# Study of the $B^{+}\to\pi^{+}\left(\pi^{+}\pi^{-}\right)$ decay in PQCD
Approach

Qin Chang1,3<EMAIL_ADDRESS>Lei Yang1<EMAIL_ADDRESS>Zhi-
Tian Zou2<EMAIL_ADDRESS>Ying Li2<EMAIL_ADDRESS>1\. Institute of
Particle and Nuclear Physics,Henan Normal University, Xinxiang 453007, China
2\. Department of Physics, Yantai University, Yantai 264005, China
3\. Institute of Physics, Henan Academy of Sciences, Zhengzhou 455004, China

###### Abstract

Based on the fitting results of the LHCb collaboration on the contributions of
various intermediate resonances to the $B^{+}\to\pi^{+}\pi^{+}\pi^{-}$ decay,
we make systematically calculate the branching fractions and localized $CP$
asymmetries of the quasi-two-body
$B^{+}\to\pi^{+}\left(\rho(770),\omega(782),\rho(1450),f_{2}\left(1270\right)\to\right)\pi^{+}\pi^{-}$
decays within the perturbative QCD (PQCD) approach. Our theoretical
predictions for the branching fractions agree well with the data within the
errors. In order to further test the framework of the three-body $B$ decays in
PQCD and the wave functions of $\pi\pi$ pair, we also calculate the branching
fractions of the corresponding two-body $B^{+}\to\pi^{+}\rho(770)$,
$B^{+}\to\pi^{+}\omega(782)$ and $B^{+}\to\pi^{+}f_{2}\left(1270\right)$
decays under the narrow-width-approximation, which are in consistence with the
experimental data. The direct $CP$ asymmetries of
$B^{+}\to\pi^{+}(\rho(770/1450)\to)\pi^{+}\pi^{-}$ decays are found to be very
small because these decay modes are tree-dominated. However, due to the large
penguin contributions from the chiral enhanced annihilation diagrams,
$B^{+}\to\pi^{+}(f_{2}\left(1270\right)\to)\pi^{+}\pi^{-}$ decay has a large
direct $CP$ asymmetry, which is also discovered by LHCb collaboration. A
relatively large $CP$ asymmetry is also expected in the
$B^{+}\to\pi^{+}(\omega\to)\pi^{+}\pi^{-}$ decay occurring via $\rho-\omega$
mixing, which would be measured LHCb and Belle-II experiments in future.

###### pacs:

13.25.Hw, 12.38.Bx

##  I Introduction

The three-body $B$ meson decays sharing a large fraction of the branching
ratio play important roles in deeply testing the standard model (SM) and
understanding the mechanism of the $CP$ violation especially the sources of
the strong phases. Experimentally, a lot of three-body $B$ meson decays have
been observed by the BaBar BaBar:2003zav ; BaBar:2005qms ; BaBar:2006hyf ;
BaBar:2007aut ; BaBar:2007hmp ; BaBar:2009jov ; BaBar:2009vfr , Belle
Belle:2002ryx ; Belle:2003ibz ; Belle:2004drb ; Belle:2005rpz ; Belle:2006ljg
; Belle:2008til ; Belle:2010wis , and LHCb LHCb:2013lcl ; LHCb:2014mir ;
LHCb:2016vqn ; LHCb:2017hbp ; LHCb:2018oeb ; LHCb:2019sus ; LHCb:2019vww
collaborations. With the steady accumulation of experimental data, the
theoretical researches for the three-body $B$ meson decays have been studied
extensively in the approach based on the symmetry principles Gronau:2005ax ;
Engelhard:2005hu ; He:2014xha , for instance, the QCD factorization (QCDF) El-
Bennich:2009gqk ; Cheng:2002qu ; Cheng:2007si ; Cheng:2013dua ; Li:2014oca ;
Chen:2023pms , the perturbative QCD approach (PQCD) Yan:2023yvx ; Li:2018psm ;
Zou:2020ool ; Chen:2002th ; Li:2018lbd ; Li:2016tpn ; Zou:2022xrr ;
Liu:2021sdw ; Yao:2022zom ; Yang:2021zcx ; Zou:2020mul ; Zou:2020fax ;
Li:2021cnd , and other theoretical methods Wang:2015ula ; El-Bennich:2006rcn ;
Chang:2019obq ; Guo:2000uc ; Cheng:2016shb . In comparison with the two-body
$B$ meson decays, the dynamic of the three-body $B$ meson decay is much more
complicated because the momentum of each final particle is a variable. In
addition, the three-body decays include both resonant and non-resonant
contributions, and how to exactly distinguish and evaluate these contributions
is important for studying the three-body $B$ meson decays.

In analyzing three-body $B$ meson decays, the Dalitz plot is a widely used and
effective method, which can be divided into different regions in term of the
characteristic kinematics. The central region in the Dalitz plot corresponds
to the nonresonant contributions, which are $\alpha_{s}$ suppressed relative
to the contributions of edges Virto:2016fbw . The edges of the Dalitz plot
correspond to the kinematical configuration where two of final particles fly
collinearly, generating an invariant mass recoiling against the third bachelor
meson. In this region, the two collinear mesons move almost in the same
direction and the bachelor recoils back. The two collinear mesons can be
viewed as a cluster, and might form resonances with different angular momenta.
Theoretically, we can refer to the three-body B meson decays in this region as
the quasi-two-body decays beyond the narrow-width approximation Virto:2016fbw
, where the interaction in the two-meson pair can be described by the
nonperturbative two-meson wavefunction Hua:2020usv ; Li:2006cva ; Yan:2023yvx
; Li:2018psm ; Zou:2020ool . Therefore, we can evaluate these quasi-two-body
decays theoretically within the QCD-inspired approaches such as PQCD and QCDF.
Experimentally, the resonance contributions are analyzed by using some
methods, such as the isobar model Herndon:1973yn ; Sternheimer:1961zz , the
K-matrix formalism Chung:1995dx  and the quasi-model independent
analysisLHCb:2019sus , among which the isobar model method is widely adopted
by BaBar, Belle and LHCb collaborations.

In the so-called isobar model, the decay amplitude can be modeled as a
coherent sum of the amplitudes of “$N$” individual decay channels
corresponding to different resonances with same spin. The decay amplitude can
be written as

$\displaystyle\mathcal{A}=\sum_{i}^{N}a_{i}\mathcal{A}_{i}\,,$ (1)

where $\mathcal{A}_{i}$ is the decay amplitude of the channel associated with
a certain resonance and $a_{i}$ is the corresponding effective coefficient
reflecting the relative magnitude and phase. From eq. (1), it can be easily
found that not only does the individual $\mathcal{A}_{i}$ provide the possible
source of $CP$ asymmetry, the interference between them also lead to $CP$
asymmetry. The effective coefficient $a_{i}$ in eq. (1) can only be determined
from the experimental data now, while the amplitude, $\mathcal{A}_{i}$, can be
calculated perturbatively by the QCD-inspired approaches such as PQCD, which
will be detailed in the next section.

In Ref. LHCb:2019sus , based on the data sample with an integrated luminosity
of 3 $fb^{-1}$ of $pp$ collisions, LHCb collaboration reported the results of
an amplitude analysis of the three-body decay $B^{+}\to\pi^{+}\pi^{+}\pi^{-}$,
where the $CP$ asymmetry effects are also taken into account. The nonresonant
component is described by the behavior of the $S$-wave contribution, The
resonant contributions are described by using the isobar model, and include
$\rho(770)$, $\omega(782)$ and $\rho(1450)$ resonances in the $\pi\pi$
$P$-wave, the $f_{2}(1270)$ resonance in the $\pi\pi$ $D$-wave and the
$\rho_{3}(1690)$ resonance in the $\pi\pi$ $F$-wave. In the
$B^{+}\to\pi^{+}(f_{2}(1270)\to)\pi^{+}\pi^{-}$ process, large $CP$ asymmetry
was measured. Further phenomenological and experimental investigations are
required to shed light on the underlying mechanism of the three-body decays
and searching for new $CP$ asymmetry effects in the three-body decays. In Ref.
Li:2016tpn , the contribution of $P$-wave state $\rho(770)$ to
$B^{+}\to\pi^{+}\pi^{+}\pi^{-}$ decay has been studied in PQCD approach, while
the possible effect of $\rho-\omega$ mixing has not been considered. The
contribution of $D$-wave $f_{2}(1270)$ resonance state was also studied in
Ref. Li:2018lbd . For completeness, we will investigate above-mentioned
$P$-wave and $D$-wave resonant contributions to the
$B^{+}\to\pi^{+}\pi^{+}\pi^{-}$ decay simultaneously in this work and the
interference effect between them will be studied. In additional, the resonant
contribution of $\omega(782)$ will be induced due to the $\rho-\omega$ mixing.

This paper is organized as follows. In section II, the theoretical framework
of PQCD approach for the quasi-two-body decay is reviewed, and the $P$ wave
and $D$ wave two-pion wavefunctions are discussed briefly. In section III, the
input parameters are collected, our numerical results and some discussions are
given in detail. Finally, we conclude with a summary in section IV. The
explicit expressions of decay amplitudes within the PQCD approach are
collected in appendix A.

##  II Theoretical Framework

The PQCD approach is based on $k_{T}$ factorization, where the small intrinsic
transverse momenta of quarks are kept in the calculation. As a result, the
end-point singularity is avoided, while the double logarithm will appear in
the QCD radiative corrections caused by the additional scale introduced by the
transverse momentum. Fortunately, these double logarithm can be resumed within
the renormalization group equation, leading to the so-called Sudakov form
factor that suppresses the long distance contributions effectively and make
the perturbative calculation reliable. Moreover, within this framework the
annihilation diagrams can be calculated perturbatively, and provide the
dominant source of the strong phase, which is required by the direct $CP$
asymmetry in SM Hong:2005wj .

