GLOBAL SYMMETRIES IN FIELD THEORY .1 Transformation of Fields and States

In field theory, groups consist of symmetry operations that leave the equations of motion unchanged in form. These may be discrete symmetries, such asP, C, T discussed inSection 2.10, or discrete internal symmetries, which will be considered inSection 3.2.5. Here we are more concerned with continuous groups. One important class is thespace-time symmetries, such as space rotations, Lorentz boosts, and translations. Another, considered in this section, isinternal symmetries, involving the interchanges of fields with similar properties, changes in their phase, etc. To formalize this, let Φa(x), a= 1,2· · ·n, be n fields (which may be spin-0, ¹₂, 1, etc.) related by a symmetry. Furthermore, consider a Lie groupGof operators U_G(β~) =e^−i~β·T~. Then, define a set ofntransformed fields

Φ⁰_a=e^−i~β·T~Φa(x)e^+i~β·T~

= Φa−i[β~·T ,~ Φa] +(−i)²

2! [β~·T ,~ [β~·T ,~ Φa]] +· · ·, (3.28) where the last form follows from the operator identity in Problem 3.5.Thus, the transfor-mation of the fields is determined by their commutators with the group generatorsTⁱ. Since we assume that the Φa are transformed into each other, one must have that

Tⁱ,Φa(x)

=−Lⁱ_abΦb(x), (3.29)

whereLⁱ_abare the components of ann×nmatrixLⁱ, which is easily shown to form ann×n dimensional representation of the Lie algebra ofG. From (3.28) and (3.29) one has that

Φ⁰_a= (e^+i~β·L~)abΦb≡U(β~)abΦb −−−−−−→

|β~|small

Φa+i~β·~L_abΦb, (3.30)

which defines how thenfields corresponding to representation Lare transformed into each other. One usually considers the case thatLis irreducible.

There is a frequently usefulmatrix notationfor fieldsAⁱtransforming under the adjoint representation⁶(3.16), in which theAⁱare reexpressed in terms of the elements of ann×n matrix

A ≡

N

X

i=1

AⁱLⁱ←→Aⁱ=Tr (ALⁱ)

T(L) , (3.31)

where Lⁱ is an arbitrary non-trivial IRREP of dimension n (usually taken to be the fun-damental or defining). It is then easy to show (Problem 3.7) that the transformation of Aⁱ →A⁰ⁱ≡(e^i~β·L~adj)ijA^j can be expressed in terms of representationLby

A → A⁰≡

N

X

i=1

A⁰ⁱLⁱ=e^+i~β·~LAe^−i~β·L~, A⁰ⁱ= Tr (A⁰Lⁱ)

T(L) . (3.32)

As described inChapter 2, if Φa corresponds to a particle, then the antiparticle field is given by or closely related to Φ^†_a. There are two possibilities for the transformations of non-Hermitian fields. One is that the fields for the particle and antiparticle are in the same IRREP, such as the pions

Φ =



 π⁺ π⁰ π⁻



, (3.33)

which transform as a triplet underSU(2) isospin. This requires that the representation is real, such as the adjoint in this example. Alternatively, the particle and antiparticle fields can be in different IRREPs, such as the kaons under isospin

Φ = K⁺

K⁰

Φ^†= K⁻

K¯⁰

. (3.34)

Then, if Φ transforms under the n representationLⁱ_n, Φ^† transforms under the conjugate representationLⁱ_n∗ =−L^i∗_n =−L^iT_n , which follow by taking the adjoint of (3.29) and using thatTⁱis Hermitian. Of course,Lⁱ_n may be real, as in theSU(2) example or for the adjoint ofSU(3).

