LEBANESE UNIVERSITY

DOCTORAL SCHOOL OF SCIENCES AND

TECHNOLOGYEDST Beirut-Lebanon

Report of an internship in a Master 2 Research

for obtaining a diploma in Master 2 Researchin Fundamental Physics option: Theoretical Physics

Virtual Compton Scattering at

MAMIa preliminary study of the Generalized

Polarizabilities of the Proton

Presented by

Mostafa HOBALLAH

Supervisor: Helene FONVIEILLELaboratoire de Physique Corpusculaire LPC

Clermont-Ferrand France

2010-2011

1

This page is intentionally left blank

2

To Mariam...

3

We meet many people in our lives, but very few are those who leave a remarkable effect

in it, an effect that lasts forever guiding us in the right direction. I want to thank Mariam

ATOUI , a very nice, kind, and aimable person, knowing her was really motivating and

exciting.

I am very pleased to have been working the last two months at the Laboratory of Corpus-

cular Physics in Clermont Ferrand in the group “ sonde electromagnetique “. Since then I

have experienced many things, some - not to say all - of which we lack back home, things

that are important and essential to the development of research, and to the improvement

of our way of thinking.

Also I would like to give thanks to Pr. Helene Fonvieille, director of research at the

CNRS, for supervising my internship, she had really been helpful, always there when I

needed anything merging me into the physics of hadrons, she is really full of information.

The internship with her opened my eyes to many new things, allowed me to meet many im-

portant people and to integrate well into the world of scientific research, thank you Helene,

you are most kind.

A special thanks goes to Pr. Abbas Nehme, for he always gave me hope and courage, he

opend my eyes to a world full of opportunities, he fixed my mind to a steady purpose, and

nothing contributes so much to tranquillise the mind as a steady purpose-a point on which

the soul may fix its intellectual eye.

A very special thank goes to my parents, for they have always been by my side. My father

and mother Ali and Sanaa have done a great job raising me in a world where my friend

was my book, they have built in me a man who learns just for the pleasure of learning, and

thinks just to feel alive. They always said that a man who doesn’t read and think only lives

one day, for all his days are alike, whereas the highly effective people are those who spend

their time learning new things.

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Contents

6

1 Introduction 7

2 Theoretical Background 8

2.1 Virtual Compton Scattering . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 The Low Energy Theorem . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Structure Functions of the proton and the Generalized Polarisabilities 11

3 Experimental Setup at Mainz 12

3.1 MAMI accelerator and the electron beam . . . . . . . . . . . . . . . . 13

3.2 The cryogenic target of hydrogen . . . . . . . . . . . . . . . . . . . . 13

3.3 The A1 hall and the 3+1 Spectrometer setup . . . . . . . . . . . . . . 14

3.3.1 spectrometer magnets . . . . . . . . . . . . . . . . . . . . . . 15

3.3.2 Particle detection system . . . . . . . . . . . . . . . . . . . . . 15

3.4 Data acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 Data analysis 18

4.1 The missing mass and the identification of the photon . . . . . . . . . 18

4.2 Removing the effect of the target walls . . . . . . . . . . . . . . . . . 19

4.3 Particles in coincidence . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.4 Rejecting the elastic events . . . . . . . . . . . . . . . . . . . . . . . . 21

4.5 Below the pion threshold . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.6 Extracting experimental events . . . . . . . . . . . . . . . . . . . . . 23

5 simulation and cross section calculation 23

5.1 Measuring cross sections . . . . . . . . . . . . . . . . . . . . . . . . . 24

5.2 Corrections and errors . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.2.1 Systematic errors . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.2.2 Statistical errors . . . . . . . . . . . . . . . . . . . . . . . . . 26

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5.3 The C++ program . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.4 The spectrometers settings . . . . . . . . . . . . . . . . . . . . . . . . 28

5.5 Fitting the structure functions and the χ2 minimisation . . . . . . . 29

5.6 Combining measurements . . . . . . . . . . . . . . . . . . . . . . . . 31

6 Conclusion 35

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1 Introduction

Few decades ago the proton was thought to be a point like particle. A study of the nuclearmagnetic moment of the proton (µ = 2.7927) proved it to have contained another parti-cles, hence the internal structure of the nucleon was not as simple as it was believed tobe. Since then, a huge part of the scientific research has been directed towards the studyof the internal structure of the nucleon.

In this report, we explain one of the experiments that was done at the Mainz Microtron(MAMI) which aimed to make a clear study of the nucleon structure. Virtual Comptonscattering off the nucleon is a good candidate to describe how the nucleon deforms underthe effect of an external electromagnetic field.

A 1 GeV beam is directed towards a cryogenic target of hydrogen, the incident electronexchanges a virtual photon of four-momentum transfer of Q2 = 0.2GeV 2/c2 with the pro-ton, the proton then emits a real photon ep −→ e′p′γ. The spectrometers of the hall A1at Mainz detect the electron and the proton where the photon is identified as the missingparticle. The electron current represents the leptonic part of the reaction, whereas thehadronic part is represented by the proton current. The virtual Compton scattering islocalised in the hadronic part where we have an incident virtual photon interacting witha proton and then emitting a real photon pγ∗ −→ p′γ.

