The Ubiquitous Photon Helicity Method for QED and QeD
R. Gastmans Instil/lie/or Theoretical Physics
Univcrsiry of uuvm, B-3030
LeIlVe!l,
Belgium
Tai TS lin Wu Gordon McKay Laborarory Harl/ard Universiry, Cambridge MA 02138, OSA
CLARENDON PRESS . OXFORD 1990
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Oxford is (I trade rnark of Oxford Vtlivers;,y Press Pubhl'/wd in Ihe V"ill'll S[(J{('S by Oxford U"iverJity Press, New York
© R.
Gastmans and Ttli TsulI \Vu 1990
All rights reserved. No part oI this publicaritJn may be reproduced, stored in a rClrit'I'aJ system, or IrallSmitted, iHtUlY form or by any means, electronic, mechanical, pholowpying. recordiltg or Oll1erwise, wilhour llie prior permission of OXford VniversilY Pres:)'.
British LibralY Cataloguing in Publicatioll Data Gaslmans, U The Ilbiqui(()us PlW{OIl : htlicil,Y method Jor Q. E. D. and Q. C. D. I. High energy photons I. Tide fl. Wu, Ta; Ts~m 539.721 7 ISBN 0-19·852043-3 Library oj Congress Cataloging in Publiu ltion Data (Dara available)
Prilued if! GreCJt Britain hy Biddle!> Ltd, Guildford & King's Lynn
.'
/,,, Nil/ell/'
thill, U.
(i.
To ProfeJsor Ronold W. P. King from T. T. W.
/
Preface I tikiul{ in the mountains with. bad shoes can be a. very painful experience. On 1.l IC' tlth~r
I'lly~ics ..
hand, it can also lead to
som ~
unexpected developmenls in particle
. l1uring the 1978 LEP Summer Study in Les IIouches, op.e of us (C:) returned frolll such a hike with a couple ugly blisters. His subsequent IIwl.wa.rd way of walking atMaded Ule atlClltion of the olher author (W), will) f<:lt the need to deliver somt' sort of ph arman:utica.J assistance. Ami ,
or
iu"vitably. the ensuing conversation turned in to a physics discussion abouL r;uliative corrections. At tha.~ time, the collider PETRA (£ositron-Elcctron Tandem Ring Ac1 ·,'I!~raLor) at DESY, Hamburg, Germany, was just beginning to come into "pl'ration, with its energy eventually reaching 46 GcV in ~he centre·of-mass t!ystcm. Since t.his energy is very much larger tha.n the eLectron mass, it i~ !:I!ident that. a systematic treatment of bremsstrahlung processes in the !:i~ll energy limit is highly desirable. Since Lhe experience of W was tllen lillli Lcd mostly to tl!e related i.>ut diffe.rent case of iJigh energy scattering and prod uction processes with fixed momentum transfer:;, G gave him a tllOl"Ough 11·d.llre on the ways of treating bremsstrahlung processes, where of course I",th ~nergy nnd momentum t ransfers are large. Aflcr LIIC LEP Summer Study, C rel.urned to Re!gium ..... hile W went to visit CERN, GeneVA, Switzerland. A few days bter, W decided to carry through carefully wh!l.t he had learned from G. He sat down early one morning ;~t his tabJe at home and !itarted to do jllSt that: dr
/JU/,)f 'A(:f:
viii
lLlIlIlCl'icll.lly wil.h ( he pwviolJi'ily known, [fllLeh IOJlger formula . A few clays lato r, he wt'OLc n.g.;till to sa.y that the nlllllcl'it:nl (:omparisons gave agreemeut lo Bille digits.
Encouraged by this IIlIrnerical agreement, W decided to tackle the much more intrica,tc case of e+e- -jo e+e-...,.. This involves, to lowest order, eight Feynman diagrams and, hence, t.hirly-six terms in the expression for the cross section. FoHowing essentia.lly the same procedure as [or the muon ease,
a very similar result, albeit a slightly more complicated one, was obtained after basically working straight for seventy-two hours. After sending this result to G, W fell justified to goof off for a few days. Together with our colleagues Frits Berend, and Ronald Klei ss (University of Leiden) and P atrick De Causmaecker and Walter Troost (University of Leuvell), we tben examined several other QED and QeD processes. Success! The cross section formulae were again a.nd again much more elegant than we
could honestly hope for previously. Nevertheless, the situation was not satisfying. Simple answers should be
obtained in simple ways. But the procedure to get these simple answers for bremsstrahlung is by no mea.ns simple. This unsa.ti sfactory state persisted for a couple of yea.rs. ?vIeanwhile, Patrick was working on his doctoral thesis ,
for which he studied the QCD bremsstrahlung processes. Travelling daily bet ween his home and the insti tute, he disposed of some three hours on the tra.in.
These trains being ra. ther crowded, he could not really work wi th
pencil and paper, but he could do some thinking . Regularly, he would come to the institute with a new idea. One day, he suggested to Walter and G to examine polarizalion amplitudes rather than calculating cross sect ions by brute force. G dismissed the idea as being too cumbersome.. . Patrick and Walter decided to wa.it, however, until G was absent to work out this new line
of thought on the sly. Using first plane polarization slates for the photons and later on photon helicity states, they ,uceeeded in obtaining the cross sections in an alternative way.
Shortly thereafter, W visited Belgium, and he was shown the results of P at ri ck and Walter. The idea of dealing with matrix elements instead of cross sections pleased him considerably, but it was fell t.hat tbe calculations could still be simplified . On the train(!) to Marseille, W had plenty of time to think about it all, and somewhere in the vicinity of Paris, he realized that the hclicity states of the phoLons led to expressions for f± which could simply be written in Lerms ofLlle helicity projection operators for the fermions (1 ± 1s)/2. From t.hen on, it was clear that the helicity method was the way to go about the bremsstrahlung problems. This book summa.rizes some ten years of resea.rch on the subject. Our
purpose is to give a pedagogical introduction to the helieity method ilIu,b'ated by many examples. Only some degree of familiari ty with Feynman diagrams and Dirac algebra is required from the reader. The book also pro· vides a.n extensive list of helicity amplitudes and cross sections for many of
i.
IJu: lIlost imp(lrt
ld,k lllil\. wrne errors hnve hcen made. We would greatly .1pprcciatc being
I'I'OllljHly inforrnoo of such errors. Hopefully, the informat.ion coutained in Lhi.'l book will be useful fDr actajle
undersLtnding of particle pllysics at high energies.
We arc \'cry grateful to our colJeagucs in the CALKUL Collaboration above for bringing this program to a good encl. We arc cspecially illdd.>lt:d to \Valter Troost for the many enlightening disCI;ssio:'ls both b efore alld during the wriling of this book. We owe special thanks to Maurice Jacob: not only did he organize the I!H8 LEP Summer Study, which led to our collaboration 011 t he helicity protllllllCd
~ralll ,
but also Chapter 12 on bcamsttahlung is based mostly on his work. In 11H: past letl years, we often discus$cd lhe problems treated in this book with our colleagues, including JOdlCIl Bartels, Johu Hell, Dirk Dmlckaert, John Ellis, Alex Grossmann, Francis Halzcn, Alberl l lofmann, Aenen ii!! Hmnpert, Harry Lehmann, Allred ~{ueller, Per Osland, Roocr10 Peccei, Paul Soding, Jacques Soffer, Hiroshi Takeda, SakUf! Yamada and Oa-Hua Zhang. We a[w J'eceived very valuable advice from Frank Mallezie and Marg:lret Owens on tIle use of the wordproCl"ssor for tIle preparation Hf tho IlIl'1I1UScript. OUf research OIl the IH,licity method has been stlpported by the United States Department of Energy and t he National Fund for Scientific Rt".search (Rdgium), where G is research director. Our collaboration across the Atlantic. was f..cilitated by a NATO Research Crant. We thank the Theory Division of CERN, Geneva. Switzerland, the Gruppc TbI.'Orie of DESY I Hamlmfg, Federal Republic of Germany, and the Centre de Phy!!ique Theorique of the eN RS, Luminy, France, for their bospitality during our stays. All this ~1tpport is gratefully acknowledged.
Id~UV f!1i
1989
and Cambridge, MA
R.G. T.T .W.
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 I I
I I
... Contents J
Introduction
1
t.l
Elementary pa.rticlcs circa 1989
I
[.2 1.3
Radiative processes . . . . . . . The importance o f high cncrgi<:::l
2 3
2 Feynman diagrams 2.1 :.1.2 2.3 2.4
7 7 14 16 18
Fcynman rules . . . . . . . . . Cross sections a.nd decay rates Exa.mple: Z ...... e+e- . . . . . Another example: e+e- -+ "f"Y
21 21 22
:1 Helicity states 3.1 F'ermiom. . . . . . .. . 3.1 Photons and gluons . . . . . 3.3 Example: e';'e- - t '11 again 3.4 Ranges of validity . . . . -1
24
27
Single bremsstraJllun,g in QED 4.1
The process e+e- ...... 1'1'1'
31
. .
31
•
35
1 .2 What jf different fermions radiate? 4. 3 The prOCCS9 c+c- --+ 1(+11-7 4.4 The process e+e- --+ e+e-..,. 4.5 Inclusion of Z-exchange
5 Single bremsstrahlung in QeD 5.1 Good to know , . . . . . 5.2 The process qq-+ 99 .. . 5.3 The process c+c- ...... 3 jch 5.4 The process q q' -+ q qlg 5.5 Other QeD processe.~ . 6
7
35 39 44
47 47 '
..
48
• •
52 55
64
Double bremsstrahlung 6.1 QED . . . 6.2 e+e- --+ <1 jets .
65
Fin ite mass effects
79
7.1 7.2 7.3 7A
79
Their occasional importance Single bremsstra.hlung An example: e+e- -- JJ+J'-r. Mass corredions for a.mplit.udes 7.4.1 Genera.l formalism
65 70
•
80 82 84 84
CON ,{,};N'I'S
xii 7.4.~
SiLlgI" 1"·""1,,1.I'I1.l>lung . . ... . All "X~'Jlpl<" --> J'+ 1'-, .. . 7.4.1· Double wlli,,,,,,r bremsstrahlung . --> I'+JC, , . 7.4.5 An example:
,,+,,-
7.4·.:1
88 91
... ... '
95
.+.-
8
97
101
The production of qual'konia 8.1 Framework. _ . . . . . . . . . . . . 8.2 150 production . . . . . . . . . . . . 8.2.1 The amplitude M( +, +, +) . 8.2.2 Tne amplitude M(+ ,+ , -). 8.2.3 The cross section for 150 production. 8.3 The produ ction of other st ates. 8.3.1 35. production. 8.3.2 'P, production. 8.3 .3 'Po production.
8.4
8.3.1
3Pl
8.3.5
'P, production.
101 106 106
108 110
· HI 112 · 112 · 114
.115
production.
116 · 118
•
Conclusi ons _ . . . . .
9 Summary of QED formulae 9.1 e+e- -> " (m. = 0) . . . . . . . . . . . 9.2 e+,,- --> 1'+ 1'- (no Z-exchc.nge; m = 0) . 9.3 e+e- -> 1'+ ,,- (with Z-exchauge; m = 0) 9.4 e+e- -> e+e- (uo Z.exchange; m = 0) 9.5 e+e- -> e+e- (with Z-exchange; m = 0) 9.6 e+e- -> , , , (m = 0) . . . . . . . . . . 9.7 e+c -> 1'+ ,,-, (no Z-exchange; m = 0) 9.8 e+.- -> 1'+ ,,-, (with Z-exchange; Tn = 0) 9.9 e+e- -> o+e- , (no Z-excbange; m = 0) .. 9.10 e+.- -; e+e- , (with Z-exchange; m = 0) . 9.11 .+e- -> TYI, (m = 0) . . . . . . . . . . . 9.12 e+e- -> 1'+ "-,, (no Z-exchange; m = 0) 9.13 .+e- -; e+,-" (no Z-exchange; Tn = 0) . 9.14 e+e- -> " , (m ~ 0) .... 9.14.1 k3 nearly parallel to p+ ..... . 9.14.2 k3 nearly parallel to p_ . .... . 9.15 e+e- -> (no Z-exchange; m of. 0) 9.15.1 k nearly paranel to p+ 9.15 .2 k nearly parallel to p_ 9.15.3 k nearly pa rallel to ih 9.15.4 k nearly paralle! to Ii9.16 e+e- -> e+e-, (no Z-exchange; rn ~ 0) 9.16. 1 k nearly parallel to p+ 9.16.2 k nearly parallel to p_ . . . • .
,,+ ,,- /
119 119 · 120 · 121 · 122
· 122 · · · · · · · · · · •
· · · · · · ·
124 125 127 130 133 137 139 145 154 154 155 157 157 159 161 164 166 166 169
I .'( IN '/ 't:N ,/,,'>'
lUll
9.16.3 k !l(~i\rly pi\]"i\JJd 1.0 ii. . 9.16.4 k nearly plll'1\lId to q_ IU7 c+e- --+ ')'"1'')'"1' (m;l 0) . . . . 9. I 7. J J.j. nell.rI y par a!l el to p+ 9.17.2 k4 Heady parallel to p_ 9.17.3 kj and kj :learly parallel to if... and p"""'-, resp. 9.17.4 kJ and k..1 nearly pRrl1.11e! to i4 ' . . 9.17.5 k~ and k" ne.'!.rl)" parallel top_ . .. !l.lS e+e- --+ p+ p- rr (no Z-exchange; m:f 0)
9.! 8.1 k~ nearly par~.lIel to 14 9.\8.2 k~ neatly parallel to p_ 9.18.3 k2 nearly parallel to ii+ 9,18.~ k2 nearly parallel to i9,18.5 ~ and k: nearly parallel to P+ iIlHI p_, resp. 9.18.6 k~ and k~ nearly parallel to p+ and if+, resp . 9.18.7 and nearly parallel to iT and ij..., re~p.
k;
k;
alld k~ nearly parallel to p_ and q. . , resp. and k~ nearly parallel to p_ and ii-, re~p . 9.l8.10k~ and k~ nearly parallel to f.. and ii-, r~p. 9.18. 11 k~ and k; nearly parallel to P... . 9.18. 1 2k~ and k~ nea.rly parallel to p'_ . 9.18. 13k~ and k; nearly parallel to iii- . 9.l8.14k~ and k~ nearly parallel to if- . 9. 19 cTe- -+ e+e-1'1' (no Z-exchange; m '" 0). 9.19 .1 k~ n\~arly parallcllo P+ 9,19.2 ~~ nearly parallel to p_ 9.19.3 k2 nearly parallel to if.... 9.19.4 ~ nearly parallello 9_ 9,19.5 fl and k~ nearly parallel to ji" and iL, rf'h~p, 9.19.6 ~ and k~ nearly parallello p+ and if...., resp. and ~; nearly parallel to ih ancl ·q:., resp. 9.19.7 9.19.8 ~ llUd ~~ nea.rly parallel to p_ and q. . , resp. 9.19 .9 ~ <md nearly parallel to jL and i-, resp. 9.19.IOk~ and ~; nearly parallel ~o (/+ and q_, resp. 9.19.11 k~ and k~ nearly parallel to p+ 9. 19.12~ and k~ nearly parallel to fL 9.19.13~ itud k~ nearly parallel to if..... 9.19.l4k~ nnd k~ nearly parallel to 1_ 9.18.8 9.18,9
k; k;
£.
l::'
10 Summary of QeD formulae 10.1 e+e- --+ (jq (no Z-exchallge) 10.2 e+e- --+ qqg (no Z-excha.nge) lO.~ eTC -+ qq, (no Z-cxchangc) 10.4 e+c- --+ gqgg (no Z-exchallgc)
172 · 175 · J 79 · 180 · 182 · 185 . '. 187 ) .1 90 · 193 · 193 · 197 · 201 .206 · 210 · 215 · 22 1 · 227 · 233 · 239 · 244 · 2-19 · 2.54 · 260 · 265 · 265 · 272 · 278 .285 .292 · 300 · 307 · 315 · 324 · 331 · 339 · 347 · 354
· 361 36" · 369
.. .
· 370 · 371 · 373
CON 'f'NN '/'8
xiv I
. \,allKc), ) . . , . . . . • . . ;' . '. 10 .5 ,,+c- -. q"iil/' 1j' (no '%,,,x'\: 10.6 e+e- -+ q"iiqq (no Z·exc lange , . . . . . . .•. ' . : : 10.7 qq' ..... qq' , . . . . . .. , , .. , . . . . . , . . . / . 10 8 -qq -+ '='q q' .. . . , . . . . . . . . . . . . . . . . . • . . qq .-; qq . . . . • . . . . , . . . . . . . . . .. . . ..• 10.9 10.10 qq ..... qq . . . . .. . . . .. . . . . . . .. . • . . . 10.Il,q -+ ,q . . . . • .. . . . . . . . . . . , . . . . . . : 10.12/Q ..... gq .. . . . . . .. . , . , . , . . ., . 10.13 9 q -+ /Q . . . . . . • . . . . . . . . . • . • . . . .
. . .
~~:;:;~:;~
.. .. . . 379 381 : : . . 385 . . . . 386
387
.. .. .. .. 388 . . . . .... .... . ..
389 390 391 392
::::::::::::::::::::::::::.::;~!
10.16 - ..... gq .. . .. . . . . . . . . . . . . . . . . . . . . • . .. 395 ,q ..... ,q . . . . . . . . . . . . . . . . . .. . . .. . . . . . . 396 10 .1 7 gq 10. 18g q - >gq . . . . . . . . . • . . . .. . .. .. : : :: ::: . . 397 . . . . . • . . . .. . . . . . . . . . . 398 10.19 qq ..... I"r . , . , . 10.20 qq ..... 9, . . . . . . . . . . • . . . .•. . . . . . . . . . 399 10.21 q" ..... gg . .. . . . . , .. .. . .. .. ,. .. . . . . . . 400 10.22n ..... qq · · . · · .. . · .. · . .. . . . . . :: . . . . . . 401 23 ]0. 9')' -+ qq . . . . . . • . . . . • . . . . .. . • .. . .. . . . 402 10. 24 99 ..... qq . . . . . . .. . . . . . . . . . . . . .. . . • . . . . 403 10.25 9 9 -+ 9 9 . . . . . . • . . . . • . . . . • . . . • . . 405 10 .26 qq' -+ qq', .. . . . .. . . . . .. .. • .. . . . . . . : : .. 406 10 .27 qq' -+ qq'g . . .. . . . • . . . . . . . . . • . . . . . : .. . . 409 10.Z8q q ..... 1j'q',..... . . . . . . . . . . . . . . . .. . 410 10.29 qq ..... qq'g . ... . . . .. . .. . . . . . • . . . . . . :::: 413 10.30 qq ..... qq-r . . . . . . . . . . . . . . , . . . , . . .. •.. .. 415 ... .. 10.31 qq ..... qqg.. . .. . . . . . . . , . .419 . .. . .... .. . 421 10.32 qq ..... q9/ ., . . . . . . . . . , . . . . . 10.33qq -+qqg...... . .. . . . . . . . .. . . . . . 425
. .
.
10.34 -yq ..... "q . . . . 10.35 "(q ..... g ,q . . . . 10 .36 gq ..... "/"/q . . 10.37 ,q -+ 9 9 q . .. . 10. 38 9 '1 -+9,'1 10.39 gq ..... ggq 10.40 Iq -+ " q 10.41 Iq ..... 9 I ii 10.42 9 q -+ "q 10 .43,q -+ 9 gq 10.44gq-+g,),q l0.45gq ..... gglj I0.46qq ..... ,,')' 10.47 ij q -+ 10.48 qq ..... gg')'
g"
. . . . .. .
.. .
.
. . . . . . . . . . , . . . . : : : : : : : : : 427 .•.• . . . . . . • . . . : .. . . .. . . . 430 ... . . . . . . . . . . . . 432 . ... ' . . . . . . . . . . . . . . . . . . . . 435
. . .. . . . . .. . . . • .. . .. : : : : : : : : 439 . 445 .. . . ...... . ... . ... . . : : : : : : 147 .. . .. .. . .. .. .. .. . .. 449
. . . • . .. . • . •. . . • . . . ..
. .... . ...... . .. . ... . :::::::
452 . . . . . . . . • . • . . . . . . . . . . . . . . . . . 455 . . . • . . . . •. . . . . . . . . . • . .. . . . 459 . . . • . . . . . . . . . . . . . .. . . . . 465 .. .. . . . . . . • . . . . . . . . . . . . , . . 467
.. . . ,
. . . . . . . . ... . . . ... ,, "" 469 . . . . . . .. . . . . . . . . . . . . . . . ,
,
,
".
...
·. . ·.
lOA!) 7jq ........ 999 10.50 'Y1 - qq-y 10.5111 - I qqg 10. 52 19 ........ qq1 10. 53 19 ........ 'iiqy
·
10.54gy ........ qq'Y JO.S5gg->qqg
.477
· 479 • 481 .184
.
· 487 · 190
10.56 99 ........ 999 10.5799 ........ JSo 10. 58 99 -> ~P(j . 10.5999 -> 3P2 (0.60 qg ........
g l5
· 496 · 50 1 · 502
· 503 , 504
0
10.6 1 q9 - ) q 3po 10.62 qg
->
· · · · · · · ·
q3P,
10.63 qg ........ q aPl 10.64 qg ........ 11'51)
•
10.65 q9 --+ q Po lO.66ljg -> 'qJP, 10.67?ig --+ q3P2 10.68i'jq-> 9 ISO 10.69 qq -> gapo 10.70 7jq ........ g 3PI 10.71 qq ..... g3P~ 3
• •
505 506 508 509 510 511 513 514
· 51S · 5 17
· · · · · · ·
[0.72g9--+9 ISo
10.7399 -> gSSl 10.7499 ........ glP! 10.7,') 99 ........ g3f'o
• •
·. .
1O.76.qg --+ 9 3Pl 1O.77gg--+ g3P,
11 P olarization 11. 1 Fermion!. . 11.2 All enmple: 11.3 Photons and I J.4 All cxamp!e:
472
51S 519 521 524 527 529 532 53 7
. . . . . . . . c+c-
--t
· 537 · 539 · 542
• •
/1 +11-
gluons . . . 9 g ........ 91S(I ..
1 Z B earnstrahlu ng 12.1 EJedron'positron linea.r coll;ders. 12.2 Na.ture of approximations . . 12.2.1 Single-pl'I.rticJe app:oximatioll 12.2.2 Externa l field approximation , 12.2.3 Important length scales. 12. 2.4 Shape of bunch 12.3 Electron wa.Vf: functions . . .
· 515
549
. . •
· 549 .
.
· 551 · 551 · 552 · .;54 .555
· 556
xvi
1.2.4
Cl'O~H Hcdioll ror (~
+ bunch
--4
t;
+ "( + hl10eh
12.4 .1 ElecLrost"tic potential . .. . . . . . 12.1.2 Initial and [inal phases . . . . . . . . 12.4.3 In tegration over transverse momenta 12.5 Energy distribution of bcams\rahlung photon 12.5.1 Mellin transform . . . . . . . . . . . 12.5.2 Residue of J«() at ( = 1 . . . . . . . 12.5.3 Ana.lytic continua.tion into the region
•
.'
• J .
.560 .560 · 561 · 564 · 568 · 569
.571 . 572
1> Re( > 0 . . . . . . . . . . . . . 12.5.4 Residue of K(O at (= O . . . . . . .
.573
12.5.5 Analytic continuation in to the region
0> Ree > -1 . . . . . . . 12.5.6 Residue of K(() at (= -1 ..
· 577 .577 _ 580 · .58l
12.5.7 Hyperbolic secant distribution 12.5.8 Summary 12.6 Discussions . . . . . . . . . . .
.
.
.
13 Outlook 13.1 The ubiquit.y of the photon . 13.2 Further developments . .. . J 3.2.1 . Supersymmetry .. . 13.2.2 Phase choice of polarization vectors. 13.2.3 Weyl-van der Waerden formalism .. 13.2.4 Quantum gravity . . . . . . . . . . . 13.2.5 Polarization vectors for mass ive spin~l 13.3 Unsolved problems . . . . . . . . . . 13.3.1 Collinear fermions and gluons 13.3.2 Relation to quaternions . . . . 13.3.3 Loops in Feynman amplitudes 13,4 Epilogue . . . . . . . . . . . . .
.583 587 · 587
..
.587 .587 · 588 · 590 pC'~rticles
.592 .593 · 596
.596 · 597
. 598 · 599
Appendix A Traces: cut-and-paste
601
Appendix B The p,'ocess -II
605
-> "( I
Appendix C Color traces
613
Bibliography
619
Author Index
632
Subject Index
645
1 Introduction With cnergy galon::, one can explore proton's core
more and more.
1.1
Elementary particles circa 1989
Of all fundamenLal interaciions in nature, Lhe electromagnetic interactions I,d,wccn charged particles wete the firl:!t Lo be studied cxtCllsivcly. T he dev,.!oprr:enl of renormalized perturbation theory by Tomonaga, Schwinger> I,'''ynman ,wd Dyson [I] in the latc 1940's made quantum clcclrodY llamk:s (q£D) into an extremely successful theory wlLich yielded maIlY \'cry precise predictions which were repeatedly confirmed by experiment, two TJTominent ",xamples being the Lamb shift [2] an d the anomalous magne.tic: mom.~nts uf Lilc electron and the muon [3]. QED, a relntivjstic quantum field t heory I m~cci on aO Abelian gauge symmetry, became !'he prototype field theo ry for Idl other interactions. Over thirty years ago , in 1954, Ya.ng a.nd Mills [4] showed that 1I0nAbelian gauge symmetries could be introduced for the description of o ther f'lIl damenta l interactiolls. At that time, however, not all ingredients were a.vailable to tum this beaut iful idea into a re alistic theory. Ten years later, in 1964, GelJ-Mann au d Zweig [5] proposed qUll.rks a!l Ute oaliic constituents of h"clrons. Shortly thereailcr, based on Lhe earlier worll of Glashol4', Wein berg and Salam [6] comtr \;cted a Yang·Y1ills theory tlf uni fi ed weak and eledromagnetic intera.ctions. ,For Lhe consLruct ion of tllis 1.!Jeory, they invoked i.he idea of Lhe Higgs mechanism ]7J, discovered II). 1964, wh iciJ al!ows one to reconcile gauge $ymme~ries with short-range forcl..'S. T he wlor degre€ of freedom, a.3sociatcd ;I.;lh ~he quarks to preserve their ordinary l,'ermi· Oirac staiistics, led Fritzsch, Cell-Mann and Leutwyler (8] to propose another Yang-Mills theory for the description of strong interaction!; of the quarks. T his t heory, called quantum drornodynarnics (QeD), together with the Glashow- Weinberg-Sala m model of weak lInd f:k:ctro m,1gnetic inter .. ctions const itutes the .slaudard model. In both Abeli,o.n and non-Abelian gauge theorie3, one necessarily RS50dates a gauge particle with every local symmetry. For qU:Jnt.nm chromodyl\i"mics, t he gauge particle is the gluolI, which interacts with the quarks . For the e\ectroweak theory, the gauge particles are the pho"ton and the interme-diate vector bosons, W and Z. It has taken a number of years for the essentia l aspects of t hese theories
t. IN'l'tW{)II(J'J'ION
10 be verified <)xI)('l'imolltl1.lly, lhe "mjor mil".lonc. being the foJ.l6'wing:
i) the discovery in 1979 or the gluon jet at PETRA of the Deutsches Elektronen-Synchrotron by Wu and collaborators [9J and ii) lhe discovery in 1983 and 1984 of the intermediate vector bosons at the pp Collider of CERN by Rubbia, Van der Meer and collaborators [10J. Presently, the missing link is the Higgs boson [7], which was introduced in lhe standard model to generate the masses. It will be a most important event when this problem is settled experimentally. At the present time, the known constituents of matter are the following:
i) neutrinos: there are at least three
(Vel V}Jlllr);
ii) charged leptons: there are at least three (e, 1', T); iii) quarks: there are at leMt five (U,d,8,C,b); and iv) gauge particles: there are at least four (-y, lV, Z, g). In this list, the particle a.nd its antiparticle are taken together as one, and internal degrees of freedom, such as spin and color, .are not explicitly counted.
It is generally believeAi that there is a sixth quark, the t quark, which however has not been seen experimentally.
1.2
Radiative processes
A radiative process or a bremsstrahlung process is any process in which one or more photons or gluons are emitted. As a rule, bremsstrahlung always accompa.nies the nonradiative process. Obviously, the cross sections for the radia.tive prucesseH are slllaller by at least one power in the coupling consta.nt: Ct
for QED or as for QCD, than for the nonradialive ones. In many cases, the radiated gauge particle goes undetected. This can oc-
cur when its energy is too small to trigger the detector or when its momentum is nearly parallel to another pa.rticle's momentum so that their tracks cannot
be resolved. Similarly, photons going down the beam pipe will in general not be detected. When this happens, one considers the radiative p~ocess as a higher order correction to the nonradiative, lower order process. Because of the sma.lIness of" and as, these corrections are t ypically in the 1- 10 %
range, bul they can be larger in specific situations. For example, initial state radiation in e+C -+ {,+I'- reduces the effective energy available for producing the final slate muons, aod, at the Z-peak, one expects the lowest order cross section to be reduced by roughly 30% [llJ. For precise experiments, the bremsstrahlung corrections must be taken into account. This is the case, [or example! in the determina.tion of the
I..~.
'l'fU; /MI'OJ['I'I1NC.'Jo; Of
IIW/lI-:N lm(!If;.~·
,
1"'1<111 IUllliuosity for r+e- storage rillgH, which iH mlUJltly done by measuring I, IH' "J'O~S section for small lingle Dhabha. scattering [12]. Dremsstrahlung nll'r1:clious also play an important role in tests of the eledroweak theory, in pHrl.icularfor t he sl.udy of the angular asymmetry in e+e- ...... /1+ p- aud ot her t)EJ) Lests at high energies . In the case of QeD, the radiative processes are qllantitatively even more important because of the larger value of as, but I hn experiments usually do not reach as high a level of accuracy a.;; in the , '.!L"C) of QED tests. Somet imes, the radiat ive processes are studied for their own sake. Withi n I, hc~ rramework of pertur bative QeD, one associates t he production of jets wilh processes in which quarks and/or gluons are emitted. In lowest order, I.wo-jet production in e+e- annihilation is described by the process e+e- ...... 'IiI, but, for thrC0-jet produclion, one has to cOlisider e+e- ...... qqg, which is It radiat ive process. For studying four.jet production, one has to know the cr()~s sections for e+ e- ...... q'lj q q , qq q' q' and q q 9 9 , the last process being ;< double bremsstrahlung process, de. The importance of radiative proccss(,.'t! is by no means restri.;ted to e+e('~Jllisions. A Iso for p'j'i wllidcl' phy~i('_~, the multi-jet phenomena are described hy gluon bremsstrahlung processes. The same is true, for exa.mple, for t he hadroprodudioo of heavy quarkonia at large transverse momenta. It is clear from thi$ 1lOltrJxhaustive list of exampJ(!$ that radiative pro('.(~~scs play an important role in high energy physics
1.3
The importance of high energies
For particle physic:;, high energy is the name of the game, at least most of the time . The higher the energy, the deeper one can probe the structure of matter. There is a very strong hope alllong physicists, that ultimately nature will reveal its simplicity at high eneJgies. At t he present moment, the maximum energy range for e+e- physics is 100 CeV (LEP aL CERN), while for pp physics oue reachet! 2 TeV (Tevatron at F'el'mi lab). T his is a very fo rtunate ci rcumstance for calculations iuvolving leptons or quarks. Indeed, the masses of Lhese particles are.so small compared to the energies involved in the collisions that they can safely be neglected in almost all case:;. Putting t he masses equal to zero in calculations certainly leads to great simplifica.tions. Yet, at the t ime when most sta ndard books on Feynman
I. IN'J'{WIJIIC,YJ'JON dil~grn.nu~
were wriLtcm,
st.and.ard
Wily
!oipec;ial nUolltiOl,l WRS given to the high .cinergy limit. As a rnsull., th(l lc!(:ilniqtJ(!s for r.R1r.ula.t.ing Fcynman dia.gra.tp.s in the HO
arc not the most. efHcicnt ones for obta.ining the high energy
limit of the cros~ sectiolls.
In the standard procedure, one adds up all the Feynman amplitudes, one takes the squared absolute value of the sum, wbich is then summed or averaged over the polariza.tions or a.ny other degree of freedom. For radiative
processes, where one deals with a large number of Feynman diagrams, this is in general a lengthy and cumbersome procedure. What we propose, alternatively, is to take advantage of the high energy limit. For massless particl~, hclicity states arc Lorentz invariant , and it ap· pears that it is much simpler to calculate first the various helicity amplitudes for a given process. \Ve will shQ\.v that, by choosing a convenient represen· tat ion for the polarization vectors of the radiated gauge particles, a sizable fraction of Feyuman diagrams gives a vanishing contribution for a. specific helicity configuration, thus simplifying the calculation of the corresponding helicityamplitude. Once the helicity amplitudes are calculated , it suffices to add their squared absolute v?lues to obtain the cross section. This method has the advantage that the calculations are performed at the level of the amplitudes, rather tha.n at the level of t.heir squa.res, which are in general much more lengthy. Furthermore, the cross sectjons are obtained. as it sum of positive quantities, which for numerical compuLations circumvents the problem of large ca.ncel· lations between contributions of opposite signs. After a summary of the standard Feynman techniques, in Chapter 2, we present the helicity method in detail in Chapters 3 through 6. More specifically, in Chapter 3, we define the helicity states for the fermions, the photons a.nd the gluons. In Chapter 4, we analyse the case of single bremsstrahlung in QED, while, in Chapter 5, we consider single bremsstrahlung in QCD. Fina.lly, in Chapter 6, we present the application of the helicity method to double bremsstrahlung in QED. In all these chapters, we illustrate the procedure by workiug out severa.l explicit examples. Even in cases where masses cannot be neglected) the helicity method, or a simple modification of it, can often be used advantageously to obtain the cross section. This somewhat mo(e complicated issue will be treated in Cha.pters 7 and 8. In Ch"pter 7, we show how to obt.ain the finite mass corrections for· spin amplitudes. In Chapter 8, the hclicity method is applied to the production of heavy qllarkonla, i.e., bound qq systems where the masses of the quarks a.re not negligible. For case of reference, we list the various hclicity (or spin) amplitudes and the cross sections for the simplest QED and QCD processes. Chapter 9 is devoted to single and double bremsstrahlung in QED , the most important processes being e+ e- -+ ",(" 1-'+ 1-'- and e+ e- with one or two additional pho· tons in the final state. Special attention is given to the spin amplitudes which
I .Y. T/f/:'/M /'()It'I'/IN(,"/',. (N' 1//(;/1 f;Nf:/um:S ,J~·tll'ri!.C
t he sihLMiolL1; where olle or l wu of Lho outgoing photol!s a.re nearly jUlrul1d lo on~ of the formiolls. We a lso lreaL lLe case of Z~exchange for the p!"On'~~c~ e.+c:- -+ /J +"-' c:+c:- ...... e+e-, e+e- -+ /J+/J-"" and e+e- --I e+e-')' . tit Chapter 10, we present the formulae for single brcmssl-rahluug ill QeD IIIHl {or the production of heavy quarkonia. Except for the heavy qnarkonia, WI· [l*,ume I.hat all quark mas.$CS can be ncgicctc"{1. Taken togel-her, these two .-11.'I.I,Lcrs may be considered as a handbook of amplitudes and cross sections fur qED and QeD proces-3CS at high cncrgieg. We ho~ tha.t they arc C/jI,~:cii:l.lly useful t.o experimentalists and theoreticians W}1O a re less concerned wil.h t heir derivation. tn Chapter 11, we explain how the helic.ity amplitudefi Cl\.n be combined I" yield cross section formulae for the Ci;l.l:te of initial st ate polarization. In (:llarter 12, we discuss the problem of oremsstrahlung in future high-energy ~·ll:dron-positron colliders-known as hca.mst rnJlluog. CO II~rary to the subji'cls di"cusscu in carlicr chaplcr~, lhe modem treatmeut of this topic sta.rted (Oltly about three year~ ago. Since our knowledge about heamst rahlung and llie related process of pair production jg al present rapidly evolving, only ~ome of the simpJest a.spects are trf'..a.ted in this book. Pina!!y, Cha.pter 13 presents ou r summary of the helicity method. We also poinl out the recent applications , vlhich were not treated in detail in lids book, and the further developments, whkh led to modified vCIsions of the heudlY method. We hope tLat the helicily method, which we expound, and the list of rormulae, which we present, will allow future, more precise invest igations or even more processes at high energi!ls. If this is the case, 'The U bi quitou~ I'hoton: will have contributed a little to a deeper understandi ng of elementa.ry par: icles and their behaviour.
,, ,
,,, ,, ,
,-, ,,,
2 Feynman diagrams It's neverthdess an cndJ~ me&! before one can assess Nature' s inventi""eness.
2.1
Feymllan rules
A Fi:yumwl {liagram [1] is a graphicz.1 rep:-e&eIl t ation of the amplitude for a given process. It is composed of three types of dernent!>! external lin~ ,
itlkrnallinc'!; and vertices. With cYcryone of these elemenh one associates a Iw\Lhema.tical expr....ssioll, and 1hc product of 1ho!Sc expressions, in an apprf.r
IIriate order (together with some overall f~torll). yields the amplitude. The Fcynman rules telll:s which mathemaLical expre&'lion is to be as,o-
t:ialt,."(\ with II. ghoen clemenL or the Feynman diagr;lm . Tbey can be derived from first principles, but we simply list the resulls. using the Minkowski mdric
+1
,. 9
0 = 9"" =
(
~
o o
~ ).
(2.1)
+ ~ + ... + b.., ,
(2.2)
-1
o o
o
-I
o
-I
III a Feynma.n dia.gram for the process UI
+ al + ... + Il ..
-4
b!
I.he iucoming particles, aI, al, ... , all, are Rhown as lines entering from the left, whereas the outgoing OUI.!:!, bt, ~, ... , bm , leave the d iagram to the right. for such lillC::!, referred to M external lines, we have the Fcynman rule~ Ii~ted i~
Table 2.l. In this table, t~e qUAntities p or k are the physical [our-momenta of tbe particles, and (1 or ,l Ia.bel the polarization dfigrecs of freedom. The spinors u(p, (7) and v(p,u) are 5Ohltiolls of the free Dir
(p - m)u(p,q)
0,
(JH m)v(p, q ) -
0,
(2.3)
where I. .. p=p", =P<J,0 - p.,. ~-
(2.4)
H.
fI'; YNMIiN IJ/AUllIiM.j' /
external line
expressIOu
particle
p,'
,
u(p,a)
incoming fermion
,
li( p, a)
incoming antifcl'mion
,.•• • ,
p,' •
n(p, a)
out'going fermion
,
"~a
v(p,a)
outgoi ng antifermion
.1.:,)'
e( k, ,\) '
incoming vedor particle
."( k , ~)
outgoing vector particle
•
'VVVVVV'
k,'
'V\IVVVV'
The 'i'-matrices in the Dirac equation satisfy the anticommu,t at ion relations
(2.5) Thel'e are several representati ons for the I'-matrices. For our purposes, the fol lo'wing representation is found to be very cbnvenient: o -y=
. (0(i _qi) 0
(011 11) 0 '
"11:::=.
!
. - 1, ?~, 3 )
1. -
(2 .6 )
where
.I)
0 ( 1 0
2 _
'
(J
-
(0 -i) 0
(2.7)
i
are the familiar 2 x 2 Pauli matrices and
11=(~
n
(2.8)
is the 2 x 2 unit matrix. T he spinors u(p,O') a.nd li( p,a ) are defi ned by
;r(p,O') = u(p, a)t,,!", where
t
(2.9)
denotes lhe adjoint, i.e., the Hermitian conjugate. T hey satisfy the
,e quat ions
;r(p,O')(p-m)
0,
v(p,u)(p+ m)
0,
(2.1 0)
e I.
,
,.·":YN MAN IWId'.:'i
whirh f01l0WH [rom {:<jns (2.3) 1I.nd the fact thaL
. ,oi·~t . i·
1J
(2.11 )
= "fl' .
W ith cxtcrual scalar lillcs, for example, fo r I·Tiggs bosons [7], one asso, ·ial.(J~ t!J~ trivial factor L Also note tha t OU f choice of assigning c: (C:") with 1Int~oi ng (incoming) vector pa.dicle$ is contrary to the usual COllvcJltio ns of mO:lL textbooks. hi this book , we deal mostly with outgoing vecto r particles, ami, in order to alleviate the notation , we preferred to omit the asLerisks ill j.lll::;c cases. For internal lincs, the Feymnan ruk!! arc listed ill Table 2.2, where m ']"!lotes the mass of the vi rtual particle under consideration . In QeD, the .!narks a nd the gluoJlS h
intcrnallioe
e:<pression
k
, k2 ~
,
J
,p
.
p
,
•
p2 _rn2 +i~
i(p+m) l!"J, + il J
p' _ m2
k
-fYI',",
k
p, a
v,b
k 'VVVVVV>
"
k2
v
'VVVVVV>
v
m'+ if
iU>+m)
'VVVVVV>
/'
virtual particle
k Z + if
.
1
Quark
gluon
6••
+ ku k,,/ml ,. -g.k.." _m +u 2
lepton
photon
+ it.
-ig,."
::;calar
massive spino!
10 Tabl" 2.3: [o'cy,ullan rules for QED and
vertex
QeD
expression
I vcrticc~ "
description
~--------1-----~------r-----
----~
photon-lepton vertex
photon-quark vertex
k
J
gluon-quark vertex
k
J P,P'l a
yr" [(p - q)p9", +(q- k) ,.9" +(k - p),9,_1 q,
1/,
!-"a
v,b
b
Uuee-gluon vertex (all moment.", t.aken to
bf!
incoming)
k,p,c
<J,d
p,C
-i9 2 [/"" r"(9_,90v -9_,90,) +r" /'d'(g.og" -9_,90,) d +r , /"'( g"pglJu -9_09po)1
four-gluon vertex (summation over e is implied)
!t.I.
1·'lIYN MAN lUI /.f\'S
11
For the vertices, we have the r"'cyllm~n rulC!l of Tll-ble 2.3 . Tn this list, 1.11(: rlttantity e (e l ) denotes the lepton (qu ark) charge, while !I is the QeD mupling constant. In QED, there is only olle ty pe of vertex, i.e., the photonrl'rmion ....ertex . Howe\'cr, because of t he non-Abelian nature of QeD [4J, Lllcre a.re a.dditional vertices involving three o r four gluons. The group structure underlying QeD is S U(3), which was first introduced ill pa.rticle physics to describe flavor symmetry among the particlcg {14} , III the context of QeD, one often uses the terminology color SU(3) [81 to lJislinguish it from the flavor S1""(3). The QeD struct ure constants rlx: Me totally antisymmetric, and
(2.12) Ttle anticommutatioll relations { T" , T'}
~ rT" + 3~ , •• '
(2. 13)
ddine the totally symmetric structure constants d,,6e. Some useful relations ;1 re
(2.14)
The only nonzero elemer.ts (up to permuta.tions) are the foUowing: 1 =
I
7:l
~
1m
=
21w
dBa = dnH
=
::;
21111; .", 21m;::; 2f3-4s
dJ38 = -d8S8
,
-1
2.;3 1 - 2
dH6
= din =-d2H = dl 5
d35~ =
-d36f;
=
-d311 • {2.15)
Finally, there are some overMI
factor~
which must be included in the
Feynman amplitudes (see Tilb:e 2.4). Fir$l of all, we uote that we Itave conservation of four-momentum llt every \'ertex. For diagrams ..... ithout closed loops) i.e., lrt.'C diagrams, this
~.
I~
fo'l';YNMAN I!IA(;UAM.I' I
/
1)r<~H<:ripl, i{)ll ddel'lllilltll'l Llle nl~)BWlltHIII a.:-tHod;tted with i Ilternn.y-fi lies, "'or do~cd loop diagrams, somo iJltt-runl momellta. wilt Bot be fixed j"i~ld one Inust insert
f dti kj(2r.)4
to integrate over all valll~s of this interna.l momentum .
Secondly, there i." an extra minus sign for diagrams in volving closed fermion loops and a rela.tive miuus sign bdween graphs which differ only by an intercha.nge of two external ldenti.cal fermion lines. In this context, a.n incoming electron is identical to an outgoing positron, and similarly for
quarks with the same flavor. Of course, for a closed fermion loop, a trace must be taken over the i-matrices appearing in the loop. Thirdly, one must incl ude a symmetry factor 1/5 to avoid overcounting of identical states in closed loops. More precisely, 5 is the symmetry number o[ the Feyr.ma.n diagram with the external lines kept fixed [1 51. For example, in the tadpole diagram o[ Fig. 2.1, there is a symmetry [actor 1/2, as the internal boson stat.e with momentum k is identical to the st.ate with momentum ~ k. For the Feynman diagram of Fig. 2.2) the symmetry fador is 1/6, eLc.
Table 2.4: Feynman rules for overall factors.
closed loop
. closed fel'miollioop
_l_Jtflk (2;r)4
- 1
between graphs with interchange relative factor (- 1) of identical fermion lines symmetry fa.ctor to avoid over-
counting of identical boson states in closed loops
conventional overall factor
1/5
-,
• !./ . /"f.;YNMA N
lWI,~:S
Fig. 2.1: Tadpole diagram with symmetry {actor 1/2.
o Fig. 2.2: Fcynm(lo diagra m wi~h symmetry factor 1/6. finally, we introduce a.n overaU fador (-i) , which is purely conventiunal , but which givcs the optical th(.'OrCln [16] its familia.r form. Let M be the invariant a mplitude for el astie scattering in t he forwar d direction const ructed ,Kcording Lo the above IUles) thell
(2. 16) with, in gCllCral, .\(0 , b,c) = a 2 + II Here,
Inl
and
P1 and Pz,
m2
S'::::'
+ (? -
2bc - 2c(~ - 2ab.
(2.1 7)
denote the mnsses of the incoming particles with momenta
(PI
+ ]>2)\ and O"Io~(S) denoles the total cross section.
'rhe }eynmfLrl rules lisLed ill Tahl('!; 2_ 1 t hrough 2.4 an" 1101, Cnlllplde~ they should be supplemented with yet another rule for diagrams involving
closed loops with gluous, the so-called ghost contrihutions [l 7]. Since we do !lot discuss such processes ill this l:iook, we simply omit tiJis extra fllie. T he procedure for writing down an amplitude is then as follows. The type of process one considers determines ihe number a.nd the nature of the
2.
I~
fo'b'YNMAN lilAGIlAMS
e+(p', (1')
I(k, A)
p-k
p -
k'
I(k', N) Fig. 2.3: Feynman diagrams for e+eexternal lines. For example, for e+e-
--+
->
II, to order .'.
,"'I) one ha.s one incoming electron
line, one incoming positron line, and two outgoing photon lines. Using the vertices a.nd the propagators which are at one's disposa,l, one then dra.\vs
all the connected Feynman diagrams which link t.he external lines together. Usually, in perturbation theory, one works to a specific order in the coupling
constant, which limits the number of vertices to be included in the diagram. With tbe help of tbe FeYllman rules which are listed, ooe writes down the
associated a,mplitllde, knowi ng that a. diagra.m has to be read against the directions of the arrows of its oriented Hnes. Thus) for e+ e- --+ 1/1 to ol'der e 2I we have the two Feynman diagrar~
of Fig. 2.3, and the amplitude reads
M
=
-e' u(p' , (T' ) [,i(k"Y):--=-~'~:2 j - ,k'+m + ,i(k,A)(p_k'l' -m'
,i(k,A)
1
,i(k', )..') "(p,o), (2 .18)
where, for any four-vector all 1 we use the notation p. = aIJ.11-' . Another example: e+C -> qq to lowest order. This time, we have only onc Feynman diagram which is given in Fig. 2.4. The corresponding a.mplitude is then sirnply
(' ') ' ( ) _( ) (' ') M=( fe, ),VP'(]'I'}up,auq,r/~vqlr,
p+ p'
(2.19)
where p + p' is the four-momentum of the virtual phot.on.
2.2
Cross sections and decay rates
To obtain the cross section formulae for a given process , one proceeds as
follows. First, one determines the amplitude M given by the Fcynman rules.
!U!.
"
G/WSS S/;'(,'TIONS AN/} J)","CA Y /IA'l't::i
q(q',r')
q(" T)
Fig . VI:
Feyn m~n
qq,
diagr?m for e+e- --+
in lowe:lt order.
I 'rum this, one calculates I.U ll , i.e., the square of its absolutp. value summed "VI:r the final state polarizations and a veraged over the init.iat staLe polarizai.it/IIM. or CQurse., whe.n polarization effects are considered, this summation ,Iud a.veraging procedure must be mocified appropriately. The summation ,,y" r polarization c](>·grecs 0: freedom is performed using the relations:
-
i, +m,
fermions,
I:>(p·a)v(p,a) -
R - rn,
antifermions,
I)!(p, 0" )U(p, 0") •
•
L ,"(k, A),-(k, A)'
,
-
-9"~
,
photons and gluons,
(2.20) l\olc tbat the formula (2.20) for the photons or gluons can only be used when the amplitude is gauge invariant, i.e., when M = f"(k, ).).M.. satisfies k"MI' :..:: O. If this is not the case, for example, when more than one gluon i~ present, special care must be taken to perform the gluon polarization sum (:()rrectly throug}l the inclusion of so-called ghost l.:ontriL'J.liullS. However, the Iwlicity amplij.ud~ method to be described in this hook nicely circumvents Ihis problem. We, therefore, do not discuss this issue here in more detllil. l-'or a. RCflttering process with two incoming part.ides find n outgoing ones, kl + k2 --+ PI + ... + p", one then writes down the multi-di{feTl.ntial crOS$ ~cctioll formul
dO' =
c?iit
x "(2"."),f2'--p-"
(2 .21)
e.
10
or
V2
am Lhe veiocitif!!4 I,hn inc:orning particles t;"k(~Jl 10 be For uJt. ra~ relatjvio'iLic par ticl(~sl 1'01 ~ 1, and, in tbe c,n), fra.me,
where Vl ami
collinear.
I'I': YNMAN f)/ACI/AMS ,
IV, - v,1 '" 2. To obtain the desired differential cross section formula, one must finally itl t(~g r;Lte over the appropriate phase space variables. For example, for kl + k'J. --+ P1 + TJ21 one finds in the high energy limit in the c.rn. frame
d" dO.
(2.22) c.m .
with s = (k , + k,)'. Similady, the deca.y rate for a particle with mass m in its rest fra.me is ctT iven
bv ,
,
dr =- -_1_1 ".,Ofl' 2m
3
3
d jii t;" (?)' ... '(2 d)'2 -" u"(k' - p, - , , . - p" ) , (2" )'21'10 r.' PnO
(2.23)
with k = (771,0), the four-momentum of the decaying particle. If r identical pa.rticles are present in the nna.l sta.te and one in tegra.tes over regions of phase space where they become in.distinguishable, one must include a symrnetry factor 1 fl'! in the cross section formula. to avoid multiple couuting of the same kinemat ical configurations.
2.3
E xam ple: Z
->
e+e-
The decay of the intennediate vector boson Z, often denoted by ZO, [6,10J into an e+ e- pairis given in lo west order by the Feynman diagram of Fig. 2.5, where the four-momenta of the particles are g iven between parentheses. In the Glashow-Weinberg-Salam model [6], the vertex is given by (2.24) where
D,
= -1
+ 4 sin' Ow and a, = -1.
The matrix 75 is defined by (2.25)
and takes the form (2 .26) in the representation of the 7,matrices given by eqns (2.6). The corresponding Feynman amplitude is then simply (2,27)
I-:X,jM!'I,~:::t
1'.1.
........+t -
17
Z(k, A)
Fig. 2.5: FeynmaII diagram for Z ...... e+c, in lowest order.
He'nee, negleding the dcr.:tron mass and usir.g momentum conserv1I.t·l0n,
e1(t..~ -I a~ ) M1
12 sin 1 Ow cm,2~: .
In
~!le
(2 .28)
Z rest frame, we then have
(2.29) Wi ~h
the relation
tl'p = J~pS(p2 - ml)O(Po) ,
J 2po
(2.30)
we can, for example , in~egrate ow".r .Jip_, yidding
e1(u;;-a~pl2 I ~ 192" '0
,
To
With e 2
::0
Sill
,0
W C()S VU'
Jd 3p.+.(, ) ( ) - - t M z -2Mzp+o 6 Mz-p+o . (2.31 ) p+o
4,,"0', we obtain '(Z - - \ __ o(v! + a;)Mz 1 - 'ee'- 48 ~I' n 2OW C()S2 u;v 1\'
(2.32)
With the cxpe~fimentli.l values for Mz and sin::! Ow [18], this gives the result J'(Z - 0 cTe-) = 0.08 CeV .
2.
IR
n;YMMAN IIJA(l/IAMS
p_ - k,
p_ - k,
Fig. 2.6: Feynman diagrams for e+.- ~ 'Y'Y, to order e'.
2.4
Another example: e+c
-+
'YT
For the process of electron· positron annihilation into two photons [19]
e+(p+)
+ e-(p_) ~ 'Y(kd + l'(k,) ,
(2.33)
we have in lowest order the two Feynman diagrams of Fig. 2.6. In the high energy limit. the corresponding ampliludes are
where we have introduced the Mandelstam variables [20]
+ p+ )'
~
2(p+ . p_),
(p- - kif
~
-2(p_ . k,),
(p_ - k,)'
~
(p-
8
-
t
(2.35) u
s +t + u
~
- 2(p_ . k,) ,
O.
H follows that
IMI' -
-41 z=
1M, + M,I'
0 .. ,0' _ ,,\ 1 •.\,
+
e' -2
v A ]· Tr b.LP-,k,hv 1'-1'"("-,k,h tu
(2.36)
:~
J.
II N()'I'IU.'II I';X II MI'I. A':
r.'1 , -
..... ')' ')'
'I'lli' first tCl'Ill i~ IMd 2 , thesccolld OlJe is I M~I~ IllJd the ~hird term is 2ReMJ M;. Till! eva!\Lat.ion (If tile tra.ccs yields
=
,(U7+;;')
(2.37)
2e"
Note t hat lhe interference term vanisiJes, as it is proportional to the expreslIi
d. _ /
_
62 " k k 1 ;r "' ]0 20
-+-
( " ' )c' (p++p_-k]-k2 )an-k- 1 a-k] ,,-
t
U
2.'/(" I -+-') 6(p++p_ -k s u
-
f
l)
'1 9(P+o+p-o-k ,o j-c-' ,pk, "'IQ
(2.38) 111111
d.
.' ("
')
(2.39) dO c .m . =2s t+~' I'l,is formtl1a is divergent for t = 0 and u = 0, i.e., in the forward and in the Ioa<:kward directions. This divergence, however, is unphys:call!.nd origina.te!! ill t.he m = 0 a.pproxima.tion which was used for t he eledron and the positron. I,:el l! (2.39) is therefore not correct for III or lui small as finite mass effects have to be taken into account. A more complete discussion of the ranges of valid ity of tbe m = 0 approximation will be presented in SlxtioJl 3.4.
, ,,
3 Helici ty states A deep emotion comes from true devotion to the f ight notion of spin in lIlotion.
:L1
Fc rmio llS
III the fClllowing chapters , we sball only work i:l the high energy limit , which III""ns that we shan neglect the fcnlliolllllasses. Later on, in Chapter 7, we NI,:!11 present the modificatiolls to the helicity method which iHC JleCessary wlwlI the finiteness of the fermio::t mass hrts to be considered . F'or massless r'TIHions, h~licity stnktl are Loreutz invarianL notions. It is the aim of this ["mil to show how much easier it i9 to calculate fir~t. the hdicity tlIr.plitudes r"r ;\ given process and Lo add their squa.red absolute values to obtain the 'T('~S section rather thall following ~he procedures of Chapler 2. Of COl1r~e, ill Chapter 2, we only trealed simple examplcs, for which the summa.tion of 1M 12 over the polarization states 'was not !Sudl a (remenclo;Js amo\lut of work. 11'lwcvcr, when a process is described by n Feynman dia.grams, it amounts til writing down n(n + 1)/2 traces. In the case of e+e- ..... el' e- / ;)" we have .j(I Fcynmall diagrams, and. if we indlldc Z-ex.change, thiii number iUCl"ca.<;es I" '/1 = 80. Clcl'lrly, am: does not want to consider 32,10 interference terms! On the other band, the proees~ t+e- --+ c+c-"n is (Iescribcd by 64 helieil.), amplitudes, of lI'}lich olle hal! can be obtained from a parity conjugation Oil ~hc other 32 amplitucies. Obviously, it is prefera.ble to con9ider 32 Hhort ")(pression s t han to evaluate 3~40 long ones. . To procCt'
u±(p)
~
v,(p)
-
l(l'F1")u,(p), ,
"I(P) -
~il±(p)(l T ')'$),
v±(p) -
!U±(p)(l ± IS),
(3.1 )
where the subscripts 011 thc spinors deuote th~ helici ties of tht" corresponding r{~nnions. F'nr tbis reason, we omit, from now 011, the explicit r~ference to
\ ~,
polari:t.atioTl la.hd~j Chapt", 2,
3.2
(J,
T)
cl.c., ..."h kh (\'PPWLI'C(J
II ,:UCI'/'Y 81'A 7'/I'S
a,.q a.rbf>i'HTlents
of the Spin91'S iu
.
Photons and gluons
Photons and g luons, being massless) are characterized by two polarization states. In this section, we introduce a. convenient representation for these helicity states [21). Let us conSIder first the case of a phot.on with four-momentum' k. An especially convenient choice of the polarization vectorsl is
(3.2) where
N = H(p - q)(p. k)(q· k)J-t , and ca/l"'(o is the tota.lly anti symmetric tensor in four dimensions \vith
(3.3) COI23
=
+1. The prime for ejl merely indica.tes that this is not, quite the polariza.tion vector to be used later On . Also, the factor 2.)2 in e'lns (3.2) is merely introduced for the sake of simplicity in later formulae. [n these expressions, we introduced two arbitrary vectors, p and q, which for convenience are taken to be light-like, i.e ., p2 = q2 = O. Note that the normalization factor N i, the same for both polarization vectors, and that
(3.4)
These polarization states can alt.ernat.ively be combined into circularly polarized states, charaderizro by
(3.5) Using t he identit,y
h",.oo> = (Jo'l,n, -'logo, + 7090'
- 7 7 90B)7"
(3.6)
we can write
l± = -NI" P A(1 ± 7')- P A 1(1 'f 75) 'f 2(p ' q) hsJ. !Similar poJarizat10n YCctOr:'H'ere introduced by tum gravity calculation .
VOrOHOY
(3.7)
122] in the context o( a quan-
,u .
I'IIO'I'ONS MOl (,'L(IONS
"
T lnl Ia.o;t term here is proportional to ;"'"(5' Si nce we work wit h massless r(lI"mions, it call frequently be omitted btlcll.u~e of Lhe conserv."ltion of an n xi;~1 current. T his is, for exa.mple, the case in pure QED and in some Lfwuries involving only veclor and/or axia l vector couplings. Accordingly, lll\: l)olarizaLion vectors Me modified 1.0 be e; so that effectively
c;-
(3 .8) T his i:1 the basic formula for t he presco t formalism in the case of photon hT(lt nru>tra hi Wlg. T here I\rc sever.. l rf;'.lSOns why t his formula leads to gre1\t sirnpiific."l.tions ill qED calculations. fl. The choice of the four-vectors p and q is still quite arlJitrary. When t he photon line is next 10 an extt>.rn a l elect ron o r posit ron line, we can exploit this freedo m of choice and take either p o r q equal to t he fermion momenblm. In t ha.t case, only one of the tet'ms will give a non vanishing
contribution. The reason is simply tha t for mll...<;sless ferllli ons
u(p)
,, =
0,
(3.0)
and similal'ly.JM v-t ype spiuors. II. If we fix the fermion helicities, either a factor 1 + 'Y~ or a factor 1 - 'Y5 appears on the fermion line i 11 the Fcy nmau diagTam. Again, th is results ill the survival of only ODe krm in the }.± formilla. C. \\Then the pboton line is next to the external electron line, there ill a. ca.ncellation of the denominator. f 'Of example,
-()/'''+'' up 2(p.k)
=
tr: ~
-N,,(p)1" A
= - N;;(p ) h("+ 1<),
(3.10)
because U( p) P= O. Of course, tlle denomi nato r (p . k) now reappears in the normalization factor N [sec cqn (2.1 2)], but only as an overall factor . This is very important when different a mplitudes have to be added . This mechanism of denominator canccl latio1l9 is to a large ex· tent r esponsible for the simplidty of t he answers obtained previously by hrute force [23]. T he case of gluon brem sstra hlung can be treated in a very similar way. However , one CAnnot drop the Pr5 Lerm in Instead, one can write
,'±.
(3. 11 )
.~ .
/"':1,/(.'/'/ ' )' S'I'ATI·;".
bUL, bee.LUst: OJ' v('ctor Cllrn-flt ClllI:->erv,d.iIJJJ, 011(' niH dl'0p the las/. L(~rJ.n l)ropurLional io jo, Thus) one (",011.11 lise df('c\.iv(·ly
(3.]2) a.nd most of the advantages
A~C
remain.
There is an additional difference with the QED case. In QED, the photon polariza.tion vectors
(It
only appear in Feynman diagrams in the combination
vectors. Nevertheless, the representation (3.12) is still useful as all scalar products can be obtained with the identity
(p' 0) =
iTr[P 11·
(3.13)
This already shows that, for example, by choosing q = p' in eqn (3.12), the sca la.r product (p' . t:) can be rna,de to vani sh if pl'l
=
O.
Also note tha.t no reference is made to a specific choice of frame for the polarization vectors. This means that the calculation of the heiicity amplitudes can be naturally performed in a covariant way. Furthermore, we only consider physical polarizations, which means we never have to worry about
the so-called ghost contributions, which were mentioned in the preceding chapter. How the whole procedure of calculating helicity amplitudes is made to work in practice will be shown by a simple example in the coming section.
3.3
Example: e+e-
->
! ! again
We already considered the process of electron -positron annihilation into two photons)
(3.14) in Section 2.4, where we calculated the CrOSS section using the standard Feynman techniques. Here, we simply want to show how the helicity amplitude
method can be applied for the same purpose. We recall the Feynman diagrams of Fig. 3.1 and the corresponding amplitudes [eqns (2.34)1 M, = M, =
(3.15)
Here, as well as in the rest of this book, we omit the photon spin indices ). in the polarization vectors
f.
Fig. 3.1: Feynm!ln diagrams for e+e- -)
,-y.
T h{:rf~
are a.Hogether 16 helicity amp\il.udes corre&pondi:)g to the two )" 'licilY ~ta.teg for the four particles. Ld us dcnoie these amplit ude8 by III(A+,A _;)q ,"\l), where'\ .. (>._) is the he!ici~y of the poshon (electIOu), ;llId ..\1 (..\~) the helicity o[ pholon 1 (2). When lhe helicities of the [erm io n~ Me the ~arr:e. AI. =- L, the hclkIt.y I-lmplitude vanishetl, because of a mismatch in the projt'Ciion operawrs (I ± 1~)/2. Henn.,
(3.16) 1;11 all ..\ , a.nd ..\~.
Let us consider M( +, -; +,,..) next. It is (."tlpecially c.ollvenieliL Lo choose
;,;-1 =
, i
4[(p+ . p_)(P • . k;)(p_ . k;)ll ,
i =- 1,2 .
(3.'17)
Ikcause of t he p roj<xtion operator (1- '·)"50)/2 which st;).nds next to lk(p_ ) ,,"Iy t he fi rst term in j+(k;) cGntri butcs. But, the substitution of I t(k,) in "'1; and M2 then produces the c.ombination v(p_) f+, \\' hich vani~he!l. Hence,
M(+,-;+ , +) = O.
(3.18)
Similarly, far M {-,+;+, +), ~he rol!:;; of the two terms in j+(k, ) are in Ler,:h,mged, and we find M(-,+i+ , +)=O, (:1.19) because of the rela.tion /C u(p_) = O. Because af parity conj<1gation , which rlivs a U helicities, we also nnd
M(-,+; -, -} = M(+, -; - , -) = O.
(3.20)
More precisely, the helicity ampliLudes in (3. 16) and (:3. 18)- (.3.20) are in fa ct propOL"tional to some power of the fen)L ian mass , which has been neglected. For a more sy~tema tj c discussit"lll af these finite mass effects, see C hapter 7.
'
.. ,Y. IIliUUlTY ,)" I'AT!S
The bc:licilJimf or tIlt: I'huLolls mill'lL l.hu:i h(~ Ol)po~ite.
M( +, -; +, -
) anJ
let us dlOo.c, ill.Lc"J of "'1"" (3.1 7) ,
N,[1>+ p- ,k,(1 +'Y.)- ,k, Substitution of these expressions in
M, =
e' 1 -I- '"Is N ,N,15(p+) 2 T
,"I,
h
p-(1 - 'Isll ·
and M, [eqns
(3 .21)
p .15)1 gives
,k,p+p_(1-1,)
x(p_- ,kJ) -
-e' -t-N,N, v(p+) ,k,
-
2c'N,N,(p+' k,)v(p+)
M, =
Considt!f '.H:xt
1>+ pI!_
,k,
1>+ p1>+
h(1 - 'Y,)
1-,(,
2 u(p_)
p- ,k,(1 - 1s)"(P- )
,k, p+(I- 1')u(P-)
o.
(3 .22)
We made repeated use of the Dirac equation for the spinors and replaced
2(p+ . p_) -2(p_' k,)
t -
-2(p+ . k,),
We find that 11l, vanishes, thus IM( +, - i +, calculaLe:
IM(+,-i+,-)l'
)1'
=
IMd',
(3.23)
which is easy t o
. e'
---, v(p+) ):, ( I - '(,),,(p _)u(1'_) ,k, (1 - 1.)u(P+) u
:: 15(1'+) ):, (1 - ,(,) (2:u(1'- )u(p_)) pol
_ 4e't/u .
(3.24)
.U. RANG};,';
()f
VAW)/ 'J'Y
27
Tlw fncL t.hat we cOllld frcely insert lile polRl"izrLLion sum fo r the spinor u(p_) 1111 ,,1 ror u(p+») i~ due to thc presence of the helidty projection operator ( ) - '""(5)/2 in tnCllc spinorial expressions. We then used
LU(p-)U(P-) ~p-,
Lv(P,)17(p+) ~'" ,
pol
pol
(3,25)
wllith converted the expression into a ~jmrlc trace. Clearl)', M(+,-;-,+) can be obtained from M(+,-;+. - ) by inter" hallging 1 +-+ 2, which amounts to l +-+ u. We thus ha.ve
1.11 1' = ~'2 [1.11(+,-;+,-)1' + IM(+, - ;-,+)I'J (3,26) 'l'11f~
factor 1/4 is due to the averaging over the fcrmiou helicities, and till; raeI.w· 2 takes into account the two othcr helicity amplitudes obtained through l'ilfiLy conjugation. T hi s result coincides of course with t he previously obI;lillcd formula (2.37). This simple example already shows !TIallY of the features encountered in Irlme complicated caBes, the most htriking one being that certain Feynma.n dil'f',m!flS gi ve vanishing contributions for a given he1ici ty configuration through :ll'propriate choices of /*. A comparison bctwt.ocn this technique and the .standard FeynmaIl techIliqlle for this process may fail to impress the rea.der. This is due to the ,implicit)' of the case which was treated. One should, however, bear in mind lh./Lt , for more complicated cases with r:lany more FeYllman dia.grams, the "iluation becomes Quite differeut. \Ve only made this pedagogical exercise to illustrate t. he basic lechniquell of the !:elicity amplitude method.
3.4
Ranges of validity
the fOr(;going presentation of tIle helicity amplitude method, we put the [('nnioll masses equal to zero and justified this approximation hy saying that '.ve only consider the high energy limit, At this poiIH·, we want t o state "lore precisely wha~ we mean by this a pproximation and determine more quantitati vely its range of validity. Every helicity {<mplitude is a function of the invariants in the process, i,e., of the scalar products of the four-momenta of the pa.rticles and, in par~icular, of the masses, What one then means by the high energy limit is II. seL of kinematic configurations where all the scalar PJouuds involving differI~nt font-100mer:ta are much larger than the squ?.red masses of the external !'art.icles. [II
\
.j.
/J I,W)f'I'Y S'/'A'I'I;8
,
c:..e-...,,(.::.E!..!.,P)"--_-.,."_- - - e+ ( E , -P)
Fig. 3.2: Kinematic configuration for e+e-
--> "{"{.
Consider, once more, the process
(3.27)
with s = (p+ + p_)', t = (1'- - k,)', u = (p_ - k,)', and let m be the lepton mass. T he high energy limit is then defined by
III ~ m',
and
lui ~ m'.
(3.28)
In practice, this means that not only the incoming energy must be sufficiently large , but also that the photons must be emitted in directions not too close to the directions of the in coming fermions. In the c.m. frame, we have the kinematic configuration depfcted in
Fig. 3.2, and the high energy approximation is valid when E ~ m, e ~ m/ E, and 1[ - 0 ~ m/E. This situation is general. The high energy approximation breaks down under two circumstances:
i) when the incoming energy is too low; ii) when nearly collinear particles are present provided they have direct
couplings to each other. The second case requires some discussion. For QED bremsstrahlung processes, it impli es that the helicity amplitude method cannot be applied without modifications wheu phot ons are emitted in directions nearly parallel to charged fermion directions. It also implies that, for example, for Bhabha scattering, e+e-
---f.
e+e- : and radiative Bhabha scattering, e+e-
-+
e+e- 1 1
the procedure of this chapter cannot be applied directly in the case of very forward scat tering, i.e., when tbe moment.um of t.he outgoing e+ (e - ) is nearly parallel to that of the incoming e+ (e- ). Also , for e+e- --> , +e-,,{ and e+e- -+ }L+j1-" the approximation fails when both the outgoing leptons arc
emitted in nearly parallel directions. On lhe other hand, nearly collinear
,"( {.
UAN(~'NH
Ot' VA/,I/)/'I'r
1I1Iult)lH! pose 110 particular prohlem ill LIH) high elle rgy limit , as pllotOIlS do 1101. lillve di rcc ~ couplillgS Lo em:!! other.
III till! QeD case, we have to consider fiuite muss efi'e('.ts wh~ n a gluoll ill lII~fI,rly collinear to a quark, an 3nLiquatk or another glllon. For example, ill 1.]11: p roces~ ~ + e- - t QQ9, we cannot neglect the quark m ass when the (rll/l"p,lir is nearly collinea.r or when the gluon is emitted in nearly the same ,l il·P(·t ioll as the quark or the .1.nLlquark. For the process e+C ...... q([99, wp have to be careflll iu addition when the two gl uons are lIcasiy parallel . 'l' ll1~rf: arc, however, no djfficult·ies with the fi nite lI\ass corrections when o ne !If t h~ gluon! i~ collinear with the incoming e+ or t,- , RS gluolls do not couple ,l i l"(~dly to electrons. Ai' a rule, L11e e1fects of.1 finite fermion mass cannot be neglected whell "" Illt) sca.lar product becomes of order m 2 in ~tead of kingof order El. In t he "i! lning Cbapters, ,I through 6. we slmll, hOW C"Cf, ignore ~h ese complicntions n ll rl continue t o work in the In = 0 li mit. Que only has to bear in HIimI 1.11,.1. the fo rmulae are inaccurate in 30ml: !Special configurations. [,ater OIl , ill (:haplcr 7, we shall discuss how theRe finite mass effects can be t.\ken into /IlTtlUllt and how the heli city < ' !aplitude method can be modificd accordingly.
4 Single bremsstrahlung in QED It 's the electron's pligh t to st'lld out light. To predicL ~hi:5 sight is within our might.
-1 .1
T he process e 1 e-
-+
"('Y·. f.
I :hll ~ idCT the process of electron-positron annihilation into three photons,
(4. 1) wlwn , t he four-momenta of the partiCles are given in the paren t he.~e$. In ]"wl:;;l o rder, there a re si x Ti'eynman diOlgrams (see Fig. 4.1 ), and the corre"l'ou(l ing amplitudes arc given by
M, M,
~
,_ ) I!
v(p+
,_
e v{p+
M,
e v(p+)
);1 2{p_ .1.:1) /-IU(P_),
J_ 2
(p+. kJ)
0"-1
-2(p_ . k, )
,_
1_2(p!.~)N
- p,+ h h -2(p~ . k1 )
2(p_.kd
p- - h
II -2(P_ . 1.:3 )
'_I ) /. - p,+ fe, /. p-- ;.,
M.
f-
-
up+
l _2(p+ , k l)
J_
, v(p+) I,
Let us denote the ,\ + (L) ill the helicity Iidicity of photon i. Just as in lhe e+eOPI>osite. Thus, for all
I,'()f ~he
P--
,_, ) /. - P-+ ft, '- p-- ft,
elllP+
Afo
+ ):3
) /. - p.+ ft, , p-- ;.,
MJ -
- ,,~
1a_ 2(p+. ka) h
-11++ ):,
2(p+ . k.)
1 _ 2(p_.kJ )
h
p-- fi1
j.
(
j.
11.1
) p- ,
:Ill
(P)
-,
, ( 1'3!'
j.
p- ).
(
)
JU P_,
2(p_ . • ,) j.,u(p_) .
(4.2)
hcl;city amplituul,..'ti by M (A ... ,.LjAI,Al.A,3 ), where of the posi tron (electron) iUld Ai, i = 1.2,3, is th ~ 'Y'Y case, the fermion helicit ies Tr.ust ne<:eAAariiy be AI , ).2, ).3,
-+
remaining hel icity amplitudes, we choose IIgain [241 the photon polarization vectors Sllch thal effeclively
,. SIN/""":
/JIU\'M ~S,/,Il AIII,II N(!
IN (II':/!
/
,(ks) -----Of-----(VVV'V'Arv
,(k,)
,(k,)
')'( k2 )
,(k,)
')'( k2 )
--...._--(V'cJV'cJV'CJ"\.,
--...--(\A"-"'''''''"'' ,(k,)
,(k,)
')'( kJ) --...._--(V'uvVV'vu
'Ik,) ----;ucvvVV>.ru
,(k,l
')'( k3)
,(k,)
Fig. 4.1: Feynman diagrams for e+ e-
--> " , .
i=1.2,3.
(4.4)
next M( +, -; +, +, +). The spinoI' v(p+) come.~ with the proj('ct ion "I,,'ral,of (1 + "15)/2 and, conseque:n tly, on ly the firs t term in I-t (.:ontribu L('.'). I ~() (I "i, I()f
Ilul l.!lel) , Lhc amplitude vanishes
at!
(4.5) M !lrl~
generally, one
f\nd~
that , for all At and )._,
M (L, L,+, +,+) ~ U(A" .1_, -, -, -) ~ O. 1"oJ' tlle amplitude M (+, -; A!.j \'()()tribute:
(1.6)
+, +. -), only the Fcynman diagrams At1 and
M(+, - ;+,+,-)
x;;(p~)
p_ h p, p_ ",.(1 -
-" ),,(p_) ,
The pha!'le of this amplitude depends on the choice of phases fot the :; pinors . As we only compute IInpola.rizcd cross s(~ctlon~ here, we ca.n dlsre· g;Jrd this QvcI'all lJh a:.:c and write
/\ s in tbe c·c- --I 7"f case, we use<.! the trick of inscrting fermion pola riza tion :
4. .I'IN(;/,!; lI(lIW.I'S'J'II AIII,IINC IN Qt;/J
:14
this bool<, the RYlllbo l '...:...) Sl.iLlldR ror
With the de/init.ion (4.4)
'lUI
mpJ(diLy sign rnodHlo It phase fa.dor',
or N;, we finally a l,tain ,:
_) -~ 2e'( p+ . p- )! (p+ . k,)[(p+ ' k3)(P_ . k31-)1} M(+1 _.1 +'+ )
•
[U (P+ , J.;)(p- .
(4,9)
k;)l'
The purpose of this exercise in Diracology was to exhibit the simplicity of the manipul ations which lead to th is result. For the belicity amplitude M( +, - : -, - , +), one finds tbat only the di· agrams M , and M, contribute, Similar manipulations then lead to
M( +
,
_._ _ ' ) ~ I
)
),...
-
? 3(
.
_e p+ p-
)! (p- . k3 )[(P+ ' k3)(P_ ' k3:lt .
[IT.=1 (p+ . k;)(p- . k,)l ~
(4.10)
The remaining helicity amplitudes are obtained by permuting the photon indi ces and by performing a parity conjuga.t ion. Hence, for the case where one su ms over the final polarizations and averages over the initial ones, one finds for the squared matrix element 3
2)(P+' k; )'
IMI' =
+ (p_ ,k;)'](p+· ki)(p- . ki )
2e6 (p+. p- ) =;-""'----;3- --
- - -- - -
(4.11)
II (l'+ ,ki)(p_ . ki ) £=1
and for the cross section
dl1 = £'(p+
+p- -
k, - k, 32(21T)5(p+' p_)
ka) IMI2 d3 k, d'k, d3 k, . kID k,o k30
(4.12 )
A few comments shou ld be made abou t this calculation as it manifests some nice features which we also find in the calculations fOl' the other processes. A -first observation is that on ly a limited number of Feynman diagrams contribute to a given he licity amplitude with our choice of po larizat ion vectors. Furthe rmore, their usc produces automatically the same denominator
for every amplitude whi ch makes the derivation of the formula fo), IM I' a trivial matter, This should be contrasted with the standa rd Feynman procedure, where most of the work to obtain this resu lt is devoted to writing the various irace resu lts over the same denomina.tor. \Vhen pola.rization effects aTe considered, one cannot disregard the phases
of t he helicity amplitudes, In Chapter 11, we show how t be heJicity amplitudes must be combined for a descript ion of initial state polari zatio n.
~ .t,
4.2
WIIA ·1' If' f)f/"l·',;r1l:;N '/, n :IIM IONS NA 1!1t1 H:Y
:35
What if different fermi ons radiate ?
IlL the two previous cases of eledron-positrou annihilation , e+C ---t T"f aud , ,1 f - ---t "f "f"f , t he choice to be made for t he photon polarization vectors was rll l,Jwr obvious. It was indeed very advantageous Lo express t(k.) in terms "r I.I!!! four -vectors p+ and p_ of the fermions enabling one to use the Dirac "~lllaiions to simplify the spinor expressions . When one deals with ~wo different charged fermion lines in t he Fcynman di,~gram s, as, fo r exampl€', in e+e- -+ 1-'+ }J-" t he choic.e of t'S is slightly tllM(; comi>licated. In order to have the denomina.t or C3ncelJations of the 1'!'I!vious sections, it is essential to use the external momenta of the fermio n Ijll'~ to which the photon is atta(:hcd. Thus, when the photo:l is rad:ated rWIll the muon line, it is advaTltagoous to express (in tcrm$ of the momenta '/; and q_ of the muons, but, for radia.tion from the electrOll line, one would prj·rer to express e in t erms of the electron momenta, pot and v-. It turns out tha.t it is perfectly pos~ible to use «1/+,11_) for the set of d ii.grams 'where the photon is attached to the mllon line and to usc t(PT'P- ) fur the remaining diagrams pruvided one ta.kes into aCcolUlt u simple phase fn d or. To see this, (onsider t.he photon to be m oving along the z-axis. As the diagrams wiLh rad i",lion from the muon Ene a.ct! from the electron lille rorm s~paratcly gauge invad:..nt ~et s, we clin make gauge transfonnations ~uch that c(p t, p_ ) and t( q+, q_) only have components in the zy-planc. B ul LlI(,n, as these vedors have the sallie nor m, they can at mO!3t differ by a. phase factor (and terms propor tional to the photon rnome;)tum arising from the gauge t rallsformations). Thus, (4.1 3) I lecatl~e
of gauge invariance, the constants (3 -= are irrelevar.t . The phase ¢ is )!;iven by the scalar vroduct of t!(P ... ,p_) a.nd t;(q ... , q_) ~ (1.1 4)
.. nd must be take!l into llCcouut. Tn Section 4.3, we treat the process e+c- .... 1'+ IC""(, and wt'; ~hal J show that th is pha:;e factor, a.lthough essential, only leads t o a. minor complication.
For the process of radiative ?I-plUr product ion,
(4.15) we have to consider the four Feynrnan diagraJru! of Fig. 4.2, with
3G
j. 8INW,I,' riIUiM.I'S'I'ltltIIl.UNU IN QI.;D
/VVV'UV->J'
'I( k) 1'-(9- )
-y(k)
Fig. 4.2: Feynman diagrams for e+e-
_e 3
M, -
2(9+'
p--
_ v(p+h" -2(p_ . k) ,fu(p-) u(q_h"v(q+) ,
_
9-l
~ I'+I' ~ -y.
}<
3
M, -
2( _e
M3 -
2(p:e~_) v(p4hu(p-lu(Q-) Itr~~. ~) "I"V(9+) ,
q+' 9-
)
v(p+) ,f-2f++kft)"I",,(p-lrr(Q-h"V(9+), - p+ '
(4.16) As explained in Section 4.2, we introduce two representations for l When the photon 'is radiated from the electron line (MI and M,), we take [24]
J.; N;'
-
Np[h
p-
}:(l 'f "1,) - ,k
P+ p- (1 ± "I')],
4[(p+, p_)(p+' k)(p_ . kll~,
(4.17)
M,l, we lake N,[.4_ .4+ }«1 'f "1,) - ,k .4- .4+(1 ± ,,)],
and, for radiation from the muon line (M3 and
I; ,,-I ",
(4.18)
't'hBY are, however, rcJak-d by CC]WI (1\.1:3) and (4 .14 ):
(4.19) For cODveniencc, let us all;O introduce the notatiou
(4.20)
wi1.!!
., +8' +t-l-t''tt.! +u':o:: O.
(4.21)
As usual, we IJOW proceed to ca.lculate the helici Ly aUiplitl.:de~. They arc !i"Hoted by M ().1> ).2; ).3, ,\., ).6 ), where ,\, :;tands for the pos: troll helicity, ).1 rllr the electron, '\3 for the p.+, '\01. for the 1-1- alld ..\$ for the r hoton. H is immediately clear tha.t the electron and po.:.'i~run hdicities must be opposite, n~ well as the muon helicitics. Hence, fo~ all '\s, M(+,+;+,-,).~)
-
M(+,+; - ,-,>'&) _ 0, (4 .22) M(-,-;- , -,A~)
_ o.
For the amplitude M(+, -;+,-,+), we find that, with ",~Iy lVl, and At'! wntrihute:
OU f
choice of ,I.,
M (+ . - ; +,-,+)
xil('l_)'l eS
--;N9 fi(p.,.hll
l-~(~
-,h-;'
2 'u(p_)u{q_h" 2(q+. k)
I -IS
2
v(q+)
r
SINUU;
1J~r:MSS,/,IlAI1WNG ,
IN Qt'/)
e, .. ,I . yNpc'·v(p+h'(h++ h-) f!+(l-'5)"(P j )u(Q-h"(1-'Ys)V(Q+) I
.'
+28 N,v(p+h,,(l - 'Y,)U(p-)u(q-h,,(P++ p-)
R-{1 - 'Y,)v(q+). (4 .23)
To simplify this result even further we rewrite, for example, the last term: j
=
2e' N, u(q-h"U>;·+ p-) 11- R+ 11- P+'Y,,(1 - ,,)u(p_) 8 v(q+) II-(l - ,,)v(p+) 3
= _ 8e N,( q+ . q_) it( q-) ,p+
11- p+ (1 -
'Ys)u(p-) 11(9+) ,4-{1- 'Y,)v(p+)
S
16.3 = --N,(q+ . q_)(p+. q_ ) 8
To obtain this result) we repeatedly inserted a summation of fermion helicities to combine products of spinor expressions. This trick was already introduced in Section 3.3 (see also Appendix A). Similar manipulations can be performed on the first term in the expression for M( +, -; +, -, +), and we find
M(+,-;+,-,+) =
. 4e'I 'N
=
S
Q
+ e i." Slv p1 " 1 (88') ,
'
(4.25)
With the definitions (-1.1 7), (4.18) and (4 .19) of Np, N, and e'·, it is easily verified t hat (4.26) SiNq + e i¢ s N p - - _i( 8 'Vq - Vp )' 1 l
I' -
39
P...
p-
". ~ (p+ . k) - (p_. k) .
(4.27)
'I'll us,
(4.28) T he evalua.tion of thc rcmaining helidty amplitudes ('an be done in a "lJwpletely analogous way. They all have a similar structure, and we refer I,lie reader to Chapter 9 for their explicit expressions. All nonvanishing helic.ity amplit udes exhibit a factoriza tion property, one r~u: lor being Ii Nq + e:;';'~ sNp , whose squared absolute value is proportional to (IIq - tip? It is amusing to !lote that formally t his factor r;oinr;ides, in the 1 1Ia.~sless limit , with the 'infrOl.red fador' obt aincd by Yennie, F'l"allt~chi and SIIHra [25], who considered the limit k _ 0 of bremsstrahlung cross sections. til our ca~e, however, the expression (v q - vp)'J is not to be evaluated at k ;; 0 sim:c Wi;! a:-e dealing with hard photon bremsstrahlung. Th:s factorization properly is fouud to bold for all single b rem~strahl\lng processes. The C
4.4
The process e+e-
--+
e+e-i
I·'or the process of raruative Bhabha scattering 112],
(4.29) we now have eight Fcynman diagrams (see Fig. Cl). The fitllt four are the
sa.me as in t he muon case, All , ... , M~, a.nd the arlditiCJ.IJall-channel diagrams iHC
given by
"
40
4. .I'IN(!/,II [}/tIIM.I'S'I'IIItIiLUNG IN Q},'D
,(k)
Fig, 4,3: Feyoman diagrams for e+e-
->
o+e-"
e:1
A-+ ;.
Mr = Tu(q-) ,l2(q_, k) 1"Il(P_)v(p+)r"v(qo.) , Ma =
::
u(q-h.lJ(p-)V(P+h"~d:~k;
jv(q+) ,
(4.30)
The relative minus sign between the first set of cliagrams, Mlo .. " M~, and tiL() second set, Ails , .. " M a, is introduced to satisfy the reqllirerncnts of Fenni MI atistics, III order to deal wit.h these additional diagrams, we introduce two new polarization vc(.:tors [24]
(4.31 )
which are agaill related to one another through a phase fador
(432) to eqn (4.19), Let MB(A" ),~; ),3, ,\., ).~) denote the heJicity amplitudes for this proCc.;SS, where ),11 '\2, '\3, X, and ),5 are respectively the helicities of the incoming: eo', lhc inmming e-, the outgoing e+, the outgoing e- and the photon . This time, however, M B ( +, +; +, +, +) is no longer zero , but picks up coni.ributions from fl1r and Ms:
.~imilar
,3
.
-2t,N+e·(>J.+x)u(q_h,,( 1 +1's)u(p_)v(p+hP(,q... + );) P+(1-1'5)V(fJ+) =
_e3e~'" [~- + eix J~; J u(p+)( 1 + 1s)U(P_ )u(q_) p_ ,'+ ( 1 -
1's)v( q+) (433)
where e'>J. is the phase factor relating
to: to £!"
i.e"
r
42 , ',iVI
=
- (,-
,-'
,
SIN(II,g IIir.FMS,I"I'I/AIII.IING IN QED
(+ ) "
The calculation for A1B( +, +; +, +, - ), with contributions from A15 a.nd M. only, is completely ana.logous , and MB ( -, -; -, -, ±) can be obtai'ncd by a parity conjugat ion. The contribution from the .-channel diagrams to lV1B ( +, -; + , -, +) has already been calculated in the 1'+ ,,- -r case, and combining with the additional cont ribu tio ns from M, and M. leads to
We want to rewrite the last spinor expression in fof8( +, - ;+, -, +) ~o that it can be combined with the first term. To this end, note tbat
and, similarly, 1
u(q-)h(l - -r,)1J(Q+) = 25 U:(q_)i>+(1 - 1,)U(P-)u(p_)P+(l-1s)V(Q+), (4,37) Combining these two results, we obtain
1
= - 2,Tr[p_ P+ 11+ 11-(1-1. )1 ss
and
,. 1\411 (+ . - ; +,- , + )
""'Xl, we ..... ant t o sim pl ify t hl' e xpressioll within lhe curly bracket . T his is ,lulu: uy sho ..... ing that
/j'N9 +e'·sNI' .J..e'·(t'N_ +ei)' tN. ) = O.
(1.10)
lu.l(rn, Wilh the dt::5 uiliol!s of N~, N" N*I~ ' X and ~') [eqns (4.17 ) (4 .19),
(Cli). (4. 32) and (4.34)1. we have
,·'i'(t'N_ +ei).tN+)
- - 4(:'-" k) {T
" -(I
+ ,,))
+'I'ip- t Po Ii, t R- ,,- Ii- A. 'C (I- ",)) } " '(p, . k)(" . k)
-N, { " -
T
(U 1) III deriving this result, we have made usc of the 'cut-and-pilSle' technique, wh:ch a:lowlI one \..0 lin; le a prodllc~ of tra.(cs involving (1 ± 15) into 1'1. single Lrl:lce by inserting p<>la rizat ion sums iu : he appropriate places. This trick was ;~lrt!.1 d y used in Section 3.3 iu t he study of e+e - --I "''''T. ImL its a.pplicability i~
quite geaeral. For more details 10 Appendix A, Ii lhen folJows thAt At 8 ( __ .+ _ +)= • , , ,
abo~t
this tecilniq!.!e, we refer the reader
~" I"N,~ +''',N'{lT>!,,_ p, A. , 2u' A-(l-1,))} xtr(q_) p+ ( 1 - 1$)U(p_)'iilp+)
A-(l '-
·f~) I!tq t-)/.u' . (4.42)
,
,f 4. .\'INI;t.r; IJIlIJMSS'/'ItAIIWNG IN QIW
11
/
and we recover the ractorir.n.i.ion propert.y for Hie helicity amplitlld~, one factor being .'N,+e'.sN., jusl as in lhe 1'+ Ie "I case. To oblaill the absolute value of this amplitude. we only have to observe that
I Tr [p- h .4+ .4-(1 - "1.)]1' h .4+ .4-(J - "1.)1
Tr [p-
4site
Tr [.4-
11+ p+ p-{1 + "(5)1
(4.43)
1
hence,
( ') }
. 3 Uti M8 (+. - ..+,-. +)=4e IsN,+ei~ s!l.r lu ss'tt' I
and
IM8(+ , -; +, - ,+)1 ' =
(4.44)
,
-2e (v, - ".) u -uu- . ss'tt' 6
This result is written in terms of the four-vectors
2 2
Vq
a.nd
vp
(4.45 ) introduced in
eqns (4.27). All the remaining helicity amplitudp.s can be computed ill the same way a.nd are listed in Chapter 9. Of course, this calculation was somewhat more in volved than in the e+e- -+ t'+ }J-1' case, but one should not forget that we had to treat twice as many F'eynman diagrams. Still, because of the fact that all helicity amplitudes factorize with a. common factor) we are able to obta.in a simple expression for the squared matri x dement
IM I'
= -eO( v, - vrflss' (s' + s") + it'({' + t") + "It'( u' + It")]1 ss' it'. (4.4.6)
Again, we note the absence of double pole tenns of the type in this expression.
4;5
05 - 1 ,
t-l, etc.,
Inclusion of Z-exchange
One of the features of the standard mode! is that, whenever a photon can be exchanged between charged fermions , the exchange of the neutral hea.vy boson Z also t akes place. The importance of Z-exchaDge was first observed at PETRA through the charge asymmetry measurement in e+e- -> /1+1'- [26]. [t can a lso be observed in high-precision meas urements.of Bhabha scattering (e+ e- -> e+e-). With increasing energy. these effects are expected to become even more important. \\'e thus want LO know the cross section formulae for e+e - -+ jl+/j- ~(and e+e- -~ e+e- "'f including Z-exchange.
I.~.
lfiGWSION ()f X· I'.·,H://A NU}o,'
Lel Lill: oollplillg of Lhe Z to the Icplolls 1)(: of the vector and axial vector I,VII", i.I:., a vertex gi\'en by (4.47) III t' :l'In~ of tl,e couplin.g co n~ taoLs v~ and {Ie of cqnl! (2.24), we 1,t1.vc
b =
(v,·-Il.)j8sin8wcos8w.
(4.48)
The propagator for a massive spin-1 particle il> gi\'en by (see Table 2.2) .910'0' - k"k"jml - , k2 _ m1 + if '
(4.49)
1>111. , for our appEcations, we can drop the k .. k~ term . T his is due to the fad Lhat it is always cOlltradcd with a. j".matrix on a fermiOIl line, where it f,iv'-'S lI. vanishing contribution ill the massless fermion case. To incorporate t ! u ~ d£cds of ~he finite width of the Z, we modify the Z·vropagator to read -ig.... P - M} + iMzr' wh~re
(1.50)
r
is the Cu!l width of the Z . For the case of e+e- ..... p+p- 'l, we have the four QED Feynman ampli· tildes M I ,M1 ,M3 and M 4 , already list.e
z~
-
,
,
$-
M~z e+ 1' M t r(i(q_h.. :a( 1 -1~)+b(I+15J:v(9i) X'(p_) J =z~,+
0,"[0(1 -,.)+
-c'+'1 I' V(P+h"[o(l w' :J-Izt·Z xU(,_)
,,) + 0(1
J21,~. ~) ,,10(1 -
0(1
+ o,)[u(p_),
+ ,,)Ju(p_) ,,) + btl
+,,)J.,(,,),
46
z.
_c J
s-
M'
,
. W [' ii(p+h"la( 1- ,),.)
z + t}
z
_
Xu(q_h"la(1
-')'5)
+ b(1 + I.)JU(P-)
- /1,- /< + b(1 + ')',)J 2(q+ . k) jv(q+).
(4.51)
It is dea.r that) if we make a specific cho ice for the fermion helicities, on ly one of the two terms in the Z·fermion vertex can contribute, either the a-tenu or the b-term. But theu, the applica.tion of the helicity amplitude method can be carried through just as in the pure QED case, the only modification being an overall factor in the matrix element. For example)
M(+ , -;+ , - ,+) == 4e'ls'N,A(a',s) +ei·sNpA(a',s')lu/(ss')} ,
(4 .52)
where N p , N, and 1> are defined by eqns (4.17) , (4.18) and (4.19) and
4xy A(x,y) = 1 + y - M"z T't'MZ r'
(4.53)
Clea.rly, when M z -> co, A(x,V) -> 1, and we recover the pure QED result. As the quantity _4(x,y) is complex, we need to know Im[ss'NpNqe- iO ] as well as Reiss' NpN,e-il>J for the calculation of the cross section. It is given by
(4.54) wbere we introduced the nolation
«0123 =
+1).
(4.55)
In exactly the sa.me way, one can trcat the inclusion of Z-exchange in the process e+e- -+ e+e-," For both cases , the helicity amplitudes and the cross sections can be found;n Chapler 9. The cross sectio n formulae , however: are
rather lengthy, whereas the helicity amplitudes are rat.her simple. If one wants to evaluate num<.".rically t.he cross section in a. gi Yen point of phase
space, it might therefore be more advantageous to evaluate separately the various helicity amplitudes as complex numbers alld to add their squared amplitudes, mther t.han c.omputing IMI' directly.
5 Single bremsstrahlung in QeD Color is a complication fo r gluon radiation: hs nicest implictl.tion is jet fragmentation.
, .1
.)
Good to know
want to a.pply ~he helidty amplitcde method for the calculation "r QeD processes. Compared to QED ca.lcuiations, there are t.hree major IIOIIl'CCS of complications.: WI'
DOW
i) There is a complication due to the appearance in the Feynman diagrams of color matrices or SUP) structure constants. We shall see that a. good choice for the polarization vectors of the gluons has the net result that une or more Feynman diagrams can be made to vanish . In this way, the helicity amplitude method reduces the amount of color algebra one has to perform Lo obtain cross section formulae. ii) As slated in Section 3.2, we have to use po!arizlltion VectOTllOf the type
for gluons with four-momentum k. If p and q are four-momenta of external ferm:olls, this implies that at most one of the terms can give "vanishing contribution by virtue of the Dirae eqmltioll . T his is because in this example p always appears in t he middle. As a consequence, we have fewer simplifications compared to the QED case. iii) T h ree- and four-gluon verticescan be presen ~ i:1. the Feymnan diagrams, generating longer exprcs;;ioJls. \Vhen this is the case, it is often useful to choose the fom-mo:neuta of other gluons in the process to pJay the role of k and II in f.. Suppose, for example, that we have three external gluons with momenta kJ, k2 and 1.3 • wit h helicities +, - and -. Suppose, furthermore, that we choose
J+(kd -
N[):l
h J3(l + 15)+ ):3
f:2 ,kl(1- "15)],
48
,j,
SIN(JJ,/i IIII.IiM,\'Sl'IIAIIWNG IN (lCI)
As giUOlls I alld:1 hn.V(!OPl)OHiU! 11(~licil.ic~J we ea.1I indcc(} t.ake
j+(k J )
=
,l-(k,,), It follows tllnt (5,3) which can a.lso be derived using the explicit expressions, viz.,
At the same time, we have
As a. result 1 we can omit any Fcynma.n diagram with a. four-gluon veItex where these three gluons appear as external ones) provided of course we are interested in the above mentioned helicity configuration . .Every t.erm in the four-gluon vertex conta.ins a scalar product of these gluon polarization vecto rs and thus the whole four-giuon vertex va.nishes.
In the following sections, we shall illustrate the use of the helicity amplitude method for QeD processes for two simple examples: q 1j -> g g and e+e- -> 3 jets, We hope to show that, at least for the simpler processes, QeD is not much more difficult than QED.
5.2
The process q q --> 9 9
For the process of quark-antiquark annihilation into
1..\\'0
gluons,
(5,6) we have to consider, in lowest order, t.he three Feynman diagrams of Fig. 5.1.
Here, i and j denot.e the color degrees of freedom for the antiquark and the quark, while a and b denote those for the gillon" As already discussed in Section 2,1, i and j take the values 1,2,3, while a and b run from 1 to 8, With the Feynman rules of Section 2.1 , we can write down the corre-
sponding amplitudes .,_(
) '(
-9 v P+
F
)
!
,p-- /<1
k, T"_2(p _ . k l )
'-( ) uk JT" p-- /<, -g v p+ ,<\ 1 ik -2(p_ ' k,) M,
-ig'v(p+ )T~'Y"u(p_ ) (k,
_: k,),f"b [2<,,(kd (k, ,«k,))
,,!. '1"111; I' /I ()( :I-:s,r; 'i q -
"(,, +-. i)
9g
g(k,. b)
ii(p .... i)
p_ -k,
g(k"b)
Fig. 5.\: Feynma.l1 diagrams for q q ..... 9 g. wlH:re we used t he rdations
(5.8) We deflote the helicity amplitudes by M(J.. .... J.._; AI. A2) with A+I L, AI loud Az the helicitics o!q, g, g(k l ) and g(k~). We immediately find the familiar i"C">Cull that, ro~ all AI and '\1,
(5.9) I :onsidef next M(+,-;
+, +). And
let us choose
l
. = I ,_. ?
(5.10)
of the ferm :oo hclici~ies, ooiy the ~ecoDd term in f'"( k;) contributes, :lIld Olle sees direclly tha.t M I and M1 give vanishing cont.ributions. With Lhis choice, we also have Ilcca.lI~
(5.11 )
.?, HIN(,'/,I: IIlli;/,{ss'I'UAIIWNG IN QeD •
I hence, lvfJ does Hot contribute cit.hcr. COIJ~cquc;nf]y,
,11(+,-;+,+) = 0,
(5.12)
By in.terchanging p+ and p_ in Ir(k,.), the same considerations lead to
J1;[(-,+;+,+) =0.
(.5.13)
A parity conjugation then tells us that also
M(+,-;-,-)=O,
M(-,+;-,-) = O.
(5.14)
Jus t as in the 8+e - --> "1"1 case, we find that the heHcities of the gauge particles must be opposite. For M( +, -; +, -), we choose
N,[A
p- ):,(1 + "1,)+ ):, p- 1>+(1 -
"IsJl.
(5.15)
With the usual notation
we have
·'(,+(k,).
e lk,))
(k, . ,+(kd)
(kl
~N,Tr [ ,.(;,
,,-(k,)) N,
-
A p-
,.(;,(1
+ "1,)]
N, = (2stuJ-t. (5 .17)
With this choice of I(k,), we find t hat M, does not contribute, and
111, = -2g
,N,N, . t (T'T")'j v(p+)
g'
- -(T'T");j v(p+) }" 81.1
/<, p- P. (p- - /<, )
p_ p+ (I -
'Ys)u(p_)
L~.
l'!lH
J'!W(.'{~SS'iq
'I
- gy
(5.18)
w,\ find
tlat 1141 and M3 a.re proportional to the sa.me spinor expression, aud
~
"'; (~')''[,(TOr);j +u(T'TO),j]
For the summation over the color degrees of freedom , we have :hal , using eqns (2.14)'
I;'l'r (1"'f"T"T')
-
~ I;T,('f'T')
-
3'
I;T' ('J"'f'T')f"k
-
-~ Dr")'
-
-61 ,
i Dr")' .\,
-
12,
••
3 •
.k
(5.19) ~o
know
16
(5.20)
0\'
L rbc'l'r (T~Td)f"M .k'
~
l\·lore relations for traces of products of color matrices can be found in i\ ppenciix C. Substitution into eqn (5.19) givt'_\
IM(+,-;+,-)I'
.,(,-9 + -+,"' ")
~ 48g -
u
4-R9~
sZ
(4t9u _~) . ~2
The expression for IM (+, -i -, +W is derived from the aoove olle by (;h{l.nging J ;..;. 2, which amounts to t +-+ u. Thus,
[M(+ , -', - , ~)I .
11
~
48g
(' ') t/.U
-9t - -III
.
(5.21 ) inte~
(5.22)
,j,
8111(.'U:
IJlU~MH!i 'I'Jt AIIWNG
IN (lCI)
,
Adding tbe pariLy (:Olljuga.te r.oTl t rjb\ll,i oJl~ aud averaging oV(~r the initial degrees of frc.;cdotTl , we finally outa.in
-8 4(t'+U') (4- - -1) IMI'=_g 9tu
3
(5,23)
"
The conclus1on from this exercise is that, apart from the complicat.ions due to the color algebra, the ent.ire calculation for q q -> 9 9 is very similar to
the calculation for the e+e- ->" case. One only has to keep in mind that the chosen representation of j.. in the gluoD case is different from the one in the photon case, where we could drop the 1<7, term.
5.3
The process e+ e-
-->
3 jets
Within the framework of perturbative QeD, the process e+edescribed by the subprocess
)0
->
3 jets is
lowest order, , Lhe Feynman diagrams a.re given by Fig. 5.2, and, corre#
Bpoodingly,
M, M,
-ee,g 0( )~I' (. ) -( ) '(k)T" -4-+ Jo . ( ) 2(p+' p-l v p+ , u p- uq- I;j2(q_. k) 1"v q+ , -ee 9 - Ii+- Jo ?( .' ) v(p+h"u(p-l u(q-h. ')( , k) I(k)T;'ju(
These expressions are very
simi1?~r
to the Feynman amplitudes 1\13 and A14
,.,+,.,-,
for the QED process e+e- -> [see eqns (4.16)]. We have again that the e+ and e- heJicities must be opposite as well as the q and q helieities. Let UB therefore proceed with tbe calculation of l'l1( +, - j +) -, +), where the order of Lhe helicities is given by e+, e- , q, qJ g, In analogy with the QED case, we take
1+ (k)
Nil -4+ 04-(1 +,,)+ 04- ,4+ Jo(1 - ,),,)}, (5 .26)
Because of the quark helicity projection operators, only the second term in f+(k) can contribute. This leads immediately to a vanishing contribution from
Ml . Th us,
g(k,a)
g(k,a)
Fig. 5.2: Feynman diagrams for e+e- - . QQ9.
".9 N . k) 2-'i jV-(p+ ) ") .( 1 - "YsU(l'_ ) ) H(+,-i+,-,+)= 8(p+ ,p_)(q+
x"(,_h,(.4++ ,10).4- .4- ,10(1 - ,,).(,+)
With the cut-and-ptl.Sl.c tCc!lIJique of Appendix A, we eJirr.i nale the reIwaled inot!x /l as follows:
-(
) '(I
P+ I
) (
)u(p_) fi(l-1')U(,-)
- "Y5 U p- -u (P_}-" ) "(1 _ IS ) \I (q_ )
-
1)
_
1 O(p+)1"
_
-8 ii(P.) /!-
p- ft .4-r,(h+ I 1<) A-(I-1,)u(q,) u(p_) ,1(1 - 1.)U(, _)
fi .P- IA++ ,k) A- (I - 1,)U(,+) u(p-) fi(I-1,)U(,-)
(5.28)
, s, .I'lNcne IJII.HMSS7'/(AII1.UNG IN Qr:I! , ,I
Choosing a ~ 1'+ and using momentum conservation; we have
A =
Thus, ,
M(+,-;+,-,+) - 8ee,gNT,j(11+ -
,q-)
U;: :;=il' 1
2ee,gT;')(p+ , q_)[(p+' p_){q+' k)(q_'
kW t , (5,30)
and, slImming over the co lor degrees of freedom, 2
IM(+,-;+,-,+)!' = 8(ee,g)' .(q+: ~(q_, k)'
(5,31)
w here 1 as usual,
•
(p+ -I- p_)',
u i'
_
(p+ _ 9-)',
(p- - q-)',
(5.32) For the amplitude M( -, +; +, -, +), we have with the same choice for ,l+: M( -,-1-;+, -
,+ ) -
ee,gN -(P+ )"( 2s T"'jV "I 1 +,)', )u (p- )
This time, we have to sjmplify the expression
_ v(p+h"( l
+ ')',)u(p_) u(p_)(1 -,')',)u(q_) u(p,. )(1 - ')',)"(9_)
xu(q-b"C4++ f) ,4_(1- ,,)v(q+l
_
1
"(".Jr'
j,- A-o,.(h.+ ,k) A-(1- O,)v(q,) u(p_)(l
_ 16( _
l'~)u(q_)
. ) "(p,) p- h-(1-o,)v(q,} p- 9II(p_ )(l - "fj)u(q_)
8t'(st;')~,
(5.34)
.'lidding
(5.35) I ',~ rronniug Lee coJor !lummalion, using eqns (2.14), we
nnd
(5.36) which wrre5ponds to the substitution p+ ...... p_ in IM(
For the
Mlpli~udes
with
oppo~ile
+, -; +, -, +)12.
quark hdicitictl, it is better to take (5.37)
Nuw, only the first term in j+ can contribute, and, beca.use of the Dirac "'Illation, M~ is made 10 vanish. The explicit calculation
leads to the simple formula (5.38) for the spin averaged squared matrix element. This formula is very similar ttl the one for e';' e- ....... P+I'--Y. if we consider radiation from t he muon lines
.. nly.
5.4
The process q q'
-+
q q' 9
(n this sL'
q(p,j)
+ q'(q, 11) _
QeD bremsstrahlung process [27], viz.,
q(P. i)
+ q'(q', m) + .q(k, (I),
(5.39)
as this case presents a. complication whiclJ was not encountered in the previ· ous examples. Here, q and q' denote quarks witb different flavors.
, ,1 . .~/N(,'J,f,' IiJtr;M,S.I'1'/(A//I,IJNG IN QCl!
56
The Fcynman M,
dil\g l'alllR
g3 _( ') -V tl I'
=
-
arc
!:IhoWIi
in Fig. 5.3; Aud the ampliLudes
r~n.d
iT""2(1'" ji+ fo "1'~ ( )-( ') Tb () k) '"I ki" P U q "!,, mn" q ,
F
~: jj(p'h"T,~ _~; ~)
l-1k'ju(p) jj(q'h,,1~. ,,(q) ,
g' - (. ') "Tb ( ) -( ') JT" A' + fo Tb ( ) - -T" P" 'j" P u q F ~"'2«1" k)"" .n" q ,
.
M,
-
,
It~, jj(p'h"Tt;u(p)jj(q'h"l~~",,(q)f'''
x [(p - p' - q + q;. <)9""
+ (q -
q' + k)"<,,
+ (p' -
k - p)"f,,], (5.40)
where, .this time, we defined
s
s'
(p+q )', (p'
+ q')' ,
t
t'
-
(p-p')',
"
,,'
(q-q')',
-
(p _ 'I')' , (5.41 ) (p' _ q)' .
As all the lines in the Feynman djagrams carry color, there is no gauge invari"nt subset of diagrams. Yet, for the application of the helicity amplitude formalism, we would like to use a polaTization vector for the gluon expressed in terms of the momenta. p and pi for .1111 and M2 ) a.nd a simjlar expression in terms of q and q' fOT M, and But how should one treat M, in this case? We shan show that tbere exists a natural way of separating M, into two contributions which can be added to Ml M, and to M, + M" yielding gauge invariant combinations, for wruch we can simply apply our technique. To this end, observe tbat the general three-gluon vertex (see Table 2.3)
"'1,.
+
(5.42) can be rewritten as
V".p(p, q, k) = ~ {Tr( .1>'"1" h., '"I,,]) + Tr( A".l"/" 7,,])
+ TrU,pl"/", "!,])}
.
(5.43) Because of the antisymmetry in (II, v, p, (1) of TT["!""."!."I."!,] and because of momeniwn conserva.tion aL the vertex 1 we have
(5.44)
57
g(k,a)
.(k,a)
,(p,j)
,(p', i)
,(p,j)
q(p',i)
,'(q,n)
q'(q',m)
q'("n)
q'(q',m)
q(p,j)
q(p', i)
,(p,j)
q'(q,n)
q'(q',m)
q'(4/ ,n)
q'(q',m)
~~--L...
.(k,a) q(p,j)
.(k,a) q(P', i)
""""""''''' g( k, a)
q'(" n)
__
q'(q', m)
Fig. 5.3: Feynman diagrams for q q'
--+
qq' g.
/ .1. SINClU,. 1III.1I M$S'I'II.AIII.IJN(; IN (Jli O ,,
fiB
Consequently, we l:i.l.1I rewrite tile Lhrcc~glllo!l vcrtex ill the forlll
and we are free to choose the sign in front of the 'Y.-matrix. The amplitude M, then becomes . 3
M,
= -
~~l' ii(p'h"T;ju(p)u(q'hVT~nu(q)foba
x ITr[(p- iih"h" 1](1 ± '(5)] + Tr[(,4- A'hv[j,'Y.](l ± 'Y,)] (5.46)
«
We denote by M(Al, A,; A3, A" A.) the helicity amplitudes for q q' -+ q g, where the helicities correspond respectively to those of q(p), q(p' ), q'(q), q'(<() and g(k). Conservation of helicity along the quark lines tells us that the belicity amplitudes vanish, unless Al = '\2 and '\3 = ).,. Suppose we calculate first M(-,-;-,-,+). For j+(k), we take all expression of the type I+(k) = N[): ii' ,4"(1 +'Y')+ A" Ii' ):(1 - 'Y5)], (5.47) where N is a. normalization factor and pfl and q" are tW9 distinct arbib·a.ry four-vectors. Clearly, ): ,l+(k)(l + 'Y5) = (5.48)
o.
Therefore, if we choose the
+ sign in front of the ')'s-mairix in the expression
_~f"
we find that its last term vanishes and that M. naturally separates into two terms. Its contribution to J\1( -, - ; -, -, +) then becomes
for
M,
"~It)
":Vis
+ M(2) 5,
- -;;w . 3
u(p'h"T;j(1 - 'Y,)u(p)u(q'hUT~n(1 - 'Y.)u(q)J""
xTr{(,4- ,4'hv[,i+,'Y.](1
+ 'Y,)l ·
(5 .49)
Clearly, M~') goes together with M, and M z , whereas M~2) should be bined with M3 and M,.
CO In-
".4-
'I'll;: I'll ()C /<;S5' q 1'('
"
q q' !I
--
Using rormu la (A.l1) of Appendix A
W(~ can rewrite the _MJI) contribution ill a form suited for its combination with Ml and M, ; . 3
- ~;"U(P')h.. ;+1
- ;t,
PTij(I-1,)u(P)u(q'h"T!.(1 -,·,)u(,)/'"
IT', T'I;;T! •• (P' )l1", rIP'( 1 - 1,)'(P) U( q'h"(i - ,. )u( q)
,
- - :',P'7");;1'i",1I(p'h"
r JI(i -1,)u(P)u(q'J-,"(1-1,)U(,)
,
+:'"(T'7"
);;'I-:'.;;(p'h,, j+ f!(I--,.)uip)u(q'h"(I-,,).(,)
3
- -SIl'i:,. k) (T'1");;T~."(P'h, "
Ii
j+ />'(1 - , , )u(p)
X"(q'h'{ 1 -1·, ).(,)
+:':,(T'T·);jT!.u(p'h, r
- -:.:, Irr·':i
jill - ,,)"(p)u(,'h"(1 -1,)u(, )
(T"'I"),;T!.ulp'h, " ,p'{ l - ,,)u{p)
xu{q'h"(l -
i~)U(q )
3
+:',,{T'T'),;T!.u(p'h, r
1/(1 - ,,)ulp)ulq'h'P -1,)U(, ) . (5.51)
M(-. -;- . - .+ ). In order to combi ne it wiLh Mil) we introd uce a factor t in ~he n:.lmerator and t he dellominMor. We "Iso make use of the relation", ,l+(l + is) = 0 and oh~aill We now re write the contribution of 1.1 ] to I
!
,5. SIN!.'I,I'; IJlIp,Iyf,~.9TnAlnUNO I N QOD
all
g" 'I" -( l'') /+ "'l k) "/ "~"(1 M I = 4'!l' ( .'/" 1") ij""w" . 2"(p'. '" ,I
=
~:,(T'T')ij1':"" ([~'€:;u(p'h" p p'(1 -
-
) (P)
,,/, 'U
'Y,)u(p)i7(q'h,,(1 - ,,/,)u(q) , (5,52)
It foJlows that 3
('
+)
~1,(Ta1")ii1'~", [P"€k) u(p'h"(p- ,k) fi(l - ')',)u(p) xi7(q'h,,(l -')',)II(q) 3
+~t,(T'T')ijT:""u(p'h"
r
p'( 1 - "Ys)u(p)
xu(q'h,,(l - "/5)U(q) , Next) we rewrite the contribution from with the manipulations on M I , We find
1\12
(5.53)
to l. l( -\ -; -, -, +) in analogy
M, = - :;,(1"1"li)T:". u(p'h" _~~ ~) ,l+(I - ,,/,)u(p) xi7(q'h,,(l -')',)u(q)
xu(q'h,,(l - ,5)U(q) -
-
~:,(TbT')ijT!n (~":l) u(p'h"(p- ftl fi(l- ,,/,)tt(p) xu(q'h,,(l - "(,),,(q)
-
~:,(1'·T')ij1':". u(p'h" 2(: k) I' r
fi(1 - ,,)u(p) xu(q'h,,(l - ,,/,)u(q)
• GI
XlJ(q'h,(1 -1~)1I(q)
xu(q')j,,(l - i$)U(q).
(5.54)
Xfi(P'h'(R- /<) il(l - ,,)u(p)u(q'h,(1 - ,,)u(,). (5.55) With the cut-and- paljlt: kdmique of Appendix A, we ca.n rewrite the spinor 1~~pres5ioTi as follows:
A :
.(p'h'(/>-)o) Ji(l - ",)u(p)u(q'h,(1 -"~lui,)
~ -(p') '(/>_ )0) il(l _ "1
u
"15
) ( ) li(p) p(1-,,)u(q') U P rr(p) ft( I I~)u(q') x,,(,')o,(1 - 7,)U(,)
4ii(P'h'(/>- I
~ 2(p. oll,'. 0)
1<) il /> ,I A"-II '-,,)u(q)ii(q') ,1(1 -,,)u(p) 1<[,
uW) A' fi. "Jf(p-
1<)(1 -,.)ul,)
x"lq') P( I -,,)u(p) . Choosing a
,t and
=:
using
rnomell~UIO
(~.56)
conservation, we hAve
A = -2.(,1) A'(I - ",)u(,) ii(,') il(1 - ,,)u(p).
15.57)
Thus, Mo+ ,14, +
Ml" ~
t" [(~·.':i
("[OT· )uT!.. -
(~ .. ':} IT'T');,T!..]
xn(p') A'(I -,.)u(q)n(q') fill - ;,)u(».
(5.58)
,j .
.... INU/./O:
IIUf,'MSS';'Ui\/IWNG I N
•
•
Simii}J.r flltttliplJlaLiol1s 011 M:I + IVf1 + MJ'l) give all ' ~na.log()t1s result, and, for the hclicity amplitude itself, We find the r.ther nice resu lt M( - , ._'I - , - , +) = ,
g3 2tt'
[ ( p.e; +) r b a
(
p'
~
k) (T T )ijT",,, -
(' +) ob p - c, II ( k) (T T )ijT",.
rJ .
'(T'·) (q"'+)T'(·~) 1 +(q.,+) (q'k)T'j T "",- (q' .k) ii TT m.
xii(I") ,4'(1- ,,(s)u(q)u(q') j/(l- "(,)u(p) -
2
3
9
[(P·<+)(TbT.) T' (p . k) ij mn
-
(P'·'+)(T·T b ) (p'. k)
'r'rn"
;j
(q"(+)T"( "Tb) 1 .' + (q"+)T'(T'Ta (q. k). if )m. - ( q' . k) ij T. mn (tt') ,l.
.
(5.59) We find that the amplitude factorizes again. The expression within the brackets is a generalization for this QeD process of the 'infrared fact.or\ which we encountered in Chapter 4 all single bremsstrahlung in QED. To proceed with the calculation of the cross section, we have to square this amplitl\de and sum over the color degrees affreedam. Using eqns (2.1 4), this leads among other things to formulae of the type
I: Tr(T'TaT'Ta) Tr(T"T') aI,
t [I: Tr{T"T'T'T') + i I: f"1'Tr(T"T'T") 1 ~b~
all -
_1
(,560)
3'
Thus,
I: IM(-,-;-, -,+)I' color
--
4g6 [(P"C+) (q.,+)' , (q"'+) (p.,+)' "'3 7 (rJ.k) - (q.k) -d (q'k) - (p.k) (p' . ,+) (p' . k)
(P'C+) (p.k)
, _ I(q'.,+) _ (q.,+) .
I
(q'.k)
2
(q·k) >
+2
(p'.,+) (rJ·k)
-
(q'.,+) , (q.,+) (P"+)'] 8 12 T2 (q'.k) (q · k) (p.k) w' 0<
(5 .61 )
Tlu: Ilix {;ollliJiJla.tions jlJ tile bracket ill'l : ~'lpl\ratcJy ga.uge invariant. To evalIIIII.<.! tll<.!Hl , olle (;RII thus use a difrerent represcntation for the gluon polar· j'/':Ltion vector in each combination separately. For example, for the fi rst 1'1): II bi II 11. tion, (p'. (.f) B~ (5 .62) (p'·I) we take
"
- ,vI"; ji
w' -
A(l
+ ,d+ /J ji ";(1 - 1.)J, (.5.63)
41(p'·,)(p'·I)(q·kJII,
wllich yields
(p', k)(p' . ,)]1 [ (,. k)
(5.64)
T llus,
B~
(p"') (5.65) (p" k)(,· I) I''or the second combinat.ion, if is obviously prefe~Rble to express f' in terms or p and q', etc. This way, all six combina Lions are readily evaluated, yielding
.z W( - ,- ;-,-,+)I'
eolor
_
[7
4g'
.,3
-
(p'.,)
(l'" IHq· I)
+7
(p.,') _ (p'P'J._ (p. k)(," k) (p. k·)(l" · k)
(q.q ') (p.q ' ) ]", +2 ~'p' .q') +'l (,. k)(," k) (p" 1)(,," I) (p. 1)(". I) It"
(560) .
All the other heEcity amplitudes call be cakulatecl ill a completely anal· ogous way. The result is simply a replacement in t he above fo rmula of the quantity s' by s, II or u'. To obtain ~he cross section, we add the squared absolute values of all the helicity amplitudes, we average over the initial and we average over the initial color spin st ates, which produces a factor degrees of freedom . which give~ a faclor k. Hence, for qq' ..... qq' g,
!,
29'[
21 +
7(p",)
,7(p·,') (p.p') (,'q') (l'" 1)(,. I) ~ (p. 1)(," k) - (p' k)(l"· k) - (q. I )(q" k)
2(p' . q') 11"· 1)(," k)
+ ,2~(~P~'~'~)",,] (p . 1)(, · k)
which is again a quite simple form1lla.
1 8
+ 81'l. + u l + U'1 tt'
(5,67)
". SIN(,'!,"" IlIINMSS'/ 'I1.AIIWNC I N Cleo
M
5.5
Other QeD processes
The other pure QCD proces,,,s of the t.ype 2 - t 3 particles are qq ~ qqg , qq - t ggy, y,9 -. ggg, and the processes obtained by crossing. They Can all be treated in the way the previous examples were hand led . The general strategy is to choose a representation for the gluon polarizations which annihilates as many Feynman diagrams as possible. In the case 9 g - t 9 9 g, where there are 25 Feynman diagrams, it is
always possible to annihilate the 10 Feynm.. n diagrams involving the four· gluon vertex, thus reducing the algebra considerably. It is amusing to note that, although no fermions are involved in this process, it is computationally advantageous [271 to introduce spinoI' algebra by rewriting the three·gluon vertex as a. trace, as we did in Section 5.4. If one expresses the g luon polariza. tion vectors in terms of the four-momenta of the gluons, one finds that many cancellations occur, ultimately leading to a simple crOSs section formula.
For the explicit formulae for these QCD processes, we refer to Chapter J O.
6 Double bremsstrahlung Wben electrons mdi:tte twice, one mllst realize; The ~pedr\lm looks 80 nice it becomes [un Lo analyse.
G,l
QED
1)'"ILle bremssLr
??,,(,,(, 20 for e+e- -+ l1'+I1--'·n· and 40 fO f e+e- -> e+e- ,,(,,(, lIut counting the Z-cxchallge dia.grarn~ for the lattl:f two processes. rvcn with symbolic manipulation programs, the use of the staudard Feynman techniques is cumbersome, alld, eventually, the resulting fOiIDulae are in <:Ollvenient for the analysis of the experiments due to their lengths. III this lIt.'d ion, we shall illustrate with the example c+l'!;- ..... ,,(,,(,,! , bow the helicity Ilmp!itudc method can be applied to yield more practic«l formulae . The pro c~ of electron-positron annillilation into four photolls [3Il,
is described by the Feynman dia.grams of Fig. 6.1.
Th~ F'~ynmall
amplitude
" + 23 other permutations of (1, 2,3,4).
(6.2)
Ii. }JO II IJ}," IJlU·;M.\·S,/,IlA II UING
61!
e+(p+)
'Y( k,) 23 other
'Y(k3)
+ pennutations 'Y(k,)
of(1,2,3,4}
'Y(k , ) e- (p_)
Fig. 6.1: Feynman diagrams for e+e-
-->
'1'1'1'1 .
The obvious choice for the photon poianzation vectors in this case is t.hat
ofeqns (4.4),
d
Ni[p+
p-
,ki(l::;: '1,)-
/<,
p+ p_(l± '1,)], (6,3)
Let M(,\+, A_; A" A" A3, A,) be the helicity amplitudes, with A+ ()._) the helicity of the positron (electron) and A" i = I, . . . ,4, the helicity of photon i. Clearly, for A+ = A_, the helicity amplitudes vanish. Moreover, for A+ =
-A_, only one of the two terms in j.;; gives a nonvanishing contribution. But, if all photon helicities are equal, either p+ or ;L necessarily appea.rs next to its corresponding spinol' in the expression for the helicity amplitude. By virt.ue of the Dirac equa.tion for massless spinors, this leads again to a. vanishing contribution. \Ve thus have to consider only the case where at leas t
one photon helicity is opposite to another photon helicity. For M(+,-;+,-,-,-), we find .\I ( +, -;
+, -, -, -) = 8e'N,N,N3 N,v(p+) 1'" h iL -,..~(P; ) /<3 p+'
p+
4
p-- /0,- h p-- /01 X p- (p _ _ k, _ k,), /02 p+ p- -2(p_ 'k,) P+ P- /:J(l-,/s)U(p-)
+ 5 other permutations of (2,3,4)
Qt;/J
/i./ .
67
+ 5 other permut ations of (2,3,4)
(6.4)
with
(6.5) Note that all fa ctors of (p+' p_) in ~he denominaLor ha.ve been cancelled, Illl well as the factors of the type (p_ -k, - ~F. By i;I. parity conjugatio:1 and/or hy permuting t.he photoll indices, one can obtain ILl! the (f'mailling helicity aillplitudes for which one photon has thc opposil.e hcJicity of the three other 1)lIol.on8 . ....o r the ampliludes of the type M(+,-;+ ,+,- ,-). the formulac arc ~omewhllt lUore lengthy due t o the fad that the pole t erms of the type (p_ - kl - k3)' do not C()llcel. For example,
M(+.-;+,+,-,-}
+!,'AU(p.) I
" '" p_ ;',(p_luu
I
+"u" '" p-
,k,( p- - h) ,k,
,k.) ;.,
P+
,k.
h h
I
+ !>n ,k, p- ,k.(p- - ,k,) h p+ /<,
+;,. h p- ,k.(p-- ;c,) /<. p.
h) (I-,,)"(p-),
(6.6)
;',;=1, ... ,4.
(6.7)
whtlre
!>;, ~ -2(p_· k;) - 2(p_· k; ) + 2(k,· k;),
To o btain the cross sectioll for e+c- ---> 11"17, wemust know the lfquared absolute value of the helicity amplitudes. For the amplitudes of the type
GH
Ii. 0011/1/,1': Iml!MSS·I'I1;1Il/.(lN(.'
.f\1 (+,
+, + ,-, -),
ICllg1.by trac()s and hanlly any simplifications occur ill the rt)xull. A much more convenient method consists in evaluatil1g the hclici ty amplitudes directly for any given point in - j
tl!i:-l ka.(IH, however, Lo
VeJ'y
phase space as complex functions of the components of the four· vectors in the process. To this end, we introduce explicit representations for the ,-matrkes a.nd the spinors [32]. Suppose we go to the e+e - c.m. frame, with the z-direction along P'+l and that we introduce the notation
k± = ko ± k"
(6.8)
for any vector kp- With the representation (2.6) for the "I-matrices, we have
(6.9) Then, we can take for any light-like vector k,
(6.10)
o o
-,j'Le-;,p·
,ff;.