From the expressions (2.93) or (2.159) on pages 23 and 35 one sees that for free fields the single particle state corresponding to Φa may be constructed by

|ai=a^†|0i ∼Φ^†_a|0i, (3.35) where in the second expression it is understood that a Fourier transformation and appro-priate projections of Dirac spinors, etc., are to be performed. Thus, the states|aiand Φ^†_a|0i transform the same way under G. This continues to hold for interacting fields as long as theTⁱcommute with the HamiltonianH. Then, the action of the generator on the state is Tⁱ|ai ∼TⁱΦ^†_a|0i= Φ^†_aTⁱ|0i+L^iT_abΦ^†_b|0i. (3.36) Assume for now that the ground state is invariant, i.e.,Tⁱ|0i= 0. Then,

Tⁱ|ai=|biLⁱ_ba=|bihb|Tⁱ|ai, (3.37) so that the representation matrix Lⁱ_ba = hb|Tⁱ|ai is just the matrix element of Tⁱ in the n-dimensional space of particles.

6The adjoint representation is real, so there is no distinction between upper and lower indices, i.e., Aⁱ=Ai.

3.2.2 Invariance (Symmetry) and the Noether Theorem

The Lagrangian densityLisinvariantorsymmetricunder a group of transformationsUG(β~) if they commute, i.e., if

L⁰ ≡UG(β~)LUG(β~)⁻¹=L (3.38) for allβ~. (A similar definition applies for invariance under discrete transformations.) Since

L⁰=L −ih

~β·T ,~ Li

(3.39) for small|β~|, invariance holds if and only if

Tⁱ,L

= 0 (3.40)

for alli.

The first part of (3.38) defines the transformation ofLwhether or not there is an exact symmetry. SinceLis a function of (Φa, ∂_µΦa), and (for a non-Hermitian field) of (Φ^†_a, ∂_µΦ^†_a) one has that

L⁰=U_G(β~)LU_G(β~)⁻¹=L(Φ⁰_a, ∂_µΦ⁰_a,Φ^0†_a, ∂_µΦ^0†_a), (3.41) where Φ⁰_a is given in (3.30), with an analogous expression for∂_µΦ⁰_a(since we are considering global transformations,β~= constant). The expressions for Φ^0†_a and∂_µΦ^0†_a are similar except that Lⁱ→ −L^i∗. It is frequently useful to considerexplicit symmetry breaking, i.e.,

δL ≡ L⁰− L 6= 0 (but small). (3.42) One can also have spontaneous symmetry breaking

L⁰=L but Tⁱ|0i 6= 0, (3.43)

i.e., the Lagrangian is invariant but the ground state breaks the symmetry (cf., the breaking of rotational invariance in a ferromagnet). Both of these cases will be considered extensively below, but for now consider an exact symmetry,

[Tⁱ,L] = 0 andTⁱ|0i= 0. (3.44) This implies degenerate multiplets of particles and definite relations between their inter-actions. It also implies conserved currents and charges according to the Noether theorem, which generalizes the result for a single complex scalar field discussed inSection 2.4.1. The Noether theorem for internal symmetries is

∂_µJ^iµ = 0, d

dtQⁱ= 0, (3.45)

where the Noether current and charge are J_µⁱ ≡ −i δL

δ∂^µΦa

Lⁱ_abΦb−i δL δ∂^µΦ^†a

(−L^i∗_ab)Φ^†_b, (3.46)

Qⁱ = Z

d³~xJ₀ⁱ(t, ~x). (3.47) Normal ordering on the fields is implied. The Noether theorem can be derived from the

Euler-Lagrange equations, in analogy with the derivation for theU(1) case inSection 2.4.1.

One can use the canonical commutation rules to show that [Qⁱ, Q^j] =ic_ijkQ^k,

Qⁱ,Φa(x)

=−Lⁱ_abΦb(x). (3.48) That is, one can identify Qⁱ=Tⁱ as a concrete construction of the generators in terms of the fields.

