Implication of the $B \to (\rho, \omega) \gamma$ Branching Ratios for the CKM Phenomenology

We study the implication of the recent measurement by the BELLE collaboration of the averaged branching fraction $\bar B_{exp} [B \to (\rho, \omega) \gamma] = (1.8^{+0.6}_{-0.5} \pm 0.1) \times 10^{-6}$ for the CKM phenomenology. Combined with the averaged branching fraction $\bar B_{exp} (B \to K^* \gamma) = (4.06 \pm 0.26) \times 10^{-5}$ measured earlier, this yields $\bar R_{exp} [(\rho, \omega) \gamma/K^* \gamma] = (4.2 \pm 1.3)%$ for the ratio of the two branching fractions. Updating earlier theoretical analysis of these decays based on the QCD factorization framework, and constraining the CKM-Wolfenstein parameters from the unitarity fits, our results yield $\bar B_{th} [B \to (\rho, \omega) \gamma] = (1.38 \pm 0.42) \times 10^{-6}$ and $\bar R_{th} [(\rho, \omega) \gamma/K^* \gamma] = (3.3 \pm 1.0)%$, in agreement with the BELLE data. Leaving instead the CKM-Wolfenstein parameters free, our analysis gives (at 68% C.L.) $0.16\leq |V_{td}/V_{ts}| \leq 0.29$, which is in agreement with but less precise than the indirect CKM-unitarity fit of the same, $0.18 \leq |V_{td}/V_{ts}| \leq 0.22$. The isospin-violating ratio in the $B \to \rho \gamma$ decays and the SU(3)-violating ratio in the $B_d^0 \to (\rho^0, \omega) \gamma$ decays are presented together with estimates of the direct and mixing-induced CP-asymmetries in the $B \to (\rho,\omega) \gamma$ decays within the SM. Their measurements will overconstrain the angle $\alpha$ of the CKM-unitarity triangle.


Introduction.
Recently, the BELLE collaboration have presented evidence for the observation of the decays B + → ρ + γ, B 0 d → ρ 0 γ and B 0 d → ωγ (and their charged conjugates) [1]. Their observation based on an integrated luminosity of 140 fb −1 lacks the statistical significance in the individual channels, but combining the data in the three decay modes and their charged conjugates yields a signal at 3.5σ C.L. [1]: This result updates the previous upper bounds [2] by the BELLE collaboration, while the upper bound from the BABAR collaboration (at 90% C.L.) [3]: remains to be updated. The experimental averages given above are defined as: and the world average [4] for the B-meson lifetime ratio: has been used in arriving at the BELLE result (1). This is the first observation of the CKM-suppressed electromagnetic penguin b → dγ transition. The CKM-allowed b → sγ transition in the exclusive decays B → K * γ was observed more than a decade ago by the CLEO collaboration [5], followed by the observation of the inclusive decay B → X s γ in 1994 [6]. Since then, data on these decay modes have been provided by a number of experimental collaborations, and the current situation is summarized in Table 1. In getting the isospin-averaged branching ratioB exp (B → K * γ), we used the following definition: and the world average (4) for the B-meson lifetime ratio. Table 1 also contains the measurements of the inclusive decay B → X s γ branching fraction, the resulting ratio of the exclusive-to-inclusive decay rates R exp (K * γ/X s γ), for each experiment separately, and their world averages, with the errors added in quadrature. The measurements from BELLE and the upper limit from BABAR on the B → (ρ, ω)γ decays given in (1) and (2), respectively, can be combined with their respective measurements of the B → K * γ decay rates to yield the following ratios: R exp [(ρ, ω)γ/K * γ] < 0.047, (BABAR) (6) R exp [(ρ, ω)γ/K * γ] = 0.042 ± 0.013, (BELLE) (7) where R exp [(ρ, ω)γ/K * γ] =B exp [B → (ρ, ω) γ]/B exp (B → K * γ). In this paper, we do an analysis of the two quantities in Eqs. (1) and (7) in the context of the SM. Table 1: The branching ratios averaged over the charge-conjugated modes (in units of 10 −5 ) of the exclusive decays B + → K * + γ and B 0 d → K 0 * γ and the inclusive decay B → X s γ taken from Refs. [4,[7][8][9][10]. The averaged branching ratios defined in (5) and the ratio of the exclusive-to-inclusive branching ratios R exp (K * γ/X s γ) are also tabulated. Fraction BABAR BELLE CLEO Average B exp (B + → K * + γ) 3.87 ± 0.28 ± 0. 