SM

    Standard Model
CPV

    CP violating
CPC

    CP conserving
BSM

    beyond the Standard Model
EFT

    Effective Field Theory
RHS

    right hand side
LHS

    left hand side
CL

    confidence level
GMFV

    General Minimal Flavor Violation
MFV

    Minimal Flavor Violation
FN

    Froggatt-Nielsen
FCNC

    flavor changing neutral current

Lessons from the LHCb measurement of CP violation in $B_{s}\to K^{+}K^{-}$

Yosef Nir1, Inbar Savoray1 and Yehonatan Viernik1

1Department of Particle Physics and Astrophysics,
Weizmann Institute of Science, Rehovot 7610001, Israel

<EMAIL_ADDRESS>
Abstract

The LHCb experiment measured the time-dependent CP asymmetries $C_{KK}$ and
$S_{KK}$ in $B_{s}\to K^{+}K^{-}$ decay. Combining with the corresponding CP
asymmetries $C_{\pi\pi}$ and $S_{\pi\pi}$ in $B\to\pi^{+}\pi^{-}$ decay, we
find that the size of $U$-spin breaking in this system is of order $20\%$.
Moreover, the data suggest that these effects are dominated by factorizable
contributions. We further study the constraints on new physics contributions
to $b\to u\bar{u}q$ ($q=s,d$). New physics that is minimally flavor violating
(MFV) cannot be distinguished from the Standard Model (SM) in these decays.
However, new physics that is not MFV can mimic large $U$-spin breaking.
Requiring that the $U$-spin breaking parameters remain below the size implied
by the data leads to a lower bound of $5-10$ TeV on the scale of generic new
physics. If the new physics is subject to the selection rules that follow from
the Froggatt-Nielsen (FN) mechanism or from General Minimal Flavor Violation
(GMFV), the bound is relaxed to 2 TeV.

##  I Introduction

CP violation in neutral meson decays has provided stringent tests of the SM
and has been a very effective probe of new physics, with examples such as the
measurement of CP violation in $K_{L}\to\pi\pi$ decays Christenson:1964fg
which led to the prediction that there is a third generation of fermions
Kobayashi:1973fv , and the measurement of time-dependent CP violation in $B\to
J/\psi K_{S}$ decays Aubert:2001nu ; Abe:2001xe  which proved that the
Kobayashi-Maskawa mechanism is the dominant source of the observed CP
violation, and excluded alternatives such as the superweak CP violation model
Wolfenstein:1974fd  and approximate CP Eyal:1998bk . For a recent review, see
Nir:2020mgy .

Recently, the LHCb collaboration provided the first observation of time-
dependent CP violation in $B_{s}$ decays LHCb:2020byh . The time dependent CP
asymmetry in $B_{s}\to K^{+}K^{-}$ decay is given by

$\displaystyle{\cal A}_{KK}^{s}(t)$ $\displaystyle\equiv$
$\displaystyle\frac{\Gamma_{\overline{B_{s}}\to
K^{+}K^{-}}(t)-\Gamma_{{B_{s}}\to K^{+}K^{-}}(t)}{\Gamma_{\overline{B_{s}}\to
K^{+}K^{-}}(t)+\Gamma_{{B_{s}}\to K^{+}K^{-}}(t)}$ $\displaystyle=$
$\displaystyle\frac{-C_{KK}\cos(\Delta m_{s}t)+S_{KK}\sin(\Delta
m_{s}t)}{\cosh(\Delta\Gamma_{s}t/2)+A_{KK}^{\Delta\Gamma}\sinh(\Delta\Gamma_{s}t/2)}.$

CP symmetry would imply $C_{KK}=S_{KK}=0$ and $A^{\Delta\Gamma}_{KK}=1$.
Combining the measurement reported in Ref. LHCb:2020byh  with the one reported
in Ref. Aaij:2018tfw , the ranges of the relevant parameters are

$\displaystyle C_{KK}$ $\displaystyle=$ $\displaystyle+0.172\pm 0.031,$
$\displaystyle S_{KK}$ $\displaystyle=$ $\displaystyle+0.139\pm 0.032,$
$\displaystyle A_{KK}^{\Delta\Gamma}$ $\displaystyle=$ $\displaystyle-0.897\pm
0.087.$ (2)

In this work we present the theoretical interpretation of this measurement,
and explore what can (or cannot) be learned from it.

Given that there are hadronic parameters playing a role, one can go in two
directions:

  * •

Assume the SM, and extract the hadronic parameters. Then we can learn about
$U$-spin breaking by comparing to the $U$-spin related parameters extracted
from $B_{d}\to\pi^{+}\pi^{-}$.

  * •

Use the consistency of the various measurements with approximate $U$-spin
symmetry to obtain constraints on new physics contributions to the $b\to
u\bar{u}s$ and $b\to u\bar{u}d$ decays.

For the latter study, we assume that, for processes that get contributions
from SM tree-level diagrams that are not CKM suppressed, the contributions
from new physics can be neglected. We allow, however, contributions of
a-priori arbitrary size and phase to flavor changing neutral current (FCNC)
processes and to CKM suppressed tree level processes. Concretely for $B_{s}$
physics, we assume that new physics contributions to $B_{s}\to J/\psi\phi$ are
negligible, while the size and the phase of new physics contributions to
$B_{s}-\overline{B_{s}}$ mixing and to $B_{s}\to K^{+}K^{-}$ are only
constrained by experimental data.

As concerns the assumption that $b\to c\bar{c}s$ decays are dominated by the
SM, note that $A(b\to c\bar{c}s)\propto G_{F}V_{cb}$. All $B$ and $B_{s}$
decay amplitudes (and, in particular, the leading semileptonic decays) are
suppressed by at least such a factor, or even stronger. If new physics
contributes significantly to this decay, then the whole consistency of the
data with the CKM picture seems accidental, which is very unlikely. Moreover,
for new physics to be comparable to the SM contribution, it needs to be
lighter than ${\cal O}({\rm TeV})$ and have tree level flavor changing
couplings to quarks. Thus, very likely it should have been directly observed
at the LHC, and affect other flavor observables significantly. In contrast,
the $b\to u\bar{u}s$ decay can get significant contributions from new physics
at the 10 TeV scale, and it is Cabibbo-suppressed compared to the charmless
semileptonic $b$ decays. The $b\to u\bar{u}d$ decay is an intermediate case,
as it has the same suppression as the charmless semileptonic $b$ decays, and
can get significant contributions from new physics lighter than ${\cal O}(5\
{\rm TeV})$.

Much – if not all – of our ability to extract lessons on the SM and on new
physics from $B_{s}\to K^{+}K^{-}$ relies on the $U$-spin relation with
$B_{d}\to\pi^{+}\pi^{-}$. Thus, in the following sections, we analyze side-by-
side the two processes. In Section II we introduce the necessary formalism and
notations. In Section III we present model independent considerations that
make our analysis as generic and as data-driven as possible. In Section IV we
use the experimental data to extract the relevant hadronic parameters, and in
Section V we find the size of $U$-spin breaking effects. In Section VI we put
bounds on the size of new physics contributions to the $B_{s}\to K^{+}K^{-}$
and to $B_{d}\to\pi^{+}\pi^{-}$ decays.

##  II Formalism and notations

We follow the formalism and notations presented in Ref. Zyla:2020zbs .

###  II.1 $B_{s}\to K^{+}K^{-}$

The neutral mass eigenstates of the $B_{s}-\overline{B_{s}}$ system are given
by

$|B_{sL,sH}\rangle=p_{s}|B_{s}\rangle\pm q_{s}|\overline{B_{s}}\rangle,$ (3)

with the normalization $|p_{s}|^{2}+|q_{s}|^{2}=1$. In terms of the dispersive
and absorptive parts of the $\Delta B=2$ transition amplitudes, we have

$\left(\frac{q_{s}}{p_{s}}\right)^{2}=\frac{M^{*}_{B_{s}\overline{B_{s}}}-(i/2)\Gamma^{*}_{B_{s}\overline{B_{s}}}}{M_{B_{s}\overline{B_{s}}}-(i/2)\Gamma_{B_{s}\overline{B_{s}}}}.$
(4)

We define the decay amplitudes,

$A^{s}_{f}=\langle f|{\cal H}|B_{s}\rangle,\ \ \
\overline{A}^{s}_{\bar{f}}=\langle\overline{f}|{\cal
H}|\overline{B_{s}}\rangle,$ (5)

and the parameter $\lambda_{f}^{s}$:

$\lambda_{f}^{s}\equiv\frac{q_{s}}{p_{s}}\frac{\overline{A}^{s}_{f}}{{A}^{s}_{f}}.$
(6)

We denote $A_{KK}\equiv A^{s}_{K^{+}K^{-}}$ and
$\lambda_{KK}\equiv\lambda^{s}_{KK}$. The parameters of Eq. (I) are given by

$C_{KK}=\frac{1-|\lambda_{KK}|^{2}}{1+|\lambda_{KK}|^{2}},\ \ \
S_{KK}=\frac{2{\cal I}m\lambda_{KK}}{1+|\lambda_{KK}|^{2}},\ \ \
A_{KK}^{\Delta\Gamma}=\frac{-2{\cal R}e\lambda_{KK}}{1+|\lambda_{KK}|^{2}}.$
(7)

Note that a consistency check is provided by
$(C_{KK})^{2}+(S_{KK})^{2}+(A_{KK}^{\Delta\Gamma})^{2}=1$.

The $B_{s}\to K^{+}K^{-}$ decay goes via the $b\to u\bar{u}s$ quark
transition. It depends on the following CKM combinations:

$\lambda^{j}_{bs}\equiv V_{jb}^{*}V_{js}\ \ \ (j=u,c,t).$ (8)

Within the SM, one can write the decay amplitudes as follows:

$\displaystyle A_{KK}^{\rm SM}$ $\displaystyle=$
$\displaystyle\hat{P}^{s}\lambda^{c}_{bs}+\hat{T}^{s}\lambda^{u}_{bs}=\hat{P}^{s}\lambda^{c}_{bs}(1+r_{s}\lambda^{u}_{bs}/\lambda^{c}_{bs}),$
$\displaystyle\overline{A}_{KK}^{\rm SM}$ $\displaystyle=$
$\displaystyle\hat{P}^{s}\lambda^{c*}_{bs}+\hat{T}_{s}\lambda^{u*}_{bs}=\hat{P}^{s}\lambda^{c*}_{bs}(1+r_{s}\lambda^{u*}_{bs}/\lambda^{c*}_{bs}).$
(9)

A few comments are in place regarding Eqs. (II.1):

  * •

Within the SM, the only source of CP violation is the CKM matrix. Hence, only
the $\lambda^{j}_{bs}$ factors are complex conjugated between $A_{KK}^{\rm
SM}$ and $\overline{A}_{KK}^{\rm SM}$.

  * •

The $r_{s}$ factor is, in general, complex. Its phase is a so-called strong
phase, which is the same in $A_{KK}$ and $\overline{A}_{KK}$.

  * •

$\hat{P}^{s}$, $\hat{T}^{s}$ (and $r_{s}$) include in them not only QCD matrix
elements, but also electroweak parameters which are neither flavor dependent
nor CP violating, such as $G_{F}$.

  * •

The $b\to u\bar{u}s$ transition has SM tree and penguin contributions. We
define $T^{s}$ as the tree contribution, and $P^{s}_{j}$ as the penguin
contribution with an intermediate $j$-quark. Then, using CKM unitarity, we
have

$\hat{T}^{s}=T^{s}+P^{s}_{u}-P^{s}_{t},\ \ \ \hat{P}^{s}=P^{s}_{c}-P^{s}_{t},\
\ \ r_{s}\equiv\hat{T}^{s}/\hat{P}^{s}.$ (10)

###  II.2 $B_{d}\to\pi^{+}\pi^{-}$

We define $q_{d}$ and $p_{d}$ for the $B_{d}-\overline{B_{d}}$ system in a
similar way to Eq. (3), and $A^{d}_{f}$, $\overline{A}^{d}_{f}$ and
$\lambda^{d}_{f}$ in a similar way to Eqs. (5) and (6). We denote
$A_{\pi\pi}\equiv A^{d}_{\pi^{+}\pi^{-}}$ and
$\lambda_{\pi\pi}\equiv\lambda^{d}_{\pi\pi}$. The parameters of the time-
dependent CP asymmetry in $B\to\pi^{+}\pi^{-}$ are given by

$C_{\pi\pi}=\frac{1-|\lambda_{\pi\pi}|^{2}}{1+|\lambda_{\pi\pi}|^{2}},\ \ \
S_{\pi\pi}=\frac{2{\cal I}m\lambda_{\pi\pi}}{1+|\lambda_{\pi\pi}|^{2}},\ \ \
A_{\pi\pi}^{\Delta\Gamma}=\frac{-2{\cal
R}e\lambda_{\pi\pi}}{1+|\lambda_{\pi\pi}|^{2}}.$ (11)

The $B_{d}\to\pi^{+}\pi^{-}$ decay goes via the $b\to u\bar{u}d$ quark
transition. It depends on the following CKM combinations:

$\lambda^{j}_{bd}\equiv V_{jb}^{*}V_{jd}\ \ \ (j=u,c,t).$ (12)

Within the SM, one can write the decay amplitudes as follows:

$\displaystyle A_{\pi\pi}^{\rm SM}$ $\displaystyle=$
$\displaystyle\hat{P}^{d}\lambda^{c}_{bd}+\hat{T}^{d}\lambda^{u}_{bd}=\hat{P}^{d}\lambda^{c}_{bd}(1+r_{d}\lambda^{u}_{bd}/\lambda^{c}_{bd}),$
$\displaystyle\overline{A}_{\pi\pi}^{\rm SM}$ $\displaystyle=$
$\displaystyle\hat{P}^{d}\lambda^{c*}_{bd}+\hat{T}^{d}\lambda^{u*}_{bd}=\hat{P}^{d}\lambda^{c*}_{bd}(1+r_{d}\lambda^{u*}_{bd}/\lambda^{c*}_{bd}),$
(13)

where $r_{d}$ is, in general, complex. The $b\to u\bar{u}d$ transition has SM
tree and penguin contributions. We define $T^{d}$ as the tree contribution,
and $P^{d}_{j}$ as the penguin contribution with intermediate $j$-quark. Then,
using CKM unitarity, we have

$\hat{T}^{d}=T^{d}+P^{d}_{u}-P^{d}_{t},\ \ \ \hat{P}^{d}=P^{d}_{c}-P^{d}_{t},\
\ \ r_{d}\equiv\hat{T}^{d}/\hat{P}^{d}.$ (14)

Note again that $\hat{P}^{d}$, $\hat{T}^{d}$ (and $r_{d}$) include in them not
only QCD matrix elements, but also flavor-universal CP conserving electroweak
parameters.

