Particles and Deep Inelastic Scattering Heidi Schellman - - PowerPoint PPT Presentation

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

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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

• f 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.

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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.

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• 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.

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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.

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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.

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The Schwarzchild radius of a black hole

R ≃ 2MG c2 (6) 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

• ff 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.

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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.

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Table of common energies in electron volts

keV 1000 eV X-rays MeV 106 eV nuclear interactions GeV 109 eV proton mass TeV 1012 eV modern accelerators The units of momentum are eV/c and those for mass are eV/c2 as one would expect if E = mc2. It is common to drop the c in the notation. If I mess up a factor of c assume it is one.

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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 xµ, the covariant form, or xµ, the contravariant form. You can convert a contravariant vector into a covariant vector by using the metric tensor gµν, which describes the geometry of your space (or time). xµ = gµνxν ≡

• ν

gµνxν

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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 xµ = (x, y, z), the metric tensor is just mij = δij =     1 1 1     (7)

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Example: Cross product and rotation in tensor notation You can define an anti-symmetric 3 dimensional tensor ǫijk such that ǫ123 = ǫ231 = ǫ312 = 1 and ǫ132 = ǫ213 = ǫ321 = −1 and the rest of the elements are zero. The cross product of two 3-vectors is then ( x × y)c = xaybǫabc (8)

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You can rotate a 3-vector by applying a rotation matrix R.

• x′

i = Rij

xj (9) R has to be a unitary matrix like the one for rotation around the z axis. Rij =     cos α − sin α sin α cos α 1     (10)

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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: mij =   1 sin2 θ   (11)

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General definition of dot product The dot product between two general 1 dimensional tensors (vectors) is: (x · y) = xµyµ = gµνxµyν (12) The ”length” of a vector is just x2 = (x · x) = xµxµ = gµνxµxν (13) which is why gµν 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.

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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 xµ = (ct, x, y, z) or (ct, x) which consists

• f 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.

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The metric tensor in space-time with no gravity is: ηµν =        1 −1 −1 −1        (14) You can see for yourself that if Xµ = (ct, x, y, z) the length of X is X2 = XµXµ = ηµνXµXν = c2t2 − x2 − y2 − z2 = c2τ 2 (15) as you would expect for the ”proper time” τ in special relativity.

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”Triple product” for 4-vectors The equivalent of a cross product uses a 4-dimensional tensor ǫµνκρ in which any element with a repeated index like ǫ1123 is zero, cyclical elements like ǫ2301 are +1 and countercyclical elements like ǫ0132 are −1 just as in the 3 dimensional case. xµ = ǫµνκρpνqκrρ (16)

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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.

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In tensor form you would define a boost tensor Λν

µ.

Λν

µ =

       γ γβ 1 1 γβ γ        (17) and do the transform as x′

µ = Λν µxν.

(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.

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x′

µ = R(1)ρ µΛν ρR(2)κ νxκ

(19) Where the R are rotation matrices with the form Rν

µ =

       1 r11 r12 r13 r21 r22 r23 r31 r32 r33        (20) and the small r’s would be the elements of the normal 3-dimensional rotation matrix.

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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) Λρ

µRν ρ =

Rρ

µΛν 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 Poincar´ e group with Boosts and 3-D rotations as subgroups.

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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 105.658367 ± 0.000004 MeV/c2 and decays with a 1/e lifetime of τ = (2.197019 ± 0.000021) × 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.

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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 Elab? The center of mass energy is just Ecm =

• XµXµ where Xµ is the sum
• f the muon and proton 4-vectors kµ and Pµ

kµ = (Elab, 0, 0,

• E2

lab − m2c4)

Pµ = (Mc2, 0, 0, 0) Xµ = (Elab + Mc2, 0, 0,

• E2

lab − m2c4)

Ecm =

• 2ElabMc2 + (M 2 + m2)c4

Note that if the muon beam energy Elab >> Mc2 or mc2, the center of mass energy is ∼ √2ElabMc2.

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• 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 Ecm. The lab frame energy E′ = Elab + Mc2 = γEcm + γβ(0) as the momentum in the cm. frame is zero. so γ = Elab + Mc2 Ecm

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• 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)

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Now let’s go to the emflaboratory frame where the muon has an energy of 500 GeV. γµ = Elab mc2 βµ =

• 1 − 1

γ2

µ

The γµ of the muon is 4732.23 so βµ is going to be very very close to one.

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In the laboratory frame the starting point is still at x(1)′

µ

= (0, 0, 0, 0) but the decay point has moved. x(2)′

µ

=        γ γβ 1 1 γβ γ        x(2)

µ

= (γcτ, 0, 0, γβcτ) (25) The time duration between the create and decay in the lab frame is ∆t = γτ = 0.01039 seconds. This is the famous time dilation effect. The distance between the creation and decay is ∆d = γβcτ = 3.11688 × 106 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.

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