Azimuthal Asymmetries in Unpolarized Semi-Inclusive DIS

Transcription

1 Azimutal Asymmetries in Unpolarized Semi-Inclusive DIS Wolfgang Käfer Pysikalisces Institut Albert-Ludwigs-Universität Freiburg

2

3 Azimutal Asymmetries in Unpolarized Semi-Inclusive DIS Diplomarbeit vorgelegt von Wolfgang Käfer Pysikalisces Institut Albert-Ludwigs-Universität Freiburg März 28

4

5 Contents 1 Introduction 1 2 Teoretical Motivation Deep Inelastic Scattering Parton Distribution Functions Polarized Distribution Functions Transverse Momentum Dependent Distribution Functions Semi-Inclusive DIS Fragmentation Azimutal Modulations in Unpolarized SIDIS Can Effect Te SIDIS Cross Section for Unpolarized Lepton - Nucleon Scattering Perturbative QCD Effects Polarized SIDIS Te COMPASS Experiment Te Beam Line Te Spectrometer Te Polarized Target Te COMPASS Trigger System Tracking Detectors Particle Identification Data Reconstruction

6 II Contents 4 Data Selection Event Selection Data Quality Kinematic Range Te Primary Vertex Beam and Scattered Muon Hadron Selection Leading Hadron Selection Particle Identification Acceptance Correction and Determination of te φ Dependence Data and MC Resolution and Binning Te Acceptance Correction Te Acceptance of te COMPASS Spectrometer Determination of te Moments of te Cross Section Results Dependence on Kinematic Variables Te cos φ Moment - te Transverse Momentum of te Quarks Te cos 2φ Moment - Comparison wit Teory Systematic Cecks Summary and Outlook 57 A Appendix 59 A.1 Te sin φ moment A.2 Tables References 74 B Deutscspracige Zusammenfassung 77

7 1. Introduction Scattering experiments ave been te most important tool to probe te structure of matter for almost a century. Since te discovery of te atomic nucleus in te famous Ruterford-Experiment via te scattering of α particles on gold foil, scattering experiments ave provided new and often surprising insigts. Altoug te existence of a substructure was known before (because of te anomalous magnetic moment of proton and neutron), te first direct evidence for a substructure of te nucleon as been found at SLAC 1 in te 196s. Altoug istorically also known as partons, te constituents of nucleons turned out to be quarks, spin 1 /2 particles carrying fractions of te elementary electric carge e. Wit te formulation of QCD [1, 2] in 1973 and te subsequent discovery of gluons at DESY 2 in 1979, our current picture of te nucleon emerged. Te nucleon is tougt to consist of tree valence quarks, eld togeter by gluons. Te gluons can furter fluctuate into so-called sea quarks, quark-antiquark pairs wic exist only on a very sort timescale. Wile it is well known tat te valence quarks carry only about alf te momentum of te nucleon, te situation concerning te nucleon spin is less clear. Te total spin of te nucleon of 1 /2 was originally tougt to be dominated by contributions of te spin of te valence quarks. However, tis assumption was ruled out by measurements of te EMC 3 experiment at CERN 4 [3]. Tis was later confirmed by oter experiments and today it is establised tat te contribution of te quark spin to te spin of te nucleon is only about 25%. Until today, it is not clear were te spin of te nucleon comes from. Anoter rater unknown quantity are te transverse momenta of te quarks inside te nucleon. Since calculations are usually made in a boosted frame, were te nucleon carries ig momentum, te momentum component of te quarks transversely to te boost direction is small and often neglected. On te oter and, te nucleon as a finite size, so te transverse momentum as to be non-zero already according to te uncertainty principle. Te investigation of azimutal asymmetries in unpolarized semi-inclusive Deep Inelastic Scattering (DIS) was originally proposed by H. Georgi and H. D. Politzer as a clean test of perturbative QCD in [4]. However, it was pointed out by R. Can immediately afterwards [5] tat azimutal modulations can also be caused by te transverse momentum of te quarks, wic is known as Can Effect. Azimutal asymmetries ave gained 1 Stanford Linear Accelerator Center 2 Deutsces Elektronen SYncrotron 3 European Muon Collaboration 4 European Organization of Nuclear Researc

8 2 1. Introduction renewed interest wit te attempt to parameterize te nucleon wit transverse momentum dependent parton distribution functions starting in te 199s [6]. Up to now, results on tese asymmetries ave been publised by te EMC Collaboration in [7, 8], te E665 experiment at Fermilab in [9] and by te ZEUS Collaboration [1] at DESY. However, tese results ave been obtained in different kinematic regions and tus are difficult to compare. Te concepts of DIS as well as te different effects contributing to azimutal modulations observed in unpolarized semi-inclusive DIS are discussed in Capter 2. After a previous attempt [11], tis tesis dedicated to te determination of tese modulations from data taken at te COMPASS 5 experiment at CERN. Te data used in tis tesis, were taken in 24 wit longitudinal (i.e. parallel to te beam axis) target polarization. An effort to extract te same modulations from data taken wit transverse target polarization is also ongoing [12]. Since te experimental conditions differ for te two polarizations, tis gives an opportunity to study systematic effects. Te COMPASS experiment in te longitudinal setup is briefly described in Capter 3. It is operating in a kinematic region similar to EMC, but as a muc larger data sample and tus can measure te dependence on te kinematic variables more precisely. Te metod used in tis tesis to extract tese modulations from COMPASS data is described in te following capters. Capter 4 gives a detailed description of te performed event selection. Since azimutal modulations are also generated by te nonuniform acceptance of te COMPASS spectrometer, a Monte Carlo simulation is needed in order to correct for tese effects. Capter 5 caracterizes te MC simulation performed for tis task and describes ow te measured angular distributions are corrected for acceptance effects and te moments of te modulations are determined. Te obtained results will be discussed in Capter 6 and an estimate of te transverse quark momentum based on te measured cos φ moment of te cross section is given. Additionally, te measured cos 2φ moment is compared wit a recent model calculation, wic estimates different contributions to tis asymmetry. 5 COmmon Muon Proton Apparatus for Structure and Spectroscopy

9 2. Teoretical Motivation Tis capter is dedicated to a brief introduction to te teoretical concepts needed in tis tesis. It is organized as follows: After a very sort introduction to Deep Inelastic Scattering in Sec. 2.1, parton distribution functions are described in Sec In Sec. 2.3, te DIS formalism is adapted to te Semi-Inclusive Deep Inelastic Scattering (SIDIS) process, were an additional adron is observed in te final state. Finally in Sec. 2.5, several sources for modulations of te SIDIS cross section wit te adron azimutal angle φ are discussed. Tis capter is based mainly on [6, 13, 14] 2.1 Deep Inelastic Scattering Deep inelastic scattering is used to gain information about te inner structure of composite particles like te nucleon. In a typical DIS reaction as depicted in Fig. 2.1 a lepton l scatters off a nucleon N to produce a adronic final state X togeter wit te scattered lepton l : l + P l + X. (2.1) Te term Deep Inelastic Scattering is used, wen te 4-momentum transfer Q of te lepton to te adron is large enoug tat te substructure of te nucleon can be re- l l' γ N X Figure 2.1: Scematic picture of te DIS process: A lepton l interacts wit a Nucleon N via te excange of a virtual poton γ. Wile te leptonic part of te interaction can be treated wit perturbative QED, te adronic part represented by te blob as to be parameterized wit a priori unknown functions.