##  III Model-independent considerations

###  III.1 $B_{s}\to K^{+}K^{-}$

The CP asymmetry in wrong-sign semileptonic $B_{s}$ decays is given by

$\displaystyle{\cal A}_{\rm SL}^{s}$ $\displaystyle\equiv$
$\displaystyle\frac{d\Gamma/dt[\overline{B_{s}}(t)\to\ell^{+}X]-d\Gamma/dt[B_{s}(t)\to\ell^{-}X]}{d\Gamma/dt[\overline{B_{s}}(t)\to\ell^{+}X]+d\Gamma/dt[B_{s}(t)\to\ell^{-}X]}$
(15) $\displaystyle=$
$\displaystyle\frac{1-|q_{s}/p_{s}|^{4}}{1+|q_{s}/p_{s}|^{4}}.$ (16)

The experimental world average of ${\cal A}_{\rm SL}^{s}$ is given by
Amhis:2019ckw

${\cal A}_{\rm SL}^{s}=(-0.6\pm 2.8)\times 10^{-3},$ (17)

implying

$|q_{s}/p_{s}|=1.0003\pm 0.0014.$ (18)

For our purposes we can then approximate $|q_{s}/p_{s}|=1$ and use

$\frac{q_{s}}{p_{s}}=\frac{M^{*}_{B_{s}\overline{B_{s}}}}{|M_{B_{s}\overline{B_{s}}}|}.$
(19)

The time dependent CP asymmetry in $B_{s}\to J/\psi\phi$ decay is given by

$\displaystyle{\cal A}_{\psi\phi}^{s}(t)$ $\displaystyle\equiv$
$\displaystyle\frac{\Gamma_{\overline{B_{s}}\to
J/\psi\phi}(t)-\Gamma_{{B_{s}}\to J/\psi\phi}(t)}{\Gamma_{\overline{B_{s}}\to
J/\psi\phi}(t)+\Gamma_{{B_{s}}\to J/\psi\phi}(t)}$ $\displaystyle=$
$\displaystyle\frac{-C_{\psi\phi}\cos(\Delta m_{s}t)+S_{\psi\phi}\sin(\Delta
m_{s}t)}{\cosh(\Delta\Gamma_{s}t/2)+A_{\psi\phi}^{\Delta\Gamma}\sinh(\Delta\Gamma_{s}t/2)}.$

The $C_{\psi\phi}$, $S_{\psi\phi}$ and $A^{\Delta\Gamma}_{\psi\phi}$
parameters depend on $\lambda_{\psi\phi}$ in a way similar to Eq. (7).

Within the SM, the $b\to c\bar{c}s$ transition has tree and penguin
contributions. In a way similar to our analysis of $B_{s}\to K^{+}K^{-}$, we
can write

$A_{\psi\phi}=\hat{T}^{s}_{\psi\phi}\lambda^{c}_{bs}(1+r_{\psi\phi}\lambda^{u}_{bs}/\lambda^{c}_{bs}).$
(21)

Here, however, the second term in parenthesis is both CKM and loop suppressed,
and can thus be neglected. We further assume that new physics effects on decay
processes with SM tree level contributions that are not CKM suppressed are
negligible. We thus obtain:

$\frac{\overline{A}_{\psi\phi}}{A_{\psi\phi}}\simeq\frac{\lambda^{c*}_{bs}}{\lambda^{c}_{bs}}.$
(22)

Eqs. (22) and (19) imply that $\lambda_{\psi\phi}$ is a pure phase:

$\lambda_{\psi\phi}=\frac{M^{*}_{B_{s}\overline{B_{s}}}}{|M_{B_{s}\overline{B_{s}}}|}\frac{\lambda^{c*}_{bs}}{\lambda^{c}_{bs}}\equiv
e^{i\phi_{s}}.$ (23)

Measurements of various CP asymmetries in $B_{s}$ decays via $b\to c\bar{c}s$
lead to the following world average for $\phi_{s}$ Amhis:2019ckw :

$\phi_{s}=0.051\pm 0.023.$ (24)

Eqs. (23) and (19) allow us to express $\lambda_{KK}$ as follows:

$\lambda_{KK}=\frac{M^{*}_{B_{s}\overline{B_{s}}}}{|M_{B_{s}\overline{B_{s}}}|}\frac{\overline{A}_{KK}}{A_{KK}}=e^{i\phi_{s}}\frac{\lambda^{c}_{bs}}{\lambda^{c*}_{bs}}\frac{\overline{A}_{KK}}{A_{KK}}.$
(25)

Using Eq. (I), we obtain

$\displaystyle|\lambda_{KK}|^{2}$ $\displaystyle=$
$\displaystyle|\overline{A}_{KK}/{A}_{KK}|^{2}=0.71\pm 0.05,$
$\displaystyle{\cal I}m\lambda_{KK}$ $\displaystyle=$
$\displaystyle\cos\phi_{s}{\cal
I}m\left(\frac{\lambda^{c}_{bs}}{\lambda^{c*}_{bs}}\frac{\overline{A}_{KK}}{A_{KK}}\right)+\sin\phi_{s}{\cal
R}e\left(\frac{\lambda^{c}_{bs}}{\lambda^{c*}_{bs}}\frac{\overline{A}_{KK}}{A_{KK}}\right)=0.119\pm
0.028.$ (26)

###  III.2 $B_{d}\to\pi^{+}\pi^{-}$

The experimental world average of ${\cal A}_{\rm SL}^{d}$ is given by
Zyla:2020zbs

${\cal A}_{\rm SL}^{d}=(-2.0\pm 1.6)\times 10^{-3},$ (27)

implying

$|q_{d}/p_{d}|=1.0010\pm 0.0008.$ (28)

For our purposes we can then approximate $|q_{d}/p_{d}|=1$ and use

$\frac{q_{d}}{p_{d}}=\frac{M^{*}_{B_{d}\overline{B_{d}}}}{|M_{B_{d}\overline{B_{d}}}|}.$
(29)

Similarly to $B_{s}\to J/\psi\phi$, the $B\to J/\psi K_{S}$ decay is dominated
by a single CKM combination and, furthermore, new physics contributions can be
safely assumed to be negligible. Consequently,

$\frac{\overline{A}_{\psi K_{S}}}{A_{\psi
K_{S}}}\simeq\frac{\lambda^{c*}_{bd}}{\lambda^{c}_{bd}}.$ (30)

Eqs. (30) and (29) imply that $\lambda_{\psi K_{S}}$ is a pure phase:

$\lambda_{\psi
K_{S}}=\frac{M^{*}_{B_{d}\overline{B_{d}}}}{|M_{B_{d}\overline{B_{d}}}|}\frac{\lambda^{c*}_{bd}}{\lambda^{c}_{bd}}\equiv
e^{i\phi_{d}}.$ (31)

Measurements of the CP asymmetries in $B_{d}\to J/\psi K_{S}$ lead to the
following world average for $\phi_{d}$ Zyla:2020zbs ; Barel:2020jvf :

$\phi_{d}=-0.768\pm 0.026.$ (32)

Eqs. (31) and (29) allow us to express $\lambda_{\pi\pi}$ as follows:

$\lambda_{\pi\pi}=\frac{M^{*}_{B_{d}\overline{B_{d}}}}{|M_{B_{d}\overline{B_{d}}}|}\frac{\overline{A}_{\pi\pi}}{A_{\pi\pi}}=e^{i\phi_{d}}\frac{\lambda^{c}_{bd}}{\lambda^{c*}_{bd}}\frac{\overline{A}_{\pi\pi}}{A_{\pi}}.$
(33)

Using the experimental values Zyla:2020zbs

$\displaystyle C_{\pi\pi}$ $\displaystyle=$ $\displaystyle-0.32\pm 0.04,$
$\displaystyle S_{\pi\pi}$ $\displaystyle=$ $\displaystyle-0.65\pm 0.04,$ (34)

we obtain

$\displaystyle|\lambda_{\pi\pi}|^{2}$ $\displaystyle=$
$\displaystyle|\overline{A}_{\pi\pi}/{A}_{\pi\pi}|^{2}=1.94\pm 0.17,$
$\displaystyle{\cal I}m\lambda_{\pi\pi}$ $\displaystyle=$
$\displaystyle\cos\phi_{d}{\cal
I}m\left(\frac{\lambda^{c}_{bd}}{\lambda^{c*}_{bd}}\frac{\overline{A}_{\pi\pi}}{A_{\pi\pi}}\right)+\sin\phi_{d}{\cal
R}e\left(\frac{\lambda^{c}_{bd}}{\lambda^{c*}_{bd}}\frac{\overline{A}_{\pi\pi}}{A_{\pi\pi}}\right)=-0.96\pm
0.06.$ (35)

##  IV The Standard Model

The values of the CKM parameters that play a role in the $B_{s}\to K^{+}K^{-}$
and $B_{d}\to\pi^{+}\pi^{-}$ decays are known from tree level decays
Zyla:2020zbs ; LHCb:2020kho :

$\displaystyle R_{uc}^{bs}$ $\displaystyle\equiv$
$\displaystyle\left|\lambda^{u}_{bs}/\lambda^{c}_{bs}\right|=0.021\pm 0.001,$
$\displaystyle R_{uc}^{bd}$ $\displaystyle\equiv$
$\displaystyle\left|\lambda^{u}_{bd}/\lambda^{c}_{bd}\right|=0.410\pm 0.025,$
$\displaystyle\gamma$ $\displaystyle\equiv$
$\displaystyle\arg\left(-\lambda^{u}_{bd}/\lambda^{c}_{bd}\right)=(67\pm
4)\degree.$ (36)

Note that, using CKM unitarity relations, we obtain

$\gamma=\arg\left(\lambda^{u}_{bs}/\lambda^{c}_{bs}\right)+{\cal
O}(\lambda^{4}),$ (37)

Neglecting the ${\cal O}(\lambda^{4})$ correction,we can rewrite the decay
amplitudes,

$\displaystyle A_{KK}$ $\displaystyle=$
$\displaystyle\hat{P}^{s}\lambda^{c}_{bs}\left[1+r_{s}R^{bs}_{uc}e^{+i\gamma}\right],$
$\displaystyle A_{\pi\pi}$ $\displaystyle=$
$\displaystyle\hat{P}^{d}\lambda^{c}_{bd}\left[1-r_{d}R^{bd}_{uc}e^{+i\gamma}\right],$
(38)

and the $\lambda_{f}$ parameters,

$\displaystyle\lambda_{KK}$ $\displaystyle=$ $\displaystyle
e^{i\phi_{s}}\left[\frac{1+r_{s}R^{bs}_{uc}e^{-i\gamma}}{1+r_{s}R^{bs}_{uc}e^{+i\gamma}}\right],$
$\displaystyle\lambda_{\pi\pi}$ $\displaystyle=$ $\displaystyle
e^{i\phi_{d}}\left[\frac{1-r_{d}R^{bd}_{uc}e^{-i\gamma}}{1-r_{d}R^{bd}_{uc}e^{+i\gamma}}\right].$
(39)

Defining

$\delta_{q}\equiv{\rm arg}(r_{q}),$ (40)

we obtain, for the CP asymmetries,

$\displaystyle C_{KK}$ $\displaystyle=$
$\displaystyle\frac{-2|r_{s}|R^{bs}_{uc}\sin\delta_{s}\sin\gamma}{1+2|r_{s}|R^{bs}_{uc}\cos\delta_{s}\cos\gamma+(|r_{s}|R^{bs}_{uc})^{2}},$
$\displaystyle C_{\pi\pi}$ $\displaystyle=$
$\displaystyle\frac{2|r_{d}|R^{bd}_{uc}\sin\delta_{d}\sin\gamma}{1-2|r_{d}|R^{bd}_{uc}\cos\delta_{d}\cos\gamma+(|r_{d}|R^{bd}_{uc})^{2}},$
(41) $\displaystyle S_{KK}$ $\displaystyle=$
$\displaystyle\frac{\sin\phi_{s}+2|r_{s}|R^{bs}_{uc}\cos\delta_{s}\sin(\phi_{s}-\gamma)+(|r_{s}|R^{bs}_{uc})^{2}\sin(\phi_{s}-2\gamma)}{1+2|r_{s}|R^{bs}_{uc}\cos\delta_{s}\cos\gamma+(|r_{s}|R^{bs}_{uc})^{2}},$
$\displaystyle S_{\pi\pi}$ $\displaystyle=$
$\displaystyle\frac{\sin\phi_{d}-2|r_{d}|R^{bd}_{uc}\cos\delta_{d}\sin(\phi_{d}-\gamma)+(|r_{d}|R^{bd}_{uc})^{2}\sin(\phi_{d}-2\gamma)}{1-2|r_{d}|R^{bd}_{uc}\cos\delta_{d}\cos\gamma+(|r_{d}|R^{bd}_{uc})^{2}},$
(42)

and for the decay rates (averaged over $B_{q}$ and $\overline{B_{q}}$),

$\displaystyle\Gamma(B_{s}\to K^{+}K^{-})$ $\displaystyle=$
$\displaystyle|\hat{P}^{s}|^{2}|\lambda^{c}_{bs}|^{2}\left[1+(|r_{s}|R_{uc}^{bs})^{2}+2|r_{s}|R_{uc}^{bs}\cos\delta_{s}\cos\gamma\right],$
$\displaystyle\Gamma(B_{d}\to\pi^{+}\pi^{-})$ $\displaystyle=$
$\displaystyle|\hat{P}^{d}|^{2}|\lambda^{c}_{bd}|^{2}\left[1+(|r_{d}|R_{uc}^{bd})^{2}-2|r_{d}|R_{uc}^{bd}\cos\delta_{d}\cos\gamma\right],$
(43)

