Published for SISSA by
Springer
Received: August 16, 2011 Accepted: September 13, 2011 Published: September 27, 2011
D.V. Matvienko, A.S. Kuzmin and S.I. Eidelman Budker Institute of Nuclear Physics, SB RAS, 11, Lavrentieva prospect, Novosibirsk, Russia Novosibirsk State University, 2, Pirogova street, Novosibirsk, Russia
E-mail:
[email protected],
[email protected],
[email protected] ¯ 0 → D ∗+ ωπ − Abstract: We suggest a parameterization of the matrix element for B decay using kinematic variables convenient for experimental analysis. The contributions of intermediate ωπ- and D ∗∗ -states up to spin 3 have been taken into account. The angular distributions for each discussed hypothesis have been obtained and analysed using MonteCarlo simulation. Keywords: B-Physics, Heavy Quark Physics ArXiv ePrint: 1108.2862
c SISSA 2011
doi:10.1007/JHEP09(2011)129
JHEP09(2011)129
¯ 0 → D ∗+ωπ − decay A model of B
Contents 1
2 The general method
2
3 ωπ-resonances
5
4 D ∗∗-resonances
8
5 Results
8
6 Decay chain simulation
13
7 Conclusion
20
A The phase integral for ω-decay
20
1
Introduction
The discovery of excited D-states (referred to as D∗∗ -states) stimulates interest in their spectroscopy and D ∗∗ → D (∗) π decay properties. There are four P -wave states, which + + + P ∗ P P ′ are usually labeled D0∗ (JjPq = 0+ 1/2 ), D1 (Jjq = 11/2 ), D1 (Jjq = 13/2 ), D2 (Jjq = 23/2 ), where J is the spin of the meson and jq is the total angular momentum of a light quark q = (u, d), which is the sum of the orbital momentum l and the light quark spin sq . In the heavy quark limit, the angular momentum jq is a good quantum number. Conservation of parity and angular momentum imposes constraints on the strong decays of the D ∗∗ to D(∗) π. Two states with jq = 1/2 decay to the D (∗) π-state in S-wave while two other with jq = 3/2 decay in D-wave. Since the decay width Γ ∼ Q2L+1 , where Q is the magnitude of the daughter particle momentum, L is the orbital momentum between decay products, and Q is small, D1 and D2∗ have small decay width of about 20 MeV, but D0∗ and D1′ are expected to be quite broad with decay width of about 300 MeV [1, 2]. A further study of these states will allow a more in-depth comparison to be made with theoretical predictions such as Heavy Quark Effective Theory (HQET) [3, 4] and QCD sum rules [5]. The last experimental studies of D∗∗ mesons were performed in B − → ¯ 0 → D0 π + π − [8] decays. These states have also been studied D(∗)+ π − π − [6, 7] and B in semileptonic B-decays [9]. Thus, understanding of their properties is significant for reducing uncertainties in the measurement of semileptonic decays and determination of the Cabibbo-Kobayashi-Maskawa (CKM) [10, 11] matrix elements |Vcb | and |Vub |. D∗∗ -states can be also produced in other hadronic B decays, e.g., B → D ∗ ωπ. Here, D∗∗ production is described by the W vertex instead of the transition Isgur-Wise functions [3], which describe these states in the D(∗) ππ modes. This channel was first observed
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1 Introduction
2
The general method
A weak B(0− ) → R(J P )1− decay amplitude (for J > 0) includes three independent terms while a strong R(J P ) → 1− 0− amplitude can have one or two independent terms. We can parameterize a decay matrix element using a set of different independent bases. In general, we can use the basis of covariant amplitudes or helicity basis etc. Since the real particles D1 and D1′ are expected to be close to the pure jq = 3/2- and jq = 1/2-states and their decays have particular orbital momenta, it is convenient to use the basis of amplitudes describing decay with fixed angular orbital momenta in the B and resonance rest frames. In this paper we use an isobar model formulation in which our decay is described by a coherent sum of a number of quasi-two-body amplitudes. The amplitudes can be subdivided into two channels. The effective Hamiltonian for Cabibbo-favored decays can be reduced to the color-favored and color-suppressed forms [14, 15]: GF ∗ HCF = √ Vcb Vud (a1 (¯ cΓµ b)(¯ uΓµ d) + C2 Hw8 ), 2 GF ∗ ˜ 8 ), HCS = √ Vcb Vud (a2 (¯ cΓµ d)(¯ uΓµ b) + C1 H w 2
(2.1)
where GF is the Fermi constant, C1 and C2 are the Wilson coefficients and Γµ = γµ (1−γ5 ). The coefficients a1 = C1 + C2 /N and a2 = C2 + C1 /N , where N is an effective number P a ˜ 8 = 1 P8 (¯ cλa Γµ b)(¯ uλa Γµ d) and H uλa Γµ b) of colors. The terms Hw8 = 21 8a=1 (¯ w a=1 cλ Γµ d)(¯ 2 (λa are the Gell-Mann matrices), involving color-octet currents, generate non-factorized contributions. The other non-factorization source is the non-factorized matrix element of the product of the color-singlet currents. It includes loop current-current terms. The colorfavored and color-suppressed channels are shown in figure 1. We show tree diagrams only, however, not all the intermediate states are described by them. In this paper we do not apply the factorization method but consider all intermediate resonant contributions up to spin 3 allowed by the momentum-parity conservation. The color-favored term receives a 1
In the case of decays with more than three particles in the final state, the term Dalitz plot is used in a general sense to refer to the distribution of the chosen degrees of freedom used to describe the decay.
–2–
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by the CLEO [12] and BaBar [13] collaborations, the latter finding an enhancement in D ∗ π mass due to the broad D1 (2430)0 -state, representing a P -wave of a D meson. Let us note that light mesons decaying to the ωπ final state (e.g., ρ(1450), b1 (1235) and their excitations) appear in the color-favored mode of this process. Thus, a possible contribution of these resonant structures to the total branching fraction can be measured. The ρ(1450)-resonance, dominant in this mode, was observed by both collaborations [12, 13], but the b1 (1235)-state was not observed in this channel. An amplitude of three-body decay can be written as a sum of the contributions corresponding to the quasi-two-body resonances [6–8]. Analysing experimental data one has to determine relative amplitudes and phases of different intermediate states. To do this, one needs the amplitudes expressed via kinematic variables convenient for Dalitz plot analysis.1 These expressions can be used for optimization of selection criteria and creation of efficient Monte-Carlo generators.