Our work here is composed of analysing the data taken from the accelerator at Mainz-Germany, eliminating all the bad events and extracting experimental cross sections fromthe good ones. We look forward to make a fit with those cross sections to calculate theproton electric and magnetic structure functions PLL−PTT/ǫ and PLT , from those we aimto extract the proton electric and magnetic Generalized Polarizabilities (GPs) αE and βM .The GPs will render us a very clear view of the spatial dependance of the polarizabilities.

We start our work by viewing the theoretical background of the VCS, recalling all thepossible diagrams, and the model in which we can extract the proton structure functions.We proceed to explain the experimental setup of the Mainz Microtron, the spectrometersof the hall A1, and the way by which data acquisition is done. Then we give a detaileddescription of the analysis procedure, defining all the cuts, and extracting VCS events.After that we will be ready to calculate the five-fold differential cross sections, find theirerror bars, and do the χ2 minimization to calculate the proton structure functions.

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2 Theoretical Background

2.1 Virtual Compton Scattering

The (e,e′γ) reaction on the proton

ep −→ e′p′γ

is a reaction where an electron interacts with a proton via a virtual photon, the finalphoton can be emitted either by the electrons giving rise to the Bethe-Heitler (BH) am-plitude, or by the proton allowing an access to the Virtual Compton Scattering (VCS)process (figure 1).

Figure 1: BH (a) and VCS (b) Feynmann diagrams

The VCS process pγ∗ −→ p′γ is probed by the spacelike photon (virtual photon) of afour-momentum transfer squared Q2, the proton then emits a real photon.

The notations for the 4-momenta of the particles are listed in the table below. The vir-tual photon exchanged in the VCS process has a four-momentum q = k − k′, while it isp − p′ in the BH process.

The Mandelstam variables are

−Q2 = q2 = 4 Ee Ee′ sin2( θe

lab

2

)

t = (q − q′)2

ands = (p + q)2

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Table 1: Notations for the four-momenta of the particlesparticle 4-momenta

Incident electron k

Scattered electron k′

Target proton p

Recoil proton p′

Initial virtual photon q

Final photon q′

The BH amplitude can be calculated exactly in QED if one knows the elastic form factorsof the proton, therefore it contains no new information on the structure of the nucleon.Knowing that light particles (electrons) are able to radiate more than heavy particles (pro-tons) do, the overall matrix element is dominated by the BH amplitude, hence making ita bit hard to analyse data and extract GPs. To avoid this, and in order to stay away fromthe BH peaks since the radiation of photons is peaked along the electron lines, one canchoose settings for the experiment in which the BH process dominates less than usual.

Using the LSZ reduction formula Guichon and Vanderhaeghen [5] found the BH matrixelement to be:

TBH =−e3

tǫ′∗

µ Lµν u(p′)Γν(p′, p)u(p)

where

Γν(p′, p) = F1(X)γν +

iF2(X)σνρ(p′ − p)ρ

2mX = (p′ − p)2

F1(0) = 1F2(0) = 1

where F1(X) and F2(X) are the Dirac form factors, and the VCS matrix element becomes:

T V CS =e3

q2ǫ′∗

µ Hµν u(k′)γνu(k)

where Hµν is the virtual Compton scattering amplitude that depends on the proton struc-ture.

The virtual Compton scattering amplitude Hµν is divided into two parts, a Born approx-imated part Hµν

B and a non-Born part that contains the higher order corrections HµνNB .

In the VCS process the polarisabilities of the proton become Q2-dependent observablescalled Generalised Polarisabilities,this intern is transformed through a Fourier transforminto a space dependent variable allowing a clear study of how the nucleon deforms underan applied electromagnetic field.

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2.2 The Low Energy Theorem

Low energy theorems (LET) play an important role in studying the properties of par-ticles. Based on a few general principles, they determine the leading terms of the lowenergy amplitude for a given reaction in terms of global model independent properties ofthe particles. Clearly, this provides a constraint for models or theories of hadron structure,unless they violate these general principles they must reproduce the predictions of the lowenergy theorem.

On the other hand, the low energy theorems also provide useful constraints for experi-ments . Experimental studies designed to investigate particle properties beyond the globalquantities and to distinguish between different models must be carried out with sufficientaccuracy at low energies to be sensitive to the higher order terms not predicted by thetheorem.

One of the best known low energy theorems for electromagnetic interactions is the theo-rem for the virtual Compton scattering developed by Guichon and Vanderhaeghen [5]. Ittells us that the expansion of the non-Born term of the VCS amplitude, which was unkownbefore, starts at order q′.

The non-Born term (HµνNB) of the VCS amplitude is a regular function of the 4-vector

q′µ, which amounts to say that HµνNB has a polynomial expansion of the form

HµνNB = aµν(q) + bµν

α (q)q′α + cµναβ(q)(q′β)2 + ...

HµνNB = aµν + bµν

α q′α + O(q′2)

where in the last step, the fact that the final photon is real has been used , and q′0 = q′.

The gauge invariance of HµνNB implies that

q′µHµνNB = 0

q′µaµν = 0

hence implying that aµν is Zero, and the first term in the expansion start at the order of q′.

