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Diffraction dissociation in pp collisions at √s=1.8 TeV
Physics Letters B 301 ( 1993) 313 -316 North-Holland
PHYSICS LETTERS B
Diffraction dissociation in pp collisions at v/s= 1.8 TeV E-710 Collaboration N.A. Amos a, C. Avila a,J, W.F. Baker a, M. Bertani b,2, M.M. Block ~, D.A. Dimitroyannis d.3, D.P. Eartly a, R.W. Ellsworth e, G. Giacomelli b, B. Gomez r, J.A. Goodman d, C.M. Guss c.~, B. Hoeneisen g, A.J. Lennox ", M.R. Mondardini b,,.4, j.p. Negret r, j. Orear b, S.M. Pruss ", R. Rubinstein a, S. Sadr c, J.C. Sanabria a,5, S. Shukla a, M. Spagnoli b, I. Veronesi b and S. Zucehelli b • Fermi National Accelerator Laboratory, Batavia, IL 6051 O, USA b Umversith dl Bologna and Istituto Nazionale di Flsica Nucleare, 1-40126 Bologna, Italy Northwestern University, Evanston, IL 60201, USA '~ University of Maryland, College Park, MD 20742, USA © George Mason University, Fatrfax, VI 22030, USA r Umversidad de los Andes, Bogota, Colombia i UmverstdadSan Francisco de Quito, Quito, Ecuador h Cornell University, Ithaca, N Y 14853, USA
Received 5 January 1993
We have studied single diffraction dissociation (/~p--.#X) in proton-antiproton collisions at x/~= 1 8 TeV, covering the ranges 3£Mx<200 GeV and 0.05<1tl <0.11 (GeV/c) 2. Parametenzmg the production to be of the form de/ (dtdM 2) = (M2x)-'exp(bt), we obtain or= I 13_+0.07 and b= 10 5+ 1.8 (GeV/c) -2. The total single diffraction dissociation cross section is 2aso= 8.1 +_1.7 mb Comparisons are made to previous lower energy data, and to an earlier measurement by us at the same energy.
We report here a study o f single d i f f r a c t i o n dissociation, ~p--,#X, at V/~= 1.8 TeV. O u r data cover the range 3 < M x < 2 0 0 G e V a n d 0 . 0 5 < Itl £ 0 . 1 1 ( G e V / c) 2. T h e a p p a r a t u s , s h o w n schematically in fig. 1, is part o f the e q u i p m e n t used to study/~p elastic scattering a n d total cross sections at the F e r m d a b T e v a t r o n C o l l i d e r [ 1-5 ]. D a t a were t a k e n in a small n u m b e r o f special r u n s with a trigger specific for this process. Referring to fig. 1, the trigger r e q u i r e d a particle ( a n t i p r o t o n ) in D e t e c t o r 2, together with at least o n e particle in the a n n u l a r c o u n t e r s R3, R4 or R5 which 1 Present address. Cornell Unlversuy, Ithaca, NY 14853, USA. 2 Present address UmversityofFerrara, 1-44100Ferrara, Italy. 3 Present address. Northwestern University, Evanston, IL 60201, USA 4 Mailing address CERN, CH-1211 Geneva, Switzerland. 5 Present address: George Washington University, Washington, DC 20052, USA. Elsevier Science Publishers B.V
i
°?
Detector 2
A Tevoiron Di~lll
A A
V l\y
A
revotron Ouodrupoll
i
I, q
2,.,./.'0
111
InteraCtion Point
Fig. 1. Schematic side view of the experiment. Detectors 2 and 4 are "Roman Pots" containing draft chambers and scintillation counters R3, R4 and R5 are annular scmtdlatton counters placed around the Tevatron beam pipe. covered the p s e u d o r a p t d i t y range 5.2 ~
Volume 301, number 2,3
PHYSICS LETTERS B
the m o m e n t u m o f each a n t i p r o t o n which d e t e r m i n e d
Mx for each event. The scattering angle, obtained from the antiproton position in Detector 4, gave the four m o m e n t u m transfer t. In our analysis, we follow previous practice (see, for example, ref. [6] ) and assume Mx and t independence in the cross section, i.e. da
dt dM2x =Af(t)g(M 2) ,
(1)
we use
f( t ) =e b` ,
(2)
g(M 2) = (M]¢) -'~ ,
(3)
and use our data to obtain b, a and the total cross section for diffraction dissociation given by dry
2trsD = 2 ~ f dt----d~x dt dM2x .
