(Advanced book classics) richard p feynman photon hadron interactions westview press (1998)

In these classic lectures, Feynman analyses the theoretical questions related to electron and photon interactions at high energies. These lectures are based on a special topics course taught by Feynman at Caltech in 1971 and 1972. The material is dealt with on an advanced level and includes discussions of vector meson dominance and deep inelastic scattering. The possible consequences of the parton model are also analyzed. ONTENTS......Page 12 EDITORS FOREWARD......Page 8 SPECIAL PREFACE......Page 10 VITA......Page 11 PREFACE......Page 17 15 GENERAL THEORETICAL BACKGROUND......Page 20 First Order Coupling......Page 22 Conservation of Current......Page 23 2nd Order Coupling......Page 24 Unitarity 2nd Order......Page 25 Proof......Page 27 End of Proof......Page 29 Research Problem......Page 30 Conservation of Current......Page 31 Remark......Page 32 Isotopic Spin, Strangeness, Generalized Currents......Page 33 Conservation of Generalized Currents......Page 35 Singularities on the Light Cone......Page 38 Vacuum Expectation of Vsub(μν)(1, 2)......Page 39 esup(+) + esup() → Any Hadrons......Page 40 Note: Annoying Point......Page 45 Commutator......Page 46 Problems......Page 47 Pion Photoproduction Low Energy (0 to 2GeV)......Page 48 Problem......Page 52 Note......Page 55 The Quark Model......Page 59 Note......Page 61 Problern......Page 66 CalcuIation of Matrix Elements......Page 67 Feynman, Kislinger and Ravndal, Phys. Rev. (1971)......Page 68 References......Page 77 tChannel Exchange Phenomena......Page 88 Comments......Page 92 sChannel Resonances......Page 93 Veneziano Fomula......Page 95 Estimates of Coupling Constants......Page 96 Electron Production of Vector Mesons......Page 99 Vector Meson Dominance Model......Page 106 φ as ss......Page 111 VDM and Photon Hadron Interactions......Page 113 Diffractive Production of ρ, ω, φ......Page 119 Other Tests of VDM......Page 125 Shadowing in Nuclei......Page 126 To Summarize the Position of VDM......Page 130 Nucleon......Page 131 In Lab......Page 132 Electromagnetic Form Factors (continued)......Page 133 Pion Form Factor......Page 135 Proton Form Factor for Positive qsup(2)......Page 138 Note......Page 139 Other Photon Processes for qsup(2)

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Libmry of Congress Catalogixlgin~Pu'Et1ica;tion Data

Feynman, Richard PhiHips

Photon-hadron tnteracrrions I Richard Feynman

p cm - (Advanced book cfassics series)

Originally published : Reading, Mass : W A Benjamin, Advanced

Book Program, 2912 (Frontiers in physics)

Includes index,

1 Photon-hadron mteract-ions 2 Hadron interacdons

1 Title XI, Series

Westview Prcss 1s a Mcmhcr of the Perscus Books Group

Cover design by Suzanne Weiser

2 3 4 5 6 7 8 9 1 0

First printing, February 1998

Find us on the World Wide Web at

h t t ~ : w w w w e ~ i e l i ~ ~ r e s s c o m

Editor's Foreword

Addison-Wesley's F~ontiers in Physics series has, since 1961, made it possible for leading physicists to communicate in coherent fashion their views of recent developments in the most exciting and active fields of physics-without

monograph Indeed, throughout its nearly forty-year existence, the series has emphasized informality in both style and content, as well as pedagogical clari-

topics they treated gradually became integrated into the body of physics knowl- edge and reader interest dwindled However, this has not proven to be the case

community has urged us to extend their life span

will keep in print those volumes in Frontiers in Physics or its sister series, Lecturre Notes ~ n d Supipkments in Physics, that continue to provide a unique account of

a topic of lasting interest And through a sizable printing, these cIassics will he made available at a comparatively modest cost to the reader

tures, the presentation in this work reflects his deep physical insight, the fresh-

pedagogical wizardq As a result, this volume will always be of fundamental

importance to anyone interested in understanding the development of quantum

hadron-hadron interactions at high energies

David Pines Urbana, Illinois

Preface

Many of us wem first in~dtlceci to the concepts of the p m n model from this b k At

p d c l e was s u p p s d tn, be a campsite of evey other pdcle The ideas and conmpts

in &is book h v e helpd pave &e way far our undersmding of the constituent name of

trra&ons which evennially led ts the Quantum Chmnndynamic (WD) &wq of qu&s and gluo'ns, & is m e of most of Feynmank b b s , the maximum bnefit is o b ~ n e d if

one has previowly studied the subjeer in some d e ~ f Fcynmank unique: p r s p t i v e can br; be a p ~ w i a t d by rwdws with a mlid b ~ k g o m d in the subject

