#### Chiral dynamics, S-wave contributions and angular analysis in \(D\rightarrow \pi \pi \ell \bar{\nu }\)

Eur. Phys. J. C
Chiral dynamics, S-wave contributions and angular analysis in D → π π ν¯
Yu-Ji Shi 0
Wei Wang 0
Shuai Zhao 0
0 INPAC, Shanghai Key Laboratory for Particle Physics and Cosmology, Department of Physics and Astronomy, Shanghai Jiao-Tong University , Shanghai 200240 , China
We present a theoretical analysis of the D− → π +π − ν¯ and D¯ 0 → π +π 0 ν¯ decays. We construct a general angular distribution which can include arbitrary partial waves of π π . Retaining the S-wave and P-wave contributions we study the branching ratios, forward-backward asymmetries and a few other observables. The P-wave contribution is dominated by ρ0 resonance, and the S-wave contribution is analyzed using the unitarized chiral perturbation theory. The obtained branching fraction for D → ρ ν, at the order 10−3, is consistent with the available experimental data. The S-wave contribution has a branching ratio at the order of 10−4, and this prediction can be tested by experiments like BESIII and LHCb. Future measurements can also be used to examine the π -π scattering phase shift.
1 Introduction
The Cabbibo–Kobayashi–Maskawa (CKM) matrix elements
are key parameters in the Standard Model (SM). They are
essential to understand CP violation within the SM and search
for new physics (NP). Among these matrix elements, |Vcd |
can be determined from either exclusive or inclusive weak
D decays, which are governed by c → d transition, for
example, c → d ν transitions. However, for a general D
decay process it is difficult to extract CKM matrix elements,
because strong and weak interactions may be entangled.
The semi-leptonic D decays are ideal channels to
determine |Vcd |, not only because the weak and strong
dynamics can be separated in these process, but also the clean
experimental signals. Moreover, one can study the
dynamics in the heavy-to-light transition from semi-leptonic D
decays. For leptons do not participate in the strong
interaction, all the strong dynamics is included in the form factors;
thus it provides a good platform to measure the form
factors. The D → ρ form factors have been measured from
D0 → ρ−e+νe and D+ → ρ0e+νe at the CLEO-c
experiment for both charged and neutral channels [
1
]. Because of
the large width of the ρ meson, D → ρ ν¯ is in fact a
quasifour body process D → π π ν¯ . The ρ can be reconstructed
from the P-wave π π mode. However, other π π resonant or
non-resonant states may interfere with the P-wave π π pair,
and thus it is necessary to analyze the S-wave contribution
to D → π π ν¯ .
In addition, the internal structure of light mesons is an
important issue in hadron physics. It is difficult to study
light mesons by QCD perturbation theory due to the large
strong coupling in the low-energy region. On the other hand,
because of the large mass scale, one can establish
factorization for many heavy meson decay processes, thus heavy
mesons like B and D can be used to probe the internal
structure of light mesons [
2,3
]. As mentioned above, D → π π ν¯
can receive contributions from various partial waves of π π .
ρ(770) dominant for D to P-wave π π decay, at the same
time, D meson can decay into S-wave π π through f0(980).
The structure of f0(980) is not fully understood yet.
Analysis of D → π π ν¯ may shed more light on
understanding the nature of f0(980). The BESIII collaboration has
collected 2.93 fb−1 data in e+e− collisions at the energy around
3.773 GeV [4], which can be used to study the semi-leptonic
D decays. Thus it presently is mandatory to make reliable
theoretical predictions. Some analyses of multi-body heavy
meson decays can be found in Refs. [
5–19
], where the final
state interactions between the light pseudoscalar mesons are
taken into account.
In this paper we present a theoretical analysis of D− →
π +π − ν¯ and D0 → π +π 0 ν¯ decays. In Sect. 2, we will
present the results of D → f0 (980) and D → ρ form
factors. We also calculate D to S-wave π π pair form factors
in non-resonance region, the π π form factor will be
calculated by using unitarized chiral perturbation theory. Based
on these results, we present a full analysis on the angular
distribution of D → π π ν¯ . We explore various
distribution observables, including the differential decay width, the
S-wave fraction, forward–backward asymmetry, and so on.
These results will be collected in Sect. 3. The conclusion of
this paper will be given in Sect. 4. The details of the
coefficients in angular distributions are relegated to the appendix.