(4)
where the integrations are carried out over all values o f t and over 2 GeV2 < M 2 < 0.05 s .
(5)
The factors o f two in eq. ( 4 ) g~ve the total diffraction dlssocmtlon cross section, which is the sum o f the diffraction dissociation o f the proton and that o f the ant~proton. These are always taken to be equal; our exp e r i m e n t only measures the dissocmtion o f the proton. The total cross section for diffraction d~ssoc m u o n (eq. ( 4 ) ) is generally written as 2aSD. Previous studies o f diffraction dissociation have been carried out at fixed target accelerators, and the C E R N ISR and SPS colhders. In a d d m o n , we have previously reported ( 3 ) a measurement o f 2 a s o = 11.7+2.3 mb at the Fermflab Tevatron Collider at x / ~ = 1.8 TeV using a different experimental technique to that reported here. Earher studies [ 3,7-16 ] have not always used cons~stent defimtions o f diffraction d~ssociation, g~vlng rise to apparent disagreement between measurements. In a d d m o n , differences in Mx coverage and sometimes indirect m e t h o d s o f measuring Mx, have also made comparisons between experiments difficult. Generally, experiments have found that the Mx dependence is consistent with M ; 2 (Le., a = 1 in eq. ( 3 ) ) , as expected by some theoretical models (see, for example, ref. [ 6 ] ). The Mx range o f the mtegra314
4 March 1993
tlon in eq. (4), given in eq. ( 5 ) , has been used in many experiments, but other ranges have also been used in the past. The t dependence o f diffraction disSOClatlon, if p a r a m e t e r i z e d as in eq. ( 2 ) , has been found to have a value o f b, in the small I t l region, o f about half o f that measured for elastic scattering at the same energy. However, some experiments have found the need for a quadratic exponential term at larger It I, and there is some evidence o f a dependence o f b on M2x, i.e., the simplification represented by eq. ( l ) may not be valid; refs. [ 7 ] contains a discussion o f these features. Our analysis is based on about 13 000 diffraction events. During data taking, we simultaneously observed elastic scattering events using a detector for the scattered proton not shown in fig. l, but described in ref. [ 1,3 ]; elastic events were prescaled by a factor o f 30, and our final elastic total was approximately 2000. In both cases, the numbers o f events are after all selection and fiduclal cuts; the fiducial cuts were m a d e in order to avoid possible detector edge effects. We used the elastic events to normalize the diffraction cross section, since we have previously measured the elastic scattering distribution and cross section at this energy [ 5 ]. Contamination o f the diffraction sample by elastic events could be determined off-line by examination o f hits in the various detectors, and was found to be essentially zero. Other background in the diffraction sample could be determined from the number o f events in the negative (non-physical) M2x region (I.e., the measured p mom e n t u m greater than the beam m o m e n t u m ) and was ~0.25%. The t acceptance o f the data was determined by the vertical placement o f the Detectors 2 and 4; the lower Mx limit was d e t e r m i n e d using Monte Carlo simulation, requiring a hit in at least one o f the three annular counters R3, R4 or Rs. O u r resuits were found to be insensitive to reasonable changes in the Monte Carlo assumptions. O u r upper M x limit o f 200 GeV was due to event statistics rather than acceptance. Our resolutmn in Mx was obtained from elastic events, where the width o f the correlation between hits in Detector 2 and Detector 4 should be zero in the absence o f resolution broadening effects. Our resolution in M2x was ~ + (64 G e V ) 2, and the resolution in t was ~ + 8 × 10 -3 ( G e V / c ) 2. We obtained the x ( h o r i z o n t a l ) and y (vertical) calibrations of the detectors using the calibration
V o l u m e 301, n u m b e r 2,3
PHYSICS LETTERS B
scintillation counter in each detector; these had accurately m a c h i n e d holes m their faces, and several accurately m a c h i n e d edges. The x calibration (based on charge d i v i s i o n ) is known with substantially less accuracy than the y calibration. The d a t a reduction steps needed to arrive at the final event sample were similar to those m our earlier work [ 1-5 ]. The parameters b and ot could each be o b t a i n e d from our data in several ways, and the results o b t a i n e d were consistent within the errors. F o r example, ot could be derived from the correlation between the x coordinates o f events in Detectors 2 and 4, together with knowledge o f the transfer functions o f the accelerator magnetic lattice between the two detectors. The value o f b could be o b t a i n e d from the x or y distributions o f events in either Detector 2 or Detector 4, as could also B (the slope o f the elastic scattering distribuu o n ) . F~g. 2 shows the observed M 2 distribution o f single diffractive events along with predicted curves for o t = l . 0 3 , 1.13, and 1.23. Fig. 3 shows the observed t d~stribution m Detector 4 along w~th predicted curves for b = 5.5, 10.5 and 15.5 GeV -2. These predicted curves are Monte Carlo generated using eq. ( 1 ), with the measured x and y resolutions folded m and taking into account the fiducml cuts; they are normalized to the total n u m b e r o f events.
