Constraints on four-fermi contact interactions from low-energy electroweak experiments

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1 Eur. Phys. J. C 5, DOI 0.007/s THE EUROPEAN PHYSICAL JOURNAL C c Springer-Verlag 998 Constraints on four-ermi contact interactions from low-energy electroweak experiments Gi-Chol Cho,a, Kaoru Hagiwara,, Seiji Matsumoto,a,b Theory Group, KEK, Tsukuba, Ibaraki 305, Japan ICEPP, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 3, Japan Received: 8 November 997 / Published online: 6 ebruary 998 Abstract. We investigate the constraints on four-ermi contact interactions from low-energy lepton-quark and lepton-lepton scattering experiments polarization asymmetries in electron muon-nucleon scattering experiments, cesium and thallium atom parity violation measurements, neutrino-nuclei and neutrinoelectron scattering experiments. These constraints are then combined by assuming the lepton and quark universalities and SU L gauge invariance of the contact interaction, which leave independent six lepton-quark and three pure-leptonic interactions. Impacts of these constraints on models with an additional Z-boson are briefly discussed. We also present updates of the low-energy constraints on the S and T parameters. Introduction Searching for an evidence of physics beyond the Standard Model SM is one of the most important subjects in the field of particle physics. Up to now, no particles other than the SM particles have been found at collider experiments. This fact implies that the mass scale of new particles should be larger than several hundred GeV. Low-energy neutral current L.E.N.C. phenomena is mediated by exchange of the photon and the Z-boson in the SM. We can parametrize new physics contributions to L.E.N.C. phenomenon as effective four-ermi contact interactions. Generally, the effective contact interactions for neutral currents among quarks and leptons can be parametrized as L NC = η ff ψ f γ µ P α ψ f ψ f γ µ P β ψ f,. f,f α,β where f,f stand for lepton and quarks, α, β = L, R denote their chirality: P LR = γ 5 /. The coefficients η lq have the dimension of mass which is often expressed as [, ], η ff = 4π 4π or Λ ff Λ ff.. The experimental limits of the scale Λ are given for some combinations of f,f, α, β and the overall sign factor a Research ellow of the Japan Society for the Promotion of Science b Present address: Theory Division, CERN, CH-, Genève 3, Switzerland, []. If the contact interactions are results of an exchange of an extra heavy neutral vector boson Z E, they are given by η ff = αg f gf β m,.3 Z E where gα f and g f β are the Z E-boson couplings to f α and f β, respectively. An exchange of a heavy boson in the schannel of the processes ff ff or ff ff can also be expressed in the form of. as long as we ignore the contribution which does not interfere with the leading SM amplitudes. In this report, we present constraints from low-energy electroweak measurements on the coefficients η ff,inorder to take advantage of the model-independent nature of the parametrization.. As a simple example, the constraints on the η ff terms are used to derive constraints on an extra Z boson parameters. Recently there have been renewed interest in the possible existence of new interactions between quarks and leptons because of an excess of events at high Q e p inelastic scattering at HERA, which was reported by H [3] and ZEUS [4] collaborations. Such an excess of events may be reproduced by introducing some lepton-quark contact interactions [5]. Therefore, it is worth examining constraints on those contact interactions from all low-energy electroweak measurements as model-independent as possible. We study in this report the following four types of lowenergy electroweak experiments; i polarization asymmetry of the charged lepton scattering off nucleus target, ii parity violation in cesium and thallium atoms, iii inelastic ν µ scattering off nucleus target and iv ν µ ν µ