GF
Heff = √
2
2 Heavy-to-light transition form factors
Feynman diagram for the D → π π −ν¯ decay is shown in
Fig. 1. The lepton can be an electron or a muon, = e, μ.
The spectator quark could be the u or d quark, corresponding
to D0 → π +π 0 −ν¯ and D− → π +π − −ν¯ . Integrating
out the virtual W -boson, we obtain the effective Hamiltonian
describing the c → d transition
Vcd d¯γμ(1 − γ5)c ν¯ γ μ(1 − γ5)
+ h.c., (1)
where GF is the Fermi constant and Vcd is the CKM matrix
element. The leptonic part is calculable using the
perturbation theory, while the hadronic effects are encoded into the
transition form factors.
2.1 D → ρ form factors
For the P-wave π π state, the dominant contribution is
from the ρ(770) resonance. The D → ρ form factors are
parametrized by [
20
]
2V (q2)
ρ( p2, )|d¯γ μc|D( pD) = − m D + mρ
ρ( p2, )|d¯γ μγ5c|D( pD) = 2i mρ A0(q2) ∗ · q qμ
μνρσ ν∗ pDρ p2σ ,
q2
¯
c
l−
ν¯l
¯
d
u/d
Fig. 1 Feynman diagram for the D → ππ −ν decays. The
lep¯
ton could be an electron or a muon, = e, μ. Depending on the
D meson, the spectator could be a u or a d quark, corresponding to
D0 → π+π0 −ν¯ and D− → π+π− −ν
¯
+i (m D + mρ ) A1(q2)
∗μ −
−i A2(q2)
∗ · q
with q = pD − p2, and P = pD + p2. The V (q2), and
Ai (q2)(i = 0, 1, 2) are nonperturbative form factors.
These form factors have been computed in many different
approaches [
21–25
], and here we quote the results from the
light-front quark model (LFQM) [
23,24
] and light-cone sum
rules (LCSR) [25]. To access the momentum distribution in
the full kinematics region, the following parametrization has
been used:
Fi (q2) =
Fi (0)
1 − ai mq22D + bi
q2
m2D
Their results are collected in Table 1. We note that a different
parametrization is adopted in Ref. [
25
], where A3 appears
instead of A0. The relation between A0 and A3 is given by
1
A0(q2) = 2mρ (m D + mρ )
A1(q2)(m D + mρ )2
+ A2(q2)(m2ρ − m2D) − A3(q2)q2 .
2.2 Scalar π π form factor and D to S-wave π π
We first give the D → f0(980) form factor parametrized as
f0( p2)|d¯γμγ5c|D−( pD) = −i F+D→ f0 (q2)
×
Pμ −
m2D − m2f0 qμ + F0D→ f0 (q2) m2D − m2f0 qμ ,
q2 q2
where F+D→ f0 and F0D→ f0 are D → f0 form factors. We
will use LCSR to compute the D → f0(980) transition
form factors with some inputs, and we refer the reader to
Ref. [
26
] for a detailed derivation in LCSR. The meson
masses are fixed to the PDG values m D = 1.870 GeV and
m f0 = 0.99 GeV [
27
]. For quark masses we use mc = 1.27
GeV [
27
] and md = 5 MeV. As for decay constants, we
use f D = 0.21 GeV [
27
] and f f0 = 0.18 GeV [
28
]. The
threshold s0 is fixed at s0 = 4.1 GeV2, which should
correspond to the squared mass of the first radial excitation of
D. The parameters Fi (0), ai and bi are fitted in the region
−0.5 GeV2 < q2 < 0.5 GeV2, and the Borel parameter M 2
is taken to be (6 ± 1) GeV−2. With these parametrizations,
we give the numerical results in Table 2.
In the region where the two pseudo-scalar mesons strongly
interacts, the resonance approximation fails and thus has to
be abandoned. One of the such examples is the S-wave partial
+ m2D −q2m2ππ qμF0D→ππ (m2ππ , q2) .
The Watson theorem implies that phases measured in π π
elastic scattering and in a decay channel in which the π π
system has no strong interaction with other hadrons are equal
modulo π radians. In the process we consider here, the lepton
pair ν¯ indeed decouples from the π π final state, and thus
the phases of D to scalar π π decay amplitudes are equal to
π π scattering with the same isospin. It is plausible that
(π π )S|d¯ c|D
∝ Fππ (m2ππ ),
where the scalar form factor is defined as
0|d¯d|π +π −
= B0 Fππ (m2ππ ),
where B0 = (1.7 ± 0.2) GeV [
10
] is the QCD condensate
parameter.