Using the experimental values of the five observables $C_{KK}$, $S_{KK}$,
$C_{\pi\pi}$, $S_{\pi\pi}$ and

$R_{\Gamma}\equiv\frac{\Gamma(B_{s}\to
K^{+}K^{-})}{\Gamma(B_{d}\to\pi^{+}\pi^{-})}=\frac{{\rm BR}(B_{s}\to
K^{+}K^{-})}{{\rm BR}(B_{d}\to\pi^{+}\pi^{-})}\frac{\tau_{d}}{\tau_{s}}=5.2\pm
0.5,$ (44)

we can obtain the values of the five hadronic parameters. Solving for the
central values of the experimental observables, we obtain:

$\displaystyle|r_{s}|=4.96\pm 0.80\,,\ \ \ \cos\delta_{s}=-0.49\pm 0.14\
(\sin\delta_{s}<0)\,,$ $\displaystyle|r_{d}|=4.64\pm 0.45\,,\ \ \
\cos\delta_{d}=-0.84\pm 0.04\ (\sin\delta_{d}<0)\,,$
$\displaystyle|\hat{P}_{s}/\hat{P}_{d}|=1.26\pm 0.11.$ (45)

Note that, since $A^{\Delta\Gamma}_{\pi\pi}$ is not measured, there is a
discrete ambiguity in $|r_{d}|$ and $\cos\delta_{d}$. We present the solution
that corresponds to a negative $A^{\Delta\Gamma}_{\pi\pi}$, which gives
$\lvert r_{d}\rvert$ close to $\lvert r_{s}\rvert$. This is not the case if
$A^{\Delta\Gamma}_{\pi\pi}$ is positive ($|r_{d}|=0.43$,
$\cos\delta_{d}=+0.10$). Once $|r_{q}|$ and $\cos\delta_{q}$ are fixed, the
sign of $\delta_{s}$ and $\delta_{d}$ are determined from the sign of $C_{KK}$
and $C_{\pi\pi}$, respectively, as they are the only observables explicitly
sensitive to $\sin\delta_{q}$ . The $1\sigma$ allowed ranges in the $\lvert
r_{q}\rvert-\cos\delta_{q}$ plane are shown in Fig. 1.

Figure 1: The $1\sigma$ allowed ranges for $\lvert r_{q}\rvert$ vs.
$\cos\delta_{q}$: (purple) $q=s$, from $S_{KK}$ and $C_{KK}$; (pink) $q=s$,
from $A^{\Delta\Gamma}_{KK}$ and $C_{KK}$; (yellow) $q=d$, from $S_{\pi\pi}$
and $C_{\pi\pi}$, for negative $A_{\pi\pi}^{\Delta\Gamma}$; (orange) $q=d$,
from $S_{\pi\pi}$ and $C_{\pi\pi}$, for positive $A_{\pi\pi}^{\Delta\Gamma}$.

A lesson to be drawn from these results is that, while in the
$B_{d}\to\pi^{+}\pi^{-}$ decay the $\hat{T}^{d}$ and $\hat{P}^{d}$
contributions are comparable, $|r_{d}|R^{bd}_{uc}\approx 1.9$, the $B_{s}\to
K^{+}K^{-}$ decay is dominated by the $\hat{P}^{s}$ contribution,
$|r_{s}|R^{bs}_{uc}\approx 0.10$.

The fact that $|r_{s}/r_{d}|$ and $|\hat{P}^{s}/\hat{P}^{d}|$ are close to
one, suggests that $U$-spin breaking effects are small. This result leads us
to investigate in detail the $U$-spin breaking in this system.

##  V $U$-spin breaking

The $B_{s}\to K^{+}K^{-}$ and $B_{d}\to\pi^{+}\pi^{-}$ are related by
$U$-spin, the $SU(2)$ symmetry under which $s$ and $d$ form a doublet.
Concretely, $U$-spin requires

$P\equiv\hat{P}^{s}=\hat{P}^{d},\ \ \ T\equiv\hat{T}^{s}=\hat{T}^{d}.$ (46)

$U$-spin breaking effects are expected to be of ${\cal O}(m_{s}/\Lambda_{\rm
QCD})\sim 0.3$. Indeed, the best fit values that we obtained,
$|\hat{P}^{s}/\hat{P}^{d}|=1.26$ and $|r_{s}/r_{d}|=1.07$ are consistent with
this expectation.

It is interesting to note that the breaking effect in
$|\hat{P}^{s}/\hat{P}^{d}|$ is much larger than the one in $|r_{s}/r_{d}|$.
The latter is, in fact, a double ratio:

$\frac{r_{s}}{r_{d}}=\frac{\hat{T}_{s}/\hat{T}_{d}}{\hat{P}_{s}/\hat{P}_{d}}.$
(47)

Factorizable contributions cancel to a good approximation (roughly,
$m_{s}/m_{b}$) in the double ratio, so the deviation from unity is affected
mainly by non-factorizable contributions Fleischer:1999pa ; Ali:2007ff ;
Beneke:2003zw ; He:2017fln . The data support the assumption that the non-
factorizable contributions are small.

To incorporate first-order $U$-spin breaking, we write

$\displaystyle A_{KK}$ $\displaystyle=$
$\displaystyle(P+p)\lambda^{c}_{bs}+(T+t)\lambda^{u}_{bs},$ $\displaystyle
A_{\pi\pi}$ $\displaystyle=$
$\displaystyle(P-p)\lambda^{c}_{bd}+(T-t)\lambda^{u}_{bd},$ (48)

where we assume small breaking, i.e. $|p/P|\ll 1$ and $|t/T|\ll 1$. Without
loss of generality, we can choose $P$ to be real. Then there are seven
hadronic parameters. Given the five observables that we use, $C_{KK}$,
$S_{KK}$, $C_{\pi\pi}$, $S_{\pi\pi}$ and $R_{\Gamma}$, we can extract five of
these seven parameters. In principle, we can extract $P$ by considering the
individual decay rates, rather than their ratio, but this has no significance
for our analysis.

When we consider the $U$-spin breaking effects to first order only, the list
of five parameters consists of two $U$-spin conserving ones,

$\displaystyle\hat{T}_{r}$ $\displaystyle\equiv$ $\displaystyle{\cal
R}e(T/P)=-3.0\pm 0.5,$ $\displaystyle\hat{T}_{i}$ $\displaystyle\equiv$
$\displaystyle{\cal I}m(T/P)=\pm 3.8\pm 0.6,$ (49)

and three $U$-spin breaking ones,

$\displaystyle\hat{t}_{r}$ $\displaystyle\equiv$ $\displaystyle{\cal
R}e(t/T)=+0.14\pm 0.07,$ $\displaystyle\hat{p}_{r}$ $\displaystyle\equiv$
$\displaystyle{\cal R}e(p/P)=+0.11\pm 0.04,$ $\displaystyle\hat{\Delta}_{i}$
$\displaystyle\equiv$ $\displaystyle{\cal I}m(t/T-p/P)=\mp 0.21\pm 0.08.$ (50)

The fourth $U$-spin breaking parameter, $\hat{\Sigma}_{i}\equiv{\cal
I}m(t/T+p/P)$, does not play a role at first order in these observables. We
learn that the $U$-spin breaking effects are of order $0.1-0.2$, so that
$U$-spin is a good approximate symmetry of this system.

We note that

$\frac{\hat{t}_{r}-\hat{p}_{r}}{\hat{t}_{r}+\hat{p}_{r}}\approx 0.12\ll 1.$
(51)

This is another way of observing that the $U$-spin breaking effects in
$\hat{P}^{s}/\hat{P}^{d}$ and much larger than those in $r_{s}/r_{d}$,
consistent with the assumption that the leading correction comes from the
$f_{K}/f_{\pi}$ factor in the factorizable contributions, and that the non-
factorizable contributions are small.

A particularly interesting combination of parameters is the following:

$R_{C\Gamma}\equiv\frac{C_{KK}\Gamma(B_{s}\to
K^{+}K^{-})}{C_{\pi\pi}\Gamma(B_{d}\to\pi^{+}\pi^{-})}.$ (52)

It has been noted by Gronau, that in the $U$-spin limit, the following
relation holds Gronau:2000zy ; Gronau:2013mda :

$\left[R_{C\Gamma}\right]_{U-{\rm spin}}=-1.$ (53)

Somewhat surprisingly, the experimental data show a strong violation of this
relation:

$\left[R_{C\Gamma}\right]_{\rm exp}=-2.8\pm 0.5.$ (54)

With our parametrization, and to first order in the $U$-spin breaking
parameters

$R_{C\Gamma}=-[1+2\hat{t}_{r}+2\hat{p}_{r}+2(\hat{T}_{R}/\hat{T}_{i})\hat{\Delta}_{i}].$
(55)

We learn that the large deviation of $R_{C\Gamma}$ from the $U$-spin limit
prediction is a consequence of (twice) three breaking parameters that add up
in the same direction.

##  VI Beyond the SM

It is not a simple task to establish, or to constrain, new physics
contributions to CP asymmetries (and even more so to decay rates) in decay
modes where there are two SM contributions that differ in their weak and
strong phases. Without any extra information, a SM interpretation of the
measurements when new physics actually plays a role is (almost) always
possible.

In the case of $B_{s}\to K^{+}K^{-}$, there is, however, extra information,
which is provided by its (approximate) $U$-spin relation with
$B_{d}\to\pi^{+}\pi^{-}$ Barel:2020jvf ; Fleischer:2010ib ; Ciuchini:2012gd ;
Fleischer:2016jbf . In the presence of new physics, a SM analysis would lead
to wrong values of $\hat{P}^{q}$ and $\hat{T}^{q}$. In particular, the new
physics contributions might mimic $U$-spin breaking effects. Thus, the fact
that a SM analysis of the experimental data led to small $U$-spin breaking
effects suggests that we can constrain the new physics by demanding that it
does not lead to spurious $U$-spin breaking that is larger than observed. Such
constraints assume that there is no cancellation between new physics effects
and genuine $U$-spin breaking effects.

To see how new physics can mimic $U$-spin breaking, we now present our
formalism for including new physics. Any contributions to the four decay
amplitudes of interest can be written as follows:

$\displaystyle A_{KK}$ $\displaystyle=$
$\displaystyle\hat{P}^{s}\lambda^{c}_{bs}+\hat{T}^{s}\lambda^{u}_{bs}=\hat{P}^{s}\lambda^{c}_{bs}(1+r_{s}\lambda^{u}_{bs}/\lambda^{c}_{bs}),$
$\displaystyle\overline{A}_{KK}$ $\displaystyle=$
$\displaystyle\hat{P}^{s}\lambda^{c*}_{bs}+\hat{T}^{s}\lambda^{u*}_{bs}=\hat{P}^{s}\lambda^{c*}_{bs}(1+r_{s}\lambda^{u*}_{bs}/\lambda^{c*}_{bs}),$
$\displaystyle A_{\pi\pi}$ $\displaystyle=$
$\displaystyle\hat{P}^{d}\lambda^{c}_{bd}+\hat{T}^{d}\lambda^{u}_{bd}=\hat{P}^{d}\lambda^{c}_{bd}(1+r_{d}\lambda^{u}_{bd}/\lambda^{c}_{bd}),$
$\displaystyle\overline{A}_{\pi\pi}$ $\displaystyle=$
$\displaystyle\hat{P}^{d}\lambda^{c*}_{bd}+\hat{T}^{d}\lambda^{u*}_{bd}=\hat{P}^{d}\lambda^{c*}_{bd}(1+r_{d}\lambda^{u*}_{bd}/\lambda^{c*}_{bd}).$
(56)

Distinguishing the SM and new physics contributions, we write:

$\displaystyle A_{KK}$ $\displaystyle=$ $\displaystyle\hat{P}^{s}_{\rm
SM}\lambda^{c}_{bs}(1+r^{s}_{\rm SM}R^{bs}_{uc}e^{+i\gamma}+r^{s}_{\rm
NP}e^{+i\theta_{s}}),$ $\displaystyle A_{\pi\pi}$ $\displaystyle=$
$\displaystyle\hat{P}^{d}_{\rm SM}\lambda^{c}_{bd}(1-r^{d}_{\rm
SM}R^{bd}_{uc}e^{+i\gamma}+r^{d}_{\rm NP}e^{+i\theta_{d}}),$ (57)

where, in $\overline{A}_{KK}$ and $\overline{A}_{\pi\pi}$, $e^{i\gamma}$ and
$e^{i\theta_{q}}$ are complex-conjugated. Without loss of generality, we can
rewrite these amplitudes as follows:

$\displaystyle A_{KK}$ $\displaystyle=$ $\displaystyle\hat{P}^{s}_{\rm
SM}\lambda^{c}_{bs}\left[1+a_{s}r^{s}_{\rm
NP}+R^{bs}_{uc}e^{+i\gamma}\left(r^{s}_{\rm SM}+b_{s}r^{s}_{\rm
NP}\right)\right],$ $\displaystyle A_{\pi\pi}$ $\displaystyle=$
$\displaystyle\hat{P}^{d}_{\rm SM}\lambda^{c}_{bd}\left[1+a_{d}r^{d}_{\rm
NP}-R^{bd}_{uc}e^{+i\gamma}\left(r^{d}_{\rm SM}+b_{d}r^{d}_{\rm
NP}\right)\right],$ (58)

where

$\displaystyle a_{s}$ $\displaystyle=$
$\displaystyle\frac{\sin(\gamma-\theta_{s})}{\sin\gamma},\ \ \
b_{s}=\frac{\sin\theta_{s}}{R^{bs}_{uc}\sin\gamma},$ $\displaystyle a_{d}$
$\displaystyle=$ $\displaystyle\frac{\sin(\gamma-\theta_{d})}{\sin\gamma},\ \
\ b_{d}=-\frac{\sin\theta_{d}}{R^{bd}_{uc}\sin\gamma}.$ (59)

