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→ϩϩϪ, from the two-photon intermediate state. The parity-violating amplitude of this process, induced by the local anomalous KϩϪ␥*␥* 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→ϩϩϪ 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 ,m͒ 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 ϫ1/2͑ 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 mcos , 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 ,m͒ 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 ͪ cos2ͬ, ͑5͒ The muon-polarization asymmetry in Kϩ→ϩϩϪ is s defined as 2 2 2 2 where (a,b,c)ϭa ϩb ϩc Ϫ2(abϩacϩbc), 4mрs 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 0ϩϪ or ϩϪ 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 рр ͉ ͉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→ϩϩϪ ϩ 2 violating amplitude of K are from the Feyn- h3͗ 6͑UQ U UQ U ͖͘, 9 man diagrams in which the ϩϪ 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 . ͑10͒ Eq. 7 . However, because of the unknown chiral perturba- 3 ⌬ ͩ ͪ tion theory parameters, the long-distance contribution to LR 1 from the B part of the amplitude, which is induced from the 00Ϫ direct KϩϪ␥*␥* anomalous transition, has never been 3
076001-2 LONG-DISTANCE CONTRIBUTION TO THE MUON-... PHYSICAL REVIEW D 65 076001
Note that each term contains two Q’s because the effec- tive Lagrangian in Eq. ͑9͒ is from the Feynman diagrams with two photons, and CPS symmetry ͓16͔ has been used to obtain this Lagrangian ͓2,15͔. From Eqs. ͑9͒ and ͑2͒,itis easy to obtain the two-photon contributions to the parity- violating form factors B and C as
2 BϭϪ ͑h Ϫ2h ϩ4h ͒, Cϭ0. ͑11͒ 9 1 2 3 FIG. 2. One-loop Feynman diagrams that give the two-photon contribution to the long-distance B part of the parity-violating am- As pointed out in Ref. ͓2͔, CPS symmetry forces the ϩ ϩ ϩ Ϫ plitude of K → induced by the vector and axial-vector contribution to C from the leading order local terms to van- resonance exchange. The diamond denotes the weak vertex, and the 0 ish. The dominant contribution to C is from the ( ) pole- full dot denotes the strong or electromagnetic vertex. type diagrams generated by the transitions Kϩ→ϩ0() 0 →ϩϪ and ( ) , via the two-photon intermediate states literature ͓20–23͔ for this task. An implementation of the FM ͑ ͒ ⌬ see Fig. 1 . This contribution to LR has been estimated by in vector couplings ͑FMV͒ was proposed in Ref. ͓13͔; this ͓ ͔ ͉⌬ ͉Ͻ ϫ Ϫ3 Lu, Wise, and Savage 2 : LR 1.2 10 for the cut seems to be an efficient way of including the O(p6) vector Ϫ р р ϩ Ϫ 0.5 cos 1.0, which is much less than the asymmetry resonance contributions to the K→␥␥ and K →␥l l ͑ ͒ L arising from the short-distance physics in Eq. 7 . processes. The basic statement of the FMV model is to use ͑ ͒ On the other hand, as shown in Eq. 11 , the contribution the idea of factorization to construct the weak vertices in- from the local terms to the parity-violating form factor B is volving the vector