# Charmless $B\to PPP$ Decays:
the Fully-Antisymmetric Final State

Bhubanjyoti Bhattacharya<EMAIL_ADDRESS>Department of Natural Sciences,
Lawrence Technological University, Southfield, MI 48075, USA  Mirjam Fines-
Neuschild<EMAIL_ADDRESS>Centre for Engineering in
Society (CES), Concordia University, 1515 Ste-Catherine St. W., Montréal, QC,
Canada H3G 2W1  Andrea Houck<EMAIL_ADDRESS>Department of Natural Sciences,
Lawrence Technological University, Southfield, MI 48075, USA  Maxime Imbeault
<EMAIL_ADDRESS>Département de physique, Cégep de Saint-Laurent, 625,
avenue Sainte-Croix, Montréal, QC, Canada H4L 3X7  Alexandre Jean
<EMAIL_ADDRESS>Physique des Particules, Université de Montréal,
1375 Avenue Thérèse-Lavoie-Roux, Montréal, QC, Canada H2V 0B3  David London
<EMAIL_ADDRESS>Physique des Particules, Université de Montréal, 1375
Avenue Thérèse-Lavoie-Roux, Montréal, QC, Canada H2V 0B3

###### Abstract

Under flavor $SU(3)$ symmetry (SU(3)F), the final-state particles in $B\to
PPP$ decays ($P$ is a pseudoscalar meson) are treated as identical, and the
$PPP$ must be in a fully-symmetric (FS) state, a fully-antisymmetric (FA)
state, or in one of four mixed states. In this paper, we present the formalism
for the FA states. We write the amplitudes for the 22 $B\to PPP$ decays that
can be in an FA state in terms of both SU(3)F reduced matrix elements and
diagrams. This shows the equivalence of diagrams and SU(3)F. We also give 15
relations among the amplitudes in the SU(3)F limit, as well as the additional
four that appear when the diagrams $E$/$A$/$PA$ are neglected. We present sets
of $B\to PPP$ decays that can be used to extract $\gamma$ using the FA
amplitudes. The value(s) of $\gamma$ found in this way can be compared with
the value(s) found using the FS states.

††preprint: UdeM-GPP-TH-23-298

##  I Introduction

The $B$-factories BaBar and Belle were built with the goal of measuring CP
violation in $B$ decays. The idea was to measure the three angles of the
unitarity triangle, $\alpha$, $\beta$, and $\gamma$, and to test the standard
model (SM) by seeing if $\alpha+\beta+\gamma=\pi$. Now, $\alpha$ is measured
using $B\to\pi\pi$ decays, and the (loop-level) penguin contribution is
removed using an isospin analysis [1]. $\beta$ is mainly measured in decays
such as $B^{0}\to J/\psi K_{S}$, which are dominated by the tree contribution.
And the standard methods of measuring $\gamma$ [2, 3, 4, 5] involve only tree-
level decays. As a result, NP can affect these measurements only if it can
compete with the tree-level SM contributions. (In principle, there could be
(loop-level) NP contributions to $B^{0}$-${\bar{B}}^{0}$ mixing, but these
effects cancel in the sum $\alpha+\beta$ [6].) Given that no new particles
have been seen at the LHC, we now know that the NP must be heavy, so its
contributions cannot compete with those of the SM at tree level. It is
therefore unsurprising that $\alpha+\beta+\gamma\simeq\pi$ was found [7].

Another way to search for NP using CP violation in $B$ decays is to measure
the same CP phase in two different ways. If the results do not agree, this
would reveal the presence of NP. An example is $\beta$. At the quark level,
the decay $B^{0}\to J/\psi K_{S}$ is ${\bar{b}}\to{\bar{c}}c{\bar{s}}$, which
has no weak phase in the SM. Similarly, the decay $B^{0}\to\phi K_{S}$
involves ${\bar{b}}\to{\bar{s}}s{\bar{s}}$, which can arise only via loop-
level gluonic and electroweak penguin contributions, and also has no weak
phase in the SM, to a good approximation. The point is that $\beta$ can be
measured using either decay [8]. The difference between the two is that, while
tree-level NP contributions are much smaller than tree-level SM contributions,
they can be of the same order as loop-level SM contributions. Thus, a
difference between the (tree-level) measurement of $\beta$ in $B^{0}\to J/\psi
K_{S}$ and its (loop-level) measurement in $B^{0}\to\phi K_{S}$ would point to
a (tree-level) NP contribution to ${\bar{b}}\to{\bar{s}}s{\bar{s}}$.
Experiments have searched for such a discrepancy, but none has been observed
[7].

In principle, this can also be done with $\gamma$. If $\gamma$ could be
extracted from decays that receive significant penguin contributions (gluonic
and/or electroweak), one could compare this (loop-level) measurement of
$\gamma$ with that of the (tree-level) methods of Refs. [2, 3, 4, 5].

In fact, methods for making a loop-level measurement of $\gamma$ were proposed
in Refs. [9, 10, 11]. They all involve charmless, three-body $B\to PPP$ decays
($P$ is a pseudoscalar meson). Under flavor SU(3) symmetry (SU(3)F), the three
final-state particles are treated as identical. The total final-state
wavefunction must be symmetric, so that the six permutations of these
particles must be considered: the $PPP$ must be in a fully-symmetric state, a
fully-antisymmetric state, or in one of four mixed states under SU(3)F.

For the measurement of the decay $B\to P_{1}P_{2}P_{3}$, the results are
usually presented in the form of a Dalitz plot. This is a function of two of
the three Mandelstam variables, say $s_{12}$ and $s_{13}$, where
$s_{ij}\equiv\left(p_{i}+p_{j}\right)^{2}$. One can then perform an isobar
analysis, which is essentially a fit of the Dalitz plot to a non-resonant and
various intermediate resonant contributions, to obtain the decay amplitude
${\cal M}(s_{12},s_{13})$ describing $B\to P_{1}P_{2}P_{3}$. In Ref. [12], it
is pointed out that one can use ${\cal M}(s_{12},s_{13})$ to construct the
amplitudes for the individual fully-symmetric, fully-antisymmetric and mixed
final states. In this way, one can study decays into final states with each of
the possible symmetries.

Ref. [12] also shows that the $B\to PPP$ amplitudes can be written in terms of
diagrams similar to those used in $B\to PP$ decays [13, 14]. The main
advantage of using diagrams to describe $B$-decay amplitudes is that it can be
argued on dynamical grounds that certain diagrams are subdominant. The neglect
of these diagrams greatly simplifies the analysis. We note that, for $B\to PP$
decays, this theoretical assumption has been borne out by experiment: decays
that are mediated by these supposedly subdominant diagrams, such as $B^{0}\to
K^{+}K^{-}$ and $B_{s}^{0}\to\pi^{+}\pi^{-}$, are indeed found to have
branching ratios considerably smaller than those of other charmless $B\to PP$
decays.

Still, we stress that this assumption does not follow from group theory.
Before putting it into practice, it must be shown that the description of the
amplitudes using the full set of diagrams is equivalent to a description in
terms of SU(3)F reduced matrix elements (RMEs). In Ref. [15], this is
demonstrated explicitly for the fully-symmetric (FS) final state in $B\to
PPP$. It is therefore justified to use a diagrammatic description of these
decay amplitudes and to neglect certain diagrams.

In the methods proposed in Refs. [9, 10, 11], these techniques are used to
cleanly extract the weak phase $\gamma$ from the FS states of various $B\to
PPP$ decays. The method of Ref. [10] is particularly interesting. It combines
information from the Dalitz plots for $B^{0}\to K^{+}\pi^{0}\pi^{-}$,
$B^{0}\to K^{0}\pi^{+}\pi^{-}$, $B^{+}\to K^{+}\pi^{+}\pi^{-}$, $B^{0}\to
K^{+}K^{0}K^{-}$, and $B^{0}\to K^{0}K^{0}\overline{K}^{0}$. These $B\to
K\pi\pi$ and $B\to KK{\bar{K}}$ decays all receive loop-level penguin and
electroweak-penguin contributions, so it is a loop-level value of $\gamma$
that is measured here. As noted above, the comparison of the tree-level and
loop-level measurements of $\gamma$ is an excellent test of the Standard
Model.

This method was applied in Ref. [16] to the measurements of the Dalitz plots
of the five $B\to K\pi\pi$ and $B\to KK{\bar{K}}$ decays by the BaBar
Collaboration [17, 18, 19, 20, 21]. However, this was a theoretical analysis:
by its own admission, it did not properly take all the errors into account.
This was improved in Ref. [22], which was a collaboration of theory and
experiment. Six possible values of $\gamma$ were found:

$\displaystyle\gamma_{1}$ $\displaystyle=$
$\displaystyle[\phantom{1}12.9\mbox{}\,^{+8.4\phantom{1}}_{-4.3}\text{
(stat)}\pm 1.3\text{ (syst)}]{{}^{\circ}}~{},$ $\displaystyle\gamma_{2}$
$\displaystyle=$
$\displaystyle[\phantom{1}36.6\mbox{}\,^{+6.6\phantom{1}}_{-6.1}\text{
(stat)}\pm 2.6\text{ (syst)}]{{}^{\circ}}~{},$ $\displaystyle\gamma_{3}$
$\displaystyle=$
$\displaystyle[\phantom{1}68.9\mbox{}\,^{+8.6\phantom{1}}_{-8.6}\text{
(stat)}\pm 2.4\text{ (syst)}]{{}^{\circ}}~{},$ $\displaystyle\gamma_{4}$
$\displaystyle=$ $\displaystyle[223.2\mbox{}\,^{+10.9}_{-7.5}\text{ (stat)}\pm
1.0\text{ (syst)}]{{}^{\circ}}~{},$ $\displaystyle\gamma_{5}$ $\displaystyle=$
$\displaystyle[266.4\mbox{}\,^{+9.2}_{-10.8}\text{ (stat)}\pm 1.9\text{
(syst)}]{{}^{\circ}}~{},$ $\displaystyle\gamma_{6}$ $\displaystyle=$
$\displaystyle[307.5\mbox{}\,^{+6.9\phantom{1}}_{-8.1}\text{ (stat)}\pm
1.1\text{ (syst)}]{{}^{\circ}}~{}.$ (1)

One solution – $\gamma_{3}$ – is compatible with the latest world average
tree-level value, $\gamma=(66.2^{+3.4}_{-3.6})^{\circ}$ [7]. The other
solutions are in disagreement, perhaps hinting at new physics. In addition, it
is found that, when averaged over the entire Dalitz plane, the effect of
SU(3)F breaking on the analysis is only at the percent level.

At this stage, the burning question is: what is the true value of $\gamma$ in
this system? The above analysis was carried out using the FS final state. One
way this question might be answered is to repeat the analysis – or perform a
different analysis to extract $\gamma$ – using a different symmetry of the
final state. The hope is that, if there are again multiple solutions for
$\gamma$, only one will be common to the two sets of solutions; this will be
the true value of $\gamma$. And if it differs from the tree-level value, this
will be a smoking-gun signal of new physics.

The formalism describing $B\to PPP$ decays with a FS final state was presented
in Refs. [12, 9, 10, 15]. However, the same formalism has not been given for
the other final-state symmetries. In this paper, we focus on $B\to PPP$ decays
in which the final state is fully antisymmetric.

