Kinematics and Parton Correlation Functions in the Parton Model. Paul McCullough Supervisors: Piet Mulders and Ted Rogers

Transcription

1 Kinematics and Parton Correlation Functions in the Parton Model Paul McCullough Supervisors: Piet Mulders and Ted Rogers August 30, 2009

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3 Abstract In the Standard Model of Particle Physics, QCD is the theory governing the strong interaction. Being that the strong interaction couples strongly, non-perturbative methods must be used. On the other hand asymptotic freedom states that over short distance scales the coupling constant is weak. Because of this we can employ both perturbative and non-perturbative methods at once. But in order for this to work we need a proper factorization formula to factor the perturbative from the non-perturbative forces. The non-perturbative forces are collected into parton distribution functions, of which several different types are used with varying degrees of approximation. The focus of this thesis is to investigate these approximations with a toy model, to see the ill-affects of using such approximations.

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5 Contents 1 Introduction 1 2 Overview of DIS A Quick Look at Elastic Scattering Structure Functions Internal Structure and Bjorken Scaling The Parton Model The Bare Vertex Parton Distribution Functions The Basics of Factorization Introduction to Factorization Wilson Lines Factorization and Kinematics Kinematics The Momentum Fraction Factorization in a Field Theory Parton Model Reduction Need for More Precise Kinematics Kinematics in a Toy Model The Integrals The Integrand Full Kinematics Standard PDF s Conclusion and Future Work 40 1

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7 Chapter 1 Introduction The Standard Model of Particle Physics is the crowning acheivement in particle physics of the 20th century. Quantum Chromodynamics (QCD) is the theory of the Standard Model describing the the strong interaction. The strong interaction describes the force felt between hadrons and their constituent particles; quarks, anti-quarks and gluons. QCD is an extension of the parton model, which first described hadrons as a collection of partons. QCD is a quantum gauge field theory based on the non-abelian gauge group SU(3). It describes color charge interactions between partons and is asymptotically free. Asymptotic freedom refers to a theory that has small coupling over short time scales, or short distances, and large coupling over large time scales, large distances. Because of the small coupling over short distance scales, one may hope to use perturbative methods to describe short distance physics in QCD (pqcd). Much of the success of pqcd is from extending the tried and true methods of perturbative QED to the more complicated non-abelian gauge theory. Like QED the perturbative calculations of cross sections is done using Feynman rules describing the interactions of quarks and gluons. The Lagrangian in QCD is L = 1 4 F A αβ F αβ A + flavors q a (i /D m) ab q b. (1.1) The quark field is q and is a Dirac spinor. Fαβ A expressed in terms of the the gluon field A A α, is the field strength tensor F A αβ = αa A β βa A α gf ABC A B α A C β. (1.2) A difference between QED and QCD that cannot be ignored is color confinement. Unlike photons and electrons which can be free particles, the color particles of QCD, quarks and gluons are never found as free particles. This 1

8 2 is from the fact that at large distances the coupling constant is so large that the partons are confined in color neutral hadrons. Thus, only hadronic states like protons, neutrons and pions are ever directly observed in nature. Over small distances the coupling constant is small and pqcd is quite appropriate. Over large distance scales on the other hand, the coupling constant is significantly larger and non-perturbative methods need to be used. In most short distance processes though, we cannot ignore the effects of confinement. This is because the initial and final states involve hadrons which are bound states described by non-perturbative physics. Therefore we must use non-perturbative methods in order to describe the internal structure of hadrons used in scattering experiments. In the early days of the standard model, before the development of QCD, data from SLAC suggested that there may be point-like constituents inside the proton. Bjorken and Feynman proposed the parton model, suggesting that hadrons were made of point-like particles, called partons. (This will be discussed more in Chapter 2.). We now know that partons consist of the quarks, anti-quarks and gluons of QCD. The parton model can be verified by probing hadrons with high energy particles, such as photons. At low resolutions one can only resolve the size of the hadron. But if the exchange of momentum is high enough, the photons can feel the internal structure of the hadron. It was the success of the parton model that led to the understanding that the strong interaction is asymptotically free. By using pqcd, one can calculate corrections to the simple parton model, and verify that QCD is indeed the correct theory of the strong interaction. In this thesis, we focus specifically on the of kinematics Deep Inelastic Scattering (DIS). DIS refers to the scattering of a lepton off of a hadron in a high energy limit, with a large exchange of momentum. For the discussion in this thesis, only electron-proton scattering will be considered. In the high energy limit the highly virtual photon emitted from the scattering electron will penetrate deep into the proton. This photon in turn interacts with an individual quark of the proton, and because of the high energy transfered, destroys the proton. This is why this scattering is called deeply inelastic scattering. An example of a DIS event taken at HERA is shown in figure 1.1. This process will serve to illustrate some of the main difficulties with perturbative calculations. Namely that the interaction involves both short range physics that can be calculated perturbatively, and long-range non-perturbative physics. The non-perturbative physics must either be parameterized by experimental data or calculated using non-perturbative techniques. Hence, in order to do calculations, it must be possible to systematically separate long and short distance physics order by order in perturbation theory. This brings us to

9 3 Figure 1.1: An example of a DIS event taken at HERA [1]. the issue of factorization. In the parton model factorization is a natural consequence. The hard scattering is simply described with pqcd, while the long distance effects are grouped together in a single function, called the parton distribution function (PDF). This PDF will prove to be an important function as it describes the structure of hadrons. There are (at least) three different classes of these functions based on certain approximations and integrations. Much of the focus of this thesis is on how these PDF s differ from one another. These functions will be able to be measured in experiment and then later used to describe hadron structure in other scattering experiments. This is called PDF universality. The DIS cross section in the parton model naturally factorizes the hard parts from the non-perturbative. In real QCD, this factorization continues to hold, but the most naive form of factorization relies on a number of kinematical approximations. In order to have a complete description of the hadron, it is desirable to have a form of factorization that avoids these approximations. Knowledge about the detailed structure of the proton is becoming increasingly more important as new particle physics experiments are planned. As an example the studies currently being planned at Jefferson National Laboratory [12], the LHC and HERA [10], will probe the limits of standard perturbative QCD methods. A new and more detailed framework of the internal structure of the proton is becoming more important for research. Current perturbative QCD methods fall short in giving a complete description of the internal structure of hadrons and much research needs to be done to solve this. In order to have a complete description of hadronic structure the differences between these PDF s will be examined using a toy model. The aim is to construct a toy model that can test any errors made by using the afore-

