PHY646 - Quantum Field Theory and the Standard Model
Even Term 2020 Dr. Anosh Joseph, IISER Mohali
LECTURE 48
Wednesday, April 15, 2020 (Note: This is an online lecture due to COVID-19 interruption.)
Topic: The Glashow-Weinberg-Salam Theory.
The Glashow-Weinberg-Salam Theory
The Glashow-Weinberg-Salam (GWS) theory gives the experimentally correct description of the weak interactions. This theory is also knows as the electroweak theory since it gives a unified description of weak and electromagnetic interactions. The existence of the electroweak interactions was experimentally established in two stages, the first being the discovery of neutral currents (exchange process involving Z boson) in neutrino scat- tering in 1973, and the second, in 1983, the discovery of the W and Z gauge bosons in proton- antiproton collisions. In 1979 Glashow, Salam, and Weinberg were awarded the Nobel Prize in Physics for their contributions to the unification of the weak and electromagnetic interaction be- tween elementary particles. In 1999, ’t Hooft and Veltman were awarded the Physics Nobel prize for showing that the electroweak theory is renormalizable. The framework of electroweak model required a fundamental scalar field, which is not generated by any gauge symmetry; it is the Higgs field. In 2012, the ATLAS and CMS collaborations at the Large Hadron Collider (LHC) experimentally detected the Higgs boson, and in 2013 the Physics Nobel prize was awarded to Englert and Brout, and Higgs for predicting the existence of the Higgs boson. Let us begin with the Higgs field; is appears as a doublet in GWS theory ! φ Φ = 1 , (1) φ2 where φ1 and φ2 are complex scalar fields. We then simply insert the field Φ into the fundamental Lagrangian of the Universe
† µ (l) µ (l) (q) µ (q) L = ∂µΦ ∂ Φ − V (Φ) + Ψ (iγ ∂µ)Ψ + Ψ (iγ ∂µ)Ψ + ··· + LΦ,Ψ. (2) PHY646 - Quantum Field Theory and the Standard Model Even Term 2020
The leptons and quarks need to be put in by hand - the ellipsis is a reminder that we have six fermionic doublets, two families, and three generations. We accept them simply as facts of Nature. 1 We simply assert that the weak hypercharge of the Higgs doublet is YH = 2 and the doublet transforms under SU(2)L × U(1)Y as
1 Φ → exp −i g0 α0 exp [−igαa T a]Φ. (3) 2
The scalar Lagrangian was designed to produce a gauge freedom. We assume the potential, also invariant under SU(2)L × U(1)Y , takes the following parametrized form
2 † † 2 V (Φ) = V0 − µ Φ Φ + λ(Φ Φ) . (4)
This is about the simplest potential possible which both obeys the required symmetry and contains a false vacuum. As a gauge choice, we are free to select a vacuum state for Φ at 0 Φg = . (5) √v 2
We have chosen this particular ground state and the corresponding hypercharge with an eye toward its interaction with the vector fields. Working from Q = T3 + Y and noting that the doublet 1 has a weak hypercharge of YH = 2 , and the downstairs component of the scalar field has a weak 1 isospin, T3 = − 2 , we find that the ground state is electrically neutral, Q = 0. In 2012, the ATLAS and CMS collaborations at the Large Hadron Collider (LHC) experimentally detected the Higgs boson with a mass of
MH = 125.6 ± 0.3 GeV, (6) giving a value of µ ' 177.6 GeV, (7) and a subsequent vacuum expectation value: v ≈ 144 GeV.
