{'fraction_non_alphanumeric': 0.050041073384446876, 'fraction_numerical': 0.021426615553121576, 'mean_word_length': 3.889223560910308, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 5, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 7, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Lessons from CLEO and FOCUS Measurements of D0-anti-D0 Mixing Parameters

If the true values of the D0-anti-D0 mixing parameters lie within the one sigma ranges of recent measurements, then there is strong evidence for a large width difference, y>0.01, and large SU(3) breaking effects in strong phases, \delta>pi/4. These constraints are model independent, and would become stronger if M_{12}/Gamma_{12}<<1 in the D0-anti-D0 system. The interesting fact that the FOCUS result cannot be explained by a large mass difference is not trivial and depends on the small D0-anti-D0 production asymmetry in FOCUS and the bounds on CP violating effects from CLEO. The large value of \delta might help explain why y ~ sin^2(theta_c).


Introduction
Recent studies of time-dependent decay rates of D 0 → K + π − by the CLEO collaboration [1] and measurements of the combination of D 0 → K + K − and D 0 → K − π + rates by the FOCUS collaboration [2] have provided highly interesting results concerning D 0 − D 0 mixing. (For previous, related results, see [3][4][5][6][7][8].) Each of the two experiments finds a signal for mixing at a level that is close to 2σ. It is not unlikely that these signals are just the results of statistical fluctuations and the true mixing parameters lie well below the experimental sensitivity. In this work, however, we interpret the experimental results assuming that their central values are not far from the true values and that D 0 −D 0 mixing has indeed been observed.
The interpretation of the results and, in particular, testing the consistency of the two recent measurements with each other, require a careful treatment of signs and phase conventions. We present the relevant model-independent formalism in section 2. In section 3 we carefully explain what parameters have the FOCUS and CLEO experiments actually measured. We emphasize that, in principle, both CLEO and FOCUS results can be accounted for even if the width difference is negligibly small. This fact was known for the CLEO result [9], but it is much more subtle for the FOCUS result.
In section 4, we analyze the theoretical implications of the FOCUS and CLEO results in a model independent framework. We do however make some reasonable assumptions.
With new physics, it is possible that there are large, CP violating new contributions to the mass difference. On the other hand, it is very unlikely that the width difference [10] and relevant decay amplitudes [11] are significantly affected by new physics. In such a framework, the measured observables depend on the mass difference x, the width difference y, two independent CP violating parameters, φ and A m , and a strong phase δ. We find that the experimental results have strong implications for the width difference y and for the strong phase δ. The qualitative features are independent of the other parameters, though the detailed quantitative results are not.
It could be that the D 0 −D 0 system is a unique example of a case where the dispersive part of the D 0 → D 0 transition amplitude is much smaller than the absorptive part, |M 12 | ≪ |Γ 12 |. (For the K 0 −K 0 the two are comparable, while for the B 0 −B 0 and B s −B s systems the situation is opposite, |M 12 | ≫ |Γ 12 |.) This situation, which is rarely discussed in the literature, is analyzed in section 5. We point out that, if this approximation is valid, the dependence on x and on the CP violating parameters can be neglected. Consequently, the FOCUS and CLEO results depend on y and δ only, and the implications become much clearer, both qualitatively and quantitatively.
Within the Standard Model, D 0 − D 0 mixing vanishes in the limit of exact SU (3) flavor symmetry of the strong interactions. For example, the sum of the contributions to the width difference from intermediate K + K − , π + π − , K + π − and K − π + states vanishes in the SU (3) limit. The fact that the one sigma ranges of the FOCUS and CLEO results constrain cos δ allows, for the first time, a calculation of this contribution based entirely on experimental data. We carry out such a calculation in section 6 and find a surprisingly large contribution to y, of order one percent.
A summary of our results is given in section 7.