10 4 2. Teoretical Motivation solved. In tis case te timescale of te interaction 1 /Q ( = c = 1 in tis tesis) is sort. Tus te quarks in te nucleon do not interact wit eac oter during te scattering reaction and te interaction can be seen as elastic scattering of leptons on free quarks. Tis can approximately be described in leading order QED wit a one poton excange between te lepton and one of te quarks of te nucleon. W/Z excange is also possible but does not contribute at te COMPASS beam energy of about 16 GeV. Te quark is emitted from te nucleon and bot te quark and te nucleon remnant fragment into adrons. Te process is caracterized by several (Lorentz-invariant) kinematical variables as defined in Tbl Teoretical predictions are often made in te center of mass system of te virtual poton and te nucleon, te so-called Gamma-Nucleon System (GNS). Tis system as te advantage, tat in te DIS regime bot proton and poton carry ig momentum, and terefore quark masses can be neglected. In te GNS, te Bjorken scaling variable x can be interpreted as te momentum fraction of te nucleon carried by te quark. Table 2.1: Definition of Kinematic Variables relevant for te DIS Process mass of te target nucleon M N 4-momentum of te incoming muon l l µ = (E, l) 4-momentum of te target nucleon P P µ = (E N, p N ) 4-momentum of te outgoing muon l l µ = (E, l ) 4-momentum of te virtual poton q q = l l 4-momentum of outgoing adron P P µ = (E, P ) neg. squared invariant mass of te virtual poton Q 2 Q 2 = q µ q µ energy transfer to te target ν ν = P µ q µ /M N Bjorken scaling variable x x = Q2 2P µ q µ fractional energy transfer of te virtual poton y y = P µ q µ l µ P µ fraction z of te energy of te virtual poton carried by adron transverse momentum of te adron w.r.t. te virtual poton P t z = PµP µ P µ q µ (not Lorentz-invariant) Te DIS cross section dσ is proportional to te product of te leptonic tensor L µν and te adronic tensor W µν dσ L µν W µν. (2.2) Wile L µν can be calculated explicitly in QED L µν = 2(l µ l ν + l µl ν l l g µν ), W µν is unknown. Symmetry arguments can be applied to reduce te number of independent components of W µν from 16 down to two, te structure functions F 1 (x, Q 2 ) and F 2 (x, Q 2 ), see e.g. [13]. An additional structure function F 3 (x, Q 2 ) is present in neutrino - nucleon scattering due to te parity violation on te weak interaction. It is important to note tat tis parameterization is model independent and follows only from tese

11 2.2. Parton Distribution Functions 5 symmetry arguments. In te parton model, wic will be discussed next, te structure functions are related to Parton Distribution Functions (PDFs). 2.2 Parton Distribution Functions In te quark-parton model quarks are treated as massless non-interacting particles, te partons, and QCD effects are ignored. In tis case te dependence of te structure functions on Q 2 vanises. Tis is known as scaling. Taking into account QCD effects, scaling does no longer old exactly, leading to te so-called scaling violation. Structure functions can be expressed in terms of parton distribution functions f q (x), wic can be interpreted as te probability f q (x)dx of finding a quark of flavor q wit momentum fraction between x and x + dx F 1 (x) = q e 2 qf q (x), (2.3) were e q is te carge of te quark wit flavor q. Te sum runs over te contributing quarks flavors, usually q = {u, ū, d, d, s, s}. Additional quark distribution functions are introduced, wen spin and transverse momentum effects are taken into account. Tese will be described in te following section. In te quark-parton model, F 1 (x) and F 2 (x) are related via te Callan-Gross relation F 2 (x) = 2xF 1 (x), (2.4) wic also is only an approximation, as soon as QCD contributions are included Polarized Distribution Functions Including spin effects but still implicitly integrating out transverse quark momentum, two additional PDFs are needed, te elicity function g(x) and te transversity distribution function (x) (flavor indices will be suppressed in te following). g(x) is te probability difference between finding a quark wit its spin parallel to te nucleon and te probability to find a quark wit its spin antiparallel to te nucleon spin inside a longitudinally polarized nucleon. (x) is defined similarly for a transversely polarized nucleon. Wile g(x) is already measurable in inclusive DIS, (x) as to be measured in semi-inclusive reactions due to its symmetry properties [6]: (x) is a ciral odd object, wic means tat it canges sign under a parity transformation. Since QED is parity conserving, a ciral odd function as to be combined wit anoter ciral-odd object in order to make te cross section parity even. Tis is e.g. possible in semi-inclusive DIS, were te oter object is a ciral odd fragmentation function like te Collins fragmentation function introduced below or in polarized proton-proton collisions via Drell-Yan processes, were te transversity distribution function enters te process twice Transverse Momentum Dependent Distribution Functions Additional PDFs appear, wen te transverse momentum k t of te quarks is taken into account. However, tey are not measurable in inclusive reactions, only in semi-inclusive

12 6 2. Teoretical Motivation reactions, again due to teir symmetry properties. Also te PDFs introduced above gain an explicit k t dependence. Of particular interest for unpolarized SIDIS is te Boer-Mulders function 1 (x, k t), wic can be interpreted as te probability difference between te two spin states of a transversely polarized quark inside an unpolarized nucleon. More details about spin and transverse dependent PDFs can be found in [6]. 2.3 Semi-Inclusive DIS In semi-inclusive DIS (SIDIS), te detection of at least one adron is required in addition to te scattered lepton l + N l + + X. (2.5) Tis provides additional information about te structure of te nucleon. Also te process is more complex, since free quarks cannot be observed, but fragment into adrons. Moreover te fragmentation process takes place at a rater low energy scale, were perturbative QCD is not applicable. Tis requires additionally models for tis fragmentation process. Usually it is assumed tat tat te scattering process and te fragmentation are independent of eac oter. Tis is referred to as factorization and as to be sown for eac process. For te case of transverse momentum dependent SIDIS, factorization as only recently been proven in [15]. Tus te SIDIS cross section can be split into tree parts (see also Fig. 2.2), namely: te probability of finding a quark wit te longitudinal momentum fraction x and possibly spin and transverse momentum, parameterized wit a PDF as described in Sec te cross section for poton-quark scattering dσ lq lq, calculated perturbatively wit QED. te probability for te struck quark to fragment into te observed adron, described by fragmentation functions D q (z ) wic will be described in te following Sec For example, considering te simplest case of unpolarized SIDIS neglecting k t dependence, te cross section is of te form dσ lp lx q f q (x)dσ lq lq D q (z ). (2.6) From tis expression, SIDIS structure functions can be defined by summing over all quark and antiquarks and integrating out internal degrees of freedom like quark transverse momentum respecting conservation laws, i.e. F (x, Q 2, z, Pt ) = x d 2 p t d 2 kt δ (2) ( p t k t P t /z )f q (x, kt 2 )Dq (z, p 2 t ), (2.7) q

13 2.4. Fragmentation 7 l l' γ N f(x) k k' D(z) X Y Figure 2.2: Scematic picture of te SIDIS process. Te two blobs represent te two elements of te interaction wic cannot be treated perturbatively: te internal structure of te nucleon described by a parton distribution function and te fragmentation process, parameterized wit a fragmentation function. were p t is an additional transverse momentum of te adron due to te fragmentation process. Te full transverse momentum of te adron is ten given by: P t = z kt + p t. (2.8) Furter integration over te semi-inclusive variables z and Pt DIS structure function F 1 (x, Q 2 ). results in te well known 2.4 Fragmentation Te fragmentation process can be described similarly to te scattering process itself. Wile for te scattering process te soft parts are parameterized wit PDFs, ere te non perturbative parts are parameterized via fragmentation functions D q (z ), wic describe te process of adronization. Tey can be interpreted as te probability for a quark of flavor q to produce a adron wit energy fraction z. Tere is obviously a vast amount of possible combinations of quark flavors and adron types. However, te number of independent fragmentation functions per adron type can be significantly reduced wit te assumption of isospin symmetry and carge conjugation invariance. Furtermore, one can distinguis te favored fragmentation function D fav (z ), were te fragmenting quark enters te adron as a valence quark, from unfavored ones D unf (z ).

