Optimization techniques in elastic proton proton collisions

LEABHARLANN CHOLAISTE NA TRIONOIDE, BAILE ATHA CLIATH TRINITY COLLEGE LIBRARY DUBLIN OUscoil Atha Cliath The University of Dublin

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O p tim ization Techniques in

E lastic P ro to n -P ro to n

C ollisions

by

A lan T. B a tes

A thesis subm itted to tlie School of M athem atics, University of Dublin,

Trinity College, for th e degree of Ph.D .

D eclaration

This thesis has not been subm itted as an exercise for a degree a t any other

university. Except where otherwise stated , the work presented herein has

been carried out by the au th o r alone. The library of Trinity College, D ublin

may lend or copy this thesis upon request. The copyright belongs join tly to

the University of Dublin and Alan T. Bates.

A cknow ledgem ents

Sum m ary

C ontents

G eneral Introd u ction

1

1

P olarization M eastirem ent

13

1

Proton-P rotoii P o la rin ie try ...

14

1.1

Analyzing Power in the

CNI

R e g i o n ... 14

2

P roton-C arbon Polariinetry ... 18

2

H elicity Single-Flip A m plitude

20

1

Models based on Regge T h e o r y ... 21

2

Experim ental D a t a ... 24

3

Bound from Positivity P r o p e r t i e s ...26

4

Spin 0-Spin 1/2 B o u n d ... 26

3

O p tim ization w ith Lagrange M ultipliers

29

1

T e rm in o lo g y ... 31

1.2

M axim ization w ith Inequahty C o n s t r a i n t s ... 34

2

MacDowell-Martin Bound; An E x a m p le ... 35

2.1

Observables and C o n s tr a in ts ... 36

2.2

O p t i m i z a t i o n ... 38

2.3

U nitarity C l a s s e s ... 39

2.4

Reconstructing the C o n s t r a i n t s ...40

4 O bservables in P ro to n -P ro to n S catterin g

43

1

Helicity A m p litu d e s... 44

2

Total Cross S e c tio n ... 48

3

Im aginary Non-Flip A m p l i t u d e ... 49

4

Elastic Cross S e c t i o n ... 50

5

Im aginary Single-Flip Amplitude ...53

6

U n ita r ity ... 54

5 O p tim ization under

and U n itarity

56

1

Lagrange Formalism

... 57

2

U nitarity C l a s s e s ... 59

2.1

and

U nitarity C l a s s e s ... 60

2.2

and

U nitarity Classes ... 61

3

R econstruction of (T ei... 64

6 B ou n d including th e Spin-A verage A m p litu d e

76

1

Lagrange Formalism ... 77

2

Unitarity C l a s s e s ... 79

2.1

and

Unitarity C la s s e s ... 80

2.2

and

Unitarity Classes

82

3

Solution of Interior Unitarity C l a s s ... 84

3.1

R e su lts... 88

4

Solution of Boundary Unitarity C l a s s ... 92

5

Interior and Boundary Unitarity C l a s s e s ... 93

5.1

Nimierical T e c h n iq u e ... 95

5.2

R e su lts... 100

7 O p tim ization in cluding atot

1

Lagrange Formalism ...108

2

Unitarity C l a s s e s ... 110

2.1

and

B'^'^

Unitarity C la s s e s ...I l l

2.2

and

B ^

Unitarity Classes ... 113

3

Solution of Interior Unitarity C l a s s ... 115

3.1

R e su lts... 121

A p p en d ices

A P artial W ave P h a se Shifts

137

B M ath em atica C ode

138

List of Figures

4.1 R egions associated w ith th e expressions of d { ^ { 0 ) in te rm s of

Jaco b i p o ly n o m ials... 46

5.1 « n , «2i u nder cTei find im itarity ; y /s = 19.5 GeV, t = —0.001 (G eV /c)^. 72 5.2 « n , Q2i wilder Op\ and u n itarity ; = 23.5 G eV ... 72

5.3 « n , a2i u nd er (Tei and u n itarity ; y /s -- 30.7 G eV ... 72

5.4 « u , f l2i u nd er and u n itarity ; \ f s = 44.7 G eV ... 73

5.5 u n d er an d u n itarity ; ^ /s = 52.8 G eV ... 73

5.6 « u , a2i u n d er a^\ an d u n itarity ; s /s = 62.5 G eV ... 73

5.7 (lii.ciix u n d er ae\ and u n itarity ; y /s = 19.5 G eV , t = —0.01 (G eV /c)^. 74 5.8 fln, 02i u n d er a n d vmitarity; ^ = 23.5 G eV ... 74

5.9 « ii,0 2 i u n d er o^x an d u n itarity ; y /s = 30.7 G eV ... 74

5.10 a f j , a2i u n der aei an d u n itarity ; = 44.7 G eV ... 75

5.11 a il, a, 2 1 u nder aei and un itarity ; = 52.8 G eV ... 75

6.1 B ehaviour of th e p o l y n o m i a l / i ( J ) ... 90 6.2 a l {k = 0 ,1 ,1 1 ,2 2 ) an d optim ized u n d e r (Jei, Im 0 + an d

u n ita rity in th e in terio r u n ita rity class; \ / s == 52.8 G eV , t =

- 0 .0 0 1 (G eV /c)2 ... 91 6.3 a l {k = 0 ,1 ,1 1 ,2 2 ) a n d a^i optim ized u n d e r a^i, Im 0 + an d

u n ita rity in th e in terior u n ita rity class; y/s = 52.8 G eV , t =

- 0 .0 1 (G eV /c)2 ... 91 6.4 ofi an d optim ized u n d er cTei, Im0_|_ a n d u n ita rity con­

s tra in ts in th e I U D u n ita rity class; y/s = 52.8 G eV , t = -0 .0 0 1 (GeV/c)*^...106 6.5 and optim ized u n d er cTei, Im 0 + an d u n ita rity con­

s tra in ts in th e I U D u n ita rity class; •y/s = 52.8 G eV , t = - 0 .0 1 (G eV /c)'^... 106

7.1 Beliaviour of h{f) over th e CNI reg io n...120 7.2 a(. {k = 0 ,1 ,1 1 ,2 2 ) a n d optim ized u n d er cTgi, Im^_,_, cTtot

an d u n ita rity in th e in terio r class; y/ s = 52.8 G eV , t = -0 .0 0 1 (G eV /c)2 ...123 7.3 a( {k = 0 ,1 ,1 1 ,2 2 ) a n d aj i optim ized u n d er cTgi, Im0_,_, atot

7.5

(I'l^

and

a ^ i

optimized under

cTei,

In i0 + ,

crtot

and u n itarity

in th e boundary un itarity class w ith

= O.lEgi;

y/s =

52.8 GeV,

t = -0 .0 0 1 (G eV /c)^...129

7.6

and

a^i optimized under

a^i, Im0_|_, cTtot and u n itarity

in th e boundary un itarity class w ith

= O.OlSgi; \ / i =

52.8 GeV,

t = -0 .0 0 1 ( G c V / c ) ^ ... 129

7.7 afi and

optimized under

Im0_|_,

atot and unitarity

List o f Tables

2.1 T h e Regge m eson exchanges w ith th e co rresp o n d in g Se a n d /g. 22 2.2 M odels for th e helicity-fiip a m p l i t u d e ...24 2.3 A nalyzing pow er d a ta from F erm ilab E 704... 25

