Labelling States

Recall that in Sec. 3.5, we introduced the notion of Cartan generators, which is the set of diagonal generators of a given group. In the last section we learned that these generators become especially useful if we want to give new labels to our particles inside a doublet⁶³object.

63In a later section we will learn that the same can be done for triplets and the conserved quantities following from SU(3)invariance.

A typical SU(2)doublet is of the form

this generator, with eigenvalue+¹₂ and a (left-chiral) electron

 0 e



an eigenstate of this generator, with eigenvalue−¹₂. These are new particle labels, which are called the isospin of the neutrino and the electron.

Following the same line of thoughts we can assign an isospin value to the right-chiral singlets. These transform according to the one-dimensional representation of SU(2), and the generators are

in this representation simply zero⁶⁴: J_i = 0. Therefore, in this one- ⁶⁴The right-chiral singlets do not trans-form at all as explained in Sec. 3.7.4.

dimensional representation, the singlets are eigenstates of the Cartan generator J3with eigenvalue zero. The right-chiral singlets, like e_R carry isospin zero. This coincides with the remarks above that right-chiral fields do not interact via the weak force. Just as electrically uncharged objects do not interact via electromagnetic interactions, fields without isospin do not take part in weak interactions.

Finally, we can assign isospin values to the three gauge fields W₊^μ, W₋^μ, W₃^μ. The three gauge fields form a SU(2)triplet

which transforms according to the three dimensional representation of SU(2). In this representation the Cartan generator J₃has

eigenval-ues⁶⁵+1,−1, 0 and therefore we assign Q3(^W₊^μ) =1, Q3(^W₋^μ) = −1, ⁶⁵This can be seen directly from the explicit matrix form of J₃in Eq. 3.121:

J3=

Take note that the triplet

⎛

⎟⎠ simply belongs to a different

basis, where J3isn’t diagonal. This can be seen as another reason for our introduction of W_±^μ.

If this is unclear, take a look at how we introduced the three gauge fields W_i^μ. They were included into the Lagrangian in combina-tion with the generatorsσiW_i^μ. This can be seen as a basis expan-sion of some objects W^μin terms of the basisσ_i: W^μ = σ_iW_i^μ = σ₁W₁^μ+σ₂W₂^μ+σ₃W₃^μanalogous to how we can write a vector in terms of basis vectors:v = v₁e₁+v₂e₂+v₃e₃. The generatorsσ_i live

in the Lie algebra⁶⁶of SU(2)and consequently our object W^μlives

66The Lie algebra is a vector space!

there, too. Therefore, if we want to know how W^μtransforms, we need to know the representation of SU(2)on this vector space, i.e. on its own Lie algebra. In other words: We need to know how the group elements of SU(2)act on their own Lie algebra elements, i.e. its gen-erators. This may seem like a strange idea at first, but actually is a quite natural idea. Recall how we defined a representation: A repre-sentation is a map⁶⁷ from the group to the space of linear operators

67To be precise: A homomorphism, which is a map that satisfies some

special conditions. over a vector space. So far we only looked at "external" vector spaces like Minkowski space. The only⁶⁸intrinsic vector space that comes

68A group itself is in general no vector space. Although we can take a look at how the group acts on itself, this is not a representation, but a realization of the group.

with a group is its Lie algebra! Therefore it isn’t that strange to ask what a group representation on this vector space looks like. This very important representation is called the adjoint representation.

Gauge fields (like W+, W−, W3) are said to live in the adjoint rep-resentation of the corresponding group. For SU(2)the Lie algebra is three dimensional, because we have three generators and therefore the adjoint representation is three dimensional. Exactly how we are able to write the components of a vector between two brackets⁶⁹, we

69v=

⎛

⎝v1

v2

v3

⎞

⎠ can write the component of W^μbetween two brackets⁷⁰, which is

70W^μ=

⎛

⎝W₁^μ W₂^μ W₃^μ

⎞

⎠

what we call a triplet . The generators in the adjoint representation are connected to the three dimensional generator we derived earlier through a basis transformation.

In the following section we move on to the "next higher" internal symmetry group SU(3). Demanding local SU(3)invariance of the Lagrangian gives us the correct Lagrangian describing strong interac-tion.

7.8 SU ( ³ ) Interactions

For three fermion fields we can find a locally SU(3)invariant La-grangian in exactly the same way we did in the last chapter for two fields and SU(2). This symmetry is not broken and the correspond-ing spin 1 fields, called gluon fields, are massless. SU(3)is the group of all unitary 3×3 matrices with unit determinant, i.e.

