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Neutral Currents and the Standard Model Lagrangian

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Neutral Currents and the Standard Model Lagrangian Neutral Currents and the Standard Model Lagrangian Nov 1, 2018 Introduction • So far, have limited Weak Interactopm discussion to exchange of W bosons (\charged current (CC) interactions") • We know that Z boson also exists • But unambiguous observation of neutral current (NC) exchange only occured in 1970's • Why was it so difficult to see? I GIM mechanism: If L diagonal in strong basis, is diagonal in weak basis ! no FCNC ! Z couples to ff pairs I NC interactions of charged particles can occur via photon exchange ! in general, at low q2, EM interactions swamp WI • Options for observing NC before the discovery of the Z: I Neutrino scattering I Parity violating effects in interactions of charged leptons I Parity violating effects in interactions of charged leptons with quarks Overview of History of Standard Model Development • Glashow,Weinberg,Salam developed unifed, gauge theory of Electroweak interactions in 1960's I Called the Weinberg-Salam (WS) model • First observation of NC's in ν and e interactions occurred after WS model proposed • NC measurements supported WS • WS model predicted: I Existence of Z I MW and MZ as function of one parameter sin(θW ) I sin(θW ) measured using ν interactions • W and Z discovered at SppS in 1982,1983 • Precision NC measurements at LEP/SLC (e+e− ! Z) starting in 1989 Today, will begin by reviewing NC measurements of the 1970's Then, on to WI Lagrangian Some Observations • Charged current interactions observed to be (V − A) couplings with university strength (once CKM matrix accounted for) • This does not mean that neutral currents must also be left-handed I And in fact, they are NOT • In original formulation of EW theory and in our discussions, we will assume neutrinos are massless (although we know now that they do have small mass) I Take as a postulate that all ν are left-handed and all ν are right-handed I Quarks and charged leptons have mass and exist both in left- and right-handed states I To full define the theory, need to measure the coupling of the neutral weak boson (the Z) to: • Left-handed ν and right-handed ν • Left-handed ` and right-handed ` • Right-handed ` and left-handed ` • Left-handed q and right-handed q • Right-handed q and left-handed q • That means we need to use all 3 options listed on page 2 in order to fully define the model Discovery of Neutral Currents: Gargamelle (I) • Gargamelle bubble chamber filled with freon • 83,000 pictures with νµ beam, 207,000 with νµ • Look for: − CC Events : νµ + N ! µ + X + νµ + N ! µ + X NC Events : νµ + N ! νµ + X νµ + N ! νµ + X • Remove bckgrnd from neutrons created in chamber walls from ν interactions (\Stars") Discovery of Neutral Currents: Gargamelle (II) • Stars show exponential fall-off along beam axis • NC event-rate flat and consistent with CC event-rate vs distance along beam axis • Event Rates: (NC=CC)ν = 0:21 ± 0:03 (NC=CC)ν = 0:45 ± 0:09 • We'll see later that these ratios agree with SM predictions • Difference in ratios for ν and ν shows that NC are not V-A Discovery of Neutral Currents: Gargamelle (III) • Also observed νµe ! νµe NC Interactions with Charged Leptons: e+e− ! µ+µ− 2 • For q << MZ , Weak Interaction matrix element much smaller than EM • Observation of Weak Interaction requires looking