# New physics search via CP observables in $B^{0}_{s}\rightarrow\phi\phi$
decays with left- and right-handed Chromomagnetic operators

Tejhas Kapoor<EMAIL_ADDRESS>Université Paris-Saclay,
CNRS/IN2P3, IJCLab, 91405 Orsay, France  Emi Kou  Université Paris-Saclay,
CNRS/IN2P3, IJCLab, 91405 Orsay, France

###### Abstract

In this paper, we investigate the time-dependent angular analysis of
$B_{s}^{0}\rightarrow\phi\phi$ decay to search for new physics signals via CP-
violating observables. We work with a new physics Hamiltonian containing both
left- and right-handed Chromomagnetic dipole operators. The hierarchy of the
helicity amplitudes in this model gives us a new scheme of experimental
search, which is different from the ones LHCb has used in its analysis. To
illustrate this new scheme, we perform a sensitivity study using two pseudo
datasets generated using LHCb’s measured values. We find the sensitivity of
CP-violating observables to be of the order of $5-7\%$ with the current LHCb
statistics. Moreover, we show that Belle(II)’s $B^{0}_{d}\rightarrow\phi
K_{s}$ and LHCb’s $B_{s}^{0}\rightarrow\phi\phi$ measurements could be coupled
within our model to obtain the chirality of the new physics.

##  1 Introduction

Currently, the only confirmed source of CP violation is the Kobayashi-Maskawa
(KM) phase present in the CKM matrix [1][2], which arises when we move the
quarks from flavour to mass eigenstate in the Standard Model (SM). However, we
expect to find more sources of CP violation owing to the observed matter-
antimatter asymmetry in the universe [3]. Thus, it is imperative to look for
CP-violating observables, especially those which are very small or zero in SM,
because if they deviate even slightly from zero (which can be checked by a
null test), it would not just be a discovery of a new source of CP violation,
it would also be a smoking-gun signal of new physics (NP).

In this article, we study the $B_{s}^{0}\rightarrow\phi\phi$ decay (where
$\phi(1020)$ is implied throughout this paper), which is a pure penguin
process. The presence of penguin quantum loop makes it an excellent probe to
search for new heavy particles and being a purely penguin-type decay keeps it
free from tree-penguin interference contamination, making it a clean
observable to search for NP [4][5][6]. The object of interest is going to be
the phase in the interference of the direct decay of $B^{0}_{s}$ mesons and
decay via mixing of $B_{s}^{0}-\bar{B}_{s}^{0}$ to CP eigenstates, which is a
CP-violating parameter. This phase is expected to be very small in SM
($-2\beta_{s}\approx O(\lambda^{2})$). In this paper, we will be presenting a
new scheme for the interference phases of different helicities within the
framework of our chosen model of study.

The objective of this article is threefold. The first is to show the power of
angular decay distribution: it can help segregate the final state when it is a
mixture of different helicities. Combining it with a $P\rightarrow VV$ type
decay ($P$-pseudoscalar particle and $V$-vector particle) gives us access to
three (helicity) amplitudes instead of one, meaning we can go beyond the
assumption of helicity-independent phases to probe three CP-violating phases,
and possibly three new indicators of NP.

Secondly, we present a new scheme for the interference phases, based on the
hierarchy of helicity amplitudes arising in our model, which is different from
the ones LHCb used in its fits [7]. We also note the fact that LHCb’s
objective is to do a null test of the interference phase, without any regard
to its origin (decay or mixing). However, we specifically assume that the weak
phase is coming from decay amplitude, not mixing amplitude. Consequently, we
change the form of helicity/transversity amplitude to include a CP-violating
decay phase. This modifies the coefficients of time-dependent part of
amplitude, which we present in Table 5. In addition, we investigate the
$B^{0}_{d}\rightarrow\phi K_{s}$ decay amplitude with our NP Hamiltonian. We
show that the Belle(II)’s $B^{0}_{d}\rightarrow\phi K_{s}$ decay measurement
along with the LHCb’s $B^{0}_{s}\rightarrow\phi\phi$ measurement can provide
the chirality of NP in our model, as long as the signs of cosine of strong
phases of these decays can be obtained from the theory.

Lastly, we perform a sensitivity study to illustrate the new scheme of
experimental analysis we are proposing. We perform a fit with two pseudo
datasets (based on two sets of result of LHCb) to calculate the sensitivities
of the CP-violating parameters, which also act as null test parameters for new
physics.

The model we choose to use in our study is that of the _Chromomagnetic dipole
operator_ _$O_{8g}$_ , which, for $\bar{b}\rightarrow\bar{s}g$ process, is
given as follows:

$\displaystyle
O_{8g}=\frac{g_{s}}{8\pi^{2}}m_{b}\bar{b}_{\alpha}\sigma^{\mu\nu}(1+\gamma^{5})\frac{\lambda^{a}_{\alpha\beta}}{2}s_{\beta}G^{a}_{\mu\nu}.$
(1)

To keep it very general, we work with both left- and right-handed New Physics
currents; this is done by adding the Chromomagnetic operator with opposite
quark chiralities (i.e. by flipping sign of $\gamma^{5}$) (see Eq. (13)). Our
choice of model is motivated by the fact that the Chromomagnetic operator has
been observed to be sensitive to contributions from various NP models
[8][9][10][11][12][13][14][15]. There are some NP models that give the same
contribution to both decay and mixing amplitudes. This causes the contribution
to cancel out in the interference phase, and they remain undetectable in
$B^{0}_{s}\rightarrow\phi\phi$ decay. However, since Chromomagnetic operator
only contributes to the decay amplitudes, a NP contribution manifesting itself
through this operator can be very well detected via this channel (the
$B_{s}^{0}-\bar{B}_{s}^{0}$ mixing amplitude has already been well constrained
by previous measurements [16][17], so we do not focus on it in this work). We
emphasise that the choice of the model is not unique, as other models could
possibly give similar contributions. Our objective is simply to show how to
fully utilise the power of angular distribution when coupled with
$P\rightarrow VV$ decays, and provide a new fit scenario, which can be used to
check for the presence of NP.

##  2 Angular decay distribution

The angular decay distribution for $B^{0}_{s}\rightarrow\phi\phi$ decay can be
described by the help of three angles as shown in Figure 1. A random choice is
made for which $\phi$ meson is used to determine $\theta_{1}$ and
$\theta_{2}$. The power of angular analysis is that it can disentangle the
final states of $B^{0}_{s}\rightarrow\phi\phi$ decay (which is a mixture of CP
eigenstates) and we get access to three (helicity/transversity) amplitudes
instead of one, meaning we can probe three CP-violating phases, and possibly
three new indicators of NP. We will neglect the contribution of scalar
$f_{0}(980)$ resonance, as it can be removed by appropriate experimental cuts
[7][18].

The amplitude for $B^{0}_{s}\rightarrow\phi\phi$ is given by

$\displaystyle\mathcal{A}(t,\theta_{1},\theta_{2},\Phi)$
$\displaystyle=A_{0}(t)\cos\theta_{1}\cos\theta_{2}+\frac{A_{\parallel}(t)}{\sqrt{2}}\sin\theta_{1}\sin\theta_{2}\cos\Phi$
$\displaystyle+i\frac{A_{\perp}(t)}{\sqrt{2}}\sin\theta_{1}\sin\theta_{2}\sin\Phi,$
(2)

where $A_{0}$ is the longitudinal CP-even, $A_{\parallel}$ is the transverse-
parallel CP-even and $A_{\perp}$ is the transverse-perpendicular CP-odd
transversity amplitude. The resulting angular decay distribution is
proportional to square of the amplitude in Eq. (2) and has 6 terms [19]:

$\displaystyle\frac{d^{4}\Gamma}{dtd\cos\theta_{1}\,d\cos\theta_{2}\,d\Phi}\propto|\mathcal{A}(t,\theta_{1},\theta_{2},\Phi)|^{2}=\frac{1}{4}\sum_{i=1}^{6}K_{i}(t)f_{i}(\theta_{1},\theta_{2},\Phi)$
(3)

The angular dependence contained in $f_{i}(\theta_{1},\theta_{2},\Phi)$ is as
follows:

$\displaystyle|\mathcal{A}(t,\theta_{1},\theta_{2},\Phi)|^{2}$
$\displaystyle=\frac{1}{4}\bigl{[}4K_{1}(t)\cos^{2}\theta_{1}\cos^{2}\theta_{2}+K_{2}(t)\sin^{2}\theta_{1}\sin^{2}\theta_{2}(1+\cos
2\Phi)$ $\displaystyle+K_{3}(t)\sin^{2}\theta_{1}\sin^{2}\theta_{2}(1-\cos
2\Phi)-2K_{4}(t)\sin^{2}\theta_{1}\sin^{2}\theta_{2}\sin 2\Phi$
$\displaystyle+\sqrt{2}K_{5}(t)\sin 2\theta_{1}\sin
2\theta_{2}\cos\Phi-\sqrt{2}K_{6}(t)\sin 2\theta_{1}\sin
2\theta_{2}\sin\Phi\bigr{]}.$ (4)

The time dependence is contained in $K_{i}(t)$ which is defined as

$\displaystyle\begin{aligned}
K_{i}(t)=N_{i}e^{-\Gamma_{s}t}\bigg{[}&a_{i}\cosh\left(\frac{1}{2}\Delta\Gamma_{s}t\right)+b_{i}\sinh\left(\frac{1}{2}\Delta\Gamma_{s}t\right)+c_{i}\cos(\Delta
m_{s}t)+d_{i}\sin(\Delta m_{s}t)\bigg{]}.\end{aligned}$ (5)

The coefficients $a_{i},b_{i},c_{i}$ and $d_{i}$ are the LHCb experimental
observables given in Table 5. $\Delta\Gamma_{s}\equiv\Gamma_{L}-\Gamma_{H}$ is
decay-width difference between the light and heavy $B_{s}^{0}$ mass
eigenstate, $\Gamma_{s}\equiv(\Gamma_{L}+\Gamma_{H})/2$ is the average decay
width and $\Delta m_{s}\equiv m_{H}-m_{L}$ is the mass difference between the
heavy and light $B_{s}^{0}$ mass eigenstate, and also the
$B_{s}^{0}-\bar{B}_{s}^{0}$ oscillation frequency. Their values are
$\Delta\Gamma_{s}=0.086\pm 0.006\text{ ps}^{-1}$ and $\Gamma_{s}=0.6646\pm
0.0020\text{ ps}^{-1}$ [18], and the oscillation frequency is constrained by
the LHCb measurement to be $\Delta m_{s}=17.768\pm 0.023\text{ (stat)}\pm
0.006\text{ (syst)}\text{ ps}^{-1}$ [20].

