A study of photon structure with special attention to the low-x region

A S tu d y o f P h o to n

S tru ctu re w ith Special

A tten tio n to th e Low-j; R egion

J. Jason W ard

D ep artm en t of Physics and A stronom y U niversity College London

Subm itted for th e degree of D o cto r o f P h ilo so p h y

ProQuest Number: 10018676

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A b stra ct

A cknow ledgem ents

First and forem ost I would like to thank David Miller for his supervision and support throughout this work. I am indebted to Bruce Kennedy and M ark Lehto who both helped to get me going w ith the analysis.

The OPAL C ollaboration are gratefully acknowledged for their hard work in building and run­ ning the detector which provides the d a ta for this thesis. Special thanks to Jim Conboy, Allan Skillman, Ed McKigney, Tony Rooke, Joerg Bechtluft, Bernd W ilkens, Steve Hillier, R ichard Nisius, Volker Blobel and Peter Hobson who have all been fine collaborators. My understanding of the theory of tw o-photon physics is much b etter after discussions with Jeff Forshaw, and Mike Seymour has been excellent with answering my HERW IG questions.

Thanks to David Munday, T ara Shears, Bill Allison and G uy Pooley, who all helped me when I was applying to various places to do a Ph.D . - all of the support and encouragement th a t I got m ade a big difference.

For financial support I would like to thank the Particle Physics and Astronom y Research Council and University College London. Also the University of London’s award of the 1994 Valerie Myerscough Prize m ade a trip to a NATO Advanced Study In stitu te possible.

During the P h.D ., and because of it, I was fortunate enough to become involved in two new activities th a t have shown ways towards having a quiet m ind. A bow to Dennis Ngo for being an excellent teacher of T a i’ Chi (Suan Yang of the T iger-C rane com bination) and an enormous thankyou to Paul Phillips and C hristine Beeston for getting me into m ountain walking.

Special thanks to Ja n Lauber, W arren M atthews, M ark Lehto (again!), M ark Pearce, Kwasi Ametewee, Nick Feast, Harvey May cock, K atrijn R aaij m akers and Philippe Persiaux, who have all provided alot of support on various occasions.

Finally, thanks to m y parents, M argaret and John W ard, and my sisters, Hayley and Michelle W ard, for their continuing Love and support.

This thesis is dedicated to all of the people nam ed above. It could not have been produced w ithout them .

C on ten ts

L ist o f Figures 8

List o f Tables 15

1 Introd u ction 17

1.1 Tw o-Photon I n te r a c tio n s ... 18

1.2 T he Photon P i c t u r e ... 18

1.3 The e7 V e r te x ... 19

1.4 7 7 Collisions at an e+e" C o l l i d e r ... 19

1.5 Deep Inelastic e7 S c a t t e r i n g ... 21

1.6 Interest in ... 23

1.7 FJ M easurem ents a t L E P ... 28

2 T h eory o f th e P h o to n S tru ctu re Function 31 2.1 P arto n D istributions of th e P h o t o n ... 31

2.2 T he Com ponents of ... 32

2.3 Vector Meson D om inance ... 33

2.4 T he Q uark P a rto n M odel ... 34

C O N T EN T S 5

2.5.1 T he D G LA P Evolution E q u a tio n s ... 37

2.5.2 P arto n D istributions at L o w - z ... 38

2.6 C h arm -Q u ark C o n trib u tio n s ... 39

2.7 F2 P aram eterisatio n s and M o d e l s ... 42

2.7.1 Glück, R eya and Vogt ( G R V ) ... 42

2.7.2 H agiw ara et al. ( W H I T ) ... 43

2.7.3 G ordon and Storrow ( G S ) ... 43

2.7.4 Drees and Grassie ( D G ) ... 44

2.7.5 Levy, Abram owicz and C harcula (LAC) ... 45

2.7.6 A urenche et al. ( A C F G P ) ... 45

2.7.7 Field, K ap u sta and Poggioli ( F K P ) ... 47

2.7.8 Schuler and Sjôstrand ( S a S ) ... 48

3 LE P and th e O PAL D e te c to r 50 3.1 L E P ... 50

3.2 T he OPAL D e t e c t o r ... 51

3.2.1 C entral Tracking D e te c to rs ... 54

3.2.2 Tim e-of-Flight ... 56

3.2.3 E lectrom agnetic C a lo rim e try ... 56

3.2.4 H adron C a lo rim e te r... 59

3.2.5 M uon D e t e c t o r ... 59

3.3 OPAL Forward D e t e c t o r s ... 60

3.3.1 Silicon T ungsten C alorim eter ( S W ) ... 60

3.3.2 Forward D etector ( F D ) ... 60

6 C O N T E N T S

4 E ven t S electio n 68

4.1 Event S e l e c t i o n ... 68

4.1.1 P r e s e le c tio n ... 69

4.1.2 F u rth er S e l e c t i o n ... 70

4.1.3 Final S e l e c t i o n ... 72

4.2 E stim atio n of B ac k g ro u n d s... 75

4.2.1 e‘^e“ —>• hadrons ... 75

4.2.2 e + e " ^ r + r “ 75 4.2.3 N o n -m ultiperipheral e+e“ ->-e‘*'e” + hadrons ... 75

4.2.4 B eam -gas e v e n t s ... 78

4.3 Trigger E ffic ie n c y ... 78

4.3.1 C alculation of E f fic ie n c y ... 78

4.3.2 E stim atio n of Efficiency from th e D a t a ... 79

4.4 D a ta Self Consistency ... 80

5 M on te C arlo S im u lation 91 5.1 7*7 F r a g m e n ta t io n ... 92

5.2 V e r m a s e r e n ... 92

5.3 F2G EN ... 92

5.4 H E R W I G ... 94

5.5 C om parison of G e n e r a t o r s ... 97

5.6 M onte Carlo Samples G e n e ra te d ... 99

6 C om parison o f D a ta w ith M o n te Carlo 102 6.1 M onte Carlo Models in th e C o m p a ris o n ... 103

C O N T E N T S 7

6.3 S u m m a r y ... 126

7 U n fold in g 127 7.1 T he Problem of M easuring 77^(2: ) ... 127

7.2 T he Forward M e t h o d ... 128

7.3 T he Inverse M ethod ... 130

7.4 D is c r e tiz a tio n ... 131

7.5 U n f o l d i n g ... 131

7.6 Unfolding Exam ples ... 134

7.6.1 Unfolding in a B i n ... 135

7.6.2 T he Unfolding T e s t s ... 136

7.7 D ata U n fo ld in g s... 143

8 S u m m ary and C onclusions 157

List o f Figures

1.1 T he m ulti-peripheral tw o-photon process labelled w ith four-vectors, z and T are th e energies of th e probing and probed photon respec­ tively, expressed as a fraction of th e beam energy... 20

1.2 C om parison of evolution of th e second m om ent of in QCD (solid line) w ith a theory in which th e coupling constant is frozen a t an initial value of Qg = 5 GeV^ (dot-dashed line)... 25

1.3 Feynm an diagram used in some QCD models of th e low-a; p a rt of th e photon stru ctu re fu n ctio n... 26

1.4 Rise in th e proton stru c tu re function m easured by th e ZEUS Col­ laboration (circles) and th e H I C ollaboration (triangles)... 27

1.5 S catter plot of M onte Carlo tagged tw o-photon events from the F2GEN generator (see Section 5.3) at Ebeam = 45.6 GeV. T he m in­ im um 7 7 m ass is 2 GeV, th e m inim um tag angle is 30 m rad and th e m inim um tag energy is 20 GeV. T he photonic parto n distrib u tio n functions from GRV (see Section 2.7.1) have been used in th e event generation... 30

