Improvements to Froissart bound from AdS/CFT correspondence
VER ´ONICA ERRASTI D´IEZ Supervisor: Rohini M. Godbole
Examiner: Edwin Langmann
Master of Science Thesis Department of Theoretical Physics
Royal Institute of Technology Stockholm 2013
Master of Science Thesis
Improvements to Froissart bound from AdS/CFT correspondence
VER ´ONICA ERRASTI D´IEZ Supervisor: Rohini M. Godbole
Examiner: Edwin Langmann
Mathematical Physics, Department of Theoretical Physics Royal Institute of Technology, SE-106 91 Stockholm, Sweden
Stockholm, Sweden 2013
Typeset in LATEX
Graduation thesis on the subject Physics for the degree of Master of Science in Engineering
TRITA-FYS 2013:47 ISSN 0280-316X
ISRN KTH/FYS/–13:47–SE c
Ver´onica Errasti D´ıez, September, 2013
Printed in India by Srivari Xerox Centre, Bangalore 2013
To
my mom, my dad and my brother Miguel, with love.
Declaration
I hereby declare that the work presented in this thesis entitled “Improvements to Froissart bound from AdS/CFT correspondence” is the result of the investigations carried out by me under the supervision of Prof. Rohini M. Godbole at the Centre for High Energy Physics, Indian Institute of Science, Bangalore, India and that it has not been submitted elsewhere for the conferment of any degree or diploma of any Institute or University. Keeping with the general practice, due acknowledgements have been made wherever the work described is based on other investigations.
September 2013 Verónica Errasti Díez
Certified
Rohini M. Godbole Professor
Centre for High Energy Physics, Indian Institute of Science, Bangalore-560012,
Karnataka, India
iii
Abstract
We review the classical derivation of the Froissart bound in S-matrix theory. We discuss what are the essential assumptions required to obtain the bound and show how the bound breaks for scattering processes mediated by massless particles.
We also review the derivation of Froissart bound from AdS/CFT, first proposed by Giddings. Further work was done by various people to improve the understand- ing.
We take into consideration the effect of the first excitation of the graviton to the scattering amplitude, not only its ground state as compared to Giddings. We further improve the estimate of the black hole horizon radius. By adding a Weyl- squared term to the usual Einstein-Hilbert gravity action, we compute the correction to Froissart bound perturbatively. This accounts for corrections coming from the consideration of a finite coupling in the dual CFT. We find a logarithmic term in the energy is added to the scattering cross-section. Such a term is usually included in QCD-based models to produce a better fit to the pp and p¯p experimental data.
It is yet to be explored whether our corrections indeed lead to a better description of the available data.
Key words: Froissart bound, AdS/CFT correspondence, Einstein-Weyl gravity.
v
Acknowledgements
The present thesis has been developed in the Indian Institute of Science (IISc), Bangalore, under an exchange programme from the Royal Institute of Technology (KTH). I would like to thank the former institute for hospitality and the latter one for giving me this opportunity.
I am happy to have had Prof. Rohini M Godbole as my supervisor in this time.
I have learnt a lot from discussions with her. Her vast knowledge of physics and remarkable clarity in conveying ideas have helped me broaden my understanding of physics. All through my stay she has taken care of me, helping me find accom- modation and granting me an extension to deepen into the topics discussed in the present work. She has also taken special interest in my future, supporting my PhD applications and broadening my options. I am indebted to Prof. Rohini M God- bole for giving me enough knowledge to start exploring physics while making sure I would discover many other aspects on my own. She has also introduced me to other colleagues of her, ensuring I would benefit from my exposure to different points of view on the same problem.
Among the collaborators I got to discuss with, I would like to emphasize the input to this thesis of Prof. Aninda Sinha, who has contributed his own ideas to this work. His door was always open for me. Very patiently, he has allowed me to scratch on his blackboard all my occurrences, helping me understand where and why I was wrong and supporting me otherwise. Prof. Aninda Sinha has an enthusiasm for physics which is contagious: it pushes his students forward and inevitably leads them to enjoy their work. I am very grateful to him for his constant help in the second part of the present text.
Prof. Edwin Langmann is responsible for the grading of this thesis. I am thankful to his constant interest in my progress and encouragement. He has been extremely helpful in presenting the contents of my work and supporting my PhD applications.
