arXiv:hep-ph/0411007v1 30 Oct 2004
A sum rule for elastic scattering
G . Pancheri
LaboratoriN azionalidiFrascati,IN FN ,Frascati,Italy
and
Y .Srivastava
N ortheastern U niversity,B oston,U SA & IN FN ,U niversity ofPerugia,Italy
and
N .Staffolani
U niversity ofPerugia,Perugia,Italy
A sum rule is derived for elastic scattering ofhadrons at high energies
w hich is in good agreem
p ent w ith experim entaldata on pp available upto
the m axim um energy s = 2T eV . Physically,our sum rule re ects the
way unitarity correlates and lim its how large the elastic am plitude can
be as a function of energy to how fast it decreases as a function of the
m om entum transfer. T he universality ofour resultis justi ed through our
earlier result on equipartition ofquark and glue m om enta obtained from
the virialtheorem for m assless quarks and the W ilson conjecture
PA C S num bers: PA C S num bers com e here
1. Introduction
C onsider the elastic scattering oftwo hadrons (A and B ) w ith the follow ing kinem atics
A (pa)+ B (pb) !
A (pc)+ B (pd)
w ith
s = (pa + pb)2 = (pc + pd)2;
t= (pa
pc)2 = ( pb + pd)2 =
(1)
~ 2;
q
2
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pd)2 = (pc
u = (pa
pb)2;
(1:1)
and let us norm alize the elastic am plitude F (s;t) so that the elastic differential cross-section and the total cross-section (for high energies) read
as
d
(1:2)
( )= jF (s;t)j2; T O T (s)= 4 = m F (s;t= 0):
dt
In the im pact param eter representation
Z
1
F (s;t)= i
p
(bdb)Jo(b
t)F~(s;b);
(1:3)
o
and the partialb-wave am plitude is given by
F~(s;b)= 1
(s;b)e2i (s;b);
(1;4)
w here the inelasticity factor lies between (0
(s;b)
1) and (s;b)
is the realpart ofthe phase shift. D irectly m easureable quantities are (a)
jF (s;t)jthrough (ddt )and (b)= m F (s;0)through T O T . In the nextsection
2,we shallobtain lowerand upperboundsfora dim ensionlessquantity Io(s)
constructed by integrating jF (s;t)jover allm om entum transfers t. U nder
rather m ild assum ptions,at high energies (s ! 1 ) it is sharpened into a
sum rule
Z
s
1
Io(s)= (1=2)
o
d
=
(dt)
dt
Z
1
(qdq)jF (s;q)j !
1:
(1:5)
o
In sec 3,we com pare these predi
pctions w ith the experim entaldata and nd
that already at the Tevatron s = 2 T eV , the integral has the value
0:98 0:03 very close to its asym ptotic lim it 1. O ur extrapolation for LH C
gives 0:99 0:03. A lso,a briefdiscussion ofthe assum ptions and an estim ate of the elastic cross-section is presented here. In sec. IV ,we present
argum ents based on an equipartition ofenergy between quark and glue derived earlier,for the universality of the above result for allhadrons m ade
oflight quarks. In the concluding section,we considerfuture prospectsand
possible applications.
2. Low er and upper bounds and the elastic sum rule
T he dim ensionless b-wave cross sections are
d2 el
= 1
d2b
2 (s;b)cos2 (s;b)+
d2 inel
= 1
d2b
2
(s;b);
2
(s;b);
(2:1a)
(2:1b)
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d2 tot
= 2[1
(s;b)cos2 (s;b)]:
(2:1c)
d2b
T he m axim um perm issible rise for the di erent cross sections allowed by
unitarity [1,2,3,4]isw hen there istotalabsorption of\low " partialwaves,
i.e.,w hen
(s;b)! 0; as b ! 0 and s ! 1 ;
(2:2)
and the \geom etric" lim it is reached
d2 tot
d2 el d2 inel
=
=
(
1=2)
!
d2b
d2b
d2b
1 (b !
0;s !
1 ):
(2:3)
M ost m odels w ith rising totalcross-sections satisfy the above[5,6,7,8,9,
10]. O ften tim es,one de nes (s;b) = e n(s;b)=2 and n(s;b) is interprpeted
as the num ber ofcollisions at a given im pact param eter b and energy s.
