ОБЪЕДИНЕННЫЙ ИНСТИТУТ ЯДЕРНЫХ ИССЛЕДОВАНИЙ

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1 I),..,.,,. II S I) ОБЪЕДИНЕННЫЙ ИНСТИТУТ ЯДЕРНЫХ ИССЛЕДОВАНИЙ Е M.V.Chizhov*. SEARCH FOR TENSOR INTERACTIONS IN KAON DECAYS AT DAONE Submitted to «Physics Letters B» * address: mih@phys.uni-sofia.bg Permanent address: Centre of Space Research and Technologies, Faculty of Physics, University of Sofia, 1126 Sofia, Bulgaria г!

2 1 Introduction The testing of the Standard Model now reaches its concluding stage. The most precise' data arc the ones from the LEP experiments with an accuracy ~ 1%. To describe these data, it is necessary to take into account the electroweak radiative corrections. The data put strict constraints on the new physics i.e. new interactions and new particles. On the other hand the low energy experiments, like particle decays, also can be used as a good test for the V-Л interactions. The recent experiments on semileptonic meson decays [1, 2] point to the possible existence of an admixture of tensor interactions [3, 4], which can help to reconcile- [5] the results of the previous experiments [6]. Unfortunately, the experimental data is poor and the experimental status of the form factors determination of the three particles semileptonic decays is obscure [7]. In this connection the construction of the ^-factory in Frascati [8] and the A'-meson decay experiments would help to check both the Standard Model and the chiral theory [9] with a good accuracy. The comparatively large energy release in the kaon decays would allow to analyze the dependence of form factors on the momentum transfer. decay Oiie of the decays, where the existence of tensor interactions in the weak processes was observed, was the radiative semilcptonic pion decay w~ * e~v~t [1]. The branching ratio B" p = (1.61 ±0.23) x 10~ 7 is smaller than the theoretical ratio /i"' = (2.41 ± 0.07) x 10~ 7 expected in the Standard Model. This fact can be: explained by the destructive interference of the standard decay amplitude with the newly introduced tensor amplitude [1, 3] Mr = cgl c F T V й( Рс )а ар (\ - 7>( Pl/ ). (1) The dimensionless constant FT describes the strength of the new interaction relative to the ordinary Fermi coupling. Its experimental value is FT И 5 x 10~ 3. At the quark level this amplitude can be described by a four fcrmion interaction of the form д*=о = G F ^ O C f T ф и< г Рф,. ф есто0{ 1 _ ^)ф и + h.c. (2) The relation between FT and // can be obtained in the framework of the relativistic quark model [3] Fr=\ h* 0.13 / r,

3 where b\ 131 MeV is the pion decay constant, in q = 310 Mev is the constituent quark mass, or by applying the QCD techniques and tjie РСЛС hypothesis [10] n 2 <0 <7f/ 0>. пл. here x = 5.7 ± 0.6 GeV~ 2 is the quark condensate magnetic susceptibility, < 0 <й 0 > = (0.24GeV) 3 is the vacuum expectation value for the quark condensate. However, the constraints following from the тг/2 decay [11], forbid the existence of tensor-interactions (2) between quarks and leptons with a coupling constant for the tensor interaction ft ~ 10~ 2, which is necessary to explain the experiment [1]. In ref. [12] an extension of the interaction (2) was proposed including also the strange quark 5=I = СЦпОс f] -^ [% ф^ {. _ ^ + h (; (;j) This interaction can be detected in the kaon decay /v'~» /~i>7. However, it is unreliable to make any physical predictions [12] on the base of the contradictory situation of the7г12 decay. This problem was solved in ref. [4], where a new tensor interaction with the coupling constant / ( ~ 0.1 was introduced T = - ^ f t ф и а ах 5 фщ ^ ф,<т рх (\ - 7 )'/v + h.c,, (4) dependent on the momentum transfer to the lepton pair Q, where V ; d(0) = <:os ^a V'J+ sin 0c Ф,. This interaction appears effectively through the exchange of massive tensor particles. In rcf. [4] a mechanism for providing mass of the tensor particles was proposed, which leads to a pole Q 2 in the interaction (4). For particle decays Q 2 is always positive and does not lead to difficulties. In scattering processes there exists a kinematic region with Q' 2 = 0. This could lead to a diffraction peak, which, however, had not been observed at the neutrino scattering experiments [13]. This difficulty can be avoided if another mass generating mechanism is used for the tensor particles. The results of ref. [4] remain valid if the interaction (4) changes its form by introducing the parameter m 2 which is of the order of the momentum transfer in the pion or kaon decay. This interaction between quarks and leptons leads to an additional tensor term /Wv in the total matrix element of the radiative semilcptonic kaon decay Л' + M = Mm + Msl) + Mr, ((>) 2