0 i Oi
i
l
i
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Ftg. 2 Dlstnbutlon o f diffraction events versus M2x. The curves are those predicted for the values o f or shown, see text for detads
4 March 1993
i
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Fig. 3. Distnbuuon o f diffraction events versus --t. The curves are those predicted for the values o f b shown, see text for detads.
The m e t h o d used to obtain our f n a l values o f a and b combines all the mformat~on into o n e j o l n t twop a r a m e t e r least squares analysis. The predicted (x2, y 2 ) , (x4, y4) and (x2, x 4 ) two dimensional distributions o f events are Monte Carlo generated for given values o f ot and b using eqs. ( 1 ) - ( 3 ) (taking into account the measured x and y resolution smearing). The total ch~-squared for the three d~stributions ~s evaluated, and then a and b are varied to find the least squares solution and statistical errors in ot and b. As a check on our x calibrations o f the detectors, these were also varied. The solutions for a and b are almost independent o f the cahbratlon constants; the minim u m ch~-squared gave cahbration constants within 5% o f those measured. O u r final chl-squared per degree o f freedom was 3.5 for all 643 data points. This high value is very probably due to a breakdown o f the assumption that Mx and t are independent in the cross section, as given in eq. (1). F o r the data points corresponding to Mx > 65 GeV the solution for b was 4 + 3 ( G e V / c ) - 2 as opposed to 10.5 _+ 1.8 ( G e V / c ) -2 when all the data are l u m p e d together. Cuts were m a d e in the data to a v m d edge effects; their effect was checked by repeated analyses using less conservative cuts. We note that at the lowest values o f Mx, the Mx distribution observed m lower energy experiments is not a simple power law, but shows resonances b u m p s near the low mass cutoff in eq. (5). If we use the most extreme a m o u n t o f those b u m p s allowed by existing data, our 315
Volume 301, number 2,3
PHYSICS LETTERS B
cross section would mcrease by 7%. Our quoted error of 18% includes this uncertainty and also the uncertainty caused by the lower Mx cutoffused in eq. ( 5 ). Our solution for the elastic slope parameter using the above procedures is B = 17.9+2.5 ( G e V / c ) -2 which agrees well with our published value of 17.0_+0.5 ( G e V / c ) -2 [5]. This gives us confidence in the results for ot and b. Our final errors are obtained by combining quadratically our systematic and statistical errors, with the major contribution being systematic. Our results are or= 1 . 1 3 + 0 . 0 7 , b = 10.5 _+ 1.8 ( G e V / c ) - : ,
2tr,a/trel = 0 . 4 9 _+0 . 0 9 . Taking tre~= 16.6-+ 1.6 m b from ref. [3] we obtain 2a, a=8.1 + 1.7 mb. This result is plotted in fig. 4, along with previous results from lower energies [ 7 15 ] and our earlier result [ 3 ] at this energy obtained using a different experimental method. We can make the following observation about our results. (i) The value of b observed for diffraction disso-
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Fig 4 The total single d]ffractton cross secuon result from th~s
experiment together with previous pp [7-12] and 0p [ 13-15] data, and our earher result [ 3 ].
316
This work was supported by the US D e p a r t m e n t of Energy, the US N a u o n a l Sctence Foundation, the Italian Mlnisterio Pubbhca lstruzlone, Colciencias and Centro lnternacional de Flsica in Colombia, and the North Atlantic Treaty Orgamzation.