_ u
D* *
d
_ u
c b
b
c
_ B0
+ D*
d
_ B0 _ d
a)
b)
Figure 1. a) Color-favored and b) color-suppressed channel.
contribution from the ωπ-resonances, e.g., ρ(1450) and b1 (1235). Since these resonances are broad, this channel allows factorization to be precisely tested [16]. The color-suppressed term receives a contribution from the D∗∗ -states, which are P - and D-wave excitations of the c¯ u states. Let us consider briefly the spectroscopy of the D-wave c¯ u excitations. We have JjPu = − − − P P P 1− 3/2 -, Jju = 23/2 -, Jju = 25/2 -, and Jju = 35/2 states. Again, as discussed above, two states with ju = 3/2 decay to the D(∗) π-state in P -wave and two other with ju = 5/2 decay in F -wave. Observable c¯ u-states with the same J P = 1+ (J P = 2− ) quantum numbers are two linear combinations of pure ju = 1/2 (ju = 3/2)- and ju = 3/2 (ju = 5/2)-states. Thus, the physical D1 and D1′ -states are as follows: |D1 > = sin ϑ1 |ju = 1/2 > + cos ϑ1 e−iϑ2 |ju = 3/2 > ,
|D1′ > = cos ϑ1 |ju = 1/2 > − sin ϑ1 eiϑ2 |ju = 3/2 > ,
where ϑ1 and ϑ2 are mixing angles. Let us discuss kinematic properties of the considered process. In the final state we have six particles, namely, D 0 and π + from the D∗+ decay, π + , π − and π 0 from the ω ¯ 0 decay. The B ¯ 0 decay is described by two invariant masses decay and π − from the B squared of the D∗ π (m2D∗ π ) and ωπ (m2ωπ ) systems, the one corresponding to resonance mass labeled as q 2 . The ω decay is described by five variables. We use invariant masses squared M02 = (P+ +P− )2 and M+2 = (P+ +P0 )2 (here Pi is a 4-momentum of the pion π i from the ω decay, i = ±, 0),2 the azimuthal angle of the π 0 in the ω decay plane, and two angles (polar θ and azimuthal φ) for a vector ~n normal to the ω decay plane. Let us note that the ω → πππ decay proceeds through two mechanisms. The first one involves an intermediate ρ-meson. Experimental studies of the e+ e− → 3π reaction have confirmed the Gell-Mann-Sharp2
The ω invariant mass squared is p2 = (P+ + P− + P0 )2 .
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_ d
D0 b
p+
x
wpp-
Figure 2. Complete visual definition of the angles for the ωπ-resonances. The angles θ and φ are defined in the ω rest frame, the angles β and ψ are defined in the D∗ rest frame and the angle ξ is defined in the ωπ rest frame.
Wagner suggestion [17] that the ω → 3π transition is dominated by this contribution. The second mechanism represents the non-resonant contribution. This contact contribution can not be excluded because interference between these mechanisms leads to a sizeable effect in the decay rate. The D∗+ decay is described by two variables. We use polar β and azimuthal ψ angles for the D 0 momentum in the D∗+ rest frame. For further applications we assume the width of the D∗ -meson to be negligible (ΓD∗+ ≈ 0.1 MeV ≪ mD∗+ − (mD0 + mπ+ ) ≈ 10 MeV). To describe the intermediate resonance decay, we use polar ξ and azimuthal ζ angles for the daughter particle momentum in the resonance rest frame. The polar angle ξ is expressed via the invariant mass squared m2D∗ π for the ωπ-states and m2ωπ for the D∗ π-states. Moreover, the matrix element does not depend on the azimuthal angle ζ for the ωπ- as well as for the D ∗ π-states. A further definition of angles depends on the decay channel. Figure 2 shows the decay scheme and definition of the angles for the ωπ-resonances. Figures 3 and 4 define these angles using momentum variables for the ωπ- and D ∗ πresonances, respectively. The notations are as follows: the variables p, Q, l, q are the four-momenta of the ω-, D ∗ -, D-meson and an intermediate resonance, respectively, while p, Q, l, q are the magnitudes of their three-momenta in the mother particle rest frames. In figures 3, 4 the directions of these momenta define angular variables θ and φ in the ω rest frame, β and ψ in the D ∗ rest frame and ξ in the resonance rest frame. In this paper each compound particle is described by a relativistic Breit-Wigner (BW) with a q 2 -dependent width. Such an approach is not exact since it does not take into account final state interactions and is neither analytic nor unitary. Nevertheless, it describes the main features of the amplitude behaviour and allows one to find and distinguish the contributions of different quasi-two-body intermediate states. Thus, the denominator of the BW propagator is: DR (q 2 ) = q 2 − m2R + imR ΓR (q 2 ). (2.2)
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D*+
y _ B0
n f q w
` q(wp)
Q(D*)
n X
x
p(w)
Y
Q(D*)
n
q
` X`` fw
Y
` yD*
X
a)
q(wp)
b)
``
Y
c)
Figure 3. Definition of the angles for the ωπ-resonances. Color-favored channel. a) The ωπ rest frame, b) the D∗ rest frame and c) the ω rest frame.
q`D**
Q(D*)
l(D)
X
x
(
p(w)
q
Q(D*)
Y
z D** a)
``
Z
Z
Z
`
X
q(D**)
)
n
p(w)
b
` X`` y * D
Y
f
w b)
l(D)
`
Y
c)
Figure 4. Definition of the angles for the D∗∗ -resonances. Color-suppressed channel. a) The D∗∗ rest frame, b) the ω rest frame and c) the D∗ rest frame.
It corresponds to the intermediate resonance R with mass mR and q 2 -dependent width ΓR . The numerator of the propagator is to be the sum over polarizations of the resonance and depends on its spin.
3
ωπ-resonances
We consider such ωπ-states, which can be combined to J P = 0− -, J P = 1+ (b1 (1235))-, J P = 1− (ρ(1450))-, J P = 2− -, J P = 2+ - and J P = 3− (ρ3 (1690))-states. Let us note that J P = 0− , J P = 2− and J P = 2+ charged states, which decay to the ωπ-final system, have not yet been observed at the present time [2]. Such states have the isotopic quantum numbers I G = 1+ . It is natural to assume that these states are members of the b and ρ-families.
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wp
Z
l(D)
b
p(w)
z
``
Z
Z
The matrix element for production of the J P = 0− intermediate state (labeled as ρ0 ) is given by: GF ∗ ¯>, MB→D = √ Vcb Vud < D∗ ρ0 |(¯ uΓµ d)(¯ cΓµ b)|B (3.1) ∗ρ ¯ 0 2 Parameterizing this amplitude in the covariant form, we have:3 GF ∗ = √ Vcb Vud MB→D gρ0 FP (q 2 )(ε∗ q), (3.2) ∗ρ ¯ 0 2
Mρ0 →ωπ = g˜ρ0 ωπ F˜P (q 2 , p2 )(v ∗ q),
(3.3)
where vµ is a polarization vector of ω, g˜ρ0 ωπ is a coupling constant and F˜P (q 2 , p2 ) is a transition form factor. The amplitude describing the ω decay comprises the contributions from the intermediate ρ-meson and 3π phase space: X p g ρππ ∆(p, P+ , P0 )(nv), Mω→3π = gωρπ (p2 ) a3π + (3.4) 2 2 D i (M )Z(M ) ρ i i i=±,0 where
ǫµνρσ P+ν P0ρ pσ (3.5) nµ = p ∆(p, P+ , P0 ) is a unit 4-vector normal to the ω decay plane and ∆(p, P+ , P0 ) is the Kibble determinant. Other notations are described in the appendix. Here and further ǫµνρσ is the Levi-Civita symbol and ǫ0123 = +1. The amplitude corresponding to the D ∗ decay is MD∗ →Dπ = gD∗ Dπ (εl).
