arXiv:0704.0404v1  [hep-ph]  3 Apr 2007
To the origin of the difference of FSI phases in
B → ππ and B → ρρ decays
A.B. Kaidalov∗ and M.I. Vysotsky†
ITEP, Moscow, Russia
Abstract
The final state interactions (FSI) model in which soft rescattering
of low mass intermediate states dominates is suggested. It explains
why the strong interaction phases are large in the Bd → ππ channel
and are considerably smaller in the Bd → ρρ one. Direct CP asym-
metries of Bd → ππ decays which are determined by FSI phases are
considered as well.
1 Introduction
There are three reasons to study FSI in B decays: to predict (or explain)
the pattern of branching ratios, to study strong interactions, and to forsee in
what decays direct CPV will be large. In view of this necessity a model for
FSI in B decays to two light mesons is suggested and explored in the present
paper.
The probabilities of three B → ππ and three B → ρρ decays are measured
now with good accuracy. The C-averaged branching ratios of these decays
are presented in Table 1 [1]. Let us look at the ratio of the charge averaged
Bd decay probabilities to the charged and neutral mesons:
Br(Bd → ρ+ρ−)
Br(Bd → ρ0ρ0)
≈ 20 , Rπ ≡
Br(Bd → π+π−)
Br(Bd → π0π0)
≈ 4 . (1)
∗kaidalov@itep.ru
†vysotsky@itep.ru
http://arxiv.org/abs/0704.0404v1
Table 1
Mode Br(10−6) Mode Br(10−6)
Bd → π+π− 5.2± 0.2 Bd → ρ+ρ− 23.1± 3.3
Bd → π0π0 1.3± 0.2 Bd → ρ0ρ0 1.16± 0.46
Bu → π+π0 5.7± 0.4 Bu → ρ+ρ0 18.2± 3.0
C-averaged branching ratios of B → ππ and B → ρρ decays.
The large difference of Rρ and Rπ is due to the difference of FSI phases
in B → ρρ and B → ππ decays (see below). In Section 2 we will determine
the differences of FSI phases of tree amplitudes which describe B → ρρ and
B → ππ decays into the states with isospins zero and two from the data
presented in Table 1. As a next step we will suggest a mechanism which
produces such phases. Once this mechanism is defined it becomes possible to
calculate FSI phases of decay amplitudes into states with a definite isospin
(not only their differences). A central question is: what intermediate states
produce FSI phases in B-meson decays into two light mesons. In the weak
decay b → uū(dd̄)d in the rest frame of a heavy quark (which is B-meson
rest frame as well) three fast light quarks are produced. Their energies are
of the order of MB/3 and momenta are more or less isotropically oriented.
The energy of the fourth (spectator) quark is of the order of ΛQCD. This four
quark state transforms mainly into multi pi-meson final state with the average
pion multiplicity about 9 (this number follows from the experimentally known
charged particles multiplicity in e+e− annihilation at Ecm = 3GeV multiplied
by 1.5 ∗ 1.5 in order to take neutral pions and third quark jet into account).
The total branching ratio of such decays is about 10−2. However such meson
state does not transform into the state composed from two light mesons
moving into opposite directions with momenta MB/2. What meson state
does transform into two light mesons can be understood from the inverse
reaction of two light meson scattering at the center of mass energy equal
to the mass of B-meson. The produced hadronic state consists of two jets
of particles moving in opposite directions. Each jet should originate from
a quark-antiquark pair produced in the weak decay of b-quark. The square
of invariant mass of a jet which contains spectator quark does not exceed
MBΛQCD and is much smaller than M
B. The energy of this jet is determined
by that of a companion quark and is about MB/2. That is why the square
of invariant mass of the second jet also does not exceed MBΛQCD. So for
B-decays the mass of a hadron cluster which transforms into light meson
in the final state should not exceed 1.5 GeV. Following these arguments in
the calculation of the imaginary parts of the decay amplitudes we will take
into account only two (relatively light) particle intermediate states for which
branching ratios of B-meson are maximal.
In Section 3 we will calculate FSI phases of tree amplitudes describing
B → ππ decays taking into account ρρ, ππ and πa1 intermediate states which
by t(u)-channel exchanges are converted into ππ. We will find that large
probability of B → ρ+ρ− decay explains about half of FSI phases of B → ππ
decays. Relatively small probability of B → π+π− decay prevents generation
of noticeable FSI phase of B → ρρ amplitudes through B → π+π− → ρρ
chain.
We will demonstrate that the strong interaction phase of the penguin
amplitude is opposite to the result of quark loop calculation, which is very
important for the value of a direct CPV asymmetry Cπ+π− ≡ C+− discussed
in Section 4. Predictions for CPV asymmetries C00 and S00 will be presented
in Section 4 as well and the value of the unitarity triangle angle α will be
extracted from the experimental data on CPV asymmetry S+−.
Subject of rare B decays is an object of intensive study nowadays and an
interested reader can find extensive list of references in a recent paper [2].
2 Phenomenology; |δπ0 − δπ2 | and |δ
0 − δ
Let us present B → ππ decay amplitudes in the so-called “t-convention”, in
which the penguin amplitude with the intermediate c-quark multiplied by
ud + VcbV
cd + VtbV
td = 0 is subtracted from the decay amplitudes [3]:
MB̄d→π+π− =
|VubV ∗ud|m2Bfπf+(0)
iδπ2 +
+ e−iγ
iδπ0 +
V ∗tdVtb
eiβPei(δ
+δ̃π0 )
, (2)
MB̄d→π0π0 =
|VubV ∗ud|m2Bfπf+(0)
iδπ2 −
− e−iγ 1√
iδπ0 −
V ∗tdVtb
eiβPei(δ
+δ̃π0 )
, (3)
MB̄u→π−π0 =
|VubV ∗ud|m2Bfπf+(0)
e−iγA2e
, (4)
where Vik are the elements of CKM matrix, γ and β are the unitarity triangle
angles and we factor out the product m2Bfπf+(0) which appears when the
decay amplitudes are calculated in the factorization approximation. A2 and
A0 are the absolute values of the decay amplitudes into the states with I = 2
and 0, generated by operators O1 and O2 (tree amplitudes), while P is the
absolute value of QCD penguin amplitude (generated by operators O3 − O6
of effective nonleptonic Hamiltonian which describes b quark decays into the
states without charm and strange quarks). δπ0 , δ
2 and δ̃
0 are FSI phases of
these three amplitudes, and it is very important for what follows that all of
them are different. It is easy to understand why δπ0 is different from δ
2 : strong
interaction depends on the isospin and is different for I = 0 and I = 2. For
example, there are definitely quark-antiquark resonances with I = 0, while
exotic resonances with I = 2 should be made from at least four quarks and
their existence is questionable. The reason why δπ0 differs from δ̃
0 is more sub-
tle. Let us consider the intermediate state made from two charged ρ-mesons
which contributes to FSI phases: Bd → ρ+ρ− → ππ. ρ+ρ− intermediate state
contribution to FSI phases can be large since Br(Bd → ρ+ρ−) is big. Both
tree and penguin induced amplitudes get FSI phases through this chain. Its
contribution to δπ0 is proportional to
(BrBd → ρ+ρ−)T/(BrBd → π+π−)T ≈
(BrBd → ρ+ρ−)/(BrBd → π+π−) ≈ 2.1, while that to δ̃π0 is proportional to
(BrBd → ρ+ρ−)P/(BrBd → π+π−)P .
How can we determine the penguin contributions to the probabilities
of Bd → ρ+ρ− and Bd → π+π−-decays? The most straightforward way
suggested in literature is to extract them from the probabilities of Bu →
K0∗ρ+ and Bu → K0π+ decays to which tree amplitudes almost do not
contribute [4, 5]1:
Br(Bd → ρ+ρ−)P =
η2 + (1− ρ)2
]2 τBd
Br(K0∗ρ+) ≈ 0.34·10−6 ,
1Contribution of tree amplitudes to these decays comes from the rescattering (Bu →
K+π0)T , K
+π0 → K0π+, and taking into account CKM suppression of the tree am-
plitudes of B → Kπ(K∗ρ) decays relative to the penguin amplitudes we can cautiously
estimate tree contribution as not more than 10% of penguin one .
