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Nuclear Physics B (Proc. Suppl.) 170 (2007) 33–38 www.elsevierphysics.com

Penguin Pollution Estimates Relevant for the Extraction of α/φ2 Jure Zupanab a

J. Stefan Institute, Jamova 39, P.O. Box 3000, 1001 Ljubljana, Slovenia

b

Department of Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia

A review of methods to extract the standard CKM unitarity triangle angle α is provided. The sizes of related theoretical errors are reviewed.

1. INTRODUCTION The determination of the standard CKM unitarity triangle angle β (φ1 ) = arg(−Vcb∗ Vcd /Vtb∗ Vtd ) from b → c¯ cs transitions (i.e. B(t) → J/ψKS and related modes) has reached an impressive precision [1] β = (21.2 ± 1.0)◦ ,

(1)

with the ultimate theory error below the percent level [2]. This sets high standards for the determinations of the other two angles, ∗ α (φ2 ) = arg(−Vtb∗ Vtd /Vub Vud ) and γ (φ3 ) = ∗ ∗ arg(−Vub Vud /Vcb Vcd ). In this review we address the theoretical errors for the methods that are used in the determination of α. These errors represent the ultimate precision at which the angle α can be measured even with unlimited statistics. The theoretical uncertainties arise because sin 2α is directly related to measured CP asymmetries only in the limit of one dominant amplitude. Focusing for the moment on B → π + π − decays for clarity, we define the “tree”, T , and “penguin”, P , amplitudes A(B 0 → π + π − ) ≡ Aπ+ π− = T eiγ + P eiδ , (2) ¯ 0 → π + π − amplitude A¯π+ π− is obwhere the B tained through γ → −γ exchange. In our notation the penguin carries only the strong phase δ, and the tree only the weak phase γ. The angle α is extracted from the indirect CP asymmetry, Sπ+ π− , where (λ) A¯ + − Sπ+ π− = −2 , λ = e−2iβ π π , (3) 2 1 + |λ| Aπ+ π− 0920-5632/$ – see front matter © 2007 Published by Elsevier B.V. doi:10.1016/j.nuclphysbps.2007.05.020

and Sπ+ π− is measured in the time dependent decay width Γ(B 0 (t) → π + π − ) ∝ Γπ+ π− [1+ (4) + Cπ+ π− cos Δmt − Sπ+ π− sin Δmt] . Here Δm is the mass splitting in the B 0 system and Cπ+ π− is the direct CP asymmetry. Expanding to first order in r = P/T one has Sπ+ π− = sin 2α + 2r cos δ sin(β +α) cos 2α.

(5)

In the limit of vanishingly small penguins Sπ+ π− = sin 2α, which receives O(r) corrections (5). The theoretical error on α therefore depends on how well we can determine the hadronic parameters r and δ. It is not possible to determine both of them from Γ(B 0 (t) → π + π − ). This is because in expression 4 we have 3 observables: Γπ+ π− , Cπ+ π− and Sπ+ π− , that depend on 4 unknowns: T, r, δ and γ. Additional input is thus required in order to determine the penguin pollution (i.e. fix the size of r). This can be obtained by using approximate symmetries of QCD: isospin, flavor SU(3) or using an expansion in 1/mb . We describe these three approaches and the resulting errors in the rest of the review. As is evident from Fig. 1 the penguin-to-tree ratios obey a hierarchy r(π + π − ) > r(ρ+ π − ) ∼ r(π + ρ− ) > r(ρ+ ρ− ). (6) For methods based on SU(3) and 1/mb expansions we expect the same hierarchy in the theory errors, while this may not be the case for methods based on isospin, as will be discussed in more detail below.