14 8 2. Teoretical Motivation For te most common case, were te quark is an up or down quark fragmenting into a pion, one as: D fav (z ) = D π+ u D unf (z ) = D π+ d (z ) = Dd π (z ) = Dū π (z ) = D π+ d (z ), (2.9) (z ) = Du π (z ) = D π d (z ) = Dū π+ (z ). (2.1) If spin and transverse momentum effects are considered, additional fragmentation functions ave to be introduced, similar to te additional PDFs. Wort mentioning ere is te Collins fragmentation function H 1 (z ), wic gives te probability difference between te two polarization states for a transversely polarized quark to fragment into an unpolarized adron. 2.5 Azimutal Modulations in Unpolarized SIDIS In leading order QED te cross section is independent of te adron azimutal angle φ, as long as transverse momentum effects are ignored. Te azimutal angle φ is defined as te angle between te lepton scattering plane spanned by incoming and scattered muon and te adron production plane, wic is spanned by virtual poton and adron momentum (see Fig. 2.3). Tus φ can be calculated using te normal vectors of te two planes in terms of te momenta of te incoming muon l, te virtual poton q = l l and te outgoing adron P via Te sign is defined as cos(φ ) = l q l q P q P q. (2.11) sign(φ ) = sign[( q l) P ]. (2.12) Since φ is invariant bot under a boost along te virtual poton axis and under spacial rotations, te above equations old already in te lab frame. Bot k t effects and QCD corrections introduce modulations of te cross section wit respect to φ. As will be sown in te following te overall cross section measured at COMPASS is of te form dσ dφ = a + a 1 cos φ + a 2 cos(2φ ) + a 3 sin φ. (2.13) were te two cosine terms receive contributions due to te Can Effect, perturbative QCD corrections and transverse momentum PDFs, wile te sine term appears because of te polarized muon beam of te COMPASS experiment. In te following, sometimes te moments dσ cos(nφ ) cos(nφ ) = (2.14) dσ of te cross section will be used. Tese can be obtained from te coefficients of Eq via cos(nφ ) = a n 2a for n >. (2.15) and analogously for sin φ.

15 2.5. Azimutal Modulations in Unpolarized SIDIS 9 P x z y q l ' l Figure 2.3: Definition of te adron azimutal angle φ. Te coordinate system is cosen in suc a way tat te Z axis is along te direction of te virtual poton, te X axis lies in te lepton scattering plane along te remaining component of te scattered muon momentum and te Y axis is cosen suc tat te coordinate system is rigt anded Can Effect As Can pointed out in [5], an azimutal modulation of te cross section is already expected in leading order QED, wen te transverse momentum of te quark is taken into account. Te QED cross section is proportional to te squares of te Lorentzinvariant Mandelstam variables s = (l + k) 2 and u = (k l ) 2, were k denotes te 4-momentum of te scattered quark dσ s 2 + u 2. (2.16) Allowing for small transverse momentum k t xp, te quark 4-momentum k can be written as k = (xp, k t cos φ q, k t sin φ q, xp ). (2.17) φ q is te azimutal angle of te struck quark, wic is not necessarily te same as te azimutal angle of te adron φ due to te additional transverse momentum p t introduced in te fragmentation process. Tis leads to a smearing effect and tus diminises te observed modulation [16]. Inserting Eq togeter wit te lepton momentum l = (E, l x,, l z ) (2.18)

16 1 2. Teoretical Motivation in Eq leads to a dependence of te cross section on φ q. Specifically, te result can be written as ( dσ x 2 (1 + (1 y) 2 ) 1 2 k ( ) 2 t dφ q Q D kt cos φ (y) cos(φ q ) + D cos 2φ q) Q (y) cos 2φ, (2.19) were te functions and D cos φ (y) = (2 y) 1 y 1 + (1 y) 2 (2.2) D cos 2φ (y) = (1 y) 1 + (1 y) 2 (2.21) ave been introduced. ( ) 2 Te Can Effect is kinematically suppressed by kt Q for cos(φ q) and kt Q for te cos 2φq term. Tese kinematic factors allow in principle to extract te mean transverse momentum of te quarks k t. Wen going from te quark to adron level, te unpolarized PDFs and fragmentation functions need to be taken into account. Assuming Gaussian distributions for te transverse momentum dependence of f q (x, k t ) and Dq (z, p t ), ( ) 1 f q (x, k t ) = f q (x) π kt 2 exp k2 t kt 2 ( ) Dq (z, p t ) = Dq 1 (z ) π p 2 t exp p2 t p 2 t (2.22), (2.23) a simplified cross section, neglecting te cos 2φ term, can be written as [17] d 5 σ dx dy dz P t dp t dφ ( ( ) P 2 exp t q ( ) Pt 2 )f q (x)dq (z )(1 + (1 y 2 )) [ ] 1 4D cos φ (y) k2 t z Pt Q ( ) Pt 2 cos φ. (2.24) Te cosine moment defined in Eq can ten be calculated using Eq to be cos φ = 2 D cos φ (y)p t z Q kt 2 ( ) Pt 2. (2.25) Tus in tis simple model one expects a linear dependence of te cosine moment on z and P t Te SIDIS Cross Section for Unpolarized Lepton - Nucleon Scattering If transverse momentum is no longer neglected, some of te symmetry arguments used to reduce te number of independent components of te adronic tensor W µν are no

17 2.5. Azimutal Modulations in Unpolarized SIDIS 11 longer valid. Tus additional structure functions arise: Te cross section for unpolarized SIDIS in terms of structure functions is given by [14] dσ dxdydzdφ dp t = α2 xyq 2 (1 + (1 y)2 )(1 + γ2 2x ) ( F UU + D cos φ (y) cos φ F cos φ UU + D cos 2φ (y) cos 2φ F cos 2φ UU ), (2.26) x wit γ = 2M Q. Te indices in F indicate beam and target polarization, or in tis case te lack tereof (U = unpolarized). Te dependence of te structure functions F = F (x, Q 2, z, Pt ) as been dropped ere for simplicity. Wile tere are several parameterizations for te cross section in terms of parton distribution functions, e.g. [14, 18, 19], Eq is model independent. As long as one stays in te pure partonmodel, were QCD effects are switced off, it is possible to obtain an expression for te structure functions in terms of PDFs and fragmentation functions similar to Equation 2.7, see again [14]. As can be seen in Tbl. 2.2, te additionally involved PDFs and fragmentation functions in te unpolarized case are te transversity (x, k t ) and te Boer-Mulders function 1 (x, k t) introduced in Sec. 2.2, eac combined wit te Collins fragmentation function H1 (z, p t ): Table 2.2: PDFs and fragmentation functions contributing to te structure functions. Te presence of te usual unpolarized PDF f(x, k t ) and te corresponding fragmentation function D(z, p t ) is due to te Can effect Structure Function contributing PDFs F UU f(x)d(z ) F cos φ UU f(x, k t )D(z, p t ), 1 (x, k t )H1 (z, p t ), (x, k t )H1 (z, p t ) F cos 2φ UU f(x, k t )D(z, p t ), 1 (x, k t )H1 (z, p t ) Flavor and adron type indices q, ave been dropped ere. Te contribution from transversity sould be very small. (x) can be obtained from measurements performed by te HERMES collaboration wit a proton target [2] and by te COMPASS collaboration on a transversely polarized Deuteron target [21]. Te Collins fragmentation function needed to disentangle (x) and H 1 (z ) can be determined from BELLE data [22]. Tese results lead to te conclusion tat, altoug (x) is non-vanising, for a deuteron target te up and down quark contributions to tis combination cancel eac oter, because of te isospin symmetry of te deuteron [21]. Little is known of te Boer-Mulders function 1 (x, k t). A model calculation [23, 24] sows tat te Boer-Mulders contribution to te cos 2φ modulation migt be of similar magnitude tan te Can effect. However, due to te scarce experimental data, tis model relies on rater strong assumptions. Predictions for COMPASS kinematics are available and will be compared to te results in Sec Te Boer-Mulders Function may also contribute to te cos φ asymmetry [14], but te size of tis effect is unknown.