Matching to the parametrization of Eqs. (IV), we have

$\displaystyle\hat{P}^{s}=\hat{P}^{s}_{\rm SM}(1+a_{s}r^{s}_{\rm NP}),\ \ \
r_{s}$ $\displaystyle=$ $\displaystyle\frac{r^{s}_{\rm SM}+b_{s}r^{s}_{\rm
NP}}{1+a_{s}r^{s}_{\rm NP}},$ $\displaystyle\hat{P}^{d}=\hat{P}^{d}_{\rm
SM}(1+a_{d}r^{d}_{\rm NP}),\ \ \ r_{d}$ $\displaystyle=$
$\displaystyle\frac{r^{d}_{\rm SM}+b_{d}r^{d}_{\rm NP}}{1+a_{d}r^{d}_{\rm
NP}}.$ (60)

For our purposes, We assume that $|r^{q}_{\rm NP}|\ll 1,|r^{q}_{\rm SM}|$, and
expand the following two ratios to first order in $r^{q}_{\rm NP}$:

$\displaystyle\frac{\hat{P}^{s}}{\hat{P}^{d}}$ $\displaystyle=$
$\displaystyle\frac{\hat{P}^{s}_{\rm SM}}{\hat{P}^{d}_{\rm
SM}}\left[1+a_{s}r^{s}_{\rm NP}-a_{d}r^{d}_{\rm NP}\right],$ (61)
$\displaystyle\frac{r_{s}}{r_{d}}$ $\displaystyle=$
$\displaystyle\frac{r^{s}_{\rm SM}}{r^{d}_{\rm SM}}\left[1+(r^{s}_{\rm
NP}/r^{s}_{\rm SM})b_{s}-(r^{d}_{\rm NP}/r^{d}_{\rm SM})b_{d}-a_{s}r^{s}_{\rm
NP}+a_{d}r^{d}_{\rm NP}\right].$

Before proceeding, we note that by factoring out of the SM contributions the
CKM parameters $\lambda^{c}_{bq}$ and $R^{bq}_{uc}$, we leave in
$\hat{P}^{q}_{\rm SM}$ and $r^{q}_{\rm SM}$ only QCD matrix elements and
flavor-universal electroweak parameters, such as $G_{F}$. Thus, in the
$U$-spin limit, we have $\hat{P}^{s}_{\rm SM}=\hat{P}^{d}_{\rm SM}$ and
$r^{s}_{\rm SM}=r^{d}_{\rm SM}$. In contrast, $r^{q}_{\rm NP}$ does include in
it, in general, flavor-dependent factors, and thus $U$-spin does not imply
$r^{s}_{\rm NP}=r^{d}_{\rm NP}$.

In the $U$-spin limit, we thus obtain from Eqs. (61)

$\displaystyle\frac{\hat{P}^{s}}{\hat{P}^{d}}-1$ $\displaystyle=$
$\displaystyle a_{s}r^{s}_{\rm NP}-a_{d}r^{d}_{\rm NP},$ (62)
$\displaystyle\frac{r_{s}}{r_{d}}-1$ $\displaystyle=$
$\displaystyle(r^{s}_{\rm NP}/r_{\rm SM})b_{s}-(r^{d}_{\rm NP}/r_{\rm
SM})b_{d}-a_{s}r^{s}_{\rm NP}+a_{d}r^{d}_{\rm NP}.$

We learn that new physics contributions can lead to
$|\hat{P}^{d}/\hat{P}^{d}|\neq 1$, and/or to $|r_{s}/r_{d}|\neq 1$ even with
no $U$-spin breaking. We recall that Eq. (IV) gives, for the experimental
central values,

$\displaystyle|\hat{P}^{s}/\hat{P}^{d}|_{\rm exp}-1$ $\displaystyle\approx$
$\displaystyle 0.26,$ $\displaystyle|r_{s}/r_{d}|_{\rm exp}-1$
$\displaystyle\approx$ $\displaystyle 0.07.$ (63)

In what follows we require that the new physics contributions do not lead to
much larger deviations from unity.

We translate the upper bounds on the new physics contributions to lower bounds
on the scale of new physics $\Lambda_{\rm NP}$. To do so, we assume that
$\Lambda_{\rm NP}\gg m_{W}$, so that the new physics can be presented by
dimension-six terms,

$\frac{g^{2}X_{bq}}{\Lambda_{\rm
NP}^{2}}(\overline{b_{L}}\gamma^{\mu}u_{L})(\overline{u_{L}}\gamma_{\mu}q_{L}),$
(64)

and we include in $X_{bq}$ only flavor dependent factors. We will compare
$g^{2}X_{bq}/\Lambda_{\rm NP}^{2}$ to the size of the SM tree level
contribution, which we take to be $g^{2}\lambda^{u}_{bq}/m_{W}^{2}$, with $g$
the weak coupling constant. Concretely, we study five classes of models:

  * •

Flavor anarchy: $X_{bq}={\cal O}(1)$.

  * •

Flavor anarchy with phase alignment: $X_{bq}=e^{i\gamma}{\cal O}(1)$.

  * •

FN symmetry: $X_{bq}=e^{i\theta_{q}}{\cal O}(|\lambda^{c}_{bq}|)$.

  * •

GMFV: $X_{bq}=e^{i\theta}\lambda^{t}_{bq}$.

  * •

Minimal Flavor Violation (MFV): $X_{bq}=\lambda^{t}_{bq}$.

###  VI.1 Flavor anarchy

We refer to new physics as being anarchic when it has neither flavor
suppression nor phase alignment. In other words, it is suppressed by its high
scale (and possibly a loop factor) but by no other small parameters,
$X_{bq}={\cal O}(1)$. In this case, we expect that

$\displaystyle r^{s}_{\rm NP}$ $\displaystyle=$ $\displaystyle r^{s}_{\rm
AN}/|\lambda^{c}_{bs}|,$ $\displaystyle r^{d}_{\rm NP}$ $\displaystyle=$
$\displaystyle r^{d}_{\rm AN}/|\lambda^{c}_{bd}|,$ (65)

where the CKM factors compensate for the $\lambda^{c}_{bs}$ and
$\lambda^{c}_{bd}$ factors that are pulled out of the parenthesis in Eqs.
(VI), and $r^{s}_{\rm AN}$ and $r^{d}_{\rm AN}$ are of the same order of
magnitude but not equal.

Examining Eqs. (62) and (VI), we learn that the largest modification will be
due to the $b_{s}$ term, which is enhanced by $(R^{bs}_{uc})^{-1}$. The
physics behind this result is simple: the SM contributions to $\hat{P}^{s}$,
$\hat{P}^{d}$, $\hat{T}^{s}$ and $\hat{T}^{d}$ are proportional to
$\lambda^{c}_{bs}$, $\lambda^{c}_{bd}$, $\lambda^{u}_{bs}$ and
$\lambda^{u}_{bd}$, respectively. Anarchic new physics will modify most
strongly the term that within the SM is the most strongly CKM-suppressed,
which is $\hat{T}^{s}$.

In the $U$-spin limit, we obtain:

$\displaystyle\frac{\hat{P}^{s}}{\hat{P}^{d}}$ $\displaystyle=$ $\displaystyle
1+\frac{s_{\gamma-\theta_{s}}}{s_{\gamma}}\frac{r^{s}_{\rm
AN}}{|\lambda^{c}_{bs}|}-\frac{s_{\gamma-\theta_{d}}}{s_{\gamma}}\frac{r^{d}_{\rm
AN}}{|\lambda^{c}_{bd}|},$ (66) $\displaystyle\frac{\hat{T}_{s}}{\hat{T}_{d}}$
$\displaystyle=$ $\displaystyle
1+\frac{s_{\theta_{s}}}{s_{\gamma}}\frac{r^{s}_{\rm AN}}{r_{\rm
SM}|\lambda^{u}_{bs}|}+\frac{s_{\theta_{d}}}{s_{\gamma}}\frac{r^{d}_{\rm
AN}}{r_{\rm SM}|\lambda^{u}_{bd}|},$

where we use the notation $s_{\gamma}\equiv\sin\gamma$, and similarly to all
phases. The ratio between the corrections to $r_{s}/r_{d}$ and
$\hat{P}_{s}/\hat{P}_{d}$ can be estimated as follows:

$\frac{|r_{s}/r_{d}|-1}{|\hat{P}^{s}/\hat{P}^{d}|-1}\approx\frac{1}{r_{\rm
SM}}\frac{|\lambda^{c}_{bd}|}{|\lambda^{u}_{bs}|}\sim 2.$ (67)

Since the corrections are comparable, the cancellation between the
contributions to $\hat{T}^{s}/\hat{T}^{d}$ and $\hat{P}^{s}/\hat{P}^{d}$ could
be accidental, so we require that $(b_{s}r^{s}_{\rm AN})/(r_{\rm
SM}|\lambda^{c}_{bs}|)\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}0.30$. Given that we consider anarchic
new physics, we further assume that $\sin\theta_{s}/\sin\gamma={\cal O}(1)$
and that $r_{\rm AN}/r_{\rm SM}\sim m_{W}^{2}/\Lambda_{\rm AN}^{2}$, where
$\Lambda_{\rm AN}$ is the high scale of the anarchic new physics. We obtain
the following bound:

$\frac{r^{s}_{\rm AN}}{r_{\rm SM}|\lambda^{u}_{bs}|}\approx
1200\frac{m_{W}^{2}}{\Lambda_{\rm AN}^{2}}\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}0.30\
\Longrightarrow\ \Lambda_{\rm AN}\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}60m_{W}\sim 5\ {\rm
TeV}.$ (68)

###  VI.2 Flavor anarchy with phase alignment

The largest spurious $U$-spin breaking due to new physics occurs when the
flavor structure is anarchic, as in Eq. (VI.1), but the new physics phase is
set at a special value. Concretely, we consider the case where the new physics
phases assume the values of

$\theta_{s}=\theta_{d}=\gamma.$ (69)

The consequences of this scenario can be straightforwardly read from Eqs.
(66):

$\displaystyle\frac{\hat{P}^{s}}{\hat{P}^{d}}$ $\displaystyle=$ $\displaystyle
1,$ $\displaystyle\frac{\hat{T}_{s}}{\hat{T}_{d}}$ $\displaystyle=$
$\displaystyle 1+\frac{r^{s}_{\rm PA}}{r_{\rm
SM}|\lambda^{u}_{bs}|}+\frac{r^{d}_{\rm PA}}{r_{\rm SM}|\lambda^{u}_{bd}|}.$
(70)

In this scenario, there is no deviation from the $U$-spin relation for
$|\hat{P}^{s}/\hat{P}^{d}|$, but the $U$-spin relation for $|r_{s}/r_{d}|$ is
violated. Thus, we require that the contribution to $|r_{s}/r_{d}|-1$ does not
exceed the experimental value, for which Eq. (IV) gives $|r_{s}/r_{d}|_{\rm
exp}-1\approx 0.07$. Estimating for the flavor-anarchic phase-aligned case
$r_{\rm PA}/r_{\rm SM}\sim m_{W}^{2}/\Lambda_{\rm PA}^{2}$, we require:

$\frac{r_{\rm PA}}{r_{\rm
SM}}\left(\frac{1}{|\lambda^{u}_{bs}|}+\frac{1}{|\lambda^{u}_{bd}|}\right)\approx
1400\frac{m_{W}^{2}}{\Lambda_{\rm PA}^{2}}\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}0.07\
\Longrightarrow\ \Lambda_{\rm PA}\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}140m_{W}\sim 11\
{\rm TeV}.$ (71)

###  VI.3 The Froggatt-Nielsen mechanism

In the FN framework Froggatt:1978nt ; Leurer:1992wg , there is a $U(1)_{\rm
FN}$ symmetry that is broken by a small spurion $\lambda_{\rm FN}$ . The small
flavor parameters – mass ratios and CKM angles – are accounted for by
different powers of $\lambda_{\rm FN}$, depending on the charges of the
relevant fields. The small parameter $\lambda_{\rm FN}$ is commonly taken to
be of the order of the Cabibbo angle, $\lambda_{\rm FN}\approx 0.2$, and
conventionally taken to carry charge $Q_{\rm FN}(\lambda_{\rm FN})=-1$ under
the $U(1)_{\rm FN}$ symmetry. For our purpose, however, where we aim to find
the parametric suppression of flavor-changing dimension-six terms, one can
relate this suppression directly to the suppression of the CKM angles (and, in
some case, quark masses). Thus, if the new physics is subject to the FN
selection rules, then the leading contributions to $b\to u\bar{u}q$ are
suppressed by $\lambda_{\rm FN}^{[Q_{\rm FN}(b_{L})-Q_{\rm FN}(q_{L})]}$,
resulting in $X_{bq}=e^{i\theta_{q}}\times{\cal O}(|\lambda^{c}_{bq}|)$.
Matching to Eqs. (VI), FN implies:

$\displaystyle r^{s}_{\rm NP}e^{+i\theta_{s}}$ $\displaystyle=$ $\displaystyle
r_{\rm FN}^{s}e^{+i\theta_{s}},$ $\displaystyle r^{d}_{\rm
NP}e^{+i\theta_{d}}$ $\displaystyle=$ $\displaystyle-r_{\rm
FN}^{d}e^{+i\theta_{d}}.$ (72)

$U$-spin implies $\hat{P}^{s}_{\rm SM}=\hat{P}^{d}_{\rm SM}$ and $r^{s}_{\rm
SM}=r^{d}_{\rm SM}$, but not $r^{s}_{\rm FN}=r^{d}_{\rm FN}$. The FN selection
rules imply that both $r^{s}_{\rm FN}$ and $r^{d}_{\rm FN}$ are ${\cal O}(1)$
(namely not suppressed by powers of $\lambda_{\rm FN}$) but not equal. Using
$U$-spin, we obtain:

