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Dalitz decay. π0 → γe+e− and the π0 electromagnetic transition form factor
Volume 233, number 1,2
PHYSICS LETTERS B
21 December 1989
D A L I T Z DECAY. n°~"/e+e- A N D T H E n o E L E C T R O M A G N E T I C T R A N S I T I O N F O R M F A C T O R H F O N V I E I L L E ", N. BENSAYAH a, j. B E R T H O T a, p B E R T I N a, B. B I H O R E A U b M C R O U A U ", G F O U R N I E R b, M G I F F O N ¢, J M I L L E R b, R. N A H A B E T I A N c, B S A G H A I b, C S A M O U R b and P VERNIN b Laboratotre de Physique CorpusculaweIN2P3-CNRS, UmversttOBlaise Pascal, F-63177Aubtbre Cedex, France b InstltutdePhvstqueNuclOmreIN2P3-CNRS, UmversttdClaudeBernard,Lyon-l, F-69622 VdleurbanneCedex, France DPhN, Centre d'Etudes Nuclbatresde Saclay, F-91191 Gzfisur-Yvette Cedex, France Received 21 July 1989
We have studied the no Dalltz decay n°~ye+e - using the pmn beam and a magnetic pmr spectrometer at the Saclay Linear Accelerator A shape analysis of the (e+e - ) mvarlant mass squared x, carried out on 32 000 events, gwes a direct measurement of the n° electromagneUc transmon form factor squared IF(x) 12,and of its slope parameter a m the expressmn F(x) = 1+ ax We find a= - 0 11 +0 03(stat ) +0 08(syst )m~-o2
We have p e r f o r m e d a high statistics experiment at the Saclay Linear Accelerator investigating both the following reactmns with stopped plons" ( n - p ~ ne+e - ) ( 1 ) and n - p - . n ° n , n°--.'/e+e - (2). In this letter we present the results on the n o form factor obtained from the study o f the n o D a h t z decay The results for the inverse plon electroproductlon process ( 1 ) are reported in ref [ 1 ] We first discuss the mot~vatmn for studying process ( 2 ) , n°-~Te+e - , and estabhsh some conventional notations If the n o were a polnthke p a m c l e , its electromagnetic decay would be entirely described by Q E D [2] To take into account an internal structure at the (n°~Y1Y2) vertex, a quantity F called the n o electromagnetic transition form factor is factorized in the a m p l i t u d e A great n u m b e r o f theoretical predictions are available for F from V D M up to recent models based on the n o quark structure (see the global review in ref [3] ) One defines F as a functmn o f the involved squared masses, F = F ( k 2, k 2, m2o), with the n o r m a l i z a t i o n F ( 0 , 0, m20) = 1 F m a y differ from 1 as one o f the photons goes off mass shell, 1.e in process ( 2 ) In this case, one conventionally writes F ( k 2, O, m ~ o ) = F ( x ) , where kl2=kl02 _ kE~-4E+E_ sinE(10) 1S the p h o t o n mass or the (e+e - ) lnvarlant mass squared, and x=k~/m,~o,Z 2
( r = 4 m 2 / m ~ o )
(dp~
dx-,,/~
X [ F ( x ) I2'
(1)
QED
(dp) 2~1 ( r)( d-x Q E D - - ' ~ X ( l - X ) 3 1+~X
~,/2 l--r/
(2)
per (n°~73,) event [2] and or= 1/137 The form factor is usually p a r a m e t n z e d by the linear expression
F ( x ) = 1 +IX,
(3)
where the slope p a r a m e t e r a is the quantity o f interest Due to the low-x enhancement o f d p / d x , see eq ( 2 ) , the effect o f a m o d e r a t e form factor slope would a m o u n t to a ~ 1% change (or less) in the integrated rate Thus it is hopeless to get any r e f o r m a t i o n on a from the D a h t z decay branching ratio, given the accuracy ( + 4%) o f the available experimental values [4,5] To d e t e r m i n e the slope o f the form factor it IS necessary to select events in a relatively hlgh-x region (roughly above 0 1 ) F r o m this starting p o i n t several experiments have been performed, extracting a e~ther
0370-2693/89/$03 50© ElsevlerSclencePubhshersB V (North-Holland)
65
Volume 233, number 1,2
PHYSICS L E T T E R S B
21 D e c e m b e r 1989