4 wheie =..JlrwOc h., +.,, _ -2tf +.V\.,. ( (7) is a QK1) correction (inner hreinsstrahluiif!,) to I lie /\' + decay constant / '/; = l(i() MeY. and» l + i' decay with tin* knon M SI) = _' is a structure dependent amplitude parametri/ed by two form factors / ', and F,\: ;"((/) is I lie phot on polarization vector with ;"</,, = 0. and ( where Q = i> <i = /) + /)/ is I he momentum transfer to the leplon pair. We ( hoose the kinematica! variables ;is the conventional quantities,r = 2/ and // = 2/J/»(//»; 4 -. 'lie dilfereulial decay width is Mere / ( = (IXI/IIIK)' 1 ili1( l 'he Dalit/ plot density /)(.r.//) is j;iveu I)V where r(-r-.'/) + I'm! ( <'.'/) (ID /»,н - /H(.r,ff), /'> /;( ' -.'/) = «? [(I + г-if.s'/) + (.r.i/) + (I - r-iv-' > /)-(.,-.../)]. / '/H.s-n = U'/N/O [(I + -),0f-' + ( '..'/) - 1л )(Г(.г /i/нг - 2«n-, /( ',//). I'snr = 2«'tlT\fi-i [(I + 1л] We introduce the following constants [с.'/)]. /' /( ' ) = «hi-1'( '.'/)../ + ( '.'/ ) + (1-1ЛП (12) 2/Wv'»i /'v. 1л = / ',. / 1- У г - / V when 1 the form factor /'V can he estimated from the 14'ЛС hypothesis and the axial anomaly as / ', = IIIK/(-\IT 2 h'n). The theory does not give exact predictions for the axial /\ and tensor / ; form factors, which should he determined ex peri menially. The explicit form of the functions //?(.. /). У/)* (r.'/) *''*( '.'/) V'(.r,//), /(.с,!/) and.l ± (.v,i/) are given in the appendix. In the radiative semileptonic pion decay ;r, j, the constant «;* = nr]./$i: 2 h'imi ~ I and the densities /)//j and />sp ji,ive almost I lie same contrihiition to the decay

