Experimental verification of the GDH sum rule on the neutron

Transcription

1 University of Ghent Faculty of Sciences Department of Subatomic and Radiation Physics Experimental verification of the GDH sum rule on the neutron Tigran Armand Rostomyan Promoter: Dirk Ryckbosch Proefschrift ingediend tot het behalen van de graad van Doctor in de Wetenschappen: Natuurkunde Academiejaar 24-25

2 Examencommissie: Prof. Dr. P. Matthys, voorzitter Faculteit Wetenschappen, RUG Prof. Dr. D. Ryckbosch, promotor Faculteit Wetenschappen, RUG Prof. Dr. H.-J. Arends Universität Mainz, Duitsland Prof. Dr. K. Heyde Faculteit Wetenschappen, RUG Prof. Dr. P. Pedroni INFN - Sezione di Pavia, Italië Prof. Dr. J. Ryckebusch Faculteit Wetenschappen, RUG Prof. Dr. L. Van Hoorebeke Faculteit Wetenschappen, RUG

3 ÆÎÁÊÎÍÅ ÀÇÊË ÊÅ Æ À ÃÁ ÇËÌÇÅ ÆÁÆ For my father: Armand Hayki Rostomyan

4 Contents 1 Introduction and Overview 1 Overview Physics motivation The GDH Sum Rule Derivation Forward Compton scattering amplitude Causality and Analyticity Crossing symmetry Dispersion relations Unitarity (Optical theorem) Low s theorem No-subtraction hypothesis GDH and other Sum Rules The generalized GDH integral The GDH sum rule for the deuteron Nucleon resonances Introduction Motivation Pion photoproduction in the GDH experiment at MAMI The Experimental Setup at MAMI The electron beam at MAMI The photon beam in A2 hall The Bremsstrahlung photon tagging spectrometer Photon Flux I

5 3.2.3 Photon Polarization The Target The unpolarized target The polarized target The GDH detector setup DAPHNE The Vertex detector Range-telescope detector MIDAS The Čerenkov detector The forward components The data acquisition Common conditions for Mainz data analysis Charged particle identification The E E and geometrical range method Stop-A particles Stop-B particles Stop-C, Stop-D, Stop-E and Stop-F particles The range-fit method Summary Neutral particle identification Common conditions of the analysis Correction for MWPC inefficiency Vertex cut conditions Cuts on polar angle Azimuthal correction Control Cylinder and the effective target length correction Tagger selection and random subtraction The integrated luminosity Converting tagger channels to photon energies Statistical errors Systematic errors GEANT simulation II

6 5 Analysis of the γ + p n + π + reaction Unpolarized total and differential photoproduction cross sections of the γ + p n + π + reaction Event selection procedure First method: One charged pion without veto for π -s Background sources and subtraction Cross section of the γ + p n + π + + π reaction Background subtraction: γ + p n + π + + π reaction Subtraction of the remaining background sources Total cross section Differential cross section Systematic errors Second method: One charged pion with π -veto Background sources and subtraction Total cross section Differential cross section Systematic errors Comparison between the methods Comparison to other existing data and theory Polarized total and differential photoproduction cross sections of the γ + p n + π + reaction Background sources and subtraction Total cross section Differential cross section Comparison between the total cross section methods Systematic errors Comparison with other existing data and theory σ 3/2 and σ 1/2 cross sections Discussion Contribution of the γ + p n+π + reaction to the GDH sum rule integral.16 III

7 6 The GDH sum rule on the neutron Unpolarized total photoabsorption cross section on the deuteron Data analysis σ tot for photon energies below 4 MeV σ tot for photon energies upto 8 MeV Systematic errors Comparison with existing data and theory Helicity dependent total photoabsorption cross section difference on the deuteron Data analysis Systematic errors Comparison with theory GDH integral for the deuteron The GDH sum rule on the neutron Summary and conclusions 131 Tabulated results 133 Bibliography 148 Acknowledgment 153 IV

8 Chapter 1 Introduction and Overview In the beginning of the 195 s and 196 s physicists discovered that nucleons are not point-like particles. It was found that the proton, which is a positively charged particle, with charge +e and mass m p, instead of having a magnetic moment µ p = eħ/2m p = 1µ N, has a magnetic moment µ p = 2.79µ N. And the neutron, which is a neutral particle, instead of having µ n =, has a magnetic moment µ n = 1.91µ N. The difference between expected and measured magnetic moments was called the anomalous magnetic moment k. This phenomenon directly derives from the internal structure of these particles. Most of the nucleon static properties, such as charge, mass and magnetic moment, are well known quantities. But the nucleon s internal structure and more specifically its spin structure are still not understood even today. In the context of the quark model the nucleons are a combination of three valence quarks, bound together by gluons. Quarks are electrically charged, point-like particles with spin 1/2. The behavior of the quarks and gluons is described by quantum chromodynamics (QCD). However, this theory is not yet able to calculate the static properties of the nucleon from its constituents. A number of deep inelastic scattering (DIS) experiments were performed to investigate the spin structure of the nucleon. It was found at CERN (EMC) in 1987 that the spin of the quarks contributes only about 3% to the total nucleon spin [1]. This was called the Spin-Crisis. Later on several experiments at CERN (EMC [2], SMC [3, 4]), SLAC (E-142 [5], 143 [6], 154 [7]) and DESY (HERMES [8, 9]) confirmed this result and went on to search for the solution. The spin contribution from so-called sea quarks (virtual quark-antiquark pairs) and the gluon clouds in the nucleon could solve this problem. The HERMES experiment has shown that the contribution of the sea quarks to the nucleon spin is negligible [1]. The other possibility is that the spin contribution of the gluon cloud will solve this crisis [11]. The virtual photoabsorption process is generally described by the spin structure functions, primarily by the polarized g 1 (x) spin structure function, where x is the Bjorken variable, which is a measure of the elasticity of the collision process: x=1 means an elastic process, < x < 1 - inelastic processes. This function is defined as the difference between the probability densities to find a quark with its polarization parallel and anti-parallel to the nucleon spin. It can be obtained from the deep-inelastic scattering cross sections of polarized leptons on polarized nucleons. From the theoretical side several sum rules for the integral of g 1 (x) over all x, based on general assumptions, have been developed in the 6 s. Two examples are the Bjorken sum rule for the proton-neutron difference g p 1(x) g n 1(x) (see section 2.1.2), which has been 1

9 verified to within 1%, and the Ellis-Jaffe sum rule for the proton and the neutron separately (see section 2.1.2), where measured data show a significant disagreement with the predicted values. There is a similar sum rule at the real photon point, called the Gerasimov-Drell-Hearn (GDH) sum rule [12, 13]. The experimental verification of the sum rules is needed to check our understanding of the underlying physical principles. The GDH sum rule is derived from fundamental physics principles (see Chapter 2.1) and can be written as: σ (ν) σ (ν) ν dν = 4π2 α m 2 k 2 S, where σ and σ are the total photoabsorption cross sections of circularly polarized photons by longitudinally polarized target particles with parallel and anti-parallel polarization orientations, respectively. The cross section difference is weighted by the photon energy ν and is integrated over all photon energies. Further, m is the mass of the target particle, S is its spin, k is its anomalous magnetic moment and α is the fine structure constant. The beauty of this sum rule is that it yields a direct link between the dynamical excitation spectrum of the target particle and the particle s static properties: the anomalous magnetic moment, spin and mass. Several physics institutes formed the GDH collaboration in The collaboration has members from Germany, Belgium (our research group at the University of Ghent), Italy, Russia, France, Japan, Sweden, and from the USA. The primary aim was the experimental verification of the GDH sum rule on the proton and on the neutron. The decision was made to perform the experiments at two places: at the microtron accelerator facility MAMI in Mainz [15, 16]: to measure in the 15 MeV < ν < 8 MeV photon energy range, and at the electron synchrotron ELSA in Bonn [17]: to measure in the 5 MeV < ν < 3 GeV photon energy range. Both are located in Germany. The upper limit of 3 GeV is not infinity, but, because of the 1/ν term in the GDH integral, the combination of these two energy ranges gives a very large part of the GDH integral, so it will be possible to observe whether the GDH integral converges or not. Other laboratories, e.g. LEGS (Brookhaven) and GRAAL (Grenoble), have also plans to perform an experiment to verify the GDH sum rule. What needs to be measured in these experiments is the helicity dependent cross section difference σ σ as a function of photon energy. Thus one requires a circularly polarized photon beam, a polarized target and a detector system which covers 4π. The technical requirements for this kind of experiments were becoming available at the beginning of 9 s and the motivation to carry out the experiments was strong enough. The circularly polarized photon beam is usually obtained from the Bremsstrahlung process of longitudinally polarized electrons. A frozen spin butanol/dbutanol target, which was cooled down up to 5 mk, was used as a polarized target. The DAPHNE detector was used as a 4π detector for the Mainz case (see Chapter 3). For the Bonn case, a new GDH detector was used. Apart from the total helicity dependent cross section difference σ σ, the Mainz experiment is also able to determine the helicity dependent cross section differences for the different partial pion photoproduction channels. This makes a comparison with the different theoretical models possible. It also gives an opportunity to study the contributions of the various nucleon resonances (see Section.2.2). This PhD work is on the experimental verification of the GDH sum rule on the neutron and on the determination of the helicity dependent cross section difference of the γ p n + π + 2

