Symmetry Groups

Symmetry plays an essential role in particle theory. If a theory is invariant under transformations by a symmetry group one obtains a conservation law and quantum numbers. For example, invariance under rotation in space leads to angular momentum conservation and to the quantum numbers j, mj for the total angular momentum.

Gauge symmetries: These are local symmetries that act differently at each space-time point xµ = (t,x) . They create particularly stringent constraints on the structure of a theory. For example, gauge symmetry automatically determines the interaction between particles by introducing bosons that mediate the interaction. All current models of elementary particles incorporate gauge symmetry. (Details on p. 4)

Groups: A group G consists of elements a , inverse elements a−1, a unit element 1, and

a multiplication rule “ ⋅ ” with the properties: 1. If a,b are in G, then c = a ⋅ b is in G. 2. a ⋅ (b ⋅ c) = (a ⋅ b) ⋅ c 3. a ⋅ 1 = 1 ⋅ a = a 4. a ⋅ a−1 = a−1 ⋅ a = 1 The group is called Abelian if it obeys the additional commutation law: 5. a ⋅ b = b ⋅ a

Product of Groups: The product group G×H of two groups G, H is defined by pairs of

elements with the multiplication rule (g1,h1) ⋅ (g2,h2) = (g1 ⋅ g2 , h1 ⋅ h2) . A group is called simple if it cannot be decomposed into a product of two other groups. (Compare a prime number as analog.) If a group is not simple, consider the factor groups.

Examples: n×n matrices of complex numbers with matrix multiplication. Hermitian: M† = M (M† = MT*, T = transpose, * = complex conjugate) Unitary: U† = U−1 Special: |S| = 1 (determinant = 1)

1 U(n) : The group of complex, unitary n×n matrices U(n) = SU(n) × U(1) for n>1 (not a simple group) U(1) The complex unit circle: U = z = exp[iϕ] , ϕ real U†U = z*z = |z|2 = 1 Gauge symmetry transformation: ψ′(x,t) = exp[−iϕ(x,t)] ⋅ ψ(x,t) Probability conserved: ψ′* ψ′ = ψ* ψ

U(1)EM describes the electromagnetic interaction. 1 boson photon A 1 quantum number charge Q

SU(n): The group of complex, unitary n×n matrices with determinant 1 Generate a unitary matrix U from a Hermitian matrix H: m U = exp[iH] = Σm H /m!

Write all Hermitian matrices using a basis set Ji = generators :

H = − Σi ϕi ⋅ Ji

Use a basis set Ji that guarantees |U|=1:

trace(Ji) = 0 2 2 i (n −1) matrices Ji → (n −1) gauge bosons A

(n−1) diagonal Ji → (n−1) quantum numbers

Gauge transformation: ψ′(x,t) = U ⋅ ψ(x,t) = exp[− Σi ϕi(x,t) ⋅ Ji] ⋅ ψ(x,t) Probability conserved: ψ′* ψ′ = (Uψ)* (Uψ) = ψ*U†Uψ = ψ* ψ SU(2) describes the (broken) Isospin symmetry of the Weak Interaction.

3 Pauli matrices σi Ji = σi / 2 3 bosons W+,W−,Z

1 diagonal Ji J3 (z-component of the isospin) SU(3) describes the (exact) Color symmetry of the Strong Interaction.

8 Gell-Mann matrices λi Ji = λi / 2

8 bosons Gluons G1,…G8

2 diagonal Ji J3, J8 (two independent colors) SU(5) is a “grand unification” group which contains the standard model.

2 SU(3)×SU(2)×U(1)Y Standard Model Color + Isospin + Hypercharge

SU(2)×U(1)Y Electro-Weak Interaction Isospin + Hypercharge J3 Y = 2 (Q − J3)

U(1)EM Electromagnetic Charge Q

SU(2)×U(1)Y breaks down to the electromagnetic U(1)EM (see p. 4).

Pauli Matrices: Gell-Mann Matrices: = Pauli matrices + rows and columns of zeros

0 1 0 1 0 0 0 1 0 0 0 σ1 = 1 0 λ1 = 1 0 0 λ4 = 0 0 0 λ6 = 0 0 1 0 0 0 1 0 0 0 1 0

0 -i 0 -i 0 0 0 -i 0 0 0 σ2 = i 0 λ2 = i 0 0 λ5 = 0 0 0 λ7 = 0 0 -i 0 0 0 i 0 0 0 i 0

1 0 1 0 0 1 0 0 0 0 0 σ3 = 0 -1 λ3 = 0 -1 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 -1

Select two independent matrices from the last row. 1 0 0 λ8 = 0 1 0 / √3 0 0 -2

3 Gauge Bosons: Conceptually, all of today’s particle theories start out with a set of fermions, for example the electron in quantum electrodynamics. Fermions are represented by a wave function ψ. One then postulates gauge invariance, i.e., the gauge transformation ψ′ = exp[−iϕ(xµ)] ⋅ ψ should leave the theory invariant. That is fine for the probability density ψ*ψ but the wave equation is not invariant. An unwanted extra term is created by the derivative of the phase factor ϕ(xµ). Derivatives are related to the 2 2 momentum operator, for example p /2me = −∇ /2me in the Schrödinger equation and

pµ = i ∂µ in the Dirac equation (448 quantum notes, p. 17). The theory can be made gauge

invariant by adding the electromagnetic four-potential Aµ to the four-momentum pµ in

order to cancel the phase derivative. The Aµ are the wave functions of the photon, the boson that describes the electromagnetic interaction. The complete gauge transformation of the wave function ψ for the electron and the Aµ for the photon becomes:

