Lecture 7 Isospin

Isospin

H. A. Tanaka Announcements

• Problem Set 1 due today at 5 PM

• Box #7 in basement of McLennan

• Problem set 2 will be posted today PRESS RELEASE

6 October 2015 The Nobel Prize in Physics 2015 The Royal Swedish Academy of Sciences has decided to award the Nobel Prize in Physics for 2015 to

Takaaki Kajita Arthur B. McDonald Super-Kamiokande Collaboration Sudbury Neutrino Observatory Collaboration University of Tokyo, Kashiwa, Japan Queen’s University, Kingston, Canada

“for the discovery of neutrino oscillations, which shows that neutrinos have mass”

Metamorphosis in the particle world The Nobel Prize in Physics 2015 recognises Takaaki lenges for more than twenty years. However, as it requires Kajita in Japan and Arthur B. McDonald in Canada, neutrinos to be massless, the new observations had clearly for their key contributions to the experiments which showed that the Standard Model cannot be the complete demonstrated that neutrinos change identities. This theory of the fundamental constituents of the universe. metamorphosis requires that neutrinos have mass. The discovery rewarded with this year’s Nobel Prize The discovery has changed our understanding of the in Physics have yielded crucial insights into the all but innermost workings of matter and can prove crucial hidden world of neutrinos. After photons, the particles to our view of the universe. of light, neutrinos are the most numerous in the entire Around the turn of the millennium, Takaaki Kajita presented cosmos. The Earth is constantly bombarded by them. the discovery that neutrinos from the atmosphere switch Many neutrinos are created in reactions between cosmic between two identities on their way to the Super-Kamiokande radiation and the Earth’s atmosphere. Others are produced detector in Japan. in nuclear reactions inside the Sun. Thousands of billions of Meanwhile, the research group in Canada led by neutrinos are streaming through our bodies each second. Arthur B. McDonald could demonstrate that the neutrinos Hardly anything can stop them passing; neutrinos are from the Sun were not disappearing on their way to Earth. nature’s most elusive elementary particles. Instead they were captured with a different identity when Now the experiments continue and intense activity is arriving to the Sudbury Neutrino Observatory. underway worldwide in order to capture neutrinos and A neutrino puzzle that physicists had wrestled with for examine their properties. New discoveries about their deepest decades had been resolved. Compared to theoretical secrets are expected to change our current understanding of calculations of the number of neutrinos, up to two thirds the history, structure and future fate of the universe. of the neutrinos were missing in measurements performed on Earth. Now, the two experiments discovered that the Takaaki Kajita, Japanese citizen. Born 1959 in Higashimatsuyama, Japan. neutrinos had changed identities. Ph.D. 1986 from University of Tokyo, Japan. Director of Institute for Cosmic The discovery led to the far-reaching conclusion that Ray Research and Professor at University of Tokyo, Kashiwa, Japan. neutrinos, which for a long time were considered massless, www.icrr.u-tokyo.ac.jp/about/greeting_eng.html must have some mass, however small. Arthur B. McDonald, Canadian citizen. Born 1943 in Sydney, Canada. For particle physics this was a historic discovery. Its Standard Ph.D. 1969 from Californa Institute of Technology, Pasadena, CA, USA. Model of the innermost workings of matter had been Professor Emeritus at Queen’s University, Kingston, Canada.

Nobel Prize® is a registered trademark of the Nobel Foundation. trademark Nobel Prize® is a registered incredibly successful, having resisted all experimental chal- www.queensu.ca/physics/arthur-mcdonald

Prize amount: 8 million Swedish krona, to be shared equally between the laureates. Further information: http://kva.se and http://nobelprize.org Press contact: Hans Reuterskiöld, Press Ofﬁcer, Phone +46 8 673 95 44, +46 70 673 96 50, [email protected] Experts: Olga Botner, member of the Nobel Committee for Physics, +46 18 471 38 76, +46 73 390 86 50, [email protected] Lars Bergström, Secretary of the Nobel Committee for Physics, +46 8 553 787 25, [email protected]

The Royal Swedish Academy of Sciences, founded in 1739, is an independent organisation whose overall objective is to promote the sciences and strengthen their inﬂuence in society. The Academy takes special responsibility for the natural sciences and mathematics, but endeavours to promote the exchange of ideas between various disciplines.

BOX 50005, SE-104 05 STOCKHOLM, SWEDEN TEL +46 8 673 95 00, [email protected] HTTP://KVA.SE BESÖK/VISIT: LILLA FRESCATIVÄGEN 4A, SE-114 18 STOCKHOLM, SWEDEN Overview

• Assign isospin values to states • Understand basic principle of “isospin symmetry” and why it works • Translate isospin symmetry into consequences for scattering amplitudes • Use Clebsch-Gordan Tables • Using phenomenology, infer isospin assignment when not known a priori What is isospin?

• In 1932 (!), Heisenberg postulated that: • protons, neutrons are up/down states of isospin 1/2 system • pions are +1,0,-1 states of isospin 1 system • strong interactions are invariant under isospin rotations • Isospin is conserved in strong interactions

Heisenberg in 1933 Why does this work? (in hindsight)

• Strong interactions are the g g same for all quarks/antiquarks • Diﬀerent quarks have diﬀerent properties, however u u d d • charge (+2/3 vs. -1/3) 2 • mass (mt=170 GeV/c ) • Since d and u quarks have such 1 1 ¯ 1 1 u 2 , 2 d 2 , 2 similar masses, interchanging ⇥ | ⇤ ⇥| ⇤ them works okay d 1 , 1 u¯ 1 , 1 • Interchanging other quarks ⇥ | 2 2 ⇤ ⇥ | 2 2 ⇤ doesn’t work as well. Assigning Isospin to multi-particle states:

Particles Individual States Combined State

p + p 1 , 1 1 , 1 1, 1 | 2 2 | 2 2 | + + p 1, 1 1 , 1 3 , 3 | | 2 2 | 2 2 1, 1 1 , 1 3 , 3 + n | ⇥| 2 2 ⇥ | 2 2 ⇥ 1 , 1 1 , 1 1, 1 n + n | 2 2 ⇥| 2 2 ⇥ | ⇥