The Gauge Boson Masses
Before the SSB of the electroweak gauge group, we have the following form of the Lagrangian involving the gauge bosons and the Higgs field
1 1 L = − (W a )2 − B2 + (D Φ)†(D Φ) + µ2Φ†Φ − λ(Φ†Φ)2, (8) 4 µν 4 µν µ µ
a where Bµ is the hypercharge gauge boson, with Bµν = ∂µBν − ∂νBµ, and Wµ are the SU(2) gauge a † 2 bosons, with Wµν their field strengths. The normalization of the λ(Φ Φ) term is conventional. The
2 / 8 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 covariant derivative acting on the Higgs doublet is
1 D Φ = ∂ Φ − igW aT aΦ − ig0B Φ. (9) µ µ µ 2 µ
0 1 Here, g and g are the SU(2) and U(1)Y couplings, respectively. The factor of 2 in the BµΦ coupling 1 comes from the Higgs multiplet having hypercharge YH = 2 . The Higgs potential V (Φ) = −µ2|Φ|2 + λ|Φ|4 induces a vev for Φ, which we can take to be real and in the lower component without loss of generality. Thus we can expand ! πaT a 0 Φ = exp 2i , (10) v √v + √h 2 2 with the vev µ v = √ , (11) λ and T a = 1 σa the canonically normalized SU(2) generators. The factors of √1 in this equation 2 2 convert between the canonical normalization of a complex scalar (Φ) and a real scalar (h). It is simplest to study this theory in unitary gauge, so we set the Goldstone bosons πa = 0, a = 1, 2, 3. Plugging in the vev we get the mass terms
2 ∆L = |DµΦ| ! 1 a a 1 0 bµ b 1 0 µ 0 = (0 v) gW T + g Bµ gW T + g B 2 µ 2 2 v
2 g0 3 1 2 ! g0 3 1 2 ! ! v Bµ + W W − iW Bµ + W W − iW 0 = g2 (0 1) g µ µ µ g µ µ µ 8 1 2 g0 3 1 2 g0 3 Wµ + iWµ g Bµ − Wµ Wµ + iWµ g Bµ − Wµ 1 2 " 0 2# v 2 2 g = g2 W 1 + W 2 + B − W 3 . (12) 8 µ µ g µ µ
± 0 1 2 3 We define four vector fields, Wµ ,Zµ and Aµ, as the following linear combinations of Wµ ,Wµ ,Wµ and Bµ 1 1 W ± = √ W 1 ∓ iW 2 , with mass M = gv; (13) µ 2 µ µ W 2
0 q g g 3 1 2 02 Zµ = p Bµ − Wµ , with mass MZ = g + g v. (14) g2 + g02 g 2 0 The fourth vector field, orthogonal to Zµ, remains massless
0 g g 3 Aµ = p Bµ + Wµ , with mass MA = 0. (15) g2 + g02 g
We will identify this field with the electromagnetic vector potential, associated with the gauge
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group U(1)EM, which in fact, came from the SSB of SU(2)L × U(1)Y → U(1)EM. Let us express the relationship between the hypercharge Y and the electric charge Q. A repre- 3 3 sentation of SU(2)L × U(1)Y has some matrix T associated with Wµ and a number Y associated with Bµ. In the fundamental representation of SU(2) we have,
! 3 1 1 1 0 T = σ3 = (16) 2 2 0 −1
For a fermion field belonging to a general SU(2) representation, with U(1) charge Y , the co- variant derivative takes the form
a a 0 Dµ = ∂µ − igWµ τ − ig YBµ. (17)
Expanding this out in terms of the mass eigenstate fields, this becomes
1 1 2 2 3 3 0 Dµ = ∂µ − ig(Wµ T + Wµ T ) − igWµ T − ig BµY 1 g0 = ∂ − ig W +T + + W −T − − ig W 3T 3 + B Y 1 , (18) µ µ µ µ g µ where T ± = (T 1 ± iT 2). The normalization is chosen so that
1 T ± = (σ1 ± iσ2) = σ±. (19) 2
To simplify expression Eq. (18) further, we define a parameter called the weak mixing angle 3 or the Weinberg angle, θW , to be the angle that appears in the change of basis from (Wµ ,Bµ) to (Zµ,Aµ) ! ! ! Z cos θ − sin θ W 3 µ = W W µ , (20) Aµ sin θW cos θW Bµ with g0 tan θ = . (21) W g Then the covariant derivative takes the form
+ + − − 3 3 Dµ = ∂µ − ig Wµ T + Wµ T − ieAµ(T + Y 1) − ieZµ(cot θW T − tan θW Y 1). (22)
We have identified the coefficient of the electromagnetic interaction as the electron charge e,
gg0 e = p , (23) g2 + g02 and Q = T 3 + Y 1. (24)
4 / 8 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 measures the electric charge. For example,
! ! 1 1 ! ! ! 0 3 0 2 − 2 0 0 0 Q = T + YL12×2 = 1 1 = − (25) eL eL 0 − 2 − 2 eL eL shows that the left-handed electron has electric charge −1.