The low energy theorem allows us to write every amplitude (BH, Born, non Born) whichcontributes to the overall matrix element as an expansion in the powers of q′. The aimof such an expansion is to reduce the complexity of the problem for low q′; If q′ is smallenough, then one can neglect the terms of orders of q′2 and higher (Although this is notalways true for some regions in θ and φ, the azimuthal and polar angles of the final photonin the center of mass frame with respect to q′).

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2.3 Cross Sections

For the reaction of the electroproduction of photons (ep −→ e′p′γ), the unpolarised crosssection is five-fold differential of the form

d5σ

dEe′ dΩe dΩ=

d3σ

dEe′ dΩe× d2σ

dΩ= Γ

d2σ

dΩ

where, in the VCS case, Γ represents the flux of the virtual photon field.

This five-fold differential cross section can be decomposed, using the low energy expan-sion, into parts that contain the Bethe-Heitler and Born amplitudes which has no criticalinformation on the GPs, and the non Born part of the cross section which contains almostall the information on the GPs, hence we can write

d5σ = d5σBH+B

+ Φq′TNB0 + O(q′2)

where TNB0 = v1.(PLL − 1

ǫPTT ) + v2.PLT

(1)

where PLL, PTT , and PLT (L:longitudinal, T:transverse) are the three structure functionsof the proton (or VCS response functions) and the phase space factor Φ is given by

Φ =(2π)5k′

32mpk√

s

where ε is the virtual photon polarisation parameter, v1 and v2 are kinematical coeffi-cients that depend on q, ε, θ, φ.

In this low energy expansion of the cross section one is able to estimate the structurefunctions PLL, PTT , and PLT if given an experimental cross section. Fitting those param-eters using a χ2 minimisation would be a straight forward procedure. One problem is thatfor the separation of PLL and PTT , the unpolarised cross section has to be measured over awide range in ε at a fixed Q2 and this is limited by the accelerator and the detector setupwhere the experiment is performed. Till now it is not possible to have a good separation ofPLL and PTT so they are measured together and only two linear combinations of structure

functions are obtained from the unpolarised cross section PLL − PTT

ǫand PLT .

2.4 Structure Functions of the proton and the Generalized

Polarisabilities

Threshold photon electroproduction off the proton allows one to measure new electromag-netic observables which generalise the usual polarisabilities. There are six ”generalisedpolarisabilities”. We ought to extract these quantities from the photon electroproductioncross sections.

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These generalized polarizabilities emerge as coefficients if the non Born amplitude of VCSis multipole expanded in terms of the final photon energy q′ taken in the limit where ittends to zero. In contrast to the real Compton scattering, the GPs of the VCS are func-tions depending on the 4-momentum transfer Q2 (functions of the virtual photon mass).

The structure functions (PLL −PTT/ǫ) and PLT are linearly dependent on the GPs, theyare linear combinations of the electric and magnetic polarisabilities αE and βM ( Guichon,Vanderhaeghen)[5]. So, measuring the structure functions of the proton will give us accessto the GPs and hence access to how the proton interacts under the effect of an externalelectromagnetic field.

3 Experimental Setup at Mainz

Since the first sights of atoms, scientists were trying to discover their constituents, buildinghuge accelerators and detectors, and generating high energy beams in hope to probe moretheir structure. The best way to study the structure of the hadrons was by studying theelectron scattering off a hadronic target. The accelerator at Mainz is capable of generatinga continuous electron beam up to energies of 1600 MeV and intensities of 100 µA. The VCSexperiment is performed in the hall A1 of MAMI, this hall is composed of a spectrometersetup to be discussed later on. Apart from the VCS experiment and the hall A1, twoother halls are hosting experiments in Mainz. Figure 2 shows an overview of the hall A1at Mainz with the three spectrometers A, B, and C.

Figure 2: Overview of the Hall of spectrometers of MAMI. The three spectrometersA, B and C rotate about a common pivot which is located on the Scattering chamber

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3.1 MAMI accelerator and the electron beam

The MAMI accelerator consists of four cascaded microtrons, an injector linac, a ther-mal source for unpolarized electrons and a laser-driven source for electrons with 80 %spin polarization. The operation principle is based on the continuous wave (cw) microtrontechnique. There the beam is recirculated many times through a normal-conduction linearaccelerating structure with a moderate energy gain per turn. Due to constant, homogenousmagnetic bending fields the length of the beam path is increasing with energy after eachturn. The magnetic fields, the radio-frequency (rf) used to accelerate the electrons, andthe energy gain per turn have to be adjusted to meet the microtron coherence condition,i.e. the condition that the length of each path is an integer factor of the rf wavelength.This microtron scheme makes efficient use of the rf power and the inherent strong longi-tudinal phase focussing guarantees excellent beam quality and stability.

In the VCS experiment at the hall A1 an electron beam of energy 1 GeV and intensityof 15 µA is used, cryogenic hydrogen as a target and the spectrometers A and B of thehall A1 to detect the proton and the electron respectively. This 1 GeV beam is not alwayssent on the same part of the target to avoid local boiling, it is made to move in a rasteringpattern (lissajoux) oscillating up and down, right and left. The position of the beam isidentified by introducing a thin aluminum screen which illuminates upon being hit by thebeam, and therefore the position is adjusted according to the illuminated parts.