Table 1 Values of the slope parameter a measured from (~t°--,Te+e - ) experiments The n ° are produced from ( n - p ) at rest except in ref [6] where the production channel is K + ~ g + n o Authors
a values (m~-02 )
Analysis
Apparatus
Saclay experiment
- 0 11 _ 0 03(stat ) 4-0 08 ( s y s t )
shape
magnetic spectrometer + drift chambers
Gumphnger et al [ 6 ]
- 0 01 + 0 08 - 0 06 (total) (star = 4-0 03)
rate
NaI spectrometer +wire chambers
F~scher et al [7]
+ 0 104-0 03(stat only)
shape
magnetic spectrometer M W P C + (~erenkov
Burger et al [ 8 ]
+ 0 02 + 0 10 (total) (stat = _+0 07 )
shape
magnetic spectrometer + spark chambers
Devons et al [ 9 ]
+ 0 01 4- 0 11 (total) (stat = _+0 05 )
shape + rate
Nal spectrometer + spark chambers
Kobrak [ 10 ]
- 0 15 _+0 10 (stat only)
shape
Llqmd hydrogen bubble chamber
Sam~os [4]
- 0 24_+0 16(total) (star = _+0 12 )
rate
L~quld hydrogen bubble chamber
from a partial rate measurement or a shape analysis o f the x spectrum. The present expertmental situation is not clear (see table l ), the measured a values range from + 0 l 0 to - 0 24 m~-o2, and the two recent measurements having the best statistical precision [ 6,7 ] give conflicting results In the experiment described here, we have proceeded to a new determination of the rc° form factor To avoid an absolute normahzation o f the experiment, the information is extracted from a shape analysis of the ( e + e - ) x spectrum Process ( 1 ) constitutes a background to the 7t° Dalltz decay events and is studied simultaneously In ref [ 1 ] we present a detailed description of the apparatus, tracking program and method used in the global analysis Fig 1 shows how the ( e + e - ) events from processes ( 1 ) and ( 2 ) are identified in the pair e n e r g y - m o m e n t u m plot We obtain 36 000 events reconstructed in the kinematical region of process ( 2 ) , constituting sample II Sample III contains 16 000 same-sign pairs clearly identified by their charge sign. The geometrical acceptance is calculated by a Monte Carlo simulation generating lepton parrs according to processes ( 1 ) and ( 2 ) and the physical background [ l l ] Indeed, in sample II the no Dahtz decay events are contaminated by the following r e a c t i o n s 66
(a) n - p - , n e + e - at rest, (b) n ° - , e + e - e + e - , l e double-Dahtz decay, ( c ) n°-,T) ,, T--,leptons via pair production and
7~-p
>
>
ne+e -
:~120
Jj ,~
+
\
7 Ld o 8o Od <: EL
40 / \
., \
/ \
40
80
\
/
/
I I I I
120 PAIR ENERGY ( M e V
)
Fig 1 Eventselectlonlnthepalrenergy-momentumplot Dashed hnes are the theoretical domains of processes ( 1 ) and (2) Samples I and II are separated along the (,J) hne of the equation P=-I 69E+261 (MeV)
Volume 233, number 1,2
PHYSICS LETTERS B
C o m p t o n scattering in the target Reaction (a) contaminates process (2) mainly because of radiative processes Reactions (b) and (c) produce two kinds of detected pairs opposite-sign [giving an e+e - background to process ( 2 ) ] and same-sign pairs This latter sample (III) cannot lead to physics measurements because of the lack of absolute normalization and the mixture o f all the involved reactions However, the Monte Carlo study of the latter events provides an indirect check o f the (e+e - ) background to process (2) As an output of the simulation, contaminations in sample II are found to be ~ 4 % from reaction (a) and ~ 4 % from reactions (b) + (c). In comparison, the "slope-dependent" part of the Dahtz decay rate computed with our final a value, contributes to ~ 6% (in negative sign) of the counts observed in this sample Radiative corrections have been included in a similar way for processes ( 1 ) and (2) [ 11 ], considering that (i) corrections need to be