2 56 G. Cho et al.: Constraints on four-ermi contact interactions from low-energy electroweak experiments electron scattering experiments. Individual experiment receives contributions of a different combination of the contact interactions. We therefore present constraints on the contact terms from each experiment separately. These constraints are then combined by assuming that the contact interactions satisfy the flavor universality and the SU L gauge invariance of the SM. This paper is organized as follows. In the next section, we review the approach of [6] that we adopt for calculating the low-energy electroweak observables with the modelindependent framework of [7]. Section 3 is devoted to survey the experimental data corresponding to the above four types of experiments. The SM predictions are given both as functions of m t and m H in the minimal SM and as a function of S and T [8] in generic SU L model. We adopt the slightly modified version of the S and T variables that was introduced in [6]. In Sect. 4 we present individual constraints on the S and T parameters in the generic SU L model, and compare the low-energy constraints with the constraints from the Z- pole experiments [9]. They update the results of [6,0,9]. In Sect. 5 we obtain constraints on the contact interactions by assuming no new physics contributions to the S and T parameters. In Sect. 6, we present constraints on models with an extra Z-boson as an example of the use of our model-independent parametrization of the low-energy data in terms of the four-fermi contact interactions. ramework The effective Lagrangian for the lepton-quark four-ermi interaction is given as follows; L lq eff = G [ C q ψ l γ µ γ 5 ψ l ψ q γ µ ψ q q=u,d C q ψ l γ µ ψ l ψ q γ µ γ 5 ψ q C 3q ψ l γ µ γ 5 ψ l ψ q γ µ γ 5 ψ q ],. where l = e, µ, τ and q = u, d. Adding to the modelindependent parameters C q and C q of [7], we introduce the parameter C 3q as the coefficient of the axial vectoraxial vector current. They can be expressed in terms of the helicity amplitude M lq of [6] as C q = M lq G M lq LR M lq RL M lq RR,.a C q = M lq G M lq LR M lq RL M lq RR,.b C 3q = M lq G M lq LR M lq RL M lq RR..c With the above identification, q -dependence of higher order electroweak effects are properly taken into account by making use of the effective Lagrangian formalism. Introducing the effective form factors ē q, ḡz q, s q, the helicity amplitude M ff of the neutral current process f α f β f αf β can be expressed as [6]; M ff q SM = { q Q f Q f [ ē q ê Γ f q ê Γ f q ] Q f I 3f ê Γ f q Q f I 3f ê Γ f q { q m I 3f Q f ŝ Z I 3f Q f ŝ ḡz0 I 3f Q f ŝ Q f ĝz [ s q ŝ ] I 3f Q f ŝ Q f ĝz [ s q ŝ ] Bff NC 0, 0 O ê q,.3 m W in the generic SU L model, where ê, ĝ Z are the MS couplings that satisfy ê =ĝŝ =ĝ Z ŝĉ which are renormalized as ê =ē m Z and ŝ = ĉ = s m Z. Q f,i 3f f = l, q denote the electric charge and the third component of weak isospin, respectively, of the corresponding fermion f α. Γ f i, Γ f i i =, and Bff NC stand for the vertex and box corrections, respectively, and their explicit forms are given in Appendix A of [6]. In the presence of the contact interactions, the complete helicity amplitude is given by the sum of.3 and the contact term; M ff q =M ff q SM η ff..4 Accordingly the coefficient C iq of the effective lepton-quark interactions can be divided into two pieces as, C iq = C SM iq C iq,.5 where the first term denotes the contribution with the generic SU L model, while the second term arises from the contact interactions. Hereafter, we denote by SM the predictions of the generic SU L model, where the form factors ḡz m Z and s m Z, or S and T parameters, are treated as free parameters. In the minimal SM, they are determined by m t and m H, whose explicit forms are given below [6,9]. The coefficients C iq SM are approximately expressed as C q SM = I 3q Q q sin θ W higher order terms,.6a C q SM = I 3q 4 sin θ W higher order terms,.6b C 3q SM = I 3q higher order terms..6c The terms C iq receive contributions from the contact interactions, C q = η lq G ηlq LR ηlq RL ηlq RR,.7a

3 G. Cho et al.: Constraints on four-ermi contact interactions from low-energy electroweak experiments 57 C q = G C 3q = G η lq ηlq LR ηlq RL ηlq RR η lq ηlq LR ηlq RL ηlq RR,.7b..7c or neutrino-quark scattering, it is conventional to introduce the model independent parameters, gα and δα α = L, R [4]; gα u α d α, δα u α d α,.8a.8b where, u α and d α are given by using the helicity amplitude as [6], q α = M νµq Lα.9a G = q α SM q α q = u, d,.9b The SM contributions are obtained from [6, 9] and the contribution from the contact interactions is q α = η νµq Lα G..0 or the neutrino-electron scattering experiments, we use the experimental data of the total cross sections for ν µ -e and ν µ -e scatterings, which are expressed in terms of the helicity amplitude as [6]; σ νe E ν σ νe E ν = m { e M νµe 4π Q = m e E ν 3 M νµe LR Q = m ee ν,.a = m { e 4π 3 M νµe Q = m ee ν M νµe LR Q = m e E ν..b Here we replace the integral over the momentum transfer Q by m e E ν. The amplitudes are then parametrized as M νµe νµe Lα =MLα SM η νµe Lα.. Next, we briefly review how we estimate the contribution of generic SU L model. In the formalism of [6], the L.E.N.C. observables are expressed in