Infrared Divergences in Perturbative Quantum Chromodynamics

Transcription

1 Master s thesis Infrared Divergences in Perturbative Quantum Chromodynamics Peter Rozing 3 November 1 Under Supervision of prof.dr. E.L.M.P. Laenen Institute for Theoretical Physics, University of Amsterdam Nikhef Theory Group

2 ii

3 Motivation In order for me to get my Master s degree in theoretical physics, I had to do a one-year project of which in the end I had to write a thesis. I chose the field of theoretical particle physics to work in. During my earlier studies, I got more and more interested in this branch of physics, especially after following master s courses on Quantum Field Theory and Field Theory in Particle Physics. The fact that one is describing physics at a very fundamental level, together with the mathematical beauty of, for instance, gauge theories and renormalization, made me realize that I wanted to learn more about theoretical particle physics, which resulted in a one-year project at Nikhef under supervision of prof.dr. E.L.M.P Laenen. The subject of this thesis is beyond introductory theoretical particle physics. It deals with infrared singularities arising in the perturbative treatment of quantum chromodynamics, the non-abelian gauge theory that describes the interaction between quarks and gluons. These singularities arise in performing loop integrals where the loop momentum gets very soft, or collinear with one of the outgoing particles. This thesis starts with some introductory comments on quantum chromodynamics which are needed in the remainder of this thesis. Chapter will then focus on finding infrared singularities without explicitly performing all loop integrations. The Landau equations and the Coleman- Norton interpretation will appear to be very easy ways to locate possible infrared-singular diagrams, while the formalism of power counting even tells something about the structure of a diagram. Chapter 3 will discuss the very important concepts of factorization and resummation. It is shown that it is possible to separate infrared safe from infrared sensitive parts in amplitudes or cross sections, and this phenomenon will lead us to the resummation of diagrams and cross sections. This will be useful in dealing with large logarithms, but also may serve as a generating functional for higher order corrections. The final chapter will be about the deeper structure of poles in scattering amplitudes. Both the general case and the case of the Sudakov form factor will be discussed. It will turn out that there are very strict constraints on the kinematic dependence of the elements of the factorization of such an amplitude, and also on the pole structure. This thesis will then be concluded with a summary. Of course, I did not do this project on my own. My supervisor, prof. Laenen, proved to be of great importance for this work to be successfully finished. When I got stuck, which happened plenty of times, he managed to explain things in a very clear way to help me out, and I thank him for that. Also, for the sake of politeness, all my friends and family who think they contributed, in any way whatsoever, to the succeeding of my thesis and my entire studies may assume to be thanked ; iii

4 iv

5 Contents 1 Introduction: A brief overview of QCD The Parton Model Quantum Chromodynamics The Gauge Group SUN Quarks and Gluons The QCD Lagrangian Dimensional Regularization and Ultraviolet Renormalization Finding Infrared Singularities 13.1 How to find Infrared Divergences Soft and collinear divergences The Landau equations The Coleman-Norton approach Power Counting General Formalism of Infrared Power Counting Finding Normal Variables for Pair Creation Calculation of the Degree of Divergence Unitarity and the Optical Theorem The Optical Theorem Unitarity and Time-ordered Perturbation Theory Unitarity in more general cross sections Summary Factorization and Resummation Factorization of Deep-Inelastic Scattering Calculation of the Structure Functions Evolution of the Structure Functions Factorization of the Drell-Yan cross section The General Formalism of Resummation Analytic Continuation of the Sudakov Form Factor The Ratio of Timelike to Spacelike Form Factor Comparison to Explicit Two-loop Calculations Summary The Structure of Poles in General Scattering Amplitudes An Operator Definition of a Factorized Amplitude The Case of the Sudakov Form Factor Non-leading Poles in the Sudakov Form Factor The Function Gα s Relating Gα s to the Splitting Function Finding Kinematic Constraints on Factorization The Eikonal Jet Functions v

6 CONTENTS 4.3. The Soft Anomalous Dimension Applying Kinematic Constraints to the Sudakov Form Factor Constraints on the Partonic Jet Function Solving for the Soft Anomalous Dimension The Three-Parton Case of Γ S Corrections in the Case of Four or More Partons Implications for the Soft Anomalous Dimension Summary Conclusions 99 A Calculations involving Eikonal Feynman Rules 13 B The Soft Matrix in Matrix Form 17 vi

