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Estimate of B(K ->pi nu nubar) from Standard Model fits to lambda_t

We estimate B(K ->pi nu nubar) in the context of the Standard Model by fitting for lambda_t = Vtd x V*ts of the `kaon unitarity triangle' relation. We fit data from epsilon_K, the CP-violating parameter describing K-mixing, and a_{psi K}, the CP-violating asymmetry in B ->J/psi K decays. Our estimate is independent of the CKM matrix element Vcb and of the ratio of Bs to Bd mixing frequencies. The measured value of B(K+ ->pi+ nu nubar) can be compared both to this estimate and to predictions made from the ratio of B mixing frequencies.

The ultra-rare FCNC kaon decays K + → π + νν and K • L → π • νν are of particular interest as these 'gold-plated decays' can be predicted in the Standard Model framework with very high theoretical accuracy.
The K → πνν decays are treated in detail in a number of papers . We list some of the key aspects of these decays.
a) The main contribution to these FCNC processes arises at small distances r ∼ 1/m t , 1/m Z ; therefore, a very accurate description for the strong interactions at the quark level is possible in the framework of perturbative QCD. This analysis has been carried out in the leading logarithmic order (LLO) with corrections to next to leading order (NLO) [1][2][3][4].
b) The calculation of the matrix element π|H w |K πνν from quark-level processes involves long-distance physics. However, these long-distance effects can be avoided by the renormalization procedure developed by Inami and Lim [5], relating the matrix element to that of the well known decay K + → π • e + ν e through isotopic-spin symmetry. Other possible long-distance contributions to B(K + → π + νν ) have been shown to be negligible [6].
c) Since the effective vertex Zds in the diagrams of Figure 1 is short-distance, these processes are also sensitive to the contributions from new heavy objects (e.g., supersymmetric particles).
A very important step in the study of K + → π + νν was achieved by the E787 experiment [7] at BNL in which two clean events were found in favorable background conditions, indicating a branching ratio of B(K + → π + νν ) = (15.7 +17.5 −8.2 ) × 10 −11 . This observation has opened the door for future more precise study of the K + → π + νν decay [8,9].
In the Standard Model, the K + → π + νν decay is described by penguin and box diagrams presented in Figure 1. The partial widths have the form: Γ(K + → π + νν ) = κ + · |λ c F (x c ) + λ t X(x t )| 2 = κ + · [(Reλ c F (x c ) + Reλ t X(x t )) 2 where The factor of 3 in the expression for κ + results from the three flavors of neutrinos (ν e , ν µ , ν r ) participating in the K + → π + νν decays. The factors F (x c ) and X(x t ) are functions corresponding to the quark loops. These functions include the Inami-Lim functions [5] and the QCD corrections that have been calculated to 10]. They depend on the variables x i = (m i /m W ) 2 with the masses of the + 2 3 quarks, m i : i = c, t. The λ i ≡ V id V * is are vectors in the complex plane that satisfy the unitarity relation: This equation describes the 'kaon unitarity triangle', which can be completely determined from measurement of the three kaon decays: K + → π • e + ν e , K + → π + νν and K • L → π • νν . This triangle is highly elongated with a base to height ratio of ∼1000.
Using the values of m c and m t in Table I, the calculations from Reference 1 yield F (x c ) = (9.8 ± 1.8) × 10 −4 and X(x t ) = (1.52 ± 0.05). The accuracy improves with increasing quark mass, and there are systematic dependences on Λ (4) MS . The c-quark contribution in (1) is smaller than the t-quark contribution, but is non-negligible. Although F (x c )/X(x t ) ∼ 10 −3 , Reλ c is much larger than Reλ t and Imλ t . (Reλ c ∼ λ while Reλ t , Imλ t and Imλ c are less than λ 5 ).