26 where G F is the Fermi coupling constant, and only the dominant terms are shown. The operators O (q) 1 and O (q) 2 , (q = u, c), are the standard four-fermion operators: and O 7 and O 8 are the electromagnetic and chromomagnetic penguin operators, respectively: Here, e and g s are the electric and colour charges, F µν and G a µν are the electromagnetic and gluonic field strength tensors, respectively, T a αβ are the colour SU(N c ) group generators, and the quark colour indices α and β and gluonic colour index a are written explicitly. Note that in the operators O 7 and O 8 the d-quark mass contributions are negligible and therefore omitted. The coefficients C and C eff 8 (µ) include also the effects of the QCD penguin four-fermion operators which are assumed to be present in the effective Hamiltonian (8) and denoted by ellipses there. For details and numerical values of these coefficients, see [11] and reference therein. We use the standard Bjorken-Drell convention [12] for the metric and the Dirac matrices; in particular γ 5 = iγ 0 γ 1 γ 2 γ 3 , and the totally antisymmetric Levi-Civita tensor ε µνρσ is defined as ε 0123 = +1.
For the b → sγ decay (equivalently the B → K * γ decays), the effective Hamiltonian H b→s eff describing the b → s transition can be obtained by the replacement of the quark field d α by s α in all the operators in Eqs. (9) and (10) and by replacing the CKM (8). Noting that among the three factors V qb V * qs , the combination V ub V * us is CKM suppressed, the corresponding contributions to the decay amplitude can be safely neglected. Thus, within this approximation, unitarity of the CKM matrix yields V cb V * cs = −V tb V * ts , the dependence on the CKM factors in the effective Hamiltonian H b→s eff factorizes, and the CKM factor is taken as V tb V * ts . Note also that the three CKM factors shown in H b→d eff are of the same order of magnitude and, hence, the matrix elements in the decays b → dγ and B → (ρ, ω)γ have non-trivial dependence on the CKM parameters.
3. Theoretical framework for the B → V γ decays. To get the matrix elements for and O (u) 2 , where the latter two are important only for the B → (ρ, ω)γ decays. One also uses the terminology of the short-distance and long-distance contributions, where the former characterizes the top-quark induced penguin-amplitude and the latter includes the penguin amplitude from the u-and c-quark intermediate states and also the so-called weak annihilation and W -exchange contributions. There are also other topologies, such as the annihilation penguin diagrams, which, however, are small. For a recent discussion of the long-distance effects in B → V γ decays and references to earlier papers, see Ref. [13].
Including the O(α s ) corrections, all the operators listed in (9) and (10) have to be included. A convenient framework to carry out these calculations is the QCD factorization framework [14] which allows to express the hadronic matrix elements in the schematic form: where F B→V are the transition form factors defined through the matrix elements of the operator O 7 , φ B,+ (k + ) is the leading-twist B-meson wave-function with k + being a lightcone component of the spectator quark momentum, φ V ⊥ (u) is the leading-twist light-cone distribution amplitude (LCDA) of the transversely-polarized vector meson V , and u is the fractional momentum of the vector meson carried by one of the two partons. The quantities T I i and T II i are the hard-perturbative kernels calculated to order α s , with the latter containing the so-called hard-spectator contributions. The factorization formula (11) holds in the heavy quark limit, i.e., to order Λ QCD /M B . This factorization framework has been used to calculate the branching fractions and related quantities for the decays B → K * γ [15][16][17] and B → ργ [15,17]. The isospin violation in the B → K * γ decays in this framework have also been studied [18]. (For applications to B → K * γ * , see Refs. [16,19,20]). Very recently, the hard-spectator contribution arising from the chromomagnetic operator O 8 have also been calculated in next-to-next-to-leading order (NNLO) in α s showing that the spectator interactions factorize in the heavy quark limit [21]. However, the numerical effect of the resummed NNLO contributions is marginal and we shall not include this in our update.