7 Chapter 1 Introduction: A brief overview of QCD Since this thesis is about infrared divergences in quantum chromodynamics, it is instructive to start with an introductory chapter about what quantum chromodynamics is and where it originates from. Quantum chromodynamics QCD is a non-abelian gauge theory that describes the strong interactions, that is, the force that binds quarks and antiquarks together in hadrons, through the exchange of gluons. It is based on the symmetry group SU3. The theory of strong interactions started out as the so-called Parton Model, which we will discuss below. Its predictions, together with some experimental results, finally led people to the non-abelian gauge theory that we know nowadays as QCD [1,, 3, 4, 5]. In this chapter, we will discuss the fundaments of the parton model and of QCD. We will discuss what they mean and how they connect to each other. We will also give some basic QCD results which we need later in this thesis. 1.1 The Parton Model The parton model [] originates from the 6 s of the previous century. Back then, the results of experiments involving proton-proton collisions had a remarkable result; people found that at energies above 1 GeV in the center of mass frame, proton-proton collisions produced a large number of pions. The remarkable thing of this, is that almost all these pions had momenta collinear with the collision axis. This led people to believe that the proton is a loosely bound assemblage of many components. In 196, an experiment at SLAC-MIT was used to put this interpretation of the proton structure to a test. People scattered electron beams with energy of GeV from a hydrogen target. The result was that there was a substantial rate for hard scattering of electrons from protons, with the total reaction rate comparable to what one would expect from quantum electrodynamics if the proton was an elementary particle. In most cases, however, there was no single proton detected after the reaction. The largest part of the scattering rate came from deep inelastic scattering DIS: the incoming electron shattered the proton and produced a final state with a large number of hadrons. This result led Bjorken and Feynman to introduce the parton model for hadron structure [13, 14]. In this model, the proton is a bound state of partons, which would be fermions quarks or antiquarks carrying electric charge. These partons could also carry another kind of charge for their bonding to each other. This model explained the SLAC-MIT result by saying that the incoming electron kicked a quark out of a proton, and after that the resulting quarks would recombine to form other hadrons. It also explained the result of proton proton collisions, in the sense that protons that collide fall into pieces, the partons, and that these would recombine to form the pions that were measured. 1

8 CHAPTER 1. INTRODUCTION: A BRIEF OVERVIEW OF QCD One remarkable feature of this parton model is what we know as Bjorken scaling. To see what this is, let us look more closely at the DIS process. A schematic representation of DIS is shown in figure 1.1. We can write the DIS cross section as follows [1] dσp, q = f 1 dξ dσ B ξp, qφ f/n ξ. 1.1 Here, dσ B is the Born cross section for the scattering of an electron from a parton, ξ is the fraction of the hadron momentum carried by the struck parton, and φ f/n is describes the distribution of parton f in hadron N. It can therefore be interpreted as the probability of finding a parton f in hadron H. As we will see later, there is no interference between different kinds of partons f, which means that these parton distributions are universal [1]. Therefore, if one can measure them in one process, one knows them for every process. k k ξp ξp + q p X Figure 1.1: Deep Inelastic Scattering. An electron or electroweak boson with momentum k kicks a parton with momentum fraction ξ out of a proton with momentum p. We can calculate the Born cross section. It is the product of a leptonic tensor L µν, and of a hadronic tensor W µν [1] dσ B ξp, q = d3 k 1 s k q Lµν k, qw µν ξp, q. 1. We are interested in the hadronic tensor W µν. We know that there is current conservation, q µ W µν =, parity and time reversal invariance W µν = W νµ. Due to these properties, the hadronic tensor has the following form W µν = g µν q µq ν W1 q x, q + p µ + q µ pν + q ν W x, q. 1.3 x x Here, we define the so-called Bjorken x [14] as the dimensionless ratio x = Q /p q. For later convenience, we also define slightly modified structure functions F 1 = W F = p qw 1.5 If we now use equation 1.1, we can identify coefficients to find a relation between the nucleon structure functions and the parton structure functions

9 1.. QUANTUM CHROMODYNAMICS F N 1 x = f F N x = f 1 1 dξ ξ F f 1 x ξ φ f/nξ 1.6 dξf f x ξ φ f/nξ 1.7 Now, we can calculate the Born diagram in DIS using Feynman rules. This will give us expressions for the parton structure functions. After some straightforward calculation, we find F f 1 z = F f z = Q f δ1 z, with Q f the electromagnetic charge of parton f. This gives us expressions for the nucleon structure functions F N x = f Q fxφ f/n x 1.8 F N 1 x = 1 Q fφ f/n x 1.9 Here, we see the Bjorken scaling arise. The structure functions of the hadronic tensor, and therefore the hadronic tensor, are only dependent on the dimensionless ratio x, not on Q. This means that the proton structure looks the same to any electromagnetic probe, no matter how hard it is struck. This scaling was indeed measured in the SLAC-MIT experiment up to 1% accuracy for Q > 1 GeV []. We also see that the nucleon structure functions satisfy F N x = xf N 1 x, which is called the Callan-Gross relation [15]. This relation is characteristic for spin- 1 fermions, and is therefore another evidence for the identification of partons with quarks. Bjorken scaling has been verified experimentally for DIS at the electron-proton collider HERA, in the experiments H1 and ZEUS. The results of these experiments for various values of the Bjorken x are combined in figure 1., together with some results of a DIS fixed-target experiment. For every value of x, the theoretical QCD prediction is depicted as a line, and one sees that QCD correctly predicts the scaling behavior of DIS. Let us go back to the implications of Bjorken scaling. We know that DIS only occurs on very rapid timescales compared to normal proton timescales, since the timescale of a reaction is roughly the reciprocal of the energy transfer, which should be large for DIS []. Because we found that the hadronic part of the DIS cross section is independent of Q, we can say that in a process like DIS, interactions between the constituents of the proton can be ignored. This means that in DIS, the partons are approximately free particles so the parton distribution functions are universal, but have strong interactions on longer timescales. This is a mystery, since interactions in quantum field theory can not just turn themselves of. This was solved, however, by the renormalization group, from which we find that there is a running coupling in non-abelian gauge theories. We will discuss this in the following sections. 1. Quantum Chromodynamics In this section, let us look at QCD as a theory of the strong interactions, and try to find any connections to the parton model discussed in the previous section. QCD describes the interactions between quarks and gluons in hadrons. Currently, we know 6 types of quarks, which we call flavors. These are the up u, down d, charm c, strange s, top t and bottom b quarks. Until now, quarks have not been observed yet as isolated particles, only as the pointlike constituents of hadrons. The hadrons are the particles that are made of multiple quarks, and we can distinguish between two different types of hadrons: the baryons, which are bound states of three quarks or antiquarks, and the mesons, which are bound states of quark-antiquark pairs [3, 5]. We will start this section by briefly discussing the gauge group of QCD, SU3. We will then discuss what this implicates for the quarks and gluons present in QCD, and derive the QCD f 3

