Electroweak unification and properties of W and Z bosons

Electroweak unificationDecays of the W bosonDecays of the Z boson

Electroweak unification and properties of W andZ bosons

Harri Waltari

University of Helsinki & Helsinki Institute of PhysicsUniversity of Southampton & Rutherford Appleton Laboratory

Autumn 2018

H. Waltari Electroweak unification and properties of W and Z bosons


Contents

In this lecture we shall

Construct a common theory for weak and electromagneticinteractions

Compute the predictions of the electroweak theory for gauge bosondecays

Discuss the experimental discovery of W and Z bosons

This lecture corresponds to chapters 15.1, 15.3 and 15.4 of Thomson’sbook.



The simple SU(2) model has problems

With the SU(2) model we are on the right track, since the interactions ofthe W -bosons with fermions come out right. However, there are anumber of problems:

The W -boson does not have electromagnetic interactions eventhough it is charged and you cannot introduce them in the same wayas for other particles

The gauge boson masses are not compatible with the gaugesymmetry and SU(2) is not confining so they are not due to thepotential energy of weak interactions (leptons have a SU(2) chargeand are free particles)

Even the fermion mass terms mf (ψLψR + ψRψL) are not compatiblewith SU(2) since left- and right-handed fields have differenttransformation rules

We shall solve the first one now and postpone the two latter ones later.



Gauging hypercharge and allowing gauge boson mixinggives the correct vertices

The solution lies in the Gell-Mann–Nishijima formula for charge:

Q = I(W )3 + Y /2⇒ Gauge weak isospin and hypercharge and then

you will have a gauge theory for electric charge, too

Hypercharge is an additive quantum number, so the first Ansatz isto associate a U(1) symmetry to it

We associate a gauge field Bµ and a gauge coupling g ′ to thesymmetry ⇒ The covariant derivative becomesDµ = ∂µ − igτ iW i

µ − ig ′ Y2 Bµ for left-handed fermions and

Dµ = ∂µ − ig ′ Y2 Bµ for right-handed fermions

We then assume that since there are two neutral spin-1 fields thatthey can mix, we have the combinationsAµ = Bµ cos θW + W 3

µ sin θW and Zµ = W 3µ cos θW − Bµ sin θW ,

where θW is the weak mixing angle (or Weinberg angle)




We associate Aµ with the photon, whereas Zµ is a neutral gaugeboson, which will couple to neutrinos

The charges for the left-handed doublet come out right if it hasY = −1

From the covariant derivative we may read that the neutrino couplesto a combination proportional to gW 3

µ − g ′Bµ

Hence cos θW = g√g2+g ′2

and sin θW = g ′√g2+g ′2

The left-handed electron then couples to the photon by12 (g sin θW + g ′ cos θW )⇒ e = gg ′√

g2+g ′2, known as the unification

condition

Notice that Y = −2 for the right-handed electron leads to the samecoupling with the photon




The current value for sin2 θW = 0.231 (which is the parameter thatcan be measured directly in a number of ways)

The kinetic terms for the gauge field have a termg2εijkεilmW jµW kνW l

µWmν

Now setting e.g. k = m = 3 we get photon components in thefour-boson vertex ⇒ a coupling W+W−γγ with the strength e2 asone would expect

Hence the SU(2)L×U(1)Y theory provides at least qualitatively theknown features of electromagnetic and weak interactions andprovides a chance to predict the outcomes of a large number ofprocesses, to which we turn next



Massive gauge bosons have three polarization states

The W -boson has a mass of roughly 80 GeV, hence it has a restframe

Since in a rest frame you cannot say, which directions are transverse,you have to have three polarization states (instead of two formassless ones, as required by the Maxwell equations)

The basis vectors for the polarization states can be chosen as

εµ+ = − 1√2

(0, 1, i , 0),

εµ− =1√2

(0, 1,−i , 0),

εµL =1

mW(p, 0, 0,E ),

where ε± are circularly polarized states, εL is the longitudinal stateand the z-axis is chosen in the direction of the momentum



We compute the W-boson decay width in its rest frame

Using Feynman rules we get the matrix element:

Matrix element for the decay W− → e−ν

M = g√2ελµ(p1)u(p3)γµ 1

2 (1− γ5)v(p4)