References
c~
ll,,,l,l
ciation ( b = 10.5+ 1.8 ( G e V / c ) - 2 ) is about half of the elastic slope B at the same energy ( B = 17.0 + 0.5 ( G e V / c ) - 2 ) , as has been found previously at lower energies. There is some indication that b becomes smaller at higher M2x, as has been found at lower energies. (it) The Mx dependence observed ( o t = l . 1 3 + 0.07, o r M~g-226±0 14)) is similar to the ~ M £ 2 observed at lower energies [ 16], and to the M ~ 2 predicted by some models. (lit) Our value for 2aSD, 8.1 + 1.7 mb, is consistent with our earlier value [3], 11.7"+2.3 mb, obtained using a completely different technique. Since they are independent measurements, we can combine them to give a value for 2aSD at x / s = 1.8 TeV of 9.4 _+ 1.4 mb. As seen m fig. 4, it is difficult to draw conclusions on the energy dependence of 2aso, although the data are consistent with an increase of 2asp with energy at roughly the same rate as the #p total cross section.
,,,,,,
• This Expetlmenl
if)
qO2
4 March 1993
[ 1] N A. Amos et al., Phys. Rev. Lett 61 ( 1988 ) 525. [2 ] N A Amoset al., Phys. Rev Lett 63 (1989) 2784 [3] N A. Amos et al., Phys. Left. B 243 (1990) 158. [4] N A. Amos et al., Phys. Lett. B 247 (1990) 127 [ 5 ] N.A Amoset al., Phys Rev Lett 68 (1992) 2433. [6l K. Gouhanos, Phys. Rep. 101 (1983) 169 [71 M G. Albrowet al, Nucl Phys. B 108 (1976) I. [8] J.W Chapman et al, Phys. Rev Lett 32 (1974) 257 [9] J Schambergeret al., Phys Rev Lett. 34 (1975) 1121 [ 10] S Bartsh et al., Phys Rev D 9 (1974) 2689 [ I 1] J.C M. Armltageet al., Nucl. Phys B 194 (1982) 365 [12]J Whltmore et al, Phys. Rep. 10 (1974) 273 [ 13 ] R.E Ansorgeet al., Z Phys. C 33 (1986) 175. [ 14] G J. Alner et al, Phys. Rep 154 ( 1987 ) 247. [15] D. Bernard el al, Phys Lett B 186 (1987) 227. [161M Bozzo, Phys. Lett. B 136 (1984) 217.
PHYSICS LETTERS B
Diffraction dissociation in pp collisions at v/s= 1.8 TeV E-710 Collaboration N.A. Amos a, C. Avila a,J, W.F. Baker a, M. Bertani b,2, M.M. Block ~, D.A. Dimitroyannis d.3, D.P. Eartly a, R.W. Ellsworth e, G. Giacomelli b, B. Gomez r, J.A. Goodman d, C.M. Guss c.~, B. Hoeneisen g, A.J. Lennox ", M.R. Mondardini b,,.4, j.p. Negret r, j. Orear b, S.M. Pruss ", R. Rubinstein a, S. Sadr c, J.C. Sanabria a,5, S. Shukla a, M. Spagnoli b, I. Veronesi b and S. Zucehelli b • Fermi National Accelerator Laboratory, Batavia, IL 6051 O, USA b Umversith dl Bologna and Istituto Nazionale di Flsica Nucleare, 1-40126 Bologna, Italy Northwestern University, Evanston, IL 60201, USA '~ University of Maryland, College Park, MD 20742, USA © George Mason University, Fatrfax, VI 22030, USA r Umversidad de los Andes, Bogota, Colombia i UmverstdadSan Francisco de Quito, Quito, Ecuador h Cornell University, Ithaca, N Y 14853, USA
Received 5 January 1993
We have studied single diffraction dissociation (/~p--.#X) in proton-antiproton collisions at x/~= 1 8 TeV, covering the ranges 3£Mx<200 GeV and 0.05<1tl <0.11 (GeV/c) 2. Parametenzmg the production to be of the form de/ (dtdM 2) = (M2x)-'exp(bt), we obtain or= I 13_+0.07 and b= 10 5+ 1.8 (GeV/c) -2. The total single diffraction dissociation cross section is 2aso= 8.1 +_1.7 mb Comparisons are made to previous lower energy data, and to an earlier measurement by us at the same energy.