Prof. Suvrat Raju agreed to give crash-courses on AdS/CFT correspondence and suggested very useful references. I am fortunate to have had him introduce me to the exciting area of gauge/gravity dualities.
During my stay in IISc I attended the Group Theory course imparted by Prof.
Sachindeo Vaidya. I learnt a lot from his insightful classes and I thank him for his suggestions and help in my search for a PhD position.
I have had the fortune of sharing office with Nirmalendu. His help and support vii
viii
have been essential to this thesis. It is with him that many mathematical details have been sorted out and it is conversing with him that I have gained considerable insight in physics. Indeed, this thesis would have not been possible without him. I very much cherish every discussion, coffee and dinner I have had with Nirmalendu.
He has become a very dear friend to me.
For invaluable help with the contour figures, I must thank my close friend Clau- dio Torregrosa. He is a source of support and energy to me and our friendship a treasure I very much value.
Specifically, this thesis and the coursework have taken place at the Center for High Energy Physics (CHEP) of IISc. I am particularly grateful to both faculties and students in the department for their welcome and constant care. In particular, I would like to thank Prof. B Ananthanarayan as chairman of CHEP.
For help with a shady gauge fixing I must thank my good friend Ananyo, in whose company many memorable nights have been spent, along with Ranjan. For long hours of heated discussions trying to solve homeworks, I want to thank my classmates Abhiram, Ranjani, Parveen and Avinash. My seniors Sandeep and Gau- rav have helped me with the fits shown in this work. Their patience and kindness have always made it easy to approach them with any doubt. I am grateful to Kir- timaan and Kallol for useful references. There are many people who have made of my stay a memorable one. In CHEP I would like to particularly thank Wrick, Nitin, Chandrasekhar, Apratim, Anjani, and Monalisa. In the Physics Department Amit, Jamshid, Rupamanjari, Debarghya and Semonti are to be acknowledged. I only have good words for them.
My stay in IISc would have not been as pleasant and exciting had I not met my friends Punya and Sathish. The nights spent with Gaussi, Senthil and Jatin are among the best memories of IISc I will take back with me. For a warm welcome upon arrival Rahul and Rati are to be thanked. Rituparno kindly lent me a bicycle with which I have wandered all around.
I am thankful to Raju for a wonderful visit to his country Nepal, I enjoyed many adventures in his company. In this trip I was warmly hosted by the excellent cook Jyoti and by Pashupati and Milan. Many thanks to Kshitiz for crazy motorcycle trips as well. I am most fortunate to have been hosted by Yogesh in Rohtak and Akul in Chennai. Their families welcomed me in their houses and beautiful parts of India were visited in their company. Specifically, for a breathtaking road trip to Agra and Jaipur my gratitude to Yogesh, Yugdeep and Rohan. Chennai and Pondicherry were explored in the fine company of Akul, Affan and Anushka.
My first three years of higher education were spent in Madrid. My dear friends and excellent physicists Jorge, Nico and Nacho were key to my interest in pursuing a Master’s degree in physics. Their company and conversations are one of the greatest joys of my student life. My friend Ana must be thanked for hosting me in Madrid in each and every of my visits. Her support and friendship always have a place in my heart wherever I go.
Last but not least, I would like to thank my family: my mom María José, my dad Antonio and my brother Miguel. My mom’s support and love know no boundary.
ix
It is in her that I find strengh in the difficult moments and it is towards her that I first turn when good news come. Through our invisible connection none of us can ever feel alone. I am most proud if I have imbibed some of this extraordinary woman’s wonderful qualities. My dad always finds time and interest to listen to my advances in physics while sharing his own experiences. His strong sense of right and wrong have, hopefully, made a good person of me. I treasure every moment we have shared. My brother Miguel is possibly the greatest force pushing me forward.
His belief that no physical problem is too difficult for me fills me with courage to attempt a solution to any given task. I shall keep on working hard to try to meet his high expectations from me, but so far he will have to be content with a Master’s degree.
