A Possibility to Measure CP-Violating Effects in the Decay K -->mu nu gamma

It is argued that a precise measurement of the transverse component of the muon spin in the decay K -->mu nu gamma makes it possible to obtain more stringent limits on CP-violating parameters of the leptoquark, SUSY and left-right symmetric models. The results of the calculations of the CP-even transverse component of the muon spin in the decay K -->mu nu gamma due to the final-state electromagnetic interactions are presented. The weighted average of the transverse component of the muon spin comprises 2.3 times 10^{-4}.


Introduction
The transverse component of the muon spin in the decay K → µνγ beyond the Standard Model is due to both the electromagnetic and CP-and T-violating interactions: where ξ EM is the contribution of the electromagnetic Final-State Interactions (FSI) and ξ odd is the contribution of the CP -odd interactions. Current limitations on the CP -violation parameters in various non-Standard models allow the transverse component of the muon spin in the decay K → µνγ to be rather large [1]: the left-right symmetric models based on the symmetry group SU(2) L × SU(2) R × U(1) B−L with one doublet Φ and two triplets ∆ L,R of Higgs bosons can give ξ odd ∼ 3.5 × 10 −3 [2], supersymmetric models-ξ odd ∼ 5 × 10 −3 [3], leptoquark models-ξ odd ∼ 2.5 × 10 −3 [4]. To extract the value of ξ odd from the experimental data, one should know the value of ξ EM exactly.
It has long been known that [5] the transverse polarization of the muon can be accounted for by the imaginary parts of the form factors parametrizing the expression for the amplitude of the decay. In this work, we compute the contribution of the electromagnetic FSI to the transverse component of the muon spin in the decay K → µνγ in the one-loop approximation (to be certain, we consider the decay K + → µ + νγ). Our calculations are performed in the framework of the Chiral Perturbation Theory (ChPT) [6].
It should be mentioned that some contributions to ξ EM were calculated in [8,9]. In contrast to the mentioned calculations, we take into account a complete set of the diagrams contributing to the imaginary part of the decay amplitude in the leading order of the ChPT.

Expression for Polarization of Muon in Terms of Helicity Amplitudes
Experimentally, the transverse component of the muon spin can be defined as follows: where N + (N − ) is the number of the produced muons whose spin is directed along(against) a beforehand specified direction of polarization. We introduce vector o specifying such direction in the case under consideration. In the kaon rest frame, it is orthogonal to the vectors q, k, and k ′ (in this frame, these three vectors are linearly dependent): a positive value of ξ implies that the projection of spin of muon on vector o is positive: s o > 0. The respective 4-vector is defined as the unit vector orthogonal to the vectors q, k, and k ′1 : or, to put it differently, where the vectors ω µ − (k, k ′ ) and ω µ + (k, k ′ ) are defined by the relationŝ The helicity amplitudes for the decay K + (p) → µ + (k)ν(k ′ )γ(q) are defined as follows: where r = ± is the helicity of the photon; s = ± is the helicity of the muon in the reference frame comoving with the center of mass of the muon and neutrino, and the amplitude M is defined by where S is the scattering matrix in the respective channel. The particles produced in the decay K → µνγ can be described by the wave function where Γ is the decay width, and the element of the phase space has the form The operator of spin s µ acts on fermion states as follows: where W µ is the Pauli-Lubanski vector andε 0 is the operator of the sign of energy. The average value of the transverse component of spin in the state |Ψ is equal to Ψ|(− s µ · o µ )|Ψ , Since where spinor v(k, N) describes the muon of momentum k and vector of spin N, the expectation value of the transverse component of the muon spin is determined by the relation where N is the normalization factor, N 2 = i,j=± |M i,j | 2 ; M r,s = M ′ r,s + iM ′′ r,s (r, s = ±) (this formula is readily obtained by isolating an infinitesimal volume of the phase space of the particles produced in the decay and employing formula (10)).
In the calculations of the helicity amplitudes we use the so called diagonal spin basis [10,11,12] formed by the vectors ω µ ± and light-like linear combinations of the vectors k and k ′ .
With the use of this basis, the helicity amplitude M r,s can be represented in a manifestly covariant form where the expression for M α (k, k ′ , q) is given by the Feynman diagrams, the polarization vectors of the photon are equal to The leading contribution to the real part of the decay amplitude is given by the tree diagrams corresponding to the Lagrangian (3) [7] (see Fig. 1). The helicity amplitudes for the decay K + → µ + νγ in the tree approximation have the form where the first index in the left-hand side denotes the polarization of the photon and the second-the polarization of the muon in the reference frame comoving with the center of mass of the lepton pair. The calculation of the imaginary parts of the helicity amplitudes is considered in the following Section. The differential probability for the decay is determined by the matrix element squared polariz.