18 12 2. Teoretical Motivation Perturbative QCD Effects Perturbative QCD introduces a dependence on cos φ already at order α s. Te relevant Feynman diagrams are sown in Fig Te calculation of te cosine modulation as first been performed by H. Georgi and H. D. Politzer in 1977 [4] and as been discussed in detail in [25], on wic tis section is based. Recently, iger order contributions were calculated in [26]. At large z, te diagrams sown in 2.4 a) are dominant, since te gluons tend to fragment into softer quarks and tus te diagrams in b) can be neglected in te ig z region. In tis limit gluon bremsstralung predicts a cos φ modulation wit a negative amplitude. As z 1 te mean cosine moment of te cross section, defined in Eq. 2.14, can be approximated by cos φ α s 2 1 zdcos (y). (2.27) Te diagrams in b), were te observed adron comes from te gluon, are most important in te low z region and give a positive amplitude, wile te contribution from c) canges its sign at z =.5. Te full expression can be found in te two publications mentioned above. Tere is also a contribution to te cos 2φ modulation, given in [27]. QCD effects are most important at Pt > 1 GeV [25, 17], so tey are expected to be small for COMPASS kinematics, were most of te statistics is at low transverse momentum. k k' k k' k k' P 1 P 2 P 3 a) P 1 P P 2 3 P 1 c) P 2 P 3 k k' k k' P 2 P 1 P 2 P 1 P 3 P 3 b) Figure 2.4: relevant Feynman diagrams for φ modulations at order α s in SIDIS: In te diagrams in a), te observed adron comes from te scattered quark, in b) from te bremsstralung gluon.

19 2.5. Azimutal Modulations in Unpolarized SIDIS Polarized SIDIS Since te muon beam used at te COMPASS experiment is naturally polarized, see Sec. 3.1, anoter angular modulation of te cross section is expected. In contrast to te Can and QCD contributions, te cross section for te beam asymmetry depends on sin φ [14]. Terefore tis effect can easily be separated from te Can and QCD contributions. Additional angular modulations arise, since COMPASS also measures wit a polarized target (see Sec ). Tese cancel due to te combination of data taken wit opposite polarization, terefore tey are not discussed ere. Te cross section relevant for tis tesis in terms of structure functions is tus: dσ dx dy dz dφ P t dp t = α2 xyq 2 (1 + (1 y)2 )(1 + γ2 2x ) ( F UU + D cos φ (y) cos φ F cos φ UU ) + D sin φ (y) sin φ F sin φ LU, + D cos 2φ (y) cos 2φ F cos 2φ UU (2.28) were D sin φ is given by D sin φ = y(1 y) 1 + (1 y) 2. (2.29) Since te focus of te analysis lies on te two cosine modulations, te sine dependence will only be discussed in App. A.1.

20 14 2. Teoretical Motivation

21 3. Te COMPASS Experiment Te COMPASS Experiment [28] was built to investigate te spin-structure of te nucleon. It is a fixed target experiment located at CERN. From 22 to 26 1 COMPASS measured wit a naturally polarized µ + beam and a polarized Deuteron target. In 27, COMPASS started taking data wit a polarized liquid NH 3 target. For bot cases, data was taken wit te target longitudinally (i.e. along te beam axis) and transversely polarized. Since tis tesis is based on data taken in 24 wit longitudinal target polarization, tis capter focuses on te setup in 24 as depicted in Fig Te Beam Line Te COMPASS experiment is located at te end of te M2 beam line [29] of te SPS 2. It uses a secondary beam created wit te SPS proton beam of 4 GeV scattering off a production target made of Beryllium. Te beam is extracted from te SPS for 4.8 s, a so-called spill, followed by a break of 12 s. In one spill, about 1 13 protons it te production target, producing mainly pions and kaons. Tese subsequently decay into (anti)muons and neutrinos. Te intensity of te final muon beam is about muons/spill. Wit te exception of a sort adron run in 24, COMPASS used a µ + beam wit an energy of 16 GeV, wic is naturally polarized due to te parity violation in te weak decay of te pions and kaons. Te polarization is momentum dependent: Since te muon beam as a rater large momentum spread of 5% [28], a measurement of te momentum of eac beam particle is required. Tis is done wit te Beam Momentum Station (BMS), located about 1m in front (upstream) of te COMPASS target. Despite of several focusing magnets, te beam is accompanied by a rater large alo: te COMPASS target as a diameter of 3 cm, but te alo witin 3 to 15 cm of te beam axis is still about 16% of te incoming flux, and about 7% of te muons are even furter away. [28]. 3.2 Te Spectrometer Te COMPASS experiment consists of two stages, te Large Angle Spectrometer (LAS) and te Small Angle Spectrometer (SAS) to allow for precise momentum measurement starting from about 1 GeV up to about 1 GeV. Eac stage as its own spectrometer magnet (SM1 and SM2). Te COMPASS experiment features different types of tracking detectors, optimized for te required rates and resolutions and a adronic calorimeter in eac stage. Te SAS also includes an electromagnetic calorimeter. A Ring Imaging CHerenkov detector (RICH) is included in te LAS, allowing separation of pions and kaons between 9 and 43 GeV. 1 apart from a break in 25 because of te SPS upgrade for te LHC 2 Super Proton Syncrotron

22 16 3. Te COMPASS Experiment µ + Beam COMPASS Spectrometer 24 top view Veto Trigger Silicon Target Straw SM1 HCAL1 Muon Filter 1 SM2 Outer Trigger Straw ECAL2 HCAL2 MWPC Inner Trigger Muon Filter 2 x SciFi z Micromegas SciFi DC GEM SciFi DC RICH 1 MWPC GEM SciFi MW1 SciFi GEM MWPC SciFi GEM Large area DC Figure 3.1: Scematic view of te COMPASS Spectrometer in 24 [28]. Middle Trigger Ladder Trigger MWPC MW2 Outer Trigger GEM Inner Trigger Ladder Trigger Middle Trigger 5 m

23 3.2. Te Spectrometer 17 Vetos Target SM1 HCAL1 K H3O π + H4I H4L H4M H4O H5L H5M µ + H5I Beam Beam SM2 HCAL2 µ Filter Figure 3.2: Detectors used for te trigger system in COMPASS (not to scale) [28] Te Polarized Target To reac ig statistics despite te small cross-section for muon scattering, COMPASS is operating wit a solid state target. Up to 24, te target system of te SMC 3 described in [3] was used. It consists of two target cells, eac 6 cm long and a diameter of 3 cm, separated by a gap of 1 cm. Te target cells can be polarized in opposite directions. Te polarizations are flipped every 8 ours for longitudinal polarization, in order to reduce systematic errors due to acceptance effects. Te acceptances for te two cells are different, due to te target magnet. It limits te angular acceptance to about 7 mrad at te upstream end of te target, wic gradually increases to 18 mrad at te downstream end. In order to maintain te polarization, te target cells are cooled down to about 5 mk. Until 26, COMPASS measured on a polarized 6 LiD target, wic can in good approximation be seen as an effective 4 2He + 2D target Te COMPASS Trigger System Te COMPASS trigger system decides, weter an interesting event as occurred. For Q 2 2 GeV 2, it is mainly based on te detection of te scattered muon by dedicated trigger odoscopes, located after tick absorbers. Tese odoscopes are combined in te so-called inner, middle, ladder and outer triggers. For te middle and outer trigger, a minimum energy deposit in one of te adronic calorimeters is required as well. A pure calorimetric trigger (wit iger energy tresold) as also been implemented. It extends te kinematic range of te COMPASS experiment to iger Q 2, were te angle of te scattered muon is so large, tat it is not detected wit te odoscopes. Te middle trigger odoscopes are also used as an inclusive trigger, witout requiring an adronic signal. To avoid fake triggers coming from alo muons, a system of veto odoscopes is installed upstream of te target. A more detailed explanation of te COMPASS trigger system can be found in [31]. 3 Spin Muon Collaboration

