Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Measurement of the π⁰ electromagnetic transition form factor Drees, Reena Meijer 1991

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1991_A1 D73.pdf [ 5.63MB ]
Metadata
JSON: 831-1.0084985.json
JSON-LD: 831-1.0084985-ld.json
RDF/XML (Pretty): 831-1.0084985-rdf.xml
RDF/JSON: 831-1.0084985-rdf.json
Turtle: 831-1.0084985-turtle.txt
N-Triples: 831-1.0084985-rdf-ntriples.txt
Original Record: 831-1.0084985-source.json
Full Text
831-1.0084985-fulltext.txt
Citation
831-1.0084985.ris

Full Text

M E A S U R E M E N T O F T H E TT° E L E C T R O M A G N E T I C T R A N S I T I O N F O R M F A C T O R Reena Meijer Drees B . Sc. (Hons, co-op) University of Waterloo M . Sc. University of B r i t i s h C o l u m b i a A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y in T H E F A C U L T Y O F G R A D U A T E S T U D I E S D E P A R T M E N T O F P H Y S I C S We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A October 1991 @ Reena Meijer Drees In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada DE-6 (2/88) In presenting this thesis i n part ia l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Co lumbia , I agree that the L i b r a r y shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publ icat ion of this thesis for financial gain shall not be allowed without my writ ten permission. Department of Physics T h e Univers i ty of B r i t i s h Co lumbia 1956 M a i n M a l l Vancouver, Canada Date: Abstract We present the result of a measurement of the 7r° electromagnetic transit ion form factor i n the time-l ike region of momentum transfer. F r o m a data sample of roughly 100,000 7T° —• e + e ~ 7 decays, observed i n the S I N D R U M I magnetic spectrometer at the P a u l Scherrer Institute (Switzerland), we measure a value of the form factor slope a = 0.02 ± 0.01 (stat) ± 0.02 (sys). Th i s measurement is consistent w i t h both the results of the recent measurement by C E L L O ( D E S Y ) i n the space-like region, and w i t h the vector meson dominance prediction of a « 0.03. Table of Contents Abstract i i Acknowledgements xiv 1 Introduction 1 1.1 Hadron S t ruc ture -Form Factors 3 1.2 P i o n F o r m Factor -Part i c le Exchange 6 1.3 F o r m Factors of Neutra l Mesons . 10 1.4 Da l i t z Decay of the TT° 11 2 Historical Background 15 2.1 Theoretical Predictions 15 2.1.1 Vector Meson Dominance M o d e l ( V M D ) 15 2.1.2 The Quark Loop Mode l 18 2.2 Previous Measurements 21 3 Experimental Setup 31 3.1 Overview - General Principles 31 3.1.1 T h e 7T° Source 32 3.2 Exper imenta l Setup 33 3.2.1 Beam and Target 34 3.2.2 S I N D R U M I Spectrometer . . . 37 3.2.3 Trigger Logic . . 39 3.2.4 Onl ine F i l t e r 40 3.3 D a t a Acquis i t ion 41 4 Offline Analysis 42 4.1 Overview 42 4.2 Detector Ca l ibrat ion 42 4.3 Pat tern Recognition and Track F i t t i n g 44 4.3.1 r - <f> F i t 44 4.3.2 z F i t 45 4.3.3 Vertex F i t . . 46 4.4 F i n a l Event Selection - Identification of Da l i t z Events 46 5 D a t a Simulation 53 5.1 Deciding the Detector Geometry 54 5.2 Generating the Part ic le Kinematics 55 5.3 Stop Dis t r ibut ion 57 5.4 Mode l l ing Detector Response 59 5.5 Trigger Simulat ion 59 6 Radiative Corrections 63 6.1 Radiat ive Corrections for the Process 7r° —• e+e~j 63 6.2 Radiat ive Corrections for the Process n~p — » • ne+e~ 76 7 Background Considerations 81 7.1 E x t e r n a l Conversion Background 82 7.2 7T° —• e + e~e + e~ Background 82 7.3 Return ing Tracks 83 7.4 Beam Buckets w i t h more than 1 7T° 83 7.5 n e + e ~ Background 83 8 F i t t i n g Procedure 87 8.1 M a x i m u m Likel ihood Technique - the Bayesian Approach 91 8.2 x2 M i n i m i z a t i o n - the Frequentist Approach 98 8.3 Summary of F i t t i n g Results 101 9 Evaluation of Systematic Errors 104 9.1 T i m e Dependent Systematic Errors 105 9.1.1 Stop Dis tr ibut ion . 105 9.1.2 Target Locat ion 106 9.1.3 Chamber Geometries 106 9.1.4 Magnetic F i e l d 107 9.1.5 Analysis Cuts 110 9.2 T i m e Independent Systematic Errors I l l 9.3 Other Systematic Errors 112 9.4 Summary 115 10 S u m m a r y and Conclusions 117 Bibliography 120 List of Figures 1.1 Feynman diagram for electron-pion scattering. T h e blob represents the extended electromagnetic structure of the pion; its form factor. T h e 7* represents the heavy v i r tua l photon which scatters the incoming electron. The process can be thought of as the sum of two processes shown; the blob contains p mesons 7 1.2 Feynman diagram for pion scattering. The blob represents the effect of the pion form factor 7 1.3 Feynman diagrams for the charge exchange reaction, a) through d) show the different reactions available to study the charged pion form factor. In reactions a) and b) the photon momentum q2 is positive; i n c) and d) it is negative 8 1.4 In a) we show the Feynman diagram for the transit ion A —* B~/*, followed by the decay of the v i r tua l photon 7* — > e+e~. In b) the same transit ion is depicted, where the particle B is specifically a photon. F igure c) shows the product ion of a neutral meson through two v i r tua l photons, one of which is nearly real (its energy is very close to its momentum). The range of momentum transfer of the v i r tua l photon used to probe the meson structure is indicated 9 1.5 T h e K r o l l - W a d a distr ibution. The surface plot shows the distr ibut ion i n x and y. . 13 1.6 T h e projection of the K r o l l - W a d a distr ibut ion in x (note the logarithmic scale - the effect of a on a linear scale is invisible i f the acceptance is uni form over a l l x). Here, the solid line shows the d istr ibut ion w i t h a = 0.0; the dashed line, the effect of a = 0.1 14 2.1 D iagram for 7r° —• e+e~j i n the V M D . The blobs represent the 2- and 3-particle couplings of the vector mesons to the in i t i a l TT and f inal state photons 16 2.2 Feynman diagram for TT° — > e+e~j in the quark loop model . One sums over a l l possible quark species and colours 19 2.3 Exper imenta l results for the form factor slopes of heavier mesons i n the region of timelike momentum transfer. The dotted line shows the range of V M D expectations. The experimental results for the 7r° form factor are also shown for comparison. The recent C E L L O results (clustered around 1991) are al l measured for spacelike momentum transfer. T h e vertical scale is the form factor divided by the square of the mass of the decaying meson, so that the results for different mesons may be compared. 28 3.1 T h e S I N D R U M I detector 35 3.2 Deta i l of the target. Also shown are the lead moderator and the inner-most wire chamber. The superinsulation around the vacuum cylinder is not shown. 36 4.1 Dis t r ibut i on i n r — z of the distance of m i n i m u m approach for e+e~ pairs. The target, moderator, a luminum support r ing , target support structure, and chamber 1 are clearly visible 47 4.2 D is t r ibut i on i n opening angle of e+e~ pairs. The sharp peak at 156° is due to photon conversion events. The broad peak at 110° is due to the asymmetrical <f> opening angle cut of the trigger 49 4.3 Dis t r ibut ion i n x and total energy of e + e~ pairs plus the neutron kinetic energy. T h e ne+e~ events populate the horizontal band at 130 M e V . T h e Dal i t z data inhabit the slanted region up to x = 1.0. T h e curving branch at smal l x and low total energy peeling away from the Da l i t z region are events w i th an extra photon which radiates away more energy. 50 4.4 Dis t r ibut ion of e+e~ pairs in the quantity EtPt and transverse opening angle. The Dal i tz events are constrained to lie i n the box between 107 and 163 M e V . The slanted bands are the ne+e~ events. The region below 107 M e V is inhabited by the radiative events 51 5.1 Two-dimensional weighting function for the s imulat ion, designed to match the stop distr ibut ion to that of the data. The m a x i m u m height is roughly 7, the average is 0.8 58 5.2 The figures i l lustrate the action of the simulated trigger. T h e histogram represents the simulated data before passing through the trigger, the points, after the trigger. In a) through c) we show the effect on the Dal i t z s imulat ion, while in d) through f) we show the n e + e ~ s imulat ion. 61 6.1 F i r s t order radiative corrections to the process 7r° —• e+e~-y. 64 6.2 T w o v i r tua l photon loop graphs, corrections to n° —> e+e~7 65 6.3 T w o dimensional surface plot showing the percentage correction to the K r o l l - W a d a matr ix element, as calculated i n [18]. T h e surface is sym-metric about y = 0 66 vin 6.4 Corrections to the K r o l l - W a d a matr ix element as a function of x, as calculated in Mikael ian and Smi th [18] 67 6.5 Verif ication of the published radiative corrections. The line are the cor-rections as calculated by Mikael ian and S m i t h [18] and the points are the result of the numerical integration using the program of Roberts and S m i t h [21] 72 6.6 Invariant mass of lepton pair versus the total energy, for the simulated events. We plot both the ne+e~ and the e+e~y events, both w i th rad ia -tive corrections 73 6.7 EtPt of lepton pair versus the transverse opening angle, for the s imu-lated events. We plot both the ne+e~ and the e+e~j events, both w i th radiative corrections. The radiative tails are clearly visible 74 6.8 T h e EtPt d istr ibut ion for the Dal i t z events. The histogram is the s im-ulat ion , the points, the data. Note that the radiative ta i l is very well fit 75 6.9 The contribution to x and (j> from the v i r tua l (solid line) and bremsstrahlung (dotted line) corrections to the total (triangles) s imulat ion. T h e total contribution of the radiative corrections is shown by the shaded his-togram 76 6.10 Feynman diagrams for the radiative corrections to the process ir~p —* ne+e". 78 6.11 Tota l energy of the lepton pair for the ne+e~ events. T h e points show the data , the histogram is the simulation. The radiative ta i l is generally well f it , although an underestimation seems likely. T h e simulation is expected to be correct to roughly 15% 80 7.1 T h e determination of the ratio of n e + e to e + e 7. Radiat ive corrections to both reactions are included 85 8.1 T h e performance of the ful l s imulation for various kinematical variables. A l l radiative corrections and backgrounds are included. T h e points are the data , the histogram, the s imulation 89 8.2 T h e performance of the ful l s imulation for various kinematical variables. A l l radiative corrections and backgrounds are included. T h e points are the data, the histogram, the s imulation 90 8.3 Graph i ca l summary of the results of the m a x i m u m l ikel ihood f i tt ing for ax (white boxes) and (dark boxes) for each of the four geometries. O n the left are the results without radiative corrections, on the right the results inc luding the radiative corrections 97 8.4 Graph i ca l summary of the results of the x 2 fitting for ax (white boxes) and a<£ (dark boxes) for each of the four geometries. O n the left are the results without radiative corrections, on the left the results inc luding the radiative corrections. 100 8.5 Summary of results for a l l four geometries, in x and <f>. The four points to the left of the double line represent the results for geometries 2, 4, 5, and 6, respectively; the point to the right indicates the error-weighted average. Its smal l error bar shows the statistical error, and the larger one shows the combined statistical error and standard error of the fluc-tuations. The results i n x and <f> are averaged to obtain the f inal results. 102 9.1 T h e result of the cal ibration on the n e + e " peak for geometry 4. T h e net change to the magnetic field is about 0.5%. O n the left, before the cal ibrat ion, on the right, after ; 108 9.2 The result of the cal ibration on the D a l i t z box for geometry 4. T h e net change to the magnetic field is about 0.5%. O n the left, before the cal ibrat ion, on the right, after 108 9.3 T h e performance of the n e + e ~ simulation (with radiative corrections) for some areas of phase space, for geometry 4. The points are the data; the histogram, the s imulation. Since the agreement between the data and the Monte Car lo is good, we are confident that we understand the action of the trigger 113 9.4 T h e performance of the ne+e~ s imulation (with radiative corrections) for some areas of phase space, for geometry 4. The points are the data; the histogram, the simulation. Since the agreement between the data and the Monte Car lo is good, we are confident that we understand the action of the trigger 114 List of Tables 2.1 F o r m factor slopes i n V M D for several neutral mesons, together w i t h the experimental results. The theoretical value for the 7r° form factor is included for comparison. See also figure 2.3 22 2.2 Summary of previous experiments to measure the form factor for the decay 7r° —• e+e~j. See also figure 2.3 29 3.1 W i r e chamber specifications 37 7.1 Backgound processes considered and their calculated contribution to the total sample, which numbers roughly 100,000 events. Conversions are modelled i n the l iquid hydrogen target, in the target walls, and i n cham-ber 1 81 7.2 T h e contribution of 7r"p —» ne+e~ expressed as a fraction of the total sample. Other background contributions are negligible. T h e quantities N provide a means for assessing the efficiency of the EtPt separation cut; they are not products of the fit, and their magnitudes are of no special importance 84 8.1 Results for ax and from m a x i m u m likelihood fitting, for each of the various geometries, w i th and without radiative corrections. Errors are statistical only 98 8.2 Results from x 2 min imizat ion , for each of the various geometries, w i th and without radiative corrections. Errors are statistical only. The x2 per degree of freedom varies from 1.2 to 1.7 for the various fits 101 8.3 Summary of the fit results for each geometry, fitting method, and spec-t r u m . Radiat ive corrections and n~p —• ne+e~ background are included. Indiv idual errors are statistical only. The summary value given is the error-weighted average, and its error takes into account the statistical error only. 103 9.1 Systematic errors from various sources. The table is d ivided into t ime-dependent errors (top) and time-independent errors (bottom) 105 9.2 Results of the magnetic field calibrations for the various runs. The mag-net was turned off and on between the various runs 109 Acknowledgements I 'd like to thank my supervisor D r . Chr is W a l t h a m for his countless hours of patient discussion, helpful suggestions and seemingly endless good humour. Thanks also to the members of the S I N D R U M group, most especially D r . Andries van der Schaaf, D r . Carsten Niebuhr , and D r . W i l h e l m B e r t l , for their help and suggestions. Invaluable comments regarding the radiative corrections from D r . Helene Fonvieille and D r . Lee Roberts were also very much appreciated. These five years of labour could not have been completed without the moral support of and frequent waterings by my fellow Barbarians Susan, G len and E r i k , as well as honorary Barbar ian Andrew. I express my graditude to the Natural . Science and Engineering Research Counc i l ( N S E R C ) C a n a d a for a 1967 Scholarship, enabling me to pursue my studies, and to the Univers i ty of B r i t i s h Co lumbia for financial support. Chapter 1 Introduction Beginning w i th the Greek philosophers, humanity has speculated about the existence of basic bui ld ing blocks of matter. The idea that matter might not be infinitely divisible, put forth by Democritus and hotly debated around 600 B C , was put aside u n t i l the recognition, dur ing the 1800's, that a l l material substances are composed of "atoms" of various different "elements". In the early days, it was thought that the atoms were indivis ible , and that , therefore, there existed about 100 different kinds which formed the basis of a l l matter. The discovery by J . J . Thompson in the 1890's [1] that different atoms could be forced to emit identical , very l ight, negatively charged "electrons", pointed to an underlying unifying structure. It was also obvious that the bulk of the mass of the atoms was therefore associated wi th positive charges. In 1911, Rutherford 's famous scattering experiment [2] showed that the positive charge was confined to a very smal l region (radius 1 0 " 1 1 cm or less) of the atom, thus giving rise to the picture of the atom as a positive nucleus surrounded by negative electrons. T h i s discovery was the start ing point for the explosive development of atomic physics that culminated w i th the establishment of quantum mechanics i n the late 1920's. D u r i n g the 1930's, the first accelerators were invented and bu i l t , leading to the discovery of a whole zoo of particles over the next 20 years. It was soon observed that one could group these particles i n families exhibit ing certain common properties. In 1964, G e l l - M a n n and Zweig [3,4] pointed out that the observed patterns could be understood i f the particles were made up of smaller constituents. Three "quarks" , called " u p " , "down" , and "strange", were enough to explain the observations. A s experimentalists bui lt larger and larger accelerators, probing the structure of matter at higher and higher energies, more quarks were added to the l ist . A s it now stands, the so-called Standard M o d e l consists of three families of two quarks each; i n order of increasing mass: \ d I \ I W i t h each family is associated a pair of " leptons", of which the electron is the most famil iar : I \ I * \ These quarks and leptons are, as far as we can te l l , pointlike. Quarks group together to form "hadrons" : three quarks make particles called "baryons" , while quark-antiquark pairs can be bound to form "mesons" . The light leptons stay single. T h e three forces (on this t iny particle scale, gravity is negligible) which operate on the nuclear scale are understood i n terms of the exchange of other particles force particle exchanged electromagnetism photon weak W±, Z° strong gluons T h e rules which govern the exchange of these particles are embodied i n the so-called "Standard M o d e l " of particle physics. T h e Standard M o d e l comes i n two parts. One explains the behaviour of the elec-tromagnetic (which, i n its marriage to quantum mechanics is called Q u a n t u m Electro-dynamics, or Q E D ) and weak forces and is hence called the Electroweak M o d e l . The other part predicts the effects of the strong force and is called, analogous to Q E D (and since gluons carry a quality dubbed "colour" by physicists), Q u a n t u m Chromodynam-ics ( Q C D ) . The Electroweak theory has proven fantastically successful in predicting, to very high accuracy, the results of a l l kinds of particle decays and collisions. Q C D , which governs how quarks stick together i n particles, is a much more difficult theory to calculate and hence is not so numerically precise i n its predictions, but i n spite of this physicists feel that they are on the right track. T h e Standard M o d e l , despite its successes, leaves many questions unanswered. It does not explain the observed family structure of the quarks and leptons. It does not explain their masses. It does not explain why we don't see quarks i n groups of 4 or more. A n d it does not include gravity. W h i l e theoretical physicists labour to invent models which simultaneously answer these questions and unify a l l the forces into one mathematical construct, experimental physicists work hard to find holes in the existing Standard M o d e l - discrepancies between the theoretical predictions and reality. Such holes may i l luminate its shortcomings and provide clues towards the creation of a better theory. It is important , then, to attack the problem of particle structure. The postulated quark structure of the hadrons and mesons should have testable consequences. 1.1 H a d r o n Structure—Form Factors In the classical method of studying particle structure, one bombards an object w i th electrons, d la Rutherford. For example, i n order to probe the charge distr ibut ion of a hadron like a proton, one measures the angular d istr ibut ion of the scattered electrons and compares it to the distr ibut ion obtained by scattering off of a point charge: do do 2^ |2 ^ dQ, dtl point Using quantum electrodynamics, the scattering from a point target, jfipoint, can be calculated to a high degree of accuracy. The function F is called the target's (in this case, the proton's) form factor, and describes its deviation from pointlikeness and hence gives an idea of the structure of the target. The form factor depends on the momentum transfer q2. If q2 is smal l then we find that F(q2) w 1; i n other words, the electron doesn't have enough energy to resolve the inner structure of the target and the target looks pointl ike. A s a simple i l lustrat ion , i f we model the proton as a motionless (static and non-relativist ic) , spinless blob wi th a spherically symmetric charge d istr ibut ion wi th m some constant (to be determined by the experiment), then the form factor is the fourier transform of the charge distr ibution : where 1.4 is the so-called dipole function. For small q2, equation 1.3 has the expansion p(r) = e (1.2) F{\q2\) = J r2p(r)e-iq'r'd3r (1.3) (1.4) F(q2) = 1 - ±\q2\(r2) + order(q4) (1.5) where (r 2 ) = / r2p(r)dr (1.6) Now, defining the form factor slope b by (1.7) we see that , i n this simple case, (1.8) so that extraction of b, by measuring the form factor at different q2 values, and fitting it w i th the dipole function and extrapolating back to q2 = 0, allows the determination of the spread of the charge distr ibut ion of our simple "pro ton" . Rea l particles, however, are neither static nor (in general) spinless, so that the analysis is more complicated than that outl ined here. The scattering electron may interact not only w i th the particle's charge, but also w i th its magnetic moment (due to its spin) , and thus, i n general, we may expect two form factors to come into play. Obviously, the relationship between the shape of the hypothesized charge distr ibut ion and the form factor slope is now not as simple as that given i n equation 1.8 above; however, i n a special reference frame (the "brick w a l l " or Breit frame) i n which no energy is transferred to the target (and the mangitude of the projectile momentum is unchanged), a connection be derived between the magnetic moment d istr ibut ion and the magnetic form factor, and between the electric charge d istr ibut ion and the electric form factor. In scattering electrons from protons it is found that , i n this frame, both form factors of the real proton are proportional to the dipole function, w i t h . ^ ( ' - o ^ y " ( i - 9 ) Hence both distributions are approximately exponential. Determinat ion of the form factor slopes allows determination of the "size" of the proton; it is found that both the magnetic and electric charge distributions have (r 2 ) « (0.8 x 1 0 ~ 1 3 c m ) 2 . In summary, then, we expect one form factor for spin 0 particles, and two form factors i f the particle has spin |. The form factors can be related i n some way to the particle 's charge and magnetic moment distributions. In the case of the proton, it turns out that a reasonable assumption seems to be that these distributions are exponential. 