10 mentioned approximations on momentum to maintain factorization. Such toy models must be consistent, and be able to mimic the varying types of PDF s. To test the relevance of certain PDF s, it is important to test their sensitivity to real observables in DIS. 4

11 Chapter 2 Overview of DIS In this section we discuss the basics of deep inelastic electron-proton scattering, p + P p + X, where X represents any final state. In general p can be any lepton, but in this thesis only electrons will be considered. The electron transfers an amount of energy, q, via a virtual photon. We define the invariant energy as, Q q 2. In the case of very large Q, the length scale probed by the electron are of order d 1/Q. But with such large exchanges of energy the proton usually shatters, resulting in a spray of hadrons represented by the final state, X, as seen in the figure 2.4. Because such short distances are being probed, and that the proton shatters, this process is called Deep Inelastic Scattering. To quantify the typical scales involved, Q 2 is generally of order 100 GeV. Hard scales of roughly this size were the standard in the classic DIS experiments performed at DESY [1]. In this case distance scales of.01 fm are being probed. Considering that the size of the proton is 1 fm in diameter, DIS experiments therefore probe the very short-scale internal structure of the proton.! Figure 2.1: The electron sees the proton as Lorentz contracted, with each parton being essentially free. 5

12 2.1 A Quick Look at Elastic Scattering 6 p p q 2 = Q 2 k k J P Figure 2.2: Lowest ordering deep inelastic scattering. The electron couples with a quark inside the proton. To lowest order in the electromagnetic coupling, the incoming electron couples directly to a quark in the proton, and the most general Feynman diagram is as shown in figure A Quick Look at Elastic Scattering We begin our examination by looking at transition currents for low energy electron-proton scattering. Lets take a look at a graphical representation first, figure 2.3. The three lines represent partons inside a hadron jet. The electron interacts with the blob, since we do not know the explicit structure of the hadron. For elastic scattering the hadron simply bounces off the electron via the electromagnetic force. For this interaction the hadron remains intact. An important feature of this type of scattering is the strong force holding the hadron together, and the electromagnetic force felt between an individual quark and the electron, and how we attempt to separate those two different phenomena. We know from QED that the electron transition current takes the following form j µ = eū(p )γ µ u(p)e i(p p)x. (2.1) The proton transition current will take another form since it does not have point-like structure, such as the electron. So the γ µ should be replaced by

13 2.2 Structure Functions 7 p p q P P Figure 2.3: Elastic Proton-Electron scattering a more general, ambiguous Γ µ. J µ = eū(p )Γ µ u(p )e i(p P )x (2.2) From basic principles we propose, Γ µ = G 1 (q 2 )γ µ + κ 2M G 2(q 2 )iσ µν q ν. (2.3) Where G 1 and G 2 are form factors, unknown functions of q 2, that reveal the form of Γ µ. We will not be able to generalize this formalism to DIS because the hadron will break up. 2.2 Structure Functions To resolve the constituents of the hadron in DIS, we must raise the energy transfered from the electron, Q 2. This will reveal the internal structure of the hadron, whereas low Q 2 only reveals the overall size of the hadron. In DIS, Q 2 is so large that the target hadron is destroyed in the interaction. This complicates the normal procedure followed in quantum field theory. Our treatment of the situation has been changed due to the nature of inelastic scattering and hadronic structure. We cannot use the same formalism we started with for the case of elastic scattering. Whereas previously we modified a typical fermion current, j µ ūγ µ u, to a hadron current, J µ ūγ µ u, by changing γ µ Γ µ and constructing the most general Γ µ with form factors. In the case where the hadron splits up it is inadequate to describe the current with a single out Dirac ū since the outgoing particle is not solitary. On the other hand the fermion current on the top of the graph is left unchanged. This is a valuable

14 2.2 Structure Functions 8 p p q P 1 P 2... P P i Figure 2.4: Inelastic proton-electron scattering. point to note, as this fact will be used to exploit DIS. Instead of continuing with the J µ previously used for elastic scattering, let us write the full expression for the differential cross section (neglecting the proton mass for now in the flux factor): dσ d 3 p = 2α2 em sq 4 p 0 Le µνw µν. (2.4) Here L e µν is the normal lepton tensor used for QED, but now we have a hadron tensor, W µν, to represent the lower half of the graph in DIS. The full expression for figure 2.2 is dσ = 1 d 3 p e 4 2s (2π) 3 2p 0 2Q 2 Tr[ /k γ µ /kγ ν ] < P, s J µ (0) X >< X J ν (0) P, s > X = d3 p ( )( e 4 1 2sp 0 Q 4 (8π) 2 Tr[ /k γ µ /kγ ν ] < P, s J µ (0) X >< X J ν (0) P, s > 4π X (2π) 4 δ 4( P + q i P i ) (2.5) ) where we group the lepton and hadron tensors in the following way W µν = 1 < P, s J µ (0) X >< X J ν (0) P, s > 4π X (2π) 4 δ 4( P + q i P i ) (2.6)

