Photon-hadron Interactions

This page intentionally left blank Andcl-son: Dasic Notions of Condcnsctl 1lqattc1-f3hysics, ARC ppl>k, lSBN 0-201-32...

Author: Richard P. Feynman

116 downloads 830 Views 16MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

Report copyright / DMCA form

DOWNLOAD PDF

U.

IF>.

Using

r e l a t f v i s t i c invariance and gauge invarimce t h i s can be written 2x1 the form

< P ~ I J ~ I P ~ ' (;21~M~l

+

l

( Y ~ ~ ~ Y ~ ) P ~ I ~ ~ >

where uf u2 a r e S-sginors describing 'the i n and out: proton 0f w m n t a pl, p2 rnolarentunt

transfer q

p2

- pl.

2

F1 and F2 are functions of q only, Other

linear coxnbinatfons of PI and F2 h w e also been defined,

lnost

w e d a r e (the

so-ealled eliectric and aragnetlc f o m factors)

clearly a s q2

+

O PI i s the charge:

Fl procon(0)

and Fe are the anoraalous m m e t i c m m n t a ,

GM(Ol

-

pp

l, PI neutron(0)

Hence GE(0)

-

O

l f o r p, O f o r n

is the t o t a l nagnetic ooment.

Ttrese have been masured by scatterdng electrons from proton@ @ i a s t i c a l l y .

The f i r s t order amplitude is

(Gk,

uj a r e electron spfnorsl

l14

Electromagnetic Form Factors 4 ne

2

The cross s e c t i o n a r i s i n g from t h i s is:

m

lab

P

By varying both .E

m d 0 we can vary q

2

and 8 hdependently, and thus

v e r i f y whether the bracket i n (22.2) does vary l i n e a r l y v i t h tanZ (812) f o r 2 fixed q , It: works well, s o both GE and C& can be extracted f o r the proton.

Data f o r GM i s m r e accurate than GE.

Far the neutron u n c e r t a i n t i e s of t h e

deuteron wave functions w k e the deeernt;mations, e s p e c i a l l y of G much ER less certain

.

Wi&er e l e c t r s i n a ~ e t i ccorrections do corn i n , e s p e c i a l l y e f f e c t s fro= the

but they a r e l a r g e l y mrderstood and a l l w a d for.

This i s an intportant but

tedious correction t o t h e d a t a t o g e t t h e Ideaf photon exchange cross s e c t i a n . Exp@rIroentel r e s u l t s a r e ussua1J-y dsscrlbed by giving deviations from a s e t of c m l e t e l y enapirical approximations, R-5y

that

l a t t e r Erurctlsn i s c a l l e d ttre dipole function GD. Most accurately know is G pHP '!

*

I t vaguely Eollowa t h e f m c t i o n GB,

116

Photon-1ZiadronInteractions

For solall q2 i t v a r i e s as about l

C-/G~ GD Thus G

M

5

1.08 f o r 1 K -q2 c 6 .

+

2

(2.6 2 . l ) (+g 1 , very c l o s e t o t h e dipole.

Above -q2 = 8

the r a t i o f a l l s below l.

f a l l s somewhat f a s t e r than 1/q4 f o r l a r g e -q2

The e q u a l i t y

.

%hn ~ ~ ~a s 1within%the accuracy of the neutron =

data. For t h e neutron CB:

a t qZ

1 is .a045 t .0043.

One can f i t with GEnZ G&(T/(I

f

L0 -c)),

Neutron-electron s c a t t e r i n g a t low energy (using e l e c t r o n s i n a t o m ) gives about

/(G /U 1 f a l l s f o r l a r g e r q Z near 2 o r 3 s o i f put ae EI? W p A q , A is perhaps .8 (large e r r o r ) . Xt is n o t pass&ble t h a t G = C /p EP MP p For t h e proton G

l

+

2

2 2 f o r a l l q , f o r a t q2 = 4M the d e f i n i t i o n of these W o G ' s i n t e r n of F F2 become i d e n t i c a l , s o unless G

Ep

w e m w t have G = G a t q2 EP HP and m 1 a t q L O. The sLnp1e r a t i o s artlong G

4

md G

m vanish

~ Thus ~ .C

G&,

EP

/(G

l

and

here (which is very unllikely) / p )

W P

-2.79 a t q 2 - + 4 M 2 P

GEp, GEn which a r e nearly r i g h t would be

expected by pure v e c t o r m s o n domintulcs (wsum%ngthe mass of p and o equal, and t h a t $I couples only weakly t o the proton).

Bonever, a l l the f o m f a c t o r s would

D b u t of the meson propagator MpZ/(Eb

be m l t i p l e s n o t of G

2

-q

2

= (1

+ 92 / .58)-l. Thus

This supposes t h a t the photon couples t o a point charge nucleon via the p ,

.

-

2 2 2 2 Its small Q behavior i s 1 1.7 Q it has t h e wrong l a r g e Q behavior Q2 i n s t e a d of I 2,S Q , It is, of course, n o t expected t h a t W continue t o

-

2 2 work a s we go t o negative Q f u r t h e r and further: frorn t h e pole a t Q

. : -SU

2 P

.

But t h e r e is l i t t l e doubt t h a t t h e l a r g e s i z e of the charge square radius (i.e.

2 the c o e f f i c i e n t -2.6 o f Q ) i s contributed t o a l a r g e e x t e n t by v i r t u a l

P f s s , ZC is not r e a l l y c l e a r why these r a t l o s do maintain t h e i r values at, 2 0 out t o $ = 2 o r 3 GeV a s they do t o LO or 202.

We c m gain so-

h s L g h t i n t o t h e rrrembg of form f a c t o r s by looking a t

the. non-relativfstic cae;cse. For a spllnless systern the forto f a c t o r is j w t the arsmatua representation of the charge density, p(@

ei9*'

d%

\p(r)

2

(23.1)

for small q2, with p(r) normalized so t h a t

Expanding i n p w e r s o f

(sinQr/Qr

p(r) 4sr dr = 1 we get

2 4nr d r

I

- Q2

t

2

16)

where r L is the mm square charge radius P

An exlpanential eer

f(Q)

-

(1

+ ~'/o~)-'.

f i n i t e not zero,

f o r the. Pharge: density gives the dipole d i s t r i b u t i o n The

qa4 depeadenee saya t h e slope a t o r i g i n must be

X t f s , however, a o t a valid concluaim t o we non-relativistic

2 views t o study large Q

.

Hevertheless the conclusiorl is e s s e n t i a l l y c o r r e c t ,

R e l a t i v i a t i c s l l y the f a c t t h a t ss q2

+

the form faetora go t o zero

mans t h a t &ere is zero probability t o find the proton alone as a point &arge, Tk&s could r e s u l t i n f i e l d thrtary e i t h e r f r m there being no %deal f i e l d e n t i t y (parton) with quantum riders of the proton, o r e l s e , i f tfiere were such the r e a l protcra is always dressed, i t k m zera q ~ l l t u d eof being an Ldeal proton. Ta w k e a r e l a t i v i s t i c analague of a thing l i k e the hydrogen atom {whose

&arge dlbstribution is e'gr)

w e irtaagine two s p i a l e s s partic_llt?s held together vSCh

a sp$;nless Haws p o t e n t i a l generated by the weak etrchiagge o f same s c a l a r =son, then (1) the non-relarlvistic waw function ahovss a e'CLr behavior, (2) the. 2 form factor varies as l /4 ~ for large Q

.

For e x m l e , l e t one p a r t i c l e A be charged aE m o s of mms m2 held weakly non relativist-icallg.

nA

4-

%,

Eke other B n e u t r a l

The w@sof the proton i e m a r

I$ a d neglecting &m zero& order the r e l a t i v e motion the i n c a a a g proton

of wmntuln pl, aeans A haa momatum arpl aztd B has m m n t m bpl (a

-

HA+% v*m%>

Finally we want a bound proton of atomatm p2 hence p a r t s of znomc?atm &pZ, bpZ. We do t h i a by h i t t i n g A with the vaty largcz 9 ultimately ta hold together p

A

lauklt

pl-p2,

&erem f a r thrt p r o m

change m1g by @Q, hence an a m m t of

mnrentm bQ awe be passed o w r t o b (by aa exchmge of the rneeoa reswnsib2e f o r iaceract..lon o f A and B,

The d i a g r w a r e

*

2 f o r large Q2 hence l / Q4 f o r the f o m llhe two propagators each give l/€)

factor.

Thus the f i r s t d i a g r m is

Clhe second

2 a t large Q

.

I f vector nresoas a r e exchanged beween a p a r t i c l e of spin 112 and one of spin zero, a 9' appears i n the numerator due t o mamatic coupling. Non r e l a t i v l s t i c a f l y i f a eystem f s mde of 3 p a r t i c l e s , the large Q2 a s m t o t i c behavior depen& not on the s i n g u l a r i t y whtrn j u s t two e m together, but r a t h e r when a l l three a r e on top of one another. Aes we s h a l l see, euch picttuces a r e too s i q l e and in&equate,

mkiag very naive loodels t a compare t o the non-rcrlativi~tie ideas-.

We a r e only They w i l l

not agree with the i n e l a s t i c scattering.

Lecture 24 Pian Fom Factor Data e x i s t s a l s o f o r the pion form factor, f a r q2

It Is beat obtagned front the reaction a f yg

-+

-c

O (spacelike)

+

nn produced by a v i r t u a l

photon (generated by elcctron s c a t t e r i n p ) , e

P By gofng t o s u f f i c i e n t energy one cgfl obta%a a region rP;here v i r t u a l pion 2 excttangs? is .most % w a r t a n t w d t h a t thfe trirtuaf, a has a s nearly zero q

-

(and is therefore aa close to its w;qs shelf m passfile).

The coupling of the

1/11

Electromapetic Form Factors v i r t u a l n t o the nucleon i s knom i n t h i s region (gNwEy5,

2

15) s o t h e only

u n h m is t h e photon-pion coupling between a r e a l n and one that is nearXy r e a l , but a l a r g e momentum trcunsfer Q, -1.2 where Fv has f a l l e n t o 0.28.

Data is a v a i l a b l e out t o q2 is jwr about exact+

This

( f o r no k n m very good reasan) the isovector p a r t of

t h e nucleon form f a c t o r

Flp-Fin which

2 a l s o f i t s a l l lower 9

2 -P

f a i r l y well t o t h e pure YDn o r p meson pole (l

.

+ % 6 ~ , vhich

It w i l l a l s o f i t -1.2,

is .32 a t Cj2

-1.2). but not a t a l l well t o t h e form f a c t o r GD of q which is too low ('15 a t Q* 2 Me have already discussed t h i s f o m f a c t o r f o r p o s i t i v e q where i t has been -4

masured i n the reaction e e-

-+

+ -.

n n.

Me have an expefimentaf, curve which is

dominated by the p pole expression (with small uslderstood corrections f o r the w-p

ft: is i n t e r e s t i n g t h a t an extrapolation s o f a r i n t o

interference, e t c . ) ,

negative Q

2

s t i l l . works reasonably well.

+

The r a t e e e--+ a1.a- measures, of course,

only the absolute square 1 ~ ~ (/ qnor ~ the ) r e a l and imaginav p a r t separately. Wowever the shape of the curve (p resonslace) is s o siwle t h a t our physical understacting m y make a separation i n t o r e a l surd lmginar)t part8 not hopeless (e.g. we wrote previously, f o r t h e p a r t near the resonance (*p arid had s o w guertsee a s t o how 'l varied with q

2

.

both portlt$vct and negative.

one believes

- that F r ( q

2

) must be

ii. m )) P P

2

) over a large range

an a n a l y t i c function of i t s variable q2 w i t h

.

2 P(q ) has so imaginary p a r t only f o r qZ inreasared.

+

Me can then check the hypathesis t h a t every-

a branch c u t along t h e p o s i t i v e r e a l q2

relation

/l(q2-mp2

Zf t h i s can indeed be done we

would have F n f o m t i o n now knom o r soon available on F,(q of 'q

2

- i.e.

t h a t Pn s a t i s f i e s a dispersion

2 ( 2 ~ ) where a t l e a s t

I FT(q2)

2

is

A check on a guess f o r tha imaginary p a r t would be t o coapute t h e real

p a r t from (24.2) f o r qZ method %s t o f i t

1 F~ I

2

3

O and check / F 1 2 ( a l t e r n a t i v e l y a much e a s i e r equivalent n

by choosing physi c a l l y appropriate a n a l y t i c expressions

2 f o r the complex f m c t i o n FE(q )).

Thent having established a reasonable

2 2 f m F, Cq ) me c a l c u l a t e s FTl(q 1 f o r negative q (where i t Is r e a l ) t o c o w a r e t o

2

- in

the p o s i t i v e q2 region 2 the main feature i s t h e p resonance and up t o r a t h e r large q no other resonance,

exper-nt.

We already s e e t h a t i t w i l l w r k well

i n f a c t not a l a r g e production, has beem found, 2

Below t h e i n e l a s t i c threshold qZ

(4v) (and i n p r a c t i c e probably

d l s t m c e above i t a l s o ) the phase of F,(q

2

some

) must, by f i r s t order u n i t a r i t y , be

the s a m as the v-n s c a t t e r h g phase eh%ft, X n f o m t i o n on t h i s is recently becoming a v a i l a b l e from p i o n - d r t u a l pion scattering analyses.

This m y h e l p

i n analyzing the d i e p e r s i m r e l a t i o n , 2 2 How should we expect F,(q ) t o go f o r l a r g e q ?

the p r o b a b i l i t y of producing f u s t two available must be very low,

nt8 B e r e

T t wt f a l l off becawe

s o w y other s t a t e s a r e

The isospin eurrent generated by the p a i r of pion8

awt be coupled t o o t h e r badrons and we require t h i s coupling n o t t o r a d i a t e .

Aa we s h a l l underatmd b e t t e r l a t e r , t h i s leads one t o expect F t o fall. as s o w parer of q

2

f o r l a r g e q2, but we do not knar the power.

Hwever, these con-

s i d e r a t i o n s make i t n o t s o s u p r i s i n g (granting the e v r i - t a l 2 P%(q ) f o r p o s i t i v e

q2) t h a t f o r negative q2 even out t o -1.2

behaviar QE. the expression

is close t a the p pole formula {which a r i s e s from (24.2) i f m s t of t h e ilaagI,nary 2

p a r t Ss near nr ). P Sum r u l e s r e s u l t front the f o l f e d n g cansideratioas. disperaLm r e l a t i o n (26.2) is v a l i d , s i n c e we know P,(O)

If we put

3

-t

i n ( 2 4 . 2 ) , we would expect ~ 2

Tf t h e wasubttacted E

2

l we. must have

~ ) (t o q go as l / q

2

timee

2

Im ~ ( q ' ~dqv2 ) ( i f Fx(q ) went more slowly than l / q we would have t o 2 m F(qt2>dqs2diverges, ae indeed we would expect f o r k F(q' ) would 2 2 probclbfy a l s o f a l l slawer t k m l / q ' 1. I f we expect F",(q 1 t o f a l l fastet:

2 than l l q (we how no s u r e a r g w a t f o r t h l s ) we would have another r e l a t i o n

called a superconvergence r e l a t i o n . We e q e c t t o have an usubtriacted dispersion r e l a t i m v a l i d because i f 2 (26.2) had an u n h o m c a s t a n t added t o the right-hmd ~3idethen Pv(q 1 could

Eiectwmagnetic Form Factors not approach zero as q

2 +

m.

In any event

I21

have the subtracted r e l a t i o n

- (24.3):

(24.2)

I n practice ( 2 4 . 4 ) i e =re useful than (24,2) t o compare F,(q

2

for

negative qZ t o i t s experimental value s i n c e uncertainties from rhe contributions f o r l a r g e q f 2 where data i s unavailable a r e mu& smaller. Me can now see using (24.4)

wtxy f ram what we already know e x p e r i ~ a t a 1 S . y

2 2 f o r Fw(q ) f o r positive q2 the value f o r negarive q should be close t o the 2 p pole value even r a t h e r f a r out. h (24.4) we know LF(Q ) is d

- there is not mueh

2 2 f o r l a r g e r q2 ( a t l e a s t no resonance t o /Pz(q ) 2 2 and ( 2 4 , S ) does not contribute integrand f o r s t i l l larger q , f o r Q

the a pole 4GeV2

even as negative a s -1.2.

Thus F,(O)

of the p pole f o m l a nearliy. fpsT/gp

-

- F,(q 2)

But F,(O)

is given by the a n a l y t i c extensirrn

is (the accident of the VDM canstrainta

1, see l e c t u r e 16) given by the p pole formula nearly, as we have seen

2 e a r l i e r ; ehus F,(q ) should a l s o be. make t h i s d l s c u s s h n quantitative.

An i n t e r e s t i n g study would be t o t r y t o 4'

One can a l s o aak i f the data on (e e"" .+ n'n-)

is accurate enough t o a l s o damhate (24.3) and t h w explain the accident t h a t

rPWn/g P

=l*

The proton form factor f a r positgvo q @'c?-+

m-'

could bc? obtained from the reaction

m e r e a r e o a t y e t any qusntitatfvo experiments but they a r e expected (kxe experimnt near threshold saw no p a i r s , indicating Q, GE a r e

mmntarily, slaall there

2

- about

there q2 z 4.

an, small (or a m l l e r ) than the &pole Eamula calcularcrd

2 2 We would expect agafn t h a t Fl(q ) md P2(q 1 am m a l y t i c

functions s a t i s f y i n g a dispersian r e l s t i a n l i k e

(Wen a w l i t u d e s can be expressed f a orore than one way, such a s h e f i c i t y

f o r the l a t t e r t h a t the s f w l e s t dispersiga relacloss a r e supposed to hold, b l a t i o n s f o r other c d i n a t i o n s mst be deduced from these,)

This t h e we have a d i f f i c u l t y . q2 < 0 by e l e c t r o n s c a t t e r i n g and f o r q2

O to 4 2 1

by e'e-

aonghilation i n t o

But haw can i t be defined e x p e r i m n t a l l y i n the region

proton-antiproton palr. cg2

.42

i a masurable experimntally for

I do not km of a physical d e f i n i t i o n .

But i t is expected

(fron f i e l d t h e o v exanples etc.) t h a t IrnF(qi2) is not zero i n thihi. region 2

.

f a c t t h a t the i n t e g r a l i n (24.5) has a threshold at ( 2 ~ )

%us

- in

no m m t of

e x p e r i m n t a l data w i l l p e d t a very d e t a i l e d t e s t of (24.5) f o r a piece of the i n t e g r m d is cmpfetcely rxnavailable, 2

P(q ) it can be defined i n no-man's tion

- but

Of course, i f a theory is availerble f a r land (O

2

tq2 c 4H ) by a n a l y t i c a l continua-

i f we had a theory it would presmably already s a t i s f y (24.5)

a i d t h e proper thing t o do would be t o compare, it d i r e c t l y t o e x p e r i m a t i n the physically a v a i l a b l e region. Thus the dispersion r e l a t f o n s f o r nucleon Zom f a c t o r a r e not of auch d i r e c t use f o r cmparinng t o e v e r h e a t .

m: FJe append here a theoretical note on t h e expected position

of t h e

2

threetrald (24.5) b e l m which I= F(q ) can be e w e c t e d t o be zero. For a charge density e-ar c l a s s i c a l physics gives F 2 a f i r s t s i n g u l a r i t y a t C$

-a

2

.

1/(q2

m

A superposition of erponentfals /-f

e3

dA

Q being 92 =

+ a')

giving a continuum

whlcb ha. (h)e-~~dh

X

of a i n g u l a r i t i e s the lowes t govewr

+h

the e l m e a t expooential t a i l .

idea is v a l i d r e l a t i v i s t i e a l l y .

Thi.

( I t h a s beea s b o m t h a t t h e position of the s i n p l a r i t y is the s m aa t h e p o s i t i o n of the s i a g u l a r i t i e s i n Feynmn d i a g r m o f a process a s i f r e a l p a r t i c l e s were

For t h e proton the v i r t u a l s t a t e N (P gradient of) e'l"r/r,

+ B+ e f i s t s

with a

E

extending out a s

Its aquare, the charge d e a s i t y involves p o l m o d a k i n

l / r times e-2wr, t h e longest exponential i s therefore

211,

and the threshold a t

-8

.Ml whicfx v i r t u a f r y

2 ~ . A =re

i n t e r e s t f a g case is one say l i k e the C , of

can emit a K (=ss m) t o be a nucLeon =ss El2, with PI2 < ;MZ.

We obtain the

sfngularlSty by studyfag the perturbation d i a g r m but can aee its p o s i t i o n

Electromagnetic Form Factors phgsicall;g t h i s way:

We ask: how sharp s r i s i n g ( e l m indepmdemt) exgonential e l e c t r i c

potential can we t o l e r a t e before we get a diw~rgeecebecause the f i e l d risea f a s t e r t h m the charge density f a l l s ?

Thus we hawe a f i e l d of mmntroa Q ( i t Is pure ileag5nary able t o make the fC: meon from the 6.

The v i r t u a l s t a t e is a p a r t i c l e of lnaea Hi!

mmntunr 0, and one of tnaes m w m n t m =ss

H

l

f a ) and a r e j m t

+ Q2 where

of mrnentlnn G l q (Breir Eran-@, u

lp.

the i n i t i a f . s t a t e i s one o f

0).

The condition of j u s t divergence i s that t h i s s t a t e J u s t has

zero energy

denodna t o r

@h&@ equation is r e l a t i v i s t i c becawe i t conserws both energy and maomnturn). This has the solutjiaa

g2 if %2

-a2 = -kmZ g

- m2

is a t q2 m +a

2

.

+

2

-

2

The threshold (orhewise the sqvsre roots a r e not positive.) 2 That i t i s b e l w the energy 4m needed t o mke a p a i r , which

was the f i r s t guess, we c a l l t h i s For M22 3: M

- m2 2 / H2~

-m

2

c9n

""anomlous" tt'brshold,

the equation cannot be solved, the s i n g u l a r i t y n m

r e a l l y comes {normal threshold) when the photon e m f i r s t d e a pair tq2

-

4m2

.

124

Photon-HadronInteractions

Mote on the quark m d e l of F e y m n , U s l i n g e r and Ravndal, The f o m f a c t o r f o r t h e proton by FKR is

(l

-1.1

- q2/4~2)-1

enp MmMaasof theproton

",

which is very poor

-

i t c u t s o f f too f a s t a t high -q

expect t h e m d e l v a l i d f o r l a r g e Q instead of 1 + 2.6 q

2

.

2

1,

For smll q

2

2

( t h e r e i s no reason t o

i t goes a s aftout 1

0

l -1.7 q or l

+ ,5

"7 q

2

In t h e m d e l the photon coupling i a direct: (processes

2 l i k e VDM a r e not considered i f they were we would add a f a c t o r (l-q /m 2

-

21-'

or

g i v l n g b e t t e r understanding). The r a t i o GE w/GH comes our 1 + q 2 / 2 2 q

2

which f a l l s too f a s t with q2 (experiment is arovod l

+ .06

q

2

1.

( h o t h e x way of deallng with t h e quark equations glving a much b e t t e r f o m f a c t o r , b u t much worse S i t s t o photon e l e c t r i c matrix e l e w n t s Ls given i n Fugiaara e t a l .

Prog, m e o r , Phys.

For t h e pion FKR would g e t (mE'

183 (1970) .l p2

,021.

2

which is p a t e n t l y absurd ( f o r e n a w l e , having a zero a t a11 odds t h e worst d i s a s t e r of t h e i r =del

qL

-

-0.08!)

It i s by

- ( i t c o w s f x m t h e ad hoc way they

d e a l t with t h e s p i n of the quark m a f f e e r e d by t h e photon) t h a t one muLd think ;Ln honesty they would include i t i n t h e i r pcrper,

Tbey di&' t because, s u r p r i s i n g l y ,

they d i d n ' t think t o c a l c u l a t e i t when they wrote t h e i r paper!

ectron-Proton Scattering. Deep lne astic Region

Electron be-

p e r d t ua t o study reactions such as e p

It is controlled by the amplitude the f a c t o r ( x

&

P

(I

ip)

(X~J

S1

is

i n which only

m e m s c novel feature i s the a b i l i t y t o vary the q

v f r t u a l photon away f r a n p2 dSscuased) t o

2

9

X

Ip) is a d a m , and ao we c m ulasure t h i s m t r i x e l e m n t (or

rather its square),

2

(LZ y P ul)

x fiere

-c et

0 ( r e d photon erperlments, *ich

o f the

we already

negative, and i n f a c t t o the Ear negative region.

very e m r g e t i e (hi& -8)

2

Xn addition,

s t a t e s x can be excited h the hfgh energy tna&ines,

A great; d e a l of i n f o m t i m Is becadng evail&sble, and there? are may theoretical

2 q u e s t i a s especially f o r the? aw large q and h i & =ss s t a t e s

X.

Ttrre? simplest e x - p e r i m ~ t st o do are those whi& do aoc require atudying the

x s t a t e s a t a l l (am over poasPble s t a t e s

X)

but j u s t study the electron beam

-

126

Photon-Hadran Irtteraetions

its deflection m d energy d e t e d n e s the =-turn dete-aee

($.U

t r a n s f e r aad

the lab system) the enare;y of x above the proton

energy l o s s Theee

-S.

were d m e f i r s t asld a r e mst c q l e t a -we discuss f i r s t therefore t h e i r

exper-a&

w i l l occupy

r e s u l t s and t h e o r e t i c a l i n t e r p r e t a t i o n (*%eh Then we s h a l l return and

&SCW@

f o r -g

lectures)

ac3.

X

.

t o look

is &m.

tbe I&, of the electrun is m m u r e d aa well

The energgy loes v = E-EQn

as its 4-mntunt

W

what we e m expect the f i n a l s t a t e s

lgke mad how they do behave experimntal3ly a s f a r

p*q

it8

2

trernafer q , The i n v a r h t a of the problem a r e thus q

lvXV where p i s the 4-mnturn

of the protan,

2

and

A l l quantities nraaaured proper

t o the hadrons a r e ezpressed a61 fuactions of these two variables,

The prob-

a b i l i t y we memare i a then proportional t o

b the cgse of mpolarized e l e c t r a ~ s(there are no e x p e r i m a t s with polarfzed electrons and protans yet, but we s h a l l d f s c w s the theory l a t e r ) , average over electron s r a r o s contributes the spur l/2 sp

yV

yli) = 2 (klpkZv

+ klvkZl, - dliv

$ yv

kf k2).

".'.

the swa and

-

ul) (u y u ) 2 F t l While the hadron s t a r e eum

gives an expression we s h a l l c a l l Ruv

t r i c p a r t of rxpressfon for any spin), (spin of p averaged, Ss alau vv S 2 (Note, the mms of the f i n a l s t a t e s which we are generatfng is e q r e e s e d i n , t e r n 2 of our variables q , v a s m

l?

+2

Hv

+ q2

2 By varying v f o r given q we can vary: H

2

q

i s negative,

and look, f o r example, f o r f i n a l s t a t e

127

EEectrcz8-Proton Scatrering. Deep Inelastk Region From a r g m m t s of gauge Invariance and r e l a t i v i t y t h e B of Che t e n s o r

(which is a l l t h a t counts f o r unpolarized e l e c t r o n , and

Ps a l l t h a t survives i f the proton i s unpolarized) a m t have the f o m

S e ~ r a pl r o p e r t i e s o f

W1, W2:

The apparent, but unwanted pole@ a t qZ =

O came from t h e way we have Tc r e m v e t h e hil3fteat pole

w r i t t e n our expression and of course cannot be r e a l . 2

-dq2vl a s q2

we must have (pep) W2

but Wl can approach a constant Wl(O,v) 2 2 2 V22q ,v)

-+

4

D er

as

q2

2 2

W2 = -9 /v +

0.

W1

+

order q4 ae q

Hence f o r small qZ, Wl

2

-r

0,

W ~ ( O . ~ ~;I

-OVWICO,vl.

2 The d i r e e t t o t a l absorption crosrs s e c t i a n f o r m I m g i n a r y photon of mss 4 r o hit

k

proton would be

arornentum of photon i n p rest agstea. We haye two p o l a r i z a t i o n cases :

Transverse,

e

i s perpendicular t o qv and t o pU

Longitudinal.

S m t i a e s c a l l e d scalar:

These two c r o s s s e c t i o n s can be used ss p a r m e t e r a i n s t e a d of W

From t h e above Eomalas

It is canvenlmt t o define

I' W2'

Photon-Hadroninteractions

128

C d i n i n g t h e e l e c t r o n s p u r t o t h e expression f o r K

irv

we o b t a i n the t o t a l

cross s e c t i o n i n the lab.

Ia p r i n c i p l e i t is poseible t o g e t both W1,

k$,

t h i s is done i n s e v e r a l eases

-

but mast accurate is W2 a d o f t e n wftrzt i s given ie W2 m d e r aarne a s s m p t i o n about Wl.

For exarclgle we have

and o f t e n d a t a is given f o r

Problem:

+ ea

without resolving them,

G

is o f t e n smll.

Show t h a t f o r proton e l a s t i c s c a t t e r i n g

2 4M W1

-

2 :G

&(v

-q

+ q 2/2H)

Lecture 26

We s h a l l now gfve a kind of p r e l i d n a r y description of t h e e x p e r f m a t a l results

- so@@d e t a i l s w i l l be d e a l t with

further l a t ~ r , Firet a useful varfable 2

.

t o discuss t h e lower energy r e s u l t s i n i s t h e mses2 of the f i n a l s t a t e Hx 2 2 M' 4 2Mv + q We s h a l l , a t f i r s t , use :M and q i n our discuesion.

< B

.

function o :f M

W,

As

of course, g e t nothing belaw M? then s l a r g e e l a s t i c p&

-

whm t h e f i n a l s t a t e is j v e t a proton proportimal corresponding t o: H = 2 t o $(q 4 2Mv) and depending on GH, GQ as we have previously s t u d i e d , This

-

contribution f a l l s a s q2 r i s e s a s 1 / ( 1 Next, f o r fixed q

2

, aa: M

- q2 1.71) 2 .

v a r i e s , we s e e e " f i r s t t f r e s o n a c e e t Mx = 1236.

129

Eiecfron-&"roton Scattering, Deep f nelastic Region a "second" one near

.a

1520 and f i n a l l y a "thir$'>irouad

l700 and some fndication

Theoretically we should s e e more resonances, e.g. l535 but

of: a fourth a t l90Q.

undodtedly t h e 'kaeeond" peak is an unresolved d x t u r e of these twa while the one near 1700 s o m t i m s c a l l e d t h e 1688 resonance is probably t h a t with four others e w a c t e d near *at

energy,

Tne reaonmce a t 1407 ( t h e Rowr rasone~nce) has not been

seen i n these experimmts ,

can be f i t t e d m B r e i t - g i p e r peaks an a background whfch

These resonaxtees

gradually dominates i n t o a smooth curve .a :M

inereaaes.

How do the resmance s t r e n g t h s vary with q2? At verg low q

2

the behavior

depends t h e o r e t i c a l l y on the angular trowanturn of the s t a t e and @ t a r t s a s an appropriate p m e r of Q (gZ

2

-q ). Tor higher qZ, however, t h e s t r e n g t h of the

m s 0 m c : e s a l l f a l l more o r l e a s a s does the e l a s t i c peak. the r a t i o (do/&)

res

/(do/dQ)elastie

with -q2 oae find.

Zn f a c t i f one p l o t s

cvrvse which rise rapidly

from -ttzreshofd (photoelectric) values t o s a t u m t i m (In tnthe v i c i n i t y of one! o r 2 jut belou) f o r q2 > about 1 GeY

For l a r g e r q

2

.

and l a r g e v t h e r e is nothing l e f t of the reaonances, 2

i t wae suggested by Bjorken that t h e funcrion vW (q ,v) (and a L o Wl(q

2

2

2 -q /2Nv on3.y.

be a function of t h e v a r i a b l e x as a f a c t i o n of

u

llx).

2~v/-~'

This f e a t u r e , chatl m -q

true, 2

vM2 (q ,v)

2

+

approaches a f a c t i o n of

There ,v))

should

(Data La o f t e n presented a l s o

This has turned a u t t o be r a t h e r accurately 2 srsch t h a t -q 12M-v = x is b p t fixed,

W,

V +

X,

F(x) Is hm as B J o r b n scaling.

kle s h a l l

have a g r e a t d e a l t o say l a t e r of i t s t h e o r e t i c a l signiffcance. Present ( S a n u a ~1972) d a t a is a v a i l a b l e in the f o l l o a n g regicms.

Resonance h g i o n Scaling b g i o n Wl, W2 Separated

I n the region w regien 12

P

w

I n my v&ri&le.

4

4, s e a l i n g is observed above

4 f o r E$

,2

-pZ>

l; VW

2

<

= (2 ' 6 )2

.

In the

is a c m a t a n t uAthin e r r o r s and s c a l e s

The d a t a f o r vM2 looks a s follows

./

vWZp&

. l 8 f. l

f a r t h e proton

.l2

f o r t h e neutron

I (vwZn&

9

a

Data f o r the neutron [by s u b t r a c t i o n from t h e deuteron) is a l s o a v a i l a b l e , The r a t i o t o W

2~

looks l i k e

i t is consistent with a s t r a i g h t l i n e ( l

-

X)

from x

O t o '75.

mere are

i n d i c a t i o n s t h a t i t c u m e s away above t h e f i n e f o r l a r g e r x and cou2d reach a value as high a s .S(?)

far x

In the region where

+

I, but higher values a r e probably ruled o u t ,

and W2 can be separated, values of R can be found

-

they v a q from O t o 0.5 with l a r g e e r r o r s s o a o d e f i n i t e trend can be seen. f f R is ass-d

R

-

-03

w n s t a n t i t is .l8

(_112}/d o r with

R S q2/v2e

s c a l i n g t h a t R -. O f o r fixed

+

,143, t h e d a t a is a l s o conrpatible with In t h e l a t t e r ease we would expect with

X.

We now begir: a r a t h e r long t h e o r e t i c a l discussion of t h e S n e l a s t i c e scattering, 2

-q /2kf

=

X,

2 I*le s h a l l f i r s t ~ S B C U Et IhB e "deep" i n e l a s t i c region -q

", v

-+

",

o r region of BJorken s ~ a l i n g . F i r s t we s h a l l discuas what is

c a l l e d the parton =del

5

-".

-p

b r i e f l y ; then we s h a l l discuss general p r o p e r t i e s of W%,

and s e e t h a t they a r e conaectcrd t o t h e carnutator of two c u r r e n t s ,

Pinally

we cam back t o t h e parton model i n w r e d e t a i l , discussing suggestions t h a t t h e partons be i d e n t i f i e d with quarks, e t c . t h i s t a a =re

- and

then discuss the r e l a t i o n of

h e t r a c t representation c a l l e d GeL1-Hann's Llght cone algebra,

m e r e then f ollowrj: miscellanesutl discuas hn, ending w i th a t t a a p t s to unders tand.

Elecsron-Pr~$~n Scarrering, Deep Inelmtic Regi'on things io other regions of the qZ,v plane (l.e.

for smsller qZ, and, or sr

resoncrnceslj etc., (actually our diacuas&~,oas will not be ro neatly organized, we

shall wander

these oubjeets).

131

Parton Mode

Farton Model The parton =del

i a t h e conceptually e a s i e s t t o understand, although i t

appears a b i t s p e c i a l , as s p e c i a l assuaptions seem t o be m d e . a b s t r a c t considerations a r e t h e r e f o r e =re the m r e e l e m n t a r y v i e w o i n t .

Nare general

s a t i s f y i n g , but f i r s t we discuss

Ln discurpafng these arattere i t is b e s t t o

keep i n mind a l l t h e p r i n c i p l e s of r e l a t i v i t y , quantum ~ l e c h a a i c s ,u n i t a r i t y , imalycicity, e t c .

One way t o do t h i s is t o work i n a coaceptual w d e l .which

s a t i s f i e s a l l these p r i n e f p l e s siatultctneousfy,

%ere is no b o r n sinrgle m d e l

which does t h i s except f i e l d theory (and t h a t m y not do i t diverge!),

m d t h a t is a very c v l i e a t e d rm&l indeed.

- all exwles

lulevartheleas we s h a l l

t r y t o s e e what f i e l d trheory d g h t suggest. f n f i e l d theory the wme function f o r a s t a t e , such a s a proton, could be given by givgn~ngthe m p l i t u d e s t a fllnd various kinds of bare f i e l d p a r t i c l e s w v i n g with various m m n t a ,

'flhese bare f i e l d p a r t i c l e s we c a l l "partons"".

Somt?ims soroe phenomaa c m be understood d i r e c t l y i n t e r n of t h i s wave f m c t i o n but usually laatrix e l e m n t s t o wave fraxctions o f o t h e r s t a t e s mst be malyzed,

The wave function, however, i s not e a s i l y transfomed from one

x e l a ~ i v i s t i csystem t o another (the

Lconim rnuat be know) became rlt repre-

s e n t s a s l i c e a t a given t i ~ t e . Certain p r o p e r t i e s art? therefore =re

easily

seen from t h e wave function i n one system than i n another.

"re deep i n e l a s t i c

s c a t t e r i n g behavior is b e s t uatdn?-rstood fro@ t h e wave function f a r the proton with extrearely high raomlmtm P ( i n +z d i r e c t i o n ) . f o r a of t h e wave Eunctlon a s P +

I n f a c t we atudy the lirnfting

a.

Tn what v a r i a b l e s w i l l t h e wave Emetion have a l i m i t f n g form?

Froar a study

of the e h a r a e t e r of t y p i c a l f i e l d h m i l t o n i a n i n a few e x w l e s , and from a study of vefy high energy i n c l u s i v e hadraa c o l l i s i o n s we c ~ n c l u d et h a t i f t h e raomntuna of the parton i s aeasured proportional. t o P as &P and t h e t r m s v e r s e maomntunr i n absolute u n i t s k then t h e a w f i t u d e s depend onZy on 5, k f o r 5 s o smll t h a t 5P is a f w &V,

- (except

as P rises

6 af order 1 h V / P a r e c a l l e d "wee'"

t h e d i s t r i b u t i o n of "we" pertons f s probably b e s t described i n absolute pZ

k

variables,

As we s h a l l s e e the "weett r r g i i o

hard t o mafyze

- but the

m i n f e a t u r e s of t h e deep i n e l s t i c s c a t t e r i n g does n o t involve them). The f a c t t h a t Er,

is f i n i t e i a

conaepuence of f i e l d theory

- (in

e, a t least:

La any obvious way, a d l r e c t

f a c t perturbation theory daes not give t h i s

r e s u l t , and therefore must n o t be r e l i a b l e here) i t is siarply guessed a t frm t h e ubiquitous reaullt of high e a e r a colllissians t h a t t h e t r m a v e r s e aromnta

-

= ( - 4 @V)

a v a i l a b l e t o the prodvcts averages about

2

.

But g r a t i n g t h i s , the &P s c a l i n g is elrpected fram f i e l d thmry ( a l t h o u e today w e could base t h i s a l s o d i r e c t l y on experiment, i t i a t h e s e a l i n g

ltlw

f o r the l o n g i t u d i n a l aramntum of the products fn very h i & energy c o l l f sSons). So=

suggestion of bow i t works is t h i s ,

i~seen t o be m d e up of two p a r t s

0%

The q f f t u d e t h a t a s t a t e af energy E

anera

E;n

E;1

$.

E2 i s domf.nated ( i n

perturbation theory) by a f a c t o r

But the z colgpanent of raowatucn of t h e parts 1s t h a t of t h e whole P S e t PI

m

5Z P, P2

m

E2P. El + f2

1.E m d P 2 + H 2 q S l?

+

the f i r s t p a r t and kl Ptfi t r m e v e r s e w ~ n t u m ,we have

Therefore A/ (E

- El

p2

+ p2,

I f HI is t h e nags of

(The P i n t h e n w r a t o r is w u a l l y absorbed 5% n o m l i z a t i m or La the form

c2.

f o r A, the e s s e n t i a l l y complicated bahavior of t h i s is a f l m c t i m of Cl,

2 2 kl f i 2, A r g m n t s l i k e these were o r i g i n a l l y used t o predict the inclueive scaling, a d they h a m been c o n f l m d by elrperimnt. F u r t h e m r e we see Cl (outside the wee region) must be positive partons a r e g o h g vigoreusly baekward negative, i t should be

/ cl[

- became (26.2)

em the rl&t hand side.

lgl

- no

not valSd i f FI is

men ( i f Cl is negafive)

- EL - E2*

( 1 - / & ~ 1 - g ~ ) P 24 P so the dencmtnator is not small but large 2 and the m l i t u d e 2s l / F 2 smiler thaa the precacding ca8e.

E

We envieaf~ethe proton of wmntusl P as beeig nrade of partens of m w n t a giP a l l sharing la various proportions F;% the mawntm of the proton, a l l 4 I l e behteen O and l ( e l s e some other w u l d have t o be negative since

1

=l).

kle s h a l l therefore think of &a t n c o d n g proton a s a box of parfons sharing the momntm and p r a c t i c a l l y f r e e ,

Another m y t o laok a t t h i s is t o take a

d y n a c v % w of the parts i n the r e a t system and ass-

f i n i t e energy of gater-

action amng p a r t s s o as time goes on they change t h e i r m m n t a , a r e created or m i h i l a t e d , etc., i a f l n i t e tims.

But m\tfiag a t large .siomntm F these t i e s

a r e d i l a t e d by the r e l a t i v i s t i c t r a x l s f o m t l o n s o as P r i s e s things chmge m r e and w r e slowly, m t i l u l t i m t e l y they appear not t o be i n t e r a c t i n g a t a l l . men the proton is h i t by the photon the interaction operator J U couples t o one nearly) ntum p2 S pl + partoa o r anather and kao&s it t o a new s t a t e of f;f the parton had

2

m we would expect a r a t e propoxtiojlonsll t o

g

Parron Model

I35

( t h e f a c t o r s 1 / 2 5 m d 1/2E a t e included because of the n o w a l i z a t i o n s of

(%l

2

I n f a e t , due t o t h e i n t e r a c t i o n energies asavmed f i n i t e , t h e

and KpV).

e n e x e loser i 8 n o t just t h a t f o r a f r e e p a r t i c l e but d i f f e r s from i t by an m k n o m (but. f i n i t e which m y vary with

X,

etc.)

Fox s p i n zero partons /&l2

-

(plu

+

P~,) (plv

+

pZv)

For erpin 112 partons

y2+ ~

1/2 s p ( v l + m) .iP

m

(2 PIw

+

2 Plv PZu

I Y ~ )

- 2 6uv(~1*q))

(27.3)

(We s h a l l c a l c u l a t e f o r s p i n 112, j u s t s t a t e r e s u l t s f o r s p i n Q,)

tkfngs easy we v L l l omit all t e r w i n q by gauge invariance. cfrc-tances

W

Thus i n

can w r i t e p

l

Ir:

in K

U'J

To m k e

because we know haw t o g e t them

put plv i n place of pZV. I n Our l i m i t i n g

= 5P

Y

t o an e x c e l l e n t approxinaatian where P

M

is

the four-momntm of t h e proton ( s t r i c t l y i t i s v a l i d f o r t h e z component only,

b u t we have seen t h a t implies i t f o r t h e t component El

-

&E; mid t h e traunsverse

coaponents being only f i n i t e a r e r e l a t i v e l y s m l l ) . Therefore i f f (x)dx

g.

a m b e r of partons with azomentum between x and &dx

each weighed

by t h e charge squared ( i n w i t n of e l e c t r o n &arge) we have

Kyy

-;

(4 P

, l

2 Pv W2 - 4 6 w v M W I)

( t h e f i r s t l/& eomen from t h e normalizing f a c t o r s 2E/2E1). we have (6~t.q)' -ni2 -6

-

Mv goes t o

with -q2

-

25 ( P e q )

+ q2 + c2

- m2 - b ;

2 Hvx t h i s becomas 2 Mv (g-X)

Xnaide the d function but as

+

-*

-

and P a q

( f i n i t e ) o r nearly

1 6 (&-X), 6 (2 MV (5-X)) = Therefore we have, i n t e g r a t i n g t h e 6 f m c t i o n , 2Mv

for large

V

and we have functions of x

2

-q /2Nv only.

I f we w e d s e a l a r partons, the formula f o r vU2 i s =changed

but Wl

0.

Zf the f x a c t i a n of the partonr, hi&a s p i n l / 2 (weighed by t h e i r @harm sgusred) a t momentum x i e v(%) we get U

2 =changed,

ZMWl

-

y(x)

f( X ) .

The r a t i o

(r

R 8

2 ' i . O

O t

i n the s c a l i n g l f d t wouZd be

In the r e g i m s t u d i e d R is of order '18 2 110 s o s c a l a r partons,

y

is l a s s than .2, not m y

A nrucfi w r e l i k e l y hypothesis is

The value of R reminbng today e x p e r i m n t a l l y bciag due Ca our h a d n g n o t enough e n e r w and

to be f u l l y i n t h e s c a l i n g l i m i t .

E.g.,

R

q Z / t would a l s o f i t

the data and give zero i n the s c a l i n g l i m i t ,

This i~a very profound c o n c l u s ~ nabout t h e s t r w t u r e of t h e u n h r l y i n g theory f o r hadrons.

We m w t mtch t o s e e i f R r e a l l y does approach zero a s the

s c a l i n g l i m i t is reached.

If charged partoas a l l c a r r i e d the f t n z d a n e a l charge 2e then charge squarred i s l f n our wits and thus we could say m l y 18% of t h e momntuae, is c a r r i e d by charged partons i n t h e proton (because i x f ( z t ) r t t m i n h r , 82X would be c a r r i e d by n e u t r a l s ,

m

. l 8 t .@l)

- the

%%.is 18% is s u r p r i s i n g l y sstall.

