Group Structure of Spontaneously Broken Gauge Theories

Transcription

1 Group Structure of SpontaneouslyBroken Gauge Theories p. 1/25 Group Structure of Spontaneously Broken Gauge Theories Laura Daniel Physics Department University of California, Santa Cruz

2 Group Structure of SpontaneouslyBroken Gauge Theories p. 2/25 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)

3 Group Structure of SpontaneouslyBroken Gauge Theories p. 3/25 Symmetries of the Lagrangian - Global Symmetry The Lagrangian of a system is given by: L = L(φ i (x), µ φ i (x)) If it is invariant under the transformation φ i (x) φ i(x) = φ i (x)+δφ i (x) where δφ i (x) = iǫ a t a ijφ j (x) where the ǫ a are spacetime independent small parameters and the t a are a set of matrices satisfying the Lie algebra of the group [t a,t b ] = if abc t c Then we say the transformation is a global symmetry of the Lagrangian.

4 Group Structure of SpontaneouslyBroken Gauge Theories p. 4/25 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 φ i (x): L = 1 2 ( µφ i ) µ2 (φ i ) 2 λ 4 [(φi ) 2 ] 2 This Lagrangian is invariant under O(N): φ i R ij φ j The potential V(φ i ) = 1 2 µ2 (φ i ) 2 + λ 4 [(φi ) 2 ] 2 with µ 2 > 0 is minimized by any constant field φ i 0 which satisfies: (φ i 0) 2 = µ2 λ

5 Group Structure of SpontaneouslyBroken Gauge Theories p. 5/25 Symmetries of the Lagrangian - SSB The constant field φ i 0 is the lowest-energy classical configuration of the system, with determined length but arbitrary direction. Choose coordinates such that φ i 0 points in the Nth direction: φ i 0 = (0,0,...,0,v), where v = µ λ Define a set of shifted fields, then rewrite the Lagrangian: φ i (x) = (π k (x),v +σ(x)), k = 1,...,N 1 L = 1 2 ( µπ k ) ( µσ) (2µ2 )σ 2 λµσ 3 λµ(π k ) 2 σ λ 4 σ4 λ 2 (πk ) 2 σ 2 λ 4 [(πk ) 2 ] 2

6 Group Structure of SpontaneouslyBroken Gauge Theories p. 6/25 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:

7 Group Structure of SpontaneouslyBroken Gauge Theories p. 7/25 Symmetries of the Lagrangian - SSB Generally, we see that although the potential is group invariant V(U(g)φ) = V(φ) the point φ 0, where V(φ) takes it minimum is not group invariant U(g)φ 0 φ 0 for some g G And although φ 0 is not invariant for all g G, it may be invariant for some g G. Those elements for which φ 0 is invariant form a subgroup H of G and, in fact, H is the little group (or isotropy group or stability group) of φ 0. U(h)φ 0 = φ 0 h H In the linear sigma model, we saw that G = O(N) and H = O(N 1).

8 Group Structure of SpontaneouslyBroken Gauge Theories p. 8/25 Symmetries of the Lagrangian - Goldstone s Theorem Regarding the appearance of massless particles in the linear sigma model (the π k ): 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 φ 0 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.

9 Group Structure of SpontaneouslyBroken Gauge Theories p. 9/25 Gauge Theories - Local symmetries Now consider symmetries of the Lagrangian in which the parameters of the symmetry transformation are spacetime dependent: ǫ a ǫ a (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.