Br(Bd → π+π−)P =
η2 + (1− ρ)2
]2 τBd
Br(K0π+) ≈ 0.59·10−6 ,
where fρ = 209 MeV and fK∗ = 218 MeV are the vector meson decay
constants, λ = 0.23, η = 0.34 and ρ = 0.20 are the CKM matrix parameters
in Wolfenstein parametrization [6], fK/fπ = 1.2 and the central values of
Br(Bu → K0∗ρ+) = (9.2±1.5)·10−6 and Br(Bu → K0π+) = (23.1±1.0)·10−6
[1] were used. The accuracy of equations (5) and (6) depends on the accuracy
of d ↔ s interchange symmetry (U -spin symmetry) of b → d(s) transition
amplitudes described by QCD penguin, however when the ratio of (5) to (6)
is calculated uncertainty factors partially cancel out and we obtain rather
stable result: instead of being enchanced as in the case of the tree amplitude
intermediate vector mesons contribution into penguin Bd → π+π− amplitude
is suppressed, (δ̃π0 )ρρ ≈ 1/2.8(δπ0 )ρρ. Taking into account that fraction of
longitudinally polarized vector mesons produced in Bu → K0∗ρ+ decays is
about 50% we get additional suppression of (δ̃π0 )ρρ by factor
Finally, phase δπP comes from the imaginary part of the penguin loop
with c-quark propagating in it [8]. In order to calculate δπP let us consider
corresponding quark diagram. The charm penguin contribution is given by
the following expression:
P = −Pc(k2) =
) + i
1− 4m
, (7)
where k is the sum of momenta of two quarks to which gluon radiated from
penguin decays: k = p1 + p2. One of these quarks forms π-meson with the
spectator quark, so neglecting spectator quark momentum in the rest frame
of B-meson we have p1 = (
). The second quark forms another π-meson
with d̄-quark radiated from penguin: p2 = x(
) where 0 < x < 1 is
the fraction of π+ momentum carried by u-quark. Substituting k2 = xm2b
into (7) and integrating it with the asymptotic quark distribution function
in π-meson ϕπ(x) = x(1−x) we obtain the value of δπP which depends on the
ratio 4m2c/m
b . In particular, for mb = 5.3 GeV and mc = 1.9 GeV (which
correspond to the masses of physical states) we obtain δπP ≈ 10o, a small
positive value. A nonperturbative calculation of δπP described in Section 3
demonstrates that the sign of δπP can be negative.
Our next task is to determine the difference of FSI phases δπ0 − δπ2 (the
large value of it is responsible for a relatively small value of Rπ). If we neglect
the penguin contribution, then from (2) - (4) we get the following expression:
cos(δπ0 − δπ2 ) =
B+− − 2B00 + 23
B+− +B00 − 23
, (8)
where Bik’s are the C-averaged branching ratios, while τ0/τ+ ≡ τ(Bd)/τ(Bu) =
0.92. Substituting the central values from Table 1 we get |δπ0 − δπ2 | = 48o.
Penguin contributions to Bik do not interfere with tree ones because α =
π − β − γ is almost equal to π/2. Taking P 2 terms into account with the
help of (6) (subtracting 0.59 and 0.30 from the first and the second lines of
Table 1 numbers describing B → ππ data correspondingly) we get:
|δπ0 − δπ2 | = 37o ± 10o . (9)
The accuracy of this 11o decrease of the absolute value of the phases difference
is determined by the accuracy of (6) and is not high. In recent paper [2] the
global fit of B → ππ and B → πK decay data was made. The tree amplitudes
of B → ππ decays were designated in [2] by T for B → π+π− and by C for
B → π0π0. According to [2] the difference of FSI phases between C and T
equals δC = −58o ± 10o, |C| = 0.37 ± 0.05, |T | = 0.57 ± 0.05 in the units
of 104 eV. The phase shift between the isospin amplitudes is determined by
these quantities:
tan(δ0 − δ2) =
3TC sin(−δC)
2T 2 + TC cos δC − C2
, (10)
and substituting the numbers we obtain:
δ0 − δ2 = 40o ± 7o , (11)
the result very close to (9). However, the same d ↔ s interchange symmetry
was used in [2] when relating B → ππ and B → Kπ decays. Fit [2] was
made in the same “t-convention” which we use (see the statement at the end
of page 3 of the paper [2]: “for simplicity, we will assume ... Ptc = Ptu”),
therefore the obtained results can be directly compared with ours.
Now let us consider B → ρρ decays. According to BABAR and BELLE
results ρ mesons produced in B decays are almost entirely longitudinally
polarized (fL(ρ+ρ−) = 0.98± 0.03[9], fL(ρ+ρ0) = 0.91± 0.4 [10], fL(ρ0ρ0) =
0.86 ± 0.12 [11]). For B decays into the longitudinally polarized ρ-mesons
we can write formulas analogous to (2) - (4) and we can find FSI phases
difference with the help of analog of (8). Substituting the central values of
branching ratios of B → ρρ decays from Table 1 we obtain: |δρ0−δ
2 | = 21o. In
order to subtract the penguin contribution with the help of (5) we should take
into account that in Bu → K0∗ρ+ decays the fraction of the longitudinally
polarized vector mesons equals approximately 50% [12], so we should subtract
0.17 · 10−6 in case of decay to ρ+ρ− and 0.08 · 10−6 for decay into ρ0ρ0. In
this way we obtain:
|δρ0 − δ
2 | = 20o+8
−20o , (12)
and the factor 2 difference between (12) and (9) or (11) is responsible for
the different patterns of B → ρρ and B → ππ decay probabilities. Let us
emphasize that while |δρ0 − δ
2 | being only one standard deviation from zero
can be very small this is not so for |δπ0 − δπ2 |.
3 Calculation of the FSI phases of B → ππ
and B → ρρ decay amplitudes
Among three amplitudes of B → ππ decays (2)–(4) only two are independent.
We will calculate FSI phases of B → π+π0 and B → π+π− amplitudes and
extract from them FSI phases of amplitudes with a definite isospin.
Our task is to take into account the intermediate state contributions
into FSI phases. As it was argued in Introduction we should consider only
two particle intermediate states with positive G-parity to which B-mesons
have relatively large decay probabilities. Alongside with ππ and ρρ there is
only one such state: πa1. So we will consider ρρ intermediate state which
transforms into ππ by π exchange in t-channel, πa1 intermediate state which
transforms into ππ by ρ exchange in t-channel and will take into account the
elastic channel B → ππ → ππ as well. This approach is analogous to the
FSI consideration performed in paper [13]. However in [13] 2 → 2 scattering
amplitudes were considered to be due to elementary particle exchanges in
t-channel. For vector particles exchanges s-channel partial wave amplitudes
behave as sJ−1 ∼ s0 and thus do not decrease with energy (decaying meson
mass). However it is well known that the correct behavior is given by Regge
theory: sαi(0)−1. For ρ-exchange αρ(0) ≈ 1/2 and the amplitude decrease
with energy as 1/
s. This effect is very spectacular for B → DD → ππ
chain with D∗(D∗2) exchange in t-channel: αD∗(0) ≈ −1 and reggeized D∗
meson exchange is damped as s−2 ≈ 10−3 in comparison with elementary
D∗ exchange (see for example [14]). For π-exchange, which gives a dominant
contribution to ρρ → ππ transition (see below), in the small t region the pion
is close to mass shell and its reggeization is not important.
We will use Feynman diagram approach to calculate FSI phases from
the triangle diagram with the low mass intermediate states X and Y (see
Figure 1). Integrating over loop momenta d4k we assume that integrals over
masses of intermediate states X and Y decrease rapidly with increase of these
masses. Then choosing z axis in the direction of momenta of the produced
meson M1 we can transform the integral over k0 and kz into the integral over
the invariant masses of clusters of intermediate particles X and Y
dk0dkz =
dsXdsY (13)
and deform integration contours in such a way that only low mass interme-
diate states contributions are taken into account while the contribution of
heavy states being small is neglected. In this way we get:
M Iππ = M
XY (δπXδπY + iT
XY→ππ) , (14)
where M
XY are the decay matrix elements without FSI interactions and
T J=0XY→ππ is the J = 0 partial wave amplitude of the process XY → ππ (T J =
(SJ − 1)/(2i)) which originates from the integral over d2k⊥.
For real T (14) coincides with the application of the unitarity condition
for the calculation of the imaginary part of M while for the imaginary T the
corrections to the real part of M are generated.
Let us calculate the imaginary parts of B → ππ decay amplitudes which
originate from B → ρρ → ππ chain with the help of unitarity condition 2:
ImM(B → ππ) =
d cos θ
M(ρρ → ππ)M∗(B → ρρ) , (15)
where θ is the angle between ρ and π momenta. For small values of θ or t
π-exchange in t-channel dominates and the calculation of Feynman diagram
2in this section the phases which originate from CKM matrix elements are omitted.