15 2.2 Structure Functions 9 L µν = 1 2 Tr[ /k γ µ /kγ ν ]. (2.7) Separated in this way, we can then just treat electron-proton scattering as virtual photon-proton scattering and restrict our considerations to the hadronic tensor, W µν. We use W µν to parametrize our ignorance of the form of the current on the other side of the graph, where the hadron breaks up. We start by constructing the most general form of W µν that we can, similar to what we did for Γ µ. The most general form is constructed of g µν and the independent momenta P and q. This can be expressed in terms of structure functions. For unpolarized scattering we write W µν = W 1 g µν + W 2 M 2 P µ P ν + W 4 M 2 qµ q ν + W 5 M 2 (P µ q ν + q µ P ν ) (2.8) Where we have intentionally omitted antisymmetric contributions to the hadron tensor because they will vanish when dotted into the symmetric lepton tensor, L e µν. In order to simplify the hadron tensor we will apply the Ward identity. From current conservation, d µ J µ = 0, it can be shown that By applying this condition to W µν we get q µ W µν = q ν W µν = 0. (2.9) q µ W µν = W 1 q ν + W 2 M 2 (q P )P ν + W 4 M 2 q2 q ν + W 5 ( (P q)q ν M 2 + q 2 P ν) = 0 (2.10) There are only two independent structure functions here. Lets look at terms individually of just P ν and q ν. First, And second, W 2 M 2 (q P )P ν + W 5 M 2 (q2 P ν ) = 0 (2.11) W 5 = (q P ) q 2 W 2 (2.12) W 1 q ν + W 4 M 2 q2 q ν + W 5 M 2 (P q)qν = 0 (2.13) W 4 = W 1 M 2 q 2 W 5 Substituting equation 2.12, W 4 simplifies to (P q) q 2 (2.14) ( ) P q 2 W 4 = q 2 W 2 + M 2 q 2 W 1. (2.15)

16 2.2 Structure Functions 10 Now we see that W µν can be expressed in terms of just two structure functions and combinations of momenta. W µν reduces to ( W µν = W 1 g µν + qµ q ν ) ( 1 q 2 +W 2 M 2 P µ P q )( q 2 qµ P ν P q ) q 2 qν. (2.16) This is the expression of the hadronic tensor in which we express the structure functions, W 1 and W 2. It is to be noted that unlike elastic scattering, there are two independent variables here, Q 2 and ν P q M. It is common to use an alternative set of independent dimensionless variables here, x B Q2 2P q (2.17) y P q P p. (2.18) The x B is called Bjorken-x. It is also standard to redefine the two remaining structure functions in the following way: F 1 = W 1 (2.19) F 2 = P q M 2 W 2 (2.20) We will explore some characteristics of the structure functions. It will be useful to perform certain contractions on W µν to extract information, and to find particular relationships among the form factors. First we will contract g µν with W µν from equation Performing the contraction yields, g µν W µν = W 1 (4 q2 q 2 ) W [ 2 P 2 2(P q)2 (P q)2 ] q 2 + q 2 = 3W 1 W 2 [ P 2 (2.21) (P q) x B ]. (2.22) Working in the massless regime and converting to the F form factors we see that g µν W µν = 3 [ F 1 F 2 ] F 2 +. (2.23) 2x B x B The contraction of P µ P ν with W µν is W 1 P µ P ν ( qµ q ν q 2 P µ P ν W µν = (2.24) g µν ) + W 2 P µ P ν (P µ P q q 2 qµ )(P ν P q q 2 qν ) (2.25) = Q2 ( F 2 ) 4x 2 F 1 B 2x B (2.26)

17 2.3 Internal Structure and Bjorken Scaling 11 We can rearrange these two equations in the convenient form F 1 = ( 2 x2 B Q 2 P µp ν g µν ) W µν 2 (2.27) F 2 = x B (12x 2 B Q 2 P µp ν g µν )W µν (2.28) Here we can clearly see the form that projection operators of F 1 and F 2 will take. These will be applied in the next chapter. 2.3 Internal Structure and Bjorken Scaling Bjorken scaling is what verified point-like structure within hadrons, and hence confirmed the parton model. This section is a precursor to the parton model. If there exists point-like structureless spin 1/2 quarks inside the proton we should be able to see them with deep inelastic scattering. Because the interaction is inelastic we will have to use our previous inelastic form factors, W 1 and W 2 to determine this. The way in which we would determine an internal structure containing point-like particles would be that for small wavelength, high energy photons the proton starts exhibiting point-like behavior. To check this we will look at the form of dσ ep ex by contracting L e µν and W µν. The leptonic tensor can be found using QED and is, L e µν = 2 ( p µp ν + p νp µ (p p m 2 )g µν ). (2.29) After a few steps and simplifications the contraction is L e µνw µν = 4W 1 (p p ) + 2W 2 M 2 [ 2(P p)(p p ) M 2 p p ] (2.30) Including the flux factor, and the phase space factor for the electron we obtain the following inclusive differential cross section for inelastic electronproton scattering, ep ex, [ ] 1 e 4 dσ ep ex = 4 ( (p P ) 2 m 2 M 2) 1/2 q 4 Le µνw µν d 3 p 4πM 2E (2π) 3. (2.31) Putting together the last pieces, ignoring electron mass and changing to the laboratory frame we have the final result dσ ep ex de dω = α 2 [ 4E 2 sin 4 θ W 2 (x B, y)cos 2 θ 2 + 2W 1(x B, y)sin 2 θ ] (2.32) 2 2