5 г.ис. Пи' densities 1ЧН-Ц} «iij«-l /'.i/j/ 'in 1 negligible.small from chir.ilily lousiderations. Therefore I lie interference Icrm t>my is not suppressed and gives a considerable contribution lo the decay rate. Such kind of a roiilrihulioii with a negative sign (destructive interference) lias been observed experimentally [1]. In the kaon decay A' + > < + i>,-. the constant и, л ss 120 is thirty times as large as the corresponding value «, r for the pion decay. This leads lo a considerable decrease in the relative contribution of /)щ and of the corresponding interference term Pi иг- In case of the radiative kaon decay lo the union /\'t * /' + " ( i*; 'In* constant is «^ ss ().."). and the leading contribution is'that of/»/«'" bis decay the interference coinrilniiion />щ.ч1) ' s essenlial due lo the large union mass, and I'lHT ls suppressed by the small const an) «^. - We can make the following conclusion: The radiative semileplonic pion decay ~,i-, is an ideal process where the tensor interaction reveals with its full strength. In the radiative scinileptonic kaon decays Кц-, the lensor interaction effect is suppressed. Moreover, in the kaon decays there; exists a strong background from the processes A'/.i. Therefore, in order Io avoid the contribution from these processes the experiments are usually made.in the narrow kinematic region?/ ~ (I + ''/). In this kinematic region the contribution to the A',.-,!-» decay from the tensor interactions is negligible, and actually tlie experiment is sensitive only lo the term.'-.'l) +. In case of t he A',,^-. decay 1 lie eont ribut ions from.s'o +,./ + and 7' arc considerable, lint they have the same distribution on tin: variable.r at у = (1 +''/<) and can not. lie separated. 1 here is still another reason for the disability to make more decisive conclusions about t he exist euce of tensor interactions while analyzing the radiative, kaon decay. As a result of the large energy release, the resonance exchange with strangeness [II]. can give a considerable contribution to the vector and axial form factors but this contribution is only roughly estimated [lo]. Therefore, in order to identify the effects due lo tensor couplings, one first lias to pin down the contribution from higher-order effects in cliiral perturbation theory. It is not an easy task to make the estimations with the required accuracy. 3 /<"/. { decay One of the brightest demonstrations of the existence of the new interactions is the A'j.i decay: A' + > 7г"/ + ;л The most general l.orent/ invariant form of the matrix element, of this decay is [Hi] M = r '"^Oc: «Ы(1 + 7 s ) It consists of a scalar, a vector and a tensor terms, where the form factors Fs, f± and /'V are functions of the squared momentum transfer to leptons Q 2 =(1 } к /V) 2

6 As should be expected from the ll'-boson exchange, there exists no compelling evidence for terms other than those of a pure vector nature. The contribution of ihe elect roweak radiative corrections to the scalar and tensor form factors is negligibly small. Therefore, the appearance of a considerable nonzero scalar and lensor form factors [l(i, 2] points to deviations from the standard V-A interaction. The term in Ihe vector matrix element with f-(q 2 ) can be reduced (using Dirac equation) to the scalar form factor: U'K ~ I'rU 1<Ы0 + 7 e )7 w(n) = -mi п(р )(1 + 7>(P/) ; lnt he same way the tensor matrix clement can be reduced to a vector and a scalar matrix elements with a distinctive dependence on the momentum of the latter: +{PK + Pr) a bu - Pi)" U(P.')(1 + 7>Ы- This leads effectively to a redefinition of / + : V = / + + {тгц/тц) FT, and F s : S = / ', + (m,/2tn K ) /_ + (P,< + P*) a (р - Vl ) /(2ml) F T- Therefore, the Dalit,/, plot density in the rest frame of the kaon is expressed through V and P {K r, Ei) = Л \V\2 + В Re(V'S) + С \S\ 2 (14) ' S = /".s + ^ / \{E,-E,) + ^-} FT. 2mк т к [ ч 2mл-J Here Л = in A - (2Е,Е - ТПА-АЯ,) - m? (Е - ^ ) В == пц т к (2Я - Д^), С = т 2 к АЕ. where Д/ч = И'»"' - Е. One can see that the study of the Л',, 3 decay mode alone cannot givp a limit on Fs- For the /\,. 3 decay, ratio т с /тк ~ 10~ 3 is small. The interference terms between the vector and the scalar matrix elements are also small. This makes it possible to separate the corresponding contributions from various matrix elements. Л characteristic feature of the tensor matrix element contribution is the existence of a minimum, in the case when the energies of neutrino E v and the positron E e are equal. This is well noticed in the cm. frame of the lepton pair. The Dalitz plot density, given in (14), can be transformed to the dilcpton cm. system? cosfl) a -^{\mlf S + ep r F T c OS 0\ 2 +pl\f + \ 2 s\n 2 0}, (15) I'K