Figure 1: Diagram which describes FSI in the decay of heavy meson MQq
into two light mesons M1 and M2. X and Y are the clusters of particles with
small invariant masses sX , sY ≤ MQΛQCD, k is 4-momentum of a virtual
particle propagating in t-channel.
for ρρ → ππ amplitude with the elementary virtual π-meson exchange can
be trusted, as it was noted above. It was already stressed that ρ-mesons pro-
duced in B-decays are almost entirely longitudinally polarized. That is why
we will take into account only longitudinal polarization for the intermediate
ρ-mesons and amplitudes of B-decays into ππ and ρLρL are simply related
MB+→ρ+ρ0 = −
MB+→π+π0 , MBd→ρ+ρ− = −
MBd→π+π− . (16)
For the amplitude of ρ+ρ0 → π0π+ transition we have:
iM(ρ+ρ0 → π0π+) = −i
g2ρππ
(p1 − k1)2 −m2π
+)(k2ρ
0) , (17)
where p1, k1 and k2 are ρ
+, π0 and π+ momenta. From the width of ρ-meson
we get g2ρππ/16π = 2.85. For the longitudinally polarized ρ-mesons in their
center of mass system we have:
+ = k2ρ
0 = − 1
(t−m2π)(1 +
) +m2ρ
, (18)
where t = (p1 − k1)2. Changing the integration variable in (15) to t with the
help of dt =
(1− 2 m
)d cos θ and introducing formfactor exp(t/µ2) with
3relative negative sign of the amplitudes follows from the expressions for transition
formfactors in the factorization approximation, see for example [15].
the parameter µ2 ∼ 1 GeV 2 we obtain:
ImMB→π+π0 = +
(m2ρ−m
g2ρππdt
16πM2B ∗ 4m2ρ
(t−m2π)(1 +
+ 2m2ρ(1 +
t−m2π
exp(t/µ2)
MB→π+π0 . (19)
For µ2 = 2m2ρ the contributions of the first two terms in square brackets
cancel, while the third term gives:
ImMB→π+π0 = −
g2ρππ
3.1MB→π+π0 , (20)
and from (4) we get:
δπ2 (ρρ) = −4.9o . (21)
Let us note that in the limit MB → ∞ the ratio Br(Bd → ρρ)/Br(Bd → ππ)
grows as M2B, that is why FSI phase δ
2 (ρρ) (and δ
0 (ρρ)) diminishes as 1/MB.
The analogous consideration of ρ+ρ− intermediate state leads to the pos-
itive FSI phase of Bd → π+π− amplitude which is enhanced relatively to
δπ2 (ρρ) according to (16):
δπ+−(ρρ) = +5.7
o , (22)
and for FSI phase of the amplitude with isospin zero in the linear approxi-
mation we get:
δπ0 (ρρ) = δ
+−(ρρ) +
δπ+−(ρρ)− δπ2 (ρρ)
. (23)
We are able to extract the ratio A2/A0 from that of C-averaged Br(Bd →
π+π−), Br(Bd → π0π0) and Br(Bu → π+π0), subtracting penguin contribu-
tion as we did deriving (9):
B̃+− + B̃00
− 1 , (24)
= 0.80± 0.09 , (25)
and, finally:
δπ0 (ρρ) = 15
o , δπ0 (ρρ)− δπ2 (ρρ) = 20o . (26)
In this way we see that B → ρρ → ππ chain generates half of the experi-
mentally observed FSI phase difference of B → ππ tree amplitudes.
It is remarkable that FSI phases generated by B → ππ → ρρ chain are
damped by Br(B → ρ+ρ−, ρ+ρ0)/Br(B → π+π, π+π0) ratios and are a few
degrees:
2(ππ) =
∗ δπ2 (ρρ) = −1.4o , δ
+−(ππ) =
∗ δπ+−(ρρ) = 1.2o ,
(A0/A2)ρρ = 1.1 , δ
0(ππ) = 2.9
o , δ
0(ππ)− δ
2(ππ) ≈ 4o . (27)
Next we will take into account ππ intermediate state. From Regge analy-
sis of ππ elastic scattering we know that good description of the experimental
data is achieved when the exchanges of pomeron, ρ and f trajectories in t-
channel are taken into account [16]. Pomeron exchange dominates in elastic
ππ → ππ scattering at high energies. For αP (0) = 1 the corresponding
amplitude T is purely imaginary and the phases of matrix elements do not
change [3]. However taking into account that pomeron is ”supercritical”,
αP (0) ≈ 1.1, we obtain the phase of the amplitude generated by pomeron
exchange 4 which cancels the phases generated by ρ and f exchanges for
I = 2. For I = 0 the sum of ρ and f exchanges produces the purely imag-
inary amplitude T and the phase of the amplitude M is due to pomeron
”supercriticallity”:
δπ0 (ππ) = 5.0
o , δπ2 (ππ) = 0
o . (28)
In paper [3] the pomeron exchange amplitude was considered as purely
imaginary. As a result though important for branching ratios phase difference
δπ0 (ππ)−δπ2 (ππ) was the same (pomeron contribution being universal cancels
in the difference of phases) it came mainly from δπ2 (ππ) negative value. In
4The amplitude of 2 → 2 process due to supercritical pomeron exchange is T ∼
(s/s0)
αP (t)(1 + exp(−iπαP (t)))/(− sin(παP (t))) = (s/s0)(1+∆)(i + ∆π/2), where in the
last expression t = 0 was substituted and αP (0) = 1 +∆ was used (∆ ≈ 0.1).
this way result for the absolute value of direct CP-asymmetry Cπ+π− was
underestimated, see below.
Finally πa1 intermediate state should be accounted for. Large branching
ratio of Bd → π±a∓1 -decay ( Br(Bd → π±a∓1 ) = (40 ± 4) ∗ 10−6) is partially
compensated by small ρπa1 coupling constant (it is 1/3 of ρππ one). As a
result the contributions of πa1 intermediate state (which transforms into ππ
by ρ-trajectory exchange in t-channel) to FSI phases equal approximately
that part of ππ intermediate state contributions which is due to ρ-trajectory
exchange. Assuming that the sign of the πa1 intermediate state contribution
into phases is the same as that of elastic channel we obtain:
δπ0 (πa1) = 4
o , δπ2 (πa1) = −2o . (29)
Summing the imaginary parts of the amplitudes which follow from (21),
(26), (28) and (29) we finally obtain:
δπ0 = 23
o , δπ2 = −7o , δπ0 − δπ2 = 30o , (30)
and the accuracy of these numbers is not high, at the level of 50%.
The analogous consideration of the real parts of the loop corrections to
B → ππ decay amplitudes leads to the diminishing of the (real) tree am-
plitudes by ≈ 30%, and we can explain the experimentally observed value
δπ0 − δπ2 ≈ 40o in our model while for ρρ final state the analogous difference
is about three times smaller, δ
0 − δ
2 ≈ 15o.
Let us estimate the phase of the penguin amplitude δπP considering charmed
mesons intermediate states: B → D̄D, D̄∗D, D̄D∗, D̄∗D∗ → ππ 5. In Regge
model all these amplitudes are described at high energies by exchanges of
D∗(D∗2)-trajectories. An intercept of these exchange-degenerate trajectories
can be obtained using the method of [17] or from masses of D∗(2007) – 1−
andD∗2(2460) – 2
+ resonances, assuming linearity of these Regge-trajectories.
Both methodes give αD∗(0) = −0.8÷−1 and the slope α′D∗ ≈ 0.5GeV −2.
The amplitude of D+D− → π+π− reaction in the Regge model proposed
in papers [18, 19] can be written in the following form:
TDD̄→ππ(s, t) = −
e−iπα(t)Γ(1− αD∗(t))(s/scd)αD∗(t) , (31)
5These amplitudes are considered as penguin due to the proper combination of CKM
matrix elements.
where Γ(x) is the gamma function.
The t-dependence of Regge-residues is chosen in accord with the dual
models and is tested for light (u,d,s) quarks [18]. According to [19] scd ≈
2.2GeV 2.
Note that the sign of the amplitude is fixed by the unitarity in the t-
channel (close to the D∗-resonance). The constant g20 is determined by the
width of the D∗ → Dπ decay: g20/(16π) = 6.6. Using (14), analog of (15),
(31) and the branching ratio Br(B → DD̄) ≈ 2 · 10−4 [20] we obtain the
imaginary part of P and comparing it with the contribution of P in B →
π+π− decay probability (6) we get δπP ≈ −3.5o6. A smallness of the phase
is due to the low intercept of D∗-trajectory. The sign of δP is negative -
opposite to the positive sign which was obtained in perturbation theory (7).
Since DD̄-decay channel constitutes only ≈ 10% of all two-body charm-
anticharm decays of Bd-meson [20] taking these channels into account we can
easily get
δP ∼ −10o , (32)
which may be very important for the interpretation of the experimental data
on direct CP asymmetry C+− discussed in the next section.
