# Quark clusters, QCD vacuum and the cosmological 7Li, Dark Matter and Dark
Energy problems

Rachid Ouyed,1 Denis Leahy1, Nico Koning1, and Prashanth Jaikumar2
1Department of Physics and Astronomy, University of Calgary, 2500 University
Drive NW, Calgary, AB T2N 1N4, Canada
2Department of Physics and Astronomy, California State University Long Beach,
1250 Bellflower Blvd., Long Beach, CA 90840, USA
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We propose a non-exotic electromagnetic solution (within the standard model)
to the cosmological 7Li problem based upon a narrow 2 MeV photo-emission line
from the decay of light Glueballs (LGBs). These LGBs form within color
superconducting, tens of Fermi in size, quark clusters (SQCs) in the
radiation-dominated post-BBN epoch. The mono-chromatic line from the
$LGB\rightarrow\gamma+\gamma$ reduces Big-Bang nucleosynthesis (BBN) 7Be by
$2/3$ without affecting other abundances or Cosmic Microwave Background (CMB)
physics, provided the combined mass of the SQCs is greater than the total
baryonic mass in the Universe. Following the LGB emission, SQC lose their
Quantum-ChromoDynamics (QCD) vacuum and decouple from the strong interaction.
Being colorless, charge neutral, optically thin and transparent to hadrons,
SQCs interact only gravitationally making them a viable cold dark matter (CDM)
candidate. The drainage (i.e. quantum tunnelling) of SQC QCD vacuum to the
external space-time (trivial QCD) vacuum offers a natural explanation of dark
energy in our model and allows for a cosmology which evolves into a
$\Lambda$CDM universe at low redshift with a possible resolution of the Hubble
tension. The connection between CDM and DE in our model supports the notion
that in-hadron confinement contains QCD condensates and disagrees with the
conventional view of (space-filling) QCD condensates. We discuss SQC formation
in heavy-ion collision experiments at moderate temperatures and the
possibility of detection of MeV photons from the $LGB\rightarrow\gamma+\gamma$
decay.

###### keywords:

Cosmology: early Universe, primordial nucleosynthesis, dark matter, dark
energy

††pubyear: 2023††pagerange: Quark clusters, QCD vacuum and the cosmological
7Li, Dark Matter and Dark Energy problems–C

##  1 Introduction

The primordial abundances of the light elements produced in the first few
minutes of the universe predicted by standard hot Big-Bang cosmology (Hoyle &
Tayler 1964; Peebles 1966; Wagoner et al. 1967) are in excellent agreement
with the abundances inferred from data (e.g. Tytler et al. 2000). BBN starts
when the deuteron (D) bottleneck is overcome at111Dimensionless quantities are
defined as $f_{x}=f/10^{x}$ with quantities in cgs units unless specified
otherwise. $k_{\rm B}T\sim 100$ keV and terminates at $k_{\rm B}T\sim 30$ keV
(redshift $z\sim 4\times 10^{8}$) due to electrostatic repulsion between
nuclei (e.g. Lang 1999); $k_{\rm B}$ is the Boltzmann constant. Subsequently,
significant amounts of D, 3H and 4He build up followed by the production of
much less-abundant elements such as 7Be. With a half-life of $\sim$ 53 days,
7Be decays into 7Li via bound electron capture, with emission of a neutrino
(e.g. Khatri & Sunyaev 2011). This cannot occur, however, until recombination
at $z\sim 1100$ when 7Be becomes singly ionized. The measured 7Li abundance is
$\sim$ 1/3 of what is expected from this process (Spite & Spite 1982) defining
the cosmological 7Li problem (see Fields 2011 for a review).

The standard theory of electromagnetic cascades onto a photon background
predicts a quasi-universal shape for the resulting non-thermal photon spectrum
(e.g. Berezinsky et al. 1990). In the case of non-thermal big bang
nucleosynthesis (BBN), cosmological constraints using this quasi-universal
shape make purely electromagnetic solutions to the 7Li problem impossible
(e.g. Kawasaki & Moroi 1995) unless injected photon energy falls below the
pair-production threshold in which case the spectral shape is very different
(see Poulin & Serpico 2015, and references therein). The effective pair
production threshold is $E_{\rm\gamma,pair}\simeq(m_{e}c^{2})^{2}/22k_{\rm
B}T\sim 12\ {\rm MeV}\times({\rm keV}/T)$ below which the double photon pair-
creation process receives a Boltzmann suppression (Kawasaki & Moroi 1995; see
also Protheroe et al. 1995); $m_{\rm e}$ is the electron mass. If injected
when BBN is over, sub-threshold photons act to post-process the abundances
computed in the standard scenario. Photons injected with $1.59\ {\rm
MeV}<E_{\gamma}<2.22$ MeV can destroy 7Be (suppressing 7Li production) without
affecting other BBN abundances, distorting the CMB background or injecting
excess entropy (Poulin & Serpico 2015; Kawasaki et al. 2020).

In this paper, we propose a model for sub-threshold photon injection that
relies on the properties of color superconducting (CSC) clusters of quark
matter that are tens of Fermis (fm) in size. These quark clusters
(QCs)222Quark clusters in our model are quark-gluon plasma assemblies of tens
of Fermi in size. Their binding energy compared to free baryons is $\sim$ 10%;
the binding energy per baryon is in the tens of MeVs. form in the early
universe and enter the CSC phase (i.e. become SQCs) in the radiation-dominated
post-BBN epoch when the universe’s temperature is in the keV range. Through
conversion of gluonic energy, via condensation, into electromagnetically
unstable light glueballs (LGBs), they emit a narrow mono-chromatic $\sim 2$
MeV line capable of reducing post-BBN 7Be by 2/3 and thus solving the 7Li
problem. The total mass of SQCs required to solve the 7Li problem exceeds that
of baryons in the Universe.

The properties of SQCs (colorless, charge-neutral, optically thin and
gradually decoupling from the strong interaction), makes them a viable CDM
candidate. The weakening of the SQC-hadron interaction is a consequence of the
loss, through quantum tunnelling, of the SQC Quantum-ChromoDynamics (QCD)
vacuum into the external (trivial QCD) vacuum333Meaning a vacuum with
vanishing QCD (quark and gluon) condensates. of space-time, which hereafter we
label “drainage". The loss of an important fraction of the gluonic content of
the SQC due to LGB decay we hypothesize may destabilize the in-SQC QCD
condensates and trigger the drainage. The leaked QCD vacuum behaves as Dark
Energy (DE) in the exterior space-time in our model and yields a two-epoch
(pre- and post-drainage) cosmology which behaves as a $\Lambda$CDM at low
redshift.

The paper is organized as follows: In §2, we derive the equations for the
destruction of 7Be by a 2 MeV monochromatic line. We show how such a line
could be produced by a lukewarm 2-flavor superconducting (2SC-like) quark
phase where a percentage ($\eta_{\rm G}$) of gluonic energy is converted to 2
MeV photons. The drainage of the SQC QCD condensates (vacuum) is analyzed in
§3 where we present the resulting cosmology with a plausible resolution of the
Hubble tension. We discuss SQCs as a CDM candidate in §4 and list our model’s
limitations and predictions in §5. We conclude in §6.

##  2 Post-BBN 7Be destruction

In our model, the CSC phase produces gluon condensation (i.e. $M_{\rm
LGB}c^{2}\sim 4$ MeV mass LGBs) which decay to $E_{0}=M_{\rm LGB}c^{2}/2\sim$
2 MeV mono-energetic photons via the ${\rm LGB}\rightarrow\gamma+\gamma$
channel. The LGBs form on a hadronic timescale when the quark cluster enters
the CSC phase and decay to photons on timescales $\tau_{\rm LGB}\sim 10^{-17}$
which is instantaneous compared to the Hubble expansion timescale (see §2.1).
This occurs in the radiation-dominated post-BBN era at time $t_{\rm G}$ when
the temperature is $T_{\rm G}$. Hereafter quantities indexed with the letter
“G" refer to values associated with the LGB-decay/photon-burst event; the
corresponding redshift is $z_{\rm G}=z(t_{\rm G})$. Subscripts “sG" and “eG"
refer to the start and end, respectively of this phase with $t_{\rm eG}-t_{\rm
sG}\sim\tau_{\rm LGB}$ such that $z_{\rm sG}=z_{\rm eG}+\delta z$ with $\delta
z<<z_{\rm G}$; effectively, $z(t_{\rm sG})=z(t_{\rm G})=z(t_{\rm eG})$.

The photon production is a delta function, $\delta(t-t_{\rm G})$, in our model
at time $t=t_{\rm G}$. As shown in Appendix A, the destruction rates for 7Be
nuclei due to such a sudden release of mono-energetic $E_{0}$ photons is

$\ln\left(\frac{Y_{\rm Be,eG}}{Y_{\rm
Be,sG}}\right)\sim-\frac{n_{\gamma}(E_{0},t_{\rm G})}{n_{\rm B}(t_{\rm
G})}\times\frac{\sigma_{\rm Be}(E_{0})}{\sigma_{\rm CS}(E_{0})}\ ,$ (1)

where $Y$ is the abundance. Here, $n_{\rm B}(t_{\rm G})$ is the universe’s co-
moving baryon number density at $t_{\rm G}$ while $n_{\gamma}(E_{0},t_{\rm
G})$ is the co-moving number density of photons from the LGB decay. The 7Be
photo-dissociation cross-section is $\sigma_{\rm Be}(E_{0})$ and $\sigma_{\rm
CS}(E_{0})$ the Compton scattering cross-section. A reduction of 7Be by 2/3
imposes that the RHS in Eq. (1) be unity (i.e. $\sim-1.1$).

Let us define $\eta_{\rm DM,sG}$ as the initial total amount in mass of SQCs
(the DM in our model) compared to baryons; i.e. before LGB decay. The
corresponding SQC co-moving number density is

$n_{\rm sqc}^{\rm com.}(t_{\rm G})=\frac{\eta_{\rm DM,sG}n_{\rm B}(t_{\rm
G})}{A_{\rm sqc}}\ ,$ (2)

with $A_{\rm sqc}$ the cluster’s baryon number. The emitted photons co-moving
number density is then

$n_{\gamma}(E_{0},t_{\rm G})=n_{\rm sqc}^{\rm com.}(t_{\rm G})\times
N_{E_{0}}=\frac{\eta_{\rm DM,sG}n_{\rm B}(t_{\rm G})}{A_{\rm sqc}}\times
N_{E_{0}}\ ,$ (3)

with $N_{E_{0}}\sim\eta_{\rm G}A_{\rm sqc}m_{\rm p}c^{2}/E_{0}$ the total
number of $E_{0}$ photons emitted per SQC. Here, $\eta_{\rm G}$ is the
fraction of the gluonic energy (i.e. of the SQC rest-mass energy) converted to
the mono-chromatic line at energy $E_{0}$; $m_{\rm p}$ is the proton mass.

Eq. (1) becomes

$\displaystyle\ln\left(\frac{Y_{\rm Be,eG}}{Y_{\rm Be,sG}}\right)$
$\displaystyle\sim-\eta_{\rm DM,sG}\frac{\eta_{\rm G}m_{\rm
p}c^{2}}{E_{0}}\times\frac{\sigma_{\rm Be}(E_{0})}{\sigma_{\rm CS}(E_{0})}$
(4) $\displaystyle\sim-\frac{\eta_{\rm G}}{1-\eta_{\rm
G}}\times\frac{\eta_{\rm DM,eG}m_{\rm p}c^{2}}{E_{0}}\times\frac{\sigma_{\rm
Be}(E_{0})}{\sigma_{\rm CS}(E_{0})}\ ,$

where, in order to get the last expression, we made use of the fact that the
SQC total mass after conversion of gluonic energy to photons is $\eta_{\rm
DM,eG}=(1-\eta_{\rm G})\eta_{\rm DM,sG}$ or $\eta_{\rm DM,sG}=\eta_{\rm
DM,eG}/(1-\eta_{\rm G})$.

Figure 1: The $\sigma_{\rm Be}/\sigma_{\rm CS}$ ratio versus photon energy.
The 7Be photo-dissociation cross-section $\sigma_{\rm Be}$ is from equation
III.8 in Ishida et al. (2014). The Compton scattering cross-section, is given
in Appendix (IV) in Kawasaki & Moroi (1995).

The 7Be photo-dissociation cross-section, $\sigma_{\rm Be}(E_{0})$, is given
by Eq. (III.8) in Ishida et al. (2014). The ratio of $\sigma_{\rm Be}(E_{0})$
to Compton scattering cross-section, $\sigma_{\rm Be}(E_{0})/\sigma_{\rm
CS}(E_{0})$, varies widely as shown in Figure 1. We see that for $2\ {\rm
MeV}<E_{0}<2.2$ MeV on average $\sigma_{\rm Be}(E_{0})/\sigma_{\rm
CS}(E_{0})\sim 5\times 10^{-4}$. If the SQC total mass after conversion of
gluonic energy to photons is the observed CDM amount, $\eta_{\rm DM,eG}\sim
5$, then $5m_{\rm p}c^{2}/E_{0}\sim 2.4\times 10^{3}$. In this case, the 7Li
problem is solved if

$\frac{\eta_{\rm G}}{1-\eta_{\rm G}}\sim 0.9\ ,$ (5)

which requires that $\sim$ 50% (i.e. $\eta_{\rm G}\sim 0.5$) of the SQCs
gluonic energy is converted to $\sim 2$ MeV photons.