2.1 Q PM and VMD predictions for • T he simple VMD and T P G/2 7

VMD predictions are from E quations 2.8 and 2.10 respectively. The Q PM prediction (E quation 2.13) are for 3 flavours, w ith = rrid = ms = 300 M eV ... 35 2.2 T he T P C/2 7 VMD prediction is shown by th e do tted line. The

L I S T OF FIGURES 9

2.3 The GRV Leading O rder param eterisatio n for four flavours (solid lines) and three flavours (dashed lines). T he lower curve of each pair is calculated a t Q ^=5.9 GeV^ and th e upper curve of each pair is calculated a t (5^=14.7 GeV^... 41

2.4 The LA C l param eterisatio n for four flavours (solid lines) and th ree flavours (dashed lines). T h e lower curve of each pair is calculated at Q^=5.9 GeV^ and th e u p p er curve of each pair is calculated at Q "=14.7 GeV"... 46

2.5 A selection of three-flavour F^i^x) param eterisations from Section 2.7. All of these curves are calculated a t Q^=14.7 G eV"... 49

3.1 A cut-away view of th e OPAL d e tec to r... 52

3.2 Side and end views of th e OPAL detector, sectioned to show th e m ain subdetector system s: central vertex cham ber, je t cham ber and z-cham bers (CV, CJ and CZ), electrom agnetic calorim eters (EB and EE), hadron calorim eters (HB and HE) and m uon cham bers (MB and M E). T he forw ard detector m odules (FD ) can also be seen. 53

3.3 Cross section through th e forward region (pre-1993) betw een 2 and 3 m etres from th e intersection region (which is to th e left of this diagram ). BP = B eam P ip e, ET = D rift C ham bers, EL = Fine Lum inosity M onitor, EE = G am m a C atcher, F P = P resam pler C alorim eter, FB = Tube C ham bers and FK = M ain C alorim eter. 61

?eam •

4.1 E stim ate of background events, (a) D istribution in Etag/Eb^ The vertical d o t-d ash ed line shows where th e m inim um tag energy cut is. (b) The d istrib u tio n in Xyis after all of th e selection cuts have been applied. T he background estim ate in th is histogram is enhanced by a factor of 10... 76

4.2 T he four m ain diagram s contributing in th e lowest order to th e process 7 7 . These processes are included in PE R M ISV. U nlabelled boson lines represent photons only... 77

4.3 Comparison of 1990 (solid), 1991 (dashed) and 1992 (d o tted ) tag distributions. Final selection cuts are represented by vertical d o t- dashed lines... 83

10 L IS T OF FIGURES

4.5 C om parison of 1990 (solid), 1991 (dashed) and 1992 (dotted) in­ variant m ass distributions. Final selection cuts are represented by vertical d o t-d ash ed lines... 85

4.6 C om parison of 1990 (solid), 1991 (dashed) and 1992 (dotted) an ti­ tag and n eu tral energy distributions. Final selection cuts are rep­ resented by vertical d o t-d ash ed lines... 86

4.7 Com parison of 1990 (solid), 1991 (dashed) and 1992 (dotted) tra n s­ verse m om entum distributions (defined in Section 4.1.3) in th e plane of th e beam and th e tag. Final selection cuts are represented by vertical d o t-d ash ed lines... 87

4.8 C om parison of 1990 (solid), 1991 (dashed) and 1992 (dotted) tra n s­ verse m om entum distributions (defined in Section 4.1.3) out of th e plane of th e beam and th e ta g ... 88

4.9 Com parison of 1990 (solid), 1991 (dashed) and 1992 (dotted) tra n s­ verse m om entum distributions (defined in Section 4.1.3) out of the plane of th e beam and th e tag. ... 89

4.10 C om parison of 1990 (solid), 1991 (dashed) and 1992 (dotted) track m ultiplicity distributions. Final selection cuts are represented by vertical d o t-d ash ed lines... 90

5.1 A representation of th e deep inelastic 67 scattering process in th e H ERW IG M onte Carlo. ISPS and FSPS are th e initial and final sta te p arto n showers respectively... 96

6.1 Tag d istributions for th e d a ta (dots) and th e M onte Carlo models in th e whole 9tag range. T he different samples are from F2GEN GRV 100% p o in t-lik e (solid), F2G EN GRV ‘perim iss(O .l)’ (dashed), H ER ­ W IG GRV w ithout th e SUE (d o tted) and HERW IG LA C l w ithout th e SUE (d o t-d ash e d )... 105

6.2 Xtrk-, ^vis and XyisFD distributions (each defined in Section 4.1.2) for th e d a ta (dots) and th e M onte Carlo models in th e low Otag range. T he different samples are from F2G EN GRV 100% po in t­ like (solid), F2GEN GRV ‘perim iss(O .l)’ (dashed), HERW IG GRV w ithout th e SUE (dotted) and HERW IG LA C l w ithout th e SUE

L IS T OF FIGURES 11

6.3 Xtrk, ^vis and XyisFD distributions (each defined in Section 4.1.2) for th e d a ta (dots) and the M onte Carlo models in th e high Otag range. T he different samples are from F2G EN GRV 100% point­ like (solid), F2G EN GRV ‘perim iss(O .l)’ (dashed), HERW IG GRV w ithout th e SUE (dotted) and HERW IG LA C l w ithout th e SUE (d o t-d a sh e d )... 107

6.4 Invariant m ass distributions for th e d a ta (dots) and th e M onte Carlo m odels in th e low Otag range. T he different samples are fro m , F2G EN GRV 100% p o int-lik e (solid), F2G EN GRV ‘perim iss(O .l)’ (dashed), HERW IG GRV w ithout th e SUE (do tted ) and HERW IG LA Cl w ithout th e SUE (d o t-d ash e d )... 108

6.5 Invariant mass distributions for th e d a ta (dots) and th e M onte Carlo m odels in th e high Otag range. T he different samples are from F2G EN GRV 100% point-like (solid), F2G EN GRV ‘perim - iss(O .l)’ (dashed), HERW IG GRV w ithout th e SUE (dotted) and HERW IG LA C l w ithout the SUE (d o t-d a sh e d )... 109

6.6 Xyisjxtrue Correlation plots from th e F2G EN and HERW IG M onte Carlo m odels in th e low Otag range. T he different samples are from F2G EN GRV 100% point-like (solid lines, closed circles), F2GEN GRV ‘perim iss(O .l)’ (dashed lines, open circles), HERW IG GRV w ithout th e SUE (dotted lines, closed squares) and HERW IG GRV w ith th e SUE (d o t-dash ed lines, open squares)... 112

6.7 Xyis/xtrue Correlation plots from th e F2G EN and H ERW IG M onte Carlo m odels in th e high Otag range. T h e different samples are from F2G EN GRV 100% point-like (solid lines, closed circles), F2GEN GRV ‘perim iss(O .l)’ (dashed lines, open circles), HERW IG GRV w ithout th e SUE (d o tted lines, closed squares) and HERW IG GRV w ith th e SUE (d o t-d ash ed lines, open squares)... 113

6.8 A n ti-ta g and n eu tral energy distributions for th e d a ta (dots) and th e M onte Carlo models in th e low Otag range. T he different sam­ ples are from F2G EN GRV 100% p o in t-lik e (solid), F2G EN GRV ‘perim iss(O .l)’ (dashed), HERW IG GRV w ithout th e SUE (dotted) and H ERW IG LA C l w ithout th e SUE (d o t-d a sh e d )... 115

12 L IS T OF FIGURES

6.10 Transverse m om entum distributions (defined in Section 4.1.3) in th e plane of th e beam and th e tag for th e d a ta (dots) and th e M onte Carlo models in th e low Ofag range. The different sam ples are from F2GEN GRV 100% p oin t-lik e (solid), F2GEN GRV ‘perim iss(O .l)’ (dashed), H ERW IG GRV w ithout th e SUE (dotted) and HERW IG LA C l w ithout th e SUE (d o t-d ash e d )... 117