18 2.3 Internal Structure and Bjorken Scaling 12 We know for the case of lepton scattering from QED that the analogous factor is 2 θ2 q2 dσ eµ eµ [cos θ ] 2m 2 sin2 (2.33) 2 So that in the limit where our electron is interacting with the individual quarks (Q 2 >> 0) the structure functions must reduce to 2W point 1 = Q2 Q2 δ(ν 2m2 2m ) (2.34) W point 2 = δ(ν Q2 2m ) (2.35) Making this comparison is to say that inelastic electron-proton scattering reduces to elastic electron-quark scattering. The telling feature of these structure functions is that they have no Q 2 or ν dependency. If we compare these results to the form factors for elastic proton scattering we notice an interesting feature. After some work it can be shown that the form factors for elastic electron proton scattering reduce to W elastic 1 = Q2 4M 2 G2 (Q 2 )δ(ν Q2 2M ) (2.36) W elastic 2 = G 2 (Q 2 )δ(ν Q2 2M ) (2.37) Where G 2 (Q 2 ) is a function only of Q 2. G 2 comes from the form factors, G 1 and G 2, and its particular structure is not important, just that it is a scaling function. In contrast to point-like scattering, these form factors cannot be rearranged to be a function of a single dimensionless variable. A mass scale is present. This mass scale is how we use elastic scattering to measure the size of the proton. In this case the mass scale is set by 0.71 GeV in G(Q 2 ), the inverse size of the proton. As Q 2 increases above this value, the form factor depresses the chance of elastic scattering, and the proton is more likely to break up. The point-like scattering structure functions do not depend on a mass scale. They are functions of the dimensionless quantity x B = Q2 2P q. This is known as Bjorken scaling, and it was the experimental verification of this that solidified the parton model, the main idea being that hadrons consist of structureless point-like particles. Figure 2.5 [13], shows the data gathered from SLAC supporting the parton model. W 2 has no Q 2 dependence implying it is a structure function of point-like scattering.

19 2.3 Internal Structure and Bjorken Scaling 13 Figure 2.5: Data taken from SLAC plotting the structure function, νw 2 vs W (an invariant quantity directly related to Q 2 ). This is for fixed value of ω = 1/x B. Below 2 GeV the structure function varies because at this scale it is elastic scattering. Beyond 2 GeV we see that the structure function is a constant with respect to W, and that the structure is now unaffected by probing energy, implying point-like structure.

20 Chapter 3 The Parton Model As a first try at understanding hadronic structure, we can try taking seriously the picture of the hadron as a collection of weakly interacting point-like particles. This is essentially the parton model picture that was considered by Feynman and Bjorken in pre-qcd days [8]. A great simplification to this problem is to think of it as two separate non-interacting forces. The electromagnetic force between the quark and electron, and the strong force holding the hadron together. We can justify this simplification because at high energy the hadron is Lorentz contracted and time-dilated by a large factor. A typical hadron size is 1 fm, so one can say that in the hadron s rest frame the constituents interact with each other on a time scale of 1 fm/c. In the center-of-mass frame of deep inelastic scattering, the boosted hadron can easily be contracted by.001 fm. From this perspective the partons interact over a much larger time scale, 1000 fm/c. The partons are essentially frozen as seen by the fast moving electron. The electron interaction is over a time scale of.001 fm/c. The interacting electron will essentially see a free quark to scatter from, without the nonperturbative forces. So that in the short-distance electron quark scattering it is a safe and useful approximation to neglect the interactions that bind the hadron. This allows us to separate the perturbative and non-perturbative forces. This caveat also suggests asymptotic freedom, a crucial characteristic of QCD. And for this reason as well we can approximate free quarks as seen from the electron, a crucial element for DIS. The description of hadrons composed of partons was very successful and led to QCD being the force describing the strong interaction. Bjorken scaling explained the data taken at SLAC [13], shown in figure 2.5, and confirmed the parton model. To be precise, the parton model makes the following assumptions: That all partons are moving parallel to the proton direction, are mass- 14

21 3.1 The Bare Vertex 15 less and on-shell That there are no interactions between partons (asymptotic freedom) Assume incoherent scattering The first assumption states that each individual parton is moving in the same direction as the parent hadron, with a certain fraction of the hadron s momentum. This is expressed in the following way k = ( ξp + ), 0, 0 t (3.1) Where ξ is the plus-momentum fraction. In the parton model we treat the cross section for proton scattering as simply the cross section for scattering off a free, massless and on-shell quark with momentum in the direction of the proton weighted by the probability density to find a parton with that momentum fraction, ξ. Therefore W µν = dξ ξ ŵµν f i (ξ) (3.2) i This expression is a statement of the assumptions behind the parton model. The ŵ µν is the tensor associated with the hard virtual photon-quark scattering. The function f i (ξ), in equation 3.2, is called the parton distribution function (PDF). This function contains all the non-perturbative information regarding the structure of the proton. The 1/ξ factor accounts for the difference in flux factor between the partonic and hadronic cross sections. Equation 3.2 is an early example of factorization between the long and short distance scales. 3.1 The Bare Vertex Because we treat DIS as a weighted quark-photon scattering the bare quarkphoton vertex should be calculated. We are only treating the proton as a single free quark, this is unrealistic but later this will generalize to include the parent process, by weighting the bare vertex as in equation 3.2. We also ignore the electron and treat just the virtual photon, since that is the only part that belongs to W µν (the leptonic tensor remains unchanged). This will later prove to be a part of the total cross section for DIS (we are calculating ŵ µν of equation 3.2). This piece, which is really pqcd, is called the hard part of this interaction, and can be calculated perturbatively. As opposed to the soft factors, which come from non-perturbative QCD inside the hadron. It is permissible to look at this interaction alone because of the timescale argument made earlier, it is also one of the assumptions in the parton

22 3.1 The Bare Vertex 16 γ q l = k + q k Figure 3.1: Bare quark electron vertex. model. Labeling the struck quark s momentum as k. If we treat figure 3.1 with Feynman rules we would start with ŵ µν = 1 8π σ d 3 l (2π) 3 2l 0 ū(l)γµ e i u(k) 2 (2π) 4 δ 4 (k + q l) (3.3) Where e i is the quark fractional charge and ŵ is written to signify this as the sub process. Using the standard completeness relations this expression simplifies to ŵ µν = 1 8π d 3 l (2π) 3 2l 0 e2 i tr[/lγ µ /kγ ν ](2π) 4 δ 4 (k + q l) (3.4) Then applying the γ trace identities and performing all the necessary contractions and simplifications reduces the expression to ŵ µν = e2 i Q 2 (kµ l ν + k ν l µ (k l)g µν )ˆx B δ(1 ˆx B ). (3.5) The x B used with ŵ µν is for the sub process, and so it was denoted as ˆx B. It is ˆx B = Q2 (3.6) 2k q The parent process x B is related to the sub-process simply by recalling that in the parton model k = ξp, such that ˆx B = x B ξ (3.7)