Easiest “highest weight” cases 35. Clebsch-Gordan coeﬃcients 1

35. CLEBSCH-GORDAN COEFFICIENTS, SPHERICAL HARMONICS, AND d FUNCTIONS !!. . . Notation: Note: A square-root sign is to be understood over every coefficient, e.g.,for 8/15 read 8/15. M M . . . − − m m !#$ ×!#$ ! ! 1 2 Assigning Isospin to multi-particle states II:+!! ' 0 3 "#$ Y1 = cos θ $ !#$ m m Coefficients !#$ !#$ ! ' ' 4π × +"#$ "#$ %#$ 1 2 + + " . . +!#$ −!#$ !#$ !#$ ! +$ +!#$ ! +%#$ +%#$ . . 1 3 iφ −!#$ +!#$ !#$ −!#$ −! Y = sin θ e +$+−!#$ !#" &#" "#$ %#$ . . 1 8π −!#$ −!#$ ! −" +!++!#$ &#" −!#" +!#$ +!#$ Particles Individual Combined State 5 3 1 +!−!#$ $#" %#" "#$ %#$ Y 0 = cos2 θ 2 4π 2 2 '+!#$ %#" −$#" −!#$ −!#$ !×!#$ %#$ " − 35. Clebsch-Gordan coefficients 1 1 1 1 1 1 +%#$ %#$ !#$ # $ ' −!#$ %#" $#" "#$ %#$ p + n ( 1, 0 + 0, 0 ) 15 −! +!#$ $#" −%#" −%#$ −%#$ , , 2 +! +!#$ ! +!#$ +!#$ Y 1 = sin θ cos θ eiφ 2 2 2 2 | | 2 $ +! −!#$ !#% $#% %#$ !#$ − 8π −! −!#$ &#" !#" "#$ | ⇥| ⇥ " %#$ ×!#$ +$ $ ! −$ +!#$ !#" −&#" −"#$ ' +!#$ $#% −!#% −!#$ −!#$ 1 15 35. CLEBSCH-GORDANY 2 = sin2 θ e2iφ COEFFICIENTS,+%#$ *!#$ ! +! +! −$! SPHERICAL−!#$ HARMONICS, + 1 1 1 3 1 2 1 1 ' −!#$ $#%!#% %#$ 2 4 2π 1, 1 , , + , −! +!#$ !#% −$#% −%#$ " +%#$ −!#$ !#&%#& $ ! + n 2 2 3 2 2 3 2 2 +!#$ +!#$ %#& −!#& ' ' | | $ ! % −! −!#$ ! "#$ | ⇥| ⇥ × AND d+!#$FUNCTIONS−!#$ !#$ !#$ $ ! +% % $ %#$ ×! +"#$ "#$ %#$ +$ +! ! +$ +$ −!#$ +!#$ !#$ −!#$ −! −! !!. . . 1 1 2 3 1 1 1 1 +%#$ +! ! +%#$ +%#$ 0 +$ ' !#% $#% % $ ! !#$ !#$ %#& !#& $ Notation: + p 1, 0 , 3 2 , 2 2 , 2 Note: A square-root sign is to be understood+%#$ ' $#" over%#"every"#$ %#$coeffi !#$cient, e.g.,for− − 8/15 read 8/15. M M . . . 2 2 3 +! +! $#% −!#% +! +! +! −%#$ +!#$ !#&−%#& −$ | | | ⇥ | ⇥ +!#$ +! %#" −$#" +!#$ +!#$ +!#$ − − +!$ −! !#!" !#% %#" +%#$ −! !#!' $#" !#$ −%#$−!#$ ! ! m 1 m 2 !×!#$! $×!#$ +! ' )#!" !#( −%#!' % $ ! +!#$ ' 3%#"!#!" −!#% "#$ %#$ !#$ 0 1 1 2 3 1 1 1 1 +$$ ! +!!' +! $#" −'!#$ !#!' ' ' ' 0 −!#$ +! %#!' −)#!" !#( −!#$ −!#$ −!#$ "#$ 1, 0 , , + , +! +! ! +! +! Y1 = cos θ $×!#$ m 1 m 2 Coefficients + n 2 2 3 2 2 3 2 2 +!#$ +!#$ ! ' ' +! −! !#" !#$ %#!' 4π +!#$ −! %#!' )#!" !#( +"#$ "#$ %#$ | ⇥ | ⇥ +! ' !#$ !#$ $ ! ' ''%#" ' −$#" % $ " ! . . | ⇥| ⇥ +!#$ −!#$ !#$ !#$ ! −!#$ ' %#" −!#!"+$ +−!#$!#% "#$ !%#$+%#$ +%#$ ' +! !#$ −!#$ ' ' ' −! +! !#" −!#$ %#!' −! −! −! 3 −%#$ +! !#!' −$#" !#$ −%#$ −%#$ . . −!#$ +!#$ !#$ −!#$ −! 1 iφ 1 1 1 3 1 2 1 1 +! −! !#( !#$ !#% ' −! $#"Y1!#$=!#!' sin θ e −!#$ −+!$+−%#"!#$$#" !#""#$ &#" "#$ %#$ . . + p , , − 8π −%#$ ' $#" −%#" −"#$ 1, 1 , 3 2 2 3 2 2 ' ' $#% −!#$'$−!#%−!#$ !! −! ' )#!" −!#(−%#!'" % $ +!++!#$ &#" −!#" +!#$ +!#$ 2 2 | ⇥ | ⇥ −! +! !#( −!#$ !#% −! −! −$ +! !#!" −!#% %#" −$ −$ | ⇥| ⇥ 5 3 1 −%#$ −! +!!−!#$ $#" %#" "#$ %#$ ' −! !#$ !#$ $ Y 0 =−! −! $#% !#%cos% 2 θ m m m 2 −$ ' !#% −$#% −% '+!#$ %#" −$#" −!#$ −!#$ 3 1 3 1 Yℓ− =( 1) Yℓ ∗ %#$−! ' !#$ −!#$ −$ 4π 2 − 2 j1j2m1m2 j1j2JM + ! −!#$ 4π ⟨ | ⟩ , , × −! −! ! ℓ m im"φ −$ −! ! J j1 j2 ' −!#$ %#" $#" "#$ %#$ 2 2 2 2 +%#$ %#$ !#$ d m,0 = Y e− # $=( 1) − − j2j1m2m1 j2j1JM 2ℓ +1 ℓ 15 − ⟨ | −⟩! +!#$ $#" −%#" −%#$ −%#$ | | +! +!#$ ! +!#$ +!#$ " Y 1 = sin θ cos θ eiφ j j j 2% $ d =( 1)m m′ d = d %#$ ×%#$ − 8π 1/2 θ 1+cosθ −! −!#$ &#" !