$\displaystyle a_{s}$ $\displaystyle=$
$\displaystyle\frac{s_{\gamma-\theta_{s}}}{s_{\gamma}},\ \ \
b_{s}=\frac{s_{\theta_{s}}}{R^{bs}_{uc}s_{\gamma}},$ $\displaystyle a_{d}$
$\displaystyle=$ $\displaystyle\frac{s_{\gamma-\theta_{d}}}{s_{\gamma}},\ \ \
b_{d}=-\frac{s_{\theta_{d}}}{R^{bd}_{uc}s_{\gamma}},$ (73)

and

$\displaystyle\frac{\hat{P}^{s}}{\hat{P}^{d}}$ $\displaystyle=$ $\displaystyle
1+\frac{s_{\gamma-\theta_{s}}}{s_{\gamma}}r^{s}_{\rm
FN}+\frac{s_{\gamma-\theta_{d}}}{s_{\gamma}}r^{d}_{\rm FN},$
$\displaystyle\frac{\hat{T}^{s}}{\hat{T}^{d}}$ $\displaystyle=$ $\displaystyle
1+\frac{s_{\theta_{s}}}{s_{\gamma}}\frac{r^{s}_{\rm FN}}{r_{\rm
SM}R^{bs}_{uc}}-\frac{s_{\theta_{d}}}{s_{\gamma}}\frac{r^{d}_{\rm FN}}{r_{\rm
SM}R^{bd}_{uc}}.$ (74)

Assuming that $s_{\theta_{q}}={\cal O}(1)$, we can obtain a lower bound on the
scale of new physics. The strongest bound can be obtained by noticing that FN
predicts a much larger deviation from the $U$-spin relations in
$|r_{s}/r_{d}|$ than in $|\hat{P}_{s}/\hat{P}_{d}|$, in contrast to the
experimental data (and to the expectations from naive factorization). The
former is enhanced over the latter by a factor of order

$\frac{|r_{s}/r_{d}|-1}{|\hat{P}^{s}/\hat{P}^{d}|-1}\approx\frac{1}{r_{\rm
SM}R^{bs}_{uc}}\sim 10.$ (75)

Thus, we require that the FN contribution to $|r_{s}/r_{d}|-1$ does not exceed
$|r_{s}/r_{d}|_{\rm exp}-1\approx 0.07$. Taking
$s_{\theta_{s}}/s_{\gamma}={\cal O}(1)$, and estimating, for FN models,
$r_{\rm FN}/r_{\rm SM}\sim m_{W}^{2}/\Lambda_{\rm FN}^{2}$, we require:

$\frac{r_{\rm FN}}{r_{\rm SM}}\frac{1}{R^{bs}_{uc}}\approx
50\frac{m_{W}^{2}}{\Lambda_{\rm FN}^{2}}\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}0.07\
\Longrightarrow\ \Lambda_{\rm FN}\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}26m_{W}\sim 2\ {\rm
TeV}.$ (76)

###  VI.4 General MFV

In the GMFV framework Kagan:2009bn , the only sources of flavor $[U(3)]^{5}$
breaking are the Yukawa matrices of the SM, but there could be new sources of
(flavor-universal) CP violation, $X_{bq}=e^{i\theta}\lambda^{t}_{bq}$. (For a
study in a related framework, see ref. Crivellin:2019isj .) Matching to Eqs.
(VI), GMFV implies:

$\displaystyle r^{s}_{\rm NP}e^{+i\theta_{s}}$ $\displaystyle=$
$\displaystyle-r_{\rm GMFV}^{s}e^{+i\theta}(1+R^{bs}_{uc}e^{i\gamma}),$
$\displaystyle r^{d}_{\rm NP}e^{+i\theta_{d}}$ $\displaystyle=$
$\displaystyle-r_{\rm GMFV}^{d}e^{+i\theta}(1-R^{bd}_{uc}e^{i\gamma}).$ (77)

$U$-spin implies $r_{\rm GMFV}^{s}=r_{\rm GMFV}^{d}$. With these replacements,
and using $U$-spin, we obtain:

$\displaystyle
a_{s}=\frac{s_{\gamma-\theta}-R^{bs}_{uc}s_{\theta}}{s_{\gamma}},\ \ \
b_{s}R^{bs}_{uc}=\frac{s_{\theta}+R^{bs}_{uc}s_{\theta+\gamma}}{s_{\gamma}}$
$\displaystyle
a_{d}=\frac{s_{\gamma-\theta}+R^{bd}_{uc}s_{\theta}}{s_{\gamma}},\ \ \
-b_{d}R^{bd}_{uc}=\frac{s_{\theta}-R^{bd}_{uc}s_{\theta+\gamma}}{s_{\gamma}}.$
(78)

and

$\displaystyle\frac{\hat{P}^{s}}{\hat{P}^{d}}$ $\displaystyle=$ $\displaystyle
1+r_{\rm GMFV}\frac{s_{\theta}}{s_{\gamma}}(R^{bd}_{uc}+R^{bs}_{uc}),$
$\displaystyle\frac{\hat{T}^{s}}{\hat{T}^{d}}$ $\displaystyle=$ $\displaystyle
1-\frac{r_{\rm GMFV}}{r_{\rm
SM}}\frac{s_{\theta}}{s_{\gamma}}\left(\frac{1}{R^{bs}_{uc}}+\frac{1}{R^{bd}_{uc}}\right).$
(79)

Assuming that $s_{\theta}={\cal O}(1)$, we can obtain a lower bound on the
scale of new physics. The strongest bound can be obtained by noticing that
GMFV predicts a much larger deviation from the $U$-spin relations in
$|r_{s}/r_{d}|$ than in $|P_{s}/P_{d}|$:

$\frac{|r_{s}/r_{d}|-1}{|\hat{P}^{s}/\hat{P}^{d}|-1}\approx\frac{1}{r_{\rm
SM}}\frac{1}{R^{bd}_{uc}R^{bs}_{uc}}\sim 25.$ (80)

Thus, we require that the GMFV contribution to $|r_{s}/r_{d}|-1$ does not
exceed $|r_{s}/r_{d}|_{\rm exp}-1\approx 0.07$. Taking
$s_{\theta}/s_{\gamma}={\cal O}(1)$, and estimating, for GMFV models, $r_{\rm
GMFV}/r_{\rm SM}\sim m_{W}^{2}/\Lambda_{\rm GMFV}^{2}$, we require:

$\frac{r_{\rm GMFV}}{r_{\rm SM}}\frac{1}{R^{bs}_{uc}}\approx
50\frac{m_{W}^{2}}{\Lambda_{\rm GMFV}^{2}}\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}0.07\
\Longrightarrow\ \Lambda_{\rm GMFV}\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}26m_{W}\sim 2\ {\rm
TeV}.$ (81)

###  VI.5 Minimal flavor violation (MFV)

In the MFV framework DAmbrosio:2002vsn , the only sources of flavor
$[U(3)]^{5}$ breaking and of CP violation are the Yukawa matrices of the SM,
$X_{bq}=\lambda^{t}_{bq}$. MFV implies:

$\displaystyle r^{s}_{\rm NP}e^{i\theta_{s}}$ $\displaystyle=$ $\displaystyle-
r_{\rm MFV}(1+R^{bs}_{uc}e^{i\gamma}),$ $\displaystyle r^{d}_{\rm
NP}e^{i\theta_{d}}$ $\displaystyle=$ $\displaystyle-r_{\rm
MFV}(1+R^{bd}_{uc}e^{i\gamma})\,.$ (82)

The consequences of MFV can be straightforwardly derived by using the analysis
of GMFV with $\theta=0$. Eq. (VI.4) implies that, in this case,

$a_{s}=b_{s}=a_{d}=b_{d}=1\,,$ (83)

and Eq. (VI.4) implies that, in the $U$-spin limit,

$\hat{P}_{s}/\hat{P}_{d}=1,\ \ \ r_{s}/r_{d}=1.$ (84)

Thus, MFV physics does not mimic $U$-spin breaking.

This result can be easily understood. For MFV physics, the new physics
contribution can be absorbed into $P_{t}$ of Eqs. (10) and (14), and thus the
MFV analysis cannot be distinguished from the SM analysis.

While the constraints from $U$-spin breaking cannot be applied to the MFV
case, one can still bound the size of the MFV contribution. Assuming $U$-spin,
we have

$\hat{T}=T+P_{u}-P_{t}-P_{\rm MFV},\ \ \ \hat{P}=P_{c}-P_{t}-P_{\rm MFV}.$
(85)

Approximating $P_{u}\approx P_{c}$, we obtain

$r={\hat{T}}/{\hat{P}}\approx{T}/{\hat{P}}+1.$ (86)

Obviously, when $|T/\hat{P}|\ll 1$, we have $r\simeq 1$, inconsistent with the
experimental result that $r\approx 4.8$. We learn that we can require that
$|T/\hat{P}|\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}3.8$, which translates into

$\Lambda_{\rm MFV}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}2m_{W}\approx 160\ {\rm GeV}.$ (87)

##  VII Conclusions

The recent measurements of two CP asymmetries in $B_{s}\to K^{+}K^{-}$,
$S_{KK}$ and $C_{KK}$, constitute the first observation of time-dependent CP
violation in the neutral $B_{s}$ system. While the theoretical analysis of
these observables suffers from hadronic uncertainties, the approximate
$U$-spin symmetry of QCD relates them to the corresponding observables in
$B_{d}\to\pi^{+}\pi^{-}$, $S_{\pi\pi}$ and $C_{\pi\pi}$. The set of five
observables - the four CP asymmetries and the ratio of decay rates,
$\Gamma(B_{s}\to K^{+}K^{-})/\Gamma(B_{d}\to\pi^{+}\pi^{-})$, provides a new
arena to test the SM, the approximate $U$-spin symmetry, and the presence of
flavor changing new physics.

We find that the measured values are consistent with $U$-spin breaking at the
level of 30% or smaller. Furthermore, the $U$-spin breaking is a factor $\sim
4$ smaller in the double ratio of matrix elements,
$(\hat{T}_{s}/\hat{T}_{d})/(\hat{P}_{s}/\hat{P}_{d})$, than in each of these
ratios separately. This result is consistent with the assumption that the
leading effect is coming from $f_{K}/f_{\pi}$ which cancels in the
factorizable contributions to the double ratio, with the remaining non-
factorizable contributions and contributions of ${\cal O}(m_{s}/m_{b})$ much
smaller.

With regard to new physics, the main tool that we use to probe it is the
observation that, in general, when interpreting the experimental results
assuming the SM, the new physics contributions will mimic $U$-spin breaking
effects. If the new physics contributions are large, they can generate large
spurious $U$-spin breaking. Assuming that there are no cancellations between
the new physics effects and genuine $U$-spin breaking effects, we found lower
bounds on the scale of classes of new physics with various flavor structures:
for flavor anarchy $\Lambda_{\rm AN}\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}5$ TeV, for FN
selection rules $\Lambda_{\rm FN}\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}2$ TeV, and for GMFV
$\Lambda_{\rm GMFB}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}2$ TeV. We further demonstrated that
new physics at a scale as high as ${\cal O}(10\ {\rm TeV})$ can generate
significant spurious $U$-spin breaking.

New physics subject to MFV does not mimic $U$-spin breaking. In fact,
analyzing the results in the MFV framework is identical to carrying out the
analysis in the SM framework. Yet, if the scale of MFV new physics is below
160 GeV, it will bring the situation closer to a single phase dominance,
suppressing the CP asymmetries to below their observed values, and is thus
excluded.

Our bounds on the scale of new physics assume that there are no cancellations
between the spurious and the genuine $U$-spin breaking effects. Such
cancellations can relax our bounds by factors of ${\cal O}(1)$. We further did
not include loop factors for the new physics contributions. Note, however,
that the relevant decay processes, $b\to u\bar{u}q$ ($q=s,d$), are not FCNC
processes, and have tree level contributions already in the SM. Finally, while
$S_{KK}$ and $S_{\pi\pi}$ depend on, respectively, $B_{s}-\overline{B_{s}}$
and $B_{d}-\overline{B_{d}}$ mixing, which are FCNC processes, we used purely
experimental data to include the neutral meson mixing parameters, and thus our
analysis is independent of the mixing mechanism.

### Acknowledgements

We are grateful to Yasmine Amhis, Yuval Grossman and Zoltan Ligeti for helpful
discussions. YN is the Amos de-Shalit chair of theoretical physics, and is
supported by grants from the Israel Science Foundation (grant number 1124/20),
the United States-Israel Binational Science Foundation (BSF), Jerusalem,
Israel (grant number 2018257), by the Minerva Foundation (with funding from
the Federal Ministry for Education and Research), and by the Yeda-Sela (YeS)
Center for Basic Research. IS is supported by a fellowship from the Ariane de
Rothschild Women Doctoral Program.