If t h e partans a r e quarks and c a r r y charges l i k e k 2 1 3 ~ 2 1 / 3 the percentage o f wmntuar c a r r i e d by t h e quslxks could be fiie;her,

We s h a l l dShicuss such a wile1 i n

Illore d e t a i l , but i t turne o u t even then i t is necessary to a s e u m s o m t h i n g e l s e n e u t r a l c a r r i e s p e r t of the momentum.

The i d e a s leading t o the s c a l i n g f o m u l a e (27,5)

e r e very r e l i a b l e ,

By

m k i n g f u r t h e r ~tssumptlonsabout the wee region we can understsnd o t h e r aspect8 of

and

kl;?.

It muat be realized we a r e now elaborating on our o r i g i n a l parton

i d e a deve1oping them f u r t h e r t o wderstand m r e f e a t u r e s of the f ( x ) curve, data f o r vW;? s e e m t o approach a constant (,32) as x

J.

ate

O which nteansr t h a t f(x)

would go as , 3 2 1 ~showing a nrem n u a e r of partons r i s i n g as

X

f a l l s i n t o the

wee regian (@U& t h a t the na&er a f wee partons is f i n i t e and independent of P,

and t h e m a n t o t a l nu*er with P),

of partons i n a stata? of raanrentuta P r i s e s logarithmically

% f a i s not e n t i r e l y uaexpected, i t is the s a m a s the distrZbutian of

products i n hadronic c o l l i ~ i o n s . The way t h i s appears t o happen can be gathered by studping the f i e l d theory equations a t high energy, and a l s o the perturbation

t h e o v of: larelshlstrdlung. i n a dk/E =

dPZ

I n the l a t t e r case n e u t r a l parricfedt a r e generated -+

dxlx distribution,

These neutral@ can generate

p a i r s eo the srnall x region contains l a r g e a d nearly equal n d e r s of p e r t i c l e a and a n t i p a r t i c l e s ,

The f i e l d equation approach suggests the

thing; and

S-

f u r a e r t h a t the law region is @aerated i n higher order perturbation from the higher m m a t a by a s e r i e s of cascades x

-c X'

-, X" going d m i a x.

fn either

cakte we conclu& the character of the pczift3 will be as a whole n e u t r a l and therefore the sby other m-

f o r proton ttnd neutron,

n e s e expectations have been made

also, f o r e x e l e L ethe , dxlx leads t o a constant cross s e c t i o n

f o r v l r t w l photons (of filred but l a r g e , negative mws sqruared),

That these

cross section8 should be a constant is elepected fram cmsidexations t h a t these photms have an aaplitudcr t o he v i r t u a l fiadrons ( l i k e the? constant cross s e c t i s n a ( t h e w g i c word 9-ron" the s a m f o r proton m d neutron.

p ) and

hadrgns give

is w a d t o "elrl>lainf' t h i s ) ,

'Thus we expect vW

a s x -c O aa 2n t o equal vW2~ ;tadeed they t u n t out t o do; both f(x) f o r p and a approach the same .32/x fox

smll a.

We a l s o lmar experimentally t h a t a t q2

The facc t h a t f ( x ) s e e m t o go as , 3 2 1 ~as x

0 -c

Q

TP

t o 32 a t 16 &V.

O i q L i e a that the t o t a l

cross s e c t i o n f o r v i r t w l photons of energy v on protans is a conetant a s energy goes t o does, x

f o r fixed w d l a r g e negative q2, j u s t as t h e r e a l photon cross s e c t i o n

We c m expect, a s long a s we a r e an the high s i d e of t h e wee region, t h a t

b/P v l t h b large enou*

Eenw the tr-vecee

t h a t , sar our I s a a e u r e m t s t e l l us, f(x)

virtual photon c r a s s sect&= $8

,321~~

i n the aealing I f a t .

And for x on the h i & s i d e of t h e wee region (-q

2

large

At t h i s point we m y well g m s s (these considerations a r e independent of t h e parton m&l) *at

happens f o r v l a r g e b u t -qL i s n o t l a r g e enough t h a t

-q 2 /ZHv is f a r enough o u t of the wee region

l &VIP.

t h e total, photo-cross s e c t i o n i s independent of v.

Ve know f o r q2

-

O

For o t h e r -qL, t h e tarreliable

2 2 YDH would give a f a c t o r -m /(q -m$

2 2 2 i n t h e amplitude o r 11/(l-q /m ) ) i n t h e P cross s e c t i o n t i m e a v independent t e r n ( l a r g e v>. %Fa i s c l e a r l y wrong f o r P

l a r g e -q 2 ( f a r from t h e p pole) f o r i t f a l l s as (-q2)-2, we have gust seen Ss experimatalliy found, above,

i n s t e a d o f (-q2)-I

But we can c e r t a i n l y guess

t h a t , f o r each -q2 t h e croas s e c t i o n i s c o n s t w . ~but dependent on -q

-

at -q

2

.

We do not a t present know C(-q

4n2e2 c(-~'),

as

2

) except f o r q2

m

2

f o r l a r g e v,

O and l a r g e

We need a good theol-y f o r t h i s functian. t h e a m b e r of partons is i n f i n i t e , t h e laonrentum contained i n

Althau*

them is f i n i t e , of course, because t h e total. nrsaatuan o f a l l p a r t a m n e u t r a l and charged is 1, ( u n i t s of P),

We have f o r t h e wmrztm c a r r i e d by a l l charged

partons weighed by e2 0

F o m d a for R @e may g e t

calculating a

fdea of haw R behaves a s we approach t h e ecafing limit by d i r e c t l y i n t h i s region,

kle work i n t h e system i n which q is

pure space-like

P

U.

q

m

(E, P,00)

=

(0, -2Px, 0, 0)

%v

4,qv - 2

Use Equation (27.4) with a l l q, put back i n t o

2

= diag. (1, 0, -l, -1)

Nar t o get ns

r(C-r)-k)).

(the expression f n the 6 function becomes ( 4

coqonent af both sides; however, i f the HIS were exact the

we want the t

z coargonent would exactly veulish ( f r m gauge invariance) .To avoid e r r o r s of t-z, v

differencer of large q w t i t i e s we take p

piu pZv +

a

t-z so 6

)1v

vmisks,

- plZ) (cZ - pZS) ( t h i ~is equivalent t o replacing the polerizatiaa

( E ~

of the photon e B = (1. 0, 0, O) by eB'

. NW

r e s u l t by gauge invarimce)

; (l,

1, 0, a); we should g e t the same

pZz rune backwards so cl

-

But pIz runs Eamard ea E 1 Plz 2(2x -{)p. 2 2 Here kL+m is the man perpendicular mnsnttlm plus the -ss2

- pZz - 2 [pZzI -C

-12

, &2* 2 p

of the partoo (at E),

a a t e v a r t h a t mans, but a t l e a s t i t i a the square of same fgnlnite energy.

as

and R a r e given by

This e x p l l c i t dewadence an mL is arrmeous.

The deviations i n the parton

w d e l of calculations u s h g f r e e partons a t e probably t o set 215 (a bbdlsxg energy correction) uncertainties oa a l l e f f e c t i v e parton Basses, the e r r o r c-B

i s assudmg i n cl

- plz

f r e e parton and a s t w r e cmplieated,

&at

E

1

Xn t h i s s p e c i a l case

,is j u s t the k i n e t i c energy of the

When operatots involve d / d t i n Schroedinger

perturbation theary t h e i r exact expreasisn is u s w l l y m r e cmplieated than j u s t the b h e t l c energy operator,

The fomula f o r B should be

The elr;pected f a l l i n g trend with l/(-q

the e r r o r s a r e large.

' l I f we estgmate *

hadronic colXisione, a d m 2

2

is s o a m l ) , we find R

B * .3

2

) is not evident i n the data, but

.25(GeV)

2

indepndent of x as i n

of the sattte order (else we could not explain why

* (2t46)/(-q2)

; f o r -q2

7 (near center of data)

1 &A/? not unreasonrrtrle c q a r e d t o the average

.l.@.

Photon-Hadr~nInreractl'ans Lecture 29

We c m get so=

q u a l i t a t i v e ideas of t h e region near x = 1 by f i r s t etudying the

e x t r e m cease x = I, the e l a s t i c form f a c t o r s , a t l a r g e coordinate system with q = (0, -Q, m m n t m of t h e proton

IJI

ii

opposite d i r e c t i o n ,

0, 0) a pure space-like vector.

(E, P, 0, 0) must have P

the proton has mmantm P'= 'b: 4 ql

vl-r

We take t h e

= (E,

-P, O I Q)

Q I 2 , a f t e r the c o l l i s i o n with z component i n t h e

For l a r g e Q, hence l a r g e P, we can describe t h e i n i t i a l

s t a t e of t h e proton a s an m l i t u d e f o r various configurations ( i n W

Pp t h e

suppose f o r example, the canflguratlon contains two non-ee

X,

kL).

Let

partone and two

wee, and draw a picture:

Before

Figure 29 ,l

low we ask t h a t the f i n a l s t a t e ( t h e ""after" "piere) t h a t is, EL f a c t o r
-

be only a proton

-P I " a f t e r s t a t e t f > ,the (complex conjugate o f )

q l i t u d e t h a t t h e a f t e r s t a t e appears i n t h e proton wave function,

But a l e f t

arovlng proton can never ( f o r l a r g e P) look l i k e the a f t e r p i c t u r e because i t contains a backward nulving parton (x2), (and any estimate of its =ss square i s 2

order P

.

requiring order

binding energy contributions t o get the ,ss2

dovn

2 t o N 1. P Thw I n s o f a r e s the proton looka l i k e " b e f ~ r e ' 9 t contributes very l i t t l e

t o e l m t i c scattering. *ere

To s c a t t e r e l a s t i c a l l y we nrust s t a r t with a configuration

one parton has nearly a l l t h e m e n t u r n , x = J and a l l the r e s t are wee,

say below l / P .

Before

Figure 29.2

Parran Model Now t h e siningle f a s t one is reversed by the i a c o a n g photon, t h e energy

is n o t changed

@U&,

and the f i n a l configuration is possible f o r a proton

beeawe wee partons (those with absolute pz f i n i t e say below order l GeV) can ga backwards a s w e l l

ss forwards.

Therefore t h e s c a t t e r i a g e l i t u d e ( f o m

f a c t o r ) is proporrlonal t o t h e square of the ~ l i t u d et h a t a proton looks l i k e '%afore1\ hence proportional t o t h e p r o b a b i l i t y t h a t the proton has a l l partons but one i n the wee region. We now s e e q m l i t a t i v e l y why t h i s should be e o m gnverse pcxnirer of 9 i n the following m n n e r .

If t h e bramW&lung a n a l y s i s of the "sea" "(dx/x)

region has any q u a l i t a t i v e r e a l i t y we can judge t h e p r o b a b i l i t y t h a t no p a r t i c l e s have t h e i r x above no o f order l/Q t o be e-= above t h i s smd c is 80-

where

n is

t h e mean n m b e r expected

e m s t a n t t o allow f o r correlritiona (unlike f i r s t order

b r e w t r a h l u n g which i s purely Poisson, perhaps each ""poton" could make two of some o t h e r o b j e c t , e t c . , s o t h a t the mm rider of p a r t i c l e s would be higher but proportional t o s o m vaguely defined "man n d e r of s t a r i s t i e a l l y indeprmdmt eventst"

).

Now we have seen, f o r smfl x a t f e a s t ,

another constant a I n xo where

X

1

- hence n,

R

gaea

the mean nvmber a b ~ v exo is a

o is s o w urnknm upper l i m i t where t h e dx/x i d e a s break down. m

%us f o r small enough xo, e-"

t i e s a power of xo,

v a r i e s as e r p (ca (h

- I n xl))

That is, t h e p r o b a b i l i t y no Lfsea'"articles

m

B

cmstant

are outside

the wee region f a l l s a s a power of Q, There m y be so=

s p e c i a l partons t o m k e up t o t a l quantum a m b e r s whieh

a r e not parts. of the s e a which mwt a l s o be i n t h e wee region, s t u d i e s of the f i e l d equations a s well

But s c a l i n g

0 t h t~h e~o r e t i c a l a r g m n t e ( r e l a t e d

( ~ 3

t o kgge-type behavior) suggest very s t r o n g l y t h a t they would show behavior the f o m

X'

dxlx where

> 0,

of

and thus t h e i r p r o b a b i l i t y t o be found i n t h e

region belaw xo would alsto f a l l a s a power o f xo, We conclude t h a t the p r o b a b i l i t y t h e proton looks l i k e our parton near \

X

l with a l l the o t h e r s below s a m low linait

X

f a l l s a s a pawer of xor say

xi.

In partPcular then f o r x wee of order l / Q , we f l n d the p r o b a b i l i t y varying as Q-, Thus the f o r a f a c t o r GM of a proton (and GE a l s o , s e e belaw) should f a l l a s a power of Q, Q - ~ , Experiraentally y is probably a l i t t l e Larger than 4,

The f a c t t h a t the f o m f a c t o r f a l l s toward zero as Q proton (as P

+ W)

has a zero a w l i t u d e t o appear

=am t h a t a

-c

a ssingle partan.

mia

m y be becawe no d n ~ z eparton has the precise quantunt n d e r s of t h e proton, but even i f i t did, a = r e cogent reason is t h a t the probability of finding a p a r t m without o t h e r f i e l d s generated by i n t e r a c t i o n fro= f t (represented by t;he s e a d x l x ) is zero, (Just m

it3

expected fram t h e theory of bre-aahlung).

our a n a l y s i s of the r a t i o R

os/ot described i n the previovs l e c t u r e c m

be aas w e l l applled t o t h e e l a a t i c piece x

*ere

2 LA + m2

2 d is the value f o r the s i n g l e parton with nearly a l l the menturn.

But t h i s r a t i o

(M

-

expressed i n t e r m of GE, GM is s i m l y

proton mass).

for large q

2

experiment.

1, We therefore find

.

Thus we expect GE/GM t o approach a constant

-

2

(L,

+ m2

215

i/k

A g r e a t p r a b l ~ a ti s f a m d when we estimate the constant from

I f GM*lrGe

disturbingly small!

2 2.79we find (liA+m2 t

Y

But data a t q2

about .6 (and f a l l i n g ? ) s o (k,

2

+ m2

&l1

.l1 ( G ~ V I ~

1

2.6 even shows GE/CM is l e s s than 1,

2

All

c ,041

Haw we can s e e haw f(x) should vary f o r x near 1, say

1 iy with y staall.

X

X.

Let is f i n d the t a t a l probabilfty that: x exceeds X. -y i . e , again f i n d one parton with nearly a l l the m m n t m ,

y whi&

kte

f(x)dx* tJt?

=Ugt

t o t a l mnrc?;ntiun of a l l

That is mat the s a w a s the probrilbility t h a t none

the olfhenit c a o t exceed y ,

can exceed x

the!

-Y

worked out a s YmY, but a 1 i t t X a coneidexation of the

problela soon s k w s t h a t Z t does vary with the s a m pswer, but with a dff f e r e n t constant i n f r o n t , o r a l t e m a t i v r ? l y one d e t a i l t o check f C,)

( m i s is w e t e a s i l y seen by C B work ~

Wenca

ot

kind of d i m n s i o n e l analysis;

out t h e case of a &x/x Poisson d i s t r i b u t i o n i n

Parron Model f o r emall y, o r

f o r x a e a r l.

That is, we expect the power law of the e l a s t i c s c a t t e r i n g t o he

l, In p a r t i c u l a r we r e l a t e d t o t h e power law o f t h e bebavior o r f ( x ) near x 3 expect f [X) a [l X) or. a l i t t l e higher, Data is not i n disagreentent with

-

this,

(Thia r e l a t i o n was f i r s t published by W e l l snd Yan.)

Can we extend these ideas t o g e t s e m understilnding of why t h e

Reeanances.

2

resmances f a l l with about the s a m power of Q f o r l a r g e Q a s does t h e e l a s t i c (Here we a r e studying t h e region l a r g e q2,2Nv

peak?

:M

+ 2Hv

= b?

-

2

3 -q2,more

precisely

Perhaps i t is b e t t e r t a franLly t u r n t h e

(-q ) f i n i t e . )

q u ~ s t i a n&out the o t h e r way rlnd ask what the f a c t t h a t the reaonancee f a l l off a s t h e e l a e t S c peak with -q

resanmces

2

tells

uct

about the wave function of these

.

I f we e x c i t e say t h e A

C

our p i c t u r e again c m a o t look lLke f i g u r e 29.1

'+

f o r the "'after" p i c t u r e could n o t be a d w d n g rapidly t o t h e l e f t becausle i t contains s o w partons going i n t h e wrong d i r e c t i o n .

a s i n f i g u r e 29.2,

We a u s t b s t e a d bave

but t h i s time fslatead of asking whether t h e last p i c t u r e

looks l i k e e protonswe k s k w i t h what amplitude doen i t leak 1Lke a A*? again expect a 6'

t o bave a sea region, e t c ,

by a f a c t o r p4'2(in

non-wee parton,

, and

Me

t o have a d i f f i c u l t y maaured

amplitude, f o r a proton 97/2)to

a p p a r with a s i n g l e

We conclude from e x p e d m n t t h e y' prob&1y equals y

can probably invent a p o s t e r i o r i a r g m n t : t o j w t i f y t h i s ,

- and

(For exaarple* the

s e a may be very e i d l a r i n the two c m e s becausse they a r e generated by t h e saw

- t h a t is,

partm

the f a c t o r =asurea

t h e chance t h i s parton is bare of non-wee p a r t i c l e s is what

- and i t is the aame parton

Fn e i t h e r ctlse.)

B e wee partons

l e f t aver from t h e o r i g i n a l r i g h t nroving protons have s n l y a c e r t a i n a a r p l i t u h t o be 2n t h e c o r r e c t proportions (of n d e r s non-9 dependent raomnta, etc.) appropriate t o a l e f t - m v i n g proton

.

3.

t o a Ll

- and a

X t i s the r a t i o of these non*

CO be

d i f f e r e n t a q l i t u d e t o b e appropriate

dependeat a q l i t u d e s (aquared) which

rlc?ter&nes t h e ultireate r a t i o o f t h e p r o b a b i l i t y of producing bLh+t o protaa a t 2

large Q

.

(Xt is always possible i n p r i n c i p l e of course t h a t f o r some p a r t i c u l a r

rescmance t h i s a~tplituldei s zero and such a resonarsee would f a l l f a s t e r with Q than t h e e l a s t i e peak, ) TZlese r e l a t i o n s of power laws f o r e l a s t i c peak, resonanees, and ~ y m p t o t i c f(x) a r e s o i n t e r e s t i n g t h a t i t is i a p o r t m t t o n o t i c e t h a t they can be expressed i n a general p r i n c i p l e (I am stron@y indebted t o J.D. Bjorken f o r d i s c w s i o n s on t h i s idea) q u i t e a p a r t from any parron theoxy i n t e r p r e t a t i o n ,

Stated i n

t h i s way they a r e seen t o be r a t h e r profound and represent a fuldaatental new property of high energ)r c o l l i s i o n s ( f o r corresponding r e l a t i o n s a r e expected m l a t i n g exclugive and i n c l u s i v e hadron reactions a t high energy) , Suppose we p l o t uW2 f o r a given l a r g e q2 against

<- -

ZHu(l-X) 2 ( X is near l and Znv * H and 2 l-Xa r e just proportional. Ve f i n d a s e t a f resanances a t appropriate Mx

2 M -q (l-X)

2 -q ). We note the s c a l e of: H

and the t a i l of t h e "background1'cum (l-x)'-'dx

NW when we increase q 2

as (q I-'

2

o r (-q2)7

(<-M2)'-ld(~;)*

2 a l l the resonances f a l l (but s t a y i n t h e a m e place' Mx ),

a s we have seen, but the sire of the '%background1' canre

a l s o f a l l s a s (q2j'.

-

( a t a given

That i a t h e "bsckground" (by background I mean t h e high x

t a l l of t h e a c a l i n g region ftrnction f ( X ) ) bear8 a fixed r a t i o t o t h e reaotlancea, P 2 I t is n o t possible by going t o higher q (or int hadronie c o l l i e l o n e t o higher s ) t o s e p a r a t e even w r e c l e a r l y t h e resonance@ from t h e backgrowd, nor do t h e resonance8 f a l l l n t o the background md get l o s t .

Put another way, the back-

ground c m be c o n s i s t e n t l y dewed ae due t o ovigrlapping reeonances o r t a i l s of reeonanees i t s e l f and a l l reeonmees u l t i m t e l y f a l l o f f t h e saw w a y (as a p m e r of q

2

2

as q

cowlsultantary

increases).

wily@

Bohr would l i k e t h i s , he would say t h e r e are two

of viewing t h e background, a s continuum singLa p a r t i c l e

s t a t e s leadfng t o ~ c a l f n gexpectations, o r arr l a r g e a d e r r a of resonanme

- and

nature conspires to make i t r ioapessible t o decide i n the interatrildiat;e r e d o n (large: H

- 2)which .i

correct.

2

)

Parton Model Lecture 30

We w i l l ar&e t h e a s s m p t i o n t h a t f o r any s t a t e (e.g. p o r A ) a9 mmntum Q inerernes the d i s t r i b u t i o n =ng

wee x

(m

f i n i t e motaenturn) approaches a dclfinitcl

Rather, =ore e x p l i c i t l y f o r aux c m e , we? a s s u m

d f s t r i b u t i m independent of Q.

t h a t the anplitude f o r t h e proton t o be a s i n g l e parton and a given d i s t r i b u t i o n aarclng were raaolkentrnn ( e , g, helm 2 &V)

and nothing i n between f a l l s somehow with

Q but the distxgbution amag the weres approaches a d e f b i t e a s y w t o t e , (e.g. a w l i t u d e f o r no w e e l q l i t u d e f o r one of nromntm 300 b V / m l i t & e f o r two a t

100 MeV and 400 MeV e t c . b e c m conbstmt r a t i o s ) ,
1, weee a t Ewmnta k ' s sf proton a t Q>= F (Q)f (kfs) P psi

( t h e index 1 i n d i c a t e s parton e w e ) and s i l n i l a r l y f o r a resonance,

Hence the

s c a t t e r i n g of a photon, which turns a r o m d the f a s t parton from Q t o Qv(e.l~;.

tn a coordinate system &ere q

U

(Q-Q\ 0 , 0, W')).

The a q l i t u d e is proportional t o

We e r ~ p i l i e i t l ysuppose t h e l a s t f a c t o r i e not zero. OP

q u m t m n-er

(e.g.

(For c e r t a i n resonance@,

Z spin) reasons i t ndght be zero, those, m y f a l l f a s t e r

w l c h Q, but we study the ones which a r e nor s p e c i a l i n t h i s way,)

The answer

mwt be r e l a r i v l s t i c a l l y S n v a r i m t , however, and thus be a function only of q2

at.Xenee

F

P

(9) Pb(Q1) is a flmction of W' only, from which we can

deduce t h a t both F (Q) end FA(Q) must be of t h e form l / ( 2 ~ ) ' / ~ (a constant P can be abaorbed i n t h e second factor).

m- aw. p where s(p, b)

-+

-2 i

A

( q 2 ) q / 2 a ( p , A)

Ja,iOfAif-E>

d

3

dapeade, on t h e s t a t e s p, A, but not an q

2

.

Photon-EdronInteractions

146

I f two s t a t e s have d i f f e r e n t powers of Cj i n t h e i r P(Q) must be orthogonal i n t h e sense j ~ ~ { l r ) f ~ ( - d3k k)

O

then they

- usually because

of

d i f f e r e n t q w n t m n a e r s , l i k e char@ strangeness o r isospin {or z angular m m n t w ) i n t h e wee s y s t e a , Ttre p r i n c i p l e discussed a t t h e end of the previous l e c t u r e very l i k e l y applies t o high energy hadxron c o l l i s i o n s also; although not our d i r e c t i n t e r e s t i n t h i s course, Z w i l l b r i e f l y o u t l n e hov i t &ght w r k there. t h+B Consider an exclusive reaction ( a t fixed o r zero mmntsan t r a n s f e r Z

G

=+

Resonance X, where W, B, C, a r e fixed,

e.m. be p and t h e outgoing wmentlm, a f C

Let t h e incaning w m n t a i n the

- - pc

pn, s

kp2 goes t o i n f i n i t y ;

snd imagine we measure the ~ s s i n gmass of the resonance by

(Ph + Pg 2

-

-MA

2

sfl-X)

=

CHBZ-~t.

+ MA2

-$fl s (1-X) .

-+

PR

Thw p l o t resonance data as l? m

2 , of course, but say a s s'~(d)d ( a is negatfve);

As s v a r i e s resonance@ s tay a t fixed

tbir s i z e goes d m v e w l i k e l y a s a power of a ,

a s expeeted f o r exclusive reactions, where a is the l e a s t negative p o s s i b i l i t y Ear

an exchmge of s o m t h i n g t h a t a c c o w t s f o r the q u m t u n w b e r changes fro= A t o Bat s i n c e we are not r e a l l y looking a t the products as t m i n c l w i v a reaction &ere,

It

5 this

t h i s s c a l i n g region near

X

f o r x near l,

must go

X

&v. Hence

only.

2

= l w f l l go i n t o the resonmces f o r large l4 only i f

e a ~ ( ~ 2 j d n=Z s'~(s(1-x))de(l-X) f m c t i o n P(.(%-1))

can be thougtxt of

has been a r g w d , (R.P. Felyn-,Fhys,

L e t t , 23 1415 (1969)) the grclbabillty should be a f m c t i o n of

C,

W

1s a function of (~(2-x))'-'

X

m l y , not o f s .

Hence t h l s

and the scaled r e s u l t goes

M

(l-x)*-'dx

(There are s i t u a r f o n s , f o r e x m l e , where C and A a r e both proton@

t h a t 10: is t e e a i c a l l p hard t o g e t t o h i & enough x t o avoid c o n t d a a t i c m of p's disintegzrareb from o t h e r hadron resona~lcesC',

etc., but these m e t e r s cannot

canerem us i n d e t a i l here; t b e r a s u l t has n a t bem r e a d i l y checked, therefore .)

r i z e what we know o r surrnfse about v? RegLon

Z)

h r g e v , laxse -q

2

2

but -q /W

x m&, cailled f@).

5

X

2

(-Q /v

2

and W1 i n the high v rct&on.

fhite, )Wl.

2ml i a r a function

of

2

v

Region 11)

Large v , fixed -q

Repion III)

2 Large v, l a r g e -q but f i n i t e

88

-+

W.

ZMv C-M~

(The l a s t f a c t o r 2Hv is the normalization from :nd

2

(-q )

.

t o dxl..

In order t h a t t h e three regions f i t together we have,

(X;

1x1

(XI; I I I )

f o r smZl

X,

for large

eq2,

f (X) goes a s a/x 2 g(-q ) goes as a.

(a

.32)

f o r x near 1, f(x) goes a s h(1-xly-l -y 4 or 5 2 2 2 2 1-1 for large h (Hx -H ) goea a s A(% -M )

M:-2,

In region I we e x p e c t s l a t t a f a l l w%tb tnereasing I n region I X we expect uslot to approach soas V

+

v as

11-0o r l/(-q 21.

f i n i t e L i m i t &pending on -q

-, (This mans vW2 i s f i n i t e , M2 is of the sme order

asl

2

2

2

(-q /v )W

I

but fs n o t equal t o i t ,)

In region ZIf ve expect os/et t o f a l l with increasing 2 2 I n the s p e c i a l region o f srrmll q2: W2 -q Wl/v + order q 4 / v ;

v as 1/v.

2

as= q ,

f a l l i n g c o e f f i c i e n t (q2)-'/

2

( a l q W).I n t h e diecussion o f the previous l e c t u r e

we saw p m e r e where y could depend on the q w t u r n n m b e r s of the wee group, like t o t a l isospio, m g u l a r lamntum e t c .

and there is a lowest one. dfscrolte; V

-y0

y

These a r e d i s c r e t e s o t h e yiVe are d i s c r e t e

Emever, t b r e is trrmsverse mlaentm which is not

could ctepend on the transverse liloentraxt of the 2

+ ylkA

f"asl~parton

s a y as

s o the r e s u l t i s

Naturally we should l i k e t o ga f u r t h e r and find o u t xnortr. about whaP qumtw

Photon-Badron Interactions n u d e r s t h e partons c a r r y .

& i t t u r n s o u t from present e x p e r i m n t s only a few

things can be l e a r n e d , but f u t u r e e x p e r l m n t s c m add i n f a m t i o n which g r e a t l y r e s t r i c t s the possibilities.

We d i s c u s s t h i s by choosing an example, t h a t

charged parcons c a r r y t h e quanturn n m b e r s of quarks.

(Zn such a model Gell-Mann's

equal time c u r r e n t tlomutation r e l a t i o n s a r e a u t o m t i c a l l y s a t i s f i e d . )

We a r e

the t h r e e quark m d e l , t h e n m b e r of quarks m y be, indeed must be,

not =king inffnite,

h e is i m d i a t e l y s t r u c k w i t h t h e question a s t o whether our explanation of s c a l i n g i a p o s s i b l e a t a l l i f t h e partons do n o t have i n t e g e r charges.

The

i d e a was t h a t the outgoing parton (now quark) could escape without f u r t h e r l a r g e (as P) i n t e r a c t i o n t o u l t i m a t e l y r r m l v e i t s e l f i n t o an outgoing bundle of

hadrons

.

But i t fs a quark (knowing no low energy hadrons a r e of charge 213 say) i t cilnnol t u r n i n t o hadrons u n l e s s i t picks up t h e e x t r a l 1 3 charge by dragglng along another parton quark.

I f t h i s e x t r a parton i s from t h e ongoing ones o f Itlomntm

of o r d e r P t h e m m n t m changes ( d t r e c t l o n B t o A) would be of o r d e r P and t h e a s u q t i o n t h a t bending f o r c e s a r e small, apparently required i n o u r d e r i v a t i o n , wauld f a i l ,

QR t h e o t h e r hand, i t is p o s s i b l e peffiaps t h a t t h e e x t r a quarks

a r e found m n g t h e wee partons, which a r e n a t u r a l t o my hadron wave f w c t i o n and s i n c e hadrons golng i n d i r e c t i o n A or B s h a r e t h e s a m wee region they could exchmgo quarks t h e r e , =king

up t h e necessary i n t e g e r s .

T h i @ is n o t c l e a r l y

s a t Z s f a c t o r y but i t is an e x c i t i n g adventure t o t r y the i d e a t h a t charged partons a r e simply quarks, s e a consequences, devise experiments t o test them; and i f they a r e found t o succeed, t o r e t u r n with more z e s t t o the q w s t i o n o f how Nature

mwt then be r e s o l v i n g t h e apparent: paradox among: a.

Partons c a r r y quark quantum numbers

b.

Wadrms do n o t

c.

S c a l i n g works,

Ln t h i s s p i r i t we look t o discovering by what m a n s we can check ( a ) .

At

the very l e a s t we a r e giving an e x a w l e of how experirnenr could lead t o f u r t h e r i d e n t i f i c a t i o n of the character of partons. To describe the parton d i s t r i b u t i o n i n a proton we would have s i x functions, u(x) d(x)

= No, of up quarks with coomntm dam

=

Similarly

<(X),

"

z(x) and

It

$1

X

to *dx

if tt

8s

art?, t h e n d e r s of anti-up, a n t i - d m and

anti-strange quarks with momntum between

X

and x+dx i n t h e proton.

The t o t a l charge on a proton is +l s o

The stranrjfeness i s zero so 1 fs(x) s(x)]dx Q.

-

-

These equations have the s o l u t i o n fu(x)

- -u(x)]dx

= 2

That i s , the n e t number of each kind of quark is j u s t the number o f t h a t k i d S-n the simple n o n - r e l a t i v i s t i c $-quark model,

(X) from 2W1 P ( o r vM2/x) is t h e nuabet of each kind weified by t h e square of t h e charge:

The observed function f

hence

Measurements on the ep s c a t t e r i n g of course give us only t h i s sum and we cannot e x t r a c t the s e p a r a t e t e r m from t h i s alone, neutron s c a t t e r i n g too,

On t h e o t h e r hand we have d a t a on

The neutron i s given by the same f o m d a except u(x)

would n6w be the nu&er of up-quarks in t h e neutron,

However, by ieospin

r e f l e c t i o n the neutron functions a r e got from chose of the proton by replacing u by d, d by u.

Thus, i f u(x) c o n t l n w s t o b e a s defined, the nuraber of

Photon-HadronInteractions

150

" u p s 8 9 n t h e proton is a l s o t h e n d e r of d m s i n t h e neutron, we have 1

f n (X)

(U(X)+ ; ( X ) 1 +

4

(d ( x ) + x ( x )

We have d a t a on both these functions,

+ $ (S (x)+F(x) )

The r a t i o fn/f

P

behaves l l k e

It has been suggested t h a t possibly f I f l-x f a l l i n g t o zero a s X + O, We n p s e e i m d i a t e l y t h a t t h i s is impossible i f charged partoas a r e quarks; s i n c e

u(x),

et^,

;;(X)

, are

c m n o t f a l l below .25,

a21 p o s i t i v e E I f

The o r i g i n a l d a t a

only went up t o &out x -. .8 and hall f a l l e n t o perhaps , 3 s o ' 2 5 was a possible fixnft a s

X -+

1 but we have s i n c e learned t h a t there w a s a ariatake i n the computer

p r o g r m and i t is c l o s e r t o .4 f o r

X

near l,

However, J u s t f o r an exercise, & a t could we deduce i f f / f was shorn t o be n P jwt 114 m x + l ? That m a n s d ( x ) f a l l s t o zero f a s t e r than u(x) a s x I. This -+

would r e s u l t i f we supposed t h a t t h e .r ( i n (q2)""y o r (l-xly-') t o t a l i s o s p i n zero is e m l l e r than the

y

f o r wee partone of

f o r i s o s p i n one (only one non-ee

Then t h e leadin@ t e r n ( l a w s t y) would require t h a t t b e f a s t parton

parton).

have the s m e i s o s p i n a s t h e proton, up-quark only (since t h e wees a r e i s o s p i n zero),

However, i n t h a t case t h e p r o b & i l i t y of e x c i t i n g t h e A resonance (of

isospfn 3/21 would f a l l f a s t e r than t h e e l m t i c

peak f o r one can only make a

A f r o a wees with i s o s p i n m e (as t h e leading quark can only c o n t r i b u t e 1/21.

Thus we would expect: t h e

resonance t o f a l l off with q

2 f a s t e r than o t h e r s

(data is i n s u f f i c i e n t t o i n d i c a t e whether t h i s is t r u e o r f a l s e i n f a c t ) . f a l l s helm 1 i t "almost P t h a t alraost s u r e l y (?) the

We can a r g w ( i f partons a r e quarks) t h a t i f fn/f surely" fafa8 t o 114 a s f o l l w s : values f o r

y

for different q u m t m n d e r a

one t h e r e f o r e dominates. we have

We f i r a t ass-

for I

-

-t

and t h a t

F1rst mong t h e s i m l e r ones of zero s t r w g e n e s s quarks

O and yl f o r I

have eventually (as x

the wee a r e = e q u a l

l) a pure L

l. Hence depending on which is smaller we o.

(Z

o r pure I

l amng the wees.

For X

t h e non-rJee quark awt be (112, 112) or u, t h i s gives i m d i a t e l y f n /If P

O

2 (charge of d ) /(charge of u j 2 = 1/4.

If I

m

1 t h e non-vee quark can be u o r d (+l)d

with weight given by the appropriate Clebsch-Gordan c o e f f i c i e n t p = (@)U, n

m (0)d -

-

I-l)u therefore

Bexi= cn the a 8 a u q t i m of unequal y ' s e i t h e r f / 4 ox 312 is the limit ( t h e r e f o r e i f below 1, i t is 1/41.

(There i s t h e p o s s i b i l i t y t h a t the aan-ee

antlquark, i n char case i f Z

O dominates f / f

n

P

quark is an

213 but t h i s p o s s l b f l i t y is

considered very unlikely f o r p and a ) , Another s t a t e m y be wees of strangeness + l and the non-ee

pure

X t is

S,

unlikely t h i s is e a s i e r than t h e non-strange cme, b u t even ea the r a t i o would be one, hence excluded. Several ntodels have been m d e wlzich p r e d i c t r a t i o 8 b e l w l but: above l / 4 . A l l of these a s s u m t h e lack of c o r r e l a t i o n of f a s t quark quantum, nu&ers and wee s t a t e charlscter which

t~rz e x p l f c f t l y

a r g m d f o r , t h a t is, they i w l y a degeneracy

yl = Y0 i n which case, af course, o t h e r values a r e possible, The reader cm show t h a t the r a t i o of GM f a r proton and neutron c a m s o u t 2

2

= -1.4 a t q 0). P n 'fhe case t h a t t h e t o t a l iangular m o w n t m among the wees is zero c~houldibe t r i e d -2 f o r i s a s p i n zero weea, for: l a r g e -q

first.

(it s t a r t s a t p /p

E

Show a l s o t h a t GM f o r the proton is p o s i t i v e f o r l a r g e -q2.

s a na s i g n

change i n necessary as q2 r i s e s , which is s a t i s f a c t o r y became e x p e r i m n t a l l y

GM f o r t h e protan never seems t o f a l l t o zera, 2 G ~ I G-r ~O and G; is p o s i t i v e a t l a r g e -q

Isaspfn one f o r t h e weas gives

.

Lecture 32

Another i n t e r e a t l n g stuity is t h e t o t a l rnorrrenturn c a r r i e d by each kind o f quark, let U

I[ufx)

+ ;(.)]X

d~: D

m

[d(x) + ;i(n)lx dx;

S

m

+ z(x)]r

be t h e t o t a l laoaentm c a r r i e d by up (plus anti-up) quiatks, e t c . =an we knaw

/x

fp(x)dx

'18, I n fa(.)&

.l2 s o

dn

Now i f we a s s w e t h a t &ere a r e no n e u t r a l partons beside quarks (e.g. no "gluons")

a l l t h e eomntwn? of t h e proton i s quarks, we would have l

U + B + S

Solving these equations we have U unreasonable,

- that

m

,76,

This r e s u l t i o c l e a r l y

314 of the momentum of t h e non-strange object proton is t o

be found i n t h e s t r a n g e q w r k s , cannot be quarks

'21, D = .03, S

- t h e r e mst be

m i s t n d i c a t e s most s t r o n g l y t h a t a l l partons sernre n e u t r a l ones carrying =menturn,

say N.

This

nwmenrw can be thought of a s i n t h e f i e l d , i f any, by which the quarks i n t e r a c t ; (possibly f o r exaatple a n e u t r a l pseudovector) which i f represented I n f i e l d theory by i n t e m d i a t e f i e l d partons a r e usually c a l l e d gluons.

I n the region near

X

4

O the functions f

P

and f

pondiag t o i n f i n i t e riders of quarks and antiquarks. is f i n i t e , so@ eq. 31.2 and diseussioa.)

behavior s o i f u(x)

diverge a s .32/x,

(The n e t n m b e r of q u a r k

and f must have the Barme P u(x) d ( x ) = dfx) , Cfct do

Further f

a / x f o r s m l T x then

-

;(X)

not think we could prove s(x) and i t s equal

:(X)

corres-

-

nuust equal t h a t value akeo

because SU(3) i s not p e r f e c t , but i t rnay n o t be f a r m a y ( i f indeed i t is not e q u a l ) , Art i n t e r e s t i n g q u a n t i t y is the raeasure of how u and d approach each o t h e r as X

4

0; although each is i n f i n i t e , , t h e d i f f e r e n c e is f i n i t e 9nd i n t e g r a b h .

Experiment gives about .l7 f o r the i n t e g r a l

/

\fp(x)

- fn(x) ldx but

0

i s l a r g e f o r i t &pends p r e c i s e l y cm the satall d i f f e r e n c e near

X

the e r m r

O (as i t is

S o w people (as f o r e x m p l e Efaschos, Wetaskopf and Kuti) have t r i e d t o guess the functions u, d, e t c . , by =king

simplifying a s s m p t i o n s h i c k a r e chosen f o r

t h e i r sinq7licity r a t h e r than t h e i r physical n e c e s s i t y ,

Such p r e d i c t i o n s may o r m y

not succeed, i f they f a i l a l l we s a y is n a t u r e is n o t s o s i w l e ,

Unlike t h e i d e a

o f the parton model we have w e d eo f a r , the additional asrsmptione made i n these

raodele have i n general na physical backing. we, give an e x m g l e of such a eheoxy here:

I n ease any of you a r e I n t e r w t e d

It: i s t h a t the wees o r dx/x region

153

Parton Model is m d e e n t i r e l y of m c o x r e l a t e d p a i r s , and t h a t on t o p of these there? a r e three "valence quarks" of tbe kind of the three quark rnodel anand the wave fuarction of these is a a i q l e product of independent pieces, aixnplicity,)

That 18, i f u(x) is t h e t y p i c a l s e a d i s t r i b u t i o n and v ( x ) t h a t o f

a valence quark we would w r i t e U

=

a+2v

f o r the proton,

U

J1

v(x) dx

*"a

f ( t o g e t 32., 2 f .

t o i n f i n i t y ( a s . 4 4 / x ) but x v(x) w m n t a c a r r i e d by quarks S and S

*"

(We do not expect such

lr

O

+

as

x

-t

Near x

-*

O i t is a(x) that. goes

0. This p r e d i c t s f o r the t o t a l

2D -U s o t h e d a t a (32 ,l) gives U = .36, B

O ( s u r p r i s e ) hence we a r e i n

S-

-18

kind of trouble ( a cannot be zero) 2

mlears t h e experilaenta2. e r r o r s allow A t any r a t e some atonrenturn (46%) must be i n gluons,

It p r e d i c t s

(fp(x) -fn(x) )dx

1

5

which is hard but n o t iorpossible t o r e c m c i l e with t h e experimnfal, -17. I t a l s o p r e d l c t s vWZa/vW 2~ f a s t e r than v.

-*

2 / 3 (hencc probably wrong) by supp~sII118t h a t a x

So Ear no d i r e c t d a t a is accurate enou* charged partonr a r e quarks,

+

1,s f a l l s

t o t e e t o r e l i & n a t e the i d e a t h a t

However f u r t h e r experinrents c m ,

We give two e x w l e s ,

neutrino arcartering and polarized e l e c t r o n and proton s c a t t e r i n g ,

*

Me erhall df s c w s the i n e l a s t i c s c a t t e r i n g of neutrinos from proton g plus anything i n d e t a i l i n the l a s t p a r t of t h i s course.

-c

u

IJe da not k n w weak

i n t e r a c t i o n theorry at3 weld, ari3 quantupl electroctyna&ca but can use tkasa experim a t s themselves t o t e s t i t (@,g, i f i n t e r a c t i o n f s n o t a point-like four fernion f n t e r a e t i m s c a l i n g w i l l f a i l i n the farm derived below)- but s w p o a i n g i t : is a l l r i g h t f o r now, we suppose the coupling with quarks i e with t h e quark quantum numbers from

is Gabibbo's angle, s i n we talsc

= 0,

2

\

.06.

(F y It (l+iy51v) ( ~ ' y , ( l + t ~ ~ ) ~ )

qmrk t o cos Be d C s i n BC

For

s t l r ~ n p lai i t y

s

where

of our rough diecueaion here

154

Photog-H~droaInteractions %an i f we s c a t t e r

ox

(z t o 3 .

quarks

to P

+ it

up quarks (swding

Thus we would expecc t o lDeasure j w t

U

+ ;Ii f

U

t o d)

the curreat

The A is just l i k e V f o r refatOviatic

were pure vector, b a t i t is V-A. p a r t i c l e s (except the

can a f f e c t an&

is not reversed for a n t i p a r t i c l e s as i t is f o r V.)

sllpx

Xf for the V-A current irnatrix ele-t

we write

i n analogy t o e l e e t r o d p d c s we f i n d t h a t tbe ganeral form by canvention is

-

(4 P P W UV2

nUV

- 4 2 6 vv 2-

2

W

% paqAw3] ~ ~ ~

(leaving out a l l tern proportional t o q the U, v end proportional t o (=ss

U

~

~

(33.2)

beeawe chey give a m l l e f f e c t s a t

2

P) ). Ml,

W2 are a s i n @ l e c t r g c i t ex*Pt ~

they contain W and GA (nearly e q w l ) ; the new Erm W cowtp from interfereace VA. 3

In the scaling re&oo vW2, 2W1, vUJ scale to three functions f2(x), f 1.(X), 2

.

flvP

= z(;

flv"

=

r;P

= 2(u

(x = -q /2Mv)

-

If partons have spin 112, f 2

+a

2 sin 6 )

~ ( d u+ cos2 e + s

sin e)

-

rlv"

+

2

d coa 6

+ a .o,2e

~ ( + d

COS

2

2

+ B s i n2 8) B

+ 9s i n2 B)

- s sin28) fin = 2(X - u cos2 B - s i n2 8) 2 2 fjYp = -2(u - CI coe 8 - s s i n B) 2 2 B s i n 8) fp -2(d r i p

= 2(;-

2 d cos 8

S

U COB

B

Thus we have many relations, f o r e x a w l e

flM

- f j vP =4(d cos2 B i s i n2 8) - r3vn =4(u cos2 e + s s i n 2 8)

P:f

+

flvP

-

*

8

-

P f;

=4(;i cos2e

See a l s o Appendix B

+

2

s i n B)

2 x fl

.

fj(r)

For the quark m d e l we get

2

Neglecting s i n 8 4

; flYP +

4

a iV" +f

f3vP

-

Sum rules I

(flYP

- flvp)dx

-

'(f3YP

+ f;Pjdx

the f i r s t is Adlet's

-

slrrn

2 (2

-2(-2

- cos2 8)

2

- cos281%-6

r u l e (net quark laorfel depenhnt)

, the

t h i r d was f i r s t

derived by Llwellyn Smgth and Gross, Distribution predictions

-

- f3Yp

r*

+

2(-(U-;)

-6(feg

2

cos t3 (4-;i)

- fen) + s i n2@

+ s i n2 B ((s-F}

(S-;}}

- (d-3))

(33.7)

the l a s t term cm probably be neglected, It gives a very c l o ~ echeck on the =del. Alrao

6 cos2ec (fep

- fen)

+ s i n2 B

((S-;)

- (d-;i))

by subtraction with the prev;toue, relation c m Ire used t o e l i & a a t e the last term. i f them was any doubt, B , . .

