Flavour Symmetry in Particle Physics

Akash JainDepartment of Physics, IISER Bhopal

November 15, 2013

Abstract

The family of Hadrons was a complete mess until mid 1900’s, when Gell-mann intro-duced his Eightfold-Way Diagrams and brought order to their clan. The diagrams mightappear to be just a fortunate chance, but they come naturally in the representation the-ory of SU3. In fact complete theory of Hadron Structure can be developed just from theargument of symmetry and group theory. My aim through this Term Paper is to establishthis relationship between Group Theory and Particle Physics, and to review the theory forobserved patterns in Hadrons.

1 Introduction

In 1917 Emmy Noether gave her famous theorem stating - ‘For every Conservation Law thereexists an underlying Symmetry and vice-versa’. The theorem implies that all conserved quan-tities we encounter in physics, say momentum, energy, angular momentum or charge, can bethought of as a result of an underlying mathematical symmetry in the action of the physicalsystem under consideration. Conversely, if we are able to identify a conserved quantity in oursystem, there must exist an underlying symmetry. Symmetries thus play a crucial role in ourunderstanding of physics. Much information about the system can be extracted just from theargument of symmetry, even when a complete dynamical theory is not available.

Symmetry, in precise language, is a mathematical operation which leaves the system invariant.In Physics one generally encounters two kinds of symmetries - Continuous and Discrete. Asname suggests Continuous Symmetries are operations which can be reached from one state toanother by continuous set of transformations, like Lorentz Transformations; on the other handDiscrete Symmetries can not be reached by continuous transformations, like Parity.

Another distinction within Symmetries that is quite common in Physics is Global and LocalSymmetry. When a mathematical operation (which leaves the system invariant) acts on allspace-time uniformly, it is called a Global Symmetry. Whereas, if the operation is space-timedependent, it is called a Local Symmetry. For example spatial rotations and translations areGlobal Symmetries, whereas twists and expansions of space-time are Local Symmetries. In thiswrite-up I will restrict myself to Global Continuous Symmetries, and especially their applicationsin Particle Physics.

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2 Group Theory

Group Theory is a branch of mathematics used to study symmetries. I shall review here quicklysome of its aspects which we will be using in subsequent sections. An extensive literature on thetopic can be found in any standard text on Group Theory or Mathematical Physics[7][8].

2.1 Groups

An abstract Group G is a set of elements (finite or infinite) with an operation (·) defined on it,which obeys:

1. ∀ u, v ∈ G, u · v ∈ G

2. ∀ u, v, w ∈ G, u · (v · w) = (u · v) · w

3. ∃ I ∈ G s.t. u · I = I · u = u ∀u ∈ G

4. ∀ u ∈ G, ∃ u−1 ∈ G s.t. u · u−1 = I

Readers can check that the set of all possible spatial rotations under operation succession (u ·wis equivalent to rotation u succeeded by w) forms a group.

Continuous and Discrete Groups: A group whose all elements can be reached by a con-tinuous change of one or more parameters (say, angle) is called a Continuous Group or a LieGroup. Otherwise if the elements of the group can be labelled by integral indices, it is called aDiscrete Group. Examples of Lie Groups includes ‘set of all real numbers over usual addition’and ‘set of orientation preserving rotations’, whereas ‘set of all even integers over addition’and ‘Parity-Identity pair of rotations’ (A group with only two elements - Identity and Parityoperator) are examples of discrete groups.

2.2 Group Representations

Every group can be represented in form of a group of invertible matrices over multiplication.This mapping is called a representation of the group. A representation D of group G is givenby:

D : G→ D(G) ⊆ GLn(F)

Where n is called the dimension of representation and F is the field over which matrices aredefined (like R or C). GLn(F) is the set of all invertible matrices.

Faithful Representations: Representations need not be 1-1; most trivial representation canthus be a map sending all the group elements to I matrix. A 1-1 representation, commonlycalled ‘Faithful Representation’, contains all the information about the original group. O3

(A ∈ GL3(R) | ATA = I) is a faithful representation for all spatial rotations, whereas SO3

(A ∈ SO3 | det(A) = 1) is a faithful representation for all orientation preserving rotations.

Fundamental Representations: A matrix group can itself be represented in form of othermatrix groups. For example SO3 is a non-faithful representation of SU2 (A ∈ GL2(R) | A†A =I, det(A) = 1). A representation formed by mapping a matrix group to itself (or another matrixgroup of same dimension) is called a Fundamental Representation of the group.

Actually corresponding to every element in SO3 there are 2 elements in SU2, thus SU2 is called(by abuse of language) a double valued representation of SO3.

2

Irreducible Representations: Given two representations D1(G) and D2(G) of a group G,you can always form a new representation by arranging them in block diagonal form:

D(G) =

[D1(G) 0

0 D2(G)

]This is fundamentally not a new representation, so we restrict ourselves to Irreducible Repre-sentations defined by - Representations which cannot be reduced to a block diagonal form by asimilarity transformation1.