Figure 1: Decay angles for the $B_{s}^{0}\rightarrow\phi\phi$ decay, where
$\theta_{1(2)}$ is the angle between the $K^{+}$ momentum in the $\phi_{1(2)}$
meson rest frame and the $\phi_{1(2)}$ momentum in the $B_{s}^{0}$ rest frame.
$\Phi$ is the angle between the two $\phi$ meson decay planes.

##  3 Search for new physics

###  3.1 CP-violating phase in the Standard Model

Before looking at how to search for NP, we must know the SM predictions [21].
We know that phase in the interference of decay with and without mixing is
almost zero in SM in $B^{0}_{s}\rightarrow\phi\phi$ decays because the KM
phase in $B_{s}^{0}$ decay amplitude cancels the one arising from the
$B_{s}^{0}-\bar{B}_{s}^{0}$ mixing box diagram (considering the dominant
$t$-quark contribution). But for a more accurate prediction of phase (to
higher orders in $\lambda$), we need to consider the contribution of $u$ and
$c$-quarks too. These contributions can arise due to QCD rescattering
$c\bar{c}\rightarrow q\bar{q}$ and $u\bar{u}\rightarrow q\bar{q}$ ($q=d,s$)
from tree operators $\bar{b}\rightarrow\bar{c}c\bar{s}$ and
$\bar{b}\rightarrow\bar{u}u\bar{s}$, respectively, and may have a contribution
up to around $20-30\%$ [22] of the dominant top amplitude. Taking into account
these contributions, the SM amplitude for $\bar{b}\rightarrow\bar{s}$ decay
for a given helicity ’$k$’ can be written as

$\displaystyle
A_{k}^{\rm{SM}}=\lambda_{t}P_{t,k}+\lambda_{c}R_{c,k}+\lambda_{u}R_{u,k}$ (6)

$\lambda_{q}=V_{qb}^{*}V_{qs}$ is the CKM matrix element. Here, while
$P_{t,k}$ arises due to gluonic penguin with a $\bar{t}$-quark in the loop,
$R_{c,k}$ and $R_{u,k}$ are the rescattering contribution. Using unitarity of
CKM matrix to eliminate the $c$-quark contribution and writing strong phases
explicitly, we get

$\displaystyle\begin{aligned}
A_{k}^{\rm{SM}}&=|V_{tb}^{*}V_{ts}|e^{-i\beta_{s}}|PR_{tc,k}|e^{i\delta_{tc,k}}+|V_{ub}^{*}V_{us}|e^{i\gamma}|RR_{uc,k}|e^{i\delta_{uc,k}}\\\
&=|V_{tb}^{*}V_{ts}|e^{-i\beta_{s}}|PR_{tc,k}|e^{i\delta_{tc,k}}\left[1+r_{k}^{\rm{SM}}e^{i(\gamma+\beta_{s})}e^{i(\delta_{uc,k}-\delta_{tc,k})}\right]\\\
&=|A_{k}^{\rm{SM}}|e^{i\phi^{\rm{SM}}}e^{i\delta_{k}^{\rm{SM}}},\end{aligned}$
(7)

where $PR_{tc,k}=P_{t,k}-R_{c,k}$, $RR_{uc,k}=R_{u,k}-R_{c,k}$, $\delta$
denote the SM strong phases,
$\beta_{s}=arg(\frac{-V_{ts}V_{tb}^{*}}{V_{cs}V_{cb}^{*}})\approx\eta\lambda^{2}$
and
$r_{k}^{\rm{SM}}=\frac{|V_{ub}^{*}V_{us}||RR_{uc,k}|}{|V_{tb}^{*}V_{ts}||PR_{tc,k}|}$.
Assuming that that the rescattering contribution is around $20-30\%$ of the
dominant penguin amplitude, we can write
$\frac{|RR_{uc,k}|}{|PR_{tc,k}|}=O(\lambda)$, as $\lambda\approx 0.22$.
Therefore, we have $r_{k}^{\rm{SM}}=O(\lambda^{3})$. This would make all the
CP-violating observables (like indirect CP asymmetry, triple product
asymmetries etc.) $O(\lambda^{3})$ (or smaller). This can be inferred from the
fact that CP violation occurs when two amplitudes interfere. Since one of the
amplitudes is much smaller than the other (by O($\lambda^{3}$)), the observed
CP-violating observables are $O(\lambda^{3})$ too. This becomes one of the key
point in our NP search: since we know the order of CP-violating observables in
SM, any observable that is larger than $O(\lambda^{3})$ would be a clear
signal of NP.

###  3.2 CP-violating phase in the new physics model:
Parametrisation

In this study, as mentioned before, we are only probing CP-violating phases in
the decay; thus, our parametrisation is done accordingly. Here, for
generality, we include both left- and right-handed currents (which could arise
from several NP models), which could give rise to new CP-violating phase(s).
Also, we assume $|\frac{q}{p}|=1$ [23].

The helicity/transversity amplitudes, with helicity/transversity ’$k$’ are
written as [24]

$\displaystyle\begin{aligned}
A_{k}(t)&=\langle(\phi\phi)_{k}|\mathcal{H}_{\rm{eff}}|B_{s}^{0}(t)\rangle=g_{+}(t)A_{k}+\frac{q}{p}g_{-}(t)\bar{A}_{k}\\\
\bar{A}_{k}(t)&=\langle(\phi\phi)_{k}|\mathcal{H}_{\rm{eff}}|\bar{B}_{s}^{0}(t)\rangle=g_{+}(t)\bar{A}_{k}+\frac{p}{q}g_{-}(t)A_{k}.\end{aligned}$
(8)

where $g_{+}(t)$ and $g_{-}(t)$ describe the time evolution of $B^{0}_{s}$ and
$\bar{B}^{0}_{s}$, respectively. Using Eq. (7) and adding a NP component, the
amplitude at $t=0$ can be written as:

$\displaystyle\begin{aligned} A_{k}(0)\equiv A_{k}&=A_{k}^{\rm SM}+A_{k}^{\rm
NP}\\\ &=|A_{k}^{\rm SM}|e^{i\delta_{k}^{\rm SM}}e^{i\phi^{\rm
SM}}+|A_{k}^{\rm NP}|e^{i\delta_{k}^{\rm NP}}e^{i\phi_{k}^{\rm NP}}\\\
&=|A_{k}^{\rm SM}|e^{i\delta_{k}^{\rm SM}}e^{i\phi^{\rm SM}}\left(1+r_{k}^{\rm
NP}e^{i(\phi_{k}^{\rm NP}-\phi^{\rm SM})}e^{i(\delta_{k}^{\rm
NP}-\delta_{k}^{\rm SM})}\right)\\\ &=|A_{k}^{\rm SM}|e^{i\delta_{k}^{\rm
SM}}e^{i\phi^{\rm SM}}X_{k}e^{i\theta_{k}},\end{aligned}$ (9)

where in the last line, we denote the quantity in the parenthesis as
$X_{k}e^{i\theta_{k}}$ and $r_{k}^{\rm NP}=\frac{|A_{k}^{\rm NP}|}{|A_{k}^{\rm
SM}|}$. The phase $\theta_{k}$ is a _mixture of weak and strong phases_.
111Notice that if we assume $\delta_{k}^{\rm SM}=\delta_{k}^{\rm NP}$, then
the phase $\theta_{k}$ would be a purely weak phase. In such a case the
interference phase in Eq. (11) would not just tell us about the presence of
NP, it would also tell us the value of CP-violating (weak) phase in the decay
amplitude. However, we work in the most general case in this calculation, as
we only wish to probe for NP.Similarly for the CP-conjugate amplitude, the
expression is ($\eta_{k}$ is the CP eigenvalue of the transversity state, with
$\eta_{\perp}=-1$ and $\eta_{0,\parallel}=1$)