5.1 In terior an d B om idary u n ita rity class co n trib u tio n s asso ciated

w ith o{)tim ization m ider th e elastic cross section a n d u n ita rity . 62 5.2 (Tel an d S ei as a function of center-of-m ass e n e r g y ... 66 5.3 (Ttot- g aiid

Ao

as a function of center-of-m ass energy...67 5.4 R esults including u p p er bo un d on | Im rsI o p tim ized u n d e r cTei

an d u n ita rity c o n stra in ts a t t = —0.001 (G eV /c)^ as a fu n ctio n of y / s... 69 5.5 R esults including u p p er b o u n d on | I m r5| op tim ized im der cTgi

6.1 The equality m ultipliers

r

and

(3

under cTei, Im 0_|_ and im itarity

constraints... 86

6.2 I Im rsI as a function of center-of-mass energy and m om entum

transfer optimized under (Tei, Im 0^. and u n itarity constraints. . 88

6.3 C ontributions from the

l U B

u n itarity classes w ith (Tei, In i0 +

and u nitarity constraints;

^/s =

52.8 GeV,

t =

—0.001 (G eV /c)^.100

6.4 C ontributions from

I

C

I LJ B

w ith

Im</)+ and u n itarity

constraints;

t —

—0.001 (G eV /c)^... 101

6.5 C ontributions from

B d I \J B

w ith (jgi, Im 0 + and u n itarity

constraints;

t =

—0.001 (G eV /c)^... 102

6.6 C ontributions from / C / U Z? w ith (Tgi, Im 0 + and u n itarity

constraints;

f

= —0.01 (G eV /c)^...104

6.7 C ontributions from

B C I U B

w ith (7ei, In i0 + and u n itarity

constraints;

t =

—0.01 (G eV /c)^...105

7.1

[In ir’sl optimized under CTeii

atot

and u n itarity inside

the interior region a t

t

= —0.001 (G eV /c)^... 121

7.2 Iln irsI optimized under fJei, Ini0 + , fjtot and u nitarity inside

the interior region

at t =

—0.01 (G eV /c)^... 122

7.3 I Im rslm ax, w ith an approxim ation for

g,

over the

CNI

region.

125

7.4 Sum m ary of the bounds on | Im rsI; at

^/s =

52.8 GeV and

G eneral Introd uction

The expression

- = - A E + A G

+ Lq + L

g

(0.1)

indicates the different contributions which sum to give the pro to n its spin of

one half. The various contributions arise from the com ponent quarks (A S ),

th(' spin of th e gluonic fields (AG) and the orbital angular m om entum of the

(juarks (Lg) and of th e gluons

{L

q

).

In the deep inelastic scattering regime

only the light flavour cjuarks

{up, down

and

strange)

contribute to th e spin

of th(' j)roton. The net helicity of the (}uark flavour

q

in the direction of the

jHoton spin, in the cjiiark parton-m odel [1, 2, 3], is given by

Ag =

j

Aq{x) dx =

j

^q^{x) — q^{x) + q ^ x ) — q^{x)'^ dx

(0.2)

P ro b in g th e p ro to n : Measurements of the cross-section differences, with

particular spin configurations of incoming leptons and target nucleons, pro­

vide information on the polarized spin structure functions. For a longitudi­

nally polarized target, in

l+p

—^

I'+ X ,

the longitudinal spin-spin asymmetry

is the quantity which is measured in polarized lepton-nucleon deep inelastic

scattering experiments. Initial leptons can be longitudinally polarized along

( ^ ) or opposite (<—) the direction of motion and nucleons are longitudinally

polarized along (=^) or opposite (<^=) the initial lepton direction of motion,

bi the Bjorken limit, or deep inelastic region.

where

is the four-moment uni transfer squared,

E

and

E'

are the energies

of the incoming and outgoing leptons, in the Lab frame, respectively and

M

is the nucleon mass. In the Bjorken limit the unpolarized structure functions,

W \{ x ,Q ‘^)

and H'

2

(x,(

5

^), are known to scale approximately [1, 2]:

Similarly the polarized structure functions,

G\{x,Q'^)

and

G

2

{x,Q'^),

are ex­

pected to scale approximately in the Bjorken limit [1, 2]:

(0.3)

—q^ =

oc ,

u = E — E '

oo ,

T h e lon g itu d in al spin-spin asy m m etry c a n be expressed in te rm s of th e u n ­ po larized an d polarized s tru c tu re ftuictions:

^

Q m E + E'cos»)

MGi-Q^G;}

" 2 £ £ ' ' { 2 H ' i s i t i ^ e < / 2 + M / 2 C O s 2 « / 2 }

w here 0 is lepton sc a tte rin g angle. T h e a sy m m e try Ay expressed in te rm s of th e v irtu a l C o m p to n sc a tte rin g asy m m etries A1 2 is

^(1 = D {Ai -\- TjAo) (0-8)

w here th e coefficients D a n d rj are know n. A nalysis of ^ || leads to th e ex­ pression [1, 2]

/l|l « D A i (0.9)

and

w here R is th e ra tio of th e lon g itu d in al to tran sv erse cross-section,

In th e q u a rk -p a rto n m odel th e polarized s tru c tu re fm iction g\ { x) can be in terp ret('d as th e difference betw een th e n u m b er den sity of q u ark s w ith spin p arallel to th e nucleon spin {q^ {x) + qHx)) an d th o se w ith sp in a n ti-p arallel {q^{x) -f q^{x)) averaged over th e q u ark flavour charges [1, 2, 3]:

w here Aq { x ) = q^{x) — q^{x) + g^(x) — (x). M easurem ents of th e lo n g itu d i­ nal a n d tran sv erse spin-spin asy m m etries, in p o larized lepton -n ucleon deep inelastic scatterin g , can lead to in fo rm atio n on th e p o larized s tru c tu re func­ tio n s w hich can b e utilized to calcu late th e co n trib u tio n from th e q u ark s to th e sp in of th e p ro to n .