10 partial channel. The main part of this PhD work are results from the Mainz experiment. Results from the Bonn experiment are also shown. The data analyses of the γ p n + π + and the γ d TOTAL channels from Mainz experiment are presented. The obtained results are the first of their kind. The analysis is performed in two steps: the development of the analysis methods for the unpolarized calibration data and the application of these methods to the doubly polarized data. The results from the unpolarized data are successfully verified with existing data. Overview The GDH sum rule and the role of nucleon resonances are treated in some details in Chapter 2. The detailed description of the Mainz experimental setup is given in Chapter 3. Particle identification methods and common conditions for the analysis of the Mainz data are discussed in Chapter 4. The analysis of the γ+p n+π + reaction on unpolarized hydrogen and γ + p n + π + on polarized butanol are presented in Chapter 5. A cross-check with Heidi Holvoet s results [42] for the γ + p n + π + + π reaction on unpolarized hydrogen is also presented here. The analysis of the total photoabsorption cross section from unpolarized deuterium data (Mainz, 1997) and of the helicity dependent total photoabsorption cross section difference from polarized deuterated butanol data (Mainz, 1998) are presented in Chapter 6. Results for the helicity dependent total photoabsorption cross section difference from deuterated butanol from the Bonn experiment are also presented there. The summary of this Ph.D. thesis and the tabulated results are presented at the end. Papers with these results are published or submitted for publications in [94, 95, 97, 98, 99, 1]. The main publications of the GDH collaboration can be found in [9, 91, 92, 93, 94, 95, 96, 97, 98, 99, 1]. 3

11 Chapter 2 Physics motivation 2.1 The GDH Sum Rule The Gerasimov-Drell-Hearn (GDH) sum rule [12, 13] gives a fundamental relation for the study of the particle spin structure via real photon absorption. It was derived by Gerasimov [12] in 1965 and independently by Drell and Hearn [13] in It relates the helicity dependence of the total photoabsorption by a target particle, σ σ, to its static properties: the mass m, anomalous magnetic moment k and spin S: σ (ν) σ (ν) ν dν = 4π2 α m 2 k 2 S (2.1) where σ and σ are the total helicity dependent cross sections for the absorption of circularly polarized photons on longitudinally polarized target particles with parallel and anti-parallel spin configurations, respectively. The cross section difference is weighted by the photon energy ν and is integrated over all photon energies. α is the fine structure constant and is equal to 1/ /2 +1 1/2 J=3/2 J=1/2 γ N σ 3/2 σ 1/2 γ N Figure 2.1: Spin configurations. In the case of the nucleon, proton or neutron, the total particle spin S = 1/2, and there are two possible spin configurations in combination with the photon spin: 3/2 (parallel) and 1/2 (anti-parallel) (Fig.2.1). The GDH integral can then be written as: σ 3/2 (ν) σ 1/2 (ν) ν dν = 2π2 α m 2 N k 2 N. (2.2) The cross section is again integrated over all photon energies. In the case of the proton k p = 1.79µ N and in the case of neutron k n = 1.91µ N. The GDH sum rule values are 24.8 µb and µb respectively. 4

12 The model independent derivation is mainly based on general fundamental physical principles: Gauge and Lorentz invariance Compton Scattering Amplitude, Low s Theorem Causality and Analyticity No-subtraction hypothesis Dispersion relations Crossing symmetry Unitarity Optical theorem The only questionable assumption in this derivation is the no-subtraction hypothesis. However, Regge theory [19] provides a strong argument for the validity of this hypothesis Derivation Forward Compton scattering amplitude The derivation of the sum rule starts by writing the Compton scattering amplitude of real photon s forward scattering by a nucleon. This scattering amplitude can be decomposed in general into six invariant functions [2]. The forward scattering amplitude (at scattering angle) is described by only two functions and can be written as: T(ν, θ = ) =< χ 2 f 1 (ν) ε 2 ε 1 + f 2 (ν)i σ ( ε 2 ε 1) χ 1 >, (2.3) where ε 1 and ε 2 are the polarization vectors for the incoming and outgoing photons respectively, χ 1 and χ 2 are the Pauli-spinors of the nucleon, σ is the Pauli matrix vector which acts on the spinor of the nucleon. f 1 (ν) is the spin independent amplitude and f 2 (ν) is the polarized amplitude which is accessible only via doubly polarized measurements. This is the most general expression which: 1. is constructed from the independent vectors ε 1, ε 2 and σ, 2. is linear in ε 1 and ε 2, 3. obeys the transverse gauge: ε 1 q 1 = ε 2 q 2 =, where q and ε are the Lorenz vectors of momentum and polarization of the photon respectively, with q q = (real photon) and ε q = (transverse polarization). 4. is invariant under rotational and parity transformations. Using right and left handed circularly polarized photons, with helicities λ = +1 and λ = 1 respectively, and assuming that the photon moves in the direction of the z-axis, the two polarization vectors may be taken as: ε R = 1 2 (ē x + iē y ), ε L = 1 2 (ē x iē y ), (2.4) and the products of the polarization vectors can be calculated for the different cases: 1 ε 1 = ε 2 = ε R ε 2 ε 1 = 1 if ε 1 = ε 2 = ε L ε 1 ε 2 5

13 Equation (2.3) then takes the form: iē z ε 1 = ε 2 = ε R ε 2 ε 1 = iē z if ε 1 = ε 2 = ε L ε 1 ε 2 T(ν, θ = ) = f 1 (ν) < χ 2 χ 1 > f 2 (ν) < χ 2 σ z χ 1 >, (2.5) where ( ) is in the case of right handed and (+) is in the case of left handed circularly polarized photons. Only two spin combinations are available when using right or left circularly polarized photons and longitudinally polarized nucleons: S = 3/2, when the spinor of the nucleon is parallel to the photon polarization, and S = 1/2, when the spinor of the nucleon is anti-parallel to the photon polarization. Then: χ 1 >= χ 2 > so < χ 2 χ 1 >= 1 (2.6) For the parallel spin configuration (S = 3/2): ( ) 1 ε = ε R and χ 1 >= χ 2 >= ( ) ε = ε L and χ 1 >= χ 2 >= 1 This means: < χ 2 σ z χ 1 >= +1 1 for { right-handed pol. Photons left-handed pol. Photons For anti-parallel spin configuration (S = 1/2): ( ) ε = ε R and χ 1 >= χ 2 >= ( 1 ) 1 ε = ε L and χ 1 >= χ 2 >= This means: < χ 2 σ z χ 1 >= 1 { right-handed pol. Photons +1 for left-handed pol. Photons Combining (2.6), (2.7), (2.8) into (2.5) the respective scattering amplitudes can be written as: { T3/2 (ν) = f 1 (ν) f 2 (ν) (2.9) T 1/2 (ν) = f 1 (ν) + f 2 (ν) or { f 1 (ν) = T 1/2(ν)+T 3/2 (ν) 2 f 2 (ν) = T 1/2(ν) T 3/2 (ν) 2 (2.7) (2.8) (2.1) In a similar way one defines the total absorption cross section as the average of the two helicity cross sections: σ T = σ 3/2 + σ 1/2 (2.11) 2 and the transverse-transverse interference term by the following helicity difference: σ TT = σ 3/2 σ 1/2 2 6 (2.12)