It is easiest to compute this in the rest frame of the W -boson, wherep1 = (mW , 0, 0, 0)

Neglecting the fermion masses (this is a good approximation to allfermions as even m2

b/m2W ' 3× 10−3) leads to

p3 = E (1, sin θ, 0, cos θ) and p4 = E (1,− sin θ, 0,− cos θ)



Orthogonality of polarization vectors gives us three terms

We shall sum over polarizations λ, but since the polarization vectorsare orthogonal MM† just reduces to the sum of squares for eachpolarization (M+M†+ +M−M†− +MLM†L)

The factor 1− γ5 reduces the leptonic part of the matrix element tojust a product of the left-chiral electron and the right-chiralantineutrino giving

u(p3)γµ1

2(1− γ5)v(p4) = E

− sin θ/2cos θ/2− sin θ/2cos θ/2

T

γµ

cos θ/2sin θ/2− cos θ/2− sin θ/2

= 2E (0,− cos θ,−i , sin θ) = mW (0,− cos θ,−i , sin θ)

noting that E = mW /2



Orthogonality of polarization vectors gives us three terms

We get

M− =gmW

2(0, 1,−i , 0) · (0,− cos θ,−i , sin θ) =

gmW

2(1 + cos θ)

M+ = −gmW

2(0, 1, i , 0) · (0,− cos θ,−i , sin θ) =

gmW

2(1− cos θ)

ML =gmW

2(0, 0, 0, 1) · (0,− cos θ,−i , sin θ) = −gmW√

2sin θ

Hence 〈|M|2〉 =g2m2

W

12 ((1 + cos θ)2 + (1− cos θ)2 + 2 sin2 θ) =g2m2

W

3

This decay width does not depend on the angles so we predict anisotropic distribution in the W -boson rest frame

Plugging this in with the kinematical factors gives

Γ(W− → e−νe) = g2mW

48π



SU(2) invariance allows us to compute the total decaywidth from one of the decays

Since all fermions are in SU(2) doublets, we haveΓ(W− → e−νe) = Γ(W− → µ−νµ) = Γ(W− → τ−ντ )

For quarks we need to take into account color and CKM-mixing, theformer giving a factor of 3 and the latter a factor of |Vqq′ |2

Since five quarks are lighter than mW , there are six possiblehadronic decay modes at the quark level

The unitarity conditions of the CKM matrix tell us that|Vud |2 + |Vus |2 + |Vub|2 = 1 and |Vcd |2 + |Vcs |2 + |Vcb|2 = 1

Overall the quark modes are equivalent to six lepton generations,

giving a tree-level prediction of Γtot = 3g2mW

16π ' 2.03 GeV andBR(W → hadrons) = 2/3, BR(W → `ν) = 1/3



Loop corrections explain the discrepancy

Experimentally we have Γtot = 2.085 GeV andBR(W → hadrons) = 67.4%

Most of the difference is explained by taking into account the NLOcorrection to the quark decays, which gives effectively a factor1 + αs/π to the Wqq′ vertices

Since αs(q2 = m2W ) ' 0.12, we get 1 + αs/π ' 1.038, which then

gives the correct branching ratio and also a total width close to theexperimental value

Altogether we have a fair understanding of the W -boson decaysbased on the electroweak theory



The W -boson was discovered in the UA1 and UA2experiments

The UA1 and UA2 experiments at CERN searched for the weakgauge bosons in pp collisions, the masses were pretty much knownby the start of the experimentThe experiments had the first ‘’4π” solid angle detectors, i.e. therewere particle detectors at nearly all directions trying to detect allproduced particlesSince the leptonic decay modes of W include a neutrino, thenegative of the sum of all momenta gave the neutrino momentum

Figure: Phys. Lett. B122 (1983) 103



The W -boson discovery was the first major result inexperimental particle physics to which University ofHelsinki contributed



The Z-boson has different couplings to left- andright-handed particles

Our next task is to compute the predictions of the electroweaktheory for the decays of the Z boson

Since Z is a linear combination of the neutral W -boson and thehypercharge gauge boson B, of which W couples to left-handedparticles and B to all, Z will have couple to both left- andright-handed particles, but with different stregths