The parameters Imλ t , Reλ t , Reλ c can be estimated within the standard unitarity triangle (UT) framework using the improved Wolfenstein parameterization [15] η,ρ, A, and λ (with Aλ 2 = |V cb |,ρ ≡ ρ(1 − λ 2 2 ) and η ≡ η(1 − λ 2 2 ) ). To O(λ 4 ) the CKM matrix is and to higher order we have The current values of these and other parameters used in this paper can be found in Table I. Using (10) and Reference 35 (see Table I), equations (6) and (8) can be naively solved to give the branching ratios for K + → π + νν and K • L → π • νν : (11) and (12) are dominated by the current uncertainties in the CKM parameters and are significantly larger than the intrinsic theoretical uncertainties. The uncertainty of |V cb | is quite significant in the evaluation of B(K → πνν ) due to the |V cb | 4 dependence. CLEO has recently measured [36] a somewhat higher |V cb | value of (46.9 ± 3.0) × 10 −3 , which would cause a significant increase to B(K → πνν ) in equations (11) and (12).
For the values of the parameters |V cb | ,ρ andη in equations (11) and (12) we adopt the more conservative approach of Reference 35. A more aggressive approach [22] for the evaluation of these errors can significantly increase the precision for B(K → πνν ). Solving equations (11) and (12) with these values gives B(K + → π + νν )| SM = (7.4 ± 1.2) × 10 −11 and B(K • L → π • νν )| SM = (2.8 ± 0.5) × 10 −11 . The precision of the outputs of the standard UT fits is dependent on the value of ξ, the SU(3) breaking correction to ∆M Bs /∆M B d . The generally accepted value of ξ is ξ = 1.15 ± 0.06; however, recent work would suggest a higher value of ξ = 1.18 ± 0.04 +0. 12 −0.0 [37] (or even as high as ξ = 1.32 ± 0.10 [38].) Given the strong dependence of equations (11) and (12) on |V cb | , we consider an estimate of B(K + → π + νν ) that is essentially independent of |V cb | . This estimate is also independent of ∆M Bs /∆M B d . It is based solely on |ε K | and a ψK , is remarkably competitive to other estimates, and has the advantage of simplicity.
In this work we directly evaluate λ t to calculate B(K → πνν ) from (6) and (8). This avoids the use of ρ andη, as has been used in previous calculations of B(K → πνν ). This approach has been discussed in the literature [23,24], but as far as we know, no calculations of B(K → πνν ) exist by this method. In order to minimize uncertainty from |V cb | , it is natural to consider |ε K | and a ψK in terms of the kaon UT 1 . We recall that is real, and λ c = V cd V * cs has a very small complex phase ϕ(λ c ) ≃ Imλ t /λ ≃ 6 × 10 −4 . The phase of V ts is ϕ(V ts ) ≃ −π + Imλ t * λ/|V cb | 2 = −π + 0.0172 = −π + 1.0 • . The phase of V td is ϕ(V td ) = −β and the angle (β K ) between λ t and λ u is This angle is very close to β, which in the SM is extracted cleanly from the precise measurement of a ψK , the CP asymmetry in B • d → J/ψK • decays: sin 2β = 0.734 ± 0.054 [39]. We use an iterative procedure, starting with β K = β, from our fit to derive Imλ t and recalculate β K = β + Imλ t * λ/|V cb | 2 . This procedure converges after one iteration since the correction to β is small. There is also a small dependence on |V cb | ; however, a 10% change in |V cb | results in only a 0.6% shift in B(K + → π + νν ), which is significantly less than the uncertainty in our result. For all practical purposes our result is independent of |V cb | . The preferred solution for β, based on other SM input, such as V ub /V cb is β = (23.6 ± 2.3) • , so we shall only consider this particular solution. The extraction of sin 2β from a ψK is also clean in models with Minimal Flavor Violation (MFV) [22,25,26]. In these models there are no new phases and all of the influences of new physics are in modifications to the Inami-Lim functions.
In the Standard Model, the apex of the kaon UT (λ a t ) is constrained by various measurements as shown in Figure 2 (without errors). The constraint from |ε K | is expressed as [10,[40][41][42] with parameters as shown in Table I. We can find the apex of the kaon UT as the intercept of the |ε K | curve with the line representing the constraint from a ψK : The apex of the kaon unitarity triangle is λ a t (no errors are shown). The circle labeled V ub is described by (24) with a radius R∼V cb V ub . The thick black lines (|εK | and a ψK ) illustrate the main constraints used in this paper. The dashed lines illustrate the constraints from K → πνν . The constraint from ∆MB d is shown as the circle centered at the origin. The inset shows the triangle (not drawn to scale).