$\displaystyle\begin{aligned} \bar{A}_{k}&=\eta_{k}|A_{k}^{\rm
SM}|e^{i\delta_{k}^{\rm SM}}e^{-i\phi^{\rm SM}}\left(1+r_{k}^{\rm
NP}e^{-i(\phi_{k}^{\rm NP}-\phi^{\rm SM})}e^{i(\delta_{k}^{\rm
NP}-\delta_{k}^{\rm SM})}\right)\\\ &=\eta_{k}|A_{k}^{\rm
SM}|e^{i\delta_{k}^{\rm SM}}e^{-i\phi^{\rm
SM}}X_{k}^{c}e^{i\theta_{k}^{c}}.\end{aligned}$ (10)

Recalling that $\arg(q/p)=2\beta_{s}\approx 2\phi^{SM}$, we finally get

$\displaystyle\frac{q}{p}\frac{\bar{A}_{k}}{A_{k}}=\eta_{k}\lambda_{k}e^{-i(\theta_{k}-\theta_{k}^{c})},$
(11)

where $\lambda_{k}\equiv\frac{|\bar{A}_{k}|}{|A_{k}|}=\frac{X^{c}_{k}}{X_{k}}$
becomes the direct CP violation measurement parameter: $\lambda_{k}\neq 1$
implies direct CP violation is present in the decay. Since in SM,
$\lambda_{k}=1$ for all helicities, the deviation of this value from $1$ (by
more than $O(\lambda^{3})$) would be a clear signal for NP, i.e.
$\lambda_{k}-1$ is a null-test parameter for NP. Another quantity that can be
used for NP search is the interference phase $\theta_{k}-\theta_{k}^{c}$. In
SM, this quantity is zero, as explained in Section 3.1. Therefore, the
deviation of this quantity from zero (by more than $O(\lambda^{3})$) would be
a signal of NP, i.e. $\theta_{k}-\theta_{k}^{c}$ is also a null-test parameter
for NP. One must note that there is one special case when neither of these two
parameters would be able to detect the presence of NP: it is the case when
$\phi^{\rm NP}_{k}=\phi^{\rm SM}$. In this case, $\lambda_{k}=1$ and
$\theta_{k}-\theta_{k}^{c}=0$ and NP cannot be detected by CP-violating
observables.

The time-dependent amplitude is given by

$\displaystyle\begin{aligned}
A_{k}(t)&=A_{k}\left[g_{+}(t)+g_{-}(t)\frac{q}{p}\frac{\bar{A}_{k}}{A_{k}}\right]\\\
A_{k}(t)&=|A_{k}^{SM}|X_{k}e^{i\delta_{k}^{SM}}e^{i\phi^{SM}}e^{i\theta_{k}}\left[g_{+}(t)+g_{-}(t)\eta_{k}\lambda_{k}e^{-i(\theta_{k}-\theta_{k}^{c})}\right].\end{aligned}$
(12)

The coefficients of the time-dependent terms in Eq. (5), obtained by using Eq.
(12), are given in Table 5. Our equation differs slightly from the one given
by LHCb, since we keep the phase $\phi^{\rm{SM}}+\theta_{k}$ separate from
$\delta_{k}^{\rm SM}$, while LHCb interprets the total phase outside the
bracket as a strong phase (even though it contains a small weak
component).222This assumption will have no effect when $\theta_{k}$ and
$\theta^{c}_{k}$ are helicity independent, since it will cancel out when we
write terms of type $A_{i}(t)A_{k}^{*}(t)$ or $|A_{i}(t)|^{2}$, and thus, our
formula and that of LHCb would exactly be the same.  This is acceptable, since
their objective is doing a null test of the phase in the interference of decay
with and without mixing. However we wish to specifically probe the presence of
a possible new weak phase in the decay amplitude, hence we keep the strong and
weak phase separate in the decay amplitude.333The mixing-induced CP violation
(i.e. the CP violation in interference of decay with and without mixing)
cannot tell us if the original source of this effect is coming from the
dynamics of $\Delta B=2$ (mixing) sector or that of $\Delta B=1$ (decay)
sector, as long as we are working with only one final state (or one pair of
CP-conjugate final state). We need information from at least one more final
state to decide unambiguously the presence of direct CP violation and/or CP
violation in mixing [25]. This changes the coefficients in Table 5 with
respect to the ones given by LHCb [7]. For simplicity of notation, we simply
call $\delta_{k}^{\rm SM}$ as $\delta_{k}$ in the rest of the paper.

##  4 Helicity amplitude analysis

###  4.1 New physics model

In this section, we apply the technique of helicity amplitude analysis to a
model with operator $O_{8g}$ with both left- and right-handed currents. As
mentioned before, our choice of operator $O_{8g}$ is motivated by the fact
that it is sensitive to NP, and that it contributes only to $B_{s}^{0}$ decay
amplitude, and not $B_{s}^{0}-\bar{B}_{s}^{0}$ mixing amplitude.

Starting from the effective Hamiltonian for $\Delta B=1$ decay, it is given by
$(q\in\\{d,s\\})$

$\displaystyle\mathcal{H}_{\rm{eff}}=-\frac{G_{F}}{\sqrt{2}}V^{*}_{tb}V_{tq}\left[\sum_{i=3}^{6}(C_{i}^{\rm{SM}}O_{i})+C_{8g}O_{8g}+\tilde{C}_{8g}\tilde{O}_{8g}\right]+\rm{h.c.}$
(13)

The operators given by $(q^{\prime}\in\\{u,d,s,c\\})$

$\displaystyle\begin{aligned}
O_{3}&=(\bar{b}_{\alpha}q_{\alpha})_{V-A}\sum_{q^{\prime}}(\bar{q}^{\prime}_{\beta}q^{\prime}_{\beta})_{V-A}\\\
O_{4}&=(\bar{b}_{\beta}q_{\alpha})_{V-A}\sum_{q^{\prime}}(\bar{q}^{\prime}_{\alpha}q^{\prime}_{\beta})_{V-A}\\\
O_{5}&=(\bar{b}_{\alpha}q_{\alpha})_{V-A}\sum_{q^{\prime}}(\bar{q}^{\prime}_{\beta}q^{\prime}_{\beta})_{V+A}\\\
O_{6}&=(\bar{b}_{\beta}q_{\alpha})_{V-A}\sum_{q^{\prime}}(\bar{q}^{\prime}_{\alpha}q^{\prime}_{\beta})_{V+A}\end{aligned}$
(14)

with the notation $(\bar{a}b)_{V\pm A}(\bar{c}d)_{V\pm
A}=\bar{a}\gamma^{\mu}(1\pm\gamma^{5})b\;\bar{c}\gamma_{\mu}(1\pm\gamma^{5})d$.

We only include the gluonic penguin operators, as
$B_{s}^{0}\rightarrow\phi\phi$ is a purely gluonic process. The operators with
tilde are obtained by changing the sign of $\gamma_{5}$ term in the definition
of operators with no tilde. We can express
$\mathord{\mathop{O}\limits^{\scriptscriptstyle(\sim)}}_{8g}$, given in Eq.
(1), in terms of operators
$\mathord{\mathop{O}\limits^{\scriptscriptstyle(\sim)}}_{3-6}$, the details of
which are given in Appendix A. Once we do this, we can write the effective
Hamiltonian as

$\displaystyle\mathcal{H}_{\rm{eff}}=-\frac{G_{F}}{\sqrt{2}}V^{*}_{tb}V_{tq}\sum_{i=3}^{6}(C_{i}^{\rm{SM}}O_{i}+C_{i}^{\rm{L}}O_{i}+C_{i}^{\rm{R}}\tilde{O}_{i}).$
(15)

Once we have the Hamiltonian, we can write the amplitude for
$B_{s}^{0}\rightarrow\phi\phi_{s}$ 444We have marked one of the $\phi$ with a
subscript ’s’, just to differentiate between two mesons for convenience in
theoretical calculations (s denoting that it contains the spectator quark).
Experiments cannot distinguish between the two. for SM, and left- and right-
handed NP currents. In the Naive Factorisation method, they are given by

$\displaystyle\begin{aligned}
\mathcal{M}^{\rm{SM}}_{\phi\phi}&=-\frac{G_{F}}{\sqrt{2}}V^{*}_{tb}V_{ts}\xi^{\rm{SM}}_{345}\langle\phi|\bar{s}\gamma^{\mu}(1-\gamma^{5})s|0\rangle\langle\phi_{s}|\bar{b}\gamma_{\mu}(1-\gamma^{5})s|B_{s}^{0}\rangle\\\
\mathcal{M}^{\rm{L}}_{\phi\phi}&=-\frac{G_{F}}{\sqrt{2}}V^{*}_{tb}V_{ts}\xi^{\rm{L}}_{345}\langle\phi|\bar{s}\gamma^{\mu}(1-\gamma^{5})s|0\rangle\langle\phi_{s}|\bar{b}\gamma_{\mu}(1-\gamma^{5})s|B_{s}^{0}\rangle\\\
\mathcal{M}^{\rm{R}}_{\phi\phi}&=-\frac{G_{F}}{\sqrt{2}}V^{*}_{tb}V_{ts}\xi^{\rm{R}}_{345}\langle\phi|\bar{s}\gamma^{\mu}(1+\gamma^{5})s|0\rangle\langle\phi_{s}|\bar{b}\gamma_{\mu}(1+\gamma^{5})s|B_{s}^{0}\rangle\end{aligned}$
(16)

with
$\xi_{345}^{p}=\xi_{3}^{p}+\xi_{4}^{p}+\xi_{5}^{p}\quad(p\in\\{\rm{SM},\rm{L},\rm{R}\\})$
are the combinations of the Wilson coefficients given by [26]

$\displaystyle\xi_{i}^{p}=C^{p}_{i}+\frac{1}{N_{c}}C^{p}_{i+1}\quad\text{(i is
odd)}\quad\quad\xi_{i}^{p}=C^{p}_{i+1}+\frac{1}{N_{c}}C^{p}_{i}\quad\text{(i
is even)},$ (17)

where $N_{c}$ are the number of colours. The Wilson coefficients and related
details of calculations are given in Appendix A.

