Heidi Schellman Northwestern Particles and Deep Inelastic Scattering Heidi Schellman Northwestern University HUGS - JLab - June 2010 June 2010 HUGS 1
Heidi Schellman Northwestern 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
Heidi Schellman Northwestern 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
Heidi Schellman Northwestern 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
Heidi Schellman Northwestern 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) → x, t ) − 1 ∂ Φ( x, t ) φ ( � x, t ) φ ( � (2) → c ∂t without changing the physical fields leads to charge conservation. This can be generalized to cover the strong and weak interactions. June 2010 HUGS 5
Heidi Schellman Northwestern 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
Heidi Schellman Northwestern The uncertainty principle The uncertainty principle ∆ E ∆ t ¯ (3) ≥ h ∆ p ∆ x ≥ h ¯ (4) ∆ L ∆ φ ¯ (5) ≥ h 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
Heidi Schellman Northwestern The Schwarzchild radius of a black hole R ≃ 2 MG (6) c 2 sets a limit on the maximum energy you can pack into a finite space. For example length scales below the Planck length of 10 − 35 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
Heidi Schellman Northwestern 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
Heidi Schellman Northwestern Table of common energies in electron volts keV 1000 eV X-rays 10 6 eV MeV nuclear interactions 10 9 eV GeV proton mass 10 12 eV TeV 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
Heidi Schellman Northwestern 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 ≡ � g µν x ν ν June 2010 HUGS 11
Heidi Schellman Northwestern 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 1 0 0 m ij = δ ij = (7) 0 1 0 0 0 1 June 2010 HUGS 12
Heidi Schellman Northwestern Example: Cross product and rotation in tensor notation You can define an anti-symmetric 3 dimensional tensor ǫ ijk such that ǫ 123 = ǫ 231 = ǫ 312 = 1 and ǫ 132 = ǫ 213 = ǫ 321 = − 1 and the rest of the elements are zero. The cross product of two 3-vectors is then y ) c = x a y b ǫ abc ( � x × � (8) June 2010 HUGS 13
Heidi Schellman Northwestern You can rotate a 3-vector by applying a rotation matrix R . i = R ij � x ′ � x j (9) R has to be a unitary matrix like the one for rotation around the z axis. cos α − sin α 0 R ij = (10) sin α cos α 0 0 0 1 June 2010 HUGS 14
Heidi Schellman Northwestern 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: 1 0 m ij = (11) sin 2 θ 0 June 2010 HUGS 15
Heidi Schellman Northwestern 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
Heidi Schellman Northwestern 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
Heidi Schellman Northwestern The metric tensor in space-time with no gravity is: 1 0 0 0 0 − 1 0 0 η µν = (14) 0 0 − 1 0 0 0 0 − 1 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
Heidi Schellman Northwestern ”Triple product” for 4-vectors The equivalent of a cross product uses a 4-dimensional tensor ǫ µνκρ in which any element with a repeated index like ǫ 1123 is zero, cyclical elements like ǫ 2301 are +1 and countercyclical elements like ǫ 0132 are − 1 just as in the 3 dimensional case. x µ = ǫ µνκρ p ν q κ r ρ (16) June 2010 HUGS 19
Heidi Schellman Northwestern 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 ) 1 where β = v √ c and γ = (1 − β 2 ) are the usual special relativity variables. June 2010 HUGS 20
Heidi Schellman Northwestern In tensor form you would define a boost tensor Λ ν µ . 0 0 γ γβ 0 1 0 0 Λ ν µ = (17) 0 0 1 0 0 0 γβ γ and do the transform as µ = Λ ν x ′ µ x 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
Recommend
More recommend