For the $B$ meson decay involving multi scales, the physics can be decomposed
in terms of the characteristic scales. It is known that there are three
characteristic energy scales in the $B$ meson decays including the electroweak
scale $m_{W}$, the scale of the B-decays $m_{b}$ and the factorization scale
$\sqrt{\overline{\Lambda}m_{B}}$ with $\overline{\Lambda}=m_{B}-m_{b}$. The
Wilson coefficients summarize the short-distance physical contributions, and
are in principle computable with the perturbative theory order by order at the
scale of $\mu\sim m_{W}$. Renormalization group equation enable us to evaluate
the dynamical effects in scaling the Wilson coefficients from $m_{W}$ to
$m_{b}$ scale. The physics between the $m_{b}$ and the factorization scale is
involved in the hard kernel $\mathcal{H}$, which is dominated by the hard
gluon exchanging process, and can be calculated perturbatively in PQCD
approach. The physics below the factorization scale is soft and
nonperturbative, and can be described by the wave function. Therefore, in the
framework of PQCD, the amplitude of the quasi-two-body $B$ meson decays
$B\to\pi\left(R\to\right)\pi\pi$ can be decomposed into the convolution
asChen:2002th

$\displaystyle\mathcal{A}=C\left(t\right)\otimes\mathcal{H}\left(x_{i},b_{i},t\right)\otimes\Phi_{B}\left(x_{1},b_{1}\right)\otimes\Phi_{\pi}\left(x_{2},b_{2}\right)\otimes\Phi_{\pi\pi}\left(x_{3},b_{3}\right)\otimes\exp[-S\left(t\right)],$
(2)

where $x_{i}$ are the momentum fractions of quarks, $b_{i}$ are the conjugate
variables of the transverse momentum of quarks, and $t$ is the largest scale
that appears in the hard kernel $\mathcal{H}$. $\Phi_{B}$ and $\Phi_{\pi}$ are
the wave functions of the $B$ and $\pi$ mesons, respectively, and
$\Phi_{\pi\pi}$ is the two-pion wave function, which involves the resonant and
the non-resonant interactions that between the two collinear $\pi$ mesons. The
exponential term is the Sudakov form factor, which can suppress the soft
dynamic effects Li:2012md ; Li:2012nk ; Li:2001ay ; Li:1997un .

Figure 1: The leading order Feynman diagrams contributing to the
$B^{+}\to\pi^{+}\left(R\to\right)\pi^{+}\pi^{-}$
($R=\rho(770),\omega(782),\rho(1450),f_{2}(1270)$) decays in PQCD approach,
where the black squares stand for the weak vertices, the purple spots on the
quark lines denote possible attachments of hard gluons, the green ellipses
represent $\pi^{+}\pi^{-}$-pair and the red ones are the light bachelor
$\pi^{+}$ mesons.

The effective weak Hamiltonian responsible for the $b\to q\bar{q}d$ transition
is written as

$\displaystyle\mathcal{H}_{eff}=\frac{G_{F}}{\sqrt{2}}\sum_{p=u,c}V_{pb}V^{\ast}_{pd}\left[C_{1}\left(\mu\right)O_{1}^{p}\left(\mu\right)+C_{2}\left(\mu\right)O_{2}^{p}\left(\mu\right)+\sum_{i=3}^{10}C_{i}\left(\mu\right)O_{i}\left(\mu\right)\right]$
(3)

with the current-current operators:

$\displaystyle
O_{1}\left(\mu\right)=\left(\overline{p}_{\alpha}b_{\beta}\right)_{V-A}\left(\overline{d}_{\beta}p_{\alpha}\right)_{V-A},\
\ \
O_{2}\left(\mu\right)=\left(\overline{p}_{\alpha}b_{\alpha}\right)_{V-A}\left(\overline{d}_{\beta}p_{\beta}\right)_{V-A},$

the QCD penguin operators:

$\displaystyle O_{3}\left(\mu\right)$ $\displaystyle=$
$\displaystyle\left(\overline{d}_{\alpha}b_{\alpha}\right)_{V-A}\sum_{q}\left(\overline{q}_{\beta}q_{\beta}\right)_{V-A},\
\ \
O_{4}\left(\mu\right)=\left(\overline{d}_{\beta}b_{\alpha}\right)_{V-A}\sum_{q}\left(\overline{q}_{\alpha}q_{\beta}\right)_{V-A},$
$\displaystyle O_{5}\left(\mu\right)$ $\displaystyle=$
$\displaystyle\left(\overline{d}_{\alpha}b_{\alpha}\right)_{V-A}\sum_{q}\left(\overline{q}_{\beta}q_{\beta}\right)_{V+A},\
\ \
O_{6}\left(\mu\right)=\left(\overline{d}_{\beta}b_{\alpha}\right)_{V-A}\sum_{q}\left(\overline{q}_{\alpha}q_{\beta}\right)_{V+A},$

the electroweak penguin operators:

$\displaystyle O_{7}\left(\mu\right)$ $\displaystyle=$
$\displaystyle\frac{3}{2}\left(\overline{d}_{\alpha}b_{\alpha}\right)_{V-A}\sum_{q}e_{q}\left(\overline{q}_{\beta}q_{\beta}\right)_{V+A},\
\ \ \
O_{8}\left(\mu\right)=\frac{3}{2}\left(\overline{d}_{\beta}b_{\alpha}\right)_{V-A}\sum_{q}e_{q}\left(\overline{q}_{\alpha}q_{\beta}\right)_{V+A},$
$\displaystyle O_{9}\left(\mu\right)$ $\displaystyle=$
$\displaystyle\frac{3}{2}\left(\overline{d}_{\alpha}b_{\alpha}\right)_{V-A}\sum_{q}e_{q}\left(\overline{q}_{\beta}q_{\beta}\right)_{V-A},\
\ \
O_{10}\left(\mu\right)=\frac{3}{2}\left(\overline{d}_{\beta}b_{\alpha}\right)_{V-A}\sum_{q}e_{q}\left(\overline{q}_{\alpha}q_{\beta}\right)_{V-A},$

where $\left(\overline{q}_{1}q_{2}\right)_{V\pm
A}=\overline{q}_{1}\gamma^{\mu}(1\pm\gamma_{5})q_{2}$, $\alpha$ and $\beta$
are the color indices, $V_{pb}V^{\ast}_{pd}$ are the products of the CKM
element, $C_{i}(\mu)$ are the Wilson coefficients, and a summation over
$q=u,d,s,c,b$ is implied. With the effective weak Hamiltonian, the leading
order Feynman diagrams for the quasi-two-body decays
$B^{+}\to\pi^{+}\left(R\to\right)\pi^{+}\pi^{-}$
($R=\rho(770),\omega(782),\rho(1450),f_{2}(1270)$) are illustrated in Fig. 1.

We consider the $B$ meson at rest for simplicity. In the light-cone
coordinate, the momenta of $B$ meson and light quark in it can be written as

$\displaystyle p_{B}=\frac{M_{B}}{\sqrt{2}}\left(1,1,0_{T}\right),\ \
k_{1}=\left(0,\frac{M_{B}}{\sqrt{2}}x_{1},k_{1T}\right),$ (4)

with $x_{1}$ being the momentum fraction of light quark. The momenta of the
$\pi\pi$ pair moving along $z$ direction and the recoiled bachelor $\pi$ meson
are respectively written as

$\displaystyle p_{2}=\dfrac{M_{B}}{\sqrt{2}}\left(1,\eta,0_{T}\right),\quad
p_{3}=\dfrac{M_{B}}{\sqrt{2}}\left(0,1-\eta,0_{T}\right),$ (5)

where $\eta=s/M_{B}^{2}$, and $\sqrt{s}$ is the invariant mass of the $\pi\pi$
pair satisfying $s=p^{2}$. The momenta of quarks in the $\pi\pi$ pair
(resonances) and the recoiled $\pi$ meson can be written as

$\displaystyle k=\left(\dfrac{M_{B}}{\sqrt{2}}z,0,0_{T}\right),\quad
k_{3}=\left(0,\left(1-\eta\right)x_{3},0_{T}\right),$ (6)

where $z$ and $x_{3}$ are the momentum fractions of quark in the $\pi\pi$ pair
and the recoiled $\pi$ meson, respectively. Besides, the momenta of two $\pi$
in the $\pi\pi$ pair have the components

$\displaystyle p_{1}^{+}=\zeta\dfrac{M_{B}}{\sqrt{2}},\ \
p_{1}^{-}=\left(1-\zeta\right)\dfrac{M_{B}}{\sqrt{2}},\ \
p_{2}^{+}=\left(1-\zeta\right)\dfrac{M_{B}}{\sqrt{2}},\ \
p_{2}^{-}=\zeta\dfrac{M_{B}}{\sqrt{2}}\,,$ (7)

where $\zeta$ represents the momentum fraction of one pion in the $\pi\pi$
pair.

The wave functions of the initial and final states are the most important
inputs in calculating the $B$ meson decays within the PQCD approach. For $B$
meson and $\pi$ meson, their wave functions have been widely studied and well
defined in the non-leptonic two-body $B$ decays. In our following calculation,
the ones given in Ref. Li:2003yj ; Wang:2019msf ; Rui:2021kbn  are used. For
the $\pi\pi$ pair, the wave functions associated with the $P$\- and $D$-wave
resonances are relatively less studied. Because the initial state $B$ and the
final state $\pi$ are pseudoscalar mesons, only longitudinal polarization
state the $\pi\pi$ pair contributes to the decay amplitude due to the
conservation of angular momentum. The $P$-wave wave function of the
longitudinal polarized two-pion pair is given as Rui:2021kbn ; Li:2003yj ;
Wang:2019msf

$\displaystyle\Phi_{P}^{L}\left(z,\zeta,s\right)=\dfrac{1}{\sqrt{2N_{C}}}\left[\omega\not{\epsilon}_{P}\phi_{P}^{0}\left(z,\zeta,s\right)+\omega\phi_{P}^{s}\left(z,\zeta,s\right)+\frac{\not{p}_{1}\not{p}_{2}-\not{p}_{2}\not{p}_{1}}{\omega\left(2\zeta-1\right)}\phi_{P}^{t}\left(z,\zeta,s\right)\right].$
(8)

where $\phi_{P}^{0}$ and $\phi_{P}^{s,t}$ are the twist-2 and twist-3
distribution amplitudes (DAs), respectively, and can be written as Rui:2021kbn