S u m r u le s : T h e B jorken sum rule [4] relates th e in teg ral over th e p ro to n an d n e u tro n spin s tru c tu re functions;

^ 9 \ { ^ , Q ^ ) d x ~ = y | l - 0 (0.13)

w here 0 3 = A n —A d is a nucleon axial coupling co n stan t som etim es expressed as th e ra tio of th e ax ial and vector coupling c o n stan t ( Ga/ Gv) of w eak de­ cays. T h e facto r ( 1 —0 (Q s/tf) ) arises from Q CD ra d ia tiv e corrections. T h is sum rule reflects th e difference in p o la rizatio n a sy m m etry in deep in elastic sc a tte rin g from p ro to n an d n eu tro n s. T h e polarized s tru c tu re fu n ctio n gi {x) is e x tra c te d for a p ro to n and a n e u tro n sep arately using different p olarized ta rg e ts. W ith m easu rem ents of ^ i(x ) one can te st th e B jorken sum rule w hich is in d e p en d en t of nucleon spin s tru c tu re details and is a fu n d am e n tal sum rule.

ag-netic an d w eak cu rren ts— an d by assum ing th e SU (3) sy m m e try in decays of th e o c te t b ary on s w ith a zero n et p o la rizatio n of th e stra n g e q u ark sea of th e nucleon;

s ' M d i = + ^ a s (0.14)

r r ( Q ' ) = ^ ‘ < ( i ) < i i = - ^ a 3 + | a s (0.15)

w here th e nucleon axial coupling co n stan ts 0 3 an d og are re la te d to th e SU (3) couplings F an d D by

a,3 = F + D , as = 3 F — D . (0.16)

T h e SU(3) couplings F a n d D describe th e (3- decays of th e b a ry o n o c te t m em bers. T h e Ellis-Jaffe sum rule p red icts r i( Q ^ ) = 0.171 ± 0.006 a t =

1 0 GeV'^ a n d a c o n trib u tio n of ap p ro x im ately 60% from th e q u ark s to th e spin of th e p ro to n .

model, created much theoretical interest [7] which lead to th e discovery of

the anom alous gluon contribtition. In the modified picture A S is replaced

by th e linear com bination A S — (3o;s/27r) A G which can be m ade sm all by

a cancellation between cjuark and gluon contributions. A new experim ental

program m e to investigate the phenomenon further also commenced. More

recent d a ta from th e Spin Muon C ollaboration

(SMC)

a t CERN

[8]

suggests

A S = 31 ±

4%

w ith th e

up

quarks providing ab o u t 83.2 ± 1.5%, th e

down

quarks abo u t —42.5 ± 1.5% and the large negative fraction of —9.7 ± 1.8%

coming from the

strange

quarks. The questions “W here does th e spin of the

proton come from? The gluons (A G )? Could it be in the orb ital angular

m om entum of the quarks

{Lg)

and the gluons (L g )?”rem ain unansw ered [9].

G lu o n C o n tr i b u tio n :

To probe the gluon polarization A G (Q ^), where

and thus m easure the gluon contribution to the p ro to n ’s spin, a high energy

polarized proton beam scattering at high m om entum transfers is required.

Studies of the gluon polarization suggest AG(Q^) ~ 0 — 2 a t

~ 1 GeV^

Collider (RHIC), Brookhaven, it is planned to use the polarized quarks of

one beam of polarized protons to probe the spin stru ctu re of th e protons in

the second beam w ith high energies

{^/s

= 50 — 500 GeV) and m om entum

(0.17)

tra n sfe rs ( p r > lO G eV /c).

T h e lo n g itu d in al spin-spin asy m m etry in th e d irec tio n of th e beam , one of th e observables p lan n ed to be m easured a t R H IC , is given by

PaPb

( i V ^ ^ - 7 V + - ) + ( i V _ _ - i V _ + )

(7V++ + 7V+_) + (iV__ + iV_+)_ (0.18) w here N ^ - , a n d N__are th e nu m b er of specific physical events observed w ith each com b inatio n of lon gitu dinal b eam p o la riz a tio n direction s an d Pa, Pb are th e p o la rizatio n s of th e beam s. T h e double sp in asy m m etry A i j , will play a v ital role in finding th e gluonic co n trib u tio n to th e p r o to n ’s sp in [10]. To p rob e th e gluon p o larizatio n , th e process,

p

+

p

^ 7 + X, or th e C^CD C o m p to n subprocess, .9 + g f/ + 7, offers good sen sitiv ity to th e gluon p o larizatio n . In th is process th e lo ng itud in al sp in-sp in a sy m m etry is d irec tly re la te d to th e gluon ])olarization. As well as Q C D C o m p to n s c a tte r­ ing, je t p ro d u ctio n probes th e gluon p o larizatio n w ith good sensitivity. T h e a s y n n n e try A n for th e j)rocess, p + p —> je ts, or th e elastic g luon-gluon svib- process, g - \ - g 9 + g-, is p ro p o rtio n al to th e sq uare of th e g luon p o la rizatio n an d th e je t ra te p ro d u c tio n is high.

exper-iinents, STA R [12] an d P H E N IX [13], an d one elastic sc a tte rin g ex p erim en t (P P 2 P P ) [14]. In th e com ing years d a ta from th e R H IC accelerato r will help Tuiravel th e Proton S p in Puzzle an d give us a deeper u n d e rsta n d in g of th e role of sp in in high energy physics.

j)Ower to be used as a polarimeter with

A P / P <

5% where

jip =

2.793 is the

proton’s magnetic moment.

O u tlin e o f th e T h esis

cross section, can be derived by including m iitarity and the elastic cross sec­

tion as constraints with the total cross section as the objective function. In

the same way a boimd on Im rs, the imaginary helicity single-flip amplitude

modihed by a kinematical factor, is derived where the system constraints are

the elastic cross section, the total cross section and the imaginary spin aver­

age non-flip amplitude, all of which are exp(^rimentally known. The Lagrange

nuiltiplier method also allows inequality constraints to be used when optimiz­

ing the system. Unitarity, appearing as an inequality, is input as a constraint

when optimizing Im rs. Many models indicate a value of ~ 0.1 for Im rs [20],

where the value of 0.1 is above the threshold value of (/Xp — l ) / 2 x 5% for

polarimetry with

A P / P < 5%.

In th(' second Chapter a Regge model calculation is used to obtain a value for

the amplitude Im

7 5

at zero momentum transfer and a synopsis of models for

the helicity-flip component is given. Experimental d ata from Fermilab E704

is presented where a 200 GeV polarized proton beam was used to measure

the analyzing power for proton-proton elastic collisions in the

CNI

region.

Other bounds on the amplitude Im rs are discussed.

work of F roissart. E n ding C h a p te r 3 is an exam ple of th e L agrange m e th o d

of o p tim iz atio n w here th e M acD ow ell-M artin b o u n d for spinless p article s is

derived.

In th e fo u rth C h a p te r th e observables, to be used as c o n stra in ts w hen o p ti­

m izing I m r5, are expressed in term s of p a rtia l wave am p litu d es. T h e o bserv­

ables are th e to ta l cross section, th e im ag inary spin average helicity non-flip

a m p litu d e, th e elastic cross section an d u n itarity . T h e h eh city re p re se n ta tio n

of Jaco b an d W ick [26] is used to express th e five helicity am p litu d es in elas­

tic p ro to n collisions as p a rtia l wave expansions. T h e observables expressed

in te rm s of helicity am i)litudes are w ritte n as p a rtia l wave series an d th e

im ag in ary helicity single-flip am p litu d e is ex panded as Taylor series in th e

C N I region.

