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Isospin:An Approximate Symmetry on the Quark LevelMolly S. [email protected], MA 02139April 27, 2004AbstractIsospin is an approximate symmetry which treats the up and down quarksas different eigenstates of the same particle. The mathematical structure fordescribing the isospin of a system is identical to that of angular momentum.We explore the implications of isospin. Specifically, we use isospin to predictthe ratios of cross-sections in pion-nucleon scattering with incredible accuracy.We also derive the Gell-Mann–Okubo formula for baryons, 2(mN+ mΞ)=3mΛ+ mΣ, which correctly predicts the mΛmass within 1%. This method ofdeveloping spurious symmetries is a powerful tool in quantum field theory.1 Introduction and MotivationHadrons are particles comprised of quarks and governed primarily by the strong,or “hadronic” force. If we have a system of hadrons, then our overall Hamiltonianis H = H0strong+ Hother,whereH0strongis due to the strong interactions—the maingoverning force for quarks and nuclei. Hotheris a perturbative term that takes intoaccount all of the other forces affecting our system. For instance, the Coulomb energyof the system is in this term because the strong force is independent of electromagneticcharge, and the electromagnetic force is about 105times weaker than the strong force.H0strongitself can be broken up into symmetry-preserving and symmetry-breakingcomponents, as we will see in §6. The symmetries we’ll be studying are based on1treating different quarks as different eigenstates of the same particle, rather than asdifferent particles themselves. These will be approximate symmetries; our goal is toget a feel for how good the approximation is. We know that perturbation theory isgood when the perturbing factor is small compared to H0. In our case, the relevantenergy scale is ΛQCD≈ 500MeV∗, and the perturbative energies will be the differencesin quark masses, as discussed in §§2.1.[3]Once we’ve developed the basic particle physics background we’ll need, we’ll seehow isospin—an approximate symmetry which holds with respect to H0strong—can beused to accurately predict scattering cross-sections for pion-nucleon scattering. We’llthen expand our model to include three quarks instead of only two. We will derive theGell-Mann–Okubo formula, which provides an extremely good approximate relationbetween baryon masses.2 Some Introductory Particle Physics2.1 QuarksAs mentioned in §1, quarks are the elementary particles that make up hadrons. We’llonly concern ourselves with the three lightest quarks in this paper, but for complete-ness, all six are summarized in Table 1. Quarks can combine to make two kindsof hadrons that we will consider in this paper. The first is the baryon, which is abound system of three quarks, such as the proton. The second is the meson, whichis a bound quark-antiquark pair, such as a kaon. Everyday matter is made up ofbaryons comprised of up and down quarks, that is, protons and neutrons. Particlescomposed of heavier quarks are much less stable because a much higher energy is re-quired to keep them intact. Quarks are confined to hadrons, so unlike leptons—suchas electrons and neutrinos—they are not observable as independent physical entities.Determining their masses is therefore much more a matter of applying theories thantaking direct measurements. Methods similar to those used in §6 have been used toput limits on the masses of the lighter quarks. [8]∗Hadrons are typically of size ∼ Λ−1QCD2Flavor Symbol ≈Mass in MeVup |u 4.2down |d 7.5strange |s 150charm |c 1500bottom |b 4700top |t 176000Table 1: Speculated quark masses.[8]2.2 Internal Quantum NumbersThe typical 8.06 student is used to dealing with external quantum numbers, like n, ,and m in Hydrogen. External quantum numbers can change—electrons in the sameatom, for instance, cannot have the same set of numbers. Internal quantum numbers,on the other hand, correspond to properties inherent to the particle in question; theyhelp us identify and label the particle. For instance, the total spin of an electronis always12,butthez-component can change. The total spin of the electron is aninternal quantum number, but the z-component is external.“Good” internal quantum numbers are conserved in all interactions. The elec-tromagnetic charge Q is a “good” example of this; an interaction in which the totalcharge of the system has changed has never been observed. Many quantum numberssimply count useful quantities. For instance, the baryon number B is always con-served. Baryons get a baryon number of +1, while antibaryons get a baryon numberof −1; everything else has B = 0. Another way of thinking of this is to say thatevery quark has a baryon number of +1/3, while antiquarks have a baryon numberof −1/3. S, or strangeness, is similar, but instead counts the number of strangequarks—in units†of −1/3. Strangeness is a “mostly good” quantum number; it isconserved in all interactions except for ones governed by the weak force.‡As we willsee in §5, the hypercharge Y ≡ B + S of a particle will be more useful than talkingabout just the particle’s strangeness. As for antimatter, a particle has the same massas its antiparticle, but opposite quantum numbers. For instance, |¯u has the samemass as |u,butaQ = −2/3, B = −1/3, and S =0. (Thebarin|¯u notes that this†An unfortunate remnant of history‡We observed this when we studied kaon decay in 8.05.3is the antiparticle.)[2, 4]2.3 InvarianceThe concept of symmetry is crucial to physics. When we say something is symmetric,we mean that it it doesn’t change under a transformation. There is a famous theoremby Noether that states that there is an inherent connection between transformations(and thus symmetries) and conservation laws. Applied to physics, this is the basisfor conservation of momentum, which arises from the fact that the origin for ourcoordinates and the directions of our axes are arbitrary—we can transform the systemby translating or rotating it, but the physics is still the same. Similarly, an abitrarytime t = 0 point leads to a conservation of energy. Charge conservation is due togauge transformations; our zero-point of A or φ is arbitrary because we can alter bothexpressions with an aribitrary f (x, t) without changing the physics of the system. [1]We can also talk about transformations in eigenspace. Suppose we


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