Electroweak Unification and Properties of W and Z Bosons

Electroweak uniﬁcation Decays of the W boson Decays of the Z boson

Electroweak uniﬁcation and properties of W and Z bosons

Harri Waltari

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

Autumn 2018

H. Waltari Electroweak uniﬁcation and properties of W and Z bosons Electroweak uniﬁcation Decays of the W boson Decays of the Z boson Contents

In this lecture we shall Construct a common theory for weak and electromagnetic interactions Compute the predictions of the electroweak theory for gauge boson decays Discuss the experimental discovery of W and Z bosons This lecture corresponds to chapters 15.1, 15.3 and 15.4 of Thomson’s book.

H. Waltari Electroweak uniﬁcation and properties of W and Z bosons Electroweak uniﬁcation Decays of the W boson Decays of the Z boson The simple SU(2) model has problems

With the SU(2) model we are on the right track, since the interactions of the W -bosons with fermions come out right. However, there are a number of problems: The W -boson does not have electromagnetic interactions even though it is charged and you cannot introduce them in the same way as for other particles The gauge boson masses are not compatible with the gauge symmetry and SU(2) is not conﬁning so they are not due to the potential energy of weak interactions (leptons have a SU(2) charge and are free particles)

Even the fermion mass terms mf (ψLψR + ψR ψL) are not compatible with SU(2) since left- and right-handed fields have different transformation rules We shall solve the first one now and postpone the two latter ones later.

H. Waltari Electroweak uniﬁcation and properties of W and Z bosons Electroweak uniﬁcation Decays of the W boson Decays of the Z boson Gauging hypercharge and allowing gauge boson mixing gives the correct vertices

The solution lies in the Gell-Mann–Nishijima formula for charge: (W ) Q = I3 + 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 is to associate a U(1) symmetry to it 0 We associate a gauge field Bµ and a gauge coupling g to the symmetry ⇒ The covariant derivative becomes i i 0 Y Dµ = ∂µ − igτ Wµ − ig 2 Bµ for left-handed fermions and 0 Y Dµ = ∂µ − ig 2 Bµ for right-handed fermions We then assume that since there are two neutral spin-1 fields that they can mix, we have the combinations 3 3 Aµ = Bµ cos θW + Wµ sin θW and Zµ = Wµ 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 gauge boson, which will couple to neutrinos The charges for the left-handed doublet come out right if it has Y = −1 From the covariant derivative we may read that the neutrino couples 3 0 to a combination proportional to gWµ − g Bµ g g 0 Hence cos θW = √ and sin θW = √ g 2+g 02 g 2+g 02 The left-handed electron then couples to the photon by 1 0 gg 0 (g sin θW + g cos θW ) ⇒ e = √ , known as the uniﬁcation 2 g 2+g 02 condition Notice that Y = −2 for the right-handed electron leads to the same coupling with the photon

2 The current value for sin θW = 0.231 (which is the parameter that can be measured directly in a number of ways) The kinetic terms for the gauge ﬁeld have a term 2 ijk ilm jµ kν l m g W W WµWν Now setting e.g. k = m = 3 we get photon components in the four-boson vertex ⇒ a coupling W +W −γγ with the strength e2 as one would expect

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

H. Waltari Electroweak uniﬁcation and properties of W and Z bosons Electroweak uniﬁcation Decays of the W boson Decays of the Z boson Massive gauge bosons have three polarization states

The W -boson has a mass of roughly 80 GeV, hence it has a rest frame Since in a rest frame you cannot say, which directions are transverse, you have to have three polarization states (instead of two for massless ones, as required by the Maxwell equations) The basis vectors for the polarization states can be chosen as 1 µ = −√ (0, 1, i, 0), + 2 1 µ = √ (0, 1, −i, 0), − 2 µ 1 L = (p, 0, 0, E), mW

where ± are circularly polarized states, L is the longitudinal state and the z-axis is chosen in the direction of the momentum

H. Waltari Electroweak uniﬁcation and properties of W and Z bosons Electroweak uniﬁcation Decays of the W boson Decays of the Z boson 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 λ(p )u(p )γµ 1 (1 − γ5)v(p ) 2 µ 1 3 2 4

It is easiest to compute this in the rest frame of the W -boson, where p1 = (mW , 0, 0, 0) Neglecting the fermion masses (this is a good approximation to all 2 2 −3 fermions as even mb/mW ' 3 × 10 ) leads to p3 = E(1, sin θ, 0, cos θ) and p4 = E(1, − sin θ, 0, − cos θ)

