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The real and imaginary part of the elastic scattering amplitude in high energy diffraction
Volume 221, number 2
PHYSICS LETTERS B
27 April 1989
T H E R E A L A N D I M A G I N A R Y PART O F T H E E L A S T I C S C A T T E R I N G A M P L I T U D E I N H I G H E N E R G Y DIFFRACTION A. M A L E C K I
Department of TheoreticalPhysics,lnsntute of NuclearPhystcs,Radzzkowskiego152, PL-31-342 Cracow,Poland Received 3 December 1988; revised manuscript received 28 January 1989
We calculate the imaginary and real part of the elastic scattering amplitude in the model of high energy diffraction based on the concept of equivalence of states. The application to p!Oscattering at the CERN Collider shows that a large comribution of the real part is consistent with the lack of a pronounced minimum in the momentum transfer distribution.
In a recent measurement at the C E R N Collider [ 1 ] o f p r o t o n - a n t i p r o t o n elastic scattering in the near forward direction a large value for the ratio of the realto-imaginary part of the hadronic amplitude p = 0.24 + 0.04 was found. Previous measurements at the Collider [2,3] at the CM energy V / s = 5 4 6 - 6 3 0 GeV have revealed the presence of a break, rather than a dip, in the m o m e n t u m transfer distribution at t = i - 0 . 9 GeV 2. / The purpose ofthis~ letter is to point out that these two facts are intimately connected to each other and an astonishingly large value o f p could already be anticipated from the lack o f a pronounced m i n i m u m in the elastic differential cross section. This will be shown on the basis o f a recently formulated model o f high energy diffraction [ 4 ]. The model introduces the concept of equivalence of states, that is, the approximate degeneracy o f certain states having d o m i n a n t internal q u a n t u m numbers corresponding to the elastic channel. The amplitude o f elastic diffraction in the model reads
T=t,+g(tav-t,),
(1)
and is to be considered in the Bjorken-type limit where the coupling constant g ~ oo and t a r - t,~0, such
that the second term in ( 1 ) remains finite ~J, ti, tav being the eigenvalue of the scattering operator in the initial state [i) and the average value o f these eigenvalues in the set of equivalent physical states
t.~= E t,i~o,i2,
(2)
J
respectively. The weighting factors ~oj in (2) are interpreted in the model [ 4 ] as wave functions o f equivalent physical states IJ ) - Constructing the states equivalent to the proton as the states build o f the proton and n partons (n = 1, 2, 3, ...) [ 5 ] one has in the impact parameter representation
I~ojl2~l~o,,(t~.... ,t,,)[z=P, f l
I~t(tj)l 2,
J=l
with I ~v(b)[2=exp( -b2/2R~)/2gR~, *on = exp ( - ~) (tT) ~/n!,
(3)
t~ being the mean number of partons and R~ is the radius of their distribution. The model is completed by assuming the func~' Eq. ( 1) originates from the definition of inelastic diffraction as a complete sum of quasi-elastic (t~v- t, being infinitesimal) transitions. The limit g ~ serves to compensate the replacement of a complete wave function expansion with the ground state contribution alone.
0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )
191
Volume 221, number 2
PHYSICS LETTERS B
tional dependence of the eigenvalues tj on the impact parameter. One takes [4,5 ] t, =i/'o (b) =irro exp( -b2/2R2)/4~zR 2,
(4)
and
IO0
27 April 1989 k
'
'
'
'
I\
I
. . . .
II /"
...a
,.A
- "
X
---
The ansatz (5) reflects the Glauber law [ 6 ] of composition of constituent profiles corresponding to point-like partons. Collecting ( 1 ) - (5) together we write the diffractive amplitude of elastic scattering as oo
T(t) =i ~ db b J o ( v / ~ t ( b ) f ' ( b ) ,
(6)
,NAR,
REAL
\\\
i 10-
G-~0.
. . . .
\,.
d
(5)
with 7~(b) = ½G6~'~)(b),
I
p-p ZLASrlc SCArrER/N
I'
=i(1-(1-Fo) fl
. . . .