We begin in Sec. 2 with a presentation of the Wigner-Eckart decomposition of
the FA $B\to PPP$ amplitudes in terms of SU(3)F reduced matrix elements. A
similar decomposition in terms of diagrams is given in Sec, 3, thereby
demonstrating the equivalence of SU(3)F reduced matrix elements and diagrams.
Various relations among the amplitude are given in Sec. 4. Sec. 5 discusses
the consequences of neglecting the $E$/$A$/$PA$ diagrams, which are expected
to be smaller than the other diagrams. Various applications of this formalism,
including the extraction of $\gamma$ and the measurement of SU(3)F breaking,
are elaborated in Sec. 6. We conclude in Sec. 7.

##  II SU(3)F Wigner-Eckart Decomposition

We begin by representing the $B\to PPP$ decay amplitudes for fully-
antisymmetric (FA) final states in terms of SU(3)F reduced matrix elements.
The amplitude for a decay process involves three pieces: a) the initial state,
b) the Hamiltonian, and c) the final state. Here, the SU(3)F representations
of the decaying $B$ mesons and the underlying quark-level transitions are
identical to those used in Ref. [15], where the FS state was studied. The
three-body final states we consider in this article are new: under the
exchange of any two of the three final-state particles, the $\left|PPP\right>$
states considered in this article are fully antisymmetric.

In this section, we perform SU(3)F Wigner-Eckart decompositions of the FA
$B\to PPP$ decay amplitudes. We adopt the notation used in Ref. [15] and
represent each element of SU(3)F by $\left|{\bf r}YII_{3}\right>$, where ${\bf
r}$ is the irreducible representation (irrep) of SU(3)F, $Y$ is the
hypercharge, and $I$ and $I_{3}$ stand for the isospin and its third
component, respectively. Note that, in general, Lie algebras are not
associative, so that the order of multiplication of elements is important.
Here we take products from left to right. We use the SU(3)F isoscalar factors
from Refs. [23, 24], along with SU(2) Clebsch-Gordan coefficients, to
construct products of SU(3)F states.

There are 16 ${\bar{b}}\to{\bar{s}}$ and 16 ${\bar{b}}\to{\bar{d}}$ charmless
three-body $B\to PPP$ decays, where $P=\pi$ or $K$. Under SU(3)F, all three
final-state particles belong to the same multiplet (an octet of SU(3)F), and
hence they can be treated as identical, so the six possible permutations of
these particles must be considered. The FA final state is antisymmetric under
the exchange of any two final-state particles. This is only possible when all
three final-state pseudoscalars are distinct, which reduces the number of
available decays to 11 for each of ${\bar{b}}\to{\bar{s}}$ and
${\bar{b}}\to{\bar{d}}$111There is a caveat here: two ${\bar{b}}\to{\bar{s}}$
and two ${\bar{b}}\to{\bar{d}}$ decays contain $K^{0}{\bar{K}}^{0}$ in the
final state. These have an FA state only if this pair can be detected as
$K_{S}K_{L}$..

For the FA final state, one wants to find $({\bf 8}\times{\bf 8}\times{\bf
8})_{\rm FA}$. The decomposition for ${\bf 8}\times{\bf 8}={\bf 27}+{\bf
10}+{\bf 10^{*}}+{\bf 8}+{\bf 8}+{\bf 1}$ can be separated into ${\bf 27}+{\bf
8}+{\bf 1}$ (total 36) symmetric and ${\bf 10}+{\bf 10^{*}}+{\bf 8}$ (total
28) antisymmetric under the exchange of the two ${\bf 8}$’s. The FA irreps of
${\bf 8}\times{\bf 8}\times{\bf 8}$ arise from the product of the
antisymmetric ${\bf 10}+{\bf 10^{*}}+{\bf 8}$ with the remaining ${\bf 8}$.
This yields an FA final state that has dimension 56 under SU(3)F. It can be
decomposed into irreps of SU(3)F as follows:

$\displaystyle({\bf 8}\times{\bf 8}\times{\bf 8})_{\rm FA}={\bf 27}_{\rm
FA}+{\bf 10}_{\rm FA}+{\bf 10^{*}}_{\rm FA}+{\bf 8}_{\rm FA}+{\bf 1},~{}~{}$
(2)

where

$\displaystyle{\bf 27}_{\rm FA}$ $\displaystyle=$
$\displaystyle\frac{2}{3}{\bf 27}_{{\bf 10}\times{\bf 8}}-\frac{2}{3}{\bf
27}_{{\bf 10^{*}}\times{\bf 8}}+\frac{1}{3}{\bf 27}_{{\bf 8}\times{\bf
8}}~{},$ $\displaystyle{\bf 10}_{\rm FA}$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{2}{3}}{\bf 10}_{{\bf 10}\times{\bf
8}}+\sqrt{\frac{1}{3}}{\bf 10}_{{\bf 8}\times{\bf 8}}~{},$ $\displaystyle{\bf
10^{*}}_{\rm FA}$ $\displaystyle=$ $\displaystyle-\sqrt{\frac{2}{3}}{\bf
10^{*}}_{{\bf 10^{*}}\times{\bf 8}}+\sqrt{\frac{1}{3}}{\bf 10^{*}}_{{\bf
8}\times{\bf 8}}~{},$ $\displaystyle{\bf 8}_{\rm FA}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{6}}{\bf 8}_{{\bf 10}\times{\bf
8}}+\frac{1}{\sqrt{6}}{\bf 8}_{{\bf 10^{*}}\times{\bf
8}}+\sqrt{\frac{2}{3}}{\bf 8}_{{\bf 8}\times{\bf 8}}~{}.$ (3)

###  II.1 SU(3)F assignments of pseudoscalar mesons

The light-quark states ($u$, $d$ and $s$) transform as the fundamental triplet
(${\bf 3}$) of SU(3)F. The antiquarks transform as the ${\bf 3^{*}}$ of
SU(3)F. The quarks and antiquarks can be assigned the following
representations using the $\left|{\bf r}YII_{3}\right>$ notation:

$\displaystyle\left|u\right>~{}=~{}\left|{\bf
3}\frac{1}{3}\frac{1}{2}\frac{1}{2}\right>~{},$
$\displaystyle-\left|{\bar{u}}\right>~{}=~{}\left|{\bf
3^{*}}{-}\frac{1}{3}\frac{1}{2}{-}\frac{1}{2}\right>~{},$
$\displaystyle\left|d\right>~{}=~{}\left|{\bf
3}\frac{1}{3}\frac{1}{2}{-}\frac{1}{2}\right>~{},$
$\displaystyle\left|{\bar{d}}\right>~{}=~{}\left|{\bf
3^{*}}{-}\frac{1}{3}\frac{1}{2}\frac{1}{2}\right>~{},$
$\displaystyle\left|s\right>~{}=~{}\left|{\bf 3}{-}\frac{2}{3}00\right>~{},$
$\displaystyle\left|{\bar{s}}\right>~{}=~{}\left|{\bf
3^{*}}\frac{2}{3}00\right>~{}.$ (4)

The pions, kaons, and the octet component of the eta meson ($\eta_{8}$) form
an octet (${\bf 8}$) of SU(3)F, while the $\eta_{1}$ is an SU(3)F singlet. The
physical $\eta$ and $\eta^{\prime}$ mesons are linear combinations of the
$\eta_{8}$ and $\eta_{1}$, constructed through octet-singlet mixing. In this
work, we avoid the complications arising from this mixing by limiting our
analysis to final states with only pions and/or kaons. The three pions and the
four kaons are as follows:

$\displaystyle\left|\pi^{+}\right>~{}=~{}\left|u\right>\left|{\bar{d}}\right>~{}=~{}\left|{\bf
8}011\right>~{},~{}~{}~{}~{}~{}\left|\pi^{-}\right>~{}=~{}-\left|d\right>\left|{\bar{u}}\right>~{}=~{}\left|{\bf
8}01{-}1\right>~{},~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\left|\pi^{0}\right>~{}=~{}\frac{\left|d\right>\left|{\bar{d}}\right>-\left|u\right>\left|{\bar{u}}\right>}{\sqrt{2}}~{}=~{}\left|{\bf
8}010\right>~{},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\left|K^{+}\right>~{}=~{}\left|u\right>\left|{\bar{s}}\right>~{}=~{}\left|{\bf
8}1\frac{1}{2}\frac{1}{2}\right>~{},~{}~{}~{}~{}~{}\left|K^{0}\right>~{}=~{}\left|d\right>\left|{\bar{s}}\right>~{}=~{}\left|{\bf
8}1\frac{1}{2}{-}\frac{1}{2}\right>~{},~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\left|\overline{K}^{0}\right>~{}=~{}\left|s\right>\left|{\bar{d}}\right>~{}=~{}\left|{\bf
8}{-}1\frac{1}{2}\frac{1}{2}\right>~{},~{}~{}~{}~{}~{}\left|K^{-}\right>~{}=~{}-\left|s\right>\left|{\bar{u}}\right>~{}=~{}\left|{\bf
8}{-}1\frac{1}{2}{-}\frac{1}{2}\right>~{}.$ (5)

###  II.2 Fully-antisymmetric three-body final states

We now construct the normalized FA $P_{1}P_{2}P_{3}$ final states within
SU(3)F. The FS final state studied in Ref. [15] could be divided into three
cases, depending on the number of truly identical particles in the final
state. For the FA state, there is only one case: in order for the FA final
state to be non-vanishing, all three final-state pseudoscalars must be
distinct from one another (e.g., $\pi^{0}\pi^{+}\pi^{-}$). We first construct
states that are antisymmetrized over the first two particles. We then add all
three combinations antisymmetrized in this way to obtain the FA state.

In what follows the state is antisymmetrized over particles that are included
within square brackets:

$\displaystyle\left|[P_{1}P_{2}]P_{3}\right>$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left[\left|P_{1}\right>\left|P_{2}\right>\left|P_{3}\right>-\left|P_{2}\right>\left|P_{1}\right>\left|P_{3}\right>\right]~{},$
$\displaystyle\left|[P_{1}P_{2}P_{3}]\right>_{\rm FA}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}\left[\left|[P_{1}P_{2}]P_{3}\right>+\left|[P_{2}P_{3}]P_{1}\right>+\left|[P_{3}P_{1}]P_{2}\right>\right]~{}.$
(6)

Note that, if any two of three (or all three) of the particles are identical
(e.g., $\pi^{0}\pi^{0}\pi^{+}$ or $\pi^{0}\pi^{0}\pi^{0}$), the three-particle
state, $\left|[P_{1}P_{2}P_{3}]\right>_{\rm FA}$, automatically vanishes.