Knarledge of f

ivp, fep, fen

a l l a r s us to obtpin

separately m d fgVP permits f u r t h e r separation i n t o 2 sin 8

U,

G, d,

+ G, 3 + d,

a, s +

+s

neglecting

Ifthe electron b e m and the proton i e polarized i n the deep i n e l a e t i c

s c a t t e r i n g we caa o b t a h a t i l l mm i n f o m t i o n , t h i s t i m about haw the partons are polrrrized i n s polarized proton.

all the charged partons have spin 312,

For definiteness we s h a l l sera-

For a p a r t i c l e a t r e s t the s t a t e =ay be described by saying the s p i n is i h saw p a r t i c u l a r space d i r e c t i o n

a.

h l a t i v i s t i c a l l y t r m s f o m e d s p i n s t a t e s can be

h s c r i b a d by a u n i t four-vector polar

satisfying

S

lu

( t h e t r a n s f o m d s is a s though i t were a

1 and ss p m 0 (pli = oaamntrtm o f s t a t e ) , GJhen U it taking mtrix e l e w n t s we can use s p u r s and sm over s t a t e s i f we replace the vetctsar)

B 13

S1

wua2 p r o j e c t i o n o p e r a t a r into the spin s t a t e

sit,

v

+ m by

the more coraplete

This s f~ a pseudowetar. li

W)

(i-iys&)/2 t o p r o j e c t

The s p i n s t a t e of p i n t h e

JV f a c t o r of

need not be t h e sane as the s p i n of p i n the

f

U

f a c t o r ; but the quantity we want

clttpendr; on s p i n aa a. 2 x 2 w t r i x (density ntatrix) one of w o s t a t e s h, one of two o u t .

However a l l the i n f o r a t i o n i s contained i f the diagonal i s k n m f o r

a l l s t a t e s , n o t anly f o r p

p ' = +z o r

but a l s o f a r p = p"

--E

f o r example (%n f a c t , these: t h r e e md p = p ? we consider only diagonal caaes p

s p i n p'

-X

W

S

X

= ((4-z)

+

a r e enough i n our ease), but f o r various

S p.

(-z)fi

Thw

WPv can then

be shorn ut be of rhe f o m

(disregarding t e r n praportional to Q Gl,

G2 a r e two new functions of q

2

U

o r qv) where W1, W2 a r e a s before and

, v,

What do we expect f o r these new functions G1, G 2 in the deep i n e l a s t i c region accclrding t o the partcln model?

Xf a parton has wmntunr pl = xP and

s p i n deseribed by wli the s c a t t e r i n g t o p2

-

pl +q is described by

Let us use f o r example t h e coordinate system where t h e photon has pure z c w a n e nt

NW

f i r s t suppose the h e l i c i t y of the protan is

-4,

thus

S

(P,

E,

0, O)/M,

Parron Model

l57

Further l e t

2

*

(charge)

h-(X) = No. of partons

*

(charge)' with h e l i e i t y

-

f o r hel.

+ and

The 6 f m c t i o n gives G(&-x)/ZPiu i n t h e s c a l i n g

lilnit,

w e have

[(P1,

W"

cl, D, @)/m

-"/m

+

with tzelicity 3. i n a proton of h e l ,

h+(x) = No, a f gartons

- i n a proton of wY

-

-

hel.

f i ~ ' / r n for hel.

-.

W@ t h e r e f o r e have

t h e r e s u l t s we already k n w 2Wl

f (xf

-

h+*-

The f i r a t i n t e g r a l gfves

xf(X).

and v?

second

f n t e g r a l gives (G1

2~'v

- 2xG2) = h+(n) - h_(.)

therefore the? Eunctbn G1

-2

X

G2 scales..

Suppose now t h a t the proton i s polarized i n + S'

W'

= (0, Q, 1, O f f o r parton wfth pol.

+X

and

X W'

d i r e e t i o n , we have

-

-s" f o r pox, -X,

k+(x) = No, of partona

*

(charge) 2 with p a l .

f

k-(x)

*

(charge)' with pol.

- x +a n proton of pol. -

-

No. of parmns

x i n a protan of pole + X.

We have an expression of the type (33,111 with h+ repkaced by k+ and h- replaced by k- from which we derive

3 therefore the function 2M vGl

+

2 2 2M v E2 scales.

2 3 Henee from (33.12) and (33.13) E?v G2 s c a l e s and Hv Cl s c a l e s and we w r i t e

2

I4 vG2

-

g1(x1.

2 Hv G2

g2 (X).

These a r e the s c a l i n g p r e d i c t i o n s f o r s p i n cases.

The term i o G2 i n (33.12) is r e l a t i v e l y n e g l i g i b l e (order l / v times term i n G1). We have

Of course, these new functions gl, g2 cannot be obtained fro% f(x) because new d i a t r $ b u t i a m a r e involved, the n d e r of partom with various spin8 is a t a t e s of various k e l i c i t g , so t h a t we &rive nothing but the scaling rulea even i f we use a s p e c i a l 111ode1l i k e qmrks, e t c .

However, i n t h a t =del we cm obtafn a

remrkable sum r u l e of Bjorken. Me can w e our proton wave f m c t i o n (described by partan) not only t o obtain predictlone f o r deep i n e l a s t i c s c a l l n g but l i k e any other weve function w e can write the ernectation value of varioua operators f o r i t ,

We have already done

'-W '.

t h a t t r i v i a l l y when we m o t e j

v

a8 the operaro

t o t a l charge on the proton t o be expressed as

(uZ)

- 7l (d-3) - 7j-l (e

NOW we n % l l d i s c w s the expectation of a leas t r i v i a l operator, the a x i a l B

decily operator f o r p

pi^'Av In>

i(;

y

-+

Y

p v 5 a

n trmeitions,

It is wrLtten empirically as

1 (G /G ) end e q e r i m e n t a l l y GA/GV A

V

1.23 2 .02.

"t

$.

i n the quark m d e l the operator .lAu is the sun over quarlcs of .r y y fr P 5

Theoretically '4-

I spin

r a i s i n g OR). Xf we change the X spln t a t h i r d mwonent we can use the r e l a t i o n

We calculate both s l d e s f o r

it

p m t m w v i n g f a s t t o the r i @ t tlith mmntraa f

and spin a' and sunr on proton s t a t e s via a spur Operator correctly projecting. the l e f t s i d e we have the correspanang sum on quark@, thus

d with m i g h t

+

1 / Z f o r u,

G

-

1/2 f o r d,

2

f ~ ~ / ~ ~ ) s ' ~Wow j 2 the . @urn

on the l e f t is exactly the i n t e g r a l over x of 2 gl, (or 2(g1+g2) depending on the? s p i n CM@)& a t we needed f o r deep i n e l a s t i c ep s c a t t e r i n g except the weighte i n the l a t t e r case were 4 / 9 f o r

U,

u,1/9 f a r d,

a and 119 f o r a ,

g

for

z, S/9 for d, X, 1/9 f o r s, f o r the neutron) whL3ie here we want 1/2 f o r u, ,; - 1/2 f a r d, z. But the difference of gl f o r proton glp mid neutron gin has weights 1 / 3 f o r U, U, - 113 f o r 4, iio r 2/3 of & a t the prtrjton (or 119 f o r

we want,

U,

Hence we have f o r the two p o k r i z a t i o n cases

On

-

L,

((33.16) is BJorbn's relation).

From these we conclude

(p, g2n)dx 0. 0 This coaes merely from the equality of the coefficient f o r the polarization z and x cases, were;

dtn

equality which would have t o hold no HItitter Mhat the weights

so we can conclude i n d i d d w l l y

1

g2dx = O

a reault e5 rotational, s

(33.18) t r y (angular etomntraa conseirvatlon)

.

Tne f i r s t relatlon is more remrkable and would r e s t simdtaneously the

quark view, and the interpretation of the weak current a s y (1.t.i~ ) not only for 5

the e l e m a t a r y p a r t i c l e s e

, Ve,

Pu"V

If

,

s

but a l s o f o r the elearentarg constitmnt.

quarks (or quark gumturn ambers on spin 112 partoas).

It@ verification, o r

f a i l u r e , would have a mast decisive e f f e c t on the direction aE future high energy theoretical physics,

Tests of the Parton Mode

Lecture 34

g u l a r raomentwet considerations ( t h a t 0 t h e proton is s p i n 1/2 and t h e parton is s p i n 2/21. This b r i n g s up a general research prablea about msngular o;xlraf;ntum and p a r t on wave f m c t i o n ~( fttneeZons of

X

and transverse raomnturn k L ) . The dependearn o f these functions on the

t r m s v e r s e w m n t a kA is not knom, and a r e very i n t e r e a t t a g tchearetically , C m my i n f o m t i o n an t h i s , o r o t h e r questions, be got by studying t h e angular

m m n t w p r o p e r t i e s of parcon wave functions? i f a y an the wave ffunetian corn

For e z a q l e , h a t r e s r r i e t i a n s

from the f a c t t h a t the t o t a l rmgular mmnturn

of the proton is j m t 1/2? Or, suppose t h e parton p r o b a b i l i t y d f s t r i b u t i o n f(x) were k n m for a d o f h e l i c i t y f o r a 15 of h e l i c l t y

+

f

312, What could you say about the d i s t r i b u t i o n

1/2?

D r e l l has suggested m o t h e r expergmat which can b e predicted by h m x e d g e of the parton d i s t r i b u t i a n .

Again we w e t h e parton as

quark w d e l t o describe

t h e idea, b u t i t can of course be analyzed the @amway i n o t h e r representations f o r parcone. p

+ i;)

The r e a c t i o a i a p

+p

-t

p+p- -4 anything.

a t very high s (e.g. P ia CH is l a r g e ) d e s a

f 66

Proton plus proton (ar il+iiep

a ~ whose r energy is

proportional t o P, and whose z colnpment of mnsentuon is of order P, as P

-i.

m.

'fhe p i c t u r e is

the 1.1 p a i r Ss m d e from the a n n i h i l a t i o n o f a parton i n proton A and anti-partan

ia proton B and t h e o t h e r partons j u s t keep gain8 on t o m&e f i n a l hadrona, the By s u i t a b l e choice of t h e energy and momntmn ( i n z d i r e c t i o n ) of

'knything".

rhe p's we can d e t e M n e xl and x2 of t h e itnnihilating partons,

(Thw if,

i n the CM, t h e morm?ntm of the ~ 0 l l i d i n gprotons is P, and t h e energy -and t o t a l z component masentm of the p 'e i r r

E

pZ, we have by conservation of atomntm

and energy

I t w i l l be e a s i e s t t o nreasme the p a i r a t l a r g e angles ( t h e angular d i s t r l -

bution is 3.

+

2 cos 8) s o no great: l o s s of intentlity occurs here, and backgrowd

hadrons which s h ~ u l dm v e primarily i n z a r e e t i o n a r e avoided. chosen h m e v e r , t h e raornenta of t h e two p ' s give d i r e c t l y P x l , P%

If 8

O La

becauaa i n such

a one-dirartntianal c o l l i s i o n of r e l a t f v f s t f c p a r t i c l e s t h e oldlnoentm of the f i n a l p a r t i c l e s is t h e s a w a s t h a t of t h e i n c i d e n t p a r t i c l e s , The cross s e c t i o n f o r t h i s is t h e crose, s e c t i o n f o r r e l a t i v i s t i c p a i r 4 a n n i h i l a t i o n ( e (l

* cos2 B

partons t o a n n i h i l a t e ; e.g.

2

}/l6 E ) tianes t h e chance G of flstding the approprfate for p

+ g,

i f partons a r e quarks:

To be v a l i d , o f coume, the wee region must be avoided ( t h a t is, P m u s t be

high mough t h a t Pxl, Px2 a r e l a r g e c q a r e d t o 1. &V)

.

One might a t f i rsc be

concerned t h a t the / P ~ / PBeate > we s t a r t with i e n o t a s t a t i o n a r y s t a t e , for

protons s c a t t e r v i a strong interactions

SO

the p o s s i b i l i t y e x i s t s t h a t the parton

d i s t r i b u t i o n s are f i r s t distullbed by ""strong" "taractions to d

e

U'S.

bef0ll.e they annihflam

But we a r e co&ng t o mderstand t h a t the "strong interactioagFg

are not so strong i n t h i s senae

- t h a t i n hadron4adron collieions

the farrt partons t h a t i n t e r a c t , but the wees.

i t is not

ThLs accomts f o r low mmentm

t r m a f e r m d s c a l i n g i n very hLgb energy hadron caljlfstons

- so

CIS

15ng

88

we

avoid t h i s wee i n t e r a c t i n g region (interactions vjia f i n i t e energies, not of order F) the p a r t a m a c t aa f r e e p a r t i c l e s nearly

((as

P

-+ m ) .

Obvfously with enough data we could see? i f a form l i k e (34.1) could work a r i f we ham

U,

G etc.

-

from neutrino s c a t t e r i n g we could t e s t i t a r e c t l y .

h e t u r e 35

Gronau has painted out that with no further data than we already have we could calculate t h i s f a c t o r C i n a t l e a s t one region, Lnw a l l the frmefions U, as/%.

u

d.

3 go

as a/rl with the same a , snd

B.

9. as

&ere as m y be close t o a (but not necessarily exactly equal, say a -8 with 8 anurll; SUS seys as = a but i t

consider (a

G,

Xf xl is sm11, we

I)

W

iiir

incorrect),

0, end then say what the e f f e c t of i t i a .

We s h a l l f i m t

Hence i n t h i s region

.24) we have

But the l e s t f a c t o r i s j u s t fep[x), so C i t s predictable %n t h i s region,

the Z W l

f o r the proton and i a Lnovn,

It s h w s t h a t Drell's experimnt could

e a s i l y serve aa a test: of m y parton =del

- and w i l l be an i n t e r e s t i n g e x p e r i m a t

t o watch* I f I) is not zero we get

so i t s e e m that there is l i t t l e e f f e c t from any reason&le B , for z l , x X 2smll

For exaraple,

Tests of the Farton Model a very weak dependence on B ,

+

I n a similar way we could describe the proton lntiprocon ease m-+u+uX t would be detemined by a f a c t a r

anything,

" l . , "

Zf f i n a l l y , the functions u, u e t c . a r e d e t e d n e d f o r the proton, the correrponding functions f o r the. picn %(X), (dif f t c u l t t o get enough energy) a

+p

e t e . ccufd be obtained Ebrougb

%(X)

+n

+ PM ''

f o r nar

Drell has suggested t h a t anather f u n d e n t a g experigletnt m y yield infornation about the character of partons.

Xt i s e++em

+

ang hedrans a t hi& energy,

W

would expect to pmduee some spln l f 2 partan p a i r of &arge ei,

I

2 (et/@) uo *ere

+- + e E? +p v-].

oo i a the cross aiectfon f o r a p h 112 p a i r prodaction (e.g.

Now t h i s pair wouM turn i n t o hadrons

- and i f the energy i s hlgh

anough,af nearly the a m energy a s tb virtutil s t a t e of a parton p a i r , i f there were a nmber af parton p a i r types of &arges el, e2 e t c . ,

crass section oe+*--,

Thus

the t a t e l

htrdrons is j u s t the total. to m&e each pair:

That is, the xirtfo approrseheer a w s t a n t , and the constant i s tbe sunn of

squares of psrton charge s

d on each type t h a t e x i s t s .

I f there a r e any

1 1 .e 2 f a r each spin zero spin O partgns they would contribute . i;a s strongly (e.g. B parton)

.

We c e l l D

e t s ' D r e l l 's conatant!'

For example i f parmnr a r e quarLa we expect

1

:e

$ + $ + $-

2/3 f o r D.

If partons mst carry i n t e g r a l charges, and be spin 112, the sm rnust be a t l e a s t

l.

I n the c a s e of quarks we would have t h e d k f f i c u l t i e s tre have disregarded

a s t o how a p a i r o f oppositely m v i n g charges of hadrons of i n t e g r a l charge,

m e r e lnay be

2 213 say can t u r n i n t o a n l p

way t o g e t the required e x t r a

S-

quarks gram a s o f t s e a of p a i r s , buc one can perhaps l e g i t l m t e l y q w s t i o n whether the above a r g m e a t s a r e e n t i r e l y c l e a r , f;nvolviag as they do an assumption t h a t the parton s t a t e s go e n t i r e l y i n t a badmn s t a r e s . t a do t h i s and orait s t a t e s o f quark quantum numbers?

1s i t r e a l l y c o n s i s t e n t It; would be most e x c i t i n g

i f the r a t i o does corn o u t 2 f 3 a s we311 i t rai@t, Ear t h e t h e o r e t i c a l qucrstions prodwed by such a simple answer would be very i n t e r e s t i n g . be worthwhile aa an

I think i t would

e x e r c i s e beforehand t o a s s m e t h e r e s u l t was = a l l y 213 and

s e e what paradoxes, i f any, would then have t o be resolved.

-

As we discussed previously t h e o r e t i c a l l y ( l e c t u r e 5 ) , the t o t a l c r o s s s e c t i o n

f o r o(e'om

any hadrons) es a function of q

2

(q2 = 48, E

€24 energy) determines

t h e vacuum expectation value of t h e product of c u r r e n t s

The vacurn s c a t t e r i n g of v i r t u a l photons by v i r t u a l hadrons is d e t e w n e d by

P.T.

< Q /( J ~ ( ~ ) J ~ lTI@> (z)

-

(qwqv

where dispersion theory t o l d ue Xm(iv)

(1

-

2

1 P

2

dyV)v(q 2 p(q 1,

+ above quantity) i s t h e f a c t o r by whieb a photon propagator I l q 2 must be

m u l t i p l i e d f o r f i r s t order e f f e c t of v f r t m l hadrons b e e ,

The low e n e r m t e s t s , l i k e e f f e e t on L a b shift depend j u s t on the towest

q

2

o r only on ivl((0) t h a t is "roday we can begln t o say something q u i t e d e t a i l e d about t h i s , f o r i n the

region (2to,) production.

2

t o J u e t above 1 &V

a t h e croas s e c t i o n is d m i n a t e d

2 Perhaps p(q ) s e t t l e s down t o D/6n f o r l a r g e

by t h e p , w, Q,

4 f a i r l y soon.

Although one d g h t suppose we d g h t have t o go deeidedliy above t h e nucleon

Tests Crf the Partsn Model

l65

p a i r c r e a t i o n , t h i s .may n o t be t r u e ; s o f t p a r t i c l e production my dominate a t a l l energies and reach its asyatptote sooner. A , Cisneroa has calculated the various hadranic contributions t o i v ' (0) , The contributions from t h e p, u and (a turn out t o be ten tims l a r g e r than

the l a r g e q2 contribution under reasonable assumptions about the l i k e l y value of 2 Dre11's constant and t h e value of q2 a t vhich p(q ) a t t a i n s i t s asymptote.

Data frem e'e-

i n t e r s e c t i n g r i n g s can be used d i r e c t l y t o obtafn t h e

contribution from the p ,

The d a t a is s i n p l y i n a e t t e d i n t h e formula

to obtain

(%at was used h e r e was the d a t a on the pion f o m f a c t o r which is alnrost tsvevthing t h e r e is i n y l v i r e u a f )

I,

p

-c

hadrone,

The 4n i n e l a s t i c i t y i s very

small even a t q Z = (.B G~v)').

+ we

In t h e case of t h e w and

a s s u m W M , which works w e l l i n t h i s type

In t h i s case tthe cantr2bution from the vector =son v = w o r QI is

of processr.

2 Prom the r e l a t i o n In (iv(q ) )

1

p(q

2

1 we deduce

For a very narrow resonance, which i s a good approxfrncrtion f o r w and 4, 2

we have p(q )

2 2 mv h-22n6(q ):m-

the value of ivf(0) i s simply

(35 -131 (This formula gives i v ' (0) P

5.3 n I O -''~V&

f o r t h e 0, i n good agreement with

what was obtained above w i n g the d a t a d i r e c t l y . ) and Q, a r e t h e r e f o r e

The contrrlbutions of t h e w

We now evaluate the cantribution from the ""tail" attained its asynrptotic v a l m D/6n a t q 2

%h' is

gives ivPt(O)

.84

X

2

qo

is as l o w se 1 GeV2 and D

10~'~ q ~ -I f:~ q .

is the q w r k value 2 / 3 we have .56

X

of p(q 2 ) a s s w n g i t has

10""~

f o r ivVt(O). We believe the

a c t u a l value of the non-vector meon emedbutiorx not t o be mueh l a r g e r than t h i s , we =sign t o i t lOOX uncertainty,

Adding the varioutt contributitoas w e get

ivy01 =t7,3ti,a)xia-2~e~-2 hadrons This corresgonda t o a correction t o the mwetaic

of the mwns of

2 For comparison, iv'(0) f o r muons i s [4n 15 M

1.1

15.3

X

10-'

&vw2#

s a the hadxonic: contributiaa is about one half a@large, Away from q2*

0, the contributions t o i(v(q 2 1 - (0)) from the t a i l of p(q 2 )

grow a s the vector =son

contributlions f a l l , s o the m c e r t a t n t i e s a r e greater,

h o t h e r expectation of i n t e r e s t from the parton w d e l is the anlyular d i s t r f butioo of hadronlic products i n high energy e'e-

-

collisions.

2 Ue expect t h a t partens of s p i n 212 arc2 produced with an (&+cos 8) rusgular dilsrrjtbutim and t h a t t h e f i n d obsemed hadroas w i l l have smll transverse nromnta r e l a t i v e to the direction 8 &n &which the partons were p r o h e e d a t hi& energy,

aou&

Iba b u r s t s of oppositely w v h hadrms ~ a r e ewected which w i l l determine

the m g f e 8,

f f &ere a r e aleo charged p a r t m s of spln O the angular d l s t r i b u t i a n

w i l l be (l

+ cos 2 e) + l Do

(1

- cos28)

f sonertarxts f o r partons of spin 112 cvrd spin O where D and D are ~ c e l % 112 2 reepectively, For q w r b rse expect of course only a ( l + cos B) d i s t r i b u t i o n ,

astic Scattering as Properties of Operators

We s h o d d now l i k e t o turn back t o a p ~ i n tof vfew, described more part i c u l a r l y i n the f i r s t Lectures of this course, of the e l e c t r o d p a d c properties of hadrons as being; raeasuremmts of properties of the current operators J (1). M For e x q l e there we s a i d we expected t h a t the corrtrnutetor [Jv(2), f (1) 1 would It vanish i f 1, 2 were outside each o t h e r s l i g h t cone, Sacmd order i n t e r a c t i o n s were described by matrix e l e m n t s of a second operator V

vU

(2, 1) which w e s h w e d

would be ezpressible as a time ordered product of the f f r s t ones J

(written

iJ

iJv(2)

1) IT)

As we have seen the e p s c a t t e r i n g m a s u r e s a function which we have w r i t t e n as

Ve now consider i t i n a .more abbstraet way.

F i r s t we note t h a t i f the s t a t e s

p and x are imagined te contain t h e i r c e a t e r of mss raglnentusl f a c t o r s , t h e

2 2 2n6(NX -(pfq) )would be m e c e s s a r y , Eor the c l a s s of s t a r e s

X'

now

man ancl

Thus K

i s t h e expectation on the protoll of the product of two operators,

V@

We have measured i t only f o r qZ

O but i t e x i s t s a l s o f o r qZ > O and of

course t h e e n t i r e function f o r a l l q2 both p o s i t i v e and negative belongs together theoretically.

For p o s i t i v e tqZ, however, &ere is a small technical point.

We

wish to a l t e r t h e d e f i n i t i o n because we do n o t wish d i a g r m which do n o t a f f e c t the proton a t a l l , thus

proton

Such t e r n s do n o t exist f o r the non-diagonal m t r i ~ elements of the product of o p e r a t a r s term < $ / Jv(-qv) Jy(q) / p > and we should l i k e t o define K ss t h e l i m i t of t h i s a s $

-"

p t o o b t a i n a s u i t a b l y useful, d e f i n i t i o n of matrix elelrnents

of t h e product operator.

We can g e t the s a w r e s u l t i f we explJlcitly saubtract

t h e contrlbutlons of t h e discomected d i a g r a m of t h e type above, write

-

~ , ()- 4 ) ~ , ( 9 ) / p > ~ ~ ~ (' d 9

Hence we

-

Then i f t h e f i n a l p is changed t o p b l i g h t l y d i f f e r e n t , t h e expression def b e d by (36,l) is coat inuous , klca would l i k e t o consider, f o r t h e o r e t i c a l pwpases, v a r i o w o t h e r current@

beside eleetromagnetie nanat) (i.e.

- we can have various $U3 generalizations

i n a n o c t e t (or

currents with d i f f e r e n t charge n d e r t l ) and a A a l currentls a s well.

Thus we l e t these indices (SU3 and a x i a l ) as w e l l ss t h e P, v i n d i c e s of space be contained i n a s i n g l e l e t t e r A ( c a l l i n g J v ( l )

A(1) e t c . ) t o save w r i t i n g

i n general a r g w n t s , aad can always go back and i n s e r t i n d i c e s a t the end, Thus i f A is any operator of our allowed s e t ( f o r our e x a q l e e , a vector o r

a x t a l vector l o c a l current i n SU3 o c t e t o r possibly s i n g l e t ) we w r i t e (obviously the diagonal proton a t a t e can be generalized i m d i a t e f y too, b u t we leave chat f o r the s t u h n t ) ,

Inelastic Scattering as Propertigs of Operafors And l a s t l y we omit w r i t i n g the l a s t t e r n Ear s h p l i c i t y , but you awt

remxnber t h a t , f o r diagonal lsatrix e l e m a t s i t i s there. %A is the diagonal r ~ a t r i xelerrrent o f t h e product of two operators,

Evidently we c m d;gflne s corresponding thing i n space:

K depends only on the d i f f e r e n c e

Since the diagonal e

q

-

2p

-

Or

We e x p l i c i t l y c a l l t h e coaponents of

2-1, its Fourier transform is KBA(q).

{V, -$l i n the system where t h e p r o t m is a t r e s t (others v i a r e l a t i v i s t i c

tranafomtion),

6)

KBA(v.

Thus

2s a t r e a t

m

-C

a11 x

2 I (6)1p>& ( E ~ - E ~ - V I

2Ex

(36.3)

Since K gives t h e product operator we can f i n d from i t the c o m u t a t o r as w e l l as t h e

ordered operator so useful i n s c a t t e r i n g .

tixntj

kle note, i n our case ( i n the r e s t system of the proton), s i n c e the, proton

Is the lowest =ss o f all, s t a t e s of b a ~ y o nn d e r one (and A dwe not: change baqon n d e r ) that

P;x

E

P

f o r a l l x.

Hance i f

no s t a t e can be reached by A(v, q)Ip> i f v

O

V .c

Q the 8 function is zero

- and (Ip>

-

a t reat)

Now t h e carnuttitor m t r i x element is defined by CBA(Z,l)

m

RgA(2, 1)

- Kh8(1*

2,

sa i t s Fourier t r m s f o m @ a t i s t i e s

We note t h a t X can be obtained from G and d e e versa because of ( 3 6 . 4 ) . %(v.

and CB*(-v,

4) -6)

-

cBA(v I

-

-CmLB(v,

6)

v >

Thus

o

(26' 6 )

6).

Thus 9.1% rar?asur&ng%(v,

w-tator

for

Q) ue a r e w a s u r i n g the Fourier trcrnsfom oE the

o f two c u r r e n t s ,

We s h a l l d-isccrss t h e cmsequenws a f t h i s i n t e r e s t i n g r e s d t i n a craomnt, but while we have these equations before u s we wish to derive a fczv Eomulm f o r the s c a t t e r i n g a q l i t u d e whlch we w i l l need l a t e r i n t h e w a r s @ , Aa we have

discussed the s c a t tering q l i t u d a f a r am inceaing photon ( v i r t u a l o r real) coupled t o J (1) say A(1) t o an outgoing one coupled t o B is detts-dned by the operator

we diacusa the e f f e c t of seaguljta ( i f there are any, 6 (2-X)

type terms) l a t e r

and omit them f o r a wbile,

If i n p a r t i c u l a r we are intezleclted i n the Eomard s c a t t e r i n g aw1itude TBA(4 00" a proton with a photon of mmnturn q we need the Pourier t r a a a f o m

(The superscrfipt F indgcates- t h a t the choice of

siw

of the i m g i n a r y part

f o r negatfve frequencies is taken accordgag t o the convention of Fewmn i n h i s

QED papers; there is a dif f e r e a t choice called u w a l amplitude.

which is

often very useful,) To take the Fourier transfofln of the f i r s t t e r n i n ( 3 6 . 8 ) we have a product o f @(t2-tl) & w e P.T.

i s i/(v+ic) and B(Z)A(1)

we have the convolutlian of these,

whose P.T. i s %A(v,

P

use

&t

hence

In the s m e way the a e c ~ n dterm is the

convolution of -%/(v-%E) (the F.T. of @(tl-tZ) and Km(-v,

NOW

6) and

4).)

Hence

inscvr-v>

where we have exp1icitl-y added possible seagull t e r n aBA which L s sinply an u n k n m ( f i n i t e order) polpemial i n v,

8, Thus

the s c a t t e r i n g m p l i t u d e is,

except: f o r a golynmial, ESfven i n t e r n of the comutator, Megatiw

V,

58 of course, not defined by experimnt,

Hawever i t can be

obcdned from nnssasuremnts of the reaction with a n t i p a r t i c l e s (using

A

f o r A)

v i a the connection implied by (36 ,IQ) m d the r e l a t i o n f o r comutatars ( r e s u l t i n g

f nelasric Scattering as Propertl'es of Operators from the f a c t t h a t

a))* -

(c,,(,

is the adjoint operator t o A)

a, - -c,

(V,

41

(-V.

so t h a t (36,101 i m l l e s

F

TBA(v.

6)

-

T&(-w*

41

h o t h e r more- usual convention (causal a ~ l i t u h )t o define the negative frequency exteasion changes the a i m of the imaggnary p a r t f o r negatlve frequencies

a, -

T;~(~,

T;~(~,

*

C

Thus (TgA(v, 9 ) )

-

a, +

c

CBA(~.

.

4)

T s (-v.

and a l s o (37.1) beeoms n m

(vhlch is j u s t as good a way af describing s c a t t e r i n g

The s i p i f i e a n e e o f F

a s T ) i n coordinate space can a m be seen.

e

(=I

given by an expression 1ik

( 3 6 . 9 ) except t h a t the sign of the +ic i n the l a s t term I s reversed,) In (37 .l) we have expressed

things i a terols of the carnutator but the l a s t t e r n caa be

m r e @%=plyexpressed a s the product operator, from (36.5) and (36.6) we have

6)

@(-v) v,(C ,,

-irC

-

-

-KABf-v,

41,the

F.T. (B(2)A(1)IT

f o r t2 2 tX,we have B(P)A(1) A(1)8(2)

- A(I)B(Z)

P.T. of eA(1)B(Z).

Hence (37.1) says

- FeT.A(1)B(2)

- A(l)B(Zf

P

[BQ2), A(1) 1 and f o r t2

have

1.0

. By retarded e m u r a t o r f o r t2 C t

c tl"V

we mean the c o w t a t o r

1:

froar. t h i s , (37.2) i s d i r e c t l y obvious. (Rmrk,

The (3(t -t ) i n 37.3 a t f i r s t s i g h t &es the r e s u l t not relet2 1 t i v i s t i e a l l y invariant, u n t i l i t f a realized t h a t the comutator Eacmr is zero Ear a p a c e l i u regions so the p l a e t2

t1 c= be t i l t e d a r b i t r a r f l y as requrfreci

172

Photon-Hrzdron Interactions

by Lorentz transformation.

This i s true a t lease i f Eke comutatox is not too

singular a t equal times which

it9

often true,

T i A defined here not r e l a t i v i e t i c a l l y

MEficulties s o m t i w a a r i s e A i n g

isvariant unless corresponding non-invar$ant

terms (called Schwinger t e r n ) a r e added i n t o the '%eagullif part of (37.3) .) Expression (37.3) serves as a general definition of the Chew amplitude, i . e , i n space-tinart lit is the retarded comutbitor, even when we do not have the diagonal e l e m n t a , or lowest s t a t e s (so the product operator is O f o r v

0).

The general definlltion of the Feyman ampiftude is the time ordered operator ( 3 6 . 8 ) . They d i f f e r by the product operator.

Note on various refations b a l l t y conditions

Crossing C

6)I *

fTM (v,

-

-

a, T',

P

T ~ ~ ( ~ .

-6)

T&

(-V,

Imginary Part G

iTa

(V,

a))* -

4,

a, - -~G,(V, a)

~,(T;

(37.8)

The crBA s a t i e f y r e a l i t y and croiilsiag relations required t o keep then

valid f a r T, GM*

(v,

Nmlg

41

m

G Z

(-\?p

-4)

-

0 s (v,

The commutator a t equal times t2

vlth respect t o v f o r a l l v, since equal t i m c ~ m u t a ton i relation

61

tl can be obtained by integrating CBA(v,

e" (t2-rl)dv/2n

6 (t2-t l).

6)

Hence Cell ELann 's

173

Inelastic Scartorirzg as Properties of Operafors becomes upon Fourier t r a n s f o m f o r our diagonal element on t h e proton, a,

a constant, Independent of

6,

This is c a l l e d a s u s r u l e .

We now r e t u r n t o our study of the f a c t t h a t i n xneaeurlag R

BA (v, Q) we a r e

m a s u r i n g t h e F,T,

of t h e c o m u t a t o r of two c u r r e n t s ,

we s h a l l ask a r e ( l ) Mhat limitations on the F.T.

As examples of q m s t i o n s

r e s u l t from t h e f a c t t h a t t h e

c o m u t a t o r vanishes outeide the l i g h t cone?

12) From e x p e r i m e n t ~ lf a c t s about

the behaviar of IZ ( e .g, Bjorken s c a l i n g ) &at

do we Ittarn about the character

of t h e c o m u t a t o r ? It behooves us t o study the general behavior of c o m u t a t o r s , and we begin

our study with the c o m u t a e a r of two a c a l a r f i e l d s of mss nn i n a syhstem without

We can express 4C1) i n t e r n s of c r e a t i o n a d m n l h i l a t i o n operators i a the usual way

Were

W

k

=

is the c o r r e c t frequency t o describe the o p e r a t o r ' s

developmat in time f o r t h e r e 1s no i n t e r a c t i m s o the enerC?;y i e ; t h a t of a f r e e particle.

The

commute with each o t h e r , and the %*'S, only a and a* do n o t

c o m t e i f they belong t o the s m e %:

We work out (37.21)

i m e d i a t e l y by fomhlng t h e c o m u t a t o r , w i n g (37.131, of

an expression l i k e (37.12).

Dropping t e r n which obviaualy c o m u t e we a r e l e f t

with:

The a c t u a l i n t e g r a l can now be done

- i t invobea

Here we w i l l j u s t do t h e s p c i a l caae m

-

Bessel functions,

0, and say what the case 'm

$ 0

174

Phorsn-Hadran Inreractions

gives, l e w i a g d e t a i l s t o the st-nt,

-

In (37.14) put

&OS@,

%

-

L for m

-

O t o get

singular 6 f w c t i o n on the 1i&t cone. For f i n i t e

we obtain &e

s h g u f a r i t y on the ligttt cone, zere

outside i t of coursle, aab a Bessel Emetion inside:

Thus we see t h a t the c-utator

f o r f r e e part?iclea is zero outside the l i g h t

c m @ and singular ( l i k e 6 f m c t i o n ) m the l i g h t cone, We helve discussed the singularit2ea of two f i e l d s , dil~cussthe ~ J n g u X a r i t i e ef r m tnuo currents. f i e l d theory J (1)

(2p+ ap)# ( l )

Also i n s t r u c t i v e i$ t a

Qne suck current ~ g h br e i n a free

@ ( l ) f o r spinor f ?ields. M an e x a w l e we w i l l leave out tlzeee gradients, e t c . asld fgnd the c-utator P

6(1)

or

h

for

"currentsW"tat a r e simply squares of a s c a l a r f i e l d

where s ( 2 , l ) i o the f r e e EgeeM c o m a t a t o r we worked out before. 2

has the s a m s i n g u l a r i t y sgn (t2-tl)6(52)

Clearly K

8s a ~ t c dbefore f o r C, but thts t i m

tnu1t;lplied bp an operator function of two pcsitions F f 2 , l ) called often a b i l o c s l operator,

This operator is needed, of course, only m the l i g h t cone.

Its matrix elements give functions of xZY-xlli

( f o r e x q l e of t2-tl f o r a

dtagonal elemnt: i n a system of a partlclie a t r e s t ) .

So i n g e w r a l the

sinlngularities along the cane a r e raodaated by a f m c t i o n of the distance from the origin of the cane, Far interactgag hadrons the e m u t a t o r of currents outeide the liglnt cone also, and non-zero only i w i d e . h w i t &as

the t r ~ s i t i o na s we cross the l i g h t cone,

ir,

elrpected to be; zero

An i n t e r e s t i n g question is

So-

s o r t of jump Is

i n slope? Or possibly i t has 6

Is i t i n value or on*

expected p r e s m b l y .

Such a question is a f m d m n t a l one

function a s the f r e e p a r t i c l e case does.

Me have the Fourier t r m a f o m e x p e r i w n t a l l y i n K

UV

(v,

$3

, B. olngular

behavior corresponds t o so= s o r t of high v, high 4 l l m i t ; so i t is the b e h a d o r a t large v, Q t h a t gives us the answer t o the question. experiaents indicate R(v, Q) s a t i s f i e s BJorken scaling function of 6

But i n t h i s region

- t h a t Is,it

58 only a

2

-q /2&,

The most otxaightfomard way t o a a l y z e t h i s is t o take the inverse t r a n s f o m of K, using the BJarken l i m i t aad aeeisrg what s l n g u l a r i t i e s i t g i m s

- then

concluding the character of the s i n g u l s r i t i e s , because they depend only on the high v, Q I i d r a r e the ame f o r the c m p l e t e function as they a r e f o r the Bjorkcm X i r n i t .

We wZl2 not t r y t o be rigorous, becauee f o r oae thllng we r e a l l y

do not know, experimentally, R f o r q2 Let Q be along the z axis.

h r g s v , QZ is nearly

V

-q2

p

0, nevertheless we can see what happens. ~

i n f a c t qZ = *MC.

~

(q>v)(qz-v) ~ v

ZMvE, ~ hence f o r

The Fourier inverse of e f m c t i o n

o f F, only would look l i k e

where s ( t ) i s the F,T, of the s t r u c t u r e f u t c t i o n f ( F ) , To put i t in anather s l i g h t l y m r e rigorous way we nors

Now omit f o r

41

xaOfDent the 2Hu (suppose we took the txmsforn of M ) and

a

note f a r l ~ r g ev, qa t h a t t o a good enough approltimatten the 6(q 2+ZWvB) can be replaced by 6 ( q 2 + 2 ~ v ~ + d ~ 2So ) . we get f o r large enough v, (q2+ Z M ~ B + ~=B(q+@p12) '

Now we can i n s e r t , i f ve. wlsh, a sgn(v) t o keep the as f o r a F.T. cf a c o r n t a t o r , m d f o r large v t h l s is j u s t sga(*BH] proton is a t r e s t ) .

Hence a s p p t o t i c a l f y

( i f the

1 76

NW

Phutoa-Hadrm Interactions

the F.T. i s easy f o r we know t h a t F.T.

sgn(t)d(s

2

2 sp(vId(q I

s o that f o r a four vector a

hence P,T.W2

noting P*x

- jf

( ~ ) d 8e-iBP*xapn(t)6(t

2

2

-R )

MC we have our previous r e s u l t

- t h a t the significance of Bjorken

ecaling is t h a t the s h g d a r i t y of the current operators has s i n g u l a r i t y on the l i g h t cone.

a,

6 fuaction-like

To be =re precise we w i l l have t o include all.

&e P P factors, etc., and define everything precisely U v gradients of 6-fwctions involved,

- there w i l l then be

The beat way t o eay i t i n a general way is

t h a t scaling shows t h a t the sfslgularitfee on the l i g h t cone of the current carnutatore a r e J w t of the s a m severity as they would be f o r f r e e p a r t i c l e fields, m i a is, of course, what we expect from the parton interpretation f o r there we a s s w d I n the f f n a l stateax(vhich beeme tlhe h t e m d i a t e s t a t e s of the c m a t a t o r ) the partons can be consfdered a@ free,

Thus there Is no surprise

i n the conclusion, i t is m l y t h a t we whLElh t o s t a t e what we have discussed (scaling) in an a b s t r a c t and general wag (as a r r a n t comatator sialngularities) with m%nimm references t a a model.

Although the s t a t e m n t s t i l l s e e m t o r e f e r

t o f r e e p a r t i c l e f i e l d s , that i a only a shorthand t o writing 8 and 8 ' functions on the l i g h t cone, Each general property of partons we assmed s a i d smathing m x e e x p l i c i t about the character of the singularity.

For exmple, t~ r~aycharged partow

are spin 112 i e t o say the a i n g u l a r i t i e s are l i k e those 6 o r t e r i s t i c of f r e e Dirac f i e l d comutatoro, r e s u l t s a r e related.

ffunctions charac-

Ea addition, vector and iai& vector

I f , f o r exanple, one adds t h a t partona a r e quarks c e r t a i n

n m r i c a l l r e l a t i a n s are implied, aa we have seen, between the s f n g d a r p a r t s of the comutators of variaus klzlds of currents.

( m a t i s t o say f o r examle, our

Inehtie Scattering as Propertiw of Operators r e l a t i o n s among feP and f i p , f i P e t c . implied by expressing everything i n terms of the s i x f m c t i o n s

U,

z, d , 3, s, z,)

Quarks, a s f r e e p a r r i c l e s , have n o t been fowd. a s t o whether the & t a i l e d views sf tbe partan =del

There a r e many questions a r e correct f o r quarks.

In

p a r t i c u l a r , the question whether a s i n g l e parton quark moving out i n recoil, need

o r need not rake i t s non-integral qwatum a m b e r with i t . In addition i t i s o f t e n very useful t o aee, vllen r e s u l t s bave been obtelned from a traodel, J u s t how mu& depends an the rnodel and whether i n f a c t the r e s u l t s carnot be s t a t e d aa a general mathematical p r i n c i p l e without: recourse t o a s p e c i f i c model, =del

In t h a t way, i f i t p r o w s l a t e r t h a t too a m y d e t a i l s of the

a r e f a u l t y , we can r t l l l begin again having learned s o w general properties

of hadrons w%thout being c a m i t t e d t o a11 t h e o t h e r s , m e r e f o r e S t i s i n t e r e s t i n g t o note t h a t t h e parton m&l is equivalent t o the general statement: (and one mueh c l o s e r t o d i r e c t elrperimental r e s u l t s ) t h a t c m u t a t o r s have 6-IXLe Zigbc-coae s i n g u l a r i t i e s

.

Light Cone A

Lecture 38

AR I n t e r e s t i n g qt1138tion mswered by Britzsch i3nd a l l - m n (1971 Coral

Gables conferenca)

Ss haw t h e s c a l i n g r e s u l t s of t h e "partons as quarks" c m

be s t a t e d i n a way wbf ch

- at the end

doesn't involve quark wave f m c t i c n s

o r operators a t a l l , We wish t o s t a t e t h a t t h e Sight cone a i n g u l s r i t i e s a r e l i k e those f a r f r e e

quark (spin 112, SU t r i p l e t ) comutatorB. 3

We man, of course, t h e l a r s s t

6' o r d type of s i a g t t l s r i t p , as we say t h e '%leadingw ( i n high frequency)

singdarity,

F i r s t we s e e what t h e s i n g u l a r i t i e s a r e libre 2 f currents were

represented by quark f i e l d s * ( X ) ,

a M r a e s p i n o r carrygag SUS i n d i c e s on which

3 x 3 rrratriees h can act, a"he conanuratror o f two splnar f i e l d s of m s m is e e s i l y worked o u t (as we did f o r Base f i e l d s ) i t is

( J u s t ss t h e propagator i s

+m

(f + m) /(p2

- m21 i n s t e a d of

l/(pZ

-

2

SB

1 so this

c o w s l i k e v i s e i n t o the commutator ( t h e r e a l p a r t of t h e propagator).

Since we a r e looking f o r the leading s i n g a a r I t y near the l f e t cone Gm Is.