10 CHAPTER 1. INTRODUCTION: A BRIEF OVERVIEW OF QCD Figure 1.: Data from the H1 and ZEUS experiments for DIS, compared to QCD predictions. lagrangian and the Feynman rules that can be deduced from it. We will end this section with a discussion on ultraviolet renormalization and its implications The Gauge Group SUN If we want to know more about QCD, we need to know something about the gauge group on which it is based, SU3. In this section, we will discuss some properties of the Lie group SUN, the continuous group of unitary N N-matrices with determinant 1. A more detailed discussion on group theory can be found in any group theory book, for instance, in [6]. In this discussion, we focussed on the subjects discussed in []. An element of a Lie group has a general form gα = e iαata. 1.1 Here, we introduced group parameters α a and, since gα is unitary, the hermitian operators t a, which are the generators of the group. In SU3, these generators are given by t a = 1 λ a, where λ a are the Gell-Mann matrices. Moreover, since the elements gα should have determinant 1, the generators should be traceless. The generators should span the space of infinitesimal group transformations, so their commutation relations must be of the form [t a, t b ] = if abc t c, 1.11 where the f abc are called structure constants. The vector space spanned by infinitesimal transformations, together with their commutation relation, is called a Lie algebra. 4

11 1.. QUANTUM CHROMODYNAMICS Let us now discuss the representation of such a Lie algebra. A representation of a Lie algebra is a set of d d hermitian matrices t a that satisfy the commutation relation of the Lie algebra. Here, d is the dimension of the representation. In general, a representation can be decomposed into a form where it is block-diagonal. These blocks, if we can not decompose them any further, are called irreducible representations. Matrices in this representation are indicated as t a r. We can also choose a basis in such a way that the trace of two matrices is proportional to the identity Tr[t a r tb r ] = t rδ ab 1.1 which together with equation 1.11 leads to a formula that shows us that the structure functions are antisymmetric f abc = i Tr [ [t a r t, ] tb r ]tc r r We will now discuss two important representations. The first one is called the fundamental, or the defining, representation. In the defining representation of SUN, the generators are just the one half the Gell-Mann matrices, t a = 1 λ a. This representation is denoted by r = F, and has dimension N. The second is the adjoint representation, denoted by r = A. This is the representation in which the matrices are just the structure constants t b A ac = if abc Since SUN has N 1 generators, as can be deduced from the requirements that SUN consists of unitary matrices of determinant 1, the dimension of the adjoint representation is N 1. Let us now look at the product of two matrices, t = t a t a. One can show that this operator commutes with all group generators [t b, t a t a ] = if bac {t c, t a }, 1.15 which vanishes due to the antisymmetry of the structure constants. This means that t is an invariant of the Lie algebra and is therefore proportional to the identity matrix t a r ta r = C r Here, C r is called the quadratic Casimir operator. Now let us go back to the constant t r from equation 1.1, and normalize it to t r = 1, as is the case in SU. Using this, we can calculate the quadratic Casimir operators in both the defining and the adjoint representation. In order to do this, let us derive a useful relation for the product of two SUN generators [9]. Consider an arbitrary hermitian N N matrix M, given by M = m 1 N + m a t a, m, m a R, 1.17 where 1 N is the N N unit matrix. Using the fact that the SUN generators are traceless, we find after calculating the trace of M that m = 1 TrM N Using this expression for m, together with the tracelessness of t a, we can find an expression for m a by evaluating [ ] Trt a M = Tr t a m 1 N + m b t b = m b Trt a t b = 1 m bδ ab = 1 m a,

12 CHAPTER 1. INTRODUCTION: A BRIEF OVERVIEW OF QCD where we used equation 1.1. The equations for m and m a now allow us to write M = 1 N TrM1 N + Trt a Mt a. 1. Let us now consider the i, j entry of the matrix M. It is given by M ij = 1 N M kkδ ij + t a klm lk t a ij [ 1 ] = M lk N δ lkδ ij + t a klt a ij = M lk δ il δ jk, 1.1 so that we can write an expression for the product of two generators t a ij ta kl = 1 δ il δ jk 1 N δ lkδ ij. 1. Let us put this equation to use. The Casimir operator in the defining representation is given by δ ij C F = t a t a ij = t a ikt a kj = 1 δ ij δ kk 1 N δ ikδ kj = 1 N 1 N δ ij, 1.3 where we used the fact that the defining representation has dimension N. We thus find the quadratic Casimir operator in the defining representation C F = N 1 N. 1.4 We can also calculate the Casimir operator in the adjoint representation. In this case we have t b At b A = ad tb A ac t b A cd = f bac f bcd = f acb f dcb = δ ad N = δ ab C A, 1.5 so that we see that the Casimir operator in the adjoint representation is given by C A = N. 1.6 The short discussion on group theory just given should be enough for one to understand all group theory appearing in this thesis. 1.. Quarks and Gluons The two important particles in QCD are the quarks and the gluons, the quarks being the constituents of hadrons and the gluons being the gauge particles that bind the quarks. In this section, we will follow [3] and [5]. QCD, as mentioned before, is a non-abelian gauge theory with gauge group SU3, which we just discussed. Since SU3 has eight generators, there should be 8 gauge fields A a µ. Furthermore, we need the quark fields to have 3 components to transform nontrivially under SU3 gauge transformations. For a given flavor, we thus have three different fields. The difference between these fields is determined by the color of a quark, so that a quark has an additional degree of freedom, which can take on the values red R, green G and blue B. This should, however, mean that the quark bound states, the hadrons, also carry color and are therefore degenerate. For 6