## References

  * (1) J. H. Christenson, J. W. Cronin, V. L. Fitch and R. Turlay, Phys. Rev. Lett. 13, 138-140 (1964).
  * (2) M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49, 652-657 (1973).
  * (3) B. Aubert et al. [BaBar], Phys. Rev. Lett. 87, 091801 (2001) [arXiv:hep-ex/0107013 [hep-ex]].
  * (4) K. Abe et al. [Belle], Phys. Rev. Lett. 87, 091802 (2001) [arXiv:hep-ex/0107061 [hep-ex]].
  * (5) L. Wolfenstein, Nucl. Phys. B 77, 375-385 (1974).
  * (6) G. Eyal and Y. Nir, Nucl. Phys. B 528, 21-34 (1998) [arXiv:hep-ph/9801411 [hep-ph]].
  * (7) Y. Nir and V. Vagnoni, Comptes Rendus Physique 21, no.1, 61-74 (2020)
  * (8) R. Aaij et al. [LHCb], JHEP 03, 075 (2021) [arXiv:2012.05319 [hep-ex]]. [arXiv:2012.05319 [hep-ex]].
  * (9) R. Aaij et al. [LHCb], Phys. Rev. D 98, no.3, 032004 (2018) [arXiv:1805.06759 [hep-ex]].
  * (10) P. A. Zyla et al. [Particle Data Group], PTEP 2020, no.8, 083C01 (2020).
  * (11) Y. S. Amhis et al. [HFLAV], Eur. Phys. J. C 81, no.3, 226 (2021) [arXiv:1909.12524 [hep-ex]] and online update at https://hflav.web.cern.ch.
  * (12) LHCb collaboration, LHCb-CONF-2020-003.
  * (13) M. Z. Barel, K. De Bruyn, R. Fleischer and E. Malami, J. Phys. G 48, no.6, 065002 (2021) [arXiv:2010.14423 [hep-ph]].
  * (14) R. Fleischer and R. Knegjens, Eur. Phys. J. C 71, 1532 (2011) [arXiv:1011.1096 [hep-ph]].
  * (15) M. Ciuchini, E. Franco, S. Mishima and L. Silvestrini, JHEP 10, 029 (2012) [arXiv:1205.4948 [hep-ph]].
  * (16) R. Fleischer, R. Jaarsma and K. K. Vos, Phys. Rev. D 94, no.11, 113014 (2016) [arXiv:1608.00901 [hep-ph]].
  * (17) R. Fleischer, R. Jaarsma and K. K. Vos, JHEP 03, 055 (2017) [arXiv:1612.07342 [hep-ph]].
  * (18) R. Fleischer, Phys. Lett. B 459, 306-320 (1999) [arXiv:hep-ph/9903456 [hep-ph]].
  * (19) A. Ali, G. Kramer, Y. Li, C. D. Lu, Y. L. Shen, W. Wang and Y. M. Wang, Phys. Rev. D 76, 074018 (2007) [arXiv:hep-ph/0703162 [hep-ph]].
  * (20) M. Beneke, eConf C0304052, FO001 (2003) [arXiv:hep-ph/0308040 [hep-ph]].
  * (21) X. G. He, S. F. Li, B. Ren and X. B. Yuan, Phys. Rev. D 96, no.5, 053004 (2017) [arXiv:1704.05788 [hep-ph]].
  * (22) M. Gronau, Phys. Lett. B 492, 297-302 (2000) [arXiv:hep-ph/0008292 [hep-ph]].
  * (23) M. Gronau, Phys. Lett. B 727, 136-140 (2013) [arXiv:1308.3448 [hep-ph]].
  * (24) C. D. Froggatt and H. B. Nielsen, Nucl. Phys. B 147, 277-298 (1979).
  * (25) M. Leurer, Y. Nir and N. Seiberg, Nucl. Phys. B 398, 319-342 (1993) [arXiv:hep-ph/9212278 [hep-ph]].
  * (26) G. D’Ambrosio, G. F. Giudice, G. Isidori and A. Strumia, Nucl. Phys. B 645, 155-187 (2002) [arXiv:hep-ph/0207036 [hep-ph]].
  * (27) A. L. Kagan, G. Perez, T. Volansky and J. Zupan, Phys. Rev. D 80, 076002 (2009) [arXiv:0903.1794 [hep-ph]].
  * (28) A. Crivellin, C. Gross, S. Pokorski and L. Vernazza, Phys. Rev. D 101, no.1, 015022 (2020) [arXiv:1909.02101 [hep-ph]].

  *[SM]: Standard Model
  *[FN]: Froggatt-Nielsen
  *[GMFV]: General Minimal Flavor Violation
  *[FCNC]: flavor changing neutral current
  *[MFV]: Minimal Flavor Violation


# Using machine learning to investigate the populations of dusty evolved stars
in various metallicities

Grigoris Maravelias1,2 contact<EMAIL_ADDRESS>Alceste Z. Bonanos1 Frank
Tramper3 Stephan de Wit1,4 Ming Yang1,5 Paolo Bonfini6 Emmanuel Zapartas1
Konstantinos Antoniadis1,4 Evangelia Christodoulou1,4 Gonzalo Muñoz-Sanchez1,4
1IAASARS, National Observatory of Athens, Greece
2Institute of Astrophysics, FORTH, Greece
3Institute of Astronomy, KU Leuven, Belgium
4Department of Physics, National and Kapodistrian University of Athens, Greece
5Key Laboratory of Space Astronomy and Technology, National Astronomical
Observatories, Chinese Academy of Sciences, People’s Republic of China
6Ballista Technology Group, Florida, USA

(2022)

###### Abstract

Mass loss is a key property to understand stellar evolution and in particular
for low-metallicity environments. Our knowledge has improved dramatically over
the last decades both for single and binary evolutionary models. However,
episodic mass loss although definitely present observationally, is not
included in the models, while its role is currently undetermined. A major
hindrance is the lack of large enough samples of classified stars. We
attempted to address this by applying an ensemble machine-learning approach
using color indices (from IR/Spitzer and optical/Pan-STARRS photometry) as
features and combining the probabilities from three different algorithms. We
trained on M31 and M33 sources with known spectral classification, which we
grouped into Blue/Yellow/Red/B[e] Supergiants, Luminous Blue Variables,
classical Wolf-Rayet and background galaxies/AGNs. We then applied the
classifier to about one million Spitzer point sources from 25 nearby galaxies,
spanning a range of metallicites ($1/15-\sim 3~{}Z_{\odot}$). Equipped with
spectral classifications we investigated the occurrence of these populations
with metallicity.

###### keywords:

Stars: massive – Stars: mass-loss – Stars: evolution – Galaxies: individual:
WLM, M31, IC 1613, M33, Sextans A – Methods: statistical

††volume: 361††journal: Massive Stars Near and Far††editors: N. St-Louis, J.
S. Vink & J. Mackey, eds.

##  1 Introduction

Although our knowledge on the modeling of stellar evolution for massive stars
has significantly advanced over the last couple of decades, we still struggle
to understand the influence of some key parameters. One such example is mass-
loss, for which the theoretical predictions are at odds with the values
derived from observations. This becomes more evident as we move away from the
main-sequence (e.g. Smith, 2014). Certain phases of evolved massive stars
display - additionally to the stellar winds (e.g. Vink, 2021) \- episodic
mass-loss activity, such as in Luminous Blue Variables (LBVs; e.g. Davidson
2020), Wolf-Rayet stars (WRs; e.g. Crowther & Walborn 2011), B[e] supergiants
(BeBRs; e.g. Kraus 2019), Yellow Supergiants (YSGs; e.g. de Jager 1998), Red
Supergiants (RSGs; e.g. Levesque 2017). We already know that the majority of
the massive stars are born into multiple systems (e.g. Kobulnicky et al.,
2014; Sana et al., 2014) which results in a significant fraction of binary
interactions (e.g. de Mink et al., 2014). However, binarity may not be the
only channel for episodic mass-loss. Moreover, it is only now that we start to
explore these populations in low metallicity environments, which can help us
understand better the underlying mechanisms (e.g. Lorenzo et al., 2022).
Simultaneously, it is those environments that seem to host superluminous
supernovae, which may be the result of the interaction of ejecta with a
complex dusty circumstellar environment formed by possible multiple mass-loss
events prior to the explosion (Neill et al., 2011; Gal-Yam, 2012).

The ASSESS project111http://assess.astro.noa.gr/ (P.I. Alceste Bonanos, see
also Bonanos et al. 2022 this volume) aims to understand the importance of
episodic mass-loss by studying a large number of evolved stars in nearby
galaxies. One of the approaches is to use machine-learning methods to classify
sources without any spectral classification, and to perform follow-up
observations to verify their nature. Given this tool we can also perform a
study of those populations across different metallicity environments. The
current paper summarizes our first results towards this goal.

##  2 The classifier

Since the classifier is presented in detail in Maravelias et al. (2022), we
provide only a brief overview in this section, so that the reader understands
the post-processing steps and the final results.

After careful search in literature we compiled a Gaia cleaned sample of 932
M31 and M33 sources with known spectral types. We grouped these into seven
classes of interest, i.e. in BSG, YSG, RSG, BeBR, LBV, (classic) WR, and
background galaxies (GAL; as a collective class of outliers). We used
photometry, and in particular consecutive color indices, from the Spitzer
([3.6] and [4.5]) and Pan-STARRS ($r,i,z,$ and $y$) bands (not corrected for
extinction as there is not enough data to perform this and the influence in
those selected bands is minimal). As this initial sample is highly imbalanced
we employed synthetic data approaches to overcome the limitations imposed by
the underrepresented classes. Although this does not completely alleviate the
problem with LBV it helps with other classes (e.g. WR). Our classifier is an
ensemble approach of three supervised machine-learning algorithms, that is
Support Vector Classification, Random Forests, and Multilayer-Perceptron (a
type of neural network). Their output results probabilities per class and
these are combined in order to obtain the final probability distribution,
while the final class is derived by using the class that corresponds to the
maximum probability.

The results from the application of this approach are fairly good. We managed
to correctly identified from the initial training sample $\sim 60\%$ to $\sim
80\%$ for BSG, GAL, and YSG, $\sim 73-80\%$ for BeBR, and $\sim 45\%$ for WR,
with the best results obtained for the RSG ($\sim 94\%$) and the worst for LBV
($\sim 28\%$ \- only from the Support Vector Classification). These results
are comparable or slightly better when compared to the results from Dorn-
Wallenstein et al. (2021), who worked with a much less homogeneous (with
respect to the labels) but more populated Galactic sample. Our overall
weighted balanced accuracy is $\sim 83\%$.

Our test sample consisted of a (limited) number of sources collected for IC
1613, WLM, and Sextans A galaxies. Due to the fact that $\sim 14\%$ of these
sources have missing values (in the selected feature space, i.e. not having
measurements across all selected Spitzer and Pan-STARRS bands), we performed
data imputation by replacing the features’ values using iterative imputer
(which models, iteratively, each feature with missing values as a function of
other features, and uses that estimate for imputation). Although overall the
missing data imputation does not affect much the results on this particular
test dataset, our results showed that we can retain a relative good prediction
score even when missing up to three features (out of the total five available,
i.e. the consecutive color indices). The accuracy obtained for this test
sample (at $\sim 70\%$) was lower than that for the M31 and M33 sample. We
attribute this discrepancy partly to photometric issues and to the total
effect of metallicity, which might affect the intrinsic colors of the sources
and extinction due to the different galactic environments. However, we still
lack a statistically significant sample to study and understand the influence
of the metallicity, a target towards which the ASSESS project is actively
working to address, both through the machine-learning applications such this
classifier as well as dedicated observational campaigns (see Munoz-Sanchez et
al. 2022; de Wit et al. 2022 in this volume).

##  3 Application and Results

After completing the training of the model using the whole available sample of
sources in M31 and M33, we proceeded with its application. We used source
lists for 25 nearby galaxies, spanning a range of 1/15 to $\sim
3~{}Z_{\odot}$, that have been compiled from Spitzer photometric catalogs, and
after Gaia screening. The classifier returns spectral types for 1156925
sources in total, including information on band completeness (the fraction of
color indices without missing data) and probability. From all these sources
only a tiny $\sim 0.5\%$ have spectroscopic classifications.

We applied a number of selection cuts in order to first examine secure
candidate sources. For this we employed a limit for band completeness so that
we keep sources with measurements across all bands (i.e. band completeness
equal to 1), which means that we excluded all sources with missing values.
Then we considered only sources with a classification probability of at least
0.51. This value corresponds to the peak of the incorrect classifications for
the M31 and M33 sources (see Fig. 6 Maravelias et al., 2022).

Given the spectral classifications and photometric information we can proceed
and plot indicative Color-Magnitude Diagrams (CMDs) to examine our results. In
Figure 1 we plot the optical $g-i$ vs. $i$ and the IR [3.6]-[4.5] vs. [4.5]
CMDs for 708 WLM sources. We can easily notice the distributions of sources as
expected in the optical with the BSG, YSG, and RSG occupying different
distinct locations. There are numerous WR though which seem to overlap with
other populations. This is also evident in the IR CMD, and it is a possible
effect of a higher rate of false positives for this class. In the IR CMD we
sse the clustering of RSG above the BSG populations as expected (c.f. Bonanos
et al., 2009). Interestingly, we identify one LBV and two BeBR candidate
sources, which shows that the classifier is able to highlight the most
promising targets.

Figure 1: An optical (left) and IR (right) CMD for WLM. We can easily notice
the distributions of BSG (blue ’X’), YSG (yellow slim triangles), RSG (red
crosses), WR (purple full triangles), GAL (black boxes), BeBR (dark violet
stars), LBV (pink pentagons). Note that the number of sources in the optical
CMD is smaller than the IR one (708 sources in total) because not all sources
have $g$ magnitudes.

Having obtained the predictions for all galaxies, we can examine the
predictions in a more global framework. We considered the fractions of the
number of predicted sources (per class) over the total number of sources in
each galaxy. In Figure 2 we present these fractions with metallicity for a
subsample of our galaxies. It is interesting to point out the increasing
(decreasing) trend of BSG (RSG) with lower metallicities consistent with
theoretical expectations that more (less) time is spend in these phases (e.g.
Ren et al., 2021; Massey, 2002). Also, WR display decreasing trend because of
the drop in the strength of the stellar winds with metallicity. YSG display a
rather flatter behavior (with the exception of IC 1613 which deserves a
revisit), while we notice far fewer BeBR than LBVs in low metallicity
galaxies.

However, we need to be careful with the interpretation of these plots, as
currently we have not taken into account the errors with respect to the
predictions that should propagate through the end fractions, as well as
completeness issues. As our samples are based on Spitzer catalogs we are
looking for (and predicting) stars with substantial amounts of dust. Although
this can be true for the majority of the RSG and the BeBR it is not necessary
for BSG (which includes main-sequence massive stars) or YSGs. We also need to
quantify the contamination. For example, the BSG class may get mixed with HII
regions, while bright AGB stars may still be present among the RSGs. However,
at the distances of the galaxies we are looking at we are actually probing the
upper part of the Hertzsprung–Russell diagram, which minimizes the
contamination by low-mass stars. Last but not least, we need to further study
in detail the influences of our selection cuts. For example Fig. 8 in
(Maravelias et al., 2022) shows that the peak probability of incorrect
classification shifts to higher probability values when examining the test
galaxies (lower-Z environments).

