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 OF 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  FORM  FACTOR  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 IN P A R T I A L F U L F I L L M E N T THE  REQUIREMENTS FOR T H E DEGREE DOCTOR OF  OF  PHILOSOPHY  in THE FACULTY OF GRADUATE DEPARTMENT OF  STUDIES  PHYSICS  W e accept this thesis as conforming to the required  THE  UNIVERSITY  standard  O F BRITISH  COLUMBIA  October 1991  @ Reena Meijer Drees  OF  In  presenting  this  thesis  in  degree at the University of  partial  fulfilment  British Columbia,  of  of  department  this thesis for or  by  his  or  scholarly her  purposes  requirements for  an advanced  I agree that the Library shall make it  freely available for reference and study. I further copying  the  agree that permission for extensive  may be granted  representatives.  It  is  by the head of my  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 p a r t i a l fulfilment of the requirements for an advanced degree at the U n i v e r s i t y of B r i t i s h C o l u m b i a , I agree that the L i b r a r y shall make it freely available for reference a n d study. I further agree that permission for extensive copying of this thesis for scholarly purposes m a y be granted by the head of m y department or by his or her representatives. It is understood that copying or p u b l i c a t i o n of this thesis for financial gain shall not be allowed without m y w r i t t e n permission.  D e p a r t m e n t of Physics T h e U n i v e r s i t y of B r i t i s h C o l u m b i a 1956 M a i n M a l l Vancouver, C a n a d a  Date:  Abstract  W e present the result of a measurement of the 7r° electromagnetic t r a n s i t i o n f o r m factor i n the t i m e - l i k e region of m o m e n t u m transfer. F r o m a d a t a 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 f o r m factor slope a = 0.02 ± 0.01 (stat) ± 0.02 (sys).  T h i s measurement is consistent w i t h b o t h the  results of the recent measurement by C E L L O ( D E S Y ) i n the space-like region, a n d w i t h the vector meson dominance prediction of a « 0.03.  Table of Contents  Abstract  ii  Acknowledgements 1  2  Introduction  1  1.1  H a d r o n S t r u c t u r e - F o r m Factors  3  1.2  P i o n F o r m F a c t o r - P a r t i c l e Exchange  6  1.3  F o r m Factors of N e u t r a l Mesons  1.4  D a l i t z Decay of the TT°  .  10 11  Historical Background  15  2.1  T h e o r e t i c a l Predictions  15  2.1.1  Vector M e s o n Dominance M o d e l ( V M D )  15  2.1.2  The Quark Loop Model  18  2.2 3  xiv  P r e v i o u s Measurements  21  E x p e r i m e n t a l Setup  31  3.1  Overview - G e n e r a l P r i n c i p l e s  31  3.1.1  32  3.2  T h e 7T° Source  E x p e r i m e n t a l Setup  33  3.2.1  B e a m a n d Target  34  3.2.2  S I N D R U M I Spectrometer . .  .  37  3.2.3  Trigger Logic .  .  39  3.2.4 3.3 4  6  7  40  Data Acquisition  41  Offline A n a l y s i s  42  4.1  Overview  42  4.2  Detector C a l i b r a t i o n  42  4.3  P a t t e r n Recognition and Track F i t t i n g  44  4.4 5  Online Filter  4.3.1  r - <f> F i t  44  4.3.2  z Fit  45  4.3.3  Vertex F i t . .  46  F i n a l E v e n t Selection - Identification of D a l i t z Events  46  D a t a Simulation  53  5.1  D e c i d i n g the Detector Geometry  54  5.2  G e n e r a t i n g the P a r t i c l e K i n e m a t i c s  55  5.3  Stop D i s t r i b u t i o n  57  5.4  M o d e l l i n g Detector Response  59  5.5  Trigger S i m u l a t i o n  59  Radiative Corrections  63  6.1  R a d i a t i v e Corrections for the Process 7r° —• e e~j  63  6.2  R a d i a t i v e Corrections for the Process n~p —»• ne e~  76  +  +  B a c k g r o u n d Considerations  81  7.1  E x t e r n a l Conversion B a c k g r o u n d  82  7.2  7T° —• e e ~ e e ~ B a c k g r o u n d  82  7.3  R e t u r n i n g Tracks  83  +  +  8  9  7.4  B e a m Buckets w i t h more t h a n 1 7T°  83  7.5  n e e ~ Background  83  +  Fitting Procedure  87  8.1  M a x i m u m L i k e l i h o o d Technique - the Bayesian A p p r o a c h  91  8.2  x  8.3  S u m m a r y of F i t t i n g Results  2  M i n i m i z a t i o n - the Frequentist A p p r o a c h  98 101  E v a l u a t i o n of Systematic E r r o r s  104  9.1  105  T i m e Dependent Systematic E r r o r s 9.1.1  Stop D i s t r i b u t i o n  . 105  9.1.2  Target L o c a t i o n  106  9.1.3  C h a m b e r Geometries  106  9.1.4  Magnetic Field  107  9.1.5  Analysis Cuts  110  9.2  T i m e Independent Systematic E r r o r s  Ill  9.3  O t h e r Systematic E r r o r s  112  9.4  Summary  115  10 S u m m a r y a n d Conclusions  117  Bibliography  120  List of Figures  1.1  F e y n m a n d i a g r a m for electron-pion scattering. T h e b l o b represents the extended electromagnetic structure of the p i o n ; its f o r m factor. T h e 7* represents the heavy v i r t u a l p h o t o n w h i c h scatters the i n c o m i n g electron. T h e process can be thought of as the s u m of two processes shown; the blob contains p mesons  1.2  7  F e y n m a n d i a g r a m for p i o n scattering. T h e blob represents the effect of the p i o n form factor  1.3  7  F e y n m a n diagrams for the charge exchange reaction, a) t h r o u g h d) show the different reactions available to study the charged p i o n form factor. In reactions a) a n d b) the photon m o m e n t u m q is positive; i n c) a n d d) 2  it is negative 1.4  8  In a) we show the F e y n m a n d i a g r a m for the t r a n s i t i o n A —* B~/*, followed by the decay of the v i r t u a l photon 7* — > e e~. +  In b) the same t r a n s i t i o n  is depicted, where the particle B is specifically a p h o t o n . F i g u r e c) shows the p r o d u c t i o n of a neutral meson t h r o u g h two v i r t u a l photons, one of w h i c h is nearly real (its energy is very close to its m o m e n t u m ) .  The  range of m o m e n t u m transfer of the v i r t u a l photon used to probe the meson structure is indicated 1.5  9  T h e K r o l l - W a d a d i s t r i b u t i o n . T h e surface plot shows the d i s t r i b u t i o n i n x a n d y.  .  13  1.6  T h e projection of the K r o l l - W a d a d i s t r i b u t i o n i n x (note the l o g a r i t h m i c scale - the effect of a o n a linear scale is invisible i f the acceptance is u n i f o r m over a l l x).  Here, the solid line shows the d i s t r i b u t i o n w i t h  a = 0.0; the dashed line, the effect of a = 0.1 2.1  D i a g r a m for 7r° —• e e~j +  14  i n the V M D . T h e blobs represent the 2- a n d  3-particle couplings of the vector mesons t o the i n i t i a l TT a n d final state photons 2.2  16  F e y n m a n d i a g r a m for TT° — > e e~j i n the quark loop m o d e l . One sums +  over a l l possible quark species a n d colours 2.3  19  E x p e r i m e n t a l results for the f o r m factor slopes of heavier mesons i n the region of timelike m o m e n t u m transfer. T h e dotted line shows the range of V M D expectations. T h e experimental results for the 7r° f o r m factor are also shown for comparison. T h e recent C E L L O results (clustered a r o u n d 1991) are a l l measured for spacelike m o m e n t u m transfer. T h e vertical scale is the f o r m factor d i v i d e d by the square of the mass of the decaying meson, so that the results for different mesons m a y be compared. 28  3.1  T h e S I N D R U M I detector  35  3.2  D e t a i l of the target. A l s o shown are the lead moderator a n d the i n n e r most wire chamber. T h e superinsulation a r o u n d the v a c u u m cylinder is not shown.  4.1  36  D i s t r i b u t i o n 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 l u m i n u m support r i n g , target support structure, a n d chamber 1 are clearly visible  47  4.2  D i s t r i b u t i o n i n opening angle of e e~ pairs. T h e sharp peak at 156° is +  due to photon conversion events. T h e broad peak at 110° is due t o the a s y m m e t r i c a l <f> opening angle cut of the trigger 4.3  49  D i s t r i b u t i o n i n x a n d t o t a l energy of e e ~ pairs plus the neutron kinetic +  energy. T h e ne e~ +  events populate the horizontal b a n d at 130 M e V .  T h e D a l i t z d a t a i n h a b i t the slanted region u p to x = 1.0. T h e c u r v i n g b r a n c h at s m a l l x a n d low total energy peeling away f r o m the D a l i t z region are events w i t h an e x t r a photon w h i c h radiates away more energy. 50 4.4  D i s t r i b u t i o n of e e~ pairs i n the quantity E Pt +  t  a n d transverse opening  angle. T h e D a l i t z events are constrained to lie i n the box between 107 a n d 163 M e V . T h e slanted bands are the ne e~ events. T h e region below +  107 M e V is i n h a b i t e d b y the radiative events 5.1  51  T w o - d i m e n s i o n a l weighting function for the s i m u l a t i o n , designed t o m a t c h the stop d i s t r i b u t i o n to that of the data. T h e m a x i m u m height is roughly 7, the average is 0.8  5.2  58  T h e figures illustrate the action of the simulated trigger. T h e histogram represents the simulated d a t a before passing through the trigger, the points, after the trigger.  I n a) through c) we show the effect o n the  D a l i t z s i m u l a t i o n , while i n d) through f) we show the n e e ~ s i m u l a t i o n .  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 t u a 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 m a t r i x element, as calculated i n [18]. T h e surface is s y m metric about y = 0  66  vin  6.4  Corrections t o the K r o l l - W a d a m a t r i x element as a function o f x, as calculated i n M i k a e l i a n and S m i t h [18]  6.5  67  Verification of the published radiative corrections. T h e line are the corrections as calculated b y M i k a e l i a n a n d S m i t h [18] a n d the points are the result of the n u m e r i c a l integration using the p r o g r a m of Roberts a n d S m i t h [21]  6.6  72  Invariant mass of lepton pair versus the total energy, for the simulated events. W e plot b o t h the ne e~ +  a n d the e e~y events, b o t h w i t h r a d i a +  tive corrections 6.7  EP t  t  73  of lepton p a i r versus the transverse opening angle, for the s i m u -  lated events. W e plot b o t h the ne e~ +  a n d the e e~j events, b o t h w i t h +  radiative corrections. T h e radiative tails are clearly visible 6.8  T h e E Pt t  74  d i s t r i b u t i o n for the D a l i t z events. T h e h i s t o g r a m is the s i m -  u l a t i o n , the points, the data. Note that the radiative t a i l is very well fit 6.9  75  T h e c o n t r i b u t i o n to x and (j> from the v i r t u a l (solid line) a n d bremsstrahlung (dotted line) corrections t o the total (triangles) s i m u l a t i o n . T h e t o t a l c o n t r i b u t i o n of the radiative corrections is shown b y the shaded histogram  76  6.10 F e y n m a n diagrams for the radiative corrections t o the process ir~p —* ne+e".  78  6.11 T o t a l energy of the lepton pair for the ne e~ +  events. T h e points show  the d a t a , the h i s t o g r a m is the s i m u l a t i o n . T h e radiative t a i l is generally well fit, although a n underestimation seems likely. expected t o be correct to roughly 15%  T h e s i m u l a t i o n is 80  7.1  T h e determination of the ratio of n e e +  to e e +  7. R a d i a t i v e corrections  to b o t h reactions are included 8.1  85  T h e performance of the full s i m u l a t i o n for various k i n e m a t i c a l variables. A l l radiative corrections and backgrounds are i n c l u d e d . T h e points are the d a t a , the histogram, the s i m u l a t i o n  8.2  89  T h e performance of the full s i m u l a t i o n for various k i n e m a t i c a l variables. A l l radiative corrections a n d backgrounds are i n c l u d e d . T h e points are the d a t a , the histogram, the s i m u l a t i o n  8.3  90  G r a p h i c a l s u m m a r y of the results of the m a x i m u m l i k e l i h o o d fitting for a  x  (white boxes) a n d  (dark boxes) for each of the four  geometries.  O n the left are the results without radiative corrections, o n the right the results i n c l u d i n g the radiative corrections 8.4  G r a p h i c a l s u m m a r y of the results of the x  97 2  fitting for a  x  (white boxes)  a n d a<£ (dark boxes) for each of the four geometries. O n the left are the results without radiative corrections, on the left the results i n c l u d i n g the radiative corrections. 8.5  100  S u m m a r y of results for a l l four geometries, i n x a n d <f>. T h e four points to the left of the double line represent the results for geometries 2, 4, 5, a n d 6, respectively; the point to the right indicates the error-weighted average.  Its s m a l l error bar shows the statistical error, a n d the larger  one shows the combined statistical error a n d s t a n d a r d error of the fluctuations. T h e results i n x a n d <f> are averaged to o b t a i n the final results. 9.1  T h e result of the c a l i b r a t i o n on the n e e " peak for geometry 4. +  102  The  net change to the magnetic field is about 0.5%. O n the left, before the c a l i b r a t i o n , on the right, after  ;  108  9.2  T h e result of the calibration on the D a l i t z box for geometry 4.  The  net change to the magnetic field is about 0.5%. O n the left, before the c a l i b r a t i o n , on the right, after 9.3  108  T h e performance of the n e e ~ s i m u l a t i o n ( w i t h radiative corrections) +  for some areas of phase space, for geometry 4. T h e points are the d a t a ; the h i s t o g r a m , the s i m u l a t i o n . Since the agreement between the d a t a a n d the M o n t e C a r l o is good, we are confident that we u n d e r s t a n d the action of the trigger 9.4  113  T h e performance of the ne e~ +  s i m u l a t i o n ( w i t h radiative corrections)  for some areas of phase space, for geometry 4. T h e points are the d a t a ; the h i s t o g r a m , the simulation. Since the agreement between the d a t a a n d the M o n t e C a r l o 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 n e u t r a l mesons, together w i t h the experimental results. T h e theoretical value for the 7r° f o r m factor is included for comparison. See also figure 2.3  2.2  22  S u m m a r y of previous experiments to measure the f o r m 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  B a c k g o u n d processes considered and their calculated c o n t r i b u t i o n to the t o t a l sample, w h i c h numbers roughly 100,000 events.  Conversions are  modelled i n the l i q u i d hydrogen target, i n the target walls, a n d i n c h a m ber 1 7.2  81  T h e c o n t r i b u t i o n of 7r"p —» ne e~ +  expressed as a fraction of the t o t a l  sample. O t h e r background contributions are negligible. T h e quantities N provide a means for assessing the efficiency of the E P t  t  separation cut;  they are not products of the fit, and their magnitudes are of no special importance 8.1  84  Results for a  x  and  from m a x i m u m likelihood fitting, for each of the  various geometries, w i t h a n d without radiative corrections. E r r o r s are statistical only 8.2  Results from x  98 2  m i n i m i z a t i o n , for each of the various geometries, w i t h  a n d without radiative corrections. E r r o r s are statistical only. T h e per degree of freedom varies from 1.2 to 1.7 for the various fits  x  2  101  8.3  S u m m a r y of the fit results for each geometry, fitting m e t h o d , a n d spect r u m . R a d i a t i v e corrections a n d n~p —• ne e~ b a c k g r o u n d are i n c l u d e d . +  I n d i v i d u a l errors are statistical only. T h e s u m m a r y value given is the error-weighted average, a n d its error takes into account the statistical error only. 9.1  Systematic errors from various sources. T h e table is d i v i d e d into t i m e dependent errors (top) a n d time-independent errors (bottom)  9.2  103  105  Results of the magnetic field calibrations for the various runs. T h e m a g net was t u r n e d off a n d on between the various runs  109  Acknowledgements  I ' d like to t h a n k m y supervisor D r . C h r i s W a l t h a m for his countless hours of patient discussion, helpful suggestions a n d seemingly endless good h u m o u r . T h a n k s also to the members of the S I N D R U M group, most especially D r . A n d r i e s van der Schaaf, D r . C a r s t e n N i e b u h r , a n d D r . W i l h e l m B e r t l , for their help a n d suggestions. Invaluable comments regarding the radiative corrections from D r . Helene Fonvieille a n d D r . Lee R o b e r t s were also very m u c h appreciated. These five years of labour could not have been completed w i t h o u t the m o r a l support of a n d frequent waterings by m y fellow B a r b a r i a n s S u s a n , G l e n a n d E r i k , as well as honorary B a r b a r i a n A n d r e w . I express m y graditude to the N a t u r a l . Science a n d E n g i n e e r i n g Research C o u n c i l ( N S E R C ) C a n a d a for a 1967 Scholarship, enabling me to pursue m y studies, a n d to the U n i v e r s i t y of B r i t i s h C o l u m b i a for financial support.  Chapter 1  Introduction  B e g i n n i n g w i t h the Greek philosophers, h u m a n i t y has speculated about the existence of basic b u i l d i n g blocks of matter. T h e idea that matter might not be infinitely divisible, put f o r t h by D e m o c r i t u s a n d hotly debated a r o u n d 600 B C , was put aside u n t i l the recognition, d u r i n g the 1800's, that a l l m a t e r i a l substances are composed of " a t o m s " of various different "elements". In the early days, it was thought that the atoms were i n d i v i s i b l e , a n d t h a t , therefore, there existed about 100 different k i n d s w h i c h formed the basis of a l l matter. T h e discovery by J . J . T h o m p s o n i n the 1890's [1] that different atoms c o u l d be forced to emit identical, very light, negatively charged "electrons", p o i n t e d to an u n d e r l y i n g u n i f y i n g structure. It was also obvious that the b u l k of the mass of the atoms was therefore associated w i t h positive charges. In 1911, R u t h e r f o r d ' s famous scattering experiment [2] showed that the positive charge was confined to a very s m a l l region (radius 1 0 "  1 1  c m or less) of the a t o m , thus g i v i n g rise to the picture of the  a t o m as a positive nucleus surrounded by negative electrons. T h i s discovery was the s t a r t i n g point for the explosive development of atomic physics that c u l m i n a t e d w i t h the establishment of q u a n t u m mechanics i n the late 1920's. D u r i n g the 1930's, the first accelerators were invented a n d b u 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 e x h i b i t i n g certain c o m m o n properties. In 1964, G e l l - M a n n a n d 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 " , " d o w n " , a n d "strange", were enough to e x p l a i n the observations.  As  experimentalists b u i l t larger a n d larger accelerators, p r o b i n g the structure of matter at higher a n d higher energies, more quarks were added to the list. A s it now stands, the so-called S t a n d a r d 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 f a m i l y is associated a pair of "leptons", of w h i c h the electron is the most familiar: I  \ I * \  These quarks a n d leptons are, as far as we can tell, pointlike. Q u a r k s group together to f o r m " h a d r o n s " : three quarks make particles called " b a r y o n s " , while q u a r k - a n t i q u a r k pairs can be b o u n d to form "mesons" . T h e light leptons stay single. T h e three forces (on this tiny particle scale, gravity is negligible) w h i c h operate on the nuclear scale are understood i n terms of the exchange of other particles force  particle exchanged  electromagnetism  photon  weak strong  W, ±  Z°  gluons  T h e rules w h i c h govern the exchange of these particles are embodied i n the so-called " S t a n d a r d M o d e l " of particle physics. T h e S t a n d a r d M o d e l comes i n two parts. One explains the behaviour of the electromagnetic (which, i n its marriage to q u a n t u m mechanics is called Q u a n t u m E l e c t r o dynamics, or Q E D ) a n d weak forces a n d is hence called the Electroweak M o d e l . T h e  other part predicts the effects of the strong force a n d is called, analogous to Q E D (and since gluons carry a quality dubbed "colour" by physicists), Q u a n t u m C h r o m o d y n a m ics ( Q C D ) . T h e Electroweak theory has proven fantastically successful in predicting, to very h i g h accuracy, the results of a l l kinds of particle decays a n d collisions. Q C D , w h i c h governs how quarks stick together i n particles, is a m u c h more difficult theory to calculate a n d 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 S t a n d a r d M o d e l , despite its successes, leaves m a n y questions unanswered. does not explain the observed family structure of the quarks a n d leptons. not e x p l a i n their masses. 4 or more.  It  It does  It does not explain why we don't see quarks i n groups of  A n d it does not include gravity.  W h i l e theoretical physicists labour to  invent models w h i c h simultaneously answer these questions a n d unify a l l the forces into one m a t h e m a t i c a l construct, experimental physicists work h a r d to find holes i n the existing S t a n d a r d M o d e l - discrepancies between the theoretical predictions a n d reality.  S u c h holes m a y i l l u m i n a t e its shortcomings a n d provide clues towards the  creation of a better theory. structure.  It is i m p o r t a n t , then, to attack the p r o b l e m of particle  T h e postulated quark structure of the hadrons a n d mesons should have  testable consequences.  1.1  H a d r o n Structure—Form Factors  In the classical m e t h o d of s t u d y i n g particle structure, one b o m b a r d s a n object w i t h electrons, d la R u t h e r f o r d . For example, i n order to probe the charge d i s t r i b u t i o n of a h a d r o n like a p r o t o n , one measures the angular d i s t r i b u t i o n of the scattered  electrons  a n d compares it to the d i s t r i b u t i o n obtained by scattering off of a point charge: do dQ,  do dtl point  ^|2  2  ^  U s i n g q u a n t u m electrodynamics, the scattering from a point target, jfi , point  can be  calculated to a h i g h degree of accuracy. T h e function F is called the target's (in this case, the proton's) form factor, a n d describes its deviation f r o m pointlikeness a n d hence gives an idea of the structure of the target. T h e form factor depends o n the m o m e n t u m transfer q . 