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Group Structure of Spontaneously Broken Gauge Theories Group Structure of Spontaneously Broken Gauge Theories Laura Daniel [email protected] Physics Department University of California, Santa Cruz Group Structure of SpontaneouslyBroken Gauge Theories – p. 1/25 Outline Symmetries of the Lagrangian • Global symmetries ◦ Spontaneous symmetry breaking of a global symmetery ◦ Goldstone’s Theorem ◦ Gauge Theories • Local symmetries (a.k.a. Gauge symmetries) ◦ Spontaneous symmetry breaking of a gauge symmetry: ◦ The Higgs Mechanism Group Structure of Spontaneously Broken Gauge Theories • Non-abelian SU(2) gauge theory (φ transforming in two ◦ different representations) Patterns of symmetry breaking ◦ EWSB (if time) ◦ Group Structure of SpontaneouslyBroken Gauge Theories – p. 2/25 Symmetries of the Lagrangian - Global Symmetry The Lagrangian of a system is given by: = (φi(x),∂µφi(x)) L L If it is invariant under the transformation ′ φi(x) φ (x)= φi(x)+ δφi(x) → i where a a δφi(x)= iǫ tijφj(x) where the ǫa are spacetime independent small parameters and the ta are a set of matrices satisfying the Lie algebra of the group [ta,tb]= if abctc Then we say the transformation is a global symmetry of the Lagrangian. Group Structure of SpontaneouslyBroken Gauge Theories – p. 3/25 Symmetries of the Lagrangian - SSB We use non-invariance of the ground state as a symmetry breaking condition. Consider the linear sigma model: set of N real scalar fields φi(x): 1 i 2 1 2 i 2 λ i 2 2 = (∂µφ ) + µ (φ ) [(φ ) ] L 2 2 − 4 This Lagrangian is invariant under O(N): φi Rijφj → i 1 2 i 2 λ i 2 2 2 The potential V (φ )= 2 µ (φ ) + 4 [(φ ) ] with µ > 0 is − i minimized by any constant field φ0 which satisfies: µ2 (φi )2 = 0 λ Group Structure of SpontaneouslyBroken Gauge Theories – p. 4/25 Symmetries of the Lagrangian - SSB i The constant field φ0 is the lowest-energy classical configuration of the system, with determined length but arbitrary direction. i Choose coordinates such that φ0 points in the Nth direction: µ φi = (0, 0,..., 0,v), where v = 0 √λ Define a set of shifted fields, then rewrite the Lagrangian: φi(x)=(πk(x),v + σ(x)), k = 1,...,N 1 − 1 k 2 1 2 1 2 2 = (∂µπ ) + (∂µσ) (2µ )σ L 2 2 − 2 λ λ λ √λµσ3 √λµ(πk)2σ σ4 (πk)2σ2 [(πk)2]2 − − − 4 − 2 − 4 Group Structure of SpontaneouslyBroken Gauge Theories – p. 5/25 Symmetries of the Lagrangian - SSB We obtain a massive σ field, but the N 1 π fields remain massless and rotate among themselves.− The O(N) symmetry is no longer manifest. Invariance of the ground state under O(N) resulted in the spontaneous symmetry breaking of the Lagrangian. We have broken the O(N) group down to O(N 1). − We can see this intuitively in the O(2) case: Group Structure of SpontaneouslyBroken Gauge Theories – p. 6/25 Symmetries of the Lagrangian - SSB Generally, we see that although the potential is group invariant V (U(g)φ)= V (φ) the point φ0, where V (φ) takes it minimum is not group invariant U(g)φ = φ for some g G 0 0 ∈ And although φ is not invariant for all g G, it may be invariant 0 ∈ for some g G. Those elements for which φ0 is invariant form a subgroup H∈of G and, in fact, H is the little group (or isotropy group or stability group) of φ0. U(h)φ = φ h H 0 0 ⇔ ∈ In the linear sigma model, we saw that G = O(N) and H = O(N 1). − Group Structure of SpontaneouslyBroken Gauge Theories – p. 7/25 Symmetries of the Lagrangian - Goldstone’s Theorem Regarding the appearance of massless particles in the linear sigma model (the πk): Goldstone’s Theorem: Spontaneous breakdown of a continuous symmetry implies the existance of massless, spinless particles. Every continuous symmetry of the Lagrangian that is not a symmetry of φ0 gives rise to a Goldstone boson, hence there were N 1 massless particles in a linear sigma model with N scalar fields.