14 Im ' A'(, ) Re ' - A(, ) + Figure 2.2: Integration contour Causality and Analyticity The condition of causality forces the analyticity of the Compton scattering amplitude in the complex plane [21]. Using Cauchy s integral theorem for a contour C (Fig.2.2), the following expression is valid: f j (ν) = 1 2πi where j {1, 2}. This contour can be split into three parts: f j (ν) = 1 + 2πi f j (ν ) ν ν dν + 1 f j (ν ) 2πi A(ν,ε) ν ν dν + 1 2πi C f j (ν ) ν ν dν, (2.13) A (, ) f j (ν ) ν ν dν, (2.14) where + = lim ε ( ν ε + + ) is the Cauchy principal value and A and ν+ε A are arcs with centers ν and and with radiuses ǫ and respectively. The integral over A is evaluated using the residue theorem and is equal to 1 f 2 j(ν). Under the No-Subtraction hypothesis s assumption (section ) the integral over A becomes. Therefore (2.14) can be written as: f j (ν) = 1 + πi f j (ν ) ν ν dν (2.15) Crossing symmetry According to crossing symmetry, the probability for a reaction is equal to that of its mirror reaction where all particles are substituted by their anti-particles. This symmetry requires that the Compton scattering amplitude has to be symmetrical for the exchange of momentum (ν ν) and polarization ( ε 2 ε 1 ) [22]. This is expressed as: f 1 (ν) = f 1 ( ν), f 2 (ν) = f 2 ( ν), (2.16) with the result that f 1 is an even and f 2 an odd function. Or f j (ν ) = f j (ν) when ν R (2.17) f j ( ν iε) = f j ( ν + iε) when ν R, ε > (2.18) 7

15 Here again j {1, 2} and ε is the radius shown on Fig.2.2. Combining (2.16) and (2.18), for ν R one can write: So, f 1 (ν) = f 1 (ν + iε) 2.16 = f 1 ( ν iε) 2.18 = f1 ( ν + iε) = f 1 ( ν) (2.19) f 2 (ν) = f 2 (ν + iε) 2.16 = f 2 ( ν iε) 2.18 = f 2( ν + iε) = f 2( ν) (2.2) and for (ε with ε > ): f 1 (ν) = f1 ( ν), f 2(ν) = f2 ( ν), (2.21) Imf 1 (ν) = Imf 1 ( ν), Imf 2 (ν) = Imf 2 ( ν) (2.22) Using this symmetry equation(2.15) is taking the form: f 1 (ν) = 1 + ( πi f1 (ν ) ν ν + f 1(ν ) ν ν f 2 (ν) = 1 + ( πi f2 (ν ) ν ν f 2 (ν ) ν ν Dispersion relations The real and imaginary parts of f j (ν) can be written as: Putting here (2.23) and (2.24) one can write: ) dν (2.23) ) dν (2.24) Ref j (ν) = 1 2 (f j(ν) + fj (ν)) (2.25) Imf j (ν) = 1 2i (f j(ν) fj (ν)) (2.26) Ref 1 (ν) = 2 π + Ref 2 (ν) = 2 π Unitarity (Optical theorem) Imf 1 (ν ) ν 2 ν 2 ν dν (2.27) Imf 2 (ν ) ν 2 ν 2 νdν (2.28) The optical theorem expresses the unitarity of the scattering matrix by relating the absorption cross section to the imaginary part of the respective forward scattering amplitude: Combining (2.1) and (2.29) one can write: Im[T S (ν)] = ν 4π σ S(ν), where S {3/2, 1/2} (2.29) Im[f 1 (ν)] 2.1 = Im[T 1/2(ν)] + Im[T 3/2 (ν)] 2 = ν 8π (σ 1/2(ν) + σ 3/2 (ν)) 2.11 = ν 4π σ T(ν) (2.3) 8

16 Im[f 2 (ν)] 2.1 = Im[T 1/2(ν)] Im[T 3/2 (ν)] 2 = ν 8π (σ 3/2(ν) σ 1/2 (ν)) 2.12 = ν 4π σ TT(ν) (2.31) Now, if one puts (2.3) and (2.31) into (2.27) and (2.28) respectively, one gets: Ref 1 (ν) = 1 4π 2 = 1 2π σ 3/2 + σ 1/2 ν 2 ν 2 ν 2 dν = σ tot ( 1 + ν2 ν 2 + ϑ(ν4 ) ) dν (2.32) Low s theorem Ref 2 (ν) = 1 4π 2 = 1 4π σ 3/2 σ 1/2 νν dν = ν 2 ν 2 ( ν (σ 3/2 σ 1/2 ) ν + ν3 ν 3 + ϑ(ν5 ) ) dν (2.33) Low s theorem [23] provides the following series for f 1 and f 2 as a function of the photon energy ν: f 1 (ν) = α m N + (α N + β N )ν 2 + ϑ(ν 4 ), (2.34) f 2 (ν) = αk2 N 2m 2 N ν + γ ν 3 + ϑ(ν 5 ), (2.35) The expansion of the unpolarized, spin-independent f 1 amplitude contains the Thomson term: α m N and the Rayleigh scattering term, which is determined by the scalar electric and magnetic polarizabilities α N and β N. The spin-dependent amplitude f 2 contains the anomalous magnetic moment k N in first, and the so called spin polarizability γ in third order of ν. The same was shown by Gell-Mann and Goldberger [24]. This theorem is valid for 1-2 spin Dirac particles and for small photon energies, where the internal particle structure is not probed: (ν m π c 2 ) No-subtraction hypothesis For the spin independent f 1 amplitude one can obtain the following sum rule by comparing the first order terms of (2.34) and (2.32): f 1 (ν = ) 1 2π 2 + σ tot (ν )dν? = α m N (2.36) This leads to a contradiction, because the total cross section σ tot is a positive quantity and so, the left part of this equation is positive, but the right part of the equation is negative. This is related to the fact that the first term in the integrand of (2.32) does not converge, since σ tot becomes an increasing function of photon energy at high energies [25]. The solution for 9

17 this problem lies in a subtraction method: at least one subtraction is necessary in the f(1) dispersion relation. This leads to the following expression: Ref1 sub (ν) = Ref 1 (ν) Ref 1 (ν = ) = α m N 2π σ tot 2 ν 2 ν 2 ν 2 dν = = 1 2π 2 + σ tot ( ν 2 ν 2 + ϑ(ν4 ) ) dν (2.37) For the f 2 dispersion relation the integrand (σ 3/2 σ 1/2 ) can be either positive or negative, so there is no sign argument as in (2.36). So there is no direct evidence that the integral does not converge. When deriving the GDH sum rule, this is the assumption that needs to be made: the integrand converges so that no subtraction is needed in the dispersion relation. This is one important motivation for the experimental verification of the GDH sum rule. Physically this means that the photoabsorption cross section becomes spin independent at high photon energies. A failure of this assumption could not be explained within the existing framework of the γn interaction. This would, for instance, imply that quarks are not point like or that the photon could strongly interact with the nucleon (for a complete review see [34]). Nevertheless, at low (resonance) energies the cross section difference is an oscillating function which can change sign according to different contributing resonances. A Regge approach [19] predicts a convergent behavior for the GDH integral GDH and other Sum Rules By comparing the second order terms for f 1 one can obtain the Baldin sum rule [26]: 1 2π 2 + σ tot ν 2 dν = α N + β N (2.38) Making an no subtraction assumption for f 2 and comparing the different order terms of (2.33) and (2.35), one gets the GDH sum rule: I GDH = + and a sum rule for the spin polarizability γ : γ = σ 3/2 σ 1/2 dν = 2π2 α k 2 ν N (2.39) The generalized GDH integral m 2 N σ 3/2 σ 1/2 ν 3 dν (2.4) There is a connection between deep inelastic scattering (DIS) experiments and the GDH sum rule [2]. Virtual photon experiments were performed in order to obtain the fraction of nucleon spin carried by the quarks. It was found that the total quark spin is only 3% of the nucleon spin [1]. The Quark Parton Model is unable to explain this Spin-Crisis. These DIS experiments used virtual circularly polarized photons with squared mass q 2 = Q 2 and energy ν and 1