2  If q is s m a l l then we find that F(q ) 2  2  w 1; i n other words, the electron  doesn't have enough energy to resolve the inner structure of the target a n d the target looks p o i n t l i k e . A s a simple i l l u s t r a t i o n , if we model the p r o t o n as a motionless (static a n d n o n relativistic), spinless blob w i t h a spherically symmetric charge d i s t r i b u t i o n p(r) = e  (1.2)  w i t h m some constant (to be determined by the experiment), then the f o r m factor is the fourier transform of the charge d i s t r i b u t i o n : F{\q \)  =  2  J  r p(r)e- ' 'd r 2  iq r  (1.3)  3  (1.4) where 1.4 is the so-called dipole function. F(q ) 2  F o r small q , equation 1.3 has the expansion 2  = 1 - ±\q \(r ) + 2  2  order(q ) 4  (1.5)  where (r ) = / 2  r p(r)dr 2  (1.6)  N o w , defining the form factor slope b by (1.7) we see t h a t , i n this simple case, (1.8)  so that e x t r a c t i o n of b, by measuring the form factor at different q values, a n d 2  fitting  it w i t h the dipole function a n d extrapolating back to q = 0, allows the determination 2  of the spread of the charge d i s t r i b u t i o n of our simple " p r o t o n " . R e a l particles, however, are neither static nor (in general) spinless, so that the analysis is more complicated t h a n that outlined here.  T h e scattering electron may  interact not only w i t h the particle's charge, but also w i t h its magnetic moment  (due  to its spin), a n d thus, i n general, we may expect two form factors to come into play. Obviously, the relationship between the shape of the hypothesized charge d i s t r i b u t i o n a n d the f o r m 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 B r e i t frame) i n w h i c h no energy is transferred to the target (and the mangitude of the projectile m o m e n t u m is unchanged), a connection be derived between the magnetic moment d i s t r i b u t i o n a n d the magnetic form factor, and between the electric charge d i s t r i b u t i o n a n d the electric f o r m factor. In scattering electrons from protons it is found t h a t , i n this frame, both f o r m factors of the real proton are p r o p o r t i o n a l to the dipole f u n c t i o n , w i t h  .  ^  (  '  -  o  ^  y  "  ( i  -  9 )  Hence b o t h d i s t r i b u t i o n s are approximately exponential. D e t e r m i n a t i o n of the form factor slopes allows determination of the "size" of the proton; it is found that b o t h the magnetic a n d electric charge distributions have ( r ) « (0.8 x 1 0 ~ 2  13  cm) . 2  In s u m m a r y , then, we expect one form factor for s p i n 0 particles, a n d two form factors i f the particle has s p i n |. T h e form factors can be related i n some way to the particle's charge a n d magnetic moment distributions. In the case of the p r o t o n , it turns out that a reasonable assumption seems to be that these d i s t r i b u t i o n s are exponential.  1.2  P i o n F o r m F a c t o r - P a r t i c l e Exchange  T h e charged p i o n is a spinless particle. We might expect, therefore, to be able to describe its electromagnetic structure by a single form factor.  T h e assumption that  works well for the p r o t o n ; namely, that the charge d i s t r i b u t i o n is exponential; does not work for the case of the 7 r . ±  In fact, the experimental d a t a fits a form factor of the  form (1.10) with m  2  w 0.56 GeV . 2  T h e form factor above corresponds to a charge d i s t r i b u t i o n of  the f o r m (1.11) the " Y u k a w a p o t e n t i a l " , as opposed to the purely exponential case as for the p r o t o n . O n e can also t h i n k of the form factor, instead of arising f r o m some extended electromagnetic d i s t r i b u t i o n , as arising from the exchange of particles. T h i s picture is an extension of the q u a n t u m picture of photon exchange m e d i a t i n g the electromagnetic interaction between two charges. In this generalization, we allow the p h o t o n to assume some mass (which it may, as long as it is for a short enough time that the Heisenberg uncertainty p r i n c i p l e is not violated). T h i s shortens its range by an exponential factor e~ , mr  a n d we recover exactly the formulae above. We note that the mass m i n equation  1.10 above is roughly m w 770 MeV,  w h i c h corresponds to the mass of the p meson.  Since the p meson has the same q u a n t u m numbers as the p h o t o n , it can be thought of as s i m p l y a heavy photon. T h e extended electromagnetic structure of the 7T* can then be described by allowing the p i o n to emit a p, w h i c h then scatters the i n c o m i n g electron. In this picture, then, the p i o n is thought of as not as a single, rice-crispy-like object, but as some fuzzy cloud of short-lived p mesons.  W e depict this graphically  i n figure 1.1. Measurement of the form factor provides i n f o r m a t i o n about this fuzzy  c l o u d : how b i g i t is, i f any other mesons other t h a n p's are present, etc.  e  e  e  e  +  P  F(q*) F i g u r e 1.1: F e y n m a n d i a g r a m for electron-pion scattering. T h e b l o b represents the extended electromagnetic structure of the p i o n ; its form factor. T h e 7* represents the heavy v i r t u a l p h o t o n w h i c h scatters the i n c o m i n g electron. T h e process can be thought of as the s u m 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 f o r m factor, since it is not possible to make a pion target. Instead, experimentalists make use of p i o n 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 o n l y i n the range of m o m e n t u m transfer examined (in practise, p i o n scattering is l i m i t e d to q < 0.2 G e V ) . 7  2  F i g u r e 1.2: F e y n m a n d i a g r a m for p i o n scattering. T h e blob represents the effect of the p i o n f o r m factor.  A n o t h e r reaction w h i c h m a y be used to study the charged p i o n f o r m factor is the socalled "charge exchange" reaction a n d its relatives, shown i n figure 1.3. T h e advantage of this reaction is that it can be used i n m a n y directions, as illustrated i n figure 1.3: given that one can make 7 r , 7r~, a n d electron beams, as well as b o t h p r o t o n a n d +  n e u t r o n targets, a m y r i a d of experimental options are possible. Different ranges of q  2  can be p r o b e d , a n d the results of the different experiments can be combined to f o r m a more complete p i c t u r e of the p i o n form factor.  F i g u r e 1.3: F e y n m a n diagrams for the charge exchange reaction, a) t h r o u g h d) show the different reactions available to study the charged p i o n f o r m factor. I n reactions a) a n d b) the p h o t o n m o m e n t u m q is positive; i n c) a n d d) i t is negative. 2  A s can be i m a g i n e d , however, this reaction, since it involves neutrons a n d protons w h i c h are themselves extended objects even less p o i n t l i k e t h a n the p i o n , provides a more indirect a n d complicated way of extracting i n f o r m a t i o n o n the p i o n f o r m factor. I n fact, the f o r m factor one is s t u d y i n g is not the same one as may be measured i n p i o n p r o d u c t i o n or electron-pion scattering experiments. In the charge exchange case, the  F i g u r e 1.4: In a) we show the F e y n m a n d i a g r a m for the t r a n s i t i o n A —* £7", followed by the decay of the v i r t u a l photon 7* —» e e ~ . In b) the same t r a n s i t i o n is depicted, where the particle B is specifically a p h o t o n . F i g u r e c) shows the p r o d u c t i o n of a n e u t r a l meson t h r o u g h two v i r t u a l photons, one of w h i c h is nearly real (its energy is very close to its m o m e n t u m ) . T h e range of m o m e n t u m transfer of the v i r t u a l p h o t o n used to probe the meson structure is i n d i c a t e d . +  f o r m factor involved describes the transition  of the i n c o m i n g p i o n a n d the target particle  A (proton or neutron) into a photon a n d the product particle B (neutron or proton). T h e form factor i n question is thus k n o w n as a " t r a n s i t i o n " f o r m factor (as opposed to the previous " s t a t i c " form factor).  It describes the electromagnetic properties of the  c l o u d of v i r t u a l particles d u r i n g the reaction n + A —• B .  1.3  F o r m Factors of N e u t r a l M e s o n s  In the particle exchange picture, processes i n v o l v i n g a single photon intermediate state such as those p i c t u r e d i n figures 1.1 a n d 1.2 are forbidden i n the case of mesons w i t h zero spin a n d charge like the 7r°, b o t h by charge conjugation invariance, a n d by conservation of angular m o m e n t u m ; the photon has a spin of 1, while the 7r° has s p i n 0. One cannot conserve angular m o m e n t u m i n allowing a 7r° to emit a single p h o t o n ; hence these particles cannot couple to single photons a n d their static f o r m factors are identically zero. However, particle transitions or decays of the k i n d p i c t u r e d i n figure 1.4 (a a n d b) are allowed, since the parent meson and its decay product can have different spins a n d charge conjugation properties. One can therefore study the electromagnetic properties of n e u t r a l mesons by s t u d y i n g their transition form factors. N o t e , however, that it is not meaningful to relate the t r a n s i t i o n form factor to the "size" of a charge d i s t r i b u t i o n , as is the case w i t h the static form factor. T h e phenomena of particle decay a n d t r a n s i t i o n are q u a n t u m mechanical i n nature, w h i c h the semi-classical picture of a n electron scattering from a charge d i s t r i b u t i o n has no way of describing. T h e t r a n s i t i o n form factor is an a b s t r a c t i o n , albeit a useful one; given some detailed m o d e l of the i n t e r n a l structure of the meson, it is possible to predict the behaviour of the f o r m factor, a n d hence to compare w i t h experiment.  In the special case where the decay product is a photon (figure 1.4b), one is exploiti n g the fact that while the neutral meson cannot couple to a single p h o t o n , coupling to two photons is allowed. It is possible to construct another allowed s i t u a t i o n , i n v o l v i n g the p r o d u c t i o n of the neutral meson by two photons, as shown i n figure 1.4c. T h i s type of p r o d u c t i o n experiment requires an e e " collider. T h e short-lived meson is identified +  by its subsequent decay products. T h e two types of reactions, decay a n d p r o d u c t i o n , complement each other since they probe the structure of the meson i n different regions of m o m e n t u m transfer.  1.4  D a l i t z D e c a y of the 7r°  T h e 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 m e t h o d by w h i c h 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 i n f o r m a t i o n , we w i l l need the decay w i d t h appropriate for a p o i n t l i k e 7r°. T h i s was first calculated by D a l i t z  [5] i n 1951 a n d also  by K r o l l a n d W a d a [6] i n 1955. T h e y isolated the pointlike part of the interaction by n o r m a l i z i n g to the decay 7r° —> 77, o b t a i n i n g , 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 ~ p a i r , n o r m a l i z e d to the 7r° mass a n d y is the energy p a r t i t i o n : +  x  =  ml.  fc±5±£  (1.15)  mlo «  2|P IIP-l(l-cosy) ml, +  (  L  1  6  )  where <p is the opening angle of the e e ~ pair, p a n d q are 4-momenta, p is the 3+  m o m e n t u m . W e also define the "energy p a r t i t i o n " a n d the m i m i n u m x value as follows: V  =  j ^ f i  (1.17)  and Ami In figures 1.5 a n d 1.6 we show the resulting distributions i n x a n d x versus y.  Equation  1.12 gives the rate of 7r° —• e e~7 normalized to the decay n° —> 77; the f o r m factor +  F is n o r m a l i z e d such that -F(O) = 1. A s i n the case of the 7r* f o r m factor, one posits a f o r m factor F of the form  or, w r i t i n g a =  „%  2  . . a n d i n casting the above i n terms of a:, F(x)  = (1 - ax)'  (1.19)  1  For s m a l l x (low m o m e n t u m transfer), we may expand F(x)  w 1 + ax + order(x ) 2  (1.20)  so t h a t , by definition 1.7, the form factor slope is given by a. W e now t u r n to more detailed a n d complicated models of the 7r° structure to o b t a i n theoretical predictions for a.  F i g u r e 1.6: T h e projection of the K r o l l - W a d a d i s t r i b u t i o n i n x (note the logarithmic scale - the effect of a on a linear scale is invisible if the acceptance is u n i f o r m over a l l x). Here, the solid line shows the d i s t r i b u t i o n w i t h a = 0.0; the dashed line, 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 h i g h accuracy by q u a n t u m electrodynamics, any calculation i n v o l v i n g hadrons is o n l y a p p r o x i m a t e . It seems that hadrons are complex objects, a n d their inner structure is not well-understood; models describing the inner structure of hadrons a n d how this manifests itself i n their electromagnetic behaviour currently give o n l y single-digit accuracy. These models take as their s t a r t i n g point the idea that the h a d r o n consists of a cloud of " v i r t u a l " pointlike particles w i n k i n g i n a n d out of existence i n accordance w i t h the uncertainty p r i n c i p l e , a n d that it is these v i r t u a l particles w h i c h interact w i t h the electromagnetic field. In this way, the models can account for the deviation from point-like behaviour a n d give a prediction for the electromagnetic f o r m factor of the hadron.  2.1.1  Vector Meson Dominance M o d e l  (VMD)  In the vector meson dominance m o d e l , 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 q u a n t u m numbers as the photon. T h e m o d e l is h i g h l y phenomenological a n d as such relies on experimental i n p u t for the n u m e r i c a l values of the couplings of the various vector mesons to b o t h the photon a n d hadron(s) being considered. T h e predictive power  Chapter  2.  Historical  16  Background  of the V M D is thus l i m i t e d by the accuracy of the experimental i n p u t .  e  r  F i g u r e 2.1: D i a g r a m for TT° —• e e~-y i n the V M D . T h e blobs represent the 2- a n d 3-particle couplings of the vector mesons to the i n i t i a l TT a n d final state photons. +  T o calculate the p i o n form factor i n this m o d e l , the p i o n " c l o u d " is w r i t t e n as a s u m over intermediate vector meson states. U s i n g figure 2.1 one obtains  [7] an expression  for F i n terms of the m a t r i x elements describing the t r a n s i t i o n from i n i t i a l p i o n to final p h o t o n v i a the intermediate vector mesons V : ^F(3 )«j:{0|j |y)(y|^|7r ) v 2  (2.1)  0  0  where j is the hadronic electromagnetic current. Since we are concerned w i t h low q , 2  the s t a n d a r d approach is to concentrate on the lowest-mass intermediate states; the p,u> a n d <(> mesons. T h e m a t r i x elements i n equation 2.1 above then reduce to the 2a n d 3-particle couplings  f^,  /</,-,, / ^ - y , / W W - Y a n d  quantities whose  magnitudes  m a y be measured experimentally i n other reactions. One thus obtains a n expression for the f o r m factor EV  2\  t  2 /  fpyfpny  to be compared w i t h the expansion for F  fury fumy  fifnf<t>*1 \  ml T h e physical u> a n d <f> mesons are mixtures of the 7 = 5 = 0 singlet (denoted here by u>o) a n d octet (here <j> ) states: 0  \u)  =  A\u ) - B\<f> )  \<f>) with A  2  + B  2  =  0  0  B|wo> +  ^ > 0  = 1, so that one may identify the slope parameter a as follows: a  fry fp-fy  "_L + E.  .™p .™$  m m  +  w l  m m  0\  _d^^on  (A V 3 /7p*7 \/3 p * \ ul 7  mm  LV  l%). ).  mm  T h e m a g n i t u d e of the factor / p - ^ / p ^ / / ^ has been measured to be of order 1, so that if one makes the further simplification that only the p contributes, one m a y o b t a i n an estimate for the slope parameter Til a ss — £ « +0.034 2  In fact, t a k i n g the full expression and using accepted a n d "reasonable" values for a l l parameters (an excellent discussion is given i n [7]), the vector meson dominance m o d e l 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 m a t r i x elements of the form  n  i.e. they involve a s u m over intermediate states. A simple a p p r o x i m a t i o n is to assume that only baryon-antibaryon states contribute to the s u m , |n) = |A/W). Since this leads  to roughly the correct experimentally measured magnitudes of the i n d i v i d u a l couplings /  a n d /n-yy theorists are fairly confident that the a p p r o x i m a t i o n is a v a l i d one. U s i n g  p w 7  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 a n order of magnitude and their signs w o u l d be completely unobtainable; hence it w o u l d not be possible to predict the sign of a. E v e n i n this case, however, the prediction for the magnitude of a stands as i n equation 2.2.  2.1.2  The Quark Loop Model  In the quark loop m o d e l , the properties of hadrons are seen as the result of the point interactions between the constituent quarks (of w h i c h the m o d e l has six, each of w h i c h come i n three different "colours") a n d the photons of the electromagnetic  field.  The  m o d e l led to the first theoretical understanding of the lifetime of the 7r°, i n 7r° —>• 77, as set d o w n by A d l e r [8] i n 1969.  In this framework, the 7r° is seen as a cloud of  quarks, a n d i n order to o b t a i n the experimentally measured value of the lifetime, one must include the effect not only of the different quark species, but also of colour. T h e v i n d i c a t i o n of this seemingly unnecessary e x t r a parameter was a m a j o r t r i u m p h of the quark model. In order to examine the other decay modes of the 7r°, one extends A d l e r ' s analysis to the case where one or b o t h of the photons are v i r t u a l , a n d hence massive. These v i r t u a l photons can then decay electromagnetically i n t o , for example, a n e e ~ +  T h e decay of the  7r°  i n t o two v i r t u a l photons  7*7* is modelled according to the  —>  7r°  F e y n m a n d i a g r a m of figure 2.2. T h e a m p l i t u d e for the decay is A(s ,si,s ',M) 0  2  =  ie (e )f^e k^k^e^F(s s ;M)  =  8ge (e )Me^ k m€ e  2  2  2  q  ulip(7  2  q  u  pa  i  pair.  p  1  a  2  [9] (2.3)  2  x C  k,  7,  e -Ty *  »  k *M  e•  F i g u r e 2.2: F e y n m a n d i a g r a m for 7r° —> e e 7 i n the quark loop model. One sums over a l l possible quark species a n d colours. +  w i t h quark mass M ; fci,&2 the m o m e n t a of the photons a n d £ 1 , 6 2 their polarizations. T h e average charge squared of the constituent quarks, i n c l u d i n g the colour factor, is given b y ( e ) . T h e form factor F is normalized such that F ( 0 , 0 ; M ) = 1. F o r the 2  purposes of e s t i m a t i o n , one assumes that the only quarks that contribute are the u,d p a i r , a n d one sets M  u  = Md = M.  1  E q u a t i o n 2.2 relates the form factor F a n d coupling constant  t o a n integral C  i n v o l v i n g the i n t e r n a l quark mass M a n d the external masses so = (kj + Jc2) ,Si = k\, 2  a n d 52 =  where C is given b y  C = [ - ^ J (2TT) [q* - M 4  2  + ie][(q + k y-M 1  2  I + ie][(q + h + k ) 2  2  —M  2  + ie]  V  (24) ' }  S t a n d a r d techniques allow for the evaluation of C a n d one can study special cases of the general expression: Generalizing equation 2.5 to real mesons, which are superpositions of six quark loops with different flavors, one obtains, instead of 2.5, 1  with g fi the pion-quark coupling; a formula which does not lead to a significantly different result. r  1. So = m^cSi  = s  2  — 0.  T h i s corresponds to the decay of the TT° into two real  photons, a n d using expressions 2.4 a n d 2.4 one may o b t a i n gM . m*o 5—j-arcsin — r 2  fry =  7r m; 2  In the l i m i t m o  2M  0  —• 0 this gives the f o r m u l a for the "triangle a n o m a l y " , w h i c h  v  was set down by A d l e r [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 < 5 < m 2  2  0  , this corresponds to the D a l i t z  decay 7r° —• e e~j. I n this case one obtains an expression for the f o r m factor +  F  M  M )  -  (1- " T f f ^ S ' }  r0-5l In the l i m i t that m„o —> 0 this becomes m  arcsm (m o/2M) J  2  2  F(s,0;M) = — s v  ff  (2.5) v  '  arcsin ^ 2M 2  7  E x p a n d i n g about s = 0, this gives  whence one can immediately identify the slope parameter a = 12M  2  Note that the quark loop model predicts that a > 0. F u r t h e r , if one chooses the quark masses M d w 200 — 300 M e V , it is possible to m a t c h n u m e r i c a l l y the p r e d i c t i o n of U i  V M D (equation 2.2). T h e two models are thus equivalent i n some sense. T h i s m a t c h i n g is k n o w n as Q U  2  duality".  T h e results of the quark-loop calculations are only weakly dependent on the i n i t i a l assumptions of m o v  —> 0, a n d are applicable to other (heavier) meson f o r m factors as  well. A g a i n , using the posited functional form of F F(q ) 2  = (1 -  V ) "  1  the V M D m a y be used to predict the form factors of the rj,j]' a n d u; the theoretical results are shown i n table 2.1, together w i t h the existing experimental d a t a . F o r heavier mesons, the form factor slope a is conventionally expressed i n dimensions of G e V  - 2  .  W e m a y convert to the more familiar dimensionless quantity by scaling the m o m e n t u m transfer b y the m a x i m u m allowed value x = <7 /<?mai) w h i c h , for the 7r° —• e e ~ 7 case, is 2  m  2  0  +  . W e see that the V M D / q u a r k loop predictions are consistent w i t h the experimental  results. Recently, a form factor has been measured i n the decay KL  e e 7 +  -  [10]; i n this  case the f o r m factor is slightly more complex, i n v o l v i n g contributions not only f r o m p, u>, a n d <f> mesons, b u t 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 f o r m [11]  where the m o d e l predicts m w m as i n the 7r° — > e e p  +  7 case, a n d |a| « 0.2 — 0.3; the T  factors are experimentally measured decay rates, included for n o r m a l i z a t i o n . Reference [10] obtains values of m = QlOtH M e V (corresponding to a = 2.7 ± 0.4 G e V " ) a n d 2  a = -0.28 ±0.13.  2.2  Previous  Measurements  W e discuss the techniques used to measure the TT° form factor i n the region of timelike m o m e n t u m transfer, using the D a l i t z decay. A brief discussion of the single spacelike measurement c a n be found i n the description of the previous experiments (experiment n u m b e r 8). T h e r e are two different approaches to measuring the f o r m factor; each leads to a different experimental design.  >  meson  Ctheory ( G e V 1.8  1  i  1.5  u 7T°  .  1.7 1.7  *)  Gezpt ( G e V  J  )  -2.6 ± 5.7 [13] -0.7 ± 1.5 [14] 1.9 ± 0.4 [15] 1.42 ± 0.21 [27] 1.7 ± 0.4 [15] 1.59 ± 0.18 [27] 2.4 ± 0.2 [15] see table 2.2  Table 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. T h e theoretical value for the 7r° f o r m factor is i n c l u d e d for comparison. See also figure 2.3.  1. O n e approach is to collect d a t a over the entire range of invariant mass x a n d to fit i t , u s i n g equation 1.13 and the s t a n d a r d 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 l o g a r i t h m i c scale of the invariant mass s p e c t r u m (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 n o r m a l i z a t i o n to the process n° —> 77.  If this process is not simultaneously m o n i t o r e d a n d  counted (effectively counting the number of neutral pions created), then one must introduce a n o r m a l i z a t i o n factor into the fit. In order to see the full range of invariant mass (which means a f u l l range of 3-momentum a n d opening angle <p, according to equation 1.15) a magnetic spectrometer is usually employed. T h i s device comprises some sort of detector capable of p o s i t i o n measurement s u r r o u n d i n g , 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 m o m e n t u m 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 n o r m a l i z a t i o n is to be measured, some way of counting either the 7r° p r o d u c t i o n , or the i n c o m i n g ir~ flux 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 a n d to measure the rate for 7r° — > e e~j i n this range, compared to the rate of 7r° —* 77. F o r +  such a measurement, one must have a way of counting the n u m b e r of TT° —• 77 events (or equivalently, the number of n e u t r a l pions produced).  T h e advantage  of this strategy is that fewer events are needed since one is concentrating o n a specific range of x; however, background identification is u s u a l l y difficult since the k i n e m a t i c a l i n f o r m a t i o n available is l i m i t e d to such a s m a l l range. T w o - a r m experiments are suited for this type of measurement, the t y p i c a l arrangement being two energy-sensitive devices ( N a l crystals, for example) at some distance from a n d defining some angle about a target. T h e r e is no magnetic field; knowledge of the energy a n d the angle is enough to specify the invariant mass of the e e~ +  p a i r (equation 1.15). A beam counter (counts the i n c o m i n g 7r~) or a  N a l crystal (counts the decay 7r° —+ 77) provides a n o r m a l i z a t i o n . W e now present a short description of the previous experiments performed to measure a, together w i t h a brief review of their salient features.  T h e i r final results are  t a b u l a t e d i n table 2.2 a n d 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 C o l u m b i a University (New Y o r k ) , u s i n g a 6 0 - M e V ir~ beam  w h i c h was slowed down by polyethylene absorbers t o stop i n a l i q u i d hydrogen bubble chamber, surrounded by a magnetic field of first 5.5, then 8.8 k G . T h e magnetic field was u n i f o r m to 4% a n d controlled to ± 2 % . E l e c t r o n track m o m e n t a were measured from photographic plates. U s i n g a sample of 3071 D a l i t z decays, the authors extracted a value for the form factor slope b y c o m p a r i n g the ratio of the n u m b e r of events seen i n r < x < 0.1 t o the number seen i n 0.1 < x < 1.0 w i t h the r a t i o obtained using the calculations of K r o l l a n d W a d a a n d Joseph. T h e y o b t a i n e d a = —0.24 ± 0.16.  R a d i a t i v e corrections t o the process 7r° —• e e~ y +  /  were not i n c l u d e d i n the analysis, nor was possible background f r o m the decay 7T° — >  e e~e e~. +  2. H. Kobrak  +  et al., 1961 [17] : a similar experiment t o the one above; a 68 M e V  p i o n b e a m was brought to rest i n a l i q u i d hydrogen bubble chamber, surrounded by a 24.7 k G field, u n i f o r m to 0.5%. T h e energy resolution was approximately 2.4%, roughly a factor of three better t h a n i n the experiment of Samios et al.. A form factor measurement was done by fitting the d a t a i n x w i t h Joseph's theory, using a sample of 7676 D a l i t z decays, resulting i n a = —0.15 ± 0.10. B a c k g r o u n d from 7T° — > e e~e e~ +  +  was not included, nor were radiative corrections.  3. S. Devons et al., 1969 [18] : a two a r m experiment performed using a 150 M e V p i o n b e a m at the Nevis cyclotron. E a c h a r m consisted of a p a i r of spark chambers m o u n t e d i n front of a s o d i u m iodide crystal, w h i c h functioned as t o t a l energy absorption spectrometers.  T h e two arms, defining a n opening angle of 120°,  surrounded a l i q u i d hydrogen target. A water Cerenkov detector rejected beam electrons, a n d a scintillation counter b e h i n d i t counted the i n c o m i n g ir~ beam. F a c i n g the two s o d i u m iodide arms was a lead plate g a m m a conversion spark chamber, to check for correlations between electron events a n d photon events.  After geometrical cuts, a sample of 2200 D a l i t z decays was obtained (997 of w h i c h h a d converted photons associated w i t h them). T h e energy a n d angle i n f o r m a t i o n was converted to invariant mass a n d a value of the form factor slope was extracted i n two ways: firstly, by fitting the invariant mass s p e c t r u m w i t h Joseph's theory a value of a =  —0.10 ± 0.09 ± 0.13 (the first error is statistical, the second,  systematic) was obtained; a n d secondly, by measuring the t o t a l rate of the D a l i t z decay a value of a = 0 . 1 1 ± 0 . 0 7 ± 0 . 1 2 was measured. T h e 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 c y c l o t r o n , using a magnetic spectrometer consisting of 3 sets of acoustic spark chambers s u r r o u n d i n g two sides a n d the b o t t o m of a l i q u i d hydrogen target to measure the electron m o m e n t u m . T h e target a n d chambers were immersed i n a u n i f o r m 3 k G magnetic field, calibrated to 0.1%.  T h e authors quoted a  m o m e n t u m resolution of 2.7% at 70 M e V / c . A s o d i u m iodide c r y s t a l m o n i t o r e d 7T° —> 77 events, p r o v i d i n g a n o r m a l i z a t i o n . T h e authors extracted a form factor a = 0.02 ± 0.10 by a fit of 2437 events over the range 0.0 < x < 0.8. R a d i a t i v e corrections, calculated according to L a u t r u p a n d S m i t h  [20], were i n c l u d e d i n  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. T h e C E R N P S  p r o v i d e d a 2.8 G e V kaon beam. T h e kaons decayed i n flight i n the apparatus, w h i c h consisted of a 4 m long decay region fitted w i t h p o s i t i o n sensitive proport i o n a l chambers i n the m i d d l e a n d at b o t h ends. T h e decay products were bent magnetically t h r o u g h a set of Cerenkov counters, scintillators, a n d spark c h a m bers, enabling m o m e n t u m determination. T h e magnetic field was calibrated to  center the  TT°  mass to 0.3 M e V / c . A total of 31,458 2  IT -—* 0  e e"7 events were +  collected, a n d the full x spectrum fit to o b t a i n a = 0.10 ± 0.03.  T h e authors  included the radiative corrections as calculated b y M i k a e l i a n a n d S m i t h  [22];  their effect is to increase the value of a b y 0.05. N o systematic error analysis was performed. T h e authors do not take into account any possible c o n t a m i n a t i o n 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, C a n a d a ) , w i t h two large s o d i u m iodide crystals preceded by sets of three wire chambers and scintillators defining a n opening angle of 60°, then 130°, a n d finally 156° about a l i q u i d hydrogen target. A 90 M e V / c ir~ beam was degraded to slow down i n the target, and resulting e e ~ pairs were stopped +  i n the N a l crystals. T h e wire chamber a n d scintillator i n f o r m a t i o n allowed for track traceback to the target. A smaller N a l c r y s t a l , faced w i t h a collimator and charged-particle-veto counters sat off to the side, m o n i t o r i n g 7r° —• 77 events from the target for n o r m a l i z a t i o n . 10,402 events from the 60° d a t a set, most of w h i c h consisted of ir~p —> ne e~ +  events, were used to check the n o r m a l i z a t i o n of the  simulation. T h e 130° sample, containing 11,736 events of w h i c h about 10,000 were 7f° —• e e~7 events, were used to find a = — COll^o®. T h e 156° d a t a were not +  used due to unforeseen levels of TT° —> 77 contamination a n d vertex reconstruction problems. T h e analysis included a thorough evaluation of backgrounds as well as radiative corrections (as formulated by Roberts a n d S m i t h [25]). 7. H. Fonvieille  et al., 1989 [26]: a magnetic spectrometer consisting of two arms,  each c o m p r i s i n g a magnet surrounded o n b o t h sides b y a series of drift chambers a n d scintillators, defined a n angle of 110° out a l i q u i d hydrogen target. T h e magnetic field p e r m i t t e d the m o m e n t u m determination of e e~ +  pairs over a n  angular range of 50° < <j> < 160°. T h e authors state a m o m e n t u m resolution of 3.5% a n d a vertex resolution of roughly 9 m m . T h e magnetic field was calibrated to 0.1%. T o eliminate background, an explicit cut of x < 0.5 was made.  Two  separate r u n s , comprising 18,346 a n d 18,353 7r° —• e e ~ 7 , yielded a = —0.021 ± +  0.036 ± 0.056 a n d 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 statistical, the second systematic). R a d i a t i v e corrections were i n c l u d e d , using an approximate m e t h o d formulated by the authors. B a c k g r o u n d calculations were also performed. 8. CELLO e e~ +  Collaboration,  1991 [27]: this was a measurement performed at the D E S Y  collider i n H a m b u r g , West Germany. U s i n g the process p i c t u r e d i n fig-  ure 1.4c (production of a n e u t r a l pion by 2 photons, one of w h i c h is almost real) they were able to measure the form factor over a very large range of m o m e n t u m transfer (from 0.5 G e V  2  to 2 G e V ) . T h e signature o f . a p i o n p r o d u c t i o n event 2  is that one of the beam electrons w i l l be scattered into the endcap calorimeter close to the beam direction by emission of a heavy v i r t u a l p h o t o n (the other beam electron is h a r d l y affected), together w i t h two clean photons i n the barrel calorimeter surrounding the entire detector from the subsequent decay ir° —> 77. U s i n g 137 events a n d fixing the pion lifetime, they o b t a i n a = 0.0326 ± 0.0026 (errors combined statistical a n d systematic). W h i l e the C E L L O result is of h i g h accuracy a n d covers a wide range of m o m e n t u m transfer, it is a measurement for spacelike m o m e n t u m transfer only. It is of interest to determine the functional form of the n° form factor over the entire range of q ; i n the 2  case of the charged p i o n , for example, m u c h effort has gone into creating a consistent picture of the form factor for b o t h negative and positive q  2  therein).  ([28] a n d the references  8  I  6  1  4 I  %  2  o s s  -  2  - 4 H  ° fi  «4-i  —O  q  -s  S-io •*-» - 1 2  A  7T  A  rj  O  7/  9  -14 -16 1960  1965  1970  1975  1980  1985  T ion 1990  1995  y e a r  F i g u r e 2.3: E x p e r i m e n t a l results for the form factor slopes of heavier mesons i n the region of timelike m o m e n t u m transfer. T h e dotted line shows the range of V M D expectations. 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 a r o u n d 1991) are a l l measured for spacelike m o m e n t u m transfer. T h e vertical scale is the f o r m factor d i v i d e d by the square of the mass of the decaying meson, so that the results for different mesons m a y be compared.  authors N.Samios et a l . , 1 9 6 1 H . K o b r a k et a l . , 1 9 6 1 S.Devons et a l . , 1 9 6 9 J . B u r g e r et a l . , 1 9 7 5 J . F i s c h e r et a l . , 1 9 7 8 P . G u m p l i n g e r et a l . , 1 9 8 7 H.Fonvieille et a l . , 1 9 8 9 C E L L O Collaboration, 1 9 9 1  # events  results  2200  a a a  2437  a =  +0.02  31,458  a =  +0.10  3071 7676  10,009 36,699 137  =  -0.24  ±0.16  =  -0.15  ±0.10  =  +0.01  ±0.11  a = a =  ±0.10 +  0.03  -0.01IQ.06  -0.11 ±0.03  a =  0.0326 ±  ±0.08  0.0026  Table 2 . 2 : S u m m a r y 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 m o m e n t u m transfer, one needs a detection system offering h i g h mom e n t u m / e n e r g y resolution a n d very good energy calibration. A m o m e n t u m uncertainty of 1 5 0 k e V o n 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 calibration, if a magnetic field is to be used, one needs to know it to 0 . 1 % at least. A l s o , since electrons lose approximately 3 0 0 k e V per c m of l i q u i d hydrogen traversed, it is desirable to know where i n the target, to 0 . 5 c m or less, the event originated. Accurate calibration of the magnetic field a n d determination of the track length i n hydrogen b o t h require h i g h statistics. T h e earliest experiments to measure a were done w i t h very low numbers of events, resulting i n h i g h statistical errors as well as h i g h systematic errors.  These experi-  ments were are also done without taking into consideration the radiative corrections o n the 7T° —• e e~7 process, corrections which are highly geometry-dependent a n d may +  be large. T h e first two experiments that d i d include the contributions due to these second-order corrections were those of Burger et a l . a n d Fischer et a l . However, the early theoretical work done o n the radiative corrections was not directly applicable to  experiments, as the calculations d i d not take into account the fact that experiments have l i m i t e d acceptance a n d do not detect a l l events w i t h equal probability. Hence a l l the early results, u p to a n d i n c l u d i n g the first high-statistics experiment performed by Fischer et a l . , are i n doubt. T h e more recent theoretical work of Roberts a n d S m i t h [25], connected w i t h the G u m p l i n g e r experiment, a n d 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 o n b o t h experiments is large. T h e situation i n the timelike region of m o m e n t u m transfer remains unclear.  Chapter 3  E x p e r i m e n t a l Setup  3.1  O v e r v i e w — G e n e r a l Principles  T h e f o r m factor predictions a n d 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 familiar unitless f o r m factor discussed i n chapter 1, we m u l t i p l y the quoted a by the mass squared of the decaying meson (in GeV),  a n d we immediately see that the f o r m factor influence o n the invariant  mass s p e c t r u m of the lepton pair increases w i t h the mass of the decaying meson. One expects the lowest effect for the low-mass 7r°. F o r the decay 7r° —> e e~7, the theoretical +  expectations o u t l i n e d i n the last chapter indicate that the f o r m factor should lead to an increase i n the number of events w i t h increasing invariant mass x.  However, one  expects a « 0.03, so that it cannot change the p a r t i a l decay rate by more t h a n 6%, even at the highest allowed invariant mass x — 1 (see figure 1.5). F o r 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 F e y n m a n +  diagrams i n v o l v i n g e x t r a internally or externally r a d i a t e d photons) p l a y an i m p o r t a n t role. These corrections also increase w i t h invariant mass, going as l n ( ^ p ^ ) , as w i l l be discussed i n chapter 7. In a d d i t i o n , there are m a n y possible sources of b a c k g r o u n d , i n c l u d i n g e e ~ pairs from 7r° —• 77 photon conversions i n the detector m a t e r i a l . +  In designing a n experiment to measure the 7r° f o r m factor, it is essential to m a x i m i z e the sensitivity to a by ensuring that the detected events cover the full range of invariant  mass. One attempts to m i n i m i z e the contributions from background processes by b u i l d ing a low-mass detection system; however, a full s i m u l a t i o n is still required to assess a l l the background to the e e~j sample. One further needs a clear u n d e r s t a n d i n g of the +  radiative corrections a n d of a l l systematic errors. T h e accuracy of the measurement hinges o n the m o m e n t u m / e n e r g y resolution of the detector. It is necessary to know to h i g h precision the acceptance of the detector. T h i s requires a thorough s i m u l a t i o n 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. factories produce high-intensity  Meson  a n d K beams by i l l u m i n a t i n g solid targets w i t h h i g h  energy p r o t o n beams, a n d 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 o b t a i n i n g the n° source have been used: 1. the reaction 7 r p —• 7r°n at rest. T h i s reaction produces roughly 6 n e u t r a l pions _  for every 10 ir~ incident on the target. 2. the reaction K  +  —+  TT TT°. +  T h e kaon is allowed to decay i n flight; 2 1 % of the time,  it w i l l result i n a 7r°. M e t h o d 1 has the following advantages over m e t h o d 2: • approximately 3 times larger y i e l d of n° per beam particle. • m u c h higher b e a m flux is attainable. • m u c h lower final state 7r° m o m e n t u m , resulting i n e e ~ pairs w i t h larger opening +  angles, m a k i n g track identification m u c h easier a n d more accurate.  • lower reaction energy, resulting i n far fewer p h o t o n conversions (background). • n o measurement of the i n c o m i n g beam m o m e n t u m is necessary t o reconstruct the event. M e t h o d 1 has these disadvantages: • a target is required, leading to m u l t i p l e scattering a n d p o t e n t i a l p h o t o n conversion background. • larger amount of b a c k g r o u n d fr o m 7r~p —> ne e~ +  process K  +  t h a n f r o m the corresponding  —» 7r e e~. +  +  If a s m a l l target is used, m u l t i p l e scattering and the n u m b e r of p h o t o n conversions i n the target w i l l be s m a l l . A l s o , the potential n~p —•* ne e~ +  ( " i n t e r n a l conversion")  b a c k g r o u n d has different k i n e m a t i c a l l i m i t s a n d can therefore be d i s c r i m i n a t e d against to a large degree b y a smart trigger a n d by offline cuts. W e cannot e l i m i n a t e i t completely, however, a n d therefore it must be s i m u l a t e d , as w i l l be discussed i n chapter 7. T h i s is not a p r o b l e m , since n~p —• ne e~ +  has been calculated (at rest) a n d mea-  sured experimentally, so that it is well understood. M e t h o d 1, t h e n , w i t h i t s h i g h 7r° flux a n d the r e s u l t i n g e e ~ pairs w i t h h i g h opening angle r e s u l t i n g i n accurate track +  reconstruction, is a good choice.  3.2  E x p e r i m e n t a l Setup  T h e f o r m 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 Institiit ( P S I ) , S w i t z e r l a n d , throughout the eighties. T h e detector has been described i n detail elsewhere [29]- [33]; we w i l l provide only a brief description of the general design a n d function. S I N D R U M I  is a magnetic spectrometer consisting of 5 concentric c y l i n d r i c a l wire chambers surr o u n d e d by a scintillator hodoscope, mounted inside a magnet solenoid. I n figure 3.1 we show a detailed d r a w i n g of the entire device. T h e spectrometer was designed for detecting rare decays i n v o l v i n g electrons i n the final state, so that low detector mass was desirable to l i m i t photon conversion background: a particle traversing a l l five c h a m bers r a d i a l l y encounters only 5.4 • 1 0  - 3  r a d i a t i o n 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 h i g h solid angle coverage: 70% of Air.  3.2.1  B e a m a n d Target  T h e 7rE3 b e a m at P S I provided approximately 1 • 10 95 M e V / c TT~ per second, at a 5  (low) p r i m a r y p r o t o n current of roughly 1/zA ( m a x i m u m 250 fiA). T h e ir~ flux that can be stopped at this energy i n a depth 1.5 c m of l i q u i d hydrogen, given that only 5 0 % of the b e a m spot strikes the small target (radius 1.9 c m , length 12 c m ) , is roughly 1 • 1 0 , 4  leading to a 7r° rate of roughly 7000 per second (1 7r° every 140 /is). T h i s resulted i n about 70 D a l i t z pairs per second i n the detector, w h i c h was as m u c h as the d a t a acquisition system could handle. A system of four quadrupole magnets focussed the beam onto a s m a l l lead cone (the " m o d e r a t o r " ) m o u n t e d i n front of a small l i q u i d hydrogen target, as shown i n figure 3.2. T h e purpose of the moderator was to slow the TT~ beam so that the particles w o u l d stop i n the first 5 c m of the target. Since the b e a m diverges r a p i d l y due to m u l t i p l e scattering after passing t h r o u g h the lead, it was necessary to place the moderator as closely as possible to the target. F o r this reason, the moderator cone was integrated into the target design a n d d i d double duty as a v a c u u m window.  F i g u r e 3.2: D e t a i l of the target. A l s o shown are the lead moderator a n d the innermost wire chamber. T h e superinsulation around the v a c u u m cylinder is not shown.  T h e target itself consisted of a m y l a r cylinder (19 m m radius) of 0.12 m m w a l l thickness, w i t h a spherical end. S u r r o u n d i n g the target was a 25 m m radius M a k r o l o n vacu u m cylinder (0.7 m m wall thickness), w i t h the moderator functioning as the v a c u u m window. Several layers of superinsulation were w r a p p e d a r o u n d the v a c u u m cylinder to prevent ice b u i l d u p on its outer surface. Particles crossing the target a n d v a c u u m cylinder i n the r a d i a l direction w o u l d traverse approximately 4 • 1 0  - 3  r a d i a t i o n lengths  of m a t e r i a l . T h e entire target system could be wheeled i n a n d out of the detector, ena b l i n g accurate repositioning between runs. T h e v a c u u m pressure a n d l i q u i d hydrogen level were controlled a n d monitored by microcomputer throughout the experiment.  3.2.2  SINDRUM I  Spectrometer  S u r r o u n d i n g the target were the 5 m u l t i w i r e p r o p o r t i o n a l chambers, w h i c h allowed for the reconstruction of the tracks of the particles passing t h r o u g h the detector.  