4 CP asymmetries of Bd(B̄d) → ππ decays
The CP asymmetries are given by :
Cππ ≡
1− |λππ|2
1 + |λππ|2
, Sππ ≡
2Im(λππ)
1 + |λππ|2
, λππ ≡ e−2iβ
MB̄→ππ
MB→ππ
, (33)
where ππ is π+π− or π0π0.
From (2) for direct CP asymmetry in Bd(B̄d) → π+π− decays we readily
obtain:
C+− = −
sinα[
2A0 sin(δ0 − δ̃0 − δP ) + A2 sin(δ2 − δ̃0 − δP )]/
cos(δ0 − δ2)−
A0P̃ cosα cos(δ0 − δ̃0 − δP )−
6In integration over cos θ the region θ ≪ 1 dominates. In this region representation
(31) is valid.
− A2P̃√
cosα cos(δ2 − δ̃0 − δP ) + P̃ 2] , (34)
where
V ∗tdVtb
P . (35)
In order to make a numerical estimate we should know the ratios A0/A2
and P/A2. The first one is given by (25) while the second one can be extracted
from the ratio Br(Bu → K0π+)/Br(Bu → π0π+) assuming d ↔ s invariance
of the strong interactions:
Br(Bu → K0π+)
Br(Bu → π0π+)
f 2KP
2|V ∗tsVtb|2
A22|V ∗udVub|2
, (36)
= 0.092(0.009) . (37)
The numerical values of A0 andA2 are given with good accuracy by factor-
ization calculation, while P appears to be 2.5 times larger than factorization
result [3]. In view of this the validity of factor fK in (36) which originates
from factorization calculation of the penguin amplitude is questinable. If
factorization of the penguin amplitudes is not assumed then the ratio fK/fπ
in (36) should be replaced by unity. In this way we get 20% larger value of
P/A2 in (37) and we will take this value of uncertainty as an estimate of the
theoretical accuracy of the determination of P :
= 0.21(0.04) , (38)
Taking into account that unitarity triangle angle α ≈ 90o and angles δ̃0 and
δP are of the order of few degrees from (34) we obtain:
C+− ≈ −0.28[1.1 sin(δ0 − δ̃0 − δP ) + sin(δ2 − δ̃0 − δP )] ≈
≈ −0.56 sin((δ0 + δ2)/2− δ̃0 − δP ) . (39)
In order to determine the lower bound on the value of C+− let us suppose
that δ0 = 37
o, δ2 = 0
o (we keep the difference δ0− δ2 = 37o, as it follows from
the data on B → ππ decay probabilities (9)), and neglect small values of δ̃0
and δP :
C+− > −0.18 . (40)
Concerning experimental number it could well happen that finally it will
be considerably below our bound. In this case the result of nonperturbative
calculation of penguin phase will be confirmed. Substituting in (39) δ0 =
30o, δ2 = −7o and δP from (32) we obtain the following central value:
C+− = −0.21 . (41)
It is instructive to compare the obtained numbers with the value of C+−
which follows from the asymmetry ACP (K
+π−) if d ↔ s symmetry is sup-
posed [21]:
C+− =
ACP (K
Γ(B → K+π−)
Γ(B → π+π−)
sin(β + γ)
sin(γ)
= 1.2(−2)(−0.093± 0.015)19.8
sin 82o
sin 60o
0.87 = −0.24± 0.04 . (42)
Let us note that one factor fπ/fK in the last equation appears from the
matrix element of the tree operator, the second one - from the matrix element
of the penguin operator. If because of nonfactorization of penguin amplitudes
we will omit the factor which appears from the penguin [5], then the numbers
in the right-hand sides of (40, 41) and (42) will become 20% smaller.
The experimental results obtained by Belle [22] and BABAR [23] are
contradictory
CBelle+− = −0.55(0.09) , CBABAR+− = −0.21(0.09), (43)
Belle number being far below (40) and (41).
For direct CP asymmetry in Bd(B̄d) → π0π0 decay from (3) we readily
obtain:
C00 = −
P̃ sinα[A0 sin(δ0 − δ̃0 − δP )−
2A2 sin(δ2 − δ̃0 − δP )]/
A0A2 cos(δ0 − δ2)−
A0P̃ cosα cos(δ0 − δ̃0 − δP ) +
A2P̃ cosα cos(δ2 − δ̃0 − δP ) + P̃ 2] , (44)
C00 ≈ −1.06[0.8 sin(δ0 − δ̃0 − δP )− 1.4 sin(δ2 − δ̃0 − δP )] ≈ −0.6 , (45)
considerably smaller than C+−. This unusually large direct CPV (measured
by |C00|) is intriguing task for future measurements since the present exper-
imental error is too big:
exper
00 = −0.36(0.32) . (46)
Belle and BABAR agree now on the value of another CPV asymmetry
measured in Bd(B̄d) → π+π− decays: Sexper+− = −0.62 ± 0.09 [22, 23]. From
this measurement the value of unitarity triangle angle α can be extracted.
Neglecting the penguin contribution we get:
sin 2αT = S+− , (47)
αT = 109o ± 3o . (48)
Penguin shifts the value of α. The accurate formula looks like:
S+− = [sin 2α(
cos(δ0 − δ2))−
A2P̃√
sinα cos(δ2 − δ̃0 − δP )−
A0P̃ sinα cos(δ0 − δ̃0 − δP )]/
cos(δ0 − δ2)−
A0P̃ cosα cos(δ0 − δ̃0 − δP )−
A2P̃√
cosα cos(δ2 − δ̃0 − δP ) + P̃ 2] , (49)
and since all the phase shifts are not big the values of cosines in (49) are
rather stable relative to their variations. For numerical estimates we take
δ0 = 30
o, δ2 = −7o and neglect δ̃0 and δP . In this way we get:
(α)ππ = 88
o ± 40(exper)± 50(theor) , (50)
where the first error comes from uncertainty in S
exper
+− while the second one
comes from that in the value of penguin amplitude, (38). Relatively large
theoretical uncertainty in the value of P̃ does not prevent to determine α
with good precision.
The relative smallness of penguin contribution to B → ρρ decay am-
plitudes allow us to determine α with better theoretical accuracy from the
experimental measurement of (S+−)ρρ just as it was done in [24]. With the
help of (5) we obtain:
)ρρ = 0.060(0.012) , (51)
where the same 20% uncertainty in extracting penguin amplitude is supposed.
Using the ratio (A0/A2)ρρ determined in (27) from the (49) neglecting strong
phases (which are much smaller than in the case of B → ππ decays) and
taking into account the recent experimental result (S
exper
+− )ρρ = −0.06± 0.18
[1] we obtain:
(α)ρρ = 87
o ± 50(exper)± 10(theor) . (52)
Let us point out that considerably larger theoretical error quoted in [4]
follows from the larger theoretical uncertainty in the value of penguin ampli-
tude assumed in that paper.
Our results for α should be compared with the numbers which follow from
the global fit of unitarity triangle [6, 7]:
αCKMfitter = (99.0+4.0−9.4)
o , αUTfit = (93± 4)o . (53)
We conclude this section with the prediction for the value of CPV asym-
metry S00:
S00 = [sin 2α(
2A0A2
cos(δ0 − δ2)) +
2A2P̃√
sinα cos(δ2 − δ̃0 − δP )−
A0P̃ sinα cos(δ0 − δ̃0 − δP )]/
2A0A2
cos(δ0 − δ2)−
A0P̃ cosα cos(δ0 − δ̃0 − δP ) +
2A2P̃√
cosα cos(δ2 − δ̃0 − δP ) + P̃ 2] = 0.70± 0.15 , (54)
a large asymmetry with the sign opposite to that of S+−.
5 Conclusions
FSI appeared to be very important in B → ππ decays.
The description of these interactions presented in the paper allows to
explain the experimentally observed difference of the ratios of decay proba-
bilities to the neutral and charged modes in B → ππ and B → ρρ decays.
Rather large absolute value of direct CP asymmetry C+− (if confirmed
experimentally) will be a manifestation of the negative sign of penguin FSI
phase in accord with nonperturbative calculation and opposite to perturba-
tive result.
We are grateful to L.V.Akopyan for checking formulas, Jose Ocariz for
recommendation to include the result for angle α which follows from CP
asymmetry (S+−)ρρ and M.B.Voloshin for useful discussion.
This work was supported by Russian Agency of Atomic Energy;
A.K. was partly supported by grants RFBR 06-02-17012, RFBR 06-02-
72041-MNTI, INTAS 05-103-7515 and state contract 02.445.11.7424;
M.V. was partly supported by grants RFBR 05-02-17203 and
NSh-5603.2006.2.
References
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(2006) 68.
[5] M. Gronau, J.L. Rosner, Phys. Lett. B595 (2004) 339.
[6] CKM fitter, http://ckmfitter.in2p3.fr.
