Non Abelian Higgs Mechanism

Transcription

1 Non Abelian Higgs Mechanism When a local rather than global symmetry is spontaneously broken, we do not get a massless Goldstone boson. Instead, the gauge field of the broken symmetry becomes massive, and the would-be Goldstone scalar becomes the longitudinal mode of the massive vector. This is the Higgs mechanism, and it works for both abelian and non-abelian local symmetries. In the non-abelian case, for each spontaneously broken generator T a of the local symmetry the corresponding gauge field A a µx) becomes massive. Example: SU) with a Higgs Doublet To illustrate the non-abelian Higgs mechanism, consider the example of SU) gauge theory coupled to a doublet of complex scalar fields Φ i x). In terms of canonically normalized fields, the Lagrangian is L = 4 F a µνf aµν + D µ Φ i D µ Φ i λ ) Φ i Φ i v, ) where D µ Φ i = µ Φ i + i gaa µ σ a ) i j Φj, D µ Φ i = µ Φ i i gaa µφ j σ a ) j i, F a µν = µ A a ν ν A a µ gɛ abc A b µa c ν. ) For v > the scalar potential has a local maximum at Φ i = while the minima form a spherical shell Φ i Φi = v /) in the C = R 4 field space; all such minima are related by the SU) symmetries to ) Φ = v. 3) Note that this vacuum expectation value spontaneously breaks the SU) symmetry down to nothing there is no subgroup of SU) which leaves this VEV invariant. Consequently, we expect all 3 vector fields A a µx) to become massive. In the process, 3 would-be Goldstone scalars should be eaten by the Higgs mechanism. Since the theory has complex or equivalently 4 real scalars, only one real scalar should survive un-eaten. Ironically, it is this un-eaten scalar σx) which is called the physical Higgs field.

2 To see how this works, let s fix the unitary gauge where Re Φ x) Im Φ x) Im Φ x), ) Φx) =, real φ r x) >, φ r x) 4) and then shift the φ r field by the VEV, φ r x) = v + σx). For v = such gauge fixing would be terribly singular, but it s perfectly OK for v and σx) < v = φ r x). In the unitary gauge, the physical Higgs field σx) is the only scalar field, the rest are frozen by the gauge-fixing conditions 4). In terms of σ, the scalar potential becomes V = λ ) Φ Φ v = λ vσ + σ ) λv = σ + λv σ3 + λ σ4 5) where the first terms is the mass term, mass = λv, while the remaining terms are selfinteractions. More interestingly, the covariant derivative of the Higgs doublet Φ becomes D µ Φ = hence µ σ ) ) ) + ig + A3 µ v + σ ) ) ) ) + ig A µ + ig i v + σ A µ i v + σ = i g A µ ia ) ) µ v + σ) µ σ i g, A3 µ v + σ) 6) D µ Φ D µ Φ = i g A µ ia µ) v + σ) + µ σ i g A3 µ v + σ) = g v + σ) A ) ) ) µ + A µ + g v + σ) A 3 µ) + µ σ). 7) The last term here is the kinetic term for the Higgs scalar σx), while the rest of the bottom line are mass terms for the vector fields and the interaction terms between the vectors and

3 the σ. Curiously, we get the same mass and similar interactions for all 3 vector fields A a µ: L g v + σ) A a µa aµ = M Aa µa aµ + g v 4 σaa µa aµ + g σ A a µa aµ ) where M = g v 4. 9) Example: SU) with a Higgs Triplet Now consider an example of a partially broken gauge symmetry, an SU) Higgsed down to a U) subgroup, or equivalently SO3) SO). This time, the scalar fields Φ a x) are real and form a triplet of the SU) rather than a doublet. Thus, L = 4 F a µνf aµν + D µφ a D µ Φ a λ Φ a Φ a v ), ) where D µ Φ a = µ Φ a gɛ abc A b µφ c, F a µν = µ A a ν ν A a µ gɛ abc A b µa c ν. ) Again, for v > the scalar potential V Φ) has a degenerate family of minima which form a spherical shell Φ a Φ a = v in the scalar field space R 3, and all such minima are equivalent by SU) = SO3) symmetries to Φ =. ) This time, this vacuum expectation value is invariant under an SO) subgroup of the SO3), or equivalently under an U) subgroup of the SU). Specifically, it s the SO) = U) generated by the T 3, the third component of the isospin T. Consequently, out of the 3 vector fields A a µ, we expect the A 3 µ to remain massless while the other fields A, µ should become massive. v 3