integrated through the acceptance cuts, ( n ) events from the inner bremsstrahlung process have to be generated "physically" so that they can be reconstructed reproducing the observed phase-space region. Our method is inspired by ref [ 12] where radiative corrections have been calculated for the annihilation process, the "inelastic" part (or inner bremsstrahlung) is included in the Monte Carlo integral, while the "elastic" part (or virtual correction) still remains a weighing factor We have verified that following this procedure, the corrections integrated without any acceptance cuts agree with the ~(x, y) result of ref. [ 13 ] We may ignore the existing controversy [ 14] about a possible two-photon loop diagrams contribution, the problem arises only in the (x~> 0.6) region where we have no events Fig 2 displays the reconstructed x spectrum for real and simulated events of sample II on which the shape analysis is carried out In the Monte Carlo distribution (including the physical background) several free parameters are adjusted by a Z 2 minimization The global fit includes all the data samples I, II, III, and the fitted parameters are (i) the n ° form factor slope a, determined from sample II, ( u ) physics parameters of the ( n - p - - , n e + e - ) process [ 1 ], determined from sample I, (nl) a relative normalization parameter N~ determined from all samples, using the Panofsky ratio as an input.
U3 FZ 0 (D
21 December 1989
16oo 1200 8oo 400
0
i
0
02
04
06
,
,
,
08
i
,
,
1
,
12
x
Fig 2 Invanant squared mass spectrum for real (full hne) and simulated (dots) events of sample II The Monte Carlo weight includes the fitted dynamics Background contaminations are taken into account, particularly, the evaluation of the contamination in sample II coming from the ( n - p - , n e + e - ) process does not depend on any model because it uses the physics extracted from the fit. We also obtain the n o form factor squared [ F ( x ) [ 2 directly at each x value, thus probing the linear behavlour of eq (3) In a first step, we have analyzed separately the two data sets of comparable statistics corresponding to two different runs taken in 1985 and 1986 Apart from very loose cuts in the tracking program, one supplementary cut is applied for x > 0 5 which eliminates 4% of the Dahtz events (region of large systematic errors) The linear behavlour o f the x ° form factor turns out to be well verified for the two data sets, yielding two slope parameter values labelled "85" and "86"" ass = - 0 02 + 0.04(stat.) __+008 (syst.) m~-o2,
(4)
as6 = - 0
(5)
2 0 + 0 03(stat ) + 0 07(syst ) m~ -2
The systematic error is computed from the three different contributions added linearly and listed below (1) relative normahzatlon uncertainty (AN~/N~ = + 3 % ) , A a l ~ _+0 05 m~-2, (2) missing mass calibration (to + 1 5 0 keV), Aa2 -~ + 0 0 1 - 0 02 m22, (3) stopping volume uncertainty ( + 1 m m vertically), Aa3 ~< _ 0 02 m ~-o2 The final result is obtained from the entire statistics (runs " 8 5 " + " 8 6 " , i e 32 000 true n o Dalitz de67
Volume 233, number 1,2
PHYSICS LETTERS B
t~ 1 X ~"
~
Ill/ltll
1
L~_
0 8
,
I
01
,
I
02
,
I
03
,
I
,
04 X
05
(m
Fig 3 Results for the ~o electromagnetic transltmn form factor squared, the error bars are statistical only The dashed line represents the envelop of the systematic error on the central value of each point (these errors are correlated from one point to the other) The straight line is the result of the hnear fit IF(x)[ 2 = 1 + 2 a x , with a = - 0 1 lm~oz
cay events used in the fit), fig 3 shows the m e a s u r e d n o form factor squared ] F ( x ) l z in the range 0 ~ < x < 0 5 We find a negative value for the slope parameter a=-0