[7] UTfit, http://utfit.roma1.infn.it.
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G.M. Gérard and W.-S. Hou, Phys. Rev. D43, (1991) 2909.
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B. Aubert et al., BABAR Collaboration, Phys. Rev. Lett. 97 (2006)
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[24] M.I.Vysotsky, Yad. Fiz. 69, (2006) 703.

  The final state interactions (FSI) model in which soft rescattering of low
mass intermediate states dominates is suggested. It explains why the strong
interaction phases are large in the $B_d\to\pi\pi$ channel and are considerably
smaller in the $B_d\to\rho\rho$ one. Direct CP asymmetries of $B_d\to\pi\pi$
decays which are determined by FSI phases are considered as well.

Introduction
There are three reasons to study FSI in B decays: to predict (or explain)
the pattern of branching ratios, to study strong interactions, and to forsee in
what decays direct CPV will be large. In view of this necessity a model for
FSI in B decays to two light mesons is suggested and explored in the present
paper.
The probabilities of three B → ππ and three B → ρρ decays are measured
now with good accuracy. The C-averaged branching ratios of these decays
are presented in Table 1 [1]. Let us look at the ratio of the charge averaged
Bd decay probabilities to the charged and neutral mesons:
Br(Bd → ρ+ρ−)
Br(Bd → ρ0ρ0)
≈ 20 , Rπ ≡
Br(Bd → π+π−)
Br(Bd → π0π0)
≈ 4 . (1)
∗kaidalov@itep.ru
†vysotsky@itep.ru
http://arxiv.org/abs/0704.0404v1
Table 1
Mode Br(10−6) Mode Br(10−6)
Bd → π+π− 5.2± 0.2 Bd → ρ+ρ− 23.1± 3.3
Bd → π0π0 1.3± 0.2 Bd → ρ0ρ0 1.16± 0.46
Bu → π+π0 5.7± 0.4 Bu → ρ+ρ0 18.2± 3.0
C-averaged branching ratios of B → ππ and B → ρρ decays.
The large difference of Rρ and Rπ is due to the difference of FSI phases
in B → ρρ and B → ππ decays (see below). In Section 2 we will determine
the differences of FSI phases of tree amplitudes which describe B → ρρ and
B → ππ decays into the states with isospins zero and two from the data
presented in Table 1. As a next step we will suggest a mechanism which
produces such phases. Once this mechanism is defined it becomes possible to
calculate FSI phases of decay amplitudes into states with a definite isospin
(not only their differences). A central question is: what intermediate states
produce FSI phases in B-meson decays into two light mesons. In the weak
decay b → uū(dd̄)d in the rest frame of a heavy quark (which is B-meson
rest frame as well) three fast light quarks are produced. Their energies are
of the order of MB/3 and momenta are more or less isotropically oriented.
The energy of the fourth (spectator) quark is of the order of ΛQCD. This four
quark state transforms mainly into multi pi-meson final state with the average
pion multiplicity about 9 (this number follows from the experimentally known
charged particles multiplicity in e+e− annihilation at Ecm = 3GeV multiplied
by 1.5 ∗ 1.5 in order to take neutral pions and third quark jet into account).
The total branching ratio of such decays is about 10−2. However such meson
state does not transform into the state composed from two light mesons
moving into opposite directions with momenta MB/2. What meson state
does transform into two light mesons can be understood from the inverse
reaction of two light meson scattering at the center of mass energy equal
to the mass of B-meson. The produced hadronic state consists of two jets
of particles moving in opposite directions. Each jet should originate from
a quark-antiquark pair produced in the weak decay of b-quark. The square
of invariant mass of a jet which contains spectator quark does not exceed
MBΛQCD and is much smaller than M
B. The energy of this jet is determined
by that of a companion quark and is about MB/2. That is why the square
of invariant mass of the second jet also does not exceed MBΛQCD. So for
B-decays the mass of a hadron cluster which transforms into light meson
in the final state should not exceed 1.5 GeV. Following these arguments in
the calculation of the imaginary parts of the decay amplitudes we will take
into account only two (relatively light) particle intermediate states for which
branching ratios of B-meson are maximal.
In Section 3 we will calculate FSI phases of tree amplitudes describing
B → ππ decays taking into account ρρ, ππ and πa1 intermediate states which
by t(u)-channel exchanges are converted into ππ. We will find that large
probability of B → ρ+ρ− decay explains about half of FSI phases of B → ππ
decays. Relatively small probability of B → π+π− decay prevents generation
of noticeable FSI phase of B → ρρ amplitudes through B → π+π− → ρρ
chain.
We will demonstrate that the strong interaction phase of the penguin
amplitude is opposite to the result of quark loop calculation, which is very
important for the value of a direct CPV asymmetry Cπ+π− ≡ C+− discussed
in Section 4. Predictions for CPV asymmetries C00 and S00 will be presented
in Section 4 as well and the value of the unitarity triangle angle α will be
extracted from the experimental data on CPV asymmetry S+−.
Subject of rare B decays is an object of intensive study nowadays and an
interested reader can find extensive list of references in a recent paper [2].
2 Phenomenology; |δπ0 − δπ2 | and |δ
0 − δ
Let us present B → ππ decay amplitudes in the so-called “t-convention”, in
which the penguin amplitude with the intermediate c-quark multiplied by
ud + VcbV
cd + VtbV
td = 0 is subtracted from the decay amplitudes [3]:
MB̄d→π+π− =
|VubV ∗ud|m2Bfπf+(0)
iδπ2 +
+ e−iγ
iδπ0 +
V ∗tdVtb
eiβPei(δ
+δ̃π0 )
, (2)
MB̄d→π0π0 =
|VubV ∗ud|m2Bfπf+(0)
iδπ2 −
− e−iγ 1√
iδπ0 −
V ∗tdVtb
eiβPei(δ
+δ̃π0 )
, (3)
MB̄u→π−π0 =
|VubV ∗ud|m2Bfπf+(0)
e−iγA2e
, (4)
where Vik are the elements of CKM matrix, γ and β are the unitarity triangle
angles and we factor out the product m2Bfπf+(0) which appears when the
decay amplitudes are calculated in the factorization approximation. A2 and
A0 are the absolute values of the decay amplitudes into the states with I = 2
and 0, generated by operators O1 and O2 (tree amplitudes), while P is the
absolute value of QCD penguin amplitude (generated by operators O3 − O6
of effective nonleptonic Hamiltonian which describes b quark decays into the
states without charm and strange quarks). δπ0 , δ
2 and δ̃
0 are FSI phases of
these three amplitudes, and it is very important for what follows that all of
them are different. It is easy to understand why δπ0 is different from δ
2 : strong
interaction depends on the isospin and is different for I = 0 and I = 2. For
example, there are definitely quark-antiquark resonances with I = 0, while
exotic resonances with I = 2 should be made from at least four quarks and
their existence is questionable. The reason why δπ0 differs from δ̃
0 is more sub-
tle. Let us consider the intermediate state made from two charged ρ-mesons
which contributes to FSI phases: Bd → ρ+ρ− → ππ. ρ+ρ− intermediate state
contribution to FSI phases can be large since Br(Bd → ρ+ρ−) is big. Both
tree and penguin induced amplitudes get FSI phases through this chain. Its
contribution to δπ0 is proportional to
(BrBd → ρ+ρ−)T/(BrBd → π+π−)T ≈
(BrBd → ρ+ρ−)/(BrBd → π+π−) ≈ 2.1, while that to δ̃π0 is proportional to
(BrBd → ρ+ρ−)P/(BrBd → π+π−)P .
How can we determine the penguin contributions to the probabilities
of Bd → ρ+ρ− and Bd → π+π−-decays? The most straightforward way
suggested in literature is to extract them from the probabilities of Bu →
K0∗ρ+ and Bu → K0π+ decays to which tree amplitudes almost do not
contribute [4, 5]1:
Br(Bd → ρ+ρ−)P =
η2 + (1− ρ)2
]2 τBd
Br(K0∗ρ+) ≈ 0.34·10−6 ,
1Contribution of tree amplitudes to these decays comes from the rescattering (Bu →
K+π0)T , K
+π0 → K0π+, and taking into account CKM suppression of the tree am-
plitudes of B → Kπ(K∗ρ) decays relative to the penguin amplitudes we can cautiously
estimate tree contribution as not more than 10% of penguin one .