T h e am p litu d e [ Im rs l is first optim ized in C h a p te r 5 w ith u n ita rity an d th e

elastic cross section expressed as inequality a n d eq u ality co n strain ts, respec­

tively. T h e bo un d, n o t a ‘s tr ic t’ bound, lim its th e value of | I m r s | in th e C NI region. T h e m iita rity c o n stra in ts divide th e solutions into different classes

w hich allows th e o p tim al solution to be selected. T h e sy stem of c o n stra in ts

in th is C h a p te r, an d subseq uent C h ap ters, is num erically solved using a com ­

b in a tio n of an aly tic calcu latio n s and m a th e m a tic a 3 .0 .

A new co n stra in t is ad d ed to th e system in C h a p te r 6, th e new co n strain t

nio in en tu in tra n sfe r in th e C oulom b N uclear Interference region. As ex p ected th e b o u n d on | IrnrsI is im proved b u t th e boim d is far from th e desired value of [ Hp — l ) / 2 X 5%. T h e u n ita rity co n strain ts play a m ore im p o rta n t role in th is sy stem of co n strain ts, th e different solution s g en era ted by th e u n ita rity c o n stra in ts are com p ared an d th e resu lta n t u p p e r b o u n d s on | Im rs | are dis­ cussed.

Chapter 1

Polarization M easurem ent

Details about the gluon polarization can be found by m easuring the double

spin longitudinal asynnnetry,

A n ,

in a particular process, ultim ately leading

to a value of the contribution from the gluons to the p ro to n ’s spin [10]. The

double spin longitudinal asynunetry is w ritten as

where

A+_,

and

N

__are the num ber of specific physical events

observed w ith each com bination of beam polarization directions and Pq,

Ph

are the polarizations of the beams. The asym m etry, to be m easured, is

dependent on the square of the beam polarization error and consequently

it is essential to have an accurate knowledge of th e beam polarization. To

])robe th e gluon polarization w ith sufficient accuracy, th e m axim um beam

polarization error

A P / P

cannot be larger th a n 5% [15]. One m ethod of

polarim etry uses the analyzing power in

pp

elastic coUisions a t sm all angles.

1

P ro to n -P ro to n P olarim etry

It is believed th a t polarization in elastic scattering vanishes a t high energy

where the am plitudes are eventually dom inated by diffraction energy. Recent

studies of hadronic scattering indicate th a t this may not be th e case [20]. For

high energy

pp

elastic scattering in the Coulomb Nuclear Interference

( CNi )

region

(t

~ —0.0012 (G eV /c)^), the analyzing power

App

possesses a small

but considerable value [27, 28].

1.1

A n a ly zin g P ow er in th e

CNI

R eg io n

The analyzing power

App

for a proton and the transverse single spin asym ­

m etry

are related through the expression

■A.ppP = A

n (

1

-

2

)

where

P

is the beam polarization and the targ et is unpolarized; for 100%

beam j)olarization the asym m etry and analyzing power are equal

The

beam i)olarization can be measured by comiting the scatters w ith th e beam

polarized up (A^^) and th en dow n (A^^) in a p o la rim e te r w ith a know^n a n a ­

lyzing pow er App-.

P

A_VP

1

A_ppA n ■ :i,3 )

_iVT + 7Vi^

T h e analyzing power App expressed in term s of th e s-ch an n el hehcity am p li­ tu d e s is [29]

•^pp ~ + 02 + 03 — 04)] (1-4)

w here do/ dt is th e differential cross

section-9 ^ .2 c. + |0 2 (s,O P + |03('S,OP + \4>4{s,t)\‘^ -|-4|(?!)5(s, i) ^ | ,

(1.5)

k is th e centre-of-m ass m o m entum and 0 ] , . . . , 0 5 are th e five in d e p en d en t helicity am p litu d es used to describe elastic pp collisions [26, 30]:

01 (•‘^1

—< + + |

0

| + + >

1

(p2{s,t)

= < -K-h 101--- > ,

03 (•Si

t)

= < H-- 101 H

>1

04 (■§, t) = < H-|0 | h > ,

05 (<5,

t) =<

-h + 101 H

>

-(1.6)

(1.7)

(1.8)

(1.9)

(

1

.

10

)

T h e helicity am p litu d es can be w ritte n as a sum of had ro n ic a n d electro ­

Tlie C oulom b p hase shift S is given by [31]

6 = —a In bt 4t - 0 .5 7 7 a (1.12)

w here a is th e fine s tru c tu re co n stan t, b is tlie nuclear slope p a ra m e te r a n d = 0.71(G eV /c)^ th e dipole Sachs form facto r p a ra m e te r. N eglecting th e a m p litu d e s 02, 04 a n d 0 i — 03, a t high energies in th e CNI region we can w rite th e analyzing pow er as

- Iin [05 + <^3)] = - 2 Im [0 ; 0+] (1.13)

w here 0+ is th e spin average non-flip am p litu d e, (0i + 0 3 ) / 2 . E xpressing th e helicity am j)litudes as 0, = 0* + {i = 1 , . . . , 5 ) , a n d neglecting th e C oulom b ])hase, th e analyzing power in th e CN i region can b e re w ritte n as

da

dt

- 2 I m a

T h e electrom agn etic helicity am p litu d es are know n, th e one-ph oton -exchan ge a m p litu d e s are given in [32], an d th e o p tical th eo rem [29, 33] gives Im0^J. oc (7t„t b u t very little is know n a b o u t th e h ad ron ic single helicity-flip a m p litu d e

Iiii^g , experim entally or theoretically, and thus the use of th e

pp

analyzing

power as a polarim eter depends on the contribution from the hadronic single

helicity-Hip am plitude.

Looking in more detail we can w rite the analyzing power in th e

C N I

region

as [33]

^

-

2

I m r

5

) ^ + 2 R e r s -

2

p l m r

5 J \ . n p / \ 2

^ ( ^ ) - 2 ( p + 5 ) ^ + 1 + p 2

w ith

(f)

2

,

and

0

i —

^ 3

not contributing where

p

= R e^+Z I m

0 4

.,

Kp-\-\ =

fip =

2.793 is the p ro to n ’s magnetic moment,

m

is the proton m ass and the

ratio rs includes a scaling by the im aginary p a rt of th e spin-average hadronic

am j)litude and l)y a kinem atical factor of

m /

^

hn(P+is,t)-

^ ^ ^

In th e CNI region when

f

~

tc,

where

tc =

—87ra-/(Jtot ~ —0.0012 (G eV /c)^ ,

(1-17)

interference between the non-hip am plitude and the single-flip am plitude is

m ost prom inent. This is reflected in the

[Kp —

2

I n ir

5

) ^ term . W hen |p| is

small, as is the case a t

^/s

~ 20 GeV [33], the m ain contribution to

App,

in

value of App in th e C N I region, 4.7% w ith I m r s = 0, is m odified by a b o u t 5.5% w hen I n ir s = ± 0 .0 5 . T h e p ositio n of th e m ax im u m of App is

w here th e C oulom b ph ase 5 is sm all a n d can be neglected for pp s c a tte rin g in th e CNI region [20]. A large bo u n d of | Im rs j resu lts in a n u n c e rta in ty on th e

m ax in nu n value of App an d to successfully use th e pp analyzin g pow er as a p o la rim eter w ith A P/ P < 5% a m axim um u p p er b o u n d of (/Up —1 )/2 x 5% ~ 4.48% on I I m r s I is p aram o u n t.