Light Cone A fgebra J(+

2

and the w s t e r n i s

s2t

where

w l l e r than the large gradient, thus

is the square i n t e r v a l from 1 t o 2 and 2 means "the leading

slnguLaritfes near the l i g h t cone are equal. '' It i s e a s i l y v e r i f i e d t h a t the- r e s u l t s we: obtain here f o r the c m a t a t o r s

of two currents each a b i l i n e a r form i n

v,

a m exactly tk s a m ðer we

suppose s ) the f i e l d s obey the usual a n t i c o m t a t l o n r e l a t i o n s a t equal, e l m @

--

T(%)$(i;2)

Y ( x ~ ) # ( x ~+)

-

- Z2) and +(Z1),+(Z2) anticommute a s i s

&'(gl

appropriate t o Fern2 p a r t i c l e s ; o r we suppose b) the spin 1112 f i e l d s abey 3 c-utation r e l a t i o n s ~(%)y(;~) F(xl) +(x2) 6 (g2 and

-

-

cornate

eu3

- 5)

+(z2)

is approprgate t o Bose p a r t f c l e s ,

Thfs i s very Iuterleratl-ng becawa i t says the ""Bose qusrrk" "del

whfch

is appropriate a t low energies i a f a no fundmental way %a contradicttlion t o '"partens

as quarlcsl%t higher enerm.

a8 808e quarks

We

can call t h i s partan ~ltoclel""prtonsr

."

Next a current of SUS type a (deseribedw ha) i s

where f o r e x a w l e f o r e l e c t r i c current h" Is diagonal say hY

-3 ) .

diag.(2/3,

- l/>,

For axial current8 change y p t o

NW when we comute

currente,

WO

r e s u l t (see reference f o r detagla),

find a s h p l i e but tedFousZy complicated

W

Tc i l h s c r a t e the idea, r-Jhich i s a l l we

intend t o do here, we t&e the ease of two electrFc curreate, and also drap several t e r n s which would vanish i f w e took s p i a averaged matrix e h m n t a , then a l l that aumives i a f ~ : ( 2 ) , $:(l)

2

Bp

(

E

(

~

~

-

l

~

~

)

~

(

~

-

,P-&

~ ~s t (~2 . 1~ ) ~ 1 aT ( 1~~ 2~1) Y ~ Y ~ Y (38.3)

(plug other t e m a dropped)

(2,l on ntutwl l i g h t cone) and h

6

is diag, (4/9, 1/9, 119)

hYhY i n our caeee.

Mow we s t i l l see the quark f i e l d s in ( 3 8 . 4 1 , but the Idea n w i s t o dis-

regard

equation (38.4) and t o suggeet t h a t e q w t i o n (38.3) ( i t s generalization

t o a r b i t r a r y currents and i n c h d f n g ola;ttted t e r n of c o ~ r s e )a r e geaerally valid.

They give the l i g h t cone singuXarities and define, on the l i g h t cone a t l e a s t

.

a s e t of new operators Va(2-1).

It Ss t h e matrix elements of these operators

which give us our s t r u c t u r e functions. No d i r e c t inrplication is m d e t h a t they can be expressed a s (38.4). Eqwtion (38.4) as well a s (38.11, (38.2) a r e j u s t s c a f f o l d i n g t o a r r i v e a t (38.3) and a r e henceforth t o be forgotten, quark operators a r e seen.

I n (38.3) no e x p l i c i t

Instead only soae new b i l o c a l operators a r e defined

(mctanfng depending on two points).

They a r e defined only on t h e l i g h t cone by

the very equations (38.3) f\

[This system would be a t r u e "algebra" i f the p r o p e r t i e s of the ,J" could now be defined independeatly,

For e x a m l e , an ideal. s i t u a t i o n ( i n f a c t a

complete theory of t h e hadrans) would r e s u l t i f equations giving the Comutatars of such

ps

could be given i n t e r n of J% and

seem possible,

A l i t t l e can be done, hcwever.

?'S

themselves,

T h i s does not

Wfth the fonn (38.4) t h e

R

e o m u t a t o r s of two :g's which have t h e i r v a r i a b l e s on t h e s a m l i g h t ray (e.ge ~ : 2 ( 3 , 4 ) f*'(l,i?)] ,

3,4,1,2

*ere

cone) again can be expressed a s m but i t is n o t much. reactions l i k e e

+p

a l l l i e on a s i n g l e generator of a l i g h t This r e l a t i o n has been a l s o hypothesized,

E t gives p r e d i c t i o n s f o r c e r t a i n two-current i n c l u s i v e -c

e

+

p+

+ p- + any

those expected fro= t h e garton =del

hadrans.

They a r e , of csurse, J u s t

i n t h e s a m conditions,

Me discuss them

later,]

I n order t o w e equation (38.3) we must take the Fourier t r a n s f o m , and f o r t h a t we can w e t h e r e s u l t (very s i m i l a r t o those we have already derived, we leave d e t a i l s t o you) i f q

(v, 0, 0, u 4- N Q

t h a t is, i t involves only t h e i n t e g r a l along t h e ray t = z camon t o t h e plane t = e(from eiv(t-z))

a d t h e l i g h t cone.

Naturally every r e s u r t of t h i s theory i s a l s o a r e s u l t of t h e "parton as quarkw "eory

f o r t h e l a t t e r is a model of t h e f a m e r ; but n o t every parton.

r e s u l t can be derived from (38,3),

I t i s possible, therefore, t h a t =any of

them a r e wrong and only (38.3) and n o t the conrplete f i e l d model may survive.

Light Cane Algebra Therefore i t is i n t e r e s t i n g t o compare the t h e o r i e s f o r varfous types of predictions raade by t h e parton a s quarks theory,

They seem t o be of t h r e e

classes. A.

Scaling, r e l a t f a n s among scalfslg functions, sum rules.

B,

Sgecial a r g m e a t a about the s t r u c t u r e futlctions derived by a r g u m n t s about hadron c o l l i s i o n s and o t h e r a r g w n t s ; l i k e dx/x behavior, t o x near l, s t r u c t u r e function

r e l a t i o n of form f a c t o r power

{I.-XI~ e t c . C, Applications t o experiments of D r e l l type e+

+P p -t any

+ e-

+

hadrans o r

hadrms, etc. p + p + Glass A a r e e x a c t l y those derlved from Sight-cone algebra, obtained from l i g h t cone algebra by d e f a u l t , discussion of how t h e a a t r i x elemeat of '?{2,l)

Class B a r e aoC

That is, they simply m o u n t t o a mtght behave,

X I is n o t a

s p e c i f i c a s s m p t i o n o f the m d e l but an attempt t o go f u r t h e r t h a t leads t o Class B r e s u l t s

-a

s e r i o u s attefDpt t o discuss the nratrix elements of t h e l i g h t

corn algebra would lead fo r e s u l t s in t h i s c l a s s . Class C a r e very i n t e r e s t i n g , a s they seem beyond (38,3) and require so= extension, even thou*

they appear evident from t h e perton view.

It f e therefore

here, t e s t i n g these t h a t a r e a l choice can be made a s t o ðer the extensions of the parton model beyond t h e expectationcl of (38.3) a r e r e a l l y sound, Tlze reason the c u r r e n t c o m a t a t o r oa t h e l i g h t cone is n o t s u f f i c i e n t f o r reactions l i k e p

+p

+

p

+ + U- + X

is t h a t we do need m t r i x eletaents l i k e

but t h i s time a s we take t h e l i r a i t

M

the anomntum of t h e c u r r e n t operator

increases we a r e a l s o changing t h e s t a t e pp; t h e r e l a t i v e mamentm of t h e two protons mwt a l a o increase,

This f e an a w h a r d l i m i t f o r a theory of operators.

It is a very i n t e r e s t i n g question,

1s t h e r e so= general a b s t r a c t way

without quark f i e l d s (partons) t o describe a l l these Class C parton p r e d i c t i o n s also?

Or a r e they perhaps wrong, m r e l i a b l e extensions of t h e idea?

And what becomes of the questian of how the partons c m @ a p a r t i n t h e proton without exhibiting quark qwntura nu&ers m n g the f i n a l s t a t e s ?

Does

current algebra h e l p us t o solve i t ? Perhaps, i t s e e m t o be t r a m l a t e d i n t o "are there any representations possible of t h e algebra (38.3) &ich do not imply quark quaatuta nu&ers atoong t h e localized s t a t e s ? t h i s anathemtiral f o m than i n physical a r g m n t s .

It may be m r e t r a c t a b l e i n

Properties of Commutators in Momentum Space

Lecture 38

We now cm@t o discuss what general properties a carnutator has t o have i n Hlomnturn space, such t h a t its Fourier trslneforrn w i l l be zero outside the l i g h t cane of configuration space. This is c l e a r l y a prolnising l i n e t o follow t o a u p p k m a t physical, i n t u i t i o n on the properties of Wvv.

At present not a great deal can be s a i d , but f o r your

i n r e r e s t i a your future research &at has been done here m y prove f m i t f u l ,

First we can

so-

obvlous r e m r k s ,

I f we multiply a comutator ( i n

space-tim) by a function G which is 1 inside sad zero outside a l i g k t cone we recover the w m u t a t a r (except, a s is the case, t h a t it is s i n g d a r right m the l @ t cone where our funct%on G is poorly defined). isfi

2 2 l6aP,V,(l/q ) (O.V.

Since the F.T.

of G

pxincfpal value) we have the convolution theoran

( p l w piece@ from the 1igbr cone). (C(q) is the F,T, of a carnutator,) Xn a similar vein, since the F,T, of we see that G mwt b v e the form

182

O

is f b n i sgn(qio)a'(q

2

183

Properties cif Commutatorsin Momentum Space

O f course

plus l i g h t cone pieces,

.wr?

can say a l s o i n (39.1) t h a t the C(u1 sf

the iategrautd is any function F(u) a t a l l , and the C t h a t c o w s out on the l e f t w i l l be the F.%',

o f a ftlnetim zero outside the l i g h t cone,

1 have n a t been

much use aE these observatiom.

able t o m&

I n using the simple views h o v e we awl: be careful of one paint.

'Z"he

f u n c t i ~ nwe multiplied Gix,tZ by, G ( x , t ) , wars &fined as l i n s i d e the l i g h t cone

and O outside; h a t i s i t exactly on the l i g b t cow? ordinarily that is such a fsrnalli region t h a t i t &as

T t i s i n d e f i d t e there, no difference in the i n t e g r a l

over space tiw of C ( x , t ) G(x,r) but i n f a c t i t make@ a great mcextaiaty because Gfx,tlhas a 6 function s i n g u l a r i t y j u s t &ere G(x,t) is poorly defined.

We can

correctly straightctn out our Eomulas f o r t h i s e f f e c t i n the following way,

kle

c a l l Ca(q) the a e m p t a r i c C(q) g i v h g just the light-cone 6 s i n g u l a r i t i e s ,

man

C(q)-Ga(q)

has no l i g h t cone s i n g u l a r i t y m d eo equat20n 139.1) f o r e x a w l e , holds

i f G is replaced by C-C,

on both eridee.

Qne c m then s i q l i f y o r rearrmge tfie

equation t a obtain one flke (39.2) but with s a m ~ d d i t i o n a lt e r n r e l a t e d t o the litght cone beethaiviar, A f sr more s u b t l e and useful observation was ~nzadeby Dyson,

i n m a y probless we a l s a know C(qf

He noticed,

O f a r c e r t a i n regions of q space (where

oo incermodiate s t a r a a may be available).

For example, f o r qO

v > O the

lawest s t a t e avall&lle f o r the f i a a l s t a t e x is the pratan i t s e l f , aroving with lasoaentlzla Q (space p a r t a t q ) hence with energy

v

hen-

and f i b w i s e v < + H -

*

&son he@ proved that the necessary a d suf f i c i e n t condf tioton f o r G(q)

veniah i n repion S

S1(q)

to

q0 < S2(q) and t o have a Povrier transform vanish

oufsi4e the l i @ t cone i a t b s t C(q) cam be written tldt

where @ vanfbshr?cr outside a regiaa R, but i s othbtmlsrst a r b i t r a r y ,

The regllon R

i s such t h a t t h e (q-u)2-s2

hyperboloid does not penetrate region S.

That i t i s a s u f f i c i e n t conditian i s easy t o see ( t h e g r e a t d i f f i c u l t y of t h e proof i s t o show i t i s necessary),

We have already seen t h a t the f r e e

p a r t i c l e c o m u t a t o r Cm(x,t) is zero a u t s i d e the l i g h t cone, s o t h e F,T. 2

2

1

(q -m 1 s p ( q

is zero o u t s i d e the l i g h t cone,

by anything, i n parCicular by P

iu-X

But i f C,,(x,t)

of

is m l t i p l i e d

( o r any superposition over U) It i s s t i l l

zero o u t s i d e the l i g h t cone, hence F.T. eiue"~m(n,f)

sgn(qo-uo ) 6 ( ( q - ~ 1 2 - d ) . 2 2 Hence superposing with weight @ ( u , s ) various cases of u and m s2 we g e t

I t i~ easy t o see, by drawing hypczrbolzls t h a t f o r our caee t h e region R

2 o f i n t e g r a t i o n where @(u,s ) does n o t vanish i s a s follows

i n s i d e two canes For a2 >

2

For s2

M!

h0k/4

U

i n s f de region bounded by M -lug/

-

u

h c i d e n t a l f y , t h e s c a t t e r i n g a q f i t u d e t h a t corresponds t o t h i s c m u t a t o r ( w i n g 337.2) becorns

where o (from s e a g u l l s ) is a polynomial i n q u ( f o r the amplitude

replace

LE(qo-aO) by i h ) . One of the d i f f i c u l t i e s I n w i n g the Dyeon representation is t h a t the function @

is n o t unique, m n y can give t h e s a m G(q).

has Been derived (I suspect not: aa rigorously

Another very s i m i l a r representatian asj

Dyaon's) by b s e r , C i f b e r t and

Sudarshm e s p e c i a l l y f o r a problern l i k e eure where C is a function only of t h e two i n v a r i a n t s

QL,

v,

i t is

185

Propertb of Commutatorsin &mefiturn S p c e This is, of course, j u s t Dyeon's representation i f the four vector

U

can be

a s s m d t o have only a time component, t h a t is, only a contponent i n t h e p

P

df r e c t i o n . There is again the expectation t h a t B(a,B) is zero o u t s i d e a more l i m i t e d

-G/Mt o

Leave our expressions i n t a c t , j u s t rentexnbexing If(a,B)

6 for

6

2 H

.

2 M , B only runs from

range i f o

In f a c t i f a

.t.

&/M, but we in

this region of a .

L t is c2ear t h a t if C is zero on t h e l i g h t cone various gradients a r e s o s e v e r a l p o s s i b i l i t i e s e x i s t with d i f f e r e n t i n t e g r a l p w e r s of v [from

&so,

zero up t o any f i n i t e v a l m ) with correrrpanding B functions; h u t we have taken the one appropriate t o 2Wl.

Equivalently we c m w r i t e our f o m with a new

definition f a r o,

I n addition, f o r our c m e , t h e a s y m e t r y i n

tlrtnas

f o r the e o m u t a t o r becorns

t h e property t h a t

T h t ~representation is very nice, and h is prob&ly tyQiq2y deter&ned by HI.

The weaicnera, h w e e r , i s t h a t no physical i n t e r p r e t a t i o n o r ezrpressfon

i n t e m of nratrix e l e m n t s is given f o r h(cr,@), T'hereEore we cannot use any physical i n t u i t i o n I n g w s f r h g haw h should behave? (other than using our h w l e d g e of how M brsheves and trorking bakward), @ay

- such a f w c t i o n f o r h(a,f3)

4.8

m a t 8s t o say we c a m o t a t any s t a g e

too 'krazy" "yeically

behave s o and fro i n t h i s mgion, e t c .

- or

i t ought ta

As we s h a l l see, t h i s i a a s e r i o u s

weakaess i n t h i s carse. We am t u r n t o s e e how h(a,B)

=WC

behave In order t o produce a function

2MWl behavlng a s we expect (see UcCWcs 3l)for l a r g e

F i r s t of a11 we have t h e s e a l i n g l%&t

Bere

= f ( X ) a Emetion of

X

only.

v

+ m.

Eq. (39.7)

V

(we t a b v

2

-q

givemt

O throughout).

.r m, - q 2 / 2 ~ v

-

x finite.

(sgn(v+BM) = I f o r

large v, s i n c e v > M).

Now i f we a s s u m b(a ,B) lnvcalved here, so

v

f a l l s away rapidly enouejrt with a only f i n i t e

c;

are

the

a c m be dropped i n the 6 function and we have

X believe i t was t h i s a r g

t with t h e Dyson retpresentacican which e i t h e r

+

OD

l e d Bjorken t o Iris s c a l i n g hypothesis, o r helped t o confilaa h i e e w p i c i o n e of

its t r u t h , qL >

o

scaling

We have here a ISOR=+ should go i n the p o s i t i v e q 4-q

2

jJe:

2

2

c m n w deternine how the f m c t i o n 2M1(q r v )

s e a l i n g region.

Letting v

+ m,

q2

-r

such t h a t

m

/22Vfv = x' is f f n i t e we have 2Wl =

(D, -xt))da which again s c a l w ; but

even m r e , from the s

NW c m we s e e t h i s rernarkablle r e s u l t from the partan model? equally well derive t h e funetion f o r qZ

; .

Can we

0 by using physical a r g u m ~ t s l We

w i l l explain q u a l i t a t i v e l y haw P t corns etbout. Por p o s i t i v e q2 we muat be concerned i n the c o m a f a t o r (36.1) with the 2 subtracted v a c u w p i e m , I n the vacurn we eaa a e p a i r s , and a t high q jwt

kle a r e concerned with hcrw t h i s p a i r production probability is tllodified by t h e presence of a proton

At f i r s t one oofght think the niodification must c o w throwgIr i n t e r a c t i o n of

187

Proparties of Commutaforsin Momentum Space the newly fnade paxtons with those i n the proton

- in

i n t e r a c t i o n about which

we know l i t t l e except t h a t i t is f i n i t e and i a not involved i n our derivation o f f(x) f o r negative q

2

exclueion principle

.

But a t high energy a: =re important effect: Ss the

- partons c a n o t be m d e i f

the proton ( i f F e d s t a t i a t i w is a s s m d ) . b i l i t y t o produce a the charge is 2/31,

Thus i n our diagram the proba-

t o the l e f t mid u t o the r i g h t a t x ' i s 4 / h U i t s (for Hmever, i t is a l t e r e d a t those x by the chance, u(x), that:

a u parton is present i n the protan. t o (4/9u(x9; i f the

they a r e already pre8ent Sn

G

Thus we have a contribution p r o p o r t i o n d

parton is w v i n g fomard i t is (4/13)~(xV. ZE a d par-

ton p a i r is produeed the probability Ss 119, mdifsed by the chance af f i n a n g

a d partoa already i n the proton, the contribution i s (1/9)d(xV, e t c . ) the e n t i r e contribution is 4/9[u(x8 f&(x")l+

1/9[d(x')+d"(xt) )l

Thus

+ 1/53 [a(x ")+;(X"

)l

o r fep(xt) ss required.

I n =king atstisties,

2

the analysis f o r q positive we aae-d

the quarks obeyad F e d

I f Bose s t a t i s t i c s a r e used i n the f o m l expression of the con-

pLeted surn over s t a t e s (34.1) we kave aoae changes,

F i r s t there is a change i n

s i g n of the F e w exclueion e f f e c t turning i n t o a Base tendency t o increase m i s s i o n beeawe a p a r t i c l e is already present,

sip

iin

men there is =other

chage in

the closed-loop diagram 4 [JJIO> when changing f rorn F e d t o Boae

s t a t i e t S e s [see, f o r exmple, R,P, Fe

, Phys,

756 (1949) 1,

Rev.

On the other Ixmd with the Botle s t a t i s t i c s sign the iaagiaary p a r t of

< O ~ J J ~ Ois> negative and i t cmnat be written ae a sum of positive p r o b a b i l i t i e s C l l

2

, but

1s minus t h a t

SW.

'Fais circurnstmce i s the b w i a of the

X

w u a l proofs of the r e l a t i o n of s p i n md ~ l t a t i s t i e s ;spin one-half p a r t i c l e s c m o t be consftltently gncerpreted ss Bose p a r t i c l a s , incerpre t a t i o n o f Base qmrlcs would

e e 3.

-I.

l%e mst s t r a i g h t f o w a r d

the "'Dxelf Constant" defined f o r the

any hctdrons ctose section waufd have the impassible value -21 3 ,

a%

course a c o w l e t e l y naive i n t e r p r e t a t i o n af quarks a s Bose p a r t i c l e s would a l s o lead t o the wrong r e s u l t t h a t the wave function would be s the i n t e r e h a n p o f two protm&. problem would be? remved.

I f quarks obey para-stetretics both of these

Me c o n t i n w our discussion of equation (39.7) now turning t o the region v

+

-, -q2 f i n i t e .

Here we expect 2W

which we have wrftten a s 2F?tW3

1

t o go a s v times a function of q

2

2Hv g(-q )I(-q

2

)

.

2

I n t h i s region c u r equation

(39.7) reads

a t f i r s t sight, for v

-+

=, we could forget -'-a

i n the 6 and obtain ZW1

9D

h(cr,o)da,

a czcnstant indepndent of q

2

asld evidently i n c o r r e c t ,

-

Elmever,

0 the "ceanstant" would, by (39.3) be f(o) which we know is i n f i n i t e , a s f o r small

% i s suggests t h a t we assume f o r smll 13 chat h ( o , ~ ) is singular.

ajx.

f(x)

X,

Sllbstitutfng i n the Antegral above gives

f o r small 8 ,

Thus

gives t h e -q g(-

2

) a s -q

2 2

dewndence of the t o t a l - v i r t u a l photon cross s e c t i o n , g e t s l a r g e approache

Finally we t u r n t o region 111: kept f i n i t e .

of MH -; '

There we expect 2MWl

only.

(~:-d)

v

+

-, large

-q

2

but

m

2Hv

- (-g 2 3

2 2 (-q2)-yh(~x -M )ZHv where h i s a fvncrion

The ( q q 2 ) ~is t y p i f i e d by t h e e l a s t i c s c a t t e r i n g (H:-M~

and y i n perhaps 4 o r 5. h

Tke I&&tof

goes ss A(H:-M~I'-'.

Far n near 1, f(n) A ~ ( l - x ) y - l and f o r l a r g e M:+~~

0)

Properties Ilf Cczmmuratavs in Nortleii;tttnSpace I n t h i s region our equation (39. '7) reads

dg &(H~-M~+~MV(B-X)-B)~(~,B>

Again a "natural" gvess (neglect

M:-$*

(40.1)

r e l a t i v e t o 2Mv(~-l) in the 6 function)

OJ

l e a d s t o t h e wmng answer:

do h (a ,l),

2Wl

a c a a s t a n t indepenrtenr of

JO But again we can f i x i t up by a more eomplicared guess f o r the behavior

*$-:M

of h(o,B) near B = I.

-

X

y

Because, from (39.9

do h(a,x)

$8

zero a8

X

+

1 (like

Therefore one way t o g e t t h e answer would bc t o aesume t h a t h(o,B)

goes t o zero near B

I, For e m q l e suppose

D(o) (l-Bjy-'

h(a,8)

D(a)da

A,

wf(*l

(Xf we =sum y &pen& on a we c m be m r e general and

g e t n e a r l y t h e s a w f i n a l r e s u l t except: wSth e o m slow logarithmic dependence@, which &ght i n f a c t be &ere i n t h i s region, eweriarrzntally) X-@

.

This is not: k s r m very w e l l

S u b s t i t u t i n g i n t o (4a.l) and i n t e g r a t i n g ( t h e B f w c t i o n s a y s

2 2 (Hx -M -a)/2Mv) we g a t

hence of t h e c o r r e c t farm i f ( c a l$l-:H

= 13

X

%is appears a t f i r s t s a t i s f a c t o r y ; h w i l l hawe the r i g h t m p p t o t i c behavfor hiy-' f o r l a r g e h i f D(o) f e l l s o f f fast enough with

U,

and

mwever, we w i l l agaLn be fooled i n our q u a l i t a t i v e expectations f o r h(X) fro= t h i s equrztion i f

W@

suppose t h a t D(U) is a sinrple funetion,

llhe operation of

i n t e g r a t i o n i n ( 4 0 . 3 ) is a very poverful sraoothing operation; f a r no siarpXf?

D(@) would we be l e a d t o guess what we k n w i s t r u e

- t h a t h ( ~ )shows a s e r i e s

of peab and v a l l e y s f o r s m l l h , t h e r e s o n a c e p e a s ,

I t f s n o t t h a t (40.3)

aays resonance p e a b a r e inqpossible

- i t is t h a t

( 4 0 . 3 ) plus a @-ss t h a t D(#)

is not too c o w l i c a t e d (i.e. does not vary ia a rapid and peculiar way) f a i l s t o do so. D(A)

To s e e t h i s take the case

g1

dS

y

4 and note t h a t (lr0,3) can be solved as

(40.4)

h(&)

J

The fourth d e r i v a t i v e of a reaonmce o s c i l l a t e s i n a very s p e c i a l nrmner

.

(For

a mueh more i n t e l l i g e n t way t o &a1 with t h e resonance regfan s e e Cornwall, Corrigan and Norton, Phys, Rev,

536, (l"31)

a d the next section.)

r i z e t h i s i n v e s t i g a t i o n , the Deser, Gilbert, fudarslhan representation

is useful a s a f o m l t o o l t o g e t dispersion r e l a t i o n s , e t c .

It is, eis of course

i t ought t o be, cs;pabfe of representing what we know of the experimental d a t a

with s o m function h(o,B). utility

- by

But i t i s disappointing i n t h a t its p r e d i c t i v e

-

gwasine, ""reasonable" hbrsv%or f o r h(~r,t?;) is not very g r e a t f i r s t

becausie we have no physical. p i c t u r e of what the function f s , lad second I t s e e m l i k e a function t h a t behaves I n a way a s cornglicated. a s ( I f n o t = r e

coqlicated

&an) the data i t s e l f .

It s e e m t h a t h ( # , @ ) Is n o t , i n f a c t , a function t h a t has much d i r e c t physical s i g n i f i c a n c e md is purely an a r t i f i c i a l expression i n a c e r t a i n m t h e m t i c c r l form of t h e f a c t t h a t the c a m t a t o r vmgsbes outside the l & t COne * It h g h t pay t o try t o Elind another representation of t h i s f a c t i n which

the k e r n e l Eunctions may have s o w d i r e c t physical representation,

Scattering i n the Deser, G i l b e r t , Sudarshan representation Given the c o m a t a t o r i n any f o m we c m find the s c a t t e r i n g amplitude

2

T(q ,v) correspandfng t o i t by w i n g the dispersion r e l a t i o n (37.2) , f t is e a y t o s e e by d i r e c t s u b s t i t u t i o n of (39.7) t h a t t h e s c a t t e r i n g kThich has 2 t h e rLght s p i n depsndence (i.e. 6uv q -q Irq v p a r t ) t o corxespond t o 2 N 1 ( c a l l i t

191

Properties af Commutators in Momentum Space This is t h e causal m p l i t u d e c o n v ~ n t i o n ,i n the Feynmn convention t h e 2 The "eeagufls" a r e a polynomfal i n q ,v,

i e sgn(v+MB) is replaced siarply by i t s .

2 This can be r e a d i l y v e r i f i e d ( s i w 1 y n o t i c e t h a t rsgn(vM8)6(q +;?Hvfl*)

-

s~R(v+MB)

2 2 2 112 where we have w r i t t e n E = +(G+M B +q ) ).

~ @ * N B ) ~ - E ~ )(6(v+MB-E)-6(v+MB+E))/ZE

& t e 1: Bftcauee we a r e expressing Tl

8bi

an i n t e g r a l with a v f a c t o r times t h e

i n t e g r a l we have r e a l l y been working with W/v and Tlv; but there is a pole i n v(%

+

0 as v

+

To avoid t h l s use i n our dispersion r e l a t i o n

0 but Tl does n o t ) ,

V1/v t o g e t fT1(q

2

,v)-T1(4

2

,O))/v.

Note 2: X t is evident Eroar (40.5) and (4Q.B) t h a t although derived front the

P

dispersion r e l a t i o n PR

V

f o r fixed Q {space p a r t of q) they a l s o s a t i s f y a

dispersion r e l a t i o n i n v f o r fixed q

2

2 (for q

Q) (a one@ s d t r a e t e d dispersion

relation)

2 ( h s d n g =a eeagulla beyond a conatcurt term absorbed i n Tl ( q ,Q).

This c m be

seen by d i r e c t s u b s t i t u t i o n of (40.6) i n t o (4Q.6a), relnembering t h a t W ( ~ ~ , - V =. ) 2

-W(q , v t ) , i.e.

t h a t h(cr,B) m-h(a,-S).

Consider

everything i s c l e a r except t h e icegrctft should be icsgn(*M6). work with v

Q (T is s ~ e t r i l c i n

V).

Then f o r q 2

Let us choose t o

O t h e r e i@ no pole i n

( s 2 + 2 ~ ~ v a + i c ) - ' i f fl is negative, the s i p of i c is a r b i t r a r y ; we may replace by sgn(v.t.4il8).

I f B > 0, sgnB

(For v negative the s i g n reversea.)

-b

= sgn(*m)

Itence we have

s o i t is s t i l l , c o r r e c t ,

I92

Photon-Hdrsn Interactions

Wl/vp is a y m e t r i c s o we? c m s e t t h e range front Q t o l&u

'- ( v + i ~ ))

-

2 -h/( - v ' ~ + ( v + ~ c ))

.

with l/(v'-(v+ik))

.t

F r w t h i s equation (40.6a) follows.

For W which c o n v e r p e Easter we expect a corresponding uneubtracred 2

d%apersion r e l a t i o n

2

for q

W2 (q , v ' )

For qZ

Q these a r e of course t h e Kronig-Kramers

2

4

.

O

formula9 r e l a t i n g r e a l and

i m g i n a r y p a r t s of the index of r e f r a c t i o n ( f o w a r d l i g h t s c a t t e r i n g ) which

is necessary i f s i g n s l s a r e not t a corn out before they go i n t o a block of s c a t t e r i n g nratsrial,

We t r y now once again (msucc;essFully a s i t w i l l turn a u t ) t o t r y t o oBtktin s o w expected l i d t a t i m s on 2MWI b e c a w e we kaow i t is a caueaf (zero outside the l i g h t cone i n space) c m u t a t a r , v a l i d not only f o r l a r g e

V

Ttris time we s h a l l use behavior

For e x a q l e we h o w f o r t h e

but f o r my xeglon.

e l a s t i c s c a t t e r i n g , i f t h e proton were a point &arge we would have f o r v > 0, 2 2W1 Z H ~ 6 ( ( p + q ) ~ - d )= 2Hv6(q +ZHv). (To g e t the c o r r e c t s

-

p o s i t i v e and negative t h i s can be w r i t t a n = r e 2 6 ( q +2m?rv)

- sgn(v-M)S(q

2

-2Mv)).

2 Mv(sgnCv+M)

c o r r e c t l y a s 2Wl

This is o b d o u s l y cauaal

- as

W@

have seen

-

Ear i t coaes from perturbation theory o f fielcls. Now in the r e a l world t h i s is a d t i p l i e d by a f a c t o r ( t h e square of t h e 2 e l s e t i c form f a c t o r ) , a f a c t i o n of qZ, say f (q 1 which f e l l s o f f gradually. 2 f o r negative q , from q2

-

Q, to behave a s (_q2)-y f o r l a r g e -q

2

.

One would

expect such m d u l a t i a n of the f w c t i o n expected f a r point-like particles r o represent some kind of s m a r i a g of t h e point and t o perhaps irayrly a lack of causality

-a

lack which must be balanced by c o r r e c t contributions from t e w

o f f t h e e l a s t i c mass s h a l l (correeponding t o o t h e r resanances, e t c ) .

qlhue, c m

we n o t tnake so= requirelnents of behavfor i n o t h e r relZions, especially non-

2 2 e l a s t i c regions by our knowledge? of f ( q ) f o r negative q ?

trnfortunately n o t

(aa Cornwall, Corrigan and Mortm, (Phys, Rev, I)3 537 (1971)) show) i t i s possible t o s t a y m t h e e l a s t i c w s s h e l l fox negative q

2

and only a f t e r the

2 behavior f o r p o s i t i v e qZ and s t i l l arrange v i r t u a l l y any f f q ) which f a l l s o f f

Properties of Commutatorsin, Momentum Space Eafrly smoothly.

2

I n p a r t i c u l a r i f f ( q ) can be w r i t t e n i n the f o m

01

f o r negative qZ, i t can b e done.

W e can s e e why t h i s i s , and a l s o g e t a

b e t t e r physical f e e l f o r constructing causal functions by the following conI t is e m l e s t t o deal with the s c a t t e r i n g Emctioas T, fox these

siderations.

t o be carnal they must be t h e R u r i e r t r a n s f o m of a retarded c o m u t a t o r and thus zero outside a. fortsard l i g h t cone, its F.T.

Let a ( x , t ) he such a function and A(q)

and l e t b(x,t) be another such fmctlon,B(qj its F.T.

Thus the con-

volution of a(x, t ) and b ( x , ~ )is obviously, by g e o e t r y , such a function (zero outsirfe Eomard l i g h t c m e ) and hence its P,T.

o r s i m l y A(q)B(q) is again

s a t i s f a c t o r y (a clausal s c a t t e r i n g function). We s e e t h a t cc&inatioas by arultiplieation (and addition of causal s c a t t e r i n g

The s h p l e s t causal s c a t t e r i n g

fuaetians a r e causal s c a t t e r i n g functions, function is

lq2 -

m

2

+ ic

(40.82

sgn q4]-1

We can generalize t h i s (multiply by eiu*' vector u

i n space time) t o find f o r any four-

that

is a c a u s a l function,

((qe)2-g+ i c

Thus the e l a s t i c s c a t t e r i n g function f o r point p a r t i c l e s

sgn( +M)I-'

(40 .IQ)

evidentfg i a causal ( a s we11 a s what you g e t by p u t t i n g -q f o r q).

tJe can

multiply t h i s by an expression l i k e (40.81, we see iq2-m2+ic

i s causal.

sgn v]-'

f(q+pjZ-d + i r sga(+~>j-'

This i s t r u e f o r any

m"

12 and hence arty s u p e r p o s i t i a a with v e i g h t

p ( u ) d p is, s o a s c a t t e r i n g amplitude l i k e (note ( q - ) ' d

is, by i t s e l f causal.

To g e t t h e c o r r e c t s

try for

= q2+2~\3

V

one need only add the

corresponding expression with u replacetd by -v; we w i l l suppose It is always; d w e awl n o t w r i t e i t : e x p l i c i t l y .

[you olight f i a d i t physically more s a t i s f a c t o r y and e a s i e r t o h t e r p r e t i f the f a c t o r is considered ss a form f a c t o r due t o v i r t u a l p a r t i c l e s l i k e a t each vertex,

p's

"Coae a t one v e r t e x would contribute a f a c t o r

CD.

&ere

id

is t h e mass squared of t h e v i r t u a l p a r t i c l e and v znea-slurerr t h e weight

of i t s contribution; g i s causal, (g(:]12,

We would then exlpect t o znultiply (40.10) by

one f a c t o r f o r each coupling; t h e r e s u l t would s t i l l be causal,

and possibly physically e a s i e r t o understand.) To g e t ul/v ( c a m a t a t o r ) from (40.11) we need only take i t s inraginary parC

In t h e region q2

G

O t h e l a s t term disappears and we a r e j u s t l e f t with

the e l a s t i c point charge s c a t t e r i n g a d t i p l i e d by a factor: f (g 2 ) given by (40.7) I t is disappointing t h a t t h e r e s t r i c t i o n s of c a u s a l i t y do a o t a f f e c t our

region of exper%aental observation p r a c t i c e t o g e t p(u)

even i f f a i r l y exact, knowledge of P(&

2

((1

< 02, F u r t h e m r e , i t w i l l be hard i n

exactly from Lnwledge o f t h e i n t e g r a l (-q2=)'Q

It Is n o t easy t o reverse the i n t e g r a l I n t o accurate

mless physical axgumnts (P dornlnance e t c .) e r e a v a i l a b l e .

Thus again we a r e t h r a m back i n t h i g p r o b l m t o understand the pracess physically ; the mthentatical properties da not help a s much a@we had hoped. 2

(We know t h a t i f f (9 ) f a l l s f a s t e r than l/$, nay a s (l/q214; then we

1

can conclude, by (4Q.14), t h a t

p(p)du

0. Such a r e l a t i o n is c a l l e d a

2 2 superconvergence r e l a t i o n -£(Q ) converges f o r l a r g e Q more rapidly Chan

.

199 .

Properriw af;Commutators in Momentum S p c e ttre f o m would suggest.

Again s l n e e f f a l l s ae (1/Q 2 ) 4 we can concjlude that.

wnp(w)dv vanish f o r n 2 2 2 expressed a s f g t q 1) with g(q 1 =

1

0, 1, 2 ,

kbternativsly, f can be

Lecture 4 1 The i d e s dfseussed a t the end of the previous l e c t u r e is i m e d i a t e l y genelcalizrible t a s c a t t e r i n g through an f a t e r ~ l e d i a t eresonance, say of mass Iv&2 , = h.

c a l l E?M-:

A point coupling would give s c a t t e r i n g a s (q 2+2Mv -h

We can m d t i p l g by m y form f a c t o r , say with a p(h,w).

+ ic

sgn(v+M)) -l.

We; are thus representing

things by a e m of a charnel rczsonances each of which has a square f o m f a c t o r

Tlre t o t a l s c a t t e r i n g fronn a l l t h i s

(S

c b a e 1 resanmce representation)

would t-hen be

The W2 which goes with t h i s (the i m g i n a r y p a r t of ?/v)

For q

2

c

0 O t h e f a s t t e r n vanishes end v: have f o r v

0 a suparpcaitlon of contr%butlons f a r each e f f e c t i v e

Bave we not gone i n a c q l e t e c i r e l e ?

3

is

O slaply

with a f o m f a c t o r

Ckzr o r i g i n a l expression giving

196

Photon-Hadron Interactions

(except f o r photon p o l a r i z a t i o n f a c t o r s ) W was of t h e f o w l

This looks j u s t l i k e (41.4).

2 The 6 is of course 6(q +~Mv-X) s o we i n t e r p r e t

f(i, -q2) a s

S

Z~G~/J(~)/X>~~ d over s t a t e s of

B

given massZ =

$+

A.

(At f i r s t t h i s would seem t o meke a function of Q ~ ,t h e space p a r t of the momentwn t r a n s f e r squared, i n s t e a d of qZ = v2 -9 2 ; but i t is t h e same because the 6 f m c t i o n r e l a t e s v t o 'Q

and i t can be e q r e s a e d e i t h e r way.)

63e s e e

2

f ( h , -q ) must be p o s i t i v e f o r -q2 > 0 ; of course, s i n c e the lowest e l a a r i c

Mx = EI (X = O) is separated from the eontinuutn a t Mx =

2 (hth. = 2 % M n ) ,

The function f (h, q2) and hence p (A, p) w i l l have a 6 (A)

cont riburion and a f t e r

t h a t the i n t e g r a l i n (41.3) w i l l s t a r t a t an I n e l a s t i c threshold hth t o *, We m y noE have gone i n a coerplete c t r c l e ,

F i r s t we now know (a) t h a t the

2

weight f a c t o r f ( h, -4 ) mst be e x p r e s s i b l e i n t h e form ((31.1) a d (b) we know what t h e f m c t i o n looks l i k e i n t h e experimentally unavailable region +q2> O (see 41.3) and, of course, t h e s c a t t e r i n g Emetion

(41.2) t h a t goes with i t ,

But do we know t h i s ?

IJe only proved t h a t the f o m (111.2) was causal; we have n o t proven t h a t every c a r n a l function must be expressfile as (41.2) and a t the m m n t we do n o t Chink we can. Since (41.2) i s causal i t mst be expressible i n t h e BGS form (39.6).

Qne

way t o do t h i s (suggested by Cornwall, Corrigan and Norton, Phys. Rev. I)3 536 (19711) is t o co&ine dens&aatora ( f t is e a s i e r t o use the Fegnma amplitudes replacing

t o gat a s i n g l e denorainator.

Thus we e a l 1 a =h@+y(1-8) m d i n t e g r a t e on a by

p a r t s t o prove 00

00

Of course t h i s can now be simplified.

I f frotn b(o,B2 we could always fSnd a

Properties of Cornmufatorsin Momentirr?~ Space r>(h,p) which would give t h i s h we would have a proof ( a s s d n g the DES repres e n t a t i o n is proved) t h a t (41.2) is a l s o a necessary f o m . cmnot be done a d

We suspect i t

i r i s poBBible t h a t (41,2), although very physical, is not

the complete expression; but o t h e r f o r m (other types of d i a g r a m o t h e r than e channels) &@fithave t o be added t o sentation,

it

- o r "abaonels"

t o g e t a c e w l e t e repre-

This is a good problem.

W for a l l q

2

i n t e r n of W f o r q

2

61

We have a f o m t h a t s u g p s t s t h e qaestion a s t o whether knowledge of the f a c t t h a t W is causal, and knowledge of

r. t o f i n d i t f a r a l l q2, v.

irs value f o r negative g' only, enables

We a r e nor now concerned with t h e p r a c t i c a l f a c t

2

t h a t knowledge; of f ( h , -q ) t o a ggven l i m i t e d e x p e r i m n t a l accuracy does n o t p e d t discovering p (?,,p) 2

f(A, -q ) f o r q2 > 0.

very well. and does n o t m t h e m t i c a l l y alone d e f i n e

Rather we suppose W

2 p e r f e c t l y koom f o r q < O and ask

f

t o what e x t e n t T is defined e v e r M e r c l , This has g r e a t i n t e r e s t f o r there a r e q u a n t i t i e s , such a s e l e c t r o ~ ~ l a g n e t i e s e l f energies, which can be defined I n t e r n o f T as i n t e g r a l s perhaps (e.g. 2 4 T(q ) d q/q2).

I f T were uniquely determined by W i n the ertperimentally

a c c e s s i b l e region we might search f o r expressions f o r these I n t e g r a l s d i r e c t l y i n t e r n of t h i s W a t q Given W(q

2

,V)

2

Q.

(Gottinghsarn farraula f o r s e l f e n e r ~ , )

2 i n the a v a i l a b l e region f-moraftntuar l i k e q 1 how m i q u e i s

2 W(q ,v) i n t h e energy-like region?

2 2 let Wa(q ,v) and Wb (q ,v) be two c a u s a l

2 functions i d e n t i c a l i n t h e q .r Q region,

t h e i r difference W

Wa

What form must i t have?

- Wb.

Let

2

Wd(q ,v)

m

study what is p o s s i b k f o r

O f o r q Z < O and hence is causal.

We s e e i m d i a t e l y

sgnv6(q2-m2) is such a function.

W

t r need nor be zero because

To g e t the most general f o m we use wnon's

theorem rJkridir says Caince i t is c a w a l ) t h a t i t mwt be e x p r e s s i b l e fn t h e fom

where @ is non-zero i n s i d e t h e region where the hyperboloid ( q - u ~ ~ mdoes s ~ not p e n e t r a t e t h e region S of q space where we know Wd vimishes.

S Le t h e

e n t i r e q2 < O region.

It is seen that every hyperbolofd cute S unless

3 % ~the m s t general Earn f o r

-

U

0.

% is

er,

*ere:

by definition

#(X)

= O f o r x < O,

Thus complete bowledge of W ~ ( ~ ' , Vi n) the experimentally available 2

q c O region p e m i t s defini tion of kfl e v e w h e r e within an a r b i t r a r y constant, 2 2 (indtependeat o f v) f m c t i o n of q f o r positive q , T f a d e t e d n e d a l s o up

t o the function

S t r i c t l y t h i s s x g w n t is not valid, gradients of S fmctfons i n space time could be vled i n @son's theorem.

Hore correctly, a t each positive qZ the

f m c t i o n W is d e t e M n e d bp the bebavior of W{q 2 ,v) f o r q2

4

O up t o an mknom

2 p o l p o d r a l i n v, the coefficients of which a r e a r b i t r a r y functions of q

.

Physical argmenta about large v as~raptoticbehavior would have t o be wed t o li&t the degree of these p o l p o a a l s .

The scaling lidt f o r

t of the farm vQl{q f a c t that W is odd t e l l s us that (41.7) a u ~ be rurd that T1 is dete-aed

up t o a function of q

fmetiou is not zero f o r q2 by W.

4

2

0 and the

X

2

1 (for v

&vcm by (41.8).

7

0)

Note tt?ls

O so Tl i a not completely deternlned f o r q

2

O

This agrees with the dispersion r e s u l t (40,6a) where a subtraction had

to be made aad m a r b i t r a e f m e t i o a IT1@

2

,0)) l e f t u n b t e w n e d .

Electromagnetic Se

We now diacuss a flew places where knowledp of W o u l d h e l p ua caJeulare the e l e c t m d p a & c p r o p e r t i e s of protons and neutrons,

Since we have masured, i n W,

t h e eleetroanagnetic coupling of protons we nd.@t herpe t o use t h e a x p e r i m n t a l knmledge t o d e t e w n e the electromagnetic energy of proton and neutron and t h e i r w a s u r e d difference, t o comare t o e x p e r i m n t ,

Aa we s h a l l s e e t h e hope

is, a t present, f r u s t r a t e d because knmledge of W f o r q2

O where i t is avail-

able is not q u l t e enough t o d e t e d n e t h e e l e c t r o m g n e t i c coupling e v e r p h e r e (T)

- the a r b i t r a r y f w c t i o n of Slnce the

rfetemtned,

EmstJer

+(a) m a t t o n e d i n the prevliom l e c t u r e is not

f o r the p-n msa difference Le only one a m b e r we

a r e fruiirtrated w t l l we Can find a t h e o r e t i c a l o r experimmtel way t o deternine T uniquely

- f o r e ~ a l a p l et o

2 deternine Tl(q ,Q) of t h e dfsperslon r e l a t i o n (40.6a)

2 f o r q .: 8. Before we discuss t h i s by f o m l mathemtics l e t As La wefl k n m , the s e l f

micallg.