terms of two form factors, ḡz 0 and s 0. They can be expressed by the S and T -parameters [8] as follows; ḡz T, s S 0.003T, where S is defined by.3a.3b S S.30δ α..4 The parameter δ α /ᾱm Z 8.7 was introduced in [6] to take care of the uncertainty in ᾱm Z. The recent estimation by Eidelman and Jegerlehner [] gives δ α = 0.03±0.09 [9]. In the minimal SM, the S and T -parameters are parametrized by m t and m H as [9], S SM x t 0.09x H 0.00x H,.5a T SM x H x t 0.003x t 0.079x H 0.08x H 0.006x 3 H,.5b where x t and x H are x t m t 75 GeV/0 GeV, x H logm H /00 GeV,.6a.6b respectively. It is convenient to introduce the following parameters; S S SSM 0 = S 0.33, T T TSM 0 = T 0.879,.7a.7b where SSM 0 and T SM 0 are the SM predictions for S and T at m t = 75 GeV, m H = 00 GeV. 3 Low-energy electroweak observables 3. Polarization asymmetry of charged lepton-nucleus scattering In this subsection, we study four experiments on charged lepton-nucleus scattering, in which two types of observables were measured. irst, polarization asymmetry A of charged lepton scattering off nucleus target A dσ R dσ L, 3. dσ R dσ L has been measured in ed-scattering at SLAC [], in ec scattering at Bates [3] and in ebe scattering at Mainz [4]. Here dσ LR denotes the differential cross section of the left- right- handed lepton scattering off target. These are Parity-odd asymmetries. Another type of the asymmetry is the polarization and charge asymmetry parameter B B dσ L dσ R dσ L, 3. dσ R where dσ RL is the differential cross section of right- left- handed negatively charged lepton scattering off nucleus target, wheres dσ RL denotes those of positively charged anti-lepton scattering. The parameter B has been measured in µ ± C scattering at CERN [5]. The measurement of the asymmetry parameter A constrains a combination of C q and C q, wheres the asymmetry B constrains the parameter C 3q. These parametrizations are valid in the mass range 60 GeV <m t < 85 GeV and 40 GeV <m H < 000 GeV

4 58 G. Cho et al.: Constraints on four-ermi contact interactions from low-energy electroweak experiments 3.. SLAC ed experiment The historic SLAC experiment [] that established parity violation in the electron-quark neutral current scattering still gives non-trivial constraints on new physics contribution. Here we quote the results of the analysis of [6]. The parameter A SLAC is expressed in [6] as A SLAC = 6G Q {C 5 u C d 34 ē Q c C u C d b 5 c, 3.3 where the terms b and c represent deviations from the valence-quark approximation. After studying the uncertainties in the correction terms b and c, the following constraints were obtained [6] for the model-independent parameters C u C d = 0.94 ± 0.6 ρ corr = C u C d = 0.66 ±.3 The theoretical prediction within the generic SU L U Y model at Q.5 GeV is given as; C u C d SM S 0.09 T, C u C d SM S T. The minimal SM predicts 3.5a 3.5b C u C d SM x t 0.004x H, 3.6a C u C d SM x t 0.003x H. 3.6b qx, qx and Dx are expressed in terms of the quark and anti-quark distribution functions in the nucleus as qx =uxdxsxcx, qx =uxdxsxcx, D = uxuxdxdx 5 sxsx 3.8a 3.8b 8 cxcx. 3.8c 5 The experiment has been performed at two different beam energies, E = 00 GeV and 0 GeV. The mean momentum transfer of the experiments may be estimated as Q 50 GeV [6]. The experimental data give the following constraints on the coefficients of C q and C 3q ; i E = 00 GeV: C 3u C 3d λ C u C d =.5 ± 0.43, λ =0.8 ± 0.04, ii E = 0 GeV: 3.9 C 3u C 3d λ C u C d =.79 ± 0.83, 3.0 λ =0.66 ± Combining the above two constraints, we find C 3u C 3d = 3.0 ± 4.9 ρ corr = C u C d =.85 ± 6.3 The theoretical prediction of the generic SU L U Y model at Q 50 GeV is given as; C 3u C 3d SM T, 3.a C u C d SM S 0.05 T. 3.b 3.. CERN µ ± C experiment The minimal SM predicts, C 3u C 3d SM x t 0.00x H, 3.3a Charge and polarization asymmetry of µ ± deep inelastic scattering off C target has been measured at CERN [5]. In the parton model, the asymmetry parameter B is given as 6G B = 5 y [ ē Q y Q C µ 3u Cµ 3d ] λ C µ qx qx u Cµ d, 3.7 Dx where λ denotes the effective µ ± beam polarization. Here we assumed that charm and strange quarks have the same effective interaction as up and down quarks, respectively: C µ c = Cµ u and Cµ s = Cµ d. The superscript µ is put to remind us that these parameters are for the µ-q scattering. All the other experiments are done for electron scatterings, and the corresponding superscript e is suppressed. C u C d SM x t 0.003x H. 3.3b The superscript µ has been suppressed since the predictions of the SM and the generic SU L U Y model are essentially the same for e and µ. The difference between the predictions for C u C d in 3.5 and 3. reflects different Q of the two experiments Bates ec experiment The polarization asymmetry parameter A of electron elastic scattering off C target has been measured at Bates [3] A Bates = 3 G Q C ē Q u C d. 3.4