23 3.1 The Bare Vertex 17 Now it is possible to extract the structure functions from ŵ µν by using the projection operators defined in Chapter 2, equations 2.27 and Contracting P µ and P ν with ŵ µν P µ P ν ŵ µν = e 2 i P µ P ν (k µ l ν + k ν l µ (k l)g µν ) δ(l2 ) 2l 0 (3.8) To fulfill finding ˆF 1, g µν must be contracted as well. = 0 (3.9) g µν ŵ µν = 2e 2 i (k q)δ(l 2 ) (3.10) = 2e 2 i (k q)δ ( (k + q) 2) = 2e 2 i (k q)δ(2k q Q 2 ) (3.11) Now apply the identity, aδ(x) = δ ( x a). This gives for our value of ˆF 1 Using equation 2.23, we see that g µν ŵ µν = e 2 i δ(1 ˆx B ) (3.12) ˆF 1 = e2 i 2 δ(1 ˆx B) (3.13) ˆF 2 = 2ˆx B ( 2e 2 i δ(1 ˆx B ) e2 i δ(1 ˆx B )) (3.14) ˆF 2 = e 2 i ˆx B δ(1 ˆx B ) (3.15) This result applies to an unrealistically free quark, and so by itself is not very useful. When we generalize to the parent process, then we will see that this result is relevant and consistent. Using the parton model equation based on the relevant assumptions, we see that the full structure functions are, after using equations 3.13 and 3.15, F 1 = i F 2 = i 1 1 x B dξ ξ ˆF 1 (ˆx B )f i (ξ) = i x B dξ ˆF 2 (ˆx B )f i (ξ) = i e 2 i 2 f i(x B ) (3.16) e 2 i x B f i (x B ). (3.17) Comparing these two results with one another gives the important Callan- Gross relation for the Parton Model F 2 (x B ) = 2x B F 1 (x B ) (3.18)

24 3.2 Parton Distribution Functions 18 p p q l = k + q P Figure 3.2: Graphically you can see how the sub-process is embedded in the parent process. Because we can do that, we can now redefine our blob of only the non-perturbative internal hadronic reactions. Now the blob represents a parton distribution function, with the pqcd parts factored out. 3.2 Parton Distribution Functions To generalize the structure functions to include the parent process there is a need to include information about the parent process, the non-perturbative forces. We defined the structure functions relative to the parton model in a way that accomplished this. F 1 = i F 2 = i 1 dξ x B ξ ˆF 1 (ˆx B )f i (ξ) (3.19) 1 x B dξf i (ξ) ˆF 2 (ˆx b ) (3.20) Here we see the PDF again. The index i runs over all types of partons (quarks of different flavors as well as gluons). This function represents the probability of finding a parton of momentum fraction ξ in the hadron. This statement is saying that the sub-process multiplied by the probability of finding a certain quark, i, with particular momentum fraction, ξ, summing over flavors and integrating over all allowed ξ is the structure function of the parent process. Graphically it would appear as is in 3.2. All the momentum fractions have to add up to 1 which constrains f i (ξ). dξξf i (ξ) = 1 (3.21) i This is another way to say that a parton with a momentum fraction exists. The bounds on the integrand is the allowed region of ξ. Because of the δ(l 2 )

25 3.2 Parton Distribution Functions 19 constraint, ξ is bounded from x B < ξ < 1. This function, f(ξ) is what is used to contain all the information of the structure of the proton, it hides our ignorance of the hadron structure. In practicality, we want to measure these functions in experiments, to use in these problems and others involving initial state hadrons. In the parton model we defined W µν as W µν = i 1 x B dξ ξ f i(ξ)ŵ µν (3.22) If we rewrite this expressing ŵ µν explicitly, after rearranging the delta functions we have W µν = e 2 1 i dξ 4Q 2 i x B ξ tr[γµ /lγ ν /k]δ(1 ˆx B )f i (ξ) (3.23) Lets check the g µν projection on our W µν of the parent process. These results should all be consistent with one another. Applying the F 2 projection operator on equation 3.2 gives F 2 = x B g µν W µν (3.24) 1 dξ = x B g µν x B ξ f i(ξ)ŵ µν (3.25) i It was previously shown that g µν ŵ µν = i e2 i δ(1 ˆx B), so the expression reduces to F 2 = dξ ξ f i(ξ)e 2 i δ(1 ˆx B )x B (3.26) i F 2 = i e 2 i x B f i (x B ) (3.27) If we compare this result with the result from equation 3.17, we see that these definitions are self consistent.

26 Chapter 4 The Basics of Factorization 4.1 Introduction to Factorization In this chapter we will look at factorization in the lowest order of deep inelastic scattering. In the next chapter we will see how approximations on momentum permit factorization. The most important result of the last chapter is equation 3.2, so we should analyze its structure in more detail. The function ˆF 2 (ξ) describes the interaction of the virtual photon with the target constituent parton, i.e. the short distance scale part of the interaction. The function f i (ξ) describes the distribution of partons with momentum fraction ξ inside the proton, and so is related to the large scale structure of the proton. The fact that these functions appear as a simple product in the integrand of equation 3.2 is called factorization, and was assumed in the parton model. Factorization in equation 3.2 is a simple consequence of the parton model assumptions. However, when we consider true QCD, field theory, a rigorous proof, order by order, will be needed to ensure that factorization is plausible [5]. The internal parton-parton interactions take place over a large time scale, are non-perturbative by nature and cannot be calculated with standard methods. If we cannot factorize these non-perturbative forces out, then they will spoil our perturbative calculations. The fact that we can calculate the hard factors perturbatively is a huge success. This allows us to measure F 2 in experiment and hence the PDF as well, using equation This can then be used in other calculations, the most simple being in F 1 of DIS, because they both clearly use the same PDF. One can see factorization in the following manner. Recall that dσ ep ex L e µνw µν (4.1) Now as I defined the above hadron tensor, there is both perturbative and 20