1.2 P i o n F o r m Factor-Partic le Exchange The charged pion is a spinless particle. We might expect, therefore, to be able to describe its electromagnetic structure by a single form factor. The assumption that works well for the proton; namely, that the charge distr ibut ion is exponential ; does not work for the case of the 7 r ± . In fact, the experimental data fits a form factor of the form w i t h m2 w 0.56 GeV2. The form factor above corresponds to a charge d istr ibut ion of the form the "Yukawa potent ia l " , as opposed to the purely exponential case as for the proton. One can also think of the form factor, instead of arising from some extended elec-tromagnetic d is tr ibut ion , as arising from the exchange of particles. T h i s picture is an extension of the quantum picture of photon exchange mediating the electromagnetic interaction between two charges. In this generalization, we allow the photon to assume some mass (which it may, as long as it is for a short enough time that the Heisenberg uncertainty principle is not violated). Th i s shortens its range by an exponential factor e~mr, and we recover exactly the formulae above. We note that the mass m in equation 1.10 above is roughly m w 770 MeV, which corresponds to the mass of the p meson. Since the p meson has the same quantum numbers as the photon, it can be thought of as s imply a heavy photon. The extended electromagnetic structure of the 7T* can then be described by allowing the pion to emit a p, which then scatters the incoming electron. In this picture, then, the pion is thought of as not as a single, rice-crispy-like object, but as some fuzzy cloud of short-l ived p mesons. We depict this graphically i n figure 1.1. Measurement of the form factor provides information about this fuzzy (1.10) (1.11) cloud: how b ig i t is , i f any other mesons other than p's are present, etc. e e + e P e F ( q * ) Figure 1.1: Feynman diagram for electron-pion scattering. T h e blob represents the extended electromagnetic structure of the pion; its form factor. T h e 7* represents the heavy v i r tua l photon which scatters the incoming electron. The process can be thought of as the sum of two processes shown; the blob contains p mesons. In practice, it is not possible to use the reaction shown i n figure 1.1 to study the form factor, since it is not possible to make a pion target. Instead, experimentalists make use of p ion scattering from atomic electrons to produce the reaction shown i n figure 1.2. T h i s reaction is very closely related to the one shown i n figure 1.1, differing only i n the range of momentum transfer examined (in practise, p ion scattering is l imited to q7 < 0.2 G e V 2 ) . F igure 1.2: Feynman diagram for p ion scattering. T h e blob represents the effect of the pion form factor. Another reaction which may be used to study the charged pion form factor is the so-called "charge exchange" reaction and its relatives, shown i n figure 1.3. The advantage of this reaction is that it can be used i n many directions, as i l lustrated i n figure 1.3: given that one can make 7 r + , 7r~, and electron beams, as well as both proton and neutron targets, a myr iad of experimental options are possible. Different ranges of q2 can be probed, and the results of the different experiments can be combined to form a more complete picture of the pion form factor. F igure 1.3: Feynman diagrams for the charge exchange reaction, a) through d) show the different reactions available to study the charged pion form factor. In reactions a) and b) the photon momentum q2 is positive; i n c) and d) i t is negative. A s can be imagined, however, this reaction, since it involves neutrons and protons which are themselves extended objects even less pointl ike than the p ion , provides a more indirect and complicated way of extracting information on the pion form factor. In fact, the form factor one is studying is not the same one as may be measured i n pion product ion or electron-pion scattering experiments. In the charge exchange case, the Figure 1.4: In a) we show the Feynman diagram for the transit ion A —* £7", followed by the decay of the v i r tua l photon 7* —» e + e~. In b) the same transit ion is depicted, where the particle B is specifically a photon. F igure c) shows the product ion of a neutral meson through two v i r tua l photons, one of which is nearly real (its energy is very close to its momentum). The range of momentum transfer of the v i r tua l photon used to probe the meson structure is indicated. form factor involved describes the transition of the incoming pion and the target particle A (proton or neutron) into a photon and the product particle B (neutron or proton). T h e form factor i n question is thus known as a " t rans i t ion" form factor (as opposed to the previous "stat ic" form factor). It describes the electromagnetic properties of the cloud of v i r tua l particles during the reaction n + A —• B . 1.3 F o r m Factors of Neutral Mesons In the particle exchange picture, processes involving a single photon intermediate state such as those pictured i n figures 1.1 and 1.2 are forbidden in the case of mesons w i th zero spin and charge like the 7r°, both by charge conjugation invariance, and by conservation of angular momentum; the photon has a spin of 1, while the 7r° has spin 0. One cannot conserve angular momentum in allowing a 7r° to emit a single photon; hence these particles cannot couple to single photons and their static form factors are identically zero. However, particle transitions or decays of the k i n d pictured in figure 1.4 (a and b) are allowed, since the parent meson and its decay product can have different spins and charge conjugation properties. One can therefore study the electromagnetic properties of neutral mesons by studying their transition form factors. Note, however, that it is not meaningful to relate the transit ion form factor to the "size" of a charge d is tr ibut ion , as is the case w i t h the static form factor. The phenomena of particle decay and transit ion are quantum mechanical i n nature, which the semi-classical picture of an electron scattering from a charge distr ibut ion has no way of describing. The transit ion form factor is an abstraction, albeit a useful one; given some detailed model of the internal structure of the meson, it is possible to predict the behaviour of the form factor, and hence to compare w i t h experiment. In the special case where the decay product is a photon (figure 1.4b), one is exploit-ing the fact that while the neutral meson cannot couple to a single photon, coupling to two photons is allowed. It is possible to construct another allowed situation, involving the product ion of the neutral meson by two photons, as shown i n figure 1.4c. Th i s type of production experiment requires an e + e " collider. The short-l ived meson is identified by its subsequent decay products. The two types of reactions, decay and product ion, complement each other since they probe the structure of the meson i n different regions of momentum transfer. 1.4 Dalitz Decay of the 7r° The lowest-energy decay of the type pictured i n figure 1.4b is the " D a l i t z decay" of the 7T°. T h i s decay has historically been the method by which to study 7r° structure, since neutral pions are easily produced at medium-energy cyclotrons w i t h ir* beams. In order to extract the form factor information, we wi l l need the decay width ap-propriate for a pointl ike 7r°. Th i s was first calculated by Da l i t z [5] i n 1951 and also by K r o l l and W a d a [6] i n 1955. They isolated the pointlike part of the interaction by normal iz ing to the decay 7r° —> 77, obtaining, to lowest order i n a , = 0.0118 (1.14) where the final answer is obtained using F(x) = 1. Here x is the invariant mass of the e + e~ pair , normalized to the 7r° mass and y is the energy part i t ion : x = fc±5±£ (1.15) ml. mlo « 2|P + IIP-l(l -cosy) ( L 1 6 ) ml, where <p is the opening angle of the e + e~ pair , p and q are 4-momenta, p is the 3-momentum. We also define the "energy par t i t i on" and the m i m i n u m x value as follows: V = j ^ f i (1.17) and Ami In figures 1.5 and 1.6 we show the resulting distributions in x and x versus y. Equat ion 1.12 gives the rate of 7r° —• e+e~7 normalized to the decay n° —> 77; the form factor F is normalized such that -F(O) = 1. As in the case of the 7r* form factor, one posits a form factor F of the form „%2 . . or, wr i t ing a = and in casting the above i n terms of a:, F(x) = (1 - ax)'1 (1.19) For smal l x (low momentum transfer), we may expand F(x) w 1 + ax + order(x2) (1.20) so that , by definition 1.7, the form factor slope is given by a. We now t u r n to more detailed and complicated models of the 7r° structure to obtain theoretical predictions for a. Figure 1.6: T h e projection of the K r o l l - W a d a distr ibut ion in x (note the logarithmic scale - the effect of a on a linear scale is invisible if the acceptance is uni form over a l l x). Here, the solid line shows the distr ibution w i th a = 0.0; the dashed l ine, the effect of a = 0.1. Chapter 2 Historical Background 2.1 Theoretical Predictions W h i l e the electromagnetic interactions between two pointlike particles can be described to very high accuracy by quantum electrodynamics, any calculation involving hadrons is only approximate. It seems that hadrons are complex objects, and their inner struc-ture is not well-understood; models describing the inner structure of hadrons and how this manifests itself i n their electromagnetic behaviour currently give only single-digit accuracy. These models take as their starting point the idea that the hadron consists of a cloud of " v i r t u a l " pointlike particles winking in and out of existence i n accordance w i t h the uncertainty principle, and that it is these v i r tua l particles which interact w i th the electromagnetic field. In this way, the models can account for the deviation from point- l ike behaviour and give a prediction for the electromagnetic form factor of the hadron. 2.1.1 Vector Meson Dominance M o d e l ( V M D ) In the vector meson dominance model , the coupling of hadrons to the electromagnetic field's photons is supposed to proceed v i a intermediate vector meson states w i t h the same quantum numbers as the photon. The model is highly phenomenological and as such relies on experimental input for the numerical values of the couplings of the various vector mesons to both the photon and hadron(s) being considered. T h e predictive power Chapter 2. Historical Background 16 of the V M D is thus l imited by the accuracy of the experimental input . r e Figure 2.1: D iagram for TT° —• e+e~-y in the V M D . The blobs represent the 2- and 3-particle couplings of the vector mesons to the in i t ia l TT and final state photons. To calculate the pion form factor in this model , the pion " c loud" is wr i t ten as a sum over intermediate vector meson states. Us ing figure 2.1 one obtains [7] an expression for F i n terms of the matr ix elements describing the transit ion from in i t i a l pion to final photon v i a the intermediate vector mesons V : ^ F ( 3 2 ) « j : { 0 | j 0 | y ) ( y | ^ | 7 r 0 ) (2.1) v where j is the hadronic electromagnetic current. Since we are concerned w i t h low q2, the standard approach is to concentrate on the lowest-mass intermediate states; the p,u> and <(> mesons. The matr ix elements i n equation 2.1 above then reduce to the 2-and 3-particle couplings f^, /</,-,, / ^ - y , / W W - Y and quantities whose magnitudes may be measured experimentally i n other reactions. One thus obtains an expression for the form factor EV 2\ t 2 / fpyfpny fury fumy fifnf<t>*1 \ to be compared w i th the expansion for F ml T h e physical u> and <f> mesons are mixtures of the 7 = 5 = 0 singlet (denoted here by u>o) and octet (here <j>0) states: \u) = A\u0) - B\<f>0) \<f>) = B|wo> + ^ 0 > w i t h A2 + B2 = 1, so that one may identify the slope parameter a as follows: "_L + E. + _ d ^ ^ o n (A LV .™$ m l m \ \/3 / p * 7 \ m l m l ) . a fry fp-fy p w 0 V 7 7 mu m%)T h e magnitude of the factor / p - ^ / p ^ / / ^ has been measured to be of order 1, so that i f one makes the further simplif ication that only the p contributes, one may obtain an estimate for the slope parameter Til2 a ss — £ « +0.034 In fact, taking the ful l expression and using accepted and "reasonable" values for a l l parameters (an excellent discussion is given i n [7]), the vector meson dominance model predicts that \a\ < 0.05 (2.2) T h e sign of a depends on the overall sign of the couplings i n the factor / ^ / V - y / . f i r r y . T h i s cannot be experimentally obtained, but may be calculated by evaluating matr ix elements of the form n i.e. they involve a sum over intermediate states. A simple approximation is to assume that only baryon-antibaryon states contribute to the sum, |n) = |A/W). Since this leads to roughly the correct experimentally measured magnitudes of the ind iv idua l couplings / p w 7 and /n-yy theorists are fairly confident that the approximation is a val id one. Us ing the resulting couplings leads to the prediction that a is positive. If, however, higher order states were to contribute significantly, then one could only calculate the couplings to w i t h i n an order of magnitude and their signs would be completely unobtainable; hence it would not be possible to predict the sign of a. Even i n this case, however, the prediction for the magnitude of a stands as i n equation 2.2. 2.1.2 T h e Quark L o o p M o d e l In the quark loop model , the properties of hadrons are seen as the result of the point interactions between the constituent quarks (of which the model has six, each of which come i n three different "colours") and the photons of the electromagnetic field. The model led to the first theoretical understanding of the lifetime of the 7r°, i n 7r° —>• 77, as set down by Ad ler [8] in 1969. In this framework, the 7r° is seen as a cloud of quarks, and in order to obtain the experimentally measured value of the lifetime, one must include the effect not only of the different quark species, but also of colour. The v indicat ion of this seemingly unnecessary extra parameter was a major t r i u m p h of the quark model . In order to examine the other decay modes of the 7r°, one extends Adler ' s analysis to the case where one or both of the photons are v i r tua l , and hence massive. These v i r t u a l photons can then decay electromagnetically into, for example, an e + e~ pair . T h e decay of the 7r° into two v i r tua l photons 7r° —> 7*7* is modelled according to the Feynman diagram of figure 2.2. The amplitude for the decay is [9] A(s0,si,s2',M) = ie2(e2q)f^eulip(7k^k^e^F(sus2;M) (2.3) = 8ge2(e2q)Me^pakim€p1ea2 x C 7, k , -y* k • T » * M e e Figure 2.2: Feynman diagram for 7r° —> e + e 7 in the quark loop model. One sums over a l l possible quark species and colours. w i t h quark mass M ; fci,&2 the momenta of the photons and £1 ,62 their polarizations. The average charge squared of the constituent quarks, inc luding the colour factor, is given by (e 2). The form factor F is normalized such that F ( 0 , 0 ; M ) = 1. For the purposes of estimation, one assumes that the only quarks that contribute are the u,d pair , and one sets Mu = Md = M. 1 Equat i on 2.2 relates the form factor F and coupling constant to an integral C involving the internal quark mass M and the external masses so = (kj + Jc2)2,Si = k\, and 52 = where C is given by C = [ - ^ I (24) J (2TT)4 [q* - M 2 + ie][(q + k1y-M2 + ie][(q + h + k2)2 — M2 + ie] V ' } Standard techniques allow for the evaluation of C and one can study special cases of the general expression: 1 Generalizing equation 2.5 to real mesons, which are superpositions of six quark loops with different flavors, one obtains, instead of 2.5, with grfi the pion-quark coupling; a formula which does not lead to a significantly different result. 1. So = m^cSi = s2 — 0. This corresponds to the decay of the TT° into two real photons, and using expressions 2.4 and 2.4 one may obtain gM . 2m*o fry = 5— j -arcs in — r 7 r 2 m ; 0 2 M In the l imi t mvo —• 0 this gives the formula for the "triangle anomaly" , which was set down by Ad ler [8] i n the first successful calculation of the 7r° lifetime. 2. so — mJo,«i = 5,62 = 0. W h e n 4 r a 2 < 5 < m 2 0 , this corresponds to the Da l i t z decay 7r° —• e+e~j. In this case one obtains an expression for the form factor F M M ) - ( 1 - " T f f ^ S ' } (2.5) m 2 r 0 - 5 l a r c s m 2 ( m f f o / 2 M ) J v ' In the l imi t that m„o —> 0 this becomes F ( s , 0 ; M ) = — a r c s i n 2 ^ v 7 s 2M E x p a n d i n g about s = 0, this gives whence one can immediately identify the slope parameter a = 1 2 M 2 Note that the quark loop model predicts that a > 0. Further , i f one chooses the quark masses M U i d w 200 — 300 M e V , it is possible to match numerical ly the prediction of V M D (equation 2.2). T h e two models are thus equivalent i n some sense. T h i s matching is known as UQ2 dua l i ty " . The results of the quark-loop calculations are only weakly dependent on the in i t ia l assumptions of mvo —> 0, and are applicable to other (heavier) meson form factors as well . A g a i n , using the posited functional form of F F(q2) = (1 - V ) " 1 the V M D may be used to predict the form factors of the rj,j]' and u; the theoretical results are shown i n table 2.1, together w i th the existing experimental data. For heavier mesons, the form factor slope a is conventionally expressed i n dimensions of G e V - 2 . We may convert to the more famil iar dimensionless quantity by scaling the momentum transfer by the m a x i m u m allowed value x = <72/<?mai) which, for the 7r° —• e + e~7 case, is m 2 0 . We see that the V M D / q u a r k loop predictions are consistent w i th the experimental Recently, a form factor has been measured in the decay KL e + e - 7 [10]; i n this case the form factor is slightly more complex, involving contributions not only from p, u>, and <f> mesons, but also from KL —• K*y followed by A '* —• />, u>, <f> transitions. T h i s leads to a quark-loop inspired prediction of a form factor of the form [11] where the model predicts m w mp as in the 7r° — > e+e 7 case, and |a| « 0.2 — 0.3; the T factors are experimentally measured decay rates, included for normalizat ion. Reference [10] obtains values of m = QlOtH M e V (corresponding to a = 2.7 ± 0.4 G e V " 2 ) and a = - 0 . 2 8 ± 0 . 1 3 . 2.2 Previous Measurements We discuss the techniques used to measure the TT° form factor i n the region of timelike momentum transfer, using the Dal i tz decay. A brief discussion of the single spacelike measurement can be found i n the description of the previous experiments (experiment number 8). There are two different approaches to measuring the form factor; each leads to a different experimental design. > results. meson Ctheory ( G e V *) Gezpt ( G e V J ) 1.8 -2.6 ± 5.7 [13] -0.7 ± 1.5 [14] 1.9 ± 0.4 [15] 1.42 ± 0.21 [27] i 1 1.5 1.7 ± 0.4 [15] 1.59 ± 0.18 [27] u 1.7 2.4 ± 0.2 [15] 7T° . 1.7 see table 2.2 Table 2.1: F o r m factor slopes in V M D for several neutral mesons, together w i th the experimental results. The theoretical value for the 7r° form factor is included for com-parison. See also figure 2.3. 1. One approach is to collect data over the entire range of invariant mass x and to fit i t , using equation 1.13 and the standard expansion of F to extract a value for a. In order to see the small effect of a, one needs high statistics over a l l x values, especially at the high end of the spectrum where the effect is the largest. However, because of the logarithmic scale of the invariant mass spectrum (figure 1.5), a huge preponderance of low x (useless for the measurement of a) events w i l l swamp the detector, unless some method of biasing the detector for larger invariant mass is implemented. Note that equation 1.13 assumes a normalizat ion to the process n° —> 77. If this process is not simultaneously monitored and counted (effectively counting the number of neutral pions created), then one must introduce a normalizat ion factor into the fit. In order to see the ful l range of invariant mass (which means a fu l l range of 3-momentum and opening angle <p, according to equation 1.15) a magnetic spec-trometer is usually employed. Th is device comprises some sort of detector capable of posit ion measurement surrounding, over as wide an angular range as possible, the 7T° source. A magnetic field bends the electrons as they traverse the detector, thus allowing for momentum determination. In order to cut out the low-x pairs, some sort of m i n i m u m opening angle requirement is usually made. If a normal -izat ion is to be measured, some way of counting either the 7r° product ion, or the incoming ir~ f lux must be installed. 