Introduction to Gauge Theories

Transcription

1 Introduction to Gauge Theories Basics of SU(n) Classical Fields U() Gauge Invariance SU(n) Gauge Invariance The Standard Model Michel Lefebvre University of Victoria Physics and Astronomy PHYS506B, spring 005 Introduction to Gauge Theories

2 Basics of SU(n) Spin and Colour Notes on SU(n) Notes on SU() Notes on SU() Colour Factors PHYS506B, spring 005 Introduction to Gauge Theories

3 Spin and Colour From deep inelastic scattering, we found evidence for the fractional charge assignment of quarks; the spin j = s = / nature of quarks; gluons; asymptotic freedom (Bjorken scaling); confinement (no quarks observed in isolation). Using the static quark model, all baryons are described as bound states of quarks, but baryon wave functions violates Pauli exclusion principle (spin-statistics theorem), e.g. the m j = / states - - ( uuu ) ( ddd ) ( sss) ++ Ω All quarks have same spin projection and same wave function, which implies a symmetric global wave function. This is not allowed for a fermion. We can solve the problem by introducing colour as an extra degree of freedom Ψ = =Φ = = = χ Ω ( J ) ( r, r, r, m m m ) ( c, c, c ) z where Φ is a totally symmetric space-spin function of the quarks, and χ is a totally antisymmetric colour function of the three quarks. At least distinct colours are needed here. PHYS506B, spring 005 Introduction to Gauge Theories

4 Spin and Colour From e + e - hadrons, there is evidence for different colours of quarks (Factor of in R). The quark model without colour predicts an infinity of multiquark states not observed in nature, e.g. states with,,4,5,7, quarks. But adding an extra degree of freedom (colour) increases the number of multiquark linearly independent states This must be controlled. QCD does it with colour confinement. First, let s look at spin. Consider a spin s = / particle A The norm Therefore A = η α mα, m =, m = α= ˆ UUˆ α= η = α = I A is left invariant by the transformation Uˆ Uˆ PHYS506B, spring 005 Introduction to Gauge Theories 4 A

5 Spin and Colour Setting m Uˆ m α β αβ = U we get UU= I ( detu) = We choose detu = ˆ Uˆ m U A = U η η U η so U is a (special) unitary matrix. Then α αβ β α αβ β Let s now consider the possible states J, J z resulting from the combination of spin / particles. Using CG coefficients, 0,0,, 0, () () = η η η η () =η η () () = η η +η η () =η η Projecting onto the m α m β space we can write () () Ψ 00 = η η η η PHYS506B, spring 005 Introduction to Gauge Theories 5

6 Spin and Colour If we apply U to both particles, we obtain () () Ψ η η η η () = UαUβ U U α β ηα ηβ () () = [ UU UU ] η η η η = [ UU UU ] Ψ 00 = ( detu) Ψ00 =Ψ Uˆ 00 U α αuβ β Uα αuβ β 00 We see that the spin singlet Ψ 00 is invariant under U as expected. Members of the spin triplet mix amongst themselves under U. Note that spin singlet but ( J ) = 0 Jz = 0 J = 0 / spin singlet = 0 z ( J ) PHYS506B, spring 005 Introduction to Gauge Theories 6

7 Spin and Colour Now let s look at colour. We assume there are possible colour states,, c j, j= often called red, green, blue. Therefore the general colour states of a quark is The norm Therefore q cj cj j= j= = c ˆ UUˆ j c Uˆ c = = I q = U is left invariant by the transformation U U Setting we get j k jk We choose detu = ˆ q Uˆ c U = U c c U c q UU= I detu = so U is a (special) unitary matrix. Then j jk k j jk k PHYS506B, spring 005 Introduction to Gauge Theories 7

8 Spin and Colour We must seek multiquark states that are invariant under U, that is colour singlets. For quark states, the state with this property is B = ε c c c c c c 6 ijk i j k i j k Projecting onto the c i c j c k space we can write Ψ = ε c c c B 6 ijk i j k If we apply U to each three quarks, we obtain ˆ U ΨB 6 () () ( detu) c c c ( detu) abc 6 a b c B ε U c U c U c ijk ia a jb b kc c =ε = Ψ =Ψ where we have used ε detu =ε U U U abc ijk ia jb kc We see that Ψ B is invariant under U. It is a colour singlet. None of the other 6 colour combinations are singlets. Ψ B is a suitable wavefunction for baryons. PHYS506B, spring 005 Introduction to Gauge Theories 8 B

