# International Summer Schools on Reaction Theory

## 2015 & 2017 editions

The 2017 lectures are based from the book:
Strong Interactions of Hadrons at High Energies" by V. Gribov, Cambridge University Press, 2009.
Click on the team number to access the material (videos, notes, exercices).

# 2017 Lectures

## Team 1: Chapter 1-2

Lecturers: A. Pilloni and A. Szczepaniak

I discuss the generic principles of reaction theory, and the symmetries of strong interactions. I introduce the relativistic definition of states as irreducible representations of the Poincare’ group. Then, I introduce the Scattering Matrix, which encodes all the informations about dynamics. Using the fundamental properties of any underlying QFT, we can prove that the S-matrix satisfies crossing symmetry, unitarity and analyticity.

## Team 2: Chapter 3-4

Lecturers: A. Jackura and M. Vanderhaeghen

In these lectures I will discuss partial wave expansions and consequences of unitarity. We will look at 2 -to -2 scattering of spinless particles, and particles with spin. We will discuss the differences in Helicity partial wave expansion and Spin-Orbit partial wave expansion. We then apply the S-matrix unitarity condition to these amplitudes and investigate their structure in the complex energy plane. Finally, we discuss some useful parameterizations of amplitudes which can be used to extract resonance information from data.

## Team 3: Chapter 5-6

Lecturers: M. Mikhasenko, M. Shepherd and E. Passemar

In the first lecture, I discuss a problem of the three particles final state. We start considering general kinematics of the decay process. Then, I introduce a customary representation of the reaction dynamics by the Dalitz plot. In the following tutorial, I post a problem to reproduce realistic Dalitz plots measured by CLEO and BaBar experiments. The second lecture is dedicated to the dispersion technique. I derive a dispersive representation of the real analytic function. As an application, we discuss the Omnes problem for the inelastic unitarity equation. In the third lecture, we focus on the physics of the partial waves at high energy. I introduce the impact parameter representation and illustrate the black disc model of particles scattering.

## Team 4: Chapter 7-8

Lecturers: V. Mathieu, J. Pelaez, C. Weiss and A. Szczepaniak

These lectures introduce the concept of Regge poles.
We introduce the Sommerfeld-Watson transformation and the Froissard-Gribov representation of partial waves.
During practicum 1, we prove, using unitarity, the factorization of residues. Practicum 2 reviews the Mathematica notebook. During practicum 3, we derive the $t$-channel quantum numbers of reactions. We also illustrate the concept of exchange degeneracy between Regge exchange on the total cross section.

# Seminars

## Light Hadrons Physics at $e^+e^-$ Colliders

Lecturer: A. Kupsc

Recent interest in the light hadrons production at e+e- colliders is driven by the aim to precisely determine hadronic contribution to anomalous magnetic moment of the muon, (g-2), in connection with the ongoing improved measurement at Fermilab. A short overview of the mechanisms for the hadron production at e+e- colliders and status of the present experiments is given. The dispersive methods for description of the simplest hadron systems like two pions in JPC=1-- state (pion vector form factor) are discussed in detail and an outlook is given for more complicated cases where three light hadrons are produced.

## Amplitude analysis at BESIII

Lecturer: J. Bennett

This talk presents an overview of amplitude analyses at BESIII, with an emphasis on the various techniques that are used to analyze experimental data, and the necessary assumptions that are made. Recent results are reviewed, with an eye toward the challenges for each analysis.

## LHCb Physics

Lecturer: T. Skwarnicki

General goals of the LHCb physics program are outlined. The limitations of the present detector are discussed, together with the upgrade program design to overcome them. Examples of measurements heavily relying on amplitude analysis are presented.

Lecturer: M. Vanderhaeghen

## Scattering from lattice QCD: formalism and results

Lecturer: R. Briceno

## Hadron-nucleus interactions at very high energies: black disc limit

Lecturer: V. Guzey

The strong interaction of a very high-energy projectile with a hadronic target reaches its maximal value determined by unitatiry of the scattering matrix. This limit, which is called the black disk limit (BDL) in the literature, corresponds to complete absorption by the target, when the total cross section is equal to twice the geometric cross section and the diffractive cross section reaches half of the total cross section. We discuss the BDL in potential scattering in non-relativistic quantum mechanics and in a field-theoretical approach to hadron-nucleus and photon-nucleus scattering. For the latter, we recapitulate main steps of the derivation presented in Gribov's seminal paper [V.N. Gribov, Sov. Phys. JETP 30 (1970) 709]. We explain that the BDL regime of photon-nucleus interaction is characterized by such dramatic signals as violation of approximate Bjorken scaling and slowing down of the energy dependence of the total cross section.