An explicit calculation of these quantities requires
knowledge of generalized light-cone distribution amplitudes
(LCDAs) [30]. The twist-3 one has the same asymptotic form
with the LCDAs for a scalar resonance [31]. Inspired by this
similarity, we may plausibly introduce an intuitive matching
between the D → f0 and D → (π π )S form factors [
7
]:
FiD→ππ (m2ππ , q2)
Fππ (m2ππ )FiD→ f0 (q2).
(9)
B0
1
f f0
It is necessary to stress at this stage that the Watson
theorem does not strictly guarantee that one may use Eq. (9).
Instead it indicates that, below the opening of inelastic
channels the strong phases in the D → π π form factor and π π
(6)
(7)
(8)
0.78 ± 0.24
scattering are the same. First above the 4π or K K¯
threshold, additional inelastic channels will also contribute. The
K K¯ contribution can be incorporated in a coupled-channel
analysis. As a process-dependent study, it has been
demonstrated that states with two additional pions may not give
sizable contributions to the physical observables [32]. Secondly,
some polynomials with nontrivial dependence on mππ have
been neglected in Eq. (9). In principle, once the generalized
LCDAs for the (π π )S system are known, the D → π π form
factor can be straightforwardly calculated in LCSR and thus
this approximation in the matching equation can be avoided.
On the one side, the space-like generalized parton
distributions for the pion have been calculated at one-loop level in the
chiral perturbation theory (χ PT) [33]. The analysis of
timelike generalized LCDAs in χ PT and the unitarized
framework is in progress. On the other side, the γ γ ∗ → π +π −
reaction is helpful to extract the generalized LCDAs for the
(π π )S system [34,35]. The experimental prospects at
BEPCII and BELLE-II in the near future are very promising.
In the kinematic region where the π is soft, the crossed
channel from D + π → π will contribute as well and this
crossed channel would modify Eq. (9) by an inhomogeneous
part. For the analogous decay of K or B mesons, it has been
taken into account either dynamically in terms of phase shifts
(in the case of the kaon decay) [36] or approximately in terms
of a pole contribution (in the case of the B meson decay) [
15
].
However, if both pions move fast, the D–π invariant mass is
far from the D∗ pole and this contribution is negligible. In
this case, the transition amplitude for the D to 2-pion form
factor can be calculated in light-cone sum rules [
7
]. This will
lead to the conjectured formula in Eq. (9).
The scalar π π form factor can be handled using the
unitarized chiral perturbation theory. In the following, we will give
a brief description of this approach. In terms of the isoscalar
S-wave states
1
|π π I=0 = √3
1
|K K¯ I=0 = √2
1
π +π − + √
6
1
K + K − + √
2
0 0 ,
π π
K 0 K¯ 0 ,
2B0 F1n/s (s) = 0|n¯ n/s¯s|π π I=0,
the scalar form factors for the π and K mesons are defined
as
√
(12)
(10)
(11)
√2B0 F2n/s (s) = 0|n¯ n/s¯s|K K¯ I=0,
where s = m2ππ . The n¯ n = (u¯u + d¯d)/√2 denotes the
nonstrange scalar current, and the notation (π = 1, K = 2) has
been introduced for simplicity. With the above notation, we
have
Fππ (m2ππ ) =
Expressions have already been derived in χ PT up to
next-toleading order [37–40]:
μi = 32πm2i2 f 2 ln mμ2i2 ,
F (s) = [I + K (s) g(s)]−1 R(s)
1
Jiri (s) = 16π 2
m2
i
μ2
1 − log
− σi (s) log
σi (s) + 1
σi (s) − 1
with σi (s) = 1 − 4mi2/s. It is interesting to note that
the next-to-next-to-leading order results can also be found
in Refs. [41,42]. Imposing the unitarity constraints, the
scalar form factor can be expressed in terms of the algebraic
coupled-channel equation
= [I − K (s) g(s)] R(s) + O( p6),
where R(s) has no right-hand cut and in the second line,
the equation has been expanded up to NLO in the chiral
expansion. K (s) is the S-wave projected kernel of
mesonmeson scattering amplitudes that can be derived from the
leading-order chiral Lagrangian:
The loop integral can be calculated either in the
cutoffregularization scheme with qmax ∼ 1 GeV being the cutoff
(cf. Erratum of Ref. [
43
] for an explicit expression) or in
dimensional regularization with the MS subtraction scheme.
