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Neutrino Electron Scattering EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH CERN-PPE/93-65 15 April 1993 Neutrino Electron Scattering y x GabyRadel and Rolf Beyer y DESY, Hamburg, Germany x CERN, Geneva, Switzerland. (to appear in Mod. Phys. Lett. A) Abstract This article reviews exp erimental results obtained from studies of neutrino elec- tron scattering and shows in particular the imp ortant input from these exp eri- ments to the improved knowledge of weak neutral currents and the con rmation of the Standard Mo del at tree level. Sp ecial emphasis is put on recent high pre- cision e-exp eriments, whose results on electroweak parameters allow, in combi- 2 nation with precise results obtained at higher Q , a test of the Standard Mo del at the level of higher order corrections. Neutrino Electron Scattering Gaby Radel and Rolf Beyer Intro duction The rst observation of a handful of e-scattering events by the Gargamelle bubble chamber [1]in 1973 was a turning p oint in elementary particle physics. The existence of weak neutral currents was demonstrated. A phenomenon which could not b e explained by the long-standing Fermi-typ e theory of weak interactions but was predicted by a combined electroweak theory, formulated by Glashow, Weinb erg, and Salam [2]: the so-called 'Standard Mo del' of electroweak interactions. Since then several exp eriments with the aim of studying the elastic scattering of neutrinos on electrons have b een p erformed, all of which added further supp ort to the Standard Mo del. The rst generation of e-exp eriments, p erformed in the 1970's, were exp eriments dedicated to the con rmation of the Standard Mo del at tree level. Integral cross sections and rst values for the 2 electroweak mixing angle, sin ,were obtained. These early exp eriments were either p erformed at W accelerators in or b eams or at nuclear reactors, where e scattering was observed for the rst e time. However, all these exp eriments su ered from rather p o or statistics, due to the very low cross section of e scattering. In 1980 the numb er of observed events by all exp eriments was in the order of 100. The situation changed in the 1980's with the second generation of exp eriments. They were counter exp eriments with large target mass dedicated neutrino detectors utilizing high intensity neutrino b eams. The statistics collected were an order of magnitude higher than for the previous generation, thus making it p ossible to measure kinematical distributions and di erential cross sections. With the last exp eriment in this series (CHARM I I [3]) electroweak parameters were measured to such an accuracy that testing the Standard Mo del at higher orders b ecame p ossible. In addition limits on quantities b eyond the Standard Mo del could b e derived. At last neutrino electron scattering nowadays is used even as an exp erimental to ol to detect solar neutrinos [4], thus testing the Standard Solar Mo del. From a theoretical p oint of view the pro cess of e-scattering is comparatively simple. In the rst, theoretical, section of this article we de ne the relevant quantities to describ e e-scattering in the theoretical frame, of the Standard Mo del at tree level. Going to higher orders we make some remarks on radiative corrections as well as physics b eyond the Standard Mo del. On the other hand, exp erimentally e-scattering is a very dicult task due the smallness of its 42 2 cross section ( 10 cm ) and to the presence of abundant comp eting pro cesses which cannot b e totally eliminated on an eventbyevent basis. Therefore sophisticated exp erimental metho ds have b een invented, whose realization we describ e in the more detailed second section. For each metho d a selected exp eriment is describ ed in detail and its results are presented. Finally we combine the results of all e-exp eriments and compare them to the results of other measurements and to the predictions of the Standard Mo del. 1 Some Theoretical Remarks Neutrino-electron scattering is a purely leptonic pro cess where a neutrino scatters o an electron by 2 the exchange of a virtual vector b oson ( g. 1). The Q range covered at presentby neutrino electron 6 2 scattering exp eriments reaches from 10 at nuclear reactors to 10 at accelerators but is always 0 small compared to the mass of the Z . 