13 1.. QUANTUM CHROMODYNAMICS instance, the pions are bound states of u or d quarks with ū or d antiquarks. This means that for each pion of a given charge, there would be nine types with the same mass, and there would be a total of 7 types of pions in nature. However, we only find 3 types of pions in nature. This is not the only problem we have. Assume there is a bound state of three u quarks. These could form a baryon with spin- 3, if all spins are in the same direction. When this happens, however, this state is symmetric under the interchange of two quarks, while the Pauli principle states that it should be antisymmetric, because quarks are fermions. One could thus say that this state does not occur in nature. However, experiment revealed the existence of the ++ baryon, which consists of 3 up-quarks and has spin- 3. Furthermore, it appeared to be a universal feature of hadrons that they are symmetric under the interchange of two quarks. The quark model thus seems to violate the Pauli principle. There is a principle that solves both problems just introduced. This principle is that hadrons should be colorless. That is, baryons should consists of a red, a blue and a green quark, and the mesons should consists of a quark of color i and an antiquark of color ī. This solves the mystery of the degeneracy of hadrons: the only possible hadrons that we find in nature are the colorless ones. But this is not all. When we combine three quarks, which can carry three different colors, we have the group-theoretical formula [6] = This means that three quarks can combine to symmetric states, states of mixed symmetry, and a singlet antisymmetric state. This antisymmetric state is the state which would turn out to be colorless. Therefore, quarks states are antisymmetric with respect to color. This means that, to satisfy the Pauli principle, they should be symmetric with respect to spin and other quantum numbers. We can show the same thing for mesons, where we have the group-theoretical formula [6] 3 3 = from which we see that there is again an antisymmetric singlet state which corresponds to a colorless meson. Besides quarks, QCD contains also gauge particles; eight different gluons. One may understand this number by remembering that in the previous section we explained that SU3, the gauge theory on which QCD is based, has 3 1 = 8 generators. Since the gluons in the QCD lagrangian to be described below appear only in the combinations A µ = A a µt a, this means that there are eight gluons. The difference between the various gluons comes, as one might expect, from their color values. Gluons do carry color, so possible singlet states are excluded. One may than verify that the different color combinations of the gluons are [5] RG, RB, GR, GB, BR, BG, 1 RR GG, 1 RR + GG BB Although we know that quarks are the building blocks of hadrons, we never managed to isolate one in experiment. The force that binds quarks together in hadrons increases with distance. Together with the principle that hadrons carry no color, we call this color confinement [3]. This makes it difficult to work with QCD, since perturbative expansions of QCD interactions are no longer convergent since the coupling is so strong. There is, however, also a phenomenon known as asymptotic freedom [3]. This means that at very short distances, quarks and gluons behave as free particles. Therefore, it is possible to treat QCD perturbatively at high energies. Since we can not treat QCD perturbatively at long distances, it is very hard to prove color confinement, but it is possible to prove asymptotic freedom. This can be done using the renormalization group, as we will show later in this chapter. 7

14 CHAPTER 1. INTRODUCTION: A BRIEF OVERVIEW OF QCD 1..3 The QCD Lagrangian We will now derive the QCD lagrangian, from which the Feynman rules of perturbative QCD are derived. The reasoning in this section comes down to the standard explanation of a non-abelian gauge theory which may be found in [1,, 3, 4]. Since QCD describes quarks, which are fermions, we start with the familiar Dirac lagrangian L = qx qx m qxqx. 1.3 To find the gauge fields, we demand that this lagrangian is invariant under the following local gauge transformations [ qx exp iα a xt a] qx, 1.31 where the SU3 group generators t a obey the following commutation relations, as in equation 1.11 [t a, t b ] = if abc t c, 1.3 with the structure constants f abc completely antisymmetric. In order for this to work, we need to add gauge fields A a µ to our lagrangian which transform in an appropriate way. We find that our Lagrangian gets the following form L = qxd qx m qxqx, 1.33 where we define the covariant derivative to be D µ = µ iga a µt a and let the gauge fields transform like A a µ Aa µ + 1 g µα a + f abc A b µ αc After inserting the covariant derivative in our lagrangian, it is invariant under the gauge transformation of equation 1.31, as it should be, but our lagrangian is not finished yet. We found that we need gauge fields in order for the lagrangian to be invariant under local SU3 gauge transformations, so we can add terms quadratic in the gauge fields that are gauge invariant. We therefore first define the non-abelian field strength to be F a µν = µ A a ν ν A a µ + gf abc A b µa c ν The transformation of the field strength can easily be derived by noting that it F µν is given by the commutator of two covariant derivatives [ Dµ, D ν ] = igf a µν t a = igf µν Using the transformations of qx and A a µ, one easily checks that the covariant derivative on the quark field transforms exactly the same as the quark field itself [ D µ qx exp iα a xt a] D µ qx, 1.37 which explains why this derivative is called a covariant derivative. This means, however, that if we write the transformation as a matrix Ux, the covariant derivative alone should transform as so that the commutator of two covariant derivatives transforms as which gives us the transformation of the field strength. D µ UxD µ U x, 1.38 [ Dµ, D ν ] Ux [ Dµ, D ν ] U x,

