Electroweak Model and Constraints on New Physics

10. EW model and constraints on new physics 1 10. ELECTROWEAK MODEL AND CONSTRAINTS ON NEW PHYSICS Revised December 2003 by J. Erler (U. Mexico) and P. Langacker (Univ. of Pennsylvania). 10.1 Introduction 10.2 Renormalization and radiative corrections 10.3 Cross-section and asymmetry formulas 10.4 W and Z decays 10.5 Experimental results 10.6 Constraints on new physics arXiv:hep-ph/0407097v1 8 Jul 2004 10.1. Introduction The standard electroweak model (SM) is based on the gauge group [1] SU(2) × U(1), with gauge bosons Wµi , i = 1, 2, 3, and Bµ for the SU(2) and U(1) factors, respectively, and the corresponding gaugecoupling constants g and g ′ . The left-handed fermion fields    νi ℓ− i ui d′i of the ith fermion family transform as doublets P under SU(2), where d′i ≡ j Vij dj , and V is the Cabibbo-KobayashiMaskawa mixing matrix. (Constraints on V and tests of universality are discussed in Ref. 2 and in the Section on the Cabibbo-KobayashiMaskawa mixing matrix.) The right-handed fields are SU(2) singlets. In the minimal model there arethree fermion families and a single ψi = and complex Higgs doublet φ ≡ φ+ φ0 . After spontaneous symmetry breaking the Lagrangian for the fermion fields is  X  gmi H ψ i i 6 ∂ − mi − LF = ψi 2MW i  
g X ψ i γ µ 1 − γ 5 T + Wµ+ + T − Wµ− ψi − √ 2 2 i X qi ψ i γ µ ψi Aµ −e i −   X g i 5 ψi Zµ . γ ψ i γ µ gVi − gA 2 cos θW (10.1) i θW ≡ tan−1 (g ′ /g) is the weak angle; e = g sin θW is the positron 3 electric charge; and A ≡ B cos θW √ + W sin θW is the (massless) ± 1 2 photon field. W ≡ (W ∓ iW )/ 2 and Z ≡ −B sin θW + W 3 cos θW are the massive charged and neutral weak boson fields, respectively. T + and T − are the weak isospin raising and lowering operators. The vector and axial-vector couplings are gVi ≡ t3L (i) − 2qi sin2 θW , i gA (10.2a) ≡ t3L (i) , (10.2b) CITATION: S. Eidelman et al., Physics Letters B592, 1 (2004) available on the PDG WWW pages (URL: http://pdg.lbl.gov/) February 2, 2008 00:19 2 10. EW model and constraints on new physics where t3L (i) is the weak isospin of fermion i (+1/2 for ui and νi ; −1/2 for di and ei ) and qi is the charge of ψi in units of e. The second term in LF represents the charged-current weak interaction [3,4]. For example, the coupling of a W to an electron and a neutrino is h   i   e Wµ− e γ µ 1 − γ 5 ν + Wµ+ ν γ µ 1 − γ 5 e . (10.3) − √ 2 2 sin θW For momenta small compared to MW , this term gives rise to the effective four-fermion interaction with the Fermi constant √ given (at tree 2 . level, i.e., lowest order in perturbation theory) by GF / 2 = g 2 /8MW CP violation is incorporated in the SM by a single observable phase in Vij . The third term in LF describes electromagnetic interactions (QED), and the last is the weak neutral-current interaction. In Eq. (10.1), mi is the mass of the ith fermion ψi . For the quarks these are the current masses. For the light quarks, as described in the Particle Listings, m b u ≈ 1.5–4.5 MeV, m b d ≈ 5–8.5 MeV, and m b s ≈ 80–155 MeV. These are running MS masses evaluated at the scale µ = 2 GeV. (In this Section we denote quantities defined in the MS scheme by a caret; the exception is the strong coupling constant, αs , which will always correspond to the MS definition and where the caret will be dropped.) For the heavier quarks we use QCD sum rule constraints [5] and recalculate their masses in each call of our fits to account for their direct αs dependence. We find, m b c (µ = m b c ) = 1.290+0.040 −0.045 GeV and m b b (µ = m b b ) = 4.206 ± 0.031 GeV, with a correlation of 29%. The top quark “pole” mass, mt = 177.9 ± 4.4 GeV, is an average of CDF results from run I [6] and run II [7], as well as the DØ dilepton [8] and lepton plus jets [9] channels. The latter has been recently reanalyzed, leading to a somewhat higher value. We computed the covariance matrix accounting for correlated systematic uncertainties between the different channels and experiments according to Refs. 6 and 10. Our covariance matrix also accounts for a common 0.6 GeV uncertainty (the size of the three-loop term [11]) due to the conversion from the pole mass to the MS mass. We are using a BLM optimized [12] version of the two-loop perturbative QCD formula [13] which should correspond approximately to the kinematic mass extracted from the collider events. The three-loop formula [11] gives virtually identical results. We use MS masses in all expressions to minimize theoretical uncertainties. We will use above value for mt (together with MH = 117 GeV) for the numerical values quoted in Sec. 10.2–Sec. 10.4. See “The Note on Quark Masses” in the Particle Listings for more information. In the presence of right-handed neutrinos, Eq. (10.1) gives rise also to Dirac neutrino masses. The possibility of Majorana masses is discussed in “Neutrino mass” in the Particle Listings. H is the physical neutral Higgs scalar which is the only remaining part of φ after spontaneous symmetry breaking. The Yukawa coupling of H to ψi , which is flavor diagonal in the minimal model, is February 2, 2008 00:19 10. EW model and constraints on new physics 3 gmi /2MW . In non-minimal models there are additional charged and neutral scalar Higgs particles [14]. 10.2. Renormalization and radiative corrections The SM has three parameters (not counting the Higgs boson mass, MH , and the fermion masses and mixings). A particularly useful set is: (a) The fine structure constant α = 1/137.03599911(46), determined from the e± anomalous magnetic moment, the quantum Hall effect, and other measurements [15]. In most electroweak renormalization schemes, it is convenient to define a running α dependent on the energy scale of the process, with α−1 ∼ 137 appropriate at very low energy. (The running has also been observed directly [16]. ) For scales above a few hundred MeV this introduces an uncertainty due to the low-energy hadronic contribution to vacuum polarization. In the modified minimal subtraction (MS) scheme [17] (used for this Review), and with αs (MZ ) = 0.120 for the QCD coupling at MZ , we have α b(mτ )−1 = 133.498 ± 0.017 and α b(MZ )−1 = 127.918 ± 0.018. These values are updated from Ref. 18 and account for the latest results from τ decays and a reanalysis of the CMD 2 collaboration results after correcting a radiative correction [19]. See Ref. 20 for a discussion in the context of the anomalous magnetic moment of the muon. The correlation of the latter with α b(MZ ), as well as the non-linear αs dependence of α b(MZ ) and the resulting correlation with the input variable αs , are fully taken into account in the fits. The uncertainty is from e+ e− annihilation data below 1.8 GeV and τ decay data, from isospin breaking effects (affecting the interpretation of the τ data), from uncalculated higher order perturbative and non-perturbative QCD corrections, and from the MS quark masses. Such a short distance mass definition (unlike the pole mass) is free from non-perturbative and renormalon uncertainties. Various recent evaluations of the (5) contributions of the five light quark flavors, ∆αhad , to the α conventional (on-shell) QED coupling, α(MZ ) = , are 1 − ∆α summarized in Table 10.1. Most of the older results relied on e+ e− → hadrons cross-section measurements up to energies of 40 GeV, which were somewhat higher than the QCD prediction, suggested stronger running, and were less precise. The most recent results typically assume the validity of perturbative QCD (PQCD) at scales of 1.8 GeV and above, and are in reasonable agreement with each other. (Evaluations in the on-shell scheme utilize resonance data from BES [36] as further input.) There is, however, some discrepancy between analyzes based on e+ e− → hadrons cross-section data and those based on τ decay spectral functions [20]. The latter imply lower central values for the extracted MH of O(10 GeV). Further improvement of this dominant theoretical uncertainty in the interpretation of precision data will require better measurements of the cross-section for February 2, 2008 00:19 4 10. EW model and constraints on new physics e+ e− → hadrons below the charmonium resonances, as well as in the threshold region of the heavy quarks (to improve the precision in m b c (m b c ) and m b b (m b b )). As an alternative to cross-section scans, one can use the high statistics radiative return events [37] at e+ e− accelerators operating at resonances such as the Φ or the Υ (4S). The method is systematics dominated. First preliminary results have been presented by the KLOE collaboration [38]. (5) Table 10.1: Recent evaluations of the on-shell ∆αhad (MZ ). For better comparison we adjusted central values and errors to correspond to a common and fixed value of αs (MZ ) = 0.120. References quoting results without the top quark decoupled are converted to the five flavor definition. Ref. [31] uses ΛQCD = 380 ± 60 MeV; for the conversion we assumed αs (MZ ) = 0.118 ± 0.003. Reference Result Martin & Zeppenfeld [21] 0.02744 ± 0.00036 Eidelman & Jegerlehner [22] 0.02803 ± 0.00065 Geshkenbein & Morgunov [23] 0.02780 ± 0.00006 Burkhardt & Pietrzyk [24] Swartz [25] Alemany, Davier, Höcker [26] Krasnikov & Rodenberg [27] Davier & Höcker [28] 0.0280 ± 0.0007 0.02754 ± 0.00046 0.02816 ± 0.00062 includes τ decay data √ PQCD for s > 2.3 GeV √ PQCD for s > 1.8 GeV 0.02778 ± 0.00016 0.02779 ± 0.00020 complete O(α2s ) converted from MS scheme 0.02737 ± 0.00039 0.02784 ± 0.00022 Kühn & Steinhauser [29] Erler [18] Davier & Höcker [30] Groote et al. [31] 0.02770 ± 0.00015 0.02787 ± 0.00032 Martin, Outhwaite, Ryskin [32] 0.02741 ± 0.00019 Burkhardt & Pietrzyk [33] 0.02763 ± 0.00036 de Troconiz & Yndurain [34] Jegerlehner [35] Comment √ PQCD for s > 3 GeV √ PQCD for s > 40 GeV O(αs ) resonance model √ PQCD for s > 40 GeV use of fitting function use of QCD sum rules use of QCD sum rules includes new BES data √ PQCD for s > 12 GeV 0.02754 ± 0.00010 PQCD for s > 2 GeV2 0.02766 ± 0.00013 converted from MOM scheme (b) The Fermi constant, GF = 1.16637(1) × 10−5 GeV−2 , determined from the muon lifetime formula [39,40], τµ−1 " × 1+  G2 m5µ = F 3 F 192π 25 π 2 − 8 2  m2e m2µ ! 3 m2µ 1+ 2 5 MW !

