 -decay theory

The decay rate Fermi’s Golden Rule density of final states (b) transition (decay) rate (c) transition matrix element (a) Turn off any Coulomb interactions

The decay rate (a) Fermi’s Golden Rule V  = weak interaction potential u = nuclear states     = lepton (  ) states Integral over nuclear volume

The decay rate (a) uPuP uDuD  “Four-fermion” (contact) interaction uPuP W uDuD  (W) Intermediate vector boson Interaction range

The decay rate (a) Assume: Short range interaction  contact interaction g = weak interaction coupling constant Assume: , are weakly interacting  “free particles” in nucleus Approximate leptons as plane waves

The decay rate (a) Assume: We can expand lepton wave functions and simplify And similarly for the neutrino wave function. Test the approximation --- deBroglie >> R therefore, lepton ,   constant over nuclear volume. (We will revisit this assumption later!)

The decay rate (a) Therefore -- the matrix element simplifies to -- M fi is the nuclear matrix element; overlap of u D and u P Remember the assumptions we have made!!

The decay rate Fermi’s Golden Rule density of final states (b) transition (decay) rate (c) transition matrix element (a)

The decay rate (b) Fermi’s Golden Rule Quantization of particles in a fixed volume (V)  discrete momentum/energy states (phase space) -- Number of states dN in space-volume V, and momentum-volume 4  p 2 dp

The decay rate (b) Do not observe ; therefore remove -dependence -- At fixed E e Assume

The decay rate (b) Fermi’s Golden Rule Differential rate Density of final states

The decay rate Fundamental (uniform) interaction strength Differential decay rate Overlap of initial and final nuclear wave functions; largest when u P  u D  a number Determines spectral shape!

E f (Q) Q-value for decay Definition of E f

d (p e ) c.f. Fig. 9.2

d (E e )

d (T e ) c.f. Fig. 9.2

Consider assumptions Look at data for differential rates - c.f., Fig. 9.3 Calculate corrections for Coulomb effects on   or    Fermi Function F(Z’,p e ) or F(Z’,T e ) Coulomb Effects -- v e velocity of electron far from nucleus

Consider assumptions Lepton wavefunctions -- In some cases, the lowest order term possible in the expansion is not 1, but one of the higher order terms!  More complicated matrix element; impacts rate!  Additional momentum dependence to the differential rate spectrum; changes the spectrum shape!

Consider assumptions Lepton wavefunctions -- “Allowed term” “First forbidden term” “Second forbidden term” etc….

Consider assumptions Lepton wavefunctions -- Change in spectral shape from higher order terms  “Shape Factor” S(p e,p )

The decay rate Fermi function Shape correction Density of final states Nuclear matrix element