Chapter 7 Unitary Symmetries and QCD As a Gauge Theory

Chapter 7

Unitary symmetries and QCD as a gauge theory

Literature: Lipkin [23] (group theory concepts from a physicist’s point of view) • Lee [24], chapter 20 (extensive treatment of Lie groups and Lie algebras in the • context of diﬀerential geometry) Interactions between particles should respect some observed symmetry. Often, the procedure of postulating a speciﬁc symmetry leads to a unique theory. This way of approach is the one of gauge theories. The usual example of a gauge theory is QED, which corresponds to a localU(1)-symmetry of the La grangian :

ieqeχ(x) ψ ψ � = e ψ, (7.1) → A A � =A ∂ χ(x). (7.2) µ → µ µ − µ We can code this complicated transformation behavior by replacing in the QED La- grangian∂ µ by the covariant derivativeD µ =∂ µ + ieqeAµ.

7.1 IsospinSU(2)

For this section we consider only the strong interaction and ignore the electromagnetic and weak interactions. In this regard, isobaric nuclei (with the same mass number A) are very similar. Heisenberg proposed to interpret protons and neutrons as two states of the same object : the nucleon: 1 p =ψ(x) , | � 0 � � 0 n =ψ(x) . | � 1 � �

125 126 CHAPTER 7. UNITARY SYMMETRIES AND QCD AS A GAUGE THEORY

We note the analogy to the spin formalism of nonrelativistic quantum mechanics, which originated the name isospin. 1 In isospin-space, p and n can be represented as a two-component spinor withI= 2 . p has thenI =| +� 1 and| �n hasI = 1 . | � 3 2 | � 3 − 2 Since the strong interaction is blind to other charges (electromagnetic charge, weak hy- percharge), the (strong) physics must be the same for any linear combinations of p and n . In other words, for, | � | �

p p� =α p +β n , | �→| � | � | � n n� =γ p +δ n , | �→| � | � | � for some α,β,γ,δ C, or, ∈

ψp N = N � =U N , (7.3) | � ψn → | � | � � � for some 2 2 matrixU with complex entries, t he (strong) physics does not change if we × switch from N to N � to describe the system. | � | � We remark at this point that this symmetry is only an approximate symmetry since it is violated by the other interactions, and is hence not a symmetry of nature. First we require the conservation of the norm N N which we interpret as the number of particles like in quantum mechanics. This yi�elds,| �

! N N N � N � = N U †U N = N N � | �→� | � � | | � � | � U †U=UU † = U U(2). (7. 4) ⇒ ⇒ ∈ A general unitary matrix has 4 real parameters. Since the eﬀect ofU and e iϕU are the same, we ﬁx one more parameter by imposing,

detU =! 1 U SU(2), (7.5) ⇒ ∈ the special unitary group in 2 dimensions. This group is a Lie group (a group which is at the same time a manifold). We use the representation,

ˆ U=e iαj Ij , (7.6)

where theα j’s are arbitrary group parameters (constant, or depending on the spacetime coordinatex), and the Iˆj’s are the generators of the Lie group.

We concentrate on inﬁnitesimal transformations, for whichα j 1. In this approximation we can write �

U + iα Iˆ . (7.7) ≈ j j 7.1. ISOSPINSU(2) 127

The two deﬁning conditions ofSU(2), Eq. (7.4) and (7.5), imply then for the generators,

ˆ ˆ Ij† = Ij (hermitian), (7.8)

Tr Iˆj = 0 (traceless). (7.9)

In order for the exponentiation procedure to converge for noninﬁnitesimalα j’s, the generators must satisfy a comutation relation, thus deﬁning the Lie algebra su(2) of the groupSU(2). Quite in general, the commutator of two generators must be expressible as a linear com- bination of the other generators 1. In the case of su(2) we have,

[Iˆi, Iˆj] = iεijkIˆk, (7.10) whereε ijk is the totally antisymmetric tensor withε 123 = +1. They are characteristic of the (universal covering group of the) Lie group (but independent of the chosen representation) and called structure constants of the Lie group. The representations can be characterized according to their total isospin. Consider now I=1/2, where the generators a re given by

1 Iˆ = τ i 2 i withτ i =σ i the Pauli spin matrices (this notation is chosen to prevent confusion with ordinary spin):

0 1 0 i 1 0 τ = τ = τ = 1 1 0 2 i−0 3 0 1 � � � � � − � which fulﬁll [σi, σj] = 2iεijkσk. The action of the matrices of the representation (see Eq. (7.6)) is a non-abelian phase transformation:

#» #» iα τ N � =e · 2 N . | � | �

ForSU(2), there exists only one dia gonal matrix (τ3). In general, forSU(N), the following holds true:

Rankr=N 1: There arer simultaneously diagona l operators. • − Dimension of the Lie algebrao=N 2 1: There areo generators of the group and • − thereforeo group parameters. E. g. in the case ofSU(2)/ ∼= SO(3) this means that there are three rotations/generators and three angles{± as} parameters.