# α mµ α2 mµ + C2 , π π2 February 2, 2008 00:19 (10.4a) 10. EW model and constraints on new physics 5 where F (x) = 1 − 8x + 8x3 − x4 − 12x2 ln x , C2 = (10.4b) 67 4 53 2 156815 518 2 895 − π − ζ (3)+ π + π ln (2) , (10.4c) 5184 81 36 720 6 and α mµ
−1 = α−1 − m  1 2 µ + ln ≈ 136 . 3π me 6π (10.4d) The O(α2 ) corrections to µ decay have been completed recently [40]. The remaining uncertainty in GF is from the experimental input. (c) The Z boson mass, MZ = 91.1876 ± 0.0021 GeV, determined from the Z-lineshape scan at LEP 1 [41]. With these inputs, sin2 θW and the W boson mass, MW , can be calculated when values for mt and MH are given; conversely (as is done at present), MH can be constrained by sin2 θW and MW . The value of sin2 θW is extracted from Z-pole observables and neutral-current processes [41,42], and depends on the renormalization prescription. There are a number of popular schemes [44–50] leading to values which differ by small factors depending on mt and MH . The notation for these schemes is shown in Table 10.2. Discussion of the schemes follows the table. Table 10.2: Notations used to indicate the various schemes discussed in the text. Each definition of sin θW leads to values that differ by small factors depending on mt and MH . Scheme Notation On-shell sW NOV sMZ = sin θW sbZ = sin θW MS MS ND Effective angle = sin θW sbND = sin θW sf = sin θW (i) The on-shell scheme [44] promotes the tree-level formula sin2 θW = 2 /M 2 to a definition of the renormalized sin2 θ 1 − MW W to all Z 2 /M 2 : orders in perturbation theory, i.e., sin2 θW → s2W ≡ 1 − MW Z MW = MZ A0 sW (1 − ∆r)1/2 M = W , cW February 2, 2008 00:19 , (10.5a) (10.5b) 6 10. EW model and constraints on new physics √ where cW ≡ cos θW , A0 = (πα/ 2GF )1/2 = 37.2805(2) GeV, and ∆r includes the radiative corrections relating α, α(MZ ), GF , MW , and MZ . One finds ∆r ∼ ∆r0 − ρt / tan2 θW , where ∆r0 = 1 − α/b α(MZ ) = 0.06654(14) is due to the running √ of α, and ρt = 3GF m2t /8 2π 2 = 0.00992(mt/177.9 GeV)2 represents the dominant (quadratic) mt dependence. There are additional contributions to ∆r from bosonic loops, including those which depend logarithmically on MH . One has ∆r = 0.03434 ∓ 0.0017 ± 0.00014, where the second uncertainty is from α(MZ ). Thus the value of s2W extracted from MZ includes an uncertainty (∓0.00054) from the currently allowed range of mt . This scheme is simple conceptually. However, the relatively large (∼ 3%) correction from ρt causes large spurious contributions in higher orders. (ii) A more precisely determined quantity s2M can be obtained from Z MZ by removing the (mt , MH ) dependent term from ∆r [45], i.e., πα (MZ ) . (10.6) s2MZ c2MZ ≡ √ 2 GF MZ2 Using α(MZ )−1 = 128.91 ± 0.02 yields s2M = 0.23108 ∓ 0.00005. Z The small uncertainty in s2M compared to other schemes is Z because most of the mt dependence has been removed by definition. However, the mt uncertainty reemerges when other quantities (e.g., MW or other Z-pole observables) are predicted in terms of MZ . Both s2W and s2M depend not only on the gauge couplings Z but also on the spontaneous-symmetry breaking, and both definitions are awkward in the presence of any extension of the SM which perturbs the value of MZ (or MW ). Other definitions are motivated by the tree-level coupling constant definition θW = tan−1 (g ′ /g). (iii) In particular, the modified minimal subtraction  (MS) scheme  introduces the quantity sin2 θbW (µ) ≡ gb ′2 (µ)/ gb 2 (µ) + b g ′2 (µ) , where the couplings b g and gb′ are defined by modified minimal subtraction and the scale µ is conveniently chosen to be MZ for many electroweak processes. The value of sb 2Z = sin2 θbW (MZ ) extracted from MZ is less sensitive than s2W to mt (by a factor of tan2 θW ), and is less sensitive to most types of new physics than s2W or s2M . It is also very useful for comparing with Z the predictions of grand unification. There are actually several variant definitions of sin2 θbW (MZ ), differing according to whether or how finite α ln(mt /MZ ) terms are decoupled (subtracted from the couplings). One cannot entirely decouple the α ln(mt /MZ ) terms from all electroweak quantities because mt ≫ mb breaks SU(2) symmetry. The scheme that will be adopted here decouples the α ln(mt /MZ ) terms from the γ–Z mixing [17,46], essentially eliminating any ln(mt /MZ ) dependence in the formulae for asymmetries at the Z-pole when written in terms of sb 2Z . (A February 2, 2008 00:19 10. EW model and constraints on new physics 7 similar definition is used for α b.) The various definitions are related by sb 2Z = c (mt , MH ) s2W = c (mt , MH ) s2MZ , (10.7) where c = 1.0381 ± 0.0019 and c = 1.0003 ∓ 0.0006. The quadratic mt dependence is given by c ∼ 1 + ρt / tan2 θW and c ∼ 1 − ρt /(1 − tan2 θW ), respectively. The expressions for MW and MZ in the MS scheme are MW = MZ A0 sbZ (1 − ∆b r W )1/2 M = 1/2W , ρb b cZ , (10.8a) (10.8b) and one predicts ∆b r W = 0.06976 ± 0.00006 ± 0.00014. ∆b r W has no quadratic mt dependence, because shifts in MW are absorbed into the observed GF , so that the error in ∆b r W is dominated by ∆r0 = 1 − α/b α(MZ ) which induces the second quoted uncertainty. The quadratic mt dependence has been shifted into ρb ∼ 1 + ρt , where including bosonic loops, ρb = 1.0110 ± 0.0005. (iv) A variant MS quantity sb 2ND (used in the 1992 edition of this Review) does not decouple the α ln(mt /MZ ) terms [47]. It is related to sb 2Z by  α b  (10.9a) sb 2Z = sb 2ND / 1 + d , π     8 αs  mt 15αs 1 1 − 1 + ln , (10.9b) − d = 3 sb 2 3 π MZ 8π Thus, sb 2Z − sb 2ND ∼ −0.0002 for mt = 177.9 GeV. (v) Yet another definition, the effective angle [48–50] s2f for the Z vector coupling to fermion f , is described in Sec. 10.3. Experiments are at a level of precision that complete O(α) radiative corrections must be applied. For neutral-current and Z-pole processes, these corrections are conveniently divided into two classes: 1. QED diagrams involving the emission of real photons or the exchange of virtual photons in loops, but not including vacuum polarization diagrams. These graphs often yield finite and gaugeinvariant contributions to observable processes. However, they are dependent on energies, experimental cuts, etc., and must be calculated individually for each experiment. 2. Electroweak corrections, including γγ, γZ, ZZ, and W W vacuum polarization diagrams, as well as vertex corrections, box graphs, etc., involving virtual W ’s and Z’s. Many of these corrections are absorbed into the renormalized Fermi constant defined in Eq. (10.4). Others modify the tree-level expressions for Z-pole observables and neutral-current amplitudes in several ways [42]. February 2, 2008 00:19 8 10. EW model and constraints on new physics One-loop corrections are included for all processes. In addition, certain two-loop corrections are also important. In particular, two-loop corrections involving the top quark modify ρt in ρb, ∆r, and elsewhere by ρt → ρt [1 + R (MH , mt ) ρt /3] . (10.10) R(MH , mt ) is best described as an expansion in MZ2 /m2t . The unsuppressed terms were first obtained in Ref. 51, and are known analytically [52]. Contributions suppressed by MZ2 /m2t were first studied in Ref. 53 with the help of small and large Higgs mass expansions, which can be interpolated. These contributions are about as large as the leading ones in Refs. 51 and 52. In addition, the complete two-loop calculation of diagrams containing at least one fermion loop and contributing to ∆r has been performed without further approximation in Ref. 54. The two-loop evaluation of ∆r was completed with the purely bosonic contributions in Ref. 55. For MH above its lower direct limit, −17 < R ≤ −13. Mixed QCD-electroweak loops of order ααs m2t [56] and αα2s m2t [57] increase the predicted value of mt by 6%. This is, however, almost entirely an artifact of using the pole mass definition for mt . The equivalent corrections when using the MS definition m b t (m b t ) increase mt by less than 0.5%. The leading electroweak [51,52] and mixed [58] two-loop terms are also known for the Z → bb̄ vertex, but not the respective subleading ones. O(ααs )-vertex corrections involving massless quarks have been obtained in Ref. [59]. Since they add coherently, the resulting effect is sizable, and shifts the extracted αs (MZ ) by ≈ +0.0007. Corrections of the same order to Z → bb̄ decays have also been completed [60]. Throughout this Review we utilize electroweak radiative corrections from the program GAPP [61], which works entirely in the MS scheme, and which is independent of the package ZFITTER [50]. 10.3. Cross-section and asymmetry formulas It is convenient to write the four-fermion interactions relevant to ν-hadron, ν-e, and parity violating e-hadron neutral-current processes in a form that is valid in an arbitrary gauge theory (assuming massless left-handed neutrinos). One has   G −L νHadron = √F ν γ µ 1 − γ 5 ν 2 × −L   i   Xh ǫL (i) q i γµ 1 − γ 5 qi + ǫR (i) q i γµ 1 + γ 5 qi , νe i     G νe 5 = √F ν µ γ µ 1 − γ 5 νµ e γµ gVνe − gA γ e 2 February 2, 2008 00:19 (10.11) (10.12) 10. EW model and constraints on new physics 9 (for νe -e or ν e -e, the charged-current contribution must be included), and G −L eHadron = − √F 2 i Xh × C1i e γµ γ 5 e q i γ µ qi + C2i e γµ e q i γ µ γ 5 qi . (10.13) i (One must add the parity-conserving QED contribution.) νe , and C The SM expressions for ǫL,R (i), gV,A ij are given in νe Table 10.3. Note, that gV,A and the other quantities are coefficients of effective four-Fermi operators, which differ from the quantities defined in Eq. (10.2) in the radiative corrections and in the presence of possible physics beyond the SM. A precise determination of the on-shell s2W , which depends only very weakly on mt and MH , is obtained from deep inelastic neutrino scattering from (approximately) isoscalar targets [62]. The ratio N C /σ CC of neutral- to charged-current cross-sections has Rν ≡ σνN νN been measured to 1% accuracy by the CDHS [63] and CHARM [64] collaborations at CERN, and the CCFR [65] collaboration at Fermilab has obtained an even more precise result, so it is important to obtain N C /σ CC to comparable theoretical expressions for Rν and Rν ≡ σνN νN accuracy. Fortunately, most of the uncertainties from the strong interactions and neutrino spectra cancel in the ratio. The largest theoretical uncertainty is associated with the c-threshold, which mainly affects σ CC . Using the slow rescaling prescription [66] the central value of sin2 θW from CCFR varies as 0.0111(mc [GeV] − 