)
.
(6.11)
Clea.rly, lhe first spinor is an eigenstate of I +"1., and the second one of 1-"15. With these formulae, it is easy to see that
(6.12) where
(6.13) Similarly,
u(k)(I - "Is)u(p-l
u(k )( l
+ "I.)u(p-)
u(p_ )(1
+7.)U( k)
u(p_)(l -7.)u(k)
2J2Ek+,
(6.14 )
li. / . QI;V
"
wJu:rc E = l~o = P-o is the bC/l.TU energy. With thEu;~t-alld-pru;te ter.r.nique dcsCI"ibcd ill Apptmdix A, we can redlln~ Itny spinor expression Lo a product of simple expressions of the above type. All hclcity amplitudes are thus expressed as (:omplc'X functions of the <"l>I1LpO:1ents of ki' i = 1, .. . ,4. III terms of these variahlc~, the helicity amplitudes bcwme
(6.15) wilh
16.16)
F(l, 2, 3, 4) = (A· u . + kn)~t2
Oy adding lht: sqcared absolute values of all helklty amplitudes 1i:ld by ;~vc[",ging over the initial spin::!, we obtain the unpolarized sqt:ared matrir. dement:
1M12 _
2e8n2{4~(k~_+kl_)J'i+k;_ +E-' [IF(I, 2, 3, 4)1 ' + IFII, 3, 2, 4)1' + WII, 4, 2,3)1' +WI2,3,1,4)1' + WI2,4, 1,3)1' + IFI:!, 4, 1,2)1']} . (6.18)
With this equntiou, it. becomes straightforward to evaluate the cross ~ec I.lon Ilumerically for any given point in phase space provided one is wiili:lg Lo introdcce complex arithmetic. In this Wily, one can also obta.i n the hclkity amplitude.s and the CtOgS lIcction formula.e for e+e- -- 11+ }, - "(1 a.nd c+e- ---) f.+e-11, which ate listed ill Chapt.er 9.
I;, /)()UII/,b'IJUf;MSS'I'R.AIILUNG
711
The study of four-jet events in e+ e- -annihilation [331 allow, one to make additional comparisons with perturbative QCD, Acc.ording to this model, one has to know the cross sections for the subprocesses e+e- - t qqgg and ,+e- -+ qijqij in lowest order. They were calculated by various authors [34) using the standard Feynman techniques: but, in this section, we want to show that the heUcity amplitude method call be used to obtain more practical formulae [or these processes. Consider first the double bremsstrahlung process [:12) (6.19) described by the Feynmall diagrams of Fig. 6.2, which yield the Feynman amplitudes
M, =
M,
t
);,
'<'2(k,. k,) '<' (k,
'j U,
+ fc-,+ /<, •
+ k, + k.)2"1
/<,- /<,- /<3 - /<,- /<J --';--CT"T')'ju(k.. h" (k, + k, + kJ)' I, 2(k,. k3) ee g'
MJ -
/<,+ );,
ee,g' (T'T") -(k) ' t ---:;-
_
(k)
11
'J
-
,l,V(k3)
xv(p+ h.,,(P-) , «,g' (T'T"j;-u(k,)-r.- /<'- /<,- );J ,l, - /<1-:: ,kJ I,u(k,) s ' (k, + k, + k3l' 2(k, . k3)
xv(p+ h.u(p- ) , Ms
ee,g'(
b ") _(
---:;- T T
'j"
)
l
fc-,+ ,k• • - /<,- );3
J
(
k, '<'2(k,. k.) "I 2(k, . kJ) ,< ,v k3)
xv(p+ h.u(p-l, ee,g' " b _ );, + /<, "- fc-,- /;3 -,-(T T )iju(k,) I'2(k, ' k,) "I 2(k,. kJ) ,l,u(k3)
xv(p+h."(p_) ,
71
,(k"j)
g(k"b)
g(k"a)
g(k"b) ,-(p_)
,(ko,i)
,(ko,i)
,(k"j)
g (k \,0, )
g(k"b)
,(k"j)
,(k"j)
g(k"a)
g(k" b)
g(k" b) , .. (p_)
,(k"i)
q(k" j)
g(k"a) g(k"b)
,(k"i) g(k"b) Fig. 6.2: Feynman diagrams for e+e-
-+
q q.q g.
."
Ii. /)(}UIJU; IJ/u;MSS'I'IlAIII.UNU
iec,g' 'f" -(
M,
•
)
(
;j" 1'+ ' .. " 1'-
)-(k) " U
ft' - fe,- fe3 or
(k ,+ k,+ k 3 )'
."
v
(k) 3
/,b,
X2(k, . k,) [2(k,· ")"" + (k, - k,),,«1 . ") - 2(k, . ")<1,,) , i<e,g'T' -( ) s ;j v p+
M.
fe, 1.- v (k) ,.U(p- )-(k) U " (k,fe,++ k;j.,++ k,)' 4,
3
feb,
X
2(k, . k,) [2(k, . '''''' + (k, - k,),,(., . e,) - 2(k, . "),,,) , (6 .20)
with" = (p+ + p_)' = 4E', E being the bea.m energy. Let M(A" A,; A3, A" A" A.) denote the helicity amplitude with AI the helicity for the e+, A, the one for e-, A3 for q, A, for q, A, the one for g(k,) and ,\. for g(k,) . As usual, we introduce explicit polarization vectors for the radiated gluons of the form
It .N,:-'
N;[A
1/ h(l T ,5)+ }.:; // A(1± ,,)),
4[(q· q')(q. k, )(q'. k;))i ,
i=1,2.
(6.21)
A very simple nonzero helicity amplitude is then M( -, +; -, +, -, -) for which we choose q = k., q' = k3 in the above expressions for ff alld N, . Of the eight Foynm"n ;<mplitudes, only three contribute and we find, after the elimination of the repeated index,
".,I( -, +; -, +, -, -) -
ee,g'N,N,_(
- 8E'(k, . k,) v P+
)
1. (
1'4
) .(
1 +,5 v k3
)-( U
)
J. ( '
)
(
k. .1'+ 1 + 7, u p-
)
(6.22)
Summing over the color degrees of freedom, it readily follows that
IM( - , +; -, +, -, -
)1' =
.,
.,'
4e '~ e2g4
3~
•, "
kL(k3 · k,)
(6.23)
"
,
)
wllCre
A = il(k;. kj ) i>j
.
(6.24)
. .' '.
lUi.
c't , .-
-~ ~
J h''/7->'
"VI; IlaV(~ gOlie to the i+(:- c.m. frame, wilh tl u! z·djr('ction along huvc introduced til(! notatioll
p+, and we (6.2,1)
fo r aU vectors k;.
All ot her nonzero helitity amplitudes for which the gluM hl!lidties ate equal have the sa.me !;Lructure. They only differ in the appearance of k;)+. kJ _ or kH instea.d of k4 _. Somewhat more complicaLed me the helidty amplit udes with opposite helicitics for the gh,OIlS. This time, it is more convenient t o choose q = k4' q' = kJ for 11 a.ndq = ks, q'= 1.:4 for i1iJl thecalcwationor M(-,+;-,+ , -, +). Five Feynman diagrams are found to contribute, yielding
>1( ' I
-,+ ,-,+,-,+ x{-
+
2 \'
). =::
V
('.('.10 I II 2 . k~ ) l
4El(k
(TQ1~)iiij( k~ )
}cz ,k.(1- /,s)v(p+)ti( p_) 1>;) }c,(1
+ ~(~)V(k3)
2(T'T');;(k, . k,) + (T'1" );;(k,. k,) (k) ,. (1 + '(k) (l:, + 1.:2 + 1.:3)2 1.1 4 1'1 1~}v J '''(p-H,kd h) ,k, ".(1 - ,,)v(P.)} .
(6.26)
We,now inlroduce the same explicil reprf'.sentatioll for the 1"-matrices and the !!pinoI9.as in Eqs(6.9), (6 .l0) ann (6.11). Tllis allows us to rewrite the helicily amplitude ill the following form:
with
71
Ii.
c,
fl -
i3
' (k, . k.) i3 'Y -,- 2g(g _ k,o) Cr
Zj,(Z;,
+
f)OIfIlU: /lIt}o)MSS,/,ltAIILUNG
(k, . k,)
•
2E(E _ k,o) <> 'Y,
+ Zi,) ,
(6.28) The summation of the color degrees of freedom then gives
.
'
, _ e'e!g'
IM(-,+,-,T,-,+)I -
(k3 . !") 48A E'(k , · k,)kl+k~_k3+k,_
x[71c, + e,I' + 91 c, - 0,1' I.
(6 .29)
The remaining nonzero he1icity ampEtudes, with opposite gluonhelicities,
can all be obtained from this equation by interchanging k, ...., k" and/or k" or by simply applying parity conjugation.
"3 ....
an
To obtaill the four-jet cross section, one must sum the squa.red a.bsolute values of the helicity amplitudes) perform the color sum, average over the jnitial lepton heiicities l symmetrize appropriately in the four-momenta of the outgoing particles, and sum over the quark flavors. For the process e+e- --> qqgg, we find
1.111.1' =
2e'AY' (L:Q}) ,3 J
D O,(I, 2,3,4) + 0,(1,2,3,4)],
where QJ is the fractional charge of the quark with flavor
0,(1,2,3,4) =
1.~3'
(6 .30)
P
k.)[7I cl + e,I' + 91c, - c,l' 1 32E'(k, . k,)ki+kLk3+k,_
J and
(6 .31 )
In eqn (6.30), the fir,t summation runs over all relevant quark flavors, while the second summation runs over all 24 permutations of kl' k21 k 3 and k4 . Consider next the process
(6.32) with q
-F q', i.e., different flavors,
In lowest order, we have the four Feynman
diagrams of rig. 6.3, with the Feynman amplitude,
" ;(k, ,;)
q(k"i)
'i(k,,;)
"(k"n) q'(1" m)
q(k"i )
'i(k"j)
i7(k~,IJ)
,(k"j)
Fig. 6.3: Fcyn man diagrams for e+e- - qifq'q'.
(r /J1}{//J/,h'IIIlIl'MSS'I'ItAJlLUNG
76
M, -
"a _ - ,k,- h- ,k, 3 mn h - 2s(k, ,k,?;jT u(k " (k, + k, + k.)' 1'"v(k.) eo;g2
xu(k.hVv(k,)v(p+h"u(p_J,
,"I, -
ee~g2
(k k) -$1'2
?
1'"'1'" -'(k) f. -I- ,k,-t f3 (k ) ij " ,an" 31'V(k k k ),1'"v " 1 +2+3 (6_33)
By inserting the heiicity projection operators in the Feyrunan amplitudes, oue finds again that the lepton and quark helicitics must be opposite to yield nonzero helicity a.mplitudes. One then proceeds to el iminate the repeated indices and to introduce explicit spinors} as was done in Section 6.1. For t.he helicity amplitudes M(A" .I,; .1 3, A"A5, A6), where Ai, i = I" " ,6, refers to the heJicitieSi of the e+, e- , q, 71, q', 7f, we find
M( +, -; +, -, -, +)
,
BTijT.':,nIQ,F(i, 2,3, 4) - QjF(4, 3,2 , I)],
M(+,-;-,+,+,-) -
B1ijT:' nl-Q,F(2,i,4,3) + QjF(3,4,1,2)],
M( -1-, --; +, -, +, -) -
BTijT::'nIQ ,F(J , 2, 4, 3) + QjF(3, 4, 2, i)],
M( +, -; -, +, - , +) -
BTiiT!n[-Q,F(2, 1,3, 4) - QjF(4, 3, 2, i )l, (6,34 )
with k<.L
F(i, 2, 3,4) = (k3 ' k,)k .. _ and
77
n =.
c 2 r/ .
(6.36)
1 .
4£2 !kl+k2+k3+k~+1'
'I'he quantities F' and H here are of course different from those of eqns (6.16) ;Iud (6.17). In the above formulae, Q, (Qj) represent t.he fractional charge ()f quark q (q'). The remaining helicity amplihldes can again be obtained by a pp lying a parity conjuga.tion. Tile case of identical quark flavors can be treated in the same way. 'INc
only have to Lake into account that there are more Feymnan diagrams and I.hat a relative minus sign has to be introduced between the Feynman amplitudes which differ hy the exchange of identical quarks. We n~el'e1y list the results:
M(+,-;+,-,-,+) -
BQ /r.~,'l'~,..[ F( 1,2,3,4) - F( 4,3 ,2, 1)] ,
M(+.-;- .+,+ ,-) -
BQ /f,; T~"[- F(2 , 1. 4, 3) + F(3, 4, 1,2)[ ,
M(+,-;+,-,+,-)
BQIlT;,T:m [F(I, 2.4,3) + F(3,4, 2, I)]
~
-1:~)i';.[F(3,
2, 4, 1) + F(I, 4, 2, 3)]} ,
BQ JiT;,T,:m 1- F(2, 1,3,4) - P( 4,3,1 , 21]
M(+,-;-,+,-,-H -
-1~jT;~I-F(2 , 3.
1, 4) - F( 4, I ,3, 2)]) ,
+ F(l, 4, 3, 2)1,
M(+,-i+,+,-.-)
~
-BQJT:'jl;~[-F(2, 3, 4, I)
,11(+,-;-,-,+.+)
~
-BQJT~,1'~IF(3,2, 1, 4) - F(4,1,2,3)].
(6.37)
To obtain the four-jet cross seetion, one musL sum all the squared ",bsolute values of the helicity amplitudes , perform the color sum, avel·age over the initial lepton helicilies, symmdrht.e appropriately in tIle rour-momenta of the outgoing particles, and slim over the quark f1a\' o~. This yields 28'
3
(2: Qi) 2: ( 16 N,F(l, 2, 3, 4) + F( 1,3, 2. 4) J
P
+F(4,2,3, 1)]F' (1,2,3. 4 )},
(6.38)
where NJ is the number of relevant quark flavors. Note tha.t th~ terms proportional to (L: Q J)2 have dropped out after momentum symmetrization.
6. /JOIII!"": JJIlI.'M.I'S'I'JiAlnIlNC
78
Including the phmm
HjHI.CC
facLul':i,
we Llll:::; obta.in fo[' the four-jet cro:::s
section:
d(J(4-jet)
o'(p+
X
+ p_ -
k, - k, - k, - k.) 128E'(2,,)8
[1M , I' + IAJ,I' 1rf'k, rF k, rFk, rFk, , klO Ie,o k"" Ie,o
with
(6.39)
IM,I' and IM,I' given byeqns (6.30) and (6.38).
This example shows again tha.t, for double brerrlsstrahlung processes, it is in general much more efficient to evaluat·e separately the various helidLy amplitudes as comPlex numbers in terms of the comp~ments of the [ourvectors in the process, ra.ther than ha.ving to consider lengthy formulae for the cross section.
One last comment should be made concerning these formulae. Because of the appearance of fadors like (k 3 · k4 ), k1+1 k2 _ , etc. in the denominators, the expressions look quite singular in colllnearconfiguratiolls, for example, k31lk'h k11p_, k,lip+. It turns out, however, that a.lso the numerators vanish in these limits, but a numerica.l evaluation of these expressions without sufficient care could quite possibly result jn a loss of numerical precision, Of course, when all jets point in different directions, away from the beam axis) no specia.l care is needed. This draws the attention to the need for a systematic treatInent of collinear configurations within the framework of the helicity amplitude method, a topic which is analysed in detail in the coming· chapler.
7 Finite mass effects The mere neglect of a mass effect
is quite suspect and often not correct.
7.1
Their occasional importance
III the previous cha.pters, we developed a general formalism for calculating l!1ultiple bremsstnlhlung in gauge thl.'Orics at high encrgiea. By consider-
ing the limit of vanishing fermion masses, we were able to obtain simple I~xprcsl>ion$ for the vnriou,:; hdicity amplitudes. For l1igh energy processes, 1I,e massless fermion limit is a good approximation unless nearly collinear I};lrlicie configllr'l.tions are encoun~ered. Tn these kinema.tical situations, finile mass corredions have to be introduced for a. correct description of the process. This can he understood as follows . Consider a phot.on 01' /I. glllon with momentum k ar.d 11. fermion with mass m and momentum p. If k is nearly parallel to p such that the scalar product (p. Ie) is of order m 2 , then, when
(p- k) appears in the denominator of a. cross section formula (sec Chapter 4), olle Ul\1st also include terms proportionallo m'l (p. k)-2 a~ tbey are of the same order of magnitude. In the preceding chaplers, :;uch terms proportiollal Lo m 2 (p . k)-2 were neglecLed. Consequently, the formulae could not be used in the case of collinear photon or gluon bremsstrahlung. This point was already emphasized in Section 3.4. In Section 7.2, we first show how the finite mass effects can be treated at the level of the croSS! section formulae in the ca.:;c of single bremsstrahlung, Mld, in Section 7.3, we present an explicit example: P.+r.- ..... Jl+ jl-i. For multiple bremsstrahlung, we find that it is essential to consider the mass corrections to the various spin amplitudes of the pt"ocess, rather than calculating these effects for the cro.:'!s section. How this can be done i~ explained ill Section 7.4. We ha.ve al.so explained in Section 3.4 why the zero fermiOll mass limil breaks down when collinear fermions are present, for example, for Bhabha scaHering in the very forward direction or for collinear muons in e+e- -+ fl + ft-"(. In principle, it should be possible to calculate t.ho: tillite mass effecLs ror these cases using the techniques of Section 7.4, but, Lo the best of our knowledge, this analysis has not been carried out systematic.ally. In this chapter, the fermions are assumed to be nol collinear.
80
7.2
7. fiN/TN MASS I:/,'FII'CTS
Single bremsstrahlung
Following the mcLiJoJ of llerend, et ai, [211, we show how it is possible, ill general, to calculate the finite fermion masS effects for single bremsstrahlung at the level of the cross section. Consider a process with an incoming fermion with momentum p. The generalizatioll to incoming anii[ermions or outgoing (anti)fermions is easy to make. The amplitude for the lower order, nonradiative process can be written in the form
Mo
(7.1 )
= A{q,)u(p) ,
where qi stands for all occurring momenta except p. Because of momentum conservation 1 we can always write an amplitude in t.his form. The nonradiative cross section) summed over all polarizations) is then given by
2: A(q;)u(p)u(p)A(q;)
IMol' =
I: A(qi)(} + rn)A(q;),
(7,2)
where m is the mass of the fermion and A(q;) = "(0 At(q;). Again, because of momentum conservation, this expression depends only on qi:
IMol'
= !o(q,).
(7,3)
We now let the particle corresponding to u(p) radiate a photon, The bremsstrahlung amplitude becomes e
ME = 2(p. k) A(q,)(p- ,k + m) ,i(k)u(p).
(7,4)
Note that the momentum pin eqn (7.4) is different from the momentum p jn eqn (7.1); in fact, it corresponds to p+k. Taking the square oft his amplitude and summing over the photon polarizations, we find, after applying some Dirac algebra,
IMBI'
=
e' - 4{p' k)' e2 m 2 - (p. k)'
e'
I: A(q,)(p-- ,k + mho{p + mh"(p-
,k + m)A(q;)
_
I: A(q,)(J,- ,k + m)A(q;)
+(p. k) L
A(q,)(,k
--
+ m)A(q,).
We areonly interested in the leading contributions when m O( m'). Thus, to leading order,
(7,5) ->
0 with (p. k) =
'1 C 2 "1 1
(p. k)'
L
+(p". k) L
_
A('I')U'- ,k)A(q,)
A(q,) ,kA(q.).
(7.6 )
Ti,e first· ~enlJ , the fmite mass correction, has a double pole in (p·k) and, being proportional to m 2 , it i9 only rele\'ant whli!l1 f is nearly parallel to p. Tht: '''~(:():Id term, however, is theanly one which Sllrvive.
(7.7) This mf'.am :,hll.t, in genern.l, the double pole t,~rms relevant for collinear pho1,011 bremsstrahlung can be obtained froIII t.he tower order, Ilonradiat;ve cross ~(·( :t ion formu la by eliminating t.he momentum p of the radiating fermion, i,(~., lIsing the variablas qi only. For Lhe ea.se where more cha.rged fcemian s U:.kc part i:J Lhe process., one b .ll to ~lJPpl("mcnt lhe zero-mass crQs~ seclioil formula. with more finite m"'~,:J rolTect ion ~ernlS, or.e for each exlernal fermion. Every lime, one shou:d (·~!'rC::;8 fo in tenus of the momema not associateJ with lhe fermion under nlllsideration. This pror:.cdurc for single p~olon bremsstrahlung is easily ger:ernli?:r:d to ll,e QeD case [27], the only complication bein~ the color structure of tile
L
IMol' ~
A(q,)(JI + m)A(q,) " fo(',),
(7.8)
where Tn tillS time is the IIlass of the C[:Juk with mOn'.entarn p. If tbis quark ;]QW en:its a gluon wil.h mUlfIcnLu:1I k, the bremsstrahbng amp~ it u
(7.10) we can simply rcrc,,"~ the awn: argumeut. lIence, summing over a.fI polilr. izations. we find
IMBI' -
'19 2m2
- 3(p. k)' . T
49 1
3(". k)
L
L
-
A(q, l( p- /0+ m)A(,,) A(,,)(,k
-
+ m)A(,,),
(7.ll)
· ....
7.
WI
li'INl1'J~
M 1\8,\' gPVliC'/,S
Fig. 7.1: FeYlill1an diagram for e+o- -+ 1'+ ,r . and the finite mass correction term, for
k nearly parallel to p,
4g'm'
- 3(p . k ),fo( qi) .
is given by
(7.12)
In the coming section, we illustrate this procedure with a simple QED examp le.
7,3
An example:
e+e- -> J1+J1-,
From the preceding sedion~ we know that in order to calculate the finite fermion lnass corrections for e+e- --+ P+Jt-'l we must first calculate the lower order cross section for t he non radiative process, i.e., e+(p+) +c(p_) -+ fl+(q+l + fl-(q-). This process is described by the Feynman diagram of Fig. 7.1, and the Feyuman amplitude reads
(7.l3) The only nonzero helicity amplitudes are
M(+,-;+,-) =
M(+,-;-,+) -
(
4 p+
.'
p (I-,s)u(p_) + p_ )2 !i(q_liP(l + ,s)v(q+)v(p+li .
2 ,(p+ - q+)' -
e (p+
+ p_)' '
(7.14 )
<11141 Umir parity c,mjugalcs. Averag ing ()V!~I' thl) initial state poJari7.a.tioIl5 awl slimming over the fiual state degrees or frccdom, we obtain
IMol2 =
+ (p_ -
2e~ (p+ - 1+)4
q_)4
(7. 15)
(p++p_/4
j?or the proce3S e+(p",,) + e-(p~) ---+ /,+ (q+) + 1'- (q_) + ')'(k), we can now n!lJsider photon emi~;;joll ill a dirediolJ close to pf-. From Seclion 7.2 , we liliOW that, ill order to obtain the nnile mass correction in this case, we have tll .;xpress IMol1 in terms of p_, q+ and fJ~:
(7.16) ;ltul the mass c.orrection term in the cross section becomes 2e~m2
(p _ _
(p+ . k)2
+ (p _ q_)4
q_)~
(q+
(7.17)
+ q_)4
where m is the electron mass. Repeating this procedure for photon emission dose to the ,L , direct.ions , we obtain all the tillite ma.:;s correction terms: _
2c ll m 2
[(p_ -
('++1_)'
_
q._) ~ + (R(p+·k)'
-
g+)4
[(P_ - q_) ~ + (p+ - q_)4 (p" + p_ )' (q_. k)' 2e6jt~
+ (p+ -
+ (p+
ih.
and
if-
q.. ) ~ + (p+ - q_)~] (p_·k),
- q.•.)4 + (p_ - q+)"-] (q_. k)'
. (7.18) lkre, /1 dellotes the muon mass. Expre::;sing this resu lt in the usual variables
S'
(q~+q_)2,
t'
_
(p __ q_)2,
1/ '
_
(p__
q+)~,
(7.19) and adding the zero-ma.ss formu la. fo r this process, we obtain the U op<Jlarized ~quared amplitude to leading order in m 2 and p.2:
IMBI'
=
1 t1+U 2 ] .' [lI2+u 2 t2+1J."~]} ,{ m' [tt2+U -2, + ++ T-~" ?
,.
(p+'
kl'
(p_ . k)'
,'(,+. kl'
('1_" k)'
(7.20) \-\le used th e same l;otation as in Scclioll 4.3, with the four-vectors Vq andlJ"
dcflued by eqns (4.27 ), vp =
p+ TC'"-;,, (p"" . k)
p-
(7.21)
.. ..
7.
~' INl'l'f;
MASS IWf'I,;(J'I'S
Bqll (7.20) is valid ["I' RII killcllmti",,1 coIIOg,,,·al.iolls without neMly collinear fernUons .
This procedure can ea-i1y be applied to all other QED and QeD single bremsst.rahlung processes. All one needs to do is to express the llonradiaLive crOSS section formula in the appropriate variables to obta.in the mass correction terms and to add the zero mass expressioll of the radiative process. \file refel' again to Cha.pter 9, where one can find a list of cross section formulae including the finite mass corrections.
7.4 7.4.1
Mass corrections for amplitudes General formalism
The technique for obtailljng the finite mass corrections, wh ich was presented in Section 7.2, works nicely for the case of single bremsstrahlung. It is,
however, not powerful enough for the case of multiple bremsstrahlung. The reason is simply that, for multiple bremsstrahlung, not just one denorninator
like (p. k) can be of order m', but also denominators of the type (1'+ k + k')', where k alld k' are photon or gluon momenta. In this section) we present a. systematic treatment [351 of the finite mass effects for multiple bremsstrahlung in which an arbitrary TIlIDlber of gauge particles are radiated nearly
parallel to fermion directions. We now find that it is essential to consider the mass corrections to the var ious spin amplitudes of the process, rather than calculating them for the cross section as was done previously. In the
zero mass limit, these spin amplitudes reduce to the helicity amplitudes we considered earlier, except for some additional amplitudes which are direct ly proportional to a fermion mass. For simplicity, we shall present the technique
in the case of QED only. Let us consider any QED process in the tree approximation , in which n photons are emitted in directions nearly parallel to a fermion direction described by the four-momentum p (later we shall consider the more gen-
eral case where additional photons are emitted in directions parallel 10 other fermion directions). Let k;, i = 1,2, . .. ,n, be the four-momenta of these collinear photons, a.nd let m be the fermion ma.s5~ i.e., p2 = m 2 . There will then be Feynn"l.an diagrams describjng the process in which fermion propa-
gato rs will have small denominators: Ll; = Ll;j
(p _ I:;)' -
m' ,
(p - k; - kj )'
_ ",' ,
(p - k, - k, - . . . - k.)' _ m' .
(7.22)
We consider the case of nearly collinear photon, such Ihal the quantities
7-4-
MIi SS (;()/ltlf;'(,"I'IONH fOil li M I'I, / 'f'/lf)h'8
ill' ki ) illH.l (ki' k J ) are of ordcl' m 2, III l.his ease , alJ the qlJautlt,ies 6. are i"~(J of oreer m 2 , IL is de.ar, therefore, thil t eve1' in the Illgh energy limit the f<'I'1I1:on mllss must be taken into account. By i:ltI'OJ,lcillg a generally pogi ~ ioncd light.like ve.ctor q, wc call write dow II the follcwillg repre~entatioll for the PObriL:htioll vectors of the photons:
),;
=
=1,
i= 1,2", .,n.
(7.23)
It'
Not.e that ail components or arc of order I, C\Cll i ll the collinear L:nit. Also, t he represent ation [or differs from the eJf{'.ctive one, eqn (3.8), which wall t:sed earlier. This is d\,;e 10 the fi'1.ct t hat, for rn3S!1ive fecmions, we do 11\1 : have r.oOScr Vll.tiOll of axial current, and . consequently, we cannot omit I.!,<: 106 t<'TlIl~ in However, for m = 0, the two representations Itce gal;ge I" i II i valent. With the. present choice for I:;, we now l'how that, in the collir:ear sit· I:ation, t he Il.mplitudes Gre at mos( of order m- Il , To this order in In, the nl.'ltri butions to the a.mpli tude will cnme only from the Feynrnan di;\.gra.ms I'OT which the collinea.r photons are attached directly next to the external f(~rw..ion with momentum p, Th:~ follows from the far.t that ollly these dia.,l{rams ha.ve all the denominators 6. Consider the case tlw.i the photons are collinear with an incoffiing fe rmion with momentum p (the remaining CII.<;(,-S of all incoming t\ntiferrni()1J or an outgoing (a nti)fermion can be treated in the same wa.y). The diagrams with I.he collinear photon~ dose to the spinor u(p) ('-ontain the expression
Ii;
l'.
A~
==
j;-
fil - ... - ,k.. ..,.. m J;
I"~
l~ .. .n
"....
+(n) - 1) other permutations of (1. , 2, . , .• n) ,
A, -
p-h+m '.'. (\ 6, ","I U]J, 1 [ .6.} 2(p · (tl)-
Olle call
Et',
, . Jcl ),'J
tt(p),
(7-25)
Vie see that (p' tt,) - O(m). show that ' itl(p) = O(m), To this end, we introduce two
ami, with our choice, eqns (7,23), for
Similarly,
(7 .24)
7.
(p . I,)
where the sign of tbat
t~ = -1,
(q·I,) ~ 0 ,
'./l," is defined by '0123 =
flN/'!'/,' MASS /II'Jo'''C'I'S
+ 1 as in eqn (4.55). It fo llows
(7.27) As the vectors p, q, tl and t2 are linea.rly independent, we can decompose kj ) i := I, .. , , n, as follows: "k"
--
(q·k;)" k) -m( ,(q . ki))] q " - (k i 'l ,)t" , - (kt)t" i-2,· P 7, ( 1 ) [(p-i (q.p )
q-p
q-p
(7.28) This relat ion, together with the fact that k? = 0, leads immediately to the result (ki - t,) = O(m) , (k i · t,) = O(m). (7.29) from the decomposition of kr, it then follows that f:iU(p) = O(m). Hence, the entire expression Al is of order m-'. It a lso follows that
where we introduced the nota.tion
(7.31) Decause of the symmetrizatjon between photons 1 and 2 conta.ined in the expression for A, we can effecti vely replace "~' ,,~, by «~, ,~I) in A,.
.
Purthermore,
N,N,5"A,Tr [.4 J> '" ,k,(l
+ .\,'Ys)I,k,.4 P ,1;, (1 -· A,/S)"(p) (7.32)
B1
T hill time, Olle easily vcrifil~ Lhat (k, . (~') _ O (m), nnd th a.t AI ;k, u(p) = O(m~). Hence, the whol!;! preceding expre::lsion i,\; of order 1 1 111 , It rauows tha.t Az is of order m - a.nd that it can effectively he replaced
I,,Y
This prOCCdJ;IC can now be r.onti nucd fOT the remain ing (JhOLons. ".,Ie "hl.;l,ia the re~mlt. t ha.t (7.34 ) i~
of order m-" a nd
(ha~
F" is given by the fo!lowiug recursion relation;
(7 .35)
with Po = 1. Ono! FI and F2 are known, i t is thus possible t o evaluate the higher ord<,..! fund ions Fr>' A vcry useful property fo r t his p urpose is the rdntion
,. ". /'1>-1 . ,'•• ,. j,"I U (P) 1'-.. - 1, ... 1'1
"..... Iv"
The remaining paIL of the dia.gram, involving the spinot str ucture A can now be t reated in the massless limit . This fIleans that if we fix the helicities for die remaining fermions, t he terms of the a.bove type will give zero whenever t he fer mion helicity operators ± ,d kill the (1 - ).11~) fador . Of Wiltse, this term already vanishe.s when some. of the collinear photolls have different hel i cilie~ . Furthermore, the relation
ip
(7.37)
HH
7. flNJ'n' h/'\'%'
1~~' i"t>C'l'S
,,,lid for ",11 i Illld j, call oftcll be u.cd Lo eliminate Lhe last term in the expression for ji~I '
The other diagrams, which do not have all the collinear photon, next to u(p) , are necessarily smaller by at least one power of m. This i, due to the fact that as soon as an acollinear photon is inserted in the string of collinea.r
photons all the fermion propagators from that point on are of order 1 in.tead of being of order m- l . To summarize, we can say that, with our choir.e of repre.entatiolJ for the cunplitudes in the collinea.r situation are at most of order n~-r\ if there are n collinear photons. To this order lU m, the relevant F'eynman diagrams are those in which the collinear photons arc attached immediately next to the
;.t·,
externa l fermion which determi'nes the collinear direction. These Feynman diagrams can casi1y be evaluated using the function FT., while the rest of the diagram can be trea.ted in the massless fermion limit. The evaluation of Fn
is r, Hther simplified using the last two relations (7.36) and (7.37). This analysis also shows that when aU (p. k,) are small, but much larger than m'2, the same set of dia.grams is the only relevant one. Tn this easel all ma.nipulations in the numerator of the Feynman diagrams can be done in t.he
massless limit, all finite mass effects being of subleading order. We shall now work out in detail the special cases of single a nd double collinear bremsstrahlung.
7.4.2
Single bremsstrahlung
In this section, we present the relevanL formulae for the case of one photon
being emitted in a collinear configuration. Suppose that this photon has a momentum
k nearly
parallel t.o the momentum
p of
an incoming fermion.
Let Aand Ap be their helicities . We already know that the relevant Feynman diagrams are those where l' stands next to nip). On the other end of the fermion line stands another fermi on spinorl which can be taken to be mass-
less. If we calculate the helicity amplitudes of the process, this spinor will + ,\''"(5), with,\' = ±J, multiplying produce a helicity projection operator from the left the expression An given in eqn (7.24). Since one photon is radiated, we need to calculate P1(k, -I) . However, Fl (k,-I) appears only in tbe CQmbination + A'''f.)FI (k,).) u~p(p), (7.38)
Hl
W
where u,,(p) is the fermion helicity state with helicity Av' It is therefore convenient to define the 'collinearity factor' P, (A; A', ,\P) by (7.39) Two comments shou ld be made about this formula. Firstly, if does not define F,P.; A', Ap) uniquely. We shall see that FIP; A', Ap) is of order ",-1 As we neglect higher order terms in
Tnl
we can choose the simplest form for
F,(A; A', Ap) . Secondly, we shall also see that, to leading order in m, the effect
r
",{
MASS COIUtHC'J'ION8 fOil t1MI'/,rl'll/)f;S
" r (I + ),')'r,)P'I(k, >.) operating Oll u~l'(p) is ttl rdain the hc.ti(:ity assignmellt "r TI.\,(p) for the case..\' =.>." Il.ud to reverse ltlC helicity of UA,(])) in the case ).,' = -.\p. In both (.;as{'..'>, the r adiation of the photon ia described by the '("l>lIincarity fador' times a lower order amplitude for which the helir.ity of tlw fermion is taken to be N. Suppose we consider first the case ).' = .>.. As
(7 .40) luuJ
(7.41) we find th;).{ the last ~erm in 1<'1 does r.ot contribute as it is m\Jltipiicd from l,he left by (1 + >.'1'5) and >.' ;: A. Hence, for aU>', F,(A;'\,A,)
~
(7 .42) To dea.l with the other helicity hwing result:
1("
t,) "
combina.lion~,
we firsl establish the foJ-
+ (k· t,) hl(l ± ,,)u(p) (7.13)
where c ;'"
~
-la PJ.. - t I.L,
(7.44)
p+
(lnd where u(p) is the ~pinor which has its spin flipped compared to u(p) . What we mean more precisely with this statement will be ma.de clear after cqn (7.47). Indeed, from tbe definition (7.26) of t2 and the identity (3.6), it follows thaL
j, ~ -iO'
I,(A f-
"A)/2(p· q),
(7 .4,\ )
ilnd [
i /2(1 ± 16)U(P) ~ T (p'q)(A""-I,J; 2
-
AH 1±O,),,(p)
± I,(l± O,)u(p) + O(m).
(J,46)
/<'JNI'J'I~
MASS' NFPh'G"J'S
+ O(m') .
(7.47)
7.
VII
!Jut then, I(k . ttl =
/, + (k ·1,)
,(,J(1 + I')U(P)
(k· t, 'fit,) h(l ± I')U(P)
As (k . I, 'fit,) = Oem) and term, of order Oem') are neglected, we can also neglect terms of order m in fll(1 ± I')U(P), which means that it can be taken in tbe In = 0 limit. On tbe other hand, the fonn ,t,(l ± IS)U(P) is a solution of the Dirac equation and an eigenvector of (1 l' I'). It must therefore be proportional to (I 'fl')U(P), where fi(p) is obtained from the spinor u(p) by flipping the hclicity. Taking into account our conventions (6. 10) and (6.11) for the fermion helicity states, tbe phase factor e;· is then completely determined and given byeqn (7.44). This establishes the quoted result, eqn (7.43). Next, we take A' = - A and A. = -).. The expression F,(k,A) is now sandwiched between the projection operators ~(l - A,,). Using the decomposition of ,/<, eqn (7.28), we see that the terms proportional to /.1 and /., do not contribute. Hence, to leading order,
'" 2(;\)
{2(p. fA) - 2N /c
Ap
[i:: ;~ "
+-.!_((p.k)_m,(q·k)) (qp) (q.p) '"
A]}
-1 {(P.f')_N[m,(q·k) I-A+2/c/J((p .k) (p·k) (q. p)
-m'i~: :j)]} '"
·- 1 {(P.")-2N(q.k)[2(p . k)-m,(Q·k)]}
(p·k)
~-
(q.p-k) . 4N(p · q)(p. k)(q · k)·
(q.p)
(7.48)
Finally, for).' = -A and Ap = A, the (p. fA) term does not contribute, and, in the decomposition of ft, only the terms proportional to 1-, and h give nonvanishiug contributions. They have, however) the effect of flipping the helicity of the spinor u(p). 'I'hl1s, by eqn (7.43),
r,r MA SS
COlUU~C'/'IONS
fOIl.
), MI'I. rl'tlm;,~·
N{~''' (k ·ll -
91
it2 )
(p. k)
2mNc;<>{ k . 11 - itz)( 9 . k) (p . k)
(7 .49)
aile:
(7.50)
In all these deri vations, one should keep in mind that the quantities rj K1.and next to the spinor u(p) and that fou (p) = Oem). Also note the relation
F,(-A; - A',-A,) ~ [F,(A;X,A,)I * , where the superscript In
-lo
*
(7.51)
denotes complex conjugation and the replacement
-Tn.
To 5ummuizc the single colliIlear brem;;strahhUig case, we find l ha ~ aU bdicity amplitudes reduce to a produt:t o f a collinearity fac.tor FI tiIneli an amplitllde for the procc;;s ill which the collinear photon if! temoved. For the ';llbwcd' amplitudes, Ap = A', thi~ lower order amplituce retains the he1iciti es of tIle fermions, but for the 'forh:d dcn' amplitudes, A~ = -).', thc helici ty of Ih! spinor u(p) i$ flipped.
7.4.3
An example: e+e-
--+
p.+j.t-,
We now illustra.te the generallech niqut'.Il of the pre<:eding section for the ca,se of
e+(p+ ) + c.-(p_) wbere
k is
-+
JL+(9+)
+ 1'-(9_) +;(k) ,
(7.52)
p_, i.e.,
nearly collinear to
kl. "" O(m) ,
(7 .53 )
where, a3 usual, we have gone lo tee c+e- c.m. frame wilh the z-axis a.long i4 alld k± = ko ± k" k1. = k",.. iky . \Vc know t hat \/1 this limit we only have to consider one Fcynman a.mplitude, viz. ,
(7.54) with
, .s'
-
(p++p _) l,
t
(q++q_)l,
t'
-
(]I+ - q+P,
"
(p- -q-)',
u' -
~
. - q-P,
lP,
(p _ _ 9+)1. (7.5.5)
7. f'lNI'/'h'
M~SS 1;1"I'I~~"I'o
\Vc also l<now tllaL tile Hpill lUlIpliiucles ill tilt: t:.ollillcar limit arc given by
M(+, - ; +. - , +)
-eFt (+; -. - )Mo(+, -; +. -),
M( +, -; -, +, +)
-eFt ( +; -, - )Mo(+, -; -, +),
M(-,+;+ ,-,+) -
-eFI(+;+.+)Mo(-,+;+,-),
M(-.+;-,+,+) =
-eF,(+;+, +)Mo(- . +;-.+) ,
M( +, +; +, -, +}
-eFI ( +; -, + }M,( +. - ; +, -),
M( +,+; -, +. +}
-eF, (+; -, + }Mo( +, -; -, +},
111(-, -; - , +, +}
-eFI (+; +, - }1I1o(-, +; -. +},
M( - ,-;+,-,+}
=
-eF,(+;+,-}Mo(-,+;+,-},
(7.56)
where the quanti t ies 1110 are obtained from the spin amplitudes M by remov· ing the photon emission parts. Not.e that in this case the first argument of PI is the last argument of M, the second one of Fj is the second one of M o, while the third one of FI is the second one of 111. These quantities Mo are the massless helicity amplitudes for e+ e- --+ Il + 11.- . Here,
"
e' _
M o(- ).,,\-)., ,).,} = :;;v(p+h"
1 + ,\,. _ ,,1 2 u(p_}u(q_h
+ ).,',. 2
v(q+} , (7.57)
:19
a.ll other helicity combinations vanish. They are easily evaluated, yielding
M o(+, -; + ,-} -
2e'(uu'}I/s' ,
M o(+,-;-,+}
-..
2e'(tt' }I/s' ,
M o(- ,+;+, - }
-
2e'(tt'}1 /s',
M o(-. + ;- ,+} -
2e'(uu'}~ / s' .
(7.58)
The amplitudes for negative helicity photon emission can then be obtained by applying a parity conjugation, together with eqn (7.51). For the evaluation of the collinearity factors F 1 ) we ha.ve p = p_ and \ve can take q = p+. If E denotes the beam energy, we find that the normaliza.·
7../. M ASS (;O/UUX.',/,/ONS fOJ11IM/'/,/'I'IIf)/';S Liou rlldor fOI>
j.+,
to leading order ill 111\ is given by
(7.59)
(7.60) /\s a result, we have cffecti vely
F,(+;+,+) _ r, (+;- ,-)
~
8N E3 k+ (p_·k ) , 4NE'k+ (2E - k_) (p- . !.) 2NmEk_kl
(p_· k)
F, (+;+,-)
~
0,
(7.61 )
/Ired
IF,(+; +, +)1' + IF, (+; - , - )1' + IF',( +; - ,+)1' + IFd +; +, - )1' (7 .62 ) This formulli is useful [or calculating the unpolarizcd cross section in the cruc k nearly parallel to p_> Using the fact that, in this collinea.r limit , 2E3'::::! 3(2E ~ k_ ), we ha.ve
(7.63)
wi th
To obtain this IC!lUlt , we added the COlltribul ions for negative heljd ty photon emission aIld averaged ovcr the initial stl\te polarizations. We Rlso used the. fRet th a.t, for k2 = 0 ,
(7.65)
7.
I)~
If one wants the an.alogou~ formula ror
I"IN/,/'I~
MASS h;'FJ.'I:;C'J'S
k along j1+1
one ca.n proceed in exactly the same way. All one has to do i~ to illtercha.nge the roles of p+ and p_, whicb implies
k+ .... k_. Hence, in this limit,
IMI'
= 2c" F( k, p+) tt'
+ ~u' , ss
(7.66)
with
For k a.long q,+, the collinearity factor F is most easily evaluated in the fra.me for whic.h ~ det.ermines the positive z-a.xis. To dist inguish the componeots of the faill"-vectors in this fra.me from the ones in the e+e- c,m. frame, we add primes to them. Tbus, k~ = ko - k~, etc., where k; is the component
of
k in
the direction of
and (7.69) Simila.r1y, for
k along if-)
we have
.
IMI' = 2e 6 F(k, q_) tt' + uu' '
(7.70)
with
F(k, q_) = (2'1_0
q k"
+ k~)k~(q_. k)'
[
(2q_O)2
Illk'!']
+ (2q_0 + k~)' + 4q:ok~
,
(7.71) where q-D denotes the Jt- energy, and the double primes refer to a. frame in which ij_ determines the positive z·axis.
The choice of these rotated frames to describe collinear bremsstrahlung along
gi.
and
if-
may seem somewha.t awkward. Their main advantage is that
they clearly exhibit the appropriate orders of I" which have to be considered . There is, however ) another reason why they are useful in practice. Here, we treated the case of e+e- -+ /1+ /1- 'Y for the sake of simplicity, but one should not forget that) in experimental studies of jet phenomena., one cannot always sepa.rate a nearly collinear gluon-quark pair inlo two jets. Hence , one may want to integrate over the gluon variables in these coll inear regions. For that pm'pose, the primed variables are much more conve nient as the experimental
cuts are readily translated in these variables.
1.4.
MAS.'.' COIW/ICTION.'i fOIl A Mi'I,I1'IJ/JI-:.'I
"
All other $iuglc bremssLl·ahlung pro(:':~~!:Ics an be treated in the same way. The same colline,uity factors occu r in the expressions for the cross ~(~ctions and only the lower order, llollnH.liaLivc part has to bc adjusted for mch process. This is a sufficiently straightforward maHer and we simply I'erer to Chapter 9 fo r .the explicit expressions of the cross sectiolls ill Lhe various collinear configura.tions.
7AA
Double collinear bremsstrahlung
When two collinear photons an: emitted, it may happen that Lhey are both Ilcarly parallel to the same fermion direction, or that each of them is nearly parallel to a different direction . In the first cJ..<;e, we a!i.sumc that kl <~nJ f~ are close to the direction o[ i/, the momentum of the incoming fermion. Wc uow huv.; to cvaluate I·~(k,., AI; kz ,A2) of Section 7.4 .1 for the various helici ty configurations. The c1!.lcula.tion proceeds exact ly as in the si:J.gie collinea.r bremsstrah!·jng case. III complete a:la!ogy with eqll (7.39), WI! o::all define a coHinearity factor 1";:('\" '\2; '\', '\p) by
(7.72) where N is the signature of the heli<:jty projec:t:on operator which is assoda ted with tile fermion spinor on the other elJd of the fen:lion line which (onnects with u.\.p(p). First of all, note that the rclat io:l
(7.73) holds, where P-~ b~fore t.he sYlnool
* denotes wmplcx conjugation and
the
replacemetlt m ..... -m. This relation is proven by rep:acing 1"5 by -'"(s. We proceed to Est the results. for the allol'lfxl arnplitm!.es, ,\' = ,\". we have F,(+ ,+ ;+,~)
-
P,(+,+; - , _· ) -
1 ,'-'11'-'1 --,-
N2 (p- k l -k2 ·q )., 2 1,.' I ( q. kI )( p. q ) hl( p- )c.) )c,
P AU + 1,)1,
1 { j 2N1 (q· kd 2N2 (p · q)
- N2 'l'r[(JJ-. }:J)
;"~2
II h(l -
)'~);} ,
7.
Dij
F,(+,-;-,-) -
I
N'(1 1 -
fl fl 2N ( 12
L
J
q'
'"
k)( 1
•
~' { N{TE'
q) " ) Ir[( ,I>-,kd,k,
l
I'
A(I-,s)J
A h(I-,5)) -
N,(q' k,)
p. q
N, { ' - fll2 2N,(p. q) rr[,k, ,II
MASS (,l'fl';C'I"S
(q . k,) }
.
(7.74) In these formulae, N, and N, are the normalization coetDcients (7.23) for the photon Jlolarization vectors, expressed in terms of k" p and the generally positioned four-vector q. Furthermore)
fl, = (p - k,)' - m' ,
L',l' =
(p - k, - k,)' - m' .
(7.75)
The allowed helicity a.mplitudes a.re now given by a sum of expressions
F,(>", A,; A', A') + FAA" A,; X,,\') times the helicity amplitude for the massJess lower order ampli t ude
ill
which the two photons a.re removed 1 while all
other particles retain t heir heiicity assignments. For the forbidden amplitudes, ,\' = -A p , we have
F,(-I-,+;+,-)
0,
F,(+,+;-,+) xTr(,k, ,k, jl A(l
+ ,.»)
F,( +, - ; +, -) . , - ., - , T. ) F2 ( T
(7.76) where e'" is given byeqn (7.44). The forbidden helici ty amplitudes are given by the sum of exp "essions F,(,\" A,; N, -N) + F,(,\" A"; N, -A') times the lower order amplitude without t he collinear photons, but with the fl ipped helicity for u(p). For the case where two photons are nearly parallel to two different fermion directions, it becomes cumbersome to give general formulae analogous to the above ones. The reason is that a given spin amplitude can become a linear combina.tion of two differenl lower order amplitudes, one for which the fermion helicities are unchanged and one for which the two fermions,
MA SS r,OIUff.·(,"{'f()fVS rolf. A M fJI.I'/'IIIJ/·:.~'
1.1. th;~l
specify the pa rnllel diroctiGII.'I, 11avc their Iwlicities fl ipped. Using the LI'(:huiqUC9 of tile roregoi ng sections, it i~, h,lwcver, straightforward t.o work out the amplitudes case by (Me.
(:o llsider once more t.he process
(7.77) ;~Ild suppose th ... t we arc interested in t.he do uble collinear limit where kl a nd k2 arc boll) nearly parallel t.o p_, i.e.,
(p_ . k;)=O(m 2 ),
k;+=O(m').
k;.l=O(m),
,=1 ,2.
(7.18)
We know that all spin amplitudes can be writt.en as a product of two radors: a helicity amplitude for e-l- e- -4 p+Il- and a coHincarily fador which W;t$ c.. lIed P'I. in eqn (7. 33). These quantities F~ depend Oil t he momenta of the collinear photons, whereas t he helkity amplitude for e"" e- -4 11-1- Jj- OOC& 110t.
It is a. s imple ma t.ter to evaluate the fllllctio:1S Fz in the e-l-e- em. frame ..... i! b the z·clil'ection along p+. They contain an arbit rary [our- vector q, which nUl conveniently be chosen eq ual \.0 p+. We then find
('2{+,"T ;+, -)
=
1<'2(+, -;+,-) -
0,
-61CmE~kl+kl_ k2J.'
08
7. nNl n' MASS IWfo'EGTS
fi(+,-;-,-)
w:ith
c_
N,N,
- 4(p_ . kd[(p_· 1.-,) + (p_ . k,) - (k, . k,)] ,
(7.80)
where, as usual, N, and N, are ihe normalization factors (7.23)for the photon polarization vedors and E is the beam energy. The first two labels of F, denote the heliciiies of photons I and 2, photon 1 being dosest to u(p_) in the Feynman diagrams. The third label is minus ihe helicity of the e+, and the last one denotes the e- helicity. In this collinear limit, we now have
M ( A,'\;'\+, A_ , '\" A,) =
.'[F,P,,, A,; -A , A) + (I
H
2)JMo('\' -A; A+,'\_) ,
(7.81) Here, the labels of M denote the helicities of the e+, c, 1'+' 1'-, "f(k:,) and , (k,). Also, Mo deno tes the nonradiative helicity amplitudes for e+e- ~ p+ p.- , whi ch we already encountered in eqns (7.57) and (7 .58). \Vith t he explicit expressions for F2 and Mo) it is now a simple matter to evaluate the wlpolarized cross section. Por kl and P-, we have
_ 2e8 It' + uu' S8'
x
1
k2
both nearly parallel to
2£
kl+k,_I.-2+I.-,- 2E - k,_ - k,_
{[I+ (2E - iF: - k,_)']IAI I' + IA,(i,2)I' + IA,(2,1)I' + IA3 1' + IA,(!,2 )I' + 1A.(2,1)1' } ,
with
A,
(7.82)
1./, M,lSS (.'Ollll/;'(;'/'IONS Jo'(JIl AMI'I,J'/'/I/J}",'i
_I_{ k!+(2E - k _)(2Ek1+ l
(p_ . k l )
Ll!Z
+ (p_ ~
Z1l)
~) [2EkHk2 . (2£ -
k1 _
-
k1_) - m 2k2_ k a ki.l. }2E] } ,
(7.83)
,1.(1,2) and
(7.84 )
i, i=l , 2.
To obt a.in the uapolarized squared ma.trix element for the collinear limi t when kl and k2 are both nearly parallel to p+, it suffices lo interchange p+ and p_ in the a.bove equations. l 'his amounts to interchanging t he subscripts + and - in lhClre formulae. As II. re~ult, Zij becomes Z;,.. The limit kJ and k2 both nearly parallel to if... is worked out in the same way, As in t he singh:. bremsstrahlung case, the quc1IJlit ies r2 ArC most easily evalua.ted in the frame where ifl- dctermlllcs the positive z-a.xis, The arbitrary foar-vedor q in P2 is Lest chosen to be q~, the four-vector obtained by applying;\ space reflection to q+, T heil,
2q.,.o
x {
[1 + (21
I0
+2~~: ~ k~+ f]IA1lll +lA;(!, 2)'2+ IA~(2, 1)1 + IA; I' + IA:(1,2)1' + IA~(2, I)I' }
with 2qiok:_k~_
(q..;. . k\ )(q+' kz )
'
2
(7 .85 )
7. fiN I on' Mil SS [, ff 'BCTS
IOU
_l_{ k:_(2q+o +
A~(1,2)
A;,
k:+)(2'1+ok~_ (q+,k , )
+('1+ ~ k
) 2
+ Z~,)
[2q+ok:_k;_(2q+o
+ k:+ + k;+) +1,Zk;+k::"~, 2q+o
A~(l, 2)
-
p.k{+k~l.
l},
(7 .86)
2q+oA:,
and
(7.87)
Analogously, for k. and k, both nearly parallel to if-, it suffices to replace in the last set of equations q+ by q_, and the primed quantities ki and Zi;
by ki' and ZIj, i.e., to eva.lua.te the components of the four-vectors kj in the rotated frame where
q_
determines the positive z-axis.
ror the mixed double collinear limit, where the photons have directions nearly paral1el to two different fermion directions, the situation is somewhat
simpler. We find that the cross section is composed of two single 'collinearity factors' and a lowest order croSS section written in the appropriate variables . . Let p and q denote any two vectors of the sct of fermion momenta, p+, p_,
q+ and q_, with p
# q. For k. and f., nearly parallel to p and if respectively,
\ve have the following simple expression for the tUlpola.rized squared matrix element in these limits:
--
IMI' =
2e
8
tt' + uu' 88'
F(k"p)F(k" g) .
(7 .88)
The collinearity fadors F [see eqns (7.64), (7.67), (7.68) and (7.71)1 arc explicitly given in Section 7.4 .3. This formula can then be used in the 12 different cases depending on the values for p and q. An analysis of the other QED double bremsstrahlung processes, etc ..... Try"! and e+.- ..... e+e-,,!,,!, shows the universality of the collinearity fadors in both the single and the double collinear situations_ Each time, only the expression for the lower order cross section has to be adjusted. We therefore find it unnecessary to present the derivation of the formulae in detail and refer, once more, to Chapter 9, where they are listed.
8 The production of qu arkonia B.1
Framework
T he applications of tIle heliciiY method, which were presented up to now, ,lealt with processes in which either the masses were completely neglected 1'1' treated to leading order . In Chaplers 4 lhrough 6, we considered QED 1\11(.1 QeD processes, in wiJicb the quark and lepton masses lVeff~ put equal I." zero, while, in Clwpter 7, we studied the fillite ma.ss effects to \eaclieg Of-
der in m 2 • This is completely justified in the high energy limit, where these ll1ilSSeS arc small compared to the cncrgic:; in the process. Tbe tedmical !'casons for the simplifications we encountered in the high energy limit is the (,():lcur~ence of the facts that helicity states arc Lorentz invariant concepts li'f massless particles and that the polarization vedors for the gauge partides which we introduced naturally combined with the heliciLy states of the fNmions, b~cause of the appearance of Lhe comuinaLiolls 1 ± I~ iu /-. The uscfuln<:si< of the hc1icity method is , howevcr , by nu mCllJl$ restricted 1.0 processes involving ollly massless (or nearly massless) particles. We have ,dready shown in Section 4.5 that the inclusion of the Z-exchange can be Lilken into account without much effort. In this chapler, we want to show I.hat, also for processes in which heavy external particles are present, the h~ licity method can be used advantageously, provided that at least some of I.b.: other edemal pa,rticlcs arc mM$le88. For I.his purpose, we present thc t:alculation of the CfQ~S ~eciions for 9 + 9 ..... 2S+ILJ + 9, where 2S .IILJ is the spectroscopic notation for a heavy quarkonium state, for example, a (cc) ~ t a t e. In this case, the mass of the quarkonium cannot be neglected a nd its I!n-eds must be included to a.ll orders. For a detailed comparison between theory and e xperimen~ in the case of JII/; hadroproollction for example, it is not sufficient to calculate direct Jltb production only. Indeed, some of the (ci') exci ted states are known to have la.rge branching ratios into the )/$ [36J. It thus becomes necessary Lo know, lor example, the production CtoSS sections for the 3 P sta.tes. \'Vithin the framework of pert urbative QeD, the hadroprodu ction of heavy qllarkonia is described by several subprol'Cf5SC~, such <1.S gg ..... 2S+1LJ. qq--> lS+ILJ, gq ..... '2s +1LJq, gq ..... 2S+lL J 'lj, qq ..... '2S+lL;g, 99 ..... 2S';"ILJ9, etc. U~iilg the s t allJ ... nl Ft:Ylllllilll It:dll!iLJllt:~, Ullt: n:
R. 'I'II/i I'IWJ)U(J'I'ION OF qUAIUWNlA
IU~
and the cro~s Rcctloll rOl'mt:l~c olle ootained were quite cumbersome. Of these subproccsHcS! the last Olle, 9 9 - t 28+IL J g, is the most r..ornpli~
cated. We IhuB present here the calculation of this process with the helicity method. For reasons of synunetry, it is useful to think of this process M 9 9 9 ---> 25+1L). We shall explain how the crossing of a gluon can be applied to yield the desired formulae for 9 9 ---> 2S+1L) g. We shall also limit ourselves to the most important cases L = S or P, i.e., S- or P~wave quarkonia. The corresponding problems of electroproduction, which involve bMically the replacement of a gluon by a photon, have been studied by Berger and Jones and by Kunszt [38J. Consider first the process
g(kl, 0)
+ g(k"
b)
+ g(k3 , c)
--->
gOp + q) + q(~p - q),
(8.1)
h)
wh ere kll k2 and kJ are the four-momenta of the gillons, while + q and ~p - q denote those of the quark and the antiquark. As we are only interested in color singlet qq states, we can omiL all the Feynman diagrams in which
only one gluon couples to the
qq system. Tbis is because the quark belongs
to a color octet. This reduces t.he number of Feynman diagrams to twelve, in lo'west order. They are given in Fig. 8.1 , where each diagram, in fact, stands for severa.l diagra.ms. The first three diagrams represent each two
diagrams: the qq lines must also be interchanged. In the last diagram, the six permutations of the gluons are implicit. We introduce the standard notation
s = (k,
+ k,)' ,
t = (k,
+ k3 )' ,
u = (k s
+ kd'.
(8.2)
For the first diagrams (s-channel), we have the amplitude
(8.3) wi th m the quark mass. The last term in this equation ;s obtained from the previous one by interchanging the quark lines, and the factor 0,./ v'3 combines the quark and t he antiquark in to a color singlet st.ate. Defining the quantity O,(p, q) by the equation
M, = u(~p+q)O,(p,q)v(~)J-q),
(8.4)
and using the formula
Tr(TdT') = 10 2 de
1
(8.5)
11.1. fJfAMf:WOIth·
'Ol
M
9(k" b)
y(k"
bl
g(k-"c)
M
9(k"b)
M
9(k"b)
+ permutations Fig. 8. 1: Feynmau diagrams for 9 + 9 _
zS+lL J
+ g.
8.
1111
'1'11/1 1' 1l0J)IJGTIUN OF qUAllI(ONIA
we have . 3 jQbe
O,(p,q)
-'9 J3 [(k, -
2$ 3
kd' (f; · t;) + 2(kl' f;)f;' -
2(k,· ti)t;"]
X [1',(1)-2 1I-2 /c3- 2m ) ,lj 2(p . k3 ) - 4(q . 1.-3 )
_ ,l;( I> + 2 II - 2 /cd 2m)')',] 2(p ' k3)
+ 4{q . k3)
(8 .6)
For the t- and u-channel amplitudes, one readily writes down sim,ilar expressIOns.
For the last diagrams) not invol ving the three-gluon vertex, one fi nds the amplitudes . 3
OD(P, q) = ~
[Cp' fil X
+ 2(q . fil+ /<1 ,lil ,l; I(p , f,) [2(p. k 1 ) + 4Cq . k1 ) ] [2Cp· k3) -
2(q . <j)+ ,lj ,'\'3] 4(q . k3 )J
+ 5 other permutations of (1,2,3 and a, b, c) .
(8 .7)
vVe now comb ine the fl'ee quarks in to a quarkonium state. This is done using the nonrelativistic bound state a.pproximation
M "" 2m,
(8.8)
where M is the quarkouium mass. It also means that for S-waves we can put q = 0, while for P-wavcs we have to retain the terms linear in q in the amplitudes, q being the relative momentum of the quarks. Following the method of Guberina, Kiihn, Peccei and Rucki [39], we introduce spin projection operators, which combine t he quark and anti quark spi ns to the appropriate singlet (5 = 0) or triplet (5 = I) states, and lake into account the nonrelativistic bound state wave function of the quarks. We refer to Guberilla et al. [39] for t.he details of the method and simply list tbe different amplitudes for quarkonium production:
At'So)
C6;M) '
1
A('Pl)
-if(..
TrltaOaC I> -
Mho + 20 I h,!l',,fJ ,
11.1. ntAM('; IVOIlK
A(' Po) =
'II! , , n[0.(o'-P"jM)(/HM)+601, ";l61rM
A('!'1 )
-
-
R'1 ( 3211"3Af 3
)1 <·1h6 t.:
j)'ll5
(8.9)
!n
lhes~
formulae, we have introduced
o
=
O,(p, O) + O,(p,O)+ O. (p,O)
D qO
e[O.(p,,) + O,(p, q)
+ O.,(p, O),
+ O.(p, q) + OD(P, ,)1
(8. 10) q=O
and the qua.ntity lk>, .... bid) is the S-wave wave fundion evalun.ted at the origin, and ~. the derivative of the P-wavc wave fuuction also e\'aluated at the origin. The quantity lk is simply related t.o lht! leptonic width of the
JS1 state through lhe formula. (8.11) willl a ~ 1/1:17, t he fine struct ure constant, and Q I the fractional cha.rge of the quar}cs. Using the quarkonium potentia.l of Hagiwara cl Id. 140\ wilh 1\ = 0.2 GeV, one finds for the (ce) system
f({ I M; = O.006(GeV)' ,
(8. 12)
wherlP. Mx is the tn<\ ~S of a 3P state. Finally, the qlJlI.ntitics CO alld ("'A ill the expressions for thc amplitudes describe the polarization states for the mlls.sivlP. spin-l and spin-2 quarkonia. Eqns (8 .9) are obtained by applying ~veral a.pproximations to a relativistic Bethe·SlIlpeter equation describil1g the production of Cjuarkonia. Because of thcsp. a~proxima.tions, it i9 llot clear how accura.te these equations arc. Nevertheless, in the a.bsencc of morc acc urate Olles, the calculations in this chapter will be based on cqns (8.9). We DOW apply the hclidty method Lo the calcula.tion of tht: production cross sections of these quarkonia states [41 ]. Let us denote by M(AI, ).1, A3) M y helieity amplitlldefor 999 - > 2S+1LJ, where the labels Ai, i == 1,2,3, refer to thc hclicities of the three glIlOl1S. Jt is sufficient to I.:onsider J\JJ( +, +, +) and M (+, +, - ) only, as all otller helicity amplitudes call be obtained by per1TI1Iting the gluons , or by a. parity conjugation, which flips all he!icitics. We prCllcllt the calculation of the ISO produdion in some detail, iJ.S it allows us
100
H,
'1'1I~:
l'IIOI!U(.''J'JON Of qIJAllIWNIA
to illustra!.e, through a sufficiently .illlple example, the basic features of the helicity technique when massive particles are present. For the other states ,
shall merely present the results of the calculation, a, the computational techniques lire almost identical. we
8.2
ISO production
8.2.1
The amplitude M( +, + , +)
For incoming gluon, wiLh positive helicities, we can choose the following polarization vectors:
(8.13) with the normalization factor
N = (2stu)-t .
(8.1 4)
This choice of polarization vectors has the advantage of being inva.riant under
cyclic permutations of (1,2,3). With the identity
4(a . b) =
,
Trip pJ,
(8.15)
one can now express all the scala.r prod ucts of four-vect ors which appear in a given amplitude in terms of s, t and u . For example,
(p. <,) -
(k, . fi)
Nsi,
(p. f,)
(k3 . ';)
Ntu ,
(p . (3)
(k, . 'j)
Nus
(8. 16)
,
"
«j'f;)
(f;·f')
«j , fi) -
1,
while many other scalar products, like (k, . ,; ), (ks '
k,
+ />, + k3 ,
s
+ t +" =
M',
(8.17)
and
2(p· k,)
=
s
+ u,
2(p. />,) = s + t,
2(p. k3 ) = t + u, (8 .18)
It.e. 1,'JO l'IU)lJfl(."1'fO""
For '-he
CMe
107
of ISu production, we /ir.sl calcu late
(8.19) l"[O(p-Mh,[, 16K M wilh ollr choice for the qlln.ntities ~, i = 1,2,3, eqns (S.l!!). The .'I-channel diagu.ras g iv~
M(+,+,+) =v'llo
o• (p, 0) --
;,,'1"
;:--;?':'c'--~ "20.'1(., - M'l)
x[(,k,- /<.)(p -, j<, - M) /; - 2Ns' /;(p -
,/<, - At );;
-/;(p-2h+At)(h-,k,)+2N,' !;(p-2/<,+MU;[. (8.20) "-tld denoting ~heir contribution to M( +, +. -) by A" .....e h~vf'!
A. =
2' 3r k
~
Ru
, Tr[(ftl231fMs(s -M1 )
~~) ~J Ii Pl~ + 2N .'It J..i ;'3 A; .h$l·
(8.21 ) OeC<\use of the"y~ appeariag ill the trll.a', Lhf'! la.s~ two t(!nns in O. were foulld to give the same colltributions as the first two terms. We can now IOubsliLute lhe expressions for J..i and j.j in A~. All
(8.22) we find that only one term in
I;
contributes, and
ig3r&:~N
A,
=
2v'3.,1,/s(s-M'?'[-(/<'- h) j<,
/<, t , )<,
p(I--,,) (8.23)
Using momentum alilservlltion, the traces can now be wriUe.1. with ).1. Jq
2ig3r~R¢N
A. = -
li;;M
t.
(8.2')
The analogous contributions from the t- and u-cha.nnel diagrams can be obtained by cyclic pcnnubtion, .\ncl
,,_ 2i93 r~o ~N A.f' A .~ A 1+/1 .. - - ..J37rM I Y ,
(S.25)
Next, Wf'; have
+ 2 cyclic pf';rmuLn.t iou':I of (1,2,3).
(8.26)
'I'II~:
H.
IUK
I'II.OI!IICTION
or CiUil/UWNIA
Indeed, br.c;'l.us(~ of tile rft in the trace, we fillo that the dflk contribution vanishes and thai each nonvauishing term appears twice. Substituting fti
[eqn (8.1 3)J then shows that the first term ill AD vanishes, and
J3~M(s - M')(t - M')
xTr[-st
h /<3 /<, /<3 /<. /<, /<3 p( I - 15)
+ 2 cyclic permutations of (1,2,3).
(8.27)
Similar manipulations to those in the A,. case then lead to the resu lt
2ig3 f"""Ro N
AD
J3r.M
=
M'stu (8 - M2)(t - M')(u - M2)'
(8 .28)
and
M(+ , +,+)
2ig3 J"'cRoN M' [
-
J3'lfM
2ig 3J""'JI.,N
kf'(st
+ tIL + "5)
(5 - M')(t - M')(u - ,11') .
J37rM
8.2.2
stu
1 - ($ _ M')(t - M')(u -
,11')
1
(8.29)
The amplit ude M( +, +, -)
This time, the third gluon has an opposite helicity, a.nd we find it convenient Lo choose
,i;(k,) -
- Nih
h
};3(1 -
,,)+ k };, /<,(1 + ')'s)J, (8. 30)
where the normalization factor N is defined by eqn (8.11) . Note that, because of the opposite helicities of gluons 2 and 3, it is perfectly possible to take l; = - E3. The minus sign in fi( k2 ) is introduced to ensure gauge equivalence with the choice of 'i rk,) in eqn (8 .1 3). Indeed , it sltffice, to anticonunute /<3 and h in eqn (8.13) to r,nd that the two expressions differ by a term proportional to fi;2) which can be omitted because of vector current conserva.t.ion.
11.2.
IS~ l'IWI!II(,"l'lON
]U!)
With the choice (8.30), /1,1: scaIOl.I' vanis h,1l.nd
pr(jdll(:l.~ ,1Il\Ollg
(p.. (i)
(kJ·t;)
_
-N:JIl,
(p·e;) -
(kl·ti) -
N:JU .
the eM's arc made to
(8 .31)
Again, many other scalar products vanish, and the eva.luation of the helicity amplitude is greatly simplified. T he ~ - ChatilleJ diagCi:l.lIls now give O~(p,O)
=
ig
3
r""'N
,,/3(, - M')
x[" 1-;(,,-2/<,+M) -t
;;(1' - 2 /<, + M)
;;-u /;(p-2
,;(1, -
,; + t
)c,+M) ,;
2/., ·1- M)
I;], (8 .32)
and
N 2 · '!Q,-O - . ~. T[ A , = 'Y ~ r U ,/" I 1'3 v3". M !Jilt ,
with the choice of
ci
I - I.
1'3 ,P7~
_ ·t
/ "
1'
,-
2 ,,3 1'3 .t/)'~j .
(3_3:!)
in eqns (8.30), we have
(8_34) hence A. :::: O. Similarly, AI = O. However, i,g:i J"'>c N s O.(p,O) ~ - ,,/3(u _ M')[/; ( p - 2
h + Ni ) ;;-- /;(1) - 2 h + U ) /;J, (8.35)
and
2i. g J /"~ ~./I,7 v'31rM - u The cakulat.ioll of AD (or vulved:
81
M2 thi~
hclicity amplitude h;
(8. 36) som~wl1a t JIlor~
in ·
110
R. '/'III: /'/WO/IC'/'(()N 01' qIIAII/WNIA
AD
-
- 2J311' M(~ _ }\1')(t _ M') '1'1'[1:, ;; ; ; +(p . ,;)
Ij I, hs
I; I; h h, + (p. 'j) I, I; I; h,1
+ 2 cyclic permutations of (1,2,3).
(8.37)
The fact that
I; h /; =1; /c, /; =,k, /i /j =
0
(8.38)
simplifies this expression considerably. Tt follows that AD
=
_ ';g3 J';'R" Tr
[(p, '5) I, /i ,ti hs
2JhM
(s -- M')(t - AI')
+(p. ti) I. Ii h
fry,
+ (p .•j) Ii Ii h hs ]
(I - M')(u - M')
_ ;9' j,b< R"N3 SU Tr J37rM
(u - M2)(S - lv(2)
[I, 1<3 1<, I, I, I, fe3 .I>{l -I . ,,) (5 - kl')(t - M')
fe, I, fe, I, h /<, fe , .1>(1 - ,s) (t - }\1')(u - lV/')
_ /<3 /<, h fe, /<, fe3 fe, P(l - ")] (u - M')(s - M')
2;9 3 J'b
s'u' (5 - AI')(I -- M')(u - M') '
(8.39)
Adding A" and AD, we obtain the helicity amplitude
2;9' J"
M(+.+,-)=
' RoN
J37rM
s'(sl + lu + us) (s-Nf2)(t-M' )(u-M')'
(8.40)
Obviously, the helicity amplitude M( +, -, +) is obtained from the above one by replacing. H u, and M(-,+,+) bys Ht.
8_2.3 T he cross section for
ISO
production
[n Sect;ons 8.2.1 and 8.2.2, we derived the nelicity amplitudes for the process
y(k,)
+ g(k,) + g(k3) .....
'So(p).
(8.41)
+ g(k,) .
(8.42)
Ultimately, we are interested in the process
g(k,)
+ g(k,) .....
'So(p)
11.:1.
'/'/!8 /' /W!JI/V,/,ION 01" (J'!'/INll S'!'tl'/'l':,'i
111
'1'1> ohlniu thc hdieil.y illHplituucl'I of I,he liltt"" proccs,;. it ~uflict:s to replace ~':l by -k~ in the ilciicity amplitudes of ~lle ronncr proccas and \.0 flip the 1ll'lidly of g luon 3, I1cncc, for 9 g ..... 18,,9, 2i.g3r~"~N
M(+ , +;-) -
j3rrM - (, ;',M
-
M(-,+;-) -
2ig 3r
CC
'liT
r
lx
114") (<<·- M')'
M')('
(, (,
u 1(st + ttl + liS) "1')(' AJ2)(u M'J) ,
RoN
../:hrM
+ ttl + 113)
SZ(st + hJ + u.s) M')(t "'P)(II M') ,
2ig"''f"'"'' 14J tV
M(+,+;+) M(+, - ; - )
M4(st
R~N
t~(st
+ lu + tis)
(s - A11 j(t -1\J 1 )(u - J' P)'
../3;r Af
(8,43)
wh ere
'k )" , u=l,-k)M
(8.44)
All the remaining hdjcHy ampliLudes can be obtl'. ined by al-'plying a parity CO!ljugM:on to ihe above ones, The cross section, aVl~raged over the i:Jiliai polarizations and color degrees of freedom, lhen reads
st + hL+US ]2 Mil M 'l )(! - ft.-P)(u - Jlf2)
w:th as
+ 54 + 14 + u~ st1l
=- g2/4~, Using the variables P = st +tu+
Q = stu,
tiS,
which are symmetrical under all permutations of s, l a.nd Lhis cross section formula in the following form:
du _ eft
8.3
(84.5)
:ra~-m MJ~
P2(M8 - 2M" P
Q(Q
(8.46) 1.1 ,
+ }-'2 + 2M2Ql Af2pp
one can rewrite
(8,47)
The production of other s tat es
In this sectiun, "Ie present the rt':slIila of the ca\cu la Lion for the P;"orlllc~joa of the other quarkonium , t·ates 3SI , · P I , 3PO, ::'Pj and aP2 ' The given helkity amplitude!; always refer to the process 99 9 --o2S+1 L } , but we have explained in Section 8,2,3 how the crossing of the third gluon should be performed.
J J~
8. ,{,III,,' I'IWIJIIC,{,ION
8.3.1
or fiUAltiiONIA
381 productioll
For this process, only the last diagram. of Fig. 8.1, uot involving the t.hrcc· gillon vertex, contribute. "Ve use the same polarization vectol'S for the gluons
as in the
ISO
case and we make use of the relation (p. <) = 0, where
f.
is the
polarization vedor for th e spin-l qua.rkonium. We then find
M(+,+.+) - O. M(+,
2gJdabc RoN M5 )3"M (5 - M2)(t - M2)(u - M')
+, _.) -
xll(t + ,,)(k,· <) - U(t
+ u)(k, .• ) -
s(i -U)(k3' <)
(8.1 8) Using the relation
Ll':llf: =
-g,lv
+ PI~p",/j\1'l ,
(8.49)
pol and summing over the gluon color degrees of freedom, we find 2 l0240,,'a1mM 5' L." I M(+,+.-)I = 9(t-M' )(u- ,wz)"
'"
(8.50)
pol
Symmetrizing lhis equaLion with. respect to s, t and u, and averaging over the initial state degrees of freedom then yields the cross section form ula:
5".",~mM [ s' 98' (t - M')'(u - M')'
dlT dt
t' + (u - M')'(8 - M')'
,,' + '(s - M')'(t - M')'
1
(8.51)
In terms of the va.riables P and Q, eqns (8.46), this l'eads
dlT
di= 8.3.2
IPI
lO"a1R.i 95'
M(P' - M'Q) (Q-M' P)"
(8. 52)
production
As in the 'Sl case, the J-, t· and u-channel diagrams do not contribute, and t he amplitude At' PI) is again proportional to dab,. We obtain
ig 3d'" R' [ . M(+,+,+)=- v:;M1 al«·<;)+a,«.<;)+a3«-<;) .oM -I- t
2b, M' (k, . <) + -
2b, \1' (k, . <) U- 1
+
21;,
1
\1' (k3 . c). ~-1
(8.53)
IU
'ff!/o,'l'ftOlJlIC"/,/o N
m' OTIIJ.;//
S'Ii1'f'K'i
will!
a,
-
(,
2(2u+s)
a, -
(t
b, -
b, -
t\.P) ,
Al1)(u
2(2) + I )
a,
b, -
2(2t +u) M' )(I A11) ,
(u
M2) ,
M'l) (s
2N [ -sW + ttl - 11M2) t - M1 S - M2
+
[_t(U 2 + su - sM') M2 / AP
2lvu
,
[-U(.9 1+ 1.3 -lMl)
2N
Ml
U
t(u? + su - sM 3)J
+
U{Sl +.'It -1.:",,111J AP
.'I
_ s(tl
+ ll1 t
I-
Ml
M~
Il -
,
W\.1 l )]
M2
.
(8 .54)
Arkr s,unmation over the polarization states of I Pi and the color stales of LI:e g1110115, we ob~ain 409607l"~aiR?
LIMCI .+. +)I' -
+ 5Q) (Q -M'P)'
M8 (_M1P
3M
pul
(8.55)
Si milar ma.nipu lat~ons give fOI" the amplitude M ( +, +, - ) it fo rmu Iii. as in !.II{' M (+ , +, +) case, but now,
2, (t - Ml)(U - M2) ,
25(t + u - 2M~J (, ·M')(t M' )(u M')' bl
[I./1,,12 + u I
2Ns~11 -
b, -
t
-
M2
S
2 Ns~1./. [
u
jl,f2
t
2
" and,
(;OUt;.Ctj\lf;;lItly,
~
·Ml
lv[2
M2
+t
I
2fols u [ 3
M'
,
I./.
I
I
1
M,l .
M'] •
(8.56)
H. '1'111,; l'IWIJIlr:'I'ION OF QUA Uf(ONIA
IH
L I M( +, +,- ) I'
10960".',,' 11" = ----::-3M:-:-'S'---'-'
pol
[
M's'
(8.57)
The cross section for' P, production then becomes:
d" dt
20m}K,' 1 3 M s 2 "' [ (-s ----;-M""2""')(C"'t--""M-;:;2""')(C"'u - M') ]' x { M 4 (s'
+ t' + u' + M4) 2stu[s' + t 4 + u' + M' (s' + t' + II') + 2MB]} + (3 - M' )(t - M2)(U - M') .
40 ".",}R:,2 _ M'DP 3M.'
+ 1\16 p' + Q(5.1vf'l -
7M' P (Q-lvf2P),
+ 2P') + 4M'Q' (8.58)
8,3,3
3pO production
The evaluation of the helicity amplitude. for 3 Po production is sr.raightfor· ward. We find
M(+ ,+,+) M(+,+ , -) -
12g 3 /""'R:,N
J3".M3
M'P (s - M2)(t - M2)(U - M') ,
493 r" Il;.IV s'[t',,' - stu(s - 6M2) - 3M' 8'(S - M')] J3d1 3 (. - M')(l- M')'(u - M')' (8.59 )
/1 .•1.
T il,.: f'JIO/JIJ( ." /'lON
m" O,/"lIt:1t S'I"A 'n :S
I I ()
du dl
,[tU(f4 _ t 1u2 + u t ) X { 8M (S- M2)1
+t + st(S4(tl-- :Pl' lvPP
4)]
1l3(U4 _ U2S~ + s· ) + (t - M' )2
+ 1M'[M 2(3t + tu + U3) -
5,tul •
,[gUll 1 ( + 3t + tu + "U3) stu + (3 _ M7 )(t _ fl.Jl )(u _ M2 ) X(-16M' ,stu + SM 4 (s' ..;.. t'
+(1 _
4;rolll~J )\13 3'
I
Q(Q _ MlP )4
+ tI')
9M'(> 1- + ~l)<"
+ I' + U '))]}
[9M1 P1 (M II _ 2M4P + p 2)
- 6M·' P' Q(2M' _ 5M t P + p2) _ P2Q'(M II
+ 2M' P _ p 2)
+2M' PG'(M' - P) + 6M' G' ] with the q uanlit i<.'!> P and
8.3.4
3Pl
Q defined in
(8. 60 )
eqnll (8.46).
production
I'or the 3 P 1 case, a somewhat lengthy, but straightforward calcula tion sbows ~hat
M(+,+,+)
~
o.
(8.61 )
Also,
M(+,+ , -)
~
4 i!l3f~k R~N
V21fM
(3
s M2 )(t - M2)2(U - M2)2
-i(k 1 • t)(s + t){st(t - 3)
+ tu{t -
u ) + us(u - ,)]
-;(k, . ,)(, + u)[,u(u - ,) + "'(u-
'l + 1.(/ - ,)IJ, (8.62)
H.
1111
where
E
'1'111'; I'IWIJ//(;'/'/ON
or (JII /II/./IONIA
is the poliLrizaliolJ Vedoi' of Lhc al )l cptl'l.l' icouium. SlIlTllIliug over its
polarization states and over the color states of the gluons, we obtain
x (is
+ t)(t + u)(u + .)lst{s +M'(t. - U)'(8t
t)'
+ tuft -
,,)' + us(u -
+ tu + uS _ .')'} .
.)'1 (8.63)
Symmetri,ing this equat.ion in 8, t and u, and multiplying with the appropriate factors, we obtain the cross se.ction formula.
du dt X
t2U'(t'
+ u')
{ M' [ (5 - M')'
+ S2) + .'t'(s' + i2)] + 11'.'(U' -c--'---:-::"'-:-"(t - M')' (u - M' )'
+2(.'t' + t'u' + "'s')(s't' + t',,' + u's' + Ai'stu)} (s - M')(t - M')(u - M')
P' [111'P'(M' - 4P) - 2Q(111' - 5/vJ"'P - P') -15M'Q'J x (Q _ M'P)' ' (864)
8,3,5
3P2
production
For the eMe of 3P, production, we lind again that
M(+, +, +) = 0,
(8.65 )
The calculation of the M( +, +, - ) amplitude is somewhat more involved. We find
;( .1.
'l'1I1~
l'IW/JU(."f'UJ/V
M(+,+,_)=_8 x
gl f
or 01'I/F;/t S·/'A'/'Jo,·...; -;JI!IN
{Sl [I(t :1:141) -
111
/Mr... (1
M1)J [t(t + U)kHI -
u(u k:J.\
st+t'U+us [(3 + I
-s(t - u)k~
t - tI)(t kl" - uk:l
,'U
"k'k!.[k . +2" ....010'..:1 l : l 13
(u {t_J) M' ) 1
(
t t-
"(u-,)
u(t + u)k1P
t
+u
tu
- -s -
t(st+tU+1t.J») su
u' *,+,u+u,»)]} , st
-k" ( 1.11.1( '1,) -u+ t - . s-
(8.66)
111 this equation, t.",P is the pola.ri~atiou tCllllor of ~he 3P, quarkonit:m, which .s .. Lisncs t he relations p"t
,.
=
0
,
t .... =
o.
(8.67)
'['u oblain t he cross section, we have to sum over the polarization slates of Llw 3 P2 sys tem Ilsing
Er."'/l<~
-= HP" .. P{Jv + p,,~pP... ) - !P",P,." .
(8.68)
pol wi~h
p..~
= -9.." + p.. p.. I.'v1~·
(8.69)
Also summing over the color degrees of freedom, we find
(8.70)
8. 'I'll II' l ' IWJ) lfCl'ION OF OUAlUIONfA
I IH
It follows that, fur " P, production, d~
4r.a1Jr.'
dt
M's' X
1 Q(Q-M'P)'
[12M'P'(M 8
-2P'Q'(7M 8
-
-
2M'P + P') - 3M'p3Q(8M8
43M' P - P')
+ M' PQ3(16M'
-
M' p
- 61P)
+ P')
+ 12M'Q']
X{M' [1 2U2(t' + 4tu + 1.1') + "'S2 (u' + {us + s') (8 - M')'
(I -
M'l'
+ 8't'(S' + 4s1 + t')] (u - M')' +12M' [3(8 3 /
+
+ 13U + u3 • + Sl3 + lu' + Us 3 ) + 1M' stuJ
+ us - M" )(si + tu + us)' [ (. _ M2)(t _ lvf')( u _ M') st + tu + us -
2(81 -I- lou
-6M' (.,t
8.4
+ Itt + itS -
MI)(;
2
.r-!
411'1
+T+~) l} . (8 .71)
Conclusions
We have shown that the application of the heJi city formalism leads to simple and useful formulaefor tbe process 9+9 -> 'S+lLJ +g, with L = S or P. For a detailed list of formulae, see Chapter J O. Tbe simplicity of our formulae, especiall y in the 3 P-cases, allows one to make more refined stud ies because of t.he savjngs in computer time. The fact that we obtained the separate helicity amplitudes for these pro· cesses, ena.bles one to consider the more complicated case of. quarkonium product ion from polarized gluons. We explain in Chapter 11 how the various helicity amplitudes should be combined in this case. Although the helicity met hod was originally designed for massless fe rmion processes, it. is seen from t hese examples> that, the formalism can a.lso be llsed advantageously in the case where massive quark pairs are produced,
9 Summary of QED formul ae /\11 Lho fo llowing (ormttlae arc presented in ~he centre-of-mass system, with 1.111,.1 energy deno~ed by 2E. Unless otherwise slaled, we take the positive zi1ld~ along 14. For an arbitrary foul'-vector k, we frcqucJltly u:,;.:: the uoLation I'" cqn, (6 .8)1
(9.1 )
kJ:. = ko±k•. 'l'11!~
indication m = 0 in the sedion headings imp!if"~~ that, throughout the l u·( ~tion, we flflglect the effects of a finite electron and/or muon mass . ConfIPq \IC[l tly, the formulae are not v(>.lid in certain kinematica.l co:lngu.ra.tions. I,'of a. discussion of the ranges of \·alidity of the approximatioll, we retcr to Sl~dion 3.4. Alternatively, the indication Tn #- 0 impliet; that the formulae ill I,IIC section are valid to leading order in the fermion mas:>p.s. The labels of the listed helicity a mplitudes always refer lo the particles in 1,I,e order in which they appear in the given process . For example, for e+ e- --+ 11+ ,F-y, the hc1icity amplitudc M( +. - j +, -, +) describes the annihila.:io:l of a positive helicity e t a.nd a. negative helicity e- into a positivc hdicily p.+, ;, negative he1icity /1- and a positive helicity "'(. As usua.l, the symbol ' ..:- ' ill to be rcad as 'an c(]uality sign modulo a phase factor'. fn this chapter , the symbol I MI~ is used to denole the square of the absolute value of M ~lImmed over the final state polarizatio!lS and averaged over the initial state IIO\arizaiions.
Process:
(9.2) Invariants: s = 2(p+ . p_),
t = -2(p+ . kd ,
(9.3)
Photon polarizations:
• i = 1,2.
(9 .4 )
I ~II
!I. SIIMMAIlY OF (JIm FOliMULA/i
Nonvanishing hdiciLy (l..lIIplit.uucs;
M(+,- ;+, - ) = J.
_ 2e'(~2+)' k._
M(+,-;- ,+)
(9.5)
_ 2e
M(-,+;+,-)
l
(~2+)' k. _
2
k' M(- ) + '. -J +) - . . ?e' , k~ 1-
Unpola.rized squared matrix clement:
IMI' =
20' (': u
+ ") . t
(9.6)
U npolarized cross section:
(9 .7)
9.2
e+e-
-+
/1+ /1- (no Z-exchange;
m=
0)
Process:
(9 .8)
Invariants: " = 2(p+ . p_) ,
(9.9)
NOllvanishing belicity amplitudes:
M (+,- ;+ .-) =
M(+,-;- ,+)
M (-,+; +,-)
, e
q_~ E
2 q:.l. e -
E
(q _+ ) q++
l
.
2 q-+
e
(q-t )l q++
E' (9.10)
-
• q-+ e
E'
1
M (-,+; -.+)
e2 q+J. (q++) , E q_+
-
e 2 -q++
E'
121
(9.11) UlltJolarized cross section:
(9.12)
9.3
e+e- -
p.+ p. - (with Z-exchange; m = 0)
l ' I'OCeISS:
(9.13) luvariants and definitions:
s = 2(p+ · p_),
4x,
A(x. y) ~ 1 + -y----;Uoit"'+'-i"M',"r .
(9.14)
Nouvanishing helicity amplitudes [for the definitions of a and h, consta nt s of the Z to the leptons, see eqns (4-.47 ) and (4.48)1:
M(+.-;+,-) -
M(+,-;-.+) "1(- .+;+.-) M(-,+;-,+)
~
elA(a2,s)q~.Et (q-H)! q_+ (l2A(ab,s)
q-.1. (q_+) E
-
1
q++
.( r
e 2 AW,·') q.t..L
E
-~
q++
(1++ q_+
coupling
c2 1A(a1 •$) ,' q++ E '
,
q-.1. q-+.•
c 2 A(ab,s) - -;E
-
tllt:
r
= e2IA (ab,s)J
-
e1JA(ab,s)J1:
-
e1IAW,·~)1
,
t,
E
q;+ .
Unpolarized squared matrix element:
(9.16) Unpolarized cross 5eclioll:
(9.17)
I~~
9. 811MMAilY 01"
qr:1i
1"OIlMULAIi
c+e- -> e+e- (no Z-exchange; m = 0)
9.4
Process:
(9.18) In variants:
s
~
2(p+ . p_ ) ,
(9. 19)
Nonvauishing helicity amplitudes:
Eg"- J(i++q~ - )!
-
M(+,+; +,+) -
4e2
M(+ , -;+ , - ) -
> ,q-J(q++ ) , e E q+_
M(+ , - ;-,+) -
e2 -q_~ (q_+ - ) E q++
t -
E
4e'- , q+-
,
q±t
2
eE q+- ' q-+
2
E
e
I
(9.20) 1
M(- ,+;+,- )
2 q~~ ) , e - (q-+ -
E
q++
r
-
'1- •.
2
e
E'
=
e2
q++
-
.. E . 4e" -
,
M(- , +; - , +)
,q:J.. (q++ e E q+_
M(- _. _ _ ) = , , ,
4e'
Eg-J.. ;(q g3 )~ ++ +-
,
E q+- '
'1+-
Unpolari7..ed squared matrix element :
(9.2 1) Unpolarized cross section:
(9.22)
9.5
e+e-
-->
e+e- (with Z-exchange;
In
= 0)
Process : .
(9.23) Inva riants and definitions:
s = 2(p+ · p_),
,,~
- 2(p+. g_) ,
123
A(%,y) = 1 + y _
'Ix"
(9.24)
Ml + iMzr
NOnVaJlishing helicity "amplitudes [for the defini t:oll5 of {J and b, the coupling cunstants of the Z to thp. leptons, ~ eqns (4.41) and (4..48)J:
M(+,+i+.+l
~
(qHIf~ _) t
,
£
~
M(+,-;+.-) -
M(+ ,-;-,+) -
-
M(- , +;+,-) -
M (-. + i - , + ) ~
M (- ,-,- , - ) ~
,+-,
,,'IA(ab, ')1-
e' [A (a"') _ 2£ A(a',I)] , .. E
q+_
2£ e' A(a',,, ) - -A(a2,l )
q+-
e 1 A(ab,s )
q_J..
E
,
(q .. ) I qT-
q.. E'
(1-1)
1
q ....
e1 IA(ab,3) 1q~- . e 2 A(ab, s)
q:.L (q_+) E
, l
q+ ...
el!A (ab,,, )! q~- , e1
[AW ..J ) _ 2E AW,t)] q+.l (f]-h- ) t q... _
e' .4 (b',,) _ 2£ A (b',. )
q,-
4 e~A (ab t) ,
.
,
E
q_+
E'
Eq_.!.. (qHf{!_ )l
E
4e' IA(ab, '11- . q,-
(9.25)
121
V. SUMMAIlY OF QlilJ fo'OUMULAE
UfJpolarizcd
IMI'
=
~qlla, rcd
mat.rix clement:
s~:,{t'u'[IA(a"s)I'+ ;:IA(a 2 ,t)i'+ 2;Re(A(a',s)A"(a',t)) +I A(b', 8)1' + ;: IA(b', 1)1' + 2S Re(A(b',S)A"(b 2 , t))] t
+2t"IA( ab, s) I' + 2s41A( ab, i)I'} .
(9.26)
Unpolarized cross section:
~~
-
:4~:{t2"'[IA(a2,s)12+ ;:IA(a 2,t)l'+ ">e(A(a',s)A"(a',t)) f
IA(b',s) I' + ;:IA(b2, ill' + ~s'Rc (A(b2,slA' (b', Il)] +2r'IA(ab,s) I'
+ 28'IA(ab, I)I'} ,
(9.27)
Process:
Invariants and definitions: s = 2(p+ . p_)
(9.29)
Photon polarizations:
N,1
i
=:
(9.30)
1,2,3.
Nonv.nishing helicity ampli t udes:
.111(+,-; +, +, - )
-e'v'sCk,_ku
-
e'v'sC(k-3+ k'3- )!
M(+,-;+ , -,+) -
- e' v'sC k,_ ku
-
e3v'sC(k2+kU~ ,
M(+ ,-;-,+, +)
-e'v'sCk,_kU. -
.111(-,+;+,+,-)
e'v'sC k3+k:i.l
-
c'v'sC(k1+ k')t 1-
,
,
e'v'sC(kl+k3_) I ,
caJ8C(k~+l'l_)t ,
M(-,+;+,-,+) -
e:lj8C k1-t k~l.
~
M(-,+;-,+,+)
e3 y'8C kl-+-kil.
-
e3v'8C(,l;~+kl_)t ,
M(+,-;+,-,-)
eJy'sCkl+k11.
-
e3J8C(L·~ . kl_)t ,
M(·,,-;-.+,-) -
cl J§Ckz+k2l.
-
eJv'8C(k~+ k2_)~ ,
(9.31) 3
M{+,-;-.-.+) M(-,+;+,-,-)
e
J8 Ck3+ k3J.
-e 3 y'8Ck1 _kj'i
~
-
e v'8C(kl+k3-)t.
~
"vsC(k,.kl_)! ,
M{-,+;-,+,-) -
-c 3 v'8Ckz_ k:i.t -
M(-,+;-,-.+)
-e3 .J8Ck3 _k
s
Uupo!arizcd squared matrix
..l.
J
"VSC(kHkJ_)! ,
-
e3...rsC(kl+~_ ) i .
e1emen~:
, L (P+ . k;)(p_ . k;H(p+ . k;)'
IMI2 =
'le6s
,
i-I
+ IP- . k;)'1 (9.32)
IT IP+ . k;)(P_ . k;) ,:1
Unpola.rizcd cross scdion:
, a' ~ (p+. k; )(p _. k; )I(P+ . k;)' do
+ (p- . k,)'1
_ J
411'2
II1p+. k; )(p_ . k;) ;=1
(9.33)
9. 7
e+e- - Ji+ Ji- 'Y (no Z-exchallge; m = 0)
Process:
' - (p+)
+ ' - (p-)
~ ,,+(q+ ) + p- (q-l
Invarianls and defiuitions:
t
=
-2(P+ -q.. ),
I' _
-2(p_ .q_),
+ 1(k).
(9.34)
9. SIIMMAIIY OF QIiD fORMULA},'
I~(j
Z:.·0 + _
-
(9.35) Photon polarizations:
(9.36) Nonvanishing belicity amplitudes:
,
-
3
e q__ IBI
[(q+.
k)~q_ . k)]"
M (+,-; - ,+,+) -
l>,l( - , +; - , + , +)
3
-
e q++ IBI [(qV 3
M(+,-;+,-,-) -
e
kJ~q- .
,
kJ
q_.cq__Zt_ B" ( q_+ (q+ . q_)qtq+ . k)( q_ . k)]} 1
3
-
e q++IBI [(q+ . k)2(q_.
kl]'
127
e' q ....· l. Z· + B'
MC-·+;+.-.-I =
M(-,+;-,+,-) =
[qHq-+(q+' q-)(q+ . k)(q_ . k)J~
,
~
,', __ 181 [(',. kl~'_' kl)'
(9. 371
(Jnpola.rized squared matrix element:
(9.381 where [see eqns (4.27)]
',=
p... (p , . kl
Ullpol1\ri~cd criJ~S
-
p(p_ . kl
.
v~
_.-
q+
(q, . kl
q-
-
(q •.
kl
.
(9.39)
section:
(9.10)
9.8
e+e -
-+ I'~
p - "( (with Z-exchangej
Tn
= 0)
Process: Invarianls and definitions:
, s' -
2(Q+,q_},
-
t' -
-2(p+ -•• ) . -2(p_
-._1.
"
=
-2(p_ - q_l.
'tt'
-
-2(p_· 9+1!
9, HIIMMAIIY OF (JIW 1"()11MU/,Ab'
[J ,
-
4xy
1 + ---:-:-::-,,-...,.,-,=-= y - M'j; + iMzr
A(x,y )
(9.42)
PhoLon polarizations:
(9.43) Nonvanishing helicity amplitudes [for the definitions of a and b, the coupling constants of the Z to the leptons, see eqns (4,47) and (4,48)]:
.~1( +, _',+) _ ,T' ) -__ e3q_~ Z,t·_ [ A(a',s) + B,A(a',·'n.1 ]Q++'I-;,(9+ 'q_)(q+ 'k )(q_ ,k)] , 1
.:. ,3 9__ [(g+
'1'(
J~
'k~(q_ 'k)], IA (a',s)+B t r1(a' , s') I,
+, _', _ ,+,+ ) =
_
c3quZ_+ [A(ab, s) + B,A(ab,s')] 11++9_+(9+ ' 9-)(Q+' k)(q_ ,k)], I
.:. ,',,+- [(1+' k~(q_, k)] t IA(ab,s) + B,A (ab,s'lI, M( - -'-,
_
, " +, , +
)=
_
e3q~l. Z_+ '1+_
[
'1 - +
9++(q+' 9-)(Q+' ")(9- ' k)
x[A(ab,s) + BtA(ab,s')]
]1
129
\1(-
,
,_
) __ c:Jq • .i Z.. _[
,+, ,+,+ -
,__
,++
'I_,('v,_)(",k)(,_,k)
]1
x[A(b',, ) + B,A(b',,')]
~ 0',++ [(", k~'_' M(+ _''''' __ ) = , " , ,
_
,
kJ
[A(b',s)
e3q_.LZ~ _yq:;:+[A(a 2,s ) + Bi A(a~,s/)J , __ [,_.(,., ,_)(,., k)(,_, k)Jt
, e q++ [(q+ , k~q_ . k)] ~ IA(a 3
..:.
M(+ _,_
+ B,A(b',")[,
2
2
,s) + BiA(a , s')I,
+ _) =_c 3q'I.LZ "+..;q::;[A (ab,s)+
, , , ,
..:... eSq_t [(q+,
B j A(ab, s')]
,,- [,++i,, ' ,_)(q, ' k)(,_, k)l t
k~(q_.
,
klr
IA(ab, s ) + BjA(ab,s')I ,
. _ _ 1 __ c3Q±J. Z -+[A«(lb,s)+ BjA(ab , s') ] M( , +, +" - [q, kl(,,_, k)I'1
,,-,i,, ' ,_)(,,'
-'- ,',,- [(,' ' '1( -
J~
k)~'_ ' k)
r ,
IA(ab, ,) + B; A(ab, ,,')[ ,
_I __ e3 q" .lZ±_IA(b1 ,s) + Bi AW,3')] ,+, ,+, .L ._
[, .. 'I-,(q,' ,-)(q, ' k)(q_, k)J '
-'-
, "'I-- [iq,' k)~'_ 'k) ]'IA(b"')+B;A(b"")I '
i9,14)
Unpolarized sq u ared matrix element:
,
IMp
=
X
e
2i'l,' k)(q_ ' k)
{q_~+ [ I AW,s) + B I AW,s')1 2 + IA(a2 ,3 )
+(q!_
+ B~A (a 2, s'WI
+ q:'+) [I A (ab, s) + B1A( ab, s'W + IA (ab, s} + Bj A(ab, s'wj i9,45)
9, .'/lMMAItY OF QED FORMULAE
130 U npolari~cd
cross section:
a'
dO" = :-::---;;--;------;-"-,, 16,,'8(q+ . k)(q_ . k)
X {q~+ [IA(b" s)
,
)
+ BJA(b' , .')1' + IA( a' , 8) + B; A( a', 8' ) I']
+( q~_ + q: +) [IA(ab, 8) + B, A(ab, s') I' + IA(ab, s) + Bi A(ab, 8')1'] •
"
+q:_ [IA(a' , s) + B,A(a', 8'l!' + IA(a', .. ) + B; A(a', ")I']} (9.46)
9.9
e+c -+ e+e-'"'( (no Z-excha nge; m = 0)
Process:
(9,47) Invariants and definitions:
s -
2(p+'p-J,
u'
-2(p_ '9 _) .
B
1+
Bo
? _
-
-2(p+ ' q_),
,ti
-
E [2 Q_, (q+ ' q_)
+ kJ.q:J. k+
-
-2(p_ , q+),
_ kj,q_J.] L
kJ.q.j.J. _ k.i.q-.l k+q+_ k_q_+'
• q-l.. ) q++q-- - q+J.
(9.48)
Photon polarizations:
J± -
N[h p- ,k(l 'F /',)-
/< h "-(1 ± /'5)J, (9.49)
9,9, ~ ... ~-
-0
1:'t "-1 (NO Z.F;X(:IIAN(;j.,','
III
= II)
131
i\OIlvaJlishing hcli dly amplitullc8:
M(+,+;+,+,+)
=
l
_4c EQtJ. D [
(q ... ,q-)
q+_
qHQ-+(q+' k)(q_ 'k )
];
- 2e?EIBol[ (q+' k)(q_. 2 ]' k) M(+, +;+, +,-) -
~
-
c:)qt.LZ_+BO' [ (q+ ' q_) Eq...-_- q++q_+(q+, k}(q_ 'k)
)lll,1 [
..'(q+"
t:
2 (q+' k)(q_ . ' )
]!
M(+,-;+ , -,+)
M(+,-i-,+,+)
=:
_
e3 q+J.Z-+B [q++q_+ I,+ . ,_)(q+. k )(g_·
- ,'g+_llll [Ig+ .
"1 (-,+;-,+ ,+)
k:lq_. k)r
k)li
]1
fl. SIiMMAIlY OF Qlm fo'OIlMU/,AI!
e3lf+.LZ:±lr [ q_+ q+_ qH(q+ · q-)(q+' k)(q_ . k)
M(+,-;-,+,-) 3
. e g_+IBI
]1
kl
[('I+' k~(q_ .
0 3 '1*+.1 Z"-+ B.
M(-,+;+,-,-) -
1
_e3q+~IBI [('1+' k~q_. k)]' M(-,+;-,+,-)
M(-,-;-,-,+)
1
4e3Elf+.LB'[
M(-,-;-,-,-)
'1+_
-
2e
3
EI B oi
(q+ ' g-) ]' q++q-+(q+ . k)(q_ . k)
[('1+ .'k~(q_.
kl
1
(9 .50)
Unpolarized squared matrix element:
I"I'
_ __
lVl
",,,11,,2,
.£rl, )
0 -e 6( vp - v q )' 001 ,.,
-
-
.:I_",IIJ2
I
-112\ ) ='d/ ~ \2ul .1:/2\ , ) ,U, "
T" I ' T'
Slltt'
1
(9.51 )
where [see eqns (4.27)] Up
p+ p= (p+ . k) - (p_ . k) ,
v, = (q+'It. k) -
'I-
(q_ . k) .
U npolarized cross section:
dl7 =
a3
2 8S'(s'
-S1T'(V -V,) p
+ 8") + tt'(t' + t") + uu'(,,' + u") .2.'tt'
(9.52)
!l.10. t+,.- .....
t+~-')'
(WI1'/f r,. f,'XCJlA Nf.'Jo,'; m
I;
IJ)
13:1
Process;
Inva.ri&l.lts &I.ld dennitions:
t' _
A(x,y )
=
1+
-2(p. · ,.) ,
'ry
v -'.'''.'j'''''+~i''M''''r .
(9.55)
I'holon polarizations:
(9.56) Nonvallhlhing bdidty am pli ludes [for the defi nitions of a ilnd b, the coupling constants of the Z lO the leptons,!Iee eqll S (4.47) and {4 .48)1:
'0
M(+,+ ;+.+. _ )=_eq+l. Z - ~ [ Ev+-
, (1+'1-,7:-" ) ]'
Q++9-+(Q+ ' k)(q_· k)
x IB;A( ab, t) + B;AI ab, I'll
-
,'I,.E·,·) [(q .... k)(q_.k) 2 j'IBOA( a b,f ) + B O., (a b,t')1, 2
3·...
9, S/IMMAIlY
134 "
M( +,-;+,-,+ ) = X
m' (J/<; /I
rOI1MULAIi
.,
q-J. ." +1 [q++q_+(q+' q_)(q+, k)(q_ ' k)], e'
[A(a', s)
+ B,A( a', s') -
B,A( a', t) - B,A (a', t') ]
l.
-
e'q __
[(1+' k)~q_ 'kl
xIA(a',s) +B,A(a', s') - B'fl(a',t) - B3A(a',t')I,
, e'q+_ [(q+,
k~(q_,
kJ
l.
IA(ab,s)+ B,A(ab,s')I,
- " _ ) __ e'Q+J.Z_+.j1_+ [A(ab,s ) + B,A(a b,s')] .M( ,1,+, ,+ 1 q+_ [1++(q+' q_)(q+ - k) (q_ ,k)]' 1
-:.. c'q_+ [(q+,
k)~q_, k)]' IA(ab,s) + B,A(ab,s')I , 3
M( _, +; _ , +, +) = _ e q:J.Z+_ [ q_ _
x [A(b', s)
-
. 3'1++ [(q+, };;)2(q_,
q++ q_ +(q+ 'q_ )(q+ ,
+ B,A(b', s') -
k)(q_ ,k)
]i
B,A(b',t) - B,A(b',t')]
k)]l
xIA(b',s) + B,A(b',s') - B,A(b',t) - B 3A(b',t')I, M(+,_;+, _ ,_) =_e
3q -.Lz.i-_ [ q++ ]~ q__ q_+(q+ , q_)(q+ ' k)(q_ 'k)
x[A(b',s)
-
q . 3 ++
+ B; A(b', s') - B;4(b', t) - B:;A(b' . 1')]
[('1+' k)~q_ 'k)]! X
IA (b', s) + B; A(b', s') - B; A( b', t) - BaA (b', t' )1,
,
-
c'Q_+
[(,+.
=. e q+ _ [(I}~ 3
, k)2(, _ . k)]' IA (,b,,) + B; A(,b.,') I,
•
.k~(q_ . k)r IA(ab,s ) + B; A(ab. /)I,
- c', -- [1,+ · k)(q_ 2 ]! . k)
/v/(- _ ' __ _ ) _ _ 2e 3 Eq+ _Z':t [8;A(ab,t)+BjA (ab, t' )] , , >, ,+-1,++,-+ (,+ · q_ )(q+. k)(q_· k)]1
•
.:. 2,'£
[(,+ .k~'_' k)]' IBiA(,b,l) + R;A (ab,n l ·
(9.57)
,Q,
SIIMMAltY Of' Q":J) FOltMU[AE
Un polarized squaw" mal.rix clement:
IMI' = X { (.
e
•
2(q+ 'k)(q_ 'k)
+ '~')
[IB,A(ab, t) + BaA(ab, 1')1' + IBiA( ab, I) + Bi A(ab, t')I']
+ BIA(a', 5') -
B,A(a', t) - B3A(a', I')!'
+IA(b" 5) + B; A(b', 5') -
BiA(b', I) - BiA( b', t')I']
+q:_ [IA(a',.)
+IA(a',.) + B; A(a', s') - BiA( a', t ) - BjA(a', t')I']
+ (qt- + q:,+) [IA(ab, 5) + H, A(ab, 8')1' + IA(ab, 8) + B; A(ab, ")1'] } . (9.58)
Un pola.rized CroSs section: "a du = 16::-,,7'.-;(-q+-.--;k7)(;O-q-_-, k:7.) X {
(s+ s~') [IB,A( ab, t) + B3A(ab, t')I' + IB; A(ab, t) + BiA(ab, t')I'] +q:_ [IA(a', s) + BI A(a', .') - B,A( a' , t) - BaA( a', t'W +IA(b', 8) + Bj A(b" 5') - B;A(b', t) - Bj A( b', t') I']
+q~+ [IA(b',.) + BtA(b', 5') - B,A(b', I) - BaA(b', t')I'
+IA(a', 5) + B; A(a', 8') + (g!_
BiA( a', I) - BiA(a', t')I']
+ q:+ ) [IA(ab, 5) + BtA(ab, 8') I' + IA(ab, s) + B; A( ab, .")1'] } (9.59)
Ji. II.
J~7
-1,.1; (rll oo U)
I' rm;ess:
Ili variants and definition!;
L!.;j _ -2(p_· '; )-2(p_ ·'j)+2(k;.';),
i,j=I, ... ,4 ,
+~(~ (2Ekl+kj.1k2.l + Zn Zi3) +6k3.1(2Ekl+k:.1k2.l + ZIlZ2t)· .
(9.61)
21
Photon pola rizntions:
,v-' ,
i= l , 2,3, 4.
- £' [32.;••;_[! ,
(9.62)
l'\onvanishing helicity amplitudes:
M(+.-;+,+,+ . -)
-4e 4 Bk._ku
- 4e4 D(kHkt)t ,
M (+ ,-; +,+,-,+)
-
_4e 4 Dk3 _ k3J..
- . , 'D (I<, + " 3- )!
M(+,-;+, - ,+,+)
=
_ 4C'1B ~ _ ku
~
1 e 4B ( k2+k~_)~ ,
M(+,-;-,+, +,+) - - 4e"Bk l _ ku.
=
4e 4 B(kl+kl_)t ,
M(- ,+; +, +.+,-)
~
4e4 Bk H k;.1
,
- 4c"B(k:+k1 _) t ,
I :lH
,I),
SUMMAR.Y OF QNfJ FORMUI,AN
n
,VI( -, +; +, +, -, +) -
1 e' kH k;;.L
-
1e' B(k5+k3_)t ,
M(-,+;+,-,+,+)
4e 4 BkHk;J.
-
4e4 B(kl+k2_)t,
M(-,+;-,+,+,+)
4e4Bk1+k~.L
-
4e'B(kt+kl_)t,
M(+,-;+,-,-,-)
4e' BkI+kl.L
-
4e'B(kr+kl_)t,
M(+,-;-,+,-,-)
4e' Bk2+ku
-
4e' B(k1+1:2_)1 ,
M(+, - ;-,-,+,-)
4e' Bk3+k3.L
-
4e'B(k~+k3_)1 ,
M(+,-;- ,-, -,+)
4 e' B k4+ ku
-
4e'B(k1+k4_)7,
M(-,+;+,-,-,-)
-4e' Bk1_k;.L
M( -, +;-,+,-,-)
-4e' B k,_k:;.L
M(-,+;-,-,+,-)
-4e'IBI: 3_ k3.l
M(- 1 +'1 I - 1 - , +)
-. 4e4B(k1+ k~)1 ~_
1
. .
4e' B(k3+kL)1 ,
- 4e'Bk,_k,.L
-
4e' B(k<+klJI ,
M(+,-;+,+,-,-)
-2e' BF(I,2,3,1)
-
2" BIF(I,2,3,4)1 e E '
M(+,-;+,-,+ ,-)
-2e' BF(1,3,2,4)
M(+,-;+,-,-,+) -
-2e' BF(1,4, 2,3) E
-
2e' BIF(!, 4, 2,:1)1 E '
M(+,-;-,+,+,-)
-2e4 BF(2, 3,1,4) E
-
0 4
M(+,-;-,+,-,+)
-2e' BF(2,4, 1,3 ) E
-
2 ., BIF(2, 4, I, 3) 1 eE'
M(+,-;- ,- ,+,+)
-2.' BF(3, 4, 1, 2) E
-
2 ' BIF(3,4, 1, 2)1 e E )
M(-,+;+,+,-,-) -
-2e'l BP'(3,4, I , 2) E
-
M( - ,+;+,-,+,-) -
-~e
BF·(2,4, 1,3) . E
-
E E
? •
-
4e 4 B(k2+ kU t ,
2 • BIF(l, 3, 2, 4) 1 e
... e
2
E
BIF(2,3,1,4)1
E
'
)
B IF(3, 4, J, 2)1 e E ' 4
2 1 BIF(2, 4, 1,3)1 e E '
130
M(-.+i+ ,-, - ,+) M (-, + ;-, +,+,-)
M(-,+i-,+,- ,+ ) M(-,+;-,-,+ , + }
=
- 2e~ lJF"(2,3, 1, 4)
E -2e
.( BP"(J, 4,2, 3)
r.;
_2e 4 nF· (1.2.,2~ E -2,
, BF'(1,2,3, 1)
-
2' BI1'(2,3, 1,111
E
e
'
-
2' BIF ( I,4,2, J II e p; ,
-
2' BI1'( 1, 3,2,1 )1 , E '
_ 2, ' IIW( I ~,3,4)1 .
E
(9.63)
I1npolarized squared mal rix element:
+w, [IF( 1,2,3, 4)1' + IF( I, 3, 2,4)1' + W(1, 4, 2, 3)1'
+ 1F(',3,l,4)1' + 1"(2 ,4, 1,3)1' + W(3,4, 1,2)1') )
.
(9.64) Uupola:ized cross sediov.:
du
=:
a 4 B'l { •
~(k-?+ + kl_}k,+k._
6<:1r" £1
+E-'[W(I ,2, J, 4)' + IF(1 , 3, 2,4)1' ~ W(1,4, 2, 3)1'
+ M
Xo (p+
F(2 , 3,1 , ' II' + 1F(2, 4, I ,3)1' + 1F(3, 4, 1,2)1'[ )
+ p_ -
,pk1 cFk2 rPka cPk~ .
kl - k2 - 4:3 - k~) --
klO k20 k30 k~ o
(9.65)
e+e- - 11- Je l l (no Z-exchange; m = 0)
9.12 P rocess:
,+(p+) + ,-(po) - ,+(,,)
-
+ ,,-(,.) + 7(k,) +7(1.,).
(9.66)
luvariants 3nd definitions:
,
"
-
2(" ,. · p_),
t -
-2(p+ . q+) ,
u
-
~2(P+
2 (q ~ ·q_),
t' -
- 2(P_ . ,,_),
,,'
-
- 2(p_ . !/+),
. q_),
110
9. "IIMMAIIY OJ' Ofo:/J FOItMULAl,'
i
B•
i = 1,2)
Wi
A a
(w+ - w,)(w_ - ,",)(w+ - w,)"(w_ - w,)" Iw+ - w" Iw- - w,llw+ - w, l lw_ - w,1
=
&
2E - k,_
+ ku.w~ + k;~ 10_
c
2E - k, _
+ ku.w:;',
el
-
(2E - k>+)w_
-
+ ka •
a, -
q_ ~+ku.
b,
2E - k, _
+ kuw: + ki~w+ -
c,
2E - k,_
+ kaw: •
el,
-
(2E - kH)W+
k'J+w:j.w_ ,
k,+w+w:
+ ku,
+ ~·i.L )
'"
q.t.J..
&, -
2E - k,_
c,
2E - k,_ -I- k;~w+,
+ k;iw+ + kuw: -
kHW+W:,
&0 •
= 1) 2)
9.
/~.
~+r-
,1
_/(t,.-
(NO Y,·/':X(:JlMU,'/·; . 'iI "" (I)
d, -
(2E - k1 +)w:
a,
~
q:.1.
-
2& -
~
2£ - k 2 _
" c,
l·tI
+ k~.1.'
+ ki.1. , ~_
+ k;.1.w_ + k2.l.w ~ -
k2+w~w_
~
b; ,
+ kl1W _ , (9.67)
I'holon polarizal:iollS;
i = 1,2. Nonva.nishing helicity amplitudes;
M(+,-j-T,- ,+,+)
=
M(+. -;-,+,+,+) _
e~ q
.1.Z+_B I 8 2
[q++q_+QJt
•
e •q__ IBBI[2('+.'_)]' \ 2 Q e~q+.1.Z_+B1B2
{q++q_+QJt
_.c q+_ IBBI[2(,+ .,_)], 1 2 Q
e4.q+.l Z_+8 \ 8 2 .Jq-::;
M(-,+;+.-,+,+) _ -
111 (-, +; -,+,+,+) -
q+_[q·HQJt
•
e q_+
IBl~ B .
[2(,+.Q q-)]!
(9.68)
Y. 811MMAIIY
J42
or (JIm
FORMULAD
1\1 ( +, -; +, -, -, -)
e1quZ~+BiB;~
lV(+,-;-,+,-,-) -
-
q+_Iq++Q]t e
' q_+IE"B I [2('1+Q. '1-)] ~
M(- ) +-1 -f- ) - , - , -) -
e4 q:J.
M (-,+;-,+, - ,-) -
Z+_ B,E;
[qHq-+QI~ t
-
M(+, -; +, -,+,-)
e q__ IB,B,1 4
[2(q+~ q_ ) ]'
4e'A {(q++q_+)t[W2 b+ (w_ - w,)el
+(q++q _+)'skHk,+(w+ , -
(p_ - k, - k,)Zs'
2Eed w, )*(w_ - w,)*(W+ - w.)(w_ - WI)
(q++q _+ )!cd (q++q_+)tw,[b+ (w+ - W.)'d] + 2Es'kl+kH w,w! (p+ - k, - k,)'S'W1W;
+
('I++ )} q_+
c'
Slkl+kHWI(W+ - w,)'(w_ - w,)*
9. I!.
~-t ,,- _
M(
/.+ W
n (NO r. ·'·:X(:/I11 NliI-,'; 'I'
_._
. +•.. +,+,
_) _ 1
-
-
'"
U)
~1{{IfHrl-+)t!U!~bl+(W_-W')C1J ( p_ k I
IS.
+
k)" 1 S
2Ec,d1 (QHq_ ... }13k 1+k H (w+ - W,)(W_ - wd(w+ - W1 )-(W_ - W2)"
+
(QHI_-t)}c 1d l 2E""kl+~'HW1W2
_(lli) t \q_+
_ (q++.) q_+
, 2
-
(Q++Q_+)hG~[bl + (W+ - wl)"dd _.(p+ - kJ - h)2,,'W1W2
8~kl+k~+Wi(WI .
2f.'(w_ (q_ + kJ
wd(w_
Wd
-wd"[w 2d l + (w. - W_ )a,!
+ k1J2S(W_ -
wd(w_ - W2)"
Ii, SIJMMAItY Of QiW
111
f()ItMUf.A~'
i
.,
1
,
+(q++q_+l'sk ,+k2+(zv+ -
2Ec,d,
l.
ZV1)(W_ l.
w,)(w+ - w,)-(w_ - tv,)" '
(Q++9-+)' wi[b, + (w+ - w,)d,] (p+ - k, - k,)'S'w,w;
, (q++q _+)lc,d, ;- 2E,' k1+ kH w., wi
Jj
Q+ + )} ci ( + q_+ s,kJ+k,+w;(w+ - wJ)(w_ - w.)
+ (::: ) l
slkl+k-Hwl(w+
~tu,)'(w
1
l
_ _ lo,)"
1
q_+)' 2E(w+ - lO,)[wid, + (WI -, w_l'a,] ( q++ (9+ + k, + k,)"(w+ - w,l(w+ - w,)"
+ (q++) t q_+
w+)a,J }
.:
2E(w_ - w,)"[c, + (tv, (q_ + k, + k,)2 S(W_ - wJ)(tV- - 1D,)-
[M( -, +; -, +, +, -)]",
,11(+,-;+,-,--, +)
\J
,
"
M(+, - ;-,+,-, +)
[M (-,+;+ ,- ,+,-)]" ,
M(-.+;+, - , -, +)
[M( +, - ; -
'1
, +, +, - )J' ,
.,
,
M(- ,+; -, + ,- ,+) = [M(+,-; +,-,+, -)J" ,
(9.69)
Un polarized squared matrix element:
l,e 8(vl>
-
, "i
t'+t"+·u' +'u"
VI,), (v" - v,,)' -'-----,-- ss'
+~[ IM( +, -; +, - , +, - )1' + IM(+, -; - , +, +, - W
+IM( -, +; - ,+, +, - )1' + IM( -, +; +, -, +, -lI'J ' (9.70) wb ere [see eqns (4.27)J p+ pVip = (p+ . k,) (p_ . k;)
,
,I
.,
i = 1,2 . (9.71 )
du =
64~3
+1,11(+. -; +. -. +. - )1' +IM(+. - ; -. +. +.-i l' +IM(-. +; -.+. +. -)I' + IM(-. +; +. -. +, 4
~6(p++p_-q+ -q_- .I-~ )
-j" }
({J ij., JJii_ IFf: d' i:, . q+o q_D klO A'1O (9. 72)
Process :
(9.73) fnvariautli a!\d ddi ni tiop.s:
s _
2(P+ ' p- L
u'
_
-2(p_· q+),
j =:
Z+-
~
1,2,
'v. SIIMMA Ity 01' (JIm /Y)ltMIII,AIl
110
(w+ - wJ)(w- - wd(w+ - w.,)'(w_ - w,)"
A
Iw+ - w, l Iw- - w1 11w+ - w211w_ - w,l '
a
b c
-
2E - k, _ + "aw:;',
d
., '.
C, -
2E -
"1- + ka'W~ ,
d, = (2E - kH )W+
i
+ k21,
(9.74) Photon polarizations:
.N.~ l
i = 1, 2.
(9.75)
~II:I.
r+,,- -
~ .......
11 ( NO Z·I~ATI/AN(,'g; III ' (I)
f\.JIIVotn iHitilig l,dkilY
_ 4e:~.E (9.+· q_)q;'J..nID~ q+ - lq++q-+Q J!
M(+,+;+,+,+, +) -
4c~E(q+·
-
c"'(Q+· q )q:. J..Z 2 +8"8" 1 1 Eql_[q++q:+QI J 2e4(q+ . q_ ) 1 I B1B~1
-
-i+. - ,+.+)
E[,+ _q. • Q)i _ e f q~.L q±.LZI _ D1 82
;
q+_ [q++q~+Q)t
,' ,B,D,I
-
_ e4q+.tZ + B)B2
fq++q_+Qlt t:4q+_ BIBl l
;
.'
'+_['HQ)I
. ,'q. +ID,ll,1[ 2('+Q
+' - + ' +l ,
•
[2{q+Qq l]'
_ efq : J.Z_+8,B2~
M(- . +;+,-,+ ,+) -
,
[2'H~_('+ ' q-l t q+ -q- +Q
M(+.- ;- ,+,T. +)
M (-
q_) [BIB21
[.+-.-+Q)!
M(+,+i+, +,-. -) -
M (+,
1<7
_ e4 q+J. Z.... _B 1 lJ 2 ..fiiti
• T.
~
q-l] i
,+- [.-+QII e41BIB21
[2 q!+q_ - ('1... . q-)] ! q+- II-+Q
M(+,-;+.-.-.- )
;
_
e4q H Z*±_8 I-B·1 .fii+i q++
.+-[,_+QII
-
ef lB1B21[2q~ _q -(q+
·q->t q, - q-+Q
9, .l'IIMMAIlY OJ! (J /{U FOIlMVI. A'"
M(+,-;- ,+, - ,- ) -
e'q+J.Z:+BjB:;
M(-,+;+,-,-,-) -
[q++q-+QJl 1
-
e'q+_IB,B,1 [2(q+Q
q-J] ,
e1 q:2..L qt.lZ+ _Bi Bi
M (- ,+; - ,+,-,-)
q+_[qHq".tQjt
e'[B,B,1 [2q~+9'I__ (q+, q+-q- +Q
-
1
'1-)]'
e'(q+ ' Q_)qHZ:'. B,B, Eq'+- ['I ++ '1 3-+ Qj!
M(-, -j-,-,+,+) -
2e1 (q+ - q_)'IB,B,1 E[q+_q_+Qj! 40' E( q+ - '1_ )q+J.Bj Bi
M (-, - ;-,-, - ,-) -
q-l--[q++q_+Q]i 4e'E(q+ 'q_ )I B,B, ]
[q+_ q_+Qjt
M(+,+ ;+, +,+, - ) 1
+
q-+)' _
( 1++
dOC; 1.
t'k1+ kHW,W;(W+ - wtl(w+ - w,)'
q++) ! dj' ( q_+ t,kll.kHW'(W_ - w,)(w+ - w,)"
(qq'I_-+)+ ! t,k1+k2+w,(w+ca'- w')(w_ - w, )(q++q_ •.)!w,[(w+ - to-.J"d" + (w, - w_) "d;j + (Pt - k, - ",)'t'w, w; (qt+q-t )! I(w, -11)+)'c; + (w_ - w,)"c;J (p_ - Ie, - k,)'t
/
,
+ (q_+)~ qH
2R(w+ - w.) o!r::i + lII~rl' l ('7+ + k! + k2 Pt'(u.l+ w. )(w+ -
qH)! 2g(w - wdo{c; + ( (q_+k +k z q~:
M( +, _0+ , , _ , + , _)
_
u.'2dil
)2l(w_ - w,)(W _ -
l
4e~A {
WI) "
} W 2)"
'
2 (qHq +)tb 2EtlkH~~2+WHW _ -W')(W+ -1/12) '
q++) ! (q_+ -,,~k-H-k~,-+-w-"(W-+-"--W-'~)"(-W-_---w-'~I' C
, __ ) l
2
,f
( +;;;-;
.92kL+kHWi(W+ -
+
,
(q~
u.'d(w_
WI)
2Ecd
11}_+),skL+kH(w+ -
W~),{W_
- WZ) ' (W+ - Wl)(W_ -Wt)
+ (q++q_+)~cd......