## Dispersive analysis of meson decays

Lecturer: E. Passemar

## Dispersive analysis of pion-pion and pion-kaon interactions: Resonance poles and Regge behavior

Lecturer: J. Pelaez

After a brief introduction to dispersion theory I show how these techniques are applied to pion-pion and pion-kaon scattering. First, to discard conflicting data sets. Second, to obtain simple but constrained parameterizations of the data. Third, to determine rigorously and in a model-independent way the existence of resonances and their parameters by analytically continuing the amplitudes to the complex plane and finding their associated poles. This is particularly important for the very wide sigma and kappa resonances. To conclude we also show that dispersive analysis of the Regge trajectories of these two resonances imply their non-ordinary (non-quark-antiquark) nature.

## Dispersion theory in hadron form factors

Lecturer: C. Weiss

## Application of Regge Theory

Lecturer: A. Blin and J. Nys

We show examples of applications of Regge theory. In order to do so, we begin by giving a summary of the tools needed, which have been taught in the previous lectures of the school. We then show specific pseudoscalar-meson production reactions, first with pion and kaon beams, then with photon beams. For the case of pseudoscalar meson beams, we systematically show the origin of the small t behaviour, the reason for removing ghost particles, the dips that appear in the observables, and the need for Pomeron cuts. We discuss exchange degeneracy and reggeization. Concerning the photoproduction processes, the relevant observables are introduced and explained. We show the connection between different representations of the amplitudes (invariant and s- and t-channel helicity amplitudes). We show the reggeons that are exchanged in neutral and charged pion production, and the implications on the forward behaviour of having an "unnatural" but very light pion exchange.

## Exploring Charmonium at the BESIII Experiment

Lecturer: R. Mitchell

Exploring Charmonium with the BESIII Experiment The charmonium system is made up of bound states of charm quarks and antiquarks in various configurations. This lecture introduces both the experimental foundations of charmonium and recent surprising developments that suggest the existence of unusual configurations of charmonium (hybrid charmonium, tetraquarks, hadronic molecules, etc.). The study of charmonium at the BESIII Experiment in Beijing, China is given special emphasis.

Lecturer: V. Guzey

Nuclear shadowing is a high-energy phenomenon that the total hadron-nucleus cross section is smaller than the sum of individual hadron-nucleon cross sections. We present the Glauber theory of nuclear shadowing and its field-theoretical generalization by Gribov [V.N. Gribov, Sov. Phys. JETP 29 (1969) 483], which is based on the space-time picture of the strong interaction. We then discuss the application of the Gribov-Glauber framework to deep inelastic scattering (DIS) on nuclear targets resulting in the so-called leading twist model of nuclear shadowing for nuclear parton distribution functions. We show how large nuclear gluon shadowing, predicted in this model, finds confirmation in the analysis of nuclear suppression of J/psi photoproduction on nuclei in ion ultraperipheral collisions at the Large Hadron Collider.

# Lectures

## Introductory Overview

Lecturer: Michael Pennington
Abstract:

## Current Experiments

Lecturer: Matthew Shepherd
Abstract:

I review recent experimental highlights in the last decade with particular emphasis on the role of reaction theory in interpreting spectroscopy data. Advance in experimental capability have provided a wealth of new data from both colliders and fixed target experiments. This data has suggested the presence of a new types of particles: candidates for hybrid mesons and tetraquarks. At the same time, this data has provided the statistical precision needed to develop more theoretically robust descriptions of the the production and decay mechanisms of these apparently new particles. A critical next step to bring these efforts together in an attempt to develop a more detailed interpretation of the data with the goal of trying to validate what have been up to now relatively simple resonance interpretations of features in the data.

## Review of Scattering Theory

Lecturer: Jonathan Rosner
Abstract:

Three lectures on scattering theory and the S-matrix are given. In the first, scattering theory is reviewed, discussing the S, T, K, and R matrices; wave packet scattering; the scattering amplitude, its partial wave expansion, and its phase shifts; and the unitarity circle. The second treats the S-matrix and related physics: one- and two-channel examples, transmission resonances, bound states, properties of S-wave amplitudes, resonances, absorption, and the inelastic cross section. The third gives some simple applications, including optical analogues, the eikonal approximation, diffractive scattering, addition of resonances, Dalitz plot applications, historical notes, and examples from electronics.

