# Discrete Spectrum Analysis of Vector OFDM Signals

Xiang-Gen Xia, , and Wei Wang X.-G. Xia is with the Department of Electrical
and Computer Engineering, University of Delaware, Newark, DE 19716, USA
(e-mail: xxia@ece.udel.edu). W. Wang is with Peng Cheng Laboratory, Shenzhen,
China (e-mail: wei_wang@ieee.org).

###### Abstract

Vector OFDM (VOFDM) is equivalent to OTFS and is good for time-varying
channels. However, due to its vector form, its signal spectrum is not as clear
as that of the conventional OFDM. In this paper, we study the discrete
spectrum of discrete VOFDM signals. We obtain a linear relationship between a
vector of information symbols and a vector of the same size of components
evenly distributed in the discrete VOFDM signal spectrum, and show that if a
vector of information symbols is set to $0$, then a corresponding vector of
the same size of the discrete VOFDM signal spectrum is $0$ as well, where the
components of the $0$ vector are not together but evenly distributed in the
spectrum. With the linear relationship, the information symbol vectors can be
locally precoded so that any of the discrete spectrum of VOFDM signals can be
set to $0$, similar to that of the conventional OFDM signals. These results
are verified by simulations.

###### Index Terms:

OFDM, VOFDM, OTFS, discrete spectrum, precoding

##  I Introduction

Vector OFDM (VOFDM) was first proposed in 2000 [1, 2, 3] and has had more
investigations in, for example, [4, 5, 6]. There have been its equivalent
systems, such as OSDM [7] and A-OFDM [8]. VOFDM converts an intersymbol
interference (ISI) channel to multiple vector subchannels where there is no
ISI across vector subchannels and in each vector subchannel, the ISI is
limited by the vector size. VOFDM is in the middle of OFDM and single carrier
frequency domain equalizer (SC-FDE) [6] and is the most general modulation
dealing with ISI. Furthermore, it was shown in [3, 6, 12, 14] that VOFDM is
good for time-varying channels due to its vector-wise demodulation.
Interestingly, it has been recently shown in, for example, [10, 11, 12, 15],
that VOFDM is also equivalent to OTFS [9, 13] for delay Doppler channels,
which has been an active research topic lately.

It is known that the discrete spectrum of a conventional OFDM signal is clear
and the information symbols located at the subcarriers are the responses at
the corresponding discrete frequencies. Thus, it is easy to avoid certain
frequrencies by setting the information symbols $0$ at these frequencies and
therefore is flexible in the design in practice. Due to the vector form in
VOFDM, the spectrum of a VOFDM signal is not as clear as that of an OFDM
signal. In this paper, we study the discrete spectrum of discrete VOFDM
signals. We first obtain a linear relationship between a vector of information
symbols and a vector of the same size of components evenly distributed in the
discrete VOFDM signal spectrum. The linear relationship is in fact the
$M$-point DFT of a modulated information symbol vector, where $M$ is the
vector size. With this linear relationship, it is easy to see that if a vector
of information symbols is set to $0$, then a corresponding vector of the same
size of the discrete VOFDM signal spectrum is $0$ as well, where the
components of the $0$ vector are not together but evenly distributed in the
spectrum. Also with the linear relationship, the information symbol vectors
can be locally precoded so that any of the discrete spectrum of VOFDM signals
can be set to $0$, similar to that of the conventional OFDM signals. Thus, it
is also flexible in a practical design to avoid certain frequencies or
frequency bands. Since the precoding is done for a vector (or vector-wisely),
when the vector size is not large, the precoding complexity is low.

This paper is organized as follows. In Section II, we derive the discrete
spectrum of a VOFDM signal and present the precoding to pre-satisfy a $0$
spectrum property of VOFDM signals. In Section III, we present some simple
simulation results to verify the theoretical results. In Section III, we
conclude this paper.

##  II Discrete Spectrum and Precoding

We first formulate a discrete VOFDM signal [1, 2, 6, 14]. Let $M$ be the
vector size and $N$ be the IFFT size. Let

${\bf x}_{k}=(x_{k}(0),x_{k}(1),\cdots,x_{k}(M-1))^{T},\,\,0\leq k\leq N-1,$
(1)

be information symbol vectors to be transmitted, where T stands for the
transpose. For the $m$th components of these $N$ vectors, $0\leq m\leq M-1$,
along the index $k$ direction, the $N$-point IFFT is

$\mbox{IFFT}_{N}(x_{0}(m),x_{1}(m),\cdots,x_{N-1}(m))$
$\stackrel{{\scriptstyle\Delta}}{{=}}(X_{0}(m),X_{1}(m),\cdots,X_{N-1}(m)).$
(2)

Let

${\bf X}_{n}=(X_{n}(0),X_{n}(1),\cdots,X_{n}(M-1))^{T},\,\,0\leq n\leq N-1,$
(3)

and

${\bf X}=(X(0),X(1),\cdots,X(MN-1))^{T}$ $=({\bf X}_{0}^{T},{\bf
X}_{1}^{T},\cdots,{\bf X}_{N-1}^{T})^{T}.$ (4)

Then, ${\bf X}$ is a discrete VOFDM signal before the cyclic prefix (CP)
insertion. Since this paper is only interested in the spectrum of a VOFDM
signal, for simplicity, we do not consider a CP for a VOFDM signal.

We now calculate the discrete spectrum of the discrete VOFDM signal ${\bf X}$
by taking its $MN$-point FFT:

${\bf y}=(y(0),y(1),\cdots,y(MN-1))^{T}=\mbox{FFT}_{MN}({\bf X})$
$\stackrel{{\scriptstyle\Delta}}{{=}}(y_{0}(0),y_{1}(0),\cdots,y_{N-1}(0),y_{0}(1),y_{1}(1),\cdots,y_{N-1}(1),$
$\cdots,y_{0}(M-1),y_{1}(M-1),\cdots,y_{N-1}(M-1))^{T}.$ (5)

Finally, let

${\bf y}_{k}=(y_{k}(0),y_{k}(1),\cdots,y_{k}(M-1))^{T},\,\,0\leq k\leq N-1,$
(6)

which are the spectrum vectors of size $M$ and their components are evenly
distributed (not consecutively located) in the discrete spectrum ${\bf y}$ of
the VOFDM signal ${\bf X}$.

As one can see that the above discrete spectrum vectors ${\bf y}_{k}$ are not
obviously related to the original information vectors ${\bf x}_{k}$ as that
for the OFDM signals, i.e., when $M=1$ and ${\bf y}_{k}={\bf x}_{k}$ for
$0\leq k\leq N-1$. We next find a relationship between vectors ${\bf y}_{k}$
and ${\bf x}_{k}$ for $0\leq k\leq N-1$ by writing out ${\bf y}$ in (5) as
follows. First, let $W_{K}=e^{-2\pi j/K}$ for a positive integer $K$. Then,
for $0\leq k_{1}\leq M-1$ and $0\leq k_{2}\leq N-1$,

$\displaystyle y(k_{1}N+k_{2})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{MN}}\sum_{n=0}^{MN-1}X(n)W_{MN}^{n(k_{1}N+k_{2})}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{MN}}\sum_{n_{1}=0}^{M-1}\sum_{n_{2}=0}^{N-1}X_{n_{2}}(n_{1})W_{MN}^{(n_{2}M+n_{1})(k_{1}N+k_{2})}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{MN}}\sum_{n_{1}=0}^{M-1}\sum_{n_{2}=0}^{N-1}X_{n_{2}}(n_{1})W_{MN}^{n_{1}k_{2}}W_{M}^{n_{1}k_{1}}W_{N}^{n_{2}k_{2}}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{M}}\sum_{n_{1}=0}^{M-1}\left(\frac{1}{\sqrt{N}}\sum_{n_{2}=0}^{N-1}X_{n_{2}}(n_{1})W_{N}^{n_{2}k_{2}}\right)W_{M}^{n_{1}k_{1}}W_{MN}^{n_{1}k_{2}}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}$
$\displaystyle\frac{1}{\sqrt{M}}\sum_{n_{1}=0}^{M-1}x_{k_{2}}(n_{1})W_{MN}^{n_{1}(k_{1}N+k_{2})},$

where Step (a) is from (2). This leads to the following relationship between
${\bf y}_{k}$ and ${\bf x}_{k}$:

$y_{k}(m)=\frac{1}{\sqrt{M}}\sum_{n=0}^{M-1}\left(x_{k}(n)W_{MN}^{nk}\right)W_{M}^{nm},$
(7)

for $0\leq m\leq M-1,0\leq k\leq N-1$. The above linear relationship is in
fact the $M$-point DFT of the information symbols $x_{k}(n)$ modulated by
$W_{MN}^{nk}$ in terms of $n$, $0\leq n\leq M-1$. From (7), we immediately
have the following theorem.

###### Theorem 1

The $k$th discrete spectrum vector ${\bf y}_{k}$ in (6) of a discrete VOFDM
signal and the $k$th information symbol vector ${\bf x}_{k}$ in (1) satisfy
the linear relationship (7), where $0\leq k\leq N-1$. If vector ${\bf
x}_{k_{0}}=0$ for some $0\leq k_{0}\leq N-1$, then vector ${\bf y}_{k_{0}}=0$.

One can easily see from (7) that for the conventional OFDM, i.e., when $M=1$,
we have ${\bf y}_{k}={\bf x}_{k}$ for $0\leq k\leq N-1$ as we have mentioned
earlier. However, when $M>1$, VOFDM is different from OFDM and if any
component of information symbol vector ${\bf x}_{k}$ is not $0$, then none of
the spectrum components in ${\bf y}_{k}$ vanishes, i.e., ${\bf y}_{k}$ is
full, for any $0\leq k\leq N-1$. Although this is the case, if some of the
frequencies are to avoid for the system, i.e., some components in ${\bf
y}_{k}$ are set to $0$, then one may precode the information symbol vector
${\bf x}_{k}$ so that (7) holds, i.e., the pre-set frequency components in
${\bf y}_{k}$ are indeed $0$. This can be obtained as follows.

Consider the last a few components of the spectrum vector ${\bf y}_{k}$ are
set to $0$ for some $0\leq k\leq N-1$, i.e.,

${\bf y}_{k}=({\bf y}_{k,1}^{T},{\bf y}_{k,2}^{T})^{T}=({\bf
y}_{k,1}^{T},0,\cdots,0)^{T},$ (8)

where ${\bf y}_{k,2}=(0,\cdots,0)^{T}$ or simply denoted by ${\bf y}_{k,2}=0$.
Let the information symbol vector

${\bf x}_{k}=({\bf x}_{k,1}^{T},{\bf x}_{k,2}^{T})^{T},$ (9)

where ${\bf x}_{k,2}$ has the same size as ${\bf y}_{k,2}$ and is to be
determined below, and ${\bf x}_{k,1}$ is the information symbol vector that
contains information symbols to transmit.