Br(Bd → π+π−)P =
η2 + (1− ρ)2
]2 τBd
Br(K0π+) ≈ 0.59·10−6 ,
where fρ = 209 MeV and fK∗ = 218 MeV are the vector meson decay
constants, λ = 0.23, η = 0.34 and ρ = 0.20 are the CKM matrix parameters
in Wolfenstein parametrization [6], fK/fπ = 1.2 and the central values of
Br(Bu → K0∗ρ+) = (9.2±1.5)·10−6 and Br(Bu → K0π+) = (23.1±1.0)·10−6
[1] were used. The accuracy of equations (5) and (6) depends on the accuracy
of d ↔ s interchange symmetry (U -spin symmetry) of b → d(s) transition
amplitudes described by QCD penguin, however when the ratio of (5) to (6)
is calculated uncertainty factors partially cancel out and we obtain rather
stable result: instead of being enchanced as in the case of the tree amplitude
intermediate vector mesons contribution into penguin Bd → π+π− amplitude
is suppressed, (δ̃π0 )ρρ ≈ 1/2.8(δπ0 )ρρ. Taking into account that fraction of
longitudinally polarized vector mesons produced in Bu → K0∗ρ+ decays is
about 50% we get additional suppression of (δ̃π0 )ρρ by factor
Finally, phase δπP comes from the imaginary part of the penguin loop
with c-quark propagating in it [8]. In order to calculate δπP let us consider
corresponding quark diagram. The charm penguin contribution is given by
the following expression:
P = −Pc(k2) =
) + i
1− 4m
, (7)
where k is the sum of momenta of two quarks to which gluon radiated from
penguin decays: k = p1 + p2. One of these quarks forms π-meson with the
spectator quark, so neglecting spectator quark momentum in the rest frame
of B-meson we have p1 = (
). The second quark forms another π-meson
with d̄-quark radiated from penguin: p2 = x(
) where 0 < x < 1 is
the fraction of π+ momentum carried by u-quark. Substituting k2 = xm2b
into (7) and integrating it with the asymptotic quark distribution function
in π-meson ϕπ(x) = x(1−x) we obtain the value of δπP which depends on the
ratio 4m2c/m
b . In particular, for mb = 5.3 GeV and mc = 1.9 GeV (which
correspond to the masses of physical states) we obtain δπP ≈ 10o, a small
positive value. A nonperturbative calculation of δπP described in Section 3
demonstrates that the sign of δπP can be negative.
Our next task is to determine the difference of FSI phases δπ0 − δπ2 (the
large value of it is responsible for a relatively small value of Rπ). If we neglect
the penguin contribution, then from (2) - (4) we get the following expression:
cos(δπ0 − δπ2 ) =
B+− − 2B00 + 23
B+− +B00 − 23
, (8)
where Bik’s are the C-averaged branching ratios, while τ0/τ+ ≡ τ(Bd)/τ(Bu) =
0.92. Substituting the central values from Table 1 we get |δπ0 − δπ2 | = 48o.
Penguin contributions to Bik do not interfere with tree ones because α =
π − β − γ is almost equal to π/2. Taking P 2 terms into account with the
help of (6) (subtracting 0.59 and 0.30 from the first and the second lines of
Table 1 numbers describing B → ππ data correspondingly) we get:
|δπ0 − δπ2 | = 37o ± 10o . (9)
The accuracy of this 11o decrease of the absolute value of the phases difference
is determined by the accuracy of (6) and is not high. In recent paper [2] the
global fit of B → ππ and B → πK decay data was made. The tree amplitudes
of B → ππ decays were designated in [2] by T for B → π+π− and by C for
B → π0π0. According to [2] the difference of FSI phases between C and T
equals δC = −58o ± 10o, |C| = 0.37 ± 0.05, |T | = 0.57 ± 0.05 in the units
of 104 eV. The phase shift between the isospin amplitudes is determined by
these quantities:
tan(δ0 − δ2) =
3TC sin(−δC)
2T 2 + TC cos δC − C2
, (10)
and substituting the numbers we obtain:
δ0 − δ2 = 40o ± 7o , (11)
the result very close to (9). However, the same d ↔ s interchange symmetry
was used in [2] when relating B → ππ and B → Kπ decays. Fit [2] was
made in the same “t-convention” which we use (see the statement at the end
of page 3 of the paper [2]: “for simplicity, we will assume ... Ptc = Ptu”),
therefore the obtained results can be directly compared with ours.
Now let us consider B → ρρ decays. According to BABAR and BELLE
results ρ mesons produced in B decays are almost entirely longitudinally
polarized (fL(ρ+ρ−) = 0.98± 0.03[9], fL(ρ+ρ0) = 0.91± 0.4 [10], fL(ρ0ρ0) =
0.86 ± 0.12 [11]). For B decays into the longitudinally polarized ρ-mesons
we can write formulas analogous to (2) - (4) and we can find FSI phases
difference with the help of analog of (8). Substituting the central values of
branching ratios of B → ρρ decays from Table 1 we obtain: |δρ0−δ
2 | = 21o. In
order to subtract the penguin contribution with the help of (5) we should take
into account that in Bu → K0∗ρ+ decays the fraction of the longitudinally
polarized vector mesons equals approximately 50% [12], so we should subtract
0.17 · 10−6 in case of decay to ρ+ρ− and 0.08 · 10−6 for decay into ρ0ρ0. In
this way we obtain:
|δρ0 − δ
2 | = 20o+8
−20o , (12)
and the factor 2 difference between (12) and (9) or (11) is responsible for
the different patterns of B → ρρ and B → ππ decay probabilities. Let us
emphasize that while |δρ0 − δ
2 | being only one standard deviation from zero
can be very small this is not so for |δπ0 − δπ2 |.
3 Calculation of the FSI phases of B → ππ
and B → ρρ decay amplitudes
Among three amplitudes of B → ππ decays (2)–(4) only two are independent.
We will calculate FSI phases of B → π+π0 and B → π+π− amplitudes and
extract from them FSI phases of amplitudes with a definite isospin.
Our task is to take into account the intermediate state contributions
into FSI phases. As it was argued in Introduction we should consider only
two particle intermediate states with positive G-parity to which B-mesons
have relatively large decay probabilities. Alongside with ππ and ρρ there is
only one such state: πa1. So we will consider ρρ intermediate state which
transforms into ππ by π exchange in t-channel, πa1 intermediate state which
transforms into ππ by ρ exchange in t-channel and will take into account the
elastic channel B → ππ → ππ as well. This approach is analogous to the
FSI consideration performed in paper [13]. However in [13] 2 → 2 scattering
amplitudes were considered to be due to elementary particle exchanges in
t-channel. For vector particles exchanges s-channel partial wave amplitudes
behave as sJ−1 ∼ s0 and thus do not decrease with energy (decaying meson
mass). However it is well known that the correct behavior is given by Regge
theory: sαi(0)−1. For ρ-exchange αρ(0) ≈ 1/2 and the amplitude decrease
with energy as 1/
s. This effect is very spectacular for B → DD → ππ
chain with D∗(D∗2) exchange in t-channel: αD∗(0) ≈ −1 and reggeized D∗
meson exchange is damped as s−2 ≈ 10−3 in comparison with elementary
D∗ exchange (see for example [14]). For π-exchange, which gives a dominant
contribution to ρρ → ππ transition (see below), in the small t region the pion
is close to mass shell and its reggeization is not important.
We will use Feynman diagram approach to calculate FSI phases from
the triangle diagram with the low mass intermediate states X and Y (see
Figure 1). Integrating over loop momenta d4k we assume that integrals over
masses of intermediate states X and Y decrease rapidly with increase of these
masses. Then choosing z axis in the direction of momenta of the produced
meson M1 we can transform the integral over k0 and kz into the integral over
the invariant masses of clusters of intermediate particles X and Y
dk0dkz =
dsXdsY (13)
and deform integration contours in such a way that only low mass interme-
diate states contributions are taken into account while the contribution of
heavy states being small is neglected. In this way we get:
M Iππ = M
XY (δπXδπY + iT
XY→ππ) , (14)
where M
XY are the decay matrix elements without FSI interactions and
T J=0XY→ππ is the J = 0 partial wave amplitude of the process XY → ππ (T J =
(SJ − 1)/(2i)) which originates from the integral over d2k⊥.
For real T (14) coincides with the application of the unitarity condition
for the calculation of the imaginary part of M while for the imaginary T the
corrections to the real part of M are generated.
Let us calculate the imaginary parts of B → ππ decay amplitudes which
originate from B → ρρ → ππ chain with the help of unitarity condition 2:
ImM(B → ππ) =
d cos θ
M(ρρ → ππ)M∗(B → ρρ) , (15)
where θ is the angle between ρ and π momenta. For small values of θ or t
π-exchange in t-channel dominates and the calculation of Feynman diagram
2in this section the phases which originate from CKM matrix elements are omitted.
