PHYSICAL REVIEW D, VOLUME 65, 076001 Long-distance contribution to the muon-polarization asymmetry in K¿\␲¿µ¿µÀ Giancarlo D’Ambrosio* and Dao-Neng Gao† Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, Dipartimento di Scienze Fisiche, Universita` di Napoli, I-80126 Naples, Italy ͑Received 9 November 2001; published 28 February 2002͒ ⌬ We reexamine the calculation of the long-distance contribution to the muon-polarization asymmetry LR , which arises, in Kϩ!␲ϩ␮ϩ␮Ϫ, from the two-photon intermediate state. The parity-violating amplitude of this process, induced by the local anomalous Kϩ␲Ϫ␥*␥* transition, is analyzed; unfortunately, one cannot expect to predict its contribution to the asymmetry by using chiral perturbation theory alone. Here we evaluate ⌬ this amplitude and its contribution to LR by employing a phenomenological model called the FMV model ͑factorization model with vector couplings͒, in which the use of the vector and axial-vector resonance ex- change is important to soften the ultraviolet behavior of the transition. We find that the long-distance contri- bution is of the same order of magnitude as the standard model short-distance contribution. DOI: 10.1103/PhysRevD.65.076001 PACS number͑s͒: 11.30.Er, 12.39.Fe I. INTRODUCTION ͉⌫ Ϫ⌫ ͉ ⌬ ϭ R L ͑ ͒ LR ⌫ ϩ⌫ , 3 The measurement of the muon polarization asymmetry in R L ϩ!␲ϩ␮ϩ␮Ϫ the decay K is expected to give some valuable ⌫ ⌫ where R and L are the rates to produce right- and left- information on the structure of the weak interactions and ␮ϩ flavor mixing angles ͓1–6͔. The total decay rate for this tran- handed , respectively. This asymmetry arises from the interference of the parity-conserving part of the decay ampli- sition is dominated by the one-photon exchange contribution, ͓ ͑ ͔͒ ͓ ͑ ͔͒ which is parity conserving, and the corresponding invariant tude Eq. 1 with the parity-violating part Eq. 2 , which amplitude can be parametrized in terms of one form factor gives ͓1,7͔: 2 2 2 2 d͑⌫ Ϫ⌫ ͒ Ϫs G ␣ 4m␮ R L 1 F ͱ 2 2 ϭ 1Ϫ ␭͑s,m ,m␲͒ ␪ 8 3 3 K s G ␣ d cos ds 2 m ␲ s M PCϭ 1 F ͒ ϩ ͒␮¯ ͒␥ ͒ K f ͑s ͑pK p␲ u͑pϪ ,sϪ ␮v͑pϩ ,sϩ , ͱ 2 2 4m␮ ͑1͒ ϫͭ Re͓ f *͑s͒B͔ͱ1Ϫ s ϫ␭1/2͑ 2 2 ͒ 2␪ where pK , p␲ , and pϮ are the four-momenta of the kaon, s,mK ,m␲ sin ␮Ϯ pion, and respectively, and s1 is the sine of the Cabibbo 2 Ϫ 2 ␮Ϯ mK m␲ angle. The sϮ are the spin vectors for the , and the quan- ϩ ͩ ͓ *͑ ͒ ͔ 2 ϩ Ϫ 4 Re f s B tity sϭ(pϩϩpϪ) is the ␮ ␮ pair invariant mass squared. s In the standard model, in addition to the dominant contri- bution in Eq. ͑1͒, the decay amplitude also contains a small ϩ ͓ ͑ ͒ ͔ͪ 2 ␪ͮ ͑ ͒ Re f * s C m␮cos , 4 parity-violating piece, which generally has the form ͓2͔ while, as a good approximation, the total decay rate can be s G ␣ M PVϭ 1 F ϩ ͒␮ϩ Ϫ ͒␮ obtained from Eq. ͑1͒: ͓B͑pK p␲ C͑pK p␲ ͔ ͱ2 ͑⌫ ϩ⌫ ͒ 2 2 ␣2͉ ͑ ͉͒2 2 ϫ¯͑ ͒␥ ␥ ͑ ͒ ͑ ͒ d R L s1GF f s 4m␮ u pϪ ,sϪ ␮ 5v pϩ ,sϩ , 2 ͱ 3/2 2 2 ϭ 1Ϫ ␭ ͑s,m ,m␲͒ d cos ␪ds 9 3 ␲3 s K 2 mK where the form factors B and C get contributions from both 2 4m␮ short- and long-distance physics. ϫͫ 1Ϫͩ 1Ϫ ͪ cos2␪ͬ, ͑5͒ The muon-polarization asymmetry in Kϩ!␲ϩ␮ϩ␮Ϫ is s defined as 2 2 2 2 where ␭(a,b,c)ϭa ϩb ϩc Ϫ2(abϩacϩbc), 4m␮рs р Ϫ 2 ␪ (mK m␲) , and is the angle between the three- Ϫ *On leave at Theory Division, CERN, CH-1211 Geneva 23, Swit- momentum of the kaon and the three-momentum of the ␮ ϩ Ϫ zerland. Email address: Giancarlo.D’[email protected] in the ␮ ␮ pair rest frame. It is easy to see that, when the †On leave from the Department of Astronomy and Applied Phys- decay distribution in Eq. ͑4͒ is integrated over ␪ on the full ⌬ ics, University of Science and Technology of China, Hefei, An- phase space, the contribution to LR from the C part of the hui 230026, China. Email address: [email protected] amplitude vanishes. 