7 where 0 is the angle between ж" and c + in the same system, ]> n Ек = у/]>1 + ш 2 к and = Ек - E T = y/q2 - In ref. [17] a method for the separation of the form factors was proposed using integration over the pion spectrum and fitting to the distribution of cosw. However, in case of I''s close to zero, the tensor contribution dependence" on cos 2 fl = 1 sin 2 t9 could be absorbed into the vector contribution, and the separation could not be provided in this way. Such a situation occurs in rcf. [17]. Meanwhile, we want to note that our formula (15) for /)(/',',,, cos0) differs from the one given in [17]. We want also to point to the lack of consistency in the definition of the form factors / + in [17], namely in the eq. (I) and in (he formula for l'(a',:t) at page 241 (in the last formula it is smaller by \/2 ). Fitting Dalitz plot to the distribution р(е ж, h' c ) leads to nonzero values of the form factors /'s//+(0) = 0.070±0.01G and /'V//+(0)l = 0.Г)3±0.10 [2]. Ли analysis was provided in terms of linearly parametrized J+(Q 2 ) = /+(0) [ I + A + Q 2 1 т\\ and constants Fs and FT- It is important to note here that, the separation of tin; scalar contribution is the most unreliable, as far as any dependence; l'r{q 2 ) () " Q' 2 or deviation from linearity in /+(Q 2 ) imitates the existence of a nonzero scalar form factor. Indeed such a dependence arises, if one relies on the now tensor interaction between quarks and leptons (5). In the framework of the relativistic quark model we can get the following relation between FT and /, ['!] i' «ii; We see that the tensor form factor FT depends on Q 2. Therefore one has to take into account the momentum transfer dependence of the form factors when analyzing experimental data. 4 Conclusion ПЛФЫК provides an opportunity to improve our knowledge about the l\ decays. In this connection we have discussed two modes of the kaoti decays К/г-г illl< ' Л' д, where possible deviation from the Standard Model could be seen. Ну kinomat ical considerations the tensor interactions can not. give a direct contribution to two particles semileptonic meson decay. Therefore we have analyzed meson decays with three particles in the final state;. The decays with more particles in the final state lead to a more complicated kinematics and include; additional form factors. We have shown, that for the Kn-, decay the; contribution from the; now tensor interaction is suppressed. More;e>vcr, the strong background from the; К и decay reduces the kinematic region available for analysis. Fortunately, the; A'ci decay i.s one, where the. presence of flu; ne;w interactions can be; detected. Analyzing tin-

8 Dalit/ plot distribution for I lie /\' + > -"<*v decay wo can very well separate; the contributions from the vector and tensor terms. 'The presence of a nonzero tensor or scalar form factor would be a signal of llie existence of new interact ions in the weak processes. Acknowledgments The author would like to thank I.A'.Avdcev and D.RKirilova for the careful reading of (lie manuscript and the 1 helpful remarks, lie also acknowledges the hospitality of the Hogoliuhov Laboratory of Theoretical Physics. JINH. Dubna. where this work has been completed. Appendix In this appendix we give an analytical expressions for the functions //i(.r.?/). к 4 2(1 -.0(1 " П) - * У) = :, > к 4 ( 0( П) х 2 (.г + у - I - г,) [.г + I/ - I - r, Sl) + (.v,!,) = (.r + i/ - I - г,) [(.г + I/ - 1)(1 - х) - г,). SI)' (.,; у) = (1 -,/ + г,) [(1 -.,-)(I -,,) + г,]. '/'(.г, ц) = 2( I - х) г ( I -.//)(..- + I/ - I) + п( I - х){2 -.. )(.г + 2i/ - 2) -rf [/ + 2(1 -.г)], /( <,.'/) = - (! - 0 ( 1 -.'/ + '/). / + ( ',.'/) = ' [( ' -(.'/-0(1 - ' )-' /].