W

see what we can expect,

mss of a point spSn 112 p a r t i c l e diverges l a g a r i t h -

2 For the e l e c t r o n b(m ) = m2

Zn i n

J

where A i a some upper cut o f f

f o r e l e c t r a d y a a d c s i f we replace the photon propagator

This bra

2

A s i @ l a r i n f i n i t y f o r the AM

i a experilnentally undefinable,

m p a t i c f o r t h e proton xrould a l s o '""proton

- by i t s e l f

electro-

not be observilble, but

'""neutron

is, i n f a c t , observable and is nzeasured t o 5 s i g n i f i c a n t f i g u r e s .

calculate i t

2

- o r even

s e e what order of arapitude i t i s ?

-

Can we

f o r e x m p l e does

our present theory say i t must be i n f i n i t e ? As long a s t h e electromagnetic i n t e r a c t i o n of t h e nucleons were v s t e r i o u s

- but now t h a t we have so=

one could always sag anything could happen

knwledige

of them we must tursver m r e s p e c i f i c a l l y , The divergence occurs from high frequeaciers and a t f i r s t it was thought the hadrons d g h t be s o f t a t high frequency and t h e electrorrtagnetic s e l f energy convergent.

But now we know f o r i n e l a s t i c s c a t t e r i n g a t l e a a t they look l i k e they of p a i n t - l i k e c o m t i t u e n t s .

a r e nra&

energy must: dfvergel

Qf course, the proton could diverge and the neutron also-

only the difference need conwrge

Wlp-Win

Does t h i s point-like behavior mean t h e

- but

t h e d i f f e r e n c e i n point-like s t r u c t u r e i s

is a l s o f i n i t e and point-like i n the s c a l i n g l i m i t .

I t is t h e coincidence

of the &-function of t h e photon prepagator aad of the e l e c t r o n propagator which &es

t h e divergent s e l f energy

- and natir we s e e t h e protonI

t h e neutron, and

proton minus neutron a l l have s i n g u l a r behavkor an the l i g h t cone s o i t a t f i r s t looks l i k e dfvergenw i s inevAt&le, Let us estimate haw much,

Since we a r e d i ~ c a s ~ ~ ti hneg h i & energ?r b e h a d o r

we can use t h e i d e a t h a t protoampartons,

Clearly t h e s e l f e n e r e d i a g r m of

rnost i w o r t a n c e a t high energy i n the s c a l i n g lidt a r e when a photon i s e ~ t t e d 2 and absorbed by the same parton therefore, ss far a s t h e divergttnt (Rn h )

-

p a r t is concerned i t i s a s i f each perton g e t s a. s h i f t i n mss p r a p o r t i o n a l t o 2 rn?eiZ~nh /mi'

&mi2

where ei, mi a r e t h e charge and mass of a parton.

m c h does t h i s change t h e mss of t h e proton? acight t r y t o c a l c u l a t e t h e change i n E the changes i n e-p 2

'nuclean

2px

-C

2

blil partons x

We caslnot honestly awe One

- PZ = M2/2P by c a l c u l a t i n g the sum of

of each parton, -=a

Hw

We would f i n d

PP(,)gnh2dK

There a r e objections t o t h i s (by the way i t i e a l s o even m r e divergent since f

ea

l f x and the x i n t e g r a l cannot be c a r r i e d t o 0).

The energy is n o t

Electromagnetic Self Energy j u s t t h e sum a f the k i n e t i c energiers of t h e partana, interaction energies a w n & the partons a r e a l s o involved,

%is Is r e f l e c t e d i n t h e dangerous Eomula con-

t a i n i a g t h e mms squared of a partan 2A

up t o now.

- a thing we s a i d was meaningless t o

The d i e t r i b u t i o a of partclns I s r changed h f i r s t order a l s o s o

2

(Et d g h t be thou&t we a r e

t h e t o t a l ro change Is not c o r r e c t l y evaluated, r i g h t t o take wave funetion

$I

1 AV!$I>

- but

of the Lagrangian

AV

- in

fox t h e perturbation on t h e B d l t a n i a n w i n g t h e o l d 2

hro a*a i s not t h e change f n the Wamiltonian, only

the Haaltoniitn there are

o t h e r e f f e c t s on the

i n t e r a c t i o n t e r n throu&h l/6Eactsrs e t c . , s o we? have nob computed t h e e f f e c t of t h e perturbation c o r r e c t l y ,) It may well be t h a t mfZ

0, o r is e f f e c t i v e l y zero (a suggestion t o me

due t o Zachariasen) due t o i n e r a c t i o n s (or ae a nnatter of p r i n c i p l e ) s o t h e 2 logarithmically divergent perturbations bi " l p n d l n g on RnA n e w r a r i s e , m y be so, but we do not know,

Thie

We mst t u r n t o w r e d e t a i l e d qurtntitativt?

analyaesl i f we a r e t o t r y t o study ?&is

further.

E l e c t r o w g n e t i s mm8 s h l f t s come from t h e emission a d r e i b b s ~ r p t iof ~~

qED f e l l us i t is of t h e photon.

P / f JP (2) J U ( l ) lT fp>6+(s21Z)d~ where 6+(s212) is t h e propagator

Hence by trzkiag Fouxier t r m s f a r t n

Me, have w r i t t e n

aa t h a t

T M LJ

42 ( T ~ +([l ~ Z / q Z l ~ 2 w ~ 1 1 / 4 )

The expression i n square brackets has the imaginary p a r t (l-v 2 /q2 )W2-Wl which i s t h e contribution froon l o n g i t u d i a a l photoas,

Zf partons a r e s p i n 1/22 i t

f a l l s f a s t e r with v thao does Tl whose imsginary p a r t W1 s i n p l y s c a l e s t o f ( x )

i n t h e Bjorken l i m i t .

Ue w i l l hereaf t e r just write T f o r T1+l (1-v 2 /q2) T -T ) / 4 2 l

and suppose the imaginary p a r t W

2 2 Wl+[l-v f q )W2-W4/6 s c a l e s t o f(x) i n the

Bjorken l i M t .

h c t u r e 43

h we have seen we can write f o r the e l e c t r o a p e t i c w s e e f f e c t

e r e OT

m

TUll

.

We must i n t e g r a t e t h i s over a l l q

2

but we have seen T is

determined by i t s behavior f o r qZ ' O (where experiment can say eoaething about i t ) sa i t is possible that mybe (43.1) can be written I n t e r n of T for q2

O only. How t h i s can i n f a c t be d m e

was shown by CottIngham.

Be showed

that i n the four-dimeasianal i n t e g r a l the contour on v could be &=gad passing any s i n g u l a r i t i e s ) Erm the r e a l l i n e v axis v

iw, w =

-oo

to

we can then write d4q

Now replace

-(U

2

2

-oo

to

(hshaw how i t Is done l a t e r ,)

a?,

dw2n9d~'

) by {-q

2

1,

,

q2

v2q2

m

(without

t o the imginary Suppose i t is t r u e ,

-u2-Q2

Q =

t o get

(Cot tin&=

formula)

Nw the quantify is a l l right f o r -qZb u t is completely unphysfcal. f o r v is

laaginary.

Me can define i t however by a n a l y t i c continuation by our disparsfon

r e l a t i o n {SO,ba), settXng

V

iw

We can carry out the i n t e g r a l s on u t o get

This then succeeds i n g e t t i n g an e x p l i c i t f o m u l a f o r the s e l f energy in f e r n of wtq2,v) i n a region accessible t a experiment. 2

by the unknam term T(q ,O)

However, we a r e f r u s t r a t e d

.

2 Z t i s e s s e n t i a l t o kn&w somthing about T(q (0) if we a r e t o be a b l e even t o deternine ðer the s e l f energy divergee.

Let us look f i r s t a t the c s n t r i -

bution f r m t h e eecand t e r n i n ((43.5) i n the e c a l i ~ gregion fraxn whlch divergences could c o m e Put -q2 = ZMvx and consider W a s a fvnetion of x and v, W(..") Tor l a r g e v cmverges t o the lMt f(x),

dx vdv

-1

- 1;

vhieh

"&e t e r n becotlles

utx,v>

o r f o r l a r g e v, where t h e square bracket f s -(2&/v)

2

14 we g e t

xf ( X ) dx, the

The v i n t e g r a l diverges logsritharically with coefficient

f r a e t ion

of trrcmentma c a r r i e d by the charged partons each wei@ed by the b a r g e squared, 2 2 Of course a cut-off of electromagnetkm is used i n (43,5), d ( q )/(+I ) is replaced by This provides a cut-off f o r our v i n t e g r a l (of order f / 2 & ) diverging as ko A

2

so the parr

f

has a eoef f i c i e n t zf(X) h. 2

However, t h e other t e r . in T(q $ 0 ) could a l s o produce a fin A' term f f i t only f a l l s aa C/(+ 2 ) a s -q2 -, G*,

2

(Zf T(q ,O)

cei(s)ds/(q

2

44

2

1 this C

1,

Z t is therefore possible t h a t these divergences from 1: ernd W cancel and t h a t the

s e l f energy (or a t l e a ~ tthe protm-neuf ron dLf ference) is f i a i t e and calculable, m e r e a r e two viesrs we c o d d take a t present,

We know of course the p*

=ss difference converges so Let us t a l k about T snd W f o r the p-n dlfferenca a t least.

In p r i n c i p l e T could be determined bp experfirnent and is therefore defined

phyaicall*y, e i t h e r of the f ollowing could happen. (a)

Equation (43.5) with t h i s T s t i l l ggves a logarithmic divergence, The re-ason is t h a t our theories a r e wrong f o r high energy; t h i s and the e l e c t r o m g n e t i c s e l f m e r g y calculatLon of QED a r e both wrong and w l l l both be fixed by t h e s a m modification of r e l a t i v i s t i c quantum mechanics a t high energ?t e o m day s o be found,

Cb)

The T is such t h a t the i n t e g r a l s converge and agree with t h e experim n t a l mas8 d i f f e r e n c e s

(c)

2

The TCq ,O) ( f o r q

2

G

.

0) is i n f a c t n o t p r e c i s e l y definable experimentally,

t h a t i t Is t o soroe e x t e n t a r b i t r a r y

- hence

that, our theory is not a b l e

t o c a l c u l a t e t h i s =ss d i f f e r e n c e p r e c i s e l y and rnust be " r e n o m 1 i z e d e t ' 1 believe i n t h i s case, i f partons a r e quarks only one r e n o m l i z a t i o n

constant, corresponding t a the e l e c t r o m ~ e t i c=ss

difference of u

and d quarks, would s u f f i c e t o make a l l the hadron s e l f e n e r m d i f f e r e n c e s converge simultaneously, As Zachariasen has suggested t o rae, t h e b e s t (.roost l i m i t i n g ) t h i n g t o

do today IS t o aseum (b] is t r u e , and may have p r e d i c t i v e value, l e a r n t h a t (a) must be s s .

This puts r e s t r i c t i o n s on possible t h e o r i e s

Xf i t leads t o a paradox o r inconsistency we

Zachariasen has shown t h a t a l l w i l l be convergent

i f t h e equal time c o w u t a t o r o f J and =del

3 vanishes

[J

P'

iy] 0. I n

the q u a r t

i t c o r ~ s g o a d st o quarks having zero r e s t =ss.

Z believe, t h i s Is a very good problem t o work an.

I myself haven"

found

enough t i e while preparing theae notes t o analyze i t i n a more elenensary o r c l e a r fashion f o r you,

2 Now can we ever hope t o g e t a t T(q ,0) experimentally o r t h e o r e t i c a l l y ?

Xt is t h e forward s c a t t e r i n g amplitude on a proton of a v i r t u a l photon of 2 mass -q

.

It would be iavolved i n a wo*lectron

forward s c a t t e r i n g e+e+p

-+

e++p v i a t h e dlagram

R i s i s not aa expergmnt t h a t can be done,

2 But knwledge of T(q ,v) a n p h e r e

EIecrromagnetie Self Etiergr would be of a s s i s t a n c e because the dispersioa r e l a t i o n s can be used t o convert i t t o knowledge of T(q

2

,Q); s t i f f no experiment suggests i t s e l f .

It is i n t e r e s t i n g t h a t T(O,O)

can be obtained t h e o r e t i c a l l y

Coapton s c a t t e r i n g from a proton ofIP a r e a l (on s h e l l ) photon,

- the forward

For Q

-+

0, v

-+

O

we have vary long wave length slow f i e l d s to whsctr, of course, the proton looks t o be simply a raassivr: point charge of =ss

M,

It s c a t t e r s then as i t would

clczssicslly [or man-rela t l v i s t l c a l l y v i a Schrddinger equation from t h e

Q2 +

-b

AeG

tern) thus (called Ray1eigh s c a t t e r i n g ) It glves 2 T(O,O) m

- %.

NOTE: Wow to r o t a t e contour to get Gottingkara f o m l a ; Use WS t e p r e s e n t a t i a n f o r T

Call E

P

mid note t h e slingufarities a r e a t v

F

Since E > BM, t h e palea below the a x i s a r e a l l f o r v > O,

;

Q - i ~ , +ic

llhe contour goes l i k e

the dotted l i n e , i t can evidently be r o t a t e d t o the i m g i n a r y axis.

Lecture 44

W e d i & k succeed i n representing AM

2

2 i n t e r m of W(q ,Q) f o r negative

2 q only, wilthout a t the s a w tim invalving ourselvee with another w h a m 2 f m c t i m T(q ,Q); and f u r t h e r each of two p a r t s i a i n f i n i t e and i t Is hard t o guess a t the difference, 2

Perbpps we should abandon t h i s and take 2

8

l a s t lnok 2

at: jwt the bI4 expressed i n t e m of W ( q ,v) for posirive and rmegaeive q

is given by

.

Zt

206

Photon-Hadron Interactions

(This is obtained by expressing

and i n s e r t i n g i n t o (42, l ) , W (v

i n the form

$6) depends

only on Q, the =pitude

of

4; and

To study the possible divergence of tf-re i n t e g r a l a t l e a s t , we go t o large 2 v and i n f a c t t o the s c a l i n g region -g 12Mv

x , hence Q =

v+m. We can write

is kinematically I.

2MM approaches the

; .

(consider 2W a function of x and v, ZW(x,v))

For l a r g e v the upper

lintit

for

X

fuaetfon f fx) f o r poeitlve x I n the scaling l i m i t ,

Far negative

i c approaches a s we have seen -f(-X); note, hotsever,that

-X

X

2 (positive q )

is not k i n e m t i e a l l y

DyndcalZy -x i s l i b t e t l i , W(x,v) e d s t a f o r x < -l but f o r large v f a l l s rapidly u n t i l there is nothing l e f t of order one. The eontributian of the scafing region gives But t h L s n l y says t h a t a dlmrgence h i & e r than l a g a r i t h a c vanishes, a Uling we expect a n p a y , Wx/Zv leading t o

Expanding the faetor ( /xf(x)(dv/v)&;

)/(2v+m)

s e e m t o give a tarn

t h i e i s not a l l , we s h a l l have t o Lnw to

2 order l / v haw M d i f f e r s from i t s Bjorken f i a t f o r bath positive and negative q ,

This suw up the probleni; experimat c m i n principle kelp m with the approach t o the limit f o r q

2

negative.

But we s h a l l have t o r e l y on theory t a

obtain the cmtribut?tans t o order I l v f o r positive q

2

before we can decide

whether d g diverges according t o present theories.

Waving f a i l e d wfth ftnrdamntal theory t o get i a f o m t i o n on etctrornagnetic

mass differences, we now turn t o nnrch cruder pictures t o diacms the possfble relatSons of 4 2 i n d i f f e r e n t terns ctf an $Uj o r $U6 multiplef. the laaguage of the quark made1 althou*

We do i t i n

mny af the r e s u l t s come from weaker

~eunuptiana, Iflire s i q l e SU3 sec. The proton, f o r e x a q l e , i n the low energy qmrk model is made af three

Electromagnetic Self Energy quarks, two

U

quarks and a d quark with t o t a l s p i n l 12 with wave function (2 uud

]p>

- udu - duu)

(+S+) symetrieed

.

(44.3)

The electromagnetic s e l f energy can be thought of as being ntade of two parts: a)

The s e l f enerp;y of the i n d i v i d u a l quarks.

We suppose & i s

t o t h e change squared o f each, thua 4a : a: a f o r u: 4: 2

proton t h i s csntr-lbutes bM f 2 N to

P

S

ifs

proportional

respect%veIy.

To a

= 9a (we s h a l l n o m l i z e a l l mass squared changes 2

2P1 of t h e proton a s a s c a l e f o r laeasaring a , t h e true. change i n M i s than P

2M a,) P b)

An interaction energy bemeen p a i rs which we take as proportkonaf

t o t h e product o f charges. r e l a t i o n of the p a i r ,

The i n t e r a c t i o n mmt depend on the wtwl s p t n

'&M w r i t e

@(l*)

i f s p i n is p a r a l l e l and B(1-y)

if

s p i n %sa n t i p a r a l l e l ; t h i s i s B(1WP) where P i s t h e s p i n exchange o p e r a t o r , We a d t i p l y by -2 f o r ud o r us, 3.4 f o r uu and + I f o r dd, ds, s d , ss p a i r s ,

call t h i s factor x

il "

The electromagnetic s e l f

operator

cam t h e r e f o r e be w r i t t e n

Thus

X t i s easy t o get the e x p e c t a t i a n of t h i s o p e r a t o r fox e v e q s l a t e .

on t h e proton t h e o p e r a t o r Bx alone does n o t s e e t h e s p i n s and gives (2(4-2-2)uud (4-2-2)udu

- (4-2-2)duu)+l.+/&

which is zero,

For the neutron, change a

t h e c o e f f i c i m t is -3 s o t h e r e is a c o n t r i b u t i o n

- 3f3,

Next we study 6yxP

therefore xf"(2uud

- udu - d u u ) + f + / g =

=1/8.t.2)uud

+ (-4-4+2)udu

+

(-4+2-4)duu)++C/&

and

Adding up the v a r i o u s c o n r r i b u t i a n s w e have f o r the proton

AM'/ZM

P

-

9a

+ 68y

U

t o d,

-

3.

We c m uldlculate the E=M s e l f naasses for evesJl p a r t i c l e of the l12

octet:

a s was done i n the previous l e c t u r e f a r the proton, the r e s u l t is:

Hence hM

We have three constants f o r four ms8 dJbfferances s o we have Ehe relation (m SSU3 r e l a t i o n ) (E+-c-)

- (p41 -

(We choose t o use P$

w h i e h f i t s well.

f o r no extremely good reason, the r e l a t i o n

AM but there l a l i t t l e t o chooole,it is -Snsi& the ewerimanta%

f f t s b e t t e r with

e r r o r f o r dk?

B@- 3 -)i. a

&Q.)

The values of the constants t h a t we get are 3a

-2.0

@(l*)

.24

B(1-r) = f .32

B

m

-78 NeV

MeV %V

p a r a l l e l spins

M V

a n t i p a r a l l e l @pins y

= -.69 HeV

The s i g n of tt 1s the oppczeice of h a t you would expect but then yort expect

+

and

it

m e t be r e n o m l i z e d , possibly t o a newtivcc value,

It man@generally

209

Electromagnetic Self Energy p a r t i c l e s wlth fewer u quarks a r e heavier, i . e . m r e p o s i t i v e l y eharged baryons are l i e t a r .

The s i g n of the 8 term from e l e e t r o s t a t l c repulsion is p o s i t i v e

as expected.

We f i n d a t t r a c t i o n of p a r a l l e l raagneto i n m s s t a t e but repulsion

of a n t i p a r a l l e l (the magnets a r e

OR

t a p o f each o t h e r ) which is c o r r e c t ; the

n e t repulsion i n the p a r a l l e l case is l e a s . GontinuFng i n t h e 56 $U6 raultiplet t o Che d e e i m t and supposing the constants a r e t h e s a w we can p r e d i c t everything:

nn2/2n (predicted) P

ss:a if-

110

G-

=2-

m

Sa

- 313(29)

- 4.72 - 1.28

an-- 3a + 3B(l+)

Tfie expexiunentarl d a t a is not good but there is a s e r i o u s discrepancy with a very recent experiment on b0

-

A

~ except ; f o r t h i s t h e s i g n s and

general order of s i z e s i s good, For pseudoscalar tnesons, noting t h a t t h e s i g n of the charges is reversad f a r i m t i p a r r i c l e s , and t h a t only a n t t s m e t r l e s p i n c o a t r i b u t e s we g e t ( c a l l B ' = @(l-y))

W e have two constants t o f i r with two p a r a m t e r a and have no predictilon,

We g e t B'

* l 4 , 3a

-2.38 again e a n f i a r z g t h a t a is negative.

close t o its value f o r t h e baryons r*rhich is what =R%

Xn f a c t a is

a e l a t i v f s t i c quark rmdel

t~-auldexpect, m g n i tudet

F u r t h e m r e the 8'

.

F i n a l l y we do t h e vector m s o n s ,

Here we need g(1-y)

*K The only d a t a on AM is t h a t f o r ' 2

is n e a r l y of the s a w order of

@(l-y)

- K*'

b f o r p a r a l l e l spina

which i a -5.7 2 1.7 HeV o r

-5.1 2 1.5 = 3a + 3b. X£ 3a = -2.38 t h i s gives B = -,9 2 5, which is P very bad, i t is of the wrong sign. X f t h e values f a r t h e o c t e t baryons a r e f2M

2

w e d we p r e d i c t bm /2M 2 1.91 P h t h i g system we a l s o have two o t h e r e f f e c t s (discussed i n Lecture 15)

a) t h e e l e c t r o m g n e t i c rnixing rnatrix between p' 3a

- 3b12,

aad wo, with off dlagonal t e r n asld wo which we fotlnd t o be

b) The a n n i h i l a t i o n term between

'fhe o f f diagoaal elernent 6 i n t h e Htasa matrix

is deterninczd by

p ,w i n t e r f e r e n c e t. be -3.7

m n i h i l s t i o n t e r n + , 5 l gives -4.2 2

energy. Sa

bm /2M

-2 &V

P

+

.9 &V.,

then corresponds t o 3e

+

.9 MeV.

Subtracting t h e

f o r t h e contribution of the s e l f

- 3b/2

t h a t b is i n f a c t p o e l t i v e and near

3.7 2 .7, 4-

% i s suggests i f

1.1 2 .4 MeV, not too con-

s 2 8 t e a t with the baryon va2ue of ' 2 4 f o r p a r a l l e l spins. rise:

This i s generally w s u c c e s s f u l .

SU6 does n o t , predsctgng b0 the *sons

- A*

things a r e very poor,

(comnpared t o 1.3 f o r baryons)

.

+

For thie baryons SU3 works but For 0.3 f o r t h e experimental 2.9 2.9.

Far pseudoscalars we g e t 3a

.X4

For t h e vector mesons the s i t u a t i o n is czonfur~td.

2

If we use mass d i f f e r e n c e s inatead of dn~/2M

P

c o n a t m t s c a m our 3a

-2.4, 8"

--1.9t @(l+)

= 6.20,

f o r the r u l e s , t h e baryon

t((1-y) = 0.98.

The predicted

211

Eleerromagnerie Self Energy .l0

- tit+ i n 0.8 s o only l i t t l e is gained here.

a l t e r e d (beeawe of t h e snrall s c a l a r s 3a = 7.0,

8'

1.0.

n

But the meson situation is

The constants c o w o u t f o r the peeudo-

mass),

For the vector m s o n s we have 3a

-4.8,

-0.3.

b

for

The B ( 1 9 ) fox baryans m y d i f f e r from b f o r vector nresoas and baxyonhj nay a l s o d i f f e r from B y o r pseu&oecalar =sons because t h e s i z e af the wave function is s o d i f f e r e n t s o the man I / r d i f f e r s .

The e l e c t r i c and

mgncttle i n t e r a c t i o n need rtot change i n the same r a t i o so i t is poseibla t h a t

b is negative f o r nreeons and p o s i t f v e f i r baryons, but i t is d i f f i c u l t t o s e e 2 why 8 s o d i f f e r e n r a s the Am gives. Wfig the v a l w of 3a should d i f f e r i n one c a e from t h e o t h e r i s n o t

2 Were It n o t f o r experiment X would deci& 3a c a l c u l a t e d f a r bar /2M is

clear, the s

P

e f o r pseudoscalar and pseudovector *ems

m d is 0.6 of t h a t f o r

9111s is because we a l g h t guess t h a t mass squared s h i f t @ due t o

baryonrrr,

s t r m g m e s a come

from a w s change i n t h e s quark.

2 The d i f f e r e n c e lin na

f o r t h e baryons i s about .40, f a r the mm088 about .2S o r 0.6 aa such, t h e self energy c o r r e c t i o n e f f e c t

- if

Thus

i t i s t o be associated a i q l y with a

proper mass chmge of t h e u quark w i l l a h o be 0.6 a s e f f e c t i v e i n IBesons a s i n baryons

.

Evidently t h e naive theory doea not work well, we do not understartd things s o well.

A m r e d e t a i l e d d p a d c a l theory 19 necessary.

case, mast p a r t i c u l a r l y the l a r g e bo AI

+

- P*

But i n m y

is most disquieting.

2 mss d i f f e r e n c e s There 58 one obsarvation ehar c m be made h e r e which suggest8 t h a t d p m t c

calculation8 rnight be posslble f o r s o m caabinatiolorus,

Hotice t h a t a is the

s e l f energy t a r n , pcrssibly involving h i & fireqmncy behavior, but f311Cy) is due to m t u a l

i n t e r a c t i o n and ought t o show no diverg;enccz?. Certain co&inatfons

of w s 8qr;tared d.iffc?rences do n o t iavolve a .

Arl these involve t h e b I energy.

They a r e

2 i s o s p i n p a r t of t h e electroaragnetic & e l f

This energy depends on t h e products of two current operators JJ each

c f which. contain8 61

O o r l, and can w k e t e r m Khose p a r t s a r e

dX

0, I o r 2 ,

-

1 I(I4-1) respectively. and IZ2 5

I n terms of Iz these go a s constant, Tz. The constant (AI

-

the A I T'

- L-

0) p a r t i s e q e r i m n t d l y l o s t i n the s t r o n g i n t e r a c t i o n s ,

1 o r Iz t e r n . a r e measured by differences proportional t o Iz, l i k e

o r p-n.

But the bI

2 t e r n e f f e c t is proportional t o :I

measured by the differences (I* ' C We a r e calculating the

&IZ

-

+ C-

- 2z0) mantioned above,

O coarponent of t h e A I

but t o i s o l a t e i t think of c a l c u l a t i n g an a r t i f i c i a l P I z i f electromagnetic current J were AIz

and is

-

+ l (instead of A I z

2 e f f e c t , of course, +2 U a t would a r i s e

1. UP

Q), A I

mm t then think o f c a l c u l a t i n g an e f f e c t of two currents l i k e J+ J-1.

.

Now we s e e why a does not a r i s e , and why i n general, &oesib&y the i n t e g r a l @

i n calculating t h i s might converge rapidly a t high v, provllded parcons a r e quarks.

For i f all, t h e current c a r v i n g fwd-ntal f

isospin 112 i t Is i w o s s i b l e f o r two J succession a t high enc;r@, contribute only t o A I

operators (partons) have

t o operate on the @amet parton i n

%us t h e v i r t u a l photon exchange d i a g r a m l i k e A

1 and a r e I q o s s i b l e f o r bZ

p

2 , only B a r e a l l w e d f o r tar rong interaction photon

.-.-.-

Baryon

parton

Large v i r t u a l lnoeaenta of the phbton is posegble i n A no m t t e r how s o f t the s t r o n g i n t e r a c t i o n s a r e

- if

the i n i t i a l rrrrmntw distribution o f t h e

partoncl involves only slow ones so can the final, s t a t e .

But i n B, i f the

virtual, photon wmntum is very h i & i t is unlikely t h a t t h e s o f t s t m n g int;ersction can p u l l the partons back together t o give rnuch o f a diagonal m p l i t u d e tlo be i n the wdieturbed baryan atate. again.

As we found. i n the previous l e c t u r e , d p m i c c s l c d a t i o n s of bT

a

2 ass

differences should be f e a s i b l e by s we know t h e o r e t i c a l l y and e x p e r i m n t a l l y &out the expected behavior of the

Electromagnetic: Self Ettergy Of p a r t i c u l a r i n t e r e s t would be a study of the

necessary m t r i x elements, 'lt

213

- no .atass difference, .c.

The e l a s t i c . t e r n t o n t r i b u t e s alarost a l l of t h e n

as a s i q l e c a l c u l a t i o n w i l l show.

- no mass

difference

The d i a g x w a r e

p/s

P, v

71

The pion form f a e t o r is d o d n a t e d by the f a c t o r on P

2

p

reaonrtnce, we t h e r e f o r e F n c l d e a

2 2 / < q -m ) a t each photon coupling s o fke -ss It

( 2 ~ ~@P 3 ~ -q )1 p - 6

(P+ 2 To order zero i n mr /$2

2

2 - aann

difference is

d4q/ (2711

4 (46 ,l1

2

UU

9

the f i r s t term i n t h e square brackets is 1, t h i s gives

S The i n t e g r a l is r e a d i l y p e r f o m d r e m a k e r f a g t h a t fn four diraensions d q

a f t e r I n t e g r a t i n g over angles. (&m210

The r e s u l t of the zero*

order caltculatlon is

3e2m 2/4n, doing the next order gives P

h t e m of &m m n t a l 4.6 &V,

Ir

2 + Am /2on thSs i s n

- no

a:

4.1 &V

a s conrpared ts the experk-

E s t i m t e s of contributions of higher i n r e m e d i a t e s t a r e s

(&%eh could be done using FKR's

m d e l ) should be small.

A s i m t l a r r e s u l t is fovnd f o r E+

+ E-

- 2z0.

It seems as though t h e

e l a s t i c t e r n &ready gives a l m s t t h e complete r e s u l t .

2 2 2 n q dq

Bstimtez, of h i @ e r

resonmces give l e s s than 20%, and the r e s u l t is i n good agreeraent with t h e data, The contributfons of t h e e l m t i c term t o s e l f energy differences o f

172 l381

b a v o n s were calculated by Grose and Fagelis, Phys, h v . Sli

3

f o r the magnetic momants, and GE and C& varying l i k e ( 1

619681 using

+ q2/.71)-2.

They get bM(elastic2

P* C'

+0.79

-

C"

6'

- 6-

-0

.".-2

::

AM(exp. )

- 1.29

MeV

+ 0,16

- 3.06

- 0.88

- 4.86

- 1.10

- 6,5

&V

1 .7

( M s t of t h i s i s due t o the charge, s i a e e there a r e factors of q f r m i n the ntagnetic part.) to +1,8 e x p e r i w n t a l , tern.

Nate & a t atl

+ C 4 C-

- 2~'

4-1.54 &V

y 44y

Ir.

1.1

( e l a s t i c ) cornpared

we e x p c t e d this should be dominated by the e l w t i ~

(kcinrcirt i n t e m d i a t e s t a t e s give less than .l and pres

ones even l e a s ,l Note t h a t other c d & w t A ~ n s(other t h AE

-

2) do not involve "a@'

(see l e c t u r e s 44 and 45, "a" i s the EM s e l f energy of a quark) such as (p-n)

- ( 9 '-3 -1,

In t h i s case one carnot skaw t h a t these dif fereacer, a l s o h v o l v e

a product of currents su& t h a t each current does not a c t on the same quark, C suggesta t h a t we may be able t o g e t a t But the f a c t t h a t "a" i s ~ R O involved

t h i s difference a l s o by estimating m t r i r e l e m n t e t o varioua know s t a t e s w i n g , f o r e x m l e , the quark ~tadel. Wwever, tkr? e l a e t i e term f a i l s u t t e r l y t h i s t i m Am(e1astfc)

+1.9, A M e q . = 4.2 2 .7.

look a t the contributions from the sealing region.

ay? To explain t h i s we Let u(x) , :(X),

s G), ;(X) be aa described i n l e c t u r e 31, then vW2/x

3

d(x)

f (X) f o r p, n, 2

, ;(X), and

iki

(Tkre neutron is obtalned from the proton by replacing u by d and vice versa, The 3'

i s l i k e the neutron but s replaces u and vice versa.

'E but u replaces d and vice versa,)

m e 3 - i s lib the

B e r e f o r e the scaling Emetion f o r (p-nl-

21.5

Electromagnetic Self Energy

- 2 ""1 is $ ( u +

(Z0

+s+a

h i & frequencies can corn i n .

- 2d - 22)

which is n o t n e c e s s a r i l y zero s o

I n the w d e l of valence quarks plus s e a i t is

zero but we do not b e l i e v e t h i s t o be likel_y, For f u r t h e r d e t a i l s on a l l these w t t e r s of EM s e l f energy s e e an a r t i c l e by W, B, Cotringham i n "H~adronic I n t e r a c t i o n s of Electrons and Pro tons ", C~umings and Osbom E d , Academic Press, N.V.

1971,

Mcturre 47

Ve now go on to discuss o t h e r e f f e c t s involving two photon couplings. The Cowton e f f e c t is t h e most c l o s e l y r e l a t e d t o what we have done,

I f the

s c a t t e r i n g is e x a c t l y i n the f a m a r d d i r e c t i o n the s c a t t e r i n g m p l i t u d e i a given by T

QV

2 (q ,v) f o r q2

thus T is the

spin

0. We previously meant the average over proton spina. averaged f o w a r d s c a t t e r i n g , we could a l s o measure f a r

s p e c i a l s p i n d i r e c t i o n s of t h e proton.

The imal5;inary p a r t o f tfte f o m a r d

s c a t t e r i n g is, of course, the t o t a l c r a s s s e c t i o n cr discussed before.

l i k e ((97

+

(E.g.

67/&jub,

a

YP

YP

or a

Yn

wtrich we have

showd resonances a t low v, a f a n o f f perhaps

and (97 4- 43/&/pb]

f o r neutrons.

The d i f f e r e n t i a l cross secegan can be f i t t e d with

For m e r g i e s from 2 t o 7 GeV we g e t around 6

16 GeY, A with A

IJ

-

8

&vd

7.4, B

P

For energies from 8 t o

is c l o s e r ; there i s some s i g n o f a quadratic term A t 2.0,

np a t 9 GeV has A

9, B

+

Et

2

(This is lnuch l i k e hadron d i f f r a c t i o n s c a t t e r i n g , @,g, 2.S,)

Thus photon d i f f r a c t i o n looks very much the

s a w a s w o d d be expected f o r hadrons except f o r t h e very lauch snraller c r o s s s e c t i o n , of course.

kle now discuss the forward s c a t t e r i n g i n mere d e t a i l including s p i n e f f e c t s . The f o m a r d amplitude w y be w r i t t e n

as a s p i n m t r i x operating between s p i n statess of t h e proton i n t h e lab. systeas,

The various laeasured q u m t i t i e s a r e expressed i n t e r n of El and E2 a s follows:

The t o t a l cross section is the imaginary p a r t of the diagonal ( i n spln) scattering

The foxward d i f f e r e f l t i a l cross section f o r =polarized

This is h-.

forward

scattering i s

The r e a l p a r t of f l c m be obtained f r m the imaginary p a r t by a dispersion r e l a t i o n , (Eq. (40.6a) f o r q2 = 0) where we use the f a c t t h a t fx(o)

2

-e

/M.

This has been evaluated (see Dammhek and GiXmn, Phye. Rev. X)1 l319 (l9705 o r Busckihorn e t a l ,

241 (11970) and the sw of the f i r s t two

Phye;, L e t t ,

t e r m of ( 4 7 . 3 ) is compared t o the expcrrimntal (do/dtIo t o see how b i g the l a s t t e r n is.

They agree within e r r o r s f o r v from 2 " 5 t o 17 GeY s o the

emtr2bution is l e e s than 10% over t h e e n t i r e range.

( B e contribution

g r e a t e a t from the f l r e t t e r n i n (47.35 above about 5 &V,

W

I f212 /v

l8

The second is 15%

a t 2 GeV and f a l l s away a t hip;her v ) ,

-

2

: p v where iiAis the 2M molazlow p a r t of the = m e t i e mrownt of the nucleon i n nuclear =ametens.

W e a l s o how, f o r small v, a s v

G =-

'=L

"1. where a

i

+

and B

-r

0, IZ(v) =

?l

(47.5)

91 II

a r e d i f f e r e n t i a l cross s e c t i o n s a t fixed t Ear in~oraing

photons polarized pergendlcular o r p a r a l l e l t o the plane of callisiczn, t = O

For

Z must be zero, of course; but within the l i d t s of experimental e r r o r

(210% f o r -t e .2, e20X up t o -t

' 6 ) i t is zera up t o t:

-0.6,

(The

average of C f a r t = .l t o . 7 is .Q2 ".Q6,)

Me have discussed the s i z e of da/dt f o r yp i n r e l a t i o n t o (VDM)pp cross section (see l e c t u r e 201, i t is twice l a r g e r than VDM expects,

The a s ~ e t r y

produces no problem f o r the correspandi-ng a s F e t r y I n t h e p c m e , i t 1s a l s o very snnall;.

'This is not w e ~ p e c t e d , s channel h e l t c i t y conservation a l a o

Electromagneiic Self Energy expects the s a r e s u l t ,

The question is:

With i n c i d e n t 1i&t i n t h e z d i r e c t l o n

with x p o b r i z a t i o n do m r e photons s c a t t e r a t a snzall angle

i n t h e d i r e c t i o n x o r i n the d i r e c t i o n y l

SQ

Ctraneve~se)/v

From t h e point of view of d i f f r a c t i o n ,

c u r r e n t s generated by t h e incident. wave must be adequate t o produce t h e c o r r e c t f o m a r d s c a t t e r e d wave t o i n t e r f e r e with t h e incident wave t o a c c o m t f o r t h e mese

l o s s of i n t e n s i t y o f t h i s wave represented by the t o t a l cross s e c t i o n .

c u r r e n t s a r e obviously l i a t e d t o the s p a t i a l region of t b e proton, and s o they produce s c a t t e r e d waves i n o t h e r d i r e c t i o n s , t h e w v a l eAt of d i f f r a c t i o n from the proton, jwt a s i n hadronic c o l l i s i o n s .

But these c u r r e n t s m a t make pure

x p o l a r i z a t i o n a t l e a s t i n t h e forward d i r e c t i o n , they a r e x d i r e c t e d c u r r e n t s . In o t h e r d i r e c t i o n s a t sm11 m g l e s we have the same i n t e n s i t y f o r d e f l e c t i o n except f o r a cos Blab, therefore E -t

m

(1

P

1

- cos 4ab) / ( l + cos

X

and y

- eZlab profection f o r x d e f l e c t i o n , 4ab,)

0 . 6 , v 2 3 . 5 where t h e d a t a is taken.

@iab-t/2v2 */2

;-

+

.03 f o r

m u s we expect: small 6, i f any,

c l o s e enough t o zero t o not be I n disagreement w i t h experiment wi&in i t s e r r o r s . r i z e , t h e Campton s c a t t e r i n g as a function of t above 2 CeV a h w s no s u r p r i s e s o t h e r than what w e can expect from d i f f r a c t i o n from the k n a m t o t a l photon absorption cross s e c t i o n . Below 2 GeV, t h e r e f o r e i n t h e r e s o n m m region, there is no d a t a ,

But i t

should be possible t e make a p r e t t y good theory of angular d i s t r i b u t i o n a d enargy variatlion by considering a s w c e s s l o n of s-chaanel resonances (mny of t h e m t r i x elienrents of which a r e kn5m from t h e study of y p * ap i n the s a m energy region, m h o m ones m y be guessed from t h e quark =del). There i s a l s o a coerputabbe n e u t r a l pion exchange t e r n

The coupling of two photons t o a neutraf pion is

from the no * 2y decay.

A l l these c a l c d a t i o n s can be c h e c k d and controlled bp f i t t i n g the c a l c u l a t e d

imaglinary p a r t of the s c a t t e r i n g t o the n i c e l y m a s u r e d t o t & cross s e c t i o n otot which shows t h e expected resonance b w s i n t h i s region of energy. YN

S c a t t e r i n g of very low Q, v is l i k e t h e s c a t t e r i n g o f radio waves, o r (if q

2

6 Q)

i t depends on the reaction of a p a r t i c l e t o nearly constant e l e c t r i c

and mag;netic f i e l d s ,

This is, of course, gives by two experimental constants,

(obtahed by nreasuremnts i n such f i e l b ) the eharge and the mgnetfc mroent ( r e s t r i c t t o s p b 112 case),

Therefore we expect the Gompton e f f e c t f o r Sow

enou& v, Q to be given e n t i r e l y i n t e r n of these constiurts. should a c t exactly a@ i f they were point p a r t i c l e s .

'.%C?! p ~ r t i ~ l e s

One can cornpute rhc e f f e c t

s e d e l a a ~ s i c a l f yo r fro= the non-relativistic approximation to the S&roedlslger equation with spfn (Pauli equartion) o r again by diagrp a r t i c l e with no i n t e r n a l excited s t a t e s ,

a s s d n g a pure

Sueh a tern is called a Barn t e w .

Thus we write T

(not averaged over spin directions of the proton) a s a sm uv from the Born t e r n and the r e s t from other diagram

B do&natsa a t Tow

The reaaon T

-

V

is t h a t i t has an energy d e n s a n a t o r

due to the i n t e m d i a t a s t a t e A of s i z e (M + v -EA) EA is M

P

(as Q

P

80

if A

%S

0) and we have a leading l / v factor which doesn't appear i n

B the rearalnlng t e r n T

.

& you expect the lestrix e l e m n t of the charge density St i s - the t o t a l charge a d

vation) ; SO aee t h a t

V

Y

is diagonal. thrrs

c xlp (Q) /p>

P a t (7

m

O

2 order Q if x is not

To show Chat the off dlisgonal nratrix e l e m n t s of other

the proton s t a t e . coqonenta of J

a l s o a proton

I

O (charge? ccmserc X / J p> M Y hence i f v , Q go t o zero together we

also go t o zero we look art q

lP>,

x [ ~ / ~Q > .cx~~[o)

3 tnatrix

e l e m n t a go t o zero.

A m r e rjgorow (but har&er t o i n t e r p r e t )

a r w m n t is &even below,

B

We now cowute Ehe lirnit of T a s v ,

9

4

O,

Xf uA is the anonralous

m m n t i n nuclear nametons, the coupling of a photon is yu

For srmll q and tq2

O t h i s l a e r s f i y worked out t o be

F"A ( y u l +5

so

EEecrrontagnetic$4Energy

while the contribution from T

--

fl(ct)

s t a r t s aa uZ nnd thus we have as v

+

O

eL/~

f

v

R

B

To show how

2 2 uA/2H

- e2

o

(47.8)

ssnrall. atore f o r d l y (at leasre f o r the f l tern) note

B

t h a t Iyvt o t a l s. well as T each separarely s a t i s f y the gauge condition qvTBv

-

hence we must have q T

Ci W'

0. NW we can w r i t e T

C

duv

+

.

B

(peQP a+ ( p q l q v b + q

2

it

pM b

kfe cannot solve this by a = +q

--

therefore s

%

peqb, b * o

W

+ pvqy) + W total h e r e must be no poles l i k e l / p * q i n a,b,c (unlike TMv

vhich hae such poles comtog from T

Clearly c

0,

apypv + b(puqv

averaged case a t l e a s t ) a s a power s e r i e s T 2 order qv

-

( f o r the symmetric @pin

-

2

-

1. NW q51 Tliv

+ qv

e

+ order q3

O requtres m

O

/peq because n s 1lp.q t e r n are alLowad.

2 peqa/q with a

$, b .r - o(paq),

c

%

oq2 a r e the only p o s s i b i l i t i e s .

2 a(p*q) and the term s t a r t s out as second d T~~~ can alao be shown t o be of

the same order imd therefore T COmpfon dtscwaion s e c Low, PPlps, Rev,

T~~~ to order 1 and v.

916 1428 (1954)

m$ Gel1

(Par a complete

n, Phya. b v .

96 1433

Forward Compton s c a t t e r i n g from non-relativis b t c Schroedinger equation The equation is (with f i r s t o r b r r e l a t i d e t i c corrections)

3%.

$

h e t d e n t amplitude is

b = Zi, 2

-

tvgi

and

d

is zero, hence the leading term cores from the

contrgbut%ng t o fl,

-+

i-

-

ig x

2.x

gi.

I n the laboratory

- $ + farm and i e -(a 2 /2M)ei*ef

Next we have the terrn a 08 operating i n second order, two

d i a g r m with energy denodnators -v and +v

220

Photon-Hadr~~ lntmctiorrs

-

2

1

+v

r gf) contributing t o f2

2 2

i

F h a l l y t h e l a s t t e m i n (47.9) with rha terln

g.2

x

d

gives

2 which c~labineawith the previous t e r n t o change rhe (1 + U&) to U

therefore have 2

Amp,

-2M &

-

t

ei

*

+

ef

- i2 2

2 - t - b -c v c*(@fx e ) A t

.

