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Home , Higgs mechanism, Mass generation

Spontaneous breaking Breaking a continuous symmetry The

The Higgs mechanism . . . or the Brout-Englert-Higgs. . . mechanism

Harri Waltari

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

Autumn 2018

H. Waltari The Higgs mechanism Spontaneous Breaking a continuous symmetry The Higgs mechanism Contents

Up to now we have gone through everything except how the particles get their . Just putting them in by hand breaks gauge invariance on which the theory is otherwise based and works well. In this lecture we shall discuss the spontaneous breaking of symmetries discuss the Higgs mechanism in gauge theories discuss the generation in the This lecture corresponds to chapter 17.5 of Thomson’s book.

H. Waltari The Higgs mechanism Spontaneous symmetry breaking Breaking a continuous symmetry The Higgs mechanism The vacuum state may break the symmetry

The idea of spontaneous symmetry breaking is that the full symmetry of a system often does not appear in nature: For instance we believe that physics is invariant under rotations, but there are all kinds of preferred directions In these cases the solutions do not exhibit the full symmetry of the theory (Lagrangian) Consider a symmetry transformation U = 1 + iG If the vacuum state is symmetric (i.e. does not change under the transformation) U|ψ0i = |ψ0i, the generator annihilates the vacuum state G|ψ0i = 0 If the theory is symmetric [G, H] = 0 but the vacuum state is not G|ψ0i= 6 0, there are degenerate states with the vacuum as H(U|ψ0i) = UH|ψ0i = E0U|ψ0i and since G|Ψ0i= 6 0, the state U|ψ0i = (1 + iG)|ψ0i is a different state from |ψ0i

H. Waltari The Higgs mechanism Spontaneous symmetry breaking Breaking a continuous symmetry The Higgs mechanism A real scalar field describes a neutral -zero particle

We shall first discuss spontaneous symmetry breaking in a very simple context. We shall look at a real scalar field: Lagrangian for a real scalar field 1 µ 1 2 2 λ 4 L = 2 ∂ ϕ∂µϕ − 2 µ ϕ − 4 ϕ

The scalar field satisfies a reality condition ϕ∗(x) = ϕ(x) Notice that this Lagrangian is not invariant under (global or local) U(1) ⇒ the field cannot be charged

We impose a Z2 symmetry under ϕ(x) → −ϕ(x), hence only even powers of ϕ allowed in the Lagrangian

H. Waltari The Higgs mechanism Spontaneous symmetry breaking Breaking a continuous symmetry The Higgs mechanism The ”wrong sign” mass term will break the symmetry spontaneously

1 2 2 λ 4 We call V (ϕ) = 2 µ ϕ + 4 ϕ the scalar potential The potential is stable (has a minimum) if λ > 0 If µ2 > 0 the interpretation is straightforward: the equation of motion is the Klein-Gordon equation and the ϕ4-term describes scalar self-interactions, |µ| is the mass of the scalar particle If µ2 < 0, the interpretation is far from straightforward QFT: Particles are excitations of the vacuum state ⇒ Find the state of the lowest energy and expand your theory around it For µ2 < 0 the lowest energy configuration is such that the field ϕ has a constant value ±p−µ2/λ (notice that ϕ → −ϕ takes from one solution to the other) Picking one solution as the vacuum state breaks the symmetry

H. Waltari The Higgs mechanism Spontaneous symmetry breaking Breaking a continuous symmetry The Higgs mechanism Expanding around the vacuum gives the theory in the broken phase

We now write ϕ(x) = v + η(x), where v = +p−µ2/λ The kinetic terms remain unchanged, but the scalar potential becomes 1 1 λ V (η) = µ2v 2 + µ2vη + µ2η2 + (v 4 + 4v 3η + 6v 2η2 + 4vη3 + η4) 2 2 4

1 2 2 λ 4 Interpretation: 2 µ v + 4 v is the energy of the vacuum state, the 1 2 3 2 2 2 2 linear term in η vanishes, 2 µ + 2 λv η = −µ η describes a particle with a mass p−2µ2 (notice: the mass term has the correct 3 sign!), λvη describes a three-scalar interaction that breaks the Z2 λ 4 symmetry and is not present in the unbroken phase and 4 η is the original four-scalar interaction The field value v is an order parameter of the phase transition, the symmetry breaking η3 interaction vanishes in the limit v → 0

