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Semi-inclusive polarised lepton-nucleon scattering and the anomalous gluon contribution
Physics Letters B 312 (1993) 166-172 North-Holland
PHYSICS LETTERS B
Semi-inclusive polarised lepton-nucleon scattering and the anomalous gluon contribution St. Giillenstern b, M. Veltri b, p. G6rnicki b, L. Mankiewicz
a
and A. Schiller a
a Institut J~r Theoretische Physik, Universitdt Frankfurt, Frankfurt, Germany b MPlfiir Kernphysik, Heidelberg, Germany Received 19 April 1993; revised manuscript received 8 June 1993 Editor: P.V. Landshoff We discuss a new observable for semi-inclusive pion production in polarised lepton-nucleon collisions. This observable is sensitive to the polarised and unpolarised strange quark distribution and the anomalous gluon contribution, provided that their fragmentation functions into pions differ substantially from that of light quarks. From Monte Carlo data generated with our PEPSI code we conclude that HERMES might be able to decide whether the polarized strange quark and gluon distributions are large. The EMC measurements of the spin dependent structure function gP (x) [ 1 ] have generated a large amount of theoretical activity, which will probably be further intensified by very recent results for gl (x) of deuterium and 3He [2,3]. One major focus of this discussion is the role played by the gluon anomaly [4-6]. We do not want to review this whole discussion but just sketch a few relevant points. It is clear that the isospin-singlet axial vector current has an anomaly. The problem is how this contribution can be split up between a point-like gluonic part and the part due to the usual quark-current. Large gauge transformations (i.e. those with a topologically number different from zero) shift contributions between these two. Thus it is possible to gauge away all gluonic contributions, but then the quark-distribution functions contain components which are incompatible with the assumptions of the parton model (e.g. quarks with large perpendicular m o m e n t u m ) . Thus to obtain a practical phenomenological model it seems necessary to allow for an anomalous gluon contribution, which, however, might be very small, if the gluons are only weakly polarized. This is our interpretation, other theoreticians propose substantially different ones. In fact the various opinions put forward vary so strongly that it seems improbable that this dispute can be settled on theoretical grounds. Therefore we advocate an observable which might be able to decide whether there is an anomalous gluon contribution, when the latter is understood (roughly speaking) as an effective descriptions of the unconventional parts of the quark distributions. A nice review of the nonperturbative aspects of the anomaly can be found in [7]. The signal we propose is constructed from semi-inclusive reactions, in which a pion is detected. We have investigated another aspect of such reactions in an earlier work [8 ] and wrote a Monte-Carlo program to analyse them in detail [9,10] for the HERMES setup [1 1,12]. We consider the two asymmetries Al and A ~÷ +~-. AI is the usual longitudinal asymmetry, A I ( X , Q2) _ crT~-arT ~ gl(x____~) cry+ + aTT Fl(X) '
(1)
and the structure functions
166
0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
Volume 312, number 1,2
PHYSICS LETTERS B
5 August 1993
F ] ( x ) = ½Z e 2 q q ( X ) ,
q g~(x, = ½ Z e 2 ( A q ( x ,
- ~AF(x))
(2)
q
are expressed in terms o f the parton distributions, 1
A F ( x ) :----- / ~ Z A a ' e g ( z ) A g ( ~ ) , x so,g
=) :_-
==l,n
q : = q+ + q +
+q-
+~_,
g := g+ + g - , A q : = q+ + ~ + - q - - ~ ' _ , Ag : = g+ - g _ ,
(3)
where the index ± refers to the relative helicity o f the parton with respect to the helicity o f the nucleon. A F ( x ) is the famous anomalous contribution. The form o f Ao"yg (2) is controversial [4-6]. We use a form which was employed in ref. [13]. Any model in which the anomaly gives a sizeable contribution in the x range up to 0.2, which seems to be suggested by the EMC data will lead to comparable results. If the anomalous contribution would be concentrated at very small x values (e.g. below 0.01 ) our analyses did not apply. The pion asymmetry A ~++" is obtained in the semi-inclusive reaction
e + N----+e' + n± + X. and is defined as A~*+=- = (1/o'r) d A a ~ + + = - / d z (1/at) da=++~-/dz '
(4)
where Aojt + +rr-
:=
n + +nO'TL
O'TI,T T
"=
0.~+
X++~Z : = crT~
+~Z -
-
-
~+ +TtO'TT
,
OTL,T T --I- O'TLTT, -
~++~+ aT?