5 G. Cho et al.: Constraints on four-ermi contact interactions from low-energy electroweak experiments 59 Here we neglected contribution from quarks other than the u and d quarks, which have been estimated to be small under the condition of the experiment [7] at a typical momentum transfer of Q = 0.05 GeV. rom the experimental data A Bates =.6 ± , 3.5 we find by using 4π/ē 0.05 GeV = 35.87, C u C d = 0.37 ± The corresponding theoretical prediction at Q = 0.05 GeV is found to be C u C d SM S T, 3.7 in the generic SU L model, and C u C d SM x t 0.000x H, 3.8 in the minimal SM Mainz ebe experiment At Mainz, polarization asymmetry of electron quasi-elastic scattering off 9 Be target has been measured [4]. The experiment was performed at the mean momentum transfer Q 0.05 GeV. In this experiment, the asymmetry parameter A Mainz can be expressed by the following combination of the model independent parameters [4]; where two coefficients C p and C n are given in terms of C u and C d as [6] C p C u C d , C n C u C d a 3.4b The theoretical prediction for C u and C d is found to be C SM u S T, 3.5a C SM d S T, 3.5b in the generic SU L model, and Cu SM x t x H, 3.6a Cd SM x t x H, 3.6b in the minimal SM. There are two accurate measurements of the weak charge in the cesium atom [8,9]; Q W Cs= { 7.04 ±.58 ± 0.88 [8], 7. ± 0.7 ± 0.88 [9], 3.7 where the first error is experimental while the second one is theoretical. Both theoretical errors come from the same estimation [0], and hence they are 00% correlated. By combining these two data, we obtain Q W Cs= 7.08 ± APV experiments in the thallium atom have been performed by Seattle [] and Oxford [] groups. The results are; A Mainz =.73C u 0.65C d.9c u.03c d, 3.9 { Q W ±.3 ± 4.0 [], 8 Tl= 0.5 ± 3.5 ± 4.0 []. 3.9 and the experimental result is given by A Mainz = 0.94 ± Here the error is obtained by taking the quadratic sum of the theoretical and experimental ones as quoted in [4]. The theoretical prediction in the generic SU L U Y model is A SM Mainz S T, 3. and that of the minimal SM is, A SM Mainz x t 0.007x H Atomic parity violation The experimental results of parity violation in atoms are often given in terms of the weak charge Q W A, Z ofthe nuclei. Using the model-independent parameter C q, the weak charge of a nuclei A, Z can be expressed as; Q W A, Z =ZC p A ZC n, 3.3 The result in the second line is obtained in [3] where Q W was extracted from the report [] of the Oxford group. In [], a part of the theoretical error that has been accounted for in [] was not included. Therefore we replaced the theoretical error of [] by ±4.0, following [3]. The two common theoretical errors are again 00% correlated. The combined result is given by Q W 05 8 Tl= 5.0 ± Theoretical predictions for the weak charges of Cs and Tl are Q SM W Cs S T, 3.3a Q SM W 05 8 Tl S 0.5 T, 3.3b in the generic SU L model. In the minimal SM, they are given as; Q SM W Cs x t 0.06x H 0.007x H, 3.3a Q SM W 05 8 Tl x t 0.09x H 0.0x H. 3.3b

6 60 G. Cho et al.: Constraints on four-ermi contact interactions from low-energy electroweak experiments 3.3 ν µ -quark scattering or neutrino-quark scattering, the experimental results are presented by using the model-independent parameters g α and δ α or equivalently q α of [4]. We examine two independent sets of experimental data; the results of all the old neutrino-nucleus scattering experiments as summarized by ogli and Haidt H [4] and the recent CCR measurement [5]. The result of [4] can be expressed as [6] = ± = ± = ± ρ corr = ± = gl gr δl δr Typical momentum transfer of these measurements is Q H = 0 GeV. The experiment of CCR collaboration has been done at slightly higher momentum transfer Q CCR = 35 GeV. The result was given for the following combination of the model-independent parameters [5]; K =.7897gL.479gR 0.096δL 0.078δR, 3.34a = ± b Taking into account of the difference of the typical momentum transfer of the two data sets, we find the following theoretical predictions for the model-independent parameters; u SM L u SM R d SM L d SM R S T, 3.35a S T, 3.35b S T, 3.35c S T, 3.35d in the generic SU L models, and u SM L x t x H, 3.36a u SM R 0.000x t 0.000x H, 3.36b d SM L x t x H, 3.36c d SM R 0.000x H, 3.36d in the minimal SM. In the above, the upper and the lower numbers in each column are the predictions at Q H = 0 GeV and at Q CCR = 35 GeV, respectively. The Q -dependences of the theoretical predictions turned out to be negligibly small. 