Although this book is dmost l8 yms otd, it still is an excellent referace, It

a m a s on the raomntenkd rmding fist of &I thecmmtwD b k s , Tke bkprovidcs

a g d undersmding of ltfte m d d from rheman who invenkd it In the "pre-QGW

or "'naive" "on m d e i tfie constituents within ha&ons were assume8 u> b hunded in the transvers dlr~tion, The pmbability of finding a pmon wifftin a high momentm

ha&on wi& a lmge msverrze momentum was assumed to fd1 like a Gaussian or an expnential W D ells us that this is not exactfy &rue and gives a p w e r law fall-off in the

transverg momentum, Beeause of this, many ""nve" "on mode8 e x p t a ~ o n s are

mabifid (in an inzportant way) by Xogarlrhmic faem, Feynmru? used t;s laugh when his pmon mdeI was refened to as "n&ve,'" and he wollld say, ""At l m t X got it ~ g h t up to

lag;&&rns." We all miss Feynman v e q much a d it is through books like this hat his

id= live on,

Vita

Richard P Feynmn

Born in 191 8 in Brooklyn, Richard P, Feyrrmm received his %,D from Princem in 1942,

&spite his youth, he played an imwmat part in the mnhattan Reject at Las AImw

World W N 11 Suwuenrfy, he taught at Gomen and at the WiEsmia ins tit^

of Twbnolqy, In 19Cj5 he r e c d v d the Nob1 Rize in Physics, along witb Sin-Imm

Tanzmaga and Jutian Schwinger* far his wark in qwntum elm@dyamies

Dr F e ~ m won his NakI Prize for suaessfufly resolving pmblents wirh the

Wry of q m t u m elee&&ynmics, He aka e r a d a m e m a G c d

fm&enral work in the of W& in~mcGons such

man play& a key mk in the devefopment of q w k theory by puwg

made1 of high energy pmmn callisian prmsses, Beyond these achievements, Dr Rynman introducd bsic new cmputa~onal Whniqws and nob~ons intophysics, above all, the ubiquiaus Feynmn S which,

perfiaps mare than my other famdim in raent scientific hisw, have thl: way

in whkh bak physiGaX qrmgses m mnceptualizd and calcufaM,

was a a m a W l y effwtive &=-re Of alI his nmerous &wads, he

was ud of the & m ~ d M&l for Tmhlng which he wan in f 972 T k

Feynman Lechrres on Physics, originally published in 1963 were described

in Scientiftc Amesican as "tough, but nourishing and full of flavor ARer 25

guide for -hers and for the it>est d ibeginning students." fn order to

un&rstan&g of physics among the lay public, Ilr: Feynman wrote The

Physicgl h w & &.E9,: Tbre Strer~ge T k o v QfLighf and mat^ fir:

numbr of advac& publieations that have k o m e class& refeerences an

Riehard Feynmm died on F e b r u q 15, r988

Editor's Foreward

Special Preface

Vita

Preface

Fast Order Coupling

8-10 QUARK MODEL OF RESONANCES

The Qu& Mdel

p%udascaiat h/ifaon Phompxduetion Nigher Energies

References

13-14 E-CHANNEL EXCHANGE PHENOMENA

tehannel Excbnge Pknomena

Comments

S-Chmnef Resomees

Estimaks af Coupling Cartsbnts

DQMIINAMCE HYPOTHESIS

Roprties of Veemz Memns

Elecmn RMuetion of Vecmr Mmarts

To Sunnmae the Position of VDM

Elecmm;lgnedc Fom Factors

Nuelmn

In b b

Pion Fom F=&r

b t o n Fom h;ac&r for Positive q2

Note

DEEP mELASTXC REGION

Xnelatic Eicxmn Nuclean SeaWng

mwv of tfie Inelastic Elec&on b m n $catering

Future Ta& of Chagd

b p Inelastic Scszt&~ng with Spin

Angular Momentum in P m n Wave Fmcdom

r Expriments Testing P m n Idea (mlQ

p + p - v+v- + Anything

EEeemn P& RducGon of

38 LIGHT CONE ALGEBRA

Efe~tFQmagne~c Self Energy

C o ~ g h m Fomu1a

Exprasion for Sdf Ewrgy in T e r n s of W Only

O&cg Elw&om@e~c Energia, Qumk Mdel

E l e c m m p e ~ c Self Mss, Q m k Mdel (continued)

I = 2 2 s Differe

F d e r Cammm& on Elm agnetie W s Differences

CornpenEffet ~ p + y p OX Y ~ I + ~n

C o m p ~ n Effmt far Very SmaIE Q , V

Forwad Gornpton Scattp;ring from Non-Relarivistic

Scbwdinger Eqw~an

Preface

The most advanced corn% in graduatl: theoretid physics at Caltmh is "*id Togih

h T b r e ~ c a l Physics," E s h yesu the professor &mm the &pie with which he will deal,

This y w (1971-73, having just come back from the 1991 Inemahonal Symposium on

Elwmn auld Pho~on I n ~ r ~ l i o n s at High Energies, held at Come11 Univemity, my own in&resr in the subjst was: mud, md f chase to malyu: the v&ous theoreticat questions

~ k d ta hat eanference, The Eeetms aerns1ves &me sa exmsive &at the deeision

waxr made to put &em in@ W k fom, with the tfioughr t h s other p p l e might atso be

inmesM Thus, the reps of the Comelt conference should be mnsidered W a compmian volume to these lectme notes The references given here are far from

comple@, but a full list of references is givert in the R w d i n g s of rhe Symposium, publish4 by the tory of Nuctw Stdies, Comdl tlnivenity, Jan