2.3 Lie Groups and Lie Algebras

As I have mentioned earlier, Groups whose elements can be reached by smooth variation ofsome continuous parameter(s) are called Lie Groups. Thus to study their properties, it sufficesto look at infinitesimal transformations near the identity, as we can always integrate many suchtransformations to get a finite element of the group.

Consider G, representation of a lie group H, whose elements are given by G(α1, α2, · · · ) ∈ Gdefined smoothly on some continuous parameters α1, α2, · · · such that u(αi = 0) = I. If wemove away slightly from the identity along a curve γ given by:

γ(t) = γ(α1(t), α2(t), α3(t), · · · )

For some parameter t such that αi(t = 0) = 0. An infinitesimal transformation near identityalong γ is given by Taylor Expansion for small t:

G(t) = G(γ(t)) = I +dG(t)

dt|t=0 · t+O(t2) (1)

g ≡v(γ)

∣∣∣∣ v(γ) =dG(t)

dt|t=0

forms a Lie Algebra corresponding to the given Lie Group G.

Lie Algebra: Lie Algebra (g, [, ]) is a vector space over a field F (say R or C) along with abracket operation [, ] defined by:

1. Bilinearity: [u+ v, w] = [u,w] + [v, w] and [au, bv] = ab[u, v] ∀ a, b ∈ F, u, v, w ∈ g

2. Antisymmetry: [u, v] = −[v, u] ∀ u, v ∈ g

3. Jacobi Identity: [u, [v, w]] + [v, [w, u]] + [w, [u, v]] = 0 ∀ u, v, w ∈ g

Infinitesimal transformation near the Identity can also be written in terms of coordinate partialsusing (Eqn. 1):

G(t) = I +∂G(αi)

∂αi|αi=0

dαidt|t=0 · t+O(t2)

also:

αi(t) = αi(0) +dαidt|t=0 · t+O(t2) =

dαidt|t=0 · t

∴ G(t) = I +∂G(αi)

∂αi|αi=0 · αi(t) (2)

1Change of basis transformation. D → D′ = SDS−1 for some S ∈ GLn(F)

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Generators of Lie Group: χi = −ι∂G(αi)

∂αi|αi=0 are called Generators of the Lie Group G.

In terms of these, infinitesimal element is given by:

G(t) = I + ιχiαi(t) (3)

Once generators in hand, one can easily reach the Lie Group by integrating many infinitesimaltransformations together:

G(αi) = limn→∞

(I + ιχiαi(t))n = eιnχiαi(t) = eιχiαi (4)

Remember αi(t) are infinitesimal parameters, while integrating infinite of them will give mefinite parameters given by

αi = limn→∞

n · αi(t)

ιχi form a basis of lie algebra g. You can verify that they are linearly independent and forma complete set. Therefore due to properties of Lie Algebra, [ιχi, ιχj ] ∈ g.

∴ [χi, χj ] = ιfijk · χk (5)

fijk are called Structure Constants of the Lie Algebra. Note that they are basis dependent; ifthe parametrization of Lie Group is changed, generators of Lie Algebra will change, and hencewill change the structure constants. One can always choose a basis where structure constantsare completely antisymmetric2, i.e.:

fijk =

fjki = fkij−fkji = −fjik = −fikj

2A fact that is not very trivial to digest. You can lookup any standard text in Lie Algebra for a detailedproof, like in Chapter 2 of Ref. [5]

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3 SU2 Representations

If a system is invariant under certain transformation group, it is said to possess that particulargroup symmetry. Whenever we have a physical quantum system with a group G of symmetriesacting on it, the n dimensional space of states H will carry a unitary representation3 of G.Why Unitary? Because Un’s corresponding lie algebra is the set of all skew-hermitian matrices,making the generators hermitian. Generators are thus Quantum Mechanical observables whichwe would like them to be if they are conserved quantities4.

As we know laws of physics do not change under ‘orientation preserving spatial rotations’, allphysical systems possess rotational (or SO3) symmetry. But it turns out that the physicalsystems carry representations of a larger isomorphic group SU2, where every element of SO3 ismapped to two elements in SU2. You must have already encountered this SU2 symmetry in thetheory of Angular Momentum and Spin in Quantum Mechanics.

Hadrons5 in Particle Physics carry an underlying SUn symmetry, basis behind which yet remainsunknown. It is this symmetry which leads to such nice geometric patterns in the arrangementof Mesons and Baryons, and is cause of the success of Eightfold-Way Diagrams and the QuarkModel. With this motivation I will start with unitary representations of the simplest groupamong SUn, i.e. SU2, relating it time to time with familiar structure of Angular Momentum.At the end, I will establish their importance in the Baryons and Mesons structure.

