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25 Structure of SpontaneouslyBroken Gauge – p. 1/ Laura Daniel [email protected] Department

Broken Gauge Theories University of California, Santa Cruz

Group Structure of Spontaneously Outline

Symmetries of the Lagrangian • Global symmetries ◦ Spontaneous breaking of a global symmetery ◦ Goldstone’s Theorem ◦ Gauge Theories • Local symmetries (a.k.a. Gauge symmetries) ◦ Spontaneous of a gauge symmetry: ◦ The Group Structure of Spontaneously Broken Gauge Theories • Non-abelian SU(2) gauge (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 under the transformation

′ φi(x) φ (x)= φi(x)+ δφi(x) → i where a a δφi(x)= iǫ tijφj(x) where the ǫa are independent small parameters and the ta are a set of matrices satisfying the 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 fields φi(x):

1 i 2 1 2 i 2 λ i 2 2 = (∂µφ ) + (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 (φ ) + 4 [(φ ) ] with > 0 is − i minimized by any constant field φ0 which satisfies:

2 (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- classical configuration of the system, with determined length but arbitrary direction.

i Choose coordinates such that φ0 points in the Nth direction: 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) (2 )σ L 2 2 − 2 λ λ λ √λσ3 √λ(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 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 , 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 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 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 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- 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 :

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 is massless because an AµA term is not gauge invariant. 2. The coupling of the photon to any field is determined by its transformation property under the . 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) with a complex scalar field coupled to itself and to an electromagnetic field:

1 2 2 2 ∗ λ ∗ 2 = (Fµν) + Dµφ + φ φ (φ φ) L −4 | | − 2

With 2 > 0, the minimum of the potential occurs at:

2 v = φ = φ = 0 r λ where φ = 0 φ 0 is the (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 . 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 acquires a mass ′ mA = gv. But the 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. After the symmetry breaking, we have one scalar field, and one massive gauge boson, still a total of four degrees of freedom.

Group Structure of SpontaneouslyBroken Gauge Theories – p. 18/25 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 φ1 which transform as a of SU = φ2 (2)  1 2 2 = (Fµν) + Dµφ V (φ) L 4 | | − where

a σ a Dµφ = ∂µ ig A φ  − 2 µ a a a abc b c F = ∂µA ∂νA + gǫ A A µν ν − µ µ ν ∗ λ ∗ V (φ) = 2φ φ + (φ φ)2 − 2

Group Structure of SpontaneouslyBroken Gauge Theories – p. 19/25 Non-Abelian Gauge Theories

For 2 > 0, the potential acquires a vev, and using the freedom of SU(2) rotations, we can write this as:

1 0 φ = √2v

The gauge boson masses arise from

a b 2 1 2 σ σ 0 a bµ Dµφ = g (0 v) A A + ... | | 2 2 2 v µ

Symmetrizing the product using σa,σb = 2δab, we find the mass term { } g2v2 ∆ = Aa Aaµ L 8 µ

Group Structure of SpontaneouslyBroken Gauge Theories – p. 20/25 Non-Abelian Gauge Theories

In the case of the spinor rep. in SU(2) all three gauge gv receive the mass mA = 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 b (Dµφ)a = ∂µφa + gǫabcAµφc

Again, when φ acquires a vev, we square the Dµφ term and find the gauge boson mass term

2 g b 2 ∆ = (ǫabcA (φ )c) L 2 µ 0

Group Structure of SpontaneouslyBroken Gauge Theories – p. 21/25 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

φc =(φ )c = Vδc 0 3 Inserting this into ∆ L 2 2 g 2 b 2 g 2 1 2 2 2 ∆ = V (ǫabcA ) = V (A ) +(A ) L 2 µ 2 µ µ  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 3 remains massless

m1 = m2 = gV, m3 = 0

With φc pointing along the 3-axis, it is still invariant under SO(2) ∼= U(1), hence two bosons acquire mass.

Group Structure of SpontaneouslyBroken Gauge Theories – p. 22/25 Pattern of Symmetry Breaking

The pattern of symmetry breaking depends on the structure of the theory, particularly the of the scalar field (see, for instance, Li:)

Group Structure of SpontaneouslyBroken Gauge Theories – p. 23/25 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 to be invariant • under a local symmetry, and this results in oneL 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)

Group Structure of SpontaneouslyBroken Gauge Theories – p. 24/25 References

1. Cheng, Ta-Pei and Li, Li-Fong, Gauge theory of elementary particles, Clarendon Press, Oxford, 1984. 2. O’Raifeartaigh, L., Group Structure of Gauge Theories, Cambridge University Press, Cambridge, 1986. 3. Peskin, Michael E. and Schroeder, Daniel V., An Introduction to Theory, Westview Press, Perseus Books Group, 1995. 4. Li, Li-Fong, of the Spontaneously Broken Gauge Symmetries, Phys. Rev. D 9, 1723 - 1739 (1974)

Group Structure of SpontaneouslyBroken Gauge Theories – p. 25/25