1Since we are working in a matrix representation ofSU(2) this statement mak es sense. The diﬀerence between the abstract group and its matrix representation is often neglected. 128 CHAPTER 7. UNITARY SYMMETRIES AND QCD AS A GAUGE THEORY

� �� 3

��

Figure 7.1: The nucleons n and p form an isospin double t. | � | �

Isospin particle multiplets (representations) can be characterized by their quantum num- bersI andI 3: There are 2I + 1 states. Consider for example once again the caseI=1/2. There are two states, characterized by theirI 3 quantum number:

I= 1 ,I = + 1 p 2 3 2 = | � . I= 1 ,I = 1 n �� 2 3 − 2 �� | �� This is visualized in Fig. 7.1, along� with the action of the operatorsτ = 1/2(τ1 iτ2): ± ± 0 0 1 0 τ p = = = n − | � 1 0 0 1 | � � � � � � � τ n = p + | � | � τ n =τ + p =0. − | � | �

This is the smallest non-trivial representation ofSU(2) and therefore its fu ndamental representation. Further examples for isospin multiplets are

I multiplets I3 + 3 1 1 p K 2He + 2 2 n K0 3H 1 � � � � �1 � − 2 π+ +1 1 π0 0   π− 1 ++ −3 Δ  + 2 + 1 3 Δ + 2 2  0  1 Δ 2 − 3 Δ−    − 2 wherem 1232 MeV andm 938 MeV.  Δ ≈ p,n ≈ AllI� 1 representations can b e obtained from direct products out of the fundamental I=1/2 representation 2 where “2” denotes the number of states. In analogy to the addition of two electron spins where the Clebsch-Gordan decomposition reads rep. 1/2 ⊗ 7.1. ISOSPINSU(2) 129

rep.1/2 = rep.0 rep.1 and where there are two states for the spin-1/2 representation, one state for the spin-0⊕ representation, and three states for the spin-1 representation, we have

2 2 = 1 3 . (7.11) ⊗ ⊕ I= 1 1 =0,1 isosinglet,I=0 isotriplet,I=1 | 2 ± 2 | However, there is an important� �� diﬀerence� betwee�� n isospin�� and spin multiplets. In the latter case, we are considering a bound system and the constituents carrying the spin have the same mass. On the other hand, pions are not simple bound states. Their structure will be described by the quark model.

7.1.1 Isospin invariant interactions

Isospin invariant interactions can be constructed by choosingSU(2) invariant interaction terms �. For instance, consider the Yukawa model, describing nucleon-pion coupling, whereL #» #» #» #» � = igN¯ τ N π= ig N¯ � τ N � π � (7.12) LπN · · which is an isovector and where the second identity is due toSU(2) invariance. Inﬁnites - imally, the transformation looks as follows: i #» #» N � =UN U=+ α τ (7.13) 2 · i #» #» 1 N¯ � = NU¯ † U † = α τ=U − (7.14) − 2 · #» #» #» #» π � =V π V=+i α t . (7.15) · #» The parameters t can be determined from the isospin invariance condition in Eq. (7.12):

1 Nτ¯ jNπj = NU¯ − τiUNVijπj.

WithV ij =δ ij + iαk(tk)ij (cp. Eq. (7.15)) and inserting the expressions forU andU †, this yields

i i τ = α τ τ + α τ δ + iα (t ) j − 2 k k i 2 k k ij k k ij � � � � � � i 2 =τ i + 2 αk[τi, τk] + (α k) i O 2 =τ i + αk2iεiklτl + (α ) � 2 �� O k � =τ + iα iε τ +τ (t ) j k { jkl l i k ij} =τ + iα τ iε + (t ) j k i { jki k ij} ! =0 (t ) = iε . ⇒ k ij −� kij �� 130 CHAPTER 7. UNITARY SYMMETRIES AND QCD AS A GAUGE THEORY

This means that the 3 3 matricest k, k=1,2,3, are given by the struct ure constants (see Eq. (7.10)). For the×commutator we therefore have

[t , t ] = ε ε +ε ε =ε ε = iε ( iε ) = iε (t ) (7.16) k l ij − kim lmj lim kmj klm mij klm − mij klm m ij where the second identity follows using the Jacobi identity. This means that the matrices tk fulﬁll the Lie algebra

[tk, tl] = iεklmtm.

Thet ks form the adjoint representation ofSU(2).

7.2 Quark model of hadrons

It is experimentally well established that the proton and the neutron have inner structure. The evidence is:

Finite electromagnetic charge radius • 15 r =0.8 10 − m � p,n� · (The neutron is to be thought of as a neutral cloud of electromagnetically interacting constituents.)

Anomalous magnetic moment • #» q #» µ=g s g = 5.59g = 3.83 2m p n −

Proliferation of strongly interacting hadronic states (particle zoo) •

p, n,Λ,Δ −,Ξ,Σ,Ω,...

The explanation for these phenomena is that protons and neutrons (and the other hadrons) are bound states of quarks:

p = uud 3 quark states. |n�=|udd� | � | � � The up quark and the down quark have the following properties 2 1 1 1 u :q=+ ,I= ,I = + ,S= ; | � 3 2 3 2 2 1 1 1 1 d :q= ,I= ,I = ,S= . | � − 3 2 3 − 2 2 7.3. HADRON SPECTROSCOPY 131

Quarks Charge Baryon number Up Charm Top 1.5 3 Mev 1270 MeV 171 000 MeV +2/3e 1/3 − Down Strange Bottom 3.5 6 MeV 105 MeV 4200 MeV 1/3e 1/3 − − Leptons Charge Lepton number

e− µ− τ − e 1 − νe νµ ντ 0 1 Table 7.1: Quarks and leptons.