3.4 ν µ -electron scattering There are three experimental results for the ν µ -electron scattering [6]. Here we use the combined data which are expressed in terms of the cross sections [6]; [ ] σ νe /E ν 0 4 cm /GeV =.56 ± 0.0, [ σ νe /E ν ] 0 4 cm /GeV =.36 ± 0.09, ρ corr = As seen from., the above cross sections expressed in terms of the helicity amplitudes, M νµe Lα. If we ignore the Q -dependence of their matrix elements, the experimental data 3.37 can be expressed in terms of the two independent helicity amplitudes as M νµe = 8.87 ± 0.35, ρ corr =0.3, 3.38 = 7.73 ± 0.36, M νµe LR in units of TeV. The theoretical predictions in the generic SU L models are M νµe SM S 0.4 T, 3.39a M νµe LR SM S 0.0 T, 3.39b and those within the minimal SM are, M νµe SM x t 0.05x H, 3.40a M νµe LR SM x t 0.0x H, 3.40b all in units of TeV. The Q -dependence of the theoretical predictions of the matrix elements M νµe Lα Q has indeed been found to be negligible between Q = m e E ν and Q = m e E ν / in., for E ν =5.7 GeV CHARM II [6]. 4 Results on generic SU L U Y model 4. Constraints on S, T parameters Here, we study the constraints on S and T parameters in the generic SU L model from the low-energy electroweak experiments listed in the previous section. iasymmetries in l-q scatterings experiments; We combine the results of the four experiments, SLAC ed, CERN µ ± C, Bates ec, Mainz ebe and find S = T ± iiapv experiments; The combined result of parity violation experiments in the cesium and thallium atoms is given by S = T ±.. 4.

7 G. Cho et al.: Constraints on four-ermi contact interactions from low-energy electroweak experiments 6 As emphasized in [3, 7], the APV experiments constrain mainly the S parameter. iii ν µ -q scatterings; rom the two data sets of the ν µ -q scattering experiments given in Sect. 3.3, the following constraint is obtained, S =0.95 ± 5.0 ρ corr = T =0.75 ±.78 T LEP/SLC Because of the strong positive correlation between the errors, only the following combination is effectively constrained; T = S ± iv ν µ -e scatterings; rom the ν µ -e scattering data in Sect. 3.4, we find S =0.035 ± 3.6 ρ corr =0.76. T =0.064 ± Summing up all the L.E.N.C. experiments, we find the following constraints on the S and T parameters, S =.44 ±.03 ρ corr =0.75, T = 0.06 ± χ min /d.o.f=3.38/3. We show in ig. our results of the individual constraints, 4., 4., 4.3 and 4.5 as well as the combined result 4.6 for δ α =0.03 []. When compared with corresponding result of [9, 0], significant improvements are found in the constraints from the APV and the ν µ -q scattering experiments. Constraints from the l-q scattering experiments are only slightly improved by including the CERN µ ± C, Bates ec and Mainz ebe experiments in the analysis. 4. Comparison with LEP/SLC data rom the updated Z shape parameter measurements by LEP/SLC, the effective charges s m Z and ḡ Z m Z have been extracted in [9]. It is assumed that the vertex functions beside the Zb L b L vertex function δ b m Z are dominated by the SM contributions and the following combination [6, 9] α s = α s m Z MS.54[ δ b m Z ] 4.7 is constrained by the hadronic decay width of the Z-boson. Then, by taking α s and δ b m Z as external parameters, the following result has been found in [9]; ḡz m Z = α s 0.8 ± s m Z = α s 0.8 ± ρ corr =0.4, 4.8a 3 4 lq APV Cs Tl ν µ q H CCR ν µ e All L.E.N.C S ig.. it to L.E.N.C. data on the S, T plane. The -σ 39% CL allowed ranges are shown separately for charged leptonquark scattering experiments, atomic parity violation, ν µ- quark scattering and ν µ-e scattering experiments. Also shown are the -σ allowed region of all the L.E.N.C. experiments as well as that of the LEP/SLC Z-pole measurements α χ min =5.4 s 0.8 δb b with d.o.f =. The above result can be expressed in terms of the S and T -parameters as follows; S = α s 0.8 ± 0.3 T = α s 0.8 ± 0.5 ρ corr =0.87, 4.9a α χ min =5.4 s 0.8 δb b Here the combination S = S 0.7δ α is determined from the s m Z measurement [9]. It is slightly different from the combination S of.4 which is constrained from the s 0 measurement at low-energies. The difference coms from the uncertainty in the hadronic corrections to the running of the effective charge ē q / s q between q = m Z and q = 0; see Appendix B of [6]. The result is shown in ig. for δ α =0.03 [] and α s =0.8. We find that the Z-pole results 4.9 are consistent with the SM predictions for x t = x H =0,S, T = 0.3, 0.88, whereas the results of the L.E.N.C. experiments 4.6 give slightly smaller S and T.