2

P roton -C arb on P olarim etry

Sim ilar to p ro to n -p ro to n elastic collisions, th e an alyzing pow er A p c for elastic p ro to n -c a rb o n sc a tte rin g in th e C N I region has a non-zero value w hich can b e

used to m easure th e p o la rizatio n of a p ro to n b eam [28, 34]. T h e an alyzing

j)ower expressed in te rm s of th e s-channel helicity am p litu d es is [35]

w here /+ + a n d /+ _ are th e helicity non-flip an d hip am p litu d es, respectively.

T h e m od ih ed helicity-flip am p litu d e, in p C elastic sc a tte rin g is

t

m a x

8

1

\ / 3 H— ( p i m r s — R e r s ) — ( p - I - t c ~ V S t c (1-18) J

(1.19)

m U

D ecom posing th e helicity am p litu d es into h adro nic a n d electro m ag n etic com ­

p o n en ts enables th e analyzing pow er to be w ritte n in te rm s th e flip a n d no n ­

flip am p litu d es [35]. In th e case of p C scatterin g , interference betw een th e

flip a n d non-flip am p litu d es is m ost p rom inent at t = tc w here

tc = ~ -0 .0 0 1 3 (G e V /c )2 (1.21)

^ to t

w ith Z — 6 for a carb o n ta rg e t [28, 35]. A lth o u g h th e spin 0-spin 1 /2 system

is in m any ways sim pler th a n th e spin 1/2 -s p in 1 /2 system , th e m axim um

value of th e p C analyzing j)ower in th e CNI region, like th e pp an alyzing

pow er, is sensitive to th e im aginary m odified helicity-flip am p litu d e I m r a n d

to use th e p C analyzing j)ower as a p o la rin ieter a n a cc u rate know ledge of

I I m r I is necessary. D ue to th e sim plicity of th e d e te c to r sy stem [34] th is

relativ e p o larim eter, w ith a th eo retically p red icted accuracy of 10 — 15%, is

one of th e can d id ate s for a p o larim eter a t R H IC .

O ne challenge is to calculate th e size of th e im ag in ary m odified helicity-

flip a m p litu d e | Imy'sj in th e case of pp collisions or | I r n r | in th e case of p C

collisions. T h e L agrange m ultiplier m eth o d , to be in tro d u ced in C h a p te r

3, is used to optim ize | I mr s j resultin g in an u p p e r b o u n d w hich lim its th e

value of th e analyzing pow er in th e CNi region. All presen t know ledge of

th e a m p litu d e | I mr s j , in th e low m o m entum tran sfer region, is presen ted in

th e n ext C h a p te r including ex perim ental d a ta from F erm ilab, R egge m odels,

C hapter 2

H elicity Single-Flip A m p litu d e

111 order to use th e p p analyzing power as a p o la rin ieter an a c c u ra te know ledge

Im r’5, based on th e p o sitiv ity p ro p erties of th e coefhcients in th e p a rtia l wave

series for th e differential cross section, is also discussed.

1

M od els based on R egge T h eory

In Regge th e o ry [36] th e sc a tte rin g am p litu d e has a variable a sy m p to tic b e­

haviour an d th is beh avio ur can be connected to a fam ily of b o u n d s ta te s a n d

resonances of different m asses an d spins [37], For pp sc a tte rin g th e re are five

in d ep en d en t helicity a m p h tu d e s [26, 30],

01

t)

= < + + |0| + + > = 0++ (^1

1)

(2-1)

02 {s, t) = < + + \4>\---> = (s, t) (2.2)

03 (s, t) = < + - 101 -F - > = 0++ (s, t) (2.3)

04 (s, 0 = < + - 101 - + > = (f>t+ (s, t) (2.4)

05 {s, t) = < + -h 101 H— > = 0 + ^ (s, t) (2.5)

T h e co n trib u tio n of a single t-channel m eson Regge pole a t a { t ) , to a n

_

|Ac—Aa| /

1

2 ^1+ ( - ! ) “• e - ' “- J r ( i . - o J («;)■-'• K s ) “-

(2.6)

where

nip is the proton mass, Sg is th e spin of th e corresponding meson

exchange and

is the minimmn vahie achieved by

on the exchange degen­

erate trajectory; Table 2.1 sliows the Regge exchanges and th e corresponding

Sg

[39]. The residues

fJ are simply related to th e coupling constants. T his

model provides a crude description of the helicity stru ctu re and

s, t depen­

dence of most two body

processes. For

{s, t) and

(/>5

(s, t) th e

leading

meson exchanges are p, uj U

2 and / [38] with

fVp (0 =

{t) =

(0 = « / (0 = 0.5 -H 0.9t

(2.7)

and the trajecto ry slope is

= 0.9.

Table 2.1: The Regge meson exchanges w ith the corresponding

Sg and

Ig.

e

le

p,

UJ

1

1

«2, /

2

For the Pom eron am plitude [38, 39],

1 / J. \ ^d^c —•^a| + |Ad —A(,|)

(a',sr

[image:38.526.56.516.34.467.2]

where the couphng is fixed by the assum ption of / dom inance and for

pp

scattering,

x p =

1.0,

A =

3.1 GeV~^ [38]. The Pom eron tra je cto ry

P

is

cvp = 1 .0 + 0.3i

(2.9)

and the trajecto ry slope for the Pomeron exchange is

a'p =

0.3. T he vertex

parity relation is [38, 39]

(

2

.

10)

while ujjper and lower vertices are related by

P . l l )

where ?/j is the intrinsic parity of particle

i.

T he signs of the tra je cto ry

contril)utions to the im aginary p art of the elastic

pp

scattering am plitudes

are

P + f — p — u + a

2

[40] and the contribution to In irs is fomid to come from

the Pom eron exchange, having the value 0.09 at zero m om entum transfer.

The spin stru ctu re of the lielicity-flip am plitiide has been investigated by

many authors. Table 2.2 lists some models and th e corresponding size of th e

helicity-flip com ponent. A review of each model is given in [20, 33].

The sign and m agnitude of Im

7 5

differs for each of the approaches m en­

tioned in Table 2.2, however the vahies suggest | InirsI < 0.1 a t RH IC ener­

Table 2.2: Models for th e helicity-flip am plitude

Model

Helicity-hip com ponent

dual quark-parton [41]

T5 = —0.06

pion exchange [42]

Im rs = 0.06

im pact picture [43]

Im rs ~ —0.06

two-gluon [44]

Im rs = 0.13

com pact diquark [45]

Im rs = 0.05 — 0.10

chiral sym m etry breaking [46]

1 Ini rs|

0.1

which, ill order to use the

p p

analyzing power as a polarim eter in th e

CNI

region with a m axim um beam polarization error of 5%, is necessary.