Contents
Contents xi
List of Figures xiii
1 Introduction 1
I Froissart bound:
the field theory approach 3
2 Lorentz symmetry 5
2.1 Lorentz transformations . . . . 5
2.1.1 Relativistic normalization of momentum states . . . . 7
2.2 Mandelstam variables . . . . 7
3 Scattering cross-section 11 3.1 S- and T -matrices . . . . 11
3.2 Total cross-section . . . . 12
3.3 Optical theorem . . . . 15
3.4 Analyticity of the scattering amplitude . . . . 16
4 Froissart bound 19 4.1 Froissart-Gribov formula . . . . 19
4.1.1 Evaluation of the contour integrals . . . . 20
4.2 Upper bound to |An(s)| from the analyticity of the scattering amplitude 23 4.3 Upper bound to |An(s)| from the unitarity of the S-matrix . . . . . 25
4.4 Comparison of the bounds to |An(s)| . . . . 26
4.5 Upper bound to the total scattering cross-section: Froissart bound . 28 4.6 Exchange of a massless particle . . . . 30
4.7 Experimental validity of the Froissart bound . . . . 31 xi
xii CONTENTS
II Froissart bound:
the AdS approach 35
5 Froissart bound from AdS/CFT 37
5.1 Einstein-Hilbert action . . . . 38
5.2 Gauge fixing . . . . 42
5.3 Boundary conditions . . . . 45
5.4 Solution to the equations of motion . . . . 46
5.5 Gauge/gravity duality: the Froissart bound . . . . 51
6 Einstein-Weyl gravity: AdS/CFT and corrections to Froissart bound 53 6.1 Einstein-Weyl gravity in 5 dimensions . . . . 54
6.2 Perturbative solution to the equations of motion . . . . 56
6.2.1 Zeroth order solution . . . . 56
6.2.2 First order solution . . . . 61
6.3 Gauge/gravity duality: the Froissart bound, the next-to-leading- order term and corrections . . . . 62
7 Discussions and outlook 65 A Appendices 67 A.1 Delta function boosted in the z-direction . . . . 67
A.2 Schwarz reflection principle . . . . 67
A.3 Legendre functions . . . . 68
A.3.1 Orthogonality and normalization of the Legendre polynomials 68 A.3.2 Legendre functions of the second kind . . . . 70
A.4 Evaluation of a useful series . . . . 71
A.5 Bessel functions . . . . 72
Bibliography 75
List of Figures
2.1 a) s-channel b) t-channel c) u-channel . . . . 8 3.1 Poles tp, up and branch points tb, ub with the associated branch cuts in
the complex t-plane for the equal mass scattering for a given, fixed s. . 16 4.1 Integration contour for (4.4) in the complex zs-plane. . . . 21 4.2 (4.65)-(4.67) fit results to σtotpp predictions (top) and to σtotp¯p (bottom). . . 32 4.3 (4.66), (4.68) fit results to σpptot predictions (top) and to σtotp¯p (bottom). . 33 5.1 Integration contours for (5.78) and (5.79) in the complex |~p|-plane. . . . 50 6.1 Integration contours for (6.45) and (6.46) in the complex |~p|-plane. . . . 60 A.1 Plot of Bessel functions of the first and second kind Jα(x), Yα(x), for
integer orders α = 1, 2, 3. . . . 73
xiii
Chapter 1
Introduction
The prime focus of this thesis is the Froissart bound, which is the statement that the total hadronic cross-section cannot grow faster than the square of the logarithm of the energy in the center-of-mass frame:
σtot. π
m2π ln2 E
E0 in the large E limit,
where mπ is the mass of the pion, E is the center-of-mass energy and E0 is some unknown energy scale. The bound is derived from the unitarity of the scattering matrix (S-matrix) and the asymptotic (large order limit) behaviour of the Legendre functions of the second kind: Ql in the large l limit.
In the context of quantum physics, unitarity restricts the evolution of quantum systems ensuring the sum of probabilities is always 1. The S-matrix describes how a physical system changes in a scattering process and therefore, it must be unitary.
Unitarity this way becomes necessary for the consistency of S-matrix theory.
Unitarity of the S-matrix straightforwardly leads to the very general result known as the optical theorem, which is one of the two cornerstones to the derivation of the Froissart bound.