;''°"kt', X \ '~" 10"2 LLI
t I ---+tn
I
,J
,
,
,
I
,
,
1#
05
I
I
,~,
e
,
I
,
,
,
,
1,5 2 It l (GeV 2]
Fxg. 1. The imaginary and real amplitude of the p15elastic scattering in the function of the squared momentum transfer It[. The parameters of the model, corresponding to the CM energy 546630 GeV, are given in the text. I m T(s, t) ~ I m T(s, 0 ) ¢ ( ~ ) , 3~oo
r(s, t)= I t[ atot(S), Crtotbeing the total cross section. In our model the scaling function reads
0
with ¢ ( t ) = (1/atot)[no e x p ( ½ R 2 t ) + a , exP(½R2t)
F( b ) =go + F, -FoF,,
(7) where
/~, (b) =rr, exp( -b2/2R2)/4zcR2~,
(8)
and a~=
lim
gaG.
(9)
Thus effectively our diffractive model is determined by the two radii Ro, R~ and the two cross sections rr0, O"1.
It should be pointed out that the imaginary amplitude (6) is characterized by only a single zero (see the solid curve in fig. 1 ) arising from the interference of the two profiles/~o and/'~ in eq. (7). In order to fill up this zero we need to introduce a real part of the amplitude. This can be done using the recipe of Martin [ 7 ] based on asymptotic scaling: Re T(s, t) = R e T(s, 0) ( ¢ + t d ¢ / d t ) , while 192
(11)
- G 2 exp (½R22t)],
where
atot =Go + Gl -o'2,
(12)
rr2 =Goa, (R 2 + R 2) - ' / 4 ~ ,
(13)
2 2l (Ro2 +R 2) =RoR
(14)
R 22
l
All the parameters of the model, together with the ratio p of the real-to-imaginary part of the hadronic amplitude in the forward direction, were established from fitting the elastic differential cross section (with properly included the Coulomb amplitude) to the experimental data [ 1-3 ]. One obtains thus for p - ~ scattering at 546-630 OeV: ~ro= 55.56 rob, el = 10.77 mb, R0=0.76 fro, R~=0.51 fro, p = 0 . 2 0 ~2. The resulting total cross section atot = 60.64 mb and the integrated elastic cross section Gx= 13.39 rob. We anticipate the discussion of inelastic diffraction [ 8 ] by
(10) ~2 The value o f z 2 for the best fit is 193 with 130 experimental points.
Volume 221, number 2 ....
PHYSICS LETTERS B
i ....
102 ~
I ....
i ....
i ....
I ....
P - P ELASTIC SCATTERING 546- 630 GeV
-, I0 c
~=~
p: 0.2
.,
.Q
.~.
.......
p-- o.o
I0-';
,0"
i: ....
I
0.5
,
,
,
I
/
,
,
,
,
f
1.5
. . . .
I
. . . .
2 Itl
I
,
,
2.5 (GeV z]
,'?"
3
Fig. 2. The pp elastic differential cross section in the function of the squared momentum transfer It[. The experimental data are compared with the results of the model described in the text, with and without the real hadronic amplitude. n o t i c i n g that the integrated cross section o f inelastic d i f f r a c t i o n ~a,f= a~. Fig. 1 presents the real a n d i m a g i n a r y parts o f the elastic scattering a m p l i t u d e while fig. 2 c o n f r o n t s the e x p e r i m e n t a l data with the results o f o u r model, with a n d w i t h o u t the real part. T h e role o f the real part in filling u p the diffractive m i n i m u m is there readily illustrated. We have r e p e a t e d the s a m e analysis in the case o f the pp ~3 elastic scattering at the ISR energy o f 53 G e V where the diffractive m i n i m u m is m u c h m o r e pron o u n c e d [ 9 ]. T h e o b t a i n e d parameters are ~o = 39.31 mb, ~, = 5.61 rob, R o = 0.70 fm, R~ = 0 . 4 1 fm, p = 0.06. It is i n t e r e s t i n g to check the u n i v e r s a l i t y o f the funct i o n ~(T). It t u r n s out that the scaling f u n c t i o n s for the ISR a n d Collider energies agree q u i t e well, w i t h i n ~a We treat diffraction in p~ and pp collisions on the same footmg Ignoring thus any odd-under-crossing contributions.