2. A n alternative strategy is to perform an "integrated measurement" by choosing a specific value (or a small range of values) of the invariant mass and to measure the rate for 7r° — > e+e~j in this range, compared to the rate of 7r° —* 77. For such a measurement, one must have a way of counting the number of TT° —• 77 events (or equivalently, the number of neutral pions produced). The advantage of this strategy is that fewer events are needed since one is concentrating on a specific range of x; however, background identification is usually difficult since the kinematical information available is l imited to such a smal l range. T w o - a r m experiments are suited for this type of measurement, the typical ar-rangement being two energy-sensitive devices ( N a l crystals, for example) at some distance from and defining some angle about a target. There is no magnetic field; knowledge of the energy and the angle is enough to specify the invariant mass of the e+e~ pair (equation 1.15). A beam counter (counts the incoming 7r~) or a N a l crystal (counts the decay 7r° —+ 77) provides a normal izat ion. We now present a short description of the previous experiments performed to mea-sure a, together w i th a brief review of their salient features. The i r final results are tabulated i n table 2.2 and presented graphically i n figure 2.3. 1. N. Samios et ai, 1961 [16] : an early magnetic spectrometer experiment done at the Nevis cyclotron at Co lumbia University (New Y o r k ) , using a 6 0 - M e V ir~ beam which was slowed down by polyethylene absorbers to stop i n a l iquid hydrogen bubble chamber, surrounded by a magnetic field of first 5.5, then 8.8 k G . The magnetic field was uniform to 4% and controlled to ± 2 % . Electron track momenta were measured from photographic plates. Us ing a sample of 3071 Dal i t z decays, the authors extracted a value for the form factor slope by comparing the ratio of the number of events seen in r < x < 0.1 to the number seen i n 0.1 < x < 1.0 w i th the ratio obtained using the calculations of K r o l l and W a d a and Joseph. They obtained a = —0.24 ± 0.16. Radiat ive corrections to the process 7r° —• e+e~/y were not included i n the analysis, nor was possible background from the decay 7T° — > e+e~e+e~. 2. H. Kobrak et al., 1961 [17] : a similar experiment to the one above; a 68 M e V pion beam was brought to rest in a l iquid hydrogen bubble chamber, surrounded by a 24.7 k G field, uniform to 0.5%. The energy resolution was approximately 2.4%, roughly a factor of three better than i n the experiment of Samios et al.. A form factor measurement was done by fitting the data in x w i th Joseph's theory, using a sample of 7676 Dal i t z decays, resulting i n a = —0.15 ± 0.10. Background from 7T° — > e+e~e+e~ was not included, nor were radiative corrections. 3. S. Devons et al., 1969 [18] : a two arm experiment performed using a 150 M e V pion beam at the Nevis cyclotron. Each arm consisted of a pair of spark chambers mounted i n front of a sodium iodide crystal , which functioned as total energy absorption spectrometers. The two arms, defining an opening angle of 120°, surrounded a l iqu id hydrogen target. A water Cerenkov detector rejected beam electrons, and a scinti l lation counter behind it counted the incoming ir~ beam. Facing the two sodium iodide arms was a lead plate gamma conversion spark chamber, to check for correlations between electron events and photon events. After geometrical cuts, a sample of 2200 Dal i tz decays was obtained (997 of which had converted photons associated w i th them). The energy and angle information was converted to invariant mass and a value of the form factor slope was extracted i n two ways: firstly, by f i tt ing the invariant mass spectrum w i t h Joseph's theory a value of a = —0.10 ± 0.09 ± 0.13 (the first error is statist ical , the second, systematic) was obtained; and secondly, by measuring the total rate of the Dal i t z decay a value of a = 0 . 1 1 ± 0 . 0 7 ± 0 . 1 2 was measured. The authors combined these two results and obtained a final result a = 0.01 ± 0.11. 4. J. Burger, R. Garland et al, 1972 [19] : yet another experiment performed at the Nevis cyclotron, using a magnetic spectrometer consisting of 3 sets of acoustic spark chambers surrounding two sides and the bottom of a l iqu id hydrogen target to measure the electron momentum. The target and chambers were immersed i n a uni form 3 k G magnetic field, calibrated to 0.1%. T h e authors quoted a momentum resolution of 2.7% at 70 M e V / c . A sodium iodide crystal monitored 7T° —> 77 events, providing a normalization. The authors extracted a form factor a = 0.02 ± 0.10 by a fit of 2437 events over the range 0.0 < x < 0.8. Radiat ive corrections, calculated according to L a u t r u p and S m i t h [20], were included in the analysis. 5. / . Fischer et al., 1978 [21], [23] : instead of using the reaction K~p —* 7r°n, this experiment employed the decay K+ —+ 7r + 7r° as the 7r° source. The C E R N P S provided a 2.8 G e V kaon beam. The kaons decayed i n flight i n the apparatus, which consisted of a 4 m long decay region fitted w i th posit ion sensitive propor-t ional chambers i n the middle and at both ends. The decay products were bent magnetically through a set of Cerenkov counters, scinti l lators, and spark cham-bers, enabling momentum determination. The magnetic field was calibrated to center the TT° mass to 0.3 M e V / c 2 . A total of 31,458 IT0-—* e +e"7 events were collected, and the ful l x spectrum fit to obtain a = 0.10 ± 0.03. T h e authors included the radiative corrections as calculated by Mikae l ian and S m i t h [22]; their effect is to increase the value of a by 0.05. N o systematic error analysis was performed. T h e authors do not take into account any possible contamination due to 7T° —* e + e~e + e~ . 6. P. Gumplinger et al., 1987 [24]: a two a r m experiment performed at the T R I U M F cyclotron (Vancouver, Canada) , w i th two large sodium iodide crystals preceded by sets of three wire chambers and scintillators defining an opening angle of 60°, then 130°, and finally 156° about a l iquid hydrogen target. A 90 M e V / c ir~ beam was degraded to slow down in the target, and resulting e + e~ pairs were stopped i n the N a l crystals. The wire chamber and scintil lator information allowed for track traceback to the target. A smaller N a l crystal , faced w i t h a coll imator and charged-particle-veto counters sat off to the side, monitor ing 7r° —• 77 events from the target for normalization. 10,402 events from the 60° data set, most of which consisted of ir~p —> ne+e~ events, were used to check the normal izat ion of the simulation. T h e 130° sample, containing 11,736 events of which about 10,000 were 7f° —• e +e~7 events, were used to find a = — COll^o®. T h e 156° data were not used due to unforeseen levels of TT° —> 77 contamination and vertex reconstruction problems. T h e analysis included a thorough evaluation of backgrounds as well as radiative corrections (as formulated by Roberts and S m i t h [25]). 7. H. Fonvieille et al., 1989 [26]: a magnetic spectrometer consisting of two arms, each comprising a magnet surrounded on both sides by a series of drift chambers and scinti l lators, defined an angle of 110° out a l iqu id hydrogen target. The magnetic field permitted the momentum determination of e+e~ pairs over an angular range of 50° < <j> < 160°. The authors state a momentum resolution of 3.5% and a vertex resolution of roughly 9 m m . The magnetic field was calibrated to 0.1%. To eliminate background, an explicit cut of x < 0.5 was made. T w o separate runs, comprising 18,346 and 18,353 7r° —• e + e~7 , yielded a = —0.021 ± 0.036 ± 0.056 and a = - 0 . 2 0 5 ± 0.032 ± 0.050, respectively, for a combined result of a = —0.11 ± 0.03 ± 0.08 (the first error is statist ical , the second systematic). Radiat ive corrections were included, using an approximate method formulated by the authors. Background calculations were also performed. 8. CELLO Collaboration, 1991 [27]: this was a measurement performed at the D E S Y e+e~ collider i n Hamburg , West Germany. Us ing the process pictured i n fig-ure 1.4c (production of a neutral pion by 2 photons, one of which is almost real) they were able to measure the form factor over a very large range of momentum transfer (from 0.5 G e V 2 to 2 G e V 2 ) . The signature of .a pion product ion event is that one of the beam electrons wi l l be scattered into the endcap calorimeter close to the beam direction by emission of a heavy v i r tua l photon (the other beam electron is hardly affected), together wi th two clean photons i n the barrel calorimeter surrounding the entire detector from the subsequent decay ir° —> 77. Using 137 events and fixing the pion lifetime, they obtain a = 0.0326 ± 0.0026 (errors combined statistical and systematic). W h i l e the C E L L O result is of high accuracy and covers a wide range of momentum transfer, it is a measurement for spacelike momentum transfer only. It is of interest to determine the functional form of the n° form factor over the entire range of q2; i n the case of the charged pion, for example, much effort has gone into creating a consistent picture of the form factor for both negative and positive q2 ([28] and the references therein). I 8 6 4 % 2 o s - 2 - 4 H s ° fi «4-i — O q - s S - i o •*-» - 1 2 - 1 4 - 1 6 I 1 A 7T9 A rj O 7/ 1960 1965 1970 1975 1980 y e a r 1985 T ion 1990 1995 Figure 2.3: Exper imenta l results for the form factor slopes of heavier mesons i n the region of t imelike momentum transfer. The dotted line shows the range of V M D expec-tations. T h e experimental results for the TT° form factor are also shown for comparison. T h e recent C E L L O results (clustered around 1991) are a l l measured for spacelike mo-mentum transfer. T h e vertical scale is the form factor divided by the square of the mass of the decaying meson, so that the results for different mesons may be compared. authors # events results N.Samios et a l . , 1 9 6 1 3 0 7 1 a = - 0 . 2 4 ± 0 . 1 6 H . K o b r a k et a l . , 1 9 6 1 7 6 7 6 a = - 0 . 1 5 ± 0 . 1 0 S.Devons et a l . , 1 9 6 9 2 2 0 0 a = + 0 . 0 1 ± 0 . 1 1 J .Burger et a l . , 1 9 7 5 2 4 3 7 a = + 0 . 0 2 ± 0 . 1 0 J.Fischer et a l . , 1 9 7 8 3 1 , 4 5 8 a = + 0 . 1 0 + 0 . 0 3 P.Gumpl inger et a l . , 1 9 8 7 1 0 , 0 0 9 a = - 0 . 0 1 I Q . 0 6 H.Fonviei l le et a l . , 1 9 8 9 3 6 , 6 9 9 a = - 0 . 1 1 ± 0 . 0 3 ± 0 . 0 8 C E L L O Col laborat ion, 1 9 9 1 1 3 7 a = 0 . 0 3 2 6 ± 0 . 0 0 2 6 Table 2 . 2 : Summary of previous experiments to measure the form factor for the decay 7r° — > e+e~j. See also figure 2 . 3 . In order to measure accurately an effect as small as the form factor slope i n the timelike region of momentum transfer, one needs a detection system offering high mo-mentum/energy resolution and very good energy cal ibration. A momentum uncertainty of 1 5 0 keV on 1 0 0 M e V electrons leads to an uncertainty of approximately 0 . 0 3 i n a. In order to achieve this k i n d of cal ibration, i f a magnetic field is to be used, one needs to know it to 0 . 1 % at least. A lso , since electrons lose approximately 3 0 0 keV per cm of l iqu id hydrogen traversed, it is desirable to know where i n the target, to 0 . 5 cm or less, the event originated. Accurate cal ibration of the magnetic field and determination of the track length i n hydrogen both require high statistics. T h e earliest experiments to measure a were done wi th very low numbers of events, resulting i n high statistical errors as well as high systematic errors. These experi-ments were are also done without taking into consideration the radiative corrections on the 7T° —• e +e~7 process, corrections which are highly geometry-dependent and may be large. The first two experiments that d id include the contributions due to these second-order corrections were those of Burger et a l . and Fischer et a l . However, the early theoretical work done on the radiative corrections was not directly applicable to experiments, as the calculations d id not take into account the fact that experiments have l imi ted acceptance and do not detect a l l events w i t h equal probabil ity. Hence a l l the early results, up to and including the first high-statistics experiment performed by Fischer et a l . , are i n doubt. T h e more recent theoretical work of Roberts and S m i t h [25], connected w i th the Gumpl inger experiment, and the approximate methods formulated by Fonvieille et a l . do take into account experimental acceptance. B o t h experiments measure a negative slope for the form factor, contrary to theoretical expectations. B o t h experiments, however, have a rather poor vertex resolution, so that the systematic error on both experiments is large. The situation in the timelike region of momentum transfer remains unclear. Chapter 3 Experimental Setup 3.1 Overview — General Principles T h e form factor predictions and results i n table 2.1 are scaled by the square of the mass of the decaying meson. In order to arrive at the more famil iar unitless form factor discussed in chapter 1, we mult ip ly the quoted a by the mass squared of the decaying meson (in GeV), and we immediately see that the form factor influence on the invariant mass spectrum of the lepton pair increases wi th the mass of the decaying meson. One expects the lowest effect for the low-mass 7r°. For the decay 7r° —> e+e~7, the theoretical expectations outl ined i n the last chapter indicate that the form factor should lead to an increase i n the number of events w i th increasing invariant mass x. However, one expects a « 0.03, so that it cannot change the part ia l decay rate by more than 6%, even at the highest allowed invariant mass x — 1 (see figure 1.5). For this reason alone, any measurement of the 7r° form factor requires large numbers of events. Further , since the decay products are an e + e " pair , radiative corrections (higher-order Feynman diagrams involving extra internally or externally radiated photons) play an important role. These corrections also increase w i th invariant mass, going as l n ( ^ p ^ ) , as w i l l be discussed i n chapter 7. In addit ion, there are many possible sources of background, inc luding e + e~ pairs from 7r° —• 77 photon conversions i n the detector mater ia l . In designing an experiment to measure the 7r° form factor, it is essential to maximize the sensitivity to a by ensuring that the detected events cover the ful l range of invariant mass. One attempts to minimize the contributions from background processes by bu i ld -ing a low-mass detection system; however, a ful l s imulation is st i l l required to assess a l l the background to the e+e~j sample. One further needs a clear understanding of the radiative corrections and of a l l systematic errors. The accuracy of the measurement hinges on the momentum/energy resolution of the detector. It is necessary to know to high precision the acceptance of the detector. This requires a thorough simulation of the experimental setup. 3.1.1 T h e 7T° Source In order to measure the 7r° form factor, an intense n° source is necessary. Meson factories produce high-intensity and K beams by i l luminat ing solid targets w i th high energy proton beams, and magnetically separating the spray of resulting decay products into meson beams. These factories are thus ideal for high-statistics measurements. T w o methods of obtaining the n° source have been used: 1. the reaction 7r _ p —• 7r°n at rest. Th i s reaction produces roughly 6 neutral pions for every 10 ir~ incident on the target. 2. the reaction K+ — + TT+TT°. The kaon is allowed to decay i n flight; 2 1 % of the t ime, it w i l l result i n a 7r°. M e t h o d 1 has the following advantages over method 2: • approximately 3 times larger yield of n° per beam particle. • much higher beam flux is attainable. • much lower final state 7r° momentum, resulting i n e + e~ pairs w i t h larger opening angles, making track identification much easier and more accurate. • lower reaction energy, resulting in far fewer photon conversions (background). • no measurement of the incoming beam momentum is necessary to reconstruct the event. M e t h o d 1 has these disadvantages: • a target is required, leading to multiple scattering and potential photon conversion background. • larger amount of background from 7r~p —> ne+e~ than from the corresponding process K+ — » 7 r + e + e~ . If a smal l target is used, mult iple scattering and the number of photon conversions i n the target w i l l be small . Also , the potential n~p —•* ne+e~ ( " internal conversion") background has different kinematical l imits and can therefore be discriminated against to a large degree by a smart trigger and by offline cuts. We cannot el iminate it com-pletely, however, and therefore it must be simulated, as w i l l be discussed i n chapter 7. Th is is not a problem, since n~p —• ne+e~ has been calculated (at rest) and mea-sured experimentally, so that it is well understood. M e t h o d 1, then, w i t h its high 7r° flux and the resulting e + e~ pairs wi th high opening angle resulting i n accurate track reconstruction, is a good choice. 3.2 Experimental Setup The form factor measurement is the last i n a series of rare decay experiments performed w i t h the S I N D R U M I spectrometer at the P a u l Scherrer Institi it (PSI ) , Switzer land, throughout the eighties. The detector has been described i n detail elsewhere [29]- [33]; we wi l l provide only a brief description of the general design and function. S I N D R U M I is a magnetic spectrometer consisting of 5 concentric cyl indrical wire chambers sur-rounded by a scinti l lator hodoscope, mounted inside a magnet solenoid. In figure 3.1 we show a detailed drawing of the entire device. The spectrometer was designed for de-tecting rare decays involving electrons i n the final state, so that low detector mass was desirable to l imit photon conversion background: a particle traversing a l l five cham-bers radial ly encounters only 5.4 • 1 0 - 3 radiat ion lengths. In order to have as high an efficiency as possible for rare decay detection, S I N D R U M I also has high solid angle coverage: 70% of Air. 3.2.1 B e a m and Target The 7rE3 beam at P S I provided approximately 1 • 10 5 95 M e V / c TT~ per second, at a (low) pr imary proton current of roughly 1/zA (maximum 250 fiA). The ir~ flux that can be stopped at this energy i n a depth 1.5 cm of l iquid hydrogen, given that only 50% of the beam spot strikes the small target (radius 1.9 cm, length 12 cm), is roughly 1 • 10 4 , leading to a 7r° rate of roughly 7000 per second (1 7r° every 140 /is) . Th i s resulted i n about 70 Da l i t z pairs per second i n the detector, which was as much as the data acquisition system could handle. A system of four quadrupole magnets focussed the beam onto a smal l lead cone (the "moderator") mounted in front of a small l iquid hydrogen target, as shown in figure 3.2. The purpose of the moderator was to slow the TT~ beam so that the particles would stop i n the first 5 cm of the target. Since the beam diverges rapid ly due to mult ip le scattering after passing through the lead, it was necessary to place the moderator as closely as possible to the target. For this reason, the moderator cone was integrated into the target design and d id double duty as a vacuum window. Figure 3.2: Deta i l of the target. Also shown are the lead moderator and the innermost wire chamber. T h e superinsulation around the vacuum cylinder is not shown. T h e target itself consisted of a mylar cylinder (19 m m radius) of 0.12 m m wal l thick-ness, w i t h a spherical end. Surrounding the target was a 25 m m radius M a k r o l o n vac-u u m cylinder (0.7 m m wall thickness), wi th the moderator functioning as the vacuum window. Several layers of superinsulation were wrapped around the vacuum cylinder to prevent ice bui ldup on its outer surface. Particles crossing the target and vacuum cylinder i n the radia l direction would traverse approximately 4 • 1 0 - 3 radiat ion lengths of material . T h e entire target system could be wheeled i n and out of the detector, en-abl ing accurate repositioning between runs. The vacuum pressure and l iqu id hydrogen level were controlled and monitored by microcomputer throughout the experiment. 