The Standard Theory of Elementary Particle Physics and Beyond

Transcription

1 The Standard Theory of Elementary Particle Physics and Beyond An Introduction in (about) 20 lectures CINVESTAV, Mexico City, Mexico 10 February-2April, 2005 Luciano Maiani, Universita di Roma La Sapienza and INFN, Roma, Italy Lecture 8, March 3,

2 5. QUARKS Quarks 1 Precursors The Eightfold way Fundamental triplets, mesons and baryons The naive (constituent) quark model Quarks 2 symmetries and currents symmetry breaking weak and electromagnetic currents Cabibbo theory The breaking of the chiral symmetry Quarks 3 GIM mechanism and the charmed quark Guessing the c-quark mass Charmed Mesons and baryons SU(2)xU(1) with two quark doublets Quarks 4 Quark masses A useful theorem The KM matrix Mixing and CP violation in B mesons 2

3 J. Letessier and J. Rafelski, Hadrons and Quark Gluon Plasma, Cambridge Monogr. Part. Phys. Nucl. Phys. Cosmol. 18 (2002). ρ( m ) = δ( m m ) i i NOTE: ρ tot = ρ( m ) + 3δ( m m π )

4 1. Precursors E. Fermi, C. N. Yang Are Mesons Elementary Particles? Phys. Rev. 76 (1949)1739. M. Gell-Mann: Symmetries of Baryons and Mesons, Phys. Rev. 125 (1962) The Fermi-Yang model: Baryons are fundamental, mesons are not N N interaction = repulsive, N-Nbar attractive, may form deeply bound states Lightest states: L=0 (S-wave) spin N-Nbar = 0 J PC = 0 -+ Isospin : 1/2 1/2 = 1 0 The 3 pions + a neutral particle - η(547)? Further states: L=0 (S-wave) Spin N-Nbar = 1 J PC = 1 -- Vector mesons : ρ(770), ω (782); in fact there is another one: φ(1020)?? A photon-like, fundamental vector field coupled to baryon number would make an attractive force for N-Nbar and a repulsive one for N-N. Gell-Mann calls it B. Tentative name: gluon (from glue) B cannot be massless like the photon. 4

5 Precursors (cont d) The Sakata Model p T = n Λ S. Sakata: On a composite model for the new particles, Progr. Theor. Phys. 16, 686, M. Gell-Mann: Symmetries of Baryons and Mesons, Phys. Rev. 125 (1962) Fundamental fields: T, Bµ Lagrangian: L = T[ i( µ g strong B µ )γ µ + M]T 1 4 B µν B µν µ2 B µ B µ B µν = µ B ν ν B µ Natural Symmetry: SU(3): T UT; U + U =1,det(U) =1 As in F&Y, an overall phase in U would be associated to baryon number conservation SU(3) symmetry is violated (at least!) by the unequal masses of the elements of T: (m p + m n -2m Λ ) 0: SU(3) SU(2) I U(1) S U(1) Q (~ 30%) (m p - m n ) 0: SU(2) I U(1) S U(1) I3 U(1) S (~ 0.1%) M = m p m n m Λ 5

6 Spectroscopy of the Sakata model: mesons T-Tbar states: SU(3) rep.= 8 1 L=0 (S-wave) Spin (T-Tbar) = 0 JPC= 0 -+ Pseudoscalar Mesons fit well (today!), including η (960) Spin(T-Tbar) = 1 J PC = 1 -- Vector Mesons fit well: ρ(770), ω (782), φ(1020), K* (892) The real problem are baryons. Simplest possibility for spin 1/2, B=1, is: (T-Tbar)T This formula entails the existence of baryons with S=+1 (K+N) but the K+N channel is flat, unlike K-N which presents several resonances. The baryon spectrum is a fatal blow for the model. Note: a K + N resonance may have been observed recently, Θ + (1540), but its relevance is anyway not comparable to that of the states Σ, Ξ, Y* 6

7 K - -p cross-section 7

8 K + -N cross-sections 8

9 The Eightfold Way M. Gell-Mann: Symmetries of Baryons and Mesons, Phys. Rev. 125 (1962) A strategic conversion: The symmetry SU(3) is suggested by the Sakata model as a (broken) symmetry of the strong interactions, which extends Isospin to include the strangeness degrees of freedom Can we find anyway a viable classification on baryons and mesons in SU(3)multiplets? can we describe the SU(3) breaking quantitatively? 9

10 Strangeness and the formula of Gell-Mann & Nishijima Hadrons make isospin multiplets, characterised by (I, I 3 ) and S; Q goes in steps of 1, like I 3, so we can put Q= I 3 + Y/2 Y is called hypercharge and is = 2<Q> (since <I 3 >=0) Y commutes with I, I 3, so it is a functions of the other conserved numbers, S and B Gell-Mann and Nishijima note that: Y = B + S so that: Q = I (B + S) 2 Y is symmetrically distributed also for the stable baryons unlike S (recall S baryons 0) Y is a good candidate to be the second commuting generator in a symmetry group of rank two (I 3 e Y). 10