resonances and then integrate out the vec- Ϫ ϩ proportional to the constant h1 2h2 4h3. Since hi’s, i tors, i.e. perform the factorization at the scale of the vector ϭ 1,2,3, are unknown coupling constants, B cannot be pre- mass. An alternative approach is to integrate out the vector dicted in the framework of chiral perturbation theory. The degrees of freedom to generate the strong Lagrangian before ͑ ͒ Lagrangian in Eq. 9 also gives rise to the two-photon con- factorization, i.e. perform the factorization at the scale of the →ϩϪ tribution to the decay KL , which has been studied kaon mass. As will be shown below, the vector and axial- in Ref. ͓15͔ within this context. However, it is not possible to vector resonance degrees of freedom play a very important use that decay to measure the unknown combination h1 role in our calculation of the one-loop Feynman diagrams of Ϫ ϩ 2h2 4h3 because the hi’s in the local terms enter the Fig. 2. →ϩϪ amplitude for the transition KL as a different linear Keeping only the relevant terms and assuming nonet sym- ϩ ϩ 3 combination h1 h2 h3. Therefore, in the following sec- metry, the strong O(p ) Lagrangian linear in the vector and tion, we have to turn our attention to phenomenological axial-vector fields reads ͓24–26͔ models and try to estimate this part of the contribution to the ⌬ asymmetry LR . f L ϭϪ V ϩ ⑀ ␣ ͑ ͒ V ͗V f ϩ ͘ hV ␣͗V ͕u , f ϩ ͖͘, 12 2ͱ2 III. FMV MODEL In order to evaluate the two-photon contribution to the B f L ϭϪ A ϩ ␣ ͓ ͔ ͑ ͒ part of the amplitude, one has to construct the local A ͗A f Ϫ ͘ i A͗A u , f ϩ ͘, 13 ϩ Ϫ 2ͱ2 K ␥␥ coupling that contains the total antisymmetric ten- sor. The lowest order contribution for it starts from O(p6)in where chiral perturbation theory ͓17͔. Therefore, the unknown cou- plings in the effective Lagrangian will make it impossible to ϩ ϩ uϭiu DUu , ͑14͒ predict this amplitude, which has been shown in the previous section. UϪirUϩiUl , ͑15͒ץDUϭ Vector meson dominance has proved to be very effective 4 in predicting the coupling constants in the O(p ) strong La- ϭ ϩϮ ϩ ͑ ͒ grangian ͓18,19͔; the unknown couplings can thus be re- f Ϯ uFL u u FR u, 16 duced significantly. However, this is not an easy task in the weak Lagrangian since the weak couplings of spin-1 reso- FR,L being the strength field tensors associated with the ex- nance with pseudoscalars are not yet fixed by experiment. ternal r and l fields. If only the electromagnetic field is ϭ ϭϪ ϭٌ Thus various models implementing weak interactions at the considered then r l eQA . We set R R ٌ Ϫٌ ϭ hadronic level have been proposed, yet it is very likely that R (R V,A), and is the covariant derivative defined mechanisms and couplings working for a subset of processes as might not work for other processes unless a secure matching ͒ ͑ ͔ ⌫ϩ͓ ץϭ ٌ procedure is provided. Nevertheless the information pro- R R ,R , 17 vided by the models can be useful to give a general picture of the hadronization process of the dynamics involved. 1 ϩ ϩ Ϫil͒u ͔. ͑18͒ץϪir͒uϩu͑ץ͑ ϭ ͓u⌫ The factorization model ͑FM͒ has been widely used in the 2