(3.6)
The factor
gD∗ Dπ (Q2 )gωρπ (p2 ) a3π
p ∆(p, P+ , P0 ) gρππ l + 2 )Z(M 2 ) 2 2 D D (M i D ∗ (Q )Dω (p ) ρ i i i=±,0 X
(3.7)
is common for all intermediate states, and ω-decay part can be expressed via the phase integral W (p2 ), presented in the appendix. The total rate for B → D ∗ ωπ decay expressed via the branching fraction BD∗+ →D0 π+ and the phase integral W (p2 ) can be presented as follows: dΓ =
6BD∗+ →D0 π+ |M |2 pQ W (p2 ) p dp2 (d cos θ dφ) (d cos β dψ) (dq 2 d cos ξ), (4π)10 m2B q 2 |Dω (p2 )|2
(3.8)
where mB is a B-meson mass, and the matrix element M describes particular dependencies for the different intermediate channels. ¯ → D∗ RJ transition, where RJ is the intermediate The matrix element for the B resonance with the integer total spin J,5 can be parameterized in terms of the amplitudes 3
Here the term (ε∗ Q) is neglected because the longitudinal currents arise far from the resonance, where they should be suppressed by transition form factor behavior. However, they also modify the angular dependence of the amplitude. Throughout this paper the longitudinal currents are neglected. 4 The coefficient a1 is expressed via Wilson coefficients, as discussed in the previous section. 5 As emphasized above, in this paper we discuss resonances with J = 1, 2, 3.
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where εµ is a polarization vector of D∗ , gρ0 = a1 fρ0 ,4 fρ0 is a weak decay constant of the ρ0 and FP (q 2 ) is a transition form factor. The strong amplitude for the ρ0 -decay is presented as follows:
with the definite angular orbital momentum L as follows: GF ∗ µνρσ ′ ∗(J) ∗ MB→D εµ εν qρ Qσ FL=J (q 2 )+ ∗ R = √ Vcb Vud gJ CJ ǫ ¯ J 2 ′ 1 ′ + im2B CJ−1 ((ε ∗(J) ε∗ )− (ε ∗(J) Q)(ε∗ q))FL=J−1 (q 2 )+ 2 fJ,J−1 (q ) ′ ∗(J) ′ ∗(J) ∗ ∗ 2 2 + iCJ+1 ((ε Q)(ε q) − fJ,J+1 (q )(ε ε ))FL=J+1 (q ) . (3.9)
′
′
εµ(J=1) = εµ ,
′
′
εµ(J=2) = εµα Qα /mB ,
fJ,J±1 (q 2 ) = and a1,0 = −1,
a1,2 = +2,
′
′
εµ(J=3) = εµαβ Qα Qβ /m2B ;
2m2B Q2 m2B − m2D∗ − q 2 + 2aJ,J±1 mD∗
a2,1 = −1,
a2,3 = +3/2,
p
(3.11)
q2
a3,2 = −1,
(3.10)
a3,4 = +4/3.
(3.12)
The parameterization of the matrix element describing the resonance decay depends on its J P quantum numbers. Thus, resonances with J P = 1− , 2+ , 3− are described by the following matrix element: ′
MRJ →ωπ = g˜J ǫµνρσ ε˜µ(J) vν∗ qρ pσ F˜L=J (q 2 , p2 ),
(3.13)
where g˜1 = gR1 ωπ , g˜2 = mR gR2 ωπ and g˜3 = m2R gR3 ωπ are appropriate coupling constants; F˜L (q 2 , p2 ) is a transition form factor and ′
′
ε˜µ(J=1) = εµ ,
′
′
ε˜µ(J=2) = εµα pα /mR ,
′
′
ε˜µ(J=3) = εµαβ pα pβ /m2R .
(3.14)
The discussed resonances with J P = 1+ , 2− are described by the following matrix element: " ′ 1 ′ ε (J) v ∗ ) − (˜ ε (J) p)(v ∗ q))F˜L=J−1 (q 2 , p2 )+ MRJ →ωπ = g˜J C˜J−1 m2R ((˜ 2 ˜ fJ,J−1 (q ) ′ (J) ′ (J) ∗ ∗ 2 2 2 + C˜J+1 ((˜ ε p)(v q) − f˜J,J+1 (q )(˜ ε v ))F˜L=J+1 (q , p ) , (3.15) where C˜J is the relative amplitude, which is in general complex, and f˜J,J±1 (q 2 ) =
2q 2 p2 q 2 + p2 − m2 + 2aJ,J±1
p
p2 q 2
,
(3.16)
where m is the charged pion mass. Then we move from the covariant amplitudes to the expressions depending on the selected angles, which are defined in the intermediate particle rest frames. 6
′
The notation ε (J ) is not related to the resonance helicity state.
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Here, CJ is the relative amplitude, which is in general complex; FL (q 2 ) is a transition form factor corresponding to the orbital momentum L; g1 = a1 fR , fR is a weak decay constant of 2 the vector resonance, when q 2 = m2R , g2 = mB gBD ¯ ∗ R2 and g3 = mB gBD ¯ ∗ R3 are appropriate ′ (J) coupling constants; ε is a convolution of the resonant polarization tensor of rank J and 6 momentum Q
4
D∗∗-resonances
fJ,J±1 (q 2 ) = f˜J,J±1 (q 2 ) =
5
2m2B p2 m2B − p2 − q 2 + 2aJ,J±1 2q 2 Q2
p , p2 q 2
q 2 + m2D∗ − m2 + 2aJ,J±1 mD∗
Results
p
(4.1)
q2
.
(4.2)
Using the technique described in the previous sections, we present the final expressions for matrix elements with different intermediate resonances. The total matrix element squared is as follows: |M |2 = |M6 + Mρ0 + Mρ(1450) + Mb1 (1235) + Mb2 + Mρ2 + Mρ3 + MD1 + MD1′ + MD2∗ + + M1− + MD2 + MD2′ + M3− |2 . 3/2
(5.1)
5/2
Here, M6 presents the non-resonant contributions to the matrix element. The amplitudes MD1 and MD1′ are as follows: MD1 = sin ϑ1 |ju = 1/2 > + cos ϑ1 e−iϑ2 |ju = 3/2 > ,
MD1′ = cos ϑ1 |ju = 1/2 > − sin ϑ1 eiϑ2 |ju = 3/2 > ,
(5.2)
where ϑ1 and ϑ2 are mixing angles and similar expressions can be used for 2− -states. The resonant matrix element can be presented as follows: g ¯ ∗ (ω)RJ g˜RJ ω(D∗ )π X GF ∗ BD CL1 C˜L2 FL1 (q 2 )F˜L2 (q 2 )PL1 L2 AL1 L2 . MRJ = √ Vcb Vud DR (q 2 ) 2 L L 1