9 Spin and Colour For quark-antiquark states, we need the anticolour vector Consider M cj c j cj cj j= j= q = q = = c c c c + c c c c + c c c c () () () () () () or, projecting onto the c j c k space Ψ M = c c + c c + c c () () () () () () which is invariant under U. It is a colour singlet. Ψ M is a suitable wavefunction for mesons. PHYS506B, spring 005 Introduction to Gauge Theories 9

10 Spin and Colour In fact, Ψ B is the only invariant qqq state; Ψ M is the only invariant quark-antiquark state; The only other invariant multiquark states contain,6,9, quarks and an arbitrary number of quark-antiquark pairs. All observed hadrons are colour singlets There are no long range confinement forces between colour singlets, e.g. pn can be separated. The nuclear force is a Van der Waals colour (strong) force. Leptons and photons are colour singlets, so they don t feel the strong force. They are not made of colour objects either, so they don t feel the nuclear force. PHYS506B, spring 005 Introduction to Gauge Theories 0

11 Notes on SU(n) t, t = if t = The commutation relation [ a b] abc c a,,..., n forms the Lie algebra of SU(n), where f abc are the structure constants of the group. The t a are the generators of the group. An element u of the group is then u = i t e ω where the ω a are real parameters that label the group element u. By convention, we will assume sum over repeated group generator indices. Consider the representation u U( u) = U If there exist a non-singular matrix M, independent of the group elements, such that U ( u) MU( u) M = 0 U ( u) 0 u n aa then U is called a reducible representation. U u = U u U u Otherwise it is an irreducible representation. In this case SU U = i T e ω a a PHYS506B, spring 005 Introduction to Gauge Theories

12 Notes on SU(n) From From UU = I we get T = T detu = we get TrT = 0 Therefore the generators representation T a are n hermitian traceless matrices. They follow the SU(n) algebra [ Ta, Tb] = ifabctc = U( uu ) implies U ( u ) U ( u ) = U ( uu ) U u U u Since we see that U* forms a complex conjugate representation. This implies that -T a * forms a representation of the generators. If there exist a non-singular matrix S such that STaS = Ta a then T a and -T a * are said to be equivalent. In this case the representation is said to be a real representation. The dimension d of a representation is the dimension of the vector space on which it acts. PHYS506B, spring 005 Introduction to Gauge Theories

13 Notes on SU(n) For d = n we have the defining or fundamental representation denoted n. In this case the generators are n n traceless hermitian matrices. There are n of them, which sets the number of ω a parameters needed to specify a group element. In the fundamental representation, SU(n) is the group of all unitary n n matrices of unit determinant. Remember the properties of commutator algebra From the Jacobi identity we get [ T, T] = [ T, T] [ at+ bt, T] = a[ T, T] + b[ T, T] T [ T T ] T [ T T ] T [ T T ],, +,, +,, = 0 f f + f f + f f = abd cde cad bde bcd ade Therefore the matrices T a defined by ( T ) a bc = if abc also satisfy the algebra of the group. They generate the adjoint representation of dimension d = n. PHYS506B, spring 005 Introduction to Gauge Theories 0

14 Notes on SU(n) Choosing the normalisation ( TT ) Tr a b = κδab the structure constants f abc are totally antisymmetric in all three indices. Of the n generators, only n are diagonal. They commute with one another, and their (real) eigenvalues are used to label group elements of an irreducible representation. An irreducible representation (multiplet) of dimension d is denoted d, though this does not generally uniquely label it. Consider an infinitesimal SU(n) transformation on a generator We obtain Ta UTU a U = + iωbtb Ta Ta +δta where δ Ta = fabcωbtc Any set of n quantities that transform under SU(n) like T a can be T T, T,, Tn denoted ( ) We can indeed define the following products S T = f ST S T= ST a abc b c a a PHYS506B, spring 005 Introduction to Gauge Theories 4