4 In the process, the Higgs mechanism should eat real scalar fields. Since we only have 3 real scalars to begin with, only one scalar should survive un-eaten the Physical Higgs field σx). To see how this works, we fix the unitary gauge Φ x) Φ x), Φ 3 x) >. 3) As usual, this gauge is badly singular at Φ =, but it s OK for the Φx) Φ. Shifting the Φ 3 x) by the VEV, we get Φ 3 x) = v + σx), where σx) is the physical Higgs scalar and also the only scalar remaining in the theory in the unitary gauge. In terms of the σx), the scalar potential becomes V σ) = λ vσ + σ ) = λv σ + λv σ3 + λ σ4, 4) where the first terms on the RHS gives the Higgs scalar mass = λv. More interestingly, the covariant derivative of the scalar triple Φ a x) becomes D µ Φ a = µ σ g A µ A µ A 3 µ v + σ = where is the cross product of two isovectors ga µv + σ) +ga µv + σ), µ σ 5) hence the covariant kinetic terms for the scalars become D µφ a D µ Φ a = µσ) + g v + σ) A ) ) ) µ + A µ. 6) As usual, the first term here is the kinetic term for the physical Higgs scalar σ, while the second term contains the mass terms for the vector fields, L M A ) ) ) µ + A µ, M = g v, 7) but only for the A µ and the A µ the third vector A 3 µx) remains massless. 4

5 The massless vector A 3 µx) is the gauge field of the un-higgsed SO) = U) subgroup of the SO3) = SU). Interpreting this gauge field as the EM field and hence the rescaled generator Q = gt 3 as the electric charge operator, we find that the physical Higgs field is electrically neutral while the massive vector fields have electric charges q = ±g. To be precise, the massive vector fields of definite charges are not the A µ and the A µ themselves but rather their linear combination W µ + = A µ ia ) µ and Wµ = A µ + ia ) µ of charges q = ±g. ) For completeness sake, let s re-express the theory at hand usually called the Georgi Glashow model) in terms of the physical fields of definite charges. Using U) covariant derivatives D µ W ± ν = µ W ± ν ± iga 3 µw ± ν, 9) we have W ± µν def = F µν if µν) ± = Dµ W ν ± D ν W µ ±, ) but F 3 µν = F µν + + g Im W + µ W ν ) where Fµν = µ A 3 ν ν A 3 µ. ) Consequently, the Lagrangian of the whole model the kinetic terms, the mass terms, and the interactions can be expressed as L = µσ) M σ σ 4 F µν F µν W + µνw µν + M W W + µ W ν λv σ3 λ σ4 + gv σ W µ + W µ + g σ W µ + W µ g F µν Im W +µ W ν) g Im W +µ W ν)). ) 5

6 General Case Let s take a closer look at eqs. 7) and 6), and focus on the mass terms for the vector fields. In both cases, we start with the kinetic terms for the original scalar fields Φ i x) or Φ a x), fix the unitary gauge, work through the algebra, and eventually obtain the kinetic term for the physical Higgs field σ, the mass terms for the vector fields or some of the vector fields and the interactions between the massive vectors and the Higgs σ. But is all we want are the mass terms for the vectors, we may simply freeze σx) : This would eliminate the interactions with the σ as well as the µσ) term, and all we would have left are the mass terms for the massive vectors. Note that freezing σx) is equivalent to freezing all the scalars at their VEVs, Φx) Φ. Consequently, to get the vector s masses we do not need to go through the details of the unitary gauge fixing, all we need are the scalar VEVs, then the kinetic terms for the frozen scalars D µ Φ D µ Φ or D µ Φ ) becomes the mass terms for the vectors. For example, for the SO3) triplet of real scalar fields from the second example D µ Φ a = gɛ abc A b µ vδ c3 = gvɛ ab3 A b µ, 3) L vector mass = Dµ Φ a) = gv) ɛ ab3 ɛ ac3 A b µa cµ = M = gv ) A µ A µ + A µa µ). 4) Likewise, for the SU) doublet of complex scalar fields from the first example, D µ Φ i = ig A a µ σ a) ij v δ j = igv ) A a µσ a) i, 5) D µ Φ i = igv A a µσ a) i, 6) L vector mass = D µ Φ i Dµ Φ i = g v = g v A a µa bµ A a µσ a) i A bµ σ b) i [ σ a σ b) = δab iɛ ab3] 6