H. Waltari The Higgs mechanism Spontaneous symmetry breaking Breaking a continuous symmetry The Higgs mechanism Spontaneous symmetry breaking exists in many systems

The previous example is the most simple case of spontaneous symmetry breaking presented in a setting The field value v is known as the of the field In a more general setting this is known as the Ginzburg-Landau theory of phase transitions, originally presented in 1950 to describe The same mathematics appears in other kinds of systems, too, ferromagnetism being one familiar example If the broken symmetry is discrete (as was in the previous example), different parts of the system may pick different solutions and there is a thin layer, where the order parameter changes from one vacuum solution to the other — these are known as domain walls

H. Waltari The Higgs mechanism Spontaneous symmetry breaking Breaking a continuous symmetry The Higgs mechanism A complex scalar field describes a charged particle

Next we shall turn to theories with continuous symmetries, as the simple example we shall consider a complex scalar field: The Lagrangian for a complex scalar field µ ∗ 2 ∗ ∗ 2 L = ∂ ϕ ∂µϕ − µ ϕ ϕ − λ(ϕ ϕ)

This Lagrangian is invariant under a global U(1) symmetry ϕ → eiqαϕ ⇒ ϕ∗ → e−iqαϕ∗, q being the charge of the field ϕ We could also gauge the symmetry, but global and local symmetries have profound differences in systems with spontaneous symmetry breaking If µ2 > 0, the equation of motion is the Klein-Gordon equation and the theory describes a scalar particle with a mass |µ|

H. Waltari The Higgs mechanism Spontaneous symmetry breaking Breaking a continuous symmetry The Higgs mechanism There is a continuous set of degenerate vacua

If µ2 < 0, the minimum energy configuration is such that 2 µ2 2 |ϕ| = − 2λ ≡ v The vacuum state is not symmetric as veiqα 6= v Notice that the symmetry transformation is a rotation in the complex plane, there is a continuous set of vacua that are connected by the symmetry transformation

H. Waltari The Higgs mechanism Spontaneous symmetry breaking Breaking a continuous symmetry The Higgs mechanism The spectrum contains a massless scalar

We use the previous prescription and expand the theory in terms of ϕ(x) = √1 (v + h(x) + ia(x)), where h(x) and a(x) are real scalar fields: 2 1 µ The kinetic terms come out as 2 ∂ h∂µh and similarly for a(x) The scalar potential becomes

1 V (h, a) = µ2(v 2 + 2vh + h2 + a2) + λ(v 2 + 2vh + h2 + a2)2 2 1 λ λ = (2λv 2)h2 + h2a2 + λv(ha2 + h3) + (h4 + a4) 2 2 4 omitting constants and linear terms in the second line √ We have a massive particle h with a mass 2λv 2 and a a, the massless particle corresponds to motion along the degenerate vacua The massless particle is known as the Goldstone

H. Waltari The Higgs mechanism Spontaneous symmetry breaking Breaking a continuous symmetry The Higgs mechanism Goldstone have not been observed

It is possible to show in general that to every generator of a continuous symmetry that is broken spontaneously (i.e. generators for which G|ψ0i= 6 0), there exists exactly one massless scalar in the particle spectrum This result is known as the Goldstone theorem and it was conjectured by Goldstone in 1961 (Nuovo Cimento 19 (1961) 154) and proved by Goldstone, Weinberg and Salam (Phys. Rev. 127 (1962) 965) In general finding massless scalars that interact with known particles is easy and constraints can be obtained from e.g. stellar cooling ⇒ No evidence for such particles The spontaneous breaking of symmetries was considered to be unviable for this reason for a couple of years

H. Waltari The Higgs mechanism Spontaneous symmetry breaking Breaking a continuous symmetry The Higgs mechanism Brout, Englert and Higgs considered scalar QED

We next turn to spontaneous symmetry breaking in gauge theories and discuss the original work of Brout, Englert and Higgs in 1964. (Phys. Rev. Lett. 13 (1964) 321/508)

The Lagrangian for scalar QED µ ∗ 2 ∗ ∗ 2 1 µν L = D ϕ Dµϕ − µ ϕ ϕ − λ(ϕ ϕ) − 4 F Fµν , where Dµ = ∂µ − ieAµ is the covariant derivative and Fµν = ∂µAν − ∂ν Aµ is the field strength tensor