.
(5)
The crucial question is what part o f the anomalous contribution is also present for semi-inclusive processes and thus A ~+ +=-. We do not try to decide this question on theoretical grounds: Strict factorisation implies that the anomaly should contribute to any channel in the same manner, but it is unclear whether factorisation applies for this case as the anomaly mixes the different m o m e n t u m scales. In the perturbative box graph the q u a r k - a n t i q u a r k pairs from the anomaly have large perpendicular m o m e n t u m which would lead to two-jet fragmentation instead of one-jet fragmentation and thus probably to a different fragmentation function. But it is unclear whether the perturbative graph has any relevance. It is possible that non-perturbative contributions dominate. 167
Volume 312, number 1,2
PHYSICS LETTERS B
5 August 1993
In view of these uncertainties we just study complementary scenarios: (1) an anomalous gluon contribution analogous to gl, i.e. we assume the anomaly to contribute proportionally to all channels, and (II) no anomalous contribution in the semi-inclusive scattering. The truth will probably lie somewhere in between, but our analysis gives then upper and lower limits for the experimental signal to be measured. (I) Anomalous contribution present in A ~++~ . In this case the differential cross section for producing a pion is given by
aT
d A a ~ ( z , x , Q 2) = ~ q e q2[Aq(x, Q2) - ( ~ t , / 2 g ) A F ( x , Q 2 ) ] D g ( z ) dz ~-~q e2 q ( x. Q 2 )
(6)
Assuming spin independence of fragmentation functions, i.e. Dqh+ = Dqh along with charge conjugation and isospin invariance the various fragmentation functions can be expressed through D, D and Ds, the favoured, unfavoured and strange fragmentation function respectively: D ( z ) = D'~+ (z) = D~- (z) = Dd + (z) = D-~- ( z ) , --
D(z) = D dit + (z) = Du~ - (z) =D-~?g+ (z) =D-dR - ( z ) ,
D s ( z ) = D7 +(z) = D f - ( z )
= D~ +(z) = n~
(z).
For a proton target (indicated by the subscript respectively superscript in ,.pa~++ ' - and At ) one obtains A,++,-
p
[ D ( z ) + - D ( z ) ] [ ~ b u ( x ) + ~Ad(x) - ~ ( a s / 2 ~ ) A F ( x ) ] + 2 D . ( z ) ~ [ A S ( x ) - ( a , / 2 ~ ) A F ( x ) l [D(z) +-D(z)l[4u(x) + ld(x)] + 2Ds(z)~s(x) ½ ~ q e 2 [Aq(x) - ((~,/2zr) A F ( x ) ] - ~ : ( z ) [AS(x) - ((ts/2~) A F ( x ) ]
½ y~qeZq(x) - ~ c ( z ) s ( x ) gl(x) -(1-~:(z)[6(x)-?(x)]'}g , ( x ) {1 + [ e ( x ) - 6 ( x )
+7(x)]~c(z) +...}
(7)
where the quantities 7(x):= -~(x):-
(c~,/2zr)AF(x) << I, 18gl (x)
AS(x) - << 1, 18gl (x)
s(x)
e(x).--
- << 1, 18El (y)
~(z) :=
D ( z ) + D ( z ) - 2Ds(z) D ( z ) + -D(z)
have been introduced and 4 " + + " - has been expanded around 7 fi = e 0. Note that the EMC data suggest that As is negative. Then all correction add up. The difference between A~ ÷ + ' - and A~ reads
A~t++lt"-p
( S gl - At =
18Fx Fl
~s-(o~s/27~)AI')(O"~O-2Ds) 18F1
D +=~
+ "'" "
(8)
The favoured and unfavoured fragmentation functions have been measured and can be parametrised by [ 14 ] 168
Volume 312, number 1,2
PHYSICS LETTERS B
5 August 1993
D(z) = 0.7 (1 - z) 1.75 Z
-D(z) = q~--~D(Z),z
(9)
with q(z) =
l+z l-z"
(10)
The strange fragmentation function is not measured yet and is assumed to be of the order of the unfavoured one. If one sets m
Ds(z) ~ D(z),
(11)
one obtains
[z/ (1
+
~c(z)~ [1/(1 +
z)]O(z) z)]D(z) = z
(12)
and
P
18Fl(X) FI (x~ -
- 1 ~
[1/(1 + zi]D(z)
"
Hence the fragmentation suppresses the signal roughly by a factor of z. N u m e r a t o r and denominator on the right hand side drop rapidly as z approaches unity. In practice one cannot evaluate the difference at high values o f z, because fragmentation into z ~ 1 pions is very improbable and one gets very high statistical errors. To increase the statistic and reduce thereby the experimental error we integrate the semi-inclusive cross sections over all z's between Zmin and 1 first and then calculate the pionic asymmetry. Here Zmin has to be chosen such that the leading pions are selected. Taking the difference between the asymmetries constructed in this way one gets A~++~ 1gEl FI
18FI
fzlmi, d z [ 1 / ( 1 + ~ i ] D ( z )
This quantity is o f course rather sensitive to the actual value of Zmin. Varying Zmin between 0.I and 0.4, fzii, d z l z l (1 + z ) l D ( z ) l f:m d z [ 1 / (1 + z) ]O(z) varies from 0.24 to 0.53. Naturally a larger Zrnin leads to in a higher signal, but is unfortunately accompanied by less statistic and therefore higher statistical errors. Thus we have taken Zmin = 0.2 which seems to be a reasonable cut to select the leading pions [15]. One can try to discriminate experimentally against pions from resonance decay, but this is not really necessary. In fact one can think o f the D ( z ) ' s as generalized fragmentation functions including also such resonance decays, i.e. as total probabilities to obtain finally a pion from a parton without regard to possible intermediate states. F o r such a broader interpretation the forms given in eqs. (9), (10) and ( 11 ) are not really applicable and one would have to deduce generalized fragmentation functions from experiments. As the additional contributions from resonance decay are not that large it is, however, possible that the resulting functions would be hardly different from the ones we use. The only requirements which are really vital for our argument are that the fragmentation functions are spin-independent, isospin-symmetric, and different for s-quarks and light quarks. To calculate the signal we used the parametrisation EHLQ set 1 (1986 updated version) [16] for the spin independent quantities and two different parametrisations of [13], namely set a) and set d) (in the following RRa and R R d resp.), given at Q2 = 4 GeV 2 and evolve them via GLAP-evolution up to the appropriate Q2 values present at H E R M E S [ 11,12 ]. We have taken into account three active flavours. The first parametrisation 169
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+ +
PHYSICS LETTERS B
5 August 1993
RRa
0,1
RR(I
f
0,08
0,06
0.04
0,02
0
Fig. 1. The asymmetry-difference A~ - A~++n- for two parameter sets from [ 13], assuming that the anomaly is not suppressed for semi-inclusive reactions. The parameter set RRa contains no large polarized gluon distribution but a large polarized strange quark distribution, while for RRb AG is large and As is small. The z-cut is 0.2.