###  4.2 Transversity amplitudes

Once we have the model clearly defined with Hamiltonian and amplitudes, we can
move on to the helicity analysis. As mentioned before, the advantage in
$P\rightarrow VV$ type decays is that the final state can be split into three
helicity states, which gives us access to three amplitudes, whose sum makes up
the total amplitude.
The general form of helicity amplitude for the process $B(M,p)\rightarrow
V_{1}(\epsilon_{1},M_{1},k_{1})+V_{2}(\epsilon_{2},M_{2},k_{2})$ is given by
555We use the convention $\epsilon^{0123}=1$. The opposite convention would
simply interchange the definition of $H_{+}$ and $H_{-}$, without affecting
$H_{0}$ [27]

$\displaystyle\begin{aligned}
H_{\lambda}&=a(\epsilon^{*}_{1}(\lambda).\epsilon^{*}_{2}(\lambda))+\frac{b}{M_{1}M_{2}}(\epsilon^{*}_{1}(\lambda).k_{2})(\epsilon^{*}_{2}(\lambda).k_{1})\\\
&+\frac{ic}{M_{1}M_{2}}\epsilon_{\mu\nu\rho\sigma}\epsilon_{1}^{*\mu}(\lambda)\epsilon_{2}^{*\nu}(\lambda)k_{1}^{\rho}k_{2}^{\sigma},\end{aligned}$
(18)

where $\lambda=\\{+,-,0\\}$ is the polarisation of final state. $a,b$ and $c$
are the _Invariant Amplitudes_. Putting the polarisation vectors for the three
different polarisations, we get

$\displaystyle\begin{aligned} H_{0}=ax+b(x^{2}-1)\quad\quad\quad H_{\pm}=a\pm
c\sqrt{x^{2}-1},\end{aligned}$ (19)

where $x=\frac{M_{B}^{2}-M_{1}^{2}-M_{2}^{2}}{2M_{1}M_{2}}$. A more convenient
basis to work than helicity basis is _transversity basis_ [19]. To go there,
we first note that by angular momentum conservation, the final state
helicities could only be $|++\rangle,|--\rangle$ and $|00\rangle$. Since
$P|++\rangle=|--\rangle$ and $P|00\rangle=|00\rangle$, we can define parity
eigenstates with eigenvalues $\pm 1$ as [25]

$\displaystyle|\parallel\,\rangle=\frac{1}{\sqrt{2}}(|++\rangle+|--\rangle)\quad\quad\quad|\perp\,\rangle=\frac{1}{\sqrt{2}}(|++\rangle-|--\rangle)$
(20)

and we’ll call these as _transversity amplitudes_ , denoted by $A_{k}$, with
$k=\\{0,\parallel,\perp\\}$, as given in Eq. (2). Clearly

$\displaystyle\mathcal{A}_{\parallel,\perp}=\frac{1}{\sqrt{2}}(H_{+}\pm
H_{-})\quad\quad\quad\mathcal{A}_{0}=H_{0}.$ (21)

Thus,

$\displaystyle\begin{aligned} A_{0}&=ax+b(x^{2}-1)\\\
A_{\parallel}&=\sqrt{2}a\\\ A_{\perp}&=\sqrt{2(x^{2}-1)}c.\end{aligned}$ (22)

Now we can compare the general helicity amplitude structure in Eq. (18) with
our $B_{s}^{0}\rightarrow\phi\phi$ amplitudes given in Eq. (16) to get the
invariant amplitudes, using which we can write the helicity and transversity
amplitudes, defined in Eq. (19) and Eq. (22), respectively.

Let us see the hierarchy of amplitudes. We compare the hierarchy predicted by
the $V-A$ structure of current with that predicted by our model (which is
based on naive factorisation method). For the case of $B^{0}_{s}$ decay,
consider the current $(\bar{b}s)_{V-A}$: The longitudinal amplitude $H_{0}$ is
the dominant one. There is a spin-flip suppression in $H_{+}$ amplitude, and
the $H_{-}$ amplitude has a chirality suppression in the light s-quark along
with spin-flip suppression [28][29]. Therefore, the expected hierarchy is
$H_{0}>H_{+}>H_{-}$ (interchange $+$ and $-$ signs for $\bar{B}^{0}_{s}$
decay) [30]. Now consider our amplitudes in Eq. (19): by putting the values of
invariant amplitudes $a,b$ and $c$, we obtain the same hierarchy as the one
obtained above, with the approximate ratio $H_{0}:H_{+}:H_{-}\sim
1:(\frac{m_{\phi}}{m_{B}}):(\frac{m_{\phi}}{m_{B}})^{2}$ [28][30]. This can
also be verified by using the numerical values of form factors from [31].

However, it is well known that this hierarchy is not observed experimentally
in $B\rightarrow VV$ decays (one of the limitations of naive factorisation
method). A large transverse polarisation was first observed in
$B_{d}\rightarrow\phi K^{*}$ [32] (and then later in $B_{d}\rightarrow
J/\psi\phi$ [17], $B_{s}\rightarrow\phi\phi$ [7] etc.) which gave rise to
intense theoretical and experimental studies of charmless $B\rightarrow VV$
decays. Several theoretical papers have been written to go beyond the naive
factorisation method and use more sophisticated tools (like QCDf, pQCD, SCET
etc.) to compute these decays more accurately [4][5][6][30]. It has been shown
in [33] that a major contributor to transverse amplitudes are the
_annihilation diagrams_ , which can explain the large fraction of transverse
amplitudes observed experimentally. However, naive factorisation method does
not take these diagrams into account; thus, we get a hierarchy that does not
match with the experiments. Although we do not go into the details of the
calculation of such diagrams as it is beyond the scope of this article, it is
worthwhile to keep this in our minds to interpret the experimental results
correctly. For the purposes of our calculation, we introduce a quantity
$\mathcal{F}$ (defined below), which contains the information about the matrix
elements, and can be calculated in the preferred choice of model.

###  4.3 New phase scheme

As mentioned before, since the total amplitude is the sum of all three
helicity/transversity amplitudes, we can actually write

$\displaystyle\begin{aligned} \mathcal{M}^{\rm
SM,Total}_{\phi\phi}&=A_{0}^{\rm SM}+A_{\parallel}^{\rm SM}+A_{\perp}^{\rm
SM}\\\ &=-\frac{G_{F}}{\sqrt{2}}V_{tb}^{*}V_{ts}(\xi^{\rm
SM}_{0}\mathcal{F}_{0}^{\rm SM}+\xi^{\rm
SM}_{\parallel}\mathcal{F}_{\parallel}^{\rm SM}+\xi^{\rm
SM}_{\perp}\mathcal{F}_{\perp}^{\rm SM})\\\ \mathcal{M}^{\rm
L,Total}_{\phi\phi}&=A_{0}^{\rm L}+A_{\parallel}^{\rm L}+A_{\perp}^{\rm L}\\\
&=-\frac{G_{F}}{\sqrt{2}}V_{tb}^{*}V_{ts}(\xi^{\rm L}_{0}\mathcal{F}_{0}^{\rm
NP}+\xi^{\rm L}_{\parallel}\mathcal{F}_{\parallel}^{\rm NP}+\xi^{\rm
L}_{\perp}\mathcal{F}_{\perp}^{\rm NP})\\\ \mathcal{M}^{\rm
R,Total}_{\phi\phi}&=A_{0}^{\rm R}+A_{\parallel}^{\rm R}+A_{\perp}^{\rm R}\\\
&=-\frac{G_{F}}{\sqrt{2}}V_{tb}^{*}V_{ts}(-\xi^{\rm R}_{0}\mathcal{F}_{0}^{\rm
NP}-\xi^{\rm R}_{\parallel}\mathcal{F}_{\parallel}^{\rm NP}+\xi^{\rm
R}_{\perp}\mathcal{F}_{\perp}^{\rm NP}).\\\ \end{aligned}$ (23)

where $\mathcal{F}^{\rm SM}$ and $\mathcal{F}^{\rm NP}$ contains the
contribution from the matrix elements for SM and NP case, respectively. The
$\xi^{p}_{k}$ ($k=\\{0,\parallel,\perp\\}$ and
$p\in\\{\rm{SM},\rm{L},\rm{R}\\}$) are combinations of Wilson coefficients,
and contain the weak phases (their actual form depends upon the model chosen
to compute the matrix elements, but it is not important for our purposes).
Notice the sign changes in the right-handed amplitude: this occurs because of
the change in sign in the axial part of the current.