Thus, u and d form an isospin doublet and combining them yields the correct quantum numbers| � for p| �and n . There are also quark-a ntiquark bound states: The pions form an isospin triplet| � while| the� η is the corresponding sin glet state (see Eq. (7.11)): | � π+ = ud¯ | � π0 = 1 u¯u dd¯ triplet states,I=1 | � √� 2 �| � −  π = d�¯u − � � �� | � | � � η = 1 u¯u + dd¯ singlet state,I=0. | � √2 | �  � � �� There are in total three known quark doublets:� u c t q=+ 2 ,I = + 1 3 3 2 . |d� |s� |b� q= 1 ,I = 1 �| �� | �� | �� − 3 3 − 2 � up/down charm/strange top/bottom

These quarks� �� can be combined� �� to give states� �� like,� e. g., Λ = uds . | � | �

7.3 Hadron spectroscopy

7.3.1 Quarks and leptons

Experimental evidence shows that, in addition to the three quark isospin doublets, there are also three families of leptons, the second type of elementary fermions (see Tab. 7.1). The lepton families are built out of an electron (orµ orτ) and the corresponding neutrino. The summary also shows the large mass diﬀerences between the six known quarks. All of the listed particles have a corresponding antiparticle, carrying opposite charge and baryon or lepton number, respectively. Stable matter is built out of quarks and leptons listed in the ﬁrst column of the family table. Until now, there is no evidence for quark substructure and they are therefore considered to be elementary. Hadrons, on the other hand, are composite particles. They are divided in two main categories as shown in the following table: 132 CHAPTER 7. UNITARY SYMMETRIES AND QCD AS A GAUGE THEORY

Quarks Flavor Other numbers Up, Down — S=C=B=T=0 Charm C = +1 S=B=T=0 Strange S= 1 C=B=T=0 − Top T =+1 S=C=B=0 Bottom B= 1 S=C=T=0 − Table 7.2: Additional quantum numbers for the characterization of unstable hadronic matter. Antiquarks have opposite values for these quantum numbers.

Type Matter Antimatter Baryons qqq ¯q¯q¯q Mesons q¯q Bound states such as qq or qq¯q are excluded by the theory of quantum chromodynamics (see Sect. 7.4). | � | � Unstable hadronic matter is characterized by the following additional flavor quantum numbers: Charm (C), Strangeness (S), Beauty (B), and Topness (T ) (see Tab. 7.2). It is important to remember that in strong and electromagnetic interactions both baryon and flavor quantum numbers are conserved while in weak interactions only baryon quantum numbers are conserved. Therefore, weak interactions allow heavy quarks to decay into the stable quark family. The quark decay channels are shown in the following table: Quark Decay products → u, d stable s uW − c sW + b cW − t bW + As we have seen, protons and neutrons are prominent examples of baryons. Their general properties can be summarized as follows: Proton Neutron Quarks uud udd | � | � Mass 0.9383 GeV 0.9396 GeV Spin 1/2 1/2 19 Charge e = 1.6 10 − C 0 C · Baryon number 1 1 32 Lifetime stable:τ 10 years unstable:τ n pe νe = 887 2s ≥ → − ± gaseous hydrogen: ionization under 1 MeV: nuclear reactors; Production through electric field 1 10 MeV: nuclear reactio ns − Target for ex- liquid hydrogen liquid deuterium periments 7.3. HADRON SPECTROSCOPY 133

The respective antiparticles can be produced in high-energy collisions, e. g.

pp pp¯pp with ¯p = ¯u¯d¯u or → | � ¯¯ pp pp¯nn with ¯n =� ¯udd� . → | � � Recall that in Sect. 4.1 we calculate the energy threshold� � for the reaction pp pp¯pp and � #» ﬁnd that a proton beam colliding against a proton target must have at least p→=6.5 GeV for the reaction to take place. | |

7.3.2 Strangeness

We now take a more detailed look at the strangeness quantum number. In 1947, a new neutral particle,K 0, was discovered from interactions of cosmic rays:

s 0 0 w + w π−p K Λ, with consequent decays:K π π−,Λ π −p. (7.17) → → → This discovery was later conﬁrmed in accelerator experiments. The processes in Eq. (7.17) is puzzling because the production cross section is characterized by the strong interaction while the long lifetime (τ 90 ps) indicates a weak decay. In this seemingly para doxical situation, a new quantum∼ number called “strangeness” is introduced. A sketch of production and decay of theK 0 is shown in Fig. 7.2. As stated before, the strong interaction conserves ﬂavor which requires for the production ΔS=0. The decay, on the other hand, proceeds through the weak interaction: Thes-quark decays vias uW −. → Baryons containing one or more strange quarks are called hyperons. With three consti- tuting quarks we can have, depending on the spin alignment, spin-1/2 ( ) or spin-3/2 ( ) baryons (see Tab. 7.3).2 There are 8 spin-1/2 baryons (octet)|↑↓↑� and 10 spin-3/2 baryons|↑↑↑� (decuplet). Octet and decuplet are part of theSU(3) multiplet structure (see Sect. 7.4).3 All hyperons in the octet decay weakly (except for the Σ0). They therefore 10 have a long lifetime of about 10− s and decay with ΔS =1, e. g. | | Σ+ pπ0, nπ+ → Ξ0 Λπ 0. →

The members of the decuplet, on the other hand, all decay strongly (except for the Ω−) 24 with ΔS =0. They therefore have sh ort lifetimes of about 10− s, e. g. | | Δ++(1230) π +p → Σ+(1383) Λπ +. → 2The problem that putting three fermions into one symmetric state violates the Pauli exclusion principle is discussed in Sect. 7.4. 3However, this “ﬂavorSU(3)” is only a sorting s ymmetry and has nothing to do with “colorSU(3)” discussed in Sect. 7.4. 134 CHAPTER 7. UNITARY SYMMETRIES AND QCD AS A GAUGE THEORY