6.11 Transverse m om en tu m distributions (defined in Section 4.1.3) in th e plane of th e beam and th e tag for th e d a ta (dots) and th e M onte Carlo models in th e high $tag range. The different samples are from F2G EN GRV 100% p o in t-like (solid), E2GEN GRV ‘perim iss(O .l)’ (dashed), HERW IG GRV w ithout th e SUE (dotted) and HERW IG LA C l w ithout th e SUE (d o t-d ash e d )... 118

6.12 Transverse m om entum distributions (defined in Section 4.1.3) out

of th e plane of th e beam and th e tag for th e d a ta (dots) and th e M onte Carlo m odels in th e low $iag range. The different samples are from E2GEN GRV 100% point-like (solid), F2G EN GRV ‘per- im iss(O .l)’ (dashed), HERW IG GRV w ithout th e SUE (dotted) and HERW IG LA C l w ithout th e SUE (d o t-d ash ed )... 119

6.13 Transverse m om entu m distributions (defined in Section 4.1.3) out of the plane of th e beam and th e tag for th e d a ta (dots) and the M onte Carlo m odels in th e high Otag range. The different samples are from F2GEN GRV 100% point-like (solid), F2G EN GRV ‘per- im iss(O .l)’ (dashed), HERW IG GRV w ithout the SUE (dotted) and HERW IG LA C l w ithout th e SUE (d o t-d ash ed )... 120

6.14 Transverse m om entum distributions (defined in Section 4.1.3) out of th e plane of th e beam and th e tag for th e d a ta (dots) and the M onte Carlo m odels in th e low Otag range. The different samples are from F2G EN GRV 100% point-like (solid), E2GEN GRV ‘per- im iss(O .l)’ (dashed), H ERW IG GRV w ithout the SUE (dotted) and HERW IG L A C l w itho u t th e SUE (d o t-d ash ed )... 121

LIST OF FIGURES 13

6.16 Track m ultiplicity, energy and energy flow d istributions for th e d a ta (dots) and th e M onte Carlo m odels in th e low Otag range. T he different sam ples are from F2G EN GRV 100% poin t-lik e (solid), F2G E N GRV ‘perim iss(O .l)’ (dashed), HERW IG GRV w ithout th e SUE (d o tted ) and HERW IG LA C l w ithout th e SUE (d o t-d ash ed ). 124

6.17 Track m ultiplicity, energy and energy flow distributions for th e d a ta (dots) and th e M onte Carlo m odels in th e high Otag range. T he different sam ples are from F2G EN GRV 100% poin t-lik e (solid), F2G EN GRV ‘perim iss(O .l)’ (dashed), HERW IG GRV w ithout th e SUE (d o tted ) and HERW IG LA C l w ithout th e SUE (d ot-d ash ed ). 125

7.1 H istogram and profile plot of Xyig and Xtme- T he HERW IG M onte Carlo has been used w ith th e GRV and w ithout th e soft under­ lying event. T he events have passed th e analysis cuts w ith tags for all of th e Otag region (50-120 m rad). T he vertical error bars on th e profile plot represent the error on th e m e an ... 129

7.2 Test unfoldings each using th e H ERW IG generator w ith th e GRV F 7 (^ ) {without th e soft underlying event) as th e unfolding M onte Carlo, (a) and (b) use th e HERW IG GRV F^(a;) without th e soft underlying event as th e “d a ta ” , (c) and (d) use th e HERW IG GRV F2 {x) with th e soft underlying event as th e “d a ta ” . Solid

lines represent th e unfolded results and th e horizontal d o tted lines represent th e GRV expectation values for each unfolded bin. . . . 139

7.3 Test unfoldings each using th e H ERW IG generator w ith th e GRV F y (^ ) th e unfolding M onte Carlo. T he “d a ta ” in each case come from HERW IG w ith the LA C l F ^ ix ). B oth th e unfolding M onte C arlo and th e “d a ta ” are w ithout th e soft underlying event. Xyis for b o th “d a ta ” and M onte Carlo is calculated without FD clusters in (a) and (b); with FD clusters in (c) and (d). Solid lines represent th e unfolded results and th e horizontal d o tted lines represent th e LA C l expectation values for each unfolded b in ... 140

14 LIST OF FIGURES

7.5 X distributions for d a ta and M onte Carlo in th e high 9tag region. T he d a ta distributions are represented by the dots. The differ­ ent M onte C arlo sam ples are from F2GEN GRV 100% point-like (solid line), F2GEN GRV ‘perim iss(O .l)’ (dashed line), HERW IG GRV w ithout th e SUE (d otted line) and HERW IG GRV w ith the SUE (d o t-d ash ed lin e)... 145

7.6 Four-flavour unfoldings of th e d a ta w ith different M onte Carlo models in th e low Otag region. T he dashed line is th e four-flavour GRV p aram eterisatio n calculated at = 5.9 GeV^, which has been included for reference only... 147 7.7 Four-flavour unfoldings of th e d a ta w ith different M onte Carlo

models in th e high Otag region. The dashed line is the four-flavour GRV p aram eterisatio n calculated at = 14.7 GeV^, which has been included for reference only... 148 7.8 C om bined four-flavour unfoldings of th e d a ta in th e low Otag region.

T he inner error bars are statistical only. T he outer error bars are th e statistical and system atic errors combined in quadrature. The broken lines are th e four-flavour GRV (dashed) and LACl (dotted) p aram eterisatio n s calculated at = 5.9 GeV^... 150 7.9 C om bined four-flavour unfoldings of th e d a ta in th e high Otag re­

gion. T he inner error bars are statistical only. The outer error bars are th e statistic a l and system atic errors combined in quadrature. T he broken lines are th e four-flavour GRV (dashed) and LACl (d o tted ) param eterisatio n s calculated at = 14.7 GeV^... 151 7.10 Com parison in th e low region of th e combined four-flavour

result of this thesis, unfolded on a log^g a; scale, w ith the th re e - flavour OPAL [34] result (w ith an estim ate of th e charm contribu­ tion added) unfolded on a linear x scale... 153 7.11 Com parison in th e high region of th e combined four-flavour

result of this thesis, unfolded on a log^g ^ scale, w ith three-flavour OPAL [34] and D E L PH I [36] results (w ith an estim ate of th e charm contribution added) unfolded on a linear x scale... 154 7.12 Com parison in th e high region of th e combined four-flavour

15

List o f Tables

1.1 Published hadronic Fo d a ta as a function of a;... 28

3.1 Sum m ary of triggers used in triggering tagged tw o-photon events, w ith typical threshold values. T he superscript S indicates a stan ­ dalone trigger and th e superscript C indicates th a t th e trigger forms p a rt of th e C EN TR L trigger described in Section 4.3.2. T he n o ta­ tion L(R) refers to th e Left (R ight) sides of O PA L... 66

3.2 Program m ed trigger conditions combining triggers from tag and hadronic activity in tagged tw o-photon events... 67

4.1 Q uality cuts applied to th e tracks, n eu tral clusters and th e track-cluster association cone. T he quantities are described in Section 4.1.2. 69

4.2 D etector statu s num ber and in te rp re ta tio n ... 70

4.3 Final selection cuts. T he quantities are described in Section 4.1.3. 73

4.4 M easured trigger efficiencies... 79

4.5 Com parison of num b er of events, integrated lum inosities and cross-sections for th e selection cuts in 1990, 1991 and 1992. T he num bers in brackets indicate th e num bers of events for th e Otag regions of 50-70 m rad and 70-120 m rad respectively. T he trigger efficiency is accounted for in th e calculation of th e cross-section for th e cuts. . 80

16 LIST OF TAB LES

5.2 M onte Carlo generated sam ples th a t have had the full OPAL de­ te cto r sim ulation (GOPAL) applied to them . The d a ta have been included a t th e b o ttom of th e tab le for comparison w ith th e M onte Carlo. T he num bers in brackets indicate th e cross-sections for the 0tag regions of 50-70 m rad an d 70-120 m rad respectively... 101

7.1 M onte Carlo samples generated by HERW IG for unfolding tests. In each entry, th e first line is th e unfolding M onte Carlo and th e second line is th e mock d ata. F u rth er explanation of this tab le is given in Section 7.6.2... 137 7.2 Four-flavour unfoldings of th e d a ta w ith different M onte Carlo

models in th e low Otag region... 149 7.3 Four-flavour unfoldings of th e d a ta w ith different M onte Carlo

17

C hapter 1

In tro d u ctio n

This thesis is a study of singly-tagged tw o-photon collisions using d a ta from th e OPAL (O m n i-P u rp o se A p p aratu s for LEP) detecto r [1] a t th e LEP (Large E lectron-P ositron) collider [2] at CERN near Geneva. T he aim of such a study is to obtain a m easurem ent of th e hadronic photon stru c tu re function, E^(T, Q^).