Other Two-Current Effects

Other quantities involving T

W

h o t h e r experimental quantity that i n m l v e s our function T

t,lv

4

p/( J ~ J ~ p3 ) ~ /

(not averaged over the spin of the p r s t m , i t involves the a n t i s $8 the byperfine s p l i t t f n g energy i n hydrogen resgon~ziblef o r the 142Q+egacycle line.

I t i s the differeace I n energy i n the gromd

8

s t a t e of a t o d c hydrogen

depending on whether the spZna of the elects011 and proton axe p a r a l l e l o r a a t i parallel,

In non-relativistic approxiwtion

it

depends on the prob&iIity t h a t

2 the electron i s on top of the proton I@(o]f i n the gratrnd etste wave EunctSon. h l a t i v i s r l e a l l y we can write

e , v,

(Rio

= Rydberg,

u , P

~na$neticntawnts of p and

Bohr m p e t a n )

The f a c t o r ( l

+ m / ~ ) - c~o w s

from reduced =ss corrections t o the Schroedinger 2

equation i n getting l+(o) 12;

1 + 3a /2 is a modificaf2on due t o the Dirac

equation,

g, %R..ti" a r e

The other factors

a13 near one and a r e due to higher

order quantum electlcody~&dccorrections.

They have been separated i n to three

factors f o r convenience of discuarp5on. g corns f rorn QED sodif ications of the

221

Pbton-l"iadronInteractions m t i o n o f t h e e l e c t r o n , d i a g r a m l i k e A below.

c o w s From t h e proton r e c o i l d i a g r m of type B i n which the f a c t t h a t the proton i s not a point charge

113

included by using m a s u r e d f o m f a c t o r s . r" c m s

fronr two photon exchmge t e m of t h e f a m C, D.

A l l theere f a c t o r s except y w h i c h depend@on a s y e t m k n m p r o p e r t i e s of t h e p r o t m have been c a l c u l a t e d t o very high accuracy

- f u r t h e r &E is

raetaaured t o a b ~ ~ r d hf iy@ accuracy,

The c o m t m t s l i k e a, 11 e t c , a r e new P knom wels enough t o d e t e d n e r"l t o about four p a r t s per raillion, Theor e t i c a l l y , the deviation due t o AB long a s F r e m i n r ~ = , certain

P

is about t h i s sam order of mgnitude.

we cannot use these accurate m a s u r e m n t s and

c a l c u l a t i o n s t o b p r o v e our knawledge of t h e constants

- or

t o put t h e problem

t h e o t h e r way i f more accurate v a l w s of t b e o t h e r constants becorn a v a i l a b l e that will t e l l

u9e an e l e c t r o m g n e t i c property of

be challenged t o c a f c u l a t e i t , w r i t e f'= f Obvioua3ly t h e proton coupling is t o

+ A,

t h e proton

Let

W

t r ~ acurrents.

- and we would

s e e what is involved, Ve s e p a r a t e t h e caaea

t h a t the photons a r e of low energy mid nromentum fro= t h e cases where they a r e high.

Where they a r e low, binding of t h e e l e c t r o n i n f n i t i a l and inte-diate

s t a t e s e t c . m a t be considered

- b u t here for law v ,

Q, t h e proton a c t s , a s we

have seen, l i k e a. p a r t i c l e of charge and raagnetic mment and thus we c m do t h i s p a r r of the c a f c t l l a t i m .

For defgniteness we do it p u t t i n g i n the exper&-

mental f o m f a c t o r s once f o r each p h o t m and i n t e g r a t e over a l l mraents, c a l l t h i s hl.

Wrlte d

AI f A2.

Now f o r high v i r t u a l moPlentwn e l e c t r o n binding,

even t h e e l e c t r o n =ss can be neglected and we can ionagine the e l e c t r o n and proton t o be f r e e and a t r e s t before and a f t e r t h e s c a t t e r i n g

- thus our two

223

Other Two-Current Efleets c u r r e n t operator

p l J~ )i

a l l t h a t is involived. a r e involved. butions i n AI

T

vv

v

jp>

f o r proton i n m d out of s a m nroanturn ( r e s t ) is

Naturally s t a t e s

X

o t h e r than pure proton Born a t a t e s

Qf course we have already c o m t e d so= high mrnenturn c o n t r i and we s h a l l have t o sllbtraet them, thus we s h a l l have not j u s t

but r a t h e r TPv

e l a s t i c where

-

elastic wv

term alone with form f a c t o r s .

C a l l t h i s TMv'

is c a i c u l a t e d from t h e Born

- TvVe l a s t l c

T vv

Then h2 is

proportional t o d i a g r a m C, D with e l e c t r o n a t r e s t i n i t i a l l y and f i n a l l y .

If Tllvt is

B y

Ear t h e proton

matrix on the proton s p i n i t depends on the s p i n f l i p amplitude

- thus

OR

t h e a n t i s y m t r i e a l p a r t of T

St v

"(proton

2 2

The f a c t o r l / ( q 1 is f o r the two photon propagators.

a t rest),

The f i r s t t r a c e

i s t h e Coonpton s c a t t e r i n g by t h e e l e c t r o n

neglecting the ntanrrctntm p aad mass o f the e l e c t r o n .

The second f a c t o r is the

coupling of ltwo photons t o t h e proton, which we do not know but which we a r e discussing; we need i t s a n e i s y = t r i c

imginary part M

where G

UV

part.

I and C? a r e functions of

be c a l l e d S1 and S2. liv

kre

wrote its

i n the form

q L and v defining t h e imaginary p a r t .

the c o w f e t e s e a t t e r f n g functions of which G

for W

I n Lecture 33

1

and G a r e the itnaginary p a r t 2

That I s , we w r i t e a fona f o r T

except S1, S2 replace G1,

G+and G1,

PV

j u s t 1Pke the above

G2 a r e i m S1,

errbstitution we can express C12 d i r e c t l y i n t e r n s of S1, S2,

Drell and Suflivan, Phys. Rev.

154 1477

Let

Im

S2.

Then by

Qne g e t s (see

61967) aod C ,N. Iddings, Phys

. Rev,

Now we can do m n y things with t h i s t o t r y t o e s t i m t e i t o r c o w a r e i t t o things measurable i n p r i n c i p l e i n ways e n t i r e l y mslogous t o our d i s -

c u s s i a a f e l e e t r o m m e t i c s e l f energy.

For an e x a p l e , we can use Cottingham'a

i d e a of r o t a t i n g t h e Contour on t h e dqo I n t e g r a l from the r e a l t o the i m g i n a r y axes, thus t h e i n t e g r a l s depend only an Sl, S2 i n the negative q2 region. F i n a l l y we can express these coaplete S1,

SZ i n t e r n of G l A 2

( t h e i r iraaginary

p a r t s ) by a dispersion r e l a t i o n , and t h u s express h2 in t e r n s of G1, C$.

Two questions coM up: F i r s t , a r e these wlinom functions, l i k e T 1 0 , ~ ' ) necessary i n the d i s -

1

persion r e l a t i o n o r a r e they a s f o r T2 without constants? q w s t i o n i s almost c e r t a i n l y h o r n , ) s c a l i n g bebavior of Gl,

C2.

(The answer t o t h i s

We can wears because we h a w t h e w p ~ t o t f c

I f t h e r e a r e s u b t r a c t i o n s , t h i s m t h o d is f r u s t r a t e d .

Second, supposing t h e r e a r e no undetterdned functions i n the dispersion r e l a t i o n and b2 can be e n t i r e l y e w r e s s e d i n tern8 of Cl, G2 i n t h e e x p e r l m t a l region; what can we do u n t i l Gl,

G2 is d i r e c t l y measured?

Zt b e c o w s a research

problem t o guess a s completely a s possible t o s e e i n what ranges of =q2, v the &l i2 s ms-ensitive

and use whatever w d e l s o r ideas a r e most r e l i a b l e t h e r e ,

W could t f y t o incorporate a l l that: is Ianm of low energy t h e e r e m and i n t e g r a l s ( l l k e I g2dx

0) t o l i m i t the p o s s i b i l i t i e s ,

At worst, c e r t a i n l i m i t a of un-

c e r t a i n t y can be e s t a b l i s h e d sknce, f o r e x a w l e , 28, which is t h e d i f f e r e n c e of up m d down s p i n partonr h+(x)

- &_(X) (see Lecture 33) cannot exceed the

sum hS(x) l- h_(x) which i s f(x) and i s w a s u r e d , t r a c e of fa T

vv

2 i a e o m way.

W

Generally p o s i t i v i t y of t h e

on any d ; t a p ~ a ls t a t e l i n r f t t h e s i z e of G1,

G2 i n t e r n of W%,

Tfiis problem should bc: pursued

And i f the answer t o t h e f i r s t questlilon i s yes and there a r e d n m

functions brought i n i n t h e fixed q

2

dispersion r e l a t i o n s ?

Then Cottfngbm'a

scheme doee nof work and we s h a l l have t o uge o t h e r mth-ods of a n a l y s i s such a s t h e f i x e d Q dispersion r e l a t i o n s t a abrain expressions on &ich we can apply our p a r t i a l physical understimding of photon badron i n t e r a c t i o n s s c a l i n g , quark nrodel, e r e . ,

- i n conjwnction with a s m n y

stap

- partans,

rules etc.,

we know (as w e l l a s possibly even the n u w x i c a l value of the n-p

that

mss differences)

t o guide us a s much a s possible t o c a l c u l a t e t h i s quantity A2 as accurately a$ possible, a d with an honest e s t i m t e of t h e possible t h e o r e t i c a l l i d t s of uncertainty.

Other Two-Current.Egecrs b c t v r e 49 Other two-current e f f e c t s I should j u s t l i k e t o add a few miscellaneous r e m r k s of s i t u a t i o n s i n

which the double operation of two c u r r e n t s is involved, namely, i n the d i s i n t e g r a t i o n of pseudoscahr =sans.

They a l l present i n t e r e s t i n g q w s t i o n s Of course m t r i x elements

f o r study; t h i s i s Beant raerely a s an introduction.

of one 3 a r e involved LR s i n g l e photon decays l i k e o

-+

1.

ny and we

have already

discussed them. Two 3% a r e obvIousIy involved i n two photon decays Like nQ q0

-+

2y.

-t

2y o r

An honest c a l c u l a t i o n af e i t h e r one of these would be very i n t e r e s t i n g

(one i n which t h e v a l i d i t y of the i l s s w t i o n s is baeked up by considerably more than t h e mere f a s t t h a t the answer agrees with t h i s one experiaent). % a t does Q l (U; + da 2s;) t o g e t SUS say? Use the quark m d e l no 2 (U; da), Q J;Z" Ji;" 1 (ij4 F)/ l L &+ l 2); t h e r a t i o of amplitudes amp nolaw 'Q 15. It say@

-

-

-

- --

-

4-2'

t h a t t h e amplitude f o r n i s high,

8t h a t

-+

2y

is r ( q

-r-

1.0

2y)

+

-

-

48-

f o r q, s o t h e r a r e should be t h r e e tims azi

Experlarentally the widths a r e r(no

width

-

-r-

2y)

+

7.2

1.0 eV, and the; p a r t i a l

The r a t i o is 11140 i n s t e a d of 3:

. 3 KeV.

Qf course, the reason f o r the a b j e c t f a i l u r e i s the very l a r g e -ss difference of r it.

and

FI

2 (mx /m

n

2

m

1/15) and we must be much more c a r e f u l of

This is where SU3 is i n d e f i n i t e

km.

- and no universal way of

&in& t h i s is

F i r s t t h e r e i s phase space: t h e general f o m u l a f o r d i s i n t e g r a t i o n of

an o b j e c t of mss nz a t r e s t i n t o two p a r t i c l e s each of whose mmatura is Q is

where K 1s t h e r e l a t i v i s t i c roatrfx elenromt. space ( i . e . i f we ass-

Xn our carte (2y) Q

m12 s o phase

K i s given by SU3) w r k a agaixlst t h e rt and the d i s -

2 crepancy is another f a c t o r of m /mn worse (i.e. MW /H

rr

'

1/540 i n s t e a d of 3/1).

Plore s e n s i b l y we should w r i t e M in its sienplest: r e l a t i v i s t i c f o m which f o r a pseudoscalar meson d i s i n t e g r a t i n g i n t o 21's of p o l a r i z a t i m el, e 2 , m a n t a kl,k2 is N

a

E

pvap

e

lil e2u

4p *

NW t h e g m s s would be t h a t a l s i d e t e d n e d by SUB. This ~ll?.ans M pea a 3 2 a s Q o r m2 ~ a t e ~ / ~ a -3t e ~, aw2/aq2 = -L %is a t l e a a t m v e s i n IPL f 40

.*.

-

.

the r i g h t d i r e c t i o n , it g h e e

0.4

a:/anZ

i n s t e a d of 3.0

.

This i s not bad b u t (a) what of the r-iaing

discrepancy astd (b) why 2

should a be given by SSU and n o t say a/m o r a/m ?

Evidently we have been dolng

3

too much e m p a r i s o n t o e x p e r i m n t and too l i t t l e thinking. Can we reason o u t f r m o t h e r things we know j u s t bow t h e r r a t e and

rl

r a t e should be compared, o r

bcw e i t h e r might be calculated o r e s t i m t e d absolutely? With regard t o the mchajniaul of t h e decay e v e r s i n c e i t was suggested by Gell-Mnn,

Sharp and Wagner, Phys,

&V.

Jcetters

8 261

(1962) i t is supposed

t o be do&nated by t h e diagrlstro going through an inre-diate

This connects i t t o t h e npy coupling constant, o r through SU

p o r w wson like

3

t o the paeudoscalar

v e c t o r photon coupling constants h general; d e t e d n e d d i r e c t l y f o r e x a w l e by t h e u

-*

ny r a t e .

Again rr

+

nny is i n t e r p r e t e d i n t h e same way a s

The pnn coupUng being k n m one e m campare t h i s with tl t h e r a t i o l"(n

+

nny)/r(n

-*

yy) with good success,

e x p e r i m a t is .l2 ( G a d e g , IPIrys. h v ,

g 501

+

yy

and p r e d i c t

Gel1

(1970)), (For two calculat&lons

corresponding t o different: choices of how t h e coupling constlznts s dewnd on t h e rnasses of the s t a t e s s e e Brown, Muncak and Singler, Phys, Rev, Letters (1369).

22 707

(1968) and Cban, @ l a v e l l %and Torgerson, Fhys. Rev.

The choice made i n the l a t t e r paper f i t s very n i c e , )

185 l754

%%is is i n t e r -

e s t l n g f o r a t f i r s t s i g h t t h e a m r a t o r is order a and deno&aator is order a

2

s o t h e order should be 137 but here m a y n w e r i c a l constants a c c w u b a t e t o o v e ~ e l n rt h i s f a c t o r and taaka the r a t i o n e a r l y 1000 tiws smaller. p i c t u r e t h e r e i s a f a c t o r i n the matrix element f o r where

is t h e i n v a r i a n t mass squared (p+

of the two pions,

+

For t h i s

-

rl + nny l i k e l/(mp2 mn:) 2 of the four-vector slrm p-)

This d i s t o r t s t h e spicsctruar away from t h e simplest Eom toward

Orher Tw-Ctrrrgnr Eflects l a r g e r p r o b a b i l i t i e s f o r l a r g e r rann

2

.

The experiments (Wrarley) a r e s o

accurate t o s e e t h i s e f f e c t , even q u a n t i t a t i v e l y , s o t h e r e can be no do&t of It is l i k e l y t h e r e f o r e t h a t no deep olysteries l i e

t h e mchanisan i n t h i s case,

I n the 2y d i s i n t e g r a t i o n a e i ther. Rowever t h e rt

is

-, of

0 +

the n is

+

+ so

-, charge

0 la a n

t h e d i s i n t e g r a t i o n is n o t allowed, s t r o n g l y ,

W'S

Consider

conjugation (which we think Ss s u r e l y s a t i s f i e d f o r decarys

a t t h i s r a t e ) (Q and no a r e charge conjugation 3.

=quires t h a t t h e a

mis

The G p a r i t y of t h e

3% does present a challenge.

, a- must

+ because

they can go i n t o 2y)

be s m e t r i c , they m w t be i n m T

added t o the X = 1 of t h e t h i r d w e o n y i e l d s only t o t a l L

O o r 2 state. 1, l, 2 , 3 a s

p o s s f b i l i t i e a , t h e r e i a no f = 0, m e r e f o r e the decay cannot occur except by Psospin is, of course, v i o l a t e d try e l e c t r a d ~ a d c s

v i o l a t l o o of i s o t o p i c spin.

s o an eleetrodpna&c v i r t u a l e f f e c t (order a i n rnatrix e l e w n t ) i s involved and the? r a t e is a f order cr

2

510

rl

swaqed by what S t would be i f

2y can compete with i t r a t h e r than being c o q l e t s l y Q -*

371 were a s t r o n g i n t e r a c t i o n ,

(The d a t a on

rr decays is Branching r a t i o X

* 1.1

23.1

'&e change i n X s p i n can be bX G parity

.

. ;

Hence the f i n a l s t a t e of t h r e e pions must have

can get ao estimate af t h e r a t i o

I part violates

O, 1, 2 but only t h e BP,

r(n

3n0)/r(n

+

a+71-n').

P,

= 1. From t h i s we

Conrbining t h r e e

s t a t e a of i s o s p i n ,none we can get f o r t h e t o t a l I spin:

tric

2,2,1,1

Pf we suppose t h e 3% a r e i n t h e i r l w e s t space wave, since they a r e Base we w i l l have f spln 3 o r 1;

by proper use of space s t a t e s t h e skew-S

S

wave m d sy-tric,

EM p e d t e only t h e l (but

t r i c X = l s t a t e s could c o w i n ) ,

Foa t h i s s t a t e

r(n

; .

3no)lr(n

-i

n'n-no)

= 3/2.

Sf

t h e r e is s m e skew s y a m t r i c

( m m e n t m dependent) space s t a t e the o t h e r Z = 1 can come i n reducing t h e 312; s i n c e i t corns i n with h i g h e r angular wmtentm i t i s probably s a t a l l e r s o t h e r a t i o m y be f a i r l y c l o s e t o 312. Bow do we c a l c u l a t e t h e r a t e

f t is 1.3 e x p e r i m n t a l l y , i~ -s

3n?

%ere does t h e i n t e r m e d i a t e photon

a c t ; can w e guess whl& i n t e r m d i a l r e s t a t e s a r e mast likely t o be i q o r t a n t ? A quilntitave e s t i m t e f o r t h i s r a t e is a problem t h a t no d i s c u s s i o n o f t h e

e f f e c t s of v i r t u a l s e l f e n e r e photon a c t i o n can f a i l t o arention.

Hypotheses in the Parton Mode

Lecture 50

We should now l i k e t o discuss what we can say about what t h e products

R would look l i k e i n deep i a c l a s t i c e e a t t e r f n g e

+p

+

e C X,

There a r e s o w

wasurerolcsnta f o r c e r t a b dcSef?lnite fSnal s t a t e s X f o r satall energies and low 2

q

of t h e v i r t w l photon (see Berkelmn, 1371 CorneZi conference),

Most

of these can be w d e r f ~ t o o dEron d t r e c t e r t m s i o n s o f our theorfefir f o r photon (qZ

0 ) reactions.

We have already d i s e a s e d pion production from v i r t u a l

pion exchrange a s y i e l d i n g info=tfon

on t h e pian f o m f a c t o r .

Xn addieion

c praductian has been s t u d i e d from v i r t u a l photons, wPth no surprises gives a f a i t account

- VBM

- s e e our theory d%scweed i n Lectures 16 t o 21 where we

2 2 = r e l y have t o replace kout by q i n the equations t h e r e yieldling a f a c t o r

m 2 l (mp'-q2) P

2 (q i s negative) r e l a t i v e t o t h e

-

O ease.

Zt i s necessary a l s o ts m k e an a s s m p t l o n of how the l o n g i t u d i n a l l y

polarized photon (possible when q

2

f 0) couples i n r e l a t i o n t o the l o a g t t u d i n a l l y

s a s s m d t h a t these a q l i t u d e s a r e r e l a t e d by a f a c t o r 2 t h e e x t r a f a c t o r q /m

P

is anansat z made because gauge

invariaace require8 chat J longfrudinal vandahea aa qL

-t

0, Thie m y be v a l i d

f o r n o t too l a r g e qZ, b u t of course i f q2 becomes r e a l l y vary l a r g e the

229

230

*

assumptions about small p + p production may begin t o f a i l . What happens a t l a r g e q2, and how i t ties on t o small q2, i n every case

€or very

v w i l l be our present concern.

W e s h a l l have t o be guided by

theory and I w i l l take t h i s opportunity t o review t h e parton model and some assumptions t h a t can be made about i t .

W e s h a l l list assumptions t h a t we can

- without today being s u r e of which a r e r i g h t and whicb wrong - j u s t t o

make

see what t h e i r consequences a r e with t h e hope t h a t experiment may later make the s e l e c t i o n (e.g.

are charged partons quarks?),

Therefore i n our list

some assumptions w i l l (perhaps) be i n c o n s i s t e n t with others.

The assumptions

w i l l come from a mixture of t h e o r e t i c a l guesses and known experimental

facts so-

- so

one might be warned t h a t i f a p a r t i c u l a r asswnption n e a t l y explains

experimental f a c t i t may not r e a l l y be a s i g n i f i c a n t confirmat

assumption might have been made with t h a t f a c t i n mind. e f f o r t w i l l be made t o derive one assumption from another n o t independent.

Finally, l i t t l e

- they are c e r t a i n l y

T h i s w i l l t h e r e f o r e unfortunately not be a m a t h e m t i c a l and

sound system, but r a t h e r a lengthy " i n t u i t i v e " o r physical discussion, For a good discussion along the same l i n e s see J . Bjorken's paper i n the 1971 Cornell Conference. General Framework W e suppose a s i n f i e l d theory, a wave function f o r a s t a t e can be given

by giving t h e amplitude f o r finding various n m b e r s of f i e l d quanta, o r partons of various momenta.

I n p a r t i c u l a r we discuss the wave functions of s i n g l e

p a r t i c f e s (eiomrttimes two p a r t i c l e s i n c o l l i s i o n ) with extremely l a r g e momentum P i n the z d i r e c t i o n (P

-c

The wave function i s being described i n Fock

QD).

space giving f o r a s t a t e the amplitude

etc.

where JIo is t h e amplitude t o f i n d no partons (usually zero); ql(pl)

is t h e

amplitude t h e r e is one parton (of such and such a type, an Lndex we are omitting)

231

Hypothese$in the Parton Model A i c h has mnrentm pl; J"Z i s t h e a w l l t u d e t h e r e a r e plS p2 e t c .

represent the vaeum s t a t e and a

p.

P

partons of Qtolltenta

For example l e t /VAC>

This can be w r i t t e n i n o t h e r ways. rL

WO

t h e operator c r e a t i n g a parton of m m n t u n

= P* /VAC> where

Then we can w r i t e the wave function e t a t e

is s o m function o f the c r e a t i o n operators,

Then we make the following assumptions : A&, -

The amplitucte t o f i n d a l a r g e

)&I on any parton f a l l s rapidly with

such Char e f f e c t i v e l y we c m i n f i r s t approximtion consider a l l Pi a r e f i n i t e (as P

+ W),

AZ. Tlne '*wave function" f o r Longitudinal m m n t m of order P, i . e . g t t = s

This r e q u i r e s some conrplicating r e m r k s t o m k e its d e f i n i t i o n c l e a r f o r t h e r e is, a s P

+

-, an

ever increasing c s n t r l b u t i o n t o $I for small

More p r e c i s e l y consider t h e density ntatrix. t o f i n d n partons of m e n t a p1 t o pn,

Let *m(plpZ...pn)

X.

be t h e m p l i t u d e

Then f o r exaraple t h e density f o r one a t

k is

That t h i s cfepends only on kL and x = kL/p when a s s w t l o n we want t o make

- with

p a r t i c l e density matrix [ l i k e dpl,,.dp,

1

X

is f i n i t e as P

a l l its generalizations.

/$i(P1pP2s * *kg .P,)

p(k,kl) depends only on hl,k;,kL/P;ki/P

wave f w c t i o n i n p

It is a l w e t but not q u i t e X%

-

is t h e

E.g. t h e one

(plp2. * k t .*P,)

e t c . , when

1

X

is f i n i t e .

6 (pi-kl) 6 fpJ-k2) )

Again t h e two p a r t i c l e density (the expectation of behaves likewise e t c .

$ ,

+

t'ki same a s saying

PLI/P, has a d e f i n i t e l i m i t .

the

The wave function

is a function of a l l of t h e mracmta including those of f i n i t e mmntuan The s c s l i n g d o e s n t t work f o r those

(which we c a l l weea).

- in

m a n a w e r of p a r t i c l e s r i a e s wlth P s o the wave f u r c t i o n JI

f a e t the f o r any fixed

n f a l l s with P ( l i k e a power of P) but t h e ' k e l a t i v e wave function" e.g., r a t i o i n which only one p a r t i c l e f o r a f i n i t e n a b e r ) is mved depends f o r finire

X,

A3. -

p mly

on

X,

pL.

I n the wave function the a q l i t u d e t o f i n d

a

232

Photon-Hadron Interations That is t o say again t h a t t h e

density matrix, e.g.

the density Ear finding p a r t i c l e s with f i n i t e (e.g.

behavior as P values of pL have a d e f i n i t e l i ~ t i n g nmbex o f such wee

-3.

m,

4-4 GeV)

and the expected

partons is f i n i t e e

& 4 .- To have continuity between A3 and A2 the mm n a e r of partons of

a given type

z&

The n d e r Ear pZ negative f a l l s o f f rapidly 80 t h a t f o r p, =

as B finite x

X$?,

negative t h e r e a r e no partane; although f o r pZ negatlve and f i n i t e t h e r e a r e some (a fixed =aunt

f a l l i n g rapidly with negative p Z ) .

(As a t r i v i a l e x a q l e of the kind of bhthavior envisaged i n the wee region where a* c r e a t e s s p a r t i c l e of consider a wave function l i k e exp (I: C ~ < ) [ O > Er l o n g i t u d i n a l mooentum h and Ck v a r i e s a s $ . / ( w - ~ ) Y ~ / ~ with W say,

-

- \fi

and a = constant.)

A%, The bE?hwior of the wees is nearly (as P of t h e d i s t r i b u t i m of t h e f a s t ( f i n i t e

X)

-3.

partoas.

a

completsly) i a d e p e n d ~ t

This again is connplicated,

I f we s t r e t c h o u t t h e v a r i a b l e pz by, f a r example, defining y

+ pz)

so f o r f i n i t e pz, y i s f i n i t e ; f o r f i n i t e n,y is &nZF

We have p a r t i c l e s a t every y fro= f l n i t e

0

Ln (

; . 03

f i n i t e RnZP-y ( f i n i t e

X)

+ 1 GeY2

+ enx.

t o Rn2P.

Rn2P

We have d r a m a graph oE t h e mean rider of p a r t i c l e s i n dye look near f i n i t e y, P

\/p:

I f we

we s e e the behavior of t h e wees; i f we look a t

we s e e t h e behavior of t h e s c a l i n g f a s t ones;

in

between i s a plateau with a E i n i t e density o f partons so the mean a m b e r of partons r i s e s a s EnP, It i s easy t o m d e r s t a n d the density but how do we mderstand the wave function?

"Phis gives t h e amplitude far every configuration, whi& &B a s e t

Hyporhesgs in rhe firton Modd of values of y f o r partons present

Ln2P

0

Y

It behaves

Bow does t h i s w l i t u d e vary with t h e p o s i t i o n of the data?

l i k e a wave funetian f a r a f i n i t e one d%mnsional drop of llqrtfd of thickness Ln2P. The a a p l i t u d e is l a r g e only i f p a r t i c l e s a r e rtlazrs o r l e a s e v e v h e n ? , with a uniform density except near t h e surfaces a t &n2P and 0 .

The behavior

a t one s u r f a c e is not s t r a n g l y influenced by what the configuration is a t the other: s u r f a c e

- they a r e i n s u l a t e d

from each o t h e r by the long

c?,

Rn2P is

never r e a l l y very l a r g e ) i n t e r w d f a r y plateau. Mathematically we a r e discuhising the prolution of W+ = Eiy f o r a rstate of

If

fined Pz (but we do n o t knnr H, of course). (e.g. l i k e

E pz a;ak

e t c . ) we want?

eigeovector of the operator W

iy

m

- - z,

F*.

qzis t h e momentum operator Cansider then t h a t

f o r a s t a t e of masn H.

H

@

is an

Now as P

+

m

2

P+% IZP) we eaa consider t h e lirnlt

(E

-

(ZPW)$

nZiy

(li1.1)

so we a r e Looking a t eigeavectors with fixed efgenvafues o f t h e operator PW a s P

+

=.

Wi,

axe a s s u d n g t h a t i t loolrcs a s i f , 8s P

d i s t i n c t l i n t i t , e x p r e s s i b l e i n x = pC/P and

is t r o d l e on t h e s m s l l mnrentm end,

p,.

This would be n i c e , but t h e r e

The equation is l i k e a cadieade, l a r g e x

genkrates staaller x through i n t e r a c t i o n t e m (I%& c o s e e ray showers).

t h e o p e r a t a r PM has s

+

turbulence equation8 , o r

Smiler matenta p i l e up m t L f new ptzenomna s e c s i n t o

change the equations (lLke viscoajtty i n turbulence, o r i o n i z a t i o n l o s s i n cos&e rays) t o f i n a l l y h t e d n e t h e wee

X

(EInlte pZZ behavior,

(The approxi-

aaatim I n showing t h a t PW depends only on x is wrong, f o r e x m p l e we c m no

longer: w r i t e

pZ,)

But by t h a t time the "'shower fkr f u l l y deveiopedf'

and t h e behavior of t h e weaa (except f o r n o m l i z a t i o n wee#) is indepenrient of t h e way i t e t a r t e d a t f i n i t e

- t o t a l strmgck

X,

of

(In the wee region

i n t e r a c t i o n e n e r g i e s a r e comparable t o k i n e t i c energies.) l'be behavior a t t h i s md Is a e o l u t i o n of t h e equation

W=O (Note the e ~ a s i a nof P , )

Tn general the operator W dioea n a t have a zero

Photon-MadronInteractions eigenvalue i f a l l boundary conditions a r e used condStion of f i n i t e n e s s a s pz equation H$

m

E@ f o r E

say the condition a t r =rely

J.

+ m. I l t

- b u t here we r e l a x the boundary

is l i k e s o l v i n g t h e Schroedinger

O when i t does not have t h i s eigenvalue by relaxing + m

and thus studying open s c a t t e r i n g s t a t e s spproxi-

t o which t h e r e a l l a r g e r behavior will. have t o be attached

- here we

mmt ultilnatelg r e a l l y a t t a c k t h e f i n i t e x s o l u t i o n of ZPW* = mL$,) It can b e s h o w t h a t s i n c e equation (51.2) is i n v a r i a n t un&r a boost

Lorenta t r a n s f o m a t i o n i n the z d i r e c t i o n by v e l o c i t y v, c a l l E i n which a l l l a r g e pZ a r e multiplied by f , t h a t JI c m be of t h e f o (so boosting does n o t change 9 , only t h e n o m l i z a t i o n ) .

This m a n s t h a t f o r

f i n i t e but smsll x the p m b a b i l i t y of one parton a t n v a r i e s a s nZ8dx/x. i n A4 corresponds t o B

lowest B s o l u t i o n we asstheory).

The

O ( f r o a e x p e r i m n t , not

Other s o l u t i a w e x i s t f o r higher B and the general s o l u t i o n i s a

l i n e a r conibinatlon of these d o s e c o e f f i c i e n t s a r e d e t e d n e d by how they f i t on t o (51.1).

S;

had hoped t o g e t a f i e l d theorqr i n t e q r e t a t i o n of b g g e

theory t h i s way but I have n o t c m p l e t e d t h e analyeis. A6. -

The d i s t r i b u t i o n of the wees is the s a m f o r a l l hadrons; Wng

a bold assumption p a r t l y guided by e x p e r i w n t ,

This is

o t h e r things i t m a n s

t h e weee a r e neutral. t o i s o t o p i c s p i n ; the wee8 f o r t h e proton look j u s t a s they do f o r the neutron.

f i e d i f f e r a c e c m only corn fro= a higher & and

hence f a l l s i n amplitude r e l a t i v e l y as P-@ (B t r i c w i ll with experiment

f;n

7

0).

I h e assumption t h a t the

Be made, (P think i t leads t o d i s a g r e e m n t

the expected r a t i o s o f n's t o K ' s f n c e r t a i n experilacmts)

f o r we imagine i n t e r a c t i o n forces a r e e f f e c t i v e i n d e t e r d n f n g t h e wee diat r i b u t i o n and such forces a r e not $Ug i n v a r i a n t .

(It s t r i k e s W*

nte a t t h i s mannear, t h a t s i n c e t h e wees a r e d e t e w n e d by

0, i , e , , a s t a t e a f zero =ss squared, and t h a t s i n c e pions have a small

nraea, t h e s t a t e of the wees m y be approximtely only pions (with kaone, of l a r g e r mass lauch reduced, hence l a r g e SU3 breaking,)

Knam pion i n t e r a c t i o n s

(perhaps described by intremediary p nzesans) might p e d t a s o l u t i o n of W$

I n t e r m a p p m p r i a t e pion base s t a t e s .

If you do t h i s 1 would suggest i t

mi@t be e a s i e r i f you work a t f i r e t with t h e

S

t r i c (inrpz) wee d i a t r i -

bution corresponding t o two f a s t h s d r m s c o l f i d i n g , r a t h e r than t h i s onesided, one-particle d i e t r i b u t i o n , )

0

23.5

Hypotheses in the Parton Mode1 A6 is a c t a w l e t e l g obvious from f i e l d theory

The a r p s u q t i m

- far

there might be so=

long range d i r e c t e f f e c t of the f a s t partons on the slow

ones i n principle.

The choice i s guided by expctrimnt (which shows t h a t the

right-mving products of hadronic co3LlisPons depend only on the right-moving i n i t i a l collidkng p a r t i c l e and not on what i t c o l l i d e d with. a s a m p tfan is wed s e e B1, a i d J, k n e c k e e t a l , Fhys, Rev. The physlcal a s s u w t i o n B 1 t h a t we

m&@

To aee how t h i s

,188 2159 (1969).

l a t e r says specifically t h a t there is

no such long range e f f e c t .

G. Continuity

demands, s i n c e the wees a r e adjacent t o the s e a (plateau

region) t h a t we a l e o have t h a t the s e a (e.g. =an n m e r s and corselationa of

rnI?kY2 where y

depends on the q u a n t a numbers

(angular mmntlmt, isospin, strangeness) c a r r i e d by the gap.

E,g, suppoee we

have a proton e t a t e and we ask f o r c e r t a i n partons s , b ,c, f o r y

y

1 and others

below y2

?J2

Perhaps t h e i r strangeness is contributim S

yl

+ 1, t h e

- 1 across t h e gap.

O s o there is a

e n t i r e proton h a s S

It 2s evident t h a t t h i s "quantura a d e r 8

carried across the gag" defined a s the qumttas a d e r s of the s t a t e d n u s Phase t o the r i g h t o f t h e gap (a,b ,c) is j u s t the sm of those t o t h e l e f t (S, t l

.

This more c q l i c a t a t d aethod of exprwslon ts zm a n t i c i p a t f o n of the

s a m i d e a f o r distr&b,butioaswhen two hadrons a r e c o l l i d i n g ,

Then i t is the

q u a n t m riders of the h a d r m taovgng t o the ri&t ntlnw the t o t a l qwntran a m b e r s of the partons t o t h e rilJht of t h e gap, A s s q t i o n A8 i s not stated, c l e a r l y , chmlgc?d.

f t is used i n two cases:

l)

We have t o say haw the gap by is

&@ion a,b,c and s,t a r e both s t a t e d

completely ancl t h e gap vldents s i ~ l became y P Increases; hence dy = Rn2P mn td the m l t u d e Earls a s . '-P

This was used i n ernslyzing the proton form f a c t o r

i n l e c t u r e 29, f a r e x a m l e .

2 ) The gap 1s i n a plateau.

there i s a l a r g e a t r e t c h of plateau,

On one s i d e or both

Here a,b,c i s flxed and s , t e t c . , is

m y t h i n g a t a l l over a rJ-ide r a g e of y up t o the o t h e r boundary (order Rn2P

236

Photon-Hdron Interactions

away) by i e k e p t fixed as %n2P rises. x,l

X

2

a r e the

X

ProbabilLty goes as (x2/i17

where

v a l u e s a t each end af the gap (used for aeyntptotle b e h w i a r

of deep l n e l a s r l c scattering n e a r

X

=

51,

Hadron-Hadron Co isions at Extreme Energies

Altlrou&tt our subject is photon-hadm i n t e r a c t i o n s we s h a l l review the a s s w t i o n s made in describdng hadron-hadron c o l l i s i o n s A energies,

+B

a t extreme

kle f i r s t leave out e l a a t i c s c a t t e r f a g and d i f f r a c t i o n dissociation

and aim t w a r d t h e l a r g e p a r t af t h e cross s e e t i o n where s e m r a f p a r t i c l e s a r e exaltted A

+

8

-+

C

+D +E +

---.

Far a hard c o l k i s i a n suppose the matenta of A, B a r e PA, PB reepectfvely

in the z d i r e c t i o n

- for e x a q l e

take eenter of mss P

f i n i t e z-vefociry v t r a s t s f o m t l o n from t h i s PA We only work with FI\,FP o r P

i .

fP,

=,

The a s p p t o t i c a l l y inco&ng wave f w c t i o n w i l l be, of course, (i.e.

before "interactdon'"

),

so= kind o f product wave functicn of A of mmncunr PI

t o r i g h t a d B of =matrun PB t o l e f t ,

Technical problem a r i s e here.

Ia f i e l d

theory t h i s e m n o t siarply be a product of the wave Emctions of each, p a r t i c l e t h a t we have been describing f o r t h a t Ss not m i q u e (for exmple suppose A contains a Fernion parton a t =maturn p, and B arcto contains one of t b s s a m

kind a t t h e s a m momatu~a,but there cannot be two In the f i e l d a s they a r e Femions).

Thw %f A is represented aa a creation operator :P

an t h e vacuum,

+A

which c r e a t e s a l l our partons, and B by :F we can define the

F:IVAC>

i n c o d n g asymptotic wave function a s

**

m e r e i s so=

trouble i n the wee regfoa where PBFa do not c o m u t e (note

*

t h a t no c r e a t i o n operators f o r f a s t partons p ^ xPA appear both i n FA and F; because A and B a r e m v i n g i n opposite d i r e c t i o n s ) ,

Actually t h i s is only

technical b e c s w e we only want t h e s t a t e a f t e r the i n t e r a c t i o n .

The problem

would a r i s e only i f we were q w n t i t a t i v e l y c a l c u l a t i n g the i n t e r a c t i o n s ; b u t now

WC

wish t o t a l k about how the wave function looks a f t e x i n t e r a c t i o n hence

say a f t e r "2nteractfon plus correction f o r overlap i n defining the i n i t i a l matate".

%C o w r l a p a f f e c t s only t h e wees, but we s h a l l as@=

the i n t e r a c t i o n

a f f e c t s only the wees a l s o .

(Ely i n t e r a c t i o n we

t h a t although P ;

/VAC> a r e both eigenfunctions of H $> =

/VAC> and :F

* * / W> is n o t .l

t h e rzffecte o f the f a c t

/

F o l l w i n g we s t a t e t h e a s s q t i o n s we s h a l l d

E /$P,

e regarding t h e i n t e r a c t i n g

wave f unc t ion. 81, -

Partons i n t e r a c t anly i f t h e i r r e l a t i v e four-mntm

-

w s m i n g they have s o w f i n i t e m s s of order L CeV.

is f i n i t e ,

This is equitrelent t o

-y f o r p o ~ i r i v ep=) t h a t partons 1, 2 i n t e r a c t only i f t h e i r r e l a t i v e y value , y l ~ Z is of order one o r smiler.

(I use L GeV off e t c .

f o r t h e general energy values a t which i n t e r a c t i o n s f a l l

Z suapect i n aeveral a p p l i c a t i o n s even a s m l i l e r value (e.g. pi,

average) m y be c o r r e c t , althouhJh possibly l a r g e r i n so=@ c i r c m t a a c e s

-

Z t of course cannot be defined p r e c i s e l y wethaut a q u a n t i t a t i v e theory.)

We umae t h i s a s s m p t i o n t o gee a t the wave function ( i n t e r n of parcon distributionsl) f o r the outgoing f i n a l s t a t e a f t e r i n t e r a c t i o n ,

The d i s t r i -

bution i n y of partons f o r the i n s t a t e s A and B have ranges of y snrall t o Ln2PA m d

- Ln2PB t o small respectively.

( e f f e c t of i n t e r a c t i o n ) over a range by

We put them together smearing things .r

l, %is srnetlring near y = O j o i n s

the p o s l t i v e and negative y regions (fro= G and B respectively)

.

Since these

regions were the s a m far both A, E (see A7) t h i s can be done most mairnply by

Wradron-HadrunCollisions st Extreme Energies j u s t extending the c o m a p l a t e a u region f r o a m e t o the other. p o s i t i o n of t h e CM Leaves no t r a c e then,

The exact

as a general p r i n c i p l e .

We a s s m e t h i s

Longitudinal t r a n s f o m t i a n s with a velocity v not too eLose t o c feave t h e (Assumption due t o

distributions of such p a r t i c l e s e s s e n t i a l l y unchanged.

C.N. Vang.) t n our a p p l i e a t i a n t h e p a r t i c l e s a r e partons, t h e t r a n s f o m t i o n a l t e r s t h e p o s i t i o n of the o r i g i n of y by &nf

1 Rn

1.t.v F

[a f i n i t e m u n t ) and tbe

assumption says t h e distribution should look t h e s a m .

Hence the s ~ a r l n g

j u s t has the e f f e c t o f extending the p l a t e a u region of A ermothly back i n t o B.

Me tni&t ask kif the srnearing of hx = X changes t h e dilstribution of partantl f o r y near Rn2P where they were determined a n t i r e l y by A .

(This region Is

c a l l e d t h e A fragmentation region, near -Rn2PB t h e B f r a w n t a t i o n region.) But near A the d i s t r i b u t i o n already s a t i s f i e s (in t h e sense t h a t a l i q u i d surface 1s nearly independent of what goes on deep below i f forces extend over f i n i t e d i s t m c e s ) the wave e q w t i o n , s o we as@-

It

it3

not chrmged,

Therefare we irnagincz t h a t the f i n a l r e a l hadron p a r t i c l e e c o w from the d i s i n t e g r a t i o n of an ""original" parron s t a t e which has the following p r o p e r t i e s .

fragmentation region (y near En2P

, i,e,

for

X

< 0 ) and l i k e

Thus t h i s is campletely described i n t e r n of t h e wave function8 f o r

s i n g l e p a r t i c l e s described i n a s s q t i o n s A 1 t o A8.

We emphasize again t h a t

our a s s m p t i o n s a r e not independent,for e x q l e , f o r B 3 t o work the p l a t e a m from each p a r t i c l e must be t h e @ a m a s A7 stays,

@e a r e n o t t r y i n g t o develop a

l o g i c a l system of asrsusptioas, but J w t s t a t e a n m b e r of mutually conraistent (or possibly i n c o n s i s t e n t

- s e e quark

assumptions l a t e r on) i d e a s ,

The p i c t u r e we a r e developing i n B3 is a wave fanetion l i k e a l i q u i d i n t h e varPabSe p wSth s u r f a c e s A, B a t which the d i s t r i b u t i o n s a r e unique but an f n t e r i o r plateau o r s e a region which s e p a r a t e s them,

%ey can be well

separated by taking P l a r g e enough, f o r they a r e separated by Rn2P is u n i v e r s a l ,

Further the r e l a t i o n a o r c o r r e l a t i o n s from one place i n y

t o another have a f i n i t e y r m g e of order one t h i s s e a i s l i k e a Mtlrkov chain weorrelated.

- which

- s o t h e general behavior

- with enougb separation i n p

in

thSngs beconre

Many obvious p r o p e r t i e s expected f o r such a &aSn can be

expected here but they w112 n o t be a l l e x p l i c i t l y s t a t e d ( f o r e x a q l e , t h e p r o b a b i l i t y t h a t t h e r e w i l l be no partan a t a l l of a given type i n a rsrnge by gwa a s e x g ( 4 h y ) f o r stollte C f o r l a r g e enough Ay, e t c . )

The ward "appropriate" 5n defining the wave fuslction i s purposely vague

f o r I a s n o t s u r e h e t h e r I am describing the f i n a l outgoing wave function a f t e r i n t e r a c t t a n when a l l the p a r t i c l e s a r e eeparating o r one i n btlrween t h e i n i r j l a l and f i n a l ,

I have not c l e a r l y reeolved aty c o n f w i o n on thLs toattar

-

but as Z oalyuste the function q u a l i t a t i v e l y i n a m n n e r described i n t h e next l e c t u r e Z have not had t o c l a r i f y i t .

On drawing the p plotcl t o deracrlLbe the partoa wave f m c t i m of asrswptions

B2 and B3 we a s a m d the wee region near y

1

O where A, B tnteract,as being

completely healed over and j u s t a srnooch continuation of a pfrlteau through y

O.