2Ea 2
2Es'k1+k2+WIWi
(q"lq_ +)iflk ... k~ 1II.(W+ -W.) ,(W_ -W2)
,++) l ab ( - q_: tkl+kH(W_ - wd(w_
('I-+)! q_1
W~) ·
ab
l'kl+kHWltui(w+ - IIJd{w+ - W2)~
_ (q_+q_.~)hLl2[b + (w+ - wd ' dJ (~+.!.) (p+ kl - k2 )2wltVi s' tt
~ ['w2b + (w_ + (q_+q_+) (p_ - kl - k2)'
WZ)C]
(I-+-I) S'
t
fI, ,\'/iMMAIIY 0/0' Qf;/I Jo'OI1MlI!.AE
M(+,-;-,+,+ , -)
4e'A {(q++'1_+)\[W'''' (p_ - k j
,
+ (w_
- w,)e,] k,)' .•'
-
2Eej d j
+
(q++q_+)}etdt _ (q++q_+)tw,[b j 2Es'k1 +k2 +WIWi
(p+ - kl -
q-+)! ( q++
e1 Sjkl+kHW1(W+ -
1
1
q_+
+
w,)'(w_ - w,)"
WI)
2E(w_ - wd"[w,d, + (IL" - w+)a,] (q_ + k, + k,)'s(w_ - w,)(w_ - w,) -
q_+) t 2E(,"+ - ,,},)le, + (w, - w_)oa,] } ( q++ (q+ + k, + .,)'s(w+ - w,)(w+ - W,)- '
M( - ,+;+,-, + , -)
4e'A { (q++q_+)l [W;h3 (p_ - kj
+(q++ q_+Pskl+k2+(w+ . +
k2)2SIWIWi
s,kJ+kHW;(W+ - wIJ(w_ -
_ (q++) ,
- wI)"d,]
dl
q++) ,
( q_ +
+ (w+
+ (w_ -
./
w , )'c,1
k,)'s'
2Ee3d3 wdCw- - w.l(w+ - W,)'(I1I_ - W,)"
(q++q _+ )l C3d, _ ~(q,-+",+",qc:-",+-,-),_W"-i,-, ' [,,,b,:..,+:....c,(w--,-+_",w..:.,!..:.)d=3] 2Es'k1+k 2+WIWi (p+ - kl - k2)2 s'Wtwi
!J,
I.Y. "I ,, - --.
+
r.",-'.,.,
(NO 7,. ~:.t:( ,'/III N(.'N:
t o! __
O)
(q++q_+)~b~ 2El 1 k1+k1 +WI(W+ - wd(w_ - wz)o
,
('++)' q_ +
,
"b,
lkH k2.dw_ - WI)(W_ - W2)4
-(:~:r t'kl~k2+u.:1W2(Wa:~Wl)(W+ _ (q++q-+)~wilb:? + (w+ - wl)d~J (p+ kl ~)2WJtvi
-W1) ·
(.!. +.!.) 5'
t'
1)
(w_-wdo r..'lJ (' + ( q++q_+ )l[Wi~+ -+(p_ - k -k )2 5' t 1
2
9. 811MMAIIY 01" QeD FO llMUU/';
If.l
M(-,-;-,-,+,- ) - 4e'A
q_+ )l ",' (q-H t,k, .,k2+tvj(w+ - w')'(tv_ - .,;;)
-
d' • q++ 1 3"2 ( q_+ ) tkJ+k2+(w_ - tv')'(w_ - w,)
+
1
q-+) , d;S + ( q++ t'kI+k,+ w jw -(w+ - W')'(w+ - w,) w,)d; + (tOl - w_)d;J (p+ - k, - k,)2t'wjw,
+ (q++q_+)iw;l(w+ -
(q++q_+)![(w, - w+)o;
•
w.)e'J
j
(p_ - k, - k,)'t
2E(w+ - w, )[c; + wid;J
q-+) '
+ ( q++
+ (w_ -
(q+
+k, + k,)'t'(w+
- w, )'(w+ - w,)
2E(w_ - wd[c' + wid;J } q++) I ( - q•.;. (q _ + k, + k,)'t(w_ - w')'(w_ - w,) ,
M(+,+;+,+,-,+) -
[M(.,.,-;- , - , +,-)]"
M(+ ,-;+,-, -, +)
[M( -,+; -, +, +, -)J' ,
M(+,-;-,+,-,+)
[M( -, +; +, -,+, -)]*, -
[M( +, -; -, +, +, -)]' ,
M(- , +;-,+,-,+) -
IM( +, -; +, -, +, - )J' ,
M(- -'- - - +) -
(M(+,+;+,+,+,-~·.
M(- , +;+ ,-,- ,+)
l
I
)
)
I
(9.76)
16:1
+l[IM( +, -; +, -, +, -)I' + IM(+, -; -, +, +, -)1' +IM(-, +; -, +, +, - )1' + IM(-, +; +,-, +, -)1' +IM( +, +; +, +, +, -)1' + IM( -, -; -, -, +, - )I'J ' wh('re [see
!',I' =
(9.77)
(4.27)]
eqllS
p+
p-
( p,·k, )- (p_·k,) ,
Vi,
q+ = (q+ ',I;;) -
9-
i
(9_ . ki »
= 1,2 . (9.78)
Uu[>oiarized cross section:
+IM(+.-; +, -,+.- W+ IM( +,-; -, +, +.-w
+IM(-, +; -, +, +, - )1' + IM(-, +; +, -, +, - )1' +IM(+, +; +, +, +, -lI' + IM(-, -; -, -, +, -)I'} "( P+
Xu
-i' P_-q+-V_-kl-kl
)lPi--tPq_cFk d3 k2 . J
Q4oq_okJOk20
(9.79)
to,
!i. SUMMA ltV Of' (JIW ,'OltMlILAg
process:
~
9.14.1
k3 nearly parallel to
p+
Photon polarizations :
;= 1,2,3.
(9.81 )
Nonvanishing helicity amplitudes:
M (+,+;+, - ,+) = e3mki~k3lo [k3+ Ek2+(p+ . k3) 2k3_
M(+,+;-,+,+)
M(+,-;+,- ,+) -
,
j'-
[r [r
e Eku 8k3 _ ' 3 k,+(p+ . k3) k3+ e E k lJ .
-
3
-
8k3_ ' kH(p+ . k,) k,+
M(-,+;+, - ,+l
e 3Ek;.L [Sk3_] , k, _(p+ . k3) kJ+
-
1
-
!
iV[( -,+ ;-,+,+)
-
M(+,-;+ , -,-)
Af(+}-j - , +, -) -
,3 E kilo [8k3_] , , k _(p+ . k3) k3+ e Eku
[r r
8k3_ ' J k1_(p+ . kJ) k3+ e Ekll.
8 ,_ ' , [ k k,_(p+ . kJ) k3+
e E
[8k,_k,_ (p+ . k,) k,+ k3+
r r !
,
M(+,-;-,+,+)
3
1 3 3 e mk + [kz_j , E(p+ . k3) 2k,+
e'E
[8k 1 _k, _ (p+ . k3) kl+k3+ 3 e E (1'+ . k,)
,
[8kk, _kk'_j' + H
3
1
3
-
'
e E [8h'l+k3- ] , (p+ . kJ ) k,_ k3+
r I
-
-
e'E (p+ . k3)
[8k 2+ k 3k,_k3+
-r
3 8k e E [ l+k3 (p+ . kJ) k,_ k3+
,
Y./ f. r+C -111("'1-0)
M (- , +., +, -
.- )
~
M(-.+i-,+.-) -
M(-,-i+. - . -)
=
=
~ E(p+ ,'m'" ["'_j! . .~) 2kJ+
:H (-. - ;-,+,-)
(9.82) llu pol;·~;i 1..£xl
3<)uiJ..("cd matrix element:
x (:::
+ ~::) .
(9.83)
tinpola.rillcu cross section:
do
=:
X
(kL"I" ... 1:2_) IH·
k'1 ~
"(p+
+p
-
_ k, _ k, _ k,)
cPr"lkcPkkJlk3 ' 2
!0
~>(J
JoG
(9.84)
9. 14.2
-
k, nearly parallel to P-
Photon polarizations:
Nt l
-
&{32ki+ k. _l t,
NOllv(luishing hd idty amplitude1!:
i::=. 1, 2,3.
(985)
Y. HIIMMtlllY Of-' QBD FORMU/,Af,'
,15G
M (+ , +; -, +, +)
, k k' [k 1!
e m
:U. '3.1
.3-
_
Ek,_ (p_ . kJ) 2k,,+
M(+,-;+,-,+)
M(+,-;-, +,+)
M(-, + ;+, - ,+)
iVI(-,+;-,+,+)
M (+,-;+, - ,-)
M (+,-;-,+,-)
M(-,+;+,-,-)
M ( -, +; -, +, '- )
M(-,- ;+,-,-)
M (- ) - 'J -
,
e3
!
'mk' k [k] , 1.1 3 1.
3-
e31n
_
Ek,_ (p_ . k3) 2k3+
k3- [k]1 ~2+
E(p_ . k3) 2k,_ e mk3_
+ -)
3
1
E(p_ . k,)
k1+ 2 []' 2k1 _ (9.86)
Unpolarized squared rnatrix element:
-::-;;;---,:-2e-=-6,:;E:..:.k:::3+;:----,-...,..,. [4E2 (2E - k3_) k3_(P_ . k3)Z
+ (?E _ -
k
)' J-
+
2
m k1_] 4E'kJ + (9 .87)
1~7
~6;;::;'''1'''':-C(;;;2'i~:-: .' k"''1:~')7:(p_' ::-":-;'" [~(',,"I + (2B I
.in =
1;'
X
~ 1I:3-
V
11:3
itJ_
k:J_)1 +
4~~,kkl, .J I:,
1J!lk1 rfJk3 .1:1_ 1.:2-),.( p++p-- k1 - k~- k)rFk 3 1 k' . ( -I 1 - +-;:11:1 _ Ie 2::1 "30 (9.88)
!l.t 5
C+C- -
J.L+ j1-'Y (no Z-exch angej
mi- 0)
1'l"Occss:
III \'a.rian~s: J
1J.15.1
-
2(p+·p-L
t =
-2(p+' q... ).
i'
-2(p_· q_),
=
u'
_
-2(p_· q+). (9.90)
k nearly parallel to 14
l'l:olon polarizations:
(9.91) Nonvanishing helicity amplitudes:
M(+.+;-,+.+) _
.'I. S/iMMA"\, Oi" (Jil l) H) "M UI, AI,'
M(+,-i+ , -, +) -
M(+,-i-,+,+ ) -
f"q+.l q++ (1'+' k) ,3 q+_
(p+ . k)
M(-,+ ;+, - ,+) -
M( - , -I-;-, + , + )
111(+, -; +,-, -)
M(+,-;-,+,- )
M( - , +i -/-, -, -)
=
[2 Lq+_q __ ] t k+
[2k_ q__ ] ~ k+q++
!I / ..,. (.I r.- _I{~ 1'-)' (NO 7.·h'X(,'/I AN( ,'~:: " , j.1l)
M (-,+i- , + ,-)
=
M i-,-i + ,-,-) -
(;:I(I __ I/".t [
(p+ ' k)
" [2k q'- f k+q++
I
,
+ -) •
-
=
,
e~m~'l q-J. [ 2k+q+_ q__ ] ~ sq_+(p+ 'k) k_
"(1'+ . k) ,
k+q+-o-fJ_+
r
(p+ 'k)
e;lmk+q __
MI- _.-
2k_
IFoIl
(29+_] t q_+
e~mkl q+J.
[2k+ q+_q__ ] ..q_+(p+, k) L
"m4 [2q~_ .s(p+ ' k) Ij_+
~
r ,
(9.92)
11111)()larizeJ squ3n. .d matrix eicmCDL:
011 11,:
m2kt] tr'1+ U.'2
,2e$Ek_ f;'l+(2F._k t ?+ sk_ (2E - 4'+ }k+(p+ ' k)'l
t
(9.93)
S'1
III1)lOIMizcd cross section:
dCT =
;:c="n"'"E". '-i-'-T::-,,, k+ ) k+ (p~. k)1
41rl(2t'
[S_<2E_k+)2 +
m~k!l sk_ (0.0' )
~U 5,2
k nearly parallel to p_
l'hoton polarizatioD9:
(9.95) NOilVanjl>hi:'lg hdicilY antplitudes:
9. HIIMMAIIY OF Qf:I! fOlWIILAF,
IBO
c"mki'l_.l [1k_ q++ q_t 3q __ (p_ . k) k+
M(+,+;-,+,+)
1i
1
3
-
e mk_q_+ [29++ i s(p_ . k) q__
-
e qt.l [2k+q++ ] i (p_ . k) k_q_+
-
[2k+ q++ q__ ] t (p_ . k) L
3
M(+,-;+,-,+)
.3
1
3
e q_.l [2k+ q_+ i (p_ . k) Lq++
M(+, -; - , +, +) -
M(-,+;+,-,+) -
c3
(p_ . k)
3 e q:.l [2k+q++ q_+] ~ q__ (p_ . k) L
e3 q-T
(p_ . k)
3
-
M(+,-;+,-,-) -
1
,
")
k_
[ 2k+q++
k_!J __
1~ •
3 e g.j..l [2k+ q++ q_+ q+_(p_ . k) L
M(-,+;-,+,+ )
M(+ -'- -"- -)
[ 2k ,.q+_ q_+ ] 1,
-
1
e q++ [2k+ q_+ i (p_ . k) k_q+_
1i .-
!I. H.
r.-tc
- /,+ 1'-" (NO
y,·,.,.:a.'lfIlN(]~:: ,,,
I
'01
U)
,
M (-.+;+.-.-)
~
c:lq--1.
(p .. k) 3
-
M (-.+; - .+.-) -
= M(-, -;+,-,-)
~
[~l'+q --t ] "
_ e (p.·k)
k.'H
[2k... 9+_Q_+ ]
f!lq.:.l [2k+q+-,,-] k_q_+
~
t
,
[2k+
q++q__ ] 1
(p+ · k)
L
_+] t
e mk.l1'+.l [2k_ qH q sq__ (p_· k) ,l'tclmk_q_+ (2 QI -tl!·
,(p. ' k)
M( - , - ; - ,+.-) =
1
k.
(11_ . k) f,3
,
q•.
t:·'mk.lq.:..1.~ [2k_ Q~+q_+) t
,~ q+ - lp-·
k}
e3 mLq++ ,(p• . k)
k+
[2q_+]!
(9.96)
,•.
Ilnpolari1.ed squared matrix clement:
(9.9 7) lJ npolarizcd
Cro3S
x
9.1 5.3
-
section:
1, ~ -:- U · '
t(
,,6 p+
.'1.'1' -
+ p_
k nearly parallel to
- q+ - q_ - k)
-
..:1 -' .r'! - .£1,. a-q+ a·q_ a- II;
I. '
1+011_0..:0
(9.98)
q....
The primed quanti lies k~ a.nd k'1. are evaluated in ~hc rotated frame wh~re '1+ deleTmincs l he positive -z-exis (sec a.heo Section 7.4 .3) ..The vector 9 in eqn (9.99) is oblained by applying a space reflect-ion to IJ+. The quanlity p. denotes t he muon mass.
111~
9. SIIMMAIlY OF iI/W {o'Ol1M U/,AE
Ph0 1.011 pola.ri:tmtiOlH;:
lV-I
(9.99)
Nonvanisbing helici ty amplitudes ; 3
111(+, - ;+ ,+, - ) -
e 1" k'lq+J.
[k~q_+]t
Eq+O(q+ , k) 8k'-q++ 3
e 1"k,+
[q+_q_+ ] t Eq+o (q+ ' k) 8
_ e3q+,oq_.1.
111(+ , -;+,- ,+)
[2k~q++q-+l t
Eq+_(q+ ' k)
k,+
M(+,-; - ,+,+)
M(- ,+;+ ,+, - ) -
M(- , +; +, - ,+)
_ _ e3q+Oq~ .J. M( - ,+;-, +,+ ) E(q+ . k)
l
[2k,-q++] ' L k,+q_+
- E(;:+Ok) [2k,- qA,~,q- -] ~
Y./5.
1:' r - -
I ' I /4- 7 (NO Z -I;X CIIA tim:; 'li
M(+ ,-; +,- ,- )
-
_
I
U)
I a:l
1!:II{.~olf- .1. [2k~q++ Jt £(9+ . k )
[ 2k~q++9_
e3q+o
~
ki.9-+
ki.
£(q+. k)
"q+0 q+.1. [2k'- q'- i 6'q+_ (Q+ . k) I4q++
M (+. - i - , +,-) -
,
3
~
M (+ .-; - , - ,+) =
M (-,+ ; + , - . - ) -
=
M (- , + ; - ,+ , -)
M ( - , + : - ,-, + )
,
e.lp k~ q_.l. [k'+q++]" 6'9+0(q+· k )
8k ~q _1
k' e3 p.... (q++q __ ] J., eq+o(q+ " k ) 8 - e'q+ oI1
tl
[2k~q_+ k~q++
E(qt . k } e3 q+o
k,
3
=
e3q-to E{q+" k )
r ,
[2.1.-'_Q__ q_ t ] t
E(q, · k ) _
= Un po larized IKluared matrix
e q+c [2k' q3"' 1' £ (9+· k ) ki.9 ... -
=
-
rr
e q+oq' .1. [2 1r' E qt- (q+ " k)
,
9++Q-t ]'
r " . [' r ki.
[2k~q:
k'-tq++
r. pk.1.q:t J.
ktq- -t
Eq-to(q-t . k ) 8k~q++ e'pk't E9+0(9+ " k )
[9+
q 8
,(
(9.1 00)
el emen ~ :
11\1.' X
+ u"J
" ,, ~
(9.10 1)
1/, SIiMMMlY
1!i1
or (JIm
/'OIlMIII,AI!'
( J11lwla.l'i1.ed <.:I'OHM ~I!d ioll:
dcr =
41r'(2q,•.o + k~) k~(q+ . k)'
(9 ,)02) ~
9.15.4
k nearly parallel to
if-
The doubly primed quantities kl and k1 are evaluated in the rotated frame where'i- determines the positive z-axis (see also Section 7.4.3). The vector q in eqn (9.103) is obtained by applying a space reflection to q_ . The quantity p. denotes the muon mass. Photon polarizations:
J±
}III):
h- h(1 ± '5) + h h- }:(I 'f ,5)] , (9.103)
Nonvanishing helicity amplitudes:
M(+,-;+ ,+, - ) =
,,
M(+,-;+,-,+)
1
,,•
. M( +, - ; -, +, +)
3
M(-,+;+,+,-)
• e /,k'''q" .1 t.L [ k"q + -± Eq_o(q_ . k) 8k~q++
1' 1
1/. 15. r,"'c .- fl+ 1'- l' (NO y, .":,'U...'IIAN(;N;III! II)
M( -,+;+ , - ,+ )
~
~
M (-.+; -, + . +) =
~
e:1q_oq"h
1'.'('1_ . k)
e3q_u E{q_·k)
-
e3q_O
E(q_·k )
3
q_0 E(q_· k ) t:
3
M (-.+;+, -.- )
M(-.+; - ,+, -)
=>
[ 2k~ '1+_ q_+ ] i k1.
e q_o
E(q_· k)
k~q_+
[2k'~q!+] i
r r .
k': q__
e q_oq_.J. Eq __ (q_ · k)
,
M (+,-;-,-, + )
k:qH
f"'1 __ (q_ . .1.: )
M (+, -;-,+.-) = =-1-uQ+ L E (q_ . k)
-
r" kl--+ ' q ]!
e3q_u1t- .I. [2.1.:"q3]t - tt
3
M (+, - ; +, -, - ) -
J65
.,
[2k~q~+ 1,;';'1_+
[2k'~qft] i I,;+.q __
[2k~q_+ .1.:" 'I
"
[2k~q+_ q_+] ~ k~
Y. .,'(IMMAII.Y OF
100
(J(o.'()
fiOI/MUME
M(-,+;-,-,+) (9.104) Un polarized squa.red matrix element:
2eGq-0 k"•
-MI'
- (2q.o + k+)k~(q . . k)' [ 4q~o
I
+ (2q.o + k~)' + 4'1·0 ' 1" • IL'k'"
J t' +s'u" (9.L05)
Unpolarized cross section:
dr; =
(9,106 )
Process:
(9.107) In va ri ants:
s
2(p+ . p. ),
t
s'
2('1+ . q.) ,
t'
-
~
9.l6.1
Ie nearly parallel to
- 2(p+ ' q+),
·U
-
-2(p+. q. ),
-2(p . . q. ),
u'
-
-2 (p. . q+J. (9 ,1 08)
p+
Photon polarizations:
(9..1 09) Non vanishing helicity amplitudes:
M(+, +;+,+,+)
=
'116• •,I~
-- ri r -., (NO z·,.;xr ,' /J IINm:j
//I
I
OJ
JG7
M(+,+:+. +,-) = =
.'1(+ , +; +.-. +) =
U (+.+;-,-I .+) "" .:..
t.'mq __ k.1 q:.1 '(p, ' k)
[2k+q~
tj t
<_q'. •
e3mk...(2q, tq~ _ ll "q_+(p , ,I.)
M(+.-;+._ .+) =
M(+.-;-.+ .+) -
M(-,+:+, +. -) =
M(-, +:+._.+) =
e'
q_+(p+·k)
k+
r
[2,Lq+_q __ J~ qi+(p+·k) I.,
e1qu
]!
eJq... _ [2k q (p., . k)
.1:+9++
,'mk'.1 q't..L [
2kt
r e',," [2<.,_, r
qt-(P+ ' k)
.'ml', [
k_ q++q~
2
'k) q+-9-+
(Pt
(p; 'k)
.I.:+q++
l
-
[2Lq".:
e
(p+·k)
[
2L 9+ q_+] t k+
r
,v,
IIIH
M(-,+;-,+,+) -
-
1
[21'_Q;t t ktl}~t
(Pt ' k) • qtt 32 (Pt ' k)
r r
2k_qt_' [
kt~t
e qt.J. , 2Lqit ' 3 [ (Pt 'k) ktq:+
M(+,-;+, - , - )
t
-
· 'qit [2k_ q+_ (Pt ' k) ktq:+
-
e qt.J. [2Lq_+ (p+ ' k) k+qtt
-
[2Lq+_q_t (pt'k) k+
M(+, - ; - ,+,-)
M(+,-;-,-,+) -
-
3
1t
.3
• 3 m k .lq-
J.
qt-(P+ 'k)
[
r
2k+ '1 Lqttq-J
e'mk+ [ 2 (p+' k) qt-q-+
1t
--l t
3 e q+.l [2k_ q+- q q+t (pt , k ) k+
M(-,+;+,-,- )
M(- ,+;-,+,-)
·"qtJ.
,,'II,I/M AltY OF (Jlc'[) I'OIlMUI-A[,
-
-
e
3
(p+ - k)
[2Lq--lt l! k+q++
.'q--q+.l [2Lq++ (p+' k) k+q:+ .3
[2LQ++'t'
q_+ (p+ ' k)
k+
3
M( - ,-;+,-,-) -
-
e mq,. _klqt.l [2k+q++ '(p+,k) k_q~+ e3 mk+ [2q++q:.-l t sq-+(Pt ' k)
r
1~
".!
,9,/fj, "t',,- _ .
"+t-r (NO
y,./>;X(.'IfItN(,'/",';
M(-,-;-,+ ,-) -
III ;
IOU
0)
['2.k+Q+_ Q__ ] t
e:lmkl.lJ+.J.. 119++(1'+ . k) "mk+ s(p+ 'k)
k_
[24-]! q_+
M (-,-;-,- ,+) _
M"[-,-;-,-,-) _ (9. 110) Unpolarizcd squa.red ma.uix clement:
[ 2 m.~·ql·~14+tl.+UJ~ k+)k+(p+' k)l ., + (2E .. k+) + ~L ~'2t'1 (9.11l)
'2t/'Ek_
!M 12 == (2R
l;npo!a.ri,zed cross section: dO"
9.16.2
=
-
k nearly parallel t o p_
pr.oton polari,zatiolls:
N- 1
_
E2132k+k_ lt,
Non'lanishing helicity a.mplittldes:
M(+.+i+,+,+) ==
(9. 1l3 )
!I. SIiMMA ItY O/<' '11';11 F/}/l.Mlf1.M :
17U
M(+,+;+ ,+,-)
M(+ ,+;+, -,+) _
e3 m k_ !2q!+q __ J t
sq+_ (p_ . I.,)
1
3
e mkl q- J. [2Lq++q_+ t sq_ _(p _ . k ) k+
};[(+ ,+; - ,+, +) -
-
e3 mLq_+
s(p_ ·k)
[
l' ,
2q++ q__
1
3
M( +,- ;+,+, -)
t e mkJ.q+-,- [ 2k q+_(p_ ' k) k+q++q_+
-
e'm L
-
'_41 t'T, ....1 , ,.., - ,.y)
[
1t
2
(p_ . k) q+_q_+
~-
e3 q-.1. (p- . k)
r?k+q++ 1 ~ ..,
3
l k_q: + J
3
-
M(+ , - ;-, + ,+) -
M(-,+;+,-,+) -
e [2k+ q!+q_ q_+(p_ . k) k_ 03 q.,-,-
[2kt q_+ (p_ . k) k_q++ 3
[2 k+ q+_ q_+ (p_ . k) k_ e
03 q:-,-
]!
[2 k+ q++ q_+ ] t q_ _(p_ . k) L 3
-
1t
r •
e q_+ [21." .q++]! (p_ . k) k_q __
r "/1.. "
I I
- , ,.1" ")
(NO X " '.'.\' / ://11 NUll';
M (-, +; -, +,+) -
,II I OJ
171
l'tk fu'\ ]!
(..1,( _~ _ -;:.l_
I
'/+_(11_' k) -L~/_+
[ ~ , .1:) [2'
.... 1(.!., - ; + , - , - ) -
-
M( I, - ; - .+,-j
,
2k+ _+q_
,,_(p_ . k )
L
e q-J. qt-(p- .
3
e q_.1
, __ (po . k)
-
eq + (p_. k)
-
~
M( , +;
, , i-) -
M (-,-; +,-. -)
~
-
2
+q++q-" [2k' q,_ (p_ . k) k:- -
3
M(-,+; -,+.- )
k+q++ k_9_+
-
M(-.+;+,-,-) -
r r r
e
clq'
(p_ .
.1 ~.)
[2k+ 9.,.+9_+] fk
[2k t Q. +] ~ I
Lq __
[2k +q-+ ] ~ Lq++
e3 [2ktqt _q___ (p_·k) k_ cJq:J.
(p_ . k)
[2k+r4+] ~ .L q:+
"
q- -l(p . k)
c3mklq
t] t
J.
[2k+4
q
k.
[
qi- (p -' k )
2.1:
r
-1.t ]!
k+'I++q_+
,'mL [2' __ q+ _( p_ . k)
~
e·'m~J.q+J.. sq--tP--- . k )
[2k_ q++ q k+
+]t
e3 rnk q ___ [2'1 ... ,(p- . k) , --
'J'
. '
9, SIIMMAIIY OF QRf) /<'0/1.1111/,118
17~
M(- , -'+-) t , ~
"'Ink.!.,!,tJ. [2k - ,,'±t
-
sq+_ (p_ . k)
k+q_+
]t
L
,3rnL
-
[2q;+q __ ] ,
.q+_(p_ . k)
, .'sq-J. [ 2k+ ]" . (p_ . k) Lq++q~+
M(- t -', , - , +)
L
03 S
-
[2k+Q+_] ' L~+
(p_ . k) e sq_J.
r r
2k+ ' q+_ (p_ ' k) k_q++q_+
M(- , - ', - , - 1 -)
3
.3.
-
[
[
(p_ . k)
2k+ Lq+_q_+
(9.114)
Unpolarized squared rnatrix element: 2 [
s+(2E-L)
+
l1~Zk~l S'4+t4+1l4 oS
k+
'"
s t
(9.115) Unpolarized cross seciion: da =
[s x
s '. + t ,
+ U, 64 (
"t'
.8
1 sk+
+ (2E - L)' + m'}3-
t l - .t l - d'k) wq+ wq_ k P+ + p- - q+ - q- k'
q+Q q-D
"0
(9.116) 4
9.16.3
k nearly parallel to ii+
The primed quantities k± and k~ are evaluated in the rotated frame where i4 deLermines the positive z-axis (see also Section 7.4.3). The vector q in eqn (9.118) is obtained by applying a space reflection to q+. Definition: (9.1 17) Photon polarizations:
(9.118)
NUHV!l.uiHiling hdidly amplil,uues;
M(+,+;+,+.,+ )
[ '] ,
,
~
e q+Oq:.1Z+_ 8k_ ' (q+ " k)- k+q~+q~ +
~
e Eq+o [32k~q __ q_+(q+ "k) ktqt+ 3
M(+,+; +,+,- )
-
c: 3 q.j.(lQ:.1 Z+_ [
q__ (q+' k )
r ,
,
8k~ ]~ kt '1+' q: ..
,
3
1-- ..." ]' e £q +0 [ 3?i.:lq = q_.,.(q+" k) I.:t'l __
~.Jrnk'.1.1.+ q- Z - [ k'+ ]7 q+o1--(Q+" k) 2 k~q++q:+
A'1(-:-,+;- , + , +) =
M(-l ", - ;+.+.-) -
M(+,-i+, "-,+) -
,
l!~mEkt
(l.,.oq- ~ ('1+"
_
k)
[2 q++ j' q__
l!
,'mk'·.i q+.1 [ _:±-=±.... k' q Eq+o(q+" k ) 8A.,1_q.....
I'hT~k±
[q+ - q-+J}
f:q+o(q+" k)
8
, [" r _=, [' r e q+oq-..:.. ~_~" 9+tQ-+ Eq~_(q+ "k ) ,1,;1+
M(+,-;-,+,+) =
AJ(-,+ ;+,+,-) -
,
e q+o [2k~q++I{~_]' Eq+_(q+ . k) k± 3
~
,
9+0111. 2k_1_+ £((1+ . k ) k+q++ e:lq+o
E(q,· k)
,
[2kC,:+o'o,
7
l!
,- - [' ]1
e mq __ k1. q 1.
k+q++
Eq+O(9+" k)
8k~q:_
e3 mk'+ 8q ... oq_+(Q+" k)
[q'+t-] I
g. SIIMMAIIY
111
or (JIW
M( - ,+;+, - , +) 3
e q+o £(,/+ . k) '' ' ( - ,-l-' .'V..l . , _ )_ I_1
+) _
[2k' q~+l} k~q+_
e'q+oq __ q:~ £(q+. k)
[2k~q'++J I k~q~+
M(+,-;+,-,-)
,
M(+,-;-,+,-)
e'q+oqu [2k~ q~+] , Eq+_(q+ . k) k~q++ 3
=e_'1'-!-+",occ E(q+ . k)
M(+,-;-,-,+)
_ e3 mq__ k'c q-~ [ k~q++ Eq+o(q+ . k) 8k~q:'+ 3
-=-_e_m....,.k:r:.t_ Eq+oq-+(q+' k)
M( -, +; +, -, - )
M( - , +;
-, +, -)
1 ' _
[2k~ q~+ k~q+
,
1~ t
[q++~-l' 8
FOIlMU[,AII
'i,/h',
~I,.
' ~r"r
.,(N01, -f:X( ;IIA N(,'/\'; ou!ll)
,11(-,+:-,-,+)
175
=
M(-,-:+ ,- ,-)
M(- ,-;-,-,+) e3 Eq+'J
q_+(q+. k) M( - , - ; -,- , - )
e, q+oq-.L z' '1--
(g, . ')
IJJlpolarized 5quared malr:>: elemenL:
t'• 4- 1/ 4 's~l'2 '
54 4-
;<
U1Ipa lari ~ ed
(9.120)
cro~~ ~cctjOl1:
[4q+o + (2q+o + "+) + 4q~"."k' 1 ~ d'" k) rrq+ o-q_ ~
,,~P,'''I
10 -
x
"A+I"+ . U~ ,54(P 't'2 + + p- - q+ - q- S
<"1-
....
I,'
q - o~
(9.121)
9.10.4
-
k nearly parallel to
ii-
The doubly primed quantities k± and k1 are evaluated in'the rota.ted frame where ii- delcnnincs the P03iLive <-a.xi~ (see abo SectiO!1 7.4.3). T!le vector q in eqn (9.123) is obtained by applying a space reflection to 9_.
... . .
1111
g. StlMMAII.Y Of' (Jim f.'(JlIMUI,A£'
Definitioll: (~.122)
Photon polarizations:
2 [32k" k" I! q-o +_
.
(9.123)
Nonvanis hing helicity ampl itudes:
M(+.+;+,+,+ )
=
• 3q -0 q'-1 Z +- [ Sk"_ q__ (q_' k) k~qHq~+
1'•
1
e Eq_o [32k~q __ ]' q_+(q_ . k) k~q++ 3
M (+, +; +, +. -)
M(+,+;+ , - ,+) _
M(+, - ;+,+ ,- )
.,
. 3 q- 0 q -.1 [2k"_ q3++ - Eq+_(q_ . k) k+q_+
M( +.-;+. -, +)
" ,
.
-
;".
• 3 q-o
. • _q++q __ [?k"
Eq+_ (q_·k)
k+
3
1
e3q~oq+1. [2k~q+ - l ' E(q_ . k) "+q __
M(+, -;-, +,+) .I
"
r 1
3
_ +_ e q-0 [2k"q3 E(g .. . k) "+q_+
r ,
177
,
~
e rnk"· q[ kll+-± q ]~ .l+L Eq_o(q_· k) 8k~q++
-
em ' k"+ [q+_q_+ E,_o('_ ' k) 8
3
M(-,+;+,+,-)
;\-1(-.+;+, -,+)
-
M(-,-:-;-,+,+ )
~
3
e
,
M(+,-. - ,+,-)
,
c~q-O.L q" [2k"- q3 +t
111(+ ,-;-, - , +) -
M(-- ,+;+,-,-)
~
-
r
r [,,,,]' , ["" , r , ['" r r k~q:+
- B(q_ . k)
cJq_o
[2k~4+q __ ki.
,Q-09-.1
2k_9'H
E(q_ . k)
I.:+q:+
_ t
-
]'
["
e q-o 2k_q+_q_+ E(q_ 'k) k~
~
~
'
k~q++
E(q_· k)
Eq_ t {q_· k)
-
>
q 09+1. [2.\:'1 q_+] r
~
M(+,-;+,-,-)
r
e 9-0 Eq_+(9-' k) e .1Q-09+.1.
£(q_. k)
_qj±q__
k'':'
[?J/I - -9-+ 1t J k~q++
e q_o
21.: 9+_9_<E (q_· k) kt _
. i t i [ k" ±-± "mk", q
8q_o(q_ . k)
8k~q+ .!
eJm kj.
[q . . q_+] t
E,_,(q_' k)
8
[2F' 1t E(q_' k) k~q __ J
e J Q- 09tJ. •
3
e
q0
r..'(q_. k)
_9+ _
[2k~q! 1.:!+-9- +
r ,
.~.
178
SIl MMAIt Y OF QRD FORMULAE
M(- .+: - ,+ ,-)
M(- ,+;-,-,+)
.1. -1. Z' 'J+_ [ k"+ k'''q q-o(q_· k) 2k~q~_q~_
M(-, - ;-,+,-)
e3mEk~ q-o( q- . k)
M(- _. - - +) ,
I
,
[2'1_+ 1' L
q'l.-
_ eJq_Oq_l.Z+ _ [
Sk~
(q_ . k)
k~q~_q~_
,
1' L
... .3 m
1t
1
_
,3Eq_o [ 32k~q-+l' (q_ . k)
k~q!_
e3q_Oq_.1Z~_ [
M(-,-;-, - ,-)
q__ (q_ . k)
Sk~ k',:,q++q~+
1'
l'
1
1
_
e' Eq_o [ 32k" ('1_ . k) k~q+_q_+
(9.124)
Unpolarized squa.red matrix element:
IMI'
= (9 .125)
Unpolarized cross section:
(9.126)
11(7 r, !,,- ..... 7"'(.,., (III1-U)
'"
j' rlwl'HI-I:
(9.127 ) 1~ 'li uil. ious:
C -
[kl~kl_~+k2_k:3+k3_rl ,
kiJ.frjJ.,
Zii
-
kH kj _
6;;
-
-2(p_ . k;) - 2(p_' k;)
6ij
-
-2(p+ ·
A,
~
A, (3.·I) -
-
koJ -
+ 2(k; · k, )' + 2«.· k;),
2(p+ . k;)
2E'
(p_ 'ka )( p- , k.)
[
i,j= 3, -l,
m'Z;:I4Zu]
k3+~'1+ + 4E16 34
'
_,_ (k:5+(2E - ka_H2Ek... - Z43) 6" (p_ . " ) +2Ek3~kH(2E-~_-k"
"",;,/2E] ,
)-m 1 k4
(p_ . k.)
A, -
2(p _ .
k~)(p_ , k.) [k3_kH k3 + k3+'~'4_k;J. .L
- (k3J.
A.(3 ,') -
B,
-
D,(3.4 ) =
2£' (p+' ks)(p+' kt )
.
2£k1+] + (p_ ,1:.. ) ,
[k k
~-
""
4-
m~ZMZU] + 4 E2A3~ ,
_ ,_ [ ••_12E - ',.)(2£k._ - Z,, ) 6~
+ D, -
-
[2E~+ Z"3 2EA34 (p_ . fe a)
m,l;3_k3.L
+ k~J.)Z3-tZU J
(p+ ' /.:J)
2EA"3 _ k~_ (2 E - k3+ - kH
2(p+.
} -
m2k.+k3.LkU/2E]
(p .. ,k.)
,
k~(p+ ' .(;4) [k3+~'1_k:lJ. + k3_k
H
- (feu
kU
+ k•.d Z3
,
g. 811MMAUY OF (JIm fOllMUME
1~II
(9.128) ~
9.17.1
k4 nearly parallel to
p+
Photon polarizations:
,lr
i
= 1,2 , 3,
)~ ,ry-l ;
(9.129)
Nonvanishing hclicity amplitudes, up to permuta.tions of photons 1,2 and 3 [for example, the helicity amplitude J\4"( +, -; -, +, +, +) is obtained from M(+, -; +, +, - , +) through the interchange kl +-+ k31:
, M(+ ,+;+,+,-,+)
2c'mkj.L ku
[
k"" . k<+ ], (2E - kH)(p+ . k,) kl+kl_k2+k2_k3_k,_ 2e'mkH k<+
IU(+,+;-,-,+ ,+) -
e'mkj.L k4.L [
E (p+· k,)
k3- k H kl+k,_kHkz_kHk._
e4mk3_kH
M(+ ,-';+,+,-, +)
M(- ,+;+,+,-, +)
1
1'
,.. IJ 17. r+r- ...... 11.,.., (III
'# 0)
M(+. -;+,+,- ,-)
-
181
2c4 (2B - kH)k:u. [ kJ_k4 ]t (Pt' .1:. ) ~·I+kl_k~ ... kl _ k3+k4+ 1
-
.+1 (+,-;-, -,+,+)
Ze (2E - .l: H
~
(2E
-
(p+' k.)
kl+kl_~tkl_k.+
-Se E J.,l "k
[
.1:3+ k, _
r ,
[
r 2
kH)(p+' k.) kl+kt_k~ +k2 _ k3_ k4'"
8,'E'k", (2£
k, _
)I':3_ [
r ,
k._
kH){P+' .1:. ) .l:1+1.:I _.l:2+.I:,_k,,+
,
4
M(-, -:- ;+,+,-,-) -
M( - ,+;-,-,+, +) :
M(+, -;-, -',+,-) -
=
2
-8c. 5 kil.
(2E
(2e
[
Al (-,-;+,+,-,-) -
=
r
kH)(p+ ' .1:.) k'l+kl_kHk2 _ k~ _k4+
8"E'~H
[
",-
kH)(p+ · k.,,) kHkl_k2+k,_k,,+
[
,
2t!4(2E - .l:4+)k3.1 k, k.. _ 1 (PI' k4 ) kl+kl_kH k,_ k3+ kH ] 2e 4 (2E - .l:H ).l: 3 _ (Pt ' kt)
Jt
.1:... _
[
r
kJ+k ,_k2+k,_kH
,
4,'Ek" [ k,+"_ (p+ .1:4 ) kltkl_kHk2_A.'3_ kh 4e 4 8kH ( k... _ ]~ (Pt' .1:. ) kltkl_kHkl_kH
k,_k._ M(-, +;- .- ,+,- ) = 4,'E>; , [ (Pt· k4 ) kltkl_k, ... kl _kl+kH ~
]!"
kJ+k4_
4,' Ek,_ [ k._ (p+' .1:.) kJ+kt_kHk,_k.t+
]1
r
,
e"mk:J.J..k;l [ .1:3 k~+ ]' E(p+· .1;.) kl+kl_ kHk2 _ kHk4_ e. 4mk 3_ kH
,'
E(p+ . k.) [kl+ kl _ kHk2 _ 1~
II. HIIMA/AII.Y Of<' (JI;() fOJtMIlLAg
.,(-,-;-,- ,+,-) -
(9.130) Unpolarized squared matrix element:
IMI'
=
(9.131) Unpolarized cross section:
do =
9.17.2
-
k4 nearly parallel to
p_
Photon polarizatjons:
;= 1, 2,3,
i = 1, . .. ,4.
(9.133)
Nonvani shing helicity amplitudes, up to pel'mutations of photons 1,2 and 3 [[or example, the helicity amplitude M(-,+; -,+,+,+) is obt"ined from Af( - , +; +, +, -, +) through the interchange k, <-+ k3J:
.,(+,+;+,+,-,+)
=
'\/(+,1-;-,-,+.+ ) =
M(I , - ;+.+,-.+}
M(+,-; r.+ ,-,-)
""
8e 4 E'2 k:J_ [ k.. (2E k. _)(p ,k. ) kl+l:l_ kHkl
M(-I-,-;-,-,+,+)
=
2c (28 - k._)k.u [ kJ+kH (p_ ,~) kl+kl_kHk2 k3_k~_
,
4
-
l'
,
4
2c (2E - k4_)kH [ k.. ]' (p_,k 4) klof kJ_k2+kl _ k4_ 4
M(- ,+; h+.-,-)
=
:=
M( - .+;-,-. 4 .+}
-
>::
,
2e (2E - k4_)kj.1. [ .l:HkH _]' (p_ . k.) kl+ k 1 _ kHk.t_ kl _ .1:,,_
2e((2E - k4_).l:3+ [ k.. (p_ . .1:.) /';1+k l _.l: H 4:2_.l:. (2E
-8e~E~kjl
[
Be· £"'l k3 _
'[
]!
k3-.\:H ]j k. _)(p_'~) kl+.I::_.l:Hkl_k3+k4 _
k..
(2£ - k4_)(P_ . .1:,,) kl+kl _ kH~_k.,_
]!
V. HilMMAI!Y 0/" Qf;/J POJ/MlnA!:
4e;EkJ~
M(+,-;-,-,+,-)
[
(p_ . Ie.)
kHkH k1+k,_k,+k1_kJ_k;_
]1
l'
1
-
4e'EkJ+ [
(p_ . k.)
k4+ k1+k,_k>+k,_k4_
M(-, +; -,- ,+, -) -
M(-,-;+,+,-,-) -
M(- , - ,'- , -
,+, -)
1
2e4mk3~kH
k,_k,_
[
(2E - k4 -)(p_' k,)
]'
k1+k,_k,+k, _kJ+kH
2e'mk 3 _k 4 _
(9.134)
Unpolarized squared matrix element:
1M I'
= (9. 135)
U npolarized cross section:
,
x6 (1'+
+ p- -
k, - k, - le3
-
Ie,)
~t~~~~~t k
k
10
Ie
20 "30
k
40
.
(9.136)
.'i. 11. " ·~r-"""r1r 1("'1: 0)
!I.17.3
185
k; and k~ nea.rly parallel ~o
p+
amI.
v-. resp.
l' IIOLon pillarizations:
J.t
~
N,!p, p_ ,k,( 1 ;o1.)- ,k, p, I>-{1 ±7,)].
/-i
-
N,I,k, h p- {1 ± 1.) + ,P- h ,k,(1 'F 0,)].
H
-
iV.!ft. p-
N-' •
j,,(1 ±
,,)+ ft,
p_
- E'!32k"k,_]! •
i=i,2,
ft.(l 'F 1.)]. ;=1, ... ,4.
(9, 137)
Nlilivanishing helicity amplituue5 , up to permutations of photons 1 a.nd 2 [(or example, the helicity amplit ude Mt +, +; -, +, +, +) is obtained from kl ( + , +; +, -, +, +) thro ugh the intcrch a.lIgtl k, ..... k~):
M(+.+;+,- , +. +) =
2e:1 mE (E 2 - k',_ )( P.. k, )(p_' k) " X
{kuk,L [k,_k._] I+ k;,k,L [k,+kH] I} . k2+
kH k4+
kl +
M(+.+i+,-.-, +) -
M(+,-;+,-,+,+) =
,
_k.t+];
4e-l.E2 [k2 _k3 (p+ . k3)(P_· k.J ) kz.tkJ+A:, _ 4
M( -,+;+,-, +,+) _
4e E1 ki.l. k2_ (p+ . k:l)(p_ . 1.<4)
4c·E2 (p+. ka)(P_ . k. )
[k:,-k4+ ] j kJ .~ k4 _
[k2+~_ k4+]! ~_k3.rk._
k3 _ k4 _
!I, SIiMMAIlY 0[<' QED FOItMULA£,
JH(J
M(+ , -;+ ,-,+,-) -
0'
{
(2E - k,_)(p+ 'k,)(p_ ' k,)
8E' k2l. kJ _ k,+ 1' k2+ [k3+k,J
- m'k;J.kuk,J. [kJ+k.._ 2EkH kJ_kH
f} ,
[k
2e'E(2E - ka;. )k2J. S_ k4-j- ] i kH(p+ ' k3)(p_ ,k,) k3+k,_
M(+,-;+,-,-,+)
1
-
2e'E(2E - k3+) [k,_kJ_kH]' (p+ ' k,)(p_ ' k,) k'tk3+k._
!vI( -, +; +, -, +, -) -
2e'£(2E - k3t)kiJ. k3_k<+ ' k1t (p+ ' k3)(P_ ,k,) kJ+k._
M (- ,+;+,-,-,+)
,
[r r
-
2<'£(2E - k3+) [k1_kS_k H (p+ ' k3)(P_ ,k,) kl+k3+ k,_
~
c' (2E - k._ )(p+ ' kJ)(p_ ' k,)
{
8E Jk;L k3_k.. '1 kl+ 3t
[k k.-l
rn'~I~k3J.k~.L
r
"E,k,+ 1
4e' E' k2J. [ k3- k.. kJ - CP+ ' k3)(P_ ,k.) k3+k.-
M(+,-; -I-,-,-,-)
-
4e' E' [kl+k3_k.. (p+ ' k3)(P- ,k,) k,_k3+k,_
1!
r 1
4<'E'k; , [k3 _k H k1+(p.j. 'k3)(P_ ,k,) k3+k,_
M(-+'+ ' , ) - t --) ,
M(- ,-; +,- ,+,-)
-
4<'£' [k1 _kJ_k H (p+ ' kJ)(P_ 'k..) kl+k3+k,_
-
e'mk;J. ku [ k3_ k,_ k,_(p+ ' k,)(p_ , k.) k3+kH
-
e'mk,_ [ k,. k3 _ ] t (p+ ' kJ)(p_ ,k.) k,_k,+
•
]!
r
.-f} ,
[k3+ k
kJ ,_ kH
,.+... - . 'Y'Y'Yi
111'1.
(III '" 0)
187
H , (-,-i +.-,-,+) =
x
{ku.kj !. ~+
[kJ+kH]!
+ khktJ. [k3 _l.: .1: ,+
.1;3_ .1;(_
t
_]!}.
k~+ kH_
(9.138) IJupolarized squAred m
IMll =
kl+k~_2(~+E.2::)1~:~ . '-4)2 ["E" + (2£ X [4E~+(2E - k(_?1 m k!_] 2
4E2k.+
1.:3+)1
+ 4~11~"_]
kf++ k?+
(2E - k:s+Pkl+kt_
.
(9.139) tJ ll poiarized crOBS section: du
=
(9. 140)
9.17.4
-
-
k3 and k-{ nearly parallel to
p+
Phoi.on pobriutions:
t, p_ />-(1 ±1,)1,
/.~
- N, I/>+ />_
r,
~
N,lh h />-(1 ±,.)+ p- p+
#1 '1'1, )].
N ,.- '
~
E2132k;+k,_I~ ,
i = 1, ... , 4 ..
,<,(1 '1'1')-
i= 1,2, j ,.,. 3, 4,
(9. 141)
9. SIiMMAIlY
lM8
or' QI~'f)
Jo'ORMULAb'
Nonvanislling hdicHy amplitudes, up to IH!1'1I1HLuLiolls of pllOtons 1 and 2
[for example, the helicil.y amplitude M( +, +; -, +, +, +) is obtained from M( +, +; +, - , +, +) through the interchange k, .... k,l: 2e'B3 k*l.l
M(+,+;+,-,+,+)
-
1
kJ+ [k3tk3-k4+k,_1'
,
M(+,+;+,-,+,-)
-
2e'IB3 1
[
k1_ kl+k3+ k3_ kHk4_
r
2e'B,(3,4)k;J. 1
k1+ [k3+k3_kHk,_1'
~l- k
2e'I B 4(3,4)1 [k k
1+ 3+ 3 -
k
4+ 4-
2e'B,(4,3)k,J.
M(+,+; +,-, -,+)
1
kl+ [k3+k3_k.l+k.-l' -
1
~l-k
2e4I B,(4,3)I[k k
k
1+ 3+ 3- 4+ 4-
2e" Bl kl.l
M(+, - ;+,-,+ ,+)
r r 1
,
kH [k3j-k3_kHk,_F
r 1
-
2e' IBd [
k,_ kH k3t k3_ kHk._
.'(2E - k3+ - k4+)Blk~.l
M(-, +; +, -, +, +)
1
EkI+ [k3+k3-kHk,_1'
r 1
-
M(+, - ;+ , - ,+,-) -
-
6'(2E - k3j- - k,+)IB1 1 [ k1_ E kJ+k3+k3_kHk,_ 2e'B,(4,3)k2.1 1
kH [k3+k3_kHk,_1'
4-
r
k,_ kz+k3+ k3_ kHk,_
l'
~'-k
2e4 IB,(4,3 )I [k k
2+ 3+ 3- 'H
1
k
2e' B,(3, 4 )k2.1
M(+,-;+,-,-,+)
1
kH [k3+k3-kHk, -1'
1
-
2e'IB,(3,4)1 [
ISO
2e 4 lJ;(3, -I )kj.1 M( - ,+;+,-,+,-)
=
-
kIt
4
2e IB
1 (a,4) 1[
M{ -,+;+,-,-.+)
,
2e 4 Bjk 2.1
,
kIt
M(-, - ;+,-,+,-) -
~
k, _ kltk3 ... k3_k.I+k.
2e 4 B;(4,3)k;.1
M(-,+;+,- ,-,-) -
[k3-+k3_k4tk4_1;
[ k3t~_k4+k4_J~
4
2e 1Bd
[
k,_
kl+~J+k3 k.t+k..
2e 4 B4(4,3)k2.1
]'.
,
kH [k3tk3_kttk4_]' 4
[
k, _
2e IB4 (4,3) 1 k2+ k3t k3_ k4+k4
t,
M(- ,-;-h-,~-,+)
-
ka [k3tk3_k4+k4 .. J3
.!.
4 2e IB .(3,411
.
[k:i+k3+~- k.t+k4-r
M (-,- i +,-,-,-)
(9.1 42)
JUU
g, SIIMMA/I.Y OJ' QNU fORMUL.4E
lJnpolari~(~d S
lTIn.trix element:
8
IMI'
+ k,_) {[1 + (21': - kat k, _ k,_ (kl_ kH kH 2E
2e kat kH
=
kH)']
IB, I'
+ IB,(3,1)1' + IB,(4,3)1' + IB31' + IB,(3,4)I' + 18,(4,3)I' } . (9 .143) Unpolarized cross section:
",' (kl_ k,_) d,,-_ 64,,' E'k3+k3_kHk,_ -kit +kH
{[ (2E - 2E kH) '] 1+
kat -
IB, I'
+ IB,(3,4)1' + IB,(4,3 )1' + IB,I' + IB.(3, 4)1' + 18.(4,3)1'}
•
x6 (p+ ~
9.17.5
+ p- -
k, - k, - k3 - k.)
~t~L~~~~ ~ k k k . ~JO ~20
'30
(9.144)
~"'O
~
k3 a.nd k4 nearly parallel to jL
Photon polarizations: i = 1,2,
,tj - Nj lfei p- h( 1 ± ')'.) + h p- fei(l
'f ,),,)] ,
j = 3,4,
i;;;;:1, ... ,4.
(9.145)
Nonvanishjng helicity amplitudes, lip to permutations of photons 1 and 2 [for example, the belicity amplitude M( +, +i -, +, +, +) is obtained from M( +, +i +, - , +, +) through the interchange k, H k,]: M~~,+;+,-,+ , +)
=
M(+,+i+,- , +,-) -
-
2e'Al(3,4)ka
2e'IA,(3,4)1 [k k
:'+
k k ]
1- 3+ 3- '1+ 4 -
t
,v.1?
r: ' ~-""'}' '}''}''}' (III';O)
M(+,+; + ,-,-, +) -
I () I
k, _lk,.k,_k.+k._I; 2e'jA.1(4, 3)1
[kl_k:H:::~+k4J
, '(2E - k,_ -
M(-.+ ;+ ,- ,+, + )
M(+ ,- ;+,-,+,-)
Mt+, -;+,-.-,+)
~_)Alkl.l. I
I
lOt
N. SIIMMA tr.Y OF QliII FOIIMULAE
M(+,-;+,-,-,-)
-
2e4A ik u . l
k,_ fk3+k3-kHk._l' l
-
2e'/A, / [
k,+ ]' k,_k3+ k3_ kHk._
<'(2E - k 3_ - k,_)Aik;.l
M (- , +; +,-,-,-)
1
Ek,_ fkJ+k3-k<+k.-l'
-
e'(2E - k3_ E
1
k.-)lAd [_
k2+ ] ., k,_k3+k3_ kHk,_
2e·A,(4,3)k;.l
M(-, - ;+,-,+,-)
1
k,_ [k3+k3-k.+k._l'
-
2e'/A,(4,3)/
[
kH k,_k3+k3_ k.+k,_
2e'A, (3,4)k;.l
1\1(-,-;+, - ,-,+)
1
k,_ [k3+k3_k.,+k._l'
t r 1
- 2e'/A.(3,4)1 [ kH k,_k3+ k3_ k,+k,_ M(-- ' + - - -) I
)
I
I
,
-
2e'Aoko 3 a 1 k, .. [k3+k3_k<+k'_1'
2e'IA 3 1[
2+
k,_k3+ kk3_ kHk._
r (9.146)
Unpolarized squared matrix element:
IMI'
=
x{
[l + CE - ~3~
-
k.::. )
'J lAd' + IA,(3, 4)1'
+ IA,(4,3)I' + IA3 1' + IA.(3,4)1' + IA,(4,3 )I'} . (9. 147)
IJlljllIlHri7.f'd ('.ross secLion:
",""=",a,,'~~;- (.1:-1+ +
.in =
6'h"2E1 k:J ... k3_k.t ... k4 _
x
{[1 + (2"- ~E
kl _
kH) kl _
-k._ )']IA'I'
+ IA,(3.411' + IA,(4.3)1' + IA,I' + IA.(3. 411' + IA.(1.3)1'} x.fl(p+
H. L8
e+e-
---f
+p_ -
k\ - k2 - k3 - k 4 )
d3 kl d'J~ rPkJ cJ3~ k
k
III
k
'10
:w
p+ p..-77 (no Z-exch a nge; m
k
'
(9.1 48)
~o
#- 0) (9.149)
I l"/i II i t iUlls:
B _
1+
E
(q+' q_)
[2 q_*
+ kuQ:.1 _
'1.1q-.!.]
kH~_
C _ (q+' , _)(q+' k,)(q_· k, ).
9.18.1
k~ nearly pa.rallel to
(9.150)
f ..
I'lloton polarizations:
N i- 1
_
E~[32ki+kj_I}.
Nonva.n ishins helid ly amplitudes:
M(+ • .,.;+.-.+.+)
~
i=1.2.
(9.l51)
9, SIIMMAIIY OF Q£D FOnMULAE
191
1
M(+, +;-,+,+,+)
-
M(+,+ ;+ ,- ,-,+)
-
-
M(+ ,+;-,+, -, +) -
M(+,-;+,-,+,+)
e'mq:.lkl1 Z+_ B [k ,t
2E(p+ ,k,) [(q+ 'kJ) (q _ , k.)J! _ e4 m k2.LQ•_ J. Z' "±_ B-
E(p+ ' k,)
r
lsc
'
k2+ k,_q++'I_+
e'mk2+q __ IBI
1'
,,
r
2E(I'+ ,k,) [(q+' k,) (q_ ,k.)l'
-
c E'I_1.Z+_ B 2 _ ' , [kz (p+' k,) Ck2+q++q_+
-
2e 4 Eq __ IBI [ k,_ (p+ - k,) _ k2+(Q+' k,)(q_, k,)
M( +, -;-,+ ,+,+) -
.4Eq+.l Z_+B [ (p+ - k,)
r 1
, 2k,_ ], C kH
r 1
-
2.' Eq+_IBI [ k,_ (p+ ,k,) kH (1+ 'k,)(q_' k,) 1
M( -,+; +,-, +,+) =
111(- ,+; -,+,+,+)
-
e' Eq±.L Z_+B [ 2k,_q_+ ] ' q+_(p+ , k,) CkHq++
2e' Eq_+ IBI [ k,_ (p+ 'k,) k2+(q+ - k,)(q_ 'k,)
1' 1
e'Eq~J.Z+_B
[2kz_Q++ l ! q-_(p+ ,k,) Ck2+Q_+ 4
-
t
2e Eq++IBI[ k,_ (p+' k,) k,t(1+ ' k,)(q_ , k, )
]!
1
'
19&
, M(+,-;+,-,-,+) ~ (2£ _
e(£2q_J.Z;. B" [8k, _Q++ ]' k,,)q __ (p+. k, ) C"ki'·,'-'+q'""_'-+ 4
4e ,E2q++IB I
(2£ M(+, -
i -, +, +,
-)
=
, [
k,_ ]' k,,)(p,. k,) kH(q,· k,)(q_ . k,)
e~(2E -
.l:2+)Q+.1Z_+B [ k,_ (p+ . k,) 2CkH Q++Q_+
]1 ,
~ ,'(2£ - k,,)q+ IBI [ k, ]' (p+' .1:,) ku(q+' kd(q _ . kl )
M(+. -i-, +.-,+)
=
,
e~ £2q+J. Z:+ B" k2~)q+_(p+·
(2E
4,' £,q_+IBI
[8.1:2_Q_+] , 1.:,) CkHtJ1 1 [
1;,,(2R - k,+)(p+. k2) .1:2+(9+' .l:1)(q_· kd .~ (-,+;+,-.+,-)
-
,
[8k2_Q_+] '
4
e E'q':'.lZ_+B ~+)q+_{p+. "',)
(2E
4,'£'q +IBI
Ckuq++
[
,
k,
]'
(2£ - k,+)(p,. k,) k,,(q,· k,)(q_· k,) M(-,+;-;-,-,-,+)
_
e4 {2E - k2+ )q~.L Z:+B· [ (p+ . k,)
]1
,
1;,_ ]' 2C~+q++q_+
~ ,'(2£ -
kH)q+ IBI [ k,]1 (p,. k,) k,,(q,. k, )(q_· k,)
M( -, +; -, +,+,-)
~
e E'q"1.Z+ _B
4
,
q++]'
[B~ (2£ - kH)q __ (p+ · .1: 1) C.I:,+q_+ 4
4e E'q++IBI [ 1;,,(2£ kH)(p+· k, ) k,,(.+. k, )(q_' k,)
]!
,
lUfJ
M(-,+;--,+, - ,+)
!I. SIIMMAII.Y
or Qh'1J
_
e'(2E - kH)q: .j.Z;_ lr [ k,_ (p+ - k,) 2Ck2+q++q_+
_
e4 (2E - k2+)q __ llll (p+' k,)
-
e'Eq_l.Z;_B· [2k,_'1++ q__ (p+' k,) CkHq_+
[
_
4
2e Eq++IBI [ (p+ . k,)
M(+ , -;-,+, -,-)
1' ,
k,_
],
k,+(q+ . I:,)(q_ . ktl
l' 1
_
e'Equ Z :+ B ' [ 2k,_q_+ q+_(p+ . 1:,) CkHq++
_
2e' Eq_+ IBI [ k,_ (p., . k,) kH(q+ . k,)(q_ . k, )
_
0' Eq: l.Z._B'
l' 1
M(- , + ;+,- , - , -)"
1
M(-,+ ; -,+,-,-)
(p+ . k,)
[
2k,_ ]' C kHq++q_+
_
2e'Eq__ IBI [ k,_ (p+ . k,) kH(q+ · k,)(q_ . k,)
M(-,-;+,-,+,-) _
o'mq_l.kil.Z+_B [ kH E(p+ . k,) 8Ck,_q++q_+
l}
l' 1
e'mkHq __ IBI
M(-, -;-,+,+,-)
_
1
e'mq+J.kil. Z_+ B [ k,+ j E(p+ . k,) 8C k,_q++q_+ e'mkHq+_IBI
l!
k,_ k2+(q+. k,)(q_ · k,) 1
M(+,-;+ ,- , -,-)
FUIlMUU6
l}
Y. /II. e+r.*
_0/1"+ /,-
i,.
(NO Z · F,X(;(fA NOl\'; 1/1
M(- ,-;+. -, - ,-) -
I- It)
e(mq_l.khE.+_I1° (2£ - kH )q__ (P-+ . Je,)
[Je~+q++ 2Ck~_q_+
l;
'"
e·m~+q++IBj
M(-,-;- . +,-.-)
-
(2); - k,, )(p• . k,) [(,•. k,)(,_ . k, )1 1 . (9. 152)
IIU llolarized squared matrix element:
IMI' = 2k2+{p+ ' kl )2{Q+ ' kd{q_· kd )( [4£2 + (2£ _ 1.:,+ )2 + ~~22~:_ ]
x
[qL +q:_+ (2~Eq+k:Jl + (2~E~k:+rl·
(9. 153)
IInpolarized cross section:
du
04181 1k1_
=
25tnr 1 E2 k2+CP+' k1 )2 (Q+' kd(q- · kd
x [4E1
9.18.2
+ (2E -
kH )l
k; nearly parallel to
+ 4~21;1~]
p_
Photon poiariz.'l.tions:
i = 1,2 .
(9.15»
U. ,l'IIMMAI(Y O{<'
IUH NOJlvani ~hing
Q~;/J ~' Ol(MULAF: ,
hcliciLY amplitudes:
J'
1
M(+ ,+;+,-,+,+) -
e'mk'J.q J.Z+_B [ k,_ (2E- k,_)(p_ ·k,) 2Ck2+Q++q_+
-
_ ______~e~'m~k~'~ - q~-=-~IB~I~--~~1 , (2E - k,_)(p_ . k,) [(q+' k,)(q _' k.))'
-
e'mq+J.k;J.Z-+B [ k,_ (2E - k,_)(p_ ' k,) 2CkH q++q_+
l'
1
M( +, +; - , +, +, +)
e'mk,_q+
IBI
1 j I
"
"
.'
'09
M(+.-;+,-,-,-)
=
""
M(-!. ,-i-,+,+ ,- )
M(-,- ;-,+.-, +)
M(-.+ ;. ,-,+.-)
=
4 e (2R - k2 )qH (p_ ,k,)
e4 E'q+J. Z_+ fJ
,
DI [
k,.
k,_(,+, 1',)(,_ 'k,)
81.:H (2E - k,_)(p_ . ,lo,) C,lo,_qH'J_ + 4
181
]'
[
]1 ,
[
=
k,. ]' (ZE - k,_)(p_' k,) k,_(q+' k,)(q_ 'k,)
=
e4 (2t.: - k1_)quZ- +n- [ kHq.::.±.-..] t q+_(p_ . k,) 2Ck l _9++
_
, ' (2E - k,_)q_+ JBJ [
_
4e
E'q+
-
(p_,~,)
e~(2E
-
k,_(q+ 'k , )(q_' k, )
kl_)q~l.Z_+B
9+-(P-, .1:,)
_
M(-,+;..l.,_._,+) _
I',.
[ k1+q_+ 2Ck,_QH
]1
"(2E-k,_)q_+JBJ [ k" (p_ ,k,) k,_(q+ ' k,)(q- - ',) e~ E2q+.L Z- + E-
(2E 4c
4
k~ _)(p_
]1
]!
,
], 8k H . I.,) CA'2_QHQ_+
E~q+_ I BI
[
[
k,.
(2E - k,_)(p_ ,k,) k,-(,+, k,)(q_ - ',)
]t
,
~.
~{lU
SI1MMAII Y ()I<' (J£U FOIIMULAE' l
M(- ,t;-,t,t,-)
-
-
.'(2E - k,_ )q:.1 Z+_IJ [ kHq++ ]' q__ (p _ ' Ie,) 2Ck,_q_+ ,4(2E - k,_)q++IBI [
(p_ . Ie,)
k>+ ]i k,_(q+ ' kJ)(q- ' kt} l
M(-,t;-,t,-,+)
-
.'E'q*-.1 Z't - B" [ 8 k 2+ ], (2E - k,_)(p_ . k,) Ck2 _q++q_+
-
40'E'Q __ IBI [ k,+ ]' , (2E - k,_)(p_ . k,) Ie,_(q+' k )(q_· k])
l
M(+,- ;+, -,- , -) -
-
0' Eq_.1 Z.. B* [2k2+q++ q__ (p_ . Ie,) Ck,_q_+
l' l
1
204E9++IBI [ k>+ i (p_. Ie,) k,_(q+' k])(q_' Ie,)
l' l
M(t,- ;-, t,-,-) -
M(-,t;t, -,-,-)
.'Eq+J.Z:+B" [2k2+q_+ q+_(p_ . k,) Ck,_q++
r r r
-
4 k2+ 2. Eq_+IBI [ (p_. k,) k,_(q+' kJ)(q_. Ie.)
-
• Eq+.1Z ±B 2k2+ 4 " '. "[ (p_ . k,) Ck,_qHq_+
-
2.'E9+_IBI [ k2+ (p_ . Ie,) k,_(q+' Ie.)(q_· Ie])
++.111( - " " ,- ) -
.4Eq* .1Z" B* [ 2k>+ (p_ . Ie,) Ck,_q++q_+
1t
l' l
-
204Eq __ IBI [ k2+ (p_ . Ie,) k,_(q+, kt}(q- . k,)
-
_ e'mqt.1k2.lZ_+B [ k,_q_+ ] ' Eq+ _(p_ . k,) . 8Ck2+q++
l.
M( -, - ; +, - , +,-)
-
. 'mk,_q_+ IBI l
2E(p_ . k,) [(q+' k,)(q_ . Ie])]'
'
, !I.IS. c ... c~ .... /I~ ~- 11 (NO Y, · ;;X(.'IIANfl/o:;
M(-,-;-, +,+,- ) -
-
III
I
U)
'ill J
c~rnq:J. J.:u.;I,+_11 £q--(V- . kll
~_q++
[
8Ck2t Q_ +
]!
e4 mk,_QHIBj M (-,- ;+ ,-. -,-)
2E(p_· k, lI(q+' k, )(, _· t, )]! '
e~mq+jkHZ: ... B· [
=
(2E
k,
]1
k, _ )(p_. k,) 2Ck"q++,_+
M (-,-;- , + ,-. -) I .
(2E - k, _)(p_' k, ) Ilq+' k,)(,_ · k']I'
(9.156) Unpolarized squared matrix element:
",_ (p_ . k, F(,+ . k, )( q_ . k,)
X[4£2 + (2E ,
,
.1:,_)'
+ 4~2~2~]
(2E'+ _ )'
x [qH+q- ... + 2E-k2 _
__ )'] + (2E, 2E ~ _
.
(
) 9.15 7
Ur:polarized cross section:
~
9.18.3
kO! nearly parallel to q",-+
The primed quantities
4+
k~.:I:
nud
~.l
are evalulLtl'!
9. 811MMAIlY Oft' QED f 'ORMULAE
in eqn (9.11;9) i. obtained by applying a 'P""" rdlcc:tionlo q+. The quantity Jj denotes tbe lTluon mass . Photon polarizations:
,i~
N,[,k,
11+ 11(1 ± 'Y,) + 11 11+
):,(1 Of 'Y,)], (9.159)
Nonvanishing helicity amplitudes:
M(+,-;+,+,+ ,-) -
k,+ f'!
e4 9+oq-.1 +_ [ ZB
M(+,-; +,-,+, +)
(q+' k,)
r 1
r k,_ l' n,_ r '
1
2k',_
Ck,+q++q_+
1
-
M( +, -;-,+ ,+,+)
-
2e'q+oq __ 1BI [
(q+: k,)
k'+(q.· k1)(q_ . k1)
f 'q+oq+.l Z_+B [
(q+ . k,)
1
Ck,+q++q_+
l'
1
k, _ k,+(q+ . kd(q-· k, )
-
204q+09+ _ IBI [
-
_ c" f19_• ~ k'· 2.l Z+- B [ k',t9++ q+Oq--(q+' k,) 8Ck,_q_+
-
e'!
(q+' k,)
1' 1
M(- , +; +, +, +, -)
M(- ,+; +,-, +,+) -
4 - e q+09+.1Z_+B [2 k2 _q_+ q+_(q+. k,) Ck2+q++
r
l' 1
4
2. 9+Oq_+ IBI [ k;_ (q+ . k,) k,+(q+' kd(q-·
kIJ
"
20J
_
_ ~4q+u9:.1.Z+_ /J
[2k~_q++ ] 1 Ck~+q_+
q- -(9+ ,kl )
_
2c~qtoqt+IBI
[
(q+' k~)
M{+,-;+,+ ,-,-) _
_
k2_ k?:+ {'1+' kl )(q_ , kJ}
"-,. Z·
[k'
]!
e #"11. 9+1. ,t n' ltq-+ (2q+u + ki+)q+_ (q+, k1 ) 2Ck~_q++
]1
('4JLk~+q_+ I D~I _ _ _..,
-
M(+,-iT,-,+,-) _
(2,+,+ 1<;+)(,+ - k,) 11,+ - k, 1(,. - I, )JI _
'
4
e Ql09 _.!.Z+ B [ 81<; ]1 (2Q.. o + .I4.~)(q+. k1 ) C~+q++q_+
4",j"g.. JBJ
[_
I,.
(2q+o + l~+)(q+· k1) k;+ (q+' kl)(q_· kl )
M(+ , -;+,-,- ,+)
_
,
_ e (2q+o + I4t)'1-.!.Z'; B- [ 14-9++ ]' 4
q__ ('1+ ' k1 ) 4
2C~tQ_ +
+ J4t)'1++181 [ k; _ 1"-,,,) ,.,+(,+-k, )I,.- k, )
_
e (2q+o
_
_ c4 (2'1+o + klt)'1t.LZ-tB [ k1_ (Q+' k, ) C k~+ql l '1_+
~ ,'(2,.. + k;+),+.IBI [ ('1+' k1 )
M(+,- ;-, + ,- ,+)
M(+,-;-, -,+,+) -
]1
k,. k~+(q+·
kIHq_·
]1
]1
kd
II
,'I. SIIMMAUY Of QgO 1"OIl.Mlfl,AI';
M (- ,+:+,+, -,-)
_
_
e'uk" [ k'2+ ]t r- 21. g" -.1 Z· J±_ .U' (2'1+0 + ~+)(q+' k.,) 2Ck,_Q++q_+
e' ",k,+q __ IBI M( - ,+;+,-,+,-) -
,11(-,+;+,- ,-,+) _
e'(?q - +0
(q+' k,)
+ ~+)q~.lZ+- B
g__ (q+ ' k, ) _
M(- ,+;-,+,-,+) -
[
k'22Ckl+g++q_+
, [ k,_q++ ] "
2Ck,+q_+
k,_
,'(29+0 + k,+)q++IB I [ ('1+' k, ) kh (q+, k,)(q_, k,) e'g't oq'-.1 Z'+- B' [ 8k'2(2g+ 0 + ~+)(g+. k,) Ck,+q++q_+
4e'q~oq __ IBI
[
,
k,_
1
c'!'q+.lk'.lZ_tB [k,+q_+ ] ' (2q+o + k,+)1+-(q+ ,k,) 2Ck,_Q++ _ _ _ _..::'2'!'k,±q-+ IB I (2g+ 0 + k,+)(q+' k,) [(g+ 'k.)(q_. k,)]l '
L .'VI( _r,
_ ., +
1
___ ) ,
,
1
- e'q+oQ_.LZ;_B" [2k;_Q++] ' q__(q+ . k,) Ck,+q_+
, _ 2e'q+oq++IBI [ kl _ ], (q+' k,) k,+(q+ . kl)(q_ 'k,)
]1
]1
(2q+o + k't)(q+ ,k,) k, +(q+, k.)(q_· k.)
M(-,+ ;- ,- ,+,+) -
]1
e'(2q+o + k,+)q+_IBI [ k, _ ]t (g+ ' k,) k't(q+ · k,)(q_ . k,) <'(2g+ 0
,11(-,+ ;- ,+,+, - )
+ k'2+ )q'+.1. Z'-+ B'
]"
'I
I ,~
, 1"
",1" - 1' 1 ( NO X . ~;X (.'II AN(.'II'; ... I uJ
M(- . -;-, +,-,- )
'in!)
,
7,: i!.'· [:'u'2_/I_+ ] ~ q+ _(I/+' '=2) C k~+ q++
4 _ C //H)/Ii.J.
:
~-
4
2e q+oq_+101 [
:
k~, ( q~' k d ( q _ 'kl)
(q+,k1 )
M (+ , - i -, - , - .+)
,
'"'Z.B·["' r
e pq- .l.
:
U
±_
q+oq- -(q+, k1 )
M(-.+ i + . - , -.- ) -
M(- ,+i-, - . - . +) -
UnpoiCll'izcd squared matrix
l±q++ 8CkLq_+
,,
£4pk2+q++I BI
2q+o{q+ ' k2 ) [(q+ , ~' I }( q_ ' kd]l
r ,
_ ", ~ o, ±.1 . Z · ± B" [ 214 _ Ck;+q ... H _+ (q+ ' .1: 2)
2£4q.,.oq+_IB I [ kL k~+ (q+ ' kd(q- ' kl ) (q+' .1:1 )
",+0, ._.1 Z·+_ B· [ 2k'1CI4 ... q++q-.~ (q+ ' .1: 2 )
M(- ,+;- .+ ,-,-) -
:
]!
, [
~
r r
2£ q+oq__ IB I 1(,+. k, ) 14,(,+· k, )(,_· k,) e
'p,.
e4pk2+q+_ IB j
2,,,(,+ · k,) U,+ · k, )(,_. 1,)1 1 .
1
r
k' Z-· + B · [ k; t q+o(q. ,,1;1) 8CkLq++Q_+ ±.1 1.1
eJ emell ~ :
, J'
,
(9.160)
~Oll
Y. SlfMMAIlY OF QE/I FORMULA/>'
Unpolarj~cd <.:r()HH
sildion:
,...
(9 .162) ~
9.18.4
k2 nearly parallel to
q_
The doubly primed quantities k~± and k~J. are evalualed in the rotated frame where'i- determines the positive z-axis (see also Section 7.4.3). The fourvector q in eqn (9.163) is obtained by applying a space reflection t.o q_. The quanti ty IJ denotes the muon mass. Photon polarizations:
(9. 163) Nonvanishing helicity ampliLudes:
JW(+.-;+.+,+,-)
=
2q_o(q_· k,) [(q+' k,)(9_ '
e'q-Oq_J.Z+_ B [ 2k~_ (q_ .•,) Ck~+q++q_+
M(+,-;+, -,+,+) -
2e'q_oq __ IBI
(q_ . k,)
1 ,
k. )I'
1t
k~_
[ k~+(q+.
k.)(q _ . k,J
l' 1
,
.'
207
MI+,-;-,+,+.+)
_
_
e1q_oquZ_+ 1J [ 2k;_ (q_ . k~) Ck1 . . 1++Q-+
2e~q--Oq+_IDI
_
]! kd
k~_
[
k~ .. (q+· ~'d(q- .
(q_. k,)
M(-, + ;+,+,+ ,-)
]!
e~.uq;'.l.k'{tZ_+ B 1--01/+_(Q_' k1 )
[ 1.:'1... Q_+ 8Ck'{_Q++
]!
t
llk2t1_+IBI ' 2'_0('_ ' k,) [I,. · k,)(,_ · k>ll ' e
,
."'1 (-, ~ ; +,-,+ ,+)
M{-, + ;-, +,+,+) = e"q_oq:.lZ+_~ q_ . (q .. k,)
_
[2k:;_1++
Ck~!IJ_1
l'
4
2e q o9++1 B I [ k!f. (q.' k,) k~+ (q+· kl )(9.· k1 ) 4
1.L , _.1 Z" t - BO e ./t1.:"'
MI+.-;+.+,-.-)
,
[2,-. + 1<;+),--(,_. k,)
1;
2+ + ... [ ""
2Ck~_,_+
l!
l!\Uk'2+ q++ I81 J\tJ ( +, - j +, -, +, -)
_
~t(2q..o + ~+)q_.l.Zt_B (q .. k,)
~ ,' ('.-. + I<;t),-- IBI [ (,_. k,)
M(+,- ;+.-.- .+) _
k~_
k~+ I,+· k,j(q_ . k,)
e'q2 oQ . .lZt _B O (21_0 + k~... )q__ (q_· k1 )
4e4q~q++ I BI (2q_o + 1.: 2+)(q·· k,)
1
[. k!J. i :2C.l1+q+-.fJ-+J
]!
, [BkJf.q++] ' Ck~i+q_ +
k~_
[ ~.,.(qt·
kd(q· . kL)
]!
g, ,\'/lMMAItY at" (Jo'D mllA/ULAI:
2U8
M(+,-;-,+,+, -)
M(+,-;- ,+,- ,+)
M(+,-;-,-,+ ,+) -
.4I'q+l-k~J.Z_+B
[
k~+
(2q_o + kq+)(q_, k,) 2Ckq_q++q_+
1~
_________e~'~I'~~~+~q2+-~IB~I________., , (2q_o
+ k~+)(q_
, k,) [(q+ 'kd(q-
,kdl!
l' 1
M (- ,+ ;+,+ ,-, - )
e'"kql qiL Z:+B" [ kq+ (2q_o + kq,.)(q_ ,k,) 2Ck~_q++q_ + e'l,k~+q+_ IBI
M (- ,+;+ ,- ,+,-)
M(-,+;+,-, -,+ )
M(-,+;-,+,+ , - ) -
e"q:'oq:.lZ+-B [8k~_q++ (2q_o + k~+)q __ (q_' k,) Ck:f+q _+
4e'q:'09++181 (2q_o
+ k'{+)(q_
[
kL
l' 1
l~
,k,) kq+(q+, k, )(q_' k.)'
'J, I ,~.
(; 1,.- _~ I' + 11- 'T 1 (NO X ·/-;X(.'I/A N(,' i'.';
I Il)
III
20'J
,
M(-, +;- ,+,-,+} = -
M(-,+;-,-,+, T) :H( +,
-j
+, - , -,-)
-
c1 (2q _ 0 + kifH It( lJ" [ k"2 ]I . _ 1. X· ':t::.:::.... (q_' k~) 2Ck'2+Q++Q_.
e~ (2q_o + k~+)q __ IB I [ k~_ ]: (q_ . !'3) k~+ (q + . kd(q-· kd
_
M(+ , - ;- ,+, '- ,-) -
.. " [" r r r r
e Wi. 1. kn Z.,._ B kH q++ (2q_O + k'1+ )q __ (Q_' k'1) 2CkJ_Q_+ e"'l·k~+ q++llll
(21_0
+ J.:~+ )(q_ . .1,;2 ) [(q+' .1:
1
e(g -0-1 q Z~.. . W q__ (q_. k2 }
)(1_ 'kd; i •
('J~." -,_++ q
Ck~+q_ +
,
2e !f_oq++ IB I [ k~_ - (q_ ' k~) . k~+(q+, C)(q- . kd 4
=
e(q~q+l.Z · +B· qt -(q- ' k7 )
[2 14_1 _+ CkY.. qtt
[
-
2e 4 q_oq_+IBI k!f_ (q_. k~) kt-.-(qt · kJ )(q_' k1
-
_ e• 111+.1. k"2l Z'_± 8" [ k"h.'1- + q-oQ ... -(q- . k2 ) SCkq_
-1 ,:
)1
r , r r ,
M(+,- ; -, - , - .+)
-
M( - , 1;+,-,-,-)
=
-
111(-,+;-,+,-, - ) = -
,
e~pki'±q_ t IBI
2,_,(,.· k,)I(,,' k,)(q.· k,}]! , • Z· 2(" =.il-oq+ "- t ll" [ ·'l(q_. k2 ) C k!f+fJ++q-+
'
2e( q oq+_IBI [ k~_ ('.' k,) "1', (,+. k, )(,_. k,j
e4 q-Qq'-.1 Z·±=----. B" [ 2k"2(q_ . k2 ) Ck~+ qHq __ ~
2,',·,,__ llJl [ (q_. k2 )
k;
k~.~(q+
,
)t
./:;){q_ . kd
,
1'
~ III
!/. IWMMAIlY OF QED FORMULAE
M(-
I
+. - - I
,
,
I
-t.) (9 .164)
U npolarized squa red matrix element: e8lBI'k~
IMI'
.
(9.165)
Unpolarized cross section:
9.18.5
-
-
kl and k2 nearly parallel to p+ and
P_. resp.
Photon polarizations:
. "l=
I? , ....
(9. 167)
211
AI( h -;+,-,+,+) -
e~mE
{
k'
('I. ' 9.)(PT · kd(p- , /e2)
kl+icz+q_+
l.,:q_l.
[k 1_k 2_q++]
H 21.9-.1 ' [k,_k,_qH
kl+k H 9-+
M( I, +;-, + , + , +) -
e~mE
r}
,
{'ug+.1 [lcl-~-q-J , k'
(q _ . 9.)(P-t . kll (p_, k))
,
1
kl+icH 9++'
,
[k,_k'-Q-+l l }
-:-' k'2.d+..l k k '1+ '1_9++ J
M( +, +;+,-, +,-)
(U; ~
M(+.+;+,-,- ,+)
e'mklJ.q:.l. kl+)(p+ . kx)(p- . k 2 )
e~mkl+q_+ (2E - kl+)(p+· k\){p_.
~
(2E
4
M(+,+;-,-,+,- ) -
12E (2E
-
M(+,-;+,-,+, -)
~
1
,
[ka'7 __ ]
Ie;)
k2 _q++
[I
r , ,
k 2-'1++ k,+k 2 ... q_+ 1-
[k l _ Q+_]'
(21-.: - kz_) (p+ · k1)(p_· /ell kJ+9-+
~
M(+,+;-,+.-,- ) -
,
1
e• m k'l.l.9- .l kz_)(p..- . k,)(p_ . k2 ) e mA)_q++
(k1+k H q_+ ] k 1 _k 1 _q++
e4mklJ.9t..l.
+]
e~mkl+q++
1
[k l +k2+Q... i kl_)(p+· kd (p-· kz ) kl _k z_9_+ [k 2+ 9 +_ ] leaH?+' kJ) (p_ . k~ ) k 3 _9_+
,
e4mki.1Q+J. [kl_k Z_q _+]) (2E - k2_)(P_ . kd (p- . /ell k\+k 2+9_+
e'mk2_9 +
(2E
k1_)(r~'
(2E
,
[kl_kHq++] l
. r
kH)(p+' k1 J(P_· k2) k l+k2_q_+ 4e 4 E 2q++
~
l_
kd(p_ ./':2) k l +9-H
4e 4 f.,'2q_.!.
(2E
[k 9 _] t
k l _k 1+q+_ ' kH)(p+' ktl(P_ · k 2) [kl+kZ_q_+
1
y,
~12
,\'(lMMAIlY
or QI?I) FORMIILM;
111(+,- ;-,+ , +,+)
M( - , +; +, - , +,+)
1
111(-,+;-, +,+,+)
-
4.'E'q" .1 [kt_k,+q++] ' (2E - k,_) (p+' kt)(p_· k,) kHk,_q_+ 1
4e'E'q++ [kt_kHq+ - ] ' (2E - k, _)(p+ . k,)(p_· k,) k1+k,_q_+ M ( +, -; -1-, -, +, -)
+ m'kl.lk2.lQt.l [kt+k-,_q_+]~} 4E
,
k, _ k2+q+t
1
2e'Eq_.l [kt _kHQ++] ' (p+ . k,)(p- . k,) k1+k,_q_+
M(+, -; +,-, -,+)
,11(+,- ; -,+,+ , - )
+m2k1.lk.'J.9:.l [kt+~,-q++ l~ } , 4E
kJ_ kH Q_+
1' 1
M(+, -;-,+ ,-,+)
-
20' EQu [k 1 _k2+q_+ (p+' k,)(p_ . k,) kHk,_q++
_k2+q--l
t 20'Eq_+ [k t (p+ . kt)(p- . k,) kt+k,_q++
, <J. f,~.
,.1- ~- _ /, j. 1' - ")' i
(NO y, -f;X(: fI A Nf ,'/1'; II!
i UJ
213
M( --,+;+,-,+,-) _ _
,
2e(Eq_t [kl_k1+Q __ ] ' , (p+' kd{p_ . k2) k , t k2 _q ... +
"
.11(-,+;+,-,-,+)
(q_. q_)(p+ . kJ)(p- . k1 )
{4E3. [kl _k2+Q_+ ]'
i
+ m2kil.kid_.L 4£ M( - , -'- ; -,
+, +, -) - (TJ,_·2 e~Eq:J. kd(p_ . 1.1) 4
2e Eq++
-
1'11 (-, +;-, +,-, +)
_
,
++]I} ,
kL "k2- Q ..kl_k1-q_+
[kl_ ~~2+q++] , ka k,_'7_+
,
[k _k 9+_]' l
(p_. kd(p_ . k2 )
k Lt k2_q++
1j+.L
H
kl+k 2_q_+
,.,
(q+' 1-)(Pt' kd(p_ . k,)
{4E"
[k,_k
H QH
]
I
q-.1 k,tk2_fJ_+
+ m 2kj_k;J.q+J. [kltk2_Q_+ ]'} , 4£
M(+ ,-;+,-.-,-)
4e(El q_J.
=
(2E
,
[kl_k2tQ++ ] '
k1_)(p t' kJ) (p_ . k~ )
kJtk2 _q .. ~
,
[kl_kHqt _],
4e~E'tq++
--
k l _k,+ '7 ++
(2£ - kz_Hp+, k1)(p_ . k,) k1+k2_q_t
M(+, -i-,+,-,-) 4{.1E~q_+
M( -,+; +,-,-,-) _
(2E
k1 _)(pt· kl )( P_'~)
(2£
4 e(','1 L:J lit.! • kl+)(p+ . kd(p_ . 4e'l £2 1 _+
(2E
[k 1 _k2/Q __ ]' k lt k1 _qH
[k '
q__ ] ; 1.-1+1.1 _1++ I_I'\"~+
1.-1 )
[1.-1_ k~+ '/ __ ] ;
k1+)(p+ . kd(p- . k2 ) kl+k2~
9. SII MM AIIY 0,.. QI:I} FOIIMVI.AN
~11
M(-.+;-.+.-.-) -
4.'E'q:.1 [k,_ k H q++ ] t (2E - kl+)(p+ . k,)(p_ . k,) kl+k,_q_+
-
4.'£'Q++ [kl _kHQ+_] ' (2E - k,+)(p+ . k, )(1'_ . k,) k,+k,_q_+
1
1
e'mk,. q:.L ' [k1_k,_q_+]' (21': - k,_)(p+ · k,)(p_. k,) ki+kHq++
M(-.-;+. - .+.-)
1
.M(-,-i+, - ,- , +)
-
e'm k,_q .. + [.kl_q __ ] ' (2E - k,_)(1'+· k,)(p_ . k,) kHq++ e'mkilq -.1 (2£ - kl+)(P+ . k1)(p_ . k,)
.
[kl +k2+q++ ]! k , _k, _q_+ 1
-
e'mk H q++ [k2+9+-] , (21': - k1+)(l'+ . k.)(p_ . k,) k,_q_+ 1
M(-,-; - .+,+,-) -
-
.'mk2.1q+.L [k1_ k,_q++] , (21': - k,_)(p.· k.)(1'_ . k,) k, +kHq_+ [k1- q+_ ]1 (2£ - k,_)(p+' k.)(p_ . k,) k,+q_+ e ,Tn k'2-Q+ +
1
M(-,-; - ,+,-. +) -
M( -, - ; +. -, -, -)
e'mki.1q+J, [kl+kH Q_+] ' (2£ - k1+)(p+' k.)(p_ · k,) k1_k,_q++
-
• 4mk1+'I_+ k2+q_(2E - k,+)(p+' k,)(p_ . k,) [k,_q++
-
, 'mE (q+ . q_ )(p+ . k,)(p_ . k,)
{
]t
• k1_k, _q_+'1 kuq_.1 [kl+ k2+q+J
• [kl+k1+q.,. +kl.l q-.1 k'1- kAl _ ll_+
]1}
'
r 1
M(-,-; - .+,-,-) -
.'mE { . [k1_k,_q++ (q+ . q_)(p+ . k.)(p_· k,) kuq+J, kl+k,+q_+
•
•
·
(9.168)
,.,
11 111
r
l ,,· - '/ '+I' - ir(NOZ.f;,tCIfANGf:: ",::;' ())
21"
(9.169) 11111",lnri1.,:J cross section:
(9170)
~1 .J~.6
k~ and k~ nearly parallel t o
p+ and (i+,
r esp.
'l'11I~ prj~ned
quantities k;± and k;.l are eva.lua.ted in the rotated b.me where '/1 ,Idermines the positive z-a.:'{is (see also Section 7.4 .3). The four-vector q ill ,'qn (9 .171) is obtained by applying a space reflection to q+. T he ql!ant;ty II ,I'~l!otes the muon mass, l'lI" lon polarizations:
It
-
N,I,k, J>. p_ (1 ±1,)+ p- h ,k, (1 '1'1,)1,
N- 1
,
-
E?(32k H .k1 _11,
jf
-
N, I;', fl+ fi(1 ±15l+
~
q:2~o13')k' - Z+ k'z_ It .
,
1\,-1
h h+ ,k, (1 '1'1,)1, (9.171)
fl, SIIA/MAIlY 01' QI!:/) FOnMII/,AE
Nonvl)ni.sJdIlg lwli city a.mplitudes:
M(+,+;+,+,+ ,-)
=
e'ml,kll.k,·, q:J. [kHk,+q++]! - 4Eq+o(2E - kt+) (p+' k,)('J+' k,) k, _k,_q_+
, -
e'ml'kl+k,± ('Jt+q--l" 4Eq+o(2E - k1+ )(p+, k,)(q+ . kz) ,
-
e4mq+oq __ kUq.j.J. [kHk,_q_+ ] ' - 2E'(p+' k,)(q .• · kz) k,_k,+q;+
1
M(+,+;+,-,+,+)
1
[k~-q+-q-+l '
4
e mq+oq __ k1 +
-
2E'qt+(p+, /;,)(q+ ' k,)
k,+
'
'1+ k' '2-Q+i- )~ [k k".)(p+' k,)(q+' k,) k,_k,+q_+
e 4 mq+okuq:1.
M(+, +; - ,+,+,+)
- E(2E -
-
e'mQ+Ok1+ [k; _qt+q_-] t E(2E - kl±)(P+' k ,)(q+' k,) k,+ ' 1
4
e mq+ok1 .Lq+J.. [kJ+k2 q- + ]' -""EC;;(2;-;E;---=-;-k~,+i(p+, k,)(qt . k,) k,_kl+ q++
M(+ ,+; +,- ,+,-)
e'mqtOkH
-
E(2E-k,t)(lJ+·k, )(Q+·k,)
'I-+ ) ~ [k'2 q+k,+ ' 1
e'mq+oq__ kuq:J. [ kHk,_ ]' - 2E'(p+ . k , )(q+ ,k,) k,_ k,+q+.,.'1_+
M( +, +; -, +, +, -)
_
e4 mQ+ok1+
[k; _q:_ ]}
2£'(p+ . k, )(1+ .•,,) k,+q++
M(+ ,+;-, - ,+ ,+)
'
e4 ml'kll.k21. q.~.L
-
4E'I+o(2E - k , +)(p+ . k,) ('1+ . k2 )
[kHk~+q_+] k,_k,_q++
1
e' ml'k,t k,+ (q+-q-+J'
1
e'I'Ek;"J.q+J. [k, _k,tq _+] ' q+o(2E-k1+)(p+.k, )(q+,k,) klt k,_9++
M(+, - ;+,+,+,-)
1
-
e'I'Ek,+ [k1 _Q+_Q_+]' q+o(2£ - k,t)(pt . k,J(qt' k,) klt '
,
1
,'(.~,
/\.I (
" I r'
_..
,1+ 11- ., ., (N () ;I, -,','X (:/I11 N U ~::
In
t
OJ
'l 17
1-,-; +,-.+,+)
M(+, - j-,+ , +,+) -
(2E
A1 (-, +j+, +, +,-) -
e" (2q+o
M(-.+;+ ,-,+,+)
+ k~+)q:;'.L
,
[kl _ k~_q_+]'
(p+ . kd(q~ · k-z )
k L +k~+q'H
,
[h_k~ _ q+_ q_+]' (p+ . kd(q+ · k:2 ) k1+ k; + 4
e (2q+o
+ k2 t )
,
4
M(- ,+:- ,+, +,+ ) _
2e q+oq: .l
[kl- ki_q++] '
(p+' kd(q+ ' k2 )
kt +k~ + q_+
4
2c q+o [k 1 _k;_1++q_] ! kl+'I,-~+ (p+' kd(q+ . kl )
M(+ ,-;+,+,- , -)
=
Jlk
e4 2+ 2q+o(p+' kd(q+' k1 )
[k1 _ q+_ q_+]j kJ+
4c Eq+oq_J..
[kJ k21/++]!
(2E - kl+)(p+ . k1) (q+· k:l)
k J+k2+q_+
1
M(+,- ;+,-,+,-) _
4
(2E
4e Eq+o [kl_k; q++q -]' k1+)(p+ . kd(q+ ' k l ) kl+k!z+
... II . .... /I MMAlly OF (J81i
"'(29+0 + k;+ )9-.l [k, _k;_q++] (p+ . k, )(q+ . k, ) k,+k;+q_+
M(+, -;+, -, - ,+)
e (29+0 + k;+) [k,_k; (p+ . k,)( q+ . k, ) k1+k,+
,
[k
,
2e'q+oq--9+J. 1 _k; _q_+]; (p , . k' )(9+ . k,) kl+kl.,. q~+
-
t
qH9-- ]'
4
M(+,- ;-, +,+,- )
l" OltMU£A~'
,
2.'Q+Oq__ [k,_k, _q+_q_+] , q++(p+ . k,)(q+ . k,) k, +k,+
Af(+,-; -, +,-,+) -
1
M (+,--;-, - , -I-, + ) _
_
e'"Ek'J.q_J. [ "I_k'+Q++] ' q+o(2 E - kH )(p+ . k,) (q+ . k,) k, +k, _9_+
,
e'I,Ek;+ [kl-9++Q- -] ' Q+O(2E - k,+)(p+ . k,)(q+ . k,) kH
M (- ,+; +,+, - , - ) -
M( - ,+; +,-,+ , -)
M( - , + ;+ .- ,- , +) -
'119 4
M( -.-;- ,+,+,-) .. c (2q+u + A-h)q:", [kl - k1_9++]! (p+ - .I:. )(q.. - k, ) 'c1+k1+Q_+
=
e4 (2q . o +
14 ... )
[4: k1-qHq--] ! 1_
(p+' k1 )(q+' 1.:1 )
'c1+k!z+
1
'\/(-.+; - , + ,-,-1) =
4e Eq+oq:J..
(2£ - kl+)(p+ - k.)(q..-·
1.:,)
[k l _k1_9++ ]! kl+k1+q_+
['-1_ 1.:1 q++q--] !
4
4e Eq+o
(2S- l -1+ )(V+-k.)(q+ -k1)
M(-.+;-.-,+.-I)
1 C }lk;.lti!_ _
= _
e: (2q+o + kl.)qu (p+ - £:.)(q+ - 1.:,) e (2q+o + k~+) (p~. k.)(q+ · 1.: 2 ) 4
M(-,-;-,-,-.+)
M( .-1;+,-.-. -) =
[~I_k2+q-+]!
2q+o(p+ - kd(q+ - ~ ) .I:, .. 14_q++
4
M(+.- i-.+.-.-) =
kl+ '4+
[k' -k~- q-+l ; kl+~ +q+ .
[kl _k~ _q+_q_ ... ]! k,+k~.
~.
.~/lMMI1ItY OF' QP;I! Io'ORMIILAE
.1\1(-,+;-,+,-,-)
M(-, + ;-,-,-,+)
M(-,-;+,+,- , -)
M(- , -; -'-, -, -, +) 4
e mq+okl+
[k~_q~
2E'(p+ . kd (q+ . k,) k,+q++
l' !
,
e4mq+oki~q+..!. [kHk~_q_ + ]' E(2£ - kl+)(p+' k,)(q+ ' k,) k,_k,+q++
,
e'mq+ok,+ [k;_q+_ q_+] ' E(2E - kJ+)(p+ - k,) (t/+ . k2 ) k,+ lVI( -, -;
+, - , - , - )
M(- -' ,
1
)
+ - -) ,
1
_
e• mq+o k·Llq- ,l
[k' 1+ k'2-9++ 1j
E(2E - kH)(P,_- k,) (q+. k, ) k,_k2+q _+J
I
!J 1/1, ..... ,- • • 1,1 I' -"{')' (N U 7,·,.;Xc.:IIANUf.';!II
A/(
. -; -,-.-,+)
0)
Z'lI
,
c4mp.ki1.).·~1.q_1. -
-
- 4Eq+o(2E - .l.:1+){JJ+ . kd(q+' k 2 )
e4mttkl+k~.~ {1++q 4Eq+o(2E
~'1+ }( P+'
[k1+k~+q++'J' kl_k~_q_+
_Jt
.4:,)(1+ ' kl )
(9.172) '
1IIl IUllllriz{.-d squa.red motrix element:
tllll'Qlarized cross section:
du =
t
a kl_k2_
256%'k1+l4 ... rE (2 E - k1+)(p+ . k, )(q+· k 2 )F
[
X tiE
2
+ (2E -
kH
2
)
'" 1
fll 1+ + 4E2.1.: _ 1 1t
(9.1 74)
9. 18.7
k~ and k~ nearly parallel to
p+ and q...,
resp.
The doubly primed quantities k'.j:l: and 1;';1. are evaluated ill the roLated frame where if- determines the positive z·axis (see also Section 7.4.3 ). The four· vcc~or q in eqll (9.175) is obtained by applying a space reflection to q_ . The (IUaIllity II dcnoL('s llJC muon mas:;. Photon polarizations:
jf -
Nlil~J
p+ p-p ±1'$) + p- p+
).~ 1(1 =F 75)] ,
(9.175)
ii. SliM MAItY OF (JED PO{(MlnAE Nonva.llj~Jdllg
hdicity arnpliLudca:
..11(+ ,+; +,+,+,-) = I
e4ml'k1+k~+ [q+-q-+ l'
-
4Eq_o(2E - kH)(p+ ' k,)(q_ . I.,) ,
[k k"
e'mq-OkU.qtl ~,-q-+ E(2E - kH)(p+ . k,)(q- . k,) k,_k~tq++
..11(+,+;+, - ,+,+)
e~mq_okl+ -
]1
[k~-q+-q- + l t k~+
E(2E - k1+)(p+ : k,)(q_ . k,) I
e"m·q_oq+_kll.q+ J. [kl+k~_q++ ] '
M(+,+; -,+,+,+)
2E2(p+. k,)(q_ . k,)
k,_k~+q~+
4
-
e mq_o<J+_kl+ 2E2Q_+(P+' k1)(Q_' k,)
[k~-q++q- - l } 1.1+
l' !
e'mq_oq+_ku.q+,L [ kHk1 2£2(p+ . k,)(q_ . I.,) k,_k¥tq++g-t
..11(+,+;+,-,+, - )
e"mq_okJ+
-
[kz
q+-1l
2E'(p+' k.)(q_· k,) k1+q-t e, mq_o k 'u.q-.L ' E(2E - "1t)(P+ · k,) (q_ . k,)
M( +,+; -,+,+,-)
-
e"mq_Ok,+ E(2E - kH)(p+ . k,)(q_· k,)
' 1
[k 1+ k"2- q++ ] , kl _ k~+q_ .
[k 2- q++ q--1 1 k2+
' 1
- e m/Lk l J.kqJ,. q~l. [k1+ k~+q++]' 4£q_o(2E - k,t)(P+' k!)(q_. k,) kl_k~_q_+ 4
M(+ ,+; - ,-, +,+)
e4mflokl+k~+ [q++Q
-
4Eq_o(2E - klt)(p+ . k,)(q_ . k,) ,
-e"floEk~lq_l
..11(+,-;+,+,+,-)
I
-1'
[k'-k~+g++ll
9_0(2E - ",+)(Pt' k,)(Q_' k,) ",+k1_Q_+ 1
-
e'floE">+ [k, _q++q__ ] ' Q_o(2E - k,+)(p+ . k,)(q_ . I.,) k,+ '
·" .' . M(
I
-/I+-II-}'Y (NO Z.I;'Xr:J/ANC:/-.'; III
1. -; +.-,+.+)
~
-
c .q-01-J. ,,'I" I
4
HI ,+;+,+ , +,-) 0\1 (- • .1- ;+,-.+,+) -
-
~
M(+ , - ; -,-r.-. - ) -
[
M(+ , -;+,- , - . -) -
1.1+_ [kkllJf
(p ... ,..1:1 )(9_ . .1.:1 )
.1:1+.1.:1'.. 9_+
1_
r r r '
e, I' k'"uq+.l. • ["1- k"1+q-+ 2q.o(p+· kd(q.· k'1) kll k1.qh e'k" IJ 2+ 2q_o(pt· 4;. )(9.. k1 )
• .
.. · 9-+ [I., .qkl+
k l _ 2.9-+ [k" kl-+k1'+9++
k .l: _1.·1_+ ]1 , [" (p+. k1){q.· Ie)) .1.:1+.1.:1'+ 2e 1.0
l•
(;4(2q.o + W-l_)q • .l. (Pt' kd(q-· k 2)
1
[.I.:I k1'.QH] t kl+k~l+q_+
e 4 (2q_o + A;'... )
[k _k1. q++ q__
(p+ ,k,)(q_, k,)
k,+k!!.
c<$}Ik!li,q
1
1.
r '
[.1.:1 _14'+1++
2q.o(pt· kd(q·· k~) k1tk:{.q-t
e pk;t 2q~(p+ . k.)(q_ . .1.7)
t
[1.:1 - 1++ 1-_] l .l.:H
2e,q-oIl+-q-J. [k ._k1_q++ fl -(JJ+ . kl)(q_·~) .l,:1+kllq~.
r ~
, [" r k l _~_qT ~q __
2e q- Q{jT~
kl·k!f_ Jt .l.:1+k1+9++9_-+
2 e ~ 1-0
4
~
kl+-k~+
(2E - .1.:1+)(1'.' kd(q_ · 1.:1 )
_ _ 2e q-OqtL (p+' k1){q.· .1.:2 )
q-+ (p+ . kd(q_·
r
[.1.: 1 _1.:1' qHq--]~
4e Eq_o
- (p+' .l:1 )(q.· k1 ) ~
[k 1_ k";_q~+
(2£ - kl +)(p+ . kJ)(q_· .('2) kl +-k~Tq_+
2e~q-{)q+_q+.L
M( 1. - ;-,+,+, - )
1I1(- ,-;-.+,-r.+)
II)
.1.:7 )
k.. k:{.
nl
,~,
SIIMMM1Y
or QI';I! FOIlMULAE
M( +, -; +, - , -, +)
M(+,-;-,+,-,+)
,
e (2q_o + k~+) [kl-~_q+-q-+l'(p+ 'kd(q-' k,) kl +k~+ 4
4. Eq-01+J. [""_ k~_ q_+ (2E - kl+)(p+, k,)(Q_ ' k,) kl+k~+q++ 4
M(+,-; - ,+,+,-)
M( + ,-;-,-,+,+)
M( - ,+;+,+,-,-)
,
M( - ,+;+,-,+,-)
_ .'(2'1_0 + k;+)q.t~ [kl_k1-q_+J' (p+' k.)(q_ . k,)
e4 (2q_o + k~+) (p+' k,)(q_' k,)
M(-,+;+,-,-,+)
k1+k;+q++
[kl_k~
q+_q_+] t kl+k1+
1
1'
!I,/II. r+ <- - . I" 1'- Y'l' (NO l,·~:X(:tltlN( ,·f;; III
M(-,+j-, t,+.- )
-
=
=
If!_q++
.\;1+4:1+q-+
)
r r r r r
e q_o 1_q++q __ " [kk" (p+ ' kd(q_ 'k~) k l +J4+ 1_
[k k"
1_ 2_1 ' + 2e' q-IIQ+-1_.1. . - (1'+' kd(q_ ,1.2 ) kak¥+r-+
(k l _J.1{ qHq q_+{p.j. ·1.:1}(q_ · kJ' kH k~+ 2e(9_0q+_
_~4 (2q_o + k2'+)q_.1 [kl_k~_q++ (p+ ' k 1)(9_ ' k~) e 4 (29_o
+ k~+)
kl+k'hq_+
J!
[k l _ 1.:2'_1 Hq- _ ] t
(Pr ,kl)(q_· k~ )
kl_k~.j.
4
,
[kl _k;_9_+]} (11+' kl )(9-' k z) k1+41+9++
=
e q_1I Q... -9-+ 2' [k k" (p+ . kd(q_ . k,) kl ... ki+
~
-
t
l - q++9- [k 2q_o(p ~· kl)(q_'~) kite ' Pk:'2t
-
M(t, -; -, - , -.t) -
.+', '"., - , - , -)
I
etpk'Lq"!. [k1-k!i+qH] 21_u(P... ' kJ)(9- ' k21 kHk¥_q_+
M( - ,ti -,-.t,+)
M( -
[k
(p+' kd(q_· k.l
M(-, +;-,t,-,+) -
M(+, - ;-,t,-, -)
II)
r. q-uq_J.. 2' •
~
-
M(t,-i +,-,-.-)
f
2e q_u9 u
r r [k"q,_q_. r
1_ 2
1_ k",+1-+ • k" [k 2q_o(p+· kt) (q_ ' k11 k\+k;_q++ e II
2.lq~.L
c~ /Ik!{±
2q-o(P+ . kll{ q. ' kl
_
)
k l+
q-oq.j.-qtJ. ,.' ' [k\+.I.'2-tQHq-+ k k:']' (Pol' kd(q_· l
~_
k~)
2c'q_o [kl_I.:~'_q~_] t (p+' k.)(q_ . k,) .I.·ak~'+q _ +
225
nu
Y. ,\'(fMMAIlY OF <11m FORMU /,A
M(-,+;-,+,-,-}
M(-,+ ; - ,-, -,+} -
M (-, - ;+, + , -, -) -
M(- , -;+, - , -, +}
M( - , -', - , -I. 1 - , +} -
M(- ,
-'+ - - -} ,
,
,
I
f:
.'i / ,¥,
r ' r - _ /." ,, -
n (NO
y"
";X(.'1f II NOI1;
til
I- U)
e m/'k j..l. k~'..1. qu 4
M(- .-i-.-.-,+) -
[kl+ klt
q-+ 11£9_0(2£ - k,+ )(p+' kt)(q_' k2) kl_k!l_q++
e4 mJlk, +k~t [q+-9 +]1 t ilqJola:i7.oo sq uared matrix
X[iEl + ('lE -~, x
,
1'
(9.176)
eh~ment:
1+
)'1+ m'lk?t
1E~kl_
1
[1 Q: O+ (2<J_1l + k2+ )1 + /:ki;, 1(q++Q--+q~_q_+), Q_o
(9,177)
1_
IJ II pOIn rizc<1 c r() $~
~c(;tion:
x ['q'-0 +(2q _0 +k"1+ )2+ II IJ'k"' 1f 2 k" Q-o 1-
9,18,8
1
.kl and k2. nearly parallc1 to p_ and q+, resp,
The primed quantitif>S k~= and .14..1. arc evaluated in the rotated (rame where '1+ de~ermines the pO$itivc z-axis (see also 5«tion i.4.3). The four-ve(:tor 9 in C
, [30" q+o -"".1+ k'2_ [!
'
(9.179)
....
y, IIIIMMAUY
m'
(JlW /<'OIlMU/,Ab'
NOllvauishillg hdicity nmplituuca :
M(+,+ ;+,+ ,+,-) =
e'1mJ1-kil.k~l.q+l.
[kl_k~+q_+]!
- 4Eq+o(2E-k,_)(p_ , k , )(qt,k,) "1+",_'1++
e'm"k l
_",+ [q+
1
q-+I'
[kl_k, q++9-+] i - 2E'qt-(p_ ' k, )(q+ ' k,) kltk,+
e''''9+oki.1.q-1.
M(+,+;+ ,-,+, +)
-
e'mq+oq_+k,_ [k'_'I++q __ ] t , 2E'9+ _(P_ ' k,)(q+, k,) k't ' 1
M(+,+;-, +,+, +) -
e' mq+oki.1.q+.L
[k,_k'_9_+] '
E(2E-k,_)(p_,k,)(q+,k,)
k",k,+9++
"'2-9+-Q-+ e'mq+ok,_ "' ]! [ £(2£ - k ,_)(p_ , k,J(q+' k,) k,+ ' 1
e"mq.,.okil.Q- l. [kt_k2_Q++] ' -£(2E-k,_)(p_,k,)(qt, k, ) k1+k,+Q_+
M(+,+;+ , - ,+,-) -
e'mq+ok ,_ £(2£ - k, _)(p_ , k,)(q+ , k,) e' mq+Ok;.lqH
M(+,+;-,+,+,-)
-2£'q+_(p_ ,k, )(q+,k,)
-
k,+
'
[k,_k,_",,-+ ]t kl+ k,+q++
e'mq+ok ,_ [k,_q:±] t 2£'(p_, k,)(q+ ,k,) k,+9+'
[kl-k~+ q++] ,
1
e mJlki.l. k~l. q-l. 4
M(+,+;-, -, +,+)
[~_q++q __ ] t
4£q+o(2£ - k, _)(p _ ,k, )(q+ ,k,) 1:" ,k,_9_+ 1
e'17lJlk,_k't [q++9--1'
'" M(+,-;+,-,+.+)
~
-
M(+ ,- ;-,+,+,+) -
_
(2q+o + k~t )q_J. [kl+.t; (p_ . kd('1+' k,) t, -k1tq-t
'1++] ~
e(
c-tC2q.tO + k2+) (p_. k1 )('1+' k~)
[ ~l+kl_q++q __ ]; k1-k;t
2.t4q+o1±.J.. - (p_ . kt}{q+· 0\;])
[kltkl
r ,
'1-+
kl_~+q++
,
2c q+o [kl+k2_ q+_q_+J1 (p_. kl )t'1+' le1 ) k,_k~+ 4
~
q+ ~2E - kJ_)(p _' kd (q+' Ie,) ~
M{-.+;+,-,+,
I)
[klt"'tq. .; ]~
c4pEk~·3:!
M(-, -/;+, +,+,-)
-
kl _kl _'1_+
e' /. Ek'H ['1+'1+.,.q-- II ,+.(2E - k,_)(I'_ . k, )iq+. <,) k._ J
'r
[I'-.!.t k'l-q-t
'1+01+ ?_I:• •
_
,
qt- (p-, kl )(Q'1 . k, ) .l:1_k2+'1+-t4
2e q+C!
[ kJ+k;_q:+
(p_ . kd( q+ . k 2 )
k 1-k2+q+_
~
Jl
4
M(-,";-;- ,T,+,-I-) -
4e Eq+ofl:J.. [k. tk2-g.. kl_)(P_' kd(q+ ' k,) kl _I4 .. '1_+
(2E
[kl+~1_q+ ~q- - Jt
4
=
M(+.- ;+,+,-.-)
~
-
' 4c EqtO (2£
L'I_k~+
k,_)(p_, kd(q+' 1. 2 )
e•JJ Ek" lJ.,qtJ..
_
]!
• 1+ , ...] '1-..- ' [k!'
j
.,
[ r
'1+0(2E - kl_ )(p_ . k 1)('1+' '\;,) 1.:,_.I4_9H. J e pEkU
q+o(2E
kl+q+ q +
kl_)(p_· kl )(q+' 1.2 )
4
M(+,-;+ .-.+,-) -
2e q+()fl_.1. (p_ . kl )(q+ . k l
-
2e 4 q+o (p_. kd(q+ ' '=2)
)
,
[k l +k2_q_+]' kl _ ~+q_+
[kl+k1_q~.~ q __ ] t kl_k~+
k l_
II. "IIMMAIlY Of' Qf:/J FOIlMUJ,M: I
_
M(+,-:+ ,-,- ,+)
[kl+k~-q++q-+l'
2e'q+oq_J.
kl_k~+
q._(p_ . k.)(q+· k,) 4
2e q+0<1_+ q+ _(/,_ . k,)(q+' k, ) _ e'(2q+o
M(+, - ; -, +, +, -)
+ k't)q+.L
(p_. I<.)(q+. k,)
[kl+k~-q++q--l! k,_k~+ 1
[kl +k,_q_+] ' k,_~+q++
I
4e' Eq+oq+.L [kl+ kLq_+ ] , (2£ - kl _)( p_· k.)(q+ . k,) kl_~+q++
/).1'(+, - ; -, +, - , +)
M(+,- : - ,-,+,+)
-
4e4 £q+o
]!
[kl+k,_q+_q_+
(2£ - "1- )(1'-' k,)(q+' k,)
"I_k"I_
4 e "k21 q-.L [ kl4 k't q++ 2q+o(p_ . k.)(q+. k,) k._k'_q _+
1'
I
e• Jt k'2+ [k Itq++q- - ]1 2qTO(P- . k.)(q+ . k,) kl _
r I
e'/,k'>q> [k It k'2+ q+t 2.1 -.1 - 2q+o(p_· "1)(Q+' k, ) kl_k,_q_+
M(-,+;+ ,+,-,-)
-
~
e J1. 2+
[k • 2q+o(p_ . k1)(q+ . k,)
Itq++q-- 2
k l_
]'
I
4e' Eq+oq+.L [kl+k, _q_+ (2E - kl_)(I'_ . k.)(q~ · k,) "I_~+q++
M(-,+; + ,-, +, - )
1'
1' I
-
40'£q+o [kl+",_9+-Q_+ (2£ - k,_)(p_ . k.)(q+· k,) kl_k,+ I
_ e'(2q+o + k't)9+.l [kl+~ q-+ ]' (p_ . k , )(Q+ ' k,) k1_k,+Q++
M(-,+; f. ,-,-,+)
I
-
e'(2q+o
+ k,+) [kl+",_q+_q_+] '
(p_ . k.)(q+ . k,)
k1 _k,+
lItH. t~ ,,- - ' /I"'/' - rr (HOZ.I-:.'i( }/lANOf:; m / U)
'"
Af (-.+;-,+.+.-) -
M(-.+i-,+ .-,+)
=
2e~q+o (p_ . kd(q+ . k 2 ) t
M{- .-:-:-,- ........ +)
[kl+J4-qHq--ll kl _k!U
JI "11.(/.+1'E'" .
Q+O(2E - ~'I _)(P_' kd (q+ ' k~ )
[k 1+ k'Hq - +1; kl-k):_q.,.-tj
,
M( -,-; + ,-,-,-) _
4e~ Eq-t oq_.l. [.l:I-t~ -q-t+l' (2£ - k1_ )(p_ . k , )(q ~ . .1:2) kl _ k~-tq_-t
4.e~Eq-to (2£
M(+,-; - ,- , -,+)
=
k1_ )(p_· kd(q-t' 1:1.)
[k1-tkZ-q+-tq-- ll kl_~+
9. SlfMMAIIY
or (}Ill!
FOIIMVI,AI>
1
J~ (-,+;-,+ , -,-)
:
-
_ e«2q+o + k't)I(.L [k'+k~_q++J' (p_ . k.)(q+· k,) k,-k;+q-+ 0'(2q+o + k;+l [kl+kLq++q __ (p_ . kd( q+ . k,) k,_ k,+
j'
1
M(-,+; - , -,- ,+)
M(- ,-;+,+, - ,- ) 1
-
e'ml,kl_k'+ [q++q--I' 4Eq+o(2E - k,_)(p_ ' k,)(q+· k, ) '
,
e• mq+o k .1.1q+.J. '
[k t- k'2- q3_± ]' 2E'Q+_(p_ . k , )(Q+' k,) k1+k,+q++
M(-, -;+,-,-,+)
-
e , mq+o k1-
2 q_+ [~3
2E'(p_ . k.) (q+ . k2) k;+q+-
t
l' 1
M(- , - ;-,+ , -,+)
-
e'mq+oknq:.l [k,_k,_q++ E (2E - k,_ )(P_ . k, )(Q+ ' k2) kl+k,+q_+
-
e'mq+ok, [k;_q++q__ ] ' £(2E - kt_)(p_ . k,)(q+ . k2) k,+ '
1
r 1
M (- , -; + ,-, -,- ) -
-
[k]- k'2- q-+ E(2E - k,_)(p_ . k.) (q+ . k, ) kl+k;+q++ e4 mq+o k 'lJ.q±l. '
e' mq+o k·1-
[k'2-q+ - q-+ E (2 E - k,_ )( p_ . k, )(q+ . k2) k,+
r
1' 1
e'mq+okl.lq' .1 [k,_ k,_q++q_+ 2E2q+_(p_ . k.)(q+ . k 2) k1+k,+
M (- ,- ;- ,+,-, - )
-
e'mq+oq_+k,_ 2-q++9-[k' 2E'q+_(p_ . k.)(q+ . k2) k,+
r
'
M(-,-;- ,-, -.+) = _
• " ,. "
[k ,.
e mlul.L"'1.L/]tJ,
1-"'7t9-+
4Eq+o(2E - k1_ )(p_ . kl1 (q+ . I., ) kit ki-q++ e4 mpk,_klt [9+-I1-_1 t
=
(9. 180)
4Eq+o(2E - k,_)(p_· k\)(q+' k~ ) .
t lU I>ola.rized
~quared
JMI' = .
matri x elemenl; 3
e
k1t k2_
2k l _~t l(2E - k1_)(p_' kd(9+'
x [4E2 + (2E - kl _)2 +
IInpolarized cross
du =
4,,;;2:£-J
"""-TV:-,,'"
.• "'O~·k~'~+~"'2'
256.' k," k" lEI"E - k, . ){p • . k, )(., ' 1<, )1' (2E -
kl_)l+ 4~:;~]
[49~ + (211+0 + k;+)1 + 4Jl:~~ 9+0""'1-
]
4
d"Jif+ tPq_ cJ3kl
•
x6 (Pt -:- p_ -
9.18.9
k,}P
tlt'CtiOll:
X(4£2 + x
]!
(}+ -
1_ - k\ - k'J)
k~ and k~ nearly parallel to
p_
(1++-9--
+ qt-q-+) cf'k 2
q+o q _0 " 10 k10
.
(9.182)
and~, rcsp.
T he doubly primed quantities ki2: a.nd k:J. Me evaJuatcd in t he rotated frame where q_ determines the positive z-axis (see also Section 7.4 .3). The fourvector q in eqn (9.1&.1) is o btained by applying a Ipace reOection to q_. The quantity jl denotes the muon mass. Phot on polarizat ions:
~ .'-0 132k"2+"'7_' '-" II
(9,183)
~,
,I'IIMMAIIY 01" !.il;/J ,..OIlMULAN
Nonvanishing hdicity amplltudcs:
M(+,+;+,+,+,-)
=
e'mjJkil.k~lq_l. _ 4Eq_o(2E-k,_)(p _ , k,)(q_.k,) e'ml,k,
1
k~+ [q++q--] ,
M(+,+;-,+, +,+)
M(+,+;+, - ,+,-)
M(+,+;-,+,+,-)
M(+, +;-,-,+,+)
z
1
e'mjJk,_k + [q+-q-+]'
M(+,-;+,+,+,-)
[k'_k~+q++'J! k1+k~_q_+
,~{,¥.
, ..1,.-
_.... fI ·f }'- 11 (NO y'_"'X(:/fA"'(,'II'; !It / tt)
M(+ ,-;~-.-, + ,+)
_ 2c 4 q_o (p_ . kd(q_ .
M( +,-;-,+,+,""":"")
=
[ kl+k~_qHq __ ] t k~)
kl_~-!f+
e (2q_o + k~t)Q+.l [kl+J:!J_q_+]! (p_. kt}(q_ · ,1;1) kl_k'{+qH 4
M( - ,+;+ ,+,+, -) ::
M(-. +;+, -.+ ,+) _
(2E
,11(-.,-;-,+,+, 1') _
2~~q-oq:J. q_._(p_ . kl )(q_·
[kH.J.;'_1!+]! ..1.:1 )
2e~q_o (p_. kd (q_·
kl_k~'+q _ +
[kl+k!J_r!+] I .1:-2 )
kl _ k!fl q__
M(+ , -;+,-,-,-) _
M( + ,-;+,-, -r, -) _
4
e (2q_o
+ !1+ )Q- J.
(p_ . kd ( q- · ~)
,
[k1+,I;:] q++]' kl ~ k!f+q_+
e~(2q-O+ k1+ ) [kl+ic!i_q ..... cf __ ]l (p_ . ..I.:l)(q_· k 2 )
kl_k~~
v.
HIiMMAlIY 01' Q81J {>'O/cMULM!
M (+, - ;+.-,-,+) -
M(+,-;-,+.+,-) -
M (+,-; - ,+,-.+)
2e"q_OQ+i [kl+k~_ q+ + q_ + ] q__ (p_ . k1)(Q_ . k,) kl_k~+
2e'q_oq++ q__ (p_ . k, )(q_ . k2 )
M(+ ,-; - ,-.+.+)
kl_k~+
[kHk,+q_+ ] I 2q_o( p_ . kd (q- . k,) kl_k2_q++
[kl+ q r_q _+]!
2g_0(p_ . kJ) (q- . k,) 4
M( - ,+;+,+. - ,-)
e jjk!fi.qtJ.
2q_o{p_ . k,)(q_ . k2)
e'"k,+
2q_o( p_ . kl)(q_ . k, )
M(-,+;+.-,+ , -) -
lH( -.+;+,-, -.+)
,
[kJ+k~_q+_q_+] t
e'"k~~q+J.
e'p.k,+
l
k1 _
[kl+k~+ q_+] t kl _k~_ q++
[k
H
q+_ q
k, _
_+] t
!ll,'o'
r
l ,. - 1'+
1'-"" ( NO Y,./-:X(,'IIANllf,'; ' " I UJ
M (-· ·, + ,-,+,~,-)
=
=
(2£
M(- ,+;-,+,-,+) =
M(- ,+;-,-, +,+) _
M(+,- ;- ,+.-,-) =
C2E
(2&
M(+ , - j-,-, - ,+)
,
M(- ,+;+,-,-,-)
=
4 e: (2q_O+ k2+ )Q.t..L [ kl+k'2_Q_+] ' (p_ . kd(q_· .1;2 ) k1_k'2+QH e((2q_o +k2+) (p_ . kd(q_ . k~)
[k +k2_q.. _Q_+]! 1
kl_k~+
II. ,'iIlMMAIlY Of QI,'/II'UIIMULAII 1
M(-,+;-.+,-,-)
-
2e'q_oq,:.1. [':I+k~_q++]' (p_. k l )( q_ . k,) k l _ k~+q_+
2e'q_o --
M( - .+;- , - . -,+)
[kl+k~ q++q__ ] ~
(p_. kd(q- · k 2 )
kl_k!{+
e1"k~.1. q:.1.
_
[kI+k~tq++] !
- 2q_o(p_ . kl)(q _· k,) kl_k~_q_+
c41-tk~+
[kI+q++
2q_o(p_ . kd(q- . k,)
kl _
'
[k1- k"2+ q-+ ] t - 4Eq_o(2E - kl_)(P_ . kd(q- . k,) kl+k~_q++ • e , mJl. k'1 1. k"-:aq+l.
M(-.-; +, +,- , - )
1.
[kl_kQ_q_+] '
e4 m q okuq+.l
M(-.-;+.-,-,+)
- E(2E - k,_)(p_ . kl)(q_· k,)
kl +kq+q++ 1
-
e'mq ok,_ [k!{_q+_q_+ ] ' E(2E - kl_)(p_ . kd(q- ' k 2 ) k~+
M(-,-;-,+,-,+)
M(- ,-;+,-.-.-) -
-
e'mq_okuq:.1. [kl _k,_q+±]l -- E(2E - kl_)(p_· k l )(1_' k 2) kItk!{tq-t 4
e mq ok1 ._
-
1
[k~_q;,_q_+] ' 2E'q_ _(p_ . kl)(q_ . k,) k2+ e"mq_oq++k , _
E(2E.-k l _)(p_·k , )(q_·k,)
1
[k1LQ++Q __ ] ' . k!{+
'
I
!l1.~
" " ,,- ....... " •. /, - i i ( NO Y.-I-:XGflAfY(,'fo,':
til
I
U)
e~m/lku.k1J.q:.J.. 4Eq_o(2E - k._ )( p_ · kd(q-·
=-
k~)
[kl_k¥+qH]! I.: H I.:1_9_+
e" 1f1l-1 k ._ k¥:j
!qHq--l} 4£q-<>(2E - k,_)(p_' kol(q-' k,) '
(9.184)
l · ll l.u larized squared mat rix- e lement:
e!!kak!i 2k, _',+[(2E k, _)(p_ . k,)(q_· k, )]'
1,111' ~
. + {2E - kl_)2 ... 4";;2~1~ 'k' ] l4El
x
[,q'_0 + (2q_0 + ' "H )' + 49~o"4'_ p2J4~ ] (q+~ q-~ + 9+-Q-+ ) ·
X
t b poJ.lI'ize
cros~
(9 .185)
sect ion:
(9.186)
!).1 8. 10
.