10 Group Structure of SpontaneouslyBroken Gauge Theories p. 10/25 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) L = ψ(x)(iγ µ µ m)ψ(x) This has a global U(1) symmetry: ψ(x) ψ (x) = e iα ψ(x) ψ(x) ψ (x) = e iα ψ(x)

11 Group Structure of SpontaneouslyBroken Gauge Theories p. 11/25 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) = e iα(x) ψ(x) The derivative term now has a complicated transformation that is not locally invariant: ψ(x) µ ψ(x) ψ (x) µ ψ (x) = ψ(x)e iα(x) µ (e iα(x) ψ(x)) = ψ(x) µ ψ(x) iψ(x) µ α(x)ψ(x) = ψ(x)( µ ψ(x) i µ α(x))ψ(x)

12 Group Structure of SpontaneouslyBroken Gauge Theories p. 12/25 Gauge Theories - Local symmetries Form a gauge-covariant derivative D µ to replace µ, such that D µ ψ(x) transforms like D µ ψ(x) [D µ ψ(x)] = e iα(x) D µ ψ(x) Accomplish this by introducing the gauge field A µ (x) and define the covariant derivative where A µ transforms like D µ ψ(x) = ( µ +iea µ )ψ(x) A µ (x) A µ(x) = A µ (x)+ 1 e µα(x)

13 Group Structure of SpontaneouslyBroken Gauge Theories p. 13/25 Gauge Theories - Local symmetries We now have a Lagrangian that is invariant under a local U(1) symmetry (gauge invariance!) L 0 = ψ(iγ µ µ +iea µ )ψ mψψ To complete the theory, make the gauge field dynamical by adding a term to the Lagrangian containing its derivatives: L A = 1 4 F µνf µν, where F µν = µ A ν ν A µ We can see this term is invariant using the transformation property of A µ or from the relation of F µν to the covariant derivative: [D µ,d ν ]ψ = ief µν ψ

14 Group Structure of SpontaneouslyBroken Gauge Theories p. 14/25 Gauge Theories - Local symmetries Combining L 0 and L A, we get the full QED Lagrangian: L QED = ψ(iγ µ µ +iea µ )ψ mψψ 1 4 F µνf µν 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: 1. The photon is massless because an A µ A µ 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.

15 Group Structure of SpontaneouslyBroken Gauge Theories p. 15/25 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: L = 1 4 (F µν) 2 + D µ φ 2 +µ 2 φ φ λ 2 (φ φ) 2 With µ 2 > 0, the minimum of the potential occurs at: µ 2 v = φ = φ 0 = λ where φ = 0 φ 0 is the vacuum expectation value (vev) of the field φ.

16 Group Structure of SpontaneouslyBroken Gauge Theories p. 16/25 Gauge Theories - SSB We can write φ in terms of the real fields φ 1 and φ 2 φ = 1 2 (φ 1 +iφ 2 ) choose 0 φ 1 0 = v and 0 φ 2 0 = 0 and define the shifted fields φ 1 = φ 1 v and φ 2 = φ 2 All this follows the procedure followed earlier for SSB of a global symmetry. So far we have one massive field φ 1 (analagous to σ) while φ 2 corresponds to the massless Goldstone boson. However...

17 Group Structure of SpontaneouslyBroken Gauge Theories p. 17/25 Gauge Theories - SSB The local gauge symmetry adds a new feature. The covariant derivative term will yield D µ φ 2 = ( µ iga µ )φ 2 = 1 2 ( µφ 1 +ga µ φ 2) ( µφ 2 ga µ φ 1) 2 gva µ ( µ φ 2 +ga µ φ 1)+ g2 v 2 2 Aµ A µ At this point, it appears the gauge boson acquires a mass m A = gv. But the interaction term gva µ µ φ 2 makes the picture a little unclear.

18 Group Structure of SpontaneouslyBroken Gauge Theories p. 18/25 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, φ 2 is removed from the theory, leaving the Lagrangian as: L = 1 4 (F µν) 2 +( µ φ) 2 +g 2 φ 2 A µ A µ V(φ) Before the symmetry breaking, we had two scalar fields and one massless gauge boson, so a total of four degrees of freedom. After the symmetry breaking, we have one scalar field, and one massive gauge boson, still a total of four degrees of freedom.