18 longitudinally polarized nucleons. These experiments are described by unpolarized F 1 (x, Q 2 ) and polarized g 1 (x, Q 2 ) and g 2 (x, Q 2 ) structure functions. As Q 2 + and ν +, the structure functions become functions only of x. The F 1 (x) and g 1 (x) structure functions can be expressed in terms of the quark distribution functions [14]: F 1 (x) = 1 2 g 1 (x) = 1 2 i=u,d,s,... i=u,d,s,... e 2 i e 2 i (q i (x) + q i (x) ) (2.41) (q i (x) q i (x) ), (2.42) where x = Q2 is the Bjorken scaling variable (x=1 means an elastic process, < x < 1 means 2mν inelastic processes), q and q are the probability densities to find quark or anti-quark i with a longitudinal polarization parallel or anti-parallel to the nucleon spin respectively, e i is the charge of the i quark. The expression is summed over all quark flavors. The g 2 (x) has no expression in terms of the quark distribution functions in the QPM [14]. The σ 3/2 and σ 1/2 cross sections are introduced as: ) 4π 2 α σ 3/2 = (F 1 (ν, Q 2 ) g 1 (ν, Q 2 ) + Q2 m N (ν Q2 2m N ) ν g 2(ν, Q 2 ) (2.43) 2 σ 1/2 = 4π 2 α m N (ν Q2 2m N ) ) (F 1 (ν, Q 2 ) + g 1 (ν, Q 2 ) Q2 ν g 2(ν, Q 2 ) 2 So, the sum of the cross sections is related only to the unpolarized F 1 structure function: (2.44) 8π 2 α σ 3/2 + σ 1/2 = m N (ν Q2 2m N ) F 1(ν, Q 2 ) (2.45) The σ 3/2 σ 1/2 cross section difference is then written as: σ 3/2 σ 1/2 = 8π 2 α m N (ν Q2 2m N ) In the case of real photons: (Q 2 = ), (2.46) becomes: ) (g 1 (ν, Q 2 ) Q2 ν g 2(ν, Q 2 ) 2 Following Anselmino [27], the following integral can be written: (2.46) σ 3/2 σ 1/2 = 8π2 α m N ν g 1(ν, ) (2.47) I A (Q 2 ) = m N + Q 2 2m g 1 (ν, Q 2 ) ν 2 dν (2.48) Changing the integration variable from ν to x we get the generalized GDH integral: I A (Q 2 ) = 2m2 N Q 2 1 g 1 (x, Q 2 )dx. (2.49) 11

19 Putting in (2.48) the value of g 1 (ν, ) from (2.47), we get: I A (Q 2 = ) = m2 N 8π 2 α + σ 3/2 σ 1/2 dν ν }{{} IGDH N (2.39) = 2π2 α m 2 N For proton and neutron this integral takes the following values: and Ip A () = k2 p 4 =.8; IA n () = k2 n 4 k 2 = k2 4. (2.5) =.912 (2.51) Ip n A () = IA p () IA n () =.112 (2.52) The difference of the polarized g 1 structure functions of proton and neutron is described in the Bjorken sum rule [28]: 1 ( g p 1(x, Q 2 ) g1 n (x, Q2 ) ) dx = 1 g A + QCDcorr. 1 ( 1 α ) s(q 2 ), (2.53) 6 g V 6 π where g A and g V are the axial and vector coupling constants of the weak interaction respectively [29, 3], and α s (Q 2 ) is the strong coupling constant [31, 32]. For the g 1 structure function of proton and neutron separately, the Ellis-Jaffe sum rules exist [33]. Assuming that the strange quarks contribution to the total spin is negligible, the Ellis-Jaffe sum rule is written: 1 g p(n) 1 (x, Q 2 )dx = g A g V ( ± g A + a ) 8 + QCDcorr. g V 3 ( ± g A + a ) ( 8 1 α s(q 2 ) g V 3 π ), (2.54) where g A g V and a 8 are known from neutron and hyperon decay [29, 3]. From the performed experiments data for the generalized GDH integral (2.49) are available down to Q 2 1(GeV/c) 2. In Fig.2.3 recent data for I A (Q 2 ) from the HERMES experiment on proton [8] are shown. The Bjorken sum rule prediction for the proton-neutron difference is also shown. On the same figure the GDH sum rule values for the proton and for the protonneutron difference are shown. The GDH sum rule value for the proton has a different sign and is in disagreement with what one would expect from a simple extrapolation to Q 2 = of the measured generalized integral for the proton. Thus, one expects that the generalized GDH integral is rapidly decreasing at the lower Q 2 values, what is very hard to explain in QCD The GDH sum rule for the deuteron The GDH sum rule can be applied to any system (nucleus, atom, molecule, etc.) which has non-zero spin [12]. Using the generalization of Low s theorem (2.35) for particles with arbitrary spin [35], one can rewrite the sum rule (2.39) in the following form: + σ (ν ) σ (ν ) ν dν = 4π 2 S ( 1 S µ Q M ) 2, (2.55) 12

20 * I(Q 2 ) GDH sum rule value (proton neutron) Bjorken sum rule (proton neutron) HERMES (proton) Q (GeV ).8 x GDH sum rule value (proton) Figure 2.3: Recent HERMES data on the proton for the generalized GDH integral [8]. where µ is the total magnetic moment of the considered system and Q, M and S are its charge, mass and spin respectively. In the case of a pointlike particle the term within brackets vanishes, since µ = S (Q/M), as predicted by Dirac s theory. When on the contrary, k = µ S Q the system possesses an M internal structure with excited states. However, the opposite is not in general true. A particle having a vanishing or very small anomalous magnetic moment k is not necessarily pointlike or nearly pointlike. In this respect, the deuteron is a particularly instructive example because it has a very small anomalous magnetic moment k d =.143µ N. This gives a very small GDH sum rule value for the deuteron: Id GDH =.65µb, which is close to zero and more than two orders of magnitude smaller than values predicted for the nucleon. On the contrary, it is well known that the deuteron has quite an extended spatial structure due to its small binding energy. The smallness of k d arises from an almost complete cancellation of the anomalous magnetic moments of the neutron and the proton whose spins are parallel and predominantly aligned along the deuteron spin direction. If we naively assume that the deuteron is formed by almost-free nucleons, then the deuteron sum rule value should be approximatively equal to the sum of the proton and neutron sum rule values: Id GDH Ip GDH + In GDH = 24.8 µb µb = 438 µb. But, due to the cancellation of the anomalous magnetic moments of the neutron and the proton, it is expected that a similar cancellation of different contributions should occur also for the sum rule integral. The reproduction of this is then a challenge for any microscopic nuclear theory of the deuteron. It has been shown in [36] that a large negative contribution to the sum rule of 413µb arises from the photodisintegration ( γ d pn) channel, which has its origin in a large negative spin 13

21 asymmetry right above break-up threshold (E γ 2.2 MeV). This contribution (in absolute magnitude) is almost equal to the sum of the neutron and proton GDH values. The resulting total predicted 25 µb value of the full GDH integral of the deuteron still overshoots the sum rule value. Thus, there is room for improvements of the theoretical framework which will allow to close the gap between the model-dependent evaluations and the sum rule prediction. The helicity dependent data on the deuteron are then interesting of their own, because of a strong anticorrelation between photodisintegration and pion production. They are also very sensitive to the relativistic effects at quite low energies which have never been tested in detail using the polarization observables. These data will then give a very precise check of the different nuclear models and can also be used to evaluate the helicity dependent cross sections on the free neutron. Due to the lack of free neutron targets, the experimental verification of the GDH sum rule for the neutron has to be performed either using a polarized deuteron or 3 He target. The Mainz and Bonn experiments that have been carried out up to now, have used a polarized deuteron target, while the feasibility of the use of a polarized, high-pressure 3 He gas cell is still under study. The use of the polarized deuteron as a substitute for the polarized neutron rests on two assumptions: a) a polarized deuteron constitutes effectively a polarized neutron target, and b) the contribution of the meson production to the spin asymmetry of the deuteron is dominated by the quasi-free process. In this framework, final state interaction effects arising from the presence of the spectator nucleon, are negligible so that the deuteron spin asymmetry can be considered as a simple incoherent sum of the contributions coming from the proton and neutron. But, first of all, the neutron is not completely polarized even inside a fully polarized deuteron target. In addition, final state interactions and other two-body effects will very likely add further complications, thus spoiling this simple idea. Moreover, expecially not far away from the pion photoproduction threshold, there is a non-negligible contribution coming from coherent π production, which certainly cannot be considered as a quasi-free process. All these factors prevent, at least in the Mainz photon energy range, a simple determination of the helicity dependent cross section difference of the neutron from the polarized deuteron data. To extract the neutron information, a reliable theoretical model which takes into account all important two-body effects is then needed. 14