3.2.2 S I N D R U M I Spectrometer Surrounding the target were the 5 mult iwire proportional chambers, which allowed for the reconstruction of the tracks of the particles passing through the detector. We summarize the attributes of the chambers in the table below. E a c h chamber consisted chamber radius (cm) length (cm) # wires wire spacing(mm) 1 3.72 9.0 224 1 2 6.40 20.0 192 2 3 19.2 58.0 512 2 4 25.6 69.0 768 2 5 32.0 80.0 1024 2 Table 3.1: W i r e chamber specifications of 2 concentric Rohacel l cylinders faced w i th K a p t o n , on the facing sides of which were evaporated t h i n layers of a luminum which functioned as inner and outer cathode planes. T h e anode wires were strung between the 2 cylinders. The wire ends were fastened to fiberglass pr inted circuit rings (containing the necessary electronics) at either end of the chamber. Chambers 2, 3 and 5 had their inner and outer cathode planes etched into strips running at +45° and —45°, respectively. The space between the R o h a c e l l / K a p t o n cylinders was filled w i th chamber gas (argon-ethane-freon mix ) . A charged particle traversing the chamber would ionize the chamber gas, and the resulting free electrons would be accelerated towards the anode i n the electric field between the cathode and anode strips. Very close to the anode, where the field was intense, secondary ionization would result i n an avalanche of electric charges, causing a signal both on the anode wire (directly) and on the nearby cathodes (by induction) . These signals were fed into a series of P C O S - I I I amplifiers, discriminators, and receivers, which processed them into a d ig i ta l readout containing the hit wires' cluster midpoint and size. W i t h the knowledge of the location of each wire, the wire numbers of the anode cluster then gave directly the x — y position of the hi t . The angled cathode strips of chambers 2, 3 and 5 enabled, i n addit ion, the determination of the z position of the hi t . T h e spatial resolution of the <f> measurement was given by the 2 m m wire spacing (CT ~ 0.6 m m ) , and the ^-resolution was determined using cosmic rays to be a ~ 0.3 m m . In addit ion to the cluster midpoint and size, the receivers generated F A S T O R signals and L A T C H E D O R signals. For both signals, chamber wires were grouped into sectors which consisted of logically O R - e d neighbouring wires. For the F A S T O R signal, only these O R signals were output. The L A T C H E D O R further required a coincidence w i t h a gate signal. The 512 wires of chamber 3 were grouped into 16 sectors of 32 neighbouring wires which were F A S T O R - e d , resulting i n a 16-bit number point ing to an address i n the Memory Lookup Uni t ( M L U ) microprocessor. T h e contents of the 2 1 6 possible addresses were 0 or 1, depending on whether the corresponding hit pattern was to be accepted or not. In this way, it was possible to test very quickly for two sectors w i t h a m i n i m u m opening angle at chamber 3. Similar ly , the L A T C H E D O R output from chambers 2-5 were fed into the Track Preselector ( T P S ) microprocessor, which compared the signal wi th stored patterns (masks) of acceptable tracks. Th i s enabled a fast track recognition i n the <j> plane. The masks were generated using simulated e + e~ pairs. A more detailed description of the operation of the M L U and the T P S can be found i n reference [34]. Mounted on the outside chamber was a cyl indrical hodoscope of 64 scinti l lator strips, each 88 cm long, 1 cm thick, and 3.3 cm wide. The ends of each strip were equipped wi th photomult ipl ier tubes, the output of which was fed into 64 "discriminator-meantimers" . These devices provided signals correlated to the time of the track's traverse of the scinti l lator. T h e 64 hodoscope time signals were subsequently fed into electronics which allowed for the selection of events wi th a given number of hodoscope hits (at least 2 separate clusters) w i t h i n a time period of 12 ns. T h e chambers and hodoscope were mounted inside an iron solenoid magnet, which provided a uni form magnetic field parallel to the beam axis (and the anode wires) of 0.33 T . Charged particles thus described helical paths inside the detector. The magnet current was monitored and recorded by microcomputer throughout the experiment. 3.2.3 Trigger Logic The scinti l lator hodoscope had a very fast response t ime, and was used to define the start of an event. Once two hodoscope hits w i th in 12 ns were detected, the online computer (a P D P 11/44) began to check the rest of the (slower) electronics. T h e M L U had to show two hit sectors in chamber 3 w i th an opening angle of more than 67.5°. T h e n at least two hits i n chamber 1 were required. The T P S selected events w i t h at least one negative and one positive track. Upon passing a l l these requirements, the event was passed to the General Purpose Master ( G P M ) , which used both the T P S and M L U information to apply a m i n i m u m <f> opening angle cut of approximately 35°. Once the event had passed this trigger stage, the data was writ ten into an event buffer on the P D P and passed to the online filter for further processing. 3.2.4 Online Fi l ter B y using the ful l wire hit information (in contrast to the trigger's use of only the L A T C H E D and F A S T O R ' s ) from al l 5 chambers and the hodoscope, the filter per-formed a more detailed track reconstruction i n the r — <f> plane (as described i n [35]) and calculated the distance of closest approach ( D C A ) of the track to the detector axis as well as its <f> emission angle at the D C A and transverse momentum pt. The emission angle was measured i n the counterclockwise direction from the positive track to the negative one. T h e filter then applied more stringent cuts, requiring at least two tracks of opposite polarity w i t h the following characteristics: 1. | D C A + , _ | < 25 m m 2. | D C A + + D C A _ | < 12 m m (the D C A was negative if the axis was inside the curve of the track) 3. 35° <<j>- -<f>+ < 260° 4. - 4 . 0 n s < < + - * _ < 1.6ns The first requirement rejected tracks which do not pass through the target, while the second ensured that a pair of correlated tracks existed. Smal l opening angle pairs were rejected, removing background and biasing the Dal i t z sample towards higher invariant mass. A further hodoscope t iming cut el iminated a l l but the very prompt tracks. 3.3 D a t a Acquisit ion D a t a for the form factor measurement were taken at P S I dur ing an dual-purpose ex-perimental r u n lasting from the end of A p r i l to the end of October 1987, w i t h a one-month cyclotron maintenance break. The long r u n period was necessary to measure the branching ratio of the rare decay 7r° —• e + e~, the results of which have been published [36]. D u r i n g this r u n , the magnetic field was set to 0.33 T and the trigger conditions set as outl ined above, and Dal i t z data were taken on four separate occasions, each time for a day or less. The detector was taken apart between these runs for repairs to the wire chambers, resulting i n four distinct data sets (labelled "geometries 2, 4, 5, and 6"), comprising a total of approximately 0.8 x 10 6 events. These raw data were subsequently passed through an offline track recognition pro-gram to reconstruct the event kinematics precisely, which we w i l l describe i n the next chapter. Chapter 4 Offline Analysis 4.1 Overview The offline analysis proceeded i n two phases: first, the raw data was read from tape and a pattern recognition program (which incorporated detailed detector cal ibration infor-mation) translated the wire and hodoscope hit information into particle 3-momentum and event vertex location; second, the final analysis looped over a l l tracks i n each event and selected the "best" pair , then applied final hard cuts (duplicating a l l cuts that went before, inc luding those made by the online system) to these tracks and finally allowed for a detailed examination of the resulting kinematical distributions. We give a brief description of each of these elements; more details (especially concerning the cal ibrat ion and pattern recognition software) can be found i n [29] - [33]. 4.2 Detector Calibration In order to translate the wire hit information into x — y — z track coordinates precise knowledge of the wire locations is necessary. It was discovered, for example, that when the fiberglass anode wire prints at each end of the chambers were glued to form rings, the resulting space at the seam was not the same as the wire spacing. A lso , the seams at either end of the chambers were rotated wi th respect to one another, inducing a slight twist i n the anode wires. Corrections for the anode "gap" and " twis t " had to be made for each of the five chambers i n order to produce the correct x,y information. The inner and outer cathode planes of chambers 2, 3 and 5 were also rotated slightly w i t h respect to one another. In addit ion, minor variations occurred dur ing the r u n period, when, on four separate occasions, the spectrometer was turned off and the inner chambers removed for repairs. U p o n their reinsertion, the chambers were slightly rotated and offset w i t h respect to one another. The relative rotations and locations of the chambers were calibrated using cosmic ray data w i t h no magnetic field (straight, throughgoing tracks). T h e following quanti -ties were determined (i = 1 ,2 ,3 ,4 ,5 ; j = 1,2,3,4; k = 2 ,3 ,5 ; / = 2,3) : • ^anodes '• * n e rotation of the jth chamber relative to chamber 5 • ^cathodes '• *he relative rotation between the inner and outer cathodes, for those chambers w i th z information • A i J , A y J : the x and y offsets of the midpoint of each chamber (relative to chamber 5) • Azl : the offset i n midpoint z relative to chamber 5 • aup down '• rotation of the anode prints at the up- and downstream end of each chamber • s'up down '• seam gap of the anode prints at the up- and downstream ends of each chamber T y p i c a l x,y offsets were on the order of 0.5 m m . The largest z offsets were 1 m m , for chamber 3; those for chamber 2 were on the order of 0.5 m m . The largest rotation of 2.4 degrees (or 1.5 wires) was found for chamber 1; a l l other rotations were less than a single wire. T h e anode seam gaps were less than 0.5 m m (different from the wire spacing), so that the twist produced in the anode over the length of the chamber was very slight. The hodoscope time signals were also calibrated (for a more detailed description of the cal ibration procedure, see reference [33]); the digital output from the discriminators had to be translated into time signals in ns and corrected for z posit ion- and amplitude-dependence. After cal ibration, a time resolution o = 315 ps per hodoscope scintil lator strip was achieved. 4.3 Pattern Recognition and Track Fi t t ing W i t h the cal ibrat ion information i n hand, accurate translation of the wire and ho-doscope hits into x,y,z and time co-ordinates was possible. The pattern recognition software fit helical tracks to this translated information. The pattern recognition and track fitting proceeded in three steps: first, wire hits were fit w i th circles i n the r — <f> plane; second, these results were combined w i th the z hit information and data fit w i t h straight lines i n the the arc length 5 versus z plane. F ina l ly , the vertex of every pair of -|— tracks was determined by extrapolating the fit tracks back into the target. A detailed description of the track fitting procedure may be found i n [37,38]; a brief description follows. 4.3.1 r - <f> F i t T h e charged particles described circular tracks in the <j> plane i n the homogeneous magnetic field, the radius of which was proportional to the transverse momentum. T h e ioniz ing particle rarely caused more than 1 wire to give a signal. T h e pattern recognition algori thm first verified that the candidate track has hits i n a l l 5 chambers as well as i n the hodoscope. It then grouped hits i n the first 3 chambers which fell w i t h i n a certain distance window into " tr ip les" . The same was done for the outer 3 chambers. Subsequently, al l triples sharing a common hit i n chamber 3 were fit w i th circles; chamber 1 hits were weighted by a factor of 4 since the wire spacing was half that of the other chambers. The resulting fit was required to have an acceptable x 2 -Those tracks for which the real chamber 1 hit differed from the circle fit projection by more than 0.5 m m were rejected, as were tracks w i t h a fit D C A of more than 25 m m . 4.3.2 z F i t The ratio of the transverse to the longitudinal momentum was constant for a charged particle traversing a magnetic field uniform in z. Therefore, i n the arc length s versus z plane, the particle track was a straight line. Once the r — <f> fit was complete and the track fit w i t h a circle, the arc length could be determined and the s — z analysis performed. The determination of the 9 angle of emission, combined wi th the results of the transverse momentum Pt from the r — <f> fit, allowed for the calculation of Pz. A t least two z hits were required for a candidate track. Each anode signal induced signals on a few neighbouring cathodes, producing smeared out cathode "clusters". A l l such clusters, on both the inner and outer cathode planes, were sought, and their centres determined. Dead and damaged cathode strips were interpolated over using the information from neighbouring strips. A l l inner and outer cathode clusters were combined into pairs, each pair defining an angle <j> which was compared to a l l anode (j> co-ordinates obtained from the wire hit information. M a t c h i n g <f> co-ordinates resulted i n the assignment of r — <f> — z coordinates to the hit . Once the track co-ordinates had been established, a straight line fit in s — z was performed. 4.3.3 Vertex Fi t T h e final step i n the event reconstruction was the determination of the event vertex. The point midway along the shortest line between the two tracks was chosen to be the event vertex. Figure 4.1 shows the resulting distr ibut ion of event vertices for e + e~ pairs i n r — z. The target is clearly visible. A lso visible are the lead moderator i n front of the target, the a luminum support r ing at (z « —110 m m , r J=S 20 m m ) , the a luminum target mounting at (z > —80 m m ) , and chamber 1 at (r « 35) m m . These structures are visible due to the large numbers of photons from 7r° decays converting into e + e~ pairs. T h e results of the track fit were written to a file, along w i th the raw data for each event. 4.4 F i n a l Event Selection — Identification of Dalitz Events The final analysis program reads the results of the reconstruction program from the file created and loops over a l l pairs of tracks wi th opposite sign, requiring that the event vertex of each combination is inside the target, and chooses the "best pa i r " on the basis of the x 2 of the vertex fit. Th i s is not a stringent test, since at this stage of the analysis, most of the events have only a single pair of proper tracks. T h e final chosen event must meet the following requirements, which duplicate (and are more stringent than) the cuts applied by the online trigger and filter and the track fitting program: • 45° < <f> < 260° : a cut which duplicates the action of the online trigger and filter at roughly 35° < <j> < 260°. • Pt > 20 M e V / c : S I N D R U M ' s transverse momentum threshold is roughly 17 M e V / c , and this requirement ensures that any systematic error w i l l be due to the 40 30 H " T I w O OH X CD > 20 10 H 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 •eotooO*aeOeiQoa ooiD>oiooa • t • • i • • • « I • 0 • • • I B a B D • • • • 0 Q • • • 0 • • • 0 • 0 o a • < • • OQDQ a •• • •• •••00 0 ">o. • • • • • • • « • ••lDODDO • ) DOCLXIDu* • L r-i n f I" I I n i - i — • • i • • • D D • • • • •• • •• » o D00| »o ••! • ooDD[TJTjDDa • •• a o POO no . • • o o • a OD oo o • • a o • DOD»•• H D D O • o • •a o a o • • 0 0 0 n 0 • 0 • oOooQQoaaoaoaorjo a •••••••[][][]• a a • DooQoQQoooiDoa ODD••••••••••DO•a . . . n . -OBOODDDO • • B • 0 1 0 0 • • • a • • • a a • o • • a I OIIDIBII D • B l gID io • B I OOG < I 0 1 B B OOODDD • • • ••ODODOooo • ODDQDoQo • • a•••••o o• a••••••a a • a•••••aa o• • O D D Ooo* a o JTTlDo • o H •o ODD • •DDDCDDo o o •o D•••••o o . • • • o • o • 0 • • • 0 B 0 < • I B • • o o a a 0 • Q Q l l • 0 • 0 « Q D • • 0 • • 0 • D O D B i a o • • • o • a oDo DO•o B i O D D i o o a DO o o o o • • • • o • ooaDQDQoaoooD>aoooOODQDiiDOo o•OOODOODOODOOODOQODDODOO0 o a • •ooODDODODQDDDDooOQQOOODDQooo o o.ioo o••••••••••••••••••••QODQD o a o ao o•••••••••••••••••••••••••O o•• •••••••••••••••••••OODDDDDDODDoo. • • D O D D D D D D D D D D D D D D D D D D D D D D D D O O O • • • • D • • • • • • • • • • • • • • • • • • • • • • • • • • • 0 ° • • 0 ••••fflmpprjDDDDDDDDDDDDDDDDoaao o •••ULIJULIJLLJDDDDDOOOOOOOOOOODODO. • oonTTTJDDDDDDDDDDDDDDQaDDQDao • o o "••CILTIDIDDDIIDDDDDDDQDDODDO • o.. 8 OOPPQDCLTIDCXJDDDDODDODD•••••• o • o • •DlinnDDDDDDDDDnDDDDDOODDQDOooii • o••••••••••••••O•••••••aODDDDa oo oDDDDDlIlDDDnDDDDDDDDODDDaooDDo. • o " • • • • • • • • • • • • • • • D D O D D O D D D O D D D . o . DDIXDDDDDDODDnODDDDDQOoaDQDi • o . o • • • • • • • • • • • • • • • • • • • • O D D • • • D • oi DODDODDDDDDODDDDOQDODDo o o•a • •••••••••••••••a a ao•o•o••ao •• O O O D D D Q D D D D D D D D « 0 ' D « • • • D • 0 0 • • 0 • « • « 0 I I I I I m T f I I I MT7T IT I TTl I i IT ITI I I I I I I I ITI I I I I l I I I -150 -130 V 110 -90 -70 -50 Z vertex position (mm) t i t • 0 D Figure 4.1: D is t r ibut ion i n r — z of the distance of m i n i m u m approach for e + e~ pairs. T h e target, moderator, a luminum support r ing , target support structure, and cham-ber 1 are clearly visible. cut rather than to our understanding of the geometry of the detector. • —300 m m < Z(5) < 300 m m : the particle track was required to lie well w i th in the region of uni form magnetic field. B y requiring the chamber 5 z hit to be well away from the edges, we define a conical fiducial volume w i t h i n which the magnetic field is uniform. • r < 19 m m , - 1 1 5 m m < z < - 8 0 m m , 0 m m < z + 104 + y/192 - r 2 : cuts which ensure that the event happened well inside the target. The quantity 104 + \ / l 9 2 — r 2 follows the curved upstream edge of the target. After the reconstruction and final analysis cuts above, the four geometries 2, 4, 5 and 6 comprise 9294, 43968, 8899, and 43464 events, respectively. In figures 4.2 through 4.4, we show distributions of some of the kinematical variables of the resulting data. There is background remaining. Figure 4.2 shows the distr ibut ion i n opening angle of the chosen e+e~ pairs. The large, sharp peak at <p « 156° is due to 7r° —» 77 events i n which both the photons convert; the resulting leptons, one from each photon, when boosted into the laboratory frame, exhibit an opening angle of approximately 156°. In chapter 6 we discuss the cuts used to eliminate these events from the data sample. In figure 4.3 is shown the invariant mass of the e + e~ pair (normalized to the 7r° mass) versus the total energy of the pair plus the neutron, Etot — Tn + E+ + E_ = 7|p+ + p_| + ml - mn + E+ + E. The events i n the band at Etot « 130 M e V are 7T~p —> n e + e ~ events; for these 3-body events, the total energy of the event is constant and equal to the in i t i a l 7r~p energy. T h e w i d t h of this band is indicative of the resolution of the detector. The Dal i t z events, i n which the photon carries away energy, populate the slanted band. Events from TT° —* e +e~7 cannot extend past x = 1.0; as x increases, the e +e~ pair carries more and more of the energy until at x = 1, E~, — 0 and the 7r° —• e +e~7 and 7r~p —• n e + e " processes merge on the plot. The ne+e~ events may be separated from the 7r° —• e +e~7 sample to a large degree by the requirement that Etot < 110 MeV, but only at the expense of high invariant mass Dalitz events. This is not so desirable. cp (e+e- opening angle) in degrees Figure 4.2: Distribution in opening angle of e +e~ pairs. The sharp peak at 156° is due to photon conversion events. The broad peak at 110° is due to the asymmetrical <j> opening angle cut of the trigger. Figure 4.4 shows the distribution of transverse opening angle against the quantity "EtPtn, 0 I I I I 1 I I I T | I EtPt = E+ + E. + \P+ + P.\ + Tn 150 130 -u CD C CD "(3 o 110 90 H 70 50 30 i i i i • i i i i i i i i i i i i i i i • ' I ' • on loooooQooaOQOOioooooooaoDDoooaooooooooaoooooiaxiii ooODOQOODOOODOOODODQODODOOODOOQDQOOaoODODOQOiOOOit DDDDDOODDDDDDDDDDDDDDDOQDDDDDDOOOQOOODDQQGDDQOOi ODDOODDDODDDDDDDOOODODDDDDOQQOODODDDDDDOQDDDOi• QOOODDDDODDODDDODDDDDDDDDDDDODODDDDDDDOODDQ"' ODDDDDDODODQDQDDDODDDDDDDDODDDDDDDDGQDDDit• OOODQQaQOOOODQaaDODDDDODODQDoDQQooooOOQi • 0 0 0 0 0 OOOOOOODOOOQOOOOOOOQOODOOOOOOOD" OOOQaODDOOOIIOGOOOlQOQOOOOOOODDODDQ' • ooaaDoaatoiiiDDoaooo-Do o DDDDDDODD • oi-ooioioiiooioiio o o D o DDOOOOQDO• •aiaoaiiooai-••ooooDODDDDDOODOo •aoati- oii•OOQODODDDDDQDDDDB oo oioo o oDOOOOOOOODDOo ••••i.•oioooDDDDDODDDDDDD-• a••o a a aaoDDDDODDDDDDQOo 'o•••••••• o••DDDDDDDDDDDQDDDDDDo ••••••DDI oil 11 M i l ••••••••••a IQQDQ DQODD _ 0 0 0 0 0 0 DDDDDDODD IDDDDODO • •D o••• • • D • • Q i 0 • • • D • . . . I D I I I I [ I I I I [ I I I I | I I I I | I I I I | I I I I 0.0 0.2 0.4 0.6 0.8 1.0 1.2 x =N[s~ / m 7T Figure 4.3: D is t r ibut i on i n i and total energy of e + e " pairs plus the neutron kinetic energy. T h e ne+e~ events populate the horizontal band at 130 M e V . T h e Da l i t z data inhabit the slanted region up to x = 1.0. T h e curving branch at smal l x and low total energy peeling away from the Dal i t z region are events w i th an extra photon which radiates away more energy. CD r—I ££) C c d OX) C c CD O H o CD > CO cd 2 6 0 -2 4 0 • 2 2 0 • 2 0 0 -1 8 0 z 1 6 0 ~_ U 0 t 2 0 -1 0 0 -8 0 -6 0 -4 0 • i i i i i I i i i i i I n i i i I n i n I i i i i i I i i i i i I i « m I « i i i i • • • • • • • • •••TXirjOOrXKXXmi**** a.-aoeooODOooaaaa. •ooDgDODQQQQQOQQoa«««««e«eoooDOOOoooo««« < • ooCKJLLCDXKXIOOaaa«««*« •ceooooooacce* - « • 0 a r X i n i X n X X 3 0 a a o a a « « « « a « o a o 0 0 Q 0 a « « a a « < « • oocnxCDDOOaaoaaa • «• • •••aooaaaaaaaa • • • • a • aorxXDOOarjoaooaaa* • «a•caaaaaaaaaaa o o o O Q O D Q O Q O O O O O Q a e . • a « e o a o a a o « a o « • • • o o o o o o r x D O O C O D o o a o « « « « « o o o o o o o o a o « « « ' « ooorjOQrjDOOOQOOOQOO* •«oODOOOoaaa««a««« PODOOODDrjQoOOQDaoQecooooooooaooooa ooaaiEDDDOCg300 00«<ioooo«oooooo««««« » • • OOODQQDDDOOOOOOOOcooooooooceooaaaa*• •••• OOOOOOOQQOOa30DDOOoOoooooe«ooooo««««« • ooooaOQQrJOQDDOODDOOooooooccooo**** • '•OOOOOCI m i l l D00DDDD0OOD0DOOOOOO««O«««« ••••noon 1 1 1 1 1 1 1 » i n n n D D o « o D o o D o D c c c « a a « « « « • • • •a • •a • •••••••OOOOaQDDQQDOODODDDDOooDoooooot • • • • ••••o«OOOOTQCH10rJODDDOOOoooo«flotIoooo««o« • • • • • • • ••••BiOOOCQiXOXIDDDDDOOOOD«ooODooo«0«oe«oo • oaaooOODOOQQDDDDODOOQDOo oo e o o « o e c o a «a•«• • ••••••oooOOOODaDDDaDDDDOODOooooooaooooo«c«««.«. «••aooQCODDGDODDDDDDD•oooooooPDooooa«« • •BOOOOPIl I I I I 0DflOOOOOODoODOoao*** • •oatoaODff lUDaQDDDQDDDDDDOOooDooooo«o-aa.