3.2 The cryogenic target of hydrogen

The integrated luminosity L is a constant of the accelerator system and the target, it per-mits us to evaluate the ability of an experiment to create enough interactions in a smallreasonable time and it is given by

L = Nb.e−/s × Acquisition time × Nb target atoms /cm2

=I

eT ρ N t

A

I(A): Beam currentNa: Avogadro’s number = 6.022 × 1023mol−1

ρ: Density of the liquid hydrogen target in g/cm3

A: Molar mass of the hydrogen = 1.008 en (g/mol)e: Electron charge = 1.6 10−19CT : Acquisition time in seconds (s)t: Thickness of the target in cm

For this, and given that the luminosity depends strongly on the number of target atoms,the liquid hydrogen target should be kept under some stable conditions in order to in-crease luminosity and reduce the energy loss of the scattered particles. The temperatureand pressure are stabilized to 21 K and 1990 mbar respectively. The constant temperatureof the target is guaranteed by a continuous pumping of liquid hydrogen and the rastering

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Figure 3: The target

motion of the electron beam. Those charecteristics are continuously monitored for a pre-cise determination of the luminosity. The dimensions of the target are specified to 5 cmin length 11.5 mm wide and 10 mm in hight. A view of the target is shown in figure 3with the specified dimensions.

3.3 The A1 hall and the 3+1 Spectrometer setup

The largest experimental hall (A1) of the MAMI accelerator complex houses three high-resolution, focussing magnetic spectrometers operated by the A1 Collaboration. The highmomentum resolution (∆p/p ≈ 10−4) together with the large acceptance in solid angle (upto 28 msr) and in momentum (up to 25%) makes this setup ideal for electron scatteringin coincidence with hadron detection. One of the spectrometers can be tilted up to anout-of-plane angle of 10, allowing for out-of-plane kinematics. A proton recoil polarimetergives, in combination with the polarized MAMI beam and a polarized helium-3 gas target,access to a broad variety of spin observables. A fourth new spectrometer (KAOS/A1),covering high momenta with a moderate path length for the detection of kaons, is currentlyin the commissioning phase. In this experiment, the only spectrometers that we use arespectrometers A and B. Figure 4 shows the spectrometers of the hall A1 with the doorsof the scintilators opend.

Figure 4: The spectrometers

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3.3.1 spectrometer magnets

The role of the spectrometer magnets is to force the charged particles entering the spec-trometer to deviate upwards in order to measure their momenta by calculating the cur-vature of their deviation and then to be detected later on. The optical system of thespectrometer A contains a quadrupole a sextupole and two dipoles, whereas the spectrom-eter B is composed only of one dipole. These spectrometers are able to reconstruct theparticle trajectory and to calculate exactly the time at which the particle is scattered andof course the type of the particle. All of this is done in a particle detection system.

3.3.2 Particle detection system

The detectors in the focal plane of the spectrometers are designed to measure the particletrack across the focal plane with an accuracy that permits to precise exactly the coordi-nates of the interaction point, momentum of the emitted particle, and the angle at whichit was emitted, then to identify the nature of the particle. To be able to do this type ofcalculations and measurements, various types of detectors had been equiped, Vertical DriftChambers VDC, two scintillator planes and a Cherenkov detector. Using those detectorswe aim to suppress, as much as we can, the background processes.

Vertical drift chambers

Figure 5: The vertical drift chambers

Resolving the particle track with a very high accuracy is very essential in analysingevents. Vertical Drift Chambers provide a precise measurement of the position and angleof incidence of both recoil electrons and knockout protons at the respective spectrometerfocal planes. This information may be combined with the knowledge of the spectrometeroptics to determine the position and angle and momentum of the particles in the target.The VDCs of each spectrometer consists of four planes, two perpendicular to the dispersiveplane, called x-planes, and they are used to determine the momentum and the out-of-planeangle of the reconstructed particle, and the other two coordinates planes, called s-planes,

15

are tilted by a few degrees.

When a charged particle passes through the VDC it ionizes the gas (argon + isobutane)and the free electrons drift towards the wires. When those electron are collected at oneof the signal wires, an electronic signal is read out. The more wires the particle hits, themore accuracy we get on the reconstructed trajectory. If one

Sometimes scintillators are used to identify and separate between protons and pions bycalculating the energy loss in the first scintillator plane (called dE plane) along with theenergy loss in the second scintillator plane (called

whenever light signal is detected an electron is identified.

3.4 Data acquisition

Data aquisition is done by a program named Aqua, this program is developed by the re-search team at the hall A1 at MAMI. Whenever a particle passes through the scintillatordetectors, Aqua registers its information as a binary format, then saves it in a file named“run” to be analysed later on. Aqua is configured to accept all the events in coincidencealong with a small fraction of the single armed events (single armed refers to the casewhen we detect a particle in spectrometer A and no particle in spectrometer B and viceversa) . Each file “run” contains half an hour of registered events which are going to beanalysed later on to extract cross sections and then GPs.