We report here a study o f single d i f f r a c t i o n dissociation, ~p--,#X, at V/~= 1.8 TeV. O u r data cover the range 3 < M x < 2 0 0 G e V a n d 0 . 0 5 < Itl £ 0 . 1 1 ( G e V / c) 2. T h e a p p a r a t u s , s h o w n schematically in fig. 1, is part o f the e q u i p m e n t used to study/~p elastic scattering a n d total cross sections at the F e r m d a b T e v a t r o n C o l l i d e r [ 1-5 ]. D a t a were t a k e n in a small n u m b e r o f special r u n s with a trigger specific for this process. Referring to fig. 1, the trigger r e q u i r e d a particle ( a n t i p r o t o n ) in D e t e c t o r 2, together with at least o n e particle in the a n n u l a r c o u n t e r s R3, R4 or R5 which 1 Present address. Cornell Unlversuy, Ithaca, NY 14853, USA. 2 Present address UmversityofFerrara, 1-44100Ferrara, Italy. 3 Present address. Northwestern University, Evanston, IL 60201, USA 4 Mailing address CERN, CH-1211 Geneva, Switzerland. 5 Present address: George Washington University, Washington, DC 20052, USA. Elsevier Science Publishers B.V
i
°?
Detector 2
A Tevoiron Di~lll
A A
V l\y
A
revotron Ouodrupoll
i
I, q
2,.,./.'0
111
InteraCtion Point
Fig. 1. Schematic side view of the experiment. Detectors 2 and 4 are "Roman Pots" containing draft chambers and scintillation counters R3, R4 and R5 are annular scmtdlatton counters placed around the Tevatron beam pipe. covered the p s e u d o r a p t d i t y range 5.2 ~
Volume 301, number 2,3
PHYSICS LETTERS B
the m o m e n t u m o f each a n t i p r o t o n which d e t e r m i n e d
Mx for each event. The scattering angle, obtained from the antiproton position in Detector 4, gave the four m o m e n t u m transfer t. In our analysis, we follow previous practice (see, for example, ref. [6] ) and assume Mx and t independence in the cross section, i.e. da
dt dM2x =Af(t)g(M 2) ,
(1)
we use
f( t ) =e b` ,
(2)
g(M 2) = (M]¢) -'~ ,
(3)
and use our data to obtain b, a and the total cross section for diffraction dissociation given by dry
2trsD = 2 ~ f dt----d~x dt dM2x .
(4)
where the integrations are carried out over all values o f t and over 2 GeV2 < M 2 < 0.05 s .
(5)
The factors o f two in eq. ( 4 ) g~ve the total diffraction dlssocmtlon cross section, which is the sum o f the diffraction dissociation o f the proton and that o f the ant~proton. These are always taken to be equal; our exp e r i m e n t only measures the dissocmtion o f the proton. The total cross section for diffraction d~ssoc m u o n (eq. ( 4 ) ) is generally written as 2aSD. Previous studies o f diffraction dissociation have been carried out at fixed target accelerators, and the C E R N ISR and SPS colhders. In a d d m o n , we have previously reported ( 3 ) a measurement o f 2 a s o = 11.7+2.3 mb at the Fermflab Tevatron Collider at x / ~ = 1.8 TeV using a different experimental technique to that reported here. Earher studies [ 3,7-16 ] have not always used cons~stent defimtions o f diffraction d~ssociation, g~vlng rise to apparent disagreement between measurements. In a d d m o n , differences in Mx coverage and sometimes indirect m e t h o d s o f measuring Mx, have also made comparisons between experiments difficult. Generally, experiments have found that the Mx dependence is consistent with M ; 2 (Le., a = 1 in eq. ( 3 ) ) , as expected by some theoretical models (see, for example, ref. [ 6 ] ). The Mx range o f the mtegra314
4 March 1993
tlon in eq. (4), given in eq. ( 5 ) , has been used in many experiments, but other ranges have also been used in the past. The t dependence o f diffraction disSOClatlon, if p a r a m e t e r i z e d as in eq. ( 2 ) , has been found to have a value o f b, in the small I t l region, o f about half o f that measured for elastic scattering at the same energy. However, some experiments have found the need for a quadratic exponential term at larger It I, and there is some evidence o f a dependence o f b on M2x, i.e., the simplification represented by eq. ( l ) may not be valid; refs. [ 7 ] contains a discussion o f these features. Our analysis is based on