1 l_+0 03(star )_+0 08(syst ) m~ "2
(6)
with a systematic error calculated as before and a 50% confidence level associated with sample II O u r result is c o m p a t i b l e with the p r e v m u s a m e a s u r e m e n t s rep o r t e d m table l, except with the one o f r e f [ 7 ] T h e stability o f the results with respect to the m a i n cuts o f the a p p a r a t u s has been verified. Finally, it has become usage to estimate the influence o f radiative corrections on the d e t e r m l n a t m n o f a A great sensitivity IS observed in our experiment, e h m m a t l n g radmtxve correctmns for processes ( l ) and ( 2 ) together would yield a - - 0.17 m ~-2, giving a shift comparable to the
68
21 December 1989
one observed in ref [ 7 ]. But u n d e r no circumstances should this effect be interpreted as a s u p p l e m e n t a r y " e r r o r b a r " on the result To conclude, our result confirms a negative value for the n ° form factor slope, what we learn from the recent experiments measuring a (see refs [6,7] and the present letter) is that one should examine carefully how systematic errors can be reduced enough to c o m p e t e with the good statistical precision o f A a = + 0.03 now achieved We wish to thank Dr. P A M Gulchon, D r J D e l o r m e and D r J M a r t i n o for discussions, and the D P H N - H E Saclay Linear Accelerator staff for their c o n t r i b u t i o n to b m l d l n g and operating this experim e n t successfully
References [ 1 ] H Fonvlellle et al, Phys Lett B 233 (1989) 60 [ 2 ] N Kroll and W Wada, Phys Rev 98 ( 1955 ) 1355 [3] L G Landsberg, Phys Rep 128 (1985) 301 [ 4 ] N P Samios, Phys Rev 121 (1961)275 [ 5 ] M A Schardt et al, Phys Rev D 23 ( 1981 ) 639 [6] P Gumphnger, Doct Thesis, Oregon State University (1987), J M Poutlssou et al, in Proc Lake Lomse Winter Institute (1987) (World Soentxfic, Singapore) [ 7 ] J Fischer et al, Phys Lett B 73 (1978) 359 [ 8 ] J Burger, Doct Thesis, Columbia University (1972)/Nevls Report 190 [9] S Devons et al, Phys Rev 184 (1969) 1356 [ 10 ] H Kobrak, Nuovo Cimento 20 ( 1961 ) 1115 [ 11 ] N Bensayab, Th~se 36me cycle no 81, Clermont-Ferrand (1988) [ 12] G Bonneau and F Martin, Nucl Phys B 27 ( 1971 ) 381, A Quenzer, Th6se d'Etat, Orsay/Universat6 de Paris-Sud
(1977) [ 13] K Mlkaehan and J Smith, Phys Rev D 5 (1972) 1763 [14] M Lambmand J Pestleau, Phys Rev D 31 (1985) 211, D Beder, Phys Rev D 34 (1986) 2071, G Tupper, Phys Rev D 35 (1987) 1726
PHYSICS LETTERS B
21 December 1989
D A L I T Z DECAY. n°~"/e+e- A N D T H E n o E L E C T R O M A G N E T I C T R A N S I T I O N F O R M F A C T O R H F O N V I E I L L E ", N. BENSAYAH a, j. B E R T H O T a, p B E R T I N a, B. B I H O R E A U b M C R O U A U ", G F O U R N I E R b, M G I F F O N ¢, J M I L L E R b, R. N A H A B E T I A N c, B S A G H A I b, C S A M O U R b and P VERNIN b Laboratotre de Physique CorpusculaweIN2P3-CNRS, UmversttOBlaise Pascal, F-63177Aubtbre Cedex, France b InstltutdePhvstqueNuclOmreIN2P3-CNRS, UmversttdClaudeBernard,Lyon-l, F-69622 VdleurbanneCedex, France DPhN, Centre d'Etudes Nuclbatresde Saclay, F-91191 Gzfisur-Yvette Cedex, France Received 21 July 1989
We have studied the no Dalltz decay n°~ye+e - using the pmn beam and a magnetic pmr spectrometer at the Saclay Linear Accelerator A shape analysis of the (e+e - ) mvarlant mass squared x, carried out on 32 000 events, gwes a direct measurement of the n° electromagneUc transmon form factor squared IF(x) 12,and of its slope parameter a m the expressmn F(x) = 1+ ax We find a= - 0 11 +0 03(stat ) +0 08(syst )m~-o2
