N_{i},$
(2.11)

where $\alpha$ and $i$ are the generation indices. Note here $U_{\alpha i}$
are the elements of the $3\times 3$ light neutrino mixing matrix which can be
expressed as $(1-\frac{\epsilon}{2})U_{\rm PMNS}$ with
$\epsilon=V^{\ast}V^{T}$ known as the non-unitarity parameter, with

$\displaystyle V_{\alpha i}\simeq M_{D}M_{N}^{-1}$ (2.12)

parametrizing the mixing between light and RHNs. Therefore, the SM gauge
singlet RHNs interact with the SM gauge bosons through this light-heavy
mixing. The light neutrino mass matrix can be diagonalized by a $3\times 3$
matrix here denoted as $U_{\rm PMNS}$

$\displaystyle U_{\rm PMNS}^{T}~{}m_{\nu}~{}U_{\rm PMNS}={\rm
diag}(m_{1},m_{2},m_{3}).$ (2.13)

As expected, a non-zero $\epsilon$ can lead to the mixing matrix $U$ becoming
non-unitary. But given the stringent constraints on $\epsilon$ [39], we can
take $U\simeq U_{\rm PMNS}$ without affecting our leptogenesis results.

Due to the light-RHN mixing, the charged-current interactions can be expressed
in terms of neutrino mass eigenstates as

$\displaystyle{\mathcal{L}_{\rm
CC}\supset-\frac{g}{\sqrt{2}}W_{\mu}\bar{e}_{\alpha}\gamma^{\mu}P_{L}V_{\alpha
i}N_{i}+\rm{h.c}.}$ (2.14)

Similarly, the neutral-current interactions in terms of the mass eigenstates
can be written as

$\displaystyle{\mathcal{L}_{\rm
NC}\supset-\frac{g}{2\cos\theta_{w}}Z_{\mu}\left[\left(\overline{\nu}_{m}\gamma^{\mu}P_{L}(U^{\dagger}V)_{mi}N_{i}+\rm{h.c.}\right)+\overline{N}_{m}\gamma^{\mu}P_{L}(V^{\dagger}V)_{mi}N_{i}\right],}$
(2.15)

where $\theta_{w}$ is the weak mixing angle. Due to these interactions, the
RHNs can decay into $\ell W$, $\nu Z$ and $\nu h$ final states. For RHNs
heavier than $W$, $Z$ and $h$, the decays are characterized as being 2 body
on-shell decays, where SM bosons will decay further into lighter SM particles.
The partial decay widths for these three processes are

$\displaystyle\Gamma(N_{i}\rightarrow\ell_{\alpha}W)$ $\displaystyle=$
$\displaystyle\frac{|V_{\alpha
i}|^{2}}{16\pi}\frac{(M_{N_{i}}^{2}-M_{W}^{2})^{2}(M_{N_{i}}^{2}+2M_{W}^{2})}{M_{N_{i}}^{3}v_{h}^{2}},$
$\displaystyle\Gamma(N_{i}\rightarrow\nu_{\alpha}Z)$ $\displaystyle=$
$\displaystyle\frac{|V_{\alpha
i}|^{2}}{32\pi}\frac{(M_{N_{i}}^{2}-M_{Z}^{2})^{2}(M_{N_{i}}^{2}+2M_{Z}^{2})}{M_{N_{i}}^{3}v_{h}^{2}},$
$\displaystyle\Gamma(N_{i}\rightarrow\nu_{\alpha}h)$ $\displaystyle=$
$\displaystyle\frac{|V_{\alpha
i}|^{2}}{32\pi}\frac{(M_{N_{i}}^{2}-M_{h}^{2})^{2}}{M_{N_{i}}v_{h}^{2}}.$
(2.16)

We will use the decay branching ratios of the RHNs into $\ell_{\alpha}W$ in
our collider analysis.

##  3 Resonant leptogenesis in general $U(1)_{X}$ scenario

The CP violating decays of the right-handed neutrinos are responsible for the
generation of lepton asymmetry in the resonant leptogenesis mechanism. The
amount of flavored asymmetry generated is proportional to the CP asymmetry
($\epsilon_{CP}$) in these decays.

$\displaystyle\varepsilon_{i\alpha}\,=\,\frac{\Gamma(N_{i}\to
L_{\alpha}H)-\Gamma(N_{i}\to\bar{L}_{\alpha}H^{c})}{\Gamma(N_{i}\to
L_{\alpha}H)+\Gamma(N_{i}\to\bar{L}_{\alpha}H^{c})}\,,$ (3.1)

The $(B+L)$-violating electroweak sphaleron processes convert this lepton
asymmetry at $T_{c}$ to baryon asymmetry, where below the critical temperature
$T_{c}\sim 149$ GeV these sphaleron processes freeze-out.

The dominant contribution to the lepton asymmetry arises from the interference
between the tree and self-energy diagrams in the $N_{i}$ decay. This
contribution can be enhanced if the intermediate state $N_{j}(j\neq i)$ is
quasi-degenerate with $N_{i}$. This resonantly enhanced $\varepsilon_{CP}$ in
terms of RHN masses and neutrino Dirac Yukawa matrix $Y_{D}$ can be expressed
as :

$\displaystyle\varepsilon_{i\alpha}\simeq\frac{1}{8\pi\left(Y_{D}^{\dagger}Y_{D}\right)_{ii}}\sum_{j\neq
i}{\rm Im}\left[\left(Y_{D}^{\star}\right)_{\alpha
i}\left(Y_{D}\right)_{\alpha j}\right]\,{\rm
Re}\left[\left(Y_{D}^{\dagger}Y_{D}\right)_{ij}\right]\frac{M_{i}M_{j}(M_{i}^{2}-M_{j}^{2})}{(M_{i}^{2}-M_{j}^{2})^{2}+A_{ij}^{2}}\,,$
(3.2)

where $A_{ij}$ denotes the regulator controlling the behaviour of decay
asymmetry in the degenerate limit $\Delta M_{ij}\rightarrow 0$. There are two
different contributions to total CP asymmetry from RHN mixing as well as from
oscillations, essentially with a similar form as above but with different
regulators.

$\displaystyle A_{ij}^{\rm mix}\,$ $\displaystyle=\,M_{i}\Gamma_{j}\,,\quad$
$\displaystyle\quad A_{ij}^{\rm osc}\,$
$\displaystyle=\,(M_{1}\Gamma_{1}+M_{2}\Gamma_{2})\left[\frac{{\rm
det}\left({\rm
Re}\left(Y_{D}^{\dagger}Y_{D}\right)\right)}{\left(Y_{D}^{\dagger}Y_{D}\right)_{ii}\left(Y_{D}^{\dagger}Y_{D}\right)_{jj}}\right]^{1/2}\,.$
(3.3)

Therefore, the total CP asymmetry is given by $\varepsilon_{\rm
tot}=\varepsilon_{\rm mix}+\varepsilon_{\rm osc}$ [18, 40, 41].

After semi-analytically solving the flavored Boltzmann transport equations,
the total lepton asymmetry produced after the sphaleron freezeout can be
written in the following form

$\displaystyle\eta_{L_{\alpha}}\ \simeq\ \frac{3}{2z_{c}K_{\alpha}^{\rm
eff}}\sum_{i}\varepsilon_{i\alpha}d_{i}\,,$ (3.4)

where $z_{c}=M_{N}/T_{c}$, $K_{\alpha}^{\rm eff}$ are the effective washout
factors in presence of $Y_{D}$ and any additional interactions (including the
effect of the real intermediate state subtracted collision terms), and $d_{i}$
are the corresponding dilution factors given in terms of ratios of thermally-
averaged rates for decays and scatterings involving $N_{i}$ (see Ref. [18] for
details).

The total baryon asymmetry $\eta_{B}$ generated from $\eta_{L}$ after taking
sphaleron efficiency and entropy dilution into account is given by

$\eta_{B}=-\frac{28}{79}\frac{1}{27.3}\sum_{\alpha}\eta_{L_{\alpha}}$ (3.5)

This is to be compared against the observed baryon asymmetry of the Universe
$\eta_{\text{BAU}}=(6.12\pm 0.08)\times 10^{-10}$ [7].

Since enough $\eta_{B}$ comparable to the $\eta_{\text{BAU}}$ can be produced
in multiple corners of the parameter space of the $U(1)_{X}$ model, we
specifically focus on maximizing the $\eta_{B}$ produced for a given RHN mass
scale $M_{N}$ and $U(1)_{X}$ gauge boson mass $M_{Z}^{\prime}$. Given the
time-complexity of the flavored Boltzmann equations, in general maximizing
$\eta_{B}$ is a formidable task. Therefore, we choose to maximize CP asymmetry
parameter $\varepsilon_{tot}$ as a function of $Y_{D}$ and $\Delta M_{N}$. We
can conveniently parameterize $Y_{D}$ using the Casas-Ibarra form

$Y_{D}\,=\,\frac{\sqrt{2}\,i}{v}U_{\text{PMNS}}\sqrt{{D}_{\nu}}O\sqrt{{D}_{N}}\,$
(3.6)

where $D_{N}=\text{diag}(M_{1},M_{2})$,
$D_{\nu}=\text{diag}(m_{1},m_{2},m_{3})$333To reduce the number of free
parameters in this study, we choose the lightest neutrino to be massless. and
$O$ is an arbitrary $2\times 3$ orthogonal matrix. When the total CP asymmetry
contribution is dominated by the mixing case, the $\Delta M_{N}$ that
maximizes $\varepsilon$ is given by $\Delta M_{N}\sim 0.5\Gamma_{N}$, where
$\Gamma_{N}$ is the average decay width of $N_{i}$-pair. However for low-scale
leptogenesis both contributions are of equal importance, which leads to a
modified relation for the optimum mass-splitting [35]

${\Delta\,M_{N,\,{\rm opt}}}\,\sim 1.23\,{\Gamma_{N}}\,,$ (3.7)

The above factor of 1.23 is obtained numerically by maximizing
$\varepsilon_{\rm tot}$ including the contributions from both regulators
$A_{ij}^{\rm mix}$ and $A_{ij}^{\rm osc}$. Since $\Gamma_{N}$ scales as
$M_{N}^{2}$ (also including the $M_{N}$ dependence of $Y_{D}$),
$\Delta\,M_{N,\,{\rm opt}}$ increases quadratically with increasing $M_{N}$.

Figure 1: Prediction of the baryon asymmetry for $\eta_{B}\geq\eta_{B}^{\rm
obs}$ in the $(M_{Z^{\prime}},M_{N})$ plane for a fixed $g_{B-L}=1\,(0.3)$ for
NH (IH) in the $U(1)_{B-L}$ case. The green region indicates optimum RHN mass
splitting ($\Delta M_{N,\text{opt}}$, see Eq. (3.7), and blue region indicates
$\Delta M_{N}=10\,\Delta M_{N,\text{opt}}$. The contours show
$\sigma_{\mathrm{prod}}$ (in ab/fb) at the $\sqrt{s}=14$ TeV LHC
(solid/dashed) and at $\sqrt{s}=100$ TeV future collider (dot-dashed, dotted).
Note : Blue region lies completely inside the green region.

Figure 2: Prediction of the baryon asymmetry for $\eta_{B}\geq\eta_{B}^{\rm
obs}$ shown in green (blue) region in the $(M_{Z^{\prime}},M_{N})$ plane for
$g_{B-L}=0.3\,(1)$ for both NH and IH in $U(1)_{B-L}$ case. The contours show
$\sigma_{\mathrm{prod}}$ (in ab/fb) at the $\sqrt{s}=14$ TeV LHC ($g_{B-L}=1$,
solid) and at $\sqrt{s}=100$ TeV future collider ($g_{B-L}=0.3$, dashed) and
($g_{B-L}=1$, dot-dashed). Note : Blue region lies completely inside the green
region.

Figure 3: Prediction of the baryon asymmetry for $\eta_{B}\geq\eta_{B}^{\rm
obs}$ shown in green (blue) region in the $(M_{Z^{\prime}},M_{N})$ plane for
$g_{R}=0.3\,(1)$ for both NH and IH in $U(1)_{R}$ case. The contours show
$\sigma_{\mathrm{prod}}$ (in ab/fb) at the $\sqrt{s}=14$ TeV LHC ($g_{R}=1$,
solid), ($g_{R}=0.3$, dotted) and at $\sqrt{s}=100$ TeV future collider
($g_{R}=0.3$, dashed) and ($g_{R}=1$, dot-dashed). Note : Blue region lies
completely inside the green region.

Figure 4: Prediction of the baryon asymmetry for $\eta_{B}\geq\eta_{B}^{\rm
obs}$ shown in green (blue) region in the $(M_{Z^{\prime}},M_{N})$ plane for
$g_{C}=0.3\,(1)$ for both NH and IH in $U(1)_{C}$ case. The contours show
$\sigma_{\mathrm{prod}}$ (in ab/fb) at the $\sqrt{s}=14$ TeV LHC ($g_{C}=1$,
solid) and at $\sqrt{s}=100$ TeV future collider ($g_{C}=0.3$, dashed) and
($g_{C}=1$, dot-dashed). Note : Blue region lies completely inside the green
region.

Figure 5:  Baryon asymmetry production in flavored case (unflavored) for
$\eta_{B}\geq\eta_{B}^{\rm obs}$ shown in green (grey) region in the
$(M_{Z^{\prime}},M_{N})$ plane for NH and different $g_{X}=1$, where
$X={(B-L,R,C)}$. The contours show $\sigma_{\mathrm{prod}}$ (in ab/fb) at the
$\sqrt{s}=14$ TeV LHC (solid) and at $\sqrt{s}=100$ TeV future collider
(dashed).