In what follows we shall use the notations and results from Ref. [15], to which we refer for detailed derivations, and point out the changes (and corrections) that we have incorporated in this analysis. The branching ratio of the B → K * γ decay corrected to O(α s ) can be written as follows [15]: where α is the fine-structure constant, m b,pole is the b-quark pole mass, and M B and m K * are the B-and K * -meson masses, respectively. The quantity ξ is the soft part of the QCD form factor T K * 1 (q 2 ) in the B → K * transition, which is evaluated at q 2 = 0 in the HQET limit. For this study, we consider ξ (K * ) ⊥ as a free parameter; its value will be extracted from the current experimental data on B → K * γ decays. Note that the quantity ξ (K * ) ⊥ used here is normalized at the scale µ = m b,pole of the pole b-quark mass. The corresponding quantity in Ref. [19] is defined at the scale µ = m b,PS involving the potential-subtracted (PS) b-quark mass [22,23], which is numerically very close to the pole mass used here.
The function C (0)eff 7 (µ) in Eq. (12) is the Wilson coefficient of the electromagnetic operator O 7 in the leading order and the function A (1) (µ) includes all the NLO corrections: where A (1) ver and A (1)K * sp denote the O(α s ) corrections in the Wilson coefficient C eff 7 , the b → sγ vertex, and the hard-spectator contributions, respectively. Their explicit expressions are given in Eqs. (5.9), (5.10) and (5.11) of Ref. [15]. The values used in the numerical analysis are collected in Table 2. Some comments on the input values are in order. The top-quark mass (interpreted here as the pole mass) has been recently updated and revised upwards by the Tevatron electroweak group [24], and the new world average m t,pole = (178 ± 4.3) GeV is being used in our analysis. The product |V tb V * ts | of the CKM matrix elements can be obtained from the estimate |V cb | = 0.0412 ± 0.0021 [25] using the relation |V tb V * ts | ≃ (1 − λ 2 /2)|V cb |, which yields |V tb V * ts | = 0.0402 ± 0.0020 for λ = 0.2224. The SU(3)-breaking effects in the K-and K * -meson LCDAs have been recently re-estimated by Ball and Boglione [26]. In this update, the transverse decay Table 2: Input quantities and their values used in the theoretical analysis. The values of the masses, coupling constants and Λ h given in the first four rows are fixed, and those of the others are varied in their indicated ranges to estimate theoretical uncertainties on the various observables discussed in the text.

Parameter
Value , has remained practically unchanged, but the Gegenbauer coefficients in the K * -meson leading-twist LCDA are effected significantly. The two Gegenbauer moments a used in the calculation of the hard-spectator contributions are now larger in magnitude, have larger errors and, moreover, the first Gegenbauer moment changes its sign. For comparison, previously, these coefficients were estimated as a ⊥2 (1 GeV) = 0.04 ± 0.04. The effect of these modifications on the QCD form factor T K * 1 (0), as well as of some other technical improvements [26], has not yet been worked out. Lastly, the first inverse moment λ −1 B,+ (µ) of the B-meson LCDA has also changed. In our previous analysis [15], we used the value λ −1 B,+ (µ sp ) = (3.0 ± 1.0) GeV −1 where the error effectively includes the scale dependence of the leading-twist light-cone B-meson wave-function φ B,+ (k, µ). In a recent paper by Braun et al. [27], the scale dependence of this moment is worked out in the NLO with the result: where (α s C F /π) ln(µ/µ 0 ) < 1 and the quantities λ −1 B,+ (µ) and σ B,+ (µ) are defined as follows: At the initial scale µ 0 = 1 GeV of the evolution, the above quantities were estimated by using the method of the Light-Cone-Sum-Rules (LCSR) and their values are presented in Table 2. At the typical scale µ sp = Λ h m b,pole ≃ 1.52 GeV (here, Λ h = 0.5 GeV is a typical hadronic scale) of the hard-spectator corrections, the first inverse moment is now estimated as: λ −1 B,+ (µ sp ) = (2.04 ± 0.48) GeV −1 . Note that, while overlapping within errors with the previously used value, the updated estimate is substantially smaller as well as the current error on this quantity is now reduced by a factor of two.
