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Introductory Reading


  • The Parton Model

    P.Hansson, KTH

    November 18, 2004


    The parton model was introduced in the late 60s by Feynman and Bjorken to address the new data

    on deep inelastic scattering of electrons on nucleons. The model views the nucleons as made up of

    point-like constituents and provides a very simple framework for calculating scattering cross sections as

    well as structure functions for the nucleons. It was very successful in describing a large set of data and

    also paved the way for particle physicists to later identify the partons with the earlier known quarks

    of QCD. This report introduces the reader to the parton model and presents the background as well

    as the most important concepts in deep inelastic scattering.

    1 Introduction

    One of mankinds oldest questions is What is the world made of?. Todays particle physics representsmankinds will to continue the search for the answer. Proposed answers has consisted of everything fromthe four elements (earth, wind, water and fire) to more modern views such as the atom. The periodic tablewas remarkable in describing chemical properties but because of its apparent structure people believedthat the fundamental building blocks of matter was still to be found. Later the nucleus was discoveredand subsequently also the nucleons. In the mid 1960s physicists introduced the concept that point-likeparticles (quarks) existed inside the proton (Gell-Mann). This idea was mainly based on the effort todescribe the rich spectrum of meson and hadron resonances that were discovered during the 1950s, whichwas interpreted as excited bound states of these point-like constituents. Later, in 1968, new data fromSLAC electron scattering experiments hinted that the proton was in fact built up out of point-like particleswhich physicists at that time called partons. The experiments proved that high-energy electrons have alarge probability of scattering with large energy transfers and into large angles. Analogous to Rutherfordsfamous experiment with the scattering of alpha particles the angular and momentum distributions of thescattered electrons were interpreted as if the protons charge distribution was localised to a number of smallscattering centres. Even though the quark model was successful in describing the variety of baryons therewas a period of time when physicists debated whether these quarks were really physical entities and notjust non-physical artifacts required by calculations. During this period Feynman introduced the so-calledparton model [2] which gave the best description of what the experiments really revealed.

    The basic idea of the parton model is that every object with a finite size must have a form factorand therefore experience a dependence on the 4-momentum transfer (Q2 dependence) from the scatteredparticle. The experiments at Stanford confirmed the anticipated[1] phenomenon called scale invariance(Bjorken scaling) which proved that scattering of high-energy electrons on the proton where independentof Q2. This implied that the proton is built up of point-like1 constituents. The coming years after theStanford experiments many experimental results indicated that these partons had the quantum numbersfrom the quark model and as of today it has become clear that the partons in Feynmans parton modelcan be identified as quarks.

    Section 2 gives an introduction to scattering theory which is useful before reading the next section whichdescribes deep inelastic scattering and the structure of nucleons. Section 4 will describe the interpretation ofthe deep inelastic scattering of electrons on protons and present evidence for Bjorken scaling. A description

    1It is not known whether they are indeed point-like or if higher resolution experiments will be able to resolve some innerstructure. This is indeed interesting but analogous to the atomic case where the inner structure of the nucleus is to first orderirrelevant when describing atomic phenomena, the point-like behaviour of the partons is a good approximation.


  • of the identification of Feynmans partons with the quarks of QCD is given in section 5. The last sectiongives a conclusion.

    2 Introduction to Electromagnetic Scattering

    This section will give a very brief explanation of the phenomenology of calculating cross-sections. Thegoal in this section is to provide the reader with some background and formalism which is necessary tounderstand the later sections. Also it will show that the basic approach of calculating the large cross-sectionformulas seen later is simpler than what meets the eye.