076001-3 GIANCARLO D’AMBROSIO AND DAO-NENG GAO PHYSICAL REVIEW D 65 076001
The determination of the above couplings in Eqs. ͑12͒ and 1 ͑ ͒ M PVϭi16ͱ2e4G f l 13 from measurements or theoretical models has been dis- A 8 A A d cussed in Refs. ͓26,25͔. In order to generate the anomalous ϩϪ␥ ␥ K * * vertex from vector and axial-vector exchange, d4q 1 given the strong Lagrangian in Eqs. ͑12͒ and ͑13͒, we have ϫ ͵ ͑ ͒4 ͑ 2Ϫ 2 ͒͑ 2Ϫ 2 ͒ to construct the nonanomalous weak VP␥ and the anomalous 2 q m q mA ␥ 3 weak AP at O(p ). By applying the factorization procedure ϫ͑ ϩ ͒¯͑ ͒␥ ␥ ͑ ͒ ͑ ͒ with the FMV model, we obtain ͑for details, see the Appen- pK p u pϪ ,sϪ 5v pϩ ,sϩ , 23 dix͒ which receives contributions only from the first term in Eq. ͑13͒. In fact the contribution from the second term is higher f L ␥͒ϭϪ 2 V ⌬ ͑ ͒ order. W͑VP G8 f ͗ ͕V , f Ϫ ͖͘, 19 ͱ2 Using the Weinberg sum rules ͓27͔ together with the Kawarabayashi-Suzuki-Riazuddin-Fayyazuddin ͑KSRF͒ sum ͓ ͔ ͓ ͔ L ␥͒ϭϪ 2 ⑀ ⌬ ␣ ͑ ͒ rule 28 , one can obtain 24 W͑AP G8 f lA ␣͕͗ ,A ͖͕u , f ϩ ͖͘, 20 1 where ⌬ϭu uϩ, and f ϭ f , m2 ϭ2m2 . ͑24͒ 6 A 2 V A V
2 3 mA ͑ ͒ l ϭ f . ͑21͒ Hence, from Eq. 21 , we have A 2 A 2 16ͱ2 f 1 f l ϭ f l , ͑25͒ is the factorization parameter satisfying 0Ͻр1.0 gener- A A 2 V V ally, and it cannot be given by the model. L ␥ ͑ ͒ Now, combining the strong S(VP ) in Eq. 12 with the with L ␥ ͑ ͓͒ L ␥ weak W(VP ) in Eq. 19 or the strong S(AP ) in Eq. ͑ ͒ L ␥ ͑ ͔͒ 13 with the weak W(AP ) in Eq. 20 , and attaching the 2 3 mV photons to the muons with the usual QED vertices, we can l ϭ f . ͑26͒ V 2 V 2 get the spin-1 resonance contribution to the B part of the 16ͱ2 f parity-violating amplitude of Kϩ→ϩϩϪ from the two- photon intermediate state. The corresponding Feynman dia- Then using the relation grams have been drawn in Fig. 2. The calculation of the contribution from the vector resonance exchange is straight- ϭ ͑ ͒ forward: lV 4hV , 27
͓ ͔ 1 which, as pointed out in Ref. 13 , is exact in the hidden M PVϭϪi32ͱ2e4G f h local symmetry model ͓29͔ and also well supported phenom- V 8 V V d enologically, we can get d4q 1 ϫ ͵ 4 2 2 2 2 PV 1 ͑2͒ ͑q Ϫm͒͑q Ϫm ͒ M ϭϪi32ͱ2e4G f h V VϩA 8 V V d ϫ͑p ϩp͒ ¯u͑pϪ ,sϪ͒␥␥ v͑pϩ ,sϩ͒; ͑22͒ K 5 d4q 1 1 1 ϫ ͵ ͫ Ϫ ͬ ͑ ͒4 2Ϫ 2 2Ϫ 2 2Ϫ 2 d is the space-time dimension generated from the integral 2 q m q mV q mA d d 2 ͐d qqqϭ(1/d)͐d qq g . Because of the logarithmic ϫ͑ ϩ ͒¯͑ ͒␥ ␥ ͑ ͒ ͑ ͒ divergence in the above equation, we do not set dϭ4 now. pK p u pϪ ,sϪ 5v pϩ ,sϩ . 