(5.3)
2
¯ 0 (R) rest frame; CL , C˜L are Here, L1 (L2 ) is the angular orbital momentum in the B 1 2 relative amplitudes defined above, PL1 L2 is the expression for the momentum dependence; AL1 L2 is the expression for the angular dependence. The expressions PL1 L2 and AL1 L2 are combined in table 1 for different intermediate states. The notations cα = cos α and sα = sin α are used. The functions fJ,J±1 (q 2 ) and f˜J,J±1 (q 2 ) used in table 1 are defined by (3.11) and (3.16) for the ωπ-resonances and by (4.1) and (4.2) for the D∗∗ -resonances. –8–
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The decay rate for the channel with D∗∗ -resonance production has a form similar to (3.8). As already mentioned, in this case the angles (θ, φ, ξ, β, ψ) differ from their analogues for the ωπ states and are described in figure 4. Here we discuss two J P = 1+ -states and a J P = 2+ -state, which correspond to P -wave in the spectroscopy of the c¯ u excitations as well as a J P = 1− -state, two J P = 2− -states and a J P = 3− -state corresponding to the + − − P P P D-wave excited c¯ u-states. Pure JjPu = 1+ 1/2 (Jju = 23/2 )- and Jju = 13/2 (Jju = 25/2 )-states decay to the D∗ π in S- (P -) wave and D- (F -) wave, respectively. As discussed above, observable J P = 1+ (J P = 2− ) states can be a mixture of pure ju = 1/2 (ju = 3/2) and ju = 3/2 (ju = 5/2) states. This fact has to be taken into account for the total amplitude construction. The parameterization of the matrix elements for all D∗∗ -states is similar to the case of the ωπ-states. However, mutual substitutions of the four-momenta p and Q and polarizations εµ and vµ have to be made. The functions fJ,J±1 (q 2 ) and f˜J,J±1 (q 2 ) for the D ∗∗ -states are as follows:
L1
L2
ρ0
P
P
ρ(1450)
S
P
P
P
D
P
ωπ
b1 (1235)
S
S
PL1 L2
AL1 L2
–9–
√ 2 √q 2 pQ mD ∗ p p −im2B q 2 p p mB q 2 pQ p i q 2 f1,2 (q 2 )p
−sθ sφ cβ sξ + sθ cφ sβ sψ − sθ sφ sβ cψ cξ
−im2B m2R
−cθ cβ cξ + sθ cφ cβ sξ − sθ sφ sβ sψ +
mB
cθ cβ
sθ sφ sβ sψ cξ + sθ cφ sβ cψ 2sθ sφ cβ sξ + sθ cφ sβ sψ − sθ sφ sβ cψ cξ +sθ cφ sβ cψ cξ + cθ sβ cψ sξ
S
D
im2B f˜1,2 (q 2 )
2cθ cβ cξ − 2sθ cφ cβ sξ − sθ sφ sβ sψ + +sθ cφ sβ cψ cξ + cθ sβ cψ sξ
P
S
m2R mB Q
P
D
−mB f˜1,2 (q 2 )Q
D
D
S
D
im2R f1,2 (q 2 )
−if1,2 (q 2 )f˜1,2 (q 2 )
−cθ sβ sψ sξ + sθ sφ sβ cψ − sθ cφ sβ sψ cξ 2cθ sβ sψ sξ + sθ sφ sβ cψ − sθ cφ sβ sψ cξ 2cθ cβ cξ + sθ cφ cβ sξ − sθ sφ sβ sψ + +sθ cφ sβ cψ cξ − 2cθ sβ cψ sξ −4cθ cβ cξ − 2sθ cφ cβ sξ − sθ sφ sβ sψ + +sθ cφ sβ cψ cξ − 2cθ sβ cψ sξ
JHEP09(2011)129
Resonance
Resonance
L1
L2
PL1 L2
AL1 L2
P
D
− 2i m3B p2 Q
sθ sφ cβ s2ξ + sθ sφ sβ cψ c2ξ − sθ cφ sβ sψ cξ
D
D
F
D
ωπ
ρ2
P
P
– 10 – ρ3
F
P
D
F
F
P
D
√i m3B m2R pQ q2
P
D
F
1 2 2 2 2 mB p Q i 2 2 2 mB f2,3 (q )p Q
F
− √i 2 m3B f˜2,3 (q 2 )pQ q
m2R m2B 2 q2
− √
m2B
F
G
F
−3/2sθ sφ cβ s2ξ + sθ sφ sβ cψ c2ξ − sθ cφ sβ sψ cξ
cθ cβ (c2ξ − 1/3) − 1/2cθ sβ cψ s2ξ − 1/2sθ cφ cβ s2ξ − −1/2sθ sφ sβ sψ cξ − sθ cφ sβ cψ (c2ξ − 1/2)
−3/2cθ cβ (c2ξ − 1/3) + 3/4cθ sβ cψ s2ξ − 1/2sθ cφ cβ s2ξ − −1/2sθ sφ sβ sψ cξ − sθ cφ sβ cψ (c2ξ − 1/2) cθ sβ sψ s2ξ + sθ cφ sβ sψ c2ξ − sθ sφ sβ cψ cξ
√ f˜2,3 (q 2 )pQ2 2
−3/2cθ sβ sψ s2ξ + sθ cφ sβ sψ c2ξ − sθ sφ sβ cψ cξ
− √i 2 m2R mB f2,3 (q 2 )pQ
−3/2cθ cβ (c2ξ − 1/3) − 1/2cθ sβ cψ s2ξ + 3/4sθ cφ cβ s2ξ −
q
2
q
√i mB f2,3 (q 2 )f˜2,3 (q 2 )pQ 2 q
√i 2 m4B p3 Q2
F
F
pQ2
sθ sφ sβ sψ + sθ cφ sβ cψ cξ
q
m3B
−1/2sθ sφ sβ sψ cξ − sθ cφ sβ cψ (c2ξ − 1/2)
9/4cθ cβ (c2ξ − 1/3) + 3/4cθ sβ cψ s2ξ + 3/4sθ cφ cβ s2ξ − −1/2sθ sφ sβ sψ cξ − sθ cφ sβ cψ (c2ξ − 1/2)
1/3(sθ cφ sβ sψ − sθ sφ sβ cψ cξ )(c2ξ − 1/5)− −sθ sφ cβ sξ (c2ξ − 1/5) + 2/3sθ sφ sβ cψ cξ s2ξ
√ p3 Q3 2
(sθ sφ sβ sψ cξ + sθ cφ sβ cψ )(c2ξ − 1/5) − 2sθ sφ sβ sψ cξ s2ξ
− √i 2 m2B f3,4 (q 2 )p3 Q2
1/3(sθ cφ sβ sψ − sθ sφ sβ cψ cξ )(c2ξ − 1/5)+
3 q
q
+4/3sθ sφ cβ sξ (c2ξ − 1/5) + 2/3sθ sφ sβ cψ cξ s2ξ
JHEP09(2011)129
b2
Resonance
L1
L2
PL1 L2
AL1 L2
S
S
-im2B m2R
−cθ cβ cξ + sθ cφ cβ sξ − sθ sφ sβ sψ +
D∗∗ 1+ 1/2
+sθ cφ sβ cψ cξ + cθ sβ cψ sξ
1+ 3/2
P
S
m2R mB p
D
S
im2R f1,2 (q 2 )
S
D
im2B f˜1,2 (q 2 )
−sθ sφ cβ sξ − sθ sφ sβ cψ cξ + sθ cφ sβ sψ 2cθ cβ cξ + sθ cφ cβ sξ − sθ sφ sβ sψ + +sθ cφ sβ cψ cξ − 2cθ sβ cψ sξ
– 11 –
2cθ cβ cξ − 2sθ cφ cβ sξ − sθ sφ sβ sψ + +sθ cφ sβ cψ cξ + cθ sβ cψ sξ
1− 3/2
D
D
D
P
D
D
D
F
D
S
P
P
P
D
P
−mB f˜1,2
−if1,2 (q 2 )f˜1,2 (q 2 ) − 2i m3B Q2 p
1 2 2 2 2 mB Q p i 2 2 2 mB f2,3 (q )Q p
p
−im2B q 2 Q p mB q 2 Qp p i q 2 f1,2 (q 2 )Q
2sθ sφ cβ sξ − sθ sφ sβ cψ cξ + sθ cφ sβ sψ −4cθ cβ cξ − 2sθ cφ cβ sξ − sθ sφ sβ sψ + +sθ cφ sβ cψ cξ − 2cθ sβ cψ sξ cθ sβ sψ s2ξ + sθ cφ sβ sψ c2ξ − sθ sφ sβ cψ cξ sθ sφ sβ sψ + sθ cφ sβ cψ cξ −3/2cθ sβ sψ s2ξ + sθ cφ sβ sψ c2ξ − sθ sφ sβ cψ cξ −cθ sβ sψ sξ + sθ sφ sβ cψ − sθ cφ sβ sψ cξ sθ sφ sβ sψ cξ + sθ cφ sβ cψ 2cθ sβ sψ sξ + sθ sφ sβ cψ − sθ cφ sβ sψ cξ