###  II.3 Three-body ${\bar{b}}\to{\bar{s}}$ and ${\bar{b}}\to{\bar{d}}$
transitions using SU(3)F

The Hamiltonian for three-body $B$ decays follows from the underlying quark-
level transitions ${\bar{b}}\to{\bar{s}}q{\bar{q}}$ and
${\bar{b}}\to{\bar{d}}q{\bar{q}}$, where $q$ is an up-type quark ($u,c,t$).
However, the unitarity of the Cabibbo-Kobayashi-Maskawa (CKM) matrix, given as

$\displaystyle\sum\limits_{q=u,c,t}V^{*}_{qb}V_{qs}=0~{}~{},~{}~{}~{}~{}\sum\limits_{q=u,c,t}V^{*}_{qb}V_{qd}=0~{},$
(7)

allows us to trade one of the up-type quarks for the other two. Here we choose
to replace the $t$-quark operators and retain only the $c$-quark and $u$-quark
operators. Thus the weak-interaction Hamiltonian is composed of four types of
operators: ${\bar{b}}\to{\bar{s}}c{\bar{c}}$,
${\bar{b}}\to{\bar{d}}c{\bar{c}}$, ${\bar{b}}\to{\bar{s}}u{\bar{u}}$, and
${\bar{b}}\to{\bar{d}}u{\bar{u}}$.

The SU(3)F representations of these operators are dictated by the light quarks
since the heavy $b$, $c$, and $t$ quarks are SU(3)F singlets. The transition
operators are given as follows:

$\displaystyle{\cal O}_{{\bar{b}}\to{\bar{s}}c{\bar{c}}}$ $\displaystyle=$
$\displaystyle V^{*}_{cb}V_{cs}B^{(\bf
3^{*})}_{\\{\frac{2}{3},0,0\\}}~{},~{}~{}~{}~{}~{}{\cal
O}_{{\bar{b}}\to{\bar{d}}c{\bar{c}}}~{}=~{}V^{*}_{cb}V_{cd}B^{(\bf
3^{*})}_{\\{-\frac{1}{3},\frac{1}{2},\frac{1}{2}\\}}~{},$ $\displaystyle{\cal
O}_{{\bar{b}}\to{\bar{s}}u{\bar{u}}}$ $\displaystyle=$ $\displaystyle
V^{*}_{ub}V_{us}\left\\{A^{(\bf 3^{*})}_{\\{\frac{2}{3},0,0\\}}+R^{(\bf
6)}_{\\{\frac{2}{3},1,0\\}}+\sqrt{6}P^{(\bf
15^{*})}_{\\{\frac{2}{3},1,0\\}}+\sqrt{3}P^{(\bf
15^{*})}_{\\{\frac{2}{3},0,0\\}}\right\\}~{},$ $\displaystyle{\cal
O}_{{\bar{b}}\to{\bar{d}}u{\bar{u}}}$ $\displaystyle=$ $\displaystyle
V^{*}_{ub}V_{ud}\left\\{A^{(\bf
3^{*})}_{\\{-\frac{1}{3},\frac{1}{2},\frac{1}{2}\\}}-R^{(\bf
6)}_{\\{-\frac{1}{3},\frac{1}{2},\frac{1}{2}\\}}+\sqrt{8}P^{(\bf
15^{*})}_{\\{-\frac{1}{3},\frac{3}{2},\frac{1}{2}\\}}+P^{(\bf
15^{*})}_{\\{-\frac{1}{3},\frac{1}{2},\frac{1}{2}\\}}\right\\}~{},$ (8)

where we have used the notation $O^{(\bf r)}_{\\{Y,I,I_{3}\\}}$ to represent
each SU(3)F operator ($O=\\{A,B,R,P\\}$). We have taken the names of these
operators and their relative signs from Ref. [26]. The weak-interaction
Hamiltonian that governs charmless $B$ decays is then simply the sum of these
four operators:

$\displaystyle{\cal H}$ $\displaystyle=$ $\displaystyle{\cal
O}_{{\bar{b}}\to{\bar{s}}c{\bar{c}}}+{\cal
O}_{{\bar{b}}\to{\bar{d}}c{\bar{c}}}+{\cal
O}_{{\bar{b}}\to{\bar{s}}u{\bar{u}}}+{\cal
O}_{{\bar{b}}\to{\bar{d}}u{\bar{u}}}~{}.$ (9)

The above Hamiltonian governs the decay of the SU(3)F triplet of $B$-mesons
[$B^{\bf 3}=(B^{+}_{u},B^{0}_{d},B^{0}_{s}$)], whose components have the same
SU(3)F representations as their corresponding light quarks. The fully-
antisymmetric three-body decay amplitude for the process $B\to
P_{1}P_{2}P_{3}$ can now be constructed easily as follows:

$\displaystyle{\cal A}_{\rm FA}(p_{1},p_{2},p_{3})$ $\displaystyle=$ ${}_{\rm
FA}\left<[P_{1}P_{2}P_{3}]\left|\right|{\cal H}\left|\right|B^{\bf
3}\right>~{},$ (10)

where $p_{i}$ represents the momentum of the final-state particle $P_{i}$.

Table 1: Amplitudes for $\Delta S=1$ $B$-meson decays to fully-antisymmetric $PPP$ states as functions of nine SU(3)F RMEs. Decay | $V^{*}_{cb}V_{cs}$ | $V^{*}_{ub}V_{us}$
---|---|---
Amplitude | $B^{{(fa)}}_{1}$ | $B^{{(fa)}}$ | $A^{{(fa)}}_{1}$ | $A^{{(fa)}}$ | $R^{{(fa)}}_{8}$ | $R^{{(fa)}}_{10}$ | $P^{{(fa)}}_{8}$ | $P^{{(fa)}}_{10^{*}}$ | $P^{{(fa)}}_{27}$
$\sqrt{2}{\cal A}(B^{+}\to K^{+}\pi^{+}\pi^{-})_{\rm FA}$ | 0 | $\frac{\sqrt{2}}{\sqrt{5}}$ | 0 | $\frac{\sqrt{2}}{\sqrt{5}}$ | $\frac{\sqrt{2}}{\sqrt{15}}$ | $\frac{\sqrt{2}}{\sqrt{3}}$ | $-\frac{3\sqrt{2}}{5}$ | 0 | $\frac{6\sqrt{3}}{5}$
${\cal A}(B^{+}\to K^{0}\pi^{+}\pi^{0})_{\rm FA}$ | 0 | $-\frac{\sqrt{2}}{\sqrt{5}}$ | 0 | $-\frac{\sqrt{2}}{\sqrt{5}}$ | $-\frac{\sqrt{2}}{\sqrt{15}}$ | $\frac{1}{\sqrt{6}}$ | $\frac{3\sqrt{2}}{5}$ | 0 | $-\frac{\sqrt{3}}{5}$
$\sqrt{2}{\cal A}(B^{0}\to K^{0}\pi^{+}\pi^{-})_{\rm FA}$ | 0 | $-\frac{\sqrt{2}}{\sqrt{5}}$ | 0 | $-\frac{\sqrt{2}}{\sqrt{5}}$ | $\frac{\sqrt{2}}{\sqrt{15}}$ | $\frac{\sqrt{2}}{\sqrt{3}}$ | $-\frac{\sqrt{2}}{5}$ | 0 | $\frac{2\sqrt{3}}{5}$
${\cal A}(B^{0}\to K^{+}\pi^{0}\pi^{-})_{\rm FA}$ | 0 | $\frac{\sqrt{2}}{\sqrt{5}}$ | 0 | $\frac{\sqrt{2}}{\sqrt{5}}$ | $-\frac{\sqrt{2}}{\sqrt{15}}$ | $\frac{1}{\sqrt{6}}$ | $\frac{\sqrt{2}}{5}$ | 0 | $\frac{3\sqrt{3}}{5}$
$\sqrt{2}{\cal A}(B^{+}\to K^{+}K^{0}\overline{K}^{0})_{\rm FA}$ | 0 | $\frac{\sqrt{2}}{\sqrt{5}}$ | 0 | $\frac{\sqrt{2}}{\sqrt{5}}$ | $\frac{\sqrt{2}}{\sqrt{15}}$ | 0 | $-\frac{3\sqrt{2}}{5}$ | 0 | $-\frac{4\sqrt{3}}{5}$
$\sqrt{2}{\cal A}(B^{0}\to K^{0}K^{+}K^{-})_{\rm FA}$ | 0 | $-\frac{\sqrt{2}}{\sqrt{5}}$ | 0 | $-\frac{\sqrt{2}}{\sqrt{5}}$ | $\frac{\sqrt{2}}{\sqrt{15}}$ | 0 | $-\frac{\sqrt{2}}{5}$ | $2$ | $\frac{2\sqrt{3}}{5}$
${\cal A}(B_{s}^{0}\to\pi^{0}K^{+}K^{-})_{\rm FA}$ | $-\frac{1}{4\sqrt{3}}$ | 0 | $-\frac{1}{4\sqrt{3}}$ | 0 | $\frac{\sqrt{2}}{\sqrt{15}}$ | $-\frac{1}{2\sqrt{6}}$ | $\frac{2\sqrt{2}}{5}$ | $-\frac{1}{2}$ | $-\frac{\sqrt{3}}{20}$
${\cal A}(B_{s}^{0}\to\pi^{0}K^{0}\overline{K}^{0})_{\rm FA}$ | $-\frac{1}{4\sqrt{3}}$ | 0 | $-\frac{1}{4\sqrt{3}}$ | 0 | $-\frac{\sqrt{2}}{\sqrt{15}}$ | $\frac{1}{2\sqrt{6}}$ | $-\frac{2\sqrt{2}}{5}$ | $\frac{1}{2}$ | $-\frac{9\sqrt{3}}{20}$
$\sqrt{2}{\cal A}(B_{s}^{0}\to\pi^{-}K^{+}\overline{K}^{0})_{\rm FA}$ | $\frac{1}{2\sqrt{3}}$ | 0 | $\frac{1}{2\sqrt{3}}$ | 0 | 0 | $\frac{1}{\sqrt{6}}$ | 0 | $-1$ | $\frac{\sqrt{3}}{2}$
$\sqrt{2}{\cal A}(B_{s}^{0}\to\pi^{+}K^{0}K^{-})_{\rm FA}$ | $\frac{1}{2\sqrt{3}}$ | 0 | $\frac{1}{2\sqrt{3}}$ | 0 | 0 | $-\frac{1}{\sqrt{6}}$ | 0 | $1$ | $\frac{\sqrt{3}}{2}$
${\cal A}(B_{s}^{0}\to\pi^{0}\pi^{+}\pi^{-})_{\rm FA}$ | $-\frac{1}{2\sqrt{3}}$ | $\frac{\sqrt{2}}{\sqrt{5}}$ | $-\frac{1}{2\sqrt{3}}$ | $\frac{\sqrt{2}}{\sqrt{5}}$ | 0 | 0 | $\frac{3\sqrt{2}}{5}$ | 0 | $\frac{3\sqrt{3}}{10}$
Table 2: Amplitudes for $\Delta S=0$ $B$-meson decays to fully-antisymmetric $PPP$ states as functions of SU(3)F RMEs. Decay | $V^{*}_{cb}V_{cd}$ | $V^{*}_{ub}V_{ud}$
---|---|---
Amplitude | $B^{{(fa)}}_{1}$ | $B^{{(fa)}}$ | $A^{{(fa)}}_{1}$ | $A^{{(fa)}}$ | $R^{{(fa)}}_{8}$ | $R^{{(fa)}}_{10}$ | $P^{{(fa)}}_{8}$ | $P^{{(fa)}}_{10^{*}}$ | $P^{{(fa)}}_{27}$
${\cal A}(B^{+}\to\pi^{+}K^{0}\overline{K}^{0})_{\rm FA}$ | 0 | $-\frac{1}{\sqrt{5}}$ | 0 | $-\frac{1}{\sqrt{5}}$ | $-\frac{1}{\sqrt{15}}$ | 0 | $\frac{3}{5}$ | 0 | $\frac{2\sqrt{6}}{5}$
${\cal A}(B^{+}\to\pi^{+}K^{+}K^{-})_{\rm FA}$ | 0 | $\frac{1}{\sqrt{5}}$ | 0 | $\frac{1}{\sqrt{5}}$ | $\frac{1}{\sqrt{15}}$ | $\frac{1}{\sqrt{3}}$ | $-\frac{3}{5}$ | 0 | $\frac{3\sqrt{6}}{5}$
$\sqrt{2}{\cal A}(B^{+}\to\pi^{0}K^{+}\overline{K}^{0})_{\rm FA}$ | 0 | 0 | 0 | 0 | 0 | $-\frac{1}{\sqrt{3}}$ | 0 | 0 | $\sqrt{6}$
$\sqrt{2}{\cal A}(B^{0}\to\pi^{0}K^{0}\overline{K}^{0})_{\rm FA}$ | $-\frac{1}{2\sqrt{6}}$ | $-\frac{1}{\sqrt{5}}$ | $-\frac{1}{2\sqrt{6}}$ | $-\frac{1}{\sqrt{5}}$ | $-\frac{1}{\sqrt{15}}$ | $-\frac{1}{2\sqrt{3}}$ | -1 | $-\frac{1}{\sqrt{2}}$ | $\frac{3\sqrt{3}}{2\sqrt{2}}$
$\sqrt{2}{\cal A}(B^{0}\to\pi^{0}K^{+}K^{-})_{\rm FA}$ | $-\frac{1}{2\sqrt{6}}$ | $\frac{1}{\sqrt{5}}$ | $-\frac{1}{2\sqrt{6}}$ | $\frac{1}{\sqrt{5}}$ | $\frac{1}{\sqrt{15}}$ | $\frac{1}{2\sqrt{3}}$ | 1 | $\frac{1}{\sqrt{2}}$ | $\frac{3\sqrt{3}}{2\sqrt{2}}$
${\cal A}(B^{0}\to\pi^{+}K^{0}K^{-})_{\rm FA}$ | $\frac{1}{2\sqrt{6}}$ | 0 | $\frac{1}{2\sqrt{6}}$ | 0 | 0 | $\frac{1}{2\sqrt{3}}$ | 0 | $-\frac{1}{\sqrt{2}}$ | $\frac{\sqrt{3}}{2\sqrt{2}}$
${\cal A}(B^{0}\to\pi^{-}K^{+}\overline{K}^{0})_{\rm FA}$ | $\frac{1}{2\sqrt{6}}$ | 0 | $\frac{1}{2\sqrt{6}}$ | 0 | 0 | $-\frac{1}{2\sqrt{3}}$ | 0 | $\frac{1}{\sqrt{2}}$ | $\frac{\sqrt{3}}{2\sqrt{2}}$
$\sqrt{2}{\cal A}(B^{0}\to\pi^{0}\pi^{+}\pi^{-})_{\rm FA}$ | $-\frac{1}{\sqrt{6}}$ | $-\frac{1}{\sqrt{5}}$ | $-\frac{1}{\sqrt{6}}$ | $-\frac{1}{\sqrt{5}}$ | $\frac{\sqrt{3}}{\sqrt{5}}$ | 0 | $\frac{3}{5}$ | 0 | $-\frac{\sqrt{3}}{5\sqrt{2}}$
${\cal A}(B_{s}^{0}\to\overline{K}^{0}\pi^{+}\pi^{-})_{\rm FA}$ | 0 | $-\frac{1}{\sqrt{5}}$ | 0 | $-\frac{1}{\sqrt{5}}$ | $\frac{1}{\sqrt{15}}$ | 0 | $-\frac{1}{5}$ | $\sqrt{2}$ | $\frac{\sqrt{6}}{5}$
$\sqrt{2}{\cal A}(B_{s}^{0}\to K^{-}\pi^{+}\pi^{0})_{\rm FA}$ | 0 | $\frac{2}{\sqrt{5}}$ | 0 | $\frac{2}{\sqrt{5}}$ | $-\frac{2}{\sqrt{15}}$ | 0 | $\frac{2}{5}$ | $\sqrt{2}$ | $\frac{3\sqrt{6}}{5}$
${\cal A}(B_{s}^{0}\to\overline{K}^{0}K^{+}K^{-})_{\rm FA}$ | 0 | $-\frac{1}{\sqrt{5}}$ | 0 | $-\frac{1}{\sqrt{5}}$ | $\frac{1}{\sqrt{15}}$ | $\frac{1}{\sqrt{3}}$ | $-\frac{1}{5}$ | 0 | $\frac{\sqrt{6}}{5}$