We  s u m m a r i z e the attributes of the chambers i n the table below. E a c h chamber consisted  chamber 1 2 3 4 5  radius 3.72 6.40 19.2 25.6 32.0  (cm)  length 9.0 20.0 58.0 69.0 80.0  (cm)  # wires  wire  224 192 512 768 1024  spacing(mm) 1 2 2 2 2  Table 3.1: W i r e chamber specifications  of 2 concentric R o h a c e l l cylinders faced w i t h K a p t o n , on the facing sides of w h i c h were evaporated t h i n layers of a l u m i n u m w h i c h functioned as inner a n d outer cathode planes. T h e anode wires were s t r u n g between the 2 cylinders. T h e wire ends were fastened to fiberglass  p r i n t e d circuit rings (containing the necessary electronics) at either end of  the chamber. C h a m b e r s 2, 3 a n d 5 h a d their inner a n d outer cathode planes etched into strips r u n n i n g at + 4 5 ° a n d —45°, respectively. T h e space between the R o h a c e l l / K a p t o n cylinders was filled w i t h chamber gas (argon-ethane-freon  mix).  A charged particle  traversing the chamber w o u l d ionize the chamber gas, a n d the resulting free electrons w o u l d be accelerated towards the anode i n the electric field between the cathode a n d anode strips. V e r y close to the anode, where the field was intense, secondary i o n i z a t i o n w o u l d result i n an avalanche of electric charges, causing a signal b o t h o n the anode wire (directly) a n d on the nearby cathodes (by induction). These signals were fed into a series of P C O S - I I I amplifiers, discriminators, a n d receivers, w h i c h processed t h e m into a d i g i t a l readout containing the hit wires' cluster m i d p o i n t a n d 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 h i t . T h e angled cathode strips of chambers 2, 3 a n d 5 enabled, i n a d d i t i o n , the determination of the z position of the h i t . T h e s p a t i a l resolution of the <f> measurement was given by the 2 m m wire spacing (CT ~ 0.6 m m ) , a n d the ^-resolution was determined using cosmic rays to be a ~ 0.3 mm. In a d d i t i o n to the cluster m i d p o i n t a n d size, the receivers generated F A S T  OR  signals a n d L A T C H E D O R signals. For b o t h signals, chamber wires were grouped into sectors w h i c h consisted of logically O R - e d neighbouring wires. For the F A S T O R signal, o n l y these O R signals were output. T h e L A T C H E D O R further required a coincidence w i t h a gate signal. T h e 512 wires of chamber 3 were grouped into 16 sectors of 32 neighbouring wires w h i c h were F A S T O R - e d , resulting i n a 16-bit n u m b e r p o i n t i n g to a n address i n the M e m o r y L o o k u p U n i 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 p a t t e r n 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. S i m i l a r l y , the L A T C H E D O R output  f r o m chambers 2-5 were fed into the Track Preselector ( T P S ) microprocessor, w h i c h compared the signal w i t h stored patterns (masks) of acceptable tracks. T h i s enabled a fast track recognition i n the <j> plane. T h e masks were generated u s i n g s i m u l a t e d e e ~ +  pairs. A more detailed description of the operation of the M L U a n d the T P S can be f o u n d i n reference [34]. M o u n t e d o n the outside chamber was a c y l i n d r i c a l hodoscope of 64 scintillator strips, each 88 c m long, 1 c m t h i c k , a n d 3.3 c m wide. T h e ends of each strip were equipped w i t h p h o t o m u l t i p l i e r tubes, the output of w h i c h was fed into 64 " d i s c r i m i n a t o r - m e a n t i m e r s " . These devices provided signals correlated to the time of the track's traverse of the scintillator. T h e 64 hodoscope time signals were subsequently fed into electronics w h i c h allowed for the selection of events w i t h 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 a n d hodoscope were mounted inside an i r o n solenoid magnet, w h i c h p r o v i d e d a u n i f o r m magnetic field parallel to the beam axis (and the anode wires) of 0.33 T . C h a r g e d particles thus described helical paths inside the detector. T h e magnet current was m o n i t o r e d a n d recorded by microcomputer throughout the experiment.  3.2.3  Trigger Logic  T h e scintillator hodoscope h a d a very fast response time, a n d was used to define the start of a n event.  Once two hodoscope hits w i t h i n 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 h a d to show two hit sectors i n chamber 3 w i t h an opening angle of more t h a n 67.5°. T h e n at least two hits i n chamber 1 were required. T h e T P S selected events w i t h at least one negative a n d one positive track. U p o n passing a l l these requirements, the event was passed to the General Purpose M a s t e r ( G P M ) , w h i c h used b o t h the T P S a n d M L U i n f o r m a t i o n to a p p l y a m i n i m u m <f> opening angle cut of a p p r o x i m a t e l y 35°.  Once the event h a d passed this trigger stage, the d a t a was w r i t t e n into a n event buffer o n the P D P a n d passed to the online filter for further processing.  3.2.4  Online Filter  B y using the full wire h i t i n f o r m a t i o n ( i n contrast to the trigger's use of only the L A T C H E D a n d F A S T O R ' s ) from a l l 5 chambers a n d the hodoscope, the filter performed a more detailed track reconstruction i n the r — <f> plane (as described i n [35]) a n d 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 a n d transverse m o m e n t u m p . T h e emission t  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, r e q u i r i n g at least two tracks of opposite p o l a r i t y 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 T h e first requirement rejected tracks w h i c h do not pass t h r o u g h the target, while the second ensured that a pair of correlated tracks existed. S m a l l opening angle pairs were rejected, removing background and biasing the D a l i t z sample towards higher invariant mass. A further hodoscope t i m i n g cut eliminated a l l but the very p r o m p t tracks.  3.3  Data Acquisition  D a t a for the f o r m factor measurement were taken at P S I d u r i n g a n dual-purpose exp e r i m e n t a l r u n lasting f r o m the end of A p r i l to the end of October 1987, w i t h a onem o n t h cyclotron maintenance break. T h e long r u n period was necessary to measure the b r a n c h i n g r a t i o of the rare decay 7r° —• e e ~ , the results of w h i c h have been p u b l i s h e d +  [36]. D u r i n g this r u n , the magnetic field was set to 0.33 T a n d the trigger conditions set as o u t l i n e d above, a n d D a l i t z d a t a were taken on four separate occasions, each time for a day or less. T h e detector was taken apart between these runs for repairs to the wire chambers, resulting i n four distinct d a t a sets (labelled "geometries 2, 4, 5, a n d 6"), comprising a t o t a l of approximately 0.8 x 10 events. 6  These raw d a t a were subsequently passed through an offline track recognition p r o g r a m to reconstruct the event kinematics precisely, w h i c h we w i l l describe i n the next chapter.  Chapter 4  Offline  4.1  Analysis  Overview  T h e offline analysis proceeded i n two phases: first, the raw d a t a was read from tape a n d a p a t t e r n recognition program (which incorporated detailed detector c a l i b r a t i o n inform a t i o n ) translated the wire and hodoscope hit information into particle  3-momentum  a n d event vertex location; second, the final analysis looped over a l l tracks i n each event a n d selected the "best" p a i r , then applied final h a r d cuts ( d u p l i c a t i n g a l l cuts that went before, i n c l u d i n g those made by the online system) to these tracks a n d  finally  allowed for a detailed examination of the resulting k i n e m a t i c a l distributions. W e give a brief description of each of these elements; more details (especially concerning the c a l i b r a t i o n a n d p a t t e r n 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 f o r m rings, the resulting space at the seam was not the same as the wire spacing. A l s o , the seams at either end of the chambers were rotated w i t h respect to one another, i n d u c i n g a slight twist i n the anode wires. Corrections for the anode " g a p " a n d " t w i s t " h a d to be made for each of the five chambers i n order to produce the correct x,y  information.  T h e inner a n d outer cathode planes of chambers 2, 3 a n d 5 were also rotated slightly w i t h respect to one another. In a d d i t i o n , m i n o r variations occurred d u r i n g the r u n p e r i o d , w h e n , on four separate occasions, the spectrometer was t u r n e d off a n d the inner chambers removed for repairs. U p o n their reinsertion, the chambers were slightly rotated a n d offset w i t h respect to one another. T h e relative rotations a n d locations of the chambers were calibrated using cosmic ray d a t a w i t h no magnetic field (straight, throughgoing tracks). T h e following q u a n t i ties were determined (i = 1 , 2 , 3 , 4 , 5 ; j = 1 , 2 , 3 , 4 ; k = 2 , 3 , 5 ; / = 2,3) : • ^anodes '• *  n e  r o t a t i o n of the j  th  chamber relative to chamber 5  • ^cathodes '• *he relative r o t a t i o n between the inner a n d outer cathodes, for those chambers w i t h z i n f o r m a t i o n • A i  J  , Ay  J  : the x a n d y offsets of the m i d p o i n t of each chamber (relative to  chamber 5) • Az  l  : the offset i n m i d p o i n t z relative to chamber 5  • up down '• a  r o t a t i o n of the anode prints at the u p - a n d downstream end of each  chamber • 'up down '• s  seam gap of the anode prints at the u p - a n d downstream ends of  each chamber T y p i c a l x,y  offsets were on the order of 0.5 m m . T h e largest z offsets were 1 m m , for  chamber 3; those for chamber 2 were on the order of 0.5 m m . T h e largest r o t a t i o n of 2.4 degrees (or 1.5 wires) was found for chamber 1; a l l other rotations were less t h a n a single wire. T h e anode seam gaps were less t h a n 0.5 m m (different f r o m the wire  spacing), so that the twist produced i n the anode over the length of the chamber was very slight. T h e hodoscope time signals were also calibrated (for a more detailed description of the c a l i b r a t i o n procedure, see reference [33]); the digital output from the discriminators h a d to be translated into time signals i n ns a n d corrected for z p o s i t i o n - a n d a m p l i t u d e dependence. After c a l i b r a t i o n , a time resolution o = 315 ps per hodoscope scintillator strip was achieved.  4.3  P a t t e r n R e c o g n i t i o n a n d Track F i t t i n g  W i t h the c a l i b r a t i o n information i n h a n d , accurate translation of the wire a n d hodoscope hits into x,y,z  a n d time co-ordinates was possible.  T h e p a t t e r n recognition  software fit helical tracks to this translated information. T h e p a t t e r n recognition and track fitting proceeded i n three steps: first, wire hits were fit w i t h circles i n the r — <f> plane; second, these results were combined w i t h the z hit i n f o r m a t i o n a n d d a t a fit w i t h straight lines i n the the arc length 5 versus z plane. F i n a l l y , 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 i n the <j> plane i n the homogeneous magnetic field, the radius of w h i c h was p r o p o r t i o n a l to the transverse  momentum.  T h e i o n i z i n g particle rarely caused more t h a n 1 wire to give a signal. T h e p a t t e r n recognition a l g o r i t h m 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 w h i c h fell  w i t h i n a certain distance window into " t r i p l e s " . T h e same was done for the outer 3 chambers. Subsequently, a l l triples sharing a c o m m o n hit i n chamber 3 were fit w i t h circles; chamber 1 hits were weighted by a factor of 4 since the wire spacing was half that of the other chambers. T h e resulting fit was required to have a n acceptable x 2  Those tracks for w h i c h the real chamber 1 hit differed from the circle fit projection by more t h a n 0.5 m m were rejected, as were tracks w i t h a fit D C A of more t h a n 25 m m .  4.3.2  z Fit  T h e r a t i o of the transverse to the l o n g i t u d i n a l m o m e n t u m was constant for a charged particle traversing a magnetic field u n i f o r m i n 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 a n d the track fit w i t h a circle, the arc length could be determined a n d the s — z analysis performed. T h e determination of the 9 angle of emission, combined w i t h the results of the transverse m o m e n t u m P from the r — <f>fit,allowed for the calculation of t  P. z  A t least two z hits were required for a candidate track. E a c h anode signal induced signals o n a few neighbouring cathodes, p r o d u c i n g smeared out cathode "clusters". A l l such clusters, on b o t h the inner and outer cathode planes, were sought, a n d their centres determined. D e a d and damaged cathode strips were interpolated over using the i n f o r m a t i o n from neighbouring strips. A l l inner a n d outer cathode clusters were combined into pairs, each pair defining an angle <j> w h i c h was compared to a l l anode (j> co-ordinates obtained from the wire hit i n f o r m a t i o n . M a t c h i n g <f> co-ordinates resulted i n the assignment of r — <f> — z coordinates to the h i t . Once the track co-ordinates h a d been established, a straight line fit i n s — z was performed.  4.3.3  Vertex Fit  T h e final step i n the event reconstruction was the determination of the event vertex. T h e point m i d w a y along the shortest line between the two tracks was chosen to be the event vertex. F i g u r e 4.1 shows the resulting d i s t r i b u t i o n of event vertices for e e ~ pairs i n r — z. +  T h e target is clearly visible. A l s o visible are the lead moderator i n front of the target, the a l u m i n u m support r i n g at (z «  —110 m m , r J=S 20 m m ) , the a l u m i n u m target  m o u n t i n g at (z > —80 m m ) , a n d chamber 1 at (r « 35) m m . These structures are visible due to the large numbers of photons from 7r° decays converting i n t o e e ~ pairs. +  T h e results of the track fit were w r i t t e n to a file, along w i t h the raw d a t a for each event.  4.4  F i n a l Event Selection — Identification of D a l i t z E v e n t s  T h e final analysis p r o g r a m reads the results of the reconstruction p r o g r a m f r o m the file created a n d loops over a l l pairs of tracks w i t h opposite sign, r e q u i r i n g that the event vertex of each c o m b i n a t i o n is inside the target, a n d chooses the "best p a i r " o n the basis of the x  2  of the vertex fit. T h 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, w h i c h duplicate (and are more stringent than) the cuts applied by the online trigger and filter a n d the track fitting p r o g r a m : • 45° < <f> < 260° : a cut w h i c h duplicates the action of the online trigger a n d filter at roughly 35° < <j> < 260°. • P  t  > 20 M e V / c : S I N D R U M ' s transverse m o m e n t u m threshold is roughly 17  M e V / c , a n d this requirement ensures that any systematic error w i l l be due to the  40  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  oOooQQoaaoaoaorjo a •••••••[][][]• a a • DooQoQQoooiDoa ODD••••••••••DO•a . . . n . -OBOODDDO • • B •  •  0 1  H  30  •  t  0 • •  • • i •  •  Q • • • 0 • •  OH  X  CD  • 0  o • <• I B  B 0  •• o o a a  «  I • 0• • • IBaBD• • • • 0  w 20 O  •a •  •• • o • •  " TI  tit  0 0 • •  0  •QQ  l l•  0 • D O D  0 • 0 o a • < • • OQDQ a •• • •• •••00 0 ">o.  0 • 0 •  •  •a  B OOODDD • • • ••ODODOooo • ODDQDoQo •  • a•••••o o• a••••••a a • • a oDo DO•o B i O D D i a•••••aa o• o o a DO o o o o • • • • o 0 « Q D • •  •  0  • • aa • o • I OIIDIBII D • B l gID io • B I OOG < I 0 1 B  0 • •  B i a o • •• o  • ooaDQDQoaoooD>aoooOODQDiiDOo o•OOODOODOODOOODOQODDODOO0 o a  • •ooODDODODQDDDDooOQQOOODDQooo o  •••••••«• o.ioo o••••••••••••••••••••QODQD o a • 0 ••lDODDO • o ao o•••••••••••••••••••••••••O o•• D • •••••••••••••••••••OODDDDDDODDoo. 0 0 ) DOCLXIDu* • • •• a o • • i L r-i n f I" I I n i-i — POO no . • • 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 • • • • • o o ••a• D • • • • • • • • • • • • • • • • • • • • • • • • • • • 0 ° • D • • •• OD oo o • ••••fflmpprjDDDDDDDDDDDDDDDDoaao 0 • D D ••a o o • •••ULIJULIJLLJDDDDDOOOOOOOOOOODODO. 0  10  >  H  • oonTTTJDDDDDDDDDDDDDDQaDDQDao • o o DOD»•• "••CILTIDIDDDIIDDDDDDDQDDODDO • o.. « HDDO • o OOPPQDCLTIDCXJDDDDODDODD•••••• o • o • • •a o a o •DlinnDDDDDDDDDnDDDDDOODDQDOooii • 0 • •• • • 0 0 0 » o D00| n • O D D o••••••••••••••O•••••••aODDDDa oo Ooo* oDDDDDlIlDDDnDDDDDDDDODDDaooDDo. • o »o ••! a o" • • • • • • • • • • • • • • • D D O D D O D D D O D D D . o . • ooDD[TJTjDDa DDIXDDDDDDODDnODDDDDQOoaDQDi • o . o • H •o ODDJTTlDo • o • •• • ••  8  • «  • • • • • • • • • • • • • • • • • • • O D D •  0  • • D •  oi  o o•a • •DDDCDDo o DODDODDDDDDODDDDOQDODDo o •o D•••••o o .• •••••••••••••••a a ao•o•o••ao •• O O O D D D Q D D D D D D D D « 0 ' D «  • •  I I I I I m T f I I I M T 7 T IT I T T l I i IT ITI I I I I I I I ITI I I I I l I I I  -150 - 1 3 0  V  110  -90  -70  -50  Z vertex position (mm)  F i g u r e 4.1: D i s t r i b u t i o n i n r — z of the distance of m i n i m u m approach for e e ~ pairs. T h e target, m o d e r a t o r , a l u m i n u m support r i n g , target support structure, a n d chamber 1 are clearly visible. +  cut rather t h a n 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 t h i n the region of u n i f o r m 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 w h i c h the magnetic field is u n i f o r m . • r < 19 m m , - 1 1 5 m m < z < - 8 0 m m , 0 m m < z + 104 + y/19 - r 2  2  : cuts  w h i c h ensure that the event happened well inside the target. T h e quantity 104 + \/l9 —r 2  2  follows the curved upstream edge of the target.  After the reconstruction and final analysis cuts above, the four geometries 2, 4, 5 a n d 6 comprise 9294, 43968, 8899, a n d 43464 events, respectively. In figures 4.2 t h r o u g h 4.4, we show d i s t r i b u t i o n s of some of the k i n e m a t i c a l variables of the resulting d a t a . T h e r e is b a c k g r o u n d r e m a i n i n g . F i g u r e 4.2 shows the d i s t r i b u t i o n i n opening angle of the chosen e e~ +  pairs. T h e large, sharp peak at <p « 156° is due to 7r° —» 77 events  i n w h i c h b o t h the photons convert; the resulting leptons, one from each p h o t o n , when boosted i n t o the laboratory frame, exhibit an opening angle of a p p r o x i m a t e l y 156°. In chapter 6 we discuss the cuts used to eliminate these events from the d a t a sample. In figure 4.3 is shown the invariant mass of the e e~ pair (normalized to the 7r° mass) +  versus the t o t a l energy of the pair plus the n e u t r o n , Et to  — T + E n  = 7|p+  T h e events i n the b a n d at E t to  «  +  + E_  + p_| + ml - m + E n  +  +  E.  130 M e V are 7T~p —> n e e ~ events; for these 3+  b o d y events, the total energy of the event is constant a n d equal to the i n i t i a l 7r~p energy.  T h e w i d t h of this b a n d is indicative of the resolution of the detector.  The  D a l i t z events, i n w h i c h the photon carries away energy, populate the slanted b a n d .  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 E t < 110 M e V , but +  to  only at the expense of high invariant mass Dalitz events. This is not so desirable.  I I I I 1 I I IT | I  0  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 "E P , t  t  n  EP t  t  = E  +  + E. + \P + P.\ + T +  n  150  i  130 -  110  u CD  C  90  H  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 • o i - o o i o i o i i o o i o i i o o o D o DDOOOOQDO• •aiaoaiiooai-••ooooDODDDDDOODOo •aoati- oii•OOQODODDDDDQDDDDB oo oioo o oDOOOOOOOODDOo ••••i.•oioooDDDDDODDDDDDD•'o• a••o a• aaoDDDDODDDDDDQOo •••a •• •••••••••••a o••DDDDDDDDDDDQDDDDDDo ••••••DDI  oil 11 M i l  CD  "(3  70  o  IQQDQ  DQODD _ 000000 DDDDDDODD  50  IDDDDODO • •D o••• • • D • • Q i 0 • • • D • . . . I D  30  I  0.0  I I I [ I I I I [ I I I I | I I I I | I I I I | I I I I  0.2  0.4  0.6  0.8  x =[s~ / m N  1.0  1.2  7T  F i g u r e 4.3: D i s t r i b u t i o n i n i a n d total energy of e e " pairs plus the n e u t r o n kinetic energy. T h e ne e~ events populate the horizontal b a n d at 130 M e V . T h e D a l i t z d a t a i n h a b i t the slanted region u p t o x = 1.0. T h e c u r v i n g branch at s m a l l x a n d low t o t a l energy peeling away f r o m the D a l i t z region are events w i t h an e x t r a p h o t o n w h i c h radiates away more energy. +  +  260  i i i i iI i i i i i In i i i In i n Ii i i i i Ii i i i i Ii « m  I« i i i i • •••TXirjOOrXKXXmi**** a.-aoeooODOooaaaa. •ooDgDODQQQQQOQQoa«««««e«eoooDOOOoooo««« <  -  • ooCKJLLCDXKXIOOaaa«««*« •ceooooooacce* - «  CD  2 4 0  •  220  •  r—I  ££)  C cd  200  -  180  z  CD  160  ~_  o  U 0  OX)  C c OH  t20  -  CD > CO  100  -  • 0arXiniXnXX30aaoaa««««a«oao00Q0a««aa«<« • oocnxCDDOOaaoaaa • «• • •••aooaaaaaaaa • • • • a • a o r x X D O O a r j o a o o a a a * • «a•caaaaaaaaaaa oooOQODQOQOOOOOQae.•a«eoaoaao«ao«••• oooooorxDOOCODooao«««««ooooooooao«««'« ooorjOQrjDOOOQOOOQOO* •«oODOOOoaaa««a««« PODOOODDrjQoOOQDaoQecooooooooaooooa ooaaiEDDDOCg300 00«<ioooo«oooooo««««« » • • OOODQQDDDOOOOOOOOcooooooooceooaaaa*• •••• OOOOOOOQQOOa30DDOOoOoooooe«ooooo««««« • • •a • • • • ooooaOQQrJOQDDOODDOOooooooccooo**** • •a '•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 « « « « • • • •••••••OOOOaQDDQQDOODODDDDOooDoooooot • • • • ••••o«OOOOTQCH10rJODDDOOOoooo«flotIoooo««o« • • • • • • • ••••BiOOOCQiXOXIDDDDDOOOOD«ooODooo«0«oe«oo • oaaooOODOOQQDDDDODOOQDOo oo e o o « o e c o a « a • « • • ••••••oooOOOODaDDDaDDDDOODOooooooaooooo«c«««.«. i ••••• «••aooQCODDGDODDDDDDD•oooooooPDooooa«« • •BOOOOPIl I I I I 0DflOOOOOODoODOoao*** • • • • •• • • • •oatoaODfflUDaQDDDQDDDDDDOOooDooooo«o-aa.««a l l oaDDQQODDQDDDDDDQDooooa«OODoooOO«««a«.i • • •• • OoaorXDOCODDODDD 0« iOoaO^^OOQDOOOOOo««ODOOOoooo««o«*aa •. • • • • i iiqOOPDDDDDDDDDDOncoacooooooocoocBOat • a a a i«ooo_3QOOODorJDooo«oca«ooooooaaao«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  80  • g f1111 l I I I I EDDDDDDDDo'• « « . « « o a o n n n D r j a o a a a .oo n n r m m iDqqqqDaDDo«. nnnnnnnnn. • C ODDODDODQ11"  -  aoDODDDOQDOOBi ooDDDDDDDDOoo*•  • • • • • 0 0  cd 60  4 0  OUDDaa•« •ea«aeoODDDDDQQaoa«•• THC|e«..< •e««ooDDDDDDQQOOO«•• • «-aoooaDODDCE30aao« t coooODDQQOrjrJO • • a  -  •  ac ••••aooaQOIIOlaaa a . c • vooooatUXDXiaoe a B• a . « •cooaaaoDTJaa 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 11111 i i i i |  30  60 90 120 150 180 210 240 270  E P (MeV) F i g u r e 4.4: D i s t r i b u t i o n of e e ~ pairs i n the quantity E P +  t  t  a n d transverse opening  angle. T h e D a l i t z events are constrained t o lie i n the b o x between 107 a n d 163 M e V . T h e slanted bands are the n e e " events. T h e region below 107 M e V is i n h a b i t e d by +  the radiative events.  