.
kJ and k2 nearly parallel to
q+
and
1_, rc:;p.
The primed (doubly primed) quantities ki:t nnd k~ J. (k~% and kjJ.) a.rc eva}':lIlcd in the rotated fram e where q. (q:.) determinC!> t he positive z·axil'l (see :01 50 Section 7.4 .3). The four· ...edor q (IJ') in eqns (9.187) is oMained by a.p. ply ing a tlpacc reflection to q+ (q_) . The quan tity p deno~cs the r:1uon mas,s. ! 'ho:..on po\arizll.l ions:
.F,
-
N,[fo, Ii+ h(1 ± ",) + h h. ,k,(1'I' ",)].
N- '
,
-
q!o!32k;i- k
It , w'
-
N, [,k, h- ;"( \ ±,,)+ Ii' h- .., (I "' ",)].
:_ll ,
f/:o[32k';+k~_ ;1 .
(9.187 )
Y. S/IMMAItY OF (J£,J) /"OItMU/,AI1 Nonva'lj~ l tillg
hc!icity amplitudes: 1
M ( +. -; +, +, +, -) -
e'I, (2q+o + k1±)k~lq--,- [k;_k~±q++] ' 4Eq_o(Q+ . k,)(q_ . k,) ki+k~_q_ +
-
e'/-,(2q+o + k;±)k,± [k;_9++q __ ] , 4Eq_o(Q+ . k,)(q_ . k,) ki+
1
1
M(+ ,-;+.+.-.+)
-
e '(2 J.' q-o + k"'±)k"l ..Lq+.l [k'1+ k"2-q-+ ] , k;_k~+q .,. +
4Eq+o(q+ . k,)(q_ . k, )
•
+ k~±) '<;+
[k,_q+_q_+]l 4Eq+o(9+' k,)(q_' k,) k,+ "/1(2q_o
j'
c"q_o(2q+o + k;+)q-.L [ k;_k,_q++ ,1 £(q+. k,) (q_ . k,) k;+k,+q_+
M(+. -;+,-.+.+)
1
-
e4 q_o(2q+o + kiT) [ki _ k,_qH q_ .. ] , k;+k,+ E(9+' k,)(q_. k,) e 4 q+o (2q-o
M(+ . - ;- ,+.+,+)
+ k"Z+ )q+.l.
£ (q+ . k,)(q_ . k,)
r 1
[k'
k"2-q-+ k;+k,+q++ 1-
•
-
M(- .+;+,+,+,-)
-
-
"q+o(2q- o + k,+) [k;_k,_q+_q_+];E(q+· k.)(q_ . k,) ki+k,+ e 4 /-, (2q+o + k,+)k2.Lq+.l .' ,,,.
4Eq_o(Q+' k,)(q_. k,)
]t
k,_k2+q_+ ki+k,_q++
['"
e',,(2q+o + k;±lk,± [k;_q+ _q_+ 4Eq_o(q+ · k,)(q_ · k,) ki+
1t 1
e'/-,(2q_o + k,+)k;j,q' -'- [k;+k, q++] ' 4£q+o(q+ . k, )(q .. . k,) ki _k,+q_+
M(- . +; +. +, -, +) -
e '(2 Jt q-o
+
r 1
k"2+ )k'1+
[ k"'l_(J+ +'l--
4Eq+O(Q+ ' k,) (q_ . 1:,)
k,+
e'q_o(2q+o + ki + )q~. J. [ki_k,_q_+]} E(q+ . k, )(q_ . k,) k;+k,+q++
M(- ,+;+,-,+,+) -
., (' ") [""
k,_k,_q+_q_+ l ~ E(q+. kd (q-· k,) k;+k,+ e q-o 2q+o + kl+
,
M(
·.+:-.+.+.+l
_
ciq,..O(:lq_1) + klt)ll: . L [!-;_ki_IJ++j' E (q±' kd (q_ . k1 ) ~+k;'+q_+
= e~q.o(2q_o + "!it) [~_k{_q++q __
q.o • {j,!" [k'1:1_*",+9++ k" j! 2E(9,· kd (q. · k,) q+o t J1.
1.1.9-1.
+q+oA!{.i. q-1. '1.0
111 (1',-; + ,-, + ,-) =
._
+)q-.1.
e (2q+o + 1::+ )(21·0 + 1: 2£(9+' kd(q • . k2 ) ' 4
M( 1. - ; -.+ , +. -)
111 (-:-,- ;- ,+,-.+)
M(+,-;-, - ,+ ,+) _
It 2.9-+
[k:. k!, q++ j! } I
~+k2_q_1
,
[k;-k;-q ~ +l' kt+k~Tq_+
t~ (2q+ll+k:,) (2q_o+e:"f") [k: k!f_QI+'1--j! 2E(9+' kt){q. , k1 )
1'11(.,-,- ; + ,- ,-,,:,-) ::::
j'
k;+~-+-
E(q.· 1:,)(9. ' k 1 )
-
,
ki+k2'+
II. "'('MMAIlY
IJ~ ' q~: D
FIJIlMfJ/,A/i
M (-.+ ;T. T,-.-)
1I1(-.+;+. - ,T, - )
1\1(-, +;+,-, -,+ )
M(- , + ;-. +, + .-)
M(-, + ; -, +,- ,+)
M( - , +; -
, - , +, +)
+ q+ok~J,.q:l. [k;_k~+q++] q-o
k~+k~_q_+ 1
1\1(,,-;+,- ,- , -) _
_
e'q+ol2q-o + k;+)q_.L [k:_k~_ q++] '
E(q+· k,)(q_. k,)
e'q+o(2q_o + k~+) E(q+' kdlq- · ..,)
k;+k~+q_+
[k:_k'_
L
Q++ q__ ]'
k:+k~+
t},
• M
,,~
" , ,,'
-.. /, I 11- '1"
(NO Z. ,.;XClIA 0"/(.'11'; III
Af( I ,-;- ~ +,-,-)
-
E(q.
[k'1_ k"2_q_+ k;+k~+ q++
-
211:1
r ,
e·y_o(2q...o + kit) (~ k"2 9+ -q-+ ]'
E(q+· k,) (q_· k, )
k'1+ k"'H
e~p(2q_o I- k2±)k;~q
..I.
4Eq+CI(Q+' k\)(q_· '=2)
M(+,-;-.-.-,+)
+ k;+ )q+J. · k,)(q_ · <,)
r!~q_ u(2q+o
_
M(+,-;-, -,+,- )
t U)
[~;:±kif_q-H Jt
*:_*1+9_ . .
e~#(2q_o + k~t )k:+ [k3_q+~q __ ] t 4£1+0(9+ ' *1)(1_ . *2) _ e~J,(2q+o -I- klt)k1~q_.l. 4£q_O(9+ ' kd ( q_·~)
[
k;.
.
J'
k l _k1'+Q_1 .
kl+k;_9++
e.~I'(2q+o"!k~t)k;'+ (k;_q;-_q_+]! "" 4F.-q_o(q+ · '=I)(q_' k, ) *1+ M(-.+;+.-,-,-)
-
.<1(-.+; - ,+,-,-)
~
M( -,+;- ,-.+. -}
-
M(- ,+;-,-,- , +) _
(; 4'1+0(29_0
[k;_ki'_q_+] •>
+ ~+)q+ ..1.
k1+~+q-+
E(1+' kd(q- . 1.2 )
e4 q+o(2q-o + ki_) E{q+ . kd(q_ .~)
[ki_k';_1+_1_ .. ],• *:+1.-3+
t:(q_o(2q~o + kl .)q:..1.
(.1:;._*;_1 . . +] t
E(q+· kd(q_ . k,)
.1:).*1+9-+
e4q_o(2q+o -I- kit) [kI-*1'-9++9--]'• E(q,. · k, )(9_' 1:,) _ e4 JJ(2q_o
.\;+tq...
+ k~... )k~l.q';'l.
4£'1+0(9+ . kd(q- . ~)
•
[~±k;_q_± Jl
r•
k;_k~+q++
(2q_o+ k2'±)k1... [k'{_q+_q_+ k;+ 4Eq+o{q±' k 1 )(q_· k1 ) t;i J1
• [k'1_ k"2tq±+ e • JJ (?-9+0 +k'1+ )k" 1L9-.1.
4Eq_o{q ... · k1)(q_·
k~)
JJ(2q..o + .1:; ... )k;+ 1Eq_o{Q+' kJ)(q_ . k,) f.4
]1
k:+k;_q_+
[k~ -q+~q--ll k; ...
(9.188)
fl . ,;'IIMMA/iY 0,.. Qr;/J FOI£MUUlh'
Ullpola.rb',cd IHjllurcd matrix element:
X
2 I/:l Jl'k2+ ' k" 1 [4q_o + (2q_o + k2+) + 4q-o ,-
(9.189)
Unpolarized cross section:
(9.190) _
9.18.11
4
kl and k2 nearly parallel to
p+
Definition s:
6;, -
B, _. B,(1,2)
_
-2(p+· k,) - 2(p+ . k,)
+ 2(k, . k,),
2E' [k k (p+ . ktl(p+ . k, ) ,- ,_1_
[k,
6;,
+
m' Z12 Z" 1 4E'6;,
(2E - kl+ )(2Ek,_ - ZI2 ) (p+ . k,)
83
- (klJ.
+ ka)Z12Z" 1 L\!12
B.(1 , 2)
mkl+klJ. [2Ek,_ - Zj, .+ 2Ek'_j . 2E6;,
(p+ . kd
(p+ . k,)
(9 .191)
•
.
v·, ;
M(+ ,+i+ ,-. +,+)
~
.
M(+,;:+,-,+,- )
2~4 EDJY;'1
~
~
2e~ r;I H3! ( ('1+ . y.)
q-~
[
(qt ' q_)
-
M(+,+:-,+,+,+)
9+-q-+ kl+kl_kHk~ _
2e4EB3Y° .l.[ ('i t'
y.)
2e~f,'IB31 [ (q+ . y.)
M(+.+ ;- , -, - ,+ ) -
M(+, +; - , +, +,-) -
~
t
qHkl ... kl.kHkJ_
q++
r
q_.k, +kl·.kHk,_ q++q-kL+k l _~+k2_
r ]1
2e·r;H~ ( 1 , 2 )q- .i [ q-T ] ('1- ' y-) 9... kll k1_ t lrk,_
2" £ 18 ,(1 ,2)1 [ (q+ . y_)
M(+ ,"T";-,+,- , + )
(9.192)
1=1,2 .
q.-.-+
.1:1 ~kl _ kHkl _
r
(q+'
y_ )
(9. ' Y-) 4
,
q+-'.1:1+,1;,_ -+--t
.1: 1+.1: 1•
2e4 EB 4 (1.2)QtJ. [ qt+(9+' y- ) q_+k1.kt_,(,'2+k,_ 2,' £ 18,( 1,2)1 [
l
r
2e~EB4 (2, 1)'1:1. [ q. t (1/+ ' Yo) 9++ kl+-':I_ k, • .l:2_
2,'£18,(2, 1)1 [
,
]!
r r .+-.--. r
• • +.-kl+'f:l_kHk, _
2e E H. (2, J)9+.1 [ q'H (9+' 9-) q_+kl+k1 _k, ... k,_
2,' £18, (2, 1)1 [ ('1+ ' y_ ) kl +k l _k,...k1 _
,
9..~/!MMAIlY
M(+,- ;+,- ,+,+)
or (JIW POIIMUI,Ah'
-
2e'EB,q+J. [ q++ (q+ . q_) q_+k,+k,_kHk,_
-
20'EIBd [
1t
l' 1
(q+' q_)
q++9--
k1+",_"'2+ k,_
2e'£B1 q_1. [
M(+, - ;-,+,+, +)
(q+' q_)
q-+ q++k,+k'_"'Hk,_
1t
l'
1
-
M(-, +; +, - , +, +)
2e'£IBd [ q+-q-+ (q+ . q_) k1+k,_kHk,_
l' 1
4
e (2E - kI+ - k,.)B,q:1. [ -
(q+ . q_)
q + 9++ k ,+k'_"'2+ k , -
,
1'
-
04(2E - kl+ - k>+ ) IBd [ q+- 9-+ (q+ . q_) kl+k,_k,+k,_
-
0'(2E - 1;1+ - k2+ )B, 9+1. [ q++ (~.~) ~+~+~-~+ ~-
1' 1
M(-, +; -,+ ,+, +)
e (2E - k1+ - ~+) ! B,I 4
-
M( +,-;+ ,-,+,-)
(q+' q_)
20" EB,(2 , J )q-1. [ -
(q+' q_ )
-
20'E IB,(2,1)1 [ (q+' q_)
q++ q_+k,+k 1_ k H k,_
q++ ] q_+kl+k,_k"k,_
M(+, - ;-, +,+, - )
t
l' 1
4
-
1t
]t , "'I+k,_k,+"',_
(q+ ' q_)
2e EIB,(1,2)1 [ '1++q-(9+'q-) k,+"',_k,+k,_
2e' BB,(2 , 1 )q+.l [ -
(q+' q_)
-
20'BIB, (2,1)1 [ (q+' q_)
1' 1
q++q- k1+k,_kHk,_
q++9- -
2e'£8,(1,2)q_1. [
M(+,-; +, -, -, +)
[
9- + q++kl+k,_k2+ k,_
q+-q-+ kl+k 1 _k,+"',_
1l' ,
1t
M( 1-. -; -, +. -, ~ ) == _
Al ( - .+;+ ,-,+,-) -
-
2c~En~(1 , 2)q+.l [ (1+ . q-)
M(-.-:-;+, -.- ,-+-)
M(-,+; - ,+,T,-) _ =
M(- .+;- .+. -.+) -
q_.
]'
qHkl+kl_k2+k1_
,
2e4 £IB,(I, 2)1 [ q+_q_+ ], (q+' q_) ka kl_khkl_
,_e'EB'2(I. -?)Q"'l . [ ], q-+ (q+' q_) qHkl+kl_k2+k2_
2,'EIB,(l,2)1 [ (9+ - q_)
_
,
q+_q + k\.,.k 1_k2j.k2 _
2e4 EB, (2, 1M.l. (q+ ' q_)
[
2e~ EBi'{1,2)q·.L [
(q.... q_)
]!
q_ , ]! qH kl+k l _kH k1_
,
q.,.+ ]' q_+k 1+k l _k H .l.:1_
4
2e E B2 0 . 2)1 [ q++q__ (q+' q_) k\+.l.:I_kH .l.: 2_
]1
.++
2e4E Bi (2,l}q:.l. [ (q+ '9- ) q_+kl+k1_kHk,_
2c4EIB~(2 , 1}1
[
(9+ ' q_)
q++q__
, ]'
]'
kl+kl_kl+kl _
4 kl + - .l.:H)lJiqu [ .++ ]'• M( +. - ;+.-.-.-) _ e (28 - (q+' q_) q_+kl+kl_kHk2 _ _
4
e (2E -
~'l+
-
k~+)JBd [
(q+ . q_)
e (2E - kl+ - k~ ... ) 8iq_.l. [ t
M(+ ,-;-, -:- . -.-)
•
9++q-]' .1.: 1 .,..1.: 1_1-,+1-,_
(q+ ' q-)
~ "(2E - k•• - k,. )18d [ (q+ '1_)
q_.,.
9+,kllkl _kHk2_
.+.... ]1
kl+kl_k-J+k~ _
, ]'
,v,
S /I .II AI All Y (J I-' Q f: /I fO It ,IIII/"H :
M( --, +:+ ,-, -.-)
M( - ,+; -, +. - , - )
It 2.'EB'(2,1)q_.l [ q++ I (q+ ,q-) q_+k1+k, _k,..kz_ J
M( - ,-,- 1, , -" -'--) ,
_ 20'EIB, (2, 1)1 [ (
1
q++q__ ] , k1+ k ,_ k2+ k,_
4
IVf(-, -;+,-,-.+)
20 £E,(I, 2)q-.L [
(~,~)
=
q++
~. ~+~-~+~-
1t
1
2.'EIB.(I,2)1 [ -
M(-,-;- ,+.+,-)
(1+''1_)
q++q__ ]', kl+k, _k2+kz _
2.'EE; (2,1 )
1
[
(
-
q_+ ], q++k, +k,_k'H k,_ 1
-
2·'£IE,(2,1) / [ q+_q_+ ]' , (q+ ' q-) k1+",_k2+"'!
M (- ,-;- .+.-, + ) -
-
2e'EB,(l,2)q+.l [ (
q_+
q,,+k1+k,_k2+~_
2e'£IB,(1.2)1 [ '1+- q-+ (q+ , q_) k1+ k ,_k2+ k,_
]t, 1
2e'EBiqu [ q++ ]' (q+ ' q_) q_+k,+k,_k2+~_
If (- , - ,, -'-" - , - I -)
J
4
-
1
20 EI B31 [ q++q-- ] , (q+' q_) kl+k,_k'H k,_
]'
!I. I,~,
t· I r
. , I' I , , - .., l"
(NO h_ 1],\'(:/1 A NUl';, "I I II)
[\4 (-,-;-.+ ,-.-)
(9.193 ) 1lll lIol arh:cd ,q uared matrix element:
2,'kl+kl_kH ,,"(,+> ,--I.:'l_(q+ + ,+, .) {[I + (2£ - k,. - k,. )"]ID", , q-)7 2£ + IB,(1,2)1' + IB,(2, 1)1' + IB,I' + 18,(1,2)1' + 18.,(2,!)1' } . (9.104) I r1ll>oIarized cross secLion:
do
=
~'('++' - + ,. -,-.)
64rrlkl + kl_~'Hk'l_ ( q+
'q-F
{[I + (''; -
<")']1 8.1'
k .. 2R
+ IB,(1, 2)1' + IB,(2, 1)1' + IB,I' + 18,(1,2)1' + \B.(', 1)1' }
,
x6(P++~L-q+-q_ -k l-
9.18. 12
- -
<1) (Pii+ rPq_ J3k tPk'l k k .
kJ and k2 nea.r1y parallel to
l
q~Dq-o
10 '2D
p_
Delio j Lions;
i,j=1,2 ,
(9,195
)
,v, ,1'IIMMAIlY ()[o' Ql:/) roIlMULAf:
A,
-
2(p_'
k:;~p_, k,)
[kl_kHki.l
+ kl+k,_k;.l + ki.l)Z"Z21]
_ (ki.l
6
'
12
A,(1,2 ) =
(9.196)
Photon polarizations:
H N;-I
N;[h p- p+(1 ± 1'5) + p+ p- ,Ie;{ l 'f 1'.)], -
i = 1,2.
E'[32k;+k;_ 14,
(9.197)
Noovan isiIing helici ty amplitudes: 1
4
M(+ ,+;+, - ,+, +) =
20 EA,q-.l [ ( ) k q++ k k q+' qq-+'1+ 1-
_. 2e' EIA3 1 [ (q+ ' q-)
M(+,+;-,+,+,+)
(q+' q-)
"k, _
q++q_ _ ] k,+k, _kHk,_
_ 2.' EA 3 q+.1.
],
I
r
1
q + ], .q++k1+ k ,_k,+k,_
_ 2."EIA31 [
1
q+-9 + ] , (q+' q_ ) k,+ k, _k,+I:,_
, 2e'EA.(1 ,2)q_.1. [ q++ ] ,. (q+' q- ) q-+I:1+ k,_ k2+ k,_
M(+, +;+, - ,+, - )
_ 2.' EII1,(I , 2)1 [ (q+ ' q- )
M(+, +;+,-,_,+) _
1
q++q__ ] ' /'1+ k, .• k,+I:,_ 1
2. '8A. (2, I )q .l [ q++ ]' (q+' q- ) ,,_+k1+ k ,_k,+k,_
. _ 2. ' EIA., (2, 1)1 [ (1+' q-)
q++q__ ]t kl+l;,_k2+k,_ 1
111(+,+; - ,+,+, -) -
2.'EA,(1,2)q+.l [ q_+ ]' (q+' q-) q++kl+k,_kH k,_ 1
2.'E IA, (1,2)1 [ q+_q_+ ] ' k1+ k,_I:,+I:,_ (q+ ' q_ )
• .Q I,~,
r+r.- - . 1'+1'-1", (NO Y,·I-:XCIIAtNII-;, Ill / U)
M (4·,·~;-, + ,-. + )
=
2c~ l!:A ;(2, 1)q+.L
[
(q+' q_) 2e t::'I A ~ (2, 1)! [ 1
-
(I] • . '1_)
:1&1
q_+ 1... +kl+ kl_ k:1+ k:1 _
]!
,
q+ - q... kl+kl_kHk1_
1. ,
4
M(+,-;+,-,+ . +)
=
e (2t: - kl _ - k2_ )A 1qH [ q.. ]' ('1t ' '7-) q_+kl+kJ kHk2_
"('P' - k,_ - Io,_ ~.j [ -
(q+ ' q_)
]1
, .. , __ kl+ kl _kH k2_
M(+.- ;- .+.+.+)
~~(-.+ ; +,- ,+,+)
=
+~ .M( - ,+. .-, -. t ')
_
M( +,-; +,-,+.-)
M(+ , - ;+ , ,- . +) -
=
2e~ EAH2, l )q_i [
(q ... q_ ) -
,,' cIA,('.l)! [ (Q+"'1-)
, q.. ], q_+kH kl_k1+k2 _
,
q,.,__ ]' . kl+ kl _kHk:1_
."
9. 8/1MMAllY Of Qf)/i }o'OIlMlJl,AE
:Uj:l
M(+,-;-,+,+,-)
2e'EA;(1,2)'IU [ -
-
M(+,-;-,+,-,+)
(9+ . '1-)
2e' E\A,(l, 2)1 [ q+-q~+ (9+· q-) k1+k, _I:2+ k,U1EA;(2, 1)'IU [
-
9-+ 'I++l:l+ k l_ k,..k,_
(9+ · 9-)
]t
1t ,
q-+ q++l:l+ k ,_k,+I.:,_
l}
l' 1
M(-,+;+,-,+,-)
-
2.'E\A,(2, 1)\ [ 9+-q-+ (9+· q-) k,+1: 1 _k,+k, _
-
2e·'EA,(2,1)q.j.,. [ 9-+ ('1+ . q-) q++kl+ k ,_k2+I.:,-
-
2e'EIA,(J,2)\ [ 9+-'1- + (q+. q_) k,+k 1 _k2+ k,_
1t
l' , 1
M(- ,+;+,-,-,+) 2.' E IA,( 1,2)1 [ -
(9+ . '1-)
'1+-'1-+ k,+k,_kHk,_
1~ , 1
M( - ,+; -,+,+,-) -
2e'EA,(2, I )q:,. [ ('1+.9-)
9++ ], q-+I:I+"I- kH k, _
1' 1
-
2e' E\A,(l, 2}1 [ 1++9- (9+·9-) 1.:1+I:, _ I.:2+ k,_
l' 1
M(-, + ;-,+,-,+) -
2e'EA,(2,I)q:,. [ (9+.9_)
2.' EIA,(2, 1)1 [ -
(q+. q_)
9++ q_+kl+k,_kHk, _
l' l' 1
q++'1-k1+k,_k2+k,_
2e' EA iq+J. [ q++ ('I+, q-) 'I_+ k Hk 1 _k k,_ H
M(+,-;+,-,-,-)
l' 1
2e'EIAd [ -
(q+. q_)
'1++q- kl+ k l_ k,+I:,_
1
AI( ,.• -
;-.+.-,-)
_
g!l711-J. [
2(;1
('1+' (1-) _
:le
EIAd [
4
(q+ ' q_)
If - + ]' ((Hkttkl_khk2_
q+ q + ], kltkl _k'j+ k2 _
M(- , +;+.-,-,-} _
111(-,+;-, + . - , -) _ t;~(2E -
,1;1_ - 1;2_ )Aiq+J.
(q+' q_)
M{-,-; + , - ,+,-) _ 2(:4 EA~(2, 1}q:.l [ (q+' q_)
M( -,-: -.+,+,-)
2,'EIA,(2,1)1 [
,
q-+
q++k1tk1_kl+k) _
kltkl _k2+k2_
=
2e EA .. (2,I)q+J. [
(q+ . q_) 4
('1+ .q_)
2,' EIA,(2, 1)1
l'
l'
[
l'
,-+ l' kHkl_kHkl_
2,'EIA.(J,2)1 [
=
q_+kl+kl_k2+k1_
2e"EA.{1,2)q:.1 [ q-+ k (q+' q_) 9++ J+kl_k1+k1_
(9+ ' 9_)
M(-. - i - ,+,-.+}
.++
[
q+-q-+
~
_
]'
,
('1+' 9_)
M(-,-;+.-,- ,+) -
q+-9-+ kl+kl_kHk~_
(q_. q_)
_
,
~4{2E - kl - k~ )I AI I [
q+
q++ q_.kHkl_k7+ k2_
]!
q++q_ _ ]' kl+k ,_kHk,_
,++
2c:4EA,(1,2)q.f.l [ (fI+' 1_) 9_+ kltkl_k'2+k,_
]1
]'
.v. .,'II MMAIIY
Of' (JIW t'(JllM UI,AE 1
M( -,- ;+,- , - ,-) _
20' "JI,;'1:L [ '1-+ ('1+ . q- ) q++kl+k,_k2+ k,_
1'
1
_ 20'E/A3/ [
q+_q_+
]'
k,+k,_k,+k,_
('1+' '1-)
M( - ,-; -, -/- ,-,-)
(9 .198) Un polarized squared matrix element:
[1-1_( 2E - k,~, -
8
/MI'
=
2e E'(q++'1 __ + q+-q-,. ) { (q+ . q_)'kHk,_k2+k, _
"-) ' )
2E
lit, I'
+ /A,( l , 2)1' + 111,(2,1)1' + IA,/' + /A,(1 , 2)I' + IA,(2, 1)I' }
.
(9.199) Unpolarized cross section:
da =
a'(qHq---/-q+-q-+) 54,,'('1+ ' q_)' kHk,_k2+k,_
{[1 -1- (2E-k,_ -k,_)')lAd' 2E
+ IA,(l, 2W -I- IA,(2, Ill' -I- IA31' -I- IA,(l, 2)1' + IA,(2 , 1)1' } , xli (p+
+ p_ -
~
9.18_13
q+ - '1_ - k, - k, )
~
kl and k2 nearly parallel to
d'ifr d3 if_ d3 k,
d'k,
1+0 q-o klO k20
.
(9 .200)
if+
T he primed quantities ki± and kLu i ::: 1,2, are evaluat.ed in the rotated frame where ifr determines the positive z-axis (see also Section 7.4.3). The [our-vector q in eqns (9 .202) is obtained by applying a space reflection to q+_ The quan tity fJ. denotes the muon mass. D efinitions:
~· Z i II
-
k'i+ k'; - - k'Ok' i.l j..l.
'
II/.'i
,1,
_.,
I'. /'
i"f (NO X-f;xr:/IANC:I\';
til
I II)
,
A'
_ (kl J.
+ k2J. 1Zi, Z21] Doj, (9.20) )
I' hololl
po larizntioll~:
,,
1\1- 1
1=1 ,2.
'1ol1va.:lishing helicity amplitudes: 1\11 ( +, -i
+, +. +, -)
M t ~ . - ;+,
'- , -,+) =
(9.202)
fl. S/iMMAUY or (JIm rOIWUI,Ag 4
c A'lfJ+..l [
M(+, -; -, +,-1- , +)
E
kI( - , +; +, -1- , - , -1- )
]
~
1
q+-q-+
kf+k~_k2+kz_
E
e'A~'(I , 2 )q~ L E
,4IA~(1, 2)1 -
+
q++ki+ki_k;+k,_
e' IAil[ -
"
]' . 1
[
,q;+.,
.,
q_+k,+kl_kHk2_
[
q++q--
1' ,
[
q +
]1
1
kl+k~_k2+ k2 _
E
l' 1
M(-, -1-;-1- ,- , +, -I-)
M(-,-I- ;-,+ ,+, +)
e'A~'q+-L
M (+,-; +, +,-,-)
-
E
e·'IA;1 [ E
q+ +k~+k~_k2 +k2_
q+_q_+ ] ki+ki_k,+k, _
t '
e'A,"(1 ,2 )q_L [ q++ ]' E lJ_+k~+J,;~_k2+ k~_
M(+ , - ;-1-, - ,+, -)
-
,'IA,(1,2)1 [ E
1', 1
'1+ +9- -
~+k~_~+~_
l'
1
M( +, - ;+, - ,-, +)
e'A,' (2, I )q_L [ q++ E '1 -+ki+ki_ k,+k,_J e'
IA,(2, 1) I E
'11.( ,. 1,. , _,/,1 II
i'l (NO
7..I~'X(.'IIAN(,'''''; lit
III}
e1!A~(2,.! )\ [ M(+,-;-,-/,-,+)
E
-
E t:'4,13·q: l E
111 (-,+;+,+,-, - ) _
-
M( - ,+;+, - .' ,-)
, ' [A;I [ E
q_+
qt+k: + k:_k~+k~ _
q"
[
,
,,,'--
k'k'F!.:' I t 1-2+2-
E 4
-
f;
]'
'
,.+ ]1 q++k;-+ki _kltkl_
q+-q-+
ki ... ki_kllk2_
e~A~. (211)q+.J. [
E
]\
q_+k; tk(_k,~k2 _
c1A;.(1 ,2)q+.J.. [
'" -
]1
7~q!'+""C;-=-..-1 \, [7,.1'11"1'.1 k' k-' H 2-,
c IAl(I , 2)1 [
M( - ,+;+,-,-,"t")
[
E
" [ A~(2, 1)[
,
k:+k: _ k2+k~_
e~ A;(1 , 2)q+l -
]t
q+ q_+
]'
,
q.+
q+tk!+kl_k1~ k2_
-
-
=
"[11>(2,1)[ [ q,_q_+ E k~+ki_k~ ~ k;_
]1 '
]'
'I
,' :IIMMAU\'
or (}I-:n
"'O/{/IIf/l, AJ,: 1
114( - -,1-;-, + . +,-)
-
,,' /1~ (2, I j,t t [' '1++ ], J'" k' k' k' k' (1-+ ' 1+ 1- H 2-
e'A~(I , 2)q:.L [
M( - , +; -, +, - ,+)
-
-
e' /A;(1,2 )/ [ qHq ] ', E kj + ki _ k~+k2_ e'
-
'1- +
q++k~+kl _ I::~+k2_
E
E
]'
,
A;q+.L [
- e'/A~I [
]}
1
9+-'1-+
]' ,
k~+ki_k2 +k~_
1
e'Ai"q-.L [ 1++ i E 1-+''-'k'k'k' ,,+ ,- 2+ 2-
M (+, - ; +, -. -, -)
1
e-'/A; l [
,11(+, - ;-,+,-, - )
q_ +kl+k~_k2+k~_
-
M ( -, +; -, - , +, +)
E
1
q++
-
q++q--
1'
E' k~+k; _k2.k2_ J '
2e' EA;'q+.L [ q_+ ]} - ('1+ - q-) 9++ k ;+k;_k,+k; _ L
2e'EIA; l [ -
(q+ - q_)
q+_q + ]' k"+k" _k,+kj_
e'A~ ( 1,2)q_.L [
111(+,-; - ,-,+,-)
E
]l
q++
q_+k, +ki_k2+k2 _
l', l
e' 1/1:,(1 ,2J/ [ -
£
q,,-+q--
k;+kl _ k,+~_
e' A~(2, 1)q-.L [ q++ E q_ +k:+kl _k~ + k~_
.11(+,-;- , -, -,+)
1
-
e' IA;(2, lJI[ q++q-]' k'k'k" E ki.;. 1- H' 1-
1I
'liS. ,. 1,
'1" 1'
11
(NOr, 1.;.H.'/ltlNI;~:."'/U)
111( - ,+; I" - , -, - ; -
c"', A'LI [ E
M(-,+i-, + , - , ) -
q+_q __
k'k'k'k' 11 1 - 111-
, -, [ ,'A",' E
]t
,
'I +- k'k'k'k' 1+ 1- h· 2_
]!
M(-, + ;-,-,+,-)
M(-, +, - , - , , + ) (9.203) III1Jl(,lnriz~d
sq uared maLrix c!ctnenl:
IMI' + IA;(J,2)1' + IA;(2, 1)1' + IA;I'
I
IA;(I,2)1' + IA;(2, I)I' } . (9.201)
l1 upolarizecl cross secLion:
du
=
0'(,,+, __ + ,+-,-+) 256.'E' k'1"'\-H k' k' 1-'""2 -
{[I + (2,,, + + 4.)'] I' k:+
lA'
"~ ...o
'
+ IA;(J ,2)1' -IA;(2, 1)1' + IA;I' + IA;(J, 211' + I A~(2, I)I' } (V.205)
~
~
1.:1 alld 1.:1 n C
9.18 .14
ii-
The dOllbly primed quantiLies ki± and k:~, i = t,2, n.re evaluated in the rotated frame where if. determines t he positive z-axis (see also Section 7.4 .3). The four-vector q in eqns (9.2(7) is obtained by applying a space reflection to q_. The quantity I' denote_, the muon mass. Definitions:
k'! ,+}-
Z!'.
k~1
'J
-
e'·k'!J.l !.l
All _
2q':'o
1
(q_ . krl(q_. k,)
A~{ 1, 2)
_ 1_ [k;'_(2q_O
i,j= 1,2,
1
[kfl kif + JJ2 z~;Z2'1 1 1-'- 4q:O~~2
+ k:'+)(2q_ok~_ + Z{',)
fj.~z
(q- . krl
+2q_ok;'_k!f_{2q_o + k','+ + k~t) + l"k~+kt;'k~L/2q-o l (q_ . kz)
A~(1 ,
"k"IT. k"U. ,-
2) =
2q-O~~2
[2 q_0 k"2- + Z"· 12 (q_ . krl
k" 1 + 2q_ 02_
(9.206)
(q_ . I:.)
Photon polarizations:
,ii' N,-l I
N;[,k; 11- h(1 ± ,),,) + II II- ,k;(1 Of ')',)J , = q'- 0.f32e',+ e' Ji 1-
i = 1,2.
I
(9.207)
Nonvanishing helicilY amplitudes:
M(+,- ;+,+,+,- ) _
e'A"4 · (2 , 1) q-J. [
E e4 IAr(2,1)1 E
q++
q-+ k"1+ 1.;"1- k"2+ k"2-
1~
211J
M(+,-;+,+ ,-,+) -
M(+,-;+ .-."T,+)
.M (+,-;-, t,T , t)
.O\1( -, +; +, t, -, + )
-
M(-,+; +,-, t , +)
M(-,t;-,t,t,t)
~"IA~(1,2)1 [ E
c~A'{q';'.L -
-
E
,
q+-q-+
J,II kl' 11:1\,'\;1 -"2+ ~ _ '.II
[
'-II
,-.
l'
'.I'-"'1 P/ 1.: # q++~1+"I ... 1_ ' _11
'
1t
.'1,
,\'/!MM~
IW OF Qlm FOUMUl,Ali
M( +, -; +, - , +, -)
M (+ , - ;+,-,-, + )
M (+ ,-; - ,-!-,+, -)
M(+ , -;- ,+,- , +)
-
e4IA~(2, 1)1 [
-
E
9+-q-+
k"1+ k"1- k"2+ A·II2-
M (+ , - ;- , -,+ ,+ )
'I"I[ E
e A,
q+_q_+ 1.11 };" k"1+ kllJ- '~2+ 2-
]t
)
M (-,+;+,-/-,- ,-) e41A~1
E
l'
,
" kll1- kll2"1". k"2k1+
'
)
M(-. +;+. -.+. - I _
i\I (-, -i-; +,
- , - , +)
M(-, -I-;--, ~ , -I"-)
_
M(-,+;-, +,-, +)
,
~
1V1 ( -, +i -, -, +, -I }
M (+,-;+, _· ,- ,-) =
M(+,-;-, + ,-,-) _
c' IAl(2. 1II [ E•
, .. q- -
kItI t II'i - 2 j::!'+ J,:!'l -
l'.
,'i/IMMA Ity or (Jr;/J rOItMUI.M;
~I ,
~1l1
1
e'A~(J, 2)'1+1 [ q-+ ]' q++ kifI+ k"1- kll2+ k"2E
M(+,-;-,-,+ , -)
e'IA~(1,2)1
_
]t k" k" k" k"
[
q+_q_+
E
1+ 1- 2+ 2-
M(+,-;-,- , -,+)
1
e'An'q' 1 +..L [ q+]' kll k" k" kll E q- - 1+ 1- 2+ 2-
M(-,+; + ,-,-, - )
e'q+_IA~1
-
[
E
q-
-
q++ ]} k"1+ kift- k"2-r-, k"21
111( -- , +.1 -
I
e'A"'q'-.l
+ - -) ,
1
,
[
E
q++
e"A~(l~ 2)q:1
1\1(- ,+;-,-,+,-)
]'
lJ q-+ k1+ k"1- kif2+ kit2-
E
qH [ q -+ k"1+ kll1- k"2+ k"2-
]t
e'IA~(I. 2)1
E e4A~(2,
M( - , +; -, - , -, +)
l)q:1
E
_
e4IA~(2, 1)1 [ E
q++ [ q-+ k"1+ k"1- klJ2+ k"2q++q__
=
8
e (q++q __
+ q+-q-+)
2E'k"1+ k"1- k"2+ k"2-
{[I + (2
]t
(9.208)
k"1+ kif1- kIf2+ k"2-
Un polarized squared matrix element:
IMI'
l!
q_O + k;'+ + 2q ... 0
k'{t)']IAnl' I
+ IA~(1,2)I' + IA~(2, 1)1' + IA~I' + IA~(1,2)I' + IA~(2,IW } . (9.209)
I JllpolllrilwJ (;fO~~ S<.'Cliou:
{[I 2567rl£4kJ.'+kJ.'_k1'+k~_ /)'4(q-++q __
+ q+-q-t )
+
(21_0 + kj't + k~t)'l IA"I' 2q_o
I
+ IA,(I, 2}1' + 1.4;{2, I }I' + IA;I' + 1.4;11,2)1' + IA;{2, I }I' } 19.210}
l'rocC:6ll:
(9.211 ) I}cfiuitions: B_1
+
Bo _ 2 _
E
(q+ ' q_)
[2q_6 + ~.l.q:J, _ khQ_J.] kH
,(:1 _
KUqt.l. _ ki.lq-l. K"2+q+_ ('1-9- + '
9. 19. 1 k; nearly parallel to fi+ Photon
polariZAtions:
t~
-
N,[p+ p_ ,k,11 T1.)-,k, p+ I>_{I ±,,)1.
'-1,2·19.213) Nonvanishing helicily arr.plitudcs:
( , +;+,+,+,+ ) = M+
C.jElq~.l.B
[ 32k2 _(Q+'q_) ( '. } k (k)1 k) q+- p+ . "'2 2+q++9-+ q+' I q_' 'I
_ ,'E'IBI [ (p+' k1 )
32k,
Iq+· q_)
,(:HQ+-9-+(Q+' kl)(Q_ ' k l )
]!
]t
2(10
~.
M(+,+; +,+,+,-) -
,~tfMMA"Y
OF <11m rOflMlIMIi
e"E(2E - k'+)";L/~ q+_ (p+ . k,)
[ 8k,_(q+ . q-) x Ie,..q++q~+(q+. k,)(q_ . kd -
_'E(2E - k,..)IBI (p+ . k,)
J!
l' 1
[
8k,_(q+ ' q-) x k,.q+_q_+( q+ ' k,)( q_ . k,)
M(+,+; +,+,-,+)
-
e'E' q+ l Z-+ B' ·0
(2E - kH)q+-(p+ . k,)
J'
1
[
8k,_(q+' q-) x k,+q++q_+(q+. k,)(q_ . k,)
r 1
-
M(+,+;+,-,+,+)
-
4e'E(q+·q_1IBol [ k,_ (2E - k,.. )(p+ . Ie,) Ie,.. (q+ . k,)( q_ . k,) e'mk2.Lqj..L Z_+ B
[k H q_+ (2E - kH)q+_(p+ . k,) 2Ck,_q++
.
J'
1
e'mk2+Q_+IBI ,
1 ,
(2E - k2+)(p+ . k,) [(q+ ' k,)(q_. kdl' e·'mkuq+.L Z+_ B [k,+q++ (2E - kH )q+-(p+ . k,) 2C k2 _q_+
M(+,+;-,+,+,+) -
l'
1
-
.'mk2+IBI qi+q-[ (2E - kH)(p+' k,) q+-q-+(q+. k,)(q_ .
M(+,+;+,+,-,-) -
e q+., Z_+BO [ 2k,_(q+' q- ) q+_(p.,. . Ie,) )'Hq++q-+ (q+ . k,)( q_ . k,) 2e'(,,+. q-llBol [ k,_ (p+, k,) k,+(q+' k,)(q_ . k.)
r 1
l' ,
e'mk2.1 q'+.1
M(+,+; +,-,-,+)
z-_+ B' [
E(p+ . k,) -
r 1
4
-
Ie.)]
k2+
8Ck 2 _q++q_+
e'mk2+Q+_ IB I 1 ,
2E(p+ . Ie,) [(q+' k,)(q_. k,)I'
'1
• .~
19. ,,1,, - ..... "t r - 71 (NO
M(-r-.+; -,+,-,+)
IH /-11)
;I,-J;XCIIJ1Nr:~;;
=
-
:l07
k,+
_ c4rnkuq:JJ,.1+J.Zt_1J" [ Eq ... _(pt . k,) 8Ck2_q++q~+
,
4
e mk1t lBI [ q++q:]' 2E(p+. k,) Q+-Q-+(1+ ' kd(q_ . kd
M(+, - ;+, -,+,+) _ e4EQ:.lQtJZt_B[ q~_(p+.
k,)
-'- 2e 4 EIB I [
]1
2k,_ Ck H 1++Q:+
1.:,_Q++1:_
(Pt ' k2) 1.:,+Q+_1_+(q+ · kl)( q• . kd
A"(+,-;-,+,+,+)
=
]1
4
e Eq±J.Z_+D [ 2k 2 _ ] (p+' k,) Ck,tq ... tq-+
_ 2,'E,,_1 81[ (p+. k, )
]1
, ,
k,k"(,,. k,)(, _ · k,)
]1
M(-, + ; T,- , + , + )
,
M(-.+;-,+,+,+) _
M(- , - ;- ,-,+. - )
4 e Eqt lZ+_B [ 21.:2_Q++] ' qt- (Pt' k-, ) Ck,±q_t 4
,
_
2e EIBI [ k2 -q}+1]' (p+' k,) 1.:2+1..-_Q __ (Q+· .1. 1 )«1_ ' kt}
_
e4 (2 E - k 2 _ }q' J.1tl.Zt-B
[_!2-__]1
gt-CPt . 1.:,)
2CkH Q++q:+
e4 (2E -
1.:2+
(p, . k,)
l! BI
U, ,\'/IMMAlty
M(t , -;t,-, - ,t) -
e' E,"qu
Z;_ Jr
(JIo'
QP:/) {o'ORMII/,Ah' 1
(8k,~q++]'
(2E - kH)q+_(p+ ,k,) CkHq_+ 1e'E'IBI
X
M( t, - ; - ,t,t, -)
_
]! k2- Q~+q-[ kHQ+_q_+(q+' kd(9- ,k, ) ,
"(2E - kH )qU Z_+ B [ k,_ ]! (p+' k,) 2C k H Q++q_+
M( t , -; -, t , -, t )
J"I (t ,-; - ,- ,t,+) 1
kH (q+'q_) [ x 2k,_ '1++'1_+ (q+ ' k, )( q_ '
kd
]'
e' mk2+(Q+' Q- lI Bol E(2E - kH)(p+ , k,) [('1+' k.)(9_'
1 ,
kIll'
e'mk;l. 9tl.Z_+Bo
k2+('J+ ,q_) x [ 2k'- 'J++9_+(q+ ' k.)(q_ ,k, ) e'mk2+(q+, q-) IBol ,
1 '
E(2E - kH)(p+, k,) [('1+ ' k.)(q_' k , )]'
M( - , +; t, - , t, -)
-
e'E'q+l.Z_+B [ 8k,_Q_+] t (2E - kH )q+_(p+ ' k,) C k,+q++
'j!
!I,/Y.
~f, - ..... ~ 'lr-71
(NU Z·fo.'l(C :I1AN(lJ..'; '"
I
tI)
2(1 \1
M(- ,+;+, -, -,+) _
''''1 (- ,+ ; -,~,+,-)
_
t!
(2E
4
•
£'lqiJ.Z... _B [8k l _ Q++]' .t2-+ )Q+_ {P+· k1 ) Ckl+q_+
4e 4 EljB j
M(- , +;-. ":",- , +)
M(+, - ;+.-.-,-)
M(+, - ; -, + . - . - )
[
= e4(2E - k~+ )q:~qtJ. Z;_D· [ q+_ (p+.~ )
k2 _
2C k1+QHQ! +
,
_
('~(U; - k1+ l jfl j [ ~_Q. .q3 ]' (p+ . k"1 ) k1tq+_q-+{Q+ ' kd(q_· k.)
_
e
4
Eq+l.Z~_B· [2kl
9+- {P... . k, )
Q"]'•
Ck1+Q_+
•
_
2e 4 .E.')B) [
=
e4Eq+J.Z:+B" [!~~_q_ +]t q+ _(p+. k 2 ) CkHq..
k2 _qf+q__ ]' (p+ ' k, ) k1+q+_q _+(q ... · kl)(q_' kJ)
]1 k2+(q ... · .I:"(q_' .tl )
- 2,'E,_+11l1 [ (p+ . .1.'1)
:H(+,-; -,-, -.+)
], ]!
,
k, - qttq-~ k 1 +Q... _q-+ (Q+· kd (q_· kll
X
=~
"'_
l!
e·mkuq+l.B" [ 2kl +(qt ' q-) q+_(p+ . k 1 ) k , _q++9_"'{9t-' k d( q_ · k d .
,'",<'+181 [ (p+ . kl )
2(,.
·q-l
q+-q-+(fJ+' k1)(q_ · kd
,
]'
. ...
!I. SIlMMAIlY 01' CJliO fOIWI!},Afo:
2711
111(-,+;-,+,-, - )
M(-,- ; + , - ,+,-)
e' mkiJ.q+.1 Z - +B [ k2+ E(p+ . k'l 8Ck,_Q++Q_+
M (-,-; - ,+,+ ,-)
l' 1
e'mkHq+_IBI
M( - , -; -, - ,+, +) _ _
111(- , -;+,-,-, -) -
e' q+~Z:+Bo [ q+_(p+' k,)
2k,_(q+' q- ) kHq++q_+(q+' k,)('1 _' k,)
2e'(q+· q-liBOI (p+' k,)
l' l' 1
[
k,_
kH (q+· k,)(q_ . k,)
e' mkiJ.q+.1 Z+_ B - [k2+Q+;' (2E - k,+)q+_(p+ . k,) 2Ck,_q_+
1
e' mk,+IBI [ q;+q_ (2E - kH )(p+· k2 ) q+_q _+(q+' k, )(q_.
e'mki.du Z :+ B "
q_+ [ (2 E - kH)q+_(p+ . k,) 2Ck,_q++
kH
l' 1
e'mkHq_+IBI 1 '
(2E - kH)(p+· k,) [('1+ . kd(q- . kdJ'
M (-, -; -, -, +, -)
0'
l' 1
Eq+J.Z:+Bo
kd
]1
271
AI(-,-;-,-.-,+)
c~E(2£
- kH)I/+J.Jr ,,-(p,' !-, }
=
Bk,_(q+' q-) [ x I.: H Q++9-+(9+ ' kd(q_ . kd
II
"E(2E - <,+)IBI (p+ . k,)
,
l' II
8k,_(.+·._ } [ x k1+ Q+_9_+(Q+ ' ktJ (q_· kd
,11(- , - ,'- , - , - , -I
e4E~q+J.H·
:=
9+_(P+·1.:1 )
~" E'I~ [
[
32k1 _(q .. · q-) KHq++q _+(q+ ' kd(q_ · ktl
32k,_(q+")
(p+ ·1.: 2 ) .1.:1+9+_1/ _+(9+ ' ,1;1)(9_' .1.: 1)
II
(9.214) l1npol ... rized squared matri:< element:
(9.215)
Cnpo:arizcd cross secLion: U'~
du =
iBI2.1.:1
256Jr1E~kH(P+, J.:.1 )Z (Q+· kJ )(q_ . /;.) Q+-9-+
X[4£2 -l.. (2E - .1.:2+)1 :-
+ 4
CE ~kl+ )
xo (p+ + p-
1 (
4n~71~] [9++9:_ + q;_q_+ + 8E1(q+. q_)
~,j.'l-- + 1+-9: + + 2(q+~t)3) 1
- q+ - q- - k\ - 1.:2 )
tPii+ d'q_ d3 k} d3k2 9+0 q_b
k k !O
20
.
(9.216)
,'1. 8/IMMAny Of' QEfI
9.19.2
~
~' Ol/MIJI,AN
~
k2 Hearly parallel to p_
Photon polarizations:
It
H
Ndl>+ p- h(1 -
N,[h
- h
Of 7,)
Pt p_(1 ± 7')]'
p- p+(l + -y,) + h p- h(1 Of -y,j] •
N .- 1
i = 1, 2 .
•
(9.217)
Nonvanishing helicity ampl itudes:
M(+,+;+.+ , +.+) =
,
e'E'q+~B [ ( k)"
q+- 1'-"
32k2+(Q+'q_) (k)(
~,-q ++ q-+
q+' , q- '
_ e'E~ I BI [
. 32k2+(Q+' q-) (p_ . k,) k,_q+ _q_+(q+· k, )(q_ . kd
k)
]'
1
Jt
,'IE(2E - k,-kj..LB
M(+,+; + , +, + ,-)
q+_ (p_ . k,)
l'
1
X
81.:,+(q+ . q-) [ k,_q++q_~(q+. k,)(q_· k1 )
e4 E(2E - k,_ )IBI
(p_ . k,) 1
X
111(+,+;+,+, - ,+)
_
8k2+(q+ . q_) ]' [ k,_q+_q_+(q+. kd(q- . k,J
_ e'Eq.,.L Z-+Bii (2£ - k,_ )q+_(p_ . k,)
] e'mki.L q:~ q.,~ Z+_ B [ k2 _ (2E - k,_)q+_(p .• . k,) 2Ck2+q++q~+
lvI( +, +; +, -, +, +)
1 ,
, _
e'mk,_IBI (2E - k,,-){p- . k,)
[
q++q' q+-q-+(q:t' k,)(q_ . k,J
]'
'.17 3
,
M(+.+: - .+.+.+)
_
c~1Iik;l..q+J.Z_+fJ [ k1 _ ]' (21:: k2 _)(p_' *1 ) 2C k, +I]HQ_+ e4 rnk,_9t-_lll l
M(+.+;+.+.-. -)
_ c~q;.,Z_+Bo [ qt--(p- . k2 )
, 2k H (q.. ' q_) ]' k2 -Q++9- +(9+ ' k1 )(9_ . k:)
,
_ 2,' (,+ · , )lll,1 [ (IL . k~ )
k,+
]'
k2_(9+ ' kJ)( q-· k L)
,
M( +, + i +, - , - , + ) "" c~mkiLqTJ. Z+_ n· [ kl_q
~~m~'2_ BJ [
q~ .9--
];
2E(p_ · k,) ,. _, _ .(,+ . k,)(q_· k,)
M(+ , + ;- ,+ , - ,+)
,.
2E(p_ · k, ) I(,+ · k,)(,_· k,)I'
M(+ ,-;+.+ .+ , - )
-
e4mkuq+J.B [ 2(',_(9+ ' q_) ]; - 9, - (P-' k2) kHq-Hq_+(ql' k:)(q_ · kl )
_ "mk,_llll [ (p_. k, )
,
2(,+· q_) q.I-Q-+(9+ · k1 )(1_'
],
kd
M{+, -;-:- , - ,+ . + ) ""
,
4
'"" 2e E DI [ (p_.
M(+, - ;- , T,+ ,. ) _ _
k,J
kHq++r], ~'2-ql q- t- (fJt- ' ktl(q_ · k1 )
4
e E'Q+!.Z_ ID [ 2k21. (p_ . 1:2 ) C kl_q~_ +q _ +
2C4 Eq ~_l fJl [ (p_ . k,)
]1
k2t-
k,_(,+ · k, H,-· k,)
];
II, HIIMMAHY OF Qlil) FOItMVI,AiI
l' l
c'Eq+J.Z_+JJ [ 2kH If -+ '1+- (p- . k,) C k"-IfH
M(- ,+ ;+ ,-, +, +)
_ 2e'E,,_+IBI [ (p_ . k,)
M(-,+;-,+,+,+) _
e' EqtjZ+_
l' 1
k>+ k,_(q+ . kd(q- . kd
B [2k
H
1' J.
q++
Ok,_q_+
q+_(I'_' k,)
l' l
-
M( +,-; +, +, -
2e' EIBI [ k,+qi+qu (1'- . k,) !c,-q+-q-+(g+. kd (q- . kd
, -)
X
k, _(,,+ . '1-) [ 2k2+Q++Q_+(q+ . k,)(q_ . kJ)
e'mk,_(q+' q_)IBol E(2E - k,_)(p_ . k,) [( q+ . kd(q- . kdJ ! 4
M(+ ,-;+ ,-, +,-)
e E'q' J. q+J. Z+_ B [ 8kH (2£ - k,_)(p_ . .1:,) Ck,_q++q:+ 4e 4 E' IBI
(2E - k, _ )(1'_ . k,)
M(+, -;+. - , - , +)
<'(2E - k,_)q+J. Z,;, q+_(p_ . k,) e 4 (2E -
k,_JIBI
(p_ . k,)
M(+,-; -,+,+, -) -
1t
'
1t
11. 19. 1 f,.-
_
,,"~;-
n'
(NO Z'/','XCIII1NUg;
II>
I
UJ
27 Ii
M(-,+;+ ,- ,+ ,- )
M(-,+;+,-,-,+ ) _
,
M{- ,+:- ,+ ,+, -)
M{-,+;-,+,-,+)
_
e~(2E - k 2_)'1+1. Z 1 - R [ k2+1++ ]' h - (Z'- . k2) 2Ck2_'1_+
_
,'(2£ - '·,_) IBI (p_ . k,)
_
e Eq:"i q+.1
4
Z+_ W[
q+- (p_ . 1:2 ) _
2kH Ck 2 _Q++'1:+
4
]1
2e EIBI [ kHq++q:_ (p_. k2! k2 _q+_q_+(1+' k,)(q_ '
,
l' kd.
Af(-,+;-,-,+,+) _
, k,_(q+.q_) ]' [ x 2k2+Q++Q_+(Q+ ' kd(q-. k 1 ) e~mk2_ ( q+ ·
1-}180 1
,'
£(2E - k,_ )(p_. k,) [(q., . k,)(, _ · k, II'
.i
~I,
Silk/MAllY OF 111m 1,'OllMII/,AI,'
M(+,-;+,-,-,-) '.'
M('-I -- ~
I
,
-I. , -
)
-)
A1 ( -, +; +, -. -
, -)
M(-
,
...
)
-1I )' -
-'- 1 -
J ~
4
_+]'
1
_
e £qtJ..Z- tB O [2k H Q Q+_(p_ ,k,) Ck,_q++
_
2e'£9_+ IBI [ k2+ (1'-' k,) k,_(qt, k.)(9_ ' k, )
-
2.' £IBI [
]t
-)
k2±q++q~
(1'_ ,k1 ) k,_q+_q-t (q+ ' k.)(q_ ,k,)
1' 1
e mk;.L qtJ..B· [ 2k,_ (9+ ' q_) ]t q+_(p_ ,k,) k2+9++Q-+(qt ' k.)(q_ 'k.) 4
M (- , +;-,-,-,+)
_ e'mk,_ IBI [ (p_, k,) M ( - , -; +, -, +, --)
M(- , - ', - ,r, -' + t -)
-
1
2(9+' q_) ]' 9t -Q-+(9+' k,)(q_' k,)
,,
!l.'9 . • -1-.,-
-.
~",,-.,. ..,
(NO
Y,·f,'XC:IIA.tU'·~::!II
OJ
/
277
M(-,-iT.-,-,-) -
- (2& M(-,-;-,+,-,-) _
-
M(-,-;-,-. +,-I -
k,_)(p_'
k,1 1(,. · k,H._ · k, III'
e"'TJlkUq:~q+lZ:'_B '
e4mk2_IBI [ qHq:_ (2£-k,_)(p_ ·k,1 •• _,_.( •• :·k, )(,_
(2E
k,_I, __(p_'
(2E
-
]1
k,.)(p_·
·k,)
k,1
[
k"
k,1 "_ (,,. k, 1(,- · k,)
"'- I.
"E(2E ,._(p_ . k,1
]1
iB '
8k"i •• · .-1
X
,'E(2E (p_ ~
]1
8k,.(•• . ,_1 ]1 [ kl_q ... lq_ dq~. kd(q .. . k l )
,,'E(,.· , -IIB,I
..
k1 _
(28 - k1 _)q-l-_(p_, k1 ) 2Ck H 9_+r- _
x
M (- , -', - - - , +1
[
[ kl -1HQ-+(Q+' kd (q_ . ~-d
];.
"'-IIBI
k,1
,
,'1.1( - . -:-,- . -,- ) _
4 e £Zq __ D' [ 32,1.:2+(9+ ' q_) ]' q+_(p_ . k2) ,(;1-QHQ--I-(Q+' k 1)(1 . . kl )
e"'E~ i BI r
-
("p~'kl)
32,1;1_(Q+ ' q_) ]; lkl-q ... -q-+(q.,... kl )(q_· · kd (9.2181
Y. SIIMMIIUY Of-- (Jt:/) FOI(MUf.Ab'
~7K
UHpulal'i:r.ud squal'ed lTIatrix clement:
(9.219)
Unpolarized cross section:
(9.220)
9.19.3
k2 nearly parallel to
The primed quantities
q..
k~±
fi+
and k;l. are evaluated in the rotated frame where
determines the positive z·axis (see also Section 7.4 .3). The four-vector q in eqn (9.221) is obtained by applying a space reflection to q+. Pholen polarizations:
N;;'
(9.221 )
"" NUllVlll1j~llillg
lJ('lieir,y
nrnrli1Iult:s:
M(+,+i+,+.+,- ) -
(2q+o
+ ~+)q+_(q+ . k:)
128,;_(.", )
x
_
_
[k~+q+lq_ ... (q+. kd(q- · 1:
1)
t! Eq!olBI
(29+0 +
k2+)( q+ . 1:2 )
]1
128,;_(",,_)
x [~+q+ _q_ +(q~. kd(q_'
M(+,...-;+,':·.- , t)
]1
kd
e4 (29+o + .1:,- )9+.1 Z, _8 0 -
-
£q+_('1+ . k, )
x[ 21:,+ k;(9+(,,") ]1 ' kd(q_ · kd e4 (2q+o + '4+)((/- ' 9_ )1 8 0 1 S(",k,)
A1(·,+i
-,+,+.+) -
(2q+o + k'+)9t_(Q .. . ~)
8k;,(,, " _)
x [ 1:'_9++Q_ ... (9+' kd(q-· kl
_
,'",l:k;.181
[
8(", ,_)
]1
(29-+0 + ~+)(q+ . k, ) q+_q_~(q+' 1:1)('1_' kd
M ( I , +;+,+,-,-)
=
e4q+0Il'+.1Z_ ... B,j [ 2~_ (q+' 9- ) - Eq+_{q~· k,) k; ... 9++9-_(Q+· k,}(9_' kl )
2e4Qto(q ....
-
]1
q.)IBOI[
B(g .... .1:,)
k; ki ... (q...- k1 ){y.· kd
]1
]1
. ..
'
fl. HIIMMAIIY Oli QI·;/! fOIlMULAH
',XII
M(+,+; - ,+,-,+) -
• Z 1i'() e 4 m k'2.1.9+.1."-+ £9+oQ+ - (9+ ' k,)
t
'2+ 9+ . 9- ) F(
[ .
x 8k,+q++q_+(q+' k.)(g_. k, )
-
e'mk'+(H·9-) 180 1 . ' 2£9+o(9+ · k,) [(9+' kt!(q_· k,))'
,
,
e'mk'iq+J.Z_+B [ k,+ ]' q+o(q+ . k,) 8C k,_ q++q_+
M(+ ,-;+, +,+ ,-)
e'mk,+q+_IB I
-
1 '
2'1+o(q+· k,)[(q+' k,)(q_. k,))'
M(+,-;+, - ,+, +) -
1
k2_q++q~
20'q+o181 [ (9+' k,)
r ,
e• q+oq-l.q+.1 , • Z'-'+- B [ 2k'2 q+ _(q+. k, ) Ck,+q++q~+
]'
k2+q+-q-+(9+· k, )(q_ · k,) 1
o'q+oq+J.Z_+ B [ 2k,_ ]' (q+ . k,) C k,+q++q_+
M(+, -;- ,+,+, +)
J' 1
-
20'q+oq+_IBI [ (q+ . k, )
k,_ k,+(q+' k,)(q_ . k,) 1
M(-,+;+,+ , -1- , -) -
e4 m "2.l.Q+J. 2+q++ " ' Z'+B [ k 'r 9+uq+-(q+ . lo,) 8Ckl_ q_+
,
04mk,+IB I [ q~ +q__ ], 29+0(q+· lo,) 9+-Q-+(q+· kt!(q- . k,J
,
- e'q+Oqt .. Z-+B [2k,_Q_+ ]' Q+-(9+· k, ) Ck,+9++
M(- ,+; +, - ,+,+)
4
-
2e q+o9_+181 [ (q+' k,)
k,_
-
20'q+oIB I [ (Q+ . k,)
1
kl+ (q+ . k.)(q ... . k,)
_ e 4 q+oq+• .. Z +- B [2k',-9++ ] 9+-(q+ ' k,) Ck,+q_+
M(-,+;-, +,+, +)
r
I
k2 9~+q-k'+9+-9-+(q+· k,)(q_ . k, )
]
!
"'. /.9. .. -I r
M(+.
'
- .," r'
o
rr
( Nf) ?,. t:,H:U Ie ,vc;,.;;
ltI
I II
'llll
-;+.+, - ,- )
M(+,-j+ , ~,+. - )
]!
, • J. Z+- B [ HI!2 e" q±oq-.lq. (2q+o + k~+)q+_(q+. k~) C k~+q+_ q~+
=
4-e~9~oIBI
(21+0
+ kh)(q+·
x[ M ( + ,~;+,- , -,+)
-
.'
k~)
]1
.1:1 qHq:_ . kd(q_ . kd
k~+ (I+_q_+ (q+
,
+._] .
- e ~ (2q+o + k;t )q .j.J.Z.j.._B· [ ~_ £.. q.. -(9t ,kl ) 2Ck~+q_+ e4 (29+o + k 2t )I B l (q+' k1 )
, M(+.-; - ,+ . + ,-) _ _
_ ei{29+o+k2+kJ.Z-lB[ k) _ ]' (9+ . k 2 ) 2Ck~ t 9-+q-+
e~(2q+o +k~+)9-t_ 18![ k~+( q+·
(q+' k2 ) M(+,-;~,+,-, + )
_
_
kl )(q_ ·
]1 kd
eiq;uq+J. Z:1 S· [Sk; q + ] 1 (2q+o + .I:'+)9t-(9+·~) Ck 2+9++ 4
4e q!oq_+ lEI (29+0 + k1+)(q+ ' k 2 )
=
k~_
[
e4mk;J.q::J.g~.lZ+_B (2q+o
,
l'
14=.._
k1+(qt' kd {q- · kd
+ k1+)g+_(qt' k2 )
[
~;+
1~
2C~_q++-q:+J
',8.
,V, S!lMMAUI' OF (JIW /"OUMULAA
_
M(-,+;+, +, - ,-)
c'mk"21. go' k'1± ]t -J. q+1. Z· J±_ IJ' [ (2q+o + k'+)g+_(fJ.' k, ) 2Ck2_Q++Q".+ 3
-
M( - ,+;+,-,+,-)
-
e'mkz+ lEI [g++'1 (2'1+0 + v,+)(q+, k,) q+_q_+(q+ , k, )(q_, ",) _
t
[8k'2_q-+ ] t (2q+o + "2+)'1+ - ('1+' ",) Ck,.Q++ I' . ' Z -+ B e. ,q±oq+.l
1
-
4e'q!oq_+IBI [ k,_ ], (2q+o+ k,+)(q+, k,) k,+(q+' kl)(q_, k l ) 1
e'(2q+o + k'+)q+l.Z:.B' [ k,_ ]' (q+ ' k,) 2C k,+ q++g-+
M( - ,+;+, -,-,+)
-
e'(2'1+o + k,+)fJ+_ IEI [ ('1+' k,)
k!(-,+;-,+,+,- )
1
k,_ ]' k2+(q+' k'),(1- ' k,J
,
-
_ e'(2q+o+ k,+)q.j.l.Z+_B [ k, q++ ] ' q+_(q+ ,k,) 2Ck,+q_+
-
e4 (29+O + V,+)IBI [ k, _qtt9 - ], (g+' k,) k,+q+-9_+ (Q+ 'k,)(q_ ,k, )
1
M(-,+;
-, +, -,+) -
_
1
e'q!oq''cqHZi B' [ 8k, t (2q+o + k,+)q+_(q+ , k,) Ck,+q++Q".+
4e'qiolBI (2q+o + k2+)(q+ ' k,) [ X
M(- ,+; - ,-,+,+l
-
-
e',.r. mk2.1q+l.Z_+B (2q+o
+ k,+)q+_(q+' k,)
[-'I,+q_+ 2Ck,_9++
r
e'mk,+q_+ lEI (2q+0+ v,+)(q+' k')[(9+' "1)('1_ 'k.)]! ' 1
e'q+oq+.lZ+_ B" [2k'_9++ ) ' '1+-('1+ ,k,) Ck,+q_+
M(+,-;+,-,-,-)
1
-
r
k'_q++Q"._ k,+q+_q_+(q+' k,)(q-, , k l )
2e'l q+olBI [ v,-q~+q-)' (q+' k,) k2+Q+_q-+(Q+' k,)(q_, k,)
"
'211:1
M(+ _' _ ,
)
,C '
"
__ )
__ e fJ+OqiJ. f,_ ... I.'J o [ '"'' ~~~ _q _ + ] ! ,
I
"
Q+ -(1+' _
M(+,-;- ,-,-,+) =
::
0
C'k~of.qH
J.: 2 )
4
2c q+oQ_ +IBI [ Jc,(9+' k2 ) k2+{Q+ ' kd(q·· U e• m "a q+l. Zo t
Do [ k'2± q+t ]1
9+09+-(9+' k2)
8Ck;.q_+
]1
4tq
(hr!k1tIBI. [ 2q+O(Q+' k~ )
kd
]1
Q+- 9- + (9+'
kd(q.·
kt }
,
M( -,+ ;+,-.-,-) = _ eiq+()q'+J.Z:tB· [__ ~k~=-___ _]' (9+' k~ )
~
Ck~+q++ q_ +
2,',."._ )E) [ (q+' k2 )
]1
kIk~+(q+ · kl )tq.·
kJ)
M(-, - ;-,-, - ,+) ,,
'2q+o ( q.~· k2){(Q+' kd(q.· kl )jS
M (- , - ;+,-, + ,-) _
e4m~j 'I+J. Z:t 8 0
Eq+oQ+_(9+·1:1 )
k;.(q. ' ,_) x [ 8~+ q++ q_ .... (q+ . kd(q . . .kd e~mkh(q+·
q. }1 8 ol
]1
lio SII MMAIIY OF (WI! I'()/t MULAfI'
M(-.-; +.- .-.-) -
c" 'm l!Jk~l.q+l
Jr
(2'1+0 + k!;.+)q+-(q~ k,) 0
,
8k,+(q+oq_) ]' [ x k!,-q++q-+(q+ ktl (q_ kd 0
0
1
e'rnEk,+IBI [ 8(q+ - q-) t (2q+o + k'o.)(q+ - k,) q+_q_+(q+ kd(q- - k , ) 0
M(- . - ;- .- .+, -) -
00'(2q+o + k'+)q+l.Z+ _ Bo E9+-(q+ - 1.:, )
,
]' k,_ (q+ - q_) x [ 2kh(q+ k,)(4_ kd 0
0
.'(29+0 + I.:;+ )(q+ - q_)IBol E(q+ - 1.:, ) ._>
]
~,-
t
[ ~4(
,VI
_,_, 0
_
_ _
1
1
..
,
e4 E'I'+0 q'+ 1. B-
)
.
!
128kUq+-q_) ]' [ x k,+q++q_+(q+ k,)(q_ k , ) 0
0
e'Eq~o l BI
1
11;,1(- •
. ' Eq+OqH B' [ 32k,_ ('It q_) ], 'I+_(q+ k,) k;+9++9-+(9+ - k, )(,,_, k,J 0
_0
,
_ _
)
_ _
I
1
)
-
0
_
e'Eq+oIBI[ 32k;_(q+-q_) (q+ k,) k,+9+ - 'I-+(9+ k,)(q_ k,J 0
0
Jt
0
(9_222)
I/./!I. r i
r"f (Nt) X · /",'X(:I/ A N( ,'1:;
• , ~ , .,
,.
III
I 0
1)l ljlu!arizl:d s'!lInnlllllli~~rix d!![JW11!.:
f,itp
cM l lll~~ _
=
2k;+(,,+, k1 )1(q .. . kd(q-· kdq+-fJ-.,.
~
X [ 4qH
, 2 m.~ ki't] + (2g+0 + *2+) + 4q_0 1 k' 2-
-(I)
:O, k ,
Q
2
~q+O
, 1+
[ :I (1++07--
3 + qt-q-t + 2(q+r..." ·"1q-r'
)l(Q±+l _ +Q+_II:++ 8 F;1(q+ .q_J)]
(9.223)
Ilupolarizcd cross section:
dO" -=
o~ I B11k2
=ccc=7""O--::,E'''''7-;-,-----,--,---256",1E2~+ (qt'
,
X
(
[4q"'l) + 2qto
k:lP(q+ ' kt)(q_· kdq, - q- t ,
2
ml~f+ ] [ .)
+ k2-j.) + 4
'k' q+o l-
q+.,.q--
:I
T
q ~ - q- I
+
2(q+ . q_)J E'
+ (2q+~q:°.l."~J 2 (q++q:_ + q+-l+ + SE'{q+ . q-l)] X~
9.19.4
•
) J34+ rflii- dlk, J'!k1 (p++p_-q+-q _ -k1 -k, k k . q+oq-o
"' 0
(9.224)
:20
-
k2 nearly parallel to q_
Tlte doubly primed quantities lifi and k!{.1 are evalualoo in the robtcd frame wl:crc q-_ tldcrmincs the positive z·a.xis (see also Section 7.4.3). The fo ur· vector q in eqn (9.225) is obtair..ed. by (l.pplying a. space reAection to q_. Photon polarizations:
fit
~
N,IA p- t,(1 'f1,)- ,k, p+ p-(!
N; I
=:
q~0[32~+k;_Jt.
± 1,)\ ,
(9.225)
KO J) va..nishing helicity a.mplit l!des:
M(~,+;+,+. +.+ )
=:
,
e~Eq .. oq~.1 B[ 32k~ (q+· q_) ]' ( k ) k ( k)( k q+- q-' ~ '~+q++q-+- q.. . I q-' d
o ,'Eq_, 181 [ (q_. k 2 )
32k, (q, . q ) kr+q+-q-... (q+ · kl )(q_ . kt)
]i
M (I,I;I,I,I,
)
J' L
1 28k~_(q+ · q_)
x [ k~+q++q_+(q+ . k1)(q _ .--kr)
,
_
e4Eq~olll l [ 128k~(q+.q_) ]' (if- . k., ) k~+q+ _ q_+(q+' k1 )(q.. ' kd
c'(2q_o + k~+ )q±J. Z_+ Bii
!l1(+,+;+,+ , - ,+)
8'1+ _ (q_ ' k,) L
]' ",-('1+ ' 1- ) [ x 2k,+q++q_+ (q+' kd(q- 'kd _
e'l (2 q 0
+ ",+ )(1+' q-l illol E(q _ ,k,)
M(+,+;+ ,-,+, +) 1
8k,+(Q+,q _ )
X
[ k~_ q++ '1_+(q+ ' k 1 )('1_ . kr)
]' 1
e'm Ek,+IDI [ 8( '1 + ''1-) ]' (2'1_0 + k,+ )('1_ ''',) '1+ - '1 -+ ('1+' kd(q- . k 1 )
!l1 (+.+; +,+.-,-) _
;11(-1, + ;+,-,-, +) L
X [
/:2+(1+ . '1-) ]' 8k'{_q+t 'l _+ (iJ+ ' k 1 )('1_ ' kr)
.4m k'+(Q+ ' '1- llBo l
U(I,
,I. I, I .
H{ I.
; !, ,- ','L + )
,1/ 1+,
- ; -, +, ... , +)
.\1( - , + ; <,\. "
, ',«1.:;',/:
)
I'J i , X,
'1 _11'/. (,/
- I, d
) _ ,, 1f11ki~q
,III
2q_o(q - k1)(('h' ,I;', )( y . k.); t' ,H(
,+;-.f, - ,+.+)
1/( ,+;
,+,t,+)
_
,
e Y_Cl'J +-l- 7.+ R [ 2k~_ql +] ' (!t -(q-, 1.-1 ) C~.J;_q_ + i
.;. 2c 1q_0181 [ (q_ . k1 )
M (+.-; + ,+ , - . -)
~
=
J..~'... q~-q-t(q-t '
C'iml4':qtl.Zi. _D" (2q
(I
I
j I J..·d [kl+qH] ' 2Ck1_'1_+
1/; q~t'I- -
J.7+)'/I _('J_' k1 )
k1)(q_,
il . .,/iA/MAItY OF QNJ) FOIIM{}I,AJI
M(+, -; +,-,+,-) e4 (2q _o + k~+)IBI (q_ . k,)
[
e q' oqHZ.~_ B" [}¥-q++ (2q_o + kq+)q+_(q_ 'k;) Ck~+q_+ 4
M( + , -; +, -, -, +)
]1
k"q 2- ++ q3 1
1'
4e"q'olBI
M(+,-;-,+,+, - ) -
Jl!/(+,-; - ,+, - , +)
JlI( + , - ;-, - , + ,+ )
e1mk~.l q+.l z_+ B [ k~/+ (2q-o + k~+)(q_ . k,) 2Ck~_.q++q_+
e1mk~+(1+-IJ]-,-I_ _ _--c,
(2'1_0
M( - ,+; +,+,-, -)
l
1'
+ k~+)(q_. k, ) [(q+ ' k,)(q_ . kIll'
,
,
M(-.- ;+,-,+ . -) _
(·~ (2q_o+ ~·1' t)/!.1 .IX-+1J [ 1.:1_9_ + ]' I/ - -(q-.
kd
_ ''(2'-0 - k~+),_+ 1131 [ (q_. kt )
M(- , - ;+ , -, - , +) _
M(- ,-;- , +. +, - )
M( - , + ; - , ~,-,+)
'2Ck~'+q+t
"'_
k'{+(qt· kd(q_ · kd
]!
]1
, ' , ' o"t.1 Z - t B[ 81." "7_ ~ {2q_o + k;t )(q . . k1 ) C k¥t qt+q_+
~
_
e4 (2q_o
+ k7t )qo2J,. q+.1Z';' B" r q . . (q_.
.:..
1.:2 )
14-
l2C k~+q+t q:!. t
e4 (2q_o + k~t ) IB l
(,_ . k, )
X
M (-,+;- , -,+, +) = _
kLq-Hq:. [ k 1t q+- 1- +(q+ . kd(q-. k l )
~4m.l4'.1q;' lZ+_B
(2q-o + krt),~+ - (q - · k2 ) 2Ck1_If_t e4 rnk">t 181
Mh - ;+,-,- ,- ) -
[k~!' h~]l
]1
]!
fl. SIIMMAIIY or (JIm fOIlMIif,AI,'
L!J(J
M(+, - ;-,+,-,-)
M(+ ,-; - , - ,-,+)
H (_ '
."V.I
-,-.+ _ _ _ ) I ,
)
1
.M(- '+. )-' ". ~ ,
1
- ,- )
.M(- " +'- ' - " - -',- )
M (-, -; -, +, + ,-)
111 (- ,-; -, - ,+,+ ) _ _
4
e q_oq+J, Z: +Bo [ Eq+_(q_' k2 )
l' ~
2kq_ (q+' q- ) A,~+q++q _ +(q+. k, l(q-' k.)
2e4 q_o(q+· q-ll Bol [ k~_ E(q_ . ", ) k~+ ( q+ . k,) (q_ . k.)
1t
:iIJl
,
8k~.(,+· ,_ ) ]' [ )( k~_q.+q_ .. (q ... · kl )(q_· k l )
_ M(-,-;-,-,+,-) =
"mf;~,IHI
[
8(" .,_)
,
(2q_o + k~... )(q_· k1 ) 9+-9- .. (q... · kd(q-· kd ~4 (29 ·o
]'
+ k'1t)qt,LZ::'tBo
Eq+_ (q_ . k 2 )
'" ( )
[
"1
]!
IJ+' 1-
AI(- ,-;-, - ,-, +) _
z
12Sk _( II+' 9_)
X
128~_ l q+·,_)
_ " E",IBI [
k~+q+_q_+ ( q .. · kd(q-'
(q_ . k1 l
A1(-,-;-,-.-.-)
=
[ k~+qHq_ ... (q .. . k1)(q _ .
~~Eq-Oq+,LB. [ 1... _(9_ . ,(:2 )
(q- . k2 )
]1 kd
32~(9 .. ·q_ )
, ]'
k2'+q++q- .~(q+. k 1 )(9_ . k 1 )
32k~_(,+· q-)
,'Eq-o'BI [
]1 kd
11+I]+-9-+ (Q+' k1 )(1I_' kd
,
]' (9226) .
.
tJ npoJ'l.riu·d squMcd matrix clement:
IMI' =
-.,,'c:·~IB,\I'c'<;'!i"-:--..-;:-.,k )q... _9_+
2k~dq_· k2 )1(q+. kl)(q_'
' ,
X [4Q _o
-t
1
ml14~ ] [ 3 3 2(11+ . 11_)3 + ( 29_0 + k")2 2+ + 4q:okj_ q++q-- + q+-9_+ + --ii- -
r
Cq_·~q~Okl+ (q! .. q-- + 1f!-9-+ + 8E'l(q .. . 9- ») ].
(9.227 )
9. .,'II MMAlIY Oi" (ililJ I"O IlM/I/,AII'
Uupola.l'it'.cd <:ros s ~w (;l ion:
(9.228) ~
9,19,5
~
kJ and k2 nearly parallel to
p+
and
p_, resp .
Photon po1arizations:
I~
- N,(,k, p- h(l ± 'Y') + h p- ,k,(l 'F 'Y.< )] , i = 1,2.
Nonv~.l1i shing
(9.229)
helicity amplitudes:
M(+ ,+;+,+,+,+)
l
4e'E"(2E-k2 _)q:j..l. [ k,_k>+ ]' q+_(p+ . kd(p- . k, ) k,+k, _q++ q_+
kI( +, +; T. +, +, -) _
4e"E'(2E-k,_) [ k, _k,+ (p+ . k,)(p_ . k,) kI+k,_q+_q_+
]t
11./9. r l ,.· . -
I'~" - 'Y "I'
( NO Z - ~:X(.'II A NUl';, III I OJ
:/1)3
,
M(+,+i+, +, -,+)
M(+,+;+,
==
4 l -1 C g ('lH- k l+)l/h[ A'l_~:H ]' q+-(p_ . kd(p_· /.:J) kHk~_q++q_+
_
1e4 b""2(2£ - kJ+) [ k,_kz+ (Pt' kd(p_ . k 2 ) kl+.1.:2 _q+_q_+
1;
~,+,+)
M(+,+;-,+,+.+)
M(+,+;T,+, -,--)
~
1e~E(q+·q_)q+.l. [ q+_(p+ . kd(p_ . k:z)
kl _ kH /':1+1.: 2 _Q++9_+
li
1\1(":",+;+,-,+,-)
(2f!
M(+,+;+, - ,-,+) =
(2E
M(+,+;-,+,+,-) _
4mkU.q· J. f; (2£ - kl+)(P+· kJ)(p_· k,)
e4 mkl+Ql+ (2E
kH)(p ... . k,)(p_ · k2)
,
[kHkH'7f±]' k,_k 2 _q:+
[1.:2+q+_] i k~_q'~+
M(+,+;-, +, -,+)
M(+ ,+; - ,-,+,+)
M(+ ,- ;+ ,+, +,- )
M(+ , - ;+, - ,+, +)
!
4e'E'q+J. [k,_kHQ_+] ' (2£ - k, +)(p+ . k,)(p_ . k,) k,+k,_q++
M(+.-; - ,+,+,+)
1
40'E'
-
[k' _k2+q+_Q_+] ' (2E - k,+)(p+ . k,)(p_ . k,) 101+10,_
-
2e'm E k i1. q+1. [ k1+"2+ ] , q+ _(p+. k,)(p_ . k,) k, _k,_q++g_+
1
.M( -, +; +, +, - , + )
-
1
4
20 mEk,+ [ k2+ ]' (p+ . k,)(p_ . k,) k,_q+_q_+ ' 1
4e'E'g+1.
[k,_k' +9_+] ' (2£ - k, _)(p+ . k,)(p_ . k,) k,+k,_q++
M(- ,+; +,-,+,+)
1
4e'E'
-
[ k'_k,+q+_g_+] ' (2E - k,_)(p+· k))(p_ . k,) k,+k,_
!I.19.
,.4- ,._
.~I' ·- -r1(Nl):1-f;X(:i1AN(lH; III/I1)
(2E
M(+,- ;-I- ,+,-,-)
=
_
M(-,-; +,-,1 , - }
-
e4m(2E-kl+)kuq: .1. [ kl_k2 _ fJ+-(P+' kd(p_· l:2) kl+k2+q++q_~
,
4
e m(2E - kl+)k2_ [ k,_ ]' (p+' kd(p_ . k 2) kHq+_q_+
"
(q+" q_)(p+' k:)(p_. k2 )
{E' >1
+lIl ·lku. k~l.q+J.
4/0,
4
.14(+,-; +,-, - , I') _
]1
q+J
!_] I
[k,_k H q
kl+k~ _ q~ I
[kl lkl_q_t]I} . k-*1+(lt_ , l
2e Eq+J.. [kl_kl±q!y]' (p+ ' kd(p- . k~) kHk, _ q~+
M(+,-;-, +,I " -)
+mlkiJ..k'~H';'.L [~·I+k,_ q~±] 4 f}
4 2e EqtJ.. M(+ , - ;-, +, -,+) _ b'+· kl )(P_ ·
[k l _k2 +q_+ ]1 ~)
k l t k , _ q++
k1 _k'l±q:..+
t} ,
:,wn
!I. •'i/ IMMAIlY O/>' QI:/I VOIlMII/.M! 1
e'm(2E - k,_)k idtL [ kl+kH ]' , q+ _(p+ . k>l(p- . k. ) k l _k2 _q++q_+
M(-,+;+,+ ,-,-)
1
_
e'm(2E - k,_ )kI+ [ kH ], (p+ . kl)(p_ . k,) k,_q+_q_+
M(- ,+;+,-, +,- )
M(- ,+;+ ,-,-, + )
M(-,+ ; - ,+,+.-)
M(-,+i-,+, -,+)
+m'kiLk;Lq+L [kl+k,_?_+]t} 4E
k l _k2+q++ 1
e'm(2E - kl+ )k;Lq+l [
M(-.+; - ,-,+,+)
q+_(p+ . k.) (p_ . k,) _
M(-I-.- ;+ ,-. - ,-)
k l_ k,_ ]' kl-/-k2+q++q_+
] e'm(2E - kl+ )k,_ [ kl _ (p+ . k.)(p_ . k,) k1+q+_q_+
1 ,
9.19.
c 1
, .'
-0
r'.-..,.., (NO r, . J:X(.'Ui1Nr,'''''; II,!
M(+ . - ;-, +.-.-)
~
'l'J7
1c~ h"l'lH
[.1. 1-1.1+9- + Jt
(2E
~-)(p1"
(2E
k1 _ )(p+ . k, )(p_ . k1 )
kh~_9++
k.)(p_· .I.,)
4c. £'1
~
M(+ ,-i - .-,+.-) -
U)
2e4mEk\~9_ .L 9t-(P+ ·kl)(p_ · J..:)
[
["' _ .(:1+11+_9_+ ] i 1;1 ~ .1.,_
kl+kh
]f
k , _~_9++q_+
2e mEk, ~ [ k2+ )t (p+ . kl)(p_' k2) k2-9+- 9-+ 4
~
011(-,+:+. - ,-.-)
-
~
M {- , + ;-. +,-,-)
-
M (-.+ ; - , -.- , +) =
• e 'E' q~• l (2E
[k 1 _k'l+'1+_q,. , Jt k,+ )(P+ . k1 )(V_ . k,) k l +k 2 _ 4
4e E2q+.L
[ki_k11if-1 (2£ - k,+)(p+ ' k.)(p_ ,k1) kl.,.kl_q~+ qTt • e'E"
(2E
•
k1+)(PT' k1 )(p_ · k 2 ) kl+k'l_q++
4e~E'I
(2E
r
[k 1_ k~Iq _ ~
r
[k 1- k2+q+ _ ]'
ku )(p+' k. )(p_ . ~) k,tkl_ q: +
•
2e 4 mE'k;J,q+.l [ kl_k1_ ]. q+_(p+ ' k.)(p_ . ..1:2) k1+k'Hq.,.i q_+
r ,
~
M( - .-; +,+.-.-)
-
M (-, -;+.-,+,-) -
2,'mEk,_ [ k,_ (p+ ' "I )(P- ,k1 ) k,+1+ - 9- +
e4mlkiJ. ku9:J, [ kltk1_ J~ 2Eqt_(p+ . kl ){p_, k2) kl _.\;2+q++q_+ e4mlkl+k2_
2E(p,' k,)(p _ . k,)I,+_,_+I! ' (2 ;:
c4mk2J.q:,L [kl_k,_q_1 ] k2 _ )(p+ . kl )(p- .~ ) kl .,.kH 9+ I i
=
[kl ,+-q-+ (2 E - k,-Hp+' k,)I.1'_ · k,) k.+ c: mk2_
, 1
r
II, .\'IIMMAItY
or Qb'O
fOIlMUI,M:
M( - ,-j+, -,-,+ ) 4 e mk1+ql.+ [k H qt -] t (2£ - kl+)(l't' k, )(p_' k,) k,_q~+
M( -,-; - ,+ ,+ , - )
e'mk2.Lq:.l. [kl _k,_q'l.t ] t (2£ - k,-)(/,+ . kl)(p_' k,) k1+kHQ:'t
.1
M( -,-; - ,+, - , +)
, M( -, - ; -,-,+,+) _
4e'E(q+, g-)q+J. [ k, _k2+ ]' qt-(P+ ' kl)(p_ ,k,) kJ+k,_q++q- t 4e'E(Q±,q_l [ kl _k2+ (p+' k,)(p- ,k,) kl+k,_qt-9-t
M(-
1
_
, .L
,.-,
M(- , -'I -
,
_
,
-
1
-)
+ - -) ,
1
M (- , -': , -
,.
]t
,
+ -) 1
1
4
_
40 £'(2E - k1+)qu [ kl_kH ]' qt-(Pt 'k, )(p_ ,k,) k1+k,_q++Q_+
_
4e'E'(2E-k lt ) [ kl k2+ (Pt ' kd(p- ,k,) k1+k,-1+- Q-t
]t
M(-,-i-,- ,-, +)
_
'k~I""'('I.I·; - k~ _ )rt+.L
1--(,'+ . kJ)(J' -
_
M (-,- ; - , -,-,- ) =
. k,)
[
kl_kH ]j k1 ... k, _I'J ...... q_ ...
4t.~El(2E-k,_) [
kl_kH (p+ ' kJ)(p_ . k, ) kH·k2_q·l_q _ t
]t
8t. 4 E3qu [ ( ' I _kH ]1 q+ _(P1 . kl )(p_ . k2) k, ... ~_ql +q_ ...
1Ilipolllfizc<"l squared r.a
2kHkz_q+_q_.,.((q.,.· q_ j (P+ . kl ){p_ . k1)jZ x
[<1E2 + (2E -
1;;1+)'
+ ;;211+J
(9.231) [!l1 pot ar ize~
eros::;
~et i o n:
df7 =
X
[-1E2 + {2E - k1+)1 +
m211+ 1
4£
I_
+ (2£ _ k: _)1 + ~22~~]
x
[4E2
x
11(q_. q-)? + q!+q:_ + q!-1:+J
" .
Y. ,~IIMMA"Y
31111 ~
9.19.6
..'
~
1.:1 'Iud k2 nearly parallel t.o
i1+
IllHl
or q8IJ FORMULA},'
U, rcsp.
The primed quantities k,± and k2.t are evaluated in the rotated frame wheNI ii+ determines the positive z-axis (see also Section 7.4.3). The four-vector q in eqIl (9.231) is obtained by applying a space reflect.ion to q+. Definition:
(9 .233) Photon polarizations:
J,' 11
1
(9.234 )
Nonvanishing helicity amplitudes:
M(+,+ ;+,+ .+,+)
2c' E(2q+o + It,+ )q:" Z+_ q__ (p+' k,)(q+ · k,)
."'I( +.+; +. +. +, -) m'q __ kl.lk,'J. [kl+k;+q++] + SE'q+o k1-It,_q:'+
M(+.+; +,+.-,+)
t}
e' (2E - k,+)(2q+o + It,t )q:"Z+q__ (p+ . kIl(q+ . k,) [
k1- k'2-
]t
.
·'J. I .'I.
~+ ,•.
_
4
r-l , -
'"t.,.
( NO y, .. 1':,\'(.'1111 fo/r;,.:; m
I
IIj
JU I
M(+,+;T. - , +,+)
AI (+,-~;- , + , + , + )
-
+ M{':",+i -,+,-, -)
~
Ek;..l.Z+ [ kl_.I4+ ] l} q+oq-kli.. J4_q++~+ '
,
_ 2c(q+Q(2E - kl+)q:.;.Zt_ [ kl _k2_ ]' q__ (p+ ' kd(q... · k2 ) kJ+k7+q+_tf- .. (;('1, 0(2£ - kl+ ) [8('1 .... q_)k1_1-1_]! q-t(P+' kd(q+ ' k 2 ) k1+~+
M( ":", +;-,-.+,-)
M(-:-,+i- ,+,+,-)
_
_ _ _ , e4 rnq ..oku.q;.J.
E(2E
=
[kl+k;_II __
KI+)(P+' .l.:1)(1t' k1 )
+] ,,
r-t
[k 2_ Q'H
q~_]l
~ ..
4Wq_+(pt' kl )(q+ · '\-2) 4
e m(2E - k1 ... )kL,q: .lZt- [ .l:1-k;t 2q+oq __ (Pt' ktl(q+ ' .1:1 ) kt+~_q++~t
e~rn(2E - kl+)k~t q+oq-t(P+ ' 1.:. )('1t' .1;2)
M{+.+ ;-,- .+,+)
k1 _kl+'i'++
e~mq __ (2'1to + ,1,;+ )ku.q:.J.. [KI+k1_IJ .. 4El(p+ ,kd(q+' .1;2) k.-A1..
e(m(2q+O + k2+)k l +
M(+,+;-, +,-,+)
-
=-
]!
]!
[(1-' q )k1_] ! 2k l +
em • 'k I.J.. k'uq+.l • [k H k'2-'1- +]' 4E11+o(2E - ht)(p+, k1)(qt' k~.d k l _J4_ qt'.. e 4m 2 kJ.+ k~± Iq+- q_ .. J~
4Eq+o(2E
k1tj(p+ ' kl )(q+ , k2)'
II, .'iliA/MAlty OJ' Q';J) "'()11I4(nAl~
[kl
1
M(+,-;+,+,+,-) -
e1ntHk~:",/+1. k;+q_+] ' -,,+-0"(2"E'"--;k-,+--;)"(p~ ,k,)(q+ , ,(;,) kJ+k,_q++
-
e4mEk~+ [kl _q+_q_+] ' q+o(2E - kJ+)(p+' k,)(q+ ' k,)
1
M(+,-;+ , -,+,+) -
0'(2'1+0 + <,< )q--q(p+ . ktl ('1+ . <,)
1
[~,, -k;-q++ ]' k, .,.k'+
M( -, + ; +, +, +, -) 1
o'm(2E - k, +).,+ [<,-q++
[('1+' q_)k,_] ,
1
-
e'm(2q+o + k;,)k,+ Eq_+ (p+ ' k,J (q+ ' k,)
-
e'(2q+o + k,+)q:j..l [k1_k; _q_+] ' (p+ ' k,)(q+ . • 2) k,+k,+9++
2k,+ 1
M(-,+; +,-,+,+)
0'(2"+0 + -
M(- ,+;-,+,+,+ )
l'
e'm (2q+o + k;+)ki.lq:.l Z+- [ k, +k,_ - 2Eq __ (p+· k,) (q+ ,k, ) k'_"'+9++q:+.
.,+)
(p+' k,)(q+ , k,)
1
[<'_k,_q+_q_+] '
.,+.,+
1
!l. !!I. " " .' . • ,. 1,. r,({'J07,-f;'.'I.('IIANtiJ>:;ItI "/II )
M{+.-; - .+.-,-) _
"'(-1, -;+.- ,+,-)
M {+,-i ":"",-.-.+)
= _ e4q__ (2E - k 1+)(2q+o + ~+)q_..I. [~'l_k~_q++]l ..:..
e\~E -kl+ )(2q+o.J-J.;t) [kl _k~_q++ q: 2t,q_+(p+. kl1(q_'
~1(+,- ;-, +,+,-)
kl _ k~_q:+
2E(p+ . ka)(q+· k, )
=
},;1)
kl+14+
,
2c 4 q+oq__ q+..I.
[ kl_~_ q_+ ]'
(p+' k.)(c/+· kl )
kl+k~+qi+
2e~q+oq
[kl _k Lq +_ q_+] I qH (P·t . kd(q+' k, ) kl ... 14+
M(+,-;-.+. - ,+} _
2e 4 q+oq ...J. [kl_k;_q_.• ] (1'+ . k.)(q~ . k,) kl+k2+q+_
1
2e4qH [J.- 1_k1_Q+_Q_+]I (p+' kd{C(+' kl ) kl+kir
M(+, -;-, - ,+,+) =
M (-, --:-i+, 1, ,-,-) -
]1.
:1111
,q, SIIMMA flY
or QIW
I'OIlMlnAl:
[,:,-!:; 1' 1
2e' q+oQ±J. '1-+ (1'+ 'k,)(q+' k,) kI+,,~+q++
M(--, +; +,-, +, - )
1
4
2e q+o [k'_k,_q+_q_+] ' (p+ ' k,)(q+ ,k,) , kI+k,+
M( -, +i + ,- , -, + ) _-,2_e_4 q,-;+..,o;-,q__-_-;-;q++(p+ - k,)(q+ ,k,)
[k, _k,_ q+ - q~+] ! kI+k,+ 1
M( -, +; -,+,+,-)
_
_ .4q__ (2E - kI+ )(2q+o + k;+ ) q~J. [k'_k;_q++] ' k, + k; + q~ + 2E(p+ - k, )(q+, "' )
_
e' (2E - k, +)(2q+o + k,+) 2Eq_+(p+ ' k,)(q+' k,)
[k' - I'Lg++ q~
1t
k,+k,+
M( - , + ; - , + ,-, +)
+
1n2 ki.lk~~ Z+_ 4Eq+OQ__
kl+k~+
[
"'_
k,_'I++ q:+ ,
[k
1 _k,+Q_+] ;c'mk'J. 'I-tJ. 2q+o(p+ ,k,)(q+ - k,) kHk2_q++
M( - , +; - , - , + , + )
1
e'm.k,+ [k,_q+_q_+ ] ' 2q+O(p+ ,k,)(q+' k,) kH kI( +, -
[k,_k;
e1 '1+oQ __ (2E - k1+)q-J. q++] t E(p+ ' k, )(q+ ,k, ) k,+k,+q: +
;+, -, - , -)
. ''1+0(2E - k,f) Eq_+(I'+ - k, )(q+ - k, )
M(+, -; -,+,-, - ) _
4
[kl-k,_qt+'I: _]' 1
kJ+k,+ 1
e (2q+o + k2+)qH ["'_k;_q_+] ' (1'+ 'k,) (q+ ' k, ) k,+k,+q++
,
04(2q+o + k,+ ) [k,_k;_q+-q- +] ' (p+ 'k,)(q+ - k,) kHk,+
]
t} '
y I !I.
,.~ " .
-.
lot,-,.., (N fJ Z . Jo,',y(:IIANI,'l-:; III/ U)
M(+,-;-,-, +,- )
_
c 4 m(2q+o+ ~+)kl.LII • .J..1.;_ 2e9 __ (P• . k\ )( q. ' ~'1 ) [
]1
k,' 1+ 1-
,
M { 1- ,-; - ,-, -.+ )
_
e4mq __ (2R-kH ) k~.lq_.L [kl_k~+q++]' 4Eq_o(p+ ' k. )(q. · k1 ) k .. I.-'_q! _
e4m(2E - kl. )k~.
.:..
4Eqtoq . • (P. · /.-.) (q+ . .1:2 ) I\/ (-, + ; + ,-.-,-}
(2E
,
[kl- A:;.q_+-]'
4 e~ Eq+oqi.-.J..
_
[kl_q.+q!]1 kl+ kHk~. q ....
kH)(P_ kd(q... · k.) 0
,
4 c~eq.o [k. _ kl_q+_q _ ~]' (2E - kH)(pt . kl )(q .. 1.-1 ) k.+~+
,
0
M (- ,+ ; , 1- ,-,-) _
_ e (2q.o + J.~+ )q-- q:.J.. [kl_.I0_Q+..-] ' (P. · kl)(q. · 1.1) kl+k~+r-_ 4
e4 (2q+o j. k1+} q. dp- .(:1)(9+ ' 1.-1 )
[kl-A1_qHq~ _]1
0
kH~+
M (- ,I;-,-,-, +) -
M (- ,-; -I, " , -. - ) -
,
M(- , -;+,-, +, -)
_
_
_ e4 m{2E - kl.)k;'.Lq_.J..Z,;_ [ ..1.:.-1.-2+ ]' 2q+09__ (p+ . kd (q+ . k2 ) k .. k~_q++ q:+ 4
e m(2E-kl+)k;. q.Oil-+(P. kd(q.· 1.-1) 0
[(9 .
q_ l 2k 1 + o
kl
,
] '
'
p, S/iMMA flY
:J Ulj
or 111m
rOILMlJl.Ali
M(-,-; +,-,-,+) -
M( - - ' - -1- - -\-) ,
1
,
,
)
2e'q+o(2£ - k,+)q_ .l Z+_ [ q__ (p+ 'k.)(I(+' k,)
!VI( - , - ; -, -, +, +)
q_+(p+ - k,)(q+ - k,) 4
-
M (- -" )
,
1
,
[8 (q+ , q_lk'_k;_] '
__ e'q+o(2£ - "1+)
e mq_.l
]1
k 1_ k'2-
kHk,+
{q+Oq __ ki.l [k:+k,_q++]!
- (p+' k,)(q+' ",)
!:,_k;+ q~+
2£'
+ - -) 1
,
4
e mQ+01f __ klt
, [I:;-I(+-q-+] '
2E'q++(p+ ,k,)(q+' k,) l\{( -, -; -, -,
+, -)
k,+
e'(2£ - "H)(2q+o + k'+)q_.lZ';' q__ (p+' k,)(q+' k,)
",_ k,_ ]t
x
[k,+",+q++q:+
_ e'(2£ - k,+)(2q+o + k2+) [2(<}+ ,q_)k1_k; ]} q_+(p+' k.)(q+' k,)
k,+-k,+
!J. 19. ~ I r -
- , ,,~,, -
"f "f (N() Y, -I·:X(.'/IA N( .. ~:: III
Id( ~ ,-j - , - , -,+)
I
0)
=
A'/(-,-;-,-,-,-) _ e4 E(2qtCl
1- 1.:;+)
1- +(P+ . ktl(q+' kz )
(9.235) IIlIpolarized squared matl"jx element:
[MI' =
e 8 k J _k2_ [4(q+. 1-)2
2k1+k2... Q+_q_+[(2£
+ q~+q:- + q! _1:+ 1 ~~I+)(P+' ktl(1+' k1 )]2
(9.23 61 IIllpoiarized cross section: ();4).;1 _).;12-
da =
[4(,+ .q_I' + q'++ " __ + q'+__q' _I
m'k'1+ 1 x [4E~+(')E-k J.,.' I~+ 4£2k _ 1 w
9. 19. 7
k~ and k~ nearly parallel to p+ and Q-, resp.
The doubly primed quauiiiies k2:i and klJ. are cvaiua~cd in the roialed frame where if.- determines the positive z-Il-xis (see also Section 7 A.3). The four· vector q in eqn (9 .239) is obtained by applying a space reflection to q_. Definition:
(9.238)
9. SIIA/MAlty or (J,:J) rOItMIILA'"
Photon
polal'i7.a.iion~:
4 -
N, l/, Ii- A(l ± 1'5) + Ii Ii- 12(1 'f 'Y5}],
N,-'
(9.239)
Nonvanishing beJicity amplitudes:
M(+,+;+,+,+,+) 1
e4 Eq_o [32(q+. q_ )kJ - k~-l' q_+(p+ . k,)(q_' k., ) kI+k~+
Jl![(+,+;+,+ ,+,-)
e 4q.t.l
M(+,+;+,+, - ,+)
M(+,+; +, -,+,+) [
kk '1- " 2+
]t
:J.I~.
,j,. - - . ,,! ,. - r r (NO Z· f:XCIlIt ,V(,' ~;;
M(+.+:-,+,+, - )
(:~ IJI(:lq_u -
-
'10
I UI
:WIJ
+ k~+)fJ __kL.l 11: .l [k U .k2'_ 11++
'J t:;~(Jl+' kL)(q . k2)
e~m(2q 0 + k~+ )kl+ -
A1(+.+: +,+,-,-)
M(+ ,+:+,-,+,-)
4E1 q_ ... (Pr
-
[ "." ];
-
c'lmQ_olJ+ _ k U .q+. l . L + 2 _ 2E1-(p+' kd(q_ . k l ) kl _"'11q++q _+
[k~l_q~ _];
-
2El(p+ . k1 )(I1-· kl
-
ehI1EkLI1.: .!,Z_t [ k1 ·· k!ft ] Q-01t- (P+' kl)(q . k 2) kLtkLq++IJ_+
-=
)
k!:+q-+
e~ mEk.!{+ l1-oqt -(P+' kd(q_· 1.: 2 )
M( f,+;-,+,+,-l
M(+,+;-, -, T,+)
k!{+
=
cimq_okJ+
Iv1 (+,+; +, - , - , +)
kl_k!{... rf!..l.
[k!f-q++q'!._]!,
· kJ)(q - . k,)
'
, ,
[2(q+ . q-)kl - 11 kl+
_
'
.
1'
:1111
V. ,'iIIMMAUY Of (JIm fO/lMlfl,Il I,'
M(+,-;+,-,+,+)
.111(+,-; - ,+,+,+)
i11(- , +;+,+,+,-) -
M(- ,+;+,+,-,+l
M(-, +;+, - , +, +)
.111(-,+; - ,+,+,+)
M (+, -;+, +, - , - )
!U!J. ,.I r " ._, ,. - ,;" ,.")" (NO Yo ·I~'.\·( ·'!I A N(.' f.:,.
.'1"1(+,--- ;+ , -, !-,-)
_
4
c ('21I_o
III
I
0)
:I! t
+ J,:~t)/I--'1-±.
(p+' kl)(q_' J..'~)
e1 (2q_n + 1.:;+) q-t{P+' k,)(q_ · k2 )
,
[kl_k~_q++]' kl+k~+q~+
[kl_k~
q++f- _] I
kl+k~+
,
"" e q_o(2E - kl+)9--9-.1 [kl_k~_q++]' E(p+· kt}(q_· k2) kl+l4'+cr-+ 4
" (+.-;+.-.-.+)
{!4q_o(2E - kit) Eq-t (pt . l:.)(q_. k~)
[k l _k 2 q++q~_]l kl+kl'+
111(+,-:-,+,+, - )
M( +.- ; - . +. -, +)
_
4
,
_ e (2q_o + 1.:1+ )9+.1 [kl_k!f_q_+] ' (P-t . 1.:1 )( q_ . k2) kltk;+q++
..:4(2q_" + )1;+) [kl_k!f q+_q_+.] 1 (Pi . kd (q_ · k l ) kH k!1+ M{+,-;- , -, I-,-'-) =
M(-,+;+,+ ,-,- )
=
{'2E
(2R
:11 ~
,V. ,l'IIMMAIIf' Of (!IW 1,'OUMUI,A8
[I.:
_ e'(2 q-o + Ie"H. ). 'ItA. t- k" 'J- q-+ (p+ .1.: )('1_' k,) k,+k!f+q++ '
M(-,+;+,-,+,-)
r 1
1
-
"(2q_O + k~+) [kl_k~_q+_q_+]' (p+ . k , )(9_ . k,) · kl+k,+
,.
e q+.l.
M( - ,+;+, - ,-,+)
(2E - k , +)(p+ . kd(q- . k,) -+ { - 4Eq_0 [kk"9]i k k" 1- 2-
X
1+
2+q++
2 ..
1/'"
1I} '
'.11
m leU ka Z_+ [ k'+"H +2 ' kk" q-09+1- ,_Q++q- + e'q_0(2E - k ,+)q __ q:l. [ I.: l_k~_q++ E(p+ . k, )(q_' k,) kl+kJ+9~+
M(- ,+:-, +.+, - )
l'
2
1
1
-
e'q_o(2E - 1.:1+) [kl-k~-q++ q~-l' , Eg_+(p+' I.: )(q_' k,) k,+k!!+
r 1
M(-,+: - ,+.-,+) -
e'(2q_o+ kJ±)q--9:1. [k 1_kL9++ (p+ ' kd(q- ' k,)
kl+k!f+q~+
e' (2 q_o + k~+) [kl-k!f_q++~_ q_+(p+. 1.: , )(9-' k, ) kl+k!f+
r ,
c'm (2E - k ,+)q __ k!fl. q:l. [ k!f+q++ - 4Eq_o(p+· kd(9- ' k,) kl+k!f_q:+
M (- ,+;-,-,+,+) -
e'm(2E - k,+)k~+ 4Eq_oq_+(p+ 'k,)( q_ ,k,)
l'
1
[kl-QH9:_]' 1
kl+
1
e q-oQ-J. 1- 2- ++ ], 2' [kk"q' (p+ ' k,)(q_ , Ie,) k, +k!f+q~+
M( + , -; + , - ,-,-)
r 1
M (+ ' -'1 -
l
-L ' I
,
-)
-'-
e q- o 1- 2-q+ 2' [kk" 3 +q- q-+(p+ ' k,)(Q_ . k,) l kl+k!f+
-
2e"q_oQ+J. k1 _k2_q_+ - (p+ ' I.:,)(q_, k, ) [kl+k!f+q++
rr 2
1
-
c 9-0 2'
,- , 9+_ q-+ [kk"
(p+ . k,)(q_ ' k,)
kl+k!f+
,
, 9. f~J,
•. f , - _
,.-j,,- 7 'Y (NO ;t,·I'XCII A N(n:; u' /(1)
M(-~,-;-,- , +. - )
=
NI(+, -;- ,-, -, +) -
M(- ,+ ;+,-,-,-)
M(-,+;-,+,-,-) -
M (-,+;- ,-,-,+) _
A1(-,-;+,+,-,-) _
M (-,- j + ,-,-, + )
:11 !J
·"
:1 14
.9. SliMMMIY OF qEV fOaMIHM,'
r l.
M(-,-;- ,+,+,-) =
_
e'Ek'" Tn uq+.l 7," '_+
k 1- k"Z±
[
k,+k~_q++q_+
q-oq.1--(p+ . k,)(q_ . k,)
l.
M(- , -;
,4mEk'; > [2(q+. q_)kl_] ' q-O'l+-(P+ . k.)(q _ . k,) k,+ e4 mq_oq±_ k"11. q±.l. [ k 1+ k"1-
- ,+,-, +)
]}
- 2E'(p+ . k,)(q_ . k,) k,_kq+q++q_+ l.
-
e'mq -0 k1+ [k"1- q3t- ]' 2E2(Pt . k, )(q_ . k,) kq+q_+ 40 Eq-oquZ_+
"r
k,_k,_ q.l--(P+· k,)(q_· k,) kl+kq+q++q_+
M (- , -;-,- ,+,+)
4·
•
[
l.
-
e'E'I_o [32 ('1+ . q_)k,_k~_]' q+_ (p+ . k,)(q_ . k,) kl+k~+
,
l.
c'm(2q_o + k't)q--kil.q - .l. [k'+k2'-q++] ' 4E'(p+ . k,)(q_ . k,) k,_k,+
AI(- , - i+,-,-,-)
-
l.
[k'-'I++q~ _ ] , 4E'q_+(p+. k,)(q_ . k,) k,+ e'm(2q_o
+ k,+)k,.
•
c mq+.L
M(-, -; -, + ,-,-) -
(2E - k,+)(p+. k,)(q_· k,) X
21. Z·- t [ {2E'k'" k 1- k"1+ q-Oq.1-k,+kq_q++q_+
r
+ q-ok;.l. [k1+k,_Q_+ ] E
k,_k~~q++
t}, l.
M(- ,-;-,-,+,-) -
2<'q_o(2E - k1+)QUZ:+ [ k,_k~_ ]' q+_ (p+. k,)(q_· k,) k,+k,+q++q~+
,
-
e'q_o(2E - k1+) [8(q+. Q_)k'_k'_] ' q_+(p+ . A,,)(q_ . k, ) k,+k~+
9. / 9.
~ ..
,.- ...
t ~ r-11
(NU r.-I~·X GII IIN Ut.;;'1I
M(-,-;- ,-,-,+) =
I
II)
31 ;j
(21£
M(-,-;-,-,-,-) =
IJ11[JO],1,rized squared matrix element.:
k!! ['(q+' (/- )2.:: 1 eIlk1-:_ .. q_+q__
+' , I q+_IJ_+
U' I +k;+Q+_q_+[(2E - k l , )(p • . kd(q- . kz)j2 2
X
[.E' + (?E - k1+ )' + 4E m Ait+ k 1
X[4q~ +
1
w
(2q_o -l
l
_
k~+)l + 1~~i~J
(9.2" )
t1npolarized cros! sedion:
dd _
9. 19.8
-
-
kJ a.nd k'2 nearly parallel to
p_ and ~,
resp.
The prirr:e<] q:,,\n liticfI k2:1: iIond 14.1 are evaluated in the rotl\ted frame where 9+ d e le~milleS the posilive z-axis (II~ 1\1110 S~tioll 7.4.3). The four-vector q
,ij,
in C'I" (9 ,211) is obtained by applying Definition:
it
SIIMMAIIY
or (JIW
f 'Ol/MUI,AI>
spaw I'd!edion to q+,
(9,243) Photon
pola.dz~tions:
(9,244)
NOJlvanishing helicity amplitudes:
M(+ . +;+,+.+,+) _
40' E'I+o"" l
Z+_
[
1
k1+k; ], q., - 'l--(p- ,k.)(q+· k,) k, _k,+'I+-I-'I_+ e'Eq+o [32 ('1+ . q_)kI+~ ] t q+_(p_ . kd('I+' k,) k , _k,+
kf(+. +;
+. +.+. - )
e4
(2E - k,_)(p_ , k, )(q+ ,k,)
x {SE2 q+ O«( l.Z+- [ k,+k, _ ]~ q-k,_khq++q~+ _ m'k;.1k;".1QH [k1 _k,+q_+] 4 Eq+o kI+ k, _ '1++
l'
1
M(+ , + i + , -I-.-.-I-)
2e'E(2'1+0 -I- k2+)q:l.Z+- [ k,+k,_ CI _ _ (P_' k.)(q+ ,k,) kl_k,+q++q~+ e'E(2q+o+ k,+) [8('1+' q_)k, +k, ] t q- +(p-' kl)('I+ ' k,) kl_k,+
t}
•
!1./!!. ,.1,' . • t· ~ ~-
..,
'"f (N U /, ./;')((.'11 II N(: N; ".
!
U)
M{+,-;+,-, +,,~)
M(+,-;-,+,+, +)
=
2E2k~J..q · J.Z+x{
q+oti--
[
11
kHI4+ k1 -k1_q++(I:+
+qtokiJ.q· ~.l. E
[k _k; 1
k1+k~+q++
4
11-/(+,+;+,+,-,-) -
kl+k;_ 4t: Eq+Of/:J. Z+_ [ (p k)( k) k k' ,
q--
- ' "1
4
-
M(+,+;+,-,T,-) -
,1tI(+,+; - ,+,T,-) -
M(T,+ ;-,+,-, +)
=
q+ "
~
1-
q-+l
2+q++q_+
,
l'
e Eq+o [32(q+. q-lkak1-l q_+{p_ . k l )(1+' k 2 ) kl_A~+ '
l
I}
.'}, SI I AlMAIl)'
or 111m
fOItMIJ/.;ll:
M(+.+;-.-.+ .+)
M( +.-;+. +.+ . -)
e'mq+okll. q" .1 Z+_ Eq+_q__ (p _ . k, )(q+ . k,)
M(+, - ;+.+.-.+)
M(+ .-;+. - .+,+) -
,
M(+ ,-;- ,+,+, +)
M(-,+;+ ,+,+ , - ) -
Jll{( -. +; +,
- , +, + ) _ _
2e~q+oq~..L q+_(p_ . kt)(q+ . k,)
[kl+k~_q~+] t k,_k~+q++
+]t
.,.-_2 :;.:c,,'9: +0 [kl+k; q3 (p_. k,)(q+. k,) k,_k,+q+_
.'1,1 9, ~
I" ,
-~, I . ' "(..,
( NO ;/'.I':.'i(,'1/ tI,.U r'~:; '"
_
tJ)
,'m
M(+. -; + ,+.-,-) -
M(+, - i+,-, +,- )
I
<:,1'1+0(2£ -kl_)I]_i. [kltkLilt 1]1 Eq+ _(p_' kd(q+' .1: 2 ) kl_ k;+II __
e~q+o(2E -
[kl+k~ - q~tq--l ~
kl _ )
EII,_(p_, kd(q-· k1 )
kl_S.~+
~
Af(+,-;+,-, -.+)
_
,
_ (::(2qI0 I' ~t)
[kl+k2_Q;tq __ ]'
IJ--(p- . k\)(I1+ 'k~)
M'(+,-;-,+.+, -) _
_
~~ (2qtO + kl .)qt.1 (p_ . k.)(q+ ,k2 )
kl_.I.1 ...
[kli ~-q--l! kl_~+qll.
M (+ ,-; , +, -,+) = X { - 1E q+vI/+J.
-[*H1<;_q-+]; ,. k' "1- H9+-
/1. SI/MMAIlY or
(l/m rOIlMULAI:
e'm(2E - k,_)k;J.'I-J. [k'+k~.qi+J! 4Eq+oQ+_(p_ . A:,) ('1+ - k,) k,_k;_q_+
M(+,-;-,-,+,+)
1
e'm(2E-k , _)k;+ [kItq~+q- -l' 4Eq+oq+_(p_ - k,)(q+ - k,) "_ 1\11( -, +i
k )'-"
[k k'
e4 'm (2E - J - va q"-.1 1+ 2+ q3t+ 4Eq+Dq+_(I'_ - kd(q+ - k,) k, _k;_
+, +, -, -)
]1
M( -, + ;+ ,-,+,-)
+
m'kiJ. .;".1. '1-J.Zt _ [ k l _ k;+ 2q+oq-k 1+~2-q++q-+ u 3 1
M(-,+; + ,-,-,+)
_
_ . 4(2'1+0 + k1+)q+J. [k,tk;_q_+]' (p_ - k,)(q+ - k,) k, _k1+q+t
1
.'(2'1+0 + ~+) [kl tk; _fJ+_q-t ] ' (p_ - k,)(q+ - k,) k,_k't
M(-,+ ; -,+,+,-)
1
M (- ,+;-, + ,-,+) -
M(+,-;+,-, - ,-)
.·q+o(2E - k,_)q:J. [k'+k; - q~tl' Eqt- (p- - k,)(q, - .-,) k,_k'tq-t. . , .'q+o(2E - k l _ ) [k, tk; _q~+q ___]' Eq+_(p_ - k,)(q+ - ") k, _k;+
]!}
'
~J
I!J.
r+,.- .• ,.., ,.-
r" (NO JI, . f:Xf:/1 A N( ,'/0;,. '" /11)
'1"1(--.+;-.-, +,+) =
11-,( .... - ;-,+.-.-) =
('2B
2e~q , oq+J.
_
[kL+k2_~~]!
IJ+.(p-· k,)((I+' k2)
2e~9_o
k L_k~_q++
[~l+.I:,.q~+]!
(p- . kd(q+ ' .(.'2) ~'L-~+q+_
111(+,-; -. - .-.+) _ _ -
e4 mk'L9_J.. 29+09+-(P- . kd(9+ . k2 )
[kL1 k;+9i+]! kl_k1.q _"1'
4
_ _ c mk1t [kHq!tq]; '2q+olJ __ (p_ . kl){q+· k1 ) kl_
M(-.+;+, . -,-) ",.
,
4
M( ,+;-,+.-.-) -
.
~
2e q+09 -q:.1. Q+ -( ll-' .l:1)(q+· ~'l)
2e~q_o IJ+ -( p-· k,)(q+' k2 )
M(- ,+; - ,-.+,-)
A1(-.+ ; - ,- .-.~ )
'-
~
[kl ... k~_qtt]' kl_~I(I __
[4-1 t ~_q++ q~ kl_k~ +
_]! k2- ]1
e4mq+okj.1. IJ_.1.Z.+ _ [ kl Eqt - q__ (p- . kd(q ~ . k1 ) k 1+k1+q+d .
!J , .'i If II MAll Y
M( - ,-;+,+,-,-)
(ifi '1'-,'fJ 1"0 nM 111. A Ii
=
em 'Ek"2.lQ-J. Z'±
M( - , -; +, -, +, -)
~_
[
q+oq--(p-, kt)(q+' k,)
k 1+ k''H k,_k;_q++q:+
e4 mEk;± _ [2(g+' q_ )kI+] q+og-+(p- 'kd(q+' k,) k1 _
1~ ~
M(-, -; +,-,- ,+)
NJ(-, - ;-,+,-,+) _ 4
e mq+okl_ 2£'Q+ _(p_ ,k,)(q+ ' k,)
M\'- - ' - , )
)
)
T'
[ k~-q~+q-- l t k,+
4e'q+oQ-.L Z ,e_ [ k, +k,_ q__ (p _ 'k.)(q+ , k,) k,_k,+q++q:+
, +)
l' ,
.4Eq+o [32(q+, q-)kl+k'-l~ q_+ (p_ 'k.)(q+' k,) kl _ k,+ M( - 1 -', -
1
~I I
e'm(2q+o + k,+)k'.Lq:.l 4E2q+_ (p _ , k,)(q+' k,)
,- , -)
e'm(2q+o + k't)k,4£'q+ _(p_ ' kd(q+ ' k,) 1
,
j
q~+l t
kI+k;+q_+
[kLq~±q-- l t k,+
2.<E(2q.,.0 + k'+)q_.LZ';' q__ (p _ ' Ie,) (qt ' k,)
M(- -'- - + ' - ) ,
[!c, _k,
,
_
e'E(2qto+ k,+J [8(q+,q_ l k1+k;]l q-t(p- 'k, )(q+' k,) k, _k,+
AI (- . - ; ~. -. -. -) ~
f: 41/1
,+ ",,
"( ,;-;,,,C-,.---.. ,,)(':--',.~),(,
V-'
., _
<,
x {8 E'Q+OQ_..L Z.+ _.
q_ _
[
)
k,t +-";
3
]'
k1_kHq-t +q _ t
_ m2k1.Lk2..LQ+J. [k i _k 2+1J'_+] 48q+o
M(- ,-i-,-,-.-)
_
k1+"1_9 ,.+
j}
4
4e Eq+oq .LZj [ kak2_ ]i ql - q__ (p_· "11 ((/+' kz) ~'1 _ k;+qH q_ +
,
e £q.o [32 (9+ . q_)kl+k'2 _], Q+ _(p_ . .4: 1 )(9+' k2 ) kl_.k~+ 4
(9.245) Unpolari zed squared m a lrix e lement:
rM I~
=
e/l;k1+ kL [4(9+' q- ? +q~ .. q:- + 2kl_k~_ q __ q_+[ (2E- kl_ )(p_· k1 )(1(+' kz)p
qt-q:+J
Unpo lari 7.cd cross section:
2 2 m ' k'1- ] [ :2 ,'I m 2± x [4£ . . ,. . (2£ - kl _) + E~~' 49+0 + (2q,.0 + kH ) + :2'k"k' ] 'I 1+ 4q+0 2_ I
(9.217)
;!~1
9.19.9
9. SliMMAIIY 01'
Q',') FOIIMIJLAIi
A~ and k~ nearly parallel to j;_ IUltilj_. res!'.
The doubly primed quantities k~± and k~L are evaluated in the rota ted frallle where ii- determines the positive z-axis (see also Section 7.4.3 ). The fourvector q in eqn (9.249) is obtained by applying a space reflection to q_. Definition:
(9.248)
Photon polarizations:
,l~ ,.. ,-1 1',
N,I;' /1- A(l ± ,,),,) + A A- h (1 'f ")'s) ). q'- 0 [3'1k'" - 2+ k" 2- )t
(9.249)
.
Nonvanishing helicity amplitudes:
M(+,+;+,+,+,+) -
2e'E(2q_o + k"t)q:.l.Zt q+_q __ (p_ . k.)(q_· k,) [ e'E(2q_o + k!ft)
[8(q+. q_)k1+k1_]!
qt-(p-, k.)(q_· k,)
Af(+ ,+; +,+,+,-)
]t
kk" 1+ 2-
k._k!f+
0'
qt-(p- . k.)(q_ . k,) x {4Eq_Oq:.l. Z+_ [ k,~+ q-kl-~tq++q-+
k,_ ]!
+m 'k"l.l k'''q 2.1 -1. 8E'lq_o
[k 1- k"2+ q3i t kl+kq_q_+ l
M(+,+;+,+, - ,+) -
40' Eq-Oq:LZt- [ kltk,_ ]' - q__ (p_ . k.) (q_ . k,) kl-k!f+q++q~+ l
e' Eq_o [32 (q+ . q_ )kltk!f_j , q_+(p_ . k.)(q_ . k,) k,_k!f+
l!}
'
!1./9. ,+r- . • •'+r-"),1' (NO X,J.;XCIIANfi1:.. '"
~1( ·~ ,'l'i+,-,+t+)
-
I
U)
'1t-(j,- . *"11(r/_ . k~) x {Ek!1.J..Q - .J..Z+_ [ kl+k~+ ] t q-Oq__ k\_k~_q++q_ +
_ q-oL'i..Lq-..L. 2E1
M(+,+;-,+ ,+,+) _
[k1_Bf
q;t]l}
£:1+£:1+q-+
e4 mq_oki.J..9+.J.. [l.:l_J.1_'1++'1_tjl 2E2q. _(p_ ' 1.:.)( 9_ . k,) k\ +k~'+
c~mq.oq,+kl_ U;:q •• {p _ 'kd( q- . 1.:2 )
[J.-;_'1+-9_+] t k{+
M(+.+:+,+, - ,-) _ 4
e q 0(2£-1.: 1_) Q+-(1'-' kd(q·' k2 )
M(+,+;+.-.+. - )
_
_
, [8(9+ .q_) klfk:f_j' kl_kq+
e~:n(2q_o -+- frI-lt )J.'i..LQ·.1
4E2q+_(p_ . kd(q_
.~)
[k,-ki_tf!.;.]1 kl+k'-l~q_+
M(-:,+ ;+,-,-,+) .,.
M(+,+:-,+ , 4, -') -
E(2£
M(+,+;-,-,+,+) -
/ .J..Q+l -e4 m'ki.J..J.-1 4Eq_o(2£
kl_)(p_' kd(q.· k'l)
e"mk'k ,_ "1+ [9+-q-+' I" (2£ k,_ )(p_· k,)(,_· k,J'
'£'. .
[J.-l_k i+Q Kt
+]!
~I..~/_ 'l++ ~
.'1 . SIIMMAII.Y
or (if:1J
!'O/fMIII,A I~
M(+,-;+,+,+,-) 1
[ kl+q~+q_-]' k
. 'm{2E - k 1_ )k~t
l
e"m ( 2q_o
M(+, - ;+,+,-,+)
' Z;.t ·t+ k" '2+ ) k l.Lq_.l
2Eqi_Q __ (P_ . k, )(q_. k, )
-
/. [ ~> ....::..! k>1.::-..:."'2,-=-_ 1_" ], kl+k~+q++q_+
e'm(2g_ 0 + k~+)k, [(g> I>' q_)k~_] t Eqt_(p_ , k,)(q_ . k,) 2k~i ' e'g 0(2E- k J -)q-.1 - Eq+ _(p_ . k,J(q_· k,)
M(+,-;+, - ,+,+)
,
_
ql+]t
[k1+k~
k,-k~tq-i
-
"9_0(2E - k1 ) k,tk~" 3 q:;,+9 [ Egt- (p- > k,)(g_ k,) "l-k!.i+
-
e'(2q 0 + k~+)q+.1 [kHk~_q-+ l ' (p_ >k,)(q_ . k,) kl_k~>t9t+
-
' «2q_o + k~t) [kltk~_q+_q_+]' (p_ . k,)(g_ - k,) kl_k~+
>
1
iV[(+ ,- ;- ,+,+,+)
1
M ( - , +; +, +,
+, -)
e'mEk~l q;'.1
-
q _-o(2E~- k,_ )(p_ k')(q_ ·
-
e'mEk~+ [k H q+- 9_+] q_o(2E - k1-)(p- . k,)(q_ . k, ) k, _
>
4e' Eq 09+.1
M( - ,+;+,-,+.+)
l-2E-' - k,_)(p_' k,)(q_. k,)
k,)
[kltk~_ q_+
kl _ k~iq++
1
1
,
l'
1
[kl+k~_q+_q_+] t
40' Eq_o
M(-,+ ;- ,+,+,+)
1' kJ_k~_ q++ _
[kl+k~+q_+
k, _ k~+
-
(2E - k, - )(p_ . k, )(q_ . k,)
-
_ . 4(2q_o + k~+)q:.1 [kl+k~_ q~ qt - (p- , kl )(q_ . k,) k,_kHq-t
-
.4(2q 0 + k~+) '1+-(1'-' k,)(q_' k,)
'
+]'
[kl+k~ q~+q k1_k;t
1
_]
~
,
""(+.-;+,"'1",-,-) ..