###  II.4 Reduced matrix elements

The 22 charmless three-body $B$ decay amplitudes (11 ${\bar{b}}\to{\bar{s}}$
and 11 ${\bar{b}}\to{\bar{d}}$) can all be written in terms of nine SU(3)F
RMEs (the $Y,I$, and $I_{3}$ indices of the operators have been suppressed):

$\displaystyle B^{{(fa)}}_{1}$ $\displaystyle\equiv$ ${}_{\rm FA}\left<{\bf
1}\left|\left|B^{(\bf 3^{*})}\right|\right|{\bf 3}\right>~{},$ $\displaystyle
B^{{(fa)}}$ $\displaystyle\equiv$ ${}_{\rm FA}\left<{\bf 8}\left|\left|B^{(\bf
3^{*})}\right|\right|{\bf 3}\right>~{},$ $\displaystyle A^{{(fa)}}_{1}$
$\displaystyle\equiv$ ${}_{\rm FA}\left<{\bf 1}\left|\left|A^{(\bf
3^{*})}\right|\right|{\bf 3}\right>~{},$ $\displaystyle A^{{(fa)}}$
$\displaystyle\equiv$ ${}_{\rm FA}\left<{\bf 8}\left|\left|A^{(\bf
3^{*})}\right|\right|{\bf 3}\right>~{},$ $\displaystyle R_{8}^{{(fa)}}$
$\displaystyle\equiv$ ${}_{\rm FA}\left<{\bf 8}\left|\left|R^{(\bf
6)}\right|\right|{\bf 3}\right>~{},$ $\displaystyle R_{10}^{{(fa)}}$
$\displaystyle\equiv$ ${}_{\rm FA}\left<{\bf 10}\left|\left|R^{(\bf
6)}\right|\right|{\bf 3}\right>~{},$ $\displaystyle P_{8}^{{(fa)}}$
$\displaystyle\equiv$ ${}_{\rm FA}\left<{\bf 8}\left|\left|P^{(\bf
15^{*})}\right|\right|{\bf 3}\right>~{},$ $\displaystyle P_{10^{*}}^{{(fa)}}$
$\displaystyle\equiv$ ${}_{\rm FA}\left<{\bf 10^{*}}\left|\left|P^{(\bf
15^{*})}\right|\right|{\bf 3}\right>~{},$ $\displaystyle P_{27}^{{(fa)}}$
$\displaystyle\equiv$ ${}_{\rm FA}\left<{\bf 27}\left|\left|P^{(\bf
15^{*})}\right|\right|{\bf 3}\right>~{}.$ (11)

The decomposition of all 22 amplitudes in terms of these RMEs is given in
Tables 1 and 2. As in the FS case [15], there are only seven combinations of
matrix elements in the amplitudes since $B^{{(fa)}}$ and $A^{{(fa)}}$, as well
as $B^{{(fa)}}_{1}$ and $A^{{(fa)}}_{1}$, always appear together:

$\displaystyle V^{*}_{cb}V_{cq}\,B^{{(fa)}}$ $\displaystyle+$ $\displaystyle
V^{*}_{ub}V_{uq}\,A^{{(fa)}}~{},~{}$ $\displaystyle
V^{*}_{cb}V_{cq}\,B^{{(fa)}}_{1}$ $\displaystyle+$ $\displaystyle
V^{*}_{ub}V_{uq}\,A^{{(fa)}}_{1}~{}.$ (12)

##  III Diagrams

In Refs. [13, 14], flavor-flow diagrams were proposed to describe two-body
$B\to PP$ decays. There are eight diagrams: $T$ (tree), $C$ (color-suppressed
tree), $P$ (penguin), $E$ (exchange), $A$ (annihilation), $PA$ (penguin
annihilation), $P_{EW}$ (electroweak penguin [EWP]), and $P^{C}_{EW}$ (color-
suppressed EWP)222Note that EWPs break isospin symmetry due to the difference
in the $Z$-boson’s couplings to up- and down-type quarks. However, in Eq.
(14), we show that the EWPs can be absorbed into six other diagrams using
redefinition rules. Therefore, the inclusion of EWPs does not impact our
analysis of equivalence between diagrams and RMEs..

###  III.1 $B\to PPP$ decays

Diagrams can also be used to describe $B\to PPP$ decays [12, 15]. These
closely follow those used in $B\to PP$ decays. For the three-body analogs of
$T$, $C$, $P$, $P_{EW}$, and $P^{C}_{EW}$, one has to “pop” a quark pair from
the vacuum. The subscript “1” (“2”) is added if the popped quark pair is
between two nonspectator final-state quarks (two final-state quarks including
the spectator). One therefore has $T_{i}$, $C_{i}$, $P_{EWi}$, and
$P^{C}_{EWi}$ diagrams, $i=1,2$. It turns out that $P$-type diagrams only ever
appear in amplitudes in the combination ${\tilde{P}}\equiv P_{1}+P_{2}$. For
the three-body analogs of $E$, $A$, and $PA$, the spectator quark interacts
with the ${\bar{b}}$, and one has two popped quark pairs. Here there is only
one of each type of diagram. Finally, for each of ${\tilde{P}}$ and $PA$, two
contributions are allowed, namely ${\tilde{P}}_{ut}$, ${\tilde{P}}_{ct}$,
$PA_{ut}$ and $PA_{ct}$, where
${\tilde{P}}_{ut}\equiv{\tilde{P}}_{u}-{\tilde{P}}_{t}$, and similarly for the
other diagrams.

All diagrams involve products of CKM matrix elements. We define

$\lambda_{p}^{(q)}\equiv V^{*}_{pb}V_{pq}~{},~{}~{}q=d,s~{},~{}~{}p=u,c,t~{}.$
(13)

The diagrams $T_{i}$, $C_{i}$, ${\tilde{P}}_{ut}$, $E$, $A$ and $PA_{ut}$ all
involve $\lambda_{u}^{(q)}$, ${\tilde{P}}_{ct}$ and $PA_{ct}$ involve
$\lambda_{c}^{(q)}$, and $P_{EWi}$, and $P^{C}_{EWi}$ involve
$\lambda_{t}^{(q)}$. In this section, we use the convention in which the
$\lambda_{p}^{(q)}$ factors are contained completely (magnitude and phase) in
the diagrams333Attention: this convention is not universal for analyses with
diagrams. Indeed, in Sec. VI, we use a different convention – there only the
magnitude of $\lambda_{p}^{(q)}$ is absorbed into the diagrams.