The precoding is that for any given information symbol vector ${\bf x}_{k,1}$,
we want to determine vector ${\bf x}_{k,2}$ such that (7) holds for ${\bf
y}_{k}$ in (8) and ${\bf x}_{k}$ in (9). And ${\bf x}_{k,2}$ is called a
precoded signal. We next show how this precoding is done.

Let

${\bf W}_{M}=\frac{1}{\sqrt{M}}\left(W_{M}^{mn}\right)_{0\leq m,n\leq M-1}$

be the $M$-point DFT matrix, ${\bf I}_{M}$ be the $M\times M$ identity matrix,
and

${\bf W}_{M}=\left({\bf W}_{M,1},{\bf W}_{M,2}\right)\mbox{ and }{\bf
I}_{M}=\left({\bf I}_{M,1},{\bf I}_{M,2}\right),$ (10)

where the numbers of the columns of submatrices ${\bf W}_{M,2}$ and ${\bf
I}_{M,2}$ are the same as the dimension of vector ${\bf y}_{k,2}$. Then, (7)
is equivalent to

$\left({\bf I}_{M,1},-{\bf W}_{M,2}\right)\left(\begin{array}[]{c}{\bf
y}_{k,1}\\\ \hat{\bf x}_{k,2}\end{array}\right)=\left({\bf W}_{M,1},-{\bf
I}_{M,2}\right)\left(\begin{array}[]{c}\hat{\bf x}_{k,1}\\\ {\bf
y}_{k,2}\end{array}\right)$ $=\left({\bf W}_{M,1},-{\bf
I}_{M,2}\right)\left(\begin{array}[]{c}\hat{\bf x}_{k,1}\\\
0\end{array}\right),$ (11)

where $(\hat{\bf x}_{k,1}^{T},\hat{\bf x}_{k,2}^{T})^{T}=\hat{\bf
x}_{k}^{T}=(x_{k}(n)W_{MN}^{kn})_{0\leq n\leq M}$ and can be thought of as a
modulated ${\bf x}_{k}$ for $0\leq k\leq N-1$.

From (11), one can see that for any given information symbol vector ${\bf
x}_{k,1}$, one can always find or solve for $\hat{\bf x}_{k,2}$ and thus a
precoded ${\bf x}_{k,2}$ can be obtained by demodulating $W_{MN}^{kn}$ from
$\hat{\bf x}_{k,2}$ such that the discrete frequency components in ${\bf
y}_{k,2}$ of the discrete VOFDM signal ${\bf X}$ in (4) are $0$.

One remark is that since the above precoding is done for a vector (or vector-
wisely), when the vector size $M$ is not large, the precoding complexity is
low. Another remark is that in the above precoding, the number of variables in
the precoded part ${\bf x}_{k,2}$ is the same as that of the number of zeros
in the spectrum vector ${\bf y}_{k,2}$. In this case, the number of free
information symbols in the unprecoded part ${\bf x}_{k,1}$ is the maximum.
However, as we shall see in the simulations in the next section, the precoded
part ${\bf x}_{k,2}$ solved from the above linear system (11) may have a
higher power, which may cause a higher peak-to-average power ratio (PAPR) in
the VOFDM signal. In order to have more degrees of freedom in designing the
precoded ${\bf x}_{k,2}$ and thus to have a smaller PAPR, less uncoded
information symbols in ${\bf x}_{k,1}$, i.e., a smaller size vector ${\bf
x}_{k,1}$, can be chosen and thus, more variables in the precoded part ${\bf
x}_{k,2}$, i.e., a larger size vector ${\bf x}_{k,2}$, can be used/allocated
to achieve the pre-set zero spectrum vector ${\bf y}_{k,2}$, i.e., to have
${\bf y}_{k,2}=0$.

##  III Simulations

In this section we present some simple simulations to verify the theoretical
results we have obtained above.

For the illustratuon purpose, we first consider the case when $M=N=4$, i.e.,
the vector size $M=4$ and the IFFT size $N=4$. In this case we set the first
information symbol vector ${\bf x}_{0}$ to $0$, then according to Theorem 1,
the frequency component vector ${\bf y}_{0}$ is $0$, whose $4$ components are
evenly distributed over the total $16$ frequency points, which can be seen
from Fig. 1.

Figure 1: Spectrum of a VOFDM signal of length $16$ with $M=N=4$ and the first
information symbol vector ${\bf x}_{0}$ is set to $0$.

We next consider the case when $M=8$ and $N=64$, i.e., the vector size is
$M=8$ and the IFFT size is $N=64$. In this case we set the first $16$
consecutive information symbol vectors ${\bf x}_{k}$ for $0\leq k\leq 15$ all
$0$. According to Theorem 1, the first $16$ consecutive frequency component
vectors ${\bf y}_{k}$ for $0\leq k\leq 15$ are all 0, which can be seen from
Fig. 2.

Figure 2: Spectrum of a VOFDM signal of length $512$ with $M=8$ and $N=64$,
and the first $16$ information symbol vectors ${\bf x}_{k}$, $0\leq k\leq 15$,
are set to $0$.

Since the precoded part ${\bf x}_{k,2}$ for a given information symbol vector
${\bf x}_{k,1}$ and a zero setting in ${\bf y}_{k,2}$ in ${\bf y}_{k}$ is
solved via the linear system (11), its power is a variable. We next set the
last two spectrum components in ${\bf y}_{k}$ to $0$ for a fixed index $k$. We
then component-wisely calculate the averaged magnitudes of the signal vector
${\bf x}_{k}=({\bf x}_{k,1}^{T},{\bf x}_{k,2}^{T})^{T}$ including the random
information symbol vector ${\bf x}_{k,1}$ with BPSK signals of amplitudes $\pm
1$ and the precoded part ${\bf x}_{k,2}$ that includes the last components of
${\bf x}_{k}$. The average is done over $1000$ random trials. The averaged
magnitudes of the signal vector ${\bf x}_{k}$ are shown in Fig. 3 component-
wisely. The averaged magnitudes of the precoded part ${\bf x}_{k,2}$ for
different vector sizes $M$ are plotted in Fig. 4, where only the last two
components in vector ${\bf x}_{k}$ are precoded.

Figure 3: Averaged magnitudes of a precoded signal ${\bf x}_{k}$ with $M=8$
and $N=64$, where the last $2$ frequency components in ${\bf y}_{k}$ are set
to $0$. Figure 4: Averaged magnitudes of the precoded part ${\bf x}_{k,2}$ for
different vector sizes $M$, where $N=64$ and the last $2$ frequency components
in ${\bf y}_{k}$ are set to $0$.

We next want to avoid the last $32$ frequencies in a discrete spectrum of
total $512$ frequency points for a VOFDM system with vector size $M=8$ and
IFFT size $N=64$. This can be done by setting the last frequency components of
the second half $32$ frequency response vectors ${\bf y}_{k}$, $32\leq k\leq
63$, to $0$. After doing so, the frequency spectrum of such a VOFDM signal is
shown in Fig. 5, where one can clearly see that the spectra at the last $32$
frequency points are $0$. For this example, according to Theorem 1 only the
last components of the second half signal vectors ${\bf x}_{k}$, $32\leq k\leq
63$, need to be precoded and the first $31$ components of each of these
vectors are free. In other words, for any of these $32$ information symbol
vectors, only the last component needs to be precoded. Also, the first $32$
information symbol vectors ${\bf x}_{k}$, $0\leq k\leq 31$, are free as well.
Its corresponding discrete VOFDM signal magnitudes of length $512$ in time
domain before CP are shown in Fig. 6, where one can see that the PAPR may be
high.

The averages of the magnitudes across the second half vectors ${\bf x}_{k}$,
$32\leq k\leq 63$, are shown in Fig. 7 component-wisely, where the first $31$
components of every vector ${\bf x}_{k}$, $32\leq k\leq 63$, and any of the
first half vectors ${\bf x}_{k}$, $0\leq k\leq 31$, all take BPSK signals with
amplitudes $\pm 1$ randomly.

Figure 5: Frequency spectrum of a VOFDM signal with the last $32$ frequency
components $0$. Figure 6: The VOFDM signal with its spectrum shown in Fig. 5.
Figure 7: Averaged magnitudes of the last $32$ information symbol vectors
${\bf x}_{k}$, $31\leq k\leq 63$, where the last components of these vectors
are precoded.

##  IV Conclusion

In this paper, we have first obtained a linear relationship between an
information symbol vector and a corresponding frequency response vector of the
discrete frequency spectrum of the VOFDM signal. It is the $M$-point DFT of
the modulated information symbol vector. From this linear relationship, it is
easy to see that a zero information symbol vector leads to a zero frequency
response vector of the spectrum of the OFDM signal. Also, information symbol
vectors can be locally precoded so that any components in the frequency
response vectors of the spectrum of the VOFDM signals can be set to $0$. Thus,
any discrete spectrum components can be set to $0$ in a VOFDM system similar
to an OFDM system. Since the precoding is done for a vector (or vector-
wisely), when the vector size is not large, the complexity is low. These
results have been verified by numerical simulations.