27 4.1 Introduction to Factorization 21 non-perturbative pieces. The hard piece being ŵ µν which can be calculated with perturbative QCD. And the non-perturbative piece, f i, which is known as the PDF. Such that And L e µνŵ µν dσ eq eq, so dσ ep ex i 1 x B dξ ξ Le µνŵ µν f i (ξ) (4.2) 1 dσ ep ex = dξ i x B ξ dσ eq eqf i (ξ) (4.3) This is how the electron quark scattering is embedded in the parent process of DIS, similar to how we embedded the ŵ µν in W µν. This is exactly how the parton model naturally factorizes. A key feature is the ability to separate the total cross section between the hard and soft factors. Then we could measure the non-perturbative pieces with experiment, which can then be used in other hadron scattering experiments. Without a factorization scheme, this is not possible. This is the nature of QCD and the coupling constant α s, which is large and leaves us with non-perturbative physics. Leaving the parton model and treating DIS as a real field theory we will see that factorization is not such a natural consequence, as it was in the parton model. In this treatment we will see exactly how certain approximations are made in order to keep factorization. Take note of the first order DIS diagram, figure 4.1, also known as the handbag diagram. This is similar to the previous figures, but now the single bubble has been divided in two. We now see the quark-electron vertex as part of the diagram, and the leptonic portion of the graph is neglected, leaving just the virtual photon. Factorization allows us to pull the hard piece from the lower bubble in figure 4.1. The upper bubble is a jet bubble. When the struck quark leaves the proton it will hadronize, meaning it will fragment into hadrons. This is confinement at work, there are no free quarks or gluons. They will always reduce into color neutral particles. This jet factor will later lead to fragmentation functions. These are similar to PDFs. Unlike PDFs though, fragmentation functions represent the probability of finding certain hadrons within a parton, where as a PDF represents the probability of finding certain partons in hadrons. These functions are not crucial for the work done in this thesis. The Φ bubble contains all the internal hadron structure. This bubble is what the PDF s are designed to describe, this is the non-perturbative physics that we want to group together as a single factorized element. Depending on how we define and group our cross-section will lead to different PDF definitions.

28 4.2 Wilson Lines 22 q l = k + q J k P Φ Figure 4.1: The standard handbag diagram for DIS. This is a tree level diagram. The non-perturbative pieces are separate and denoted by the blobs Φ and J. 4.2 Wilson Lines So far in this thesis I have only considered tree level diagrams. Just as in QED as we consider higher order diagrams we will begin to deal with divergences. We will need to regularize these divergences in order to make sensible QCD predictions. The way in which this is treated is by grouping the divergent contributions into the PDF. By defining our PDF in such a way, we avoid having infinities in our calculations. This is very convenient, because then all we have to do is to measure these PDF s in experiment and not deal with such divergences. Also, defining a proper way of grouping these divergences will prove to be very useful for factorization. By doing so an extra factor will be added to the previous definition of PDF, this factor is called the Wilson line. This Wilson line will also be used to test the universality of the PDF s, a very important property if one wants to use PDF s for other hadron scattering experiments. I will continue to treat the hard factor to tree level, the Born approximation, but it is still useful to show how these extra contributions disentangle from the hard scattering into the PDF with Wilson lines. And it is quite all right to treat the nonperturbative portion this way as well. The process is to group the divergent contributions into the PDF, the non-perturbative piece of our graph. In this sense it does not make sense to add piece by piece order by order, as this is non-perturbative. Because of this rather large coupling constant there is no converging sum, and it does not make sense to think of the PDF perturbatively. We begin by looking at

29 4.2 Wilson Lines 23 types gluon emission from the Φ bubble. Consider this first example, which can be neglected. A gluon is emitted and attaches to the struck quark. This example can be completely neglected by simply including it within the Φ bubble. There is no reason why we cannot simply define it this way. The next example is an emitted gluon attaching to the struck quark after the hard interaction. This particular contribution has been shown to contain 1/ɛ divergences in dimensional regularization [5]. Because this is non-perturbative, there is no sense in looking at this order by order. So our final result should be for an arbitrary number of emitted gluons. Our final result will also have the added benefit of disentangling this emissions from our hard factor, allowing us to keep the perturbative and non-perturbative factorized at higher orders. Reference [2] for a Ward Identity proof of Wilson Lines.

30 Chapter 5 Factorization and Kinematics 5.1 Kinematics Before we begin with the field theory treatment of DIS, the kinematics should be properly explained. Throughout the rest of this thesis I will define momentum in light-cone coordinates. Light-cone coordinates define two axis, on the perimeter of the light-cone. The + component points along the positive x direction along the light cone, while the minus component is in the x direction. For the following vector, a = [a 0, a 1, a 2, a 3 ], in lightcone coordinates would be a = (a +, a, a 1, a 2 ) = ( (a 0 + a 3 ) 2, (a0 a 3 ) 2 ; a t ) (5.1) Where a 1 and a 2 remain the same, and are called the transverse components, denoted by a t. The only other thing we need to know about light-cone coordinates is how to take the dot product of them. It is easy enough to show that a b in light-cone coordinates is = a + b + a b + a t b t (5.2) Now that we see how to use light-cone coordinates lets see how the momentum in DIS is defined. Different texts will define these in slightly different ways. As I have stated earlier, I define the incoming hadron jet s momentum with a capital P. The struck quark with a k, and the incoming photon with q. The hadron jet is q + k = l. The particles we know that are on-shell are constrained by their mass. 24