22 2.2 Nucleon resonances Introduction An important aspect of our knowledge about the nucleon structure is the understanding of its excitation spectrum. The excited states of the nucleon are called nucleon resonances. Pion photo- and electro-production and pion-nucleon scattering are invaluable tools in the study of these states. In these reactions the target nucleon can be excited to a resonant state, which then can decay to the nucleon ground state via the strong interaction by the emission of mesons. The parameters of the nucleon resonances, such as mass, decay widths and decay amplitudes can then be accessed. Due to the strong nature of the interaction the lifetimes of the resonances are very small (typically 1 23 s) and their widths are correspondingly large. Typical widths of nucleon resonances are on the order of 1-3 MeV/c 2 so that most resonances overlap with their neighbors. Most of the characteristics of the resonances have been derived from partial-wave or isobarmodel analysis of the different available data sets on (electromagnetic) γn and (hadronic) πn scattering. Several predictions have also been made within the framework of quark models (see e.g. [37, 38, 39]). This considerable effort has led to a fair understanding of several nucleon resonances. Nevertheless, the precision with which the resonance properties are known, is still not satisfactory. This is evident from the large uncertainties given by the Particle Data Book [4], where all known baryon resonances are listed together with their properties. This is in particular true for the resonances that occur at the second resonance region and higher excitation energies (see Fig.2.5), where a substantial overlap among the different resonances occurs. Many predicted states are not sufficiently well established and many properties of the observed states (e.g. coupling constants, branching ratios, helicity amplitudes) are only poorly known. The contributions of the resonances which occur with a reasonable strength in the MAMI energy region ( (P 33 (1232)), P 11 (144), D 13 (152) and S 11 (1535)) are shown in Figure.2.4. Their peak position, width and relative strength are illustrated. The -resonance is the only important one in the so-called first resonance (or -resonance ) region (that is for excitation energies below 5 MeV). The D 13 state is the dominating one in the second resonance region (excitation energies from 5 MeV to 1 GeV), with overlapping smaller contributions from P 11 and S 11 (Fig.2.5). The classical notation is used here for the nucleon resonances: K 2I2J (W), with K = S, P, D, F,... for l =, 1, 2, 3,.... I and J are the isospin and the total angular momentum of the resonance, respectively, W stands for the mass of the resonance in units of MeV/c 2 and l is the relative orbital angular momentum of the decaying nucleon-pion pair. As an example, the first excited state of the nucleon, the well known resonance is notated: P 33 (1232), corresponding to an I = 3/2, J = 3/2 state at 1232 MeV/c 2 and decaying into a nucleon-pion pair with relative orbital angular momentum of l = 1. The nucleon ground state is P 11 (938). Resonances with odd l(p, F,...) have positive, resonances with even l(s, D,...) have negative parity. In partial-wave analysis, and also in the isobar-model analysis, the physical observables are written in terms of helicity amplitudes. The helicity amplitudes are in turn written as partialwave expansions of electromagnetic multipoles. By parameterizing the amplitudes fitted to 15

23 a.u P 33 (1232) 15 D 13 (152) 1 5 S 11 (1535) P 11 (144) E γ (MeV) Figure 2.4: The contribution (taken from the analysis [44]) of the nucleon resonances to the γn πn total cross section for excitation energies below 1 GeV. the experimental data, one can obtain the strength of the various multipoles that contribute to the studied process. Specific nucleon resonances are then related to the specific multipoles Figure 2.5: The contribution (taken from the analysis [44]) of the nucleon resonances to the γn πn total cross section for excitation energies between 5 MeV and 1.6 GeV. Left: most important contributions; right: less important contributions. 16

24 Photon Photon Pion Pion Resonance L Multipole J P l Multipole 1 E1 1/2 - E + S 11 (1535) 1 E1 3/2-2 E 2 D 13 (152) 1 M1 1/2 + 1 M 1 P 11 (144) 1 M1 3/2 + 1 M 1+ P 33 (1232) 2 E2 3/2 + 1 E 1+ P 33 (1232) 2 M2 3/2-2 M 2 D 13 (152) Table 2.1: The correspondence between electromagnetic pion multipoles and resonances in γn πn reactions. and the knowledge of the different multipoles allows to pin down the resonance contributions and characteristics. In single pion photoproduction (γn πn) the multipoles are defined as follows. As the total angular momentum J is conserved, one has: L ± 1 2 = J = l ± 1 2, where L is the multipolarity of the photon. The relative angular momentum of the outgoing particles is denoted by l. The parity of the final state is ( 1) l+1 since the intrinsic parity is positive for the proton and negative for the pion. Parity conservation implies that: for electric transitions, and: ( 1) L = ( 1) l+1 ( 1) L+1 = ( 1) l+1 for magnetic transitions. In table 2.1 the correspondence between electromagnetic pion multipoles and resonances for the first excited states that occur in the MAMI energy range is shown. Here the electric multipoles are denoted by E and the magnetic multipoles by M, the photon multipoles by EL and ML and the pion multipoles by E lx and M lx, where x is + when J = l + 1 and - when 2 J = l 1. 2 The cross sections can be decomposed as a sum of the contributions given by the different multipoles (see next section). From fits of the existing experimental data and using isobarmodel calculations the different multipoles can be then obtained. From these parameters the resonance properties are then deduced. Examples of such partial-wave analysis are [43], [44], HDT [45]. The comparison of these models with experimental data nicely illustrates the present status of our knowledge of the multipoles (e.g. Fig.2.8 and Fig.2.9) Motivation Up to now the determination of the multipoles has been performed using mostly unpolarized data from γ(π)n πn reactions. In this case, the cross section can be decomposed as an incoherent sum of the contributions given by the different multipoles: σ = E o+ 2 + M E M M E (2.56) 17

25 Figure 2.6: Multipole decomposition of the total unpolarized cross sections of γp pπ (top) and γp nπ + (bottom) channels in the -resonance region (HDT model). Figure 2.7: Multipole decomposition of the total polarized cross sections of γ p pπ (top) and γ p nπ + (bottom) channels in the -resonance region (HDT model). From the equation above, it is clear that only the few largest multipoles can be safely accessed in this case. The smaller ones can not be determined in a reliable way. This situation changes when one goes to the double polarization observables: to the γ N πn case, which can be accessed using a circularly polarized photon beam and a longitudinally polarized nucleon target. These observables were unmeasured up to now, but the recent technological developments in the polarized-beam and polarized-target techniques have improved this situation, giving the possibility to access them. For example, the helicity dependent total cross section difference ( σ = σ 3/2 σ 1/2 ), where the 3/2(1/2) corresponds to the (anti)parallel γ-nucleon spin configuration, can be expressed as: σ = E o+ 2 + M E M E 1+ M 1+ 3 M E E 2 M (2.57) Here the sensitivity to smaller contributions is enhanced by the change of sign of some multipoles and by the presence of some interference terms between different multipoles. As an example of this effect let as look at the simplest case by considering the total cross section of the γp pπ and γp nπ + channels in the resonance region. The multipole 18

26 decomposition of the total unpolarized cross section of γp pπ ( γp nπ + ) channel within this energy range is shown at the top (bottom) of Fig.2.6. In the γp pπ case, only the M 1+ multipole plays a significant role, while in the γp nπ + case, M 1+ and, close to the pion photoproduction threshold, E + multipoles are visible. The multipole decomposition of the helicity dependent total cross sections of γ p pπ (on the top) and γ p nπ + (at the bottom) channels in the -resonance region are shown in Fig.2.7. The sensitivity to the E + multipole is enhanced by a change of its sign and the E 1+M 1+ interference term gives access to the E 1+ multipole. Due to a similar reason, also the contribution of higher multipoles is not negligible. Plotted are the multipole decompositions of the HDT model, but this behavior does not significantly change when a different multipole model is taken. A very large data sample is now available below 5 MeV. For this reason and due to the fact that the -resonance has a dominant contribution in this energy region, we have now a good understanding of -resonance properties. As an illustration of this the new helicity dependent data [95] (from the GDH collaboration) for the γ p pπ channel in the -resonance region are compared with the (magenta), (green) and HDT(blue) models in Fig.2.8. A good agreement between the data and the different models can clearly be seen. In the second resonance region and above, as shown in Fig.2.5, on the contrary, there are several resonances overlapping with each other. This fact and the small amount of available data (with very scarce polarized data) make the separation of the different resonance contributions much more difficult. For this reason larger uncertainties are still associated to the properties of these higher resonances. As an example, the helicity dependent differential cross section data [93] (from the GDH collaboration) for the γ p pπ channel in the second resonance region in comparison with (yellow) and (blue) theoretical predictions, that were available when the data were published, are shown in Fig.2.9. In contrast to what was happening in the -resonance region, a significant disagreement is now present in the second resonance region. These new data have provided important fresh input to tune both and calculations. In order to better reproduce these data, the M 2 and E 2 multipoles of, for instance, should be multiplied by 1.11 and.81 times respectively. Additional single pion photoproduction channels need to be measured in order to perform a reliable isospin decomposition of the different multipoles. Due to the large branching decay ratios of the higher resonances into ππn and ηn channels, also precise measurements of these reactions, using different polarization observables, are then also needed to get a reliable multipole determination using a coupled channel approach. These studies will also give a better insight into the GDH sum rule by determining the contributions of the different partial channels to the sum rule integral. The different partial wave analyses give reasonably adequate predictions for the contributions of the single pion channels. Few model calculations have been proposed for the double pion channels. There is the Karliner evaluation [46], the Coersmeier estimate [47] and the calculation by L vov and Petrunkin [48]. For the contribution of the double pion photoproduction to the GDH sum rule on the proton they predict 65 µb, 83 µb and 46 µb, respectively. The large differences between these again call for an experimental constraint and more precise model calculations. These arguments gave the motivation for the second goal of the GDH experiment, i.e. the measurement of the doubly polarized cross sections of all single and double pion photoproduction channels. Such data were not available yet and they put new constraints on the existing 19