#" "#$ m ,m −+! −m,m!#$ !#%m, $#%m %#$ !#$ +% % "$ d 1 =cosθ d =cos d 1 = + ′ − ′ ′ 0,0 1/2%#$,1/2 ×!#$ +$ 1,1 $ ! −$ +!#$ !#" −&#" −"#$ What are the amplitudes for decay? ' +!#$ $#%− −−!#% −!#$ −!#$ +%#$ +%#$ ! +$ 1 +$15 2 2 Δ $×%#$ ,#$ +%#$ +!#$2 !#$ !#$ %2 $ 2!iφ 1/2 +%#$ *!#$θ ! +! sin+!θ +,#$ ,#$ "#$ Y = sin θ e 1 −$!−!#$ ' −!#$ $#%!#% %#$+!#$ +%#$2 !#$ −!#$ +! +! +! d 1/2, 1/2 = sin d 1,0 = +$+%#$ ! +"#$ +"#$ 4 2π − −+%#$2 −!#$ !#&−%#&√2 $ ! −! +!#$ !#% −$#% −%#$ +%#$ −"!#$ !#" !#$ %#!' +$+!#$ %#, &#, ,#$ "#$ %#$ +!#$ +!#$ %#& −!#&1 cos'θ ' % +!#$ +!#$ %#" ' −$#" % $!' d 1 = − $×!+!+%#$ &#, −%#, +%#$ −+!%#$−!#$+%#$ ! −!#$ +%#$ !#""#$−!#$ %#!' ' ' ' ' 1, 1 +% % $ %#$ ×! +!#$− −!#$2!#$ !#$ $ ! +$ −!#$ !#, !(#%" $#" +"#$+%#$ − "#$%#$ !#$'%#$!#& -#$' !#& $ ! ! $+! *!#$$ &#, !#%" −$#" ,#$ "#$ %#$ !#$ −!#$ +!#$ !#$ −!#$ −! −! & + + + + +%#$ +! +!#$! +−%#$!#$ -#$'+%#$!#& −!#$'−!#& ' *%#$ $#, −!)#%" !#" +!#$ +!#$ +!#$ +!#$ !#$ !#$ -#$' −!#& −!#$' !#& % $ ! $×$ +& & % − + +$ ' !#% $#% % $ ! −%#$ +%#$ !#$' −!#& -#$'−!#& −! −! −! −!#$ −!#$ %#& !#& $ +$ +$ ! +% +% +$ −%#$ !#%" (#%" $#" $#"+%#$ ' $#" %#" "#$ %#$ !#$ +! +! $#% −!#% +! +!+−!!#$ !$#%"+! "#!& ' −%#!' +!#$ −%#$ !#" !#$ %#!' −%#$ +!#$ !#&−%#& −$ +$ +! !#$ !#$ & %$ ' *!#$ !)#%" −%#%" −!#" !#"+!#$,#$+! "#$%#"%#$−$#"!#$ +!#$ +!#$ +!#$ +! +$ !#$ −!#$ +$ +$ +$ −!#$ −!#$ %#" ' −$#" % $ +$ −! !#!" −!#%! *%#$ %#"&#%"−$,#,' $#" −!#!' −!#$ −!#$ −+!#$%#$−−!#$! !#!'−%#$ +!#$$#"!#"!#$−!#$ %#!' −$ −$ −%#$−!#$ ! +$$ ' %#!& !#$ $#, +! ' )#!" !#( −%#!' % +$! −%#$ !&#%" $,#,' +$#"!#$!#!'' %#"!#!" −!#%−!#$ −%#$ "#$!#$ !#$ %#$% !#$ ! ! +! *! &#, ' −%#, & % $ ! ' −!#$ !)#%" %#%" −!#" −!#" × +$$ ! %#$ −!#$ !#$−!#$ % ' *$ %#!& −!#$ $#,' ++!! $#"+! −+!#$! +!!#!' ' −'! *!#$ '!$#%" −"#!& −!#$' %#!'+! %#!',#$ −"#$)#!"%#$ !#(− −!#$ −!#$− −!#$ +! +! ! +! +! −$ *%#$ !#%" −(#%" $#" −$#" −%#$ −%#$−%#$ −%#$−%#$ ! +$ −! !#!& %#!' %#, !#"+! −! !#" !#$ %#!' ! +! ' %#, !#" −!#!&−%#!' +!#$ − %#!' )#!" !#( ! ' !#$ !#$ $ ! ' ' −%#$ $#, !)#%" !#" + ' *! %#, −!#" −!#!& %#!'''&%$%#" ' −$#"!'% $−! −!#$! &#, −!#%"−−$#"!#$,#$' "#$%#" −!#!" −!#% "#$ %#$ −! *$ !#!& −%#!' %#, −!#" ' ' ''' ' +! !#$ −!#$ ' ' ' −! +! !#" −!#$ %#!' −! −!−$ +!#$−! !#,−!(#%" −$#"%#$−"#$+! −!#!'"#$ −$#" !#$ −%#$ −%#$ +$ −$ !#,' !#!' $#, $#" !#" −! −%#$ &#, %#, ,#$ !#$ ! %#" $#" "#$ +! −! !#( !#$ !#% +! −! )#%" $#" !#!& −!#!'' −!−!#"$#" !#$ !#!' −$ −!#$ %#, −&#, −,#$ − − ' ' !)#%" ' −$#, −! ' '!#")#!" −!#(−%#!' % $ −%#$ ' $#" −%#" −"#$ ' ' $#% '$−!#% −! *! ! )#%" −$#" !#!&!#!' −!#" & % $ ! −$ −%#$ ! 3/2 1+cosθ−! +θ! !#( −!#$ !#% −!−$ *$−! !#,' −!#!' $#, −−$$#"+!!#"!#!"−!−!#%−! %#"−! −−!$ −$ −%#$ −! ! d 3/2,3/2 = cos 2 2 ' ! !#$ !#$ $ +! −$ !#!& %#!' !%#,! !#"$#% !#% % − 1+cosθ 2 ' −! %#, !#" −−!#!&− −%#!' 3/2 m 1+cosmθ mθ 2 ! ' %#, !#" !#!& %#!' & % $ d Y −= √=(3 1) Ysin ∗ d 2,−2!= ' !#$ −!#$ −$ − − −− $ ' !#% −$#% −% j1j2m1m2 j1j2JM 3/2,1/ℓ2 − −2 ℓ 2 2 −$ *! !#!& −%#!' %#, −!#" −$ −$ −$ ⟨ | ⟩ # −! $−! ! ℓ 4π m imφ −$ −! ! J j j 3/2 1 cos θ θ 2 1+cosθ d = Y e ' −$ %#!& !#$ $#, =( 1) 1 2 j j m m j j JM d = √3 − cos d 2,1 = sin θ m,0 ℓ − − − 2 1 2 1 2 1 3/2, 1/2 2 2 − 2 2ℓ +1 −! −! &#, '&−%#, % − ⟨ | ⟩ − 1+cos" θ −$ ' %#!& −!#$ $#, −% −% 3/2 1 cos θ θ √6 2 2 2 d = (2 cos θ 1) −! −$ !