3.1 spin j representation of SU2

For all positive integers n, there exists an n dimensional unitary representation of SU2 calledspin j (n = 2j + 1), given by:

spin j : SU2 → U2j+1 j =

0,

1

2, 1,

3

2· · ·

(6)

The nomenclature is derived from Quantum Mechanics, but has more general sense to it. spin jdepicts the representation of SU2 for 2j + 1 dimensional state space H. Let me pick up threeappropriate6 generators J j1 , J

j2 , J

j3 for corresponding spin j representation. Commutator re-

lationships between them are given by:

[J jl , Jjm] = ιεlmnJ

jn

where, εijk is the fully antisymmetric Levi-Cevita Tensor, given by:

εijk =

1 for cyclic permutations of 1,2,3−1 for anti-cyclic permutations of 1,2,30 if any two indices are same

Since none of the generators commute, I can diagonalize any one of them at a time, which isconventionally chosen to be J j3 . Thus basis states of H are eigenvectors of J j3 . Action of J j1and J j2 will give some non trivial combinations of basis states, which are not much of interest.What are of interest are step-up and step-down operators, given by:

J j± =1√2

(J j1 ± ιJj2) (7)

3Representation in the unitary group Un.4By Noether’s Theorem5Baryons and Mesons are collectively termed as Hadrons.6Such that the structure constants are completely antisymmetric.

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Note that:[J j3 , J

j±] = ±J j±

[J j+, Jj−] = J j3

If I consider a basis state |φ〉 with eigenvalue of J j3 as m:

J j3 |φ〉 = m|φ〉

J j3(J j±|φ〉) = (m± 1)J j±|φ〉

Thus, step-up and step-down operators alters the eigenvalues of J j3 by ±1. Basis states of spin j

can be labelled7 with the eigenvalues of J j3 (denoted by mj) and j:

|j,mj〉 mj = j, j − 1, · · · ,−j (8)

These are commonly called ‘Spin (2j + 1)-tuple’. If we have any one basis state, all others canbe reached by J j± operators.

3.1.1 Fundamental Representation of SU2

Fundamental Representation of SU2 is the map from SU2 to SU2 itself, i.e. spin 12 . Generators

for spin 12 are given in terms of well known Pauli Matrices:

J12i =

1

2σi (9)

σ1 =

[0 11 0

]; σ2 =

[0 −ιι 0

]; σ3 =

[1 00 −1

]And, step-up and step-down operators are:

J12+ =

[0 10 0

]; J

12− =

[0 01 0

]

As spin 12 representation contains full information about SU2, all the higher dimensional rep-

resentations can be reached from it using Tensor Product.

3.2 Tensor Product of Representations

This is the generalization of what is called the Addition of Angular Momentum in QuantumMechanics. If I have a composite system of two independent components both carrying SU2

representations spin jA and spin jB respectively, its states are denoted by:

|jA, jB,mjA ,mjB 〉 = |jA,mjA〉 ⊗ |jB,mjB 〉

We have total (2jA + 1)(2jB + 1) such states. A theorem in Representation Theory8 says thatthe tensor product of two SU2 representations spin jA and spin jB can be decomposed into thedirect sum of irreducible representations spin j given by:

j = |jA + jB|, |jA + jB − 1|, · · · , |jA − jB| (10)

mj = mjA +mjB (11)

7I have skipped the explicit proofs. See Chapter 3 of Ref. [5] for details.8See section 9.4.1 of Ref. [6] for details.

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With respective basis states:

|jA, jB, j,mj〉 =∑

mjA+mjB

=mj

C(mjA ,mjB ; j,mj) · |jA, jB,mjA ,mjB 〉 (12)

mj is the eigenvalue of J j3 operator for corresponding representation. C’s are called Clebsch-Gordan coefficients tabulated in Review of Particle Physics (1982), numerical values of whosedoes not bother us much at this point.

There is a nice diagrammatic way to handle these representations which might not be so illu-minating at the moment, but it finds its importance in higher groups SUn, and give base forEightfold-Way Diagrams in Particle Physics. spin j representation is diagrammatically repre-sented on an axis of mj values, with base states marked on it. J j± takes one state to anotherby step-up and step-down operation.

Figure 1: Diagrammatic Representation of spin j

Now consider (spin jA ⊗ spin jB). The tensor product is generally written in symbolic form asfollows:

(2jA + 1)⊗ (2jB + 1) = (2|jA + jB|+ 1)⊕ · · · ⊕ (2|jA − jB|+ 1) (13)

Draw one of the representation (say jA). Now draw the other representation (say jB) on each ofthe states of the first, overlapping it with mjB = 0 (See Fig 2). Forget the jA representation; fora corresponding value of mj you get a multiplet of jB states. Take a state from each availablemultiplet, and form the highest j representation, spin |jA + jB|. Respectively from residualstates, take one state each and form the second highest representation, and so on until you runout of states. You will see that the state decomposes as expected and will furnish you 2jB + 1representations.

States with some particular value of mj in the resultant representations are actually linearcombinations of states from parent representations with mjA +mjB = mj , coefficients of whichare given by Clebsch-Gordan coefficients (see Eqn. 12).