1.31), where mc is the effective mass which is numerically close to the MS mass m b c (m b c ), but their exact relation is unknown at higher orders. For mc = 1.31 ± 0.24 GeV (determined from ν-induced dimuon production [67]) this contributes ±0.003 to the total uncertainty ∆ sin2 θW ∼ ±0.004. (The experimental uncertainty is also ±0.003.) This uncertainty largely cancels, however, in the Paschos-Wolfenstein ratio [68], NC σ N C − σν̄N R− = νN . (10.14) CC − σ CC σνN ν̄N It was measured recently by the NuTeV collaboration [69] for the first time, and required a high-intensity and high-energy anti-neutrino beam. A simple zeroth -order approximation is 2 2 Rν = gL + gR r, (10.15a) 2 gR , r (10.15b) 2 2 R − = gL − gR , (10.15c) 2 + Rν = gL where 2 gL ≡ ǫL (u)2 + ǫL (d)2 ≈ 1 5 − sin2 θW + sin4 θW , 2 9 February 2, 2008 00:19 (10.16a) 10 10. EW model and constraints on new physics Table 10.3: Standard Model expressions for the neutral-current parameters for ν-hadron, ν-e, and e-hadron processes. At tree level, ρ = κ = 1, λ = 0. If radiative corrections are included, C ρN bνN (hQ2 i = −12 GeV2 ) = 0.9978, κ bνN (hQ2 i = νN = 1.0086, κ 2 −35 GeV ) = 0.9965, λuL = −0.0031, λdL = −0.0025, and λdR = 2 λuR = 7.5 × 10−5 . For ν-e scattering, ρνe = 1.0132 and κνe = 0.9967 (at hQ2 i = 0.). For atomic parity violation and the b SLAC polarized electron experiment, ρ′eq = 0.9881, ρeq = 1.0011, κ′eq = 1.0027, b b κeq = 1.0300, λ1d = −2 λ1u = 3.7 × 10−5 , λ2u = −0.0121 and λ2d = 0.0026. The dominant mt dependence is given by ρ ∼ 1 + ρt , while κ b ∼ 1 (MS) or κ ∼ 1 + ρt / tan2 θW (on-shell). Quantity ǫL (u) ǫL (d) ǫR (u) ǫR (d) gVνe νe gA C1u C1d C2u C2d Standard Model Expression  − 2κ bνN sb2Z + λuL 3   1 N C ρνN − + 1 κ bνN sb2Z + λdL 2 3   2 N C ρνN − κ bνN sb2Z + λuR 3   1 N C ρνN κνN sb2Z + λdR b C ρN νN  1 2 3   ρνe − 1 + 2b κνe sb2Z 2   ρνe − 1 2   ρ′eq − 1 + 4 b κ′eq sb2Z + λ1u 2 3   1 2 ′ ρeq − κ b′eq sb2Z + λ1d 2 3   1 κeq sb2Z + λ2u ρeq − + 2b 2   ρeq 1 − 2b κeq sb2Z + λ2d 2 2 gR ≡ ǫR (u)2 + ǫR (d)2 ≈ 5 sin4 θW , 9 (10.16b) CC /σ CC is the ratio of ν and ν charged-current crossand r ≡ σνN νN sections, which can be measured directly. (In the simple parton model, ignoring hadron energy cuts, r ≈ ( 1 + ǫ)/(1 + 1 ǫ), where ǫ ∼ 0.125 3 3 is the ratio of the fraction of the nucleon’s momentum carried by antiquarks to that carried by quarks.) In practice, Eq. (10.15) must be corrected for quark mixing, quark sea effects, c-quark threshold effects, non-isoscalarity, W –Z propagator differences, the finite muon mass, QED and electroweak radiative corrections. Details of the neutrino spectra, experimental cuts, x and Q2 dependence of structure functions, and longitudinal structure functions enter only February 2, 2008 00:19 10. EW model and constraints on new physics 11 at the level of these corrections and therefore lead to very small uncertainties. The CCFR group quotes s2W = 0.2236 ± 0.0041 for (mt , MH ) = (175, 150) GeV with very little sensitivity to (mt , MH ). The NuTeV collaboration finds s2W = 0.2277 ± 0.0016 (for the same reference values) which is 3.0 σ higher than the SM prediction. The 2 = 0.3000 ± 0.0014, discrepancy is in the left-handed coupling, gL 2 which is 2.9 σ low, while gR = 0.0308 ± 0.0011 is 0.6 σ high. It is conceivable that the effect is caused by an asymmetric strange sea [70]. A preliminary analysis of dimuon data [71] in the relevant kinematic regime, however, indicates an asymmetric strange sea with the wrong sign to explain the discrepancy [72]. Another possibility is that the parton distribution functions (PDFs) violate isospin symmetry at levels much stronger than generally expected. Isospin breaking, nuclear physics, and higher order QCD effects seem unlikely explanations of the NuTeV discrepancy but need further study. The 2 extracted gL,R may also shift if analyzed using the most recent set of QED and electroweak radiative corrections [73]. The laboratory cross-section for νµ e → νµ e or ν µ e → ν µ e elastic scattering is dσνµ ,ν µ G2 me Eν = F dy 2π  νe 2 νe 2 × (gVνe ± gA ) + (gVνe ∓ gA ) (1 − y)2    νe2 y me , − gVνe2 − gA Eν (10.17) where the upper (lower) sign refers to νµ (ν µ ), and y ≡ Ee /Eν (which runs from 0 to (1 + me /2Eν )−1 ) is the ratio of the kinetic energy of the recoil electron to the incident ν or ν energy. For Eν ≫ me this yields a total cross-section   G2F me Eν 1 νe νe 2 νe νe 2 (gV ± gA ) + (gV ∓ gA ) . σ= 2π 3 (10.18) The most accurate leptonic measurements [74–77] of sin2 θW are from the ratio R ≡ σνµ e /σ ν µ e in which many of the systematic uncertainties cancel. Radiative corrections (other than mt effects) are small compared to the precision of present experiments and have negligible effect on the extracted sin2 θW . The most precise νe experiment (CHARM II) [76] determined not only sin2 θW but gV,A as well. The cross-sections for νe -e and ν e -e may be obtained from νe by g νe + 1, where the 1 is due to the Eq. (10.17) by replacing gV,A V,A charged-current contribution [77,78]. The SLAC polarized-electron experiment [79] measured the parity-violating asymmetry A= σR − σL , σR + σL February 2, 2008 00:19 (10.19) 12 10. EW model and constraints on new physics where σR,L is the cross-section for the deep-inelastic scattering of a right- or left-handed electron: eR,L N → eX. In the quark parton model A 1 − (1 − y)2 , (10.20) = a + a 1 2 Q2 1 + (1 − y)2 where Q2 > 0 is the momentum transfer and y is the fractional energy transfer from the electron to the hadrons. For the deuteron or other isoscalar targets, one has, neglecting the s-quark and antiquarks,     1 3 5 3GF 3GF 2 C1u − C1d ≈ √ − + sin θW , a1 = √ 2 4 3 5 2πα 5 2πα (10.21a)     3GF 1 1 9GF a2 = √ C2u − C2d ≈ √ sin2 θW − . (10.21b) 2 4 5 2πα 5 2πα There are now precise experiments measuring atomic parity violation [80] in cesium (at the 0.4% level) [81], thallium [82], lead [83], and bismuth [84]. The uncertainties associated with atomic wave functions are quite small for cesium [85], and have been reduced recently to about 0.4% [86]. In the past, the semiempirical value of the tensor polarizability added another source of theoretical uncertainty [87]. The ratio of the off-diagonal hyperfine amplitude to the polarizability has now been measured directly by the Boulder group [86]. Combined with the precisely known hyperfine amplitude [88] one finds excellent agreement with the earlier results, reducing the overall theory uncertainty to only 0.5% (while slightly increasing the experimental error). An earlier 2.3 σ deviation from the SM (see the year 2000 edition of this Review) is now seen at the 1 σ level, after the contributions from the Breit interaction have been reevaluated [89], and after the subsequent inclusion of other large and previously underestimated effects [90] (e.g., from QED radiative corrections), and an update of the SM calculation [91] resulted in a vanishing net effect. The theoretical uncertainties are 3% for thallium [92] but larger for the other atoms. For heavy atoms one determines the “weak charge” QW = −2 [C1u (2Z + N ) + C1d (Z + 2N )]   ≈ Z 1 − 4 sin2 θW − N . (10.22) The recent Boulder experiment in cesium also observed the parityviolating weak corrections to the nuclear electromagnetic vertex (the anapole moment [93]) . In the future it could be possible to reduce the theoretical wave function uncertainties by taking the ratios of parity violation in different isotopes [80,94]. There would still be some residual uncertainties from differences in the neutron charge radii, however [95]. The forward-backward asymmetry for e+ e− → ℓ+ ℓ− , ℓ = µ or τ , is defined as σ − σB AF B ≡ F , (10.23) σF + σB February 2, 2008 00:19 10. EW model and constraints on new physics 13 where σF (σB ) is the cross-section for ℓ− to travel forward (backward) with respect to the e− direction. AF B and R, the total cross-section relative to pure QED, are given by R = F1 , (10.24) AF B = 3F2 /4F1 , (10.25) where    e2 ℓ2 F1 = 1 − 2χ0 gVe gVℓ cos δR + χ20 gVe2 + gA gVℓ2 + gA , (10.26a) e ℓ e ℓ e ℓ F2 = −2χ0 gA gA cos δR + 4χ20 gA gA gV gV , tan δR = G χ0 = √ F h 2 2πα M Z ΓZ , MZ2 − s sMZ2 i1/2 ,
2 2 2 2 M Z − s + M Z ΓZ (10.26b) (10.27) (10.28) √ and s is the CM energy. Eq. (10.26) is valid at tree level. If the data is radiatively corrected for QED effects (as described above), then the remaining electroweak corrections can be incorporated [96,97] (in an approximation adequate for existing PEP, PETRA, and TRISTAN data, which are well below the Z-pole) by replacing χ0 by χ(s) ≡ (1 + ρt )χ0 (s)α/α(s), where α(s) is the running QED coupling, and evaluating gV in the MS scheme. Formulas for e+ e− → hadrons may be found in Ref. 98. At LEP and SLC, there were high-precision measurements of various Z-pole observables [41,99–105], as summarized in Table 10.4. These include the Z mass and total width, ΓZ , and partial widths Γ(f f ) for Z → f f where fermion f = e, µ, τ , hadrons, b, or c. It is convenient to use the variables MZ , ΓZ , Rℓ ≡ Γ(had)/Γ(ℓ+ ℓ− ), σhad ≡ 12πΓ(e+ e− )Γ(had)/MZ2 Γ2Z , Rb ≡ Γ(bb)/Γ(had), and Rc ≡ Γ(cc)/Γ(had), most of which are weakly correlated experimentally. (Γ(had) is the partial width into hadrons.) O(α3 ) QED corrections introduce a large anticorrelation (−30%) between ΓZ and σhad [41], while the anticorrelation between Rb and Rc (−14%) is smaller than previously [100]. Rℓ is insensitive to mt except for the Z → bb vertex and final state corrections and the implicit dependence through sin2 θW . Thus it is especially useful for constraining αs . The width for invisible decays [41], Γ(inv) = ΓZ − 3Γ(ℓ+ ℓ− ) − Γ(had) = 499.0 ± 1.5 MeV, can be used to determine the number of neutrino flavors much lighter than MZ /2, Nν = Γ(inv)/Γtheory (νν) = 2.983 ± 0.009 for (mt , MH ) = (177.9, 117) GeV. There were also measurements of various Z-pole asymmetries. These include the polarization or left-right asymmetry ALR ≡ σL − σR , σL + σR February 2, 2008 00:19 (10.29) 14 10. EW model and constraints on new physics where σL (σR ) is the cross-section for a left-(right-)handed incident electron. ALR has been measured precisely by the SLD collaboration at the SLC [101], and has the advantages of being extremely sensitive to sin2 θW and that systematic uncertainties largely cancel. In addition, the SLD collaboration has extracted the final-state couplings Ab , Ac [41], As [102], Aτ , and Aµ [103] from left-right forward-backward asymmetries, using f B AF LR (f ) = f f f σLF − σLB − σRF + σRB f σLF f + σLB f + σRF f + σRB = 3 A , 4 f (10.30) where, for