As the scalar potential is the same, there is again a symmetric phase with µ2 > 0 and a broken phase with µ2 < 0 2 µ2 In the broken phase the minimum of the potential is at |ϕ| = − 2λ

H. Waltari The Higgs mechanism Spontaneous symmetry breaking Breaking a continuous symmetry The Higgs mechanism It is possible to gauge the away

In gauge theories you have the freedom of choosing the gauge — in the case of U(1) this means that you may rotate the axes of the complex plane freely It is possible to impose such a gauge condition that you eliminate any motion along the minimum of the potential, this eliminates the Goldstone boson Hence the Goldstone is an unphysical degree of freedom in gauge theories Such a gauge choice is called the unitary gauge Gauging away the Goldstone boson is known as the Brout-Englert-Higgs mechanism — this will have further consequences

H. Waltari The Higgs mechanism Spontaneous symmetry breaking Breaking a continuous symmetry The Higgs mechanism The gauge field will acquire a mass in the broken phase

We’ll now expand the covariant derivative:

µ ∗ µ ∗ µ ∗ µ ∗ 2 ∗ µ D ϕ Dµϕ = ∂ ϕ ∂µϕ + ie(A ϕ ∂µϕ − ∂ ϕ Aµϕ) + e ϕ ϕA Aµ

In the broken phase the first term produces the canonical kinetic terms for the scalar fields and the last term becomes a mass term for the gauge field (mass being ev) in addition to containing the regular µ two scalar - two interactions and a hA Aµ term not present in the symmetric phase The second term contains interactions between the scalar field and µ the photon but also a mixing term evA ∂µa (for h this cancels) implying that the Goldstone and the original photon are not the proper degrees of freedom

H. Waltari The Higgs mechanism Spontaneous symmetry breaking Breaking a continuous symmetry The Higgs mechanism The unitary gauge eliminates the Goldstone boson

The kinetic term for the Goldstone, the mixing term and the mass term can be combined to

1 1 1  1 2 ∂µa∂ a + evAµ∂ a + e2v 2AµA = e2v 2 A + ∂ a 2 µ µ 2 µ 2 µ ev µ

0 1 Choosing the gauge in a way that Aµ = Aµ + ev ∂µa eliminates the Goldstone boson This gauge choice makes the Goldstone the third polarization state of the massive gauge boson

H. Waltari The Higgs mechanism Spontaneous symmetry breaking Breaking a continuous symmetry The Higgs mechanism Summary: The Higgs mechanism

The gauge symmetry is broken by the vacuum expectation field of a charged scalar field In the broken phase the gauge boson is massive and we may choose a gauge, where the Goldstone boson is ”eaten ” to become the longitudinal polarization state The h boson is physical and has a mass term with the correct sign In the broken phase there are new interaction terms that are not 3 µ present in the symmetric phase (h and hA Aµ), these break charge conservation and are proportional to v

H. Waltari The Higgs mechanism Spontaneous symmetry breaking Breaking a continuous symmetry The Higgs mechanism In the Standard Model SU(2)×U(1) is broken by the Higgs mechanism

We would want to use the Higgs mechanism to break the electroweak symmetry so that the weak gauge bosons become massive In order to break a symmetry we must choose a scalar field that is charged under the gauge group The vacuum expectation value breaks all of the quantum numbers that are nonzero for the corresponding field ⇒ to get unbroken U(1)em we should have a electrically neutral field involved The minimal solution is to take a SU(2) doublet with hypercharge Y = ±1, in the Standard Model the convention is Y = +1

H. Waltari The Higgs mechanism Spontaneous symmetry breaking Breaking a continuous symmetry The Higgs mechanism The Standard Model uses the Ginzburg-Landau potential

We introduce the Higgs field as

φ+ ϕ = φ0

Also in the Standard Model we assume that the scalar potential is V (ϕ) = µ2ϕ†ϕ + λ(ϕ†ϕ)2 With µ2 < 0 the minimum of the potential is at |ϕ|2 = −µ2/2λ and we assign it to the real part of the neutral component, i.e.