15 Notes on SU(n) Using the antisymmetry of f abc we get the identity ( R S) T = ( R T) S S T is indeed a SU(n) scalar δ S T =δs T + S δ T = ω S T + S ω T = from which we can show that 0 Therefore we can write In general T T T = T = ki n a a= n a a= where d is the dimension of the representation. For U in SU(n), we can have Then TrT = TrT = kd T b = UTU TrT = Tr UTU UTU = TrT b a a a a PHYS506B, spring 005 Introduction to Gauge Theories 5

16 Notes on SU(n) Let s choose a = α corresponding to one of the n diagonal generators. Then d a = = α α where ( t i α ) i i= n d Tr = Tr a = α = i a= i= TrT TrT t T T n t kd Therefore Finally we obtain the important result ( n ) d T T T = ( tα ) I d i i= are the eigenvalues of T α We also note that the normalization condition can be written as Tr ( TT ) =κδ a b ab ab t α i i= d =δ where ( t are the eigenvalues of T α ) α, any one of the n i diagonal generators PHYS506B, spring 005 Introduction to Gauge Theories 6

17 Notes on SU() In SU() there are n = generators. In the fundamental representation, we choose the Pauli matrices σ a = σ σ = 0 σ = 0 i σ = 0 T a a, a =,, They have the normalization and they verify the Lie algebra 0 i 0 0 Tr TT a b = δab [, ] T T = iε T a b abc c where ε abc is totally antisymmetric with ε =. Note that T is the only (n = ) diagonal generator. Its eigenvalues t label each state of a multiplet. Setting ± ± we obtain [ T T ] T T it, = ± T ± ± T t T t raises by lowers by + therefore PHYS506B, spring 005 Introduction to Gauge Theories 7

18 Notes on SU() The action of T ± can be represented by T T + t Each irreducible representation of SU() is characterized by an integer p which corresponds to a multiplet of dimension d p = 0,,, d = + p We can represent graphically the irreducible SU() multiplet by states on the t axis 0 p = 0 t p = t t p = Note that T a T a * does not change the spectrum of states. Indeed, there exist a S such that ST S = T a This can be verified in with a S a = T = σ In general this change implies t t. We see that d d and are equivalent PHYS506B, spring 005 Introduction to Gauge Theories 8

19 Notes on SU() ( n ) Remember that in general where T α is diagonal with eigenvalues ( In the case of SU() we have : : : T This is the familiar = 0I = 0 d i= T T T = tα I d i t α ) i = ( ) ( + ) = 4 T I I T = ( ) + ( 0) + I = I J = j j+ j = 0,,,,, PHYS506B, spring 005 Introduction to Gauge Theories 9

20 Notes on SU() The product of irreducible representation can also be performed graphically = = = 4= 5 = = PHYS506B, spring 005 Introduction to Gauge Theories 0

21 Notes on SU() In SU() there are n = 8 generators. In the fundamental representation, we choose the Gell-Mann matrices λ a T a = λ a a =,,, i λ = 0 0 λ = i 0 0 λ = i λ 4 = λ 5 = i λ 6 = 0 0 λ 7 = 0 0 i λ 8 = i They have the normalization Tr TT a b = δab PHYS506B, spring 005 Introduction to Gauge Theories

22 Notes on SU() and they verify the Lie algebra [, ] T T = if T a b abc c where f abc is totally antisymmetric with nonvanishing elements f = f = f = f f f f f f = 56 = 46 = 57 = 45 = 67 = T, T = 0 There are n = diagonal generators. They commute, so [ ] 8 Their eigenvalues t and t 8 are used to label each states of a multiplet. We can define T T ± it U T ± it V T ± it ± ± 6 7 ± 4 5 Note that when considering flavour SU(), it is useful to define Y T Also note that T, T and T form an SU() sub-algebra. Therefore f abc = ε for abc,, abc 8,, PHYS506B, spring 005 Introduction to Gauge Theories