19 Group Structure of SpontaneouslyBroken Gauge Theories p. 19/25 Non-Abelian Gauge Theories We can generalize gauge theories and the Higgs mechanism to non-abelian gauge symmetries. Consider an SU(2) gauge theory with a complex doublet of scalar fields φ = ( φ 1 φ 2 ) which transform as a spinor of SU(2) L = 1 4 (F µν) 2 + D µ φ 2 V(φ) where D µ φ = ) ( µ ig σa 2 Aa µ φ F a µν = µ A a ν ν A a µ +gǫ abc A b µa c ν V(φ) = µ 2 φ φ+ λ 2 (φ φ) 2

20 Group Structure of SpontaneouslyBroken Gauge Theories p. 20/25 Non-Abelian Gauge Theories For µ 2 > 0, the potential acquires a vev, and using the freedom of SU(2) rotations, we can write this as: φ = 1 2 ( 0 v ) The gauge boson masses arise from D µ φ 2 = 1 2 g2 (0 v) σa 2 σ b 2 ( ) 0 A a v µa bµ +... Symmetrizing the matrix product using {σ a,σ b } = 2δ ab, we find the mass term L = g2 v 2 8 Aa µa aµ

21 Group Structure of SpontaneouslyBroken Gauge Theories p. 21/25 Non-Abelian Gauge Theories In the case of the spinor rep. in SU(2) all three gauge bosons receive the mass m A = gv 2. If instead we take φ to be real-valued under SU(2) and transform according to the vector rep, the appropriate covariant derivative is (D µ φ) a = µ φ a +gǫ abc A b µφ c Again, when φ acquires a vev, we square the D µ φ term and find the gauge boson mass term L = g2 2 (ǫ abca b µ(φ 0 ) c ) 2

22 Group Structure of SpontaneouslyBroken Gauge Theories p. 22/25 Non-Abelian Gauge Theories Using the freedom in SU(2) to pick a direction for the vev, choose it to point in the 3 direction Inserting this into L φ c = (φ 0 ) c = Vδ c3 L = g2 2 V 2 (ǫ abc A b µ) 2 = g2 2 V 2( (A 1 µ) 2 +(A 2 µ) 2) In the case of the vector rep. in SU(2) the gauge bosons corresponding to generators 1 and 2 acquire masses, while the boson corresponding to generator 3 remains massless m 1 = m 2 = gv, m 3 = 0 With φ c pointing along the 3-axis, it is still invariant under SO(2) = U(1), hence two bosons acquire mass.

23 Group Structure of SpontaneouslyBroken Gauge Theories p. 23/25 Pattern of Symmetry Breaking The pattern of symmetry breaking depends on the structure of the theory, particularly the group representation of the scalar field (see, for instance, Li:)

24 Group Structure of SpontaneouslyBroken Gauge Theories p. 24/25 Summary Physical Lagrangians have internal symmetries (they are invariant under certain group transformations) Invariance of the ground state is a condition for spontaneously breaking symmetries of L A Gauge theory is one in which we require L to be invariant under a local symmetry, and this results in one gauge boson per generator of the local symmetry group Spontaneous symmetry breaking of a gauge theory (The Higgs Mechanism) involves: 1. SSB a global symmetry of the system (the symmetry that the scalar field φ transforms under) 2. Choose a gauge to remove Goldstone bosons (and timelike components of gauge bosons) from the theory We can categorize spontaneous symmetry breaking for various groups (and various representations of the groups)

25 Group Structure of SpontaneouslyBroken Gauge Theories p. 25/25 References 1. Cheng, Ta-Pei and Li, Li-Fong, Gauge theory of elementary particles, Clarendon Press, Oxford, O Raifeartaigh, L., Group Structure of Gauge Theories, Cambridge University Press, Cambridge, Peskin, Michael E. and Schroeder, Daniel V., An Introduction to Quantum Field Theory, Westview Press, Perseus Books Group, Li, Li-Fong, Group Theory of the Spontaneously Broken Gauge Symmetries, Phys. Rev. D 9, (1974)