31 5.1 Kinematics 25 This leaves us with equations like as well as for the virtual photon P 2 = M 2 (5.3) (k + q) 2 = M 2 J (5.4) (P k) 2 = M 2 X (5.5) (5.6) q 2 = Q 2 (5.7) Where M J is the jet mass. MX 2 is the remnant mass, the mass of the incoming hadron jet after it is struck and loses a quark, and M 2 is simply the incoming hadron s mass. With these constraints we can make some simple statements about these momenta. First, we consider the virtual photon and proton to be in a frame where they have no transverse momentum, and where the proton is highly boosted in the + direction. Then we can define P = (P +, M 2 2P +, 0 t) (5.8) Furthermore, the virtual photon momentum can be written in the form, q = ( xp +, Q 2, 0). (5.9) 2xP + We have defined q in this way, to be some fraction of the incoming hadrons momentum, but in the opposite direction. This is defined in such a way to preserve the definition stated earlier, q 2 = Q 2. And the jet momentum takes the simple form: l = (k + q) = (k + xp +, k + Q2 2xP +, k t) (5.10) In the parton model when we thought of the partons, we said they were frozen. In this view the partons would move parallel to the hadrons motion with some fraction of the total hadron s momentum. This set up the natural approximation on k, in a massless limit consistent with the parton model. The hat denotes this particular approximation. ˆk = (ξp +, 0, 0 t ) (5.11) The virtuality of k is small compared to the + component which justifies setting it to zero. This component also must be zero in any massless treatments of the quark due to the light-cone dot product. In a later section

32 5.1 Kinematics 26 we will see how ξ is further simplified in this approximation. Because I am particularly concerned with these approximations on momentum, and how they affect our overall cross section, k will often be treated exactly. This is a trivial definition, k = (k +, k, k t ), but will be used later. These definitions will often be referred to in the remainder of this thesis. In the Parton Model, where particle masses are neglected, the jet factor is a delta function, proportional to δ(l 2 ) The Momentum Fraction A common approximation is made on momentum fraction which I want to discuss. We said that x B is the momentum fraction, ξ, without any justification. This is safe to make in the limit of a certain approximation. If we assume no jet mass, as can be done for high energy scattering, we see that 0 (k + q) 2 = 2k q + q 2 = 2ξP q Q 2 (5.12) In this approximation we see that ξ = x B = Q2 2P q (5.13) This is reasoning behind the convenient labeling of x B = Q2 2P q. In the case where no approximation is made, no longer can we say ξ = x B. In the case where we keep kinematics exact, ξ = x N. This is called the Nachtman x. It is not difficult to show that Nachtman x relates to the Bjorken x by x N = 1 + 2x B (5.14) 1 + 4x2 B M 2 Q 2 Note that for small 4x2 B M 2 then x Q 2 N x B. Again we will look at the jet mass equation, this time M J 0. And looking at the components M 2 J = (k + q) 2 = 2(k + x N P + )(k + q )k 2 t (5.15) k + = M 2 J + k2 t 2(k + q ) + x NP + (5.16) Since we are treating the momentum exactly, the x that appears is Nachtman x. What we see is that k + = x N P + + O ( Λ 2 ) (5.17) Q Which shows that indeed, k + = xp + is indeed an approximation within the Parton Model. And that there is an O ( ) Λ 2 Q that is neglected when approximating massless hadrons and quarks. We want to see what this small piece contributes, specifically towards the remnant mass.

33 5.2 Factorization in a Field Theory Factorization in a Field Theory A basic requirement for the standard factorization formula, like equation 3.2, is the use of kinematical approximations. To see this note how the hadronic tensor would appear in a field theoretical treatment. Consider the handbag diagram, figure 4.1, which corresponds to this hadron tensor. W µν = i e 2 i 4π d 4 k (2π) 4 Tr[γν J(k + q)γ µ Φ(k, P )] (5.18) In this case Φ and J represent the lower and upper bubbles respectively of the handbag diagram. They are correlators representing the internal structure of the proton for Φ, and the fragmentation function for the jet factor, J. Φ contains information of the PDF. J contains the information of the fragmentation function, specifically how our struck quark fragments back to hadrons. Lets take a look at the kinematical variables and see what approximations we can make. We will be looking at the arguments of J and Φ, which are k + q = l and k, P, respectively. k + q looks like Which is of order k + q = (k + ξp +, k + Q2 2xP +, k t) (5.19) ( Λ2, Q, Λ) (5.20) Q Now turning to the magnitude of k and P individually we see P (Q, Λ2 Q, 0 t) (5.21) k (Q, Λ2, Λ) (5.22) Q This evaluation of magnitude lets me safely approximate the arguments of J in a Di basis as (0, q, 0 t ) ˆl, taking k 0 when compared to q in l. I also make a similar approximation on k, ˆk (xp +, 0, 0 t ). On account of these magnitudes we can say in a dirac basis expansion, Φ(k, P ) = γ µ Φ µ + σ µν Φ µν (5.23) J(k + q) = γ µ J µ + σ µν J µν (5.24) the dominating pieces are γ Φ + and γ + J, by a power suppression of Λ2 Q. Once we expand in a Dirac basis, keeping only the dominant terms, we can rewrite the trace of the hadronic tensor as the following

34 5.2 Factorization in a Field Theory 28 W µν = i e 2 i 4π d 4 k (2π) 4 Tr[γµ γ + γ ν γ ]J (k + q)φ + (P, k) (5.25) This equation is valuable. This is before any harmful kinematical approximations have been made, so that this result is still generally a valid assumption. So far all the approximations used are fine. These are normal approximations to make and are power suppressed. At this point it is common to utilize different approximations strictly within the J and Φ + pieces to factorize the integral. We will continue further to see approximations used in standard factorization in QCD. At this point we cannot apply the same approximations we previously made. This is because of the internal virtualities in J and Φ +. Small changes in momentum lead to very large changes in the virtual propagators in J and Φ + due to the denominator of correlators. So a more careful analysis of the momentum are needed to ensure safe approximations. Inside J and Φ + there are pieces of the form 1 l 2 m 2 (5.26) If we keep the ˆl approximation we had before, this will create changes of order Λ 2 in the denominator. Therefore we must keep the l + component of momentum. And a similar argument is made for k. The approximations commonly made are l (l +, q, l t ) = (l +, ˆl, l t ) (5.27) k (x B P +, k, k t ) = (ˆk +, k, k t ) (5.28) From now on, we assume a small boost preformed that sets l t = 0. The above approximations are made on the l and k + components. Using these pieces only produces a small order difference in the denominator of the virtual propagators of magnitude ( Λ2 Q )2, which is acceptable. Our hadronic tensor now takes the form W µν = i e 2 i 4π d 4 k (2π) 4 Tr[γµ γ + γ ν γ ]J ( l)φ + (P, k) (5.29) This is quite nice from a factorization standpoint, with these approximations in place the integral is totally factorizable. Clearly J is only dependent on the integrated variables k t and k and Φ + only depends on k +. We can finalize factorization by expressing,