27 April 1989
a n error o f 2 0 - 3 0 % , u p to z = 110. However, for higher v a l u e s o f T, a n d especially i n the region o f the s e c o n d m a x i m u m , their d i s c r e p a n c y b e c o m e s s u b s t a n t i a l which m e a n s that at the I S R energies the a s y m p t o t i c b e h a v i o u r o f scattering a m p l i t u d e has n o t yet b e e n reached. At p r e s e n t o n e c a n n o t n e i t h e r exclude n o r c l a i m that the scaling sets already o n at the C o l l i d e r energy. Nevertheless, the a p p r o x i m a t e scaling at small a n d m e d i u m T allows us to treat the M a r t i n recipe with c o n f i d e n c e which is a posteriori c o n f i r m e d b y a good a g r e e m e n t o f o u r value o f p with that o b t a i n e d i n the analysis restricted to the C o u l o m b - s t r o n g interference region [ 1 ] o n l y ~4. ~4 The best fit to the near forward direction data [ 1 ] using the exponential (in t) imaginary amplitude and the corresponding Martin real part yields: p=0.23, fftot=60.4 mb and the slope b= 15.0 GeV -2. When using instead the standard parametrization with a constant ratio of the real-to-imaginary part one obtains: p=0.23, atot= 60.3 rob, b= 15.7 GeV -2.
References [ 1 ] UA4 Collab., D. Bernard et al., Phys. Lett. B 198 (1987) 583. [2] UA4 Collab., M. Bozzo et al., Phys. Lett. B 155 (1985) 197. [3 ] UA4 Collab., D. Bernard et al., Phys. Lett. B 171 (1986) 142. [4] E. Etim, A. Matecki and L. Satta, Phys. Lett. B 184 (1987) 99. [ 5 ] H.I. Miettinen and J. Pumplin, Phys. Rev. D 18 (1978) 1696. [ 6 ] R. Glauber, Lectures in theoretical physics, Vol. 1, eds. W.E. Brinm and L.G. Dunham (Interscience, New York, t959) p. 315. [7] A. Martin, Lett. Nuovo Cimento 7 ( 1973 ) 8 l 1. [8] A. Matecka et al., to appear. [91 K.R. Schubert, Tables of nucleon-nucleon scattering, in: Landolt-B6rnstem, Numerical data and functional relationship in science and technology, New Series, Vol. 1/ 9a (Springer, Berlin, 1979).
193
PHYSICS LETTERS B
27 April 1989
T H E R E A L A N D I M A G I N A R Y PART O F T H E E L A S T I C S C A T T E R I N G A M P L I T U D E I N H I G H E N E R G Y DIFFRACTION A. M A L E C K I
Department of TheoreticalPhysics,lnsntute of NuclearPhystcs,Radzzkowskiego152, PL-31-342 Cracow,Poland Received 3 December 1988; revised manuscript received 28 January 1989
We calculate the imaginary and real part of the elastic scattering amplitude in the model of high energy diffraction based on the concept of equivalence of states. The application to p!Oscattering at the CERN Collider shows that a large comribution of the real part is consistent with the lack of a pronounced minimum in the momentum transfer distribution.
In a recent measurement at the C E R N Collider [ 1 ] o f p r o t o n - a n t i p r o t o n elastic scattering in the near forward direction a large value for the ratio of the realto-imaginary part of the hadronic amplitude p = 0.24 + 0.04 was found. Previous measurements at the Collider [2,3] at the CM energy V / s = 5 4 6 - 6 3 0 GeV have revealed the presence of a break, rather than a dip, in the m o m e n t u m transfer distribution at t = i - 0 . 9 GeV 2. / The purpose ofthis~ letter is to point out that these two facts are intimately connected to each other and an astonishingly large value o f p could already be anticipated from the lack o f a pronounced m i n i m u m in the elastic differential cross section. This will be shown on the basis o f a recently formulated model o f high energy diffraction [ 4 ]. The model introduces the concept of equivalence of states, that is, the approximate degeneracy o f certain states having d o m i n a n t internal q u a n t u m numbers corresponding to the elastic channel. The amplitude o f elastic diffraction in the model reads
T=t,+g(tav-t,),
(1)
and is to be considered in the Bjorken-type limit where the coupling constant g ~ oo and t a r - t,~0, such
that the second term in ( 1 ) remains finite ~J, ti, tav being the eigenvalue of the scattering operator in the initial state [i) and the average value o f these eigenvalues in the set of equivalent physical states
t.~= E t,i~o,i2,
(2)
J
respectively. The weighting factors ~oj in (2) are interpreted in the model [ 4 ] as wave functions o f equivalent physical states IJ ) - Constructing the states equivalent to the proton as the states build o f the proton and n partons (n = 1, 2, 3, ...) [ 5 ] one has in the impact parameter representation
I~ojl2~l~o,,(t~.... ,t,,)[z=P, f l
I~t(tj)l 2,
J=l
with I ~v(b)[2=exp( -b2/2R~)/2gR~, *on = exp ( - ~) (tT) ~/n!,
(3)
t~ being the mean number of partons and R~ is the radius of their distribution. The model is completed by assuming the func~' Eq. ( 1) originates from the definition of inelastic diffraction as a complete sum of quasi-elastic (t~v- t, being infinitesimal) transitions. The limit g ~ serves to compensate the replacement of a complete wave function expansion with the ground state contribution alone.