28 Note that we retain the loop momentum q in the Feynman integral of the above equation only because we are con- Fortunately, one will find that the logarithmic divergences in cerned about the leading order two-photon contribution to Eqs. ͑22͒ and ͑23͒ cancel each other, provided that the rela- the B part of the amplitude, and the Feynman integrals re- tions ͑24͒ and ͑27͒ are satisfied. Of course, any violation of lated to the external momenta are obviously of higher order these relations will lead to divergent results; a renormalized with respect to Eq. ͑22͒. On the other hand, we do not con- procedure is thus needed, and further uncertainty will be in- sider the diagrams generated by the weak anomalous volved. In this paper, we are concerned only about the lead- KϩϪV␥ and KϩϪVV couplings, because either they ing order two-photon contribution to the B part of the ampli- have no contributions to the B part of the amplitude or they tude. Therefore, it is expected that Eqs. ͑24͒ and ͑27͒ can be are higher order. Likewise, the axial-vector resonance contri- regarded as good approximations for this goal. Neglecting 2 bution is m in Eq. ͑28͒, whose effect is smaller than 5%, we have
076001-4 LONG-DISTANCE CONTRIBUTION TO THE MUON-... PHYSICAL REVIEW D 65 076001
m2 We calculate this amplitude in a phenomenological model M PV ϭ ͱ ␣2 A ϩ 8 2 G8 f VhV ln called the FMV model. The muon polarization asymmetry V A 2 ⌬ mV LR has been estimated up to the factorization parameter , which is not given by the model, but may be determined ϫ͑ ϩ ͒¯͑ ͒␥ ␥ ͑ ͒ pK p u pϪ ,sϪ 5v pϩ ,sϩ . phenomenologically in the future. We have established that ͑29͒ the background effect may obscure the standard model pre- diction but large new physics effects can still be tested. Comparing Eq. ͑29͒ with Eq. ͑2͒, and using ACKNOWLEDGMENTS s G ϭ 1 F ͑ ͒ G8 g8 , 30 ͱ2 We would like to thank the CERN TH Division for its generous hospitality and Suzy Vascotto for carefully reading one will get the two-photon contribution to the parity- the manuscript. D.N.G. is supported in part by the INFN of violating form factor B as Italy and the NSF of China. This work was supported in part by TMR, EC Contract No. ERBFMRX-CT980169 m2 (EURODA⌽NE). 2␥ϭ ͱ ␣ A ͑ ͒ B 8 2 g8 f VhV ln . 31 m2 V APPENDIX: NONANOMALOUS WEAK VP␥ AND →ϩϪ ͉ ͉ ANOMALOUS WEAK AP␥ VERTICES From the measured KS decay rate, we fixed g8 ϭ ͓ ͔ IN THE FMV MODEL 5.1 12,2 . Note that f V and hV can be determined from the phenomenology of vector meson decays, as shown in Ref. The ⌬Sϭ1 nonleptonic weak interactions are described ͓ ͔ ͉ ͉ϭ ͉ ͉ϭ 2 ϭ 2 26 , f V 0.20, and hV 0.037. Thus, using mA 2mV , by an effective Hamiltonian we have ␥ Ϫ G ͉B2 ͉ϭ2.16ϫ10 3. ͑32͒ H⌬Sϭ1ϭϪ F Q ϩ ͑ ͒ eff VudVus* ͚ Ci i H.c. A1 ͱ2 i Now from Eqs. ͑3͒, ͑4͒, and ͑5͒, we can get the asymmetry ⌬ 2␥ LR contributed by B : in terms of Wilson coefficients Ci and four-quark local op- erators Q . If we neglect the penguin contributions, justified ⌬ ϭ ͉ 2␥͉ϭ ϫ Ϫ3 ͑ ͒ i LR 1.7 B 3.6 10 33 ͓ ͔ by the 1/NC expansion 22 , we can write the octet-dominant H⌬Sϭ1 for integrated over the full phase space, and piece in eff as