ψ′ = exp[−iϕ] ⋅ ψ ћ,c = 1 −1 µ A′µ = Aµ + q ∂µ ϕ ∂µ = ∂/∂x q = electric charge = −e for the electron

pµ = i ∂µ − q ⋅ Aµ pµ = momentum operator = generalized “covariant” derivative

With these definitions one has gauge-invariance for the probability density ψ*ψ and for the Dirac equation:

ψ′*ψ′ = ψ∗ψ µ µ γ p′µ ψ′ = me ψ′ ⇒ γ pµ ψ = me ψ

In similar fashion one can incorporate the electromagnetic interaction into the

Schrödinger equation using the energy and momentum operators E and p : 2 E = +i ∂/∂t − q ⋅ Φ E ψ = p /2me ψ µ p = −i ∂/∂x − q ⋅ A pµ = (Ε,−p), x = (t,x), Aµ = (Φ,−A)

For larger groups the momentum operator and the gauge-invariant derivative involve a i product of several group generators Ji and the corresponding gauge bosons A µ : i 2 pµ = i ∂µ + g ⋅ Σi Ji ⋅ A µ g = generalized charge, g /4π = coupling constant

There is a parallel to the definition of the covariant derivative Dµ in general relativity: ν ν ν λ Dµ A = ∂µ A + Γ µλ A

4 The first term describes the change of Aν, the second term the change of the coordinate system from one space-time point to the next. The wave equation for the gauge bosons, i.e.. the generalization of the Maxwell equations, can be derived by forming a gauge-invariant field tensor using the generalized

derivative. For non-Abelian symmetry groups (e.g. SU(2), SU(3) ) the wave equations become more interesting because they imply interactions between the gauge bosons themselves, not just a mediation of the interaction between fermions.

Symmetry Breaking: Gauge symmetries are often broken, but in a subtle way. The fundamental equations are symmetric, but the ground state wave function breaks the symmetry. An analog is a ferromagnetic ground state, where the rotational symmetry of the electromagnetic interaction is broken by a specific spin orientation of the ferromagnet (“spontaneous symmetry breaking”). When a gauge symmetry is broken the gauge bosons are able to acquire an effective mass, even though gauge symmetry does not allow a boson mass in the fundamental equations.

The breakdown of the electro-weak symmetry SU(2)×U(1)Y converts the 4 massless gauge bosons A1,A2,A3, and B of the symmetric theory into the 3 massive bosons W+,W−,Z of the weak interaction and the massless photon A of the electromagnetic interaction (which represents the remaining symmetry). The conversion can be broken up into two steps. First, combinations with well-defined isospin J3 = ±1 are formed:

− W + = (A1 ± i A2) /√2 3 Then, A is mixed with the gauge boson B of U(1)Y :

3 Z = cosϑW −sinϑW ⋅ A A sinϑW cosϑW B

The resulting observed particles are the Z and photon A. The weak mixing angle ϑW is determined by the ratio of the coupling constants g for SU(2) and g′ for U(1)Y :

tan ϑW = g′/g

5 SU(3) Color: 8 gauge bosons Gi are associated with the 8 generators Ji = λi / 2. The bosons are the 8 gluons that mediate the strong interaction. Define three color quantum numbers, plus three complementary colors for antiparticles: R = red G = green B = blue R = G+B = cyan G = R+B = magenta B = R+G = yellow The quark-gluon interaction transfers color from

qG one quark to another (see the diagram). Therefore, GRG gluons carry two color indices, which can be assigned to rows and columns of the Gell-Mann matrices λi : RGB

R * * * 0 1 0 qR

G * * * for example GRG : 0 0 0 = (λ1 + i λ2)/2 B * * * 0 0 0

The SU(3) color symmetry forces all hadrons to be white, i.e., all color quantum numbers are zero . That can be achieved with the following combinations of quarks q and anti-quarksq:

ik qq = 1/√3 [qRqR + qGqG + qBqB] = 1/√3 Σik δ qi qk meson, symmetric

ijk qqq = 1/√6 [ +qRqGqB − qBqGqR = 1/√6 Σijk ε qi qj qk baryon, antisymmetric

+qBqRqG − qGqRqB when exchanging color indices of +q q q − q q q ] G B R R B G two particles.

even odd permutation

SU(3) “Eightfold Way”: This is a SU(3) symmetry completely different from color SU(3), and it is only approximate. It describes the assembly of the up, down, and strange quarks into hadrons. Instead of three quark colors one has three quark flavors. This symmetry leaves out the other three quarks (charmed, bottom, and top) and ignores mass differences between the quarks.

6 Supersymmetry: Space-time has two basic symmetries, translation and rotation. The rotational symmetry can be divided into spatial rotations and mixed space-time “rotations”, the Lorentz transformations. The combination of all these symmetries forms the Poincare group. There is only one extension of the Poincare group that is consistent with basic principles, and that is supersymmetry. In addition to the normal space-time coordinates one introduces anti-commuting coordinates, which cannot be described by ordinary numbers. Supersymmetry requires that each particle is paired with a super- partner, a fermion with a boson and vice versa. Supersymmetric theories are even more constrained than gauge theories, and divergent terms in a perturbation expansion tend to be cancelled out because two super-partners contribute with opposite signs. None of the superpartners of existing particles have been found yet, which means that supersymmetry is certainly broken. Nevertheless, it looks promising as a symmetry at very high energies, where it allows a unification of the three coupling constants of the standard model into one universal coupling constant. Figure: Extrapolation of the inverse coupling constants α−1 towards higher energies µ for the standard model (left) and for its super-symmetric extension (right). All coupling constants become equal at the same energy, and that energy is not too far from the Planck energy, the fundamental energy scale obtained from ћ,c, and the gravitational constant.

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