2

E x p e r im e n ta l D a ta

The analyzing power in th e

CNI

region has been m easured w ith th e 200 GeV

/ c

polarized proton beam facility a t Fermilab. For the first tim e a t high ener­

gies polarizations effects have been observed in th e

CNI

region. The use of a

j)olarized beam and a recoil sensitive scintillator targ et have m ade the detec­

tion possible. In previous experim ents w ith unpolarized beam s and polarized

[image:40.526.72.520.65.374.2]

Ta-ble 2.3 [47] suggests th a t the analyzing power in the

CNI

region is small and

j)Ositive, and the d a ta agrees w ith the theoretical prediction of a purely

CNI

analyzing power originating from the interference between the hehcity single-

hip am plitude and the helicity non-flip am plitude. Analysis of th e d a ta [33]

indicates a positive value of 8 — 30% for In irs.

Table 2.3: Analyzing power d a ta from Fermilab E704.

— t

range

(GeV/c)2

< - t >

(GeV/c)2

j ^ p p

(%)

1 . 5 0 X 1 0 - ^

-

4 . 0 0 X 1 0 “ ^ 2 . 8 8 X 1 0 - ^ 4 . 4 6 ± 3 . 1 6 4 . 0 0 X

-

1 . 2 5 X 1 0 - 2 8 . 3 0 X 1 0 - 3 3 . 1 1 ± 1 . 0 9 1 . 2 5 X 1 0 - 2 _ 2 . 2 5 X 1 0 - 2 1 . 7 5 X 1 0 - 2 2 . 6 2 ± 1 . 0 1 2 . 2 5 X 1 0 - 2 - 3 . 2 5 x 1 0 - 2 2 . 7 3 X 1 0 - 2 3 . 1 7 ± 1 . 0 7 3 . 2 5 X 1 0 - 2 - 4 . 2 5 x 1 0 “ 2 3 . 6 8 X 1 0 - 2 2 . 1 7 ± 1 . 3 9

4 . 2 5 x 1 0 - 2 - 5 . 0 0 X 1 0 - 2 4 . 7 5 X 1 0 - 2 0 . 2 7 ± 2 . 7 7

The P P 2 P P experim ent, approved by RHIC, plans to complete a detailed

study of elastic

p p

scattering using polarized proton beam s w ith center-of

3

B ou n d from P o sitiv ity P ro p erties

A fundam ental consequence of un itarity is th a t the absorptive unpolarized

differential cross section for the elastic scattering of particles of a rb itra ry spin

nuist obey the representation [48]-[50]

da -^ ° °

= ^ ( 2 n + l ) c „ ( s ) P „ ( c o s ^ ) ,

c„(s) > 0

‘ n= 0

(X ^ ( I n i ( / ) i ) ^ . (2.12)

i

P n iCOS

0)

is a Legendre polynom ial whose argum ent is th e cosine of the center-

of-niass scattering angle and the absorptive differential cross section satisfies

^ ^ ( . , 0 ) > ! ^ ( M < 0 ) (2. 13)

which k'ads to a bound on Im rs given by [51]

I m r 5 < 2 . 5 . (2.14)

This result lim its the size of the analyzing power

App

to 4.7% ± 13.1% at

small

t

and w ith an upper boimd of 2.3 th e required value of 5% for the

beam polarization accuracy cannot be obtained b u t th e bound lim its the

value th e analyzing power can take in the

CNI

region.

4

Spin 0-S p in 1 /2 B ound

A stu d y of bounds on the single helicity-hip am plitude

may provide im-

in the

CNI

region. The optim ization technique of Lagrange multipUers, ex­

tended by Einhorn and Blankenbecler [21] to include equality and inequality

constraints, is used to derive bounds on th e modified single helicity-fiip am ­

plitude I n i05, based on un itarity and experim ental quantities, where

Ini05(s,^) =

In i0 5 (s,i)

(2.15)

and

- t ~ 0

k Im 4>^{s,t)

Hodgkinson [52] used

ae\

and the slope

g

to drive a bound on th e helicity-fiip

am plitude for spin 0-spin 1/2 collisions;

I ' * ! ? ® " - ® ' " !

with

^

(

Traei 1 / 3

\

60

crt^ot/

where /++ and /+ _ are the helicity non-flip and flip am plitudes respectively.

Tlie optinuun

| I m r 5| < 2.3

(2.18)

is obtained if a similar bound for

pp

collisions is assumed over the energy

range

y/s =

50 — 500 GeV. This upper boimd on [ I mr sl is significantly

In the following chapters the variational formalism of Einhorn and Blanken­

becler [21] is introduced. A munber of equality and inequality constraints

for

p p

elastic scattering in the

CNI

region are found, and with the variational

method of Einhorn and Blankenbecler an upper bound on | ImrsI in the

CNI

C hapter 3

O p tim ization w ith Lagrange

M u ltip liers

To optimize a function subject to constraints, equality and inequality con­

straints, the met hod of Lagrange m ultipliers can be employed [21]-[25]. The

m ethod is used to derive bounds on the helicity single-flip am plitude in elas­

tic

p p

scattering w ith m iitarity constraints, appearing as inequalities, and

various experim ental (luantities, appearing as equality constraints. Such ex­

perim ental cjuantities are the to tal and elastic cross sections, and th e slope

Tlie F ro issart b o u n d [19] was th e first b o u n d on th e a sy m p to tic b eh av io u r

of th e to ta l cross section a t high energy (s —>• oc);

(Ttot < C log^ (s/so ) (3.1)

w here is th e center-of-m ass energy and sq is a c o n sta n t. Since th e resu lt of

F ro issart m any developm ents of th e m e th o d of o b ta in in g b o u n d s on s c a tte r­

ing a m p litu d es have b een m ade [53] - [67] , ran g in g from spinless sc a tte rin g

to nucleon-nucleon sc a tte rin g an d sc a tte rin g of p articles of a rb itr a r y spins.

In th e following C h a p te rs th e L agrange n uiltip lier m e th o d of o p tim iz a tio n is

used to b o u n d th e im ag in ary helicity single-flip a m p litu d e I m

05

in elastic pp

collisions. T h e pp sy stem is a spin 1/2 -sp in 1 /2 system w ith five in d ep en d en t

helicity am p litu d es, two non-flip, two double-fiij) a n d one single-flip. Com -

j)ared to th e spin

0

-spin

0

system or th e spin

0

-spin

1/2

system , th e n um ber

of helicity am p litu d es is g re a te r and th e alg eb ra following o p tim iz a tio n can

presen t som e challenges. T h e derivation of th e b o u n d is based on u n itarity ,

an a ly tic ity in th e L eh m an -M artin ellipse an d on p olynom ial b eh av io u r, w ith

no d epen dence on th e o re tic al models.

In th is C h a p te r, before deriving b o u n d s in th e p p system , th e basic

con cepts a n d term inology of th e o p tim iz atio n tech niqu e is in tro d u c e d [

21

]

[25] along w ith a descriptio n of th e co nd ition s req u ired to m axim ize a

o b ta in th e M acD ow ell-M artin b o u n d [68] in spinless sc atterin g .