3.3 Isospin

The theory developed above looks similar to the theory of Angular Momentum in QuantumMechanics. Reason being that the theory of Angular Momentum is the result of rotational SU2

symmetry in our system. There are also other systems that possess SU2 symmetry, for instancethe theory of Isospin in Nuclear Physics.

Similarity in the masses of Neutrons and Protons and their coexistence in nuclei led people tobelieve that they are merely different states of the same particle- nucleon, more or less the sameway electrons have two spin states- ↑ and ↓. This new property was termed Isospin, and wasbelieved to manifest the charge difference between the two. I will however refrain from the topicof charge, until we reach Quark Model.

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Figure 2: Tensoring of Representations

Isospin is also considered to be a consequence of an underlying SU2 symmetry same as spin.Nucleons carry (I = 1

2) fundamental representation of Isospin group, with I3 values 12 (proton)

and −12 (neutron). Protons and Neutrons thus form a Isospin Doublet.

There are other hadrons which carry Isospin symmetry. Some of them are:

Isospin Baryons Mesons

Singlet (Λ),(Ω−) (η),(η′)

Doublets (n,p+),(Ξ−,Ξ+) (K0,K+),(K−,K0)Triplets (Σ−,Σ0,Σ+) (π−,π0,π+)Quadruples (∆−,∆0,∆+,∆2+)

3.3.1 Bound States of Nucleons

Nucleons possess the fundamental representation (spin 12) of isospin, thus all higher representa-

tions can be reached from their composite, commonly known as nuclei. Tensoring two nucleonsgives two irreducible representations I = 1, 0: 2⊗ 2 = 3⊕ 1.

Figure 3: Two nucleon bound states.

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I = 0 (p-n antisymmetric state) is the ground state of two nucleon system, commonly knownas Deuterium. I = 1 are the excited states, which includes p-p (di-proton), n-n (di-neutron)and n-p symmetric (excited deuterium) states.

Similarly all the higher nucleon bound states can be constructed from the symmetry argument,stability or instability of whose vests on energy considerations. I have given below the threenucleon bound states, most of which are not found in nature.

2⊗ 2⊗ 2 = (3⊕ 1)⊗ 2 = 4⊕ 2⊕ 2

Figure 4: 3⊗ 2 and 1⊗ 2 diagrams for three nucleon bound states. Angular brackets signify thelinear combinations of the arguments and its permutations.

By calculating the Clebsch-Gordan coefficients it can be verified that no two states are thesame, though they appear to be at first. 〈nnp〉 are the states of Tritium (3H1) and 〈npp〉 arethe states of a Helium Isotope (3He2); (ppp) and (nnn) are non physical.

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4 SU3 Representations

Next we move to SU3 symmetries. In the current context, they find their prominent applicationin Particle Physics in form of the Eightfold-Way diagrams. The representation theory for SU3

can be developed more or less in the same way we did for SU2, but the mathematical juggleryinvolved gets much complicated. I will try to keep the things fairly simple and understandableon the cost of mathematical preciseness, for which you must spare me. You can find a detailedtreatment of lie algebra representation theory in Ref [5], where most of the loopholes I will beleaving in my discussion are covered.

SU3 has dimension n2−1 = 8, thus its lie algebra su3 (3×3 traceless hermitian matrices) have 8generators. I will denote them by Ji. Commutation relations between the generators are givenby:

[Ji, Jj ] = ιfijkJk (14)

Recall that the structure constants fijk are completely antisymmetric, and its non vanishingcomponents (along with their permutations) are given by:

f123 = 1; f458 = f678 =

√3

2

f147 = f165 = f246 = f257 = f345 = f376 =1

2

With these relations one can easily verify that out of all generators only J3 and J8 commute.This is not surprising because we have two independent 3× 3 traceless diagonal matrices, andwe choose to diagonalize J3 and J8. Similar to step-up and step-down operators in SU2, herewe have:

T± = J1 ± ιJ2; T3 = J3

V± = J4 ± ιJ5; V3 =

√3

2J8 +

1

2J3

U± = J6 ± ιJ7; U3 =

√3

2J8 −

1

2J3

Surprisingly they form three sets of SU2 sub-algebras:

[T3, T±] = ±T±; [T+, T−] = T3

[V3, V±] = ±V±; [V+, V−] = V3

[U3, U±] = ±U±; [U+, U−] = U3

But not all three of them are independent, as U3 and V3 are linear combinations of T3 and J8.T3, V3 and U3 are observables with eigenvalues denoted by t, v and u respectively, which arerelated by:

t = v − u (15)

These values can be raised and lowered by T±, V± and U± operators. Due to this interdepen-dence of observables, there is a degeneracy in our system, i.e. we have different states withsame values of t, v and u.

Same as SU2 we can give an expression for the dimension of representation in terms of maximaleigenvalues; which I will state here without a proof. If I choose to pick the base state withmaximal eigenvalue v = jv, dimension of representation is given by:

n = (|jv|+ 1)(2|t|+ 1)(2|u|+ 1) = (|jv|+ 1)(2|t|+ 1)(2|jv − t|+ 1) (16)

where v and u are other two quantum numbers for that particular state.