The four EWP diagrams, $P_{EW1,2}$ and $P^{C}_{EW1,2}$, are not really
independent: their addition only has the effect of redefining other diagrams.
The following redefinition rules can be used to absorb the four EWP diagrams
into six other diagrams:

$\displaystyle T_{1}$ $\displaystyle\to$ $\displaystyle T_{1}+P^{C}_{EW1}~{},$
$\displaystyle T_{2}$ $\displaystyle\to$ $\displaystyle T_{2}-P^{C}_{EW2}~{},$
$\displaystyle C_{1}$ $\displaystyle\to$ $\displaystyle C_{1}+P_{EW1}~{},$
$\displaystyle C_{2}$ $\displaystyle\to$ $\displaystyle C_{2}-P_{EW2}~{},$
$\displaystyle({\tilde{P}}_{ut}+{\tilde{P}}_{ct})$ $\displaystyle\to$
$\displaystyle({\tilde{P}}_{ut}+{\tilde{P}}_{ct})+\frac{1}{3}(P^{C}_{EW1}+P^{C}_{EW2})~{}.$
(14)

Note that, before redefinition, $T_{i}$, $C_{i}$, ${\tilde{P}}_{ut}$, and
${\tilde{P}}_{ct}$ each involve only a single product of CKM matrix elements,
$\lambda_{p}^{(q)}$. After redefinition, this is no longer true.

There are therefore in total ten diagrams, namely $T_{1,2}$, $C_{1,2}$,
${\tilde{P}}_{ct}$, ${\tilde{P}}_{ut}$, $E$, $A$, $PA_{ct}$, and $PA_{ut}$.
The decomposition of all 22 amplitudes in terms of the diagrams is given in
Tables 3 and 4.

Table 3: Amplitudes for $\Delta S=1$ $B$-meson decays to fully-antisymmetric $PPP$ states as a function of the three-body diagrams (${\bar{b}}\to{\bar{s}}$ diagrams are written with primes). Decay | $V^{*}_{cb}V_{cs}$ | $V^{*}_{ub}V_{us}$
---|---|---
Amplitude | ${\tilde{P}}^{\prime}_{ct}$ | $PA^{\prime}_{ct}$ | ${\tilde{P}}^{\prime}_{ut}$ | $PA^{\prime}_{ut}$ | $C^{\prime}_{1}$ | $C^{\prime}_{2}$ | $T^{\prime}_{1}$ | $T^{\prime}_{2}$ | $E^{\prime}$ | $A^{\prime}$
$\sqrt{2}{\cal A}(B^{+}\to K^{+}\pi^{+}\pi^{-})$ | $\sqrt{2}$ | 0 | $\sqrt{2}$ | 0 | $-\sqrt{2}$ | 0 | 0 | $\sqrt{2}$ | 0 | $-\sqrt{2}$
${\cal A}(B^{+}\to K^{0}\pi^{+}\pi^{0})$ | $-\sqrt{2}$ | 0 | $-\sqrt{2}$ | 0 | 0 | $\frac{1}{\sqrt{2}}$ | $\frac{1}{\sqrt{2}}$ | 0 | 0 | $\sqrt{2}$
$\sqrt{2}{\cal A}(B^{0}\to K^{0}\pi^{+}\pi^{-})$ | $-\sqrt{2}$ | 0 | $-\sqrt{2}$ | 0 | $-\sqrt{2}$ | 0 | $\sqrt{2}$ | 0 | 0 | 0
${\cal A}(B^{0}\to K^{+}\pi^{0}\pi^{-})$ | $\sqrt{2}$ | 0 | $\sqrt{2}$ | 0 | 0 | $\frac{1}{\sqrt{2}}$ | $-\frac{1}{\sqrt{2}}$ | $\sqrt{2}$ | 0 | 0
$\sqrt{2}{\cal A}(B^{+}\to K^{+}K^{0}\overline{K}^{0})$ | $\sqrt{2}$ | 0 | $\sqrt{2}$ | 0 | 0 | 0 | 0 | 0 | 0 | $-\sqrt{2}$
$\sqrt{2}{\cal A}(B^{0}\to K^{0}K^{+}K^{-})$ | $-\sqrt{2}$ | 0 | $-\sqrt{2}$ | 0 | $-\sqrt{2}$ | 0 | 0 | $-\sqrt{2}$ | 0 | 0
${\cal A}(B_{s}^{0}\to\pi^{0}K^{+}K^{-})$ | $-\frac{1}{\sqrt{2}}$ | $\frac{1}{\sqrt{2}}$ | $-\frac{1}{\sqrt{2}}$ | $\frac{1}{\sqrt{2}}$ | 0 | $-\frac{1}{\sqrt{2}}$ | $\frac{1}{\sqrt{2}}$ | $-\frac{1}{\sqrt{2}}$ | $\sqrt{2}$ | 0
${\cal A}(B_{s}^{0}\to\pi^{0}K^{0}\overline{K}^{0})$ | $-\frac{1}{\sqrt{2}}$ | $\frac{1}{\sqrt{2}}$ | $-\frac{1}{\sqrt{2}}$ | $\frac{1}{\sqrt{2}}$ | 0 | $\frac{1}{\sqrt{2}}$ | 0 | 0 | 0 | 0
$\sqrt{2}{\cal A}(B_{s}^{0}\to\pi^{-}K^{+}\overline{K}^{0})$ | $\sqrt{2}$ | $-\sqrt{2}$ | $\sqrt{2}$ | $-\sqrt{2}$ | 0 | 0 | 0 | $\sqrt{2}$ | $-\sqrt{2}$ | 0
$\sqrt{2}{\cal A}(B_{s}^{0}\to\pi^{+}K^{0}K^{-})$ | $\sqrt{2}$ | $-\sqrt{2}$ | $\sqrt{2}$ | $-\sqrt{2}$ | 0 | 0 | $-\sqrt{2}$ | 0 | $-\sqrt{2}$ | 0
${\cal A}(B_{s}^{0}\to\pi^{0}\pi^{+}\pi^{-})$ | 0 | $\sqrt{2}$ | 0 | $\sqrt{2}$ | 0 | 0 | 0 | 0 | $\frac{3}{\sqrt{2}}$ | 0
Table 4: Amplitudes for $\Delta S=0$ $B$-meson decays to fully-antisymmetric $PPP$ states as a function of the three-body diagrams. Decay | $V^{*}_{cb}V_{cd}$ | $V^{*}_{ub}V_{ud}$
---|---|---
Amplitude | ${\tilde{P}}_{ct}$ | $PA_{ct}$ | ${\tilde{P}}_{ut}$ | $PA_{ut}$ | $C_{1}$ | $C_{2}$ | $T_{1}$ | $T_{2}$ | $E$ | $A$
${\cal A}(B^{+}\to\pi^{+}K^{0}\overline{K}^{0})$ | -1 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1
${\cal A}(B^{+}\to\pi^{+}K^{+}K^{-})$ | 1 | 0 | 1 | 0 | -1 | 0 | 0 | 1 | 0 | -1
$\sqrt{2}{\cal A}(B^{+}\to\pi^{0}K^{+}\overline{K}^{0})$ | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | 0
$\sqrt{2}{\cal A}(B^{0}\to\pi^{0}K^{0}\overline{K}^{0})$ | -2 | 1 | -2 | 1 | 0 | -1 | 0 | 0 | 0 | 0
$\sqrt{2}{\cal A}(B^{0}\to\pi^{0}K^{+}K^{-})$ | 0 | 1 | 0 | 1 | -1 | 0 | 0 | 0 | 2 | 0
${\cal A}(B^{0}\to\pi^{+}K^{0}K^{-})$ | 1 | -1 | 1 | -1 | 0 | 0 | 0 | 1 | -1 | 0
${\cal A}(B^{0}\to\pi^{-}K^{+}\overline{K}^{0})$ | 1 | -1 | 1 | -1 | 0 | 0 | -1 | 0 | -1 | 0
$\sqrt{2}{\cal A}(B^{0}\to\pi^{0}\pi^{+}\pi^{-})$ | -3 | 2 | -3 | 2 | -1 | -1 | 2 | -2 | 3 | 0
${\cal A}(B_{s}^{0}\to\overline{K}^{0}\pi^{+}\pi^{-})$ | -1 | 0 | -1 | 0 | -1 | 0 | 0 | -1 | 0 | 0
$\sqrt{2}{\cal A}(B_{s}^{0}\to K^{-}\pi^{+}\pi^{0})$ | 2 | 0 | 2 | 0 | 0 | 1 | -2 | 1 | 0 | 0
${\cal A}(B_{s}^{0}\to\overline{K}^{0}K^{+}K^{-})$ | -1 | 0 | -1 | 0 | -1 | 0 | 1 | 0 | 0 | 0

###  III.2 Equivalence of RMEs and diagrams

In Sec. II, it was shown that the 11 ${\bar{b}}\to{\bar{d}}$ decay amplitudes
can be expressed in terms of nine RMEs, two of which contain
$V_{cb}^{*}V_{cd}$ while seven others contain $V_{ub}^{*}V_{ud}$. In the
previous subsection, we have seen that the same 11 ${\bar{b}}\to{\bar{d}}$
decay amplitudes can also be expressed in terms of ten diagrams. By comparing
the expressions for the amplitudes of the 11 ${\bar{b}}\to{\bar{d}}$ decays,
it is possible to express the nine RMEs in terms of the ten diagrams. These
expressions are

$\displaystyle V_{cb}^{*}V_{cd}B_{1}^{{(fa)}}$ $\displaystyle=$ $\displaystyle
2\sqrt{6}\left({\tilde{P}}_{ct}-PA_{ct}\right)~{},$ (15) $\displaystyle
V_{cb}^{*}V_{cd}B^{{(fa)}}$ $\displaystyle=$ $\displaystyle\sqrt{5}P_{ct}~{},$
(16) $\displaystyle V_{ub}^{*}V_{ud}A_{1}^{{(fa)}}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{3}}{2\sqrt{2}}\left(8{\tilde{P}}_{ut}-8PA_{ut}-3T_{1}+3T_{2}+C_{1}+C_{2}-8E\right)~{},$
(17) $\displaystyle V_{ub}^{*}V_{ud}A^{{(fa)}}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{5}}{8}\left(8{\tilde{P}}_{ut}-3T_{1}+3T_{2}+C_{1}+C_{2}+E-3A\right)~{},$
(18) $\displaystyle V_{ub}^{*}V_{ud}R_{8}^{{(fa)}}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{15}}{4}\left(T_{1}-T_{2}-C_{1}-C_{2}+E-A\right)~{},$
(19) $\displaystyle V_{ub}^{*}V_{ud}R_{10}^{{(fa)}}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{3}}{2}\left(T_{1}+T_{2}-C_{1}+C_{2}\right)~{},$ (20)
$\displaystyle V_{ub}^{*}V_{ud}P_{8}^{{(fa)}}$ $\displaystyle=$
$\displaystyle\frac{1}{8}\left(T_{1}-T_{2}+C_{1}+C_{2}+5E+5A\right)~{},$ (21)
$\displaystyle V_{ub}^{*}V_{ud}P_{10^{*}}^{{(fa)}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2\sqrt{2}}\left(T_{1}+T_{2}+C_{1}-C_{2}\right)~{},$
(22) $\displaystyle V_{ub}^{*}V_{ud}P_{27}^{{(fa)}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2\sqrt{6}}\left(T_{1}-T_{2}+C_{1}+C_{2}\right)~{}.$
(23)

Note that, in all of these relations, it naively appears that both sides
involve the same products of CKM matrix elements. However, this is not really
true – as noted above, once the EWP contributions have been removed by
redefining the other diagrams, these other diagrams no longer involve a well-
defined product of CKM matrix elements.