There are two common parametrizations of the vertex between thespinors, either gZ (cLγ

µ 12 (1− γ5) + cRγ

µ 12 (1 + γ5)) and

gZ (cV γµ − cAγ

µγ5), where gZ =√g2 + g ′2

The two representations have the connection cV = cL + cR ,cA = cL − cR



The covariant derivative gives the couplings cL and cR

We may write the kinetic terms as iψLγµDL

µψL + iψRγµDR

µψR ,

where DLµ = ∂µ − igτ iW i

µ − ig ′ Y2 12×2Bµ and DRµ = ∂µ − ig ′ Y2 Bµ

For RH fermions Y = 2Q so for up-type quarks Y Ru = 4/3, for

down-type quarks Y Rd = −2/3 and for leptons Y R

` = −2

Since Bµ = Aµ cos θW −Zµ sin θW , we find the couplings of Z to RHfermions to be −g ′Q sin θW = −gZQ sin2 θW so cR = −Q sin2 θW

For LH fermions we have Q = I3 + Y /2⇒ Y = 2(Q − I3)

Hence the Bµ term gives a contribution−g ′(Q − I3) sin θW = −gZ (Q − I3) sin2 θW

The W 3µ term gives a contribution gI3 cos θW = gZ I3 cos2 θW

Altogether we have gZ (I3 − Q sin2 θW ) so cL = I3 − Q sin2 θW



We get the couplings by plugging in the quantum numbers

We may easily compute the couplings for all of the fermions:

Fermion cL cR cV cAe−, µ−, τ− −0.27 0.23 −0.04 −1/2νe , νµ, ντ 1/2 0 1/2 1/2u, c , t 0.34 −0.16 0.19 1/2d , s, b −0.42 0.08 −0.35 −1/2

You may notice that for neutrinos the coupling structure is the same asin charged current weak interactions (since only W 3 contributes) andthat the parity-breaking axial-vector couplings are always ±1/2 as theycome from weak isospin.



The decay of Z to left-handed particles is nearly that ofthe W

Since also Z is a massive spin-1 particle, the decay is very similar tothe W decay

The changes are the coupling, where g/√

2→ gZcL and the massmW → mZ

Hence 〈|ML|2〉 = 23c

2Lg

2Zm

2Z

For right-handed particles the kinematics are the same, only thecoupling changes cL → cR

There is no interference between the left- and right-handedamplitudes and hence 〈|M|2〉 = 2

3 (c2L + c2

R)g2Zm

2Z



The decay widths are different for charged leptons andneutrinos

You can also write the width in terms of cV and cA asc2V + c2

A = 2(c2L + c2

R)

Altogether we have Γ(Z → f f ) =g2ZmZ

48π (c2V + c2

A)

Since c2V + c2

A are different for different particle types, the branchingratios are not universal

The branching ratio to neutrinos is roughly twice that to chargedleptons, and decays to down-type quarks are favored compared toup-type quarks

With the SM particle content the Z -boson branching ratio tocharged leptons is 10% (3.3% per flavor), to neutrinos it is 21% andto hadrons 69% (taking into account the QCD vertex correction)



UA1 claimed the discovery of the Z boson based on ahandful of events

Z was discovered from the pp → e+e− mode in 1983:



Nowadays the Z boson is a ”standard candle” in particlephysics

The Z mass has been measured accurately at LEP, mZ = 91.19 GeV

The leptonic modes can be seen clearly from the background inhadron colliders

The Z → e+e− has been used to calibrate the calorimeters, whichwas crucial to achieve a good mass resolution for the Higgs



Summary

The combined theory of electromagnetic and weak interactions isbased on the gauge symmetry SU(2)L×U(1)Y

The neutral gauge bosons mix to form the observable gauge bosonsAµ and Zµ

The couplings come out right if e = gg ′/√g2 + g ′2, known as the

unification condition

The W - and Z -bosons were discovered in 1983-84 at CERN

The predictions of the electroweak theory lead to non-trivialbranching ratios for the Z -boson — which are experimentallyconfirmed


Date post:	15-Feb-2022
Category:	Documents
Upload:	others
View:	4 times
Download:	0 times

Electroweak unification and properties of W and Z bosons

Documents