## References

  * [1] X.-G. Xia, “Precoded OFDM systems robust to spectral null channels and vector OFDM systems with reduced cyclic prefix length,” in Proc. of ICC, vol. 2, pp. 1110-1114, New Orleans, LA, USA, Jun. 18-22, 2000.
  * [2] X.-G. Xia, “Precoded and vector OFDM robust to channel spectral nulls and with reduced cyclic prefix length in single transmit antenna systems,” IEEE Trans. Commun., vol. 49, no. 8, pp. 1363–1374, Aug. 2001.
  * [3] X.-G. Xia, Modulated Coding for Intersymbol Interference Channels, New York, Marcel Dekker, Oct. 2000.
  * [4] C. Han, T. Hashimoto, and N. Suehiro, “Constellation-rotated vector OFDM and its performance analysis over Rayleigh fading channels,” IEEE Trans. Commun., vol. 58, no. 3, pp. 828–837, Mar. 2010.
  * [5] P. Cheng, M. Tao, Y. Xiao, and W. Zhang, “V-OFDM: On performance limits over multi-path Rayleigh fading channels,” IEEE Trans. Commun., vol. 59, no. 7, pp. 1878–1892, Jul. 2011
  * [6] Y. Li, I. Ngebani, X.-G. Xia, and A. Host-Madsen, “On performance of vector OFDM with linear receivers,” IEEE Trans. Signal Process., vol. 60, no. 10, pp. 5268–5280, Oct. 2012.
  * [7] N. Suehiro, C. Han, T. Imoto, N. Kuroyanagi, “An information transmission method using Kronecker product,” in Proc. IASTED Int. Conf. Commun. Syst. Netw., pp. 206–209, Sept. 2002.
  * [8] J. Zhang, A. D. S. Jayalath, and Y. Chen, “Asymmetric OFDM systems based on layered FFT structure,” IEEE Signal Process. Lett., vol. 14, no. 11, pp. 812–815, 2007.
  * [9] R. Hadani, S. Rakib, M. Tsatsanis, A. Monk, A. J. Goldsmith, A. F. Molisch, and R. Calderbank, “Orthogonal time frequency space modulation,” in Proc. IEEE Wireless Commun. Netw. Conf. (WCNC), San Francisco, CA, USA, pp. 1-6, Mar. 2017.
  * [10] P. Raviteja, Y. Hong, and E. Viterbo, “OTFS performance on static multipath channels,” IEEE Wireless Commun. Lett., vol. 8, no. 3, pp. 745-748, Mar. 2019.
  * [11] Y. Ge, Q. Deng, P. C. Ching, and Z. Ding, “OTFS signaling for uplink NOMA of heterogeneous mobility users,” IEEE Trans. on Commun., vol. 69, no. 5, pp. 3147-3161, May 2021\.
  * [12] X.-G. Xia, “Comments on ’The transmitted signals of OTFS and VOFDM are the same’,” IEEE Trans. Wireless Commun., vol. 21, no. 12, p. 11252, Dec. 2022.
  * [13] Y. Hong, T. Thaj, and E. Viterbo, Delay-Doppler Communications: Principles and Applications, Elsevier, London, UK, 2022.
  * [14] X.-G. Xia, VOFDM Systems,
https://www.eecis.udel.edu/~xxia/VOFDM_PPT.pdf.

  * [15] I. van der Werf, H. Dol, K. Blom, R. Heusdens, R. C. Hendriks, and G. Leus, “On the equivalence of OSDM and OTFS,” Signal Processing, vol. 214, (2024) 109254. Available on-line, Sept. 14, 2023, via https://doi.org/10.1016/j.sigpro.2023.109254


# Dark matter via Baryogenesis: Affleck-Dine Mechanism in the Minimal
Supersymmetric Standard Model

K. El Bourakadi1,2<EMAIL_ADDRESS>M. Ferricha-Alami1,2
<EMAIL_ADDRESS>Z. Sakhi1,2<EMAIL_ADDRESS>M. Bennai1,2
<EMAIL_ADDRESS>H. Chakir1,2<EMAIL_ADDRESS>1Quantum Physics and
Magnetism Team, LPMC, Faculty of Science Ben M’sik,
Casablanca Hassan II University, Morocco
2 Centre of Physics and Mathematics, CPM, Rabat, Morocco

###### Abstract

We conducted an investigation into Affleck-Dine baryogenesis within the
context of D-term inflation, specifically focusing on its relationship with a
recent reheating formalism. It was found that by considering a specific
reheating temperature, the observed baryon asymmetry can be accounted through
Affleck-Dine baryogenesis. Additionally, the majority of gravitinos are
inferred to be generated from the decay of the next-to-lightest supersymmetric
particle, with Q-balls potentially serving as a source of gravitinos via NSP
decay. The temperature at which decay occurs depends on the charge of the
Q-balls, which is determined by the fragmentation of the Affleck-Dine
condensate. Remarkably, the gravitino mass required for dark matter aligns
naturally with the theoretical gravitino mass.

##  I Introduction

Affleck-Dine baryogenesis [1, 2] is a notable method for explaining the baryon
asymmetry in the universe. This mechanism involves the utilization of a scalar
field that possesses baryon number. Within the framework of the Minimal
Supersymmetric Standard Model (MSSM), there exist numerous scalar fields known
as flat directions, which are composed of squarks (and sleptons). In the limit
of supersymmetry (SUSY), the scalar potential associated with these flat
directions becomes zero, but it is altered by the effects of SUSY breaking and
higher order operators [3]. Within the Minimal Supersymmetric Standard Model,
there exist numerous flat directions that lack classical potentials. These
flat directions are only subject to modifications through soft supersymmetry
breaking mass terms and non-renormalizable terms. The presence of these flat
directions gives rise to several intriguing consequences in cosmology,
including baryogenesis and Q-ball formation. In the Affleck-Dine (AD)
mechanism for baryogenesis, a complex field associated with a flat direction
(referred to as the AD field) possesses a large value during the period of
inflation. Subsequently, as the mass of the AD field exceeds the Hubble
parameter H, it begins to oscillate. During this time, the AD field acquires a
velocity in the phase direction due to the presence of a baryon number-
violating operator, leading to an efficient generation of baryon number [4].

In the context of primordial inflation [5, 6, 7, 8, 9, 10, 11, 12, 13, 14],
scalar fields characterized by shallow potential energy can achieve non-zero
vacuum expectation values (VEVs) [15, 16, 17, 18, 19, 20, 21, 22],
supersymmetric extensions of the standard model incorporate numerous scalar
fields known as flat directions, which exhibit a vanishing scalar potential
under the condition of exact supersymmetry. D-term potential is employed in
the construction of supersymmetric inflation models. Within these D-term
inflation models, supergravity effects do not introduce mass terms of the
order of $O(H^{2})$, where $H$ is the Hubble parameter. Instead, the AD field
in these models possesses only a soft supersymmetry-breaking mass term at the
scale of the weak force. Consequently, the AD field can acquire a substantial
expectation value during inflation, facilitating the success of AD
baryogenesis [23, 24, 25, 26].

The nature of dark matter (DM) remains a fundamental inquiry in cosmology,
particularly regarding the particle physics origin of the dark matter particle
[27]. Within the framework of the Minimal Supersymmetric Standard Model and
its extensions, the identity of the dark matter candidate is intrinsically
linked to the nature of supersymmetry breaking [28]. In scenarios of gravity-
mediated SUSY breaking, the neutralino emerges as a natural candidate for the
lightest SUSY particle (LSP). Conversely, in gauge-mediated SUSY breaking
(GMSB), the gravitino assumes the role of a natural LSP candidate [29]. In
GMSB (gauge-mediated supersymmetry breaking) models, there are two potential
mechanisms for the production of a gravitino dark matter density. Firstly, if
the reheating temperature $T_{re}$ is sufficiently high (around $10^{7}GeV$)
and the gravitino mass ($m_{3/2}$) is approximately $0.1GeV$, gravitinos can
be generated through thermal scattering [30]. Alternatively, gravitinos may be
produced through the late decay of next-to-lightest supersymmetric particles
(NSPs). However, this latter process is subject to strong constraints imposed
by big-bang nucleosynthesis (BBN) [31].

The formation of Q balls during the Affleck-Dine baryogenesis process is a
widely recognized phenomenon [32, 33]. Notably, in the context of gauge-
mediated supersymmetry breaking, Q balls with sufficiently large charges (Q)
exhibit stability against decay into baryons (nucleons). Consequently, these Q
balls have the potential to serve as dark matter in the universe and may be
detectable. The baryon asymmetry observed in the universe is attributed to the
baryon numbers that are not retained within the Q balls [34]. In the realm of
scalar field theory, a Q-ball refers to a non-topological soliton that
possesses a conserved global charge [35]. A Q-ball solution arises when the
energy minimum occurs at a non-zero value of the scalar field while
maintaining a fixed charge Q. During its rotation, the condensate of the AD
field experiences spatial instabilities, leading to the formation of spherical
lumps known as Q balls [36]. Q balls charge essentially corresponds to the
baryon number associated with the AD field [37]. The study of the formation of
large Q-balls has primarily been conducted through analytical approaches
employing linear theory [38, 39], as well as numerical simulations conducted
on one-dimensional lattices [32]. However, both methods rely on the assumption
that the Q-ball configuration is spherical, which prevents us from accurately
determining whether the Q-ball configuration is truly achieved. In this work,
we consider the AD mechanism in light of the D-term inflation to discuss the
observed baryon asymmetry that lies in AD baryogenesis. The recent formalism
that aims to derive the reheating temperature as a function of the
inflationary parameters to constrain the reheating temperature according to
Planck’s data, relating the baryon asymmetry to the reheating temperature
provides an alternative way to study the observed value of $n_{B}/s,$ NSPs
abundance and the gravitino dark matter mass according to observation. The
paper is organized as follows: In Sec. II we present the Affleck-Dine
mechanism and derive the general form of the baryon number density. In Sec.
III we discuss the D-term inflation in the AD mechanism. In Sec. IV we provide
the basic formalism for deriving the reheating temperature in D-term inflation
then study the observational constraints on Baryogenesis. In Sec. V we study
the Q-ball decay and the abundance of NSP to constrain gravitinos dark matter.
In Sec. VI we conclude.

##  II  Affleck–Dine Mechanism in MSSM

Let’s take a brief look at the Affleck-Dine mechanism [40, 41] within the MSSM
in this section. The MSSM’s superpotential is presented in [42].

$W=\gamma_{U}QUH_{U}+\gamma_{D}QDH_{D}+\gamma_{E}LEH_{D}+\mu H_{U}H_{D},$ (1)

The superfields $H_{U},H_{D},U,D,L,E,$ and $Q$ respectively represent Higgs
doublets, up-type and down-type right-handed quarks, left-handed lepton
doublets, right-handed charged leptons, and left-handed quark doublets. The
supersymmetric theories of the field space encompass numerous directions where
the D-term contributions to the scalar potential vanish entirely. The scalar
potential along these directions is derived solely from F-terms and SUSY-
breaking contributions. Among these ”flat” directions, there is a captivating
category consisting of the SUSY partners of the SM fermions, such as squarks,
sleptons, and Higgs fields.