The Noether currents are also useful for explicitly broken symmetries. The Noether charges are no longer time independent, but one can use the canonical commutation rules to show that the commutation rules in (3.48) still hold provided the charges and fields are evaluated at equal times. For example, suppose L=L⁰+L¹, where onlyL⁰ is invariant,

Tⁱ,L⁰

= 0 (alli), Tⁱ,L¹

6

= 0 (somei). (3.49)

Then, the change inLis related to the divergence ofJ~, δL=L⁰− L=h

−i~β·T ,~ L¹i

=−β~·∂^µJ~_µ. (3.50) This immediately implies

∂^µJ_µⁱ =i Tⁱ,L

⇒ Z

d³~x ∂^µJ_µⁱ =−i Tⁱ, H

, (3.51)

where the last step assumes that any symmetry breaking is in the mass and interaction terms (i.e., that kinetic energy terms are canonical). The (non-conserved)Tⁱ and∂^µJ_µⁱ are evaluated at the samet. Equation (3.51) implies that

ha(pa)|Tⁱ|b(pb)i=i(2π)³δ³(~p_a−~p_b)ha(p_a)|∂^µJ_µⁱ|b(p_b)i

E_b−E_a . (3.52)

This relation is useful when states a and b are not related by the symmetry and are not degenerate in the symmetry limit. One then has that the leakage of Tⁱ|bi into |ai is pro-portional to the symmetry breaking.

The Complex Scalar

As a first example, consider an IRREP relating n complex scalars φa, a = 1· · ·n, trans-forming as

Tⁱ, φa

= −Lⁱ_abφb under some Lie algebra that will be determined from the symmetries of the Lagrangian density,

L=

n

X

a=1

(∂_µφ_a)^†(∂^µφ_a) + non-derivative terms. (3.53) The Noether currents are

J_µⁱ =iφ^†_aLⁱ_ab←→∂_µ φ_b, (3.54) wheref←→∂_µg≡f(∂µg)−(∂µf)g. The derivative (kinetic energy) terms are invariant under the groupU(n) =SU(n)×U(1). Under anSU(n) transformation

φ→e^i~β·L~φ, φ^† →φ^†e^−i~β·L~, (3.55)

where φ is the n-component column vector (φ1 φ₂· · ·φ_n)^T, φ^† is the row vector (φ^†₁ φ^†₂· · ·φ^†_n), andLⁱ is the fundamental representation matrix Lⁱ_n of SU(n). The SU(n) invariance is obvious with this matrix notation

L^KE≡(∂µφ)^†(∂^µφ)→(∂µφ)^†e^−i~β·~Le^+i~β·~L(∂^µφ) = (∂µφ)^†(∂^µφ). (3.56) L^KEis also invariant underU(1) transformations,φ→e^iβIφ.U(n) is the maximal possible symmetry group of the system; depending on the mass and interaction terms the symmetry may be smaller.

Including mass terms

L= (∂µφ)^†(∂^µφ)−φ^†µ²φ, (3.57) whereφ^†µ²φ=φ^†_aµ²_abφ_b andµ² is ann×nmatrix with elementsµ²_ab. The Hermiticity ofL requires thatµ² is Hermitian. The eigenvectors and eigenvalues ofµ² correspond to states of definite mass and to their mass-squares, respectively. For now, however, let us assume that µ²is already diagonal,µ²= diag µ²₁µ²₂· · ·µ²_n

. Under a group transformation, φ^†µ²φ→φ^†e^−i~β·~Lµ²e^+i~β·~Lφ, (3.58) so the requirement for invariance is that

e^−i~β·~Lµ²e^+i~β·L~ =µ² (3.59) for allβ, which is equivalent to~

hβ~·L, µ~ ²i

= 0. (3.60)

Equation (3.60) determines what subgroup ofU(n) survives. For example, if all of the masses are the same,µ² =µ²₁I, then L is invariant underU(n). If all the massesµ²_a are different, then the symmetry is reduced to

U(1)ⁿ=U(1)1×U(1)2× · · · ×U(1)n, (3.61) where onlyφa transforms nontrivially underU(1)a, i.e.,

φ_a→e^iβaφ_a, φ_b→φ_b forb6=a. (3.62) For the intermediate case of

µ²= diag





µ²₁· · ·µ²₁

| {z }

n1

µ²₂· · ·µ²₂

| {z }

n2





 (3.63)

with n1 fields of massµ1 and n2 of mass µ2, the symmetry group isU(n1)×U(n2), with the first (second) set of fields transforming underU(n1) (U(n2)).