JHEP09(2011)129
2+ 3/2
P
(q 2 )p
Resonance
L1
L2
PL1 L2
AL1 L2
P
P
√i m3B m2R Qp 2
cθ cβ (c2ξ − 1/3) − 1/2cθ sβ cψ s2ξ − 1/2sθ cφ cβ s2ξ −
D ∗∗ 2− 3/2
q
2
– 12 – 3− 5/2
D
P
mR mB − √ Qp2 2 q
sθ sφ cβ s2ξ + sθ sφ sβ cψ c2ξ − sθ cφ sβ sψ cξ
F
P
− √i 2 m2R mB f2,3 (q 2 )Qp
−3/2cθ cβ (c2ξ − 1/3) − 1/2cθ sβ cψ s2ξ + 3/4sθ cφ cβ s2ξ −
P
F
D
F
F
F
D
2
q
− √i 2 m3B f˜2,3 (q 2 )Qp q
m2B
F
G
F
−3/2cθ cβ (c2ξ − 1/3) + 3/4cθ sβ cψ s2ξ − 1/2sθ cφ cβ s2ξ − −1/2sθ sφ sβ sψ cξ − sθ cφ sβ cψ (c2ξ − 1/2)
√ 2 f˜2,3 (q 2 )Qp2
−3/2sθ sφ cβ s2ξ + sθ sφ sβ cψ c2ξ − sθ cφ sβ sψ cξ
√i mB f2,3 (q 2 )f˜2,3 (q 2 )Qp 2
9/4cθ cβ (c2ξ − 1/3) + 3/4cθ sβ cψ s2ξ + 3/4sθ cφ cβ s2ξ −
q
2 q
√i 2 m4B Q3 p2
F
F
−1/2sθ sφ sβ sψ cξ − sθ cφ sβ cψ (c2ξ − 1/2)
q
m3B
−1/2sθ sφ sβ sψ cξ − sθ cφ sβ cψ (c2ξ − 1/2) sθ cφ sβ sψ cξ s2ξ − 4/15sθ cφ sβ sψ cξ +
+1/3sθ sφ sβ cψ (c2ξ − 1/5) − cθ sβ sψ sξ (c2ξ − 1/5)
√ 2 Q3 p3
(sθ sφ sβ sψ cξ + sθ cφ sβ cψ )(c2ξ − 1/5) − 2sθ sφ sβ sψ cξ s2ξ
− √i 2 m2B f3,4 (q 2 )Q3 p2
sθ cφ sβ sψ cξ s2ξ − 4/15sθ cφ sβ sψ cξ +
3 q
q
+1/3sθ sφ sβ cψ (c2ξ − 1/5) + 4/3cθ sβ sψ sξ (c2ξ − 1/5) Table 1: Summary of momentum and angular distributions for different intermediate states, which are described in this paper.
JHEP09(2011)129
2− 5/2
2
−1/2sθ sφ sβ sψ cξ − sθ cφ sβ cψ (c2ξ − 1/2)
6
Decay chain simulation
xL+1 |hL (x0 )| 0 , L+1 x |hL (x)|
(6.1)
L (L + n)! −i i(x− πL ) X 2 (−1)n e (2ix)−n x n!(L − n)! n=0
(6.2)
BL (x) = where hL (x) =
is a spherical Hankel function, x = kr, x0 = k0 r, k, k0 are the magnitudes of the daughter particle three-momentum in the mother particle rest frame for the case when the resonance four-momentum squared is equal to q 2 and m2R , respectively, and r = 1.6 GeV −1p is a hadron scale. According to our normalization, these functions are equal to one, when q 2 = mR . Another common normalization gives BL (x) = 1 for x = 1. The Blatt-Weisskopf functions corresponding to L discussed here are given below for convenience: B0 (x) = 1, s
B1 (x) =
B2 (x) =
s
B3 (x) =
s
B4 (x) =
s
1 + x20 , 1 + x2 (x20 − 3)2 + 9x20 , (x2 − 3)2 + 9x2 x20 (x20 − 15)2 + 9(2x20 − 5)2 , x2 (x2 − 15)2 + 9(2x2 − 5)2 (x40 − 45x20 + 105)2 + 25x20 (2x20 − 21)2 . (x4 − 45x2 + 105)2 + 25x2 (2x2 − 21)2
(6.3)
If we consider one angular variable only, the distributions can be the same for different resonant hypotheses. Efficient separation between resonances is possible, when all angular variables are taken into account. This statement is demonstrated in figures 5 and 6 for the ωπ- states and in figures 7 and 8 for the D∗ π-states. As mentioned above, J P = 1+ P -wave and J P = 2− D-wave states are a mixture of pure states. However, for demonstration purposes we consider and show angular distributions for pure states. Moreover, we use a simple relativistic quark model of mesons to estimate constant ratios CJ−1 /CJ
– 13 –
JHEP09(2011)129
To demonstrate the angular distributions for each intermediate resonance in the D∗ ωπ final ¯ 0 → D ∗+ ωπ − events according to the phase space distribution state, we generate 2 × 106 B using the qq98 program package [18]. For a further study we fill profile angular spectra with the appropriate weight density functions for each resonant hypothesis, which have been obtained above. A description of each vertex includes transition form factors. Since it is not yet possible to obtain these form factors from rigorous theoretical calculations, we rely on the simple phenomenological Blatt-Weisskopf model [19, 20]. For L > 0 this simple form factor suppresses growth of the matrix element with final particle momentum. The Blatt-Weisskopf functions BL (x) are chosen as follows:
20 10 0 -1
Arbitrary Units
Arbitrary Units
Arbitrary Units
30
8 6 4 2
-0.5
0
0.5
0 -1
1
-0.5
0
0.5
Arbitrary Units
Arbitrary Units
Arbitrary Units
6 4
0 -1
1
-0.5
0
0.5
5
0 -1
1
2
-0.5
0
0.5
0 -1
1
0
0.5
1
c3)
4
2
4 3 2 1
2 -2
0
2
0
φ
-2
0
2
0
φ
-2
8 6 4
0
2
φ
d3) Arbitrary Units
d2) Arbitrary Units
d1) Arbitrary Units
-0.5
cosβ
Arbitrary Units
4
4 3 2
4
2
1
2 0
4
cosβ
Arbitrary Units
6
1
6
c2)
8
0.5
cosξ
Arbitrary Units
Arbitrary Units
Arbitrary Units 0.5
0
b3)
7.5
c1)
0
-0.5
b2) 10
cosβ
Arbitrary Units
0 -1
1
2.5
0
4
cosξ
10
-0.5
6
2
b1)
20
1
a3)
cosξ
30
0.5
cosθ
2
0.5
0
-2
0
e1)
2
0
ψ
-2
0
e2)
2
0
ψ
-2
0
2
ψ
e3)
Figure 5. Simulated angular distributions for the ωπ-resonances. The figures a1), b1), c1), d1), P e1) correspond to the J P = 0− (ρ− = 1− 0 ) intermediate state; a2), b2), c2), d2), e2) — J (ρ(1450)− )-state; a3), b3), c3), d3), e3) — J P = 1+ (b1 (1235)− )-state.