The AD field potential rise from soft SUSY breaking terms and
nonrenormalizable terms [43, 44]. Therefore, we present the superpotential
that elevates a level direction $\phi$ as

$W=\frac{\lambda_{n}}{nM^{n-3}}\phi^{n}.$ (2)

Here, $\phi$ denotes the superfield that encompasses the flat direction $\phi$
and its corresponding fermionic partner, while $M$ refers to a large scale
that functions as a cut-off, and $\lambda_{n}\sim O(1)$ denotes the Yukawa
coupling. One can express the scalar potential of the AD field along a flat
direction as [45]

$V(\phi)=\left(m_{\phi}^{2}-c_{H}H^{2}\right)\left|\phi\right|^{2}+\frac{1}{M^{2n-6}}\phi^{2n-2}+\frac{m_{3/2}}{nM^{n-3}}\left(a_{m}\phi^{n}+h.c\right).$
(3)

The potential terms that are proportional to the soft mass squared
$m_{\phi}^{2}$ and the gravitino mass $m_{3/2}$ are a result of SUSY breaking
at the true vacuum. The Hubble parameter $H$ in the expression reflects the
impact of $SUSY$ breaking, triggered by the inflaton’s finite energy density
[41]. Here, $c_{H}$ denotes a real constant which approximately is of the
order unity. The calculation of the baryon asymmetry is relatively
straightforward. The baryon number density correlates with the AD field, as
follows:

$n_{B}=\beta i\left(\dot{\phi}\phi-\phi^{\ast}\dot{\phi}^{\ast}\right).$ (4)

The AD field has a corresponding baryon charge $\beta$, which is limited to a
maximum of $1/3$. The equation describing the AD field’s motion in an
expanding universe can lead to deriving the time evolution of the baryon
number density

$\dot{n}_{B}+3Hn_{B}=2\beta\frac{m_{3/2}}{M^{n-3}}Im\left(a_{m}\phi^{n}\right),$
(5)

the baryon comoving density is expressed as:

$n_{B}(t)=\frac{1}{a(t)^{3}}\int^{t}a(t^{\prime})^{3}2\beta\frac{\left|a_{m}\right|m_{3/2}}{M^{n-3}}\sin\theta
dt,$ (6)

the generated baryon number density at the oscillation time $t_{os}$ [45]

$n_{B}(t_{os})=\frac{2\left(n-2\right)}{3\left(n-3\right)}\beta\delta_{eff}\left|a_{m}\right|m_{3/2}\left(H_{os}M^{n-3}\right)^{2/\left(n-2\right)},$
(7)

here $\delta_{eff}\equiv\sin\theta\left(=O(1)\right).$ Upon completion of the
reheating process following inflation, this results in the following baryon
asymmetry:

$\frac{n_{B}}{s}=\frac{1}{4}\frac{T_{re}}{M_{p}^{2}H_{os}^{2}}n_{B}(t_{os}),$
(8)

the equation incorporates the reheating temperature $T_{re}$, $s$ as the
entropy density of the universe, and $M_{p}=2.4\times 10^{18}GeV$ as the
reduced Planck scale. It can be expressed using $H_{os}\simeq m_{\phi}$ as
[15]:

$\frac{n_{B}}{s}\approx
10^{-10}\times\beta\left(\frac{1TeV}{m_{\phi}}\right)\left(\frac{T_{re}}{10^{9}GeV}\right),$
(9)

for $n=4,$ and for the case $n=6$,

$\frac{n_{B}}{s}\approx
10^{-10}\times\beta\left(\frac{1TeV}{m_{\phi}}\right)^{1/2}\left(\frac{T_{re}}{100GeV}\right).$
(10)

These predictions on the baryon asymmetry by the Affleck-Dine mechanism look
promising. However, the Affleck-Dine condensate tends to be vulnerable to
spatial perturbations, leading to the formation of $Q$-balls, which are
nontopological solitons. The presence of $Q$-ball formation does not change
the projections regarding the baryon asymmetry, provided that any generated
$Q$-balls are unstable and disintegrate prior to the onset of big-bang
nucleosynthesis.

##  III Dynamics of the inflaton field in AD mechanism

Taking into account the possibility that D-terms provide the dominant
contribution to the vacuum energy [46, 47, 48], we consider D-term inflation
[49, 50] which incorporates a neutral chiral superfield, $S$, and superfields
$\psi_{\pm}$ possessing charges of $\pm 1$ under a $U(1)$ symmetry [51].

$V=\left|\kappa\right|^{2}\left(\left|\psi_{+}\psi_{-}\right|^{2}+\left|S\psi_{+}\right|^{2}+\left|S\psi_{-}\right|^{2}\right)+\frac{g^{2}}{2}\left(\left|\psi_{+}\right|^{2}-\left|\psi_{-}\right|^{2}+\xi^{2}\right)^{2}.$
(11)

In this scenario, we have added a positive Fayet-Iliopoulos (FI) term,
$\xi^{2}$, and the potential’s global minimum, which preserves supersymmetry,
occurs at $S=0$, $\psi_{+}=0$, and $\psi_{-}=\xi$. Additionally, there is a
local minimum at $\psi_{+}=\psi_{-}=0$ for $\left|S\right|>S_{c}\equiv
g\xi/\kappa$. By making a $U(1)_{FI}$ phase rotation, we can assume that $S$
is real. Introducing $\sigma\equiv\sqrt{2}S$, the potential of $\sigma$,
including a 1-loop correction, is provided for
$\sigma>\sigma_{c}=\sqrt{2}g\xi/\kappa$ by [52]

$V\left(\sigma\right)=\frac{g^{2}\xi^{2}}{2}\left(1+\alpha\ln\left(\frac{\kappa\sigma^{2}}{\Lambda^{2}}\right)\right),$
(12)

here $\alpha=g^{2}/16\pi^{2}$ and $\Lambda$ is a renormalization mass scale
that does not affect physical quantities. When the AD scalar $\phi$ has an
initial value of $\phi_{i}\sim O(M)$, the potential is mainly influenced by
the F-term of the AD field. As a result of supergravity effects, a mass term
of $O(H^{2})$ arises for the inflaton, causing it to quickly roll down to its
actual minimum. Hence, inflation is not feasible under these circumstances. To
create a successful inflationary model, the value of $\phi_{i}$ must be less
than $\phi_{c}$, which is defined as [53]:

$\phi_{c}=\sqrt{2}\left(\frac{g}{\sqrt{2}\lambda}\xi^{2}M^{n-3}\right)^{\frac{1}{n-1}}.$
(13)

For $\phi_{i}\leq\phi_{c}$, the universe is dominated by the D-term of the
inflaton field. It can be demonstrated simply that the slow roll criterion for
$\sigma$ is met with $\epsilon\ll\eta$. The slow roll parameters are defined
as
$\epsilon=\left(M_{p}^{2}/2\right)\left(V^{\prime}(\sigma)/V(\sigma)\right)^{2}$
and $\eta=M_{p}^{2}V^{\prime\prime}(\sigma)/V(\sigma),$ knowing that the
period of inflation will last for a specific e-folds number given as
$N\simeq\frac{1}{M_{p}^{2}}\int_{\sigma_{end}}^{\sigma_{k}}\frac{V}{V^{\prime}}d\sigma$.

##  IV Reheating and Baryogenesis

###  IV.1 Reheating Mechanism

Following the end of inflation, there needs to be a reheating stage where the
energy stored in the inflaton field is transformed into a plasma comprising of
relativistic particles. After this stage, the conventional radiation-dominated
evolution of the early universe takes over. The energy from the vacuum that
drove inflation gets converted into potential and kinetic energy, which is
then utilized by the condensates of fields oscillating around their various
minima. In the previously presented model, the vacuum energy, represented by
$V=g^{2}\xi^{4}/2$, is currently being transported by some combination of the
$\psi_{-}$ and $S$ fields, in each oscillating condensate the relative portion
of the energy depends on their effective masses after inflation, that are
respectively given by $\lambda\xi$ and $\sqrt{2}g\xi\ $[51]. The universe
maintains a low temperature until $H$ falls below the decay width of either
the $S$ or $\psi_{-}$ fields. At this point, one of these fields will undergo
decay, releasing its accumulated energy and causing the universe to undergo a
process of reheating.

During the initial stages, the quasi-de-Sitter phase for $N_{k}$ e-folds of
expansion is powered by the inflaton field. The scale of the comoving horizon
decreases proportionally to $\sim a^{-1}$. Once the accelerated expansion ends
and the comoving horizon begins to expand, the phase of reheating commences.
Following an additional $N_{re}$ e-folds of expansion, all the energy
contained within the inflaton field has been entirely dissipated, leading to
the formation of a hot plasma having a reheating temperature of $T_{re}$.
Subsequent to that phase, the Universe undergoes radiation domination for an
additional $N_{RD}$ e-folds of expansion before its transition into a state of
matter domination.

In cosmology, we detect perturbation modes that are of a similar magnitude to
that of the horizon. To illustrate, Planck determines the pivot scale at
$k=0.05Mpc^{-1}$. The comoving Hubble scale $a_{k}H_{k}=k$ is linked to the
present time scale relative to when this mode exited the horizon in the
following form [54],

$\ln\left(\frac{k}{a_{0}H_{0}}\right)=\ln\left(\frac{a_{k}}{a_{end}}\right)+\ln\left(\frac{a_{end}}{a_{re}}\right)+\ln\left(\frac{a_{re}}{a_{eq}}\right)+\ln\left(\frac{a_{eq}H_{eq}}{a_{0}H_{0}}\right)+\ln\left(\frac{H_{k}}{H_{eq}}\right).$
(14)

Quantities denoted by a subscript $\left(k\right)$ are computed at the moment
when the horizon is exited. Other subscripts correspond to different eras such
as the end of inflation $(end)$, reheating $(re)$, radiation-matter equality
$(eq)$, and the current time $(0)$. knowing that
$\ln\left(\frac{a_{k}}{a_{end}}\right)=N_{k},$
$\ln\left(\frac{a_{end}}{a_{re}}\right)=N_{re}$ and
$\ln\left(\frac{a_{re}}{a_{eq}}\right)=N_{RD}.$

Knowing that $g^{2}\xi^{2}$ dominates in Eq. (12) which leads to
$V^{\prime}/V=\alpha/\sigma$ and $V^{\prime\prime}/V=-\alpha/\sigma^{2},$ the
Hubble parameter during inflation is derived as

$H_{k}=\pi M_{p}\sqrt{\frac{12\alpha A_{s}(1-n_{s})}{(6+\alpha)}},$ (15)

with the scalar amplitude $\ln(10^{10}A_{s})=3.089_{-0.027}^{+0.024}$
according to recent results [55]. For specific values of $\alpha,$ $H_{k}$ is
determined from $(1-n_{s}).$ The inflaton field at the end of inflation can be
estimated considering $\left|\eta\right|=\left|\eta_{end}\right|\simeq 1$ will
lead to