Figure 2: The fractions, of the predicted class members over the total sample
size for each galaxy, with metallicity.

##  4 Conclusions

We developed a machine-learning classifier based on optical (Pan-STARRS) and
IR (Spitzer) photometry, using M31 and M33 sources as training sample, with an
overall weighted balanced accuracy of $\sim 83\%$. Then we applied this
classifier to 25 nearby galaxies (with $1/15-\sim 3~{}Z_{\odot}$) to get
spectral classification (including Blue/Yellow/Red Supergiants, B[e]
Supergiants, Wolf-Rayet, Luminous Blue Variables, and background galaxies) for
about 1.1 million sources. Selecting some initial quality cuts for the output
class probability and data availability we explore the population statistics
with respect to the metallicity. We identify potentially interesting trends,
such as the increase/decrease of Blue/Red supergiants and drop of WR with
decreasing metallicity. However, to fully exploit these results further
processing is needed to correctly account for predictions errors, completeness
issues, and influences of quality cuts. Nevertheless, this catalog is an
excellent tool for identifying interesting objects and prioritizing targets
for observations, especially for JWST.

## Acknowledgments

We acknowledge funding support from the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme (Grant
agreement No. 772086).

## References

  * Bonanos et al. (2009) Bonanos, A. Z., et al. 2009, AJ, 138, 1003
  * Bonanos et al. (2022) Bonanos, A. Z., et al. 2022
  * Crowther & Walborn (2011) Crowther, P. A., & Walborn, N. R. 2011, MNRAS, 416, 1311
  * Davidson (2020) Davidson, K. 2020, Galaxies, 8, 10
  * de Jager (1998) de Jager, C. 1998, A&A Rev., 8, 145
  * de Mink et al. (2014) de Mink, S. E., Sana, H., Langer, N., Izzard, R. G., & Schneider, F. R. N. 2014, ApJ, 782, 7
  * de Wit et al. (2022) de Wit, S., Bonanos, A. Z., Tramper, F., Yang, M., Maravelias, G., Boutsia, K., Britavskiy, N., & Zapartas, M. 2022
  * Dorn-Wallenstein et al. (2021) Dorn-Wallenstein, T. Z., Davenport, J. R. A., Huppenkothen, D., & Levesque, E. M. 2021, ApJ, 913, 32
  * Gal-Yam (2012) Gal-Yam, A. 2012, Science, 337, 927
  * Kobulnicky et al. (2014) Kobulnicky, H. A., et al. 2014, ApJS, 213, 34
  * Kraus (2019) Kraus, M. 2019, Galaxies, 7, 83
  * Levesque (2017) Levesque, E. M. 2017, Astrophysics of Red Supergiants
  * Lorenzo et al. (2022) Lorenzo, M., Garcia, M., Najarro, F., Herrero, A., Cerviño, M., & Castro, N. 2022, arXiv e-prints, arXiv:2207.09700
  * Maravelias et al. (2022) Maravelias, G., Bonanos, A. Z., Tramper, F., de Wit, S., Yang, M., & Bonfini, P. 2022, arXiv e-prints, arXiv:2203.08125
  * Massey (2002) Massey, P. 2002, ApJS, 141, 81
  * Munoz-Sanchez et al. (2022) Munoz-Sanchez, G., Maravelias, G., Bonanos, A. Z., Tramper, F., de Wit, S., & Yang, M. 2022
  * Neill et al. (2011) Neill, J. D., et al. 2011, ApJ, 727, 15
  * Ren et al. (2021) Ren, Y., Jiang, B., Yang, M., Wang, T., Jian, M., & Ren, T. 2021, ApJ, 907, 18
  * Sana et al. (2014) Sana, H., et al. 2014, ApJS, 215, 15
  * Smith (2014) Smith, N. 2014, ARA&A, 52, 487
  * Vink (2021) Vink, J. S. 2021, arXiv e-prints, arXiv:2109.08164

Miriam Garcia I have two questions: a. What about using spectroscopic data in
the machine-learning method? and b. In IC 1613 we have detected an LBV-
candidate and a WO star, which I do not see in your results.

Grigoris Maravelias As far as the first question, we want to maximize the
number of targets for which we want to have predicted spectral types. The
largest lists of stars can come from photometry. In supervised machine
learning you need the input data and their corresponding classes for proper
training. When you apply the classifier you have to do it in on a dataset
which has the same input (in our case photometry in the form of color indices)
as your training. That’s why we tried to get as many spectral types as
possible from literature and link them to photometry. If we would like to use
other input data to train (e.g. spectra, spectral parameters) we should have
the exact same features to apply to. And unfortunately there is not such a set
(and if there was, e.g. spectra, the sample size would be much more limited in
contrast to photometry).
For the second question, I actually presented only a part of our results,
based on some quality cuts including those without any missing data. However,
this would open the discussion about how we handle cases with missing values
and I would redirect you to the paper as this demands much more time for
discussion. It is most possible that these sources can be found in the other
parts.

Griffin Hosseinzadeh Is it possible to train the algorithm with lower
metallicity samples?

Grigoris Maravelias With the current use of Spitzer and Pan-STARRS photometry
only M31 and M33 have significant samples. The Magellanic Clouds are not
covered by Pan-STARRS and the inclusion of other surveys would include more
systematics. However, this will change as we do have a large observing
campaign to obtain spectra in many of these galaxies (along with others of
course, e.g. see Marta Lorenzo’s work on Sextans A), so that we can re-train
the classifier in the future with more data.


[a]Physics Division, Los Alamos National Laboratory, Los Alamos, NM, 87545,
USA [b]Chemistry Division, Los Alamos National Laboratory, Los Alamos, NM,
87545, USA [c]Center for Experimental Nuclear Physics and Astrophysics, and
Department of Physics, University of Washington, Seattle, WA 98195, USA
[d]Nuclear Engineering and Nonproliferation Division, Los Alamos National
Laboratory, Los Alamos, NM, 87545 USA<EMAIL_ADDRESS>
# Variable Energy X-ray Fluorescence Source

S.R. Elliott,Corresponding author  E.M. Bond  B. Dodson  G. Rusev  R.
Massarczyk  S.J. Meijer  M. StortiniPermanent Address: Physics Department,
Yale University, New Haven, CT, 06511, USA  C. Wiseman

###### Abstract

We detail the design of a variable energy, x-ray fluorescence source using a
low activity (1.8$\times 10^{6}$ dpm) 99Tc $\beta$ source that irradiates thin
foils. By rotating the source among foils of Ti, Zn, Nb, Ag, and Au, the
device produces x rays between 4 and 70 keV at a rate near 1 Hz. When the
source is placed in a storage position, the external radiation is non-
detectable. The design of the shielding and rotation mechanism permits use in
vacuum and at liquid nitrogen temperature. The design is intended for the
study of the low energy response to radiation impinging upon Ge detector
surfaces. The source will be useful for understanding the detector response in
large-scale Ge arrays such as Majorana and LEGEND.

X-ray fluorescence (XRF) systems, Solid state detectors, Cryogenic detectors

##  1 Introduction

Low-background experiments searching for rare processes using Ge detectors
(e.g. double beta decay) have made significant contributions looking for new
physics at low energy [1, 2, 3, 4, 5, 6, 7, 8], especially following the
development of p-type, point-contact detectors [9, 10, 11]. Models of the
background contributions at energies below a few keV, however, have
significant uncertainty. Data from the Majorana Demonstrator, for example,
show that 210Pb is present as evidenced by the observation of its 46-keV
$\gamma$ ray in the spectrum, as well as the observation of 210Po $\alpha$
emission at 5.3 MeV. Since 210Po is a daughter of 210Pb (Fig. 1), and both,
210Po and 210Bi, have shorter half-lives, some fraction of the Po must be
supported by the Pb decay. While the 46-keV $\gamma$ ray can penetrate the
insensitive layer that encloses most of the detector surface to approximately
1 mm, $\alpha$ particles cannot. However, the passivated layer surrounding the
point contact, and separating the HV-biased n+ surface, is only of the order
of a $\mu$m thick permitting $\alpha$ observation. Since both radiations are
observed, it is assumed that at least some of the 210Pb must irradiate the
passivated surface. Although the spectrum includes the 46.5-keV $\gamma$ line
from 210Pb, it does not observe the associated 10.8-keV x ray [1].
Furthermore, the 46-keV level decay transition has a large internal conversion
coefficient leading to electron emission with energies near 30 keV, which are
also not seen. These non-observations indicate an incomplete understanding of
the response of the passivated surface to radiation, such as low energy
$\gamma$s or electrons, that create charge near the surface.

By scanning the surfaces of a Ge detector with x-rays of various energies, the
depth-related effects of energy deposit as a function of position can be
studied. This motivates our design of a variable energy x-ray fluorescence
source, with energies between $\sim$4-70 keV in a tunable device that can be
operated in vacuum and at liquid nitrogen temperatures. Details on the design
of this source are presented in this paper.

Figure 1: 210Pb Decay Chain.

Although x-ray fluorescence sources have been commercially available in the
past, we were unable to find one that met our present needs. As a consequence,
we chose to design and build a custom source for this purpose. The literature
is rich in discussion of x-ray fluorescence and we referred to a number of
studies that guided us as we optimized the x-ray production. Reference [12] is
an old summary the field of x-ray fluorescence and its use for analysis. The
IAEA report [13] discusses the relative fluorescence rates due to $\alpha$,
$\beta$, $\gamma$, and x-ray excitation. Meijer and Unger [14] estimated x-ray
production from a number of sources and targets for fluorescence analysis.
Reference [15] describes a 1-MeV electron beam to excite La x rays to image
human coronary arteries and Refs. [16, 17, 18] studied the performance of an
x-ray fluorescence emission from a 244Cm source exciting target material for
the Mars Pathfinder program.

##  2 Device Simulation

Our design concept employs a $\beta$ source radiating carefully chosen foils
to produce x-ray fluorescence. By assembling a wheel of differing material
foils that can be spun relative to the radioactive source, a variety of
possible x-ray energies can be produced. We selected a group of 5 foils to
provide high-rate $K_{\alpha}$ emissions somewhat evenly distributed between
$\sim$4-70 keV. Our simulations indicated that a $\beta$ source would produce
more x rays than an $\alpha$ or $\gamma$ source in agreement with Ref. [13]

The ideal source for our needs is a pure $\beta$ decay to a stable daughter
with a long half-life and an endpoint as low as possible, but high enough to
ionize the Au K shell. We considered two primary options, 14C (5730 yr, 156
keV) and 99Tc (2.1$\times 10^{5}$ yr, 297 keV). Our simulations indicated that
14C does not produce significant $K_{\alpha}$ emission in Au. Therefore, we
choose 99Tc as our $\beta$ source.

The key design parameters include the radioactive source, its activity, the
foil thicknesses, and the source-foil housing materials. The figure of merit
used to optimize the device was the predicted percentage of spectrum counts
residing within the $K_{\alpha}$ peak for each foil. The resulting energy
spectra with their associated x-ray and background rates differ for each foil.
Therefore, it was also necessary to design the device holding the source and
foils to maximize the x-ray rate, while minimizing background. We simulated
99Tc radiation of the foils to determine the thickness that maximized the
$K_{\alpha}$ production rate. Prior to obtaining a 99Tc sample, we compared
the simulation to a measurement of an available 14C $\beta$ source irradiating
a Ti foil. The simulation result was within 20% of the measured value
indicating that the simulation worked well enough to guide our design choices.

We simulated foils of Zn ($K_{\alpha}$ = 8.639 keV), Ag (22.163 keV), and Au
(68.804 keV) to establish the appropriate thickness, however, the final
selection of foils included Ti and Nb to provide energies of interest. For the
three foils simulated (Zn, Ag, Au), the optimum thicknesses were found to be
80 $\mu$m, 80 $\mu$m, and 250 $\mu$m, respectively. Readily available
commercial foils come in pre-determined thicknesses and our final selections
were chosen to be close to these simulation optima.

The simulation software used for this study was Geant4 [19]. Given that low-
energy electromagnetic physics was the primary interest here, we used the
physics list “Shielding$\\_$LIV”. A number of electromagnetic processes that
were important to our studies are turned off by default in Geant4, and had to
be enabled. It was also found that the Bearden data [20] in Geant4 simulated
the correct x-ray energies much better than the default fluorescence data.

##  3 Device Design

The construction materials for the device were chosen to minimize background
arising from bremsstrahlung from the stopping $\beta$ rays. The design is a
graded-Z disk separating the source and foils. By surrounding the source with
low Z material, bremsstrahlung is limited and $\beta$s that do not impinge on
the foil are blocked. Surrounding this low-Z region, is a higher-Z material to
shield any remaining bremsstrahlung. The design, shown in Figs. 2 and 3 has an
Al structure contained within W. Tungsten (4.5$\times 10^{-6}$ m/(m K)) and Al
(21$\times 10^{-6}$ m/(m K)) have coefficients of linear expansion similar
enough to address cryogenic engineering issues. Plastic could have been a good
choice for the low-Z shield otherwise. The dimensions of the Al and W were
determined by simulation to reduce background to less than 10% of the signal
rate in the lowest rate configuration with the Au foil.

Starting from the x-ray emission end, the device consists of a W foil holder,
an Al shielding disk, a W shielding disk, and a source holder consisting of
two thin Al pieces. These parts are shown in Fig 2. A drive shaft passes
through the source holder to attach to the foil holder. When the shaft is
turned the foil holder and shield disk rotate, while the source holder remains
stationary. Therefore the choice of irradiated foil can change without
altering the x-ray target spot.