With these definitions, the kinetic terms in the Lagrangian for Zµ and the photon Aµ are
1 2 1 2 1 2 µ L = − F − Z + m Z Zµ, (26) kin 4 µν 4 µν 2 Z with 1 MZ = gv, (27) 2 cos θW and
Zµν = ∂µZν − ∂νZµ,Fµν = ∂µAν − ∂νAµ. (28)
We see that the couplings of all of the weak bosons are described by two parameters: the well- measured electron charge e, and a new parameter θW . The couplings induced by W and Z exchange will also involve the masses of these particles. However, these masses are not independent, since it follows from Eqs. (13) and (14) that
MW = MZ cos θW . (29)
Already there is an unambiguous prediction from the theory: the W ± bosons should be lighter than the Z boson. All the effects of W and Z exchange process, at least at tree level, can be written in terms of the three basic parameters e, θW , and MW . Upon using the experimentally measured masses of W and and Z bosons we get the value for the Weinberg angle ◦ θW ≈ 28.17 . (30)
Let us note that the specific value of the Weinberg angle is not a prediction of the Standard Model; it is an open and unfixed parameter. This parameter is constrained and predicted through −1 other measurements of Standard Model quantities, for instance, θW = cos (MW /MZ ). Currently we do not know why the measured value of θW is what it is - we simply accept it as a given fact of our Universe.
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The Lagrangian for Gauge Bosons
A straightforward calculation gives the following Lagrangian for the gauge bosons
1 1 1 1 L = − F 2 − Z2 + m2 ZµZ − (∂ W + − ∂ W +)(∂ W − − ∂ W −) gauge 4 µν 4 µν 2 Z µ 2 µ ν ν µ µ ν ν µ 2 + − h + − + − +mW Wµ Wµ − ie cot θW ∂µZν(Wµ Wµ − Wν Wν )
+ − − + + + − + i +Zν(−Wµ ∂νWµ + Wµ ∂νWµ + Wµ ∂µWν − Wµ ∂µWν )
h + − + − −ie ∂µAν(Wµ Wν − Wν Wµ )
+ − − + + − − + i +Aν(−Wµ ∂νWµ + Wµ ∂νWµ + Wµ ∂µWν − Wµ ∂µWν ) 2 2 1 e + − + − 1 e + − + − − 2 Wµ Wµ Wν Wν + 2 Wµ Wν Wµ Wν 2 sin θW 2 sin θW 2 2 + − + − 2 + − + − −e cot θW (ZµWµ ZνWν − ZµZµWν Wν ) + e (AµWµ AνWν − AµAµWν Wν ) 2 h + − − + + − i +e cot θW AµWµ Wν Zν + AµWµ ZνWν − Wµ Wµ AνZν . (31)
µ+ ν− λ The Feynman rules can be read off from this Lagrangian. For example, the vertex W (p1)W (p2)Z (p3) µν λ with all momenta incoming is given in Fig. 1 and it has the contribution −ie cot θW [g (p1 − p2) + νλ µ λµ ν g (p2 − p3) + g (p3 − p1) ].
µ+ ν− λ Figure 1: The vertex W (p1)W (p2)Z (p3) with all momenta incoming. It has the contribution µν λ νλ µ λµ ν −ie cot θW [g (p1 − p2) + g (p2 − p3) + g (p3 − p1) ].
α β µ+ ν− 2 2 αµ βν The Z Z W W u vertex is given in Fig. 2 and it has the contribution ie cot θW [g g + gανgβµ − 2gαβgµν].
The Higgs Lagrangian
Now let us return to the field h. Even in unitary gauge (πa = 0, a = 1, 2, 3), this field, known as the Higgs boson, is still present. Note that while Φ, the Higgs doublet, has charges under the weak- and hypercharge-gauge groups, the Higgs boson h does not. Expanding out the Lagrangian, we find
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α β µ+ ν− 2 2 αµ βν αν βµ Figure 2: The Z Z W W u vertex. It has the contribution ie cot θW [g g + g g − 2gαβgµν]. that the terms involving the Higgs boson are
2 2 2 1 2 mh 3 g mh 4 LHiggs = − h( + mh)h − g h − 2 h 2 4mW 32 mW h 1 h2 1 +2 m2 W +W µ− + m2 Z Zµ + m2 W +W µ− + m2 Z Zµ , (32) v W µ 2 Z µ v W µ 2 Z µ where √ mh = 2µ. (33)
Using 2m sin θ v = W W , (34) e e µν the Feynman rule for a Higgs boson interacting with two W bosons (see Fig. 3) is i mW g . sin θW 2 e mZ µν e µν For a Higgs boson and two Z bosons is (see Fig. 4) i g = i 2 mW g . sin θW mW sin θW cos θW
Figure 3: The Feynman rule for a Higgs boson interacting with two W bosons.
2 2 2 2 The Higgs mass is mh = 2λv = 2µ is unrelated to other three parameters e, sin θW and mW . 0 Note that we started with four parameters µ, λ, g, g and ended up with four: e, θW , mh and mW .
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Figure 4: The vertex corresponding to a Higgs boson and two Z bosons.
References
[1] M. E. Peskin and D. Schroeder, Introduction to Quantum Field Theory, Westview Press (1995).
[2] M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press (2013).
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