$\sigma_{end}=\frac{\sqrt{\alpha}M_{p}}{\eta_{end}},$ (16)

while its value at the beginning of inflation satisfies

$\sigma_{k}=M_{p}\sqrt{\frac{\alpha(6+\alpha)}{3(1-n_{s})}}.$ (17)

In this direction, one would drive the inflationary e-folds as

$N_{k}=\frac{\sigma_{k}^{2}}{2M_{p}^{2}},$ (18)

and the potential at the end of inflation in the following way
$V_{end}=V(\sigma_{end})$. In addition to Eq. (14), a second relation of the
reheating e-folds can be derived by tracking the energy density at the end of
inflation and the reheating phase as,

$N_{re}=\frac{1}{3\left(1+\omega_{re}\right)}\ln\left(\frac{\rho_{end}}{\rho_{re}}\right),$
(19)

here $\omega_{re}$ is the equation of state corresponding to the reheating
phase, The reheating temperature is determined by the final energy density
$\rho_{re}=\left(\pi^{2}/30\right)g_{re}T_{re}^{4},$ with $g_{re}$ is the
number of relativistic species at the end of reheating. On the other hand, in
Ref. [56] they estimated the energy density at the end of inflation by
$\rho_{end}=\left(3/2\right)V_{end}.$ Assuming that entropy remains conserved
from the end of reheating until the present time, one can establish a
relationship between the temperature at the end of reheating $T_{re}$ and the
CMB temperature $T_{0}$ taking into account $N_{k}$ the number of e-foldings
during inflation, the Hubble parameter $H_{k}$ and the value of the potential
at the end of inflation $V_{end}$ [56, 57],

$T_{re}=\left[\left(\frac{43}{11g_{re}}\right)^{\frac{1}{3}}\frac{a_{0}T_{0}}{k}H_{k}e^{-N_{k}}\left[\frac{3^{2}\cdot
5V_{end}}{\pi^{2}g_{re}}\right]^{-\frac{1}{3\left(1+\omega_{re}\right)}}\right]^{\frac{3\left(1+\omega_{re}\right)}{3\omega_{re}-1}}.$
(20)

It can be observed that when $N_{re}$ is greater, $T_{re}$ becomes smaller and
conversely. This means that reheating occurs more rapidly and effectively,
resulting in higher temperatures, which aligns with expectations.

###  IV.2 Observational Constraints on Baryogenesis

The cosmic accelerating expansion is caused by the inflaon’s energy dominance
during the first stages of the universe’s evolution, which leads to the
dilution of other matter contents and their asymmetry that existed before
inflation [58]. Conversely, the abundance of the baryon asymmetry after the
Big Bang Nucleosynthesis has been confirmed as $n_{B}/s=9.2\times 10^{-11}$
through the observation of CMB [55] and measurements of the primordial
abundances of light elements [59].

Following the end of inflation, the system progresses and experiences changes
in the overall energy density and Hubble constant until a time when the
supergravity components in the potential equation (3) governing the AD
direction reach a level of equivalence with the $-H^{2}\left|\phi\right|^{2}$
term. When $H$ is approximately equal to $m_{3/2}$, the AD field begins to
oscillate near the minimum. It is crucial to note that during this phase, all
terms in the potential for $\phi$ have similar magnitudes, including those
that conserve and violate baryon (or lepton) numbers. Furthermore, the
evolution of the $\phi$ direction as it oscillates around zero results in
significant CP-violating phases. In these circumstances, the AD mechanism
results a baryon-to-entropy ratio which is obtained in the previous section.

Figure 1: Spectral index dependence of baryon number Considering baryon
asymmetry with both $n=4$ (top curves) and $n=6$ (bottom curves), we fixed
D-term potential parameter $\alpha$ to take $1.2$ and $1.4$ values.

In Fig. (1) we present the behavior of baryon number as a function of the
spectral index. We study the $n_{B}/s$ evolution for some fixed values of the
equation of state parameter $\omega_{re}$ and baryon asymmetry number
parameter $n$ corresponding to the AD mechanism, in the case of the D-term
inflationary potential considering both $\alpha=1.2,1.4.$ Note that the black
line corresponds to $\omega_{re}=-1/3$, the red line corresponds to
$\omega_{re}=-1/4$, blue line corresponds to $\omega_{re}=-1/6$, the green
line corresponds to $\omega_{re}=0$, the purple line corresponds to
$\omega_{re}=1/6$ and the yellow line corresponds to $\omega_{re}=1/4$. The
vertical light orange region represents Planck-2018 bounds on $n_{s}$ and the
red dashed line represents a precision of baryon number $\approx 10^{-10}$
from CMB experiments. We observe that for the case of $n=4$ only lines with
$\omega_{re}=-1/3$ and $-1/4$ tend towards the central values of $n_{s}$ for
$n_{B}/s\lesssim 10^{-10}$ considering $\alpha=1.2,$ while for $\alpha=1.4$
the cases with $\omega_{re}=-1/6$ and $-1/4$ provide the best consistency with
data. Howerver, for $\omega_{re}=-1/3$ and $0$ are partialy consistent for
$n_{B}/s\gtrsim 10^{-10}$ and $n_{B}/s\lesssim 10^{-10}$ respctevly. On
another hand, when we consider $n=6$ and $\alpha=1.2,$ the line with
$\omega_{re}=-1/6$ still provide good consistency for all our chosen interval
of $n_{B}/s$, while $\omega_{re}=-1/4$ presents fine compatibility only when
$n_{B}/s\gtrsim 10^{-10}.$ Moreover, $\omega_{re}=0$ is the best compatible
case for $\alpha=1.4.$ From this analysis we can conclude that the observed
reheating temperature from CMB data provides a wide range of possible values
of the baryon asymmetry number far away from the central value
$n_{B}/s=9.2\times 10^{-11},$ which affects the predicted mass of gravitino
dark matter from the AD mechanism as we will discuss in the next section.

##  V Q-Balls and Dark Matter

When the reheating temperature that occurred after inflation is as high as
$10^{10}GeV$, the thermal processes can generate enough gravitinos to
potentially serve as a dark matter for a wide mass range in certain
supergravity models [60]. It is worth noting that the decay of the next-to-
lightest supersymmetric particle could also play a role in generating the
lightest supersymmetric particle gravitino abundance [61, 62]. A $Q$-ball is a
type of coherent state of a complex scalar field that exists and remains
stable thanks to the conservation of a particular global U(1) quantum number
[63]. Within the MSSM framework, the standard baryon and lepton numbers may
serve as the conserved quantity for $Q$-balls composed of squarks and
sleptons, respectively. It is worth exploring whether these entities could
have emerged in the early universe and if their lifespan is sufficiently long
to allow them to persist until today and potentially contribute to dark matter
[35, 64]. If $Q$-ball formation takes place during the inflationary period
before the reheating process is finished, then the fluctuation will become
non-linear once the Hubble parameter reaches a certain threshold [45],

$H=H_{non}=\frac{m_{\phi}\left|K\right|}{2\alpha},$ (21)

where $\alpha$ is adopted in previous analyses to have values around
$\alpha\simeq 30-40,$ $K$ is the parameter involved in the one-loop correction
of the scalar potential of $\phi$ at the time of $Q$-ball formation [38, 39].
At the time when the Hubble parameter reaches $H_{non}$, the baryon number
density of the condensate reaches,

$n_{B}(t_{non})\simeq
n_{B}(t_{osc})\times\left(\frac{H_{non}}{H_{osc}}\right)^{2}.$ (22)

It is possible to estimate the charge carried by an individual Q-ball as [45],

$\displaystyle Q$ $\displaystyle\simeq$ $\displaystyle\frac{4}{3}\pi
R_{Q}^{3}\times n_{B}(t_{non})$ (23) $\displaystyle\sim$ $\displaystyle
3\times
10^{-3}\times\frac{2\left(n-2\right)}{3\left(n-3\right)}\beta\delta_{eff}\left|a_{m}\right|\left(\frac{m_{3/2}}{m_{\phi}}\right)\left(\frac{M}{m_{\phi}}\right)^{\frac{2\left(n-3\right)}{\left(n-2\right)}}$

When a $Q$-ball interacts with particles in a thermal bath, a portion of its
charge may evaporate. The amount of charge that evaporates is predicted to be
around $\Delta Q=O(10^{18})$ [39]. Therefore, if the charge of a $Q$-ball is
greater than $\Delta Q$, the $Q$-ball could potentially survive the
evaporation process. The abundance of NSP following the decay of Q-balls (up
until the decay of NSP) can be expressed as [65]:

$Y_{N}(T)\simeq\left[\frac{1}{Y(T_{d})}+\sqrt{\frac{8\pi^{2}g_{\ast}(T_{d})}{45}}\left\langle\sigma\upsilon\right\rangle
M_{p}\left(T_{d}-T\right)\right]^{-1}.$ (24)

While if annihilation is not efficient after the production of NSPs via Q-ball
decay, the resulting abundance is expressed as [65]:

$Y_{N}=3\times
10^{-10}\left(\frac{N}{3}\right)\left(\frac{f_{B}}{1}\right)\left(\frac{n_{B}/s}{10^{-10}}\right),$
(25)

we should bring to mind as well $\rho_{c}/s\simeq 1.7\times
10^{-9}\left(h/0.7\right)^{2}GeV.$ The notion that Affleck-Dine baryogenesis
could explain both the baryon asymmetry and dark matter through $Q$-ball decay
was abandoned in favor of neutralino in the MSSM [66]. In the case of $n=6$
Affleck-Dine baryogenesis, experimental evidence doesn’t exclude the
possibility that dark matter particle mass must be of $1GeV$, it appears to be
somewhat unnatural and improbable in the context of gravity-mediated SUSY
breaking.

Figure 2: Variation of the NSPs abundance as a function of the gravitino mass
considering different index spectral values and reheating temperature in
$GeV.$ The left plot corresponds to $n=4$, and the right plot corresponds to
$n=6$.

