Heidi Schellman Northwestern Particles and Deep Inelastic Scattering Heidi Schellman Northwestern University HUGS - JLab - June 2010 June 2010 HUGS 1
Heidi Schellman Northwestern Cross sections Cross sections are the observables for scattering. Let’s look at this from a common sense point of view first. We measure a given reaction rate between a beam of A and a target made up of B ’s which gives a final state X . A + B → X What determines the rate? • the beam flux F A which is the number of A hitting the target/cm 2 /sec. • the volume of target hit by the beam = a × t where a is the area and t is the target thickness June 2010 HUGS 2
Heidi Schellman Northwestern • the number of B per cubic centimeter N B = ρ B ∗ N Av /m B ( AMU ) Here ρ B is the density of the material , N Av is Avagadro’s number and m B is the mass of B in atomic mass units. These are all quantities which depend on your experimental setup. You would expect the reaction rate to scale as: R ( sec − 1 ) F A ( cm − 2 sec − 1 ) × a ( cm 2 ) × t ( cm ) × ρ B ( gr/cm 3 ) = × N Av ( AMU/gr ) × m b ( AMU ) × σ • here σ is a quantity which does not depend on the incoming beam, or the target size, but only on the nature of A and B . It has units of cm 2 . This is the cross section. June 2010 HUGS 3
Heidi Schellman Northwestern Cross section For the strong interaction at nuclear energies, the nucleus acts like a black disk. You hit it or you don’t. If you miss it, no interaction if you hit it, you are certain to have an interaction. The cross section for proton-antiproton scattering is around 7 × 10 − 26 cm 2 or 7 × 10 − 30 m 2 The area of a proton of radius 1 fm is π × 10 − 30 m. So the area over which 2 circles of radius 1 fm could hit each other is quite close to the proton-antiproton scattering cross section we observe. One note - there is a nuclear physics unit - the barn - which corresponds to (10 fm) 2 or 10 − 28 m . We use barns instead of cm 2 most of the time. This concept can be generalized to other interactions, which aren’t so easy to describe as black disks. June 2010 HUGS 4
Heidi Schellman Northwestern The cross section can be differential (you don’t integrate over all of the variables.) dσ/dθ or dσ/dp T are very common quantities to study. June 2010 HUGS 5
Heidi Schellman Northwestern What sets the cross section for an object? The cross section can be small for two reasons 1. The coupling can be small - even a direct hit may not result in much interaction. 2. The effective area for the interaction is very small (as in the weak interactions) June 2010 HUGS 6
Heidi Schellman Northwestern Cross section for colliding beams of particles The above was for a beam of particles hitting a solid target. High energy physicists now frequently collide bunches of particles with other bunches of particles in an accelerator. Here they define a quantity called the luminosity L which is similar to the factors above for a fixed target experiment. R ( sec − 1 ) = σ L Imagine that I have a bunch of N + protons and a countercirculating bunch of N − anti-protons. I would expect the reaction rate to grow if: • N + or N − increases • f the frequency with which the bunches cross through each other increases June 2010 HUGS 7
Heidi Schellman Northwestern • A = s x ∗ s y , the transverse size of the beams decreases (as that makes the particles more likely to find each other). You can see how this maps onto the other definition of luminosity... This is how luminosity turns out to be defined L = f N + N − 4 πs x s y June 2010 HUGS 8
Heidi Schellman Northwestern Luminosity Luminosity is a measure of how often protons/antiprotons get close enough to interact N N + - L = f s x s 4 π y f = beam crossing frequency 2 µ sec 11 n= protons/bunch 10 s = transverse beam size = 0.0001 m L ~ 10 31 crossings/cm 2 /sec June 2010 HUGS 9
Heidi Schellman Northwestern Experimental cross section Say I want to measure the cross section a certain type of nucleus using an electron beam? My detector doesn’t cover all angles, so I lose electrons outside the detector. This is called acceptance, A . My detector doesn’t always work perfectly either so I lose some more electrons. This is called efficiency ǫ . And there can also be backgrounds. Very infrequently some other particle like a photon or a quark or a muon can fake an electron in my detector. This is background, B . The rate I actually observe is R ( e − + N → e − + X ) = A ( e − ) ǫ ( e − ) L σ + B June 2010 HUGS 10
Heidi Schellman Northwestern Where L is the luminosity and contains information about the beam flux and target thickness. Fortunately I can normally estimate ǫ using the data and A and B using some theoretical models. I can also measure the variation of the rate of X production as a function of, say, the transverse momentum component of the e − . dR � T ) dσ ( e − + N → e − + X ) = L dp T A ( p T ) ǫ ( p T → p ′ + B ( p ′ T ) dp ′ p T T Here I’ve included smearing from the produced p T to a measured p ′ T in the efficiency. June 2010 HUGS 11
Heidi Schellman Northwestern Generic Scatter A + B → C + D C A B D June 2010 HUGS 12
Heidi Schellman Northwestern General form for the cross section in terms of the matrix element (2 π ) 4 � |M| 2 S σ = B ) 2 − ( m A m B c 2 ) 2 ( P µA P µ � � � i c 4 ) d 4 p i δ 4 ( P µ A + P µ B − p µ 1 ....p µ � δ ( p µ i p iµ − m 2 N ) (2 π ) 3 i Where P A and P B are the 4-momenta of the incoming particles and m A and m B are their masses. The term at the end in brackets is the Phase Space. S is a spin factor. A measurement of the cross section is thus a measurement of the matrix element and can be used to test theoretical models and determine matrix element parameters. June 2010 HUGS 13
Heidi Schellman Northwestern Averages over initial states Cross sections are defined with an average over possible initial states and a sum over possible final states. One often does not know the initial spins of the particles so one averages over the possibilities. | ← | → | ← | → >, >, >, > (1) ← ← → → Averaging over initial state Imagine that I am scattering electrons and positons. If my experiment has unpolarized beams, then the current of → positrons is 1/2 of the total positron current and the current of ← electrons is also 1/2 of the total electron current. June 2010 HUGS 14
Heidi Schellman Northwestern So the total cross section will be σ ( ← ) + σ ( ← ) + σ ( → ) + σ ( → σ T = (1 / 2) ∗ (1 / 2) ∗ ) ← → ← → One averages over the initial spin states if one doesn’t know them. June 2010 HUGS 15
Heidi Schellman Northwestern Reminder about identical/non-identical processes 1. If two Feynman diagrams are absolutely identical, they are part of the same scattering amplitude and get summed before the squaring of the matrix element. 2. If there are different initial state spin configurations, those are different diagrams and you need to average over them in calculating the cross section (unless the incoming particle are polarized and you want to study a particular spin configuration). 3. If there are different final state spin configurations for a given initial state, those get summed over just like other different final states. June 2010 HUGS 16
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