Hadron resonances

This is the Geii-Mann-Okubo formula. And it works very well. For exam-ple, it implies

(11.28) Putting in (in MeV)

MN = 940 Mr; = 1190 M3 = 1320 (11.29) gives

(exp. 1115) (11.30)

compared to the experimental value of 1115, the difference is less than 1%.

Considering that isospin breaking is bigger than this, it is much better than we have any right to expect.

11.3 Hadron resonances

Particles like the baryons and mesons that participate in the strong interac-tions are generically called hadrons. The baryons and mesons that we have discussed are the lightest hadrons. But there are also an enormous number of excited states of these light states that can be produced in particle collisions but decay back into the light states so quickly that they appear only as en-hancements in the scattering cross-section. The first hadron resonance to be discovered was the

.6.,

which shows up as a very large enhancement in the 1rP scattering cross-section at about 1230 MeV for angular momentum 3/2.

The resonance appears in all the charge states, from

+

^{2 to} -1, so this is a spin 3/2, isospin 3/2 state. It is part of a 10 of SU (

3).

All the other states in the 10 have now been observed.

L).O L).++

--~*--- ~*0 --~*+-­

HI-*

n-(11.31)

174 CHAPTER 11. HYPER CHARGE AND STRANGENESS When Gell-Mann first discussed SU(3), then- had not yet been observed.

Gell-Mann was able to predict not only its existence, but also its mass. Let us repeat his calculation.

Again, we assume that H ^M

s

transforms like the 8 components of an octet, and compute

(11.32) where we have called the decuplet wavefunction B* and

(11.33) is the general decuplet state. We have already done the tensor analysis, and in this case, we can immediately write down the result just by thinking. Since there is only one reduced matrix element, the matrix elements of all octet op-erators are proportional (component by component). Thus the matrix element we want is proportional to the matrix element of the generator, T8, and thus to the hypercharge, Y. This means that we predict equal spacing for the isospin representations

M~· - M!:i = M=.· - ^M~· = Mn- - M=.· (11.34) Experimentally, in MeV,

M~· = 1385 M=.· = 1530 (11.35) The spacings are nearly equal, and the average is about 150, thus we expect the n- at about 1680. Gell-Mann was even able to predict the fate of the n-. With the predicted mass, it could not decay into two lighter particles conserving strangeness and baryon number. The lightest pair of particles with baryon number 1 and strangeness -3 is the K- and 3° (or

K

^{and s-) with}

a total mass of about 1815. Thus Gell-Mann predicted that the n- would look not like a resonance, but like a weakly decaying particle, decaying into states with strangeness -2, B1r and AK-. Sure enough, the n- was seen (first in bubble chamber photographs) with a mass of 1672 MeV. This was very convincing evidence that SU (

3)

is a good approximate symmetry of the strong interactions.

11.4 Quarks

Today, this may seem rather trivial, because we now know that all of these strongly interacting particles are built out of the three light quarks, u, d and

11.4. QUARKS

s, transforming like the 3 of

SU(3).

d u

s

I

The baryons can be built of three quarks because 3 0 3 0 3 = 10 EB 8 EB 8 EB 1 To see this, note that 3 0 3

=

6 EB

3,

so

3 0 3 0 3

=

(6 0 3) EB (3 0 3) and 6 0 3 looks like

uij vk =

~

( uij vk

+

^{uik vj}

+

^{ukj vi)}

+~ (

Eikf.Eemn umj vn

+

Ejkf.Efmn uim vn)

175

(11.36)

(11.37)

(11.38)

(11.39)

which is 10 EB 8. Thus both the 10 and the 8 of baryons that we have al-ready seen can be built out of 3 quarks (the I also appears, in higher angular momentum states).

The corresponding antiquarks transform like a

3

(11.40)

The mesons are built out of quark plus antiquark. Since 3 0

3

^{= 8 EB 1,}

this is either an octet like the ^1r, K, ^'TJ states we have already seen, or a singlet, like the TJ¹ (actually, because of the medium strong interactions, the ^'TJ and TJ' mix slightly).

176 CHAPTER 11. HYPERCHARGEAND STRANGENESS The quarks have spin 1/2, and as we will see later, carry another property, color, that is essential for the understanding of the strong interactions. They also have baryon number 1/3 because three quarks are required to make a baryon. The u and d quarks have zero strangeness, and thus their hypercharge is 1/3. The s quark has strangeness -1, andY= -2/3. For the quarks, the

Quarks were originally introduced as a mathematical device, a shorthand for doing SU(3) calculations. Today, though we know that quarks are real and we have a detailed understanding of many aspects of the strong interactions.

Still, however, there is much that we cannot calculate. We are often forced to fall back on symmetry arguments.

I can't resist doing one more example of an SU(3) relation from the early days. The octet of spin 1/2 baryons have magnetic moments. Unlike the elec-tron magnetic moment, however, the baryon moments cannot be calculated just from their masses and quantum electrodynamics. They depend on the internal structure of the particles. But we can use SU(3) to say a lot about them .. The crucial observation is that the operator that describes the mag-netic moment, whatever it is, must be proportional to

Q,

the electric charge of the quarks. It is therefore an SU(3) octet operator, and we can use the Wigner-Eckart theorem. We expect

J-t(B) =

aTr(B

Bt Q)

+

f3Tr(Bt B Q) (11.42) Thus we expect 6 relations among the 8 magnetic moments (there is actually a 9th, because it is a transition magnetic moment that is responsible for the electromagnetic decay, :E⁰ -+ A-y). In fact, all the magnetic moments can be calculated in terms of J-t(P) and J-t(N). These predictions were first worked out by Sidney Coleman and Shelly Glashow in 1961.

11.4. QUARKS 177

Problems

ll.A. What would the Gell-Mann-Okubo argument tell you about the masses of particles transforming like a 6 of SU(3)?

ll.B. Compare the probability for ~

+

production in 1r0 P -+ ~

+

with the probability for E*⁰ production inK-P -+ E*0, assuming SU(3) sym-metry of the S-matrix.

ll.C. Use the

SU(3)

argument discussed in the chapter to repeat the calculation of Coleman and Glashow, predicting all the spin 1/2 baryon mag-netic moments in terms of

p,(P)

and

p,(N).

Chapter 12

Im Dokument Lie Algebras in Particle Physics (Seite 192-197)