Fig. (2) shows the evolution of the abundance of NSPs as a function of
gravitino mass knowing that $m_{\phi}\simeq m_{3/2}\left|a_{m}\right|$ and
$\left|a_{m}\right|\approx 10$, from the figure we observe that the gravitinos
mass has a weak effect on the behavior of $Y_{N}.$ However, increasing the
reheating temperature causes a decrease in the abundance of NSPs. On the other
hand, for the case of $n=6,$ the NSPs abundance decreases quickly for
$m_{3/2}\leq 20GeV,$ while for $n=6$ the abundance of NSPs stabilizes when
$m_{3/2}\gtrsim 2GeV.$ Moreover, the most compatible value of reheating
temperature is given by$\ T_{re}\simeq 10^{7}GeV$ since it coincides with the
central value of $n_{s}=0.9649$ which equivalently predicts values of NSPs
abundance around $Y_{N}\simeq 10^{-11}$ for both $n=4$ and $n=6$ that
corresponds to the AD mechanism. When the RH sneutrino NLSPs possess suitable
characteristics, the decay of the $d=6$ Q-balls into RH sneutrinos could
potentially generate gravitino dark matter while remaining consistent with Big
Bang Nucleosynthesis, even in cases where the gravitino mass $m_{3/2}$ exceeds
$1GeV$, while if the fragmentation of the condensate results in only
positively charged Q-balls, the preservation of B-conservation and R-parity
conservation indicates that the gravitino mass should be approximately $2GeV$
[67]. To achieve a successful scenario with RH sneutrino NLSPs in this mass
range, two conditions must be met: firstly, the MSSM-LSPs resulting from
Q-ball decay should be able to decay into RH sneutrinos before
nucleosynthesis, and secondly, the RH sneutrinos should be able to decay into
gravitinos without violating constraints imposed by nucleosynthesis and free-
streaming [68]. This could be accomplished if the RH neutrinos possess
enhanced Yukawa couplings through the see-saw mechanism [69].

When the annihilation takes place following the production of NSPs, the
resulting abundance of NSPs for temperatures significantly lower than $T_{d}$
can be expressed as follows:

$Y_{N}(T)\simeq\left[\sqrt{\frac{8\pi^{2}g_{\ast}(T_{d})}{45}}\left\langle\sigma\upsilon\right\rangle
M_{p}T_{d}\right]^{-1}.$ (26)

Assuming this situation, the production of all dark matter gravitinos must
occur through the decay of NSPs. When the constraint on $Y_{N}$ is applied
based on the the equivalence of $\rho_{NSP},\rho_{LSP}$ and $\rho_{DM}$, we
can derive the following:

$Y_{N}(T)\simeq 3\times 10^{-12}\left(\frac{3\times
10^{-8}GeV^{2}}{\left\langle\sigma\upsilon\right\rangle}\right)\left(\frac{1GeV}{T_{d}}\right)\left(\frac{10}{g_{\ast}(T_{d})}\right)^{1/2}\sim
4\times 10^{-12}\left(\frac{100GeV}{m_{DM}}\right),$ (27)

to achieve $T_{d}\sim 1GeV$, a substantial annihilation cross-section of h
$\left\langle\sigma\upsilon\right\rangle\simeq 10^{-8}-10^{-7}GeV^{2}$ is
necessary, where $m_{DM}$ represents the mass of the dark matter particle. For
such a high level of annihilation cross-section, the neutralino NSP must not
be Bino-like but rather Higgsino-like [60]. When $T_{re}$ is less than
approximately $10^{7}GeV$, $Q$-balls possessing baryon numbers ranging from
$10^{12}$ to $10^{18}$ will decay subsequent to reheating and maintain the
effective Enqvist-McDonald correlation that links the energy densities of
baryonic matter and dark matter [70, 71],

$\left(\frac{\Omega_{N}h^{2}}{0.11}\right)\approx\left(\frac{m_{NSP}}{1.5GeV}\right)\left(\frac{Nf_{B}}{3}\right)\left(\frac{\Omega_{b}h^{2}}{0.02}\right).$
(28)

Figure 3: Dark matter mass variation as a function of the baryon asymmetry
number.

We study the behavior of the dark matter mass as a function of the baryon
asymmetry in Fig. (3), the mass of dark matter is decreased when we consider
higher values of $n_{B}/s.$ From the predicted value of baryon asymmetry, one
would estimate dark matter masses around $m_{DM}\simeq 1GeV.$ However, our
analysis proves that the reheating temperature has a noticeable consequence on
the baryon asymmetry which can slightly modify the range of dark matter rang
for consistency with a wider number of models.

Figure 4: Variation of the dark matter mass as a function of spectral index
$n_{s}$, the black line corresponds to $n=4$ and the red line corresponds to
$n=6.$

Fig. (4) represents the evolution of the dark matter mass with respect to the
spectral index, the central orange region corresponds to the observed value of
spectral index $n_{s}=0.9649\pm 0.0042.$ From the $m_{DM}-n_{s}$ plane we
observe that CMB experiments predict lower bounds on the dark matter masses,
for the $n=4$ case dark matter masses are predicted to be $m_{DM}\gtrsim
100MeV$, while for $n=6$ the lower bound on the dark matter masses is given as
$m_{DM}\gtrsim 100eV$. For both AD model cases, even if the dark matter can
reach a higher range of masses according to recent Plancks data, the gravitino
dark matter mass arising from NSP decay must be below $1GeV$ to satisfy the
constraints imposed by BBN on NSP decay. The least restrictive upper bounds on
BBN are obtained for stau and sneutrino NSPs, which permit gravitino masses
ranging from $0.1$ to $1GeV$.

##  VI Conclusion

The estimation of baryon asymmetry in recent advancements relies on whether
the AD field experiences early oscillation due to thermal effects at a high
reheating temperature. Furthermore, achieving the appropriate baryon asymmetry
is possible through the reduction of baryon asymmetry with the aid of early
oscillation. Additionally, Affleck-Dine baryogenesis remains a feasible and
compatible explanation for both gravitino dark matter and baryon asymmetry,
with the majority of gravitinos being generated through NSP decay.
Consequently, the nature of the NSP and the constraints imposed by BBN and CMB
play a crucial role in determining the conclusions related to NSP decay, which
is an important subject of study.

In summary, the previously proposed scenario involving Affleck-Dine
baryogenesis was reexamined with a focus on the abundance of Q-balls in
connection to gravitino dark matter. If Q-balls cannot withstand evaporation,
the production of gravitinos becomes reliant on the decay of NSPs, with their
abundance determined by the thermal relic density. However, if Q-balls can
survive evaporation, NSPs are generated as decay products of Q-balls.
Consequently, these persistent Q-balls offer a new potential avenue for
gravitino dark matter through NSP decay.

In this work, we have shown that the baryon asymmetry can be alternatively
constrained from plancks data using recent formalism that was proposed to
study the reheating phase. Moreover, the derived baryon number can provide a
good tool to discuss the NSP abundance from Q-balls decay. On another hand,
one can estimate gravitino dark matter by taking into account previous
findings on NSP decay.

In conclusion, our findings indicate that both baryon asymmetry and gravitino
dark matter can arise from the decay of Q-balls within the gauge-mediated MSSM
framework, while satisfying the constraints imposed by BBN. In order to
advance the model, a comprehensive numerical investigation of Affleck-Dine
condensate fragmentation and Q-ball decay is required. This area of study will
be the focus of forthcoming research efforts.