We can also write the total amplitude for different transversities. However,
before doing that, we use the simple argument that the _contribution from the
Chromomagnetic dipole operator ( $O_{8g}$) is suppressed in transverse penguin
amplitudes_ (originally pointed out in [30]). Since we do not expect
contributions from annihilation diagrams here, the hierarchy we discussed
previously gets stronger and NP contribution becomes totally dominated by the
longitudinal component. This would imply $\mathcal{F}^{\rm
NP}_{0}\gg\mathcal{F}^{\rm NP}_{\parallel,\perp}$. Hence, we can ignore the NP
transverse components, and the only transversity amplitude that would change
in the presence of NP manifested via Chromomagnetic operator would be

$\displaystyle A^{\rm Total}_{0,\phi\phi}=A^{\rm SM}_{0}(1+r^{\rm
L}e^{i(\omega_{\rm L}+\sigma)}-r^{\rm R}e^{i(\omega_{\rm R}+\sigma)}),$ (24)

where we parametrise the NP contribution as follows:

$\displaystyle\frac{\xi^{\rm{L}}_{0}\mathcal{F}^{\rm
NP}_{0}}{\xi^{\rm{SM}}_{0}\mathcal{F}^{\rm
SM}_{0}}=r^{\rm{L}}e^{i(\omega_{\rm{L}}+\sigma)}\quad\quad\quad\frac{\xi^{\rm{R}}_{0}\mathcal{F}^{\rm
NP}_{0}}{\xi^{\rm{SM}}_{0}\mathcal{F}^{\rm
SM}_{0}}=r^{\rm{R}}e^{i(\omega_{\rm{R}}+\sigma)}.$ (25)

$\omega_{\rm L,R}$ are the weak/CP-odd phases and $\sigma$ is a strong/CP-even
phase. Recalling the definition of interference phase from Eq. (11) and
putting Eq. (24) in it, we can write

$\displaystyle\frac{q}{p}\frac{\bar{A}^{\rm Total}_{0,\phi\phi}}{A^{\rm
Total}_{0,\phi\phi}}=\lambda_{0}e^{-i(\theta_{0}-\theta_{0}^{c})}=\frac{1+2\cos\sigma(r^{\rm
L}e^{-i\omega_{\rm L}}+r^{\rm R}e^{-i\omega_{\rm R}})}{1+2r^{\rm
L}\cos(\omega_{\rm L}+\sigma)-2r^{\rm R}\cos(\omega_{\rm R}+\sigma)}.$ (26)

Therefore, only $\lambda_{0}$ and $\theta_{0}-\theta_{0}^{c}$ would get
contributions from NP, while other transversities CP-violating parameters
would assume their SM values, i.e
$\theta_{\parallel}=\theta_{\parallel}^{c}=\theta_{\perp}=\theta_{\perp}^{c}=0$
and $\lambda_{\parallel}=\lambda_{\perp}=1$.

As we can see, the five theoretical parameters ($r^{\rm L,R},\omega_{\rm L,R}$
and $\sigma$) cannot be determined, as we do not have sufficient observables
(only $\lambda_{0}$ and $\theta_{0}-\theta_{0}^{c}$). Nevertheless, an
observation of non-zero value of $\lambda_{0}-1$ and/or
$\theta_{0}-\theta_{0}^{c}$ would clearly indicate the presence of NP.

Let us now compare the phase scheme that LHCb used in their fit to ours.
Before comparing with our parametrisation, we note that the interference phase
in LHCb is defined as

$\displaystyle\frac{q}{p}\frac{\bar{A}^{\rm Total}_{k,\phi\phi}}{A^{\rm
Total}_{k,\phi\phi}}=\eta_{k}\lambda_{k}e^{-i\phi^{\rm LHCb}_{k}},$ (27)

where $k$ is the transversity and $\eta_{k}$ is the CP parity of the
transversity state. Comparing Eq. (27) with Eq. (11), we find $\phi^{\rm
LHCb}_{k}\equiv\theta_{k}-\theta^{c}_{k}$. LHCb uses the following two
different fit configurations:

  * •

LHCb helicity-dependent (HD) scheme:
$\phi^{\rm LHCb}_{0}=0$, $\lambda_{k}=1$ $\forall k$ ($\phi^{\rm
LHCb}_{\perp}$ and $\phi^{\rm LHCb}_{\parallel}$ are the CP-violating fit
parameters).

  * •

LHCb helicity-independent (HI) scheme:
$\phi=\phi_{k}^{\rm LHCb}$ $\forall k$, $\lambda=\lambda_{k}$ $\forall k$
($\phi$ and $\lambda$ are the CP-violating fit parameters).

The new fit configuration we are proposing is

  * •

NP manifested via Chromomagnetic operator:
$\phi_{\perp}^{\rm LHCb}$=$\phi_{\parallel}^{\rm LHCb}$=0 or equivalently
$\theta_{\parallel}=\theta_{\parallel}^{c}=\theta_{\perp}=\theta_{\perp}^{c}=0$,
$\lambda_{\perp}$=$\lambda_{\parallel}$=1
($\phi_{0}^{\rm LHCb}$ and $\lambda_{0}$ are the CP-violating fit parameters).

The LHCb fit configuration does not match to ours, and a new fit of LHCb data
with this new scheme based on our model would be very interesting.

##  5 Left or right: $B^{0}_{d}\rightarrow\phi K_{S}$ decay

A decay very similar to $B_{s}^{0}\rightarrow\phi\phi$ decay is the
$B_{d}^{0}\rightarrow\phi K_{s}$ decay, since at the quark level, both contain
a $\bar{b}\rightarrow\bar{s}s\bar{s}$ decay. We thus expect the weak
interaction to be the same in both the decays, while strong interaction may
differ. In this section, we investigate how the experimental results of
$B_{d}^{0}\rightarrow\phi K_{s}$ complements the
$B_{s}^{0}\rightarrow\phi\phi$ results, within the left- and right-handed
Chromomagnetic operator model.

We start with the $B_{d}^{0}\rightarrow\phi K_{s}$ decay. The phase in the
interference of decay with and without mixing in SM is $2\phi_{1}$, where
$\phi_{1}$ is the unitary triangle angle. However, NP contributions may
deviate its value from $\phi_{1}$, and what we measure experimentally should
then be called $2\phi_{1}^{\rm eff}$. Using Eq. (42), we can thus write

$\displaystyle\frac{q}{p}\frac{\bar{\mathcal{M}}^{\rm Total}_{\phi
K_{s}}}{\mathcal{M}^{\rm Total}_{\phi K_{s}}}=-\lambda_{\phi
K_{s}}e^{-2i\phi_{1}^{\rm
eff}}=-e^{-2i\phi_{1}}\left(\frac{1+\hat{r}^{\rm{L}}e^{i(-\hat{\omega}_{\rm{L}}+\hat{\sigma})}+\hat{r}^{\rm{R}}e^{i(-\hat{\omega}_{\rm{R}}+\hat{\sigma})}}{1+\hat{r}^{\rm{L}}e^{i(\hat{\omega}_{\rm{L}}+\hat{\sigma})}+\hat{r}^{\rm{R}}e^{i(\hat{\omega}_{\rm{R}}+\hat{\sigma})}}\right),$
(28)

where the negative sign is present as $\phi K_{s}$ is a CP-odd state.
Rearranging and rationalising the RHS, we get

$\displaystyle\lambda_{\phi K_{s}}e^{-2i(\phi_{1}^{\rm
eff}-\phi_{1})}=\frac{1+2\cos\hat{\sigma}(\hat{r}^{\rm
L}e^{-i\hat{\omega}_{\rm L}}+\hat{r}^{\rm R}e^{-i\hat{\omega}_{\rm
R}})}{1+2\hat{r}^{\rm L}\cos(\hat{\omega}_{\rm L}+\hat{\sigma})+2\hat{r}^{\rm
R}\cos(\hat{\omega}_{\rm R}+\hat{\sigma})}$ (29)

We can now compare Eq. (26) and Eq. (29). As mentioned before, we assume the
weak interaction contribution from NP to be the same for both the decays, thus
making $\omega_{\rm L,R}=\hat{\omega}_{\rm L,R}$. In addition, we assume that
$r^{\rm L,R}$ and $\hat{r}^{\rm L,R}$ are small and positive. This implies
that the sign of the strong interaction (coming from the ratio of matrix
elements $\frac{\mathcal{F}^{\rm NP}_{0}}{\mathcal{F}^{\rm SM}_{0}}$ and
$\frac{\mathcal{F}^{\rm NP}_{\phi Ks}}{\mathcal{F}^{\rm SM}_{\phi Ks}}$) is
contained in the terms $\cos\sigma$ and $\cos\hat{\sigma}$, respectively. In
addition, we see in Eq. (26) and Eq. (29) that the right-handed contribution
from NP has opposite signs for the two cases in the denominator. Therefore, if
we can theoretically predict the sign of $\cos\sigma$ and $\cos\hat{\sigma}$,
we can tell the chirality of NP in the following two cases:
Case 1: Only left-handed NP is present ($r^{\rm R}=\hat{r}^{\rm R}=0$)
Taking the ratio of real and imaginary parts of Eq. (26) and Eq. (29), and
expanding in $r^{\rm L}$ and $\hat{r}^{\rm L}$, we get

$\displaystyle\begin{aligned} \tan(\theta_{0}-\theta_{0}^{c})&\approx 2r^{\rm
L}\sin\omega_{\rm L}\cos\sigma+{\rm O}((r^{\rm L})^{2})\\\ \tan(2\phi^{\rm
eff}_{1}-2\phi_{1})&\approx 2\hat{r}^{\rm L}\sin\hat{\omega}_{\rm
L}\cos\hat{\sigma}+{\rm O}((\hat{r}^{\rm L})^{2})\end{aligned}$ (30)