T h e D a l i t z events inhabit a boxlike region between 107 < E Pt t  < 163, where the two  extremes correspond to the cases where the neutron is e m i t t e d parallel a n d antiparallel to the e e~  p a i r , respectively. If we apply the further restriction that E P  +  t  can eliminate ne e~ +  mass events.  t  < 170, we  events to a high degree, without the loss of the h i g h invariant  I n order to assess the efficacy of these cuts, we must simulate b o t h  the 7T° —• e e ~ 7 a n d the 7r~p —>• n e e ~ processes. T h e b a c k g r o u n d also needs to be +  +  simulated. W e t u r n now to a discussion of the simulation procedure.  Chapter 5  Data  Simulation  In order to extract a parameter of the size of the form factor slope from the d a t a , one must u n d e r s t a n d the d a t a very well. In order to achieve this u n d e r s t a n d i n g , we use the C E R N package G E A N T for s i m u l a t i o n of the detector response. A f u l l description of this package a n d its implementation i n our M o n t e C a r l o s i m u l a t i o n c a n be found i n [39] a n d [40]; we give here a brief outline of the steps involved. T h e m a i n steps i n s i m u l a t i n g the d a t a are as follows: 1. Choose the detector geometry, the magnetic field value, a n d the process to be modelled. M a k e any restrictions on the particle kinematics. 2. A s s u m i n g that the Tt~p atom is at rest, the energy available to the subsequent reactions is E = m - +m x  p  — B( - ) v P  = 1077.83941 M e V (the b i n d i n g energy of the  •K~p system is approximately 0.4 k e V [41]), assign the 4-momenta of the resultant particles according to the appropriate m a t r i x element a n d Lorentz transformations, w i t h i n the k i n e m a t i c a l restrictions imposed i n step 1. 3. Decide o n the location of the interaction. 4. Pass the particles through the detector, s t a r t i n g from the interaction point a n d ending when the particles leave the sensitive volume of the detector. M o d e l the p r o d u c t i o n of secondary particles such as photons ( due to electron bremsstrahlung) a n d electrons ( from photon pair p r o d u c t i o n ) a n d pass these through the detector as well.  5. A s the particles lose energy i n the various detector p a r t s , sensitive a n d n o n sensitive, m o d e l the signal produced i n the sensitive ones. 6. W r i t e the resultant response into a d a t a file identical i n format to that of the real data. 7. R u n the trigger s i m u l a t i o n programs. R e a d each event f r o m the d a t a file a n d either discard or accept i t . W e now discuss these steps i n more detail, h i g h l i g h t i n g the most i m p o r t a n t assumptions.  5.1  D e c i d i n g the D e t e c t o r  Geometry  T h e basic setup of the S I N D R U M spectrometer, as discussed i n the previous chapter a n d i n reference  [33] (and the references contained therein) was assumed to be fixed  a n d conforming to the specifications. T h e slight chamber misalignments discussed i n chapter 4 were taken into account. However, since the effect of the anode wire p r i n t gaps a n d twists resulted i n less than a single wire difference i n a l l but the most extreme cases, it was decided not to model t h e m , but to include t h e m only as corrections d u r i n g the track reconstruction of the data. T h e p o s i t i o n of the target was determined on a r u n - b y - r u n basis by e x a m i n i n g the d i s t r i b u t i o n of event vertices, a n d checking the location of the target edge. A t y p i c a l d i s t r i b u t i o n is p i c t u r e d i n figure 4.1. T h e 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 u p s t r e a m . T h e magnetic field was set to 3.313 k G throughout the experimental r u n , a n d the solenoid current monitored at 5 m i n u t e intervals d u r i n g the r u n . T h i s value was recorded i n the d a t a file of each accepted event. T h e measured magnetic field value  was estimated to be accurate to about 1% (using a field m a p produced i n a n 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 a n accuracy of less t h a n 0.5%.  In order to achieve this, the s i m u l a t i o n  was first r u n using the n o m i n a l 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. T h e s i m u l a t i o n was then redone using the corrected field, i n order to account for any changes i n acceptance. T h e magnetic field was u n i f o r m to better t h a n 1% w i t h i n the chosen fiducial volume inside the chambers. Since no accurate field m a p was made, we assume a u n i f o r m field for s i m u l a t i o n purposes, a n d then apply a fiducial cut d u r i n g the analysis.  5.2  G e n e r a t i n g the Particle K i n e m a t i c s  Some m i n i m a l requirements were imposed on the i n i t i a l lepton kinematics i n order to cut d o w n on the amount of computer time required to track the events. These requirements were set below the physical detector thresholds, so that the detector, trigger, a n d filter s i m u l a t i o n a n d subsequent analysis w o u l d determine the event acceptance.  The  cuts imposed were the following: • T h e i n i t i a l electron m o m e n t a should lie w i t h i n S I N D R U M , a n d s h o u l d be large enough to allow the lepton to hit the hodoscope. T h i s l i m i t e d the s i m u l a t i o n to p r o d u c i n g events w i t h 27° < 0 < 135° 12 M e V / c < P  t  Here 6 is the l o n g i t u d i n a l angle of the emitted electron, a n d P is its m o m e n t u m t  i n the x — y plane.  • Since the trigger a n d filter applied a cut i n transverse opening angle, we further restricted the s i m u l a t i o n to generate events l y i n g w i t h i n the region: 15° <4> < 260° t  • T h e above restrictions o n transverse m o m e n t u m a n d opening angle result i n a n effective restriction o n 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 m a t r i x element over this variable, we set the a d d i t i o n a l restriction x > 0.001 e x p l i c i t l y i n order to reduce the computer time needed for the calculation. W e considered the following reactions: 1. 7T° —> e e"7 according to the m a t r i x element set out b y K r o l l a n d W a d a [6], +  w i t h the f o r m factor slope set to zero. 2. 7T —• e e 77* a n d 7T° —• e e~77, the first order radiative corrections to the +  0  _  +  above process, as calculated by Roberts a n d Smith.[25]. 3. TT~p — > ne e~ +  w i t h first order radiative corrections n~p —* ne e~~f a n d ir~p —• +  ne e~7*, as formulated by Fonvieille et a l . [26]. +  4. 7T° —> e e ~ e e ~ using the m a t r i x element derived b y 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 +  p r o d u c t i o n i n a specified area of the detector. 6. 7T° —• 77 where one or b o t h of the photons were forced to undergo C o m p t o n scattering or pair p r o d u c t i o n . 7. TT~p —• n-f where the 129 M e V photon is forced to undergo pair p r o d u c t i o n .  In the case of the processes 4 t h r o u g h 7, the cuts imposed o n the p r i m a r y leptons o u t l i n e d above were modified. F o r the process 7r° —» e e e e ~ , we required only that +  _  +  at least one e e " c o m b i n a t i o n fulfilled the restrictions listed. F o r the p h o t o n conversion +  b a c k g r o u n d 5, 6 a n d 7 we waived a l l restrictions a n d generated the events i n Air w i t h a l l possible m o m e n t a . A more complete discussion of the m o d e l l i n g of the background processes 3 to 6 c a n be found i n chapter 7.  5.3  Stop D i s t r i b u t i o n  T h e events were generated i n the l i q u i d hydrogen target according to stop d i s t r i b u t i o n found for the d a t a . T h e event o r i g i n , as calculated by the reconstruction p r o g r a m , was p l o t t e d for each real event i n a n r — z projection. T h i s was digitized a n d used as the d i s t r i b u t i o n function for the generation of the simulated events. T h e stop d i s t r i b u t i o n was taken to be r a d i a l l y symmetric a n d identical for each of the 4 r u n periods, for each process. D u r i n g the final analysis, the d a t a a n d s i m u l a t i o n vertex d i s t r i b u t i o n s i n the r — z plane of b o t h the processes 7r° —> e e ~ 7 a n d 7r~p —+ ne e~ +  +  for each of the 4 r u n  periods, were plotted a n d digitized. B y d i v i d i n g the stop d i s t r i b u t i o n of the d a t a by that of the s i m u l a t i o n , 8 sets of (2-dimensional) weights were generated. These weights were applied to each M o n t e C a r l o stop d i s t r i b u t i o n , so that the s i m u l a t i o n w o u l d m a t c h the d a t a as closely as possible. T h e importance of this m a t c h i n g w i l l be discussed i n chapter 9. A t y p i c a l weight d i s t r i b u t i o n can be seen i n figure 5.1. Note that the weights are quite close to 1 for most regions of the target.  F i g u r e 5.1: T w o - d i m e n s i o n a l weighting function for the s i m u l a t i o n , designed to m a t c h the stop d i s t r i b u t i o n to that of the d a t a . T h e m a x i m u m height is roughly 7, the average is 0.8.  5.4  Modelling Detector  Response  Once the detector geometry has been defined a n d entered into the G E A N T package, the p r o g r a m w i l l step each of the particles t h r o u g h the various detector p a r t s , m o d elling its energy loss along the way. T h e p r o d u c t i o n of secondary particles t h r o u g h p a i r p r o d u c t i o n , bremsstrahlung, and C o m p t o n scattering is a u t o m a t i c a l l y performed. These secondary particles are also traced t h r o u g h 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, w h i c h then takes care of the specific detector response. W e note here that the complex process of gas i o n i z a t i o n , i o n drift, a n d subsequent amplification a n d avalanche near a chamber wire has been a p p r o x i m a t e d by recording the energy lost by the electron i n the chamber a n d assigning it to the nearest wire. T h e response of the cathodes is a p p r o x i m a t e d by a G a u s s i a n 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 a m p l i t u d e . T h e hodoscope signals are determined by assigning the energy lost i n a p a r t i c u l a r scintillator by the particle to that scintillator. T h e energies assigned to the anodes, cathodes, a n d hodoscope scintillators are translated i n t o d i g i t a l output identical i n format to the output of the electronics a n d w r i t t e n to a file w h i c h 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 a n d G P M ) a n d filter were being designed a n d the appropriate masks were being developed d u r i n g the i n i t i a l construction of the spectrometer, trigger s i m u l a t i o n programs were w r i t t e n .  T h e i r purpose was to test the  correctness and completeness of the masks (which were produced by s i m u l a t i n g e e ~ +  pairs) a n d hence t o verify the performance of the online r — <f> p a t t e r n recognition. T h e programs were w r i t t e n b y W . B e r t l a n d H . P r u y s of the S I N D R U M I collaboration. T h e programs take as i n p u t a raw d a t a file w i t h the format generated b y the electronics. Different trigger conditions m a y be set d u r i n g a n i n i t i a l i z a t i o n phase. W e set the conditions appropriate to the D a l i t z experiment outlined i n chapter 3, a n d pass the simulated d a t a through the trigger. T h e results are shown i n figures 5.2a) t h r o u g h f ) , where we i l l u s t r a t e the action of the trigger b o t h o n the simulated D a l i t z a n d ne e~ +  d a t a , as a function of m o m e n t u m a n d transverse opening angle. W e see that the trigger cuts harder o n the ne e~  d a t a t h a n o n the D a l i t z d a t a : 7 8 % of the n e e ~ events are +  +  lost i n the trigger, compared to 5 5 % of the D a l i t z events. T h e higher the track m o m e n t u m , the more likely it w i l l be lost i n the trigger. T h i s is because the masks were p r o d u c e d b y simulated e e~ pairs from 7r° decay; such pairs w i l l always have a smaller +  m o m e n t u m range t h a n the e e~ +  pairs from Tt~p — > ne e~. +  T h i s was a n error. H a d  this cut been less stringent, a more thorough analysis of the n~p —• n e e " d a t a w o u l d +  have been attempted. L o o k i n g at the positron a n d electron transverse m o m e n t u m dist r i b u t i o n s , we see further that the positive tracks are cut harder t h a n the negative tracks; the masks, being simulated themselves, were not quite s y m m e t r i c 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 m o +  m e n t u m range. T h e trigger s i m u l a t i o n also applies a cut i n transverse opening angle; however, f r o m figure 5.2c) a n d 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 d a t a a n d s i m u l a t i o n is poor at this point is not w o r r y i n g , since m u c h more stringent cuts w i l l be made later o n 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, a n d not due to the poorly-understood trigger.  T h e reason for s i m u l a t i n g the  action of the trigger i n the first place was to verify the reduction i n the ne e~ +  sample.  F i g u r e 5.2: T h e figures illustrate the action of the simulated trigger. T h e histogram represents the simulated d a t a before passing through the trigger, the points, after the trigger. In a) t h r o u g h c) we show the effect on the D a l i t z s i m u l a t i o n , while i n d) t h r o u g h f) we show the ne e~ s i m u l a t i o n . +  In a d d i t i o n to the online trigger, the d a t a also passed t h r o u g h the online  filter.  T h i s stage of the d a t a 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 d a t a t a k i n g , every 4 t h event was w r i t t e n to tape by the online filter,  regardless of its being accepted or not. I n this manner, it is possible to test the  efficiency of the later analysis cuts: we find that no events w h i c h were not passed by the filter found their way into the final d a t a sample.  Chapter 6  Radiative C o r r e c t i o n s  6.1  R a d i a t i v e C o r r e c t i o n s for the Process 7r° —> e e +  7  R a d i a t i v e corrections are processes such as those p i c t u r e d i n figures 6.1 a n d 6.2, p r o cesses involve more t h a n two photons. It is less likely that these events w i l l take place; radiative corrections, w i t h F e y n m a n diagrams i n v o l v i n g e x t r a vertices, are smaller by a fa 1/137. Note that the radiated photons p i c t u r e d i n figure 6.1 m a y be either external (hence i n p r i n c i p l e detectable) or internal. In b o t h cases, they change the m o m e n t u m of the electrons a n d hence modify the shape of the invariant mass s p e c t r u m .  Since  the slope parameter is itself a second-order effect, one w o u l d expect that the effect of these r a d i a t i v e corrections could be appreciable, and hence they must be i n c l u d e d i n any analysis that expects to extract a value for a. In the case of the decay 7r° —> e e~7, the contributions to the m a t r i x element +  corresponding to the diagrams shown have been evaluated exactly by M i k a e l i a n a n d S m i t h a n d others [20,22,25] a n d the resulting corrections to the K r o l l - W a d a f o r m u l a 1.11 t a b u l a t e d [22], as shown i n figures 6.3 a n d 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 c o n t r i b u t i o n to the f o r m factor, while a later paper 2  [44] comes to the conclusion that these graphs have an e x t r a factor of ^f- associated w i t h t h e m , a n d can hence be safely ignored. Since a is defined i n terms of the invariant mass x, one could i n p r i n c i p l e apply the  F i g u r e 6.2: T w o v i r t u a l p h o t o n loop graphs, corrections to 7r° —> e e +  7.  0.M  2.77 %  0.99  -33.51 % F i g u r e 6.3:  T w o dimensional surface plot showing the percentage correction to the  K r o l l - W a d a m a t r i x element, as calculated i n [18]. y = 0.  T h e surface is s y m m e t r i c about  F i g u r e 6.4: Corrections to the K r o l l - W a d a m a t r i x element as a function of x, as calculated i n M i k a e l i a n a n d S m i t h [18].  results of M i k a e l i a n a n d S m i t h ' s calculations i n x directly to the simulated, uncorrected s p e c t r u m as a m u l t i p l i c a t i v e factor. T h e radiative corrections expressed as a percentage change over the " b a r e " spectrum are roughly linear i n x over the effective range of the d a t a , so t h a t the resulting value for a increases by a p p r o x i m a t e l y 0.05. However, this naive approach to the radiative corrections is not correct.  A s one does not actually  simulate any r a d i a t i v e events using this m e t h o d , the i m p l i c i t a s s u m p t i o n is that the acceptance a n d detection efficiencies for the radiative events are exactly the same as for the bare events. T h e r e is no a priori reason for this to be so; the radiative events can be 4-body decays a n d as such are k i n e m a t i c a l l y quite different f r o m the bare 3-body D a l i t z decays. A n o t h e r serious drawback of this simplistic m e t h o d is t h a t , since one does not simulate any radiative events, it is not possible to check whether the bare events plus the corrections actually m a t c h the d a t a i n any k i n e m a t i c a l region other t h a n x — y space.  T h e applied m u l t i p l i c a t i v e factor cannot create four-body decay  events i n the regions forbidden to the three-body bare events. In s u m m a r y , it is not possible to gain any insight into the kinematical behaviour of the radiative corrections w i t h o u t generating some. Before m o v i n g on to discussing the techniques used to generate radiative events, we first make a few observations concerning the diagrams shown. T h e radiative corrections p i c t u r e d are those of second order i n a only. Higher order corrections of course exist, b u t each added p h o t o n line suppresses the a m p l i t u d e of the event by a factor of y/a so that one usually considers only the leading order terms p i c t u r e d . T h e diagrams divide themselves u p n a t u r a l l y into two classes; " v i r t u a l " or " i n t e r n a l " , a n d " b r e m s s t r a h l u n g " or " e x t e r n a l " radiative corrections. V i r t u a l radiative corrections are those i n w h i c h the e x t r a p h o t o n is reabsorbed onto one of the leptons a n d is therefore not experimentally detectable. T h e b r e m s s t r a h l u n g corrections lead to free photons w h i c h are i n p r i n c i p l e detectable. B o t h the v i r t u a l a n d bremsstrahlung corrections separate into a convergent  set of integrals a n d a divergent group. W h e n one performs the integration numerically, the u s u a l technique employed to cope w i t h these divergent integrals is to introduce a p h o t o n cutoff energy A , below w h i c h the integral diverges, a n d above w h i c h it converges. A d d i n g the divergent part of the bremsstrahlung c o n t r i b u t i o n to the v i r t u a l corrections, one finds that the divergent parts of each cancel exactly (independent of the value of A ) , a n d the t o t a l c o n t r i b u t i o n is finite. In order to simulate radiative events, the results derived by M i k a e l i a n a n d S m i t h are not sufficient as they are couched i n terms of the two degrees of freedom x a n d y. T h e v i r t u a l corrections a n d the bare process may be fully described by only two k i n e m a t i cal variables, b u t 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.  T h i s variable has been integrated over to derive the surface i l l u s t r a t e d  i n 6.3. W e therefore use the code developed by L . Roberts [25] w h i c h performs the i n tegrations n u m e r i c a l l y over phase space variables. T h e code separates the v i r t u a l a n d b r e m s s t r a h l u n g corrections a n d calculates the m a t r i x element for each separately. T h e cutoff energy A is set to a s m a l l 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 t u a l corrections, a n d hence may be safely used to cancel the divergent part of the v i r t u a l corrections. If A is set too h i g h , one i n effect ignores bremsstrahlung events w i t h low E^. T h e integrations to be performed are quite complex; the v i r t u a l corrections involve n u m e r i c a l evaluation of a five-dimensional integral while the bremsstrahlung corrections are eight-dimensional. T h e code uses a generalized M o n t e C a r l o integration technique to evaluate the integrals. T h e range of each integration variable is d i v i d e d up into s m a l l cells ( the so-called integration g r i d ) a n d the c o n t r i b u t i o n of each cell to the t o t a l integral is estimated on the basis of the fraction of r a n d o m l y d r a w n points that  fall i n i t . These r a n d o m l y drawn points are converted i n t o the event kinematics; i n this way, the integration package also functions as an event generator. A s the accuracy of the desired integral is increased, a refining a l g o r i t h m repartitions the integration space i n t o smaller a n d smaller cells, focusing i n o n steep areas of the function.  O n e must  verify that this p a r t i t i o n i n g and refining a l g o r i t h m does not introduce any bias i n t o the result a n d skew the d i s t r i b u t i o n of generated events. In the case of the bremsstrahlung corrections, for example, the eight variables of integration are the energy, cos 8, a n d <f> angle of the bremsstrahlung p h o t o n , the energies of the p o s i t r o n a n d electron, the opening angle between the electron a n d p o s i t r o n , a n d the cos 8 a n d <j> angle of the electron. T h e p r o g r a m first models the 2-body decay of the i n i t i a l TT° into a v i r t u a l photon and a bremsstrahlung photon, a n d then allows the v i r t u a l photon to undergo a 3-body decay into e e~j. +  Obviously, the d i s t r i b u t i o n of  events should be u n i f o r m i n the cos 8 a n d <j> angles of the bremsstrahlung photon; the i n i t i a l 2-body decay of the 7r° does not favour any p a r t i c u l a r direction. If these d i s t r i b u t i o n s are i n any way skewed, the subsequent cuts m o d e l l i n g the detector acceptance w i l l throw out or accept too m a n y events a n d give an incorrect final event d i s t r i b u t i o n . Note, however, that this skewing w i l l not lead to a n incorrect estimation of the integral! T h i s is because the integrated function is flat i n these v a r i ables; every point along the axis contributes the same amount to the final answer a n d 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. T h e r e was an error i n the p a r t i t i o n i n g / r e f i n i n g a l g o r i t h m of the integrating package w h i c h produced holes a n d spikes i n the d i s t r i b u t i o n of cells. R a t h e r t h a n rewrite the integrating package, it was decided s i m p l y to re-randomize these variables after the integration was  finished  but before the subsequent detector cuts, since, as pointed out above, the final answer w o u l d not be sensitive to this operation.  