M(+,-; +,-,+,-) -
M(+ , ;+,- .- ,+) _
M(+.- i - ,+,-,"") -
M( .,.. ,-;-,-,+,+) _
q+_(p_ . kd (q_ . k~)
fI , .... IIMMM/,y
01' q/';U FOltMU/,At'
M(- , + ;+, +,-,-)
M(-,+ ;+,- ,+,-)
M(-,+;+,-,-,+)
LH(-,+; - ,+,+, -)
M(-,+;-,+,-, + )
M( - ,+;-,-,+,+) _
q-oki.d- ~Z';'_ Eq__
e (29_0 + k~+ q+_(p_ ,kl)(q_, k,) 4
M (+,-;+,- , -, - )
)q-.L
[
"1- k~
kl+k~+q++q_+
[k1+k~_qt+ l i k)_k~+q_+
i} ,
1
'
!J.I!J. ,· J r ·
" ' !"- 1'r (NO i:· /I X( !JfIlNC.'I;';",
I n)
,
_
M (h - ; - ,- ,-,+)
e4mEk~.l.q+J. [1.:1+k2'tll_+] , q_o(2E -' kL_)(p_· 1.:1)(q_· k2) I.:l-I.:~_q+± 4
e mEk2t 9_0(2£ -
~,,_}(p_,
[k +9+- 9-+]! 1
k, ){q_ ' k, )
k1_
M(-, +;+,-,-,-)
,
4
!H (-,+:- ..... ,-,-)
= _ e q_o(2E - kl _) I} ' _ [kltk;-9it]' Eqt_(p_, kl )(9_ ' !'2 ) !'I-kl'+9-t
e~9_o(2E -
k 1 _)
[kh.L1 G.t9u]!
Eqt_(p_ · kd(q_ . k7)
M(-.+ i- , - .+. - ) -
kl_k'{t
e4 m(2(j_o + k!J, )kjiq l.Z';'_ 2Eq+_q__ hJ_ . k.)(q_ . k1 ) [
k1- k"2-
]'
e4 m(2q_o + kfbY'L- [ (1t. q_)k'{_] I Eqt-(p _ . kl )(9_ . k2 ) 2k~t M( - .+; .• -.-. +) =
e4 m(2E - 1.: 1_ )ku 9:l. 4Eq_oQ+_ (p_ . kd(q- . k~ )
[kli k~.
]1
=
M(-, - ;+,+, , -) _
1
_ e'm 1.:1 1. kl 19+.L [k1_ k!f+9- . . 4£9. 0(2E k 1_ )(p_ . I.:.)(q_ . k1 ) kltkl'_qH c4 m 1 k l _k 2+ [9+-q-+l~ 4E._,(2E k,_)(p_· k,)(q_ . k, ) '
,
j,
I),
e4r/l.q_okl.LfJ~.L
M(-,-;+,-, -,+)
[kl-k~-q-+l! k1+k~+q++
- £(2£ - k , -)(l'- 'kd(q - " k,) _
M (- ,-; - ,+,+,-)
SIIMMAIIY OF QI)n fOIlMII/,M)
-
[k~-q+-q-+ l!
,4",q ok , _ £(2£ - k, _)(1'_ ' kd( q_ ' k,)
k~+
k 1+ k"2+ ] e, m (2 EJ - k 1- )k'"2.1 q-J. Z _·1__ ( - 2q-o
e'm(2E -- kt-)krt ( (q+, q_) kl+]' q-oq+- (p- 'k, )(q_ ,k,) 2k, _
M(- , -;-, + ,-,+)
,
e'm(2q_o + k~+ ) k,_ [k~ q}+qu] ' 1£'Q+_(1'_ " kd(q- , k,) k~+
,11(- ,-;- , - ,+, +)
_
2e4 q-0 (2E - k 1 )q - .1 Z't - [ k 1+ k":2 g+ _q __ () _ , k,)(q_ ' k,) k,_k~+q++q _+
1
1'
1
."q_o(2£ - k,_) [8(Q+' q _ )k, +k~ ] ' k,_k~+ q+_(p_, kd(q- , k,)
M( -
, -"+ , , - , - -) )
em 4 g+_ (p _ ' kJ)(q- ,k,)
{k' [k1-2-q+ k" + q O l tq
Q-oq--
4. 4£q_oq J.Z+ _ ,[
3
k 1+ k"2+
[
k, _ k~_q++q_+
k,+k~
- q-- (p- ' kd(q- ,k,) k, _ k~+q++Q:+
.4 Eq_o q-+(p- ,k,)(q_ ,k,)
[32('1+'
]
kl+k~+q_+
2E'
Ek~2J..q-.l '" Z'+
M(-,-; -, -,+ , -)
)
l' 1
q-)k' + k~-ll
k, _kr+
t
1}} '
!
"
hl(-,-;-,-,-, +) -
(h-(,'- 'kl)(q_· .1.-.)
x { 4Eq_Oq_J.Z'; _ [_ k,~+kl_ 1__
11( -, ;--. -,-,-)
kl_kHg++q_+
11
c4 E(2q_o + k1-t)q_J.Z,; q+_(p_' k, )(, _· k, )
=
e~E(2q_0+ kqT ). q+_(p_ . k':l )(q_ . k 2 )
[8(q.... q-~=I+kl11 ~_k 2+
(9.250)
lillpolari:r.ed squared l:::latrix clement:
Ilnpola.rired cross section:
X
[4E1
+ (2E -
xo~ (P... + p- 9.1 9.1 0
-
.1: 1_ )2
+
n;;i-l
4
1+
q l - q- - kl - k, )
-
[4 Q::' 0 + (2 q_0 + k;+)2+
tPii+ Iflq tPkl rFk2 1+oQ-o
k\ and k2 llca,rly parallel to
k
k
10
'w
ii+ and if-,
.
~2kI~
4q_0
2-
1
(9.252)
resp.
The primed (doubly primed) quantitie.~ k;:i: and k;J. (k2't and k2.l. ) Me eval uated in ~ h e rotated frame where ~ (q- l determines the positive .i.-axis (see also Section 7.4.3). The fo ur-vector q (q') in eqns (9.25'1) is obta.ined by applying a space reflection to q+ (q_). Defi l.l itioll : (9 .253)
y,
3:12
,\'I IMM~
flY 01-' QI>IJ FUUMIJ/,AN
Photon poLarizaLioJlH:
, 132k"2t k"2- It q-o. No nvanishing helicity amplitudes:
kf( t. +; +, +, t, +)
kf ( t , +; t , t, +, _.)
M( +,-I-;+,-I-,-,+)
M( +,+;+,-,-I- , +)
M(t , t;- , +,+,+)
,
(9,254)
9 J9
,1, - _. ,.1,,- ., ')' (NO 7,.,.;,H 'IIA Nm,';
II'
I
U)
q_ ,(2q_o + k;+) Cq+· k,)Cq_· k,) I 28(q+ 'q.)k:.k;' X
[
k'1+ k"1+
M(+ ,+;-,+.+ ,-) = I!.1- "2. '."
];
[
'u e•mq_oll;l'"
M(+,+;-.-.+.+) =
etlnlkl±~t
q+oq_ t(2q_o + k~+)(q+ . kl)(q_ , Ie,)
M(+.- i+, +,-.- ) =
=
, [(qt ' q_)]' 2
[U""
O,]1 e•In C ""'1+0 +k'I t )1·"· uq 1. .... , - "'2tqt+ 4Eq_cq+_(qt' .li)(q_· k,) k1+kl'_q_+ e' m(2q_o + ki+ )kl'+ 4Eq_cq+_(qt · k,)(q _· k2 )
[J.:\_q!~ q__ ]1 kit
!I, ,\'IIMMAIIY Of>' liNII rOnMULAI;
M( +,-;+,+, - ,+)
M(+,-;+,-,+,+)
M(+ ,-;-,+,+,+ )
M(-,+;+,+,-/-,-)
M(- ,-/-; -/- ,-/-, - ,+)
M(-,-/-;+, - ,+,+) _
,"q-o(2Q+O + k;+) [k; E(9+' k, )(9-' k,)
M(- ,+; -,+,+,+) .4 q+o(2q_o
-/- k,+) [ k; Eq+_(q+, k.)(q_ ,k,)
9. /9, ,.'O"
- .r~' - 'T'T
( /'10 Y..f:,\'(;IIA/'IUf:; w III)
M{+,-i -.+.-. -) .,.
M(+,-;+, -, -.-)
4
=
4e Eq+o9_0Q_J. 9+_(9+ . q- )(9+ . kd(9- . h,) l
4c' f;Q+o9_o Q+-(9+' q-)('7I-' kd(9_' k2 ) M ( +,-i~, - .-,+)
_
M(+ , - ;-. -,+. +)
M(-.+;+.T . - ,-) -
" E(q+ . kl )(9_' k,)
[ki- k~_Ql+ k~+k;+q_ +
,
l'
[J.;-~-rl.+q--ll ..l~+I.'2+
{2,,,,_,, __ [II, e; ,1-]1 Ql- _
k~+k;+q_+
9, ",(IMMAIlY Of (J(;V f!JaMUI,A'"
M(-,+ ;+,- ,+, -)
M{-, +; +,- ,-,+)
M( -, +; -, +,+,-)
L
kj
M(- ,+;- ,+,- , +)
[ k' k" q 1 + 2+ -+
1\1 (- , +;-, -, +,+)
1\1(+ ) - 'I -I-' 1 -
,
-
,
-)
04q+o(2q_o + k~+) Eq+_(q+ ' kt)(q_· k,)
M (+,-; -,+ . - , -)
k~ q~+l'
. ,+, -)
, "( "f , -"-
=
_
]l [14'-r4+ qu]1 ['J
I 1.11'l.I· )" . l~:'±_ .." ' , , .("~'I-~~~
'JI ,Ij_ " 1_"-;1
4/.,'Q+O"+_(II-I- . I.' !l(f/_' k1) L~_k';+q_+
e~rn (2q_1,I + .1.:2'+ )J.:~ t
k~+
01£'q,01.,.- ('7+ · kd(q-·
.1.:1 )
e4 m{2(J+o + k~+).t;t 4 Eq-o(qt . kt} (q_' -':,)
[k~ _ q __ q_ ... ] ;
M(+, - ; , ,- ,+) -
M( - .+i+, - .-,-)
M (-,~; - ,
, 1"- )
_
_
e"q±o(2q_o + k~t ) qtJ. E(q .,.· .1.: 1)(9-' .1.: 2 )
,
[kl_k'i_q_t] ' ki+k'l+q_+
,
[kltk2_'1_+] '
_ e(m(2q_o + kl'+)!-j J, 9i! 4Eq,0('7_ . .1.: 1)(9_ ' k,) kl_.l.:1'+'7++ 1'!1 m (2q_O - k1'... )kl+ 4E9iO(9+' .1.:1 )(9_ ' k1 )
M (-.-; -.-r- , - , -)
k~ .~
[1.::;
e~m 2 k'· 1.1 J.:"", 2_ - .1
9+- q-+ ] 1 kr~
Z·t
,v..'ilIMMAIlY 2(., 4 '"
.M( -,-;+, -,-,+) -
OF OIW FOIIMUI,AII'
:.! 1,/. '/ . L(J_UI\'I.lf/_l/.J+_
]1
k'k" 1+ 2[
+, -)
111(- , - ,' - '" .,
e4mq+ok~+
-
[8(q+ ,q_)kU! q_+(2q_o + k~+ ) (q+ . kJ) (q_ ' k,) k;+ j
_
-
M(- ,-;-,-, +,+)
1
iV!( -, -; +, -
, -, -)
e4mq_ok~l.q _ l.Z;_ [ k;+kq_ ]' q+oq--(q+ ' ktl(q_' k,) k;_k~+q++q~+
. 'mq_oki+ [2('1+' q_ )k~ ] } '1+oq-+(q+ ' kJ)(q_ . k,) k~+ M\'- , -')
~
+ - -) ,
1
1
M (- ) -'1 !
,
+ -) ,
4C4q+ Oq_Oq_~Z';'_ [ k;_k~_ ]' '1--(q+ . k, )(q_ ,k,) k; + k~+ '1++q~+
,'q+oq-o [32('1+ ' q-)k;-k~- l} '1-+(q+ ' k.)(q_ ' k,) k;+k~+
·~ I !I
,I,
(NO l. · /o.X(.'I/ANfjf,; "'
." ' . I r"· ')"1
M( -,-;- ,-,-,+)
i\1( -.-;-, -,-,-)
I
tI)
k'k" 1_ "l_
=
[
_ _ 2e 4q_o(2q+o+
~t)q-.lZt_
q__ (q+ ' k\)(q_' ~"~) =
[
]! kj_kl_
kI_k1+ q++q~+
]1
e'9_0(2q_o + k~+) [8(9+" q_)k~ _ k~_]! fJ _,(q .. "kl)(9_ok1) k~+k1+ (9. 255)
tllll,o!a.rizt:d squared matrix dement:
[
2
,1
x 4q ... o + (211;'0 + '/:1 +)
'k"']
hl+ ~ "'1 m 2. + 4In1.0 2"A '/:' ][4q_o + (211_0 + k2 ... ) + 4 ~ k!j 1_ If-o _
(9.256) IJllpoJarized t:t"08S section :
(9.207)
9.19.11
•
•
kl and 1.:2 nearly parallel to
p_
Definilio ns:
i,j = I,2 ,
Y.
SIlMM~/{'Y
OP' (JIm
r()IlMII/,~",
_1_ [k,_ (2 J' - klt)(U,k,_ ~_X,,) C.'12 (p+ . kd
1l,(J,2)
2Ek, _k,_(2E - k ,+ - k2+) - m1k2+kiLkU/2E] + (p+ . k,) . ,
2(p+ . k2(p+. k,) [k,+k,_k a
lJ3
+ kl _kz+ka _ (k lJ.
+ kU )Z12 Z21] M
,
Un
(9.258) Photon polarizations :
N,-'
i = 1.,2 .
•
Nonvanishing helicity amplitudes:
M(+,+;+,+,+,+)
=
40' E,8,q:~ [
q__
1
q++ ]' .r-+k1+ k,_k,+k,_
4e 4 EIBd !
4
4. E,8,(2, I )q-i-.l [ 3 q:H • ]' q-q_+k,+k,_ k2+k,_
"'1(+, +; +, +, +, -)
4e'EIB,(2, 1)1
,11(+.+;+,+,-,+) 4e' E I,8,( 1, 2)1 1
2e'EB3q+L [
.111(+. +;+,-,+.+)
(q+ . q_)
-
2.'EIB3 (q+' q_)
1
[
q-+ ], qHkl+k, _k2+ k,_ q+ - q- +
k,+k , _k2+ k,_
]
~ •
(9.259)
!I, I'I,
,,+,,- .- r'c-1'7
I
(NO i1-f,'XC'II11/"'r:Kj HI
U)
M(+,+ ;- ,+,+,+) -=
M(+,+;+,+,-,-) ~
:W(t-,+;...,.-,-, + ,-)
- kl+ - k2+)1 B d [q+ _q__ kl-.kl_kHk2_1~ ,
2t:~ (2£,
2t:tr;B~(1, 2)q: .l -
[
(q+ . 1-)
2,'EIB. (J,2)1 [ -
(q+ ' q_)
q-+ __ q++k1+kt _ k2+}:2-
Jl
,j'
. q._,_+ kJ+kl _kz+k1_
M(+,+,+ , , ,+)
,
2,'EIB.(2, 1)1 [ -
M(+,+;- ,+, +,- )
q,-.-+ k 1 -..lr 1_k H k2 _
2c~EB4(1,2)q:.l
(q.,. -:q=1-~
:=
-
Al(+, + i -, + , -,+ )
(4+' q-)
-
[
l'
]l q~+.l.'l+kl_kh·kl_ q~ -l-
2e 4 EI 13.( 1, 2)lq++q __
,'
(9+' q_) [q+_q_ +lr 1-l-k 1_k1 +k1 _)'
2(:4£8.(2 ,1)q: .;. [ q~+ (q+' q_) q~+kl+kl_kHk~_ 4 2e E l 8 1 (2, j )I'1'++q- :_"-"", '
(q+ . q_) Iq+_q_+kl+kl _lv:!+k2_11
1
2c EB 1q_.l [ q~i - (1-' 1_) q:.,.].·!+kl_kHk2_ 4
-
2e 4EIB 1 Iq++q __
,
J'
M(+,- ;-,+, +,+)
~/;, E 1J, '1-1- [ -
-
('1+. q- )
26' E I Bd ('1+. '1 - )
[
L
l
q__ __
]'
q+_q_+ ], k,+k, _ k2+", _
46'£B,(2, l)q~1- [
M(--,+;+,+ ,+,- )
'1-+
q+~!:,.,k,_kHk,_
q++ q~+k1+k,_k,+k2 _
,,
4:::6:....'£= 1B:.:o.'(2"-,:.L !..: 1),-1---,-
[q+_ q_+k,+k,_ k2+ k,_ I' 4e £B;(1,2kt1- [
~-
---;--] I [ ~q_+ q++ k, +k, _ k2+ k, _
,'(2E -
"1+ - k2+)B,q:1(q+ . q_ )
[ . q~+ q" +k1+k, _k2+k2_
- k1+ - kH ) IB, lq++q __
,,
(q+. '1- ) [q+-(Hk H
k, _ k2+ k,-l'
,
(q+. q_) [q+_q_+k,+k, _k,+k, _l' 26'EB,(1,2)QH [ . ('1+. '1- )
,
q!+ q~+k,+k,_k,+ k,_
2e'EIB,(1,2)I'I++'I __
,,
('1+ . ti- ) [q+ _q_+k,+kl_k,+kz_l'
l' ,
U ' EB,(2, 1 )'IH [ q!+ (q+. q_) q~+k1+k,_k2+k,_ 26' £ 18,(2, I )I'I++q--
M(+, -; +,-,-,+)
I
+.. ]' ,
e'(2E - k,+ - k,+) IBd [ '1+ q (q+. '1-) k1+ k,_k2+ k , _
6 4 (2£
M( + , - ;+,-,+,-)
]'
~+~+~-~+~-
(q+. q-) -
, +, +, +)
q++
.'(2E - k,+ - k,+)8,q:1-
M( -,+;+,-,+,+)
M( -, +; -
,
4
kl( -, +; +, +, - , +)
l'
L
]1
l'
1
oJ
/ '1. ,. I"
M( ·I .
, ".,.
Y" (,V(J 7" f:,\'{ '/III NI N,', 111 /11)
'; - ,+...J, -)
'1- + '/1,kl+"' I_ k1 ,k , _
(1/ 1 "/)
:=
2," :1 11,{". [)I [ (q+ ' q_)
-
II
l
~,i/-://l('1..1)'I+1
I
,+-q-+ j • ~"l+kl_k2 ... k2_
,-+
4
M(+ . - ; -.+ , -.+)
-
2e £'Bz(I,2)q+-l [ (9+ 'q_) q+_k1+kl _kHkz _
--'- 2t:~ El lJ~ ( 1 , 2)1 [ q+-q-+ . (q+ ' '1 _) kl+kl_k2+~l_
M(+. - ; - , - ,+,+ )
-
4 4e E Ball+ .!. [ (/ ~ + q__ q~+kl+1'I_k2+kl
II
,
I'
1!
M(-.+ ; •. + .- .- ) 4c 4EID:JI -
M(- , + i + .-, +, -)
=-
~I
[q.. _q- .. '\;I +.1:1 _1"Hk2_ 1? 2e 4 E Di ( I , 2)Q+J.
(q ~ 'q _ )
[q+>kl +.l: I_'\;'I+1.:2_,1
-II .
4
-
M( - ,+;+ ,-, -,+) -
!
9 +
2e ElfJ,( 1, 2)1 [ 11+ q + (q+ . q_ ) kHk._~_kl _
2,' £8;(2. l)q: ,
[
(q+ ' q_)
,
,_ _
)'
qi+kl+kl _ J.q t kz_
l' ,
2,'£18,(2. 1) 1 [ M (- , +;-.T,t ,-).r..:
(q_ ' q )
q.-q-+ kl _.l"I _k2+ kl_
2f,~EB;( 1 , 2)q+_ (9+'Q- )
[
~+kHkl_kHk2
2e~E I B2{l. 2)[9++q __
II
kl ( - , + ; -, +,
-, + )
M {t , - ; t, -, - , -)
M{+,- ;- ,+,- ,- )
M {+ ,-;-, - ,t, - )
EI B, {l, 2)1 l [qt- q-t k1tk l _ k2+ k,-l' 40'
-
--
'
l' !
4e'EB,{2, 1 )q-! [
M{+, - ;-, - , - , t )
~-
q+t ~t~ t ~ - ~.~ -
4e'EiB,{2, 1)1 !
,
[qt-q- tkl tkl- kHk2- 1'
,
2e' E Bi q ~ ! [ q-t 1' (qt'
111(- ,+;+,- ,- ,-)
l'
1
. =
111 (-- ,+;- ,+ ,-, -)
2e' E IBd [ '(qt' ,,_)
qt-q-+ k1tkl_k,. k,_
'
1
2e' EB;,,: , [ q'l-t t (qt' q-) q~tkl+kl_k2+k, _ 2e' EI8, lq++Q __
9/'
.1,
.r l ,
')')· {N"7. . f;,Ii('IIAIi(:~;,,,,/(J)
All ,- ;+. - .+.-) _ 'lr~/W~('J..l)l/ll [ (q+' q-l
2e ~ EI/J~(2,
-
111( - , - ;+. - .-,+) -
.
~
0/( - . - ;
• f . 1.-)
-
,11(- ,-;-.-,+,+)
=
9_1Iq"'_9__kl+k,_kl+k2_1t
2e·E[J~(2,1 )q.. _
[
••• q++k l +k , _kl+k1_
2e 4 EIB.(2,1)1 [ 9+-9-+ (q+' '1_) k,_k, _kHkl_
2e~EB~(1.2)q+J
[
=
2(:"(2£ - k,.,. - 1.:11 )8,q_.i. [
]!
]!
g..
I]~+k:+k, k1... 1.:1 _
9__
2e 4 (2£ - kl+ -
]!
]!
,.. fl_+k,+k, _kHk1_
2"£')8, (1.2») [ , •• , .. (q.,.. q_) k,..k,_k1or.t, ..
k,..,lID,1
tq~_q_+kHkl_k1+k,_Jt ' 2e Hc.'8'3Q-.1 [ , q++ (q+' 9_) q:+A'I+k'_~'l+kl _
2t:" E 8 3 11]++q ..
M{- ,-;-,+, -. - ) -
]!
2e4 EIB4 ( 1. 2)lq .. +q __
-
_
•
2,'/;B;(1 ,2nu [ qt. (1]+ ' q_) r-+ kl .. k ,_ k 2+ kl_
(q+ . q_ )
~
M(- .-;+. - ,-.-)
f/~ .. I:,+A·I_kl+k2_
1)lq++q __
(q+ . q_)
-
M(-,-;-,+.-. +)
(9+ '
]l
fitt
]!
l!
g, Silk/MAllY
M(- I - 1' - , -
,
or (Jf; f)
{o'OIlM Uf-AI>'
+ -) 1
4e'EI8,(I , 2)1
.,
[q+_q_ +k,+k 1 _ kHk, _ F
4e' EBi(2,1)q+.L. [ q++ ~~ + ~+~-~+~-
]l
4e'EIB, (2, 1)1 1 '
[q+-4-+ kl+k , _kHk, _]'
_ M (- , -'I , - , - 1 -) ,
4£' EBi(l-l [ q++ , ] l q.,_ q~+kl+k, _ kH k, _
4e'EIBd
(9,260)
Unpolari zed squared matrix element:
1MI'
20
=
8
E' [4(q+ , '1_)'
j
+ (I.l-+q:- + q~-q::+l
i 1•
kJ+k1_kH k'- 'J+ _q-+(q+' q-)'
)( { [1
+
CE - ~'; -
,j, k,+ ) ') IBd'
+ IB,(I , 2)1'
+ IB, (2, 1)1' + IB31' + IB.(I , 2)1' + IB,(2, ] )1' }
,i , (9,26] )
Unpol arized cross section:
du =
0-"
[4 (q+ , q- ?
+ q~+q:: _ + q-l--q:+l
64".2k1+ k1 _ kHk,_q+_q_ +(q+ ' q-)2
)( {[I + CE - ~'~ - kH )'] IBd' + 18, (1, 2)1' + 18,(2, ill' + IB,I' + 18.(t, 2W + 18 4 (2, 1)1' } • x D (p+
+ p_ -
q+ - q_ - k, - k,)
d3 ih- ff'if- d3 k, ff'k, , q+o q-o klO k,o
(9,262)
Ikliniliol\s:
i,j=l,2,
Al .,,~ {t, 2)
-
2Ez [k k (p_. kd (p_. k1 ) 1+ H
1
m Z l1 Z n ]
+ 4E~ An
_ _1_ [kH (2E - k l _)(2 Ekzt - Zzd ( 06. 12 ])_. k l )
_I 2Ek1+kH (2E - k l _
-
.1;2 _ ) -
m7k.l_ku.ki.J../2E]
(p_ . k,)
~
""' 2(p_.
k:;l(p_ . k
1
)
[kl _ kHk;.l.
+ kl+k~_ ;';.1
(9263) l'IIOLoll polarizations:
N,- 1
-
E "32k · k,· Ii l '..-
NOJivanishing heli city
i=1, 2.
I
a:nplitude~ :
M(+,-I-jT,+, + ,+) = ~
4e 4 E AJq" .l.
--;----"'-'=":--:--:.•'1. _ !q_ +q_+ kl +k l _ k~+k2 _1' 4e. 4 E!All
Iq+_q __ kl_ k, _ k2+k?_l t'
Af (+.+; +,+, + ,-) =
.'\e'IEA 1 (2,l)q: .1
,
q+_ ! q++ q_+k l +kl_~ ~ ~_J' 4e" EIA l (2, 1)i
(9 .264)
9. 8/IMMAItY OF Cllm FOItMU/, AfI'
M (+ ,+;+,+,-,+) -
4.' BA,( I, 2)q+.L q+_ [q++q_+kJ+ k l _ k2+k,_]} 4<'EIA,(1,2}1 ' , I.q+_q_+kl+ k._kH k,_1' 2<' EA 3 q+.L [ q;+ ]} (9+' q- ) q:+k1+ k._ k 2+ k,_
M(+, +i +, -, +,+}
2<' £ IA3 Iq++1--
M(+ ,+; -, +,+,+} -
2e'£.4 3 1+1. [ q- + ]} (q+ . 9_) Q++k1+ k._ kHk,_
_ 2.' E I.4 3 1 [ 9+-q-+ (q+ . q_) kH k._k2+ k, _ AI (+, +; +, +, - , -}
_
]1
2<'(2E-k,_-k, }Aiq:.L q+_ !q++q_+kI+k,_ k, .•. k., _] l
_
2<4(2E - k._ - k, )I A.] [q+_q_+ k'1+ k. _k2+ k,_ ]'' ' 2<'EA; (1,2)q+.1 [ q~+ (q+ ' q_ ) q~+ k,+k,_k'Hk,_
M(+,+; +, - ,+,-)
]!
2e' £ IA,(1, 2)l q++q--
M(+,+ ;+, - , - ,+} _
2e'EA:(2, 1)1+.1- [ q~+ (q+ ' q_) q~ +kHk,_k,+k, _
]1
2.' £ ]A,(2, 1)Iq++q-2e'EA, (1,2 )q+.l- [ q_+ (9+' q-) q++k,+k, _k,+k,_
M(+, +; - ,+, +,-) _
2e'~IA,(1: 2}1 \q+ '
q-)
r, ~+-:-+, 1t , LIC'+"l-"H"'··J
]!
·
A1(~-,+;--,+,-,+)
I' ,
_ 2,'EIA,(2, 1) 1 [ q+-q-+ (9..-" q_) kJ+kl _kz+kz_
M(+,-;+,+,+ , -) _ q+_ [qHq_+kL+kL_kH
J..'1_lt
4,'EIA, (2,1)1 [Ii'+- q_+kl +k1 _ k2+k2_11"
MI+ _' ,
,
1
~1
+ - +) _ ,
4c~
EA 4 (1, 2)q'
.L
,
,
q+_ [q++q_+kl+kl_kHk~_ l J 4e4EIA~(1,2)1
M(+,-;+,-,+,+)
__
64(2£ - kL_ - k2 _)A 1q_.L [
q?±kl±kl_kHk~_
(q+" q_)
e4 (2£ -
q!±
k~_)IAJ t q+±q __
,,
k"l_ -
(q± " q_) !q±_q_+k1±k1 _ kz.,..k 2 _1 l
111"(+,-;-,+,+,+)
_
_
e (2E - k l _ - k 2 _)A .q_..L [ q-. (q+" q_) qtt k l+ k l_ kHk2_
e (2£ - 1.1_
-
k2 _)!Ad [. q±-~-+ _
(q±" q_)
M(-,+;+,-,+,+) _ _
M(-,+;-,+,+,+) _
k1 .,..k l _k 2 ±kz _
2e1EA1q:..L [ q_+ )t (q+' q_) qtt k lt'l·1_k'l+k'l_
[-q+:c~-+-'--l'
2, +EIAd (q+ ' q_) 1..-.+k 1 _k h k 2 _
2e-tEA 1 q:.L [ -4± (q+' q_) q~+k1+kl_kHk~_
I!
I'
,
4
4
I'
,
I'
I),
.I'liMMAlty or (Jim fOIlA/III,A li
.11(+,-;+,+,-,-)
M(+, - ;+, - , +, - )
-
[q+_q_+k,+k,_ k2+A,,_ It'
'
-
2e' EA;( I, 2)q+J. [ q~+ t (q+ ,q_) q~+kl+k,_k2+k,_
1
2e'E IA,(1,2)lq++q _
M(-I-,- ;-I-,- , -,+)
111 (-1-,-;-,-1-,-1-,-)
M (+ , - ; -,t , - ,+)
M (-,+;+, - ,+, - )
M(-,+;+,-,-,+)
!I.UI, ,. ~ r"
, • ,
Ir
'
'1 '1 (NO 1.. }:X( : 1/11 N( ;fo,'; '"
I
II}
. M(- ,-; -, +,+,-)
M( , ri - .+,-,+) -
(q+ ·q_)[ql__ q ~ +kl~kl _ kHk~ _ I!'
2e~EA'l(1,2)q;'.L [ (q+ 'q_)
,
q!+ ]' q~+kl ... kl_k2+k~_
2e 4 EIA~(l, 2) [q++q __
Af(- .+ :- .-.+.-)
~
'1c"1 EA~q+.L
[
q+_ [q+_q_+kl+k , _kHk2_ 1 ~ <\e~
~
EIA31
[Q+ _q_+k l ... k l _kH k2 _li
'
4
q!+ ]' M(t , i +, , - , -) = 2c EAjq_. [ (q+' q_) q~ll'ltkl _ kHk'l _
.'1 ( I •
,t, ,-)
.'1(-.+;+.-.-.-) =
e.~ (2E - kl _
kl_)Ajq:.l [ q-+ (1 .. ' q_) q++kl_kl_k2... k~ _ -
_ "(2 E - k._ - k,-JI Ad [ (q+ , q_) M( ,!;
,+. -,-)
_
,._q_+. kllk, k1 , k7 '
]1
,
c~ (2E - k l _ - k2 _)Aiq: ... [ _ q! _ _ ]' (9+ ' q_) q:+kl+l'l_kakl_
e. 4 (2E - k1 _
-
l'
,
-
k'l _) AJ!qHq--
(q+ ' q_) (q+_q_+kl+kl_k2+k2_ lt '
.'I. SIIMMA II)' III" IJI','1J f'OIWIII,Ii/-:
Al (- ,+; -. - .+. - I -
_ ------:1 c:' /~ A~( 1, ~Iq+l.
q+- [qH'I-+" ,+k,-kHkz_l} ::.E.!:.IA2'1-'-.:(I""2:1.1 1 ['I+_q_+/;I+k, - k2+k2- 1'
__
M(-, +;- .-,-, +I -
.:.:.4e=--'
-,--1- - - - , -
•
4e'EA;(2. 1Iq+J.
1 ,
(/+_q_+l;l+k, _ k,+k,_I'
M( - , - ; +. -, +, - I
_
2e' £04,(2, 1Iq:~ [ (q+ ' q_)
-
2e' EIA,(2 , 1)1 [ q+ - q- + ('1+ ' '1- I kl+ k, _ k,+ k, _
2.' EA,( L 2Iq:~
1'1 (- )- ;+,-,- ,+)
l'
q-+ !f++k l +k, _k2+ k, _
(q+ ' 'i-)
•
'J '•
l' 1
[
q- + k QH i+ k ,_ k2+ k ,-
_
40'EIA, (1, 211[ q+_ q_+ ] ('1+ ' q-I k1+ k' _ /;2+ k, _
_
2e'EA. (2, l )q:~ [ q~+ ]' q~+ kl+ k l_k2+k, _ ('1+ ' q-I
t 1
M ( -, - ; - , +. +. - I
2e' EIA ,(2, 1)1,,++Q- -
1\1( - - ' - -I- -
•
I)'
I
,
+1 1 •
(Q+ ' q_ 1[q+_q _+kltkl _k2+k" I'
Al (-, - ;- ,-, +.+ ) -
20' (2E - k1 _
-
k2 _ ) Al q_~ 1
'1+_ [q++ Q_+k 1+k 1_k,+k2 _1'
2e' (2E - k 1 _
-
};,-II;1 11 1
(q+ _q_+kJ+kl_k2+k,_I'
'
'IL'!
,I,
. ,. 1,. Y1(NOX.I':.\"f'IIAN t :/-,.,,,/UJ
fl./( .,-;
j , -,-. ~.)
M(-.-;-,+.- .-)
:;;:
2(:4
£11;q':.1 [
(q ... . q_)
_]!
q!t f/:tkltkJ_k2t k2_
2e 4 E IA3 Iq... tq _ _
"
(
M( -, -;-.- ,+ , - ) _
2)!J... .1
,
qt- lq+±1-+kH k, k2_k1_1' 4e-tEIA 1 (1,2)1
,.
[q+_q_ ~ kl+~·I _k2tk2_J'
4e 4 E Aj(2,1)1+.1
M(-. - ;- ,-,-,+) _
qt- [1f±+'I_tkl+ kl_J:1-+ 4e~E I A2(2,
M(-,-;- ,-, - ,-) -
k'l _lt
1)1
,
q+_ [qt t q_ +"'ltk,_k~ .. k~_l' 4
4e E IA d
-'-
[qt_q_tkl+ kl _ k1 .. k2_1f·
tJnpolarized squared n:atrix element:
2(:,.6£" !4(1+ . q_)4 .!. q! .. q~ _ + q~- q~.:.l kltk1 _ k~ _ kl _ q+_q __ (q+ . q_ P x
\[1 + (2£- ';'E
- k,_
rJ
lAd' +1",(1.2)1'
+ IA~(2, 1)12 .j lA312 + IA 1 (1. 2)j1 + IA4 ,:2, I)i~ } . (9,266)
9. .,'IIMMAUY OF (JIm FOItMlnAi':
UnpOiCLl'iz,l'd truss xe(:tioll:
d"
x
{[I + e iE - r] IA,I' E
k,_
-
+ IA,(I, 2)1' + IA,(2, IW + IA,I' + IA,(I, 2)1' + IJI.(2, I)I' } •
xb (p+
9.19.13
+ p_
- q+ - q_ - k, - k,)
I;; and k~ nearly parallel to
dJ;hcPi/-d'k,d' k, q+o q-o klO k,o
.
(9. 267
)
q+
The primed qua.ntit ies ki± and kh , i = 1,2, are evalua.ted in the ro ta.ted
frame where ih det.ermin es the positive z-axis (see also Section 7.4.3). The four·vec to!' q in eqns (9.269 ) is obtained by _,.pplyi ng a Sl)ace reflection to q+. Defini tions:
kJi+ k'j -
-
A'I
-
2(q+· k,)
k'· '.' i.l A"jl.
i,j= l , 2,
,
+ 2(g+ . k, ) + 2(1<, . k,) ,
2qto (q+' k, )(q+ ' k,)
[1/
k'
1- 2-
m2z;,z~' l
+ 4q~oL'>;,
..!, [k;_(2q+o + kj +)(2q+ok,_ + Z;,) 6 12
A~(] , 2)
mk~+k~.L
2q+06 ;,
(q+·k,)
(9.268)
9. 19.
,.+,.- _ ,,"c- -y., (NO Z·~',\TIIAN(:~:"III/IJ)
1.
= I? ,• .
(9.269)
NOllvanisl:ing helicity amplitude:!:
M(+,+;+,+,+,+) -
M(+, T;+ ,+,+,-)
~
-
M(+,+;+,+,-,+) -
~
2 t: ~ A'lq-~ ~ Z+
q--I~i~q:+ki+kj_k~+kl_ l! "IA;I [ 8(,,· ,-) q+-
k: +.1.';_*"1+ *"1-
2t: 4 A;-( 1 ,2)q:~ z+_ :I k'1+ I.'1_ I.'11 k'1 · •• 9-- [ 9++q-+ J
,'IA',(!,2)1 [ 8(,, · q- ) q_+
k;+k;_k1+kl_
2c 4 A1- (2, I )q':.1.2+_ 3 k'1+ ~19-- [9++9_.;.
r r
,1:'1+ It'1_.' i
,'IA;(2,1)1 [ 8(,, · q- ) .l1+k: _ 4.~I4 _
q-+
?_c4A'J9_,!. ~ Z +_
M(+,+;- ,+.+.+ )
-. --
I
9__ [q++ q: +k:+k:_k~+k~_I > ~
" IA;I [ 8(,,· ,_) ]1 kj+k:_~+kl_
9-+ '
M(+.+;+.-,-.-) -
M(+,+;-,+,+,-)
r ,
'
2e·lq.~~-+_ 44'· • Z
r
q__ [q.j.f!f-+kl+ki_kl+k;_lt
"IA;I [ 8(,,· q-) 1_+ k:"k:.kl,.kl_
U ' A~ ( 1,2)9:.1. Z+_ ~
r
J It'1+ k' q-- I9++9-+ 1- k'1+ .1:'2_ It
-
'·IA~(1.2) 1 [ 9_ +
8(q+·,_) kj+k:·~lk;'
"
:1/ l I , I :
, I,
, I )
M (+,-; +, -\- ,+ , - )
M ( I , -;+,+, -
Af(+ , -- ; +,-
,+)
,+, +)
M (+,-;-- ,+, -I-,+)
M ( - ,-I-;+,+, -I-, -)
M (-, + :-I-, +, - ,+)
:. / ' .'1.\1..1
". .' " . I 'I I "",~
In
I,' 'f I /,' I
(J,..
t,!vn "'(J}(.1/f 11. i/:
"'"'.'. .H(
• ' ,
1 ,1 \'(1 ,' /\1 II I \',:J
, 1;1 .
'I ,
, 1, 1 ) ('II
if{I,
i -, I ,-, )
.11(1,
i -I .
, of ,
.1I ( ~ ,
; r,
, , I) -
)
'I )
,
M(I .-;-. ·j . l. -)
Af t I .
.1' ...
",/")
,1,,11',~ I :l2{1 . 2 )
g
I
,
.'
'II III I t I
.1 ~l l
t-I '
: 1
fl . •~(IMMAIlY Of.' 1/1','1) i ,'OHMlil, AN
l' 1
2<' EA!,q_.l [ q;± M(t , - ;-,-,t,t) = - (q±' q_) q:±k:±k:_k\±k\_ -
M (-, t;t, t ,- , -)
e'IA;!qttq__ E [q+_ q_±k:±k:_k\+k\_ll • 2e'EA;·q:.l [
(~:~)
)
M(-.t; t. - . -.+)
e'IA,(2,1)1 E
M( - .t;--.t,t. - )
M (- ,+; - , t .-, t )
M( - . t ;-.- , t .+)
q~±
' ]'
~±~±~-ij±ij-
e'IA;lq+tq--
A1( -, +i +,-, +, -
1
2e'EAj(2, 1)q:l (qv q-)
.
IH( :-,-;+.-.- ,-) =
Af{+,-;-,+,-,-) 4
- *
' {(+ - '" ""
-
..
..:.. ,- ,)
-
2e E IAji [ q+-q-+ (q .. 'q_) k~+k: _ k~+~_ 2e
(q .. 'q_)
.01.1(-, t-; - .+, -,-)
M( - ,+;
- .-,+.-)
' qi+
, 12
q:t-k:+k:_~,kl_J
etl A~(l , 2)lq+ I q __ M(+, - :-,-.-,+)
l!
:1110
9, ,\'PMMAnY OF QN/! I-'OllMUI.M:
M( - , +: -, -, -, +)
_ e' A~(2, J )lft.l.
ft1 (-,-:+ ,-,- , + ) -
e'IA~(2, 1)1 [
q+-q-+ k~+kl_k~+k~_
2e'A~'(2,
e'IA~,(l, 2)1 [
,
e4 1A~1 [
+ -) -
,
q__ [q++Q':+k:+ k:_ ~+k,_l'-
e'IA~(2, 1)1 [ 8('1+' q-) q_+
,
l'
,
_
,
8(q+' q-)
2c4 A'"q-.l.Zt_
, ,- ,- ,- ) M(- , -'+
I
1
1
kl+kl_k~+k2_
q-+
,
kt+kl_k2+k~_
q__ [q++q::+k;+k;_k~+kU'
-
M(- -' - - - + ) I
l'
1
8(q+' q-)
20' A;q_.I.Z;
M(- , -:-,-,+,+)
1
l )q_.I.2.. _
2e'A"(1 4 ) 2)q_ .1 Z· +_
q-+
M( - , -'I - , -
l' 1
E
M(-,-;+ , -,+,-) -
q-+ q++k;+kj_k~+k;_
E
-
[
k~+k~_k~+k~_
l'
1
1t
Y. /Y.
~ .. f ~ .-.
,.·t r ·· .,"( (N() /l·,.:XC:II/i NUi',';
lit
Ill) '.
,~
", .
_~.::" I
.M(-.-;-.-,-,-) :::;
tl -J."IJ t' _
1.1 I" k' k' (, 't+ ,.1 - -t~I+"'I - 2t 2-
I! (9.270)
Ilupolarized squared matrix elemeul:
IMI' =
,q:-
eA ['l (q+' q-F + q! + q!_q:+] 2E'lk\+ k:_k~_k~_ q+_q_+ x {
.~ Cqi \~:: + k~+ ) 2]IA;
[I
0
2
+ IA~(i I 2)1'l
+ IA;(2, III' + IA; I' + IA~( 1 , 2)1' + IA~(2, I)I'}
_ (9.271 )
cro~ .~
Unpolati7ed
x
section:
{[I+
(3}+o
~2:~: + k~+ )l]IA;I~
+ IA;(1,2)1' + 1~;(2, 1)1' + IA;I' + I A~(1 , 2)1' -IA~(2, I)I'} ~ x6 (p_
9.1 9.14
+ p_
-" - ,_ - k, - k, )
rPihcPq cfJk1 cfik2
- -.
q+o q_r;. k10 kon
(9.272)
k~ and k; DeMly p""Uel to li-
T he douhly primed quantities k:± and ~~. i = 1,2, are evaluated in t he rotated frame where i- determin~ the positive z-axis (sec also S(.'CtiOll 7.4.3). T he four vedor q in eqns (9.274) is obta.ined by applying a space ref!er.tion to q_. Defin itions:
Z~I. .J
-
k~' .+ k'!J-
- k"' k'! . j. JJ.,
!I . .;'IIMMAIIY 0[0' QI.;J) [o'OIlMIII.M: 'J ,
[
-'1-0
~ [k~_(2q_O
+ '1f/_I)Ll , , A"J'J
1- , -
( q_' k\ )( q-. k) 2 A~(J. 2)
,u" 'n, ] 1It~~~t
1.;" kif
+ k;'+)( 2q_Ok~_ + Z;',)
/::"{,
(q_. k,)
k" (?q + 2'1- 0 k"1-2- -0 + k" 1+ -'' k" '2+ ) .1- m'k" 2+ k'" 11. k" 21. /?(I "" -0 ]
(q_.k.,) m,
( q- " k, )( q- ' k) 1
[kif1+ kif,- k"a + k", - kifH k"2J.
(k "iJ. ''k" 2 l . )Z" 12 Z" 2\ ] /::'''12
[2'1 -0 k"2- + Z'"12 ('1- . kd
+ 2'1 -0 k"] 2('1- . kz)
(9.2n)
.
Phololl polarizations:
J;
Ni[h 11- A(1 ± "is) +
h h-
,kit 1 T "is)J . (9.274)
NOIlvaaish ing heJicity amplitudes:
11(+.+;+,+.+.+)
=
M(+.+;+, +. +. - )
/." k"2[q 3++ q3-+ k"1~. k"1 -~+ _
c'IA~(J.2)1 [ q+-
M( +,+;+,+. -,+)
-
Ii•
8(q+ .q-}
kIt1+ kit1- kll2+ kll2-
]t
2. 4 A~·(2.1 )'1:.1 Z+_ [q3+ ~ q3- +k"1+ k"1- k"2+ k"2- I! 1
-
e'IA~(2, 1)1 [~(q+. q- ) ] ' lJ+-
kJf+k~/_kq+kq_
'1.1'1. ,I,.
., I,
17 (NO
X-~;Xf:I/I\N(:"", ' 1I
A/(i-.-;+, - . +.+)
M(+,+;+,-!-, -,-)
=
I
tlj
1(:11.. +(';'I"'~ k"H · J.:II,~ k"H
k"~~ I!
2,'A"' I , '_ . Z -,-
-
1-' 'f ~ I
,S
[I -+ Ie'.'1+ i" 1_ A:" l+ kit ~-
_ ,'IA;I [ 8i,+' ,_) ]1 J::'+kJ,'_k!1+kQ_
q+-
M(+ ,-;+,-,+,-J
~
2e 4 AZ( I, 2)q: .1 Z 1
, " A;(l ,2)1 [ q+-
M(l,+; +,-,-,+) -
2c t A~(2, 1 )q" .1 Z+ _
I,'++
,l
- oj.
k'.'1+ k"1-"'./-1 IJI L."ll "'2-
_ ,'IA:(2, 1)1 , +M( I,
;+,+,+,-) -
]! k;'.,.kr_k1'+k2_ Bi",,)
[
8i,+' q) k"1+ Ie"1-· kit11 k"1_
e4A~· (2,1)q_..I. [ Eq+_ e~IA~(2,
];
q~+ q_+k;/~ J..~'~k!.f+k!f_
1) q+..-q __
, q-.,., kit1+ k"1- k"2+ k,,[i' Elq .,.,~
111(+,-; +,+, - ,+)
M( -1 ,-;+.-,+,+l
E: Iq.,.- q -+ k"1+ kit1_ 1.:"1+ Ie"1- It'
, ]'
9. 811MMAny OF (JIm 1'()ItMUI.Afo;
M(+ ,-;-,+,+,+)
M ( - , +; + , +, +, -)
M( - ,+;+,+, - ,+)
M(- ,+;+, -,+, +)
M(+,-;+,+,-,-) _
e4A~·q_.L
Eq+_
[
1!
q++ 3
,,
" k"1- k"2+ k" E [q+-q-+ k1+ 2- J'
M(+, -; +,-,+, - )
,v I .'J.
,'r
o ,
AI (~ . , -;+ ~ -,-,+ )
_ =
l'
E, (q+-q-+~I '."+ k"1 - k"2+ k"2- J'
M (+,-;-,+,+, -)
M(../-
o
'
../- - -)
.,-,-,,,
,
M(+. - i-,-, + ,+ )
M {-. + ; + , + , - . - )
M (-, + i + .-,-, - )
,
, ' jA;(2,1 )1 [ q+-,-+ k"I+ k"J-2+2 E k" k"M( -. + ; +,- , - . + )
l'
'
9, 8/ I MMAUY OF (JIW FOUMU/,AIi'
M(-,+;-,+,+,-)
M( -, +;
- , +, - , +)
M (-, + ;-,-,+,+)
e' A~q:L [ -
Eq+_
q;+
q_+k~/+ k~'_k~+ k~_
1t
"I
e'IA~lq++q __
M (+,- ;+, -, - , -)
M(+,-;-,+, -,- ) e 'IA"I I [
E,
1
q+-q-+ t , k"1+ k" '1- kif H k" '2-
e'A~( l , 2)q+J.
E
I
q_+
[q++ k Jl1+ k"1- k"~H k"2-
M(+ -'- - - +) I
,
,
I
l
e'I A~:r2,
E
1)1
q+-q-+ [ k"1+ k/l1- kif2+ kit2-
t
1
I
1
M (-, +i+.-,- ,-)
.
_ r~ A""I,I:!..J:.!. [ _
q+]• If -- pi k" 1.;" ' I+ kit J_2+2_
F,
e, q+_ IA"II E
M (-, +;- ,+ ,-,-)
-
,
, -,
e1 A'''q' £'q .~ -
r [ "-' r [
q'H kIt k" k" k"
q --I+
l -
H~-
q - 1- /.:",+1_2+1_ kif kif k"
e"!Ailq+ +q__
~
,'
E [12 +- q- +J.."1+ kit1- k"2+ 1.;") - I' M (-,+;- . -,+,-)
-
U
e'A " (l , 2)q.,l. q++ ' [ , , " " .It ,It Eq+_ q_+k1+kl_"'Hk2_
et lA ~( 1,2)19++ 9. E[1+-9-+ k"k" 1+ 1- k"k!'J 2+ 2- } '
e'u A({2, 1)Q-' J. E.q+_
M{- ,+; -,- ,-, +) -
[
q++ ,
"" '" q_+k,_ k,_k7~~_
e"IA~( 2 , 1)lq..,.9--
--r,
~
Elq +- q- +k"1+ k"1_ 1.:"2_ 1.:",_ I' 2e'.4~- ( 2,
M (-,- i- ,..L,+.-)
1)9+.I.Z; _
k\_+ q3-+ kit1+ 1;'.'1 _ ~+. k!!_II 4
e A:(2, 1)] [ 8(q+,q.) q+kf.kf.kl'+kl'_
M(- ,-i-. + ,-, + ) -
-
2e 4 AZ"{ 1, 2)9·..1.1:1': . [q3++ q:-+ kit1+ k "1- k"H· k:J- I~
,' IA;{l ,2)1[ q+-
M (- ,-; - ,-,+,T) -
r
1.
2e 4 .4"q I - .. Z ..· -
r
' :} :} k"1+ k"1 _ k"1+ k"2_ ]! 8+·q·+
"IA;I [ q+-
11
8(,+ .q_) k"1+ I.", •. kit k"2_ J
8(,+ · , - ) k"I~ k"1- k"~~ JJi-
r r
x{
[I +Cq-O +2:~: + k~+
) ']
IA~I' +IA~(l, 2)1'
+ IA~(2, 1)1' + IA~I' + IA~(1,2)1' + IA:;(2, I)I' } .
(9.276 )
Unpolar ized c.ross section :
X
IA;'I' + IA~(!, 2W + IA~(2, 1)1' + IA~12 + IA~(I, 2W + IA~(2, 1)1'}
x v<'i( p+ + p_ - q+ - q_ - k 1
-
k) J ,
3
ii+ rPif- rPk, t'k, q+o '1-0 klO k,o
•
(9.277 )
10 Summary of QeD formulae 1\11 1,11t: fl!Ilow ing formulae nrc prCl!t'lItca ill the ccntrc-of-mafiS system, with ('"ud ('lwrEU' detwled by 2E. Unle!ls Olh~rwise slaled, we take tile positive;:illollg fi+. For an itl':,itnHY four-vector k, we frequent!y usc the notat ion !i"~' 1)<jIlS (G .8)J ;,~is
kr
0;;
ko ± k•.
k-L = k~.
+ ik~ .
( 10.1 )
T!.:-nllgilfJllL tIl is ch",ptcl', we m~gk:ct tIle effects of ? fin :te electron ::.nd/ol' '{lin I'll lIlf\.SS, PXC('pt ill tile Secl,io:ls IO .57thto\lgh 10.77, where we pn:seat the (''''1IlIlla!' ror tho production of heavy quarkonia. Consequently, the i'ormuiae ;' 1'" :.ot valid in certnia kit:cmatical configuratious. For a discussion of the r;ILg,e.' of n.!:ciit,y of the approximatioll, we refer to Section 3.4. \Ve also men Lillil thdt , for tilE' QeD procl~SSCS in th:s cha.pt.er, be pos:iiblc colltriolltiMIS rll!() to .%:".exch,t!lge are no t ta ken -inlo acr.ount. The labcis of the listf!d he:icil)' amplitudes ;JW1l,Ys refer to the particles ill Olt' or"!~it.i've b:.'lidty tJ and a positive helkity 9, As usual, the I':ymbo] '.:.' is 1.0 be Icad (1.S 'an equality sign rr.o(lu lo a "liMe factor' . III th:s dl ...lpter, t. he HYllltJol l:1!11;5 usc..! :0 denote ,iJe ~'-Iu a ~e of the ahsolute "due of M slIm'ned Ill'(,r the fi nal stat e dc!:rces of freedom (JXll ari?>a~ioll and color) ::nd i)1'eragcd ,)\'CI' tho! ini tia I stv.te dcgree5 of fr(''C(loU\, The sum mat ion m-e!' rt';>cated color in,jit:c~ is aJw.:\ys implied.
10.1
e I· e- ......
q q (no Z-ex change)
I'roce:;s:
(10.2) Invariant-s: 'U
N(,:wanishing hdidly ampli ttlrlC!l:
:11(+.-;+, - )
ee fJ, fJ+ ... , ., E
(qh) I q- +
= -2(pt , IJ-).
(10.3)
SIIJ/MAUY OJi (JI:/J i"O/WII/.M.'
III
:171l
M(- ,+ :+ , - )
Un polarized squared rnaLrix element:
,,' I," I' = 6e' e"2 t' + $ .II.J
(10.5)
7
Unpolari zed c W Ss section:
d(j 'Q' t " + udt = 6r.a J . '
10.2
e+e- -> qqg (no Z-exchange)
Process:
Invari.wts and defini tions: u
Z+ _ = q++q __ - q+.L q-.L •
Gluon pola.rizations:
; ± - N!.4_ .4+
,k(1 ; ,s}+ ,k
Ii+ ..1-(1 ± ,5) ], (10.9)
Nonvan is hing helicity amplitudes :
JI·f (+, - :+, - ,+) M(+ ,- : -, + ,+) M( -, + ;+ , - ,+ } M(-,+:- ,+ .+) -
I(J.~ .
,. 1,.
-ii'l; ( NOX. I\X{,'IIIINlif,1
M(+,-;-,+,- ) _ M( -,+i + ,- , - ) -
M(-,+;- , .... ,-)
I,,,'I-. (q.· ._)(,•. k)(,_· kll! ' (l O.IO)
=
~qllil.rl'd "YO'l
absolute v,-d ues of the l10 llvilnishillg hclicity 'l mplitudes, summed the fillal slale color degrees of freedom:
1"'(1,-;+,-, +)1' -
IM (-,...J...;-,+.- )12 -
8(u q g)1q2 _ (q_ . k )(q_· k) ,
I M ( + , -;-,~, +)I' -
I M (-,~; + , - ,-) I '
-
8(ee'1g)'ql_ (q,. k)(q _ · k) ,
IM(-, + i + . -,+W -
IM( +, -;-,+,-)1' -
8(UqY )2q:± (q•. k)(q_. k) ,
IM(-,+ ;-, +.+)I' = IM(+,-;+,-,-)I'
8(ee gyfql •. (q,. k)(q _ . k) '
(lO. I I) Unpolari:>:ed squared matrix elem!'nt :
(10.12) Unpolarized cross 6cclion:
10 .3
c+e-
~ qq '}
( no Z-ex<:hange)
Process: "(p,) + ,-(p_) ~
'itq"j) + q(,_, i) + ,(k) .
illvariants aDd definHions: $
-
2(p+·p_),
1
_
-2(p. · ,. ),
t' _ -2 (p_. q_ ),
(10.14)
:m
III, ,,'/W,\/AIt\' or (,C'J) /i()/w/nAN
Z_+ = q- +
(10.16) Nonvanishing helicity amplitudes:
M(+,-;+,-,+) M(+ -' - -1- --1-) _ }
1
1
'\,
!q++q_+(q+ ''1- )('1+ - k)(q_ ' k)]S '
eJ 8'.iqu Z_ + B
"
M(-, +;+,-,+) M( - ,+; -, +,+) "'/(+,-:+ , -, -) M(+,-;-,+,-) M(-,+;+, - ,-) .M(-,+;-,+,-) _
!qt+ q-+(q+ ,q_)('1+ . k)(q_. k)]i ' e3 8"q' ~ Z';' B"
(10 .17)
Squared a.b:'301ute va.lues of Lhe nonvanishing helicity amplitudes, summed over the fin al state color degrees of freedom:
1,11(+,-:+, - ,+)12 _ iM(-,+:-,+,-ll' -
IM( ' T
1 -
. -,+,+)"1 1
_ IM(-,+;+,-,-)I'
6e 6 IBI'Q: _ (q+ 'k)(g_ 'k) , 6e6 I BI'q~
(q+' k)(q_ . k) ,
:n:1
1114. ,.• ,. -. "i'/illl (Nil 7,. h.'i(:/Ill Nr:,.:)
I M(--.+:~.-.+) I '
_ IM(+. -:-. I·.·"II'
IM(-.+:-,+.+) I'
~
IM(+.-:+.-.-)I'
(icflllJfq:. -
(". k) (, .. . k) ,
"'1811 1 ~ ue 1+ -
(". k)(, .. · k) . (10.18)
I l llpol ....i;..ed s'lu.ucd ml).trix clement:
IJ'll l' -_
-
3e'( VI' - Q)' IV,
t2
' l'J+'+" u U
,,'
(10.19)
wllf're [see equ5 (4.27)]
q.. q. ".~ (,, · k) - (, .. ·k)
P.. pv = , (p,. k) (p .. . k) ,
(10.20)
Hll po!ar ized cross sectioll:
do = (10.21)
e-e- _ 7jqg 9 (11 0
10.4
Z ~ e xch an ge)
I'rocess:
(1 0.22) Odinilions:
i=I, .... 4, i,i=1, .. . ,4 ,
c
~
D
-
CL
-
1
jWI -
U!,I' lw, -
iJ}~llwl
-
tQ~ l l w2
-
tOJ It~
- w(1 '
1
z z
I ~1 1
"34 -
Z Z 11 ~ L 4 :t2
+ (k
1 •
k4)Zj1{Zi1 .l. Zi~) (Zl ~Z~ - Z\4ZJ1) 2E(E _~)
+(k , . kJ )ZI4(ZlZ + ZJZ)(Zi2Zj'L2E(E - k,, )
Zi4Zj2) •
:171
1(/. SII MM Alll' 0 /0' (Je l) /,'O /l.M III. M :
ti,
-
Xl<x:;.,(XI1
+ X,,)(Z;, + Z j,)
(k, . 1.-,)Z,,(Zi , + Z j,)(Z12Z,. - Z",Z3') + 2E(£ - k30 )
+(k,: kJ )Z14(Z'2 + Z")(Zj,Z,,
- Zj,Zj,)
2£(E - k
-
,
IZ"Z., - z,Jz.,j'
+ (k,. kJ)Z;,(Zi, + Zi3)(Z"Z43 -
Z"Z,,)
2£(E - k,o)
+ (k, . k,.)Z'3(Z" + Z,,)(Zi ,Z' 3 - Zi,Z;,) 2E(£ - k,o)
d, -
Z13Z;,(Z"
+ z.,)(Z i, + Zi3)
.,.· (k, . k3)Z;,(Zj , + Zi3)(Z"Z" -
Z13Z,,)
.,.· (k,· k.)Z,3(Z" + Z.,)(Z;,Z~, -
Z;3Z,,)
2E( E - k,o)
2E(£ - k30)
C3
-
Iz" z" - z" Z3II'
+
(k, . k,)Z3 ' (Z" -I- Z,,)(Z2,Z34 - zi.z,,) 2£(£ - k30)
.,.· (k,· I.-,)Z:;,(Z;, -I- Z")(Z"Z,, -
Z"Z3.)
2£(E - k. o)
d3 -
Z;,Z3I(Z;,
+ Z3')(Z" + Z,.,)
... (k,. I.-,)Z" (Z,, + Z,,)(Zi,Z;. - Z;4Z,, ) • 2£(£ - k3O)
.,.· (k, . k3)Z;,(Z;, + Z,j,)(Z"Z" 2£(E -
<.,
-
IZ Z '"
'43 -
Z"Z31)
1.-'0)
Z Z j',+( k,. k3)Z4I(Z,.+Z,,)(Zi,Z.;3- Z'3 Z;,, ) " J4I 2E(E _ k.,o) (k,· k.,)Z;3(Z" + Z~,)(Z2JZ43 - Z"Z,,) + 2E(E - k3O) ,
\,
I(I. J.
,.',
"i'l!l~1
(NO 1- . f,'Xf'IItiNU1-:J
(k2 • kJ )Z.I1(Zll + Z"lJ )(7.:iIZ~~ - Zi3Z;,) 2£(8 k.10 )
+.
(k l • k4 )ZiJ(Z:i\
+
+ Z~1)(Z2IZt3~o)
2E(E
Z23Z~d
.
(10.23)
(:IUOl: polariza~iool!:
i=I,2. (10.24)
,
'1( J
. _ ,+,+, ,+,+ )
__ 4u,g'C I.
"'1"'' '1-;.
M(+. ; +,-, +,-) _ M(~. -; +.-, -, +)
I.
_
M(+.- ;-.+.+,-) _
(kH)' (U.·I-"llI1}·(U.'.3-w~) · I.
"'J+-
2eeqg2C[cl(T~:t~);j
WJ -
W~
+dJ(T b1'a)'J]
E( 1.:1+1.:1+ )3( ~'J "kH 1t Iw, [lw;
2cc q g'C[c;(Ta]'b).j
+ d5('/,bTd);
E(kl+k,+P(k3 +k H l [UI1Pw~ 2ec 7 g'Ch('1'o,/,b)'i /. d2(,!'b'l'a);,j
E( kl+ k,_)3( k3+kH )~ [w, !'w;
III . .\' /lMM~ Ill' ()f<' (j(.'}) /.'O/W/IU/i
:1711
2ee,g' C [e,; (,roo,!,!),; + d. (1"1" )'j I E(k1+ k2+)3 ( kHk4+)~ 110,1'10;
]\1(+, -; - I +, -, +)
2ee,g'C [c.,(1"T') ,; + d'I(1"1'"),;]
M (-, + : + , -, -I-,·_)
E( k1+k2+ )'(k3t k<+)t Iw, 1"v3
2ee,g'C[c;(1"T')'j + d;(T'P),,]
M( -, -I- :+,-, - , -I-)
-- E(kl+kz+)'(k3+k4+)~lw, l'w3 2ee,g'C[c,(TaT ~),;
M (- ,+ ;-,-I- ,+,-)
+ d3 (T'T'),jl
E( kl+ k2+ )3(k3+k<+)~ Iw,l'w"
2ce,g'C[c;(T'T'),; + d~(T;Ta)'j]
,
.M(-,-I-:-,+,-,+)
E( kJ+k2+)'( kJ+kH) f Iw, I'w"
M(+,-: +,-, -, - ) =
x [(1"1"),j( W, - w,l'( w, - w,)" - (1"1"),; (101 - W3)"( tv2 - W,r ]. M( -
L
_ , _
'1,
,
+ __ ) ,
,
,
(k'k +)' (WI -( ",,)(W3 ).- "") 1
=
_ 4ee,g'C' k~H k~'H
x[(1"1"),, (wl - W3)"(W, -
'3+
W3 -
'W-t
W,r - (1"1"),;(W, - W,),(W, - W,)"],
x [(T'1"),,(wl - W3)"(W2 - W,)" -- (l"1"),,(wl - W,)'(W2 - 10,)"]. (l0.25) Squared absolute values of the nonvanishing helicity amplitudes, summed over the final state color degrees of freedom:
IM( +, -
; +, -,+, +)1' = IM( - , +; -, +, -,-)1"
11/.4. ,,-' , -- - <j Ill' !! ( NO
Y,.f.'.'·(.'/lA N( : ~:)
I M(+,-;- , + ,~ ,+ lI' ~
IM(-, +;+,-,-, -II'
IM(-._;+._, + ,-:-)jl _ lM(+ ,-i - , +,-,-W
1-'1(- ,+;-,+,+,+)1' - IM(+.-;+, -,-,-)I'
IM (+.- ;+.-. +,-W
IM ( .... , - :+,-, - ,
=
'W =
IM(+, - ;-, +,+,-II'
~
IM( - ,+;-,+,-.+;I'
IM(-,+;-, +,+,-W
IM(-,+; +,-, -,+)I'
:Ji1
/II
IM( +, -- ; - , -I ,-, +)1'
SIi A/MAIII'
, I,
IM( - ,+ ;I,
()'.' (j( : f)
'.'()IIMIIJ.M.'
W
3E'(k1+k,+)6(kJ+k .. l' lw,
- 1O,I' I1O"' lw,J2 . ( 10.20)
Un poJ;:uized sqllitred lllil.tri x element:
[91 e ,9 k' k' k k 1 ·'! I' _ 1."'4D{8(kl++ki-+k~++kU 1+ H J+ 4+ 11
-
3
t
+91w, -
lV, 1'1w, -
1 + 3E' ( k, +k,+)6(k3+k.. l'lw,
x
WI
_
W3
1'1 W:z -
W-:
I'
tv31 ' - Iw, - tv,1' IW3 - tv.d.'] - w,I'
71c, + dd' + 91c, - dd' + 71c2 + d, l' + 91c,. - el,l' [ Iw, I'l w, l' Iw,I" IW 3[2
+71eJ + el3 1' + 91 C3 -
d3 1 ' _1_ (Ie,
Iwd'lw,I'
+ d;1' + 91c, - (I, I' ]} Iw,I' lw,I" (10.27)
UnpolarizC'd cross section:
I
+3£2 (k ,+kH
)6(k" k4+Plw, -
w,l'
+ dII' + 91c, - dI I' +.:.7ccJc::..,':"+,d,::',--I'i+, 9col",c'c--,-d-",1,-' 11I1,I' ltv,I' Iw,I'lw31' + 71c3 + d3 1 ' + 9ic3 - d,l' + 71e, + d,I' +91G. - d'I'] } Iw,I"lw.d' Iw,I' lw,I'
x [ 71",
, + p_ -
xli (p+
k, - k, - k3 -- k,)
~~~~~~~~ . klO k" k"" k. o
(10.28)
,
\
fII,!i
,.1,
l ' I'\'I'I~~s
"I"'/I/(N(}/' .~:\'f·I/ i1 I1' r:/I·)
{dilfl'l'l'llt (pwr'k J1i.lvors):
( 1 (J .~!))
1)dillilioIl9:
i,j=i, ... , 4, 7 "'r" ij oro .. 'lE2(k , _k2.,.kJ+J.,-!+ )t '
1"(1,2,3,'1) -
k~.i (kJ
·
k,, )~_
1].
[k2_ Zi~,(Z31 + 1:'3. ) + klJ.kj.iZz~,(Z;4 + 7,;:.1 E. k 10 kJ-(E, kll,) (10.:111 )
Nonvil.t1ishiug hejieity a.mplitudes:
AI( 7, - ; +, -, -, -)
-
111(+,-:.,-,-,-.+) .'H (+,-;-,t, +,-)
_C,q1 B(e 9 F( 1,2,4,3)
-I c~ V(3, 4, 2, I )J '
-cl 8 (c, F( I, 2.3,4) - e~F(4, 3. 2, 1)1.
- -el DJ-c
9
F(2, 1, 'I. 3) + c~F(3, 4, I, ').)1,
M(+, -:-,+ ,-,+) -
_ C.q2 B[-c qF(2,
M (-, t;+ ,-,+.-) -
-el 8 [-c F "{2, 1,3,4) -
M( -,.,-; t , , - , ...... ) -
-r.l8[ - cq F"(2, 1,4,3) + e~F"(3. 4,1,2)1,
M(-,+;-,+, t , - ) -
-eg"l BJeqF"( I, 2, 3,") - c~r (4,3 ,2, l )J ,
9
1,3,4) - e;F(4, 3, 1,2J1, c~F"(4, 3,1,2)] ,
M(-.+; - ,t,-, ...... ) = -eg 1 B [eq F "( 1,2,4,3) -1 c~F " (:j , 4 , 2,1)1. (10,3 1)
III, ,'iIl AlMJlIlI' OF (WI) FIIJ/,\III/,Af.J
Squ
or
the IHH1Yllni slii llg !It'li riLy i!..Hlplitudcii, Sl\ llll1 H~d over the fintl! sLa te color degrees of rre(:dOlll:
IM( +,-;+,-,+, - )i' =
IM( - , +; - ,+ , -,+)I'
+2e,e;Re(F(I,2 ,4,3 )F"(;3,1, 2, 1)) ], 1111(+,- : -1-,-,-,+)1' -
-
IAJ (- , + ; - , +,+, -)l'
2e'g'B'[e~IF( I ,2, 3 ,4)1 ' + e;'1F(4,3,2,1)1' -2e,e:,Re(F(I , 2,3,1)F"(4,3,2,1)) ] ,
IM(+,-;- , +,+, - )I' -
-
IM(-,+;+, - , - ,+W 2e'g'B' [c;IF(2, 1,4,3)1' + e7IF(3,4, 1,2) 1" -2e,e;Re(F(2 , 1,4,3)F"(3 , 4, 1,2))],
1111(+,-;-, +, - ,+) 1' _
IM( - , + ;+,-,+,-) I'
2e'g'lJ'[e;W(2, 1, 3,4)1' + e;'1F(4, 3, 1,2)1' +2e,e; Re(F(2, 1,3 ,4)P"(4,3,1,2)) ] ,
(10.32)
Unpolarized sq uared matrix element:
IM I' = e'g"}f' x
{e;[IF(I, 2, 3,4) 1' + IFll , 2, 4, 3)1' + 1F(2, 1, 3,4)1' -I- 1F(2, 1, 4,3)1'] +e;' [1F(3, 4, 1,2)1 ' + IF (1, 3, 1,2)1' + 1F(3, 4, 2,1)1' -I- IF (4 , 3, 2, 1)1 ']
+2c,e~R" [F(l, 2, 4, 3)F"(3, 4, 2,1) - P( I, 2, 3, 4) r"O( 4,3 , 2, I ) +1"(2,1,3 ,4)F"( 4,3, 1,2) .- F(2, 1,4, 3)F" (3 , 4, I , 2)J) _
(J 0.33)
III ';,
,I t'
,
'/'/'1 'I ~
,'f} :,; - ~:s I 'ff
I NI" /' I
,,
------_._._---(\
X{!J}lI/"( :,2,3, 'IW -I ]F{l,l,-1,:I) 2 -I- P'll, t,:I"t W +
1 1' (:l)1 , 1,a) I~1
-'Q7ilF(3, 4, 1,2)1' I 1"(1, 3, 1,2)1' + IF(3, ' ,2, 1)1' + 1F( 4,3,1, !)I'I +~(J iqj Ih:[ r~(J, 2, 4, 3)r (3.
~ 1'(2, I, 3, 4)r
' (4, 3, 1,2)
P(2, I , 4, 3)F'P, 4, I , 2)I }
(10.:14 )
10,6
(o:+e- _ q7jqq (no
Z~e xc han ge)
I' roccss (idl:!nlic;t\ quark flavors ):
( 10.3,) Ikfillj~ i ons;
i,j"':::"t, _,., 4 , I
n ,
" (:_, ?K, 3>,I I
M( I,
41-.."'2 (k l I kll k;\+ kH -
[k H 2:.. (Z;)1
k4.l ( k~.k4 ) k~ _
; I, 1,-, - ) -
M (+, -; +, -, +, - )
"
),
-
+ Za.c.)
E -k20
kl1
i-a I Z~3( 2;'1- 2.;4) 1 k3_(/;,' -klO)
. (10.3G)
-ee~D1!;'i~IF(2,~, 4 ,1)- F ( I ,
"'9' n{:Z;;T~.JF(l, 2,.1, 3) + 1'"(3, ,. 2, I)] - T!.JTi'.. IF'(3 ,2, 4, 1) f· {o'(1," , 2, 3}]} ,
M( -, -; +,
-, - , --r)
-
t..1!,!I~ 81~T,~~[P( 1 , 2, 3, 1) - ('1 , 3,2,1): ,
/II.
8/ I ~fMA/n'
01" qr.'I) /"O/WU/,AI;'
M(+ ,-;-,+ ,+,-) -
ee,g' B1~j'l':,,, [/"(2, I , 4, 3) - F(3, 4, 1,2) J ,
M (+, -;-, +, - , -i-)
ee,g' n{ 1ijT!" [F(Z, 1,3,4) + F ( 4,3, j , 2)J
- T'~iT;':. [P(2, 3,1,4) + F(4, 1,3, 2)J} , M"(+)-;-, - ,+,+)
ee,g' BT,~jT,~[F(3 , 2,1,4) - P( 4,1 ,2, 3)J,
M( - ,+; +,+ ,-,-)
ee,g' BT'~iT,~[F'(3, 2, 1, 4) - F"( 4,1,2,3)]'
M( -, +;+,-,+,-)
ee;B{T,jT!n !F" (2, 1,3,4) -I- 1"' (4,3,1, 2)J
-T!iTi~ I F'(2, 3, 1, 4)
+ P"( 4, 1,3,2)]} ,
M(-,+ ;+ ,-,-,+)
-
ee,g' BT;j1~~n IF' (2, 1, 4, 3) - P" (3, 4, 1, 2)] ,
M(-, + ;-,+,+,-)
-
ee,g' BT;jT!n lP" (4,3,2,1) - F '( l, 2, 3, 4)J ,
M( - ,+; - ,+, -,+) -
-ee,g' B{ T,j1;;'".[F"(J, 2, 4., 3) + r(3, 4, 2, I)J
-T'~iT;~IF" (3,2,4, I) + F"( 1,4 , 2,3)]} ,
M(-,+;-,-,+, )
ee,g' B1;:'j1i':.IF"(I, 4,3, 2) - F ' (2,3,4, 1)1. (1O.37)
Squared absol uLe values of the nonvanisbing helicity amplitudes, sumnled over the fill,1 stale color degrees of freedom :
[.M(+, - ;+,+, - ,- )i' = [M(-,-I-;-, -,+, +)i' 2e'e;9'
B'{ [F (Z, 3, 4, I )[' + [F(l, 4, a, 2) [' - 2R.e[ F(Z, 3, 4, 1}r(I, 4, 3,2)1} ,
"-,
IM(+,-;+,- ,+,-) I ~
...
IM (-.-t;
,i ,
,I) I~
x (1F(l, 2, 1, 3)1' + W(3,4, 2, : lI' + IF'(3, 2, 4, 1)1' + .Pi I, 4, 2, 3)1' +2R.i P( I, 2, 4, 3)1"(3, 1, 2,1) + 1'(3 ,1, 2,1)F'( I, 1, 2, 3)]
+~ [lei(F(l",<,3) + 1'(3,4,2, 1J)(F' (3,.,2, 1) + 1"' (1, 1,2.3))1}. 1,11(-,-;+,-,-,+)1' _
1114(-, +; -, '.+, · )1' 2C 1C;!/B Z{IF(l,2 ,3, 4W
+ I F('I, :l,2, 1 )1~
-2R. iF(I, 2,3.' )F· (4.3,2, 1)J} , IM(+.-; - .+.+.-)l~
- lM(-,-:-;+.- .- ,---W _
2e2e~lB2{1F(2, l,4,3)i1 T
F(3 , 1,1,2)1 1
-'lie: F(2 , I ,4, 3)F' (3. 1. 1,211} , 1.11'(-,-;-,-,-,-11' -
x (1F(2. 1,3.4)1' +
IM (- ,+; +.- . +.-)I'
IF(".3, 1,2)1' + IF('. 3, 1,111' + 11'(4, 1,3,2;1'
+2Rc [F(2. 1. 3, 4 )r (4, 3, 1.2) + 1"(2, 3, !, 4 )F~( 4, 1,3 ,2)1
+~Rei (F(2 , I . 3,4H
1'(4 ,3,1, 2))(r I2, 3, 1, 1)
~ F' ( 1. 1,3,2))]} ,
III. SIIMMAIIF Of<' (J(.'/!
,,M (+ I
1
- ' __ ,I, 1
,
)
)
,L
r
)I'
-
f<'()IlMlnAl~
IM(-,+ ; + , +, - ,-) I'
_ 2e'";9' 8'{ W(3, 2, 1,4 ll' + I1"( 4, 1,2, :l)I'
-2Re[F(3,2, 1, 4)r(4,1 , 2, 3)J}.
(JO.38 )
Unpoluri zed squared ma.t rix clement:
", X{IF(I , 2, 3,'I) I'
+ IF(J , 2,4 , 3)1'+ IF (2,1 , 3,4)1' + IF(2,1,4,3)1'
+ IF (4, 3,2, 1) 1' + 11"(3,4, J ,2 )1' + 1F(3, 4, 2, 1}1' + 1F(3, 2,4, 1)1' +1F(1, 4,2 , 3)I' + 11"(4.3 , 1,2)[' + IF(2, 3, 1, 1)1 ' + 1F(4, 1,3 ,2)1' + IP(2, 3, 4, 1)1' + IF ( 1, 4, 3,2)1'
+ IP(3 , 2, ],4 )1' + IF( 4, 1,2,3)1'
+2Rc[F(I,2, 4,3)F" (3,4, 2, 1) - F( I,2 , 3,4)F"(4,3, 2,1) +1"(2, 1, 3, 4)F" (4, 3, 1,2 ) - 1"(2,1,4, 3)F'"(3, 4,1 , 2) +1"(3,2, 'i, ] )F' (l, 4, 2, 3) - 1"(2,3,4, I )F' ( I, 4, 3, 2) +1"(2, 3, 1,4 )1""(4 , 1,3,2 ) -P(3,2 , 1,1)1'''('1, 1, 2, 3) 1 +3" [1"(2,1,3 ,4) + F(4, 3,],2)1[1"'(2 , 3,1 , 4) + F' O, 1,3,2)1
+ ~[F( 1,2,4 , 3) + F(3,4,2, J)j[F'p,2,4, 1) + F' (1, 4, 2,3)1]} . (10 .39)
'I ,J'
I/J. '1. '1'1'"
. l l llj1o\,u'il."'/ "I'I IS.~ ~''''l, i(ln:
x
{!F(l. 2,3 .4)1' + W( I, z. 4, 3)1'
T
-l-W(4,3, 2, 1W -I- iF(3, 4, 1, 2),1
+ jF{3, 4, 2, J )I~ + IF(J, 2, 4, I)
!F(Z. I. 3,·1)['
+ 1"(2, 1,4,3 )1'
-I-1F(1, 4, 2, 3W + 1F'(4, J, I, 2)!1 + IF(2 . 3, 1, 4) 2 + iF(4, t
IF(2, 3, 4,1 W-+ IF(l, 4, 3, 2W + IF(3,2,
,:me [/;'( 1,2,4, 3)P-(3, 4, 2,:) -
"!
I. :1, 2W
,,4W+ IF(4, 1,2, 3W
F ( l , 2. 3, 1.)F-(4,3, '2, I)
+1-"(2, t,:I,tJ)F"(4,3, 1,2) - F('2, 1" j ,3)F'(3, 4, 1,2)
+ /'0.2 .4, I)P · (1.4,2,3) - P('.3. 4, 1),,"(1,4 ,3,2)
- P(2, 3, I,1)F"Ct, 1. 3, 2) - F(3, 2, L 4 )P' (4, 1,2, 3)
+ ~jP(2, I: 3, 4) + F(4, 3, I, 2)][F"(2,3, 1,4) + F"(4, I, :~, 2)] +5 [F(I,2.4,3)
+ F(3,4,2 .I )j[F"(3,2,4, I) + F"(1.4,2,31J]} t I O.4U)
10.7
q(/_qr/
Process
(dificr~ Ij L
G!lark lI;\vors) :
(10.'11) Delill j lions: s = 2(1'+ ' p_),
NOll vanishillg IJelici\.y
(10.42) amr!it()(k~:
M(+,+;+ ,+) ==
:!~rI
III. SIiMMMI.Y 0/0'
quo /o'OIlI./lI/,AJo:
M (+,-;+ , - ) .11(- , +; - ,+) (10.43) Squared absolute va lues of Lhe nOllvanishing helicity amplitudes, summed over the init ial a.nd final slate color degrees of freedom:
IM(+, +;+,+)I' 1,11(+, _;+, _)1' Unpolal' i z~d
9+-
IM(-,+;- ,+)I'
=
'I (10.44)
squared matrix element:
IMI' Unpolarized
£" 329'+,
IM (-, -; -,-W
CrOss
=
~g' 8' ~ U'
(10.45)
sedion:
(1 0.46 )
10 .8
qq -; q' q'
Process (di ffe ren t quark 6<1"ors ):
q(p+, i)
+ 9(P- ,j) --> q'( q+. m) + '1'(9- , n) .
( 10.47)
rovari anLs: s = 2(p+ . p_ ) ,
(10.48 )
Nonvanishing hclicity ampli t udes: 1
qE+-!- ('1++) ' qt -
_\J -M (+ , _.+ I ,
9
M (+, -; -,+ )
'T~·T·nm q-~ 9 E
M (-, +; +,- )
'T?T" '1
rUR
I)
g 'T~T'nm q:~ E I)
.
M(-,+; -,+)
9
'T'.T" q+.L I J 8 m. l.... L
'
(q.q++ ,.- ), 1
1
(9+)' q++
1
1
(q++ ), q +_
(10.49)
/1'/ ~ '1 'f
/U.!J.
Sqlt;II'(~{1 ahsolule
. "I','r
LIII;
o f Ule lloll vnl\i.~h jIl R IlI'l kity amplitudns, surnmed mhlr (J(!g l'CCjj o f f\'cc·(lolll iJr li:(' iui l irli IIll\.lc a nd tue /inal state:
va.lues
,
IM{-,-:+,-) i == IM(- .+:-.+)i' 2
1.'vf(+,-;-,+JI
2
=
_ 29. 9++
E' •
,
E;·
I M( - ,+;+,-)I~ - 2lt"J
(10.50)
Ih:polarized sq"..lared ma lrix clement :
4
IM/' = -g l Illipolatizcd cross
(l'
ti ~
"1",
(10.5\)
••
~cction :
(10.52)
10.9
qq-qq
Process (icent iC1l1 (JlliU"k f1avorll):
(l0.53) 1:1 variants:
t = - 2(p, .q.).
(10.51)
Nonvan ishiug hclicity amplitudes:
M(+. ,:+.+) = 4l
BQt J. ,Jq+..- l +
+ T:ir:.i] [ T~.T:j 9+gtt
M( + ,-;-, + )
M( - , +:+.- ) _
2g2~;T:'Jq~..1. ( t"JH ~r) l
M (-. +: - .+)
_ -2lT:.;T: j Q: .J.
M (- ,-:- . - )
_ 4g' Eg, l ,!q+,q·t -
(q!+)l '+.
[T:';T:; + r:,T:.J] 9+-
q+t
(10.55)
III. SIIMMAIIY ()/O' (1(:/) [o'()f/MIILAN
Squareu iLbsului,<.: VilJUf:S of the nOJlvtlllishillg' hdicit,y amplitll(Jes, SlLnltn(!tl
over lhe co lor degrees of freedom of the iuitial sla,l,c and the filial sLnLe:
W - 8g'~ (s'
IAf(+, -1-; -to, -I- )1'
IM( -, -; - , -
IM ( -1-, - ; + , _ ll'
IM( - ,-I-;-, +) I'
t'u'
- ~t,,) 3
=
IM (+, -;- ,+) I' = IM( -,-I-; -I-,-JI'
(10.56)
Unpolarized squared matrix element:
MI' = ~ , Igg
[S4 -I- (' + u'
(10.57)
t' u'
Un pola.rized cross section:
da = dt
4,.-,,1 9
[s' + /,1 -I-
U'
8]
,'i'u' - 3tu
(10.58)
qq -> qq
10.10
Process (identical quark flavors): q(p.,.,i)
+ q(p_,j) -. q(I+,m) + q(g_,n).
(10.59)
Invariants:
(10.60) Nonvanish ing helicity amplitudes:
M(+,+;+,+) = M( +,- ;+,-) 1
M (+ , -j-,+) -
'T.' .T'
9
1)
q-J. (q+-)' '
"m E•
q++
M(-,-I-i-l-, - )
M(-,+;-, + ) M (-,-;-,-)
(10.61)
,
ItUI.
"(II -. "r'1
. S(III:ll'!~d
IIV(~l'
all/iolll'.,l:
the wi or
(If lilc nO :lvl tlli ~hiIIK lu·lkil.y att:pliludC!l, summed of rl'{.'{dn1l1 of lhf' illiLia l .~jatc a:ld the fina.J stale:
vnl\l(~
(kgn~:s
1.11(+.+,+.+)1'
~
IM(-,-,-,-II'
- ' IM( +, -, +, -I I' - IM(-,+,-.+)I' _ 89•• s2tz
IM(+.-,-.+))' = 1,11(-.+,+.-)1'
89
4
[, u - R] - sl
" •
'"""'i .
3
(10.62)
I1I1[)u!
(:0.63) li:l polarized cross
~ection:
dd _
41fCl~
dt - 9,,41 2
+ •• . . u - 8,] 3'stu . [., ~
(10.64)
I' rocess:
,(k) H(p. i) - o(k') Pmitive z·a.xis: along
+ q(P.i) .
(10.65)
k.
In\'.a.rjanL~:
' ~ 2(k·p).
t ~ -2(k·
If),
h=
-2(k·p· ) .
( l O.66)
Pi:oton po!a: izatiolls:
(10.67j Nonvanishing helicity amplitudes:
M(+,+;+,+) M(+,-;+,-)
=
:1IJ1l
//1, ,;IIMA/Aftl' Of' (I(;I! I'IJIlMII/,A/';
M(-,+ ;- , +)
-e~6iiP~ (E~J t
M(-,-;-,-) =
-e28iP~ q lp'..
"
(SE) t
(10.68)
p~
Squared absolute values of the nonvanishing hclicity amplitudes, summed
over the color degrees of freedom of the initia.l state and the final state:
IM(+,+;+,+)I'
~ IM(-,-;-,-W - -12e'?u'
IM(+, -:+,-)i'
IM(- , +; -, -I- )1'
1
u s
-12e, -.
(10.69) ,
Un polarized squared matrix element:
IMI' = -2.',
2
,
s su -I- u
( lO.70)
Unpolarizet! cross section:
d" dt
10.12
.)
'Q' 5' + u'
= - ..or-a
J
s3
u
.
(10.71)
-yq-tgq
Process:
"I(k) Positive z-axis: along Invari ants:
+ q(p, i) ---> g(k', a)+ q(p',j) .
(10.72)
k. t= - 2(k·k'),
u
= -2 (k' p').
(10.73 )
Phot.on and gluon polarizations:
,i"±(k) -
N [fI
P ,k( l ± "Is) - l j/ p(l + "Is») ,
l L(k')
Nf/:,
Ii p (l ± 7s) + " fI ,k'( J 'l' 1S»),
-
,
(2stuJi.
(10.74 )
Nonvunishing helicity a.mplitudes:
,vi.
M(+ , +;+,+) = e,gTjip'_
(BE) 1"+ t
;1Il1
IiU3, !l¥ -1'1
M(+ , -:+.-) =
('~g'l;i11~ (E~+)!
M(-,+;-,+ ) -
eq9Tj,p';.
=
M(- , -;-,-)
'1"f.P'l
eqg )1;/_
(f:~Jt (BE') , 11'+
( 10.75)
Squared ab.501ule values o f the nonvMishi~g heJicily amplhudes, summed over the color d~g:'ces o f freedom of the initial st~te And the final state:
1"'(+. +; +, +)1'
-
IM(-, -;-,-)I'
-
M(+,-;+,-)I'
-
IMI-,+;-,+)I'
-
_16t;lg2 ~ 4
1t'
- Hie:g1 ~.
(10.76)
••
I:npo:ari2'.cd squared matrix elemenl:
(10.77) Cupolarized cross seclion:
BlfOcrsQi 3
8
2
+ till
,s3 U
(10.7S)
ProCCliS:
g(k, a) Pos:l i v~ z-a:o; ill: Along Invariants:
+ ,(p, i) ~ o(k') + q(p' ,j) .
(10.79)
k, t __ 2(k.k').
(10.80)
Cluon and photon polarizations:
_
,
(2stu)~.
(10.81)
.
'
III. HI IMM AIlY
Nonvfllli:dlilig IlcJi,ciiy amplit.udes;
or (ICI)
Jo'()IWUI,AI~
,
•P'i (8E)' 1"+
111(-/-,+ ;+,+ ) -
e,gTjip'_
M(+,-;+, - )
e,gT;ipj, ( E1"+
,11(-, +; - ,+)
e.gTj;p~ (E~J} ,
2 )
p' lvI( - , - ; -, -) = ',gT;i rI~
~
(SE)' 1"+
(IO.S 2)
Squared absolute values of the nonvani,hing heli city amplitudes, summed, over the color degrees of freedom of the initial state and t.he fiua l state: s
IM( +, +; +, +)1' _ IM(-,-;-, -)I' _ -16e'9'q u' 11I1(+, - ;+, - }1' = IM(- ,+;-,+)J'
,.
- 16e;g' -. s
(10.83)
U npoladzed squared matrix element:
IlV-../:" I'.. ==
,s' + 'u'
(10.84)
Q'[oS ' ·"I u_'
( 10.85)
1, 9 --e 3
q
su
Un polarized cross section:
du dt
gq
10 .14
--t
1:'O'Cts
3
s' u '
gq
Process :
g(k, a} Positive z·a.>..-1s: along In varia.nts:
s=2(k·p) ,
+ q(p, i) -> g(k' , b} + q(p',j ).
(10.86)
k. t = - 2(k · k'),
u=-2(k · p') .
(10.87)
G!uon polarizations:
I±(k'} 11'-1 _
N[,k' p' #(1 ± ')'5) + p j/ ,k'(l 'f ')'5)], (2stu)i.
(10.88)
10. IS.
1;; --'''lq
I'iUIiVUllilltdllg twlkiiy Il.lIlpl iLUUIll<:
,'" (U·)l [:u:.'('rRT~)J,-p'_(TbT'}jil,
M(+. + ;+,+) =
-2l!:f ..r.:
M ( +, -; +, -) _
-g'p~ Co;~i') t [2E(1"T');o - pc(T'T');,1 '
M( -,+ :-,+) -
-y'p:;
M(-,-: -, -)
-2g';~ (~)l :2 E(T'T');;- PC(,/,',/,');;I .
p_
1'+
(E~); :2E(T"T');; -
PC(T',/,");,j,
(10.89) SQua.red absolute va lues of thl': nonvanishing helicity ampEtudes, summed over the color dl':grees of fr('edom of tln: in itial ,tate aud Lhe final stale:
IM(- ,+,+,+)I' - W(-,-:-,-)I' = ' 8y' [;: IM(+,-: +,-)I'
=
IM(-,+:-,+)I'
= 4By' [;: -
::j, ~~l .
(lo.nO)
Unpoiurb.led squared mairix element:
IMI2-= g'*(.s2 + t/)
[-,- - -'-] . t~ 9su
(10.9 1)
[2. - 9.\u· ~]
(10.92)
1(k) " 'i(p,i) ~ 1(k') + q(p',j).
(10.93)
Unpolarizeci cross section:
d(T dt
10,15
, 7[
~
=
1i"()~ .,2+ u .
2
51
t~
'yq
Pro Cf"!i.S:
PQsitive ':'Axis: along
k.
Inwlriants:
t=-2(k·k').
(10.94)
PhotO!1 polarizations:
.'\,-1 _
(2.'1 ~u ) t.
(10.95)
/II, 811 .II hi A It Y 0 f'
Ill) IJ
J.'I!ll .II (lid N
Nonvanishiug hdicity (l,mplituc]es:
"( +,+;+,-/- ) M M( -'r} - ' -l1
I
,
-)
e',0., <" p'';:. pi
(BE) p' t ,.
e'~ q ' J'p'1.
- 2- )~ ( Ep+
-
-
,
(8E) -- " 11'+
p~ e 2 8··-
M(-,-;-, - )
tJ
1)
p'-
(10,90)
Squa.red absolute values of the nonvani shing helicity amplitudes] summccj-.....
over the colo r degrees of freedom of the initial 'state and the final state: , s )1' - -12eq - , IM(+, +; +, +)1' , I IM( - , -'1 __ u
IM(+,-;+,-)I' = IM(-, +; - ,+)I'
-
- l')e' ... Ii :". S
-
( [0.97)
Unpolari zed squa.red matrix element: ,
IMI' =
-8e' ,
8
+U su
2
(10.98)
Unpotarlzed cross section:
d" = - 81rCt2Q~ s' elt
'
+ u'
(10.99)
S3"
,q -+ gq
10 . 16 Process :
1'(k) + q(p,i) P osit.ive z· axis: alo ng
->
g(k',a)
+ q(p',j).
(10.100)
k.
Invariants:
s=2(k·p),
t = -2(k' k'),
It
= -2(k . p').
(10,101)
Photon and gluon polarizations:
NIl.:" NI):'
J!
f/(l '1'7') -
P,I/ 1.:(1 +1',)],
,(>(1 ± 1',) + "
po P(l 'f 1'5)1 , (10,102)
:wr,
fO.f'l. 1I'i--'"''
M(+.~;+,T)
=
,11(- , - ', - , -) =
(10.103)
S(IUarcd absoluh: values of the nonvanishing helicity Il.:nplitucles, summed uver the co lor degrees of freedom of the in.i lia.1 stale a.nd the final state:
.,
-' 1"(-,-;-, - )1' - -16e-g • u'
IM(+,-;+,-)I' -
1.1((-,+;-,+)1'
u - -16e;l-· ,
(10. 104)
lJnpolarizcu squared matrix element:
__
Mil
8
~
.<;2
+ UZ
= _ _ e-gZ,,---,C:"J q su
(10.105)
UnpoliHizcd cross secLion:
d, d,
8:rrCtCl'sQ} 3
S2 _
u2
(10.10G)
..,lU
Process:
g(k, ,) + qlp, i) ~ 11k') Positive z-axi~: along
+ ii(p',j) .
(10. 107)
k.
Invarian~s:
.~:::
2(k.p),
, = -2(k . k'),
u = -2( k·p' ).
(10.108 )
Gluon polarizations:
, " '(k) -
Nip
f '(k') -
NIl<'
N- 1
Ii ,kll±o,)+ I<
j! 1>(1
p P'(I±")-,, Ii
(2stu)t.
;0,,)1.
/<'(1 TO,)], (10.109)
:l!)O
III, SIIMMAIIY Of (}CO fOItMII/,A/;
(\lonvalli,ShillP;
hdidl.y amplitudes: 1
(BE)' 1"+
M ( +, +; +, +) -
p" -e,gT,; 1"~
M( +, - ; +, - )
-e,g1ijp~ (E~J l
M( - ,+;-,+) -
-e,gT:jp~ (e!J
i
.
M(- , -' - , -)
(10 ,110)
Squared a.bsolute vaJues of the nonvanishing helicity amplitudes, summed-", over th e color degrees of freedom of the init.ia.l st.ate a.nd the final state:
IM(+,+;+,+)i'
, ,s
IM(-, - ; -,-)I'
IM(+,-;+,-)i'
=
IM(-,+;-,+W
, ,,'
- l6e-g -
, ,u
-I Geq9 -
s
(10.111)
l1npolarized squa.red matrix element:
,IM I' = -
I , , 52
3e qg
+ ,,'
(10,112)
su
Unpolariz.ed cross section :
dq
dt 10.18
= -
1 1rO' 3
Q' S ' T, 0'5
J
S3
I 2 ,
(10.113)
U
gq-+gq
Process:
g(k, a) Positive z~ ax.i$: along
rn varianLs: S
= 2(k ' p),
+ q(p,i) --> g(k', b) + q(p',j).
(10. 114)
k. t = - Z(k' k'),
u= -2 (k ' p' ),
(10,115)
Gluon polarizations;
J±(k') N- I
N[,k' -
f/ PCI ± '/5) + p f/
(2stu)l,
,k'(1 =f '/5») , (10 ,11 6)
liJ. f~.
V'/
')1
M(+,+; .... ,+} :::::
,
(E0)' [p'_(T'T');;-2£(T''I''J;,j,
M(+,-;+,-) -
-9',1,
M( - ,+; - ,-) _
-lp~ (E~)t rp'_ (T"T~)'i-2E(T~TQ)'J~ '
,11(-,-; - , - ) -
-2g'~~ (~)! [p~(J'T')';-2E(T'T·)',I. (10.117)
S(IUareC absolute values oJ the llOllvanish iug hd:city amplitudes, summed over the color Jegrees of fre-edom of the inilial state and the n:lal ij~ate:
IM( - ,+;~,+) I' ~ IM(+,-;+,-)iZ
=
1,11(-,-; - , - )1' - ' S9' [;: -
[::
IM(-,+;-,+)[2 - 18l
;~l
'
-~:].
(1 0.118)
Uu?olarized squarec r.1atrix dement:
(10. 119) Unpoial'ized cross section:
de
dt = lI"O}
1.0,19
2 .1
+u ,,1
2
[It1 -
'l
95U
.
(10.120)
liq - 'Y'Y
Proccsst
,(p+,i) + ,(p_,j) ~ o(k,)
+ o(k,).
(10.121)
fnval"iantll: .~ = 2(1'~ '
p_),
(111.122)
Pholon poJari."ations:
(10.123)
I(), S/iMMAIIY 01,' (i()/) fo'O/W/lI,M:
NOllvanislliug Iidicil,y a111 llli Ludcs :
M(+, - ;+, - ),
ku.
2
2e,Oi;-k ' 1-
M(+ , - ; -, +) k;~
2,
M(- ,+;+,- )
2e q V i j-k '
M( - ,+;-,+) =
2e,ci;-k
,-
k i~
2
(10,121)
,-
Squared absolute values of the nouva.nishing hcl icity amplitudes, summed over the co lor degrees of freedom of the inilia1 stale:
IM (+, - ;+,
_)1'
IM(+, - ;-,+)I'
=
~,
IM(-,+;-,+)I' -
12e;
IM(-,+;+,-)I'
I 12e: -, u
(JO,125)
Unpolarized squared matrix element:
,,' 1,,1-11' = 3~ e,"t' + tu
(10, 126)
Unpolarizcd cross section:
d" _ 2,,-,,'Q} dt
10 ,20
38'
I'
+ ,,'
(10, 127)
Itt
q q ---+ 9 I
Process:
(10.128) Invariants:
s = 2(p+ ' p_) ,
1= -2(p+' k,),
u =
-2(p+ ' k, ) ,
(10,129)
Gluon and photon pola.rizations;
N
(2stur~ ,
(10130)
flU:!
'I 'I
'!J .,
M(+,-;+,-) .11(+ , -; -.+) M(-,+;+,-) .11(-,+; -.+ )
(10 ,131)
.<;
Sqtla;e-ci absoluLe values of the nonvani~h i ng hclicity amplitudes, su:nmed ')ver the color degrees of ireec!.orn or the in it.i,,1 ~ ~ nLe and the fiual state:
M(+, - i+ . -W
=
I M(-,+;-,+) I~ :: ;6e!l¥ ,
IM(-.- ;- ,+)'
~
IM(-,+;+.-)I' - 16e~l':. u
(10,132)
Unpola.:i?.ed S
CTOlS
se<'tion:
de _ S,;!,llosQj " + 111 dl 9s~ ttl
10 .2 1
(10,134 )
7j q --+ 9 9
P:o('; CSS:
( 10,135 )
Invariants; t= - 2(p.~ ·kd . Gluo~
(10.136)
polarizll.tiollf;:
£=1.2 , (10,137)
1011
S/iMMAIIY 0'" 11(,'/1 /"(}//M/lf,AJo:
/11
NOl1va.ni:illi ug /u:lici ly (\,n lplit ndes:
M (+, - ;+,-)
_9' k21 [k 1+(1"1") ,). · ... k2+ (T'T')"] I;, Ek '+
,11(+ ,- ;- , -1)
ku -9., Ek !;
1+
[k 1+ (T'T ') i} -r'k 2+ (T 'T') i j] )
t_ &·
M(- ,+; +, - )
-g' ;~~+ [k, +(T'T' )'j
+ k,+(T'l")'i ] ,
111 (- ,+;-,+)
k;~ [k 1+ (1" '''') - g 2 Ef.:?,+ J, ij
+ k'2+ (T'T')] 11
'
1,10.138 )
Squa.red absolute values of the nonvanishing helicity ampJ it lHl e~1 summed over the coiar degrees of freedom of the initi a.l sta.te and the final slaLe: --,
IM( +, -; +, - )1'
IM( -, +; - , +)1'
48g'
IM (+,-; -, +)I' = IM(-,+; + , -)i" - 489' Unpolarized
~ qua.rcd
[~!)u st~l ..
'
[~> ::].
,,
(10, (3 9)
mat.rix dement:
-IMI' = -8g<(t' + ,,') [43
9tu
- -1] .'
(10. 140)
Unpolal'i zcd c!'oss section: dfY SfTo} t2 + u 2 -
10,22
1"'(
-+
(1 0.141)
If q
ProCt-'ss:
,(kd + ,(k,) Positive. z-axis: along lnvc? riants:
-+
q(q', i)
+ q(q,j) .
(10. 142)
j;,. t = - 2(k, . q'),
U=
-2(k, .q) .
(10. 143)
Photon pola. rizations:
N [,4 N -
Ii'
):,(1 ± ,5) - ):, ,4
(2s tu )- I.
.4'(1 T ,5)1, (10.144)
N'IIIVill ii"b iug IldirilY nll'[Jiillilil'>!: M (+,-;~ ,- )
M(-.-i+,-)
_ -2.:6" (::)!
M(-.+; - , I)
=
2e!,siJ
(::r .
(10.1451
Squared nbsolllt~ values of the nOlivAnish ing helici~)' ampliludes, summl'!
IJ"' (+,-; - ,-)rl
..:.
IM(+,-;-, +)I' = (I"pol,\riZe<1 squarec.l ma.trix
1111(-.+;-,+)(2 M(
=
12e~ ~.
.+;+,-)1' -
(10.146)
e1c:ncl1 ~ :
IMI' = 6t.~, t
1
-I' u'l
(10.117)
til
I1ll polariz(·c.I cross scct;on: (10.148)
10.23
91
~ 1jq
I' roccss:
9(k.. a) + o(k,) ~ q(q', i) + q(q,)) . PosiLive z-axis: a.lon&
( 10.149)
fl'
Inva.riants:
, = -2(k, . q'),
u=-2(k l "q).
(10.150 )
Gluon and phokm polarizaLiom::
J"(k,) = Nih A' ",(I±o,)+ "d' h(I+-,,)],
(10 .151)
".
10'l
/fl. iill MMMI Y ()Io' (J(.'/I fOIlMII/,AI:
NOllvanisllillg IleliciLy .utlplitudcs: 1
kl( +, -; +, - )
2cq ,gT"
(!:)' , 1
M(+, - ; - ,+)
- 2e,gTj;
(:~) ,
1
M( - ,+;+ , -) _
-2e q gT ", J'
(q+) , q_
M(-, + ; - ,+) =
(10 152) '\
Squared absolute values of the non vanishing helicity amp litudes , summed over the <.:olor degrees of freedom of the initial state and the fina.l state:
1111(+, -; +, _)1' 1111(+,_; _,+)1'
u IM( - ,+;-,+)I' _ 16e'g' , t' =
IM(-,+;+,-l l'
= 16e'g'!, q
u
(10.153)
Unpoiarized squared matrix element:
,- , 1•M I - e, 9
,
,t- + u,
"
"
(10.154)
tu
Unpolal'ized cross section :
(10,155)
10 .24
gg~qq
Process:
g(k" a) Positive z-axis: along Illvariants:
+ g(k"
b)
->
q(q', i)
+ q(q,j) ,
(10.156)
klo
s = 2(k, . k,),
t = - 2( k, ' q') ,
u =
-2(k, , q) .
( 10, 157)
Gluoll pola.riza.tions:
N["j [II
1/
ft,(1 ± '5) + ,k;
A' AU 'f 1.)J,
, = I , ?... ~
l.
(10. 158)
M(+,-i+. -) "" M(+,-; - ,+)
-~ (;:)' [,+(T"T');,
.11(-.+;+.-) -
y;c, (,_ )1 [,+(T"T');,+,_(T'T");,J q+
~
M(-,+;-,+)
;. 9-(1 ',/,"),.J
'
. (10.150)
Sqllil.red a.hsolute values of H:e Ilollvanishing helicity amplitudes, summe'd "lint" the color degrees of :reedOlfl of tLe initial state ,Lnd t.he final ~t!\.te:
IM(+,-;+, - )I' IM(+, - ; -·.+)I'
IM(- ,+;-,+W - 489 4
IM(- ,+; +, -)I'
~
-
[~~
-
::] ,
489~ [~ 9u
fl. 52
(10.160)
[lllpol?_rized MjU fl.rr~cl matrix elemen j,:
(10.161) Unpolerized cross section:
3r.o} (' - t.!~
do
dt 10.25
99
~
=
8
32
[49lu -
II
.!j2
(10 .162)
.
99
. Pl'Ocess:
g(klo a) Positive ~-axis: along Inva.riants:
+ g(k"
b)
--t
g(1.:3 , c)
+ g(k"
d).
(10.163)
kl ' (10.!.4)
Gluon polarizations:
trl. HI/MAlMO'
N
or (}(.'II/·'()IIMU/,A/,'
(2slut t .
-
.'
(!0 .165)
Nonvan ish ing helic.ity amplitudes:
M( .+;+. +)
-
29' ;:
(-ur~"'f"'m + sr f;=) ,
, I
/
dm
,11(+,-;+, - )
2g1 ~ ( - u f~b'ffl jcdm
M(+,-;-,+) -
2/ s~ (I r bm rJ.m + sr"" f~dm)
M( - ,+;+ , - ) -
29' _1 (tr·mfMm
+ sr=fbdm ) ,
,
M(- , +;-,+) -
29''=(_uj"1rm f ,d", + sr dm f'=) st
1
!\If(-, - ; -, -) -
2g2~ (_u j Uhn p :dm + sfo,dm jbcITI ) _ (10. 166) tu
+ Spulm f'xrrl) ,
.
,
,
Sit
Squared absolute values of t he nOl\vanishing helicity amplitudes, summed over the color deg rees of freedom of the initial state and the final state:
IM(+,+;+,+)I' - IM(- ,-;- ,-)i' IM(+,-;+,-)i' IM(+,- ;- ,+)I'
IM(-,+; - ,+)I' =
-
1449'
s'(s'
+, t' + u') t
?
,
U·
,,'(5' + I' + u 2 ) -
IM(-,+; +,-)I'
5'1'
1449' 1449 4
/'(5' + t' ,
,
5 U
'
+ u') '
(10.167) Unpolarized squared matri x element: --, _ ~
• (5'+/4+ u4)(s'+I'+u')
IMI - 89
5
2' t U,
.
(10.168)
Unpolari zed cross section :
(54 + I' + ,,'J(5' + t' + ,,') dl = - 8..'t'u'
dCT
9".a~
(10.169 )
(10.170)
q q'
10.26 l ' I'Oc(:~s
--I
q g'i
(di tfcrcr.t qu
(10.171 ) t ll\la.ria!l t~
.'
and dcfiniliOI:s:
'(p.-P-), t' _
Z_-..
.
- 2(p . ,.),
q__ '1+_ - q:J.. I(+J.'
(10.172)
I'hotoll polarizations:
( 10.173) NOllvauishiug heJici ty amplitudes:
M (t,t; t,t,
+l
-
M( +, t ; t , t, - ) M (t, -;t,-,+)
~
49~ eT~ .T~ . D
EqtJ..
""~J,
. l'
[II:_Q+\'I .. _1_ ... (q ... · k)J.
!:/eT~.rQ . B ' m. n,
2leTn Ta.8 """1
. z'
q+J. + E (L1 .... qi_q:!. ... (q+ . k)]t '
, 'Itl,q - _ .
_ l' [ LqHq+_q~+ (q+ . k)l'
,
'['''.'1''' B ' { q+] M(-'+;-'+l+) .. 22 g e "" ") (j... J. k - q+- q-+ (1+k. ) :\:1 (+, - ; -1, - ,-) -
M( - .+;-, +. -)
~
~
2 Q .. • q++ 2g e.TmiTnjfl q+J. k (. k) - q+-q-+ q+
[
2 1"" '~. B" gem'''J
2
quq
.,
J.
[LqH'I+ _q ~+(q_·
1'
l'
k)j2
•
" " ""
.,
1(1, S/lA/MAltY OF (/(il) FOItMlnAN
; Of;
,11(- -'- - +) )
t
1
,
., )
)
(10.174).
!
Squared absolute values of the nonvan ishing helicity amplitudes, summed ' over the color degrees of freedom of the initial state and the final state:
:,
,11(- , -;-,- ,- )
1111(+,+;+,+,+)1'
=
IM(-, - ;-,-,-)I'
=
=
32g"e' IBI' E'
Lq_+(q+' k)'
,
\
/
IM(+,+; +,+,-)I' - IM(-,-; -,-,+)I' -
8g'e' IBI'(q+ · q-)' E'Lq_+(q+' k)
iM(-, +;-, +,-)i1
8g4 e'IB I'Q:_ Lq_+(q+ ,k) '
IM (- , +: - ,+ ,+)i2 = IM(+, -;+, - ,-)I'
8g'e'IB I'q1. + Lq ~ +(q+ . k ) '
IM(+, - ;+,-,+W
(10.175) Unpolarize"d squared matrix element:
2 < '(Q Q')' 8' + s" + u' + u" 1J>.'fl' _ - -'99 c /v+ + /v _ it'
(10,176)
where
v_ :;:;:;
pq- " (p_ . k)(q_· k)
(10,177)
Un polarized cross section:
->
q q'.9
Process (different quark flavors):
q(p+, i ) +4(1'_ , j ) ...., q( q+, m)
+ '1'('1-, n) + g( k, a) .
(10.179)
l c varian ts a.nd definitions:
s -
2(1'+'
p_) ,
t
t'
i
,
(lO .17S)
q q'
1
.,.1,
"
d" =
10.27
1
u -2(1'_ ' q-),
u'
-2(p+ 'q_),
/J -
J
e Iefl_(/+_ ( "'+ 9+. ,.")Ii•
[k +q+- (T'T') '''' ~'V
+ Z · (k,, _)Z(k,q )~ .(1"1") 1 2(q_. k) "" "J
-Z· (k q )~ (T'T' ) 'j
•
+
"J
r,u
( JO. :80)
( ;luon !)olarizalions:
(10.18 1) Nonvanishing liClicilY
amplittHl~s:
M(+,+; r, r,+) -
4g 3BE9-t.l. , q.... -.jqHq · +
M{+,+; +,+,-) -
2IfB'(q, . ,_)q+" Eq+ _ v'9++Q-+ 2g~ Bq~... .L ql-.L
M(+,-;+,-.+) -
,
q+-)qHq: +
M(-,+; ·' .+.+) -
_ 2g 3 E9-tl. (q+_) t
M(+ . - ; +,- , -) -
-
M( - ,7; --,+,-)
-
q--
1- ~
2g:lB'q+.1
('++
q-~-
q-+
2g 3 B'g....L q -.l. 02 q+-Vq
,
r
1/1, S /I M MMI, Y I! i" (I (,'J) VI! It M IJI. AI;
111H
M(-,-;-,-,+) -
29" B( q+ ' q- )q+J. Eq+_,jq++q_+ '
M(-,-;-,-,-) -
49 3 B'Eq+l q+-Jq++q-+
"
(1 0,182)
Squared absolute values of the nonvanishing helicity amplitudes, summed "
over the color degrees of freedom of the initial state and the ftnal state:
,
IM( -, -; -, - , _)1'
IM(+,+;+,+,+)I'
, /
IM(+,+; +,+,-)12
IM(-,-;-,-,+ll'
IM( +, -; +, -, + )1'
IM(-,+;-,+,-)I' -
IM( -,+:- ,+,+)I'
=
IM(+,-;+,-,-)I' -
'i
4g6C('1+ ' '1_ )2 E'q+_q_+
~.
4g"Cq~+
, q+-q-+
(10, 183) Unpo ial'i zed squared matrix element:
(10184) Unpo larized cross section:
d" =
"'~ [ 7(p+' q_ ) 108,,' (p+' k)(q_ 'k )
7(p_ . q+) (p+ 'q+) + (p _ . k)(q+ . k) - (p+ ' k)(q+·
(p- 'q- ) - (p_. k)(q_. k) "
2(p+· p-)
k)
2(q+ . q- )
+ (p+' k)(p_ ' k) + 7"Cq-=+"",k,,"·)7"(qO=_L,k'"")
1
,
,
10 .287]q-7/ (/"I t'r
« lilfcrr:lli (IUark flu\'ol's): , (p,,;) +q (p_ , j)
~
g(", on ) + q'(q_ , n) + 1(k ) .
(1 0.180)
IHYMiants and definitions:
.,
"
-
2(p+ . p_ ) ,
t
2(,,· , _),
t' -
Z.. _ -
B
~
- "2 (Pt 'q+ ),
u
-
-2(P+ · q-) ,
-2(p_ . q_),
u'
-
-2( p_ . q+ ) ,
-
q- -+ 9+- - q:J.q;.L,
q++q-- - q:l.q-.L,
Z_,
(10.187)
=
1'11111011 polarizations:
N' - l
=
E2 :32k+k_ }}.
.\Jonvi'.nishing he:icit.y a:nplitudc&:
M(- ,+ ;+,-,+) M(-, + ;- , + , + ) M h - ;+, - ,- ) 111 (+ , - ; - ,+, -) M(- ,+ ;+, - ,-)
M( -, + ; - ,+, - )
( 10.288)
1/1, SIIMMAIlY
1111
qUI! /"O/lM UI,AI:
{)/O'
S(J ll i~J'(!d abso lute valllc';s of th e lionvanis hillg hdidt,y ;LOIpli t.uc!es, slln\lw~d ovcr lL<.: colo I' dcgrt!(~s of freedom of the inilia.l ::Iiaie a.ud lhe I1nal slale:
IM(+ ,- ;+,-,+ll'
lM(-,+;-,+,-)i'
=
=
9: (_IBI' , k+k_ ,q+, q- l
8g" e
8g4e'~ IBI'
1111(+,-;-,+,+)1'
IM(-,+;+,-,-)I'
IM(- , +; +, -, +ll'
IA'I (+, - ; -, +, - )1'
8~il e 2
IM(+,-;+,-, - )i'
8g'e'
IM(-,+;- ,+,+)I'
=
k+k_(q+ , q_) , q' IBI' -+ k+L(q+' 'I-l'
,
q:'+IBI' k+k_(q+ ' 9_) (1 0,190)
Unpo!a.ri"ed squared matrjx element: t ' + I"~ + ' + " 2 u 1l tV IA'I I' -_ -- 9-g 4 e'(Q tV P - Q')' 9 ss'
,
(10.191)
where Isee eqns (4.27)J
v.=
p+ p(p+'k) - (p_.k)'
v -
q+
(10.192)
, - ('1+ . k)
Unp o laJ:ized c ross sect ion:
\
d" =
(10,1 93)
10 .29
qq .....
7/ q' g
Process (different quark flavors) : (10.194) Invariants and defini tions: s
.'
-
2(p+' p_) ,
t' -
-2(p_ · '1-),
u' =
-2(p_ .q+),
/lU~!J.
'i '/ -
~' V' /1
111 n" · l
n _
1
(2 k+k _) t
=
4/ .• +'1+_
[2(T47'b). 'J'h + 4/ _+L - q: l. kJ. T~.IT"T6) . 'J " ", (q_ .1.:) 'J ".,. _ ({h
L - qt.l. kJ. r.b.( Tbp) 'J
k) (q+ ..
8' _
1 (2 k_k_)t
.
- CJ_ J..lI+J. ,
[2(T~T~)."~ '" I.
n'~
I
q_ lK- -q-l. kir.t.(Tarb) (q_. k) IJ nm
nm. '
1
- q++k_ - q+J. kl. T~.(TOT· ) (q_ .k) 'J " ....
C I[2
(q+. q )
~ 3 - k, L - (" . 1')("
'k)
+
,
7q+_ qq+' k)
71_+
+[""(, . . k) +
2q++ k+(,+ , k)
2q__ T
1
L (,., k) . (10.195)
GhlOII polarizations:
(10.196) N Ollv"'llj~hil\g
hclicily amplilUces:
-"(+,-; +,-,+)
-
g3 B q_ l Z+_
(q+ ' /[ )Jq++ q +
I
1\4 (+.-;-,+,+) _
M(-,T;+,-,+)
J. Z,-) (,++)1
MI-.+;-,.,+) _
93Bq" 1 1/_ - q+. /1-
_
gJB'q_.l.Z;
M(+ , -;+, - ,- )
M(+,-;-.+,-) _
(
q··- q·I·'1
q-+
)
gJB'ql.l.Z:j 1--('1+ . '1-)
('.t)1 q- I
.
(II_I)'
10, SIIMMAUY Or Q(,'/) rOIlMlII,A8
'11'[
M(- , +;+ ,-, -) g3 B'q"-.1 Z-+-
,11(- , +;- , +, -) =
(10.197)
Squared absolute values of the nonvanishing helicity amplitudes, summed
over tne color degrees of f,eedom of the initial state and the final state:
29'Cq:'_ (q+,q-)'
!M(+,-;+,-,+)I' - IM(-,+;-,+,-)I' IM (+, -; - , +, +l l'
_ IM'(--,+; +, - ,-)I'
IM(-,+;-,+,+l l'
- ; - ,+, - )1'
IM( +,
IM(- ,+;+, - ,+)I' =
_
/
2g6 (;q1 _
(q+.q-)' 296C q:'+ (q+ ' q-)'
IM(+,- ;+, - ,-)I'
Un polar ized squared matrix element.:
IMI'
=
29' [
27
7(1'+' 9+) (p+' k)(q+· k)
-
7(1'_ 'q-) (1'+' p+) (1' _ 'k)(q_ ' k) - (1'+ . k)(p+ 'k)
+
(q+'q_) (q+' k)(q_· k)
+
2(p+·q_) (1'+' k)( q_ . k)
+
2(p_ 'Q+)] \ (1'_ . k)(q+ . k) (10 .199 )
Un polarizcd cross section :
dCT =
x
t'
+ t" + ,,' + u'" 2 ,
S oS
"
6 (p+
+p- -
q+ - q- - k)
If'ih d'q_ d'k Q+OQ-3
ko' (10.200)
I/I"YI1, 'I II ~ 'I 'n
'I n.30 ]'roCl:ll!l
IJ (/ --+ f]f], (idcutical quark Ilflvorl'l):
q(p.,i) + , (p.,j) _ q(,., m)
+ q(q., ,,) + o(k).
(10.201)
!lIvilriunls and defini tions: =
2(p+,p_),
s' =
2(Q+'1_),
.!I
t' -
-'(p.. q.),
B =
(10.202)
Photoll polarizuliollS:
(10.203) Nonvallitdling helicit.), a.m plitudes:
M(+,+;+,+,--) x 1- .1 1'""'" "" nl [ q-+
M(+,-;+ , -,+)
M(-.+;+,-,+)
_
1+ .1
r"r"] '"
q++
mj
"1. ,',' /lI\l ll I :l/n · "f,' (! {.'f) ,"()!{ft{ lJ /, /J/';
r t·t
III ( , I ;
, I, I)
_.
.! (I '.' ( ' ~ .,
' If
/ / '/ '"
. '/
"U
,"
"J
(I •
+.l
l
l' 1
., - ........ ..
M(+,- ;+,-, -)
M(+ ,-;-, +,-) ;\.-1 ( - ,
+; +, - , - ) /
M(-,+;-,+,-) M( - - , - , - , + ) )
M (-,- ; - , - , -}
qt]1 1 B'}' [L'i++'i-+ ('I+' k/
' g2 ~q
~
\
x [qHT,~,;T~j _ q- 1 T~;I:~j l q+-
( 10204)
'I--
Squared .'tbsolute values of the nonvanishing helicity a.mp lit udes, summe~ over the color deg rees of frc"edom of the initi al sta.te and the 'f1nal stat.e: '
IM(+,+;+,+,+)I' = IM( -,- ;- ,- ,- W
IM(·-t-:+;+,+, - )I'
=
IM(-, ..-;--, - ,+)1'
1·1vf(+ , -;+,- ,+)I'
=
IM(-,+;-,+,- )I'
A/ 1 I .
• 1.1 li'
IA/'·I;I .
II'
1" 1-. +;+ .- . ")1' - 1-"' (+. ;- .+.-Jl'
8g.e;l fi l'l
=
o
qt -q: ~
~'_ q , +Q- - (9 + ' k)
•
IM(-,+;-.+.+W - IM (-, -:+.- .-W (10 ,2051 11ullol.".ril.t.'
!11
2
=
+ UrI)
-Utl'(ll'
_
j(.~l + "rIH.!s'_ tt' - llU')] ,
(IO .20G)
WhN('!SCC Nlns (,1.27)]
p, - (p_ ' kl
O
-- -
p-
(,,_,k) '
(/'_' k)'
( lU .:W7 )
<117 =
( IlJ .20S)
10.3 1
qq-qqq
I'J'lJ( '~~ (j d cn~jcal
q(p+. iJ
quark Oavors):
+ q(p_,))
....... q(q, t m) + q«({ , Il) + g(k, a).
l 11v«l"il1llls and d (~li llhion s:
, _ 2(p.· p ).
t -
-2(p-, · '1-),
t' -
-2Ip_ '1-) •
(10.200)
.. .. . I]
,.
~
Iti. S/IM .II ~ /I Y OF (W/I /<'Ol/.M InA N
III
B, -
,
il '
-
/
1
1
1.,
B, -
,
-Z"(k, g+ )T:,,(T'T a)m; + k+'I+_ TL(7'"T')mj
B; _
".
1 [Z(k ,q+)Z"(k, q-) (TaTb)Tb , k+[Lg+_(g+' k)J} 2(q_' k) n, mJ
- Z(k, q+)T~;(T'Ta)mj
c, _ ~_
[_4_ +
3 k+k _
2('1+ ' '1 -)
_
(q+' k)(q_ · k) q-+ - k+(g_ . k)
1
c, - 3
[
+ k.,.q+_ T,~;(TaT')mj 1'
q+-
\
L(q.,. · k) 7'1++
7'1__
+ k_(q+. k) + L(g_· Ie)
1,
2('1+ . q_)
q__ k+k_ + (q+ . k)( q_ . k) - k.. (q _. k) q++ - "+ig+ . k)
C,
1'
1[ 20
- 9' - k+k_ -
10(g+ . q-)
('1+ ' k)(g_ . k)
+
7q_+ k+(g_ . k)
+
7'1+ _ 1L (qT . k) j ,
q--
+ k_(q_ . k) (10.2 10)
.'
/f1 ..1I.
'1'1
• q'l!/
(:1 11 01 1 pll l l~ri:t. l\ !.io u~:
;'
=
/VI" /"
h(1 ±7,)+
h /1.
t(1 '1'7.)], (1 0,211 )
NOII V
M{+,+i +, +, +)
__
r
3 4g E q±iBl _ q:i R1 ] Jq l fq-+ l q... q-'
H M{+ ,+; + , +,- l _ _ 2g3{~+ .q_) [1':'l : _ q:.!.B~l EJqHq-+
111 (-1, ; 1,-, + ) , ... ,+ ) -
qu
- 1..
?9'8 q' " -1. q+_
M ( -I ,
q+_
Jq_+q! +
I
293 D~q:iqL
q--V ! .. 9-+ ' Q
M(-, +;+, - ,+)
Ml-,+ ;- ,+,+)
(,,.-.. )1 "-) ! ( ,.,
=
M(+.-;+,-,-) _
,
3
M(+, - ;- , T, - ) ,11(-.+;+,-,-) _ M(-, +;-,+,-)
M(-,-;-,-,-:-) _
_ 29 .B!.tq_.l. ('. ' ) ' fJ- q... + 29 3B q_ iq 1i
2 q- - Jq~+q-+ '
2g' B; IJ+J.q:?i
9+-/9++9:; 3
_ 29 (q., .q-) [qf .. SI _ q_J.B2] E Jq~+qq.. q-J
M(- , -;-,-,-) _
4g E
[q t iB:
J9++q-+
11+-
,Q-iB2] , '1--
( IO.2 12j
1,'1
II .';
,l..) l pl; ~ r,·d ...1): :111 1111" V;I.]. II":-: (I\'CI"
';,! .11 ,1/ , 1 Ii ).
(J
I-'
/-'OU ,II tJ ,. \ 1"
(,!( '"
Iwl ic il.,)' ill llp! iLlldc·. . , Sl ln l llH'. i \.111' c(ll lll' dl'l' n·t::; o f fn'('d'>ltl of 1.114' ilt il. i,1i :,1.;1.1.(' ,1 1111 III( ' fi n;"! :-; \.; t{., '; "
1!II( I .+ ;·j .
e,-l lI'
Il tlItVOIlli::l l i ll .I'.