To calculate a probability density function (PDF) for λ a t , we follow the Bayesian approach of References 43, 44, and 22. Let f (x) be the PDF for x, where x is a point in the space of (β K , |ε K | ,B K , m t , m c , λ, α s , η cc , η ct , η tt ). Equations (14) and (15) define the mapping from x to λ a t . Through these equations and f (x), we derive f (λ a t ), the PDF for λ a t . f (x) depends on the PDF's for the components of x. We assume that the component PDF's are independent from one another except for the small dependence of η cc on m c and α s (discussed below). The component PDF's are taken from Table I. Figure 3 shows the PDF for λ a t . We find the following central values: For B(K + → π + νν )| SM we obtain from Equations (6) and (16): R + λ 2 = (7.07 ± 1.03) × 10 −11 (17) The three largest contributions to the uncertainty are due toB K (0.69 × 10 −11 ), m c (0.44 × 10 −11 ) and a ψK (0.49 × 10 −11 ). The probability distribution for B(K + → π + νν )| SM is presented in Figure 4.
In obtaining the results of equation (17) we have accounted for the correlations between |ε K | (one of the inputs for determining λ a t ), F (x c ) and X(x t ) through the variables x c , x t , and Λ MS ) are given in Reference 1, from which we have parameterized Table 1 to get: Equation (19) is accurate to 0.7% for α s in the range 0.116 to 0.122 [45]. The expression for |ε K | (and the determination of the apex, λ a t ) has a dependence on x c and  Figure 4: The PDF for B(K + → π + νν )|SM , obtained from the measurements of |εK | and a ψK . The 95% C.L. upper limit is 8.9 × 10 −11 and 95% C.L. lower limit is 5.6 × 10 −11 .
x t through the Inami-Lim functions S 0 (x c ), S 0 (x t ) and S 0 (x c , x t ). In addition, the NLO correction η cc has the following dependence [45]: The largest correlation through m c causes both endpoints of the vector describing B(K + → π + νν ), λ a t and Reλcf F (xc)

X(xt)
to move in similar directions, so that the uncertainty on the length of the vector is smaller than the uncertainties in either endpoint. Inclusion of the correlations due to x c , x t and Λ (4) MS reduces the uncertainty in B(K + → π + νν )| SM by ∼20%.
For K • L → π • νν we obtain from (8) and (16): The four largest contributions to the uncertainty are due tô B K (0.37 × 10 −11 ), a ψK (0.23 × 10 −11 ), m c (0.16 × 10 −11 ) and m t (0.08 × 10 −11 ). The results of these new calculations (17) and (22) of K → πνν branching ratios from fits to λ t are in a good agreement with the calculations based on the standard unitarity triangle variables (11) and (12) but are free of uncertainties in |V cb | and are independent of ∆M Bs /∆M B d . The main source of uncertainty in (17) and (22) is the lattice calculation ofB K = 0.86 ± 0.15. (We note that some lattice calculations using domain-wall fermions [18,46,47] find values ofB K that are 10-15% lower than the recent world average [37,48] that we use in Table I.) If future lattice QCD calculations [49] can significantly reduce the uncertainty inB K , an improvement in B(K → πνν )| SM will be possible.
Given the difficulty of assigning PDF's to theoretical uncertainties, we explore the influence of a more conservative scanning technique on the uncertainty in B(K + → π + νν )| SM . We determine λ a t again from only |ε K | and a ψK , using gaussian errors for all quantities exceptB K and m c , which are scanned throughout their ranges: 0.72<B K <1.0 and 1.2< m c <1.4. ForB K =0.72 and m c =1.4, which maximizes B(K + → π + νν ), the 95% CL upper limit is B(K + → π + νν )| SM < 9.9 × 10 −11 . For B K =1.00 and m c =1.2, which minimizes B(K + → π + νν ), the 95% CL lower limit is B(K + → π + νν )| SM > 5.0 × 10 −11 . These limits are not much worse than those derived from Figure 4.