1

T erm inology

O b j e c t i v e f u n c ti o n : T h e function th a t we w ant to optim ize is called

th e objective function. T his fu nctio n dep en d s on a set of real variables

X i , X2, ■.. ,Xn, d en oted by

f { x ) = f { x i , X 2 , . . . X n ) . (3.2)

T he o bjective fun ction is som etim es nam ed th e cost or p e n a lty function.

C o n s t r a i n t s : We consider equ ality co n strain ts an d in eq u ality co n strain ts.

Ecjuality c o n stra in ts are w ritte n as

/ a ( x ) = 0 a = 1,2, . . . p . (3.3)

In equ ality c o n stra in ts are w ritte n as

gfsix) > 0 , l3 = 1, 2, . . . q . (3.4)

Any p o in t x = (a^i, X2, ■ ■ ■ x„) th a t satisfies th e c o n stra in ts is called a feasible

p o in t and th e set of such p o in ts is called th e feasible set S.

T a n g e n t c o n e : T h e set of all (u n it leng th ) half-lines h, o rig in atin g a t a

point Xq in S an d ta n g e n t to a curve in S is called th e ta n g e n t cone to S a t

D iffe re n tia ls :

Given a function / , we denote its gradient vector a t

xq

by

/'(.To). Given any vector

v, th e linear functional

f ' { x o, v) = ( f ' {xo) , v)

is

called th e first differential of / and is denoted by

n a f

S f = f { x o , v) = {f ' {xo), v) =

V i - ( 3 . 5 ) i = l ^

Similarly, the second differential of / a t xo is defined by the q u ad ratic form

/ " ( x o , = E E ■ ( 3 6 )

i=i j = i

R e g u la r p o in ts o f S:

Let

X

q

be a feasible point and let A: be a unit vector

satisfying

( / ' ( x o ) , A - ) = 0 , Vq . ( 3. 7 )

If every

k satisfying Ec[ii.

( 3 . 7 )

lies in the tangent cone

C

a t

xq, then

Xq is a

regular point of

S.

N o r m a l p o in ts o f S:

If the gradients

f' {xo) are linearly independent,

xq

is a norm al point. Every norm al point is a regular point.

1.1

M axim ization w ith E quality C onstraints

The stan d ard m ethod of Lagrange m ultipliers determ ines all local m axim a

(or m inim a) th a t are regular points. It is sum m arized by the following two

T heorem 3.1

Let X

q

be a regular point of S and let

xq

be a local maximum

of f i x ) on S.

(i) Then there exists multipliers

Aq

such that the auxiliary function

L = f + f ^ Xafa

a = l

has a vanishing gradient

d L

L' {xq)

= —

— = 0

z = 1, 2 , . . . n .

(3.8)

U X i

(li) For a maximum

L " { x o , h ) < 0

(3.9)

for all h in the tangent cone at

xq

.

(ill) I f Xo IS normal, the multipliers

Aq

are unique.

T heorem 3.2

I f Eqn. (3.8) is satisfi,ed and if L"{xQ,h) is strictly negative

fo r all h in the tangent cone at xq, then

x q

is a local maximum of f [x) .

hi practice the theorems are used as follows; Solve the

n

gradient equa­

tions,

L'[xq)

= 0, for

xq

as a fimction of the unknown multipliers Aq. The

solutions

Xq = . x o ( A q )

are inserted into the constraint functions

/ q ( x o )

and

the multipliers are chosen to satisfy the constraint conditions

f a ( x o ) =

0.

1.2

M axim ization w ith Inequality C onstraints

T h e definitions of no rm al p o in ts an d reg u lar p o in ts ex ten d to in e q u ality con­

stra in ts if we divide these into interior c o n stra in ts [3 in I {xq), a n d b o u n d a ry

co n strain ts /j in B {xq), defined by

/(xo) = {^|i?/3(xo) > 0}

(3.10)

B{xq) ^ {(3\g0{xo) = {)} .

(3.11)

C bnsider th e m ax im izatio n of f { x ) su b ject to th e co n strain ts

U x ) = Q,

a = l , 2 , . . . p ,

(3.12)

ggi^c)

> 0 ,

0 = l , 2 , . . . q .

(3.13)

For any feasible Xq, let /(x q ) be th e set of indices (3 for w hich ,9/j(xq) > 0 an d

B{xq) be tho se for whicli g^ixo) =

0.

T h e following tw o th eo rem s o u tlin e

th e co nd ition s necessary to optim ize w ith in eq u ality co n strain ts.

T h e o r e m 3 .3 Let xq he a regular point and a local m axim um of f m the

feasible set S. Then

(%) There exists multipliers X^, and >

0

such that the auxiliary function

P Q

L = f + 1 ^ 0 9 0

a = l 0 = 1

has a vanishing gradient

(li) If 6

G

I{xq) we m ay choose fi/s =

0;

we ma y ignore any inequality

cojistramt f o r which gis{xo) >

0.

( i l l ) Let S\ be the subset of S f o r which g^ix) =

0

f o r all j3

G

B{ xq) f o r which

/i/j > 0.

Then

L " (x o ,/))< 0

(3.15)

f o r all h in the tangent cone o f S\ at Xq.

(iv) I f Xqis a n o r m a l point, the m ultipliers are unique.

T h eorem 3.4

If Eqn. (3.14) satisfied and if

L " { x o , h ) < 0

(3.16)

f o r all h in the t a n g e n t con e a t Xq, th e n t q is a local m a x i m u m o f f { x ) .

In Tlieoreins 3.3 and 3.4 the inequahty constraint

for which the corre­

sponding multiplier

fiff

is positive, effectively is an equality constraint.

2

M acD ow ell-M artin B ound; A n E xam p le

Mac'Dowell and Martin found a lower bound on the logarithmic derivative

w here th e lo garith m ic d erivative g is given by [69]

(3.18)

Using th e m e th o d of L agrange m ultipU ers th e M acD ow ell-M artin b o u n d can be ob tain ed . For equal m ass elastic sc a tte rin g th e center-of-m ass energy y/s

an d th e m o m en tu m tra n sfe r t are w ritte n as

w here k is th e center-of-m ass m o m en tu m an d 8 is th e center-of-m ass s c a tte r­ ing angle.

2.1

O bservables and C onstraints

For identical or equal m ass spinless sc a tte rin g th e to ta l a n d ab so rp tiv e elastic cross sections have p a rtia l wave expansions [69]

v 's = V 4k'^ + 4rn2 (3.19)

an d

t = —2k^ (1 — cos^^) (3.20)

<^tot —

~j^

^ (2/ + 1) Q/ (3.21)

an d

where ai is the im a g in a ry p a rtia l wave ampUtiide. The lo g a rith m ic derivative

g has the p a rtia l wave expansion [69]

( ^ ) X 3 ^ E ( 2 ' Da , . (3.23)

O nr aim is to constrain the lo g a rith m ic derivative, therefore g is the objective

fim c tio n . The constraints are the to ta l and absorptive elastic cross section

phis the p o s itiv ity constraint

Ui = ai - af > 0 (3.24)

w hich is a direct conseqiience o f u n ita rity [26, 69, 70].