11 SU(3) generators Le 8 Gell-Mann matrices 0 ;λ 4 = ;λ 5 = 0 i 0 ; i 0 0 λ 1,2,3 = σ1,2,3 0 λ 6 = ;λ = i ;λ = i U = e i! a" a # a ; # a = 3 3, complex, hermitian; Tr(# a ) = 0 SU(3) SU(2) I U(1) Y λ 1,2,3 = I 1,2,3 ;λ 8 = Y [λ i,λ j ] = 2if ijk λ k (i, j,k =1,2...8) λ 0 = {λ i,λ j } = 2d ijk λ k (i, j,k = 0,1,...8) f ijk = structure constants of the Lie algebra; completly antisymmetric; d ijk = completly symmetric; the numerical vlues are given in Gell-Mann s article. Tr(λ i λ j ) = 2δ ij Two fundamental reps (1) : 3, 3-bar: 3 and 3-bar are not equivalent (unlike SU(2) why?) dim of the Regular Rep. (2) = 8 (λ (3) ) α,β = [(λ (3) ) α,β ]* = (λ α,β ) * (1) we can construct all reps with their tensor products (2) the rep. which transforms like the generators themselves 11

12 Symmetry breaking We write the strong interaction Lagrangian (e=0, G=0): L strong = L sym + L sb in the Fermi-Yang-Sakata model, L sb is just due to the unequal masses: M = m p m n m Λ = m m 1 p n 2 M = m 0 λ 0 + m 8 λ 8 + m 3 λ 3 ; m T = (m p + m n + m Λ ) /3 1GeV L sb = T M p M n 0 T 0 0 M! 1 + m + m 2m 1 p n Λ 6 0 Under SU(3), M transforms like a superposition of the diagonal components of an octet. Thus, neglecting isospin violation we are led to the octet hypothesis: L sb! m + m + m 1 p n Λ

13 Invariant Operations in SU(3) Vector of rep. 3 Vector of rep. 3-bar: Note: x i (Ux) i = U k i x k y i (U k i ) * y k = y k (U + ) i k = (yu + ) i y i x i (yu + Ux) = (yx) = in var iante That is : δ i j = invariant The second invariant operation: completly antisymmetric tensor: ε ijk U i i' U j j' U k k' ε i' j'k' = detuε ijk = ε ijk That is : ε ijk ( or ε ijk ) = invariant 13

14 The irreducible tensors (IT) 1) {i T(n,m) = T 1,i 2,...i n } {i,i { j1, j 2,..., j m };T 2,...i } n j { j, j2,..., j m } δ i = 0 2 quantum numbers: corrisponds to rank = 2 of SU(3) 2) T(n,0) T(0,m) = T(n,m) [T(n 1,0) T(0,m 1) dim(n,0) dim(0,m) = dim(n,m) + dim(n 1,0) dim(0,m 1) dim(n,m) = (n +1)(m +1)(n + m + 2) 2 Fundamental Reps. : (1,0)= 3 (0,1) = 3-bar Regular Rep.: (1,1)= 8 Decuplet: (3,0) = 10 Antidecuplet: (0,3)= 10-bar n=m : self conjugate rep. 14

15 Decomposition of 3, 8, 10 Isospin of u and d= +/- 1/2 Hypercharge of s=y Hypercharge of u and d = y+1 Trace(Y)=0 Y=-2/3 These quantum numbers fix the decomposition in isopin and hypercharge of the Irr. Tensors. 8 = x a x b - 1/3δ b a = (I=1, Y=0); (I=0, Y=0), (I=1/2,+1), (I=1/2, -1) 10 =Symm(x a x b x c )=(I=3/2, Y=1);(I=1, Y=0);(I=1/2, Y=-1); (I=0, Y=-2) Note: TrY=0 u(+1/2 1/ 3 ) x = d( 1/2 1/ 3 ) ; s(0 2 / 3 ) 15

16 S=Y +1 (497.7) Mesons: J PC =0 -+ Masses in MeV (493.7) (135.0) 0-1 (139.6) η' (547.3) (957.8) (139.6) (493.7) (497.7) -1-1/2 0 +1/2 +1 I 3 16

17 S=Y +1 K *0 K *+ (892) Mesons: J PC =1 -- Masses in MeV 0 ρ ω 0 φ (782.6) (1020) ρ + (770) -1 ρ 0 K *- K *0-1 -1/2 0 +1/2 +1 I 3 17

18 Baryons with J P =1/2 + Masses in MeV S N (939.6) P (938.3) +0 Λ 0 (1115.7) -1-2 Σ - (1197.4) Σ - (1192.6) Σ + (1189.4) Ξ - (1321.3) Ξ 0 (1314.8) -1-1/2 0 +1/2 +1 I 3 18