At this point we can define a quantity
$\Sigma\equiv[\tan(\theta_{0}-\theta_{0}^{c})\tan(2\phi^{\rm
eff}_{1}-2\phi_{1})]$. As we defined $r^{L}$ and $\hat{r}^{L}$ to be positive
and the Chromomagnetic operator leads to $\omega_{L}=\hat{\omega}_{L}$, we
obtain the relation

$\displaystyle sign(\Sigma)=sign(\cos\sigma\cos\hat{\sigma})$ (31)

Case 2: Only right-handed NP is present ($r^{\rm L}=\hat{r}^{\rm L}=0$)
Taking the ratio of real and imaginary parts of Eq. (26) and Eq. (29), and
expanding in $r^{\rm R}$ and $\hat{r}^{\rm R}$, we get

$\displaystyle\begin{aligned} \tan(\theta_{0}-\theta_{0}^{c})&\approx-2r^{\rm
R}\sin\omega_{\rm R}\cos\sigma+{\rm O}((r^{\rm R})^{2})\\\ \tan(2\phi^{\rm
eff}_{1}-2\phi_{1})&\approx 2\hat{r}^{\rm R}\sin\hat{\omega}_{\rm
R}\cos\hat{\sigma}+{\rm O}((\hat{r}^{\rm R})^{2})\end{aligned}$ (32)

Thus, in this case, we find an opposite relative sign with respect to the
left-handed model:

$\displaystyle sign(\Sigma)=-sign(\cos\sigma\cos\hat{\sigma})$ (33)

Hence, if the experiments show non-zero CP violating phase results, one can
test the chirality of the NP contribution by combining the
$B_{s}^{0}\rightarrow\phi\phi$ and $B_{d}^{0}\rightarrow\phi K_{s}$ decay
measurements, along with the relative sign of $\cos\delta$ and
$\cos\hat{\delta}$, which might be obtained theoretically. This conclusion is
summarised in Table 1.

$\cos\sigma$ | $\cos\hat{\sigma}$ | $\Sigma$ | NP chirality
---|---|---|---
+ | + | + | LH
+ | – | + | RH
+ | + | – | RH
+ | – | – | LH
– | + | – | LH
– | – | – | RH
– | + | + | RH
– | – | + | LH
Table 1: Table demonstrating the chirality of NP arising from different
combinations of signs of $\cos\sigma$, $\cos\hat{\sigma}$ and
$\Sigma\equiv[\tan(\theta_{0}-\theta_{0}^{c})\tan(2\phi^{\rm
eff}_{1}-2\phi_{1})]$, under the assumption that only left-handed or right-
handed NP is present. $\sigma$ and $\hat{\sigma}$ denote the strong phase
difference between NP and SM in $B_{s}^{0}\rightarrow\phi\phi$ and
$B_{d}^{0}\rightarrow\phi K_{s}$ decays, respectively (see Eq. (25)).
$\theta_{0}-\theta_{0}^{c}$ and $2\phi^{\rm eff}_{1}$ are the phase in the
interference of decays with and without mixing in
$B_{s}^{0}\rightarrow\phi\phi$ and $B_{d}^{0}\rightarrow\phi K_{s}$ decays,
respectively. $\phi_{1}$ is the unitary triangle angle.

##  6 Sensitivity study with the new fit configuration

In this section, we illustrate an analysis with the new phase scheme we
proposed in Section 4.3. To start, let us list all the possible fit parameters
before considering any model assumptions:
$(|A_{0,\perp,\parallel}|^{2},\delta_{0,\perp,\parallel},\theta_{0,\perp,\parallel},\theta^{c}_{0,\perp,\parallel}$,$\lambda_{0,\perp,\parallel}$).
First, using the relation $|A_{0}|^{2}+|A_{\perp}|^{2}+|A_{\parallel}|^{2}=1$,
we remove one of the amplitudes, e.g. $|A_{\parallel}|^{2}$. Next, we notice
that in Table 5, the phases always appear as combinations of
$\theta_{k}-\theta_{k}^{c}$ and $\psi_{i}-\psi_{j}$ where
$\psi_{k}\equiv\theta_{k}+\delta_{k}$. For example, the combination
$\theta_{k}^{c}+\delta_{k}$ can be rewritten as
$\psi_{k}-(\theta_{k}-\theta_{k}^{c})$. Now, let us use the results of our
model introduced in the Section 4.3: Chromomagnetic operator contributes
predominantly to longitudinal polarisation, giving us
$\theta_{\parallel}=\theta_{\parallel}^{c}=\theta_{\perp}=\theta_{\perp}^{c}=0$.
Then, the arguments of trigonometric functions in Table 5 can be expressed by
the three parameters,
($\theta_{0}-\theta_{0}^{c},\delta_{\parallel}-\delta_{\perp},\delta_{\parallel}-\delta_{0}-\theta_{0}$).
As explained in Section 3.2, the first parameter is the phase in the
interference of decay with and without mixing, which is a CP-violating
quantity, while the last two contain strong phases. Thus, only the first one
can be used for a null test. Finally, our model also imposes
$\lambda_{\perp,\parallel}=1$. As a result, we are left with 6 parameters to
fit

$(\lambda_{0},\theta_{0}-\theta_{0}^{c},\delta_{\parallel}-\delta_{\perp},\delta_{\parallel}-\delta_{0}-\theta_{0},|A_{0}|^{2},|A_{\perp}|^{2})$

Only the first two can be used for a null test: $\lambda_{0}\neq 1$ and/or
$\theta_{0}-\theta_{0}^{c}\neq 0$ is a clear signal of new physics.

To illustrate the fit, we first construct two pseudo datasets by using the
LHCb best-fit values, denoted as Data HI and Data HD for the LHCb helicity-
independent and helicity-dependent fit, respectively. The details of the
statistical procedure applied in this study are given in Appendix B.

Our fit results are shown in Table 2, and the correlation matrices are given
in Appendix C. We note that the results using Data HI and Data HD agree
relatively well. The obtained uncertainty of $\sigma(\lambda_{0})=6-7$% and
$\sigma(\theta_{0}-\theta_{0}^{c})=5-6$% with the currently available LHCb
statistics (5 $\rm fb^{-1}$) may be used as an indication for future studies.

| Data HD | Data HI
---|---|---
Fit Parameter | Central Value | $\sigma$ | Central Value | $\sigma$
$\lambda_{0}$ | 0.978 | 0.058 | 0.984 | 0.070
$|A_{0}|^{2}$ | 0.386 | 0.025 | 0.385 | 0.032
$|A_{\perp}|^{2}$ | 0.287 | 0.018 | 0.288 | 0.036
$\theta_{0}-\theta^{c}_{0}$ | -0.002 | 0.055 | 0.066 | 0.053
$\delta_{\parallel}-\delta_{\perp}$ | -0.259 | 0.054 | -0.261 | 0.056
$\delta_{\parallel}-\delta_{0}-\theta_{0}$ | 2.560 | 0.071 | 2.589 | 0.079
Table 2: Fit results based on our model assumptions, i.e. longitudinal
component dominance for NP contributions coming from Chromomagnetic operator
($\theta^{c}_{\parallel}=\theta_{\parallel}=\theta^{c}_{\perp}=\theta_{\perp}=0$
and $\lambda_{\parallel}=\lambda_{\perp}=1$).

##  7 Conclusions

In this article, we investigate a new physics search with the CP violation
measurements of the $B_{s}\rightarrow\phi\phi$ decay. The large statistics of
the LHCb experiment allows one to perform the time-dependent angular analysis
of this decay channel. Such an analysis gives access to the information of the
helicity amplitudes, which are sensitive to different types of NP effects. In
the LHCb analysis, two types of NP scenarios have been investigated, called
helicity-dependent and -independent assumptions. In this work, we propose a
new search scenario based on the NP model induced by the left- and right-
handed Chromomagnetic operators, producing a new quark level $b\to s\bar{s}s$
diagram with an extra source of CP violation. Using the fact that the NP
coming from this type of operator is dominated by the longitudinal amplitude,
we derive a new scheme of phase assumptions which can be tested by the LHCb
experiment. The same NP effects can manifest itself in the time-dependent CP
asymmetry measurement of $B^{0}_{d}\rightarrow\phi K_{s}$ decay. We found that
Belle(II)’s $B^{0}_{d}\rightarrow\phi K_{s}$ decay measurements could
complement LHCb’s $B^{0}_{s}\rightarrow\phi\phi$ measurement to obtain the
chirality of NP operator, under the condition that the signs of the strong
phases of these decays can be predicted by the theory. Finally, we present a
sensitivity study of the CP-violating parameters of our proposed model in
order to illustrate how the fit can actually be performed. We show that on top
of the two CP-violating parameters, there are four extra parameters to be
fitted simultaneously: two amplitudes and two phases. The theoretical
predictions for these extra parameters depend heavily on the models describing
the strong interaction. On the other hand, a non-zero measurement of the
former two CP violating parameters can be interpreted immediately as a signal
of NP. Our sensitivity study shows that LHCb with current statistics can
determine these two parameters at $5-7\%$ precision. These numbers are
obtained using two pseudo datasets and they might not reflect the reality,
though, the sensitivities obtained could be used as an indication for future
studies. Even though the current measurements do not show a clear signal of
NP, further theoretical and experimental efforts would shed more light on
these results, and would pave the way for future studies.