IM (-,-;---, - .-)i'
=
~ 1(\'f" f'.~ ' [_~ + _C~ _ q+ +'1- -
'/+-,/- +
-I- q+ - 'I- + -
C,[,,++'I __
2(q+ · '1-
ll] ,
1;1/(+.+; +,+,-)1' = IM(- ,-; - ,-,+)i' 'I!I" ('1+..; "1_ )' [
=
},,-
-I- --.£'-- _ ("' ['1++ '1 __ + '1 ,-- '1-;
C,
q++rt--
'11·-1-+
IM (-I-.- ;+ ,--. -I-W
IM(+ , - ; -,+,+)I' _ 1.11(- , +;+.-,-)[ ' -
4gi,C ,(/ '1+-1 .. + " 49"('--'!,q+ _
q + i ,fj --
"C'f/:" + '.1J!.__
IM( - . +; +. -. + )1' _ IM (+, -; -, +, _ ll' =
'_1 J1] , ;
q+ -i·fj+-'i - +lf--
IM( -, +;--, + . - )I' -
1,\/ (-. +;- .+.+)1'
- 2(1,
q+ +ft-j (l6C 1 l'++
IM (+,- ;+, - .- li'
_:.I
(10.2 I J)
q+- q- +
l rl l p{)l(lri ~ , .~d 'l qlIarcd I ll
iT' =
1;1
_
2[/ 27
{[_'7(I'/')('1_ +-'1- )
il)-
(1'+ .
'1 -l
+
'k )
7(/,- - q+ ) _ (1'+ - '!±l..__ (1'_ - ,.) ('1+ - k) (1)+ - q(q+ - k)
2(1'+
p- l
+
(1'+ ,k)(l'- - k)
(1'- -j') (q- - I;) 7(1'+' 1+) -I- [ (1'-" - () (q+ - k)
(1'-- ' "1+) (1'-- ' k)(q+ . /, )
_
],<, + ,"
('1+ - k )( q_ - k-)
(1'- - q+ )
(T'+ - J,) (,,_. k) - (1'_ ,1,)('1+ - k) 2(q+ - '1-)
1,,' -I- ,"
+ (1+ - ':) ('1- - I.e) -
(p+ - q- ) + (p+ - I.: )(q_. k)
+ u·' + 1/" tf'
(p+ - 1-- )
+ (]'-_' /,j(q _ . Ie) -I- (p+ ,1:)(1'_ . k)
~ :I
2('7+' '1- )
itp- - q-)
2(1'+ '1'- )
1[
+
\
10(7'.- 7'-) _ 1O(1f+''1- ) (p+' k)(p~ · k) (q+' k)(q_ . k)
+ ('
u n'
..
(f-- - 'i-)
(1'-1- 'q+ )
+ (p+
-+ /'
. k)(q_1 ' k) -I-
(I'~
, 1:)('1- - k)
1(S ' +S")(ll' -I- ll'/- S," ) } fl"""
'
(10 .214)
/11.'1'
I 'I II "
'/ '1
'1'1 r
oJ ,'1 j'I,'" I , I. ,-,',
,~'" I 1" 11:
-[{I'
,f"
'II)
q
(I' , A: )('I.' ~(P+ 'il . )
:l(I I ' Q ) ] ..;1+S"~lll ~ -t-IJ.' ~ Ip+' (-')(1'. , k) ·1 Iq+ ' k)(q. . !-) t1'
(/'-'1/-)
(r:-: k)( ,, ~-k) + 7(" ",
"I,)
I
[ (," . k )(q. · (-.)
, T
'{(p '(1_ ) (1'-' ' )1'1 · ' k )
2(11 , . P ) (I'I . k)(I'_' k)
X
10.32
(p+ 'q-)
I,'. ' ' )('1.' k) 2(q~' '1-)
+ (q+ · 1.-)(11 _ ' k)
' [ (JI- ' '7"T") (P+' Ij -) -- :1 (1'-' kH~I + ' k)- I -(r+ ' 1. )('1 _ . k )
_
---(f'I '/I ,I) -(I'. ' (-11'1.' q
+ (fI,
"
{p
' q- }
(p .I.) (q , . k)
1,,2+ :;/2 + t Z -+ -l'2 lI?l'
(P+ ' (]+) (1)-'(1-) I .,""'''' ' k )( q" ' k) (/L' I.- )(q_' k)
I O(p+ ' p_) _ 10(1]-; ' q_) ] ($' (p_ . k)(p , .I.:) (11_ - 1.-)( '/_ ' k)
'1 PI
-
+ $fl)( Il' + U1J' -SS'») Il'IIU'
,)QJii+ rf'ii_- dJk - .
+/)- - q"T" - q- - ,.; -
1(+01]-0
( IO.2I.i)
ko
ijq~q q 'l
1' I"UCeSH (iCP.fl Li(:a.l q u ,irk fl/'_ VO l'S): (10,:!J6) !11 ~;u'ia!lls
, "
aud ddi r.i l iOl'lS;
..
2(/, ... 'p_),
/
-
-2(,,_ . q.,) ,
" -
..
2(q~ ' If_).
/'
-
-2(p_ '
11-),
u' -
- :.!(p ~
. q.),
'1(p _ . II-d ,
D "" I t- E(2A:.Ic_Il • I k kj q- .!. - k-; k.!.q: j. ) '1.+' k )(q_ . k ) , ( IO.n7) P hulon !)oln rintions:
( IO.2 IS)
/II . SIIMMA flY Of" tJ('O VOUMII/,AH
Nonvallishing hclicity amplitude!;;
8g2eqT::JoTi'!n Eq+.1. lJ
M(+ ,+;+,+,+) =
.
i
,
q, _ [2k,k-q++q_+J'
M(+,+:+,+, " )
-
-
4g 2 eq T:: j Ti':r.q+.l.(Cf+' g_)B-
1
•
Eq+_ [2k+LqHq-+F
M(+,-;+,-,+) M(+ , -;-,+, +)
-/2g'e,Q:l.q+l.B [Z-+T;jT,~m - 2T" ·T? (q+' q_) OJ ,m
q+_[k+k_q++q::+)~
1
.,f2g2 eq T/jT:m q+l. Z_+B
)
(q+' q_)[k+k_q++q_+lt '
ivf(- ,+;+,-,+) kfh+; - ,+,+) M(+,-;+,-,-) 111(+,-;-,+,-) M(-,+;+, - ,-) M( -,+; - ,+, - ) M( -, - ;-,-, +)
V'igZe, q 1i!jT:;m q.t..l Z:+B -
(q+' q_ )[k+,Lq++q_+lt '
'2'
.,
v~geqq_ l q+.L B"
[z*-+ T'T'nm_2'J'a.r.a 1
\
ij
q+_[k+Lq++q':+J!
(q+' q_)
4g2eqT~jT;';"q+J.(q+·
nJ
,m
q_)B L
Eq+_ [2k+ k_Q++q_+l' M (- ,-;-,-,-) =
8g'leqT;:'jTttnEq+.LB'" i
.
(1O.2j9)
q+- [2k+ k-'l++q-+ l' Squared absolute values of the nonvanishing helicity amplitudes, summed over the color degrees of freedom of the initial state and the final state:
IM(+,+; +, +,+)I' - 114(•
IM(+, +; +, +, _)1'
-' - _ _ )1'
'"
1
IM( - , -; - , -, +)1'
III ..Y,Y.
'Ill -. '1 '1 II
1M(+,-;+ , -.+)I ' =
8 9
~
IM(-,+;··.
I,
q'- 11):' , ['
~ 1
~i k .... Jc
,M(+,·· ;- .+,+W
(q+' (J-)
)
I'
+
8
M(-,+;+,-.- )I' -
~
M( - .+;+,-,+lI' - IM(+,-;-.+,-)I'
Z -+ +
z·-t
3q+_ q_ ... (Q+· y_)
q;
4 1
1,
ltill
9 e~ k ... k_(q+ . q_)'
q: . 1BIZ
8 41
9 e~ k .... k_ (q+ . q_)'
~
IM(+, -; +,-,-)I'
1,11(-.+;-,+.+)1' -
8
+
2 'l- -q-+
.2qi+lll~[! 2 Z ... +Z" ... . . ( . ) + + . kof k _ l q+ 41_ q+ _q_+ 3q..._q_+ (q• . q
(:0.220)
= g e,
LJllpola rizcd sCJ'.l,ued matrix element:
I·uti'(,,1 _ where
u'~) + ~(u' + u")( ss' + ttl -
1It/)] , (10.221)
[800 cqns (4.27)1
"' --
p+
(p•. k) -
p-
(10.222)
(p •. k)'
Un?Qlarized crass section:
dn =
+fm'(u' xb~(p ...
10.33 Procctl~
.l..
+ u") + ~(u'"
p_ - q+ - q_ - k)
u")(s.'i'
(p- dJ q...
q-
cPr: .
q... o q-{) ~
uu'>]
(!0.223)
'jq ~qqg
(idcntic;d q l:Ark flavors);
fill'+, i) + q(p_ ,i) -- q(q" m) + q{q_, n) + g{k, a). Invariants Rnd J cfil1iLions:
s _
+ tt' -
2(PI ' p_},
(10.224)
If), SUMMAR\, OJi QCIJ JlOIlMUI,AI,'
,
,
,
, I .
./
~,
;
,
,1
!
j
,, ,
,
.~
(10,225)
,( ;JIIOIl pO:l1rir.alious :
.",,-1
_
E2[32k+k_ I~.
(10,226)
NOllvanis!ling helidty amplitudes: 49JD 2 Eqh jY1(+ ,+ : -~,+,+)
-
M (+.+; +,+,-) = M( +.-, + ,-.+) _ M(+ . -;-,+,+ ) l'vf (
q+.jqhfJ-+' 9Jn~q ' ..LZ+ _ Z_~
Eq+ _q __ ';qH q~~
g3 q_..L
[B1Z+_ + 2B2fJ
\Iq++ q + (q+' Q-)
. L q' .l.l
'J.,.-q- +
J
g3B 1Q+J.Z_+ (q+ . q ).jq++ q +
, +i+,-,+)
111(-,+:-,+,+) M(+ ,-; +,-, -) .<1 (-;-, -;- , +,-) j\.1( - ,+;+,-,-) -
l' B'q+J.. Z:+ (q_+)' q.,. - (q+. q-)
g3R'qj ..lZ: + (9+ . q ).jQH 1- + ' 3
M(- ,+; , ! , - ) _ .'11(-, - : - ,-,+) -
9 9".1
[ BIZ~ _ + 2B 2Q: ..L9+.1]
)9++q-+ (1+' q-)
1+- 1-+
g~ B 2 Q_.1 Z.+_7.:+
£'q+-IJ--
M(-,- ; -, - , - ) =-
q+-
J9-+q!+ '
49 38 2£'q1'..1.
(10,227)
q+- y'1"T"+Q- .,.
Squ3red absolute v;tlues of tIll;' non vanishing helici ty amplitudes, summed over t he coior degrees of freedom of the initial state and the fina l state:
IM(+,I;+,+, +)I' = IM(-.-;-,-, - Jj' =
169 6C I E" q+-1- +
,'i1I MM~
/II.
tty Of (J(.'I! fO lW/!/,A/:
IM (+,+ ;+, + ,- )I' -
, IM(- ,-; -,-,+ )I -
IM (+, - ;+, - ,+)I' -
IM(- ,+; - ,+,- )I'
IM (+, - ; - , +, +)1'
-
"gOC q' IM( - ,+;+, -,- )I' - - , ±- , (q+ ,q- )
IM(- ,+ ;+ ,- ,+)I'
-
IM(+, - ;-, +, - )I' -
4g6 C,( q+ ' q- )' E'Q+ - 9- +
2.C 9 , q-2 + (q+ -q-)
,
IM . . (+ '-I' +' __ , 'I'
IM(- ,+; -, +,+) I' -
J
"
Unpolarized squared matrix element:
6 IMI' __ 29_ {[ 7(p _ ' q- )
-
27
(1' _ ' k)( q_ 'k)
+
7(p+ -q+ ) _ (1'+' 1'- ) (p+ ' k)(q+ 'k) 7(1'-+"-' - 7k')-!-: (p-_'-, j';'7 c)
2(p_ '9+) - (q+ ' k)(q _ 'k) + (1'_ 'k )(q+ 'k) (9+ - q-)
+
7(p+'p- ) [ (1'+' k) (I'- 'k)
+
2(p+ 'q-) + (1'+' k)(q _ 'k)
1/' + /"
\ + u' + u" 55'
7(q+'q_) (p- , q- ) (p+ 'q+ ) (q+ 'k) (q_ ' k) - (1'_ 'k)(q _ , I,) - (1'+ - k)(q+' k)
'q+) 2(P+' q-) 15' + 5" + u' + u" +(1' _2(1'_ + -,--"-"'-i'c.,-'-=-'--;-c 'k )(q+ ' k) (1'-1' 'k) (q_ - k) tt'
_~ [ (p+ q+) 3 (1'+' k)(q+ 'k) _
+
(p- -q- ) .L (1'+'1'- ) (1'_ 'k)(q_ ' k) , (7'+ ' k) (J>- 'k)
1
l O(p_, q+) _ 10(1'+ - q- ) (u' (1'_ ' k)(q+ ' k) (1'+ - k)(q_ 'k )
+
(Q+ '9 -) '
(q+' k)(q __' k)
+ u")( 5S' + it' ss'tt'
Uti ) } ' (10,229)
/U"•.,~ "
"flf ""'
""HI
71 'I
t ~ upolari~<"fl nu~ M'("l :OI1:
Q-1 I[
I / (1
1
IORJ"Zs
=
-
7(p_"/J_) (p_ " k)(q_ ".I.)
(y+"q-) (q" ' kl (q. · k)
,[ T
7(/I"r'l/ i ) (ll +"P-) + (p+ "k)(,,+ "k) - (p+ " k)(p_ "k)
" Z(P_ "q+) + (p .. k)(" . kl
+
7(p, · p_1 (p_. kl(p . . I)
-'
2(p+·q_ ) 1/2 (p,. k)(, • . k)
;-::'"'7.\-;:,="
2(1'.... "q- ) l"~"' +(p . . k)(q,· I) + (p_, k)(,_· k)
-3
[(P"") (p,. k)(q" 'kl
_
,,'
7(q,' q.) (p_. ,_) (p, . ,,) (q,. I )(q.-T) - (p . . k )(,. . I) - (p , . 1)(,,· I) 2(p_ "q+)
I
+112 '1 U1 +ll t2
+
(p_ .q.) (p .. k)(, . . k)
+
(p,.p .) h'.· I}(" • . I)
1
lO(p_" q_) _ lO(P.,. "q- ) (u2 (p • . kl(,,· I) (p, ' k )(q • . I)
+ sa + u1 + U'7
+
It'
(" . ,. ) (q, . k)(,.· I )
+ ut2 )(u.. + tt' -
,,'tt'
,I rPq+
(10.230)
q+oq-O /.:o
10 .34
'(q
~
n
lfU')}
q
P rocess:
(10.231) Positive z· axIs: along kl " Ddi:litions:
.
t
=
2'
'J ,'"
( ' 0.232)
Photon !x>larizatiuus:
i = '2,3 ,
N,-1
=
4 E[P~ ki+(p'" ki)l~ ,
i=2,3.
(10.233)
1/1, "liMA/AUI' Oi" ()CI) i"OIlMIII,AN
1211
NOilVclllisliiug hdicity
alllplitlld(!~:
M(+,+;+,+,+) = M(+,+ ;+,-,+) M(+ ,+; - ,+ ,+) I
./
M( + , -; +,+, - ) M( - , + ;+, - ,+ ) M(-,+;-,+,+) }If(+ ,-;+,-,-)
M(+, - ; -, +, -)
\
M( -,+;- ,-,+) 1
M( - , - ;+ , - , - )
3 -
'Z'(k
') [
- e,O;jPJ.'3 , P
(1" k, k3)(., k) E3 "k . p_ 2+ 3+ p' 2
1'
l' 1
M (-, -;- , + ,-)
') -e,36;jP.I.'Z·(k > "p
'' '( __ , ___ ) =
_
~ Y.J
,
f
,
)
[
(p' , k,) k) E 3p_'3k2+ .k3+ (' p. 3
4e;5;jE'P'.1.
.1. '
(10,234)
[Ep'_k,+k3+(p" k,)(p" k3 )] , Squared absolute values of the nonvanishing helicity amplitudes, summed over the color degrees of freedom of the initial state and the final stale:
IM(+,+;+,+,+)I' _ IM(-,-;-,-,-)I' IM(+,+;+,-,+)I' _ IM(- ,- ;- ,+, - )I'
48e 6q E3 p'+
k,+k 3 +(F' , k,)(p' , k3 ) , Ge~p~(p" k,)' £3p'_ kJ+(rl ' k3) ,
"
1,11 (+.+;-, +.-)1'
-
1.11(-.-;+.+.-)1'
,
IM(-,-i +, ·_ ,·_)I
IM(-.+;-.-.+)I'
IM(-.+; >. - .+)1' - IM(+.-;-.+.-)I' 1"' (-.+;-,+.+)1'
~
IM(+.-;+.-.-)I'
Unpoliu·i7.ed squared matrix element:
,
L(P' k,)(P·· 1;)I(p , I ,)' ~ (p'. k. )'1
1/1.111 -
2e:(p.p') ,=1
J
-
(lO.2.16)
ITrp .I,)(P' . k;) i=l
Unpolarh:ed cross section:
, do
_
'Q'( "') L(p·k, )(p'.I,)\(p.I, )' +(p'.k, )'1 () I P I' . "" 16r.~El
~-----,,---------------k, IIp' . I;)
ITrp.
('0.237 )
10.35
-yq -.. 9"Yq
l'rocess:
7(1,) Positive z axis; along
+ q(P. i) ~ g(l,. b) + ,(k,) + q(p·.il.
(l0.238)
k"
Ddiuitions;
Z(k'"p')
:=;.
I ;+P'-
- k"ilPj,
i=2,J.
(10.239)
-, /(1 ,
,,(I AI AI A II Y IIr <)(.'/1
/o'U II M U I.A II'
Gluon l:lnd phol,o ll politrization.!:i:
,r±(k.)
Ndji " ,k,(1 ± ,,),,) - ,k, j/ p(l 'F ")',)),
f
;"(k,) -
N,[ ,k, ji p(l ± ")'.) +
,l±(k,)
N, [P' " ,k,(l 'F ,,),,) - ,k, j/ p(1
j/ ,k,(1 'F ")',)1,
± ")'5)1 ,
)
,, ( 10,240)
M(+, +;+",+) M(+,+ ;+,-,+) .I
M (+,+;-,+,+)
"
E.p-' e?T~ I 2g , j;P.1 [ k2+ k't(P" k,)(p' , k,)
M(-, +;+, - ,+) -
ge!T}';p.iZ(k"p') [EP'.:'k2+(:'+k,){P" k3 )]
M(-,+;-,+,+)
ge;T},k;.1 Z(p', k,) [ -
i:, _
\
1~
M(+, - ;+,+,-)
~ 1
k>+ ], Ep_k3t(P" k,)(p" k3)
t
M(+ ,- ;+,-,-) M(+,- ;-, +,-) ,11(-,+;-,-,+)
ge;Tj,k2.lZ"(p', k,) [ k2t ]' k, _ EP'_k3t(p' , k,)(V' k3)
, III , .~.~.
"(11-- r17'1
;11 [-,-;+,-,-)
M(- , -;-,-,-) =
(1U.241)
'jqllared Ilbsobte values of the non vanishing helicity ampiii.udes, summed "V(:1' the color degrees of freedom of the initial state and the final sta.te:
IM(+,+;+,+,+)1 2 -
IjH{ - , - ; - , _. • ~)l2 -
IM( +, +; +, - , + )1'
IM(-,-;-,+,-)I'
-
IM(+.+;-,+,+)I' -
IM(-,-; +,-,-)i'
64g 2 e:S 3 p+ h+k3+(p" k2)(P' ·l'J) , 89·jc~P'+(p'. k2)~ E.1Y~k3 +(pI·
=
k3) ,
8g 2e;p+(p' . h3. )~
EJp_h+(P" k2 )
16g 2 e:£'p+1''! k2+kH(P' . k·l)(p' . k3 ) ,
IM(+,-;+,+, - )I' -
IM(-,+;-,-,+)I' -
IM(-,+;+,-,+) I' -
q + 3f IM(+,-;-, +,-)I' EP'_ k2+(P" k1 )
IM(-,+; -, +,+)I' -
,
8g2e4p'. kl
IM(+,-;+,-,-)I' -
,
2 8g 2 ei4 p'+ 1.2+
E/UH(P" k, ) .
(lU.242)
Unpolarized squared matrix element:
, I " I' M
-
L(P' k,)(p' . k;)[(p· k,)' + (p' . k,)'[ 8 , ' ( ') E'"-'_ _, -_ _ _ _ _ __ 3"9 cq p. p :I ntp k;)(P" k, )
(10.243)
;e:o I
Unpolarized cross secLion:
;=1
"(U P k + l'-!P . - 2k -)d X 3 , cPk~ d3~ . Po k10 k30 3
P.
(10.244)
1/1 . .\·//MMliIlY Of ' QI.'/J rOIWUI,A8
10.36
,,
.IJ q -> TY q
Process:
g(k l , a) + q(p, i) ..... '1(k,) Positive z·axis: along k1 . Definit.i ons:
+ '1(k3)+ q(P' ,j) .
-
, ki--P.l"k iJ.., P+
Z(k"p') -
k,+ p~ - ki.lp~,
Z( p,, ki )
(10.245)
i = 2,3.
( 10.246)
Gluon and photon polarizations:
)
/"(k3)
-
N 3[,k,
N1- I
-
E'[32p~p~]t ,
fi
p(1 ± "fs) -
N,-l -
4E[P~k2+(p' . k,)]t ,
1 N3
4Ef/J~ k3+(P' . k3)] ! .
,
Ii " .ka(1 'F '15»),
(10.247)
Nonvanishing hclic;ty a mplitudes:
M(+, +;+, +,+) =
'"
4ge'T··E'p'· '1 J' 1.
l' 1
11{
1
- ,+;+, -, +} -
M(- ,+; -,+,+) _
"Z(k'3, P 'J [ Ep:k2+(P',k,)(p'.k k3+ ge,'T'j'P.l
) 3
14(+,-; + ,-,-)
__ 9,.;'li~i~:~1.7.· (!~ 1
k,.
A:1! [
~'1-
1~'I(kH (P"
]j
kz)(p' · k.l)
M(+, -:- ,+,-) -
ge:'ljip~Z'(k3,p') [EP~k1+(:~~'2)(PI' kJ)] t
M(-, +;-,-,+)
2 'Ta ,. [ -
gt.q
E'p' ]' JiP.!. k2 +kl+( p" k2)(P" k:d
IH ( - , - ; 1,-. -)
M(-,- :-, +, -) (10.248) Squared absolute values of the nonvanishing helicity amplitudes, surmtlcd f)Ver the color degret!s of free
-
IM(I , +: ·I ,-.+)I' -
IM(-,-:-,-,-)I'
IM(+,+;+,-,+W
IM(-, - : - , +,-)I' -
-
64g2c,r £ 3';'+ • k1 ... kl+ (p' · 1: 2)(1'" kJ) '
8,q1 e:p'+ {p" .1:2 )2 E Jp'_I.-J+(p' · .1: 3 )
,
892e:pt.lp' · k,)2
IM(+, +: - , + ,~ )I' -
IM( -', - : +, -, -) 1' - E3 '- k2+ (p" p
IM (+, - ; +,+,-w -
M( -.+;-, - ,+J I2
IM(- ,+: +, - ,+) ., -
IM(+ ,-:-,+,-)I'
1"'(-, +: - ,+, +)1' -
IM ( -1-, - j +, - , _ )11
~
k~)
,
169~ C 4 Ep' p'l 1 +-
1: 2... k)+ (P'· k1 )(P" 1:3 ) '
Sle4, p'+k"3+
t..'p_ .l:1 dJi . 1.:1 )
,
' PH Sg2 C·lq p+ Ep'_k,+ (p' . k,) .
(10.219 )
Unp()[al'ized sq uared matrix element:
,
L:(P' k,)(p' . k,)I(p, k, )'
IMj2 -
~l e: (p. pI) ; ><-1
+ (p' . k;)'1
3
rTfp.k,)(p' . k, ) ;;1
.
( 10.250 )
". /II. -,,(lMMAIIY OF
quu
f OIl MU I,A/,
Ullpolitri;t,ed cross section : 3
'Q.I ( ') :L(P· k, IIp' . ki)[(p . k;)' + (p' . k;)'l asa / p' P !.;i=:.!'_ _ --,.---_ _---c_ _ _ __
d'7 -
12".'E2
J
IT (p . I.:i)(p' . ki ) 1= 1
(10.25 1)
10,37
)
'(q --> ggq
Process:
I'(k,)
+ q(p, i) ~ 9( ""
Positi ve z~ axjs: along Definitions:
b) + g( 1.:" c) -1- q(l'" j) .
(10.252)
k1 • i , j = 2,3,
i = 2,3.
(10.253)
Photon and g illon polarizations;
,t;±
N'[h
Ii
\"
,{1(1 'f "IS) - j/ ,{1 h(l ± 1',)1,
N,J
(10.254)
Nonvanishing helicity amplitudes: 3 rI' [ 8 1! M(+ '+;+'+'+) = p'k k '(k~ 1.) , ., ( k) ( k) + 2- 3- "~3 1"-"2+'3+ pl. " p"',
2g' e
, " (
"" _
)_
9'e,1"i Z( k, ,1")
[k1 _ Z ' ( .1:,. k3 )2{p', k3)(T~T")ji
X
_,+ + _) =
M(+ o
,
"
~ ,".', qf'J..
1"+.l: 2 _1.-3 _(lC2'
,11(-+"+-+).," X
(,'" k,) ]' E 3P'!k2+k3+{JI ' ~'2 )
+ k3 _Z · (k,l. k~ 1Z (pi, k1)(1"Th)j.1'
E~ -'/'-
[ ~'l)
X [,1,:1_ Z( k:, kJ )Z' (P', kJ)('f)T")ji
.
[
r, . ,+, + - 2P'+ k:1_k~_(~ . .1:3 )
k 1+kJ +(}I ' k2 ){p"
1t
~'3) _
+ k". Z{ kJ' k, )Z -(p', ~·l)(T'T·)iol
gJ e.P.iZ(k3 ,P')
[
- 2P'_ " l_k3. (k1 · '=3)
k" Ep'!k}+(p'· k, )(p" k3)
]!
[1.2 _ Z(k1, .1:3 )2 ' (p', k3)(T~T")J' + k,. Z( 1.3 ,1;1 )Z-(p'. k ~)( 1"'70 ),;1 '
:H(-+'-+-)"" a'e,p'.i. Z(k~,p·) [ kH , , ., 2p+kl _kJ_ (k~· 4)) Ep'!k;)+(rt· ~'1 )(TI' kl
)
]!
x [k 2 _ Z( kl' kJ )Z' (p', .l:3 )(1,or<),1 + k,. Z( .1:3. 1:,)Z' (p', k, )( T
M ( -, -; +, -. -) X
'
[1.-2 _ Z' ( 1.:"
-=
91eqp~Z - (k~,P')
2p+ k1 _ kJ _
[
(l;-:- k3)
.1: 3 )2'(1", k3)(T·TC )ji -I
.(:3 _
"
Z' (kJ, k1)Z(p', k, )(T'''1~)Jil •
1." ]' EP"k,,(,;" k,)(,;' k,)
+ kJ_ Z · (kJ,k!)Z(p',k1l('.f"T~»),] ,
M (-+"--+):r. r/t-yr/i. , , ,. Ptk2_ k3 _(k," kJ )
I
[
- 2';,k,. k,.(k," k,)
x [kz_ZV (k~,k3)Z{p',1.:3)(1'bT"}),
]t ,
M(+ _" _ 1 _) _AlC~pJ.Z·(Ji"J'P') , '
1.-1+ EP'!kJ_-(p" l:1 )( P'- , kJ)
I
[
,
E';. ]' kH-k3+ (p" k2 ){P' "l'3)
x k,. Z-(k" k,)Z(P', k,)( T'T')" - k,.Z ' (k" k,)Z(p', k,)(1"1')"
I'
III.
811M .lJ AIII' O F C,C'I) FoI/Mlll,M;
M(--- ' - - - )= 2g'e,rJ,[ E' . "" )/+k,_k3_(k,. k,) p'_k2+kH(P" k,)Ui· k3) X
[k, _Z"(k"k3 )Z(p',k3 )(T'T') j;
]1
+ k3 _ Z' (k3 ,k,)Z(p',k,)(T' T ' )j;]
.
(10.255) Sq uared absolute values of t be Ilonvanishing helicity amplitudes ) summed over the color degrees of freedom ot the initial sta.t e and t.he final st.ate:
1Jl.'I(+,+;+, +,+)I'
=
IM( - ,-; - ,-,- )I'
_ 329· ,;E3 [9k2+ (p" k,)
+ 9k3+(p"
k,) - p~(k, . k3 )]
3k",k3+( k, . k,) (p" ",)(1'" k,)
IM(+,+;+, - ,+)I'
=
IM(-, - ;- ,+,- )I'
_ 4g' e;(p" k,)' r9k2+(}i ' k,) + 9k3+(p" /';, ) - p~('" . k 3 )] 3£3)I_k,+(k,· k,)(I'" k3)
IM(+,+; -,+, +)I'
=
IM( - ,-;+, - ,- )I'
.1g'e;(p · /" f [9k 2t (p" k, ) + 9k3 t (p'· k,) - P~(k2' /;,)] 3E3 p_k,.(k2 · k,)(p" ",)
1'\;/ (+, _;+, +, _) 1'
=
IM(-,+; - ,- ,+)I'
8g"e;Ep? [9/';2+(;/' /';3) + 91.:>t(p' · /,;,) - P~( k2' 3k,+I.:3+ (k,· k, )(p" /';,)(p" ",)
/,;,lJ
, TrIM,
~1- ' ~lr'l
· IA/(· ,·-;, ,-,+)1' - 1-'1 (+,-;-,+, _· iI' 91.: 2:(" . k:\ ) + !.l k~ ... (ll . kl )
P+ (kl ' k3 )j
-
3EI'.kHlk,· k, )( p" k, ) I"(-,~;-,+ ,+ )I'
IMI+ ,-;~,- ,-)I'
-
_ 4g~e:k~+ [9h.. (p'·
Dk 3..,.(,I · k~ ) - p'+(k 2 . k3 )] 3Ep~kH ( k2' ka)(p" k3) k3 )
'"T
(10.256)
IIIl !,olmizcd s quared rua.trix element:
,
Dp -k,l( P' - k,)i(p - k,J' + (P' . k,)'J X~i'~'
____~,,-______________
(1 0. 257)
ITt, -kiH p' - k,) ;=1
lillpo );.ri'1.N] cross sect ion;
uluQj
O(p - k,l(I' - k,)
+ 9(p . k,)(P' - k, ) -
721f1E2
Ip, p')(k,
- k,)
(k:z.~)
,
L;lp' k, )(p' . ki )l(p - k.)' + (I" . ki)'J
,
I1 (p·k.)(P'-k,) ,., (10.258 )
P~ocess :
9(k". ) + ,(p,i) ~ g(k"b ) +o lk,)
+ ,(p',j).
(10.259)
Posit ive z-lU"is; ;dong kl'
hH'ar:ants ar.d clefir.i lioll5: .,
_
2(k r · p) ,
t' =
- 2(P ' II),
-u' _
'- 2(k2 'P).
Ifi. Si /MMAIlY ()f (1(,'/1 f'r)/(MIILM:
13(;
Zip', r.",'J
p+' k' 1_
-
-
P"k 1. i.l, i = 2,3,
(10.260)
Gillon and photon polari za.tions:
j±(k3) -
N,If! ,p);,(l 'f 1'5) - /<3 ,p'
.N-l ,
-
E'[32p~p~lt ,
N,'
-
4E[P~k2+(p"
•N3-
-
4E[P~k3+(p" kJ)lt,
'
p(1
± 1'5)),
k,)Jt , (10.261)
Nonvanishing helicity amp litudes:
M(+, +;+,+,+) = -
4g'e,E'p'.i. p~k,_IEp~k2+k,+(p'. k,)(p"
[Z ' (p',k,)(T"T');d
X
M{+,+;+, - ,+) -
g'e,p';Z(p',k,) [ -
p'+k,_ X
1
kJ )),
p~k;~(TbT');;l, 1
(p'.}';,) ]' E3p'!kHk3+(P" k3)
[Z(k"p')(T'T')ji -
p~k'+ (TbT')jil
'
, M(+,+ ;-,+,+) _
g'c,P'.i:Z(kJ,P') [ p~k,_ X
[ZIP', k,)(T'T')jd p:Lku (T 'T')ji]
2g e,p'J. [ P'+k,_ X
,
1
2
M(+, - ;+ ,+,-) _
(p'. k3) ]' E3 p':k,.kJ+(p'. k,)
EP'_ ], kHk,+(p" k,)(p" k,)
[Z'(p', k,)(T'T')i' 1-
p~k;J.(T'T')jil
'
III ,'I.~.
!I'I
• #"t 'I
"h+,+,-,+)
,11 (_.,+;-, +,+ )
-
k,_iBp"'kHk,,(P" k,)(p"
[Z ' (kl,pl)( T41'~)ji - 1J~kH ( TbTQ)i,l
X
M(+ , - ;+,- , - ) "'"'
kz_ [Ep'! 1.:2 + kh (p' . kz )(p' . kJ) j ~ X
M~+,-;-, + .-)
[Z( k"p' )(T'T');; - pck,+(T'T')j,[
g2e q pi.Z ·(k 3,P')
-
k,lIt
P'~k1_
- -
A~J '"
[
Ep'!kH(p' · k2)(P'· k.,)
x [Z(P', k2)( PT~)ji
2g2eqp~
M(-,+;-,-,+) _
[
P'+k 2_
1t
+ p~ k1J (TbT" Ii' J
1t
Ep'kl! kJ_(p'· k2 )(p'· ka)
x [Z(p\k2l(T"rtb; + p~k2J.(T~T")jd '
M(- , -', + , - , - ) _
1
._g1 e9 P'J.Z· (k3,[/) [., (t!. kJ) t lI... k2 _ /'. :Jp'!k2 + kJ • (p' . '-"2) X
M(-,-;-,t,-) -
[z-(,,', k'l )(T~Tb)li + p~ ,1;;..1. (T~T" ),;] g~Cq[lJ.ZW(P',1.:2 )[ p'.~k2
,
(p'.k 1) ]' F;Jp':kHhJ' (p'. /;;3)
[Z' (k2 ,pl)(PT")ji - P:.k2 (T"P)ji] ,
X
M(-,-,-,-,-)
,
+
49 , e.q E'~ .JJ'J..
1"+k2 _[ErI_k2+k3+(P'· k~ )( pl. kJ)I~ X
[Z(p' , k2 )(T"T6 )J'
+ p~ k1..:. (T~T")j, 1 (10.262)
Iii . .o;IIMMAlt.Y Or ((UfJ I 'OJ/MUI,AI, SqlH!.f'cd Ll-I)SO!llj,(~ values of the nonvil.fliHldtlg hdidLy ampiitLHh::J, 1')ull1l1l(~d over Lhe color degrees of freedom of the IIlII,in.l staLe and the lim!.! state:
IM(+, +; +, +, +)1'
IM(-,-;-,-,-)I' .,
329";E'[18(1"· k,) + 91"-"2+ - Ptk,-J 3k,+k,_k3+(p" k,)(p" k3)
IM(+ ,+;+ ,-,+)I'
IM( -, -; - , +, - W 49'0;(1'" k,)'[18(1'" k,) + 9p'-"2+ - p+k,-J 3E'p,-k,_k,+(P' . k3)
IAf( +, + ; -, +, + )1'
"
IM(-,-;+, - ,-)I' 49'":(1'" .3)'[18(p'· k,) + 9p'_"'2+ - p+k,-J 3£'p'_k2+"'_(1'" ",)
IM(+, -;+,+, -)I' -
IM(-, +;- , -,+)I' . 8,q'e;E1"~[18(p'· k,)
-I- 91"_k2+ -
p,+-",-]
3k2+",_k3 +(p" k,)(p' . 1:3 )
IM(-, -I-;+ , - ,-I-)I'
IM( +, - ; -, +, _)1' 4g'e;k5+[18(jI· .,) + 9p'-k2+ - 1"+"'-] 3Ep'-"Hk,_(p' . k,)
IlvI( -, +; - , +, -I-)1'
IM(+, - ;+,-,-)I' 4g'e'P II 2+ j18(p'· k 2 ) + 91" - k 2+ - p'+ k2- J 3EP'_k,_k3+(P' - k3) (10.263)
Ullpolarized squared matrix element: 3
IMI'
4
,_ tt') 2::(p - k,)(p'. k,)[(p' kd' + (p'. k,)'] uu ,,;;'-"_ _--;,--_ _ _ _ __ __ 72(k, ·k,) 3 IJ(p. k;)(p' . k;)
'(9 " 9
= 9 e,
.,
S8 T
i=l
(10.264)
f/L19.
!1 'I .... II V'I
do
_
, Q'I 5 761r~
?~"t
.=1 (10.265)
10.39
9q~99q
J' roC(:s:;:
(10.266)
.
!,(J~i~ivc
z-axis: along kl . IJdinllio::ls; i,j,-2,3, i = 2,3,
i
=
2,3,
F . '0P't.4[PCk,.-2(P'.k,)+P'. k><+2(P'.k,) !J k2_ k3_
+ k,.(rI· k, ) + 'HlP'· k, )] + 3G ['hlP'· ',)17.',. + 2(p'. ',)J (k:l' k3)
p'+
k,_(k-z· k3)
+ I.:~+{"P" }.-J)[p'_kH + 2(p" k~)l + Z"P'_lkH (p'. k3) + k3+ (p, k~ )l]. 1.:3
(k2 ·k3 )
1.:. _kJ_ (10.267)
Cleon polarizaliolls;
;;=
N,JR R'
/.f . Ni'
)<,(1 ± ",) + '"
N,[,.;,,, Ji(l
Ii 1'(1 'F ",)J,
± .,,) + J/ P )<, (1 'FO,)[,
. 4E[P~k,_(P'. k,)[ t ,
i = 2,3.
(10.268)
III, ,'I11,IWilllf' VI" qCII! /o'Ol(MII/,i\ Ii
NOllval1isliillg hclidLy a,J:Jpli Lndes :
111(+ -I-' -l1
X
")
+ , T' ) -_
2g31'~ :2
[
P'+k,_k3_
.8
3
p~k2+ k3+(P"
k,)(p" k3)
]l
{(1"1"T');; Z(k" "3)Z-(1" , k,)Z-(1", k,) (k, ' k3) +(T"T'T');; Z(k" k, )Z-(p' , ",)Z'(p', k3) (k, ' k,)
+(T'T'T')j,2p~ k;1. Z" (p' , ",) _ (1"T'T');/~ k;JZ ( Ie" k,)Z'(p', k,) (k, ' k,)
-\
+(T'T'1");,2I'1. k;J. Z' (p', k,) _ (T'T'T');/1. k3.l Z(k" "3)Z'(p', k,) } (k"k3 )
M(+,+ ;+ . -,+) =
93p~Z(k"p') 2p'Jk,.k3 -
[
,
(p',k,) J~ E'p';:kH k3+(p" k3)
,
1
.. ;
{ (1'"T'T')'; Z· ("" k3) Z(p', k,) z (1", k3) (k"k3)
X
"
+(T'T'T');; Z '( "3. k,)Z(p', k,)Z(p', k3) (k"k,)
+(T'1"T');,2p~ k,J. Z(p' , k,) _ (T'1"T")j/~ "21. Z -( "3 , Ie, )Z(1". k3) (k" le3 )
+(T'T"T');,2p~ k3.1 Z (p', k, ) _ (T'T'T")', p',L k3.1 Z· ("" "3)Z(p', ", ) } ,
Ili( +,
+; _, +, +)
X {
(.k.--' k3)
.
p'1 Z(k3' p') [--;.d'(Pc.'-;-'.:::k3'1-)-:--;--:- ]I 2p';k' _"3_ E3 p':!k,."k3+(p" Ie,)
= g3
(T"T'T');, Z· ("'."3)(Z(P', k,)Z(P' , k3)
,
(-'" '",,:l)
+(T"T'T');; Z'( k3, ", )Z(p', ",)2(1" , I,,) • (k"k3) +(T'T"1");,2p';, k2.1Z(p', k3) _ (T'T'T');, p',L k2.l Z' (k 3 , k,)Z(p' . h'3) (k, ' k3) .
+( T'1'"T');,2p'; k3.1Z(p' , k,) } _ (1"1"1") p1 kuZ'(k" k3)Z{p', k,)} J - . (k"k,) •
,
'" "'(·.-; 1. + . -) = x
-li'I',
[
/.;,1-
p;k,,_kl _ kuk:;Jil. k~i{v ' .t...,)
]1
{ rr-'J . 'I>C}J' Z(J:" k, }Z"(P·.,l:1 )7.°(//, k:l) (kl . .1.'3) +(T"1'7
'I,. Z(k, . ',)Z·(P', kYZ·(P'. k, ) (', · k,)
+('J ot p'l'<)j,2P'1.. k;J. Z' (p' 1£3) _ {""'-r cP)j/J.. k-i.1, Z( ,(:3, ,I:,)Z"(1, *3) (k2 · "3)
-:-(T·:r-1·~»). 2p'J kiJ. Z"(P', I.',) _ (rcTT)J/~ kjl. Z (I.'~, . e3l?· (p' , (k,·k,)
M( . . • ;~,_.+) _ 9'P',7.(k"P') [ k,. 2p1k2 _k, _ Ep'!~ .... (p'· k,)(P" .e3) X
1:,)} ,
ji
{ ('r.'1·~r)Ji Z(kl' k3)Z' (ll , k, )Z' (P' , k3) (k, . k,)
+:T"l"'1" )"
Z( k" k, ) Z, (P' , k, )Z·(P'. k,) U:2 . .e3)
_ (T'1'"1"'),.2p', .;, Z · (p·. ") _ (T'T" ]'") j , k;, Z( '" k,)Z·(p' . k,1 J' (k, . k3) + ( 1~/"1"),.2P', ';' Z· (P', k, ) _ ('I'T'T" ),/' k;, Z( k, •• ,)Z·(P' •• ,)} (.,·k,) ,
M (- .+ ;- .+.-) = g'';,Z(.,.P') [ k,. 1/ k 2I4 ,_k,_ S"ok" (v ' k,)(P" k,)J x
{ (1'"1"1')" lI( k, . ") Z· (P'.', )Z· (". ") (k",I;3)
+(1" 1"'1")" Z( k,. k, )Z ·(". k, )Z ·(". k, ) (k,· ,1:3)
-(T~rT~ )J,·2p~ kiJ. Z' (p' . .4:3) _ (T~rcT~)J; p~ k,zj, Z( "=3. k'l }Z*(p', k3) (k, . k,)
+(T"T"1")J. 2p:'" k~J. Z· (1/, kl )
_
(T'T'T~)j.P'.1 kj.l Z{ k1 • k3 )Z"{P' , k1) } (,(:".1.-3 )
,
III. 8 11MMAitY OF qe() fOUMUI,AlI
M (+ .-;+, -,-) =
!h~Z"(k"P') 2p~k, _ k3_
[
, kH Ep':'k,+(p" k,)(P"
k,i
]t
x { (Ta T'T' );i Z"(k" ka)Z(;i, k,)Z(p', ka ) (k,.ka) ,
+ (T'T' T')'i Z "(k" k,)Z(1, k,)Z(1, 1'3) , (k"k,) +( T'T'T');,2p';k u Z(;>' , ka) _
(TbT'T");;P~ ka Z "( k3, 1::,)Z(1, k,) (k, ' ka)
+(T'T'Tb);,2p~ ku Z( p' , k,) _ (T'TbTa);iP'jkuZ"( k"
k,)Z(1 , k,)} \ (k"k,) ,
M I+ _'_ -'- _) , , , ' " X
g'p'~Z'(k3,P') 2p"f.k'_"a_
[
k3+ Ep'!'.k2+ (p" k,)(p" "3 )
]t
{ (TOT ' T 'Li Z' (k" k3)Z(p', k,)Z(p' , k3) (10, , k,) +(T"T'1");i Z ' (k" k,)Z(p' , k,)Z(p' , k3 ) (k, ' k,)
+(Tb T'T');i2p~kaZ(p', 1.'3) _ (T'T' T " );ipjkuZ"(k" k,)Z(1 , k,,) (k, . k, )
+ (T'TaT')ii 2p~ ku Z(p', 10,) _ (T'T ' T '); i p~ k'l Z"( k" ka) Z(p', k,)} (k, , loa) ,
M ( - , +'1 - - +) -~,
X
gap~ ,2k P+ ',-"a-
[
Ep' k'tkJ+(P" k,)(p" I.,)
l}---
{ (T'T'T' );; Z · (I." k,}Z(p' , k,)Z (p', k3) (k, , loa) , +(1"1"1");; Z"("3 , k, )Z(p' , k,)Z(p', k3) (k" ka)
+(1"1"T');i2p~ ka Z( p', k, ) _
(1"1"1'");/i kuZ' (k" k,)Z(1, k~) (k, . 1::3 )
+(T'1"Tb );i2p~ k31. Z(p', k,)
( T'Tb Ta);'p~ k3~ Z"( k"
_
k3)Z(1, (I., , k3)
k,)} ,
A1( --,-;+.- ,-} =
9::'jl;·(~~lt ll) 214.1.:1- ka_
[_ ._ J/I _'J";,) ]1 1_:"'(' /.:'J I L::I~ (,,, • k, )
x { (" -'/'.1 OC)I' /, ( 1:" 1.:;1);";"(/1', 1.-:) z· (,I , .l~ I)
(,=".4::1)
+(1'7'T'I"
',IZ' (V, ',IZ' (1". ',)
Z( t"
(k,·k,1
+('1'''''7'lj,2p~ ';l Z' (p', k,) _ {T'T'T"I'/' k;lZ{k" k,IZ' (p', 1,1 {ll' kJ}
+('I
k,IZ' (p' , ',I} (k 1 ·1.:3 )
M(-,-;-,+,-J ~ g~p'iZ'(k,.P'J [ 2I4k,_1.: 3 _
(p'·l·,)
EJ~k~_k3"' (P"
kJ)
,
]t
x {(T'rT"),; Z( k" k,IZ' (II, k, )Z'{v, k,1
(l, . !,) +( 7"T'7"I;; Z( !" k, )Z' (p', t, IZ ' (P' , !,I (k, · k, )
+(T'T'T" J,,2p~ k;, Z' {p', !,J _ (7"7"7"1/' k;, Z( '" k,)Z ' (p', k,1 (k,· k,1
+(1'1"7"I,;2p~ kil Z' (p' . !,I _ (1'1'7"1/ ' k;, Z{k" k,)Z' (p', k,1} (k,·k,)
M(- , - ;-, -,-I =
2g'p',
[
t4k, _I.)_
E'
P_kHkJ"(p" k'l )(P"
,
]1 .tal
x {WT'T')" Z, ( k" k,IZ(p', ' ,IZ(p', k,1 (k, . k,1 + (7"T"1')" Z'( k"
',IZ{",,', )Z{P', k,) {k,·k,1
+(TiT.P)JL2~; k2J Z(p'. k.:) +('J""T" 'I" ),,2p';. k;n Z( p', k,)
_ ('rT~r )Ji p~ ka Z( 1:,:). 1.·,)2 (p'. 1.'3) (k,· k, )
(rrT"')J' rli. k3J. ZO!k" k3)Z( p'. ~'l) } (.1. 1 ' 1.,,) . (10.2691
'"
III. SIW Mil It Y or' IWO 1,'()ltM InA II' S
over the color degrees of freedom of the iniLild staLe and lht: fiJli.d slnle: -
IM(-,-;-,-,-)I'
-
4g6E3F kHk3+(P' . k,)(P"
IM(+, +; +, - , +) 1' -
IM( -, - ; -, +, - )1'
-
g6(p' , 10,)' F 2E3p'_ k3+([1 . k3) ,
IM(+, +; -,+, +)1'
-
IM( - , - ; +, -, - )1'
-
g6(p' . 1.:3 )' F 2 EJp'_ "H(P' . k,) ,
IM(+,-; + ,+ ,-J I'
-
.\ IM( -, -1-; -. - . _1_)1' kHk3+ (P' , k,) ([1 . k3) ,
IM( -I-,+ ;-I- , +,+W
kJ5 '
gGEp"F
IM(- . -I-; -I- , -.-I-JI' -
IM(+,-;-. -I-,-)I' -
g'k'3± F 2Ep'.k2+(p'· k.,)'
IM(- .-I-; - . -I- , -I- )I' -
1,11(+.-;+.-.-)1'
g6P2+ F 2Ep'. 1.:3+(p' . 1.:3) .
-
,
..,
(10.270J Unpolrnizecl squared ma.trix element: 3
• L:(P' ki)(p'· k,)[(p' ki )' -I- (p'. I.:;)'J IMI' = .!L '-"=""-'----,,--------108 3
"
. ,
.
fI(p . k,)(p" k, )
.,
i=l
x{
lOr . ') _ 9 [(po kd(p'· 1.:,) -I- (I" k,)(p" kd P P (k, . /';,)
+ (p . k,)(p'. k3) + (p. k3)(l"· k,)
"
(k, ·k3 )
+ (p. k,)(l"·
1.: , )
-I- (l" "d(l"· k3) ] +
(k, . k3) x
_ /
81
(p. p')
[(I" 1.:,)(1" . ",JI(p· 1.:,)(1" . k3)
-\- (p , k3) (P' . k,)) (k, . k,)(k , . k3)
-I- (I" k,)(1i . 1.:,)1(1" k3)(P' , k,) -I- (1' . k, )(p' . k3 )J (k, . 1.:,)(1.:,. k3)
+(p , k3)(p' . k3)1(P' 1.:, )(p' . k,) -I- (p . k, )(1" (k" k3 )(k,· 1.:3 )
. kdJ
l} . (10.271)
{llIp()lnri"l!~d t:ro,;~ S,~~iUlI:
,
tiff
, L (;' . !;)(rI . k'; H(,' . !;J' + h' . '; )'; { . 1
H!I'
I 72811"Z '
=
:I
._--
I O(p· ,I)
(V' k')JI(I" k, )(;, · k;) -9
,.,
l(p,
1.: 1 )(P'
. k2 ) + (p. k~l(p' , kl ) + (p.
k2 ){P' . kJ) - (p' "J){p' . 1.:2 )
(I, ·1,)
(k, . k,)
(p .•,)(",. I,) + (". k';(p'· k,)] + ('·1' k.))
+
l(p, kJ(p'· k,H(p' k,Hp'· ") + (I"
81
(1"1")
k,Hp'· k,li
(I,·k,)(k ,·!,)
+ (,,",,)(,,,. I,H(p' k,)(rI' k,UJLI,)(V', k,)J (I, ,1,)(1" k, )
+ (I" k,)(P" k,JI(p' k,)(p" ") + (V' k,)(p', k,)J]} 1..·3Hk~,
(k,. "(k
XU
I+P -
Procus:
k
1
k3)
,)tfJkl d:lfJ d~; ,.
2- ,-P
(10.272)
k'1O 1.:30 Po
,(k,) + '(p, i) -, ,( k,)+ ,( ',) + .(P', j) .
Positive z,~xis: along
(10.273)
k"
Defiaitions:
i : 2,3.
{I O.274}
Photon polAJ"iztl.tions:
/'(k,) ~
N,I"'"
,k,(l'f ,,)- !;
",,'(J± ,,)J,
i=:l ,3.
(10.275)
/II. ";IIMMAIlY Or' qCiJ FOUMUL,u,' NOllV~llishiJ1g
hdicil.y llillplitudes:
M( +. +; +.+,+) =
l' , l' L
M(+.+; +,- ,+)
36 "Z'k ') [ (1", k,) e, 'jp~ \ -,,1' E3 p'3k2+ k3+(P" k3)
M(+,+;-,+, +)
') [ (p" le3 ) k) e,3.5ij p "Z(k ~ '3 , P £' p_'''k42+ k3 + (' P . "2 4
M(+ , -;+, +, -) M( -, +;
+. --, +)
Af(-, +;
- , +,+) -
M(+,-; -,+ ,-) 1'd(-,
+;-, -,+)
M(- , -', +, -
M (- - ' I
1
I
, -)
+ -) I
M(- , -', - , - , -) =
(10,276)
Squared absolute values of the nonvanishing hdicity amplitudes, surnmed ove, Lhe color degrees of freedom of the initial state and the final state:
1111(+, +; +, +, + )1'
I M(-,-; ~ ,-,-)I'
IM(+ ,+ ;+, -,+)I'
I M(- , -;-, + , ~ W
k2+k3+(P" k,)(p" k3 ) ' 6e~p+(p" k,)'
£3 p'-- k3+ (p' , k3) ,
IMI ; ', +; -, ; ', " II' = 1.11(-,-;+, ·· , - W jM( ",-;+,+,-11' M(-, f-;
t-,-.-:-W
-
-
IM(- ,+;- ,-,+W -
-
M(+, -;-.-,-)I'
IM(-,+;-,+,+II'
IMI+,-;+.-.-II'
-
~
~ ~~ f ~JI~(1/ . ~:;j)'j gl,I_~'H(P"
~'l)'
l:.!ei Ep+ll! kl+k3+(pl, k2 )(p" kl ) ' 6e~6 p'.+i(;3+ "
Cp'_k,+(p'· k,1 '
6etP+ 6 , k2 H EP'_ k, +(p' . k,1 . (10.2771
111 .poiariz<..oC squareci matrix element;
,
Dp , k,)IP' . k,)!(p' k,)' + (p' . k,)' 1
IM!2 - 2e:(p.p') '=.'----,,--------
(! 0.278)
IIi•. k,)(p' . k,J
,e:: I I II I polarized cros~
section:
,
"Q' ( 'P') Dp, !·,IIp' · k,IIIP' k,r'
du _
(
,p 16;t"1,£'2
!.~
,
+ (p'. k,)'1
IIip , koJ(p'· •. )
,.,
(10279)
1(k,) + ;;(p, i) 'g(k" b) + ,(k,) + ;lp',). Positive z·axis: nlong
k\.
Dcnni tiouo::
Z(ki,p') Gluon and pholon
k,+P'_ -
polarizat;on~:
k:i.p~ ,
. .) '\ . t=p"
(10.281)
III , ,I'IIMMA /l.y
Nil
-
4E[p~k3+(P" k3)]t .
Nonva,nishing helicity amplitudes:
4g e~ Tl~ E2p'.i
M (+, +;- ,+, +) M(+,- ;+, + , - ) M( -, +;+,-,+ ) .M (-, +; - ,+,+) M (+ , -;+ , - , - )
1v} (+ , - ;- ,+ , - ) M (-,+;-,-,+)
M (- , - ; + , - , - )
M (-,-;- ,+, - )
0,.. Q(.'O i"O/l.MII/,M:
(10.282)
'\
i\1 (-, - ; - , -,-) =
( 10.283 )
'-:' !II,II"I'11 idl'll)) lIte va.!uc'i or .he t!OIlVlI.ui:'lhing !Idicity B.mplitude~. ~umtllcd ,,\","r liu.l color ticgr<..1.!3 of frc(.'drllll or ~h c initial sla~c and the": final state:
"( +.+;+.+. +)1' M( +.+;+.
, -I )
2
1"(+.+; -, +, e)l'
IM(-. -;-.-, -)l' -
-
IM( -, - ; - , +, -
-
1.11(-, - ;+,-.-)1' -
)I' =
"(+. -; -,+,-)1' - 1-"(-, +;- ,-,+)1'
61gZe; £3 p'. kHkJ+ (r/' k, ){Jt " kl )'
8g 2 e;p'. (P' "k2 ): EJv_kl l (p' " kl ) '
8g1 c;P'_(r/" kl )1 El"P'_ J:a (f/ ' k,}'
16g1c"Et.I''2 i + = k1... kl+(p'" k)) (p'" kl )
•
IM(-, +;·I ,-,+)i' - 1.'1(+.-;-,+.-)1' IM(- , ..... ;-, +,+)I '
V IM(·I. -;+, - .-)I' - £,0,.(1"· k, ) . B,q2 C: +·t:i1
=
(10 .284) l ll'l)oiarizcd lI<1ua:"cd ma.trix clement:
,
Dp · k,)(p'. k,l/(p· k, )' ·1 (F"
k,)'1 "' - - - - - - illp· k,)(p' .k,,)
,1111 _ ~gle:(p"p')~
(10.285)
,-, Un poJa rizclJ cros~ !:Icctio,,:
dn
_
'Q' ( 0'..,.0
/
, .) L (P· k,Hp' · k;l l(p · k,)'
p" p ~",,-,_
12;r'£l
,
+ (1". k, )'1
II (p· k,)( p'· k;) .:: 1
(10.286)
10.42
gli ~ n1i
Proccss:
(10.287)
III. "" I MMM'.Y 01" (JUI! F(}II.MIII,M:
-
Positive z·axis: ;doag kl' Dcftnili oll s:
Z ([J,, k) <, -
p+I k';- - P/L& kU,
Z (k; ,p') =
k ' - k" ';+p_ ';~p~
,
i = 2,3.
( 10.288)
Gl uon and photon polari zations:
}±(k;) N,·-
N;[.k; jJ j./(1 + 'Ys) -
4E[p~k;+(p' · k.)11 ,
1
"jJ' ,k;(1
'f 75)1,
t. = ? .... , 3 .
, (10.289)
Nonvan ishing helici ty amplitudes:
M( +, +; +, +, +) M(+,+;+.-,+) NI( +,+;-, +,+) M( +, - ;+, +,-) NI(-, +; +, - , +)
1' 1
;\-1(- , +;-, +,+)
ge;T;jk;LZ(P', k,) [ k,+ k,_ El''-k3+(p' . k,)(p' . ",,)
M(+ .-;+, - ,-)
ge~T;'kaZ ' (P' , k, ) [ kH t k,_ Ep'- k3+ (p' . k,)(p' . k,) ,
-
1
'.
~
M(4 , - ;-,+,-) = !I'".'l 0'Jl1J.'X' I (1':t'/'')[ 'i7l k (I k)(p' ·'3 k) "I~·H·l··~
M{- ,-;-,-, +)
-
~.
'
[
r
"'II-
2!/C q };JI J. kHk;,.~(p'· k,)(p'· k"l ) I.
1
'T' 'X ' II3,P') [ E3~k,+kl+(:I' II'" I,) k~)
M{-,-; + , -,-) -
!/e~ iiP.!'
M(- ,-;- . -:-.-) -
ge 7
MI-,-;-,-,-) -
'T';jPJ.'Z' IIl.P') [ Ell"3k'(I" k I,)(;I' _ 21
31
]t
r r
kl )
4 ge!I''('.~. E'rI.J. '2
1'0,290)
[£p'_IH",.,(p" I,)(P" I,)[l '
Squan:d absolute values of the non vanishing he:idly amplitudes, summed over lile color degrees of freedom of the initial st~le and the final stale;
IM(+,I; 1.1.,)1' -
(M(-.-;-.-.-)[' -
MI+, +; +, -, + )1' = IM(-, -; -,+,-)['
.
,
649'e~.Pp
k:,a.k3+(P' . k, )(P' . 1:3) , 8g1e:p~(p'
-
. k1 )2 £3]1'_1:31 (pl. k's ) ,
~
8g'e:p... (p'· k3)' EJp_h.(pI - 1.-2 )'
[MI I ,+;-,+.+)'
-
M(-.-;+.-,-)['
1011(+,-; +,+, -)'
-
16g1 e:Ep... p! lM(-, - i-,-,+W 1,.1,_(1'" I,)(p" 1,)'
1011 (-,+;+, -,+)['
-
8g'et~ k' I + 3+ IM(+, - i-,+ ,-)11 = Ep'_ I" (p' , I,) ,
lM(- , +;- , +,+ }12
-
[M(+.-;+. -,-)I' - Ep'_ 1,_(1" , h,) ,
Sg2c!p'-+ k5~
(1 0.291) Unpolarized squared matrix element:
, L;(p. l'lIp' ·1·,)[lp· I,)' ''[' 1.'~J
-
+ (v"
1,)'[
8 , e • (P' P') ""=1.' _ __ _ __ 3"9 - _ _-;-_ 3 f
[1( p. I,)(p', I,) iel
( [0,'92)
III. Sli M AI A/I Y Of' QC/J I,'O/l.M II I,MI'
Unpolaril.(;u cross section: 3
,. 2:: (P' k;)(p' . k,)[( p . k;)' + (p' . kil'l Q [(p . p ) ,,'_""_ _--,;-_ _ _ _ _ _ __ 12rr'E' 3 k,)(p' . k;) '"
""
_
OS"
mp·
(IO.2!J3)
10.43
;q-.ggq
Process:
'Y(k,)
+ q(p, i)
Positive z·axis: along Definitions:
~ g(k" b)
+g(k3 , c) + q(p' , j),
(10.294)
kl _
Z(k"k j )
i,j
= 2,3,
Z(p',k, )
i
= 2,3,
i = 2,3 .
(J 0.295)
Photon a.nd gluon polarizations:
I;±
N,[,k, " ,(/(I 'f'Y,) - jJ,(/ ,k,(1 ±"I')]'
/.t
NolA P li(1 ± "Is) + j/ " A( I 'f "1,)1,
---
i = 2,3 .
Nonvanishing helieity amplitudes:
(10.296)
.HI
I' I- - +1 ""
I
,
I "
,
c-7.1 '
-,!J·lr:,,JI1;t, (k~~'l._ [-.-'C7.~(,,-/,,.'::;-'~). 'If/1-~:1_~':1_(~'1' ~':I}
1·,·:I,('kHk'~(J"· 1.::1)
x [,,,.7,-(k" k, )lI(.', j., )('1"'1")" + k,.7.' ( k" k,) Z(p', k,)(T'T'),, ] , ,\I ( I', +; - , or . +) =
[k,. Z' (k"
x
2;:~!~i~(t:;:L) [E3p~~~ ~J~31p' .k
2
!,)Z(p', ,-,)(T'T'j"
+ k,. Z' ( k"
, )]
l
k,) Zip', k,)(T'T'),;] ,
"'(+' _'+'+'_)~,p.k~_k3_ 9',,>,. [ E;,I. ( i;l· 1:,) kH-k)+(p" k1){p"
k3)
11
Ik3 - Z(k.l , k2 )Z' (p', k 1 )(rtr")'J + !'l- 2( 1:2 , k,)Z· (p', k3 )('J '~1 ~)'J 1 '
x
M(- t ,+ _ +)= 9 ',~Z'k V'- ~ J, >') , , "
2>',k,
'>.(', -k,)
[
kl+ f:p'!k,,(p' - k,)(p' - k,)
]1
x [k,. Z( k" k,)Z ' (,l, k,)(7'7")" 1 k,J( k" k,) Z' (p', ,-,)('1",/,'),;] ,
M( ,I; ,1, +) =
glt..P'~Z(k2''')
[ ., k1+ 2>"k,.k,.(k, - k,) Ep'!k,,(v - k,)(v - k,)
)( [1.'3_ Z( 1: 3, k2 )Z"(p', k2 )(T~T"')jj M(
. I,
,t,
,
+ kz- Z( 1.1, kJ )Z' (pi
I
kJ)(T'T"),; 1 I
- ) - g",p.Z ' (k,,>,) [ k" -2p,k,.k,. {k,-k,) Ep'!k,, (V",)(V-k,j
x [k,. Z'( k" k,) Z(p', k, )(7"1")"
+ k,. Z' (k"
'1( ' ) ~g'-;'"!"",'I";",,-Z-: ' (;Ck,.,,c,l",,,) /1'+--+--" , , " -2p', k, k, (k, -k,) x [k,.Z-(""-,)Z(I", k,) (T'1'"),,
[
11 1;
k,)Z(p', k,)(T"1")"
I'
k.. ]1 Ep'!k,,(p-k,)(V·k, )
+ k,.Z' (k"k,)Z(P' ,k,)(T"1"),,]
,
1(1 , SIIMMA II I' Of qe'f) /,'O/Wlfl,M: .
M (- -' - -) , , t"
=
l'
1
g'e,7'l Z'(k",)/) [(/1, k,) 2p'"k,_k,_(k" kJ) g'lJ"!k,+k3+(p" k,)
x [k,,_Z(k 3 , k,)Z'(p', k,)(1"T')'j t k,_Z(k" k3)Z~(P', k3 )(7"1"),j) , i\1( - _ '_ t _)= g'eqP1Z'(k"p') [ (p',k,) , , " 2p'+k,~ k3_ (k2' k3) E'p':!k H k3+(p" k3)
]t
x [k3 _Z(k3 ,k,)Z"(p',k,)(T'1"),j t k,_Z(k"k3 )Z"(p',k,)(T'T')ij) , 1
M( - _, - - _) = 2g'eq )tL [ E3 ], , , " l''tk,_k,_(k" k,) p'_k,+k3+(p" k,)(p" k3) X
[k3_ Z"( k" k,)Z(p', k,)(1'b1"),j t k,_ Z'(k" k3) Z(p', k3 )(1"1");;] . (10,297)
Squ2..red absolute values of the non vanishing helicity an1plitudes, summeu over the color degrees of freedom of the initial state a.nd the final state;
1"'1(t,+; t ,t,+)I' = IM( - ,-;-,-,-) I' 32(l'e;E3 [9k H (p" k3) t 9k3 +(p', k,) - p,+(k" k3) ) 3kH k3 +(k" k,)()f' k,)(p' , k3)
IM(t,t;t,-,t)i' = IM( - ,-;-,t,-)I' _
49'e;(p', k,)' [9k,+(p" k3) t 9k3+(p" k,) - p'+(k" k3 )]
3E3 p'--kJ+(k" k3 )(p" k3)
IM(+,+;- ,t,+)i' = IM(-,-;t, - , - )I' .
...----
494e;(P" k3 )' [9k H (P" k,) t 9k3 +(p', k,) - p't(k, , k3 ) ]
3E'p'--k2+(k" k3 )(p" k,)
IM(+,-; +,t,-)I' = 1111(_,+; _ ,_,+)1' 89'e;Ep'3 [91:2+(1'" k3) t 9k3+(p'· k,) - p,+ (k" k3)]
3kHk3+(k" k3 )()f' k,)(P" k3)
IAJ(-,+i-l-,-,-I-W == IM (+,-;- , ·I,···)I"J 1.tj1,:;kJ+ [9/,:.• +(11. k:d -I- !)k:1+h/· ,I,:·Z) - ' 1/+( 1."2 . kJ)j
3 {1:;O':h(k'J • k:J)(p" kl )
IMh-;-.+,+).' _
1,11(+.-;+.-.-)1'
~
494e~ kj+ [9'\-1+ (1'" kJ) + 9kJ+(II·
):2) -
P~(k2' kJ )]
3Ep'-k H (k 2 · k-J)(p'. kJ} (10.298) I i 1Jpoli1.ri~ecl sql:.arcd matrix dCnl('nt:
IfI'W
=
, L(p . k,)(P' . h, )!(p . k,)'
+ ()1 . k.1'1
,
(10.299)
..,
IT (P' k.)(p' · k;J
IJ npolarj;,;ed cross 8ectiol~ :
du
;
o-1aQJ
9( p· kz)(p'· kJ)
+ 9(1"
7217 2 £2
kJ)(p' · k1 )
-
(p. p')(k, . k3)
(kz ·k~)
,
L (P' k.)(p' . kd[(p · kd'
+ (P' . k;J'[
)( ~!-----;-,-------IT (P' k, )(p"
k.J
;:!
(10.300)
P :-oCCS 9 :
(10.301 ) Positive z-axis: along fl ' 1n~·a.riants a.nd de[lIj ~ io n:5;
,
"
-
2(k 1 ·p),
t
~
2(P' . k,J •
t' -
~
-2(k,· k,).
"
-
-2(k,p') .
-2(p . p') •
,,'
-- -2(k,· p).
'Ir,1i
I II
k"i+P_'
Z(k; , p')
-
k" ' "iJ.P.l
,,'IIMMAIIY
or (} (: /J
1"OIWUI, AII'
i = 2.3.
I
(J 0.302)
Gluon and photon polari:tations:
N,-'
-
4E [p',.k2+(p'. k,) ]t,
-1
"NJ
(10.303)
No nvanishing helicity amplitudes:
)\1[(+.+;+,+.+) =
p"k, _[El/_k2+kJt(p' . k,)(p" k3)] 1
fp~ ki.1 (T"T').j + Z' (p', k,)(T'T"k] ,
X
1
M(+ .+;+, - ,+)
_ g'e,PLZ(p', k,) [ (p'·k,) ]' p'.;.koE3P':kHkJ+(p'. kJ) X
fl, kH(T"1");; -
Z( k" p' )(T'T" );; J '
J'
1
g'e,p'LZ(kJ,p' ) [ (1'" k3V M( +" c; - ,+ , +) =. p'+k,_ E3 p'.!k h k3+(p'· k,) [p'Un (T ' T '),;
X
M(+, - ;+,+,-)
2g'eqp~ [ p+k,_ X
+ Z(P' , k,)(T'T');;]
£1"_ kH k3+(p" k,)(p" k3)
[p~";~(r'T');;
,
1'
l.
+ Z "(p', k,)(T '1'") ;;]
,
Ifi
n
II 'i - . 1J"t"if
M ( -~';'i-,-,+)
=
, .'I1C"lj 7.(£':1.111 [ __ . k:l -+ ]' /''+1''11~1f:!£·1+ (}1· k1)(p" 1.: 3)
X Ip.lki.l ('I ~'I '~)Lj + Z-(,l, k.:!)(TtT4)ij] , .0\1(- "-' -
. +, -)
-
k,_1 ',·ri.?kH
-
1.:2_[E~k'Hk3-+ (II· k1)(p" k3)] ~
)( [pI k2 _(TOTb)'J _ M(-,-;-,+,-) -
g~e,p~z.(£'3'p'){ ",. k,_
Z(k2,pl)(T~Ta )iJl '
It
k3 .. Eri.?k" (V · k, )(P' . k, )
[p~k1i(T·T~);J + Z(p', k,)(T~T"),j] ,
X
M(-,+;-,-,+) -
k,)11
Ip~kH(T"Tb)/j - Z ' (k2'P'}(1'~T")ij] ,
X
M ( - .-;+.-. -)
k,. I" . k,)IP"
2g","'i [
r/+ .',-
Er/_ k,.k,.(P' · ',)(1'" k,)
r
)( [p~k1.l(rT')ij"t Z(pl, ~){T~T")'JI,
, M {-.- ; +.-.-) -
.'"p:'Z' (k,.r/) [ (j1. k,) ]' p+k2 _ £3p'!khk:s+()J· kl ) X
M(- , -;-,+.-) =
fp~k~J.(T"T")i} - Z ' (pl,k2)(T~T")iJ] ,
r
g'"p', Z' lp·. " ) [ (p'. k,) P~k2_ E3p~kl+k:l"' (p'· k.,) X
.
M( - - ,' - , - - ) )
[P~kH(TOT~),j - Z· (kZ)p')(T~T")jJl . 4g' e 1 E'pJ.
- p', k,_IEr/_kHk,+IP" ',)IP" k,))l X
[PLkU(T'"T6),j + Z(P',k,)(1·T")i.iJ . (10.304)
III. S/I,\1MAUY
o/" qeD
/,,()/(M/lI,M:
Squared a.bsol ute values of the Ilonva,nisllillg hdki!.y amplitu
. IM(+,+;+,+,+)i' = IM(-, -; - ,-,-)I' 3294.;E'[18(1" , k,) + 91'~ kH - p'ck,-l 3kH k,. _k,+(p'· k,)(p" k3)
IM(+, +;+,- ,+) I'
IM(-,-;-,+, - )1'
IM(+,+;-,+,+)I'
IM( -, -; +, -, -
)1'
4g'e;(p" k3 )'[18(p'· k,)
+ 9p'-kH
-
P'+kl-l
3E3p'-k H k,_ (p" k,)
IM (+, -; +, + , _ )1'
IM(-,+;-, -,+ )I' 89'e;£1'': [18(1''· k,) -I- 9p~k2+ - p'ck,-l 3k2+k,_k3+ (p" k,)(p" k3)
IM(-,+;+,-, +) I'
IM(+,-; -,+, -) I' 4g4e;k5+ [lS(p'· k, ) + 91''-k,+ - p'ck,- l 3E1',-k2+k,_(p" k,)
jM(-,+; _ , +,+) 1'
IM( +, -,; +, -, _ )1' 4g'e;ki+[JS(p" k,)
+ 9p'-kH
-
J''ck, - l
3Ep'-k, _k3+(p" k3) (10.305)
/
Unpolarized squared matrix element:
, _
IMI'
" (9 '
= 9 eq
+ 9 ' _ it') 2:: (1"
ss uu 7? (kl . k,)
-
~'~~l~_
ki)(p' . ki )[(p· ki )' _
+ (1"
. k;)'J
o--'-'_ _ _ _ _ __
n(1' . k;)(p' . k;) 3
j:::: 1
(10.306)
I II, Jlul:, ri1.I't I ('!u!<,~ sf'\:liull:
Ju
_
(lO.30n
10.45
99-999
Process: (10308)
l'osiLive z-a.xis: along Definitions:
fl'
+ 2(p , k,.)
p = :~o&.___ 4 [ p~ -':1.,.
k~_
!J
+ kH(P"
-':3) ..... k3+(P' - 1.-2)] (k z · ~-3)
~ pi k3~ '
k3 _
+ 36 [k2+(P" P'+
+k3 +(p'· k3)[P'_k~+ + 2(1/' k2 )1 + 2p' kJ_ ( k l
·
+ 2(p' , kl )
k3)
k2)[]I_k3+ + 2(p'· kJ)1 kz_(k2' k3)
[h.. (p' · I.-J l + 1.-3... (7)' .1.-1)1], /';1_k3_ . . -__ 'J ,3. 1,)
Z(k" k,)
-
ki+kj_ - k:.lkJJ .
Z(p', k, )
-
,. k'd, p~ki_ -P.l
i = 2,3,
Z(k"p')
-
k,... P-' - k-,.lP.l' I
i=2,3 .
,
(lO.309)
Cluon polarizalio:ls:
i=2,3.
(la.3l0)
III, .... II MAItlIIY ()f.' Q(:{) "'ONMIII.M:
NO Il V;tlJisilillg lwlkil.y cllllp li l..udcH:
2g"l'~
111(+ , +;+ , + , +) = [ ' /i" , p';",_k, _ p'_k2+k,+(P" k,)(p' , kJ)
]t
x {- (T'T'T');' Pl. kj,L Z(k" k,)Z' (p' , k,) (k"k,)
-(TaT' T') ,1,L k;l Z("3, k,)Z' (P', k3) 'J (k"k,) , +(T'T'T' ),;2p,L kj,L Z'(l", k,)
+ (T'T'T") ;j Z(k 3 , k,)Z ' (1, k,)Z' (p', ",) ,
+(T'T'T')'j2pl. k;,L Z' (p', k,)
(k, · ",)
+ (T'T'T');; Z( k"
'\
kJ)Z ' (p', k,)Z' (p', ",) } (",·k,) ,
M(+,+;+,-,+) = g'p:!Z(k" p' ) [ (p'.k,) 21'1",_k3E'p~k2+k3+(p" k,)
]t
x {- (T"T'T');/~ "," Z ' (k" k3)2(1, k,) (k,·k,)
_(T'T'T,);/ik2J.Z' (k" k,)Z(1, k, ) (k, · k,)
+(T'T"'l");j2p~k3J.Z(p' , k,) + (T'T'Y");; Z' (k3, k,)Z(p', 1.,)2(,,', k,) (k, . k3)
+(T'T'1'b)ij2p~ ka Z(p', k,) + (T'T ' T ");; Z' (k"
M(+, + ;- ,+,+) =
g'p~Z(k"p') 2l'~k2_k3 _
[
k,)Z(p', k,)2(P', k3)} (k,·k,) ,
(r/ ' k3) E'p"!kH k3+(P' . k,)
]1
x {_ (T'1'bT')/~ k3J. Z' (k" k3)Z(P', k,) (k,·k,)
-(1"1"T'),/,1. k23,Z' (k" k,)Z(1, ",) (k, ' k3)
+(1"T'Y")'i 2p~k3J.Z(p', k,) + (T'T'T'),; Z' (k" k, )Z( p', k,)Z(p', k3) (k,·k,)
+(T'T'T') 'j2p~ kHZ(p', k3) + (T' l"l"
),; Z ' (k" k3)Z(p', "2)Z(P' , k3) } (k"k,) ,
"'CI. - ;,... +,-) x
=
~ll~!-
}4~''l_k:l-
{-('/""I~'I'C}'J p'J. kjJ. Z( k
1•
[ _.
~_!,),' _ _ ].1
!wlthl(/I·kj)(I, · kJ)
k3)Z "{p',
(k,· k,)
k~)
~
-(T-'/'''T');J P'.l k; J, Z( .1: 3 • k~ )Z' (pi I ka) (k2 '.l:3)
1.(T'T'T'),;2,,~
I. (1"'PT' );; Z( k" k, )Z' (,;, k,) Z· (v', k,)
+ (,/"'1"1");;21", ki, Z· (,I, <,)
+ (1'T' 1");; Z( k"
M(-,+;+, - , 'I) = X
(k,·k,)
.'';iZ«"v') [ 21'';.1:2_.1: 3_
k, ) Z· (,;. k,)Z' (p', k,) } (k,.k,) ,
k,_
Ep':!k 2... (p'· .1:2)(11. k:l)
];
{_(1'.T~·''C );/'J. kiiJ.. Z( k'l) k~) Z· (l/, k1 ) (k,·k,)
-('1'<1""'1")" p~ kL Z( .1:3• .1:1)ZO(p'. kJ ) (k, . <,)
-(1"7"1');,2p~ "i, Z · (p', k,) I. ('J"T
+(T'7"1"),,2,;,
M( -. + ; -.+,+) =
+ (1'1"7"),; Z( k"
9':-;Z(k"P') [ .
.l1k,.<,.
E~k"
k,.
k,) Z' I';. ',)Z' (1", <,) } «,·k,) •
(1'" k,)(p" k, )
]1
x { _ (1"1"T' );;';' ki, ZI<" k,)2' (p', k,)
.
(k,· k,)
_ (rTCT~)iJ 1'4 ki.l Z( k:). ~) z· (p' k-J) (k,·k,) I
+(1"'rT")iJ2p~ kj.l. z' (p',.l: 1 ) + (T~TCT<1),j Z( .1.::), /.;2 )2 ' (p'. k 2 )Z"(p', kJ ) (k,· k,)
1.(T'7"T');,2p~ ki, Z· (p', k,) + (T"T'T'),; Z( k" (-,)Z' (,;, k,) z' (,;. k,) } (kl·k J )
,
iii, .... OMMA IlY 01-' i}i:iJ /o'i)ItMII/,AIi
M(+, - ; +, - ,-) = X
g,1J'~Z"(k,,]J')
[
2rJ~k,_k3_
kH lo/I'!kH(p" k,)(1'" 103 )
{_ (T'1'b T ,)/li kJl- Z"( "" k3 )Z(p', k,) (k, ' kJ)
]1
,
_(T'''j''Tb);/~ loa 2"("3, k,)Z(I" , k3) (10, ' 103 )
+(1'bT'T');;2p~ ku Zip', k,) + (T'T'Ta);; Z o( 103 , k,)Z(p', ",)Z(p', 103) (k, . "3)
+(T' T"T' ),j 2p;'k u Z(p', kJ)
)\1 ( +,-j - ,+1-) X
=
+ (T'T'T" );i ~.'(k"
g3p' Z'(k
31P 21'~k,_k3_ .L
') [
k3 )Z(P', ",)Z(p' , k3) }.\ , (k,·k 3 )
1
k3+
£1''':k,+('[I' k,)(p" "3)
]'
{-(T'TbT');/~ k3 l-Zo(k" ",)Z(]>' , k,) (k, ' "3)
-(TaT' T')/~ ka Z "(k3' ",)Z(1", 103 ) (10, . k3)
+(TbT"T')'j2p~ kH ZiP', 10,) + (T'T' T");; Z ' ("3 , ",)Z(p', k,)Z(p', k,) (k, . k3)
+(T'T'T');j2p~ ka Z(p', k3) + (T'T'Ta);; Zo(k"
M( - ,+;-, -, + ) =
i'zli
p';k,.k3 _
lr
Ep~
k3)Z(p', k,)2 (p', k3)} (k,·k 3 )
k H k3+(zI' Ic,)(p" k3)
,
]1
x {_ (T"TbT,) /~kuZ '(k" k3 )Z(p', k,) (k, . k3) -(TarT'J,; 1".1 kn Z' (k3 ' ", )Z(p', k,) (Ic, · le 3 )
+(TbT" T');j2p~ ku Z(p' , Ie,) + (1"1"1'");; Z'(k3, /c,)Z(p', ",)Z(p', k3) (k,.k 3 )
+(T'T"T')'j2p~ loa Zip', k3) + (T'T'T" )'i Z "(""
"3)2(,;, k,)Z(zI, k3)} (k"k3) ,
IIII,~,
!I 'I
'1111'/
' ''(_ ,_1+,_._) = !1',I.% ' (k""I) [ .. _ _ . 'J.lr.t.kJ_t',' I_ x
IJI~''')
g:l,l'k~~~~ldll'
)1 A'l)
{-(,f"''f'~'f'C)'J p'_k.j.l Z( k1 • k:,) Z· ttl . k~) (kl ' kJ)
-(T'T'T')" V. k;,Z(k,. k,)Z -(V. k,) (k,·k,)
+(1"ToT')'12]1.l ki_ Z"(p', 4:1 ) + (r-rc'r-),) Z( k,) . ~)Z ' ~p', k1 )Z"(P'. kJ) (A-:: .!'.,)
~WT'T'),,2V' k;,Z' (P', k,) + (r
,11 (-,-1-,+.-)
D
"V',Z ' (k"V) [ 2,r~
k1 _ kJ _
(p'·k,)
E'Jrt:I':-nkH(P' , 1.3)
•
)'
('I'" 'J '~1 ")i) I)~ kl,;, Z (1.:1, k,) Z ' (II', k,) (k,·k,)
X {
_ (T'T' T'),,"" k;. Z(k" k, )Z' (P', k,) (k,· k,) I
(T'rT'),,2p~ ki, Z, (,U, ) + (T'T'T")" Z(k,. k,)Z -(V. k,)Z ' (P', k,) (k,·k,)
+(T"I "'{")"2,,, k;, Z' (p', k,) + (T'T'T' ),' Z(>"
M(-, -; , _, ) =
k,)~- (V' k, )Z' (p', k,)} ~ k1 ' .t.)
29'V,_ [ E' )1 p';k1 _kl _ P~kl .. k3"(P" k1 )(p" k,,)
x {_ (T"T'T')./"kUZ' (k,.I»Z(II, I,) (k,'k,) _ (T'T'T'),,'>: k" Z-( k" k,)Z(V. k,) (!,·k,) I (T'1"T'),,2P'i ku Z(p', k, ) + (T'T'T" ),j Z' (k" .,)Z(P', k,)2(1I. k,) (k,·k,)
.('/'T"'I'),,2p~ k.. ZiP'. k,) + (T'T'r)1i z'rk" k,)Z(P', k-,) Z(P', (k·,· k,)
!.,)} .
(10.311)
,
Iii Sqllill'C
the
.... II M,\/Ally m' (WI! li lllWrll. Af;
1«'licil.y "a.mpli tude:.;,
nOUVtl.li iHh i' l).!;
SH IIl!1l<'d
ov~ r t he colo \' degr ees of freedom of t he Ini!.itl."! stOLle a.nd the Il ll:l.! !:iLat(~:
49 6E"P .J
•
IM(+,+;+,+, +)i'
-
IM(-, -; -, -, - )1' k,.k,+(P'· ",)(P" k,) ,
IM(+,+;+,-,+)I'
-
IM( -, -; -, +, - )1' 2E'p'-k3+(p"
g6(p' . k,)' F k,) ,
g6(p' . k, )' F 2E'p'-"H (P" k,) ,
]M(+, +; -, +, +)1'
IM(- ,- ;+,- ,-) ]'
IM( +, - ; +,+, _ )1'
IM(- , +; -, -, +)1' - k2+ k, +(p' , k,l(P' , k3 ) ,
gGEp~F
IM( - ,+;+,-,+)I' _ IM(+,-;- ,+,-)I'
iM(-,+;-, +,+) I'
IM(+, -;+,-,-lI'
,
"-
2Ep'-kZ+(p' . k,) , g"q+F 2Ep'-k3+ (p' . k3) . (10.312)
Unpoladzed squared matrix element:
, L (P ' k,)(I'" k, )I(p, k,)'
6
+ (1" . k;)'J
IMI' = JL ,,'-'-'.- - --;------108 3 IJ(p . I.:;)(p' . k,)
;=1
><
{lot.. ') _ 9 [(p, k.)(p'· k,) + (1" k,)(r'· ",) p
(k, . /",)
/1
+ ()I' ",)(p"
"3) + (1" k3 )(p'· k,)
(k,.k 3 )
+ (p . k3)(p' . k, ) + (p . k,)(p' . k,) ] + .,.-8.:..::1"7 (k, . k,) ><
[(1"
(p , p')
k.)(p' · k.) [(p· ",)(p'. " , ) + (p. k,)(1'" k,)J (", . k,)(k, . k,) .
(p . k,)(p' . 1.:')](1" k,)(p' . k,) + (p. '" )(p' . k,)J + (k, . k,)(k,· k,)
+ (p . "3)(1'"
", )[(p' k, )(p': ",) + (p. k, )(p', kllJ ]} (k, . k')(~l . ",)
. (10 .313)
I(J./(,.
t('1
1Ii;,
'")1'1
1',lp\)l'l.ri~I'iIITI I!\'''
,In
l'4·,·!.iu:l:
,
L:(," k, lil" . j., II II" .,
_ 17287r~
(11' k,
lII(t"
,.;), + (I" . k, )'1 ~")(1'" .1;,)
xl.O( . ') _ 9[(p·k ' )(I', . k, ) , (p·k,)I/".j·,) l(PP (kl'~'21 •
(,,' k, )(,,'· k3) - (p' k3)(p'· k1 ) (k,·k,)
+ (p. k,)(p' . k, ) + (". k, )(1" Ik, . j., )
X
. k, )]_ ~S--:'" (1'.,,)
l'(1'
k,)(p' . k,)I(,,· k, )(p' . k,) + (I" k,lIp . k,) ) ( k\ ·1;2)(1.'1' 1..1)
+ (I"
i,)(p' . k,J:(p. k,)(p' . k,) + (I" kdlp . k,)1 (k, . k1 )(k 2 · kJ)
t (I' . k, lip' . ",)f(p' k, )(1" • k, ) + :1" k, )(p' .
k,)I]}
(1. 1 ' ks)(k, · 1:3 ) xlJ
10.46
4
I
{k\ + 1' - k1 ~ k3 - p)
rP kl J:3 1:3 Jlji kwk-Jf)~
.
(10.3 14 )
qq ~ Tn
Process:
i'holvll polarizations:
' 1\"i-
~onvaFli~ hi ng
'
-
E'13"k . - .+ k·,-,d '
hdir:ly <1Il:pliiudt..'tS:
.'1(+,-;+ , +, - ) _ "" (1,-;\, -,1) -
i
=:
1,2,3.
( 10.3 16)
4fi(j
M(+,-:-,+,+)
[k,+I: ,_l:",,/,;,,_k3 +k.1 _lt
M( -,+:+ ,+, - )
J8e~6iJkJ+k3J. . l' [k,+ kl _ k2+k, _kJt k3 _ l'
.11(-,+;+,- ,+)
.;se~lii; k2+ k;l [kJ+kl._k'l tk2_ k3+ k.1 _]t '
M (-,+ ;-, +,+)
v'8e~bijkl+ ki.L [k ,+kl _k2+k,_k3+ kJ_ lt '
'
"
"
.:,. ·1 J.
.,
y8e~8i)kl + kl.L
M (+, -;+ , - , - )
J
\
i
M(+,-:-,+,-) M(+, - ; - , - ,+)
[k l +"I_ "2t k,_k3+k3 _ll
'
.;se~8ijkl_ kil.
M( - , -I-;+, - , - )
[kJtkl_ kz+k2_ k3+ k3_1 t
,,
1
J8e~.si)'2_ k;l.
M(-,+;-,+,-) -
(10.317) Squa.rec1 absolute vi\lues of t he nonvanjshing heli ci ty a.mplitudes , summed over the color degrees of freedom of t.he in itial state:
IM( +, -; +, +, - )1'
IA/( -, +; -
IM(+,-;+, -,+)I'
1il1( - , +; - , +, - )f
IM (+,-; - ,+,+W
IM( - ,+; +,+,-)I'
-
, -, +ll'
IM'(-, +; +, -, - )1' IM( +, -;
-
24e,s k'J k' +"I_k,+k,_
24e,6 ..,' k l+k,_k3+"3_ -
-, -, +)1'
1111(_,+;+, _,+)1' - IU( +, -; -, +, _)1' -
6 ?4·e .., q J.:21-
A:2t k,_ k3+ k3 _
24e6, k', h
kl +kl_k2+ k,_ 24e~ki+
kHk'_/,;3+ k3_
, , , , ,
,-
,'I,
.1
IM( - ··>: -.+,+) I'
~
M ( I,
; I.
,
JI'
=
/..''1+ /"', _ ~':I + kJ _
'
(10 .31S) 11,lpolill'iv,(;(1 S(I " ,1I'C(I'Il!lll'ix c\cJncrH:
( : 0 . 31~ )
,., UIIJlolarh:cd era,s seclio:l:
(10.320)
10 .47
(jq-Ig-y)'
Pl0CesS :
Photon po l<1.~iza.tiolll>:
i
,,., ,
=
E'13"" • .J' ) ~' +'-
i = 1,2,3.
:"lonv~ llishi:l~ helicity amplillHics:
f ,+.-)
~
M(-r, -i+ . - . ..:...)
=
M(+. - ;
[4-,1 '·l_/..·H};~_kHkJ_l~ , v'Sge:T;j k1_ .1:2.1.
= 2.;),
(1 0.322)
III . .'UI,liM/IIO' 0/0'
V
M(-, +;+,+ , - ) M(-,+;+, - ,+) M( - , +;- ,+,+)
(J(.'J)
100/1< ' .,,',''' '. ' .• "1 . ij ll.:\t 11:1.1
[k1+k1 _ k2+k2_kJ+k3 _1t '
ySge:TljkHki·.l [kI+k1_k2+k,_I'3+ k3_1; ,
- -)
VSge;T,";kl+kU [k ,tk,_ k' t k,_k3+ k3_1 t '
M(+, -;-,+ ,-)
.j8ge;T;"; k' t kn [k,t k,_ k 2+ k,_k3 +k,_ l l '
,11(-1-' -' + • "
1
M(+, - ; - , - ,+)
I,'(}HMUI,A i'.'
.j8ge~T;jk3tkU
[k1+kl_k2+k2 _ k3t k 3-1 i
'
lvI( - , +; +1-, - .)
J8ge~Tijkt_ki.l [kl t kl_"2+k,_k3t k3-1! '
M(-,+;-,+,-)
J8ge~Tijk2-k2.l [k ,t k, _k,.,k,_k3t k3 _1! '
VSge;T;jk3 _ k;~
(10.323)
Squared c.bsoluLe values of the non vanishing helicity amplitudes! summed over the color degrees of freedom of the ini tial state and the final state:
, -- , +11'
IM(+,-;+,+,-ll'
IM ( - , +; -
IM( +, - ; +, - , +11'
IM(-,+;-,+,-)I'
IM( +, - ; -, +, +11' _ IM( - , +;
IM(-, +; +, +, _ )1' IM'( - ,+;+, -,+) I'
+, -, _)1 '
IM(+, - ;- ,-,+)I' IAf(+,-;- ,+, -) I' -
32g'e 4q 1.-'3 k lt k, _k2t k, _ ' 329'e~kL
kltkl _k3+ k3_
'
3292e:k[ _ kHk,_k3+ k3_ ' 'J? ' ell4k"3+ '-9
kI+kl_k2tk,_ ' £.2 3?g'e4 ql\.'2+
III~II
~IIIJ
ii'I'·' !Jn
,
IM (-,+ ; -,+,+) I -
IM(+ ,--; ", ._ ,
-w=
kHkl_ks_ks.. (00 .324)
I! upolari7.ed
squa.~ed
matrix clc:Jlcu ~:
,
_ ]6 ',' E' I:(p+' k;)(p_· k, )I( p+· k; )' + (p-. k;)'1 IM)1 = g q " ' - " . ! ' - - - ; - - - - - - - - --
9
' II lp+' k; )( p_· k; )
l:n polti.rizcd cross gection:
dq
_
(10.325)
,
. 'Q' I: (P•. k; )(p_ . k;)I(p_· k,)' + (p_ . k,),1 Gsa I ";.,,'_ _ _, _ _ _ _ __ _ __ 1811"' 3 II (p+' k;)(p_· k;) .: 1
(]O.326)
qq-tgg-y
10 .48 Process:
(J 0.327) invnriants ilnd tlcfinitious: u
-
-2(p+·~),
A "" [kl+ki_kl.,.kl_k3+k3_lt.
(10.328 )
Gluo n a..."ld photon polariza lio Ds:
I'(k,)
~
N,II<, p- P. (] ±7,)+ P. P- 1',(1 '1'7' )1.
i =l,2,3 ,
(10.329)
... .
III. ,'UIMMAItY OF (JCJ) I"OIlMULAI,'
171l
No nvanb:lhing hclicit.y amplitudes:
M(+,-;+,+,-) !V!( +, -; +, -, +)
M(+ , - ; - , + ,+) J2g2e,k3+ k31. [Z (1'"1") .. A(k,.k,) " 12 . OJ
_L
M( - , + ;-,+,+)
J2g'e,kt+ k il. [Z (1'"1") .. A(k, . k,) 12 oJ
+Z
M(+,-;+,-, - )
J2g'e,kl+kll. [Z: (T"T')" A(k, . k,) " 'J
+ Z· (T'P) "
M( -, +; +, +, - )
r
Z (1"1'") .. ]
oJ,
21
'\
M(-,+;+,-,+)
21
(1"1'") .. )
oJ,
21
'J
M(+,-;-,+, - ) M( -'- - ' - '
1
,
,
"-J
, -,
M {- ,+ ;+, - , - ) M(-, +; - ,+, - ) M(-,+;-,-,+)
- J2g2e,k3_ k31. [Z> (TaT') " "12 'J A(kl . k, )
+ Z'
21
(1',"",,) .. ] 1
'J
.
(10.330) Squared absolute values of the nonvanishing helidty amplitudes, summed Over the color degrees of freedom of the initial state and the final state:
IM(+,-;+,+,-jI'
= I!VJ{-,+;-,-,+ )I'
11/.111. ijq ..... Y!l"(
I M (+,-;+,-,+)'~ =
-
IM(+.-,-.+.+!I'
~
,M(-. +i - . +. - W 89"~e~k'l+A1_
3A~( kl' k~) [9k\<.kz _
+ 91r1_.I;H
-
, 2(k: . k~)I ,
IM(-,+; +.-,-),'
IM(-.+;+.+.-)I' - IM(+ .-;-.-,+)I' =
8g~e:k~k:!_
3/12(lr
• 1
. k,) [9k H k2 _ + 9k 1_kH - 2(kl . k.,)] •
IM(-.+,- .- .+)I' = IM(+.-;-.+,-)I'
IM(-.+,-,+.+)I' - IM(+.-;+,-.-)'
Unpol;lrizcd squan.:d maLrix
2• ,
elem~nt:
,
L: (p+' k, )lp_ . k,)[(p+' 'i)' + (p - . k;)'1
IMI' = :7.:,' (9U' +9tUl' - sil'i-"'--- ,,-------.-II1 p+ . ,;)(p- . 1;) j :: I
(10.332)
Unpolarizcd cross section:
(1(1
=
, q''.
~$a
108/1"2
9'" +
n .. '
, ,
L:(p+ . 1; )(1'- . ki )i(p+ . k. )'
+ (1'- . ki)')
::ruU -S8 !"i_'.!'_ _ _:;-_ __ __ _ _ __
ss'
J
II11'+ .1;)(1'- . ki) ;:: I
(10.333)
1n
/II, S/IMMAny OF (JCI) f'()IWULAiI'
10.49
7jq -> ggg
Process:
1j(p+, i) + q(p_, j) -, g(k" a)
+ g( k"
b) + g(k3 . c) ,
(J 0,334)
Definitions:
F -
l'
-80 - 4 kl+ " - + • 1- k2+ 9 (k" k,)
! -::...+-,--;,k"c'-::.. '",3:!:.+ +-",k'C+c.:k7:' (1:, 'k3)
+ k2+1:3 - + k,_ k3+ ] + 18 [ 1:1+1:]_(k2+k3_ + 1:,_k3 +) (k,· k3) (k, ,k,)(I:, ,1:3 )
+
k2+k, _ (I:3+k]~
+ 1:3-1:1+)
(k, ,k,)(k" k3)
k3+k3 - (I:,+I:,- + I:I_k2+) ] + ' (k, 'k3)(k" k3) (10.335)
Gluon polarizations:
'- 1 ? 3
~
-
1'""
•
(10,336)
Nonva,nish ing helicity amplit udes:
M( +. -; + , + , -) =
2V2(k l . A,,)(k, ' k3)(I:, ' ' 3) X
[(T"T'T' )i j Z:;]_ _ (T"T"T' )ij Z;] k3+k,_ 1:2+" -(T'T"T')i ' Z32 'k3+ k 2-
+ (T'T'T")i ' Zi, "
1+ k 2-
lII. j Y,
ii'/ .... il/l/i
M (+,-;+,-,+)
filU k,_ J.·u Y. 1:1 X:I~ "ll
~ 2J'i(k, , k,)(k.,·:-k,,)(kC-',,'-;.k-:,) x
[(7'.'1'.'I '~);J
Zj, _ k3tkl_
+(TCrTO )i) Z;3
k1+k3_
~
-
(T~rr)ij
Zi, /:,+ /:,_
(TcT~Ta)ij
Z,'3 k1+ k3_
1
g3C kl_k1.LZ1J Z3~Zn 212(k, ,k,)(k, . k,)(k,· k,)
x [ (TATP)ij
Zi, /:3+};'_
_ (T" T oTb );j Z2'_ k2+k,_
-(T7"~);i Zi-l _ (T)Te'f"')ij Zi2 /:3,
. (T'T'~ ;'- ) i J
-:-
M(-.+;-.+.-) _
k2 _
'-:, ... k 2 _
Z;, k - ,'T'~T' ;'" ) iJL Z;" !. k2- 3~1+~ 3_
1
2v'2fk1 ' k1 )(kl . k3)(k" k3) X
[(T"T"T~)ij Z~I
k3tkl_
_
(T"T
k, ... k, _
"-(rr=T~)'J k Z~ + (TOrr-).) k Zi2 3+ 2_
+(T"'J'''rt )'J Z-i:t
~~k~_
M( ,+i +, - ,-) -
1+ 2-
-
(~T~T"),j
ZiJ k'+};3 _
1
fiC k2tk;.LZI3 ZJ1~1 212(k, . k,J (k, . ',)Ik, · k,) X [(T-TT )ii Z3'
k3+kl_
-rPTOTC)'J Zi1
k3+/:2_
_ (rcTdT6)'i
_ P"''PP)i1 Zil k1+ kl_
+ (T6T~/')ij Zi2
klTk, _
Zi, _ (r'FT")'j Zi3 ].
kHJ..J _
k1+ k3_.
iii. 811MMAitY
2J2(k, . k,)(k, . k3 )(k.,· k3) x
[(1'"1'"1");;
Z3'
k3+k, _
-(1"1"1"),; Z;, _ k3+ k,_
+(1"1"1"),;
kHk,_
+ (T'T'T')'j
Zjz kl+ k,_
Z" - (1"1"1"),; kl+k,_ Zi3
kHk,_
l,
Z31 _ (1"1"1"),; Z21 k3 .,.k l _ kHk l _
-(T'T'T'),; Z32
k"k, _
+(T'T'T')'j
Z"
kH k 3 _
+ (1"1"1");;
ZI2 kl+k,_
- (1"1"1"),) Z13
kl+ k,_
1
g3Ck,.ka Zi,Z32 Z;1 2J2(k , . k,)(k, . k3)(k,· k3 )
x [(T"T'T' )'j Z31
k3 +k,_
_ (1"1"1"),;
Z'I
kHk l _
-(1"1"1"),; Z32
+ (1"1"1")') Z'2
+(1"1"1")"
_ (1"1"1"),; Z" kl+ k,_
k3+k,_
M(+ , -;-,- , +)
_ (1"1"1"),; Z2'
g3Ck 1+ku Z i,Zj,Zil 2J2(k 1 • k,)(k 1 • k 3 )(k,· k,)
x [(1'"1"1',),;
M(+ ,-; - , +, -)
fOIlMUJ.AJi
g3C kl + Ie; 1 Z1:17,:1'2/:11
M(-,+;-,+,+)
M (+ ,- ;+,-,-)
or-Clef)
Z"
kH k3 _
kl+k,_
1
g'C k3+ k,J. Zj3Zj,Z;1 2.J2(k , . k,)(k, . k3l(k, . k3)
Z31
_ (1"1"1")'; Z"
- (1"1"1"),; k Z~
+ (1"1"1"),; k Z~
x [ (1"1"1',),;
kHk l _
3+ 2-
k2+ k,_
h- 2-
.,:", ,;
"( 'k
M(-,+;+,-,-)
I ••
•/ .
'/.
'/ .
;;-;iiYc' ~ I - ~~I:I":I~"~I
= 21i(k-: 1,,)(1', ~;i(~· k,) x
[(l'«'P~1'C),j
X31 _ k3+k,_
_ {TbT"T~h
Z32 k3+ k1 _
+(T
Zn
k1+~_
(T~Terb)ii
Zn
kHk , _
+ (Tbr
kJ+ k~ _
-
(T~TbT4)ij
Z13
kHk, _
1
.. 2,/2(" . k, )( k, . k,)(',. k, ) x
[('rT~T<)ij &'1 _ (TaT
A1( - , .;-,-,.)
-
~ + k, _
k3+kt_
Z31 _ (TbTCTlI),j Z,~~ ks+k2_ k, f k2 _.
Z~3
kHk3_
_ (T' r IiT"-Jij
ZIJ
kl+k3_
1
Z./2(k, . k2)(k, . kJ) (h· k3)
x
[(T4T~TC)ij
(Ta1'cT~),j
Z:n
_
Z32
+ (T'T'T" ),; , I! , 1_
k 3 -/-k 1 _
-(T'T"T' )';k ,.~ H
2-
Z1'
kz_kl_
ZI2
Squared ab~olute vabcs of the nonvanishing hclicity ampHludes, sUI:1Tned over lhe color degrees of freedom of the initial state and the final ~tate:
IMh-; +,+,-11'
-
IMi-, +, -, -, +)1'
g6 kLF , kHk ,_kHk1_ g6k1 F k,.k,_k;j.kJ _
IM(+,-;+,- ,+W
=
IM(-,+;-,+,-)I'
-
IM(7,-;-,+,+)I'
-
IM( ,+; +,-,-W
-
,.
yf'kl F ka k2_k3+k3_
, ,
/0, .\'(lMM/IIIY Of'
Q(,'J)
FO/{MIII.AIi
., "
IM(-, +;+,+,-)1' _ IM{+,-;-,-,+)I'
=
IM(-,+;+,-,+lI' - 1M(+, -; -, +, - ) 1'
g6ki+F = """"";:--::-<:L..,-- , kH k._ k3+ kJ _
IM(-,+;-,+,+)I' - IM(+, - ;+, -,-)I'
=
kl+k,_k,+k,_ '
gOP F k2±k,_k3+k3_
T
-";-- '''!-+-;--·
•
,
(10.338) Unpolarized squared matrix: element: 3
6
L: (p+ ' k;l(p- . k;)[(p+. k;)'
+ (p_
. k,)' ]
]MI' = 2~ ~;="-'----'-3- -- - - -- -
II (p+ . k;){p_ . k;) ;=1
x{
IO(P' ) _ 9 [(p, . . k.)(p_ . k,) + (p+ . k,)(p_· k.) + p. (k • . k,)
+ (Pt' k,){p_ . k3) + (Pt· k3)(P_
. k,)
(k,.k 3 )
+ (Pt' kJ)(p_ . k.) + (11+ . k.) (p_ . k3) 1+
81
(p+ 'P-)
(k,·k 3 )
x [(Pt' k. )(1'- . k.) [{I'+' k,)(p_ . k3) + (p+ . k3)(P_ . k,)] (k • . k,)(k • . k3)
+ (p+' k,)(p_ . k,)J{pt' k3}(P_ . k. ) + (p .• . k,)(p_ . kJ )] {k • . k,){k, . k3}
(Pi . kJ)(p_ . k3 )[(p+ . k,)(p_ . k, ) + (p+ . k,}(p_ . k.) ]]} + (k • . kJ)(k, , kJ) • . {I 0.339) Unpolarized cross section: 3
J
c/" -
as
64811"
L:(Pt ' k; )(p_ . k,)[(p+ . k;)'
+ (p_
. k;)']
1=1
= ' - - - - - - - , 3 ; - - - - - -- -
(Pi . p-lII{p+ . k;){p_ . k;) i=l
x {IO( . ) _ 9 [{Pt . k.){p_ . k,} + (Pt' k,)(p_ . k.) P+ p(k • . k,)
+ (1'+ . k,}{p_ . k3) + (Pt' k3}{P_ . k'L (k,.kJ )
~
,
,
177
+ (" + ' k;,)(,,_
,,401)
+ (b-: ~'I )(" . .to:'J.]
(k, . A:;,)
+
HI
(1,+ ' /1_)
[(P_' k,)(p_· k,){(1,,' ",)(,,_ . k,) + (p,. I")(p_, k,)1 (I., . k,)( k, . k,)
X
+(p,. <,)(p- . .,H(p,· <'l(p- . k,)
I ('" .
<, l(p:' 1,)1
(k l . k2 )(k 1 · kJ )
Ip,·I,)(p . !,lI(p • . <,lIp· . !,) I (p, ·1,)(._ ·
+
(k, . .l'J)(k, . );3) '4
X",(/)++p_ -k, - k'l-k;,)
cJl k, d3 k, cf3 ~:J ,. k '
.
<,)ll} ~
110.340)
11;10 '20";:50
10. 5 0
11
qq-r
--t
P roce~s:
+ ,(k,) - , q(,', .) + q("n + 7(1,) . along kJ . , (k,)
Positive z·axis: Defin itions:
(10.311 )
I-'I:ol"on polarizations:
,,IV"
~onvi\nishjng
fl4( I
,
helicity ampli tudes:
,+;+, - , +) ""
(10.313)
/II. SliMA/AII)' II/<' qUI! 1'()/lMIII,M :
2":~liij(q+q~f t '/~.~!l
M( t,t ;-,+,+)
Eq+q~
[('I' q' )(q. k.)]l ,,_,,~ ('I' .
M (+, - ; + , - ,+)
q+q~
q+,,~
-
M( -,+; +,-,+)
- 2e;SijQ-t- [ ') 3 -
,
Al( +, - ; + , _. , - )
2e3q 0jj(J+' [
~e qtJ ij q+
M( +, - ;- ,+,-) M(- ,+;+,-, -)
q_q~(q .
k,)(q" k3)
(qg' ) kJ )(q" k3)
q_q~(q .
=
2e~6ijq~q.1
r
2 e;c1;}(q+q~
q~(q . 'I' )
_g_(q.k 3 )(q' ·k3 )
q+q~
_
,
]1
]l
+'1.1'1',:) [('I' '1')('"
Eq+q~
,
k3 ) ] q_q~(q" k3)
[<". '1')('1" q_ q~ (q.
,
k3 ) ] t k3) (10.34'1)
Sqnared a.bso lu te values of the nonvanishing helicity amp litud es) slimmed over t he color degrees of freedo m of t he fina l staie:
IM(+,+;+, - , +)i' 1,11(+,+; --,+, +)1' IM(+, - ; -1- , -, -I- )1'
IM(- , - ; -, +,-)i' =
IM (-, - ;+, - , - )I' IM( - , +: - , +, _ )1 '
12c~(q.
, ,,
1
2e;6iJ(q+,,~ + qi q~)
",
'\
'1-('1''1') ]' q~ (g. k3 )(q" k3)
Eq+q~
M(- -' - -'- -)
,
r ]t r
2 "Ii [ (q . 'I' ) - e, ,jq+ q _q~(q . ka)(q" kJ)
M(-,+; - ,+,-)
i
1
-
_ 2e~6ijqlqJ. [
M( - ,- ;+, - , - )
r r
,
1
('1''1') k3)(q" k3)
q _ q~ ( q.
('I' 'I')
q+q~
. k,)
k3)(g" k3)
q~ ( q.q') '1_ (q . k3 )( q' . k3)
[
M(-,+;-,t, + )
1
q~(q'
2e~0;jg~ql [
lvf(+, - ; - ,+ ,+)
1
q- ( q' . q)
q ij(ll.q.L 2c '.I " [
"
'1 ' )(<1'. k3 )'
E~q+q_q+q'-
12e~(q. '1' )('1' k3 )'
E"Q+Q- '7tq'-
12e6q ({• 2 (a, . q'1,
IM( . I; I, • )' IM(-,1'; "I'-"IW =
I~f (~·,··;-,·~,-)
IM( -,+;-,+,+)!' -
IM( -I ,
.
, I -
:-l,-,-W -
(H(/~lq· ~',')((I" k: l )
,
,.,e(,· t,)(,' "',) . 12e;q7(q·q')
,.,e(,· k,)I,', k,) . (10.3"5)
l!n[lolari7.cd
~qtl ll.r(:d
malrix elemt:Llt:
,
L:(,' k,)(,'· k,)I(, · k,),' ·, (i· k,)'J 1M ~ = 6e:(q· q') ~""'----;:,---J1(,' k,)(,' . ,.,)
110.3<6)
.", 1
Unpolarh:ed cross section:
, do
3o
_
'Q'(
,9'1
"(1" l
-1"1
i-1
16~ 'l £1
x~
10.51
') L:I,· k;ll,'· k,II(,· k,)'
~
--t
,
+ (,'. k;)'J _
TIl,· k,)(,'· k,) }-q-q ' -
k3 ) J!3qcP;;d3/~ 9010 k3l)
.
(10.:111)
qq 9
O{k,) , 7{k,) - 'i(,'.i)
iJusitivc z-axis: along k,
+ ,("j) ~ 9Ik,,<).
(1O.3"S)
.
Definitions:
" r, ( q,'q)
=
..
IJtq ,_ - fJJ.fJ,!,
( ' 0.3,'9 )
Photon and gluon pohu ilit.tiolls:
)"'(1-,) -
N,IA A'
';'(1 ±O,)-
h A A'(I To.)J,
-. iii. S/i MAIAlII ' OF CWO 1'()/W(lI,A/-:
.II ,
I '"
-
'1- 1 ",
-
( 10.350)
No nvCl.nishi ng heli cit.y am pl itudes:
M( +, +; +,-,+ ) M(+ ,+ ;-,+, +) -
2ge;Tf;( q.,q~ + qJ.q';)
[('I . '1')('1' . k,)
Eq+q~
q_ q~(q·k,)
2ge; T)',(q+q+ + q1 q~) [(q. 'l')( q , k'r Eq+q~ q_ q ~( q" "0 ) q_ ( q . q' )
_ ?g _ e'Te J. [ q ii (/l.q ,.
JlI (+ ,- ;+ , - , +)
q~ ( q.
'1+'1+ ')
I ... ge q2Tcjil/1S
,
IH(+,-i- ) T,+)
11-1(-, + ; +,- ,+ ) -
-2ge'T~q+
M(- , +; - , +.+) -
2ge2T~q'
[
' )'
,
J'
+
29,'T' q' , l' +
[
] i2
r i
(q . g') . k3)(q" k3)
q _ q ~( q
]1
q_q~(q '
(')
, ! q. q 1 k3)(q" k3)]
[
]'
, (q.' q) k3)(q" k,)
q_q~ (q.
M(+.-;-,+,-)
-
- 2ge'T,'I [ • q JI +
M(-.+;+.-.-)
-
29";T)';'I';'I~ q+q"
(q . q')
[
q~ ('I' 'I') q ('I' k3)(g" k3)
]1
r
, '1. <1 .('1 ' k3 )(q" kg)
;11'(-.+; - .+. - )
g', ;;,," g" q- (g. g' ) . 2'7""[ • qd+ q~(q. k3)(q" 1.-3 )]
M (- . - ; + . - , - )
2g e;T;';(q+'I+ + q.lq';) [( q . q'l( q' k3) Eq+q+ q.q~(q" k3)
M (-,-;-.+,-)
_
'\
k,) (q" k, )
q~ ( q .q') '1-('1 ' k,)(q" k3)
•
i
q+rt+
,11 (+, -; +, - , - )
t
r
2ge;15';(q+'I+ + g~g~ ) [ (". q') (q' . k3 ) ] I !;;q.,.'I" q-q~ (q. k3) (10.351)
~.!lIan:(1 .. h~ulull· valul'lI nf LIlt' U,IIIVUlli... hill)( Iwlidly 1\tIlJlliuul.'.~, ~\lIHlll t :d "v('r ~ltI' n.lor (1.,!I,n:.... ur rn~'tluIU ut LJII' ilill.! h!;\Ln;
M(+.+,+,-,+W -
,r
J(jy2 e;(q'I['}(ql k'" l'~ " .:. q+q-q+q-
M(- , - ; -, +, - li' -
16g2~;(q .
IM(+.+;-,+,-)I' - ,M(-,-;+,-,-)I' 1,111+.-;+.-.+)1' - IMI- ,+;-.+, -)I' 1,11(+·-;-,-.+11'
IM(-,+;+,-. - W
1,1/(-,+ ;+.-.+11' - IMI+. -,-, +.-l!'
q')(q.
E~q t q
~'3)'
q'l'l'
16g 2c:q2 (q. tI) 'I+'I:,,(q. k3)(q" .1:3)' 16g~ eS¢:( q . 'I'}
_
qdt {q . k3)(q" .l'J)'
=
1Gg'c;q!(q . q') ,_.c(q · k,)(q" 10,)' 1692e;q:;(q . q')
IMI-.-,-,+,+ )I' ::: lM(+,-i+,-.-)I" .. q_q')q-. k }(q'--'k )' 3 J (10.352) U npolarize
,
L:lq· k,)(.' . k,)i(,' k.I' - Iq'· 1;,1'1 lA'1[2
=
Sgl e:(q' q') C'-:.!'_ _ -;,_ __ _
(10.353)
ITI.· k,lI.'· k,) i..:: I
Unl>ola rize
'Q'(
dq
=
oso '
I
,
•
,
,. L:! •. k,lI,'·k,l!( •. q . q 1 ial
-
,
k,I'+(q··k, I'j
IT (,·k,)(.' . k,) i-I
(10.354)
lO.u2
I9-
q (rt
Proces:s:
'Ik,) + a(k,. b) P05itivc ;HIJ'; ;8: along
q(,',;)
+ .("
j) + ,(k,) .
(10.355)
kl .
Definitioa:s:
Z(,',,) _
(10.356)
/11. SilMMAlIl' or (/1..'0 rOIl.M Ii /,M: P1L o j,o ll I-LII d gl1l011 ]>olll.r izatiolls:
1- 1
1\ 1
N,-' N3-
1
-
4E[q+q~ (q. q')]} ,
= 4[(q· q')(q . ka)(q' . ka))t .
(JO.357)
Nonva.n ishing heJi city amplitudes; 1
2ge;TJi(q+ J~ + q~q'i.) [(q. q') ('l" k3) ] ' Eq+q,+ '1_ q~(q. k3)
M (+,+ ;+ ,-,+ ) M (+, +;-, +,+) M(+,-;+,-,+) -
2ge;Tji«I+'I~ + ql.q~ ) Eq+q~ b _ 2ge'T ,. q i iQ1.QJ.
1
[('I' q')(q. k3) ]' q _ q~ ('I'
q- (q.' q)
[
q~(q. k3 )(
q+q+
. k,)
!
]
k,) !
• M (+, -;-,+,+)
2ge;Tj,q'cqi. [ q~(q q') ]' q+q+ q-(q. k3)(q" k3 )
M( - ,+;+,-,+)
-
-2ge~Tt· q+ [
M( - ,+;-, +, +)
-
2 e'l'b , 9 , j,q+
[
r
(q q') . k3)( q' . k3)
q_q~ (q
q_'l~(q.
('l . 'I') k3)( q" 1.:3
- !,
)J
r 1
M(+ , - ;+ , - ,- )
2 'Tb , [ (q . 'I' ) ge, jiq+ q-q~ ('l.k3) ( q'.ka)
M(+, - ; - ,+, - )
-
2 'T b [ (q . q') - ge, j''l+ q-q_q' ' ( k3,q·, )I' k ) .
M( -, +; +, -, - ) -
2 b ,. [ ]} 2geJ,iqJ.QJ. q, (q. q, ) q+q,+ q_( q . k3) (q' . 1.:3 )
r 1
fOJj::. 111
" 1'11
., ','/', . , [___ .2:.: ( q ., q )_ ] 1
,,~
M( -,+ ;-, +, - )
_ ~.I~f.J._ . )II!I '!~
1/~. (fJ' ~:: I)( fJ" kj
1/11/'1
'l.rJ('~'I1H(I+ '/~ + fIlJ/;} elf ,Iff.
_
M( - , - ,' 1 I - , - )
2ge:1t,(,/~ q~ + q~11J
=
M( - , -;- ,...I.., -)
Eq+q+'
)
[(9' q')(q. k31] 1 Q- Q'...(1'· 1.:3)
kJl]'
[(9' 9')(9' · q_q'-(q' 1.:3)
(10 ..153) Squared
IU(+ ,+;+, -, +)!' -
IM(-,-;-, +,-) I' -
IM(+ ,+; - , +, +)I' -
IM(-, - ;+ , - , -W
IM(+,-; +, - ,+)I'
IM( -,+; -,+, - ).I" -
~
IM(+ ,-; - , +,+)i' -
IM(-, +; +, - , -) I'
-
-
IM(+,- ; -,+, -) 1' -
IM( -, +; - , +, + )1' -
IM( +,-; +, - , -) I' -
I M(-,+ ; +,-, ~ )I'
IGg'e:(q' q')(q'. 1.:3 )2 £,tlf+q_q'~rt'-
16gle~(q.
q') (1 ' 1.:3)1 £.•4rt-rq- 1+q_ "
1692e;Q:(q . q') 9+Q+(q' 1.:3 )(1" kg)' 1692e;q~(q.
q')
q+q+(q' k3 )(q" k3) ' JG92c~q~ ( q.
q')
q- qc(,, ' k, )(q" k,) , 1692e~q~(q .
q')
q_qc(q ' k,)(q" k,)' (10.359)
t:npolarized squared matrix element:
,
2:::Cq' k, )(q' · k,)[(, , k,)' + (q' . k,)'J 1JVJ 11 =
S/e~(q· q') i ~ J
(10.300)
3
Ill q' k,) (q" k,) Unpolui?..cd cross section:
du
"'"
,
Dq' k,)(q', k,)I(q ' k;)' ... (q" k;j'1 o""'Q='C!Q'!J",(q~, ..q-,-') ,.,"-'_ _~_ _ _ _ _ _ __
= 4r.2
£2
-
3
I1iq k,)(q'
k,) (10,361)
· '. ~~~
IfI
10.53
.'I 1 1~ IMAII. Y
0,.. 11( .'1) l"OUMI/f.A/O'
;.9 -; '1 (J!I
1) r(.)(":CI:iS: 1'(k,)
+ g(k2.b) ---> q(q'.i) + q(q,j) + g(k3,C).
Positive z-a.x is: along Definitions:
2('1 ,'1')
( l0362)
k'l . Z(q', 'I)
-
Z(kJ, 'I') -
k3+q~ -
Glll on and photon polarizatio ns:
Ni"
.. 4[('1' q')(q. k3)(q'· k3))t .
k'lqL· (10.3G3 1.,
(10.361)
No nvan ishing helicity amplitudes: 1
(q'·k,)
]'
[ q~q,,!(q' '1')('1' le3)
x lq~Z"(q' , q ) Z(k3,q)(TbT');' +q_Z"(q,q')Z(k 3 ,q')(T'T')j, ] ,
M(+,-;+,-, +) =
92e,q~Z(q',q)
qdtk3+ [q.q~(q. 9')(q· "3)(Q" k3l)t
x [q~Z'(k3' q)(T'T')ji - ql.Z"(k 3 ,q')(T'Tb )j'] •
1/1.:'.1. i fl '" iJ'IlI
M(+,-;
-,+.+J X
" "~~:... (~hL'I " _ _-:-"-".c
[q~ Z · (k3. q)('/'~r)ji - q~Z- (k:.l.q,)('['e'['~);.] •
,
M(-,+;+,-,+) = X [ , ' Z(q'"
M"( _, +;
_, +, +)
=
k3+ (q:q~( q. q'){q. 1.'J)(q' "k3)]'
)Z' (k" , )(T'1" )i;+ ,_ Z(" q')Z' (k" ,')(T'T')i>'1 '
.9"e~q"T,
_ k3 t
X
["Z(,'")Z ' (k,,,I(1"1"),;
X [,'
"
I'
Z ' (q'. q)Z( k,., )(N')j>' + q_ Z' ( q, ,') Z( k" ,'I(T'T')j'
+"+ _ - I
9 "'11]+
k3+ Iq:q~(q· q'j(q' k3)(q" k3)]l
!lc~q:;Z- (IJ.q')
=
1
,." k,. [,_q, (q . q')(q . k, )rq' . k,)I'
[q~Z(kl,q)~rTaLi - Il_Z(1.:3.q')(T
111(- .+;-.+.-) X
,
['1~ Z· ('1'. q) Z(k3. q)(Tbp/ji + q. ZO(q. q')Z( kl' q')(T"Tb)j'] ,
",.
X
.
k,. [q~q?(,. ,')(,. k,)(," k,)11
M(+ , - ;",+.-J = _
M( -
+ ,_Z(q,q'IZ' (k"q')(T'T'),,1
g', 'l " t
M( + .-;+ ,-. - ) ....:
X
~q~q~(q. '1')('1" k:.l)(q'· 1.:3};t
=
,.q,k,+ I,_,~(,· q')(q. k,){q'. k,)I!
h~ Z( k3. q )('['hp) ji
M(-, -; +, -, -)
xi"
g2e~q:Z' (q','1)
-
9.1. Z(.I'J, q')(T"rII)ji] ,
~
Z( ,'. ,) Z' ( k", I(T'1" )" + ,_ Z( q, ,')Z' (k" ,')('1'T'I,;
I'
/II. 8/IMMAUY Of'
q(:f) I"()/W(!{,AI:
kI(- , -;-,+ , -) =
x [q~ Z( g', If )Z'( k3, q)(TbT'};i + q_ Z( q, q'}Z'( k3; q')(T'Tb);i] , (10.365)
Squared absolute values of the Donvanishing helicity amplitudes, summed over the color degrees of freedom of the initial state and the final state:
IM(+,+;+,-,+)I'
=
IM(- ,-;-,+,-)I'
8g'e;(1" k 3 }' [9q~(q. k3)
+ 9q+(q'· /';3)
- k3+(1 '
q'l]
3E4q+q_q~q~k3+
IM(+,+;-,+,+l!'
=
IM(-, - ;+,-,-)I'
Sg<e~(q· k3)' [gq~(q.
/';3)
+ 9q+(1', k3) -
k3+(q· q'l]
3E'q+q_ q~q~k3+
IM(+,-;+, - ,+)I'
=
IM(-,+;-,+, - li'
4g'e~q~ [gg~ (q . k3) + 91+ (1' . k3l - k3+( q , q')] 3q+q,+k3 +(q· k3 }(q" k3 )
IM(+,-; - ,+,+) I'
=
IM(- ,+i +,-, -) I'
8g'e~q,! [91,+(q·
/,;3)
+ 9q+(q"
k3) - "'3+(q. q'}]
3q+q'+"'3+(q · k3}(q" k3)
IM(-,+;+, - ,+)i'
=
IM(+,-i-,+,-)I'
8g'e;q! [9q+(q. k3)
+ 91+(1"
3q_q~ (q'
IM(-, +;-,+,+)i'
=
k3) - k3+(q. k3 )(q" k3)
q'lj
IM(+, - i+,-, - l l'
8g4e;q~ [9q,+(q' k3)
+ 9q+(q'· /,;3) -
3q_q~k3+(q.
/,;3+(q. q')]
k3}(q" k3) (10,366)
'" 1;11 [lOlal'll':I:( 1 ~qlli.I.fI:(1 tlllllri x d eil lell":
, Dq· k.)(q' · k,)J(,'
',r' + (0'. k,)'1
IMI' x 19(,· k, )(,'· .,) + 9(,· k,)(,'· ',) - (q. ,')(k,. k,)I . (10.367 ) U ItPQlarizcd cross section:
, L:(q. k,)(q'· k,)I(, ' ki )' + (,'. ki)'1 do
_
x [9(,· !-,)(,'. k,)
+ 9(,· k,)(,'. k, ) - (,. ,')(!;. ' ,)1 (10.358)
10.54
9 9-qq7
Process:
9(k"a ) + 9(k" b) ~ q(,',;)
Positive z·axis: along
+ ,("j) + 7(k,).
(10.369)
k, .
Invariants and definitions: =
- 2(k1' q).
u' -
-2(", . q'),
u
s' = 2(,·,'),
Z(q,q') "" q+q~ -q~q~, GJuon and photon polarizalions:
Z(q', q)
=
q~q_
- q';ql.. '
( l O.370)
1(/. SIIMMAIIY Of (J(:I! 1,'OIlMII/,M:
,,-1
,-. '1
(10.371) Nonvan ishing hdicity amplitudes:
M(+ ,+; +,-,+) =
M(+.+; - ,+,+) _
g2 0,(q+9\-
+ q.iq~)
(q. k3) g_q~(q.
