Particles and Deep Inelastic Scattering

Transcription

1 Particles and Deep Inelastic Scattering Heidi Schellman University HUGS - JLab - June 2010 June 2010 HUGS 1

2 Course Outline 1. Really basic stuff 2. How we detect particles 3. Basics of 2 2 scattering 4. Quark model of the prton 5. General models - Structure functions and QCD 6. Parton Distribution Functions June 2010 HUGS 2

3 The very very basics 1 This is a somewhat random list of the ground rules for particle physics. Particles are identical Elementary particles are identical except for kinematic properties such as their momentum, spin or position - if they have a distinguishing property they are a different kind of particle. This leads to interesting symmetries of their wave functions under exchange. Particles of spin 1/2, 3/2... (fermions) have wave functions which are anti-symmetric under exchange while those of spin 0,1,2... (bosons) have symmetric wave functions under exchange. The hypothesis that particles are identical thus leads to very strong constraints on the wave functions. One way of explaining why they are identical is to consider particles as being excitations in a field. June 2010 HUGS 3

4 Noether s Theorem: Noether s theorem (also known as Noether s first theorem) states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. Restated in physics terms a symmetry of a system implies that something is conserved and breaking that symmetry breaks the conservation law. Examples are: 1. Translational symmetry implies momentum conservation. 2. Rotational symmetry implies conservation of angular momentum. 3. Time invariance implies energy conservation. June 2010 HUGS 4

5 4. Gauge invariance - the fact that the vector potential in electromagnetism can be modified by the addition of derivatives of a scalar field Φ: A( x, t) A( x, t) + Φ( x, t) (1) φ( x, t) φ( x, t) 1 c Φ(x, t) t (2) without changing the physical fields leads to charge conservation. This can be generalized to cover the strong and weak interactions. June 2010 HUGS 5

6 The speed of light in vacuum is a constant The speed of light in vacuum is a constant, c. This leads, through a long series of arguments, to the principles of special relativity and an additional constraint, Lorentz invariance, under which quantities of different types, scalars, momentum vectors, electromagnetic tensors, transform in a well defined way under changes of reference frame. June 2010 HUGS 6

7 The uncertainty principle The uncertainty principle E t h (3) p x h (4) L φ h (5) This imposes stringent constraints on what you can observe and, if you want to measure something very small (small x) requires a higher and higher energy probe. June 2010 HUGS 7

8 The Schwarzchild radius of a black hole R 2MG c 2 (6) sets a limit on the maximum energy you can pack into a finite space. For example length scales below the Planck length of meters are probably off limits as any particle energetic enough to probe that scale is energetic enough to collapse under its own gravitation and become a black hole before it probes anything. June 2010 HUGS 8

9 Relativistic Kinematics Units The particles we study have integer charge in units of the electron charge and we use electromagnetism to accelerate them. For this reason, the electron-volt (ev), is our standard unit of energy. June 2010 HUGS 9

10 Table of common energies in electron volts kev 1000 ev X-rays MeV 10 6 ev nuclear interactions GeV 10 9 ev proton mass TeV ev modern accelerators The units of momentum are ev/c and those for mass are ev/c 2 as one would expect if E = mc 2. It is common to drop the c in the notation. If I mess up a factor of c assume it is one. June 2010 HUGS 10

11 Special Relativity Tensor notation First let me introduce tensor notation. We will often be dealing with mathematical objects which involve several Lorentz indices and we need a convenient form equivalent to 3-vectors in normal Euclidean space. In general, a vector like object can be written as x µ, the covariant form, or x µ, the contravariant form. You can convert a contravariant vector into a covariant vector by using the metric tensor g µν, which describes the geometry of your space (or time). x µ = g µν x ν ν g µν x ν June 2010 HUGS 11

12 there is an implicit sum over any index which appears once in the covariant part and once in the contra-variant part so I won t show the sum again. The metric tensor can be very simple, or very complex. Example: Normal vectors in 3-space For normal 3-space vectors x µ = (x, y, z), the metric tensor is just m ij = δ ij = (7) June 2010 HUGS 12

13 Example: Cross product and rotation in tensor notation You can define an anti-symmetric 3 dimensional tensor ɛ ijk such that ɛ 123 = ɛ 231 = ɛ 312 = 1 and ɛ 132 = ɛ 213 = ɛ 321 = 1 and the rest of the elements are zero. The cross product of two 3-vectors is then ( x y) c = x a y b ɛ abc (8) June 2010 HUGS 13

14 You can rotate a 3-vector by applying a rotation matrix R. x i = R ij x j (9) R has to be a unitary matrix like the one for rotation around the z axis. R ij = cos α sin α 0 sin α cos α (10) June 2010 HUGS 14

15 Example: Metric on the surface of a sphere If one is looking at the surface of a sphere of unit radius with coordinates (θ, φ) where θ is the polar angle, the metric tensor is: m ij = sin 2 θ (11) June 2010 HUGS 15

16 General definition of dot product The dot product between two general 1 dimensional tensors (vectors) is: (x y) = x µ y µ = g µν x µ y ν (12) The length of a vector is just x 2 = (x x) = x µ x µ = g µν x µ x ν (13) which is why g µν is called the metric - it defines the length. Note that, for the dot product to work,your vectors need to be defined at the same point, otherwise the metric on even the 2-sphere becomes pretty useless. June 2010 HUGS 16