example, σLF is the cross-section for a left-handed incident electron to produce a fermion f traveling in the forward hemisphere. Similarly, Aτ is measured at LEP [41] through the negative total τ polarization, Pτ , and Ae is extracted from the angular distribution of Pτ . An equation such as (10.30) assumes that initial state QED corrections, photon exchange, √ γ–Z interference, the tiny electroweak boxes, and corrections for s 6= MZ are removed from the data, leaving the pure electroweak asymmetries. This allows the use of effective tree-level expressions, ALR = Ae Pe , AF B = (10.31) 3 Ae + Pe , Af 4 1 + Pe Ae where f Af ≡ f 2gV gA f2 (10.32) f2 g V + gA , (10.33) and g fV = f gA =   (f ) t3L − 2qf κf sin2 θW , √ ρf √ ρf t3L . (10.33b) (f ) (10.33c) Pe is the initial e− polarization, so that the second equality in Eq. (10.30) is reproduced for Pe = 1, and the Z-pole forward(0,f ) backward asymmetries at LEP (Pe = 0) are given by AF B = 43 Ae Af (0,q) where f = e, µ, τ , b, c, s [104], and q, and where AF B refers to the hadronic charge asymmetry. Corrections for t-channel exchange (0,e) and s/t-channel interference cause AF B to be strongly anticorrelated with Re (−37%). The initial state coupling, Ae , is also determined through the left-right charge asymmetry [105] and in polarized Bhabba scattering at the SLC [103]. The electroweak radiative corrections have been absorbed into corrections ρf − 1 and κf − 1, which depend on the fermion f and on the renormalization scheme. In the on-shell scheme, the quadratic mt dependence is given by ρf ∼ 1 + ρt , κf ∼ 1 + ρt / tan2 θW , while in MS, bb ∼ 1 + 32 ρt ). In the MS scheme ρbf ∼ κ bf ∼ 1, for f 6= b (b ρb ∼ 1 − 43 ρt , κ February 2, 2008 00:19 10. EW model and constraints on new physics 15 √ b/4b s 2Z b the normalization is changed according to GF MZ2 /2 2π → α c 2Z . √ 2 (If one continues to normalize amplitudes by GF MZ /2 2π, as in the 1996 edition of this Review, then ρbf contains an additional factor of ρb.) In practice, additional bosonic and fermionic loops, vertex corrections, leading higher order contributions, etc., must be included. bℓ = 1.0013, For example, in the MS scheme one has ρbℓ = 0.9981, κ ρbb = 0.9861, and b κb = 1.0071. It is convenient to define an effective f f angle s2f ≡ sin2 θW f ≡ κ bf sb 2Z = κf s2W , in terms of which g V and g A √ are given by ρf times their tree-level formulae. Because g ℓV is very (0,ℓ) (0,b) (0,c) small, not only A0LR = Ae , AF B , and Pτ , but also AF B , AF B , (0,s) AF B , and the hadronic asymmetries are mainly sensitive to s2ℓ . One finds that b κf (f 6= b) is almost independent of (mt , MH ), so that one can write s2ℓ ∼ sb 2Z + 0.00029 . (10.34) Thus, the asymmetries determine values of s2ℓ and sb 2Z almost independent of mt , while the κ’s for the other schemes are mt dependent. LEP 2 [41] has run at several energies above the Z-pole up to ∼ 209 GeV. Measurements have been made of a number of observables, including the cross-sections for e+ e− → f f¯ for f = q, µ− , τ − ; the differential cross-sections and AF B for µ and τ ; R and AF B for b and c; W branching ratios; and W W , W W γ, ZZ, single W , and single Z cross-sections. They are in agreement with the SM predictions, with the exceptions of the total hadronic cross-section (1.7 σ high), Rb (2.1 σ low), and AF B (b) (1.6 σ low). Also, the SM Higgs has been excluded below 114.4 GeV [106]. The Z-boson properties are extracted assuming the SM expressions for the γ–Z interference terms. These have also been tested experimentally by performing more general fits [107] to the LEP 1 and LEP 2 data. Assuming family universality this approach introduces three additional parameters relative to the standard fit [41], describing the γ–Z interference contribution to the total hadronic and leptonic tot and j tot , and to the leptonic forward-backward cross-sections, jhad ℓ asymmetry, jℓfb . For example, tot jhad ∼ gVℓ gVhad = 0.277 ± 0.065 , (10.35) which is in good agreement with the SM expectation [41] of 0.220+0.003 −0.014 . Similarly, LEP data up to CM energies of 206 GeV were used to constrain the γ–Z interference terms for the heavy quarks. The results for jbtot , jbfb , jctot , and jcfb were found in perfect agreement with the SM. These are valuable tests of the SM; but it should be cautioned that new physics is not expected to be described by this set of parameters, since (i) they do not account for extra interactions beyond the standard weak neutral-current, and (ii) the photonic amplitude remains fixed to its SM value. Strong constraints on anomalous triple and quartic gauge couplings have been obtained at LEP 2 and at the Tevatron, as are described in the Particle Listings. February 2, 2008 00:19 10. EW model and constraints on new physics 16 The left-right asymmetry in polarized Møller scattering e+ e− → is being measured in the SLAC E158 experiment. A precision of better than ±0.001 in sin2 θW at Q2 ∼ 0.03 GeV2 is anticipated. The result of the first of three runs yields sb 2Z = 0.2279 ± 0.0032 [108]. In a similar experiment and at about the same Q2 , Qweak at Jefferson Lab [109] will be able to measure sin2 θW in polarized ep scattering with a relative precision of 0.3%. These experiments will provide the most precise determinations of the weak mixing angle off the Z peak and will be sensitive to various types of physics beyond the SM. e+ e− The Belle [110], CLEO [111], and BaBar [112] collaborations reported precise measurements of the flavor changing transition b → sγ. The signal efficiencies (including the extrapolation to the full photon spectrum) depend on the bottom pole mass, mb . We adjusted the Belle and BaBar results to agree with the mb value used by CLEO. In the case of CLEO, a 3.8% component from the model error of the signal efficiency is moved from the systematic error to the model error. The results for the branching fractions are then given by, B [Belle] = 3.05 × 10−4 [1 ± 0.158 ± 0.124 ± 0.202 ± 0] , (10.36a) B [CLEO] = 3.21 × 10−4 [1 ± 0.134 ± 0.076 ± 0.059 ± 0.016] , (10.36b) B [BaBar] = 3.86 × 10−4 [1 ± 0.090 ± 0.093 ± 0.074 ± 0.016] , (10.36c) where the first two errors are the statistical and systematic uncertainties (taken uncorrelated). The third error (taken 100% correlated) accounts for the extrapolation from the finite photon energy cutoff (2.25 GeV, 2.0 GeV, and 2.1 GeV, respectively) to the full theoretical branching ratio [113]. The last error is from the correction for the b → dγ component which is common to CLEO and BaBar. It is advantageous [114] to normalize the result with respect to the semi-leptonic branching fraction, B(b → Xeν) = 0.1064 ± 0.0023, yielding, R= B (b → sγ) = (3.39 ± 0.43 ± 0.37) × 10−3 . B (b → Xeν) (10.37) In the fits we use the variable ln R = −5.69 ± 0.17 to assure an approximately Gaussian error [115]. We added an 11% theory uncertainty (excluding parametric errors such as from αs ) in the SM prediction which is based on the next-to-leading order calculations of Refs. 114,116. The present world average of the muon anomalous magnetic moment, aexp µ = gµ − 2 = (1165920.37 ± 0.78) × 10−9 , 2 (10.38) is dominated by the 1999 and 2000 data runs of the E821 collaboration at BNL [117]. The final 2001 data run is currently being analyzed. The February 2, 2008 00:19 10. EW model and constraints on new physics 17 QED contribution has been calculated to four loops (fully analytically to three loops), and the leading logarithms are included to five loops [118]. The estimated SM electroweak contribution [119–121], aEW = (1.52 ± 0.03) × 10−9 , which includes leading two-loop [120] µ and three-loop [121] corrections, is at the level of the current uncertainty. The limiting factor in the interpretation of the result is the uncertainty from the two-loop hadronic contribution [20], −9 + − ahad µ = (69.63 ± 0.72) × 10 , which has been obtained using e e → hadrons cross-section data. The latter are dominated by the recently reanalyzed CMD 2 data [19]. This value suggests a 1.9 σ discrepancy between Eq. (10.38) and the SM prediction. In an alternative analysis, the authors of Ref. 20 use τ decay data and isospin symmetry (CVC) to obtain instead ahad = (71.10 ± 0.58) × 10−9 . This result implies µ no conflict (0.7 σ) with Eq. (10.38). Thus, there is also a discrepancy between the 2π spectral functions obtained from the two methods. For example, if one uses the e+ e− data and CVC to predict the branching ratio for τ − → ντ π − π 0 decays one obtains 24.52 ± 0.32% [20] while the average of the measured branching ratios by DELPHI [122], ALEPH, CLEO, L3, and OPAL [20] yields 25.43 ± 0.09%, which is 2.8 σ higher. It is important to understand the origin of this difference and to obtain additional experimental information (e.g., from the radiative return method [37]) . Fortunately, this problem is less pronounced as far as ahad is concerned: due to the suppression µ at large s (from where the conflict originates) the difference is only 1.7 σ (or 1.9 σ if one adds the 4 π channel which by itself is consistent between the two methods). Note also that a part of this difference is due to the older e+ e− data [20], and the direct conflict between τ decay data and CMD 2 is less significant. Isospin violating corrections have been estimated in Ref. 123 and found to be under control. The largest effect is due to higher-order electroweak corrections [39] but introduces a negligible uncertainty [124]. In the following we view the 1.7 σ difference as a fluctuation and average the results. An additional uncertainty is induced by the hadronic three-loop light-by-light scattering contribution [125], = (+0.83 ± 0.19) × 10−9 , which was estimated within a form aLBLS µ factor approach. The sign of this effect is opposite to the one quoted in the 2002 edition of this Review, and has subsequently been confirmed by two other groups [126]. hadronic effects at three-loop order i h