+++
-0.02
10 -1 X
describes a large sea polarisation and no gluon polarisation at the reference point. The gluon polarisation enters only due to evolution and its contribution remains therefore small. In the second parametrisation the sea is weakly polarised and the gluons strongly, with the unpolarised distribution as an upper bound. We show the signal resulting from the two parametrisations (RRa and RRd) at Q2 = 10 GeV 2 in fig. 1. It was evaluated at the HERMES x-bins and the averaged Q2-value in each x-bin for HERMES kinematics. The error bars reflect the projected statistical error in a standard HERMES run (600h, 40mA). The error bars and the averaged Q2 values were estimated [15] with PEPSI [9,10]. It is obvious that the signal is of the same order for both parametrisations and is hardly detectable with the accuracy reachable in a standard HERMES run. To resolve it, the statistical error would have to be reduced at least by a factor of 2. (II) No anomalous contribution present in A ~++~-. Without any anomalous gluon contribution in the semiinclusive scattering and with the same assumptions about fragmentation functions as above one obtains for a proton target now --
A~++,- = [ D ( z ) + D ( z ) ] [ 4 A u ( x ) [D(z) +-D(z)]([4u(x)
+ ~Ad(x)] +2Ds(z)~as(x)
+ ~d(x)) + 2Ds(z)~s(x)
= ~ q e 2 A q ( x ) - l~C(z)as(x) ½y~qe~q(x)- ~c(z)s(x) 1
½ ~ q eqz [Aq (x) - (~s/Zn) AF (x) ] + 6 (~s/2n) AF (x) - ~ c (z)As (x)
-
F,(x)gl(x)(1
½Zq e2qq(x) - ~c(z)s(x) +67(x)-Jc(z)6(X))l___~(z_~(.~ ~ g[ l ( 1x ) + (e(x)-5(x))x(z)+67(x)+...]
where the same notation as above is used and a ' + + ' the difference between ~.p~++ ' - and A~ becomes
A.++~p
s g, ([z/(l+z)]D(z)'~ -A~-
18FIF~
[1/(l+z)]D(z)]-
,
has been expanded again around 7
as ([z/(l+z)]D(z)'~
1--~
[1/(l+~)--~(~i]
(15) 5 = e
0. Now
6(c~dZn)aF. ÷
18F~
'
(16)
or taking the asymmetries for the z-integrated quantities, An+ + , -
.., +
s
g,
- A ~ - 1 8 F , F] 6 (C~s/2n)AF 18El
fz]mi. fzlmi dz [ l / ( 1 -{- ~- i ]O~z)
As - ~
dz[1/(l+z)]n(z)/ (17)
The first two terms are the same as in the previous scenario eq. (14), whereas the third has to be compared with the gluon contribution above. One sees immediately that now the gluon contribution is substantial larger, 170
Volume 312, number 1,2
PHYSICS LETTERS B
5 August 1993
+ + 0.!
* I~IL,
+ A=0
i
0.08 p
A = [),; A= I
0.08
0.06 0.04
~- ,,
___~__. ,,
0.04
0.02 0.02 0 -0.02
-o.o
10 - I
lo -1
X
Fig. 2. The asymmetry-difference At - A ~ + +n- as in fig. 1, assuming that the polarised gluons do not contribute at all to semi-inclusive pion production.
Fig. 3. The scenario with a large gluon polarisation (RRd) and a suppression factor 2 for the anomalous contribution.
namely by the factor 6 f~m~. d z [ 1/ ( 1 + z ) ] D ( z ) / f~mi, dz [ z~ ( 1 + z ) ] D ( z ) ~ 18, when the quantities are again integrated over z with a Zmin = 0.2 which l e a d s t o fZ~mi,dz[z/(1 +
z)]D(z)/ flmm d z [ 1 /
(1 +
z)]D(z)
0.34.