$\displaystyle\phi_{P}^{0}\left(z,\zeta,s\right)$ $\displaystyle=$
$\displaystyle\frac{3F_{P}^{\parallel}\left(s\right)}{\sqrt{2N_{C}}}z\left(1-z\right)\left[1+a_{p}^{0}C_{2}^{3/2}\left(1-2z\right)\right]P_{1}\left(2\zeta-1\right),$
$\displaystyle\phi_{P}^{s}\left(z,\zeta,s\right)$ $\displaystyle=$
$\displaystyle\frac{3F_{P}^{\perp}\left(s\right)}{2\sqrt{2N_{C}}}z\left(1-2z\right)\left[1+a_{p}^{s}\left(1-10z+10z^{2}\right)\right]P_{1}\left(2\zeta-1\right),$
$\displaystyle\phi_{P}^{t}\left(z,\zeta,s\right)$ $\displaystyle=$
$\displaystyle\frac{3F_{P}^{\perp}\left(s\right)}{2\sqrt{2N_{C}}}z\left(1-2z\right)\left[1+a_{p}^{t}C_{2}^{3/2}\left(1-2z\right)\right]P_{1}\left(2\zeta-1\right),$
(9)

with the Legendre polynomial $P_{1}\left(2\zeta-1\right)=2\zeta-1$ and the
Gegenbauer polynomial $C_{2}^{3/2}\left(t\right)=3\left(5t^{2}-1\right)/2$.
The Gegenbauer moments $a_{p}^{0,s,t}$ in Eq. (9) are left to be determined by
combining experimental data. For the time-like form factor
$F_{P}^{\parallel}(s)$, we employee the vector-dominance model. Then,
including the $\rho-\omega$ interference, it can be parameterized as Refs.
LHCb:2019sus ; Wang:2016rlo ; Li:2017obb ; BaBar:2012bdw ,

$\displaystyle F_{P}^{\parallel}\left(s\right)$ $\displaystyle=$
$\displaystyle\left[GS_{\rho}\left(s,m_{\rho},\Gamma_{\rho}\right)\frac{1+c_{\omega}RBW_{\omega}\left(s,m_{\omega},\Gamma_{\omega}\right)}{1+c_{\omega}}+c_{\rho(1450)}RBW_{i}\left(s,m_{\rho(1450)},\Gamma_{\rho(1450)}\right)\right]$
(10) $\displaystyle\times$ $\displaystyle\frac{1}{1+c_{\rho(1450)}}\,,$

where $s=m^{2}(\pi\pi)$ is the two-pion invariant mass squared,
$m_{\rho,\omega,\rho(1450)}$ and $\Gamma_{\rho,\omega,\rho(1450)}$ are the
masses and decay widths of the corresponding resonances, respectively.
$c_{\omega,\rho(1450)}$ are the weight factors relative to the
$\rho\left(770\right)$ resonance. For the excited-states, only
$\rho\left(1450\right)$ contribution is taken into account. The functions
$GS_{\rho}$, $RBW_{\omega}$ and $RBW_{\rho(1450)}$ describe the mass lineshap
of each resonance state respectively.

Based on the analysis of $B^{+}\to\pi^{+}\pi^{-}\pi^{+}$ decay in Ref.
LHCb:2019sus , the central values of the Gegenbauer moments $a_{p}^{0,s,t}$ in
the two-pion DAs and the magnitudes of the weight factors $c_{\omega}$ and
$c_{\rho(1450)}$ can be determined as

$\displaystyle a_{p}^{0}=-0.7,\quad a_{p}^{s}=0.8,\quad
a_{p}^{t}=-1.3,\quad\lvert c_{\rho(1450)}\rvert=0.11,\quad\lvert
c_{\omega}\rvert=0.01.$ (11)

As for the lineshap function, the Gounaris-Sakurai mass lineshape
$GS_{\rho}\left(s,m,\Gamma\right)$ is employed to describe the
$\rho\left(770\right)$ resonance contribution LHCb:2019sus , and is given as

$\displaystyle
GS_{\rho}\left(s,m_{\rho},\Gamma_{\rho}\right)=\frac{m_{\rho}^{2}\left(1+D\Gamma_{\rho}/m_{\rho}\right)}{\left(m_{\rho}^{2}-s\right)+f\left(s\right)-im_{\rho}\Gamma(s)},$
(12)

with the functions defined as

$\displaystyle f\left(s\right)$ $\displaystyle=$
$\displaystyle\Gamma_{\rho}\frac{m_{\rho}^{2}}{p_{0}^{3}}\left[p_{1}^{2}\left[h\left(s\right)-h\left(m_{\rho}\right)\right]+\left(m_{\rho}^{2}-s\right)p_{0}^{2}\frac{\mathrm{d}h}{\mathrm{d}s}\bigg{|}_{\sqrt{s}=m_{\rho}}\right],$
$\displaystyle h\left(s\right)$ $\displaystyle=$
$\displaystyle\frac{2}{\pi}\frac{p_{1}}{\sqrt{s}}\log\left(\frac{\sqrt{s}+2p_{1}}{2m_{\pi}}\right),$
$\displaystyle\frac{\mathrm{d}h}{\mathrm{d}s}\bigg{|}_{\sqrt{s}=m_{\rho}}$
$\displaystyle=$ $\displaystyle
h\left(m_{\rho}^{2}\right)\left[\left(8p_{0}^{2}\right)^{-1}-\left(2m_{\rho}^{2}\right)^{-1}\right]+\left(2\pi
m_{\rho}^{2}\right)^{-1},$ $\displaystyle D$ $\displaystyle=$
$\displaystyle\frac{3}{\pi}\frac{m_{\pi}^{2}}{p_{0}^{2}}\log\left(\frac{m_{\rho}+2p_{0}}{2m_{\pi}}\right)+\frac{m_{\rho}}{2\pi
p_{0}}-\frac{m_{\pi}^{2}m_{\rho}}{\pi p_{0}^{3}},$ (13)

where the $p_{1}$ is the momentum of one daughter from $\rho(770)$ resonance,
and is equal to $p_{0}$ when $s=m_{\rho}^{2}$. For the $\omega(782)$ and
$\rho(1450)$ resonances, we follow Ref. LHCb:2019sus  and adopt the
relativistic Breit-Wigner (RBW) line shape ParticleDataGroup:2022pth , which
is given as

$\displaystyle RBW_{R}(s)=\frac{m_{R}^{2}}{m_{R}^{2}-s-im_{R}\Gamma(s)},$ (14)

where $R=\omega(782)$ and $\rho(1450)$, and $m_{R}$ is their nominal mass.
$\Gamma(s)$ is the mass-dependent width for the general case of a spin-$L$
resonance and is given as

$\displaystyle\Gamma(s)=\Gamma_{R}\left(\frac{\mid\vec{q}\mid}{\mid\vec{q}_{R}\mid}\right)^{2L+1}\left(\frac{m_{R}}{\sqrt{s}}\right)X_{L}^{2}(\zeta),$
(15)

where $\mid\vec{q}\mid$ is the momentum of one of $\pi$ in the $\pi\pi$ pair,
and is equal to $\mid\vec{q}_{R}\mid$ when $s=m_{R}^{2}$. The values of
$\Gamma_{R}$ and $m_{R}$ corresponding to the $\omega(782)$ and $\rho(1450)$
can be found in Ref. ParticleDataGroup:2022pth . $X_{L}(\zeta)$ is the Blatt-
Weisskopf angular momentum barrier factor Blatt:1952ije

$\displaystyle L=0:\;\;X_{L}(\zeta)=1,$ (16) $\displaystyle
L=1:\;\;X_{L}(\zeta)=\sqrt{\frac{1+\zeta_{0}^{2}}{1+\zeta^{2}}},$ (17)
$\displaystyle
L=2:\;\;X_{L}(\zeta)=\sqrt{\frac{9+3\zeta_{0}^{2}+\zeta_{0}^{4}}{9+3\zeta^{2}+\zeta^{4}}},$
(18)

with $\zeta=r|\vec{q}|$ with $r$ being the effective meson radius, and
$\zeta_{0}$ being the value of the $\zeta$ when $s=m_{R}^{2}$. It should be
noted that in Refs.Wang:2016rlo ; Li:2017obb ; BaBar:2012bdw , the authors
adopted the Gounaris-Sakurai mass lineshap for describing the contribution of
$\omega(782)$.

For the $D$-wave function of $\pi\pi$ pair, similar to the case of $P$-wave
$\pi\pi$ pair, only the longitudinal polarization state contributes to the
decay amplitudes. The longitudinal component of $D$-wave function of $\pi\pi$
pair is similar to the $P$-wave case, which is explicitly written as
Zou:2020ool ,

$\displaystyle\Phi_{D}^{L}\left(z,\zeta,s\right)=\sqrt{\frac{2}{3}}\dfrac{1}{\sqrt{2N_{C}}}\left[\not{p}\phi_{D}^{0}\left(z,\zeta,s\right)+\omega\phi_{D}^{s}\left(z,\zeta,s\right)+\frac{\not{p}_{1}\not{p}_{2}-\not{p}_{2}\not{p}_{1}}{\sqrt{s}\left(2\zeta-1\right)}\phi_{D}^{t}\left(z,\zeta,s\right)\right].\
\ \ $ (19)

The twist-2 and twist-3 DAs are

$\displaystyle\phi_{D}^{0}\left(z,\zeta,s\right)$ $\displaystyle=$
$\displaystyle\frac{9F_{D}^{\parallel}\left(s\right)}{\sqrt{2N_{C}}}a_{D}z\left(1-z\right)\left(2z-1\right)P_{2}\left(2\zeta-1\right),$
$\displaystyle\phi_{D}^{s}\left(z,\zeta,s\right)$ $\displaystyle=$
$\displaystyle-\frac{9F_{D}^{\perp}\left(s\right)}{4\sqrt{2N_{C}}}a_{D}\left(1-6z+6z^{2}\right)P_{2}\left(2\zeta-1\right),$
$\displaystyle\phi_{D}^{t}\left(z,\zeta,s\right)$ $\displaystyle=$
$\displaystyle\frac{9F_{D}^{\perp}\left(s\right)}{4\sqrt{2N_{C}}}a_{D}\left(2z-1\right)\left(1-6z+6z^{2}\right)P_{2}\left(2\zeta-1\right),$
(20)

where $P_{2}\left(2\zeta-1\right)=1-6\zeta+6\zeta^{2}$ is the legendre
polynomial, and the Gegenbauer moment $a_{D}=1.1$ is determined based on the
analysis in Ref. LHCb:2019sus . The $D$-wave time-like form factors
$F_{D}^{\perp}$ and $F_{D}^{\parallel}$ also satisfy the relation
$F_{D}^{\perp}/F_{D}^{\parallel}\approx f_{D}^{T}/f_{D}$, and
$F_{D}^{\parallel}$ can also be described by the RBW lineshape shown in Eq.
(14).