15 1.. QUANTUM CHROMODYNAMICS Since F a µν transforms according to F a µν exp [ iα a xt a ] F a µν exp [ iα a xt a ], we see that the simplest gauge invariant kinetic term for the gauge fields is the trace of F a µν, since the trace is cyclic. We thus find the following lagrangian L = qxd qx m qxqx 1 4 Tr [ F µν F µν ], 1.4 where we used the abbreviation F µν = F a µν ta. We are not finished, since the inverse gauge field propagator we would find for this lagrangian is a matrix which is not invertible. Therefore, we add a gauge fixing term to the lagrangian L = qxd qx m qxqx 1 4 Tr [ F µν F µν ] λ ξ A, 1.41 which allows us to define gauge field propagators for various choices of λ and ξ µ. This gauge fixing term, however, poses new problems, since a mismatch between numbers of degrees of freedom arises from it [, 1, 3]. To solve this problem, we add so-called ghost fields, first introduced by Feynman, DeWitt, Faddeev and Popov [16, 17, 18]. These are unphysical fields since they are anticommuting scalar fields obeying Fermi-Dirac statistics, and they precisely cancel the problematic terms generated by the gauge fixing term. Adding the ghost fields to equation 1.41, we have L = qxd qx m qxqx 1 4 Tr [ F µν F µν ] λ ξ A + ξ µ c a µ δ ad gf abd A µ b cd = L Dirac + L Gauge Fields + L Gauge Fixing + L Ghost, 1.4 where c d and c a are the ghost and antighost fields, respectively. Equation 1.4 is our definitive QCD lagrangian, and the second line makes clear that it is constructed from various terms, namely the bare Dirac lagrangian, the lagrangian describing the gauge fields, a gauge fixing lagrangian and a ghost lagrangian. From equation 1.4, we can calculate the Feynman rules of QCD, which we can use to calculate Feynman diagrams in perturbative QCD. These Feynman rules are given in, among others, the appendix of [] Dimensional Regularization and Ultraviolet Renormalization As, for instance, Quantum Electrodynamics, Feynman diagrams in perturbative QCD may contain singularities. We may distinguish between two types of singularities: ultraviolet singularties, coming from the high-momentum limit of the loop integrals, and infrared singularties, coming from the zero-momentum limit of the loop integrals [, 1]. It is a well-established result that the ultraviolet singularties in Feynman diagrams may be absorbed into the various parameters appearing in the lagrangian of the theory one is considering, by calculating counterterms and cleverly choosing Z-factors multiplying the parameters of the lagrangian. The ultraviolet singularties are then absorbed into the parameters of the lagrangian, and the theory one is considering is rendered ultraviolet finite. This technique is called renormalization, and for a more detailed discussion on this subject one should consult, for instance, [1, ]. We will only discuss it briefly since we assume that the reader is familiar with it. The counterterms of an ultraviolet divergent diagram are calculated using dimensional regularization. This technique changes the number of space-time dimensions to n = 4 ǫ, which allows us to do loop integrals that are divergent in n = 4 dimensions. The result will then contain a term proportional to 1/ǫ, which is clearly a singularity when one goes back to n = 4 by taking the limit ǫ. These singularties are then absorbed into the parameters of the lagrangian. Clearly, when doing ultraviolet renormalization, ǫ >, since the integral becomes singular in the highmomentum limit due to the fact that the power of the momentum in the numerator containing the integration volume is larger than or equal to the power of momentum in the denominator. When we work in n dimensions, we need to consider the mass dimension of the coupling g. In n = 4 one may easily verify that the mass dimension of g is, using the fact that the mass 9

16 CHAPTER 1. INTRODUCTION: A BRIEF OVERVIEW OF QCD dimension of the lagrangian is 4. In n = 4, one then finds that the mass dimension of g should be ǫ, and therefore, when working in dimensional regularization, we need to replace g gµ ǫ, where µ is a mass scale known as the renormalization scale [1]. This scale will turn out to play an important role in the analysis of a renormalized theory. The definition of µ defines the renormalization scheme. The most simple renormalization scheme one can think of, is one in which µ is chosen to be an independent parameter that is the same for every divergent integral. This renormalization scheme is called the minimal subtraction MS scheme. In this thesis, however, we will use the modified minimal subtraction MS scheme, which differs from the MS-scheme by a factor exp[ γ E + ln4π], where γ E is Euler s constant [1]. The reason for this definition has to do with the result of a dimensional regulated integral. Such results are proportional to 1 ǫ γ E + ln4π, 1.43 so that by using the MS-scheme, the contributions γ E and ln4π are absorbed in the definition of the renormalization scale µ, and the dimensional regulated integral will be only proportional to 1/ǫ. The introduction of a renormalization scale µ with the dimension of mass will lead us to a very powerful equation, called the renormalization group equation [1,, 4]. We will make very much use of it in the rest of this thesis, so we will derive it, along with its implications, here. Let us first look at a non-renormalized vertex function. Since it is not renormalized, it is independent of the renormalization scale µ, but we know that we can renormalize it by multiplying it by a Z-factor. We get µ d dµ Γn = µ d [ n/ Z φ gµ ǫ Γ n r p i, g r, m r, µ ] =, 1.44 dµ where the subscript r denotes whether or not a quantity is renormalized. We can perform this differentiation to find where we defined [ nγg + µ µ + βg ] + mγ m Γ n r =, 1.45 g r m r γg = µ µ ln Z φ 1.46 βg = µ g r µ 1.47 mγ m = µ m r µ Equation 1.45 is what we call the renormalization group equation. From now on, for simplicity, we will not write the subscript r anymore since we will always be dealing with renormalized quantities. The important points for us, however, are the equations 1.47 and They describe the variation of the mass m and coupling g with respect to µ. This means that the coupling and quark mass in QCD depend on the scale one is using. Let us now try to quantify this. We first turn to βg, which we defined as [1, 4] βgµ = µ µ gµ = µ µ Z 1 g g The function βg is a perturbative function, and satisfies an expansion of the form [1] 1