8 6 G. Cho et al.: Constraints on four-ermi contact interactions from low-energy electroweak experiments We can combine all the L.E.N.C. data with the above LEP/SLC data, and find for δ α =0.03, S= α s ± 0.3 T= α s ± 0.4 ρ corr =0.86, 4.0a α χ min =0.8 s δb 0.005, 4.0b with d.o.f = 5L.E.N.C.3LEP/SLC = 6. The combined fit does not change significantly from the previous result of [9] mainly because of the dominant role of the Z-pole data. 5 Constraints on contact terms In this section, we study the constraints on the contact interactions from all the L.E.N.C. data. 5. Constraints on general contact interactions In the general framework of the contact interactions, each L.E.N.C. experiment constrains certain combinations of the contact terms. rom the SLAC ed experiments, two combinations of the model independent parameters Cq e and Cq e are constrained. Assuming the SM prediction 3.6, the experimental data 3.4 lead the following constraints; Cu e Cd e = 0.7 ± 0.6 ρ Cu e Cd e corr = = ±.3 Here and in the following, we adopt the SM predictions for m t = 75 GeV and m H = 00 GeV and δ α = 0.03 []. Effects of small changes in the SM predictions for different values of m t and m H or for S and T can easily be accounted for by using the parametrizations given in Sect. 3. The CERN µ ± C experiment is used to extract the model independent parameters C µ q and Cµ 3q. rom the experimental data given in Sect. 3, we find the following constraints on the linear combination of C µ q and Cµ 3q ; C µ 3u Cµ 3d =.5 ± 4.9 C µ u ρ corr = Cµ d =.74 ± 6.3 The Bates experiment on ec scattering constrains the following combination; C e u C e d =0.05 ± The Mainz experiment on ebe scattering constrains the combination A Mainz =.73 C e u 0.65 C e d.9 C e u.03 C e d 5.4 = ± 0.9. The two APV experiments give; Q W Cs = 376 C e u 4 C e d =0.96 ± 0.9, 5.5 Q W Tl = 57 C e u 658 C e d =.58 ± By combining the two sets of the ν µ -q scattering data, we find u L = ± u R = ± 0.05 d L = ± d R = ± ρ corr = The contact term contributions to the model-independent parameters C iq and q α are found in.7 and.9, respectively. rom the ν µ -e scattering experiments, 3.37, we find M νµe M νµe LR = ηνµe = ηνµe LR = 0.5 ± 0.35 = 0.03 ± 0.36 ρ corr =0.3, 5.8 in units of TeV. The two pure-leptonic contact interactions are constrained directly by these experiments. 5. Constraints on SU L invariant contact interactions Without any further assumptions, there are 08 leptonquark contact couplings η lq. In this subsection, we combine the individual constraints by taking the following assumptions; i lepton universality ii quark universality invariance for SU L singlet ex- iii SU L change η eq = ηµq = ητq ηlq, 5.9 η lu = η lc = η, lt η ld = η ls = η, lb 5.0a 5.0b ηαl lu = ηαl, ld η ν lq Lβ = ηlq Lβ. 5.a 5.b

9 G. Cho et al.: Constraints on four-ermi contact interactions from low-energy electroweak experiments 63 Six lepton-quark contact terms remain as independent parameters; { η,η lu LR,η lu LR,η ld RR,η ld RL,η lu RR lu. 5. Let us first examine how the above six contact interaction terms are constrained from each low-energy experiment. We express the model-independent parameters, C iq, q α and M lq in terms of ηlq with the above assumptions. We find the following constraints on the various combinations of the contact terms: Here η lq is measured in units of TeV. SLAC ed experiment: 0.5η lu 0.657ηlu LR 0.39ηld LR 0.39ηlu RL ηrr lu 0.5ηld RR = 0.86 ± CERN µ ± C experiment: 0.06η lu 0.4ηlu LR 0.06ηld LR 0.5ηlu RL ηrr lu 0.5ηld RR =.4 ± Bates ec experiment: η lu 0.5ηlu LR 0.5ηld LR ηlu RL 0.5ηRR lu 0.5ηld RR =0.5 ± Mainz ebe experiment: 0.455η lu 0.0ηlu LR 0.80ηld LR 0.390ηlu RL ηrr lu 0.545ηld RR = 0.44 ± APVCs: η lu 0.47ηlu LR 0.59ηld LR ηlu RL 0.47ηRR lu 0.59ηld RR =0.040 ± APVTl: η lu 0.465ηlu LR 0.535ηld LR ηlu RL 0.465ηRR lu 0.535ηld RR =0.04 ± In 5.3 and 5.4, looser constraints are dropped. rom the result of ν µ -q experiments 5.7, the following three contact terms are constrained; η lu ηlr lu ηlr ld = 0. ± = 0.06 ± 0.83,ρ corr = = 0.47 ±.4 where, the contact terms are given in units of TeV. By adding all data of low-energy lepton-quark experiments given in , we find η lu ηlr lu ηlr ld ηrl lu ηrr lu ηrr ld = 0.8 ± = 0.08 ± = 0.0 ±.43, 5.0a =.47 ± 4.00 =.39 ± 3.53 =.76 ± ρ corr = , 5.0b 0.97 χ min/d.o.f=.8/7, 5.0c in units of TeV. We give below the three eigenvectors of the 