Other 3 α had = (−1.00 ± 0.06) × 10−9 . Correlations contribute [127], aµ π with the two-loop hadronic contribution and with ∆α(MZ ) (see Sec. 10.2) were considered in Ref. 128, which also contains analytic results for the perturbative QCD contribution. The SM prediction is atheory = (1165918.83 ± 0.49) × 10−9 , µ (10.39) where the error is from the hadronic uncertainties excluding parametric ones such as from αs and the heavy quark masses. We estimate its correlation with ∆α(MZ ) as 21%. The small overall discrepancy between the experimental and theoretical values could be due to fluctuations or underestimates of the theoretical uncertainties. On the February 2, 2008 00:19 18 10. EW model and constraints on new physics other hand, gµ − 2 is also affected by many types of new physics, such as supersymmetric models with large tan β and moderately light superparticle masses [129]. Thus, the deviation could also arise from physics beyond the SM. Note added: After completion of this Section and the fits described here, the E821 collaboration announced its measurement on the anomalous magnetic moment of the negatively charged muon based exp on data taken in 2001 [130]. The result, aµ = (1165921.4 ± 0.8 ± 0.3) × 10−9 , is consistent with the results on positive muons and appears to confirm the deviation. There also appeared two new + − evaluations [131,132] of ahad µ . They are based on e e data only and are generally in good agreement with each other and other e+ e− based analyzes. τ decay data are not used; it is argued [131] that CVC breaking effects (e.g., through a relatively large mass difference between the ρ± and ρ0 vector mesons) may be larger than expected. This may also be relevant in the context of the NuTeV discrepancy discussed above [131]. 10.4. W and Z decays The partial decay width for gauge bosons to decay into massless fermions f1 f 2 is
GF M 3 Γ W + → e+ νe = √ W ≈ 226.56 ± 0.24 MeV , 6 2π (10.47a) 3
CGF MW √ Γ W + → ui dj = |Vij |2 ≈ (707.1 ± 0.7) |Vij |2 MeV , 6 2π (10.47b) h i 3 CGF MZ i2 i2 √ Γ (Z → ψi ψ i ) = gV + gA (10.47c) 6 2π    300.4 ± 0.2 MeV (uu) , 167.29 ± 0.07 MeV (νν),
≈ 383.2 ± 0.2 MeV (dd) , 84.03 ± 0.04 MeV e+ e− ,   375.8 ∓ 0.1 MeV (bb) .  For leptons C = 1, while for quarks C = 3 1 + αs (MV )/π +  1.409α2s /π 2 − 12.77α3s /π 3 , where the 3 is due to color and the factor in parentheses represents the universal part of the QCD corrections [133] for massless quarks [134]. The Z → f f widths contain a number of additional corrections: universal (non-singlet) top quark mass contributions [135]; fermion mass effects and further QCD corrections proportional to m b 2q (MZ2 ) [136] which are different for vector and axial-vector partial widths; and singlet contributions starting from two-loop order which are large, strongly top quark mass dependent, family universal, and flavor non-universal [137]. All QCD effects are known and included up to three-loop order. The QED factor 1 + 3αqf2 /4π, as well as two-loop order ααs and α2 self-energy corrections [138] are also included. Working in the February 2, 2008 00:19 10. EW model and constraints on new physics 19 3 , on-shell scheme, i.e., expressing the widths in terms of GF MW,Z incorporates the largest radiative corrections from the running QED coupling [44,139]. Electroweak corrections to the Z widths are then i2 incorporated by replacing g i2 V,A by g V,A . Hence, in the on-shell scheme the Z widths are proportional to ρi ∼ 1 + ρt . The MS normalization accounts also for the leading electroweak corrections [48]. There is additional (negative) quadratic mt dependence in the Z → bb vertex corrections [140] which causes Γ(bb) to decrease with mt . The dominant effect is to multiply Γ(bb) by the vertex correction 1 + δρbb , m2 where δρbb ∼ 10−2 (− 1 t2 + 1 ). In practice, the corrections are 2M 5 Z included in ρb and κb , as discussed before. For 3 fermion families the total widths are predicted to be ΓZ ≈ 2.4968 ± 0.0011 GeV , ΓW ≈ 2.0936 ± 0.0022 GeV . (10.48) (10.49) We have assumed αs (MZ ) = 0.1200. An uncertainty in αs of ±0.0018 introduces an additional uncertainty of 0.05% in the hadronic widths, corresponding to ±0.9 MeV in ΓZ . These predictions are to be compared with the experimental results ΓZ = 2.4952 ± 0.0023 GeV [41] and ΓW = 2.124 ± 0.041 GeV (see the Particle Listings for more details). February 2, 2008 00:19 20 10. EW model and constraints on new physics Table 10.4: Principal Z-pole and other observables, compared with the SM predictions for the global best fit values MZ = 91.1874±0.0021 GeV, MH = 113+56 −40 GeV, mt = 176.9±4.0 GeV, αs (MZ ) = 0.1213 ± 0.0018, and α b(MZ )−1 = 127.906 ± 0.019. The LEP averages of the ALEPH, DELPHI, L3, and OPAL results include common systematic errors and correlations [41]. The heavy flavor results of LEP and SLD are based on common (0,q) inputs and correlated, as well [100]. s2ℓ (AF B ) is the effective angle extracted from the hadronic charge asymmetry, which has (0,b) some correlation with AF B which is currently neglected. The values of Γ(ℓ+ ℓ− ), Γ(had), and Γ(inv) are not independent of ΓZ , the Rℓ , and σhad . The mt values are from the lepton plus jets channel of the CDF [6] and DØ [9] run I data, respectively. Results from the other channels and all correlations are also included. The first MW value is from UA2, CDF, and DØ [141], while the second one is from LEP 2 [41]. The first MW and MZ are correlated, but the effect is negligible due to the tiny MZ error. The three values of Ae are (i) from ALR for hadronic final states [101]; (ii) from ALR for leptonic final states and from polarized Bhabba scattering [103]; and (iii) from the angular distribution of the τ polarization. The two Aτ values 2 and are from SLD and the total τ polarization, respectively. gL 2 gR are from NuTeV [69] and have a very small (−1.7%) residual anticorrelation. The older deep-inelastic scattering (DIS) results from CDHS [63], CHARM [64], and CCFR [65] are included, as νe are well, but not shown in the Table. The world averages for gV,A νe dominated by the CHARM II [76] results, gV = −0.035 ± 0.017 νe = −0.503 ± 0.017. The errors in Q , DIS, b → sγ, and and gA W gµ − 2 are the total (experimental plus theoretical) uncertainties. The ττ value is the τ lifetime world average computed by combining the direct measurements with values derived from the leptonic branching ratios [5]; the theory uncertainty is included in the SM prediction. In all other SM predictions, the uncertainty is from MZ , MH , mt , mb , mc , α b(MZ ), and αs , and their correlations have been accounted for. The SM errors in ΓZ , Γ(had), Rℓ , and σhad are largely dominated by the uncertainty in αs . Quantity Value Standard Model Pull mt [GeV] 176.1 ± 7.4 176.9 ± 4.0 −0.1 MW [GeV] MZ [GeV] ΓZ [GeV] 180.1 ± 5.4 80.454 ± 0.059 80.390 ± 0.018 2.4952 ± 0.0023 2.4972 ± 0.0012 80.412 ± 0.042 91.1876 ± 0.0021 91.1874 ± 0.0021 February 2, 2008 00:19 0.6 1.1 0.5 0.1 −0.9 10. EW model and constraints on new physics 21 Table 10.4: (continued) Quantity Γ(had) [GeV] Γ(inv) [MeV] Γ(ℓ+ ℓ− ) [MeV] σhad [nb] Re Rµ Rτ Rb Rc (0,e) AF B (0,µ) AF B (0,τ ) AF B (0,b) AF B (0,c) AF B (0,s) AF B (0,q) s̄2ℓ (AF B ) Ae Aµ Aτ Ab Ac As 2 gL 2 gR gVνe νe gA QW (Cs) QW (Tl) Γ(b→sγ) Γ(b→Xeν) 1 (g − 2 − α ) 2 µ π ττ [fs] Value Standard Model Pull 1.7444 ± 0.0020 499.0 ± 1.5 1.7435 ± 0.0011 501.81 ± 0.13 — — 20.804 ± 0.050 20.785 ± 0.033 20.750 ± 0.012 20.751 ± 0.012 83.984 ± 0.086 41.541 ± 0.037 20.764 ± 0.045 0.21638 ± 0.00066 0.1720 ± 0.0030 0.0145 ± 0.0025 0.0169 ± 0.0013 0.0188 ± 0.0017 0.0997 ± 0.0016 0.0706 ± 0.0035 0.0976 ± 0.0114 0.2324 ± 0.0012 0.15138 ± 0.00216 0.1544 ± 0.0060 0.1498 ± 0.0049 84.024 ± 0.025 41.472 ± 0.009 — 1.9 20.790 ± 0.018 −0.7 0.01626 ± 0.00025 −0.7 0.21564 ± 0.00014 0.17233 ± 0.00005 0.1032 ± 0.0008 0.0738 ± 0.0006 0.1033 ± 0.0008 0.23149 ± 0.00015 0.1472 ± 0.0011 0.9347 ± 0.0001 0.30005 ± 0.00137 0.03076 ± 0.00110 0.30397 ± 0.00023 0.03007 ± 0.00003 −0.507 ± 0.014 −72.69 ± 0.48 −0.5065 ± 0.0001 −73.19 ± 0.03 −116.6 ± 3.7 3.39+0.62 −0.54 × 10−3 4510.64 ± 0.92 290.92 ± 0.55 February 2, 2008 0.5 −2.2 −0.9 −0.5 0.8 1.9 1.2 0.5 −0.4 −0.8 0.1439 ± 0.0043 0.925 ± 0.020 −0.040 ± 0.015 1.1 −0.1 1.5 0.142 ± 0.015 0.136 ± 0.015 0.670 ± 0.026 0.895 ± 0.091 1.1 1.0 0.6678 ± 0.0005 0.9357 ± 0.0001 −0.0397 ± 0.0003 −116.81 ± 0.04 (3.23 ± 0.09) × 10−3 4509.13 ± 0.10 291.83 ± 1.81 00:19 −0.8 −0.5 0.1 −0.4 −2.9 0.6 −0.1 0.0 1.0 0.1 0.3 1.6 −0.4 22 10. EW model and constraints on new physics 10.5. Experimental results The values of the principal Z-pole observables are listed in Table 10.4, along with the SM predictions for MZ = 91.1874 ± 0.0021 GeV, MH = 113+56 −40 GeV, mt = 176.9 ± 4.0 GeV, αs (MZ ) = (5) 0.1213 ± 0.0018, and α b(MZ )−1 = 127.906 ± 0.019 (∆αhad ≈ 0.02801 ± 0.00015). The values and predictions of MW [41,141]; mt [6,9]; the QW for cesium [81] and thallium [82]; deep inelastic [69] and νµ -e scattering [74–76]; the b → sγ observable [110–112]; the muon anomalous magnetic moment [117]; and the τ lifetime are also listed. The values of MW and mt differ from those in the Particle Listings because they include recent preliminary results. The agreement is 2 from NuTeV is currently showing a large (2.9 σ) excellent. Only gL deviation. In addition, the hadronic peak cross-section, σhad , and the A0LR from hadronic final states differ by 1.9 σ. On the other hand, (0,b) AF B (2.2 σ) and gµ − 2 (1.6 σ, see Sec. 10.3) both moved closer to the SM predictions by about one standard deviation compared to the 2002 edition of this Review, while MW (LEP 2) has moved closer by 0.8 σ. Observables like Rb = Γ(bb)/Γ(had), Rc = Γ(cc)/Γ(had), and the combined value for MW which showed significant deviations in the past, are now in reasonable agreement. In particular, Rb whose measured value deviated as much as 3.7 σ from the SM prediction is now only 1.1 σ (0.34%) high. (0,b) Ab can be extracted from AF B when Ae = 0.1501 ± 0.0016 is taken from a fit to leptonic asymmetries (using lepton universality). The result, Ab = 0.886 ± 0.017, is 2.9 σ below the SM prediction† , and also B 1.5 σ below Ab = 0.925 ± 0.020 obtained from AF LR (b) at SLD. Thus, (0,b) it appears that at least some of the problem in AF B is experimental. (0,b) Note, however, that the uncertainty in AF B is strongly statistics dominated. The combined value, Ab = 0.902 ± 0.013 deviates by 2.5 σ. It would be extremely difficult to account for this 3.5% deviation by new physics radiative corrections since an order of 20% correction to κ bb would be necessary to account for the central value of Ab . If this deviation is due to new physics, it is most likely of tree-level type affecting preferentially the third generation. Examples include the decay of a scalar neutrino resonance [142], mixing of the b quark with heavy exotics [143], and a heavy Z ′ with family-nonuniversal couplings [144]. It is difficult, however, to simultaneously account for Rb , which has been measured on the Z peak and off-peak [145] at LEP 1. An average of Rb measurements at LEP 2 at energies between 133 and 207 GeV is 2.1 σ below the SM prediction, while (b) AF B (LEP 2) is 1.6 σ low. † Alternatively, one can use Aℓ = 0.1481 ± 0.0027, which is from LEP alone and in excellent agreement with the SM, and obtain Ab = 0.898 ± 0.022 which is 1.7 σ low. This illustrates that some of the discrepancy is related to the one in ALR . February 2, 2008 00:19 10. EW model and constraints on new physics 23 The left-right asymmetry, A0LR = 0.15138 ± 0.00216 [101], based on all hadronic data from 1992–1998 differs 1.9 σ from the SM expectation of 0.1472 ± 0.0011. The combined value of Aℓ = 0.1513 ± 0.0021 from SLD (using lepton-family universality and including correlations) is also 1.9 σ above the SM prediction; but there is now experimental agreement between this SLD value and the LEP (0,ℓ) value, Aℓ = 0.1481 ± 0.0027, obtained from a fit to AF B , Ae (Pτ ), and Aτ (Pτ ), again assuming universality. Despite these discrepancies the goodness of the fit to all data is excellent with a χ2 /d.o.f. = 45.5/45. The probability of a larger χ2 is 45%. The observables in Table 10.4, as well as some other less precise observables, are used in the global fits described below. The correlations on the LEP lineshape and τ polarization, the LEP/SLD heavy flavor observables, the SLD lepton asymmetries, the deep inelastic and ν-e scattering observables, and the mt measurements, are (5) included. The theoretical correlations between ∆αhad and gµ − 2, and between the charm and bottom quark masses, are also accounted for. Table 10.5: Values of sb 2Z , s2W , αs , and MH [in GeV] for various (combinations of) observables. Unless indicated otherwise, the top quark mass, mt = 177.9 ± 4.4 GeV, is used as an additional constraint in the fits. The (†) symbol indicates a fixed parameter. Data All data sb 2Z s2W 0.23120(15) 0.2228(4) αs (MZ ) MH 0.1213(18) 113+56 −40 All indirect (no mt ) 0.23116(17) 0.2229(4) 0.1213(18) 79+95 −38 Z pole (no mt ) 0.23118(17) 0.2231(6) 0.1197(28) 79+94 −38 LEP 1 (no mt ) 0.23148(20) 0.2237(7) 0.1210(29) 140+192 −74 SLD + MZ 0.23067(28) 0.2217(6) 0.1213 (†) 43+38 −23 AF B + MZ 0.23185(28) 0.2244(8) 0.1213 (†) 408+317 −179 MW + MZ 0.23089(37) 0.2221(8) 0.1213 (†) 67+77 −45 MZ 0.23117(15) 0.2227(5) 0.1213 (†) 117 (†) DIS (isoscalar) 0.2359(16) 0.2274(16) 0.1213 (†) 117 (†) QW (APV) 0.2292(19) 0.2207(19) 0.1213 (†) 117 (†) polarized Møller 0.2292(42) 0.2207(43) 0.1213 (†) 117 (†) elastic νµ (νµ )e 0.2305(77) 0.2220(77) 0.1213 (†) 117 (†) SLAC eD 0.222(18) 0.213(19) 0.1213 (†) 117 (†) elastic νµ (νµ )p 0.211(33) 0.203(33) 0.1213 (†) 117 (†) (b,c) February 2, 2008 00:19 24 10. EW model and constraints on new physics The data allow a simultaneous determination of MH , mt , sin2 θW , (5) and the strong coupling αs (MZ ). (m b c, m b b , and ∆αhad are also allowed to float in the fits, subject to the theoretical constraints [5,18] described in Sec. 10.1–Sec. 10.2. These are correlated with αs .) αs is determined mainly from Rℓ , ΓZ , σhad , and ττ and is only weakly correlated with the other variables (except for a 10% correlation with m b c ). The global fit to all data, including the CDF/DØ average, mt = 177.9 ± 4.4 GeV, yields MH = 113+56 −40 GeV , mt = 176.9 ± 4.0 GeV , sb 2Z = 0.23120 ± 0.00015 , αs (MZ ) = 0.1213 ± 0.0018 . (10.50) In the on-shell scheme one has s2W = 0.22280±0.00035, the larger error due to the stronger sensitivity to mt , while the corresponding effective angle is related by Eq. (10.34), i.e., s2ℓ = 0.23149 ± 0.00015. The mt pole mass corresponds to m b t (m b t ) = 166.8 ± 3.8 GeV. In all fits, the errors include full statistical, systematic, and theoretical uncertainties. The sb 2Z (s2ℓ ) error reflects the error on s2f = 0.23150 ± 0.00016 from a fit to the Z-pole asymmetries. The weak mixing angle can be determined from Z-pole observables, MW , and from a variety of neutral-current processes spanning a very wide Q2 range. The results (for the older low-energy neutral-current data see [42,43]) shown in Table 10.5 are in reasonable agreement with each other, indicating the quantitative success of the SM. The largest discrepancy is the value sb 2Z = 0.2358 ± 0.0016 from DIS which is 2.9 σ above the value 0.23120 ± 0.00015 from the global fit to all data. Similarly, sb 2Z = 0.23185 ± 0.00028 from the forward-backward asymmetries into bottom and charm quarks, and sb 2Z = 0.23067 ± 0.00028 from the SLD asymmetries (both when combined with MZ ) are 2.3 σ high and 1.9 σ low, respectively. The extracted Z-pole value of αs (MZ ) is based on a formula with negligible theoretical uncertainty (±0.0005 in αs (MZ )) if one assumes the exact validity of the SM. One should keep in mind, however, that this value, αs = 0.1197 ± 0.0028, is very sensitive to such types of new physics as non-universal vertex corrections. In contrast, the value derived from τ decays, αs (MZ ) = 0.1221+0.0026 −0.0023 [5], is theory dominated but less sensitive to new physics. The former is mainly due to the larger value of αs (mτ ), but just as the hadronic Z-width the τ lifetime is fully inclusive and can be computed reliably within the operator product expansion. The two values are in excellent agreement with each other. They are also in perfect agreement with other recent values, such as 0.1202 ± 0.0049 from jet-event shapes at LEP [146], and 0.121 ± 0.003 [147] from the most recent lattice calculation of the Υ spectrum. For more details and other determinations, see our Section 9 on “Quantum Chromodynamics” in this Review. The data indicate a preference for a small Higgs mass. There is a strong correlation between the quadratic mt and logarithmic MH February 2, 2008 00:19 10. EW model and constraints on new physics 25 terms in ρb in all of the indirect data except for the Z → bb vertex. Therefore, observables (other than Rb ) which favor mt values higher than the Tevatron range favor lower values of MH . This effect is enhanced by Rb , which has little direct MH dependence but favors the lower end of the Tevatron mt range. MW has additional MH dependence through ∆b r W which is not coupled to m2t effects. The strongest individual pulls toward smaller MH are from MW and A0LR , (0b) while AF B and the NuTeV results favor high values. The difference in χ2 for the global fit is ∆χ2 = χ2 (MH = 1000 GeV) − χ2min = 34.6. Hence, the data favor a small value of MH , as in supersymmetric extensions of the SM. The central value of the global fit result, MH = 113+56 −40 GeV, is slightly below the direct lower bound, MH ≥ 114.4 GeV (95% CL) [106]. The 90% central confidence range from all precision data is 53 GeV ≤ MH ≤ 213 GeV . Including the results of the direct searches as an extra contribution to the likelihood function drives the 95% upper limit to MH ≤ 241 GeV. As two further refinements, we account for (i) theoretical uncertainties from uncalculated higher order contributions by allowing the T parameter (see next subsection) subject to the constraint T = 0 ± 0.02, (ii) the MH dependence of the correlation matrix which gives slightly more weight to lower Higgs masses [148]. The resulting limits at 95 (90, 99)% CL are MH ≤ 246 (217, 311) GeV , respectively. The extraction of MH from the precision data depends strongly on the value used for α(MZ ). Upper limits, however, are more robust due to two compensating effects: the older results indicated more QED running and were less precise, yielding MH distributions which were broader with centers shifted to smaller values. The hadronic contribution to α(MZ ) is correlated with gµ − 2 (see Sec. 10.3). The measurement of the latter is higher than the SM prediction, and its inclusion in the fit favors a larger α(MZ ) and a lower MH (by 4 GeV). One can also carry out a fit to the indirect data alone, i.e., without including the constraint, mt = 177.9 ± 4.4 GeV, obtained by CDF and DØ. (The indirect prediction is for the MS mass, m b t (m b t ) = 162.5+9.2 −6.9 GeV, which is in the end converted to the pole mass). One obtains mt = 172.4+9.8 −7.3 GeV, with little change in the sin2 θW and αs values, in remarkable agreement with the direct CDF/DØ average. The relations between MH and mt for various observables are shown in Fig. 10.1. Using α(MZ ) and sb 2Z as inputs, one can predict αs (MZ ) assuming grand unification. One predicts [149] αs (MZ ) = 0.130 ± 0.001 ± 0.01 for the simplest theories based on the minimal supersymmetric extension of the SM, where the first (second) uncertainty is from the February 2, 2008 00:19 26 10. EW model and constraints on new physics Figure 10.1: One-standard-deviation (39.35%) uncertainties in MH as a function of mt for various inputs, and the 90% CL region (∆χ2 = 4.605) allowed by all data. αs (MZ ) = 0.120 is assumed except for the fits including the Z-lineshape data. The 95% direct lower limit from LEP 2 is also shown. inputs (thresholds). This is slightly larger, but consistent with the experimental αs (MZ ) = 0.1213 ± 0.0018 from the Z lineshape and the τ lifetime, as well as with other determinations. Non-supersymmetric unified theories predict the low value αs (MZ ) = 0.073 ± 0.001 ± 0.001. See also the note on “Low-Energy Supersymmetry” in the Particle Listings. One can also determine the radiative correction parameters ∆r: from the global fit one obtains ∆r = 0.0347 ± 0.0011 and ∆b rW = 0.06981 ± 0.00032. MW measurements [41,141] (when combined with MZ ) are equivalent to measurements of ∆r = 0.0326 ± 0.0021, which is 1.2 σ below the result from all indirect data, ∆r = 0.0355 ± 0.0013. Fig. 10.2 shows the 1 σ contours in the MW − mt plane from the direct and indirect determinations, as well as the combined 90% CL region. The indirect determination uses MZ from LEP 1 as input, which is defined assuming an s-dependent decay width. MW then corresponds to the s-dependent width definition, as well, and can be directly compared with the results from the Tevatron and LEP 2 which have been obtained using the same definition. The difference to a constant width definition is formally only of O(α2 ), but is strongly enhanced since the decay channels add up coherently. It is about 34 MeV for MZ and 27 MeV for MW . The residual difference between working consistently with one or the other definition is about 3 MeV, i.e., of typical size for non-enhanced O(α2 ) corrections [54,55]. Figure 10.2: One-standard-deviation (39.35%) region in MW as a function of mt for the direct and indirect data, and the 90% CL region (∆χ2 = 4.605) allowed by all data. The SM prediction as a function of MH is also indicated. The widths of the MH bands reflect the theoretical uncertainty from α(MZ ). Most of the parameters relevant