To the origin of the difference of FSI phases in $B\to\pi\pi$ and $B\to\rho\rho$ decays

TL;DR

This paper proposes a model where soft rescattering explains the difference in FSI phases between B to pi pi and B to rho rho decays, accounting for observed phase variations and CP asymmetries.

Contribution

It introduces a dominant soft rescattering FSI model to explain phase differences and CP asymmetries in B meson decays, providing a unified explanation for experimental observations.

Findings

Large FSI phases in B_d→ππ decays

Smaller FSI phases in B_d→ρρ decays

Predicted direct CP asymmetries in B_d→ππ decays

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 $B_d\to\pi\pi$ 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.

Equations

$$R_{\rho }\equiv \frac{\mathrm{Br}({\mathrm{B}}_{\mathrm{d}}\to {\rho }^{+}{\rho }^{-})}{\mathrm{Br}({\mathrm{B}}_{\mathrm{d}}\to {\rho }^0{\rho }^0)}\approx 20,R_{\pi }\equiv \frac{\mathrm{Br}({\mathrm{B}}_{\mathrm{d}}\to {\pi }^{+}{\pi }^{-})}{\mathrm{Br}({\mathrm{B}}_{\mathrm{d}}\to {\pi }^0{\pi }^0)}\approx 4.$$

$$M_{{\bar{B}}_d\to {\pi }^{+}{\pi }^{-}}$$

$$M_{{\bar{B}}_d\to {\pi }^0{\pi }^0}$$

$$M_{{\bar{B}}_u\to {\pi }^{-}{\pi }^0}=\frac{G_F}{\sqrt{2}}|V_{ub}{V}_{ud}^{\ast }|m_B^2{f}_{\pi }{f}_{+}(0)\left\{\frac{\sqrt{3}}{2\sqrt{2}}{e}^{-i\gamma }{A}_2{e}^{i{\delta }_2^{\pi }}\right\},$$

$$\mathrm{Br}(B_d\to {\rho }^{+}{\rho }^{-}{)}_P={\left(\frac{f_{\rho }}{f_{K^{\ast }}}\right)}^2{\left[\lambda \sqrt{{\eta }^2+(1-\rho {)}^2}\right]}^2\frac{{\tau }_{B_d}}{{\tau }_{B_u}}\mathrm{Br}(K^{0\ast }{\rho }^{+})\approx 0.34\cdot 10^{-6},$$

$$\mathrm{Br}(B_d\to {\pi }^{+}{\pi }^{-}{)}_P={\left(\frac{f_{\pi }}{f_K}\right)}^2{\left[\lambda \sqrt{{\eta }^2+(1-\rho {)}^2}\right]}^2\frac{{\tau }_{B_d}}{{\tau }_{B_u}}\mathrm{Br}(K^0{\pi }^{+})\approx 0.59\cdot 10^{-6},$$

$$P{e}^{i{\delta }_P^{\pi }}=-P_c(k^2)=\frac{1}{3}\ln (\frac{M_W^2}{m_b^2})+i\frac{\pi }{3}(1+\frac{2m_c^2}{k^2})\sqrt{1-\frac{4m_c^2}{k^2}},$$

$$\cos ({\delta }_0^{\pi }-{\delta }_2^{\pi })=\frac{\sqrt{3}}{4}\frac{{\mathrm{B}}_{+-}-2B_{00}+\frac{2}{3}\frac{{\tau }_0}{{\tau }_{+}}{B}_{+0}}{\sqrt{\frac{{\tau }_0}{{\tau }_{+}}{B}_{+0}}\sqrt{B_{+-}+B_{00}-\frac{2}{3}\frac{{\tau }_0}{{\tau }_{+}}{B}_{+0}}},$$

$$|{\delta }_0^{\pi }-{\delta }_2^{\pi }|=37^o\pm 10^o.$$

$$\tan ({\delta }_0-{\delta }_2)=\frac{3TC\sin (-{\delta }_C)}{2T^2+TC\cos {\delta }_C-C^2},$$

$${\delta }_0-{\delta }_2=40^o\pm 7^o,$$

$$|{\delta }_0^{\rho }-{\delta }_2^{\rho }|={20^o}_{-20^o}^{+8^o},$$

$$\int d{k}_{0}d{k}_z=\frac{1}{2M_B^2}\int d{s}_{X}d{s}_Y$$

$$M_{\pi \pi }^I=M_{XY}^{(0)I}({\delta }_{\pi X}{\delta }_{\pi Y}+i{T}_{XY\to \pi \pi }^{J=0}),$$