The m & ~ d is d 4 t with on an advanced level; far ins-=, kmwEedge of the theory of hadron-hadran interactions is assumed I have tried to analyze in detail where

we stand theoretically today The ueament is somewhat uneven; for example, I should

have l&& trr study the &wry of the decay of the in more detail than I was able to do,

On the o&er hand, there are long discussions of vector m a o n dominmce and af inelastic scattering The possible consequences of the parton model are fully discussed

Time did not pernit me to complete the original plan which was to include the

Wry of weak interaction currenrs which are so closely related to elecmmagnetic cments

Many thanks must g o t Muro Cisneros whoedited, corrected,and extended the

l e c ~ e s from my class now Wilhout his effort, this book would not have k n possible

X afso wish to &ank Ms Helen Tuck for typing the leemre notes

Photon-Hadron

Interactions

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Theoretica Background

Lecture l

One very powerful way of experLmentally i n v e s t i g a t i n g t h e s t r o n g l y

i n t e r a c t i n g p a r t i c l e s (hadraas) fs t a look a t the@, t o probe them with a Imm p a r t i c l e ; i n p a r t i c u l a r t h e photon (no o t h e r i s E u x m a s w e l l ) This per&ta a much f i n e r c o n t r o l of v a r i a b l e s , and probably decreases t h e

t h e o r e t i c a l c a p f e x i t y o f trhts i n t e r a c t i o n s , F"or example Sn an ordinary hadron-hadron coLlSsion l i k e np -, s p we a r e h i t t i n g two unknoms t o g e t h e r ,

2

and f u r t h e r , we can only vary t h e energy, we cannot vary t h e (3 of t h e pion which mast be mw2 In f a c c a "pion f a r off its aass s h e l l " may be a

m a n i n g l e s s - o r a t l e a s t h i & f y conrplicated i d e a , C?n t h e o t h e r hand in

y + p "P p "t n we knaw t h e y Is s i n g l e and d e f i n i t e , and we can vary t h e cgL

of t h e y by using v i r t u a l y % svi, for e x m p l e , e l e c t r o n s c a t t e r i n g

We a r e a a s s d n e ; t h a t we do knaw t h e photon, QED has been checked s o

c l o s e l y t h a t we know t h a t i f t h e p h o t m propagator were off by a f a c t o r of

t h e form (1 - qZ/h2)-1 then h exceeds 4 o r 5 &V The w l i t u d e s a r e k n w n

t o about 5% for q a s high a s ( 1 &V) For t h e r e s t of t h i s course we

s h a l l a a s w QED is e x a c t There Is a l r e a d y evidence, a s we s h a l l sear, t h a t

i n v i r t u a l photon-hadmn c o l l i s i ons t h e photon a c t s nomaf Ly (i .e , obeys QED e x p e c t a t i o n s ) up t o A of 6 t o B EeV,

At any r a t e we s h a l l suppose QED exact - where we m a n by QED t h e standard

i n t e r a c t i o n theory f o r e l e e t r o n e , muGns and photons Exact, b u t i n c o q l e t e -

f o r hadrons a r e charged and i n t e r a c t a l s o with t h e QED system He d i s c u s s

f i r s t : how we s h a l l a s s w e we can d e s c r i b e t h i s i n t e r a c t i o n

Since e L is s m a l l i t is natural, t o d e s c r i b e t h e i r i n t e r a c t i o n i n a s e r i e s

of o r d e r s i n e , One photon exchange, two photon exchange, e t c ( f t might

be thought t h a t t o d e s c r i b e t h i s coupling we s h a l l have t o have so= d e t a i l e d dynamical theory of t h e hadrons - u l t i m t e l y , of course, yes - b u t some

t h i n g s can be s a i d i n general r e s t r i c t i n g t h e m t x i x elements whatever t h e underlying hadran dynaaics - and i t is t h e s e r e s t r i c t i o n s we seek i n t h i s

l e c t u r e )

Tfie no coupling ease p r e s e n t s no problem The f a c t o r g i v i n g t h e m p l i t u d e

t h a t a hadron system goes from an incoming s t a t e In, in>, t o an outgoing

Tfie Beate In, Ln> m a n s a s t a t e tirt-ilch f a r i n t h e p a s t is a s p p t a t i c a l l y

f r e e s t a b l e hadrons (otilble i n s t r o n g i n t e r a c t i o n s only, e.g no i s "stable'f) described by momenta, and h e l i c i t i e s , a l l contained via t h e index n , The s t a t e

General Theoretical Bmkground

The g e n e r a l coupling of e l e c t r o n s and hadroas is represented by t h e

The electron-photon system goes from s t a t e N t o M, t h e hadtons from n i n t o

m o u t , liJe suppose t h e only i n t e r a c t i o n p o s s i b l e is by t h e exchange of a

p h a t m - and t h i s phaton i s c h a r a c t e r i z e d by a p o l a r i z a t i o a p, mamentm q:

That is t o say (supposing we could memure t h e amplitllde) we? d e f i n e i n a given elrpertnent t h e q u a n t i t y

This is dane by renoving f r m t h e measured m p l i t u d e t h e k n a m (by QED theory)

f a c t o r s

I f i s then our f i r s t s u p p o s i t i o n t h a t t h i s q u a n t i t y 2 ( depends

ii

only on t h e s t a t e s m, n of t h e hadron system m d only t h e v i r t u a l =meaturn

q and p o l a r i z a t i o n of t h e v i r t u a l photon, 3 , depends i n no way an how

t h e photon was made (eg whether by v ' s o r e l e c t r a n 8 o r on t h e angles and

e n e r g i e s of t h e e l e c t r o n f o r f i x e d q and phaton p o l a r i z a t i o n )