#$ !#$ & d j = − msin m j d 2,0 = j sin θ 1,1 2 % 3/2d, 3/2 −=(2 1) 2 ′ d = d4 %#$ ×%#$ − −$ −! !#$ −!#$ −& 1/2 θ 1+cosθ −m ,m − m,m m, m +% % $ d 1 =cosθ d =cos d 1 = 3/2 ′ 3cosθ −1 θ ′ 1 cos θ′ 2 3 0,0 −$ −$! 1/2,1/2 1,1 d = − cos d 2 = − − − sin θ d = +%#$sin+θ%#$cos θ ! +$ +$ 2 2 1/2,1/2 2, 1 1,0 − 2 2 2 ,#$− − 2 " $×%#$ 2 +%#$ +!#$!#$ !#$ % $ ! 1/2 θ 1 sin θ 3/2 3cosθ +1 θ +,#$2 ,#$1 "#$cos θ 2 1 cos θ 2 3 2 1 d = sin d = d 1/2, 1/2 = sin d 2, 2 = − d 1, 1 = − +!#$(2 cos+θ%#$+1)!#$ d−0!#$,0 = +!cos θ+! +! 1/2, 1/2 1,0 − − 2 +$+%#$2 !− +"#$ +"#$2 − 2 2 − 2 − − 2 − √2 # $ +%#$ −!#$ #!#" !#$ %#!'$ +$+!#$ %#, &#, ,#$ "#$ %#$ 1 cos θ +!#$ +!#$ %#" ' −$#" % $!' d 1 = +!+%#$ &#, −%#, +%#$ +%#$ +%#$ !#$ %#$ !#" !#$ %#!' ' ' ' ' 1, 1 − Figure 35.1: The sign convention is that of Wigner (Group Theory, Academic Press, New York,− 1959),+ also used− by Condon and Shortley (The − 2 Theory of Atomic Spectra, Cambridge Univ.+$ Press,−!#$ New!#, York,!(#%" 1953), Rose$#" (Elementary Theory of Angular Momentum+%#$, Wiley,−%#$ New!#$' York,!#& 1957),-#$' !#& and Cohen (Tables of the Clebsch-Gordan Coe+!ffi*!#$cients,&#, North American!#%" −$#" Rockwell ,#$ Science"#$ Center, %#$ Thousand !#$ Oaks, Calif., 1974). The coefficients & +!#$ −!#$ -#$' !#& −!#$'−!#& here have been calculated using computer programs' *%#$ written$#, − independently!)#%" !#" by+ Cohen!#$ and+!#$ at+ LBNL.!#$ +!#$ !#$ !#$ -#$' −!#& −!#$' !#& % $ ! $×$ +& & % − + −%#$ +%#$ !#$' −!#& -#$'−!#& −! −! −! +$ +$ ! +% +% +$ −%#$ !#%" (#%" $#" $#" +! −!#$ !$#%" "#!& ' −%#!' +!#$ −%#$ !#" !#$ %#!' +$ +! !#$ !#$ & %$ ' *!#$ !)#%" −%#%" −!#" !#" ,#$ "#$ %#$ !#$ +! +$ !#$ −!#$ +$ +$ +$ −!#$ −!#$ %#" ' −$#" % $ −! *%#$ &#%"−$,#,' $#" −!#!' −!#$ −!#$ −!#$ −!#$ −%#$ +!#$ !#" −!#$ %#!' −$ −$ +$ ' %#!& !#$ $#, +! −%#$ &#%" $,#,' $#" !#!' −!#$ −%#$ !#$ !#$ % +! *! &#, ' −%#, & % $ ! ' −!#$ !)#%" %#%" −!#" −!#" %#$ −!#$ !#$−!#$ % ' *$ %#!& −!#$ $#, +! +! +! +! −! *!#$ !$#%" −"#!& ' %#!' ,#$ "#$ %#$ − − +$ −! !#!& %#!' %#, !#" −$ *%#$ !#%" −(#%" $#" −$#" −%#$ −%#$−%#$ −%#$−%#$ ! +! ' %#, !#" −!#!&−%#!' ' −%#$ $#, !)#%" !#" ' *! %#, −!#" −!#!& %#!' &%$!' −! −!#$ &#, −!#%"−$#" ,#$ "#$ −! *$ !#!& −%#!' %#, −!#" ' ' ''' −$ +!#$ !#,−!(#%" $#" −"#$ −"#$ +$ −$ !#,' !#!' $#, $#" !#" −! −%#$ &#, %#, ,#$ +! −! )#%" $#" !#!& −!#!' −!#" −$ −!#$ %#, −&#, −,#$ ' ' !)#%" ' −$#, ' !#" −! *! )#%" −$#" !#!&!#!' −!#" & % $ ! −$ −%#$ ! 1+cosθ θ −$ *$ !#,' −!#!' $#, −$#" !#" −! −! −! −! d 3/2 = cos 3/2,3/2 2 2 +! −$ !#!& %#!' %#, !#" 1+cosθ 2 ' −! %#, !#" −!#!& −%#!' 3/2 1+cosθ θ 2 ! ' %#, !#" !#!& %#!' & % $ d = √3 sin d 2,2 = − − − 3/2,1/2 − 2 2 2 −$ *! !#!& −%#!' %#, −!#" −$ −$ −$ # $ 3/2 1 cos θ θ 2 1+cosθ ' −$ %#!& !#$ $#, d = √3 − cos d 2,1 = sin θ 3/2, 1/2 2 2 − 2 −! −! &#, '&−%#, % − 1+cosθ −$ ' %#!& −!#$ $#, −% −% 3/2 1 cos θ θ √6 2 2 2 d 1,1 = (2 cos θ 1) −! −$ !#$ !#$ & d 3/2, 3/2 = − sin d 2,0 = sin θ 2 − − − 2 2 4 −$ −! !#$ −!#$ −& 3/2 3cosθ 1 θ 1 cos θ 2 3 −$ −$! d = − cos d 2 = − sin θ d = sin θ cos θ 1/2,1/2 2, 1 1,0 2 2 2 − − 2 −" 2 3/2 3cosθ +1 θ 2 1 cos θ 2 1 cos θ 2 3 2 1 d 1/2, 1/2 = sin d 2, 2 = − d 1, 1 = − (2 cos θ +1) d 0,0 = cos θ − − 2 2 − 2 − 2 2 − 2 # $ # $