In the latter scheme, the meson loop function gi (s) is given
by
1 − log
− σi (s) log
σi (s) + 1
σi (s) − 1
1
Jiri (s) ≡ 16π 2
= −gi (s).
The expressions for the Ri are obtained by matching the
unitarization and chiral perturbation theory [
44,45
]:
R1n(s) =
23 1 + μπ − μ3η + 16fm22π 2Lr8 − Lr5
r 2m2K + 3m2π
+8 2Lr6 − L4 f 2
+ 8f s2 Lr4 + 4f s2 Lr5
(13)
(14)
(15)
(16)
(17)
(18)
(20)
m2π
− 288π 2 f 2 1 + log
m2π
− 72π 2 f 2 1 + log
m2K
+ 72π 2 f 2 1 + log
m2
i
μ2
m2
η
μ2
m2
η
μ2
m2
η
μ2
,
16Lr6 4m2K + m2π + 32fL2r8 m2K + 3 μη
2
+ f 2
9s − 8m2K
18 f 2
Jηrη(s) + 43fs2 J Kr K (s).
Here the Lri are the renormalized low-energy constants, and
f is the pion decay constant at tree level. The μi and Jiri are
defined as follows:
(21)
(22)
(23)
(24)
(25)
R1s(s) =
√
23 16fm22π 2Lr6 − Lr4 + 8f s2 Lr4
R2n(s) = √12 1 + 8fL2r4 2s − 6m2K − m2π + 4fL2r5 s − 4m2K
16Lr6 6m2K + m2π + 32fL2r8 m2K + 3 μη
2
+ f 2
R2s(s) = 1 + f 2 4Lr5 s − 4m2K
8Lr4 s − 4m2K − m2π + f 2
16Lr6 4m2K + m2π + 32fL2r8 m2K + 3 μη
2
+ f 2
m2K
+ 36π 2 f 2 1 + log
m2
η
μ2
With the above formulas and the fitted results for the
lowenergy constants Lri in Ref. [
45
] (evolved from Mρ to the
scale μ = 2qmax/√e), we show the non-strange π π form
factor in Fig. 2. The modulus, real part and imaginary part
are shown as solid, dashed and dotted curves. As the figure
shows, the chiral unitary ansatz predicts a form factor F n
1
1.5
1.0
with a zero close to the K¯ K threshold. This feature has been
extensively discussed in Ref. [
46
].
3 Full angular distribution of D → π π ν¯
In this section, we will derive a full angular
distribution of D → π π ν¯ . For the literature, one may consult
Refs. [
47, 48
]. We set up the kinematics for the D− →
π +π − ν¯ as shown in Fig. 3, which can also be used for
D0 → π +π 0 ν¯ . The π π moves along the z axis in the D−
rest frame. θπ + (θ ) is defined in the π π (lepton pair) rest
frame as the angle between z-axis and the flight direction
of π + ( −), respectively. The azimuth angle φ is the angle
between the π π decay and lepton pair planes.
Decay amplitudes for D → π π ν¯ can be divided into
several individual pieces and each of them can be expressed
in terms of the Lorentz invariant helicity amplitudes. The
amplitude for the hadronic part can be obtained by the
evaluation of the matrix element:
Aλ =
i GF
N f0/ρ √2
Vc∗d μ∗(h) π π |c¯γ μ(1 − γ5)d| D ,
(26)
where μ(h) is an auxiliary polarization vector for the lepton
pair system and h = 0, ±, t , N f0/ρ = √λq2βl /(96π 3m3D ),
βl = 1−mˆ l2 and mˆ l = ml / q2. |Vcd | is taken to be 0.22 [
27
].
The functions Ai can be decomposed into different partial
waves,
mmπρπ 11 ++ ((RR||qq0||))22 ,
and the Blatt–Weisskopf parameter R = (2.1 ± 0.5 ±
0.5) GeV−1 [
49
].