Notations and Formalism
We investigate neutral D decays. The two mass eigenstates, |D 1 of mass m 1 and width Γ 1 and |D 2 of mass m 2 and width Γ 2 , are linear combinations of the interaction eigenstates: The average mass and width are given by The mass and width difference are parametrized by Decay amplitudes into a final state f are defined by It is useful to define the complex parameter λ f : The processes that are relevant to the CLEO and FOCUS experiments are the doubly-Cabibbo-suppressed D 0 → K + π − decay, the singly-Cabibbo-suppressed D 0 → K + K − decay, the Cabibbo-favored D 0 → K − π + decay, and the three CP-conjugate decay processes.
We now write down approximate expressions for the time-dependent decay rates that are valid for times t < ∼ 1/Γ. We take into account the experimental information that x, y and tan θ c are small, and expand each of the rates only to the order that is relevant to the CLEO and FOCUS measurements: physics, the picture of CP violation is simplified since there is no direct CP violation. The effects of indirect CP violation can be parametrized in the following way [12]: (2.9) Here R and R m are real and positive dimensionless numbers. CP violation in mixing is related to R m = 1 while CP violation in the interference of decays with and without mixing is related to sin φ = 0. The choice of phases and signs in (2.9) is consistent with having φ = 0 in the Standard Model and δ = 0 in the SU (3) limit (see below). We further define (2.10) With our assumption that there is no direct CP violation in the processes that we study, and using the parametrizations (2.9) and (2.10), we can rewrite eqs. (2.6)−(2.8) as follows:

CLEO and FOCUS Measurements
The FOCUS experiment [2] fits the time dependent decay rates of the singly-Cabibbo suppressed (2.12) and the Cabibbo-favored (2.13) modes to pure exponentials. We define Γ to be the parameter that is extracted in this way. More explicitly, for a time dependent The above equations imply the following relations: Let us define A prod as the production asymmetry of D 0 and D 0 : Then The one sigma range measured by FOCUS is The interpretation of this measurement simplifies when the following two facts are taken into account: (i) The E687 data [13] suggest that A prod is small for FOCUS, of order 0.03.
(ii) The CLEO data [1] suggest that R m is not very different from one (see below). Actually, CLEO implicitly assume that this is the case in their analysis by using Evaluating (3.3) to linear order in the small quantities A prod and A m yields The CLEO measurement [1] gives the coefficient of each of the three terms (1, Γt and (Γt) 2 ) in the doubly-Cabibbo suppressed decays (2.11). Such measurements allow a fit to the parameters R, R m , x ′ sin φ, y ′ cos φ, and x 2 + y 2 . Fit A of ref. [1] quotes the following one sigma ranges: 3 R = (0.48 ± 0.13) × 10 −2 , We would like to point out that the interpretation of the FOCUS and CLEO results in terms of y, x, φ, δ and A m is almost independent of our assumption that there is no CP violation in decay. To understand this point, let us parametrize CP violation in decay in the following way: (3.8) Experimentally, we have the following constraints on the asymmetries in the Cabibbofavored [14], singly-Cabibbo-suppressed [15][16][17] and doubly-Cabibbo-suppressed [1] decays: For FOCUS, eq. (3.6) would be corrected by terms of order A CP (K − K + )A prod and A CP (K − K + )A m , which are negligible. For CLEO, the results in eq. (3.7) have been 3 CLEO quote a range for y ′ . It is obvious however that, with our conventions, their range applies to y ′ sign(cos φ) or perhaps to y ′ cos φ. Since the one sigma range is | cos φ| > ∼ 0.8, the difference between these two possibilities is unimportant for our purposes.
obtained allowing for CP violation in decay. There is however another subtle aspect of direct CP violation where our theoretical assumption does play a role. In the presence of new CP violating contributions to the decay amplitudes, the CP violating phases in λ f are not necessarily universal. Therefore, the use of a single phase φ in eq. (2.9) and consequently in eqs. (3.6) and (3.7) is valid only in the absence of direct CP violation.