⌬ ϭ3.0͉B2␥͉ϭ6.5ϫ10Ϫ3 ͑34͒ G LR H⌬Sϭ1ϭϪ F * Q ϩ ͑ ͒ eff ͱ VudVusCϪ Ϫ H.c., A2 for Ϫ0.5рcos р1.0. 2 2 From Eqs. ͑33͒ and ͑34͒, if the factorization parameter Ӎ1, which is implied by naive factorization, we find that with the long-distance contributions from B2␥ can be compared Q ϭ ͑¯ ␥ ͒͑¯ ␥ ͒Ϫ ͑¯ ␥ ͒͑¯ ␥ ͒ with the short-distance contributions given in Refs. ͓2,4͔.In Ϫ 4 sL uL uL dL 4 sL dL uL uL . ͑ ͒ Ref. ͓13͔, Ӎ0.2–0.3 is preferred by fitting the phenomenol- A3 →␥␥ →␥ ϩ Ϫ ogy of K and KL l l . In this case, the contribu- Q tions from Eqs. ͑33͒ and ͑34͒ might be small, although not The bosonization of the Ϫ can be carried out in the FMV model from the strong action S of a chiral gauge theory. If fully negligible. Generally, 0Ͻр1.0; therefore, this uncer- we split the strong action and the left-handed current into tainty may make it difficult to get valuable information on ϭ ϩ J ϭJ 1 ϩJ 2 the structures of the weak interaction and the flavor-mixing two pieces S S1 S2 and , respectively, the ⌬ ϩ→ϩϩϪ QϪ operator is represented, in the factorization approach, by angles by measurement of LR in K . However effects larger than 1% would be a signal of new physics. Q ↔ ͓͕͗J 1 J ͖͘Ϫ͗J 1 ͗͘J ͘Ϫ͗J 2 ͗͘J ͔͘ Ϫ 4 , 2 2 1 , ͑A4͒ IV. CONCLUSIONS ϭ Ϫ We have studied the long-distance contribution via the with ( 6 i 7)/2; for generality, the currents have been two-photon intermediate state to the parity-violating B part supposed to have nonzero trace. of the amplitude of Kϩ→ϩϩϪ. Within chiral perturba- In order to apply this procedure to constructing the fac- 3 tion theory, one can calculate it up to an unknown parameter torizable contribution to the O(p ) nonanomalous weak combination, which is shown in Eq. ͑11͒. At present, we VP␥ Lagrangian, we have to identify in the full strong action have no way of estimating this unknown combination, and it the pieces that can contribute at this chiral order. We define, is therefore impossible to determine this parity-violating am- correspondingly, plitude and to predict its contribution to ⌬ by using chiral LR ϭ ϩ ͑ ͒ perturbation theory alone. S SV␥ S2 , A5
076001-5 GIANCARLO D’AMBROSIO AND DAO-NENG GAO PHYSICAL REVIEW D 65 076001
͑ ͒ where SV␥ corresponds to the first term in Eq. 12 , and S2 The explicit term that will give a contribution in our calcu- corresponds to the leading order ͓O(p2)͔ effective Lagrang- lations has been written in Eq. ͑19͒. L ␥ ian 2 in chiral perturbation theory for the strong sector. For the anomalous weak AP vertex, the corresponding Evaluating the left-handed currents and keeping only left-handed currents are terms of interest we get
␦ SV␥ f V ϩ ␦ ϭϪ ٌ ͑ ͒ SA␥ f A ϩ u Vu , ϭϪ 2 ␦ ͱ mAu Au, l 2 ␦l ͱ2 ␦S f 2 2 ϭϪ ϩ ͑ ͒ u uu. A6 ␦l 2 ␦S 1 1 WZWϭ⑀␣ ͭ L ϩ ϩ R ϩ ͮ ͑ ͒ F␣ U F␣U,u uu , ␦ 2 A8 Then the effective Lagrangian in the factorization approach l 16 2 is
␦ ␦ ␦ SV␥ S2 SV␥ L fact͑VP␥͒ϭ4G ͫͳ ͭ , ͮ ʹ Ϫ ͳ ʹ W 8 ␦ ␦l ␦ ͑ ͒ l l where the first term is from the f A part in Eq. 13 , and the second term is from the Wess-Zumino-Witten Lagrangian ␦ ␦ ␦ S2 S2 SV␥ ͓30͔. An equation similar to Eq. ͑A7͒ will be obtained, and ϫͳ ʹ Ϫ ͳ ʹ ͳ ʹͬϩH.c. ͑A7͒ ͑ ͒ ␦l ␦l ␦l the explicit term for our purpose has been shown in Eq. 20 .
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