T he d a ta used in this analysis were taken in th e years 1990-1992 and correspond to 45.93 p b “ ^ of e'*'e“ in teg rated luminosity.

As an introduction it is acknowledged th a t th e photon has a hadronic structure. A sim ple picture for this stru c tu re is presented, followed by th e m ethod used to stu d y this stru ctu re at an e+ e" collider. T he photon stru c tu re function is then introduced and th e m otivation for m easuring it is given.

18 CH APTER 1. I NT RO DUC TI ON

1.1

T w o-P hoton Interactions

In classical electrodynam ics th e photon is described by th e linear Maxwell equa­ tions. However, in quantum m echanics, th e photon is not a photon all of th e tim e. A photon of energy can flu ctu ate into a virtu al sta te of a charged particle pair, by th e un certain ty principle. If nipair is th e m ass of th e charged particle pair, then th e lifetim e of th e state is given by

2E

(1.1)

TOp«.V

assum ing <K E^ and h = c = 1. This tim e increases as th e photon energy increases. Therefore, an in teractio n of two photons becomes possible because one photon can in teract w ith one of th e charged particles in th e state th e oth er photon has fluctuated into.

1.2

T he P h oton P ictu re

T he virtu al sta te of th e charged p article pair can be either a charged lepton- an tilep to n pair (/"'■/“ ), a q u ark-antiquark pair {qq ) or a m assive p air (e.g. W '^ W ~ ) . A cut-off param eter po m ay be introduced to separate th e range of 7 —>• çç fluctu­ ations into low- and high-virtuality states. Such a separation is necessary because th e low -virtuality state is in th e regim e of non-perturbative QCD physics. The vector m eson dom inance (VM D) m odel approxim ates th e range of 7 —> gq fluc­ tu atio n s below Po by a sum over low -m ass vector meson states. Each meson state can be w ritten as |V). po sets th e m inim um transverse m om entum of th e qq p air in th e high-virtuality p e rtu rb a tiv e p a rt of 7 —>■ çg fluctuations (th e so-called ‘p o in t-lik e’ com ponent). This s ta te can be w ritten as \qq). T he com plete photon wave-function for low-m ass states [3] is th en

b ) = Cbarellbare) A ^ | V) -f ^ C, -f ^ Q |/+/ ). (1.2)

1.3. T H E Eq V E R T E X 19

1.3

T he 87 V ertex

By th e u n certain ty principle an electron of energy Eb can fluctuate into a v irtu al electron-photon state. This is illu strated in th e top vertex of Figure 1.1. If th e electron is scattered w ith energy E[ into a solid angle elem ent dfl, at angle 6i to th e in itial electron direction, after producing a photon of energy zEb th en th e flux for such a process is given by [4]

( U )

aem is th e electrom agnetic coupling constant and which is by deflnition th e negative of th e four-m om entum squared of th e photon, is given by

= _ g2 ^ 4EbE[ sin^ (1.4)

where th e m ass of th e electron has been neglected. It should be noted th a t th e above flux factor peaks at small values of and small photon energies.

1.4

77 C ollisions at an e^e Collider

T he only way of studying photon stru c tu re a t current e'^'e" colliders is to use an e7 vertex from one electron (positron) to produce a nearly-real photon th a t will display th e stru c tu re described in Section 1.2. A second photon of high virtu ality from th e eq vertex of th e opposing positron (electron) can be used to probe th e s tru c tu re of th e nearly-real photon. T he m ulti-peripheral tw o-photon process is illu strated in Figure 1.1. It is custom ary to label th e four-vector of th e v irtu al probe photon as q and th e four-vector of th e nearly-real photon as p. T he invariant kinem atic variables

91_____ (1 5 )

2 p - q Q^ + P^ + W ^ ^ ^

20 C H A P T E R 1. INTRODUCTION

1.5. DEEP IN EL A S T IC Ey S C A T T E R IN G 21

can be defined w ith reference to th e four-vectors in Figure 1.1. VF is th e invariant mass of th e tw o-photon system . T w o-photon collisions where one photon is v irtu al and one is real will be called ‘7*7 collisions’. T he * m arks th e highly v irtu al photon.

1.5

D eep Inelastic

e j

Scattering

W hen ^ 4 GeV^ and ~ 0 th e tw o-photon collision can be regarded as deep-inelastic electron-photon scattering, where th e bare probe photon couples to a quark inside th e nearly-real photon resulting in an hadronic final state. E xperi­ m entally, deep inelastic e j scattering is observed w ith singly-tagged events, where th e probe photon has its determ ined from th e energy (E[) and angle (^1) of th e scattered electron (called th e ‘ta g ’) and ~ 0 is ensured by requiring th a t th e positron is not seen in th e detector (called th e ‘a n ti-tag ’ condition). Therefore, from now on, E[ and 9i will be called Etag and Otag respectively.

In th e single-tag regime. E quation 1.5 simplifies to

and X, called ‘Bjorken x ’ or ‘x g / to differentiate it from th e Feynm an x variable, can be in terp reted as th e fraction of th e fo ur-m o m en tu m of th e nearly-real photon carried by th e struck quark. From th e a n ti-ta g condition, the nearly-real photon is approxim ately collinear w ith th e beam (p • pi 2:: 2P ^ r where r is th e energy of th e probed photon as a fraction of th e beam energy) and

= ( 1.8)

T otal D ifferen tial C r o ss-S e c tio n

The am plitude for e'*'e“ —>-e‘*'e“ X shown in Figure 1.1 can be w ritten as

22 C H A P T E R 1. IN TRO DUC TI ON

The j ’s are th e electrom agnetic currents of th e leptons. T he R^°‘ term relates to th e coupling of th e two photons to th e final sta te X . For leptonic final states th e cross-section can be obtained from E quation 1.9 by an exact QED calculation. The cross-section for a hadronic final state cannot be calculated so exactly because it involves th e theory of QCD which is not as predictively powerful as QED.

The photons rad iated from th e incoming leptons are in either a transverse (T ) or longitudinal (L) polarisation state. T he to ta l differential cross-section will therefore contain four sub cross-sections (Tt t, <^ll and two interference term s, ttt and t t l- T he subscripts refer to th e polarisations of the first and second photon respectively. T he to ta l differential cross-section for unpolarised lepton beam s is [5]

da ,

— = LtT {O 'TT + ^1 <^LT + ^2 ^TL + ^1^2 d r

where

+ 2^1^2 t~t t cos2</> + 2‘\JCi(l + (2(1 4- C2) ttl cos(j) (1.10)

= I f . ( . . n )

which uses th e variables defined in Figure 1.1. ^ is th e angle between th e scat­ tering planes of th e electron and positron in th e 7 7 centre-of-m ass frame. T he term s Lt t i ^1 and cg are calculable in QED. T he interference term s vanish after integration over (j). In th e lim it when the second photon is real {P^ = —p^ = 0) th e only cross-section term s to survive are a j T and aiT- One can now relate these term s to th e usual construction of th e cross-section in term s of stru ctu re functions.