for terms not allowed by EM ! Parity Violating Effects • Easiest signature: e+e− ! µ+µ− angular distribution NC Interactions: Quark-Lepton Interactions • Look for interference between weak (NC) and EM scattering amplitudes • First unambiguous measurement from e-Deuteron scattering: e(polarized) + d(unpolarized) ! e + X • Measure A ≡ (σL − σR)=(σL + σR) • General form using parton model 2 2 2 A=Q = a1 + a2 1 − (1 − y) = 1 + (1 − y) for isoscalar target, a1 and a2 constant • Measuring A as fn of y allows determination of a1 and a2 • These constants depend on quark and lepton couplings to Z Polarized eD Scattering (I) • Polarization obtained from laser optical pumping of Gallium Arsenide (photoemission of e) • Can change circular polarization of laser to change polarization (two methods) Polarized eD Scattering (II) • Good agreement with SM predictions • Provides estimate of the one parameter of the model: sin(θW ) I To understand this statement, we need to build up the SM description of EW interactions Building the SM Lagrangian (WS Model) • Start with CC interactions 1 − γ J = νγ ( 5 )e = ν γ e µ µ 2 L µ L y Jµ = eLγµνL • Can write these 2 currents in terms of raising and lowering operators of weak isospin: A new SU(2) quantum number µ 0 1 0 0 χ ≡ τ = τ = L e− + 0 0 − 1 0 L y Jµ = χLγµτ+χL Jµ = χLγµτ−χL • Since these are 2 components of an SU(2) triplet, there must also be a 3rd component 0 J = χLγµτ3χL • Can J 0 be the Weak Neutral Boson (the Z)? No! (see next page) Why isn't J 0 the Z? • We know there are RH WNC: I ν,ν NC scattering rate not consistent with V-A I eRD scattering not zero • How can this be? • In addition to WI, there is EM, which is also NC • If we unify WI and EM, have 2 neutral currents and can create Z and γ from linear combinations of these • Expand our gauge group to include both: SU(2)L × U(1) 0 I Two coupling constants g and g I Four gauge bosons: 1 2 3 Wµ , Wµ , Wµ SU(2))L triplet coupling g 0 Bµ U(1) singlet coupling g The Unified Gauge Interaction Lagrangian (I) • Boson fields: 1 1 L = − F~ · F~ µν − f f µν gauge 4 µν 4 µν F~µν = @µW~ ν − @ν W~ µ + gW~ µ × W~ µ fµν = @µBν − @ν Bµ • Lepton fields: I Want to couple to left-handed e and ν: ν = 1 (1 − γ )ν ν e L 2 5 χL ≡ L where 1 eL = 2 (1 − γ5)e 1 I No RH ν: χR ≡ eR = 2 (1 + γ5)e LH members are weak iso-doublets and the RH charged leptons are weak iso-singlets. There is no RH neutrino • We'll come back to the quarks later The Unified Gauge Interaction Lagrangian (II) • For Strong Interactions we saw B + S Y Q = I3 + = I3 + 2 2 • Postulate a similar \weak hypercharge" and require same relation to hold. For quarks YL = −1 YR = −2 (constructed to give the leptons the right charge) Y Q = I3 + 2 Q(ν ) = 1 + −1 = 0 ν e L 2 2 χL ≡ L 1 −1 Q(eL) = − 2 + 2 = −1 1 −2 χR ≡ eR = 2 (1 + γ5)e Q(eR) = 0 + 2 = −1 • This choice has additional advantage that by giving all members of a multiplet the same Y we have [I3;Y ] = 0 and both are simultaneouly observable Q is a conserved quantum number! The Unified Gauge Interaction Lagrangian (III) • Lepton terms in LaGrangian (kinetic plus interaction): µ 0 Y L = χ iγ @µ + ig Bµ χR + leptons R 2 µ 0 Y ~τ χ iγ @µ + ig Bµ + ig · W~ µ χL L 2 2 • Note: Need to introduce the Higgs to add mass terms. We'll postpone that discussion! • The neutral interaction terms in the LaGrangian are from Bµ and W3 µ 0 Y µ 0 Y τ3 LNC = − χ γ g Bµ χR + χ γ g Bµ + g (W3)µ χL R 2 L 2 2 µ 0 Y = − χγ gI3(W3)µ + g Bµ χ 2 where I3 = τ3=2 and