This i s a s t r o n g assumption It has been v e r i f i e d most completely for

t h e case of proton form f a c t o r masurements, b u t is o f t e n a s s w d i n checking equipment, comparing r e s u l t s fro= one lab t o another e t c We assume i t

We emphasize then t h a t (q) is an e x g e r i a e n t a l l y defined q w a t i t y -

d e f i n a b l e i n p r i n c i p l e f o r all q

Me f i n d i t convtmient t o d e f i n e a new matrix J defined i n a non-mixed r e p r e s e n t a t i o n a s

ay(q) Ju (q) d q + higher order

The l comes f r m t h e zero order ( h e r e we saw the S matrix i s r e a l l y t h e

u n i t faaerix i n an xed representation)

Unitarity requires t o f i r s t order (d - a(-q)* transpose)

o r , @Lam a (q) is a r b i t r a r y , = J J Le h a m i t i a n , Since all q are

h a d r a w , o r e l s e note the following discmeion

It is n o t , s t r % c t l y , true t h a t from Bq (1.5) can be c m p l e t e l y obtained from expeziment That is because a (q) is not cmpletelry a r b i t r a v

P

When i t come f r w the usual diagram r u l e s i t always s a t i s f i e s qUap(q) - 0

Thus one component of a (the one i n direction $1 is always miseing (unless

General Thearefical Background 7

q2 = 0) and thus one compment of J t h e component i n t h e d i r e e r i o n q is

&@sing, We f i n i s h t h e d e f l n i t i o r t of JP by ehoosing q J = 0

I.l U This we do i n t h e following way F i r s t , f o r q2 m 0, a f r e e proton of

p o l a r i z a t i o n e t h e coupling is- e J (q) - b u t f o r a f r e e photon e 18 undefined -

by an m d e r l y i n g f i e l d theory then ( l ) , = 0 i f 1 and 2 a r e space

l i k e aeparated (spbofizecf by X1 @ 2 1 you wish you may a s s m e t h i s -

but i t i s very i n t e r e s t i n g t h a t we can prove i t from our a s s a p t i o n t h a t t h e

s t r e n g system i n t e r a c t s with QED ( s u b j e c t t o s y s t e a a t i c e r r o r s i n proof due

t o t a c i t assumptions This w y n o t be an i a p o r t a n t p o i n t b u t i t i s f n t e r e s e i n g

s o X will waste your clme by proving i t )

This i e v i a dfagrams of t h e type whose m p l i t u d e depends on a computable

f a c t o r from t h e leptons" times a matrix elelnent depending on t h e two a o m n t a and p o l a r i z a t i o n s of t h e N o v i r t u a l photons -1/2 VUv (g1 q 2 ) A8 defined

t h i s is s m e t r i c a f - i n ql +-, q2, u c-t v, f o r Bose statistics a m g photons does n o t p e M t us t o d i s t i n g u i s h t h e phatons s o no o t h e r f u n c t i o n can b e

experimentally defined, Using an undxed r e p r e s e n t a t i o a , and coordinate

space ( v i a double Fourier t r a n s f o m ) we can r e p r e s e n t t h i s m p l i t u d e as

+ - 4 "

* ~ n p a r t I of t h i s course ' ~ e p t o n s f t w i l l mean s-, e , ,

and c m , a s shorn i n QED theory, be g e n e r a l l y w r i t t e n a s t h e m a t r i x element (batwean i n and o u t photon-lepton s t a t e s ) of t h e tione ordered product of t h e

o p e r a t o r a y ( l ) and a (2) (symbolized Since a r b i t r a q a(l)a(Z) can

IJ

be made V is a q e r i m n t a l i l y defined,

Ct v

I wish now t o prove a n m b e r o f t h i n g s , a o i t w i l l be m z e convenient

eo r e a t r i e t t h e i n t e g r a l t o t l > t 2 and w r i t e t h i s and t h e f i r s t two o r d e r s a s

Evidently one can w r i t e an e n t i r e a a r i e s af f u n c t i o n s t o e v e r i n e r e a s i n g order

+ This gives t h e r e s t r i c t i a n f u s i n g a a )

+ obtained by w r i t i n g T T = l and expanding t o 2nd order The 2nd i n t e g r a l

(L@ over a l l tl, tq X t can be s p l i t i n t o a p a r t t l t2 and a p a r t t2 c tl

General Theoretical Backgroltnd

w r i t e

t o g e t

Now t h e f i r e t f a c t o r must vanish, f o r we could t a k e t h e case t h a t a(21, & ( l )

c o m a r e f i r s t ( f a r example one photon from art, e l e c t r o n , another f r m a muan i n

I w e a t o r d e r ) hence we s u r e l y always ~ u s t have

d e t e m i n i n g t h e r e a l p a r t a f V (1, 2 ) In a d d i t i o n I n g e n e r a l we mm& have

P V (caking a d j o i n t of l a s t term)

