Lecture 6 Isospin • What Is Isospin? • Rota�Ons in Isospin Space • Reac�On Rates • Quarks and Isospin • Gell-Mann-Nishijima Formula

Home , Isospin

Lecture 6 Isospin • What is Isospin? • Rotaons in Isospin space • Reacon rates • Quarks and Isospin • Gell-Mann-Nishijima formula

FK7003 208 SU(2) – Isospin

Isospin introduced based on the observaon that:

mp = 0.9383 GeV and mn = 0.9396 GeV

Heisenberg proposed in 1932 the idea that the proton and neutrons are two states of the same parcle, the nucleon.

Later we could observe several mulplets of parcles with similar masses but diﬀering charges: {p,n} {π +,π 0,π −} {Ξ−,Ξ0} {Σ+,Σ0,Σ−} m ≈ 0.94 GeV ≈0.140 GeV ≈1.32 GeV ≈1.19 GeV Similar with diﬀerent spin states of the same parcle, the nucleon, the pion, … € € € € But the masses are NOT exactly the same within a muplet.

FK7003 209 This lead to the idea that protons and neutrons would be the up and down states of the same parcle.

A general nucleon = an arbitrary combinaon of up and down state of this new “internal” spin: ⎛ α⎞ ⎛ 1⎞ ⎛ 0⎞ N = ⎜ ⎟ = α⎜ ⎟ + β⎜ ⎟ ⎝ β⎠ ⎝ 0⎠ ⎝ 1⎠ And the proton and neutron: ⎛ 1⎞ ⎛ 0⎞ p = ⎜ ⎟ n = ⎜ ⎟ ⎝ 0⎠ ⎝ 1⎠ This new, abstract internal spin is called “Isospin” , behaves in the same way as spin €

€ € rd An isospin state is wrien as: where I,I3 I is the total Isospin and I3 is its 3 component. 1 1 ⎛ 1⎞ 1 −1 ⎛ 0⎞ p = = ⎜ ⎟ n = = ⎜ ⎟ 2 2 ⎝ 0⎠ 2 2 ⎝ 1⎠ € This is only notaons, but the interesng physics arises from Heisenberg postulate: The Strong interacon is invariant under rotaon in Isospin space. € € From Noether’s theorem, it means that isospin is conserved by the strong interacon.

FK7003 210 Isospin Mulplets

Assume that if the EM interacon could be turned oﬀ the masses of the parcles inside the same mulplet would have the same mass. 1 1 p = m = 0.9383 GeV 2 2 p 1 −1 n = m = 0.9396 GeV 2 2 n € π + = 11 0.140 GeV € π 0 = 10 0.135 GeV

− € π = 1−1 0.140 GeV € Isospin I3 The number of states in a mulplet is given by 2I+1 €

So we can deduce the total isospin from the number of states in the mulplet.

If we know the Isospin we can deduce the number of states in the mulplet.

FK7003 211 Usefulness of Isospin

Invariance of the strong force under isospin transformaon If a reacon/decay is governed by the strong interacon, all other reacons/ decays that are the results of its rotaon in isospin space also exist and have rate/lifeme that can be deduced from it.

If one parcle is found in Nature within a certain mulplet, then all other members of the mulplet must exist must exist too, with very close mass.

Conservaon of Isospin in reacons governed by the strong interacon The total isospin before a reacon/decay is equal to the isospin aer the

reacon/decay (applies to both I and I3)

FK7003 212 Usefulness of Isospin (2)

Isospin is only useful when dealing with the strong force.

The theory of strong force works very well for high energy interacons (q2>> 1GeV)

At low energy it is too strong to allow the use of perturbaon theory Much more difficult to compute hadron masses from first principles Or to compute reacon rates from first principles.

Easier to make predicons at very high energies when working with quarks

When working with hadrons, oen phenomenological models

Isospin is a powerful tool to understand parcle masses and rates

FK7003 213 Rotaon in Isospin State

Consider a rotaon in isospin space around the y-axis ⎛ 1⎞ ⎛ 0⎞ It transforms isospin up (I3=+1) into isospin down (⎜ ⎟ I3=-1): ⎜ ⎟ ⎝ 0⎠ ⎝ 1⎠ ⇒ transforms a proton into a neutron and vice-versa.

€ €

Generally transforms I I3 ⇒ I −I3

π + = 11 π 0 = 10 π − = 1−1 => π+ transforms into π - and vice versa. 1 1 1 −1 p = n = => p transforms into n and vice versa. € 2 2 € 2 2 € € € What is the Isospin rotated equivalent of reacon: ? (a) p + p →d + π +

€ €

€

FK7003 214 What is the representaon of Deuteron in Isospin Space?

Consider a state made of two nucleons n+p, total isospin is either I=0 or 1

I=0 1 ⎛ 1 1 1 1 1 1 1 1 ⎞ 00 = ⎜ − − − ⎟ Iso-singlet 2 ⎝ 2 2 2 2 2 2 2 2 ⎠ 1 00 = (pn − np) 2 €

€

FK7003 215 What is the representaon of Deuteron in Isospin Space?

Consider a state made of two nucleons n+p, total isospin is either I=0 or 1

I=0 1 ⎛ 1 1 1 1 1 1 1 1 ⎞ 00 = ⎜ − − − ⎟ Iso-singlet 2 ⎝ 2 2 2 2 2 2 2 2 ⎠ 1 00 = (pn − np) 2 € 1 1 1 1 1 ⎛ 1 1 1 1 1 1 1 1 ⎞ 1 1 1 1 I=1 11 = 10 = ⎜ − + − ⎟ 1−1 = − − Iso-triplet 2 2 2 2€ 2 ⎝ 2 2 2 2 2 2 2 2 ⎠ 2 2 2 2 1 11 = pp 10 = (pn + np) 1−1 = nn 2 € € €

€ € €

FK7003 216 What is the representaon of Deuteron in Isospin Space?