N ow let us consider bounds for the dim ensionless integralIo(s) de ned
in Eq.(1.5). T he lower bound is easily obtained
Z
Z
1
Io(s)
1
(qdq)j= m F (s;q)j
o
(qdq)= m F (s;q);
(2:4a)
o
w hich upon using Eq.(1.3) leads to
Z
Io(s)
Z
(qdq) (bdb)Jo(qb)[1
(s;b)cos2 (s;b)];
(2:4b)
so that we have nally
Io(s)
1
(s;0)cos2 (s;0)
1
(s;0):
(2:4c)
T he upper bound requires m ore input[23]. Ifwe assum e (an ugly technical
assum ption) that sin2 (s;b) does not change sign (to leading order in s),
then one has the follow ing upper and lower bounds
(1 +
K
)
ln(s=so)
Io(s)
1
(s;0); (K > 0):
(2:5)
T hese boundshave been obtained incorporating (i)unitarity,(ii)positivity,
(iii) correct behavior near b = 0 and (iv) the asym ptotic behavior for
b! 1 .
Som e usefulrem arks: (1) For hadrons(notquarks and glue),the lowest
hadronic state has a nite m ass (m > 0),hence there is a nite range of
interaction. T hus,in the lim it ofboth b and s going to 1 ,we have
1
(s;b)cos2 (s;b) !
0; (s;b)sin2 (s;b) !
0;
(2:6)
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faster than an exponentialin b. (2) T he higher m om ents
Z
In (s)=
(dt)( t)n jF (s;t)j; (n = 1;2;:::);
(2:7)
are dim ensionaland go to zero in the asym ptotic lim it. T hus,they are less
usefulthan the zeroeth m om ent.
From Eq.(2.5),we obtain the sum rule as s ! 1
Io(s)! 1; as s !
1 :
(2:8)
3. C om parison of the sum rule w ith experim ental data
T he integral I0(s) should rise from its threshold value 2ja0jk ! 0,
w here a0 is the S-wave scattering length (com plex for pp) and k is the C M
3-m om entum ,to its asym ptotic value 1 as s goes to in nity. In Fig.(1),we
show a plot ofthis integralfor available data [11,12,13,14,15,16,17,18]
on pp and pp elastic scattering forhigh energies[19]. H ighestenergy data at
p
s = 1:8 T eV forpp from the Ferm ilab Tevatron [11],give an encouraging
value of0:98
0:03 dem onstrating that indeed the integralis close to its
asym ptotic value of1. W e expect it to be even closer to 1 at the LH C (our
extrapolation gives the value 0:99 0:03 for LH C ).
4. U niversality of the sum rule
It can be show n that the centralvalue ofthe inelasticity (s;0)! 0 at
asym ptotic energies s ! 1 for allhadrons m ade oflight quarks. H ence,
we have the universalresult[23]that IA B (s) ! 1 as s ! 1 ,w here A ;B
are either nucleons or m esons m ade of light quarks. T he reasons are as
follow s:
(i) For nucleons as wellas light m esons,halfthe hadronic energy is carried
by glue. In Q C D such an equipartition ofenergy is rigorously true[20,24]
for hadrons m ade ofm assless quarks ifthe W ilson area law holds.
(ii)Ifwe couple (i)to the notion thatthe rise ofthe cross-section isthrough
the gluonic channel,w hich is avour independent,the asynptotic equality
ofthe rise in allhadronic cross-section autom atically em erges.
5. conclusions
O ur (dim ensionless) sum rule re ects the fact that unitarity strongly
correlates the fall o in the m om entum transfer to the m agnitude of the
scattering am plitude athigh energies. Its satisfaction by experim entaldata
5
I0 (s)
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1
0.8
0.6
0.4
pp
pp
LHC
0.2
0
1
10
10
2
10
3
10
4
s [GeV]
p
Fig.1.A plotofI0 (s)vs. s using experim entaldata [11,12,13,14,15,16,17,18].
T he last point is our extrapolation for LH C .
at the highest energy con rm s our initial hypotehsis that the rise in the
totalcross-section as a function ofthe energy is indeed proportionalto the
fallo in the m om entum transfer. A s a by product,we nd that the ratio
el
! (1=4), w hich is again in very good agreem ent w ith data at the
tot
p
highest Tevatron energy s = 2 T eV .
W e also nd universality. T hat is, asym ptotically, IA B ! 1 for any
hadrons A ;B m ade oflight quarks. T hese m ay be testable at future LH C
and R H IC m easurem ents w ith heavy ions (or by other m eans [21].
C urrently, we are extending sim ilar considerations for one particle inclusive cross-sections.
5.1.Acknow ledgem ents
It is a pleasure to thank Paolo G irom iniand A llan W idom for fruitful
discussions. W e also take the opportunity to thank B ill G ary and other
organizers of this m eeting for their kind hospitality and for m aking this
conference stim ulating and enjoyable.
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[24] \D ispersive techniques for (s); R had (s) and instability of the perturbative
vacuum ",Y .Srivastava,S.Pacetti,G .Pancheriand A .W idom ,Published in
eC onfC 010430:T 19,2001.A lso,in \Stanford 2001,e+ e physics at interm ediate energies"165-168,e-PrintA rX iv: hep-ph/0106005.