Figure 35.1: The sign convention is that of Wigner (Group Theory, Academic Press, New York, 1959), also used by Condon and Shortley (The Theory of Atomic Spectra, Cambridge Univ. Press, New York, 1953), Rose (Elementary Theory of Angular Momentum, Wiley, New York, 1957), and Cohen (Tables of the Clebsch-Gordan Coeﬃcients, North American Rockwell Science Center, Thousand Oaks, Calif., 1974). The coeﬃcients here have been calculated using computer programs written independently by Cohen and at LBNL. Scattering:

• Scattering is a general concept of “something in”, “something out” S A + B + C + D + E . . . . → W + X + Y + Z . . .

• What can we say about what “S” will do? • Conservation laws: Historically, had no idea what a • Energy “pion” is, for example. • Momentum • Angular Momentum What conservation/symmetry • ? rules apply? • Isospin: perhaps it is conserved in strong interactions. • We can then say (more about) what can happen and what can’t happen. • Complication: initial state can have more than one isospin value Amplitudes and Cross Sections:

• In Quantum Mechanics, the amplitude for a transition A→B via some scattering process S is given by the product: A S B • The probability for the transition | is| given⇥ by the absolute magnitude squared of the amplitude:

2 P = A S B A S B = A S B B S† A = A S B ⇥ | | ⇤⇥ | | ⇤ ⇥ | | ⇤⇥ | | ⇤ |⇥ | | ⇤|

• This is related to the “cross section”: in general, the cross section carries an additional “phase space” parameter associated with Fermi’s Golden Rule. For now, we can “equate” cross section with probability.

• Isospin symmetry is a statement that whatever transitions are eﬀected by S, total isospin (total and component) is conserved. Bound state of two nucleons

• We have four possible way for two nucleons to bound • How does this work? 1 1 =1 0 2 ⇥ 2 2 2=31 ⇥ • “two isospin 1/2 objects combine to form an isospin 1 (“isotriplet”) and an isospin 0 (“isosinglet”). • What is the isotriplet? 1 , 1 1 , 1 1, 1 | 2 2 ⇤| 2 2 ⇤ ⇥ | ⇤ 1 1 , 1 1 , 1 + 1 , 1 1 , 1 1, 0 2 | 2 2 ⇤| 2 2 ⇤ | 2 2 ⇤| 2 2 ⇤ ⇥ | ⇤ ⇥ 1 , 1 1 , 1 1, 1 | 2 2 ⇤| 2 2 ⇤ ⇥ | ⇤ • The isosinglet? 1 1 , 1 1 , 1 1 , 1 1 , 1 0, 0 2 | 2 2 ⇤| 2 2 ⇤| 2 2 ⇤| 2 2 ⇤ ⇥ | ⇤ ⇥ nucleon-nucleon scattering

1 , 1 1 , 1 0, 0 1, 1 p + p d + + isospin | 2 2 ⇤| 2 2 ⇤⇥| ⇤| ⇤ 0 1 1 1 1 p + n d + 2 , 2 2 , 2 0, 0 1, 0 | ⇤| ⇤⇥| ⇤| ⇤ n + n d + 1 , 1 1 , 1 0, 0 1, 1 | 2 2 ⇤| 2 2 ⇤⇥| ⇤| ⇤