$$\mathrm{Im}M(B\to \pi \pi )=\int \frac{d\cos \theta }{32\pi }M(\rho \rho \to \pi \pi )M^{\ast }(B\to \rho \rho ),$$

$$M_{B^{+}\to {\rho }^{+}{\rho }^0}=-\sqrt{\frac{18.2}{5.7}}{M}_{B^{+}\to {\pi }^{+}{\pi }^0},M_{B_d\to {\rho }^{+}{\rho }^{-}}=-\sqrt{\frac{23.1}{5.2}}{M}_{B_d\to {\pi }^{+}{\pi }^{-}}.$$

$$iM({\rho }^{+}{\rho }^0\to {\pi }^0{\pi }^{+})=-i\frac{g_{{\rho }_{\pi \pi }}^2}{(p_1-k_1{)}^2-m_{\pi }^2}(k_1{\rho }^{+})(k_2{\rho }^0),$$

$$k_1{\rho }^{+}=k_2{\rho }^0=-\frac{1}{2m_{\rho }}\left[(t-m_{\pi }^2)(1+\frac{m_{\rho }^2}{2E_{\rho }^2})+m_{\rho }^2\right],$$

$$\mathrm{Im}{M}_{B\to {\pi }^{+}{\pi }^0}$$

$$\mathrm{Im}{M}_{B\to {\pi }^{+}{\pi }^0}=-\sqrt{\frac{18.2}{5.7}}\frac{g_{\rho \pi \pi }^2}{16\pi }\frac{m_{\rho }^2}{4M_B^2}3.1M_{B\to {\pi }^{+}{\pi }^0},$$

$${\delta }_2^{\pi }(\rho \rho )=-4.9^o.$$

$${\delta }_{+-}^{\pi }(\rho \rho )=+5.7^o,$$

$${\delta }_0^{\pi }(\rho \rho )={\delta }_{+-}^{\pi }(\rho \rho )+\frac{A_2}{\sqrt{2}{A}_0}\left[{\delta }_{+-}^{\pi }(\rho \rho )-{\delta }_2^{\pi }(\rho \rho )\right].$$

$${\left(\frac{A_0}{A_2}\right)}^2=\frac{{\tilde{B}}_{+-}+{\tilde{B}}_{00}}{\frac{2}{3}{B}_{+0}\frac{{\tau }_0}{{\tau }_{+}}}-1,$$

$$\frac{A_0}{A_2}=0.80\pm 0.09,$$

$${\delta }_0^{\pi }(\rho \rho )=15^o,{\delta }_0^{\pi }(\rho \rho )-{\delta }_2^{\pi }(\rho \rho )=20^o.$$

$${\delta }_2^{\rho }(\pi \pi )=\frac{5.7}{18.2}\ast {\delta }_2^{\pi }(\rho \rho )=-1.4^o,{\delta }_{+-}^{\rho }(\pi \pi )=\frac{5.2}{23.1}\ast {\delta }_{+-}^{\pi }(\rho \rho )=1.2^o,$$

$$(A_0/A_2{)}_{\rho \rho }=1.1,{\delta }_0^{\rho }(\pi \pi )=2.9^o,{\delta }_0^{\rho }(\pi \pi )-{\delta }_2^{\rho }(\pi \pi )\approx 4^o.$$

$${\delta }_0^{\pi }(\pi \pi )=5.0^o,{\delta }_2^{\pi }(\pi \pi )=0^o.$$

$${\delta }_0^{\pi }(\pi a_1)=4^o,{\delta }_2^{\pi }(\pi a_1)=-2^o.$$

Full text

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 $B\to\pi\pi$ and three $B\to\rho\rho$ 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 $B_{d}$ decay probabilities to the charged and neutral mesons:

[TABLE]

Table 1

[TABLE]

$C$-averaged branching ratios of $B\to\pi\pi$ and $B\to\rho\rho$ decays.

