Power corrections in e+ e- -->pi+ pi-, K+ K- and B -->K pi, pi pi

CLEO-c measurements of the timelike form factors F_pi, F_K at \sqrt{s}=3.671 GeV provide a direct probe of power corrections (PC's) at energies near m_B. PC's in F_pi, F_K and B \to K pi, pi pi are separated into perturbative and soft parts. In F_pi, F_K the latter are \ge O(10) larger. A PC fit to the B \to K pi, pi pi data also yields a \ge O(10) soft-to-perturbative hierarchy for the QCD penguin PC's. Hence, both can be attributed to dominance of the soft-ovelap between energetic (approximately) back-to-back collinear partons, and consistency of the B\to K pi, pi pi fit with the Standard Model appears to be naturally realized. The CP asymmetries S_{K_s pi^0}, C_{K_s pi^0} are well determined, providing a clean test for new physics.

Much effort has gone into the theoretical description of B decays into light meson pairs. Apart from being of interest in QCD, the issue has important implications for new physics search strategies which rely on comparing decay rates and CP asymmetries in different final states. The decay amplitudes can be organized into expansions in powers of 1/m b . The leading power (LP) contributions are calculable in QCD factorization (QCDF) [1] in terms of universal non-perturbative quantities. Numerous leading power predictions for B → M 1 M 2 decays are in gross conflict with the data. In B → Kπ, ππ the direct CP asymmetry A π + π − is too small, A K + π − is too small and of wrong sign, A K + π − ≈ A K + π 0 , contrary to observation, and the branching ratios Br K 0 π 0 , Br π 0 π 0 are too small. A possible explanation is that certain PC's are of same order as or larger than their LP counterparts and have large strong phases, due to non-perturbative effects.
Continuum e + e − → M 1 M 2 light meson cross sections at √ s ≈ 3.7 and 10.58 GeV at the charm and B factories provide a direct probe of PC's in the timelike vectorcurrent matrix elements M 1 M 2 |qγ µ q|0 . Perturbative calculations of PC's on the light-cone contain IR logdivergent terms of the form α s (µ h )(1/ √ s) n ln m ( √ s/Λ), signaling the breakdown of short/long-distance factorization. Λ represents a physical IR cutoff on the longitudinal momentum of, e.g., a valence quark in the convolution integrals of light meson light-cone distribution amplitudes (LCDA's) with hard scattering amplitudes. We therefore divide the PC's into perturbative and non-perturbative parts (soft overlaps), where the former are defined by imposing Λ, µ h > ∼ 1 GeV, and fits to the data give ranges for the latter. For example, the vector-current form factor PC's are written as δF = δF pert. + δF n.p. . In this letter we focus on PC's in e + e − → π + π − , K + K − and B → Kπ, ππ. Details, and results for V P , V V final states will be given elsewhere [2].
The calculable LP contributions arise at twist-2 in the LCDA's, and fall like 1/s [4]. We obtain where the errors due to variation of the first two LCDA Gegenbauer coefficients [5], and the scale at which they and α s are evaluated, √ s to √ s/2, are added in quadrature. Thus, the form factors must be dominated by PC's. A dual-resonance model calculation of F π [6], which includes the first three ρ resonances explicitly, is consistent with the twist-2 result for √ s > 3 GeV [7]. Evidently, duality has set in, implying sensitivity to PC's. δF K,π enter at 1/E 2 , or twist-4 perturbatively, and to first approximation fall like 1/s 2 . We obtain δF pert.
K,π from convolutions of two twist-3 valence quark LCDA's with the tree-level hard-scattering amplitudes (twist-4 valence quark LCDA's contribute negligibly). The model parameters of [5] are employed for the LCDA's. Perturbative higher Fock state effects are of same twist and order of magnitude, and therefore would not alter our result for the soft PC's. Fig. 1 shows the ranges obtained as the cutoff Λ in the divergent terms is varied from √ s to 1 GeV. Λ is roughly the lowest gluon virtuality allowed. Ranges for asymptotic LCDA's are also shown. There are large accidental cancelations between asymptotic and non-asymptotic effects at lower Λ. The magnitudes of each separately therefore give a better indication of the size of perturbative effects for Λ near 1 GeV. Comparing the asymptotic plots to Eq. 1, it is clear that the dominance of the PC's in F K,π is due to their soft parts, Similar soft enhancement would account for F π (m J/Ψ ) ≈ 0.10, as extracted from J/Ψ decays [8].