– 14 –
JHEP09(2011)129
5
0
-0.5
a2)
10
0 -1
0 -1
1
cosθ
a1)
-0.5
5 2.5
cosθ
0 -1
10 7.5
7.5 5
10 7.5 5 2.5
2.5 0 -1
Arbitrary Units
Arbitrary Units
Arbitrary Units
10
-0.5
0
0.5
0 -1
1
-0.5
0
0.5
10 7.5 5
0 -1
1
-0.5
0
0.5
0.5
4
0 -1
1
2
-0.5
0
2
2
0
-2
0
2
4
0
-2
0
2
φ
d3)
4 3 2
0
ψ
1
6
φ
Arbitrary Units
Arbitrary Units
e1)
2
0.5
2
1 0
0
cosβ
d2)
2
-2
-0.5
c3)
3
φ
4
0
0 -1
1
4
d1) Arbitrary Units
0.5
1
0
5 2.5
Arbitrary Units
4
1
7.5
cosβ
Arbitrary Units
Arbitrary Units
6
0.5
10
c2)
8
0
cosξ
Arbitrary Units
6
c1)
-2
-0.5
b3)
8
cosβ
0
5
b2) Arbitrary Units
0
7.5
0 -1
1
2 -0.5
10
cosξ
b1)
5
1
2.5
cosξ
10
0.5
a3)
2.5 0.5
0
cosθ
Arbitrary Units
Arbitrary Units
Arbitrary Units
5
0 -1
-0.5
4 3 2 1
-2
0
e2)
2
0
ψ
-2
0
2
ψ
e3)
Figure 6. Simulated angular distributions for the ωπ-resonances. The figures a1), b1), c1), d1), − e1) correspond to the J P = 2+ (b− (ρ− 2 ) intermediate state; a2), b2), c2), d2), e2) — 2 2 )-state; − − a3), b3), c3), d3), e3) — 3 (ρ3 (1690) )-state.
– 15 –
JHEP09(2011)129
7.5
2.5
Arbitrary Units
0 -1
1
a2)
10
0
4
cosθ
a1)
-0.5
6
2
cosθ
0 -1
8
1.5 1
Arbitrary Units
Arbitrary Units
Arbitrary Units
2
2
1
3
2
1
0.5 0 -1
-0.5
0
0.5
0 -1
1
-0.5
0
0.5
cosθ
-0.5
1
1
0.5
2
1
0.5 0 -1
-0.5
0
0.5
0 -1
1
-0.5
0
0.5
cosξ
1
0
0.5
0.5
0 -1
1
-0.5
0
0
0.8 0.6
2
0.4
0
φ
0.5
1
1
0.5
0.25 -2
0
2
0
φ
-2
0
2
φ
d3) Arbitrary Units
Arbitrary Units
Arbitrary Units
0.75
0.5
cosβ
d2)
1
0
0.75
d1) 1
0.5
2 1.5 1 0.5
0.25 0
-0.5
c3)
0.2 -2
0 -1
1
Arbitrary Units
Arbitrary Units
Arbitrary Units
0.25 0
0.5
1
c2)
0.5
1
2
cosβ
c1) 0.75
0.5
b3)
1
cosβ
1
0
cosξ
Arbitrary Units
2
-0.5
-0.5
b2) Arbitrary Units
Arbitrary Units
b1)
0 -1
0 -1
1
cosξ
-2
0
e1)
2
0
ψ
-2
0
e2)
2
0
ψ
-2
0
2
ψ
e3)
Figure 7. Simulated angular distributions for the P -wave D∗∗ -resonances. The figures a1), b1), + P c1), d1), e1) correspond to the JjPu = 1+ 3/2 narrow state; a2), b2), c2), d2), e2) — Jju = 11/2 broad state; a3), b3), c3), d3), e3) — JjPu = 2+ 3/2 narrow state.
– 16 –
JHEP09(2011)129
1
0.5
a3) Arbitrary Units
1.5
0
cosθ
a2) Arbitrary Units
a1) Arbitrary Units
0 -1
1
cosθ
Arbitrary Units
Arbitrary Units
Arbitrary Units
Arbitrary Units
2
3
1
0.75
2
0.5
1
6
4
2
1 0.25
0.5
0 -1
1
-0.5
0
0.5
3 2
0.5
0
0.5
0 -1
1
-0.5
0
0.5
3
2
-0.5
0
0.5
0 -1
1
-0.5
0
0.5
-0.5
0
0.5
-0.5
0
0.5
1
b4)
1
0 -1
4
2
-0.5
0
0.5
0 -1
1 cosβ
-0.5
Arbitrary Units
1.5
1
0
0.5
1
cosβ
c3)
c4)
0.8 0.6 0.4
2
1
0.2
0.2 0
2
0
-2
0
0
2
φ
0.5
1.5
1
0.5
0
e1)
2
0
0
2
-2
0
0
2
0.6 0.4
0
2
4 3 2 1
-2
0
ψ
e2)
φ
d4)
0.2
ψ
-2
d3) Arbitrary Units
Arbitrary Units
1
0
φ
d2)
1.5
-2
-2
φ
Arbitrary Units
-2
d1) Arbitrary Units
4
cosξ
0.5
0
6
0 -1
1 cosξ
1.5
1 cosβ
0.4
0
1
2
c2) Arbitrary Units
Arbitrary Units
c1)
0.6
0.5
0.5
cosβ
0.8
0
cosθ
b3)
1
0.5 0 -1
0 -1
Arbitrary Units
1
-0.5
a4)
1
b2) Arbitrary Units
Arbitrary Units
b1) 1.5
0 -1
1 cosθ
1.5
1 cosξ
cosξ
2
0.5
0.5
1
-0.5
0
a3) Arbitrary Units
Arbitrary Units
a2)
1
0 -1
-0.5
0
2
-2
0
ψ
e3)
2
ψ
e4)
Figure 8. Simulated angular distributions for the D-wave D∗∗ -resonances. The figures a1), b1), − P c1), d1), e1) correspond to the JjPu = 1− 3/2 broad state; a2), b2), c2), d2), e2) — Jju = 25/2 narrow − P state; a3), b3), c3), d3), e3) — Jjpu = 2− 3/2 broad state; a4), b4), c4), d4), e4) — Jju = 35/2 narrow state.
