As usual, these notes are intended for use by class participants only, and are not for circulation. Week 8: Lectures 15, 16 Masses for Vectors: the Higgs mechanism April 6, 2012 The momentum-space propagator free field Green function) for a gaugefixed massless vector field, Eq. 14) is G µν k) = 1 k 2 + iɛ g µν + kµ k ν ). 26) k 2 + iɛ) The propagator for Proca vectors, is readily found from the Lagrange density, Eq. 18): G µν k) = 1 k 2 + iɛ g µν + kµ k ν m 2 ). 27) Note that k µ G µν k) = 0, consistent with the automatic transversality of the Proca field. On the other hand, the second term in this propagator grows quadratically in the momentum, k. This is in contrast to the order zero behavior of massless propagators in covariant gauges, Eq. 26). Looking ahead to the perturbative expansion for theories with vector particles, we can see that Proca fields will lead to special challenges, because loop integrands will grow rapidly in general. For power counting purposes, the factor 1/m 2 acts like a coupling with negative dimensions of mass, giving a power counting similar to that of nonrenormalizable theories. The way we meet the challenges of the Proca propagator is to make our massive vector fields out of massless vector fields, while preserving the underlying gauge invariance of the physical predictions of the theory as we shall see this is consistent with gauge fixing. This procedure is called spontaneous symmetry breaking. 19
The mode of spontaneous symmetry breading implemented in the Standard Model is illustrated in the simplest context by the Abelian Higgs model. In this model, we couple a massless vector Maxwell) field to a charged i.e. complex) scalar field with a wrong sign mass term. The complete density is given by with L U1) = 1 4 F µνf µν +id µ [A])Φ) id µ [A])Φ) V Φ 2 ) D µ [A] = µ + iga µ, V Φ 2 ) = µ 2 Φ 2 + λ Φ 4. The potential is illustrated in the figure, and the sign of the mass term results in a minimum away from Φ = 0. In fact, the potential is minimized along a circle in complex Φ space see the figure), whose position is determined by V Φ 0 ) = 0 Φ 0 = µ 2 /2λ. The vacuum state of the Φ field can be anywhere around the circle. The two degrees of freedom in Φ can be thought of as rotations around the circle, which don t change the potential, and translations in the radial direction, which climb the walls of the potential. Expanding the field around the minimum, the latter correspond to massive excitations, the former to massless excitations. To make these degrees of freedom more transparent, we reparameterize the scalar field standard complex form as a phase times a modulus. Both are position-dependent This form will suggest a gauge transformation: ) v + ηx) Φx) = e iζx)/2v. 2 In this form, we have traded the complex field Φ for two real fields, ζx) and ηx). Although ζx) appears in an exponential, at the end of the day the Lagranian depends on it only through its derviatives. v is the vacuum expectation value of Φ; ζx): the phase, and ηx) measures fluctuations about the minimum at Φ = exp[iζx)] v/ 2). 20
We now re-express the Lagrangian in terms of ζx) and ηx) L U1) = 1 4 F µνf µν + 1 2 id µ[a]v + η)) id µ [A]v + η)) V v + η) 2 /2) + µ ζ) 2 + 2g v + η) µ ζ)a µ ζx) is a massless field, and couples linearly to A. The classical vacuum of our Lagrange density is at η = 0 with any constant ζ this is the spontaneous symmetry breaking, referring to the symmetry of the potential under changes in ζx). The kinetic term, ζ) 2, however, increases the energy whenever ζ changes from point to point. Thus the true minimum energy corresponds to ζ equal to a constant, any constant, but only one constant, everywhere and at all times. Whatever vacuum state the system chooses breaks phase invariance, even though the Lagrange density remains completely invariant. The presence of a term that is proportional to A and v ζ suggests that the two fields mix, and indeed this is what happens. The net effect of ζ is to add a mass and equivalently an extra degree of freedom to the hitherto massless vector field A µ, as we shall see. 21