As explained later in §5, the value of $\eta_{\rm G}$ can be lower (less
gluonic energy shed by SQCs) while still solving the 7Li puzzle. This is
because in addition to losing gluonic energy via LGB decay, SQCs will lose
their QCD condensates so that if $\eta_{\rm DM,0}\sim 5$ in today’s
universe444The subscript “0” refers to values at redshift $z=0$. then
$\eta_{\rm DM,eG}$ must have been higher due to drainage (see first bullet
point in §5).

###  2.1 The mono-chromatic $\sim 2$ MeV line

The QCD phase diagram is complex (e.g. Rüster et al. 2004; Alford et al. 2008;
Baym et al. 2018) and in principle one cannot exclude the existence of a CSC
phase where gluon condensation would yield electromagnetically unstable $X$
“particles" (here LGBs), as required in our model. Figure 2 shows a
hypothetical phase diagram with the dashed line depicting a possible
trajectory leading a quark cluster from its state at birth (in an unpaired
phase) to the CSC phase. We require an unpaired phase at $\mu<\mu_{\rm csc}$
which bridges the hadronic phase and the CSC phase. I.e. the CSC phase is
accessed, at low-$T$, from low density unpaired phase with a first-order line
separating the two phases. Once in the CSC phase, a cluster becomes a SQC and
produces LGBs in the radiation-dominated post-BBN epoch at keV temperatures.

In this paper, we will use the neutral 2SC phase as a reference CSC phase. It
has the interesting property of converting a percentage ($\eta_{\rm G}=3/8$)
of its gluonic energy to LGBs (i.e. gluonic condensation) at low temperature
with a subsequent decay to a mono-chromatic line via the ${\rm
LGB}\rightarrow\gamma+\gamma$ channel (see Appendix B for details). In the 2SC
phase, an LGB with mass $M_{\rm LGB}c^{2}\sim 4$ MeV would decay to two
$E_{0}=(M_{\rm LGB}c^{2}/2)\sim 2$ MeV photons on timescales of $\tau_{\rm
LGB}\sim 10^{-17}$ s. We set the CSC quark chemical potential at $\mu_{\rm
csc}=500$ MeV wth a corresponding number density $n_{\rm csc}=\mu_{\rm
csc}^{3}/\pi^{2}\sim 10^{39}$ cm-3. Thus the density inside555Not to be
confused with the SQC co-moving density given in Eq. (2). an SQC is $n_{\rm
sqc}=n_{\rm csc}$ which is about ten times nuclear saturation density when it
enters the CSC phase at time $t_{\rm G}$.

Once the cluster crosses into the CSC phase, the first order transition
proceeds on hadronic timescales. During the transition, and because of latent
heat released, a SQC gets heated to a temperature $k_{\rm B}T_{\rm
sqc}\sim\Delta_{\rm csc}^{2}/\mu_{\rm csc}$ where $\Delta_{\rm csc}$ is the
CSC superconducting gap; here, $\Delta_{\rm csc}$ and $\mu_{\rm csc}$ are in
units of MeV (see Eq. 21). The resulting LGB energy is $E_{\rm
LGB}=\sqrt{(M_{\rm LGB}c^{2})^{2}+(k_{\rm B}T_{\rm sqc})^{2}}$. LGBs cannot
form at temperatures exceeding the melting temperature which is of the order
of the LGB’s rest-mass energy; $k_{\rm B}T_{\rm LGB,m}\sim M_{\rm LGB}c^{2}$.
Thus we must ensure that $T_{\rm sqc}<T_{\rm LGB,m}$ in addition to
$E_{0}=E_{\rm LGB}/2<2.2$ MeV where the 2.2 MeV upper limit keeps the line
below the Deuteron photo-ionization threshold (see §1). In general for
$\Delta_{\rm csc}^{2}/\mu_{\rm csc}<2$ MeV, or $\Delta_{\rm csc}<31.6\ {\rm
MeV}\times(\mu_{\rm csc}/500\ {\rm MeV})^{1/2}$, these conditions are
satisfied while guaranteeing a narrow $\sim 2$ MeV line. $E_{\rm LGB}\simeq
M_{\rm LGB}c^{2}\sim 4$ MeV is achieved for a reasonable range in $\mu$ as
shown in Figure 3 (see Appendix B for details).

Figure 2: A possible QCD phase diagram; temperature versus quark chemical
potential. The dashed curve (“1 to 2") depicts the cooling path of the quark
cluster (formed at “1" with $T_{\rm QCD}\sim 150$ MeV) traversing the unpaired
phase and entering the CSC phase, at keV temperature, from the low density
phase (“2"). A first-order line separates the unpaired phase from the CSC
phase releasing heat (the vertical “2 to 3" arrow). A SQC gets heated to a
temperature not exceeding the LGB melting temperature $k_{\rm B}T_{\rm
sqc,m}\sim M_{\rm LGB}c^{2}$ (see §2.1). The $\sim 4$ MeV LGBs decay to the 2
MeV narrow photon line via ${\rm LGB}\rightarrow\gamma+\gamma$. “$1^{\prime}$"
shows the initial state of a NS core making its way to “2" via cooling an
compression (see §5 and Appendix B).

###  2.2 The SQC size and baryon number

The SQC has to be optically thin to the 2 MeV photons. To first order, the
photon mean-free-path is $\lambda_{\rm csc,\gamma}=1/(n_{\rm csc}\sigma_{\rm
q\gamma})\sim 10^{2}$ fm$/n_{\rm csc,39}$ with $\sigma_{\rm q\gamma}\sim
10^{-28}\ {\rm cm}^{2}$ the Thomson cross-section for photon scattering off up
and down quarks. We set a typical SQC radius to be $R_{\rm
sqc,thin}\sim\lambda_{\rm csc,\gamma}$ with a corresponding baryon number
$A_{\rm sqc}\sim 10^{6}$.

Figure 3: LGB mass versus quark chemical potential ($\mu$) for $\Lambda_{\rm
QCD}=245$ MeV (the scalar parameter of QCD; see §2.1 and Appendix B for
details). The curves from top to bottom are for $\Delta/\mu=0.1,0.09,0.08$,
respectively. The two dashed lines show the $4.0\leq M_{\rm LGB}c^{2}({\rm
MeV})\leq 4.4$ range yielding a mono-chromatic photon line in the $2.0\leq
E_{0}({\rm MeV})\leq 2.2$ range.

##  3 SQCs and Dark Energy (DE)

It is not unreasonable to conjecture that the loss of a substantial fraction
($\eta_{\rm G}$) of gluonic energy by the SQC could modify the QCD condensates
potential within the SQC and trigger drainage of the SQC vacuum into the
exterior (trivial QCD) vacuum. To investigate the implications of drainage, we
adopt the scenario that the ground-state which fills up space-time is empty
and that QCD condensates are supported within the SQCs as suggested for
hadrons (e.g. Brodsky et al. 2012, and references therein). The in-hadron
condensates has the advantage of removing the extremely large (a factor of
$\sim 10^{46}$) contribution to the cosmological constant predicted when using
the conventional view of (space-filling) QCD condensates. The novel idea
introduced in this paper, namely the leakage of the in-SQC QCD condensates
into the surrounding space-time vacuum, could offer an explanation of Dark
Energy (DE) and yield a cosmology which may help solve the Hubble tension
while yielding a $\Lambda$CDM universe at low redshifts.

The total amount of QCD vacuum energy stored in the quark clusters is
$\rho_{\rm QCD}^{\rm vac.}V_{\rm sqc}^{\rm tot.}$ with $\rho_{\rm QCD}^{\rm
vac.}$ the density of the QCD condensates and $V_{\rm sqc}^{\rm tot.}$ the
total volume occupied by the SQCs which is constant in time (and assumed to
not change on LGB decay). Drainage means that the density inside the SQC
decreases from

$\rho_{\rm sqc,eG}=(1-\eta_{\rm G})\rho_{\rm csc}\ ,$ (6)

at $t=t_{\rm G}$ (at the end of the LGB phase denoted by “eG") to

$\displaystyle\rho_{\rm sqc,0}$ $\displaystyle=(1-\eta_{\rm G})\rho_{\rm
csc}-\rho_{\rm QCD}^{\rm vac.}$ (7) $\displaystyle=\rho_{\rm
csc}\times(1-\eta_{\rm G}-f_{\rm V})$ $\displaystyle=\rho_{\rm
sqc,eG}\times\left(1-\frac{f_{\rm V}}{1-\eta_{\rm G}}\right)\ ,$

at redshift $z=0$ with the assumption that the drainage timescale is a
fraction of the age of the universe (see below). The parameter $f_{\rm
V}=\rho_{\rm QCD}^{\rm vac.}/\rho_{\rm csc}$ is a measure of the contribution
of the QCD condensates to the SQC rest-mass energy. At the start of the LGB
phase (denoted by “sG"), $\rho_{\rm sqc,sG}=\rho_{\rm csc}$.

We can get a rough estimate of the time $t_{\rm tun.}$ it would take the SQC
QCD vacuum to leak into the exterior space-time trivial QCD vacuum. A rigorous
calculation would follow for example Coleman (1977) and is beyond the scope of
this paper. We consider instead a simple tunnelling problem across a square
barrier with height given by the expectation value of the QCD vacuum and width
$L\sim R_{\rm sqc}$. The corresponding tunnelling probability is then $P_{\rm
tun.}\sim e^{-R_{\rm sqc}/\delta}$ where $\delta$ is the penetration depth
(e.g. Cohen-Tannoudji et al. 2006). The tunnelling timescale is $t_{\rm
tun.}\simeq(R_{\rm sqc}/c)\times(1/P_{\rm tun.})$ giving us

$t_{\rm tun.}\sim\frac{\delta}{c}\times xe^{x}\ ,$ (8)

where $x=R_{\rm sqc}/\delta$. With $\delta$ of the order of a Fermi (which is
not unrealistic), solutions with tunnelling timescales of a few billion years
($1\ {\rm Gyr}\leq t_{\rm tun.}\leq 4$ Gyr) require $R_{\rm sqc}\sim 88.5$ fm
consistent with our model where optically thin SQCs with $R_{\rm
sqc}<10^{2}/n_{\rm csc,39}$ fm are necessary to solve the cosmological 7Li
problem. Thus, the drainage occurs on astrophysical timescales so that $t_{\rm
G}<<t_{\rm tun.}$ or in terms of redshift, $z_{\rm G}>>z_{\rm tun.}$ with
$z_{\rm tun.}$ the characteristic drainage redshift; this also validates our
assumption in deriving Eq. (7).

The time evolution of the density inside the SQC we can then write as

$\displaystyle\rho_{\rm sqc}(t)$ $\displaystyle=(1-\eta_{\rm G})\rho_{\rm
csc}-\rho_{\rm QCD}^{\rm vac.}(1-e^{-(t-t_{\rm G})/t_{\rm tun.}})$ (9)
$\displaystyle=\rho_{\rm csc}\left(1-\eta_{\rm G}-f_{\rm V}(1-e^{-(t-t_{\rm
G})/t_{\rm tun.}})\right)$ $\displaystyle=\rho_{\rm
sqc,eG}\left(1-\frac{f_{\rm V}}{1-\eta_{\rm G}}(1-e^{-(t-t_{\rm G})/t_{\rm
tun.}})\right)\ .$

The equation above incorporates the key parameters in our cosmology namely,
$\rho_{\rm csc},\eta_{\rm G},f_{\rm V}$ and $t_{\rm tun.}$ which are all
fundamentally related to QCD.

The DM density in our model is the SQC density $\rho_{\rm sqc}(t)$ averaged
over the volume of the universe. Its time evolution, with $V_{\rm univ.}(t)$
being the Hubble volume at time $t$, is

$\rho_{\rm DM}(t)=\begin{cases}\frac{\rho_{\rm sqc}(t)V_{\rm sqc}^{\rm
tot.}}{V_{\rm univ.}(t)}\ &{\rm if}\quad t>t_{\rm G}\ ({\rm or}\ z<z_{\rm
G})\\\ \frac{\rho_{\rm csc}V_{\rm sqc}^{\rm tot.}}{V_{\rm univ.}(t)}\ &{\rm
if}\quad t\leq t_{\rm G}\ ({\rm or}\ z\geq z_{\rm G})\ ,\end{cases}$ (10)

with the resulting cosmology analyzed in Appendix C.