The c m u t a t o r i s z e r o o u t s f d e trhe l i g h t cone, Xnaitrfe I b e l i e v e we can mke

i t a r b i t r a w (although s o w l i t t l e f u r t h e r study of epecia2 c a s e s i a necessary

t a v a r i f y t h i s ) hence we deduce

v y y ( l p 2) m J p ( l I Jy(2) (2 99)

1 is i n foetrard light cone of 2

We have alnnost proved Eq (2,5) but n o t f o r a y tl > t2, anly f a r tl

i n s i d e t h e l i g h t cone of 2 The d i f f e r e n c e is very i w o r t a n r : because

Eq, (2 , S ) , t o be r e l a t i v i s t i c a l l y i n v a r i a n t r e q u i r e s

o u t s i d e t h e light: cone, E q (2 ,S) a l s o i a a a t u r a l if hadirons coma f r w any

rurderlying f i e l d theory, f o r then our p i c t u r e o f coupling i f Cl > t p can

Lep t a n s Wadrons

- c u t

be c u t a t a t between t2 and t2; t h e f i r s t coupling is J ( l ) and t h e aecoad

is J,(2), s o we g e t the product, But one be averse t o a e s d n g t h a t

s t r o n g i n t e r a c t i o n s can be deser%bed by a c m p l e t e s e t of s t a t e s (and t h a t t h e

complete @ e t can be trrlren as [ R , i n > ) a t any a r b i t r a r y tim Nevertheless

if we continue our study of t h e r a q u i r w a t e r f o r consistency with QED t o 4th order we can do i t ,

Proof

P

% i s point not: important, but we do include the proof f o r completeness,

If (2.5) were r i g h t then 2 would be

i t is eome polyoomial in ay ( l ) , (av(

(Rant naw on we o&t t h e p o l a r i z a t i o n $adices - they always go i n an obvious,

way with the p o s i t i o n i n d i c e s ) ?he f i r s t order is made t o agree with Jir(l)

80 i n gemrixl w e can w r i t e CU(12) represents t h e devlatlon of V(l.2) fro= J(1) J I Z ) ) f o r tr t 2 )

NW &D f o d n g T T t 5 check u n i t a r i t y the e r p - % h a f a c t o r s go out TO 2nd and 41;h order i n a t h e U ( 1 2 3) t e r n does not e n t e r To second order we have

s o we conclude U i s hennitian U e v c r v h e r e , and U ( 1 2) O i f

aa before

NW I n 4th order we have

General Theoretical Background I I

Uf12) U(34) a(1) a(2) a (3) a(4) : &CD = ABCD

U(13) V(24) & ( l ) a ( 3 ) a(2) a ( 4 ) : ACBD AF, B)D + ABCD

U(141 U(23) a ( l ) a(4) a(2) a(3) : ADSC A C ~ , q C + AB@, + ABCD

U(23) U(14) a ( 2 ) a ( 3 ) a(1) a(4) : BCAD = BE, A ~ D + p, ~ C + DABCD

U(24) U(l3) a ( 2 ) a ( 4 ) a ( l ) a(3) BDAC B@, 4~ f E, ~ D + CABED, 4 + ABCD

U(34) U(12) s(3) a ( 4 ) a ( l ) a ( 2 ) CDAB - C[D, A]B + E, ~ D + BACE B +

+ A @ , B ~ D + ABCD (2.12) and t h e f f r s c term i s

Now we g e t mny obvious r e l a t i o n s For example t h e c o a f f f e i e n t of ABCD mwt

be zero (take ease a l l 4 p o t e n t i a l s c o m u t e ) , We b e l t e v e t h a t t h e v e c t o r

p o t e n t g a l a is an a r b i t r a r y f u n c t t o n of space and time We can t h e r e f o r e

chooee It t o be d i f f e r e n t fro= zero only I n f o u r s m a l l regions of space time around t h e p o i n t s 1, 2 , 3 and 4; c a l l t h e s e regions ol, a2, 0 3 , 0 4 For

o a r s p e c i a l i n t e r e s t h e r e t a k e t h e case t h a t t h e v a r i a b l e s have t h e follawtn-ng

l f ght cone propertgee

CB* D] - 0 - [D,

Only D, C] and [A, g $ 0

1 is outside the l i g h t cone of 2, 3 is outside the l i g h t cone of 4 Omitt5ng

t h e AI3CD t e r n which we noted was indepandently zero and c o l l e c t i n g what 18

l e f t i n this caee we g e t

Only t h e thArd term need t o be turned erovnd t o E, ~ C + BE, g E, g then a11 t h e term a r e cwf f i e i e n t s of F, AD o r CB B, 4 and muat a l l vanish But v l t i m a r e l y we a r e l e f t with U(24) U(13) E, AI] [B, o r f t n a l l y

$&ace the c o m a t a t o r s a r e s u f f i c i e n t l y general, X t h i n k we can crsnclude t h e

i n t e g r a l w i l l be zero only i f the integrand f a and U(13) a O even i f 1 is

o u t s i d e t h e l i g h t cone of 3,

End of Proof

We can t h e r e f o r e conclude

I n the proofs of (2.9) and (2.16) we have assuned t h a t a(il t l ) and

t l ) a ( f 2 , t2g a r e s u f f i c i e n t l y general functions of 1 and 2 (when