The spin-0 final state has only one polarization state and
the amplitudes are
i MD ( f0, 0) = N1i
i MD ( f0, t ) = N1i
√λ
q2
F1(q2) ,
q2
m2D − m2f0 F0(q2) ,
(29)
(30)
with N1 = i GF Vc∗d /√2. For mesons with spin J ≥ 1, the
π +π − system can be either longitudinally or transversely
polarized and thus we have the following form:
i MD (ρ , 0) = −
αLJ N1i
2mρ
q2
× (m2D − m2ρ − q2)(m D + mρ ) A1
A2 ,
i MD(ρ , ±) = −βTJ N1i (m D + mρ ) A1 ± m D√+λmρ V ,
i MD(ρ , t ) = −αLJ i N1
√λ
q2
A0.
The αLJ and βTJ are products of the Clebsch–Gordan
coefficients
αLJ = C1J,,00;J −1,0C1J,−0;1J,0−2,0 · · · C12,,00;1,0,
βTJ = C1J,,11;J −1,0C1J,−0;1J,0−2,0 · · · C12,,00;1,0.
For the sake of convenience, we define
1
i MD(ρ , ⊥ /||) = √ [i MD(ρ , +) ∓ i MD(ρ , −)],
2
i MD(ρ , ⊥) = −iβTJ √2N1
√λV
,
i MD(ρ , ||) = −iβTJ √2N1 (m D + mρ ) A1 .
(34)
Using the generalized form factor, the matrix elements for
D decays into the spin-0 non-resonating π π final state are
given as
A00 =
At0 =
1
N2i mππ
1
N2i mππ
√λ
q2 F1ππ (m2ππ , q2) ,
m2D − m2ππ F0ππ (m2ππ , q2) ,
q2
N2 = N1 Nρ ρπ /(16π 2), with ρπ = 1 − 4m2π /m2ππ .
The above quantities can lead to the full angular
distributions
d5
dm2ππ dq2d cos θπ+ d cos θl dφ
3
= 8 I1(q2, m2ππ , θπ+ )
+I2(q2, m2ππ , θπ+ ) cos(2θ )
+I3(q2, m2ππ , θπ+ ) sin2 θ cos(2φ)
+I4(q2, m2ππ , θπ+ ) sin(2θ ) cos φ
+I5(q2, m2ππ , θπ+ ) sin(θ ) cos φ
+I6(q2, m2ππ , θπ+ ) cos θ
+I7(q2, m2ππ , θπ+ ) sin(θ ) sin φ
+I8(q2, m2ππ , θπ+ ) sin(2θ ) sin φ
+I9(q2, m2ππ , θπ+ ) sin2 θ sin(2φ) .
For the general expressions of Ii , we refer the reader to the
appendix and to Refs. [
48,50
] for the formulas with the S-,
P- and D-waves. In the following, we shall only consider the
S-wave and P-wave contributions and thus the above general
expressions are reduced to:
I6 = 4 83π sin2 θπ+ Re[ A|1| A1⊥∗] + mˆ l2 41π Re[ At0 A00∗]
+mˆ l2 43π cos2 θπ+ Re[ At1 A01∗]
√3
I7 = 4 4√2π
3
+ 4√2π
I8 = 2βl 4√√23π sin θπ+ Im A00 A1⊥∗
3
+ 4√2π
I9 = 2βl 83π sin2 θπ+ Im A1⊥ A|1|∗ .
sin θπ+ cos θπ+ Im A10 A1⊥∗ ,
sin θπ+ Im[ A00 A|1|∗] − mˆ l2Im[ At0 A1⊥∗]
sin θπ+ cos θπ+ Im[ A01 A|1|∗] − mˆ l2Im[ At1 A1⊥∗]
Since the phase in P-wave contributions arise from the
lineshape which is the same for different polarizations, the I9
term and the second line in the I7 are zero.
3.1 Differential and integrated decay widths
Using the narrow width approximation, we obtain the
integrated branching fraction:
B(D− → ρ0e−ν¯) = (2.24 ± 0.09) × 10−3/(2.16 ± 0.36)
×10−3(LFQM/LCSR),
(37)
(38)
B(D− → ρ0μ−ν¯)
= (2.15 ± 0.08) × 10−3/(2.06 ± 0.35) × 10−3(LFQM/LCSR),
(39)
B(D¯ 0 → ρ+e−ν¯)
= (1.73 ± 0.07) × 10−3/(1.67 ± 0.27) × 10−3(LFQM/LCSR),
(40)
where theoretical errors are from the heavy-to-light transition
form factors. These theoretical results are in good agreement
with the data [
27
]:
B(D− → ρ0e−ν¯ ) = (2.18+−00..1275) × 10−3,
B(D− → ρ0μ−ν¯ ) = (2.4 ± 0.4) × 10−3,
B(D¯ 0 → ρ+e−ν¯ ) = (1.77 ± 0.16) × 10−3.