Our model offers a resolution of the Hubble tension (see Kamionkowski & Riess
2022 for a recent review and references therein) and can be understood as a
consequence of a CDM universe converting into a $\Lambda$CDM universe at
$z_{\rm tun.}$. Figure 4 shows that $H_{0}\sim 73$ km s-1 Mpc-1 can be
obtained for a range in $\eta_{\rm G}$ and $f_{\rm V}$ values with a drainage
characteristic redshift $2<z_{\rm tun.}<6$ (i.e. $1<t_{\rm tun.}({\rm
Gyr})<4$). Our cosmology yields a universe which is younger than the
$\Lambda$CDM universe with an age of $\sim 13$ Gyrs with $\eta_{\rm G}$ and
$f_{\rm V}$ of the order of tens of percents each. It remains to be shown
whether our cosmology is in agreement with other cosmological data and
measurements which the flat $\Lambda$CDM model explains extremely well (see
e.g. metrics and tests suggested in Schöneberg et al. 2022). Furthermore, we
caution that the details of how the QCD vacuum mixes with the space-time
vacuum and how it evolves while preserving flatness remains to be understood.
Nevertheless, being a vacuum the “DE" component in our model obeys an
equation-of-state with parameter $w=-1$.

Figure 4: The Hubble constant $H_{0}$ as a function of $z_{\rm tun.}$ (the
drainage characteristic redshift) in our model for different values of
$\eta_{\rm G}$ and $f_{\rm V}$. The $H_{0}=73$ km s-1 Mpc-1 value is shown as
the horizontal line. The resulting age of the universe is shown for each case
and is younger than the $\Lambda$CDM universe.

##  4 SQCs as cold dark matter (CDM)

Evidence for CDM is abundant but its nature remains unknown despite many
theoretical investigations and dedicated experiments which have yet to detect
any associated particle (e.g. Garrett & Gintaras 2011; Schumann 2015; Bertone
& Hooper 2018; Kisslinger & Das 2019; Oks 2021; Arbey & Mahmoudi 2021). In our
model, the 7Li problem is solvable if the total mass in SQCs is of the order
of the measured CDM value. In addition, SQCs are colorless, cold, optically
thin, electrically neutral and would make an ideal CDM candidate if they
decouple from the strong force (or interact minimally with baryons) following
drainage.

In the pre-LGB and pre-drainage epoch (i.e. at $z>z_{\rm G}$), SQCs should
interact with hadrons. The SQC-baryon interaction cross-section is
$\sigma_{\rm sqc}=\pi R_{\rm sqc}^{2}\times(\tau_{\rm crossing}/\tau_{\rm
conv.})$. Here, $\tau_{\rm crossing}=R_{\rm sqc}/v_{\rm sqc}$ is the baryon
SQC crossing time and $\tau_{\rm conv.}$ the conversion timescale of a baryon
to the CSC phase; $v_{\rm sqc}$ is the quark cluster’s speed. The rate of
baryons swept by a SQC in the pre-drainage era is $\sigma_{\rm sqc}n_{\rm
B}v_{\rm sqc}\sim(A_{\rm sqc}/\tau_{\rm conv.})\times(n_{\rm B}/n_{\rm csc})$
with $A_{\rm sqc}=\frac{4\pi}{3}R_{\rm sqc}^{3}n_{\rm csc}$. Using $n_{\rm
B}(T)=\eta_{\rm B}n_{\gamma}(T)$ and $n_{\gamma}(T)\sim 6.2\times 10^{32}$
cm${}^{-3}\times T_{\rm MeV}^{3}$, the rate is $\sim(3.8\times
10^{-16})\times(A_{\rm sqc}/\tau_{\rm conv.})\times(\eta_{\rm B,-9.2}/n_{\rm
csc,39})\times T_{\rm MeV}^{3}$; $\eta_{\rm B}\simeq 6.1\times 10^{-10}$ is
the baryon-to-photon ratio. The integrated rate is found from $\int\sigma_{\rm
sqc}n_{\rm B}v_{\rm sqc}\ dt$ with $T_{\rm MeV}\sim(1\ {\rm s}/t)^{1/2}$ and
is highest at temperatures of tens of MeV when the quark clusters form
following the cosmic QCD phase transition at $t\sim 10^{-4}$ s. For conversion
timescales exceeding $\sim 10^{-13}$ s, the SQC will not grow in size in the
pre-LGB era will not interact with diffuse baryons thereafter. Although
$\tau_{\rm conv.}$ exceeds the strong interaction timescales, it is not
unrealistic because the conversion is a two-step process with the baryon first
deconfining and crossing the unpaired quark phase before reaching the CSC
phase. In particular, the unpaired phase may act as an energy barrier that
requires that hadronic matter is injected with energy in order to be
deconfined.

We have established in §3 that a cosmology which resolves the Hubble tension
while yielding an age of the universe of $\sim 13$ Gyr requires $\eta_{\rm
G}+f_{\rm V}<30\%$ and $2<z_{\rm tun.}<6$ (i.e. $1<t_{\rm tun.}({\rm
Gyr})<4$). I.e. the fraction of strongly interacting (i.e. non-drained) SQCs
in today’s universe is in the range $2.3\times 10^{-6}<e^{-1/H_{0}t_{\rm
tun.}}<0.04$. This assumes that drainage of $\sim 10$% of the SQCs condensate
is enough to induce decoupling. Because drainage timescale is likely to exceed
the typical timescale for the formation (and collapse) of structure in the
early universe, we expect primordial halos to be dominated by post-LGB but
non-drained SQCs. Baryons will fall into these halos but as we showed above,
these diffuse baryons will not interact with SQCs whether drained or not. We
expect a change (a “phase transition") in structure growth past $t=t_{\rm
tun.}$ because: (i) a percentage of the SQCs mass is converted to DE and; (ii)
SQCs decouple from the strong force. It is beyond the scope of this paper to
understand the true impact of these effects on subsequent structure growth and
only state-of-the-art CDM simulations which incorporate the drainage of SQCs
mass as described in details in Appendix C can help elucidate this aspect of
our model.

In summary, SQCs are transparent to photons and would interact weakly with
baryons from early times even before drainage begins. They could constitute a
viable CDM candidate and may be responsible for the formation of mini-halos
and of the larger structures in the universe (Navarro, Frenk & White 1996;
Abel et al. 2002). Once halos form and virialize, SQCs will interact the
strongest with stars and in particular with matter at the highest density,
i.e. neutron stars (NSs). However, if $t_{\rm tun.}$ is a small fraction of
the age of the universe then, by the time compact stars start to form in the
universe, SQCs would have decoupled from the strong interaction (see §3). In
other words, a NS is unlikely to convert by SQC capture. If the cores of NSs
can access the CSC phase via cooling and mass accretion this would have
interesting implications to astrophysics (see Eq. (14)).

##  5 Discussion

Here we briefly discuss some limitations and distinctive features of our model
and leave these as future investigations.

  1. 1.

The evolution of the SQC and of the DM densities: We first recall, from Eq.
(9), that the density inside an SQC evolves from $\rho_{\rm csc}$ to
$(1-\eta_{\rm G})\rho_{\rm csc}$ after the LGB phase. Drainage further
decreases the SQC density to an asymptotic value of $(1-\eta_{\rm G}-f_{\rm
V})\rho_{\rm csc}$ at $t>>t_{\rm tun.}$ (which we associate with $z=0$). The
minimum possible value of an SQC would be given by the bare quarks masses or,
$\sim 90$% of their original mass. However, we find $\eta_{\rm G}$ and
$\eta_{\rm V}$ to be of the order of tens of percents each (see §3), and an
SQC would lose a fraction of its mass to DE over the age of the universe.
Meaning that the DM content in today’s universe is a fraction of that in the
pre-BBN era according to our model.

Averaging the total SQC mass over the Hubble volume gives us the DM density in
our model. The ratio of total amount of DM to that of the baryonic matter
evolves from a maximum value of $\eta_{\rm DM,sG}$ before LGB decay to
$\eta_{\rm DM,eG}=(1-\eta_{\rm G})\eta_{\rm DM,sG}$ after the LGB decay to
photons just before drainage starts. At full drainage, which we associate with
$z=0$, the ratio is

$\eta_{\rm DM,0}=(1-\eta_{\rm G}-f_{\rm V})\eta_{\rm DM,sG}=(1-\frac{f_{\rm
V}}{1-\eta_{\rm G}})\eta_{\rm DM,eG}\ .$ (11)

Assuming $\eta_{\rm DM,0}\sim 5$ as measured today and extrapolating back to
the LGB era, it means that the amount of DM content is larger than what we
used in §2 where we set $\eta_{\rm DM,eG}=\eta_{\rm DM,0}\sim 5$ (i.e. when
taking $f_{\rm V}=0$).

  2. 2.

The 7Li problem revisited: The solution to the cosmological 7Li problem
presented in §2 did not take into account drainage of the SQC condensate
following LGB decay as discussed above. From Eq. (11), $\eta_{\rm
DM,sG}=5/(1-\eta_{\rm G}-f_{\rm V})$ and when plugged in Eq. (4), Eq. (5)
becomes

$\frac{\eta_{\rm G}}{(1-\eta_{\rm G}-f_{\rm V})}\sim 0.9\ ,$ (12)

which can be rounded off to give

$\eta_{\rm G}\sim\frac{1-f_{\rm V}}{2}\ .$ (13)

I.e. a smaller percentage of the gluonic energy of the SQCs converted to LGBs
could resolve the 7Li problem. E.g. for $f_{\rm V}>2/5$ we need $\eta_{\rm
G}<3/8$ which is less than the maximum $\eta_{\rm G}\sim 3/8$ expected from
the 2SC-phase. For $f_{\rm V}=0$ we recover the $\eta_{\rm G}\sim 0.5$ value
we arrived at in §2.

  3. 3.

The CSC phase and Neutron stars: Figure 2 shows a suggested pathway, starting
at point “$1^{\prime}$", a NS core could take to enter the CSC phase since
conversion following SQC capture is suppressed (see §4). A NS born with (or
which acquires through evolution) a core in the unpaired phase could
transition to the CSC phase by a sequence of cooling (to keV temperature via
the URCA process; e.g. Paczyński 1972) and compression (to $\mu_{\rm csc}=500$
MeV via mass accretion). Take a NS with a core making up a fraction $\eta_{\rm
c}$ of the total mass. The energy released from conversion of gluonic
condensation (e.g. LGBs) to photons is $(\eta_{\rm c}M_{\rm NS}/m_{\rm
p})\times(\eta_{\rm G}m_{\rm p}c^{2})=\eta_{\rm c}\eta_{\rm G}M_{\rm
NS}c^{2}$. Comparing this to the NS binding energy $\frac{3}{5}GM_{\rm
NS}^{2}/R_{\rm NS}$, we conclude that NSs with compactness parameter

$\frac{M_{\rm NS,\odot}}{R_{\rm NS,6}}<0.11\times\frac{\eta_{\rm
c}}{0.1}\times\frac{\eta_{\rm G}}{0.1}\ ,$ (14)

may be completely obliterated in the process; the NS mass and radius are in
units of solar mass, $M_{\odot}$, and $10^{6}$ cm, respectively. NSs with
higher compactness parameter would loose mass leaving behind a pure CSC core.
In this latter case, the conversion to a CSC star puts a constraint on
$\rho_{\rm csc}$ due to the black hole limit $2GM_{\rm NS}/c^{2}<R_{\rm csc}$
with $R_{\rm csc}$ the radius of the CSC star. With $\rho_{\rm csc}R_{\rm
csc}^{3}\sim\rho_{\rm NS}R_{\rm NS}^{3}$ this gives $\rho_{\rm csc}<10^{16}\
{\rm g\ cm}^{-3}/M_{\rm NS,\odot}^{2}$ which is consistent with the $\mu_{\rm
csc}\sim 500$ MeV (i.e. a 2SC-like phase) adopted in our model. Thus, if some
NSs follow a path as suggested in Figure 2, the resulting photon fireball may
have interesting implications to explosive astrophysics.

  4. 4.

SQC-hadron decoupling: If an important percentage of the quarks condensate
$(\bar{q}q)$ is lost during the drainage then it is not unreasonable to assume
that the quarks within the SQC become undressed and should in principle
decouple from the strong interaction. They would still rely on gluons to
remain bound while exhibiting minimal interaction with baryons. The exact
details of this decoupling remain to be worked out.

If our model is correct, it allows a unique connection between cosmology and
the QCD condensates. An estimate of the parameter $f_{\rm V}$ from
cosmological observations is an indication of the contribution of the QCD
condensates to the mass of hadrons.

  5. 5.

SQCs in today’s detectors: The tunnelling being stochastic, DM at any given
time is a mixture of non-tunnelled and tunnelled SQCs in our model. However,
if $t_{\rm tun.}\leq 1$ Gyr, only a tiny fraction ($<e^{-1/H_{0}t_{\rm
tun.}}\sim 10^{-6}$) of SQCs would have preserved their QCD condensates in
today’s universe. The majority would interact only gravitationally and would
thus evade detection in current DM experiments. Instead we propose that our
model can be validated by experiments which can access the 2SC phase at
temperatures below the LGB melting temperature; i.e. at $<M_{\rm LGB}c^{2}\sim
4$ MeV (see Appendix B). The Compressed Baryonic Matter experiment (CBM) at
FAIR explores the QCD phase diagram in the region of high baryon densities
(representative of neutron star densities) and moderate temperatures using
high-energy nucleus-nucleus collisions (e.g. Ablyazimov et al. 2017; Senger
2020). We note that the 2SC phase carries $M_{\rm LGB}c^{2}\sim 4$ MeV LGBs
(i.e. can solve the cosmological 7Li problem) at baryonic density as low as a
few times that of nuclear matter (see Figure 3) and that LGBs form on strong
interaction timescales and decay to MeV photons on timescales of $\sim
10^{-17}$ s. Thus, experiments such as the CBM at FAIR could be used to detect
the photons (albeit a very smeared line) from the LGB decay. However, it is
not clear if and how CBM could explore the temperature regime we are
interested in.