##  4 Results and Discussion

In this section, we compute the baryon asymmetry production in three scenarios
with different $U(1)_{X}$ charges (described in Sec. 2), with the flavored
Boltzmann transport formalism as described in previous section [18]. Firstly,
we show the effect of mass-splitting on the predicted baryon asymmetry in the
$U(1)_{B-L}$ case as a function of $(M_{Z^{\prime}},M_{N})$ for
$g_{B-L}=1\,(0.3)$ for NH (IH) scenario. For these chosen values of the gauge-
couplings, we choose to restrict $M_{Z^{\prime}}\geq 6$ TeV, to be consistent
with the dilepton bounds from LEP-II [38]. The colored regions indicate the
parameter space for $\eta_{B}\geq\eta_{B}^{\rm obs}$. The green region
indicates optimum RHN mass splitting $\Delta M_{N,\text{opt}}$ and blue region
indicates $\Delta M_{N}=10\,\Delta M_{N,\text{opt}}$. We clearly find that
setting the $\Delta M_{N}$ to the optimum value gives the maximum parameter
space consistent with the condition $\eta_{B}\geq\eta_{B}^{\rm obs}$. On
increasing the $\Delta M_{N}$ by a factor of 10, we notice the parameter space
reduces while still being a subset of the maximum parameter space allowed for
optimal $\Delta M_{N}$. This is a general feature of the baryon asymmetry
production for all $U(1)_{X}$ charges. Hence, we will hereafter set $\Delta
M_{N}$ to the optimum value in all cases. Furthermore, we also include the
collider sensitivity of the relevant parameter space in
$(M_{Z^{\prime}},M_{N})$ plane. The contours show $\sigma_{\mathrm{prod}}$ (in
ab/fb) at the $\sqrt{s}=14$ TeV LHC (solid/dashed) and at $\sqrt{s}=100$ TeV
future collider (dot-dashed, dotted).

We now plot the predicted baryon asymmetry $\eta_{B}$ in the
$(M_{Z^{\prime}},M_{N})$ plane for a fixed $g_{X}=(0.3,1)$ for both NH and IH
in the $U(1)_{X},\,{X=(B\text{-}L,R,C)}$, shown in Figs. 2,3 and 4 for optimal
$\Delta M_{N}$. For $U(1)_{B\text{-}L}$ in Fig. 2, successful leptogenesis is
possible for $M_{N}>0.5(0.8)$ TeV for NH(IH) and $g_{B-L}=0.3$. Upon
increasing the gauge coupling $g_{B-L}=1$, the parameter space shrinks due to
increased washout effects and dilution. In this case, successful leptogenesis
is only possible for $M_{N}>1.5(1.8)$ TeV for NH(IH), with
$\sigma_{\mathrm{prod}}^{\text{NH}}=2$ ab at LHC and
$\sigma_{\mathrm{prod}}^{\text{IH}}=10$ fb at future collider.

For $U(1)_{R}$ in Fig. 3, it can be seen that successful leptogenesis is
possible for $M_{N}$ as low as $0.2$ TeV for both NH and IH if $g_{R}=0.3$,
while for $g_{R}=1$, $M_{N}>0.2(1.2)$ TeV for NH(IH) is required. For all
these cases, $\sigma_{\mathrm{prod}}$ at LHC reaches $\sim 1$-$1.5$ ab.
Similarly for $U(1)_{C}$ in Fig. 4, successful leptogenesis is possible for
$M_{N}>0.4(0.6)$ TeV for NH(IH) and $g_{C}=0.3$, while for $g_{C}=1$,
$M_{N}>1.3(1.6)$ TeV for NH(IH) is required. The $\sigma_{\mathrm{prod}}$
reach at LHC for NH and IH for $g_{C}=1$ is around $\sim 1$ ab.

In addition to studying the flavored case, we also compare our results with
the unflavored regime for all three $U(1)_{X}$ scenarios as shown in Fig. 5.
In this figure, we plot the prediction for the baryon asymmetry production in
the flavored case (shown in red) and unflavored case (shown in grey) in the
$(M_{Z^{\prime}},M_{N})$ plane. We have chosen to set the respective gauge
coupling in each case to be unity and with ordering set to NH. It can be
clearly seen that in all cases, the unflavored treatment usually overestimates
the parameter space for required baryon asymmetry. For eg., successful
leptogenesis in $U(1)_{B\text{-}L}$ for unflavored case is possible for
$M_{N}>0.3$ TeV, while in the proper treatment involving the flavored
transport equations, $M_{N}>1.5$ TeV is required. Hence, flavor effects play
an important role in determination of $\eta_{B}$ generated through resonant
leptogenesis [29].

##  5 Conclusion

We have studied the generation of baryon asymmetry through the resonant
leptogenesis for the $U(1)_{X}$ extension of the SM. We numerically solve the
flavored Boltzmann transport equations to calculate the total baryon asymmetry
generated. After maximizing the $\eta_{B}$ for given $M_{N}$ and
$M_{Z^{\prime}}$, we show that the three different cases considered in this
work can naturally explain the observed baryon asymmetry of the Universe in
the large portion of the available parameter space. We find that the
$U(1)_{C}$ case offers successful leptogenesis in a larger portion of the
parameter space as compared to $U(1)_{B\text{-}L}$ and $U(1)_{R}$. We also
perform a comparative study between the flavored and unflavored leptogenesis,
showcasing the impact and importance of the flavor effects at play. Finally,
we have also studied the collider prospects for all these different scenarios.
We find that although HL-LHC might not be able to probe all the relevant
parameter space, $\sqrt{s}=100$ TeV future collider can access these regions,
thereby offering an opportunity to test resonant leptogenesis in the
$U(1)_{X}$ model.

## Acknowledgements

GC thanks Arindam Das and Bhupal Dev for discussions and for collaboration
during the early stages of this work. The work of GC is supported by the U.S.
Department of Energy under the award number DE-SC0020250 and DE-SC0020262. GC
also acknowledges the Center for Theoretical Underground Physics and Related
Areas (CETUP* 2024) and the Institute for Underground Science at SURF for
hospitality and for providing a stimulating environment, where this work was
finalized.

Note Added: While we were finalizing this work, Ref. [42] appeared, primarily
based on an earlier version of this draft. But they have not included the
flavor effects, which are known to be important for TeV-scale leptogenesis
[29]. Neither did they perform a scan of the $(M_{N},M_{Z^{\prime}})$
parameter space for collider tests as presented here.

##  Appendix A Appendix

In this section, we provide the reduced cross sections used in this work for
calculating the $\eta_{B}$ using flavored Boltzmaan transport equations. The
form of
$\hat{\sigma}=\frac{8\sigma}{s}[(p_{\text{in}}.p_{\text{out}})^{2}-m_{f}^{4}]$
where $\sigma$ is the total cross section in center of mass frame for the
processes participating in the resonant leptogenesis, $p_{\text{in}}$ and
$p_{\text{out}}$ are the momenta of incoming and outgoing state particles. The
reduced cross sections involving Higgs $(h)$ have been obtained following
[43].

  * (a)

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

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

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

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

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

  * (b)

$\ell h\to\ell h$ process mediated by $N$ in the $s$-channel and $t$\- channel

$\displaystyle\hat{\sigma}(\ell h\leftrightarrow\ell h)$ $\displaystyle=$
$\displaystyle\frac{2\alpha^{2}\pi~{}m_{N_{1}}^{2}}{\sin^{4}\theta_{W}M_{W}^{4}s}\Bigg{\\{}\sum_{i=1}^{2}\frac{m_{N_{i}}^{2}}{m_{N_{1}}^{2}}(m_{D}^{\dagger}m_{D})_{ii}^{2}\Bigg{[}\frac{s}{m_{N_{i}}^{2}}+\frac{2s}{m_{N_{1}}^{2}\mathcal{D}_{i}}+$
(A.3)
$\displaystyle\frac{s^{2}}{2m_{N_{1}}^{4}\mathcal{D}_{i}^{2}}-\Bigg{(}1+2\frac{s+m_{N_{i}}^{2}}{m_{N_{1}}^{2}\mathcal{D}_{i}}\Bigg{)}\log\Bigg{(}1+\frac{s}{m_{N_{j}}^{2}}\Bigg{)}\Bigg{]}+2\frac{m_{N_{2}}}{m_{N_{1}}}$
$\displaystyle\mathcal{R}e\Bigg{[}(m_{D}^{\dagger}m_{D})_{12}^{2})\Bigg{]}\Bigg{[}\frac{s}{m_{N_{1}}^{2}\mathcal{D}_{1}}+\frac{s}{m_{N_{1}}^{2}\mathcal{D}_{2}}+\frac{s^{2}}{2m_{N_{1}}^{4}\mathcal{D}_{1}\mathcal{D}_{2}}-$
$\displaystyle\frac{(s+m_{N_{1}}^{2})(s+m_{N_{1}}^{2}-2m_{N_{1}}^{2}m_{N_{2}}^{2})}{(m_{N_{1}}^{2}-m_{N_{2}}^{2})\mathcal{D}_{2}}\log\Bigg{(}\frac{s}{m_{N_{1}}^{2}}+1\Bigg{)}$
$\displaystyle-$
$\displaystyle\frac{(s+m_{N_{1}}^{2})(s+m_{N_{1}}^{2}-2m_{N_{1}}m_{N_{2}})}{(m_{N_{1}}-m_{N_{2}})m_{N_{1}}^{3}\mathcal{D}_{2}}\log\Big{(}\frac{s+m_{N_{1}}^{2}}{m_{N_{1}}^{2}}\Big{)}\Bigg{]}$
$\displaystyle-$
$\displaystyle\frac{(s+m_{N_{1}}m_{N_{2}})(s+m_{N_{1}}m_{N_{2}}-2m_{N_{1}}^{2})}{(m_{N_{1}}-m_{N_{2}})m_{N_{1}}^{3}\mathcal{D}_{2}}\log\Bigg{(}\frac{s+m_{N_{1}}m_{N_{2}}}{m_{N_{1}}m_{N_{2}}}\Bigg{)}\Bigg{\\}}~{}~{}~{}$

where
$\mathcal{D}_{i}=\frac{(s-m_{N_{j}}m_{N_{1}})^{2}+m_{N_{j}}m_{N_{1}}\Gamma_{ij}^{2}}{m_{N_{1}}^{2}(s-m_{N_{i}}m_{N_{1}})}$
is the off-shell part of the propagator.

  * (c)

$\ell\ell\leftrightarrow hh$ process mediated by $N$ in the $t$\- channel

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

  * (d)

Pair production of $N$ from different initial SM charged fermions:

$\displaystyle\hat{\sigma}(f\bar{f}\leftrightarrow
N_{i}N_{i})=\frac{g_{X}^{4}x_{\Phi}^{2}s^{2}}{12\pi}\Bigg{[}\frac{(1-\frac{4m_{N_{i}}^{2}}{s})^{\frac{3}{2}}}{(s-{m_{Z}^{\prime}}^{2})^{2}+{\Gamma_{Z}^{\prime}}^{2}{m_{Z}^{\prime}}^{2}}\Bigg{]}(C_{A}^{2}+C_{V}^{2})$
(A.5)

where the vector and axial-vector couplings are given in Tab. 2 considering
$x_{\Phi}=1$. This cross section should be averaged over the color factor for
quarks.

Type of fermion | vector coupling $(C_{V})$ | axial vector coupling $(C_{A})$
---|---|---
| $\frac{q_{f_{L}}+q_{f_{R}}}{2}$ | $\frac{q_{f_{L}}-q_{f_{R}}}{2}$
charged lepton $(\ell)$ | $-\frac{3}{4}x_{H}-1$ | $\frac{1}{4}x_{H}$
up-type quarks $(q_{u})$ | $\frac{5}{12}x_{H}+\frac{1}{3}$ | $-\frac{1}{4}x_{H}$
down-type quarks $(q_{d})$ | $-\frac{1}{12}x_{H}+\frac{1}{3}$ | $-\frac{1}{4}x_{H}$
Table 2: Vector and axial-vector couplings of different SM charged fermions
with $Z^{\prime}$ where the axial vector couplings vanish for $x_{H}=0$ case.

  * (e)

For $i\neq j$, RHN pair production cross section is

$\displaystyle\sigma(N_{i}N_{i}\leftrightarrow N_{j}N_{j})$ $\displaystyle=$
$\displaystyle\frac{g_{X}^{4}x_{\Phi}^{4}\sqrt{(s-4m_{N_{j}}^{2})(s-4m_{N_{i}}^{2})}}{72\pi\\{(s-m_{Z^{\prime}}^{2})^{2}+\Gamma_{Z^{\prime}}^{2}m_{Z^{\prime}}^{2}\\}}$
(A.6)
$\displaystyle\Bigg{[}(s-4m_{N_{j}}^{2})(s-4m_{N_{i}}^{2})+12\frac{m_{N_{i}}^{2}m_{N_{j}}^{2}}{m_{Z^{\prime}}^{4}}(s-m_{Z^{\prime}})^{2}\Bigg{]}$

For the other case,

$\displaystyle\hat{\sigma}(N_{i}N_{i}\leftrightarrow N_{i}N_{i})$
$\displaystyle=$
$\displaystyle\frac{g_{X}^{4}x_{\Phi}^{4}}{128\pi}\Bigg{[}\frac{(s-4m_{N_{i}}^{2})^{3}}{3s((s-m_{Z^{\prime}}^{2})^{2}+m_{Z^{\prime}}^{2}\Gamma_{Z^{\prime}}^{2})}+\frac{(s-4m_{N_{i}}^{2})}{sm_{Z^{\prime}}^{4}(s-4m_{N_{i}}^{2}+m_{Z^{\prime}}^{2})}\Big{\\{}m_{Z^{\prime}}^{2}(s-4m_{N_{i}}^{2})^{2}+$
(A.7) $\displaystyle
2(m_{Z^{\prime}}^{2}-2m_{N_{i}}^{2})^{3}+s(8m_{N_{i}}^{4}+3m_{Z^{\prime}}^{4})-4m_{N_{i}}^{2}(m_{Z^{\prime}}^{4}+4m_{N_{i}}^{4}+m_{Z^{\prime}}^{2}m_{N_{i}}^{2})\Big{\\}}+$
$\displaystyle+\Big{\\{}\frac{(3m_{Z^{\prime}}^{2}-4m_{N_{i}}^{2})(s-4m_{N_{i}}^{2})^{2}+m_{Z^{\prime}}^{4}(3s-20m_{N_{i}}^{2})+2m_{Z^{\prime}}^{2}(m_{Z^{\prime}}^{4}+8m_{N_{i}}^{4})}{sm_{Z^{\prime}}^{2}(s-4m_{N_{i}}^{2}+2m_{Z^{\prime}}^{2})}\Big{\\}}$
$\displaystyle\log\Big{(}\frac{m_{Z^{\prime}}^{2}}{s-4m_{N_{i}}^{2}+m_{Z^{\prime}}^{2}}\Big{)}\Bigg{]}~{}~{}~{}~{}~{}$