TI.li8 is physically what 1 want and lea& t o 62 (next l e c t u r e ) with

Hadron-Hadron Cotlisions af Extreme E~ergr'es u n i f o d t y i n t h e y defined there f o r physical p a r t i c l e s rsnd i n a c c o r h c e with invariance under f i n i t e v e l o c i t y t r a n s f o m t l a n s which applies t o p a r t i c l e s , But t e c h n i c a l l y t h e cume in the y space y = %n d g h t s h w a b m p near y = 0, a b w p which m v e s when we m k e a f i n i t e Lorenrz transfo-tion,

which is f%ne s i n c e wave f m e r i o n s need n o t be r e l a t i v i s t i c But i t a w e be a b a p

i n v a r i a n t s (X thank F. M e r r i t t f o r p o i n t i a g t h i s o u t ) . (90 c ~ n s t r u c t e dt o

have no p h y s i c d e f f e c t a s a b m p i n the f i n a l r e a l hadron

d i s t r i b u t i o n C2.

It is a "theoretical a r t i f a c t " due t o earelesaness i n finding

t h e ri&t no-lization

and d e f a i t i o n of v a r i a b l e s f o r the wave functian.

Rn

r e s u l t cannot r e a l l y be s m o t h i n t h e s c a l e of y would n o t be s m o t h (near y

0) i f Z had a r b i t r a r i l y chosen t o w e

f a r dyldy ' i s not c o n s t m t

y ' = Ogn

The

.

Latcture 53

We naw go on t o describe what t h e products &&t

look l i k e i n hadronic

m l l i s i o a s (astill leaving o u t d i f f r a c t i o n diseociation).

We have, of course,

no q?lantitative way t o g e t from the wave function described i n p a r t m e t o t h e wave f m c t i o n described i a outgoing r e a l hadrons, t h a t i n y space the r e l a e i o n of: p a r t m hadrone

+

partons described i n A .

-+

But we s h a l l s i ~ l p l ya s s m e

hadrons is muck l i k e t h e r e l a t i o n

Later on we s h a l l have t o describe t h e

produces expected from s t a t e s wfifch m l i k e that: i n B3, have gaps i n them, f o r e x q l e , a s t a t e with J u s t two parteas a t opposite ends of the y w a l e , separated by 2P, W assume ( t h e "conrplement" of B3):

partons

It h a s been ptoposed t h a t t h i s "sea correspondmg t o a parton gap" be

i t s e l f a gap.

This w i l l not be i n c g n s i s t e n t with what we w i l l say next (C2),

and which we w e i n hadron c o l l i s i m s ,

But i t does n o t seem reasonable by

i t s e l f t o ale i f liadron~m k e a un;ivereaI s e a a s 82 supposea

- f o r Z think

that

m a n s t h a t i f there were any d f s t r i b u t i o n a f two lmpbs of hadrons with a gap between, they would w k e a s e a of partons, s o i f there i s no s e a o f partons t h e r e is no separated l-e any r a t e i t i s

n?y

of hadrons, but there must be a hadron s e a ,

At

s t r o n g b e l i e f t h a t there is i n f a c t such a s e a Sn t h i s case

m d not a gap i n hadron m m n t a corresponding t o t h e gap i n parton w m n t a , A63.

A. CSsneros points o u t t h e two outgoing I m p s c a r q l n g oppoeite

+

hadronic quantusn n d e r s (in the case e e-

-t

hadrons) would generate a dipole

s t r o n g current ulhich would tend t o r a d i a t e s o f t e r hadtrons i n t o Low x,

To pre-

vent t h i s r a d i a t i o n becorns fnereasing1y m r e d i f f i c u l t a s t h e energy increases, (as f o r m - b o d y exchange r e a c t i o n s ) .

Gny exclusive m - l u q p r o b a b i l i t y w i l l

f a l l a s a p w e r of energy r e l a t i v e t o tha t o t a l inclusive reaction i n which the r a d i a t i o n , generating an i n t e r n a d l a t e plateau, is penair fed, Now f o r a wave function a s i n B3 we can iraagine the various partons d i s i n t e g r a t i n g more o r Less l i k e i n Cl but n o t r e a l l y independently, tholse a t the ends of the y raage deternine the. hadrons t h e r e , and those i n t h e c e n t e r a f f e c t i n g the hatirons i n the center, but i n a universal manner independent of y i n t h i s region,

Thus

W@

again g e t a hadron d i s t r i b u t i o n l i k e t h a t i n C 1

w i t h t h r e e regions, but the plateau may be a new and d i f f e r e n t d i s t r i b u t i o n

but a s e a nevertheless,

( m e t h e r the two plateau regions, t h e one i n C1

corresponding t o an i n i t i a l parton gap, and t h i s one f o r wave function B3, a r e t h e s a m o r n o t is a d i f f i c u l t problem I have not y e t been able t o decide, We s h a l l

call t h e assamptian t h a t they a r e equal C6, s e e l e c t u r e $5.)

This

a e s m p t i o n can be m d e by J u s t repeating t h e wording of Cl j u s t changing t h e naolcs of t h e s e a , o r i t can be put i n m o t h e r t o t a l l y equivalent way.

H~dron-HIrdronCollisions at Extreme Energies There i s a possible confusion here bemeen t h e "'initiaLWwave

Remrk;

function of C1 and the "appropriate" wave function of 83.

C l i n ece-

c o l l i s i o n s is j u s t a f t e r t h e i n t e r a c t i o n with the photons

- t h e parron

p a i r is j u s t t r e a t e d , like a

** ac a

There s t i l l remains tlrne f o r i n t e r a c t i o n s (via t e r w

i n t h e Wm%Itonim) t o a c t before we reach t h e partan representation

of t h e "final" h i t i a l s t a t e fumction i n t h e sense of B3,

- i.e.

before we reach the '"appropriate"

wave

%%is i n t e r a c t i o n converts t h e i n i t i a l f a s t

parton i n t o two o r Illore, and these aggan a r e broken up e t c . , i n a c s c a d e fashion naitkfng profound changes, f o r e x m p l e by f i l l i n g i n the gap i n tfie low pZ region md c r e a t i n g so= s o r t of parton plateau i n t h e f i n a l outgoing s t a t e ""appropriate" t o t h e i n i t i a l s t a t e CL, The reason no such extensive m d i f i c a t f o n is raade i n going front the i n i t i a l s t a t e i n a h a d r m c o l l i s i a n t o t h e f i n a l appropriate s t a t e B3 i s t h i s , "Prhe d i s t r i b u t i o n of the f a s t s a t i s f i e s W*

( n o n e e e ) partons i n the i n i t i a l s t a t e already

EJI s o l i t t l e disturbance is worked t h e r e by t h e m i l t a n i a n ,

Only i n t h e w t u a l l y overlappfng wee cegicras does the f u r t h e r a c t i o n of M linodify t h i n g s ( t o slnooth o u t t h e plateau).

This f a t r u e n o t became d i s t a n t partons have no e f f e c t , b u t r a t h e r b e c a w e

they have a rmiversal a f f e c t ,

Eere (and i n Cl) y c m be more p r e c i s e l y defined

f o r t h e hadrons a r e an t h e i r mss s h e l l and have a d e f i n i t e mass. y

gn(E9,)

an

We take

say i n L n G (&anlge of s c a l e m a n s j u s t a

change An o r i g i n of y ) F

C3. -

P u t t i n g t h i s a11 together i t naems that. i n a hadron c o l l i s i o n A

m y t h i n g i n t h e GM systera p l o t t i n g x

+B

-J.

P /P f o r x negative the d i s t r i b u t i o n of produces Z

The i d e a t h a t t h e rlglrt mrovers depend only on A and t h e l e f t Hlovers only li k e 8 is c a l l e d l i d t i n g ( I ,e. a s P suggested by C.N. Yaag e t a l , &ys, Rev,

; .69)

framntation.

188 2159

It: was

(1969) but a t t h e time i t

was supposed t h e e regions separated and had ao m m m i c a t i o n , but i n f a c t

there i s a s e a bemeen,

This s e a , hotrrever, we suppose is universal and

(although i t l o g i c a l l y could) we a s s m e i t c a r r i e s no information from t h e r i g h t t o l e f t region. kle have o d t t e d t h e d i f f r a c t i o n d i s s o c i a t i o n but i t is evident t h a t i f i t is added. i n a d e f i n i t e percentage t o the i n e l a s t f c w i l l not change our

conclur3ion,

However, the f r a c t i o n t h a t e l a s t i c s c a t t e r i n g is of the t o t a l

cross s e c t i o n does not Beem t o be universal ( f o r e x a q l e , f o r pp a t l a r g e P aeg/ctot

%

.25 whereas f o r 'n

p i t is c l o s e r t o . l 7 , s e e G. Giacomelli i n

Proceedings h t e r d a r n Conference an Bleaentary P a r t i c l e s , North Holland Press (2971)).

Thua lim%ting f r a e e n t a t i o n cannot be absolutely exact.

It

is probably generally nearly c o r r e c t ; perhaps i n a f u t u r e more exact under-

standing i t w i l l be t r u e f o r p a r t s of the c o l l i s i o n characterized by a o m other p a r m e t e r (e,g. impact, paratweter) b u t when i n t e g r a t e d over t h i s p a r a w t e r i t i s no longer exact f o r d i f f e r e n t cases give variocls d i f f e r e n t r e l a t i v e

weights t o t h e various values of the p a r a m t e c ,

Nevertheless even with t h i s

evidence a g a i n s t its p e r f e c t universal v a l i d i t y we continue to ancllyze i n a naive and s h p T e way leaving refinements t o some f u t u r e date, m e r e a r e a n m b e r a f a d d i t i o n a l conclusions m d e by assumning the b r k o v i a n fdea and extending our ideas such a s about gaps A8 from parton t o hadron wave functions.

M e s h a l l not discuas the= i n d e t a i l f o r hadron

c o f l i s f o n s i s n o t our caain s u b j e c t but give s o m e a w l e z ; .

We a s e m a ,

analogously t o A8 t h a t

(right

whiag

p a r t i c l e q u a n t m a m b e r s of A ncFnus quantum n d e r s of

a l l hadrons t o t h e ri&t of t h e gap), A+B

121

-c

For e x a w l e , f o r t h e exclusive c o l l i s i o n

C+D s o the outgoing s t a t e is pure C t a r i & r , B t o Ieft, the gap is

~ +fin2 ~ 1 PB/ 1

an4 1 P

~ = Pens ~and~ e-aby = ( 4 ~ ~ =~ B-'~ 1where ~ ' a depends

on t h e quantum nu&ers of A-C,

should go a s s-~('-')

Looked a t fuon a Regge point of view t h i s

whose a depends on the quantm. nunabera exchanged i n t h e

t channel, which is the @ama s A-C,

Thus a is i d e n t i f i e d with 2 f l - a ) ( o r

whatever the c o r r e c t pawer law of e n e r a f a l l - o f f turns o u t t o be) m d we mke a contact with t h e theory of exelusive r e a c t i o n s ,

(The s a w goes i f C is

two p a r t i c l e s l i k e n f p of fixed t o t a l mass2 n o t n e c e s s a r i l y a t resonance. X do not know of examples o f f resonance where t h i s pawer law has been checked

-

again we s e e the universal p r i n c i p l e t h a t going t o higher energy does not. l i f t a resonance ever higher against '"on-reamant'$

badground

- the

l a t t e r can

always a l s o be thought of a s taiLa of o t h e r resonanceo,) Again applying t h i s t o the case t h a t C only is near t h e end of its range, i o nearly 1 but D is anything, even m a y p a r t i c l e s , we s e e we a r e

so x

generating a gap of en(l-nc)

and the q l i t u d e goes a s e-aPn('-xc' a(l-xc )a-ldxc with a

The d i s t r i b u t i o n of xc is then d(l-xcla

Thie r e s u l t , though "'legally" t r u e a s P a nearly unobservable. collision. of rnllss from x

xc

-*

1,

2-2o.

is i n p r a c t i c a l eases

For example suppose C i s a proton produced i n a p+

Protons a l s o come from d i f f r a c t i o n d i s s o c i a t i o n of a reeonanee of

% say

-

m,

= (I-X~)~.

going t o proton W and pion.

(E -p )/$ t o x P P

-

and mmnturn of the proton

9;n

)/S

(as P + -1, where Ep, pp are the energy P the r e s t f r a m of the resonmce, Although t h e

(E +p P

This s p i l l s protans over a raage

l a t t e r is l e s s than one, t h e r e is a vary 811li311 gap (of range .98 t a l f o r %2

2.16) f r e e of d f f f r a c t i o n generated procons i t is too small t o i s o l a t e

experlentally.

Z f a gross p l o t is made f o r xc not s u f f i c i e n t l y near l

various nu&ers o f d i s s o c i a t i o n proton& a r e included and the v a r f a t i a n of numbers appaars f a r from ( l - ~ ~ 1 ' ' ~ ' d x ~ . This d i f f i c u l t y does not a r i s e f o r pions when t h e i n c i d e n t p a r t i c l e i s a proton,

Lecture 54

I should l i k e t o m k e a few coments about, our "conclu6sion" fraza C2, PSrst slnccz t h e m a n number of p a r t i c l e s goes as dxlx i n t h e smll region and continues across x

O a s dp/f

i t is evident t h a t the t o t a l n u d e r of particles

of a given kind ( t h e m u l t i p l i c i t y f o r t h a t kind) r i s e s logarithmlclal1y with fLnP o r with Ilns.

This i s a l s o obviow i n tha a r e a of the y pilot where the

plateau expands logarithmically with s.

But the plateau region is ( s t a t i s t i c a l l y )

n e u t r a l , f t s average f o r any a d d i t i v e qucsntm n u d e r such as charge, t h i r d component o f isospin, bargon nu&er, hypercharge, z cawonent o f angular

mmnturn e t c . , must be zero (because i f not

would give a Rns dependent

&/X

value fox one of these fixed conserved quantuna riders). t h e cascade i d e a of how the plateau is f a m e d , 1 -

dx (Number of a* a t x

- Number

of n- a t

X]

We expect t h i s from

Thus such i n t e g r a l s a s

conver*

t o nurabers which a r e

O c h a r a c t e r i s t i c of the p a r t i c l e A ( i n i t i a l l y taaving t o t h e r f g h t ) independent of P a s P

-t

=.

i n t e g r a l Ear

Independently carrespanding "&eft numbers" l i k e t h e s&m -1 t o O can be deflned, tJhich should iiepend on B.

X

I n p a r t i c u l a r then we. can d e f i n e d e f i n i t e q u a n t m nuoibers ( f o r the a d d i t i v e quantum n m b e r s ) f o r t h e right-rrroving particless3,by simply adding the t o t a l nuniber f o r a l l f o r kihich

X

0) i n t h e CM system.

O ( i . e . pZ

This nuaber w i l l vary from event t o event, of course, but we want t h e s t a t i s t i c a l

expected man over many events.

a constant a s s

+ we

This "mean r i @ t quanturn rider" w i l l approach

Thus we can t a l k of the "right mean 3-isospfn" o r the

"right m a n strangeness".

I t is evident t h a t these mean r i g h t q u a n t m n w b e r s

must, under t h e i d e a s of C2, be t h e same a s those of t h e i n c o d n g right-moving particle.

The plateau region does n o t l e t any qwntum nmbexs s l i p through i t .

( I f f o r a x m p l e we take a s p e t r i c c o l l i s i o n A nmbar conservation and must be t h a t of b. t o ntake A

+B

+A

then by o v e r a l l quantum

t r y the r i g h t q u a n t m a m b e r s (and t h e l e f t )

S

But by l i m i t i n g f r a ~ e a t a t i o nreplacing the l e f t A by B

does nog change t h e d i s t r i b u t i o n of r i g h t movers hence they

s t i l l carry t h e quantrna n u h e r s of A,) 'I'hus f n t e r e s t t n g l y a s P

+

the right-moving p a r t i c l e s t n the mean carry

t h e energy, t h e atowntuna ( d n w a constant), the 3 i s a s p i n , strangeness, baryon number, z angular w-mentum, e t c . , of t h e incoming r i g h t mover. NOB: -

We show t h a t , disregarding q m n t i t i e s of order 1/P, the difference of

t o t a l energy E and t o t a l ~ o m e n t wP of t h e p a r t i c l e e moving t o t h e ri+t is a constant D

.P

C(ci

- pZi)

(independent of P a s P

-t

"10

and i n f a c t i f t h e plateau For f i n i t e

is universal, t h e same constant D f o r every p a r t i c l e A).

positive) the difference from one hadranic p a r t i c l e i s c (p

2 2 .(m

A

X

(say

- Px

-p

)/2Px (m i s the m ~ of s the hadron) which is of order 1/P and t h e r e f o r e

negligible.

I"fie main contribution cornea from x near zero where the d i s t r i b u t i o n

of a p a r t i c u l a r type is edplc, hence the contribution t o

E

- p of

these is

Hadron-Hdron Collisions af Extreme Enargitls

c

j

( E - P ~ ) ~ P ~ / E~

.

m

The i n t e g r a l gives

so D

C

P ,"""Q

i n the plateau.

I f t h e p l a t e a w a r e universal, the constant II i s universal

and m y be e a s i l y calculated i n t e r n of already nreasured quzmtities, Xbua i n a c o l l i s i o n of 8 and B each p a r t i c l e i s converted i n t o a t r a i n of p a r t i c l e s rnoving i n i t s own d i r e c t i o n .

The " t r a i n A"

the q u a a t m numbers

"S

of t h e p a r t i c l e A and its energy (by the cansertvation of energy) but has l o s t a c e r t a i n m o e n t m D i n the i n t e r a c t f a n , i t i r s held back a b i t by the i n t e r (Such a f i n i t e moment= t r a n s f e r

a c t i o n , A and B each l o s e D t o the o t h e r ,

is, of course, c o n s i s t e n t and understandable i f only wees i n t e r a c t i n the collision.) (For the wave function of a s i n g l e hadron described i n a s s m p r i o n s

A2 t o A6, t h e t o t a l aomentunt of t h e partons is, of course P, the t o t a l ntomentm of t h e s t a t e , but t h e t o t a l energy energy E

is not the t o t a l

C i

P because of i n t e r a c t i o n energy which compensates t h e expected

f l n i t e excess of Cci above CpzI.) As a f i r e t s t e p t o describe these things f o m a l l y , we a r e t r y i n g t o describe t h e s t a t e /Ain right' Bin left' ( h e r e "h riight" Beans having very l a r g e p o s i t i v e Iongftudinal momntum: P, and "in l e f t " m a a s -P) i n t e r m of autgaing hadron s t a t e s the a w t r f x ,

- an e l e w n t

of

O f course the mst l i k e l y thing is t h a t the two p a r t i c l e s

do n o t c o l l i d e , making simply

/aut

Bout

left>.

Ve wish t o d e a l with

t h e wave functions i f they c o l l i d e , s o we w r i t e as usual S a r e speaking of the T la;atrk, o u t l i n e our ideas.

l

+ iT

and we

kfe s h a l l not n o m l i z e i t c o r r e c t l y , but j u s t

F o m l l y t h i s wave function can be given i n t e r m of the

amplitude t o f i n d various outgoing hadrons.

I f c* is the (formal) operator

t o represent the c r e a t i o n a f so@ kind of a hadron (kind, transverse momntm pk, and l ~ n g l t u d i n a lmowntum p a r e i n d i c e s of c*) we can represent such s t a t e s by

XIVAC,where

d i s c w a i n g how X looks,

X is some operator function of the c*.

We have been

let M be an operator t o c r e a t e a p l a t e a u

r r p i c a l u a i v e r s a l plateau f o r s o w range of x around O

- say x

- say

a

-.2 t o +.2

Photon-HadronInteractions [ t h e exact way the plateau of M c u t s off f o r f i n i t e x i s arbitrary; its choice a f f e c t s the d e f i n i t i o n of G ~ , Gdefined ~ l a t e r , but the f i n a l operator Next we w r i t e X a s G ~ G % where G R is t o mrrdify

X i s n o t dependent on t h i s ] .

the s e a on the r i g h t ( f o r x r 0. X t involves c r e a t i o n operators c

*

t o add

p a r t i c l e s t o (and beyond) t h e plateau operator M, and m a i h i l b t i o n o p e r a t o r s c t o take p a r t i c l e s o u t (which were put i n by our a r b i t r a r y choice 05 how the 3 O ( t h a t i s the meaning af the R). L * Likewise G is an operator function of c , c s n l y f o r x < Q. The operators L R

plateau M i s defined) but a l l these f o r x

G ,G c o m u t e s i n c e they contain operators of d i f f e r e n t p a r t i c l e s (soate s i g n s must be adjusted f o r Fermi p a r t i c l e s ) , R,

L)

Thus we w r i t e

G ~ G % IVAC, A B

where we have w r i t t e n [M-plateau3 = H/VAC>. on the p a r t i c l e A, e t c .

(54.3)

R

The operator GZ depends only

I f you want things t o look even n i c e r w r i t e the l e f t

s i d e i n terms of operators too, say d* which c r e a t e incantag p a r t i c l e s , and then have

R*% L* VAC, = ~ , " ~ i / ~ - ~ l a r e a u ,

TdA

* is equivalent ( i n t h i s two-body equation a t l e a s t ) t o R* B but i n an odd representation i n which dA a c t s on the vacuum and CA on

Thus t h e operator d: G:,

the i4-plateau s t a t e . A rasearch problem which f s very intportant, and v i r t u a l l y u n k n m

t h e o r e t i c a l l y , is ( t h e very r a r e ) c o l l i s i o n s a t e x t r e w e n e r m i n which the r e l a t i v e moaenta t o t h e o r i g i n a l d i r e c t f o n .

p a r t i c l e s come out a t

Fox:

example, proton-protm e l a s t l c s c a t t e r i n g a t f i n i t e angle, e.g. go0, where t i s the same order a s s a s

S

-r"

m.

What kind of physical view accounts f o r these

c o l l i s i o n e , I s h a l l not discuss ideas which have been t r i e d here, f o r our s u b j e c t is photons, but s h a l l only Coment t h a t nothing is c l e a r l y understood and you can s t a r t from s c r a t c h on your own. have t o be abandoned o r quantified?)

(For example w i l l assumption B1

(You s t a r t by looking f i r s t roughly a t

the experimental r e s u l t s t o r e m & e r q u a l i t a t i v e s a l i e n t f e a t u r e s t h a t d g h t need explanation,) nts:

P

By assuming t h a t the wee region is t h e same f o r each hadron and t h a t

Hadron-Hdmn Collisionsat Extreme Energies only vees i n t e r a c t have we not a s s m e d t h a t a l l t o t a l c r o s s s e c t i o n s a or a PP =P e t c . a r e equal, c l e a r l y contrary t o f a c t ? 1 have n o t thought t h i s out c l e a r l y but have always supposed t h a t t h e p a r t of t h e wave function which does i n t e r a c t (which i s always i n f i n i t e s i m a l conrpared t o t h e p a r t where they go past each o t h e r without i n t e r a c t h g ) could s t i l l have soare a o m l b z a t i o n r e l a t e d t o the t o t a l c r o s s s e c t i o n f o r t h a t p a r t i c u l a r c o l f i s i o n without being inconsistent: with o t h e r ideas. each G

A

Ta the forrnal expression above, f o r example,

could carry a numerical c o e f f i c i e n t gA proper t c A. 2 2

t o t a l c r o s s s e c t i o n s proportional t o gAgg,

00

'This would ntalse

a s is s a i d , f a e r a r f z a b l e ,

It

m y be. t h a t the previous a s s m p t i o n s do n o t i a p l y t h a t the totall c r o s s s e c t i o n s

a r e n e c e s s a r i l y equal, but r a t h e r perhaps t h a t they a r e f a c t o r i z a b l e , imply f o r example, t h a t a

- \/I;-

/b

PP \ ns

etc.

I C would

W e do not have any evidence on

whether t h i s is t r u e .

Tn these s t u d i e s we have made no r e m r k s which p e m i t us t o understand transverse =ownturn behavior (except t o say t h a t transverse nrromenta i a hadronic c o l l i s i o n s a r e limited, a r e s u l t taken d i r e c t l y from rrxperirnent].

Obviously

l o t s of i n t e r e s t i n g t h e o r e t i c a l questions r e m i n , such a s what functfon is t h e trcinsverw mmntuar d i s t r i b u t i o n , how does i t d i f f e r f o r various values of o r f o r n%

arid K's?

X,

Now should exclusive cross sectiona vary with t , e t c . ?

This e n t i r e realm of phenomena has been l e f t out of our a n a l y s i s , an e x c e l l e n t f u t u r e opportunity f o r advance l i e s here,

Hadronic States in Deep

astic Scattering

Lecture 55

We a s s u m t h a t i n the o r i g i n a l f i e l d Ea<oniain describing hadrons i n t;em of partons t h e r e a r e t e m givlrrg t h e coupling of partons wllth the

vector p o t e n t i a l of t h e quclntunn e l e c t r o w g n e t i c f i e l d .

We s h a l l assuvle i n

the s p i r i t of .olinIaum e l e c t r o ~ g n e t i ccoupling t h a t they couple i n t h e sixnplest m y expected Eram t h e propagation operators v i a gauge inrrariance.

That is we

IBSUIEIB:

That is a l ~ at h e coupling t h a t would be vtllid i f they were i d e a l f r e e p a r t i c l e s , This coupling Is not unique i f t h e partons a r e spin. 2 o r higher, but f o r t h e present t h i s w i l l not coneem us f o r we s h a l l suppose partons a r e e i t h e r s p i n O o r s p i n 112,

AZthou*

we a r e i n danger of not having t h e moat

general caae we s h a l l nevertheless explicitly next take the working hypothesis (suggested, of course, a s we have discussed by experixaent on vWZ and W1 and not a p r i o r i by theory) t h a t

(a i s an index f o r the kind of partan).

250

251

Final Hadronic Smrm in Deep l ~ e l ~ z sSc~tterr'ng ti~ We have seen how t h e assumptions AP-A8 plus these two D1 m d D2 l e a d t o the s c a l i n g expectations f o r t h e deep i n e l a s t i c seattergng ( l e c t u r e 27) m d Urnever, here w e s h a l l

there i s no reason t o repeat a l l t h a t h e r e again.

discuss what we can say about the products i n photon c o l l i s i o n s , i n p a r t i c u l a r we begin with the deep i n e l a s t i c ep s c a t t e r i n g regian cg2

-2Mvx.

Mv

tm of the proton, q t h a t of the v i r t w l photon) s o

P*q

virtual

V

photon energy i n laboratory (proton at r e s t ) system* kt us use the coordinate

-

system wfth t h e v i r t u a l photon purely spacelike qufi (0, -ZPx,O,O) PMm(P,P,O,O) 2 92 4 p 2 x 2 , 2 Hv 4P x. Then a s a r e s u l t of our avsvmptions t h e parton wave function before and a f t e r the c o l l i s i o n s looks l i k e :

Imdiately

A f tl?r

&up l i n g

That Is, one parton (say type a) moving t o the l e f t , the r e m i n i n g partons mvin,ng t o r i g h t j u s t a s i n o r i g i n a l proton, l e s s the individual parton a of mmentum x.

The r e l a t i v e p r o b a b i l i t y of t h i s p i c t u r e is e t n a ( x )

is t h e nunber of partons of type cr wSth pz/P

fi

X

where na(x)

i n the origilzsl proton s t a t e .

Then:

.

The t o t a l cross s e c t i o n i n t h i s sca3e

is a superp o s i t i o n of cases of d i f f e r e n t typee a of partons with weights,

on t h e character of the c o l l i s i o n through x ;

wa(x)

kla depending

2

e;na(x)/T

eg ng(x) ; l3

the s m of the weights being l.

h e obvious consequence of t h i ~and our other a s s m p t i o n s is t h a t i n t h i s system

{for fixed x a s we very F,

o r i f you l i k e v). The longitudinal m o a n t a w i l l s c a l e as P, %.e, i f they a r e s t a t e d i n

u n i t s of P a s say 1-tP t h e d i s t r i b u t i o n s w i l l be independent of P a s P (depend only on

Q),

We expect aZso near q equal zero t o find a dq/n behavior, Q

+

For p o s i t i v e

(night movers) we expect. i t behaves likrtt t h e universal H-plateau charaetlerirstic

of a wave function l i k e B3, which we know from hadron colZisione,

Pox negative

1-t wcs axe i n the ""plateau regian of i n i t i a l parton gap" Mefined i n connectfan

252

Photon-Hadron Interactions

with Cl),

We have not assmecl these two p l a t e a w a r e t h e s a w s o t h e c o e f f i c i e n t

of dr1ll-r need n o t be the same.

1 E they a r e not we s h a l l have trollible defining

what happens i n t h e t r m s i t i o n regian

- i t cannot

Leads t o the same c o e f f i c i e n t f o r plus and minus

go simply a s Cipz/c f o r t h a t

n. We s e e however t h a t t h i s

question is t o some e x t e n t an a r t i f a c t of aur p a r t i c u l a r choice of caardinate system.

Note t h a t the s t a t e described here a s "immdiatelp a f t e r coupling" i s

an i n i t i a l parton s t a t e ( i n the sense discussed i n t h e xemrk following the discussion on Cl)

- there

s t i l l must be i n t e r a c t i o n s from t h e Hamiltonian

before i t becomes the '%appropriatem outgoing wave function.

%is w i l l produce

cascading a f t h e left-moving parton i n t o t h e gap smearing the wee region i n t o negative

Q

and making l a r g e readjustments f o r the r i & t - m v i n g

(because, f o r f i n i t e

X,

system a l s o

they a r e no longer t h e c o r r e c t s o l u t i o n of W$ = EJf

s i n c e one parton is missing), This a l l appears q u i t e complicated and i t is d i f f i c u l t t o make firm predictions,

However, we might cant inue to assume i n t e r a c t i o n s a r e each Limited

i n range on a r a p i d i t y p l o t I n gaps, e t c .

- although t h e r e

a r e m n y of them possible f i l l i n g

But we s h a l l t r y t o adhere t o t h e grincllple a t l e a s t t h a t the

parton mavfng t o the l e f t

d e t e m i n e a t h e f i n a l hadrons t o t h e l e f t

and lgke-

wise f o r t h e r i g h t . We put t h i s idea f o m l l y i n t o t h e following a s s m p t i o n ,

a g e n e r a l i z a t i o n of Cl, 62 (we w r i t e i t independently, f o r i t may n o t be t r u e while t h e s p c i a l case Cl o r C2 m y be).

C5. I n the c e n t e r of massi system ( o r one m v i n g l o n g i t u d i n a l l y a t m y v e l o c i t y not near c)

(Lfkewise , exchanging l e f t and r i g h t . ) Assumption CS, i f i t were r i g h t and t h i s c o n t i n u i t y i n iLndPZ/c, would seem t o suggest t h a t both plar;eatts f i t tagether.

I arn not s u r e of myself

h e r e but s h a l l put i t down as an e x p l i c i t a s s m p t i o n which would, i f i t i s t r u e , r e w v e a l l our d i f f i c u l t i e s t h a t of hadron-hadxon c o l l i s i o n s .

- the

dtl/rl

region i s always universal,

233

Final Hadronic States in Deep Ittelastic Scatrcrifrg

This a s m t i o a i s a t present, very weakly based and m y e a s i l y be wrong

- it i s an i n t e r e s t i n g conjecture,

We now make a more d e t a i l e d discussion of our expectations f o r the l e f t -

moving p a r t i c l e s , (1 have p r o f i red g r e a t l y from conversations with A. Cisneros on these m t t e r s , )

For these p a r t i c l e s a v a r i a b l e more convenient than rt (which

goes d m t o -x) i s z

- rl/x

-p,/Px

= P*p/Psq the f ractton that the l e f t Since t h i s

nnovhg p a r t i c l e ' s moraenturln i s of the t o t a l left-moving nroaentuta.

is z = P.p/Peq i t is t h e energy of t h e p a r t i c l e i n t e r n of t h e energy v of t h e photon i n the laboratory system, how t h e v i r t u a l photon f r a g m n c s ,

X t is t h e proper v a r i a b l e f o r seeing

Of course a@ v

4 m,

x fixed, t h e d i s t r i b u t i o n s

i n z s c a l e a s we have s a i d ,

If we could be s u r e t h a t only a partoa of type a c m e out (which by t h e

way, can be much more n e a r l y done f a r neutrZno s c a t t e r i n g

- i n the quark m d e l ,

neutrino scattering can l e a d t o unique quarks t o t h e l e f t ) t h e d i s t r i b u t i o n t o the l e f t would be unique

- say

a function of r only D : ( , )

the p r o b a b i l i t y of n y K a t zl,z2 depends on

z l ~ Z as

7ite

Da (z 1"z )

- and

Dty(z).

"fhe n a b e r of v % with a given z i s

these functions do i n no wily depend on

X.

m e y do not

depend on x because the hadrons t o t h e l e f t d e ~ e n donly on the parton t o t h e l e f t ( a ) axld the adjacent wees from t h e hadron ( i f a t a l l )

- and

these

l a t t e r a r e m i v e r s a l and unaffected by the ternoval of t h e parton a t x from the proton,

The l a t t e r does not a f f e c t l e f t - m v i n g h a d r o n ~ f o r its r e l a t i v e

mmntrura t o l e f t - m v i n g hadrons i s not f l n i t e but g r w s a s P

4 0,

TIze a c t u a l d i s t r i b u t i o n seen a t a given x w i l l depend on x because the r e l a t i v e p r o b a b f l i t i e s of produchtng d i f f e r e n t kinds of partons or w i l l depend on x,

The a c t u a l dSstributions DD(x,z) w i l l be the weighted average f o r each

where the weights wa(x) proportional here t o eeLna(x) a r e defined by

These functions D (z), o r equivalently t h e i r c r e a t i o n operator

R on

K-plateau ( i f 65) isolates sontethinlg charaetaristic: of gartons and, i f our

254

Photon-Hadron Interactions

a s s w t i o n s a r e a l l c o r r e c t , therefore very fundamntal. indeed,

Me s h a l l

discuas l a t e r a s p e c i f i c parton -&l (quarks) a s well a s s o w p r a c t i c a l questions about t h e possible e x t r a c t i o n of t h e i s o l a t e d R (z) f r o a experilaent, a s well a s t h e p o s s i b i l i t i e s of finding t h e u

D

may behave,

(2)

(X)

by s p e c i a l guesses about hou

To raa t h e p o s s i b i l i t y of s p e c i a l f u n c t i o m c h a r a c t e r i s t i c

of each kind of parton i s a very i n t e r e s t i n g p o s e i b i l f t y , and one t h a t could be an entrance t o a path i n t o t h e h e a r t of t h e m c h a n l s w of strong i n t e r actims, These same functions D

(2)

- for

w i l l appear i n c e r t a i n o t h e r experilsents

e x a m l e , of course, i n deep neutrino proton

i.

p

+ products

experimnts.

The

a n a l y s i s is nearly t h e saae a s h e r e except t h a t t h e f m d m n t a l coupling m y be d i f f e r e n t so although n Agein i n the e*e-

(X)

a r e the s a a the weights wo(x) c o w o u t d i f f e r e n t l y ,

collision the a s s q t i m D2 says our i n i t i a l s t a t e

is j u s t a p a i r , parton o and a n t i p a r t o n

2 R L C6 t h e f i n a l s t a t e would be C ea (Da D;

with weight eo2,

+ :Da

a

Thus, ass&ng

/ ~ - ~ l . a t e a uagain > producing

hadrozls i n any one d i r e c t i o n characterized by the dlstributioln

where we sum an a over parrcns and a n t i p a r t o n s , I f , f o r e x a w l e , i n a o m experio~sntwe could be s u r e t h a t a c e r t a i n parton a erne o u t t o t h e l e f t say, then a s we have seen we would expect t h a t t h e t o t a l "leeft-mvbg quantrrrn rider" "he

sm, f o r so= a d d i t i v e quasltm

a m b e r of t h a t n d e r f o r a l l hadrons arrving t o t h e l e f t averaged over a l l events) w u l d be t h a t of the parton a ,

Thus i n p r i n c i p l e we could define

o r d e t e d n e I n tern of e ~ p e r i m n tt h e q u a n t m n m b e r s of the partons, the s t a t e i s not pure we s h a l l have t o Imm something of: the weights d

e t h i s useful

- but there are s o

\(X)

If to

different: kinds of e x p e r i w n t s

possible t h a t i n p r i n c i p l e the wa(x) can be d e t e m i n e d a s well. a s t h e o v e r a l l q m n t m a m b e r s of the partons, The p a r t i c l e s t o t h e r i g h t u n d e r h e p i n e l a s t i c e-p s c a t t e r i n g corn f r o n Erawentation of a proton with one cr parton of wroentw f r a c t l o n say E (p_a,x) ( E ) .

They a r e evidently not very fundamental.

X

remved,

But i t is c l e a r

t h a t the sam kind of f h a Z s t a t e r e a u l t a (on both s i d e s , l e f t and r i @ t ) i n Rrell"

experiment p

+p

J-

p

+ + 1-1- + any

hadrons s o t h e products i n t h i s

255

Final Hadrank States in Deep Inelastic Scattering e x p e r i m n t can be e n t i r e l y expressed i n t e r n of these E (p-cr,x)

and hence

has been worked out) i n t e r n of the products for deep

(supposing n,(x) ep s c a t t e r i n g .

"

We

leave i t f a r you t o w r i t e t h e e x p l i c i t r e l a t i o n s and t o

suggest p r a c t i c a l experfmn ts t o t e s t your ideas.

Our f i n a l hadron s t a t e is according t o our assunptions

NOTE:

L

V,'"' 4,

R

1%

(p-~,3

plateau,

,.I i s t h e operator f o r a l e f t partan a , and EB

where B

moving f r a g m n t s .

for the right(p-a,x> But t h i s can be considered a s a lanemnic only f o r the

expression i s probably ilnpossible f o r one operator M allways. would seem t o prevent us from w r i t i n g quanturo n u a e r s of two quarks (or

+ B)

a

Cz

plateau, which would have t o t a l

whi& is impossible t a w r i t e i n termss

of the hadron operators having m l y i n t e g r a l quantufo nmbers. to J, Wndula f o r pointing t h i s o u t , )

Because nothing

(1 a a indebted

A v a l i d m t h e m a t i c a l representation

f o r these ideas is an e x c e l l e n t problem. The reader shauld be warned t h a t a n d e r of these s c a l i n g predictions

f o r s p e c i a l produets of r e a c t i o n s may only hold a t much higher energies than t h a t a t which s c a l i n g f o r t h e t o t a l cross s e c t i o n (vW2 and Ml) s e t s I n , This warning r e s u l t s from t h e o r e t i c a l experience with a n m b e r of e x a w l e s of a n a l o g ~ u st h e o r e m i n n o n - r e l a t i v i s t i c quantum nogchanis where the sum works we11 before the i n d i v i d u a l t e r n do,

This is becautlre i f c e r t a i n i n t e r -

a c t i o n s a r e disregarded I n working t h e t o t a l p r o b a b i l i t y by assuming c e r t a i n s t a t e s only a r e "entered" subsequent gnteractions naay not change the t o t a l p r o b a b i l i t y t h e s t a t e was ' % ~ ? ~ e r e db"u t atay r e d i s t r i b u t e t h a t p r o b a b i l i t y over d i f f e r e n t f i n a l s t a t e s than were expected,

In t h e s p e c i a l case of s w f l

X

t h e predictions a r e e s p e c i a l l y sitaple,

F i r s t consider t h e r i g h t s i d e ( o r i g i n a l proton).

Here we? have a parton d-is-

t r l b u t i o n j u s t l i k e t h a t of a proton with only a very low x parton rernoved aad the wees disturbed (by i n t e r a c t i o n with t h e plateau developing from the left). a proton

Thus a l l t h e partons o r any s u b s t a n t i a l x a r e exactly l i k e t h a t of

- and we

(at least for z

can expect the saate d i s t r i b u t i o n of hadrms t o c o w out X)

a s do coae out f o r a hadron c o l l i s i o n of a proton, say

or $(E).

Hence f o r n small ER

Jil

(p-a,x> p' Next f o r amall x a l l na(x)dn go a s Ca &/X where Ca is a constant, ~o

t h a t wafx)

small

C,/):

CB

m

y

approaches a constaat y

B

Next call. llL t h e mixture :D

X.

r

parton, each wei@&witb f o r smll

X,

weight ya.

-

independent of x f o r

z ~ ~ D o: f t h e d i s t r i b u t i o n s f o r each

Our hadron d i s t r t b u t i o a becoma thus,

nearly

That is, f o r small m t h e proton f x a p e n t e i n r o a f o r a independent of x and the ss=

a s i t does f o r a hadron c o l l i s i o n .

E r a p n t a i n a unfversal way independent of X

Bnd the v i r t u a l photon a l s o X.

Sfnce we have afisawd t h e low

region the sam f o r a l l hadrons, t h e Col and yap and heace By do not depend

on the p a r t i c l e s t r u c k by t h e photon f n o m l i z e d t o t h e t o t a l cross-section f o r c o l l f s i a n , of course).

A sntall,

X

photon and

rt

hsdron behave j u s t l i k e the

c o l l i s f o n of two hadrons, each f r s g m n t s i n its m c h a r a c t e r i s t i c way,

That

of t h e photon f s independent of X,

2 For f t n i t e q, negative q we can s t i l l use our system of coordinates i n rJhtch q has only a space component Q.

exrzept i f Q

0.

It fs c l e a r h e r e hor~evelrt h a t only x near zero can be

a f f e c t e d by t h e *ocon

They a r e ,

Q; t h a t i s only t h e wees a r e e f f e c t e d .

however, a f f e c t e d i n a very conrplicated way f o r i n t e r a c t i o n i s important i n t h e wee region.

%a cannot t h e r e f o r e p r e d i c t whet vi3.3, happen t h e r e , but we

can note (a) t h a t i t is the s m f o r every hadroll A, A

+y

-c

products f o r

we have the @amweea f o r every hadron accordLng t o A&, and (b) t h e fragm n t a t i o n of the f i n i t e m i n t h e above system is c h a r a c t e r i s t i c of partons of system A only, f o r only t h e wees a r e e f f e c t e d by t h e photon.

In consequence of (a) the products on t h e l e f t rJhillh can be Ctczscribed

2 &ere Peq

i n t e r n of z

P Is t h e proton f o u r + ~ n t u n r , p is t h a t of a

product and q t h a t of the photon, f o r f i n i t e z m v distribution D

Y19

+

=, is so= kind of a

Z ( ~ ) . The d i s t r i b u t i o n c l e a r l y depends on

the v i r t u a l

257

Fhail Hadronic States in Dwp fnelastie Scatterkg QE the photon, b e e a w e the c o q l i c a t e d i n t e r a c t i o n s of the wees depend

=ss

on t h i s nrawntusn.

I n the o t h e r d i r e c t i o n ( t h e v a r i a b l e (q*p/q*P) the proton

f r a p e n t s i n t h e B a E way a s i t does f o r hadron c o l l i s i o n s , tions hold f o r q2

These eonsidera-

O a l s o , of course, but our coordinate system is inconvenient

f a r such a c a s e , kfe could a l s o use the c e n t e r of =ss

system f o r any fin.nite q

Conservation of energy and momentm meaQs t h e v i r t u a l photon (P'

2

+ m,qZ

finite)

i n t e r a c t s only wlth t h e wee partons of t h e t a r g e t proton ( o r hadran G).

This

i n t e r a c t i o n i s conrplicated but produces t h e s a w d i s t r i b u t i o n f o r any hadron f o r given q

2

.

The h a d r m behaves a s i t always does where

turbed whether by another hadron o r by a photon,

i t s wees a r e dis-

P o m a l l y our f i n a l hadran

s t a t e is

%us a s f a r a s h i @ energy i n e l a s t i c c o l l i s i o n e a r e concerned t h e (virtuill o r r e a l ) photon s e t s j u s t l i k e a hadron inasmuch as It appears t o have

itcl

2

2

own (q dependent, o r q 4 1 £ r a m a t a t i o n products, i n its;

d i r e c t i o n , the hadron f r a p e n t i n g a l a o i n Its c h a r a c t e t i s t l c way, IPhis af course makes a n i c e union with the- i d e a of vector =son

dodnance,

t h a t a f r e e photon (q2-0) has a c e r t a i n reasonable p r o b a b i l i t y t o be a v i r t u a l vector meson and a s such would behave i n hadrotl c o l l i s i o n s l i k e a hadron, note now

W@

We

s h a l l n o t have t o d e t e d n e with what prabab2lity i t looks l i k e

a badran and how t h i s varieJLee with q 2 , f o r i n any event i t , a s a whole, should a r t j u s t l i k e a hadron does i n v

In t h e c e n t e r of a r e te-

-S

+

-,a"

~olli~~eeion~r

p i c t u r e (and a l s o kn t h e spacelike q f i p r e ) t h e r e

of coupling i n which t h e p h ~ t o nE f r e t divides i n t o partons on the

way %n3 f o r e x a m l e one f a s t one slow, and these slow partons i n t e r a c t o r lsaxnihilate with t h e wee partons of the hadron.

Thus t h e p i c t u r e t h a t the

inco&ng photon loofts with some m p l f r u d e l i k e partons i t s e l f i s r e i n s t a t e d . lLsi g

2

r i s e s (and c e r t a i n l y where

X

2

= -q 12Mv is f i n i t e ) t h e contribution sf

aueh d2agram £falls away and only t h e d i r e c t coupllng t e r n of photon s c a t t e r i n g

a parton of the hadran reraain important.

F i n a l l y we a h a l l mateh our f i n i t e qZ region t o our smll we have done before we s h a l l suppose when

2

-

V +

.D

region.

ks

v is very l a r g e and -q2 l a r g e but

-q / ~ H vsmall the 1 i ~m ty be r a b n i n e i t h e r order:

r e s u l t e i t h e r from our f i n i t e q2, v

X

- i.e.,

formula o r from

X

we can get t h e

f i n i t e , but small,

fomula. mus (55.1) mwt agree with (55.2) f o r Large q

2

.