Contribution of FSI to Imaginary Part of Decay Amplitude
The imaginary part of the amplitude for the decay K → µνγ in the leading order of the perturbation theory is described by the diagrams in Fig. 2. We take into account the diagrams in Figs. 2g, h omitted by the authors of [8] in spite of the fact that they are of the same order of magnitude. We employ the Cutkosky rules [13] to replace the propagators with the δ functions. Thus we obtain the expression for the imaginary part of the amplitude in terms of the integrals: where N i is the product of the remaining propagators in the respective diagram and T i are the respective tensor structures (label i specifies the diagram in Fig. 2, i =a÷i); in the case of the diagrams in Figs MeV -is the mass of the π 0 meson). The computations of the diagrams in Fig. 2 are made with the REDUCE package. These diagrams are calculated exactly, no approximation is used.
The calculated imaginary part of the amplitude of the decay K → µνγ takes the form where -is the contribution of the diagrams in Figs. 2a, 2b, 2c, 2d, 2g, 2h; -is the contribution of the diagrams in Figs. 2e and 2f and -is the contribution of the diagram in Fig. 2i. Tensor structures E i have the form xλǫ , and the coefficients in the above expressions are given by θ function in formula (29) isolates the kinematic domain in which the imaginary part of the diagram in Fig. 2i does not vanish; Substituting the expressions (22)-(29) in formula (14), we represent the transverse muon polarization in the form where ±N). Since the imaginary parts of the amplitudes under consideration are much less than the respective real parts (M ′′ r,s < < M ′ r,s ), the denominator of the expression (34) is determined by the equation (19). The coefficients c IB n , c V n , c A n , c

Discussion of Results and Conclusion
The transverse component of the muon spin in the decay K → µνγ is plotted in Figs. 3 and 4 as a function of the kinematic variables x and y. As is seen, it varies through the range (0 ÷ 7) × 10 −4 and the the weighted average is equal to (the notation see in formula (14) ξ EM = x min dx dyΞ where the lower limit of the integration with respect to x, x min = 0.1, corresponds to the cutoff energy of the photon ≃ 25MeV . The accuracy of the result ≃ 20% is determined by the accuracy of the ChPT in order O(p 4 ) at these energies. Note that ξ EM is negative in sign over all Dalitz plot (positive direction is given by the vector o introduced in Section 2.) The values of the parameters F V and F A used in our plots are: F A = 0.042 and F V = 0.095; these values predicted by CHPT coincide with those used in [8,17].
The transverse polarization (which is twice the muon spin) agrees well with the results presented recently [17] and disagree with [8] and [9]. The point is that the authors of [8,9] took into account only a part of the diagrams contributing to the transverse polarization of the muon. Our results show that the diagram estimated in [9] does not give a leading contribution to the imaginary part of the amplitude and the maximum value of the transverse polarization of the muon is overestimated in [8] by an order of magnitude. However, it should be emphasized that our results sustantiate the conclusion made in [9]: "An experimental evidence of P T = 2ξ at the level of 10 −3 would be a clear signal of physics beyond the SM," -in spite of the fact that the analysis performed in [9] is incomplete. Our results contradict to the conclusion of [8].
Thus an observation of the transverse spin of the muon of the order 10 −3 in the experiments [14,15,16] would signal CP and T violation because the background CP -even effect does not exceed 7 × 10 −4 and its average value is not over 3 × 10 −4 . Experiments of this sort can be a good tool for testing the above-mentioned non-Standard models.