to ν-hadron, ν-e, e-hadron, and e+ e− processes are determined uniquely and precisely from the data in “model-independent” fits (i.e., fits which allow for an arbitrary electroweak gauge theory). The values for the parameters defined in Eqs. (10.11)–(10.13) are given in Table 10.6 along with the predictions 2 and of the SM. The agreement is reasonable, except for the values of gL ǫL (u, d), which reflect the discrepancy in the recent NuTeV results. February 2, 2008 00:19 10. EW model and constraints on new physics 27 Table 10.6: Values of the model-independent neutral-current parameters, compared with the SM predictions for the global best fit values MZ = 91.1874 ± 0.0021 GeV, MH = 113+56 −40 GeV, mt = 176.9 ± 4.0 GeV, αs (MZ ) = 0.1213 ± 0.0018, and νe solution, α b(MZ )−1 = 127.906 ± 0.019. There is a second gV,A νe νe given approximately by gV ↔ gA , which is eliminated by e+ e− data under the assumption that the neutral current is dominated by the exchange of a single Z. The ǫL , as well as the ǫR , are strongly correlated and non-Gaussian, so that for implementations we recommend the parametrization using gi and θi = tan−1 [ǫi (u)/ǫi (d)], i = L or R. θR is only weakly correlated with the gi , while the correlation coefficient between θR and θL is 0.27. Quantity ǫL (u) ǫL (d) ǫR (u) ǫR (d) 2 gL 2 gR θL θR gVνe Experimental Value 0.326 ±0.013 −0.441 ±0.010 −0.175 +0.013 −0.004 −0.022 +0.072 −0.047 SM Correlation 0.3460(2) −0.4292(1) −0.1551(1) 0.0776 0.3005±0.0012 0.3040(2) 0.0311±0.0010 2.51 ±0.033 0.0301 2.4631(1) 4.59 +0.41 −0.28 −0.040 ±0.015 −0.507 ±0.014 −0.0397(3) C1u + C1d C1u − C1d 0.148 ±0.004 −0.597 ±0.061 0.1529(1) −0.5299(4) 0.62 ±0.80 −0.07 ±0.12 −0.11 −0.21 −0.02 −0.01 −0.03 0.26 5.1765 νe gA C2u + C2d C2u − C2d nonGaussian −0.05 −0.5065(1) −0.0095 −0.0623(6) 0.95 −0.55 −0.57 −0.26 −0.27 −0.38 (The ν-hadron results without the new NuTeV data can be found in the previous editions of this Review.). The off Z-pole e+ e− results are difficult to present in a model-independent way because Z-propagator effects are non-negligible at TRISTAN, PETRA, PEP, and LEP 2 energies. However, assuming e-µ-τ universality, the low-energy lepton e )2 = 0.99 ± 0.05, in good agreement with asymmetries imply [98] 4(gA the SM prediction ≃ 1. The results presented here are generally in reasonable agreement with the ones obtained by the LEP Electroweak Working Group [41]. We obtain higher best fit values for αs and a higher and slightly more precise MH . We trace most of the differences to be due to (i) the February 2, 2008 00:19 28 10. EW model and constraints on new physics inclusion of recent higher order radiative corrections, in particular, the leading O(α4s ) contribution to hadronic Z decays [150]; (ii) a different evaluation of α(MZ ) [18]; (iii) slightly different data sets (such as the recent DØ mt value); and (iv) scheme dependences. Taking into account these differences, the agreement is excellent. 10.6. Constraints on new physics The Z-pole, W mass, and neutral-current data can be used to search for and set limits on deviations from the SM. In particular, the combination of these indirect data with the direct CDF and DØ average for mt allows one to set stringent limits on new physics. We will mainly discuss the effects of exotic particles (with heavy masses Mnew ≫ MZ in an expansion in MZ /Mnew ) on the gauge boson self-energies. (Brief remarks are made on new physics which is not of this type.) Most of the effects on precision measurements can be described by three gauge self-energy parameters S, T , and U . We will define these, as well as related parameters, such as ρ0 , ǫi , and b ǫi , to arise from new physics only. I.e., they are equal to zero (ρ0 = 1) exactly in the SM, and do not include any contributions from mt or MH , which are treated separately. Our treatment differs from most of the original papers. Many extensions of the SM are described by the ρ0 parameter,   2 ρ0 ≡ M W / MZ2 b c 2Z ρb , (10.51) which describes new sources of SU(2) breaking that cannot be accounted for by the SM Higgs doublet or mt effects. In the presence of ρ0 6= 1, Eq. (10.51) generalizes Eq. (10.8b) while Eq. (10.8a) remains unchanged. Provided that the new physics which yields ρ0 6= 1 is a small perturbation which does not significantly affect the radiative corrections, ρ0 can be regarded as a phenomenological parameter which multiplies GF in Eqs. (10.11)– (10.13), (10.28), and ΓZ in Eq. (10.47). There is enough data to determine ρ0 , MH , mt , and αs , simultaneously. From the global fit, ρ0 = 0.9998+0.0008 −0.0005 , 114.4 GeV < MH < 193 GeV , mt = 178.0 ± 4.1 GeV , αs (MZ ) = 0.1214 ± 0.0018 , (10.52) (10.53) (10.54) (10.55) where the lower limit on MH is the direct search bound. (If the direct +0.0010 limit is ignored one obtains MH = 66+86 −30 GeV and ρ0 = 0.9993−0.0008 .) The error bar in Eq. (10.52) is highly asymmetric: at the 2 σ level one has ρ0 = 0.9998+0.0025 −0.0010 and MH < 664 GeV. Clearly, in the presence of ρ0 upper limits on MH become much weaker. The result in Eq. (10.52) is in remarkable agreement with the SM expectation, ρ0 = 1. It can be used to constrain higher-dimensional Higgs representations to have vacuum expectation values of less than a few percent of those of the February 2, 2008 00:19 10. EW model and constraints on new physics 29 doublets. Indeed, the relation between MW and MZ is modified if there are Higgs multiplets with weak isospin > 1/2 with significant vacuum expectation values. In order to calculate to higher orders in such theories one must define a set of four fundamental renormalized parameters which one may conveniently choose to be α, GF , MZ , and MW , since MW and MZ are directly measurable. Then sb 2Z and ρ0 can be considered dependent parameters. Eq. (10.52) can also be used to constrain other types of new physics. For example, non-degenerate multiplets of heavy fermions or scalars break the vector part of weak SU(2) and lead to a decrease in
the value of MZ /MW . A non-degenerate SU(2) doublet ff1 yields a 2 positive contribution to ρ0 [151] of CGF √ ∆m2 , 8 2π 2 (10.56) where ∆m2 ≡ m21 + m22 − 4m21 m22 m1 ln ≥ (m1 − m2 )2 , m21 − m22 m2 (10.57) and C = 1 (3) for color singlets (triplets). Thus, in the presence of such multiplets, one has 3GF X Ci √ ∆m2i = ρ0 − 1 , 8 2π 2 i 3 (10.58) ′
where the sum includes fourth-family quark or lepton doublets, bt ′ t̃
E0
or E − , and scalar doublets such as b̃ in Supersymmetry (in the absence of L − R mixing). This implies X Ci i 3 ∆m2i ≤ (85 GeV)2 (10.59) at 95% CL. The corresponding constraints P on non-degenerate squark and slepton doublets are even stronger, i Ci ∆m2i /3 ≤ (59 GeV)2 . This is due to the MSSM Higgs mass bound, mh0 < 150 GeV, and the very strong correlation between mh0 and ρ0 (79%). Non-degenerate multiplets usually imply ρ0 > 1. Similarly, heavy Z ′ bosons decrease the prediction for MZ due to mixing and generally lead to ρ0 > 1 [152]. On the other hand, additional Higgs doublets which participate in spontaneous symmetry breaking [153], heavy lepton doublets involving Majorana neutrinos [154], and the vacuum expectation values of Higgs triplets or higher-dimensional representations can contribute to ρ0 with either sign. Allowing for the presence of heavy degenerate chiral multiplets (the S parameter, to be discussed below) affects the determination of ρ0 from the data, at present leading to a smaller value (for fixed MH ). February 2, 2008 00:19 30 10. EW model and constraints on new physics A number of authors [155–160] have considered the general effects on neutral-current and Z and W boson observables of various types of heavy (i.e., Mnew ≫ MZ ) physics which contribute to the W and Z self-energies but which do not have any direct coupling to the ordinary fermions. In addition to non-degenerate multiplets, which break the vector part of weak SU(2), these include heavy degenerate multiplets of chiral fermions which break the axial generators. The effects of one degenerate chiral doublet are small, but in Technicolor theories there may be many chiral doublets and therefore significant effects [155]. Such effects can be described by just three parameters, S, T , and U at the (electroweak) one-loop level. (Three additional parameters are needed if the new physics scale is comparable to MZ [161]. ) T is proportional to the difference between the W and Z self-energies at Q2 = 0 (i.e., vector SU(2)-breaking), while S (S + U ) is associated 2 with the difference between the Z (W ) self-energy at Q2 = MZ,W and 2 Q = 0 (axial SU(2)-breaking). Denoting the contributions of new physics to the various self-energies by Πnew ij , we have new Πnew W W (0) − ΠZZ (0) , (10.60a) 2 MW MZ2
Πnew MZ2 − Πnew α b (MZ ) ZZ ZZ (0) S ≡ 2 2 2 4b s Zb cZ MZ

new 2 MZ2 Πnew b c 2Z − sb 2Z ΠZγ MZ γγ − − , (10.60b) c Z sb Z b MZ2 MZ2
2 − Πnew (0) Πnew MW α b (MZ ) W W WW (S + U ) ≡ 2 4b s 2Z MW

new 2 MZ2 Πnew c Z ΠZγ MZ b γγ − . (10.60c) − sb Z MZ2 MZ2 α b (MZ ) T ≡ S, T , and U are defined with a factor proportional to α b removed, so that they are expected to be of order unity in the presence of new physics. In the MS scheme as defined in Ref. 46, the last two terms in Eq. (10.60b) and Eq. (10.60c) can be omitted (as was done in some earlier editions of this Review). They are related to other parameters (Si , hi , b ǫi ) defined in [46,156,157] by T = hV = b ǫ1 /α , S = hAZ = SZ = 4b s 2Z b ǫ3 /α , U = hAW − hAZ = SW − SZ = −4b s 2Z b ǫ2 /α . (10.61) A heavy non-degenerate multiplet of fermions or scalars contributes positively to T as ρ0 − 1 = 1 − 1 ≃ αT , 1 − αT (10.62) where ρ0 is given in Eq. (10.58). The effects of non-standard Higgs representations cannot be separated from heavy non-degenerate February 2, 2008 00:19 10. EW model and constraints on new physics 31 multiplets unless the new physics has other consequences, such as vertex corrections. Most of the original papers defined T to include the effects of loops only. However, we will redefine T to include all new sources of SU(2) breaking, including non-standard Higgs, so that T and ρ0 are equivalent by Eq. (10.62). A multiplet of heavy degenerate chiral fermions yields 2 X t3L (i) − t3R (i) /3π , (10.63) S=C i where t3L,R (i) is the third component of weak isospin of the left-(right-)handed component of fermion i and C is the number of colors. For example, a heavy degenerate ordinary or mirror family would contribute 2/3π to S. In Technicolor models with QCD-like dynamics, one expects [155] S ∼ 0.45 for an iso-doublet of