27 Figure 2.8: The total helicity dependent cross section of γ p pπ channel in the -resonance region in comparison with (magenta), (green) and HDT (blue) models. Figure 2.9: The helicity dependent differential cross section of γ p pπ channel as a function of the CM angle at second resonance region in comparison with (yellow) and (blue). theoretical models and partial-wave analyses for the nucleon multipoles and its resonant structure. Technically this has been achieved, for the first time, by the GDH experiment at the Mainz MAMI tagged photon facility Pion photoproduction in the GDH experiment at MAMI The goal of the GDH collaboration at Mainz was the measurement of the helicity dependent cross section of all single and double pion channels on the nucleon. The following reactions are the ones available in the Mainz photon energy range (115 MeV < E γ < 8 MeV). γ p pπ nπ + pπ + π nπ + π pπ π pη γ n nπ pπ nπ + π pπ π nπ π nη γ d... pn dπ o The data taking on the proton was completed at Mainz in The target polarization was 75%. The different partial channel analyses of these data have been completed. The corresponding results are published in [91, 93, 94, 95, 96, 97, 98]. The analysis of the γ p nπ + reaction from 2 MeV to 8 MeV are presented in Chapter 5. These data are published in [95, 97, 98]. A test data taking with a deuterated butanol (neutron) target was completed in the same year, with 3% target polarization. The complete data taking on deuterated butanol (neu- 2

28 tron) was completed in Mainz in 23. Due to a new doping material, this time the target polarization was around 7%. The different partial channel analysis from these runs are still under way. 21

29 Chapter 3 The Experimental Setup at MAMI For the experimental verification of the GDH sum rule the helicity cross section difference σ 3/2 σ 1/2 as a function of incoming photon energy needs to be measured. Thus one requires a circularly polarized photon beam, a longitudinally polarized nucleon target and a detector system which covers almost 4π angular acceptance. The GDH collaboration decided to perform experiments at the MAMI microtron accelerator facility in Mainz [15, 16] and at the ELSA electron synchrotron in Bonn [17]. Photons in the energy range from pion photo-production threshold (about 15 MeV) up to 8 MeV, with about 75% polarization, are available at MAMI. At ELSA photons between 5 MeV and 3 GeV with similar polarization can be obtained. In this chapter the complete experimental setup in the Mainz A2-hall is described: the polarized electron beam (Section 3.1), the polarized photon beam (Section 3.2), the target (Section 3.3), the detector setup (Section 3.4) and the data acquisition (Section 3.5). 3.1 The electron beam at MAMI The Mainz Microtron MAMI was built in 1979 at the nuclear physics department of the Johannes Gutenberg University in Mainz. The first version of this accelerator was MAMI-A1 with a maximum electron energy of 14 MeV. Next was MAMI-A2 in 1983 when the accelerator started to operate with 183 MeV. In 1991 MAMI-B was built, producing electrons up to 855 MeV. Another upgrade to MAMI-C (1.5 GeV) is planned for 25. The GDH experiment was carried out at MAMI-B, so we further discuss only that accelerator. MAMI is a continuous wave electron accelerator which consists of a 1 kev electron source, a 3.5 MeV Linear Accelerator (LINAC) and three Race-Track Microtrons (RTM). In Fig.3.1 the accelerator setup and the various experimental halls to which the beam can be delivered are shown. In order to create a polarized electron beam, a special electron source is used. Circularly polarized 83 nm laser light from a Titanium-Sapphire laser induces photo-electric emission of linearly polarized electrons from a strained GaAs.95 P.5 -crystal [49] photo-cathode. The polarization of these electrons is typically around 75%. 1 kev electrons are leaving the electron source and enter the first accelerator stage: the LINAC, which accelerates them up to MeV. These electrons are then injected into a cascade of three RTMs. 22

30 A A A Tagger X RTM e LINAC RTM RTM C A A B Figure 3.1: MAMI accelerator floor plan. Fig.3.2 illustrates the principle of the RTM. Each RTM consists of one LINAC section and two large dipole magnets, each deflecting the beam by 18. After the primary injection the electrons follow the race-track, they are accelerated each time they pass through the LINAC and deflected at each turn by the dipoles with a radius which is dependent on their energy. After a certain number of turns the beam is extracted and transported to the next RTM. An energy selection is possible by extracting the beam after a specific number of recirculations in the third microtron. By this method the beam energy E is given by E= n 7.5 MeV, n=1,..., 45 circulations. Table 3.1 shows the main parameters of the RTMs. MAMI delivers a high stability continuous beam and currents up to 1 µa can be reached. During the unpolarized test photons with 855 MeV were used. During the actual GDH experiment, with polarized photons, two energy settings were used: 525 MeV and 855 MeV. The reason for this is discussed in section In order to reduce systematic errors originating from a fixed beam polarization the direction of the beam polarization was switched every second. The beam polarization directions were monitored and included into the collected data stream in order to verify that both polarization directions are equally distributed. Fig.3.3 shows the yield from the separate beam polarization directions. It is clear that both polarization directions are equally present. 23

31 LINAC Figure 3.2: Schematic diagram of the Race-Track Microtron. RTM1 RTM2 RTM3 Input energy (MeV) Output energy (MeV) No. of recirculations Energy gain/recirculation (MeV) Table 3.1: MAMI RTM parameters. Number of events x P e P e Scaler number (a) N(P e )/N(P e ) Time (a.u.) (b) Figure 3.3: (a) Number of events in the Scalers for both polarization directions. (b) Ratio of the normalized rates of the polarization directions as a function of time. 24

32 Faraday cup Tagger Electronics C v STAR -counter FFW DAPHNE & MIDAS Target Magnet Radiator 2m Figure 3.4: A2 hall at MAMI. 3.2 The photon beam in A2 hall The Bremsstrahlung photon tagging spectrometer The GDH experiment was performed in the A2 hall at MAMI. Fig.3.4 shows an overview of the main components of the experimental setup. Real photons are obtained from the electron beam through Bremsstrahlung radiation in a thin ( 1 3 radiation lengths) radiator which is placed at the entrance of a tagging spectrometer (Fig.3.5). In the Coulomb field of radiator nuclei the electrons radiate Bremsstrahlung photons. Produced photons are collimated towards the target and the detectors. Radiator materials used during the GDH experiment were nickel and Vacoflux (polarized iron). For the experiment it is necessary to know the energy of the photons that induce the reaction. Therefore the photons were tagged by means of the Glasgow-Mainz Bremsstrahlung Photon Tagging Spectrometer [5, 51], tagger for short, which was built by the Glasgow and Mainz groups and has been operational since several years at MAMI. The tagger consists of a large dipole magnet and an array of 353 focal plane scintillation detectors. A large dipole magnet behind the radiator deflects the primary electron beam. The deflection radius of the electron depends on its energy. The electrons that radiated a photon with higher energy have less energy and are deflected more. They are detected by the scintillation detectors. As is shown in the zoomed part of Fig.3.5 neighboring scintillation detectors overlap, which is used to generate coincidences between them and to suppress random background from neutrons, photons or multiply scattered electrons from the primary beam. Coincidences between two neighboring scintillators give rise to a tagger signal. This leads to a total of 352 possible hit tagger channels. Electrons that did not produce a Bremsstrahlung photon are dumped into the Faraday cup, which is calibrated to monitor the electron beam current. The tagger is calibrated to exactly correlate each tagger signal with the energy of the deflected electron E e (with about 25