4 Data analysis

In this section, we are going to list the technics by which event selection is done, weknow that there are many background events that do not contribute to the cross sectionsthat we want to calculate, like the electroproduction of pions. For this we use a programnamed “COLA”, which is also developed by the research team at Mainz, to preservethe photon electroproduction events and suppress the contribution of the backgroundprocesses. “COLA” is the program which does event reconstruction from the raw data ofthe detectors. For each event, COLA calculates a number of physics variables that can beused for physics analysis.

4.1 The missing mass and the identification of the photon

In the VCS experiment we observe two main processes, the first one is the electropro-duction of the photons ep −→ e′p′γ and the second one is the electroproduction of pionsep −→ e′p′π0. We know that the Low energy theorem applies in a region in

√s where we

have only photon electroproduction events i.e a region below the pion threshold, hence weneed to only select those events.

In the hall A1, and in the VCS experiment, the only spectrometers that were used arethe spectrometers A and B, in the spectrometer A we detect the proton, while in B wedetect the electron. Using only those two spectrometers we are not able a priori to knowwhich of the electroproduction events is happening, so from a kinematical point of viewwe calculate the missing 4-momentum pmiss and the missing mass Mx in the reaction andidentify the missing particle.

We have:

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with cuts/Mx2 full scale

−1000 0 1000 2000 0

1000

2000

3000

4000

Cou

nts

mmiss2 [MeV2/c4]

Figure 8: missing mass squared

pmiss = (ke + pp) − (k′

e′ + p′p′)

and the missing mass of the non detected particle is given by

Mx2 = (p0

miss)2 − (~pmiss)

2.

In order to select the photon electroproduction events, we should select a missing mass ofzero, since the final photon is on the mass shell. To do that, we make a cut in the missingmass plot near zero with a lower limit of −0.001GeV 2 and upper limit of 0.002GeV 2.

In figure 8 we can see that there is a peak centered at zero which signifies the photon.

4.2 Removing the effect of the target walls

with cuts/VertexZ

−20 −10 0 10 20 0

500

1000

1500

2000

Cou

nts

z [mm]

No Cuts/VertexZ

−50 0 50 0

50

100

150

200

Cou

nts

[103 ]

z [mm]

Figure 9: Zvertex with and without the cut.

The liquid hydrogen target is bounded by a metallic alloy wall of 9µm that contains it,the electron beam passes the target in its longest dimension. While passing the walls, thebeam interacts with the material of the walls causing some reactions that we do not want

19

in our analysis, therefore we use the program COLA to remove the effect of the targetwalls. COLA generates a plot (called Zvertex) of the registered events as a function of thelength of the target, hence a cut can be made such that it removes the effect of the targetwalls.

The cut on the Zvertex is made such that |Z| < 18mm where Z represents the axis thatpasses throught the longest dimension of the target (see figure 9).

4.3 Particles in coincidence

In analysing the stored data, we look for particles that are detected in coincidence i.e par-ticles that correspond to the same reaction. Let t0 denote the initial time of interaction,tA and tB be the times at which detectors A and B detect the proton and the electronrespectively, then we have

tA = t0 + ∆tAtB = t0 + ∆tB

where ∆tA and ∆tB are the time needed for both particles to travel from the point ofinteraction to the point where they get detected at the spectrometers.

We define TAB as the coincidence time and it is given by

TAB = tA − tB

In order to get two particles in a real coincidence we should get TAB ≈ 0 up to someconstant since ∆tB << ∆tA, and if we want to strict TAB to zero, real coincidences willnot appear. This approximation will insure us that an electron detected at A correspondsto a proton detected at B, whereas if |TAB | > 0 we have what is called a false (or random)coincidence, and the particles corresponding to this detection are of no use.

To select a good coincidence between the two spectrometers, we apply a cut in coinci-dence time near zero, we select |TAB | < 3ns hence insuring that almost all the detectedparticles are in coincidence. This of course is not sufficient enough, since there will be alsosome random coincidences under the peak. Those are also subtracted by reweighting theevents with a negative weight which is proportional to the percentage of random eventsunder the peak. When one applies negative weights to particles, one is subtracting, usingan indirect way, some of them.

Figure 10 represents the coincidence time plot between the two spectrometers A and Bwhere teh random coincidences are subtracted, true coincidences are marked by a peakat zero. Figure 11 shows the coincidence time plot before subtracting the random events

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with cuts/Tcoinc

−30 −20 −10 0 10 20 30 −10

0

10

20

30

40

Cou

nts

[103 ]

TAB [ns]

Figure 10: Coincidence time between the two spectrometers A and B.

where they appear as a flat distribution under the peak.

No Cuts/Tcoinc

−50 0 50 0

50

100

Cou

nts

[103 ]

TAB [ns]

Figure 11: Coincidence time between the two spectrometers A and B without sub-tracting the random coincidences.

One important thing is that sometimes we detect particles in coincidence that do notcorrespond to a photon electroproduction event, i.e they correspond to elastic scatteringof electrons off the protons, but those events are removed by another cut which rejects allthe elastic events.