1, 1 1, 1 1, 1 S 1, 1 A 35. Clebsch-Gordan coeﬃcients 1 | ⇤ ⇥ | ⇤ ⇤ | | ⌅ ⇥ 1 [ 1, 0 +35.0, CLEBSCH-GORDAN0 ] 1, 0 1 [ 1, COEFFICIENTS,0 + 0, 0 ] S 1, 0 SPHERICAL1 A HARMONICS, 2 | ⇤ | ⇤ ⇥ | ⇤ 2 ⇤ | ⇤ | | ⌅⇥ 2 AND d FUNCTIONS !!. . . 1, 1 1, 1 1, 1 S 1, 1 A Notation: | Note: ⇤ A square-root sign⇥ is to| be understood⇤ ⇤ overevery| |coeﬃcient,⌅ e.g.,for 8/⇥15 read 8/15. M M . . . − − m m !#$ ×!#$ ! ! 1 2 +!! ' 0 3 "#$ • Amplitudes:Y1 = cos 1:1/θ √2:1 $ !#$ m m Coefficients !#$ !#$ ! ' ' 4π × +"#$ "#$ %#$ 1 2 + + " . . +!#$ −!#$ !#$ !#$ ! +$ +!#$ ! +%#$ +%#$ . . • Probability/Cross1 3 section:iφ 1:1/2:1 −!#$ +!#$ !#$ −!#$ −! Y = sin θ e +$+−!#$ !#" &#" "#$ %#$ . . 1 8π −!#$ −!#$ ! −" +!++!#$ &#" −!#" +!#$ +!#$ 5 3 1 +!−!#$ $#" %#" "#$ %#$ Y 0 = cos2 θ 2 4π 2 2 '+!#$ %#" −$#" −!#$ −!#$ !×!#$ %#$ " − +%#$ %#$ !#$ # $ ' −!#$ %#" $#" "#$ %#$ 15 −! +!#$ $#" −%#" −%#$ −%#$ +! +!#$ ! +!#$ +!#$ Y 1 = sin θ cos θ eiφ 2 $ +! −!#$ !#% $#% %#$ !#$ − 8π −! −!#$ &#" !#" "#$ " %#$ ×!#$ +$ $ ! −$ +!#$ !#" −&#" −"#$ ' +!#$ $#% −!#% −!#$ −!#$ 1 15 Y 2 = sin2 θ e2iφ +%#$ *!#$ ! +! +! −$!−!#$ ' −!#$ $#%!#% %#$ 2 4 2π −! +!#$ !#% −$#% −%#$ " +%#$ −!#$ !#&%#& $ ! +!#$ +!#$ %#& −!#& ' ' $ ! % −! −!#$ ! "#$ × +!#$ −!#$ !#$ !#$ $ ! +% % $ %#$ ×! +"#$ "#$ %#$ +$ +! ! +$ +$ +%#$ +! ! +%#$ +%#$ −!#$ +!#$ !#$ −!#$ −! −! +$ ' !#% $#% % $ ! +%#$ ' $#" %#" "#$ %#$ !#$ −!#$ −!#$ %#& !#& $ +! +! $#% −!#% +! +! +! +!#$ +! %#" −$#" +!#$ +!#$ +!#$ −%#$ +!#$ !#&−%#& −$ +$ −! !#!" !#% %#" +%#$ −! !#!' $#" !#$ −%#$−!#$ ! !×! $ +! ' )#!" !#( −%#!' % $ ! +!#$ ' %#"!#!" −!#% "#$ %#$ !#$ +$$ ! ' +! $#" −!#$ !#!' ' ' ' −!#$ +! %#!' −)#!" !#( −!#$ −!#$ −!#$ +! +! ! +! +! +! −! !#" !#$ %#!' +!#$ −! %#!' )#!" !#( +! ' !#$ !#$ $ ! ' ''%#" ' −$#" % $ ! −!#$ ' %#" −!#!" −!#% "#$ %#$ ' +! !#$ −!#$ ' ' ' −! +! !#" −!#$ %#!' −! −! −! −%#$ +! !#!' −$#" !#$ −%#$ −%#$ +! −! !#( !#$ !#% ' −! $#" !#$ !#!' −!#$ −! %#" $#" "#$ ' ' $#% '$−!#% ! −! ' )#!" −!#(−%#!' % $ −%#$ ' $#" −%#" −"#$ −! +! !#( −!#$ !#% −! −! −$ +! !#!" −!#% %#" −$ −$ −%#$ −! ! ' −! !#$ !#$ $ −! −! $#% !#% % m m m Y − =( 1) Y −! ' !#$ −!#$ −$ −$ ' !#% −$#% −% j1j2m1m2 j1j2JM ℓ − ℓ ∗ ⟨ | ⟩ −! −! ! ℓ 4π m imφ −$ −! ! J j j d = Y e =( 1) 1 2 j2j1m2m1 j2j1JM m,0 2ℓ +1 ℓ − − − − ⟨ | ⟩ " % j m m′ j j %#$ ×%#$ θ 1+cosθ d =( 1) − d = d +% 1 1/2 1 m′,m − m,m′ m, m′ % $ d 0,0 =cosθ d =cos d 1,1 = − − +%#$ +%#$ ! +$ +$ 1/2,1/2 2 2 $×%#$ ,#$ +%#$ +!#$!#$ !#$ % $ ! 1/2 θ sin θ +,#$ ,#$ "#$ 1 +!#$ +%#$ !#$ −!#$ +! +! +! d 1/2, 1/2 = sin d 1,0 = +$+%#$ ! +"#$ +"#$ − − 2 − √2 +%#$ −!#$ !#" !#$ %#!' +$+!#$ %#, &#, ,#$ "#$ %#$ 1 cos θ +!#$ +!#$ %#" ' −$#" % $!' d 1 = +!+%#$ &#, −%#, +%#$ +%#$ +%#$ !#$ %#$ !#" !#$ %#!' ' ' ' ' 1, 1 − − + − − 2 +$ −!#$ !#, !(#%" $#" +%#$ −%#$ !#$' !#& -#$' !#& +! *!#$ &#, !#%" −$#" ,#$ "#$ %#$ !#$ & +!#$ −!#$ -#$' !#& −!#$'−!#& ' *%#$ $#, −!)#%" !#" +!#$ +!#$ +!#$ +!#$ !#$ !#$ -#$' −!#& −!#$' !#& % $ ! $×$ +& & % − + −%#$ +%#$ !#$' −!#& -#$'−!#& −! −! −! +$ +$ ! +% +% +$ −%#$ !#%" (#%" $#" $#" +! −!#$ !$#%" "#!& ' −%#!' +!#$ −%#$ !#" !#$ %#!' +$ +! !#$ !#$ & %$ ' *!#$ !)#%" −%#%" −!#" !#" ,#$ "#$ %#$ !#$ +! +$ !#$ −!#$ +$ +$ +$ −!#$ −!#$ %#" ' −$#" % $ −! *%#$ &#%"−$,#,' $#" −!#!' −!#$ −!#$ −!#$ −!#$ −%#$ +!#$ !#" −!#$ %#!' −$ −$ +$ ' %#!& !#$ $#, +! −%#$ &#%" $,#,' $#" !#!' −!#$ −%#$ !#$ !#$ % +! *! &#, ' −%#, & % $ ! ' −!#$ !)#%" %#%" −!#" −!#" %#$ −!#$ !#$−!#$ % ' *$ %#!& −!#$ $#, +! +! +! +! −! *!#$ !$#%" −"#!& ' %#!' ,#$ "#$ %#$ − − +$ −! !#!& %#!' %#, !#" −$ *%#$ !#%" −(#%" $#" −$#" −%#$ −%#$−%#$ −%#$−%#$ ! +! ' %#, !#" −!#!&−%#!' ' −%#$ $#, !)#%" !#" ' *! %#, −!#" −!#!& %#!' &%$!' −! −!#$ &#, −!#%"−$#" ,#$ "#$ −! *$ !#!& −%#!' %#, −!#" ' ' ''' −$ +!#$ !#,−!(#%" $#" −"#$ −"#$ +$ −$ !#,' !#!' $#, $#" !#" −! −%#$ &#, %#, ,#$ +! −! )#%" $#" !#!& −!#!' −!#" −$ −!#$ %#, −&#, −,#$ ' ' !)#%" ' −$#, ' !#" −! *! )#%" −$#" !#!&!#!' −!#" & % $ ! −$ −%#$ ! 1+cosθ θ −$ *$ !#,' −!#!' $#, −$#" !#" −! −! −! −! d 3/2 = cos 3/2,3/2 2 2 +! −$ !#!& %#!' %#, !#" 1+cosθ 2 ' −! %#, !#" −!#!& −%#!' 3/2 1+cosθ θ 2 ! ' %#, !#" !#!& %#!' & % $ d = √3 sin d 2,2 = − − − 3/2,1/2 − 2 2 2 −$ *! !#!& −%#!' %#, −!#" −$ −$ −$ # $ 3/2 1 cos θ θ 2 1+cosθ ' −$ %#!& !#$ $#, d = √3 − cos d 2,1 = sin θ 3/2, 1/2 2 2 − 2 −! −! &#, '&−%#, % − 1+cosθ −$ ' %#!& −!#$ $#, −% −% 3/2 1 cos θ θ √6 2 2 2 d 1,1 = (2 cos θ 1) −! −$ !#$ !#$ & d 3/2, 3/2 = − sin d 2,0 = sin θ 2 − − − 2 2 4 −$ −! !#$ −!#$ −& 3/2 3cosθ 1 θ 1 cos θ 2 3 −$ −$! d = − cos d 2 = − sin θ d = sin θ cos θ 1/2,1/2 2, 1 1,0 2 2 2 − − 2 −" 2 3/2 3cosθ +1 θ 2 1 cos θ 2 1 cos θ 2 3 2 1 d 1/2, 1/2 = sin d 2, 2 = − d 1, 1 = − (2 cos θ +1) d 0,0 = cos θ − − 2 2 − 2 − 2 2 − 2 # $ # $