By analyzing the 11 ${\bar{b}}\to{\bar{s}}$ decays, one can similarly
establish a corresponding set of expressions relating the diagrams to the RMEs
for ${\bar{b}}\to{\bar{s}}$ decays.

This demonstrates the equivalence of diagrams and SU(3)F for the fully-
antisymmetric $PPP$ state.

##  IV Amplitude Relations

Since all 22 decay amplitudes can be expressed in terms of seven combinations
of RMEs, the amplitudes must obey 15 independent relationships in the SU(3)F
limit. These relationships can be found as follows. The 11
${\bar{b}}\to{\bar{s}}$ decay amplitudes can be expressed in terms of the
seven combinations of RMEs – there must be four relations among these
amplitudes. A subset of these relations can be obtained by considering
processes related by isospin symmetry, while the remaining can be found using
the full SU(3)F symmetry. The process can be repeated for the 11
${\bar{b}}\to{\bar{d}}$ decays generating four additional amplitude relations.
The remaining seven relations follow from the application of U-spin symmetry
that relates ${\bar{b}}\to{\bar{s}}$ decays to ${\bar{b}}\to{\bar{d}}$ decays.

###  IV.1 ${\bar{b}}\to{\bar{s}}$ Decays

The 11 ${\bar{b}}\to{\bar{s}}$ decays (see Table 1) include four $B\to
K\pi\pi$ decays, two $B\to KK{\bar{K}}$ decays, four $B_{s}^{0}\to\pi
K{\bar{K}}$ decays, and one $B_{s}^{0}\to\pi\pi\pi$ decay. Each decay
amplitude can be expressed as a linear combination of seven RMEs. Therefore,
these amplitudes must satisfy four relationships. We find that the four $B\to
K\pi\pi$ decays and the four $B_{s}^{0}\to\pi K{\bar{K}}$ decays each satisfy
one quadrangle relationship, while two additional quadrangle relationships
span multiple types of decays. These relations are:

  1. 1.

$B\to K\pi\pi$:

$\displaystyle\sqrt{2}{\cal A}(B^{+}\to K^{+}\pi^{+}\pi^{-})_{\rm FA}+{\cal
A}(B^{+}\to K^{0}\pi^{+}\pi^{0})_{\rm FA}$ $\displaystyle=$
$\displaystyle\sqrt{2}{\cal A}(B^{0}\to K^{0}\pi^{+}\pi^{-})_{\rm FA}+{\cal
A}(B^{0}\to K^{+}\pi^{0}\pi^{-})_{\rm FA}.$ (24)

  2. 2.

$B_{s}^{0}\to\pi K{\bar{K}}$:

$\displaystyle\sqrt{2}{\cal A}(B_{s}^{0}\to\pi^{0}K^{+}K^{-})_{\rm
FA}+\sqrt{2}{\cal A}(B_{s}^{0}\to\pi^{0}K^{0}\overline{K}^{0})_{\rm FA}$
$\displaystyle=$ $\displaystyle-~{}{\cal
A}(B_{s}^{0}\to\pi^{-}K^{+}\overline{K}^{0})_{\rm FA}-{\cal
A}(B_{s}^{0}\to\pi^{+}K^{0}K^{-})_{\rm FA}.$ (25)

  3. 3.

$B^{0}\to K\pi\pi,B_{s}^{0}\to\pi K{\bar{K}}$, and $B^{0}\to KK{\bar{K}}$ or
$B_{s}^{0}\to\pi\pi\pi$:

$\displaystyle{\cal A}(B^{0}\to K^{0}\pi^{+}\pi^{-})_{\rm FA}-{\cal
A}(B^{0}\to K^{0}K^{+}K^{-})_{\rm FA}$ $\displaystyle=$ $\displaystyle{\cal
A}(B_{s}^{0}\to\pi^{-}K^{+}\overline{K}^{0})_{\rm FA}-{\cal
A}(B_{s}^{0}\to\pi^{+}K^{0}K^{-})_{\rm FA}~{},$ (26)
$\displaystyle\sqrt{2}{\cal A}(B^{0}\to K^{+}\pi^{0}\pi^{-})_{\rm
FA}+\sqrt{2}{\cal A}(B_{s}^{0}\to\pi^{0}K^{+}K^{-})_{\rm FA}$ $\displaystyle=$
$\displaystyle{\cal A}(B_{s}^{0}\to\pi^{-}K^{+}\overline{K}^{0})_{\rm
FA}+\sqrt{2}{\cal A}(B_{s}^{0}\to\pi^{0}\pi^{+}\pi^{-})_{\rm FA}~{}.$ (27)

###  IV.2 ${\bar{b}}\to{\bar{d}}$ Decays

The 11 ${\bar{b}}\to{\bar{d}}$ decays (see Table 2) include seven $B\to\pi
K{\bar{K}}$ decays, one $B^{0}\to\pi\pi\pi$ decay, two $B_{s}^{0}\to K\pi\pi$
decays, and one $B_{s}^{0}\to{\bar{K}}K{\bar{K}}$ decay. Again, each decay
amplitude can be expressed as a linear combination of seven RMEs, so that
there must be four amplitude relationships. We find two quadrangle
relationships among these amplitudes:

$\displaystyle{\cal A}(B^{0}\to\pi^{+}K^{0}K^{-})_{\rm FA}-{\cal
A}(B^{0}\to\pi^{-}K^{+}\overline{K}^{0})_{\rm FA}$ $\displaystyle=$
$\displaystyle{\cal A}(B_{s}^{0}\to\overline{K}^{0}K^{+}K^{-})_{\rm FA}-{\cal
A}(B_{s}^{0}\to\overline{K}^{0}\pi^{+}\pi^{-})_{\rm FA}~{},$ (28)
$\displaystyle\sqrt{2}{\cal A}(B^{0}\to\pi^{0}K^{+}K^{-})_{\rm FA}-{\cal
A}(B^{0}\to\pi^{+}K^{0}K^{-})_{\rm FA}$ $\displaystyle=$
$\displaystyle\sqrt{2}{\cal A}(B^{0}\to\pi^{0}\pi^{+}\pi^{-})_{\rm
FA}+\sqrt{2}{\cal A}(B_{s}^{0}\to K^{-}\pi^{+}\pi^{0})_{\rm FA}~{}.$ (29)

In addition, all seven $B\to\pi K{\bar{K}}$ decays satisfy one amplitude
relationship, while another relationship involves multiple different
amplitudes. As these relationships are not particularly enlightening, we do
not present them here.

###  IV.3 U Spin

The final states in six ${\bar{b}}\to{\bar{s}}$ decays in Table 1 and six
corresponding ${\bar{b}}\to{\bar{d}}$ decays in Table 2 do not involve any
$\pi^{0}$s. Each pair of corresponding ${\bar{b}}\to{\bar{s}}$ and
${\bar{b}}\to{\bar{d}}$ decays is related by U-spin reflection
($d\leftrightarrow s$). The six pairs are:

  1. 1.

$B^{0}\to K^{0}\pi^{+}\pi^{-}$ and $B_{s}^{0}\to\overline{K}^{0}K^{+}K^{-}$,

  2. 2.

$B^{0}\to K^{0}K^{+}K^{-}$ and $B_{s}^{0}\to\overline{K}^{0}\pi^{+}\pi^{-}$,

  3. 3.

$B_{s}^{0}\to\pi^{-}K^{+}\overline{K}^{0}$ and $B^{0}\to\pi^{+}K^{0}K^{-}$,

  4. 4.

$B_{s}^{0}\to\pi^{+}K^{0}K^{-}$ and $B^{0}\to\pi^{-}K^{+}\overline{K}^{0}$,

  5. 5.

$B^{+}\to K^{+}\pi^{+}\pi^{-}$ and $B^{+}\to\pi^{+}K^{+}K^{-}$,

  6. 6.

$B^{+}\to K^{+}K^{0}\overline{K}^{0}$ and
$B^{+}\to\pi^{+}K^{0}\overline{K}^{0}$,

where the first (second) decay is ${\bar{b}}\to{\bar{s}}$
(${\bar{b}}\to{\bar{d}}$). In each pair, amplitude terms multiplying
$V^{*}_{cb}V_{cs}$ and $V^{*}_{ub}V_{us}$ in the ${\bar{b}}\to{\bar{s}}$
process equal amplitude terms multiplying $V^{*}_{cb}V_{cd}$ and
$V^{*}_{ub}V_{ud}$ in the ${\bar{b}}\to{\bar{d}}$ process (up to an overall
negative sign arising from the order of final-state particles [29]; see Tables
1 and 2). Thus, one can write relations among the ${\bar{b}}\to{\bar{s}}$ and
${\bar{b}}\to{\bar{d}}$ decay amplitudes involving CKM matrix elements.

However, there is another relationship between U-spin pairs that is more
useful experimentally [30, 29, 31]. It is

$\frac{A_{s}}{A_{d}}\,\frac{{\cal B}_{s}}{{\cal B}_{d}}=-1~{},$ (30)

where

$\displaystyle{\cal B}_{d}$ $\displaystyle=$ $\displaystyle|{\cal
A}(\bar{b}\to\bar{d})|^{2}+|{\cal A}(b\to d)|^{2}~{},$ $\displaystyle{\cal
B}_{s}$ $\displaystyle=$ $\displaystyle|{\cal
A}(\bar{b}\to\bar{s})|^{2}+|{\cal A}(b\to s)|^{2}~{},$ $\displaystyle A_{d}$
$\displaystyle=$ $\displaystyle\frac{|{\cal A}(\bar{b}\to\bar{d})|^{2}-|{\cal
A}(b\to d)|^{2}}{|{\cal A}(\bar{b}\to\bar{d})|^{2}+|{\cal A}(b\to
d)|^{2}}~{},$ $\displaystyle A_{s}$ $\displaystyle=$
$\displaystyle\frac{|{\cal A}(\bar{b}\to\bar{s})|^{2}-|{\cal A}(b\to
s)|^{2}}{|{\cal A}(\bar{b}\to\bar{s})|^{2}+|{\cal A}(b\to s)|^{2}}~{}.$ (31)

${\cal B}_{d}$ and ${\cal B}_{s}$ are related to the CP-averaged
${\bar{b}}\to{\bar{d}}$ and ${\bar{b}}\to{\bar{s}}$ decay rates, while $A_{d}$
and $A_{s}$ are direct CP asymmetries. The CP-conjugate amplitude ${\bar{\cal
A}}(\bar{b}\to\bar{q})$ is obtained from ${\cal A}(b\to q)$ by changing the
signs of the weak phases. These relations hold for all final symmetry states
for all U-spin reflections.