  * (f)

Pair production of $N$ from Higgs $(h)$ in the $s$-channel:

$\displaystyle\hat{\sigma}(hh\leftrightarrow
N_{i}N_{i})=\frac{{g_{X}^{4}}x_{\Phi}^{4}}{12\pi}\frac{(x-4)^{\frac{3}{2}}(1-\frac{4{m_{\Phi}}^{2}}{xm_{N_{i}}^{2}})^{\frac{3}{2}}\sqrt{x}}{cy+(x-y)^{2}}$
(A.8)

where $x=\frac{s}{m_{N_{i}}^{2}}$,
$y=\frac{m_{Z^{\prime}}^{2}}{m_{N_{i}}^{2}}$,
$c=\frac{\Gamma_{Z^{\prime}}^{2}}{m_{N}^{2}}$ and $\Gamma_{Z^{\prime}}$ is the
total decay width of $Z^{\prime}$. Reduced cross section for $\Phi$ pair
production from RHN in $t-$channel and $u-$channel processes:

$\displaystyle\hat{\sigma}(N_{i}N_{i}\leftrightarrow
hh)=\frac{Y_{N}^{4}}{8\pi}\Bigg{(}1-\frac{4m_{N_{i}}^{2}}{s}\Bigg{)}\Bigg{[}\frac{s-4m_{N_{i}}^{2}}{2m_{N_{i}}^{2}}+\Bigg{(}1-\frac{4m_{N_{i}}^{2}}{s}\Bigg{)}^{\frac{3}{2}}+\log\Bigg{(}\frac{s-\sqrt{s(s-4m_{N_{i}}^{2})}}{s+\sqrt{s(s-4m_{N_{i}}^{2})}}\Bigg{)}\Bigg{]}~{}~{}~{}~{}~{}$
(A.9)

## References

  * [1] Particle Data Group collaboration, _Review of Particle Physics_ , _PTEP_ 2022 (2022) 083C01.
  * [2] P. Minkowski, _$\mu\to e\gamma$ at a Rate of One Out of $10^{9}$ Muon Decays?_, _Phys. Lett. B_ 67 (1977) 421.
  * [3] T. Yanagida, _Horizontal gauge symmetry and masses of neutrinos_ , _Conf. Proc. C_ 7902131 (1979) 95.
  * [4] M. Gell-Mann, P. Ramond and R. Slansky, _Complex Spinors and Unified Theories_ , _Conf. Proc. C_ 790927 (1979) 315 [1306.4669].
  * [5] R. N. Mohapatra and G. Senjanovic, _Neutrino Mass and Spontaneous Parity Nonconservation_ , _Phys. Rev. Lett._ 44 (1980) 912.
  * [6] S. L. Glashow, _The Future of Elementary Particle Physics_ , _NATO Sci. Ser. B_ 61 (1980) 687.
  * [7] Planck collaboration, _Planck 2018 results. VI. Cosmological parameters_ , _Astron. Astrophys._ 641 (2020) A6 [1807.06209].
  * [8] M. Fukugita and T. Yanagida, _Baryogenesis Without Grand Unification_ , _Phys. Lett. B_ 174 (1986) 45.
  * [9] 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.
  * [10] S. Davidson and A. Ibarra, _A Lower bound on the right-handed neutrino mass from leptogenesis_ , _Phys. Lett. B_ 535 (2002) 25 [hep-ph/0202239].
  * [11] W. Buchmuller, P. Di Bari and M. Plumacher, _Cosmic microwave background, matter - antimatter asymmetry and neutrino masses_ , _Nucl. Phys. B_ 643 (2002) 367 [hep-ph/0205349].
  * [12] K. Moffat, S. Pascoli, S. T. Petcov, H. Schulz and J. Turner, _Three-flavored nonresonant leptogenesis at intermediate scales_ , _Phys. Rev. D_ 98 (2018) 015036 [1804.05066].
  * [13] A. Pilaftsis and T. E. J. Underwood, _Resonant leptogenesis_ , _Nucl. Phys. B_ 692 (2004) 303 [hep-ph/0309342].
  * [14] A. Pilaftsis, _CP violation and baryogenesis due to heavy Majorana neutrinos_ , _Phys. Rev. D_ 56 (1997) 5431 [hep-ph/9707235].
  * [15] M. Drewes, B. Garbrecht, P. Hernandez, M. Kekic, J. Lopez-Pavon, J. Racker et al., _ARS Leptogenesis_ , _Int. J. Mod. Phys. A_ 33 (2018) 1842002 [1711.02862].
  * [16] A. Pilaftsis and T. E. J. Underwood, _Electroweak-scale resonant leptogenesis_ , _Phys. Rev. D_ 72 (2005) 113001 [hep-ph/0506107].
  * [17] F. F. Deppisch and A. Pilaftsis, _Lepton Flavour Violation and theta(13) in Minimal Resonant Leptogenesis_ , _Phys. Rev. D_ 83 (2011) 076007 [1012.1834].
  * [18] P. S. B. Dev, P. Millington, A. Pilaftsis and D. Teresi, _Flavour Covariant Transport Equations: an Application to Resonant Leptogenesis_ , _Nucl. Phys. B_ 886 (2014) 569 [1404.1003].
  * [19] P. S. B. Dev, M. Garny, J. Klaric, P. Millington and D. Teresi, _Resonant enhancement in leptogenesis_ , _Int. J. Mod. Phys. A_ 33 (2018) 1842003 [1711.02863].
  * [20] T. Appelquist, B. A. Dobrescu and A. R. Hopper, _Nonexotic Neutral Gauge Bosons_ , _Phys. Rev. D_ 68 (2003) 035012 [hep-ph/0212073].
  * [21] 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 [1011.4769].
  * [22] C. Coriano, L. Delle Rose and C. Marzo, _Vacuum Stability in U(1)-Prime Extensions of the Standard Model with TeV Scale Right Handed Neutrinos_ , _Phys. Lett. B_ 738 (2014) 13 [1407.8539].
  * [23] S. Oda, N. Okada and D.-s. Takahashi, _Classically conformal U(1)’ extended standard model and Higgs vacuum stability_ , _Phys. Rev. D_ 92 (2015) 015026 [1504.06291].
  * [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 (2016) 115038 [1605.01157].
  * [25] A. Davidson, _$B-L$ as the fourth color within an $SU(2)_{L}\times U(1)_{R}\times U(1)$ model_, _Phys. Rev. D_ 20 (1979) 776.
  * [26] R. E. Marshak and R. N. Mohapatra, _Quark - Lepton Symmetry and B-L as the U(1) Generator of the Electroweak Symmetry Group_ , _Phys. Lett. B_ 91 (1980) 222.
  * [27] W. Buchmuller, C. Greub and P. Minkowski, _Neutrino masses, neutral vector bosons and the scale of B-L breaking_ , _Phys. Lett. B_ 267 (1991) 395.
  * [28] S. Blanchet, Z. Chacko, S. S. Granor and R. N. Mohapatra, _Probing Resonant Leptogenesis at the LHC_ , _Phys. Rev. D_ 82 (2010) 076008 [0904.2174].
  * [29] P. S. B. Dev, P. Di Bari, B. Garbrecht, S. Lavignac, P. Millington and D. Teresi, _Flavor effects in leptogenesis_ , _Int. J. Mod. Phys. A_ 33 (2018) 1842001 [1711.02861].
  * [30] J. A. Casas and A. Ibarra, _Oscillating neutrinos and $\mu\to e,\gamma$_, _Nucl. Phys. B_ 618 (2001) 171 [hep-ph/0103065].
  * [31] L. Basso, A. Belyaev, S. Moretti and C. H. Shepherd-Themistocleous, _Phenomenology of the minimal B-L extension of the Standard model: Z’ and neutrinos_ , _Phys. Rev. D_ 80 (2009) 055030 [0812.4313].
  * [32] P. Fileviez Perez, T. Han and T. Li, _Testability of Type I Seesaw at the CERN LHC: Revealing the Existence of the B-L Symmetry_ , _Phys. Rev. D_ 80 (2009) 073015 [0907.4186].
  * [33] Z. Kang, P. Ko and J. Li, _New Avenues to Heavy Right-handed Neutrinos with Pair Production at Hadronic Colliders_ , _Phys. Rev. D_ 93 (2016) 075037 [1512.08373].
  * [34] P. Cox, C. Han and T. T. Yanagida, _LHC Search for Right-handed Neutrinos in $Z^{\prime}$ Models_, _JHEP_ 01 (2018) 037 [1707.04532].
  * [35] 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 [2112.09710].
  * [36] B. Dutta, S. Ghosh and J. Kumar, _A sub-GeV dark matter model_ , _Phys. Rev. D_ 100 (2019) 075028 [1905.02692].
  * [37] A. Das, N. Okada and D. Raut, _Enhanced pair production of heavy Majorana neutrinos at the LHC_ , _Phys. Rev. D_ 97 (2018) 115023 [1710.03377].
  * [38] A. Das, P. S. B. Dev, Y. Hosotani and S. Mandal, _Probing the minimal U(1)X model at future electron-positron colliders via fermion pair-production channels_ , _Phys. Rev. D_ 105 (2022) 115030 [2104.10902].
  * [39] M. Blennow, E. Fernández-Martínez, J. Hernández-García, J. López-Pavón, X. Marcano and D. Naredo-Tuero, _Bounds on lepton non-unitarity and heavy neutrino mixing_ , _JHEP_ 08 (2023) 030 [2306.01040].
  * [40] P. S. B. Dev, P. Millington, A. Pilaftsis and D. Teresi, _Kadanoff–Baym approach to flavour mixing and oscillations in resonant leptogenesis_ , _Nucl. Phys. B_ 891 (2015) 128 [1410.6434].
  * [41] J. Klarić, M. Shaposhnikov and I. Timiryasov, _Uniting Low-Scale Leptogenesis Mechanisms_ , _Phys. Rev. Lett._ 127 (2021) 111802 [2008.13771].
  * [42] A. Das and Y. Orikasa, _Resonant leptogenesis in minimal $U(1)_{X}$ extensions of the Standard Model_, 2407.05644.
  * [43] M. Plumacher, _Baryogenesis and lepton number violation_ , _Z. Phys. C_ 74 (1997) 549 [hep-ph/9604229].


# Mock Maass Forms Revisited

Yingkun Li  and  Christina Roehrig  Fachbereich Mathematik, Technische
Universität Darmstadt, Schlossgartenstrasse 7, D–64289 Darmstadt, Germany
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this paper, we use theta integrals to give a different construction of mock
Maass forms studied by Sander Zwegers. With this method, we construct new
real-analytic modular forms, whose Fourier coefficients are logarithms of
algebraic numbers in a real quadratic field.

##  1\. Introduction

In [Coh88], Cohen shows that the function

$\varphi_{0}(\tau):=\sqrt{v}\sum_{\begin{subarray}{c}n\in\mathbb{Z}\\\ n\neq
0\end{subarray}}T(n)\,e^{2\pi
inu/24}\,K_{0}(2\pi|n|v/24)\quad(\tau=u+iv\in\mathbb{H})$

is a weight 0 Maass form, i.e. it transforms as a modular form of weight 0 on
$\Gamma_{0}(2)$ up to some explicitly given multiplier and it is an
eigenfunction of the weight 0 Laplace operator $\Delta_{\tau}$ defined in
(1.1). Here, $K_{0}$ denotes a modified Bessel function of the second kind,
which makes immediately clear that the second property holds (see (2.18) for
the exact definition). As an arithemtic function, $T(n)$ is defined for any
$n\equiv 1\ (\operatorname{mod}24)$ and describes the excess of the number of
inequivalent solutions $(x,y)$ of Pell’s equation $x^{2}-6y^{2}=n$ with
$x+3y\equiv\pm 1\ (\operatorname{mod}12)$ over the number of inequivalent
solutions with $x+3y\equiv\pm 5\ (\operatorname{mod}12)$.

One can also establish the relation

$\sum_{\begin{subarray}{c}n\in\mathbb{Z}\\\ n\equiv 1\
(\operatorname{mod}24)\end{subarray}}T(n)\,q^{|n|/24}=q^{1/24}\sigma(q)+q^{-1/24}\sigma^{\ast}(q),$

where $\sigma$ and $\sigma^{\ast}$ are the $q$-hypergeometric functions
defined by

$\displaystyle\sigma(q)$ $\displaystyle:=\sum_{n\geq
0}\frac{q^{n(n+1)/2}}{(1+q)\dots(1+q^{n})}=1+\sum_{n\geq
0}(-1)^{n}q^{n+1}(1-q)\ldots(1-q^{n})$
$\displaystyle=1+q-q^{2}+2q^{3}-2q^{4}+q^{5}+\ldots,$

respectively

$\displaystyle\sigma^{\ast}(q)$ $\displaystyle:=2\sum_{n\geq
1}\frac{(-1)^{n}q^{n^{2}}}{(1-q)(1-q^{3})\dots(1-q^{2n-1})}=-2\sum_{n\geq
0}q^{n+1}(1-q^{2})(1-q^{4})\ldots(1-q^{2n})$
$\displaystyle=-2q-2q^{2}-2q^{3}+2q^{7}+2q^{8}+2q^{10}+\ldots.$

These functions can be rewritten as indefinite theta functions. Zwegers
[Zwe12] used this to put Cohen’s example in a more general framework and
interpreted $\varphi_{0}$ as the “harmonic” part $\Phi^{c_{1},c_{2}}_{a,b}$ of
a real-analytic modular form $\hat{\Phi}^{c_{1},c_{2}}_{a,b}$, where harmonic
means that the function vanishes under the modified Laplace operator

(1.1)
$\tilde{\Delta}:=\Delta_{\tau}+\frac{1}{4},~{}\Delta_{\tau}:=v^{2}(\partial_{u}^{2}+\partial_{v}^{2}).$

In general, Zwegers showed that $\tilde{\Delta}\hat{\Phi}^{c_{1},c_{2}}_{a,b}$
is the difference of two theta series $\theta^{(1,1)}$ depending on negative
lines $c_{1}$ and $c_{2}$ respectively (see Theorem 3.3 for details).