The SU (3) F diagrammatic representation gives a convenient general classification of the B → Kπ, ππ amplitudes [9]. TheB → Kπ amplitudes are The CKM factor λ p = V pb V * ps , and there is a sum over p = u, c. The B − (B 0 ) Br's are given by |A| 2 (τ B 0 /τ B − |A| 2 ) in our normalization.
T (a 1 ) and C (a 2 ) are the color-allowed and color-suppressed 'tree' amplitudes. P p (a 4,6 ), P p EW (a 7,9 ), and P C ,p EW (a 8,10 ) are the QCD, electroweak penguin (EWP), and color-suppressed EWP amplitudes, respectively. In ππ the corresponding quantities are primed ('). These amplitudes consist of LP parts (the corresponding QCDF coefficients a i [1] are given in parenthesis) T LP , C LP , etc., and PC's δT , δC, etc.. P E ,p EW and A are EWP and 'tree' weak-annihilation (WA) PC's.
The Kπ PC fit is dominated by |δP p |e iδp , |δC|e iδC , |δT |e iδT . The strong phases are defined relative to the corresponding naive-factorization amplitudes. Allowing for a soft contribution that is an order of magnitude greater than its perturbative part leaves A negligible. The electromagnetic u, c-loop penguin contractions are negligible in the EWP PC's, so we drop their 'p' superscripts. The approximate relation (up to an unknown SU (3) F breaking factor) δP C EW ≈ 3/2 C 9 /C 1 δC, and the order of magnitude relations (equalities in the one-gluon exchange approximation) δP EW ∼ 3/2 C 10 /C 1 δC and δP E EW ∼ 3/2 C 9 /C 2 A relegate the EWP PC's to a minor role. The C i are Wilson coefficients for the ∆B = 1 effective Hamiltonian [1]. In terms of the measured uncertainties, the impact of δP C EW on Br A K + π − is found to be < ∼ 2.5σ, 1.5σ, 1.5σ, respectively. The impact of the remaining EWP PC's and A on all observables is < 1σ. The ππ PC fit is dominated by The 'tree' WA amplitude E plays a significant role in π + π − , and the ππ EWP PC's are negligible.
The ππ P ′ c fit in Fig. 3b is multiplied by f K /f π for comparison with Fig. 3a. This time, LP . However, f K /f π δP ′ c appears to be larger than δP c , in accord with δF π > δF K . This conclusion is reinforced if |δP ′ u /δP ′ c | is bounded from above, e.g. ≤ 3, see Fig. 3b. C ′ /T ′ is plotted in Fig. 3d. The lower bound on its magnitude increases from roughly 0.15 to 0.25 for |δP ′ u /δP ′ c | ≤ 3. The predicted ranges for A π + π 0 , A π 0 π 0 are [−0.  Fig. 2 eliminate the dominant leading log µ b scale dependence (∝ C 1 α s /π) in δP p, pert. . Products of twist-2,3 × twist-3 K,π valence quark LCDA's are included in the convolutions with the hard-scattering amplitudes. Again, higher Fock-state effects would not alter our conclusions.
Our results are summarized in Fig. 4. The largest real and imaginary contributions to δP (′) p ,pert. come from Fig. 2b. The QCD dipole operator Q 8g (Fig. 2e) and QCD penguin WA (Q 3,..,6 in Fig. 2a) contributions are a factor of 2-3 smaller in magnitude, and real. All three are dominated by diagrams in which a gluon is not attached to the B. They contribute at twist-3 (1/m b ). (Other contributions to δP (′) p ,pert. , e.g. Q 3,..,6 in Fig. 2c and Fig. 2d, are negligible.) In the limit Λ → Λ QCD they would give rise to two energetic outgoing quarks, and soft-overlaps for both light mesons, as indicated by a quadratic dependence on log m B /Λ. This is also the case for δF pert.
