Lecture 6 Isospin. What is Isospin? Rota4ons in Isospin space Reac4on rates Quarks and Isospin Heavier quarks FK

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1 Lecture 6 Isospin What is Isospin? Rota4ons in Isospin space Reac4on rates Quarks and Isospin Heavier quarks FK

2 SU() Isospin Isospin introduced based on the observa4on that: m p = GeV and m n = GeV Heisenberg proposed in 93 the idea that the proton and neutrons are two states of the same par8cle, the nucleon. Later we could observe several mul4plets of par4cles with similar masses but differing charges: { p,n} π +,π 0,π { } { Ξ,Ξ 0 } { Σ +,Σ 0,Σ } m 0.94 GeV 0.40 GeV.3 GeV.9 GeV Similar with different spin states of the same par4cle, the nucleon, the pion, But the masses are NOT exactly the same within a mu4plet. In the case of the proton / neutron the mass difference was (wrongly) assigned to the difference in charge. FK

3 This lead to the idea that protons and neutrons would be the up and down states of the same par4cle. A general nucleon = an arbitrary combina4on of up and down state of this new internal spin: N = α β = α 0 + β 0 And the proton and neutron: p = 0 n = 0 This new, abstract internal spin is called Isospin, behaves in the same way as spin An isospin state is wri^en as: component. I,I 3 where I is the total Isospin and I 3 is its 3 rd p = = n = 0 This is only nota4ons, but the interes4ng physics arises from Heisenberg postulate: The Strong interac4on is invariant under rota4on in Isospin space. From Noether s theorem, it means that isospin is conserved by the strong interac4on. = 0 FK

4 Isospin Mul4plets Assume that if the EM interac4on could be turned off the masses of the par4cles inside the same mul4plet would have the same mass. p = n = m p = GeV m n = GeV π + = π 0 = 0 π = 0.40 GeV 0.35 GeV 0.40 GeV The number of states in a mul4plet is given by I+ Isospin I 3 So we can deduce the total isospin from the number of states in the mul4plet. If we know the Isospin we can deduce the number of states in the mul4plet. FK

5 Usefulness of Isospin Invariance of the strong force under isospin transforma8on If a reac4on/decay is governed by the strong interac4on, all other reac4ons/ decays that are the results of its rota8on in isospin space also exist and have the same rate/life4me If one par4cle is found in Nature within a certain mul4plet, then all other members of the mul4plet must exist must exist too, miss very close mass. Conserva8on of Isospin in reac8ons governed by the strong interac8on The total isospin before a reac4on/decay is equal to the isospin acer the reac4on/decay (applies to both I and I 3 ) FK

6 Usefulness of Isospin () Isospin is only useful when dealing with the strong force. The theory of strong force works very well for high energy interac8ons (q >> GeV) At low energy it is too strong to allow the use of perturba8on theory Much more difficult to compute hadron masses from first principles Or to compute reac4on rates from first principles. Easier to make predic4ons at very high energies when working with quarks When working with hadrons, ocen phenomenological models Isospin is a powerful tool to understand par8cle masses and rates FK

7 From Previous Lecture Ques4on Find the matrix represen8ng a rota8on by π around the y- axis. The rota4on vector is: θ = πy The corresponding rota4on matrix in angular momentum space is: U(θ) = cos(θ /) i( u θ σ )sin(θ /) U(θ) = cos(π /) i( u y σ )sin(π /) = iσ y U(θ) = iσ y = i 0 i = 0 i 0 0 χ = Spin up par4cle (z direc4on) 0 Uχ = = 0 => Spin up par4cle is transformed to spin down Can Transform between possible spin states equivalent to rota4ng coordinate system FK7003 3

8 Rota4on in Isospin State A rota4on in isospin space around the y- axis We remember from the previous lecture that this rota4on transforms a spin up into a spin down: 0 transforms a proton into a neutron and vice- versa. Generally transforms I I 3 I I 3 θ = πy 0 π + = π 0 = 0 π = => π + transforms into π - and vice versa. (a)p + p d + π + What is the Isospin rotated equivalent of reac8on:? FK

9 Rota4on in Isospin State A rota4on in isospin space around the y- axis We remember from the previous lecture that this rota4on transforms a spin up into a spin down: 0 transforms a proton into a neutron and vice- versa. Generally transforms I I 3 I I 3 θ = πy 0 π + = π 0 = 0 π = => π + transforms into π - and vice versa. (a)p + p d + π + What is the Isospin rotated equivalent of reac8on:? (b)n + n d + π From the point of view these reac8ons (a) and (b) are the same => Same reac8on rates! Well verified experimentally. FK

10 What is the representa4on of Deuteron in Isospin Space? Consider a state made of two nucleons n+p, total isospin is either I=0 or I= Iso- triplet = 0 = = pp 0 = + = ( pn + np) = nn I=0 Iso- singlet 00 = 00 = ( pn np) Which one represents the deuteron? If the deuteron has I= then we expect addi4onal pp and nn states, We do not observe these states in Nature => I=0 for the deuteron FK7003 4

