Why we believe there?s a strong force. Why Colour? Why not something alongs

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Why we believe there?s a strong force. Why Colour? Why not something alongs

Coleman College, US has reference to this Academic Journal, Why we believe there?s a strong force. Why Colour? Why not something alongside no inappropriate mental imagery Probing the Colour Force The study of simple massive quark bound states So Far?not speculated what holds the proton together. Also Serious Mystery: W- (sss) W- is J = 3/2 So the spin wave function can be: |?1 ? 2 ? 3> c(spin)f(flavour) = |s1s2s3> | ? 1 ? 2 ? 3> COMPLETELY SYMETRIC!! Slides available at: www-pnp.physics.ox /~huffman/ Must have another part so that wave function Y = c(spin)f(flavour)j(space) = symetric Not allowed in consideration of Fermions! Not only is colour needed, we know it must have an antisymetric wave function in addition to it must be in a singlet state alongside zero net colour.

 Malkin, Marc Coleman College www.phwiki.com

 

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One of the more interesting things is the ?width? in the mass of the J/y. On a mass resonance: Breit-Wigner: e+ e- Solve Hydrogen atom but alongside a reduced mass of me/2: Get first bound state of -6.8eV. Can do the same in consideration of the strong force. Use this potential in consideration of the quarks in addition to fit so that aS in addition to F0. Discover: aS = 0.3 in addition to F0 is 16 tons! Why so narrow? Spectroscopic notation: n2s+1LJ J/y ? 13S1 cc0 ? 13P0 or 23P0 l?n is true only in consideration of 1/r potential 1). Gluons are spin 1 in addition to massless like photons 2). Gluons have parity -1 One gluon forbidden: c cbar is colour singlet; gluons have colour charge Two gluons: P in addition to C problem Three gluons allowed but it is now suppressed by a factor of as6

J/y has JPC = 1 like the photon What is the Isospin of the J/y? We already know that I3=0 because of the Quark composition. Because I3 ? I, we know that I = 0,1,2,3?. Integer not « integer. Look at the decays of the J/y so that I-spin eigenstates: J/y ? r+p-, r0p0, r-p+ in almost equal proportions Both rho in addition to pion have Isospin I = 1 So the J/y could have I = 0,1, or 2 only. Use the 1×1 Clebsh-Gordon table so that settle the matter. yes no no How the Beit-Wigner width is measured: E DEbeam In Theory In Fact ? Conservation of Probability 3 3.15 Why This Shape? Would like so that motivate this alongside a relativistic wave equation. To do this try using E2=p2c2+m2c4 Operator equivalents of E in addition to P: So the QM equivalent of Energy in addition to momentum conservation is: This is called the Klein-Gordon equation.

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With: Solutions so that K-G equation: For now, ignore negative E solutions. Throw in a potential (But can we stay fully relativistic??) Na‹ve Solution Becomes: If we assume V is a real function you can multiply the K-G equation by f* in addition to multiply the complex conjugate of the K-G equation by f. The potential, V, drops out. You then get a continuity equation that you can interpret as some sort of conservation of probability Suppose that V is a purely imaginary constant V = iG Assume G is << (p2c2+m2c4) in addition to expand the second term: Note that we no longer have Probability conservation!!! In the Rest Frame of the particle p=0 in addition to we are left with: Transform into ?Energy? space: And The Breit-Wigner form emerges. Probability per unit time of a transition from initial state |i> so that final state |f> is constant. Call it W Mfi is the ?matrix element?, in QM you recognize it as r(E) is the ?density of states available at energy E?. Non-relativistic! Understanding the: dLIPS

How many QM ?states? are there in a volume ?V? up so that a given momeum ?P?? y x z Dx Dpx How big is this? Understanding dLIPS Want so that increment these values by the smallest amount possible in addition to be assured I have crossed so that a new state: 6 ? dimensions are needed in QM so that describe a particle. Three in space x0, y0, z0; in addition to three in momentum px0, py0, pz0 The smallest term in this group is the DxDp term? the uncertainty principle tells us its size! 6 dimensions!! Understanding dLIPS Smallest distinguishable volume ? one state!

Understanding dLIPS Fine in consideration of the ith particle.? But suppose we have N total particles in the final state (and we only need so that worry about the final state because in any experiment we take great pain so that put the initial particles into a single, well-defined state). Ni+Nj = wrong!! These states are more like dice! Or calculation of specific heat of a crystal. How many possible combinations of two distinguishable pairs are possible? Understanding dLIPS Turns out the Matrix Element has Volumes that will cancel alongside these volume elements. Is Lorentz Invariant!!! Understanding dLIPS Now we need so that add up all of our dN?s so that get the total. Anticipate the 1/Vn terms from the matrix element. Integration is over all possible values of the ith momentum. But we do not have independent momenta, if all but one of the momenta is known, the last one is also known!

Understanding dLIPS And this is perfectly fine?just remember so that apply energy in addition to momentum conservation at the end. But using properties of the Dirac delta Function we can re-cast this equation in the following form (and explicitly include energy in addition to momentum conservation. Needed so that keep Lorentz inv. Understanding dLIPS We do have Energy in addition to Momentum conservation. So, in consideration of example, in CM frame alongside total Energy W. Note the additional factors

Malkin, Marc Managing Editor

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