The starting point for detailed analysis of D → π π ν¯ is
to obtain the double-differential distribution d2 /dq2dm2ππ
after performing integration over all the angles
0.4
0.6
1.0
0.8
GeV
where apparently in the massless limit for the involved lepton,
the total normalization for angular distributions changes to
the sum of the S-wave and P-wave amplitudes
dq2dm2ππ = | A00|2 + | A01|2 + | A||| + | A1⊥|2.
1 2
In Fig. 4, we give the dependence of branching fraction on
mππ in the D− → π +π −e−ν¯e process. The solid, dashed,
and dotted curves correspond to the total, S-wave and
Pwave contributions. For the S-wave contribution, there is no
resonance around 0.98 GeV, and theoretically, this should be
a dip.
Due to the quantum number constraint, the process D¯ 0 →
π +π 0 ν¯ receives only a P-wave contribution and D− →
π 0π 0 ν¯ is generated by the S-wave term.
To match the kinematics constraints implemented in
experimental measurements, one may explore the generic
observable with m2ππ integrated out:
O =
(mρ +δm )2
(mρ −δm )2
dO
dm2ππ dm2ππ .
We use the following choice in our study of D → π π ν¯ :
δm =
ρ .
In the narrow width-limit, the integration of the lineshape is
conducted as
dm2ππ |Lρ (m2ππ )|2 = B(ρ0 → π −π +) = 1.
(41)
However, with the explicit form given in Eq. (29), we find
that the integration
is below the expected value. On the other hand, the integrated
S-wave lineshape in this region is
dm2ππ |Lρ (m2ππ )|2 = 0.70
dm2ππ |L S(m2ππ )|2 = 0.37,
which is smaller but at the same order. Integrated from mρ −
ρ to mρ + ρ , we have
B(D− → ρ0(→ π +π −)e−ν¯)
= (1.57 ± 0.07) × 10−3/(1.51 ± 0.26) × 10−3 (LFQM/LCSR),
(51)
Differential decay widths for D → π π ν¯ are given in Fig. 5,
with = e in panel (a) and = μ in panel (b). The
q2dependent ratio Rπμπ/e is given in panel (c). Errors from the
form factors and QCD condensate parameter B0 are shown
(49)
(50)
(55)
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
q2 GeV2 (a)
Fig. 5 Differential decay widths for the D− → π +π − ν¯ with = e
in (a) and = μ in (b). The q2-dependent ratio Rπμπ/e as defined in
Eq. (55) is given in (c). The dashed and dotted curves are produced
using the LFQM and LCSR results for D → ρ form factors. Errors
from the form factors are shown as shadowed bands, and most errors
cancel in the ratio Rπμπ/e given in (c)
2 1.5
V
e
G
30 1.0
1
2
dq 0.5
B
d
0.0
2 1.5
V
e
3G1.0
0
1
2
dq 0.5
B
d
0.0
as shadowed bands, and most errors cancel in the ratio Rπμπ/e
given in panel (c).
3.2 Distribution in θπ +
We explore the distribution in θπ + :
d3
dq2dm2π π d cos θπ +
π
= 2 (3I1 − I2)
Compared to the distribution with only P-wave contribution,
namely D → ρ (→ π π ) ν¯ , the first two lines of Eq. (56)
are new: the first one is the S-wave π π contribution, while
the second line arises from the interference of S-wave and
P-wave. Based on this interference, one can define a forward–
backward asymmetry for the involved pion,
π
AFB ≡
0
√3
= 2
1
−
0
−1
(2 + mˆ l2)Re[ A00 A10∗] +
d3
d cos θπ + dq2dm2π π d cos θπ +
3√3
2
mˆ l2Re[ At0 At1∗].
We define the polarization fraction at a given value of q2
and m2π π :
(1 + mˆ l2/2)| A00|2 + 3/2mˆ l2| At0|2 ,
d2 /(dq2dm2ππ )
(1 + mˆ l2/2)(| A01|2 + | A||| + | A1⊥|2) + 3/2mˆ l2| At1|2
1 2
d2 /(dq2dm2ππ )
FS (q2, m2ππ ) =
FP (q2, m2ππ ) =
,
(58)
and also
FL (q2, m2π π )
(1 + mˆ l2/2)| A01(q2, m2π π )|2 + 3/2mˆ l2| At1|2
= (1 + mˆ l2/2)(| A0| + | A||| + | A⊥| ) + 3/2mˆ l2| At1|2
1 2 1 2 1 2
.