24 18 3. Te COMPASS Experiment Tracking Detectors A large variety of tracking devices is used at COMPASS. Tey can be divided into tree groups: Very Small Area Trackers (VSAT), Small Area Trackers (SAT) and Large Area Trackers (LAT). Teir resolution in space and time is optimized to te particle flux in different parts of te detector, wic varies over more tan 5 orders of magnitude. Te tracking detectors are usually grouped togeter into tracking stations. A tracking station typically consists of one VSAT, one SAT and one LAT. Te outer region, were te fluxes are rater low, is covered by te LAT, te intermediate region by te SAT and te part closest to te beam by te VSAT. Table 3.1: Tracking Devices used at COMPASS VSAT SAT LAT Scintillating Fibers, Silicon Micro strips MICROMEs GAseous Structure (MICROMEGAS), Gas Electron Multiplier (GEM) Drift Cambers, Straw Tubes, Multi Wire Proportional Cambers (MWPC) Particle Identification Muon Identification Muons are identified exploiting te fact, tat muons are te only carged particles wic can pass troug a large amount of material. For tis reason, tere are tick walls of concrete or iron at te end of eac stage of te spectrometer, te muon filters. Particles wic cause its in bot parts of te muon walls, wic are wire cambers located directly in front of and beind te filters, are ten identified as muons. Also information from te two adronic calorimeters can be used to distinguis muons from adrons, since teir energy deposit wit respect to teir momentum is muc lower tan te energy deposit of adrons. Hadron Identification Pions and kaons can be identified using te RICH detector, wic is located downstream of SM1. Te RICH utilizes te fact tat a particle wit a velocity β iger tan te speed of ligt in te medium wit refractive index n emits potons wit a caracteristic angle Θ C to its direction of motion: cos Θ C = 1 βn = 1 n 1 + m2 P 2. (3.1) In te second equation Θ C is expressed in terms of momentum P and mass m o te particle. Tus it is possible, to determine te particle mass by measuring Θ C in addition to te momentum obtained from te tracking. It can be seen from Eq. 3.1, tat te minimal velocity for Cerenkov radiation is β min = 1 n P min = m n 2 1. (3.2)

25 3.3. Data Reconstruction 19 An upper experimental limit Θ C,max exists for β 1. Different particle types can not be distinguised, as soon as β is close to one for bot teir masses and bot particles emit Cerenkov radiation wit an angle of almost Θ C,max. Tus it is difficult to separate electrons and pions, since electrons emit Cerenkov radiation wit te maximum angle over te wole momentum region covered by te COMPASS Experiment, wile pions emit Cerenkov radiation wit almost Θ C,max from about 8 GeV. Te COMPASS RICH is operated wit C 4 H 1 as radiator gas, wic as a refractive index of about n C4 H 1 1 =.15. In tis case, te lower tresolds for Cerenkov radiation are approximately 2.5 GeV for pions, 8.9 GeV for kaons and 17 GeV for protons. Te upper limit to separate electrons and pions is about 8 GeV, pions and kaons can be separated up to about 43 GeV. 3.3 Data Reconstruction Te data is digitized directly on te front end cards, temporarily stored on read-out buffers and merged on event builders [32]. Ten it is transfered to CASTOR [33], te CERN Advanced STORage manager, were it is again buffered on disks until it is finally written to tape. After te transfer of te acquired data to tape, te raw data is reconstructed using te COmpass Reconstruction and AnaLysis software (CORAL) [34], wic is based on C++ and ROOT [35]. At tis stage, te pysical information, like tracks and momenta of particles or te position of vertices is determined. Bot te track and vertex reconstruction are based on a Kalman fit [36], wic is a widely used approac in ig energy pysics for suc tasks. More details about te track and vertex reconstruction can be found in [37, 38]. Events containing at least one vertex are stored in mini Data Summary Tapes (mdsts), wic can be processed wit anoter software tool called PHAST (PHysics Analysis Software Tools) [39]. Wit PHAST, interesting pysical events for a particular analysis can be selected and pysical quantities calculated. Te desired information can ten be written into a ROOT tree, so tat te final steps of te analysis can be performed easily. Tis analysis requires a Monte Carlo (MC) simulation, wic simulates bot te pysical interaction and te detector response. It will be described in Sec Here te events are generated wit LEPTO [4]. Afterwards te detector response is simulated wit COMGEANT (COMPASS GEometry ANd Tracking). COMGEANT [41] is te detector simulation tool for te COMPASS experiment, based on GEANT 3.21 [42], a program used to simulate te passage of elementary particles troug matter. Te output of COMGEANT is ten similar to te raw data, so it can be reconstructed and analyzed wit CORAL and PHAST. Additional information coming from te MC generator is still present in te mdsts, wic allows to estimate detector properties like acceptances or resolutions.

26 2 3. Te COMPASS Experiment

27 4. Data Selection Since COMPASS as a broad pysics program and tus covers a large kinematic region, not all te recorded events are suitable for a DIS analysis. Terefore a data sample is needed wic is enriced wit interesting events. Also certain quality criteria ave to be fulfilled. Te event sample used in tis analysis is based on a precut sample for 24 data, originally produced for te extraction of g 1 [43] for Q 2 > 1. GeV 2. However, only te data taken in te weeks W26 and W27 are used. Tere are two reasons for tis: Firstly, RICH identification is used to reject electrons, an option wic was only introduced in a newer version of CORAL. At present, only part of te data 1 as been reconstructed using tis newer CORAL version. Secondly tere were minor canges to te spectrometer during te weeks W28-W38 [44], wic were not taken into account in te presently existing MC simulation. Te weeks W22/W23 were not used, since during tis time te beam conditions were very different from te nominal ones because of problems in te SPS [45]. Neverteless, tere is still plenty of statistics and te error of te measurement is dominated by te intrinsic uncertainty of te detector simulation. Tis capter contains a detailed description of te performed event selection. 4.1 Event Selection Data Quality To avoid false modulations due to canges in te spectrometer performance, several criteria (e.g. te number of tracks and te number of vertices per event) are monitored for every spill and are used to reject eiter complete runs or spills, were te spectrometer performance varies. Tese spills are grouped togeter in bad spill lists. Tese lists as well as furter information can be found in [46]. Furter, a reasonable quality of te reconstructed tracks is ensured wit a cut on χ 2 red = χ 2 N Hits N FitPar < 1 (4.1) for eac track. N Hits is te number of its associated wit te track, N FitPar = 5 te number of fitted parameters. Te five parameters determined for eac track are te positions X, Y, te derivatives dx /dz, dy /dz, and te ratio q / P of carge q and 1 te data taken in te periods W22, W23, W26, W27, W38, W39, W4