about 13 000 diffraction events. During data taking, we simultaneously observed elastic scattering events using a detector for the scattered proton not shown in fig. l, but described in ref. [ 1,3 ]; elastic events were prescaled by a factor o f 30, and our final elastic total was approximately 2000. In both cases, the numbers o f events are after all selection and fiduclal cuts; the fiducial cuts were m a d e in order to avoid possible detector edge effects. We used the elastic events to normalize the diffraction cross section, since we have previously measured the elastic scattering distribution and cross section at this energy [ 5 ]. Contamination o f the diffraction sample by elastic events could be determined off-line by examination o f hits in the various detectors, and was found to be essentially zero. Other background in the diffraction sample could be determined from the number o f events in the negative (non-physical) M2x region (I.e., the measured p mom e n t u m greater than the beam m o m e n t u m ) and was ~0.25%. The t acceptance o f the data was determined by the vertical placement o f the Detectors 2 and 4; the lower Mx limit was d e t e r m i n e d using Monte Carlo simulation, requiring a hit in at least one o f the three annular counters R3, R4 or Rs. O u r resuits were found to be insensitive to reasonable changes in the Monte Carlo assumptions. O u r upper M x limit o f 200 GeV was due to event statistics rather than acceptance. Our resolutmn in Mx was obtained from elastic events, where the width o f the correlation between hits in Detector 2 and Detector 4 should be zero in the absence o f resolution broadening effects. Our resolution in M2x was ~ + (64 G e V ) 2, and the resolution in t was ~ + 8 × 10 -3 ( G e V / c ) 2. We obtained the x ( h o r i z o n t a l ) and y (vertical) calibrations of the detectors using the calibration
V o l u m e 301, n u m b e r 2,3
PHYSICS LETTERS B
scintillation counter in each detector; these had accurately m a c h i n e d holes m their faces, and several accurately m a c h i n e d edges. The x calibration (based on charge d i v i s i o n ) is known with substantially less accuracy than the y calibration. The d a t a reduction steps needed to arrive at the final event sample were similar to those m our earlier work [ 1-5 ]. The parameters b and ot could each be o b t a i n e d from our data in several ways, and the results o b t a i n e d were consistent within the errors. F o r example, ot could be derived from the correlation between the x coordinates o f events in Detectors 2 and 4, together with knowledge o f the transfer functions o f the accelerator magnetic lattice between the two detectors. The value o f b could be o b t a i n e d from the x or y distributions o f events in either Detector 2 or Detector 4, as could also B (the slope o f the elastic scattering distribuu o n ) . F~g. 2 shows the observed M 2 distribution o f single diffractive events along with predicted curves for o t = l . 0 3 , 1.13, and 1.23. Fig. 3 shows the observed t d~stribution m Detector 4 along w~th predicted curves for b = 5.5, 10.5 and 15.5 GeV -2. These predicted curves are Monte Carlo generated using eq. ( 1 ), with the measured x and y resolutions folded m and taking into account the fiducml cuts; they are normalized to the total n u m b e r o f events.
0 i Oi
i
l
i
0005 I
I
I
S000
i
w
0.010 I
•
-----
(2- 1.13
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I
.. I
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°,,
w soo o
E 7
10C
"
I
0
I
I
I0000 20000 Mx2 ( G e V 2 )
I
50000
Ftg. 2 Dlstnbutlon o f diffraction events versus M2x. The curves are those predicted for the values o f or shown, see text for detads
4 March 1993
i
i
~
.~_-
i
1
0
1500
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30C
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b-105
....
b " 15 5 - - "
I 0 (~5
--
I
\
I
I
I
0 I10
I
-t (GeV/c)2
Fig. 3. Distnbuuon o f diffraction events versus --t. The curves are those predicted for the values o f b shown, see text for detads.