19 S 0 Δ (1232) (1530) (1385) (1672) Predicted: /2-1 -1/2 0 +1/ /2 I 3 19

20 Mass relations for the stable Baryons Choose a basis of tensors (B (a) ) i j corresponding to definite isospin and hypercharge. Form the matrix: B i j = ψa (T (a) ) i j, where ψa are the fields of the eight baryons: B j i Σ Λ Σ + p 6 = Σ Σ0 2 + Λ n 6 Ξ Ξ 0 2 Λ 6 Form invariant tensors with B, B-bar and λ 8 Develop in the field bilinears: you get the general expression of M (8). i B j Σ Λ 6 = Σ Ξ Σ + Σ0 2 + Λ Ξ 0 6 p n 2 Λ 6 20

21 M (8) = 3{aTr(Bλ 8 B) + btr(bbλ 8 )} = at 1 + bt 2 T 1 = [NN + ΣΣ ΛΛ 2ΞΞ] T 2 = [ 2NN + ΣΣ ΛΛ + ΞΞ] Note: coeff. of (a-b) is proportional to Y coeff. of (a+b) + const. is proportional to C 2 =-Y 2 /2+2I(I+1): N Σ Λ Ξ (a-b) (a+b) C Add a costant mass, eliminate the 3 parameters: Gell-Mann Okubo relation 1 3Λ + Σ (N + Ξ) = side = 1128 MeV 2 0 side = 1135 MeV!!! 21

22 Decuplet M = M 0 + ay Masses are equispaced, 2 parameters and 4 masses. Δ (1232) (1385) (1530) (1672) Δ=153 Δ=145 Δ=142 22

23 Pseudoscalar Mesons Gell-Mann Okubo η = 4K π 3 1 st side = 547 MeV 2 nd side = 615 MeV??? squared masses? There is octet- singlet mixing! η 2 = 4K 2 π 2 The G-M O relation gives the mass of 8>, but there are other 2 unlnowns: the mass of 0> and a non-diagonal mass. Trovo sin 2 θ = M 8 η η' η linear mass formula: sinθ= 0.41, θ = 27 0 quadratic mass formula: sinθ= 0.19, θ = η(547) >= cosθ 8 > +sinθ 0 > η'(958) >= sinθ 8 > +cosθ 0 > 1 st side = GeV 2 2 nd side = GeV 2? better? 23

24 Gell-Mann Okubo Vector Mesons octet-singlet mixing: ω(783) >= cosθ V 8 > +sinθ V 0 > ω 8 = 4K * ρ = 933MeV ω 2 8 = 4K *2 ρ φ(1020) >= sinθ V 8 > +cosθ V 0 > sin 2 θ V = M 8 ω φ ω linear: sinθ= 0.80, tanθ=1.3, θ = 53 0 quadratic: sinθ= 0.77, tanθ=1.2, θ = 50 0 = 0.863GeV 2 A simple result: the angle is close to the ideal mixing angle, for which: ω = cosθ sinθ = φ = sinθ cosθ = tanθ = 2;θ =

25 Quarks! Basic fermions of spin 1/2: an SU(3) tripletto: u d ; s Nota: I 3 Y Q S +1/2 +1/3 +2/3 0 1/2 +1/3 1/ /3 1/3 1 Y such that q-qbar has integer hypercharges Q=I 3 +1/2Y the s quark carries the negative unit of strangeness Mesons = q-qbar = as before. Baryons: Sakata used basic fields with B=1(p,n, Λ). Now, we can choose the baryon numer of the triplet as appropriate: B( q)= 1/3 Baryons= (qqq) (qqq) has only negative strangeness, and contains only the observed reps: 1, 8, 10 Three quarks for Master Mark! 25

26 Qaurk Composition of the Baryons Octet: I Y (S) partic. [ud]u, [ud]d 1/2 1 0 p, n [su]u, [su]d, [sd]u, [ud]s 0,1 0-1 Σ, Λ [su]s, [sd]s 1/ Ξ Decuplet: {uuu}, {uud}, {udd}, {ddd} 3/2 1 0 Δ {uus}, {uds}, {dds} Y* {uss}, {dss} 1/ Ξ* {sss} Ω 26

27 Quarks! (continua) Octet Baryons: [3 3 ]A 3=3-bar 3=8 1= B [ab ]c and we have to subtract the Trace (= singlet). In formulae, we get a tensor with mixed symmetry : B {[ab]c} = N(B [ab]c 1 6 ε abcε def B [de ] f ) = = N( 2 3 B [ab ]c 1 3 B [bc]a 1 3 B [ca ]b ) = N 3 [(B [ab ]c + B [cb ]a ) + (B [ab ]c + B [ac ]b )] the same symmetry applies to the spin wave function: 1/2 1/2 1/2 1/2 the overall wave-function (spin * SU(3)) is symmetric the same holds (trivially) for the decuplet: Δ ++ =u u u What happened to Fermi statistics!!!??!! 27