Updating the analysis presented in Ref. [15], and using the experimental results on the branching ratios for the B → K * γ and B → X s γ decays given in Table 1, the phenomenological values of the soft part of the QCD form factor are: ξ γ and B ± → K * ± γ branching ratios and from the ratioR exp (K * γ/X s γ), respectively. The QCD form factorT K * 1 (0) differs from its soft partξ [19], which in our notation is given in Eq. (5.13) of Ref. [15]. However, the updated input parameters reduce this correction, yielding typically a correction of 2 − 4% only, in contrast to about 8% previously. Thus, the QCD transition form factorT K * 1 (0) now differs only marginally from its soft part, and is estimated as follows: The central value of the QCD form factor (16) extracted from the current data has remained unchanged compared to the previous estimateT K * 1 (0) = 0.27±0.04 (see Eq. (5.25) of Ref. [15]), but the error is now reduced by a factor 2, mostly due to the reduction of the uncertainty on the input parameters. It remains an interesting and open theoretical question if improved theoretical techniques for the calculation of the transition form factor T K * 1 (0) could accommodate this phenomenological result.

4.
Results for B → (ρ, ω) γ decays and comparison with the BELLE data. This part is devoted to an update of the theoretical predictions for the B → ργ and B 0 d → ωγ branching ratios, and their comparison with the BELLE data. Results for the direct and mixing-induced CP-violating asymmetries in these decays, the isospin-violating ratio in the B → ργ decays, and the SU(3)-violating ratio in the neutral B 0 d → ρ 0 γ and B 0 d → ωγ decays are also presented.

Branching ratios.
We now proceed to calculate numerically the branching ratios for the B ± → ρ ± γ, B 0 d → ρ 0 γ and B 0 d → ωγ decays. The theoretical ratios involving the decay widths on the r.h.s. of these equations can be written in the form: where m ρ and m ω are the masses of the ρ-and ω-mesons, ζ is the ratio of the transition form factors, ζ =T ρ 1 (0)/T K * 1 (0), which we have assumed to be the same for the ρ 0 -and ωmesons, and S ρ = 1 and 1/2 for the ρ ± -and ρ 0 -meson, respectively. To get the theoretical Table 3: Input parameters and their values used to calculate the branching fractions in the B → ργ and B 0 d → ωγ decays. The parameters entering in the B → K * γ part in Eqs. (17) and (18) are given in Table 2.
branching ratios for the decays B → ργ and B 0 d → ωγ, the ratios (17) and (18) should be multiplied with the corresponding experimental branching ratio of the B → K * γ decay.
The theoretical uncertainty in the evaluation of the R th (ργ/K * γ) and R th (ωγ/K * γ) ratios is dominated by the imprecise knowledge of ζ =T ρ 1 (0)/T K * 1 (0) characterizing the SU(3) breaking effects in the QCD transition form factors. In the SU(3)-symmetry limit, (3)-breaking effects in these form factors have been evaluated within several approaches, including the LCSR and Lattice QCD. In the earlier calculations of the ratios [15,28], the following ranges were used: ζ = 0.76±0.06 [15] and ζ = 0.76 ± 0.10 [28], based on the LCSR approach [29][30][31][32][33] which indicate substantial SU(3) breaking in the B → K * form factors. There also exists an improved Lattice estimate of this quantity, ζ = 0.9±0.1 [34]. In the present analysis, we use ζ = 0.85±0.10, given in Table 3 together with the values of the other input parameters entering in the calculation of the B → (ρ, ω) γ decay amplitudes.