## References

  * [1] Affleck, I., & Dine, M. (1985). A new mechanism for baryogenesis. Nuclear Physics B, 249(2), 361-380.
  * [2] Dine, M., Randall, L., & Thomas, S. (1996). Baryogenesis from flat directions of the supersymmetric standard model. Nuclear Physics B, 458(1-2), 291-323.
  * [3] Kasuya, S., & Kawasaki, M. (2014). Q-ball dark matter and baryogenesis in high-scale inflation. Physics Letters B, 739, 174-179.
  * [4] Kawasaki, M., & Takahashi, F. (2001). Adiabatic and isocurvature fluctuations of Affleck–Dine field in D-term inflation model. Physics Letters B, 516(3-4), 388-394.
  * [5] Liddle, A. R. (1998). An introduction to cosmological inflation. High energy physics and cosmology, 260.
  * [6] El Bourakadi, K., Ferricha-Alami, M., Filali, H., Sakhi, Z., & Bennai, M. (2021). Gravitational waves from preheating in Gauss–Bonnet inflation. The European Physical Journal C, 81(12), 1144.
  * [7] El Bourakadi, K., Bousder, M., Sakhi, Z., & Bennai, M. (2021). Preheating and reheating constraints in supersymmetric braneworld inflation. The European Physical Journal Plus, 136(8), 1-19.
  * [8] El Bourakadi, K., Asfour, B., Sakhi, Z., Bennai, M., & Ouali, T. (2022). Primordial black holes and gravitational waves in teleparallel Gravity. The European Physical Journal C, 82(9), 792.
  * [9] Sakhi, Z., El Bourakadi, K., Safsafi, A., Ferricha-Alami, M., Chakir, H., & Bennai, M. (2020). Effect of brane tension on reheating parameters in small field inflation according to Planck-2018 data. International Journal of Modern Physics A, 35(30), 2050191.
  * [10] K. El Bourakadi, Z. Sakhi and M. Bennai, Preheating constraints in $\alpha$-attractor inflation and gravitational waves production, Int. J. Mod. Phys. A 37, 2250117 (2022).
  * [11] Bourakadi, K. E., Koussour, M., Otalora, G., Bennai, M., & Ouali, T. (2023). Constant-roll and primordial black holes in f (Q, T) gravity. arXiv preprint arXiv:2301.03696. Physics of the Dark Universe (2023).
  * [12] Bourakadi, K. E., Sakhi, Z., & Bennai, M. (2023). Observational constraints on Tachyon inflation and reheating in f (Q) gravity. arXiv preprint arXiv:2302.11229.
  * [13] Bourakadi, K. E., Sakhi, Z., & Bennai, M. (2023). Cosmological Signatures from the Post-Inflationary Epochs. arXiv preprint arXiv:2303.04875.
  * [14] Bourakadi, K. E. (2023). Exploring the impact of Tree Level Higgs Potential on Reheating in f (Q) Gravity. arXiv preprint arXiv:2303.02517.
  * [15] Allahverdi, R., & Mazumdar, A. (2012). A mini review on Affleck–Dine baryogenesis. New Journal of Physics, 14(12), 125013.
  * [16] Ferricha-Alami, et al. Mutated hybrid inflation on brane and reheating temperature. The European Physical Journal Plus, 2017, vol. 132, p. 1-10.
  * [17] Ferricha-Alami, M., et al. Smooth hybrid inflation on brane constrained by Planck data. International Journal of Modern Physics A 29.26 (2014): 1450146.
  * [18] Ferricha-Alami, M., Safsafi, A., Bouaouda, A., Zarrouki, R., & Bennai, M. (2015). Kähler potential braneworld inflation in supergravity after Planck 2015. International Journal of Modern Physics A, 30(34), 1550208.
  * [19] Ferricha-Alami, M., Mounzi, Z., Jdair, O., Naciri, M., Bennai, M., & Chakir, H. (2017). Randall–Sundrum II model from small field inflation in light of planck data and reheating temperature. Moscow University Physics Bulletin, 72, 425-432.
  * [20] Ferricha-Alami, M., Jdair, O., Chakir, H., & Bennai, M. (2018). Logarithmic Potential Braneworld in Light of the Recent Experiment Observation. Revista Cubana de Física, 35(2), 86-90.
  * [21] Mounzi, Z., Ferricha-Alami, M., Safsafi, A., & Bennai, M. (2016). Braneworld Inflation in Supergravity with a Shift Symmetric Kähler Potential. Journal of Astrophysics and Astronomy, 37, 1-10.
  * [22] Mounzi, Z., Ferricha-Alami, M., Safsafi, A., & Bennai, M. (2017). Observational constraints on a hyperbolic potential in brane-world inflation. Gravitation and Cosmology, 23, 84-89.
  * [23] Gaillard, M. K., Murayama, H., & Olive, K. A. (1995). Preserving flat directions during inflation. Physics Letters B, 355(1-2), 71-77.
  * [24] Odintsov, S. D., & Oikonomou, V. K. (2016). Gauss–Bonnet gravitational baryogenesis. Physics Letters B, 760, 259-262.
  * [25] Oikonomou, V. K., Pan, S., & Nunes, R. C. (2017). Gravitational baryogenesis in running vacuum models. International Journal of Modern Physics A, 32(22), 1750129.
  * [26] Odintsov, S. D., & Oikonomou, V. K. (2016). Loop quantum cosmology gravitational baryogenesis. Europhysics Letters, 116(4), 49001\.
  * [27] Feng, J. L. (2010). Dark matter candidates from particle physics and methods of detection. Annual Review of Astronomy and Astrophysics, 48, 495-545.
  * [28] Bertone, G., Hooper, D., & Silk, J. (2005). Particle dark matter: Evidence, candidates and constraints. Physics reports, 405(5-6), 279-390.
  * [29] Fujii, M., & Yanagida, T. (2002). Natural gravitino dark matter and thermal leptogenesis in gauge-mediated supersymmetry-breaking models. Physics Letters B, 549(3-4), 273-283.
  * [30] Gómez-Vargas, G. A., López-Fogliani, D. E., Muñoz, C., & Perez, A. D. (2020). MeV-GeV $\mathit{\gamma}$-ray telescopes probing axino LSP/gravitino NLSP as dark matter in the $\mathit{\mu\upsilon}$SSM. Journal of Cosmology and Astroparticle Physics, 2020(01), 058.
  * [31] Kawasaki, M., Kohri, K., Moroi, T., & Yotsuyanagi, A. (2008). Big-bang nucleosynthesis and gravitinos. Physical Review D, 78(6), 065011\.
  * [32] Kusenko, A., & Shaposhnikov, M. (1998). Supersymmetric Q-balls as dark matter. Physics Letters B, 418(1-2), 46-54.
  * [33] Kasuya, S., & Kawasaki, M. (2000). Q-ball formation through the Affleck-Dine mechanism. Physical Review D, 61(4), 041301.
  * [34] Kasuya, S., & Kawasaki, M. (2014). Q-ball dark matter and baryogenesis in high-scale inflation. Physics Letters B, 739, 174-179.
  * [35] Coleman, S. (1985). Q-balls. Nuclear Physics B, 262(2), 263-283.
  * [36] Kusenko, A., & Shaposhnikov, M. (1998). Supersymmetric Q-balls as dark matter. Physics Letters B, 418(1-2), 46-54.
  * [37] Kasuya, S., & Kawasaki, M. (2014). Baryogenesis from the gauge-mediation type Q-ball and the new type of Q-ball as the dark matter. Physical Review D, 89(10), 103534.
  * [38] Enqvist, K., & McDonald, J. (1998). Q-balls and baryogenesis in the MSSM. Physics Letters B, 425(3-4), 309-321.
  * [39] Enqvist, K., & McDonald, J. (1999). B-ball baryogenesis and the baryon to dark matter ratio. Nuclear Physics B, 538(1-2), 321-350.
  * [40] Affleck, I., & Dine, M. (1985). A new mechanism for baryogenesis. Nuclear Physics B, 249(2), 361-380.
  * [41] Dine, M., Randall, L., & Thomas, S. (1995). Supersymmetry breaking in the early universe. Physical Review Letters, 75(3), 398.
  * [42] Nilles, H. P. (1984). Supersymmetry, supergravity and particle physics. Physics Reports, 110(1-2), 1-162.
  * [43] Ng, K. W. (1989). Residual F-terms and the Affleck-Dine mecahnism for baryogenesis. Nuclear Physics B, 321(2), 528-540.
  * [44] Dine, M., Randall, L., & Thomas, S. (1996). Baryogenesis from flat directions of the supersymmetric standard model. Nuclear Physics B, 458(1-2), 291-323.
  * [45] Fujii, M., & Hamaguchi, K. (2002). Nonthermal dark matter via Affleck-Dine baryogenesis and its detection possibility. Physical Review D, 66(8), 083501.
  * [46] Casas, J. A., & Munoz, C. (1989). Inflation from superstrings. Physics Letters B, 216(1-2), 37-40.
  * [47] Casas, J. A., Moreno, J. M., Munoz, C., & Quirós, M. (1989). Cosmological implications of an anomalous $\mathit{U\ (1)}$: inflation, cosmic strings and constraints on superstring parameters. Nuclear Physics B, 328(1), 272-291.
  * [48] Stewart, E. D. (1995). Inflation, supergravity, and superstrings. Physical Review D, 51(12), 6847.
  * [49] Halyo, E. (1996). Hybrid inflation from supergravity D-terms. Physics Letters B, 387(1), 43-47.
  * [50] Binétruy, P., & Dvali, G. (1996). D-term inflation. Physics Letters B, 388(2), 241-246.
  * [51] Kolda, C., & March-Russell, J. (1999). Supersymmetric D-term inflation, reheating, and Affleck-Dine baryogenesis. Physical Review D, 60(2), 023504.
  * [52] Panotopoulos, G. (2005). D-term inflation in D-brane cosmology. Physics Letters B, 623(3-4), 185-191.
  * [53] Kawasaki, M., & Takahashi, F. (2001). Adiabatic and isocurvature fluctuations of Affleck–Dine field in D-term inflation model. Physics Letters B, 516(3-4), 388-394.
  * [54] Liddle, A. R., & Leach, S. M. (2003). How long before the end of inflation were observable perturbations produced?. Physical Review D, 68(10), 103503.
  * [55] Aghanim, N., Akrami, Y., Ashdown, M., Aumont, J., Baccigalupi, C., Ballardini, M., … & Roudier, G. (2020). Planck 2018 results-VI. Cosmological parameters. Astronomy & Astrophysics, 641, A6.
  * [56] Cook, J. L., Dimastrogiovanni, E., Easson, D. A., & Krauss, L. M. (2015). Reheating predictions in single field inflation. Journal of Cosmology and Astroparticle Physics, 2015(04), 047.
  * [57] Dai, L., Kamionkowski, M., & Wang, J. (2014). Reheating constraints to inflationary models. Physical review letters, 113(4), 041302.
  * [58] Takeda, N. (2015). Inflatonic baryogenesis with large tensor mode. Physics Letters B, 746, 368-371.
  * [59] Cooke, R. J., Pettini, M., Jorgenson, R. A., Murphy, M. T., & Steidel, C. C. (2014). Precision measures of the primordial abundance of deuterium. The Astrophysical Journal, 781(1), 31.
  * [60] Bolz, M., Buchmüller, W., & Plümacher, M. (1998). Baryon asymmetry and dark matter. Physics Letters B, 443(1-4), 209-213.
  * [61] Feng, J. L., Rajaraman, A., & Takayama, F. (2003). Superweakly interacting massive particles. Physical review letters, 91(1), 011302\.
  * [62] Feng, J. L., Su, S., & Takayama, F. (2004). Supergravity with a gravitino lightest supersymmetric particle. Physical Review D, 70(7), 075019.
  * [63] Kusenko, A., & Shaposhnikov, M. (1998). Supersymmetric Q-balls as dark matter. Physics Letters B, 418(1-2), 46-54.
  * [64] Kusenko, A. (1997). Solitons in the supersymmetric extensions of the standard model. Physics Letters B, 405(1-2), 108-113.
  * [65] Seto, O. (2006). Affleck-Dine baryogenesis and gravitino dark matter. Physical Review D, 73(4), 043509.
  * [66] Enqvist, K., & McDonald, J. (2000). The dynamics of Affleck–Dine condensate collapse. Nuclear physics B, 570(1-2), 407-422.
  * [67] Doddato, F., & McDonald, J. (2012). New Q-ball solutions in gauge-mediation, Affleck-Dine baryogenesis and gravitino dark matter. Journal of Cosmology and Astroparticle Physics, 2012(06), 031.
  * [68] Ishiwata, K., Matsumoto, S., & Moroi, T. (2008). Gravitino dark matter with weak-scale right-handed sneutrino. Physical Review D, 77(3), 035004.
  * [69] Doddato, F., & McDonald, J. (2011). Comment on Gravitino Dark Matter and Baryon Asymmetry from Q-ball Decay in Gauge Mediation (arXiv: 1107.0403). arXiv preprint arXiv:1107.1402.
  * [70] Shoemaker, I. M., & Kusenko, A. (2009). Gravitino dark matter from Q-ball decays. Physical Review D, 80(7), 075021.
  * [71] Roszkowski, L., & Seto, O. (2007). Axino dark matter from $Q$-balls in Affleck-Dine baryogenesis and the $\Omega_{b}$-$\Omega_{DM}$ coincidence problem. Physical review letters, 98(16), 161304.


[a]Chiara Pinto for the ALICE Collaboration

# (Anti)(hyper)nuclei production in small collision systems measured with
ALICE at the LHC

###### Abstract

The production mechanism of (anti)nuclei in ultrarelativistic hadronic
collisions is under intense debate in the scientific community. The
description of the experimental measurements is currently based on two
competing phenomenological models: the statistical hadronization model and the
coalescence approach. Light (anti)nuclei have been extensively measured in
small collision systems, namely pp and p–Pb collisions, with ALICE at the LHC.
Recent results on the (anti)deuteron production measured in jets and in the
underlying event, and in high-multiplicity pp collisions at a center-of-mass
energy $\sqrt{s}$ = 13 TeV are discussed in the context of phenomenological
models, as they provide new insights on the nucleosynthesis process.

