Back to Gauge Symmetry. The Standard Model of Par0cle Physics

Transcription

1 Back to Gauge Symmetry The Standard Model of Par0cle Physics

2 Laws of physics are phase invariant. Probability: P = ψ ( r,t) 2 = ψ * ( r,t)ψ ( r,t) Unitary scalar transformation: U( r,t) = e iaf ( r,t) U 2 = U * U = e iaf ( r,t) e iaf ( r,t) = 1 ψ ( r,t) ψ ( r,t) = Uψ ( r,t) = ψ ( r,t)e iaf ( r,t) P = ψ ( r,t) 2 = ψ * ( r,t) ψ ( r,t)u * U = ψ * ( r,t)ψ ( r,t) = P

3 Local Unitary Phase Transformation: Let ψ (x,t) ψ (x,t)e iaf (x,t). Schroedinger's equation for a free particle: 2 2m 2 ψ (x,t) x 2 = i ψ (x,t) t 2 ( iaf (x,t) ψ (x,t)e ) x 2 = i 2m ( iaf (x,t) ψ (x,t)e ) t This no longer has the same form of the Schroedinger eq. Carry out the differentiation. It is straight forward but messy.

4 Need to introduce new fields? Vector and scaler fields: B x and B 0 ( in one dimension B x and B 0 ) Notice, I changed the name from A and V to B x and B 0 for a very important reason - to be revealed. Which transformation as: B x B x = B x + f x B 0 B 0 = B c f is an arbitrary function of x and t. 1 2m i x g 2 1 c B x f t Schroedinger s equa0on. ψ (x,t) = i t g1b 0 ψ (x,t)

5 Charge changing weak interactions only involve left handed doublets. Include them in wave function. ψ (x,t) ψ (x,t)χ L χ L,l : leptons, χ L,q : quarks, χ Ll, χ Lq are column matrices for left handed leptons and quarks respectively. Examples: χ L,l = ν e e L 1 0 L = ν Le, 0 1 L = e L χ L,q = d u L 1 0 L = d L, 0 1 L = u L

6 SU2 Gauge Transformation! SU(2) transformation: ψ (x,t) ψ (x,t)e i ε i τ = ψ (x,t)χ L e i 1 3 ε ν (x,t)τ ν τ ν are 2 2 matrices which operate on 2 1 column matrix χ L = (χ Ll, χ Lq ) Requiring invariance under SU(2) phase transformation introduces the 3 SU(2) fields: W µ +, W µ, W µ 0 with weak coupling charge g 2 U1 Gauge Transformation! Compare with ψ (x,t)ξ R,L ψ (x,t)ξ R,L e iaf (x,t). ξ R,L are either right or left singlets, e.g. e R, e L 3 neutral fields : B µ, Coupling charge : g 1

7 How are these fields and charges related to the observed fields and charges of the electromagne0c and weak interac0ons?

8 Combining U(1) L SU(2) L U(1) L SU(2) L transformation: ψ L (x,t) ψ L (x,t)e iβ+i ε i τ B µ and W µ 0 act together on charge neutral L leptons and quarks. Since SU(2) L only couples to L U(1) R transformation: ψ R (x,t) ψ R (x,t)e iβ(x,t) Only B µ couples to R.

9 SU2 L W 0 and B can interfere at the amplitude level since they involve the same two ini0al and final states.. e L e L g 1 Y L B µ + g 2 W µ 0 and g 2 B µ g 1 Y L W µ 0 Z µ and A µ U1 R e R e R e R e R B µ

10 Electro- Weak Interference e W µ 0 u e Z µ u e u e u e B µ u e A µ u e u e u

11 Weak Mixing Parameter A µ = cosθ W W µ 0 + sinθ W B µ Z µ = sinθ W W µ 0 + cosθ W B µ g 2 g 1 = tanθ W A µ = g 2 W µ 0 + g 1 B µ g g 1 2 Z µ = g 1 W µ 0 g 2 B µ g g 1 2 sin 2 θ W e 2 = 1 g g 2 2 or 1 α = α g1 α g2 M W M Z = cosθ W M W c 2 80 GeV M Z c 2 91 GeV

12 The Strong Interaction What is the quantum of the strong interaction? The range is finite, ~ 1 fm. Therefore, it must be a massive boson.

13 Gauge Transformations! SU(3) transformation: ψ (x,t)ξ ψ (x,t)ξe i αi T i = ψ (x,t)ξe 1 ξ = r b g T ν are 3 3 matrices 8 α ν (x,t)t ν G i ν 8 strong QCD color fields: rg, rg, rb, rb, gb, gb, gg,bb, rr (-1 since they are not all independent)

14 The quantum of color is the gluon. Color Force Field Strong charges come in types labeled r, g, b for red, green and blue. (E&M only has one kind of charge) Both quarks and gluons posses color charge. (photons carry no electric charge.)

15 The Higgs The U(1)XSU(2) Lagrangian is only gauge invariant if the masses of all the Fermions and Bosons in the wave equa0on have zero mass. Masses of Bosons (e.g. M W and M Z ) can be very large! Mass of leptons are not zero (some are quite large, e.g. top quark) Need addi0onal field to make the electroweak wave func0on guage invariant. Predict new massive boson field, i.e. Higgs field φ.

16 Higgs Par0cle Standard Model: Gauge Symmetry Field quanta (Bosons) and interac0ons with par0cles (Fermions) All masses (field Bosons, par0cle Fermions) violate gauge symmetry All par0cles had no mass just a^er the Big Bang. As the Universe cooled and the temperature fell below a cri0cal value, an invisible force field called the Higgs field was formed together with the associated Higgs boson. The field prevails throughout the cosmos: any par0cles that interact with it are given a mass via the Higgs boson. The more they interact, the heavier they become, whereas par0cles that never interact are le^ with no mass at all.