− See Peskin & Schroeder, Cheng & Li, etc. for proofs of Goldstone’s theorem. Group Structure of SpontaneouslyBroken Gauge Theories – p. 8/25 Gauge Theories - Local symmetries Now consider symmetries of the Lagrangian in which the parameters of the symmetry transformation are spacetime dependent: ǫa ǫa(x) → This is a local symmetry of the Lagrangian, also known as a gauge symmetry. Gauging the symmetry complicates the transformation of the derivative term and requires the inclusion of a new vector field to keep the Lagrangian gauge invariant (that is, invariant under the local, spacetime dependent symmetry). This new vector field, called the gauge field introduces dynamics into the theory. Group Structure of SpontaneouslyBroken Gauge Theories – p. 9/25 Gauge Theories - Local symmetries As an example, requiring gauge symmetry of the Dirac free electron theory introduces dynamics into the theory. Consider the Lagrangian for the free-electron field ψ(x) µ = ψ(x)(iγ ∂µ m)ψ(x) L − This has a global U(1) symmetry: ′ − ψ(x) ψ (x)= e iαψ(x) → ′ ψ(x) ψ (x)= eiαψ(x) → Group Structure of SpontaneouslyBroken Gauge Theories – p. 10/25 Gauge Theories - Local symmetries Now gauge the symmetry by replacing α with α(x), so the transformations become: ′ − ψ(x) ψ (x)= e iα(x)ψ(x) → ′ ψ(x) ψ (x)= eiα(x)ψ(x) → The derivative term now has a complicated transformation that is not locally invariant: ′ ′ iα(x) −iα(x) ψ(x)∂µψ(x) ψ (x)∂µψ (x) = ψ(x)e ∂µ(e ψ(x)) → = ψ(x)∂µψ(x) iψ(x)∂µα(x)ψ(x) − = ψ(x)(∂µψ(x) i∂µα(x))ψ(x) − Group Structure of SpontaneouslyBroken Gauge Theories – p. 11/25 Gauge Theories - Local symmetries Form a gauge-covariant derivative Dµ to replace ∂µ, such that Dµψ(x) transforms like ′ −iα(x) Dµψ(x) [Dµψ(x)] = e Dµψ(x) → Accomplish this by introducing the gauge field Aµ(x) and define the covariant derivative Dµψ(x)=(∂µ + ieAµ)ψ(x) where Aµ transforms like ′ 1 Aµ(x) A (x)= Aµ(x)+ ∂µα(x) → µ e Group Structure of SpontaneouslyBroken Gauge Theories – p. 12/25 Gauge Theories - Local symmetries We now have a Lagrangian that is invariant under a local U(1) symmetry (gauge invariance!) µ = ψ(iγ ∂µ + ieAµ)ψ mψψ L0 − To complete the theory, make the gauge field dynamical by adding a term to the Lagrangian containing its derivatives: 1 µν A = FµνF , where Fµν = ∂µAν ∂νAµ L −4 − We can see this term is invariant using the transformation property of Aµ or from the relation of Fµν to the covariant derivative: [Dµ,Dν]ψ = ieFµνψ Group Structure of SpontaneouslyBroken Gauge Theories – p. 13/25 Gauge Theories - Local symmetries Combining and A, we get the full QED Lagrangian: L0 L µ 1 µν QED = ψ(iγ ∂µ + ieAµ)ψ mψψ FµνF L − − 4 The dynamics of the theory come from requiring gauge invariance of the Lagrangian, that is, by requiring the Dirac free electron theory to be invariant to local U(1) transformations. Note the following features of the theory: µ 1. The photon is massless because an AµA term is not gauge invariant. 2. The coupling of the photon to any matter field is determined by its transformation property under the symmetry group. 3. The QED Lagrangian does not have gauge field self-coupling. Group Structure of SpontaneouslyBroken Gauge Theories – p. 14/25 Gauge Theories - SSB Now let’s combine local gauge invariance and spontaneous symmetry breaking: The Higgs Mechanism Consider an Abelian U(1) gauge theory with a complex scalar field coupled to itself and to an electromagnetic field: 1 2 2 2 ∗ λ ∗ 2 = (Fµν) + Dµφ + µ φ φ (φ φ) L −4 | | − 2 With µ2 > 0, the minimum of the potential occurs at: µ2 v = φ = φ = 0 r λ where φ = 0 φ 0 is the vacuum expectation value (vev) of the field φ. | | | | Group Structure of SpontaneouslyBroken Gauge Theories – p. 15/25 Gauge Theories - SSB We can write φ in terms of the real fields φ1 and φ2 1 φ = (φ1 + iφ2) √2 choose 0 φ 0 = v and 0 φ 0 = 0 | 1| | 2| and define the shifted fields ′ ′ φ = φ v and φ = φ 1 1 − 2 2 All this follows the procedure followed earlier for SSB of a global ′ symmetry. So far we have one massive field φ1 (analagous to σ) ′ while φ2 corresponds to the massless Goldstone boson. However... Group Structure of SpontaneouslyBroken Gauge Theories – p. 16/25 Gauge Theories - SSB The local gauge symmetry adds a new feature. The covariant derivative term will yield 2 2 Dµφ = (∂µ igAµ)φ | | | − | 1 ′ ′ 2 1 ′ ′ 2 = (∂µφ + gAµφ ) + (∂µφ gAµφ ) 2 1 2 2 2 − 1 2 2 ′ ′ g v µ gvAµ(∂µφ + gAµφ )+ A Aµ − 2 1 2 At this point, it appears the gauge boson acquires a mass ′ mA = gv. But the interaction term gvAµ∂µφ2 makes the picture a little unclear. Group Structure of SpontaneouslyBroken Gauge Theories – p. 17/25 Gauge Theories - SSB We now make a particular choice of gauge, called the unitary gauge to eliminate the massless Goldstone boson from the theory. Use the local U(1) gauge symmetry to choose α(x) so that φ(x) is real-valued at every point x (see Cheng & Li, p242). ′ With this choice, φ2 is removed from the theory, leaving the Lagrangian as: 1 2 2 2 2 µ = (Fµν) +(∂µφ) + g φ AµA V (φ) L −4 − Before the symmetry breaking, we had two scalar fields and one massless gauge boson, so a total of four degrees of freedom.
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