23 Notes on SU() We obtain the following commutation relations [ T, T ] [ T, U ] [ T, V ] =± T ± ± ± = U± ± =± V± Therefore we have [ T8 T± ] [ T, U ] [ T, V ], = 0 =± U 8 ± ± 8 ± =± V± T+ raises t by and does not affect t T lowers t by and does not affect t U t t + lowers by and raises by U t t raises by and lowers by V t t + raises by and raises by V t t lowers by and lowers by U + T V t 8 V + U T + t The action of T ±, U ± and V ± can be represented as PHYS506B, spring 005 Introduction to Gauge Theories

24 Notes on SU() Each irreducible representation of SU() can be characterized by a set of two integers (p, q) which correspond to a multiplet of dimension d. q p p = 0,,, q = 0,,, d = + p + q + p+ q ( )( ) PHYS506B, spring 005 Introduction to Gauge Theories 4

25 Notes on SU() Graphically, the irreducible SU() multiplets show up as states in the t t 8 plane. The boundaries are hexagons of sides p and q, which collapses into triangles if p or q vanishes. t 8 q p t 8 q t q q q ( p, q) = ( 0, q) t p q ( p, q) p p t 8 p p ( p, q) = ( p,0) t PHYS506B, spring 005 Introduction to Gauge Theories 5

26 Notes on SU() Starting from a (p, q) boundary containing (p+q) sites, the representation can be graphically obtained with the following rules: the boundary sites are singly occupied; the next inward layer is doubly occupied; the next inward layer is triply occupied, etc; a triangle shaped (or a dot) layer is reached; the next layers have the same occupancy as the previous one. If you did it right the resulting number of states will be ( )( ) d = + p + q + p+ q Here are the first smallest irreducible representations t 8 t ( pq, ) = ( 0,0) ( pq, ) = (,0) t 8 t PHYS506B, spring 005 Introduction to Gauge Theories 6 ( pq, ) = ( 0,) t 8 t Notice that the triangles for the fundamendal representations are equilateral.

27 Notes on SU() Here are three more examples: t 8 t 8 t t ( pq, ) = (,) 8 ( pq, ) = (,0) 0 PHYS506B, spring 005 Introduction to Gauge Theories 7

28 Notes on SU() 7 t 8 t Note that each state can be labeled by t, t 8 and d, where d is the dimension of the sub-multiplet of the SU() sub-algebra. ( pq, ) = ( 5,) PHYS506B, spring 005 Introduction to Gauge Theories 8 48

29 Notes on SU() Note that d* is equivalent to d only if p = q. In particular and are not equivalent ( n ) Remember that in general d T T T = tα I d i i= where T α is diagonal with eigenvalues ( t α ) i Either t or t 8 can be used, as shown in the following example: T = 0I = = ( ) + ( 0) + = T I I 8 4 = + + I = I 8 4 T = ( ) + ( 0) + I = I = ( ) + + ( ) = 8 4 I PHYS506B, spring 005 Introduction to Gauge Theories 9 I

30 Notes on SU() The product of irreducible representations can also be done graphically. For example, = 6 = = PHYS506B, spring 005 Introduction to Gauge Theories 0

31 Notes on SU() We find the following results = 6 = 8 = = and so on. It is remarkable that among the low-lying configurations of quarks, only quark-antiquark and qqq can belong to a colour singlet! All hadron states and physical observables are colour singlets It turns out that SU() is the only (compact semi-simple) Lie group that satisfies the following requirements: because there are colours, a quark must be represented by a triplet and quark and antiquark states are different; quark-antiquark and qqq can be singlets; qq and qqqq cannot be singlets. PHYS506B, spring 005 Introduction to Gauge Theories

32 Colour Factors We have seen that a quark is a colour SU() triplet : = = = q c c q q c In matrix notation we have j j j j= j= 0 0 c = red 0 c = blue c = green c c red R q c= c = c blue B c c green G Also, an antiquark is a colour SU() triplet * c c R = = = = * * anti-red * * * * q cj cj q c c c anti-blue B j= * * c c anti-green G PHYS506B, spring 005 Introduction to Gauge Theories