9 References [1[ V. X. Holotov (I nl.. I'hys. Lett. В21.Ч (1990).408. [2] S. Л. Akiinenko il ill., Phys. Lett. H259 (l!)i)l) 225. [:i] Л. Л. Poblagucv. I'liys. LetI. B238 (1!)!)()) 108. [ 1] M. V. Chizliov. Mod. Phys. Lett. AS (1!)!)3) 27.~»:j. [5] Л. Л. Poblaguev. I'hys. Lett. H28fi (I!W2) Hi!). [fi] 1'. Depommier (I nl.. I'liys. Lett. 7 (1903) 285; Л. Stetz г./ «/.. Nucl. Phys. ИШ (11)78) 285; Л. Bay г./ «/.. I'liys. Lcll. HI 74 (l!)8(i) 115; L. K. Piiloncn л/ и/.. I'liys. Hcv. Lett. 57 (1!)8(>) M02; S. Kgly f./ «/.. I'liys. l-dt. И17Г) (l!)8fi) 97; H222 (1989) ГШ. [7] C. Donaldson d «/.. l'liys. Rev. 1)9 (1971) '2900; V. K. Hirulov id nl.. Slid. I'liys (1981) 1: Y. Cho r.t nl., I'liys. Rev. 1)22 (1980) [8] (i. Vignola, l'roc. of I lie Workshop on Physics and Detectors for 1)ЛФМК, «1. Ci. Pandicri. Fiascali [!)].J. Mijneris, С Kckcr and.1. (Jasscr, The ПЛФЛ'Н Physics Handbook, «Is. L. Maiani c.t nl., 1-Va.scal.i [10] V. M. Melyaev and I. I. Kogan, Phys. Lett. U280 (1992) [II] M. H. Voloshin, i'liys. Lett.. H28:t (1992) 120. [12]! :. Cabridli, Pliys. Lett,. 1Ш1 (l!)!k!) 409. [1-Ч].VI. Joiik(!r id ul I'hys. LeU.!)!)]} (1981) 205;. V. McrRsrna rl nl., I'liys. Lett. Ml В (1984) 129; 1). Crirefiat a id., Phys. Lett, B247 (1990) НИ. [I'll I). Yu. Bardin and К. Л. Ivanov, Sov..J. Part. Nud. 7 (1970) 280. [I')] CJ. Kr.krr i.l nl., Slid. Phys. 1Ш1 (1989) 311; Cl. Iv-.ker a nl., l'liys. Lett. B223 (1989) 425; I. F. Donoftliiic i.l nl., Phys. R. ( ;v. 1)39 (1989) 1947; M. Pras/alowic.x. and (.'. Valencia, Nud. Phys. 1Ш1 (I9!)O) 27. [10] II. Sieincr i.l at., Phys. Lett. B30 (1971) 521. [17] II. Braun i.l. ill., Snd. 1'hys. B89 (1975) 210. Received by Publishing Department on November 10, 1995.

10 Чижов М.В. Поиск тензорных взаимодействии в каопнмх распадах в DAONE Е Рассмотрены полулентонпые каопные распады А' и K jr Используется ciimbiii общий вид матричных элементов для этих распадов. Дополнительные члены мсмуг возникнуть в результате новых тензорных взаимодействии кварков и леитонов. Такие члены были обнаружены в недавних экспериментах. Идеальной возможностью "для подтверждения этих результатов могут явиться высоко прецизионные эксперименты в DAffcNE. Работа выполнена и Лаборатории теоретической физики им.н.н.боголюбова ОИЯИ. Препринт Объединенного института ялерпых исслелонании. Дубна Chizhov M.V. Е Search for Tensor Interactions in Kaon Decays at DA<1>NE The semileptonic kaon decays K l2y and K^ are considered. We use the most general forms of the matrix elements for these decays. Additional terms could arise as a result of new tensor interactions between quarks and leptons. Such terms have been detected in recent experiments. The high precision experiments at ОАФЫЕ may be ideal to confirm these results. The investigation has been performed at the Bogoliubov Laboratory of Theoretical Physics, JINR. Preprint of the Joint Institute for Nuclear Research. Dubna. I'W5

11 Макет Т.Е.Поиеко Подписано в печать Формат 60 у. 90/16. Офсетная печать. Уч.-нчдлистов 0,99 Тираж 460. Закач Цена 594 р. Ичдак'льекии отдел ОГгьединенпою института ядерных исследований Дубна Московской области