Before o p tim iz in g the lo g a rith m ic derivative it is useful to re w rite the

scattering am plitudes as dimensionless am plitudes. We define the norm alized

diniensionless to ta l cross section Aq = (A’^/47t) atot, the norm alized absorptive

elastic cross section = (/c^/47t) cr^, and the norm alized dimensionless

lo g a rith m ic derivative g^ = (A’^S(Ttot/47r) p, where, in the high energy lim it^

Ao = + 1) ~ (3.25)

/ I

Eei = ' ^ { 2 l + l ) (3.26)

/ I

and

.go = E (2^ + 1) ^ (^ + 1) (3.27)

______________________________I___________________ I

' i n p r o to n -p ro to n s c a tte rin g th e to ta l and e la s tic cross sections are n o rm a h z e d b y th e

T h e eq uality co n strain ts, an d Eg;, are expressed as

a ^ 0 ~ 2 / g; an d 3 ^el

2

^

I of

respectively, w here a a n d f3 are eq u ality m ultipliers. T h e in eq u ality o r uni-

ta rity c o n stra in t Ui = ai — of > 0 is expressed as

(2/ + 1)

XiUi

~

2l\iUi

an d by definition th e ineq uality m ultip lier A/ n m st be zero or positive

[21]-[25].

2.2

O p tim iza tio n

T lu ’ aux iliary function w ith th e lo garithm ic derivativ e as th e ob jectiv e func­

tion is in troduced:

L = + O ' — 2 / g; + (3.28)

2

IXi (^ai - af^

w here A; > 0. To optim ize th e system we d ifferen tiate th e au x iliary fun ction

L w ith resp ect to th e im aginary p a rtia l wave am p litu d e ai, to first order-

d L

dai

= 2 r - 2al + 2/A, + A,) ai

an d second

order-c n

daf -41{13 + X,)

(3.29)

For an m in im um we requ ire d L / d a i = 0 an d d ^ L / d a J > 0, th is leads to th e

con ditio n j3 < —\ i w ith

^ j2 ^

2{(3 + \ i ) 2{j3-\-Xi) 2 { f 3 - \ - \ i )

2.3

U n ita r ity C lasses

W hen op tim izin g th e system it is n a tu ra l to divide th e p a rtia l waves in to two

classes [21] - [25]. For each u n ita rity in eq uality th e re are tw o classes, I an d

B:

= { J \ U i > 0 , Xi = {)} , = { J \ U i = 0 , X i > 0 } , (3.32)

I is called th e in terio r u n ita rity class an d B is called th e b o im d a ry u n ita rity

class. T h e in terio r u n ita rity class becom es

/ ^ = { / | 0 < a, < 1, A, = 0} (3.33)

an d th e l)o un dary u n ita rity class splits into two subclasses;

— » B^'° = { l \ a , = 0 , X i > 0 } (3.34)

B^^ = { l \ U i > 0 , X i > 0}

Interior U n ita rity Class

T h e in e q u ality m u ltiplier A; is equal to zero an d th e im ag in ary p a rtia l wave

am p litu d e is

w here r i = a/{2\(3\) > 0 a n d rg = \/{2\l3\) > 0. T h e m axim u m I for positive

p a rtia l waves is

an d th e m inim um I is Imin = 0.

B ou n d ary U n ita rity Class

T h e b o u n d a ry u n ita rity class = { / 1 a; = 0 , A/ > 0} is n o n -em p ty w hen

I > L = Unax find th e re is no co n trib u tio n from th e p a rtia l wave am p li­

tu d e 0/ in th is m iita rity class. T h e o th e r b o u n d a ry u n ita rity class =

{ / 1 a/ = 1, A; > 0} is no n -em p ty w hen I < 0 a n d is em p ty w hen I > 0 an d

consecjuently th e re is also no co n trib u tio n from th is u n ita rity class.

2.4

R econ stru ctin g th e C onstraints

In th is case we are in terested in co n trib u tio n s from th e in terio r u n ita rity class

/ ^ '. T h e norm alized dim ensionless to ta l cross section Aq is re c o n stru c te d by

s u b s titu tin g th e expression for th e im agin ary p a rtia l wave a m p litu d e ai, given

ai = ri -

T2

P

(3.36)

in E q u atio n (3.36), into

L

Ao = 2 Y ^ l a i (3.38)

/=i

to give

Ao = 2 j 2 l ( r , - r 2 l ‘^ ) ^ ^ (3.39)

^ ^

^

2r2

for large I. S im ilarly th e norm alized dim ensionless ab so rp tiv e elastic cross

section Eg/ an d lo g arith m ic derivative can be reco n stru cted ;

Ee/ ~ ^ ^ , (3.40)

3 T2

(3.41)

T h e logarithm ic derivative cj w ritte n in term s of th e n orm alized dim ensionless

logarithm ic derivative go is

9 =

(3.42)

.‘’■ O 'tot

or

an d su b stitu tio n of E q u atio n s (3.39) and (3.41) in to th is expression for g

leads to th e bo u n d

9 > 1 ^ .

(3.44)

Ss T2

T h e ra tio r i / r2, w here

ri

4

, V . {3.45)

is sim ply found by solving E q u atio n s (3.39) an d (3.40). T h e m inim ized log­

arith m ic derivative w ith th e r i / r2 = 4v4o/(3E e;) is w ritte n as

1 cr,

(3.46)

97T S CTei

and in th e high energy lim it, k ^ \ f s / 2 , th e M acD ow ell-M artin b o u n d

9 > — (3.47)

0071 (Tel

is o btam ed.

In deriving th e lower bo u n d on g we have only considered leading ord er

/ term s. If lower order I te rm s are included th e com plete M acD ow ell-M artin

bound

1

1

C hapter 4

O bservables in P ro to n -P ro to n

S catterin g

1

H elicity A m p litu d es

For the elastic scattering of two protons at CM energy

^ / s

and CM m omen­

tu m

k —

\ / s —

A m ? j 2 ,

there are sixteen helicity am plitudes which under the

following relations [26, 30];

P a r i t y c o n s e r v a tio n

(A'iAii.#.!A,A

2

) = ( - J ) " - " ( - a ; - Ail 01 - A, - A

2

)

(4.1)

T im e re v e r s a l in v a ria n c e

( a ; a ' 1 I A

1

A

2

) = ( - i r " (A

1

A

2

I

| a ; a ' )

(4.2)

I d e n tic a l p a r tic le s c a tte r in g

{ \ ' M 4 >

[A.Ai) = ( - 1 ) - '- ' ( a ;a ;i 0 iajA ,)

(4.3)

S y m m e tr y p r o p e r t i e s

,

di^{0) = di^_,{e)

(4.4)

reduce to two non helicity-flip am plitudes, two double helicity-Hip am pli­

tudes, and one single helicity-flip am plitude where

X = X \ — X2, = X \ —

The non helicity-fiip am plitudes

(p\

and ^

3

, the double helicity-flip am pli­