Spontaneous symmetry breaking Breaking a continuous symmetry The Higgs mechanism The Higgs mechanism . or the Brout-Englert-Higgs. mechanism Harri Waltari University of Helsinki & Helsinki Institute of Physics University of Southampton & Rutherford Appleton Laboratory Autumn 2018 H. Waltari The Higgs mechanism Spontaneous symmetry breaking Breaking a continuous symmetry The Higgs mechanism Contents Up to now we have gone through everything except how the particles get their masses. 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 mass generation in the Standard Model 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) Uj 0i = j 0i, the generator annihilates the vacuum state Gj 0i = 0 If the theory is symmetric [G; H] = 0 but the vacuum state is not Gj 0i 6= 0, there are degenerate states with the vacuum as H(Uj 0i) = UHj 0i = E0Uj 0i and since GjΨ0i= 6 0, the state Uj 0i = (1 + iG)j 0i is a different state from j 0i H. Waltari The Higgs mechanism Spontaneous symmetry breaking Breaking a continuous symmetry The Higgs mechanism A real scalar field describes a neutral spin-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 @ '@µ' − 2 µ ' − 4 ' The scalar field satisfies a reality condition '∗(x) = '(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 '(x) ! −'(x), hence only even powers of ' 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 (') = 2 µ ' + 4 ' 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 '4-term describes scalar self-interactions, jµj 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 ' has a constant value ±p−µ2/λ (notice that ' ! −' 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 '(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 particle physics setting The field value v is known as the vacuum expectation value 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 superconductivity 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 = @ ' @µ' − µ ' ' − λ(' ') This Lagrangian is invariant under a global U(1) symmetry ' ! eiqα' ) '∗ ! e−iqα'∗, q being the charge of the field ' 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 jµj 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 j'j = − 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 '(x) = p1 (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 p We have a massive particle h with a mass 2λv 2 and a massless particle a, the massless particle corresponds to motion along the degenerate vacua The massless particle is known as the Goldstone boson H. Waltari The Higgs mechanism Spontaneous symmetry breaking Breaking a continuous symmetry The Higgs mechanism Goldstone bosons 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 Gj 0i= 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 ' Dµ' − µ ' ' − λ(' ') − 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 j'j = − 2λ H. Waltari The Higgs mechanism Spontaneous symmetry breaking Breaking a continuous symmetry The Higgs mechanism It is possible to gauge the Goldstone boson 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 ' Dµ' = @ ' @µ' + ie(A ' @µ' − @ ' Aµ') + e ' 'A 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 photon 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.