The large difference of $R_{\rho}$ and $R_{\pi}$ is due to the difference of FSI phases in $B\to\rho\rho$ and $B\to\pi\pi$ decays (see below). In Section 2 we will determine the differences of FSI phases of tree amplitudes which describe $B\to\rho\rho$ and $B\to\pi\pi$ 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 $b\to u\bar{u}(d\bar{d})d$ 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 $M_{B}/3$ and momenta are more or less isotropically oriented. The energy of the fourth (spectator) quark is of the order of $\Lambda_{QCD}$. 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 $E_{cm}=3$GeV 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 $M_{B}/2$. 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 $M_{B}\Lambda_{QCD}$ and is much smaller than $M_{B}^{2}$. The energy of this jet is determined by that of a companion quark and is about $M_{B}/2$. That is why the square of invariant mass of the second jet also does not exceed $M_{B}\Lambda_{QCD}$. 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 $B\to\pi\pi$ decays taking into account $\rho\rho$, $\pi\pi$ and $\pi a_{1}$ intermediate states which by $t(u)$-channel exchanges are converted into $\pi\pi$. We will find that large probability of $B\to\rho^{+}\rho^{-}$ decay explains about half of FSI phases of $B\to\pi\pi$ decays. Relatively small probability of $B\to\pi^{+}\pi^{-}$ decay prevents generation of noticeable FSI phase of $B\to\rho\rho$ amplitudes through $B\to\pi^{+}\pi^{-}\to\rho\rho$ 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 $C_{\pi^{+}\pi^{-}}\equiv C_{+-}$ discussed in Section 4. Predictions for CPV asymmetries $C_{00}$ and $S_{00}$ will be presented in Section 4 as well and the value of the unitarity triangle angle $\alpha$ 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; $|\delta_{0}^{\pi}-\delta_{2}^{\pi}|$ and

$|\delta_{0}^{\rho}-\delta_{2}^{\rho}|$

Let us present $B\to\pi\pi$ decay amplitudes in the so-called “$t$-convention”, in which the penguin amplitude with the intermediate $c$-quark multiplied by $V_{ub}V_{ud}^{*}+V_{cb}V_{cd}^{*}+V_{tb}V_{td}^{*}=0$ is subtracted from the decay amplitudes [3]:

[TABLE]

[TABLE]

[TABLE]

where $V_{ik}$ are the elements of CKM matrix, $\gamma$ and $\beta$ are the unitarity triangle angles and we factor out the product $m_{B}^{2}f_{\pi}f_{+}(0)$ which appears when the decay amplitudes are calculated in the factorization approximation. $A_{2}$ and $A_{0}$ are the absolute values of the decay amplitudes into the states with $I=2$ and 0, generated by operators $O_{1}$ and $O_{2}$ (tree amplitudes), while $P$ is the absolute value of QCD penguin amplitude (generated by operators $O_{3}-O_{6}$ of effective nonleptonic Hamiltonian which describes $b$ quark decays into the states without charm and strange quarks). $\delta_{0}^{\pi}$, $\delta_{2}^{\pi}$ and $\tilde{\delta}_{0}^{\pi}$ 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 $\delta_{0}^{\pi}$ is different from $\delta_{2}^{\pi}$: 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 $\delta_{0}^{\pi}$ differs from $\tilde{\delta}_{0}^{\pi}$ is more subtle. Let us consider the intermediate state made from two charged $\rho$-mesons which contributes to FSI phases: $B_{d}\to\rho^{+}\rho^{-}\to\pi\pi$. $\rho^{+}\rho^{-}$ intermediate state contribution to FSI phases can be large since ${\rm Br}(B_{d}\to\rho^{+}\rho^{-})$ is big. Both tree and penguin induced amplitudes get FSI phases through this chain. Its contribution to $\delta_{0}^{\pi}$ is proportional to $\sqrt{({\rm Br}B_{d}\to\rho^{+}\rho^{-})_{T}/({\rm Br}B_{d}\to\pi^{+}\pi^{-})_{T}}\approx\sqrt{({\rm Br}B_{d}\to\rho^{+}\rho^{-})/({\rm Br}B_{d}\to\pi^{+}\pi^{-})}\approx 2.1$, while that to $\tilde{\delta}_{0}^{\pi}$ is proportional to $\sqrt{({\rm Br}B_{d}\to\rho^{+}\rho^{-})_{P}/({\rm Br}B_{d}\to\pi^{+}\pi^{-})_{P}}$.

How can we determine the penguin contributions to the probabilities of $B_{d}\to\rho^{+}\rho^{-}$ and $B_{d}\to\pi^{+}\pi^{-}$-decays? The most straightforward way suggested in literature is to extract them from the probabilities of $B_{u}\to K^{0*}\rho^{+}$ and $B_{u}\to K^{0}\pi^{+}$ decays to which tree amplitudes almost do not contribute [4, 5]111Contribution of tree amplitudes to these decays comes from the rescattering $(B_{u}\to K^{+}\pi^{0})_{T}$, $K^{+}\pi^{0}\to K^{0}\pi^{+}$, and taking into account CKM suppression of the tree amplitudes of $B\to K\pi(K^{*}\rho)$ decays relative to the penguin amplitudes we can cautiously estimate tree contribution as not more than 10% of penguin one .:

[TABLE]