33 to Faraday cup Scintillators e - Focal plane Primary beam Electron beam Spectrometer magnet Photon beam Radiator Figure 3.5: The Glasgow-Mainz tagger. 2 MeV energy resolution). With the knowledge of the energy E of the primary electron beam the energy of the radiated photon can be calculated as: E γ = E E e (3.1) This means that the energy of each individual photon that induces a nuclear reaction can be determined. Fig.3.6 shows the typical Bremsstrahlung shape of a spectrum as a function of tagger channel and the same, converted to energies. The highest channel corresponds to the highest electron energy, which in turn corresponds to the lowest photon energy. The zeros in the spectra are due to dead tagger channels. The normal tagged photon energy range for an electron beam of 855 MeV lies between 115 and 795 MeV. For the GDH experiment the tagger was operating with a photon intensity of 1 7 s Photon Flux For the data normalization during cross section calculations one needs a precise knowledge of the number of tagged photons which reach the photonuclear target. This number is different from the amount of electrons that produced a photon and that are detected in the focal plane detectors. In order to obtain a well defined beam spot, the photon beam is collimated behind the radiator. Because of the angular spread of the Bremsstrahlung radiation part of the photons are cut away. Moreover, some of the detected electrons are not secondary Bremsstrahlung electrons, but are coming from Møller scattering in the radiator. Also, there is some background in the tagger from the primary electron beam. These two last effects are smaller than the first one and they are treated by inserting a correction for the Møller electrons and by the time correlation (random subtraction), respectively. 26

34 Number of events x Number of events x Tagger channel (a) Photon energy (b) Figure 3.6: Tagger Bremsstrahlung spectrum as a function of tagger channel (a), and as a function of photon energy (b). The highest tagger channels (E γ < 2 MeV) are switched off. The Bremsstrahlung process can be characterized by a polar opening angle θ c, which encloses half of the radiated photons. This angle is energy dependent: θ c = m e E, (3.2) where m e is the electron mass. Photons with large θ will be stopped in the collimator and will not reach the target. The lower the energy of the primary beam, the wider the spread of the Bremsstrahlung photons is, and thus the fraction of photons that survive the collimation is lower. The ratio between the number of photons that reached the target N γ (E γ ) and the number of electrons detected in the tagger N e (E γ ) is called the tagging efficiency ǫ tagg (E γ ): ǫ tagg (E γ ) = N γ(e γ ) N e (E γ ) (3.3) N e (E γ ) (see Fig.3.7) is readily obtained from the tagger scalers which contain the total yield per tagger channel. Due to the collimators, lower primary beam energy corresponds to lower tagging efficiency. The tagging efficiency also depends on the radiator material and on the focusing of the primary electron beam on it. This last factor is not constant over a long period of time. Therefore, it is necessary to continuously monitor the efficiency during data-taking. In this experiment this was done by using lead-glass and pair detectors. The lead-glass detector has a cubic shape ( cm 3 ) and 1% photon detection efficiency. But there is a limit on the intensity that the detector can stand: 1 5 γ/ s, while the experiment should run at 1 7 γ/ s intensities. By using this detector the photon flux can be determined in a straightforward way, but it can not be used during the experiment. Hence, during the normal data-taking another method has to be used. For this purpose the pair detector was used, which uses the detection of e + e pairs as a monitor for the photon flux. It was installed in front of the lead-glass detector (Fig.3.8). It is designed to have a low photon 27

35 ε tagg E =855 MeV E =525 MeV E γ (MeV) Figure 3.7: Tagging efficiency for 525 MeV and 855 MeV primary electron beam. efficiency such that it can stand higher intensities. The lead-glass detector is used to calibrate its efficiency during dedicated low-intensity runs. Once the efficiency of the pair detector is known, one can monitor the tagging efficiency during the normal high-intensity runs. The pair detector consists of three thin plastic scintillator detectors (P1-3) and a thin copper converter foil (to have a low count rate in the scintillators). A photon that hits the copper foil can produce an electron-positron pair or an electron via Compton scattering. In the measured energy range pair production has a much higher cross section than Compton scattering. The P1 P3 e - Cu P2 e + Lead-glass Figure 3.8: The pair and the lead-glass detectors. 28

36 .6 εpair effective convertor thickness :.614 +/.2 cm E γ (MeV) Figure 3.9: Detection efficiency of the pair detector with the fit and the determined effective convertor thickness. produced electrons can be detected by the coincidences between P2 and P3. To reduce the background, the detector P1 in front of the copper foil is used as an anti-coincidence with P2 and P3. At regular intervals the lead-glass photon detector, which is positioned at the end of the experiment s beam line, is moved into the direct (low-intensity) photon beam. The pair detector count rate for a given photon flux will depend on its detection efficiency and on the thickness of the convertor foil. The pair detector efficiency ǫ pair (E γ ) can be determined by making the ratio of the yield N pair (E γ ) in the pair detector and the yield N Pb glass (E γ ) in the lead-glass detector at low intensity photon beam, which acts as a normalization detector. The pair detection efficiency can be parameterized in the following way: ǫ pair (E γ ) = N pair(e γ ) N Pb glass (E γ ) = d e+ e eff σ (E γ ) (3.4) The energy behavior of ǫ pair is governed by the cross section for e + e photoproduction. In this case the known e + e yield from 1 cm copper is used to obtain the shape of the energy distribution. The absolute value of the efficiency is controlled by the effective thickness d eff. From the fit of the parameterization to the measured ratio the effective thickness can be extracted. Fig.3.9 shows the detection efficiency of the pair detector together with the fit from which the effective convertor thickness can be calculated. During normal data taking runs without leadglass detector, the pair count rate is divided by the parameterization as stated above, with the d eff value as obtained in a normalization run with the lead-glass detector: N γ (E γ ) = N pair (E γ ) d eff σ e+ e (1cm Cu) (E γ), (3.5) which enables the determination of the photon flux with the pair detector. 29

37 P γ /P e 1.8 E =525 MeV.6 E =855 MeV E γ (MeV) Figure 3.1: The ratio P γ /P e between the photon and the electron polarizations as a function of photon energy for two values of primary electron energy E Photon Polarization During the Bremsstrahlung process the longitudinal polarization of an electron is converted into circular polarization of the emitted photon. According to [52], the fraction of the polarization that is transferred to the photon depends on the primary electron energy E and on the energy of the emitted photon E γ. The ratio between the photon and the electron polarizations (P γ /P e ) is given by: P γ 4E γ E Eγ 2 = (3.6) P e 4E 2 4E γ E + 3Eγ 2 At E γ = E, when the entire energy of the electron is transferred to the photon, one has P γ /P e = 1, which means that the electron polarization is completely transferred to the photon. Fig.3.1 illustrates the energy behavior of the polarization transfer for two different E settings. At lower transferred energies the photon takes only part of the electron polarization. In order to obtain a sufficiently high degree of polarization in the low photon energy region, two different primary electron energy settings were used: 855 and 525 MeV. Once the electron polarization is known, the photon polarization can be immediately calculated: 4E γ E Eγ 2 P γ = P e 4E 2 4E (3.7) γe + 3Eγ 2 The polarization of the primary electron beam is measured using Møller scattering of the 3

38 Coincidence Møller trigger 48 counters Primary beam E e E e Electron beam Radiator (Møller target) Figure 3.11: The Møller polarimeter. longitudinally polarized electrons on the atomic electrons of a magnetized ferromagnetic foil [53]. For this purpose the tagger is simultaneously used as a tagger for the Bremsstrahlung photons and as a Møller polarimeter with a separate electronic chain and readout system [54]. The Vacoflux foil was used as a Bremsstrahlung and as a polarized Møller radiator. Vacoflux foil is a FeCo-alloy (49% Iron, 49% Co, 2% Va), which can be magnetized to P t = (8.1 ±.2)%. It is very thin to allow high polarization, which means low yield of Bremsstrahlung photons. To overcome this, the foil is rotated by α = 25 with respect to the beam axis. Two sets of tagger channels have been selected to detect electron pairs in coincidence as a two-arm spectrometer. A good Møller event is the coincidence between two arms with the sum of the electron energies equal to the primary beam energy. The experimental setup is shown in Fig The polarized Møller cross section can be written as a function of the unpolarized cross section: ( ) pol ( ) ( unpol dσ dσ = 1 + ) a jk p j t p k e, (3.8) de e de e j=x,y,z k=x,y,z where a jk is called the analyzing power [53] and is calculated from QED [55], p j t and p k e are the magnetized target and the electron beam polarization Cartesian components respectively. z, x and y are the beam, horizontal and vertical axes, respectively. The Møller asymmetry A Møller is the following combination of the cross sections for parallel 31