We now discuss the difference in the hadronic parameters involving the ρ 0 -and ωmesons. It is known that both mesons are the maximally mixed superpositions of theūu anddd quark states: |ρ 0 = (|dd − |ūu )/ √ 2 and |ω = (|dd + |ūu )/ √ 2. Neglecting the W -exchange contributions in the decays, the radiative decay widths are determined by the penguin amplitudes which involve only the |dd components of these mesons, leading to identical branching ratios (modulo a tiny phase space difference). The W -exchange diagrams from the O A entering in B ± → ρ ± γ and ǫ (0) A in the B 0 d → ρ 0 γ decay have been estimated in the factorization approximation for the weak annihilation (and W -exchange) contribution, but this is expected to be a good approximation in the heavy quark limit, where the O(α s ) non-factorizable Table 4: Updated theoretical estimates of the functions ∆R(ρ/K * ) and ∆R(ω/K * ), and the ratios of the branching ratios R th (ργ/K * γ) and R th (ωγ/K * γ) defined in Eqs. (17) and (18), respectively. The third and fourth rows give the branching ratios B th (B → ργ) and B th (B 0 d → ωγ) (in units of 10 −6 ) and direct CP asymmetries in the B → ργ and B 0 d → ωγ decays, respectively. corrections are found to be suppressed in the chiral limit [13]. Moreover, their magnitudes can be checked experimentally through the radiative decays B ± → ℓ ± ν ℓ γ, as emphasized in Ref. [13]. These and the other parameters needed for calculating the branching ratios in the B → (ρ, ω) γ decays are given in Table 3, where we have also given the default ranges for |V tb V * td | and the CKM-Wolfenstein parametersρ andη obtained from the recent fit of the CKM unitarity triangle [25].
The individual branching ratios B th (B → ργ) and B th (B 0 d → ωγ) and their ratios R th (ργ/K * γ) and R th (ωγ/K * γ) with respect to the corresponding B → K * γ branching ratios are presented in Table 4. Note that in our estimates there is practically no difference between the B 0 d → ρ 0 γ and B 0 d → ωγ branching fractions, as the two differ only in the signs of the weak-annihilation contributions in the decay amplitudes, but these contributions given in terms of the parameters ε where the current experimental values of the B → K * γ branching ratios given in Table 1 have been used in arriving at the result (19). These theoretical estimates, carried out in the context of the SM, are in the comfortable agreement with the current BELLE measurements (1) and (7).

CP-violating asymmetries.
The direct CP-violating asymmetries in the decay rates for B + → ρ + γ and B 0 d → (ρ 0 , ω) γ decays and their charged conjugates are defined as follows: .
The explicit expressions for the first two of these asymmetries in terms of the individual contributions in the decay amplitude can be found in Ref. [15] and the one for the last, A dir CP (ωγ) may be obtained from A dir CP (ρ 0 γ) by obvious replacements. Their updated values in the SM, taking into account the parametric uncertainties and adding the various errors in quadrature, are presented in Table 4. The main contribution to the errors is coming through the scale dependence and the uncertainty in the c-to b-quark mass ratio, which is a NNLO effect. A complete NNLO calculation will certainly be required to reduce the theoretical errors. It should be noted that the predicted direct CP-asymmetries in all three cases are rather sizable (of order 10%) and negative. This differs from our earlier estimates [15,28], worked out for A dir CP (ρ ± γ) and A dir CP (ρ 0 γ), where the explicit expressions were erroneously typed and used in the numerical program with the incorrect overall sign.