This large difference is due to the fact that the gluons now contribute to At but not to ..pan++"- . Obviously this leads to a much higher signal for parametrisation with a strong gluon polarisation than for those with a large strange quark polarisation and no or a small gluon polarisation only. Fig. 2 shows the result for the two scenarios of [13], with large and small gluon contribution respectively. Obviously a large gluon component ( R R d ) would now be easy to detect, while the signal for RRa is hardly statistically significant. Let us also note that the pure gluon signal is z-independent, a property which would be checked by the experiment. At this point it is important to note that the expected absolute experimental errors in the relevant m e d i u m x region (0.04 ~< x ~< 0.3) are smaller than for A~ alone as the average pion multiplicity is there close to 1 (1.0 for Z m i n = 0.1 and 0.6 for Zmin = 0.2) such that A~ and A ~++"- count the same events (up to those o f interest). The statistical errors are thus not independent and when the difference is taken they cancel partially. Probably none of the extreme scenarios described above is realistic. A part of the gluon anomaly will be present also in the semi-inclusive channels. The actual fraction will certainly depend on the dynamics, i.e. x, Q2 and z values. To estimate to what extend the gluon anomaly has to be suppressed "on average" in the semi-inelastic channel such that the signal is still visible and for sake of simplicity we take a constant fraction of the anomaly 2 into account in the following. Then the signal build up out o f the z-integrated pionic asymmetry is given by
A•+ p
q-r~-
-t-
-- A t --
18FI
S ( lridZ' ,'l lff-FiF1 fZlin d z [ i / ( 1
+ z~3"]
- ~
f lZmin d z [ 1 / ( 1 +
6(1 - 2) + 2
. fzlmin d z [ 1 / (1 + z ~ ( ~ , ]
z)]O(z) (18)
1
The signal for RRd-like parametrisations is shown in fig. 3 for different values of 2. It is clear from the figure that a 50% suppresion is sufficient to keep the signal detectable in a standard HERMES-run in the case of large gluon polarisation ( R R d - p a r a m e t r i s a t i o n ) . Finally we want to mention that if the anomalous contribution to pion production is not suppressed it should be seen in experiments like those of ref. [ 17]. Such ( i m p r o v e d ) experiments are therefore complementary to the 171
Volume 312, number 1,2
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proposed analyses from HERMES. Together they should be able to decide whether there is a large anomalous gluon contribution or not for not too small x-values, i.e. (x >/ 0.01 ). L.M. thanks A. von Humboldt Stiflung and A.S. thanks the MPI f/Jr Kernphysik for support. This work was partly supported by the KBN grant 2-0224-91-01. We also acknowledge helpful discussions with S. Bass.
References [1] [2] [3] [4] [5] [6] [7] [8]
EMC Collab,, J. Ashman et al., Nucl. Phys. B 328 (1990) 1. SMC Collab., B. Adeva et al., Phys. Lett. B 302 (1993) 533. W. Meyer (El41 Collab.), talk given at the Meeting of the German Physical Society (Mainz, 1993), and to be published. R.D. Carlitz, J.C. Collins and A.H. Mueller, Phys. Lett B 214 (1988) 229. G. Altarelli and G.G. Ross, Phys. Lett. B 212 (1988) 391. L. Mankiewicz and A. Schfifer, Phys. Lett. B 242 (1990) 455. A.E. Dorokhov, N.I. Kochelev and Yu.A. Zubov, Int. J. Mod. Phys. A 8 (1993) 603. L.L. Frankfurt, M.I. Strikman, L. Mankiewicz, A. Sch~ifer, E. Rondio, A. Sandacz and V. Papavassiliou, Phys. Lett. B 230 (1989) 141. [9] L. Mankiewicz, A. Sch~ifer and M. Veltri, Comput. Phys. Commun. 71 (1992) 360. [10] M. Veltri, M. Diiren, L. Mankiewicz, K. Rith and A. Sch/ifer, Physics at HERA Workshop, Vol. 1, (DESY, 1992) p. 447. [11 ] K. Coulter et al., HERMES proposal, DESY/PRC 90/1 (1990). [12] M. D/iren and K. Rith, Physics at HERA Workshop, Vol. 1 (DESY, 1992) p. 427. [13] G.G. Ross and R. G. Roberts, The gluon contribution to polarized nucleon structure functions, preprint RAL-90-062 (1990); there are several misprints in this paper, please note that B should be 200 instead of 2 for the scenario with a large gluon contribution. [14] J.J. Aubert et al., Phys. Lett. B 160 (1985) 417. [ 15 ] M. Veltri, PhD Thesis, University of Heidelberg (1992) MPIH-V 18-1992. [16] E. Eichten, I. Hinchliffe, K. Lane and C. Quigg, Rev. Mod. Phys. 56 (1984) 579; 58 (1985) 1065; updated version (1986). [17] FNAL E581/704 Collab., D.L. Adams et al., Phys. Lett. B 261 (1991) 197.