Stru ctu re F u n ction s

The transverse and longitudinal photon stru c tu re functions are defined as

"'"I

and

= (1.13)

L6. I N T E R E S T IN 23

T he m ore commonly used stru c tu re functions are F i { x , Q^) and T7(a;, Q‘^) which are defined as

= (1.14)

and

(a;, (1.15)

T he e7 cross-section can now be w ritten as

dcr^^_S7 167ra^_E?T_{em b}

dxdy Q" [(1 - y) F^{x, Q^) + xy-^FJ{x, <3=>)] . (1.16) Events are strongly peaked tow ards small tag angles and high tag energies, as was discussed in Section 1.3. Therefore y is small (see E quation 1.8) and (1 — y). It can be seen from E quations 1.15 and 1.12 th a t T^(a;, Q^) > xFi[x,Q'^) so th a t

(1 - y ) F2 {x,Q^) > x y^F i( x, Q' ^) and

d(Te'y ^ I d w a l ^ E ^ r

dxdy ( l - y ) f ? (2:,Q:^). (1.17)

T he stru ctu re functions defined above can either be QED stru c tu re functions, from reactions of the type 7 7 -4- /■*■/“ , or hadronic stru c tu re functions from reactions of th e ty p e 7 7 -4- hadrons . T he muonic F ^ i x , Q^) has been m easured by OPAL [6], CELLO [7] and D ELPH I [8]. All of these m easurem ents agree well w ith QED predictions. This work is only concerned w ith th e hadronic photon stru ctu re function.

1.6

Interest in

T he theory behind th e photon stru ctu re function will b e given in more detail in C h ap ter 2. To m otivate th e m easurem ent of th e hadronic photon stru ctu re function, its interesting features are sum m arised below.

H igh Q2

24 CH AP T E R L INTRO DUC TIO N

for this behaviour in F2 is therefore a test of p ertu rb ativ e QCD. If the coupling

constant is frozen at an in itial value of then F2 bends asym ptotically to a

constant th a t is independent of [9]. This is illu strated in Figure 1.2, which has been adapted from [10]. T h e difference between a fixed coupling and a running coupling becomes greater a t larger .

Low

T he evolution of th e photon stru c tu re function from 1 GeV^ to ~ 5 GeV^ is not well understood. T h ere is no apparent need for a p o in t-lik e com ponent of F2 for sm aller th a n ~ 1 GeV^, b u t th e transitio n betw een th e hadronic

shape and th e point-like shape of F2 appears to be com plete at ~ 5 GeV^.

This tran sition region contains a significant n o n -p e rtu rb ativ e com ponent and is therefore difficult to calculate. Some theorists argue th a t one can sta rt evolving F2 from values of less th a n 1 GeV^ [11, 12] and correctly predict F2 for

> 5 GeV^. O thers argue th a t this is not possible [13]. Some d a ta do exist, but th ere is controversy on th e validity of this data. Clearly, new m easurem ents in this region are valuable.

Low X

1.6. I N T E R E S T IN F7 25

A s y m p t o t i c v al u e f o r f i x e d c o u p l i n g

0.25

Fixed c o u p l i n g

QCD

0.2

0.15

0.1

,3

.2 4

10' 10

10 10'

(GeV^)

26 C H APT ER 1. I NT RO DUC TI ON

q u a r k

b o x 7

Jmnmnnr

g l u o n

l a d d e r '^ T T T T T T T

\Tinnnnnr

PHOTON REMNANT

7

1.6. I N T E R E S T IN F7 27

<N

<N

<N

2

- Q^ = 1.5Ge\^

. G RV94

- Q^ = 3.0Gey^ \ ç 7 = 4.5G eV‘

\ Q \h1) = 3 .5 G eV ' = S.OGeV

1

2

( HI ) = 6 .5 G eV

1

2

1

&=25GeY \

Him 1 iim i 1 niiiii 1 m ull i ni

ç7 ( H \) = l 5 0 G e \

2

Q ( H h = 2 0 0 G e \

1

=500G e

2

1

HI (prelim) NMC(95) E665

= 2000G e =5000G e

10 ho ho ho 10 ho 'ho ho lo ho ho ho 'i

28 CH AP T E R 1. INTRO DUC TIO N

1.7

F

2

M easurem ents at LEP

As th e beam energy of an e+e" collider increases, greater values of W can be reached, so th e m inim um value of x for a given decreases (see E quation 1.7). Although th e event ra te for a given tagging range would decrease w ith increasing beam energy, th e accessible values w ithin th a t tagging range increases (see Equations 1.3 and 1.4).

Table 1.1 shows m ean values and a:-ranges of singly-tagged hadronic tw

o-Collider Coll. {Q‘ XGeV'O (and range)

x-rcinge (and No. bins)

Ref.

PE T R A PLU TO 2.4 (1.5-3) 0.016-0.700 (3) [24]

4.3 (3-6) 0.03-0.80 (3) [24]

9.2 (6-16) 0.06-0.90 (3) [24] 5.3 (1.5-16) 0.035-0.840 (6) [24] 45.0 (10-100) 0.1-0.9 (4) [25]

TASSO 23.0 (7-70) 0,02-0.98 (5) [26]

JA D E 24.0 (10-55) 0.10-0.90 (4) [27] 100.0 (30-220) 0.1-0.9 (3) [27]

P E P T P C/2 7 0.7 (0.5-1.0) ₍₄₎ [28]

1.3 (1.0-1.6) ₍₄₎ [28]

5.1 (4-7) 0.02-0.74 (3) [28]

20.0 0.196-0.963 (3) [29]

TRISTA N AMY 73.0 (30.0-110.0) 0.125-0.875 (3) [30] TO PAZ 5.1 (3-10) 0.010-0.20 (2) [31] 16.0 (10-30) 0.20-0.78 (3) [31] 80.0 (45-130) 0.06-0.98 (3) [31] VENUS 40.0 (20-75) 0.09-0.81 (4) [32] 90.0 (45-240) 0.19-0.91 (4) [32]

LEP OPAL 5.9 (4-8) 0.001-0.649 (3) [33, 34]

14.7 (8-30) 0.006-0.836 (4) [33, 34, 35] D ELPH I 12.0 (4-30) 0.001-0.847 (4) [36]

0.001-0.350 (3) [36]

Table 1.1: Published hadronic F7 d a ta as a function of x.

1.7. F ; m e a s u r e m e n t s a t LEP 29

linear e+e" collider {Ebeam = 250 GeV) will achieve even lower-a: and higher values [23].

A n x -Q ^ kinem atic plot of M onte Carlo tagged tw o -p h o to n events at L E P l is shown in Figure 1.5. It clearly shows th e effect of certain necessary cuts, such as m inim um 7 7 m ass, m inim um tag energy and m inim um tag angle, on th e d istri­ b u tio n of events.

It will become clear in the next chapter th a t th ere are theoretical uncertainties in th e low-a; behaviour of th e photon stru ctu re function. Since a m easurem ent of F2 at LEP extends to lower x th an any previous F2 m easurem ent, this thesis

30 C H A P T E R 1. I N T R O D U C T I O N

>

O 1

W 10 3 a

10

10

-I— I—I 11111 -i— I— I— 1 1 1 1 1 1 --- 1— I— I— 1 1 1 1 1 1 ---1— I— I— I I 1 1 1

LEPl

/ \

DIS

10-4

\/

10-3 10-2

■, I

10-1

w_YY

^B jorken

Figure 1.5: Scatter plot of Monte Carlo tagged two-photon events from the F2GEN generator (see Section 5.3) at E b e a m = 45.6 GeV. The minimum 7 7 mass

31

C h ap ter 2

T h eory o f th e P h o to n S tru ctu re

F unction

This ch ap ter begins by considering how can be constructed from parton d istri­ bution functions. Various forms of these distributions will be presented. The d eterm in atio n of th e hadronic and point-like com ponents of F7 , using Vec­ to r Meson D om inance (V M D ), th e Q uark P arto n Model (Q PM ) and quantum - chrom odynam ics (QCD) calculations will th en be discussed. B oth DGLAP and B FK L evolution are considered. T he charm quark contribution to F7 is consid­ ered, followed by a review of all of th e available param eterisations and models.