Figure 1: Diagram which describes FSI in the decay of heavy meson MQq
into two light mesons M1 and M2. X and Y are the clusters of particles with
small invariant masses sX , sY ≤ MQΛQCD, k is 4-momentum of a virtual
particle propagating in t-channel.
for ρρ → ππ amplitude with the elementary virtual π-meson exchange can
be trusted, as it was noted above. It was already stressed that ρ-mesons pro-
duced in B-decays are almost entirely longitudinally polarized. That is why
we will take into account only longitudinal polarization for the intermediate
ρ-mesons and amplitudes of B-decays into ππ and ρLρL are simply related
MB+→ρ+ρ0 = −
MB+→π+π0 , MBd→ρ+ρ− = −
MBd→π+π− . (16)
For the amplitude of ρ+ρ0 → π0π+ transition we have:
iM(ρ+ρ0 → π0π+) = −i
g2ρππ
(p1 − k1)2 −m2π
+)(k2ρ
0) , (17)
where p1, k1 and k2 are ρ
+, π0 and π+ momenta. From the width of ρ-meson
we get g2ρππ/16π = 2.85. For the longitudinally polarized ρ-mesons in their
center of mass system we have:
+ = k2ρ
0 = − 1
(t−m2π)(1 +
) +m2ρ
, (18)
where t = (p1 − k1)2. Changing the integration variable in (15) to t with the
help of dt =
(1− 2 m
)d cos θ and introducing formfactor exp(t/µ2) with
3relative negative sign of the amplitudes follows from the expressions for transition
formfactors in the factorization approximation, see for example [15].
the parameter µ2 ∼ 1 GeV 2 we obtain:
ImMB→π+π0 = +
(m2ρ−m
g2ρππdt
16πM2B ∗ 4m2ρ
(t−m2π)(1 +
+ 2m2ρ(1 +
t−m2π
exp(t/µ2)
MB→π+π0 . (19)
For µ2 = 2m2ρ the contributions of the first two terms in square brackets
cancel, while the third term gives:
ImMB→π+π0 = −
g2ρππ
3.1MB→π+π0 , (20)
and from (4) we get:
δπ2 (ρρ) = −4.9o . (21)
Let us note that in the limit MB → ∞ the ratio Br(Bd → ρρ)/Br(Bd → ππ)
grows as M2B, that is why FSI phase δ
2 (ρρ) (and δ
0 (ρρ)) diminishes as 1/MB.
The analogous consideration of ρ+ρ− intermediate state leads to the pos-
itive FSI phase of Bd → π+π− amplitude which is enhanced relatively to
δπ2 (ρρ) according to (16):
δπ+−(ρρ) = +5.7
o , (22)
and for FSI phase of the amplitude with isospin zero in the linear approxi-
mation we get:
δπ0 (ρρ) = δ
+−(ρρ) +
δπ+−(ρρ)− δπ2 (ρρ)
. (23)
We are able to extract the ratio A2/A0 from that of C-averaged Br(Bd →
π+π−), Br(Bd → π0π0) and Br(Bu → π+π0), subtracting penguin contribu-
tion as we did deriving (9):
B̃+− + B̃00
− 1 , (24)
= 0.80± 0.09 , (25)
and, finally:
δπ0 (ρρ) = 15
o , δπ0 (ρρ)− δπ2 (ρρ) = 20o . (26)
In this way we see that B → ρρ → ππ chain generates half of the experi-
mentally observed FSI phase difference of B → ππ tree amplitudes.
It is remarkable that FSI phases generated by B → ππ → ρρ chain are
damped by Br(B → ρ+ρ−, ρ+ρ0)/Br(B → π+π, π+π0) ratios and are a few
degrees:
2(ππ) =
∗ δπ2 (ρρ) = −1.4o , δ
+−(ππ) =
∗ δπ+−(ρρ) = 1.2o ,
(A0/A2)ρρ = 1.1 , δ
0(ππ) = 2.9
o , δ
0(ππ)− δ
2(ππ) ≈ 4o . (27)
Next we will take into account ππ intermediate state. From Regge analy-
sis of ππ elastic scattering we know that good description of the experimental
data is achieved when the exchanges of pomeron, ρ and f trajectories in t-
channel are taken into account [16]. Pomeron exchange dominates in elastic
ππ → ππ scattering at high energies. For αP (0) = 1 the corresponding
amplitude T is purely imaginary and the phases of matrix elements do not
change [3]. However taking into account that pomeron is ”supercritical”,
αP (0) ≈ 1.1, we obtain the phase of the amplitude generated by pomeron
exchange 4 which cancels the phases generated by ρ and f exchanges for
I = 2. For I = 0 the sum of ρ and f exchanges produces the purely imag-
inary amplitude T and the phase of the amplitude M is due to pomeron
”supercriticallity”:
δπ0 (ππ) = 5.0
o , δπ2 (ππ) = 0
o . (28)
In paper [3] the pomeron exchange amplitude was considered as purely
imaginary. As a result though important for branching ratios phase difference
δπ0 (ππ)−δπ2 (ππ) was the same (pomeron contribution being universal cancels
in the difference of phases) it came mainly from δπ2 (ππ) negative value. In
4The amplitude of 2 → 2 process due to supercritical pomeron exchange is T ∼
(s/s0)
αP (t)(1 + exp(−iπαP (t)))/(− sin(παP (t))) = (s/s0)(1+∆)(i + ∆π/2), where in the
last expression t = 0 was substituted and αP (0) = 1 +∆ was used (∆ ≈ 0.1).
this way result for the absolute value of direct CP-asymmetry Cπ+π− was
underestimated, see below.
Finally πa1 intermediate state should be accounted for. Large branching
ratio of Bd → π±a∓1 -decay ( Br(Bd → π±a∓1 ) = (40 ± 4) ∗ 10−6) is partially
compensated by small ρπa1 coupling constant (it is 1/3 of ρππ one). As a
result the contributions of πa1 intermediate state (which transforms into ππ
by ρ-trajectory exchange in t-channel) to FSI phases equal approximately
that part of ππ intermediate state contributions which is due to ρ-trajectory
exchange. Assuming that the sign of the πa1 intermediate state contribution
into phases is the same as that of elastic channel we obtain:
δπ0 (πa1) = 4
o , δπ2 (πa1) = −2o . (29)
Summing the imaginary parts of the amplitudes which follow from (21),
(26), (28) and (29) we finally obtain:
δπ0 = 23
o , δπ2 = −7o , δπ0 − δπ2 = 30o , (30)
and the accuracy of these numbers is not high, at the level of 50%.
The analogous consideration of the real parts of the loop corrections to
B → ππ decay amplitudes leads to the diminishing of the (real) tree am-
plitudes by ≈ 30%, and we can explain the experimentally observed value
δπ0 − δπ2 ≈ 40o in our model while for ρρ final state the analogous difference
is about three times smaller, δ
0 − δ
2 ≈ 15o.
Let us estimate the phase of the penguin amplitude δπP considering charmed
mesons intermediate states: B → D̄D, D̄∗D, D̄D∗, D̄∗D∗ → ππ 5. In Regge
model all these amplitudes are described at high energies by exchanges of
D∗(D∗2)-trajectories. An intercept of these exchange-degenerate trajectories
can be obtained using the method of [17] or from masses of D∗(2007) – 1−
andD∗2(2460) – 2
+ resonances, assuming linearity of these Regge-trajectories.
Both methodes give αD∗(0) = −0.8÷−1 and the slope α′D∗ ≈ 0.5GeV −2.
The amplitude of D+D− → π+π− reaction in the Regge model proposed
in papers [18, 19] can be written in the following form:
TDD̄→ππ(s, t) = −
e−iπα(t)Γ(1− αD∗(t))(s/scd)αD∗(t) , (31)
5These amplitudes are considered as penguin due to the proper combination of CKM
matrix elements.
where Γ(x) is the gamma function.
The t-dependence of Regge-residues is chosen in accord with the dual
models and is tested for light (u,d,s) quarks [18]. According to [19] scd ≈
2.2GeV 2.
Note that the sign of the amplitude is fixed by the unitarity in the t-
channel (close to the D∗-resonance). The constant g20 is determined by the
width of the D∗ → Dπ decay: g20/(16π) = 6.6. Using (14), analog of (15),
(31) and the branching ratio Br(B → DD̄) ≈ 2 · 10−4 [20] we obtain the
imaginary part of P and comparing it with the contribution of P in B →
π+π− decay probability (6) we get δπP ≈ −3.5o6. A smallness of the phase
is due to the low intercept of D∗-trajectory. The sign of δP is negative -
opposite to the positive sign which was obtained in perturbation theory (7).
Since DD̄-decay channel constitutes only ≈ 10% of all two-body charm-
anticharm decays of Bd-meson [20] taking these channels into account we can
easily get
δP ∼ −10o , (32)
which may be very important for the interpretation of the experimental data
on direct CP asymmetry C+− discussed in the next section.