33 Colour Factors In order to carry the colour force, gluons carry colour and anticolour blue red red, anti-blue colour conservation at vertex So in principle there are 9 possible colour-anticolour combinations. We can therefore have the following gluon multiplets * = 8 If the gluon singlet existed, it could appear as a free particle. It would carry a long range force which would couple to baryons, approximately proportional to mass. No extra contribution to gravity has been found by experiment. This often heard argument is hokey... Gluons are in the octet multiplet. As we will see later, gauge invariance predicts gluons to be of the adjoint representation of SU(), which is the octet multiplet. Like quarks, they cannot then be observed as free 8 particle: g = z a a a= PHYS506B, spring 005 Introduction to Gauge Theories g

34 Colour Factors In matrix notation We can built each gluon octet members in the colour basis 0 0 g g g 0 0 z z z 0 0 etc. z = 8 g a c λ a c which yields i 6 i 7 ( 8 ) g = RB+ BR g = RG GR 5 g = RB BR g = BG+ GB i g = RR BB g = BG GB g = RG+ GR g = RR+ BB GG 4 6 Since gluons carry colour, they will interact with one another, unlike photons that do not. PHYS506B, spring 005 Introduction to Gauge Theories 4

35 Colour Factors The strength of the electromagnetic force between charged particles is governed by the coupling constant α, or the fundamental electric charge g e = e : Heaviside-Lorentz system Likewise, the strength of the chromodynamic, or strong, force between colour charged particles is governed by the strong coupling constant α s, or the fundamental colour charge g s : We now state the Feynman rules for tree-level diagrams in QCD: External lines: incoming quark outgoing quark incoming antiquark outgoing antiquark incoming gluon outgoing gluon a, µ a, µ u p s c u v p s c v p s c g = 4πα PHYS506B, spring 005 Introduction to Gauge Theories 5 ε µ ε µ (, ) ( p, s) c (, ) (, ) ( p) z a ( p) z * a e g = 4πα s s

36 Colour Factors Propagators: quark-antiquark gluon Vertices: quark-gluon three-gluon four-gluon a, µ bv, a, µ ( + m) i q/ q m ig δ PHYS506B, spring 005 Introduction to Gauge Theories 6 µ v q ab ig s λγ a gf g k k + g k k + g k k bv, c, λ µ bv, s abc µ v λ vλ µ λµ ν µλ νρ µρ νλ µ λρ λ µρ ig s fabe fcde g g g g + f f g g g g ace bde v v + f f g g g g ade cbe µλ ρv ρλ µ v a, µ d, ρ a, µ k k k c, λ

37 Colour Factors Consider the q q = ud QCD interaction u( p, s ), c u( p, s ), c a, µ q bv, u( p, s ), c u( p, s ), c p+ p = p+ p4 q = p p = p p ig µ µ vδab () v im = u c igs λaγ u c u 4 c 4 ig s λbγ u c q M = gc where we have the colour factor s F µ γ ( 4) γ u u u µ u q CF = c λac c4 λac PHYS506B, spring 005 Introduction to Gauge Theories 7

38 Colour Factors Comparing with the QED result for e - µ - scattering we infer the following strong potential energy (in the non-relativistic limit) α V ( r) = C = C r s s F F Since in the electromagnetic case we have gs 4πr α g e r 4π V r = QQ = QQ we see that the colour factor is like the product of the colour charges of the two quarks. The colour factor depends in which colour multiplet is the incoming quark pair. Let s use the notation then ( in) p c ( in) ( out) p c ( out) q :,, q :, p, c q :,, q : 4, p, c 4 4 = = = β, c jc k cj ck jk cj ck jk jk where the β jk are the Clebsch-Gordan coefficients for SU() needed to build a given multiplet. PHYS506B, spring 005 Introduction to Gauge Theories 8 r

39 Colour Factors Now, = = = β, 4 4 c jc k cj ck jk cj ck jk 4 jk Since the incoming and outgoing quark pairs are necessarily in the same multiplet, we have used c j c k = c j c 4k Therefore implies = = β =,,,4,4 jk jk,, 4 = Now consider the total colour operator () T(, ) = T(, 4) T T = T T T = T( 4) T ( Then ) T = T + T ( ()) ( ) () T = T + T + T T therefore T T is proportional to I. PHYS506B, spring 005 Introduction to Gauge Theories 9