0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )
191
Volume 221, number 2
PHYSICS LETTERS B
tional dependence of the eigenvalues tj on the impact parameter. One takes [4,5 ] t, =i/'o (b) =irro exp( -b2/2R2)/4~zR 2,
(4)
and
IO0
27 April 1989 k
'
'
'
'
I\
I
. . . .
II /"
...a
,.A
- "
X
---
The ansatz (5) reflects the Glauber law [ 6 ] of composition of constituent profiles corresponding to point-like partons. Collecting ( 1 ) - (5) together we write the diffractive amplitude of elastic scattering as oo
T(t) =i ~ db b J o ( v / ~ t ( b ) f ' ( b ) ,
(6)
,NAR,
REAL
\\\
i 10-
G-~0.
. . . .
\,.
d
(5)
with 7~(b) = ½G6~'~)(b),
I
p-p ZLASrlc SCArrER/N
I'
=i(1-(1-Fo) fl
. . . .
;''°"kt', X \ '~" 10"2 LLI
t I ---+tn
I
,J
,
,
,
I
,
,
1#
05
I
I
,~,
e
,
I
,
,
,
,
1,5 2 It l (GeV 2]
Fxg. 1. The imaginary and real amplitude of the p15elastic scattering in the function of the squared momentum transfer It[. The parameters of the model, corresponding to the CM energy 546630 GeV, are given in the text. I m T(s, t) ~ I m T(s, 0 ) ¢ ( ~ ) , 3~oo
r(s, t)= I t[ atot(S), Crtotbeing the total cross section. In our model the scaling function reads
0
with ¢ ( t ) = (1/atot)[no e x p ( ½ R 2 t ) + a , exP(½R2t)
F( b ) =go + F, -FoF,,
(7) where
/~, (b) =rr, exp( -b2/2R2)/4zcR2~,
(8)
and a~=
lim
gaG.
(9)
Thus effectively our diffractive model is determined by the two radii Ro, R~ and the two cross sections rr0, O"1.
It should be pointed out that the imaginary amplitude (6) is characterized by only a single zero (see the solid curve in fig. 1 ) arising from the interference of the two profiles/~o and/'~ in eq. (7). In order to fill up this zero we need to introduce a real part of the amplitude. This can be done using the recipe of Martin [ 7 ] based on asymptotic scaling: Re T(s, t) = R e T(s, 0) ( ¢ + t d ¢ / d t ) , while 192
(11)
- G 2 exp (½R22t)],
where
atot =Go + Gl -o'2,
(12)
rr2 =Goa, (R 2 + R 2) - ' / 4 ~ ,
(13)
2 2l (Ro2 +R 2) =RoR
(14)
R 22
l
All the parameters of the model, together with the ratio p of the real-to-imaginary part of the hadronic amplitude in the forward direction, were established from fitting the elastic differential cross section (with properly included the Coulomb amplitude) to the experimental data [ 1-3 ]. One obtains thus for p - ~ scattering at 546-630 OeV: ~ro= 55.56 rob, el = 10.77 mb, R0=0.76 fro, R~=0.51 fro, p = 0 . 2 0 ~2. The resulting total cross section atot = 60.64 mb and the integrated elastic cross section Gx= 13.39 rob. We anticipate the discussion of inelastic diffraction [ 8 ] by
(10) ~2 The value o f z 2 for the best fit is 193 with 130 experimental points.