  6. 6.

The stability of the CSC phase: Our findings seem to hint at the standard
neutral 2SC phase (adopted in this proof-of-principle paper) as the
unspecified CSC phase. However, the 2SC phase may be unstable at small
$\Delta$ values due to the mismatch in the up and down quarks chemical
potential (Huang & Shovkovy 2004). It suggests that either the 2SC phase is
stable in the regime of chemical potential ($\mu$) and the superconducting gap
($\Delta$) values we used or that another stable 2SC-like phase exits in
nature and remains to be identified. Another possibility are collective
excitations (i.e. phonons) of CSCQNs which would decay directly to photons via
phonon-to-photon conversion channels. Mono-chromatic photons of energy $1.59\
{\rm MeV}<E_{\gamma}<22.2$ MeV would result if the CSCQN radius is $544.6\
{\rm fm}<R_{\rm CSCQN}\sim hc/E_{\gamma}\sim 753.6$ fm. Physically, the
decaying particle is not an LGB but is instead a long-wavelength mode which is
resonant with the quark cluster containing it. In this scenario, higher
resonant modes should also be excited and these could affect the very
sensitive D abundance which is not desired. Superconducting strings (Haber &
Schmitt 2018) is an interesting avenue to explore in this context. If such
domain walls could form during the cosmic QCD phase, one could imagine a
scenario where they would evolve into 2SC-dominated strings in the post-BBN
era. It remains to be shown that LGBs as described in our model could form in
this case.

  7. 7.

Matter-Antimatter annihilation and SQC size: We hypothesize that each quark
cluster is born with an anti-matter deficit of $\eta_{\rm B}\simeq 6.1\times
10^{-10}$ meaning that there is one extra baryon per $\eta_{\rm B}^{-1}$
quark-antiquark pairs; $\eta_{\rm B}$ is the baryon-to-photon ratio. After
annihilation (on timescales of $1/n_{\rm csc}\sigma_{\rm annih.}\sim 10^{-13}$
s with $\sigma_{\rm annih.}\sim$ mbarn), a cluster has only baryons left in
it; this assumes that annihilation does not destroy the cluster and instead it
reduces it into a pure baryon cluster of radius $R_{\rm sqc,f}\sim R_{\rm
sqc,thin}$ where “f" stands for final. Here, $R_{\rm sqc,thin}\sim 10^{2}$ fm
is the typical size of a SQC set by the photon mean-free-path in the 2SC
phase. In other words, we claim that the maximum size of the “shrapnel" of the
annihilated much bigger parent cluster to be of the order of $R_{\rm
sqc,thin}$ (see §2.2). In this case, a cluster’s birth radius can be obtained
from $n_{\rm csc}R_{\rm sqc,thin}^{3}/n_{\rm csc}R_{\rm sqc,b}^{3}=\eta_{\rm
B}$ which gives $R_{\rm sqc,b}\sim 10^{5}$ fm$/n_{\rm csc,39}$. Some
constraints and implications to consider in the future include: (i)
Annihilation should also yield pions. These would decay on weak-interaction
timescales and if their mean-free-path turns out to be much smaller than that
of photons they may affect the $\sim 10^{2}$ fm SQCs; (ii) While cluster
formation (when the universe has aged such that its temperature is in the tens
of MeV) is followed rapidly by annihilation, we must avoid re-creation of
matter-antimatter pairs. I.e. ensure that pair-creation timescales exceed the
Hubble expansion timescale; (iii) In the picture we propose here, SQCs would
require a formation mechanism which is quite different from how cosmic
strange-quark nuggets (Witten 1984) and Axion quark nuggets (Zhitnitsky 2003)
form.

##  6 Conclusion

In this proof-of-principle paper, we have proposed that a color
superconducting (CSC) phase of lukewarm QCD matter could offer a non-exotic
solution to the cosmological 7Li, the CDM and the Dark Energy enigmas. The
narrow 2 MeV photon line which destroys 7Be in the radiation-dominated post-
BBN epoch we attribute to gluonic condensation (i.e. light glueballs or LGBs)
in the CSC phase and its electro-magnetic decay modes (§2). The detailed
properties of the CSC phase remain to be scrutinized although a neutral 2SC-
like phase with a superconducting gap $\Delta\sim 0.1\mu$ is hinted at.

CDM, according to our model, consists of colorless, charge neutral, optically
thin cosmic quark clusters in the CSC phase (SQCs) with $R_{\rm sqc}\sim 100$
fm in size and baryon number $A_{\rm sqc}\sim 10^{6}$. They decouple from the
strong interaction and interact only gravitationally thus evading detection in
currents DM experiments. If SQCs could be produced in experiments such as the
Compressed Baryon Experiment at FAIR (see bullet point #5 in §5), they could
be detected via the MeV photons from $LGB\rightarrow\gamma+\gamma$ decay
giving support to our model.

The transparency of SQCs to the strong interaction is due to drainage of the
QCD condensates (vacuum) within the SQCs into the trivial QCD vacuum of the
exterior space-time which yields Dark Energy. As drainage proceeds, our
cosmology gradually transitions from a non-DE to a DE ($\Lambda$CDM-like)
universe at moderate redshift while allowing for a possible resolution of the
Hubble tension (§3).

Our model does not introduce new physics to solve the 7Li, DM and DE problems
but instead makes use of still uncertain properties of QCD phases and its
vacuum properties. We are not the first to propose a connection between QCD
in-hadron condensates and cosmology. It has been argued based on empirical
properties of hadrons that confinement is a pre-requisite for retaining
condensates inside hadrons, which then largely eliminates the problem of the
smallness of the cosmological constant. Our model, while evading rigorous
calculation for an efficacious idea, is in line with such arguments, while the
concept of drainage of the in-SQC vacuum, and the decoupling of SQCs from the
strong force, is indeed a new idea (see §3) that may turn out to have other
useful physical applications.

## Acknowledgements

R.O., D.L. and N.K. acknowledge the support of the Natural Sciences and
Engineering Research Council of Canada (NSERC). P.J. is supported by a grant
from the Natural Science Foundation PHY-1913693. We thank R. Rapp for
interesting discussions on various aspects of the paper. R.O. thanks P.
Serpico for brief comments on Appendix A.

## Data Availability

No new data were generated or analysed in support of this research.

## References

  * Abel et al. (2002) Abel T., Bryan G. L., Norman M. L., 2002, Science, 295, 93
  * Ablyazimov et al. (2017) Ablyazimov T et al. 2017, Eur. Phys. J. A 53 60
  * Alford et al. (2001) Alford, M.,G. Rajagopal, K., Reddy, S. & F. Wilczek, F. 2001, Phys. Rev. D64, 074017
  * Alford et al. (2008) Alford, M. G., Schmitt, A., Rajagopal, K. & Schäfer, T. 2008, Reviews of Modern Physics, 80, 1455
  * Arbey & Mahmoudi (2021) Arbey, A. & Mahmoudi, F. 2021, Progress in Particle and Nuclear Physics, 119, 103865
  * Baym et al. (2018) Baym, G., Hatsuda, T., Kojo, T. et al. 2018, Reports on Progress in Physics 81, 056902
  * Berezinsky et al. (1990) Berezinsky V. S., Bulanov S. V., Dogiel V. A., Ginzburg V. L. & Ptuskin V. S., 1990, “Astrophysics of Cosmic Rays’", Amsterdam, Netherlands: North-Holland (1990) 534 p
  * Bertone & Hooper (2018) Bertone, G. & Hooper, D. 2018, Reviews of Modern Physics 90, 045002
  * Brodsky et al. (2012) Brodsky, S. J., Roberts, C. D., Shrock, R. & Tandy, P. C. 2012, Phys. Rev. C85, 065202
  * Carmeli, et al. (2006) Carmeli, M., Hartnett, J. G. & Oliveira, F. J. 2006, Foundations of Physics Letters, 19, 277
  * Cohen-Tannoudji et al. (2006) Cohen-Tannoudji, C., Diu, B. & Laloe, F. 2006, Quantum Mechanics. (Hermann and John Wiley & Sons, Paris, France)
  * Coleman (1977) Coleman, S. R. 1977, Phys. Rev. D15, 2929; Erratum, Phys. Rev. D16, 1248(E) (1977)
  * Fields (2011) Fields, B. D. 2011, Annual Review of Nuclear and Particle Science, 61, 47
  * Garrett & Gintaras (2011) Garrett, K. & Gintaras, D. 2011, Advances in Astronomy, 2011, 968283
  * Haber & Schmitt (2018) Haber, A. & Schmitt, A. 2018, Journal of Physics G Nuclear Physics 45, 065001
  * Hoyle & Tayler (1964) Hoyle, F. & Tayler, R.J. 1964, Nature 203, 1108
  * Huang & Shovkovy (2004) Huang, M. & Shovkovy, I. A. 2004, Phys. Rev. D70, 051501(R); Phys. Rev. D 70, 094030
  * Ishida et al. (2014) Ishida, H., Kusakabe, M. & Okada, H. 2014, Phys. Rev. D90, 8, 083519
  * Iwamoto (1982) Iwamoto, N. 1982, Ann. Phys., 141, 1
  * Kamionkowski & Riess (2022) Kamionkowski, M & Riess, A. G. 2022, Ann. Rev. Nucl. Part. Sci. 2023, 1
  * Kawasaki & Moroi (1995) Kawasaki, M. & Moroi, T. 1995, ApJ, 452 506
  * Kawasaki et al. (2020) Kawasaki, M., Kohri, K., Moroi, T., et al. 2020, J. Cosmology Astropart. Phys., 2020, 048. doi:10.1088/1475-7516/2020/12/048
  * Khatri & Sunyaev (2011) Khatri, R. & Sunyaev, R. A. 2011, Astronomy Letters, 37, 367
  * Kisslinger & Das (2019) Kisslinger, L. S. & Das, D. 2019, International Journal of Modern Physics A, 34, 1930013
  * Komatsu et al. (2011) Komatsu, E. et al., 2011, Astrophys. J. Suppl. 192, 47
  * Lang (1999) Lang, K.R., Astrophysical Formulae, Volume II: Space, Time, Matter and Cosmology, Springer-Verlag, Berlin 1999
  * Navarro, Frenk & White (1996) Navarro J. F., Frenk C. S., White S. D. M. 1996, ApJ, 462, 563
  * Oks (2021) Oks, E. 2021, New Astronomy Reviews, 93, 101632
  * Ouyed & Sannino (2001) Ouyed, R., & Sannino, F. 2001, Phys. Lett. B. 511, 66
  * Paczyński (1972) Paczyński, B. 1972, Astrophys. Lett.11, 53
  * Peebles (1966) Peebles, J. P. E. 1966, ApJ, 146, 542
  * Planck collaboration (2018) Planck collaboration 2018, A&A 641, A6
  * Poulin & Serpico (2015) Poulin, V. & Serpico, P. D. 2015, Phys. Rev. Lett. 114, 091101 (PS15)
  * Protheroe et al. (1995) Protheroe, R.J., Stanev, T. & Berezinsky, V.S. 1995, Phys. Rev. D, 51(8), 4134
  * Rischke et al. (2001) Rischke, D. H. Son, D. T. &. Stephanov, M. A. 2001, Phys. Rev. Lett.87, 062001
  * Rüster et al. (2004) Rüster, S. B., Shovkovy, I. A. & Rischke, D. H. 2004, Nuclear Phys. A, 743, 127
  * Ryden (2016) Ryden, B. 2016, “Introduction to Cosmology", 2nd Edition (Cambridge University Press, Cambridge)
  * Sannino et al. (2002) Sannino, F., Marchal, N. & Schäfer, W. 2002, Phys. Rev. D66, 016007
  * Schumann (2015) Schumann, M. 2015, European Physical Journal Web of Conferences, 96, 01027
  * Schöneberg et al. (2022) Schöneberg, N., Abellán, G. F., Sánchez, A. P., Witte, S. J., Poulin, V. & Lesgourgues, J. 2022, Phys. Rep.984, 1
  * Senger (2020) Senger, P. 2020, International Journal of Modern Physics E, 29, 2030001
  * Shovkovy (2004) Shovkovy, I. A. 2004, lectures delivered at the IARD 2004 conference, Saas Fee, Switzerland, June 12-19, and at the Helmholtz International Summer School and Workshop on Hot points in Astrophysics and Cosmology, JINR, Dubna, Russia, Aug. 2-13 [arXiv:nucl-th/0410091]
  * Spite & Spite (1982) Spite, F. & Spite, M. 1982, A&A,115, 357
  * Svensson & Zdziarski (1990) Svensson, R. & Zdziarski, A. A. 1990, ApJ, 349, 415
  * Tytler et al. (2000) Tytler, D., O’Meara, J. M., Suzuki, N. & Lubin, D. 2000, Phys. Scripta T85, 12
  * Wagoner et al. (1967) Wagoner, R. V., Fowler, W. A. & Hoyle, F. 1967, ApJ, 148, 3
  * Witten (1984) Witten, E. 1984, Phys. Rev. D30, 272
  * Zhitnitsky (2003) Zhitnitsky, A. R. 2003, J. Cosmology Astropart. Phys., 2003, 10

##  Appendix A Post-BBN 7Be destruction

In Poulin & Serpico (2015, and references therein), the photon-generating
particle $X$ decays exponentially on timescale $\tau_{\rm X}$ and forms at
early time in the universe at $t$ $(<<\tau_{\rm X})$. In our model, the $X$
particle (i.e. the LGB) forms and decays to a narrow $E_{0}\sim$ 2 MeV photons
instantaneously in the radiation-dominated post-BBN era at time $t_{\rm G}$
when the quark cluster enters the CSC phase; here “G" stands for gluons as a
reference to the LGB era (see §2). Thus, we have a delta function production
at time $t_{\rm G}$; $\delta(t-t_{\rm G})$. The source injection term (e.g.
Eq. (2) in Poulin & Serpico (2015) and §IV in Kawasaki & Moroi (1995)) becomes

$S(E_{\gamma},t)=n_{\gamma}(E_{\gamma},t)\times\delta(t-t_{\rm
G})\times\delta(E_{\gamma}-E_{0})\ ,$ (15)

with $n_{\gamma}(E_{\gamma},t)$ the co-moving photon density at time $t$;
$\delta(E_{\gamma}-E_{0})$ is the spectral shape factor for a monochromatic
line at $E_{0}$.