## Dalitz and VanHove Plots

Lecturers: Tim Londergan and Vincent Mathieu
Abstract:

For more than 60 years, the Dalitz Plot has been an invaluable tool in analyzing reactions with three particles in the final state. Several Nobel Prizes in physics have been awarded, for which the Dalitz Plot played a major role in the reaction analysis. We review the Dalitz Plot and its use in reaction theory. First, we summarize Dalitz’s original introduction of this graphical method in 1953. It used non-relativistic kinematics, and was employed to analyze decays of “tau” and “theta” particles (these are now known to be kaons). Then, we introduce the standard relativistic kinematics used in modern Dalitz Plots. We show how resonances appear in Dalitz Plots and how their masses, widths and spins can be extracted. For high-energy reactions such as B decays, we show how a transformation called the “square Dalitz Plot” can simplify analyses. We end by discussing some current issues in reaction theory for which Dalitz Plots are again proving useful.

In the text files, the 4x3 first columns correspond to (E,px,py,pz) of particles 1(Eta), particle 2(Pi) and particle 3(P). The last two columns are s12 and s23. Units are GeV. The events are in the center-of-mass frame of the reaction. The Mathematica notebook reads the data from the text files, displays the Dalitz and Van Hove plots, performs cut in the Van Hove angle and show the mass projections with and without the cut.

## AmpTools

Lecturer: Matthew Shepherd
Abstract:

This talk presents narrative for a tutorial on the AmpTools software package. An example, which is distributed in the links below, is discussed in detail.

## Principles of the S-Matrix

Lecturer: Michael Doring, Cesar Fernandez-Ramirez and Igor Danilkin
Abstract:

Lecture III:
Resonances are the poles in the unphysical Riemann sheets of the amplitude. In this talk I present, using a Mathematica notebook, a simple coupled-channel K-matrix calculation that shows how the amplitudes are analytically continued from the real (physical) axis --where experimental data are obtained--, to the first (physical) Riemann sheet and to the other (unphysical) Riemann sheets. Once this continuation is done I show how we can "catch" the poles and their impact on the physical amplitude under different scenarios.
Lecture IV: "Dispersive approach, Unitarity, Pion vector form factor"
The constraints from quantum field theory play a prominent role in modern hadron physics. These are analyticity, unitarity and crossing symmetry. In this lecture, I will discuss selected examples that illustrate the usefulness of them. In particular, I will focus on Omnes-Muskhelishvili problem and pion vector form factor. At the end of my talk, the numerical implementation of the Omnes function will be discussed in detail.

## Dispersion Relations: Light Quark Masses from Eta Meson Decay in Pions

Lecturer: Emilie Passemar
Abstract:

## Quark Models

Lecturer: Elena Santopinto
Abstract:

Abstract:

## Computing QCD on a Lattice

Lecturer: Jozef Dudek
Abstract:

Discussion of techniques to determine hadron scattering amplitudes and their resonant singularities from the discrete spectrum of states computed in lattice QCD.

## Two Photon Production of Mesons

Lecturer: Michael Pennington
Abstract:

Lecturer: David Horn
Abstract:

Setting the background for the dominant ideas in strong interaction of the early 1960s, we outline the major concepts of the S-matrix theory. An independent theoretical development was the emergence of hadron duality in 1967, leading to a realization of the Bootstrap idea by relating hadron resonances (in the s- channel) with Regge pole trajectories (in t- and u-channels).

Lecturer: Paul Hoyer
Abstract:

A review of the duality phenomena observed in hadron and lepton induced processes, together with their interpretations and implications. Includes 4 exercise problems.

## Bound states in Field Theory

Lecturer: Paul Hoyer
Abstract:

The basics of bound states in field theory. The Hamiltonian operator formulation of bound states defined at equal time of the constituents. The Schrödinger and Dirac equations give the Born terms (lowest order in $\hbar$) of a loop expansion, even though they contain all powers of the coupling $\alpha$. Born level relativistic bound states as candidates for the |in> and |out> states of an S-matrix for hadrons. Includes 4 exercise problems.

## Measuring Transverse Size with Virtual Photons

Lecturer: Paul Hoyer
Abstract:

The virtuality of the photon exchanged in electron scattering measures the transverse size of the coherent scattering region in the target. The two-dimensional Fourier transform from transverse momentum to transverse coordinate space is done in a specific frame. The results of analyses using data on hadron form factors are reviewed. Analogous analyses can be done also with inelastic electroproduction data. Includes 3 exercise problems.

## Applying Regge Theory

Lecturer: Peter Landshoff
Abstract:

Regge theory successfully correlates a wide variety of hadronic reactions. In particular, it provides an excellent description of the highly accurate proton-proton and proton-antiproton elastic scattering data taken over the wide range of centre-of-mass energies, from that of the CERN Intersecting Storage Rings to that of the Large Hadron Collider (some 400 times greater). It involves the exchanges of the known families of particles, and that of a new object called the pomeron, which may represent glueball exchange. Extending the analysis to deep inelastic ep scattering calls for a second pomeron. The variation with $Q^2$ of its contribution is well described by perturbative QCD.

