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 † 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 † 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 √ 2 2 and their coupling remains g. Using standard conventions g2 G M 2 = F√ 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