We have p e r f o r m e d a high statistics experiment at the Saclay Linear Accelerator investigating both the following reactmns with stopped plons" ( n - p ~ ne+e - ) ( 1 ) and n - p - . n ° n , n°--.'/e+e - (2). In this letter we present the results on the n o form factor obtained from the study o f the n o D a h t z decay The results for the inverse plon electroproductlon process ( 1 ) are reported in ref [ 1 ] We first discuss the mot~vatmn for studying process ( 2 ) , n°-~Te+e - , and estabhsh some conventional notations If the n o were a polnthke p a m c l e , its electromagnetic decay would be entirely described by Q E D [2] To take into account an internal structure at the (n°~Y1Y2) vertex, a quantity F called the n o electromagnetic transition form factor is factorized in the a m p l i t u d e A great n u m b e r o f theoretical predictions are available for F from V D M up to recent models based on the n o quark structure (see the global review in ref [3] ) One defines F as a functmn o f the involved squared masses, F = F ( k 2, k 2, m2o), with the n o r m a l i z a t i o n F ( 0 , 0, m20) = 1 F m a y differ from 1 as one o f the photons goes off mass shell, 1.e in process ( 2 ) In this case, one conventionally writes F ( k 2, O, m ~ o ) = F ( x ) , where kl2=kl02 _ kE~-4E+E_ sinE(10) 1S the p h o t o n mass or the (e+e - ) lnvarlant mass squared, and x=k~/m,~o,Z 2
( r = 4 m 2 / m ~ o )
(dp~
dx-,,/~
X [ F ( x ) I2'
(1)
QED
(dp) 2~1 ( r)( d-x Q E D - - ' ~ X ( l - X ) 3 1+~X
~,/2 l--r/
(2)
per (n°~73,) event [2] and or= 1/137 The form factor is usually p a r a m e t n z e d by the linear expression
F ( x ) = 1 +IX,
(3)
where the slope p a r a m e t e r a is the quantity o f interest Due to the low-x enhancement o f d p / d x , see eq ( 2 ) , the effect o f a m o d e r a t e form factor slope would a m o u n t to a ~ 1% change (or less) in the integrated rate Thus it is hopeless to get any r e f o r m a t i o n on a from the D a h t z decay branching ratio, given the accuracy ( + 4%) o f the available experimental values [4,5] To d e t e r m i n e the slope o f the form factor it IS necessary to select events in a relatively hlgh-x region (roughly above 0 1 ) F r o m this starting p o i n t several experiments have been performed, extracting a e~ther
0370-2693/89/$03 50© ElsevlerSclencePubhshersB V (North-Holland)
65
Volume 233, number 1,2
PHYSICS L E T T E R S B
21 D e c e m b e r 1989
Table 1 Values of the slope parameter a measured from (~t°--,Te+e - ) experiments The n ° are produced from ( n - p ) at rest except in ref [6] where the production channel is K + ~ g + n o Authors
a values (m~-02 )
Analysis
Apparatus
Saclay experiment
- 0 11 _ 0 03(stat ) 4-0 08 ( s y s t )
shape
magnetic spectrometer + drift chambers
Gumphnger et al [ 6 ]
- 0 01 + 0 08 - 0 06 (total) (star = 4-0 03)
rate
NaI spectrometer +wire chambers
F~scher et al [7]
+ 0 104-0 03(stat only)
shape
magnetic spectrometer M W P C + (~erenkov
Burger et al [ 8 ]
+ 0 02 + 0 10 (total) (stat = _+0 07 )
shape
magnetic spectrometer + spark chambers
Devons et al [ 9 ]
+ 0 01 4- 0 11 (total) (stat = _+0 05 )
shape + rate
Nal spectrometer + spark chambers
Kobrak [ 10 ]
- 0 15 _+0 10 (stat only)
shape
Llqmd hydrogen bubble chamber
Sam~os [4]
- 0 24_+0 16(total) (star = _+0 12 )
rate
L~quld hydrogen bubble chamber