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J. Zupan / Nuclear Physics B (Proc. Suppl.) 170 (2007) 33–38 PT 

Π Π







1 2

_ |A 00 |

_ |A +− |

1 2

Ρ Ρ

Ρ Π

2θ

Π Ρ 0.2

0.4

0.6

2. ISOSPIN METHODS 2.1. Measurement of α from B → ππ The standard methods for obtaining α from B → ππ, ρρ and ρπ all use isospin. The measurement of α from B(t) → ππ or longitudinal ρρ decays relies on the triangle construction due to Gronau and London [3], shown in Fig. 2. The triangles visualize the isospin relations between decay amplitudes. Furthermore, the observable 

sin(2αeff ) = Sπ+ π− / 1 − Cπ2+ π− is directly related to α through 2α = 2αeff −2θ, with θ defined in Fig. 2. The construction relies on the following assumptions: (i) that ΔI = 5/2 operators are not present in the effective weak Lagrangian (these can arise from eg. ΔI = 2 electromagnetic rescattering of two pions), (ii) that there are no isospin breaking corrections and (iii) that A+0 is a pure tree, so that the bases of the two triangles in Fig. 2 coincide. The ΔI = 5/2 effects have not been analysed in detail in this context. One expects them to be parametrically small, suppressed by electromagnetic coupling αem and thus O(1%). The other two corrections are related to isospin breaking due to the different u and d quark masses and charges. Isospin breaking has several effects relevant for the α determination: (i) the basis of operators in

|A 00 | _ |A +0 |= |A +0 |

0.8

Figure 1. The sizes of penguin-to-tree ratios r obtained from an isospin decomposition (first plotted/dark estimates for π + π − and ρ+ ρ− ) or using SU(3) (all others) with data from [1]. The very loose bounds on r(ρ± π ∓ ) which follow from an isospin analysis are not shown.

|A +− |

Figure 2. The Gronau-London construction, with only one of four possible triangle orientations shown. The subscripts i in Ai and A¯i denote the pion charges, eg. A+− = π + π − |H|B 0 , while A¯i refer to the CP conjugated modes.

the effective weak Hamilton is extended to include electroweak penguin operators (EWP) Q7,...,10 , (ii) the π 0 mass eigenstate no longer coincides with the isospin eigenstate leading to π 0 − η − η  mixing (and similarly to ρ0 − ω mixing for the vector mesons), (iii) the reduced matrix elements between states in the same isospin multiplet may no longer be related simply by SU(2) ClebschGordan coefficients. Not all of these effects can be bounded at present. In the literature only the first two have been analyzed in significant detail [4–6]. The effect of EWP is known quite precisely. Including EWP still gives two closed triangles as in Fig. 2. The triangle bases, however, are now at an angle to each other since A0+ , A¯0+ are no longer only pure tree. Neglecting small contributions from Q7,8 operators [7,5], while relating Q9,10 to the tree operators [7,8], gives rise to the following shift, ΔαEWP , in the extracted value of α

ΔαEWP

3 =− 2



C9 + C10 C1 + C2



sin(β + α) sin α sin β (7) = (1.5 ± 0.3 ± 0.3)◦ ,

where the first error is from the experimental uncertainty on α and β, while the second is an estimate of the error due to nonzero Q7,8 matrix elements. Exactly the same shift due to EWP

J. Zupan / Nuclear Physics B (Proc. Suppl.) 170 (2007) 33–38

holds for the ρρ system and between the T and T tree amplitudes in B → ρπ. The effect of π 0 − η − η  mixing is known less precisely. Because the π 0 wave function is composed of I = 1 and small I = 0 terms, |π 0  = |π3  + |η +  |η  , with  = 0.017 ± 0.003, and  = 0.004 ± 0.001 [9], there are additional  contributions from A(B 0 → π 0 η ( ) ) with the consequence that the triangles in Fig. (2) no longer close [4]. Varying the phases of these amplitudes, while constraining the magnitudes from experiment leads to the following bound on the resulting bias [5] |Δαπ−η−η | < 1.6◦ .

