Lecture 4: Antiparticles & Virtual Particles

1. Lecture 4:Antiparticles & Virtual Particles • Klein-Gordon Equation • Antiparticles & Their Asymmetry in Nature • Yukawa Potential & The Pion • The Bound State of the Deuteron • Virtual Particles • Feynman Diagrams Useful Sections in Martin & Shaw: Chapter 1

2. The Schrodinger Equation i = Ae(px-Et) ℏ ∂ ∂ -iℏp iℏ  & Note that: ∂x ∂t ∂ p -iℏ∇ E iℏ So define: ∂t ∂  ℏ i ∇2 ∂t 2m Schrodinger Equation (non-relativistic) Aei(kx-t) Free particle E = p2/2m 

3. The Klein-Gordon Equation To make relativistic, try the same trick with E2 p2c2 + m2c4 ∂  1 m2c2 ∇2   c2 ∂t ℏ2 Klein-Gordon Equation Aexp(- Et) i For every plane-wave solution of the form ℏ (positive E) Aexp( Et) i There is another solution of the form ℏ first proposed by de Broglie in 1924 (negative E)

4. The Dirac Equation 3 n=1 ∂ ∂ -iℏ   cn mc2  iℏ  ∂xn ∂t Dirac Equation     still have positive and negative energy states but now also have spin!  Try again, but attempt to force a linear form: Where n and  are determined by requiring that solutions of this equation also satisfy the Klein-Gordon equation andneed to be 4x4 matrices and

5. Dirac Hole Theory E Nowdays we don’t think of it this way! Instead we can say that energy always remains positive, but solutions exist with timereversed (Feynman-Stukelberg) 0 .. How do you prevent transitions into ''negative energy" states? Dirac ''Hole" Theory ''sea" of negative energy states

6. Antimatter Antimatter Anderson 1933

7. Where is the Antimatter ?? Where’s the Antimatter ??? The Earth Spontaneous combustion is relatively rare The Moon Neil Armstrong survived The Planets Space probes, solar wind... Ouside the Solar System Comets... Another Part of the Galaxy Cosmic Rays... Other Galaxies Mergers, cosmic rays... Larger Scales Diffuse ray background

8. The Yukawa Potential m2c4 ∇2 ℏ2 whose solution is g2 e-r/R 4 r e2 1 4 r  V(r) =  V(r) er Yukawa Potential So this gives us a new ''charge" g and an effective range R hmmm... sounds like the ''neutron-proton" problem n p whatever keeps them together must be very strong and short-ranged  n p note that if m=0, we would have the equivalent of an electromagnetic potential: For a static solution, Klein-Gordon reduces to ∇ in this case the solution is where R ℏ/mc

9. The muon ''meson" EM ''carrier" of electromagnetic field = photon (massless boson) Strong nuclear force ''carrier" of field must be some massiveboson R  10-15m  1fm ℏc = 197 MeV fm  mc2 = 100 MeV e = 0.511 MeV ''lepton" p = 938 MeV ''baryon" Yukawa (1934) -meson (muon)Anderson & Neddermeyer (1936) m = 105.6 MeV ! ...but a fermion, doesn’t interact strongly (looks like a heavy electron) ''Who ordered that ?!"(I. I. Rabi)

10. The pion Marietta Blau Don Perkins Cecil Powell -meson (pion), m=140 MeVPowell et al. (1947)

11. Bound State of the Deuteron (from Bowler) reduced mass ''Bohr Condition" assume mp ≃ mn M, so  = (MM)/(M+M) = M/2 also take p ≃ℏ/r (de Broglie wavelength) ℏ2 g2 Mr2 4r ED =  exp(mcr/ℏ) m2c2 g2mc Mx2 4ℏx E = e-x The Bound State of the Deuteron n p ED = p2/ 2 + V(r) let: x mcr/ℏ