 0  hϕi = √ v/ 2

with v real (in the SM v = 246 GeV) 1 τ1, τ2 or τ3 − 1 do not annihilate this, but τ3 + 2 1 does ⇒ three (w) Goldstones, I3 + Y /2 the only conserved charge

H. Waltari The Higgs mechanism Spontaneous symmetry breaking Breaking a continuous symmetry The Higgs mechanism The W - and Z-bosons are massive in the unitary gauge

We choose the unitary gauge so the Higgs field becomes

1  0  ϕ = √ 2 v + h(x) The mass terms for the gauge bosons come from the part of the covariant derivative, where the gauge field is squared

 0 −   0 +  1 gW3 − g B gW gW3 − g B gW (0 v+h) + 0 − 0 8 gW −gW3 − g B gW −gW3 − g B  0  1 × = (v + h)2(g 2W +W − + (gW − g 0B)2) v + h 8 3

We see that the W -boson gets a mass gv/2 and also the Z-boson is massive, after normalizing the state properly the mass is pg 2 + g 02v/2, the photon gets no mass term

H. Waltari The Higgs mechanism Spontaneous symmetry breaking Breaking a continuous symmetry The Higgs mechanism

The Standard Model predicts the ratio mW /mZ

2 2 mW g 2 From the masses we immediately get 2 = 2 02 = 1 − sin θW mZ g +g 2 Since sin θW can be measured independently, this is one test of the symmetry breaking mechanism 2 2 2 We have 1 − sin θW ' 0.77 and mW /mZ ' 0.78, the difference can be accounted by one-loop corrections to the gauge boson masses If the Higgs was not a SU(2) doublet, this ratio would be completely different

H. Waltari The Higgs mechanism Spontaneous symmetry breaking Breaking a continuous symmetry The Higgs mechanism The couplings are proportional to masses

You can also notice that the mass terms imply a hW +W − and hZZ coupling, which are proportional to the gauge boson masses The Higgs decays to these channels were found to be compatible with expectations, so this is a strong indication that the particle discovered is involved in electroweak symmetry breaking These couplings are dictated by the gauge symmetry — there is no freedom

H. Waltari The Higgs mechanism Spontaneous symmetry breaking Breaking a continuous symmetry The Higgs mechanism masses can be generated with Yukawa interactions

Since left- and right-handed are doublets and singlets under SU(2), the mass term mψψ = m(ψLψR + ψR ψL) breaks the gauge symmetry The Higgs field can help: Since it is a doublet, you may contract it with the left-handed fermion doublet to get a singlet, which can be combined with the right-handed singlet to a SU(2) invariant term Using the doublet L = (ν `−)T and the doublet Q = (u d)T and the right-handed singlets, the following gauge invariant combinations can be found

L = y`(L · ϕ)`R + yd (Q · ϕ)dR + h.c.

When expanded around the vacuum state, we get mass terms y√`v `` 2 and y√d v dd ⇒ the fermion masses are related to the couplings by 2 √ mf = yf v/ 2

H. Waltari The Higgs mechanism Spontaneous symmetry breaking Breaking a continuous symmetry The Higgs mechanism The charge-conjugated Higgs doublet can generate masses for up-

We were able to generate masses for charged and down-type quarks The masses are free parameters, they do not come from any symmetry (but are allowed by the symmetries) We still need a mass term for up-type quarks, that can be achieved c ∗ by using the charge-conjugated Higgs doublet ϕ = −iσ2ϕ c Now we can write yu(Q · ϕ )uR , which is gauge invariant and gives a similar mass term for up-type quarks Dirac masses can be generated for neutrinos with ϕc if right-handed neutrinos are included

H. Waltari The Higgs mechanism Spontaneous symmetry breaking Breaking a continuous symmetry The Higgs mechanism The Higgs couplings to fermions are proportional to masses

The mass terms imply that the Higgs couples√ to fermions with a coupling proportional to the mass yf = 2mf /v The Higgs couplings are flavor conserving (flavor diagonal)

The largest coupling is to the top quark, yt ≈ 0.9, while the smallest −6 is the electron Yukawa coupling ye ≈ 3 × 10 , the reason for this hierarchy is not known

H. Waltari The Higgs mechanism Spontaneous symmetry breaking Breaking a continuous symmetry The Higgs mechanism Summary

The vacuum state of the system can break a symmetry, this leads to either domain walls (discrete symmetries) or Goldstone bosons (continuous symmetries) In gauge theories the Goldstone bosons can be gauged away The Higgs mechanism can generate masses to gauge bosons, the masses are given in terms of the parameters of the gauge group The Higgs field can also generate fermion masses but they are free parameters The Higgs couplings to other particles are proportional to the masses of the particles

H. Waltari The Higgs mechanism