Eq+q\-
l' 1
[
g')(q" k3)
x [Z'(g,q')(T' Tb)j'
+ Z"(q'.q)(TbT')j, j .
r 1
,9'O,Z(
M(+. -; +,-.+)
x [q+q~(T'1'b);i - q~q~(1"T')j. j •
M(+ ,-;- ,+,+) -
g'e,Z( q, 'I') [ q~ q+9+ q- (q. '1')('1' ka)(q': k3)
1' 1
x [q~q~(T'Tb);, - q_q~(T'T');,j •
M(-.+; -t-, -, +)
-
M(-.+ ;- ,+,+) -
92 e,q+ 1 [q_qc (q· q')(q . k3 )(q" k3)1' x [Z(q,q')(T'T b);, -t- Z(q'.q)(TbT')j ,j. g'leqq~
[H~(9'
1
g')(g. k3 )(q" k3Jl'
x [Z(q. q')(T'Tb);,
M(+,-; +, -.-)
-
+ Z( 'I'. q)(TbT');,j ,
9'e,
1
g')(q' k3)(q" k3)1' x [Z"(g, q')(T"T b);, + Z"(q' , q) (T'T');, j ,
,
11/..14 . !J!J..... if 'I"f
~ .!I!)
,fl ~ r9~tL_. _ _
M (+ , - ; -.+ .-) =
[,- ,1-(,· ,')(', . ",H';' .,)[1 x [z ' (, .1 )(1"1·);; + Z-(,'. , )(1"T" ),,[ •
M( -,+; + ,- .-) =
•' .,Z'(",, ) [ 9+q~
q'
r r
9- (q· q')(q . kJ)( ql . kJ)
)( [q~q~ (TOTb)j, _ q_q~ (ThT°);;l •
.'\11(-,;-;-, +. -) -
.'.,Z·(,·. ,) [ " ..,
,~( , .
,
,
,'H, ' k,)( , "
k,)
x [q+ q~ (T"r)j; - q~q~(T! ""')" J '
M(-,-; + ,-,-) -
g 2et (qt q't
+ 11.q',L )
E"," X
M(-.-;-. + ,-) -
X
,_ ,~
(,.
[Z(, .o')(T"T')"
92t!q(q+q't+9:iq~J Eq+9+
, (q. kJ)
[
[
q')(q"
]2
k,)
+ Z(,·. ,)(N ' ),,] •
(q' · k3) q- 9'- (q · q')(q ' k3)
]t
[Z(, .,')(T' 1"),; + Z(, ' ,ql(N'),,] .
( 10.372)
Squz.red absolute values of the non vanishing helicilY ampli tude!, summe d over the color degrees of freedom of the initial state and the fin al state ;
[M(+ . + ;-,'. + )[' -
[M (- . -;+.-.- II'
M (+ . - ;+ . - .+) [' -
[M (- .+;-.+. - )['
1UU
/II.
SIiMMAlty Of QUI)
1"OIlM(/I,Al~
IM(+,-;-,+,+)I' - IM(-,+; +,-,-)I'
IM(+, -; -, +, _ JI'
IM(- ,+;+, - ,+W
4g"o'q'
3q_q~(q . k:)(q" IM( - , +; -, +, +)1'
k3) [9q+q'_ + 9q_q~. - 2(q. q')j ,
IM(+, -;+, -,-)I' 4 • , " 9 -,q+
[, 3q_ q~(q' k3 )(q" k3) 9q+q_
')] + 9g_q+'-( 2 q' q
.
(10.373) Un polarized squa.red matrix element : 3
~:tq . ki )(q'· ki )[(q· kil'
4 2
e, (glt'+9"v:_ .,"') i~1 96..
IMI' = 9
+ (q'. kil') (10.374)
3
II (q. ki)(q' . ki ) i=l
Unpolarized cross ·section: 3
a'aQ' S f ga' + 9uu ' - sst 76811"
d"
~(q . ki)(q'· k;)[(q . k i )'
+ (g' . k;)']
i- I
~----~3~---------------
Il(q. ki)(q'· k;)
..j,,
i=l
(10.37.5)
10 .55
99-->qq9
Process:
g(k a) + g(k" b) "
Positive z~axis: along
->
kl .
Defiultions:
Z(q,g')
Z(g', q)
Z (q, k3 )
Z(k3 ,q)
Z(q',k3 )
g'+ k3-
-
'ok3..l) q.J.,
Z(k,1, q')
(10.376)
1\
10.55. jlfI " "i'l1l
~
1}J!I"
+ (/j.(q' ,koJ)
A'J)
.1.: 3+
+ q+q'_ + q-Q't] + :IG 2
[q+lt'tJ/-(I/I . k~) + q~(q . k3JJ
(q. q')
'2k3+
+ q_q~ [q+(qJ ',k~) + q'-(q' k~)J + (9' kJ)(q" 2kJ _
Gluon
poll\fiza~io;I~:
ji±
~
N;[,j' A 1;(1 ± "~I + I; A A'( 1 T "~I],
, w'
-
1E[q.q~(,,' q')[t,
N- J
,
-
1£[9+9:'(Q' q')]i ,
Ii
-
.'1,[1, A h'(1 ± ,,)+ h' h I,P T ,,)],
,
-
4[ (q· q'j(q . k3 )(q'· k3)1j .
IV·'
"
k3 )(q+q± + q_q,- )]. ks+k3 _ (10.377)
i=1,2,
(10.378)
Nnn\'arushing helicity ampliturle.:;:
M(+,+,+,- , +) ; -
9'(q..~ + qd")Z"(q.q'IZ"(q',q)(q'· I 2Eq+<,+ [q.q'.(q· if)'(q'
',11
',Ii
x {(T'T'T');, Z(k;~ _ (T' T'7");; Z(k" qIZlq', k,) k3+q_ kH q_q'+k3_
+(T'T"T')j; Z(q, k,1
_ (T'T'T'I;; Z(k"
q+k 3 _
q?Z(q, k,)
k3 +Q_Q+k3 .,
+(T' T'T') Z(q', k,1 + (T'T'T') Z(k'd)} " q+'k 3-
J'
,, 1\;3+q _
'
I ii. SIIMMAIlY OF QeD rOIlMU /,A/,
1 \) ~
M(+ ,+; -,+,+) _ _
g'(q+q++qlq~)Z' (q, (i')Z' (q',q)(q ' k3) ~ 2Eq+'I+ rq_ q~ (q · q')3( q" k3) ]!
x {(ya T 'T C);/(k3 ,q) _ (T a1" 1', ).Z(k 3 ,q)Z(q',k3 ) kJ+ q-
k3+ q_ q',k,_
J,
+(1" T OT ');; Z(q, k,) _ (T ' T' T O) .. 2(k3, 1')Z(q, k3) 9+ k3 J' k3+q~ q+I:,_ +(7" T"T');; 2\9', k,) 9+k3-
+ (T'T'T');; Z(k3 ,9')} k3+ q~
M(+ ,-; + ,-.+) = _9'Z'(q',q) [ q. q' 2q+ (q . q'),(q' k,)(q' . 1:,)
,
]!
x { (TaT'T ');; Z' (k3 , q} _ (TaT'T') Z ' (k" q)Z' (q', k,) /;3+9-
)1
+(T'T.T,)Z'(q,k 3 ) )1
q+
k
9+k3-
3Z
'(q,q') [
2q+
(r'T' 1,a)Z' (k"q')Z"(q,k3 )
'3-
+(T'T 01");; Z' ~q'. k3 ) M (+,-;-,+.+) = 9
_
k3+q_ q~l. k3_
..
,I
k3+q~q+k3_
+ ercr'TO); , Z ' ( k"
k3+9~
q-q~
(q . q' )'(q ' k3)(q" k3)
. J
q' ) } ,
]t
x {(T'T'T');; Z~(k3' q) _ (TaT'T');; Z ' (k 3 , q)Z' (q', ",) "3+1_ k3 >q. q+k3-
+(1"1"1");; Z' (9, k3) _ (T'1"1'") .. Z' (1:3 , q')Z'(q, k,) q+k3 -
"
k3+q~q+k,_
+(1" 1'"1'6) .. Z'(q', k3) -'" (T'1~T") Z ,( 1:3, q')} ) 1 / k ' )1 q+ 3k3+q~ ) M(-, + ;+,-, +) _
g'q+ Z( q,q' )Z(q', q) 2 rq- q~(q . q')'(q. k, )( 9' . k,)J1
x { (TaTbT ');; Z' ( 1:" q) _ (T'T ' T b),/' (k3, q)Z"(q', k,) kJ+q-
k3+q_q+k3 _
+(T'T"T');; Z' (q, k,) _ (T'T ' r a). Z -(k" q')Z-(q, k3) 9+ k," k3+q~q+k3+(T'TaT');;
Z· ( k ) I
\q ,
+ 3'I'k
3
+ (T'TbTa)
,
,2' (k3 , q ) } )'k" 3+q _
/lU,'J , !I!J"
Ij'lIl
. M(-,+;-,+,") = _
!i!'(~7.('/. 'l)X('I"'I) :l [t] _(/~ (1/ . 1/' P(~; :-J,:~ )(~I' -:- k~)]l (1';:1> If) _ (,f"'l'c,!,b) .. Z · P':3, If) Z · (q', kJ) k;l+q_ }' kHq_q~k3_
X {( '1'" 'j"b1'<) ji ,,-
+P'~T'QT~)i' Z · (q. k:"l) _ (TtT"r~ )j, Z4( k3. q')Z4(q, k:d q_k3_
k3+IJ'- Q+J..-3_
+(1''1'"'1''),; Z'( 9', ") + (T"T'T") q~~kJ_
M(+ , -;+ ,-,-)
_ _
. Z· ('" ")} ). kHq~ ,
93q~Z·(q,q')Z · (qJ, q )
2[q_q'-(I}' (JI )3(q. k3)(1}1. k3)]t
{ (T"T'T')'; 1.('",) _ (T"T'T');; Z(",,)7.(,',k,) k3 ... q_ ~'3+q_q~k3_
X
+(1"1'""")" Z(9.',) _ (T'T'T") ' Z(k",')Z("k,) q,k3 _
+(T7"T'),; Z(,', k,) q'tk3-
M (+, _ ; _, +, _) _
khq~q+k3_
)'
+ ('1'''''''1'")
. Z(k",,) } J'
kJ+q~
,
g3 q+Z"(q,q')Z "(q',q)
2[,_" (, . ,')'(, . k,)(,' . k,)[ I x { (T' 1"T'),; Z(k",) _ (T"'1"T')' Z(',. ,)Z(q', ") k3+q_
k3+q_q~k3_
J'
+('1"1'"'1"),; Z(" ',) _ (T'T'T")" Z( k" ,')Z(" ") q._k3 _
+(T'T"T'), ; Z(q' , k,) ,,'+ k3 . M(-,+ ; +. - , - )
_
93Z "~(q, q') 2q.
X
[
kHq~
J'
I} . k:!-
+ (1'T'1'") Z(k"
J', '")} ' 3+9 _
, " (q. ,')'(,. k,)(," k,)
]1
{(T"T''!'')', Z(k",) _ (T"T'T') Z( '" , )Z( q', k,) k3+q_
)'
~':li 9 .· q~~-
+(T'''"T')" Z(q,k,) _ (T" ' ''T")Z(k,,'')Z(q,k,) q+k3 _
+(T'T"T') ..2(9', ")
",'k + 3-
I'
+ (T'T'T')
k3+q~q_k3 _
Z(I" q')}
"3+4_ k'
,Ill'!
IH( - ,+i -, +,-)
=
g"Z "'(q', q) 2r/t-
"
"
-j
!
,
, l
, ~
'J i
+ (T'T"1'"J ,q'J} , +( T'T'l") J'Z"(q',k3) 'k J. k I 1 q+ "3'3+Q_ Z"(k 3
Squa.red absolute values of the nonvanishing helicity amplitudes, summed over the color degrees of freedom of the initial stale and the final state:
IM(+,+;+,+ ,-JI' 1111(+. +;+ ,- ,+)1'
IM( -, -; - , - , +)1' IM(-.-;-.+,-J1Z -
IM(+, -; +, +, _)1' - IM(-,+; - ,-,+) I'
4g 6 ( q , k3 )' F E4q+q_trq~
1
g.(q' , k3 )' F E"'q+(!_(!t q'-
1
., IM(+.-: +,-,+II _
,,/"/: ,.Ilt;/~(fJ' ~·J)(fJ'-:J.::;)' 9Gq~ ,..
IM(-,-;+,+,-II' -
M(+,-;-.-.+)'j
1,11(-. -;+,-. +)1'
M(+,-:-,+,-I' - --,-.T(g'-7~t'iI'( q_q_ q . "3 q""k'i' '3 "
2
V
(10.380)
U:lflolMized squared matrix element :
,
IMI' -
g
L(q· k,)(,'· k,ll(q· ki l' + I,'· I;)'} '
'lSS
.. ,
,
II(,· <;I:q"
,. , x {'0(,. q' ) _ 9
k,1
f(" "ltq" k,1 + .
(q. k,J(q'. k, I
( k\ . k,)
(q "-,Jlq' ·'-,1 + (q . k,j(,'· ,,) + (k, · k, )
+ Iq 'k,)(q"k,)+ (q .k,)(q" k, )] (k, · ' -')
x
+
SJ (q. q' )
r(q . ", )(q' . !', HI•. k,)(q' . <,) + (q . !,)(q' . !-,)} ..
(k l
.
k~)(kl
. kJ)
+(q. ',Jlq'. ,,)I(q' k, )(q'· k,) + (q.
«, . <,) (k,. k, )
(q . k,)iq" <,)1(,'
+
k,)(q" k, )1
<, JI.'. I,) + (,. <,)(q"
( kl . k3 )(k1 • 1:3 )
k, )}]} .
( JO.3S] )
/fI
t ) II po l(l. l' i :l.(~ d
.... ",\/ ,\/ A III' 0 f (WI! fO /(.M( II.M:
no!'s St,(,t.ioll: 3
2:)'1 ' k; )(q" k;)[(q ' k, )' + (q" k,)'1 O'~ i= l <1608,,' = - - ---. 3- - - -- -- il(q . k;)(q' , k;) 1= 1
x {
r(q, k,)(q' , k,) + (q. k,)(q"
lO t , ') _ 9 q q _
k,)
,
,
(k, ' k,)
"
(q ' ,1;')('1', k3) T
+ ('I' "3)('1"
k,)
(k"k3)
+ ('I' k3)(q"
k,) + ('I' k,)(q', k3)] (k, . k,)
+
8] (q. q' )
x [(q, k.)(1" ",)[(g' ",)(q', "3) + (q ' k3)(q" k,)1 (k, 'k,)(k, 'k3) , (9' ",)(9', k, )I(9' k,)(q' · k, ) + (9' k.)(q" k3 )1 .,. (k, ' k,)(k, ' k3)
+ ('I' k3)(q"
k, )[(q' k,)(q' , k,) + (1' k,)(q" (., 'k3)(k" k3)
x6' (k, + k,-k J -q-
d3 j;J d3 qd3 k , '30 go qo
q.
,,
kIll]}
'
1. ;, ','
(10,382)
.,
1 I ,
'1'1 ...... g'1'1
10.56 Process:
g(k" a) + g( k" b) - g(k3' C) Posit.i ve z-ax is: a.long
+ g(k., d) + g(k"
e) ,
( 10.383)
kl '
Defini t ion~:
F =
108 [2("3' k, ){k3 · k, ) ,k3+k3_kHk,_
+ 2{k3' ",)(k3 , ks) + 2("3' k,)(k, ,k,) k3+k3_k. _k5+
k3+ kHk. _ks_
+ 2(k3' k.)(k., k, ) + 2(k3' k5 )( k" k52 + 2(k3 ','::""")",,(k"'·'.,...'.:.:;k,,"-) ~~+~ - ~
~~-~+~
~-~~~ -
~IJ 7
/O.JIJ. 1/ II - . , II /I
IJ =
Xv. IUI[Tr (1"1'.'J "" I"'r~ )]- X!H A"',u-Ic,,_ kHk._
Iln[Tr(rT~1'(Ter)1
- k z~ ImIT, (r-T'1"T'T' ») + k z~ ImIT,(r-T'1"T'T')) 4+
$_
~
Z,53 I m(Tr (7"T'T"T~7''')J ku /(3_
+ ZH ImllY (T"1"1'-?,'C'],J)] _ kr..,.k._
+
... ,_
Z:uZ43.Lm[Tr (T·T'T·TJT~ )] k3+-lc s_Ic H k3-
+ Z31Z~31m1Tr (r-Tcp~TJr
Z31 Z~s Im(' I'r ('J"'J'4T~'J'
Z43Z.I\41mlTr ( T"'['-IIT~TeT~)] -""="i"'-i'--S:-~'-'~-ll
k3+k~_k:H-k3_
k4+k3_k~+~'4 _
kHk4_kHkS_
_ Z3~Z511m[Tr (,r"T"'1"Te'J'J)] k3+kS_k5 ... k._
+ Z4~Z$,,1 mlTr (T"T"TiT"T")] kHk~_kHk3_
(to.384)
Cl1,J,llll polarizations:
,
f. "
N l-
l
-
N I I)::4 ;'t. };,(1 ± 15) + ,kJ };$ .k((I TIS)]
-
1E[k._lcs _(k 4 'ks)]t .
!
j:/' - ,v,I", '" ",(1 ± 0.,) + ", h ",(1'1' 0,»),
,v-' ,
-
4Elk H kS+ (k• . lcs)]t )
,Ii = N,r", '"
",(I ± 0') + ", " , h(l 'f 7.)1.
N- '
,
-
'[(k, · ., )(k,· k, )(k,. k, »)l,
Ii
=
N. lf; ", h(1±7,)+ h ", " . (1 'f7,»),
,v-'
•
=
4E [ks _k._(/r3 · k.);t
If
-
N,I,k, ".
,
I
",(1 ±O,) + ", " , ",(1 TO,)),
N-' = 4Elk3_ '~s_( 1:l' k,\)] ! .
(10.385)
III. SUMMA flY OF qUI) (o'(JUMUUlh:
Nouva.rtishing hclidiy amplitudes:
M(+ ,+;+,+,+) M(.,.., +; +, +, -) -
___~g~3B~Z£~~Z~'3~Z~;~J(~k~3_·k~'~)~~ 1 E 2 k3_k._(kJ • k.,) [k.,+k~+(k, . ks )I'
M(+,+ ;+ , - ,+) M( 'I-,+;- ,+,+)
E2kJ_ks_(k3' k, ) [~i+k>+(k4' k,)J'1
'
[(k.' ks )3] t
2g'BZ~JZ'3 -
'
E'kJ_(kJ · k.)(k 3 • k,)
••.,.k....
93 B" k,_ Z"Z"ZJ.
M(+,-; +, +,-) -
1.-3_(k3· •• )(k3 · k,)
1 ,
[k~+j:G+(k. · k,PJ'
g3 B" k._ Z"Zs3 Zl,
M(+ . - ;+,-,+)
1 ,
M(+,-;-,+,+) M(-,+;+,+,-) M(-,+ ;+,-,+) M(-, +;- ,+ .+)
2!f'B ok;1k u.u Z"Z" ,, '3_ k4_k,_(k3' k.)(k3 · ko) [kHkS+ (k, · k5 )1' .
29 B k5J. k~J. k,J. Z"Z'5 3
M(+,-;+ . -,-) M( +, - ; -, +, -)
M(+,-;-.-,+) M ( -, +;
+, -, -)
M(- .+;-.+, -)
1 ,
1O,51i,
II!I -IIY!I
M(-,+;-,-,+)
M(-,-;+,'-,-) =
M(-,- ;-,+,-) _
E2kl_ k~_(k3 ' k~ ) [k4+k~ .. (k4 ·
M (-,-;-,-,+) =
M( - ,-;-,-,-J =
'/ '/ 2g '8· /~~:1""'63
,'
kl»]'
g3B'Z;3ZSl ZMk,) , k1 )
E~kl_kG_(kJ' k6) ;kl+kH(k1 ' k~)]~ , 8g 3 BEZZ43 ZS3
(10,38G)
Squared absolute values of the nonvanislling hc1icity amplitudes, summed over the color degrees 0: freedom of liJe initial slale and the final state:
IM(+,+;+,+,+)I' IM(+, -;
I,+, - W
IM(-,-;-,-,-)1 1 -
IM(-,-.-,-,+)I'
1696 F F;4
(I" k,)(I" 1,)(1" I;)'
g6 F(J.:3 ' kd 3 = E4(!:3' kb)(k4' k,:,) , g6F(k~'
IA1(+. + i + .-,+W -
IM(-,-;- ,+,-)i' -
ksY E4(k3' k"4)(k4 ' k~)'
IM(+,+;-,+,+W -
IM(-,-; +,-,-W -
g6F(k1 , kS )3 E4(k3 . l'4')(kJ , 1.5) ,
IM( -, +; - , - ,+)1'
lJ.Fk 1 (kJ' k4)~k3' 1.$)(1. 4 ' ks ) ,
2
IM( .;-,- ;+,+,-) 1 -
-
IM(,',-;+, - ,+)I' -
IM(-,+;-,-,-)i'
IM(+, -; -, +, -)1' -
M(-,+;+, -,-lf -
=
IM(-,+;+,+,-)11 - IM( t , -; -, - , +)1' -
,-
4 g6Fk,-
(I" .,)(k" 1,)(1" ko)' g8Fk~
(I, ·1,)(1,,1,1(1, ,koJ' 9BFkt+ (I" 1<.)(1" I,)(k, "';) , gGFJ..·4
'± IM(-,+ ;+ . -,+) I' = IM(+,- ;-,+,-W (I,. ',)(1, , 1,)(1, · .,) '
IM(-,+;-,+,+)I' -
IM(+,-;t,-,-lI'
g6Fk H4 = (I, . k4)(k3' k.~)(k4 'ks) '
(10,387)
III, SIIMMAUY OF QC/J ,,'()IIMIIJ.M:
r.{I{1
UupoJa.l'i:;,cJ Mllla,nod maLrix el(!rnent: 6
IMI' -
279 16
'2)k; . kj)'
<~J~_ _
.:.;i
Il( k;.kj) i<j
+(k, . k,)(k 1 ,k,)(k,. k3 )(k3 • k,)(k.,· k,) +(k , . k,)(k, . k,)(k, . k,)(k3' k.)(k3 · k,) +(kl . k,)(I.-, . k,)(k,. k3)(k3' k,)(k, · k,) +(kl . k,)(I.-, . k,)(k,· k,)(k 3 · 1.-,)(1.-3 ' k5) +(k, . k3)(k, . 1.-,)(1.-,. k3)(k,· k,)(k., . k5) +(k , . k3)(k, . I.-.)(k,. k.)(k, . k,)(k3 • k,) +(k,· k3)(k, . k,)(k, . k3)(k,· k,)(k, . k,) +(k, . k3)(k, . k,)(k,. k. )( k,· 1.-,)(1.-3 ' k. )
+(1.-, . k.)(k 1 • k,)(k, . k3)(k, · 1.-,)(k3 • k,)]. (10.388)
+Ck, . k,)Ck , . k.)Ck,· k,)(k,· k,)Ck. · k,)
+Ck, . k,) Ck, . k. )Ck, . k. )(k,. k,)(k, · k,)
+Ck,· k,)Ck,· k,)Cb,· k,1{k, · k,llk, · k.,)]
(10.38'1)
10 .5 7
99 _
ISO
Process:
( 10.3'10) Inva.ria.nts and clcfinit iom:
(10.391)
1/1. SI/,\/MA II.Y 0/0' (leI! /o'OIlMII},AII G luon pnli\rhmtiUlls (Lhc four·vt:dor k iii
H.
g"" ll('ra.lly p<")sil,ioned foul'-vedo l'
wiLh k' = 0):
P ±(k,) -
NI):, 1):,(1 '1' I's) + I,,k ,k1(1 ± ,,)],
(10.392) Nonvaoishing heJicity amplitudes [the definition of the quantity Ho is given in eqn (8.11)]:
M(+,+) 111(-, - ) =
(10.393)
Squared abso lut.e values of the non vanishing helicity amplitudes, summed over the color degrees of freedom of the ini tial sLale:
8g 4
m
jM(-I-,+) I'= IM(,-)I'= 31f I \1' .
(1 0.394)
Unpo1arized squared matdx element:
IMI'
4 p2
=
g"'O . 481l'M
(10.395)
Uopola.riz.ed cross section: a =
10.58
99
->
7r20'~R5
2
3M' 6(8 - M ).
(10.396)
3Po
Process:
g(k"a)+g(k"b)
--+
3po(p).
(10.397)
AI' .
(10.398)
In variants and definitions: -' = 2(k, . k,) =
p'
=
GIuo n po.1arizations (the four- \'e~tor k is a generally positioned foul'-vedo r wi t h k' = 0):
,C" (kd -
N fl,,k h(l 'f i s) + ,k, ): ):,(1 ± IS)],
4[( k, . k,)(k, · k)(k,· k)j t .
(10.399)
N\)llvalli~hil:g
IIdidly ltfl:plill1dl'/o [1. 1l(' ddilli!.ion
i,l 1t~ 11U1~l!Lil)'
WI i ~ given
ill (:(P (o.J'l)]:
( +, _) __ ni9~UJb~~ , ( ) t- __ M(- ,_)_ 10.100 , v'3;r M3 abso lute values of tnt.: non .... anisiliug helicity amplit udes , ':)ummed " M
Hquil.~ed
()vcr the color degrees of froodom of the initial slate: (10A01)
Unpo larizcd squared rnaLrix. element:
(10.102) Ur:poJarizcd cross section: ( 10.403)
Process:
(10.101 ) Invariallts and definiEons:
(10.10.' ) Gluo :'l pola.rizat ions (Lhe fou r·vector k is a. generally POBiLiollcd four-vector with k1 -:::: 0):
(10.106) Nonvanish ing helicity amplitudes [lhe Jenllition of lilC quantity R~ is ~i ven in cqn (8 .12) and t he quantity (.(tfJ cenotes the polariza.tion le',1 sor of the JP:!]:
W(+, -) -- 1 6i9lfl~N26Q~fn{j {2k10' kI S I(k 1 .k)' + (k 1 .k)'] ..j-xlvI3
J
+ ':2 M'k" k-C
+2M2kl", k,,(k\ - k 2 · k) _ :i(.olw" k 1}'k1"k' [2k\o( kl - k'l' I,:)
+ Mlk,,]}, ,
/(I.
' Jr;t'OP
J6i Y
M( - , +)
+ 2M'k,.kp(k, - k,· k)
{2k'ok,p
H/lUk/AUY OF' QC/J FOIIMU/,A/,
I(k, . k)' + (k, .k)'] + ~M'k"kp
+ ifomk"'k'vP
[2kIP(kl - k,· k)
+ M'kpJ}. (10.407)
Squared absolute values of the non vanishing helicity am;>litude9, summed over the color degree.s of freedom of the initial state and summed over the 3P2 pola.rizations:
(10408)" Unpolarized squared matrix element:
IIWI' -- 9'M ,,1v1 3 •
(10.409)
Un polarized cross section: q
10 .60
q9
-+
= J 6rr'a}R? '( _ M')
M5
0
S
(10.410 )
.
q ISO
Process:
q(p+,i) + g(k,a) -
q(q+,j)
+
(lOAll)
ISO (p).
Invariants and definitions:
s = 2(p+ . k) , S
+t +u =
u = - 2(k· (1+) ,
p' = M' .
(10.412)
Gluon polarizations:
(10.413) Nonvanisbing helicity amplitudes [the definition of the quantity in eqn (8.11)1:
M(+,+;+)
Ro is given
-= _ 'I.1{1U(j'/'A~.
M(+, - ;+)
t - M 'l
[
l; II l'
N
:hrMQ+_
M(-,+; -) _
29' 11,,'1;'1++ [ E t - Ml 3'/1"Mq+_
.111( -,-;-) =
_,g't RMl.Tf;[ 31fMq+_ B'
(10. 414)
Squa.~cd
absolute valuea of the lloDvAnishing heli city ampl itudes, summc
1,11(+,+; - )1' -
IM(-,-;-)I'
11.1(+,-;+)1' -
IM(-,+;-)I'
-
8g 6 Rls 2 3'/!'Mt(t - M3)1 ,
-
8gGRlu 2 3lrMt(t - Af'l)2'
(10.115)
Unpolarizcd squared matrix clement: ( 10.116)
Uopolarizcd crolls acetiol} :
do
10.61
211'a~.RJ(s2
+,,2) dt ::::- .- 9MI11 /( t - ,"(2 )2 .
(1 0.417)
q(p"i)+9(k,,) - q(",j )+'P,(p).
(10.118)
qg ~ g3P.
Process:
JnvaJiauls and defill itions:
s+t+u=p'l=M l
.
'(10.119)
Gluon polarizations:
( 10.120)
III. SIIMMAIIY
or QC'f)
fOltMI//,Ab'
NOJJval1ilihing It(~licity ttmplit.lldcs (the ddillil.ioll of Uw quaJJtity R~ lR giv(~1l
in
"'I" (8.12)1 : 2ig'R;T}~s(t
M(+,+;+) -
(t - M')' [311'M3£q+ _I'1
'
4ig'R!"TJj(t - 3M')Q++ [ E (t - M')' 3dl'Q+_
M(+, - ;+)
4ig3R!"Tj~(t - 3M')Q++ [ E (t - M'), 3r.lvf3qt_
M( - ,+; - )
M (-,-;-)
- 3M') ,
l! ,
1
'i
"
2ig'R!"Tj';s(t - 3M')
=
(10..121 )
1 .
(t - M')' [3"M3£q+_I' Squared absolute values of the non vanishing helicity amplitudes, summed evel' the color degrees of freedom of the ini'tial sta.te a.nd .the final state:
IM(-, -; - )1'
IMC +, +; +ll'
IM(+,-;+)I' = IM(-,+;-)I'
_~2g6R;'S2 ( t
_ 3M')'
31r M 3 t(t - 111')' 32g6 11';'u'(t - 3M'f 3"kJ3t(t - ,11,).,
(10.422)
Un polarized squared matrix element:
-----.r> __ 2g'111'(s' + u')(t Ik I .-
3M')'
(10.423 )
9:orft13t(t _ M')'
Unpolarizcd cross section:
+ u')(1 - 3M')' 9Af's't(t - M')'
81T"~R'.'(s'
da
dt
= _.
(10.424)
Process: q(p+,i)
+ g(k,a) ->
'1('1." j) -I- 3p,(p) ,
(10.425)
Inva.riants and definitions: s = 2(p+ . k),
s+t+u=p'=M'.
u = -2(k ''1+)
,
(10.426)
Gluon. polarizations:
(10.427)
r,U1
'/tl - "I:I/,:
/II./il'..
hdicily Il.l1!pli!.lld,~~ [llll' rlpliui ti ,.u !If lite qUill/l.ity H', il; given ($.12) ~lIJ the four-vcl"lot r d~!ltul!'s t lll~ pulil.rizat:on or llJC ;'Jl: !:
Nlwval\i~llil\g . ill Clpl
M(-, +; - l -
kPJ2-...hrMEq+_
(t x
M ( - , - ;-) -
[(I - uHp+ . f) - (.~ + t)(9+' t) (!
x
2it"i'')6 k<>p+.'Jlj'ht6j ,
M1 )2.jr.MEq._
[0 + u)(P... · t) + (3 -
f)(q_· f) - 2it· O-. sk"p+,Sq+-,q ] .
(1 0.428) Squa.ted a.bsolute values of the nonva. ~ ishillg helicity amplitud
IMh+;+)i'
~
IM(-, -;- ll'
IM(+,-;+)I'
~
IM( -, +; --)I' -
649 6 Rn'J~t + 2,,;uM 2) 7fM3(t -
M~14
64,96 R?{ 1J.2 t + 2su;M1)
7rArJ(i
M2)4
(10129) Uu pola.rizcd squurcd matrix eleme;1t:
(100130) Unpolz.rized cross section:
{lO," ')
GOB
III, Sl l MMAlty OF (l(.'O l 'Olt MU/,tIIl'
q 9 -+ q Jp2
10.63 Process:
(10.432) fnva.rja.nts aod definitions:
3
+I + " =
(10.433)
1" = M' .
Gluon pola.rizations:
(10.434) 'onvanish ing helicity amplitudes [the dcfiniLioll of t he quantity R; is given in eq n (8.1 2) dnd the quantity (0.13 denoies the pola.rizat ion tensor of t.he 3P2 1:
M( +, +; +)
4ig'R:., Tj~{·P [-2M]! =
-
s(t _ .M')'
1I't3
,
{(I + tt) P+.P+p
+2(sl + su - tM') p+oq+p + (s' + tz)q+oq+p
M(+, - ;+)
+ 2;[(1 - u) P+o - (. + t)q+o]c"v,pkl'p~q~ } ,
M(- ,+', -)
3
-
4ig R;Tj;
[-2M]i {(t' + u')P+.PW ,,13
+2(5U + lu - tM')p+.q+p + (5
+t )' q+oqW
4i.g;llt!,.'/~~/."r' [ ,. 'l.M]1 {( )' 1+"1'+1" M~P 7ft'
M(-,-;-) - .9(£ _
+ 2(3£ + SIl - tM'l1J+<>q+J3 -2il(t + u)P+a
+ (3 -
+ (S2 + t2)q+"Q+(1
t)q+alt""pftk"p~q:}. (10.435)
Squared absolute values of the nOllvanishing helicity amplitudes, summed over the color degrees of freedom of the initial state and the final sta.te and over the polari7.ation sUites of the 3Pl:
IM(+,+;+) I' ~ IM(- , -;-)I'
IM(+,- ;+)I' -
Ullpo!arj",cd squanxl
IMI2
= -
IM(- ,+;-)I'
ma~rix
clement:
91fM:~:R? Ml)~ [(5'
! u')(t'
+ 6M 4 ) + 12stuM2 ]
(10.137)
Ullpoiarizccl cross scctiO:1: dff _ dt
10,64
If:i7rifsR~
9MJ.92t(t _ M2)4
gg
[(s2
+ 1I1) ({1 + 6M 4) + 1:J3tuM~ I .
(lU.4J8)
~ qlSO
Process:
q(p+,i)+g(k,a) InvarianLs and definit.ions:
-t
q(q ... ,j) + lSO(p).
(10.4:19)
s= 2(pt,k), s+t+tI=p1=M~.
(10.440)
Gluon polarizations:
r -
Nit A+ fl+(1 Of "(')+ p. A+
,,(I
± ",>II, (10.441)
....
nlO
III,
NOllvalll~ hi ng
,;'II MMAI/Y OF QUO rOf/MULA/!!
hcJicity amplitudes [the ddillitioll of tlte ·quantity Ro is given
in cqll (8,11)]:
1' 1
M(+,+;+)
4g'R., Tlj [ E3 t - M' 3.. Mq+_
M (+,-;+)
_ 2g RoT;jq++ [ E t - M' 3"Mq+_ _
M (- ,+; - )
l} l}
29311oTl)q++ [ E t-M' 37rMq+_ 3
M( - , - ;- ) =
1' 1
3
_ 49 R.,T;j [
t-M'
E3 37<Mq+_
(
(10 .442)
Squared absolute values of the nonvanishing helici ty amplitudes, summed over the color degrees of freedom of the initia.l sta.te an d the final sta.te:
8g" R5s' 3.. Mt(t - M')' '
[M( - , - ; - )[' [M(+,-;+ )[' =
[M (-, + ;- )['
:l.-.Mt(t - M' )'·
(10.443)
Unpolarized squared matri x element:
6R'
90 ('" ') [ 1~1 [ ' = - 18.. Mt(t_M,), ·sTU.
(10.444)
Un polarized cross section:
+ u')
d"
2;ro:}R~(s'
dt
9M s't(t - M')' .
(1 0.445)
P rocess :
q( p+ ,i)
+ g(k,a) -->
q(q+,j)
+ 3po(p).
(1 0.446)
In variants and defini tions:
s =2(p+, k ), s+t+u=p' =M' .
(1 0.447)
Gluon polarizations:
,t:±
N [}: 11+ 1>+(1 'f "/5) + p+ 11+ }:(I ± "/5)], (10.448)
~
I!
, r\()llViln ilitd ng hdkily a[n)l!ll.udll~ !ttl!' .tdillj(.joll of tIl(' (jll
iii
C\]iI
(8, I 2)1;
M(+, +;+ ) M (+,-;+)
'li!J:J /Wlij,~{t
-
(l- Nf2)1 [311' M 3Eq+_ lt'
_
4ig3RiT,j(t-3M2)q..-+ [
(t
M2)1
E 3r.M.J q+_
4ig J R1T,j(t-3 M 2)tJ++ [ E (t - M2)1 31<M3 q+_
M(-,+;-) M(-,-;-)
- :U.jl) ,
~
]1 ]1
2igJ RiT,js(t _ J]lfl ) (t -
Jl.{ 2)2
[J'/r MJ Eq+_lt
(lO A49 1
Squared absolute values of the nonvanishing helicity amplitudes, summed Qv('!r the color degrees of frt.-edom of tbe initial state an d the f~md sLa Le:
IM(+,+; +II'
~
IM(-,-;-II'
~
32g6R!( s l(t - :J M~)~ 3;rAPt(t - M~)4
IM(+,-;+II'
~
IM(-, +; -II'-
32 g6R?u 1(t - 3,\(2)2 31r A.Pt( t Ml)4
(10.1501
U!:polarized squared matrix e! emen ~ :
- 2-,g,-'"-lf.",'1 -,(''C',,+Ci''~'1,,(t~-~3"M=-'LI' I" f l~=• 911' M 3 t(t _ M1 )4
(1 0.<51I
Unpolarized cross sect:on:
8'lr(llR?(3' + u')(t - J.l\fI)' dt = 9APsZt( t M2)~
(10A;2)
i1(p+,i)+g(k,a) ..... q(q.,j) + Jp1(p).
(l U,153)
dO'
Process:
u1Variants a.nd definitions: s = 2(p+·k),
s+t+u_p2-'-M2,
(1CA.,,)
Gluon polarizations:
F'
Nlft A, P·,·(l 1' 1,) + fl. A+ ,k(l±o,)1. (to .455 )
III. HIIMMAI()' OF QC/J FOItMIII.AN
NOIlvanislJing hclicity arnplitudes {the ddill iLioH of ttlc quant.ity R'I is giveH in eqn (8.12) and the four-vedor, denotes 1.1,,, polari"ation of the "P,I:
M (+, +;+)
iy'8g' R; Ti~ [(t (I - M'),j"MEq+_
-
+ u)(Pv ,)
+(., - I)(q+· <) + 2'i<""""kop+~q+-r<'] , M( +.-;+)
iv'8g3 R'T1 " (I - M')'j" MEq+ _
-
-($
M(-, +; -)
iv'8g 3 R', Til M'),j7rll1Eq+_
-(., + I)(q+ . , ) -
[(1 - u)(p+ . , ) i
+ I)(q+ ' <) + 2i
(I
M (-, - ;-) -
'...
i v'8g'R;
u)(p+ . t)
2;,oP"ko p+.e Q+, "]
1ii
(I - M')'j7C lvlEq+ _
+(" -
[(I -
[(I + u )(p+· ,)
1)(/+' , ) - 2;,oP"kaP+M+-r<'] .
(10.456 )
Squared absolut.e values of the nonvanishing helicity amplitudes, summed over t he color degrees of freedom of the initial state and the final state and over t he polarization states of the 'P,:
IM( +,+;+)I' IM(+.-;+)I'
=
IM( - .-;-)I'
64g6 R,?(,,'t + 2.", M') 7rM3(1 - Ml)'
IM(-,+;-)I'
64.q6R,?(u't + 2suM' ) 1f M3( 1 - M')' (10.457)
Unpolarized squa.red matrix element :
[(5' + u'}/ + 4suM']
IMI' =
(10.458)
Unpo !arized cross section:
d" dt
3.11>1'5'(1 - M')'
"
[($' + u')t + 4suM']
.
( [0.459)
i ,
.1 ,,
~,
7J!I-+ 7j 3p'l
10.6 7 ['roccss:
q(p+. i) + y(A:, (I) __ 1'(q-+ ,j} + :lp,(p}.
(10.160)
Invaria.nts nnd definilions:
(10.461) Gluon polari1.at:ons:
(10.462) Nonva.nishing helicity ampHtude:l {the deflll.ition of the quantity ~ is given in eqn (8. 12) and ~hc q UMl.ity f.(>!J denotes t he polariulion tensor of the 3P2 J:
M(+,+;+i -
M(+.-;+ ) =
"ig3R;,~("'.8 [-2M]! {( . )' - .,0 _ M2)2 'lrt' I ..,.. u P-+oP+8
4ig3Jr(T!jetJ [-2.~J]! { ' u(t
Af2 P
ll't3
(t
+ 1J
,
)P+oP+o
+2(.m + tu - tM2)p+oq+~ + ($ + t)'q+"q-+o
M( - ,+;-) -
4i9:JR'tT;~f"fJ u(t-lvbp
+2($u
+ tu -
[-2M]l {'(t +u)p+oP+,O , Jrt 3 tMl )p+"q-+o
+ (s + /)'q+oq.o
/0 , HIiMMAilY OF QUI) fOJtMULAB
514
1:/.)' , 13 1! {(t + u)'p+.p+p
4ig 3 R~ T'-(·p [- 2M
- 5(t _
M( -, -; -) -
-/-2(sl
+ Btl -
-2i[(1
+ u)p+. + (5 -
IM')p+.q+p
+ (s' + I')q+oq+p
t)q+.J<"".pk"p~q+} _ (10.463)
Squared absolute values of the Donvanishing helicity amplitudes, summed over Lhe color degrees of freedom of t he initial state and the final state and over the polarization states of the JP2:
IM(+,+;+)I'
=
IM(-,-; - )I'
-,
64 96 H.I'/. ( , , "M' ) , ,,3 ( '{ _')' S 1 -I- 6$luM + 6u 3 rrw t t -lv,
" --
IM(+,-:+)I' - IM(- ,+:-lI' 649" II!? 3". M3t(t _ M')'
(' , " u t + 6sluM + 65
M ') (10.464)
Unpola.rized squa.red ma.trix element: __
Mf2 =
1J
4g6 R"
'. - 9"t(1 _ 111')'
[(5' + u')(t' + 6111 4 ) + 1251u1l1']
(10.465)
U npolarized cross section:
16..«1 RY'
du
dt = 10.68
- 9s'l( 1 - ,11')4
[is'
+ u')(I' + 6M') + 12stuM'] ,
(10.466)
qq .... g ISO
Process:
(10467)
Invariants and dennitions: 5
= 2(p+ ' p_) ,
5
+1+u =
t=-2(p+,k),
p' = M' _
u = -2(p_ ' k), (10.468)
rll f.
(10.469) ~or.vanishing'
helicity amplitudes [the definition of the 4uaulHy l/Q is given
in eqn (8.11 )]:
M{.f,-;+)
- _?3SRoTi~kJ. lOll]
M(-.+;+)
-
g3Ro Tii k i [ 2,L ,. - .~f2 31f'Mk+
-
_9
M(+ , -;-)
]!
[ 2k+ 3lf M k_
r
l/ 7 lo ijk.1 [ 21:_ S M2 3r.Alk+
t
,
M(- ,+; - ) -
g3 RoT,jki [ 2k+ ] s Nf2 3;rM.L
1
[10.470)
Squared ab!lOiule values of the non vanishing heiicity amplitudes, summed over the colo: degrees of fK"(:dolll of ~he initia.l 'Il&te and the final st .... ~e :
IM(+,-; +)I' IM( - ,
M[-,+;-)I'
-
L+)I' - IM(+,-;-)I' -
31f'Ms(8 _ Ml)l'
89 8 mt]
(10.471)
Un!>olaJized squared matrix element:
[10.472) Unp'llnri1.cd
10.69
cro~~
scction:
qq~93PO
ProCf'-SS:
(IOA74 )
, !J HI
/I},
SliMMAI1Y Of<' (J(.'I! 1,'OIWIII.IIP.'
I nvOLl'ialll,H and <.Jclillitiol!s:
s = 2(p+ ' p_) , $
t = - 2(p+ . k) ,
u =
-2(p _' k),
+ t + u = pl = M' ,
(10.475)
G Inon polarizations:
(10 .476)
NOllvanishing helicity amplitudes [the definition of the quantity If, is given in eqn (8.12)):
M(+ ,- ;+ ) " 'R'I T'ij (S - 3•,\ 1')k" J. (. - M')'
~~g
M(-,+;+l
_ _ 4i9'R~T;j (s - 3Afl)kl. [ k+ M(+, - ;-) (8 - M' )' 6" M'L
l!
M (-,+;-) =
! (10.477)
:1
1 'j
,
"
Squared absolute val ues of the nonvanishing helieity amplitudes, summed over the color degrees of freedom of the initial state and the final state:
IM(+,-;+)I'
IM(-, +; -)i'
IM(+, -; - )I'
=
IM(-,+; +)I'
32g6 R7(s - 3M' )',,' 3".M3.(8 - M')'
329' R:?(s - 3M')'t' (10.478) 311' M3 s( s - M')' .
Unpolarized squared matrix clement:
-
1M I' --
16g"R"(s - 3M')'(t'
+ ,,')
--"--::-:,'C-'-:--:::--.,--,-:,'=::7-:--'2i1r MOs(s - M' )' .
(10.479)
Unpolari zed cross section:
dq dt =
64".,,~ R;'(s
- 3M')'(t' + u')
27J\f3s 3 (s - M' )'
(10.480)
1
ij(r'""i)
+ q(1'- , j) .....
II1VAriants a.tld definitions:
!J
<:JUOIl
+ 1 + u -= p1
.II(~'IIl)
,= -2 (PT · k) ,
(lOAB!)
+ :J/'I (P), u
= -2(p_. k),
= M"l .
(IO AS2)
polarizll.~ion8:
j' = N [ft p_ h(1 ±-,,)+ p+ p- ft(1 '1'7,)[ . (10.'183) Nonvi!lI ishilig helicity II.ffiplitudes [t he defi nition of the quantity Jr. is g iven ill cqn (S.J~) and the four-vector (denote! the polarization veclorof the JPd:
,11(-.-;-) =
- E(,
2ig 3 R'I T,jk.l.
[
M' )' J.Mk , k. (u-,)(p+·,)
-(.~ + tJ( p . £) - 2i,a~6p.aP_a ~·..,(,s1 '
M(-,+i+) -
-£(3
2i93 Fr. "li~ ki
[
(
M"l)1.,firAfl.:.L ($+u)p_·tJ
+($ - I)(p- . £) - 2i~atl"'&p+ .. P_/l k..,·(.&1 '
M(-,-;- ) -
2igllr. Tg·k
£(3
11"f"l);~\1 CE [(3 + u)(p+ . l)
T(S -I)(p_ . t ) - 2it"O"llp+(ov_p k..,(sj ,
M(-.+; - ) -
E (s
2igl R'\ T,~k.i [ M"~)1.j~MCL (u - S}(PI . ()
- (!J + l)(p_· l)
+ 2it."P-np+.,.p ,k.,(&].
(10 .184 )
Squared abllOlule va.:ues of the nonvaui.shillg l:.elicity amplitudes, summed o\'er the co1or de,!\:rees o f fct:edom of the initial state and the filLal state and over the polarization states of the 3Pl : 6<:1g11 Irl(su1 + 2tuM2 )
)M(+.-;+)I' -
[M(-,+ ; - )[' -
[,11(+ . -;-)' -
Mg&.n;'l(st 2 + 2luM2) [M (-, +;T)I' = ;rAP(s - .M 2)4
.:t"A1J(s _
,\"f"l ) ~
(10.185)
niH
or Q(;U VOI/.J\1I1I,Mi
I II. SIIMMAIlY
Un polarized squared matrix clement:
IMI'
=
32 6 R" 9
,
91TM'(s - M' )'
Is(t'
+ It') + 4tuM']..
(10.486)
. (,
+ u') + 41 uM'I .
(10.487)
Unpolarizt::d cross section:
d"
12S"<>1R~
-dt = 9M's'(s _ M')' Is t
"
'.
Process:
(10,488) Invariants and definitions: 3
= 2(p+' p_ ),
s
+t +u =
t = -2(p+. k),
u
= -2(p_ ' k) ,
p' = ,112 ,
(10.489)
Gluon polarizations:
( 10,490) Nonvanishillg belicity amplitudes Ithe definition of the quantity R', is given in eqn (8. 12) and the quantity <"'" denotes the polar izat.ion tensor of the I: ,
M( +, -; +) =
aP
.g'R; T;jkJ.. [ 32M .'(s _ M')' ;rk~L -2(su
+ tu -
l' {' + , 1
sM')p+.p_P
(3
U )p+.p+p
+ (. + t)'P_.P_P
'P,
I
",
"
,,
1
.,,
1
~
M( -, +; +)
-
;93 R' Ta·k" l 'P
s'(: -']M~)'
[
1t d+k~, {(s + U)'P+.P+P 32M
-2(3t + iu - sM')p+.p_ P + (s'
+ t')p_.P_P
:~ "
"
M(+,-;-)
"'"
i9;)U~'liik.l.I ..(1
--'
8 2 (0$
-
:l:~M
[
,.
M~P
_ . _ _ .u
]! ,+"U,
'/I"klk_
{(.~
-':'2 (311 +!u - .'JAfJ)p+"p_p + (s
M(-.+;-)
-
)P+<>P+fJ
+ t)1 p_op_ p
ig-' ft'1 T·-,k- c ail [ 32M ]' ' -3 1 (8 ·J .i1)2 1l"k+k: {(S+U)'P+oP+:3 M - 2(st + tu - SM2)PtaP_P
+ (81 + (l)p- oP- fJ
Squared absolute vahles of the nonvanishing helicily amptiludes, summed over lhe co:or degrees of freedom of the jniti,d ~tate and the final state and over the polarization states of the 3g:
IM(+,-;+)I' = IM(-,+;-)I' _
6 49
G
31rMls(s
M AP)4
(.s "\I
4 ), + 6sfuA
IMh+;+)I' - IM(+, - ; - )1'
Unpolarized squared malrix eleu-.ent (10.493)
Unpolarized cross section: (1 0.194)
10 .72
gg~ 9180
Process:
(10.495)
III. 8/1MMA Ity Of' (J(.'I) fO/(MIJI,A!I'
l'osiLive z-axis: along
kl .
inva.ria.nts and defini tioIls: t = -2 (k, . k3)'
s+t+u=p'=M'.
" =
-2(kl . k3), (10.496)
Giuon polariza.tions:
' ..
M-J
=
1,
(10.497)
Nonvanishing helicity amplitudes [the defini tion of the quantity Ro is given in eqn (8.11)]:
M(+ , +;+) -
i.1 3.fI<, /" b' s'( _,t + tu + us) E' [247rMk3+ k3_]t (s - M')(t - M')( u - 111')'
-
i93 RoJ"1x 1I14(8t -I- tu + us) E'[241f/\1k3+k3 _]1 (s - 111')(1 - M' )(u - 111') ,
M (+ ,+i-) M (+,-i+ )
-
M (-,+i+) -
i93 R.,f"" u'( st + lu + us) E' [247r1l1k3+k3_]1 (s - M' )(t - M' )(u - 111')' ig3J?;,/ob' t'(s'! + lu + us ) E'[247rMk3+k3_lt (s - A(2 )(t - M')(u - M' )'
M (+ , - ; -) -
ig 3 R.,f"1x I' (st + tu + u.o) E'[24rr1l1k3+k3_lt (s - M')(t - M')(u - M' ) ,
M (- ,+;- ) -
i9 3 f1<,f""' u'(st + tu -I- u_,) E'[241rMk3+ k3_1! (s - M 2)( t - M')(u - M')'
M(- ; - ; +) -
ig' Ro/'1x M' (s t + tu -I- us) E'[2471 Mk3+ k3_ ]t (s - M') (I - M' )(u - M' ) ,
--
;9 3 R.,/'" s'(st + tu + ".o) £,[24rrMk3+k3_1! (s - M')(t - M')(u - 111')-
M( -,- ; - )
(10.498)
Sqllll1'cd l\1Jsolul.l! Yallll:H of lilt) l\(J I IV'~ lIjl! h ill)o( IlI'lkily arllpJitud{)~, ~ulluJlcd ()Vcr the color dcg n..'1.~ of frct..Jolll or I.hf: ill ilinl M 1atlJ alld the final ~lale:
IM(+, +: +) :'
~
IM(-,-:-)I'
-
16g6m ,~ 4 (.~ t + tu + u.~)~ 1I"Mlltu[(/i W)(l M~) ( u W ))2 ,
:M(+,+: -)1' - IM(-,-:+)I' 169~ mMiI(st
-
IM(+,-: +)I' -
"1!"
M stu[( s -- /I'PH t - 1\.1'4 )( H
_
M 2)j"l ,
IM(-,+:-)I' 16g11 ~u 4 (S,t
-
IM(-,+:+)I' -
+ tu + US)2
'If
M stuI(s
+ ttt + US)l
.11')(1- W)(u - .11')1' '
IM(+,-:-)I' (100499)
Unpolarized squared matrix element:
M ~ _ 96R l (Mil, + .,.1 + t4 + ttl)(3t + ttl + U3? I I -8.M,'" I(,-.l-I')('-M')(,,-M'II'
(10.500)
Unpolarized cross section:
dq 7rCt~Rl (M8 + 8 ~ + t 4 + 1.11)( 8 t + tu + UIl)1 dt = '2MIl~t;; [(;-= M~)(t - M~){u - M2)jl
(10.501)
Process:
9(k"a) + 9(k"b) - 9«"c) + '5',(p),
(10,502)
Positive z-axill: along k\. Invariants and definit ioos:
(10.503)
<-:(11011
IloinriJl,jl,Lio II S:
"
Nih PI ,k,(l ± '"Is) + ,k, ,k, h(1 Of '"I,)], ( I O.50~)
, Nonvan ishing helicity a.mplitudes [the definit ion of the quantity Ro is gjve~l in cqn (S.II) a nd the rour-vector< denot~.s the polarization veGior of the 35, J: 9" R"d"'J"M
M(+ ,+; +)
s
E'1241rk3+k3_1t (3 - M' )(t - JI12)(U - M') X
It(t
+ u)(I: , · <) -
u(t + u)(k,· c) ,,
g3 Rod"
M(+, - ;+)
E '1241r/;3+ k3_Jt X
M(-,+; +) -
It(s - u)(k, . c)
t (s - M')(t - M') (u - 1\1')
+ tt(s + u)(I,,· <)
i'R"d,b, J"M
1l
E'[241rk3 +k3 _jt (s - M' )(t - M')(u - M')
x [tis
M!+ , , - ,' - ) -
' J"M
g3
+ t )(I: , · <) + u(s -
Rod,be J"M
t)(k,. <)
'U
E'[241rk3 .,.k3 _J ! (s - M')(t - M')(u - I'vI') X
It(.
+ /)(k,
. <)
+ U(8 -
t) (k,· c)
M (-.+;-)
,
_
M'4)(l _ 111'/)(1(
9 3~/Ja'>-:..fM
M(-.-;- ) -
3
E" :24dl+~"3_Jt (3 X
M2)(t - M2 )(U
M2)
[t(t + u)(k, . f) - U{t ~ u)(k, · () +3(t ~ u)(ka · t) - 2i( l
+ 'j)(""1~J.·,<>k2fJk3-T(~ 1
.
(10.505) Squa red absolute values of the non vanishing helici~y amplitudes, summed over the color degrees of freedom of the initial state a.nd the final Slate and over the polil.riza.:ion states of the :~I :
IM(+.+;+)I' -
160g6 RJM s 2 IM(-.-; -)I' = 9r. [(' /HZ)(u - M')P ,
IM(+.-; +)I' IM (-.+; ~ ) I'
M (- ,-r;-) '
-
IM(+. -;-) '
1609GU~Mt l
-
9r.1(, - ,11' )(. - M')I" IG0g6 ,mMu: 9.1(, - ,11')(1 - M')J' · ( 10.506)
-
Un polarizeJ squared mal rix clemenl:
5g~ H~M [","_=,,'~'_== 367."
[(l
1"'I1)(u
M2)1~
u'
+ 1(' - M')(u " - -'1')]' + [(,
(10.507)
Unpobrizcrl t:ros)j section:
d. dl
-
571"0:~mM [ 98 2
,,1 I(t - ;I1/l)(tJ
".,l )P u'
I'
+[ ($ - M~)(u - A'I1 Jl~
1('
1\112 )( l
M'I]' ].
(10.508)
III, 811MMAUYOrqcI! I'OUMUI,Ah'
r)~1
10.74
qq
--->
9
IPI
Process;
g(k"a)-\-g(k"b)
---t
g(k3'C)
+ 'P,(p).
(10.509)
Positive z-axis: along kl . Invariants and definitions:
s = 2(k, . k-,) , s
,,
. ;!
1= -2(k, . k3) ,
+ t + 11 = p' = lVI' ,
P = sl + tu
u = -2(k,· k3)'
+ us,
Q=stu.
(10.510)
Gluon polarizations;
N !,k, ,k3
;,,(1 'f /5) + h
/c3,k,(I/,5 )], ,
(10.511) Nonvanishi ng heiicity amplitudes [the definition of the quantity I~ is given in eqn (8.12) ",nd tbe four-vector < denotes the polarization vecto r of the 'P, [:
ig 3 Ridabc 3 2 E'[2.,,-Mk3+ k3_lt (5 - .111 )'(1 - M')(u - .11')
M(+,+i+) = X
{t(: ~~ [tM" -
(811
+ t')M 4 -
stuM' + 2s'tu]
- (k",) [11M" - (st -\- u 2)M4 - sl'"M' -\- 2.'111 1 1L - A12
ig3 If!, do" t E2[21i'Mk3+ k3_lt (s - M')(I - M2)'(u - JV[')
M(+,-i+) x
{~~ ~S' ruM" - (sl -\- u').M' -
sluM' -\- 2st'11J
+ S-M2 (ks . <)
sluM'
[sM' _ (tu + 8'lM'
-
+ 251 211J
l
II).
n
"y _.
!J 11'1
"
M (-,+ ;+ ) =
""' )( u
+ (ka · d [s M6 _ (tu + S~ ) M4 _ S CU.~of2 + 2stu'] 3
M1
_ 2i( u _ lv(1)M2 CoP'll kla k,pkhf6 } ,
M(T , - ', - I'
,
;9. 3R' d~'x;
-
M ')(' x { (k\ . (~ [tM 6 t - M·
+ (k3' l ) ,s -
M'
+2-i(u. M (-, + ;- )
=
u
,\-P )(u
t~)M~
M~ ) 1
_ sluM'
+ 2stu 2]
[.5 M6 _ (ttl + .,1 )M 4 _ sl uM1
+ 2stu2]
_
(s u -!-
M~)M·llnO"'&k\ .. k20kJ~CS} ' t
ig3R~d~k
Ell~bfM k3+kJ-J~
X{~~ ;)2 ru MS -
($ {s t
M~ )
M2)(t ·· AflP (U
+ u2)M~
- stuM?
+ ;~ ~~)2 [sM 6 - (tu + s l)ll'[4
+ 2sflu]
- stu M 2 + 2st 1uJ
-2i(t - M'JM,,"O-'6 k\okl;1kh(6} ,
M( - , -;-) _
s
ig3 1l',d~6<
El[2~M~3+k3.1 ~ X {
(1.; \ . l)
t- Ml
[t Me _
(s {S!.l
M 2)2(t - A","")"(u'-----;M"")
+ t 1 )j\.j4
_ sluM 2
+ 2s1 tu]
. '
.
/IJ. I>IIMMIIIlY
i9: 1R~ dabc
M(+,+;-)
Of
1
E'[327rMk;,+k3 _Jt (8 - 111')(1- M')(u - M') x{2(k, . C)(S - u) [3tM' _ (su -I- 3t')M' t - M'
-I-
2( k .,)( t - s) [ 2
U -
-
1\1 2
3uM' - (st -I- 3u')JVI'
2(k,.,)(u- t)[ ' 35M' - (tu -I- 3s')M' s -M'
-8iM',o/h' k,.k,ft k3,Q }
M (-, - ;+)
I.J(;J) I"()"MUI.A/~
+ U(st -I- 2«') 1
+ s(tu + 25') ]
,
i 93 R'l dab,
-
+ 1(5'" + 212)]
1
E'[321TMk3+k3 _J1 (s - M')~t - M')(u - M') X {2( "\ '~)~2- u) [3tM' - (Su
+ 2(k~'~)~1; s) [3tt A{4 _
+ 3t')M' + t(su + 2t')1
(51 + 3u')M'
+ u(st + 2,,')]
- 2(k: ·~)t2- t) [35M' - (tu -I- 35')M' -I- s(lt< + 25')] (10.5 12) Squared absolute values of t.he nonvanishing belicity amplitudes) summed over the color degrees of freedom of the :initial state and the final state and
over the polarization states of the 'PI:
[M(+,-I-;-I-l!' =
x [
[M( +, + ; -
)1'
'1
. '.',
[M(-,-; - )i'
3lrM[(s - 11>[2)(t - M')(" -
M'
"
M'W
2stu(s' + }\1') + (5 _ M')(I - M')(u - /'.'[2)
1
IM(-,-;+W 64096 R;' M 7 [- M'(st + ttt + us) + 5stuJ 3".[(8 - ,11')(1 - M')( 11 - M')]'
, IM(+,-; + )I ' -I M(- , +;-')I '
31rM[ (.~ X
[lV/4
M1)(l
+
Ml )(U
A·P )j2
1 2stu(t - M4) (. - M')(, - M')(u - M')
1
M(-,+; +)I' _ IM(+.-;-)I'
(10.513) Unpolarized squl'ired ma.:rix element: 5g 6 R?
1 3< M ul(,:, ----' M"'''')(o-,-_-'.MO""")("u-,;:;""")o,l'
x {M~(.'l2
~ /2 + u 2 + M~)
....- 2"tuls4 + t4 T u 4 + M~ (S1 + {~+ til) + 2MB] } . (05 - AfJ)(t - M2 )(u - Ml) .
(10 ,514)
l":lpolarized cross .set:tion;
dO' dt =
4Or. o~R7
_ M IDF
+ Mt. p 2 + Q(5M8 - 7/,,[CP + 2pl ) + 4M~Qz (Q - M1P)3
3Msl
(10,515)
Process:
g{k J, a)
+ g(k 2 ,b)-'I
g(k3, C)+ 3Po(p).
(10,5 16 )
Positive z-axis: along k! . Invariants and definitions: ,~
P =.-;t + tUTUS,
- 2(>,. k, ),
Q=
Sl tl ,
(10.517)
/11. Sf/MMAlty Of QCf) fUIlMULAI>
Gluon
pOhtri~ilLions:
(10.518) Nonvanishing helicity amplitudes [the definition of the quantity R', is given in eqn (8.12)]:
M(+,+;+) - M(-,-;-)
(
i g3R;!""
S'[t',,2 - stu(s - 6M') - 38'M'(s - M')] E'[61rM3k3+k3_lt (s - M2)(t - M2)'(u - M')'
M(+,+ ;-) = M(-,-;+) =
;9 3 R~r'" [ E2
1
3M' t st + ttt + ttS 211'k3+k3_ (8 - M')(t - M')(u - M')'
M(+,-;+ ) = M(-,+;-) ig"R;1""t'[s"U' - stu(t - 6M2) - 3t'M'(t - M'l]
E'[6"M3k3+k3_lt
(s - 111')2(t - 111')(1£ - M')'
M( - ,+;+) = M(+,-;-)
-
;g 3 11;1'"
,,'[,'/' - slul u - 6M') - 3u' M'( u - M'l] E' [6d{'kJ +k3 _]t (s - /If')'(t - A'!'l'(u - lvJ2) (10 .519)
Squared absolute values of the nonvanishing helicity amplitudes, summed over the color degrees of freedom of the initial state and the final state:
IM(+,+;+)I'
=
1111(_,_;_)[' 64g 6 11;'5 [PQ + 38' M' P + 2sM'QI' "ll:Ptu (s - M')'(t - M')'(u-M2).,
i,
1.11(+,+;-)1' - IMh -';+II'
IM(+,-;+)I' - IM(-'+; '-)I' [PQ ..:.. 31 2 Ail P + ZlM:lQP :i"!\.P,H/ (7-- = .'Hl)4(i M')1(1.i Af 2).I'
64g GR~~l -
IM( -, +; 'ell' - IM( +, -; - 11' 64g6 R;l1J. [PQ + 3tl~ M2 P + 2uM1QF /f,\13!;t (.~ - Ml)t(t - M2)~(U - ,',11)2'
(:0.520)
Unpolarized sq uared mM.rix elcr.1ent:
!;lIP
=
;r:\f3Q[;n:r 1I1 2P)1
{9M~r4(Ms -
4
2M P + Pl)
_6f\1l p3Q(2M8 _ 5M 4P + p2) _ p2Q2( M8
+ 2M4 P _ p2)
+2M2 PQ3(M 4 _ P) -I- 6M"Q"j.
(10 .521)
Ullpolarized cross section:
du
dt
_
4:rrct 3 RI2
' Sl ,~lMJQ(Q _ M~P ) '1
[
9M< 'P4(M8_U""?+pl)
_6M2 p3Q(2M s _ 5M 4 P -I- p2) _
P~Ql(M8
+ 2Af4P _
p2)
(10.,22)
10,76
gg-->g'Pl
Process:
g{kJ, a) -I- 9(k1•b) Posi tive z-axis : along
-+
g(kJ' c)
+ 3p, (p) .
(lO.523 )
k\ .
lnvari
:} -I- l
+ to =
p~ = M'.! •
P=st-l-I,U-!-us,
Q=
sill..
(10 ..')21)
GIua;}
po la. rj ~a.L i DJ(s:
(10.525) NOllvnnishing helicity amplitudes [the definition of the quanti ty R; is given in egn (8.12) and the four-vector' denotes the polarization vectoro[ the SP,]:
ig 3 R'1jolbt; .-s.~ 2E2,J,rM ($ - M2)(t _ M2)2(u _ M2)2
M"(+,+;+) =
X {
(k, . ,) (s -I- t) (st( t - s) -I- tu (t - u) -I- S1l( U - s) ]
ig3R'lf""
M(+, - ;-I-) -
t
2E'V"M (s - .I\1 2)'(t - M')(u - M 2)1
x{ (k,· ')(1 -I- 11) [sits - t) -I- tu(u - t ) -I- su(u - 5)] -(k,· <)(8 +1 ) [st (s - t ) -I- tutu -
i9 3 R; p:tc
lvf(-,+;-I-) -
tJ -I- su(s - ")1
u
2E 2V 1r M (8 - M2 )2 (t -lvJ2)2( U - M') x
{(k, .'l et + ttl Ist (t -
5) + tu(t - u) + sues
-(k3' e)(s + u) [st (s - t) -I- tU(i - u)
- ttl]
+ sues -
ti)]
/tl.11i. !/!I"
1/"1',
.11( - .- ;-) =
t.'}:'U'I/"W"
It
'l.1!;"1.j;"M (.., -
X{ (kl . f)(l
,\1-;)1(1. -"'M""''')'''(-u---M=')
+ ll)[st(l - 09) + lu(l- U) + 5U{S - U)]
-(kJ · c)(s - u) [sl(s - t) -r tu(t
jl;}( - , - j - )
-
193 R~r'x 2E 1 v:rM X
M(- , -;-) _
+ su(s -
1£)]
t
(3 -
M?)2(t - M?)(u - ,\,[2)2
{(kl . e)(1 + It) [sl(s -(kJ
- ~)
· ~)(3
I)
+ tutu -!) + su(u - .,)]
+ i) [st(.!' -!) + IU(u -!) + .51((8 -
ig~ 1ft/"k
'l.E2 V:rM (.s
u)]
.'I
)lPH!
lv(2)2(U
M1)2
X{ (k,· c)(s + t)[.>t(t - $) + tu(t - Ii) + su(tJ - s)] +(k:z. t )(s + tI) [SII(U -
.~)
+ tU(tl -
+2ft<>P-,..l"kl<,>"k~4k:htdt - u){st + tu
t) + st(t - $)]
+ US _
2
o9 )}.
(10.526) Squared absolute values of tIle Ilonvdllishing heliciLy
IM(+,+;+)I' - 1.11(-.-;-)1' -
96ge R? .s2 "~\1""(;-'---'"CI1"';'::)11t - M~ )~(u X {(s / t)(1
M~)4
+ u)(u -/ s)[st(s - t? + tll(t -
U)2 + .su(s -
T.lll
r,:l ~
It!, StlMMAflY OF q( .'li fOIl.MUI,AI,'
IM(+,-;+)I' _ IM(- ,+;-)I' -
roM'(s - IH' )' (t - M')'(u - M')' X
{($
+ t)(t + ti)(U + S)[st(S - t)' + tll(t - u)' + sues - ul']
+M'(s - u)'(st
+ tu + us _ I ' )' }
,
IM(-,+;+)I' - IM(+,-;-)I' ( \
" M'(s - M,) ,·(t - MZ )' (u - ,11')' X
{(s + t)(t + 1/)(" + s)lst(s - t)' + tu(t - tI)' + sues - 1/)'J +M'(s - t)'(st + tu + us _ tI')'} .
(10.527)
Unpolarized squared maLrix elemen t: 6
- -, _ 3g R? P' [M' P'(M' - 4P) - 2Q(M8 - 5M'P _ P') 1,111 - "M' (Q _ M'P )'
loM'Q'J (10.528)
UnpoJari zed cross section:
d" dt
-
121Ta}R;" P'[M'P' (M' - 4P) - 2Q(M' - 5M'P - P' ) -15M'Q'J MSs' (Q _ M'P )' (10.529)
Process:
9(10" a)
+ g(k"
b)
-4
g(kJ , c)
+ 3p,(p).
(10.530)
Positive z-axis: along k1 . InvarianLs and definitions:
s = 2(k, . k,) ,
t = - 2(k, . k,) ,
s + t + u = p' = M' , Gluon polar izat ions:
u
P = st + tu + uS ,
= - 2(k, . ks), Q = stu.
(10.531)
,
,
111.77_ JII/ --, U;'I~
5:1:1
j " (', ) ) '(k, )
- NI.kJ ~
~l ~a(! T "Yr.) I
h fo, fo,1 I ± ,,)1.
Nit, to h ll ±",) + t,
t , fo,11 ",,)1.
t;l[32k3.,. kJ_l~ .
,v- ' -
110.532)
NOJlvanisldng hclicity Amplitudes /the definition of thc quallliLy R~ is givcn i:l cqn (S.l 2) And ~he quautiLy eo{J denotes the polariz
M(-,+;+)
=
ig 3 Jr. r~t"'/1 [ 2M - E-t(s-M2)(/-M2){u-M2) 7rk3+kl _
{~2 fi{i kl:\f2) -
u(u
+ st+tu+u.s [(9 -
t -
X
,'"
-jl.'J"
-
k2~\1'.1) 1 [I.'I~(tJH1 -
+ u(2u -
S)k2il )
1 2'I(..,,"n ;""-k"e[. (u {t_s) lu I 1 :} lfJ t(t M 2) - l + u - -:;- -
(l2(u-.sL _ u- t _ U(u. - NP)
-
su) - kduM2 - .!Ii)]
U)(U'I<1 - uh.. )(t kw - 1I ;"1P)
(t(2t - s)k1P
- k 2fJ
.r
"k"" [k (.it(ll - t) '.1.3 211 ( 11') UUl
-~(,,,,P
]1
tl
+ u') 1 + "(I' j _ M'.l k30 kJC t(st+tU+tIS)) su
tu _ U(st s
+ S - -51.!t
-
+tu +US» )]} , st
u(sl + tu + US)) .st
,n III, SIIMM.4IlY OF QUI! FOIlMUI,AiI'
-uk,. (t(21 - U)k, ~ - s(2; - u)k3., )
[k 1}1 (8tel2(t - "') U) _ ,_st _ t +s
-'-2' , "P" 'kP UJ~!JPIl r.:;1 I'i'Z 3 I
- hJ
U
I
1
+ I') k'ok,~ + ,,'(s' lJ _ M'
t(st+iU+U S)) su
\
s i s--,(_sl--,:_t_"+-,----U8-,-) )] } +',p k ( i'(S - u) -8+t - -s(; - NI') u lu
ig 3 RI
M (+, - ;-) =
rabce:~{J
-E'(s-M,)(t'~NI')(u-lVf2)
x {,u,[ t(t _k,"/11') tu + u,s + sl +stu.
[
" -) ')) + s(s k_ •M') 1[
lu - t - 8)(lkl • '
+ sk3a)(tk,~ + skaa )
"
,
u'(s'
+ /')
1
-uk,.(,\2t - u)k , P-s(2s - u)k3,o)+ u-/11' k'ok' a -
k,,,,vkP[k ( S'(t - u) ' 2U:/J.II(JCo ;. K:2 3 1/3 ( M') - t t t ~
+k'0
((l(S - U)
si
+5 -
( H'l) - 8 $5-1)-:1.
~ -
i(Si + iU+US)) Sit
1.£
+ ,t -
5t
-1£ -
B(st + iU+US))]} t 'U
,
M (-, +;-) =
;g'R; j"',QO [ 2M E'(s - M2)(t - M')(u - 1"12) "k3+k3_
u(" _ M') + S(s k_ "ji NI') k,O "M {t,[k," 3
X
+ US [(t + st +Iu slu
C
'- ) st
1'
+ k3" ( sM' -- tU)I
s - 1<)(U<,. + sk,.)(uk,O + S,,"a)
'
c +JI;3JJ
O·
! .
("'(S - I) s(s M'l )
k"k'k'['. (Ul(t-J) I '2 3 "1 ,(1 t(t _ M -l)
-
- s
.
+ u - -."t -
t 1- tI
-
tll II
-
.("
"'»)]}
+ Iu +ttl
'
t(.'lt+tU+tlJ)) SU
tU+US))]}.
-k, (t2(~_S) _u~t_tu _ tl(st·1 fj U(U "'P) s sf
(10.533) Squared absolute values of the nonvan.isLing Lelicity ;\mplj~udes, su mmed over the color degrees of frl."Cdom of t.Le initial slate !\lid l~le final s·.a~e and over the polariza tion sta.tes of ~Le sP2 :
W(+,t;+)[' - [M (-, .. ; - )[' Mgt Kt
%A1JQ(Q
[lo '1'P'(3 '2 \ r'l \1') i\{'lP)~'':'' II -4"a - +! '
-123(5 - 3M')M 4p~ - 3(25,.,1 _ 333M 1 + 8M( )M6 p3Q
I fl . .'i IIMMllln' 0[0' QUO VOIIM 111.11 h'
IMh -; +)I'
IM(-,+;-)I'
=
64g'R;'
lfM3Q(Q - M' P)'
,
.,
[12M" P'(31' _ 41M'
+ M') '.
-121(t - 3M')M' p' - 3(25t' - 33tM' + 12{t' - 41M' - 31VJ')M' P'Q + (SI'
-2{1' - 5tM' - 30M')P' Q'
+ (29t'
+ SM')M6 p l Q
+ 9tM' -
- 51tM'
15M") M' P'Q'
+ 18M" )M' PQ3
+2(1- llM' )P'Q' - (9t - llM'lM'Q'L IM(-,+;+)iZ = IM(+,-;-)I' 6
64g R;'
1f M3Q (Q - M'P)"
112M8 P'(3 u' _ 4uM'
- 121.1(1.1 - 3M'lM'P' - 3(251.1' - 3JuM'
+12(1.1' - 4uM' - 3M") M' P'Q
+ M') + 8M') M ' P3Q
+ (81.1' + guM'
_ 2(<<' - 5uM' - 30M')P'Q' + (29,,' - SluM'
.,'
- 15M' )M'P' Q'
+ l SM')M' PQ'
+2(" -- IlM' )P'Q3 - (9" - IlM 2 )M'Q'j.
IMI' =
[
.~
I
7rM 3Q(Q ~M2P)' 12M'P'(M -2M'P + P') - 3M' P'Q(8 M S
-
S
M' P
+M' PQ3(16M' - 61P)
+ 4 P') -
2P'Q'(7 M8 _ 43M' P _ P' )
+ 12M'Q' j.
(10.535)
Unpo larized .cross section:
du
dt
4 'R" ""s ,
\' PM '( 8 - 2M 'P +P') M3 s'Q(Q-M'P)' [12J1 _3M' P 3 Q(8 M 8
-
lvI" P + 4P') - 2P'Q'(7M 8
+ M' PQ3(16M' - 61P)
l
(10.534)
Unpola ri zcd squared matrix element :
g6 R"
j
+ 12M'Q'j.
-
43M" P - P' ) (lO.5~J6)
11 Polarization Bc,lm polarization gives information on Lh~ electron'::; inclinatioll to behave with aberration.
111 the preceding chapters, we expounded a convenient way to calculate helil;ity 'lfllplitudt.'!l for vuriuus proc(.:~~cs at bigh t.1H.:rgks. 111 this chapter, we show how those }lelidty amplitudes must be combim::d for the description of processes in which arbitrary spin polarizations oecu r. We first treat the case of fermion polarization in detail and then allaly~e the polarization effects for phoLOns and gluolls. Throughout this chll.pter, we again neglect the finite
mass effects for simplicity.
1 1.1
Fermio ns
The positive helicity S!i\.tes for fermions, introdu~d in Chapter 3 [eqns (3. 1)], describe particles with spill pol flcizalioll along the direction of lnotioll, whereas nega.tive helicity states have their spin polarization opposite to the direction of motion. This can be seen as follows. A fermion at rest, with mass m, is described by the spinor u(O) which satisnes the Dirac equation
(o'-I)u(O)=O.
(11.1)
WHh our representation of the "-malriceg, eqns (2.G), we lJavc two ilJdependent solutions,
1 "+(0) = v'2
(
1) ; 0
(11.2)
and (1 1.3)
Bot h spinors an' eigenstates of E~ , the spin-operator in the z-direction, where f is gh'en by
- (iiO) 0
E=
(j
,
(IlA)
I I.
I'()l,~ IUX~
'I'/ON
Le.,
'.
(11.5 ) . ",;
Similarly, one finds that (u+ ± u_)/0 are eigenstates of E., while ("+ ·i,,-)/0 are eigenstates of E,:
± "
(11.6) . ~
How does one describe fermions moving along the z-direction with, for example) spin polarization along the positive x- a~i~? It suffices to boost the states .(11.2) and (11.3) in the z-direction, which tra.nsfo rms the spinors u± into hclicity eigenstates once the fel'mjon mass in neglected. Such a boost
does not affect the spin in the x-direction. Hence, the amplitude for spin in the positive >:-direction ("r = + 1) is simply given by the following linear combination of helicity amplitudes M (±): 1
M(s. = +1) - - [M(+ ) -IM(- )J.
o
"
·,·
(11.7) "
Obviously, one also finds that
- o1
111(s: = -I)
M(s,
=
+ 1)
~
[lvl(+) +iM( - )J,
~
[M(+ ) -iM(-)J .
v2
M(s, = -1)
=
[M(+ )- M(-)J,
.~ (11.8)
The same analysis can he repeated for the antifermions, described by the
.,"j: t~
.~
v-type spinol's. Now,
( 11.9) and
. (I ) "
"- (0) =
1
0
0
-~
.
· ..
~
(11.10)
are the two jndependent solutions of the Dirac equation
hO + l)v(O) =
O.
(11.11 )
;,
.-.
II.I!. AN
~.',\'/1 MI'I,,,,·:,I ,. -
Thmllgh LIt<:
""J,II'
S1\IlI(l hOl)~!" 1,llIly !lr;~ Lr;~II",r"rll,,'d
illl,,, hdidl,y
~ig
Hilt
BOW,
E, v_ :r- t:+ v'2 E
U_
~ i~l .
• v'2
-
±" :r: tI,
-
.L
v'2
V_
T iv+
v'2
(11.12)
To describe transversely polariz.ed antife:-mious, we thus have to use the following combinations of helicity ampl itudes:
M(s", = +1)
I
= v'2 I
[M H + M(+lI ,
v'2 (MH -
M(:;" == -1)
M(+)]
Al(.!I~
= +1)
-
; , [M(-) +iMI+)[,
M ("
= -1)
-
; , [MH- iM(+l:
(11.13)
Since the expressions iiu and vv are invariant under rotations, it follows tha.t the states (u+ -iL)/v'2 and (v_ - v+)/.,fi descrihe the spin statc~ with 9" = + 1. In t he same way, one can construct the other transverse spi n ~lates involving t he spinors IT or V. T he results of ~h i s sect ion are sllmrruuized in Table 11.1.
When e 1 ond e- beams are kept citculatillg in storage rings for long periods of time, the particles tend to l.lign their spins alon!!, the direction of t he rr.agndic field which keeps them in circ\I!
(I !.Ii) for the (:ru;c whl~re the e- end e- spin p o l ari zat ~or.s a:e thus diredeclll. lo:lg the x-direclion. As usual, we take the po~i tive z-clirection along ih .
/I , l'OI.A llI1.II'1'ION
Table 11 . /: Spinor combinaLions for
Sz:
u+
=
+1
+ tJ._
== -1
Sy
= +1
u+ - u_
u+
+ itL
SO!':
J2
J2
u+ -u_
u+ +u_
J2
J2
v _ + v+
tJ_ -
J2 v_ -v+
J2
tmllH VCl'Se
v+
u+
+ v+
-1
LU_
J2
+ iu_
1.1+ - ilL
J2
J2 1)_
=
u+ -
J2
J2 iL
8 11
pola.l'izalion.
+ iv+
'l L -
lV+
J2
J2
v_ + iih
'v_ - zv+
J2
J2
J2
Let M(A" .\,; A3, A,) be the helicity amplitudes for this process, with A, (A,) the helie;!y of the e+ (e) and A, (A,) that of the It+ (1'-). The nonvauishing hclicity amplitudes are given by [see eqns (9.10)1:
M( -I-,-;+, - ) =
M(+,-;- , +) M(-,+; - ,+) -
e' q;'~
E
M (-,+; +, -) = e'
q~l. E
1
(q++) , q_+
(q_+) r. q++
(11,15)
Denoting s, '= + 1 by i and Sz = - 1 by 1, we know from the preceding scction t hat t he amplitude for sx(e+) = +1 is given by (1 J.l6) It then follows that, for sr.(e+)
= +J
and s,(e- ) = +1,
(ILl?)
II . !:. n~
AN ~;XI1MI'I,Jo."."'·+r- ·· '{I'/I -
M ( +,+;A:I, A ~ ) = M ( -,
i ll tile" b i~11 ( ~II (~l"gy lintil. '1 '1 1(' nlJll::iniHg {"o rn h illilLi(llI ~ ur ~ pill pull1 l·i:.mI.iol1 (:;111 be wurkccJ ul,;l ~i[])
il;~dy
--j
A:I , A~) ..:.
(j
rcad~
=
M {l,l ; ,\:, ,>'~ )
~
liM(-,+;'\"A,)+M(+,-;'\"A, )].
M(T, T;).3,)..I)
Let t.;s concentrate on tIle important case lor which sr(e+) .~.r(e - ) = - I . There a.re or.ly two amplit'..Jdes, j.e.,
1l.1l
(1 LiS) +1 ar.d
'l' ( ), () i] q++
•
q_+
,"/ ( T,.l;+,-) =
e 2£
M (T,1;-, +) -
2e~ [q+..l.(:::f +q_..l. (:::)t]
q+..l.
q_+
+ q-..l.
q++
(11.19)
d
" (,';. IM(T,j;+,-)I' ~ I M(1, j ;-,+)I' ~ E'
+qU ,
(l!.20)
S UHl:n.ing oyer the helicit ies of the ot:tgoi:J.g /1+ and p.- , we have
du O 1) dt
=
T." a 2
2
81','6 (q+~
. T"
2
( 11.21)
q+,),
with
t = -2(p.+ . q.,.).
(11.22)
Thi~
formula supposes compk-tc ~jJ ill polarization
du dl
_
p , du{1l)
""
16E~ lq+o
at
T."0'1'1
'-t P (I _ P) [dU{TT) + dU(!))] + (1 _ p),du (!1) dt dt dl 2 + q_t -
(
"(1
~ ·"I
1 - 2P; 1.1..." - q+~ J.
(1\.23)
with
du(!!) _ du(J I) _ ca' ( , dt
-
dl
- 8E6 q+J
, )
+ q+:
.
(I!."I)
J I. N!I,A/UZA 'I'I1JN
CI,carty, whell P
= 1/2, Le. , no polarization, W(: r<~c()vcr the unpolarizeu Cl'OtiS
section of eqll (9.12).
11.3
Photons and gluons
Throughou t this book, we used polariza.tion vedors for incoming photons and gluons of t he form
. ~l , j
(11.25) where k is the four-momentum of the partide and k2 and k3 are taken to be light-like. Let us first convince ourselves that eqn (11.25) corresponds to a positive helicity sLaLe for f'"+ and a negative helicity sta.te for (.--. The four-vector potential of a photon or gl uon with momentum k" IS given by 4.1'( X' k) -
,
,
1
- [(211" )32ko]I
['"(k )e-i(k.r)
+ <"(k)ei(k.r1] .
, :i
• \
(11.26 )
Let us take k- = (E, 0, 0, E), i.e_, motion along the .-axis. To the expression (IL26) then correspond the following c.om ponents of the electric. field: 1
E.(x; k)
[(~~31' [Re(f;(k)) sin(A: . x) -
lm(c;(k)) cos(k· x)]
E.(x;k)
[(~~31 ~ [Re(E~(k)) sin(k· x) -
Im(C; (k)) cos(k· x)] (IL27)
From eqn (l1.25), .it follows that
(11 .28)
A simple calculation shows that
4N' E' [(kJrk,_ - k'r k3_)'
+ (k"k3 _ -
k,.k,_)'] = ~,
(11.29)
as
(1l.30) Hence, writing
1
.
<
--SinO
Ji
I - -cos8
Ji
' '
(lUll)
"
fl..Y.
/'I/()'/'(}N8 AN/) (.'/,fI()N.'·
wehitvc
D,(r; k) -
I'; [ (2lT)3
]! co~ ((k .:z:)+6), 11",,((k. ...-)+, -")2 .
D [ -(2;rp.
c
(11.32)
E~
loud EJI are found to be e(j l:al a.nd Lhe phase difference between Elf a nd Es is ~"Cn to be -r.J2 . This is precisely the definition of positive helicity, also called righL·handed circular I'oiuizal ion. (In optics, the convention is , for SOIDC rcason, just the other way around: the case of (11 .32) is (;a.!led lefHanded circular polarization {43] .; The same manipulat.iom lead lo lhe conclusion that (;"- denote~ photon~ or gl uons with negative r.elicities or right-handed circular polarizations. How do we dC$("ribc the c;..se of linear polnrization? Consider the combi· The amplitudes of
nitt\oo
(11.33)
where
(ILl") with p" ~ (1,0,0,
1),
q" ~ (1,1,0,0),
(11.35)
It follows that
(lUG) and
E~(x; k) _ _ [ 2£ ]' ,in(l ' x) (211"P , ( l U7) Thu3, (."11 iii found to describe the eaSel of linelM pola.rization dong the
x-axIs. In practic.e, we want to perform the ca1cula~ions with tr.e polarization vectors f "+' given by eqn (11.'25). To go from f - ± to ql:, we usc the relat ion [sec cql1 (4.13)] (11.38)
J I.
J'(I/,~ II (~II'/'ION
with e±iO' --
-(l'±' q
(.~ ) ,
( 1l.;!D)
and t he quantHies IJ± are irrelevant because o f gauge in variance. The phCUiC facto r eiCt is easBy evaluated:
e'" = -N N;rr[p It I<
1<, h
1«1 - "I" )] (11.40)
with ( 11 .41)
LcL M( +) and M( -) be the helicity amplilude.3 for" gi ven process calculated with the polarization vectors ,"±. The ampli tude for the process invo lving a photon or a gl uon with linear polariza.t ion along the x-nxis I S then gi ven by
M(x) = _ k2.lZ~3M( + )
+ k2-,Z,3J1:(-)
2k2+ [k,_k3 _(k, . k3 )J'
(11.42)
=
ReplacilJg qll by q'lJ, (1,0 , 1,0) in tbe abo ve formul ae, we obtain the descript ion of lin ear polariza.tioll along the y-ax is. In tha.t. case, the phase
rador becomes
e and
ill"
=-
ik2.lZ23 1 k't[2k,_kJ _(k, . k3 )1'
•
_ i[k2.l Z"M(+) + k;-,Z'3 M (-)J 11(Y) - 1 2k2+[k,_k3 _(k,. kJ )!,
I
(11.43)
(1 1.44 )
is t he correspondi ng ampli tude . \Vhen the motion of the photon o r g luon lakes place along the negat ive z-axis, we choose as in eG n (l 1.3 4), but witb p" = (1,0,0,1)- Analogous to the a,bove Ct', the phase factor for linear polarization along the x-axis t.hen becomes
,;±
(IL4.5) with ( 11,46) Taking q~ ::: (1,0 1 1) 0) we obtilin photons or gluo ns with linear poJarization along the y-axis, and the relevant phase factor is (11.47)
<,
1',,1,11' 11.2: llelieil.y f:()!Hl , il!,.ti oll ~ rul' !i ll/'II I' p(ll ll ri~,:.t i t)!I. 'l' h f~ q!llLll~iLie~ tr,
k. = + E
k,
11.4
~-E
-
_. M(x)
M (,)
e'o M ( _) + e- i :> 111(-)
~ j"' M (+)
v"i
f:Y M ( + )
i
i
+e-;O'M(_) ;
v"i
+ e-' PM( v"i
)
e;P'M(+)
+ e-;P' M( - ) v"i
An ex a mple : 9 9 _ 9 J50
b Section 10.72, we presented the he!icity ampli t udes for the process (11.48) wi Lh the following choice of polarization vectors for t he /l;!noIlS:
(11.49 ) By taking the positive z-axis along
kJ , we obtailled the IlonvanisiliIlg helici ty
amp litudes in eqas (10.4 98).
Suppose we w"nt to caJc ulale t he cross sect ion fo r the case of incomir:g gluons which a.re both linearly polarized in c!~rectio n s perpe n dicul
II . !'()/,A II.1Z/I'I'ION
glunn 2 we have to usc ,,;~', given by cqn (I I A 7) . They are simply e ia .
k3.L
=
e1rJJ
[kHk,_l !
, (11.50)
e i {J'
ik3 1
-
ie-it/>
["3t k3-1 f
wi t h r/> the azimuthal angle of k3 . From eqns (10.498) , it then follows that
,
._.~
HM(+,+;+) -e2;6M(+,-;+)
Jl1(x , y;+) =
9' Rof"'>«st
+ t ·u + us) [M' + .' + u'e';· - t'e-';" ]
2E'[241TMk3+k3_1t(s - M')( t - M')(u - 1\1 2 ) M(x,y;-)
t[M (+. + ;- ) - o"" M(+.-;-)
93 Rof'''(st + tll + us)
1M' + s' - I',,';. + ·u'.-';/o]
2E'[%rMk3+ k3_1~(. - M')(t - M')(u - M') (11.5 1) Summing over t he color degrees of freedom of the bitial state and the final st;Lte, we
find
IM(x.v;+)i' = IM(x,y;-)I' 4gGR& (st IT
M stu[( 8
-
+ tu + us)'
M')( t - M 2)( u
-
M')],
x { [IW1+s'+(u'-t')cos(2r/»]'
+ (u'+t')'sin'(2r/»} , (11.52)
and the polarized cross section, averaged over the initial state color degrees of freedom and summed over the tinal state degrees of freedom, reads
du
dt
"ifsRi
(st+tU+U8)' 2M8 tu [is - M2)(t - M')(u - 1\1')], 3
x{ IM'+s'+(u' - t')coS(2r/» ]' +(u' +t')'sin'(2,pl}. (11.53)
1i'17
Th is nx am p 1(1 W I\~ pa rt jell Iad y ~ i 111 pill I /\Ji I hc' d ifrc'!'c'u t hdi ci LY illllpi i l UdC8 (',)ul(1 be nkcly mmbined. TlIi!! wn.~ dllc~ iu parl lo lhc fact that the color ~LrucLuJ"c for cadllldidty M!lJlliLudc: Willi givCll by a!t overall fa.ctor III appl ications to other processes, OllO mny encounter more complicated siLuaLiolLs where the helicity. amplitudes ate linear comb ination s of cEfferO;lt rolor lJlatrices. In that CMe, it is prohably be5L to resort to cor:lplex il.riUllne~ic. Oue first evalua.tes a.11 the coefficients of the color matrices in each amplitude as complex numbers depending on t he components of the four·mon~cota in the process. MuHiplying the Coefficients with the appropriate phase factors then yields the amplitudes for the polari~ed sl'atLering proce~s, and the correspondillg cross section is obtai ned by n.dding the squared ab!';olute v:t.lu~,.s of ~he polarization amplitudes, summed and/or averaged over the color degrees of freedom.
r lx.
'.
,.
1
12 Beamstrahlung Photons mediate,
but
electron~
radiate:
The loss of coU!lLing rale make,s people Iulminate.
12 .1
Electron-positron linea r coHider s
1n this book, the helid ly method is applil.-u Lo a variety of proce5ses. These processes are conwmienlly described in terms of Feynman diagrams, and as a rc:ouH all fonnulae obtained so fa.r, including lhose of Chapters 9 and 10, are ea.ch proportional to powers of the relevant cou pling constants. Such applications of the helicily method arc by now rornpletely understood, altho'Jgh the calculations remain to be worked out fOI" prol.'csses involving more particles, arld f(l rther simplifications can be achieved in some case-s. As an c"ilmplc, tIle fet!uiLs {lO.SI2) for the prodllction u! t.llC lp} heavy q~".rkonium state through 9 9 ....... 9 lPI have been o bt.ained only r C~~ lltl y. the previous version being signifkantly morc cOInl-'licaled. T hese are not t he only known applications of the helicity method. How· ever, t hp.:>c other applica~ions have not yet reached a state of complet ioll comlMrable to those already described. In t.his chapter, which is essentially the Ia.st chapter of the book, we alteml-'t to present one of these other applications, namely that \.0 beamstrahlung. Beamstrahlullg refers to the bremsstrah lung due to the in terpenetra tioll of two bunches of elect rons ar:d rositror.~. lts importance to physics comes a.bout V, Ilsing ~u;JeH;ond u dil\g i\~~el erating o_vit.ies. Since the energy loss due to synchrotron radiation is propo rtional to t he fourth power of lhe energy of the circulating eleclroIl and positron bunches, it is unlikely for the ccntre-or-mass energy a t LEP to exceed signilicalltly this 200 CeV. In order LO reach the TeV regime, Wilh cireui;),!' coUiding machincs,;:\ larger ring is necessary. Such a projC<:t he.s [JO~ been planned, and is quite unlikely. SinC\~ tf:.: limiting factor is synchrot ron radiation, 1'1 possible e-ltcrr.ali ve is ~he electron.po~itron linear collidcr, which co!1sists of two linear accc\er
'"
'.
.
.' . 13. IJI!AMS'I'nAI//'(INU
the electron butlch alld Lhe positl'On bUTlcll illi.erpenetrate cadi other only
once. This is to be contrasted with circul",. collid"rs, such as LEP, PETRA at DESY and PEP and SPEAR at SLAC, where the electron and positron bunches cross each other many times, even in one millisecond. The single-pass nature of e+e- linear colliders imposes strong constraints on the design of such accelerators. After Doe collision, the energies contained in the electron and the positron bunches are completely lost, since no practical way of recovering them has been found. Accordingly, the number of particles in each bunch is limited by power requirement, 'which is typically laken to be 100 or 200 megawatts, r.omp.",;,ble to that of LEP. On the other hand, the potentially interesting annihila.tion cross sections decrease with increasing energies as the square of the centre-oC-mass energy_ Therefore, with the power limitation) a. useful luminosity, which must compensate the decrease of the ,'lllnihilatioll cross sect ions , can be reached only by making the radius of the bunches exceedingly small: typically less than one micron,
often milch less [15J. During the last few years, there are discllssions of Ihree TeV e+e- linear colliders:
1. CLIC (CERN Linear C.ollider) at CERN, with a centre·ol-mass energy of 2 TeV; 2. TLC (TeV Linea·r .collider) at SLAC, wit.h the same c.entre·of-mass energyj and 3. Super, with a centre-ai-mass energy of 10 TeV.
As knowledge about linear colliders increases, the parameters for CLIC and TLC change. On the other hand, those for Super, which is an accelerator for the far future, remain al the values given by Richter [46J. The main parameters are
J = 100 Hz, R(radius of bunches) = 5
A,
N(number of particles io each bUllch) = 3 x 10', L.(length 01 bunch) = 0.3 I,m.
(12.1)
With these parameters, tbe number density of the electron and positron bunches is aboul 1027 Icm" i.e., the density i. roughly that of water. Such very high densities have the following consequence. Although synchrotron radia.tion is avoided by using linear accelerators, there is nevertheless slgninc;1lli photon radiation due to the interpenetration of such very dense bunches. This photon radiation is called beamstrahlung [47J, and is studied
111..11.
NA'f"IJIU:(JF AI'I'JI ()XI.J.{ ATION8
filiI
ill this chfl.pl,r.r ill ~h() high energy limit. ror Mil III: )jiltlplc: (;aSCH. Tile [ll"esenLII.' I,ioll fo11oWA that o( .Taeal> aut! WI! [4~1 .
12 .2
N ature of approximations
Simi:ar to mM)' other problems in field theory, beamstrahhu:g i:s best siudied using Feynman diagr«m~ a~ llc~(;f i bcd in Chaptel' 2. It is ho we:ver not feasible to apply the the method of Fcymnil.n dia.grams directly. As give:l for example by (12.1)' there are alwa.ys many millions of eledrons and positrons in each bunch. No method ha<> ever been developed to st.udy the behaviour of a Feynman diagram involving millions of incoming elect ron lines a nd millions of inwrning p ositron Iiue:!. Therefore, by the nature of Lhe problem of bcamstrabluog , i1pproximations must be inLroduced at the beginning. fn ~his section, these approximations are discussed. 1n the future, if a decision is ever reached to construct an eledroli.positroll linear collider in the TeV regime, these approximations will probably be :·emo ....ed o r a~ leasL rerl'l.cf'd by bet.ter onl!s. Where feasible, we shall at lclIlp:, here to give a discussion of how ~hesc improvements may be carried OtIL.
12.2 .1
Single.particle
approxima~ion
Consider the interpenetration of an electron bUllch and tt positron bunch. If the two bunches are accurately cen tred with respec ~ to each otllt:r, then there is a mutual focusing eff(.'"(:L that lends to reduce the tra.nsverse sizes of the hunches. This eff~t is usefu l because it increases t}le lumillosi~y oC the electron-positrou t.:ollider. The mutuaJ focusing is Ileces ~arily a collective efTed hnd hence very dif· ficuH to extract from the feynman diagra.ms. Thus, the first assumption to be: imposed is that the mutual focusing effect is small. In order to put this effed of mutual focusing on a more (juantitative basis, consider an electron penetrating through a positron bUIlch (or equivalently a. posit ron through an electron bunch). While a si ngle electron does not disturb the positron bunch in any significant way, Lhe posi tron bunch acts as a for.using lens for the electron. Thus., there is a foca.l length, taken at the distance R, the mean radius (If the positron bunch. In accelerator physics, the disruptiQn factor D is delined [49] e~sea tia lly as the ratio of the bunch length L t tQ its focal length: (12.2)
where re is the classical radius of the electron, which i ... equal to ~.818 x \0-15 m , and 1" is the ratio vf the beam energy 10 lIJC electron masS. For the Super (flachine of Rkhter,·( "" 107 and, henle, D = 0.1 .
(12.3)
It. 1J";AMS'I'ItA IIWNG
., .'
OUf first as~ulTlpl.ion that the mutual CO('.lIl'Iitlg effect is small can uc rest.ated in the form that the focal length is milch IOllger than the bunch length, I.e.,
D«1.
(12,4 )
This is well satisfied by the Super machine. When the relation (12.4) holds, the electron and the positron bunches are not significantly distorted by each other. Under this circnmstance, it is justified to apply the single-particle approximation. More precisely, the problem of beamstrahlung<:an be replaced by the problem of a single electron penetrating and being deflected by the collective effect of a positron bunch (or equivalently, a positron by tbat of an elect ron bunch) . For the most important case of the radiation of a single photon , the process under consideration is
e
+ bunch -> e + 'Y + bunch .
(12.5)
Clearly, this single-particle approximation already simplifies the problem great1y. Let us discuss briefly how this condition (12.4) of small disruption factor may be removed. The issue is essentially the determination of the charge distribution of a bunch as seen by a.n electron or positron of the other bunch. It is feasible to carry out this determination classically, through the electromagnetic intcracl.ion between the bunches, and such a classical approximation is
likely to be quite
'. 1
,
.~
approximation can again be a.pplied. The main novel feature is tha.t different electrons (positrons) enc.ounter dHferent c harge distributions of the positron
(electron) bunch.
12.2.2
External field approximation
For the process (12.5), it is natural to use the external field approximation, i.e. , the approximation of replacing the bunch by an external field . This is most easily accomplished by a Lorentz transform to the coordinate system where the bunch is at rest, arid then replace the bunch by an external electrostatic polential. Rela.tive motion of the positrons (O J' electrons) in the bunch is neglected. . \Vith t.his external field approxima.t.ion, the Feynman diagra,m for the
process (12.5) is now the one shown in Fig. 12.1 , where each cross denotes an interaction with the external electrostatic potential. There are of course many additional Feynman diagrams with interIlal photon lines describing radiative corrections, but these additional diagrams will not be studied here.
There is a fundamental difference betwccn the Feynman diagram depicted in Fig . 12.1 and those of the earlier chapters of this book, namely, that of Fig. 12.1 stands for an infinite sct of Feynman diagrams. Let n; be the number of crosses, i .e., interactions with the external field , to the left of the
~ i, 'i"
!
au!.
NATIf/u:m' AI'/'/WX/MATIONS
I'
Fig. 12.1: Feynman diagram for e-
+ bunch
---10
e-
+ '1" + bUllch.
Cf7 vertex of Fig. 1.2.1, a.nd ItJ be the corresponding number Lo the right. [In the figure, 11; = 5 alld Itf = 4.J Then, both fI, awl Tlf need to be summed over all integers 0, 1, 2, 3 .. . , with the exceptioll of n; = 111 = O. Because of this sum, the beamstrahillng cross section is not simply proportional to some power of the fine struclure constant, contrary ~o, for example, all the CbSe$ gi·... en iu Chaj)ters 9 and 10. Indeed, it is not a po!ynomiltl ill tbe fine struclure coll~tant either. Consider the exceptional term 11; = = 0 jusi mentioned. Since it represents Lhe process C ---10 e-1 without allY external fidd, alld. hence , must be zero due to energy-momentum conservation , it is also correct to include this term. With this additional term that contributes nothing, n; and TIl are summed independently over all non-negntive integers. These sums have very simple interpretations. Thc slim over t); mCilJi9 tha.t tjle incoming electron plane wave should be replaced by the electron wave function i~ the presence Qf the externa.l electrostatic potential, and similarly the sum over nf mea-ns the replacement of the outgoing electron plane wave by a similar wave fuuction with the external potential. Forlunately, the dete.rmioation of the necessary \'>'avc functions at high energies in the presence of the external potential was completely solved in the fifties [050,511 a.nd are in a form suitable for the study of beamstra.hlung. The (~xt ernal potential cannot be trallslationaHy in ...ariant, since it is due to an electron or positron bunch of finite extent. Therefore, it is more natural and simpler to express these wave functions using: coordinate space rather than momentum space. In other words, the ma.trix element for thc Feynmp_n diagram, or rather the infinite set of Fcynman diagrams, of Fig. 12.1 is most simply expressed as a.n integral over the product of the wave functions m coordinak space:
n,
(12.6)
where, a.:; mentioned a.bove, W, and l/Jf are the initial and final electron wave functions in the presence of the external potent ial due to the positron bUllCh.
, Ie. IJBAMSTnAII{,IING
.
..
~
12.2.3
Importan t length scales
Before attempting to evaluate t.his matrix element, it is necessary to undel'· stand t he various important parameters associated with the process ( 12.1 ), and to express the regime of interest to be studied in terms of these para m~ eters.
Three characteristic lengths a re particularly relevant. The first one is the correlation length L ,. The notation here is to use a bar to,denote a length in the la boratory system (which is also the c. m. system) where the incoming electron and positron have opposite momenta, w11ile the correspond ing quantity without a bar means the length in the system where the positron (electron ) bunch is at rest. Thus, L, and the bunch length I., a re both taken in the laboratory system, while
.
",
<.
( i2.7)
and
where the virtual electron lengt h L , = "II.. will be defi ned in the next paragraph. As a lready mentioned , when an elect ron (positron) passes through the op positely moving positron (electron) bunch, it is deflected and focused by its electromagnetic inter •.ction wi t h the bunch as a wbole. The correlation length L.; is defined as the dis Lance it travels for a. deflection in angle of 1/" computed on the basis of a transverse distance.R and an a.verage cha.rge density over Lb . The second charaderistic length is the bunch length I.,. The third one is the virtual electron length Le1 that is the distance over which coherent radiation by the impinging electron is al lowed before it ent.ers the bunch. It is simply (12.8) L, = "I/m = "IT,/o , where tn is the electron mass. It may be instructive to give the approximate values of these charp..deristic lengths for the Super machine as descr ibed by (12.1 ). They are
10- 3 m ,
L,
2xlO- lO m ,
L.
2
Lb
3xlO-'m ,
L,
3 m,
L, -
4x lO - r,m ,
L,
40 m.
X
(12.9)
It is observed in this case that
(12.10) For the study of beamstrahlung at hi gh energies, Lc is not really an appropriate characteristic length t because it is adually not a relevant correla.tion length. The releva nt correlation lengt h is instead [52] (12.11)
,
IfU!.
NIi'/'IffU;; OF A/'I'IWXIMJ1'f'1 0NH
(12. 12)
This reJa~ioll can bu seen most readily fl~ !oJ!ows. During t.he transverse ddlc(".tion of the electron as s~n ill I.he laboratory syst.em, the milSS of the electron is the beam energy E, not the rest mass m. Since E :P m, this (".()rre1at:on length I.. is expected 10 be independen t of m. In o ther worus, since I~ is dearly unrelated to allY correlation length, to must be of the form L!-~ r,~, where the power a i~ determilLl.:d so tha.t the power dcpcnuence,> of L~ and Le on m cancel out. Since Le i~ defined t.o be the distance travel!ed for a defleclion in angle of 1/1 = mlE, L. is proponional to m. On the ot.her hand , it follows from eqn (12.8) thai
-J...=Ejm. ,
(12.13)
The required cancellation of the dependence on m yields imm edia~ely the result of eqn (12.12). For tbe SUj)er machine, it follow s from the numerical ...·atucs of eqns (12.9) that -lc=5xlO ro, t< = 0.05 m. [ 12.14)
-,
Thus, io this case,
(12.15) Note that (12.1.') Implies (1 2.1 0), but (12.10)doc~ not nec€ss:uily imply the relations (1 2.15). Throughout the remainder of this chapter, the inequalities (12.15) will be assumed to hold.
12.2.4
Shape of bunch
It rema.:ns to choose t he class of charge distrib\l~ions for the bunch. The actual charge distributions are very complicated and depend 0:1 the detailed design of tbe accelerating strudures of the Iirlcar collider. Since we do !lot wish to work with only the most idealized charge distribution, we reach t he following compromise. The transverse distributiOIl of the charge is taken to be \luiform and rota.· tionally symmetric. In olher words, the charge distr ibution in the transverse T!I-plane is taken to he a constant (which, however, depen ds on the loneiludinal varia.ble z) when x 2 +!I~ < R2, where the radius R is il:dcpendent of z. Porx 2 +y2 > R2,chargesareabsent. The loo~ituclinal di:::Ilribution of the charge, however, is allowed to Vil.ry in a fairly arbitra.ry way. To avoid unnecessary complications, this IOllgitudinal cbarge distribution is not allowed to ue zero for any finite value of.t. This innocent condition is amply satisfied for rea.listic bunches. For large values of !z!, the charge distribution is assumed to decrease exponenti ...lIy. Thb is also quite realistic. For techn:cal rcaSOLlS, longitudinal distri butions that
12. IJI;'AMS'I'nAIII,IINC,'
a.pproach 7.~ro for large Izi a.~ a GauSSiR.ll, Of lIlorc generally raster Lila-II a.ny expollential, are excluded from consideration . In a.ny ea.se l such J'apitl approa.cbes to zero are believed not to occur for realistic bunches.
::
At t.he time when this chapter is being written, tbe case of the trallSVOl'sally nonuniform bunch has not been solved . This case is expected to havo interesting features. IT tbe bunch is rotat ionally symmetric, then an electron that go~.s through the buncb at the axis of symmclry is not.. dellecled. At t he other extreme, an electron that travels at a large di stance from the axis experiences very little electromagnetic. force and is thus deHecled by only a small angle. There is therefore a distance from the axis where the deflection
angle is maxima1. At this distance, the matrix element, suitably defined: for the process (12. 1) is especially large, and the method of treatment to be discussed here is not adequate. We can look forward to many interesting developments in the theory of beamstrahlung in the next few years.
12_3
Electron wave fun ctions
The matrix element for the Feynman diagram of Fig. 12.1, after summation over all possible numbers of interactions with the' externa.l fie ld , is given by eqn (12.5). In order to evaluate th is expression for the matrix element, we need a know ledge of ",,(x) and ",,(x), the initi,,! and final electron wave functions which arc so lutions of tbe Dirac equation in the presence of t he external field. . Even though the external potentials are relatively simple for the bunches described in Section 12.2.4, it is nevertheless not possible to solve the Dirac equation explicitly. Fortunately, because of the very high energy of the incoming electron, especia.lIy in the frame where the bunch is at resL, the so-called high energy approximation [50,51) developed in the fifties cau be used for beamstrahlung. In a pplying this higb energy approxi mation , the fol lowing point is especially important. As specified by the condition (12.15), the bunch length L, is much looger than the correlation length f,. This implies that the angle of deflection of the electron by the posit.ron bunch must be laken into ac.count and cannot be neglected. [n terms of the high energy approximation, this means tha.t it is not sllfficicnt to use the leading approximation [50]; rather the next-order approximation [511 mnst be employed. The high energy
The formalism is actually quite simple. Let the spin-O wave function ",(x), which satisfies the Klein-Gordon equation [53)
{[E - eV(x)I'
+ \7 2 -
m
2
}",(X) = 0,
(12.16)
., .'
"
I!! ..~.
~:U:(."I'/W N
WA
n: FIIN(."/"lONS
,,(if) ~ c' ''I~ A (i).
(12.17)
Suustitutioll iuto eqn (12. 16) then gives
{[E - eV{i'W - 1V'1}(i'W + 2iV'r/(£)' \1 + i [V':2t}(.i')] + \1:2 - m2 }A(:l.1 = O. (12.18) lu as much as the wriling oftJ!(i) in terms of A(.i) and 1)(£) is not unique, it is permissible Lo &plil eqn (12 .18) into two eqllations. For the present. purpose, it is especially convenient to choose the following splitling: (12.19) and
(12.20) This choke has the advautage thnt the phase 1/(1) is completely determitwu byeql] (12.19), since the 1l.mpl iiude A(i) does not appear there. T he equatiom (12.19) and (12.20) a re exact. For application to beamstrahlung where the energy E is very large, both the phase T}(X) and the amplitude A(£) can be expa.nded in powers of the momentum k = (El - m 2 )L This will be carried out later on in this set:l.ion for the phase. QIH:C this higlJ cllcrgy approximation is ulldcrstood for the Klein-GordolJ equation, it can also be used for the Dirac equation . Here, the electro!"! wave functiun satisfies
IE - eV(X)
+ is · \l
- mt1lw(X) == 0,
(12.21)
where f3 is simply;O and C; is as usual ~/1. Once again, l/>(i') is expressed in the form of eqn (12.17)
(12.22) except lhai here A(i') is a [our-componelli spillor. It rernaius 10 choose convenient equations for the pil
[E - eV(i) - if· \IT)(.i')
+ iii· \l -
mPjA(i')
~
O.
(12.23)
1£, IJ8,IM,I"I'IlAII/,UNU
With "(X) knowII, this determines Ihe &rnplil,ll.l" 11(£). We proceed to dde"mine approximately from eqn (12,19 ) the pi""" '1(x), common to the KleillGordon equation and the Dirac equation. Let k be the momentum of tile electron, so that the phase of the incident wave fUD ction is k' Without loss of generality, let this momentum k be in the "''z-plane, For application 10 beamstrahlung, where z is the direction of motion of the electrons in the bun'ch , the case of interest is k, » k., ... , Expand '1(x) to second order in the sense that [50,5 1] ~
x,
(12,21) in t.he limit of large k. Then, as z
'1o(x)
->
--+ -CX>l
the boundary conditions are
(12,25 )
and
0
The gradient of '1(:ii), as given by eqn (12,24) is ~
V'TJ
=
k + V''1o
1
+ k V''11 ,
(12,26)
and hence
IV'"I' _
k'
+ 2k, V'110 + zk ' '11 + IV'1/OI'
k'
+ 2k ~:o + (2~~ + IV''lol' + 2k. ~~ )
(12,27)
Substitution into eqn (12,19) then gives
- 2EcV Since E = k
+ c2V ' -
2k 01/0 Dz - (0'11 2 OZ
110 ) + IV'r/o I' + 2k. 0ox
+ O(k-1), the ter ms of order k and order I
= 0,
give respectively
iJ'lo = -eV
oz
and a'll =
az
-k iJryo _ • ax
~
2
(12,29)
-
[( 0'10) 2
ox
(12,28)
(or/o) ']
+ ay
(12,30)
It is now merely a. ma.tt.er' of integrating over z to get 11(X) to the desired accuracy, With the boundary conditions (12.25), the result is
'I
(x)
I = k'x-S(x , y,z)+ 2k[= dz'{2k'(Z_Z,)oev~~,Y'Z')
- [as(~:, z') r - [oS(~:'Z') ll
(12,31 )
"
,
WhOTI!
S(x,y,z) =
j'
(12,32)
(1::' f:V(.r. , IJ,z'),
-00
The tefl/IS in cqn (12,31) llave simple interpretations. The first one, k' i, is that of the incident wave fllrlCtion. The flcconn one, -S(x,y,z), i& Lhe additional phase Rhi ft rOt a d.:..ssical tr ....jectory parallel to the z-axi .... The last term, proportiolll\,1 to k- I , is the correction to the phase shift due to the bendin g of the cIMsic;}.1 trajectory in the external field . Since the hunch length Lb is much larger than the correlation length e~, this 1/ k term must be kept; otl'f'"Iwise the coherence is much overestimated. This explicit solution (12.31) shows clearly that the high energy ;,pproximation is .~imp ler than the WKB approximativn [54] or phYllical optics. It only remains lo determine the amplitude A(i) . For the Klein ·Gordon case, the leac.ing-order approximation to eqn (12,20) is simply
iiA
iiz
~ 0,
(12.33)
because 'V17(X} ,.... f. give5 the ouly piece proportional to f. A compariwll with the iocidclIl plane wave exp(ik· i) leads lo the leading-order a.pproximation A(X) = 1. Similarly, for the Dirac case, it follows 'from the helil:ity mdhod that, with m neglected and the illcidcnt plum: wave in a hclicity !$late, the leading-order approximation is A(£) '"" heliciLy state along the classical trajeclory.
( 12.34)
III other word~, the ceviation of ~he classical trajectory from the !in~ parallel to the z-axis needs to be taken inlo account, just M it. 113 ;u ~qu (12.31). Explicitly, jf the incident plaue wa.ve i5
i/'inC(i) = eik.iuo I
(1 2.35)
wllere Uo is a. spinor describing the electron, then eqn (1 2.34) is
A(x,y, z) =
{I+ 21),/h -!\7.lS(x,y, z)]}
( 12.36)
Ill) ,
where the I'ubscript 1. refer:; to the transverse x-y directions_ A tirst-ordcr pcrturbative ca\culat io:"J gives a slightly more complicated expression
A(x , y, .. ) = {1-21k[ih-'V_S(x,y , z)
, (0' + oy~D')
+ [ ,.., dz'
ax]
S(x)y,z')
]1
Ilo·
( 12.:17 )
BeamsLra.hlung calcula.tions have mostly been based on the simpler approximat ion (12_36), bt'1:ausc the addi t ionnl term in (12.37) is spin indcpl;.ndent and hence contribules little in the case where the disruption factor is small. As already dis(".II~~ed in S\~ction J 2.2.1, this condition of small dj~ruption factor is n<x"d,,'d in order lo apply the single-particle approximation that leads to the Feyuman diagram of Fig. 12.1.
12.4
Cross section for e + bUlldl
-->
e + 'Y + bUllch
It is the purpof>e of this section to obtain au explicit expression for the cllcrgy
distribution of the beamstrahlung photon on tbe bMis of the electron wave functions of Section 12.3. In the next section, this explicit expression is to be evaluated asymptotically under the condition (12.15). The bunch shape is the one described in Section 12.2..4, namely it is longitudinally .nonuniform but
~ransversely
uniform. The simplest example of such a cha rge distribution
IS
8ech'(2z/ L, ) .
( 12.38)
Such a distribution, with exponential decrease for large
Iz!, is
actually quite
a good appmxirnation for realisti c bunches in electron-positron linear collicl-
when complications such as those due to head-tail interaction are under control. In fact, the distribution (12.38) is often a better approximation than a. Ga.ussiiln.
Cl'S
12.4.1
Electrostatic potential
When the relat ive motion of the positrons in the bunch is neglected, t hen the external potential experienced by t he incident electron is elecLrostatic in the fran:e where the bunch is a t rest. Thus the Dirac equation for the e lectron
indeed takp.s the form of eq n (12.21). In this frame where the bunch is at rest, let Lb be the nominal bunch length. The average charge density is then (12.39)
Let this bunch length Lb be used as the scale along the bunch, taken to be in the z-direction, then we define a normalized charge density p by expressing the actual charge density po as
po(z)
=
Ne
( z)
(12.40)
L R' P 1 -.Jb . 1r /,
The electrostatic po(cnliaJ for such a charge dist ribution is
V(x,y ,z ) =
1 L,R' xJ Ne
1f
00
-00
(Z')
dz'p -
L,
dx' dy' [(x - x')' + (y - v')'
+ (z - Z')'r t , + yl').
(12 .41)
< R2. Since R is very much smaller than L, (by nearly ten orders of magnitude in the case of the Super machine for example), the charge distribution p(z' / Lb) can be replaced very accura.teiy by p(z/ Lo) in the expression for the difference where the
Xl
and y' integrations a.re over' a. circle
X '2
I! ../.
t : IW.'iS
.~·J,:I
!'/'fON
nm r ... ~u,It'1I
~.
, -i ,. I bu""11
V(z,y,z ) - V(O,O,z). With this approxiulIlliou, V(X ,71 , Z) .:(;1
i~ givell
hy, fol'
+y1:5 1(1 , 1
l'ia.z +y2 ( ' ) eV(r. ,y, z ) = eV(O,O , z) + Lb Jil P Lit '
(12.42)
where, by eqn (12.41),
though the two terms 011 the right-hand side of eqn (12.42) arc comparable in magnitude, t.he first term is independent of z a.nd II ~nd, hence, leads to a constant phase shift in the matrix clemcnt. Such a constant phase shift is of no consequence and, therefore, it is sufficient to use ~~ve D
IVa r' z , V(" ,) = -Lb 0.~P(L-)' It
12.4.2
(12.44)
Initial aud fmal pbases
Wilh eqn (12.22) applied to initial and final wave functions.
(12.45) the matrix element of eqn (12.6) is
(12.46) where a.nd the z-y integration is again over the cirde Xl + y l < R2. The reason why it is unnecessary to consider r > R i~ that the trallsversc si'le of ~ he elecLron beam is also taken to he R and , hence, t here is no incident electron where r
> R. In order to make usc of eqn (12.31) to write down the phases tP;(i) and
f/(il for t he electrostatic po tential (12.3 1), it is convenient to define
T() -
T(IJ -
l_ l_
d(, pte')
,
d('[T((')I',
I e. IJE! AMS'/"II.AIII,IINU
5Wl
VW TJ({) TJ (~)
-
l~ deT(e'),
f f
dt p(e),
~'h(e)12,
fe~ de TJ (e') ,
VJ(t)
,
"
(12.48)
where (= z/L b •
(12.49)
Note t bat because of tbe way p(~) is norma lized, T(O has the property that T(~) -> 1 as ( -> 00. This implies a simple relation between T(O and TJ(O , namely (12.50) Since the inc..oming e lectron momentum is along the z-d irection 1 iLfollows
from eqn (12.31) t hat 4>.(x) is simply -' (-)
Y';
x =
For the phase leading to
k
'i Z -
~/(x),
N
Cf
r' [
R'
T
(z) 2 Lb + k.
, NCfL~1'( Z l] R' Lb'
,.
(12.51)
it is necessary to reve·rse the z·axis in eqn (12.31),
( 12.52) where
SJ(X, y, z) = 1~ dz' eV(x, y, z'),
(12.53)
and kj has been taken t9 be in the xz·plane. By eqn (12 .44) and the defini· tions (12.48), here (12.54) and
It.~.
CIWSS .~·/,:(J'/'/{)N FOil" I- hltr./t - . " I 7 I hllrh
SubstiLuliOri of {'(J Ul! (I'VH) ILnd (I:.LI).'i) inlo tlln (1l'fillil. ioil (12.<17)
giVt! s
(12.56) The first t hree terms
Of]
the right.-hand
~jde
of
eqll
(12 .56) , namely (12.57)
reach a maximu m \'alue at
(12.58)
y =
Yo
R'
= -2Nak..,~·
Tllis point. lies wi,t.hin the range of t.he z·y integration when
. kw , < (2No)' (kJz + k..,'..c), + R . Ullder this circumstance, these x-y integratiom can be carried out
(12.59) wi~h
the
method of stalionary phase to give for the quantity MI' of eqn (12 .46) ( 12.60)
where
1>o(Z} = ¢(xo, Yo, zl·
(12.61 )
In order to get an explicil formula for this ¢o(z), it is useful to express UJ and T which appear in cqn (12. 56), iii terms of U and 1'. This is most
" easily accomplished by difJerenlialing the definitions (12.48):
-
[-T«()
+ 2U«() - (I'
(12.62)
(12.63) Therefore,
1j«(J
~
-1'«() + 2U«() - (+ C,.
(12.64 )
'"
. I~,
IJII'AM;''I'IiAlII.I1N(.'
and (12,(i!j )
where
C'l' and Cu are two constants. These constants are given by (12,56)
and (12,57)
"
,
but these explicit expressions are not needed,
Let X be the fraction of incident energy carried by I,he outgoing photon. Thus, and (12.68) k..,=Xk i kj = (I -X }ki' "
In terms of this quantit.y X, the longitudinal moment.um transfer is
ki
-
kj' - k.." = 2X(J
~ X}k i [X kj, + (I -X}(k;, + k~y) + X'",'].
(12.59) .1
Here, the last term proportional to m' is kept only for the purpose of avoiding a logarithmic divergence, which becomes log L, as we shall see later. The substitution of eqns (J2.58) and (12.64)- (12.69) into eqn (12.56) gives
¢o(z}
=
2X(1 ~ X)k, {I(k jz
-2[k..,. (k j •
+ k..,x}' + k;,]X' T(;)
+ k,,) + k~,]X U(L) + (k~, + k:" + X'm')
L} + q,oo, (12.70)
where ¢oo is an irrelevant consLant given
12.4.3
by
Integration over transverse momenta
Beginning with the expression (12 .60 ) for tbe matrix element for the beamstrablung process (12.5), we calculate the square of the matrix element and sam over spin. It is assumed that the electron and the positron bunches are both rota.t.ionally symmetrical and uniform in tbe tra.nsverse directions up to
a radius R, and, furth ermore, that they are perfectly aligned so that the axes of the bunches coincide. Further integrations over the transverse momenta. of the outgoing electron a.nd outgoing photon then give the energy distribution ·
"
"
or lhe bcalJ1!1LI'uhlulig l'ilOLOIl I!.~
[n]' .,(X»)("(.I -
n 21f?~} 2Na
J (X) ::::;
,
xl
"I] ,
;~ , . , [ Xm'(, x ; _GOdz _Xldz F(z,zlexp 12( I - X lk,·
(12.72)
where $(.\') is the .spin factor
,
(X) = 2 -2X +X' 2(I-XI
(12.73 )
and F (z, .. ') involves the integration over t.tan.werse momenta:
F(z,;;')
==:.
x
{i(k"
+k,.IX ,(
L) - k·"II(k,. I ""~IX ,( ; ) - k,, :
H;,IX 'C:.l-'j[X<) -1]),
(12 .74 )
with
(12.701 If the spin of the. electron were zero, thcn s(X) would simply be 1. In eqn (12.72), lhe two z and Zl integrations come fmm the matr~x element a.nd its wmplex conjugate. The region of integration for eqn (12.74) is given
by (12.59). It is the purpose of this section to carry out the integration Oil the righthand side of eqn (12.74) in order to geL a simple expression for F( z, Zl ). By eqll (12.70) for ¢!o, the exponent in cqu (12.74) is
where
A B =
C
2{l
-
XL, X )k,
[T(-=-)-T("lJ Lh Lb '
L.
IU(-=-)-u(.:'-)I 1Jb ["", "
(1
X }k;
1 -
2X(1
X )k; (,-z').
(12.77)
U. I)CAMS'I'IlAIlWNU
506
With this notation, t.he quantity F(z,z') take, the form
(.) (z') a -T --F(z, t') = -,. {X" T L~ Lb 8A
-x HL) + T(~)1a~ + :c} 1"0("/)'
(12.78) .
where
(12.79) still with the region (IZ.59) of integration, Byeqn (12 ,76) , this Po is
Fo("/) =
(2".)-' X
Jdk,x dk,.d"-r,d"-r, + (k,y + "-r,l')A
expi{l(k,< + "-r.)'