(8)

These two examples of isospin breaking effects show that even though not all of the isospin breaking effects can be calculated or constrained at present, the ones that can be are of the expected size ∼ (mu − md )/ΛQCD ∼ α0 ∼ 1%.

(9)

2.2. Measurement of α from B → ρρ The isospin analysis in B → ρρ follows the same lines as for B → ππ, but with three separate isospin relations, one for each polarization. However, in B → ρρ the longitudinal component completely dominates so that only one isospin relation is relevant. Another difference from the ππ system is that ρ resonances have a non-negligible decay width. The two ρ resonances can then also form an I = 1 final state [10], which affects the analysis at O(Γ2ρ /m2ρ ), but it is possible to constrain this effect experimentally [10]. As before, the theoretical error on the extracted value of α is given by isospin breaking effects. While the shift due to EWP is exactly the same as in ππ, there are also novel effects due to ρ − ω mixing [5]. A relatively large O(1) effect is expected near the ω resonance in the invariant mass of the final π + π − pair, however the integrated effect of the ω resonance is < 2% and is thus of the expected size for isospin breaking. There are also other isospin effects, e.g. that the couplings between ρ+ , ρ0 and π + π 0 , π + π − final states may not be given by SU(2) Clebsch-Gordan coefficients. All the effects that can be estimated now, how-

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ever, are found to be of the expected size of O(1%) [5]. 2.3. B → ρπ The Snyder-Quinn method exploits the fact that ρ± π ∓ and ρ0 π 0 are intermediate resonances in the B → π + π − π 0 three body decay [11]. From ¯ 0 Dalitz plot anala time dependent B 0 and B ysis one can therefore measure the magnitudes ¯ 0 ) → ρ± π ∓ , ρ 0 π 0 and relative phases of the B 0 (B amplitudes from the overlaps between different ρ resonance bands. This additional information has a beneficial consequence that only an isospin relation between the penguin amplitudes is needed (in contrast to the Gronau-London triangle constructions in B → ππ, ρρ, where relations between tree amplitudes are also used). Isospin breaking effects are thus expected to be P/T suppressed [5]. (This would not be true for the pentagon analysis [12–14], though.) The exception to this rule are EWP whose presence also modifies the isospin relation between the penguin amplitudes. This correction is expected to lead to the largest bias in the extracted value of α since its size is roughly of the isospin breaking times a tree amplitude, while the other isospin breaking is expected to have a size of the isospin breaking times a penguin amplitude. However, since EWP are directly related to the tree amplitudes, they can easily be taken into account similarly as in B → ππ. Other isospin breaking effects are then expected to be further P/T ∼ 0.2 suppressed. For instance, using SU(3) relations the shift in α due to π 0 − η − η  mixing is estimated in [5] to be |Pρη +  Pρη | |T | (10) ≤ 0.024 + 0.069 ≤ 0.1◦ .

|Δαπ−η−η | =

This result relates to only one possible source of isospin breaking but it does nevertheless show that the suppression exists. 3. SU(3) METHODS 3.1. B → ρρ We do not discuss the B → ππ system in this context since the penguin pollution is significant, and thus any SU(3) based method is expected

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J. Zupan / Nuclear Physics B (Proc. Suppl.) 170 (2007) 33–38