39 P e (%) Time (a.u.) Figure 3.12: Electron beam polarization measured with Møller polarimeter as a function of time over a total period of 15 hours. Each data point corresponds to a measurement of 4 hour. ( ) ( dσ magnetized foil and beam polarization de e and anti-parallel polarizations A Møller = ( ( ) ( ) dσ de e dσ de e ) ( dσ de e + dσ Most of the a jk are small and the asymmetry can be approximated as: ) dσ : de e de e ) (3.9) A Møller a zz P t P e cosα, (3.1) where P t and P e are the magnetized foils and electron beams polarization degrees, respectively, and α is the angle between these polarization directions. The analyzing power a zz = -7/9. From an experimental point of view this asymmetry can be measured by detecting events for two relative orientations of the magnetized target foil and electron beam: A Møller = N N N + N, (3.11) where N is the number of the events with parallel and N with anti-parallel polarization directions. Since α is well known from the orientation of the radiator foil relative to the beam line, P t is also a well determined parameter and A Møller is measured experimentally, P e can be determined: A Møller P e = (3.12) a zz P t cosα The measured asymmetry A Møller is very small (.2) and statistically significant results (P e = 73 ± 2%) could be obtained after 4 hours of measurement. Fig.3.12 shows the beam 32

40 Polarising magnet Target cell Refrigerator Target cell DAPHNE Figure 3.13: Target in the polarization (top) and in the frozen-spin (bottom) mode. polarization measured with the Møller polarimeter as a function of time. Each data point corresponds to a measurement of 4 hour. 3.3 The Target The polarized data, of which the analyses are described in this thesis, were taken on Butanol and deuterated Butanol targets. For detector calibration and reliable analysis procedures evaluation reasons also calibration data using an unpolarized photon beam and unpolarized target materials (hydrogen and deuterium) were taken. First, in the next section (3.3.1), a short overview of the unpolarized targets is given. In section a more detailed explanation of the polarized target working principles and properties is given The unpolarized target The unpolarized liquid-target was constructed in the beginning of the nineties and was used for calibration runs. It was also used in the previous DAPHNE experiments at MAMI which studied the photoabsorption and photoproduction processes on hydrogen, deuterium, 3 He, 4 He targets (see e.g. [56]) between 1992 and The mylar target cell is a 25 cm long cylinder with 2.15 cm radius. A Gifford-McMahon refrigerator [57] reduces the temperature of the cooling gas ( 4 He) to 2.5 K, which is enough to liquefy hydrogen, deuterium, 3 He and 4 He. The pressure and the temperature in the refrigerator were monitored during the experiment. The density of the hydrogen target was.78 g/cm 3. It was.162 g/cm 3 for the deuterium target. The density was stable to within.5%. 33

41 P T (%) Data run number Figure 3.14: Target polarization over 15 hours The polarized target The horizontal polarized frozen-spin target was built in the beginning of the nineties by groups from Bonn, Nagoya and Bochum [58, 59]. It is operating at low temperatures to keep the target polarization for a long period. This target is the first target which uses the internal holding coil principle in a horizontal refrigerator. It has an important advantage: all the target setup components are sitting at backward angles, which allows a nearly 4π acceptance for the detection of the produced particles. A solid pure hydrogen target can not be polarized. Therefore another type of target had to be used. The target material had to fulfill some conditions: short polarization build-up times, high hydrogen density, resistance against radiation damage and unpolarized non-proton ingredients. For the GDH sum rule measurements on the proton a butanol (C 4 H 9 OH) target, doped with porphyrexide as a paramagnetic substance, was chosen. The density of the butanol material is about.94 g/cm 3. The dilution factor, which is the fraction of the polarizable nucleons in the material, is.135. It has high ( 8%) maximal polarization for the hydrogen protons. The presence of spin-less C and O nuclei, which will not be polarized, will not cause any target asymmetry. It also has a short build-up polarization time, long relaxation time and good radiation resistance. Butanol is packed into spherical beads which are placed in a cylindrical container. The container is 2 cm long and has a diameter of 2 cm. The effective filling factor of the target beads in this container is 63%. For the NGDH experiment, where the GDH sum rule on neutrons is measured, the deuterated butanol (C 4 D 9 OD) target was used. This target is similar to the butanol target. The 34

42 target container is the same as for the butanol target. The density of this material is about 1.19 g/cm 3. The dilution factor is.238. This target type also has high maximal polarization value with presence of unpolarized C and O nuclei. The effective filling factor of the target beads in this container is again 63%. Due to the low magnetic moment of nucleons a high (>1 T) magnetic field and low temperatures (<2 mk) are required for their polarization. The principle of Dynamic Nuclear Polarization (DNP) allows to weaken this conditions. Via DNP high polarization levels can be reached: the protons in the butanol target can be polarized up to P T = 8%. In this technique the target material is doped with paramagnetic impurities. The free electrons in these impurities can easily be polarized up to nearly 1% using only 2.5 T magnetic field at a 3 mk temperature. By microwave irradiation ( 7 GHz) the electron polarization is transferred to the nucleon spins. Once the nucleons are polarized, the polarization has a long relaxation time at low temperatures (<7 mk). The target setup consists of a horizontal dilution refrigerator, a target container, a holding coil, a super-conducting magnet and a microwave system. The whole detector system is put on a rail system. In this way the detector system is able to move away from the target and the polarizing super-conducting magnet is placed over it (Fig.3.13, upper case). Using 2.5 T magnetic field and 7 GHz microwaves the target is polarized at a 3 mk temperature. Then the internal super-conducting holding coil, which is placed around the target volume, is switched on. The target temperature is brought down to 5 mk with a 3 He/ 4 He dilution refrigerator. Under these conditions a holding magnetic field of only a.42 T is required to retain the polarization for a long time. In this frozenspin mode (Fig.3.13, lower case) a relaxation time of more than 2 h can be obtained for a butanol target. The holding coil itself is made very thin (less than 1 mm), so that it does not influence the outgoing particles. The holding coil is built as a solenoid and its magnetic field has sufficient homogeneity for a Nuclear Magnetic Resonance (NMR) measurement, which was used for continuously monitoring of the nucleon polarization during the experiment. The error on the measured polarization is about 2%. The large magnet is then removed and the detector system is placed back to its place. The relaxation time was not always used to the limit: in order to keep a sufficient target polarization during the data taking, the target was repolarized approximately every 48 hours. In order to suppress systematical effects, from time to time the target polarization direction was switched. In Fig.3.14 the target polarization as a function of time is shown. 3.4 The GDH detector setup The primary aim of the experiment in Mainz was the measurement of the helicity dependence of the total polarized photoabsorption cross sections on proton and neutron, from which the GDH sum rule for proton and neutron can be calculated. The secondary aim of this experiment was to study the resonance properties, for which reason measuring of the helicity cross sections of all partial pion photoproduction channels (see page 2) appearing in the Mainz photon energy range (115 MeV < E γ < 8 MeV) was necessary. All non hadronic, electromagnetic background reactions then needs to be suppressed. For this purposes a detector with large (almost 4π) angular and momentum acceptance is 35

43 FFW Daphne v Midas Cerenkov Star Figure 3.15: The GDH detector setup needed. Also this detector should have good particle identification efficiency for all particles listed in the reactions on page 2. The larger the acceptance of the detector, the smaller the contribution from extrapolations and so, the smaller the systematical errors. The detector setup (fig.3.15) used in Mainz satisfies all our requirements for this kind of experiment. The DAPHNE detector had already proven its value [6] and was used as the central detector. It covers a large part of the angular acceptance. For polar angles it covers 21 < θ < 159 range. To increase the angular acceptance in forward direction the MIDAS detector was inserted in DAPHNE, which covers 7.5 < θ < 16.5 angular acceptance. Downstream of this, the Čerenkov detector was placed, which was used to suppress the large amount of electromagnetic background. Further, forward components where installed. These components where designed to cover the very forward 2 < θ < 5 angles DAPHNE The DAPHNE (Détecteur à grande Acceptance pour la PHysique photonucléaire Experimentale) detector was designed and built in the beginning of the 9 s by the Saclay (France) and Pavia (Italy) groups at the Commissariat à l Energie Atomique in Saclay [6]. It was designed for photoreaction experiments on light nuclei with multi-particle (up to 5 particles simultaneously) DE EE FE DS ES FS CE BE A CS BS MWPCs Figure 3.16: DAPHNE 36