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Direct CP violation in singly Cabibbo-suppressed D-meson decays

The LHCb and CDF collaborations reported a surprisingly large difference between the direct CP asymmetries, Delta A_CP, in the D0 ->K+ K- and D0 ->pi+ pi- decay modes. We show that this measurement can be plausibly explained within the standard model under the assumption of large penguin contractions matrix elements and nominal U-spin breaking. A consistent picture arises, accommodating the large difference between the decay rates, and the measured decay rates of the D ->K pi modes.


Introduction
The D 0 → K + K − and D 0 → π + π − decays are induced by the weak interaction via an exchange of a virtual W boson and are suppressed by a single power of the Cabibbo angle. Direct CP violation in singly Cabibbo-suppressed (SCS) D-meson decays is sensitive to contributions of new physics in the up-quark sector, since it is expected to be small in the standard model: the b-quark penguin amplitudes necessary for interference are down by a loop factor and small Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, and there is no heavy virtual top quark which could provide substantial breaking of the Glashow-Iliopoulos-Maiani (GIM) mechanism. Naively, one would thus expect effects of order O( We define the amplitudes for final state f as 1 Note that the possibility of CP violation by tree amplitudes has already been pointed out in [1].
Here A T f is the dominant tree amplitude and r f the relative magnitude of the subleading amplitude, carrying the weak phase φ f and the strong phase δ f . We can now define the direct CP asymmetry as where the last equality holds up to corrections of O(r 2 f ). LHCb and CDF measure a time-integrated CP asymmetry. The approximately universal contribution of indirect CP violation cancels to good approximation in the difference The measurements of LHCb, ∆A CP = (−0.82 ± 0.21 ± 0.11)% [2], CDF, ∆A CP = (−0.62 ± 0.21 ± 0.10)% [3], and inclusion of the indirect CP asymmetry A Γ [6,7], lead to the new world average (including the Babar [4], Belle [5], and CDF [8] measurements) ∆A CP = (−0.67 ± 0.16)% [3]. We show that the large difference of SCS branching ratios, Br , implies large penguin matrix elements, which in turn account for the large value of ∆A CP .

A consistent picture
The starting point of our analysis is the weak effective Hamiltonian The Wilson coefficients of the tree operators Q pp 6 , and the chromomagnetic operator Q 8g , can be calculated in perturbation theory [9]. The hadronic matrix elements are of nonperturbative nature and will ultimately can be computed using lattice QCD [10]. We will estimate their size using experimental data.
A leading power estimation of the ratio r LP f ≡ |A P f (leading power)/A T f (experiment)|, using naive factorization and O(α s ) corrections, yields r LP . This is consistent with, yet slightly larger than the naive scaling estimate. We expect the signs of A dir K + K − and A dir π + π − to be opposite, if SU (3) breaking is not too large; so for φ f = γ ≈ 67 • and O(1) strong phases we obtain ∆A CP (leading power) = O(0.1%), an order of magnitude smaller than the measurement. However, from SU (3) fits [12,13,14,15,16] we know that power corrections can be large. To be specific, we look at insertions of the penguin operators Q 4 , Q 6 into power-suppressed annihilation amplitudes. The associated penguin contractions of Q 1 cancel the scale and scheme dependence. A rough estimate of their size leads to r PC | for the insertion of Q i , with an uncertainty of a factor of a few. Larger effects are very unlikely [11]. Again assuming O(1) strong phases, this leads to ∆A CP (r f,1 ) = O(0.3%) and ∆A CP (r f,2 ) = O(0.2%) for the two insertions. Thus, a standard-model explanation seems plausible.
This conclusion receives further support from data. The large difference of SCS branching ratios translates into a ratio of amplitudes (normalized to phase space) of , whereas the amplitudes would be equal in the SU (3) limit. This has often been interpreted as a sign of large SU (3) breaking. On the other hand, the ratio of the Cabibbo-favored (CF) to the doubly Cabibbosuppressed (DCS) amplitude is A(D 0 → K − π + ) ≈ 1.15 × A(D 0 → K + π − ), after accounting for CKM factors, in accordance with nominal SU (3) breaking of O(20%). This value is affirmed by the fact that the experimental amplitudes satisfy the sum rule relation This expression would vanish in the U -spin limit and receives correction quadratic in U -spin breaking. An inspection of the effective Hamiltonian (4) shows that the combination P of penguin contractions of Q ss 1,2 and Q dd 1,2 proportional to V cb V * ub is U -spin invariant, while P break , the combination of penguin contractions contributing to the decay rates vanishes in the U -spin limit. P break contributes with opposite sign to the two SCS decay rates, and P gives rise to a nonvanishing ∆A CP . Guided by the considerations above, we perform a U -spin decomposition of the amplitudes to all four (CF, SCS, DCS) decays, and fit these amplitudes to the data (branching ratios and CP asymmetries) [17]. There we also provide an exact definition of the amplitudes and a translation between the U -spin decomposition and the operator picture.
Our main point is that under the assumption of nominal U -spin breaking, a broken penguin P break , which explains the difference of the D 0 → K + K − and D 0 → π + π − decay rates, implies a ∆U = 0 penguin P that naturally 2 yields the observed ∆A CP . The scaling P break ∼ U P together with our fit to the branching ratios alone yields P break ∼ T /2 (see Fig. 1), leading to the estimate Figure 1: The results of our fit. Solid, dashed, and dotted lines denote one-sigma, two-sigma, and three-sigma contours, respectively. Left panel: A fit to the branching ratios only yields P break ≡ sd P ∼ T , assuming nominal U -spin breaking. T is the tree amplitude. The lower bound of P/T avg in the middle panel is directly related to the large difference of decay rates for the SCS modes. (T avg is the average value of T from the fit). It translates into the upper bound on ∆A CP -the fit results can naturally accommodate the measured value (right panel).
for U ∼ 0.2. This is consistent with the measured ∆A CP assuming O(1) strong phases. A fit to the full data set including CP asymmetries confirms this naive estimate (see Figure 1), showing that large penguin contraction matrix elements together with nominal U -spin breaking lead to a consistent picture accommodating all data on decay rates and CP asymmetries of all four (CF, SCS, DCS) modes [17].