There are six U-spin relations of this kind. Two additional U-spin amplitude
relations connect several ${\bar{b}}\to{\bar{s}}$ and ${\bar{b}}\to{\bar{d}}$
decays. Since these additional relations are of no particular interest, we do
not present them here. Along with the four ${\bar{b}}\to{\bar{s}}$ and four
${\bar{b}}\to{\bar{d}}$ decay amplitude relations, of which one pair [Eqs.
(26) and (28)] is related by U-spin reflection, this makes a total of 15
independent relations. This is consistent with the fact that 22 decay
amplitudes are all expressed as a function of seven combinations of SU(3)F
matrix elements.

##  V Neglect of $E$/$A$/$PA$

We have seen in the previous sections that FA $B\to PPP$ decays can be written
in terms of SU(3)F RMEs or in terms of diagrams, and that these descriptions
are equivalent. However, the diagrammatic description does provide an
additional useful tool.

When diagrams were introduced to describe $B\to PP$ amplitudes [13, 14], it
was noted that the description in terms of diagrams provides dynamical input.
In particular, the diagrams $E$, $A$, and $PA$ all involve the interaction of
the spectator quark. As such, they are expected to be considerably smaller
than the $T$, $C$, and $P$ diagrams, and can therefore be neglected, to a
first approximation. This reduces the number of unknown parameters and
simplifies the analysis considerably. It must be stressed that this does not
follow from group theory – it is dynamical theoretical input. Even so,
experimental measurements are consistent with this approximation; the
branching ratios of processes that proceed only through $E$/$A$/$PA$ are
indeed considerably smaller than those that are described by $T$/$P$/$C$.

With this in mind, it is likely that the $E$/$A$/$PA$ diagrams can be
neglected in $B\to PPP$ decays. The neglect of these diagrams leads to
relationships among the SU(3)F RMEs:

$\displaystyle B_{1}^{{(fa)}}$ $\displaystyle=$
$\displaystyle\frac{2\sqrt{6}}{\sqrt{5}}B^{{(fa)}}~{},$ $\displaystyle
A_{1}^{{(fa)}}$ $\displaystyle=$
$\displaystyle\frac{2\sqrt{6}}{\sqrt{5}}A^{{(fa)}}~{},$ $\displaystyle
P_{8}^{{(fa)}}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{3}}{2\sqrt{2}}P_{27}^{{(fa)}}~{}.$ (32)

The above relations reduce the number of combinations of RMEs in SU(3)F.
Because $B_{1}^{{(fa)}}$ and $B^{{(fa)}}$ always appear with $A^{{(fa)}}_{1}$
and $A^{{(fa)}}$, respectively [Eq. (12)], the first and second relations only
lead to a reduction of the number of RMEs by 1. An additional reduction by one
RME can be attributed to the third relation. The total number of RMEs upon
neglecting $E$/$A$/$PA$ diagrams is then 5, down from the original 7.

This leads to two additional relations among the ${\bar{b}}\to{\bar{s}}$
amplitudes, and similarly for the ${\bar{b}}\to{\bar{d}}$ amplitudes. For
${\bar{b}}\to{\bar{s}}$, the additional relations are:

  1. 1.

${\cal A}(B_{s}^{0}\to\pi^{0}\pi^{+}\pi^{-})_{\rm FA}=0$, i.e., the decay
$B_{s}^{0}\to\pi^{0}\pi^{+}\pi^{-}$ is pure
$E^{\prime}/A^{\prime}/PA^{\prime}$. This simplifies Eq. (27) into a triangle
relationship.

  2. 2.

${\cal A}(B^{+}\to K^{+}\pi^{+}\pi^{-})_{\rm FA}={\cal A}(B^{0}\to
K^{0}K^{+}K^{-})_{\rm FA}+2{\cal
A}(B_{s}^{0}\to\pi^{-}K^{+}\overline{K}^{0})_{\rm FA}$.

For ${\bar{b}}\to{\bar{d}}$, they are:

  1. 1.

${\cal A}(B^{+}\to\pi^{+}K^{+}K^{-})_{\rm
FA}=cA(B^{0}\to\pi^{+}K^{0}K^{-})_{\rm FA}$
$+\sqrt{2}{\cal A}(B^{0}\to\pi^{0}K^{+}K^{-})_{\rm FA}$,

  2. 2.

${\cal A}(B^{0}\to\pi^{+}K^{0}K^{-})_{\rm FA}+{\cal
A}(B_{s}^{0}\to\overline{K}^{0}\pi^{+}\pi^{-})_{\rm FA}$
$~{}~{}~{}=~{}\sqrt{2}{\cal A}(B^{0}\to\pi^{0}K^{+}K^{-})_{\rm FA}$

##  VI Applications

In Sec. III, we established a one-to-one correspondence between SU(3)F RMEs
and flavor-flow diagrams for the fully-antisymmetric $PPP$ state. By
expressing all 22 ${\bar{b}}\to{\bar{s}}$ and ${\bar{b}}\to{\bar{d}}$ decay
amplitudes in terms of both RMEs and diagrams, we showed that these approaches
are equivalent. In this section, we go beyond the demonstration of this
equivalence and explore predictions that can be tested experimentally.

###  VI.1 Observing decays to the $PPP$ fully-antisymmetric state

To obtain the fully-antisymmetric final state for a given $B\to PPP$ decay,
one proceeds as follows [12]. For the decay $B\to P_{1}P_{2}P_{3}$, one
defines the three Mandelstam variables
$s_{ij}\equiv\left(p_{i}+p_{j}\right)^{2}$, where $p_{i}$ is the momentum of
each $P_{i}$. Only two of these three are independent. Say the $B\to
P_{1}P_{2}P_{3}$ Dalitz plot is given in terms of $s_{12}$ and $s_{13}$. One
can obtain the decay amplitude ${\cal M}(s_{12},s_{13})$ describing this
Dalitz plot by performing an isobar analysis. Here the amplitude is expressed
as the sum of a non-resonant and several intermediate resonant contributions:

${\cal M}(s_{12},s_{13})={\cal N}_{\rm
DP}\sum\limits_{j}c_{j}e^{i\theta_{j}}F_{j}(s_{12},s_{13})~{},$ (33)

where the index $j$ runs over all contributions. Each contribution is
expressed in terms of isobar coefficients $c_{j}$ (magnitude) and $\theta_{j}$
(phase), and a dynamical wave function $F_{j}$. ${\cal N}_{\rm DP}$ is a
normalization constant. The $F_{j}$ take different forms depending on the
contribution. The $c_{j}$ and $\theta_{j}$ are extracted from a fit to the
Dalitz-plot event distribution. With ${\cal M}(s_{12},s_{13})$ in hand, one
can construct the fully-antisymmetric amplitude. It is given simply by

$\displaystyle{\cal M}_{\rm FA}(s_{12},s_{13})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{6}}\left[{\cal M}(s_{12},s_{13})-{\cal
M}(s_{13},s_{12})-{\cal M}(s_{12},s_{23})\right.$ (34) $\displaystyle\hskip
56.9055pt\left.+~{}{\cal M}(s_{23},s_{12})-{\cal M}(s_{23},s_{13})+{\cal
M}(s_{13},s_{23})\right]~{},$

where one uses the relationship
$s_{12}+s_{13}+s_{23}=m_{B}^{2}+m_{P_{1}}^{2}+m_{P_{2}}^{2}+m_{P_{3}}^{2}$ to
express the third Mandelstam variable in terms of the first two. For any
three-body decay for which a Dalitz plot has been measured, one can extract
the fully-antisymmetric amplitude in the above fashion. The Dalitz plane can
be divided into six regions by three lines of symmetry; along each line of
symmetry, there is a pair of Mandelstam variables that are equal. It is
sufficient to construct ${\cal M}_{\rm FA}$ in only one of these six regions
as the other five regions do not contain additional information due to the
fully-antisymmetric nature of ${\cal M}_{\rm FA}(s_{12},s_{13})$. In a similar
vein, one can construct the fully-antisymmetric amplitude for the CP-conjugate
process, $\overline{{\cal M}}_{\rm FA}$ from its measured Dalitz plot.

The fully-antisymmetric amplitudes for the process and its CP conjugate are
not directly observable as these contain unknown phases. However, one can
construct the following three linearly independent observables using these
amplitudes:

$\displaystyle{\cal X}_{\rm FA}(s_{12},s_{13})$ $\displaystyle=$
$\displaystyle|{\cal M}_{\rm FA}(s_{12},s_{13})|^{2}+|\overline{\cal M}_{\rm
FA}(s_{12},s_{13})|^{2},~{}$ $\displaystyle{\cal Y}_{\rm FA}(s_{12},s_{13})$
$\displaystyle=$ $\displaystyle|{\cal M}_{\rm
FA}(s_{12},s_{13})|^{2}-|\overline{\cal M}_{\rm FA}(s_{12},s_{13})|^{2},~{}$
$\displaystyle{\cal Z}_{\rm FA}(s_{12},s_{13})$ $\displaystyle=$
$\displaystyle{\rm Im}\left[{\cal M}_{\rm FA}^{*}(s_{12},s_{13})\overline{\cal
M}_{\rm FA}(s_{12},s_{13})\right].~{}$ (35)

For any given decay, the observables ${\cal X}_{\rm FA},{\cal Y}_{\rm FA},$
and ${\cal Z}_{\rm FA}$ depend on the position in the Dalitz plot and are
related to the CP-averaged decay rate, the direct CP asymmetry, and the
indirect CP asymmetry. While ${\cal X}$ and ${\cal Y}$ exist for any three-
body decay with three distinct particles in the final state, ${\cal Z}$ is a
meaningful physical observable only for decays in which the final state is
flavor neutral, such as $K^{0}K^{+}K^{-}$ or $\pi^{0}K^{0}\overline{K}^{0}$.

###  VI.2 Confronting data

As we saw in the previous section, when the $E$/$A$/$PA$ diagrams are
neglected, the amplitudes can be written as functions of five combinations of
RMEs. Four are proportional to $V^{*}_{ub}V_{uq}$, and the fifth is a linear
combination of pieces proportional to $V^{*}_{ub}V_{uq}$ and
$V^{*}_{cb}V_{cq}$. However, when it comes to using this parametrization to
describe actual data, this counting must be reexamined. This is because some
observables measure CP violation, which is sensitive to the weak phases of the
CKM matrix. The RMEs proportional to $V^{*}_{ub}V_{uq}$ and $V^{*}_{cb}V_{cq}$
do not contribute equally to these observables. So, the number of RMEs that
can be probed by the data is actually six, five proportional to
$V^{*}_{ub}V_{uq}$ and one proportional to $V^{*}_{cb}V_{cq}$.