In this paper, we will use the theta integral approach as in [FK17, CL20] to
extend these results and construct a preimage $\tilde{\vartheta}$ of one such
theta function $\theta^{(1,1)}$ under the operator $\tilde{\Delta}$.
Furthermore, the harmonic part Fourier coefficients of $\tilde{\vartheta}$ are
logarithms of algebraic numbers in real quadratic fields, similar to the ones
studied in [CL20].

######  Theorem 1.1.

Let $L$ be an anisotropic lattice of signature $(1,1)$ and
$\theta^{(1,1)}_{L+h}(\tau,t)$ the theta function defined in (2.12) for $h\in
L^{*}/L$. Then there exists $\tilde{\vartheta}_{L+h}(\tau,t)$ such that

(1.2)
$\tilde{\Delta}\tilde{\vartheta}_{L+h}(\tau,t)=\theta^{(1,1)}_{L+h}(\tau,t)$

for all $t>0$, and whose harmonic part coefficients $\tilde{c}_{t}$ are given
in (3.17). Furthermore, we have
$\Phi^{c_{1},c_{2}}_{h,0}=\hat{\vartheta}_{L+h}^{c_{1},c_{2}}(\tau)=\tilde{\vartheta}_{L+h}(\tau,t_{2})-\tilde{\vartheta}_{L+h}(\tau,t_{1})$
when $c_{i}=c(t_{i})$ via (2.7) with $\hat{\vartheta}$ defined in (3.1).

######  Remark 1.2.

Similar results should also hold in the isotropic case, as in [Li17].

######  Remark 1.3.

In Examples 3.8 and 3.9, we show that our result goes beyond the work of
Zwegers and produces a new type of modular form.

The paper is organized in the following way. We recall preliminary notions of
theta functions in Section 2. Section 3 contains our main construction. The
first half rephrases the construction of Zwegers. The second half extends it
and proves Theorem 3.7, which implies Theorem 1.1. The last section proposes a
generalization of the construction in Section 3, and some future directions.

Acknowledgement: The authors are supported by the Deutsche
Forschungsgemeinschaft (DFG) through the Collaborative Research Centre TRR 326
_Geometry and Arithmetic of Uniformized Structures_ , project number
444845124.

##  2\. Preliminaries

In this section, we recall preliminary notions of quadratic space, unary theta
functions and theta kernels that will be used in constructing mock Maass forms
later. As usual, we denote $\mathbf{e}(z):=e^{2\pi iz}$.

Let $L$ be an even, integral lattice of rank $n$ with a quadratic form $Q$ and
associated bilinear form $B(\cdot,\cdot)$, and denote $L^{*}\subset
V:=L\otimes\mathbb{Q}$ its dual lattice. When $Q$ is positive definite, we
have the theta function

(2.1) $\theta_{L+h,p}(\tau):=\sum_{x\in L+h}p(x)\mathbf{e}(Q(x)\tau)\quad(h\in
V),$

where $p:V(\mathbb{R})\to\mathbb{C}$ is a harmonic polynomial of degree $d$.
We omit the subscript $p$ when it is the constant 1. The theta function is a
modular form of weight $d+n/2$ and a congruence subgroup of
${\mathrm{Mp}}_{2}(\mathbb{R})$. The vector-valued modular form

(2.2) $\Theta_{L,p}(\tau)=(\theta_{L+h,p}(\tau))_{h\in L^{*}/L}$

transforms with respect to the Weil representation $\rho_{L}$ on
${\mathrm{Mp}}_{2}(\mathbb{Z})$. See [Bor98, Sch09] for more information about
$\rho_{L}$. When $Q$ is negative definite, we set

(2.3) $\theta_{L+h,p}(\tau):=v^{d+n/2}\theta_{L^{-}+h,p}(-\overline{\tau}),$

where $L^{-}$ denotes the lattice $L$ with the quadratic form $-Q$. When $L$
has rank 1, the polynomial $p$ is necessarily linear and we denote

(2.4)
$\theta_{L+h}^{(\epsilon)}:=\theta_{L+h,x^{\epsilon}},~{}\Theta_{L}^{(\epsilon)}:=\Theta_{L,x^{\epsilon}}$

for $\epsilon\in\\{0,1\\}$.

When $Q$ is not definite, i.e. of signature $(n^{+},n^{-})$ with $n^{\pm}\geq
1$, one can attach to $L$ a two variable theta function in the following way.
Let ${\mathcal{D}}$ be the Grassmannian of negative $n^{-}$-planes in
$V(\mathbb{R})\cong\mathbb{R}^{n^{+}}\oplus(\mathbb{R}^{n^{-}})^{-}$ with
$\mathbb{R}^{n}$ the definite $\mathbb{R}$-quadratic space with quadratic form
$Q((x_{1},\dots,x_{n})):=\sum_{1\leq i\leq n}x_{i}^{2}/2$. For each
$Z\in{\mathcal{D}}$ and $\lambda\in V(\mathbb{R})$, let $Z^{\perp}\subset
V(\mathbb{R})$ be its orthogonal complement and
$\lambda_{Z},\lambda_{Z^{\perp}}$ the projections of $\lambda$ such that
$Q(\lambda)=Q(\lambda_{Z^{\perp}})+Q(\lambda_{Z})$. Then the Siegel theta
function given by

(2.5) $\theta_{L+h}(\tau,Z):=v^{n^{-}/2}\sum_{\lambda\in
L+h}\mathbf{e}\left(Q(\lambda_{Z^{\perp}})\tau+Q(\lambda_{Z})\overline{\tau}\right)$

is a real-analytic modular form in $\tau$ of weight $\frac{n^{+}-n^{-}}{2}$.
More generally, one can add a harmonic polynomial $p$ of degree $d^{+}$, resp.
$d^{-}$, in the coordinates of ${Z^{\perp}}\cong\mathbb{R}^{n^{+}}$, resp.
$Z\cong(\mathbb{R}^{n^{-}})^{-}$, to define $\theta_{L+h,p}$, which will be a
real-analytic modular form of weight $\frac{n^{+}-n^{-}}{2}+d^{+}-d^{-}$.

This is a special case of the theta series considered by Borcherds. In Theorem
4.1 in [Bor98], the explicit transformation behavior is given. For example,
when $V$ is indefinite of dimension 2, it will have signature $(1,1)$ and
$V(\mathbb{R})$ is isometric to the real hyperbolic plane
$V_{0}(\mathbb{R})=\mathbb{R}^{2}$ with quadratic form $Q_{0}$ given by

(2.6) $Q_{0}((x_{1},x_{2})):=x_{1}x_{2}.$

Let $B_{0}$ be the associated bilinear form. The symmetric space
${\mathcal{D}}$ of $V_{0}$ is parametrized by $\mathbb{R}^{\times}$ via

(2.7) $\begin{split}c:\mathbb{R}^{\times}&\to{\mathcal{D}}\\\
t&\mapsto\mathbb{R}\cdot Z^{-}_{t},~{}\end{split}$

where we denote

(2.8) $Z^{\pm}_{t}:=\left({t}{},\pm{t^{-1}}{}\right)\in V_{0}(\mathbb{R}).$

The space ${\mathcal{D}}$ has two connected components, and we denote by
${\mathcal{D}}^{+}$ the component parametrized by $\mathbb{R}^{\times}_{+}$
under $c$. For any $\lambda=(\lambda_{1},\lambda_{2})\in V_{0}(\mathbb{R})$,
we can write it as

(2.9)
$\lambda=\lambda_{t}^{+}Z_{t}^{+}-\lambda_{t}^{-}Z_{t}^{-},~{}\lambda_{t}^{\pm}:=\frac{B_{0}(\lambda,Z_{t}^{\pm})}{{2}}=\frac{\pm\lambda_{1}t^{-1}+\lambda_{2}t}{{2}}\in\mathbb{R},$

which satisfy the equations

(2.10)
$(\lambda^{+}_{t})^{2}+(\lambda^{-}_{t})^{2}=\frac{\lambda^{2}_{1}t^{-2}+\lambda^{2}_{2}t^{2}}{{2}},~{}(\lambda^{+}_{t})^{2}-(\lambda^{-}_{t})^{2}=\lambda_{1}\lambda_{2}=Q_{0}(\lambda).$

For a lattice $L\subset V=V_{0}$, the theta function $\theta_{L+h,p}$ can be
explicitly written down as

(2.11)
$\theta_{L+h,p}(\tau,t)=v^{d^{-}+1/2}\sum_{\begin{subarray}{c}\lambda\in
L+h\end{subarray}}p(\lambda_{t}^{+},\lambda_{t}^{-})\mathbf{e}\left((\lambda_{t}^{+})^{2}\tau-(\lambda_{t}^{-})^{2}\overline{\tau}\right).$

Note that the only harmonic polynomials on
$V(\mathbb{R})\cong\mathbb{R}^{1,1}$ have degree $(d^{+},d^{-})$ with
$d^{\pm}\leq 1$. We denote

(2.12)
$\theta_{L+h}^{(d^{+},d^{-})}(\tau,t):=\theta_{L+h,x^{d^{+}}y^{d^{-}}}(\tau,t)$

where $(x,y)\in\mathbb{R}^{1,1}$. When $t=t_{0}\in\mathbb{R}^{\times}$
satisfies $t_{0}^{2}\in\mathbb{Q}^{\times}$, we have a rational splitting
$V=U\oplus U^{\perp}$ with $U$ positive definite such that
$U^{\perp}(\mathbb{R})=c(t_{0})$, and $P\oplus N\subset L$ is a finite index
sublattice with $P:=L\cap U$ and $N:=L\cap U^{\perp}$ definite lattices. The
theta function $\theta_{L+h}^{(d^{+},d^{-})}$ has the following corresponding
splitting

(2.13)
$\theta^{(d^{+},d^{-})}_{L+h}(\tau,t_{0})=\sum_{\begin{subarray}{c}\mu\in
P^{*}/P,~{}\nu\in N^{*}/N\\\ P\oplus N+(\mu,\nu)\subset
L+h\end{subarray}}\theta^{(d^{+})}_{P+\mu}(\tau)\theta^{(d^{-})}_{N+\nu}(\tau)$

for $d^{\pm}\in\\{0,1\\}$.

Suppose $L$ is anisotropic. Then

(2.14) $L=L_{\mathfrak{a},M}=M\mathfrak{a}$

for some fractional ideal $\mathfrak{a}$ of a real quadratic field $F$ with
quadratic form $Q_{\mathfrak{a},M}={\mathrm{Nm}}/(AM)$,
$A={\mathrm{Nm}}(\mathfrak{a})$ and $M\in\mathbb{N}$. Its dual lattice is
$\mathfrak{a}\mathfrak{d}^{-1}$ with $\mathfrak{d}$ the different of $F$. The
isometry between $V(\mathbb{R})$ and $V_{0}(\mathbb{R})$ is given by

(2.15) $\begin{split}\iota:V(\mathbb{R})&\to V_{0}(\mathbb{R})\\\
\lambda&\mapsto\frac{1}{\sqrt{AM}}(\lambda,\lambda^{\prime}).\end{split}$

Using this isometry, we can give the following explicit expression of
$\theta_{L+h,p}(\tau,t)$ as in (2.11)

(2.16) $\theta_{L+h,p}(\tau,t)=v^{d^{-}+1/2}\sum_{\lambda\in
L_{\mathfrak{a},M}+h}p\left(\frac{\lambda
t^{-1}+\lambda^{\prime}t}{2\sqrt{AM}},\frac{-\lambda
t^{-1}+\lambda^{\prime}t}{2\sqrt{AM}}\right)\mathbf{e}(Q_{\mathfrak{a},M}(\lambda)u)\mathbf{e}\left(-\pi\frac{(\lambda
t^{-1})^{2}+(\lambda^{\prime}t)^{2}}{AM}\right).$

The splitting in (2.13) happens when $t=t_{0}\in\mathbb{R}^{\times}$ satisfies

(2.17) $t_{0}^{2}\in F^{1}:=\\{\alpha\in
F^{\times}:{\mathrm{Nm}}(\alpha)=1\\}.$

For later purpose, we will also require the $K$-Bessel function $K_{0}(x)$,
which is defined by

(2.18) $K_{0}(x):=\int^{\infty}_{0}e^{-x\cosh(T)}dT,~{}x>0$

and satisfies the second order differential equation

$\left(x\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial}{\partial
x}-x\right)K_{0}(x)=0,$

with the asymptotics

(2.19) $K_{0}(x)\sim\sqrt{\pi/(2x)}e^{-x}$

as $x\to\infty$. For convenience, we also set

(2.20) $K_{0}(0):=0$

and state a simple lemma that will be useful later.

######  Lemma 2.1.

For any $a\in\mathbb{R}$, we denote
$K_{0}(x;a):=\mathrm{sgn}(a)\int_{|a|}^{\infty}e^{-x\cosh(T)}dT$. Then

(2.21) $\lim_{x\to\infty}\frac{K_{0}(x;a)}{K_{0}(x)}=0.$

###### Proof.

The claim is trivial for $a=0$ and we only need to prove it for $a>0$. In that
case, $b:=\sinh(a/2)>0$ and a change of variable
$u=\sqrt{x}\sinh(T/2),du=\sqrt{x}\cosh(T/2)\,dT/2=\sqrt{u^{2}+x}\,dT/2$ gives
us

$\displaystyle\sqrt{x}e^{x}K_{0}(x;a)$
$\displaystyle=\sqrt{x}\int_{a}^{\infty}e^{-2x\sinh(T/2)^{2}}dT=2\int^{\infty}_{\sqrt{x}b}e^{-2u^{2}}\frac{du}{\sqrt{u^{2}/x+1}}<2\int^{\infty}_{\sqrt{x}b}e^{-2u^{2}}du.$

In particular, we have the bound

(2.22) $|K_{0}(x;a)|<\frac{e^{-x(1+2b^{2})}}{xb}.$

Taking the limit $x\to\infty$ and applying (2.19) finishes the proof. ∎

##  3\. Mock Maass Forms

In this section, we will use the suitable integrals of Siegel theta kernels to
construct higher depth analogues of the mock Maass forms constructed by
Zwegers. In particular, we will express Maass’ and Zwegers’ constructions as
special examples (see Section 3.1).