[TABLE]

where $f_{\rho}=209$ MeV and $f_{K^{*}}=218$ MeV are the vector meson decay constants, $\lambda=0.23$, $\eta=0.34$ and $\rho=0.20$ are the CKM matrix parameters in Wolfenstein parametrization [6], $f_{K}/f_{\pi}=1.2$ and the central values of ${\rm Br}(B_{u}\to K^{0*}\rho^{+})=(9.2\pm 1.5)\cdot 10^{-6}$ and ${\rm Br}(B_{u}\to K^{0}\pi^{+})=(23.1\pm 1.0)\cdot 10^{-6}$ [1] were used. The accuracy of equations (5) and (6) depends on the accuracy of $d\leftrightarrow s$ interchange symmetry ($U$-spin symmetry) of $b\to d(s)$ 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 $B_{d}\to\pi^{+}\pi^{-}$ amplitude is suppressed, $(\tilde{\delta}_{0}^{\pi})_{\rho\rho}\approx 1/2.8(\delta_{0}^{\pi})_{\rho\rho}$. Taking into account that fraction of longitudinally polarized vector mesons produced in $B_{u}\to K^{0*}\rho^{+}$ decays is about 50% we get additional suppression of $(\tilde{\delta}_{0}^{\pi})_{\rho\rho}$ by factor $\sqrt{2}$.

Finally, phase $\delta_{P}^{\pi}$ comes from the imaginary part of the penguin loop with $c$-quark propagating in it [8]. In order to calculate $\delta_{P}^{\pi}$ let us consider corresponding quark diagram. The charm penguin contribution is given by the following expression:

[TABLE]

where $k$ is the sum of momenta of two quarks to which gluon radiated from penguin decays: $k=p_{1}+p_{2}$. One of these quarks forms $\pi$-meson with the spectator quark, so neglecting spectator quark momentum in the rest frame of $B$-meson we have $p_{1}=(\frac{m_{b}}{2},\frac{m_{b}}{2})$. The second quark forms another $\pi$-meson with $\bar{d}$-quark radiated from penguin: $p_{2}=x(\frac{m_{b}}{2},-\frac{m_{b}}{2})$ where $0<x<1$ is the fraction of $\pi^{+}$ momentum carried by $u$-quark. Substituting $k^{2}=xm_{b}^{2}$ into (7) and integrating it with the asymptotic quark distribution function in $\pi$-meson $\varphi_{\pi}(x)=x(1-x)$ we obtain the value of $\delta_{P}^{\pi}$ which depends on the ratio $4m_{c}^{2}/m_{b}^{2}$. In particular, for $m_{b}=5.3$ GeV and $m_{c}=1.9$ GeV (which correspond to the masses of physical states) we obtain $\delta_{P}^{\pi}\approx 10^{o}$, a small positive value. A nonperturbative calculation of $\delta_{P}^{\pi}$ described in Section 3 demonstrates that the sign of $\delta_{P}^{\pi}$ can be negative.

Our next task is to determine the difference of FSI phases $\delta_{0}^{\pi}-\delta_{2}^{\pi}$ (the large value of it is responsible for a relatively small value of $R_{\pi}$). If we neglect the penguin contribution, then from (2) - (4) we get the following expression:

[TABLE]

where $B_{ik}$’s are the $C$-averaged branching ratios, while $\tau_{0}/\tau_{+}\equiv\tau(B_{d})/\tau(B_{u})=0.92$. Substituting the central values from Table 1 we get $|\delta_{0}^{\pi}-\delta_{2}^{\pi}|=48^{o}$.

Penguin contributions to $B_{ik}$ do not interfere with tree ones because $\alpha=\pi-\beta-\gamma$ is almost equal to $\pi/2$. 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 $B\to\pi\pi$ data correspondingly) we get:

[TABLE]

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 $B\to\pi\pi$ and $B\to\pi K$ decay data was made. The tree amplitudes of $B\to\pi\pi$ decays were designated in [2] by $T$ for $B\to\pi^{+}\pi^{-}$ and by $C$ for $B\to\pi^{0}\pi^{0}$. According to [2] the difference of FSI phases between $C$ and $T$ equals $\delta_{C}=-58^{o}\pm 10^{o}$, $|C|=0.37\pm 0.05$, $|T|=0.57\pm 0.05$ in the units of $10^{4}$ eV. The phase shift between the isospin amplitudes is determined by these quantities:

[TABLE]

and substituting the numbers we obtain:

[TABLE]

the result very close to (9). However, the same $d\leftrightarrow s$ interchange symmetry was used in [2] when relating $B\to\pi\pi$ and $B\to K\pi$ 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 … $P_{tc}=P_{tu}$”), therefore the obtained results can be directly compared with ours.