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4.1 Diagrammatic Representation

I will skip the math, but what is of more importance to us is the diagrammatic illustrations ofthe SU3 representations and tricks to handle them. Extending our ideas developed for SU2, weadd another dimension to our diagrams as we have two independent observables this time. Weneed to depict 3 SU2 sub-algebras in a 2-D diagram, so a triangle will be the most obvious andsuitable choice.

Figure 5: D(0, 4) or D(15): Non-degenerate 15 Dimensional SU3 Representation.

In Fig. 5, I have a 15 Dimensional Representation, with step-up and step-down operators indi-cated. This is actually one of the simplest cases because for v = jv, |t| = 0; such representations(and also |t| = jv) are called non-degenerate representations, as they show no degeneracy. Ingeneral |t| can take any value from 0 to jv, and we can have degeneracy in our system. I haveshown in Fig. 6, a 42 dimensional degenerate representation with jv = 5

2 and t = 1. Note theshape which is a semi-regular hexagon, with two set of equal side lengths.

There is a more natural way to denote these diagrams other than jv and t. Define:

α = 2|t|β = 2|jv − t|

n = (α+ 1)(β + 1)(α+ β + 2)

2(17)

α and β are nothing but the top-most and bottom-most sides of the semi-regular hexagonrespectively. The representation is called D(α, β) or D(n) (if α > β) or D(n) (if α < β).

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Figure 6: D(2, 3) or D(42): Degenerate 42 Dimensional SU3 Representation.

Degeneracy is depicted by multiplets at the particular position in diagrams. There is an easyalgorithm to plot it:

1. Plot your representation without degeneracy.

2. Mark states of outermost semi-regular hexagon layer as non-degenerate. Next inner semi-regular hexagon layer is doubly degenerate and so on, until you reach a triangle.

3. States inside the ‘triangle’ will have same degeneracy as the periphery of triangle.

Figure 7: D(5, 2) or D(81): Degeneracy in diagrammatic representation.

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4.2 Fundamental Representation of SU3

Fundamental representation(s) of SU3 are the map to SU3 itself. There are two fundamentalrepresentations in this case: D(1, 0) (or D(3)) and D(0, 1) (or D(3))

Figure 8: D(3) and D(3): Fundamental Representations of SU3.

Generators of the fundamental representations are given by Gell-mann matrices:

Ji =1

2λi (18)

λ1 =

0 1 01 0 00 0 0

; λ2 =

0 −ι 0ι 0 00 0 0

; λ3 =

1 0 00 −1 00 0 0

λ4 =

0 0 10 0 01 0 0

; λ5 =

0 0 −ι0 0 0ι 0 0

λ6 =

0 0 00 0 10 1 0

; λ7 =

0 0 00 0 −ι0 ι 0

; λ8 =

1 0 00 1 00 0 −2

Readers might want to check the corresponding commutation relations. Same as SU2, all thefurther representations can be reached by tensoring these fundamental representations.

4.3 Tensor Product of SU3 Representations

Method we adopt for tensor product is again diagrammatic. Similarly like SU2, overlap oneof the diagrams on all the states of the other diagram (Caution! You do not have anotherdimension at your ease to do that). Look at the outermost outline of the resultant figureyou get, a hexagon, sides of which will give the corresponding values of α and β. Plot thecorresponding diagram, and omit the used states from your overlapped figure. Now again lookat the outermost outline after omission, new values of α and β will give you another diagram.Continue doing this until you are out of states; these are the resultant diagrams after tensoring.In Fig. 9 I have depicted following tensor product:

D(2, 1)⊗D(0, 2) = D(2, 3)⊕D(3, 1)⊕D(1, 2)⊕D(2, 0)⊕D(1, 0)

15⊗ 6 = 42⊕ 24⊕ 15⊕ 6⊕ 3

Similar steps can be adopted to reach all SU3 representations from the fundamental represen-tations D(3) and D(3).

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Figure 9: Tensoring of Representations. D(2, 1) ⊗ D(0, 2) gives D(2, 3) ⊕ D(3, 1) ⊕ D(1, 2) ⊕D(2, 0)⊕D(1, 0)

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5 SU4 and Higher Representations

5.1 SU4 Representations

We will not be discussing SU4 representations mathematically, but it is not hard to guess whatwe expect based on our earlier experience. There should be 42−1 = 15 generators, out of which3 will be diagonal (we can have 3 independent 4× 4 traceless matrices). Rest 12 non-diagonalgenerators will give rise to 6 step-up and 6 step-down operators. Hence we will get 6 SU2 sub-algebras (denoted by T,U, V,W,X, Y with their eigenvalues t, u, v, w, x, y), out of which only3 will be independent. Again due to commutating generators we will have degeneracy in thesystem.

Figure 10: Diagrammatic Representations of SU4 - Tetrahedron, Octahedron and 14-hedron.Inner details have been skipped for visual simplicity.