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## Neutral Pion Photoproduction

Lecturer: Vincent Mathieu
Abstract:

The parametrization of the amplitudes for the reaction $\gamma p \to \pi^0 p$ in terms of Regge exchanges in the $t-$channel is explained step by step. The data and the Mathematica file used to perform the fit are given.

## Unitarity in 2 and 3 Body Final States

Lecturer: Ian Aitchison
Abstract:

The aim is to provide a simple introduction to how the tools of “the S-matrix era” - i.e. the constraints of unitarity, analyticity and crossing symmetry - can be incorporated into analyses of final state interactions in two- and three-hadron systems. The main focus is on corrections to the isobar model in three-hadron final states, which may be relevant once more as much larger data sets become available.

## Production Dynamics of Axial Vector Meson in the 1 to 2 GeV Mass Range

Lecturer: Edmond Berger
Abstract:

## The Past and the Future

Lecturer: Geoffrey Fox
Abstract:

This talk will give a broad overview, incl. role of hadrons, hadron reaction phenomenology, past achievements and future goals.

## Scattering Theory and Light-Front Quantization

Lecturer: Stanley Brodsky
Abstract:

A primary question in hadron physics is how the mass scale for hadrons consisting of light quarks, such as the proton, emerges from the QCD Lagrangian even in the limit of zero quark mass. If one requires the effective action which underlies the QCD Lagrangian to remain conformally invariant and extends the formalism of de Alfaro, Fubini and Furlan to light-front Hamiltonian theory, then a unique, color-confining potential with a mass parameter $\kappa$ emerges. The actual value of the parameter $\kappa$ is not set by the model -- only ratios of hadron masses and other hadronic mass scales are predicted. The result is a nonperturbative, relativistic light-front quantum mechanical wave equation, the {\it Light-Front Schr\"odinger Equation } which incorporates color confinement and other essential spectroscopic and dynamical features of hadron physics, including a massless pion for zero quark mass and linear Regge trajectories with the identical slope in the radial quantum number $n$ and orbital angular momentum $L$. The same light-front equations for mesons with spin $J$ also can be derived from the holographic mapping to QCD (3+1) at fixed light-front time from the soft-wall model modification of AdS$_5$ space with a specific dilaton profile. Light-front holography thus provides a precise relation between the bound-state amplitudes in the fifth dimension of AdS space and the boost-invariant light-front wavefunctions describing the internal structure of hadrons in physical space-time. One can also extend the analysis to baryons using superconformal algebra -- $2 \times 2$ supersymmetric representations of the conformal group. The resulting fermionic LF bound-state equations predict striking similarities between the meson and baryon spectra. In fact, the holographic QCD light-front Hamiltonians for the states on the meson and baryon trajectories are identical if one shifts the internal angular momenta of the meson ($L_M$) and baryon ($L_B$) by one unit: $L_M=L_B+1$. We also show how the mass scale $\kappa$ underlying confinement and the masses of light-quark hadrons determines the scale $\Lambda_{\overline{MS}}$ controlling the evolution of the perturbative QCD coupling. The relation between scales is obtained by matching the nonperturbative dynamics, as described by an effective conformal theory mapped to the light-front and its embedding in AdS space, to the perturbative QCD regime. The data for the effective coupling defined from the Bjorken sum rule $\alpha_{g_1}(Q^2)$ are remarkably consistent with the Gaussian form predicted by LF holographic QCD. The result is an effective coupling defined at all momenta. The predicted value $\Lambda^{(N_F=3)}_{\overline{MS}} = 0.440 m_\rho = 0.341 \pm 0.024$ GeV is in agreement with the world average $0.339 \pm 0.010$ GeV. We thus can connect $\Lambda_{\overline{MS}}$ to hadron masses. The analysis applies to any renormalization scheme.

## The LHC Inverse Problem

Lecturer: Gordon Kane

Suppose there are some discoveries at LHC – then the problem is how to interpret them, what do they imply – without a theory, that’s hopeless. There are always many interpretations. The goal is to get a low scale theory and then presumably underlying theory at Planck scale. We will “guess” the underlying theory – but even then must test it or will never get consensus. Our guess for a top-down theory will be a well motivated compactified M-theory – starting at 11D, and at Planck mass in the resulting 4D theory. I’ll try to make it clear why string/M theories are good physics and testable.