The W foil holder has 5 through holes, 10.626 mm in diameter, centered on
0.5-mm-deep mounting wells of 14.5 mm diameter. The 10-mm diameter foils
reside in these wells and the foil holder is bolted to the bottom of the W
shielding disk trapping the foils next to the Al shielding disk. The Al
shielding disk is 45.5 mm in diameter and 5 mm thick. It has 5 collimation
slits for $\beta$ passage from the source to the foils. The W shielding disk
is 55 mm in diameter but has a 46-mm diameter inner well to enclose the Al
shielding disk. The overall thickness of the W shielding disk is 12 mm with
the well depth being 5 mm to match the thickness of the Al shielding disk.
Slots for $\beta$ passage through the W shielding disk match those in the Al
shielding disk. The slots have varying dimensions chosen to provide similar
x-ray emission rates. The source holder consists of two Al plates, one 3 mm
thick and one 1 mm thick, which form a sandwich trapping the source. The
thicker piece has a 1.1-mm deep well feature, 10.09 mm in diameter, to house
the thin titanium foil with the 99Tc deposit (not to be confused with the Ti
target foil). The 1 mm piece has a through hole, 6.5 mm diameter, that allows
emitted $\beta$s to enter the shielding slots. Tabs on the source holder and
foil holder permit a secure arrangement, when desired. A fully shielded sixth
position places the 99Tc in a safe position with no detectable emanated
radiation.

Figure 2 shows an exploded view of the device. The photos in Fig. 3 show the
source parts and the assembled device attached to a stepper motor via the
shaft. Along the shaft is an infrared shield to minimize radiative heating of
the device when it is in use near a Ge detector. Also shown is a G-10 shaft
adapter to minimize heat conductance down the shaft.

Figure 2: An exploded view of the device.

Figure 3: Left: The foil holder, W shielding disk, Al shielding disk and the 1
mm slice of the source sandwich. Right: The final assembled device.

##  4 Source Preparation

The 99Tc electrodeposition method was developed based on a procedure by Voltz
[21], using an electrodeposition cell that has been previously described [22].
A 2.77-mL aliquot of the 99Tc solution (obtained from Eckert & Ziegler [23];
2.22$\times 10^{7}$ dpm in 5 g, 0.001 M NaOH) was dried to $\sim$0.2 mL and
transferred to the electrodeposition cell, followed by 1.35 mL of the
electrodeposition solution (2 M (NH4)2SO4, adjusted to pH = 1). The
electrodeposition was carried out for 60 minutes at 0.1 A using a 51 $\mu$m Ti
foil as the cathode. The resulting black deposit was dried on a hotplate set
to 110 C and assayed by beta analysis: 1.77$\times 10^{6}$ dpm for a 15%
yield. To reduce the risk of contamination, the deposit was covered by a
transparent 2 $\mu$m PET film with an 8 $\mu$m thick acrylic adhesive from
PolyK Technologies LLC.

##  5 Results and Measurements

We measured the x rays from the source with a FAST-SDD® silicon drift detector
from Amptek Inc. [24]. The detector has a thickness of 0.5 mm and collimated
to an area of 50 mm2. It is placed in a vacuum in a casing with a 0.5-mil-thin
beryllium window and cooled at 220 K. The signals from the detector were
processed by a stand-alone digital pulse processor and the Apmtek data
acquisition program. We measured the efficiency of the detector with a set of
Eckert & Ziegler standard sources of 54Mn, 57Co, 75Se, 109Cd, 133Ba, 139Ce,
153Gd, and 241Am.

The efficiency ($\epsilon$) for detecting x rays with this detector as a
function of energy ($E$) and distance ($D$) is shown in Fig. 4. The curves
through the data points are an empirical parameterization to aid in
interpolating between data points. The results of this interpolation are given
in Table 1 where the quoted uncertainty includes both contributions. The
parametric formulas are,

$\epsilon(E)=\sum_{k=0}^{5}a_{k}\cdot\mathrm{log(E)}^{k}$ (1)

and

$\epsilon(D)=\delta\cdot e^{\eta D}+\kappa\cdot e^{\lambda D}$ (2)

with the parameter fit values given in the Appendix. The measured points in
the efficiency curves are known rather well (0.5% or better). The efficiency
uncertainty for our data, however, is dominated by the uncertainty in the
placement of the source relative to the detector as the geometry for our
device was not optimum for this purpose. Our placement estimate is 2$\pm$1 mm
for the Ti and Zn foils, and 5$\pm$1 mm for the others. We used the position
dependent parameterization to propagate this position uncertainty to estimate
the efficiency uncertainty.

Figure 4: Left: the efficiency curves for the detector as a function of energy
for 5 separation distances. Right: The efficiency for our 5 lines of interest
as a function of distance.

Figure 5 shows example spectra. All spectra show signs of the $K_{\alpha}$ and
$K_{\beta}$ lines, though the Au result is weak due to the low efficiency and
reduced production rate of the higher energy x ray. Table 1 summarizes the
measured results with a listing of the x-ray production per $\beta$ decay of
the 99Tc. The detector efficiency is not precise at the low energy Ti x ray,
as there were no efficiency calibration points used below 5.9 keV, the
efficiency here was extrapolated, rather than interpolating. The detection
efficiency also changes rapidly in this region, together resulting in lower
confidence in the efficiency value at this energy.

Figure 5: Example spectra from Ti, Zn, Nb and Ag (left), where the $K_{\alpha}$ and $K_{\beta}$ lines are visible and Au (right), where K${}_{\alpha_{2}}$, K${}_{\alpha_{1}}$ and K${}_{\beta_{1}}$ are seen.  Table 1: The measurement parameters and results. See text for discussion of the Ti results and concerns regarding efficiency. The line energies are from Ref. [25] where we use the experimental values. Foil | Thick. | Kα Ener. | Kβ Energy | Live | Counts | Efficiency | x ray | Rate/99Tc Decay
---|---|---|---|---|---|---|---|---
| ($\mu m$) | (keV) | (keV) | Time (s) |  | (%) | Rate (Hz) |  $\times 10^{-5}$ (Hz)
Ti |  050 |  04.511 |  04.932 |  00571 | 707$\pm$27 | 0.16$\pm$0.02 | 800$\pm$100 | (2600$\pm 300)$
Zn |  075 |  08.639 |  09.572 |  05979 | 738$\pm$27 | 7.16${}^{+1.41}_{-1.17}$ | 1.72${}^{+0.35}_{-0.29}$ | (5.84${}^{+1.17}_{-0.98})$
Nb | 100 | 16.615 | 18.622 |  03602 | 411$\pm$20 | 9.11${}^{+1.88}_{-1.55}$ | 1.25${}^{+0.27}_{-0.22}$ | (4.25${}^{+0.90}_{-0.75})$
Ag |  075 | 22.163 | 24.942 |  08088 | 986$\pm$31 | 5.31${}^{+1.13}_{-0.92}$ | 2.30${}^{+0.49}_{-0.40}$ | (7.78${}^{+1.67}_{-1.37})$
Au | 259 | 68.804 | 77.980 | 49880 |  046$\pm$70 | 0.17$\pm 0.03$ | 0.54$\pm 0.12$ | (1.84$\pm 0.42)$

##  6 Summary

We have constructed and tested a variable x-ray source using $\beta$ ray
fluorescence producing an x-ray emission rate of about 1 Hz between 4 and 70
keV. A low radioactivity 99Tc deposit provided the $\beta$ rays and when this
activity is placed in its storage position, the external radiation is non-
detectable. The device was designed to be used in vacuum at 80 K and is
intended to be used to explore the low energy response of a Ge detector
passivated surface to radiation.

## Acknowledgements

We thank Tom Burritt for his machining expertise. We gratefully acknowledge
the support of the U.S. Department of Energy through the LANL Laboratory
Directed Research and Development (LDRD) Programs for this work. This material
is based upon work supported by the U.S. Department of Energy, Office of
Science, Office of Nuclear Physics under Federal Prime Agreement LANLEM78, and
under award number DE-FG02-97ER41020.

## Appendix: Efficiency Parameters

Table 2 gives the values for the various constants for the equation for
$\epsilon(E)$ and Table 3 gives the constants for $\epsilon(D)$.

Table 2: The parameters for $\epsilon(E)$ resulting from the fits with Eq. 1. | 2 mm | 22 mm | 42 mm | 62 mm | 82 mm
---|---|---|---|---|---
a0 | -97.07 | -93.27 | -131.6 | -140.6 | -144.9
a1 | 151.8 | 139.4 | 204.3 | 218.4 | 226.0
a2 | -91.4 | -82.35 | -126.5 | -135.9 | -141.8
a3 | 27.45 | 24.16 | 38.81 | 41.89 | 44.10
a4 | -4.14 | -3.563 | -5.939 | -6.435 | -6.832
a5 | 0.2485 | 0.2096 | 0.3606 | 0.3919 | 0.4195
Table 3: The parameters for $\epsilon(D)$ resulting from the fits with Eq. 2. | Ti | Zn | Nb | Ag | Au
---|---|---|---|---|---
$\delta$ | 0.008194 | 17.11 | 12.83 | 7.562 | 0.00710
$\eta$ | -0.3373 | -0.194 | -0.195 | -0.199 | -0.0319
$\kappa$ | 0.3108 | 0.799 | 0.4543 | 0.249 | 0.234
$\lambda$ | -0.1543 | -0.0341 | -0.0340 | -0.0336 | -0.1938

## References

  * [1] I. J. Arnquist, others., Exotic dark matter search with the Majorana Demonstrator (2022).  arXiv:2206.10638.
  * [2] I. J. Arnquist, others., Search for solar axions via axion-photon coupling with the Majorana Demonstrator (2022).  arXiv:2206.05789.
  * [3] I. J. Arnquist, others., A search for spontaneous radiation from wavefunction collapse in the Majorana Demonstrator (2022).  arXiv:2202.01343.
  * [4] J.-W. Chen, et al., Atomic ionization by sterile-to-active neutrino conversion and constraints on dark matter sterile neutrinos with germanium detectors, Phys. Rev. D 93 (2016) 093012.
  * [5] H. B. Li, et al., Limits on spin-independent couplings of wimp dark matter with a p-type point-contact germanium detector, Phys. Rev. Lett. 110 (2013) 261301\.
  * [6] Z. Y. Zhang, et al., Constraints on sub-gev dark matter-electron scattering from the CDEX-10 experiment (2022).  arXiv:2206.04128.
  * [7] Y. Wang, et al., First experimental constraints on wimp couplings in the effective field theory framework from CDEX (2022).  arXiv:2007.15555.
  * [8] C. E. Aalseth, et al., CoGeNT: A search for low-mass dark matter using p-type point contact germanium detectors, Phys. Rev. D 88 (2012) 012002\.
  * [9] P. N. Luke, F. S. Goulding, N. W. Madden, R. H. Pehl, Low capacitance large volume shaped-field germanium detector, IEEE Trans. Nucl. Sci. 36 (1989) 926.
  * [10] P. S. Barbeau, J. I. Collar, O. Tench, Large-mass ultralow noise germanium detectors: performance and applications in neutrino and astroparticle physics, JCAP 09 (2007) 009.
  * [11] R. Cooper, D. Radford, P. Hausladen, K. Lagergren, A novel hpge detector for gamma-ray tracking and imaging, Nucl. Instrum. Meth. A 665 (2011) 25–32.  doi:https://doi.org/10.1016/j.nima.2011.10.008.
URL https://www.sciencedirect.com/science/article/pii/S0168900211018985

  * [12] V. Valkovic, A. Markowicz, N. Haselberger, Review of recent applications of radioisotope excited x-ray fluorescence, X-Ray Spect. 22 (1993) 199.
  * [13] C. G. Clayton, et al., Radioisotope x-ray fluorescence spectrometry, Tech. Rep. 115, International Atomic Energy Agency, Vienna, 1970 (1970).
  * [14] H. Meijer, E. Unger, On the application of radioisotope x-ray fluorescence analysis for the solution of environmental and industrial problems, J. Radioanal. Chem. 32 (1976) 413.
  * [15] H. L. Manning, R. E. Shefer, R. E. Klinkowstein, C. A. Mistretta, A $k_{\alpha}$ dual energy x-ray source for coronary angiography, Med. Phys. 18 (1991) 880.
  * [16] R. Rieder, H. Wänke, T. Economou, A. Turkevich, Determination of the chemical composition of martian soil and rocks: The alpha proton x-ray spectrometer, J. Geophys. Res. 102 (E2) (1997) 4207.
  * [17] J. Brückner, G. Dreibus, R. Rieder, H. Wänke, Refined data of alpha proton xray spectrometer analyses of soils and rocks at the mars pathfinder site: Implications for surface chemistry, J. Geophys. Res.-Planets 108 (E12) (2003) 1.
  * [18] G. M. Perrett, Calibration of the mars science laboratory alpha particle x-ray spectrometer for analysis of visible elements and light invisible components, Ph.D. thesis, The University of Guelph (2015).
  * [19] S. Agostinelli, et al., Geant4: a simulation toolkit, Nucl. Instrum. Meth. Phys. Res. A 506 (2003) 250.
  * [20] J. A. Bearden, X-ray wavelengths, Rev. Mod. Phys. 39 (1967) 78.
  * [21] R. Voltz, M. I. Holt, Electrodeposition of 99Tc from aqueous solution, J. Electrochem. Soc. 114 (1967) 128.
  * [22] J. R. Fitzpatrick, E. Bond, A. Slemmons, D. Vieira, Preparation of americium targets for nuclear chemistry experiments at dance, J. Radioanal. Nucl. Chem. 276 (2008) 561.
  * [23] Eckert & Ziegler, BEBIG GmbH, Robert-Rössle-Str. 10, 13125 Berlin, Germany.
  * [24] Amptek, Inc, 14 DeAngelo Drive, Bedford, MA 01730 USA.
  * [25] R. D. Deslattes, et al., X-ray transition energies (version 1.2), database access date: Oct. 25, 2022 (2005).
URL http://physics.nist.gov/XrayTrans