###  3.1. Mock Maass Theta Functions I

Let $L=\iota(L_{\mathfrak{a},M})\subset V_{0}(\mathbb{R})$ be the even,
integral, anisotropic lattice as in (2.14), and
$\Gamma_{L}\subset{\mathrm{SO}}(L)$ the subgroup of the discriminant kernel
consisting of totally positive units in $\mathcal{O}_{F}^{\times}$ congruent
to 1 modulo $M\sqrt{D}$. Then $\Gamma_{L}$ is isomorphic to $\mathbb{Z}$. Let
$\varepsilon_{L}>1$ be its generator. Given $t_{2}>t_{1}>0$ and denote
$c_{i}:=c(t_{i})\in{\mathcal{D}}$, we consider

(3.1)
$\hat{\vartheta}^{c_{1},c_{2}}_{L+h}(\tau):=\int^{t_{2}}_{t_{1}}\theta_{L+h}(\tau,t)\frac{dt}{t}.$

This is a real-analytic modular function in $\tau=u+iv\in\mathbb{H}$. When
$t_{2}/t_{1}\in\Gamma_{L}$, we can evaluate (3.1) by unfolding. For $h\not\in
L$, we can unfold the integral to obtain

$\displaystyle\hat{\vartheta}^{c_{1},c_{2}}_{L+h}(\tau)$
$\displaystyle=\frac{\log(t_{2}/t_{1})}{\log\varepsilon_{L}}\int^{\varepsilon_{L}}_{1}\theta_{L+h}(\tau,t)\frac{dt}{t}$
$\displaystyle=\sqrt{v}\frac{\log(t_{2}/t_{1})}{\log\varepsilon_{L}}\sum_{\lambda\in\Gamma_{L}\backslash
L+h}\mathbf{e}\left(Q_{0}(\lambda)u\right)\int^{\infty}_{-\infty}\mathbf{e}\left(\frac{(\lambda
e^{-\nu})^{2}+(\lambda^{\prime}e^{\nu})^{2}}{2}iv\right)d\nu$
$\displaystyle=2\sqrt{v}\frac{\log(t_{2}/t_{1})}{\log\varepsilon_{L}}\sum_{\lambda\in\Gamma_{L}\backslash
L+h}\mathbf{e}\left(Q_{0}(\lambda)u\right)\int^{\infty}_{0}e^{-2\pi
v|Q_{0}(\lambda)|\cosh(T)}dT.$

Using (2.18), we can evaluate

(3.2)
$\hat{\vartheta}^{c_{1},c_{2}}_{L+h}(\tau)={2\sqrt{v}}\frac{\log(t_{2}/t_{1})}{\log\varepsilon_{L}}\sum_{\lambda\in\Gamma_{L}\backslash
L+h}\mathbf{e}\left(Q_{0}(\lambda)u\right)K_{0}(2\pi v|Q_{0}(\lambda)|).$

When $h\in L$, we have the additional constant term

${\sqrt{v}}\frac{\log(t_{2}/t_{1})}{\log\varepsilon_{L}}\int^{\varepsilon_{L}}_{1}\frac{dt}{t}={\sqrt{v}}\log(t_{2}/t_{1}).$

This recovers the construction of Maass cusp forms by Maass [Maa49]. For
$c_{1}=c(1)$ and $c_{2}=c(\varepsilon_{L})$, we denote

(3.3) $\vartheta_{L+h}(\tau):=\hat{\vartheta}^{c_{1},c_{2}}_{L+h}(\tau).$

More generally, we can evaluate the Fourier expansion of
$\hat{\vartheta}^{c_{1},c_{2}}_{L+h}$ without unfolding as

$\displaystyle\hat{\vartheta}^{c_{1},c_{2}}_{L+h}(\tau)$
$\displaystyle=\sum_{\lambda\in
L+h}\int^{t_{2}}_{t_{1}}\phi_{\tau}(\lambda^{+}_{t},\lambda^{-}_{t})\frac{dt}{t}={\sqrt{v}}\sum_{\lambda\in
L+h}\mathbf{e}\left(Q(\lambda)u\right)\int^{t_{2}}_{t_{1}}\mathbf{e}\left(\left((\lambda^{+}_{t})^{2}+(\lambda^{-}_{t})^{2}\right)iv\right)\frac{dt}{t}$
$\displaystyle={\sqrt{v}}\sum_{\lambda\in
L+h}\mathbf{e}\left(Q_{0}(\lambda)\tau\right)\int^{t_{2}}_{t_{1}}e^{-4\pi
v(\lambda^{-}_{t})^{2}}\frac{dt}{t}={\sqrt{v}}\sum_{\lambda\in
L+h}\mathbf{e}\left(Q_{0}(\lambda)\tau\right)\int^{t_{2}}_{t_{1}}e^{-\pi
vB_{0}(\lambda,Z^{-}_{t})^{2}}\frac{dt}{t}.$

This is Zwegers’ function $\hat{\Phi}^{c_{1},c_{2}}_{h,0}$ in Definition 2.3
of [Zwe12], and we have shown that it possesses an integral representation
given in (3.1). For general $a,b\in L^{*}$, we can express
$\hat{\Phi}^{c_{1},c_{2}}_{a,b}$ as a linear combination of
$\hat{\vartheta}_{\tilde{L}+\tilde{h}}$’s for a suitable sublattice
$\tilde{L}\subset L$. See 3.5 for an example.

Now define for $c_{i}=c(t_{i})$

(3.4)
$\begin{split}\hat{\vartheta}^{c_{1},c_{2},+}_{L+h}(\tau)&:=\sqrt{v}\sum_{\lambda\in
L+h}a_{\lambda}(t_{1},t_{2})\mathbf{e}(Q_{0}(\lambda)u)K_{0}(2\pi|Q_{0}(\lambda)|v),\\\
\varphi^{c_{0}}_{L+h}(\tau)&:=\sqrt{v}\sum_{\lambda\in
L+h}\beta_{t_{0}}(\lambda\sqrt{v})\mathbf{e}(Q_{0}(\lambda)u)\end{split}$

with the coefficients $a_{\lambda}$ defined by

(3.5)
$a_{\lambda}(t_{1},t_{2}):=\begin{cases}\frac{1}{2}\left(1-\mathrm{sgn}(\lambda_{t_{1}}^{+}\lambda_{t_{2}}^{+})\right)&\text{
if }Q_{0}(\lambda)<0,\\\
\frac{1}{2}\left(1-\mathrm{sgn}(\lambda_{t_{1}}^{-}\lambda_{t_{2}}^{-})\right)&\text{
if }Q_{0}(\lambda)>0,\\\ \log(t_{2}/t_{1})&\text{ if }\lambda=0,\end{cases}$

and the function $\beta_{t}$ defined by

(3.6) $\beta_{t_{0}}(w):=\begin{cases}e^{-2\pi
Q_{0}(w)}\int^{\infty}_{t_{0}}e^{-4\pi(w_{t}^{-})^{2}}\frac{dt}{t}&\text{ if
}w_{t_{0}}^{-}w_{t_{0}}^{+}>0,\\\ -e^{-2\pi
Q_{0}(w)}\int_{0}^{t_{0}}e^{-4\pi(w_{t}^{-})^{2}}\frac{dt}{t}&\text{ if
}w_{t_{0}}^{-}w_{t_{0}}^{+}<0,\\\ 0&\text{ if
}w_{t_{0}}^{-}w_{t_{0}}^{+}=0,\end{cases}$

a function on $V_{0}(\mathbb{R})\backslash\\{0\\}$.

From the definition of $\beta_{t}$ in (3.6), one can check that

(3.7) $\begin{split}e^{-2\pi
Q_{0}(w)}\int^{t_{2}}_{t_{1}}e^{-4\pi(w^{-}_{t})^{2}}\frac{dt}{t}&=a_{\lambda}(t_{1},t_{2})K_{0}(2\pi|Q_{0}(w)|)+\beta_{t_{1}}(w)-\beta_{t_{2}}(w)\end{split}$

for $w\in V_{0}(\mathbb{R})$. Using this relation, Zwegers decomposed
$\hat{\Phi}_{h,0}^{c_{1},c_{2}}=\hat{\vartheta}^{c_{1},c_{2}}_{L+h}$ into a
harmonic and a non-harmonic part in the following way.

######  Theorem 3.1.

[Theorem 2.6 in [Zwe12]] In the notations above, we have

(3.8)
$\hat{\vartheta}^{c_{1},c_{2}}_{L+h}(\tau)=\hat{\vartheta}^{c_{1},c_{2},+}_{L+h}(\tau)+\varphi^{c_{1}}_{L+h}(\tau)-\varphi^{c_{2}}_{L+h}(\tau).$

######  Remark 3.2.

It is easy to check that the coefficients $a_{\lambda}$ have the following
simpler form

(3.9) $a_{\lambda}(t_{1},t_{2})=\begin{cases}1&\text{if }\lambda\neq 0\text{
and }t_{1}^{2}<\left|\frac{\lambda_{1}}{\lambda_{2}}\right|<t_{2}^{2},\\\
\frac{1}{2}&\text{if }\lambda\neq 0\text{ and
}t_{i}^{2}=\left|\frac{\lambda_{1}}{\lambda_{2}}\right|\text{ for
}i\in\\{1,2\\},\\\ \log\frac{t_{2}}{t_{1}}&\text{if }\lambda=0,\\\
0&\text{otherwise.}\end{cases}$

The function $\hat{\vartheta}_{L+h}^{c_{1},c_{2}}$ also satisfies nice
properties under differential operators. Suppose $w=(w_{1},w_{2})$ with
$Q_{0}(w)=w_{1}w_{2}\neq 0$, then

$\mathrm{sgn}(w_{t}^{-}w_{t}^{+})=\mathrm{sgn}((w_{2}t)^{2}-(w_{1}t^{-1})^{2})=\mathrm{sgn}(\log(t^{2}|w_{2}/w_{1}|))$

and we can rewrite

(3.10)
$\begin{split}\beta_{t_{0}}(w)&=\mathrm{sgn}((w_{2}t_{0})^{2}-(w_{1}t_{0}^{-1})^{2})\int^{\infty}_{t_{0}|w_{2}/w_{1}|^{1/2}}e^{-\pi|Q_{0}(w)|(t^{2}+t^{-2})}\frac{dt}{t}\\\
&=\frac{1}{2}K_{0}(2\pi|Q_{0}(w)|,\log(t_{0}^{2}|w_{2}/w_{1}|)).\end{split}$

A straightforward calculation gives the identity

$\displaystyle(v\partial_{v}^{2}+\partial_{v}-x^{2}v)\left(K_{0}(xv;a)\right)=x\sinh(a)\,e^{-xv\cosh(a)}.$

Using (3.10) it is then easy to check that $\beta_{t_{0}}$ satisfies

(3.11)
$(v\partial_{v}^{2}+\partial_{v}-4\pi^{2}Q_{0}(\lambda)^{2}v)\beta_{t_{0}}(\lambda\sqrt{v})=2\pi\lambda_{t_{0}}^{+}\lambda_{t_{0}}^{-}e^{-2\pi
v((\lambda_{t_{0}}^{+})^{2}+(\lambda_{t_{0}}^{-})^{2})}$

for any $\lambda\in V_{0}(\mathbb{R})\backslash(c(t_{0})\cup
c(t_{0})^{\perp})$. This equation proves the following theorem of Zwegers.

######  Theorem 3.3.

[Theorem 2.7 of [Zwe12]] In the notation above, we have 111 Note that the
unary theta series as defined here and the one defined by Zwegers differ by a
factor of 4, i.e. $4\theta_{L+h}^{(1,1)}(\tau,t)$ equals
$\vartheta_{h,0}^{c}(\tau)$.

(3.12)
$\begin{split}\tilde{\Delta}\varphi^{c_{i}}_{L+h}(\tau)&=2\pi\theta_{L+h}^{(1,1)}(\tau,t_{i}),\\\
\tilde{\Delta}\hat{\vartheta}^{c_{1},c_{2}}_{L+h}(\tau)&=2\pi(\theta_{L+h}^{(1,1)}(\tau,t_{1})-\theta_{L+h}^{(1,1)}(\tau,t_{2})).\end{split}$

######  Remark 3.4.

When $t_{0}^{2}\in F^{1}$, the function $\theta_{L+h}^{(1,1)}(\tau,t_{0})$ is
in $S_{3/2}\otimes v^{3/2}\overline{S_{3/2}}$ by (2.13).

######  Example 3.5.

In Remark 2.2 and Section 6 of [Zwe12] it is re-proved that Cohen’s function
$\varphi_{0}$ is a Maass form of weight 0 by relating it to
$\hat{\Phi}_{a,b}^{c_{1},c_{2}}$ for suitable $a,b,c_{1},c_{2}$.

Zwegers considers the quadratic space $(\mathbb{Z}^{2},Q^{\prime})$ with
$Q^{\prime}((m,n))=\frac{1}{2}(3m^{2}-2n^{2})$. Further, the indefinite theta
series depends on the characteristics $a=\left(\begin{smallmatrix}1/6\\\
0\end{smallmatrix}\right)$ and $b=\left(\begin{smallmatrix}1/6\\\
1/4\end{smallmatrix}\right)$.