2.1

P arton D istrib u tion s o f the P h o to n

In th e intro d u ctio n th e photon stru c tu re function was presented as p a rt of th e

67 cross-section, which is related to th e m ulti-peripheral e'*‘e~ —)■ e + e ' -f hadrons cross-section by a simple flux-factor. We know, however, th a t th e photon some­ tim es consists of partons, so it is n a tu ra l to consider F ^ as a sum of distributions of p arto n s in th e photon. T his is central to th e theory th a t follows. In Leading O rder (LO),

32 C H A P T E R 2. T H E O R Y OF THE PH O T O N S T R U C T U R E FUN CTI ON

T he qi{x ,Q^) = ' ÿ ( x , Q ^ ) condition is assum ed to hold and so Equation 2.1 simplifies to

F^{x,Q^) = 2xf^e^qy{x,Q^).

2 = 1

It is often m ore convenient to work w ith th e singlet and non-singlet quark d istri­ butions, qs{xj Q^) and Q^) respectively. These are the quark d istributions th a t are used to determ ine F2 param eterisations (see Section 2.7).

ql {x, Q^) = 2 ^ q ] (

2

3

1 = 1

U f

t=l

where

(e^) = — (2. 5)

i= i

The construction of F2 in next-to-leading order (NLO) [37, 11] from th e pho­

tonic parton d istrib u tio n functions is m ore com plicated. It involves th e gluon distribution, of the photon, unlike th e LO case.

2.2

T he C om ponents of

It is usual to sep arate th e photon stru c tu re function into a hadronic and a point­ like p art

Fq(x, Q^) = F [ ^ { x , Q^) + Q2). (2.6)

2.3. V E C T O R MESON DOM IN AN CE 33

2.3

Vector M eson D om inance

Photons are known to behave like hadrons when interacting w ith o ther hadrons [39]. In th e Vector Meson D om inance (VM D) picture, at low 4 -m o m en tu m -sq u ared transfers, th e interaction of a photon w ith hadrons is dom inated by th e exchange of vector mesons which have th e sam e q u an tu m num bers as th e photon. T here­ fore, th e hadronic p art of is usually chosen according to VMD. T he photon couples to th e vector mesons p, w, cj) and J/'tp resulting in

FHAD _ pVMD _ Ç p v (2.7)

where / y /47r are determ ined from d a ta to be 2.20 for p°, 23.6 for w, 18.4 for (j) and 11.5 for J / ^ [39]. f v from E quation 2.7 is related to cv from E quation 1.2 by Cy = Anaeml f v ' This hadronic p a rt is com pletely analogous to hadron behaviour, as th ere is no increase w ith log (it exhibits Bjorken scaling) and th e æ-shape is not calculable in p ertu rb atio n theory.

T he p arto n distributions in th e p meson are experim entally unknow n, so it is assum ed th a t th e p d istributions are th e sam e as those in th e pion. The pion stru c tu re function for approxim ately x > 0.2 is known from experim ental results from th e Drell-Yan [40] production of p-pairs in pion-nucleon scattering.

T he sim plest VMD estim ate [5, 9] th a t can be derived from pion stru c tu re function m easurem ents is

= 0.2 c«e„(l - x). (2.8)

O ne m ight alternatively construct th e photonic p arto n d istributions as

P { x , p^) = K j A ( z , /i^) (2.9)

w here p = q^{= 'p ) ot 1 < k, < 2 and is a very low resolution scale ( ~ 0.3 GeV^). The K p aram eter is introduced [5] to deal w ith theoretical am bi­ guities, especially higher order gluonic corrections.

A lternatively, one can fit to th e low-Q^ d ata. T P C/ 2 7 fitted th eir d a ta at = iG eV ^ [28] w ith

34 C H A P T E R 2. T H E O R Y OF THE PH OT ON S T R U C T U R E F UN CTION

and obtained A = 0.22, B = 0.06, a = 0.31 and 6 = 2.5. This is illu strated on a linear-a; scale in Figure 2.1, w ith th e sim ple VMD estim ate of Equation 2.8, and on a logjo X scale in Figure 2.2.

2.4

T h e Quark Pcirton M odel

Before QCD was developed, a tte m p ts at trying to determ ine th e properties of F2 and F2 cam e from calculations based on th e free quark-parton m odel and

light-cone algebra. In th e quark-parton m odel (Q PM ) th e stru ctu re functions are calculated by treatin g th e quarks as free particles w ithout strong interactions. In th e lim it of light quark masses {mf/Q"^ 1) th e stru ctu re functions can be w ritten [41, 42]

3 a U f

{x - h ( l - a:) )log

(m f 4- P ^ x { l — æ)) — 2(1 — 3a^ -j- 3cr ) -f- m ^(l -f 2ar — 2x'^)

{mf — P^{x^ — a;)) (2 . 11)

= — E ef x^(l - X). (2.12) ^ i=l

W hen P^ = 0, E quation 2.11 reduces to th e P^ independent form ula ( I - X 3 a

2 = 1

-f- 8a;(l — (t) — 1]. (2.13)

2.4. T H E Q U A R K PA RT ON MODEL 35

S 0.6

QPM (Q =14.7 GeVO

QPM (Q^=5.9 GeV^)

TPC/2yVM D

Simple VMD

0.5

0.4

0.3

0.2

0.1

0.8 0.9

0.1 0.2 0.3 0.4 0.5 0.6 0.7

X

Figure 2.1: QPM and VMD predictions for . T he simple VMD and T P C/ 2 7

36 CH A P T E R 2. T H E O R Y OF THE PH O T O N S T R U C T U R E FUNCTION

0.6

FKP (p" = 0.3) + TPC/2y VMD

FKP (p* = 0.5) + TPC/2y VMD

TPC/2yVMD

0.5

0.4

0.3

0.2

0.1

2.5. QCD CALCULATIONS 37

2.5

QCD C alculations

Q uarks can rad iate gluons and gluons can produce q u ark -an tiq u ark pairs, so th e Q PM calculation m ust be modified. If a valence quark (one of th e quarks from th e

7 —>• çg vertex) is struck and it has not ra d ia ted a gluon th en th e kinem atics of th e h ard scattering are not affected. W hen one or m ore gluons are ra d ia ted from th e valence quark, they carry away some of th e fo u r-m o m en tum from th a t quark. If th e valence quark is still struck it will th en be seen to have a sm aller fo u r- m om en tum th a n it had initially. If a sea quark (one of th e quarks from a gluon —>■ qq vertex) has been struck it will be seen to have a sm aller fo u r-m o m en tu m th a n th e gluon it was produced from.

QCD predictions for F2 can take several different approaches. One can proceed by

using th e operator product expansion and renorm alisation group equations (O P E - RG E) [43, 44], evolution equations [45] or Feynm an diagram s in th e leading log approxim ation [46, 47, 48, 49].