««-oaDDQQODDQDDDDDDQDooooa«OODoooOO«««a«.i OoaorXDOCODDODDD 0« i ••••• • • • •  • • • • • • • • a l l • • • i a a a • • • • > • • • • i oaO^^OOQDOOOOOo««ODOOOoooo««o«*aa • . • iiqOOPDDDDDDDDDDOncoacooooooocoocBOat • i«ooo_ 3 Q O O O D o r J D o o o « o c a « o o o o o o a a a o « o « • • • • • • COXDOOODODOOQoa«o«caooooaaaaa a a a • • • • •tfflDOOOCPDDDOOOOo««a •aaaaaaaoaaia•a• •oODUDDgDODDOOOOaa••a••aoooaooaoo•••• •oQDQrxDQOQODOaa<i««« • •aoaaaaaaoaaia* • • t D O f J H U g C D D O O D O a a « • « a a a o o o o a o o o o o a a a • • oorjrjrjrjQDogQoooaaa • • •caaccooaaaaaooa • • • Q O d H U X D D D D D O P O • " « « ••ooooaooaoooa• a OJILQXH3DDDDQ «>«««• • eaeeooorjQaoaoaa • • • DtJLOlITrinnriDDaoca •«««acoOD0QQaaa«a •pfi n n]••••••••• c«aa «««««cooanrjQaooa a • g f lIIII EDDDDDDDDo'• « « . « « o a o n n n D r j a o a a a . n n r m m n n n n n n n n n . - • C O D D O D D O D Q 1 1 " aoDODDDOQDOOBi oo 1111 iDqqqqDaDDo«. • • • • • 0 0 OUDDaa•« THC|e«..< ooD DD D oo*• •ea«aeoODDDDDQQaoa«•• •e««ooDDDDDDQQOOO«•• • «-aoooaDODDCE30aao« t coooODDQQOrjrJO • • a ac ••••aooaQOIIOlaaa a . c • vooooatUXDXiaoe a B • a . « • c o o a a a o D T J a a a • • 1 1 1 1 1 1 1 1 i i 1 1 1 1 1 1 1 1 i i M | n i n J n u 1 1 M 1 1 1 1 1 i i i i | 30 60 90 120 150 180 210 240 270 E P (MeV) Figure 4.4: Dis t r ibut ion of e + e~ pairs in the quantity EtPt and transverse opening angle. T h e Da l i t z events are constrained to lie i n the box between 107 and 163 M e V . The slanted bands are the n e + e " events. The region below 107 M e V is inhabited by the radiative events. T h e Da l i t z events inhabit a boxlike region between 107 < EtPt < 163, where the two extremes correspond to the cases where the neutron is emitted parallel and antiparallel to the e+e~ pa ir , respectively. If we apply the further restriction that EtPt < 170, we can eliminate ne+e~ events to a high degree, without the loss of the high invariant mass events. In order to assess the efficacy of these cuts, we must simulate both the 7T° —• e + e~7 and the 7r~p —>• n e + e ~ processes. The background also needs to be simulated. We t u r n now to a discussion of the simulation procedure. Chapter 5 D a t a Simulation In order to extract a parameter of the size of the form factor slope from the data, one must understand the data very well . In order to achieve this understanding, we use the C E R N package G E A N T for simulation of the detector response. A fu l l description of this package and its implementation i n our Monte Car lo s imulation can be found in [39] and [40]; we give here a brief outline of the steps involved. The m a i n steps i n simulating the data are as follows: 1. Choose the detector geometry, the magnetic field value, and the process to be modelled. Make any restrictions on the particle kinematics. 2. Assuming that the Tt~p atom is at rest, the energy available to the subsequent reactions is E = m x - +mp — B(v-P) = 1077.83941 M e V (the b inding energy of the •K~p system is approximately 0.4 keV [41]), assign the 4-momenta of the resultant particles according to the appropriate matr ix element and Lorentz transforma-tions, w i t h i n the kinematical restrictions imposed i n step 1. 3. Decide on the location of the interaction. 4. Pass the particles through the detector, starting from the interaction point and ending when the particles leave the sensitive volume of the detector. M o d e l the product ion of secondary particles such as photons ( due to electron bremsstrahlung) and electrons ( from photon pair production ) and pass these through the detector as well . 5. A s the particles lose energy i n the various detector parts , sensitive and non-sensitive, model the signal produced i n the sensitive ones. 6. Wri te the resultant response into a data file identical i n format to that of the real data. 7. R u n the trigger s imulation programs. Read each event from the data file and either discard or accept i t . We now discuss these steps in more detail , highlighting the most important assump-tions. 5.1 Deciding the Detector Geometry The basic setup of the S I N D R U M spectrometer, as discussed i n the previous chapter and i n reference [33] (and the references contained therein) was assumed to be fixed and conforming to the specifications. The slight chamber misalignments discussed i n chapter 4 were taken into account. However, since the effect of the anode wire print gaps and twists resulted in less than a single wire difference in a l l but the most extreme cases, it was decided not to model them, but to include them only as corrections during the track reconstruction of the data. The posit ion of the target was determined on a run-by-run basis by examining the d istr ibut ion of event vertices, and checking the location of the target edge. A typical d is tr ibut ion is pictured i n figure 4.1. The target was found to be i n the same position (to w i t h i n 2 m m ) for the 3 later geometries; for geometry 1 it was 4 m m further upstream. The magnetic field was set to 3.313 k G throughout the experimental r u n , and the solenoid current monitored at 5 minute intervals dur ing the r u n . Th is value was recorded i n the data file of each accepted event. The measured magnetic field value was estimated to be accurate to about 1% (using a field map produced i n an earlier experiment using the S I N D R U M detector [31]); however, as w i l l be discussed i n chapter 9, we desire an accuracy of less than 0.5%. In order to achieve this, the s imulation was first r u n using the nominal value of 3.313 k G , whereupon the resultant events were used to calibrate the magnetic field for each of the four separate r u n periods. The s imulation was then redone using the corrected field, i n order to account for any changes i n acceptance. The magnetic field was uniform to better than 1% wi th in the chosen fiducial volume inside the chambers. Since no accurate field map was made, we assume a uniform field for s imulat ion purposes, and then apply a fiducial cut during the analysis. 5.2 G e n e r a t i n g t h e P a r t i c l e K i n e m a t i c s Some m i n i m a l requirements were imposed on the in i t ia l lepton kinematics i n order to cut down on the amount of computer time required to track the events. These require-ments were set below the physical detector thresholds, so that the detector, trigger, and filter s imulat ion and subsequent analysis would determine the event acceptance. The cuts imposed were the following: • The i n i t i a l electron momenta should lie w i th in S I N D R U M , and should be large enough to allow the lepton to hit the hodoscope. Th is l imited the s imulation to producing events w i th 27° < 0 < 135° 12 M e V / c < Pt Here 6 is the longitudinal angle of the emitted electron, and Pt is its momentum i n the x — y plane. • Since the trigger and filter applied a cut i n transverse opening angle, we further restricted the s imulation to generate events ly ing w i t h i n the region: 15° <4>t< 260° • The above restrictions on transverse momentum and opening angle result in an effective restriction on the m i n i m u m invariant mass of the e + e~ pair of roughly 0.01. Since the simulation performs a numerical integration of the matr ix element over this variable, we set the addit ional restriction x > 0.001 expl ic it ly i n order to reduce the computer time needed for the calculation. We considered the following reactions: 1. 7T° —> e +e"7 according to the matr ix element set out by K r o l l and W a d a [6], w i t h the form factor slope set to zero. 2. 7T0 —• e +e _77* and 7T° —• e+e~77, the first order radiative corrections to the above process, as calculated by Roberts and Smith.[25]. 3. TT~p — > ne+e~ w i th first order radiative corrections n~p —* ne+e~~f and ir~p —• ne+e~7*, as formulated by Fonvieille et a l . [26]. 4. 7T° —> e + e~e + e~ using the matr ix element derived by M i y a z a k i [42]. 5. 7T° —• e +e~7 where the photon was forced to undergo C o m p t o n scattering or pair product ion i n a specified area of the detector. 6. 7T° —• 77 where one or both of the photons were forced to undergo Compton scattering or pair production. 7. TT~p —• n-f where the 129 M e V photon is forced to undergo pair product ion. In the case of the processes 4 through 7, the cuts imposed on the pr imary leptons outl ined above were modified. For the process 7r° —» e + e _ e + e ~ , we required only that at least one e + e " combination fulfilled the restrictions l isted. For the photon conversion background 5, 6 and 7 we waived a l l restrictions and generated the events i n Air w i th a l l possible momenta. A more complete discussion of the model l ing of the background processes 3 to 6 can be found i n chapter 7. 5.3 Stop Distr ibution The events were generated i n the l iquid hydrogen target according to stop distr ibut ion found for the data. The event origin, as calculated by the reconstruction program, was plotted for each real event in an r — z projection. Th i s was digitized and used as the distr ibut ion function for the generation of the simulated events. The stop distr ibut ion was taken to be radial ly symmetric and identical for each of the 4 r u n periods, for each process. D u r i n g the final analysis, the data and simulation vertex distributions i n the r — z plane of both the processes 7r° —> e + e~7 and 7r~p —+ ne+e~ for each of the 4 r u n periods, were plotted and digitized. B y div id ing the stop distr ibut ion of the data by that of the s imulat ion, 8 sets of (2-dimensional) weights were generated. These weights were applied to each Monte Car lo stop distr ibut ion, so that the simulation would match the data as closely as possible. The importance of this matching w i l l be discussed in chapter 9. A typical weight distr ibution can be seen i n figure 5.1. Note that the weights are quite close to 1 for most regions of the target. Figure 5.1: Two-dimensional weighting function for the s imulat ion, designed to match the stop d istr ibut ion to that of the data. The m a x i m u m height is roughly 7, the average is 0.8. 5.4 Model l ing Detector Response Once the detector geometry has been defined and entered into the G E A N T package, the program w i l l step each of the particles through the various detector parts , mod-ell ing its energy loss along the way. The production of secondary particles through pair product ion, bremsstrahlung, and Compton scattering is automatical ly performed. These secondary particles are also traced through the detector. Once a particle reaches a volume defined as "sensitive" by the user (for example, a chamber wire or hodoscope cell), control is passes to the user's subroutine, which then takes care of the specific de-tector response. We note here that the complex process of gas ionizat ion, ion drift , and subsequent amplif ication and avalanche near a chamber wire has been approximated by recording the energy lost by the electron i n the chamber and assigning it to the nearest wire. T h e response of the cathodes is approximated by a Gaussian response curve over 7 nearest neighbours on each side of the central strip. T h e cathode strips are simulated to be 100% efficient (no dead strips are simulated). A noise signal is added to the signal amplitude. The hodoscope signals are determined by assigning the energy lost in a particular scintillator by the particle to that scinti l lator. The energies assigned to the anodes, cathodes, and hodoscope scintillators are trans-lated into d ig i ta l output identical i n format to the output of the electronics and written to a file which can subsequently be read directly by the offline analysis programs. 5.5 Trigger Simulation W h e n the online trigger ( T P S , M L U and G P M ) and filter were being designed and the appropriate masks were being developed during the in i t i a l construction of the spec-trometer, trigger s imulation programs were written. The i r purpose was to test the correctness and completeness of the masks (which were produced by s imulating e + e~ pairs) and hence to verify the performance of the online r — <f> pattern recognition. The programs were wri t ten by W . B e r t l and H . Pruys of the S I N D R U M I collaboration. T h e programs take as input a raw data file w i th the format generated by the elec-tronics. Different trigger conditions may be set dur ing an ini t ia l izat ion phase. We set the conditions appropriate to the Da l i t z experiment outl ined i n chapter 3, and pass the simulated data through the trigger. The results are shown i n figures 5.2a) through f), where we i l lustrate the action of the trigger both on the simulated Da l i t z and ne+e~ data, as a function of momentum and transverse opening angle. We see that the trigger cuts harder on the ne+e~ data than on the Dal i t z data : 78% of the n e + e ~ events are lost in the trigger, compared to 55% of the Dal i tz events. The higher the track mo-mentum, the more likely it w i l l be lost i n the trigger. Th i s is because the masks were produced by simulated e+e~ pairs from 7r° decay; such pairs w i l l always have a smaller momentum range than the e+e~ pairs from Tt~p — > ne+e~. Th i s was an error. H a d this cut been less stringent, a more thorough analysis of the n~p —• n e + e " data would have been attempted. Look ing at the positron and electron transverse momentum dis-tr ibut ions , we see further that the positive tracks are cut harder than the negative tracks; the masks, being simulated themselves, were not quite symmetric w i t h respect to charge. A g a i n this is exaggerated i n the n e + e ~ sample, because of the larger mo-mentum range. T h e trigger s imulation also applies a cut i n transverse opening angle; however, from figure 5.2c) and f) we can see that the simulation's cut is not the same as the real trigger's. T h e fact that the agreement between the data and s imulation is poor at this point is not worrying , since much more stringent cuts w i l l be made later on i n the offline analysis. A n y systematic errors due to the event selection w i l l then be due to these cuts, and not due to the poorly-understood trigger. The reason for s imulat ing the action of the trigger i n the first place was to verify the reduction i n the ne+e~ sample. Figure 5.2: T h e figures i l lustrate the action of the simulated trigger. The histogram represents the simulated data before passing through the trigger, the points, after the trigger. In a) through c) we show the effect on the Dal i t z s imulation, while i n d) through f) we show the ne+e~ s imulat ion. In addit ion to the online trigger, the data also passed through the online filter. T h i s stage of the data acquisition software is not modelled, since, again, later cuts i n the analysis w i l l duplicate i n a more stringent way the requirements imposed by the filter. D u r i n g the data taking, every 4th event was writ ten to tape by the online filter, regardless of its being accepted or not. In this manner, it is possible to test the efficiency of the later analysis cuts: we find that no events which were not passed by the filter found their way into the final data sample. Chapter 6 Radiative Corrections 6.1 Radiative Corrections for the Process 7r° —> e + e 7 Radiat ive corrections are processes such as those pictured i n figures 6.1 and 6.2, pro-cesses involve more than two photons. It is less likely that these events w i l l take place; radiative corrections, w i th Feynman diagrams involving extra vertices, are smaller by a fa 1/137. Note that the radiated photons pictured in figure 6.1 may be either external (hence i n principle detectable) or internal . In both cases, they change the momentum of the electrons and hence modify the shape of the invariant mass spectrum. Since the slope parameter is itself a second-order effect, one would expect that the effect of these radiative corrections could be appreciable, and hence they must be included in any analysis that expects to extract a value for a. In the case of the decay 7r° —> e+e~7, the contributions to the matr ix element corresponding to the diagrams shown have been evaluated exactly by Mikae l ian and S m i t h and others [20,22,25] and the resulting corrections to the K r o l l - W a d a formula 1.11 tabulated [22], as shown in figures 6.3 and 6.4. We note that these analyses neglect the (form-factor dependent) corrections shown i n figure 6.2. One analysis [43] claims that these diagrams make a large contribution to the form factor, while a later paper 2 [44] comes to the conclusion that these graphs have an extra factor of ^f- associated w i th them, and can hence be safely ignored. Since a is defined in terms of the invariant mass x, one could in principle apply the Figure 6.2: T w o v i r tua l photon loop graphs, corrections to 7r° —> e + e 7. 0.M 2.77 % 0 . 9 9 -33.51 % Figure 6.3: T w o dimensional surface plot showing the percentage correction to the K r o l l - W a d a matr ix element, as calculated i n [18]. T h e surface is symmetric about y = 0. Figure 6.4: Corrections to the K r o l l - W a d a matr ix element as a function of x, as calcu-lated i n Mikae l ian and S m i t h [18]. results of Mikae l ian and Smith 's calculations i n x directly to the simulated, uncorrected spectrum as a mult ipl icat ive factor. T h e radiative corrections expressed as a percentage change over the "bare" spectrum are roughly linear i n x over the effective range of the data , so that the resulting value for a increases by approximately 0.05. However, this naive approach to the radiative corrections is not correct. A s one does not actually simulate any radiative events using this method, the impl ic i t assumption is that the acceptance and detection efficiencies for the radiative events are exactly the same as for the bare events. There is no a priori reason for this to be so; the radiative events can be 4-body decays and as such are kinematical ly quite different from the bare 3-body Dal i t z decays. Another serious drawback of this simplistic method is that , since one does not simulate any radiative events, it is not possible to check whether the bare events plus the corrections actually match the data i n any kinematical region other than x — y space. The applied mult ipl icative factor cannot create four-body decay events i n the regions forbidden to the three-body bare events. In summary, it is not possible to gain any insight into the kinematical behaviour of the radiative corrections without generating some. Before moving on to discussing the techniques used to generate radiative events, we first make a few observations concerning the diagrams shown. The radiative corrections pictured are those of second order in a only. Higher order corrections of course exist, but each added photon line suppresses the amplitude of the event by a factor of y/a so that one usually considers only the leading order terms pictured. T h e diagrams divide themselves up natural ly into two classes; " v i r t u a l " or " in te rna l " , and "bremsstrahlung" or "external" radiative corrections. V i r t u a l radiative corrections are those i n which the extra photon is reabsorbed onto one of the leptons and is therefore not experimentally detectable. T h e bremsstrahlung corrections lead to free photons which are i n principle detectable. B o t h the v i r tua l and bremsstrahlung corrections separate into a convergent set of integrals and a divergent group. W h e n one performs the integration numerically, the usual technique employed to cope wi th these divergent integrals is to introduce a photon cutoff energy A , below which the integral diverges, and above which it converges. A d d i n g the divergent part of the bremsstrahlung contribution to the v i r tua l corrections, one finds that the divergent parts of each cancel exactly (independent of the value of A ) , and the total contribution is finite. In order to simulate radiative events, the results derived by Mikae l ian and S m i t h are not sufficient as they are couched i n terms of the two degrees of freedom x and y. The v i r t u a l corrections and the bare process may be fully described by only two k inemat i -cal variables, but the 4-body bremsstrahlung corrections require one more parameter, x-y, the invariant mass of the two photons, i n order to specify the event kinematics completely. Th i s variable has been integrated over to derive the surface i l lustrated i n 6.3. We therefore use the code developed by L . Roberts [25] which performs the i n -tegrations numerical ly over phase space variables. The code separates the v i r tua l and bremsstrahlung corrections and calculates the matr ix element for each separately. The cutoff energy A is set to a small value well below the experimental energy resolution. T h i s corresponds to asserting that the bremsstrahlung events w i t h photons of < A cannot be experimentally distinguished from the v i r tua l corrections, and hence may be safely used to cancel the divergent part of the v i r tua l corrections. If A is set too high, one i n effect ignores bremsstrahlung events w i th low E^. T h e integrations to be performed are quite complex; the v i r tua l corrections involve numerical evaluation of a five-dimensional integral while the bremsstrahlung corrections are eight-dimensional. The code uses a generalized Monte Car lo integration technique to evaluate the integrals. The range of each integration variable is divided up into smal l cells ( the so-called integration grid ) and the contribution of each cell to the total integral is estimated on the basis of the fraction of randomly drawn points that fal l i n i t . These randomly drawn points are converted into the event kinematics; i n this way, the integration package also functions as an event generator. As the accuracy of the desired integral is increased, a refining algorithm repartitions the integration space into smaller and smaller