Under the identification $(m,n)\mapsto 3m+\sqrt{6}n$ we have
$(\mathbb{Z}^{2},Q^{\prime})\simeq(\mathfrak{a},Q)$ with
$\mathfrak{a}=3\mathbb{Z}+\sqrt{6}\mathbb{Z}$ an integral ideal in the ring of
integers $\mathcal{O}$ of $F=\mathbb{Q}(\sqrt{6})$ and $Q={\mathrm{Nm}}/6$.
The sublattice $2\mathbb{Z}^{2}$ is then isometric to
$L:=L_{\mathfrak{a},2}=2\mathfrak{a}=6\mathbb{Z}+2\sqrt{6}\mathbb{Z}$. Its
dual lattice is
$\mathfrak{a}\mathfrak{d}^{-1}=\frac{1}{2}\mathbb{Z}+\frac{\sqrt{6}}{4}\mathbb{Z}$,
where $\mathfrak{d}=2\sqrt{3}\mathcal{O}$. We split

$\displaystyle a+\mathbb{Z}^{2}$
$\displaystyle=\left(a+2\mathbb{Z}^{2}\right)\sqcup\left(a+\binom{1}{0}+2\mathbb{Z}^{2}\right)\sqcup\left(a+\binom{0}{1}+2\mathbb{Z}^{2}\right)\sqcup\left(a+\binom{1}{1}+2\mathbb{Z}^{2}\right)$
$\displaystyle\simeq\left(\frac{1}{2}+L\right)\sqcup\left(\frac{7}{2}+L\right)\sqcup\left(\frac{1}{2}+\sqrt{6}+L\right)\sqcup\left(\frac{7}{2}+\sqrt{6}+L\right).$

We thus consider the theta series associated to $L=L_{\mathfrak{a},2}$. Let
$t_{1}=1$, $t_{2}=\varepsilon:=5+2\sqrt{6}$. For

(3.13) $h\in\\{1/2,7/2,1/2+\sqrt{6},7/2+\sqrt{6}\\}$

and $i=1,2$, the theta function $\theta^{(1,1)}_{L+h}(\tau,t_{i})$ vanishes
identically since the automorphism $\lambda\mapsto\lambda^{\prime}$ on $L+h$
creates a minus sign. That means
$\hat{\vartheta}^{c_{1},c_{2}}_{L+h}=\hat{\vartheta}^{c_{1},c_{2},+}_{L+h}$ is
a Maass form by Theorem 3.3. Furthermore, we have the identity

(3.14)
$\hat{\vartheta}^{c_{1},c_{2}}_{L+h}=\hat{\vartheta}^{c_{1},c_{2},+}_{L+h}=\frac{\vartheta_{L+h}}{4}=\frac{\vartheta_{L+h+3+\sqrt{6}}}{4}$

for all $h$ in (3.13).

Notice that if $\lambda\in L+h$ for $h$ in (3.13), then
$\mathfrak{b}=(2\lambda)$ is an integral ideal of $\mathcal{O}$ with norm
congruent to 1 modulo 24. Conversely given such an ideal $\mathfrak{b}$, we
can find a unique generator $\beta$ such that
$1\leq\beta/\beta^{\prime}<\varepsilon^{2}$ and $\beta/2\in L+h$ for an $h$ in
(3.13). In case $\beta/\beta^{\prime}=1$, i.e. $\beta\in\mathbb{Q}$, then
$\lambda=-\varepsilon\beta/2$ also gives non-zero value for
$a_{\lambda}(1,\varepsilon)$. We list these ideals with norm less than 100
together with the generator and value $a_{\lambda}(1,\varepsilon)$ below.

Table 1. Values of $a_{\lambda}(1,\varepsilon)$

$\begin{array}[]{c|c|c|c|c|c}&&\frac{1}{2}&\frac{7}{2}+\sqrt{6}&\frac{7}{2}&\frac{1}{2}+\sqrt{6}\\\
\text{Norm}&\beta=2\lambda&&&\\\ \hline\cr
1&1,-\varepsilon&\frac{1}{2}&\frac{1}{2}&&\\\ \hline\cr\hbox{\multirowsetup
23}&1+2\sqrt{6}&&&&1\\\ &-\varepsilon(1-2\sqrt{6})&&&1&\\\
\hline\cr\hbox{\multirowsetup 25}&7+2\sqrt{6}&&1&&\\\
&-\varepsilon(7-2\sqrt{6})&1&&&\\\
&-5,5\varepsilon&&&\frac{1}{2}&\frac{1}{2}\\\ \hline\cr\hbox{\multirowsetup
47}&7+4\sqrt{6}&&&1&\\\ &-\varepsilon(7-4\sqrt{6})&&&&1\\\
\hline\cr{49}&7,-7\varepsilon&&&\frac{1}{2}&\frac{1}{2}\\\
\hline\cr\hbox{\multirowsetup 71}&-(5+4\sqrt{6})&&&1&\\\
&\varepsilon(5-4\sqrt{6})&&&&1\\\ \hline\cr\hbox{\multirowsetup
73}&13+4\sqrt{6}&1&&&\\\ &-\varepsilon(13-4\sqrt{6})&&1&&\\\
\hline\cr\hbox{\multirowsetup 95}&1+4\sqrt{6}&1&&&\\\
&-\varepsilon(1-4\sqrt{6})&&1&&\\\ &\varepsilon(11-6\sqrt{6})&&&1&\\\
&-(11-6\sqrt{6})&&&&1\\\ \hline\cr\hbox{\multirowsetup
97}&-(11+2\sqrt{6})&&&&1\\\ &\varepsilon(11-2\sqrt{6})&&&1&\\\
\hline\cr\end{array}$

Also, the map $\lambda\mapsto-\varepsilon\lambda^{\prime}$ identifies
$\hat{\vartheta}^{c_{1},c_{2},+}_{L+\frac{1}{2}}(\tau)$ (resp.
$\hat{\vartheta}^{c_{1},c_{2},+}_{L+\frac{7}{2}}(\tau)$) with
$\hat{\vartheta}^{c_{1},c_{2},+}_{{L}+\frac{1}{2}+\sqrt{6}}(\tau)$ (resp.
$\hat{\vartheta}^{c_{1},c_{2},+}_{{L}+\frac{7}{2}+\sqrt{6}}(\tau)$). Using
these, we have

$\displaystyle\sqrt{24}^{-1}\hat{\vartheta}^{c_{1},c_{2},+}_{L+\frac{1}{2}}(24\tau)$
$\displaystyle=\dots+W_{-95}(\tau)+\frac{1}{2}W_{1}(\tau)+W_{25}(\tau)+W_{73}(\tau)+\frac{1}{2}W_{121}(\tau)+\dots,$
$\displaystyle\sqrt{24}^{-1}\hat{\vartheta}^{c_{1},c_{2},+}_{L+\frac{7}{2}}(24\tau)$
$\displaystyle=\dots+W_{-95}(\tau)+W_{-71}(\tau)+W_{-47}(\tau)+W_{-23}(\tau)$
$\displaystyle\quad+\frac{1}{2}W_{25}(\tau)+\frac{1}{2}W_{49}(\tau)+W_{97}(\tau)+\dots,$

where $W_{n}(\tau):=\sqrt{v}\mathbf{e}(nu)K_{0}(2\pi|n|v)$. We can now write
Cohen’s function as

(3.15)
$\begin{split}\varphi_{0}(\tau)&=\hat{\vartheta}^{c_{1},c_{2},+}_{L+\frac{1}{2}}(\tau)-\hat{\vartheta}^{c_{1},c_{2},+}_{L+\frac{7}{2}}(\tau)-\hat{\vartheta}^{c_{1},c_{2},+}_{L+\frac{1}{2}+\sqrt{6}}(\tau)+\hat{\vartheta}^{c_{1},c_{2},+}_{L+\frac{7}{2}+\sqrt{6}}(\tau)\\\
&=\frac{1}{2}\left(\vartheta_{L+1/2}(\tau)-\vartheta_{L+7/2}(\tau)\right).\end{split}$

###  3.2.  Mock Maass Theta Functions II

In this section, we will further decompose the function
$\hat{\vartheta}^{c_{1},c_{2}}_{L+h}$ into the difference of two functions
that depend only on $c_{1}$ and $c_{2}$ respectively.

For $c_{0}=c(t_{0})$, consider

(3.16)
$\begin{split}\tilde{\vartheta}^{c_{0}}_{L+h}(\tau)&:=\int^{t_{0}\varepsilon_{L}}_{t_{0}}\theta_{L+h}(\tau,t)\log(t)\frac{dt}{t}={\sqrt{v}}\sum_{\lambda\in
L+h}\mathbf{e}\left(Q_{0}(\lambda)u\right)\tilde{\beta}_{t_{0}}(\lambda\sqrt{v}),\\\
\tilde{\beta}_{t_{0}}(w)&:=e^{-2\pi
Q_{0}(w)}\int^{t_{0}\varepsilon_{L}}_{t_{0}}e^{-4\pi(w^{-}_{t})^{2}}\log(t)\frac{dt}{t},\end{split}$

where $L\subset V_{0}(\mathbb{R})$ is an even, integral, anisotropic lattice.
Now for $\Lambda=\Gamma_{L}\cdot\lambda\in\Gamma_{L}\backslash L+h$, we define

(3.17)
$\tilde{c}_{t_{0}}(\Lambda):=\begin{cases}\frac{\log(t_{0}^{2}\varepsilon_{L})}{2}+\log\varepsilon_{L}\cdot\mathbb{B}_{1}\left(\frac{\log(|\lambda/\lambda^{\prime}|t_{0}^{-2})}{2\log\varepsilon_{L}}\right)&\text{
if }Q_{0}(\Lambda)\neq 0,\\\
\frac{1}{2}\log(\varepsilon_{L})\log(t^{2}_{0}\varepsilon_{L})&\text{ if
}Q_{0}(\Lambda)=0.\end{cases}$

Here $\mathbb{B}_{1}:\mathbb{R}/\mathbb{Z}\to\mathbb{R}$ is the first periodic
Bernoulli polynomial defined by $\mathbb{B}_{1}(x)=x-\frac{\delta_{x\neq
0}}{2}$ for $0\leq x<1$. Note that $Q_{0}(\Lambda)=0$ if and only if $\lambda$
is trivial since $L$ is anisotropic. Then we have the following result.

######  Proposition 3.6.

For any non-trivial $\lambda_{0}\in L+h$ and $n\in\mathbb{Z}$, denote
$\lambda_{n}:=\lambda_{0}\varepsilon_{L}^{n}$. Let
$\Lambda:=\Gamma_{L}\cdot\lambda_{0}$. Then we have

(3.18)
$\sum_{n\in\mathbb{Z}}\tilde{\beta}_{t_{0}}(\lambda_{n}\sqrt{v})=-\log\varepsilon_{L}\sum_{n\in\mathbb{Z}}\beta_{t_{0}}(\lambda_{n}\sqrt{v})+\tilde{c}_{t_{0}}(\Lambda)\cdot
K_{0}(2\pi|Q_{0}(\lambda_{0})|v).$

###### Proof.

Denote $m:=Q_{0}(\lambda)$. From (3.11), we have

$\left(v^{2}\partial_{v}^{2}+v\partial_{v}-4\pi^{2}m^{2}v^{2}\right)\left(\sum_{n\in\mathbb{Z}}\beta_{t_{0}}(\lambda_{n}\sqrt{v})\right)=2\pi
v\sum_{n\in\mathbb{Z}}(\lambda_{n})_{t_{0}}^{+}(\lambda_{n})_{t_{0}}^{-}e^{-2\pi
v(((\lambda_{n})_{t_{0}}^{+})^{2}+((\lambda_{n})_{t_{0}}^{-})^{2})}.$

On the other hand, we have

$\displaystyle\left(v^{2}\partial_{v}^{2}+v\partial_{v}-4\pi^{2}m^{2}v^{2}\right)\tilde{\beta}_{t_{0}}(\lambda_{n}\sqrt{v})=-2\pi
v\log\varepsilon_{L}((\lambda_{n-1})_{t_{0}}^{+}(\lambda_{n-1})_{t_{0}}^{-})e^{-2\pi
v(((\lambda_{n-1})^{+}_{t_{0}})^{2}+((\lambda_{n-1})^{-}_{t_{0}})^{2})}$
$\displaystyle-2\pi
v\log(t_{0})\left(((\lambda_{n-1})_{t_{0}}^{+}(\lambda_{n-1})_{t_{0}}^{-})e^{-2\pi
v(((\lambda_{n-1})^{+}_{t_{0}})^{2}+((\lambda_{n-1})^{-}_{t_{0}})^{2})}-((\lambda_{n})_{t_{0}}^{+}(\lambda_{n})_{t_{0}}^{-})e^{-2\pi
v(((\lambda_{n})^{+}_{t_{0}})^{2}+((\lambda_{n})^{-}_{t_{0}})^{2})}\right)$
$\displaystyle-\frac{1}{4}\left(e^{-2\pi
v(((\lambda_{n-1})^{+}_{t_{0}})^{2}+((\lambda_{n-1})^{-}_{t_{0}})^{2})}-e^{-2\pi
v(((\lambda_{n})^{+}_{t_{0}})^{2}+((\lambda_{n})^{-}_{t_{0}})^{2})}\right).$

Therefore applying the operator
$v^{2}\partial_{v}^{2}+v\partial_{v}-4\pi^{2}m^{2}v^{2}$ to the left hand side
of (3.18) gives us

$\displaystyle\left(v^{2}\partial_{v}^{2}+v\partial_{v}-4\pi^{2}m^{2}v^{2}\right)\left(\sum_{n\in\mathbb{Z}}\tilde{\beta}_{t_{0}}(\lambda_{n}\sqrt{v})\right)$
$\displaystyle=-2\pi
v\log\varepsilon_{L}\sum_{n\in\mathbb{Z}}(\lambda_{n})_{t_{0}}^{+}(\lambda_{n})_{t_{0}}^{-}e^{-2\pi
v(((\lambda_{n})_{t_{0}}^{+})^{2}+((\lambda_{n})_{t_{0}}^{-})^{2})}$

and

$\displaystyle\left(v^{2}\partial_{v}^{2}+v\partial_{v}-4\pi^{2}m^{2}v^{2}\right)\left(\sum_{n\in\mathbb{Z}}\tilde{\beta}_{t_{0}}(\lambda_{n}\sqrt{v})+\log\varepsilon_{L}\sum_{n\in\mathbb{Z}}\beta_{t_{0}}(\lambda_{n}\sqrt{v})\right)=0.$