The dependence of the direct CP-asymmetry on the CKM unitarity-triangle angle α is presented in the left frame in Fig. 1. We note that the CP-asymmetries are calculated with the strong phases generated perturbatively in O(α s ) in the QCD factorization approach. In particular, they do not include any non-perturbative rescattering contribution. We recall that for the CP-asymmetries in non-leptonic decays, such as in B → ππ, current data point to the inadequacy of the perturbatively generated strong phases [25]. In radiative decays B → (ρ, ω)γ, such long-distance effects enter via the penguin amplitudes P  In the hadronic language, they can be modelled via the hadronic intermediate states, such as B ± → ρ ± ρ 0 → ρ ± γ, B ± → D * ±D * 0 → ρ ± γ, etc. Their relative contribution at the amplitude level was estimated for the decay B − → ρ − γ as |P c /P t | ≃ 0.06 [13], with |P u | ≪ |P c |. A recent model-dependent estimate [35] of the long-distance contribution in B 0 → ρ 0 γ via the intermediate D + D − state, B 0 → D + D − → ρ 0 γ, puts the relative contribution of the long-distance (LD) and short-distance (SD) contributions to the decay widths as Γ LD /Γ SD ≃ 0.3, using the lowest order result for Γ SD . Taking into account that the next-to-leading order contributions in Γ SD , updated in this paper, result in an enhancement by a factor of about 1.7, and noting further that the perturbative charm-penguin contribution should be subtracted from Γ LD to avoid double counting, the remaining rescattering contributions are very likely below 10%. However, one can not exclude an enhanced charm-penguin contribution at this rate and the CP-asymmetry A dir CP (ρ 0 γ) could beinfluenced from such long-distance contribution. Charm-penguin enhanced effects can be also tested in the Dalitz pair reaction B 0 → ρ 0 γ * → ρ 0 e + e − through measurements of the Stoke's vector components [35].
We now discuss the time-dependent (or mixing-induced) CP-asymmetry in the B 0 d (t) → (ρ 0 , ω) γ andB 0 d (t) → (ρ 0 , ω) γ decays. Below, the equations for the B 0 d -meson decays into the final state with the ρ 0 -meson are presented. Similar quantities for the decays with the ω-meson production can be obtained by the obvious replacement: ε The time-dependent CP-asymmetry in the decays of neutral B 0 d -mesons and its CPconjugate involves the interference of the B 0 d −B 0 d mixing and decay amplitudes and is given by [36]: where ∆M d ≃ 0.503 ps −1 is the mass difference between the two mass eigenstates in the B 0 d −B 0 d system. For getting explicit formulae for C ργ and S ργ , it is convenient to introduce the quantity: where In terms of λ ργ , the direct and mixing-induced CP-violating asymmetries can be written as follows: Thus, the direct CP-violating asymmetry C ργ is expressed by Eq. (6.6) in Ref. [15] while the mixing-induced CP-violating asymmetry S ργ in NLO can be presented in the form: where A (1)t R and A u R are the real parts of the NLO contributions to the decay amplitudes entering Eq. (23). It is easy to see that, neglecting the weak-annihilation contribution (ε (0) A = 0), the mixing-induced CP-asymmetry vanishes in the leading order. However, including the O(α s ) contribution, this CP-asymmetry is non-zero.
The dependence on the CKM unitarity-triangle angle α of the mixing-induced CPasymmetry for the B 0 d -meson modes considered is presented in Fig. 1 (right frame). The dashed lines show the dependence in the LO while the solid lines correspond to the NLO result. Thus, fixing the parameters to their central values, one notices a marked effect from the NLO corrections on both S LO ργ and S LO ωγ . However, including the errors in the input parameters, the resulting allowed values for S NLO ργ and S NLO ωγ are rather uncertain. This is worked out by taking into account the SM range α = (92 ± 11) • [25], and the numerical values for these asymmetries in the leading order and including the O(α s ) corrections are as follows: Thus, the ±1σ ranges for the mixing-induced asymmetries in the SM are: −0.04 ≤ S NLO ργ ≤ 0.05 and −0.01 ≤ S NLO ωγ ≤ 0.09. They are too small to be measured in the near future. Hence, the observation of a significant (and hence measurable) mixing-induced CP-asymmetries S ργ and S ωγ would signal the existence of CP-violating phases beyond the SM.