6 6 covariance matrix with the smallest errors: 0.353η lu 0.67ηlu LR 0.85ηld LR 0.35ηlu RL 0.65ηRR lu 5.a 0.85ηld RR =0.04 ± η lu 0.394ηlu LR 0.38ηld LR 0.8ηlu RL 5.b 0.044ηRR lu 0.9ηld RR = 0.5 ± η lu 0.76ηlu LR 0.009ηld LR 0.88ηlu RL 5.c 0.70ηRR lu 0.047ηld RR = 0.36 ± The most accurate constraint 5.a is essentially follows from the APV measurements 5.7 and 5.8. The second most accurate constraint 5.b is essentially obtained by the ν µ -q scattering data 5.7 or 5.9. inally the ν µ -e scattering experiments constrain the pure-leptonic contact interactions η ll. We find in units of TeV ; η ll = 0.5 ± 0.35 ηlr ll = 0.03 ± 0.36 ρ corr = As a reference, the minimal SM all η ff = 0 gives an excellent fit to all the data χ SM/d.o.f=6.9/ Therefore, low-energy electroweak experiments do not require any new interactions. In the actual fit, we used the original data, 3.33 and 3.34, instead of 5.7 to retain accuracy. This explains that the d.o.f. in 5.0c is not 6 but 7. Equation 5.7 should be read as approximate constraints since the ν µ-q data expressed in terms of u α and d α slightly deviate from the Gaussian

10 64 G. Cho et al.: Constraints on four-ermi contact interactions from low-energy electroweak experiments Table. The hypercharge Y and the extra U E charge Q E of the left-handed quarks and leptons in Z χ,z ψ,z η and Z ν models field Y 4Qχ 7/5Qψ Q η Q ν ν, e ν c e c 3 4 u, d u c d c Discussion In this paper, we have studied the constraints on the four- ermi contact interactions from low-energy electroweak experiments. rom the polarization/charge asymmetry measurement of charged lepton-nucleus scattering experiments, atomic parity violation experiments, and from neutrino-quark scattering experiments, constraints on the lepton-quark contact interactions were obtained, while neutrino-electron scattering experiments constrain the lepton-lepton contact interactions. By assuming the flavor universality and the SU L gauge invariance of the contact interactions, those constraints are parametrized conveniently as the constraint 5.0c for the six lepton-quark contact terms and 5. for the two pureleptonic contact terms. By using our result, it is easy to examine consequences of models of new physics that affect low-energy electroweak observables. As an example, we briefly study constraints on an extra Z-boson in E 6 models. The models contain two additional new neutral gauge bosons, one is the SO0 singlet Z ψ, and the other one is the SU5 singlet Z χ. In general the two gauge bosons are mixed, and the lighter one Z E is given by the following linear combination; Z E = Z χ cos β E Z ψ sin β E. 6. Mixing angle β E = 0,π/, tan 5/3 and tan 5 correspond to Z χ,z ψ,z η and Z ν models, respectively. ollowing the Particle Data Group notation [], the hypercharge Y and the extra U E charge Q E of the left-handed quarks and leptons are given by Table. Then, expressing the contact terms by U E gauge coupling g E and the extra gauge boson mass m ZE as, = αg f gf β m, 6.a Z E g f L = g EQ f E, 6.b η ff g f R = g EQ f C E, 6.c TeV /m ZE g E Z ψ Z η Z χ Z ν Z ψ 95% CL upp. limit σ limits best fit 3 π/ π/4 0 π/4 π/ β E ig.. Constraints on ge/m Z E from the low-energy electroweak measurements. The -σ allowed region and the 95% CL upper bounds are given for the Z E models that are characterized by the mixing angle β E; see 6. m ZE GeV Z ψ Z η Z χ Z ν Z ψ 0 π/ π/4 0 π/4 π/ β E ig % CL lower limits of the extra Z boson mass m ZE as a function of the mixing angle β E we can find the constraints on each extra Z-boson model. We show in ig. the constraints on ge /m Z E for each model parametrized by the mixing angle β E. In the figure, the region of ge /m Z E < 0 is unphysical. The 95% CL upper bounds are obtained under the constraint ge /m Z E > 0. The reason for the appearance of the peak at β E = π/ Z ψ model can be easily understood as follows; in the Z ψ model, all left-handed fermions have the same U E charge, and hence the couplings are Parity conserving, which makes the most rigorous constraints from APV experiments useless. The set of six couplings, η lq, is also less constrained by APV measurements at β E π/5, thus the another peak is appeared. To estimate the limit on the extra Z-boson mass, the extra U E coupling g E should be fixed. Assuming ge = g Y =ē m Z / c m Z =0.70, we show the 95% CL lower limit on the extra Z-boson mass in ig. 3.