Figure: the potential in the Φ-plane. V! ) Re!) Im!) 2! = µ 2" The acquisition of a mass by the vector field is made manifest in the socalled unitary gauge, in which ζx) disappears. That is, we make the local gauge transformation defined by the phase of the scalar field, A µx) =A µ x)+ µ ζx) Φ x) = e iζx)/2v Φx). The Lagranian in unitary gauge F 2 is gauge invariant) is then given by id µ [A])Φ) id µ [A])Φ) = µ η) 2 + g2 2 η2 A 2 + g 2 vηa + g 2 v2 A 2 V Φ 2 ) = µ4 2λ + µ2 η 2 + µ λη 3 + λ 4 η4. 22
The presence of the massless field, whether absorbed by a Higgs mechanism or not, is an example of the Goldstone theorem, which states that spontaneous symmetry breaking implies the presence of a massless scalar field, a Goldstone boson, here ζx). In the Higgs mechanism, the Goldstone field ζx) is absorbed into the gauge field, which acquires a Proca-like mass, m 2 A = gv 2. In this context, ζ is often referred to as a would-be Goldstone boson The Higgs Mechanism in the Standard Model The Gauge theory the underlies the Standard Model is a direct product of three theories, each with its own set of vector fields: SU3) SU2) L U1) A µ a B µ b C µ, where the subscript L on the SU2) L refers to its coupling to fermions. The fields are commonly referred to as gluons, A µ a, 1 = 1... 8, weak vector bosons, B µ b, b =1, 2, 3, and the hypercharge U1) field, Cµ. The fermions of the standard model fall into three generations : k = 1, 2, 3, of left- and right-handed quark fields, q L,Rx), k) and left- and right-handed lepton fields, l k) L,Rx), L, R project Weyl spinors, but in the Standard Model picture, all underlying fields are Dirac, ψ L = 1 2 1 γ 5) ψ, ψ R = 1 2 1 + γ 5) ψ. The SU2) L content of the generations is a series of two-component weak isospin, I = 1/2 representations, ul ) cl ) tl ) νe,l ) νµ,l ) ντ,l ) d L s L b L e L µ L τ L The corresponding right-handed fields always SU2) L singlets. 23
Given this assignments of transformation under SU2) L, the covariant derivatives for quarks and leptons are: q k) L : D µ = µ + ig s A µ,a T a D) σ b + igb µ,b 2 + Y ig 2 C µ q k) R : D µ = µ + ig s A µ,a T D) a + ig Y 2 C µ l k) L : D µ = µ + igb µ,b σ b 2 + ig Y 2 C µ l k) R : D µ = µ + ig Y 2 C µ, where both left- and right-handed quark fields only) couple to gluons in the same fashion. The left-handed fields couple to the weak vector bosons according to the two-dimensional defining representation of SU2) L, and each field couples to the weak U1) vector via its hypercharge, Y, which we ll specify shortly. Spontaneous symmetry breaking is implemented in the Standard Model through a scalar Higgs field, which is an SU2) L I = 1 doublet. That 2 is the Higgs field transforms according to a two-dimensional representation of SU2) L, just like the left-handed fermions, although as a scalar field it does not possess handedness, Φ= φ + This completes the full set of fields for the Standard Model. φ 0 ). 24
As we noted above, electric charges are determined by a combination of hypercharges and SU2) L representations, specifically, Q EM = Y 2 + ISU2) L 3. This is how the values of hypercharge are determined. The list of hypercharges, Y, SU2) L and S3) representations are then U1) Y ) SU2) SU3) L 1/3 2 3 R 4/3, 2/3 1, 1 3 L 1 2 1 R 2, 0 1, 1 1 Φ 1 2 1 q k) q k) l k) l k) As in the abelian Higgs model, the Standard Model Lagrangian has a built-in vacuum expectation v, whose value at the minimum of the potential follows from an opposite-sign mass term, L Higgs = id µ [A]) Φ ) idµ [A]) Φ ) V Φ 2 ) V Φ 2 ) = µ 2 Φ 2 + λ Φ 4. Just as in the abelian model, v = µ/ 2λ. The mass content of the vectors is found by choosing unitary gauge, based on the parameterization of the Higgs field as an SU2) L phase times a specific I 3 = 1/2 field, chosen to represent the modulus, measured relative to the vacuum expectation value, v/ 2, Φx) =e iζ σ 2v 0 v+ηx) 2 ) The Higgs Lagrangian in unitary gauge is then L Higgs = 1 8 [ g 2 B 2 1 + B 2 2)+g C gb 3 ) 2] v+η) 2 + 1 2 µη) 2 µ 2 η 2 vλη 2 λ 4 η4 +const. 25
Vector bosons of definite mass and EM charge are found by combining the quadratic terms on the vector field with the Maxwell and Yang Mills Lagrange densities for C and B µ b, respectively, with the results, W µ ± = Bµ 1 B µ 2 ) 2 Z µ = C µ sin θ W + B µ 3 cos θ W γ µ = C µ cos θ W + B µ 3 sin θ W sin θ W = g g2 + g 2 M W = vg 2, M Z = M W cos θ W, M γ =0 This is the basis of our current description of weak and electromagnetic, as well as strong, interactions. 26