28 22 4. Data Selection momentum P, all at a fixed Z position 2 [37]. Also a cut on χ 2 red, vertex < 2, wic for a vertex wit N track tracks is given by χ 2 red, vertex = χ2 vertex 2N track 3, (4.2) [38] is applied for te primary vertex. Additionally, tracks wic leave te detector before te first spectrometer magnet are excluded from tis analysis by demanding te last it Z last to be beind SM1 (Z last > 3.5 m) Kinematic Range In order to select DIS events, Q 2 > 1 GeV 2 is required. A cut on te invariant mass W > 5 GeV avoids te resonance region of te cross section. Additionally te relative energy loss of te lepton y is restricted to.1 < y <.85. Te lower cut is to ensure good resolution in y and to eliminate elastic scattering events, wile te upper cut discards events were radiative corrections become important. Furtermore, te sum of te relative energies z i of te adrons i in te event as to fulfill <.85 (4.3) adrons i z i to exclude decay products of exclusively produced mesons (see Fig. 4.1), since tese processes are not included in te MC simulation Te Primary Vertex If a beam particle is found and assigned to a vertex, te vertex is called a primary vertex. It is required to be inside te target. For tis reason, te distance of te vertex to te target axis must be smaller tan 1.4 cm. Also te target material as a tendency to settle down, so tere is less material in te upper part of te target cell tan in te lower one. Terefore, a cut on te Y coordinate of te primary vertex of Y < 1. cm is applied. In some events, more tan one primary vertex is reconstructed. In tis case, only te so called best primary vertex is taken, wic is te one wit te most outgoing tracks or in case of equal number of tracks te one wit te best χ 2 red. In addition to te detection of te incoming and outgoing muon, at least one additional outgoing track, considered as a adron candidate, is required. All tese particles ave to pass additional cuts described below to be considered in te analysis Beam and Scattered Muon Bot te incoming and te outgoing muon ave to be reconstructed to be able to calculate te DIS variables. If more tan one positively carged outgoing muon is 2 te coordinate system in te Lab system is oriented as follows: te nominal beam direction defines te Z axis, te Y axis is oriented upwards and to make te coordinate system rigt anded, te X axis points towards te left side, wen looking along te direction of te beam. Te origin is at te nominal target position, as sown in Fig. 3.1

29 4.1. Event Selection 23 dn/dz all exclusive events.5 1 z all = i zi Figure 4.1: z all distribution. Te peak comes from te decay products of exclusively produced mesons (mainly pions from ρ decays). Te red line indicates te cut on z all. found, te event is discarded. In addition, information of te two Muonwalls (MW) located after tick absorbers (see sec ) is used to searc for muons not identified as suc wit te reconstruction software. If te track of te adron candidate as at least four its in MW1 or at least six its in MW2, it is considered to be a muon and te event is discarded as well. To ensure te reliability of te muon identification te muon track as to ave traversed an amount of detector material corresponding to at least 3 radiation lengts X. Te momentum of te beam muon as to be between 12 and 2 GeV and te extrapolated beam track as to cross bot target cells to equalize te fluxes in te upstream and downstream cell. In some events, te scattered muon goes troug te ole of te first absorber and does not cross te second absorber and tus as not penetrated enoug material to be identified as a muon. If an additional positive muon is created in te interaction, tis is falsely identified as te scattered muon. Tis leads to an incorrect calculation of all kinematical variables and tus tese events ave to be rejected. A set of cuts based on geometrical considerations as been introduced in [47] to remove tese events. Te tracks of all adron candidates are extrapolated to te downstream end of te spectrometer at Z = 5 m. Ten te extrapolated positions X e and Y e in te plane ortogonal to te beam axis must lie outside te following regions: (Xe 45 cm) 2 + Y 2 e < 1 cm (4.4) if te last it of te track Z last was before SM2 (Z last < 2 m) or (Xe 35 cm) 2 + Y 2 e < 15 cm or X e 55 cm < 13 cm and Y e < 3 cm (4.5) if te last it was after SM2. Events were a adron candidate ad a it after te last muon absorber (Z = 38 m) were rejected as well.

30 24 4. Data Selection ] -1 dn/dz [cm Z [cm] Figure 4.2: Z position of te primary vertex witout cuts (black curve) and after all cuts, in particular te requirement to be inside te target (red dotted curve). Y [cm] Y [cm] X [cm] X [cm] Figure 4.3: cuts(rigt). distribution of primary vertices in te X Y plane before cuts (left) and after

31 4.2. Hadron Selection 25 After all cuts, about 4 million DIS events wit at least one adron surviving te cuts described below remain. An overview over te effect of eac cut is given in Tbl. 4.1 at te end of tis capter. 4.2 Hadron Selection Apart from te scattered muon, one or more additional outgoing particles ave to be detected. Tese adron candidates ave to fulfill additional requirements in oder to suppress contamination by muons and electrons. To reduce te number of muons treated as adrons, a minimum energy deposit in one of te adronic calorimeters is required. As depicted in Fig. 4.4, te tresold is 4 GeV for HCAL1 and 5 GeV for HCAL2. Furtermore, te adron is discarded if it as clusters in bot adronic calorimeters. For te same reason, a cut on te amount of traversed material (in radiation lengts) X /X < 1 is applied. To reject electrons, RICH identification is used, wic will be described in section 4.4. Electrons could in principle also be identified using te electromagnetic calorimeter in te SAS, but ECAL2 was not fully operational during te 24 run. For tis reason ECAL2 was not used in te analysis. Furtermore a minimum polar angle of te adron in te lab system of θ,lab > 2 mrad is applied. Te reason for tis cut is to remove a peak in te φ distribution at φ (see Fig. 4.6), wic is tougt to come from potons converting to electronpositron pairs. To ensure tat te track really comes from te primary vertex, te first it Z first of te adron track as to be in front of te first spectrometer magnet (Z first < 3.5 m). As can bee seen in Fig. 4.5, tere are several tracks were tis is not te case. Tere are two main reasons for tis: Te first is te way tracking is performed in te COMPASS experiment [37]. In a first stage, partial tracks are fitted in regions wit only weak magnetic fields, were te tracks are essentially straigt lines. In a second step, te track is bridged between tese segments using a track dictionary, and te final track parameters are obtained by fitting te complete track. So if te first track segment is missing, because tere are not enoug its, or searced at te wrong position because of problems in te bridging procedure, te track will only start after SM1. Te oter main reason is of course, tat te track could also pysically start after SM1, if te particle is created from a neutral particle, e.g. poton conversion to an electron-positron pair. Finally, a minimum transverse momentum (w.r.t. te poton direction, cf. Fig. 2.3) Pt >.1 GeV is required, to ensure φ is well defined. In total, a bit more tan 6.5 million adrons pass te cuts above, again an overview over te effect of eac cut is presented in Tbl. 4.2 at te end of tis capter. 4.3 Leading Hadron Selection Te Can effect is a prediction made for struck quarks. Tese quarks are mostly contained in te adron wit largest energy fraction z. Terefore a subsample enriced wit tese quarks is created containing only te adrons wit largest z. To ensure

32 26 4. Data Selection E [GeV] E [GeV] P [GeV] P [GeV] 5 Figure 4.4: Correlation between measured energy in te calorimeter and te reconstructed momentum for HCAL1 (left) and HCAL2 (rigt). Te red lines indicate te minimum required energy deposit for te particle to be accepted as a adron. Note tat most of te adrons end up in HCAL1. ] -1 [cm dn dz first SM1 2 1 RICH HCAL [cm] Z first Figure 4.5: Te position of te first it of eac track. If te first it is after SM1, te particle is rejected.