H. Waltari Electroweak uniﬁcation and properties of W and Z bosons Electroweak uniﬁcation Decays of the W boson Decays of the Z boson Orthogonality of polarization vectors gives us three terms

We shall sum over polarizations λ, but since the polarization vectors are orthogonal MM† just reduces to the sum of squares for each † † † polarization (M+M+ + M−M− + MLML) The factor 1 − γ5 reduces the leptonic part of the matrix element to just a product of the left-chiral electron and the right-chiral antineutrino giving

T − sin θ/2  cos θ/2  µ 1 5  cos θ/2  µ  sin θ/2  u(p3)γ (1 − γ )v(p4) = E   γ   2 − sin θ/2 − cos 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

H. Waltari Electroweak uniﬁcation and properties of W and Z bosons Electroweak uniﬁcation Decays of the W boson Decays of the Z boson Orthogonality of polarization vectors gives us three terms

We get gm gm M = W (0, 1, −i, 0) · (0, − cos θ, −i, sin θ) = W (1 + cos θ) − 2 2 gm gm M = − W (0, 1, i, 0) · (0, − cos θ, −i, sin θ) = W (1 − cos θ) + 2 2 gmW gmW ML = (0, 0, 0, 1) · (0, − cos θ, −i, sin θ) = − √ sin θ 2 2

2 2 2 2 2 g mW 2 2 2 g mW Hence h|M| i = 12 ((1 + cos θ) + (1 − cos θ) + 2 sin θ) = 3 This decay width does not depend on the angles so we predict an isotropic distribution in the W -boson rest frame Plugging this in with the kinematical factors gives 2 − − g mW Γ(W → e νe ) = 48π

H. Waltari Electroweak uniﬁcation and properties of W and Z bosons Electroweak uniﬁcation Decays of the W boson Decays of the Z boson SU(2) invariance allows us to compute the total decay width from one of the decays

Since all fermions are in SU(2) doublets, we have − − − − − − Γ(W → e νe ) = Γ(W → µ νµ) = Γ(W → τ ντ ) For quarks we need to take into account color and CKM-mixing, the 2 former giving a factor of 3 and the latter a factor of |Vqq0 |

Since ﬁve quarks are lighter than mW , there are six possible hadronic decay modes at the quark level The unitarity conditions of the CKM matrix tell us that 2 2 2 2 2 2 |Vud | + |Vus | + |Vub| = 1 and |Vcd | + |Vcs | + |Vcb| = 1 Overall the quark modes are equivalent to six lepton generations, 2 3g mW giving a tree-level prediction of Γtot = 16π ' 2.03 GeV and BR(W → hadrons) = 2/3, BR(W → `ν) = 1/3

H. Waltari Electroweak uniﬁcation and properties of W and Z bosons Electroweak uniﬁcation Decays of the W boson Decays of the Z boson Loop corrections explain the discrepancy

Experimentally we have Γtot = 2.085 GeV and BR(W → hadrons) = 67.4% Most of the diﬀerence is explained by taking into account the NLO correction to the quark decays, which gives eﬀectively a factor 0 1 + αs /π to the Wqq vertices 2 2 Since αs (q = mW ) ' 0.12, we get 1 + αs /π ' 1.038, which then gives the correct branching ratio and also a total width close to the experimental value Altogether we have a fair understanding of the W -boson decays based on the electroweak theory

H. Waltari Electroweak uniﬁcation and properties of W and Z bosons Electroweak uniﬁcation Decays of the W boson Decays of the Z boson The W -boson was discovered in the UA1 and UA2 experiments

The UA1 and UA2 experiments at CERN searched for the weak gauge bosons in pp collisions, the masses were pretty much known by the start of the experiment The experiments had the ﬁrst ‘’4π” solid angle detectors, i.e. there were particle detectors at nearly all directions trying to detect all produced particles Since the leptonic decay modes of W include a neutrino, the negative of the sum of all momenta gave the neutrino momentum

Figure: Phys. Lett. B122 (1983) 103

H. Waltari Electroweak unification and properties of W and Z bosons Electroweak unification Decays of the W boson Decays of the Z boson The W -boson discovery was the first major result in experimental particle physics to which University of Helsinki contributed

H. Waltari Electroweak unification and properties of W and Z bosons Electroweak unification Decays of the W boson Decays of the Z boson The Z-boson has different couplings to left- and right-handed particles

Our next task is to compute the predictions of the electroweak theory for the decays of the Z boson Since Z is a linear combination of the neutral W -boson and the hypercharge gauge boson B, of which W couples to left-handed particles and B to all, Z will have couple to both left- and right-handed particles, but with diﬀerent stregths There are two common parametrizations of the vertex between the µ 1 5 µ 1 5 spinors, either gZ (cLγ 2 (1 − γ ) + cR γ 2 (1 + γ )) and µ µ 5 p 2 02 gZ (cV γ − cAγ γ ), where gZ = g + g