Now let us consider $B\to\rho\rho$ decays. According to BABAR and BELLE results $\rho$ mesons produced in $B$ decays are almost entirely longitudinally polarized ($f_{L}(\rho_{+}\rho_{-})=0.98\pm 0.03$[9], $f_{L}(\rho_{+}\rho_{0})=0.91\pm 0.4$ [10], $f_{L}(\rho_{0}\rho_{0})=0.86\pm 0.12$ [11]). For $B$ decays into the longitudinally polarized $\rho$-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 $B\to\rho\rho$ decays from Table 1 we obtain: $|\delta_{0}^{\rho}-\delta_{2}^{\rho}|=21^{o}$. In order to subtract the penguin contribution with the help of (5) we should take into account that in $B_{u}\to K^{0*}\rho^{+}$ decays the fraction of the longitudinally polarized vector mesons equals approximately 50% [12], so we should subtract $0.17\cdot 10^{-6}$ in case of decay to $\rho^{+}\rho^{-}$ and $0.08\cdot 10^{-6}$ for decay into $\rho^{0}\rho^{0}$. In this way we obtain:

[TABLE]

and the factor 2 difference between (12) and (9) or (11) is responsible for the different patterns of $B\to\rho\rho$ and $B\to\pi\pi$ decay probabilities. Let us emphasize that while $|\delta_{0}^{\rho}-\delta_{2}^{\rho}|$ being only one standard deviation from zero can be very small this is not so for $|\delta_{0}^{\pi}-\delta_{2}^{\pi}|$.

3 Calculation of the FSI phases of $B\to\pi\pi$

and $B\to\rho\rho$ decay amplitudes

Among three amplitudes of $B\to\pi\pi$ decays (2)–(4) only two are independent. We will calculate FSI phases of $B\to\pi^{+}\pi^{0}$ and $B\to\pi^{+}\pi^{-}$ 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 $\pi\pi$ and $\rho\rho$ there is only one such state: $\pi a_{1}$. So we will consider $\rho\rho$ intermediate state which transforms into $\pi\pi$ by $\pi$ exchange in $t$-channel, $\pi a_{1}$ intermediate state which transforms into $\pi\pi$ by $\rho$ exchange in $t$-channel and will take into account the elastic channel $B\to\pi\pi\to\pi\pi$ as well. This approach is analogous to the FSI consideration performed in paper [13]. However in [13] $2\rightarrow 2$ 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 $s^{J-1}\sim s^{0}$ and thus do not decrease with energy (decaying meson mass). However it is well known that the correct behavior is given by Regge theory: $s^{\alpha_{i}(0)-1}$. For $\rho$-exchange $\alpha_{\rho}(0)\approx 1/2$ and the amplitude decrease with energy as $1/\sqrt{s}$. This effect is very spectacular for $B\to DD\to\pi\pi$ chain with $D^{*}(D^{*}_{2})$ exchange in $t$-channel: $\alpha_{D^{*}}(0)\approx-1$ and reggeized $D^{*}$ meson exchange is damped as $s^{-2}\approx 10^{-3}$ in comparison with elementary $D^{*}$ exchange (see for example [14]). For $\pi$-exchange, which gives a dominant contribution to $\rho\rho\to\pi\pi$ 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 $d^{4}k$ 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 $M_{1}$ we can transform the integral over $k_{0}$ and $k_{z}$ into the integral over the invariant masses of clusters of intermediate particles $X$ and $Y$

[TABLE]

and deform integration contours in such a way that only low mass intermediate states contributions are taken into account while the contribution of heavy states being small is neglected. In this way we get:

[TABLE]

where $M_{XY}^{(0)I}$ are the decay matrix elements without FSI interactions and $T_{XY\to\pi\pi}^{J=0}$ is the $J=0$ partial wave amplitude of the process $XY\to\pi\pi\;\;(T^{J}=(S^{J}-1)/(2i))$ which originates from the integral over $d^{2}k_{\perp}$.