We've emphasized that our estimate uses only a ψK and |ε K | . Nevertheless, it is interesting to consider how the measurements of ∆M B d and |V ub | would constrain λ a t . Here we will use the more aggressive treatment of |V cb | errors (see Table I) in order to obtain the smallest errors on B(K + → π + νν ). From the following relations: and using the approximations of (9): V * tb ≈ 1, V us = λ, V ud ≈ (1−λ 2 /2), and V cb ≈ −V ts , we convert the equations above into: These two equations describe two circles whose intersections contain the apex of the kaon UT (see Fig. 2), and are correlated somewhat through V cb . Similar to the case of |ε K | , with large uncertainties fromB K , there are large uncertainties in the extraction of λ a t from the ∆M B d and |V ub | constraints, with large uncertainties from f 2 B dB B d , |V ub | and |V cb | . The uncertainty on the constraint from B-mixing may be significantly improved by the addition of ∆M Bs , once the situation with ξ is resolved (this will be further improved once ∆M Bs is actually observed). Using the Bayesian procedure described earlier and the parameters in Table I, the PDF for λ a t derived solely from the constraints of ∆M B d and |V ub | is shown in Fig. 5. We see that this PDF does not constrain the kaon UT apex as well as a ψK and |ε K | . Combining all four constraints, we get the PDF for B(K + → π + νν ) shown in Fig. 6, which is only slightly more precise than Fig. 4. analysis we obtain The CKM matrix appears to be the dominant source of CP violation. However, some models [50] allow for a significant contribution of new physics to B(K → πνν ) while preserving the equality between sin 2β as measured from a ψK and global CKM fits. A crucial test of the CKM description will be to compare β derived from B(K → πνν ) to that from a ψK [12,[27][28][29]. The most important new information on the CKM matrix will be measurements of B(K + → π + νν ) [9] and B(K • L → π • νν ) [51] to 10% precision. The combination of these, in context of the SM, will determine sin 2β to 0.05 [30], competitive with the current uncertainty on sin 2β . The comparison of this angle obtained from B(K → πνν ) with that from a ψK will provide a very strong test of the SM description of CP-violation.
Another critical test of the SM will be the direct comparison of B(K + → π + νν ) to either ∆M Bs /∆M B d , which in the SM both directly measure |V td | , or to evaluations of B(K + → π + νν )| SM such as this work. Currently, the E787 measurement of B(K + → π + νν ) = (15.7 +17.5 −8.2 ) × 10 −11 is consistent with the SM expectation, but the central experimental value exceeds it by a factor of two. To date there is only a limit on ∆M Bs > 14.4ps −1 (95% C.L.)[52], but it is likely to be observed soon. Until ∆M Bs is observed, this limit can be used to set an upper limit on B(K + → π + νν )[1]. A recent calculation of this limit [17] gives B(K + → π + νν )| SM < 13.2 × 10 −11 , which is below the central experimental value [7]. This work used a value of ξ = 1.15 ± 0.06, whereas a higher value of ξ would raise this upper limit. Our work is an estimation of B(K + → π + νν )| SM based solely on |ε K | and a ψK and is not dependent on |V cb | or ∆M Bs /∆M B d . Our 95% C.L. upper limit is 8.9 × 10 −11 with the largest systematic error of this approach coming fromB K . The uncertainty from our prediction is comparable to the expected experimental uncertainties that might be achieved in the future measurements of K + → π + νν [8,9]. An experimental measurement significantly larger that determined from ∆M Bs /∆M B d or our 99% C.L. limit of B(K + → π + νν )| SM < 10 × 10 −11 will be a strong indication of new physics.

ACKNOWLEDGMENTS
We would like to thank P. Cooper, G. Isidori, D.E. Jaffe, P. Mackenzie, U. Nierste, L. Okun and A. Soni for useful discussions and comments and W. Marciano for the original stimulation to perform this type of calculation. This work was in part supported by the U.S. Department of Energy through contract #DE-AC02-98CH10886, in part by the Fermilab Particle Physics Division and in part by IHEP. L.G.L is grateful to the Fermilab Administration and Particle Physics Division for their hospitality and support for his stay at Fermilab during the preparation of this work.

NOTE
During the final preparation of this work for publication we found that Reference 56 considered fitting for the apex of the UT from the CP-violating data only (|ε K | and a ψK ), as we do. However, Reference 56 used (ρ,η), which is dependent on |V cb | and is not as suitable for analysis of K → πνν .