One can also add quartic interaction terms, L= (∂µφ)^†(∂^µφ)−φ^†µ²φ−X

abcd

λ_abcdφ^†_aφ_b φ^†_cφ_d, (3.64) whereλ^∗_abcd=λ_badc (from Hermiticity), and Bose symmetry projects out the parts ofλ_abcd that are symmetric inacand inbd. In general,⁷this reduces the symmetry toU(1), but there

7Even theU(1) would be broken in the presence of terms likeφ⁴+φ^†4or (φ+φ^†)φ^†φ.

could be a higher symmetry for specific λ’s. For example, fullU(n) invariance is restored for

λ_abcd=λδ_abδ_cd, µ²=µ²₁I

⇒ L= (∂µφ)^†(∂^µφ)−µ²₁φ^†φ−λ φ^†φ²

. (3.65)

The Hermitian Scalar

Hermitian scalars,φ_a=φ^†_a, transform as Tⁱ, φ_a

=−Lⁱ_abφ_b. (3.66)

Consistency requires thatLⁱ =−L^i∗, i.e., the representation is real. The Lagrangian density L=1

2(∂µφ_a) (∂^µφ_a) + non-derivative terms (3.67) implies Noether currents

J_µⁱ =−i(∂µφ_a)Lⁱ_abφ_b (3.68) (the second term in (3.46) is absent for Hermitian fields). The special case

L= 1 2

h(∂µφa)²−µ²φaφa

i−λ(φaφa)² (3.69) isO(n) invariant (Problem 3.8).However,

L=1

2(∂_µφ_a)²−1

2φ_aµ²_abφ_b

−κ_abcφ_aφ_bφ_c−λ_abcdφ_aφ_bφ_cφ_d

(3.70)

has no symmetries at all in general.

Complex Scalar in a Hermitian Basis

A complex scalar field φ can always be written in terms of two Hermitian scalars as in (2.91) on page 23,φ= (φR+iφ_I)/√

2, whereφ_R,I are Hermitian. For complex fields that are in the same representation as their adjoints, such as in (3.33), it is almost always simpler to rewrite the theory in terms of Hermitian fields. In the case such as (3.34) that the complex fields and their adjoints transform separately, it is still sometimes useful to go to a Hermitian basis (especially for formal manipulations), although the complex basis is usually simpler for explicit calculations. The relation between the bases is straightforward, but can be confusing.

Supposeφis an n-component complex field transforming with representation matrices Lⁱ_φ. It is then convenient to introduce the complex 2n-component vector Φ =

φ φ^†

, with half of the components redundant. Φ transforms under the reducible representation

Lⁱ_Φ=

Lⁱ_φ 0 0 −L^i∗_φ

. (3.71)

One can introduce 2n Hermitian fields φ_aR, φ_aI, by φ_a = ^√1₂(φaR +iφ_aI), and the 2n-component real vector φ_h =

φ_R φ_I

. The two bases are related by the unitary transfor-mation

whereIis then×nidentity. Hence, the representation matrices for the symmetry generators in the Hermitian basis are

which are manifestly imaginary and antisymmetric for HermitianLⁱ_φ.