Making use of Eqs. (4) and (5) in Poulin & Serpico (2015) with the destruction
of element $A$ only (i.e. $\gamma+A\rightarrow P$) gives the abundance as

$\frac{dY_{A}(t)}{dt}=-Y_{A}(t)\int dE_{\gamma}\sigma_{\gamma+A\rightarrow
P}(E_{\gamma})\times\frac{S(E_{\gamma},t)}{\Gamma(E_{\gamma},t)}\ ,$ (16)

where $\Gamma(E_{\gamma},t)$ and $\sigma_{\gamma+A\rightarrow P}(E_{\gamma})$
are the photon interaction rate and the photo-dissociation cross-section,
respectively. The interaction rate is for all processes including Bethe-
Heitler pair-creation on nuclei, double photon pair-creation, scattering off
thermal background photons and Compton scattering. Red-shifting is negligible
because the rates of electromagnetic interactions are faster than the cosmic
expansion rate.

Inserting Eq. (15) into Eq. (16) and integrating over energy gives

$\frac{d\ln{Y_{A}(t)}}{dt}=-n_{\gamma}(E_{0},t)\times\delta(t-t_{\rm
G})\frac{\sigma_{\gamma+A\rightarrow P}(E_{0})}{\Gamma(E_{0},t)}\ .$ (17)

This assumes a single scattering ($N_{\rm scat}=1$, as in our model) is enough
to make the $E_{0}$ photon no longer dissociate a 7Be nucleus. Otherwise
$\Gamma(E_{0},t)$ should be divided by $N_{\rm scat}$.

Dissociation starts at redshift $z_{\rm sG}$ and ends abruptly at redshift
$z_{\rm sG}=z_{\rm eG}+\delta z$ with $\delta z<<z_{\rm sG}$; subscripts “eG"
and “sG" stand for end and start, respectively, of the photon burst from LGB
decay in the post-BBN era. After integration over a very small time $t$ with
$t_{\rm sG}<t_{\rm G}$ and $t_{\rm eG}>t_{\rm G}$ but $z(t_{\rm sG})=z(t_{\rm
eG})=z(t_{\rm G})$ one obtains:

$\log{\left(\frac{Y_{\rm Be,eG}}{Y_{\rm
Be,sG}}\right)}=-n_{\gamma}(E_{0},t_{\rm G})\ \frac{\sigma_{\rm
Be}(E_{0})}{\Gamma(E_{0},t_{\rm G})}\ ,$ (18)

where $n_{\gamma}(E_{0},t_{\rm G})$ is the co-moving number density of $E_{0}$
photons from LGB decay. Here, $\sigma_{\rm
Be}(E_{0})\equiv\sigma_{\gamma+A\rightarrow P}(E_{0})$ while $Y_{\rm
Be,sG}\equiv Y_{A}(t_{\rm sG})$ and $Y_{\rm Be,eG}\equiv Y_{A}(t_{\rm eG})$
are the corresponding 7Be abundances.

With $E_{0}<2.2$ MeV, Bethe-Heitler pair-creation on nuclei (see Eq.(4-356) in
Lang 1999) is negligible and so is pair creation off the CMB background
photons when $T_{\rm CMB}$ is a few keV or less; the interaction rate is
dominated by Compton scattering (CS). Thus $\Gamma(E_{0},t_{\rm
G})\simeq\Gamma_{\rm CS}(E_{0},t_{\rm G})=n_{\rm e}(t_{\rm G})\sigma_{\rm
CS}(E_{0})$ with the CS cross-section given in Appendix (IV) in Kawasaki &
Moroi (1995). Furthermore, because at $T_{\rm CMB}<20$ keV positrons are
negligible (e.g. Appendix C in Svensson & Zdziarski 1990), the total electron
density is of the order of the baryon number density, $n_{\rm e}(t_{\rm
G})\sim n_{\rm B}(t_{\rm G})$. The destruction rates for 7Be nuclei due to a
sudden release of mono-energetic $E_{0}$ photons at $t_{\rm G}$ becomes

$\ln\left(\frac{Y_{\rm Be,eG}}{Y_{\rm
Be,sG}}\right)\sim-\frac{n_{\gamma}(E_{0},t_{\rm G})}{n_{\rm B}(t_{\rm
G})}\times\frac{\sigma_{\rm Be}(E_{0})}{\sigma_{\rm CS}(E_{0})}\ .$ (19)

##  Appendix B Light glueballs (LGBs) in the 2SC phase

The spectrum in the 2SC state is made of 5 massive gluons with a mass of the
order of the gap $\Delta$, 3 massless gluons and gapless up and down quarks in
the direction 3 (blue) of color; in this appendix we use QCD natural units
with $\hbar=c=1$. The 3 massless gluons in the 2SC phase do not interact with
the gapless blue quasi-particles and the quasiparticles from the green and red
paired quarks decouple from the low-energy confined $SU_{c}(2)$ phase. The 3
massless gluons in the 2SC phase bind, or are confined, into LGBs when the
temperature is below the melting temperature $T_{\rm LGB,m}\sim M_{\rm LGB}$
(Ouyed & Sannino 2001, and references therein). I.e. LGBs melt when the
temperature exceeds the confinement value (Sannino et al. 2002). Below $T_{\rm
LGB,m}$ is the $SU_{c}(2)$ phase which effectively constitutes a partial
confinement within the 2SC phase.

The LGB mass at $T\leq T_{\rm LGB,m}$ is $M_{\rm LGB}\sim\hat{\Lambda}_{\rm
c}$ with $\bar{\Lambda}_{\rm c}$ the intrinsic scale associated with the
$SU_{c}(2)$ theory. It is the confining scale of the $SU_{c}(2)$ gluon-
dynamics in 2SC and when $\Delta<<\mu$, the one loop relation gives (Rischke
et al. 2001)

$\bar{\Lambda}_{\rm
c}=\Delta\exp{\left(-\frac{2\sqrt{2}\pi}{11}\times\frac{\mu}{g_{\rm
s}\Delta}\right)}\ ,$ (20)

with $g_{\rm s}=\sqrt{4\pi\alpha_{\rm s}}$ and $\alpha_{\rm
s}=\frac{12\pi}{(33-2n_{\rm f})\ln{(\mu^{2}/\Lambda_{\rm QCD}^{2})}}$ the
$SU_{c}(3)$ coupling constant evaluated at $\mu$; $\Lambda_{\rm QCD}$ is the
scalar parameter of QCD and $n_{\rm f}$ the number of relevant quark flavors.
For $\Delta=0.1\mu$ and for $\Lambda_{\rm QCD}=245$ MeV, for example, we get
$M_{\rm LGB}\sim 4$ MeV for a range in quark chemical potential as shown in
Figure 3.

Figure 3 shows that $M_{\rm LGB}\sim 4$ MeV is naturally obtained for
$\Delta=0.1\mu$ and for $\Lambda_{\rm QCD}=245$ MeV.

Once created, LGBs are stable against strong interactions but not with respect
to electromagnetic processes. They couple to two photons via virtual quark
loops and decay on timescale $\tau_{\rm LGB}\sim 5.5\times 10^{-14}\ {\rm
s}\times(M_{\rm LGB}/\rm MeV)^{-5}$ (Ouyed & Sannino 2001).

The pairing energy density released during the unpaired-to-2SC phase
transition is $Q_{\rm pairing}=\mu^{2}\Delta^{2}/\pi^{2}$ (e.g. Alford et al.
2001; Shovkovy 2004). Thus during the transition, to a first approximation a
SQC gets heated to a temperature

$T_{\rm sqc}=\frac{Q_{\rm pairing}}{n_{\rm csc}}\sim\frac{\Delta^{2}}{\mu}\ .$
(21)

LGBs would not melt during the unpaired-to-CSC heating phase if
$\Delta^{2}/\mu<M_{\rm LGB}$ allowing for the $LGB\rightarrow\gamma+\gamma$ to
occur (see §2.1).

##  Appendix C Our cosmology

Let us write again the time evolution of the density inside a SQC as given in
Eq. (9):

$\displaystyle\rho_{\rm sqc}(t)$ $\displaystyle=(1-\eta_{\rm G})\rho_{\rm
csc}-\rho_{\rm QCD}^{\rm vac.}(1-e^{-(t-t_{\rm G})/t_{\rm tun.}})$ (22)
$\displaystyle=\rho_{\rm sqc,eG}\left(1-\frac{f_{\rm V}}{1-\eta_{\rm
G}}(1-e^{-(t-t_{\rm G})/t_{\rm tun.}})\right)\ ,$

with $\rho_{\rm sqc,eG}=(1-\eta_{\rm G})\rho_{\rm csc}$ the SQC density at the
end of the LGB/photon-burst phase. As noted earlier, the equation above
incorporates the key parameters in our cosmology namely, $\rho_{\rm
csc},\eta_{\rm G},f_{\rm V}$ and $t_{\rm tun.}$ which are all fundamentally
related to QCD.

The time evolution of the DM density given in Eq. (10) can be expanded as

$\rho_{\rm DM}(t)=\begin{cases}\frac{\rho_{\rm sqc}(t)V_{\rm sqc}^{\rm
tot.}}{V_{\rm univ.}(t)}=\frac{\rho_{\rm sqc}(t)}{\rho_{\rm
sqc,0}}\times\frac{\rho_{\rm sqc,0}V_{\rm sqc}^{\rm tot.}}{V_{\rm
univ.,0}}\times\frac{V_{\rm univ.,0}}{V_{\rm univ.}(t)}\ &\\\
\quad\qquad\qquad\qquad\qquad{\rm if}\quad t>t_{\rm G}\ ({\rm or}\ z<z_{\rm
G})&\\\ \frac{\rho_{\rm csc}V_{\rm sqc}^{\rm tot.}}{V_{\rm
univ.}(t)}=\frac{\rho_{\rm csc}}{\rho_{\rm sqc,0}}\times\frac{\rho_{\rm
sqc,0}V_{\rm sqc}^{\rm tot.}}{V_{\rm univ.,0}}\times\frac{V_{\rm
univ.,0}}{V_{\rm univ.}(t)}\ &\\\ \quad\qquad\qquad\qquad\qquad{\rm if}\quad
t\leq t_{\rm G}\ ({\rm or}\ z\geq z_{\rm G})\\\ \end{cases}$ (23)

where the subscript “0" refers to the current age of the universe at $z=0$
(i.e. at $t=t_{0}$); recall that $t_{\rm tun.}$ is a fraction of $t_{0}$ which
simplifies our equations. $V_{\rm sqc}^{\rm tot.}$ is the total volume
occupied by SQCs, which is constant in time, and $V_{\rm univ.}(t)$ the Hubble
volume at time $t$.

