Wi→ f =2π!

φ f Vint φi2 dndE

e−,νe( )

Beta  decay:  spectrum  and  life:me    

wave function of the parent nucleus

product wave function of the daughter nucleus, electron, and antineutrino

!pe−,E

e−

!pν ,Eν

!pD,ED

!pD +!pe−+!pν = 0

TD +Te− +Tν =Q = E

dndE

e−,νe( ) = V 2

4π!6c3pe−2 E −E

e−( )21− mv

2c4

E −Ee−( )

2 dpe−

The  expression  for  the  density  of  final  states  of  en  electron  emiUed  with  a  given  energy  and  momentum  (integrated  over  all  angles)  is:  

Vint ≈ gδ!rn −!rp( )δ !rn −

!re−( )δ !rn − !rυ( )O(n→ p) zero-range  

Wi→ f pe−( )dpe− =

M fi' 2

2π 3!7c3F ZD, pe−( ) pe−2 E −E

e−( )21− mv

2c4

E −Ee−( )

2 dpe−

Depends on nuclear wave functions

F Z, pe−( ) = 2πη

1− e−2πη, η ≡ ±

Ze2

!ve−

Fermi function: positive (negative) sign used for β- (β+) decay

Deviations around the endpoint due to nonzero neutrino mass…

KATRIN  neutrino  experiment  hUp://www.katrin.kit.edu    

hUps://www.youtube.com/watch?v=dmmVb779NP4    

If  we  assume  that  the  matrix  element  does  not  depend  on  w,  and  amer  taking  out  the  strength  g  of  the  weak  interac:on,  one  obtains:    

fT = 0.693 2π 3!7

g2me5c4 M ' fi

2 , M ' fi ≡ gM ' fi

€

f ZD ,w0( ) = F ZD , w2 −1( ) w2 −1 w0 − w( )21

w0

∫ wdw f-function

€

w = Ee − /mec2where                                                            and  w0    is  the  reduced  max.  electron  energy.    

electrons  

positrons  

On  a  more  fundamental  level,  beta  decay  of  hadrons  can  be  viewed  as  the  transforma:on  of  one  type  of  quark  to  another  through  exchange  of  charged  weak  currents  (W  bosons  carry  net  charges;  Z  boson  is  neutral  -­‐  it  is  the  source  of  neutral  weak  current).  

n  

p   e- νe  _  

W-

e- ν  

e- ν  

Z0  

The  flavor  of  quarks  is  conserved  in  strong  interac:ons.  However,  weak  interac:ons  change  flavor!    

u→ d + e+ +νed→ u+ e− +νe

Beta decay: microscopic view

Three generations of matter

There  are  three  "sets"  of  quark  pairs  and  lepton  pairs.  Each  "set"  of  these  par:cles  is  called  a  genera:on,  or  family.  The  up/down  quarks  are  first  genera:on  quarks,  while  the  electron/electron  neutrino  leptons  are  first  genera:on  leptons.  

up  type  (q=+2/3e)  

down  type  (q=-­‐1/3e)  

neutral  

charged  (q=-­‐e)  

mass  

par:cles  of  higher  genera:ons  to  decay  to  the  first  genera:on  (everyday  maUer  is  made  of  par:cles  from  the  first  genera:on)    

http://phys.org/news/2012-12-particles-suffice-nature.html

12 matter particles suffice in nature?

The  result  of  the  sta:s:cal  analysis  is  that  the  existence  of  further  fermions  can  be  excluded  with  a  probability  of  99.99999  percent  (5.3  sigma).  The  most  important  data  used  for  this  analysis  come  from  the  recently  discovered  Higgs  par:cle.  

hUp://journals.aps.org/prl/abstract/10.1103/PhysRevLeU.109.241802    

When  a  quark  beta-­‐decays,  the  new  quark  does  not  have  a  definite  flavor.  For  instance  (Cabibbo  1963):   u→ d ' = d cosθc + ssinθc Cabibbo angle θc=13.02º

However,  the  observed  weak  transi:ons  are  between  quarks  of  definite  flavor.  In  general,  the  strong-­‐interac:on  quark  eigenstates:  

€

ud"

# $ $ %

& ' ' cs"

# $ $ %

& ' ' tb"

# $ $ %

& ' '

€

ud'"

# $ $ %

& ' ' cs'"

# $ $ %

& ' ' tb'"

# $ $ %

& ' '

are  different  from  weak  interac:on  eigenstates:    

A  quark  of  charge  +2/3  (u,c,t)  is  always  transformed  to  a  quark  of  charge  -­‐1/3  (d,s,b)  and  vice  versa.  This  is  because  the  transforma:on  proceeds  by  the  exchange  of  charged  W  bosons,  which  must  change  the  charge  by  one  unit.    

Beta decay: mass and weak eigenstates

d 's 'b '

!

"

###

$

%

&&&=

Uud Uus Uub

Ucd Ucs Ucb

Utd Uts Utb

!

"

####

$

%

&&&&

dsb

!

"

###

$

%

&&&

Cabibbo _Kobayashi-Maskawa (CKM) matrix

For  nuclear  beta-­‐decay,  we  are  mainly  concerned  with  the  transi:on  between  u-­‐  and  d-­‐quarks.  As  a  result,  only  the  product      enters  into  the  process.  

GV =GF cosθc

Kobayashi  and  Maskawa  generalized  the  Cabbibo  matrix  into  the  Cabibbo–Kobayashi–Maskawa  matrix  (or  CKM  matrix)  to  keep  track  of  the  weak  decays  of  three  genera:ons  of  quarks  

The current determination:

The  choice  of  usage  of  down-­‐type  quarks  in  the  defini:on  is  purely  arbitrary  and  does  not  represent  some  sort  of  deep  physical  asymmetry  between  up-­‐type  and  down-­‐type  quarks.    

Beta decay: CKM Matrix

weak eigenstates mass eigenstates