The Strong Interaction
What is the quantum of the strong interaction?
The range is finite, ~ 1 fm.
Therefore, it must be a massive boson.
Relativistic equation for a massive particle field.Scaler (Klein-Gordon) equation:
E2 − p2c2 −m2c4 = 0
− 2 ∂2Φ∂t2
+ 2c2∇2Φ−m2c4Φ = 0
∇2Φ− m2c2
2 Φ = 1c2
∂2Φ∂t2
+ 0
Compare with Schroedinger equation
- 2
2m∇2Ψ = i ∂Ψ∂t
Yukawa theory of the strong interaction
Relativistic equation for a massive particle field.
− 1c2
∂2Φ∂t2
+ ∇2Φ − m2c2
2 Φ = 0 Steady state ∂2Φ∂t2
= 0 Φ(r , t)→φ(r )
Add source term gδ r − 0( ) ∇2φ − m2c2
2 φ = gδ r − 0( )
Away from r = 0 1r2
∂∂r
r2 ∂φ∂r
⎛⎝⎜
⎞⎠⎟− m
2c2
2 φ = 0 ⇒ φ ∝− gre−
mcr= g
2
re−r/R
Yukawa potential: φ(r)= g2
r e−r/R
Exercise: verify φ is a solution.
Spherically symmetric solution φ ∝ − gre−
mcr= g
2
re−r/R
Photons field: m = 0 φ ∝ − gre−0r = g
2
r. g2 =
e2
4πε0
Strong field: R 1.5 fm=1×10−15 m.
Exercise: Predict the mass of the Yukawa particle.
R =
mc=
hc2πmc2 mc2 =
hc2πR
=1240 eV-nm
2π ×1.5 ×10−6 nm= 123 MeV
1937
• µ lepton (muon) discovered in cosmic rays.
• Mass of µ is about 105 MeV.
• Ini@ally assumed to be Yukawa's meson but it was too penetra@ng.
• Meanlife: ~ 2.2 µs this is too long for a strongly interac@ng object – or is it?
Lattes, C.M.G.; Muirhead, H.; Occhialini, G.P.S.; Powell, C.F.; Processes Involving Charged Mesons Nature 159 (1947) 694;
Motivation In recent investigations with the photographic method, it has been shown that slow charged particles of small mass, present as a component of the cosmic radiation at high altitudes, can enter nuclei and produce disintegrations with the emission of heavy particles. It is convenient to apply the term "meson'' to any particle with a mass intermediate between that of a proton and an electron. In continuing our experiments we have found evidence of mesons which, at the end of their range, produce secondary mesons. We have also observed transmutations in which slow mesons are ejected from disintegrating nuclei. Several features of these processes remain to be elucidated, but we present the following account of the experiments because the results appear to bear closely on the important problem of developing a satisfactory meson theory of nuclear forces. (Extracted from the introductory part of the paper.).
Discovery of Pi Meson 1946 • Charged π meson (pion) discovered in cosmic rays. • The previous μ produced from π decays via π→ µ+ ē.
π µ
Proper@es of pions Spin of pion S = 0. Parity of Pion: P = -1
Pion mass: mc2(π ± ) =140 MeV mc2(π 0 ) :mc2 =135 MeV
Pion decay:
π+ → µ+ +νµπ - → µ− +νµ
⎫⎬⎪
⎭⎪τ = 26×10−9s.
µ+ → e+ +νe +νµµ− → e− +νe +νµ
⎫⎬⎪
⎭⎪τ = 2.2×10−6s.
π 0 →γ + γ τ = 8×10−17 s.
• Strongly interac@ng par@cles are called hadrons.
• Quarks are the fundamental objects of strong interac@ons.
• Quarks have spin ½ and are described by the Dirac equa@on.
• Quark wave func@ons are quantum states of a 6-‐dimensional “flavor” symmetry SU(6) whose mathema@cal descrip@on is similar to the descrip@on 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 rota@ons in flavor space.
• Flavor is a strongly broken symmetry!
Strong Interac@ons (Rohlf Ch. 18. p502)
Color Force Field • The quantum of color is the gluon.
• 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.)
V ∝1r
Electrosta@c interac@on
q q
qqq q
quark-‐quark interac@on
q q
V ∝Ar+ Br
Energy in a flux tube of volume v:V = ρv = ρar = Br
Large r
Small r
q qV ∝
Ar
V ∝
Ar+ 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.
Mass of the nucleon: Mc2 ~ 1000 MeV.
Mass of quark: muc2=1.5-‐4 MeV mdc
2=4-‐8 MeV
Where does the nucleon mass come from?
modest resolu@on: cons@tuent quarks
high resolu@on: current quarks, an@quark pairs, and gluons
2 / 3
Y=B+S
Iz
−2 / 3
−1 / 2 −1 / 2
u
s
ds
du
Y=B+S
Iz
−2 / 3
−1 / 2 −1 / 2
2 / 3
The fundamental SU3 mul@plets. Gell-‐Mann, Neiman (1963)
Ψ =ψ (space)ψ (spin)ψ (color)ψ (flavor)
π 0 ∝ uu − ddπ− ∝ du π + ∝ ud
η ~ η8 ∝ uu + dd − ssK+ ∝ usK 0 ∝ ds
K 0 ∝ dsK− ∝ ds
ψ (color)∝ RR + BB +GG
SU(3) flavor mul@plets and their wave func@ons in flavor for the simplest mesons
in which the quarks are in a rela@ve s state (l=0) and spins an@-‐aligned (j=0)
Mesons are composed of quarks-antiquark pairs.
Baryons are composed of three quarks. SU(3) flavor mul@plets and their wave func@ons in flavor for the simplest baryons in which the quarks are in a rela@ve s state j=1/2 and l=0
p ∝ u ↑ u ↓ d ↑ +u ↓ u ↑ d ↑ −u ↑ u ↑ d ↓ + all permutations.
uud
uus
udd
dds
dss uss
uds
ψ (color)∝ RGB − RBG + BRG − BGR +GBR −GRB
The Lowest State in SU(4) u,d,s,c quarks
Quark-‐Quark Poten@al Discovery of J/Ψ
BNL p + p→ e+ + e− + X
SLAC e+ + e− → e+ + e− , µ+ + µ−
Charmonium
Charmonium Produc@on
States of charmonium
Construc@ng hadrons from quarks.
Decay interaction
Decay interaction
weak
Vacuum polariza@on. Running coupling constant. Rohlf P502
Running coupling constant.
αS ≈12π
33 − 2n f( ) ln k2
Λ2
⎛
⎝⎜⎞
⎠⎟
Λ ≈ 0.2 GeV/c
Convert to distance:
αS ≈12π
33 − 2n f( ) lnRΛ
2
r2
⎛
⎝⎜
⎞
⎠⎟
RΛ ≈ λΛ = 6 fm.
Compare with electromagnetic: α ~ 0.01 Beginning to converge!
Running strong coupling constant
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