1 @ @ @ @ @ @ () () @ @ @R @R @ @ e e (e) (e) @ @ @ @ e e @ @ @R @ @ 0 W Z @ @ W @ @ @R @ e e @ @ @ @ @ @ @ e e e e @ @ @R @R @ @ @ @ (a) (b) (c) Figure 1: Feynman diagrams for the processes of neutral current (NC) e-scattering (a), and charged current (CC) e-scattering via the exchange of a W -Boson (b,c). e Kinematics The kinematics of elastic neutrino electron scattering is fully describ ed by a single variable; for instance by , the angle of the outgoing electron with resp ect to the neutrino b eam. Let E and E be e e the energies of the incoming neutrino and outgoing electron resp ectively, m the electron mass, and e y = E =E b e the fractional energy loss of the neutrino in the lab oratory system. With the assumption e of m E and the small angle approximation for cos one nds e e e 2 E =2m (1 y ); (1) e e e 2 and, as 0 y 1, the exp erimentally imp ortant constraint: E < 2m : e e e This means the outgoing electron is scattered in extremely forward direction, which is used exp eri- mentally to subtract, on a statistical basis, background events, whichhaveamuch broader distribution 2 in E . On the other hand this signature makes it imp ossible to measure directly the y -distribution e e of e-reactions, as reachable resolutions are just in the same order of magnitude as the kinematical b ound. Cross sections The mo del indep endent e ective neutral currentinteraction Lagrangian can b e written as [5]: p NC L =2 2G ( )(g e e + g e e )+ h:c: (2) F L L L L L R R R e 2 In the limit of small Q , propagator e ects can b e neglected. L; R denote the chirality of the fermion. The same can b e expressed in terms of the axialvector and vector couplings of the neutral vector b oson e e to the electron. The relation b etween the chiral couplings and g and g is given by V A 2g = g + g and 2g = g g (3) L V A R V A From the e ective Lagrangian the di erential cross section | often called y -distribution | is readily calculable: ; 2 d 2G m m y e e F 2 2 2 = E g + g (1 y ) g g : (4) L R L;R R;L dy E 2 The (1 y ) -term originates from angular momentum conservation, and suppresses backward scattering (y = 1) of (anti)neutrinos o (left-)right-handed electrons. For high neutrino energies (E m ) the e term linear in y is negligible. 2 g 1 A a ν− µe σ/Eν b 0 νµe 10 νee ν− ee -1 − νµe νµe ν− ee -2 ν 1 ee ×10-42 cm 2 GeV -1 -3 -3 -2 -1 0 1 0 0.2 0.4 0.6 0.8 1 2 g sin Θ V W Figure 2: Dependence of neutrino-electron cross section on electroweak parameters. a): contours in 2 the g g plane. The contours correspond to four ideal measurements with sin = 0.23. b): Cross V A W 2 section as a function of sin . Note the logarithmic scale. W To obtain the cross section of -scattering, where the charged currentinteraction contributes to e the scattering amplitude, one has simply to substitute g by g +1: L L 2 The Standard Mo del relates the coupling constants and the electroweak mixing angle sin and W the relative coupling strength of the neutral current (NC) with resp ect to the charged current (CC): 2 2 g = sin and g = (sin 1=2): (5) R W L W 2 The resulting cross section dep endence on sin is shown in g. 2b. Obviously the neutral current W coupling constants cannot b e determined from a single measurement. Even a measurementof ( e) and ( e) leaves a fourfold ambiguity for the coupling constants. A measurement of electron-neutrino electron scattering cross sections only partly resolves the problem. A twofold ambiguity remains. This + problem can b e solved using measurements fo the forward-backward asymmetry in e e -collisions. 2 2 These exp eriments yield g g [6]. V A Radiative corrections When considered in the context of the electroweak p erturbation theory, the expression for the cross sections (4) is only true in rst order of the coupling constant, i.e. in Born approximation (tree level). Exp eriments have, however, achieved an accuracy that makes it necessary to consider also contributions of higher order in the p erturbation series (radiative corrections). Including corrections prop ortional to G in the matrix elements, the di erential cross section of neutrino-electron scattering (one-lo op F cross section) can b e written as [7]: ew QE D d d d e e e = + : (6) d y d y d y The second term corresp onds to pure QED corrections, while the rst term is obtained from the tree-level result by replacing: 2 2 2 e 2 G ! G (Q ) and sin ! (Q )sin : F F e W W 3 e 2 2 (Q ) and (Q ) are correction functions which dep end on the unknown parameters of the Standard e Mo del, like the mass of the top quark m , and on the renormalization scheme applied.
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