from a partial rate measurement or a shape analysis o f the x spectrum. The present expertmental situation is not clear (see table l ), the measured a values range from + 0 l 0 to - 0 24 m~-o2, and the two recent measurements having the best statistical precision [ 6,7 ] give conflicting results In the experiment described here, we have proceeded to a new determination of the rc° form factor To avoid an absolute normahzation o f the experiment, the information is extracted from a shape analysis of the ( e + e - ) x spectrum Process ( 1 ) constitutes a background to the 7t° Dalltz decay events and is studied simultaneously In ref [ 1 ] we present a detailed description of the apparatus, tracking program and method used in the global analysis Fig 1 shows how the ( e + e - ) events from processes ( 1 ) and ( 2 ) are identified in the pair e n e r g y - m o m e n t u m plot We obtain 36 000 events reconstructed in the kinematical region of process ( 2 ) , constituting sample II Sample III contains 16 000 same-sign pairs clearly identified by their charge sign. The geometrical acceptance is calculated by a Monte Carlo simulation generating lepton parrs according to processes ( 1 ) and ( 2 ) and the physical background [ l l ] Indeed, in sample II the no Dahtz decay events are contaminated by the following r e a c t i o n s 66
(a) n - p - , n e + e - at rest, (b) n ° - , e + e - e + e - , l e double-Dahtz decay, ( c ) n°-,T) ,, T--,leptons via pair production and
7~-p
>
>
ne+e -
:~120
Jj ,~
+
\
7 Ld o 8o Od <: EL
40 / \
., \
/ \
40
80
\
/
/
I I I I
120 PAIR ENERGY ( M e V
)
Fig 1 Eventselectlonlnthepalrenergy-momentumplot Dashed hnes are the theoretical domains of processes ( 1 ) and (2) Samples I and II are separated along the (,J) hne of the equation P=-I 69E+261 (MeV)
Volume 233, number 1,2
PHYSICS LETTERS B
C o m p t o n scattering in the target Reaction (a) contaminates process (2) mainly because of radiative processes Reactions (b) and (c) produce two kinds of detected pairs opposite-sign [giving an e+e - background to process ( 2 ) ] and same-sign pairs This latter sample (III) cannot lead to physics measurements because of the lack of absolute normalization and the mixture o f all the involved reactions However, the Monte Carlo study of the latter events provides an indirect check o f the (e+e - ) background to process (2) As an output of the simulation, contaminations in sample II are found to be ~ 4 % from reaction (a) and ~ 4 % from reactions (b) + (c). In comparison, the "slope-dependent" part of the Dahtz decay rate computed with our final a value, contributes to ~ 6% (in negative sign) of the counts observed in this sample Radiative corrections have been included in a similar way for processes ( 1 ) and (2) [ 11 ], considering that (i) corrections need to be integrated through the acceptance cuts, ( n ) events from the inner bremsstrahlung process have to be generated "physically" so that they can be reconstructed reproducing the observed phase-space region. Our method is inspired by ref [ 12] where radiative corrections have been calculated for the annihilation process, the "inelastic" part (or inner bremsstrahlung) is included in the Monte Carlo integral, while the "elastic" part (or virtual correction) still remains a weighing factor We have verified that following this procedure, the corrections integrated without any acceptance cuts agree with the ~(x, y) result of ref. [ 13 ] We may ignore the existing controversy [ 14] about a possible two-photon loop diagrams contribution, the problem arises only in the (x~> 0.6) region where we have no events Fig 2 displays the reconstructed x spectrum for real and simulated events of sample II on which the shape analysis is carried out In the Monte Carlo distribution (including the physical background) several free parameters are adjusted by a Z 2 minimization The global fit includes all the data samples I, II, III, and the fitted parameters are (i) the n ° form factor slope a, determined from sample II, ( u ) physics parameters of the ( n - p - - , n e + e - ) process [ 1 ], determined from sample I, (nl) a relative normalization parameter N~ determined from all samples, using the Panofsky ratio as an input.