Non-degenerate representations diagrammatically take form of a regular tetrahedron in 3-D,which has 6 edges (corresponding to SU2 sub-algebras) and 4 vertices. Degenerate representa-tions can take form of semi-regular octahedron and 14-hedron.

Fundamental representations are the map from SU4 to SU4 itself, diagrammatically which arenothing but regular unit-sided tetrahedrons with a base-state at each vertex.

Figure 11: Fundamental Representations of SU4

Same as earlier cases, all representations of SU4 can be reached from fundamental represen-tations. If your 3-D visualization is strong, you might be able to tensor these fundamentalrepresentations to get structures given in Fig. 10.

5.2 Higher SUn Representations

Even diagrammatic representations of SUn for n > 4 are hard to tackle, because diagrams aren − 1 dimensional, and at best we can comprehend 3-D objects. Though the theory remainssimilar it is of less use, and we have to go back to rigorous group theoretical formalism which

15

I will not be discussing here. We can still get some information about our SUn system bystudying its SU3 sub-algebras and hence projecting it to our familiar 3-D.

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6 Quark Model

Now when we have assembled all the required mathematical tools, we are ready to discuss thePhysics part this paper is supposed to cover. There is an apparent9 global SUn symmetryin particle physics called ‘Flavour Symmetry ’, much similar to the rotational symmetry inQuantum Mechanics, which governs the pattern Baryons and Mesons families exhibit. Butunlike rotations, symmetry involved here is more of a hypothesis posed by the ‘Quark Model ’,which best explains the similarity in hadrons found in nature.

By the mid of 19th Century the Hadron zoo had grown into a complete mess. Much like theelements in the periodic table, family of hadrons was growing huge, and was awaiting theirown periodic table. Finally in 1961, Gell-mann introduced his ‘Eightfold-way Diagrams’, andarranged Hadrons in weird figures based on their charge and strangeness. Though the figuresmight appear completely empirical to someone taking his first course in Particle Physics, theyappear naturally from Group Theory and are nothing but the SU3 representation diagrams.This gave a strong sign of an underlying SU3 symmetry in the system, which was soon realizedby Gell-mann and Zweig, who independently gave the ‘Quark Model ’ for Hadrons in 1964.

Inspired from nucleons which were understood as Isospin Doublet, fundamental representationof SU2 which generated all the nuclei occurring in nature; it was proposed that there existsa fundamental SU3 Triplet termed Quarks, which generated all the Hadrons. The feasibilityor limitations of the model can be looked-up in any standard text on Particle Physics, I willconcentrate here on how does it explain the Hadron structure. On the course we shall notnecessarily trace the historical development, and will assume all quarks we need are known. Weshall inherit following facts from theory:

1. Antiparticles have all the quantum numbers with sign reversed compared to their corre-sponding particles.

2. Only possible quark combinations are qq (mesons and anti-mesons), qqq (baryons) andqqq (anti-baryons), where q is the antiparticle of q.

3. u, c, t have charge 23 , while d, s, b have charge −1

3 in electronic units.

6.1 2 Quark Model - Up (u) and Down (d)

Although the original idea was proposed with 3 quarks, it is worth looking at 2 quark system uand d to start with. Not only it is simpler as it exhibits SU2 symmetry, it will give us an insighton how can we generalize our theory to 4 and higher quark systems. Initially known hadrons(non-strange) were just composed of u and d quarks. Inspired from nuclear systems, u andd can be considered as an Isospin Doublet10 with I3 values ±1

2 respectively. Correspondinglyu and d form a Isospin Doublet with I3 = ∓1

2 respectively. These will serve as fundamentalrepresentations for this model.

Figure 12: u and d quarks. Fundamental representation of 2 quark model.

9The symmetry is apparent/approximate and not exact. I shall come to this point towards the end.10Right now we have no reason to believe that this Isospin is the same Isospin we encountered in nuclear case,

but as we will see soon it is.

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Charge in this model is nothing but a mere shift of origin:

q = I3 +1

6for quarks; q = I3 −

1

6for antiquarks

We can define quantities like Up-ness and Down-ness to measure the content of u and d quarksin our composite system.

U = I3 ±1

2; D = −I3 ±

1

2(+ for quarks, - for antiquarks)

Positive value of U and D tells the number of u and d quarks in the system, while negativevalues signify the presence of corresponding anti-particles. Similar to isospin, U , D and q alsoadds up in composite systems.

Mesons: As we know Mesons are qq bound states, they can be formed by q ⊗ q.

2⊗ 2 = 3⊕ 1

Figure 13: u and d quark Meson states.

We get an Isospin triplet (π−,π0,π+) and an Isospin singlet (η0), all of which are observed innature.

Baryons: Baryons are one step complicated than mesons as they are qqq bound states. Theycan be reached by q ⊗ q ⊗ q.

2⊗ 2⊗ 2 = (3⊕ 1)⊗ 2 = 4⊕ 2⊕ 2

Figure 14: u and d quark Baryon states. Angular Brackets signify linear combinations of thearguments and its permutations.