4.3.
Isospin-violating ratio. The charge-conjugated isospin-violating ratio is defined as follows: The explicit NLO expression in terms of the vertex, hard-spectator and weak-annihilation contributions to the decay amplitude can be found in Ref. [15]. The dependence of this ratio on the angle α is shown in the left frame in Fig. 2. With the improved input, the updated result is: Thus, the isospin violation in B → ργ decays is expected to be small in the SM. The reason for this lies in the dependence ∆ ∝ ε (±) A ) 2 , α s ], and we have used the current knowledge of the angle α from the CP-asymmetry in B → ππ decays and the indirect unitarity fits, yielding α = (92 ± 11) • [25].

The weighted factors in ∆
A )/2. In our approximation,ε A = 0 and, neglecting tiny corrections ∼ (F ∆ε A ) 2 , the final expression is greatly simplified: The dependence of this ratio on the angle α is shown in the left frame in Fig. 2. In the SM, with the input parameters specified above, this ratio can be estimated as: This value is an order of magnitude smaller than the isospin-violating ratio (30) in B → ργ decays due to the suppression of the weak-annihilation contributions in the decays of the neutral B-meson. In this case, the neglected subdominant long-distance contributions may become important. They can be estimated in a model-dependent way. In any case, the result in (35) should be improved by including the contributions of the penguin operators and the NNLO corrections. The ratio ∆ (ρ/ω) in the SM is also too small to be measured. Both the ratios ∆ and ∆ (ρ/ω) are sensitive tests of the SM, and as argued in Refs. [28,37] for the isospin-violating ratio ∆, their measurements significantly different from zero would reveal physics beyond the SM.

Determination of |V
To extract the value of |V td /V ts | from the B → (K * , ρ, ω) γ decays, we use the ratioR th [(ρ, ω) γ/K * γ], which can be rewritten within the SM as follows: where the error in r (ρ/ω) th takes into account all the parametric uncertainties except in ζ and |V td /V ts | which are treated as free variables. Applying this equation to the BABAR upper limit (6) and the BELLE experimental range (7), the product ζ |V td /V ts | can be restricted as follows: where the bound is at 90% C.L., following from the BABAR data. At present, the error in (38) is dominated by the experimental uncertainty. Using the range ζ = 0.85 ± 0.10 for the ratio of the transition form factors, one gets the following constraints on the CKM matrix element ratio |V td /V ts |: where the lower limit from the BABAR data (6) corresponds to 90% C.L. In arriving at these numbers, the theoretical and experimental errors were considered as uncorrelated.
Taking this correlation into account, the BELLE data yields the range 0.16 < |V td /V ts | < 0.29, which is much larger than but in agreement with the SM range |V td /V ts | = 0.20±0.02. The dependence of the ratioR th [(ρ, ω)γ/K * γ] on |V td /V ts | is shown in the right frame in Fig. 2. The solid curve corresponds to the central values of the input parameters, and the dashed curves are obtained by taking into account the ±1σ errors on the individual input parameters inR th [(ρ, ω)γ/K * γ] and adding the errors in quadrature. The current measurement for this quantity is also shown in this figure. Experimental error is currently large which renders the determination of |V td /V ts | uncertain. However, in the long run, with greatly increased statistics, the impact of the measurement ofR exp [(ρ, ω)γ/K * γ] on the CKM phenomenology, in particular the profile of the unitarity triangle, will depend largely on the theoretical accuracy of the ratio ζ. Note that using |V td /V ts | = 0.20 ± 0.02, the estimates (37) and (38) result into the lower limit ζ > 0.81 (at 90% C.L.) from the BABAR data and the range 0.71 < ζ < 1.19 from the BELLE measurement. These inferences are not precise enough to distinguish among models of SU(3)-breaking. We hope that with the first measurement ofR exp [(ρ, ω)γ/K * γ] having been already posted [1], the ratio ζ will receive a renewed theoretical effort, in particular from the lattice community.