cells, focusing i n on steep areas of the function. One must verify that this part i t ioning and refining algorithm does not introduce any bias into the result and skew the distr ibution of generated events. In the case of the bremsstrahlung corrections, for example, the eight variables of integration are the energy, cos 8, and <f> angle of the bremsstrahlung photon, the energies of the positron and electron, the opening angle between the electron and positron, and the cos 8 and <j> angle of the electron. The program first models the 2-body decay of the in i t ia l TT° into a v i r tua l photon and a bremsstrahlung photon, and then allows the v i r t u a l photon to undergo a 3-body decay into e+e~j. Obviously, the d istr ibut ion of events should be uniform in the cos 8 and <j> angles of the bremsstrahlung photon; the i n i t i a l 2-body decay of the 7r° does not favour any part icular direction. If these distributions are in any way skewed, the subsequent cuts modell ing the detector acceptance wi l l throw out or accept too many events and give an incorrect final event d istr ibut ion. Note, however, that this skewing w i l l not lead to an incorrect estimation of the integral! This is because the integrated function is flat i n these var i -ables; every point along the axis contributes the same amount to the final answer and favouring one point over another makes no difference. In the case of the bremsstrahlung corrections, it was found that the cos 8, <j> distributions were not flat. There was an error i n the part i t ioning/re f in ing algorithm of the integrating package which produced holes and spikes i n the distr ibut ion of cells. Rather than rewrite the integrating package, it was decided s imply to re-randomize these variables after the integration was finished but before the subsequent detector cuts, since, as pointed out above, the final answer would not be sensitive to this operation. We r u n the code i n an attempt to reproduce the results quoted by Mikae l ian and S m i t h ; see figure 6.5. The corroboration is not fantastic, but out to x w 0.5 (where most of the data is) the agreement was deemed acceptable. It must be noted that generating graphs such as those shown i n 6.5 requires many hours of computer time. We find the total contribution to the matr ix element for 7r° —• e+e~j to be i\ 0t = To + r r c = r 0 + r v + r 6 = (6.2499 ± 0.0026) • 1 0 - 8 + ( -0.9873 ± 0.0004) • 1 0 - 8 + (1.0341 ± 0.0005) • 1 0 - 8 (6.1) The errors are statistical and are estimated by the Monte Car lo integrating package on the basis of the number of points drawn per integration region. The radiative corrections contribute a total Trc « 0.75% to the total decay w i d t h , i n agreement w i t h the results of Mikae l ian and Smi th . We add the radiative events to the bare events generated according to the K r o l l - W a d a distr ibut ion i n the proportions given by equation 6.1, and verify that these radiative events (the bremsstrahlung photons are tracked and allowed to interact wi th matter) now fill out the tails of the distributions as expected. In figure 6.6 we plot the energy of the lepton pair against their invariant mass, and figure 6.7 shows the total energy-momentum versus the opening angle of the lepton pair . The radiative events are clearly visible below the kinematical region allowed the bare events. In figure 6.8 we see that the ta i l fits the data very well . It is instructive to plot the effect of the radiative corrections on the bare events, especially for the x and <f> distributions. In figure 6.9 we show the relative contri -but ion to the invariant mass and opening angle spectrum from both the v i r t u a l and bremsstrahlung radiative corrections. We see that the corrections have a greater impact on (j> than they d o o m . Th is is presumably due to the fact that the extra photons being Figure 6.5: Verif ication of the published radiative corrections. The line are the correc-tions as calculated by Mikael ian and S m i t h [18] and the points are the result of the numerical integration using the program of Roberts and S m i t h [21]. 150 130 H > 0) 110 H c m ^ 90 CD CD o 70 H 50 30 i i i i I i i i i I i i i i I i i i i I i i i i I i i i i • • • L O Q G D D D D O D D D B D D O D D O O O D D D D I I D O D L O G D D I I L T D M • I< • • • - • n n n n n n n Q g Q Q Q g i i n n n n n n n n n n n n n n n n n n n n - n R R . n n - . . c n n o Q C 0••••••••••••••••OODDDDDODDoDODDDDoDODa••oon eo•• — ~ nmrjrxiDOoo.. •••I • - DDL • orjrjn •••0 •DDDDDO D D D D O D O ODDDDDDDDD] J lUjJODDbDOODD« innnnnnnnim • . JODDDDDOOOOD o o o a • • O D D O O O O O O A O I A O D D O O D O D O O O O O O A A A O B O O O u u u a < • • • • | oof O D D • 0 0 ODD • O D D • • • • °DCJ o • •• i i i D I D D O O D D i ••••OODOODD• nODDCOrjQDi 3D • D • D • a • JDDOD [•••DO • 0 0 • 0 • D • 0 • • B 0 I • • I I I I I | I I I I | I I I I | I 0.0 0.2 0 . 4 0.6 X =\ s" i i I i i i i I i i i i 0.8 1.0 m 2 1.2 7T Figure 6.6: Invariant mass of lepton pair versus the total energy, for the simulated events. We plot both the n e + e " and the e + e " 7 events, both w i th radiative corrections. i——I • I—I CD ft o > C c d 260 240 220 200 180 160 140 120 100 80 60 40 i i i i I i i i i i I i i i I i i i • nnnnnnnlTT ) • • • • • J-OOTID OOTJDDD •••ODDon ODDdDDoo ••••ODD ••OOCIo DO PDDorjooo D D Q D D D O O O o O o D o O g l T l D D D D D D o o o OO[IJD0[]0D[]D<IOB 8 D Q Q Q Q D D D D o n D • • • • • • • • • • D O D O ' 0•) III IQODQaQoo D Q H J D D D O O O O • • •' "•••••••DDD 0 o • • • • • • • • O D D D D • • oDDDOOQOODDD•o •••CODODDDDOOO• • ••••ODOooOi 3DODD0D o B O O I D D O D O O O O S B JOT) • • 0 o o a a _ 1 D D D D B o• 3 D D D D D Q B O D J3O0OODOB0 •••••••Do JDDDDDDDDo U D D O O D D D O jODODOo 3 0 D D D D D DDDDo l l n o • • I_J_L • ' i ' I ' ' ' ' I ' i ' ' I I D l • • • 0 i 0 B • I 0 0 B I 0 B • I • < D O " • • I a • > A • • • 11 B • I • • I • • D • D o a • a • a 6 • B I D 0 0 • • 1 • t o o n a o 0 o • • i o a o o • • • D D O D D H D D O D O A • J•D D DO Q • • • D D Q D D D D I • D D O D D D O • • •oOoODDi • B • • • • • • • • < ••DDODDDO•• • ••DODDDD'" •ooDDQODDi• ••OODQODO• I | I I I I | I I I I | I I I I | I I I I | I I I I | I I ™ .r-r, 210 240 I I I | I I I 30 60 90 120 150 180 E.P. (MeV) it L TT 270 L Figure 6.7: EtPt of lepton pair versus the transverse opening angle, for the simulated events. We plot both the ne+e~ and the e +e~7 events, both wi th radiative corrections. T h e radiative tails are clearly visible. Chapter 6. Radiative Corrections 7 5 1 0 0 0 ' ' 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Figure 6.8: T h e EtPt d istr ibut ion for the Dal i tz events. The histogram is the s imulat ion, the points, the data. Note that the radiative ta i l is very well fit. emitted do not transfer much energy, but do transfer momentum, and w i l l therefore kick the leptons a bit further apart. -1000 i -2000 • 1 • • • 1 ' ' 11' 1 111 1 1 1 1 1 1 1 111 1 1 1 3000 .u H11111111111111 < H111111111111H111111 .2 .4 .6 .8 - 5 0 0 1 -1000 i i i l l l l i i l n m i l i t l i i i i n i l i l n i i !• " I " " I' I invariant mass x IIIIIIIII|IIIIIIIII|IIIIIIIII|IIIIIIIII|IIIIIIIII|IIIIIIIII|IIIIIIIII|IIIIIIIII 20 40 60 80 100120140160180 opening angle 0 Figure 6.9: T h e contribution to x and </> from the v i r tua l (solid line) and bremsstrahlung (dotted line) corrections to the total (triangles) s imulation. T h e total contribution of the radiative corrections is shown by the shaded histogram. 6.2 Radiative Corrections for the Process 7t~p —> n e + e Radiat ive corrections to the internal conversion process ir~p — » ne+e~ must also be considered, as they result in lepton pairs leaking down into the Dal i t z decay region and contaminating i t . Th i s is especially important at higher values of x , since the two processes approach one another in energy as the lepton invariant mass increases. We expect higher contamination at higher x. Since this is the region where the sensitivity to a is highest, it is imperative that the ne+e~ radiative corrections be understood. N o exact theoretical calculations have been done for the ne+e~ radiative corrections, so approximate methods must be used to determine the shape of the radiative spectra. T h e approximate methods used for the ne+e~ process are quite general and equally applicable to the Dal i t z decay. A more complete discussion of the method used is given i n references [45,46]; we note here that the v i r tua l corrections shown i n figure 6.10 can be implemented by a mult ipl icat ive weighting factor (which is less than 1) carried along wi th each "bare" event: da da ^ dx virtual dxo w i t h a U 2 13 (mWx) 28 \ where one neglects the (small) dependence of the v i r tua l corrections on y. T h e bremsstrahlung corrections pictured in figure 6.9 are modelled by allowing one of the leptons to emit a photon according to the probabil i ty d istr ibut ion (expressed i n the centre of mass of the emitted photon) rE-mc 1 + 6(x,y)brems = / P(q) dq Jo w i t h Afq\A(, q , q2 q where 7r \me 2 / The bremsstrahlung photon is emitted wi th energy q from a lepton of energy E. One imposes a threshold energy cut ^ > A ( A « 0 . 1 M e V ) so that only those photons visible Figure 6.10: Feynman diagrams for the radiative corrections to the process Tr~p — > ne+e~. by the apparatus are modelled. Every event is forced to undergo bremsstrahlung. Its probabil ity, as calculated by the equation above, is carried along as a weighting factor (again, always less than 1). T h i s approximate method is based on the assumption that the radiative corrections to ir~p —• ne+e~ correspond to those calculated in [46] for the process e + e~ —• hadrons. Its accuracy is estimated by the authors to be roughly 10%-15%. In figure 6.11, we show the results of carrying the weighting factors along. Note that the shape of the (clearly visible) radiative ta i l is generally well simulated, although a slight underestimation of the ta i l seems to be occurring. ' 11 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 n 1 1 1 1 1 111111 n i n 111 i [ 111 11 11111 11 111111111111111 1111111 11111 111 100 110 120 130 140 150 160 t o t a l e n e r g y (MeV) Figure 6.11: Total energy of the lepton pair for the n e + e ~ events. The points show the data , the histogram is the simulation. The radiative ta i l is generally well fit, although an underestimation seems likely. The simulation is expected to be correct to roughly 15%. Chapter 7 Background Considerations A n y process which produces e+e~ pairs may be considered a background to the Dal i t z decays. Table 7.1 shows a list of a l l such processes. A major source of background are 7r° decays involving photons, where the 7 undergoes Compton scattering or pair production to produce other electrons. S I N D R U M is a low-mass detector, so the probabi l i ty of photon conversion i n the wire chambers is very low. More probable are conversions i n the target's a luminum support r ing , in the lead degrader i n front of the target, and i n the l iqu id hydrogen and M y l a r housing of the target. Another possible source of background are so-called "returning tracks" ; electrons from beamline muon decays process contribution to total sample 7T° —> e+e~e+e~ < 10 events 7T° —> e +e~7 (compton) < 20 events 7T° —• e +e~7 (pair prod.) < 10 events 7T° —• 77 (pair prod.) < 5 events 7T° —• 77 (2x pair prod.) < 10 events 7T° —> 77 (compton 4- pair prod.) < 15 events 7 T - p —> wy (pair prod.) < 5 events n~p —¥ ne+e~ fa 2500 events Table 7.1: Backgound processes considered and their calculated contribution to the total sample, which numbers roughly 100,000 events. Conversions are modelled i n the l iqu id hydrogen target, i n the target walls, and in chamber 1. 7.1 E x t e r n a l C o n v e r s i o n B a c k g r o u n d E x t e r n a l conversion events, i n which a photon from a pion decay 7r° —• 77 or 7r° —> e +e~7 or from n~p —* nj converts into an e+e~ pair or results i n a Compton electron due to interaction w i t h matter inside the detector, are a potential source of spurious e + e~ pairs. Such events occur in huge numbers i n the detector's massive parts , such as the a luminum target support r ing and lead degrader clearly visible i n figure 4.1. T h e constraints on z and r discussed i n chapter 4 remove v i r tua l ly a l l of the support r ing and degrader conversions. A further requirement that the inner two chambers exhibit two and only two hits greatly reduces events w i th extra electrons (Dal i tz events i n which the photon converts, and 7r° —• 77 events in which both photons convert). Th i s cut eliminates events w i th an extra electron track of momentum greater than about 2-6 M e V / c (depending on the event origin and emission angle). T h e m i n i m u m opening angle requirement practical ly eliminates the conversions from ir~p —• 717, since the 129 M e V photon produces e+e~ pairs w i th opening angles smaller than 20°. The 7T° —* 77 conversions are also removed by this cut, although to a lesser extent since the photon is less energetic. We demonstrate this by simulating and then analyzing these processes to find the contributions given in the final column of table 7.1. 7.2 7T° —• e + e ~ e + e " B a c k g r o u n d T h e largest contribution from the double Dal i t z decay is expected from "crossed pa i rs " , where each lepton comes from a highly asymmetrical decay of one of the v i r tua l photons. T h e remaining low energy electrons should be visible as hits in the inner chambers, so we expect that mult ip l i c i ty cuts on the inner two chambers w i l l result i n clean events. In order to check this , double Dal i tz events are generated according to the distr ibut ion functions given i n [42] and passed through the analysis. After the requirement that chambers 1 and 2 have only two hits each, we find that the contamination due to the double Da l i t z decay is negligible. 7.3 Returning Tracks In any pion beam, there wi l l be muon and electron contamination emanating from the product ion target. T h e requirement that the e+e~ pair taken as a good event originates w i t h i n the target eliminates the beam electrons. However, i f a beam muon decays v i a p, —» vve i n the hodoscope and the resulting electron passes through the target and continues back to the hodoscope on the opposite side, it mimics an e+e~ pair w i th an opening angle of 180 degrees. These "returning tracks" are a potential source of background. One eliminates most of them at the online trigger stage by requiring that the two hodoscope signals occur at the same time. The few remaining returning tracks, visible at EtPt = m M = 106 and <j> = 180° in figure 4.4 (page 51), are removed by a requirement on the total e + e~ momentum; Ptot > 10 M e V / c . 7.4 B e a m Buckets with more than 1 TT° Roughly 1 i n every 10 8 beam buckets w i l l contain two neutral pions. We therefore expect about 100 of these in our sample. Th i s w i l l result in only one Dal i t z pair , so that background from this source is negligible. 7.5 ne+e~ Background The theoretical ratio of the number of ne+e~ events to the number of e + e ~ 7 events is approximately 1. Th is ratio is reduced by the online trigger and filter, as discussed i n chapter 5: the trigger cuts on m a x i m u m track curvature and therefore imposes a m a x i m u m momentum cut; since the n e + e ~ events have higher e+e~ momenta, these events are reduced i n number. We fit for the resulting ratio of n e + e to e + e 7 #ne+e~events #ne+e~ + #e+e~yevents inc luding radiative corrections to both processes. The results of the fit are given i n table 7.2 below, and a typical example is shown i n figure 7.1. geometry N(ne+e~) iV(e +e"7) R 2 9230 11810 0.237 ± 0.007 4 ' 52207 62087 0.235 ± 0.003 5 16495 18188 0.228 ± 0 . 0 0 7 6 16998 53178 0.218 ± 0 . 0 0 4 Table 7.2: T h e contribution of n~p —* ne+e~ expressed as a fraction of the total sample. Other background contributions are negligible. The quantities N provide a means for assessing the efficiency of the EtPt separation cut; they are not products of the fit, and their magnitudes are of no special importance. In order to remove as many of the ne+e~ events from the Da l i t z sample, kinematical constraints must be imposed. F r o m figure 4.4, we see that the Dal i t z events inhabit a boxlike region from 107 < EtPt < 163 M e V . B y requiring EtPt < 170 M e V , we eliminate most of the 7r~p —• ne+e~ events. There is st i l l some contamination due to ne+e~ radiative events, in which one of the electrons loses energy by emission of a bremsstrahlung photon, thus populat ing the region below Etotai = 136 M e V allowed to the "pure" n~p —> ne+e~ process. These events must be simulated. T h e code used to simulate these events is approximate, accurate to roughly 20%. A brief discussion of the approximate treatment of the radiative corrections has been given i n chapter 6. We may then add the ne+e~ s imulation to the e+e~j for any set of cuts ( in part ic -u lar , the cuts designed to eliminate the ne+e~ above) by (1 — R) (R) total simulation = , T . , r x ( # e+e~7 after cuts)+—— • - x ( # ne+e~ after cuts) N(e+e~j) v N(ne+e-) v ' -20 0 20 40 60 80 100 m i s s i n g energy (MeV) Figure 7.1: T h e determination of the ratio of ne+e to e + e 7. Radiat ive corrections to both reactions are included. where R and N are given in table 7.2. After al l the cuts (those listed i n chapter 4, the cut on EtPt, and the mult ip l i c i ty cuts on the inner chambers) we f ind that approximately 2% of the total simulated data set is due to ne+e~. W h i l e this is a smal l contamination, it must be included, since the invariant mass distr ibut ion of the ne+e~ sample is radical ly different from that of the process TT° —* e+e~j. Chapter 8 Fi t t ing Procedure We divide the ful l data set into the four separate run periods (for each of which the chamber geometry was slightly different, as discussed in chapter 3) which we refer to as "geometries 2, 4, 5 and 6". For each geometry we simulate double this number of events ( including radiative corrections) as well as double the corresponding expected number of n e + e ~ events ( including radiative corrections). Recal l that other background contamination is negligible. The analysis as outl ined i n this chapter is performed for each separate geometry, and the results compared for consistency. F i n a l l y the results are combined and we quote an average value for a. For completeness we list the cuts that are applied to both the s imulat ion and the data in the final analysis: 1. 45° < 4>t < 260°, a cut which duplicates the action of the trigger and filter stages of the data acquisition hardware and eliminates photon conversion background. 2. Pt > 20 M e V , a duplication of S I N D R U M ' s transverse momentum threshold. 3. —300 m m < Z(5 ) < 300 m m : requiring the track to lie w i t h i n the region of uni form magnetic field. 4. r < 19 m m , —115 m m < z < —80 m m , 0 m m < z + 104 + y/192 — r2 : cuts which ensure that the event happened well inside the target. 5. Ptot > 10, which eliminates the rest of the n decays. 6. O n l y 2 hits are allowed in each of the inner two chambers, a cut which eliminates any further background process involving extra electrons or photons, such as 7T° —• e + e - e + e ~ , external conversion, and pair production. 7. Etot + Ptot < 170 to separate the 7r° —• e + e~7 from the ir~p —• ne+e~ process. App l i ca t i on of these cuts results i n 5837 events for geometry 2, 35873 events for geom-etry 4, 5957 events for geometry 5, and 26520 events for geometry 6, giving a total of 74187 7T° —> e + e~7 events w i th a contamination due to ne+e~ of roughly 2%. In figures 8.1 and 8.2 we show the performance of the ful l s imulation ( including Dal i t z and ne+e~) for various kinematical variables. The agreement w i th the data is excellent. We may now proceed to extract a. F r o m chapter 1, the most obvious method of extracting a is to fit for the parameter i n the invariant mass spectrum x, since then we may extract a directly. A lso , as discussed i n chapter 6, the radiative corrections are expected to have a small effect. If we denote the distr ibut ion of the data by F , and the Monte Car lo by / , then we can write the fitting procedure schematically by F(x) = (1 + 2ax) x f(x) We denote the a obtained by fitt ing i n x by ax. As a double check, it is useful to consider the analysis of the distr ibut ion i n <j>, where, as seen i n i n chapter 6, we expect the radiative corrections to have an appreciable effect. We can extract a slope parameter b: F(<£) = ( l + 2 ^ ) x / ( « £ ) Now b and ax are not the same, although b is related to ax i n a measurable manner. We obtain the relationship between ax and b by creating an artif icial " d a t a " sample; we set i i in iiiln i in i nil i h II 11nnIM ni n 11L n11 n11if ^ |||111if 0 20 40 60 80 100 120 140 P t ( + ) ( M e V / c ) 0 20 40 60 80 100 120 140 R ( - ) ( M e V / c ) Figure 8.1: The performance of the ful l s imulation for various kinematical variables. A l l radiative corrections and backgrounds are included. The points are the data , the histogram, the simulation. Figure 8.2: T h e performance of the ful l s imulation for various kinematical variables. A l l radiative corrections and backgrounds are included. The points are the data , the histogram, the simulation. a to some non-zero value i n the K r o l l - W a d a matr ix element, r u n the Monte Car lo , and generate a <f> spectrum. We then fit for b. Repeating for a number of a values allows us to map out the relationship between the two values. The relationship is linear over the range of a we consider (from -0.1 to 0.1): a = 1.65(1)6 + 0.0000(3) (8.1) We denote the a value obtained by fitting i n <f> and applying the relation above by O0. T h e bracketed numbers in equation 8.1 indicate the statistical error on the last digit . B o t h the statistical and systematic errors on b must be transformed to obtain the correct errors on a^. We employ two different fitting procedures for the parameters ax and a^, which are now outl ined. The. two procedures stem from different philosophies of statistical analysis, and use different sets of mathematical tools to arrive at the final result. For each fitting method, we fit w i th and without the radiative corrections, i n order to assess their contribution to the final result. If the analysis is properly done, we expect the final answers (16 of them, i n ax and from each of 2 fitting methods and 4 separate data sets) to be consistent w i th each other. 