U^†U=UU^†=1 det U=1. (7.138) As usual for Lie groups we can write this as an exponential func-tion⁷¹

71As already noted in Sec. 2.4, Capital Roman letters A, B, . . . are always

summed from 1 to 8. U=e^iTAθA. (7.139)

The defining equations 7.138 of the group require, as for⁷²SU(2), the

72See 3.80 and the following text plus equations, where the basis was given by the 2×2 Pauli matrices.

generators to be Hermitian and traceless

T_A^† =T_A (7.140)

tr(T_A) =0 (7.141) A basis for those traceless, Hermitian generators is, at least in one

representation, given by eight⁷³3×3 matrices, called Gell-Mann ⁷³It can be shown that in general for SU(N)there are N²−1 generators. The number of generators is often called the rank of a group.

The generators of the group are connected to these Gell-Mann

ma-trices, just as the Pauli matrices were connected to the generators^{74 74J}i =^σ₂ⁱ, see Eq. 3.82 and the explana-tions there.

of the SU(2)group via TA = ¹₂λA. The Lie algebra for this group is given by

[T_A, TB] =i f^ABCT^C, (7.143) where we adopted the standard convention that capital letters like A, B, C can take on every value from 1 to 8. f^ABC are called the struc-ture constants of SU(3), which for SU(2)were given by the Levi-Civita symbol_ijk. They can be computed by brute-force

computa-tion, which yields^{75 75}This is not very enlightening, but we

list it here for completeness.

f¹²³=1 (7.144)

where all others can be computed from the fact that the structure constants f^ABC are antisymmetric under permutation of any two indices. For example

f^ABC = −f^BAC= −f^CBA. (7.147) All other possibilities, which cannot be computed by permutation, vanish.

Analogously to what we did for SU(2)in Sec. 7.2, we introduce triplets of spin ¹₂ fields

Q=

and exactly as for SU(2)we get new labels for the objects inside this triplet, which we will discuss in the next section.

To make the Lagrangian

L =i ¯Q∂_μγ^μQ−QmQ^¯ (7.149) locally SU(3)invariant, one again adds coupling terms between the spin ¹₂fields and new spin 1 fields. The derivation is analogous as for SU(2), but the computations are quite cumbersome, so we just quote the final Lagrangian⁷⁶

76Remember: the sum over capital letters(A, B, C, ...)runs from 1 to 8 where f^ABC are the structure constants of SU(3)that already ap-peared in the commutator of the generators. Furthermore, D_αis defined as

D_α=∂α+igT^CG_α^C (7.152) where T^Care the generators of SU(3)defined at the beginning of this section. As you can check every term here is completely analogous to the SU(2)case, except we now have different generators with different commutation properties.

7.8.1 Color

From global SU(3)symmetry we get through Noether’s theorem new conserved quantities, which is analogous to what we discussed for SU(2)in Sec. 7.7. Following the same lines of thought as for SU(2) tells us that we have 8 conserved quantities, one for each genera-tor. Again, we can only use the conserved quantities that belong to the diagonal generators as particle labels. SU(3)has two Cartan⁷⁷

77Recall, Cartan generators=diagonal

generators. generators ¹₂λ₃and ¹₂λ₈. Therefore, every particle that interacts via the strong force carries two additional labels, corresponding to the eigenvalues of the Cartan generators.

The eigenvalues⁷⁸of ¹₂λ3= ¹₂

78The eigenvectors are of course

⎛

Forλ8= ₂^√1₃

Therefore if we arrange the strong interacting fermions into ⎠.

triplets (in the basis spanned by the eigenvectors of the Cartan gener-ators), we can assign them the following labels, with some arbitrary spinorψ:

3). This means something of the form idea comes from the fact that if we add the three colors, i.e.

⎛

we get a state with charge zero (a colorless state), because

λ₃

which is analogous to sunlight that contains all colors of light, but is colorless, nonetheless.

Completely analogous to what we did for SU(2)we assign the color-charge zero to all SU(3)singlets, which are then particles that do not interact via the strong force. Formulated differently: They

are colorless. In addition, one can use the (8-dimensional⁸⁰) adjoint ⁸⁰The adjoint representation of SU(3) is 8 dimensional, because we have 8 generators.

representation of SU(3)to assign color to the gauge fields G^μ_A, i.e.

the gluons, completely analogous to how we assigned isospin to the W-Bosons in Sec. 7.7.