Using the theoretical framework introduced above, we then calculate the decay
amplitudes of $B^{+}\to\pi^{+}\left(R\to\right)\pi^{+}\pi^{-}$
($R=\rho(770),\omega(782),\rho(1450),f_{2}(1270)$) decays. Their explicit
forms are given in the appendix A. With these obtained amplitudes, we can
further evaluate the observables of the decay considered in this work. The
differential branching fraction of the quasi-two-body $B$ meson decay is El-
Bennich:2009gqk

$\displaystyle\frac{\mathrm{d}^{2}\mathcal{B}}{\mathrm{d}\zeta\mathrm{d}\omega}=\tau_{B}\frac{\lvert\vec{p_{1}}\rvert\lvert\vec{p_{3}}\rvert}{32\pi^{3}M_{B}^{2}}\lvert\mathcal{A}\rvert^{2},$
(21)

where

$\displaystyle\lvert\vec{p_{1}}\rvert=\frac{\lambda^{1/2}\left(\sqrt{s},m_{\pi},m_{\pi}\right)}{2\omega},\quad\lvert\vec{p_{3}}\rvert=\frac{\lambda^{1/2}\left(M_{B},m_{\pi},\sqrt{s}\right)}{2M_{B}},$
(22)

with
$\lambda\left(a,b,c\right)=\left[a^{2}-\left(b+c\right)^{2}\right]\left[a^{2}-\left(b-c\right)^{2}\right]$.

##  III Numerical results and discussion

Table 1: The values of input parameters. The decay constants of intermediate resonance are given in Ref. Cheng:2010yd ; Cheng:2011fk ; Li:2000zb , while other parameters are given in PDG ParticleDataGroup:2022pth . Masses (GeV) |  $m_{B}=5.279$ , $m_{\pi}=0.140$ , $m_{\rho_{770}}=0.775$ , $m_{\omega}=0.783$ ,
---|---
$m_{\rho_{1450}}=1.465$ , $m_{f_{2}}=1.276$
Decay constants (MeV) |  $f_{B}=210\pm 20$ , $f_{\rho_{770}}=216\pm 3$ , $f_{\rho_{770}}^{T}=165\pm 9$ , $f_{\omega}=187\pm 5$ ,
$f_{\omega}^{T}=151\pm 9$ , $f_{f_{2}}=102\pm 6$ , $f_{f_{2}}^{T}=117\pm 25$
Decay widths (MeV) |  $\Gamma_{\rho_{770}}=149.1$ , $\Gamma_{\omega}=8.49$ , $\Gamma_{\rho_{1450}}=400$ , $\Gamma_{f_{2}}=186.7$
Wolfenstein parameters |  $A=0.826^{+0.018}_{-0.015}$ , $\lambda=0.22500\pm 0.00067$ ,
$\bar{\rho}=0.159\pm 0.010$ , $\bar{\eta}=0.348\pm 0.010$
Table 2: The $CP$-averaged branching ratios for $B^{+}\to\pi^{+}(R\to)\pi^{+}\pi^{-}$ ($R=\rho(770)$, $\omega(782)$, $\rho(1450)$, $f_{2}(1270)$ ) decays. The numbers in the “PQCD” column correspond to our theoretical results within PQCD approach, and the ones in the “Exp.” column are the experimental date LHCb:2019sus .  Decay modes | PQCD | Exp.
---|---|---
$B^{+}\to\pi^{+}\left(\rho\left(770\right)\to\right)\pi^{+}\pi^{-}$ | $10.47_{-3.63-1.06-0.94}^{+4.79+1.42+1.11}\times 10^{-6}$ | $8.44\pm 0.09\pm 0.06\pm 0.38\times 10^{-6}$
$B^{+}\to\pi^{+}\left(\omega\left(782\right)\to\right)\pi^{+}\pi^{-}$ | $7.83_{-3.56-1.53-0.41}^{+4.14+1.35+0.41}\times 10^{-8}$ | $7.60\pm 0.46\pm 0.15\pm 0.61\times 10^{-8}$
$B^{+}\to\pi^{+}\left(\rho\left(1450\right)\to\right)\pi^{+}\pi^{-}$ | $6.68_{-2.49-0.94-0.61}^{+3.00+1.18+0.68}\times 10^{-7}$ | $7.90\pm 0.46\pm 0.30\pm 2.89\times 10^{-7}$
$B^{+}\to\pi^{+}\left(f_{2}(1270)\to\right)\pi^{+}\pi^{-}$ | $1.27_{-0.78-0.22-0.18}^{+0.69+0.17+0.15}\times 10^{-6}$ | $1.37\pm 0.05\pm 0.11\pm 0.21\times 10^{-6}$
Table 3:  The direct CP asymmetries parameters (in units of $\%$). Other legends are the same as those of Table 2.  Decay modes | PQCD | Exp.
---|---|---
$B^{+}\to\pi^{+}\left(\rho\left(770\right)\to\right)\pi^{+}\pi^{-}$ | $1.5_{-2.2-2.9-0.0}^{+2.1+0.1+0.0}$ | $0.7\pm 1.1\pm 0.6\pm 1.5$
$B^{+}\to\pi^{+}\left(\omega\left(782\right)\to\right)\pi^{+}\pi^{-}$ | $40.6_{-9.8-8.0-0.6}^{+10.8+0.9+0.8}$ | $-4.8\pm 6.5\pm 1.3\pm 3.5$
$B^{+}\to\pi^{+}\left(\rho\left(1450\right)\to\right)\pi^{+}\pi^{-}$ | $-1.4_{-1.2-1.1-0.0}^{+1.2+0.3+0.0}$ | $-12.9\pm 3.3\pm 3.6\pm 35.7$
$B^{+}\to\pi^{+}\left(f_{2}(1270)\to\right)\pi^{+}\pi^{-}$ | $73.6_{-13.0-19.4-1.7}^{+6.7+3.3+1.7}$ | $46.8\pm 6.1\pm 1.5\pm 4.4$

The values of input parameters used in the numerical calculation, such as
masses, decay constants, decay widths and the Wolfenstein parameters of CKM
matrix, are summarized in Table 1. Using these parameters, we present our
numerical results of the $CP$ averaged branching fractions and the CP
asymmetries, which are listed in Tables. 2 and 3, respectively. In the
calculation, three kinds of theoretical uncertainties are evaluated as has
been shown in the second columns of Tables. 2 and 3. The first error is caused
by the parameters in the distribution amplitudes of the $B$ meson, $\pi$
meson, and the $\pi\pi$ pair, such as the decay constants, shape parameter and
the Gegenbauer moments. The second error is caused by the higher order
radiative corrections and higher power corrections, which are evaluated by
varying the $\Lambda_{QCD}=\left(0.25\pm 0.05\right)~{}{\rm GeV}$ and the
factorization scale $t$ from $0.8t$ to $1.2t$. The last error arises from the
uncertainties of the CKM matrix elements.

From Table 2, it can be found that our theoretical results for the branching
fractions of $B^{+}\to\pi^{+}\left(R\to\right)\pi^{+}\pi^{-}$
($R=\rho(770),\omega(782),\rho(1450),f_{2}(1270)$) decays in the PQCD approach
are in good consistence with the experimental data reported by the LHCb
collaboration within errors. It should be noted that the theoretical errors
are dominated by the nonperturbative parameters, such as the Gegenbauer
moments in the wave function of the $\pi\pi$ pair. In fact, the theoretical
description of the two-pion wavefunction is still in the stage of the
modeling, the forms determined through the phenomenological models have to be
used in the calculation, because the result based on the first principle is
absent. As a result, the significant uncertainties caused by the hadronic
inputs are unavoidable for now, more efforts are required for improving
precision of the theoretical predictions.

It is well known to us that the $\omega(782)$ meson decaying directly to two
pions is prohibited strictly due to the conservation of the $G$ parity,
however this $G$-parity violating process can occur via $\rho-\omega$ mixing
effect. Some works indicate the potential importance of the $\rho-\omega$
mixing effect between the $\omega(782)$ and $\rho(770)$ resonances
Wang:2015ula ; Guo:2000uc ; Cheng:2016shb . In order to describe such effects,
a model is proposed to directly parameterize the interference between these
two contributions LHCb:2019sus . In this work, such effect is also considered
in a similar way via Eq. (10). Because the contribution associated with
$\omega(782)$ is induced only by the $\omega-\rho$ mixing effect, the
branching ratio of the
$B^{+}\to\pi^{+}\left(\omega\left(782\right)\to\right)\pi^{+}\pi^{-}$ is
expected to be far smaller than that of decay with intermediate resonance
$\rho(770)$ in principle. It can be clearly seen from the numerical results
given in Table 2, ${\cal
B}(B^{+}\to\pi^{+}\left(\omega\to)\pi^{+}\pi^{-}\right)/{\cal
B}(B^{+}\to\pi^{+}\left(\rho\to)\pi^{+}\pi^{-}\right)\sim{\cal O}(10^{-2})$.