17 1.. QUANTUM CHROMODYNAMICS g3 βg = ǫg 4π n= g n, b n 1.5 4π where the linear term in g comes from the fact that in dimensional regularization, the coupling comes with a factor µ ǫ, as we discussed earlier. It is now straightforward to calculate βg to one loop, since we can just calculate all Feynman diagrams and from there calculate Z g. We will not do that here, but just give the result in n = 4 dimensions to lowest order in g [1] βg = g3 11CA + n F 16π = b g 3, where n F is the number of quark flavors. Let us now go to equation If we insert our expression for βg, we end up with a simple differential equation for gµ. We can solve it, and find [1] gµ = 16π b lnµ /Λ, 1.5 where Λ is defined by Λ = µ exp[ 8π /b gµ ], and b = 1/16π [11C A n F ]/3. This means that we find the following expression for the running coupling α s µ = 4π b lnµ /Λ We thus see that for large µ, the coupling goes to zero. The renormalization scale µ, however, has the dimension of mass, so we may think of it as some momentum scale. We then find the famous concept of asymptotic freedom: for very high momentum scales, that is for very short distance scales, quarks behave like free particles. Let us now look at equation This is also a differential equation that is not hard to solve. We find [4] [ mµ = mµ µ exp dλ ] µ λ γ mgλ We see that, besides the running coupling, we also have a running mass in QCD. We know that γ m g is a perturbative quantity, just like βg, so it vanishes for very large µ. Therefore, for µ, we find that the quark mass goes to zero. In our perturbative treatment of QCD, we are forced to work with large µ to have a small coupling, and in this region, the masses of the light quarks for instance, u and d, vanish. The heavy quarks like t, however, still have some mass, but in this thesis we will only consider the u and d quarks so that we may treat all our quarks as massless fields. Now that we discussed both the parton model and QCD, we can ask ourselves if they are consistent with each other. In the parton model, we found that at high energies, partons behave like free particles. This was a mystery for a time, but in the 7 s non-abelian gauge theories, of which QCD is an example, were discovered. It turned out that these are the only theories in 4 dimensions which have asymptotic freedom, which could be calculated from the renormalization group. This explained the high energy behavior of partons. Apparently, quarks were bound in hadrons by gauge fields vector bosons, which we now call gluons. There is also the phenomenon of Bjorken scaling in the parton model. This also appeared in QCD, but not exactly: since the coupling only vanishes for infinite momenta, there are always some corrections to Bjorken scaling due to the emission of high momentum gluons. This was, however, confirmed by experiments in the 7 s, which showed that Bjorken scaling is only an approximate scaling, since there are violations corresponding to the evolution of the parton distribution functions. This can also be seen in figure

18 CHAPTER 1. INTRODUCTION: A BRIEF OVERVIEW OF QCD With the basis of QCD we just discussed at hand, we will now start with our discussion on infrared divergences in QCD. 1

19 Chapter Finding Infrared Singularities In the introduction we gave a brief overview of what Quantum Chromodynamics is and does. We will now start with discussing the subject of this thesis, namely the infrared singularities arising in perturbative QCD. We know that the evaluation of Feynman diagrams in gauge theories may generate singularities, when an integration over loop momenta is performed. The divergences associated with the high-momentum limit of these integrations are discussed in the previous chapter, and we will assume for the rest of this thesis that ultraviolet renormalization has been carried out, so that our theory is free of ultraviolet singularities and we take ǫ <. In QCD, we find besides the mentioned ultraviolet divergences also infrared divergences, of which there are two types: divergences associated with very soft loop momenta, called the soft divergences, and divergences associated with loop momenta collinear to one of the external particles, called collinear divergences. In this chapter, we will introduce them, show how we can find them, and discuss some of their implications..1 How to find Infrared Divergences In this section, we will first give an example of infrared divergences. We will show the difference between collinear and soft divergences and we will discuss how they arise. After that, we will discuss some ways of tracking down these divergences without explicitly calculating integrals..1.1 Soft and collinear divergences We will start by calculating a typical vertex with loop momentum k in massless scalar φ 3 theory []. It is given in figure.1. Using the familiar Feynman rules in scalar theory [], we find the following integral in scalar theory I = g 3 µ 3ǫ d n k 1 π n k + iǫp 1 k + iǫp + k + iǫ..1 We can calculate this integral using familiar techniques that can be found in, for instance, [1,, 3], and we find a function that is ultraviolet finite but contains infrared poles I = igµ ǫ 1 g 4πµ ǫγ1 B ǫ, 1 ǫ + ǫ q 4π q,. iǫ ǫ where Bα, β is the Euler Beta function [1], defined as Bα, β = 1 dyy α 1 1 y β 1 = ΓαΓβ Γα + β..3 The function I clearly diverges for ǫ. Let us now consider the QCD-diagram of figure.1. Note that one could imagine a second one-loop contribution, namely, a self energy diagram on 13