By definition, FS + FP = 1.
In Fig. 6, we give our results for the S-wave fraction FS
(panel (a)), longitudinal polarization fraction FL in P-wave
contributions (panel (b)) and the asymmetry AπF B (panel
(c)). Only the curves for the light lepton e are shown since
the results for the μ lepton are similar. These observables and
the following ones are defined by the integration over m2π π ;
for instance,
FS (q2) =
dm2π π [(1 + mˆ l2/2)| A0| + 3/2mˆ l2| At0|2]
0 2
dm2π π d2 /(dq2dm2π π )
(59)
(60)
and likewise for the others.
3.3 Distribution in θl and forward–backward asymmetry
Integrating over θπ + and φ, we have the distribution:
d3
dq2dm2ππ d cos θl =
3 m2
= 4 ˆ l | At0|2 + | At1|2
3π
4 d cos θπ+
× (I1 + I2 cos(2θl ) + I6 cos θl )
3
+ 2 cos θl
×
Re[ A|1| A1⊥∗] + mˆ l2Re[ At0 A00∗ + At1 A01∗]
3
+ 4
3
+ 8 (1 + mˆ l2) + (1 − mˆ l2) cos2 θl
1 − (1 − mˆ l2) cos2 θl | A00|2 + | A01|2
×(| A||| + | A1⊥|2).
1 2
(61)
FL0.6
1.2
1.0
0.8
0.4
0.2
0.0
0.00
0.05
LFQM
LCSR
B 0.2
π AF
Fig. 7 Same as Fig. 5 but for
the asymmetry AlF B in the
D → π π ν¯
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
q2 GeV2 (a)
Fig. 6 Same as Fig. 5 but for the S-wave contributions (a) and the
longitudinal polarizations in P-wave contributions (b) to the D → π π ν¯ ,
and the forward–backward asymmetry AπF B (c). Notice that, for the
AπF B , there is a sign ambiguity arising from the use of Watson theorem.
These diagrams are for the light lepton e, while the results for the μ
lepton are similar
LFQM
LCSR
Since the complex phase in the P-wave amplitudes comes
from the Breit–Wigner lineshape, the coefficient cφs
vanishes.
Numerical results for the normalized coefficients using the
two sets of form factors are shown in Fig. 8. The coefficients
bφc and bφs contain a very small prefactor, 3√3/(32√2π ) ∼
0.037, and thus are numerically tiny, as shown in this figure.
The cφc is also small due to the cancellation between the | A⊥|2
and | A|||2.
3.5 Polarization of μ lepton
In this work, we also give the polarized angular distributions
as
0.0020
Fig. 8 Same as Fig. 5 but for
the normalized coefficients in
the φ distributions of the
D− → π +π − ν¯ . The left
panels (a, c, e) are for the light
lepton e, while the right panels
(b, d, f) are for the μ lepton
0.0
– 0.2
– 0.4
μ
A
– 0.6
Fig. 9 Same as Fig. 5 but for the polarization distribution of D →
π π μν¯μ. Theoretical errors are negligible
+ I2(λμ) cos(2θl ) + I3(λμ) sin2 θl cos(2φ)
=
Aλμ (q2, m2ππ ) =
and we show the numerical results in Fig. 9.
3.6 Theoretical uncertainties
Before closing this section, we will briefly discuss the
theoretical uncertainties in this analysis. The parametric errors in
heavy-to-light transition form factors and QCD condensate
parameter B0 have been included in the above. As one can
see, these uncertainties are sizable to branching fractions and
other related observables, but are negligible in the ratios like
Rπμπ/e. This is understandable, since most uncertainties will
cancel in the ratio.
For the heavy-to-light form factors, we have used the
LCSR and LFQM results. In LCSR, the theoretical accuracy
for most form factors is at leading order in αs . An analysis of
Bs → f0 [
26
] has indicated the NLO radiative corrections to
form factors may reach 20%. The radiative corrections are,
in general, channel-dependent but should be calculated in a
high precision study. It should be pointed out that radiative
corrections in the light-front quark model is not controllable.