Examples:(a) Consider theU(1) group acting on a single complexφ→exp(+iβ)φ, so that L_φ= 1. Aand the representation matrices are therefore

A= 1

while a finiteU(1) transformation is just a rotation φ_R

(b) Consider two complex fields φ = φ⁺

φ⁰

transforming as a doublet under SU(2), Lⁱ =τⁱ/2. (The superscripts look ahead to applications to the Higgs.) In the (reducible) 4-dimensional Hermitian basis φ_h = (φ1 φ3 φ2 φ4)^T, where φ⁺ = (φ1+iφ2)/√

Now consider the case ofnfermionsψ_a, with

L= ¯ψ_ai6∂ψ_a−ψ¯_am_abψ_b= ¯ψi6∂ψ−ψmψ.¯ (3.77) In the second form,ψis then-component column vector (ψ1ψ2· · ·ψ_n)^T andm=m^† is an n×nHermitian matrix⁸ that can be taken to be diagonal.ψandψ^† transform as

Tⁱ, ψa

=−Lⁱ_abψb, Tⁱ, ψ_a^†

= +L^i∗_abψ^†_b (3.78) under a symmetry transformation, and the corresponding Noether current is

J_µⁱ = ¯ψ_aγ_µLⁱ_abψ_b. (3.79)

8Generalized fermion mass terms involvingγ⁵or, equivalently, non-Hermitian matrices, are considered inProblem 3.32andChapter 8.

As in the scalar case, the symmetry group is determined by L. One has

L → L⁰= ¯ψi6∂e^−i~β·~Le^+i~β·~Lψ−ψe¯ ^−i~β·~Lme^+i~β·L~ψ. (3.80) The kinetic term is invariant underU(n) =SU(n)×U(1), but the invariance condition for the mass term is

e^−i~β·~Lme^+i~β·L~ =m↔h

β~·L, m~ i

= 0. (3.81)

The full U(n) is maintained for n degenerate masses, m = m1I, while the symmetry is reduced toU(1)ⁿ forndistinct masses.

3.2.3 Isospin andSU

(3)

Symmetries SU(2)Isospin

Isospin is an approximate symmetry of the strong interactions, broken by∼1%. The break-ing is ultimately due to theu−dquark mass differences. This is usually viewed as intrinsic to the strong interactions when discussing QCD, though the masses are actually associated with the electroweak sector. There is a comparable breaking from electromagnetism. We first describe a simple model of isospin in terms of the nucleons and pions. Introduce the nucleon (proton and neutron) and pion fields

ψ= (funda-mental) and triplet (adjoint), respectively,⁹ underSU(2),

Lⁱ_ψ=τⁱ

2, Lⁱ_π

jk=−i_ijk. (3.83)

The diagonal generator is T³ (L³_π is diagonal in the π^±,0 basis). Consider the Lagrangian density

. Theg_π term is theYukawa interaction between the pion and nucleon.¹⁰ The second (matrix) form in (3.84) makes it especially easy to read off the Feynman rules for the ¯ppπ⁰, ¯nnπ⁰, ¯npπ⁻, and ¯pnπ⁺ vertices, namely,

−g_πγ⁵, +gπγ⁵,−√

2gπγ⁵, and−√

2gπγ⁵, respectively.

We should emphasize that experimentallyg_πis very large: the experimentalπNcoupling

9The convention for the π^± fields in (3.82) is common in particle physics, and convenient because π⁺ = (π⁻)^†. However, the Condon-Shortley phase conventions usually employed for states in rotational multiplets in quantum mechanics (and in standard Clebsch-Gordan coefficient tables) would instead require the conventionπ^±=∓(π1∓iπ2)/√

2 (Problem 3.12).

10We use the term Yukawa interaction in a generalized sense, i.e., for any 3-point interaction between a spin-0 and spin-¹₂ particles.

G_πobserved from low energyπN andN Ninteractions, which may differ fromg_πby strong interaction corrections, is ∼13.05(8) (e.g., Gorringe and Fearing, 2004). Therefore, (3.84) should be viewed as a model to illustrate symmetry considerations and not as a serious perturbative field theory for the strong interactions. L⁰ is SU(2) and reflection invariant (theγ⁵ is because the pions are pseudoscalar); using (3.32),

ψ→e^+i~β·~τ2ψ, ψ¯→ψe¯ ^{−i~β·~τ2}, Π→e^i~β·~τ2 Πe^{−i~β·~τ2}, (3.85) from which both ¯ψγ⁵Πψ and Tr Π² are invariant underSU(2). The invariance of the pion self-interaction is further discussed inProblem 3.15.