33 4.4. Particle Identification 27 dn/dφ φ Figure 4.6: φ distribution witout (black line) and wit (blue dotted line) a cut on θ,lab. Te correlation between θ,lab and φ will be discussed in Sec tat te detected leading adron really is te leading one, it as to fulfill additional requirements: If te missing energy z miss = 1 z all = 1 z i (4.6) adrons i in te event is bigger tan z of te leading adron, it is possible, tat te real leading adron was a neutral particle wic did not leave a track in te spectrometer. Te signature of a neutral particle would be a cluster in one of te calorimeters wit no track assigned. Tus te two adronic calorimeters are searced for suc clusters wit energy E Cluster wit E Cluster + 2 E Cluster > E and no assigned track. In tis equation E is te energy of te leading adron, obtained from momentum and assigned mass. Te resolution E Cluster for te two adronic calorimeters is given by [28]: E HCAL1 E HCAL1 [GeV] = E HCAL2 E HCAL2 =.59 2 E HCAL1 [GeV] (4.7).66 2 E HCAL2 [GeV] +.52 (4.8) If no suc cluster is found, te leading adron is accepted, else rejected. Additionally, te energy fraction z lead of te leading adron as to be bigger tan z lead >.25. As can be seen in Fig. 4.7, for z lead <.25 te leading adron is often not correctly identified. After all tese cuts, about 1.3 million leading adrons remain to be used in tis analysis. 4.4 Particle Identification Altoug te analysis was done for a sample containing all kinds of (carged) adrons, i.e. mainly pions and some kaons and protons, RICH information was used to identify

34 28 4. Data Selection dn/dz z Figure 4.7: MC: Distributions of identified leading adrons (black curve), correctly identified leading adrons (dased-dotted green curve) and incorrectly identified leading adrons (blue dotted curve). Te vertical red curve indicates te cut z >.25 for te leading adron sample. te adron type, wenever possible. Te φ distribution itself relies only on te measured momentum and tus is independent of te assigned mass. However, for correct calculation of te adronic energy fraction z, te particle mass as a small impact. As described in Sec , te RICH allows to reconstruct te velocity β from te measured angle Θ C of te Cerenkov radiation. Since te momentum can be obtained from te tracking, it is possible to determine te particle mass. To allow identification, te particle as to ave a momentum inside te range, were te RICH allows identification and te RICH as to ave found a Cerenkov ring belonging to te track, see Fig Since te refractive index of te RICH radiator gas canges slowly in time due to atmosperic pressure and temperature variations, te minimum tresolds P min for Cerenkov radiation are calculated for every event. In order to ensure a reasonable number of Cerenkov potons, te actual minimal required momenta ave to be at least P min +.5 GeV for pions and P min + 1. GeV for kaons and protons. Electrons always emit Cerenkov radiation in te momentum range covered by te COMPASS spectrometer. During te reconstruction wit CORAL, for eac track wit RICH information, a likeliood L is calculated for mass ypoteses of pions, kaons, protons, electrons and muons. Also a likeliood L BG for te case were te ring is coming from background in te RICH (electronic noise, oter particles in te event, etc.) is calculated. Te muon likeliood cannot be used, since te RICH is incapable of distinguising between pions and muons, due to teir similar masses. More information about adron identification wit te RICH can be found in [48, 49]. In tis analysis,

35 4.4. Particle Identification 29 [mrad] Θ C electron pion kaon proton [GeV] P Figure 4.8: Te correlation of te reconstructed Cerenkov angle wit te measured momentum for pions, kaons, protons and electrons. Te pion band is suppressed by a factor of 1 and te proton band is enanced by a factor of 6, also te criteria for te identification are different tan te ones used for te analysis. te particle is assumed to be of te most likely type, if te maximum likeliood L max fulfills L max > C 2ndmax L 2ndmax and L max > C BG L BG, (4.9) were L 2ndmax is te second largest likeliood. Te factors C 2ndmax and C BG depend on te particle type and are listed in Tbl. A.1 in App. A.2. If no identification wit te RICH is possible, te adron is assumed to be a pion. If te adron candidate is identified as an electron, it is discarded.

36 3 4. Data Selection Table 4.1: Event Statistics Cut rejected vertices remaining vertices [%] vertices after cut total number of primary vertices at least two outgoing particles scattered muon found and unique best primary vertex Inside target beam momentum X/X χ 2 red < 1 for muon tracks < y < Q 2 > 1. GeV 2, W > 5 GeV equal flux vertex χ 2 red < z all extrapolation and Z last Cuts adron passing all cuts (cf. Tbl. 4.2) *was already applied in te precut sample Table 4.2: Hadron Statistics Cut rejected adrons remaining adrons[%] adrons after cut adron candidates Z first before SM Z last after SM χ 2 red < X/X HCAL cuts RICH ID electron Pt θ,lab final adrons final leading adrons

37 5. Acceptance Correction and Determination of te φ Dependence Tis capter describes ow te modulations of te cross section are determined from te measured count rates. In order to avoid false modulations due to te detector acceptance, a MC simulation is used to correct detector acceptance effects. Terefore a brief introduction to Monte Carlo simulation in general and te tools used for te MC simulation in te COMPASS experiment is given in Sec. 5.1 as well as a sort comparison between data and MC. Since te moments of te cross section are determined using a fit to binned istograms, te coice of binning is explained in Sec Te acceptance correction applied to te measured count rate to compensate te non-uniform acceptance of te apparatus will be described in Sec. 5.3, leading to a discussion of te acceptance in Sec Finally, all te steps necessary to extract te moments of te cross section are summarized in Sec Data and MC Altoug te term Monte Carlo (MC) in principle only describes te use of random numbers to solve an integral numerically, te term MC simulation in te context of particle pysics usually implies te use of tese tecniques to simulate pysical processes and te detector response to tese. Te use of MC simulations is te only viable way to estimate te performance of te complex detector systems used in particle pysics. In order to compare measured distributions wit simulated ones, te simulation as to include several steps. Tis often referred to as a MC cain and is depicted scematically in Fig. 5.1: First, te pysical reaction is simulated using a MC generator, e.g. LEPTO [4] or PYTHIA [5]. In a second step, te detector response of a particular experiment is simulated. Te output of tese detector simulation tools as to be in a similar format tan te measured data, so tat in te last step te MC data can be reconstructed analogously to data. For tis tesis, te scattering process is simulated wit LEPTO using only leading order matrix elements and te detector response wit COMGEANT [41], te detector simulation tool for te COMPASS experiment. Afterwards te simulated data is reconstructed wit CORAL and ten te same event selection as for real data can be performed for te reconstructed MC data using PHAST. In te following, te term generated MC (data) will be used for te output of te MC generator alone, wile te term reconstructed MC is used for te output of te full MC cain, including detector simulation and reconstruction. To ensure tat te MC simulation describes te apparatus properly, a good agreement between real data and reconstructed MC is mandatory. Fig. 5.2 sows te x, Q 2, y, z

38 32 5. Acceptance Correction and Determination of te φ Dependence MC generator LEPTO Detector simulation COMGEANT Reconstruction CORAL Figure 5.1: Te MC cain for te COMPASS experiment. and Pt distributions for bot real and MC data. Te agreement is mostly satisfactory, apart from a strong disagreement for Pt 1 GeV. Tis disagreement for large Pt is expected, since te QCD contributions described in Sec are important tere, but are not included in te MC simulation. Tis illustrates tat, since it is not possible to include all possible processes in te MC simulation, differences between real and MC data are possible. Disentangling tese effects from effects due to non-optimal detector description can be very difficult. 5.2 Resolution and Binning In general, a resolution for an observable x can be estimated from MC by calculating te difference between te generated x gen and te reconstructed quantity x rec : x = x gen x rec. (5.1) Since te modulations in te azimutal angle φ are determined using fits to binned istograms, te resolution in φ is important: Te number of bins as to be a compromise between te need to ave enoug bins to fit a function wit four parameters (Eq. 5.7 to te istogram and te requirement, tat te bin widt is considerably larger tan te angular resolution of te measurement to avoid smearing effects. A limit for te number of bins is also given by te available data: te statistics in eac bin as to remain large enoug for a meaningful measurement. In order to obtain te dependencies for various kinematical variables, te sample is furter divided into several kinematic bins. Te obtained resolution for φ is sown in Fig. 5.3, leading to a coice of 18 bins in φ. For te kinematic binning, te situation is sligtly more complex, since te resolution depends on te value of te variable. Also te bins are not equidistant but cosen in suc a way tat eac bin contains a similar number of events. However, eac kinematic bin is large enoug tat smearing effects are negligible. For eac kinematical bin, te azimutal angle φ is calculated according to Eq and 2.12 and filled into a istogram of 18 equidistant bins ranging from π to π.