W e r u n the code i n an attempt to reproduce the results quoted by M i k a e l i a n a n d S m i t h ; see figure 6.5. T h e corroboration is not fantastic, but out to x w 0.5 (where most of the d a t a is) the agreement was deemed acceptable.  It must be noted that  generating graphs such as those shown i n 6.5 requires m a n y hours of computer time. W e find the t o t a l c o n t r i b u t i o n to the m a t r i x element for 7r° —• e e~j to be +  i\ t 0  =  To + r  =  r + r + r  =  (6.2499 ± 0.0026) • 1 0  0  r c  v  6  - 8  + ( - 0 . 9 8 7 3 ± 0.0004) • 1 0  - 8  + (1.0341 ± 0.0005) • 1 0  - 8  (6.1)  T h e errors are statistical a n d are estimated by the M o n t e C a r l o integrating package o n the basis of the number of points d r a w n per integration region. corrections contribute a total T  rc  «  T h e radiative  0.75% to the total decay w i d t h , i n agreement  w i t h the results of M i k a e l i a n and S m i t h .  W e a d d the r a d i a t i v e events to the bare  events generated according to the K r o l l - W a d a d i s t r i b u t i o n i n the proportions given by equation 6.1, a n d verify that these radiative events (the b r e m s s t r a h l u n g photons are tracked a n d allowed to interact w i t h matter) now fill out the tails of the d i s t r i b u t i o n s as expected. In figure 6.6 we plot the energy of the lepton p a i r against their invariant mass, a n d figure 6.7 shows the total energy-momentum versus the opening angle of the lepton p a i r . T h e radiative events are clearly visible below the k i n e m a t i c a l region allowed the bare events. I n figure 6.8 we see that the t a i l fits the d a t a very well. It is i n s t r u c t i v e to plot the effect of the radiative corrections o n the bare events, especially for the x a n d <f> d i s t r i b u t i o n s . In figure 6.9 we show the relative c o n t r i b u t i o n to the invariant mass a n d opening angle s p e c t r u m f r o m b o t h the v i r t u a l a n d b r e m s s t r a h l u n g radiative corrections. W e see that the corrections have a greater i m p a c t o n (j> t h a n they d o o m . T h i s is presumably due to the fact that the e x t r a photons being  F i g u r e 6.5: Verification of the published radiative corrections. T h e line are the corrections as calculated by M i k a e l i a n a n d S m i t h [18] and the points are the result of the n u m e r i c a l integration using the program of Roberts and S m i t h [21].  150  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< • • • - •nnnnnnnQgQQQgiinnnnnnnnnnnnnnnnnnnn-nRR.nn-..  •0••••••••••••••••OODDDDDODDoDODDDDoDODa••oon eo•• — cnnoQC ~ nmrjrxiDOoo.. •••I J • - DDL • orjrjn lUjJODDbDOODD« •••0 •DDDDDO D D D D O D O O D D D D D D D D D ] JODDDDDOOOODuou o u ao< a • innnnnnnnim •.  130 H  •ODDOOOOOOAOIAODDOODODOOOOOOAAAOBOOO • • • •  > 0)  i i i D I D D O O D D i  ••••OODOODD• nODDCOrjQDi | oof • 0 0 3D  110 H  ODD  ODD  • • • °DCJ o • ••  cm  ^  CD  • ODD  90  •  CD  70 H  • D • D •  o  a •  JDDOD  50  [•••DO • 0 0 • 0 • D • 0 • • B  30  0  I  0.0  I  • • I  I  I  I  | I  I  0.2  I  I  | I  0.4  X  I  I  I  | I  0.6  =\ s"  i  i I i i i i I i i i i  0.8  m7T  1.0  1.2  2  F i g u r e 6.6: Invariant mass of lepton p a i r versus the t o t a l energy, for the simulated events. W e plot b o t h the n e e " a n d the e e " 7 events, b o t h w i t h r a d i a t i v e corrections. +  +  i i i i I i ii i i I  260  Li i i I i i i  • nnnnnnnlTT  l  l  I_J_L •  ) • • • • •  J OOTID OOTJDDD •••ODDon ODDdDDoo ••••ODD ••OOCIo DO PDDorjooo -  240 i——I  220  ' i ' I ' ' ' ' I ' i '  no • •  I I  I  D l  '  • <  DO"  • • • 0 i 0B • I00 B I0B•  DDQDDDOOOoOo D oOglTlDDDDDDooo OO[IJD0[]0D[]D<IOB DQQQQD DD D o n D• •••••••••DODO'  200  8  0•) III IQODQaQoo D Q H J D D D O O O O • • •'  •  I—I  "•••••••DDD 0 o •  180  • ••••••ODD DD••  oDDDOOQOODDD•o •••CODODDDDOOO•  CD  160  o  140  • ••••ODOooOi 3DODD0D o B O O IDDODOOOOSB JOT) • • 0 o o a a _ 1 D D D D B o• 3 D D D D D Q B OD  ft  J3O0OODOB0 •••••••Do JDDDDDDDDo UDDOODDDO  120  jODODOo  • • I a • > A• • • 11 B • I • • I ••  30DDDDD  DDDDo  >  C  100  D • Do a • aB I D0 0 • •  1  • toon •aao 0 o • • i 6 •o a o o • • • D D O D D H D D O D O A•  80  J•D D DO Q • • •DDQDDDDI • DDODDDO • •  cd  •oOoODDi  60  •B••••••••< ••DDODDDO••  • ••DODDDD'" •ooDDQODDi•  ••OODQODO• TT  40  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  30  60  ™ 90  .r-r, 180 120 150  E.P. (MeV) it  210 240 270  L  F i g u r e 6.7: E Pt of lepton p a i r versus the transverse opening angle, for the s i m u l a t e d events. W e plot b o t h the ne e~ a n d the e e~7 events, b o t h w i t h r a d i a t i v e corrections. T h e r a d i a t i v e tails are clearly visible. t  +  +  Chapter  6.  Radiative  1000  ''  Corrections  7 5  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  F i g u r e 6.8: T h e E P d i s t r i b u t i o n for the D a l i t z events. T h e histogram is the s i m u l a t i o n , the points, the d a t a . Note that the radiative t a i l is very well fit. t  t  e m i t t e d do not transfer m u c h energy, b u t do transfer m o m e n t u m , a n d w i l l therefore kick the leptons a b i t further apart.  •  1  •••  1  ' '1' 1  3000  11111111111111111  iiilllliilnmilitliiiinililnii  !•  " I " "  I'  I  -500 1  -1000 i -2000  H11111111111111 < H111111111111H111111 .u  .2  .4  .6  .8  i n v a r i a n t mass x  -1000  IIIIIIIII|IIIIIIIII|IIIIIIIII|IIIIIIIII|IIIIIIIII|IIIIIIIII|IIIIIIIII|IIIIIIIII  20 40 60 80 100120140160180  opening angle 0  F i g u r e 6.9: T h e c o n t r i b u t i o n to x a n d <> / from the v i r t u a l (solid line) a n d bremsstrahlung (dotted line) corrections to the total (triangles) s i m u l a t i o n . T h e t o t a l c o n t r i b u t i o n of the r a d i a t i v e corrections is shown b y the shaded histogram.  6.2  R a d i a t i v e C o r r e c t i o n s for the Process 7t~p —> n e e +  R a d i a t i v e corrections t o the internal conversion process ir~p — » ne e~ +  must also be  considered, as they result i n lepton pairs leaking down into the D a l i t z decay region  a n d c o n t a m i n a t i n g i t . T h i s is especially i m p o r t a n t at higher values of x , since the two processes approach one another i n energy as the lepton invariant mass increases. W e expect higher c o n t a m i n a t i o n 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~ r a d i a t i v e 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 a n d equally  +  applicable to the D a l i t z decay. A more complete discussion of the m e t h o d used is given i n references [45,46]; we note here that the v i r t u a l corrections shown i n figure 6.10 c a n be implemented b y a m u l t i p l i c a t i v e weighting factor (which is less t h a n 1) carried along w i t h each " b a r e " event: da  da  ^  dx virtual  dxo  with a  U  2  13  28 \  (mWx)  where one neglects the (small) dependence of the v i r t u a l corrections on y. T h e bremsstrahlung corrections p i c t u r e d i n figure 6.9 are modelled by allowing one of the leptons to emit a p h o t o n according to the p r o b a b i l i t y d i s t r i b u t i o n (expressed i n the centre of mass of the emitted photon) rE-m 1 + 6(x,y)brems = / Jo  c  P(q) dq  with Afq\ (,  q ,  A  q  2  q where 7r  \m  e  2/  T h e bremsstrahlung p h o t o n is emitted w i t h energy q from a lepton of energy E. O n e imposes a threshold energy cut ^ > A ( A « 0 . 1 M e V ) so that only those photons visible  Figure Tr~p — >  6.10: Feynman ne e~. +  diagrams  for  the  radiative  corrections  to  the  process  by the apparatus are modelled. E v e r y event is forced to undergo bremsstrahlung. Its probability, as calculated by the equation above, is carried along as a weighting factor (again, always less t h a n 1). T h i s approximate m e t h o d is based on the assumption that the radiative corrections to ir~p —• ne e~ correspond to those calculated i n [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 c a r r y i n g the weighting factors along. Note that the shape of the (clearly visible) radiative t a i l is generally well simulated, although a slight underestimation of the t a i l seems to be occurring.  111111  ' 11 1 1 1 1 1 1 1 1 1 1 1 1  111 11111  11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  n  11111  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) F i g u r e 6.11: T o t a l energy of the lepton p a i r for the n e e ~ events. T h e points show the d a t a , the h i s t o g r a m is the simulation. T h e radiative t a i l is generally well fit, a l t h o u g h a n u n d e r e s t i m a t i o n seems likely. T h e simulation is expected to be correct t o roughly 15%. +  Chapter 7  B a c k g r o u n d Considerations  A n y process w h i c h produces e e~ pairs may be considered a background to the D a l i t z +  decays. Table 7.1 shows a list of all such processes. A m a j o r source of background are 7r° decays i n v o l v i n g photons, where the 7 undergoes C o m p t o n scattering or p a i r p r o d u c t i o n to produce other electrons.  S I N D R U M is a low-mass detector, so the p r o b a b i l i t y of  photon conversion i n the wire chambers is very low. M o r e probable are conversions i n the target's a l u m i n u m support r i n g , i n the lead degrader i n front of the target, a n d i n the l i q u i d hydrogen a n d M y l a r housing of the target. A n o t h e r possible source of b a c k g r o u n d are so-called " r e t u r n i n g t r a c k s " ; electrons from beamline m u o n decays  contribution  process 7T° 7T° 7T° 7T° 7T° 7T° 7T  —> e e~e e~ —> e e~7 (compton) —• e e~7 (pair prod.) —• 77 (pair prod.) —• 77 (2x p a i r prod.) —> 77 (compton 4- pair prod.) p —> wy (pair prod.) +  +  +  +  -  n~p —¥ ne e~ +  < < < < < < <  to total sample 10 events 20 events 10 events 5 events 10 events 15 events 5 events  fa 2500 events  Table 7.1: B a c k g o u n d processes considered a n d their calculated c o n t r i b u t i o n to the total sample, w h i c h numbers roughly 100,000 events. Conversions are modelled i n the l i q u i d hydrogen target, i n the target walls, a n d i n chamber 1.  7.1  External Conversion Background  E x t e r n a l conversion events, i n w h i c h a photon from a p i o n decay 7r° —• 77 or 7r° —> e e~7 or from n~p —* nj converts i n t o a n e e~ +  +  pair or results i n a C o m p t o n electron  due to interaction w i t h matter inside the detector, are a potential source of spurious e e ~ pairs. Such events occur i n huge numbers i n the detector's massive p a r t s , such +  as the a l u m i n u m target support r i n g a n d lead degrader clearly visible i n figure 4.1. T h e constraints o n z a n d r discussed i n chapter 4 remove v i r t u a l l y a l l of the support r i n g a n d degrader conversions.  A further requirement that the inner two chambers  exhibit two a n d only two hits greatly reduces events w i t h e x t r a electrons ( D a l i t z events i n w h i c h the p h o t o n converts, and 7r° —• 77 events i n w h i c h b o t h photons convert). T h i s cut eliminates events w i t h an e x t r a electron track of m o m e n t u m greater than about 2-6 M e V / c (depending on the event origin a n d emission angle). T h e m i n i m u m opening angle requirement p r a c t i c a l l y eliminates the conversions from ir~p —• 717, since the 129 M e V p h o t o n produces e e~ +  pairs w i t h opening angles smaller t h a n 20°. T h e  7T° —* 77 conversions are also removed by this cut, although to a lesser extent since the p h o t o n is less energetic. We demonstrate this by simulating a n d then a n a l y z i n g these processes to find the contributions given i n the final c o l u m n 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 c o n t r i b u t i o n from the double D a l i t z decay is expected f r o m "crossed p a i r s " , where each lepton comes from a highly a s y m m e t r i c a l decay of one of the v i r t u a l photons. T h e r e m a i n i n g low energy electrons should be visible as hits i n the inner chambers, so we expect that m u l t i p l i c i t y cuts on the inner two chambers w i l l result i n clean events. In order to check this, double D a l i t z events are generated according to the d i s t r i b u t i o n functions given i n [42] a n d passed through the analysis. A f t e r the requirement that  chambers 1 a n d 2 have only two hits each, we find that the c o n t a m i n a t i o n due to the double D a l i t z decay is negligible.  7.3  R e t u r n i n g Tracks  In any p i o n b e a m , there w i l l be m u o n a n d electron c o n t a m i n a t i o n e m a n a t i n g f r o m the p r o d u c t i o n 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 b e a m m u o n decays v i a p, — » vve i n the hodoscope a n d the resulting electron passes t h r o u g h the target a n d continues back to the hodoscope on the opposite side, i t m i m i c s a n e e~ +  pair with  an opening angle of 180 degrees. These " r e t u r n i n g t r a c k s " are a p o t e n t i a l source of background. O n e eliminates most of them at the online trigger stage b y r e q u i r i n g that the two hodoscope signals occur at the same time. T h e few r e m a i n i n g r e t u r n i n g tracks, visible at E P t  t  = m  M  = 106 a n d <j> = 180° i n figure 4.4 (page 51), are removed b y a  requirement o n the t o t a l e e ~ m o m e n t u m ; P +  7.4  tot  > 10 M e V / c .  B e a m Buckets w i t h more t h a n 1 TT°  R o u g h l y 1 i n every 1 0 beam buckets w i l l contain two n e u t r a l pions. 8  W e therefore  expect about 100 of these i n our sample. T h i s w i l l result i n only one D a l i t z p a i r , so that background from this source is negligible.  7.5  ne e~ +  Background  T h e theoretical ratio of the number of ne e~ +  events to the n u m b e r of e e ~ 7 events +  is a p p r o x i m a t e l y 1. T h i s ratio is reduced by the online trigger a n d filter, as discussed i n chapter 5: the trigger cuts o n m a x i m u m track curvature a n d therefore imposes a m a x i m u m m o m e n t u m cut; since the n e e ~ events have higher e e~ +  +  m o m e n t a , these  events are reduced i n number. W e fit for the resulting ratio of n e e +  to e e 7 +  #ne e~events +  #ne e~  +  +  #e e~yevents +  i n c l u d i n g radiative corrections to b o t h processes. T h e results of the fit are given i n table 7.2 below, a n d a t y p i c a l example is shown i n figure 7.1.  geometry  iV(e e"7)  N(ne e~)  2 4 ' 5 6  R  +  +  9230 52207 16495 16998  11810 62087 18188 53178  0.237 0.235 0.228 0.218  ± 0.007 ± 0.003 ±0.007 ±0.004  Table 7.2: T h e c o n t r i b u t i o n of n~p —* ne e~ expressed as a fraction of the t o t a l sample. O t h e r background contributions are negligible. T h e quantities N provide a means for assessing the efficiency of the E Pt separation cut; they are not products of the fit, a n d their magnitudes are of no special importance. +  t  In order t o remove as m a n y of the ne e~ events from the D a l i t z sample, k i n e m a t i c a l +  constraints must be imposed. F r o m figure 4.4, we see that the D a l i t z events inhabit a boxlike region from 107 < E P t  t  < 163 M e V . B y requiring E P t  eliminate most of the 7r~p —• ne e~  events.  +  to ne e~ +  < 170 M e V , we  t  There is still some c o n t a m i n a t i o n due  radiative events, i n w h i c h one of the electrons loses energy b y emission of a  bremsstrahlung p h o t o n , thus p o p u l a t i n g the region below E tai = 136 M e V allowed to to  the " p u r e " 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. W e m a y then a d d the ne e~  s i m u l a t i o n to the e e~j for any set of cuts (in p a r t i c -  +  +  u l a r , the cuts designed to eliminate the ne e~ +  total simulation  (1 — R) = , . , r x ( # e e~7 after N(e+e~j) +  T  v  above) b y (R) cuts)+—— • - x ( # ne e~ N(ne+e-) +  v  after  cuts) '  -20 0 20 40 60 80 100 m i s s i n g e n e r g y (MeV) F i g u r e 7.1: T h e determination of the ratio of ne e to b o t h reactions are included. +  to e e +  7. R a d i a t i v e corrections  where R a n d N are given i n table 7.2. After a l l the cuts (those listed i n chapter 4, the cut on EP, t  t  a n d the m u l t i p l i c i t y cuts o n the inner chambers) we f i n d that approximately 2 %  of the t o t a l simulated d a t a set is due to ne e~. +  W h i l e this is a s m a l l c o n t a m i n a t i o n , i t  must be i n c l u d e d , since the invariant mass d i s t r i b u t i o n of the ne e~ sample is r a d i c a l l y +  different from that of the process TT° —* e e~j. +  Chapter 8  Fitting Procedure  W e d i v i d e the full d a t a set into the four separate r u n periods (for each of w h i c h the chamber geometry was slightly different, as discussed i n chapter 3) w h i c h we refer to as "geometries 2, 4, 5 a n d 6". F o r each geometry we simulate double this n u m b e r of events ( i n c l u d i n g radiative corrections) as well as double the corresponding expected n u m b e r of n e e ~ events ( i n c l u d i n g radiative corrections). R e c a l l that other background +  c o n t a m i n a t i o n is negligible. T h e analysis as o u t l i n e d i n this chapter is performed for each separate geometry, a n d the results compared for consistency. F i n a l l y the results are combined a n d we quote an average value for a. F o r completeness we list the cuts that are applied to b o t h the s i m u l a t i o n a n d the d a t a i n the final analysis: 1. 45° < 4> < 260°, a cut w h i c h duplicates the action of the trigger a n d filter stages t  of the d a t a acquisition hardware and eliminates photon conversion background. 2. P > 20 M e V , a d u p l i c a t i o n of S I N D R U M ' s transverse m o m e n t u m threshold. t  3. —300 m m < Z ( 5 ) < 300 m m : requiring the track to lie w i t h i n the region of u n i f o r m magnetic field. 4. r < 19 m m , —115 m m < z < —80 m m , 0 m m < z + 104 + y/19 — r 2  ensure that the event happened well inside the target. 5. Ptot > 10, w h i c h eliminates the rest of the n decays.  2  : cuts w h i c h  6. O n l y 2 hits are allowed i n each of the inner two chambers, a cut w h i c h eliminates any further background process involving e x t r a electrons or photons, such as 7T° —• e e e e ~ , external conversion, a n d pair p r o d u c t i o n . +  -  +  7. E t + Ptot < 170 to separate the 7r° —• e e ~ 7 f r o m the ir~p —• ne e~ +  to  +  process.  A p p l i c a t i o n of these cuts results i n 5837 events for geometry 2, 35873 events for geometry 4, 5957 events for geometry 5, a n d 26520 events for geometry 6, g i v i n g a t o t a l of 74187 7T° —> e e ~ 7 events w i t h a c o n t a m i n a t i o n due to ne e~ +  +  of roughly 2 % .  In figures 8.1 a n d 8.2 we show the performance of the full s i m u l a t i o n ( i n c l u d i n g D a l i t z a n d ne e~) +  for various kinematical variables. T h e agreement w i t h the d a t a is  excellent. W e m a y now proceed to extract a. F r o m chapter 1, the most obvious m e t h o d of extracting a is to fit for the parameter i n the invariant mass s p e c t r u m x, since then we m a y extract a directly. A l s o , as discussed i n chapter 6, the radiative corrections are expected to have a s m a l l effect. If we denote the d i s t r i b u t i o n of the d a t a b y F , a n d the M o n t e C a r l o b y / , then we c a n write the fitting procedure schematically b y F(x)  = (1 + 2ax) x f(x)  W e denote the a obtained b y fitting i n x by a . x  A s a double check, i t is useful to consider the analysis of the d i s t r i b u t i o n i n <j>, where, as seen i n i n chapter 6, we expect the radiative corrections to have a n appreciable effect. W e c a n extract a slope parameter b: F(<£) = ( l + 2 ^ ) x / ( « £ ) N o w b a n d a are not the same, although b is related to a i n a measurable manner. W e x  x  o b t a i n the relationship between a a n d b by creating an artificial " d a t a " sample; we set x  i i in iiiln i in i nil i  0  20 P  t  40 (+)  h II 11nnIM ni n 11L n11 n11if ^|||111if  60  80  100 120  ( M e V / c )  140  0  20 R  40 (-)  60  80  100 120  140  ( M e V / c )  F i g u r e 8.1: T h e performance of the full simulation for various k i n e m a t i c a l variables. A l l radiative corrections a n d backgrounds are included. T h e points are the d a t a , the histogram, the simulation.  F i g u r e 8.2: T h e performance of the full simulation for various k i n e m a t i c a l variables. A l l radiative corrections and backgrounds are included. T h e points are the d a t a , the histogram, the simulation.  a to some non-zero value i n the K r o l l - W a d a m a t r i x element, r u n the M o n t e C a r l o , a n d generate a <f> s p e c t r u m . We then fit for b. Repeating for a number of a values allows us to m a p out the relationship between the two values. T h e 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)  W e denote the a value obtained by fitting i n <f> a n d a p p l y i n g the relation above by O0. T h e bracketed numbers i n 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 o b t a i n the correct errors on a^. W e employ two different fitting procedures for the parameters a  x  a n d a^, w h i c h  are now o u t l i n e d . The. two procedures stem from different philosophies of statistical analysis, a n d use different sets of m a t h e m a t i c a l tools to arrive at the final result. F o r each fitting m e t h o d , we fit w i t h a n d without the radiative corrections, i n order to assess their c o n t r i b u t i o n to the final result. If the analysis is properly done, we expect the final answers (16 of t h e m , i n a  x  and  from each of 2 fitting methods a n d 4 separate  d a t a sets) to be consistent w i t h each other.  