techni-fermions, assuming NT C = 4 techni-colors, while S ∼ 1.62 for a full techni-generation with NT C = 4; T is harder to estimate because it is model dependent. In these examples one has S ≥ 0. However, the QCD-like models are excluded on other grounds (flavor changing neutral-currents, and too-light quarks and pseudo-Goldstone bosons [162]) . In particular, these estimates do not apply to models of walking Technicolor [162], for which S can be smaller or even negative [163]. Other situations in which S < 0, such as loops involving scalars or Majorana particles, are also possible [164]. The simplest origin of S < 0 would probably be an additional heavy Z ′ boson [152], which could mimic S < 0. Supersymmetric extensions of the SM generally give very small effects. See Refs. 115,165 and the Section on Supersymmetry in this Review for a complete set of references. [115,165]. Most simple types of new physics yield U = 0, although there are counter-examples, such as the effects of anomalous triple gauge vertices [157]. The SM expressions for observables are replaced by 2 MZ2 = MZ0 2 2 MW = MW 0 1 − αT √ , 2 S/2 2π 1 − GF MZ0 1 √ , 2 1 − GF MW 0 (S + U ) /2 2π (10.64) where MZ0 and MW 0 are the SM expressions (as functions of mt and MH ) in the MS scheme. Furthermore, 1 M3β , 1 − αT Z Z 3 βW , ΓW = M W 1 Ai = Ai0 , 1 − αT ΓZ = (10.65) where βZ and βW are the SM expressions for the reduced widths 3 and Γ 3 ΓZ0 /MZ0 W 0 /MW 0 , MZ and MW are the physical masses, and Ai (Ai0 ) is a neutral-current amplitude (in the SM). February 2, 2008 00:19 32 10. EW model and constraints on new physics The data allow a simultaneous determination of sb 2Z (from the Z-pole asymmetries), S (from MZ ), U (from MW ), T (mainly from ΓZ ), αs (from Rℓ , σhad , and ττ ), and mt (from CDF and DØ), with little correlation among the SM parameters: S = −0.13 ± 0.10 (−0.08) , T = −0.17 ± 0.12 (+0.09) , U = 0.22 ± 0.13 (+0.01) , (10.66) and sb 2Z = 0.23119 ± 0.00016, αs (MZ ) = 0.1222 ± 0.0019, mt = 177.2 ± 4.2 GeV, where the uncertainties are from the inputs. The central values assume MH = 117 GeV, and in parentheses we show the change for MH = 300 GeV. As can be seen, the SM parameters (U ) can be determined with no (little) MH dependence. On the other hand, S, T , and MH cannot be obtained simultaneously, because the Higgs boson loops themselves are resembled approximately by oblique effects. Eqs. (10.66) show that negative (positive) contributions to the S (T ) parameter can weaken or entirely remove the strong constraints on MH from the SM fits. Specific models in which a large MH is compensated by new physics are reviewed in [166]. The parameters in Eqs. (10.66), which by definition are due to new physics only, all deviate by more than one standard deviation from the SM values of zero. However, these deviations are correlated. Fixing U = 0 (as is done in Fig. 10.3) will also move S and T to values compatible with zero within errors because the slightly high experimental value of MW favors a positive value for S + U . Using Eq. (10.62) the value of ρ0 corresponding to T is 0.9987 ± 0.0009 (+0.0007). The values of the b ǫ parameters defined in Eq. (10.61) are b ǫ3 = −0.0011 ± 0.0008 (−0.0006) , b ǫ1 = −0.0013 ± 0.0009 (+0.0007) , b ǫ2 = −0.0019 ± 0.0011 (−0.0001) . (10.67) Unlike the original definition, we defined the quantities in Eqs. (10.67) to vanish identically in the absence of new physics and to correspond directly to the parameters S, T , and U in Eqs. (10.66). There is a strong correlation (80%) between the S and T parameters. The allowed region in S − T is shown in Fig. 10.3. From Eqs. (10.66) one obtains S ≤ 0.03 (−0.05) and T ≤ 0.02 (0.11) at 95% CL for MH = 117 GeV (300 GeV). If one fixes MH = 600 GeV and requires the constraint S ≥ 0 (as is appropriate in QCD-like Technicolor models) then S ≤ 0.09 (Bayesian) or S ≤ 0.06 (frequentist). This rules out simple Technicolor models with many techni-doublets and QCD-like dynamics. An extra generation of ordinary fermions is excluded at the 99.95% CL on the basis of the S parameter alone, corresponding to NF = 2.92 ± 0.27 for the number of families. This result assumes that there are no new contributions to T or U and therefore that any new families are degenerate. In principle this restriction can be February 2, 2008 00:19 10. EW model and constraints on new physics 33 relaxed by allowing T to vary as well, since T > 0 is expected from a non-degenerate extra family. However, the data currently favor T < 0, thus strengthening the exclusion limits. A more detailed analysis is required if the extra neutrino (or the extra down-type quark) is close to its direct mass limit [167]. This can drive S to small or even negative values but at the expense of too-large contributions to T . These results are in agreement with a fit to the number of light neutrinos, Nν = 2.986 ± 0.007 (which favors a larger value for αs (MZ ) = 0.1228 ± 0.0021 mainly from Rℓ and ττ ). However, the S parameter fits are valid even for a very heavy fourth family neutrino. Figure 10.3: 1 σ constraints (39.35%) on S and T from various inputs. S and T represent the contributions of new physics only. (Uncertainties from mt are included in the errors.) The contours assume MH = 117 GeV except for the central and upper 90% CL contours allowed by all data, which are for MH = 340 GeV and 1000 GeV, respectively. Data sets not involving MW are insensitive to U . Due to higher order effects, however, U = 0 has to be assumed in all fits. αs is constrained using the τ lifetime as additional input in all fits. There is no simple parametrization that is powerful enough to describe the effects of every type of new physics on every possible observable. The S, T , and U formalism describes many types of heavy physics which affect only the gauge self-energies, and it can be applied to all precision observables. However, new physics which couples directly to ordinary fermions, such as heavy Z ′ bosons [152] or mixing with exotic fermions [168] cannot be fully parametrized in the S, T , and U framework. It is convenient to treat these types of new physics by parameterizations that are specialized to that particular class of theories (e.g., extra Z ′ bosons), or to consider specific models (which might contain, e.g., Z ′ bosons and exotic fermions with correlated parameters). Constraints on various types of new physics are reviewed in Refs. [43,91,169,170]. Fits to models with (extended) Technicolor and Supersymmetry are described, respectively, in Refs. [171], and [115,172]. The effects of compactified extra spatial dimensions at the TeV scale have been reviewed in [173], and constraints on Little Higgs models in [174]. An alternate formalism [175] defines parameters, ǫ1 , ǫ2 , ǫ3 , ǫb (0,ℓ) in terms of the specific observables MW /MZ , Γℓℓ , AF B , and Rb . The definitions coincide with those for b ǫi in Eqs. (10.60) and (10.61) for physics which affects gauge self-energies only, but the ǫ’s now parametrize arbitrary types of new physics. However, the ǫ’s are not related to other observables unless additional model-dependent assumptions are made. Another approach [176–178] parametrizes new physics in terms of gauge-invariant sets of operators. It is especially February 2, 2008 00:19 34 10. EW model and constraints on new physics powerful in studying the effects of new physics on non-Abelian gauge vertices. The most general approach introduces deviation vectors [169]. Each type of new physics defines a deviation vector, the components of which are the deviations of each observable from its SM prediction, normalized to the experimental uncertainty. The length (direction) of the vector represents the strength (type) of new physics. Table 10.7: 95% CL lower mass limits (in GeV) from low energy and Z pole data on various extra Z ′ gauge bosons, appearing in models of unification and string theory. ρ0 free indicates a completely arbitrary Higgs sector, while ρ0 = 1 restricts to Higgs doublets and singlets with still unspecified charges. The CDF bounds from searches for p̄p → e+ e− , µ+ µ− [183] and the LEP 2 e+ e− → f f¯ [41,184] bounds are listed in the last two columns, respectively. (The CDF bounds would be weakend if there are open supersymmetric or exotic decay channels.) Z’ ρ0 free ρ0 = 1 CDF (direct) LEP 2 Zχ 551 545 595 673 Zψ Zη 151 379 146 365 590 620 481 434 ZLR ZSM 570 822 564 809 630 690 804 1787 Zstring 582 578 − − One of the best motivated kinds of physics beyond the SM besides Supersymmetry are extra Z ′ bosons. They do not spoil the observed approximate gauge coupling unification, and appear copiously in many Grand Unified Theories (GUTs), most Superstring models, as well as in dynamical symmetry breaking [171,179] and Little Higgs models [174]. For example, the SO(10) GUT contains an extra U(1) as can be seen from its maximal subgroup, SU(5) × U(1)χ . Similarly, the E6 GUT contains the subgroup SO(10) × U(1)ψ . The Zψ possesses only axial-vector couplings to the ordinary fermions, and its mass is generally less p p constrained. The Zη boson is the linear combination 3/8 Zχ − 5/8 Zψ . The ZLR boson occurs in left-right models with gauge group SU(3)C × SU(2)L × SU(2)R × U(1)B –L ⊂ SO(10). The sequential ZSM boson is defined to have the same couplings to fermions as the SM Z boson. Such a boson is not expected in the context of gauge theories unless it has different couplings to exotic February 2, 2008 00:19 10. EW model and constraints on new physics 35 fermions than the ordinary Z. However, it serves as a useful reference case when comparing constraints from various sources. It could also play the role of an excited state of the ordinary Z in models with extra dimensions at the weak scale. Finally, we consider a Superstring motivated Zstring boson appearing in a specific model [180]. The potential Z ′ boson is in general a superposition of the SM Z and the new boson associated with the extra U(1). The mixing angle θ satisfies, M 2 0 − MZ2 Z 2 tan θ = 2 1 , MZ ′ − M 2 0 Z1 where MZ 0 is the SM value for MZ in the absence of mixing. Note, 1 that MZ < MZ 0 , and that the SM Z couplings are changed by the 1 mixing. If the Higgs U(1)′ quantum numbers are known, there will be an extra constraint, g2 MZ2 , (10.68) θ=C g1 MZ2 ′ where g1,2 are the U(1) and U(1)′ gauge couplings with g2 = q √ 5 sin θ W λ g1 . λ ∼ 1 (which we assume) if the GUT group breaks 3 directly to SU(3) × SU(2) × U(1) × U(1)′ . C is a function of vacuum expectation values. 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