+ 0 1, 1 1 , 1 1, 0 135., 1 CLEBSCH-GORDAN COEFFICIENTS, SPHERICAL HARMONICS, + n + p | ⇤| 2 2 ⇤⇥| ⇤| 2 2 ⇤ AND d FUNCTIONS 0 + 1 1 1 1 !!. . . + p + n 1, 0 , 1, 1 , Notation: 2 2 Note: A square-root2 2 sign is to be understood over every coeﬃcient, e.g.,for 8/15 read 8/15. M M . . . | ⇤| ⇤⇥| ⇤| ⇤ − − 0 m m + n + p 1 1 !#$ ×!#$ !1 1 ! 1 2 1, 0 , 1, 1 +!!, ' 0 3 "#$ 2 2 2 2 Y1 = cos θ $ !#$ m m Coefficients 0 | ⇤| ⇤⇥|!#$!#$⇤| ! '⇤ ' 4π × +"#$ "#$ %#$ 1 2 + + " . . + p + n +!#$ −!#$ !#$ !#$ ! +$ +!#$ ! +%#$ +%#$ . . 1 1 1 1 1 3 iφ 1, 1 , 1−,!#$0 +!#$, !#$ −!#$ −! Y = sin θ e +$+−!#$ !#" &#" "#$ %#$ . . 2 2 2 2 1 8π | ⇤| ⇤⇥| ⇤| −!#$⇤−!#$ ! −" +!++!#$ &#" −!#" +!#$ +!#$ 5 3 1 +!−!#$ $#" %#" "#$ %#$ Y 0 = cos2 θ + 0 2 4π 2 2 '+!#$ %#" −$#" −!#$ −!#$ !×!#$ %#$ " − + n + p +%#$ %#$ !#$ # $ ' −!#$ %#" $#" "#$ %#$ 15 −! +!#$ $#" −%#" −%#$ −%#$ +! +!#$ ! +!#$ +!#$ Y 1 = sin θ cos θ eiφ 2 $ 1 2 2 1 +! −!#$ !#% $#% %#$ !#$ − 8π −! −!#$ &#" !#" "#$ 3/2, 1/2 + 1/2, 1/2 3/2, 1/2 1/2, 1/2 " %#$ ×!#$ +$ $ ! −$ +!#$ !#" −&#" −"#$ 3 3 3 3 ' +!#$ $#% −!#% −!#$ −!#$ 1 15 | ⇤ | ⇤⇥ | ⇤ | ⇤ Y 2 = sin2 θ e2iφ +%#$ *!#$ ! +! +! −$!−!#$ ' −!#$ $#%!#% %#$ 2 4 2π −! +!#$ !#% −$#% −%#$ " +%#$ −!#$ !#&%#& $ ! 1 2 2 1 +!#$ +!#$ %#& −!#& ' ' $×! % −! −!#$ ! "#$ A = 3/2, 1/2 + 1/2, 1/2 S 3/2, 1/2 1/2, 1/2 +% % $ %#$ ×! +!#$ −!#$ !#$ !#$ $ ! 3⇥ | 3⇥ | 3| ⇤ 3| ⇤ +"#$ "#$ %#$ ⇤ ⇤ ⇥ ⇤ ⇤ ⇥ +$ +! ! +$ +$ +%#$ +! ! +%#$ +%#$ −!#$ +!#$ !#$ −!#$ −! −! +$ ' !#% $#% % $ ! +%#$ ' $#" %#" "#$ %#$ !#$ −!#$ −!#$ %#& !#& $ +! +! $#% −!#% +! +! +! +!#$ +! %#" −$#" +!#$ +!#$ +!#$ −%#$ +!#$ !#&−%#& −$ M3/2 = 3/2, 1/2 S 3/2, 1/2 M1/2 = 1/2, 1/2 S 1/2, 1/2 +$ −! !#!" !#% %#" +%#$ −! !#!' $#" !#$ −%#$−!#$ ! !×! $ +! ' )#!" !#( −%#!' % $ ! +!#$ ' %#"!#!" −!#% "#$ %#$ !#$ | | ⇥ | | +$$⇥ ! ' +! $#" −!#$ !#!' ' ' ' −!#$ +! %#!' −)#!" !#( −!#$ −!#$ −!#$ +! +! ! +! +! +! −! !#" !#$ %#!' +!#$ −! %#!' )#!" !#( ⇥2 ⇥2 • Amplitude expressed+! ' !#$ in !#$terms $ !of ' ''%#" ' −$#" % $ ! −!#$ ' %#" −!#!" −!#% "#$ %#$ A = M3/2 M1/2 two “underlying” transitions.' +! !#$ −!#$ ' ' ' −! +! !#" −!#$ %#!' −! −! −! −%#$ +! !#!' −$#" !#$ −%#$ −%#$ 3 3 +! −! !#( !#$ !#% ' −! $#" !#$ !#!' −!#$ −! %#" $#" "#$ ' ' $#% '$−!#% ! −! ' )#!" −!#(−%#!' % $ −%#$ ' $#" −%#" −"#$ −! +! !#( −!#$ !#% −! −! −$ +! !#!" −!#% %#" −$ −$ −%#$ −! ! ' −! !#$ !#$ $ −! −! $#% !#% % m m m Y − =( 1) Y −! ' !#$ −!#$ −$ −$ ' !#% −$#% −% j1j2m1m2 j1j2JM ℓ − ℓ ∗ ⟨ | ⟩ −! −! ! ℓ 4π m imφ −$ −! ! J j j d = Y e =( 1) 1 2 j2j1m2m1 j2j1JM m,0 2ℓ +1 ℓ − − − − ⟨ | ⟩ " % j m m′ j j %#$ ×%#$ θ 1+cosθ d =( 1) − d = d +% 1 1/2 1 m′,m − m,m′ m, m′ % $ d 0,0 =cosθ d =cos d 1,1 = − − +%#$ +%#$ ! +$ +$ 1/2,1/2 2 2 $×%#$ ,#$ +%#$ +!#$!#$ !#$ % $ ! 1/2 θ sin θ +,#$ ,#$ "#$ 1 +!#$ +%#$ !#$ −!#$ +! +! +! d 1/2, 1/2 = sin d 1,0 = +$+%#$ ! +"#$ +"#$ − − 2 − √2 +%#$ −!#$ !#" !#$ %#!' +$+!#$ %#, &#, ,#$ "#$ %#$ 1 cos θ +!#$ +!#$ %#" ' −$#" % $!' d 1 = +!+%#$ &#, −%#, +%#$ +%#$ +%#$ !#$ %#$ !#" !#$ %#!' ' ' ' ' 1, 1 − − + − − 2 +$ −!#$ !#, !(#%" $#" +%#$ −%#$ !#$' !#& -#$' !#& +! *!#$ &#, !#%" −$#" ,#$ "#$ %#$ !#$ & +!#$ −!#$ -#$' !#& −!#$'−!#& ' *%#$ $#, −!)#%" !#" +!#$ +!#$ +!#$ +!#$ !#$ !#$ -#$' −!#& −!#$' !#& % $ ! $×$ +& & % − + −%#$ +%#$ !#$' −!#& -#$'−!#& −! −! −! +$ +$ ! +% +% +$ −%#$ !#%" (#%" $#" $#" +! −!#$ !$#%" "#!& ' −%#!' +!#$ −%#$ !#" !#$ %#!' +$ +! !#$ !#$ & %$ ' *!#$ !)#%" −%#%" −!#" !#" ,#$ "#$ %#$ !#$ +! +$ !#$ −!#$ +$ +$ +$ −!#$ −!#$ %#" ' −$#" % $ −! *%#$ &#%"−$,#,' $#" −!#!' −!#$ −!#$ −!#$ −!#$ −%#$ +!#$ !#" −!#$ %#!' −$ −$ +$ ' %#!& !#$ $#, +! −%#$ &#%" $,#,' $#" !#!' −!#$ −%#$ !#$ !#$ % +! *! &#, ' −%#, & % $ ! ' −!#$ !)#%" %#%" −!#" −!#" %#$ −!#$ !#$−!#$ % ' *$ %#!& −!#$ $#, +! +! +! +! −! *!#$ !$#%" −"#!& ' %#!' ,#$ "#$ %#$ − − +$ −! !#!& %#!' %#, !#" −$ *%#$ !#%" −(#%" $#" −$#" −%#$ −%#$−%#$ −%#$−%#$ ! +! ' %#, !#" −!#!&−%#!' ' −%#$ $#, !)#%" !#" ' *! %#, −!#" −!#!& %#!' &%$!' −! −!#$ &#, −!#%"−$#" ,#$ "#$ −! *$ !#!& −%#!' %#, −!#" ' ' ''' −$ +!#$ !#,−!(#%" $#" −"#$ −"#$ +$ −$ !#,' !#!' $#, $#" !#" −! −%#$ &#, %#, ,#$ +! −! )#%" $#" !#!& −!#!' −!#" −$ −!#$ %#, −&#, −,#$ ' ' !)#%" ' −$#, ' !#" −! *! )#%" −$#" !#!&!#!' −!#" & % $ ! −$ −%#$ ! 1+cosθ θ −$ *$ !#,' −!#!' $#, −$#" !#" −! −! −! −! d 3/2 = cos 3/2,3/2 2 2 +! −$ !#!& %#!' %#, !#" 1+cosθ 2 ' −! %#, !#" −!#!& −%#!' 3/2 1+cosθ θ 2 ! ' %#, !#" !#!& %#!' & % $ d = √3 sin d 2,2 = − − − 3/2,1/2 − 2 2 2 −$ *! !#!& −%#!' %#, −!#" −$ −$ −$ # $ 3/2 1 cos θ θ 2 1+cosθ ' −$ %#!& !#$ $#, d = √3 − cos d 2,1 = sin θ 3/2, 1/2 2 2 − 2 −! −! &#, '&−%#, % − 1+cosθ −$ ' %#!& −!#$ $#, −% −% 3/2 1 cos θ θ √6 2 2 2 d 1,1 = (2 cos θ 1) −! −$ !#$ !#$ & d 3/2, 3/2 = − sin d 2,0 = sin θ 2 − − − 2 2 4 −$ −! !#$ −!#$ −& 3/2 3cosθ 1 θ 1 cos θ 2 3 −$ −$! d = − cos d 2 = − sin θ d = sin θ cos θ 1/2,1/2 2, 1 1,0 2 2 2 − − 2 −" 2 3/2 3cosθ +1 θ 2 1 cos θ 2 1 cos θ 2 3 2 1 d 1/2, 1/2 = sin d 2, 2 = − d 1, 1 = − (2 cos θ +1) d 0,0 = cos θ − − 2 2 − 2 − 2 2 − 2 # $ # $