4.4 Rejecting the elastic events

The process of scattering an electron off a proton may also involve elastic scattering, i.ethe final particles are only the electron and the proton. These events happen regularly inany experiment like the one we are studying. As for how they appear in the plots createdby COLA, we can recognise them as a very dense cluster of points in a region in the plot ofthe scattering angle and momentum of the proton for example. This clustering simplifies

21

then we substitute this value as the center of mass energy of the photon electroproduction

process√

s = Ep + Eγ =√

m2p + q′2 +

√

q′2 = 1073MeV where we find q′ = 126MeV/c.

To be able to make a full study of the photon electroproduction cross section usingthe low energy theorem, we should restrict our analysis to a region in

√s where we do

not have a pion production. For that we take an upper limit for the momentum of themissing particle to be slighly below the pion threshold q′ = 120MeV/c and a lower valueof q′ = 80MeV/c (figure 13).

4.6 Extracting experimental events

The low energy expansion of the photon electroproduction cross section depends mainlyon five parameters q, q′, ǫ, θ, φ. In the experiment, we only have the possibility of coveringa small region in ǫ since we are limited by the experimental setup, hence binning in ǫ hasno meaning and is not done. The TNB

0 in equation (1) contains the kinematical variablesv1 and v2, those variables have a strong dependance on cosθ and φ, hence when choosingthe bins that we want to extract cross sections from, we should pay attention to this de-pendence. In the plot of the cosθ vs φ we choose to take 36 bins in φ and 40 bins in cosθwhich makes a total of 1440 bins (figure 14).

with cuts/CosThetaCM vs PhiCM−140−0.5−100o 0o 100oφγ*cos(θγ

* )

Fig ure 1 4 :A p lo t o f co sv sw i th a ll th e spec i fi ed c uts

A f or tr an b ased p r ogr am n am ed PAW (Physics An alysis Wor ksta tion )[7 ] can extr act th enu mb er of exp er im ental events f r om each b in in cosθ an d φ an d p u ts th em in a fi le to b er ead later wh ile calcu latin g cr oss section s.

5 s i mu l at i o n an d c ro s s s e c t i o n c al c u l at i o n

I n th is section we ar e goin g to d escr ib e th e p r o ced u r e we u sedto calcu late th e exp er im en -tal cr oss section s an d th eir er r or b ar s, th en we will talk ab ou t th e χ2 m in im ization an d

2 3

how we fit the structure functions of the proton.

5.1 Measuring cross sections

In order to calculate the experimental cross sections we must have the number of exper-imental events in a certain solid angle ∆Ω5 = ∆Ee′∆Ωe∆Ω for a given luminosity sincethe experimental cross section is given by

d5σexp =Nexp

Lexp ∆Ω5

From COLA we can have the number of experimental events and the value of the ex-perimental luminosity, but what we have is not sufficient to calculate the experimentalcross sections, since we have no idea of how we can have a good approximation of thesolid angle. To do this, we use a simulation of the type Monte Carlo which can simulatethe same process ep −→ e′p′γ under the same conditions of the MAMI accelerator. To beconsistent, we fix a total cross section for the ep −→ e′p′γ process and start generatingevents exactly in the same phase space distribution of the experimental ones. Applying thesame cuts and binnings to the simulated plots we finally obtain the number of simulatedevents per bin along with the simulated luminosity.

In eahc bin we have:

d5σsim =Nsim

Lsim× 1

∆Ω5

then ⇒ d5σexp =Nexp

Lexp× Lsim

Nsim× d5σsim

(2)

The term in the LET expansion of the cross section which contains the GPs is mostlysensitive to the polar and azimuthal angles of the final photon(θ, φ), so we use the plot ofcosθ vs φ from COLA and from simulation to extract values for the number of events perbin. The extraction of the number of events per bin is done using PAW.

The simulated cross section d5σsim in equation (2) is the cross section of type Bethe-Heitler and Born. An existing FORTRAN based program is used to calculate it alongwith other variables (v1, v2,Φq′) that are going to be used when fitting the structure func-tions.

Figure 15 below shows the comparison between the simulated and experimental plots ofsome variables, one can notice the good agreement between both plots along with someslight differences that we can justify as a GPs effect.

24

Figure 15: Comparison between simulated (blue) and experimental (red) plots; Ver-texZ: plot of the number of events with respect to the Z axis of the target, M2 photonregion: plot of the missing mass, CosThetaCM vs PhiCM: cosθ vs φ, φ: azimuthalangle of the final photon, CosThetaCM: cosine of the polar angle of the final photon.epsilongamma: photon polarization.

5.2 Corrections and errors

As we all know, in experimental physics one is never sure of what he calculates, andany variable calculated without an error bar is meaningless, so in order to have crosssections that make sense, we should calculate their error bars. We have two types oferrors, systematic and statistical errors, the systematics are errors that have to do withthe detector itself whereas the statisticals are related to the limited number of events.

5.2.1 Systematic errors

In quantum electrodynamics, a correction due to the interaction of a charged particle withits electromagnetic field to the value of some physical quantity or to the cross section ofsome process is called a radiative correction, it may be considered a result of the emissionand absorption of virtual photons or creation of electron-positron pairs. This correction isrepresented as a power series in the fine structure constant α = 1

137. This correction must

be taken into account along with the systematic correction on the experimental luminosity.