2 i s i n t h e l i g h t cone of 1 ) Me b e l i e v e t h i s t o be true, X t i s l e f t t o those i n t e r e s t e d in Gnare rigorouhl proof@ t o v e r i f y thSe, f o r example by

t r y i n g t o c o n s t r u c t a b a s i s ,

Lecture 3

We found i n the previous l e c t u r e

Vliv(l, 2) = Jv(2) f o r tl t 2

f u r t h e m r e

General Tlzeorerical Background

This means t h e a r l g i n a l s m e t r i c VIIV(l, 22) can b e v r l t t e n

The l a s t terns corns because t h e r e could be a 6 ( t l - t 2 ) t e r n , o r by r e l a t i v i t y

4

a 6 (1-2) term o r g r a d i e n t s thereof leading t o j u s t a c o n s t m t : o r polynomial

under Fourller t r a n a f o m ,

These s e a ~ j l l t e w would m a n t h a t t h e a b s t r a c t f a m f o r T would be l s k e

thus adding a l o c a l term a t one space time p o i n t , but second o r d e r i n a ( l )

a s appears i n QED f o r the i n t e r a c t i o n with a s c a l a r p a r t i c l e , f o r e x a m l e

Of course, i n s t e a d i t could c o n t a i n g r a d i e n t s , a s F (1) F ( l ) f o r exaaple,

v a r i a b l e s , Such a form is of course, a l s o , t h e i m e d i a t e result: of

supposing a l o c a l f i e l d theory f o r hadrons

Research Problem

% a t experiments could b e s t e s t a b l i s h e x i s t e n c e o r non-existence

of s e a g u l l s ?

I n QED f o r s p i n 1 / 2 t h e r e i s no s e a g u l l , f o r s p i n 0 t h e r e l a (but

with M e m r Duffin matrices t h e r e is not: - r e s o l v e this!) Since a l l

q u a n t i t i e s a r e defined by experiment t h e r e a l t t y of suck s e a g u l l s f a r

14 Photon-Hadron Interactions

eome s p e c i a l technical d i f f i c u l t i e s comfng from the highly divergent nature

of some of these expressione eo mathematical r i g o r requires a l i t t l e more

a t t e n t i o n I n p r a c t i c e they give trouble only i n the vacuum expectation

(of, f o r example V ) and i n no other problem and eo they can be avoided beet

lJ v

by disregarding them (They have been analyzed by Schwinger, and a r e c a l l e d

t r i v i a l Schwinger t e r m )

Conrervation of Current

We suppose now t h a t current i e conserved i n the more conventional sense

t h a t we w i l l take i t t o be t r u e t h a t quantum electrodynamics cannot determine

a (1) completely, but a gradient a (1) + V ~ ( 1 ) ( ~ ( 1 ) i e an a r b i t r a r y function,

J,, (1) Jv(2) 0(tl-t2) J,,(l) Jv(2) + 0(titl) J,, (2) J,,(l)

C I T

We have a l e o t o d i f f e r e n t i a t e the 0 with respect t o Thw when = 0,

General Theoreticat Background

appear on the 6 (l, 2 ) , b u t t h e poSnt we want t o make i s Chat the e q m k time

c o m u t a t o r of charge d e n s i t y and c u r r e n t l s d e t e m i n e d by serngulls, I n

p a r t i c u l a r , i f as may people (@,g, e l l - a n a l have sauggestcitd s e a g u l l s vanish

(by analogy t o QED where t h e coupling i s purely $A y $ d ~ CO B s p l n I f 2 EIald

tr li

t h e r e a r e no s e a g u l l diagr-) we would have

w e have no d i r e c t t e e t of t h i a y e t kltbough we do have t e s t s of V V (1, 2) 0)

bt YV

b m r k

Sehwiager has painted o u t t h a t ( 3 8 ) is i q o a e i b l e Becawc: i f Jo P

and Jv - 1, 2 , 3 i s v r i t t e n on t h e v e c t o r 3 i t says

Therefore takf ng divergence

because p - 3 g But 8 is t h e operaroz Hp-pH where X i s t h e

U & l t o n i a n of t h e aystem ( a s s a n g t h e r e is one-generally t h e energy of

s t a t e o p e r a t o r ) OK

Naw t h e v s e u w e x p a c t a t i o n v a l w , b a t t h e energy o f the i n t e = d i a t e

e n t i r e l y aesociated wfth the vacuurn problem mid could be reawed We can indeed

have the analol3y of QED; %Q seagullat a d have equation ( 3 , 8 ) s a t i s f i e d f o r ever-y problem except the vacurn, The precise s t a t e m n t is that (3.8) holds

i f fronr the c m u t s t o r you subtract i t e vacurn expectation value t i m s r unit

F r m the very low race of K' -* no + y e r d + y + n (alt;bou& C@ + n + y #.e East enaugfi) we cmclude Cltat weak interactions a r e invalved here, Thus we

think J has zero nzatrix elements bemean etaces of different strangeness,

h m g a e t s of d i f f e r e n t isoepin we can descrlbe the r e s u l t by saying J has

p a r t s of X spin f n 0 (isoscalar), f: I (isovector), 1 2 (i~aoteaaor) etc

and w e appropriate Glt?bsch Gordtvr coefficianta t o r e l a t e ampxitudes amng diffetctnt multiplets Slnee J daee riot change charge i t nnust only LnvoLve

the 3rd emponeat, i f it: i e isovector for e x a ~ p l e , The f a c t that proton a d