The other cornerstone is the assumption of certain analytical properties of the scattering amplitude. Specifically, we must demand that the singularities of the scattering amplitude don’t lie within the branch cut of the Legendre functions of the second kind Ql previously mentioned. This is equivalent to considering scat- tering processes which are mediated only by massive intermediate particles. As a consequence of this, there does not exist a Froissart bound for Quantum Electrody- namics (QED), where the force carrier is the massless photon. Instead, we expect a gauge theory like Quantum Chromodynamics (QCD), which is mediated by massive particles, to admit a Froissart bound.
We also have to demand that the discontinuity of the scattering amplitude ac- cross its branch cuts has a polynomial upper bound in the center-of-mass-energy.
Although this is not regarded as a very demanding assumption, it is convenient to keep in mind that the assumption makes it possible to violate the Froissart bound without necessarily violating unitarity [1].
1
2 CHAPTER 1. INTRODUCTION
A mathematically rigorous derivation of the bound in the context of QCD is still missing. Solving QCD exactly is difficult. However, the popular conjecture that there exists a gauge-gravity duality can be used to enhance the understanding of the problem. The conjectured duality relates one weakly(strongly)-coupled theory of gravity (amenable to study within perturbation theory) to a strongly(weakly)- coupled Conformal Field Theory (CFT) [2]. Efforts are being made so that the strongly-coupled theory resembles QCD in the proper limit.
In our work, we exploit the duality and re-derive the Froissart bound for some large N CFT using the mathematical machinery corresponding to the gravity dual.
Here N refers to the degree of the symmetry group SU (N ) of the CFT. In more details, one starts by placing a point mass in the Infra-Red (IR) boundary of the AdS5 space (5-dimensional Anti-de-Sitter spacetime). For a sufficiently large mass, a black hole is expected to form. The black hole perturbs the background metric in such a way that, in the linear approximation and long-distance limit, the Froissart bound is saturated. This is seen from the comparison of the geometrical cross- section of the black hole with the cross- section of the scattering event in the dual CFT, in the spirit of previous works [3–5].
We further propose to consider the addition of a Weyl-squared term in the usual Einstein-Hilbert gravity action and study the consequences of such changes on the Froissart bound. The Weyl-squared term leads to higher (than two) derivative terms in the gravity equations of motion and its presence becomes relevant in the context of string theory.
This thesis is arranged in two parts containing 3 chapters each:
• The Froissart bound in field theory:
Chapter 2 introduces Lorentz symmetries and presents a Lorentz invariant formalism to describe two to two particle scattering events. Chapter 3 contains the basics of S-matrix theory. Two fundamental principles of S-matrix theory are discussed here: the unitarity of the S-matrix and the analiticity of the scattering amplitude. In chapter 4 we make use of the tools in the two previous chapters to derive the Froissart bound in the spirit of Froissart’s original work [6]. However, we avoid the discussion of the substractions in his original work, since it has later on become clear that this is indeed not necessary [7].
• The Froissart bound in AdS5:
Chapter 5 reviews the derivation of the Froissart bound in AdS5 via the AdS/CFT correspondence. Chapter 6 contains original work related to chap- ter 5. In this chapter, we modify the derivation outlined in the previous chapter to include a Weyl-squared term in the action. We also improve the approximations of previous works [3,4] to find subleading terms to the Frois- sart bound. Chapter 7 discusses the new results of chapter 6 and points out open questions for future works.
We work in the natural units c = ~ = G = 1, where c is the speed of light in vacuum, h = 2π~ Planck’s constant and G is the gravitational constant.
Part I
Froissart bound:
the field theory approach
3
Chapter 2
Lorentz symmetry
Poincaré group is the group of isometries of Minkowski (flat) spacetime. Poincaré transformations consist of translations, rotations and Lorentz boosts. The metric of the Minkowski spacetime is invariant under any such translation, rotation, Lorentz boost or combination of them. Any two inertial observers are related by Poincaré transformations and the laws of physics are identical on any inertial frame.
Any relativistic field theory on flat spacetime has Poincaré group as its symme- try. The lagrangian of any such relativistic field theory is invariant under the action of the Poincaré group. In a quantum picture, one paticle states are classified using the Casimirs of this group: mass and Pauli-Lubanski operators. Poincaré invariance is also reflected in the measurables of the theory, such as scattering cross-sections.