The m e t h o d used to obtain our f n a l values o f a and b combines all the mformat~on into o n e j o l n t twop a r a m e t e r least squares analysis. The predicted (x2, y 2 ) , (x4, y4) and (x2, x 4 ) two dimensional distributions o f events are Monte Carlo generated for given values o f ot and b using eqs. ( 1 ) - ( 3 ) (taking into account the measured x and y resolution smearing). The total ch~-squared for the three d~stributions ~s evaluated, and then a and b are varied to find the least squares solution and statistical errors in ot and b. As a check on our x calibrations o f the detectors, these were also varied. The solutions for a and b are almost independent o f the cahbratlon constants; the minim u m ch~-squared gave cahbration constants within 5% o f those measured. O u r final chl-squared per degree o f freedom was 3.5 for all 643 data points. This high value is very probably due to a breakdown o f the assumption that Mx and t are independent in the cross section, as given in eq. (1). F o r the data points corresponding to Mx > 65 GeV the solution for b was 4 + 3 ( G e V / c ) - 2 as opposed to 10.5 _+ 1.8 ( G e V / c ) -2 when all the data are l u m p e d together. Cuts were m a d e in the data to a v m d edge effects; their effect was checked by repeated analyses using less conservative cuts. We note that at the lowest values o f Mx, the Mx distribution observed m lower energy experiments is not a simple power law, but shows resonances b u m p s near the low mass cutoff in eq. (5). If we use the most extreme a m o u n t o f those b u m p s allowed by existing data, our 315
Volume 301, number 2,3
PHYSICS LETTERS B
cross section would mcrease by 7%. Our quoted error of 18% includes this uncertainty and also the uncertainty caused by the lower Mx cutoffused in eq. ( 5 ). Our solution for the elastic slope parameter using the above procedures is B = 17.9+2.5 ( G e V / c ) -2 which agrees well with our published value of 17.0_+0.5 ( G e V / c ) -2 [5]. This gives us confidence in the results for ot and b. Our final errors are obtained by combining quadratically our systematic and statistical errors, with the major contribution being systematic. Our results are or= 1 . 1 3 + 0 . 0 7 , b = 10.5 _+ 1.8 ( G e V / c ) - : ,
2tr,a/trel = 0 . 4 9 _+0 . 0 9 . Taking tre~= 16.6-+ 1.6 m b from ref. [3] we obtain 2a, a=8.1 + 1.7 mb. This result is plotted in fig. 4, along with previous results from lower energies [ 7 15 ] and our earlier result [ 3 ] at this energy obtained using a different experimental method. We can make the following observation about our results. (i) The value of b observed for diffraction disso-
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experiment together with previous pp [7-12] and 0p [ 13-15] data, and our earher result [ 3 ].
316
This work was supported by the US D e p a r t m e n t of Energy, the US N a u o n a l Sctence Foundation, the Italian Mlnisterio Pubbhca lstruzlone, Colciencias and Centro lnternacional de Flsica in Colombia, and the North Atlantic Treaty Orgamzation.
References
c~
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ciation ( b = 10.5+ 1.8 ( G e V / c ) - 2 ) is about half of the elastic slope B at the same energy ( B = 17.0 + 0.5 ( G e V / c ) - 2 ) , as has been found previously at lower energies. There is some indication that b becomes smaller at higher M2x, as has been found at lower energies. (it) The Mx dependence observed ( o t = l . 1 3 + 0.07, o r M~g-226±0 14)) is similar to the ~ M £ 2 observed at lower energies [ 16], and to the M ~ 2 predicted by some models. (lit) Our value for 2aSD, 8.1 + 1.7 mb, is consistent with our earlier value [3], 11.7"+2.3 mb, obtained using a completely different technique. Since they are independent measurements, we can combine them to give a value for 2aSD at x / s = 1.8 TeV of 9.4 _+ 1.4 mb. As seen m fig. 4, it is difficult to draw conclusions on the energy dependence of 2aso, although the data are consistent with an increase of 2asp with energy at roughly the same rate as the #p total cross section.
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[ 1] N A. Amos et al., Phys. Rev. Lett 61 ( 1988 ) 525. [2 ] N A Amoset al., Phys. Rev Lett 63 (1989) 2784 [3] N A. Amos et al., Phys. Left. B 243 (1990) 158. [4] N A. Amos et al., Phys. Lett. B 247 (1990) 127 [ 5 ] N.A Amoset al., Phys Rev Lett 68 (1992) 2433. [6l K. Gouhanos, Phys. Rep. 101 (1983) 169 [71 M G. Albrowet al, Nucl Phys. B 108 (1976) I. [8] J.W Chapman et al, Phys. Rev Lett 32 (1974) 257 [9] J Schambergeret al., Phys Rev Lett. 34 (1975) 1121 [ 10] S Bartsh et al., Phys Rev D 9 (1974) 2689 [ I 1] J.C M. Armltageet al., Nucl. Phys B 194 (1982) 365 [12]J Whltmore et al, Phys. Rep. 10 (1974) 273 [ 13 ] R.E Ansorgeet al., Z Phys. C 33 (1986) 175. [ 14] G J. Alner et al, Phys. Rep 154 ( 1987 ) 247. [15] D. Bernard el al, Phys Lett B 186 (1987) 227. [161M Bozzo, Phys. Lett. B 136 (1984) 217.