General 'rheorerieal Background

neutron have charges + l, O already shows t h a t J Is not pure i s o s c a l a r

independent of isoapln, nor pure isovectox (where t h e charges would have t o be opposite) but contains a l i n e a r co&inatIon of these two, No expsrixnent s e e m

t o r e q u i r e X 2 , but I do not know him p r e c i s e l y o r extensively t h i s ha5 been tested Recently so= evidence w a s claimed f o r the need of an 5 2 coqorxent

i n comparing y p + n ; ' ~ and y N i %-p a t energies near the d resonance but i t

appears that c o r r e c t i a a s f o r d a u t e r m s t r u c t u r e ( f o r The y neutron r a t e is

i n f e r r e d fram y D data) *re i n c o r r e c t l y analyzed

Host t h e o r i s t s today a s a a e h1 .p O o r &X: P L only f o r J , (This i s

1.1

evidently s fumdamntaf question because i t t e l l s s o m t h i n g of haw 3 is

" u l t t m t e l y '"coupled ; f o r f u r t h e r s t r o n g i n t e r a c t i o n s c m s e r e n g X-spin

cannot a l t e r & i s rule-+@ see ""ln'>tbroulgtr t h e strong d p m i c couplling i n

t h i s respect a t Least because the strong coupling consemas t h i s Z s p i n

eharactar , )

Having a v a i l a b l e a ~ a t r i x elements cmlJ in> f o r a v a r i e t y of s t a t e s n (and m)

v

a l l s f the s a m X-@pin m l t l p l e t pe&t;s an@ by appropriate l i n e a r c m b i a a t i o n

always t o i s o l a t e t h e pieces due t o t h e i s o s c a l a r and i s a v e c t o r p a r t s e p a r a t e l y

5

Thus we can define matrix e l e m n t a and therefore operators f a r J (q) nnd J " ( ~ )

But f o r the vector we could a l s o c a l c u l a t e (via Cl&sch Cordm c c e f f i c i e n t s )

m t r i x elements b e t ~ e o ! ~ s p e c i f i c s t a t e s of other cmponents of the vector current jW(g) o r (with i s o s p i n + l o r -1) I n t h i s way new kinds of

U;

currencsare defiaable

"Illis would J u s t be an e x e r c i s e i n Cleibech Clcrdan c o e f f i c i e n t s , but we think so= of these currants a r e p h y e i c a w important a l s o , Ffe think t h e current ~''(~1 is the nonatrangenaes e h a ~ g i n g nonparity v i o l a t i n g parr of

v

W& i n t e r a c t i o n (an a s e w t i o n k n m ss WC) This leads t o a suggestian

by which these extensions gf current: a r e w e f u l i n a p w e r f u l t h e o r e t i c a l w a y

e i t h e r t h a t these currents c a m from soae apeastor i n a f i e l d theory underlying

hadrons, o r t h a t hadrona a r e suck t h a t weak perturbation f i e l d s could be coupled (we w i l l w e the l a t t e r hypothesis), We can expect, f o r any two points and components of current ~ ~ ( 1 ) and ~ t ( 2 ) t h a t they c o m t e i f 1, 2 a r e apace

P

Like separa ted

llhis La a new a r s r t v t i o n , We a r e trying to Lnduce new laws md restrictLonrs

on J and the hadron system We know - 0 f o r 1 x 1 where J,

V our e l e c t r o w g n e t i c current, is an Lsovector and isoacalar, Ustag isospin only how f a r is i t possible t o go t o prove say t h a t the tsovector p a r t , o r the

+ e t c c m a t e l Xn t h e realm of isospia what we ass- here 1s t h a t

the apace-like carnutation law i s t r u e not only f o r t h e t o t a l curreat

J~ + Jv3 but ale0 f o r the i s o s c a l a r p a r t alana with i t s e l f , the isoveeror

p a r t alone with i t s e l f , the i s o s c a l a r part: with the iaovector p a r t and

generalizations fox the Leovector p a r t with d i f f e r e n t I spin compment

d i r e c t i o a s

We caa a l s o have the s c a t t e r i n g of an 3larogPnary a-type vector "photont'

t o a b-type photon governed by

That is, extend the concept of vector potential a (1) t o contain mather

the case i f I spin which we k n w is exactly conserned Consider the s c a t t e r i n g

of an I - l p a r t i c l e v i a JV' ?he charge of the f i n a l s t a t e is one higher

U

than the i n i t i a l , s o

General Theoretical Ba~kground

shows s must be J y (g) Hence t h e e q u a l time commutation r e l a t i o n r e s u l t s ,

( t h i s a s s m e s t h e r e a r e no s p e c i a l t e r n i n 8(x1, E2) &ich would i n t e g r a t e

o u t , a q u e s t i o n we s h a l l s e e r e l a t e d t o s e a g u l b again)