0 0 Figure 7.2: Sketch of the reactionπ −p K Λ and the decays of the neutralK andΛ. Tracks detected in a bubble chamber (a).→ Feynman diagrams for the production and the Λ decay (b). Notice thatS(K 0) = 1, K 0 = d¯s andS(Λ) = 1, Λ = uds . Source: [8, p. 140]. | � | � − | � | �

Spin-1/2: Octet Spin-3/2: Decuplet Baryon State Strangeness Baryon State Strangeness p(938) uud 0 Δ++(1230) uuu 0 | � | � n(940) udd 0 Δ+(1231) uud 0 | � | � Λ(1115) (ud du)s 1 Δ0(1232) udd 0 + | − � − | � Σ (1189) uus 1 Δ−(1233) ddd 0 | � − | � Σ0(1192) (ud+du)s 1 Σ+(1383) uus 1 | � − 0 | � − Σ−(1197) dds 1 Σ (1384) uds 1 0 | � − | � − Ξ (1315) uss 2 Σ−(1387) dds 1 | � − 0 | � − Ξ−(1321) dss 2 Ξ (1532) uss 2 | � − | � − Ξ−(1535) dss 2 | � − Ω−(1672) sss 3 | � − Table 7.3: Summary of the baryon octet and decuplet. 7.3. HADRON SPECTROSCOPY 135

Figure 7.3: Bubble chamber photograph (LHS) and line diagram (RHS) of an event showing the production and decay ofΩ −. Source: [25, p. 205].

The quark model, as outlined so far, predicts the hyperon Ω − = sss as a member of the spin-3/2 decuplet. Therefore, the observation of the production,| � | �

+ 0 K−p Ω −K K , → and decay,

0 0 0 Ω− Ξ π−,Ξ Λπ ,Λ pπ −, → → →

of the Ω− at Brookhaven in 1964 is a remarkable success for the quark model. A sketch of the processes is given in Fig. 7.3. Note that the production occurs via a strong process, ΔS=0, while the decay is weak : ΔS =1. | |

7.3.3 Strong vs. weak decays

Generally speaking, strong processes yield considerably shorter lifetimes than weak processes. Consider, for instance, the following two decays,

Δ+ p+π 0 Σ+ p+π 0 → 24 → 11 τ = 6 10 − s τ = 8 10 − s Δ · Σ · 1 1 uud uud + u¯u + dd¯ uus uud + u¯u + dd¯ | �→| � √2 | � | �→| � √2 | � � � �� (strong) � (weak). � 136 CHAPTER 7. UNITARY SYMMETRIES AND QCD AS A GAUGE THEORY Quarks, u, d, s.

s = 1�

q q n

s = 0�

Figure 7.4: Sketch of the possible spin conﬁgurations for quark-antiquark bound states. Theq¯q pair is characterized by orbital excitationsl (rotation) and radial e xcitationsn (vibration). Source: [8, p. 141].

The final state is identical in both decays but the lifetime is much longer for the weak process. Since the final state is equal, this difference in lifetime must come from a difference in the coupling constants. Forτ 1/α 2 whereα is a coupling constant: ∼

αweak τΔ 7 = 2.7 10 − . α ∼ τ · strong � Σ

7.3.4 Mesons

Mesons are quark-antiquark bound states: q¯q . In analogy to the spin states of a two- electron system (and not to be confused with| th� e isospin multiplets discussed on p. 128), the q¯q bound state can have either spin 0 (singlet) or spin 1 (triplet) (see Fig. 7.4). Radial vibrations| � are characterized by the quantum numbern while orbital angular m omentum is characterized by the quantum number l. The states are represented in spectroscopic notation:

2s+1 n lJ wherel = 0 is labeled byS,l=1byP and so on. A summary of then=1, l = 0 meson states is shown in Tab. 7.4. A summary of the states withl 2 can be found in Fig. 7 .3.4. ≤

7.3.5 Gell-Mann-Nishijima formula

Isospin is introduced in Sect. 7.1. The hadron isospin multiplets forn=1, l = 0 are shown in Fig. 7.6. This summary leads to the conclusion that the chargeQ of an hadron with 7.3. HADRON SPECTROSCOPY 137

Figure 7.5: Summary of mesons from u, d s quarks forl 2. Cells shaded in grey are well established states. Source: [8, p. 143]. ≤ 138 CHAPTER 7. UNITARY SYMMETRIES AND QCD AS A GAUGE THEORY

Mesons(n=1, l = 0) 1 3 1 S0 (spin 0) 1 S1 (spin 1) π+(140) ud¯ ρ+(770) ud¯ π−(140) ¯ud ρ−(770) ¯ud �| �� | �� π0(135) 1/√2�dd¯ u¯u ρ0(770) 1/√2�dd¯ u¯u + − + − K (494) u¯s K∗ (892) u¯s �| � � �| � � K−(494) �¯us K∗−(892) �¯us 0 | � 0 | � K (498) d¯s K∗ (896) d¯s 0 | � 0 | � K¯ (498) ds¯ K¯ ∗ (896) ds¯ η(547) 1/ √6 u¯u+dd¯ 2s¯s φ(1020) =ψ s¯s ∼ � � − 1 −� | � � η (958) 1/ √3 u�¯u+dd¯+s¯s ω(782) =ψ 1/√2�u¯u+dd¯ � � � 2 ∼ � � � � � Table 7.4:� Summary ofn=1, l = 0 meson states. �

baryon numberB and strangenessS is given by B+S Q=I + 3 2

which is called Gell-Mann-Nishijima formula. As an example, consider the Ω− hyperon where 0 + (1 3)/2= 1. − −