In order to further test the framework of the quasi-two-body $B$ decays in
PQCD approach and the wave functions of two-pion pair, we also calculate the
branching fractions of two-body $B^{+}\to\pi^{+}R$ decays by using the
obtained results for the corresponding quasi-two-body
$B^{+}\to\pi^{+}\left(R\to\right)\pi^{+}\pi^{-}$ decays and the experimental
data for $R\to\pi^{+}\pi^{-}$ decays. It is known that under the narrow-width-
approximation (NWA). a branching fraction of the quasi-two-body decay can be
decomposed as

$\displaystyle\mathcal{B}\left(B^{+}\to\pi^{+}\left(R\to\right)\pi^{+}\pi^{-}\right)\simeq\mathcal{B}\left(B^{+}\to\pi^{+}R\right)\times\mathcal{B}\left(R\to\pi^{+}\pi^{-}\right),$
(23)

The branching fractions of $R\to\pi^{+}\pi^{-}$ ($R=\rho\,,\omega$ and
$f_{2}$) decays have been measured with high precision, and the experimental
results averaged by PDG are ParticleDataGroup:2022pth

$\displaystyle\mathcal{B}\left(\rho\to\pi^{+}\pi^{-}\right)$
$\displaystyle\simeq$ $\displaystyle 100\%,$ (24)
$\displaystyle\mathcal{B}\left(\omega\to\pi^{+}\pi^{-}\right)$
$\displaystyle=$ $\displaystyle(1.53^{+0.11}_{-0.13})\%,$ (25)
$\displaystyle\mathcal{B}\left(f_{2}\to\pi\pi\right)$ $\displaystyle=$
$\displaystyle(84.2^{+2.9}_{-0.9})\%.$ (26)

The experimental results for ${\cal B}(f_{2}\to\pi^{+}\pi^{-})$ can be
obtained by using the relation ${\cal
B}(f_{2}\to\pi^{+}\pi^{-})=2/3\times{\cal B}(f_{2}\to\pi\pi)$ required by the
conservation of $G$ parity. Then, using these experimental data and our
results for the corresponding quasi-two-body
$B^{+}\to\pi^{+}\left(R\to\right)\pi^{+}\pi^{-}$ decays listed in Table 2, we
can finally obtain

$\displaystyle\mathcal{B}\left(B^{+}\to\pi^{+}\rho\right)$ $\displaystyle=$
$\displaystyle\frac{\mathcal{B}\left(B^{+}\to\pi^{+}\left(\rho\to\right)\pi^{+}\pi^{-}\right)}{\mathcal{B}\left(\rho\to\pi^{+}\pi^{-}\right)}=\left(10.47^{+0.00+7.32}_{-0.00-5.64}\right)\times
10^{-6},$ (27) $\displaystyle\mathcal{B}\left(B^{+}\to\pi^{+}\omega\right)$
$\displaystyle=$
$\displaystyle\frac{\mathcal{B}\left(B^{+}\to\pi^{+}\left(\omega\to\right)\pi^{+}\pi^{-}\right)}{\mathcal{B}\left(\omega\to\pi^{+}\pi^{-}\right)}=\left(5.12^{+0.37+3.87}_{-0.43-4.29}\right)\times
10^{-6},$ (28) $\displaystyle\mathcal{B}\left(B^{+}\to\pi^{+}f_{2}\right)$
$\displaystyle=$
$\displaystyle\frac{\mathcal{B}\left(B^{+}\to\pi^{+}\left(f_{2}\to\right)\pi^{+}\pi^{-}\right)}{2/3\times\mathcal{B}\left(f_{2}\to\pi\pi\right)}=\left(2.26^{+0.08+1.79}_{-0.04-2.09}\right)\times
10^{-6},$ (29)

where the first error corresponds to the experimental error and the second one
corresponds to the theoretical error given in Table. 2. In comparison with the
experimental results ParticleDataGroup:2022pth ,

$\displaystyle\mathcal{B}\left(B^{+}\to\pi^{+}\rho\right)$ $\displaystyle=$
$\displaystyle\left(8.3\pm 1.2\right)\times 10^{-6},$ (30)
$\displaystyle\mathcal{B}\left(B^{+}\to\pi^{+}\omega\right)$ $\displaystyle=$
$\displaystyle\left(6.9\pm 0.5\right)\times 10^{-6},$ (31)
$\displaystyle\mathcal{B}\left(B^{+}\to\pi^{+}f_{2}\right)$ $\displaystyle=$
$\displaystyle\left(2.2^{+0.7}_{-0.4}\right)\times 10^{-6},$ (32)

it can be found that our theoretical results are in agreement with
experimental data within the uncertainties.

At last, we turn to discuss the results of the direct $CP$ asymmetries in
these decays, which are presented in Table. 3. It can be found from the table
that the direct $CP$ asymmetries of
$B^{+}\to\pi^{+}(\rho(770,1450)\to)\pi^{+}\pi^{-}$ decays are very small,
which are consistent with the experimental data given by LHCb. It is known
that the direct $CP$ asymmetry is mainly induced by the interference between
the tree and penguin contributions. However, the decays
$B^{+}\to\pi^{+}(\rho(770,1450)\to)\pi^{+}\pi^{-}$ are dominated by the tree
contributions from $b\to du\bar{u}$ process with large Wilson coefficients,
and the contributions of annihilation diagrams associated with penguin
operators with $q=u$ and $d$ significantly are canceled by each other, which
can be clearly seen from Eq. (A.1). For the
$B^{+}\to\pi^{+}(f_{2}(1270)\to)\pi^{+}\pi^{-}$ decay, it has a large $CP$
asymmetry as large as $73.6\%$ due to the fact that the penguin contributions
from chiral enhanced annihilation diagrams no longer cancel each other out.
Our theoretical results agree with the analysis of LHCb collaboration within
errors. Similarly, a relatively large $CP$ asymmetry is expected also by PQCD
in the $B^{+}\to\pi^{+}(\omega\to)\pi^{+}\pi^{-}$ decay, which could be tested
in LHCb and Belle-II experiments in future.

##  IV Summary

In this paper, motivated by the analysis of $B^{+}\to\pi^{+}\pi^{-}\pi^{+}$
decay made by LHCb collaboration, we investigated the
$B^{+}\to\pi^{+}(R\to)\pi^{+}\pi^{-}$ decays with the P-wave $\rho(770)$,
$\omega(782)$, $\rho(1450)$ and D-wave $f_{2}(1270)$ intermediate resonant
states within the framework of PQCD approach, in which the effect of
$\rho-\omega$ mixing are considered. The branching fractions and direct CP
asymmetries of these decays are calculated, our numerical results are
summarized in Tables 2 and 3. Our results for ${\cal
B}(B^{+}\to\pi^{+}(R\to)\pi^{+}\pi^{-})$ ($R=\rho(770)$, $\omega(782)$,
$\rho(1450)$, $f_{2}(1270)$), as well as the extracted ${\cal
B}(B^{+}\to\pi^{+}(\rho(770),\omega(782),f_{2}(1270)))$, are consistent with
the experimental data within errors. The theoretical results for the direct
$CP$ asymmetries of tree dominated
$B^{+}\to\pi^{+}(\rho(770/1450)\to)\pi^{+}\pi^{-}$ decays are very small,
while the ones of tree and penguin dominated
$B^{+}\to\pi^{+}(\omega/f_{2}\to)\pi^{+}\pi^{-}$ are relatively large. Most of
them agree well with the LHCb data except for
$B^{+}\to\pi^{+}(\omega\to)\pi^{+}\pi^{-}$ decay, which occurs only via
$\rho-\omega$ mixing effect. More theoretical and experimental efforts are
needed for clarifying the underling mechanism of such special decay mode.

## Acknowledgment

This work is supported by the National Natural Science Foundation of China
(Grant Nos. 12275067, 12305101, 11875122, 12375089), Natural Science
Foundation of Shandong province (Grant no. ZR2022MA035 and ZR2022ZD26),
Natural Science Foundation of Henan Province (Grant No. 225200810030), and
Excellent Youth Foundation of Henan Province (Grant No. 212300410010).

## Appendix A:Decay amplitudes

The decays amplitudes of decay modes studied in this work are given as

$\displaystyle\mathcal{A}\left(B^{+}\to\pi^{+}\left(\rho_{770}/\rho_{1450}\to\right)\pi^{+}\pi^{-}\right)=$
$\displaystyle\frac{G_{F}}{2}\Big{\\{}V_{ub}^{*}V_{ud}\big{[}C_{1}(M_{VP,P}^{LL}+W_{VP,P}^{LL}-W_{VP,V}^{LL})+C_{2}M_{VP,V}^{LL}$
$\displaystyle+(\frac{C_{1}}{3}+C_{2})(F_{VP,P}^{LL}+A_{VP,P}^{LL}-A_{VP,V}^{LL})+(C_{1}+\frac{C_{2}}{3})F_{VP,V}^{LL}\big{]}$
$\displaystyle-
V_{tb}^{*}V_{td}\big{[}(C_{3}+C_{9})(M_{VP,P}^{LL}+W_{VP,P}^{LL}-W_{VP,V}^{LL})+(C_{5}+C_{7})$
$\displaystyle(M_{VP,P}^{LR}+W_{VP,P}^{LR}-W_{VP,V}^{LR})+(\frac{C_{3}}{3}+C_{4}+\frac{C_{9}}{3}+C_{10})(F_{VP,P}^{LL}$
$\displaystyle+A_{VP,P}^{LL}-A_{VP,V}^{LL})+(\frac{C_{5}}{3}+C_{6}+\frac{C_{7}}{3}+C_{8})(F_{VP,P}^{SP}+A_{VP,P}^{SP}$
$\displaystyle-
A_{VP,V}^{SP})+(-\frac{C_{3}}{3}-C_{4}+\frac{3}{2}C_{7}+\frac{1}{2}C_{8}+\frac{5}{3}C_{9}+C_{10})F_{VP,V}^{LL}$
$\displaystyle+(-C_{3}+\frac{1}{2}C_{9}+\frac{3}{2}C_{10})M_{VP,V}^{LL}+(-C_{5}+\frac{1}{2}C_{7})M_{VP,V}^{LR}$
$\displaystyle+\frac{3}{2}C_{8}M_{VP,V}^{SP}\big{]}\Big{\\}}.$ (A.1)
$\displaystyle\mathcal{A}\left(B^{+}\to\pi^{+}\left(\omega\left(782\right)\to\right)\pi^{+}\pi^{-}\right)=$
$\displaystyle\frac{G_{F}}{2}\Big{\\{}V_{ub}^{*}V_{ud}\big{[}C_{1}(M_{VP,P}^{LL}+W_{VP,P}^{LL}+W_{VP,V}^{LL})+C_{2}M_{VP,V}^{LL}$
$\displaystyle+(\frac{C_{1}}{3}+C_{2})(F_{VP,P}^{LL}+A_{VP,P}^{LL}+A_{VP,V}^{LL})+(C_{1}+\frac{C_{2}}{3})F_{VP,V}^{LL}\big{]}$
$\displaystyle-
V_{tb}^{*}V_{td}\big{[}(C_{3}+C_{9})(M_{VP,P}^{LL}+W_{VP,P}^{LL}+W_{VP,V}^{LL})+(C_{5}+C_{7})$
$\displaystyle(M_{VP,P}^{LR}+W_{VP,P}^{LR}+W_{VP,V}^{LR})+(\frac{C_{3}}{3}+C_{4}+\frac{C_{9}}{3}+C_{10})(F_{VP,P}^{LL}$
$\displaystyle+A_{VP,P}^{LL}+A_{VP,V}^{LL})+(\frac{C_{5}}{3}+C_{6}+\frac{C_{7}}{3}+C_{8})(F_{VP,P}^{SP}+A_{VP,P}^{SP}$
$\displaystyle+A_{VP,V}^{SP})+(\frac{7}{3}C_{3}+\frac{5}{3}C_{4}+2C_{5}+\frac{2}{3}C_{6}+\frac{1}{2}C_{7}+\frac{1}{2}C_{8}+\frac{1}{3}C_{9}$
$\displaystyle-\frac{1}{3}C_{10})F_{VP,V}^{LL}+(C_{3}+2C_{4}-\frac{1}{2}C_{9}+\frac{1}{2}C_{10})M_{VP,V}^{LL}$
$\displaystyle+(C_{5}-\frac{1}{2}C_{7})M_{VP,V}^{LR}+(2C_{6}+\frac{1}{2}C_{8})M_{VP,V}^{SP}\big{]}\Big{\\}}.$
(A.2)