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We get an Isospin quadruplet (∆−,∆0,∆+,∆2+) and two linear combinations of Isospin dou-blet (n0,p+). The Isospin Doublet is what we see commonly around us, which provides thefundamental representation for all nuclei (see Sec. 3.3.1).

Similarly one can form 2 quark representation of anti-baryons:

Figure 15: u and d quark anti-baryon states.

6.2 3 Quark Model - Up (u), Down (d) and Strange (s)

Now we are ready to add another quark to the game, s or strange quark. In the beginning of thehunt for new elementary particles, some strange (so were they named) hadrons were observed,which could be explained by inclusion of another particle to our set of quarks and going toSU3 instead of SU2. u, d, s form the fundamental representation for this model, out of whichsub-algebra u-d reduces to our SU2 Isospin algebra.

Figure 16: D(3) and D(3): Fundamental Representations of SU3.

Charge in this model is defined to be

q =2

3(t+ v)

Whereas U and D are revised to be:

U =2

3(t+ v)± 1

3; D =

2

3(u− t)± 1

3(+ for quarks, - for antiquarks)

Similar to these we define a new quantity Strangeness:

S =2

3(−v − u)± 1

3(+ for quarks, - for antiquarks)

+ve value of S tells the number of s quarks in our composite system, while -ve value tells thenumber of s anti-quark11. Putting S = 0 we will land back in u-d sub-algebra, and aboveformulas for U , D and q will reduce to their earlier SU2 forms.

Mesons: Now it is an easy job to reach Mesons (qq); just find 3⊗ 3.

3⊗ 3 = 8⊕ 1

11Caution! In Physics literature generally −S is termed as strangeness. It is just a matter of historicalconvention.

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Figure 17: u, d and s quark Meson states.

They are generally called Meson Octet and Singlet. What we have just derived is nothing butGell-mann’s first Eightfold-Way diagram for Mesons. It is more conventional to draw it on qand (−S) axes as follows:

Figure 18: Meson Octet in Eightfold-Way representation.

Baryons: Baryon states (qqq) in 3 quark Model are given by (See Fig. 19):

3⊗ 3⊗ 3 = (6⊕ 3)⊗ 3 = 10⊕ 8⊕ 8⊕ 1

They take form of Baryon Decuplet, Octet and Singlet. These are Eightfold-Way diagrams forBaryons, which can also be expressed in terms of q and −S (Fig. 20).

Figure 20: Baryon Decuplet and Octet in Eightfold-Way representation.

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Figure 19: u, d and s quark Baryon states. 3⊗ 3 = 6⊕ 3; 6⊗ 3 = 10⊕ 8; 3⊗ 3 = 8⊕ 1.

Symmetric linear combination of the two Octets formed after tensoring gives the Gell-mann’sBaryon Octet, while the singlet is not observed in nature.

Similarly anti-baryons (qqq) in 3 quark model can be reached by:

3⊗ 3⊗ 3 = (6⊕ 3)⊗ 3 = 10⊕ 8⊕ 8⊕ 1

Figure 21: Anti-baryon Decuplet and Octet in Eightfold-Way representation.

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6.3 4 Quark Model - Up (u), Down (d), Strange (s) and Charm (c)

In 1974, a new quark c or Charm was discovered12. Soon after its discovery, whole new rangeof hadrons having charm content were reported. Quark Model had to be modified just slightlyto include this new member. Same as we went from SU2 to SU3 by adding s quark, inclusionof c modifies the underlying symmetry to SU4. (u, d, s, c) and (u, d, s, c) form the fundamentalrepresentations for this model:

Figure 22: Fundamental Representation for 4 quark model.

Charge in this model will be given by:

q =1

2(t+ v + w + x)± 1

6(+ for quarks, - for antiquarks)

Whereas up-ness, down-ness, strangeness and charmness are given by:

U =1

2(t+ v − y)± 1

4; D =

1

2(u− t− w)± 1

4(+ for quarks, - for antiquarks)

C =1

2(w + y + x)± 1

4; S =

1

2(−u− v − x)± 1

4(+ for quarks, - for antiquarks)

Mesons: Mesons, i.e. bound states of qq take form of 14-hedron (15-tuplet) in this model. Iwill not depict the overlapping because it is too messy but the tensoring goes as:

4⊗ 4 = 15⊕ 1

Figure 23: 4 quark meson states. Meson 15-tuplet and singlet.

12Not exactly c was discovered, but its bound state cc in form of J/ψ meson.

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Ground states of all observed u, d, s and c Mesons arrange themselves nicely in the above 3-Dstructure.

Baryons: Baryons states in 4-quark model form 20-tuplets (tetrahedron and octahedron) andquadruplet (tetrahedron). Though quadruplet is not found in nature.

(4⊗ 4)⊗ 4 = (10⊕ 6)⊗ 4 = 20⊕ 20⊕ 20⊕ 4

Figure 24: 4 quark Baryon states. Baryon 20-tuplets and quadruplet. Quadruplet is howevernot found in nature.