20 CHAPTER. FINDING INFRARED SINGULARITIES p 1 p 1 q k q k p p Figure.1: Vertex with incoming momentum q, outgoing momenta p 1 and p, and loop momentum k, in scalar theory left diagram and QCD right diagram. one of the external legs, but as one can easily check, such diagrams in a massless theory vanish in dimensional regularization. After a short calculation of the vertex correction, we find where Γ µ q, ǫ = ieµ ǫ ūp 1 γ µ vp Fq /µ, ǫ,.4 Fq /µ, ǫ = α s 4πµ ǫγ π C 1 ǫγ1 + ǫ [ 1 F q iǫ Γ1 ǫ ǫ + 3 ] ǫ is a scalar function, as it should be, since any other kind of function would spoil the gauge invariance of the QED-process of quark pair production. Obviously, QCD processes should not spoil gauge invariance. Just as I, the function F also diverges for ǫ. Moreover, the two equations. and.5 look very much alike for the leading double pole 1/ǫ, which can be seen from the expansion of the function Γx for small x. We will now try to understand where these poles come from. Consider again the integral leading to equation. I d n k 1 π n k + iǫp 1 k + iǫp + k + iǫ..6 We work with a Feynman integral found from a diagram in scalar theory since we are, for the moment, only interested in the denominator of such integrals. We now introduce so-called lightcone coordinates defined by the product of two spacetime vectors a and b according to a b = a + b + a b + a b 1. Assuming that p 1 and p are in opposite directions, with p 1 in the plus-direction and p in the minus-direction, we use the so-called eikonal approximation [], that is, we neglect k compared to the inner products of k with p 1 and p. The eikonal approximation means that one only keeps the denominator term that has the strongest singularity when, for instance, a loop momentum becomes soft, so we keep the product of a vanishing and a non-vanishing momentum component and drop the product of two vanishing momentum components, since the former is dominant over the latter. Applying the eikonal approximation to our integral, we find I dk + dk d k k + k k iǫ p+ 1 k + iǫp k+ + iǫ..7 Using that q = p 1 + p, so that q = p 1 p in a massless theory, and extracting this from the denominator gives 1 The profit of lightcone coordinates is that after choosing the momentum in question to be in one specific direction, the four-momentum has only one nonzero component instead of two. 14

21 .1. HOW TO FIND INFRARED DIVERGENCES I 1 q dk + dk d k k + k k iǫ k + iǫk + + iǫ..8 From this integral, we can identify three regions of logarithmic divergence [1]. Let us make this explicit. All components of k µ are of order q, and we scale them with a variable λ which vanishes at the points where I becomes singular. In the first region, we scale all components of k µ with λ, k µ λ q. This is called the soft region, since all components of k µ vanish. In the second region, we scale k with λ, k + also with λ and k with λ. The third region is the same, but with k + and k switched. The second and third regions are called collinear regions, since in this regions the component of k µ parallel to p 1 or p remains finite, while the others vanish. Summarized, we use the following scalings k ± q k λ q, k µ λ q k λ q Here, the left column are the collinear regions, and the right column is the soft region. Equation.8 gets, in all the three regions, the following form after rescaling I 1 d 4 λ q λ 4..9 We see that this integral diverges logarithmically when λ vanishes. We thus conclude that the 1/ǫ poles found earlier come from loop momenta that are very soft, or from loop momenta collinear with one of the outgoing momenta. The 1/ǫ poles come from loop momenta that are very soft, and at the same time are collinear with one of the outgoing momenta..1. The Landau equations We have now identified singularities for one particular vertex. One may wonder if it is possible to find some criterium for finding singularities that is applicable to any process. In order to find this, let us turn for a moment to the definition of Feynman parameters [] 1 A 1 A...A n = 1 dα 1... dα n δ Applying this to our equation.6, we find where D is given by I = i α i 1 n 1! [α 1 A 1 + α A α n A n ] n..1 d n k 1 dα 1 dα dα 3 δ 1 3 i=1 α i π n D 3,.11 D = α 1 k + α p 1 k + α 3 p + k + iǫ = 3 α i li..1 Here, we have defined line momenta l i, which are nothing more than just the momenta along every line in the Feynman diagram. We also note that I is ultraviolet finite, so that the poles arise due to the vanishing of D. Let us now look at the complex k µ plane. This is pictured in figures.a to c. Normally, propagator poles are distributed like in figure.a. If we want to calculate an integral in this plane, we can use Cauchy s theorem [8], which states that we may always deform the integration contour as we like, as long as we do not cross a singularity. This is done in figure.b. The 15 i=1