35. CLEBSCH-GORDAN COEFFICIENTS, SPHERICAL HARMONICS, AND d FUNCTIONS !!. . . Notation: Note: A square-root sign is to be understood over every coeﬃcient, e.g.,for 8/15 read 8/15. M M . . . − − m m !#$ ×!#$ ! ! 1 2 +!! ' 0 3 "#$ Y1 = cos θ $ !#$ m m Coefficients !#$ !#$ ! ' ' 4π × +"#$ "#$ %#$ 1 2 + + " . . +!#$ −!#$ !#$ !#$ ! +$ +!#$ ! +%#$ +%#$ . . 1 3 iφ −!#$ +!#$ !#$ −!#$ −! Y = sin θ e +$+−!#$ !#" &#" "#$ %#$ . . 1 8π −!#$ −!#$ ! −" +!++!#$ &#" −!#" +!#$ +!#$ 5 3 1 +!−!#$ $#" %#" "#$ %#$ Y 0 = cos2 θ 2 4π 2 2 '+!#$ %#" −$#" −!#$ −!#$ !×!#$ %#$ " − +%#$ %#$ !#$ # $ ' −!#$ %#" $#" "#$ %#$ 15 −! +!#$ $#" −%#" −%#$ −%#$ +! +!#$ ! +!#$ +!#$ Y 1 = sin θ cos θ eiφ 2 $ One More Example: +! −!#$ !#% $#% %#$ !#$ − 8π −! −!#$ &#" !#" "#$ " %#$ ×!#$ +$ $ ! −$ +!#$ !#" −&#" −"#$ ' +!#$ $#% −!#% −!#$ −!#$ 1 15 Y 2 = sin2 θ e2iφ +%#$ *!#$ ! +! +! −$!−!#$ ' −!#$ $#%!#% %#$ 2 4 2π −! +!#$ !#% −$#% −%#$ " +%#$ −!#$ !#&%#& $ ! 0 +!#$ +!#$ %#& −!#& ' ' + n + p $×! % −! −!#$ ! "#$ +% % $ %#$ ×! +!#$ −!#$ !#$ !#$ $ ! +"#$ "#$ %#$ +$ +! ! +$ +$ +%#$ +! ! +%#$ +%#$ −!#$ +!#$ !#$ −!#$ −! −! 1 1 1 1 +$ ' !#% $#% % $ ! +%#$ ' $#" %#" "#$ %#$ !#$ −!#$ −!#$ %#& !#& $ 1, 0 2 , 2 1, 1 2 , 2 +! +! $#% −!#% +! +! +! +!#$ +! %#" −$#" +!#$ +!#$ +!#$ −%#$ +!#$ !#&−%#& −$ | ⇤| ⇤⇥| ⇤| ⇤ +$ −! !#!" !#% %#" +%#$ −! !#!' $#" !#$ −%#$−!#$ ! !×! $ +! ' )#!" !#( −%#!' % $ ! +!#$ ' %#"!#!" −!#% "#$ %#$ !#$ +$$ ! ' +! $#" −!#$ !#!' ' ' ' −!#$ +! %#!' −)#!" !#( −!#$ −!#$ −!#$ +! +! ! +! +! 2 1 1 2 +! −! !#" !#$ %#!' +!#$ −! %#!' )#!" !#( 3 3/2, 1/2 + 3 1/2, 1/2 3 3/2, 1/2+! ' !#$ !#$3 1/ $2, ! 1'/2 ''%#" ' −$#" % $ ! −!#$ ' %#" −!#!" −!#% "#$ %#$ | ⇤ | ⇤⇥ | '⇤+! !#$ −!#$| ' ' ' ⇤ −! +! !#" −!#$ %#!' −! −! −! −%#$ +! !#!' −$#" !#$ −%#$ −%#$ +! −! !#( !#$ !#% ' −! $#" !#$ !#!' −!#$ −! %#" $#" "#$ 2 1 1 ' ' $#%2 '$−!#% ! −! ' )#!" −!#(−%#!' % $ −%#$ ' $#" −%#" −"#$ A = 3/2, 1/2 + 1/2, 1/2 S 3/2, 1/2 −! +! !#( −!#$1/!#%2, −!1/−2! −$ +! !#!" −!#% %#" −$ −$ −%#$ −! ! ' −! !#$ !#$ $ −! −! $#% !#% % 3⇥ | 3⇥ | 3| m ⇤m m 3| ⇤ ⇤ ⇤ ⇥ ⇤ Y − =( 1) Y⇤ −! ' !#$ −!#$ −⇥$ −$ ' !#% −$#% −% j1j2m1m2 j1j2JM ℓ − ℓ ∗ ⟨ | ⟩ −! −! ! ℓ 4π m imφ −$ −! ! J j j d = Y e =( 1) 1 2 j2j1m2m1 j2j1JM ⇥2 ⇥2 m,0 2ℓ +1 ℓ − − − − ⟨ | ⟩ A = M3/2 M1/2 " j j j % 3 3 m m′ %#$ ×%#$ θ 1+cosθ d =( 1) − d = d +% 1 1/2 1 m′,m − m,m′ m, m′ % $ d 0,0 =cosθ d =cos d 1,1 = − − +%#$ +%#$ ! +$ +$ 1/2,1/2 2 2 $×%#$ ,#$ +%#$ +!#$!#$ !#$ % $ ! 1/2 θ sin θ • What about: +,#$ ,#$ "#$ 1 +!#$ +%#$ !#$ −!#$ +! +! +! d 1/2, 1/2 = sin d 1,0 = +$+%#$ ! +"#$ +"#$ − − 2 − √2 +%#$ −!#$ !#" !#$ %#!' 0 + +$+!#$ %#, &#, ,#$ "#$ %#$ 1 cos θ +!#$ +!#$ %#" ' −$#" % $!' d 1 = + p + n +!+%#$ &#, −%#, +%#$ +%#$ +%#$ !#$ %#$ !#" !#$ %#!' ' ' ' ' 1, 1 − − + − − 2 +$ −!#$ !#, !(#%" $#" +%#$ −%#$ !#$' !#& -#$' !#& 0 +! *!#$ &#, !#%" −$#" ,#$ "#$ %#$ !#$ + n + p & +!#$ −!#$ -#$' !#& −!#$'−!#& ' *%#$ $#, −!)#%" !#" +!#$ +!#$ +!#$ +!#$ !#$ !#$ -#$' −!#& −!#$' !#& % $ ! $×$ +& & % − + −%#$ +%#$ !#$' −!#& -#$'−!#& −! −! −! +$ +$ ! +% +% +$ −%#$ !#%" (#%" $#" $#" +! −!#$ !$#%" "#!& ' −%#!' +!#$ −%#$ !#" !#$ %#!' +$ +! !#$ !#$ & %$ ' *!#$ !)#%" −%#%" −!#" !#" ,#$ "#$ %#$ !#$ +! +$ !#$ −!#$ +$ +$ +$ −!#$ −!#$ %#" ' −$#" % $ −! *%#$ &#%"−$,#,' $#" −!#!' −!#$ −!#$ −!#$ −!#$ −%#$ +!#$ !#" −!#$ %#!' −$ −$ +$ ' %#!& !#$ $#, +! −%#$ &#%" $,#,' $#" !#!' −!#$ −%#$ !#$ !#$ % +! *! &#, ' −%#, & % $ ! ' −!#$ !)#%" %#%" −!#" −!#" %#$ −!#$ !#$−!#$ % ' *$ %#!& −!#$ $#, +! +! +! +! −! *!#$ !$#%" −"#!& ' %#!' ,#$ "#$ %#$ − − +$ −! !#!& %#!' %#, !#" −$ *%#$ !#%" −(#%" $#" −$#" −%#$ −%#$−%#$ −%#$−%#$ ! +! ' %#, !#" −!#!&−%#!' ' −%#$ $#, !)#%" !#" ' *! %#, −!#" −!#!& %#!' &%$!' −! −!#$ &#, −!#%"−$#" ,#$ "#$ −! *$ !#!& −%#!' %#, −!#" ' ' ''' −$ +!#$ !#,−!(#%" $#" −"#$ −"#$ +$ −$ !#,' !#!' $#, $#" !#" −! −%#$ &#, %#, ,#$ +! −! )#%" $#" !#!& −!#!' −!#" −$ −!#$ %#, −&#, −,#$ ' ' !)#%" ' −$#, ' !#" −! *! )#%" −$#" !#!&!#!' −!#" & % $ ! −$ −%#$ ! 1+cosθ θ −$ *$ !#,' −!#!' $#, −$#" !#" −! −! −! −! d 3/2 = cos 3/2,3/2 2 2 +! −$ !#!& %#!' %#, !#" 1+cosθ 2 ' −! %#, !#" −!#!& −%#!' 3/2 1+cosθ θ 2 ! ' %#, !#" !#!& %#!' & % $ d = √3 sin d 2,2 = − − − 3/2,1/2 − 2 2 2 −$ *! !#!& −%#!' %#, −!#" −$ −$ −$ # $ 3/2 1 cos θ θ 2 1+cosθ ' −$ %#!& !#$ $#, d = √3 − cos d 2,1 = sin θ 3/2, 1/2 2 2 − 2 −! −! &#, '&−%#, % − 1+cosθ −$ ' %#!& −!#$ $#, −% −% 3/2 1 cos θ θ √6 2 2 2 d 1,1 = (2 cos θ 1) −! −$ !#$ !#$ & d 3/2, 3/2 = − sin d 2,0 = sin θ 2 − − − 2 2 4 −$ −! !#$ −!#$ −& 3/2 3cosθ 1 θ 1 cos θ 2 3 −$ −$! d = − cos d 2 = − sin θ d = sin θ cos θ 1/2,1/2 2, 1 1,0 2 2 2 − − 2 −" 2 3/2 3cosθ +1 θ 2 1 cos θ 2 1 cos θ 2 3 2 1 d 1/2, 1/2 = sin d 2, 2 = − d 1, 1 = − (2 cos θ +1) d 0,0 = cos θ − − 2 2 − 2 − 2 2 − 2 # $ # $

• The very close masses of the p and n is hard to accept as a coincidence • Other particle systems (p, K, D, etc.) have nearly degenerate masses • Heisenberg postulated: • these are multiplets of SU(2) “isospin” analogous to angular momentum • strong interactions are invariant under rotations of isospin • Today, we understand this due to the near degeneracy of the u,d masses • Same “algebra” to determine relations between decay and scattering rates • add component isospin to determine total isospin of the state • match total isospin components before and after to determine which channels are “allowed” by conservation of isospin. • Isospin is somewhat of an “accidental” property • extension to the strange quark with SU(3) doesn’t work as well • we will see SU(2) and “isospin” again in a more fundamental context