Eqtxatlon (4.6) and its g e n e r a l i z a t i m t o t h e much wider group SU3 x SU3

arc! Gelf-&nnfs equal tim comulirtor r e l a t i o n s , They r e p r e s e n t t h e f i r s t guessed d p m i c a l property of hadrons t h a t is nor simply a consequence of

r e l a t i v i s t i c quantum mechanics g e a e r a l principles,

We c m a l s o d e s c r i b e t h i s from t h e p o i n t of view of a property of t h e

t h e s m of t h e f P r s t two, t h e l a s t one is (38~~flly cmputcible and is c l e a r l y a

f i r s t o r d e r hadronic nracrix e l e a e n t of a c u r r e n t J i n thLs cage J~ i t s e l f

AS we shall, show i n a m m n t ( 4 7 ) is equivalent CO (4.6) i f no s e a g u l l s e x i s t ;

i f they do e x i s t ( 4 7 ) 18 t r u e b u t (S 6 ) has t o be madifled (4.7) is t h e

m e s e m y be obtained from n o t i n g t h a t the g e n e r a l i z a t i o n of a gauge t r a n a f o m t i o n

a' -r a + P X is, i n the group aa -t aa + qVXa + (X X ay)' Supposing T[d is

unchanged by such a t r a n s f o m t i o n we Efnd

a s a f u n c t i o n a l relation-or c a l l f n g 6T/Sa(l) t h e f u n c t i o n a l d e r i v a t i v e one

e a s i 1 y deduces

f o r afll X ( l ) , s o i n t e g r a t i n g by p a r t s we have

kthen T is w r i t t e n i n a p w e r s e r i e s i n a (4.2) and is s u b s t i t u t e d i n t o (4.111, zero and f i r s t o r d e r t e r n g i v e (4.9) and (4.10)

Since idsospin is e x a c t l y cclinserved (4.91, (4.16) mwt be e x a c t l y s a t i s f i e d when r e s t r i c t e d t a trhe t h r e e s p i n components of v e c t o r c u r r e n t s , % a t of

S U which i a only ""almost" s a t i s f i e d ? G11-Mnn h a s proposed t h a t $U3, although

3

nor: e x a c t l y s a t i s f i e d f o r t h e e n t i r e hadronic system may be more and w x e

accurately s a t i s f i e d as s h o r t e r s h o r t e r space-tim i n t e r n a l s a r e invalved, That is how i t would behave i f an wderlying f i e l d theory had propagator

gradient terms s a t i s f y i n g SU3, but mass-like terms v i o l a t i n g SUy (E g 9 q + + Gmq where m i s a nonSU3 i n v a r i a n t matrix, q a r e quark operators) If J'

is a strangeaeas changing c u r r e a t , h a d n g f o r e-plf? a w t r i x e l e w n t beWeen

A, N then v .Ja(l) $ O b e c w s e A, have d i f f e r e n t m a s e s (If A , N a t r e e t

we f i n d

(eeagufl t e r n have been i m o r e d )

These r e l a t f o n s a r e 05 very g r e a t I n t e r e s t because they a r e a o n l i a e a r

requiring absolute s c a l e s , Thus ( i f valid) they can serve as supplying

absolute s c a l e d e f i n i t i o n s t o t h e currents s o t h a t t h e r u l e t h a t weak

i n t e r a c t l m of hadrons i s V + A ( r a t h e r than V - .7A) is d e f l n d l e m d

therefore t e s t a b l e , Thls p a r t i c u l a r t e s t has bean m d e by Adler and

Weisberger uslng fCAC t o take t h e pion coupling ah, a mmure? of t h e divergence

of the a x i a l current We d i s c w s how @-**at m r e d i r e c t t e s t s c m be made

by neutrino scattering l a t e r on i n t h e course ( i n P a r t XI),

zero i f 1 is outeide the l f g h t cone of

2 , nonzero i n s i d e What kind of s i n g u l a r i t y does i t have j u s t passing through, o r n e a r t h e l i & t cone, For f r e e f i e l d s of any noass i n f i e l d

2 theory t h e c m u t a t o r has a 6(sI2) type s i n g u l a r i t y a c r o s s t h e cone There

i 8 experimental evidence from i a e X m t i c s c a t t e r i n g experilnents of e l e c t r o n s

General Theoretical Backgragnd 23

We note there is no mass r e a o m l i z a t i o n of the photon, the pole is s t i l l a t

so 4ne i b measures the vacuum polarization correction due t o hadross i n such

predminantly low enerll;y QED problem as the h& S h i f t e t c

2

I h e imeginary p a r t of v(q ) f o r q2 O is the " v i r t u a l photon ltfetima"

ltnd gives the r a t e o f production o f hadrons i n (say) an electron-position

e o l l i a i o n , Because the i w g i n a r y p a r t of the m p l i t u d e represents a l o s s i n probability t h a t a photon r e m i n e a photon

2

+ m l + 4 ~ e ~ & v ( ~ ~ ) i.e Prob 1 + 4ne i (v-v*)

X t is therefore dgreetly accessible t o experiment, The r e a l p a r t is r e l a t e d

t o tbe i m g i n a r y p a r t by a dispersion r e l n t i o n Therefore hadronic vacurn

polarization e f f e c t s ( t o ordere2) could be c-letely detemined a f t e r s u i t a b l e experimnts a r e done, We w i l l discuss t h i s matter i n d e t a i l i n the aexr Lecture

Consider the process 2 + e- + hadrons iln s t a t e ra out, It is gmeretad

2

l a e i

iE2y ,p1) 7 'moa, / '$ (4) 1