Rotations and boosts comprise the Lorentz group, which is a subgroup of the Poincaré group. The Lorentz group does not contain translations and hence is an isotropy subgroup of the isometry group of Minkowski spacetime. Since we will only consider translation invariant theories, it will suffice to discuss the Lorentz group and its relevant implications.
In this chapter we begin by describing the Lorentz transformations and Lorentz invariant particle states. In the context of scattering of particles, we introduce Mandelstam variables. Mandelstam variables are defined as functions of the four- momenta of the particles involved in the scattering event. These variables are invariant under the action of the Lorentz group and it is convenient to describe scattering processes of two to two particles in terms of them.
2.1 Lorentz transformations
In this section we introduce Lorentz transformations. This leads us to introduce four-vectors as well.
Consider two inertial observers O and O′. The observer O′ is moving with a uniform velocity ~β with respect to O. In the frame of O, spacetime events are labelled by coordinates xµand in the frame of O′, by coordinates x′µ. The coordi- nates xµ and x′µ are related by Lorentz transformations. Lorentz transformations
5
6 CHAPTER 2. LORENTZ SYMMETRY
are four-dimensional transformations given by
x′µ= Λµνxν, (2.1)
with µ, ν = 0, 1, 2, 3. The transformation matrix Λµν is given by
Λµν =
γ −γβx −γβy −γβz
−γβx 1 + (γ − 1)ββx22 (γ − 1)βxββ2y (γ − 1)βxββ2z
−γβy (γ − 1)βxββ2y 1 + (γ − 1)ββy22 (γ − 1)βyββ2z
−γβz (γ − 1)βxββ2z (γ − 1)βyββ2z 1 + (γ − 1)ββ2z2
, (2.2)
where γ = 1 − β2−1/2 and β ≡ |~β|.
The spacetime interval of events, defined as
ds2= ηµνdxµdxν, η = diag(1, −1, −1, −1), (2.3) is invariant under this transformation:
ηµνdxµdxν = ηµνdx′µdx′ν (2.4) The 4 × 4 diagonal matrix ηµν is the metric of the 4-dimensional flat Minkowskian spacetime.
The “norm", defined by the “dot product"
xµxµ= ηµνxµxµ (2.5)
is also invariant under the transformation (2.1). So, the transformation matrices (2.2) are elements of the group SO(3, 1). This group is generated by six generators Ji and Ki (i = 1, 2, 3) satisfying the Lie algebra
[Ji, Jj] = iεijkJk, [Ji, Kj] = iεijkKk, [Ki, Kj] = −iεijkJk,
εijk is the Levi-Civita symbol. (2.6) Ji’s generate the spatial rotations about the i-th spatial direction and Ki’s are the generators of Lorentz boost. Any arbitrary group element can be expressed as a combination of rotations and boosts.
The invariance of the dot product (2.5) leads to the relation
ΛµσηµνΛνρ= δσρ. (2.7)
Any quantity vµ which transforms as xµ under the action of Lorentz group is a four-vector. So if
vµ→ v′µ = Λµνvν, (2.8)
then vµ is a four-vector. Owing to relation (2.7), the norm of any four-vector vµ
vµvµ= ηµνvµvν (2.9)
is always invariant under any Lorentz transformation. From here on, we will refer to any quantity invariant under Lorentz transformation as Lorentz invariant. Note that scalars and dot products of two four-vectors are always Lorentz invariant. This information will be used in later sections.
2.2. MANDELSTAM VARIABLES 7
2.1.1 Relativistic normalization of momentum states
In this subsection, we discuss the Lorentz invariance of one-particle states in mo- mentum space.
Let us denote two different one-particle states in configuration space by |xi and
|yi. Let these states be elements of an orthonormal set:
hx|yi = δ(3)(x − y) . (2.10)
Two different one-particle states in momentum (Fourier) space are denoted by
|p) and |q). The normalization of these states can be obtained by taking the Fourier transform of the orthonormality condition (2.10):
(p|q) = (2π)3δ(3)(p − q) . (2.11) But this normalization is not Lorentz invariant. This can be easily seen by applying a boost in the z-direction. Using (2.2) and (2.8) for βx = βy = 0 and βz = β, we get
Ep′′ p′x p′y p′z
=
γ 0 0 −γβ
0 1 0 0
0 0 1 0
−γβ 0 0 γ
Ep
px py
pz
=
γ (Ep− βpz) px py
γ (pz− βEp)
. (2.12)
Under the Lorentz boost (2.12), Dirac delta function transforms as δ(3)(p − q) → δ(3) p′− q′= Ep′
Epδ(3)(p − q) (2.13) (for details see appendixA.1). Hence (2.11) is not Lorentz invariant.