The solutions to the differential equation
$(v^{2}\partial_{v}+v\partial_{v}-4\pi m^{2}v^{2})f=0$ are of the form

$f=b\cdot I_{0}(2\pi|m|v)+c\cdot K_{0}(2\pi|m|v)$

with $I_{0}$ and $K_{0}$ the modified Bessel functions and $b,c$ constants.
Since $\beta(\lambda_{0}\sqrt{v})$ and $\tilde{\beta}(\lambda_{0}\sqrt{v})$
both decay as $v\to\infty$, we have

$\sum_{n\in\mathbb{Z}}\tilde{\beta}_{t_{0}}(\lambda_{n}\sqrt{v})+\log\varepsilon_{L}\sum_{n\in\mathbb{Z}}\beta_{t_{0}}(\lambda_{n}\sqrt{v})=c\cdot
K_{0}(2\pi|m|v).$

Using the bound (2.22) in the proof of Lemma 2.1 and equation (3.10), we have

$\lim_{v\to\infty}\frac{\sum_{n\in\mathbb{Z}}\beta_{t_{0}}(\lambda_{n}\sqrt{v})}{K_{0}(2\pi|m|v)}=0.$

On the other hand, a change of variable gives us

$\displaystyle\sum_{n\in\mathbb{Z}}$
$\displaystyle\tilde{\beta}_{t_{0}}(\lambda_{n}\sqrt{v})=\sum_{n\in\mathbb{Z}}\int_{t_{0}|\lambda^{\prime}_{n}/\lambda_{n}|^{\frac{1}{2}}}^{t_{0}|\lambda^{\prime}_{n-1}/\lambda_{n-1}|^{\frac{1}{2}}}e^{-\pi
v|m|(r^{2}+r^{-2})}\log(r|\lambda_{n}/\lambda^{\prime}_{n}|^{\frac{1}{2}})\frac{dr}{r}$
$\displaystyle=\frac{1}{2}\log|\lambda_{0}/\lambda^{\prime}_{0}|K_{0}(2\pi|m|v)+\log\varepsilon_{L}\sum_{n\in\mathbb{Z}}n\int_{t_{0}|\lambda^{\prime}_{n}/\lambda_{n}|^{\frac{1}{2}}}^{t_{0}|\lambda^{\prime}_{n-1}/\lambda_{n-1}|^{\frac{1}{2}}}e^{-\pi|m|v(r^{2}+r^{-2})}\frac{dr}{r}.$

The second sum can be evaluated as

$\displaystyle\sum_{n\in\mathbb{Z}}$ $\displaystyle
n\int_{t_{0}|\lambda^{\prime}_{n}/\lambda_{n}|^{\frac{1}{2}}}^{t_{0}|\lambda^{\prime}_{n-1}/\lambda_{n-1}|^{\frac{1}{2}}}e^{-\pi|m|v(r^{2}+r^{-2})}\frac{dr}{r}$
$\displaystyle=\sum_{n\geq
1}\left(\int_{0}^{t_{0}|\lambda^{\prime}_{n-1}/\lambda_{n-1}|^{\frac{1}{2}}}e^{-\pi|m|v(r^{2}+r^{-2})}\frac{dr}{r}-\int^{\infty}_{t_{0}|\lambda^{\prime}_{-n}/\lambda_{-n}|^{\frac{1}{2}}}e^{-\pi|m|v(r^{2}+r^{-2})}\frac{dr}{r}\right)$
$\displaystyle=\frac{1}{2}\sum_{n\geq
1}\left(K_{0}(2\pi|m|v,\log(t_{0}^{-2}|\lambda_{n-1}/\lambda_{n-1}^{\prime}|))-K_{0}(2\pi|m|v,\log(t_{0}^{2}|\lambda_{-n}^{\prime}/\lambda_{-n}|))\right).$

We now choose $\lambda_{0}$ such that
$t_{0}^{2}\leq|\lambda_{0}/\lambda_{0}^{\prime}|<t_{0}^{2}\varepsilon_{L}^{2}$,
which implies that $t_{0}^{-2}|\lambda_{0}/\lambda_{0}^{\prime}|\geq 1$,
$t_{0}^{-2}|\lambda_{n}/\lambda_{n}^{\prime}|>1$ and
$t_{0}^{2}|\lambda_{-n}^{\prime}/\lambda_{-n}|>1$ for all $n\geq 1$. Applying
Lemma 2.1 and equation (2.22) gives us

$\displaystyle\lim_{v\to\infty}K_{0}(2\pi|m|v)^{-1}\sum_{n\in\mathbb{Z}}n\int_{t_{0}|\lambda^{\prime}_{n}/\lambda_{n}|^{\frac{1}{2}}}^{t_{0}|\lambda^{\prime}_{n-1}/\lambda_{n-1}|^{\frac{1}{2}}}e^{-\pi|m|v(r^{2}+r^{-2})}\frac{dr}{r}=\frac{1}{2}\begin{cases}1&\text{
if }t_{0}^{2}=|\lambda_{0}/\lambda_{0}^{\prime}|,\\\ 0&\text{
otherwise.}\end{cases}$

This finishes the proof. ∎

Now, we can define the harmonic generating series

(3.19)
$\tilde{\vartheta}^{c_{0},+}_{L+h}(\tau):=\sqrt{v}\sum_{\Lambda\in\Gamma_{L}\backslash
L+h}\tilde{c}_{t_{0}}(\Lambda)\mathbf{e}(Q_{0}(\Lambda)u)K_{0}(2\pi|Q_{0}(\Lambda)|v).$

As an immediate consequence of Proposition 3.6, we have the following result.

######  Theorem 3.7.

Let $c_{0}=c(t_{0})$ with $t_{0}>0$. Then the real-analytic modular form
$\tilde{\vartheta}^{c_{0}}_{L+h}$ has the decomposition

(3.20)
$\tilde{\vartheta}^{c_{0}}_{L+h}(\tau)=\tilde{\vartheta}^{c_{0},+}_{L+h}(\tau)-\log(\varepsilon_{L})\varphi^{c_{0}}_{L+h}(\tau)$

into the sum of a harmonic and a non-harmonic part. Furthermore for
$c_{i}=c(t_{i})$, the mock Maass theta function
$\hat{\vartheta}^{c_{1},c_{2},+}_{L+h}$ defined in (3.4) and its completion
$\hat{\vartheta}^{c_{1},c_{2}}_{L+h}$ can be written as

(3.21)
$\begin{split}-\log(\varepsilon_{L})\hat{\vartheta}^{c_{1},c_{2},+}_{L+h}(\tau)&=\tilde{\vartheta}^{c_{1},+}_{L+h}(\tau)-\tilde{\vartheta}^{c_{2},+}_{L+h}(\tau),\\\
-\log(\varepsilon_{L})\hat{\vartheta}^{c_{1},c_{2}}_{L+h}(\tau)&=\tilde{\vartheta}^{c_{1}}_{L+h}(\tau)-\tilde{\vartheta}^{c_{2}}_{L+h}(\tau).\end{split}$

Under the differential operator $\tilde{\Delta}$, we have

(3.22)
$\tilde{\Delta}\tilde{\vartheta}^{c_{0}}_{L+h}(\tau)=-\log(\varepsilon_{L})2\pi\theta^{(1,1)}_{L+h}(\tau,t_{0}).$

###### Proof.

Equation (3.20) follows directly from Proposition 3.6. To prove the first
equation in (3.21), we first check that the constant terms are both
$-\log(\varepsilon_{L})\log(t_{2}/t_{1})$. Then it suffices to check

(3.23)
$-\log(\varepsilon_{L})\sum_{n\in\mathbb{Z}}a_{\lambda_{n}}(t_{1},t_{2})=\tilde{c}_{t_{1}}(\Lambda)-\tilde{c}_{t_{2}}(\Lambda)$

for all non-trivial $\lambda_{0}\in L^{*}$, where we have denoted
$\lambda_{n}:=\lambda_{0}\varepsilon_{L}^{n}$ and
$\Lambda:=\Gamma_{L}\cdot\lambda_{0}$. We can choose a unique $\lambda_{0}$
within the orbit $\Lambda$ such that

$t_{1}\leq|\lambda_{0}/\lambda_{0}^{\prime}|^{1/2}<t_{1}\varepsilon_{L}.$

Let $m:=Q_{0}(\lambda_{0})\in\mathbb{Q}^{\times}$ and $N\in\mathbb{N}_{0}$
such that

$t_{1}\varepsilon_{L}^{N}\leq t_{2}<t_{1}\varepsilon_{L}^{N+1}.$

Suppose $|\lambda_{N}/\lambda_{N}^{\prime}|\leq t_{2}$. Then
$t_{2}<t_{1}\varepsilon_{L}^{N+1}\leq|\lambda_{0}/\lambda_{0}^{\prime}|^{1/2}\varepsilon_{L}^{N+1}=|\lambda_{N+1}/\lambda_{N+1}^{\prime}|^{1/2}$,
and $t_{1}\leq|\lambda_{n}/\lambda_{n}^{\prime}|^{1/2}\leq t_{2}$ precisely
when $0\leq n\leq N$. The two equalities could only hold for $n=0,N$, and only
one holds when $N=0$ since $t_{2}>t_{1}$. Therefore, we have

$\displaystyle\sum_{n\in\mathbb{Z}}a_{\lambda_{n}}(t_{1},t_{2})$
$\displaystyle=\\#\\{n\in\mathbb{Z}:t_{1}<|\lambda_{n}/\lambda_{n}^{\prime}|^{1/2}<t_{2}\\}+\frac{1}{2}\\{n\in\mathbb{Z}:t_{i}=|\lambda_{n}/\lambda_{n}^{\prime}|^{1/2}\text{
for }i=1\text{ or }2\\}$
$\displaystyle=N+1-\frac{1}{2}\left(\delta_{t_{1}^{2}=|\lambda_{0}/\lambda_{0}^{\prime}|}+\delta_{t_{2}^{2}=|\lambda_{N}/\lambda_{N}^{\prime}|}\right).$

On the other hand, we have

$\displaystyle-$
$\displaystyle\frac{\tilde{c}_{t_{1}}(\Lambda)-\tilde{c}_{t_{2}}(\Lambda)}{\log\varepsilon_{L}}=\frac{\log(t_{2}/t_{1})}{\log\varepsilon_{L}}-\left(\mathbb{B}_{1}\left(\frac{\log(|\lambda_{0}/\lambda_{0}^{\prime}|t_{1}^{-2})}{2\log\varepsilon_{L}}\right)-\mathbb{B}_{1}\left(\frac{\log(|\lambda_{N}/\lambda_{N}^{\prime}|t_{2}^{-2})}{2\log\varepsilon_{L}}\right)\right)$
$\displaystyle=\frac{\log(t_{2}/t_{1})}{\log\varepsilon_{L}}-\left(\frac{\log(|\lambda_{0}/\lambda_{0}^{\prime}|t_{1}^{-2})}{2\log\varepsilon_{L}}-\frac{\delta_{t_{1}^{2}\neq|\lambda_{0}/\lambda_{0}^{\prime}|}}{2}\right)+\left(\frac{\log(|\lambda_{N}/\lambda_{N}^{\prime}|t_{2}^{-2})}{2\log\varepsilon_{L}}+\frac{\delta_{t_{2}^{2}\neq|\lambda_{N}/\lambda_{N}^{\prime}|}}{2}\right)$
$\displaystyle=N+\frac{1}{2}\left(\delta_{t_{1}^{2}\neq|\lambda_{0}/\lambda_{0}^{\prime}|}+\delta_{t_{2}^{2}\neq|\lambda_{N}/\lambda_{N}^{\prime}|}\right)=N+1-\frac{1}{2}\left(\delta_{t_{1}^{2}=|\lambda_{0}/\lambda_{0}^{\prime}|}+\delta_{t_{2}^{2}=|\lambda_{N}/\lambda_{N}^{\prime}|}\right)$

since $1\leq
t_{1}^{-1}|\lambda_{0}/\lambda_{0}^{\prime}|^{1/2}<\varepsilon_{L}$ and
$\varepsilon_{L}^{-1}<t_{2}^{-1}|\lambda_{N}/\lambda_{N}^{\prime}|^{1/2}\leq
1$. The case $t_{2}<|\lambda_{N}/\lambda_{N}^{\prime}|$ can be handled
similarly. The second equation in (3.21) follows from (3.8), (3.20) and the
first equation in (3.21). Finally, equation (3.22) follows from (3.20) and
(3.12). ∎

######  Example 3.8.

Let $L=L_{\mathfrak{a},M}$ be the lattice in Example 3.5 and recall that we
chose $t_{1}=1$ and $t_{2}=\varepsilon=5+2\sqrt{6}$. For $i=1,2$ and $h$ in
(3.13), the function $\theta^{(1,1)}_{L+h}(\tau,t_{i})$ vanishes identically,
and $\tilde{\vartheta}^{c_{i}}_{{L}+h}=\tilde{\vartheta}^{c_{i},+}_{{L}+h}$ is
a Maass form. We compute the Fourier expansions of the theta series
$\tilde{\vartheta}^{c_{i},+}_{{L}+h}$ given in (3.19) by calculating
$\tilde{c}_{i}(\Lambda)$ as defined in (3.17). Note that
$\Gamma_{L}=\langle\varepsilon_{L}\rangle$, where
$\varepsilon_{L}=\varepsilon^{2}$. We thus have

$\displaystyle(\log\varepsilon_{L})^{-1}\tilde{c}_{1}(\Lambda)$
$\displaystyle=\frac{1}{2}+\mathbb{B}_{1}\left(\frac{\log(|\lambda/\lambda^{\prime}|)}{4\log(\varepsilon)}\right),$
$\displaystyle(\log\varepsilon_{L})^{-1}\tilde{c}_{2}(\Lambda)$
$\displaystyle=1+\mathbb{B}_{1}\left(\frac{\log(|\lambda/\lambda^{\prime}|)}{4\log(\varepsilon)}-\frac{1}{2}\right).$