Turning to diagrams, the first thing is that we cannot redefine diagrams to
absorb the EWPs, since that mixes pieces involving different CKM factors.
Instead, we do the counting as follows. When $E$/$A$/$PA$ are neglected, there
are ten diagrams. $T_{i}$, $C_{i}$, and ${\tilde{P}}_{ut}$ involve
$\lambda_{u}^{(q)}$, ${\tilde{P}}_{ct}$ involves $\lambda_{c}^{(q)}$, and the
four EWP diagrams involve $\lambda_{t}^{(q)}$. Here, it is important to use a
different convention for the diagrams than that used in Sec. III.1. Here, the
diagrams contain only the magnitudes of the $\lambda_{p}^{(q)}$; the phase
information, including minus signs, is explicitly written as a factor
multiplying the diagrams. The key point now is that, just as was the case in
$B\to PP$ decays [32, 33, 34], the EWP diagrams are related to the tree
diagrams. Taking the ratios of Wilson coefficients $c_{1}/c_{2}=c_{9}/c_{10}$,
which holds to about 5%, the simplified form of these relations is

$P_{EWi}=\kappa T_{i}~{}~{},~{}~{}~{}~{}P^{C}_{EWi}=\kappa C_{i}~{},$ (36)

where

$\kappa\equiv-\frac{3}{2}\frac{|\lambda_{t}^{(q)}|}{|\lambda_{u}^{(q)}|}\frac{c_{9}+c_{10}}{c_{1}+c_{2}}~{}.$
(37)

These are the same EWP-tree relations as hold for the FS state; see Ref. [9].
With this, there are six independent diagrams, of which two –
${\tilde{P}}_{ct}$ and ${\tilde{P}}_{ut}$ – always appear together as a linear
combination.

###  VI.3 Extracting $\gamma$

####  VI.3.1 $B\to K\pi\pi$ and $B\to KK{\bar{K}}$ decays

The method proposed in Ref. [10] and carried out in Refs. [16, 22] uses the FS
states of three $B\to K\pi\pi$ and two $B\to KK{\bar{K}}$ decays. They are
$B^{0}\to K^{+}\pi^{0}\pi^{-}$, $B^{0}\to K^{0}\pi^{+}\pi^{-}$, $B^{+}\to
K^{+}\pi^{+}\pi^{-}$, $B^{0}\to K^{0}K^{+}K^{-}$, and $B^{0}\to
K^{0}K^{0}{\bar{K}}^{0}$ (with both $K^{0}$ and ${\bar{K}}^{0}$ identified as
$K_{S}$). These are chosen because the amplitudes can be expressed as
functions of only five combinations of diagrams (and not six).

However, this method cannot be applied to the FA states, since there is no
such state for $B^{0}\to K^{0}K^{0}{\bar{K}}^{0}$. Of the six $B\to K\pi\pi$
and $B\to KK{\bar{K}}$ decays listed in Table 1, two are not used in the above
method: $B^{+}\to K^{0}\pi^{+}\pi^{0}$ and $B^{+}\to K^{+}K^{0}{\bar{K}}^{0}$.
While the first decay clearly has an FA state, the second decay has one only
if the $K^{0}{\bar{K}}^{0}$ in the final state is detected as $K_{S}K_{L}$.
While this may be possible experimentally, it is not easy, so we will not
include this decay.

In this case, the amplitudes for the five decays $B^{0}\to
K^{+}\pi^{0}\pi^{-}$, $B^{0}\to K^{0}\pi^{+}\pi^{-}$, $B^{+}\to
K^{+}\pi^{+}\pi^{-}$, $B^{0}\to K^{0}K^{+}K^{-}$, and $B^{+}\to
K^{0}\pi^{+}\pi^{0}$ are functions of six diagrams, so there are 12 unknown
theoretical parameters: six magnitudes of diagrams, five relative strong
phases, and $\gamma$. And there are a total of 12 observables: the CP-averaged
decay rates (${\cal X}_{\rm FA}$) and direct CP asymmetries (${\cal Y}_{\rm
FA}$) for the five decays, and the indirect CP asymmetries (${\cal Z}_{\rm
FA}$) of $B^{0}\to K^{0}\pi^{+}\pi^{-}$ and $B^{0}\to K^{0}K^{+}K^{-}$. With
an equal number of observables and unknown theoretical parameters, $\gamma$
can be extracted from a fit, albeit with discrete ambiguities.

Now, it is expected that
$|{\tilde{P}}_{uc}|\simeq\lambda^{2}|{\tilde{P}}_{tc}|$, where
$\lambda\equiv\sin\theta_{C}\simeq 0.22$, so it is not a bad approximation to
neglect ${\tilde{P}}^{\prime}_{uc}$. If one does this, there are now ten
unknown theoretical parameters, which will reduce the discrete ambiguity in
the extraction of $\gamma$. (In this case, it is possible to add a theoretical
parameter parametrizing the breaking of SU(3)F; see discussion below.)

####  VI.3.2 General analysis

To date, methods to extract $\gamma$ from $B\to PPP$ decays have focused
mainly on $\Delta S=1$ $B\to K\pi\pi$ and $B\to KK{\bar{K}}$ decays. However,
there are many more decays, including $\Delta S=0$ processes and/or $\Delta
S=1$ $B_{s}^{0}$ decays. Looking at Tables 1 and 2, and eliminating those that
(i) contain $K^{0}{\bar{K}}^{0}$ in the final state and (ii) vanish when
$E$/$A$/$PA$ are neglected, we see there are a total of 17 $B\to PPP$ decays
that have an FA final state. All of these are functions of the same six
diagrams, so there are in total 12 unknown theoretical parameters. If/when the
Dalitz plots of these decays are measured, we have the potential to perform a
fit to the data with many more observables than unknown parameters. (Of
course, this also holds for the FS final state.) We will probably be able to
extract $\gamma$ with no discrete ambiguity.

###  VI.4 SU(3)F breaking

In this entire discussion, it has been assumed that SU(3)F is a good symmetry.
However, we know that SU(3)F is in fact broken, and these breaking effects
will inevitably affect the extraction of $\gamma$. In some cases, it is
possible to include new theoretical parameters in the fit that measure the
size of SU(3)F breaking. Now, the fits are performed at a specific point in
the Dalitz plot. But there is evidence that, when one averages over the entire
Dalitz plot, the size of SU(3)F breaking is significantly reduced.

As described above, the amplitudes of the FS states of the three $B\to
K\pi\pi$ and two $B\to KK{\bar{K}}$ decays used in the analysis of Ref. [22]
are functions of five effective diagrams. As such there are ten unknown
parameters. But there are 12 observables. In light of this, the $B\to
KK{\bar{K}}$ amplitudes were multiplied by an additional SU(3)F-breaking
parameter $\alpha_{SU(3)}$. It represented the fact that, for these decays,
one must pop an $s{\bar{s}}$ pair from the vacuum, while in $B\to K\pi\pi$
decays, a $u{\bar{u}}$ or $d{\bar{d}}$ pair is popped. In Ref. [22], it was
found that, while the value of the magnitude of $\alpha_{SU(3)}$ could be
sizeable at a given point in the Dalitz plot, it could also have either sign.
When averaged over the entire Dalitz plot, it was found that the effect of
SU(3)F breaking was only at the percent level.

A similar technique can be used for the FA $B\to PPP$ states. The number and
type of SU(3)F-breaking parameters that are added to the amplitudes depend on
how many more observables there are than unknown theoretical parameters. But
in principle, it should be possible to add such parameters and see if, as was
the case above, the size of SU(3)F breaking is actually reduced when averaged
over the entire Dalitz plot.

Another technique for testing U-spin breaking by averaging over the Dalitz
plot was discussed in Ref. [15] for the fully-symmetric final state. This
technique is to apply Eq. (30) to two decays that are U-spin reflections of
each other, in the presence of U-spin breaking. In terms of the Dalitz plot
observables of Eq. (35), Eq. (30) can be rewritten as

$\displaystyle-\frac{{\cal Y}_{\rm FA}({\bar{b}}\to{\bar{s}})}{{\cal Y}_{\rm
FA}({\bar{b}}\to{\bar{d}})}$ $\displaystyle=$ $\displaystyle Y_{\rm FA},~{}$
(38)

where $Y_{\rm FA}$ is a real number that captures the amount of U-spin
breaking. Under perfect U-spin symmetry $Y_{\rm FA}=1$, however, its measured
value may be $Y_{\rm FA}>1$ or $Y_{\rm FA}<1$ depending on the Dalitz plot
point. Averaging over the Dalitz plot one can then test the amount of U-spin
breaking in these decays. This technique can be applied to test U-spin
breaking in the six U-spin-related pairs of decays listed in Sec. IV.3.

##  VII Conclusions

Recently, the CP-phase $\gamma$ was extracted from observables associated with
the Dalitz plots of $B^{0}\to K^{+}\pi^{0}\pi^{-}$, $B^{0}\to
K^{0}\pi^{+}\pi^{-}$, $B^{+}\to K^{+}\pi^{+}\pi^{-}$, $B^{0}\to
K^{+}K^{0}K^{-}$, and $B^{0}\to K^{0}K^{0}\overline{K}^{0}$ [22]. These decays
all receive significant loop-level gluonic and/or electroweak penguin
contributions, and so could be affected by NP. The presence of this NP would
be revealed by a difference between the (loop-level) value of $\gamma$ found
here and the value found using a standard method involving only tree-level
decays [2, 3, 4, 5].

In three-body charmless $B\to PPP$ decays, there are six possibilities for the
final state: a fully symmetric state, a fully antisymmetric state, or one of
four mixed states. The analysis of Ref. [22] used the FS state and found six
possible values for $\gamma$. One value agrees with that measured
independently using tree-level decays, while the other five are in
disagreement and hint at the presence of NP. In order to determine which of
these is the true value of $\gamma$ in this system, one must extract $\gamma$
from a second set of $B\to PPP$ decays, this time using a different symmetry
of the final state. There may again be multiple solutions, but the true value
of $\gamma$ will be common to both analyses.

In this paper, we present the formalism describing charmless $B\to PPP$ decay
amplitudes in which the final-state particles are all $\pi$’s or $K$’s, and
the final state is fully antisymmetric. This can be used to perform analyses
for extracting $\gamma$. In FA states, there are no identical particles in the
final state; there are 11 ${\bar{b}}\to{\bar{s}}$ and 11
${\bar{b}}\to{\bar{d}}$ $B\to PPP$ decays of this type. (But note that four
decays have $K^{0}{\bar{K}}^{0}$ in the final state. These have an FA state
only if this pair can be detected as $K_{S}K_{L}$.) We write all 22 amplitudes
in terms of seven combinations of nine SU(3)F reduced matrix elements. We also
present the 15 relations among the amplitudes, some of which can be tested
experimentally.

The amplitudes can also be written in terms of eight combinations of ten
diagrams. By comparing the expressions for the amplitudes in terms of RMEs and
diagrams, we are able to write the RMEs as functions of diagrams. This
demonstrates the equivalence of diagrams and SU(3)F. Diagrams also provide
dynamical input: the three diagrams $E$, $A$, and $PA$ all involve the
interaction of the spectator quark and are expected to be considerably smaller
than the other diagrams. If $E$/$A$/$PA$ are neglected, we find two additional
relations among each of the ${\bar{b}}\to{\bar{s}}$ and
${\bar{b}}\to{\bar{d}}$ amplitudes.

We show how the FA amplitudes can be measured through an isobar analysis of
the Dalitz plots. The analysis of Ref. [22] cannot be applied to FA states
(since $B^{0}\to K^{0}K^{0}\overline{K}^{0}$ has no FA state), so we describe
other sets of $B\to PPP$ decays that can be used to extract $\gamma$ using the
FA amplitudes. Finally, we discuss how SU(3)F breaking is reduced when it is
averaged over the entire Dalitz plot.

Acknowledgments: This work was financially supported by the National Science
Foundation under Grant No. PHY-2013984 (BB, AH) and by NSERC of Canada (AJ,
MFN, DL).

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