7.4 Quantum chromodynamics and colorSU(3)

The quark model, as discussed so far, runs into a serious problem: Since the quarks have half-integer spin, they are fermions and therefore obey Fermi-Dirac statistics. This means that states like

++ 3 Δ = u↑u↑u↑ ,S= 2 � � where three quarks are in a symmetric� state (have identical quantum numbers) are for- bidden by the Pauli exclusion principle. The way out is to introduce a new quantum number that allows for one extra degree of freedom which enables us to antisymmetrize the wave function as required for fermions:

++ Δ = ε u↑u↑u↑ N ijk i j k ijk � � � � where is some normalization c onstant and the� quarks come in three diﬀerent “colors”:4 N q | 1� q q1,2,3 = q2 . | �→| � |q � | 3�   4The new charge is named “color” because of the similarities to optics: There are three fundamental colors, complementary colors and the usual combinations are perceived as white. 7.4. QUANTUM CHROMODYNAMICS AND COLORSU(3) 139

��

Figure 7.6: Summary of hadron isospin multiplets.n=1, l=0. Source: [8, p. 147].

Since color cannot be observed, there has to be a corresponding new symmetry in the Lagrangian due to the fact that the colors can be transformed without the observables being aﬀected. In the case of our new charge in three colors the symmetry group isSU(3), the group of the special unitary transformations in three dimensions. The Lie algebra of SU(3) is

T a,T b = if abcT c

where, in analogy to Eq. (7.10),f abc� denotes� the structure constants and where there are 8 generatorsT a (recall thato=N 2 1=8, see p. 127) out of whichr=N 1=2are diagonal. − − a 1 a The fundamental representation is given by the 3 3 matricesT = 2 λ with the Gell- Mann matrices ×

τ1 τ2 τ3 0 0 1 0 1 0 0 i 0 1 0 0 λ1 =   λ2 =  −  λ3 =   λ4 = 0 0 0 �1�� 0� 0 �i0�� 0 �0 ��1 � 0        −  1 0 0  0 0 0  0 0 0  0 0 0         0 0 i   0 0 0  0 0 0 1 0 0 − 1 λ = 0 0 0 λ = 0 0 1 λ = 0 0 i λ = 0 1 0 . 5   6   7   8 √   i0 0 0 1 0 0i0 − 3 0 0 2 −         140 CHAPTER 7. UNITARY SYMMETRIES AND QCD AS A GAUGE THEORY

One can observe that these matrices are hermitian and traceless,

a λa† =λ a Trλ = 0.

Furthermore, one can show that

Tr λaλb = 2δab � � and

1 λa λb = 2 δ δ δ δ (Fierz identity). ij kl il kj − 3 ij kl � � The structure constants ofSU(3) are given by 1 f = Tr ([λ , λ ]λ ) abc 4i a b c and are antisymmetric in a, b, and c. The numerical values are

f123 = 1 √3 f =f = 458 678 2 1 f =f =f =f =f =f = 147 156 246 257 345 367 2 fabc = 0 else.

As in the case ofSU(2), the adjoint representat ion is given by the structure constants which, in this case, are 8 8 matrices: × (ta) = if . bc − abc The multiplets (again built out of the fundamental representations) are given by the direct sums

3 3¯ = 1 8 (7.18) ⊗ ⊕ where the bar denotes antiparticle states and

3 3 3 = 1 8 8 10 . (7.19) ⊗ ⊗ ⊕ ⊕ ⊕ The singlet in Eq. (7.18) corresponds to the q¯q states, the mesons (e. g.π), while the singlet in Eq. (7.19) is the qqq baryon (e.g. p,| n�). The other multiplets are colored and can thus not be observed.’Working| � out theSU(3) potential structure, one ﬁnds that an attractive QCD potential exists only for the singlet states, while the potential is repulsive for all other multiplets. 7.4. QUANTUM CHROMODYNAMICS AND COLORSU(3) 141

The development of QCD outlined so far can be summarized as follows: Starting from the observation that the nucleons have similar properties, we considered isospin andSU(2) symmetry. We found that the nucleonsn andp correspond to the fund amental representations ofSU(2) while theπ is given by the adjoin t representation. To satisfy the Pauli exclusion principle, we had to introduce a new quantum number and with it a new SU(3) symmetry of the Lagrangian. This in turn led us to multiplet structures where the colorless singlet states correspond to mesons and baryons.