to have large errors. The penguin pollution on the other hand is small for B → ρρ, as is evident from Fig. 1. This small penguin pollution makes, somewhat surprisingly, the method based on the SU(3) symmetry as theoretically clean as the isospin analysis [15]. The reason is that SU(3) symmetry is used to directly bound P/T , while the isospin construction involves relations between the tree amplitudes as well. The isospin breaking to these larger amplitudes thus also enters into the corrections. One further advantage of the SU(3) approach over the isospin analysis is that one does not need a measurement of Cρ0 ρ0 , which is not yet available. (Because of the absence of this measurement the isospin analysis at present leads only to a broad constraint of |α − αeff | < 22.4◦ at 95% CL [16].) The basic idea of the method based on SU(3) flavor symmetry is to relate the ΔS = 0 B 0 → ρ+ ρ− decay, in which tree and penguin amplitudes have CKM elements of similar size T ∼ ∗ ∗ Vub Vud , P ∼ Vcb Vcd , to the ΔS = 1 decays in ∗ , which the tree is CKM suppressed, T  ∼ Vub Vus  while the penguin is CKM enhanced: P ∼ ∗ . Thus P  /T  is 1/λ2 enhanced over P/T Vcb Vcs and can be used to bound P/T . In Ref. [15] the authors used the penguin dominated process K ∗0 ρ+ for this purpose. The relation to the penguin in B 0 → ρ+ ρ− is then (restricting the discussion only to the decays into longitudinally polarized states)   |Vcs |fK ∗ P 2 , (11) |AL (K ∗0 ρ+ )|2CP−av = F |Vcd |fρ where fK ∗ and fρ are the K ∗ and ρ decay constants, and the parameter F is equal to 1, if SU(3) breaking can be factorized, and if colorsuppressed EWP and penguin annihilation contributions are neglected. In this limit expression (11) yields P/T = 0.10 ± 0.02 showing that penguin pollution is small. Taking a conservative range 0.3 ≤ F ≤ 1.5 and including pre-ICHEP06 experimental errors on the sine and cosine components of the CP asymmetry to fit for the two unknowns, α and the strong phase δ, the authors +9.1 +1.2 of Ref. [15] find α = [91.2−6.6 (exp)−3.9 (th)]◦ , where the last error is due to the variation in F and represents the limiting theoretical error of

this approach. It is comparable to the theoretical error in the isospin analysis. 3.2. B → ρπ The Snyder-Quinn method uses interference of ρ resonances in B → π + π − π 0 to extract the information on the relative phase [11]. A potential problem is that the ρ resonance bands do not overlap head-on and one is sensitive to the resonance tails. This can be avoided by using just the ρ± π ∓ final states and the SU (3) related modes [17]. Similarly to (2), the tree and penguin contributions are first defined according to their weak phases A(B 0 → ρ± π ∓ ) = eiγ T± + P± .

(12)

In total there are |T± |, |P± |,   8 unknowns:  arg P± /T± , arg T− /T+ and α, but just 6 observables: the decay width, and the direct and indirect CP asymmetries for each mode. Additional information on penguin contributions can be obtained from the SU(3) related ΔS = 1 modes B 0 → K ∗+ π − , K + ρ− and B + → K ∗0 π + , K 0 ρ+ , in which the penguins are CKM enhanced and the tree terms CKM suppressed compared with the ρ± π ∓ final state. This gives at 90% CL [17] 0.16 ≤ |P+ /T+ | ≤ 0.24, 0.12 ≤ |P− /T− | ≤ 0.29,

(13)

where in order to relate the ΔS = 1 and ΔS = 0 channels, annihilation like topologies have been neglected. Since the penguin contributions are relatively small, the error introduced because of the SU(3) breaking on the extracted value of α is also expected be small, of order O(δSU(3) P± /T± ). A Monte Carlo study with up to 30% SU(3) breaking on penguins for instance gives biases in the extracted value of α of ∼ 2◦ [17], but further analyses are called for. Finally, note that unlike the Snyder-Quinn approach the method of using SU(3) symmetry determines α only up to discrete ambiguities. In order to resolve these ambiguities an additional assumption is made in [17], namely arg(T− /T+ ) < 90◦ , in agreement with factorization theorems. The improvement on the knowledge of α gained through this constraint has been further explored in [18].