– 17 –
JHEP09(2011)129
Arbitrary Units
a1)
0 -1
1 cosθ
cosθ
Arbitrary Units
0
Arbitrary Units
-0.5
Arbitrary Units
0 -1
2 (q 2 )F ˜ 2 (q 2 ) p mb m4 F˜ 2 (q 2 ) + bD f˜1,2 D Γb1 (q 2 ) = p 1 b1 S 4 Γb , 2 (m2 ) p0 1 mb1 + bD f˜1,2 q2 b1
(6.4)
where p0 is the magnitude of the ω momentum in the resonance rest frame, when q 2 = m2b1 . Here, we use the fact that the experimental ratio of the amplitudes with L = 2 and L = 0 is about 0.3 [2] and thus the constant bD ≈ 53. For a q 2 -dependent width of the pure D1 and pure D1′ we consider the decay to the D∗ π: ΓD1 ,D1′ (q 2 ) =
mD1 ,D1′ 2 Q ′, p Γ F˜D,S (q 2 ) 2 Q0 D1 ,D1 q
(6.5)
where Q0 is the magnitude of the D∗ momentum in the resonance rest frame, when q 2 = m2D1 ,D′ . For a q 2 -dependent width of the ρ(1450) we consider its decays into the a1 (1260)π 1 and ωπ modes: Γρ(1450)
4 2 2 2 2 2 2 mρ(1450) mρ(1450) F˜S (q ) + bD f˜1,2 (q )F˜D (q ) k(a1 ) Γ + = (1 − a) p 2 (m2 k0(a1 ) ρ(1450) m4ρ(1450) + bD f˜1,2 q2 ρ(1450) ) p q 2 ˜ 2 2 p3 +a F (q ) 3 Γρ(1450) , (6.6) mρ(1450) P p0
p p where the parameter a = 2/5, when q 2 > ma1 + mπ− [22], and a = 1, when q 2 ≤ ma1 + mπ− , k(a1 ) ispthe momentum of the a1 in the ρ(1450) rest frame, k0(a1 ) is the same momentum, when q 2 = mρ(1450) . Here, we use the fact that the experimental ratio of the amplitudes with L = 2 and L = 0 in the a1 (1260) → ρπ decay is about −0.06 [2] and accept the same value for the ρ(1450) → a1 (1260)π decay. Thus, we can estimate the constant bD ≈ 182. Obviously, it is impossible to analyse spectra without knowledge of the relative phases in the amplitude. Thus, in figure 9 we show some typical distributions for different relative
– 18 –
JHEP09(2011)129
and CJ+1 /CJ in (3.9), which are responsible for relative contributions of amplitudes with different orbital momenta in the total matrix element [21]. The constant ratios for all discussed states are chosen roughly as follows: CJ−1 /CJ = 3/2, CJ+1 /CJ = 2. For Dalitz plot analysis, interference between resonances should be taken into account. For a one-dimensional distribution, an interference term for resonances, which decay to the same final state, can cancel out after integration over other variables. However, in a real experiment such cancellation can disappear due to the nonuniform detection efficiency, so that a finite interference term can be observed. For resonances, which decay to the different final states ωπ and D∗ π, the interference term cannot be neglected. For demonstration purpose we show distributions between b1 (1235)− and pure D10 as well as ρ(1450)− and ′ pure D10 . However, there is possible interference between the resonant and non-resonant structures. ′ For simulation we use BW functions for b1 (1235)− , ρ(1450)− , D10 and D10 . Thus, the q 2 -dependent widths have to be obtained. For a q 2 -dependent width of the b1 we consider the dominant decay to the ωπ [2]:
Arbitrary Units
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1
0.5
2.5 2 1.5 1
0 -1
-0.5
0
0.5
0 -1
1
cos ξ1
-0.5
4 3 2 1 0
0.5
1
cos ξ2
a2) b1 D1
Arbitrary Units
Arbitrary Units
a1) b1 D1
0
4 3 2 1
-2
0
2
0
φ1
-2
0
2
φ2
b2) ρ′ D1′
b1) ρ′ D1′
Figure 9. Demonstrative interference distributions. The figures a1), a2) correspond to the distributions over angles cos ξ1 and cos ξ2 for interference between the b1 (1235)− and pure D10 ; b1), b2) correspond to the distributions over angles φ1 and φ2 for interference between the ρ(1450)− and ′ pure D10 . The subscripts 1 and 2 correspond to the ωπ- and D∗∗ -resonances, respectively.
phases ∆ϕ, such as 0, π/2, π, 3π/2 and the distribution without interference. The relative constant amplitudes between resonant matrix elements squared are chosen of one order of magnitude for the b1 (1235)− and D10 for simplicity and one order of magnitude smaller ′ for the D10 than for the ρ(1450)− according to experiment [13]. Although small, the interference effects are not negligible.
– 19 –
JHEP09(2011)129
0.5
7
Conclusion
Acknowledgments This work was supported in part by the RFBR grants 11-02-112-a, 11-02-90458-a, and grant DFG GZ: HA1457/7-1.
A
The phase integral for ω-decay
In this section we present the phase integral W (p2 ) at the decay rate defined by (3.8). The integral Z (√p2 −m0 )2 Z M2 + max ∆(p, P+ , P0 ) 2 2 2 W (p ) = π gωρπ (p2 )× dM0 dM+2 2 2 p (2m+ )2 M+ min 2 X g ρππ × a3π + (A.1) 2 2 D i (M )Z(M ) ρ i i i=±,0
is a standard phase space factor for ω-decay [23]. Here, the Kibble determinant ∆(p, P+ , P0 ), which zeros determine the phase-space boundary, is presented as follows: 2 p pP+ pP0 ∆(p, P+ , P0 ) = pP+ m2 P+ P0 , pP0 P+ P0 m20
where m and m0 are the charged and neutral pions masses, respectively; the range limits of M+2 are M+2 min M+2 max
=
(+−) (E+
=
(+−) (E+
+
(+−) 2 E0 )
+
(+−) 2 E0 )
q q (+−) 2 (+−) 2 2 − ( E+ − m + E0 − m20 )2 , q q (+−) 2 (+−) 2 2 − m − E0 − ( E+ − m20 )2 ,
– 20 –
(A.2)
JHEP09(2011)129
¯ 0 → D∗+ ωπ − decay, in which a total amplitude is a We have described a model of the B sum of contributions of different intermediate states. In our study we consider different resonant contributions to the matrix element, such as light ωπ hadrons with the spinparities of J P = 0− , 1± , 2± , 3− , and heavy-light hadrons, which are excitations of the c¯ u states in P - and D-waves. All resonances are described by the relativistic Breit-Wigner factors. The resonant matrix elements are parameterized in the angular basis, which is convenient for the experimental Dalitz plot analysis and is natural from the physical point of view. Monte-Carlo simulation based on the obtained expressions has been performed. The angular distributions obtained for the listed above intermediate states and their interference effects are demonstrated.