39 5.2. Resolution and Binning 33 1/N dn/dx x ] -2 [GeV 2 1/N dn/dq Q [GeV ] 1/N dn/dy y 1/N dn/dz z ] -1 [GeV 1/N dn/dp t P t [GeV] Figure 5.2: x, Q 2, y, z and P t distributions for data (black circles) and MC (blue squares).

40 34 5. Acceptance Correction and Determination of te φ Dependence ] -1 [rad dn/d φ φ [rad] φ, rec Figure 5.3: Difference between reconstructed and generated φ. Te two dotted red lines indicate te cosen bin widt., gen 5.3 Te Acceptance Correction Te acceptance A of te COMPASS Spectrometer as a function of φ is obtained for eac kinematic bin from te generated and reconstructed angular distributions via A(φ ) = N rec(φ ) N gen (φ ), (5.2) were N gen (rec) (φ ) is te number of generated (reconstructed) adrons in a given φ bin. Wile for te reconstructed adrons te same cuts are applied tan for real data, te only requirement on te generated data is tat te events lie in a similar kinematic region (Q 2 > 1 GeV 2,.9 < y <.9 and for te acceptance corrections for leading adrons z >.25) tan te real data. No leading adron selection is performed on te generated distributions, since te number of reconstructed adrons as to be a subsample of te number of generated ones. Since te leading adron selection described in Sec. 4.3 does not always identify te correct leading adron, tis would not be te case anymore, if for te generated φ distribution only real leading adrons are considered. A(φ ) can be interpreted as te probability for a adron emitted under te angle φ to be detected in te spectrometer and to survive te cuts described in te previous capter. So strictly speaking tis quantity sould be called a pseudo acceptance, since it depends on te cuts applied in te event selection. N rec (φ ) follows a binomial

41 5.4. Te Acceptance of te COMPASS Spectrometer 35 distribution, since te particle is eiter reconstructed or not, and tus te error of te acceptance function is A(φ )(1 A(φ )) σ A (φ ) =. (5.3) N gen (φ ) Te corrected counting rates N corr (φ ) are ten obtained from te measured rates N meas (φ ) as N corr (φ ) = N meas(φ ) (5.4) A(φ ) wit te error (te φ dependencies are suppressed ere for simplicity) (σnmeas ) 2 ( σa ) ( A σ Ncorr (φ ) = N corr + = Ncorr + A N meas N meas AN gen ) 2, (5.5) wic follows wit Gaussian error propagation. Te MC simulation contains more statistics tan te data sample, so tis error is dominated by N meas. In order to improve te acceptance correction, an additional correction for te efficiencies for middle and outer trigger odoscopes was performed, since bot triggers were found to ave inefficient slabs. To tis end, te scattered muon track in te real and te reconstructed MC data was extrapolated to Z = 4 m. Ten te extrapolated radial positions X Ext and Y Ext distributions are compared to te distributions obtained from data. In case of strong disagreement, efficiencies are introduced in te MC data to compensate. Tis is exemplarily sown in Fig. 5.4 for te middle trigger in te period W27. Since it is muc faster to introduce te efficiencies offline, tey were not included directly in te MC simulation. Also tese inefficiencies are time dependent, so MC wit different settings for te two different periods would ave been necessary. Since te effects on bot kinematical and φ distributions were found to be small, and MC production is time consuming, tis was not necessary. Te obtained acceptances for upstream and downstream cell are exemplarily sown in Fig. 5.5 for a small, medium and large x bin and will be discussed in te next section. 5.4 Te Acceptance of te COMPASS Spectrometer As can be seen in Fig. 5.5, te acceptance depends strongly on te kinematic bin and also on te target cell: Te acceptance of te downstream cell is iger tan for te upstream one. Wile in te low x region, te acceptance decreases towards small φ, for larger x te situation is inverse and te acceptance drops strongly wit larger φ. In order to understand tis beavior, studies on generated MC data were performed. It turned out, tat tese effects are due to te differences in angular acceptance: Te acceptance in te polar angle of te adron in te lab system θ,lab increases from 7 mrad for te upstream end of te target to 18 mrad at te downstream end because of te limited acceptance of te target magnet. As can be seen in Fig. 5.6 (top row), an upper bound on θ,lab causes mainly a loss of events at large φ, wile a lower cut causes a loss at small φ (bottom row). Tis can be understood wit Fig. 5.8:

42 36 5. Acceptance Correction and Determination of te φ Dependence [cm] Y Ext [cm] Y Ext [cm] X Ext [cm] X Ext Figure 5.4: distributions of te positions X Ext and Y Ext of te extrapolated muon track for data(left) and MC before te correction(rigt). Te black box at 2 cm < X < 3 cm, Y < cm indicates an inefficient slab. φ is te azimutal angle of te adron, in te coordinate system, were te poton direction 1 defines te Z axis. Considering only te case, were te adron is emitted in te lepton scattering plane, tere are two cases: φ = corresponds to te adron going towards te scattered muon, wile φ = π means tat te scattered adron moves away from te scattered muon. As can bee seen in Fig. 5.8, θ,lab tends to be larger for φ = π tan for φ =. Tus te acceptance for events wit larger φ is lower in te upstream cell, were te maximal accepted angle is smaller tan for te downstream cell. Since a large polar angle typically means large x, tis is especially important in te large x bins. Due to te different angular acceptances tis loss starts at lower x for te upstream cell tan for te downstream one. Tis can be seen in te middle plots of Fig. 5.5, were te acceptance at large φ decreases more for te upstream cell tan for te downstream cell. Tis cell dependence is te main reason, tat te analysis, in particular te acceptance correction, is performed for upstream and downstream cell separately. 5.5 Determination of te Moments of te Cross Section For reasons given in te previous paragrap, te wole analysis is done for te upstream and downstream cell separately. In order to extract te dependence of te amplitudes of te modulations for te kinematical variables x, y, Q 2, z, P t, te data is divided into about 1 kinematic bins for one of te kinematic variables 2, wile te dependence on te oter variables are integrated out. Ten, since COMPASS measures wit a polarized 1 te definition of φ is depicted again in Fig. 5.7 for convenience. 2 Te binning can be found in Tbl. A.2 in te Appendix

43 5.5. Determination of te Moments of te Cross Section 37 acceptance acceptance φ φ acceptance.15 acceptance φ -2 2 φ acceptance acceptance φ -2 2 φ Figure 5.5: Acceptances of te COMPASS spectrometer for positive adrons in te first (top row, x <.8), fourt (middle row,.2 < x <.26 ) and eigt (bottom row,.8 < x < 1) x bin eac for upstream (left) and downstream cell (rigt).

44 38 5. Acceptance Correction and Determination of te φ Dependence dn/dφ dn/dφ φ dn/dφ dn/dφ φ φ φ Figure 5.6: generated φ distribution wit different cuts on θ,lab : <7 mrad (top left, acceptance at te upstream end), θ,lab < 18 mrad (top rigt, acceptance at te downstream end), 2 mrad <θ,lab < 7 mrad(bottom left) and 2 mrad < θ,lab < 18 mrad (bottom rigt).

45 5.5. Determination of te Moments of te Cross Section 39 P z y x q l ' l Figure 5.7: Definition of te adron azimutal angle φ as already depicted in Fig Beam axis Hadron wit ~ Virtual direction Scattered muon 1 2 > Hadron wit ~ Incoming muon Figure 5.8: A scematic picture of te scattering process in te lab frame for te case, wen te adron is produced in te lepton scattering plane. If te adron goes in te same direction as te scattered muon, wit respect to te virtual poton, φ = and te polar angle in te lab frame is smaller tan for te case, were te adron goes in te opposite direction.