J. Zupan / Nuclear Physics B (Proc. Suppl.) 170 (2007) 33–38 ∓ 3.3. α from a± 1π While an isospin analysis is practically im∓ possible for a± since one either has to deal 1π with resonance overlaps in 4-body final states or with pentagon relations, the extension of the SU(3) method to a1 (1260)± π ∓ final states is fairly straightforward and similar to the case of ρπ [19]. An additional complication is that the K1A state, which belongs to the same SU(3) multiplet as a1 , is an admixture of two mass eigenstates. As before the method works much better if P/T is small. If P/T 1 one can use bounds on penguin pollution, otherwise general SU(3) fits are possible (but in this case no suppression of SU(3) errors is expected). Because of the K1A mixing one needs branching ratios for at least 3 SU(3) related modes to arrive at bounds: either Γ(B 0 → K1+ (1400)π − ) and Γ(B 0 → K1+ (1200)π − ), or Γ(B + → K10 (1400)π + ) and Γ(B + → K10 (1200)π + ), and in addition Γ(B 0 → + + 0 a− → a+ 1 K ) or Γ(B 1 K ). At CKM2006 the first measurement of the time dependent B(t) → ∓ a± 1 π decay width was reported by BaBar [20].

4. 1/mb EXPANSION METHODS An interesting way of using a 1/mb expansion to obtain precise constraints in the ρ¯, η¯ plane using the time dependent B → π + π − decay is presented in Ref. [21,22]. While not leading to a direct constraint on α, which is the main focus of this review, it offers an alternative way of using the same data as in the isospin analysis to obtain an information on the CKM unitarity triangle and is thus worth reviewing. A crucial observation by the authors of this approach is that the penguin, P , and tree, T , parameters in (2) already contain the absolute values of CKM el∗ ements |Vub Vud | and |Vcb∗ Vcd | respectively. They define a new ratio where the dependence on the CKM elements is explicit r˜ 1 P =r=  r˜,  T 2.5 η¯2 + ρ¯2

(14)

with ρ¯, η¯ the parameters in the Wolfenstein parametrization of the CKM matrix. One can then get a precise constraint in the ρ¯, η¯ plain, if the hadronic parameters r˜ and the strong phase

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δ are known to some reasonable degree. Changing the variables from ρ¯, η¯ to tan β, η¯, since β is already well determined from B 0 → J/ψKS0 , one gets (here S = Sπ+ π− ) √ 1 + cot βS − 1 − S 2 (1 + r˜ cos δ). (15) η¯ = (1 + cot2 β)S It is important to note that the unknown hadronic parameter r˜ enters only in the form 1 + r˜ cos δ. In QCD factorization its value is r˜ = 0.107 ± 0.031, where the error is the combined error from the input parameters and the (model dependent) estimates of uncertainties due to subleading corrections. The error on η¯ is thus small, at the order of a few percent. In the original proposal [21,22] the strong phase δ was taken to be small, so that cos δ → 1 with corrections only of second order in δ. These corrections are small if charming penguins are perturbative (in QCD factorization δ = 0.15 ± .25 [22]). Alternatively one could determine the strong phase δ from Cπ+ π− simultaneously with η¯ in (15). A 1/mb expansion was used also in Ref. [23] in a different way, as an additional constraint on the isospin analysis. 5. CONCLUSIONS In conclusion, we see that the isospin breaking effects that have been analysed are of the expected size. In certain cases they can be further suppressed, for instance in the Snyder-Quinn method of extracting α from B → ρπ the unaccounted for isospin breaking is expected to be P/T suppressed. And finally, and somewhat surprisingly, in modes with small penguin amplitudes (B → ρρ, ρπ) the SU (3) methods lead to theoretical errors of the same order as the isospin breaking in the isospin analyses. This is because in the SU (3) methods the effect of the SU (3) breaking error is suppressed by the small penguin amplitudes. REFERENCES 1. Summer 2006 HFAG average, available at www.slac.stanford.edu/xorg/hfag/ 2. M. Ciuchini, M. Pierini and L. Silvestrini, Phys. Rev. Lett. 95 (2005) 221804; H. Boos,

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