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:

E2 − p2c2 − m2c4 = 0

∂2Φ − 2 + 2c2∇2Φ − m2c4Φ = 0 ∂t2

m2c2 1 ∂2Φ ∇2Φ − Φ = + 0 2 c2 ∂t2

Compare with Schroedinger equation

2 ∂Ψ - ∇2Ψ = i 2m ∂t Relativistic equation for a massive particle field.

1 ∂2Φ m2c2 ∂2Φ   − + ∇2Φ − Φ = 0 Steady state = 0 Φ(r,t) → φ(r) c2 ∂t2 2 ∂t2

m2c2 Add source term gδ (r − 0) ∇2φ − φ = gδ (r − 0) 2

mc 2 2 − r 2 1 ∂ ⎛ 2 ∂φ ⎞ m c g  g −r/R Away from r = 0 ⎜ r ⎟ − φ = 0 ⇒ φ ∝ − e = e r2 ∂r ⎝ ∂r ⎠ 2 r r

g2 Yukawa potential: φ(r) = e−r/R r

Exercise: verify φ is a solution. mc g − r g2 Spherically symmetric solution φ ∝ − e  = e−r/R r r

2 2 g −0r g 2 e Photons field: m = 0 φ ∝ − e = . g = r r 4πε0

Strong field: R  1.5 fm=1×10−15 m.

Exercise: Predict the mass of the Yukawa particle.

 hc hc 1240 eV-nm R mc2 123 MeV = = 2 = = −6 = mc 2πmc 2π R 2π × 1.5 × 10 nm 1937

• µ lepton (muon) discovered in cosmic rays.

• Mass of µ is about 105 MeV.

• Inially assumed to be Yukawa's meson but it was too penetrang.

• Meanlife: ~ 2.2 µs this is too long for a strongly interacng 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.

µ π Properes 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:

+ + ⎫ + e+ ⎫ π → µ +νµ ⎪ −9 µ → +νe +νµ ⎪ −6 ⎬ τ = 26 ×10 s. ⎬ τ = 2.2 ×10 s. π - → µ− +ν µ− → e− +ν +ν µ ⎭⎪ e µ ⎭⎪

π 0 → γ + γ τ = 8 ×10−17 s. Strong Interacons (Rohlf Ch. 18. p502)

• Strongly interacng parcles are called hadrons.

• Quarks are the fundamental objects of strong interacons.

• Quarks have spin ½ and are described by the Dirac equaon.

• Quark wave funcons are quantum states of a 6-dimensional “flavor” symmetry SU(6) whose mathemacal descripon is similar to the descripon 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 rotaons 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.) Electrostac interacon

1 V ∝ r quark-quark interacon

q q

q q

q q q q Small r

q q A V ∝ r

Large r

Energy in a flux tube of volume v: V = ρv = ρar = Br

A V ∝ + Br r A V ∝ + Br A  .05 GeV-fm B ~ 1 GeV/fm r 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.

2 2 Mass of quark: muc =1.5-4 MeV mdc =4-8 MeV Where does the nucleon mass come from?

modest resoluon: high resoluon: constuent quarks current quarks, anquark pairs, and gluons The fundamental SU3 mulplets. Gell-Mann, Neiman (1963)

Y=B+S Y=B+S s 2 / 3 2 / 3 d u −1 / 2 −1 / 2 Iz Iz −1 / 2 −1 / 2 s −2 / 3 u −2 / 3 d Mesons are composed of quarks-antiquark pairs.

SU(3) flavor mulplets and their wave funcons in flavor for the simplest mesons

in which the quarks are in a relave s state (l=0) and spins an-aligned (j=0)

K 0 ∝ ds K + ∝ us η ~ η ∝ uu + dd − ss π 0 ∝ uu − dd 8 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 mulplets and their wave funcons in flavor for the simplest baryons in which the quarks are in a relave s state j=1/2 and l=0 udd uud

uds dds uus

dss 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 Potenal Discovery of J/Ψ

BNL p + p → e+ + e− + X

SLAC e+ + e− → e+ + e− , µ+ + µ− Charmonium

Charmonium Producon

States of charmonium Construcng hadrons from quarks. Decay interaction Decay interaction

weak Vacuum polarizaon. Running coupling constant. Rohlf P502 Running coupling constant.

12π αS ≈ Λ ≈ 0.2 GeV/c ⎛ k2 ⎞ 33 2n ln ( − f ) ⎜ 2 ⎟ ⎝ Λ ⎠

Convert to distance:

12π αS ≈ RΛ ≈ λΛ = 6 fm. ⎛ R2 ⎞ 33 2n ln Λ ( − f ) ⎜ 2 ⎟ ⎝ r ⎠ Running strong coupling constant

Compare with electromagnetic: α ~ 0.01 Beginning to converge!