Consider a state made of two nucleons n+p, total isospin is either I=0 or 1

If the deuteron has I=1 then we expect addional pp and nn states, We do not observe these states in Nature => I=0 for the deuteron

FK7003 217 Rotaon in Isospin State

€ €

Generally transforms I I3 ⇒ I −I3

€ € (b) n + n →d + π − From the point of view these reacons (a) and (b) are the same => Same reacon rates! € Well veriﬁed experimentally. FK7003 218 € Isospin and Reacon Rate

Consider the 3 reacons: (a)p + p →d + π + (b)n + n →d + π − (c)p + n →d + π 0

1) Verify that I3 is conserved in each reacon. 2) Esmate the rao of the rates of (a):(b):(c) for idencal experimental condions, same energies momenta, etc…, the € € only diﬀerence being the parcles interacng€ .

• Sum I3 for each side of (a): ½ + ½ = 0 + 1 (b): - ½ - ½ = 0 – 1 (c): ½ - ½ = 0 + 0

• What happens with the rates? 1 1 1 1 (LHS-a) = 11 (RHS-a) 00 11 = 11 2 2 2 2 1 1 1 1 00 1 1 1 1 (LHS-b) − − = 1−1 (RHS-b) − = − 2 2 2 2 € € 1 1 1 1 1 00 10 10 (LHS-c) − = [10 + 00 ] (RHS-c) = 2 2 2 2 2 € € FK7003 219 € € Isospin and Reacon Rate (cont’d)

+ (a)p + p →d + π (b)n + n →d + π − (c)p + n →d + π 0

• What happens with the rates? 1 1 1 1 (LHS-a) = 11 (RHS-a) 00 11 = 11 € 2 2 2€2 € 1 1 1 1 00 1 1 1 1 (LHS-b) − − = 1−1 (RHS-b) − = − 2 2 2 2 € € 1 1 1 1 1 00 10 10 (LHS-c) − = [10 + 00 ] (RHS-c) = 2 2 2 2 2 € € Isospin is conserved in the strong interacon so for the reacon (c) € the transion from € |00> to |10> is not allowed.

Only the inial states (|10>) can lead to the d+π0 state

⇒ This yields the rao of amplitudes: Ma:Mb:Mc=1:1:1/√2

⇒ Raos of rates: |Ma|:|Mb|:|Mc|=1:1:½

FK7003 220 Non Conservaon of Isospin with the EM and Weak Interacons

u γ

π0 u γ

EM Decay of π0 Weak Decay of Λ

Initial state: |I I3 >=|1 0> Initial state: |I I3 >=|0 0>

Final state: |I I3 >=|0 0> Final state: |I I3 >=| ½ ½ >+ | 1 -1>

The EM and Weak Interacons do not conserve Isospin.

The W±, Z0 and γ and the leptons are assigned Isospin state= |00>

FK7003 221 Quarks and Isospin

The inial 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: 1 1 1 1 u ≡ d ≡ − 2 2 2 2

For the an-quarks: € € The (-) sign in front of d will not maer 1 1 1 1 u ≡ − d ≡ − for us in this course 2 2 2 2

And the other quarks carry no isospin: |00> € €

FK7003 222 Quarks and Isospin (2)

Let’s reformulate some Isospin states from the quarks:

1 1 1 1 0 1 1 ⎛ 1 1 1 1 1 1 1 1 ⎞ π + = ud = − = −11 π = (uu − dd) = 10 = ⎜ − + − ⎟ 2 2 2 2 2 2 ⎝ 2 2 2 2 2 2 2 2 ⎠

I3 − 1 1 1 1 π = du = − − = 1−1 π - 0 + 2 2 2 2 π π € € 3 pions with approximately the same mass of 140 MeV What about the following state: € 1 1 ⎛ 1 1 1 1 1 1 1 1 ⎞ (uu + dd) = 00 = ⎜ − − − ⎟ 2 2 ⎝ 2 2 2 2 2 2 2 2 ⎠

This corresponds to another parcle with I3=0 with mass 548 MeV and Strangeness=0 η €

FK7003 223 Why does Isospin work?

Not a stupid queson!

Think in terms of quarks: mu≈3 MeV and md≈5 MeV to be compared with the masses of : m ≈ 135-140 MeV π - π0 π+

0 1 π + = ud π − = du π = (uu − dd) 2

The quark masses are essenally neglible compared to the pion mass itself. € € € The pion masses are dominated by binding energy of the quarks and quark moon

If mu> 1 GeV >> md 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 essenally a symmetry of interchange of u and d quarks.

€ € FK7003 224 Gell-Mann-Nishijima Formula

1 Empirically we observe that: Q = I3 + (B + S + C + B) 2

rd where Q is the charge, I3 is the 3 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

FK7003 225 Summary

• Isospin is a good symmetry of the strong interacon – Hadron masses, reacon/decay rates respect the Isospin symmetry – The strong interacon is invariant under SU(2) transformaons 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. – Sll SU(3) symmetry can be useful to enumerate the right list of expected hadrons – The strong interacon is not invariant under u,d,s SU(3) transformaons

FK7003 226