But from (2.13), one can readily see that Epδ(3)(p − q) is a Lorentz invariant quantity. Therefore we define
|pi =q2Ep|p), (2.14)
such that the normalization
hp|qi = 2Ep(2π)3δ(3)(p − q)
is Lorentz invariant. The numerical factor 2 is due to a convention.
2.2 Mandelstam variables
Our aim in the next chapter is to discuss scattering processes. In the context of the scattering of two particles to two particles, the discussion is best done in terms of Mandelstam variables. In a scattering process, the dynamical variables are the
8 CHAPTER 2. LORENTZ SYMMETRY
Figure 2.1. a) s-channel b) t-channel c) u-channel
energies, momenta and angles of the particles involved. Mandelstam variables are particular functions of these quantities such that Lorentz invariance is manifest.
In this section, we define the Mandelstam variables s, t and u for the equal mass 1 + 2 → 3 + 4 scattering process. We show that only two of them are independent.
We also introduce the concept of scattering channels.
Consider the scattering process 1 + 2 → 3 + 4. For simplicity, assume all the particles have the same mass m. We denote their four-momenta by Pi, where i = 1, 2, 3, 4. Here, i is the label of the particle and is not the Lorentz index, which will be denoted by greek letters. We use the metric signature (+, −, −, −). The conservation of four-momenta leads to
E1+ E2 = E3+ E4,
p1+ p2= p3+ p4. (2.15)
The Mandelstam variables are defined as
s = (P1+ P2)2, t = (P1− P3)2, u = (P1− P4)2.
(2.16) All of them are not independent variables, since there is a constraint
s + t + u = 4m2. (2.17)
Therefore, in order to describe the equal mass process 1 + 2 → 3 + 4 only two independent variables are needed.
In the center-of-mass frame, p1 = −p2 and p3 = −p4. Consequently, E1+ E2 = 2qm2+ p21,
E3+ E4 = 2qm2+ p23. (2.18) The conservation of enegry (2.15) along with (2.18) yields
p1= p3= p. (2.19)
Let θs be the angle between p1 and p3 and zs= cos θs. In the center-of-mass frame, the Mandelstam variables can be expressed in terms of E1, E2, p and zs as
s = (E1+ E2)2, t − 2p2(1 − zs) , u = −2p2(1 + zs) .
(2.20)
2.2. MANDELSTAM VARIABLES 9
Again, combining (2.18), (2.19) and (2.20), we can rewrite s as
s = 4p2+ m2. (2.21)
In following sections where we discuss the scattering process 1 + 2 → 3 + 4, we will consider (s, t) or (s, zs) as the independent variables at convenience.
s, t and u can also be understood as channels: different possible scattering events leading to 1 + 2 → 3 + 4 processes where the interaction involves the exchange of only one intermediate particle. In the s-channel (see figure2.2.a), particles 1 and 2 produce an intermediate particle that eventually splits into 3 and 4. In the t-channel particle 1 and particle 2 interact via the intermediate particle and produce particle 3 and 4 respectively (see figure2.2.b). The u-channel is the t-channel with the role of the particles 3 and 4 interchanged (see figure2.2.c).
The Mandelstam variables (2.20) are for the s-channel. The similar for the t- and u-channels can be written by a simple permutation.
Chapter 3
Scattering cross-section
In this chapter we introduce the concept of scattering cross-section. Consider the scattering process 1 + 2 → f, with f any final state. The cross-section is the effective surface area seen by particles 1 and 2. When two particles scatter, the scattering cross-section is the key to understand the interaction between them.
