Download The Strong Interaction

The Strong Interaction
What is the quantum of the strong interaction?
The range is finite, ~ 1 fm.
Therefore, it must be a massive boson.
Yukawa theory of the strong interaction
Relativistic equation for a massive particle field.
Scaler (Klein-Gordon) equation:
E 2 − p 2 c2 − m 2 c 4 = 0
− 2
∂2 Φ
2 c2 ∇2 Φ − m 2 c 4 Φ = 0
+

∂t 2
∇2 Φ −
m 2 c2
1 ∂2 Φ
Φ= 2 2 +0
2

c ∂t
Compare with Schroedinger equation
2 2
∂Ψ
∇ Ψ = i
2m
∂t
Relativistic equation for a massive particle field.
1 ∂2 Φ
m 2 c2
2
− 2
+∇ Φ−
Φ=0
2
2
c ∂t

Add source term gδ ( r − 0 )
Away from r = 0
∂2 Φ
Steady state
=0
2
∂t


Φ(r, t) → φ (r )
m 2 c2
∇ φ−
φ = gδ ( r − 0 )
2

2
1 ∂ ⎛ 2 ∂φ ⎞ m 2 c 2
r
φ=0
⎟⎠ −
2 ∂r ⎜⎝
2
∂r
r

Yukawa potential:
⇒
g −
φ ∝− e
r
g2 −r/R
φ (r) =
e
r
Exercise: verify φ is a solution. mc
r

g2
=
r
e− r/R
Spherically symmetric solution
g −
φ ∝− e
r
g −0 r g 2
Photons field: m = 0 φ ∝ − e
=
.
r
r
mc
r

g2
=
r
e− r/R
e2
g =
4πε 0
2
Strong field: R  1.5 fm=1 × 10 −15 m.
Exercise: Predict the mass of the Yukawa particle.
R=

hc
=
mc 2π mc 2
mc 2 =
hc
1240 eV-nm
=
= 123 MeV
2π R 2π × 1.5 × 10 −6 nm
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 + + ν µ ⎫⎪
⎬
−
π → µ + ν µ ⎪⎭
π0 →γ +γ
τ = 26 ×10−9 s.
τ = 8 ×10−17 s.
µ + → e+ + ν e + ν µ ⎫⎪
⎬
−
−
µ → e + ν e + ν µ ⎪⎭
τ = 2.2 ×10−6 s.
Strong Interac@ons (Rohlf Ch. 18. p502) • 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! 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.) Electrosta@c interac@on 1
V∝
r
quark-­‐quark interac@on q
q
q
q
q
qq
q
Small
r q
q
Large
A
V∝
r
r Energy in a flux tube of volume v:
V = ρv = ρar = Br
A
V ∝ + Br
r
A
V ∝ + Br
r
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 mdc2=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
The fundamental SU3 mul@plets. Gell-­‐Mann, Neiman (1963) Y=B+S Y=B+S 2/3
d
−1 / 2
s
−1 / 2
−2 / 3
s
u
−1 / 2
2/3
−1 / 2
Iz Iz u
−2 / 3
d
Mesons are composed of quarks-antiquark pairs.
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) K 0 ∝ ds
K + ∝ us
η ~ η8 ∝ uu + dd − ss
π ∝ uu − dd
0
π − ∝ du
π + ∝ ud
K − ∝ ds
K 0 ∝ ds
Ψ = ψ (space)ψ (spin)ψ (color)ψ (flavor)
ψ (color) ∝ RR + BB + GG
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
udd
uud
uds
dds
dss
uus
uss
p ∝ u ↑ u ↓ d ↑ + u ↓ u ↑ d ↑ − u ↑ u ↑ d ↓ + all permutations.
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
(
⎛k ⎞
33 − 2n f ln ⎜ 2 ⎟
⎝Λ ⎠
2
)
Λ ≈ 0.2 GeV/c
Convert to distance:
αS ≈
12π
( 33 − 2n f )
⎛ RΛ2 ⎞
ln ⎜ 2 ⎟
⎝r ⎠
RΛ ≈ λΛ = 6 fm.
Running strong coupling constant Compare with electromagnetic: α ~ 0.01
Beginning to converge!