Putting Eq. (22) into Eq. (23) we get

$\rho_{\rm DM}(t)=\begin{cases}\rho_{\rm DM,0}\frac{V_{\rm univ.,0}}{V_{\rm
univ.}(t)}\times\frac{\left(1-\frac{f_{\rm V}}{1-\eta_{\rm
G}}\left(1-e^{-(t-t_{\rm G})/t_{\rm tun.}}\right)\right)}{\left(1-\frac{f_{\rm
V}}{1-\eta_{\rm G}}\right)}&\\\ \quad\qquad\qquad\qquad\qquad{\rm if}\quad
t>t_{\rm G}\ ({\rm or}\ z<z_{\rm G})&\\\ \rho_{\rm DM,0}\frac{V_{\rm
univ.,0}}{V_{\rm univ.}(t)}\times\frac{1}{(1-\eta_{\rm G})-f_{\rm V}}\ &\\\
\quad\qquad\qquad\qquad\qquad{\rm if}\quad t\leq t_{\rm G}\ ({\rm or}\ z\geq
z_{\rm G})\\\ \end{cases}$ (24)

with $\rho_{\rm sqc,0}=(1-\eta_{\rm G})\rho_{\rm csc}-\rho_{\rm QVD}^{\rm
vac.}$, $\rho_{\rm sqc,eG}/\rho_{\rm sqc,0}=(1-\frac{f_{\rm V}}{1-\eta_{\rm
G}})^{-1}$ and $\rho_{\rm csc}/\rho_{\rm sqc,0}=((1-\eta_{\rm G})-f_{\rm
V})^{-1}$; see §3. Also, $\rho_{\rm DM,0}=\rho_{\rm sqc,0}V_{\rm sqc}^{\rm
tot.}/V_{\rm univ.,0}$ is the DM density at $z=0$. The jump in the value of
$\rho_{\rm DM}$ at $t=t_{\rm G}$ in the equation above is due to the fact that
a fraction $(1-\eta_{\rm G})$ of the DM is lost to the 2 MeV radiation before
the start of the drainage era at $t_{\rm eG}$ (see §2).

Eq. (24) can be expressed in terms of redshift as

$\rho_{\rm DM}(z,z_{\rm tun.})=\begin{cases}\rho_{\rm
DM,0}(1+z)^{3}\times\frac{\left(1-\frac{f_{\rm V}}{1-\eta_{\rm
G}}\left(1-e^{-\frac{(1+z_{\rm
tun.})^{3/2}}{(1+z)^{3/2}}}\right)\right)}{\left(1-\frac{f_{\rm
V}}{1-\eta_{\rm G}}\right)}&\\\ \quad\qquad\qquad\qquad\qquad{\rm if}\quad
z<z_{\rm G}\sim z_{\rm eq.}&\\\ \rho_{\rm
DM,0}(1+z)^{3}\times\frac{1}{(1-\eta_{\rm G})-f_{\rm V}}\ &\\\
\quad\qquad\qquad\qquad\qquad{\rm if}\quad z\geq z_{\rm G}\sim z_{\rm eq.}&\\\
\end{cases}$ (25)

where we emphasize the dependency of $\rho_{\rm DM}$ on the drainage
characteristic redshift $z_{\rm tun.}$ to help differentiate our model from
the $\Lambda$CDM cosmology. Because $t_{\rm G}<<t_{\rm tun.}$ (or $z_{\rm
G}>>z_{\rm tun.}$) and most of the time between $t_{\rm G}$ and $t_{\rm tun.}$
is in the matter dominated era, we approximate $t/t_{\rm tun.}\sim(1+z_{\rm
tun.})^{3/2}/(1+z)^{3/2}$ and $V_{\rm univ.,0}/V_{\rm univ.}(t)\sim(1+z)^{3}$.
Our model does not depend critically on $z_{\rm G}$ and by setting $z_{\rm
G}\sim z_{\rm eq.}$, where $z_{\rm eq.}$ is the redshift at matter-radiation
equality, we can simplify the equations without changing the final results.

The redshift of matter-radiation equality in our model is estimated by writing
$\rho_{\rm DM}(z_{\rm eq.},z_{\rm tun.})=\rho_{\rm r,0}(1+z_{\rm eq.})^{4}$,
with $t<<t_{\rm tun.}$ (i.e. $z>>z_{\rm tun.}$) so that $\rho_{\rm DM}(z_{\rm
eq.},z_{\rm tun.})\simeq\rho_{\rm DM,0}(1+z_{\rm
eq.})^{3}\times(1-\frac{f_{\rm V}}{1-\eta_{\rm G}})^{-1}$. This yields

$\displaystyle 1+z_{\rm eq.}$ $\displaystyle\sim\frac{\rho_{\rm
DM,0}}{\rho_{\rm r,0}}\times\left(1-\frac{f_{\rm V}}{1-\eta_{\rm
G}}\right)^{-1}$ (26) $\displaystyle=\frac{\Omega_{\rm DM,0}}{\Omega_{\rm
r,0}}\times\left(1-\frac{f_{\rm V}}{1-\eta_{\rm G}}\right)^{-1}$
$\displaystyle=\frac{\omega_{\rm DM}}{\omega_{\rm
r}}\times\left(1-\frac{f_{\rm V}}{1-\eta_{\rm G}}\right)^{-1}\ ,$

with $\omega_{\rm DM}=\Omega_{\rm DM,0}h^{2}\simeq 0.120$, $\omega_{\rm
b}=\Omega_{\rm b,0}h^{2}\simeq 0.0224$ and $\omega_{\rm r}=\Omega_{\rm
r,0}h^{2}\simeq 4.18\times 10^{-5}$ as measured by Planck (Planck
collaboration 2018); $h=H_{0}/100$ km s-1 Mpc-1 is the dimensionless Hubble
constant in today’s universe. As expected, $z_{\rm eq.}$ in our model is
larger (i.e. requires an increase in the radiation to compensate for larger DM
in the past) than in the case of the pure $\Lambda$CDM cosmology which does
not capture the converted (by LGB decay and vacuum drainage) component of the
DM.

Finally, our cosmology can be described by

$\displaystyle H$ $\displaystyle(z,z_{\rm tun.})=H_{0}\times$ (27)
$\displaystyle\times\sqrt{\Omega_{\rm r,0}(1+z)^{4}+\Omega_{\rm
b,0}(1+z)^{3}+\Omega_{\rm DM}(z,z_{\rm tun.})+\Omega_{\Lambda}}$

with $\Omega_{\rm DM}(z,z_{\rm tun.})=\rho_{\rm DM}(z,z_{\rm tun.})/\rho_{\rm
c}^{0}$ and $H_{0}^{2}=8\pi G\rho_{\rm c}^{0}/3$ where $\rho_{\rm c}^{0}$ is
today’s critical density; $\Omega_{\Lambda}=1-\Omega_{\rm r,0}-\Omega_{\rm
b,0}-\Omega_{\rm DM,0}$.

The co-moving sound horizon and the co-moving angular-diameter distance to the
surface of last scatter are (Ryden 2016)

$r_{\rm s}(z_{\rm tun.})=\int_{z_{ls}}^{\infty}\frac{c_{\rm
s}(z)dz}{H(z,z_{\rm tun.})};\quad D_{\rm A}(z_{\rm
tun.})=\int_{0}^{z_{ls}}\frac{cdz}{H(z,z_{\rm tun.})}\ .$ (28)

The sound speed of the photon-baryon fluid at $z\geq z_{ls}$ is $c_{\rm
s}(z)=c/\sqrt{3(1+R(z))}$ with $R(z)=(3/4)(\omega_{\rm
b}/\omega_{\gamma})/(1+z)$ and $\omega_{\gamma}=2.47\times 10^{-5}$.

The Hubble constant $H_{0}$ is found by solving $\theta_{\rm s}D_{\rm
A}(z_{\rm tun.})-r_{\rm s}(z_{\rm tun.})=0$ with $\theta_{\rm s}=1.041\times
10^{-2}$ the angle subtended by the sound horizon (Planck collaboration 2018).
For the $\Lambda$CDM one finds $H_{0}\sim 67.3$ km s-1 Mpc-1 ($h\sim 0.673$;
by solving $\theta_{\rm s}D_{\rm A}-r_{\rm s}=0$ using standard $\Lambda$CDM
cosmology). In our case, $H_{0}\sim 73$ km s-1 Mpc-1 ($h=0.73$) can be
obtained for a range in $\eta_{\rm G}$ and $f_{\rm V}$ values (see examples in
Figure 4) with a drainage characteristic redshift $2<z_{\rm tun.}<6$ (i.e.
$1<t_{\rm tun.}({\rm Gyr})<4$); see §3.


#  Standard Model Predictions for Rare $K$ and $B$ Decays
without $|V_{cb}|$ and $|V_{ub}|$ Uncertainties111Submitted also to the
DISCRETE2020-2021 Conference

Andrzej J. Buras and
TUM-IAS, Lichtenbergstr. 2a, D-85748 Garching, Germany
Technical University Munich, Physics Department, D-85748 Garching, Germany,
E-mail

###### Abstract:

The persistent tensions between inclusive and exclusive determinations of
$|V_{cb}|$ and $|V_{ub}|$ weaken the power of theoretically clean rare $K$ and
$B$ decays in the search for new physics (NP). We demonstrate how this
uncertainty can be practically removed by considering within the SM suitable
ratios of various branching ratios. This includes the branching ratios for
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$, $K_{L}\rightarrow\pi^{0}\nu\bar{\nu}$,
$K_{S}\to\mu^{+}\mu^{-}$, $B_{s,d}\to\mu^{+}\mu^{-}$ and $B\to
K(K^{*})\nu\bar{\nu}$. Also $\varepsilon_{K}$, $\Delta M_{d}$, $\Delta M_{s}$
and the mixing induced CP-asymmetry $S_{\psi K_{S}}$, all measured already
very precisely, play an important role in this analysis. The highlights of our
analysis are 16 $|V_{cb}|$ and $|V_{ub}|$ independent ratios that often are
independent of the CKM parameters or depend only on the angles $\beta$ and
$\gamma$ in the Unitarity Triangle with $\beta$ already precisely known and
$\gamma$ to be measured precisely in the coming years by the LHCb and Belle II
colaborations. Once $\gamma$ is measured precisely these 16 ratios taken
together are expected to be a powerful tool in the search for new physics.
Assuming no NP in $|\varepsilon_{K}|$ and $S_{\psi K_{S}}$ we determine
independently of $|V_{cb}|$:
$\mathcal{B}(K^{+}\rightarrow\pi^{+}\nu\bar{\nu})_{\text{SM}}=(8.60\pm
0.42)\times 10^{-11}$ and
$\mathcal{B}(K_{L}\rightarrow\pi^{0}\nu\bar{\nu})_{\text{SM}}=(2.94\pm
0.15)\times 10^{-11}$. This are the most precise determinations to date.
Assuming no NP in $\Delta M_{s,d}$ allows to obtain analogous results for all
$B$ decay branching ratios considered in our paper without any CKM
uncertainties.

##  1 Introduction

The rare $K$ and $B$ decays and the quark mixing being GIM suppressed in the
Standard Model (SM) and simultaneously being often theoretically clean are
very powerful tools for the search of New Physics (NP) [1]. Unfortunately the
persistent tensions between inclusive and exclusive determinations of
$|V_{cb}|$ and $|V_{ub}|$ (see e.g. [2, 3, 4, 5, 6]) weaken this power
significantly. As recently reemphasized by us [7] this is in particular the
case of the branching ratios for K-meson decays and the parameter
$\varepsilon_{K}$ that exhibit stronger $|V_{cb}|$ dependences than rare $B$
decay branching ratios and the $\Delta M_{s,d}$ mass differences.

To cope with this difficulty one can consider within the SM suitable ratios of
two properly chosen observables so that the dependences on $|V_{cb}|$ and
$|V_{ub}|$ are eliminated [8, 9, 7]. While in [8, 9] $B$ physics observables
were considered, our analysis in [7] was dominated by the $K$ system and its
correlation with rare $B$ decays and $B_{s,d}^{0}-\bar{B}_{s,d}^{0}$ mixing.
In this manner we could construct 16 $|V_{cb}|$ independent ratios that were
either independent of the CKM parameters or only dependent on the angles
$\beta$ and $\gamma$, that can be determined in tree-level processes. Having
one day precise experimental values for the ratios in question and also
precise values on $\beta$ and $\gamma$ will hopefully allow one to identify
particular pattern of deviations from SM expectations independently of
$|V_{cb}|$ and $|V_{ub}|$ pointing towards a particular extension of the SM.

This note summarizes the main results of [7] and is arranged as follows. In
Section 2 we will simply list the 16 ratios in question and briefly discuss
their first implications. In Section 3 we report on inconsistencies between
the determinations of $|V_{cb}|$ from $\varepsilon_{K}$, $\Delta M_{d}$ and
$\Delta M_{s}$. In Section 4, combining these ratios with the assumption of no
NP contributions to $|\varepsilon_{K}|$, $\Delta M_{s}$, $\Delta M_{d}$ and
$S_{\psi K_{S}}$, we will present SM predictions for all branching ratios
considered by us that are most accurate to date. Our note closes with a short
outlook in Section 5.

##  2 $|V_{cb}|$ and $|V_{ub}|$ Independent Ratios

As four basic CKM parameters we will use

$\boxed{\lambda=|V_{us}|,\qquad|V_{cb}|,\qquad\beta,\qquad\gamma}$ (1)

with $\beta$ and $\gamma$ being two angles in the UT. Their determination from
mixing induced CP-asymmetries in tree-level $B$ decays and using other tree-
level strategies is presently theoretically cleaner than the determination of
$|V_{ub}|$. A recent review of such determinations of $\beta$ and $\gamma$ can
be found in [1, 10, 11].