8.1  M a x i m u m L i k e l i h o o d Technique — the Bayesian A p p r o a c h  Bayesian analysis is performed by combining prior i n f o r m a t i o n about the parameters of the m o d e l (denoted by the vector 8) w i t h the information f r o m the d a t a sample into the "posterior d i s t r i b u t i o n " . M o d e l parameters are then estimated by m a x i m i z i n g the posterior w i t h respect to the parameters 6. T h e p r i o r i n f o r m a t i o n about the parameters, denoted  incorporates a l l p r e v i -  ously k n o w n facts about the parameters; it may consist of previous results, subjective  bias, or any c o m b i n a t i o n thereof. W h e n no prior i n f o r m a t i o n is available a n d / o r subjective bias is not appropriate, one uses a "noninformative" or " u n i f o r m " p r i o r , w h i c h assigns equal p r o b a b i l i t y to a l l values of 6. T h e i n f o r m a t i o n encapsulated i n the d a t a enters v i a the " l i k e l i h o o d f u n c t i o n " f(8\x), w h i c h expresses the p r o b a b i l i t y of observing the d a t a , given certain values of the model parameters. O n e can t u r n this around; this p r o b a b i l i t y m a y 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 o n i n t u i t i o n a n d not o n any f o r m a l m a t h e m a t i c a l principles; however, it seems an eminently reasonable assumption to make a n d leads to very useful results. Once a p r i o r has been chosen and the likelihood function calculated, the posterior d i s t r i b u t i o n is given by n(8\x) — normalizing  constant  x 7r(8)f(8\x)  (8.2)  W h i l e the p r i o r incorporates the beliefs about 8 before the sample is observed, the posterior reflects the u p d a t e d beliefs about the parameters after the experiment has been done.  Bayesian parameter estimation defines the most likely parameter values  as those w h i c h m a x i m i z e the posterior d i s t r i b u t i o n . Note that since derivatives w i l l be taken, the n o r m a l i z i n g constant drops out and is not i m p o r t a n t . Note also t h a t , */ one assumes a uniform  prior,  Bayesian m a x i m u m l i k e l i h o o d estimation consists of  m a x i m i z i n g the l i k e l i h o o d function, i n other words, Bayesian likelihood parameter  estimation  are mathematically  and Frequentist  maximum  equivalent.  C r e d i b l e 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 i m i t s 9{, 9j, where (8.3)  Let us now move o n to specifics. U s i n g m a x i m u m l i k e l i h o o d considerations, it is fairly straightforward to derive a simple expression for a i n terms of the first- a n d second-order moments of the spectrum to be fit. W e present here an extended version of the argument found i n reference [19]. W e begin by calculating the l i k e l i h o o d function. T h e l i k e l i h o o d Ck of observing n independent events i n a category k is Poisson distributed: T)  nk  T h e t o t a l l i k e l i h o o d of observing a d a t a sample w i t h M categories is then  fc=i  n  *  !  Hence l n £ = - ^2pk + ^riklnpk k k  -  ^ln(ra !) k f e  Defining the category k to be the interval Xk < x < Xk + S i n the experimental spectrum f(x)  , we m a y write the p r o b a b i l i t y pk of finding an event i n the category k as Pk = / Jx  w  f(x)dx  f(x )6 k  k  for 8 s m a l l enough. S u b s t i t u t i n g , we find ln£  =  - £ { / k  k  J x k  =  - / J o  f c  fornix  f c  = / o ( x ) ( l + 2ax).  f c  Thus  ,  f (x)(l 0  J o  k  /(x)dx+^n ln{/(x )(5}-^ln(n !) k k  N o w , i n terms of the parameter a, f(x) \nC = -  f(x)dx}+^2n \n{f(xk)S}-J2H^.) k  + 2ax)dx  + J2nkHfo(xk)(l+2axk)6}-J2Hnk\) k  ,  .  k  (8.4)  W e now have the l i k e l i h o o d function for the d a t a i n terms of the parameter a. W e forget the results of a l l previous experiments a n d choose a u n i f o r m p r i o r . O u r posterior is  then equal to the likelihood function, a n d the most likely value for a m a y be found by m a x i m i z i n g w i t h respect to a: d\nC  0  da  Hence f ^ m a x  /  =  xf {x)dx 0  2^7  n-kXk + 2axk  Since a is s m a l l , we approximate i 1  1 + 2ax  —  2ax  giving t / ./o  ^max  «  xf (x)dx 0  V n ^ z ^ l — 2a:rt) k  k O b s e r v i n g that / fo(x)dx  k  = No, the number of M o n t e C a r l o events i n the s p e c t r u m ,  we m a y rewrite this i n terms of moments: Nx 0  0  (jfjxo  =  Nx -  =  x-2ax  2aNx  2  (8.5)  2  N o w , since this experiment was performed without any absolute n o r m a l i z a t i o n , we do not know  that is, we have no way of c a l i b r a t i n g iVo, the number of M o n t e C a r l o  events generated, to N, the final number of D a l i t z decays seen. A l l is not lost, however: the n u m b e r of events seen experimentally must be N  =  J f (x)(l+  =  J fo( )dx  =  N +  =  N (l  2ax)dx  0  x  0  0  + 2a J  xf (x)dx 0  2aN x 0  + 2ax ) 0  0  (8.6)  w h i c h quickly gives a simple expression for  S u b s t i t u t i n g into equation 8.5 gives  1 + 2axo => x  0  XQ — x  =  (1 + 2ax )(x  «  x — 2ax + 2axoX  0  — 2ax )  2  — 2a(x  2  2  2  — XQX)  a n d hence 1  XQ  /  2 \x  ( XQ - X \ i . - ( *° l + x[ _ _ l+X \ ~=- — \X* — XQX  N  iVo  (8.7)  — xox  2  to first order.  -  0  -  (8.8)  0  0  W e m a y now establish the classical " l - c r " (a = 0.32) credible region for a. In order to do this, we need the likelihood function, w h i c h is the exponential of equation 8.4. In order to simplify the calculations, we observe that equation 8.4 m a y be w r i t t e n as ln£  =  -  /  f (x)(l 0  + 2ax)dx+  J o  + X> ln{(5} - £ l n ( n k k f c  =  —  k  fp(xk)(l .  f c  + 2axk) +  J2^\n{f (x )}+ k 0  k  J2n ln{(l k k  +  2ax )} k  !)  k  rik l n { ( l + 2axk)} + constant  (8-9)  where we have a p p r o x i m a t e d the integral by a discrete s u m , a n d have separated the terms that depend o n a from those that do not. W h e n we exponentiate b o t h sides, the constant t e r m w i l l s i m p l y adjust the a m p l i t u d e of the resulting curve. W e shall ignore i t , choosing instead to define the a m p l i t u d e by the requirement that  JCda  = l1  (8.10)  T h e l i k e l i h o o d function, as given by equation 8.9, is simple to calculate numerically. T h e l i k e l i h o o d function is sharply peaked, a n d m a y be a p p r o x i m a t e d by a n o r m a l  d i s t r i b u t i o n i n the parameter a. T h i s is convenient, since, for a n o r m a l d i s t r i b u t i o n , equation 8.3 leads us to (8.11) In other words, the s t a n d a r d deviation of the n o r m a l d i s t r i b u t i o n defines the 68.3% credible region of the parameter. W e may therefore easily find the s t a n d a r d deviation by t a k i n g the second derivative of the likelihood function: 0  Note that we must be careful about the n o r m a l i z a t i o n : the quantity  is i m p l i c i t l y  included as a parameter i n the above equations. T h 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) a n d 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 m a y s i m p l y substitute x by <f> i n the above derivation to extract the m a x i m u m likelihood parameter estimation for the (j> case.  T h e two parameters a  x  and  are  not the same, as pointed out at the beginning of this chapter, b u t we m a y convert f r o m one to the other using equation 8.1. W e apply the analysis to b o t h the corrected a n d uncorrected M o n t e C a r l o spectra. C o n t a m i n a t i o n from ir~p —> ne e~ +  F i g u r e 8.3 a n d table 8.1 summarize the resulting  is included.  a n d a for the various geometries. x  W e do not show the results for the normalizations since they are of.no real interest; a l l geometries a n d fitting procedures give consistent results. T h e four separate geometries give consistent results. F i t t i n g i n x a n d <f> also leads to consistent results. Note that the radiative corrections have a m u c h larger effect i n <f> t h a n i n x , a n d that we do not o b t a i n consistent results u n t i l we include them!  no radiative  with radiative  corrections  corrections  0.30  0.30 f-  0.25 -  0.25 -j 0.20 -i  0.20^ r  0.15 -  0.15-I  r cd0.10^  cdO.10 0.05 -J  0.05  0.00  0.00  -0.05  -0.05  -0.10  -0.10 m aximum  likelihood fitting  J  maximum  likelihood  fitting  F i g u r e 8.3: G r a p h i c a l s u m m a r y of the results of the m a x i m u m likelihood fitting for a (white boxes) a n d (dark boxes) for each of the four geometries. O n the left are the results w i t h o u t radiative corrections, on the right the results i n c l u d i n g the radiative corrections. x  geometry 2 4 5 6  without radiative a  corrections  x  0.140 0.031 0.031 0.027  ±0.026 ±0.010 ± 0.025 ±0.012  0.166 0.121 0.141 0.123  with radiative a  corrections  x  ±0.018 ±0.007 ± 0.018 ±0.008  0.023 0.008 0.076 0.008  ± 0.026 ±0.010 ± 0.025 ±0.012  - 0 . 0 0 3 ± 0.018 0.002 ± 0 . 0 0 7 0.098 ± 0.018 0.020 ± 0.008  Table 8.1: Results for a a n d from m a x i m u m likelihood f i t t i n g , for each of the various geometries, w i t h a n d without radiative corrections. E r r o r s are statistical only. x  8.2  x  2  M i n i m i z a t i o n — the Frequentist A p p r o a c h  T h e s t a r t i n g point for x  2  m i n i m i z a t i o n is once again the l i k e l i h o o d function.  assume that each measured d a t a point n  k  = f(x ) k  If we  has an associated error o~ = y n t , k  /  1  a n d we further assume that this measurement error is independently r a n d o m a n d is distributed around n observing n  k  k  i n a G a u s s i a n manner, then we m a y write the l i k e l i h o o d of  events i n category k i n terms of the parameter a as  where S is the w i d t h of the category. T h e total likelihood over M categories is then  A s before, we choose the most likely parameters a a n d -j^ as those w h i c h m a x i m i z e this l i k e l i h o o d function. M a x i m i z i n g the l i k e l i h o o d function is equivalent to m i n i m i z i n g the negative of the log l i k e l i h o o d function:  W e m a y drop the constant factor of 1/2 a n d M In 6; they w i l l not affect the m i n i m i z a t i o n . W e then m i n i m i z e the so-called " x " function 2  X  2  _ " =  /  /(x  f c  )-^/ (xO(l+2flxQ| 0  2  W e assume that the number of entries njt in each category is governed by Poisson statistics. The standard deviation a 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  2  W e use a s t a n d a r d iterative a l g o r i t h m w h i c h chooses different values of the parameters a and  calculates the chi-squared, a n d uses this i n f o r m a t i o n to steer itself towards  the m i n i m u m x  2  a  n  d the associated parameter values.  In order to assess the errors on the parameters thus found, the s t a n d a r d practise is to p e r t u r b the values of the parameters slightly a n d observe the change i n defines some 2-dimensional confidence region i n the parameter space.  A\  2  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. T o evaluate this, we h o l d each of the parameters fixed, i n t u r n , a n d find the amount by w h i c h we must vary the other to induce a change i n x fact that a x  2  of 1.  T h e magic number 1 appears due to the  d i s t r i b u t i o n w i t h one degree of freedom (the single parameter) is the  2  square of a n o r m a l l y d i s t r i b u t e d quantity: Ax l-cr level), Ax  2  2  < 1 happens 68.3% of the time (the  < 4 happens 95.4% of the time (the 2-cr level), etc.  A l l of this  x  2  statistical analysis is s t a n d a r d fare, a n d we make use of the P L O T D A T A [48] analysis package f r o m T R I U M F , w h i c h w h i c h allows basic interactive fitting a n d p l o t t i n g . M a n y advanced statistical analysis packages exist, capable of h a n d l i n g scores of parameters a n d complicated fits (most notably M I N U I T ) ; however, since ours is a fairly simple p r o b l e m i n v o l v i n g 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 d a t a is d i v i d e d u p into the 4 geometries.  E a c h d a t a set is fit i n x a n d <f> (converting v i a equation 8.1), w i t h a n d  w i t h o u t radiative corrections. T h e fit results are s u m m a r i z e d i n table 8.2 a n d i n figure 8.4. W e note that the results i n x a n d <j> are consistent, a n d t h a t , 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  with radiative corrections  0.30  0.30  0.25 -  0.25  0.20  0.20  0.15 -j  0.15  co 0.10 -_  co 0.10 -j  0.05 -  0.05 -  0.00  0.00  -0.05 -3  boxes : fit i n x b l a c k b o x e s : f i t i n <f>  O-  -0.05 -0.10  -0.10 X  2  X  fitting  fitting  2  F i g u r e 8.4: G r a p h i c a l s u m m a r y of the results of the x fitting for a (white boxes) a n d 0.$ (dark boxes) for each of the four geometries. O n the left are the results without radiative corrections, o n the left the results i n c l u d i n g the radiative corrections. 2  geometry 2 4 5 6  without radiative a  corrections  x  ,. 0.088 ± 0.074 0.034 ± 0 . 0 2 7 -0.012 ±0.070 0.012 ± 0 . 0 3 6  x  with radiative a x  0.211 0.169 0.230 0.072  ±0.117 ±0.045 ± 0.097 ± 0.052  -0.050 ±0.068 - 0 . 0 0 3 ± 0.027 0.005 ± 0.074 - 0 . 0 0 8 ± 0.038  corrections a</> -0.038 ±0.102 0.010 ± 0 . 0 3 6 0.167 ± 0 . 1 0 1 0.039 ± 0.054  Table 8.2: Results f r o m x m i n i m i z a t i o n , for each of the various geometries, w i t h a n d w i t h o u t radiative corrections. E r r o r s are statistical only. T h e x per degree of freedom varies from 1.2 to 1.7 for the various fits. 2  2  8.3  S u m m a r y of F i t t i n g Results  T h e two tables 8.1 a n d 8.2 are s u m m a r i z e d i n table 8.3 a n d figure 8.5 below. W e o b t a i n consistent results between a l l the d a t a sets a n d the fits i n x a n d i n (j>. T h e statistical error of the <f>fiti n x  2  * larger; this is not surprising since we expect a somewhat s  washed-out effect i n <f> due to the fact that x a n d <j> are not uniquely correlated.  X geometry  a  x  2  maximum a  minimization at  2 4 5 6  -0.067 ±0.034 -0.005 ±0.018 0.003 ± 0.036 -0.012 ±0.019  - 0 . 0 4 6 ± 0.045 0.012 ± 0 . 0 2 3 0.171 ± 0 . 0 4 7 0.042 ± 0.028  all  - 0 . 0 1 4 ± 0.012  0.032 ± 0 . 0 1 6  likelihood  x  0.026 0.007 0.076 0.009  ±0.036 ±0.017 ± 0.029 ±0.019  0.019 ± 0 . 0 1 1  - 0 . 0 0 4 ± 0.048 0.002 ± 0.020 0.094 ± 0.032 0.021 ± 0 . 0 2 1 0.023 ± 0 . 0 1 3  Table 8.3: S u m m a r y of the fit results for each geometry, fitting m e t h o d , a n d spectrum. R a d i a t i v e corrections a n d iv~p —• ne e~ background are i n c l u d e d . I n d i v i d u a l errors are statistical only. T h e s u m m a r y value given is the error-weighted average, a n d its error takes into account the statistical error only. +  W e also see that the two fitting procedures also give consistent central values. T h e error obtained using m a x i m u m l i k e l i h o o d is however m u c h smaller t h a n the correspondi n g statistical error f r o m the x fit, a n effect especially pronounced i n the <f>fits.Since 2  we have not been able to find any flaw i n either the m a x i m u m likelihood or x  2  analysis,  we have adopted a more robust method by w h i c h to estimate the statistical errors. We generate several h u n d r e d "new" d a t a spectra by allowing the d a t a spectrum i n x a n d (j> to vary o n a b i n - t o - b i n basis, w i t h i n its 1-a error bars. W e fit this "new" d a t a , by b o t h methods, using the s i m u l a t i o n , a n d tabulate the resulting values for a a n d the n o r m a l i z a t i o n . T h e distributions i n a a n d the n o r m a l i z a t i o n are G a u s s i a n , a n d we take their central value a n d their s t a n d a r d deviations as the true a a n d statistical error of  Chapter 8. 02-  Fitting  Procedure 02 0.1  cd  o.o -0.1  I  1  I  I  cd  h  0.0 -0.1H -02  -02  X* fit i n f>  X* fit i n x  02  02 0.1 cd  Cd  o.o  0.0  -O.H  -0.1 H  -02  -02 maximum likelihood fit i n x  Cd  ii  maximum likelihood fit i n $  0.10  0.10  0.05  0.05-j  0.00  I * *  Cd  0.00  -0.05  -0.05  -0.10  -O.W  F i g u r e 8.5: S u m m a r y of results for a l l four geometries, i n x a n d <f>. T h e four points to the left of the double line represent the results for geometries 2, 4, 5, a n d 6, respectively; the point to the right indicates the error-weighted average. Its s m a l l error bar shows the statistical error, a n d the larger one shows the combined statistical error a n d standard error of the fluctuations. T h e results i n x a n d <f> are averaged to o b t a i n the final results.  the measurement.  O u r results are now independent of fitting m e t h o d a n d 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 m i d w a y between those obtained f r o m the m a x i m u m likelihood a n d x  2  analyses.  C o m b i n i n g the results of a l l the geometries, t a k i n g an error-weighted average over the x a n d <f> spectra, we o b t a i n the 4 results noted i n the last line of table 8.3. combine the m a x i m u m likelihood result w i t h the x take the t o t a l statistical error to be the x by a l l methods are s i m i l a r . T h e fluctuation  i n the x  2  2  fluctuation  2  We  result by simple averaging, a n d we  error, since the statistical errors obtained error is taken to be the average of the  ^ d m a x i m u m likelihood methods. We o b t a i n : a a  x  0  = 0.003 ± 0 . 0 1 1 ± 0 . 0 2 8  (8.12)  = 0.027 ± 0 . 0 1 3 ± 0 . 0 5 4  (8.13)  W e t u r n to an evaluation of the systematic errors i n the next chapter.  Chapter 9  E v a l u a t i o n 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 b o t h the transverse m o m e n t u m P a n d the transverse opening t  angle <f> enter directly into the calculation of the invariant mass s p e c t r u m x a n d the t  opening angle d i s t r i b u t i o n <f>, a n y t h i n g w h i c h affects these quantities w i l l have a direct impact on the measurement of the form factor slope. T h e systematic errors may be either time dependent (arising from the variation i n the detector setup f r o m r u n to run) or time independent (due to problems w i t h the analysis or the s i m u l a t i o n ) . T h e scatter i n the results obtained i n the last chapter (shown i n figure 8.3 a n d indicated i n the results 8.12,8.13) gives a m i n i m u m value for the combined run-dependent systematic errors. W e double-check the magnitude of these errors by v a r y i n g the quantities 1. stop d i s t r i b u t i o n 2. target location 3. chamber locations 4. magnetic field 5. analysis cuts a n d n o t i n g the effect on the result. T h e results of our systematic error analysis are tabulated i n table 9.1.  error source  X  magnetic field stop distribution target location  0.015 0.005 0.005 « 0.010 < 0.003  chamber A x , A y , Az, A<f> analysis cuts  t  d> 0.020 0.003 0.005 « 0.035 < 0.003  chamber construction hardware trigger ne e~ e e ~ 7 radiative corrections  0.003 0.003 0.010 0.001  0.003 0.003 0.010 0.001  total  0.022  0.047  +  +  Table 9.1: Systematic errors from various sources. T h e table is d i v i d e d into time-dependent errors (top) a n d time-independent errors (bottom). 9.1 9.1.1  T i m e D e p e n d e n t Systematic E r r o r s Stop Distribution  T h e deeper inside the target an electron is generated, the more m a t e r i a l (and distance) it must traverse to reach the hodoscope and trigger the electronics.  Those events  w h i c h originate at s m a l l target r a d i i , therefore, tend to have higher m o m e n t a t h a n those w h i c h occur at the outer edges of the target.  Since the form factor slope is  directly dependent on the m o m e n t u m d i s t r i b u t i o n of the electrons, it is v i t a l that the stop d i s t r i b u t i o n be correctly modelled. Since we know the final stop d i s t r i b u t i o n , we can adjust the s i m u l a t i o n to m a t c h the d a t a as closely as possible using the scheme of weights o u t l i n e d i n chapter 5. B y adjusting these weights to the point where the stop d i s t r i b u t i o n s become statistically different, we conclude that the error i n a due to i m p r o p e r knowledge of the experimental stop d i s t r i b u t i o n is as t a b u l a t e d i n 9.1; we see no significant systematic error.  9.1.2  Target L o c a t i o n  T h e experimental stop d i s t r i b u t i o n is measured w i t h respect to the centre of S I N D R U M . T h e p o s i t i o n of the target must also be fixed relative to this p o i n t . In other words, specifying the stop d i s t r i b u t i o n does not fix the target location a n d hence the stop d i s t r i b u t i o n m a y 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 w i l l traverse less hydrogen, a n d have a correspondingly higher m o m e n t u m t h a n those events coming f r o m the back of the target. M i n i m u m - i o n i z i n g electrons lose approximately 300 k e V for every centimetre of l i q u i d hydrogen traversed; a 0.5 c m shift i n the target, therefore, can induce a 150 k e V shift i n the mean track energy. If we arrange the cuts on the target vertex a n d the l o n g i t u d i n a l emission angle i n such a way as to remove most of the events c o m i n g from the very tip of the target, we m a y m i n i m i z e this systematic error. B y m o v i n g the target by 2 m m i n the s i m u l a t i o n a n d a n a l y z i n g the result, we o b t a i n the systematic error quoted i n the table. It is s m a l l .  9.1.3  Chamber  Geometries  A s discussed i n chapters 4 a n d 5, S I N D R U M ' s wire chambers were calibrated for the run-dependent x,y,z  offsets a n d rotations relative to chamber 5, as well as for the r u n -  independent anode print gaps a n d twists. T h e offsets a n d chamber rotations represent a m u c h larger effect t h a n the anode print gaps a n d twists. These calibrations were determined using cosmic rays. T h e rotations are accurate to roughly 1 • 1 0 a n d the x,y,z  offsets to 0.5 m m . T h e largest r o t a t i o n a n d x,y  - 4  mrad,  offset was found for  chamber 1; 40 m r a d 0.9 m m , respectively. T h e largest z offset (1.8 m m ) was measured for chamber 3.  In comparison, the largest twist was measured for chamber 3 (0.19  m m ) , a n d the anode print gaps were determined to vary from < 0.1 m m (chamber 1)  to 0.5 m m (chamber 4). W e redo the track fit w i t h these geometry calibrations set to their l i m i t i n g cases for the r u n periods for w h i c h the calibrations are a m a x i m u m (the "worst case"). E x t r a c t i n g the variation of a w i t h respect t o 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 positioning is the largest source of systematic error.  