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 $B\to\pi\pi$ decay amplitudes which originate from $B\to\rho\rho\to\pi\pi$ chain with the help of unitarity condition 222in this section the phases which originate from CKM matrix elements are omitted.:

[TABLE]

where $\theta$ is the angle between $\rho$ and $\pi$ momenta. For small values of $\theta$ or $t$ $\pi$-exchange in $t$-channel dominates and the calculation of Feynman diagram for $\rho\rho\to\pi\pi$ amplitude with the elementary virtual $\pi$-meson exchange can be trusted, as it was noted above. It was already stressed that $\rho$-mesons produced in $B$-decays are almost entirely longitudinally polarized. That is why we will take into account only longitudinal polarization for the intermediate $\rho$-mesons and amplitudes of $B$-decays into $\pi\pi$ and $\rho_{L}\rho_{L}$ are simply related 333relative negative sign of the amplitudes follows from the expressions for transition formfactors in the factorization approximation, see for example [15].:

[TABLE]

For the amplitude of $\rho^{+}\rho^{0}\to\pi^{0}\pi^{+}$ transition we have:

[TABLE]

where $p_{1}$, $k_{1}$ and $k_{2}$ are $\rho^{+}$, $\pi^{0}$ and $\pi^{+}$ momenta. From the width of $\rho$-meson we get $g^{2}_{\rho\pi\pi}/16\pi=2.85$. For the longitudinally polarized $\rho$-mesons in their center of mass system we have:

[TABLE]

where $t=(p_{1}-k_{1})^{2}$. Changing the integration variable in (15) to $t$ with the help of $dt=\frac{M_{B}^{2}}{2}(1-2\frac{m_{\rho}^{2}}{M_{B}^{2}})d\cos\theta$ and introducing formfactor $exp(t/\mu^{2})$ with the parameter $\mu^{2}\sim 1\;GeV^{2}$ we obtain:

[TABLE]

For $\mu^{2}=2m_{\rho}^{2}$ the contributions of the first two terms in square brackets cancel, while the third term gives:

[TABLE]

and from (4) we get:

[TABLE]

Let us note that in the limit $M_{B}\to\infty$ the ratio $Br(B_{d}\to\rho\rho)/Br(B_{d}\to\pi\pi)$ grows as $M^{2}_{B}$, that is why FSI phase $\delta_{2}^{\pi}(\rho\rho)$ (and $\delta_{0}^{\pi}(\rho\rho)$) diminishes as $1/M_{B}$.

The analogous consideration of $\rho^{+}\rho^{-}$ intermediate state leads to the positive FSI phase of $B_{d}\to\pi^{+}\pi^{-}$ amplitude which is enhanced relatively to $\delta_{2}^{\pi}(\rho\rho)$ according to (16):

[TABLE]

and for FSI phase of the amplitude with isospin zero in the linear approximation we get:

[TABLE]

We are able to extract the ratio $A_{2}/A_{0}$ from that of $C$-averaged ${\rm Br}(B_{d}\to\pi^{+}\pi^{-})$, ${\rm Br}(B_{d}\to\pi^{0}\pi^{0})$ and ${\rm Br}(B_{u}\to\pi^{+}\pi^{0})$, subtracting penguin contribution as we did deriving (9):

[TABLE]

[TABLE]

and, finally:

[TABLE]

In this way we see that $B\to\rho\rho\to\pi\pi$ chain generates half of the experimentally observed FSI phase difference of $B\to\pi\pi$ tree amplitudes.

It is remarkable that FSI phases generated by $B\to\pi\pi\to\rho\rho$ chain are damped by ${\rm Br}(B\to\rho^{+}\rho^{-},\rho^{+}\rho^{0})/{\rm Br}(B\to\pi^{+}\pi,\pi^{+}\pi^{0})$ ratios and are a few degrees:

[TABLE]

[TABLE]

Next we will take into account $\pi\pi$ intermediate state. From Regge analysis of $\pi\pi$ elastic scattering we know that good description of the experimental data is achieved when the exchanges of pomeron, $\rho$ and $f$ trajectories in $t$-channel are taken into account [16]. Pomeron exchange dominates in elastic $\pi\pi\to\pi\pi$ scattering at high energies. For $\alpha_{P}(0)=1$ 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”, $\alpha_{P}(0)\approx 1.1$, we obtain the phase of the amplitude generated by pomeron exchange 444The amplitude of $2\to 2$ process due to supercritical pomeron exchange is $T\sim(s/s_{0})^{\alpha_{P}(t)}(1+exp(-i\pi\alpha_{P}(t)))/(-\sin(\pi\alpha_{P}(t)))=(s/s_{0})^{(1+\Delta)}(i+\Delta\pi/2)$, where in the last expression $t=0$ was substituted and $\alpha_{P}(0)=1+\Delta$ was used ($\Delta\approx 0.1$). which cancels the phases generated by $\rho$ and $f$ exchanges for $I=2$. For $I=0$ the sum of $\rho$ and $f$ exchanges produces the purely imaginary amplitude $T$ and the phase of the amplitude $M$ is due to pomeron ”supercriticallity”:

[TABLE]

In paper [3] the pomeron exchange amplitude was considered as purely imaginary. As a result though important for branching ratios phase difference $\delta_{0}^{\pi}(\pi\pi)-\delta_{2}^{\pi}(\pi\pi)$ was the same (pomeron contribution being universal cancels in the difference of phases) it came mainly from $\delta_{2}^{\pi}(\pi\pi)$ negative value. In this way result for the absolute value of direct CP-asymmetry $C_{\pi^{+}\pi^{-}}$ was underestimated, see below.