The $|V_{cb}|$ independent ratios constructed by us have the general power-
like structure

$R_{i}=C_{i}[\sin\gamma]^{p_{1}}[\sin\beta]^{p_{2}},$ (2)

with the coefficients $C_{i}$ either being constants or being very weakly
dependent on $\beta$ and $\gamma$. Whenever the ratios are not already in the
form of (2), we derive an approximate power-law expression, with non-integer
exponents. The latter are indeed fitted to describe as power-law functions of
parameters some more complicated exact expressions, with the best possible
accuracy, which is $\lesssim 2\%$. The dependence on $|V_{us}|$ is negligible,
the angle $\beta$ is already known from $S_{\psi K_{S}}$ asymmetry with
respectable precision and there is a significant progress by the LHCb
collaboration on the determination of $\gamma$ from tree-level strategies
[12]:

$\boxed{\beta=(22.2\pm
0.7)^{\circ},\qquad\gamma=(65.4^{+3.8}_{-4.2})^{\circ}\,.}$ (3)

Moreover, in the coming years the determination of $\gamma$ by the LHCb and
Belle II collaborations should be significantly improved so that precision
tests of the SM using our strategy will be possible.

The 16 $|V_{cb}|$ independent ratios in question are given as follows

$\boxed{R_{0}(\beta)=\frac{\mathcal{B}(K^{+}\rightarrow\pi^{+}\nu\bar{\nu})}{\mathcal{B}(K_{L}\rightarrow\pi^{0}\nu\bar{\nu})^{0.7}}=(2.03\pm
0.08)\times 10^{-3}\left[\frac{\sin
22.2^{\circ}}{\sin\beta}\right]^{1.4}={(2.03\pm 0.11)}\times 10^{-3}\,.}$ (4)
$\boxed{R_{\rm SL}=\frac{{\cal B}(K_{S}\to\mu^{+}\mu^{-})_{\rm SD}}{{\cal
B}(K_{L}\rightarrow\pi^{0}\nu\bar{\nu})}=1.55\times
10^{-2}\,\left[\frac{\lambda}{0.225}\right]^{2}\left[\frac{Y(x_{t})}{X(x_{t})}\right]^{2}\,,}$
(5)

where $X(x_{t})$ and $Y(x_{t})$ are well known one-loop functions with
$x_{t}=m_{t}^{2}/M_{W}^{2}$ [1]. Then, we have

$\boxed{R_{1}(\beta,\gamma)=\frac{\mathcal{B}(K^{+}\rightarrow\pi^{+}\nu\bar{\nu})}{\left[{\overline{\mathcal{B}}}(B_{s}\to\mu^{+}\mu^{-})\right]^{1.4}},\qquad
R_{2}(\beta,\gamma)=\frac{\mathcal{B}(K^{+}\rightarrow\pi^{+}\nu\bar{\nu})}{\left[{\mathcal{B}}(B_{d}\to\mu^{+}\mu^{-})\right]^{1.4}}.}$
(6)

In particular the ratio $R_{1}$, implying the correlation in Fig. 2 should be
of interest in the coming years due to the improved measurement of
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ by NA62, of $B_{s}\to\mu^{+}\mu^{-}$ by
LHCb, CMS and ATLAS and of $\gamma$ by LHCb and Belle II. Explicity we have

$\displaystyle R_{1}(\beta,\gamma)$ $\displaystyle={(55.24\pm
2.48)}\left[\frac{\sin\gamma}{\sin
67^{\circ}}\right]^{1.39}\left[\frac{G(22.2^{\circ},67^{\circ})}{G(\beta,\gamma)}\right]^{2.8}\left[\frac{230.3\,{\rm
MeV}}{F_{B_{s}}}\right]^{2.8},$ (7) $\displaystyle R_{2}(\beta,\gamma)$
$\displaystyle={(8.29\pm 0.40)}\times
10^{3}\left[\frac{\sin(67^{\circ})}{\sin\gamma}\right]^{1.41}\left[\frac{190.0\,{\rm
MeV}}{F_{B_{d}}}\right]^{2.8},$ (8)

where $F_{B_{d}}$ and $F_{B_{s}}$ are weak decay constants and

$G(\beta,\gamma)=1+\frac{\lambda^{2}}{2}(1-2\sin\gamma\cos\beta)\,.$ (9)

Furthermore, there are also the following $|V_{cb}|$ independent ratios

$\boxed{R_{3}(\beta,\gamma)=\frac{\mathcal{B}(K_{L}\rightarrow\pi^{0}\nu\bar{\nu})}{\left[{\overline{\mathcal{B}}}(B_{s}\to\mu^{+}\mu^{-})\right]^{2}},\qquad
R_{4}(\beta,\gamma)=\frac{\mathcal{B}(K_{L}\rightarrow\pi^{0}\nu\bar{\nu})}{\left[{\mathcal{B}}(B_{d}\to\mu^{+}\mu^{-})\right]^{2}}\,,}$
(10)

which are given by

$R_{3}(\beta,\gamma)=(2.17\pm 0.09)\times
10^{6}\left[{\frac{\sin\gamma\sin\beta}{\sin(67^{\circ})\sin(22.2^{\circ})}}\right]^{2}\left[\frac{G(22.2^{\circ},67^{\circ})}{G(\beta,\gamma)}\right]^{4}\left[\frac{230.3\,{\rm
MeV}}{F_{B_{s}}}\right]^{4}\,,$ (11) $R_{4}(\beta,\gamma)=(2.79\pm 0.13)\times
10^{9}\left[{\frac{\sin(67^{\circ})}{\sin\gamma}\frac{\sin\beta}{\sin(22.2^{\circ})}}\right]^{2}\left(\frac{190.0\,{\rm
MeV}}{F_{B_{d}}}\right)^{4}.$ (12)

In view of future improved results from NA62 and Belle II, of particular
interest are the ratios

$\boxed{R_{5}(\beta,\gamma)=\frac{\mathcal{B}(K^{+}\rightarrow\pi^{+}\nu\bar{\nu})}{\left[\mathcal{B}(B^{+}\to
K^{+}\nu\bar{\nu})\right]^{1.4}},\qquad
R_{6}(\beta,\gamma)=\frac{\mathcal{B}(K^{+}\rightarrow\pi^{+}\nu\bar{\nu})}{\left[\mathcal{B}(B^{0}\to
K^{0*}\nu\bar{\nu})\right]^{1.4}}.}$ (13)

Explicitly

$R_{5}(\beta,\gamma)={(2.69\pm 0.51)}\times
10^{-3}\left[\frac{\sin\gamma}{\sin
67^{\circ}}\right]^{1.39}\left[\frac{G(22.2^{\circ},67^{\circ})}{G(\beta,\gamma)}\right]^{2.8},$
(14) $R_{6}(\beta,\gamma)={(9.07\pm 1.23)}\times
10^{-4}\left[\frac{\sin\gamma}{\sin
67^{\circ}}\right]^{1.39}\left[\frac{G(22.2^{\circ},67^{\circ})}{G(\beta,\gamma)}\right]^{2.8}\,,$
(15)

with $G(\beta,\gamma)$ defined in (9). Next, in view of future LHCb and Belle
II measurements, of particular interest are the ratos

$\boxed{R_{7}=\frac{\mathcal{B}(B^{+}\to
K^{+}\nu\bar{\nu})}{{\overline{\mathcal{B}}}(B_{s}\to\mu^{+}\mu^{-})},\qquad
R_{8}=\frac{\mathcal{B}(B^{0}\to
K^{*0}\nu\bar{\nu})}{{\overline{\mathcal{B}}}(B_{s}\to\mu^{+}\mu^{-})}}$ (16)

with

$(R_{7})_{\text{SM}}={(1.20\pm 0.17)}\times 10^{3}\left(\frac{230.3\,{\rm
MeV}}{F_{B_{s}}}\right)^{2}\,,$ (17) $(R_{8})_{\text{SM}}={(2.62\pm
0.25)}\times 10^{3}\left(\frac{230.3\,{\rm MeV}}{F_{B_{s}}}\right)^{2}\,.$
(18)

Turning our attention to $\varepsilon_{K}$ and $\Delta M_{s.d}$ we define two
$|V_{cb}|$ independent ratios

$\boxed{R_{9}(\beta,\gamma)=\frac{|\varepsilon_{K}|}{(\Delta
M_{d})^{{1.7}}},\qquad R_{10}(\beta,\gamma)=\frac{|\varepsilon_{K}|}{(\Delta
M_{s})^{{1.7}}}\,.}$ (19)

The explicit expressions for them read

${R_{9}(\beta,\gamma)=6.405\times 10^{-3}\,{\rm ps^{1.7}}\left(\frac{\sin
67^{\circ}}{\sin\gamma}\right)^{1.73}\left(\frac{\sin\beta}{\sin
22.2^{\circ}}\right)^{0.87}\bar{R}^{\epsilon}_{d}},$ (20)
${R_{10}(\beta,\gamma)=1.516\times 10^{-5}\,{\rm
ps^{1.7}}\left(\frac{\sin\gamma}{\sin
67^{\circ}}\right)^{1.67}\left(\frac{\sin\beta}{\sin
22.2^{\circ}}\right)^{0.87}\left(\frac{G(22.2^{\circ},67^{\circ})}{G(\beta,\gamma)}\right)^{3.4}\,\bar{R}^{\epsilon}_{s}}.$
(21)

Here

$\bar{R}^{\epsilon}_{d}=\left(\frac{214.0\,{\rm
MeV}}{\sqrt{\hat{B}_{B_{d}}}F_{B_{d}}}\right)^{3.4}\left(\frac{2.307}{S_{0}(x_{t})}\right)^{1.7}\left(\frac{0.5521}{\eta_{B}}\right)^{1.7}\,,$
(22) $\bar{R}^{\epsilon}_{s}=\left(\frac{261.7\,{\rm
MeV}}{\sqrt{\hat{B}_{B_{s}}}F_{B_{s}}}\right)^{3.4}\left(\frac{2.307}{S_{0}(x_{t})}\right)^{1.7}\left(\frac{0.5521}{\eta_{B}}\right)^{1.7}\,.$
(23)

Finally, there are also the following four ratios

$\boxed{R_{11}(\beta,\gamma)=\frac{\mathcal{B}(K^{+}\rightarrow\pi^{+}\nu\bar{\nu})}{|\varepsilon_{K}|^{0.82}}=(1.31\pm
0.05)\times 10^{-8}{\left(\frac{\sin\gamma}{\sin
67^{\circ}}\right)^{0.015}\left(\frac{\sin
22.2^{\circ}}{\sin\beta}\right)^{0.71},}}$ (24)
$\boxed{R_{12}(\beta,\gamma)=\frac{\mathcal{B}(K_{L}\rightarrow\pi^{0}\nu\bar{\nu})}{|\varepsilon_{K}|^{1.18}}=(3.87\pm
0.06)\times 10^{-8}{\left(\frac{\sin\gamma}{\sin
67^{\circ}}\right)^{0.03}\left(\frac{\sin\beta}{\sin
22.2^{\circ}}\right)^{0.9{8}},}}$ (25)
$\boxed{R_{q}=\frac{\mathcal{B}(B_{q}\to\mu^{+}\mu^{-})}{\Delta
M_{q}}=4.291\times 10^{-10}\
\frac{\tau_{B_{q}}}{\hat{B}_{q}}\frac{(Y_{0}(x_{t}))^{2}}{S_{0}(x_{t})},\qquad
q=d,s\,.}$ (26)

The first two of these formulae express explicitly the fact that combining on
the one hand $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ and $\varepsilon_{K}$ and
on the other hand $K_{L}\rightarrow\pi^{0}\nu\bar{\nu}$ and $\varepsilon_{K}$
allows within the SM to determine to a very good approximation the angle
$\beta$ independently of the value of $|V_{cb}|$ and $\gamma$. This is one of
the most interesting new results of our paper. The CKM independence of $R_{q}$
has been pointed out already in [8].

There are two first interesting consequences of these $|V_{cb}|$ independent
relations. First, combining the LHCb and Belle II results we find [7]

$(R_{7})_{\rm EXP}=(3.86\pm 1.48)\times 10^{3}.$ (27)

The central value is by a factor of 3.2 larger than the SM prediction (17) but
due to large error in the experimental $B^{+}\to K^{+}\nu\bar{\nu}$ branching
ratio the tension is only at $1.8\sigma$.

The second result is for $R_{s}$ in (26) for which one finds [9]

$(R_{s})_{\rm SM}=(2.042^{+0.083}_{-0.058})\times 10^{-10}{\rm
ps},\qquad(R_{s})_{\rm EXP}=(1.61^{+0.19}_{-0.17})\times 10^{-10}{\rm ps},$
(28)

that is a $2.1~{}\sigma$ tension.

While the ratios presented here are independent of $|V_{cb}|$ and $|V_{ub}|$
they depend on the hadronic parameters which enter in particular the ratios
involving $|\varepsilon_{K}|$, $\Delta M_{d}$ and $\Delta M_{s}$. This brings
us to the next important topic.