17 CDF Tevatron (Fermi Lab) 1 TeV 1 TeV M H > 170 GeV/c 2

18 LHC CERN 7 TeV 7 TeV

19 Electroweak The Standard Model ( ) L L = 4 1 W µν W µν 4 1 B µν Bµν + Lγ µ µ g 1 2 τiw µ g Y 2 B µ ( ) R + i µ g 1 2 τiw µ +Rγ µ i µ g Y 2 B µ ( g Y 2 B µ )φ V φ ( ) ( G 1 LφR + G 2 LφR) + h.c. Strong - QCD L = q ( iγ µ µ m)q g( qγ µ T a q )G a µ G µν a G a µν Symmetry breaking Weak + Electromagnetic A µ g W 0 µ + gb µ, Z µ gw 0 µ gb µ, W 1 2 µ, W µ

20 The Standard Model Electroweak: L = 4 1 W µνw µν 4 1 B µν B µν + Lγ µ ( µ g 1 2 τiw µ g Y 2 B ) µ L ( ) R + i µ g 1 2 τiw µ +Rγ µ i µ g Y 2 B µ ( g Y 2 B µ )φ Vφ ( G 1 LφR + G 2 LφR) + h.c. Strong: L = q ( iγ µ µ m)q g( qγ µ T a q )G a µ G a µν µν G a

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22 The Strong Interaction What is the quantum of the strong interaction? The range is finite, ~ 1 fm. Therefore, it must be a massive boson.

23 Gauge Transformations! U(1) transformation: ψ (x,t) ψ (x,t)e iaf (x,t) =ψ (x,t)e iβ(x,t) A, V Electromagnetic scalar and vector potentials. SU(2) transformation: ψ (x,t) ψ (x,t)e i ε i τ = ψ (x,t)e i 1 3 ε ν (x,t)τ ν τ ν are 2 2 matrices. W +, W, W 0 Weak fields. SU(3) transformation: ψ (x,t) ψ (x,t)e i αi T i = ψ (x,t)e 1 8 α ν (x,t)t ν T ν are 3 3 matrices G ν 8 strong QCD color fields: rg, rg, rb, rb, gb, gb, gg,bb, rr (-1 since they are not all independent)

24 Strong Interac0ons (Rohlf Ch. 18. p502) Strongly interac0ng par0cles are called hadrons. Quarks are the fundamental objects of strong interac0ons. Quarks have spin ½ and are described by the Dirac equa0on. Quark wave func0ons are quantum states of a 6- dimensional flavor symmetry SU(6) whose mathema0cal descrip0on is similar to the descrip0on of angular momentum. The flavors, denoted u, d, s, c, b and t. are components of a flavor vector in a 6 dimensional space. Perfect SU(6) symmetry would imply all quarks have the same mass energy and the magnitude of its SU(6)- vector would be independent of the rota0ons in flavor space. Flavor is a strongly broken symmetry!

25 The quantum of color is the gluon. Color Force Field Strong charges come in types labeled r, g, b for red, green and blue. (E&M only has one kind of charge) Both quarks and gluons posses color charge. (photons carry no electric charge.)

26 Electrosta0c interac0on V 1 r

27 quark- quark interac0on q q q q q q q q

28 Small r q q V A r Large r Energy in a flux tube of volume v: V = ρv = ρar = Br V A r + Br

29 V A r + Br A.05 GeV-fm B ~ 1 GeV/fm Note: when r~1 fm, the energy is ~ 1 GeV. This is the field energy in the flux tube which accounts for most of the mass of the hadron.

30 Mass of the nucleon: Mc 2 ~ 1000 MeV. Mass of quark: m u c 2 =1.5-4 MeV m d c 2 =4-8 MeV Where does the nucleon mass come from? modest resolu0on: cons0tuent quarks high resolu0on: current quarks, an0quark pairs, and gluons

31 The fundamental SU3 mul0plets. Gell- Mann, Neiman (1963) Y=B+S Y=B+S s d 2 / 3 u 1 / 2 2 / 3 1 / 2 1 / 2 1 / 2 s 2 / 3 I z u 2 / 3 d I z

32 Mesons are composed of quarks-antiquark pairs. SU(3) flavor mul0plets and their wave func0ons in flavor for the simplest mesons in which the quarks are in a rela0ve s state (l=0) and spins an0- aligned (j=0) π 0 uu dd π du K 0 ds K + us η ~ η 8 uu + dd ss π + ud K ds K 0 ds Ψ = ψ (space)ψ (spin)ψ (color)ψ (flavor) ψ (color) RR + BB + GG

33 Baryons are composed of three quarks. SU(3) flavor mul0plets and their wave func0ons in flavor for the simplest baryons in which the quarks are in a rela0ve s state j=1/2 and l=0 udd uud dds uds uus dss uss p u u d +u u d u u d + all permutations. ψ (color) RGB RBG + BRG BGR + GBR GRB

34 The Lowest State in SU(4) u,d,s,c quarks

35 Quark- Quark Poten0al Discovery of J/Ψ BNL p + p e + + e + X SLAC e + + e e + + e, µ + + µ

36 Charmonium Charmonium Produc0on States of charmonium

37 Construc0ng hadrons from quarks.

38 Decay interaction

39 Decay interaction weak

40 Vacuum polariza0on. Running coupling constant. Rohlf P502

41 Running coupling constant. α S 12π ( 33 2n f ) ln k2 Λ 2 Λ 0.2 GeV/c Convert to distance: α S 12π ( 33 2n f ) ln R Λ 2 r 2 R Λ λ Λ = 6 fm.

42 Running strong coupling constant Compare with electromagnetic: α ~ 0.01 Beginning to converge!