11 Consider the 3 reac4ons: Isospin and Reac4on Rate (a)p + p d + π + (b)n + n d + π (c)p + n d + π 0 ) Verify that I 3 is conserved in each reac4on. ) Es4mate the ra4o of the rates of (a):(b):(c) for iden4cal experimental condi4ons, same energies momenta, etc, the only difference being the par8cles interac8ng. Sum I 3 for each side of (a): ½ + ½ = 0 + (b): - ½ - ½ = 0 (c): ½ - ½ = What happens with the rates? (LHS- a) (LHS- b) (LHS- c) = (RHS- a) 00 = = (RHS- b) 00 = = [ ] (RHS- c) 00 0 = 0 FK7003 4

12 Isospin and Reac4on Rate (cont d) (a)p + p d + π + (b)n + n d + π (c)p + n d + π 0 What happens with the rates? (LHS- a) (LHS- b) (LHS- c) = (RHS- a) 00 = = (RHS- b) 00 = = [ ] (RHS- c) 00 0 = 0 Isospin is conserved in the strong interac4on so for the reac4on (c) the transi8on from 00> to 0> is not allowed. Only the ini4al states ( 0>) can lead to the d+π 0 state This yields the ra4o of amplitudes: M a :M b :M c =::/ Ra4os of rates: M a : M b : M c =::½ FK

13 Non Conserva4on of Isospin with the EM and Weak Interac4ons u γ π 0 u γ EM Decay of π 0 Initial state: I I 3 >= 0> Final state: I I 3 >= 0 0> Weak Decay of Λ Initial state: I I 3 >= 0 0> Final state: I I 3 >= ½ ½ >+ -> The EM and Weak Interac4ons do not conserve Isospin. The W ±, Z 0 and γ and the leptons are assigned Isospin state= 00> FK

14 Quarks and Isospin The ini4al symmetry between the proton and the neutron is related to their quark contents and the similar masses between the u and d quarks. The Isospin states for the quarks are: u d For the an4- quarks: u d The (- ) sign in front of d will not ma^er for us in this course And the other quarks carry no isospin: 00> FK

15 Quarks and Isospin () Let s reformulate some Isospin states from the quarks: π + = ud = = π 0 = (uu dd) = 0 = + π = du = I 3 = π - π 0 π + What about the following state: 3 pions with approximately the same mass of 40 MeV (uu + dd) = 00 = This corresponds to another par4cle with I 3 =0 with mass 548 MeV and Strangeness=0 η FK

16 Why does Isospin work? Not a stupid ques4on! Think in terms of quarks: m u 3 MeV and m d 5 MeV to be compared with the masses of : π - π 0 π + m MeV π + = ud π π 0 = = du (uu dd) The quark masses are essen8ally neglible compared to the pion mass itself. The pion masses are dominated by binding energy of the quarks and quark mo8on If m u > GeV >> m d for example then the Isospin symmetry would break down. The proton (uud) and (udd) would have very different masses and the rates of: (a)p + p d + π + (b)n + n d + π would be very different. Isospin symmetry is essen4ally a symmetry of interchange of u and d quarks. FK

17 Gell- Mann- Nishijima Formula Empirically we observe that: Q = I 3 + (B + S + C + B) where Q is the charge, I 3 is the 3 rd component of the Isospin, B Baryon number, S, C, B are the Strangeness, Charmness and Beauty of the hadron. The Meson Nonet The Baryon Octet Strangeness FK

18 Example: how can we put things together to derive the quantum numbers of a new par4cle Σ c + Λ + c + π 0 The decay occurs with a life4me smaller than 0 - s. The par4cle Λ + c is an isosinglet with quark content udc. ) What is the interac4on governing the above decay? ) What is the quark content of the? 3) Quantum numbers? 4) Is isospin conserved in the reac4on? Σ c + 5) Does the have isospin partners? What is their quark content? Σ c + Σ c + 6) What is the minimum mass of? FK

19 Σ c + Λ + c + π 0 Life4me indicates that this reac4on is governed by the strong interac4on FK

20 The Other Quarks Hadrons with s, c and b are also formed. The large mass difference makes observables generally sensi4ve (=> not conserved) to interchange between a light and a heavy quark. FK7003 5

21 The Meson Nonet Beyond SU() The hadrons can be organised in the plan with strangeness and Isospin. It appears that there is a SU(3) symmetry of interchange of the 3 quarks u, d, s The mass of the s quark is significantly higher and this symmetry is only useful to guess what hadrons to expect. In terms of rate predic4ons it does not work due to the mass difference. The Baryon Octet Strangeness I 3 FK7003 5

22 Summary Isospin is a good symmetry of the strong interac4on Hadron masses, reac4on/decay rates respect the Isospin symmetry The strong interac4on is invariant under SU() transforma4ons in isospin space. This is due to the fact that the mass difference between u and d quarks is negligible compared to the binding energy of all hadrons. The other quarks are much heavier Observables are no longer invariant under the interchange with a u or a d quark. S4ll SU(3) symmetry can be useful to enumerate the right list of expected hadrons The strong interac4on is not invariant under u,d,s SU(3) transforma4ons FK