##  3 $|V_{cb}|(\beta,\gamma)$ from $|\varepsilon_{K}|$, $\Delta M_{d}$ and
$\Delta M_{s}$

When one day the angle $\gamma$ will be measured precisely we will be able to
answer the question whether all these SM predictions for the ratios in
question can be satisfied by the data simultaneously. We have just seen two
possible tensions. But in fact, we can do more already today. Namely, we can
ask the question, whether the SM can describe the very precise data for
$\varepsilon_{K}$, $\Delta M_{s}$ and $\Delta M_{d}$ with the same values of
$|V_{cb}|$, $\beta$ and $\gamma$. Also, theory in this three cases is in a
good shape due to progress from LQCD groups over many years and in the case of
$\varepsilon_{K}$ due to the progress in [13].

To answer this question we considered in our paper the following hadronic
parameters

$F_{B_{d}}\sqrt{\hat{B}_{B_{d}}}=214.0(39)\,{\rm MeV},\quad
F_{B_{s}}\sqrt{\hat{B}_{B_{s}}}=261.7(38)\,{\rm MeV}\,$ (29)

in the case of $\Delta M_{d,s}$ and $\hat{B}_{K}=0.7625(97)$ in the case of
$\varepsilon_{K}$. The latter one is the PDG average, while the values in (29)
are the averages of $2+1$ and $2+1+1$ LQCD results quoted by PDG obtained in
[9]. The result of this exercise is shown in Fig. 1.

We find that with the chosen hadronic parameters it is not possible to obtain
simultaneous agreement with the data on $\varepsilon_{K}$, $\Delta M_{d}$,
$\Delta M_{s}$ and $S_{\psi K_{S}}$, used to determine $\beta$, within the SM
independently of the value of $|V_{cb}|$ and $\gamma$. While these tensions
are still moderate they could hint for some NP at work. In this context a
precise measurement of $\gamma$ and the improvements on hadronic parameters
will be important. The consequences of these findings are discussed in detail
in [7]. But the message is clear. The ratios involving $\varepsilon_{K}$,
$\Delta M_{d}$, $\Delta M_{s}$, that is

$R_{9},\qquad R_{10},\qquad R_{11},\qquad R_{12},\qquad R_{d},\qquad R_{s},$
(30)

are not expected to agree simultaneously with the future data for the set of
hadronic parameters in (29). This lead us to propose a different strategy for
SM predictions for rare $K$ and $B$ decays summarized in the rest of this
writing.

Figure 1: The values of $|V_{cb}|$ extracted from $\varepsilon_{K}$, $\Delta
M_{d}$ and $\Delta M_{s}$ as functions of $\gamma$ for different values of
$\beta$. $|V_{cb}|$ extracted from $\Delta M_{d}$ is independent of $\beta$.

##  4 Improved SM Predictions for rare $K$ and $B$ Decays

While the ratios listed above are useful in the context of the testing of the
SM and in the search for NP, they are not as interesting as the observables
themselves. Therefore, assuming in addition no NP in $\varepsilon_{K}$,
$\Delta M_{d}$ and $\Delta M_{s}$ and in the mixing induced CP-asymmetry
$S_{\psi K_{S}}$, these ratios allowed to obtain $|V_{cb}|$ independent SM
predictions for a number of branching ratios [7]. As these four quark mixing
observables are very precisely measured and theoretically rather clean, the
resulting predictions obtained in this manner turned out to be the most
precise to date. We report on these predictions in Table 1.

Decay | Branching Ratio | Decay | Branching Ratio
---|---|---|---
$K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ | $(8.60\pm 0.42)\times 10^{-11}$ | $B_{s}\to\mu^{+}\mu^{-}$ | $(3.62^{+0.15}_{-0.10})\times 10^{-9}$
$K_{L}\rightarrow\pi^{0}\nu\bar{\nu}$ | $(2.94\pm 0.15)\times 10^{-11}$ | $B_{d}\to\mu^{+}\mu^{-}$ | $(0.99^{+0.05}_{-0.03})\ \times 10^{-10}$
$K_{S}\to\mu^{+}\mu^{-}$ | $(1.85\pm 0.10)\times 10^{-13}$ | $B^{+}\to K^{+}\nu\bar{\nu}$ | $(4.45\pm 0.62)\times 10^{-6}$
|  | $B^{0}\to K^{0*}\nu\bar{\nu}$ | $(9.70\pm 0.92)\times 10^{-6}$

Table 1:  Present most accurate $|V_{cb}|$ independent SM estimates of the
branching ratios considered in the paper. The $\gamma$ dependence is either
very small or absent. See [7] for details.

However, in view of the inconsistencies in different determinations of
$|V_{cb}|$ seen in Fig. 1 to obtain these results we did not perform on
purpose the usual global fit of observables which would require in addition
the input on $|V_{cb}|$ from tree-level decays in contradiction with the main
strategy of our paper. We concluded therefore that it would be a bad idea to
assume, as done in global fits, that NP is absent simultaneously in
$\varepsilon_{K}$, $\Delta M_{d}$, $\Delta M_{s}$ and $S_{\psi K_{S}}$.
Therefore, to obtain SM predictions for rare Kaon decays we only assumed the
absence of NP in $\varepsilon_{K}$ and $S_{\psi K_{S}}$. To obtain predictions
for $B_{s}\to\mu^{+}\mu^{-}$ and $B_{d}\to\mu^{+}\mu^{-}$ we assumed,
following [8], the absence of NP in $\Delta M_{s}$ and in ($\Delta
M_{d}$,$S_{\psi K_{S}}$), respectively but not simultaneously. In our view
this strategy for finding SM predictions is presently more powerful than any
global fit which would include decays like $B\to K\mu^{+}\mu^{-}$, $B\to
K^{*}\mu^{+}\mu^{-}$ and $B_{s}\to\phi\mu^{+}\mu^{-}$ that seem to exhibit
significant contributions from NP. Moreover, these local assumptions of the
absence of NP, with the goal to obtain SM predictions, are weaker than made in
global fits. But one should be aware of the fact that the SM predictions in
Table 1 that are based on hadronic parameters in (29) are likely not be true
simultaneously in view of the results in Fig. 1. But this is precisely what we
are after!

Figure 2: The correlations of
$\mathcal{B}(K^{+}\rightarrow\pi^{+}\nu\bar{\nu})$ with
$\overline{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})^{1.4}$ (left panel) and with
${\mathcal{B}}(B_{d}\to\mu^{+}\mu^{-})^{1.4}$ (right panel) as given in (7)
and (8), for different values of $\gamma$ within the SM. The ranges of
branching ratios correspond to $38\leq|V_{cb}|\times 10^{3}\leq 43$ and
$20^{\circ}\leq\beta\leq 24^{\circ}$. The SM area corresponds to the one in
Table 1. The gray area represents the present experimental situation.

##  5 Summary and Outlook

We have summarized the main results of our recent paper [7], where following
and extending significantly the strategies of [14, 15, 16, 17, 18, 8, 9, 19],
we have proposed to search for NP in rare Kaon and $B$-meson decays without
the necessity to choose the values of the CKM elements $|V_{cb}|$ and
$|V_{ub}|$, that introduce presently large parametric uncertainties in the
otherwise theoretically clean decays $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$,
$K_{L}\rightarrow\pi^{0}\nu\bar{\nu}$, $K_{S}\to\mu^{+}\mu^{-}$,
$B_{s,d}\to\mu^{+}\mu^{-}$, $B\to K(K^{*})\nu\bar{\nu}$, in the parameter
$\varepsilon_{K}$ and $\Delta M_{s,d}$. The 16 $|V_{cb}|$ independent ratios
of branching ratios and mixing observables, depending only on the angle
$\beta$ and $\gamma$ in the UT, will certainly play an important role in the
search for NP as soon as the branching ratios in question and the angle
$\gamma$ will be precisely measured. Several plots of these ratios and also of
correlations between various branching ratios can be found in our paper. We
present here only two correlations in Fig. 2 which could turn out in the
coming years to be the smoking guns of NP. For the set of hadronic parameters
in (29) we have also identified inconsistencies between the determinations of
$|V_{cb}|$ from $\varepsilon_{K}$, $\Delta M_{d}$ and $\Delta M_{s}$. This led
us to propose a novel strategy to obtain SM predictions for rare $K$ and $B$
decay branching ratios which is based on locality rather than globality as
explained in Section 3.

##  Acknowledgements

We would like to thank Pietro Baratella, Jean-Marc Gérard and Peter Stangl for
discussions. A.J.B acknowledges financial support from the Excellence Cluster
ORIGINS, funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation), Excellence Strategy, EXC-2094, 390783311. E.V. has been partially
funded by the Deutsche Forschungs-gemeinschaft (DFG, German Research
Foundation) under Germany’s Excellence Strategy- EXC-2094 - 390783311, by the
Collaborative Research Center SFB1258 and the BMBFgrant 05H18WOCA1 and thanks
the Munich Institute for Astro- and Particle Physics(MIAPP) for hospitality.

## References

  * [1] A. J. Buras, Gauge Theory of Weak Decays.  Cambridge University Press, 6, 2020.
  * [2] M. Bordone, N. Gubernari, D. van Dyk, and M. Jung, Heavy-Quark expansion for ${{\bar{B}}_{s}\rightarrow D^{(*)}_{s}}$ form factors and unitarity bounds beyond the ${SU(3)_{F}}$ limit, Eur. Phys. J. C 80 (2020), no. 4 347, [arXiv:1912.09335].
  * [3] M. Bordone, B. Capdevila, and P. Gambino, Three loop calculations and inclusive Vcb, Phys. Lett. B 822 (2021) 136679, [arXiv:2107.00604].
  * [4] G. Ricciardi, Theory: Semileptonic B Decays and $|V_{xb}|$ update, PoS BEAUTY2020 (2021) 031, [arXiv:2103.06099].
  * [5] Y. Aoki et al., FLAG Review 2021, arXiv:2111.09849.
  * [6] D. Leljak, B. Melić, and D. van Dyk, The $\overline{B}$ → $\pi$ form factors from QCD and their impact on $|V_{ub}|$, JHEP 07 (2021) 036, [arXiv:2102.07233].
  * [7] A. J. Buras and E. Venturini, Searching for New Physics in Rare $K$ and $B$ Decays without $|V_{cb}|$ and $|V_{ub}|$ Uncertainties, arXiv:2109.11032.
  * [8] A. J. Buras, Relations between $\Delta M_{s,d}$ and $B_{s,d}\to\mu^{+}\mu^{-}$ in models with minimal flavour violation, Phys. Lett. B566 (2003) 115–119, [hep-ph/0303060].
  * [9] C. Bobeth and A. J. Buras, Searching for New Physics with $\overline{\mathcal{B}}(B_{s,d}\to\mu\bar{\mu})/\Delta M_{s,d}$, Acta Phys. Polon. B 52 (2021) 1189, [arXiv:2104.09521].
  * [10] S. Descotes-Genon and P. Koppenburg, The CKM Parameters, Ann. Rev. Nucl. Part. Sci. 67 (2017) 97–127, [arXiv:1702.08834].
  * [11] A. Cerri, V. V. Gligorov, S. Malvezzi, J. Martin Camalich, and J. Zupan, Opportunities in Flavour Physics at the HL-LHC and HE-LHC, arXiv:1812.07638.
  * [12] LHCb Collaboration, R. Aaij et al., Simultaneous determination of CKM angle $\gamma$ and charm mixing parameters, arXiv:2110.02350.
  * [13] J. Brod, M. Gorbahn, and E. Stamou, Standard-Model Prediction of $\epsilon_{K}$ with Manifest Quark-Mixing Unitarity, Phys. Rev. Lett. 125 (2020), no. 17 171803, [arXiv:1911.06822].
  * [14] G. Buchalla and A. J. Buras, $\sin 2\beta$ from $K\to\pi\nu\bar{\nu}$, Phys. Lett. B333 (1994) 221–227, [hep-ph/9405259].
  * [15] A. J. Buras, Precise determinations of the CKM matrix from CP asymmetries in B decays and $K_{L}\rightarrow\pi^{0}\nu\bar{\nu}$, Phys. Lett. B 333 (1994) 476–483, [hep-ph/9405368].
  * [16] G. Buchalla and A. J. Buras, $K\to\pi\nu\bar{\nu}$ and high precision determinations of the CKM matrix, Phys. Rev. D54 (1996) 6782–6789, [hep-ph/9607447].
  * [17] A. J. Buras, F. Parodi, and A. Stocchi, The CKM matrix and the unitarity triangle: Another look, JHEP 0301 (2003) 029, [hep-ph/0207101].
  * [18] A. J. Buras, D. Buttazzo, J. Girrbach-Noe, and R. Knegjens, ${K}^{+}\to{\pi}^{+}\nu\overline{\nu}$ and ${K}_{L}\to{\pi}^{0}\nu\overline{\nu}$ in the Standard Model: status and perspectives, JHEP 11 (2015) 033, [arXiv:1503.02693].
  * [19] M. Blanke and A. J. Buras, Emerging $\Delta M_{d}$-Anomaly from Tree-Level Determinations of $|V_{cb}|$ and the Angle $\gamma$, Eur. Phys. J. C79 (2019), no. 2 159, [arXiv:1812.06963].