As two particles approach each other, there is a chance that they pass without disturbing each other. In this case the final state of the particles is the same as the initial state. Another possibility is that when the particles meet they interact with each other. This interaction happens by exchange of a third particle, producing a different final state. The scattering cross-section is proportional to the probability of the happening of these interactions. The initial state of the particles is related to the final state by the scattering matrix or the S-matrix and the probability of the transition is given in terms of the S-matrix elements.
We start this chapter by introducing the S-matrix and stating the various prop- erties of it which are relevant for our discussion. We write the S-matrix in terms of the transition matrix T , which is given in terms of the Lorentz-invariant scattering amplitude A. We close the chapter with a discussion of the analytic structure and properties of A.
3.1 S- and T -matrices
Let us consider a many body scattering process. Let |ii be the initial state of the particles and |fi be the final state. The scattering matrix (S-matrix) is such that
| hf|S|ii |2 is the probability of |fi being the final state, given an initial state |ii.
|ii must be understood as being a state of two wavepackets in the far past, while
|fi is the many-particle (or many-wavepacket) final state in the far future. The wavepackets are taken to be localized in configuration space, so that they can be independently constructed.
The probability of ending up in some final state is 1:
X
f
| hf|S|ii |2= 1. (3.1)
11
12 CHAPTER 3. SCATTERING CROSS-SECTION
Since this is true for all initial states |ii, the S-matrix is unitary:
S†S = I. (3.2)
We can split the S-matrix as
S = I + iT. (3.3)
The identity operator I accounts for the two particles passing each other without interaction. On the other hand, the transition matrix T represents the interaction of the particles. For the study of the interaction, only T is of relevance.
In the following, we restrict ourselves to scattering processes with only two particles in the initial state. In momentum space the T -matrix is related to the Lorentz invariant matrix element (or scattering amplitude) A (1 + 2 → f) as
hf|T |p1p2i = (2π)4δ(4)
N
X
n=3
Pn− P1− P2
!
A (1 + 2 → f) , (3.4) where N − 2 is the total number of outgoing particles. In this expression, the factor (2π)4 comes from the Fourier transform and the four-momenta conservation is ensured by the Dirac delta function.
3.2 Total cross-section
In this section we compute the total cross-section for the equal mass 1 + 2 → 3 + 4 scattering process. We consider scattering events which are symmetric about the collision axis in the center-of-mass frame. This total cross-section will be used in the next section to derive the optical theorem.
The transition probability is defined as P1+2→f ≡ | hf|T |p1p2i |2
= h(2π)4δ(4)PfPf − P1− P2
i2
|A (1 + 2 → f) |2. (3.5) The square of the Dirac delta function is not mathematically rigorous and can be rephrased as
h(2π)4δ(4)PfPf− P1− P2
i2
= (2π)4δ(4)PfPf − P1− P2
R
d4x, (3.6) whereR d4x is to be understood as the space-time volume for the particle interaction withPfPf = P1+ P2. From (3.5) and (3.6) it follows that
P1+2→f = (2π)4δ(4)PfPf− P1− P2Rd4x|A (1 + 2 → f) |2. (3.7) The transition rate per unit volume is defined as ddP4x and can be extracted from (3.7):
ddP4x = (2π)4δ(4)PfPf − P1− P2
|A (1 + 2 → f) |2. (3.8)
3.2. TOTAL CROSS-SECTION 13
We define the differential cross-section as dσ ≡ Φ1 dP
d4x. (3.9)
Φ is the flux of incoming particles. We will discuss Φ in details later in this sec- tion. For the time being, it will suffice to say that it depends only on initial-state parameters.
Integrating (3.9) the final-state three-momenta yields the total cross-section:
σ = 1
Φ Y
f
Z d3pf (2π)32Ef
dP
d4x. (3.10)
Using (3.8) we get σ = Φ1 QfR d3pf
(2π)32Ef (2π)4δ(4)PfPf − P1− P2
|A (1 + 2 → f) |2. (3.11) If we restrict ourselves to the case with only two particles in the final state (i.e.