7 = g v A a µa bµ δ ab since A a µa bµ is symmetric in a b. = M Aa µa aµ for M = gv. 7) This recipe freezing Φx) Φ to find the vector masses applies to any kind of gauge theory with scalars in any kinds of multiplets. Indeed, consider a general gauge symmetry G with generators ˆT a and gauge fields A a µx) a =,..., dimg)). Let scalars Φ α x) belonging to some multiplet m) of G develop non-zero vacuum expectation values Φ α. Then the covariant derivatives of these scalars D µ Φ α x) = µ Φ α x) + iga a µx) T a m)) α β Φβ x) ) become in the unitary gauge D µ Φ α x) = D µ Φ α + terms involving the physical scalars 9) where D µ Φ = iga a µx) Tm)) a α β Φ β. 3) In eq. 9), the terms involving the physical scalars and the physical scalar fields themselves depend on the details of the unitary gauge fixing. On the other hand, the covariant derivatives of the VEVs 3) depend only on the VEVs themselves. Moreover, such derivatives are linear functions of the vector fields with constant coefficients, so their squares become quadratic mass terms for the vectors, D µ Φ α D µ Φ α = iga a µ Φ ) β T a β m) α igabµ Tm)) a α γ Φ γ = A a µa bµ g Φ β T a m) Tm)) b β γ Φ γ by a b symmetry of the A a µa bµ 3) = Aa µa bµ g Φ β { T a m), T b m)} β γ Φ γ. In other words, L vector masses = M V ) ab A a µ A bµ, 3) 7

8 where the mass matrix for the gauge fields obtains as M V ) ab = g Φ β { T a m), T b m)} γ β Φ γ g Φ { T a m), T b m)} Φ. 33) To be precise, eq. 33) applies to Higgs VEVs belonging to a single multiplet of complex scalars. For a multiplet of real scalars, there is an extra factor due to different normalization of the VEVS, and for several Higgs multiplets with non-zero VEVs, the general formula is complex Higgs multiplets ) M ab V = g Φ { Tm) a, T m) b } Φ + g Φ m) real Higgs multiplets Φ m) Φ { T a m), T b m)} Φ. 34) In general, such mass matrix is not diagonal, and we need to diagonalize in order to find the physical vector masses. For example, in the Glashow Weinberg Salam theory of the weak and EM interactions it s explained in the next set of notes the mass matrix mixes the SU) gauge field W 3 µ and the U) gauge field B µ, and the mass eigenstates are the massless EM field A µ and the massive neural field Z µ involved in the weak interactions. An additional complication of the GWS theory or any other theory with non-simple gauge group G = G G are different gauge couplings g for different factors G. In this case, the g factor in eq. 34) for the mass matrix element M ) ab should be replaced with ga) gb) where ga) is the coupling of the gauge group factor containing the generator T a, and likewise for the gb). Thus, the most general formula for the vector mass matrix stemming from the Higgs mechanism is complex Higgs multiplets ) M ab V = ga) gb) Φ m) Φ { Tm) a, T m) b } Φ + real Higgs multiplets Φ m) Φ { Tm) a, T m)} b Φ. In my notes on the GWS theory we shall see how this works in detail, and how the gauge couplings affect the eigenstates of the mass matrix. 35)