Now, introduceSU(2) breaking byL=L⁰+L¹, where L¹=−ψτ¯ ₃ψ=− ψ¯_pψ_p−ψ¯_nψ_n

. (3.86)

L¹thus represents a splitting between the proton and neutron masses,

m_p=m+ m_n=m−, (3.87)

which parametrizes contributions both from the quark masses and from electromagnetism.¹¹ It is straightforward to show that

T³,L¹

= 0,

T^1,2,L¹

6

= 0. (3.88)

Thus, SU(2) is broken to U(1)T³, corresponding to a conserved charge T³ with values t³_π± = ±1, t³_π0 = 0, and t³_p = −t³_n = 1/2. Actually, L is invariant under U(1)T³×U(1)B, where the secondU(1) corresponds to a conserved fermion (or in this case, baryon) number B_p,n = 1, Bπ = 0. The conservation ofB and T³ is equivalent to that of B and electric chargeQ, whereQ=T³+B/2 when restricted to the fields in this example.L¹transforms as the T = 1, T³ = 0 component of an irreducible tensor operator. One can therefore use the Wigner-Eckart theorem for relations between its matrix elements, in exact analogy to broken rotational invariance in quantum mechanics.

SU(3)Symmetry

SU(3) is an approximate global symmetry of the strong interactions that extends theSU(2) isospin subgroup. It is valid at the ∼25% level for masses, but works better for relations between couplings and matrix elements. SU(3) was proposed independently by M. Gell-Mann and Y. Ne’eman in the early 1960s to account for the fact that the low lying mesons and baryons could be associated in octets (the eightfold way). As described in Section 3.1.2,SU(3) has eight generators. The fundamental representation matricesLⁱ₃=λⁱ/2 and structure constants are listed in Tables 3.1and 3.2. SU(3) has rank 2, so two generators, T³ and T⁸, can be simultaneously diagonalized with the Hamiltonian. Empirically, these correspond to the strong interaction quantum numbers by theGell-Mann-Nishijima relation

Q=I3+Y

2, I3=T³, Y= 2

√3T⁸, (3.89)

where Q is electric charge, I₃ is the third component of isospin, Y = B +S is strong hypercharge,B is baryon number, andSis strangeness (S = 0 for the pions and nucleons).

11For simplicity we ignore isospin breaking terms in the interactions or fromµ²

π^±−µ²_π0. The latter are easily shown to come only from electromagnetism to leading order, and not from the quark mass differences.

The low-dimensional representations of SU(3) are the 1,3,3^∗,6,6^∗,8,10,10^∗, and 27, where the 1,8 (adjoint), and 27 are real and the others complex. The observed light hadrons can be assigned to the 1,8,10, and 10^∗. It is convenient to display the states on weight diagrams, with the axes corresponding to I3 and Y. Then, the other generators of the Lie algebra describe transitions from one state to another. For example, the lowest lying J^P = ¹₂⁺ baryons¹²andJ^P = 0⁻ mesons (J is the spin andP is the intrinsic parity) both transform under the adjoint (octet) representation, as shown inFigure 3.1. In the absence of SU(3) breaking, the states in each octet would be degenerate. Each consists of two isospin doublets withY=±1, and one isotriplet and one isosinglet withY= 0. The baryon fields, which annihilate states withB= 1 and strangeness S=Y −B, are given by

Σ^±= 1

√2(ψ1∓iψ₂), Σ⁰=ψ₃, Λ =ψ₈ p= 1

√2(ψ4−iψ5), n= 1

√2(ψ6−iψ7) Ξ⁰= 1

√2(ψ₆+iψ₇), Ξ⁻= 1

√2(ψ₄+iψ₅),

(3.90)

while the pseudoscalar meson (B= 0) fields are π^± = 1

√2(φ1∓iφ₂), π⁰=φ₃, η =φ₈ K^± = 1

√2(φ4∓iφ5), K⁰ K¯⁰

= 1

√2(φ6∓iφ7).