f = 3, 4), we can express the scattering amplitude in terms of the Mandelstam variable s and zs, introduced in the previous section. In this case the total cross- section in the center-of-mass frame is given by
σ = 1 Φ
Z d3p3 (2π)32E3
Z d3p4 (2π)32E4
(2π)4δ E3+ E4−√
sδ(3)(p3+ p4) |A (s, zs) |2. Integrating over p3, we get
σ = 1 Φ
Z d3p4 16π2
1
E3E4δ E3+ E4−√
s|A (s, zs) |2, where Ei2 = p24+ m2i. For m3 = m4= m, the above reduces to
σ = 1 Φ
Z d3p4 16π2
1 p24+ m2δ
2qp24+ m2−√ s
|A (s, zs) |2. (3.12) This integral is best done in the spherical coordinates
d3p4 = dp4p24dΩ, dΩ = d (cos θs) dϕ, in which
σ = 1 Φ
Z d (cos θsdϕ) 16π2
Z
dp4 p24 p24+ m2δ
2qp24+ m2−√ s
|A (s, zs) |2. (3.13) Using
Z
dxδ (f (x)) =X
i
|f′(xi) |−1,
14 CHAPTER 3. SCATTERING CROSS-SECTION
where the prime denotes derivation with respect to x and xi is such that f (xi) = 0, we can integrate over p4 to obtain
σ = 1
Φ
Z d (cos θs) dϕ
16π2 |A (s, zs) |2 p24 p24+ m2
qp24+ m2 2p4
m2=s4−p24
= 1
Φ
Z d (cos θs) dϕ 16π2
p4
√s|A (s, zs) |2.
Momentum conservation demands that pi = p for all i = 1, 2, 3, 4. For scattering processes symmetric about the collision axis,
σ = 1 Φ
p 8π√
s Z
d (cos θs) |A (s, zs) |2. (3.14) The flux Φ is the product of the probabilities to find particles 1 and 2 per unit volume multiplied by the relative velocity between these particles:
Φ = |φ1(x) |2|φ2(x) |2vrel. (3.15) Using the relativistic normalization of the states in (2.14) and the definition of relative velocity for two particles,
Φ = 4E1E2|v1− v2|. (3.16) The velocities can be expressed in terms of the relativistic energy and momentum of the particles as
E = γm, p = γmv,
)
v = p
E. (3.17)
From (3.15) and (3.17) it follows that
Φ = 4|E2p1− E1p2|.
In the center-of-mass frame the above reduces to Φ = 4p√
s. (3.18)
From (3.14) and ( 3.18) we obtain the desired result:
σ = 1 32πs
Z
d (cos θs) |A (s, zs) |2. (3.19)
3.3. OPTICAL THEOREM 15
3.3 Optical theorem
In this section, we relate the forward scattering amplitude A (1 + 2 → 3 + 4) to the total cross-section of the scatterer. This relationship is known as the optical theorem. The optical theorem is a straightforward consequence of the unitarity of the S-matrix and is one of the two main pillars to our derivation of the Froissart bound in the next chapter.
It can be easily seen from(3.2) and (3.3) that
2ImT = T†T. (3.20)
Inserting a complete set of intermediate states, hp3p4|T†T |p1p2i can expressed as
hp3p4|T†T |p1p2i =Y
i
Z d3qi
(2π)32Ei hp3p4|T†| {qi)}i h{qi} |T |p1p2i . (3.21) From (3.4), (3.20) and (3.21) it follows that
2ImA (1 + 2 → 3 + 4) = Q
i
R d3qi
(2π)32Ei(2π)4δ(4)(PiQi− P1− P2) A∗(p3p4 → {q}i) A (p1p2 → {q}i) , where Qi is the four-momenta of the i-th intermediate state particle. If we restrict ourselves to the special case of forward scattering, the above reduces to
2ImA (1 + 2 → 1 + 2) = Y
i
Z d3qi
(2π)32Ei (2π)4δ(4) X
i
Qi− P1− P2
!
|A (p1p2 → {q}i) |2, (3.22)
and using (3.11) we get
σ = 1
Φ2ImA (1 + 2 → 1 + 2) . (3.23)
In the center-of-mass frame the initial particle flux Φ is given by (3.18) and we can express the scattering amplitude A in terms of the Mandelstam variable s and zs. For forward scattering the Mandelstam variable t vanishes, as can be readily seen from (2.16). Using (2.20), this sets zs= 1. The total cross-section σ can then be expressed as
σ = 1 2p√
sImA (s, zs= 1) . (3.24)
This result is known as the optical theorem.