##  1 Introduction

High-energy hadronic collisions at accelerators create a suitable environment
for the production of light (anti)(hyper)nuclei. The production of light
(anti)(hyper)nuclei has been extensively measured, in recent years, at various
collider experiments, from the Alternating Gradient Synchrotron (AGS) [1, 2,
3], to the Relativistic Heavy-Ion Collider (RHIC) [4, 5, 6, 7], and to the
Large Hadron Collider (LHC) [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
20]. However, the production mechanism of light (anti)nuclei is not completely
clear. There are two competing models used to describe the hadronization
process: the thermal-statistical model and the coalescence approach. In the
Statistical Hadronization Model (SHM) [21], hadrons are produced by a source
in thermal and chemical equilibrium and their abundances are fixed at the
chemical freeze-out. This model provides an excellent description of the
measured hadron yields in central nucleus–nucleus collisions using the grand-
canonical ensemble treatment [22]. For small systems, such as pp and p–Pb
collisions, the light nuclei production can be described using a different
implementation of this model based on the canonical ensemble, where exact
conservation of quantum numbers is required [23]. Significant deviations in
this case are observed between the experimental data and the canonical SHM
predictions [19]. The SHM is a macroscopic model, hence the mechanisms of
hadron production and the propagation of loosely-bound states through the
hadron gas phase are not addressed by this model.

On the other hand, the production of light (anti)nuclei can be modelled via
the coalescence of protons and neutrons that are close by in phase space at
the kinetic freeze-out and match the spin, thus forming a nucleus [24]. In the
coalescence model, the production probability of a nucleus with mass number
$A$ is proportional to the coalescence parameter $B_{A}$, which can be
calculated from the overlap of the nucleus wave function and the phase space
distribution of the constituents via the Wigner formalism [25].
Experimentally, the coalescence probability $B_{A}$ is accessible through the
ratio between the invariant momentum distribution of a nucleus with mass
number $A$ and the square of the invariant momentum distribution of protons:

$B_{A}={\biggl{(}\dfrac{1}{2\pi
p^{\mathrm{A}}_{\mathrm{T}}}\dfrac{\mathrm{d}^{2}N_{\mathrm{A}}}{\mathrm{d}y\mathrm{d}p_{\mathrm{T}}^{\mathrm{A}}}\biggr{)}}\bigg{/}{\biggl{(}\dfrac{1}{2\pi
p^{\rm
p}_{\mathrm{T}}}\dfrac{\mathrm{d}^{2}N_{\mathrm{p}}}{\mathrm{d}y\mathrm{d}p_{\mathrm{T}}^{\mathrm{p}}}\biggr{)}^{A}},$
(1)

where the labels $A$ and p refer to the nucleus and the proton, respectively.
The invariant spectra of the (anti)protons are evaluated at the transverse
momentum of the (anti)nucleus, divided by the mass number $A$:
$p_{\mathrm{T}}^{\rm p}$ = $p_{\mathrm{T}}^{\mathrm{A}}$/$A$. In such a model,
neutrons and protons are assumed to have the same production spectra, since
both belong to the same isospin doublet.

In this paper, recent results on the production of light nuclei measured with
ALICE at the LHC are discussed in the context of the production models. In
particular, two recent results are highlighted in the following Sections, for
their particular importance in testing the models: the measurement of the
production of deuterons in high multiplicity (HM) pp collisions, and the
measurement of the production of (anti)deuterons in and out of jets in pp
collisions. In the former case, by combining the measurements of the
production of (anti)nuclei and the femtoscopic measurement of the source
radius in the same collision system and energy, it is possible to compare the
coalescence parameter $B_{2}$ with parameter-free coalescence predictions. In
the latter case, by comparing the (anti)deuteron coalescence probability in
jets and that out of jets, it is possible to test the coalescence models,
since smaller phase-space distances (and hence larger coalescence
probabilities) are expected to characterize the hadrons produced in jets.

##  2 Testing coalescence models in high-multiplicity pp collisions

In HM pp collisions at a center-of-mass energy $\sqrt{s}$ = 13 TeV, the
coalescence models can be tested by combining the measurement of the
production of light (anti)nuclei [18] with the precise measurement of the
source radius with femtoscopic techniques [26]. In panels (a) and (b) of Fig.
1, the transverse momentum spectra of protons and deuterons, respectively,
needed to obtain the coalescence parameter $B_{2}$ with Eq. 1, are shown.
Using the formalism described in Ref. [27], it is possible to compare the
experimentally measured coalescence parameter $B_{2}$ with parameter-free
coalescence predictions, where the only ingredients needed are the emission
source size, which is measured for this specific dataset as shown in panel (c)
of Fig. 1, and the deuteron wave function [28]. As shown in panel (d) of Fig.
1, several wave functions for deuterons have been tested. The Gaussian wave
function provides the best description of the currently available ALICE data,
despite the Hulthen one would be favoured by low-energy scattering
experiments.

Figure 1: Transverse momentum distributions of protons and deuterons measured
in HM pp collisions at $\sqrt{s}=$ 13 TeV are shown in panels (a) and (b),
respectively [18]. The $\langle m_{\rm T}\rangle$ scaling of the source size
measured in HM pp collisions at $\sqrt{s}=$ 13 TeV is shown in panel (c) [26].
Panel (d) shows the comparison between measurements and theoretical
predictions for the coalescence parameter $B_{2}$ for (anti)deuterons as a
function of $p_{\rm T}/A$ [18]. Theoretical predictions are obtained using
different wave functions to describe nuclei: Gaussian (yellow), Hulthen
(blue), $\chi$EFT (gray) and two Gaussians (green).

##  3 Testing coalescence models through production of nuclei in and out of
jets

Recent preliminary results on the deuteron production in jets and outside jets
in pp collisions at $\sqrt{s}$ = 13 TeV with ALICE have shown that, despite
the production of deuterons in jets is about 10$\%$ of the total production
[17], the coalescence probability in jets is one order of magnitude larger
than that outside jets. This study has been carried out dividing the azimuthal
plane in three regions, identified by the angular distance with respect to the
leading particle, namely the particle with the highest transverse momentum in
each event (with the condition $p_{\rm T}^{\rm{lead}}>$ 5 GeV/$c$) and with
pseudorapidity $|\eta|<$0.8. Three equal-size regions, 120∘ wide, are
consequently defined: the one around the leading particle (Toward), the one
back-to-back to it (Away), and the one transverse to both of them
(Transverse). The Toward and Away regions contain contributions from the
leading and recoil jets in addition to the underlying event, while the
Transverse region is dominated by the underlying event [29]. The transverse
momentum distribution of deuterons in jets is obtained by subtracting the
underlying event (Transverse region) from the Toward region, resulting to be
in agreement with the in-jet deuteron spectra obtained with the two-particle
correlation method [17], while the underlying event transverse momentum
distribution corresponds to the Transverse region one. The coalescence
parameters $B_{2}$ in jets and in the underlying event are obtained as a
function of $p_{\rm T}/A$ using Eq. 1 and shown in Fig. 2. The observed
enhancement of the in-jet $B_{2}$ is in agreement with the coalescence
picture, and it can be interpreted as due to the reduced distance in phase-
space between nucleons in jets compared to hadrons outside of jets. These
results will be compared with predictions from simple coalescence and from a
reaction-based deuteron production model. In the former case, the transverse
momentum distributions of nucleons are generated using PYTHIA 8 with Monash
2013 tune, while in the latter case (anti)deuterons are generated by ordinary
nuclear reactions between the nucleons produced in the collision with
parametrized energy-dependent cross sections tuned on available experimental
data.

Figure 2: Coalescence parameter $B_{2}$ in jets (full circles) and in the
underlying event (empty circles) as a function of the transverse momentum per
nucleon $p_{\rm T}/A$.

## References

  * [1] E878 Collaboration, Phys. Rev. C 58 (1998) 1155-1164.
  * [2] E802 Collaboration, Phys. Rev. C 60 (1999) 064901.
  * [3] E864 Collaboration, Phys. Rev. Lett. 85 (2000) 2685–2688.
  * [4] STAR Collaboration, Phys. Rev. Lett. 87 (2001) 2623011–2623016.
  * [5] PHENIX Collaboration, Phys. Rev. Lett. 94 (2005) 122302.
  * [6] BRAHMS Collaboration, Phys. Rev. C 83 (2011) 044906.
  * [7] STAR Collaboration, Nature 473 (2011) 353 [Erratum: Nature 475 (2011) 412].
  * [8] ALICE Collaboration, Nature Phys. 11 no. 10, (2015) 811–814.
  * [9] ALICE Collaboration, Phys. Rev. C 93 no. 2, (2016) 024917.
  * [10] ALICE Collaboration, Eur. Phys. J. C77 no. 10, (2017) 658.
  * [11] ALICE Collaboration, Phys. Rev. C 97 no. 2, (2018) 024615.
  * [12] ALICE Collaboration, Nucl. Phys. A 971 (2018) 1–20.
  * [13] ALICE Collaboration, Phys. Lett. B 794 (2019) 50–63.
  * [14] ALICE Collaboration, Phys. Lett. B 800 (2020) 135043.
  * [15] ALICE Collaboration, Phys. Rev. C101 no. 4, (2020) 044906.
  * [16] ALICE Collaboration, Eur. Phys. J. C80 no. 9, (2020) 889.
  * [17] ALICE Collaboration, Phys. Lett. B 819 (2021) 136440.
  * [18] ALICE Collaboration, JHEP 01 (2022) 106.
  * [19] ALICE Collaboration, Phys. Rev. Lett. 128 25 (2022) 252003.
  * [20] ALICE Collaboration, Phys. Lett. B 754 (2016) 360-372.
  * [21] A.Andronic, P.Braun-Munzinger, J.Stachel, H.Stöcker, Phys. Lett. B 697 (2011) 203-207.
  * [22] A.Andronic, P.Braun-Munzinger, K.Redlich, J.Stachel, Nature 561 (2018) 321–330.
  * [23] V. Vovchenko, B. Doenigus, H. Stoecker, Phys. Lett. B 785 (2018) 324.
  * [24] Butler, S. T. and Pearson, C. A., Phys. Rev. 129 (1963) 836.
  * [25] R. Scheibl and U. W. Heinz, Phys. Rev. C59 (1999) 1585-1602.
  * [26] ALICE Collaboration, Phys. Lett. B 811 (2020) 135849.
  * [27] K. Blum, M. Takimoto, Phys. Rev. C 99 4 (2019) 044913.
  * [28] F. Bellini et al., Phys. Rev. C 103 (2021) 014907.
  * [29] ALICE Collaboration, JHEP 04 (2020) 192.