Volume 221, number 2 ....
PHYSICS LETTERS B
i ....
102 ~
I ....
i ....
i ....
I ....
P - P ELASTIC SCATTERING 546- 630 GeV
-, I0 c
~=~
p: 0.2
.,
.Q
.~.
.......
p-- o.o
I0-';
,0"
i: ....
I
0.5
,
,
,
I
/
,
,
,
,
f
1.5
. . . .
I
. . . .
2 Itl
I
,
,
2.5 (GeV z]
,'?"
3
Fig. 2. The pp elastic differential cross section in the function of the squared momentum transfer It[. The experimental data are compared with the results of the model described in the text, with and without the real hadronic amplitude. n o t i c i n g that the integrated cross section o f inelastic d i f f r a c t i o n ~a,f= a~. Fig. 1 presents the real a n d i m a g i n a r y parts o f the elastic scattering a m p l i t u d e while fig. 2 c o n f r o n t s the e x p e r i m e n t a l data with the results o f o u r model, with a n d w i t h o u t the real part. T h e role o f the real part in filling u p the diffractive m i n i m u m is there readily illustrated. We have r e p e a t e d the s a m e analysis in the case o f the pp ~3 elastic scattering at the ISR energy o f 53 G e V where the diffractive m i n i m u m is m u c h m o r e pron o u n c e d [ 9 ]. T h e o b t a i n e d parameters are ~o = 39.31 mb, ~, = 5.61 rob, R o = 0.70 fm, R~ = 0 . 4 1 fm, p = 0.06. It is i n t e r e s t i n g to check the u n i v e r s a l i t y o f the funct i o n ~(T). It t u r n s out that the scaling f u n c t i o n s for the ISR a n d Collider energies agree q u i t e well, w i t h i n ~a We treat diffraction in p~ and pp collisions on the same footmg Ignoring thus any odd-under-crossing contributions.
27 April 1989
a n error o f 2 0 - 3 0 % , u p to z = 110. However, for higher v a l u e s o f T, a n d especially i n the region o f the s e c o n d m a x i m u m , their d i s c r e p a n c y b e c o m e s s u b s t a n t i a l which m e a n s that at the I S R energies the a s y m p t o t i c b e h a v i o u r o f scattering a m p l i t u d e has n o t yet b e e n reached. At p r e s e n t o n e c a n n o t n e i t h e r exclude n o r c l a i m that the scaling sets already o n at the C o l l i d e r energy. Nevertheless, the a p p r o x i m a t e scaling at small a n d m e d i u m T allows us to treat the M a r t i n recipe with c o n f i d e n c e which is a posteriori c o n f i r m e d b y a good a g r e e m e n t o f o u r value o f p with that o b t a i n e d i n the analysis restricted to the C o u l o m b - s t r o n g interference region [ 1 ] o n l y ~4. ~4 The best fit to the near forward direction data [ 1 ] using the exponential (in t) imaginary amplitude and the corresponding Martin real part yields: p=0.23, fftot=60.4 mb and the slope b= 15.0 GeV -2. When using instead the standard parametrization with a constant ratio of the real-to-imaginary part one obtains: p=0.23, atot= 60.3 rob, b= 15.7 GeV -2.
References [ 1 ] UA4 Collab., D. Bernard et al., Phys. Lett. B 198 (1987) 583. [2] UA4 Collab., M. Bozzo et al., Phys. Lett. B 155 (1985) 197. [3 ] UA4 Collab., D. Bernard et al., Phys. Lett. B 171 (1986) 142. [4] E. Etim, A. Matecki and L. Satta, Phys. Lett. B 184 (1987) 99. [ 5 ] H.I. Miettinen and J. Pumplin, Phys. Rev. D 18 (1978) 1696. [ 6 ] R. Glauber, Lectures in theoretical physics, Vol. 1, eds. W.E. Brinm and L.G. Dunham (Interscience, New York, t959) p. 315. [7] A. Martin, Lett. Nuovo Cimento 7 ( 1973 ) 8 l 1. [8] A. Matecka et al., to appear. [91 K.R. Schubert, Tables of nucleon-nucleon scattering, in: Landolt-B6rnstem, Numerical data and functional relationship in science and technology, New Series, Vol. 1/ 9a (Springer, Berlin, 1979).
193