+[k,,(k,r + k,r) + "-r,(k" + "-r,)) 8 + (k;, + k;')C}. (12.S0) '1
It is now dear that a more natural variable is
l
(12.81) With this varia.ble, the quantity 1"0("") of eqn (12.80) is
1, ,
.J .' X
expi l(kT<
+ kTY)' A + (kTr"-rr + kTy"-r,)B + (k~r + k~y)C) , (12.82)
where the region of integration (IZ ..59) is simply
.'
.'
kTr + kT' < independent. of ,
[,L'
Fo(z , ,) = -
(2NQ)' R '
(12.83)
Therefore,
4AC2"._ B' { I
[,(2Na),4AC-B'l} R 4C .
- exp ,
(12,84)
It remains t o apply the differential operator on tbe right-hand side of
eqn (12,78) to tbis expression for Fo(z, z'). Only after differentiation, the
II:,~,
(;I((}SH ,'W(/'/'/ON Jo'(}//.: ,I hIli'" .. , r. I- ") 1 h,.!wll
expl icit vlllues of II, f), cqns (12.77) imply Lhil.t
8
c:
11.:>
1
a,
2X(1 _ X)k,
v
-1
1~lfI~
givcn hy
[ , (')' X r Lb
[
(1 2.77) CArl be used.
a
vA - 2XT
(L~') aB 0 01 + Be (,' ) 8
,'),0
UUl,
OJ
8,' ~ 2X l1- X )k, X'~(L, V,;,-2X , L, aB+ac
(12.85) Therefore,
Trus opt.'I'ator is to be applied to the q\lantity Fo(z, :;;') of eqn (12.84) to give F(ZtZ'), which js then subslhutcd into eqn (12.72 ) to yield the energy distri· bution of lile be.ams tra.hlung photon. for this purpose, integration by parts can be used 1.0 deal wiLh the last term on t he right·hand side of eqn (12.86), OIlCC with respe.:t to z and once with rc~pccL to z'. Since t he ranges of integration for both variables are the entire real line, the integrated terms vanish. Moreover, because ofthe structul'f~ ofthe integralld in eqn (12, 72), the result of the integrat ions by part s is proportional to m Z and is hellce negligible. With this approximation , F( z, z') is given by
F (z,z') = V
1 {
B'J} .
( 12.87)
(")[';." drpi cxp 1i fMC-B' ] - - ; - l"
(12.88)
[.(2N«)' 1AC -k4C
x vA 4;tC _ B~ 1 - exp ~
A more convellient form is
F(z , z')::: -,;X'[ -- T (' -) 4C
L6
-
,
G
Lb
'
4e
-
with
~ W J{
(2Ncw),. '
Fl
(12.89)
we define
W(I,(') ~ T(I) - Til') - (I - (,)-'[U(() - U((')j',
(12.90)
Ie.
f>68
/J,;AMSTIIAIIWN(,'
then it is rCMJily verified that this functioll has an interesting symmetry
property ill thaL it call be expressed equally well in terms of the functions 'Ij(O and Uf(~) of eqns (12.48):
(12.91 ) as a consequence of eqlls (12 .64) and (12 .65). With this quantity W, the substitution of eqIls (12.77) a.nd (12.88) into eqn (12.72) gives the energy distribution of the beamstrahlung photon as
leX ) =
-;a [2Na R ]' .(X) 2(ZT.')k; x
f
]00_= dz ]=dz' (z -00
[ (Z) L. -
Z')- 1 T
T
(Z')]' L,
1,
d1Iryexp{i2(I~X)k,[m2(z-Z')+'IL,W(;b ;)]}, (12.92)
where w is given by eqn (12.89) , k, is the energy of the electron or positron under consideration, X is the fractional energy of the radiated photon (i.e., the photon energy divided by k,) and s(X) i. given by eqn (12.73). In the remainder of this chapter, this quantity l eX) is to be evaluated under the assumption (12 .1 5). In terms of this leX), the average fractional beamstrahlung energy 108s is
6=
12.5
l'
(12.93)
dX X I(X ).
Energy distribution of beamstrahlung photon
The expression (12.92) actually depends on the quantities k; and L. only through the ra.tio L./k;. It is therefore the same in the laboratory as in the frame ,. .·here the bunch is at rest. To make this explicit, it is only necessary to scale z by L, . If ~ is scaled by (2No:/ R)', then
leX) =
;a A3.(X)
1: 1: de (Ie
x
l'
=
Xm'L. 2(1 - X)k,
(e -
dl111 exr[ill 1
0- 1 [T(e) -
*')J'e'P(H')
~XA3W(e,n],
(12.94)
where
f3
(12.95)
and
(12.96)
-:-.1
.
. -1
.
Je.!J. UNlm(;Y IJISnW}/{'{'wN OJ<' IJh'A MS'I'UA 1/1.IIN(,' 1'1/0TO,v
Tn order to evaluate the r ighl-hl'l.l ~ d Hi d'1 o f I~qll ( 1:l.lItI ) a ~yrn r>loticalJy for la.rge values of A, it il:luccc~sn.ry to know tile I'I~giu1J where W ({, {I) i~ positivtJ. It follows from the Schwarz ineqmdity
(12.97) that
[U(,) -
U(OI'
~
(( - ()[T(O -
T(e')I·
(12 .98)
If the observlltion is mnde t ha t the longitudinal charge dis tri butio n is nowhere l.l.!rO jn~jde au)' realistic bUllch , theu "T (O is a strictly iucreasiug functi on of { inside the bunch , ancl for eqn ( 12.98) the equali ty sign hold 5 only for ~ = ~I. Therefo re, ( 12.99) sign of W((,f) = sign of (e - (). This properly (12.99) is of cenLral importallcc ill the development of this section.
12.5.1
Melli n transform
Wc follow LI,c proced ure of J acob and Wu [18]. In order to dderml ne the a~ymptolic behaviour of J(X) with the conditions (12.15), we employ t he mc~hod of Mellin transforms, which has been used to analyse Lhe high energy behaviour of Feynml1n diap,ran1s [55,56]. The Mellin transfo rm of (12.94) is to be carried out with respect to the .... ariable A, defined by eqll (12.96) . Thus, we consi(ier the photon spectrum all a function of A with the panmetcr X, rather than a f unction of X with the parameter A. With this in mind , we write (12.100) which defines K(A). Of course, T(X ) and K(A) are actually fuu clioIlS of hath X and A. With this notation, defille the Mellin transform
( 12.1 01) Since, as !jeer} [rom C(jn (12.94),
/« A) ~ O(A)
(12.102)
K(A) ~ O(A')
( 12. 103)
as A --I ee, aJld as A _ 0 , the K«() as defined byeqn (12.101) is ana.lytic for
3> Re ( > L
(12.104 )
Ie. IJEA MS1'IlA 1I &ffNC!
670
The residue of ](0 at ( = 1 give, the leadiug beh"vio~r of ](A), and hence of [(X), in the limit « Lb. This residue i. computed in the next section. Higher-order terms, including the end and nonuniformity effects, are to be extracted from the behaviour of the analytic continuation of K(O at \ = 0 and ( = - 1 - see Sections 12.5.4 and 12.5.6, respectively. [The behaviour of K( 0 near ( = 3, on the contrary, is of no interest. It gives merely the behaviour of J (X) for the opposite limit t, ~ Lb.J ,-, Remaining in the region (12.104) , we ca.n substitute (12.94) into (12.101) to get an explicit expression for K«() . The A integral is
e,
'1" o
dAA- ' -( A3eXp[i~
2X 1- X
,! ,
'<
. A3W(~ , nl
, ,
!ie-··(/6r(1- (/3) ['1 -5iei'(/6r(1 --
1
~XXW(e,erl+C/3
for
(13)[-~ 1 ~XX W(~,e')r1+(/3
W(~, n > 0,
for W(e,f) <
o.
(12.105) Because of (12.99), this can be rewri tten as
f
dA A-1-( A' exp [i1] 1: \ . A3 W(U'i] =
!isg(~
11
- ()e- i .«(/6)sg«(-(')r(1 _ (/3) ['1 2X IW(U')lj-1H/3. I-X (12.106)
i·1 j
The substitut.ion into eqn (12.101) then gives , usinll eqns (12.100) and (12.91) ,
K«() = !r(1- (/3) [1 ~XX
x
i: i: d~
de
r'H/3
Ie - el- 1 17(O - r(tlJ'
2X ]-1+(/3 1 (3 + 0- f(1 - (/3) [ 1 _ X X [r(e)
-
"
", 1' " de 1 de Ie _. - 00
-00
(,[-I
r«(')I' ei~ (H') e -;·«(/6)sg(H') IW(e,{,)I-1+(/3, (12.107)
Its behaviour near (= 1,0, -1 is to be determined in succession.
1
It.S.
F.Nl-.'IlUY /)/H'/'IUlUI'I'ION OJ' IJh'AM.''''/'UAIIWNr; I' /IOTON
ii7 !
Residue of K(O at ( = 1
12.5.2
At ( = 1, I( ( () has a. si mple pole, LeL RI be ita residue, Le., (12.108)
The contri bution to Litis resid ue comeS" from the vicinity of ~ =
~' .
Let (12.109)
(12.11 0)
For p.
.~m alJ,
it follows from eqns (12.18) that
(12.111)
and hence, from (:qn (12.90) , (12.112)
T herefore) in t he vicinity of (= 1, K «() is approxima.tely given by
K«() 1
x [ 12 1 "I'p«()'
~ and the
r~idue
r(l) [
4X
IX
]-! 3",_1_j- d("lp(n]"', \- I
-~
]-'-'" (12.113)
Rl is (12.114)
One recognizCl) here ~e .. er31 of llle filctors entering the expression for radiat:oll 'during bunch crossing' [.i7,5HJ.
·
. ,"
&72
If.
12.5.3
IJI~A
M.I''I'ltA'' WN(J
Analyt.ic contimmtion into the region 1 > Ite ( > 0
The considerations of Section 12.5.2 indicate 1,I:.t ~ and I' of egns (12.109) and (12,110) are more appropriate variables. In terms of these v~ri"blcs, the quantity K(() of (12.107) is
j{(C) = (3
+ (r
X [T(~
i
r (1 -
+ 1'/2) -
J (/3) [ 1 ~ X 2)(
T(Z - It/2))"
-1+(/3
1 d~1 00
00
- 00
-00
e;j)p e-;,((/B)sg.
dJ1.
I-J'r'
jl1'(e, J1.W1+(/3 , (12,115)
where
(12.116) is an odd function of I' and has the property from (12 .112) that
(12.117) for small J1.. Before we can study the behaviour of K(() in the vicinity of ( = 0, we must rewrite the right-band side of (12.115) in such a. wa.y t hat it makes sense -in that vicinity. This can be accomplished in a variety of ways; we choose one that facilita.tes further analytic continuation t.o near ( :::; -1. Ou:: procedure involves I"ewriting the 11- integral usjng suitable contours. '/lie assume tha.i p(O is analytic so that analytic continuation to complex values of I' is possible, at l.e ast for small values of IlmJ1. l· Consider the following factor in the integrand of (12.115):
J1.-i e-;«/6 [vV(e",,)[-H(/3 for J1. > 0, { -1'-l '<(/6[-WCe,/<)[-t+(/3 for e
( 12.118) I'
< O.
Starting ,·. . ith the a.bove expression for p > 0, we continue analytically to f1, < O. The results are, using (12.J J 7),
(12.119) an d
(12.120) We attempt to write the function L(-/l')' as defined by egn (12.118), as a linear combinat.ion of the two different contributions given by (12.119) a.nd (12.120): (12.121)
ft. S.
r.NK/U:Y /JIS '/'/U/lII'I'/ON
m
1U.'A dfS·/, /tAIJ I./INI.' I'/W'/'ON
(.'I~ ( :·J =: I .
SillCC
W({, II)
i8
(12.122)
an odd rllHctioll of IJ, it follow" from (12.118)-(12.121) that (12.123)
Eqn5 (12.122) a nd (12.123)givt!
_ -i~
C a.nd
(12.124 )
_ .21r{/3 sin(1I'(/3) Cl - C 'r"
(12.125)
5107 ..
Note that C J and ('2 are finite as ( ..... 0 , but not for ( -I -1. T his property turns Ollt to be imvortant. Using eqn (12TH) together ....
r
H/
K(O = (3+ (t ' f(l- (/3) [I 2X
X
x IT((
+ p/2) -
T(i - p/2)}' e'"
'
L: dZ 10 dpp-'
e-"'" !I"(i, p )t'H/' , (12.120)
where, for the }i integration , contour integrations:
Ie
is defmed by the weighted average of t wo
(12.127) In (12.127), t he cn ntours of integrat ion C+ and C_ arc along l!Ie feal axis except for an indentation at the origin into the upper and lower half planes rC!IJledively, as shown on Fig. 12.2. Eqll (12.126) gi ves the desired analytic continuation of X«) into the enlarged region 3 > Re( > 0 (i2. 128)
except for the pole at ( = 1 already studied in Section 12.5.2. \.Ve can now proceed to the study of the singularity <1t ( = O.
12.5.4
Residue of ]((0 at ( = 0
While the residue of J{(O at ( = I comes [rom the region of integration in the vicin:ty of ~ = f, that at ( = 0 comes from the region of I.u ge 1(1.
/2. Ilf)AMS'I'UAII/, IlN(; ';
,,
I'-plane
.,, '.
I' -plane
,. ,,
Fig. 12.2: The contours of integration C,. and C in the I'-plane. Having in mind the hyperbolic secant distribution (12.38), we assume more generally that decreases exponentially as ( ..... 00 and ( ..... -00 . More explicitly, let
pro
as ~
as
----t
e---+
- 00
I
(12.129) 00)
e,
where aI, a2, Cl and Cz are four positive constants. By a tra.nslation on the values of Cl and Cz can be made equal; however, such a choice has no particula.r advantage. Similar to (12.108), let ~ be the residue of K(C) at (= 0, i.e., ~=lim(K((). (-0
(12.130)
In this section, we calculate Ro. Let Tl and j¥, be the T and fV corresponding to
Pl(O ~ C, e"e, while
T2
and
l¥,
(12.131 )
correspond to
p,(O
=
c,e-"e.
(12.132)
Then from (12 .126), near ( = 0,
K(()
~
[(,(0
+ f(,(O,
(12.133)
. ' ",
flU.
fjN/<;JWY mS'I"/WllI'!'/ON m' IJNAMS'I"11I1JII./lNI: l'II01'ON
where
7(,(0 _
I-Xl' de('3" lc-+-I +5Ilc_I )d/.1./.1. -, fiX -co
xh({ + Iltl)
- Tl(( - pI2Wc;{hl [ IV1(~, JlW 1 H/J
(12.131)
ilnd
Kz(O
""
1
;x f'" d~a1+ +~k_ X
)di././J-l
X["'2(~ + 11/2) - T2({- p/2)12e'''~[iV2(('fl)tlH/3. Since T<2«() can be obtained from -Kl(O by the repla.ce.ments (11 (.:1 - I C1, it is sufficieat to concentrate on ](1(()From eqns (12.48) and (12.90), the 1'1 of eqn (12 .! .31) leads to
-
" (0
- .. (/2
and
(c!/al)e-'(,
T,(O =
21(c~/aJ)e2"'E 1 1 1
u.«)
(cJ/a~) e",E,
-
(12.135)
(12.136)
and
e)= c? [eza)E_e2~'{'_ " '2a1
wre
alee
2
C)
(e~'~-e..)e)'l
.
(12.137)
The important point here is that the right-hand side of eqll (12.137) is of the form of ", product e2
e
K1«) . . . (- 1 12X - X (~I 3lc+
+!
()dIH!1;PJl/a, sinhI'. 3lc_ pC(jshfl-8tnh~,
(12. 138)
for ( small. This exhibit.s explicitly the pole Slruc~ure at , = O. Since the value of C 1can be changed by shining e, it cannol appear in lhe residue al (= 0; indeed, it does not. Let
E((J) = (~ (
3 lc+
+~ I )d1l eiP", 3 lc_
9inhp. po cosh p ~il1h I' '
(12.139)
then the residue of K{() at (= 0 is
(12.140)
J2.
flI:~
M.I'TUA III. UNC,'
Ii rcmaill" t.o ovo.lllo.to t.his function E(fj} for slII,,1I fi. Sillce fj i8 positive, we c.an close the contour of integra.Lion in the UPP<'W ~t(\.lf plane to get (12.111 ) where Yn is the nth positive zero of y - tan y =
o.
(12.1 42)
The first few zeros are at [59)
y,
4.493409458,
y, -
7.725251838,
Y3 =
10.904121 66,
(12.143)
etc. From (12.141), it is clear that E«(3) is " decreasing function of fj for positive (3. For large n, y. is asymptotically given by (12.144)
We know tbat [60)
.,
I)" + ~t' e-~'n
-
(e-r... 1 !)
, "
'1=0
h~ -log(l - e- O')
-
210g 2 - log«(3,,)
+ 0«(3)
+ 0«(3) ,
(12.145)
,F,
where II> is defined on p. 27 [eqn (1)], is the hypergeometric fun ction [po 30, eqn (10)], kg is defined on p. 1I0 [.qns (12) and (13)), all of the Bateman Manuscript Project [60]. Therefore,
E«(3) = 2(-log(3 + K.) as (3
-->
+ 0«(3)
(12.146)
0+, where K, is defined by J(,
= -bgrr
00
=1
2
r.
+ 210g2 - 2 + :2::(-' Yn
-1.71315032.
2
n
)
+1
(12 .147 )
Finally, the substitution illto eqn (12.140) gives
R., =
_I:;.
X (log 4fP _ 2J{,). lLllLl
(12.148)
/f.5.
I~Nh'IU;l'
12.G.G
/J/STfU/)/i"/,/(}N (If' IJI':A ,\/ST/IAII/./l NC,' 1'l/m'ON
Allalytic cOIlt,iun;diu!l
iul~o
t.lw region 0 > Rc ( >-1
III order (0 s()lve the proulmn of Lh~ dr( ~d. or the dcnsity gradienl for a distribution snch as (12 .38), we need ],0 stutly ~h c behaviour of J«(() near' ( '"" -\. For this purpose, we must rcwrite t be right·haud side of eqn (12.12(j) in such a way that it makes sense in lha! vicinity. Si m ilar to the clt'vciopment of Section 12.5.3, the method of deforming ihe contml[ of integration i~ Lo be u;;e:d. Unlike that o[ Soction 12.5.J, thill deformation is applied to the integration. Fi rst, we a::;sume that the domain of analyticity in is large enough to per mit Lite deformatioil of th~ coutour. However, the re~ult to be obtained in Section ]2.5.6 can be wri ttfm in a way that this analyticity is not essential. Siuce thc singularity of K(O at ( = 0 is a pole, no~ a branch point, and t he result (12.UO) on ~ is of the form of two separa.le cOlitrib;rLiOfll! from the two ends of the b·,.II1Cb, thl! a~1 a !ytic conblu ation can be cl\rried O'Jt via eithcr ~ h c upper hi11f plane or the lowr,r n2.lf plane. The required analytk cout inuation is per formed bi' merely rewriting I( () , eC]l1 (J2 .1 2G), in the form
e
e
x[r(( + }l12) - r(Z -ll / 2W e,h e-i~06 [W(e, pJl- 1+(/3 , (12.119) when! the comour C can be, ror r.xample, any of the four shown in Fig. 12.3. Except for the simple pol es al ( "" 1 and ( = 0, equ (12. 149) is val id in the fur~her enlarged region 3>Re(>-I. ( 12.150)
12.5.6
Residue of K() at ( = - 1
Once again, K () has a simple pole at (::: -I. Similar to eqll:; (12.108) and (12.130), let R_l be the residue R_I = ( li_ m _1(
+ I)K(().
(12.151)
T he origin Q£ this pole is exceptionally ~im ple: for ( near-l, OW II integration is approximately, by (12. 127),
(1 2.152)
57~
12. II NAMS'I'ItAIII.lINU
Fig. 12.3: Four possible choices of the contour C of integration. The horizontal straight portions of the four contours are on the rea.l axis.
fll.S,
f:Nf':J((,'}' fJ{S'f'{(IIIIU'ION {)f.' m:A/otS'l'Il1lf1f,fl NO f'/fO'!'ON
i.711
where Co = C_ - (\ i~ a ~111,dl C:Ollnt,{'ITlol:kwisl' I:in:!c arollnd t,he ol'igin, 'Therefore, by eqll (12 ,119 ), t.lll: I'mliilue is
Ii -, = -i.)3]'(,) [ 2X ]-"'~J'i d -, 4 'I X -""'> p.}i .
~
C
Co
In order to evaluate the integration over}i, it b integrand for limal! jJ [571. Since
nl!rc~,:;ary
to expand the
, (~ + p./2) =, + ~'T'jl + ~T"J'~ + 48 'T"'p.l + 3~4 'Tm'J.I.~ + O(J.l.5). 1
(12,154)
wi Lh ~ = T(~),
(12.155)
etc" it foHows that
( 12.155)
=
TIl
1 ,,:1 + 1920'}J 1 ""!!. + v~( JI ') , + 24'1 j1
(12.157)
"1'«( + p/2) - "1'«( - p/2) = ~2 /" _, d/l [,(~ + 1I/2W .=
,2p. + 1..(TT" + TI'l)J13 12
+ 960 _1_(, T"" + 4T'T'" _~ 3T":l)J1S + O(Jii). . (J2.158)
The lillolititlltion of both eqns (12,157) and (12.158) ir.to thfl definition of W(~,I-I) , eqn (12 .116), with (12.90) then gives W«(,p) =
'r«+ p/2) -1'«( -
p/2) - I,-' [U(h 1,/2) -
uet - p/2))' (12.159)
ant! hence, [rom eqns (1 2.153) and (12_156),
R_l =
-iJ! r(v [. ?X ]-4 /3 [d~- [ d/lj.I. [T r2 + -2,'T"'/1 1 2 X + O(/I~ )I 4::>r 1lck, 1
T"] -4/3[ 1 [l:l
Xj1-1 _
- 1 3T'T'" j ,"2 Jll 45 Tr2
+ O(Jl~)] .
( 12.160)
Ie,
580
IJI':AMS'I'IIA (II.UNC:
Since r' ::: p({L it i.'1 now ~triLigltLforwi1rd 1,0 Ilkk ollL the reljiJuc at /1, = 0 to gel (he result
R_l =
L
I~ r(1) ~x r1/3 35/6fcd~ p({)-8/3[3p«()p"m -
4P'({)'j.
(12.161) This differs from the previously obtained results [57,61,621 in 'an essential way, namely the contour of integration is not the rea.l axis, but, for example,
any of the paths shown in Fig. 12.3. With such a path of integration, the integral is well defined, and no cutoff is needed. In some circumstances, it is possible to deform the contour C back to the real axis at the expense of subtracting and adding some terms. For example, if the condition (12.129) is strengthened to be
as
~ ~
- co, (12.162)
as
~
-;.
00,
then the subtraction of the integrand calculat.ed on t.he basis of the leading behaviour makes it possible to use the real axis as the contour of integration. Such a procedure makes it unnecessary to assume analyticity in ~. However; the resulting formula for R_I is not elegant and also not especially informative. Eqn (12.161) can be simplified slightly by an integration by parts, Because of the contour C, the integrated terms are zero and the result is
(12.163)
..
Although the integra.nd is positive on the real axis, R-l may be positive or nega ~ lve.
12,5,7
Hyperbolic secant distribution
As an example, we apply our results to the hyperbolic secant distribution (12.38):
pro = sech'(20 =
4e«
(I +e'<)"
(12.164)
Comparison wi th (J2.J29) shows that (12. 165 )
It follows from (12.114), (12.140) and (12.163) that, for the present case, 3
=
R 1
_I [ 8 1(1 - X)]'i [r(1 )I'
?
-"
SX'
3'
"
IU.
":N~:IWY
J)fS'f'/UllfI'I'JON (JIo' IJI\'A MS'/,UA I//./INf,' /'1l0'f'(JN
I -X
X iLl
=
!:!..)
f;( X
t-X'2iJ~'
_1-[I-Xj4/312If3[ruw. 511"
'"
4X
3
([ 2.166)
Since R_I is negative, contrary to previous conjecture (61[, it may· be worthwhile to give a derivation of (12.166). For the hyperbolic secant charge distribution (12.164), the R_I of (12.163) is
=.3.. r( ' ) [1 - Xl~/3 22 / 3 35/ 6 J 15 :I 4X -I.
([2.[67)
.' ]-'1' [(I(1 -+.I)el]' =!cd{ [(1 +~)2 eq3
([2.168)
R
- I
where
'-I
Define a function of a complex variable f/ by
(12.169)
Thell, L\ (II) lias a
~imple
pole at '1 = 0, and the
de~ired
LJ is
(12.170) For Re'1 > 0, the coillour C, as showlJ for example in Fig. 12.3, can be replaced by the real axis, and thus t he evaluation of LI is straightforwa.rd :
(12.171 ) Thi~
shows expli citly the pole structure at 7j = 0, and also that il is po!;itivc for '1 > O. However, analytic continuation to fl.egati~ values of '1 gives
(12.172) which il) negative. The le~son is that we have to be very careful in providing re:.iablc arguments iloout even the sign of higher·order corrections.
12.5.8
Summary
When the correlation length lc is much shorter than t he nominal bunch Jellgth L 6 , the energy spectrum of the bcamslr
by
.
a 2 - 2X + X~ [Lb J(X) - - - -· R, 2:;1 X (~
to j, + flo + -R_, Lb
(12.173)
1£. B/,AMS'I'IlAIlLUNG
682
where R llo and lL, are given by eqll' (12 .114), (12.140) and (12.163), " respectively. The extra factor present in the Dirac case has been explicitly wriLLen down.
It is instructive to rewrite this resldt in terms of the actual charge density po(z) rather than the normalized charge density p(z/ L&). They are related byeqn (12.40). The result is \ .. a 2-2X + X, I-X
"
..
fiX ) - 27:'
4/3 3-'/6 [k ]'/3 ~dz [epo(zlt8/3[ep'(z)]' } , [1 Xl '4X 7r'R' Jc
+~r(1) ,
_'i_
0
(12.174)
!
,
",,
where, by eqn (1 2.129),
lim p~(z)/po(z),
.%--00
"
(12. 175)
Examples of the contour C of integration are shown in Fig. 12.3. The important point to be noticed in eqn (12.174) is that the nominal bunch length L6 does not appear anywhere on the right-hand side. This is the way it should be, because there is nothing to prevent. us from using 2L6 or ~ Lb , for example, instead of L6 in all the intermediate steps. The result (12.173), or equivalently (12 .174), takes the form of tm'ee terms. Roughly speaking, the nrst term gives the beamstrahJung during beam crossing, while the seocnd term gives the effects of the bunch ends. Both terms can be sizeable for very high energies. The value of the third term depend. on the variation of charge densities fo(z) and p~(z) along the bunch. Contrary to the claims of Chen and Yokoya [61], it i. not large. At least for the special case of the hyperbolic secant bunch, this term is negative, reducing slighily ~he beamstrahlung energy loss. In obtaining the present
l'e~mlt ,
we have assumed an exponent iitl decrease
in lhe charge density far away from the centre of the bunch. This is as good a. description as, if not beLler than, a Gaussian bunch for realistic cases. Nevertheless, it may be asked what the corresponding result looks like for a Gaussian bunch. Unfortuna.tely, the a.nSwer is that it :s very much more complicated, and hence, probably of little practical use.
It.fi.
12.6
DISCUSS IONS
Disc ussio ns
So far as methodology ill cOl\n~rncd, l he mlliJl lesson ~hat one learns from studying beamstrahlong in 'reV elect ron-positron linear colliders is t.he BaI;:!e one as from the other chapt ers of this book, n amely, Feynman diagrams can be used with grf!at a.d-vantage. T hi5 point is worth emphasizing becallsf!, while everybody agrees that Feynman diagrams are UsefllJ for the calculation of radiative processell ~\l(:h as t hose listed in Chapte rs 9 ftnd 10, t here has been great. resistance in the use of Pf!yn man diftgrams for beamsLrahlung a nd relalcd processes. Indeed, until very Te<'.ently, Feynma.n diagrams have been used only in t he work of Jacob and Wu. Fortunately, this situation is changing, and others arc now in the process of adopting such methods. T he study of the radiation rrom a charged particle in a mllgnetic field has a long history [63J. Much cmph~is has been pll!.ced on the case of an infinite uniform magnetic field. In this case, it is very difficuh, if not impoBsib:e, to make efficient use of feynman diagrams. In vicw of the large and excellent literature on this case, it was natural to make use of t his knowledge to deal wi th Lhe caseof beamslrahl u ng in high·energy electron-positron colliders [64]. T his is perhaps the original reason why Feynman diagrams were not used at the beginning [6.51. By now , however, there is 110 reason wh aLsoever not to \lSe the powerful tool provided by FeYllman. The study of radiative processes for high·energy electron-positron linear colliders is still in its infan cy. In this chapt er, we have cOII~idcr ed only the process (1 2.5), which is analogous to e+ + e- --+ e+ + e- + "1- Similarly, in analogy with c+ + e- --+ e .... + e- +...,. +...,., (12.176 ) 'Y+e±
-->
e+
+e- + e±.,
(12.l77)
(J 2.l78) we can list the following processes which in volve dectron and/or po~itron bunches: (12.179 ) e-* + bunch --+ e-* + 'Y + .., + bunch,
1
+ bunch --+ e+ + e- + hunch,
(12.180)
and
( 12.181) T here are of course ma llY, many other simila.r processf'"~ or interest. The f"eYlllllan diagrams fOf (12.180) and (12.181) are shown il l Fig. 12.4 and fi g. 12.5. Of these examples, 60 far only (12.180) ha9 been 9tudied [66J. For actual appl ications, the process (12.181) is especially import
I f!,
"f·:AM.',, "nU1.IIUf NU
kJ
~::
:
~
"
•
k'J
Fig. 12.4: FeynInan diagram for pair production by an photon in a hunch of electrons (positrons).
Fig. l2 .5: Feynman diagram for pair production by an electron in a bunch of posi trons. stjm uia.te the inter-est in these and other processes involving electron a.nd/or positron bunches.
We conclude this chapter with a discussion of the implication of these studies of beamstrahlung alld pair production in the design and construction
of future electron-positron linear colliders with energies of 1 TeV on I TeV or higher. Such a discussion is net::cssarily preliminary - it is conceiva.ble
that somebody may come up with a totally new and unexpected idea that changes our th inking on such accelerators. Nevertheless , a d iscllss ion at this
time may still be of interest . As already mentioned in Section 12.1 , a power limitation of 100 or 200 megawatts together wi t h the necessi ty of useful luminosity make it esseotial to have very small bunch radii. This in turn leads to significant beamstrah-
lung energy loss. Fortunately, this energy loss, although varying from event to event, dOf'..5 not cause too much complication in the design of the detectors and in the a.na.lysis of the experimental data. The main effect is rather that a higher-energy accelerator is needed for a given desira.ble centre-of-mass en-
/.~
Ii.
OJ,';' '/1:,.. ;/, INS
('q~y, For "X ~Ullj1! ' " i( III<' ilVl'I'a ~,·l,,· j 'lIlli l. r;ddllll~~
;Ii
1\
,'III'rgy ! .)K~ h.
!7
%,
w!,id, 1'('iI~':'II;~hl.· vi\h!!~, t l14'lI, II) ;~('hi(·'-'t,:.:! T,'V ill til<' /'I':lll,. ,-()r·n'i\>!'~ "lIeq~y.
lilt: (:I\{'rg i . ~ u f the ,-],-droll ;1.11(1 pOllit l'tlll 11.:aI/IS ~h()III( 1 hI'! aoout 1.'2 TeV,
T he .:;.rcnLio:1 of clcl;t ro(L·po~il, ron pilirs i()Gi hy th.: b.:OI.mstr<:.hlung photon SCClnS La ca.use mud, 1I 1(,r(' I.rouhlf'.. [)f'.cOl.lIse bo~h eieclro us aud positrons OI.re c reated, lhere is ItO simple way ~u d cllccl them lI,way from the focllsing dCI'iccs, At present, t l,c implication of the presence of 3t:cll pair9 Oil the (Ics ign of the final focus has IIOt bcclI completely IInd c::r~rood. AlLhollg b the complica.tions from bot h beamstrahl ung linu pair produc. tiun a rc by themselves not ratal, it must be rCfIlt:lI1hcrcd tha.t the design of the multi-TcV elcclron-ptlsitrOll linenr co lliders such as CLle aud TLC is marginal even without these ex t ra dif!iclillies. It t herefo re seem s tha~, if Lhe pow('r li,lliLn.tion of aoout 200 megal'.'ana remains impos{'Il, it is rMlle r unlikely that , for the clcct ro n·po:;it ro n system, ".n effective cen Lre-of·mass ell· e rgy or 2 'reV with a useful luminosi ty can be readw(1 before Lhe beginning of t he next cc ntury.
,
13 Outlook The use of hclicity is not a.n eccentricity: It is pure simplicity which brings felicity.
13 .1
The ubiquity of the photOll
T he ubiquity of the photoll is a well-known fact in particle physiOi. Photons are necessarily present whenever charged partides la.ke part in a. process. The sa.me is true for gluons when strongly interacting parlicles collide or decay. Photons and gJuons being the carriers of the electromagnetic force and tilt: sLrong force, tr.cy playa fundamental role in particle physics and their ubiquity only WIderscores tl'..js fact . 11 is lhere fore essen~ial to study cMciully the processes in which they take part. To undersland their behaviour is to IInderstand two fun da.mental fOl'ces in nature and the helici~y method i/:l a key ingredient in this exploration. We hope that, by now, the reader is convinced that the helicity method is an efficient way for ohtaining cross section formulae in the higli-encrgy limit. !\evcrtheles~, we learn from the examples studied in the previous chaplenl that LIle final result of the calculat ion is often much simpler than the intermediate formul.v: which had to be worked out. This is beca1,:se the s~arling poin~ of the calculation usually consish of mauy Feynmll.n djagrams. As a result, we strongly suspect t hat there is still room for ir:1provemer:t in the hdicity ruclhod. In this chapter, we present a series of recent developments of th!;' hclicity method, which can be regarded either as extensions of the method to pitysit:al situ a tions \loi previollsly covered in this book, or as attempts to streamline the calculations even f\lrther. ,",Ve a.!so discuss some unsolved problerrut and present them a..~ challcngtl; to the reader.
13.2
13.2.1
Further d evelopme nt s
Supersymmetry
Supersymmetry [671 is a symmetry which rela.tes bosons anJ fermions. III most supersymmelric theories, the number of fermionjc Jcgrccs of freedom is equa.l Lo the number of bosonic degrees of freedom. The simplest supersymmetric model, the WClIs-Zurnino modcl, dcscribe$ a m1Uj~ive Majorana
(;1,
fermioJl .:1l'incr.,; i\.rc a. sc.:dar lid.! ;\lu)
(J/I'/'(,(}O/,
it pHeudoHCalnl'
!idd. WI~
Lhus have two fcrmil)nic degrees of freedom, the i,WQ spin sta.tes of tht: rOJ'l llioll , and two bosonic degrees of freedom. 1n the case of unbroken slipersYll lllldry. the bosons and the fermion., ha.ve the same mass. SupersymOlct.ry can provide relations among scattering arnpliLudcl:) ill " voiving one kind of particle and scattering amplitudes invol~ing its stip e 1'-
partners. This idea was exploited by Parke and Taylor [68\, wbo cmbcci,kd ' QCD in a SO(2) extended ,upersyrnrnetri c version of QCD. This larger tb,,· ory encompasses, besides still morC particles, the usual gluons (g), massleRs spin-~ gluinos (A) and ma.ssless complex scalar gluons (¢). Becau,e of ' "persymmetry Ward identities 169], they derived tha.t the helicity amplitu
( 13.l) where the sl.lbscripts on 9 and ,\ denote the helicities of the particles and
where 9 9 are obtained through crossing. In this way, t.h e cross section for the gluonic process could be calculated from the scalar amplitudes only. For processes like g 9 -+ 99 g and 9 9 -+ 9 9 fJ g, other relations of the type ( 13.J ) also exist. For exam ple, (13.2) where ki } i = 1, ... ,5, denotes the four-momentum of gluon i. Again, the purely gluonic process is expressed in terms of an associated process involving scalars, whi ch is simpler to calculate. For more deta.ils of this methorl, we refer to Parke and Taylor 168], where also the cross sections for g 9 --, 9 9 9 and 9 9 --> 9 9 9 9 are expl ici tly listed. Additional supersymmetry relations for the 9 9 --> 9 9 9 9 helicityampli· tudes were also deri ved by Kuoszt liOl.
13.2.2
Phase choice of polarizat ion vectors
Throughout this book , we made extensive use o f the photon po!arizaLion
vectors of the form
(13.3) , which introduced two light -like reference four-vectors p
1-"// IITlII: /I 11/".' III'.' I. ()/' .II I'X·I:"
f.'1.:t.
«("hl,{1
Ih)u( k)
\li'II(~I)('1 !
··(.;i·;,(k)'
fi{qh,(1 +,r,)u(k)
(1:1.1 1
/2u(k)(1 - , ,)"(1)' Siltisfy all the rt.'"!,]uirements
(,+· (-)=-1. Hence, LllC expressions (1.4) (;[In also be us~d in lhe oLlculation of hdi"i!,y am!>litudes. To this end, we s imply ohserve lhat
.I'(k,,)
(1 - i'.~)ll{k)u{q)(l
- -
+ IS)
/2u(,)(!
(1 + "Hk)'('iI(J -
- If
T (1
+ 75)U(q)U"( k)(1
- 75)
+ 15)U(')
,oJ + (1
-,, }"(,, );;( k)(1 /2 "(k)(1 -7,)U(,)
+ "~I
photon is radial(.-d from a fermion Iinc, one sho\lld t ... ke one of tJ,(' extern ... 1 four-vector,;; of the fermion,;; to play the role of the rcfcrel"!c('; fOl lt"· ve::tor q. 111 t h i~ way, olle again encounkrs the usual simplificAt.ions i;J ~l\ll sp:nor algebra, similar lo what happened in Chapte rs 4- 6 . When radiation from two different fermion lines occu rs, such as e+eiI.
11+ 11-7, one would like to lISC
(13.7) This relaliOIl is to he compared with eqn (4.J.3): the main ui:Tet('nce is l:u; absence of t he ll. phase factor relating e:l: (k, q) to ~ j.(k, p). As a conseque nce, the bookkeeping of the phase fadors fo r thl' different conld butions to a spec:ific helicilY amplitude is somewhat simplified: il is now done automaticlllly tkough the 1I0fllJiLJizIl.t ion factor of c* in eqns ( IA ), which is cOlilplex. This kdwiquc can also be uS(ld for QeD caklllalioJls, and Xli, Zhal:g and Chang 17l] np!>lied it t o Ihe evaluation of lhe hdicity alTlpliLlIde~ for qq ---> q' q 9 .9 , wh ich is it douhle bremsstrahlung process. Other applicatiolls of t his technique have b~n presc nt~ by G\miOiL ~_nd Kunszl, who calculated the he\icily amplitudes for gg _ qqq'fI and fo!" 9 9 _ 1/ q where f denotes any Egh~ lepton [72]. In the SMile way, I w:i: / ZO + TI jds, n = 0,1 . 2, slich as , for eX
a,
I.Y. o If'n 001..
mention f.he work of (:lLllioll rwd · Kalillow~ki [7:1], who study the process 9f1 - t 9 9 9 9 a.nd the amdy:-i s by J(UIIszt [701 or Ute pro('(':-: . . Finally, we
(\.1:-10
99---;99Qq·
13.2.3
Weyl-van der Waerden formalism
The helicity states for fermions are in fact described by two' componclii. spinors because of the helicity projection operators (1 ± "{5)/2. For I.his reason, one can nicely reformulate the heEci ty method using the Weyl-van cler \Vaerden forma.lism [751, which explicitly refers to the Ilonvauishing com-
ponents of the spinors only. Thus, the helicity spinors u+(p) = v_ (p) of eqn (6.10) and ofeqn (6.1l) are rep laced by ,,+(p) = v_(p)
ll_
(p) =v+(p)
---; PA, (13.8)
where
(13.9)
The undotted indices refer to the lIpper components of the Dirac spinor, whereas the dotted indices refer to the lower components . The raising and lowering of indices is done with Lhe anti symmetric tensor fA,; = .
,AB
= (
0
-I
~ ),
(13.10)
I.e.,
pA
I
--
pBIlA
v'P+ I
PA = CilBpB
v'P+
(-:~ )
(~r )
(13.11)
H then follows from eqns (1.9) and (1.11) that u+(p) = v_(p)
->
PA'
(13.12) in this formalism.
One also defines a spinor product
< pq >=p"q
A
=
P.lq+ - p+q.l
=
vp+q+
'
(13.13)
I.Y 2.
1"/!Jn 'II~: 1!
m : l!/-."I.(JI'M f:N"/"S
which lHdisfi.:s 1.lm
tll( ~ pmpt~I·I. i(~~
< 1)(/ >
·1
1< pq> I'
<
fIJI
>=
0,
(13.14)
= 2(p · ,).
As a result,
(13. 15) Similarly, the matrix
(13.16)
with k1 = 0, has the following nonvanisttillg matrix elemelLts
(13.17) One then readily finds thaL
(13.18) elc.
Also, for the
pho~oll
polariza.tion vectors (1.6), we have that
(J+(k,,))AB -
h < kq >
.
k'~qi:1
'
h
()+(k,q)t. -
< kq > q),ka ,
(r(k,,)),B -
· q
(I-(k,q)).. -
· ·..... 1B'
while all oLller components vanish.
.,fj
hk
,i 1.:8
,
(13.19)
/.Y.
oun.oo/(
\OVith this f()rmalism, one can now rcfofmul/\Le the entire hclicity mcl.hotl . All helicity amplitudes arc ul timately cxprc~!icd in terms of spino( prod·
uds (13.13), which are closely related to ollr quantities Zi j, defineu ill eqn (6.13). This program has been carried out by Be rends and Giele [76], where th" transla.tion of fourwcomponenl Dirac expressions in terms of W~yl-van del' \~'aerden expressions is extensively discussed . Applications of this technique to the purely gluonic processes 9 9 --> 9 g , 9 9 --> 9 9 9 and 9 9 --> 9 9 9 9 anu to the ne utrino cOllllting processes e+e- --t vii...,. and e+e- - ) 1l1/,1 can be found in Berends et al. 1761.
13.2.4
Qllant1Un gravity
Quantum gravity is the gauge t.heory of gravitational interactions. Being a. ~auge theory, its perturbative treatment is quite similar to t hat of QED or QeD. The main difference, from the point of view of calc.uln.tio ns, is t.he fact that the graviton is a gauge particle with spi n 2. Jt is described by a polarization tensor (/H'(k) , which satisfies the following conditions:
<W(k) = f""(k),
ei.>Jl
=::
0,
(13.20)
Using tbe polarization vectors f~(k), which we introduced for the photons or the gluons, it is possible to construct explicit representations for the graviton polarization tensor. H suffices to note that
(13.21) satisfies all the requirements of eqns (13.20). The quantity
O(k) Z(p)
ZIp)
Fig. 13.l: f'cynman diagrams for Z - . e+ e - ...,.
c'I,ocl /'
+ tl1 + til + 1,'1 sst
32
[
,"
~
x S- -
(p+' kHp • . k) ('I.' k)(q • . k) 2
l1
v + (p .. k)( •. . k) + (p+' k)( •• . k) t
,-
+ -C-.:iT::-" (p+' kH.+· k) u
l1
+ (p.' k)(.,
. k)
1.
(13.23) In thi~ formula., the qua:ltity ... is related t.o G, Newlon 's gr1'lvilaiiu n1l.1 canstan:, through,..2 = 32~G. tlnc.llhe invariants.:l, s', t, t', II a.r1l.l t,' are defined as ulIua1 by
, " 13.2.5
-
2(P+ . p. ),
I
-
2(.+· • . ),
I'
-
-2(p+ · q... ).
u
-
- 2(p, .•• ),
~
-2(p. ' •. ),
u' .
-2cp . . q+ l. (13.24)
Polarizar.ion vectors for massive spin-! partid c:s
It is possible to a.d;o.pt the helk.ity melhod to lh~ desc.:ri[Hion of p rocc:'>~s involving massiv(· spin-l particles. Ho .....ever, a.s we have not wo:ked out a general method for .bis purpose, .....e sha.ll merely illustrate a. possible way oi dealing willI llI(1ssive spir:-! part:cles. To this end, W~ choose the process
(1:1.25) with the F'eynman diagrams depicted in Fig. 13. [. The co:respo"ding Feymr:an t.mplituclcs a re
.
'
/.1.
-
-c'i1(p_)
l(k)21;_+~)
M, _
- e'iI(I'_)
l"(p)[a(I-"f5)+b(I+Is)1 ~/'+-kJc) " p+ .
M,
1'(I,)[<>(I-I,)+b(1
OU-n,OOf(
+ "I.)J" (I +), I
/(k)v(p+), ,
( 13.26) ..
wiLh c~(p) Lhe polarization vector of the Z-boson. The relation of the quantities a and b with the paramet.ers of the standard model are gi\'en in eqns (2.24 ). First, we choose the following polarization vectors for the photon:
(13.27) It follows that the hclicity amplitudes are
M p-(! +I5)V(P+),
M(-,+, +) -
2e'bNiI(p_) ,l"(p)(i>++
M(+ , -,-)
2c'aNu(p_) 1>+(p-+ ; .. ) 1"(I')(I- Is )v(p+),
M(- ,+,-) -
2e'bNu(p_) j"(p)(p++ I ) p-(!+ 'Y5)V(P+)· (13.28)
A massive spin·l pa.rticle has three polarization states. following three expressions
/;(1') -
+ IS) + P- h
N [p' p+ p-(l
1 [
M~
Mz P - (p. p_)
p- 1,
Consjder the
fi(l - IS)],
(13 .29)
I
with I
p =p++
k
(p+. k) - ( )p_.
(13.30)
1"1'-
The expressions fo(p), i = 1,2,3, satisfy the rel •.tions (1" <:) = O. Furthermore, they a.re orthogonal to one a.,lother, i.e.,
(c;'c;)=o,
i f j,
(13.31 )
j
I ,i
•
IY.N.
J.'illI'rlilill m:Vf,·/. O/'Mf.:N'I'....
and t hey nrc
Jlol'nL;lli~cd:
((; ·(,)=-1 ,
(13.32)
i = 1,2,3.
These expressions ar~ thus possible polari~a ti on !'tates for the Z·par ticle . It then suffices to evaluate the hclidty amplitudes, eqrls (13.28) , [or each ,li (p) , i = 1,2,:i, eqns (13.29). !l.nu to osqull f C the absolute values of the resulting ex pressions to ohtain the cross sect ioll . ,\ simple calculation shows t hat Ute helicity amplitude M(+, -, +J yields the following contributions upon insertion 0: ,LiCp):
M 1(+.-,+) =
0,
M,(.,-,+) -
'J( -" k)- U p- " I - "'f.~ IJ',p+ , (p+.N;p_
M,(-,-,+)
" ' (P'p-)
~
L:I-\-I ( pol '
_(
) '(
)')
O.
(13.33)
__ )I,_16'<"IP'p-)' +" - Cp, . k)(p_ . k)
113.34)
T he remaining helicjty amplitudes are read ily {'.valuat ed in the same w~.y. The unpbhlrized squared matrix clement , su mmed over Lhe final state polar· iZ1l.tion (Iegrees of freedom and averaged over the polarization!' of the Z· boson, then bewme5 :781
J6e4 (a2+b1) (p.p+ )1+(p.p_)2 3 (p,. k)(p_ . k )
polarization vectors describing ~he spin st ales of the Z were used by Bohm and Sack [79J , who study the related process e+C ..... 12. Of course, this example is 110 simple tLa t one could equa lly wo!'l1 use the standard techniques of sum ming over the polariu. tion states of the Z . How· cVo!'r, for more complkated situations, the method we present ed here coul d prove to be !TIMe collvenient. For example, if the p<Jlarjzation vec to r of the Z appears in a. FeyrunaJl diagram in rI. dosed fermion loop, our meLhod a llows onc to ca.lcula.te the associated trace and to express the result in terms of fo ur-momenta only. Of course, the p rice to be paid :5 thal Olle must calculate the trace thrice. The way of introducing poln_r:~ati on vectors for massive spin· l pltfticJes, like we did iu CqllS (13.29), is hy no means unique. An alterna tive method, using massive spinors t o define c" i = 1,2,3 , has been presented by PCl.5-
Ver),
~imilar
(13_35)
sa.rino [80J.
13.3
Unsolved problems
Regarding beamstrahlung and associatc-:d topics such as pail' production, there are numerous unsolved problems as a.!ready discussed ill Cba.pter t 2, Many of t hese unsolved problems are critical for the actual design of a fllture multi-TeV eledron-positron . collidcr. fn t his section, we list. a llumb( ~ J' of oiher impo rtant and unsolved problems; they are not diredly relaied beamstrahlung.
.
,0
.
13.3.1 I. ,
,
4, :c.
to' al
11
Collinear ferrninus and gluons
In Chapters 3- 6, we d eveloped the helieity method in the high-energy lim where the fermion masses ~vere neglected. In Section :3.4, we stat.ed t he ra!lg of validity for this a.pproximation: the energy of the incoming particles mil be su"ffkielltly large and no collinear part.icles sho uld he present. In C hapter 7) we showed how the corrections du e to a finite fermion rna could he Laken inLo account at the level of lhe spin amplitudes. This allow, us to t reat. t he case of photon or gl uo n emission in direcLions nearly CO l1 ilH~ t.o a fermion d irection to lea.ding order in m/ E. Here, In deno tes the lermi( mass and E is the incoming energy. The helic ity rnethod developed there does not. allow us to describe I.. ca.ses of nearly col linear fel'mioIlS or nearly collinear gluons. In Section 3. \YC gave some examples of configurations where difficulties of this kinel ad~ It is not unlikely that a .more or less simple modification of the helici met.hod can be devised wh ich would permit t.he treatm ent o f these sped configurations, but it remains to be worked out. To trea.t the case of collinear fermjons, Berends, Daverveldt and Kleiss [f propose to eva.luate tbe spin amplitudes [or a given process exactly, i.· without neg lecling the fe rmion masses . To this en d , t.hey first rewrite t . . maSS1Ve spll10rs as
/
1
\2 (p . ko)
6)
/
1
\ 2(p· ko)
(,p+rn )I'-(ko),
('f;+m) ,k,u-(ko) ,
( L1.3
where lL(k-o) is a negative helicity massless spinor. The four-vectors ko a: kl must. be genera.lly positioned and sa.tis fy t.he relat.ions
7)
he ,
Dy
kJ
=
0,
(ko · kd = O.
( 13.3
The spin ampli t udes can th en be expressed in t.erms of tue c.om ponents of t d ifferen t four-momenta in the usual way, and the cross section is obtained adding the squared absolute values of the spin amplitudes.
/ ,1..1
liNSfl/.\'f.:/J l'IIIJ/II./·:.l/,<;
T!Jis lpduliqll" h;1Ji IH~~II 'lI'pli"d I... III<' clIit'lIllIl.ioll of llw !:n)~s ~,~<:1.i'!II fOl" r,+/;- -> (;"1"( ·-t+(- (f "" " ,/1,1) I•.v t h.·sl' alLl.lwl·.~ illld hy MHil1l and Milrlilll:;; [82jlo (,+(:- -. 1'+('-")' for lHIl~(>il1).\ r"rll1iol1~ lll~ar thl' IJI";~III diJ'l,d,ioh. It. is d(~ar, howeH'J", LIIil.I,. wiLli tl.i~ IIIdllod , 1.I.c fOrillUlac arc uot si!lIpl(' ;L~ 1IHl.T1y l!Iil~S t(~rlll::; arc n:L.]ill{,d which (:oll!d ill fa.ct be negled('(1.
13.3.2
Relation to qUFJ.ternions
In Lhe hclicity method, one frequently deals wilh cxpressions of the type (13.3H) Althocgh we I:se four -dimensional ..,-matrices, the problem is essentially two-dirr.ens!ollalone. Indeed, we have that [sec cqJlS (6.10),(6 .1 1)1
u.(p)
~
( XO+ )
"-(pi
~ ( x-° )
iI
(IJ.:I!l)
wjth
1(, .. .. . (1) ff+
- - pnn+p'/7}
v ,,-
~
_1_(",,1 _
JP+
0
P'O)(0) I
(13AU)
The a -Illatrices are the familiar Pauli matrices, given iu eqns (2.7). With om representation (2 .6) of ihe T-malrices, we have
Using eqn (13.41) in the evaluation of A+ , eqn (13.38), we find that the four· dimensional matrix multiplication reduces to a two-dimeIlsional one because of the vanishing of the lower compollcnt:; of u+ , cqu (13.39) , We are thus left with problem of evaluating
This problem can be form ulated in a different way by introducing cJuater· !lions. II quaternion q [83J i,; a linear combiuutioll of the type
(13.-13) ..... here tile tJllil.ntities qi, i = 0, ... ,3, are ordinary (complex) numbers autl where the imaginary units e., 1 = 1, ... ,3 , satisfy the following multiplication rules
(13.-11)
13, OI!'J'WOl\
the tot"tlly antisymmctric tensor ill a dimensions (Cl2a = +l). It is immediately seen that the matrices -ia obey the same multiplication rules os the units e;. We can thus represent the expression ko 11 +k' if by the
with
(;iJk
quaternion q) if we identify qo =
ko ,
Similarly, the expression ko 11 -
(13,45)
k· if can be represented by q,
where
q is
the
(quaternionic) conjugate quaternion:
(13.46) The problem of the evaluation of the matrix product (13.42) is thus reduced to a multiplication of quaternions of the type (13.47)
It. could very well be that, for the evaluation of long expressions of the type (13.42), a computer would need less time to perfol'm the associated quaternionic multiplications t han for the stra.ightforward two -dimensional matrix mul tiplications.
13.3.3
Loops in Feynman amplitudes
The heJic,ity method produced some remarkable simplifications ill the calculation of bremsstrahlung processes in the high-energy limit. This was achieved trough the introduction , in a covariant way, of explicit polarization vectors
for the radiated gluons and/or photons, It seems naturai to suppose that similar ideas could be fruitful also for the calcula.tion of loop corrections at high energies and large momentum
t ransfers , The external field approximation used in Chapter 12 is in fact derived from multiloop Feynman diagrams [84]. However, this is a case where the transverse momentum t.ransfer is small and, hence, qualitatively very different fforn our concerns here. In Appendix B, we present the calculation of the process 'Y 'Y -> 'Y 'Y, which
involves a loop integral in the lowest order Feynman amplitudes, Through a. good choice for the photon polarization vectors, we could circumvent the problern of ultraviolet divergences ..."hich a.re normally encountered with the standard procedures. From this' example, it appears that Borne amount of
simplification occurs and this is presumably also t rue for other loop calculations.
Although the helicity method reduced the numher of terms for this example, we did not succeed in developing a useful technique bosed on the helicity method for the Feynman integrals themsel ves. We tried out several ideas , on the matter, but we must admit tbat we could not come up with anything we dare to present: all our efforts produced more complica.iions
than simplifications .. ,
f,Y.~ .
13.4
"'·/'/I.lJ(:IIf-:
E pilog ue
' When a macroscopic object is illlaninal(!o by an electromagnetic wave, the resulting lield is, in gencwl, fairly cornpJ icl\Lcd and not easily calculated. If the wavel ength of t he ill umi nating electromagnetic wave is small compared
with ~he d-imensions of the objeot, however, geomclrical opt ics tah~s over, ilnd reflect ion an d refraction become a suitab le and sufficient description for many situations. As a. concrete example, ld us imagiue that we are in a. room equipped with a. red light hulb at one instant ami with a blue 1i g~lt bulb at a later time. 1l is a wmmon experience that , a.s the light bt;\b is changed, the objects in the room change thei r colo($ but not their shapes and sizes. This implies, in particular, tha.t the scat~erin g cross section does not depend on the light freqt:Cllcy. T:1al l.he shapes and ~iz('S are indcpcllden l of Lhe wavelength simplifies the physics~and our daily life-greatly. Just imagine trying to sit in a (;hair that has different shapes for dj jrcrent color rays ill the sunbeam!' [85J Compared with the classical case, the quantum situation is much more subtle and imeresting. Indeed, both claM-ically an d quantum mechanically, SilllplificatioTls can be e xpected and do occur when tlle wavelength of t-he incident p:uticle is JIlIu:h smC'.llcr ~h alJ the sizes of the interacting objects. In (he d assical case, geometricalll.nd scaling rules apply, albeit wit h du~ compiicatioill! from caust ics including foci and shadow bounclaries. T he si~uatjOIl which is both physically imporlillit and well understood is t he ela~tic an d diffractive scattcrir.g of haclrons at very high energies. Here, the behaviour of tlle total cross sections (obtained throug:l the opt ical theo· rem) and elastic differential cross sections was predicted theoreti cally !S6] an d later verified experimeuLally ill the case of proton .antip roton scattering [87J. In particu;ar, bo~h the total Cf OSS section and the ra.tio of t he integrat ed elastic cross ~ection to t he tolall:ross sC{".tion increase wi t h incre~ing enc:gy. The situa.tion t.re;:.~ in the present book is a second case where grea.t simplifica.tions occur when t he centre-of· mass energy is much larger t han t he rfll':t m;L~Ses of the scattering a.ud prod1lced particles. This sitnp~i fi c:ation was linl found by direct calcula.t ion, as mentioned i:l the Preface, and later understood through the helicity method, which is the central subject here. H is natural to specu l1l.te that t.~ere are otl:er situat ions in rutide physics where similar simplifications occur . Til is speculation m ust be true. "'. everthp,. less, at the moment , it is by no meall$ dear where to look. Let us conclude this book with a possible suggestion. It is the best one we can t!::ink of; whether it will work remain., to be seen. III the firs t instance giWll above, the io tal energy is very lare;e., but the trJl_osvcrse momentum transf(!rs are not large compared wilh the :-est masses of t.he particles. til t he !lecond instance, t he subject of this ·book, both t:1e tota.l energy And the transverse momentum tra.mfcr::; A(e very large compared with the rest masses and are of t he wille order ofma gnilude in most caseA. An
600
I~ .
Of/'t'1.00f(
intermediate in st.a.ncc, between these two, tll1~g(~Ht~ itself, where it i ~ pcrhi\p~ not unreasona.ble to also expect simpLificaliol1~, In this intermediate case, the total energy is very la,rge compared wi t h the transverse momentuOl transfers, which are in turn I,nuch larger than the rest masses of the particles. For parlicle physics, t.hjs case ca.n perha.ps be ex-
pected to be important at the SSC (5.uperconducting S.upcr Q.ollider) in Tex •• and the LHC (Large Hadron Q.ollidcr) at CERN, where the centre-of"mas. energies a.re respectively 17 and 40 GeY , while the transverse momentum
trallsfers •.re very roughly of the order of 100 GeV /e. Since this case is intermediate, fea.tures of both tbe first and the second instances are expected to appea,r. A very preliminary step in this direction wa.s taken a. few years ago [88J. If ottr speculation is right, we can look forward to new fa.sdnating developments in partide physics. Felix qui potuit re'r um cognosc.ere causas!
•
Appendix A Traces: cut-and-paste Manipula.tions of
~p ill or
c.xprc!'tsion:s and
tt
of .....-matrices life key iugre-
dients for all effi cient application of the heJicity amplitude method. ~hny formulae are known in the literature [89] for rewriting products of tra ceR of the type Tr[ ....·rl'" .. :Tr[ ... 'Y" .. .] or similarly for spiMr expressions. Unfor tunately, it is not always easy to remember these formulae as they often diff('T from case lo t:asc. III the heJicity amplitude method, one dCAls with expressions involving
(he projection operators (I ± 1'~)/2 and contractions of t he type ;k wilh k1 = O. For these situations, it is often pOl;sible to factorize a givel1 cxpres~ion or coldvcrsely to combine a product of expressioils into a sir.gle. onc. It. is the pmpose of this appendix to show how this 'cut-and-paste' technique CClll easily be implemellted without Lhe need of n:lIlcmhering more or l~s complicated formulae. The starting poinl in this game is the obf;lcrvation that, for a massless spinor u(p),
(I ±O,)u(p)u(p) (J 'f1,) -
( I ±o,)(2:u(p),,(p))(1 '1'7,) pol
~
'(I ± 1,) p.
(A.I)
1n this W?l.'y, a. product of spinor expTe~sions can DC combine
rr :::
~ \u(p) j ,1(I-O.)u(,)"(')
This
P p(! +-(,)"(p).
(A.2)
pro~rty
is Ilseful for genera.ting overall factors consisting of scalar products. If, in U!e previous example, b = c with bZ = 0, the l.h.s. is equal to
2(b· p)ltlft p,! A(1+ -,,)1,
(A ..3)
which contains a. factor (b· Pl, wlll::reas ill the r.h .s. none of the Lwo factors do. Of course, it is hidden in some way in the r.h.s.
... ..
M'I'I;NIJIX A.
'J'/l.ACgS: (;U'J'-ANfJ-I'AS'J'1!)
This technique is also very i.l.scful for simplifying expressions which contain repeated indices, for example~
M = u(kd"l_(t - "I,)u(k,)u(k,J-y"(1
+ I,Ju(k.) ,
(A.4)
where k, and k3 are light-like vectors. Obviously, M
=
U(k,h,,(1 - 15)u(k-,)u(k,)(1 ,,(k,)(1
_
4 u(k,h" /:, ,k,"I",,(k,) ,,(k,)(1 + ,),),,(k3 )
_
16(k,. k3) U(kl)(J u(k,)(1
k) 16(k"""3 2u(k, )(1
-
u(k.)(J
+ 15)u(k3)u(k3h"(1 + 1,)u(k.) + ,),),,(1,,)
+ 1,)u(k,) + "I5)u(k3)
+ 1,)u(k,)u(k3)(1 -
"I,)u(k,) Tr[,k,(J + 15) ,k3(1 - 15)]
+ "I,)u(k¥(k,)( 1 -
1,)u(k,).
(A.5)
This expression no longer contains repea.ted indices. In th is examp le, things worked out nicely because the combination 1-'5 in the first. spinor expression matched the 1 + /s combination in the second one_ What if the I 15 factor is changed into J - "I,? Take
+
M' = U(k,h,,( 1 - 1,)u(k,)u(k,J-y"(J -,,)u(k, ) ,
(A.6)
and let us perform similar ma.nipulations, but with an extra four-vector, a,
as follows M' = u(k, )-y,,(I-'(5)U(k,)u(k,) p(I- 1,)u(k3 )u(k,)-y"(I-"I5)1l(k4 ). u(k,) p(1 - "I,)u(k3)
(A.7) Clearly, we must choose a
M' = =
If kl is also
M'
=
_
light ¥ !jke~
f:.
~~2) k3 . Then,
4 u(k,)-r" /<, fi /<31'''(1 - "Is)u(k·d u(k,) p(J - "IS)U(k3) .
-8 li(k,) ,ka
P /:,(1 -
"I,)u(k ..) . u(k,) p(1 -"I5)u(k3)
(A.S)
we can choose a = kl' and rewrite M'
-16 (k, . k,) u(k,) /:,(l-'5)u(k,) u(k,) ,k,(l -,s)u(k,) -16(k , . k3) u(k,) ,k,(1 - 1')1l(k.)u(k3) h(l-1'5)U(k,) Tr[f:, ,k, (1 - 1'5) h(J + "15) ,k,J I
(k, _k,) u(kd ,k,(l - "Is)u(k,)u(k3) };,(J - '(,)u(k,).
(A.9)
ill ilgaill rossili]!: to di!lIirifLle tho repeal/xl illdio•.':S, [,u~ oll ly at t he expense of ill~rod \Jf~ing 1\ fncLor (kJ . k 2 ) in t he uellominil.lor. One could also choose a = k4 • wit h k~ = 0, in which case one obtains
One sees thal
i~
(A.10) but now, (k.)· k~ ) is present in the cenominalor. H lhusappea.rs that there are several ways or rewrit ing M ' , an
It ca.n be deoomposc..-o. in the 16 independent combin,'ltions of "(-matrices; 11., '7,., [1", 1~ ; , 1,,7~ a.nd ~i'; which is a. 4 x 4 malrix.
(A.13) Multiplying both !\ides of Lhis equation wit.h the 16 independent 7-matrices alld LakiJlg the trace, one finds
S = 0,
T'''' = 0,
A" = 0 ,
(A ] 4)
lIence. ~"T,!~'
which completes the proof.
P fi 1(1 ± ~,)I ~ 2"
V' ,
(A.15)
)
Appendix B The process "y "y -> "y "y T he: usefulness of the helicity method is by 110 means rCR lrictC'd to caic ut.Hions involving tree diagrams. \;) this appendix, we illustra.te it ~ applicability to a simple process described , in l owes~ ord£! r, by a set o! one-loop Feynrnan diagrams: (B.l) O(k, ) + O(k,) - O(k,) + O(k, ) . There
i,'
,11,
J(:~, 1)·[/; ;. ,I.,(p+ A ) J,ip+ ;"+ ,',) J;(j>+ ,', )]
J X (P2 + ic]f(p -.. k4 }2 + It:l [(p '' kl ,11,
-
i,' X
M,
+ k~P + ilJl(p + kiF -
J(~~.T'['<; j> #J>+ ft,) ,<;(,(>+ ft ,- ft,) l'(l>+ h )1
1 [p1 + ifll(p ~ k3)' + iell(1) + k, - k, }2 + ic] [(p + k,)~
i,'
ill '
+ if]'
J(~~, Trl!; ,, ;'U+ /<,) / . (1)+ h+ h) I;(i>+ h )] 1
X
[p2 + il][(p + klF
+ 1(.][(:0 + kl + k2)2 T
it:l! (p + kd 2 + ill(8.2)
The sta.ndard procedure [or calcula.ting one-loop Feynman dia.grams COlisists of fi rst com uining the denomi n ators inlO a single denoruiJ!i~Lor_ This is done by repea tedly u~jllg lhe trick of introducing extra integrat:on va.ria.bles, ea.lled Feyn man paramcLers 1911: i
1"(0 + /3) ('
AI>[]IJ = 1'(0)\'(,8)
Jo
2;0-1(1 - x)ll-l dX(Ax + 80 X)Jo+D-
(D.3)
11/ 'J'J':NfJ/ .\' 11. '1"111,;
/'U.O(.'/~',i'.''';
I(k , ) ~i(k2)
I(k,)
Fig. B.2: Feynman diagrams for 1+ "i
-->
"i + 7·
1'"r ";1
1l/'NJ,vOIJ II.
Tlif. 1'/lI}{''';.\S.,.1'
tiltt
''J'J
/? + '2(,1 . " ) I /, t if ,
(13.4 )
W1Jcre, in general , ~ Lc four-vedor /( i~ a li!:"ar combination of lhe fourvec~ors in the proces!!. The <[uant i:y L is tll.'11 a sCal M quanl ity lep consists in slJiftir.g the ilJtegratioll va.riab!c
(H.5)
p - tp-f(,
which makes the der:(tIninator 1\ (\lI;clion of p2 unly, i.e.,
(ll.H) In this case, t!tC shiH (0.3) can be performed witho ul paybg at:entiou h' evcntt:al ~u l'face terr:1S as the integrals (B .2) <'.;c only logarithmically d ivergC:lt.
After lhcsc maJlipu]"tions, the expression M I ,
fO I'
cxa.ll1ple, readl!
6:(1 - z)
XT(p3"+~,("I~z~)~I'=Y(~I~x~)~+~u==~("~Y~)}'+"=·')~· '
IIl. 7)
with
(lI.S) aud (B.Y )
We /lOW pn)((!ecI to calculll.te tJ!e various hel:ci~y amplil;Jdcs ~or tl::5 IJrnC~~ denok.'d by M (~ll ~'l; A:J,A.). Suppose we fi:st eva ludlc M( +, +; -, + ). To ~ hi8 cr.d, we chO()~
J,t" J,~ '
Ii
N/,k~
h ):d I + 7~)-t
= Ni h h h(J ..
It -
~,h
;.. (1 - "(,)) .
+ ,,)+ /<, /<, 1'.(1 .. ,,)} ,
NI I.,k, ,k,( 1 + ,,)+ ;., h M' - , ,)}. Nt):.. ).1 ,kJ(1
+ "'f~ )+
ft~
):. ):~(J
- "Is)],
ID.IO)
1108
with
(B.ll) Note that photons 3 a.nd 4 have the same polariza.tion vectors, which is possible because of their opposite hcli cities. Inserting eqns (B.10) and (B.ll) into t he expression (B.7) for)M" we find that its contribution to Al( +, +; -, + ) reads
M I(+,+;-,+) = ie' x
J(~:)' 10'
d"
10' dy 10' dz6z(1 -z)
[TI+) +1'HJ
[p' + z(1 -
1
I
z) [sy(1 - x)
+ ux(1 -
y)! + if]'
,
(B.12)
with
(,'t~)' Tr l,k,
1'11+)
x
1'11- 1
-
,ks ,k,(i>- F) ,k,
/<, ,k,(i>- F+ ,k,)
,k3,k1 ,k4(P- !(+ ,kl+ /<,) h ;'3 /<.(i>- !<+ h )(l+-;,lj,
($t~)' Tr [,k, /<3
h(i>-
I(l /<, /<'
/<3(i> -
1<+ /<,)
X /<, /<1 ,k3(i>- ,((+ /
.
(B.13) Vile no'.\.' 'want to perform the p-integral. As the numerator of e xpres~ sion (B.12) contains four powers of p and the denominator only eight powers of p, \',Ie can expect a logarithmic divergence in the integra.l. vVe thu s have to
regularize the expression (B.12). 1f we take a regularization procedure which respects Lorent z covariance, we can replace
(B.14) where the coefficient A depends on the chosen regulariza.tion scheme. I3y symmetric integration, we can omit all the terms with odd powers in p. T he replacement (B.14) in eqns (B.l3) then shows that, with our choice (B.lO) of polarization vectors, the (p2)2 and pZ terms vanisb. To see this, it suffices to use the formulae
1'0 j, "
I
jl. J-y"
-2
J /J ,c " ,6.
(B.15)
AJ'I 'NN{)L~
11.
'I'II~: l'II(}(.W8.~·..,1
-. 11'
tJ(J!I
Moreover ,
, --
T (+)
(B.16) With the formula
(a> 2), (B.I7) we obtain
Md-.+; -,+)
" l' l' 11
Suty{l-y)2(I-z)' . =-dxdydz 16;1"20 I) 0 [sy( l -x)+u:t(l-y)+acjl -
"
t
12::-1;'
(B.18)
Similar ma.nipulations Oil M2 And M3 yield
(B.19)
Adding these Um:c contributions and taking into account t he factor 2 of eqn (D.2), then leads to (B .20)
For lhe calculation of the helicity amplitude M( t. +i +, +), we choose
AI'I'J.,'NI!IS II. '1'1/1,' l 'I/ ()( :/'.'SS1T-11
it
, (1J.21) J
\vith
IV
=
(2stu)-t .
(0.22).
Again, ihe divergent (p2)2 Lerm~ are fo ulld to vanish, but, this Lime, the 1" terms contribute. They are readily evaluated using (Q > 3). (B.2:1) One is then left ,..... ith the integnLls over the Fey nrna. n pa.ra.meters , which ar0.
slightly more complicated for t his helicity ampl itude. The result is
e' M(+ ' +;+'+)=-21[2
{t2 2s' +,,' [In- (U) U- t (Ut ) +J } . t +.- 'J+-:;-10 0
A (;onveni ent cho ice of polarization vedors for A;f( +, +;
(B.21)
-, - ) is
(B25) where the normalization facior N is given by eqn (0.22). Thi s choice of ,'s a.gain eliminates t he (p2)'2 terms, and one obtains t he simple result
e' M(+,+;-,-)= ;;-;;. L,1fM
(B26)
All the remaining helicity amplitudes can be obtained from egos (B.20), (B.21) and (B.26) through cros~ing. For example, M(+,-,-;+, - ) is obtained from M(+ , +;+,+) by replacing k, H -k.,. As a result, $ H U [see eqns (B.8)J a nd the logaritlun5 in eqn (8.24) become complex. We fmel
M(+ ,-; +, -) .:. - e' { S2+t' In 2..,.'2 2H2
,(-t) s-t - - In (-t) +1 S
. [" + t' In
+2.11"
u,2
oS
U
(-t) s -t]} - . ii
U
(B.27)
1I1'1'~.'N /)/.\'
II.
'l'/1J.. 1'IIOI'/';'-;SI"
'll
A COIlI!'!.-:I' E.,Lol' )wlidly ;ul\p);ll!d('~ 1'", j,).i," !!n""-'~'" il1dll
Tlwy
WI TI! OO t.:"Ii lil '.1
hy iutl'(ulllril'J,!;
I'lu)joll
1",l ilri;-:'lliwl
'If'dors ill a IIO I H'U
V
licity method is also applil'ilhk
J,rOLl.'~~e~
ill\'olvillg dosr:d-Iool' 1,1"YUIIl,HI Jiagrams. For lhel)j'O(:cS~ "( + "( - I 1 .... "(, we fOlllld that Lile div;'fi!:ftll(""I 'I'ul, lem, (Ulsocia Lr.d with the iYllcgrab over the vir~ ual monlcuLa , cou ld 1,,\ Ii il'l 'ly eircumve.lltl'O by A. clever dlOiceo( polari.-:ati01;' vectors for tJw ph"IIlII~, It)
.'.'
Appendix C Color traces In this appendix, we ii5l tile color traces which arc u('t)dQ{] fot' Lhe (·valuation
of the ~qua.rerl hclicity a m p lilud('~ f;ummcd We !ir:;l briefly ouLlinc the method.
OVClf
the colur d l lSn~ of rn~·
Tlw SU(3) color matrices T~, a::: 1, ... ,8, are i!cfmil.i;1. 1i aud sal. i ~ry T'(1· ) ~O.
(C.I)
Togdhc. with tile 3 x J unit m'l.trix 11 , Lhey f.OlIstil ulc ,. ha~i~ f\!:' matrices . Any 3 x:l rna.t rix J\1 can be written in th e fOfm
~Iw
:\ x ;\
, A?plying this relation to lne matrix Af , wl lose
el er:lC1 Il~:lre
M ,} = O~i 6~i ' wi~h
P ILnd q tixeC. Iwmbel's from the set (1,2,3), we End
lJpi /if} = 2L: T,j ~
L (8"kO~,T'A) + ~8p~6,j ,
(eA I
k.t
• T his relation is very useful for
t~e
evaluation of coJOI' t ra.ees involvillg a ~ltml1latio!l OW'!!::I ...olof inllex. ror cXl'lmplf>, let 0, a.ml n~ be ~_ny (:olllbi uatiol1 of the co!or matrices, Then,
LT,(T'O,T'O,) ~ ~
L i.i;I',~
LT,~(O, )j;I;~(O,)" ~
~Tr(OdTr(01)
-
~Tr(OlO,) .
(C.ff)
nclatior. (C.5) can ... Iso be used when the matrices 1'''' appear in dilrcl"clI ~ trac~.
For example,
• In Doth
(C.6) and (C.i), the resulting expressior.s contain fe wer color ml!.trices and are thus ea.s:<:r to evaluate, ...Ve now !1st the; color lra.ces wh:ch are needed for t llC summation over t ltc colo r degrees of freedom ill the squared helici ty amplitudes. ODe can also derive ma.ny other equation::; vyexploiting the invaria!J(:e of the trac~ under <:yelie pcr mllta.tion of LIt(: color lw).lrices. In what follows, ll!c su mma.~jon over the repeated color indio!~ a,b, ... ,e is alway~ understood. CIl.3eS,
,oI/'/'fo.'N/JlX " , ('(I /,on ,/,IIA('~:S
01 1
C .1
2 color matrices Tr (1'"1'") = 4 ,
C .2
(C.S)
4 color m a trices 1'r (1"]") Tr (T'T') = 2, Tr (T"T'T'T') =
-~ , 3
Tr (T'T'T'T') = 16 , 3
C,3
(C,!))
6 colo,r mat ric es Tr (1"1"T') Tr (7"1" 1" ) =
-~ ,
Tr (T'T'T' ) Tr (1"1"'1") =
~,
Tr (T'T'T'T') Tr (T'T' ) =
~,
1'r (T"T'T"T') Tr (1"1") = -
~,
Tr (T'T'T'T') Tr(T'T') =
j,
Tr(1'"T' 1'"1") 1'r(1"T') =
-~,
Tr (T'T' T'1") T r (1"1") =
~,
Tr (7"T'1"'1"T'T') = 10 , 9
Tr (T'1"T' 1""1"T') =
~,
'1'1' ('I"' 'J ,~, f" 'I ,j" J'"
'/" )
'l'r ('I ''''J~'I "',/,"'1 ""1 "') ::.:.: ". ~ •
T, (1"'1''''f'~'f~''I'A1'~) "" - ~ "
H'
«'. 111)
C.4
8 co lor matrices T, ('1"7"1'1") T,· (1"T' 1'T) ~ ~ , 3
T,(1"7"1"T')T,(T'r'T'7<) = -~,
Tr(T"7,f,rTJ)Tr{r-T~T~Td) _
-!,
Tr (T~'1,f,1'·,!,J) Tr ('r,!,crtlT·) =
!,
Tr (7'''TIJ 'J'"1"z) Tr (T"TtirD'j'") = _
~,
'I'r (r-T~TeT") Tr (TQT J J'C7"') ==
3
3
~, G
Re[l, (7'"1'rr')[ IC,[T, (r'7"7'1")J =
:~ ,
Re[T, ('f'T'rT')[ Rc[T' (T'7"T'T') J =
-~ ,
Rei'"
('I"T"1 "I"IJ
Rcrr, (l"'1"7"1")J = - ~ ,
ReIT. ('1"1"'1<-r")[ It.), (l"'T"r"r')J = -
~,
11'4'1,·p" I· 'I"I"')J n.[T, (T'1"7"1')J = -~ , 3
.
Rerl ,· ('1"j''f<'I'''1J n.[T,·(l"'T'7'T')J = 23 . 12
(e.ll)
Il/'I'l,:/VI)JA' (:. (:OI,Of! 'I'I(A(.'I';,'I'
/il(j
C.5
10 color matrices ImlTr (T'T'7"'T'T')] ImlTr (T'1'''1''T'T')J =
" , ~~ 7
ImITr(T'T'T'T"T')J Im[Tr(T'T'T'T' T')J = -1 2' "
IrnITr(T'T'T' T'T' )Jhn[Tr(T"T'T'T'T')J = -I'Z ' 1 Irn[Tr(TaT'T'T'T')J Im[Tr(1'"T'T'T'T')J = 12' Im[Tr (T'T'T'T'T')J Im[Tr (T'T'T'T'T') J = lIZ' 7 Im[Tr (T'T'T'T"T')J ImlTr (1"1"1"1"1")1 = 12' Im[Tr (T'T'T'T'T' )JImIT!' (T"T'T'1'"T')J = - 172 ' 1 [m[Tr (TUT'T'T'T')J lm[Tr (T'T'T'1"T')] = -1 2 ' 1
Im [Tr(T'T'T' T'T')! Im[Tr(T'T'TdT"T')J = 1 2' 7 Im [l'r(T'T'T'T'T') ) hn[Tr(T'T'T"T'T')] = -1 2' ImlTr (T"T'T'T'T')] ImlTr (1"1'':£''1''1'')] = 0, Im[Tr (T'1"T'Td1") ]ImITr (T'T'T'T'T')] =
-I~ ,
ImITr(T'T"T' T"T'l] ImITr(T'TdT'T'T')J = /Z, ImlTr (T"T"T' T"T')I lIn[Tr (1"1"1"1"1" )] =
a,
7 II11[Tr(1"1"1"1"1")1 ImITr(T'T'T'T'T')1 = 12'
Im ITr(T'T'T'T'T') Jlm[Tr(T'T'T'T'T')J = -lIZ' Im[Tr(T'T'T'T'T ')] (m ITr(T'T'T'T'T')] =
1~'
/
11/'1'1-.'."1111.1"
,: I 'I
I 'I n/Ill 1'11,1,'/', '(
1'! '
JmITr p ... " ....r'I"" I ... )]
JIHITr('l~"' ...""'I ... '/ .... )I . . . ~ 17'1 '
Inll'I'r ('I'~ 'lol>-l ...'I'J'r)J I,urrr(" ... '/ ... ·, .. ,/"·.,.' )j
>=. --
jl,! '
hnlTr (1"' T~'J" l,.j'I" JJ hll[Tr (7""'-'/""/ ~'l'J)J = - II,! '
ImlTT 11 '" 1'j,T"1'd'rli Illl'Tr (7"' ·J~1 ~-T:l>r~)i ,
~
2-J 2 '
(C, 1:.1 )
,.
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IIIlJ1. /(JWiA I'll Y
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Author Index Achterberg, O. 2, 44 , 70, 620, (f24 , 626 Ackermann, H. 2, 70, 620, 625 Adeva, B. 44, 65, 623, 625 Aguilar- Bellitcz, M. 17) 101 , 6'22, 627 Akulov , V .P. 587 , 629 Albert , D. 595 , 629 AI.ksan, R. 44, 624 Alexander, G. 2,44, 620, 624 Ali, A. 70, 626 Allaby, J. V. 65, 62f Allison, J. 2,44,65, 70, 621, 623, 625- 6 Almeida., !". 44, 624 Altilrelli , G. 2, 622 Althoff, M . 44, 6!!4 Amaldi , U. 550, 627 Am brus, K. 65, 625 Ape!, \N.D . 44 , 623 Armitage, J. 2, 621 Ar mstrong, B. 17,101,622, 627 Arnison , G. 2, 16, 599, 621, 630 Ash, W. W. 65, 624 Astbury, A. 2, 16, 599, 621, 630 Aube rt , B. 2, 16 ,599,621,630 Azernoon, T. 2, 620 Bacci, C. 2, 16, 599, 621, 630 Backer, A. 2,44, 70,620, 624, 626 Bagnaiil, P. 2, 16, 621 Baier , R. 101 , 627 Barer, V.N . 583 , 628 Baines, J. 65, 625 Baksoy, L.A. 65, 62·1 Ball , A .H. 44, 65 , 70,623, 625- 6' Ba.mford , G. 44, 70, 623, 626 Band, H.R. 65 , 624 Banerjee, S. 44 , 623
Banner, M. 2, 16 , 6el Darber, D.P. 2, 4/1, 65, 620, V'JJtJ, 6e5
Barlow, RJ . 2,4'1 ,65,70,620, 62:J, 626 Barnett, R.M. 17, 101 , 622, 627 ll arreiro, F. 2, 44 , 70,620,624, 626 Bartel, W. 2, 44, 65, 70 , 621, 6"23, 6125-6 Battiston, R. 2, 16,599, 621, 6,90 Bauer, G. 2, 16, 621 Becker, L. 65, 6125 Beelter, U. 2, -14 , 65 , 620, 623, 625 Behrend , H.J. 44, 6e3 Bei , G. O. 65, 625 Bell , A. 44 , 70 , 623, 626" Bell, J .S. 580,583, 628·-9 Bell , K.W. 2, (If!0 Bell, M. 580, 583, 628- 9 Bella, G. 44, 624 Ren da, H. 2, 620 Berdllgo, J. 14 , 65, 62,1, 625 [lerends, F.A. 23, 31, 41, 65, 68, 70, 80, 81, 592, 596, 623,
625-6, 629- 30 Berger, Ch. 2,44,70, 620, 624- 5 Berger, £.L. 102, 627 Berghoff, G. 44, 65, 623, 625 Bernabei , R. 599, 630 Bethke, S. 44, 65 , 70,623, 625-6 Fleuselinck, R. 44, 624 Bezaguet, A. 2, 16,599,621,630 [lhahha, H.J. 3, 39 ,65,6212 Binnie, D.J'vl . 2,44, 620, 624 Bj orken, J.D . 569, 628 Bl ankcnbecler, R. 571, 583,628-9 Blobel, V. 2, 44,70, 6eO, 624, 6e6 moch , Ph. 2, 16 ,621 Bock , R.K. 2, 16, .599,621,630
,. WI'I/{} If 1{II JI t:x
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Dollf, r-,' . 13 , tif!! HonA.udi, F. 2, Hi, fiel norer~ K. 2, H" 621 nurghini, M. 2, 16, 62 r Dorn, :\1. 543, 627 Uouchez, J. 44, 6iN Bourrcly. C. 599, 690 Bowcoc.k, T.J.V. 2, 16,599, agl, 680 Uowdcry, C. 44 , 65, 70, 6f!3, 6!!5- 6 Bozzo, M. 599, 690 OrM.cin i, P.L. 599, 630 OrallJclik , R. 2, 620 Brandt, S. 2, 44, 70, 620, 624 , 6i6 Uransoll, J .G. 2, 44, 6~, fieO, 613, 6~5
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BraulilIChwcig, W. 2, 41 , 6f4 Brillouin, L. 559, 6!8 Rron , J. 2. 6fO Droul, I 2.9,619 Diihring, It. 2. 6£0 Rldkma.n, D. 2, 65, d20, 62!) Ourgcr, J . 2 , 16, 44, 70, 61!()- 1. Cf4
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Huschhorn, C. H, 623 Bussey, P.J . 44, 62/
C.... urt:ra., n. 17, )01, 6!!f!, fie7 CRivelli, M . 2, Ifi, [I!J!J, 62J, 6:/0 C.mpbcll, A..1. 11 , fi:!,f Cander, T. 2, fi~1 Capell, M. 41 , 6!!;J Carbonara... F. liml. Ii:'" CArnesecch i, G. 011 , (j~,"J Carrara, R. 599, 6.'/11
CllI'rull , T . 'l. IIi, [.~J ~) . (;::1. Ij.'1fI C.-.rlwri,l!;hl, S. I•. ~1 , Iii•. M11 .~ CiI!llullur.·, 11.J. '1, ti!!O Cll.~l.;d .li , II.. [)im, fJ'."ffJ (!ill:t., P. 2, Ifi, .'",!JC), fi!!l, (i:1f1 (.""un. .. I"II , ""1 ' . I'..... J...,
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626 C hang, L. 588.. !). 6:J!I Chen, Ch. H. 6:!.'/
Chen, G.F. 65, (;25 Chen, 11.5.2,11, G5, 6:!O,
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Chen , M. 2 , 44, 6-5, 6~U, 0:1:1. rj~:~ Chen, M.L.14 , 65 , 6e:l, Ii:!:; Chen, M. Y. 44, 65. Ii!!.'/. (i:!..~ Clu' n, P. 580-2, 628 Cheng. C.P. 2, 44, 65,6£0, (j:!:I, fj~!i,
Cher:g, H. M3, 5."i6, ~(j!J, r)~ I :{ 62830 Chicfo.ri , G. 599, 6,'/0 Chinowsky, \V. 2, 6f!O Chisholm. J.s.n. 601, (j,'11 COOIlet., J.-C. 2, 16, 6JlI·622 Citrin, J. 65, 6£5 Ghrobanck , D. 44 , 6P9 ('1lU, Y.S. 2, 620 Cit~olill, S. 2, 16,599,621, 610 Clare,
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Clark. A.C. 2, 16, 6ff C larke, D. 2, 44, Brl, 70, 621, 1i:!:I. li£5 fi Clear ..... atn. S.H. 65, 624
Cline, D. 2, 16, 62J Cnops, A.M. 599, 630 CochCl, C. 2, 16,599,621,630
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Colas, J. 2, 16 , 5!HI, 6~1, 6,'10 Colas, P. 44, 623 Conforti, G. 17, 101 ,622,6£7 Conta, C. 2, 16, 622 Coombes, R.W. 65, 624 Corden, M. 2, 16, 599,621, 630 Cordier, A. 44, 629 Cords, D. 2, 44, 65, 70, 621, 623, 625- 6 Costantini, V. 611, 631 Cozzika, G. 44, 624 Crawford, R.L. 17, 101, 622, 627 Criegee, L. 2, 44, 70, 620, 624 - 5 d'Agostini, G. 44, 628 Daiton, J .B. 14, 624 Dallman, D. 2, 16, 599, 621, 680 Danekaert, D. 68 , 70, 625 D'Angelo, S. 599, 630 Darriulat, P. 2, 16, 622 Darvill, D.C. 2, 621 Dau , D. 2, 16, 621 Daum, H .J. 2,44,70,620,624 , 626 Daverveldt, P.R. 596 , 6'.10 Davier, M. 44,623 DeBeer, M. 2, 16,599, 621, 630 de Boer, W. 44, 623 De Causmaecker, P. 22-3, 31, 41,
55,65,68,70,80-1 ,84,6226 Deffur, E. 44, 65, 623, 625 Dehne, Ch. 44, 62,f Dehne, H.C. 2, 70, 620, 625 Delfi no, M.C. 65, 625 Della Negra, M. 2, 16, 599 , 621, 6:10 Demoulin, M. 2, 16,599,6121,630 Denegri, D. 2, 15,599, 621, 690 Derikum , K. 2, 44, 70, 6'20, 61N, 626 Desalvo, R. 599, 630 De Tollis , B. 611,631 Deuter, A. 44, 624 Dcvenish, R. 2, 620
J)illil." "I,o, lJ. 2, If>, ~!)g, 62 1, 6.10 lJi Ciaccio, A. 2, 15, 621 Dieckmann, A. 65, 625 Diehlmann, K. 44, 624 Diekmann, A. 2, 620 Dietrich, G. 65, 625 Di Lella, L. 2, 16, 622 Dines- Hansen , J. 2, 16, 622 Dirac, P.A .M. 7, 18, 622 Dittmann, P. 2, 44, 70 , 621, 623, 626 Dobrzynsk i, L. 2, 16 ,599,621,6311 Dornan, P.J. 2, 44, 620, 624 Dorsaz, P.-A. 2, 16, 622 Dowell, J.D. 2, 16, 599, 621 , 630 Downie, N.A. 2, 620 Drago, E. 599, 630 Drell, S.D. 571,583,628- 9 Drumm, H. 2, 44, 70,621 ,623,626 Dueros, Y. 44, 624 Duerdoth, I. 2, 6f!1 Duerdoth, I.P. 44, 65,70,623, 6256 Duinker, P. 2, 44, 65, 620, 6'23, 625 Dyson, F.J. 1, 7,619 Edwards, M. 2, 16, 599, 621, 630 Eggert, K. 2, 16,599, 621, 6090 Eichler , R. 2,44,70, 621, 623, 626 Eichler, R.A. 17,101,6122,627 Eisenberg, Y. 2, 44, 6'20, 624 Eisenhandler, E. 2, 16, 599, 62 1, 690 Ellis, N. 2, J 6,599, 621, 6'30 Ellis, R.I<. 70, 626 Elsen, E. 2, 44, 65, 70, 621, 623, 625-6 Engelmann, R. 2, 16, 622 Engler, J. 44 , 623 Englert, F. 1- 2,9,619 Erhard, P. 2, \6, 599, 62 1, 630 Eskreys, A, 2, 44, 70 , 620, 624-·5 Fabricius, K. 70, 626 Faddeev, L.D. 13, 6!!2
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625- 6 (:'eng, Z.Y. 44, 65, 6!!9, 625 Fenner, H. 44. 52:i Fenofeldl, 11. 2, 65,620, 6£5 , Fc:K:fcld t . H.S.14 , 6!3 Fcynman, R.P. 1, 7,605,619,631 field. J .H. 44,6113 Finch . A. 65, 625 Fincke, M. 2. 16 , 621 Fi$cher, JI.'M. 2, 44, 620, 6£~ F'\"lo, H. 11 , 6£'; Fliiggc, G. 2, 44, 6£0, 6t3 ~~ohrmann , R. 2, 44 , 6!!O, Mi Fong. D. 2, 44. 65. 620, 629. 625 Font.a.ine, G. 2, 16 • .~99 , 6~/ , 630 Ford, \V.T. 65 , 62'; foster, a. 2, 44 , 6£0, 6!!~ F06tt.... , F. 2, 44, 65 , 70, 6~1, 623, 625- 6 Fournier, D. 44, 6ft3 Fournier. J .P. 599, 6,90 Franke, G . 2, 44 , 70 , 6£0, 62';-5 Frnternali. M. 2, 16, 62£ Fra.utschi, S.C. 39, 623 Freeman , J. 2, 6fO Frey, R. 2, 1(;, 599. 6f1 , 690 Fries, D.C. 44 , 6ts Fritzscb, H. 1, 11, 619 Froidevau x, D. '! , 16, 62£ friihwirlh, R. 2, 16,599 ,621, 690 ~'lIe8, 'tV . H , 6f3 Fuk ushima, M. 2, -i4 , 65 ,620, 6!!9, 6£5 fumag&lli , G. 2. 16, 6£!! GAbriel , W. 2, 70, 620, 6£5 Gaerncrs, K.J.F. 70, 6£6 Caido~ , A. 44, 62';
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70,80,8'1, lOr" 622- .", (i~!n 7 Gather, K. 2, 44, 6f!U, 611/ Cccr, S. 2, 16, ~99, 6£1, (i.'lf) Gcll·Mann, M. I, I I, IU 9, li!:!:ifi.fj Gem..el, II. 2, "1\ , 70, (i!!ll, li!!/ .~ Ceorge, Il4.4 , 6£'; Gerke, Ch. 2, H. 70. 62tJ, (;!1~ .'1 Gettner , r-.I.W. 65 , lie,; Ghcslluicre, C. 2, 16, ri!I!I , fi!!/, (Uti Chez, P. 2, Hi, ::'99, 6'2! , (i.'/{) Giboni, K.L. 2, Hi, 59!) , 6:2 1, Ii:U} Gih!lOn, W .IL 2,16, 59!/, I)~I , r;.'lo Gide, W. 592, 6119 Gildemeister, O. 2, iti, Ii!!! Giraud- Heraud , Y.:!, IG, .')!J!I, l i:!l, 630 Givernaud . A. 2, 16, 599 , 1i!!I, (j;1f} C lashow, S. 1, 16, 619 Glasser, !le. 2, 44 , 65 , 70, (i!!fJ, 6'~ - 6
Glauber, R. 553 , 556 ,6£7 Gnat, Y. 4i1 , 6!f C-.odda rd. M C. 2, 4.'i , 70,621, fi!!.'J,
6£6 GodNre, G.P. 65, 6£-1 Godine<:, A. 2, 16.599, 6BI, 69/J Goggi , V.G. 2, 16, 6££ Goldberg, M. 44, 6£.( Goldschmidt-Clennont, Y. 65, 6!!,f Gol'fan(l, Y\t.A. 587, 629 Gordon, W. 0556, 6£8 Cos.sling, C. :.1 , 16, (j£~ Gott:lchalk, n. 65, tie,; Gottschalk, T.D. 70, 6£6 GrAyer, G. 2, 16, 599, 6~1, 630 Crigull , ll. 2, 70, 6fO, 6!5 Crindhammcr , G. 11, 623
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J.t-'. 4t1; 629 Groom, l),,," 65, 625 Grossetete, 13. 44, 624 Grosse- Wiesmann, P. 44, 623 Grote, H. 2, 16, 622 Grunhaus, j, 44,624 Grupen , C, 2,44, 70 , 620, 624, 626 Guberina" G. 104, 627 Gunderson, B, 44 , 623 Gunion, J,r'. 589-90, 629 Guo, J,C. 2, 65, 620, 625 Gural.nik, G,S, 1.. 2,9,619 Gu\brod, F, 70, 626 Gutierrez, p, 2, 16, 599, 621, 690 Hagen, C.R, 1- 2,9,619 Hagiwara, K. 17,101, 105,622, 6e7 Haguenauer, M, 599, 6,90 Hahn, B, 2, 16, 622 Haidan, R. 599 , 690 Haidt, D. 2, 44, 65 , 70 , 621, 629, 625- 6 Haissi n,ki, J. 44, 62,1 Hamilton, W.R, 597 , 690 Hamon, 0, 44, 624 Hanni , H. 2, 16, fie2 Hansen, j,R. 2, 16, 622 Hansen, P. 2, 16, 622 Han,I-Kozanecka, T , 2, 16,599,621,
690 Hariri, A, 2, 620 Harnew, N, 2, 16, 622 Hart, J,C. 2, 620 Harting, D. 44, 65, 623, 625 Hartmann, H. 2,44, 620, 624 Hassard, .J, 2, 621 Hassard, J.F. 44, 70 , 623, 626 Hayes, ICG, 17, 101 , 622, 627 Haynes, W,J , 2, 16, 599, 6111, 690 Hebbeker, T. 44,65, 629, 625 Hedgecock , R, 2, 621 I·leintze, j . 2, 44, 65, 70, 621, 623, 625-6 Heillzelmann , G. 2, 44, 65,70,621 , 61lS, 6fJ5-6
JJ<:JI('lIhralld, K.I1. 44, 65, 70, 62:1, 625-6 Helm, M. 2, 621 Heltsley, B.K. 65, 6fJ5 Hepp, V. 70, 6fJ6 Hemandez, J ..I. 17,101 ,622,627 Herten, G. 2, 44, 6.5, 620, 623, 625 Hertzberger , L,O. 2, 16,599, 621, 690 Heuer, R.D. 2, 44, 65,70,621,629, 6fJ5-6 Heyland, D. 2, 620 Higgs, P.W. 1-2,9, 619 Hilger, E. 2, 44, 620, 624 Hillen, W. 2, 44, 620, 624 Himel, T. 2, 16, 550 , 622, 627 Hinchliffe, l. 17,101,622, 627 Ro, M.e. 2,44,65, 620, 625 Hodge<;, C. 2, 16, 599, 621, 630 Hoffmann, D. 2, 16,599, 621, 630 Hoffmann, H. 2, 16,599, 6fJl, 690 Hohler, G. 17, 10] , 622, 627 Holder, M. 2, 620 HoJJebeek, K 551, 627 Holthuizen, D ..l, 2, , 16, 599, 621, 690 Homer, R .,J. 2, 16,599, 621, 630 Honma, A. 2, 16,599, 6fJl, 630 Hopp, G. 44, 629 lIs", H,K. 2, 620 Hsu , T.T. 2, 620 Hughes, G. 2,44,65,70, 621 , 6fJ3, 6fJ5-6 Ht~tschig, B. 2, 44, 620, 624 Humpert, B. 101, 627 Hungerbiihler, V. 2, 16, 622 Hurst, R.B. 65, 6fJ4 Huttunen, J. 65, 6fJ5 lllingworth, J. 2, 620 I1yas, M.M. 44, 65 , 623, 625 lmori, M. 2, 6fJI
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l';lf:iJl, O. 55fi, fi'ZB Kll'iss, R. 20, 3l, 41, 65, 68, 70, 80, 84,589, 596, 623, 625 6, 629- 30 Klepikov, N.P. 583, 628 KloYning, A. 2, 44, iO , 6M, 6!!4-S Knies, C. 2,41,6.5,70,620,62/-5 Knop, G. 2, ·14, 620, 62'; Koba.yash.i, T. 2, 44, 65, 70, 621, 6£3, 626 Kodl, W. 2,44,620, 62'; Koene, B. 599, 6'10 Kofoed- lIausen , O. 2, 16, 622 Kogan, E. 2, 620 Kolanoski, H. 4'1, 624 Komamiya, 5. 2, 14, 65, 70 , 621, 629, 6£,5-6 Kooijma.n, D. 44, 65, 623, 625 Kopke, L. 44, 61./ Koppilz, B. 2, 44, 70, 6!:!O, 62/, 6t6
Kalinowski, J. 590, 629
Korbach, W. 2, (feO
Kanzaki, J. 44,61),70, 629, 625-6 Kapitza, H. 2, 44, 70 , 620, 624, 626
Korner, J.G. 70, 626 Koshiba,1-1. 2, 44, 65 , 70, 62] , 62S, 625-6 Kolthaus, R. 14 , 623 Kotll, U. 2, 44, 6fO, 624 Kovacs, F. 44, 62~ Kow
Jacobs, IJ. 14, (J2~
Jadach, S. 44, 62~ Jank. W. 2, 16,599,621, fl90 Jenkins, C. 41, 62'; Jenni, P. 2, 16, 622
Jiang, D.Z. 44, 65, 629, 625 Jocksch, A . 14, 624 Johnson, J.R. 65, 625 Jone;., D. 102 , fir? Jone~, W .G. 2, 620 J005, P. 2, 44,620, 621 Jora~. G. 2, l6, 521 Journe, V. 44, 629 Junge, H. 11, 65, 624-5
Kapusta, F. 41, 6~4 Karimaki., V. 2, \6,599. 621, 630 Karplus, R. 605, 631 KarsllOll, U. 2, 620 Kat kov, 1. 583, 6!?8 Kawaba1l\, S. 2, 17, H, 70, 101, 6!?f-S, 626- 7 Kawamot o, '1'. 65, 625 Kaye, HB. 65, 624 KP.eler , R. 2,16,599.621 , 630 Kellog, R.G . 2, 44, 70, 620, 624,
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Kren:!., W. 2,44,65,620,623,625 Iol', r-,'L 44, 62! Kruse, U. 44, 623 Kryn , D. 2, 16,599,621,690 Klick, H. 44, 62.{ Kuhn, J.B. 104, 627 Kunszl , Z. 7Q, 102, 588- 90, 626-7,
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612:;- 6 Leu, P. 2, 44 , 620, 624 Leuchs, K. 2, 16, 621 Leucbs, R. 2, 16,599, 621, 630 Leutwyler, H. 1,11,619 Leveque, A. 2, 16, 599, 0'21, 630 Lewendel, B. 44,70, 624, 626 Li, J . 2, 620 Li, Q.Z. 2, 44,65, 620, 623, 625 Lierl , H. 44, 62.1 Likhtman, E.P. 587, 629 Lillestol, E. 2, 44 , 70 , 620, 624-5 Lillethun, E. 2, 620 Linglin, D. 2, 16, 599, 621, 630 Linssen, L, 599, 630 Livan , M. 2, 16, 6122 Lloyd, S.L. 2, 620 Locci, E. 2, 16,599 , 621,630 Loebinger, F. 2, 621 Loebinger, F.l<. 44 , 65, 70, 623, 625- 6 Loh, E.C. 65 , 625
!',ilir, II. 2,11, 6!!O, 624 LohrmiLTlll, K 44,) 624 London , C. 41, 624 Loret, M. 2, 16, 621 LOllcatos, S, 2, 16, 622 Lu, M. 2, 620 Lubelsmeyer, K. 2,44 , 620, 624 Luckey, D. 2, 44, 65, 620, 623, 625 Luers, D. 44, 623 Liihrsen , W . 2, 44, 70 , 620, 624, 626 Luit, E. J . 44, 65 , 623, 6.2 5 Liike, D. 2, 44, 620, 624 Lynch, C.R. 17, 101,622, 62 7 Lynch, ILL. 2, 620 Ma, C.M. 2, 620 Ma, D.A. 2, 620 Macbeth, A. 2, 621 Macbeth, A.A. 44 , 65, 70 , 623, 625·6 Madsen , B. 2, 16, 622 Mallik , U. 44, 624 Malosse, J .-J . 2, 16, 6121 Mana, C. 44, 65, 592 , 597, 623, 625, 629-630 Man delstam , S. 18, 622 Mani , P. 2, 16, 622
Manley, D.M. 17,101 , 622, 627 Mansoulie, B. 2, 16, 622 Mantovani, G.C. 2, 16, 622 Mapelli , L. 2, J6, 622 Marciano, W.J . 595 , 629 Marini , A. 65, 624
Markiewicz ) T.W. 2) 16 ,599, 621) 630 Ma rshall, R. 2, 44 , 65, 70, 621, 623, 625- 6 Marti n, A.D. JO,) , 120, 627 Martinelli , C. ?, 622 Martinez, M. 592,597,629-30 Martyn, H.-U. 2, 44, 620, 624 Maruyama, T . 65, 6125 Maschuw, R. 2, 620
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Merkel, B. 2, 16, 622 Mermikidcs , M. 2, 16, 62£ Merola, L 599, 630 Mcrtiens, U.D. 2, 70, 520, 625
Messner, I-LL. 65, 624 :\1cyer, H. 2, H, 70, 620, 62!, 626' Meyer, H.J. 2, 41, 70, 6~O, 624, 6U6' Meyer, J. 44, 624 Meyer, O. 2, 70 , 620, 626 Meyer, O.A. 135, 624 Meyer, T. 2, 44, 6£0, 6f!3 Michalowski, S.J. 65, 62! Michelsen, U. H, 70 , 6':11-5 Mikenberg, G. 2, 620
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62:i-6 Nyt':, .J, f)5 , 625 Olx:rlack, H. 4-1, 623 Odaka, S. 11, 70, 62.?) 626 Ogg, M. 2, 620 Oldham, S.J. 70, £J'in OliYe, KA. 17, 101, 62£, 621 0lscn, J .IV!. 44, 624 Olsson, J. 2, 44, 65, 70,6£1, 6!J:J,
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Rosskamp, P. 44 , 624 Rossler, M. 2, 70, 620, 626 Itost, M. 2, 44 , 70. 620, 6£4, 6£6 Roth, F. 2, 620 Rothenberg, A. 2, 16, 621! Roussarie, A. 2, 16, 622 Ruhbia, C. 2, 16,599, 6£1 , 6:]1 Rubio, A. 65, 625
Ruhio, J. A. H, 629 Rucki, R. 101,104, 627 Riihmer, W. 2, 620
Itiisch, R. 2, 620 Rushton, f..l. 14-, 624 Rykaczcwsld, H. 2, 44, 65, 620, 623 Sack , T h. 5!)'5, 629 Sildoulet, B. 2, 16, .599,621, 6S1
Sajot, G. 2, 16, 599, 621, 631 Sillam , A. 1, J6, 619 $alicio, J. 44,6,),629,623 Salmon, C.L. 2, 6f!O Salvi, G. 2, I G, 599,621,631 Salvini, G. 2, 16,599,621, 6S1 Sander, H.G . 2, 44, 6f!O, 6!Lf SanguineHi, G . .'j90, 630 Sass,.1. 2. Hi, 599, 62f , 6,'JJ Sato, A. 2, 44 , 05, 70, 62f, 623,626 Salldr.1 1. 2, (i2fJ ,aXO S Il. I)S . • . ."" ).). " ) " ) , ""7" J... " "f' SC;H!', L\I1 . Ij~, Ii!!'; Sell fl', M . 7tJ, (i~(i S.:li3dll'f, .I. 2, Iii, fi:J:JMi!J
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6 Schll1ilH, G. 4.." 62.1 Schmitt, J. 70, 6'2(1 Sdllniti':. D. 2, H , fi!!O, (i!!4 Scnmliscr, P. 2, 62() SdllleekJoLh, U. {if'i, (i:J/; Sdmci(\n, n. 1'1, litl.'j Sch nell, W. 5fiO, 6!!'l Schroder, V. 41 , (iEY Schubert, Kn.. 17, lOl , (i.'!:J, li!!7 Schultz yo u DriJL~ig, A. 4, I;~(! Sclmlz, I. 014, 6.5, 6fNI, fi:!.5 SchiitLc, W. 44, 624 Schwinger, .1. 1, 7,511:1, fiOri, (jUI, 6£8, 691 Sciacca, G. 599, 630 Scott, W. 2,16,599,6£1, r;.'11 Scroek) R.R. 17, 101, 6'2!J, (i!!7 Sechi-Zorn, 3. 2, 44, 65, 70, fi:]IJ,
62';,626 SedgeOOef, J. 2, 620 Sette, C. 599, 630 Shah, T.P. 2, 16, 599, 621 , fiJI Shambroom, V,,I.D. 65, 6!!.( Shapira, A. 2, 620 Siebke, H. 4-1 , 6t!4 Siegrist, J. 5.')0, 621
Siegrist, J. L. 2, 16,622 Sindt, II. 44, 6£.'1 Siuf1Illl. K. 2, 44, G5, 620, 6fd:J, (i!.!.~ Sirlin, A. 601, 631 Sivers, D. 70, 626 Shapira. A. 620 Ska rd, J.A . 2, 44,70,620, BiN 5 Skard, J.A.J. 65, 625
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Skuja, A. 2, 44, 70, 620, 024, 626 Smith, D. 2, 16, 62.1 Smith, J.G . 65, 624 Smith, KM. 44, 624 Soding, P. 2, 620 Soffer, J. 599, 630 Sokolov, A.A . 539,583,627- 9 Speiser, D.R.. 11, 622 Spiro, M. 2, 16, 599 , 621, 631 Spitzer, H. 2, 44, 70 , 620, 62.{, 626 Spitzer, J. 65, 626 Steffen, P. 2,44,65 , 70, 621 , 629,
6'25- 6 Steiner, H.M. 2, 16, 622 Stella., B. 2, 44, 70 , 620, 62.{, 626 Stephens, K. 2, 65, 70,621 , 6f!5- 6 Steuer, M. 44 , 65 , 629, 625 Stimpil, G . 2, 16, 622 Stirling, W.J. 589, 629 Stocker, F. 2, 16, 622 Strauss, J. 2, 16, 599, 621, 681 St reets, J. 2, 16, 621 Strupperi ch, K. 44, 624 SU, S.·Q . 592, 629 Suda, T. 2, 621 Sumorok, K. 2, 16,599, 621, 631 Suura, H. 39, 61!H Suzuki, M. 17, 101 , 622, 627 Swartz, M. 2, 16, 622 Swider, G.M. 44, 65 , 623, 625 SZOIlSCO, F. 2, 16,599, 6121, 631 Takeda, H. 2, 44, 65, 70 , 621, 629, 626 Takeshita, T. 65, 625 Tang, H. W. 2, 44, 65 , 620, 628,625 Tang, L. G. 2, 620 Tao, C. 2, 16,599,621, 691 Tarski, J. 11,622 Taylor, T.R.. 588, 629 Teiger, J. 2, 16, 622 Ternov, I.M. 539, 583 , 627- 9 Terraoo, A .E. 70, 626 Teuchert, D. 44, 65, 62:!, 625
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Thomson, J.C. 44, 624 Tigner, M. 549, 627 Timm, U. 2, 44, 70, 620, 624, 626 Timmer, J. 2, 16,599, 6e1, 691 TimmerIJlans, J. 599, 630 Ting, S.C.C. 2, 44, 65, 620, 629, 625 Tomonaga, S. 1,7, 619 Tornqvist, N.A. 17, 101, 6'22,627 Totsu.ka., Y. 2,44,65,70 , 621 , 623, 625- 6 Tovey, S. 2, 16, 622 Trines, D. 2, 44 , 620, 624 Tripp", T.G. 17, 101,622,627 Troost, \V. 22- 3,31,41,65,68, 70, 80,84 , 105, 622-3, 62S-7 Trower, W.P. 17, 101, 6112,627 Tscheslog , E. 2, 16,599,621, 631 Tung , K. L. 2, 44, 65, 620, 6f!9, 625 Tuomioiemi, J. 2, 16,599, 621, 631 Tylka, A.J. 44,624 Van der Meer, S. 2, 16, 621 van der Waerden, B.L . 590, 629 Van Eijk, B. 2, 16, 621 van Neerven , W.L, 592, 629 Vannini, C. 599, 680 Vannucci, F. 2, 44, 65, 620, 625 van Staa, R. 2, 44 , 70 , 620, 624, 626 van Swol, R. 599, 630 Veillet, J .J. 44, 624 Velasco, J. 599, 630 Vercesi, V. 2, 16, 622 Vermaseren, J .A.M. 70, 626 Vialle, J.-P. 2, 16,599, 621 , 631 Visco, F. 599, 630 Vismara, G. 599, 691 Volkov, D. V. 587, 629 voo Goeler, E. 65 , 624 von Holtey, G. 599, 680 von Krogh , J. 2, 44 , 65, 70 , 621, 629, 625-6
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Wacker, K. 2, 6!!0 Wagman, C .S. 17, 101, 622, 6$7 Wagner, A. 2, 44, 65, i O, 621, 623, 625-6 Wagner , S. 65, 6/!5 Wagner, W. 2,4-1, 70, 620, 62", 626 Wahl, H.D. 2, 16,599,621, 631 WalJraff, W. 2,44, 620, 62" WlI.loschek, P. 2, 4-4, 70, 620, 624, 6£6
Wang, M.Q. 14, 623 WaTlg. X.R. 2, 620 Wang, V.X. 65, 625 \Vard, J .C. 588, 629 Warming, P. 2, H , OS, 70, 621, 623, 6£·5- 6 Walal1abe, Y. 2,44 ,70 , 621, 629,
626 Watkins, P. 2, 16,599, 6!lI, 6$1 Weber, G. 2, 44 , 6,), 70, 621, 62:1, 625- 6 Wedemeyer, R. 2, 44, 620, 6£4 Wei , P.S. 2, 620 Weidbe rg, A. R. 2, 16, fi$!f! Wei nberg, S. 1, 16, 619 Weinstein, It. 65. 624 Welch, C . 2, 70, 620, 626
'vVeleh, C.E. 44, 611" Wenniger, H. 44 , 70,623, 626 WenL7.cl, G. 559, 628 \Vermes, N. 2, 14,620, 62" Wt:$s, J. 587, 629 Weyl, n. 590, 629 Whi le, ),,1. 2, 41 , 65, 620, 6£3, 625 Whittaker, J.D. 65, 625 Wiik, B. H. 2, 44, 619, 62" \Villrodt., J . 70, 626 W ilson, J. 2 , 16 , 59!), {j21, 631 WilM>n, It. 2, 16, Gel Winicr, C.G. '.!, 101 , 70, (iSO, 624,
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Xi, J.P. 2, 620 Xiao, e h. 44, 624 Xic, Y.G. 2, 16,621 Xu, z. 583-9, 629 Xuc, S.T. 44, 62./ Yamada, S. 2,4-4.6.5: 70, 621, 623, 65!5-6 Yanagisawa.. C. 2, 44, 70,621,628,
626 Yang, C.N. I, 11, 619 Yll.ng, P.C. 2, 620 Yarker, S. 2, 620 Yen, W.L. 2, 621 YClUlie. n.R. 39, 623 Yokoya, K. 580- 2 , 6£8 YOgt , C .P. 17, 101, 62£, 627 Youngman , C. 2, 620 Yu, C.C. 65, 625 Yu , X.H. 2, 620 Yvert , ~. 2 ,1 6 , 599,621, 6:11 Zaccone, H. 2, '16. 622 Zachara. M. 44, 65, 62"-5 Zakrzewski, J .A. 2, 16, 622 Zdarko, R.W. 65, 625
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Zech, G. 2,44,70, 620, 624, 626 Zeller, W. 2, 16, 6£2 Zeng, Y.Q. 65 , 625 Zhang, D.-H. 588-9, 629 Zhang, N.L. 2, 65, 620, 625 Zhang, Y. 44, 623 Zhll, R. Y. 2,44,65, 620, 623, 625 Zhu, Y.C. 44, 623 Zimmerman, W. 2, 44, 70, (WO, 624 ,
626 Zobernig, G. 2, 619-20 Zorn, G.T. 2,44,65,70, 620, 6246 Zumino, B. 587, 628 Zurfluh, E. 2, 1.6, .599, 621, 691 Zweig, G. 1, 619
Subject Index Abelian gauge symmetry I aCC"A':lerator
linear 519- 50, 583-5 Beam luminosity 3, 65, 550, 581 - 5 beamstrahlung 549 85 photon energy 564- 5, 567- 8 , 581-
2 Bhabha. scattering 3, 28, 44 , 79, 122-1\ bremsstrahlung correction 2- 3 double 3, 65-78 in Qe D 70-8 in QED 65 9
gluon 3, 23-4 mass cff«.ts in 79-100 multiple 19 proce&S 2, 29 ai ngle in QeD 47 64 in QED 31-46
bunch charge distribution 555, 560 , 580-
2 correlation length M4-5, SS9, 581 c ross ing 571 , 582 length550-2,554-5 ,55960,581 mean raclim 55 1 pnrnn111tcf s .ibO :
in e+e- ...... JJ ' p-
:1 , 41, liS 5M, IiB!j r.ollilll:llrity far. t o r H9- !)S (', ()IIi Il!'tl( jlMtit:l' ;!l 28--!), 78-9, 596-
cue
7
color degree of frw.om 1- 2,9,62 ma.t rices 11 , 613-17 trM:elI of 11 , 51 , 62, 6 14-17 cross ,ec~ions 11 ·16 Oecay ra les 16 Dirac cquAlion 8, 23, 537-8, 557-
60 d is ru ption fador 551 Eled ron cla.ssical radius 551 , 554 virtual · lengt.h 554 ele<:trou-positron a.nnihilation into e+e- 3 , 28. 44 , 19 into e+e-l+ l- 597 into c+c-.., 28. 39-014,46,597 into e+e-..,.., 21 , 65, 69 , 100 into.., .., 14, 18-1 9,24-7,35 . 43 into "1"'11 31-5 into "1"f'Y'Y 65-9, 100 into Il+/c 2- 3, 44,82-3,539-
42 into p+ P-"1 28, 35--9, 45--6, 55, 79,82- 4,91-5 into 1'+P- '''(1 65 . 69, 9; - 100 inlo VV"( 592 inlo v Ii "f "! 592 into qq 3, 14-1 5 int o q'j 9 3, 29,52-5 into q q9 9 3, 29, 70- 4 into qqqq 3,70 ,77-8 into qqq'q' 3, 7,1-7 into Z'Y 595 int o 2 jets 3 into 3 jelH 3,1 8,52 5 into 4 jets 3, 70-8 ele<:troweak Lheory 1-2
S'II'.lIi(."" INIJf:X
Permi·Dirac: sta.l.isiics
J
Peyoman diag"ams 7 - 19,583 for e- +bunch--> e- ')"+bunch 553 for e*+bunch --4 e+e-e±+buf]ch 583-4 for e+e- --> e+e-", 39-41 for c+e- -> ')"") 14, 18,24-5 for e+e- --> 111 31- 2 for e+ e- -> 65-6 for e+ c- --> 1'+ 1'- 82 for e+ e- -> p+ ''-, 36, 45 - 6 for e.+e- -+ qq 14- 15 for e+e- -> qqg 52-3 fore+ e- -> '1<;99 70- 2 for e+c -> qqq'q' 75- 6 for 9 9 9 -> 2S+1L J 102- 3 for ,+bunch-> e+ e- +bunch 5834 for" -> ')", 605-6 for q q - . 9 9 48- 9 for q q' -> q q' 9 56- 7 for Z --> e+e- 16-17 for Z -> e+ e- ')" 593-4 loops in 12,598,605-11 Feynman parameters 605 Feynman rules 7-14.
,,1',
Gamma-matrices 8-9) 16 1 68 traces of 601 - 3 gauge inva.riance
15
gauge particle 1-2 geometrical optics 599 Glashow- Weinberg-Salam model 12, 16 g luino 588 gluon 1- 2 four - vertex 10- 11,47 three- vertex 10- 11,47,56,58 graviton 592- 3 gravity 592- 3 f{elicity amplitudes 4, 2 1 for e+C --> e+e- 122- 4 for e+e- -> e+c(+e- 597
ror( : I{ ~- -) c+(~ -,:l!)
"", 1:10
G, I (j(;- 78, 597 for e+c- ---t c+e-"l''f 115- 5:1 , 265- :168 for c+e - -> 7, 25- 7, 119·-20 for e+e - -> "1/')" 31 ·-4,121 ·
5) 154- 7 forc+e- -> ,)"-1/,65-9,137-· 9, 179- 93 for ,+e- -> p+ ,,- 82 ,92 , 120-
1, 540 for e+e- -> ,,+,t-,)" 37- 9,91 3,125-:10,157-66 for e+c- -> I'+P -", 97-100 ,
139-45, 193-265 for e+ e- -+ // /J "I 592 for .+e- -> v V-II 592 fo r .+e- --> qq 369-"10 for e+e- --> qqg 52-5, :170- 1 for e+e- -> qq, 371-3 for e+ e- --> q Ii 9 9 70- 4, 3738 for e+e- ..... qqqq 77,381 - 5 for e+e- --> qqq'q' 76-7,379Sl for e+e- --> Z -,595 , for 9 9 --> 9 9 403-4, 588, 592 for 9 9 o'9?_
-->
9 9 9 496- 501, 588,
for 9 9 --> 9 9 9 9 588, 590, 592 for g.g --> 9 ' So 111,519- 21 for 9Y -> 93S1 [)21- 3 for 9 9 -+ 9 1p, 524- 7 for 9 9 --> 9 3PO 527- 9 for 99 -+ 9 "PI 529- 32 for 9 9 -+ .9 3p, 532-6 for 9 9 --> 'S0 501- 2 for 99 -+ "Po 502-3 for 9 9 --> 3p, 503-4 for 99 --> q q 402-:J for 99 --> qqg 490-6 for 9 9 ..... q q 9 9 590 for 9 9 -+ q q'"f 487-90 for gg -> qqfL 589
.';/i/llf:f.''I' I NIJK\,
I le'li ,-il,}, (nml'.] ) arllplilllllt'!; for !f!f '• • 'j 1/ il' lf' ;jA!1 f"I' g-, - . ifl' -iiJl ::! (o r 91 ..... 'i'lfl "lH 7
fur [J"Y--I qq,,(-18 1 :J for "1 ..... ..,.., 607- Jl for"! "'f - . Ii q ,toO- I for .,.., ..... Ii 'I g 119-81 for i i -+ i1q,) 471- 9 for yq ..... qg 392--3 for yq - 'l1391 - 2 for U q - . q 9 9 ·139-45 ror 9 q ...... IJ 9 "f 435- 9 for gl} --+ Q1 "1430 '2 for 9 f] ..... q ISo 504- 5 fo r 9 q --I q 3PO 50.')-{j for gq _ . q 3PI 506- 7 for 99 - t q~P2 5089 for 9 Ii ...... ij!l 396- 7 forgq l'ij1:J9j- 6 rOt gq .. • li99 45965 for gij ...... qU"1·155 - 9 for gli --I fir....,· 449-32 for.qq ...... ([150 509- 10 for g"if
for
->
gq .....
q 3PO 510- 1 q3Pt 511-12
for !(ii - I -qlP1 513 14 for -,q ....... q9 :190-1 for -yq - 9i' :H!9- 90 -9094325 fll' 'Yq ..... qY1427- 9 f
fur ') q .-, Ii "Y :193- 4 fol' 'l q --t q99 452-5 fur "11 --t q91447- 9 fllr 'i ii - - t 7i'Y'i 445- 7 fur 'I 'I --+ 9 q 387- 8
for '19 ..... qqy 4]5-19 f"l" '/1/ - - qq-r 413- 15 ror 'Iii ..... 99 48 52,399- 4.00 for q'if - !J"I, 398-9 r"r f/ fj - ' ggg472 7
r' lf 1/1/ fur f. /r for ror fur fllr for for
for [or fnr
for [or
' Y.'I" tI(i~1 71 'Iii -~ !/'T" ;11;7 !) 'Iii - . 1I .1/ (; r.r;!J qii - . gl,c,'u .'J 11 I t, '1ii ..... ,t/:!/'o ,il!'. IIi qr; --I Y:!/'1 .117 tHo qlj ..... .I/!/~l .'iIN 1!1 q q -> .II 7. r,1i!J q 7j --t "'f "I :1!17 H qii ..... 7/1 tl Gr, 7 qfl ..... qq :IHK \J ((q --I q ii!1 421 ."i qq - . qlj"l ~ H) 11 qq ..... qfj /, .",i'l!)
for fir; - ,,' ii' :nw 7 {orqq ...... q''f/ .IJ~l tJ 2 for q7j -> (/' ij'.'IY ."i~n for I( q - . I/Ii' ' '( -10!J IlJ fo r q Ii --I h !j$!J
for q q' ---. q q' :IR5 Ii forqq' ..... '1q'9M·)--(ja ,~Oj) Ii fo r qq'_qq'-y tlW'i (j for Z --t e+ c 7 .')9:1 !'. (or g9Y ..... ISO }OG 10 for !}!lg ...... lSI J 12 f(.lr 999--1 11'1 11111 fo r 99g--l 3Poll~ for g!}g ..... :JP1 115
I.... IIi
forggg ..... JPlll{,i- IH
sti\te!l 4. 21-9 for (ermio:l~ 21-2, 6R for gluons 23-4 fo r photons 22 3 for quarb 55 Higgs boson 2, 9 Higgs mcch;misUl 1 higll energy limit 4
validity of 19, 27·9 inlnU'ed factor 39, 62, 592
intermediate vector boson 2
Jet 2-3 Klein·Cordon equation
556~9
S{//JJIIC'I'INIJNX
11~8
LEI' 3, 549-50 lepton 2 LHC 600 Mandelstam variables 18 mass effects 19, 28-9, 79- 100 for amplitudes 84- 100 for double bremsstrahluog 957 for e+e- -. p,+ p,--{ 82-4, 91-5 for e+e - -+ p,+p,-"{"{ 97- foO for single bremsstrahlung 80-2, 88-91 Mellin transform 569-70 metric 7 Neutrino 2, 592 non-A belian gauge symmetry 1, 11 Optical theorem 13, 599 Pauli matrices 8, 597-8 PEP 43, 65, 550 PETRA 2,44,65,550 photon 1 helicity see heLicit,y photon-photon scattering 605-11 pp collider 2-3 polarization 537-47 degrees of freedom 7 for fermions 537- 9 for gluons 542-5 for gravitons 592 for massive spin-l particles 5935 for photons 542-5,588-9 , 591 in e+e- -+ p,+ p,- 539-12 in 9 9 -+ 9 'So 545- 7 linear 543-5 sum 15, 112, 117 vector 4,22-4,35,47-8,64 phase choice of 35, 588-90 QCD 1-3,9-11,101 structure consta.nts 11
QED 1-3 , 9
qllll.rk 1···3 qUMkonia 3, 101-18,549 production 101-18 quatcrnions 597-8 RegularizaLioll 608
Schwarz inequality 569 SPEAR 550 SpIll
factor 565 , 582 opera~or 537 spinor 8, 21, 68, 537-9 product 590 t\V(}-component 590-1 SSC 600 standard model 1-2 Supor 550-1, 554-5, 560 supersymmetry.587-8 symmetry local 1 factor 12- 13, 16 synchrotro n l'adi"tion 549-50 Tevatron 3 TLC 550, 585 Wess-Zumina model 587-8 Weyl-van der Waerden formalism 590- 2 WKB-n.pproximation 559 tv-particle 1- 2 Yang-Mills theory I Z-decay 15--17, 593-5 Z-exchange 44-6, 65, 121- 1, 12730, 133-6 Z-particle 1-2, J6 Z-propagator 45
Z-IVidth 45
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