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