Construction of QCD Lagrangian We now take a closer look at thisSU(3) transformation of a color triplet,

q1 q1� q1 a igsαaT q = q q� = q� = e q =U q , (7.20) | �  2  → | �  2   2  | � q3 q3� q3       whereg s R is used as a rescaling ( and will be used for the perturbative expansion) of the group∈ parameterα introduced previously. The reason of introducing it becomes clear in the context of gauge theories. In analogy to the QED current,

µ µ jQED =eq e ¯qγq, we introduce the color current 5, which is the conserved current associated with the SU(3) symmetry, jµ =g ¯qγµT a q a=1 8. (7.21) a s i ij j ··· In the same spirit, by looking at the QED interaction, int = j µ A =eq ¯qγµqA , LQED − QED µ e µ yielding the vertex, Aµ q q where we can see the photon – the electrically unchargedU(1) gauge boson of QED –, we

5The Einstein summation convention still applies, even if the color indexi andj are not in an upper and lower position. This exception extends also to the color indices a, b, ... of the gauge ﬁelds to be introduced. There is no standard convention in the literature, and since there is no metric tensor involved, the position of a color index, is merely an esthetic/readability problem. 142 CHAPTER 7. UNITARY SYMMETRIES AND QCD AS A GAUGE THEORY postulate an interaction part of the QCD Lagrangian of the form,

int = j µAa =g ¯qγµT a q Aa , (7.22) LQCD − a µ s i ij j µ which translates in the vertex (which is not the only one of QCD as we shall see),

a Aµ qj qi

a Now there are 8SU(3) gauge bosonsA µ for QCD : one for each possible value ofa. They are called gluons and are themselves colored. Continuing with our analogy, we deﬁne the covariant derivative of QCD 6,

a a Dµ =∂ µ + ig sT Aµ, (7.23) and state that the QCD Lagrangian should have a term of the form,

˜ = ¯q(Di/ m)q. (7.24) LQCD −

Up to this point, both QED and QCD look nearly identical. Their diﬀerences become crucial when we look at local gauge symmetries. Such a transformation can be written,

a igsαa(x)T q(x) q�(x) =e q(x) , (7. 25) | �→| � | � and we impose as before that the Lagrangian must be invariant under any such transformation. This is equivalent of imposing,

6 Note thatD µ acts on color triplet and gives back a color� triple��t;∂ µ does� not mix the colors, whereas the other summand does (T a is a 3 3 matrix). × 7.4. QUANTUM CHROMODYNAMICS AND COLORSU(3) 143

c c c c c Making the ansatzA µ� =A µ +δAµ where δA µ Aµ and expanding the former equation c | | � | | c to ﬁrst order inδA µ (the term proportional toα a(x)δAµ has also been ignored), we get,

c c a 2 2 c c a ! 2 2 a c c igsT δAµ + igs(∂µαa(x))T +i gs T Aµαa(x)T =i gs αa(x)T T Aµ T cδAc =! (∂ α (x))T a + ig [T a,T c]α (x)Ac , ⇒ µ − µ a s a µ or, renaming the dummy indices and using the Lie algebra su(3),

T aδAa = (∂ α (x))T a g f T aα (x)Ac T a µ − µ a − s abc b µ ∀ a a c A � =A ∂ α (x) g f α (x)A . (7.26) ⇒ µ µ − µ a − s abc b µ like in QED non-abelian part � �� Eq. (7.26) describes the (infinitesimal) gauge transformation of the gluon field. In order for the gluon field to become physical, we need to include a kinematical term (depending on the derivatives of the field). Remember the photon term of QED, 1 photon = F F µν F =∂ A ∂ A , LQED − 4 µν µν µ ν − ν µ where the last is gauge invariant. As we might expect from Eq. (7.26), the non-abelian part will get us into trouble. Let’s look at,

δ(∂ Ac ∂ Ac ) = ∂ ∂ α +∂ ∂ α g f α (∂ Ac ∂ Ac ) µ ν − ν µ − µ ν a ν µ a − s abc b µ ν − ν µ g f (∂ α )Ac (∂ α )Ac . − s abc µ b ν − ν b µ � � We remark that the two ﬁrst summands cancel each other and that the third looks like theSU(3) transformation unde r the adjoint representation. We recall that,

a q q � = (δ + ig α T )q (fundamental representation) i → i ij s a ij j b B B � = (δ + ig α t )B (adjoint representation) a → a ac s b ac c respectively, where,

tb = if = if . ac − bac abc

a Hence, ifF µν transforms in the adjoint representation ofSU(3), we should have,

δF a =! g f α F c . µν − s abc b µν We now make the ansatz,

F a =∂ Aa ∂ Aa g f Ab Ac , (7.27) µν µ ν − ν µ − s abc µ ν 144 CHAPTER 7. UNITARY SYMMETRIES AND QCD AS A GAUGE THEORY

and prove that it fulﬁlls the above constraint.

δF a =δ(∂ Aa ∂ Aa ) g f δ(Ab Ac ) µν µ ν − ν µ − s abc µ ν = g f α (∂ Ac ∂ Ac ) g f (∂ α )Ac (∂ α )Ac − s abc b µ ν − ν µ − s abc µ b ν − ν b µ g f (∂ α )Ac + (∂ α )Ac g f g f α Ae Ac g f α Ab Ae , − s abc − µ b ν ν b µ �− s abc − s bde d µ� ν − s cde d µ ν Using, � � � �

e c c e b e fabcfbdeαdAµAν =f abefbdcαdAµAν =f acefcdbαdAµAν,

and

f f f f = (iT a )(iT d ) (iT d )(iT a ) = T a,T d = if T c , aec dbc − acb dec ec cb − ec cb eb adc eb we get the desired result. � � a We check ﬁnally that a kinematic term based on the above deﬁnition ofF µν is gauge invariant :