The contributions of the
$n\bar{n}=\frac{1}{\sqrt{2}}\left(u\bar{u}+d\bar{d}\right)$ and the
($s\bar{s}$) quark components to the decay amplitude of the
$B^{+}\to\pi^{+}\left(f_{2}\left(1270\right)\to\right)\pi^{+}\pi^{-}$ are
written as

$\displaystyle\mathcal{A}_{n\bar{n}}\left(B^{+}\to\pi^{+}\left(f_{2}\left(1270\right)\to\right)\pi^{+}\pi^{-}\right)=$
$\displaystyle\frac{G_{F}}{2}\Big{\\{}V_{ub}^{*}V_{ud}\big{[}C_{1}(M_{TP,P}^{LL}+W_{TP,P}^{LL}+W_{TP,T}^{LL})+C_{2}M_{TP,T}^{LL}$
$\displaystyle+(\frac{C_{1}}{3}+C_{2})(F_{TP,P}^{LL}+A_{TP,P}^{LL}+A_{TP,T}^{LL})\big{]}$
$\displaystyle-
V_{tb}^{*}V_{td}\big{[}(C_{3}+C_{9})(M_{TP,P}^{LL}+W_{TP,P}^{LL}+W_{TP,T}^{LL})+(C_{5}+C_{7})$
$\displaystyle(M_{TP,P}^{LR}+W_{TP,P}^{LR}+W_{TP,T}^{LR})+(\frac{C_{3}}{3}+C_{4}+\frac{C_{9}}{3}+C_{10})(F_{TP,P}^{LL}$
$\displaystyle+A_{TP,P}^{LL}+A_{TP,T}^{LL})+(\frac{C_{5}}{3}+C_{6}+\frac{C_{7}}{3}+C_{8})(F_{TP,P}^{SP}+A_{TP,P}^{SP}$
$\displaystyle+A_{TP,T}^{SP})+(C_{3}+2C_{4}-\frac{1}{2}C_{9}+\frac{1}{2}C_{10})M_{TP,T}^{LL}$
$\displaystyle+(C_{5}-\frac{1}{2}C_{7})M_{TP,T}^{LR}+(2C_{6}+\frac{1}{2}C_{8})M_{TP,T}^{SP})\big{]}\Big{\\}}\,,$
(A.3)
$\displaystyle\mathcal{A}_{s\bar{s}}\left(B^{+}\to\pi^{+}\left(f_{2}\left(1270\right)\to\right)\pi^{+}\pi^{-}\right)=$
$\displaystyle-\frac{G_{F}}{\sqrt{2}}V_{tb}^{*}V_{td}\big{[}(C_{4}-\frac{1}{2}C_{10})M_{TP,T}^{LL}$
$\displaystyle+(C_{6}-\frac{1}{2}C_{8})M_{TP,T}^{SP})\big{]}\,,$ (A.4)

respectively, where $G_{F}$ is the Fermi constant, $V_{ij}$ are the Cabibbo-
Kobayashi-Maskawa matrix elements. In above formula, the functions
$F_{VP,P}^{LL},M_{VP,V}^{LR},A_{TP,P}^{SP}$, $W_{TP,T}^{LL},....$ are the
individual decay amplitudes corresponding to different currents and diagrams;
$F$, $M$, $A$ and $W$ correspond to the factorable emission diagram, the non-
factorable emission diagram, the factorable annihilation diagram, the non-
factorable annihilation diagram, respectively; the subscript $P$, $V$ and $T$
denote the pseudoscalar, vector and tensor states, respectively; The
superscripts $LL$,$LR$,and $SP$ denote $(V-A)(V-A)$, $(V-A)(V+A)$ and
$(S-P)(S+P)$ currents, respectively. The explicit expressions of these
functions can be found in the Ref. Li:2021cnd .

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# The Solar Meridional Circulation and Sunspot Cycle Variability

###### Abstract

We have measured the meridional motions of the magnetic elements in the Sun’s
surface layers since 1996 and find systematic and substantial variations. In
general the meridional flow speed is fast at cycle minima and slow at cycle
maxima. We find that these systematic variations are characterized by a
weakening of the meridional flow on the poleward sides of the active (sunspot)
latitudes. This can be interpreted as a inflow toward the sunspot zones
superimposed on a more general poleward meridional flow profile. We also find
variations in the meridional flow which vary from cycle-to-cycle. The
meridional flow was slower at both the minimum and maximum of cycle 23
compared to similar phases of cycles 21, 22, and 24. Models of the magnetic
flux transport by a variable meridional flow suggest that it can significantly
modulate the size and timing of the following sunspot cycle through its impact
on the Sun’s polar magnetic fields. We suggest that the meridional flow
variations observed in cycle 23 contributed to the weak polar fields at the
end of the cycle which then produced a weak cycle 24 and the extraordinary
cycle 23/24 minimum.

HATHAWAY & UPTON MERIDIONAL FLOW AND THE SUNSPOT CYCLE Corresponding author:
D. H. Hathaway, Mail Code ZP13, NASA/Marshall Space Flight Center, Huntsville,
AL 35812 USA<EMAIL_ADDRESS>
11affiliationtext: Space Science Office, NASA Marshall Space Flight Center,
Huntsville, Alabama, USA.22affiliationtext: Department of Physics and
Astronomy, Vanderbilt University, Nashville, Tennessee, USA.33affiliationtext:
Center for Space Physics and Aeronomy Research, University of Alabama in
Huntsville, Huntsville, Alabama, USA.

##  1 Introduction

The sunspot cycle minimum between cycles 23 and 24 (cycle 23/24 minimum) was
exceptional compared to others in modern times. In December of 2008 the
13-month smoothed sunspot number reached its lowest level since July of 1913
and the smoothed number of spotless days in a month reached its highest level
since August of 1913. In September of 2009 geomagnetic activity, as measured
by the aa index, reached record lows (since measurements began in 1868) while
galactic cosmic rays, as measured by neutron monitors, reached record highs
(since measurements began in 1953).

Since cycle 23/24 minimum in late-2008 we have seen the rise of cycle 24 as
the smallest sunspot cycle in 100 years. This provides the simple answer to
the question: What caused this extraordinary sunspot cycle minimum? This deep
and extended minimum was caused by the typically delayed start of a small
cycle. Statistically, small cycles start late and leave behind a long cycle
and a deep minimum (Hathaway et al., 1999) (c.f. minima preceding cycles
12-15). Two effects contribute to this: the actual start times for small
cycles are delayed (Hathaway et al., 1994) and the Waldmeier Effect
(Waldmeier, 1935) (in which small cycles rise more slowly) moves the date of
cycle minimum between overlapping cycles.

This explanation raises an obvious follow-on question: What caused cycle 24 to
be so small? This can be attributed to the weak polar fields built up during
cycle 23. Models of the Sun’s magnetic dynamo suggest that the Sun’s largely
dipole magnetic field at cycle minimum is the seed for the magnetic field that
erupts in the form of sunspots after amplification by the Sun’s differential
rotation (Babcock, 1961; Leighton, 1969).

Based on these models, Schatten et al. (1978) proposed that the strength of
the polar fields at sunspot cycle minimum should be a good predictor of the
amplitude of the following sunspot cycle maximum. Recently, Muñoz-Jaramillo et
al. (2013) confirmed (by using counts of polar faculae as a proxy for the
polar fields prior to 1976) that the polar fields themselves are indeed well
correlated with the amplitude of the following sunspot cycle maximum. Sunspot
cycle predictions based on the polar fields at minimum have successfully
predicted the last three cycles (Schatten et al., 1978; Schatten and Sofia,
1987; Schatten et al., 1996) as well as cycle 24 (Svalgaard et al., 2005).

Wang and Sheeley (2009) have noted that the strength of Sun’s axial dipole is
more closely attuned to dynamo theory and may be measured more accurately than
the polar fields for previous cycles. The axial dipole largely determines the
interplanetary magnetic field near cycle minima and this field can be derived
from historical geomagnetic measurements (Svalgaard and Cliver, 2005;
Rouillard et al., 2007). In fact, Wang and Sheeley (2009) suggest that it is
this connection that makes the geomagnetic aa index at its minimum such a good
predictor for the amplitude of following cycle (Ohl, 1966; Hathaway et al.,
1999).

This explanation for the cause of a weak cycle 24 minimum raises yet another
follow-on question: Why did cycle 23 produce such weak polar fields and axial
dipole? Dynamo models (Babcock, 1961; Leighton, 1969) and models for the
transport of magnetic flux in the solar photosphere (DeVore et al., 1984; Wang
et al., 1989; van Ballegooijen et al., 1998; Schrijver and Title, 2001;
Baumann et al., 2004) produce the polar fields through the emergence of tilted
bipolar active regions followed by the poleward transport of the magnetic
flux. The strength of the polar fields depends on the details of both the
sources (the total magnetic flux and tilt of the active regions) and the
transport processes (the meridional flow and the non-axisymmetric cellular
convective flows). Either the sources or the transport processes, or both,
must vary to give variable polar fields and the consequent solar cycle
variability.

##  2 Producing the Sun’s Axial Dipole

Both the active region sources and the transport processes that produce the
axial dipole moment are evident in magnetic butterfly diagrams (Fig. 1). These
diagrams are produced by averaging the Sun’s photospheric magnetic field over
longitude during each solar rotation to show the magnetic field as a function
of latitude and time. The butterfly wings in these diagrams show the emergence
of magnetic flux in the active latitudes with a predominance of one polarity
at higher latitudes and the opposite polarity at lower latitudes due to the
Joy’s Law tilt of active regions (Hale et al., 1919). This polarity pattern
alternates from hemisphere-to-hemisphere and cycle-to-cycle reflecting Hale’s
Law (Hale et al., 1919).