Finally $\pi a_{1}$ intermediate state should be accounted for. Large branching ratio of $B_{d}\to\pi^{\pm}a_{1}^{\mp}$-decay ( ${\rm Br}(B_{d}\to\pi^{\pm}a_{1}^{\mp})=(40\pm 4)*10^{-6}$) is partially compensated by small $\rho\pi a_{1}$ coupling constant (it is $1/3$ of $\rho\pi\pi$ one). As a result the contributions of $\pi a_{1}$ intermediate state (which transforms into $\pi\pi$ by $\rho$-trajectory exchange in $t$-channel) to FSI phases equal approximately that part of $\pi\pi$ intermediate state contributions which is due to $\rho$-trajectory exchange. Assuming that the sign of the $\pi a_{1}$ intermediate state contribution into phases is the same as that of elastic channel we obtain:

[TABLE]

Summing the imaginary parts of the amplitudes which follow from (21), (26), (28) and (29) we finally obtain:

[TABLE]

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 $B\to\pi\pi$ decay amplitudes leads to the diminishing of the (real) tree amplitudes by $\approx 30\%$, and we can explain the experimentally observed value $\delta_{0}^{\pi}-\delta_{2}^{\pi}\approx 40^{o}$ in our model while for $\rho\rho$ final state the analogous difference is about three times smaller, $\delta_{0}^{\rho}-\delta_{2}^{\rho}\approx 15^{o}$.

Let us estimate the phase of the penguin amplitude $\delta^{\pi}_{P}$ considering charmed mesons intermediate states: $B\to\bar{D}D,\bar{D^{*}}D,\bar{D}D^{*},\bar{D^{*}}D^{*}\to\pi\pi$ 555These amplitudes are considered as penguin due to the proper combination of CKM matrix elements.. 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 $D^{*}(2007)\;$–$\;1^{-}$ and $D^{*}_{2}(2460)\;$–$\;2^{+}$ resonances, assuming linearity of these Regge-trajectories. Both methodes give $\alpha_{D^{*}}(0)=-0.8\div-1$ and the slope $\alpha_{D^{*}}^{\prime}\approx 0.5GeV^{-2}$.

The amplitude of $D^{+}D^{-}\to\pi^{+}\pi^{-}$ reaction in the Regge model proposed in papers [18, 19] can be written in the following form:

[TABLE]

where $\Gamma(x)$ 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] $s_{cd}\approx 2.2GeV^{2}$.

Note that the sign of the amplitude is fixed by the unitarity in the $t$-channel (close to the $D^{*}$-resonance). The constant $g^{2}_{0}$ is determined by the width of the $D^{*}\to D\pi$ decay: $g^{2}_{0}/(16\pi)=6.6$. Using (14), analog of (15), (31) and the branching ratio $Br(B\to D\bar{D})\approx 2\cdot 10^{-4}$ [20] we obtain the imaginary part of $P$ and comparing it with the contribution of $P$ in $B\to\pi^{+}\pi^{-}$ decay probability (6) we get $\delta^{\pi}_{P}\approx-3.5^{o}$666In integration over $\cos\theta$ the region $\theta\ll 1$ dominates. In this region representation (31) is valid.. A smallness of the phase is due to the low intercept of $D^{*}$-trajectory. The sign of $\delta_{P}$ is negative - opposite to the positive sign which was obtained in perturbation theory (7).

Since $D\bar{D}$-decay channel constitutes only $\approx 10\%$ of all two-body charm-anticharm decays of $B_{d}$-meson [20] taking these channels into account we can easily get

[TABLE]

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 $B_{d}(\bar{B}_{d})\to\pi\pi$ decays

The CP asymmetries are given by :

[TABLE]

where $\pi\pi$ is $\pi^{+}\pi^{-}$ or $\pi^{0}\pi^{0}$.