9.1.4  Magnetic Field  T h e value of the magnetic field enters directly into the calculation of the electron m o m e n t u m d u r i n g the track  fitting.  W e can calibrate the magnetic field using either  the ne e~ d a t a or the 7r° —> e e~7 data. W e note that since the magnet was turned off +  +  a n d o n each time that the spectrometer was taken apart, we might expect 4 different calibration values.  C a l i b r a t i o n o n t h e ne e~ +  peak  W e choose a n energy variable i n which the ne e~ d a t a is as sharply peaked as possible. +  T h e appropriate k i n e m a t i c a l variable, from the considerations outlined i n chapter 3, is the t o t a l energy of the final state Etot — E  +  + .EL + T  n  a quantity w h i c h was also useful i n checking the accuracy of the ne e~ radiative correc+  tions. W e perform a one-parameter fit using x - n i i n i m i z a t i o n to extract a m o m e n t u m 2  c a l i b r a t i o n . T h e result of one of the fits is shown i n figure 9.1. T h e results for the 4 different r u n periods are tabulated i n table 9.2. T h e error quoted is statistical only.  —  100  111 n 111M11111111111111111 i n 111111111111  i  110  120  130  140  total energy  150  160  (MeV)  [ 1111111111111111111111111111111111111 n 1111111111  100  110  120  130  140  150  160  t o t a l e n e r g y (MeV)  F i g u r e 9.1: T h e 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 c a l i b r a t i o n , o n the r i g h t , after. +  1  60  80  1  1  I  1  100  1  1  I  1  120  '  1  I  1  1  140  1  I'  i  1  160  180  60  r'r  | i—i 80  E  i | i i  i | i i i | i  100  120  t o t  +P  t o t  i i | i i i  140  160  180  (MeV)  F i g u r e 9.2: T h e result of the calibration o n 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 calibration, o n the r i g h t , 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 +  W e m a y also calibrate using the 7r° —• e e~j d a t a , a procedure w h i c h m a y 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 c a l i b r a t i o n . W e perform a one-parameter x - i i i 2  m  n  m  z a  t i o n o n the quantity E P , t  t  discussed i n  chapter 3. T h e 7r° —» e e~7 process is restricted to a b o x w i t h very sharp edges, a n d +  it is the location of these edges w h i c h determines the fit. T h e fit is, however, sensitive to changes i n the ne e~ +  contamination a n d to a d d i t i o n of the radiative corrections,  so that a comparison w i t h the n e e ~ calibration results is nice to have. T h e results +  are shown i n figure 9.2 a n d tabulated i n table 9.2. W e o b t a i n consistent results; this  geometry 2 4 5 6  e e +  7  1.0020 1.0024 1.0032 1.0036  calibration ±0.0010 ±0.0005 ±0.0010 ±0.0007  ne e +  1.0012 1.0028 1.0012 1.0068  calibration ±0.0010 ± 0.0005 ±0.0010 ± 0.0007  Table 9.2: Results of the magnetic field calibrations for the various runs. T h e magnet was t u r n e d off and o n between the various runs. verifies o u r i n i t i a l guess that the calibration is independent of energy, a n d hence c a n be a t t r i b u t e d to a simple scale factor on the magnetic field. Notice that the changes to the field are s m a l l ; o n the order of 0.5%. T h i s is entirely consistent w i t h the accuracy of the magnet current monitor installed d u r i n g the experimental r u n . A r m e d w i t h these magnetic field scaling factors, we r e t u r n to the M o n t e C a r l o s i m u l a t i o n a n d reset the field for each r u n period. T h i s is a necessary step since the detector acceptance is a function of magnetic field. I n order to o b t a i n the resulting error i n a, we r u n through the analysis once w i t h the magnetic field set to its l i m i t .  W e o b t a i n the result shown i n table 9.1. A further source of error is the nonuniformity of the magnetic field. A n accurate field m a p was not made for S I N D R U M I, although one existed for the previous i n c a r n a t i o n of the detector [31]. F r o m this m a p , we estimate that w i t h i n the central region of the chambers ( w i t h i n a cone denned b y excluding the outer 10 c m of chamber 5) the magnetic field is u n i f o r m to better t h a n 1%. W e m a y verify this b y repeating the magnetic field c a l i b r a t i o n for different fiducial volumes of the detector; we see a variation consistent w i t h the above assertion. However, the statistics are not sufficient for this to be a n accurate m e t h o d of m a p p i n g the field. W e apply a fiducial cut a n d calibrate o n the interior region.  9.1.5  Analysis Cuts  In order t o 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 o b t a i n ; we s i m p l y record the exact value of the k i n e m a t i c a l variable i n question before m o d e l l i n g the detector response, a n d then plot the difference between the exact "remembered" value a n d the reconstructed value.  T h i s is of course the resolution of the s i m u l a t i o n , not of the  a c t u a l detector, b u t provides a useful estimate. S I N D R U M ' s m o m e n t u m (and energy) resolution is approximately 5% at 100 M e V . T h e angular resolution is approximately 3.5°. T h e quantity E Pt t  controls the amount of ne e~ c o n t a m i n a t i o n i n the final sample; +  a 5% resolution, together w i t h the estimation that the radiative t a i l of the ne e~ +  process is understood to no better t h a n 1 5 % leads us to the systematic error due to ne e~ +  c o n t a m i n a t i o n 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 t h a n the one the trigger makes. T h u s the final result  is d o m i n a t e d by the error associated w i t h this cut rather t h a n by the error due to the p o o r knowledge of the action of the trigger. B y v a r y i n g the opening angle cut by the stated resolution a n d finding 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 m o m e n t u m threshold is a function of the hodoscope radius a n d the vertex p o s i t i o n . I n order to m i n i m i z e systematic error due to uncertainty i n these quantities we a p p l y an explicit requirement of P  > 20 M e V / c .  t  V a r y i n g this  q u a n t i t y by 10% results i n no appreciable systematic error.  9.2  T i m e Independent Systematic Errors  Run-independent systematic errors w i l l increase the overall error o n the final result quoted i n 8.12,8.13 a n d shown i n figure 8.5. We elaborate briefly on their evaluation. 1. Chamber construction  : T h e chambers exhibit twists a n d wire spacing irregular-  ities, w h i c h are constant from r u n to r u n . These are s m a l l i n comparison to the aforementioned offsets a n d rotations, a n d hence they are not modelled but i n cluded i n the track reconstruction of the d a t a only. E l i m i n a t i n g these corrections gives negligible systematic effect. 2. Hardware  trigger  : T h e 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 a n d charge-asymmetric m o m e n t u m cut. T h e ir'p —• ne e~ d a t a , w h i c h have a larger +  electron m o m e n t u m , are reduced i n number by a factor of roughly 3, a n d therefore provide a good test of our trigger s i m u l a t i o n p r o g r a m . W e r u n this s i m u l a t i o n on the n e e ~ M o n t e C a r l o ; if subsequent results m a t c h the real d a t a well, we may +  be confident that the trigger is properly understood. F r o m figures 9.3 a n d 9.4, we see that the action of the trigger is understood, at least for the n e e ~ of geometry +  4. S i m i l a r results are achieved for geometries 2 a n d 5. W e are confident that the trigger is understood, a n d that the systematic error induced b y i t is s m a l l . 3. Contamination  from n e e " : T h e radiative ne e~ events contaminate the D a l i t z +  +  sample. These events must be well understood. F r o m figure 9.4h) we see that the long radiative t a i l is accurate to w i t h i n roughly 15%. B y increasing the number of ne e~ +  t a i l events b y 15% a n d reanalyzing the d a t a , we o b t a i n the systematic  error shown i n the table. 4. Radiative  corrections  : T h e results of the numerical integra-  to TT° —• e e~j +  t i o n of the m a t r i x elements of the n° —• e e~*y radiative corrections have been +  checked against published semi-analytical values [22], a n d agree t o w i t h i n 5%. F u r t h e r m o r e , the radiative t a i l i n E Pt t  is very well modelled.  error induced b y the uncertainty i n the 7r° —• e e~j +  T h e systematic  radiative corrections must  be s m a l l .  9.3  O t h e r Systematic  Errors  O t h e r sources of systematic errors have been investigated a n d found t o be negligible. These include • Possible  error due to the multiplicity  cuts on the inner chambers.  T h e possible  source of error depends o n how the s i m u l a t i o n decides o n wire h i t s . T h e energy loss as calculated b y G E A N T is deposited directly into the cathode a n d anode strips of the detector.  T h i s discrete deposited energy is converted i n t o a G a u s -  sian response function a n d then transformed into a cluster of cathode signals, w h i c h are converted into wire hits b y the analysis package d u r i n g the track r e construction. W e have checked that the real d a t a a n d s i m u l a t i o n produce s i m i l a r  m l n n m . . I  •I  *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  cos# emission  10  20  P  30 t  40  (+)  50  0.5  60  1.0  angle  70  (MeV/c)  80  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  P  30 t  (-)  40  50  60  70  (MeV/c)  80  F i g u r e 9.3: T h e performance of the n e e ~ simulation ( w i t h radiative corrections) for some areas of phase space, for geometry 4. T h e points are the d a t a ; the h i s t o g r a m , the s i m u l a t i o n . Since the agreement between the d a t a a n d the M o n t e C a r l o is good, we are confident that we understand the action of the trigger. +  T 111 u 1 n 11 m  0.0  m  111 m  m  1111111 j 11111  j 11 m  u  11111111111111111111111111111111111111  1111  -1.0  0.2 0.4 0.6 0.8 1.0 1.2  liniilHitilllllHllllllHlllll  iilni  0.0  0.5  1.0  energy p a r t i t i o n Y  i n v a r i a n t mass x llmillllll  -0.5  IIIIIIIIII  pitBilll|IIIUIIII|lllllllll|lllllllll|lllllllll|lllllllll|lllllllll|llllll|!  20 4 0 60 80 100120140160180  opening angle 0  F i g u r e 9 . 4 : T h e performance of the n e e ~ s i m u l a t i o n ( w i t h radiative corrections) for some areas of phase space, for geometry 4 . T h e points are the d a t a ; the h i s t o g r a m , the s i m u l a t i o n . Since the agreement between the d a t a a n d the M o n t e C a r l o is good, we are confident that we understand the action of the trigger. +  cluster sizes, a n d that hence the analysis thresholds for deciding true wire hits are correct. W e are confident that we do not miss real hits. • Possible  errors due to misalignment  of the hodoscope. Since the outermost c h a m -  ber (number 5) was mounted onto the inside of the hodoscope, it is h i g h l y 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, fine-tuning its location is unnecessary. • Errors  due to binning.  W h e n fitting for a i n x a n d (j> we b i n the d a t a into 100  a n d 90 bins, respectively. Since b o t h the resolution i n x a n d i n <f> is m u c h smaller t h a n the size of one b i n , we expect no systematic effect f r o m "edge" events being misassigned to neighbouring bins. We have checked that increasing the b i n size by a factor of two does not affect the result of the fit. • Possible  dependence on detector <f> quadrant.  W e divide the d a t a u p into four  samples according to the quadrant of emission of the electron to study the effect on a. W e see no systematic effect.  9.4  Summary  Table 9.1 summarizes the contributions to the t o t a l systematic error f r o m various sources. Note that the s t a n d a r d 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. W e see that the error o n a  x  is d o m i n a t e d by the uncertainty i n the value of the magnetic field; this is  not s u r p r i s i n g since x is a linear function of the m o m e n t u m . O n the other h a n d , since the opening angle is only i n d i r e c t l y dependent on the magnetic field (the  acceptance  i n <j> is weakly dependent on the magnetic field since the trigger also cuts o n track t  curvature),  is relatively free of this systematic error. However aj, is more sensitive  to the calibrations of the chamber rotations a n d offsets t h a n is a . x  O u r two results are  thus a  x  a<t>  = 0.003 ± 0 . 0 1 1 ± 0 . 0 2 2 = 0.027 ± 0 . 0 1 3 ± 0 . 0 4 7  T a k i n g the error weighted average of the two central values a n d quoting the smaller errors, we o b t a i n our final result a = 0.02 ± 0 . 0 1 ± 0 . 0 2  C h a p t e r 10  S u m m a r y and C o n c l u s i o n s  T h e basic difficulty w i t h an experiment a t t e m p t i n g to measure the ir° f o r m factor v i a the D a l i t z decay is the l i m i t e d range of m o m e n t u m transfer available to probe the p i o n structure. T h e v i r t u a l p h o t o n used to study the p i o n structure is l i m i t e d to energies below the p i o n mass.  It is difficult to resolve the quark structure of the meson at  such low energies, a n d the results of previous experiments (table 2.2) are, accordingly, inconclusive. W e point out that the result we o b t a i n is also consistent w i t h a structureless 7T°, a l t h o u g h it represents a substantial improvement i n the u n d e r s t a n d i n g of systematic errors. C o u l d the error bars be reduced i n a future experiment? D a t a for the f o r m factor measurement were taken over only three days. It w o u l d be a very simple matter to reduce the statistical errors by a factor of 3 by r u n n i n g for a m o n t h ; however, the reduction of the systematic errors is not so simple. In our experiment, the availability of a large number of events, spread over m u l t i p l e runs, combined w i t h our fits for a i n two spectra using two different methods, provides m a n y double checks a n d is essential to our understanding of the systematic errors, the largest of w h i c h were the magnetic field, the chamber calibrations, a n d the uncertainty i n the Tr~p —> ne e~ +  radiative t a i l .  A n accurate field m a p might have been helpful i n reducing the error associated w i t h the field's n o n - u n i f o r m i t y ; however, since the dependence of a o n the value of the field is linear, a n d since we use the same d a t a set for the c a l i b r a t i o n a n d 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 w h i c h causes the systematic error. A field m a p w o u l d have reduced the need for f i d u c i a l chamber cuts and w o u l d therefore have increased the n u m b e r of events. T h e uncertainty i n the relative positions of the chambers proves to be the largest source of error i n this measurement. T h e measurement of a using the <f> d i s t r i b u t i o n is especially sensitive to the relative chamber rotations. T h e reconstructed z m o m e n t u m is quite sensitive to the z positions of the chambers, a n d has a considerable influence o n the d e t e r m i n a t i o n of a . x  T h e chamber geometries were c a l i b r a t e d u s i n g cosmic rays,  and it is difficult to see how these calibrations could have been i m p r o v e d . T h e uncertainty i n the ir~p — » ne e~ +  radiative t a i l is a systematic error that could  be reduced i n future experiments. Because the trigger cut so heavily o n the ne e~ +  data,  a detailed u n d e r s t a n d i n g of this process is difficult to achieve. T h e acceptance of the detector for the ne e~ +  d a t a is very sensitive to the magnetic field a n d to the chamber  geometries. C o m b i n e d w i t h a slight difference i n trigger acceptance between electrons and positrons of h i g h m o m e n t u m , this leads to the observed charge a s y m m e t r y i n the TT~p —* ne e~ +  d a t a (see figures 9.3 a n d 9.4). W h i l e this a s y m m e t r y is reasonably well  u n d e r s t o o d for geometries 2, a n d 4 a n d 5, for geometry 6 it is m u c h more pronounced and is not well s i m u l a t e d . It is not surprising, then, that the r a d i a t i v e t a i l of this process is u n d e r s t o o d to no better t h a n 20%. A more careful choice of trigger conditions, plus a d d i t i o n a l r u n n i n g w i t h a reversed magnetic field w o u l d have aided considerably i n a n u n d e r s t a n d i n g of this i m p o r t a n t background process. It w o u l d be difficult, but not impossible, to improve the systematic a n d s t a t i s t i c a l errors by 30 or 5 0 % . W h e t h e r this is necessary, however, is not clear, since, by t i m e invariance, we expect the f o r m factor for p i o n decay 7r° —* 77* to be no different from the f o r m factor for p i o n p r o d u c t i o n 77* —• 7r°. B y using p i o n p r o d u c t i o n , one not only probes the p i o n at large (negative) m o m e n t u m transfer, but also avoids a l l of the  systematic errors o u t l i n e d above (introducing, of course, others).  T h 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 t h theoretical expectations. In short, t h e n , our result of a = 0.02 ± 0.01 ± 0.02 serves to clear u p past discrepancies, a n d brings the measurement of the 7r° form factor i n the timelike region of m o m e n t u m transfer into line w i t h b o t h the theoretical expectations a n d the recent (spacelike) C E L L O result.  It seems that the structure of the n e u t r a l p i o n can be  understood i n the framework of the S t a n d a r d M o d e l .  Bibliography  [1] J . J . T h o m p s o n : Rays of Positive  Electricity,  Longmans G r e e n , L o n d o n , 1913  [2] E . R u t h e r f o r d : P h i l . M a g . 21, 669 (1911) [3] M . G e l l - M a n n : P h y s . L e t t . 8, 214 (1964) [4] G . Zweig: C E R N R e p o r t 8 4 1 9 / T h 412, 1964 [5] R . H . D a l i t z : P r o c . P h y s . Soc. A 64, 667 (1951) [6] N . M . K r o l l a n d W . W a d a : P h y s . R e v . 98, 1355 (1955) [7] G . B a r t o n a n d B . G . S m i t h : Nuovo C i m e n t o 36, 436 (1965) [8] S. L . A d l e r : P h y s . R e v . 177, 2426 (1969) [9] L I . A m e t l l e r et a l . : N u c l . P h y s . B228, 301 (1983) [10] N A 3 1 C o l l a b o r a t i o n , P h y s . L e t t . B240, 283 (1990) [11] L . B e r g s t r o m et a l : P h y s . L e t t . 131B, 229 (1983) [12] L . G . Landsberg: P h y s . R e p . 128, 301 (1985) [13] A . K o t l e w s i , preprint C o l u m b i a University, N e w Y o r k (1973) [14] M . R . J a n e et a l . : P h y s . L e t t . 5 9 B , 103 (1975) [15] Y u . B . B u s h n i n et al., P h y s . L e t t . 7 9 B , 147 (1978); R . I. D z h e l y a d i n et al., P h y s . L e t t . 8 4 B , 143 (1979); ibid 8 8 B , 379 (1979); ibid 9 4 B , 548 (1980); ibid 102B, 296 (1981)  [16] N . P . Samios: P h y s . R e v . 1 2 1 , 275 (1961) [17] H . K o b r a k : N u o v o C i m e n t o 2 0 , 1115 (1961) [18] S. Devons et a l . : P h y s . R e v . 1 8 4 , 1356 (1969) [19] J . B u r g e r : D o c t o r a l Thesis, C o l u m b i a University, N e w Y o r k , 1972 [20] B . E . L a u t r u p a n d J . S m i t h : P h y s . R e v . D 3 , 1122 (1971) [21] J . Fischer et a l . : P h y s . L e t t . 7 3 B , 359 (1978) [22] O . K . M i k a e l i a n a n d J . S m i t h : P h y s . R e v . D 5 1763 (1972); ibid D 5 , 2890 (1975) [23] L . Rosselet et a l : P h y s . R e v . D 1 5 , 574 (1977) [24] P . G u m p l i n g e r : D o c t o r a l Thesis, Oregon State University, P o r t l a n d , 1986; J . M . Poutissou et a l . : P r o c . of the Lake Louise W i n t e r Insitute (1987), p u b l . W o r l d Scientific [25] L . R o b e r t s a n d J . S m i t h : P h y s . R e v . D 3 3 , 3457 (1986) [26] H . Fonvieille et a l . : D o c t o r a l Thesis, L ' U n i v e r s i t e Blaise P a s c a l - C l e r m o n t II, C l e r m o n t - F e r r a n d , France, 1989, a n d P h y s . L e t t . B 2 3 3 , 65 (1989) [27] C E L L O C o l l a b o r a t i o n , Z. P h y s . C 4 9 , 401 (1991) [28] S. R . A m e n d o l i a et. a l : P h y s . L e t t . 1 4 6 B , 116 (1984) [29] C h . G r a b : A Search for the Decay / i —• e e e ~ , D o c t o r a l Thesis, Universitat +  +  +  Z u r i c h , 1985 [30] N . K r a u s : T h e R a r e Decay fi  +  1985  —• e u u e e~, +  ti  e  +  D o c t o r a l Thesis, Universitat Z u r i c h ,  [31] W . B e r t l et a l : N u c l . P h y s . B 2 6 0 , 1 (1985) [32] S. E g l i : T h e R a r e Decay TT  +  — » e e e~v , +  +  e  D o c t o r a l Thesis, U n i v e r s i t a t Z u r i c h ,  1987 [33] C . N i e b u h r : A Search for the R a r e Decay TT° - • e e , D o c t o r a l Thesis, E T H +  _  Z u r i c h , 1989 [34] W . B e r t l et a l . : N u c l . Inst. & M e t h . 2 1 7 367 (1983) [35] A . van der Schaaf et a l : N u c l . Inst. & M e t h . A 2 4 0 , 370 (1985) [36] C . N i e b u h r et a l . : P h y s . R e v . D 4 0 , 2796 (1989) [37] R . E i c h l e r : S I N D R U M note 15, internal m e m o , unpublished (1983) [38] J . F . C r a w f o r d : N u c l . Inst. & M e t h . 5 7 , 237 (1983) [39] R . M e i j e r Drees, M S c thesis, U n i v e r s i t y of B r i t i s h C o l u m b i a , B . C . , 1987 [40] R . B r u n , F . B r u y a n t , M . M a i r e , A . C . M c P h e r s o n , a n d P . Z a n a r i n i , G E A N T 3 package, C E R N D a t a H a n d l i n g D i v i s i o n , D D / E E / 8 4 - 1 [41] M . L e o n a n d H . Bethe, P h y s . R e v . 1 2 7 , 636 (1962) [42] T . M i y a z a k i a n d E . Takasugi: P h y s . R e v . D 8 , 2051 (1973) [43] G . B . T u p p e r , T . R . Grose, a n d M . A . Samuel, P h y s . R e v . D 2 8 , 2905 (1983) [44] M . L a m b i n a n d J . Pestieau, P h y s . R e v . D 3 1 , 211 (1985) [45] M . N . B e n s a y a h , Thesis, L ' U n i v e r s i t e Blaise P a s c a l - C l e r m o n t I I , C l e r m o n t F e r r a n d , France, 1989 [46] G . B o n n e a u a n d F . M a r t i n : N u c l . P h y s . B 2 7 , 381 (1971)  [47] J . O . Berger: Statistical  Decision  Theory : Foundations,  Concepts,  and  Methods,  © 1 9 8 0 , Springer-Verlag, New Y o r k [48] J . L . C h u m a , P L O T D A T A package, T R I U M F © 1 9 9 0 [49] W . H . Press, B . P . F l a n n e r y , S. A . Teukolsky, a n d W . T . V e t t e r l i n g : Recipes : the Art of Scientific Cambridge, U . K .  Computing,  Numerical  © 1 9 8 9 , C a m b r i d g e U n i v e r s i t y Press,  

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:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0084985/manifest

Comment

Related Items