Similarly, anti-baryon diagrams are nothing but a inversion about origin of these diagrams.

6.4 Complete Quark Model

Near 1980’s final two quarks, top (t) and bottom (b) were added to the family, rising the totalnumber to 6. Till now it should be obvious that the corresponding symmetry will rise to SU6.As I have mentioned earlier, for any n > 4 diagrammatic representation of SUn is not muchfruitful, so Eightfold-Way Diagrams meet a limitation. But if we have some insight of higherdimensions, we can project out SU4 sub-algebras out of SU6 and get some information aboutthe system; after all at best we have only 3 quark bound states. But this procedure is not muchworth either, because t and b quarks are too heavy and rarely form any bound state.

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7 Summary

I will recapitulate what have we accomplished in more general notation. SUn, set of all n × nunit determinant unitary matrices, form a Lie Group. If a physical system is carrying SUnsymmetry, its state space H will carry a unitary representation of SUn. For example, quantummechanical systems possess SU2 symmetry corresponding to Angular Momentum and Spin, andthus carry a 2j + 1 dimensional unitary representation of SU2. Here j is a quantum numberthat corresponds to angular momentum state or spin of the system.

Similarly Hadrons in Particle Physics possess SU6 flavour symmetry. Basis states of the funda-mental representations corresponds to the six quarks u, d, s, c, t, b and six antiquarks u, d, s, c, t, b.Composite systems of these 2 fundamental representations gives rise to all Mesons (qq) andBaryons (qqq,qqq). SU3 ⊂ SU6 sub-algebra corresponding to quarks u, d, s gives rise to theGell-mann’s Eightfold way diagrams which provided basis for the Quark Model of hadrons.Similarly SU4 ⊂ SU6 sub-algebra corresponding to quarks u, d, s, c gives rise to the pyramidal3-D Eightfold-Way diagrams.

The final impression I want to make here is that the complete theory of Hadron Structure (orflavour symmetry) can be developed just from the argument of symmetry and group theory.This exhibits a strong correlation between Physical Systems and Mathematics, even strongerexamples of which can be found scattered all over in Physics.

8 Discussion

Concept of Symmetry is not new to physicists. It is the physics analogue to the word Beautiful.Ellipticity of the bound orbits, or the conservation of Energy-Momentum, or the spherical struc-ture of cosmic bodies, all are consequences of some underlying symmetry. The same argumentof symmetry led people to believe in existence of a Theory of Everything and Super-symmetry.But it sometimes makes one wonder - why is the world so symmetric or ordered, it could havebeen a lot more messier. A pure mathematical topic Representation Theory of Unitary Matricesand a pure physical theory of Hadrons in Particle Physics - the correspondence between the twois both surprising and elegant.

We have certain cases of exact SUn symmetries in Physics, i.e. state vectors can not be distin-guished with each other apart from the generator’s eigenvalues, for example Gauge Theories andLorentz Transformations. But the symmetry we have at hand, flavour symmetry of Hadronsis an approximate one. Reason being the mass of quarks which is way different from eachother, and one can easily distinguish between quarks on basis of their masses. Apart from themasses too, quarks come in 2 varieties - charge 2/3 and −1/3, which again makes the symmetryapproximate.

Even if approximate, symmetry in Hadrons is of much theoretical importance. After all evenperiodic table of elements is an approximate symmetry, but a whole branch of chemistry isdedicated to its study, and no one can argue upon its importance. Eightfold-Way Diagrams arealso important in their own right; they classify hadrons in a coherent fashion and help us studytheir properties and structure.

My discussions in this term paper are mainly directed towards the Eightfold-Way Diagrams andflavour symmetry, but in a similar fashion one can develop a theory for any symmetry whatso-ever. In the discussion we had in Section 3.2, about SU2 symmetries and their representations,just change of the terminology will fetch us the familiar quantum mechanical theory of angularmomentum and spin. Local Symmetries (or Gauge Symmetries), which I have not touched at

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all here, can also be developed on the same grounds except that the transformations will now bespace-time dependent. This will demand stronger ‘symmetry’ in the system, and we will havestricter conservation laws. For instance theories of fundamental forces - Electromagnetism (U1),Weak Interactions (SU2) and Strong Interactions (SU3) which comprises the Standard Model,have Gauge Symmetries. All of them have their corresponding charge which is conserved locallyat every space-time point.

References

[1] David Griffiths, Introduction to Elementary Particle Physics.

[2] Francis Halzen, Alan D. Martin, Quarks and Leptons.

[3] S. D. Joglekar, Mathematical Physics - Advanced Topics.

[4] Krishna Kaipa, Lecture Notes on Matrix Groups, Lie Algebras, SU2 and SO3.

[5] Howard Georgi, Lie Algebra in Particle Physics.

[6] Peter Woit, Quantum Mechanics for Mathematicians: Fall 2012 Course Notes.

[7] Dummit Foote, Abstract Algebra.

[8] Arfken, Mathematical Methods for Physicists.

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