0556-2821/2002/65͑7͒/076001͑6͒/$20.0065 076001-1 ©2002 The American Physical Society GIANCARLO D’AMBROSIO AND DAO-NENG GAO PHYSICAL REVIEW D 65 076001 predicted. As mentioned above, when we integrate over ␪ without any cuts in Eq. ͑4͒, only the contribution from the B part of the amplitude will survive. Therefore, it would be interesting to calculate this part of the parity-violating ampli- ⌬ tude, and estimate its contribution to the asymmetry LR using phenomenological models. The paper is organized as follows. In Sec. II, we briefly reconsider the two-photon long-distance contributions to FIG. 1. Feynman diagrams that give the two-photon contribu- ϩ ϩ ϩ Ϫ tion to the long-distance C part of the parity-violating amplitude of K !␲ ␮ ␮ within chiral perturbation theory. In order to ϩ ϩ ϩ Ϫ ⌬ K !␲ ␮ ␮ in chiral perturbation theory. The wavy line is the evaluate the asymmetry LR from the long-distance B part of photon. The diamond denotes the weak vertex, the full dot denotes the amplitude, models are required. So in Sec. III we intro- the strong or electromagnetic vertex, and the full square in ͑b͒ de- duce a phenomenological model involving vector and axial- notes the local ␲0␮ϩ␮Ϫ or ␩␮ϩ␮Ϫ couplings. vector resonances, called the FMV model ͑factorization model with vector couplings͒ from Refs. ͓13,14͔, for this Fortunately, the form factor f (s) in Eq. ͑1͒ is now known. task. Section IV contains our conclusions. In fact chiral perturbation theory dictates the following de- composition ͓7͔: II. CHIRAL PERTURBATION THEORY s ␲␲ In this section we reexamine the two-photon contributions ͑ ͒ϭ ϩ ϩ ͑ 2 ͒ ͑ ͒ ϩ ϩ ϩ Ϫ f s aϩ bϩ wϩ s/mK . 6 !␲ ␮ ␮ m2 to the parity-violating amplitude of K in chiral K perturbation theory. There are local terms that can contribute ␲␲ Here, w denotes the pion-loop contribution, which leads to to the amplitude, which can be constructed using standard ϩ ͓ ͔ a small imaginary part of f (s) ͓7,12͔, and its full expression notation 5 . The pion and kaon fields are identified as the can be found in Ref. ͓7͔. This structure has been accurately Goldstone bosons of the spontaneously broken SU(3)L ϫ tested and found correct by the E856 Collaboration, which SU(3)R chiral symmetry and are collected into a unitary ϫ ϭ 2ϭ ͱ ⌽ Ӎ also fixes aϩϭϪ0.300Ϯ0.005, bϩϭϪ0.335Ϯ0.022 ͓8͔. 3 3 matrix U u exp(i 2 / f ␲) with f ␲ 93 MeV, and Recently the HyperCP Collaboration also studied this chan- ␲0 ␩ nel in connection with the study of CP-violating width 8 ϩ ϩ Ϯ Ϯ ϩ Ϫ ϩ ␲ charge asymmetry in K !␲ ␮ ␮ ͓9͔. This channel will K ͱ2 ͱ6 be further analyzed by E949 ͓10͔ and NA48b ͓11͔. It is known that, within the standard model, the short- 1 ␲0 ␩ ␲Ϫ Ϫ ϩ 8 0 distance contributions to M PV in Eq. ͑2͒ arise predominantly ⌽ϭ ␭ ␾͑x͒ϭ K . ͱ2 • ͱ2 ͱ6 from the W-box and Z-penguin Feynman diagrams, which ͩ ͪ carry clean information on the weak mixing angles ͓2͔. The 2␩ Ϫ 0 8 authors of Ref. ͓4͔ generalized the results in Ref. ͓2͔ beyond K K¯ Ϫ ͱ6 the leading logarithmic approximation: for the Wolfenstein ␳ Ϫ р␳р ͉ ͉ϭ parameter in the range 0.25 0.25, Vcb 0.040 ͑8͒ Ϯ ϭ Ϯ 0.004, and mt (170 20) GeV, Thus, at the leading order, the local terms contributing to the ϫ Ϫ3р⌬ р ϫ Ϫ3 ͑ ͒ !␲ ϩ Ϫ ! ϩ Ϫ ͓ ͔ 3.0 10 LR 9.6 10 7 decays K l l and KL l l can be written as 2,15 Ϫ р ␪р with the cut 0.5 cos 1.0. Hence, the experimental de- is G ␣ ␮ ϩ͒ץ␮ ϩϪץ L ϭ 1 F 2¯␥ ␥ ␭ 2 termination of B and C would be very interesting from the ␹ f ␲ l ␮ 5l͕h1͗ 6Q ͑U U UU ͘ theoretical point of view, provided that the long-distance ͱ2 contributions are under control. ␮ ϩ ␮ ϩ UQU ͒͘ ץU Ϫ ץϩh ͗␭ Q͑UQ The dominant long-distance contributions to the parity- 2 6 ͒ ͑ ␮ 2 ϩ͒ץ␮ ϩϪץϩ!␲ϩ␮ϩ␮Ϫ ϩ ␭ 2 violating amplitude of K are from the Feyn- h3͗ 6͑UQ U UQ U ͖͘, 9 man diagrams in which the ␮ϩ␮Ϫ pair is produced by two- ͓ ͔ photon exchange 2 . Since these contributions arise from where ͗A͘ denotes Tr(A) in the flavor space, and Q is the nonperturbative QCD, they are difficult to calculate in a re- electromagnetic charge matrix: ⌬ liable manner. The contribution to the asymmetry LR from the long-distance C part amplitude, whose Feynman dia- 2 grams are shown in Fig. 1, has been estimated in Ref. ͓2͔ 00 with the cut Ϫ0.5рcos ␪р1, which indicates that it is sub- 3 stantially smaller than the short-distance part contribution in 1 ͑ ͒ Qϭ 0 Ϫ 0 .