δ F a F µν = 2F µνδF a = 2g f α F µνF c = 0. µν a a µν − s abc b a µν = fcba µν a � � − =Fc Fµν �� Finally, we get the full QCD Lagrangian, 1 = F a F µν + ¯q(Di/ m )q, (7.28) LQCD − 4 µν a − q

a with D/ andF µν deﬁnded by Eqs. (7.23) and (7.27) respectively. This Lagrangian is per construction invariant under localSU(3) gauge transformatio ns. It is our ﬁrst example of a non-abelian gauge theory, a so-called Yang-Mills theory.

a Structure of the kinematic term From the deﬁnition ofF µν, Eq. (7.27), we see that,

F a F µν = ∂ Aa ∂ Aa g f Ab Ac (∂µAν ∂ νAµ g f AµAν), µν a µ ν − ν µ − s abc µ ν a − a − s ade d e will have a much richer� structure than in the ca�se of QED. a a µ ν ν µ First, we have – as in QED – a 2-gluon term ∂µAν ∂ νAµ (∂ Aa ∂ Aa ) corresponding to the gluon propagator, − − � �

µ, a ν, b k

gµν = δab. (7.29) − k2 7.4. QUANTUM CHROMODYNAMICS AND COLORSU(3) 145

Then we have a 3-gluon term g f Ab Ac (∂ Aν ∂ νAµ) yielding a 3-gluon vertex − s abc µ ν µ a − a

� b � Aν(k2) a c Aµ(k1)A λ(k3)

=g f [g (k k ) +g (k k ) +g (k k ) ]. (7.30) s abc µν 1 − 2 λ νλ 2 − 3 µ λµ 3 − 1 ν

b c µ ν Finally we have also a 4-gluon term gsfabcAµAν ( gsfadeAd Ae ) yielding the 4-gluon vertex − − � � b k Ac (k ) Aν( 2) λ 3 a d Aµ(k1) Aρ(k4) 2 = ig s [fabefcde(gµλgνρ g µρgνλ) +f adefbce(gµνgλρ g µλgνρ) +f acefbde(gµρgνλ g µνgρλ)] − − − − (7.31)

Unlike in QED, gluons are able to interact with themselves. This comes from the fact that the theory is non-abelian. As a consequence, there is no superposition principle for QCD: the ﬁeld of a system of strongly interacting particles is not the sum of the individual ﬁelds. Thence, there is no plane wave solution to QCD problems, and we cannot make use of the usual machinery of Green’s functions and Fourier decomposition. Up to now there is no known solution.

7.4.1 Strength of QCD interaction

In QED, when we take a term of the form,

� e e � � � � � � � � � � � � � � � � � 146 CHAPTER 7. UNITARY SYMMETRIES AND QCD AS A GAUGE THEORY where the denotes some other part of the Feynman diagram, the expression is proportional toe 2 = 4πα. In the case of QCD, we have a few more possibilities. We look at the generalSU(n) case. The QCD result can be found by settingn=3. First, for the analogous process to the one cited above : j j

� a a � � g T gsT � � s ij jk � � � � i k � � � � � � � � � � a a � � � � � 2� a a � which is proportional tog s�TijTjk = 4παsCF δik, where � n2 1 C = − , (7.32) F 2n is the color factor, the Casimir operator ofSU(n). To ﬁnd it, we used o ne of the Fierz identities (see exercises), namely, 1 1 T a T a = δ δ δ δ ij jk 2 ik jj − n ij jk � � 1 1 n2 1 = nδ δ = − δ . 2 ik − n ik 2n ik � � Next we look at, i i

� � � g T a g T b � � s ij s ji � � � � � � a b � � � � � � � � j j � � � � � 2� a b ab � which is proportional tog s�TijTji = 4παsTF δ , where � 1 T = . (7.33) F 2 To ﬁnd it, we used the fact that, 1 Tr T aT b = δab. 2 � � 7.4. QUANTUM CHROMODYNAMICS AND COLORSU(3) 147

Finally we investigate the case where, b b

� � � � � gsfabc gsfdbc � � � � � � a d � � � � � � � � c c � � � � � 2� ad � which is proportional tog s�fabcfdbc = 4παsCAδ , where �

CA =n. (7.34) To ﬁnd it, we used the relation, f = 4iTr T a,T b T c , abc − that we have shown in the beginning of this�� section.� � 4 1 In the case of QCD,C F = 3 ,TF = 2 ,CA = 3. From the discussion above, we can heuris- tically draw the conclusion that gluons tend to couple more to other gluons, than to quarks. At this stage, we note two features speciﬁc to the strong interaction, which we are going to handle in more detail in a moment :

Conﬁnement : At low energies (large distances), the coupling becomes very large, • so that the perturbative treatment is no longer valid, an the process of hadronization becomes inportant. This is the reason why we cannot observe color directly. Asymptotic freedom : At high energies (small distances) the coupling becomes • negligible, and the quarks and gluons can move almost freely.