As each cycle progresses, new tilted dipoles emerge at progressively lower
latitudes so that the higher latitude following polarity flux progressively
cancels the preceding polarity flux that previously occupied those latitudes.
The poleward transport of the high latitude polarity is evident in Fig. 1 in
the form of unipolar streams that move poleward with time. As each cycle
reaches its maximum phase the sunspot zones extend to their lowest latitudes
with leading polarity in close proximity to the equator. This allows for the
diffusive transport of leading polarity flux across the equator and
cancellation with the (opposite polarity) leading polarity flux in the other
hemisphere.

Figure 1: A magnetic butterfly diagram produced from data acquired by the
National Solar Observatory. This shows the emergence of magnetic flux with the
systematic separation of polarities in the active latitude bands as well as
the poleward transport of this flux.

Figure 2: The Sun’s axial dipole moment (black line) as measured at the Wilcox
Solar Observatory for the last 40 years. The smoothed sunspot number is shown
in red. This shows that the dipole reverses at about the time of cycle maximum
and reaches its maximum at about the time of cycle minimum. The amplitudes of
the following cycles decrease as the amplitudes of the dipole moment decrease.

The polarity of the high latitude flux is opposite to that of the polar fields
at the start of the cycle so the emergence and poleward transport cancels the
polar fields from the previous cycle and then builds up opposite polarity
polar fields over the remainder of the sunspot cycle. This is seen in the
axial dipole moment (Fig. 2) calculated from the latitudinal distribution of
the surface magnetic field. The dipole reversals occur at about the time of
cycle maximum and the axial dipole moment that is built up at the end of the
cycle is well correlated with the strength of the following cycle (Wang and
Sheeley, 2009; Muñoz-Jaramillo et al., 2013).

This process could potentially lead to catastrophic behavior for solar cycle
amplitudes. Big sunspot cycles have more magnetic flux emerging in active
regions which would produce stronger axial dipoles along with increasing cycle
amplitudes. Likewise, smaller sunspot cycles should produce a string of cycles
with diminishing amplitudes. While the sunspot record does show periods of
increasing and decreasing cycle amplitudes, these trends are always limited to
4-5 cycles as part of the 100-year Gleissberg Cycle (Gleissberg, 1939). This
suggests that either the transport processes or the Joy’s Law tilt of active
regions (or both) must somehow change systematically with sunspot cycle
amplitude to stem the tide of increasing or decreasing cycle amplitudes.

A study of active region tilt by Dasi-Espuig et al. (2010) indicates that the
tilt may indeed vary with sunspot cycle amplitude. They found a tendency for a
smaller proportionality between active region tilt and latitude in bigger
cycles. This effect would help to modulate the polar field production by
leaving less of an excess of the higher latitude polarity during big cycles as
shown by Cameron et al. (2010) and Jiang et al. (2011).

Another mechanism for modulating the polar fields is to modulate the
latitudinal transport. This transport is facilitated by two processes – a
random walk by the rapidly evolving convection pattern and the direct
transport by the poleward meridional flow. The convection pattern includes
granules with velocities of $\sim 3000$ m s-1, lifetimes of $\sim 10$ minutes,
and diameters of $\sim 1$ Mm and supergranules with velocities of $\sim 300$ m
s-1, lifetimes of a day, and diameters of $\sim 30$ Mm. In one year, a random
walk by supergranules should have $\sim 400$ “steps” of 15 Mm giving a
displacement of $\sim 300$ Mm The meridional flow velocity is only $\sim 10$ m
s-1 but this direct flow also gives a displacement of $\sim 300$ Mm over the
course of a year. While these simple calculations suggest that these two
processes have similar impact, it should be noted that the diffusive effect of
the supergranules gives both poleward and equatorward motions and the
magnitude of the diffusivity still remains uncertain (Hagenaar et al., 1999).

In this paper we report on our measurements of the meridional motions of the
magnetic elements. We find variations in the strength and structure of the
meridional flow that may help to explain in general how the polar fields are
modulated and in particular how they were modulated during cycle 23. These
variations may have contributed to the production of the weak axial dipole
moment during cycle 23 which, in turn, caused the small amplitude for cycle
24.

##  3 Measuring the Meridional Flow

Measurements of the meridional flow can be made by a variety of techniques
including direct Doppler, local helioseismology, and feature tracking.
Advantages and disadvantages of the different techniques are discussed in the
Appendix. We choose to use magnetic feature tracking because is not masked by
large systematic signals and it measures the motions of the features of
interest for magnetic flux transport – magnetic field elements.

Our measurement technique is described in more detail in Hathaway and
Rightmire (2010), Hathaway and Rightmire (2011), and Rightmire-Upton et al.
(2012). We acquired full-disk magnetograms from the ESA/NASA Solar and
Heliospheric Observatory Michelson Doppler Investigation (SOHO/MDI) from May
1996 through March 2011 and from the NASA Solar Dynamics Observatory
Helioseismic and Magnetic Imager (SDO/HMI) from April 2010 through July 2013.
The SOHO/MDI magnetograms (Scherrer et al., 1995) were individual or 5-minutes
averages obtained every 96 minutes except for a data gap (the SOHO summer
vacation) from June to October 1998. The SDO/HMI magnetograms (Scherrer et
al., 2012) were averaged over 12-minutes and obtained every 60 minutes.

We projected each full-disk magnetogram onto a Mercator projection grid in
heliographic longitude and latitude. We extracted long, thin strips of data
with longitudinal widths of $105^{\circ}$ and latitudinal heights of
$2^{\circ}$ centered on the central meridian at a series of latitudes. These
strips were cross-correlated with similar strips from magnetograms obtained 8
hours later and 8 hours earlier. The offset in longitude and latitude giving
the highest correlation gives the differential rotation and meridional flow.
The average meridional flow profile obtained with the SDO/HMI data is shown in
Fig. 3.

Figure 3: The average meridional flow profile obtained with data from the
SDO/HMI instrument acquired between April 2010 and July 2013. Meridional flow
profiles used in previous surface flux transport models are shown with the
dotted (Wang et al., 1989), dashed (van Ballegooijen et al., 1998) and dashed-
dotted (Schrijver and Title, 2001) lines.

This meridional flow profile is problematic for most surface flux transport
models – it indicates that the poleward motions of the magnetic elements peak
in mid-latitudes and extend right to the poles. The early surface flux
transport models (Wang et al., 1989) employed a meridional flow that had
maximum poleward velocities immediately adjacent to the equator with a slow
fall-off to higher latitudes in each hemisphere (dotted line in Fig. 3.).
Later models (van Ballegooijen et al., 1998; Schrijver and Title, 2001) used
profiles that do match ours up to latitudes of $40-50^{\circ}$ but then have
little or no poleward flow at latitudes above $75^{\circ}$ (dashed and dashed-
dotted lines in Fig. 3).

The latitudinal structure of the meridional flow can significantly alter the
strength and structure of the polar fields it produces in these models. The
continued poleward flow at high latitudes tends to make the fields more
tightly concentrated at the poles. A peak flow velocity at very low latitudes
keeps opposite polarity elements from canceling across the equator.

##  4 Meridional Flow Speed Variations

We calculated the average profile for each 27-day rotation of the Sun and fit
each of them to orthogonal polynomials in $\sin B$ where $B$ is the
heliographic latitude. The fits to the meridional flow are dominated by the
terms of order 1 and 3.

Fig. 4 shows the history of the fit coefficients for the meridional flow
profiles. Here the coefficients found with SOHO/MDI for cycle 23 (Hathaway and
Rightmire, 2010) are now augmented by those we find with SDO/HMI data for
cycle 24 and by those found with NSO/Kitt Peak data for cycles 21 and 22 by
Komm et al. (1993a) using a similar method. (Komm et al. (1993a) correlated
square areas at $\sim 24$ hour time lags. We both excluded areas of strong
field associated with sunspots.) The variations in the meridional flow speed
over the course of each cycle are substantial. The meridional flow is fast at
cycle minima and slow at cycle maxima as was previously noted by Komm et al.
(1993a).

Figure 4: The meridional flow speed history from 1980 through mid-2013 is
represented in terms of the coefficients (S1 and S3) of the polynomials fit to
the profiles. Results for individual Carrington rotations are shown in black
for data from SOHO/MDI and in blue for the new results from SDO/HMI. Both sets
of data are shown with $2\sigma$ error bars. The data points (with $1\sigma$
error bars) for the 2-year intervals from 1980 to 1990 from Komm et al.
(1993a) are also shown in black. The smoothed sunspot number is shown in red
as a reference for comparing the meridional flow speed variations with the
phases of the sunspot cycles.

Fig. 4 also shows variations in the meridional flow speed from cycle-to-cycle.
The meridional flow was slower at both the preceding minimum (1996) and the
maximum (2000) of cycle 23 when compared to the minima and maxima of both the
earlier cycles (21 and 22) and the later cycle (24).

##  5 Meridional Flow Structure Variations

The fit coefficients shown in Fig. 4 help to characterize the amplitude of the
variability but do not fully describe the meridional flow variations. Cameron
and Schüssler (2010) noted that the variations in the S1 term shown in Fig. 4
could be explained in terms of the previously reported changes to the
meridional flow profiles derived from local helioseismology (Zhao and
Kosovichev, 2004; Gizon, 2004; González Hernández et al., 2008, 2010) and
interpreted as inflows toward the active latitudes. This behavior is seen in
Fig. 5 by showing each individual profile in a color-coded image.

The nature of the slow-down of the meridional flow at cycle maxima is clearly
evident in Fig 5. The slow-down is seen as a weakening of the poleward flow on
the poleward sides of the active latitudes (lighter shades of red in the north
and blue in the south. This can indeed be interpreted as an inflow toward the
active latitudes that is superimposed on the average poleward flow profile as
suggested by Cameron and Schüssler (2010), noted earlier from helioseismology
(Zhao and Kosovichev, 2004; Gizon, 2004; González Hernández et al., 2008,
2010), and from one of our earlier studies of magnetic element motion
(Hathaway and Rightmire, 2011).

Figure 5: The meridional flow profile history from May 1996 to July 2013 is
represented by showing the flow profiles from each 27-day solar rotation as
color-coded vertical strips. Shades of blue represent southward flow while
shades of red represent northward flow. Black dots indicate the location of
the active latitudes as given by the latitudinal centroid of the sunspot group
area in each hemisphere.