22 CHAPTER. FINDING INFRARED SINGULARITIES contour is deformed in such a way that the integral does not feel the poles, and the result of the integration will be finite. In equation.11, however, we are not dealing with a normal propagator, and poles may be distributed in some other way. A possibility is that the poles are distributed like in figure.c. Here, it is not possible to deform the contour away from the poles, since we should cross a singularity. This integration would thus give us singularities. Situations like this, in which we cannot deform the integration contour away from the poles, are called pinches [1]. Figure.: Integration contours in the complex k µ plane. The poles are indicated by the black dots, and the integration contour is indicated by the red arrowed line. How do we know if there are pinches in an integral? Let us take a closer look at D. D is quadratic in momenta, so we can think of it as a parabola. Also, the poles which correspond to D = are very close to each other, so we may say that they coincide. This means that an integral has pinched singularties if is satisfied. Using equation.1, we find k µ D = at D =.13 D = k µ i α i k µ l i = i α i l i ǫ is =,.14 where ǫ is is called an incidence matrix, which is +1 if l i is in the same direction as k, -1 if l i is in the opposite direction to k, and elsewhere. It is not hard to find when the function D vanishes. For D to vanish, each of its terms should vanish, which obviously happens when α i = or when the line momentum goes on shell, li = m, where, of course, in a massless theory one would write li =. Together with equation.14, we see that we find a pinch, and therefore a possible divergence in an integral, when l i = m i or α i =.15 and i α i l i ǫ is =.16 These equations are called the Landau equations [1], and if they are satisfied, the integral can be singular. To check the Landau equations, let us apply them to equation.6, so that D is given by equation.1. It is straightforward to see that the Landau equations give α 1 k µ α p 1 k µ + α 3 p + k µ =

23 .1. HOW TO FIND INFRARED DIVERGENCES One solution is exactly the soft region we discussed before k µ =, Another two solutions give us the collinear regions α α 1 = α 3 α 1 =..18 This is exactly what we would expect. k = ζp 1, α 3 =, α 1 ζ = α 1 ζ.19 k = ζ p, α =, α 1 ζ = α 3 1 ζ The Coleman-Norton approach We now have a technique to track down pinches, but there exists an easier technique, which is due to Coleman and Norton []. First, we make two identifications α i l µ i = xµ i, From this we see that we may write x µ i as α i = x i l i x µ i = x i v µ i, with vµ i = 1, l i li..1. We recognize that we can interpret x µ i as a four-vector describing the free propagation of a classical on-shell particle with momentum l i. Now, consider again the left diagram of figure.1. We can contract some lines of the loop. Diagrams constructed in this way, are called reduced diagrams [1]. The possible reduced diagrams from the left diagram of figure.1 are shown in figure.3. Figure.3: Reduced diagrams. The diagrams in the first row have one line contracted, diagrams in the second row have two lines contracted, and the diagram in the third row has all three lines contracted. We see that only the first two diagrams in the first row and the diagram in the third row describe the propagation of a classical particle. The other diagrams do not, since classical free particles 17

24 CHAPTER. FINDING INFRARED SINGULARITIES do not leave and come back. Let us now take a closer look at the classical diagrams. The first has x µ p 1 k xµ k =, since after these two paths one arrives back at the starting point. Using equation., we find that α p 1 k µ α 1 k µ =, which is the Landau equation for α 3 =! The second diagram gives in exactly the same way the Landau equation for α =. Finally, the third diagram gives us k µ = α = α 3 =, which is the soft solution to the Landau equations. What do we conclude from this? Clearly, reduced diagrams which describe classical propagation of free particles give solutions to the Landau equations. Therefore, if we find reduced diagrams which describe classical particles, we have found a pinch. This approach by Coleman and Norton is a very powerful device for tracking down pinches.. Power Counting In the previous section, we discussed how we can track down pinches, necessary requirements for singular integrals. We found a very powerful way to find these pinches, namely the approach by Coleman and Norton, but these pinches are just a necessary condition for singularities. Integrals with pinches may be finite, so we need something more to tell wether or not an integral is going to produce a singularity. We will use power counting for this purpose, a technique that we will introduce in this section by following [3]...1 General Formalism of Infrared Power Counting Let us start with the general formalism of power counting, and apply it to the process of pair creation as an example. First, we imagine that a pinch is on a surface normal to the integration contour [1]. We are interested in the behavior of the integral near this pinch surface. The integration variables are separated into two sets here, namely intrinsic coordinates and normal coordinates. The normal coordinates are the ones the integral is singular in, and the intrinsic coordinates are just to parameterize the pinch surface. We now go through the following steps. 1. As before, we redefine each normal coordinate κ j in terms of a scaling variable λ, which vanishes at the poles. This gives κ j = λ aj κ j.. We expand each denominator term to lowest order in λ, k i κ j, λ m i = λai fκ j. Note that step corresponds to doing the eikonal approximation. After these two steps, the resulting integral is called the homogeneous integral. For a pinch surface S, the homogeneous integral is now proportional to λ ns, with the degree of divergence n S given by n S = i a i j A j + S I,.3 where S I is a possible power of λ from momentum factors in the numerator these factors may arise, for instance, from three-gluon vertices. We now have three possible kinds of behavior of our integral at the pinch surface. For n S >, the integral is finite for λ, for n S = our integral may diverge logarithmically, and for n S < our integral may diverge as a power. Finally, we should check if we can find any pinches in the homogeneous integral. For pinches other than the one we just analyzed, we have to repeat the formalism. If we can t find any, we are done. Let us first, as an example, go back to equation.6. We first look at the region of soft divergence of this integral. In this region, we use k µ as normal coordinates and scale them as k µ = λk µ. Using the same approximation as after equation.6, the denominator terms become [11] k λ p 1 k = p + 1 k λ p + k = p k+ λ. 18