2.5.1 T he D G L A P E volution E quations

T he cross-section for th e hard scattering process will depend on th e scale of th e v irtu al probe photon and on th e m om entum fraction distrib u tio n , D ( x , Q ‘^), of th e partons in th e real photon at this scale. T he evolution equation for th e p arto n density D ( x , t ) m ay be w ritten as

t — D{x.,i) = — P {x) 0 D{x.,t). (2.14)

T he convolution integral is defined as

a { x ) ® h [ x ) = [ - a ( - \ b { y ) . (2.15)

3^ y \yj

38 C H A P T E R 2. T H E O R Y OF THE PH O T O N S T R U C T U R E FUNCTION

partons in th e branching process, E quation 2,14 has to be generalised to a coupled set of evolution equations of th e form

(2.16)

The parto n sp littin g functions P i j { x ) have th e physical in terp retatio n , to first order in CKg, of being th e probability of finding p arto n % in a parto n j with a fraction x of th e m om entum of th e parent parton. T he lowest order approxim ation to th e splitting functions [16] are

P q q { z ) = Cf

P q g { ^ ) = Cf [ I — z Y

1 + (1 - ^ ) 2

Pggi^^^f) = + ( k i i ) + , ( i _ + - { n C A ~ 4 n j T R ) S ( l - z ) (2.17)

where C f = 4 /3 , T r = 1 / 2 and Ca = 3. T he plus prescription on th e singular p arts of these sp littin g functions is defined under th e integral sign as

f dx f { x ) [g{x)]+ = f dx ( f { x ) - f { l ) ) g { x ) . (2.18)

•/ 0 V 0

This removes th e singularity of th e integrand in th e evolution equations at z = 1, corresponding to th e emission of a soft gluon. The rem aining singularities at z = 0 are outside th e range of th e integration, so all of th e integrals are finite.

2.5.2

P arton D istrib u tion s at Low-x

2.6. C H A R M - Q U A R K CON TRIB U TION S 39

T he D G LA P equations are derived keeping only th e leading log term s in Q^. T he log(l/a;) term s are neglected. In th e æ —>■ 0 lim it th e log(l/a:) term s becom e im ­ p o rtan t. One can still use th e DGLAP evolution equations for low-a:, provided log lo g (l/æ ). T he tre a tm e n t of th e low-a; region corresponds to a resum ­ m ation of term s proportional to log to all orders in p ertu rb a tio n theory. Considering th e gluon distrib u tio n only, G { x ,t) , such a scenario yields [50]

where

( - )

However, th e values m ay not be large (especially a t low-æ), so it is m ore useful to resum term s proportional to Og log(l/a;) to all orders. This is done by th e B alitsk y -F ad in -K u raev -L ip ato v (BFKL) equation [14, 15]. A simple derivation of th e B FK L equation and th e low-a: behaviour is given by M ueller [51]. M ueller actually calculates th e low-z behaviour of th e wave function of a hadron ra th e r th an in term s of stru c tu re functions. The inclusive gluon distribution g at small-a; in a quarkonium wave function is

g{x,b'^) oc bx~^ (2.21)

where A = 4 log 2 Nas/'ïï. N is th e num ber of colours. T he A num ber is about 0.5 for Ofg ~ 0.2, leading to an approxim ate divergence at low-a:. T he p aram eter b is related to th e scale by a Fourier transform ation.

2.6 Charm —Quark C ontributions

40 CH AP TE R 2. T H E O R Y OF THE PH OT ON S T R U C T U R E EUNCTION

approxim ation to th e charm quark contribution to th e photon stru ctu re function for Q^< 100 GeV^ is m ade by sum m ing th e contributions from the QPM processes of 7*7—)■ cc (direct) and >y*g -> cc (resolved).

D irect Q PM p rocess

Since th e QPM direct process is commonly used when including charm in F ^ param eterisations (see Section 2.7), it is briefly described. It is calculated via th e lowest order B eth e-H eitler process [55, 56] and is given by

Q^)\direct = ^ ^ (2.22)

where ec= 2/3 is th e c h arm -q u ark electric charge and

w{z, r) = z [/){—! + 8z (l — z) — Arz{l — z)}

2 I _ \ 2 I o _ 2 _ 2

4-{z^ + (1 - + 4 rz (l - 3z) - log 7- ^

i — p (2.23)

w ith

T he charm contribution is added for (3 > 0 {W^ > 4ml) and is zero for ^ < 0 (W^ < 4m^), and therefore incorporates th e charm mass threshold. The effect of adding in this charm contribution to F^(a;) is visible in Figure 2.3.

R esolved Q P M p rocess

T he expression for th e Q PM contribution to F^ from th e resolved process j*g cc is given by [54]

F2,ci^^Q^)\resolved = ^ ^ j ^ (2.25)

2.6. C H A R M - Q U A R K CON TRIB UTI ON S 41

ü 0.6

GRV (iif = 4)

GRV (n^ = 3)

0.5

0.4

0.3

0.2

0.1

42 CH APTE R 2. T H E O R Y OF THE PHOT ON S T R U C T U R E FUNCTION

2.7

F

2

P aram eterisations and M odels

T he form ulation of th e D G LA P equations is convenient for obtaining analytical solutions [54] for th e evolution of p arto n distributions. No prediction is m ade for the size and shape of th e functions themselves. A common m ethod of making a theoretical prediction of th e photon stru ctu re function is to choose a reference scale Ql , param eterise th e p arto n distributions at th a t scale and th en evolve those distributions num erically using th e DGLAP equations. Q^) can then be constructed a t a given and a global num erical fit to th e d a ta perform ed to determ ine th e best values for param eters.

This section presents some of th e basic features of th e available theoretical pre­ dictions of F2 .

2.7.1 Gluck, R eya and Vogt (GRV)

These authors have produced p arto n distributions of th e proton [57] and the pion [58] th a t have been generated from a valence-like stru ctu re at a common, very low resolution scale. T he deep-inelastic scattering d a ta have been reproduced, in p articu lar th e HERA results on F2 [21, 22] are in excellent agreem ent w ith the

GRV prediction. This success provides m otivation for using a low startin g scale in th e case of th e photon.

T he GRV photonic p arto n distributions [11] are in LO and NLO {D IS ^ factorisa­ tion scheme). The evolution sta rts at Ql = 0.25 GeV^ (LO) and Q l = 0.3 GeV^ (NLO). T he photonic in p u t d istributions are purely from VMD. They are taken to be those from E quation 2.9, w ith /F) ~ x^{l — x Y being th e valence-like (i.e. a > 0) inputs from [58] and [5]. Only one param eter, k, is left to be fixed from a least squares fit to th e F2 d a ta [24, 25, 26, 27, 28, 30]. The best k, values

were found to be /«lo = 2 and /«atlo = 1.6, w ith good agreem ent of th e resulting p aram eterisations w ith th e d ata.

2.7. P A R A M E T E R I S A T I O N S A N D MODELS 43

This p aram eterisation is used to generate some of th e M onte Carlo samples used in this thesis (see C hap ter 5). T he th ree and four flavour param eterisations are illu strated in Figure 2.3 for two different values.