we have used the fact that I3 = 0 for χR Changing Basis • We have two neutral fields: A and B3 • Before we introduce the Higgs, both are massless. They can mix I Higgs will give mass to one of these states, breaking degeneracy I But the massive state is a linear combination of A and B3 • We can identify one of the neutral bosons as the photon I This state must remain massless when Higgs introduced • Can identify which combination is the photon: it must couple to charge: Y Q = I + 3 2 The Weinberg Angle θW • We have two couplings: g and g0 • Can always express the ratio as g tan θW = g0 • Then g sin θW = pg2 + g02 g0 cos θW = pg2 + g02 • And our LaGrangian becomes: µ 0 Y LNC = − χγ gI3(W3)µ + g Bµ χ 2 p 2 0 µ Y = − g + g 2 χγ sin θW I3(W3)µ + cos θW Bµ χ 2 • Now we can pick out the piece that couples to charge and identify it with the photon The photon, the Z and the W ± • Define photon field as piece that couples to charge Aµ = Bµ cos θW + (W3)µ sin θW • The Z is the orthogonal combination Zµ = −Bµ sin θW + (W3)µ cos θW • Because photon couples to charge, we can relate e to the couplings and θW : 0 e = g sin θW = g cos θW • The W ± bosons are W ± iW W ± = 1 p 2 2 and their coupling remains g. Using standard conventions g2 G M 2 = Fp W 8 2 • sin θW is a parameter to be measured (many different techniques) 2 sin θW ∼ 0:23 How about the quarks? • Follow same prescription as for the leptons 5 • Wµ coupling is left handed: γµ(1 − γ )=2 , B coupling is left-right symmetric: γµ I Left handed weak isodoublets, right handed weak isosinglets I Y value for multiplets chosen to enforce Q = I3 + Y=2 L fermion Q I3 YL YR 1 ν` 0 2 −1 - 1 ` −1 − 2 −1 −2 2 1 1 4 u; c; t + 3 + 2 + 3 + 3 1 1 1 2 d; s; b − 3 − 2 + 3 − 3 Predicted Z Couplings to Fermions • The Z current specified by Zµ = −Bµ sin θW + (W3)µ cos θW • Together with the LaGrangian from page 18 this gives (with some math) Z 3 2 EM Jµ = Jµ − sin θW jµ • The neutral weak coupling is NOT (V-A) but rather 5 CV γµ + CAγmu(1 − γ ) 2 • Values of CV and CA can be calculated from sin θW • Weak NC vector and axial vector couplings are: f Qf CA CV 1 1 ν 0 2 2 1 1 2 e -1 − 2 − 2 + 2 sin θW 2 1 1 4 2 u 3 2 2 − 3 sin θW 1 1 1 2 2 d − 3 − 2 − 2 + 3 sin θW Counting Parameters 0 • Three parameters of the model: g, g and sin θW • Can replace them with any three measured quantities that depend on these parameters • Want to use the 3 best measured parameters I e: Known from EM interactions I GF Known from µ decay 2 I sin θW or MZ 2 • Before LEP: sin θW as measured in ν-scattering, polarized electron scattering and e+e− ! my+µ− was parameter of choice • Today, MZ is measured to much higher precision • We'll see next Tues how the Z gets its mass and why in the SM it is predicted in terms of the other SM parameters 2 • We'll also see next Thurs that sin θW receives radiative corrections and comparisons of different measurement methods tests loop corrections in EW theory 2 • These sin θW measurements also probe existence of possible new massive particles 2 sin θW from (anti-)Neutrino-electron scattering • Can express scattering rates in terms of GF • νµ scattering only through NC, νe through both CC and NC 2 GF meEν 2 2 σ(νµe ! νµe) = L + R =3 2π e e 2 GF meEν 2 2 σ(νµe ! νµe) = L =3 + R 2π e e 2 GF meEν 2 2 σ(νee ! νee) = (Le + 2) + R =3 2π e 2 GF meEν 2 2 σ(νee ! νee) = (Le + 2) =3 + R 2π e Best fit to experimental data: 2 sin θW = 0:234 .
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