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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.
• [email protected] assumed to be Yukawa's meson but it was too [email protected].
• Meanlife: ~ 2.2 µs this is too long for a strongly [email protected] 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
[email protected] 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 [email protected] [email protected] are called hadrons.
• Quarks are the fundamental objects of strong [email protected].
• Quarks have spin ½ and are described by the Dirac [email protected].
• Quark wave [email protected] are quantum states of a 6-‐dimensional “flavor” symmetry SU(6) whose [email protected] [email protected] is similar to the [email protected] 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 [email protected] in flavor space.
• Flavor is a strongly broken symmetry!
Strong Inte[email protected] (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 ∝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 [email protected]: [email protected] quarks
high [email protected]: current quarks, [email protected] 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 mu[email protected]. 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 [email protected] and their wave [email protected] in flavor for the simplest mesons
in which the quarks are in a [email protected] s state (l=0) and spins [email protected]‐aligned (j=0)
Mesons are composed of quarks-antiquark pairs.
Baryons are composed of three quarks. SU(3) flavor [email protected] and their wave [email protected] in flavor for the simplest baryons in which the quarks are in a [email protected] 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 [email protected] Discovery of J/Ψ
BNL p + p→ e+ + e− + X
SLAC e+ + e− → e+ + e− , µ+ + µ−
[email protected] hadrons from quarks.
Decay interaction
Decay interaction
weak
Vacuum [email protected]. 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