Theoretical Interpretation
We now assume that the true values of the various mixing parameters are within the one sigma ranges measured by FOCUS and CLEO. That means in particular that we hypothesize that D 0 −D 0 mixing is being observed in the FOCUS measurement of y CP and in the CLEO measurement of y ′ cos φ. The combination of these two results is particularly powerful in its theoretical implications.
Let us first focus on the FOCUS result (3.4). We argue that it is very unlikely that this result is accounted for by the second term in (3.6). Even if we take all the relevant parameters to be close to their one sigma upper bounds, say, |x| ∼ 0.04 (we use Fig. 3 of ref. [1] to extract this upper bound), | sin φ| ∼ 0.6, |A m /2| ∼ 0.4 and A prod ∼ 0.03, we get y CP ∼ 0.01, about a factor of two too small. We can make then the following model independent statement: if the true values of the mixing parameters are within the one sigma ranges of CLEO and FOCUS measurements, then y is of order of a (few) percent. Note that this is true even in the presence of CP violation, which does allow a mass difference, x = 0, to mimic a deviation from the average lifetime. Practically, we can take the FOCUS result to be given to a good approximation by This is a rather surprising result. Most theoretical estimates are well below the one percent level (for a review, see [18]). These estimates have however been recently criticized [19,20].
We will have more to say about this issue in section 6.
Second, we examine the consistency of the FOCUS and CLEO results. The two most significant measurements, that of y cos φ in eq. (4.1) and that of y ′ cos φ in eq. This requirement allows us to make a second model independent statement: if the true values of the mixing parameters are within the one sigma ranges of CLEO and FOCUS measurements, then the difference in strong phases between the D 0 → K + π − and D 0 → K − π + decays is very large. For δ = 0 we get y ′ /y = 1 instead of the range given in eq.
(4.2). To satisfy (4.2), we need, for example, The result in eq. (4.3) is also rather surprising. The strong phase δ vanishes in the SU (3) flavor symmetry limit [21]. None of the models in the literature [9,22,23]  Model, D 0 − D 0 mixing and D 0 decays into K + K − , π + π − and π ± K ∓ are described to an excellent approximation by physics of the first two generations. Consequently, the Standard Model makes a clean prediction that any CP violating effects in these processes are negligibly small. We can thus safely set φ = 0 and R m = 1. The statements below hold in any model where CP is a good symmetry in the relevant processes.
It is important to realize that the choice of φ = 0 is equivalent to choosing |D 1 (|D 2 ) to be the CP-odd (CP-even) state, |D − (|D + ). This can be seen from eq. (2.9). It gives We define the CP-odd state as the mass eigenstate that does not decay into K + K − . Indeed, we now have In the CP limit, a non-zero value of y CP (see eq. (3.4)) requires unambiguously that the width difference is large: The fact that y > 0 is preferred suggests that the CP-even state has a shorter lifetime, that is |D +,− = |D S,L where S and L stands for 'short' and 'long' lifetimes, respectively.
This important result holds in the CP limit model independently. Then, to leading order in |M 12 /Γ 12 |, we have: x/y = 2 |M 12 /Γ 12 | cos φ 12 , A m = 4 |M 12 /Γ 12 | sin φ 12 , φ = − 2 |M 12 /Γ 12 | 2 sin 2φ 12 . The third equation requires that | sin δ| is not large and consequently narrows the range for δ even further, The conclusion of our discussion here is that if the D 0 −D 0 system provides a (unique!) example of |M 12 | ≪ |Γ 12 |, then the FOCUS and CLEO measurements determine y to be at the few percent level and the strong phase δ is well above π/2.

Implications for the Width Difference
The value of the phase δ has important implications for another aspect of our study, that is the width difference. The contributions of the four charged two-body states, to Γ 12 , the absorptive part of the transition amplitude D 0 |H|D 0 , can be written as which leads to the following contribution to y: There are two points that we would like to extract from eq. (6.3). First, in the SU (3) limit, BR(D 0 → K − K + ) = BR(D 0 → π − π + ) = √ R BR(D 0 → K + π − ). The phase δ defined in (2.9) vanishes in the SU (3) limit which is consitent with the fact that y 2c = 0 in this limit. Second, we can use the measured branching ratios for the four decay modes and the value of the phase δ as fitted to the CLEO and FOCUS results to estimate y 2c .
We use [25] BR(D 0 → K − π + ) = (3.83 ± 0.09) × 10 −2 , interactions has to be modified. In particular, there are two independent pieces of evidence that the strong phase in D → K ± π ∓ decays is very large, δ > ∼ π/4 and perhaps δ ∼ 3π/4 (while δ = 0 in the SU (3) limit): (i) Either a negative sign for cos δ or large x and large sin δ are necessary to make the signs of the mixing parameters measured by FOCUS and by CLEO consistent with each other.
(ii) cos δ far from its SU (3) limit value of one implies that some contributions to the width difference are at the percent level.
We also discussed the possibility that in the D 0 − D 0 system |M 12 /Γ 12 | ≪ 1, in contrast to the neutral B meson systems. In such a case, the D 0 − D 0 system is not sensitive to new physics, even if new physics dominates M 12 . In particular, CP is expected to be a good symmetry regardless of whether there are large CP violating contributions to M 12 . The above statements about large SU (3) breaking effects become even sharper in this case.
A much clearer picture would emerge if the accuracy of the measurements improves and, in particular, if the mixing parameters are measured separately in the D 0 and D 0 decays. For example, the FOCUS collaboration has summed over the D 0 → K + K − and D 0 → K + K − modes, but there is much to learn from comparing them to each other.
Explicitly, we obtain from eq. (3.1): A difference between the fitted decay width of the two CP conjugate modes will provide important information on the CP violating parameters.