π,K , which depend quadratically on log √ s/Λ. Therefore, the relative importance of non-perturbative PC's in δP (′) p should be similar to Eq. 3, given the proximity of energy scales. Indeed, comparison of Figs δC (′) pert. is due to Q 1 in Fig. 2c. The diagrams again enter at twist-3 (1/m b ). In the Λ → Λ QCD limit only the spectator quark light meson would be produced via a soft-overlap, as indicated by a linear dependence on log m B /Λ. Therefore, we expect δC (′) , n.p. /δC (′) , pert. to be smaller than the QCD penguin ratio. According to Figs. 3c, 4d (for low Λ) this may well be the case, but the errors are too large to reach a definitive conclusion. We emphasize that the similarity of Eq. 5 to Eq. 3 supports the importance of the charm-loop [14], dipole, and WA PC's in the full QCD penguin amplitudes, and increases our confidence that the B → Kπ, ππ 'puzzles' can be accounted for as in the fits. Given the penguin PC soft dominance, the origin of f K /f π δP ′ c > δP c is the same as δF π > δF K , a larger ππ soft-overlap. That certain soft PC's are of same order as their LP counterparts should be viewed as accidental, rather than a breakdown of the 1/E expansion. The latter would be signaled by a violation of power counting among the PC's themselves.
The asymptotic LCDA plots in Figs. 4a,b imply that the charm quark loops can yield large strong phases in the full penguin PC's. Their presence may be indicated by the fits for δ (′) c in Figs. 3a,b. However, a sizable strong phase could also originate from soft-overlaps in light meson production, independently of such loops, explaining the large strong phase in C. The existence of such a mechanism would be confirmed by a nonvanishing strong phase difference between the LL and LT  (L=longitudinal, T =transverse) polarization amplitudes in e + e − → ρ + ρ − at the Υ(4S). This can be searched for with the current B factory data sets [2,15]. The asymptotic LCDA plots for δP ′ u ,pert. and δP ′ c ,pert. in Fig. 4c support the possibility, allowed by the PC fits, that the total up and charm penguin PC's can have a large hierarchy, e.g., δP (′) u /δP (′) c < ∼ 3. The perturbative analysis also implies that they can have large strong phase differences. Similar considerations in B → φK imply that a large negative shift in the timedependent CP -asymmetry relative to sin 2β is possible e.g., δS φKs ≈ −(0.10 − 0.15) [2].
The QCD dipole operator contribution to δP p ,pert is also plotted in Fig. 4a. Eq. 5 then implies that the total Q 8g PC could be of order P p LP , see Fig. 3a. It is noteworthy that this is an order of magnitude larger than its LP matrix element. Therefore, the size of new CP violating dipole operator contributions currently probed could be O(10) smaller than expected based on LP analyses.
Continuum e + e − → V P, V V cross sections at √ s ≈ 3.7 and 10.58 GeV and penguin-dominated B → V P, V V data can be accounted for if soft-to-perturbative PC ratios vary from O(1) to O(few) [2], rather than O(10) as for P P . This appears to shed light on the "ρ − π puzzle" (and the smaller "K * − K puzzle"), i.e., how to simultaneously understand the π as a qq state like the ρ, and a nearly massless Nambu-Goldstone boson. In a physical picture of the π (and K) that addresses this puzzle [16], the valence q andq are always a distance < 1/Λ QCD apart. However, r π ≈ r ρ ≈ 1/Λ QCD is accounted for by a much larger soft cloud of higher Fock states, attributed to a zitterbewegung-like motion of the tightly bound valence quarks. This may account for the large P P soft-overlaps, as well as the larger ππ soft-overlaps because the π approximates a Nambu-Goldstone boson more closely than the K.
Note added: After completion of the work reported here we were informed by L. Silvestrini that his group has also carried out a Kπ PC fit, with similar predictions for S Ksπ 0 , C Ksπ 0 , which has since appeared [17]. For an earlier discussion by them, see M. Pierini in [14].