172
PHYSICS LETTERS B
Semi-inclusive polarised lepton-nucleon scattering and the anomalous gluon contribution St. Giillenstern b, M. Veltri b, p. G6rnicki b, L. Mankiewicz
a
and A. Schiller a
a Institut J~r Theoretische Physik, Universitdt Frankfurt, Frankfurt, Germany b MPlfiir Kernphysik, Heidelberg, Germany Received 19 April 1993; revised manuscript received 8 June 1993 Editor: P.V. Landshoff We discuss a new observable for semi-inclusive pion production in polarised lepton-nucleon collisions. This observable is sensitive to the polarised and unpolarised strange quark distribution and the anomalous gluon contribution, provided that their fragmentation functions into pions differ substantially from that of light quarks. From Monte Carlo data generated with our PEPSI code we conclude that HERMES might be able to decide whether the polarized strange quark and gluon distributions are large. The EMC measurements of the spin dependent structure function gP (x) [ 1 ] have generated a large amount of theoretical activity, which will probably be further intensified by very recent results for gl (x) of deuterium and 3He [2,3]. One major focus of this discussion is the role played by the gluon anomaly [4-6]. We do not want to review this whole discussion but just sketch a few relevant points. It is clear that the isospin-singlet axial vector current has an anomaly. The problem is how this contribution can be split up between a point-like gluonic part and the part due to the usual quark-current. Large gauge transformations (i.e. those with a topologically number different from zero) shift contributions between these two. Thus it is possible to gauge away all gluonic contributions, but then the quark-distribution functions contain components which are incompatible with the assumptions of the parton model (e.g. quarks with large perpendicular m o m e n t u m ) . Thus to obtain a practical phenomenological model it seems necessary to allow for an anomalous gluon contribution, which, however, might be very small, if the gluons are only weakly polarized. This is our interpretation, other theoreticians propose substantially different ones. In fact the various opinions put forward vary so strongly that it seems improbable that this dispute can be settled on theoretical grounds. Therefore we advocate an observable which might be able to decide whether there is an anomalous gluon contribution, when the latter is understood (roughly speaking) as an effective descriptions of the unconventional parts of the quark distributions. A nice review of the nonperturbative aspects of the anomaly can be found in [7]. The signal we propose is constructed from semi-inclusive reactions, in which a pion is detected. We have investigated another aspect of such reactions in an earlier work [8 ] and wrote a Monte-Carlo program to analyse them in detail [9,10] for the HERMES setup [1 1,12]. We consider the two asymmetries Al and A ~÷ +~-. AI is the usual longitudinal asymmetry, A I ( X , Q2) _ crT~-arT ~ gl(x____~) cry+ + aTT Fl(X) '
(1)
and the structure functions
166
0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
Volume 312, number 1,2
PHYSICS LETTERS B
5 August 1993
F ] ( x ) = ½Z e 2 q q ( X ) ,
q g~(x, = ½ Z e 2 ( A q ( x ,
- ~AF(x))
(2)
q
are expressed in terms o f the parton distributions, 1