U3 FZ 0 (D
21 December 1989
16oo 1200 8oo 400
0
i
0
02
04
06
,
,
,
08
i
,
,
1
,
12
x
Fig 2 Invanant squared mass spectrum for real (full hne) and simulated (dots) events of sample II The Monte Carlo weight includes the fitted dynamics Background contaminations are taken into account, particularly, the evaluation of the contamination in sample II coming from the ( n - p - , n e + e - ) process does not depend on any model because it uses the physics extracted from the fit. We also obtain the n o form factor squared [ F ( x ) [ 2 directly at each x value, thus probing the linear behavlour of eq (3) In a first step, we have analyzed separately the two data sets of comparable statistics corresponding to two different runs taken in 1985 and 1986 Apart from very loose cuts in the tracking program, one supplementary cut is applied for x > 0 5 which eliminates 4% of the Dahtz events (region of large systematic errors) The linear behavlour o f the x ° form factor turns out to be well verified for the two data sets, yielding two slope parameter values labelled "85" and "86"" ass = - 0 02 + 0.04(stat.) __+008 (syst.) m~-o2,
(4)
as6 = - 0
(5)
2 0 + 0 03(stat ) + 0 07(syst ) m~ -2
The systematic error is computed from the three different contributions added linearly and listed below (1) relative normahzatlon uncertainty (AN~/N~ = + 3 % ) , A a l ~ _+0 05 m~-2, (2) missing mass calibration (to + 1 5 0 keV), Aa2 -~ + 0 0 1 - 0 02 m22, (3) stopping volume uncertainty ( + 1 m m vertically), Aa3 ~< _ 0 02 m ~-o2 The final result is obtained from the entire statistics (runs " 8 5 " + " 8 6 " , i e 32 000 true n o Dalitz de67
Volume 233, number 1,2
PHYSICS LETTERS B
t~ 1 X ~"
~
Ill/ltll
1
L~_
0 8
,
I
01
,
I
02
,
I
03
,
I
,
04 X
05
(m
Fig 3 Results for the ~o electromagnetic transltmn form factor squared, the error bars are statistical only The dashed line represents the envelop of the systematic error on the central value of each point (these errors are correlated from one point to the other) The straight line is the result of the hnear fit IF(x)[ 2 = 1 + 2 a x , with a = - 0 1 lm~oz
cay events used in the fit), fig 3 shows the m e a s u r e d n o form factor squared ] F ( x ) l z in the range 0 ~ < x < 0 5 We find a negative value for the slope parameter a=-0
1 l_+0 03(star )_+0 08(syst ) m~ "2
(6)
with a systematic error calculated as before and a 50% confidence level associated with sample II O u r result is c o m p a t i b l e with the p r e v m u s a m e a s u r e m e n t s rep o r t e d m table l, except with the one o f r e f [ 7 ] T h e stability o f the results with respect to the m a i n cuts o f the a p p a r a t u s has been verified. Finally, it has become usage to estimate the influence o f radiative corrections on the d e t e r m l n a t m n o f a A great sensitivity IS observed in our experiment, e h m m a t l n g radmtxve correctmns for processes ( l ) and ( 2 ) together would yield a - - 0.17 m ~-2, giving a shift comparable to the
68
21 December 1989
one observed in ref [ 7 ]. But u n d e r no circumstances should this effect be interpreted as a s u p p l e m e n t a r y " e r r o r b a r " on the result To conclude, our result confirms a negative value for the n ° form factor slope, what we learn from the recent experiments measuring a (see refs [6,7] and the present letter) is that one should examine carefully how systematic errors can be reduced enough to c o m p e t e with the good statistical precision o f A a = + 0.03 now achieved We wish to thank Dr. P A M Gulchon, D r J D e l o r m e and D r J M a r t i n o for discussions, and the D P H N - H E Saclay Linear Accelerator staff for their c o n t r i b u t i o n to b m l d l n g and operating this experim e n t successfully
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