where (+−)
=
M02 − 2m2 , 2M0
(+−)
=
p2 − M02 − m20 2M0
E+ E0
(A.3)
are the energies of π + and π 0 from ω decay in the π + π − rest frame; 1 + (mω r)2 p 1 + ( p2 r)2
(A.4)
is the form factor which restricts too fast growth of the width Γω (p2 ) with p2 , so that Γω (p2 ) → const as p2 → ∞ (here r is a hadron scale) [24]; the quantity gωρπ (m2ω ) ≃ 16 GeV −1 [25–27]; the a3π and gρππ amplitudes are assumed to be real constants and, thus, a3π = (0.01 ± 0.23 ± 0.25) × 10−5 MeV −2 [23] and gρππ ≃ 6 [23, 25, 27]; 2 2 2 ), Dρ±,0 (M±,0 ) = M±,0 − m2ρ±,0 + imρ±,0 Γρ±,0 (M±,0
(A.5)
where M−2 = p2 − M+2 − M02 + m20 + 2m2 , 2 ) Γρ±,0 (M±,0
mρ±,0 = M±,0
2 ) k±,0 (M±,0 k±,0 (m2ρ±,0 )
!3
Γρ±,0 ,
(A.6)
ki is an absolute value of pion momentum in the π + π − rest frame for i = 0, in the π + π 0 rest frame for i = + and in the π − π 0 rest frame for i = −; Z(M±,0 ) = 1 − is1 Φ(M±,0 ,
p
p2 )
(A.7)
is the factor taking into account the interaction of the ρ and π mesons in the final ω decay state, where the parameter sp 1 = 1 ± 0.2 corresponds to the prediction of [28], where the specific form of the Φ(M±,0 , p2 ) function can be found. The couplings gωρπ and gρππ are the same for ρ±,0 because of isotopic invariance. As emphasized in the text, the angle ξ is related to the intermediate resonance mass by a simple expression. Finally, let us give here the p2 -dependent width of the ω-meson taking into account the 3π and π 0 γ modes [2]: 2 (p2 ) m2ω (p2 − m20 )3 gωπγ W (p2 ) Bω→3π Γω + 2 B Γ . Γω (p ) = 2 (m2 ) ω→πγ ω W (m2ω ) p (m2ω − m20 )3 gωπγ ω 2
(A.8)
A form factor gωπγ (p2 ) has a similar form (A.4) [23] and gωπγ (m2ω ) ≃ 0.7 GeV −1 [27].
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gωρπ (p2 ) = gωρπ (m2ω )
References [1] A.F. Falk and M.E. Peskin, Production, decay and polarization of excited heavy hadrons, Phys. Rev. D 49 (1994) 3320 [hep-ph/9308241] [SPIRES]. [2] Particle Data Group collaboration, K. Nakamura et al., Review of particle physics, J. Phys. G 37 (2010) 075021 [SPIRES]. [3] N. Isgur and M.B. Wise, Weak transition form factors between heavy mesons, Phys. Lett. B 237 (1990) 527 [SPIRES].
[5] A. Le Yaouanc, L. Oliver, O. Pene, J.C. Raynal and V. Morenas, Uraltsev sum rule in Bakamjian-Thomas quark models, Phys. Lett. B 520 (2001) 25 [hep-ph/0105247] [SPIRES]. [6] Belle collaboration, K. Abe et al., Study of B − → D∗∗0 π − (D∗∗0 → D(∗)+ π − ) decays, Phys. Rev. D 69 (2004) 112002 [hep-ex/0307021] [SPIRES]. [7] BABAR collaboration, B. Aubert et al., Dalitz plot analysis of B − → D+ π − π − , Phys. Rev. D 79 (2009) 112004 [arXiv:0901.1291] [SPIRES]. ¯ 0 → D0 π + π − decays, [8] Belle collaboration, A. Kuzmin et al., Study of B Phys. Rev. D 76 (2007) 012006 [hep-ex/0611054] [SPIRES]. [9] BABAR collaboration, B. Aubert et al., Measurement of semileptonic B decays into orbitally excited charmed mesons, Phys. Rev. Lett. 103 (2009) 051803 [arXiv:0808.0333] [SPIRES]. [10] N. Cabibbo, Unitary symmetry and leptonic decays, Phys. Rev. Lett. 10 (1963) 531 [SPIRES]. [11] M. Kobayashi and T. Maskawa, CP violation in the renormalizable theory of weak interaction, Prog. Theor. Phys. 49 (1973) 652 [SPIRES]. ¯ 0 → D∗+ ρ′ − , ρ′ − → ωπ − , [12] CLEO collaboration, J.P. Alexander et al., First observation of B Phys. Rev. D 64 (2001) 092001 [hep-ex/0103021] [SPIRES]. ¯ 0 → D∗+ ωπ − , [13] BABAR collaboration, B. Aubert et al., Study of the decay B Phys. Rev. D 74 (2006) 012001 [hep-ex/0604009] [SPIRES]. [14] K.G. Wilson, Nonlagrangian models of current algebra, Phys. Rev. 179 (1969) 1499 [SPIRES]. [15] A.N. Kamal, A.B. Santra, T. Uppal and R.C. Verma, Nonfactorization in hadronic two-body Cabibbo-favored decays of D0 and D+ , Phys. Rev. D 53 (1996) 2506 [hep-ph/9503390] [SPIRES]. [16] Z. Ligeti, M.E. Luke and M.B. Wise, Comment on studying the corrections to factorization in B → D(∗) X, Phys. Lett. B 507 (2001) 142 [hep-ph/0103020] [SPIRES]. [17] M. Gell-Mann, D. Sharp and W.G. Wagner, Decay rates of neutral mesons, Phys. Rev. Lett. 8 (1962) 261 [SPIRES]. [18] D. Jaffe, QQ Documentation Sources, http://www.lns.cornell.edu/public/CLEO/soft/QQ. [19] J. Blatt and V. Weisskopf, Theoretical nuclear physics, John Wiley and Sons, New York U.S.A. (1952).
– 22 –
JHEP09(2011)129
[4] M. Neubert, Heavy quark symmetry, Phys. Rept. 245 (1994) 259 [hep-ph/9306320] [SPIRES].
[20] F. Von Hippel and C. Quigg, Centrifugal-barrier effects in resonance partial decay widths, shapes and production amplitudes, Phys. Rev. D 5 (1972) 624 [SPIRES]. [21] M. Wirbel, B. Stech and M. Bauer, Exclusive semileptonic decays of heavy mesons, Z. Phys. C 29 (1985) 637 [SPIRES]. [22] CMD2 collaboration, R.R. Akhmetshin et al., Study of the processes e+ e− → ηγ, e+ e− → π 0 γ → 3γ in the c.m. energy range 600-1380 MeV at CMD-2, Phys. Lett. B 605 (2005) 26 [hep-ex/0409030] [SPIRES]. √ [23] M.N. Achasov et al., Study of the process e+ e− → π + π − π 0 in the energy region s below 0.98 GeV, Phys. Rev. D 68 (2003) 052006 [hep-ex/0305049] [SPIRES].
[25] V.M. Braun and I.E. Filyanov, QCD sum rules in exclusive kinematics and pion wave function, Z. Phys. C 44 (1989) 157. [Sov.J.Nucl.Phys.50:511,1989], [26] M. Lublinsky, gωρπ reexamined, Phys. Rev. D 55 (1997) 249 [hep-ph/9608331] [SPIRES]. [27] J.L. Lucio-Martinez, M. Napsuciale, M.D. Scadron and V.M. Villanueva, ω → ρπ transition and ω → 3π decay, Phys. Rev. D 61 (2000) 034013 [SPIRES]. [28] N.N. Achasov and A.A. Kozhevnikov, Signature of the triangle singularity in the reaction e+ e− → π + π − π 0 , Phys. Rev. D 49 (1994) 5773 [SPIRES].
– 23 –
JHEP09(2011)129
[24] N.N. Achasov, M.S. Dubrovin, V.N. Ivanchenko, A.A. Kozhevnikov and E.V. Pakhtusova, A fresh look at φ-ω mixing, Sov. J. Nucl. Phys. 54 (1991) 664. [Yad.Fiz.54:1097-1108,1991], SJNCA,54,664;