8.1 M a x i m u m L i k e l i h o o d T e c h n i q u e — t h e B a y e s i a n A p p r o a c h Bayesian analysis is performed by combining prior information about the parameters of the model (denoted by the vector 8) w i th the information from the data sample into the "posterior d i s t r ibut ion" . M o d e l parameters are then estimated by maximiz ing the posterior w i t h respect to the parameters 6. The prior information about the parameters, denoted incorporates a l l previ -ously known facts about the parameters; it may consist of previous results, subjective bias, or any combination thereof. W h e n no prior information is available and /o r sub-jective bias is not appropriate, one uses a "noninformative" or "un i f o rm" prior , which assigns equal probabi l i ty to a l l values of 6. The information encapsulated in the data enters v ia the " l ikel ihood function" f(8\x), which expresses the probabil i ty of observing the data, given certain values of the model parameters. One can t u r n this around; this probabi l i ty may be identified as being the likelihood of each of the parameters being the true value, given the data. It must be noted that this reversal-and-identification is based solely on intu i t ion and not on any formal mathematical principles; however, it seems an eminently reasonable assumption to make and leads to very useful results. Once a prior has been chosen and the l ikelihood function calculated, the posterior d istr ibut ion is given by W h i l e the prior incorporates the beliefs about 8 before the sample is observed, the posterior reflects the updated beliefs about the parameters after the experiment has been done. Bayesian parameter estimation defines the most likely parameter values as those which maximize the posterior distr ibution. Note that since derivatives w i l l be taken, the normal iz ing constant drops out and is not important . Note also that, */ one assumes a uniform prior, Bayesian m a x i m u m likelihood estimation consists of max imiz ing the l ikel ihood function, i n other words, Bayesian and Frequentist maximum likelihood parameter estimation are mathematically equivalent. Credible regions, or confidence intervals, on the parameters describe the uncertainty of the result of the estimation. A 100(1 — a ) % credible region for 8 is defined by the l imits 9{, 9j, where n(8\x) — normalizing constant x 7r(8)f(8\x) (8.2) (8.3) Let us now move on to specifics. Using m a x i m u m l ikel ihood considerations, it is fairly straightforward to derive a simple expression for a i n terms of the first- and second-order moments of the spectrum to be fit. We present here an extended version of the argument found i n reference [19]. We begin by calculating the l ikel ihood function. The l ikel ihood Ck of observing n independent events i n a category k is Poisson distr ibuted: T)nk The total l ikel ihood of observing a data sample w i th M categories is then fc=i n * ! Hence l n £ = - ^2pk + ^riklnpk - ^ l n ( r a f e ! ) k k k Defining the category k to be the interval Xk < x < Xk + S i n the experimental spectrum f(x) , we may write the probabil ity pk of finding an event in the category k as Pk = / f(x)dx w f(xk)6 Jxk for 8 smal l enough. Subst i tut ing, we find l n £ = - £ { / f(x)dx}+^2nk\n{f(xk)S}-J2H^.) k J x k k k = - / / ( x ) d x + ^ n f c l n { / ( x f c ) ( 5 } - ^ l n ( n f c ! ) J o k k Now, i n terms of the parameter a, f(x) = / o (x ) ( l + 2ax). Thus f o r n i x , , . \nC = - f0(x)(l + 2ax)dx + J2nkHfo(xk)(l+2axk)6}-J2Hnk\) (8.4) J o k k We now have the l ikel ihood function for the data i n terms of the parameter a. We forget the results of a l l previous experiments and choose a uniform prior . Our posterior is then equal to the l ikelihood function, and the most likely value for a may be found by maximiz ing w i t h respect to a: d\nC 0 da Hence n-kXk f ^ m a x / xf0{x)dx = 2^7 + 2axk Since a is smal l , we approximate i 1 — 2ax 1 + 2ax giving t ^ m a x / xf0(x)dx « V n ^ z ^ l — 2a:rt) ./o k k k Observing that / fo(x)dx = No, the number of Monte Car lo events i n the spectrum, we may rewrite this i n terms of moments: N0x0 = Nx - 2aNx2 (jfjxo = x-2ax2 (8.5) Now, since this experiment was performed without any absolute normal izat ion, we do not know that is, we have no way of cal ibrating iVo, the number of Monte Car lo events generated, to N, the final number of Dal i tz decays seen. A l l is not lost, however: the number of events seen experimentally must be N = J f0(x)(l+ 2ax)dx = J fo(x)dx + 2a J xf0(x)dx = N0 + 2aN0x0 = N0(l + 2ax0) (8.6) which quickly gives a simple expression for Subst i tut ing into equation 8.5 gives 1 + 2axo => x0 = (1 + 2ax0)(x — 2ax2) « x — 2ax2 + 2axoX2 and hence XQ — x — 2a(x2 — XQX) 1 / XQ -(8.7) N ( XQ - X \ l + x0[ - 0 _ _ (8.8) 2 \x2 — xox i . - ( *° - l+X0\ ~=- — iVo \X* — XQX to first order. We may now establish the classical " l - c r " (a = 0.32) credible region for a. In order to do this , we need the l ikelihood function, which is the exponential of equation 8.4. In order to simplify the calculations, we observe that equation 8.4 may be writ ten as l n £ = - / f0(x)(l + 2ax)dx+ J2^\n{f0(xk)}+ J2nkln{(l + 2axk)} J o k k + X> f c ln { (5 } - £ l n ( n f c ! ) k k = — fp(xk)(l + 2axk) + rik l n { ( l + 2axk)} + constant (8-9) k . k where we have approximated the integral by a discrete sum, and have separated the terms that depend on a from those that do not. W h e n we exponentiate both sides, the constant term w i l l s imply adjust the amplitude of the resulting curve. We shall ignore i t , choosing instead to define the amplitude by the requirement that 1 JCda = l (8.10) The l ikel ihood function, as given by equation 8.9, is simple to calculate numerically. The l ikel ihood function is sharply peaked, and may be approximated by a normal distr ibut ion i n the parameter a. Th i s is convenient, since, for a normal d istr ibut ion, equation 8.3 leads us to In other words, the standard deviation of the normal d istr ibut ion defines the 68.3% credible region of the parameter. We may therefore easily f ind the standard deviation by taking the second derivative of the l ikelihood function: Note that we must be careful about the normalization: the quantity is impl i c i t ly included as a parameter in the above equations. Th i s quantity is not of especial interest; i n the above, we have set the normalizations to their most likely value (according to equation 8.8) and have then calculated the subsequent error on a. We note further that the two parameters are correlated, and the correlation is given by equation 8.6. One may simply substitute x by <f> i n the above derivation to extract the m a x i m u m likel ihood parameter estimation for the (j> case. The two parameters ax and are not the same, as pointed out at the beginning of this chapter, but we may convert from one to the other using equation 8.1. We apply the analysis to both the corrected and uncorrected Monte Car lo spectra. Contaminat ion from ir~p —> ne+e~ is included. Figure 8.3 and table 8.1 summarize the resulting and ax for the various geometries. We do not show the results for the normalizations since they are of.no real interest; a l l geometries and f i t t ing procedures give consistent results. The four separate geometries give consistent results. F i t t i n g i n x and <f> also leads to consistent results. Note that the radiative corrections have a much larger effect in <f> than in x, and that we do not obtain consistent results unt i l we include them! (8.11) 0 no radiative corrections with radiative corrections 0.30 0.25 -0.20^ 0.15 -cdO.10 0.05 -J 0.00 -0.05 -0.10 0.30 f- 0.25 -j 0.20 -i r 0.15-I r cd0.10^ 0.05 J 0.00 -0.05 -0.10 m a x i m u m l ike l ihood f i t t ing m a x i m u m l ike l ihood f i t t ing Figure 8.3: Graphica l summary of the results of the m a x i m u m likelihood f i tt ing for ax (white boxes) and (dark boxes) for each of the four geometries. O n the left are the results without radiative corrections, on the right the results inc luding the radiative corrections. without radiative corrections with radiative corrections geometry ax ax 2 0.140 ±0 .026 0.166 ±0 .018 0.023 ± 0.026 - 0 . 0 0 3 ± 0.018 4 0.031 ±0 .010 0.121 ±0 .007 0.008 ±0 .010 0.002 ± 0 . 0 0 7 5 0.031 ± 0.025 0.141 ± 0.018 0.076 ± 0.025 0.098 ± 0.018 6 0.027 ± 0 . 0 1 2 0.123 ±0 .008 0.008 ± 0 . 0 1 2 0.020 ± 0.008 Table 8.1: Results for ax and from m a x i m u m likelihood f i t t ing, for each of the various geometries, w i th and without radiative corrections. Errors are statistical only. 8.2 x2 Minimizat ion — the Frequentist A p p r o a c h T h e start ing point for x 2 min imizat ion is once again the l ikel ihood function. If we assume that each measured data point nk = f(xk) has an associated error o~k = y / n t 1 , and we further assume that this measurement error is independently random and is distr ibuted around nk i n a Gaussian manner, then we may write the l ikel ihood of observing nk events i n category k i n terms of the parameter a as where S is the w idth of the category. The total l ikelihood over M categories is then As before, we choose the most likely parameters a and -j^ as those which maximize this l ikel ihood function. M a x i m i z i n g the l ikel ihood function is equivalent to min imiz ing the negative of the log l ikel ihood function: We may drop the constant factor of 1/2 and M In 6; they w i l l not affect the min imiza -t ion. We then minimize the so-called " x 2 " function 2 _ " / / ( x f c ) - ^ / 0 ( x O ( l + 2 f l x Q | 2 x We assume that the number of entries njt in each category is governed by Poisson statistics. The standard deviation a2 of a Poisson distribution is equal to the mean fi; in the limiting case of large data samples, the number of entries observed approaches (* and hence a = y/rtk. Note that for n* less than about 20 the Poisson distribution becomes noticeably skewed, and this approximation is no longer valid. One must then calculate / i , given that is the most likely value of the Poisson distribution. Since we have only 2 or 3 bins (at the extreme ends of the distributions) out of a total of 100 that have n* < 20, and so the approximation was deemed adequate. X = We use a standard iterative algorithm which chooses different values of the parameters a and calculates the chi-squared, and uses this information to steer itself towards the m i n i m u m x2 a n d the associated parameter values. In order to assess the errors on the parameters thus found, the standard practise is to perturb the values of the parameters slightly and observe the change i n A\2 defines some 2-dimensional confidence region i n the parameter space. However, we are not interested i n the confidence region of the 2 parameters jointly, but i n the confidence region of each of the parameters by themselves. To evaluate this , we hold each of the parameters fixed, i n t u r n , and find the amount by which we must vary the other to induce a change i n x2 of 1. The magic number 1 appears due to the fact that a x2 d istr ibut ion w i th one degree of freedom (the single parameter) is the square of a normal ly distributed quantity: Ax2 < 1 happens 68.3% of the time (the l-cr level), Ax2 < 4 happens 95.4% of the time (the 2-cr level), etc. A l l of this x2 statistical analysis is standard fare, and we make use of the P L O T D A T A [48] analysis package from T R I U M F , which which allows basic interactive fitting and plott ing. M a n y advanced statistical analysis packages exist, capable of handl ing scores of parameters and complicated fits (most notably M I N U I T ) ; however, since ours is a fairly simple problem involving only two parameters and no pathological functions, the basic package is quite sufficient. A s i n the case of m a x i m u m likelihood fitting, the data is divided up into the 4 geometries. E a c h data set is fit in x and <f> (converting v i a equation 8.1), w i th and without radiative corrections. The fit results are summarized i n table 8.2 and i n fig-ure 8.4. We note that the results i n x and <j> are consistent, and that , once again, the radiative corrections are needed for this consistency. no r a d i a t i v e c o r r e c t i o n s w i t h r a d i a t i v e c o r r e c t i o n s 0.30 0.25 -0.20 0.15 -j co 0.10 -_ 0.05 -0.00 -0.05 -3 -0.10 boxes : f i t i n x b l a c k boxes : f i t i n <f> X2 f i t t ing 0.30 0.25 0.20 0.15 co 0.10 -j 0.05 -0.00 -0.05 -0.10 O-X 2 f i t t ing Figure 8.4: Graph i ca l summary of the results of the x2 f i tt ing for ax (white boxes) and 0.$ (dark boxes) for each of the four geometries. O n the left are the results without radiative corrections, on the left the results including the radiative corrections. without radiative corrections with radiative corrections geometry ax ax a</> 2 ,. 0.088 ± 0.074 0.211 ±0 .117 -0 .050 ± 0 . 0 6 8 - 0 . 0 3 8 ± 0 . 1 0 2 4 0.034 ±0 .027 0.169 ±0 .045 - 0 . 0 0 3 ± 0.027 0.010 ± 0 . 0 3 6 5 - 0 . 0 1 2 ±0 .070 0.230 ± 0.097 0.005 ± 0.074 0.167 ±0 .101 6 0.012 ±0 .036 0.072 ± 0.052 -0 .008 ± 0.038 0.039 ± 0.054 Table 8.2: Results from x2 min imizat ion , for each of the various geometries, w i th and without radiative corrections. Errors are statistical only. The x2 per degree of freedom varies from 1.2 to 1.7 for the various fits. 8.3 S u m m a r y of Fi t t ing Results T h e two tables 8.1 and 8.2 are summarized i n table 8.3 and figure 8.5 below. We obtain consistent results between al l the data sets and the fits i n x and i n (j>. T h e statistical error of the <f> fit i n x2 * s larger; this is not surprising since we expect a somewhat washed-out effect i n <f> due to the fact that x and <j> are not uniquely correlated. X2 minimization maximum likelihood geometry ax at ax 2 - 0 . 0 6 7 ±0 .034 -0 .046 ± 0.045 0.026 ±0 .036 -0 .004 ± 0.048 4 - 0 . 0 0 5 ±0 .018 0.012 ±0 .023 0.007 ±0 .017 0.002 ± 0.020 5 0.003 ± 0.036 0.171 ±0 .047 0.076 ± 0.029 0.094 ± 0.032 6 - 0 . 0 1 2 ±0 .019 0.042 ± 0.028 0.009 ±0 .019 0.021 ±0 .021 all - 0 . 014 ± 0.012 0.032 ±0 .016 0.019 ±0 .011 0.023 ± 0 . 0 1 3 Table 8.3: Summary of the fit results for each geometry, fitting method, and spectrum. Radiat ive corrections and iv~p —• ne+e~ background are included. Indiv idual errors are statistical only. The summary value given is the error-weighted average, and its error takes into account the statistical error only. We also see that the two fitting procedures also give consistent central values. The error obtained using m a x i m u m l ikel ihood is however much smaller than the correspond-ing statistical error from the x2 fit, an effect especially pronounced i n the <f> fits. Since we have not been able to find any flaw i n either the m a x i m u m likelihood or x2 analysis, we have adopted a more robust method by which to estimate the statistical errors. We generate several hundred "new" data spectra by allowing the data spectrum i n x and (j> to vary on a b in- to -b in basis, w i th in its 1-a error bars. We fit this "new" data , by both methods, using the s imulat ion, and tabulate the resulting values for a and the normalizat ion. The distributions i n a and the normalizat ion are Gaussian, and we take their central value and their standard deviations as the true a and statistical error of Chapter 8. Fitting Procedure 02-cd o.o -0.1 -02 02 I I I 1 h 0.1 cd o.o -O.H -02 0.10 0.05 Cd 0.00 -0.05 -0.10 X* fit i n x maximum likelihood fit in x 02 0.1 cd 0.0 -0.1H -02 02 Cd 0.0 -0.1 H -02 0.10 0.05-j Cd 0.00 -0.05 -O.W X* fit i n f> I * * i i maximum likelihood fit in $ Figure 8.5: Summary of results for al l four geometries, i n x and <f>. T h e four points to the left of the double line represent the results for geometries 2, 4, 5, and 6, respectively; the point to the right indicates the error-weighted average. Its smal l error bar shows the statistical error, and the larger one shows the combined statistical error and standard error of the fluctuations. The results i n x and <f> are averaged to obtain the final results. the measurement. O u r results are now independent of fitting method and spectrum, as shown i n table 8.3. T h e central values are essentially unchanged. W e note that this error estimation produces results midway between those obtained from the m a x i m u m likel ihood and x2 analyses. Combin ing the results of a l l the geometries, taking an error-weighted average over the x and <f> spectra, we obtain the 4 results noted i n the last line of table 8.3. We combine the m a x i m u m likelihood result wi th the x2 result by simple averaging, and we take the total statistical error to be the x2 error, since the statistical errors obtained by a l l methods are s imilar . The fluctuation error is taken to be the average of the fluctuation i n the x2 ^ d m a x i m u m likelihood methods. We obtain: ax = 0.003 ±0 .011 ± 0 . 0 2 8 (8.12) a 0 = 0.027 ±0 .013 ±0 .054 (8.13) We turn to an evaluation of the systematic errors in the next chapter. Chapter 9 Evaluation of Systematic E r r o r s Systematic errors are due to the incorrect or incomplete knowledge of the behaviour of the detector. Since both the transverse momentum Pt and the transverse opening angle <f>t enter directly into the calculation of the invariant mass spectrum x and the opening angle d istr ibut ion <f>, anything which affects these quantities w i l l have a direct impact on the measurement of the form factor slope. The systematic errors may be either time dependent (arising from the variation in the detector setup from r u n to run) or time independent (due to problems wi th the analysis or the simulation) . T h e scatter i n the results obtained i n the last chapter (shown i n figure 8.3 and indicated in the results 8.12,8.13) gives a m i n i m u m value for the combined run-dependent systematic errors. We double-check the magnitude of these errors by varying the quantities 1. stop d istr ibut ion 2. target location 3. chamber locations 4. magnetic field 5. analysis cuts and noting the effect on the result. The results of our systematic error analysis are tabulated i n table 9.1. error source X d> magnetic field 0.015 0.020 stop distribution 0.005 0.003 target location 0.005 0.005 chamber A x , A y , Az, A<f>t « 0.010 « 0.035 analysis cuts < 0.003 < 0.003 chamber construction 0.003 0.003 hardware trigger 0.003 0.003 ne+e~ 0.010 0.010 e + e~7 radiative corrections 0.001 0.001 total 0.022 0.047 Table 9.1: Systematic errors from various sources. T h e table is divided into time-dependent errors (top) and time-independent errors (bottom). 9.1 T i m e Dependent Systematic Errors 9.1.1 Stop Distr ibution T h e deeper inside the target an electron is generated, the more mater ia l (and distance) it must traverse to reach the hodoscope and trigger the electronics. Those events which originate at smal l target rad i i , therefore, tend to have higher momenta than those which occur at the outer edges of the target. Since the form factor slope is directly dependent on the momentum distr ibut ion of the electrons, it is v i ta l that the stop d istr ibut ion be correctly modelled. Since we know the f inal stop d istr ibut ion , we can adjust the s imulation to match the data as closely as possible using the scheme of weights outl ined i n chapter 5. B y adjusting these weights to the point where the stop distributions become statistically different, we conclude that the error i n a due to improper knowledge of the experimental stop distr ibut ion is as tabulated i n 9.1; we see no significant systematic error. 9.1.2 Target Locat ion T h e experimental stop distr ibut ion is measured w i th respect to the centre of S I N D R U M . T h e posit ion of the target must also be fixed relative to this point. In other words, specifying the stop distr ibut ion does not fix the target location and hence the stop d istr ibut ion may be moved relative to the ends of the target. If, for instance, the target is shifted slightly downstream, those events generated at the upstream end wi l l traverse less hydrogen, and have a correspondingly higher momentum than those events coming from the back of the target. Min imum- ion iz ing electrons lose approximately 300 keV for every centimetre of l iquid hydrogen traversed; a 0.5 cm shift i n the target, therefore, can induce a 150 keV shift in the mean track energy. If we arrange the cuts on the target vertex and the longitudinal emission angle i n such a way as to remove most of the events coming from the very tip of the target, we may minimize this systematic error. B y moving the target by 2 m m in the s imulation and analyzing the result, we obtain the systematic error quoted in the table. It is small . 9.1.3 C h a m b e r Geometries As discussed i n chapters 4 and 5, S I N D R U M ' s wire chambers were calibrated for the run-dependent x,y,z offsets and rotations relative to chamber 5, as well as for the r u n -independent anode print gaps and twists. The offsets and chamber rotations represent a much larger effect than the anode print gaps and twists. These calibrations were determined using cosmic rays. The rotations are accurate to roughly 1 • 1 0 - 4 m r a d , and the x,y,z offsets to 0.5 m m . The largest rotation and x,y offset was found for chamber 1; 40 m r a d 0.9 m m , respectively. The largest z offset (1.8 mm) was measured for chamber 3. In comparison, the largest twist was measured for chamber 3 (0.19 m m ) , and the anode print gaps were determined to vary from < 0.1 m m (chamber 1) to 0.5 m m (chamber 4). We redo the track fit w i th these geometry calibrations set to their l imi t ing cases for the r u n periods for which the calibrations are a m a x i m u m (the "worst case"). Ext rac t ing the variation of a w i th respect to this change i n each of the calibrations leads to the results tabulated i n 9.1. T h e uncertainty i n the relative chamber posit ioning is the largest source of systematic error. 