## Acknowledgments

We would like to express our gratitude to François Le Diberder for his careful
reading of the manuscript and help in doing the statistical analysis. We would
also like to thank Franz Muheim for the helpful correspondence.

##  Appendix A The Chromomagnetic Dipole operator

The Chromomagnetic operator for $\bar{b}\rightarrow\bar{s}g$ is given by

$\displaystyle
O_{8g}=\frac{g_{s}}{8\pi^{2}}m_{b}\bar{b}_{\alpha}\sigma^{\mu\nu}(1+\gamma^{5})\frac{\lambda^{a}_{\alpha\beta}}{2}s_{\beta}G^{a}_{\mu\nu}.$
(34)

This operator is written only for on-shell gluons. But while dealing with
amplitudes, we need matrix elements, and the corresponding one is given by
($X=\phi,K_{s}$)

$\displaystyle\langle\phi
X|O_{8g}|B_{s}^{0}\rangle=\frac{\alpha_{s}}{\pi}\frac{m_{b}}{q^{2}}\bar{b}_{\alpha}\gamma^{\mu}\not{q}(1+\gamma^{5})\frac{\lambda^{a}_{\alpha\beta}}{2}s_{\beta}\bar{q}^{\prime}_{\rho}\gamma_{\mu}\frac{\lambda^{a}_{\rho\sigma}}{2}q^{\prime}_{\sigma}.$
(35)

with $q$ being the gluon momentum. Approximating $q^{\mu}=\sqrt{\langle
q^{2}\rangle}\frac{p_{b}^{\mu}}{m_{b}}$, and using the fierz identity
$\frac{\lambda^{a}_{\alpha\beta}}{2}\frac{\lambda^{a}_{\rho\sigma}}{2}=\frac{1}{2}(\delta_{\alpha\sigma}\delta_{\beta\rho}-\frac{1}{3}\delta_{\alpha\beta}\delta_{\rho\sigma})$
and dirac equation, we can write the matrix element corresponding to $O_{8g}$
operator in terms of $O_{3-6}$ as

$\displaystyle O_{8g}=\frac{\alpha_{s}}{4\pi}\frac{m_{b}}{\sqrt{\langle
q^{2}\rangle}}\left[O_{4}+O_{6}-\frac{1}{3}(O_{3}+O_{5})\right],$ (36)

where $\langle q^{2}\rangle$ is average value of $q^{2}$, varying between
$\frac{m_{b}^{2}}{4}$ and $\frac{m_{b}^{2}}{2}$ [34][35].

Thus, we can write the effective Wilson coefficients for operators $O_{3-6}$
as

$\displaystyle\begin{aligned}
C_{3}^{\rm{L(R)}}&=-\frac{1}{3}\frac{\alpha_{s}}{4\pi}\frac{m_{b}}{\sqrt{\langle
q^{2}\rangle}}\mathord{\mathop{C}\limits^{\scriptscriptstyle(\sim)}}_{8g}\\\
C_{4}^{\rm{L(R)}}&=\frac{\alpha_{s}}{4\pi}\frac{m_{b}}{\sqrt{\langle
q^{2}\rangle}}\mathord{\mathop{C}\limits^{\scriptscriptstyle(\sim)}}_{8g}\\\
C_{5}^{\rm{L(R)}}&=-\frac{1}{3}\frac{\alpha_{s}}{4\pi}\frac{m_{b}}{\sqrt{\langle
q^{2}\rangle}}\mathord{\mathop{C}\limits^{\scriptscriptstyle(\sim)}}_{8g}\\\
C_{6}^{\rm{L(R)}}&=\frac{\alpha_{s}}{4\pi}\frac{m_{b}}{\sqrt{\langle
q^{2}\rangle}}\mathord{\mathop{C}\limits^{\scriptscriptstyle(\sim)}}_{8g}.\end{aligned}$
(37)

##  Appendix B Statistical procedure

The LHCb experimental observables $(a_{i},b_{i},c_{i}$ and $d_{i})$ are given
in Table 5: they are the LHCb observables. The only available information from
LHCb is the result of fit of those measurements to the theory parameters
($|A_{k}|^{2},\delta_{k},\phi_{k}$), given in [7]. Therefore, in our study, we
first construct pseudo dataset, i.e. the central values and the covariance
matrices for the LHCb observables, from this available information. The
covariance matrix is obtained by using

$\displaystyle V_{ij}^{-1}=N\int\
\left.\left(\frac{\partial\hat{f}(x)_{\vec{v}}}{\partial
v_{i}}\frac{\partial\hat{f}(x)_{\vec{v}}}{\partial
v_{j}}\frac{1}{\hat{f}(x)_{\vec{v}}}\right)\right|_{\vec{v}=\vec{v}^{*}}\ dx$
(38)

where

  * •

$\hat{f}$ is the normalised probability distribution function, which in our
case is the angular decay distribution given by Eq. (3). Integration over $x$
represents integration over the complete phase space and time.

  * •

$\vec{v}$ is the vector of LHCb observables ($a_{i},b_{i},c_{i}$ and $d_{i}$)
that LHCb measures.

  * •

$\vec{v}^{\;*}$ is the values of $\vec{v}$ obtained by using the best fit
values of the theoretical parameters obtained by LHCb [7]. Note that there are
two fits performed by LHCb with the so-called helicity-independent and
-dependent assumptions, and we use both to construct two pseudo datasets.

  * •

$N$ is the number of events.

Finally, using this pseudo dataset, we perform a $\chi^{2}$ fit using
$\vec{v}_{i}$ with our model assumptions, which we call
$\vec{v}_{i}^{\rm\;model}$:

$\displaystyle\chi^{2}=\sum_{i,j}(\vec{v}_{i}^{\rm{\;model}}-\vec{v}_{i}^{\;*})V^{-1}_{ij}(\vec{v}_{j}^{\rm{\;model}}-\vec{v}_{j}^{\;*}).$
(39)

##  Appendix C Correlation matrices

| $\theta_{0}-\theta^{c}_{0}$ | $\delta_{\parallel}-\delta_{\perp}$ | $\delta_{\parallel}-\delta_{0}-\theta_{0}$ | $|A_{0}|^{2}$ | $|A_{\perp}|^{2}$ | $\lambda_{0}$
---|---|---|---|---|---|---
$\theta_{0}-\theta^{c}_{0}$ | 1.00 | 0.01 | -0.33 | 0.00 | 0.00 | -0.03
$\delta_{\parallel}-\delta_{\perp}$ | 0.01 | 1.00 | 0.38 | -0.11 | 0.13 | -0.01
$\delta_{\parallel}-\delta_{0}-\theta_{0}$ | -0.33 | 0.38 | 1.00 | -0.24 | 0.23 | -0.03
$|A_{0}|^{2}$ | 0.00 | -0.11 | -0.24 | 1.00 | -0.72 | -0.67
$|A_{\perp}|^{2}$ | 0.00 | 0.13 | 0.23 | -0.72 | 1.00 | 0.49
$\lambda_{0}$ | -0.03 | -0.01 | -0.03 | -0.67 | 0.49 | 1.00
Table 3:  Correlation matrix based on our model assumptions, i.e. longitudinal component dominance for NP contributions from Chromomagnetic operator ($\theta^{c}_{\parallel}=\theta_{\parallel}=\theta^{c}_{\perp}=\theta_{\perp}=0$ and $\lambda_{\parallel}=\lambda_{\perp}=1$). Pseudo dataset used: Data HD. | $\theta_{0}-\theta^{c}_{0}$ | $\delta_{\parallel}-\delta_{\perp}$ | $\delta_{\parallel}-\delta_{0}-\theta_{0}$ | $|A_{0}|^{2}$ | $|A_{\perp}|^{2}$ | $\lambda_{0}$
---|---|---|---|---|---|---
$\theta_{0}-\theta^{c}_{0}$ | 1.00 | -0.02 | -0.37 | 0.03 | -0.04 | -0.01
$\delta_{\parallel}-\delta_{\perp}$ | -0.02 | 1.00 | 0.40 | -0.06 | 0.07 | -0.04
$\delta_{\parallel}-\delta_{0}-\theta_{0}$ | -0.37 | 0.40 | 1.00 | -0.19 | 0.21 | -0.04
$|A_{0}|^{2}$ | 0.03 | -0.06 | -0.19 | 1.00 | -0.85 | -0.76
$|A_{\perp}|^{2}$ | -0.04 | 0.07 | 0.21 | -0.85 | 1.00 | 0.65
$\lambda_{0}$ | -0.01 | -0.04 | -0.04 | -0.76 | 0.65 | 1.00
Table 4: Correlation matrix based on our model assumptions, i.e. longitudinal
component dominance for NP contributions from Chromomagnetic operator
($\theta^{c}_{\parallel}=\theta_{\parallel}=\theta^{c}_{\perp}=\theta_{\perp}=0$
and $\lambda_{\parallel}=\lambda_{\perp}=1$). Pseudo dataset used: Data HI.