From (2) for direct CP asymmetry in $B_{d}(\bar{B}_{d})\to\pi^{+}\pi^{-}$ decays we readily obtain:

[TABLE]

where

[TABLE]

In order to make a numerical estimate we should know the ratios $A_{0}/A_{2}$ and $P/A_{2}$. The first one is given by (25) while the second one can be extracted from the ratio ${\rm Br}(B_{u}\to K^{0}\pi^{+})/{\rm Br}(B_{u}\to\pi^{0}\pi^{+})$ assuming $d\leftrightarrow s$ invariance of the strong interactions:

[TABLE]

[TABLE]

The numerical values of $A_{0}$ and $A_{2}$ are given with good accuracy by factorization calculation, while $P$ appears to be 2.5 times larger than factorization result [3]. In view of this the validity of factor $f_{K}$ 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 $f_{K}/f_{\pi}$ in (36) should be replaced by unity. In this way we get $20\%$ larger value of $P/A_{2}$ in (37) and we will take this value of uncertainty as an estimate of the theoretical accuracy of the determination of $P$:

[TABLE]

Taking into account that unitarity triangle angle $\alpha\approx 90^{o}$ and angles $\tilde{\delta}_{0}$ and $\delta_{P}$ are of the order of few degrees from (34) we obtain:

[TABLE]

In order to determine the lower bound on the value of $C_{+-}$ let us suppose that $\delta_{0}=37^{o},\delta_{2}=0^{o}$ (we keep the difference $\delta_{0}-\delta_{2}=37^{o}$, as it follows from the data on $B\to\pi\pi$ decay probabilities (9)), and neglect small values of $\tilde{\delta}_{0}$ and $\delta_{P}$:

[TABLE]

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) $\delta_{0}=30^{o},\delta_{2}=-7^{o}$ and $\delta_{P}$ from (32) we obtain the following central value:

[TABLE]

It is instructive to compare the obtained numbers with the value of $C_{+-}$ which follows from the asymmetry $A_{CP}(K^{+}\pi^{-})$ if $d\leftrightarrow s$ symmetry is supposed [21]:

[TABLE]

Let us note that one factor $f_{\pi}/f_{K}$ 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

[TABLE]

Belle number being far below (40) and (41).

For direct CP asymmetry in $B_{d}(\bar{B}_{d})\to\pi^{0}\pi^{0}$ decay from (3) we readily obtain:

[TABLE]

[TABLE]

considerably smaller than $C_{+-}$. This unusually large direct CPV (measured by $|C_{00}|$) is intriguing task for future measurements since the present experimental error is too big:

[TABLE]

Belle and BABAR agree now on the value of another CPV asymmetry measured in $B_{d}(\bar{B}_{d})\to\pi^{+}\pi^{-}$ decays: $S_{+-}^{exper}=-0.62\pm 0.09$ [22, 23]. From this measurement the value of unitarity triangle angle $\alpha$ can be extracted. Neglecting the penguin contribution we get:

[TABLE]

[TABLE]

Penguin shifts the value of $\alpha$. The accurate formula looks like:

[TABLE]

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 $\delta_{0}=30^{o}$, $\delta_{2}=-7^{o}$ and neglect $\tilde{\delta}_{0}$ and $\delta_{P}$. In this way we get:

[TABLE]

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 $\tilde{P}$ does not prevent to determine $\alpha$ with good precision.

The relative smallness of penguin contribution to $B\to\rho\rho$ decay amplitudes allow us to determine $\alpha$ with better theoretical accuracy from the experimental measurement of $(S_{+-})_{\rho\rho}$ just as it was done in [24]. With the help of (5) we obtain:

[TABLE]

where the same $20\%$ uncertainty in extracting penguin amplitude is supposed. Using the ratio $(A_{0}/A_{2})_{\rho\rho}$ determined in (27) from the (49) neglecting strong phases (which are much smaller than in the case of $B\to\pi\pi$ decays) and taking into account the recent experimental result $(S_{+-}^{exper})_{\rho\rho}=-0.06\pm 0.18$ [1] we obtain:

[TABLE]

Let us point out that considerably larger theoretical error quoted in [4] follows from the larger theoretical uncertainty in the value of penguin amplitude assumed in that paper.

Our results for $\alpha$ should be compared with the numbers which follow from the global fit of unitarity triangle [6, 7]:

[TABLE]

We conclude this section with the prediction for the value of CPV asymmetry $S_{00}$:

[TABLE]

a large asymmetry with the sign opposite to that of $S_{+-}$.

5 Conclusions

FSI appeared to be very important in $B\to\pi\pi$ decays.

The description of these interactions presented in the paper allows to explain the experimentally observed difference of the ratios of decay probabilities to the neutral and charged modes in $B\to\pi\pi$ and $B\to\rho\rho$ 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 perturbative result.

We are grateful to L.V.Akopyan for checking formulas, Jose Ocariz for recommendation to include the result for angle $\alpha$ which follows from CP asymmetry $(S_{+-})_{\rho\rho}$ 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.