9.1.4 M a g n e t i c F i e l d T h e value of the magnetic field enters directly into the calculation of the electron momentum during the track fitting. We can calibrate the magnetic field using either the ne+e~ data or the 7r° —> e+e~7 data. We note that since the magnet was turned off and on each time that the spectrometer was taken apart, we might expect 4 different cal ibration values. C a l i b r a t i o n o n t h e ne+e~ p e a k We choose an energy variable i n which the ne+e~ data is as sharply peaked as possible. T h e appropriate kinematical variable, from the considerations outl ined i n chapter 3, is the total energy of the final state Etot — E+ + .EL + Tn a quantity which was also useful in checking the accuracy of the ne+e~ radiative correc-tions. We perform a one-parameter fit using x 2 -n i in imizat ion to extract a momentum cal ibrat ion. The result of one of the fits is shown in figure 9.1. The results for the 4 different r u n periods are tabulated in table 9.2. The error quoted is statistical only. — i 111 n 111M11111111111111111 i n 111111111111 [ 1111111111111111111111111111111111111 n 1111111111 100 110 120 130 140 150 160 100 110 120 130 140 150 160 total energy (MeV) total energy (MeV) Figure 9.1: The result of the calibration on the n e + e " peak for geometry 4. T h e net change to the magnetic field is about 0.5%. O n the left, before the cal ibrat ion, on the r ight , after. 1 1 1 I 1 1 1 I 1 ' 1 I 1 1 1 I ' 1 6 0 8 0 100 120 140 160 180 i r'r | i—i i | i i i | i i i | i i i | i i i 6 0 8 0 100 120 140 160 180 E t o t + P t o t ( M e V ) Figure 9.2: The result of the calibration on the Dal i t z box for geometry 4. The net change to the magnetic field is about 0.5%. O n the left, before the cal ibration, on the right , after. C a l i b r a t i o n o n t h e 7r° —• e + e 7 p e a k We may also calibrate using the 7r° —• e+e~j data, a procedure which may be preferable since we w i l l calibrate the energy range i n which we are interested, thus obviating the need to worry about a possible energy dependence of the cal ibration. We perform a one-parameter x 2 - m i n i m i z a t i o n on the quantity EtPt, discussed i n chapter 3. The 7r° —» e +e~7 process is restricted to a box wi th very sharp edges, and it is the location of these edges which determines the fit. The fit is, however, sensitive to changes i n the ne+e~ contamination and to addit ion of the radiative corrections, so that a comparison w i th the n e + e ~ calibration results is nice to have. The results are shown i n figure 9.2 and tabulated in table 9.2. We obtain consistent results; this geometry e + e 7 calibration ne+e calibration 2 1.0020 ±0.0010 1.0012 ±0 .0010 4 1.0024 ±0 .0005 1.0028 ± 0.0005 5 1.0032 ±0 .0010 1.0012 ±0 .0010 6 1.0036 ±0 .0007 1.0068 ± 0.0007 Table 9.2: Results of the magnetic field calibrations for the various runs. The magnet was turned off and on between the various runs. verifies our in i t i a l guess that the cal ibration is independent of energy, and hence can be attr ibuted to a simple scale factor on the magnetic field. Notice that the changes to the field are smal l ; on the order of 0.5%. Th is is entirely consistent w i t h the accuracy of the magnet current monitor installed during the experimental r u n . A r m e d w i t h these magnetic field scaling factors, we return to the Monte Car lo s imulat ion and reset the field for each run period. Th i s is a necessary step since the detector acceptance is a function of magnetic field. In order to obtain the resulting error in a, we r u n through the analysis once w i th the magnetic field set to its l imi t . We obtain the result shown i n table 9.1. A further source of error is the nonuniformity of the magnetic field. A n accurate field map was not made for S I N D R U M I, although one existed for the previous incarnation of the detector [31]. F r o m this map, we estimate that w i t h i n the central region of the chambers (within a cone denned by excluding the outer 10 cm of chamber 5) the magnetic field is uniform to better than 1%. We may verify this by repeating the magnetic field cal ibration for different fiducial volumes of the detector; we see a variation consistent w i th the above assertion. However, the statistics are not sufficient for this to be an accurate method of mapping the field. We apply a f iducial cut and calibrate on the interior region. 9.1.5 Analysis Cuts In order to assess the sensitivity of a to the values of the final cuts, we need to know the detector resolution of these quantities. These are simple to obtain; we s imply record the exact value of the kinematical variable i n question before modell ing the detector response, and then plot the difference between the exact "remembered" value and the reconstructed value. Th i s is of course the resolution of the s imulat ion, not of the actual detector, but provides a useful estimate. S I N D R U M ' s momentum (and energy) resolution is approximately 5% at 100 M e V . The angular resolution is approximately 3.5°. T h e quantity EtPt controls the amount of ne+e~ contamination i n the final sample; a 5% resolution, together wi th the estimation that the radiative ta i l of the ne+e~ process is understood to no better than 15% leads us to the systematic error due to ne+e~ contamination stated i n table 9.1. Because the trigger is not understood well enough, we apply a final cut i n transverse opening angle more stringent than the one the trigger makes. Thus the final result is dominated by the error associated w i th this cut rather than by the error due to the poor knowledge of the action of the trigger. B y varying the opening angle cut by the stated resolution and f inding the resulting change i n the form factor slope, we arrive at the conclusion that the systematic error due to this cut is negligible. T h e lower transverse momentum threshold is a function of the hodoscope radius and the vertex posit ion. In order to minimize systematic error due to uncertainty i n these quantities we apply an explicit requirement of Pt > 20 M e V / c . Vary ing this quantity by 10% results i n no appreciable systematic error. 9.2 T i m e I n d e p e n d e n t S y s t e m a t i c E r r o r s Run-independent systematic errors w i l l increase the overall error on the final result quoted i n 8.12,8.13 and shown in figure 8.5. We elaborate briefly on their evaluation. 1. Chamber construction : The chambers exhibit twists and wire spacing irregular-ities, which are constant from r u n to run . These are smal l i n comparison to the aforementioned offsets and rotations, and hence they are not modelled but i n -cluded i n the track reconstruction of the data only. E l i m i n a t i n g these corrections gives negligible systematic effect. 2. Hardware trigger : The hardware trigger removes a large number of events. It applies an opening angle cut i n the transverse plane, as well as a non-linear and charge-asymmetric momentum cut. The ir'p —• ne+e~ data, which have a larger electron momentum, are reduced in number by a factor of roughly 3, and therefore provide a good test of our trigger s imulation program. We r u n this s imulation on the n e + e ~ Monte Car lo ; i f subsequent results match the real data well , we may be confident that the trigger is properly understood. F r o m figures 9.3 and 9.4, we see that the action of the trigger is understood, at least for the n e + e ~ of geometry 4. S imi lar results are achieved for geometries 2 and 5. We are confident that the trigger is understood, and that the systematic error induced by it is small . 3. Contamination from n e + e " : The radiative ne+e~ events contaminate the Dal i t z sample. These events must be well understood. F r o m figure 9.4h) we see that the long radiative ta i l is accurate to wi th in roughly 15%. B y increasing the number of ne+e~ t a i l events by 15% and reanalyzing the data , we obtain the systematic error shown i n the table. 4. Radiative corrections to TT° —• e+e~j : The results of the numerical integra-t ion of the matr ix elements of the n° —• e+e~*y radiative corrections have been checked against published semi-analytical values [22], and agree to w i t h i n 5%. Furthermore, the radiative ta i l i n EtPt is very well modelled. T h e systematic error induced by the uncertainty in the 7r° —• e+e~j radiative corrections must be smal l . 9.3 Other Systematic Errors Other sources of systematic errors have been investigated and found to be negligible. These include • Possible error due to the multiplicity cuts on the inner chambers. The possible source of error depends on how the simulation decides on wire hits . T h e energy loss as calculated by G E A N T is deposited directly into the cathode and anode strips of the detector. This discrete deposited energy is converted into a Gaus-sian response function and then transformed into a cluster of cathode signals, which are converted into wire hits by the analysis package dur ing the track re-construction. We have checked that the real data and s imulation produce s imilar *jfr + i Q £ l i | i i i i i i i i i | i i i i i i i i i | i i i i i m f r j 1.0 -0 .5 0.0 0.5 1.0 cos# emission angle •I m l n n m . . I i i t l n m m J in11m11111ii11111111111111111 ii 11111[ ii i n 111111 m 111111 6 50 100 150 200 250 300 transverse c/> 10 20 30 40 50 60 70 80 P t (+) (MeV/c) 10 20 30 40 50 60 70 80 P t (-) (MeV/c) Figure 9.3: T h e performance of the n e + e ~ simulation (with radiative corrections) for some areas of phase space, for geometry 4. The points are the data; the histogram, the s imulat ion. Since the agreement between the data and the Monte Car lo is good, we are confident that we understand the action of the trigger. T 111 u 1 n 11 m m 111 m m 1111111 j 11111 j 11 m u 1111 0.0 0.2 0.4 0.6 0.8 1.0 1.2 invariant mass x 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1.0 -0 .5 0.0 0.5 1.0 energy partit ion Y l l m i l l l l l l l i n i i l H i t i l l l l l H l l l l l l H l l l l l i i l n i IIIIIIIIII p i t B i l l l | I I I U I I I I | l l l l l l l l l | l l l l l l l l l | l l l l l l l l l | l l l l l l l l l | l l l l l l l l l | l l l l l l | ! 20 40 60 80 100120140160180 opening angle 0 Figure 9 . 4 : T h e performance of the n e + e ~ s imulation (with radiative corrections) for some areas of phase space, for geometry 4 . The points are the data; the histogram, the simulat ion. Since the agreement between the data and the Monte C a r l o is good, we are confident that we understand the action of the trigger. cluster sizes, and that hence the analysis thresholds for deciding true wire hits are correct. We are confident that we do not miss real hits . • Possible errors due to misalignment of the hodoscope. Since the outermost cham-ber (number 5) was mounted onto the inside of the hodoscope, it is highly unlikely that the hodoscope was moved or rotated i n any way relative to i t (calibrations were done relative to chamber 5). Furthermore, since the hodoscope was used only as a trigger condition, not as part of the track reconstruction, f ine-tuning its location is unnecessary. • Errors due to binning. W h e n fitting for a i n x and (j> we b i n the data into 100 and 90 bins, respectively. Since both the resolution i n x and i n <f> is much smaller than the size of one b i n , we expect no systematic effect from "edge" events being misassigned to neighbouring bins. We have checked that increasing the b in size by a factor of two does not affect the result of the fit. • Possible dependence on detector <f> quadrant. We divide the data up into four samples according to the quadrant of emission of the electron to study the effect on a. We see no systematic effect. 9.4 S u m m a r y Table 9.1 summarizes the contributions to the total systematic error from various sources. Note that the standard deviation of the scatter (indicated i n the results 8.12, 8.13) is i n agreement w i t h the magnitude of the time-dependent errors. We see that the error on ax is dominated by the uncertainty i n the value of the magnetic field; this is not surprising since x is a linear function of the momentum. O n the other hand, since the opening angle is only indirectly dependent on the magnetic field (the acceptance i n <j>t is weakly dependent on the magnetic field since the trigger also cuts on track curvature), is relatively free of this systematic error. However aj, is more sensitive to the calibrations of the chamber rotations and offsets than is ax. O u r two results are thus a x = 0.003 ±0 .011 ± 0 . 0 2 2 a < t > = 0.027 ± 0 . 0 1 3 ±0 .047 Tak ing the error weighted average of the two central values and quoting the smaller errors, we obtain our final result a = 0.02 ± 0 . 0 1 ± 0 . 0 2 Chapter 10 S u m m a r y and Conclusions The basic difficulty w i t h an experiment attempting to measure the ir° form factor v i a the Da l i t z decay is the l imited range of momentum transfer available to probe the pion structure. T h e v i r tua l photon used to study the pion structure is l imited to energies below the p ion mass. It is difficult to resolve the quark structure of the meson at such low energies, and the results of previous experiments (table 2.2) are, accordingly, inconclusive. We point out that the result we obtain is also consistent w i t h a struc-tureless 7T°, although it represents a substantial improvement i n the understanding of systematic errors. C o u l d the error bars be reduced i n a future experiment? D a t a for the form factor measurement were taken over only three days. It would be a very simple matter to reduce the statistical errors by a factor of 3 by running for a month; however, the reduction of the systematic errors is not so simple. In our experiment, the availabil ity of a large number of events, spread over mult iple runs, combined wi th our fits for a i n two spectra using two different methods, provides many double checks and is essential to our understanding of the systematic errors, the largest of which were the magnetic field, the chamber calibrations, and the uncertainty i n the Tr~p —> ne+e~ radiative ta i l . A n accurate field map might have been helpful i n reducing the error associated w i th the field's non-uniformity; however, since the dependence of a on the value of the field is l inear, and since we use the same data set for the cal ibration and for the extraction of a, the use of the average value of the field should not induce a large systematic error. It is the change i n acceptance due to the nonuniformity of the field which causes the systematic error. A field map would have reduced the need for f iducial chamber cuts and would therefore have increased the number of events. The uncertainty i n the relative positions of the chambers proves to be the largest source of error i n this measurement. The measurement of a using the <f> d istr ibut ion is especially sensitive to the relative chamber rotations. The reconstructed z momentum is quite sensitive to the z positions of the chambers, and has a considerable influence on the determination of ax. The chamber geometries were calibrated using cosmic rays, and it is difficult to see how these calibrations could have been improved. The uncertainty i n the ir~p —» ne+e~ radiative ta i l is a systematic error that could be reduced i n future experiments. Because the trigger cut so heavily on the ne+e~ data, a detailed understanding of this process is difficult to achieve. The acceptance of the detector for the ne+e~ data is very sensitive to the magnetic field and to the chamber geometries. Combined wi th a slight difference i n trigger acceptance between electrons and positrons of high momentum, this leads to the observed charge asymmetry i n the TT~p —* ne+e~ data (see figures 9.3 and 9.4). W h i l e this asymmetry is reasonably well understood for geometries 2, and 4 and 5, for geometry 6 it is much more pronounced and is not well simulated. It is not surprising, then, that the radiative ta i l of this process is understood to no better than 20%. A more careful choice of trigger conditions, plus addit ional running w i th a reversed magnetic field would have aided considerably i n an understanding of this important background process. It would be difficult, but not impossible, to improve the systematic and statistical errors by 30 or 50%. Whether this is necessary, however, is not clear, since, by t ime invariance, we expect the form factor for pion decay 7r° —* 77* to be no different from the form factor for pion production 77* —• 7r°. B y using pion product ion, one not only probes the pion at large (negative) momentum transfer, but also avoids a l l of the systematic errors outl ined above ( introducing, of course, others). Th i s has recently been done by the C E L L O collaboration, who find a value for a of 0.0326 ± 0.0026 [27], i n excellent agreement w i th theoretical expectations. In short, then, our result of a = 0.02 ± 0.01 ± 0.02 serves to clear up past dis-crepancies, and brings the measurement of the 7r° form factor i n the timelike region of momentum transfer into line w i th both the theoretical expectations and the recent (spacelike) C E L L O result. It seems that the structure of the neutral pion can be understood i n the framework of the Standard Mode l . Bibliography [1] J . J . Thompson: Rays of Positive Electricity, Longmans Green, London , 1913 [2] E . Rutherford : P h i l . M a g . 21, 669 (1911) [3] M . G e l l - M a n n : Phys . Lett . 8, 214 (1964) [4] G . Zweig: C E R N Report 8 4 1 9 / T h 412, 1964 [5] R . H . Dal i t z : Proc . Phys . Soc. A 64, 667 (1951) [6] N . M . K r o l l and W . Wada : Phys . Rev. 98, 1355 (1955) [7] G . Bar ton and B . G . Smi th : Nuovo Cimento 36, 436 (1965) [8] S. L . Adler : Phys . Rev. 177, 2426 (1969) [9] L I . Amet l ler et a l . : N u c l . Phys . B228, 301 (1983) [10] N A 3 1 Col laborat ion , Phys . Lett . B240, 283 (1990) [11] L . Bergstrom et a l : Phys . Lett . 131B, 229 (1983) [12] L . G . Landsberg: Phys . Rep. 128, 301 (1985) [13] A . Kot l ews i , preprint Co lumbia University, New York (1973) [14] M . R . Jane et a l . : Phys . Lett . 59B, 103 (1975) [15] Y u . B . B u s h n i n et al., Phys . Lett . 79B, 147 (1978); R . I. Dzhelyadin et al., Phys . Lett . 84B, 143 (1979); ibid 88B, 379 (1979); ibid 94B, 548 (1980); ibid 102B, 296 (1981) [16] N . P . Samios: Phys . Rev. 1 2 1 , 275 (1961) [17] H . K o b r a k : Nuovo Cimento 20 , 1115 (1961) [18] S. Devons et a l . : Phys . Rev. 184 , 1356 (1969) [19] J . Burger : Doctoral Thesis, Co lumbia University, New York , 1972 [20] B . E . L a u t r u p and J . Smi th : Phys . Rev. D 3 , 1122 (1971) [21] J . Fischer et a l . : Phys . Lett . 7 3 B , 359 (1978) [22] O. K . Mikae l ian and J . Smi th : Phys . Rev. D 5 1763 (1972); ibid D 5 , 2890 (1975) [23] L . Rosselet et a l : Phys . Rev. D 1 5 , 574 (1977) [24] P . Gumpl inger : Doctoral Thesis, Oregon State University, P o r t l a n d , 1986; J . M . Poutissou et a l . : Proc . of the Lake Louise Winter Insitute (1987), publ . W o r l d Scientific [25] L . Roberts and J . Smi th : Phys . Rev. D 3 3 , 3457 (1986) [26] H . Fonvieille et a l . : Doctoral Thesis, L 'Univers i te Blaise Pascal - Clermont II, Clermont-Ferrand, France, 1989, and Phys . Lett . B 2 3 3 , 65 (1989) [27] C E L L O Col laborat ion , Z. Phys . C 4 9 , 401 (1991) [28] S. R . Amendo l ia et. a l : Phys . Lett . 1 4 6 B , 116 (1984) [29] C h . G r a b : A Search for the Decay / i + —• e + e + e~, Doctoral Thesis , Universitat Zur i ch , 1985 [30] N . K r a u s : The Rare Decay fi+ —• e+utiuee+e~, Doctoral Thesis, Universitat Zur ich , 1985 [31] W . B e r t l et a l : N u c l . Phys . B 260 , 1 (1985) [32] S. E g l i : T h e Rare Decay TT + — » e+e+e~ve, Doctoral Thesis , Universitat Zur i ch , 1987 [33] C . Niebuhr : A Search for the Rare Decay TT° - • e + e _ , Doctoral Thesis, E T H Zur ich , 1989 [34] W . B e r t l et a l . : N u c l . Inst. & M e t h . 217 367 (1983) [35] A . van der Schaaf et a l : N u c l . Inst. & M e t h . A 240 , 370 (1985) [36] C. Niebuhr et a l . : Phys . Rev. D 4 0 , 2796 (1989) [37] R . Eichler : S I N D R U M note 15, internal memo, unpublished (1983) [38] J . F . Crawford : N u c l . Inst. & M e t h . 57 , 237 (1983) R . Mei jer Drees, M S c thesis, University of B r i t i s h Co lumbia , B . C . , 1987 [39] [40] [41] [42] R . B r u n , F . Bruyant , M . M a i r e , A . C. McPherson , and P. Zanar in i , G E A N T 3 package, C E R N D a t a Handl ing Div is ion , D D / E E / 8 4 - 1 M . Leon and H . Bethe, Phys . Rev. 127 , 636 (1962) T . M i y a z a k i and E . Takasugi: Phys . Rev. D 8 , 2051 (1973) [43] G . B . Tupper , T . R . Grose, and M . A . Samuel, Phys . Rev. D 2 8 , 2905 (1983) [44] M . L a m b i n and J . Pestieau, Phys . Rev. D 3 1 , 211 (1985) [45] M . N . Bensayah, Thesis, L 'Univers i te Blaise Pascal - Clermont II, Clermont-Ferrand , France, 1989 [46] G . Bonneau and F . M a r t i n : N u c l . Phys . B 2 7 , 381 (1971) [47] J . O. Berger: Statistical Decision Theory : Foundations, Concepts, and Methods, ©1980, Springer-Verlag, New York [48] J . L . C h u m a , P L O T D A T A package, T R I U M F ©1990 [49] W . H . Press, B . P . Flannery, S. A . Teukolsky, and W . T . Vetterl ing: Numerical Recipes : the Art of Scientific Computing, ©1989, Cambridge University Press, Cambridge , U . K . 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0084985/manifest

Comment

Related Items