A third type of uncertainties resides in the scalar π π
form factor. In this work, we have used the unitarized results
from Refs. [
44,45
], where the low-energy constants (Li s) are
obtained by fitting the J /ψ decay data. A Muskhelishvili–
Omnès formalism has been developed for the scalar π π form
factor in Ref. [18]. Compared to the results in Ref. [
18
], we
find an overall agreement in the shape of the non-strange
π π form factor, but the modulus from Ref. [
18
] is about
20% larger. This would induce about 40% uncertainties to
the branching ratios of the D → π π ν¯ , while the results
for the ratio observables are not affected.
Finally, the Watson theorem does not always guarantee
the use of Eq. (9), the matching of D → π π form factor and
D → f0 form factors. As we have discussed in Sect. 2, such
an approximation might be improved in the future.
4 Conclusions
In summary, we have presented a theoretical analysis of the
D− → π +π − ν¯ and D¯ 0 → π +π 0 ν¯ decays. We have
constructed a general angular distribution which can include
arbitrary partial waves of π π . Retaining the S-wave and
Pwave contributions we have studied the branching ratios,
forward–backward asymmetries and a few other observables.
The P-wave contribution is dominated by ρ0 resonance, and
the S-wave contribution is analyzed using the unitarized
chiral perturbation theory. The obtained branching fraction for
D → ρ ν, at the order 10−3, is consistent with the available
experimental data, while the S-wave contribution is found
to have a branching ratio at the order of 10−4, and this
prediction can be tested by experiments like BESIII and LHCb.
The BESIII collaboration has accumulated about 107 events
of the D0 and will collect about 3 fb−1 data at the
center-ofmass √s = 4.17 GeV to produce the Ds+ Ds− [
51,52
]. All
these data can be used to study the charm decays into the f0
mesons. In addition, sizable branching fractions also indicate
a promising prospect at the ongoing LHC experiment [53],
the forthcoming Super-KEKB factory [
54
] and the
underdesign Super Tau-Charm factory. Future measurements can
be used to study the π –π scattering phase shift.
Acknowledgements We thank Jian-Ping Dai, Liao-Yuan Dong,
HaiBo Li and Lei Zhang for useful discussions. This work is supported
in part by National Natural Science Foundation of China under Grant
Nos. 11575110, 11655002, Natural Science Foundation of Shanghai
under Grant No. 15DZ2272100 and No. 15ZR1423100, by the Young
Thousand Talents Plan, and by Key Laboratory for Particle Physics,
Astrophysics and Cosmology, Ministry of Education.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
Appendix A: Angular coefficients
In the angular distribution, the coefficients have the form
I1 = (1 + mˆ l2)| A0|2 + 2mˆ l2| At |2 + (3 + mˆ l2)/2(| A⊥|2 + | A|||2)
I2 = −βl | A0|2 + βl /2(| A⊥|2 + | A|||2),
I3 = βl (| A⊥|2 − | A|||2), I4 = 2βl [Re( A0 A|∗|)],
I5 = 4[Re( A0 A∗⊥) − mˆ l2Re( At A|∗|)],
I6 = 4 Re( A|| A∗⊥) + mˆ l2Re( At A0∗) ,
I7 = 4 Im( A0 A|∗|) − mˆ l2Im( At A∗⊥) ,
I8 = 2βl [Im( A0 A∗⊥)], I9 = 2βl [Im( A⊥ A|∗|)].
(A1)
Substituting the expressions for Ai into the above equation,
we obtain the general expressions
I1(q2, m2ππ , θπ+ )
|YJ0(θπ+ , 0)|2
J=0,...
= (1 + mˆ l2)|A0J |2 + 2mˆ l2|AtJ |2
+2
+2mˆ l2 cos(δtJ − δtJ )| AtJ || AtJ |
+
× 2 cos(δ ⊥J − δ ⊥J )| A ⊥J|| A ⊥J | ,
× | A0J A|J| ∗| sin(δ0J − δ|J| ) − mˆ l2| AtJ A ⊥J ∗| sin(δtJ − δ ⊥J ) ,
I8(q2, m2ππ , θπ+ ) = 2βl
I9(q2, m2ππ , θπ+ ) = 2βl
J =1,... J =1,...
× YJ−1(θπ+ , 0)YJ−1(θπ+ , 0)| A ⊥J A|J| ∗| sin(δ ⊥J − δ|J| ) .
(A10)
(A2)
(A3)
(A4)
(A5)
(A6)
(A7)
(A8)
(A9)
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