17 Special Relativity Metrics in general relativity or in strange coordinate systems can get pretty hairy but the vectors and metric for normal space-time in the absence of large masses are much simpler. You can define a Lorentz 4-vector x µ = (ct, x, y, z) or (ct, x) which consists of the time coordinate and the x, y, z coordinates of a normal 3-vector. The coordinates are normally numbered 0-3 with 0 being the time-like coordinate and 1-3 indicating the space-like x, y and z. It is common to use the indices µ, ν, κ, ρ for the indices of Lorentz 4-vectors. June 2010 HUGS 17

18 The metric tensor in space-time with no gravity is: η µν = (14) You can see for yourself that if X µ = (ct, x, y, z) the length of X is X 2 = X µ X µ = η µν X µ X ν = c 2 t 2 x 2 y 2 z 2 = c 2 τ 2 (15) as you would expect for the proper time τ in special relativity. June 2010 HUGS 18

19 Triple product for 4-vectors The equivalent of a cross product uses a 4-dimensional tensor ɛ µνκρ in which any element with a repeated index like ɛ 1123 is zero, cyclical elements like ɛ 2301 are +1 and countercyclical elements like ɛ 0132 are 1 just as in the 3 dimensional case. x µ = ɛ µνκρ p ν q κ r ρ (16) June 2010 HUGS 19

20 Transformations on 4-vectors The simplest Lorentz transformation between two frames moving relative to each other is a boost along the z axis. All other situations can be reduced to this one by an appropriate set of rotations. From special relativity you know that if the relative velocity of the two frames is v, a position (x, y, z) at time t transforms as: ct = γ(ct + βz) x = x y = y z = γ(+βct + z) where β = v c and γ = 1 (1 β2 ) are the usual special relativity variables. June 2010 HUGS 20

21 In tensor form you would define a boost tensor Λ ν µ. Λ ν µ = γ 0 0 γβ γβ 0 0 γ (17) and do the transform as x µ = Λ ν µx ν. (18) In general you can do a transform along an arbitrary direction by doing a rotation to the frame where the motion is along the z axis, followed by the boost followed by a rotation to whatever direction you want in the new frame. June 2010 HUGS 21

22 x µ = R(1) ρ µλ ν ρr(2) κ νx κ (19) Where the R are rotation matrices with the form R ν µ = r 11 r 12 r 13 0 r 21 r 22 r 23 (20) 0 r 31 r 32 r 33 and the small r s would be the elements of the normal 3-dimensional rotation matrix. June 2010 HUGS 22

23 Note that the tensor notation keeps track of the ordering of the boosts and rotations which is important as in general 3-D rotations and boosts do not commute. R(1) ρ µr(2) ν ρ R(2) ρ µr(1) ν ρ (21) Λ ρ µr ν ρ R ρ µλ ν ρ (22) As you probably already know the 3-D rotation matrices form an algebraic group - called SO(3). So do the boosts. And the boosts and 3-D rotations combined form a larger group, the Poincaré group with Boosts and 3-D rotations as subgroups. June 2010 HUGS 23

24 Example: Muon decay in flight Fermilab used to have a 500 GeV beam of muon particles. The muon is a heavier version of the electron and has a mass of ± MeV/c 2 and decays with a 1/e lifetime of τ = ( ± ) 10 6 s into an electron, a muon neutrino and an electron anti-neutrino. This beam is aimed at a stationary hydrogen (proton + electron) target. June 2010 HUGS 24

25 Center of mass 1. What is the center of mass energy of the muon proton system if the proton is at rest and the muon has energy in the lab frame E lab? The center of mass energy is just E cm = X µ X µ where X µ is the sum of the muon and proton 4-vectors k µ and P µ k µ = (E lab, 0, 0, P µ = (Mc 2, 0, 0, 0) X µ = (E lab + Mc 2, 0, 0, E 2 lab m2 c 4 ) E 2 lab m2 c 4 ) E cm = 2E lab Mc 2 + (M 2 + m 2 )c 4 Note that if the muon beam energy E lab >> Mc 2 or mc 2, the center of mass energy is 2E lab Mc 2. June 2010 HUGS 25

26 2. What is the γ of the muon-proton center of mass system in the lab frame? We can get this from the energy in the CM which is E cm. The lab frame energy E = E lab + Mc 2 = γe cm + γβ(0) as the momentum in the cm. frame is zero. so γ = E lab + Mc 2 E cm June 2010 HUGS 26

27 3. How far does a 500 GeV muon which decays after the average lifetime τ in the center of mass travel in the laboratory? In the muon frame the muon is created at space-time position: x (1) µ = (0, 0, 0, 0) (23) and decays at the same position after a typical time τ at x (2) µ = (cτ, 0, 0, 0) (24) June 2010 HUGS 27

28 Now let s go to the emflaboratory frame where the muon has an energy of 500 GeV. γ µ = E lab mc 2 β µ = 1 1 γµ 2 The γ µ of the muon is so β µ is going to be very very close to one. June 2010 HUGS 28

29 In the laboratory frame the starting point is still at x µ (1) = (0, 0, 0, 0) but the decay point has moved. γ 0 0 γβ x (2) µ = γβ 0 0 γ x (2) µ = (γcτ, 0, 0, γβcτ) (25) The time duration between the create and decay in the lab frame is t = γτ = seconds. This is the famous time dilation effect. The distance between the creation and decay is d = γβcτ = meters. This means that despite the short lifetime of the muon, a beam of 500 GeV muons will, on average, go 3117 km before decaying. This means you can store muons in an accelerator. June 2010 HUGS 29