In our case we account for the radiative corrections and the error on the luminosity forthe photon electroporduction process is a 2% change of the experimental luminosity. Tobe honest, we should stress the fact that almost all this error comes from the radiative

25

corrections, since we can measure a stable experimental luminosity. This will not changemuch our calculated cross sections since this term is considered mostly as a normalizationfactor multiplied by the total coss section.

5.2.2 Statistical errors

Probability distributions are widely used primarily in experiments which involve count-ing. The sampling errors which occur in counting experiments are called statistical errors.Statistical errors are one special kind of error in a class of errors which are known as ran-dom errors. In counting the number of experimental and simulated events (Nexp,Nsim),the statistical error for this number in a certain bin is given as ∆Nexp =

√

Nexp and∆Nsim =

√Nsim respectively. What is essential for us is mainly the error on the cross

section itself, so we should see how the error propagates in equation (2).

Since we have no statistical errors on experimental and simulated luminosities, and ofcourse the simulated cross sections are exactly calculated, then we are left with errors onthe number of events and we have

∆d5σexp = d5σexp ×√

1

Nexp+

1

Nsim

This error will be calculated for each bin in cosθ and φ and will be added to the associatedcross section.

5.3 The C++ program

Calculating cross sections is done in a C++ program prepared during this training period.The first task in this program is to read several input files of the number of experimentalevents per bin and the number of simulated events per bin, also it reads a file generated bythe FORTRAN based code that contains the simulated cross sections of the type Bethe-Heitler and Born and the other variables needed for the fit (v1, v2,Φq′). Then it calculatesthe cross sections and their errors and puts them in output files where each file containsthe cross sections calculated for different bins in cosθ. The simulated and experimentalluminosities do not change from bin to another, so there is no need to read them from filesand they are introduced as floats in the program itself.

In this C++ code, some condtitions were applied to restrict the cross section calculation.

• We excluded bins that contain less than 20 events for both simulated and experimentalones, since calculating the cross section in a bin with too small number of events willincrease its statistical error.

• There are some regions in θ and φ where we have a high sensitivity to the GPs, regionswhere the Bethe-Heitler and Born term in the cross section is not very large compared to

26

the term that contains the GPs. A test is done to identify this region. First of all we havein our hands two theoretical ways in which we can calculate the photon electroproductioncross section, first we have the LET expansion which gives us the cross section as twoparts (Bethe-Heitler and Born plus a term which contains the GPs) and the dispersionrelations (DR) formalism which give exactly the total cross section and it is done by Bar-bara Pasquini[8].

VCS q2-02-inp. Definition of good bins for the LEX fit

-150

-100

-50

0

50

100

150

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

cosθCM

φ (d

eg)

Figure 16: The fine bins (blue dots are the bins in which we have a remarkable GPeffect); Red: acceptance of the ou-of-plane setting, green: acceptance of the in-planesetting.

In both cases, we calculate the effect of the GP term relative to the Bethe Heitler andBorn term. Let d5σ

Tdenote the total photon electroproduction cross section calculated

using the dispersion relation, A and B be the ratios of the GP term to the Bethe-Heitlerand Born term from the LET expansion and the dispersion relation respectively, then fromthe LET expansion we have (using (1))

C =Φq′CMHNB

0

d5σBH+B

and from the DR formalism we have

D =d5σ

T − d5σBH+B

d5σBH+B

where the term d5σT − d5σ

BH+Brepresents the term sensitive to the GPs in the DR

27

formalism.

In order to choose the good bins we require that |C − D| < 0.02 i.e the difference issmaller that 2%. Using this we will restrict our study to a certain number of bins. Figure16 is a plot which shows the good bins (in blue color) which we can use to extract theGPs throught the LET.

5.4 The spectrometers settings

In this experiment, two different spectrometers settings have been used, the first settingis the in-plane setting where the spectrometers are placed in the plane parallel to lep-tonic plane, and the second setting is the out-of-plane setting where the spectrometersare slightly tilted out of the leptonic plane by a certain angle. The cross sections arecalculated for both settings and plotted using PAW. Figures 18 20 are the plots for thecross sections in the selected bins.

In figures 17 19 18 20, the cross section is plotted along with two curves, a red curve whichcorresponds to the Bethe-Heitler cross section, and a green curve which corresponds tothe Bethe-Heitler cross section along with the term which contains the GPs where certainvalues are set for the structure functions.

Figure 17: cross sections in pbMeV −1sr−2 for the in-plane setting, plotted for allthe bins in the acceptance

28

Figure 18: cross sections in pbMeV −1sr−2 for the in-plane setting plotted only forthe blue bins

5.5 Fitting the structure functions and the χ2 minimisation

To fit the structure functions (PLL − 1

ǫPTT ) and (PLT ) in equation (1) we are going to

use the method of χ2 minimisation. The χ2 is given by:

χ2 =N∑

i=1

[yi − f(ϕi, ϑi)]2

∆y2i

yi = d5σexp(ϕi, ϑi)

f(ϕi) = d5σBH+Bi + Φq′i(v1(ϕi, ϑi).G + v2(ϕi, ϑi).H)

∆yi = ∆d5σexp(ϕi, ϑi)

where G = PLL − 1

ǫPTT and H = PLT

To find the values of G and H we calculate the partial derivatives of χ2 with respect tothose variables and set them to zero, we have

∂χ2

∂G=

∂χ2

∂H= 0

To find the errors ∆G and ∆H of G and H respectively we should calculate the secondderivative of χ2 with respect to those parameters and construct the inverse error matrix

E−1 where E−111

=∂2χ2

∂G2, E−1

22=

∂2χ2

∂H2and E−1

12= E−1

21=

∂2χ2

∂G∂H=

∂2χ2

∂H∂G.

