Time reversal Isospin Symmetry
The strong forces on u and d quarks are the same, as far as we know but the (smaller) electromagnetic forces are different because the quarks have different electric charges.In addition, the d quark is a few MeV/c2 heavier than the u quark, as we discuss later. This isospin symmetry, as it is called, is a good approximation. Families of particles like (6.1) and (6.2) are called isospin multiplets. Isospin quantum numbers • In order to formulate isospin symmetry mathematically, we first introduce three new quantum numbers, that are conserved in strong interactions. 1)Hypercharge Y is defined by
are the baryon number, charm bottom and top. 2)Also introduce Q- em charge,
3)We define isospin
Note; mathematics of I, I3 similar to mathematics of spin quantum numbers:
Simple quark model. Allowed quantum numbers: Isospin and Strangeness
For baryons allowed quark states are, and strange quark does not contribute to isospin An example: sigma baryons isotriplet
• The A member of triplet can be produced e.g. in reaction: and then decays in one of the ways:
In weak interaction: lifetime 8x10-10 s If we discovered in these reactions and do not know its quantum numbers, we can find from conservation laws that for
And hence :
Since I3=1, we predict two more states with I3=0 and I3=-1, also discovered Two families of the lightest mesons (including u,d,s quarks) for quark-antiquark pair 0- and 1-
Parity for L=0, and the spin S = 0 or S=1 in quark model
Comment on u–anti u, d-anti d, and s-anti s states
• For these states it can be matematically shown (see appendix C) that states with definite isospin quantum numbers are:
• The state. Has the same quantum numbers as mix with and can mix These states: 6. 23 and mixture 6.24 ab, correspond to The light baryons in quark model
Spin/ Parity.: (L=0) and
Baryon magnetic moments prediction from adding Dirac’s quarks moments (in simple quark model)
The agreement with the 3-quark simple model of baryons is not exact: there is an admixture of other states with the same quantum numbers: e.g. 1) not simple S states (like for deutron), or 2) pentaquark states or something else? The u, d quark mass splitting (1)
In the simple quark model we effectively assume that there are only valence quarks in the hadrons: quark-antiquark pairs in mesons, and 3 quarks in baryons (or 3 antiquarks in antibaryons) We assume no gluons, no see of quark antiquark pairs, no exotic states.
In this case if we try to calculate the masses of the quarks, we will get the so called “constituent” masses, (see e.g. constituent quark masses calculated when fitting the magnetic moments of quarks in Chapter 6):
ms = 510MeV/c2, mu,d = 336MeV/c2. (6.32)
Constituent masses of the quarks, listed in 6.32, are “effective” quark masses, including for instance the mass of the relativistic gluon cloud emitted by these quarks. Not surprisingly the “constituent” masses are (significantly) larger than the more realistic for many calculations “bare” quark masses, which should not take into account e.g. the relativistic mass of gluon clouds, emitted by quarks. Such “bare” masses (only of the order of few MeV for u and d quarks), are usually called “current algebra” quark masses, and they will be discussed later in this course.
Here we estimate the mass splitting of the u and d constituent quarks. We get a reasonable value for the mass splitting, comparable to the one obtained in more sophisticated calculations, which are not discussed here. The u, d quark mass splitting (2) We will use the measured masses of Σ particles to estimate the quark mass splitting, and assume that the quark mass difference arise from the quark mass differences, and from electromagnetic interactions between pairs of quarks (see for details M&S 4th Chapter 6)
We have:
where eq are the quark charges, δ is a constant, and M0 is the contribution to the Σ masses arising from the strong interactions between the quarks which is assumed to be the same for all pairs of quarks. From these equations we get:
This estimate is within a factor of 2 from more sophisticated calculations. Hadron mass splittings
The mass differences between different members of a given supermultiplet are conveniently separated into the small (neglected in this estimate) mass differences between members of the same isospin multiplet and the large mass differences between members of different isospin multiplets For the 3/2+ decuplet this gives:
2 where ms − mu,d ≈ 170 MeV/c from (6.32) The L = 0 heavy quark states (mesons)
Although in the above discussion we have only considered states containing the ‘light quarks’ u, d and s, it is straightforward to extend these ideas to include the ‘heavy quarks’ b and c (see M&S 6th Chapter) Here we have a very brief discussion of states containing heavy quarks, mostly giving names for mesons and baryons. For more details see M&S. The L = 0 heavy quark states (baryons, examples)
Extensive data exist only for baryons containing a single heavy quark. We shall therefore concentrate on the predictions for these states.
The spectroscopy of hadrons with non-zero charm and bottom is still very much a developing area of experimental research Color
The quark theory of hadrons as we saw is very successful, but appears to contradict the Pauli principle. The apparent contradiction between the quark model and the Pauli principle was resolved in 1964 by Greenberg, who argued that in addition to space and spin degrees of freedom, quarks must possess a new quantum number: a color. The total wavefunction is then written as the product of a) a spatial part ψ(r), b) a spin part χ, and c) a color wavefunction χC :
In addition the color confinement hypothesis is stated, that hadrons can only exist in states, called color singlets, which have zero values for all color charges, while quarks, which have non-zero color charges, can only exist con- fined within them. As we shall see shortly, this explains why hadrons have integer electric charges, while fractionally charged combinations like are forbidden, in accordance with experimental observation. Color charges and confinement The basic assumption of the color theory is that any quark q = u, d, s, ... can exist in three different color states χC = r, g, b, standing for ‘red’, ‘green’ and ‘blue’, respectively. Just as the spin states χ = α, β correspond to different values of the spin component S3, the color states χC correspond to different values of two of the color charges called the color hypercharge and the color isospin charge. They are denoted Yc and , and their values for the single quark states χC = r, g, b are listed in Table 6.13(a). Their values for other states composed of quarks and antiquarks then follow by using the they are additive quantum numbers, like electric charge, whose values for particles and antiparticles are equal in magnitude but opposite in sign
Again, we assume that according to confinement hypothesis, only states with zero color quantum numbers can exist:
We next consider the combinations of m quarks and n anti-quarks that are allowed by the confinement condition (6.41). Only the compbinations below (see M&S for details) can exist according to the confinement: Confinement
We assume that according to confinement hypothesis, only states with zero color quantum numbers can exist:
We next consider the combinations of m quarks and n anti-quarks that are allowed by the confinement condition (6.41). Only the combinations below (see M&S for details) can exist:
One can show (see M&S Chapter 6) that the only antisymmetric combination for the color part of the wave function (you can check that it is antisymmetric):