4 CP asymmetries of Bd(B̄d) → ππ decays
The CP asymmetries are given by :
Cππ ≡
1− |λππ|2
1 + |λππ|2
, Sππ ≡
2Im(λππ)
1 + |λππ|2
, λππ ≡ e−2iβ
MB̄→ππ
MB→ππ
, (33)
where ππ is π+π− or π0π0.
From (2) for direct CP asymmetry in Bd(B̄d) → π+π− decays we readily
obtain:
C+− = −
sinα[
2A0 sin(δ0 − δ̃0 − δP ) + A2 sin(δ2 − δ̃0 − δP )]/
cos(δ0 − δ2)−
A0P̃ cosα cos(δ0 − δ̃0 − δP )−
6In integration over cos θ the region θ ≪ 1 dominates. In this region representation
(31) is valid.
− A2P̃√
cosα cos(δ2 − δ̃0 − δP ) + P̃ 2] , (34)
where
V ∗tdVtb
P . (35)
In order to make a numerical estimate we should know the ratios A0/A2
and P/A2. The first one is given by (25) while the second one can be extracted
from the ratio Br(Bu → K0π+)/Br(Bu → π0π+) assuming d ↔ s invariance
of the strong interactions:
Br(Bu → K0π+)
Br(Bu → π0π+)
f 2KP
2|V ∗tsVtb|2
A22|V ∗udVub|2
, (36)
= 0.092(0.009) . (37)
The numerical values of A0 andA2 are given with good accuracy by factor-
ization calculation, while P appears to be 2.5 times larger than factorization
result [3]. In view of this the validity of factor fK in (36) which originates
from factorization calculation of the penguin amplitude is questinable. If
factorization of the penguin amplitudes is not assumed then the ratio fK/fπ
in (36) should be replaced by unity. In this way we get 20% larger value of
P/A2 in (37) and we will take this value of uncertainty as an estimate of the
theoretical accuracy of the determination of P :
= 0.21(0.04) , (38)
Taking into account that unitarity triangle angle α ≈ 90o and angles δ̃0 and
δP are of the order of few degrees from (34) we obtain:
C+− ≈ −0.28[1.1 sin(δ0 − δ̃0 − δP ) + sin(δ2 − δ̃0 − δP )] ≈
≈ −0.56 sin((δ0 + δ2)/2− δ̃0 − δP ) . (39)
In order to determine the lower bound on the value of C+− let us suppose
that δ0 = 37
o, δ2 = 0
o (we keep the difference δ0− δ2 = 37o, as it follows from
the data on B → ππ decay probabilities (9)), and neglect small values of δ̃0
and δP :
C+− > −0.18 . (40)
Concerning experimental number it could well happen that finally it will
be considerably below our bound. In this case the result of nonperturbative
calculation of penguin phase will be confirmed. Substituting in (39) δ0 =
30o, δ2 = −7o and δP from (32) we obtain the following central value:
C+− = −0.21 . (41)
It is instructive to compare the obtained numbers with the value of C+−
which follows from the asymmetry ACP (K
+π−) if d ↔ s symmetry is sup-
posed [21]:
C+− =
ACP (K
Γ(B → K+π−)
Γ(B → π+π−)
sin(β + γ)
sin(γ)
= 1.2(−2)(−0.093± 0.015)19.8
sin 82o
sin 60o
0.87 = −0.24± 0.04 . (42)
Let us note that one factor fπ/fK in the last equation appears from the
matrix element of the tree operator, the second one - from the matrix element
of the penguin operator. If because of nonfactorization of penguin amplitudes
we will omit the factor which appears from the penguin [5], then the numbers
in the right-hand sides of (40, 41) and (42) will become 20% smaller.
The experimental results obtained by Belle [22] and BABAR [23] are
contradictory
CBelle+− = −0.55(0.09) , CBABAR+− = −0.21(0.09), (43)
Belle number being far below (40) and (41).
For direct CP asymmetry in Bd(B̄d) → π0π0 decay from (3) we readily
obtain:
C00 = −
P̃ sinα[A0 sin(δ0 − δ̃0 − δP )−
2A2 sin(δ2 − δ̃0 − δP )]/
A0A2 cos(δ0 − δ2)−
A0P̃ cosα cos(δ0 − δ̃0 − δP ) +
A2P̃ cosα cos(δ2 − δ̃0 − δP ) + P̃ 2] , (44)
C00 ≈ −1.06[0.8 sin(δ0 − δ̃0 − δP )− 1.4 sin(δ2 − δ̃0 − δP )] ≈ −0.6 , (45)
considerably smaller than C+−. This unusually large direct CPV (measured
by |C00|) is intriguing task for future measurements since the present exper-
imental error is too big:
exper
00 = −0.36(0.32) . (46)
Belle and BABAR agree now on the value of another CPV asymmetry
measured in Bd(B̄d) → π+π− decays: Sexper+− = −0.62 ± 0.09 [22, 23]. From
this measurement the value of unitarity triangle angle α can be extracted.
Neglecting the penguin contribution we get:
sin 2αT = S+− , (47)
αT = 109o ± 3o . (48)
Penguin shifts the value of α. The accurate formula looks like:
S+− = [sin 2α(
cos(δ0 − δ2))−
A2P̃√
sinα cos(δ2 − δ̃0 − δP )−
A0P̃ sinα cos(δ0 − δ̃0 − δP )]/
cos(δ0 − δ2)−
A0P̃ cosα cos(δ0 − δ̃0 − δP )−
A2P̃√
cosα cos(δ2 − δ̃0 − δP ) + P̃ 2] , (49)
and since all the phase shifts are not big the values of cosines in (49) are
rather stable relative to their variations. For numerical estimates we take
δ0 = 30
o, δ2 = −7o and neglect δ̃0 and δP . In this way we get:
(α)ππ = 88
o ± 40(exper)± 50(theor) , (50)
where the first error comes from uncertainty in S
exper
+− while the second one
comes from that in the value of penguin amplitude, (38). Relatively large
theoretical uncertainty in the value of P̃ does not prevent to determine α
with good precision.
The relative smallness of penguin contribution to B → ρρ decay am-
plitudes allow us to determine α with better theoretical accuracy from the
experimental measurement of (S+−)ρρ just as it was done in [24]. With the
help of (5) we obtain:
)ρρ = 0.060(0.012) , (51)
where the same 20% uncertainty in extracting penguin amplitude is supposed.
Using the ratio (A0/A2)ρρ determined in (27) from the (49) neglecting strong
phases (which are much smaller than in the case of B → ππ decays) and
taking into account the recent experimental result (S
exper
+− )ρρ = −0.06± 0.18
[1] we obtain:
(α)ρρ = 87
o ± 50(exper)± 10(theor) . (52)
Let us point out that considerably larger theoretical error quoted in [4]
follows from the larger theoretical uncertainty in the value of penguin ampli-
tude assumed in that paper.
Our results for α should be compared with the numbers which follow from
the global fit of unitarity triangle [6, 7]:
αCKMfitter = (99.0+4.0−9.4)
o , αUTfit = (93± 4)o . (53)
We conclude this section with the prediction for the value of CPV asym-
metry S00:
S00 = [sin 2α(
2A0A2
cos(δ0 − δ2)) +
2A2P̃√
sinα cos(δ2 − δ̃0 − δP )−
A0P̃ sinα cos(δ0 − δ̃0 − δP )]/
2A0A2
cos(δ0 − δ2)−
A0P̃ cosα cos(δ0 − δ̃0 − δP ) +
2A2P̃√
cosα cos(δ2 − δ̃0 − δP ) + P̃ 2] = 0.70± 0.15 , (54)
a large asymmetry with the sign opposite to that of S+−.
5 Conclusions
FSI appeared to be very important in B → ππ decays.
The description of these interactions presented in the paper allows to
explain the experimentally observed difference of the ratios of decay proba-
bilities to the neutral and charged modes in B → ππ and B → ρρ decays.
Rather large absolute value of direct CP asymmetry C+− (if confirmed
experimentally) will be a manifestation of the negative sign of penguin FSI
phase in accord with nonperturbative calculation and opposite to perturba-
tive result.
We are grateful to L.V.Akopyan for checking formulas, Jose Ocariz for
recommendation to include the result for angle α which follows from CP
asymmetry (S+−)ρρ and M.B.Voloshin for useful discussion.
This work was supported by Russian Agency of Atomic Energy;
A.K. was partly supported by grants RFBR 06-02-17012, RFBR 06-02-
72041-MNTI, INTAS 05-103-7515 and state contract 02.445.11.7424;
M.V. was partly supported by grants RFBR 05-02-17203 and
NSh-5603.2006.2.
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