40 Colour Factors Now, ( ()) (), 4 T, = T = T + T + T 4 T but () () T = c * c c T c = I c * c ( λ ) Likewise a j k j a k j k a jk jk jk = Ic λ ac 4 T = Ic λ c a 4 a Therefore we obtain T T =, 4 T T, = I c λac c4 λac ( ()) ( ) = ICF = T T T Since each quark is in the colour triplet state, we have ( ()) 4 = = = T T T I PHYS506B, spring 005 Introduction to Gauge Theories 40

41 Colour Factors The possible multiplets for the quark pair is given by T = C F = = In the triplet * case, we have 4 Therefore In the sextet 6 case, we have Therefore T C F ( 6) = 0 = = I I * = 6 We conclude that in the triplet state the quarks attract each other, while they repel each other in the sextet state. Neither state can be observed free in nature, but pairs of quarks occur in baryons, which are singlet totally antisymmetric states. This means that pairs of quarks in baryons must be in an antisymmetric state, which is the case when they are in the triplet state, ie when they attract each other! This is not a proof, but we see that the colour potential is favorable for binding when quarks are in a singlet configuration. PHYS506B, spring 005 Introduction to Gauge Theories 4

42 Colour Factors Similarly we can consider the u( p, s ), c u( p, s ), c a, µ q bv, v ( p, s), c v p, s, c p+ p = p+ p4 q = p p = p p ig µ µ vδab () v im = u c igs λaγ u c v c ig s λbγ v( 4) c4 q M = gc where we have the colour factor s F qq = ud QCD interaction µ γ u v γ v( 4) u µ q CF = c λac c λac 4 PHYS506B, spring 005 Introduction to Gauge Theories 4

43 Colour Factors Comparing with the QED result for e - µ + scattering we infer the following strong potential energy (in the non-relativistic limit) α V ( r) = C = C r s s F F gs 4πr The negative sign comes from the fact that the vertex (both in QED and QCD) does not carry the sign of the charge (electric or colour). Again, the colour factor depends in which colour multiplet is the incoming quark pair. Let s use the notation ( in) p c ( in) ( out) p c ( out) q :,, q :, p, c q :,, q : 4, p, c 4 4 then, = = c c c c = β c c jk * j k j k jk j k jk, 4 = 4 = c c c c = β c c jk * j 4k j k jk j k jk PHYS506B, spring 005 Introduction to Gauge Theories 4

44 Colour Factors To obtain the colour factor in terms of the total colour operator, we proceed as for the quark-quark case, but care is needed when dealing with the anticolour basis representation of the SU() generators. As before, for the quark triplet we obtain T = Ic λac a But for the antiquark triplet * we must use the generators of the complex conjugate representation But ( ) * ( *) = = λ 4T c c c T c I c c a 4j k j a k 4j k a jk jk jk λ =λ λ =λ therefore * T a a a a 4T = I c c λ = Ic λ c so we finally obtain a 4j k a kj a 4 jk T T =, 4 T T, = I c λac c λac 4 ( ()) ( ) = IC F = T T T PHYS506B, spring 005 Introduction to Gauge Theories 44

45 Colour Factors The quark is in the triplet state, and the antiquark in the triplet * state () * ( ) T = T = I T = T = I 4 4 = The possible multiplets for the q-qbar pair is given by 8 C F = 0 = C F 8 = = 6 In the singlet case, we have T = 0: In the singlet 8 case, we have T = I: 4 4 We conclude that the force is attractive in the colour singlet case, but repulsive for the octet. Again, this is not a proof, but it shows that the colour potential is favorable for binding a quark and an antiquark in the colour singlet configuration (which corresponds to colour singlet mesons found in nature). It is not favorable to the existence of coloured mesons. We notice that the strong potential energy operator can be written as ˆ () α both for the quark-quark and quark-antiquark s Vs ( r) = T T r interactions. This is the analogue of the spinspin interaction, or the isospin-isospin interaction. PHYS506B, spring 005 Introduction to Gauge Theories 45