6. Current and potential impact ofR exp [(ρ, ω) γ/K * γ] on the CKM unitarity triangle. In this part we present the impact of the B → (ρ, ω) γ branching ratio on the CKM parametersρ andη. For this purpose, it is convenient to rewrite the ratiō R th [(ρ, ω)γ/K * γ] in the form in which the dependence on the CKM-Wolfenstein parametersρ andη is made explicit: Here, the function G(ρ,η, ε) encodes both the LO and NLO contributions: and the functions G i (i = 0, 1, 2) are defined as follows:  30 Table 6: The input parameters used in the CKM-unitarity fits. Their explanation and discussion can be found, for example, in Ref. [41]. The parameter η 1 is evaluated at the scale of MS mass m c (m c ) = 1.30 GeV.
Numerical values of the real and imaginary parts of the functions G i (i = 0, 1, 2), and the parametric uncertainties, are given in Table 5. The three rows in this table correspond to the decays B ± → ρ ± γ, B 0 d (B 0 d ) → ρ 0 γ, and B 0 d (B 0 d ) → ωγ, respectively. It should be noted that the function G(ρ,η, ε) (42) is related with the dynamical function ∆R, introduced in Ref. [15] to account for the weak-annihilation and NLO corrections, with: G(ρ,η, ε) = R 2 t (1 + ∆R). To undertake the fits of the CKM parameters, we adopt a Bayesian analysis method. Systematic and statistical errors are combined in quadrature. We add a contribution to the χ 2 -function for each of the input parameters presented in Table 6. Other input quantities are taken from their central values given in the PDG review [38]. The lower bound on the mass difference ∆M Bs in the B 0 s −B 0 s system is implemented using the modified χ 2 -method (as described in the CERN CKM Workshop proceedings [39]), which makes use of the amplitude technique [40]. The B s ↔B s oscillation probabilities are modified to have the dependence P (B s →B s ) ∝ [1 + A cos(∆M Bs t)] and P (B s → B s ) ∝ [1 − A cos(∆M Bs t)]. The contribution to the χ 2 -function is then: where A and σ A are the world average amplitude and error, respectively. The resulting χ 2 -function is then minimized over the following parameters:ρ,η, A, Further details can be found in Ref. [41]. We present the output of the fits in Table 7, where we show the 68% C.L. ranges for the CKM parameters A,ρ andη, the angles of the unitarity triangle α, β and γ, as well as sin(2φ i ) with φ i = α, β, γ, and ∆M Bs . The allowed profile (at 95% C.L.) of the unitarity triangle from the resulting fit is shown in Fig. 3 as shaded region. Here we also show the 95% C.L. range of the ratioR exp [(ρ, ω) γ/K * γ] =B exp [B → (ρ, ω) γ]/B exp (B → K * γ), which is used as an input in the fits now. We find that the current measurement ofR exp [(ρ, ω) γ/K * γ] is in comfortable agreement with the fits of the CKM unitarity triangle resulting from the measurements of the five quantities (R b , ǫ K , ∆M B d , ∆M Bs , and a ψK S ). The resulting contour in theρ −η plane practically coincides with the shaded region, and hence not shown. We conclude that due to the large experimental error onR exp [(ρ, ω) γ/K * γ], but also due to the significant theoretical errors, the impact of the measurement of B → (ρ, ω)γ decays on the profile of the CKM unitarity triangle is currently small. How this could change in future is illustrated by reducing the current experimental error onR exp [(ρ, ω) γ/K * γ] by a factor 3, which is a realistic hope for the precision on this quantity from the B-factory experiments in a couple of years from now. The resulting (95% C.L.) contours are shown as dashed-dotted curves, which result in reducing the currently allowedρ −η parameter space. This impact will be enhanced if the theoretical errors, dominated by ∆ζ/ζ, are also brought under control.