35 5.3 Parton Model Reduction 29 e 2 { i dk d 2 } { k t 4π (2π) 4 Φ+ ( k, P ) Tr[γ µ γ + γ ν γ ] } dl + J ( l) W µν = i (5.30) with a simple change of integration variables, dk + = dl +. This takes the nice form of {PDF} H {JetFactor} (5.31) The PDF, times the hard factor, accounting for the quark electron scattering and lastly the jet factor, or fragmentation function. It is exactly these approximations made on momentum that allows us to have a usable factorization formula. This is how the standard PDF is defined. This is an important result, as it is needed to be able to factor the long distance non-perturbative physics and the short distance hard physics in DIS to be useful. These approximations affect the dependencies of Φ and J that allow us to split up the integral in this way. We will see later how these approximations affect our overall cross section. In the previous section we took the parton model to a field theoretical representation and performed a standard factorization at tree level. In reality this is extended to all orders. We then examined certain approximations that make DIS in a field theoretical description factorizable, and redefined the PDF. 5.3 Parton Model Reduction I want to show how this is equivalent with the parton model results. This can be done in a few steps. First, substitute, dk d 2 k t (2π) 4 Φ+ ( k, P ) = f i (ξ), (5.32) which I can by definition, and J = 2π l δ( l 2 ). Now our hadron tensor takes the form, W µν = e 2 i dl + Tr[γ µˆ/lγ ν γ ]2πδ( l 2 )f i (ξ). (5.33) 4π i I put l back into the trace. Let me examine the delta function now using the identity δ( x a ) = aδ(x). after applying an identity we see that δ( l 2 ) = δ(2 l + l ) = 1 2 l δ( l + ) (5.34) δ( l 2 ) = 1 2 l ξp + δ(1 x B ξ ) (5.35)

36 5.4 Need for More Precise Kinematics 30 With this and a change of integration variables, d l + = dξp +, the hadron tensor becomes W µν = e 2 i dξ 4π 2 l i ξ Tr[γµˆ/lγ ν γ ]2πδ(1 x B ξ )f i(ξ) (5.36) From here we are almost at the parton model result. Next I will multiply W µν by ˆk + ˆk. The last step would be to show ˆl ˆk+ = Q2 + 2, which you can check for yourself. Finally we arrive at the final conclusion, same as equation 3.23 of the parton model. W µν = i e 2 1 i dξ 4Q 2 x B ξ Tr[γµˆ/lγ νˆ/k]δ(1 x B ξ )f i(ξ) (5.37) 5.4 Need for More Precise Kinematics There are three main types of PDF s. Each type of PDF refers to a combination of approximations on momentum and integration. We have seen the scheme of kinematical approximations in the derivation of factorization at tree level. That is the starting point of two commonly used PDF s, the standard fully integrated PDF and the transverse momentum dependent PDF (TMD PDF). The standard PDF is defined in equation 5.32, with both the minus component and the transverse component integrated out. These PDF s only depend on the longitudinal momentum fraction variable, x B and the hard scale, Q 2. The other components of momentum are integrated out and thus this PDF is not dependent on the virtual and transverse momentum. This the most standard PDF, it contains the least precise kinematics. The next type of PDF is the TMD PDF. The TMD PDF is very similar to the standard PDF except this PDF leaves the integral over k t undone, allowing the PDF to resolve transverse momentum details in DIS. This PDF still ignores the contribution from the virtual component of momentum. This type of PDF is more accurate and used to determine k t dependent effects. The last type of PDF is the fully unintegrated PDF, or parton correlation function (PcF). This PDF is unique in that it keeps all components of momentum exact. This PDF leaves both the minus and transverse component integrals undone, which is also why this can be called the fully unintegrated PDF. This is naturally the most troublesome of the PDF s to apply, but also the most telling. Each type of PDF is defined in a way such that it is

37 5.4 Need for More Precise Kinematics 31 factorizable. For any PDF to be useful it also must have its own rigorous factorization theorem. As of this writing, PcF s have not been fully factorized. For deep analysis of DIS cross sections it is argued that PcF s are needed, specifically when looking at the underlying process. The relevance of keeping the virtual component of momentum is seen in the following analysis, by looking at the sensitivity of M X to k. Let me define k = (k +, k2 + k 2 t 2k +, k t) (5.38) and use the following approximations Accordingly k + x B P + (5.39) M 2 X = (P k) 2 = M 2 + k 2 2P + k 2P k + (5.40) Because of the approximations I drop the first and last term. Plugging in my kinematical values this reduces to MX 2 = k 2 k2 + kt 2 x B (5.41) x B MX 2 = (x B 1)k 2 kt 2 (5.42) After a few extra steps one will obtain the following expression. M 2 X = (1 x B )2P + k k 2 t (5.43) k Λ is of order 2 x B. So the first term is of order (1 x B) P + x B Λ 2 and the second term of order Λ 2. From this it is evident that both terms are of equal value for moderate x B, and at small x B the first term strongly dominates. This explicitly shows strong k dependence of M X. When looking specifically at the underlying process this k component should not be ignored.