    2.1 Spin-less Scattering

    The most straightforward way of discussing this subject is to start from the very basic level. In thiscase it is non-relativistic perturbation theory. Assume that we know solutions to the Schrodinger equationH0 = En. The goal is to solve the Schrodinger equation for a particle moving in an interaction potentialV (x, t):

    (H0 + V (x, t)) = i

    t. (1)

    We know that any solution can be written as



    an(t)n(x)eiEnt. (2)

    We want to find the unknown coefficients an(t). We can substitute eq. 2 into eq. 1 and integrate over thevolume. If we assume that the particle is in a particular eigenstate i of the unperturbed Hamiltonian ashort time before the interaction and that the interaction has ceased a short time later (1st approximation)we can make use of the orthonormality of the solutions and end up with an integral for the coefficient af

    Tfi af (t) = it




    f (x)eiEf t


    V (x, t)[



    , (3)

    which can be written as

    Tfi = i

    d4xf (x)V (x)i(x). (4)

    To obtain a physically meaningful quantity we define the transition probability per unit time as

    W = limT


    . (5)

    If we integrate over a set of initial (i) and final states (f) we obtain the famous Fermis golden rule

    Wfi = 2|Vfi|2(Ei). (6)

    To describe a relativistic electron in an electromagnetic field we must use the Klein-Gordon equation todescribe the electron. The Klein-Gordon equation becomes

    ( + m2) = V with V = ie(A + A) e2A2. (7)

    The amplitude (see section 4) is (neglecting the term e2A2)

    Tfi = i

    f ie(A + A)i(x)d

    4x = ... = i

    jfi Ad4x, (8)

    where jfi is the electromagnetic current in figure 1,

    jfi ie(f (i) (f )i). (9)


  • Figure 1: Transition currents in a Feynman diagram.

    Figure 2: Feynman diagram for spin-less e e.

    The electron has four-momenta pi,pf and wave solution (free particle) i(x) = Nieipix,

    jfi = eNiNf (pi + pf )ei(pipf )x. (10)

    Even though there exists no spin-less leptons, it provides us with a fundamental understanding of scatteringof leptons against targets and this will be important for understanding the idea of deep inelastic scatteringand all the fancy stuff later.

    For example, to obtain the spin-less e scattering we simply need to identify the potential Awith the field from the muon. That is, replace A with 1

    q2j2 , where q is the momentum transfer and j


    is the current associated with the muon (fig.1). The scattering amplitude becomes [5]

    Tfi = iNANBNCND(2)44(pD + pC pB pA)M, (11)iM = (ie(pA + pC))




    (ie(pB + pD)), (12)

    where M is called the invariant amplitude. The lowest order Feynman diagram is shown in figure 2 wherethe factor i g

    q2is called the photon propagator.

    We are now in a position to translate the transition amplitude W into a very useful quantity knownas the cross-section for a specific process. It is the observable we need to compare the calculations withexperiments. The cross-section can be interpreted as the area seen by the incoming particles and be writtenas

    d =|M|2

    FdQ, (cross section = W / Flux * N(Final states)) (13)

    where dQ is the Lorentz invariant phase space. The physics resides in the invariant amplitude M.


  • 2.2 Spin and Relativistic Scattering

    To move on we need to include relativistic electrons and the fact that electrons are spin particles. Theresult obtained in section 2.1 therefore needs to be modified. The electrons now have to satisfy the Diracequation instead of the Klein-Gordon equation. An electron is described by the four-component wavefunction = u(x)eipx and the equation to solve is (compare with eq. 7)

    (p m) = 0V , 0V = eA. (14)

    Using equation 8 the scattering amplitude is

    Tfi = i

    jfi Ad4x, jfi = eufuiei(pfpi)x. (15)

    Compare this with equation 9 in section 2.1 for the spin-less electron. In principle we need only to repeatthe procedure in section 2.1 to obtain the scattering amplitudes for an electron scattering on a lepton(eg. muon or electron). However, practically this is non-trivial since it involves summing over all spinstates. This calculation is simplified when considering the non-relativistic region since the spin directiondoes not change. This can be understood because the electrons are mostly interacting with their electricfield at these energies which cannot flip their spin direction. In the relativistic region one has to carryout the summation over the spins. To obtain the unpolarized2 cross-section (in the following case fore e) we define the tensors Le and Lmuon which separates the sum over the muon and theelectron spins

    Le =1



    [u(k)u(k)][u(k)u(k)], (16)

    and similar for the muon vertex. The invariant amplitude becomes

    |M|2 = e4

    q4Le L

    muon . (17)

    If we switch to the process ee+ + by exchanging the appropriate momentum and charges3 andcarry out the tedious summation over the spins we arrive at (using = e2/4)

    d(ee+ +)d


    4s(1 + cos2 ),

    s center-of-m


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