(3.91)

Theη⁰, which also hasI3=Y = 0, is anSU(3) singlet. Theπ, K, η, η⁰ system is referred to as anonet= 8 + 1.

Y

I

3

n p

Σ

⁻

Σ

⁰

, Λ Σ

⁺

Ξ

⁻ ₋₁

Ξ

⁰

1

−1 1

Y

I

3

K

⁰

K

⁺

π

⁻

π

⁰

, η π

⁺

K

⁻ ₋₁

K ¯

⁰

1

−1 1

Figure 3.1

Weight diagrams for the

J^P

=

¹₂⁺

baryon and

J^P

= 0

⁻

meson octets. The anti-baryons are in a separate octet, while the mesons and their antiparticles are in the same octet.

Analogous to (3.82) one can define 8-component vectors ψ and φ, so that ψ →

12A baryon (meson) is a hadron with half-integer (integer) spin. A hyperon is a baryon with nonzero strangeness (but no heavy quantum numbers such as charm).

exp (i~β·~L_adj)ψ, φ → exp (i~β·L~_adj)φ under an SU(3) transformation. The extension of (3.84) to the baryon and pseudoscalar meson octets is

L⁰= ¯ψ_i(i6∂−m)ψ_i+1 inde-pendentSU(3)-invariant meson-baryon interactions, known as the “F” and “D” couplings, which are, respectively, antisymmetric and symmetric in theSU(3) indices. This can be un-derstood as follows. An invariant coupling is just a singlet component of the direct product of the fields in the interaction term. ForSU(2) the meson-baryon interaction in (3.84) involves the direct product of two doublets and one triplet, (2×2)×3 = (1 + 3)×3 = 3 + [1 + 3 + 5], where the IRREPs are labeled by their dimensionality 2I+ 1. The singlet only occurs once in the decomposition, so there is only one invariant. For SU(3), however, there are two independent ways to form an invariant from 8×8×8,

8×8 = 1 + 8 + 8 + 10 + 10^∗+ 27

Similarly, there are now two invariant meson self-interaction terms, associated with the singlet and symmetric octet components of 8×8.

L⁰ can be written in matrix notation, using

M ≡

SU(3)Breaking

The degeneracy of theSU(3) multiplets is only good to around 25%, though the predictions for couplings and amplitudes are typically much better. The Gell-Mann-Okubo (GMO) ansatz is that the breaking can be described by an operator that transforms as an octet, i.e.,

L=L⁰+₈L⁸, (3.98)

whereL⁰is a singlet (i.e., invariant),8 is a small coefficient, andL⁸transforms as the 8^th component of an octet of operators,Lⁱ, i= 1· · ·8:

U_GL⁰U_G⁻¹=L⁰, U_GL⁸U_G⁻¹=

e^i~β·~Ladj

8jL^j (3.99)

or equivalently,

Tⁱ,L⁰

= 0, Tⁱ,L⁸

=− Lⁱ_adj

8jL^j =if_i8jL^j. (3.100) When first postulated, the actual form ofL⁸ was not known, only its transformation prop-erties. The power of such an ansatz is that it allows matrix elements ofL⁸ to be related by SU(3) in terms of one or more parameters that can be measured (or calculated in a more de-tailed theory). To illustrate this, recall the Wigner-Eckart theorem forSU(2), which relates the matrix elements of an irreducible tensor operator¹³ T_q^k, which carries angular momen-tum (or isospin) kand z- componentq, between statesα1 and α2 with angular momenta andz componentsj1,2and m1,2:

hα2 j2m2|T_q^k|α1 j1 m1i=hα2j2kT^kkα1 j1ihj2 m2 |k q j1 m1i. (3.101) The double-barred quantity is the reduced matrix element, which depends on the operator and states, but is independent of m₁, m₂, and q, while the second quantity is a Clebsch-Gordan (CG) coefficient, which leads to selection rules, m2 = q+m1 and |k −j1| ≤ j2≤k+j1, and relations between the nonzero matrix elements. [An excellent table of CG coefficients can be found in the Review of Particle Properties (Patrignani, 2016, or their website).] Similarly, forSU(3) the matrix element of an octet operator between two octets can be written in terms of two quantities that depend on the dynamics and the f and d symbols, which are analogs of the CG coefficients.¹⁴ Thus,

−8hi|L⁸|ji=α if_i8j+βd_i8j, (3.102) where αand β are proportional to₈. The symmetry-breaking term in the Hamiltonian is

−₈R

d³~xL⁸(x), implying that the shift in the mass m_Br of baryonB_r due to theSU (3)-breaking is

∆mBr ∼ hB_r| −8L⁸|B_ri 2mBr

, (3.103)

where the 2mBr in the denominator is from our covariant normalization convention. There-fore,

mB_r =mB₀+c^r_fmα+c^r_dmβ, (3.104) where mB₀ is a common (SU(3)-invariant) mass, c^r_f,d are the “CG” coefficients obtained from if andd by taking the appropriate linear combinations of indices,mα∼α/(2mB0),

13 T³, T_q^k

=qT_q^k,

T¹±iT², T_q^k

=p

(k∓q)(k±q+ 1)T_q±1^k .

14One can generalize the CG coefficients to arbitrary SU(3) representations using isoscalar fac-tors(de Swart, 1963; Patrignani, 2016).

and similarly form_β. Ignoring isospin breaking, there are four masses,M_N ≡(Mp+Mn)/2, MΣ, MΞ,andMΛ, which can be expressed in terms of three parameters,m_B0,α,β. The latter cannot be predicted a priori, but there is one linear relation, theGell-Mann-Okubo relation (Problem 3.20)

M_Ξ+M_N

2 = 3M_Λ+M_Σ

4 +O ²₈

, (3.105)

where the result holds to leading order in 8. Experimentally, the GMO relation works extremely well. The individual masses in GeV,

M_N ∼939, M_Ξ∼1318, M_Σ∼1193, M_Λ∼1116, (3.106) indicate SU(3) breaking at the 20% level, but the left- and right-hand sides of (3.105) are 1129 and 1135, respectively, equal to better than 1%. Similar formulae apply to other low-lying hadronic states, such as the pseudoscalar (J^P = 0⁻) mass-squaresµ²_π, µ²_K, µ²_η and the lowest vector (J^P = 1⁻) meson mass-squares, µ²_ρ, µ²_K∗, µ²_φ. (One must include the effects of octet-singlet mixing between theη andη⁰ and between theφandω. SeeSection 5.8.3.)

The situation is simpler for the J^P = ³₂⁺ states, which transform as a 10 (decuplet).

There is only one invariant of the form 10×8×10^∗, so there is only one reduced matrix element inh10|L⁸|10i, leading to a linear spacing in the masses as one goes to lowerY. This works very well, as can be seen in Figure 3.2. In fact, the Y=−2, S=−3 baryon Ω⁻ was not known at the timeSU(3) was proposed. The prediction of its existence and mass from the GMO ansatz was a great triumph forSU(3).

Y

I3

∆⁻ ∆⁰ ∆⁺ ∆⁺⁺ 1230

Σ^∗− Σ^∗0 Σ^∗+ 1385

Ξ^∗− Ξ^∗0 1530

Ω⁻ 1672 (predicted)

−1

−2 1

−³2 3

−¹2 1 2

−1 2 1

Im Dokument The Standard Model and Beyond (Seite 111-123)