Fig. 5 also shows substantial differences between cycle 23 and our new results
for cycle 24. The weakening of the meridional flow on the poleward sides of
the active latitudes is hardly discernible in cycle 24. We also find polar
counter-cells (equatorward flow at high latitudes) with the MDI data during
cycle 23 that are not yet seen in HMI data during cycle 24. Cycle 23 appeared
to have a counter-cell in the south that extended down to $60^{\circ}$ in
mid-1996. By 2001 the counter-cell boundary moved poleward of our $75^{\circ}$
observation limit. At about this time a counter-cell is seen to dip below this
limit in the north and then proceed to grow to its maximum extent in 2006-2007
after which it shrinks but is still apparent in MDI data in 2010. It is worth
noting that during the 1-year overlap between MDI and HMI from April of 2010
to March of 2011 this northern counter-cell was visible in the MDI data but
not seen in the HMI data even at $85^{\circ}$ north (Rightmire-Upton et al.,
2012).

##  6 Possible Effects on the Polar Fields

The effects of these meridional flow variations on the polar fields and the
axial dipole need to be determined. Several previous studies have
characterized the effects of simply changing the amplitude of the meridional
flow. In addition, Jiang et al. (2010) and Cameron and Schüssler (2012) have
recently examined the effects associated with variations in the shape of the
meridional flow profile that are very similar to those found here (inflows
toward the active latitudes).

The parametric study of Baumann et al. (2004) used the meridional flow profile
of van Ballegooijen et al. (1998) (dashed line in Fig. 3) and varied the peak
velocity from 0 to 30 m s-1 with a fixed diffusivity of 600 km2 s-1. They
found that the polar fields increased in strength as the meridional flow
amplitude increased from 0 to 8 m s-1 — more high latitude, following polarity
flux is carried to the poles and the meridional flow itself counters the
diffusion away from the poles. As the meridional flow amplitude increases
above 8 m s-1 the polar fields become weaker — the higher velocities prevent
opposite polarities from canceling across the equator so that the net flux
carried by the meridional flow decreases. (This result is similar to what was
found earlier by Wang et al. (1989)).

The switchover in the sensitivity of polar field strength to meridional flow
speed variations at 8 m s-1 makes any conclusions about the effects of the
observed variations difficult based on this study. The average flow speed of
11 m s-1 is very close to this switchover point and the switchover point
itself must depend on the diffusivity and meridional flow profile used in the
calculations.

The structural changes to the meridional flow profile (the slowdown above the
active latitudes) should have a direct and negative effect on the poleward
transport of the high latitude following polarity but have little effect on
the flux cancellation across the equator. In general, this should modulate the
sunspot cycle amplitudes – big cycles would have stronger inflows that would
produce weaker polar fields than expected from the increase of active region
sources. In fact, we find (Fig. 5) that this inflow was stronger in cycle 23
than it is in cycle 24. Jiang et al. (2010) and Cameron and Schüssler (2012)
show that variations like this may indeed provide a nonlinear feedback on the
amplitudes of the solar cycles and may even be a key ingredient in determining
the amplitudes of solar cycles. Although the variations they employ are
significantly stronger than those we find in cycles 23 and 24 (neither cycle
shows any actual equatorward flow within the active latitudes) their results
do suggest that the meridional flow slowdown poleward of the active latitudes
in cycle 23 contributed to producing the weak polar fields.

##  7 Conclusions

The exceptional depth of cycle 23/24 minimum and the exceptional length of
cycle 23 can both be attributed to the small amplitude of cycle 24 – small
cycles typically start late and leave behind low minima (Hathaway et al.,
1999). The small amplitude of cycle 24 can be attributed to the weak polar
fields produced prior to cycle 23/24 minimum by the activity and flux
transport of cycle 23 – weak polar fields produce weak cycles (Muñoz-Jaramillo
et al., 2013). The strength of the polar fields (or the axial dipole moment)
is largely determined by a combination of three processes: the total magnetic
flux emerging in active regions, the characteristic tilt of those active
regions, and the transport (meridional flow and diffusion) of that emerging
flux.

Cycle 23 was significantly smaller than cycles 21 and 22 that preceded it.
This change in active region sources alone can be a significant source of the
change in polar fields. Dasi-Espuig et al. (2010) found that there are changes
to the characteristic tilt of active regions as a function of cycle amplitude.
However, they did not examine cycle 23 for comparison with other cycles. Here
we examined changes in the meridional flow measured from the motions of the
small magnetic elements that populate the Sun’s surface. We found variations
in the meridional flow that may have contributed to producing the weak polar
fields in cycle 23 and may play a more general role in modulating the
amplitudes of the solar cycle.

We found that cycle 23 was characterized by a slower meridional flow and that
this variation in the meridional flow was primarily due to a slowdown of the
meridional flow at the latitudes poleward of the sunspot zones. This slowdown
is not seen in cycle 24 – a much weaker solar cycle. This suggests that this
is a characteristic feature of solar cycles: the weakening of the meridional
flow is greater in bigger cycles. Cameron and Schüssler (2012) suggest that
this may be a key nonlinear process needed to modulate the amplitudes of the
solar cycles.

We suggest that this change in the meridional flow contributed to the
production of the weak polar fields at cycle 23/24 minimum. While this
suggestion is supported by the modeling work of Cameron and Schüssler (2012),
more detailed modeling work with the actual meridional flow profiles is needed
for confirmation.

##  Appendix A Flow Measurement Methods

Measurements of the meridional flow can be made by a variety of techniques
including direct Doppler, time-distance helioseismology, and magnetic feature
tracking. The different techniques give information about flows at different
depths and about motions of different features. Both the direct Doppler and
time-distance helioseismology methods must first characterize and remove
systematic signals larger than the meridional flow signal itself. Magnetic
feature tracking is not subject to such large systematic errors and has the
distinct advantage of measuring the motions of the features of interest for
magnetic flux transport – magnetic field elements.

The Doppler signal due to the axisymmetric meridional flow is masked by the
much larger signal due to the convective blue shift and by
instrumental/scanning artifacts. The convective blue shift is produced by the
correlation between brightness and radial flow velocities in granules. It
gives a blue shift at disk center that falls off toward the limb and,
depending on spectral line, becomes a red shift near the limb itself. The
convective blue shift signal typically varies by $\sim 500$ m s-1 from disk
center to limb and the signal is known to vary locally in the presence of
magnetic field elements. Various methods have been devised to separated the
convective blue shift signal from the meridional flow signal (Snodgrass, 1984;
Ulrich et al., 1988; Hathaway, 1996) but all require an accurate measurement
of the convective blue shift and all assume that this signal does not vary
with latitude. Imaging artifacts, which (like the convective blue shift and
meridional flow signals) remain relatively fixed on the image can also
introduce systematic errors to the measurement of the meridional flow. Fig. 6
shows examples of these Doppler signals to illustrate the difficulties
entailed in making direct Doppler measurements of the meridional flow. While
this method is sensitive to the meridional flow at high latitudes, it is
insensitive to the meridional flow near the equator.

Figure 6: The direct Doppler signals associated with measuring the meridional
flow are shown here with the same scaling. The convective Blue Shift (left)
has a dynamic range of 500 m s-1 and is show here without any local variations
due to the presence of magnetic field. Imaging artifacts (center, from HMI)
can have a dynamic range of 200 m s-1. The meridional flow (right) has a
dynamic range of only 10-15 m s-1.

Local helioseismology measures the meridional flow by determining small
differences in the sound travel time between north-to-south and south-to-north
moving acoustic waves. Time-Distance helioseismology does this by correlating
signals observed at points separated in latitude. Duvall and Hanasoge (2009)
recently reported on a previously unnoticed systematic center-to-limb
variation in acoustic wave travel times. The source of this signal is still
uncertain but it is clearly seen as an apparent flow away from disk center
that has affected all previous measurements of the meridional flow using this
method. The signal itself can be characterized by measuring the signal away
from disk center along the equator as was done by Zhao et al. (2012). Their
characterization of this signal indicates that it can be several times larger
than the meridional flow signal itself (see their Fig. 2.). As with the direct
Doppler method, this method requires an accurate measurement of a systematic
signal and assumes that this signal does not vary with latitude. This method
is sensitive to meridional flow near the equator and it provides important
depth information but it becomes more uncertain at high latitudes. Even with
high resolution data these measurements rarely extend poleward of
$60^{\circ}$. Furthermore, when using meridional flow profiles from
helioseismology one must chose the appropriate depth for the flux transport
profile.

The feature tracking method using the small magnetic elements has significant
advantages over either direct Doppler or local helioseismology. It can be used
to measure the meridional flow from the equator up to at least $85^{\circ}$
latitude (Rightmire-Upton et al., 2012). It does not require characterizing
and removing any substantial systematic signals and it directly measures the
motions of the features of interest for magnetic flux transport – the magnetic
elements. Conversely, the motions of the magnetic elements are not given by
the motions of the surface plasma. The rotation rate of the small magnetic
elements (Snodgrass, 1983; Komm et al., 1993b) is significantly faster than
the surface Doppler rate – indicating that the magnetic elements are anchored
further down in the surface shear layer.

We should note that Dikpati et al. (2010) have suggested that magnetic element
feature tracking is subject to a large systematic signal associated with the
“diffusion” produced by supergranules. They argue that the diffusive transport
of magnetic elements away from the active latitudes should give a fictitious
flow velocity away from the active latitudes with this method and they
estimated the magnitude of the effect using a 1D (latitudinal) transport model
in which there were no magnetic elements per se. We (Hathaway and Rightmire,
2011) investigated this possibility by transporting magnetic elements in both
latitude and longitude using only supergranular flows and then attempted to
measure this associated fictitious meridional flow. We found that it must be
less than the 1-2 m s-1 noise level in our measurements (Fig. 10 of that
paper). Furthermore, the meridional flow variations we find using magnetic
feature tracking are in the form of an inflow toward the active latitudes -
not the outflow that would be produced according to Dikpati et al. (2010).

###### Acknowledgements.

The authors were supported by a grant from the NASA Living with a Star Program
to Marshall Space Flight Center. The HMI data used are courtesy of the
NASA/SDO and the HMI science team. The SOHO/MDI project was supported by NASA
grant NAG5-10483 to Stanford University. SOHO is a project of international
cooperation between ESA and NASA.

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