Since $\varepsilon_{L}$ preserves the set $L+h$ for all $h$ given above, it is
sufficient to compute $c_{i}(\Lambda)$ for one representative
$\Lambda\in\Gamma_{L}\backslash L+h$. To evaluate the periodic Bernoulli
polynomials, note that

(3.24) $\mathbb{B}_{1}(x+n)+\mathbb{B}_{1}(-x)=0$

for all $x\in\mathbb{R},n\in\mathbb{Z}$. Using table 1, we obtain for $i=1,2$
and $h$ as in (3.13)

(3.25)
$\tilde{\vartheta}_{L+h}^{c_{i},+}(\tau)=i\frac{\vartheta_{L+h}(\tau)}{8}.$

Now it is easy to see that

$\displaystyle\varphi_{0}$
$\displaystyle=-(i\log\varepsilon_{L})^{-1}\left(\tilde{\vartheta}^{c_{i}}_{L+\frac{1}{2}{}}-\tilde{\vartheta}^{c_{i}}_{L+\frac{7}{2}{}}-\tilde{\vartheta}^{c_{i}}_{L+\frac{1}{2}+\sqrt{6}}+\tilde{\vartheta}^{c_{i}}_{L+\frac{7}{2}+\sqrt{6}}\right)$
$\displaystyle=-(\log\varepsilon_{L})^{-1}\left(\tilde{\vartheta}^{c_{2}}_{L+\frac{1}{2}{}}-\tilde{\vartheta}^{c_{2}}_{L+\frac{7}{2}{}}-\tilde{\vartheta}^{c_{2}}_{L+\frac{1}{2}+\sqrt{6}}+\tilde{\vartheta}^{c_{2}}_{L+\frac{7}{2}+\sqrt{6}}\right)$
$\displaystyle\quad+(\log\varepsilon_{L})^{-1}\left(\tilde{\vartheta}^{c_{1}}_{L+\frac{1}{2}{}}-\tilde{\vartheta}^{c_{1}}_{L+\frac{7}{2}{}}-\tilde{\vartheta}^{c_{1}}_{L+\frac{1}{2}+\sqrt{6}}+\tilde{\vartheta}^{c_{1}}_{L+\frac{7}{2}+\sqrt{6}}\right).$

######  Example 3.9.

Let $F$ be the same as in Example 3.5. Take $L=L_{\mathfrak{a},M}$ with
$\mathfrak{a}=\mathcal{O}=\mathbb{Z}+\sqrt{6}\mathbb{Z}$ and $M=2$. Then
$L^{*}=\frac{1}{2}\mathbb{Z}+\frac{\sqrt{6}}{12}\mathbb{Z},\varepsilon_{L}=\varepsilon^{2}=(5+2\sqrt{6})^{2}=49+20\sqrt{6}$,
and it is easy to check that

$\theta^{(1,1)}_{L+h}(\tau,t_{0})=-\frac{\sqrt{6}}{24}\eta(\tau)^{3}\overline{g(\tau)}v^{3/2}$

for $h=\frac{1}{2}+\frac{\sqrt{6}}{12}$ and $t_{0}=1$, where

$g(\tau)=q^{1/48}(1-23q^{11}+25q^{13}-47q^{46}+49q^{50}-71q^{105}+73q^{111}-95q^{188}+97q^{196}+O(q^{200}))$

is a modular form of weight $3/2$. Using the isomorphism

$\Gamma_{L}\backslash(L+h\sqcup
L+h^{\prime})\cong\\{\mathfrak{b}=(\beta)\subset\mathcal{O}:{\mathrm{Nm}}(\beta)\equiv-5\bmod{48}\\},~{}[\lambda]\mapsto(2\sqrt{6}\lambda),$

with $h^{\prime}=\frac{1}{2}-\frac{\sqrt{6}}{12}$, we can write

$\displaystyle\sqrt{12}^{-1}$
$\displaystyle\tilde{\vartheta}_{L+h}^{c_{0},+}(48\tau)=\dots+\left(3\log\varepsilon+\log\frac{29+6\sqrt{6}}{25}\right)W_{-235}(\tau)$
$\displaystyle+\left(\log\varepsilon+\log\frac{155+28\sqrt{6}}{139}\right)W_{-139}(\tau)+\left(2\log\varepsilon+\log\frac{55+14\sqrt{6}}{43}\right)W_{-43}(\tau)$
$\displaystyle+\log\frac{7+2\sqrt{6}}{5}W_{5}(\tau)+\left(4\log\varepsilon+\frac{55-6\sqrt{6}}{53}\right)W_{53}(\tau)$
$\displaystyle+\left(3\log\varepsilon+\log\frac{155-48\sqrt{6}}{101}\right)W_{101}(\tau)+\log\frac{151+10\sqrt{6}}{149}W_{149}(\tau)+\dots.$

##  4\. Action under $\tilde{\Delta}$.

In this section, we will generalize the mock Maass theta functions studied in
3.2, and study their images under the differential operator $\tilde{\Delta}$.
In the notation of Sections 3.1 and 3.2, consider

(4.1)
$\hat{\vartheta}^{c_{1},c_{2}}_{L+h}(\tau;P):=\int^{t_{2}}_{t_{1}}\theta_{L+h}(\tau,t)P(\log
t)\frac{dt}{t}=\int^{\nu_{2}}_{\nu_{1}}\theta_{L+h}(\tau,e^{\nu})P(\nu)d\nu$

for any polynomial $P(X)\in\mathbb{C}[X]$. This defines a real-analytic
modular function in $\tau$. When $P$ is the constant function, this is just
the function $\hat{\vartheta}^{c_{1},c_{2}}_{L+h}(\tau)$ defined in (3.1).
When $P(X)=X$ and $t_{i}=\varepsilon_{L}^{i-1}t_{0}$, this is the function
$\tilde{\vartheta}^{c_{0}}_{L+h}$ from (3.16). Just as the function
$\hat{\vartheta}^{c_{1},c_{2}}_{L+h}$ in Section 3.1, the generalization
$\hat{\vartheta}^{c_{1},c_{2}}_{L+h}(\tau;P)$ has the following Fourier
expansion

(4.2) $\hat{\vartheta}^{c_{1},c_{2}}_{L+h}(\tau;P)={\sqrt{v}}\sum_{\lambda\in
L+h}\mathbf{e}\left(Q_{0}(\lambda)\tau\right)\int^{t_{2}}_{t_{1}}e^{-\pi
vB_{0}(\lambda,Z^{-}_{t})^{2}}P(\log t)\frac{dt}{t}.$

When $t_{2}=\varepsilon_{L}t_{1}$, we simply write
$\hat{\vartheta}^{c_{1}}_{L+h}$ for $\hat{\vartheta}^{c_{1},c_{2}}_{L+h}$.

To understand the action of $\tilde{\Delta}$ on this function, we first record
a standard differential operator relation satisfied by the Siegel theta
kernel.

######  Lemma 4.1.

In the notation above, we have

(4.3)
$4\tilde{\Delta}\theta_{L+h}(\tau,t)=(t\partial_{t})^{2}\theta_{L+h}(\tau,t).$

###### Proof.

One can show that the function $f(\tau,t)$ defined by

$\begin{split}f(\tau,t)&:=\phi_{\tau}(\lambda_{t}^{+},\lambda_{t}^{-}),~{}\phi_{\tau}(x,y):=\sqrt{v}\mathbf{e}\left({x^{2}}{}\tau-{y^{2}}{}\overline{\tau}\right),\end{split}$

satisfies the identity

$\displaystyle(t\partial_{t})^{2}f(\tau,t)$ $\displaystyle=\left(\left(2\pi
v\right)^{2}\left((\lambda_{1}t^{-1})^{2}-(\lambda_{2}t)^{2}\right)^{2}-4\pi
v\left((\lambda_{1}t^{-1})^{2}+(\lambda_{2}t)^{2}\right)\right)f(\tau,t)$
$\displaystyle=4\left(\left(4\pi
v\lambda_{t}^{+}\lambda_{t}^{-}\right)^{2}-2\pi
v\left((\lambda_{t}^{+})^{2}+(\lambda_{t}^{-})^{2}\right)\right)f(\tau,t).$

On the other hand, a straightforward calculation yields

$\tilde{\Delta}\phi_{\tau}(x,y)=\left((4\pi vxy)^{2}-2\pi
v(x^{2}+y^{2})\right)\phi_{\tau}(x,y).$

By the definition of the theta series in (2.11), we deduce that (4.3) holds. ∎

Now the image of $\hat{\vartheta}^{c_{1},c_{2}}_{L+h}$ under the operator
$\tilde{\Delta}$ is given by the following lemma.

######  Lemma 4.2.

We have

$4\tilde{\Delta}\hat{\vartheta}^{c_{1},c_{2}}_{L+h}(\tau;P)=(t\partial_{t}\theta_{L+h})(\tau,t_{2})P(\nu_{2})-(t\partial_{t}\theta_{L+h})(\tau,t_{1})P(\nu_{1})\\\
\quad-\left(\theta_{L+h}(\tau,t_{2})P^{\prime}(\nu_{2})-\theta_{L+h}(\tau,t_{1})P^{\prime}(\nu_{1})\right)+\hat{\vartheta}^{c_{1},c_{2}}_{L+h}(\tau;P^{\prime\prime}).$

######  Remark 4.3.

Note that
$t\partial_{t}\theta_{L+h}(\tau,t)=-8\pi\theta^{(1,1)}_{L+h}(\tau,t)$.

###### Proof.

We apply Lemma 4.1 to obtain

$\displaystyle 4\tilde{\Delta}$
$\displaystyle\hat{\vartheta}^{c_{1},c_{2}}_{L+h}(\tau;P)=4\int^{\nu_{2}}_{\nu_{1}}\tilde{\Delta}\theta_{L+h}(\tau,e^{\nu})P(\nu)d\nu=\int^{\nu_{2}}_{\nu_{1}}(\partial_{\nu}^{2}\theta_{L+h}(\tau,e^{\nu}))P(\nu)d\nu$
$\displaystyle=(t\partial_{t}\theta_{L+h})(\tau,t_{2})P(\nu_{2})-(t\partial_{t}\theta_{L+h})(\tau,t_{1})P(\nu_{1})-\int^{\nu_{2}}_{\nu_{1}}(\partial_{\nu}\theta_{L+h})(\tau,e^{\nu})P^{\prime}(\nu)d\nu$
$\displaystyle=(t\partial_{t}\theta_{L+h})(\tau,t_{2})P(\nu_{2})-(t\partial_{t}\theta_{L+h})(\tau,t_{1})P(\nu_{1})-\left(\theta_{L+h}(\tau,t_{2})P^{\prime}(\nu_{2})-\theta_{L+h}(\tau,t_{1})P^{\prime}(\nu_{1})\right)$
$\displaystyle\quad+\hat{\vartheta}^{c_{1},c_{2}}_{L+h}(\tau;P^{\prime\prime}).\qed$

We can now specialize $P$ and $c_{i}$ to obtain the following corollary, which
immediately follows from equation (3.22), where the image of
$\tilde{\vartheta}^{c_{i}}_{L+h}$ is given, and Remark 4.3.

######  Corollary 4.4.

For $c_{i}=c(t_{i})$ with $t_{i}>0$, we have

(4.4)
$\begin{split}4\tilde{\Delta}&\left(\hat{\vartheta}^{c_{1},c_{2}}_{L+h}(\tau;X)+\frac{\log
t_{1}}{\log\varepsilon_{L}}\tilde{\vartheta}^{c_{1}}_{L+h}(\tau)-\frac{\log
t_{2}}{\log\varepsilon_{L}}\tilde{\vartheta}^{c_{2}}_{L+h}(\tau)\right)=\theta_{L+h}(\tau,t_{1})-\theta_{L+h}(\tau,t_{2}),\\\
\tilde{\Delta}&\left(\hat{\vartheta}^{c_{0}}_{L+h}(\tau;X^{2})-4\log(t_{0}^{2}\varepsilon_{L})\tilde{\vartheta}^{c_{0}}_{L+h}(\tau)\right)=-2\log\varepsilon_{L}\theta_{L+h}(\tau,t_{0})+2\vartheta_{L+h}(\tau).\end{split}$

Using the same procedure as in Section 3.2, one can calculate the Fourier
coefficients of the harmonic part of $\hat{\vartheta}^{c_{0}}_{L+h}(\tau;P)$
for any polynomial $P$. From (4.4), we see that in order to construct
$\tilde{\Delta}$-preimage of $\theta_{L+h}$, one would need to construct
$\tilde{\Delta}$-preimage of $\vartheta_{L+h}$, which seems to require new
ideas. It would be very interesting to study the Fourier coefficients of such
preimage.

## References

  * [Bor98] Richard E. Borcherds.  Automorphic forms with singularities on Grassmannians.  Invent. Math., 132:491–562, 1998.
  * [CL20] Pierre Charollois and Yingkun Li.  Harmonic Maass forms associated to real quadratic fields.  J. Eur. Math. Soc., 22(4):1115–1148, 2020.
  * [Coh88] Henri Cohen.  q-identities for Maass waveforms.  Invent. Math., 91(3):409–422, 1988.
  * [FK17] Jens Funke and Stephen S. Kudla.  Mock modular forms and geometric theta functions for indefinite quadratic forms.  J. Phys. A, 50(40):404001, 19, 2017.
  * [Li17] Yingkun Li.  Harmonic Eisenstein series of weight one.  In L-functions and automorphic forms, volume 10 of Contrib. Math. Comput. Sci., pages 171–184. Springer, Cham, 2017.
  * [Maa49] Hans Maass.  Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen.  Math. Ann., 121:141–183, 1949.
  * [Sch09] Nils R. Scheithauer.  The Weil representation of SL2(Z) and some applications.  Int. Math. Res. Not., 22(2009):1488–1545, 2009.
  * [Zwe12] Sander P. Zwegers.  Mock Maass theta functions.  Q. J. Math., 63(3):753–770, 2012.