As an example, of typical QCD calculation, we sketch the calculation of the

Gluon Compton scattering

g(k)+q(p) g(k �) +q(p �). → There are at ﬁrst sight two Feynman diagrams coming into the calculation,

k, a k�, b k, a k�, b + p, i p, i k+ p, l p�, j p k �, l� p�, j − 148 CHAPTER 7. UNITARY SYMMETRIES AND QCD AS A GAUGE THEORY

which yields the following scattering matrix element,

2 1 b a i fi = ig ¯u(�p)/ε∗(k�) /ε(k)u(p)T T − M − s p/+ k/ m jl li � − 1 b a +¯u(�p)/ε(k) /ε∗(k�)u(p)Tjl Tl i . (7.35) p/ k/� m � � � − −

We start by checking the gauge invariance ( fi must vanish under the substitution ε (k) k ): M µ → µ

2 b a a b i � = ig ¯u(�p)/ε(k�)u(p) T T T T , − Mfi s ji li − jl� l�i � � where

b a a b b a c T T T T = T ,T = ifbacT = 0! ji li − jl� l�i ji ki � � � So we need another term, which turns out to be the one corresponding to the Feynman diagram,

k, a k�, b

p� p, c − p, i p�, j

The calculation of the gluon-gluon scattering goes analogously. We need to consider the graphs,

�+ �+ �+

7.4.2 QCD coupling constant

To leading order, a typical QED scattering process takes the form, 7.4. QUANTUM CHROMODYNAMICS AND COLORSU(3) 149

e−(p)e −(p�)

γ(q)

2 2 withq = (p� p) 0. − ≤ In the Coulomb limit (long distance, low momentum transfer), the potential takes the form,

α 1 11 V(R)= R� 10− [cm]. (7.36) − R me ≈

1 WhenR m e− , quantum eﬀects become important (loop corrections, also known as vacuum≤ polarization), since the next to leading order (NLO) diagram,

+ e− e starts to play a signiﬁcant (measurable) role. This results in a change of the potential to,

α 2α 1 ¯α(R) V(R)= 1 + ln + (α 2) = , (7.37) − R 3π m R O − R � e � where ¯α(R) is called the eﬀectiveoupling. c

We can understand the eﬀective coupling in analogy to a solid state physics example : in an insulator, an excess of charge gets screened by the polarization of the nearby atoms. + Here we createe e− pairs out of the vacuum, hence the name vacuum polarization.

1 As we can see from Eq. (7.37), the smaller the distanceR m e− , the bigger the observed “charge” ¯α(R). What we call the electron chargee (or the≤ ﬁne structure co nstantα) is the limiting value for very large distances or low momentum transfer as shown in Fig. 7.4.2.

For example the measurements done at LEP show that, ¯α(Q2 =m 2 ) 1 > α. Z ≈ 128 In the case of QCD, we have at NLO, the following diagrams, 150 CHAPTER 7. UNITARY SYMMETRIES AND QCD AS A GAUGE THEORY

�(R) �(Q2)

1/137 1/137

R Q2

Figure 7.7: Evolution of the eﬀective electromagnetic coupling with distance and energy (Q2 = q 2). −

+ + + �q ¯q �� screening antiscreening antiscreening . We can picture the screening/antiscreening phenomenon as follows,

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � + � + � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � R α (R) R α (R) �⇒ s � �⇒ s � Figure 7.8: Screening and antiscreening.

For QCD, the smaller the distanceR (or the bigger the ene rgyQ 2), the smaller the observed coupling ¯αs(R). At large distances, ¯αs(R) becomes comparable with unity, and the perturbative approach breaks down as we can see in Fig. 7.4.2. The region concerning conﬁnement and asymptotic freedom are also shown. 7.4. QUANTUM CHROMODYNAMICS AND COLORSU(3) 151

�s (R) ��s(Q�) con�nement

asymptotic freedom

R �QCD ≈ 200 MeV Q�

Figure 7.9: Evolution of the eﬀective strong coupling with distance and energy (Q2 = q 2). −

Theβ-function of QCD In the renormalization procedure of QCD, we get a diﬀeren- 2 tial equation forα s(µ ) whereµ is the renormalization scale,

∂α µ2 s =β(α ) (7.38) ∂(µ2) s α α 2 α 3 β(α ) = α β s +β s +β s + , (7.39) s − s 0 4π 1 4π 2 4π ··· � � � � � � with

11 2 2 β = n n = 11 n (NLO) (7.40) 0 3 c − 3 f − 3 f 17 5 1 n2 1 β = n2 n n c − n , (NNLO) (7.41) 1 12 c − 12 c f − 4 2n f � c �

wheren c is the number of colors andn f is the number of quark ﬂavors. These two numbers enter into the calculation through gluon respectively quark loop corrections to the propagators.

7 We remark at this stage that unlessn f 17 ,β 0 > 0, whereas in the case of QED, we get, ≥

4 βQED = <0. (7.42) 0 − 3

This fact explains the completely different behavior of the effective couplings of QCD and QED. To end this chapter, we will solve Eq. 7.39 retaining only the first term of the power

7As of 2009, only 6 quark ﬂavors are known and there is experimental evidence (decay witdth of the Z0 boson) that there are no more than 3 generations with light neutrinos. 152 CHAPTER 7. UNITARY SYMMETRIES AND QCD AS A GAUGE THEORY expansion ofβ. ∂α β µ2 s = 0 α2 ∂(µ2) − 4π s

∂αs β0 2 2 = ∂(lnµ ) αs − 4π 2 2 αs(Q ) lnQ dαs β0 2 2 = d(lnµ ), αs − 4π � 2 � 2 αs(Q0) lnQ 0 and hence,

2 1 1 β0 Q 2 = 2 + ln 2 .( 7.43) αs(Q ) αs(Q0) 4π Q0

2 2 We thus have a relation betweenα s(Q ) andα s(Q0), giving the evolution of the eﬀective coupling. A mass scale is also generated, if we set,

1 2 2 2 = 0 α s(Λ ) = . αs(Q = Λ ) ⇒ ∞

Choosing Λ =Q 0, we can rewrite Eq. (7.43) as,

2 4π αs(Q ) = Q2 . (7.44) β0 ln Λ2