11 G. Cho et al.: Constraints on four-ermi contact interactions from low-energy electroweak experiments 65 Table. Constraints on g E/m Z E and m ZE from the low-energy electroweak experiments for the four representative extra Z-boson models. We assumed g E = g Y = 0.70 to obtain the 95% CL lower limits of m ZE. Z E β E ge/m Z E χ min 95% CL limits TeV ge/m Z E m ZE Z χ ± TeV 470 GeV Z ψ π/.0 ± TeV 40 GeV Z η tan 5/ ± TeV 340 GeV Z ν tan ± TeV 90 GeV Constraints on ge /m Z E from the low-energy electroweak experiments for the four representative extra Z-boson models are listed in Table. The 95% CL upper limits of ge /m Z E and the 95% CL lower limits of m ZE for ge = g Y = 0.70 are also given in the Table. As compared to [8], Z χ and Z η models are more severely constrained by updated low-energy electroweak experiments. Acknowledgements. The authors wish to thank V. Barger, K. Cheung, D. Haidt, K. Tokushuku and D. Zeppenfeld for stimulating discussions. The work of K.H. is supported in part by the JSPS-NS Joint Research Project. The work of G.C.C. and S.M. is supported in part by Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture of Japan. References. E. Eichten, K. Lane and M.E. Peskin, Phys. Rev. Lett Particle Data Group, Phys. Rev. D H collaboration, Z. Phys. C ZEUS collaboration, Z. Phys. C G. Altarelli, J. Ellis, G.. Giudice, S. Lola and M.L. Mangano, hep-ph/970376; K. Babu, C. Kolda, J.M.-Russell and. Wilczek, Phys. Lett. B ; V. Bager, K. Cheung, K. Hagiwara and D. Zeppenfeld, Phys. Lett. B ; N. Di Bartolomeo and M. abbrichesi, Phys. Lett. B ; M.C. Gonzalez-Garcia and S.. Novaes, Phys. Lett. B ; A.E. Nelson, Phys. Rev. Lett ; W. Buchmüller and D. Wyler, Phys. Lett. B ; S. Godfrey, Mod. Phys. Lett. A ; L. Giusti and A. Strumia, hep-ph/ K. Hagiwara, D. Haidt, C.S. Kim and S. Matsumoto, Z. Phys. C , C E 7. J.E. Kim, P. Langacker, M. Levine and H.H. Williams, Rev. Mod. Phys M.E. Peskin and T. Takeuchi, Phys. Rev. Lett , Phys. Rev. D K. Hagiwara, D. Haidt and S. Matsumoto, hepph/970633, to be published in Eur. Phys. J. C 0. J.L. Hewett, T. Takeuchi and S. Thomas, hep-ph/960339, to appear in Electroweak Symmetry Breaking and Beyond the Standard Model, ed. T. Barklow, et al., World Scientific. S. Eidelman and. Jegerlehner, Z. Phys. C C.Y. Prescott et al., Phys. Lett. B P.A. Souder et al., Phys. Rev. Lett W. Heil et al., Nucl. Phys. B A. Argento et al., Phys. Lett. B P.A. Souder, in Precision tests of the standard electroweak model, ed P. Langacker, World Scientific 995, D.H. Beck, Phys. Rev. D M.C. Noecker, B.P. Masterson and C.E. Wieman, Phys. Rev. Lett C.S. Wood et al., Science S.A. Bllundell, J. Sapirstein and W.R. Johnson, Phys. Rev. D P.A. Vetter et al., Phys. Rev. Lett N.H. Edwards et al., Phys. Rev. Lett J.L. Rosner, Phys. Rev. D ; J.L. Rosner, hep-ph/970433, submitted to Comments on Nuclear and Particle Physics 4. G.L. ogli and D. Haidt, Z. Phys. C K. Mcarland et al., hep-ex/97000, submitted to Phys. Rev. Lett. 6. CHARM collaboration, Z. Phys. C ; L.A. Ahrens et al., Phys. Rev. D ; CHARM-II collaboration, Phys. Lett. B W.J. Marciano and J.L. Rosner, Phys. Rev. Lett , EPhys. Rev. Lett P. Langacker and M. Luo, Phys. Rev. D