##  Appendix D $B^{0}_{d}\rightarrow\phi K_{S}$ decay

The amplitude for $B^{0}_{d}\rightarrow\phi K_{S}$ for SM, left-handed NP and
right-handed NP case respectively, can be written as

$\displaystyle\begin{aligned} \mathcal{M}^{\rm{SM}}_{\phi
K_{s}}&=-\frac{G_{F}}{\sqrt{2}}V^{*}_{tb}V_{ts}\hat{\xi}^{\rm{SM}}\mathcal{F}^{\rm
SM}_{\phi K_{s}}\\\ \mathcal{M}^{\rm{L}}_{\phi
K_{s}}&=-\frac{G_{F}}{\sqrt{2}}V^{*}_{tb}V_{ts}\hat{\xi}^{\rm{L}}\mathcal{F}^{\rm
NP}_{\phi K_{s}}\\\ \mathcal{M}^{\rm{R}}_{\phi
K_{s}}&=-\frac{G_{F}}{\sqrt{2}}V^{*}_{tb}V_{ts}\hat{\xi}^{\rm{R}}\mathcal{F}^{\rm
NP}_{\phi K_{s}},\end{aligned}$ (40)

where $\hat{\xi}^{p}(p\in\\{\rm{SM},\rm{L},\rm{R}\\})$ are combination of the
Wilson coefficients, which contain weak phases, and their exact form depends
upon the model chosen to evaluate the matrix elements. Like for the case of
$B^{0}_{s}\rightarrow\phi\phi$ decay, the variables $\mathcal{F}_{\phi
K_{s}}^{\rm SM}$ and $\mathcal{F}_{\phi K_{s}}^{\rm NP}$ contain all the
information about the matrix elements. Note that ${K}_{0}$ is a flavour
eigenstate, which, by Kaon oscillation, oscillates between $K_{0}$ and
$\bar{K}_{0}$ and we see the mass eigenstate $K_{S}$ in detectors.

Now the total amplitude, which is the sum of all three amplitudes, can be
written as

$\displaystyle\mathcal{M}^{\rm{Total}}_{\phi
K_{s}}=\mathcal{M}^{\rm{SM}}_{\phi
K_{s}}\left(1+\frac{\hat{\xi}^{\rm{L}}\mathcal{F}^{\rm NP}_{\phi
K_{s}}}{\hat{\xi}^{\rm{SM}}\mathcal{F}^{\rm SM}_{\phi
K_{s}}}+\frac{\hat{\xi}^{\rm{R}}\mathcal{F}^{\rm NP}_{\phi
K_{s}}}{\hat{\xi}^{\rm{SM}}\mathcal{F}^{\rm SM}_{\phi K_{s}}}\right).$ (41)

Using similar parametrisation for NP as in Eq. (25) (but putting hats to
differentiate from $B_{s}^{0}\rightarrow\phi\phi$ case), we get

$\displaystyle\mathcal{M}^{\rm{Total}}_{\phi
K_{s}}=\mathcal{M}^{\rm{SM}}_{\phi
K_{s}}\left(1+\hat{r}^{\rm{L}}e^{i(\hat{\omega}_{\rm{L}}+\hat{\sigma})}+\hat{r}^{\rm{R}}e^{i(\hat{\omega}_{\rm{R}}+\hat{\sigma})}\right).$
(42)

##  Appendix E Coefficients of time-dependent terms

Table 5: Coefficients of the time-dependent terms and angular functions used
in Eq. (5). Amplitudes are defined at $t=0$.

i | $N_{i}$ | $a_{i}$ | $b_{i}$ | $c_{i}$ | $d_{i}$ | $f_{i}$
---|---|---|---|---|---|---
1 | $|A_{0}|^{2}$ | ${(1+\lambda_{0}^{2})}/{2}$ | $-\lambda_{0}\cos(\theta^{c}_{0}-\theta_{0})$ | ${(1-\lambda_{0}^{2})}/{2}$ | $-\lambda_{0}\sin(\theta^{c}_{0}-\theta_{0})$ | $4\cos^{2}\theta_{1}\cos^{2}\theta_{2}$
2 | $|A_{\parallel}|^{2}$ | ${(1+\lambda_{\parallel}^{2})}/{2}$ | $-\lambda_{\parallel}\cos(\theta^{c}_{\parallel}-\theta_{\parallel})$ | ${(1-\lambda_{\parallel}^{2})}/{2}$ | $-\lambda_{\parallel}\sin(\theta^{c}_{\parallel}-\theta_{\parallel})$ | $\sin^{2}\theta_{1}\sin^{2}\theta_{2}(1{+}\cos 2\Phi)$
3 | $|A_{\perp}|^{2}$ | ${(1+\lambda_{\perp}^{2})}/{2}$ | $\lambda_{\perp}\cos(\theta^{c}_{\perp}-\theta_{\perp})$ | ${(1-\lambda_{\perp}^{2})}/{2}$ | $\lambda_{\perp}\sin(\theta^{c}_{\perp}-\theta_{\perp})$ | $\sin^{2}\theta_{1}\sin^{2}\theta_{2}(1{-}\cos 2\Phi)$
4 | ${|A_{\parallel}||A_{\perp}|}/{2}$ | $\begin{array}[]{c}\sin(\delta_{\perp}-\delta_{\parallel}+\theta_{\perp}-\theta_{\parallel})\\\ -\lambda_{\perp}\lambda_{\parallel}\sin(\delta_{\perp}-\delta_{\parallel}+\theta^{c}_{\perp}-\theta^{c}_{\parallel})\end{array}$ | $\begin{array}[]{c}\lambda_{\perp}\sin(\delta_{\perp}-\delta_{\parallel}+\theta^{c}_{\perp}-\theta_{\parallel})\\\ -\lambda_{\parallel}\sin(\delta_{\perp}-\delta_{\parallel}+\theta_{\perp}-\theta^{c}_{\parallel})\end{array}$ | $\begin{array}[]{c}\sin(\delta_{\perp}-\delta_{\parallel}+\theta_{\perp}-\theta_{\parallel})\\\ +\lambda_{\perp}\lambda_{\parallel}\sin(\delta_{\perp}-\delta_{\parallel}+\theta^{c}_{\perp}-\theta^{c}_{\parallel})\end{array}$ | $\begin{array}[]{c}-\lambda_{\perp}\cos(\delta_{\perp}-\delta_{\parallel}+\theta^{c}_{\perp}-\theta_{\parallel})\\\ -\lambda_{\parallel}\cos(\delta_{\perp}-\delta_{\parallel}+\theta_{\perp}-\theta^{c}_{\parallel})\end{array}$ | $-2\sin^{2}\theta_{1}\sin^{2}\theta_{2}\sin 2\Phi$
5 | ${|A_{\parallel}||A_{0}|}/{2}$ | $\begin{array}[]{c}\cos(\delta_{0}-\delta_{\parallel}+\theta_{0}-\theta_{\parallel})\\\ +\lambda_{0}\lambda_{\parallel}\cos(\delta_{0}-\delta_{\parallel}+\theta^{c}_{0}-\theta^{c}_{\parallel})\end{array}$ | $\begin{array}[]{c}-\lambda_{0}\cos(\delta_{0}-\delta_{\parallel}+\theta^{c}_{0}-\theta_{\parallel})\\\ -\lambda_{\parallel}\cos(\delta_{0}-\delta_{\parallel}+\theta_{0}-\theta^{c}_{\parallel})\end{array}$ | $\begin{array}[]{c}\cos(\delta_{0}-\delta_{\parallel}+\theta_{0}-\theta_{\parallel})\\\ -\lambda_{0}\lambda_{\parallel}\cos(\delta_{0}-\delta_{\parallel}+\theta^{c}_{0}-\theta^{c}_{\parallel})\end{array}$ | $\begin{array}[]{c}-\lambda_{0}\sin(\delta_{0}-\delta_{\parallel}+\theta^{c}_{0}-\theta_{\parallel})\\\ +\lambda_{\parallel}\sin(\delta_{0}-\delta_{\parallel}+\theta_{0}-\theta^{c}_{\parallel})\end{array}$ | $\sqrt{2}\sin 2\theta_{1}\sin 2\theta_{2}\cos\Phi$
6 | ${|A_{0}||A_{\perp}|}/{2}$ | $\begin{array}[]{c}\sin(\delta_{\perp}-\delta_{0}+\theta_{\perp}-\theta_{0})\\\ -\lambda_{\perp}\lambda_{0}\sin(\delta_{\perp}-\delta_{0}+\theta^{c}_{\perp}-\theta^{c}_{0})\end{array}$ | $\begin{array}[]{c}\lambda_{\perp}\sin(\delta_{\perp}-\delta_{0}+\theta^{c}_{\perp}-\theta_{0})\\\ -\lambda_{0}\sin(\delta_{\perp}-\delta_{0}+\theta_{\perp}-\theta^{c}_{0})\end{array}$ | $\begin{array}[]{c}\sin(\delta_{\perp}-\delta_{0}+\theta_{\perp}-\theta_{0})\\\ +\lambda_{\perp}\lambda_{0}\sin(\delta_{\perp}-\delta_{0}+\theta^{c}_{\perp}-\theta^{c}_{0})\end{array}$ | $\begin{array}[]{c}-\lambda_{\perp}\cos(\delta_{\perp}-\delta_{0}+\theta^{c}_{\perp}-\theta_{0})\\\ -\lambda_{0}\cos(\delta_{\perp}-\delta_{0}+\theta_{\perp}-\theta^{c}_{0})\end{array}$ | $-\sqrt{2}\sin 2\theta_{1}\sin 2\theta_{2}\sin\Phi$

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