Inverting this matrix we have E =1

det(E−1)E−1, where E−1

11=

∂2χ2

∂H2, E−1

22=

∂2χ2

∂G2and

29

Figure 20: cross sections in pbMeV −1sr−2 for the out-of-plane setting plotted onlyfor the blue bins

The values for the structure functions, their errors and the minimum χ2 for the in-planesetting are:

PLL − PTT

ǫ= [26.79 ± 37.18]GeV −2

PLT = [−3.98 ± 2.22]GeV −2

χ2min = 24.8361 for 26 degrees of freedom

and for the out-of-plane setting we have:

PLL − PTT

ǫ= [25.26 ± 3.05]GeV −2

PLT = [−26.12 ± 6.13]GeV −2

χ2min = 50.368 for 42 degrees of freedom

The error ellipses for both settings are shown in figures 21 and 22

5.6 Combining measurements

We have calculated the values of the structure functions for two different settings, so toget a final value for them, we have to combine the two measurements and find the values

31

Figure 21: The error ellipses for the in-plane setting

of the structure functions for all the registered events.

We denote by Xi and Yi [i=1(first measurment),2(second measurment))] the values ob-

tained from the measurments for PLL− PTT

ǫand PLT respectively, and by PLL− PTT

ǫand

PLT the values for the structure functions for the combined measurments. By referring toreference [9] we have,

PLT =

2∑

i=1

YiWi

2∑

i=1

Wi

with

Wi =1

∆Y 2i

and

32

Again we must stress the fact that this is only a preliminary analysis. The values thatwe have calculated are marked by hand in figure 23

preliminary

preliminary

Figure 23: World data on the structure functions PLL−PTT

ǫ(top) and PLT (bottom)

deduced from VCS analysis. LET and DR results are indicated by squares and filledcircles respectively. The inner (outer) error bars are statistical (total). The curvesare theoretical calculations done using fixed parameters. Our values of the structurefunctions are marked by a pink point bounded by a pink rectangle.

34

6 Conclusion

My work in this internship was based on the photon electroproduction experiment (ep −→e′p′γ) done in the hall A1 of the Mainz Microtron, at a beam energy of 1GeV and a four-momentum transfer of 0.2GeV 2. As a first step I had to extract the experimental crosssections and find their error bars, then I used a χ2 minimization to fit the structure func-tions of the proton.

In this experiment, the electron scatters off a proton target and gets detected after thatin coincidence with the proton in the spectrometers B and A respectively of the hall A1,the photon is identified as the missing particle and is selected by a cut in the missing massquared plot.

The photon electroprodution reaction permits us to study the physics concerning thenucleon. In this scheme, virtual compton scattering is the best candidate to study thegeneralized polarisabilities of the proton and give a view of how the proton deforms underan external electromagnetic field. The low energy theorem gave us the axioms on whichwe based our work and calculations, we need not use any other model since we worked ina region below the pion threshold, a region where we can apply the low energy expansionand see a generalised polarisabilities effect.

Althought what we have done in this training period is just a preliminary analysis, we

have obtained reliable values for the structure functions PLL − PTT

ǫand PLT .

This training period has opened my mind to the new world of scientific research, it hasset my foot to the path of ultimate knowledge, in particular, to the field of experimentalparticle physics. During my internship, I had the chance to visit the nuclear physicsinstitute in Mainz and meet the research team there. After that, I went back to work withthe group at Clermont Ferrand, developing tools and creating programs to calculate crosssections. While doing this I used a new operating system called LINUX and learned mostof its commands.

35

References

[1] J. ROCHE - Diffusion Compton virtuelle a MAMI et mesure des polarisabilites

generalisees du proton, these du DAPNIA, 21 decembre 1998.

[2] P. JANSSENS et al, NIM A 566 (2006) 675.

[3] L. LYONS - Statistics for nuclear and particle physicists, Cambridge university press(1993)

[4] P. JANSSENS - Double-polarized virtual Compton scattering as a probe of the proton

structure. Thesis at the unversity of GENT, 2007.

[5] P.A.M GUICHON and M.VANDERHAEGHEN - progress in particle and nuclear

physics. Vol.41, 1998.

[6] Image gallery of the Hall A1 at Mainz:http://wwwa1.kph.uni-mainz.de/A1/gallery/.

[7] Physics Analysis Workstation:http://wwwasd.web.cern.ch/wwwasd/paw/.

[8] D.DRECHSEL, B. PASQUINI, M. VANDERHAEGHEN - Dispersion relations in real

and virtual Compton scattering Phys. Rept. 378 (2003) 99-205.

[9] PHILIP R. BEVINGTON - Data reduction and error analysis for the physical sciences

McGraw-Hill, Inc. (1969).

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