%is i s

easily done,

t h e r e s u l t s agree i f only we add t h e r e s u l t : DYsq2 = Dr f o r l a r g e g

2

.

mat

is:

2

The f r a p n t a t i o n produces of a photon of l a r g e -q becam independent of

2 2 -q a s -q r i s e s ,

(We a r e i n a l l czzsaaj, n o m l i z i a g t o the t o t a l cross s e c t i o n

which is varying, as l / q 2 , of course.)

Partons as Quarks

Lecture 56

We could now go on t o discuss various ntadels of what quantm n u d e r s partons carry, but we s h a l l l i m i t ourselves t o one e x a w l e , the one t h a t i s aost interesting,

The student should t r y other examples, suck a s the Sakata

nradef, t o see whether we t a n eliminate them by experivrents now done, or proposed, We s h a l l suppose the charged partom c m @ i n s i x v a r i e t i e s , three plus their antiparticles,

The three called u,d,s carry the quantum nu&ere of the

three quark ~ t a t e s(of the lw energy quark m d e l ) .

El..

This we s u m a r i z e by:

, Most o f our previous a s s w p t i o n s

were guided, o r we thought they were guided by f i e l d theory o r considerations based on hi& energy experimnts, guess,

Tkfs, of course, is not, i t i s an inspired

But tt i s a l s o contrary t o what can be t r u e i f the f i e l d ttreol-y i s too

ordinary,

For i a such a theory there would be a b m e s t a r e of quark nmber

me (and non-integral charge i n a localized wave packet) and, i n view of the cansemation of quark n u d e r , sone eigenstate of the system of quark nwaber one is expected.

I n other words we expect t o see r e a l p a r t i c l e s with quark

quantum n u a e r s .

They have not been seen.

they have very large =ss

- but t h i s makes

It is possible t o imgin~1:t h a t

i t very bard t o take a l l the

259

previous a s s m p t i o n s about a l l parton i n t e r a c t i o n s l i m i t e d t o t h e GeV region, etc.

There may be some way

t o reconcile a l l tlnis

intriguing; of t h e o r e t i c a l problms.

- i t is one of

t h e moat

I n order t o emphasize i t 1 w i l l make

another unneteeaary assumption t h a t I w i l l n o t use but I add t o remind you of t h e problem,

-

E2.

I f you p r e f e r t o replace i t by

"'physical quarks have high mass", go ahead

- you s t i l l have t h e o r e t i c a l work

t o do reconciling i t with E l and t h e r e s t of our a a s m p t i o n s , slay be wrong

- one of

Of c m r s e E l

the most important e x p e r i m n t a l jobs of the f u t u r e i s

t o f i n d out whether E l is indeed c o r r e c t o r impossgble and s o we should work o u t a s rnany t e s t a b l e consequences of i t a s p o s s i b l e ,

So f a r we have, a s

discussed i n l e c t u r e 32, corn t o t h e conclusion from expertmeat t h a t i n any ease :

.

E3,

%at they may be l i k e we do not now

know, except perhaps they m y not be vector (because the @ , p degeneracy i s not l i f t e d )

.

Although t h e problem of reconciling E l , E2 and f f e l d theory sl?igPtt be very d i f f i c u l t , i t appears ( a t l e a s e a t f i r s t s i g h t ) t o be n o t a t a l l d i f f i c u l t t o reconcile El, E2 and t h e o t h e r a s s m p t i o n s we have e x p l i c i t l y aade about parton and hadron d f a t r i b u t i o n s ! A c a r e f u l review of our assumptions shows t h i s ,

as already mentioned, i n

There i s possible doubt,

B1 ( i n t e r a c t i o n only between partons of small

r e l a t i v e y) but t h i s i s only used t o m k e the l a t e r asstznrptions = r e

plausible

and nraybe replaced by t h e s e l a t t e r aaswptfonap, A place where t h e r e Is an e s p e c i a l l y i n t e r e s t i n g , but n o t I n c o n s i s t e n t ,

conclusion is i n connectlon with '%ight+ov%ng q u a n t m nmbars" "iscussed

a

i n l e c t u r e 54) f o r the parton f a c t f a n Da and which should aow be non-integral, These n m b e r s a r e defined s t a t i s t i c a l l y a s t h e average over a l l events

-

although each event must give i n t e g r a l values t h e average, of course, need n o t be,

For e x a m l e i f we know one quark (and no antiquark) is s e n t t a t h e f f g h t ,

the mean n u d e r of basyon l e s s antibaxyons found on the r i g h t should (at extremely high e n e r m , a t Least) be

+

113.

The a r g u e n r s leadizleg t o t h i s concfusion a r e not obviated by the f a c t t h a t t h e partons do not have i n t e g r a l quantunr a m b e r s .

Xmgine f o r a very

Parterrs as Quarks l a r g e %nP t h a t the quarks are d i s t r i b u t e d i n the ( f h a l s t a t e ) wave function i n a long plateau:

( t h e l e f t end of the plateau is generated by a long chain

*

il:

of cascades v i a t e r n s l i k e a a a from t h e i n i t i a l s i n g l e pubirk)

Then i n turning i n t o f i n a l hadrons various bundles of quarks go together t o make l e g i t i m a t e hadron quantum numbers.

I n doing so they take combinations

of quarks over a f i n i t e range o f y a s i l l u s t r a t e d above.

(The o v e r a l l t o t a l

t r i a l i t y must be zero, of course, f o r the i n i t i a l s t a t e has zero t r i a l i t y ; our i l l u s t r a t i o n t h e i n i t i a l s t a t e has baryon n u d e r one,)

in

Me assume t h a t

t h e r e i s s o w non-zero p r o b a b i l i t y p e r dy t o pick of 3 quarks ( o r 3 antiquarks1 t o m k e a baryon ( o r antiibaqyon). t o pick up s t r a n g e quarks.

Likewise t h e r e should be a f i n i t e p r o b a b i l i t y

Pt is then seea t h a t a s a s t a t i s t i c a l matter with

a s u f f i c i e n t l y Long plateau kn y ((sufficiently long Wrkovian chain) t h e quark number (and strangeness) becows randomized m d the c e n t r a l region of t h e plateau is n e u t r a l oa t h e average i n these v a r i & l e s ,

This means t h a t the

r i g h t and l e f t mean q u a n t m n u d e r s of t h e f i n a l kadrons approach a constant t h a t depends a s we have supposed on t h e i n i t i a l r i g h t o r l e f t quark character. Pt doesn't m k e any difference exactly where you c u t t h e p l a t e a u i n deciding which is r i g h t o r which l e f t , a s long a s yclu c u t sowwhere n w r t h e ntlddle.

%is result: is s o i n t e r e s t i n g and Its e x p e r i m n t a l v e r i f i c a t i o n would represent such a d i r e c t measure of the supposed non-lntegral q u w t m numbers of t h e parton quarks t h a t we should say s o w wards about its possible v e t i fication,

F i r s t i n e l e c t r o n production i n general we do n o t have a s i n g l e

type of quark t h r m t o the l e f t ea t h e beauty of t h e result: would be confuged by having t o f i r s t know t h e wa(xl by ane e t h o d o r m o t h e r (see m t h o d s below). On the other hand f o r x near I we have been l e d by e x p e r i m n t ( r a t i o of

vWZn t o vWZp, l e c t u r e 31) t o suppose t h a t only t h a t is t r u e , near x

U

quarks survive

- hence i f

1 our left-moving quark may be a pure u quark.

h o t h e r way t o insure pure quarlcs is by neutrino s c a t t e r i n g which we discuss below, Secondly, although i t m y be easy t o f o m pions i n the plateau region

i t m y be harder t o f o m R's and s t i l l =re of t h e i r m s s e a ? ) ,

d i f f i c u l t t o f o r a baqona (becawe

If: t h i s is t h e case we should need a very lmng plateau

indeed t o g e t equilibriuan i n baryon a d e r , althou& hyperon charge a g h t be e a s i e r aad i s o s p i n e a s i e s t of a l l .

I n the e n e r m r m g e s a v a i l a b l e t o experi-

a n t therefore I would expect t h e quantum n d e r r u l e s t o work b e s t f o r iaospin, next f o r hypereharge, and b a v o n n h e r last ( i .e. requiring the l a r g e s t enerl3y)

.

The e a s i e s t would be i s o s p i n , l e f t - w v f n g z isospln: C IZiHiB

For e x q l e we would expect t h a t the

t h e @urnof each product hadron . s o e n s t o the

l e f t ( f o r e x a ~ l ei n the s y s t m with q pure @spacelike, o r myba the center of M -S), w i l l per c o l l i s i o n , f a r a g i v m x be given by

t h a t is, i t w i l l risae from zero f o r sm1l x (where u(x) r i s e sitrove

+ 112

( o r f a l l below

- 112)

etc,)

;(X)

and approach C 112 f o r

X

, never

near 1 (&ere,

we think u(x) d o ~ n a t e s ,) This r e l a t i o n can serve as can a n e e r of o t h e r s of t h i s type I n one of two ways,

In one case we ~ i g j n ttpuppose u(x), ;(X) e t c . , already knom frorn

eomc? other atethod a n s l y s f s of p f u r t h e r an.

-b

p

- such as neutrino -t

+ B B"" C anything,

scattering explained in l e c t u r e 33

- or

o r by perhapis one of t h e equsrtions developed

I n t h a t ease (36.1) is a q u a n t i t a t i v e prediction t o t e s t the

m s i s t e n c y of a l l the i d e m t h a t partons are! quarrks.

Alternatively

it

cm

i t s e l f be used t o give f u r t h e r i n f o m t i o n on t h e s i x separate functions u(x),

;(X)

etc., s o t h a t they m y be separately dete*ned,

Thieae could then

be cowared t o the r e s d t s of o t h e r raethods of ~ [ e t t i n gthem, but do n o t serve d i r e c t l y a s a check of the m d e l , Nonewr, even wlthout coaplete separation of a l l the functions a r e l a t i o n such a s (55.1) m y be used t o cheek t h e quark model because of the existence l l o f surrr m l e s (Equation 31.2) such a s (u(x) L(x))dx 2, (d(x) a(x))dx

-

-

so the integral, over x of the a m r a t o r of (56.1) should be 7/18.

I do not know a t t h i s tgne which kind of exper-ntal

infomtion w i l l

becorn available f i r s t and so my generat discussion s u f f e r s from a canfusfon Sn m a l y z i n g these t h e o r e t i c a l expectatlonra

a s t o which corns f i r s t the

m

1

horse o r c a r t .

There a r e a very l a r g e n&er

of such r e i a t i o n s whose

I w i l l therefore merely i n d i c a t e

organization is thereby m d e d i f f i c u l t .

some of t h e general r e l a t i o n s expected from our theofy and leave the choice of the b e s t way t o use them o r contbine them t o cowitre t o e x p e r i m a t up t o you,

Lecture 57

There is m o t h e r way t o insure t h a t m e g e t s a pure quark of one type r e c o i l i n g i n t h e f i n a l s t a t e , and t h a t is with deep i n e l a s t i c n e u t r i n o o r antiaeutrgnfno s c a t t e r i n g , a s we d i s m s s e d i n l e c t u r e 33.

We a s s m e of course,

weak coupling with t h e hadron p a r t of the weak current given t h e usual GJ*J U 1.l i n t e r n of q u a r b a s Cabibbo suggested, That is, we e x p l i c i t l y a s s m e :

We s h a l l a s e m f o r our d i s c w s i o n t h a t t h e current i n t e r a c t i o n is polnt-like, but t h a t i s a B a t t e r f o r e x p e r i m n t t o decide,

This very

i n t e r e s t i n g question i s , however, beside the scope of our discussion;

after

i t is d e t e d n e d t h e s a m kinds of questions and r e m a r h w i l l apply t o products generated by t h e aforementioned curneat operator.

I n aall cases

we can m a l y z e things a s i f they were the effect: of a v i r t u a l W meon f i e l d W of m a n t m q (generated by the lepton) coupled with t h e current: i n E4, li U J u s t as before (Equation 33.2) the t o t a l square rnatrix eleronent of J can Ir

be expressed i n t e r m of W2, W1, Wg s o t h e products, s u m d over spins and angles, a t l e a s t , c m a l s o be s o aslalyzed, each be s p l i t i n t o p a r t i a l Wl,

W2,

dlscuss oaXy t h e s c a l i n g region.

W

3

We have seen vW;! i o r e l a t e d t o W

- -

f i P - f y ( j u s t the s c a l e d function WV

WZ, WWg can

f o r s p e c i a l types of products.

r e l a t i o n should hold f a r products a b o ,

protons) was J u s t purely u(x).

That is t h e total, W1,

But =re

We

l and t h e

Lnteresting we noticed

ZHW3 f o r a n t i n e u t r i n o s c a t t e r i n g on

I-

Hence the s a m holds f o r t h e products, i e . ,

the p r o b a b i l i t y of a product i n M1, ( d e t e d n e d a s t h e appropriate c o e f f i c i e n t i n t h e v a r i a t i o n of cross s e c t i o n with laboratory n e u t r i n o angle klteepag q ,

V

fixed) minus t h a t i n W3 f o r ;p e c a t t e r i n g a r e purely the products f o r a u quark

being knocked backwards (Eomarda is a proton l e s s a d q w r k with p r o b a b i l i t y 2 cos BC

- the

.O6),

Thus by studying the products t o t h e l e f t i n t h i s cooibination we a r e

2 l a t t e r can i n f i r s t approxinration be negleczed a s s i n BC is only

studying the fragnaentacion products, Du(z), e~cpectedof one s i n g l e quark, a u quark i n f a c t . Ey choosing o t h e r ca&inations we can s e l e c t r e c o i l quarks of d i f f e r e n t types.

For example, f i P + f i p f o r neutrinos on protons gives pure

and should give various praducts with p r o b a b i l i t y ~;(z);(xl; t r i b u t i o n independent of

X,

the t o t a l cross s e c t i o n being

fiP-fgP f o r neutrinos on protons gives 2 d quark, s i n BC

a.

quarko

a f i x e d dis-

;(X),

Again

a pure d quark (cos

2

-94

.06 s quark),

From these d i s t r i b u t i o n s t h e t o t a l quantunt n m b e r s of the q m r k s can be deteracained. can n o t ,

Xn t h i s wag alone c e r t a i n m d e l s can be eliminated, but o t h e r s

For e x a w l e we cannot d i s c i n g d s h t h e quark model from t h e three-

t r i p l e t w d e l where t h e r e a r e 9 partons (and 9 antipartons) i n s e t s of three. A, B, and C each s e t having three s t a t e s l i k e quarks wi th various i n t e g e r

qwntum numbers,

Thus what would c o w out i n t h e experiment on fa-fg f o r

up s c a t t e r i n g , where we expect a pure u quark i n t h e q w r k rrrsdel, would, i n the t h r e e - t r i p l e t m d e l , be a -k l12 i s o s p l n parton b u t e i t h e r of type A o r

B o r C with equal p r o b a b i l i t y s o the mean charge on o t h e r n m b e r s can be l 1 3 i n t e g r a l and j u s t equal t o t h e u value, f o r t h a t is how t h e i n t e g e r charges f o r A, B, C were chosen.

(Other experiments, such a s e'e-

-t

anything o r

c o r r e l a t i o n s of l e f t and r i g h t s i d e d i s i n t e g r a t i a n s might d i s t i n g u i s h these models, )

There a r e a l a r g e nuniber of predictions i m p l i c i t i n the r e l a t i o n t h a t the d i s t r i b u t i o n of a given final hadron i n the l e f t (photon) d i r e c t i o n ( f o r convenience not n o m l i z e d ) is i n general given f o r deep i n e l a s t i c ep s c a t t e r i n g by: (Equation 55.1)

265

Bartans as Quarks where DU(z) e t c . , a r e t h e d i s t r i b u t i o n s of t h e product i n question f o r pure There a r e s i x f m c t i o n s i n general

up quarks, e t c .

80

analyze unless u(x) e t c . , were a l l alreadly a v a f l a b l e ,

i t is d i f f i c u l t t a

Howver, by taking

c e r t a i n co&inatfma of Eaewuremnts fewer functions axe involved,

%

i l l u s t r a t e t h i s with an e x a q l e ,

+

=4

Suppose we ask t o produce a etc.

WCg

c a l l t h e d i s t r i b u r i ~ n'D

By isospin r e f l e c t i o n t h e p r o b a b i l i t y a

t h a t a d produces a produces a

S-;

(z)

y i e l d s a n+ is t h e same a s

and by charge conjugation again t h e s a E a s t h a t a

In t h i s way

W-,

U

and :D

(x,z)

s e e f o r n production t h r e a r e r e a l l y only

WQ

t h r e e independent functions

In fact i f

kre

4"

measure t h e nu&er of n a n u s t h e a d e r of

R-

at a

given z i t (Equation 57.1) a l l reduces t o one function: 4"

D" (X,z>

- 'D

". ( x , ~ ) = A(z)

- V1 (d(x)-a(x)

(u(x)-;(x))

( f o r v i r t u a l y on proton) where A(z)

-

-k

:D

(x)

- :D

Thus we expect t h e d i s t r i b u t i o n ( p r o b a b i l i t y

(z).

as a function of z) t o be the same f o r a l l 4

(u(x)-;(X))

1

- $d(x)-8(x)),

X.

Aa we vary x we can deternine

within a constant.

This is j u s t a s i n 56.1, but

we do n o t have t o e a s u r e over a l l z t o i n t e g r a t e , and m a a u r e o t h e r p a r t i c l e s A =re

a s well,

wrzsurenaent of

3.

W

md

71-

a t soare convenient z would be enough.

The absolute c o e f f i c i e n t can be d e t e w n e d i n two ways, e i t h e r from t h e sum

-

r u l e s (Equations 31.21, o r by the hypothesis t h a t a s 4 a s ~ u ( x ) fep(x)

known function a s n

+

l.

X

-, 1 only u ( x ) s u m i v e s

Additional information would

came from t h e eame eltperiment on t h e neutmn, of course, (we g e t

g1

(U-;))

4

(dd)

+

e

The sum of the number of

W+

and

n- does not give

about the d i s t r i b u t i o n s , but we can roughly p r e d i c t its

US X

m& t h a t iet new dependence

The expression i n c u r l y brackets is t h e same a s fep(x) except f o r t h e c o e f f i c i e n t of t h e l a s t t e r n (which should be s l w l y 1 Is probably small ( f o r not only should

S

However, t h a t t e r n

,H be l e a s t h m say

i n a proton

U,:

4 r e l a t i v e t o $1 l but a l s o u is enhanced by F s o the d i s t r i b u t i o n of n' plus nis probably n e a r l y i n d e p e n d a t of x, and i f no-lized

t o fePfx) depend@ on

Arturo Gianeros haa suggested an, hypothesis which we explain i n w r e d e t a i l below, which a m u n t s t o a s e w n g t h a t near z

1 t h e f m c t i o n a ;D,(z)

f a l l o f f with variouhl powers of (I-z) and i n p a r t i c u l a r chat a s z

+

+

function :D

i s much l a r g e r than

+

o r :D

:D

.

-t.

l the

This .lakes t h e c o e f f i c i e n t s

of (57,3), (57.4) equal. a s z -, l, Thus i t m a n s t h a t ia t h i s region t h e p r o b a b i l i t y of f i n d i n g a n' a s we vary x is a d i r e c t measure of ufx) and t h e p r o b a b i l i t y of fgnding a n- m a s u r e e ;(X)

+

+ a(.),

d(x) i n t h e s a a e e a l e ,

This i s s t i l l another suggestion of how the i n d i v i d u a l determination of the f m c t i o n s na(x) m y be f a c i l i t a t e d . pe-ts

Xn f a c t , t h i s hypothesis, i f t r u e ,

a d e t e d n a t t o n of the s i x functions u(x), 5(x) e t c . (up t o an

o v e r a l l a u m r i c a l canstant) , by memuring the dia t r i b u t i o n functions f o r charged m s o n s only near z proton i s

l, f o r both proton asld neutron.

used a s a t a r g e t t h e n

(X)

Sf only t h e

cannot be d a t e m i n e d without =king

.trzemuremnts o f neutral. plebsons which is d i f f i c u l t experinrttntallg, Me can do eintilar things f o r t h e production of o t h e r p a r t i c l e s , f o r example K-mesons

.

Here t h e r e a r e s i x independent functions Df(z),

o t h e r s a r e obtained by f s o s p i n r e f l e c t i o n o r charge conjugation.

D 5 $ - 6' -

-d

D$

.

the (For e x m l e ,

The student e m v e r i f y t h a t f o r u on protons

+(2)

i f we m a s u r e p a r t i c l e s a t a given z t o t h e l e f t , i f N 4-

K e t c . , we f i n d t h e following r e s u l t s : on one f w c t i o n , with t h e s a w

m o t h e r coIlrbination (*ich

X

i s t h e n m b e r af

The i s o t o p i c s p i n difference depends

dependent c o e f f i c i e n t as before:

does not require R'

and

a l s o f a c t o r i z e s i n t o one f w c t i o n OE x , one of z:

p to

be distinguished)

P a r t m as Quarks

Ttre suxtl of a l l four depend@ s t r i c t l y on two functions o f z :

but the last f a c t o r i n curly brackets is l i k e l y t o be close t o fepfx) f o r any z , F i n a l l y t h e fourth r e l a t i o n , illrrolving two funetions is

(a measure of hyperckarge, but unequally s e n s i t i v e t o stxmge- a d non-stirange

quarks s i n c e we do a o t a s s m e SUj invariance). Cisneros' aassmption E(i(be1ow) h e r e a m @t h a t , a s z survive, c a l l them a , @ respectively. as z

+

-, 1,

4-

3.

only DU a d D@

Then a rseasure of K mesons t o t h e l e f t

1 is a dSLrect marsure of v a r i o m c d i n a t i o r a s of the u,d functions.

In t h i s , a s i n a l l cases, data on vn gives a d d i t i o n a l infornation, change ufx)

d(x) and

;(X)

*-+

Z(x) fi t h e f o m u l a a ,

Me have n o t discussed t h e r i g h t d i s t r i b u t i o n s , but t h e r e a r e r e l a t i o n s h e r e too f o r various e x p e r i m n t s ,

I?ftz mgneion only one as an e x a q l e , ""

Llewellyn Smith' s sum r u l e (Equation 33.6) neglecting s i n2 BC f;p(x)-f;p 4(ap{x)

- fen(x))

works f o r every x a a a t o t a l cross seetlton,

-

k now

s e e t h a t t h e o b j e c t s produced a t t h e right ( i n the hadran f r a v n t e t i c m region) a r c the s a w on both a i d e s f o r t h e proton but n o t f o r t h e neutron f o r &a en experiment we nrwt observe the ieoepin r e f l e c t e d products.

-

If

t h i s is done, the r e l a t i o n holds f o r t h e p a r t i a l cross s e c t i o n s f a r any products t o the ri&t i f the l e f t producta a r e n o t observed. Finally

CO

b r i n g a l l t h e hypotheses we have- made &out pertons together

i s t o one l i s t we note f i n a l l y our suggestion t h a t the $"ifep suggests t h a t when a protan h a s a quark near x

mmatuol, t h a t quark Is a

U

quark.

3

r a t e s strongly

1 and a remainder of snslllf,

We t r y t o genara2ltze t h i s t o any barnon

of t h e 56 nrultiplet i n $U6 language a m i n g our $U6 only quatlitativaly, n a t e z a ~ t l yq w n e i t a e l v e l y .

Photon-Hadran Interaetiom

268

A %&er

of t h e deciroet: 4, h a s a s w f l e r probability of having a quark

near x = 1 than does t h e o c t e t . f a r t o t a l c r o s s sections,

We have already discussed t h e i a p l i c a t i o n s

X t has i q l i c a t i o n s f o r products a l s o , of course.

klcz a l s o a s s u m t h a t i f a s t a t e i s a pure quark t o t h e l e f t i t has an

m l i t l x d e t o be a b a q o n t o the l e f t with z n e a r l y 1 which is proportional the the chance t h e baryan contains a f a s t quark near m e r e a r e them w a y implleations Eor products

X

= Z of the s a w kind,

- and we have ~ t i o n e dso=

t h a t came from u(x) being 1arlger than a l l t h e o t h e r s a s x

-+

I,

There a r e

o t h e r s , of another type, f o r e x a q l e (Claneroa, p r i v a t e comunication) i n eke 4"

e e

-

-c

hrrdrons s i n e e 5u haa four times t h e p r o b a b i l i t y of ad the chance o f

producing a proton with x n e a r l ( i n t h e c e n t e r of =ss)

and anything, e l s e is

f o u r times the probabi l i t y of producing a neutron, I f &S 5s c o r r e c t we shauld l i k e t o a s s u w solnething analogous f o r t h e The analogoos a s s m p t i o n Fa t h a t when one quark takes m e t of the

=ems.

mwatufn i t i s of a type t h a t t h e low e n e r m quark =del t o be m b e o f .

Cbe t h e r e f o r e (Zn agreemeat with Zisneros) aserne:

( f o r t h e charge conjugate i t i a it8

Q

supposes t h e meon

@

near x = l and the remainder

.

We have b u i l t a very t a l l house of cards asking s o m a y weakly-based eonJectures one upon t h e o t h e r and a g r e a t d e a l anay be wrong.

(Probably

- s a m p l a t e a u f o r gap and hadron - b u t i f i t were wrong i t does not a l t e r t h e t h r w t of any of t h e o t h e r s - j u s t i n t h e operator expreesions the weakest is C6

we s h a l l have t o be c a r e f u l t o u l ~ et h e rgght plateauh),] tdevertheless t h i a is the b e s t guess I can riitaake

ROW

- and we- can t r y t o use them has warking hypotheses.

Probably t h e g r e a t e s t challenge t o e x p e r i w n t and theory is t o get soracr, evidence o f quark qumurturn n u d e r s i n high-energy collisions,

The low-energy

quark ausdel, good a s i t is, i a n o t enough, t h e r e i s always l i n g e r i n g doubt that: the r e g u l a r i t i e s observed have so=

e n t i r e l y d i f f e r e n t bask8 o r arc?,

i n part, accidental,

The establSshment of evidence f o r the quark model

( m d we have i n d i c a t e d very nrany ways

- both

i n the l a s t few l e c t u r e s a s w e l l

a s e a r l i e r -(Llewellyn S ~ t ' sh sum r u l e (Equation 33.61, t h e s p i n sum r u l e f o r glp-gIn

e t c . ) by high energy e n p e r i m n t s would c o n f i w a t once the r e a l i t y

of the r e g u l a r i t i e s i n t e r p r e t e d by the low energy quark aaotlel.

This would

m k e f i r m a e m j e c t u r e of deepest s i g n i f i c a n c e t o understanding high energy physics

-

t h e intgortance of quark q u a a t m n m b e r s .

Supposing f o r a m m n t t h i s i s done, t h a t h e next s e r i o u s question w i l l become t h e o r e t i c a l

- what

e x a c t l y is t h e r e l a t i o n of t h e quark q u a l i t i e s a t

high energy and, t h a s e a t low e n e r m .

"I"heW"prtans a s q u a r b " "del

does not

iarply t h e low energy model ( i . e . why a r e the wave f m c t i o n s n o t m r e c o q l i c a t e d involving quark antiquark p a i r s ) n o r v i c e versa. would n o t b e understood. courage

- you mi&t

confirmed,

A t present t h e i r relation

To s t a r t working on t h i s n m w i l l take a l i t t l e

waste your time

- mybe

partons a s quarks w i l l not be

I f you do s t a r t , p o s s i b l y one place t o s t a r t might be t o t h i n k

about low energy m a t r i x eferoents l i k e d

-c

p

+y

i n a fahit m v i n g systetn i n

which a l l ( o r some) m m n t a a r e of t h e o r d e r P s o parton wave f m c t i o n s can be used,

(We have one r e l a t i o n of t h i s kind i n Bjorken"

sslun

r u l e f a r g,

Equation 3 3 - 1 6 . ) F i n a l l y i t should be noted t h a t even i f our house of cards s u r v i v e s and proves t o be r i g h t , we have n o t thereby proved t h e e x i s t e n c e a f partons.

'Ilhe

final, r e s u l t of o u r c o n s i d e r a t i o n s has been t o d a a c r i b e ehe r e s u l t of t h e

-

operation of a c u r r e n t on a proton s t a t e .I /p> ( f o r l a r g e v , -q2 ZMur) a s U a l i n e a r corobination of o p e r a t o r s l i k e D ~ E ~ ( ~ - ~ , ~ ~ M /c Yr eAact i,n g f i n a l outgoing hadron s t a t e s only,

It might be wise t o follow t h i s o u t f o m a l l y

without mentioning partons (analogous t o t h e way Gel1-Mann and F r i t z s c h d e s c r i b e parton r e s u l t s f o r t o t a l i n c l u s i v e c r o s s s e c t i o n s i n tevma of c a r n u t a t i o n r u l e s f o r q u a n t i t i e s , c u r r e n t s , defined i n general whether partans

""exist" o r r o t ) . From t h f s p o i n t a f view t h e partons would appear as an unnecessary s c a f f o l d i n g t h a t was used i n b u i l d i n g our house of cards. On t h e o t h e r hand, t h e partons would have been a u s e f u l psychological guide a s t o what r e l a t i o n s t a expect

- and

i f they continued t o s e r v e t h i s

270

PIZO~OB-R~TOP~ Intemctioll~

way to produce other valjid expectations they would of course b e d n to becclr~e "'real'"

pposiibly as real as any other thczoretjlcal structure invented to

bscrjlbe lature. A t any rate we shall see.

to,

It is good to have sowthing to look forvard

Appendix A. The sospin of Quark Fragmentatior Products

The dfseussion (Lecture 56) Leading t o the idea t h a t a d d i t i v e quark quantum n d e r s could appear a s mean t o t a l quantum n d e r s of p r ~ d u c t sm v i n g i n one d i r e c t i o n i s s u r p r i s i n g

- e s p e c i a l l y when i t

is

noted t h a t what holds f o r 3-isospin holds a l s o f o r any o t h e r c o q o n e a t such a s l-isospin o r 2-isospin (although of course i n p r a c t i c e they a r e nearly inrpossible t o measure).

It looks l i k e m isaospin 112 o b j e c t

could be represented by a group a f i s o s p i n l obJects (e.g. pions) which a t f i r s t s e e m irapossible

- except

-

t h a t we have an i n d e f i n i t e

n m b e r of such o b j e c t s , It i s t h e r e f o r e of i n t e r e s t t o make a very simgle s p e c i a l =the-

m t i c a l -del,

t o show t h a t indeed such things can be done i n p r i n c i p l e .

This i s e s p e c i a l l y i a p a r t a n t when i t is r e a l i z e d t h a t our previous attempts a t m a t h e m t i c a l f o m u l a t i o n cannot be copletcr; and m w t be looked a t a s mere m e m n i c s (see note i n Iazeture SS on the D and E operators).

This l i t t l e ntodel may h e l p by its exalngle t o l e a d t o c o r r e c t

pososfble formal expressians of our ideas. I n t h i s model suppose quarks c a r r y only fsospfn 112 m d hadrons a r e

271

only pions of isospin 1

- made

of quark antiquark p a i r s .

Zmgine t h a t we s t a r t with some current a n n i h i l a t i o n (analogous t o eie- but i n more general. isospin d i r e c t i o n ) i n i t i a l l y d i s i n t e g r a t i n g i n t o a p a i r of quarks Qa,

GB

(a,@ a r e SU2 i n d i c e s fixed i n t h i s problem.

I a e d i a t e l y a f t e r interaction:

-P

P

Q@

N quark p a i r s i n s i n g l e t s t a t e

a f t e r M m i l t m i a n operates;

m k e s hirdrons

o f type

Next the a c t i o n of the Wamiltonian f s t o produce p a i r s of quarks i n a s i n g l e t s t a t e uniformly spaced i n y space

- a t y p i c a l one i s qihi

The number o f such p a i r s M is then proportional

d on a l l I, equally.

t o Rn2P whlch we take t o be very l a r g e ,

( h e could a l s o ass-

the

n u d e l : d i s t r i b u t e d v i a Poisson with a man M e t c . , but we avoid compli-

- choose N

cations which only serve t o confuse our point

fixed.)

Next t h i s row of: quarks is a a s m d t a convert t o pions by a simple Thus ( i n

r u l e , each pion is f o ~ by d a p a i r adjacent i n t h e y space. figure) I f the f i r s t new s h g l e t p a i r had index 1, the next f i r et n

3 e t c . the

f o m d front an antiqmrtr index fl and a quark, index f ;

igi

next by an antiquark index

1,

and a qusrk index

3;

- the

- etc. ?.

To describe the i s o s p i n a t a t e of a n we we an isospin 3-vector

+

Thus i f the s is a n e u t r a l pion wo we have V with only a z cowoneat,

(0,0,1).

4-

For a a we have

1 ?=(l,f,O) 42-

etc.

antiquark of index y and a quark of index 6 fom a vector

if

The amplitude t h a t an 7~

characterized by

i s then proportianal t a the yd m t r i x element of the two-

by-rwo ~ o a t r i xoeif where

U

a r e the Pauli m t r i c e s ,

m i t e t h i s a s .

(WE! work i n r e l a t i v e m p l i t u d e e and p r o b a b i l i t i e s l a v i n g overall.

n o m l i z a t i m r o the end. )

Bus t h e t o t a l amplitude t o f i n d t h e n ' s i n d i r e c t i o n s ?l$2. ..qH i s Amp =

2

........ ~ * ? ~ l o,>the

< ~ / o l . ? ~ 1 ie>i j l ~ = 5 ~ 1<~~%l a * $ ~

i,.j.**

SW

on

273

Appendix A. The lsogpin of Quark Fragmanrati~nProducts i, j e t c . , being became the newly forrnd

(TQ

pairs are i n singlet states.

Waving t h e a w l i t u d e (fn an $U2 i n v a r i a n t f o m , of course) we can ask m a y questions,

The r e l a t i v e p r o b a b i l i t y of f i n d i n g any eonffguration

is the square

(A2 1

h e r e p a , p g a r e 2x2 density matrices corresponding to the s t a t e s ,

= a**;,

say

a i s 112 i n z,

b**%.

p@ 21 .

I.>,

For s t a t e

p@

/o%
=

112, Az = 112, Ax

Thus i n t h a t case a

0, A

Y

0. If. a is a e t a t e

spinning i n the d i r e c t i o n of a u n i t vector then A/a is t h a t u n i t vector. The expected v a l u e o f z isoapin is e v i d e n t l y A /2a. Now suppose we observe the i s o s p i n character of only a l i m i t e d n d e x of pions, e

I n f a c t we

ng over the elzaracrer a f the r e s t ,

s h a l l do two cases; s m i n g over a l l t o g e t the n o m l i z a t i o n of our p r o b a b i l i t y , and s

n number k

-lanything,

ng over a l l but one, a s i f we s t u d i e d products

h any case a sum over an unobserved n mans a

on V over 3 perpendicular values, s p b a l i z e d by

-

We need t h e f o m u l a

which is e a s i l y v e r i f i e d , Now f i n d t h e n o m l i z a t i o n

3

1 VV I 2"

over Vl,

P(Vl,. ,VN).

p@ is c m v e r t e d from b i a s % t o 3b-o4% by (A3).

2 i t is converted t o 3 b4-a.5.

smming

' *VE

Next sum Y

2

and

N

Continue f o r N terms t o 3Nb+(-l) o.B whose

t r a c e wiCh pa gives (suppose t r ( l ) = 3)

2 - 3Nab + (-l)N (A-%) N For l a r g e N t h i s f s a l m s t e x a c t l y 3 ab and s o we s h a l l divide by t h i s t o get n o m l i z e d p r o b a b i l i t i e s , Chat the kth pion is o f type Yt i s

Thus t h e n o m l i r e d p r o b a b i l i t y

except Vk

4

Now again we can sum over V1,

g2 up

t o *k-l

converting p

; and a l s o sum i n an analagoup way f i r s t on

(a

gH,

B

t o 3k-'b+(-l)k-1 -*

then VN-l

etc. to

Ihw t h e n e t is ( s c a l e t r ( 1 )

?k+k+lt o convert pa t o 3N-ka+(-l)N'k(o.~).

-

from which m y question about the one p a r t i c l e d i s t r i b u t i o n function can

now be answered.

For exemple, sum over a l l p o s s i b i l i t i e s f o r

gk

and check

n o m f i z a t i o n (for large N I , The mean p r o b a b i l i t y t h e kth neson ie n cr.?

5.

-

(ax+Ioy) i n (As).

+ i s PkzC

For n- put a -3=

42-

+

-

-

m a n i s o s p i n of t h e kLh hadron is P ~ " :P 3.

"placed

o-V

by - i o

only the one coming from p

42"

obtained by s e t t i n g

(ox-Lay).

Thus t h e

a term l i k e (A53 with

...

o o l u s %a ox. Note f i r s t using Y Y t h i s is 2a and hence gives

Ilkewise t h e t e r n i n a * t only c o n t r i b u t e s i f a i s i n t h e z d i r e c t i o n . kie f i n d

%is

r e s u l t canfirats a l l our expectations,

F i r s t I f k is I n t h e middle -t-

of the plateau, o f o r d e r N/2, n o t near e i t h e r end, then P' which is very snra2S. C pkli

-

-

-

%

5

small i f e i t h e r k i s small (i.e. near In the f o m r case neglecttng

the B end) o r near N ( i . e . near the a end). we have

i f k i s EinSte, near the g a d , a r e s u l t t h a t depends only on 8 , the quark a t t h a t tmd, and p r a c t i c a l l y ( a s N a t the other end, result, 2

N/2

%%us tbe plateau region has becornrr n e u t r a l .

plrW can only avoid being

t s m of order 3-'

- P"

m)

n o t a t a l l on the quark

(Evkdently f o r k near N we have the e x a c t opposite

l)

Appendix A. The lsospin of Quark Fragnzentet-ion Products F i n a l l y t h e t o t a l z-isospin q u a n t m n m b e r of a l l t h e pions t o the l e f t is

the z-conrponenf q m n t m n u e e r of the l e f t - m v i n g quark!

is t o be taken frm k

The sum on k

l t o near N12 t o hold only t h e l e f t m v e r s , the

contribution from t h e a-dependent t e r n a s only of order 3-N'2,

and the

sus is of the s a m order; the same as if we e u m d the l e f t t e r n above to i n f i n i q ,

Obviously t h e r e s u l t is i n s e n s i t i v e t o exactly where i n

t h e plateau we s t o p , i t is only necessary t h a t we s t o p the sum on k a t s o m point f a r from e i t h e r end, (One might f u r t h e r n o t i c e t h a r

neglecting

for e v e q k.

'+ Pk - ZP; o r-

l e zero,

This is a consequence of the f a c t t h a r

the quark has t o t a l Isospin 112, and n o t higher.

We leave

t o t h e reader

t o show t h a t t h i s genexalizea t o t h e r e a l case and eould i n principle becow a t e s t of the t o t a l i e o s p i a character of the partons. 3.

N;~)

-

+ N;~)

% a t is

0

- 2 ~ ; ~ is) zero f o r any r

t o the l e f t i f the l e f t m v e r s

c o w from a s i n g l e parton (of any kind o r superpoeition) and i f the isospfn of partons i s e i t h e r zero o r me-half

,l

Appendix B. A Test of Partons as Quarks

A Test of Partons a s Quarka

Following J, D, Bjorken we note t h a t the sum of nautrino and anti-neutrino cross-sections can be predicted f a i r l y c l o s e l y i n t e r n s of q a n t i t i e s alreaGZy known,

Sfnce nteaeuremnts of these t o t a l cross

s e c t i o n s a r e tha e a s i e s t t e s t of quark quantunr a m b e r s , we give m analyfiis here, Let us w a s u r e a l l cross s e c t i o n s i n u n i t s of Gs/2n where G i s the FemS constant and s the square of t h e eenter of mss energy.

Par

nucleans, then, our u n i t is GENJn where E is the laboratory energy. The t o t a l c r a s s s e c t i o n of a neutrino wirh a s p i n 112 p a r t i c l e Ss 2,

With a n a n t i p a r t i c l e i t i s 213,

Hence on a proton the cross s e c t i o n

t h e f a c t o r x c o d n g because t h e crose s e c t i o n v a r i e s with neutrons we replace d by

U,

6.

For

e t c . , s o the m a n neutrino cross s e c t i o n of

Appendix B, A Test tlf Partons as Quarks

0

t h e anti-eutrlno

cross s e c t i o n is

0

Since ;/G

3,

a r e p o s i t i v e , but undout3tedly l e s s than d,

U,

we s e e t h a t

aust be s u b a t a n t i a l l y l e s s than 1 b u t g r e a t e r than 113. The sum 28 l

Emever, i n t e g r a t i n g t h e a m of (3l,3) and (31,4) we have

Experinentally t h i s i n t e g r a l i s 0.31 s o i f we c o d d f o r g e t t h e i n t e g r a l /x(s&)dx

we would have o G

mwt aurely be l e s s than much l e s s ,

and

rf+a

It would be hard ts -age

t e r n produce m r e than a 10% e f f e c t . t e s t of our parton quark =del:

34

*

9 F (.31)

0.74.

Bur

8 4

and, when wei&ed by X , s u r e l y

t o lnake i n c l m i o n of the l a s t Thus we have a very strjbngenr

o - 6 c m n o t exceed 0.74 rrcnd y e t a l m s t

s u r e l y c e m o t f a l l below 0.74 bp noore than 18%. One can a l s o c a l c u l a t e upper lialts f o r oVP .t

.'.

m .

and ova

+ gvn

s e p a r a t e l y (u6tlng o t h e r proportions o f fep and fen}; they a r e 0.64 and 0.84 reepectivelp. m e s e n u m r i c a l e s t i m t e s w s t be revised by a few percent, f o r we 2 have neglected s i n BC.

course, b u t T,D.

meg a r e v a l i d only a t tzsyerptotic energy, of

h e has pointed out t h a t e l e c t r o n data i n d i c a t e s Chat

t h i s should only reqtrire a few GeV,

This page intentionally left blank

Anam1aus threshold,

122-523

B

Bjorken" s m rule,

158

C

Carnutator function,

29-30

Contpton scattering $44 $H,

225-220

Conservation of current,

5, 13-15

Conservation of generalized currents,

18

Cotcinghirn?f omula ,

201-203

Deep inelastic scattering with spin,

155-159

I)

Beser, Gflbert, Sudarsfsan representation, 185-185

9,

102-108

ftfspersion relations for pion f o m factor,

"119-122

Wson representation,

183-384

Elastic tern In self energy,

213-215

Electromagnetic eurtent (defined for hadran@)

5

Diffractive productian of p, W ,

E

Electromagnetic self energy, quark mdel, F

Final hadrone in high eaerg;y photon col1isions,

250-258

Index

Form factors, high, energy,

Gell-peyg; foggual time comutation Generalized currents eonservatioa, Wigh energy hadran-hadron colfisions, IXtJperffne splitting in hydrogen, Impact: parameter viemoint,

Inelastic electron nucleon scattering, Inelastic neutrino-nucleon scattering, Zsaspia o f quark fragments, Light cone, singularittea on, Light cone algebra, Light cone behwlor of comutatcrs, Limiting frapentation Elewellyn Smlth and Gross s m rule, Mamentm carried by qrtark~, Neutrino-nucleon total cross sections Nucleon electromagnetic Som factors Operators in inelastic scattering, Particle Erawntation, Parton frasentation, Partan model

fndex

Partan model hypotheses Partons as quarks,

Low energy pion photoproduction Parton wave function Pfon absorption Pion form factor, dispersion relations, Pion photoproductfon, Propagator function, P~eudnecalarmeson photoproduction, h i s h energy,

Q

Qwnk fragmentation Quark fragmentation products, Quark fragntents, iaospin of, Quark model, dpamfcal relativistic, dynmical non-relativistic,

R

Regge pole fomula,

Re~~onances, Resonancee in scaling function,

S

s-channel resonances, S matrix, Scattering in the DGS representation,

Index

T

Second-order electromagnetic coupling,

7

Second order unitarity,

8-10

Shadowing in nuclei,

108-112

Singulariries on the light cone,

21-22

Spin scaling funetfans,

156-157

Spin structure functions,

l55

t-channel exchange phenomena, Time ordered product,

V

Vacurn expectation o f comatator,

24-25

Vacurn expectation of time-ordered product product 25

W

Vacurn polarization due ta hodrons

22, 164-165

Valence quarks

152-153

W H in yM-Z)at

96-102

Vector meson dominance model,

89-113

Vector mesons,

82-85, 88-89

Veneziano f omula,

78-73

Mee partons,

232, 234

Interactions I

Fundamental Interactions

Affective Interactions

Fundamental Interactions

Designing Interactions

Empowering Interactions

Phytochemicals: Nutrient-Gene Interactions

Phytochemicals: Nutrient-Gene Interactions

Beneficial Plant Microbe interactions

Virus-Insect-Plant Interactions

Particle-Lung Interactions

Spacecraft-Environment Interactions

Aerosol-cloud-climate interactions

Spacecraft-Environment Interactions

Laser-Plasma Interactions

Fluid-Structure Interactions

Interactions in ultracold gases

Protein-Solvent Interactions

Atom-photon interactions

Stockley's Drug Interactions

Membrane Protein-Cytoskeleton Interactions

Drug-Nucleic Acid Interactions

Classroom Interactions in Literacy

Sound-Flow Interactions

Modulational interactions in plasmas

Drug-Membrane Interactions

Laser-Plasma Interactions