2.7.2

H agiw ara et al. (W H IT )

These are a set of six LO photonic p arto n distributions (W H IT l to W H IT6) [54] which have system atically different gluon contents. T he evolution starts at Ql = 4 G eV ^ The d a ta [24, 25, 26, 27, 28, 29, 30, 31, 32, 34] at 4 < Q^< 100 GeV^ are fitted to determ ine th e free param eters. N ot all of th e experim ental d a ta points are used. Firstly, th e d a ta at ) lower th a n 4 GeV^ are om ited. Secondly, bins are accepted only if

( g2 ^

Xl ower bin edge > ^ _j_ ^ l|^ m a x p ’ (2 .2 6 )

where is th e experim ental cut on th e visible invariant m ass of th e hadronic final state. This is on th e grounds th a t th e bins th a t fail this condition m ight suffer from large sy stem atic uncertainties in th e unfolding procedure (see C hapter

7)-T he charm quark contributions to F2 are calculated by th e Q PM (w ith rric =

1.5 GeV) at Q^< 100 GeV^, w ith contributions from b o th th e direct and resolved processes (see Section 2.6).

2.7.3

Gordon and Storrow (G S)

These are LO and NLO [ M S factorisation scheme) p aram eterisatio n s [37]. The evolution starts a t Qq = 5.3 GeV^, which is th e average of th e low-Q^ PLU TO d a ta [24]. The d a ta [24, 25, 26, 27, 30, 60] are p artially used to flt th e 5 free param eters given below. T h e high d a ta [30, 27, 60] were fitted for n / = 4 flavours. Two sets of p aram eterisations are given, corresponding to two different assum ptions on . T he p aram etric form of th e LO distrib u tio n s are

44 C H A P T E R 2. T H E O R Y OF TH E PH OTO N S T R U C T U R E FUNCTION

for the singlet and non-singlet sectors and

Ql) = C) (2.28)

or

Qo) = gi^^ O) -f ^ P g q ( x ) 0 m^, m^). (2.29) Po

T he quark masses m ^ (= m^) and are tre a ted as bounded free param eters. T he VMD p a rt is tre a ted as in E quation 2.9 and hence incorporates a k. factor. T he p aram eters B and C are associated w ith th e sea and gluon sectors of th e pion respectively, which are of th e form given by [61]. For th e PL p a rt, th e lowest order B eth e-H eitler form [55, 56] is used. C harm is tre a ted w ith th e same way as for light flavours, w ith rric = 1.5 GeV. T he second term in Equation 2.29 represents a com ponent to th e gluon d istrib u tio n estim ated from B rem sstrahlung off th e singlet quarks.

T he NLO distributions have been constructed w ithout a new flt to th e d a ta by enforcing th e same E^(a;, Qq) as in th e LO case, together w ith assum ptions on flavour decom position.

2.7.4

D rees and G rassie (D G )

This is a LO param eterisation [62] th a t avoids the tw o-com ponent decom position of E quation 2.6. Param eterisations for th e singlet, non-singlet and gluon distri­ butions are obtained using th e full solution of th e LO inhomogeneous evolution equations [45] which are free of divergences [63]. The input d istributions are cho­ sen a t (Jo = 1 GeV^ such th a t th e only d a ta th a t was available a t th e tim e [64], a t = 5.9 GeV^ , are reproduced. T he weakness of this p aram eterisation is th a t only 7 d a ta points were available a t one value, so they were a poor constraint to th e assum ed 13 free param eters. A = 0.4 GeV throughout.

2.7. pq P A R A M E T E R I S A T I O N S A N D MODELS 45

2.7.5

Levy, A bram ow icz and Charcula (LAC)

T he aim of these LO param eterisations [65] was to apply th e approach of Drees and Grassie to fu rth er available m easurem ents of th e photon stru c tu re function. T he evolution starts at Qq = 4 GeV^ for sets 1 and 2 and Qq = 1 GeV^ for set 3. T he evolution was carried out for 4 flavours, b u t th e charm contribution to th e photon stru ctu re function was only taken into account when > 4m^, w ith rric = T5 GeV. This threshold appears as a discontinuity in th e F ^ ix ) distributions (see Figure 2.4). T he x dependence of th e in p u t quark distributions a t Qq is assum ed to be

xq{x) = (2.30)

for each of th e four flavours and th e gluon distribution at is assum ed to have th e form

xG{ x) = CgX^^{l — x)^^. (2.31)

T here are a to ta l of 12 free param eters. No a tte m p t is m ade to flt th e QCD scale p aram eter A which is assum ed to be A = 0.2 GeV. The two term s in th e quark d istributions are intended to reflect th e point-like and hadronic p a rts of photon stru ctu re. This was th e first a tte m p t to determ ine th e gluon d istrib u tio n in th e photon. No physical constraints are placed on the quark flavour decom position and on th e gluon density. Vogt points out [66] how this approach leads to un­ physical results (e.g. s(æ, Q^) > d{x, Q^) in some regions) and wild reactions of th e fitted gluon density on fluctuations and offsets in th e d ata. This dem onstrates how th e present d a ta do not provide useful constraints on g'^.

This p aram eterisation is used to generate some of the M onte Carlo sam ples used in th is thesis (see C hapter 5).

2.7.6

A urenche et al. (A C F G P )

46 CH AP T E R 2. T H E O R Y OF TH E PH OT ON S T R U C T U R E FUNCTION

1

0.9

LACl(iif = 4)

LACl(n^ = 3)

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

■2 1

10 10

2.7. f; P A R A M E T E R I S A T I O N S A N D MODELS 47

evolution between Qq = 0.25 GeV^ and Ql = 2 GeV^, they exactly correspond to th e distributions in th e pion determ ined [68] at 2 GeV^. No fit to th e d a ta is perform ed, and no a tte m p t is m ade to explain th e d a ta at ~ 1 GeV^, where higher tw ist contributions m ay be non negligible.

At = 2 GeV^ an rz/ = 4 evolution begins which generates a charm distribution, b u t in th e calculation of this distribution is not used. In stead th e expression of [69] is used which correctly accounts for the charm m ass threshold. AjYs{n/ = 4) is fixed to 200 MeV.

2.7 .7

Field, K ap u sta and P oggioli (F K P )

A different approach to calculating F ^ is by th e direct sum m ation of Feynm an diagram s [46, 47], where leading log ladder diagram s are sum m ed to all orders in as{Q^). F K P used th is m eth o d [48, 49] to calculate F ^ they introduced a phenomenological cu t-o ff (pj) in th e pt of th e quarks a t th e ^ qq vertex. This separates th e hadronic and point-like com ponents of th e photon stru ctu re function. Furtherm ore, K ap u sta [70] detailed a derivation of various contributions to th e photon stru c tu re function in th e pertu rb ativ e region. This all-order QCD approach by F K P was p aram eterised in an AMY p ap er [30] and is assum ed to apply for all Pt > Pt- A separate p art for th e hadronic com ponent m ust be added to th e cross-section for pt < p?, which is not provided in th e FK P model. T he hadronic com ponent is usually assum ed to be p aram eterised by th e T P C/ 2 7

form ula of E quation 2.10. T h e ‘F K P + T P C/ 2 7 V M D ’ construction is illu strated in Figure 2.2, for two values of Q^ and two values of p°.

48 C H AP T E R 2. T H E O R Y OF T H E PH O T O N S T R U C T U R E FUNCTION

2.7.8

Schuler and Sjostrand (SaS)

This is th e m ost recent set of param eterisations of th e photonic p arto n d istribution functions [71]. Schuler and Sjostrand point out th e problem s of th e F K P m odel and then provide a solution to these problem s w ith th eir own p arto n d istribution functions.

T he photon stru ctu re function is sep arated into p ertu rb ativ e (‘anom olous’) and n o n -p e rtu rb ativ e (hadronic) contributions. T he anomolous p a rt is fully calculable and depends on x, Qq, A and P^.

2.7. F7 P A R A M E T E R I S A T I O N S A N D MODELS 49

Leading Order

Leading Order

W H IT l

WHIT3

WHIT6 =

L A C l -E

LAC2

LAC3

ta 0.9 ta 0.9

■ 1111________ I____ I___ I 111 J I I I I I n

%T0.9

0.8

0.7 0.6 0.5 0.4 0.3

0.2 0.1

0

-II111 "1—1—i—_{rri ii| “ 1 T 1 1 n L Ü 1 .1 111|}

---

—

—

----

—

—

G S l

ACFGP :

=

GRV

GRV

Ë

DG

:

GS

0.7

Ï- T 0.6 E - -E

E - T 0.5

L _0.4

0.2

— — 0.1 E- ^

-1 ijjJ---1-- 1—1 .1 1 1 II1---1-- 1—L i..t 1 r _{0 “■} ■ ■ ■ 1___ 1_1_I 1 1 1 1 1 1___ 1_1_1 1 1 I r

10 10-1 10-2 10