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Lecture 7 Isospin

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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? • Heisenberg noticed that protons and neutrons are very close in mass p 938.272 MeV/c2 p 1 , 1 → | 2 2 ⇥ n 939.565 MeV/c2 n 1 , 1 ⇥ | 2 − 2 ⇤ • We now know that the pions also closely spaced: π+ 139.570 MeV/c2 π+ 1, 1 → | ⇥ π0 134.977 MeV/c2 π0 1, 0 → | ⇥ 2 π- 139.570 MeV/c π− 1, 1 ⇥ | − ⇤ • Likewise the Delta Resonances: ++ 3 3 + 3 1 ++/+/0/ ∆ , ∆ , Δ - 1232 MeV/c2 → | 2 2 ⇥ → | 2 2 ⇥ 0 3 1 3 3 ∆ , ∆− , ⇥ | 2 − 2 ⇤ ⇥ | 2 − 2 ⇤ The Hypothesis • 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 | π+ + p 1, 1 1 , 1 3 , 3 | | 2 2 | 2 2 1, 1 1 , 1 3 , 3 π− + 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 , , + , −! +!#$ !#% −$#% −%#$ " +%#$ −!#$ !#&%#& $ ! π + n 2 2 √3 2 2 3 2 2 +!#$ +!#$ %#& −!#& ' ' | | $ ! % −! −!#$ ! "#$ | ⇥| − ⇥ × AND d+!#$FUNCTIONS−!#$ !#$ !#$ $ ! +% % $ %#$ ×! +"#$ "#$ %#$ +$ +! ! +$ +$ −!#$ +!#$ !#$ −!#$ −! −! !!. 1 1 2 3 1 1 1 1 +%#$ +! ! +%#$ +%#$ 0 +$ ' !#% $#% % $ ! !#$ !#$ %#& !#& $ Notation: π + 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 +$$ ! +!!' +! $#" −'!#$ !#!' ' ' ' 0 −!#$ +! %#!' −)#!" !#( −!#$ −!#$ −!#$ "#$ 1, 0 , , + , +! +! ! +! +! Y1 = cos θ $×!#$ m 1 m 2 Coefficients π + n 2 2 3 2 2 √3 2 2 +!#$ +!#$ ! ' ' +! −! !#" !#$ %#!' 4π +!#$ −! %#!' )#!" !#( +"#$ "#$ %#$ | − ⇥ | − ⇥ +! ' !#$ !#$ $ ! ' ''%#" ' −$#" % $ " ! . | ⇥| − ⇥ +!#$ −!#$ !#$ !#$ ! −!#$ ' %#" −!#!"+$ +−!#$!#% "#$ !%#$+%#$ +%#$ ' +! !#$ −!#$ ' ' ' −! +! !#" −!#$ %#!' −! −! −! 3 −%#$ +! !#!' −$#" !#$ −%#$ −%#$ . −!#$ +!#$ !#$ −!#$ −! 1 iφ 1 1 1 3 1 2 1 1 +! −! !#( !#$ !#% ' −! $#"Y1!#$=!#!' sin θ e −!#$ −+!$+−%#"!#$$#" !#""#$ &#" "#$ %#$ . π + 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
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  • The Quark Model
    The Quark Model • Isospin Symmetry • Quark model and eightfold way • Baryon decuplet • Baryon octet • Quark spin and color • Baryon magnetic moments • Pseudoscalar mesons • Vector mesons • Other tests of the quark model • Mass relations and hyperfine splitting • EM mass differences and isospin symmetry • Heavy quark mesons • The top quarks • EXTRA – Group Theory of Multiplets J. Brau Physics 661, The Quark Model 1 Isospin Symmetry • The nuclear force of the neutron is nearly the same as the nuclear force of the proton • 1932 - Heisenberg - think of the proton and the neutron as two charge states of the nucleon • In analogy to spin, the nucleon has isospin 1/2 I3 = 1/2 proton I3 = -1/2 neutron Q = (I3 + 1/2) e • Isospin turns out to be a conserved quantity of the strong interaction J. Brau Physics 661, Space-Time Symmetries 2 Isospin Symmetry • The notion that the neutron and the proton might be two different states of the same particle (the nucleon) came from the near equality of the n-p, n-n, and p-p nuclear forces (once Coulomb effects were subtracted) • Within the quark model, we can think of this symmetry as being a symmetry between the u and d quarks: p = d u u I3 (u) = ½ n = d d u I3 (d) = -1/2 J. Brau Physics 661, Space-Time Symmetries 3 Isospin Symmetry • Example from the meson sector: – the pion (an isospin triplet, I=1) + π = ud (I3 = +1) - π = du (I3 = -1) 0 π = (dd - uu)/√2 (I3 = 0) The masses of the pions are similar, M (π±) = 140 MeV M (π0) = 135 MeV, reflecting the similar masses of the u and d quark J.
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  • Invariance Principles
    Invariance Principles • Invariance of states under operations implies conservation laws – Invariance of energy under space translation → momentum conservation – Invariance of energy under space rotation → angular momentum conservation • Continuous Space-Time Transformations – Translations – Rotations – Extension of Poincare group to include fermionic anticommuting spinors (SUSY) • Discrete Transformations – Space Time Inversion (Parity=P) – Particle-Antiparticle Interchange (Charge Conjugation=C) – Time Reversal (T) – Combinations of these: CP, CPT • Continuous Transformations of Internal Symmetries – Isospin – SU(3)flavor – SU(3)color – Weak Isospin – These are all gauge symmetries Symmetries and Conservation Laws SU(2) SU(3) 8 generators; 2 can be diagonalized at the same time: rd I3 … 3 component of isospin Y … hypercharge SU(3) Raising and lowering operators Y = +1/2√3 Y = +1/√3 SU(3) SU(3) Adding 2 quarks SU(3) Adding 3 quarks Quantum Numbers • Electric Charge • Baryon Number • Lepton Number • Strangeness • Spin • Isospin • Parity • Charge Conjugation Electric Charge Quantum Numbers are quantised properties of particles that are subject to constraints. They are often related to symmetries Electric Charge Q is conserved in all interactions Strong ✔ Interaction Weak ✔ Interaction Baryon Number Baryon number is the net number of baryons or the net number of quarks ÷ 3 Baryons have B = +1 Quarks have B = +⅓ Antibaryons have B = -1 or Antiquarks have B = -⅓ Everything else has B = 0 Everything else has B = 0 Baryons = qqq = ⅓ + ⅓ + ⅓
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