12. m2c2 g2mc Mx2 4ℏx ED =  e-x [ ] g2 4ℏc 22 1 x2 1 x m M m M = Mc () ( )e-x g2 4ℏc m M m M ( )e-x 1 x 1 x2 ( )2 > g2 4ℏc m M 1 x ex ( ) > g2 4ℏc ( ) > 140 MeV 938 MeV (2.718) g2 4ℏc e2 1 4ℏc 137 > s compare with  0.4 The Stong Coupling Constant for a bound state to exist, ED < 0 this is a  minimum when x=1

13. Virtual Particles >  Heisenberg uncertaintyE t ℏ ''virtual particle" What does ''carrier of the field" mean ?? Note: the time it would take for the carrier of the strong force to propagate over the distance R is t  R/c soR ~ℏc/E if we associate E with the rest mass energy of the pion, then R ~ℏ/mc which is what enters into the Yukawa potential ! This implies we are ''borrowing" energy over a ''Heisenberg time"

14. ''Field Lines" p n Strong Nuclear Force (finite range) ''Field Lines" +  EM (infinite range)

15. e+ e+ p1 p3 e+ e+ p1 p3 q q p2 p4 e- e- p2 p4 e- e- x t Leading order diagrams for Bhabha Scattering e+ + e e+ + e Feynman Diagrams Feynman Diagrams

16. x t e+ e+ p1 p3 e+ e+ p1 p3 q q p2 p4 e- e- p2 p4 e- e- Leading order diagrams for Bhabha Scattering:e+ + e e+ + e Some Rules for the Construction & Interpretation of Feynman Diagrams 1) Energy & momentum are conserved at each vertex 2) Charge is conserved 3) Straight lines with arrows pointing towards increasing time represent fermions. Those pointing backwards in time represent anti-fermions 4) Broken, wavy or curly lines represent bosons 5) External lines (one end free) represent real particles 6) Internal lines generally represent virtual particles

17. x t e+ e+ p1 p3 e+ e+ p1 p3 q q p2 p4 e- e- p2 p4 e- e- Leading order diagrams for Bhabha Scattering:e+ + e e+ + e Some Rules for the Construction & Interpretation of Feynman Diagrams 7) Time ordering of internal lines is unobservable and, quantum mechanically, all possibilities must be summed together. However, by convention, only one unordered diagram is actually drawn 8) Incoming/outgoing particles typically have their 4-momenta labelled as pn and internal lines as qn 9) Associate each vertex with the square root of the appropriate coupling constant, x , so when the amplitude is squared to yield a cross-section, there will be a factor of xn, where n is the number of vertices (also known as the ''order" of the diagram)

18. x t and e+ e+ p1 p3 e+ e+ p1 p3 q q p2 p4 e- e- p2 p4 e- e- Leading order diagrams for Bhabha Scattering:e+ + e e+ + e Some Rules for the Construction & Interpretation of Feynman Diagrams 10) Associate an appropriate propagator of the general form 1/(q2 + M2) with each internal line, where M is the mass of mediating boson 11) Source vertices of indistinguishable particles may be re-associated to form new diagrams (often implied) which are added to the sum Thus, the leading order diagrams for pair annihilation ( e- + e+ + ) are:

19. ''Play Catch ?" The ''play catch" idea seems to work intuitively when it comes to understanding how like charges repel.

20. ''Play Anti-Catch ?" But what aboutattractiveforces between dissimilar charges?? Are you somehow exchanging ''negative momentum" ???! The best I can offer: Note from Feynman diagrams (and later CPT) that a particle travelling forward in time is equivalent to an anti-particle, going in the opposite direction, travelling backwards in time. e+  Feynman-Stuckelberg interpretation is that the photon scatters the electron back in time! .. e- The ''play catch" idea seems to work intuitively when it comes to understanding how like charges repel.

21. Higher Order Diagrams So this basically a perturbative expansion in powers of the coupling constant. You can see how this will work well for QED since ~ 1/137, but things are going to get dicey with the strong interaction, where s~ 1 !! More Bhabha Scattering...

22. Richard Feynman .. (Baron) Ernest Stuckelberg von Breidenbach zu Breidenstein und Melsbach