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

H. Waltari Electroweak uniﬁcation and properties of W and Z bosons Electroweak uniﬁcation Decays of the W boson Decays of the Z boson

The covariant derivative gives the couplings cL and cR

µ L µ R We may write the kinetic terms as iψLγ DµψL + iψR γ Dµ ψR , L i i 0 Y R 0 Y where Dµ = ∂µ − igτ Wµ − ig 2 12×2Bµ and Dµ = ∂µ − ig 2 Bµ R For RH fermions Y = 2Q so for up-type quarks Yu = 4/3, for R R down-type quarks Yd = −2/3 and for leptons Y` = −2 Since Bµ = Aµ cos θW − Zµ sin θW , we ﬁnd the couplings of Z to RH 0 2 2 fermions to be −g Q sin θW = −gZ Q sin θW so cR = −Q sin θW

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

Hence the Bµ term gives a contribution 0 2 −g (Q − I3) sin θW = −gZ (Q − I3) sin θW 3 2 The Wµ term gives a contribution gI3 cos θW = gZ I3 cos θW 2 2 Altogether we have gZ (I3 − Q sin θW ) so cL = I3 − Q sin θW

H. Waltari Electroweak uniﬁcation and properties of W and Z bosons Electroweak uniﬁcation Decays of the W boson Decays of the Z boson 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 cA e−, µ−, τ − −0.27 0.23 −0.04 −1/2 νe , νµ, ντ 1/2 0 1/2 1/2 u, c, t 0.34 −0.16 0.19 1/2 d, s, b −0.42 0.08 −0.35 −1/2 You may notice that for neutrinos the coupling structure is the same as in charged current weak interactions (since only W 3 contributes) and that the parity-breaking axial-vector couplings are always ±1/2 as they come from weak isospin.

H. Waltari Electroweak uniﬁcation and properties of W and Z bosons Electroweak uniﬁcation Decays of the W boson Decays of the Z boson The decay of Z to left-handed particles is nearly that of the W

Since also Z is a massive spin-1 particle, the decay is very similar to the W decay √ The changes are the coupling, where g/ 2 → gZ cL and the mass mW → mZ 2 2 2 2 2 Hence h|ML| i = 3 cL gZ mZ For right-handed particles the kinematics are the same, only the coupling changes cL → cR There is no interference between the left- and right-handed 2 2 2 2 2 2 amplitudes and hence h|M| i = 3 (cL + cR )gZ mZ

H. Waltari Electroweak unification and properties of W and Z bosons Electroweak unification Decays of the W boson Decays of the Z boson The decay widths are different for charged leptons and neutrinos

You can also write the width in terms of cV and cA as 2 2 2 2 cV + cA = 2(cL + cR ) 2 gZ mZ 2 2 Altogether we have Γ(Z → f f ) = 48π (cV + cA) 2 2 Since cV + cA are different for different particle types, the branching ratios are not universal The branching ratio to neutrinos is roughly twice that to charged leptons, and decays to down-type quarks are favored compared to up-type quarks With the SM particle content the Z-boson branching ratio to charged leptons is 10% (3.3% per flavor), to neutrinos it is 21% and to hadrons 69% (taking into account the QCD vertex correction)

H. Waltari Electroweak uniﬁcation and properties of W and Z bosons Electroweak uniﬁcation Decays of the W boson Decays of the Z boson UA1 claimed the discovery of the Z boson based on a handful of events

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

H. Waltari Electroweak uniﬁcation and properties of W and Z bosons Electroweak uniﬁcation Decays of the W boson Decays of the Z boson Nowadays the Z boson is a ”standard candle” in particle physics

The Z mass has been measured accurately at LEP, mZ = 91.19 GeV The leptonic modes can be seen clearly from the background in hadron colliders The Z → e+e− has been used to calibrate the calorimeters, which was crucial to achieve a good mass resolution for the Higgs

H. Waltari Electroweak uniﬁcation and properties of W and Z bosons Electroweak uniﬁcation Decays of the W boson Decays of the Z boson Summary

The combined theory of electromagnetic and weak interactions is based on the gauge symmetry SU(2)L×U(1)Y The neutral gauge bosons mix to form the observable gauge bosons Aµ and Zµ The couplings come out right if e = gg 0/pg 2 + g 02, 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-trivial branching ratios for the Z-boson — which are experimentally conﬁrmed

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