A F ( x ) :----- / ~ Z A a ' e g ( z ) A g ( ~ ) , x so,g
=) :_-
==l,n
q : = q+ + q +
+q-
+~_,
g := g+ + g - , A q : = q+ + ~ + - q - - ~ ' _ , Ag : = g+ - g _ ,
(3)
where the index ± refers to the relative helicity o f the parton with respect to the helicity o f the nucleon. A F ( x ) is the famous anomalous contribution. The form o f Ao"yg (2) is controversial [4-6]. We use a form which was employed in ref. [13]. Any model in which the anomaly gives a sizeable contribution in the x range up to 0.2, which seems to be suggested by the EMC data will lead to comparable results. If the anomalous contribution would be concentrated at very small x values (e.g. below 0.01 ) our analyses did not apply. The pion asymmetry A ~++" is obtained in the semi-inclusive reaction
e + N----+e' + n± + X. and is defined as A~*+=- = (1/o'r) d A a ~ + + = - / d z (1/at) da=++~-/dz '
(4)
where Aojt + +rr-
:=
n + +nO'TL
O'TI,T T
"=
0.~+
X++~Z : = crT~
+~Z -
-
-
~+ +TtO'TT
,
OTL,T T --I- O'TLTT, -
~++~+ aT?
.
(5)
The crucial question is what part o f the anomalous contribution is also present for semi-inclusive processes and thus A ~+ +=-. We do not try to decide this question on theoretical grounds: Strict factorisation implies that the anomaly should contribute to any channel in the same manner, but it is unclear whether factorisation applies for this case as the anomaly mixes the different m o m e n t u m scales. In the perturbative box graph the q u a r k - a n t i q u a r k pairs from the anomaly have large perpendicular m o m e n t u m which would lead to two-jet fragmentation instead of one-jet fragmentation and thus probably to a different fragmentation function. But it is unclear whether the perturbative graph has any relevance. It is possible that non-perturbative contributions dominate. 167
Volume 312, number 1,2
PHYSICS LETTERS B
5 August 1993
In view of these uncertainties we just study complementary scenarios: (1) an anomalous gluon contribution analogous to gl, i.e. we assume the anomaly to contribute proportionally to all channels, and (II) no anomalous contribution in the semi-inclusive scattering. The truth will probably lie somewhere in between, but our analysis gives then upper and lower limits for the experimental signal to be measured. (I) Anomalous contribution present in A ~++~ . In this case the differential cross section for producing a pion is given by
aT
d A a ~ ( z , x , Q 2) = ~ q e q2[Aq(x, Q2) - ( ~ t , / 2 g ) A F ( x , Q 2 ) ] D g ( z ) dz ~-~q e2 q ( x. Q 2 )
(6)
Assuming spin independence of fragmentation functions, i.e. Dqh+ = Dqh along with charge conjugation and isospin invariance the various fragmentation functions can be expressed through D, D and Ds, the favoured, unfavoured and strange fragmentation function respectively: D ( z ) = D'~+ (z) = D~- (z) = Dd + (z) = D-~- ( z ) , --
D(z) = D dit + (z) = Du~ - (z) =D-~?g+ (z) =D-dR - ( z ) ,
D s ( z ) = D7 +(z) = D f - ( z )
= D~ +(z) = n~
(z).
For a proton target (indicated by the subscript respectively superscript in ,.pa~++ ' - and At ) one obtains A,++,-
p
[ D ( z ) + - D ( z ) ] [ ~ b u ( x ) + ~Ad(x) - ~ ( a s / 2 ~ ) A F ( x ) ] + 2 D . ( z ) ~ [ A S ( x ) - ( a , / 2 ~ ) A F ( x ) l [D(z) +-D(z)l[4u(x) + ld(x)] + 2Ds(z)~s(x) ½ ~ q e 2 [Aq(x) - ((~,/2zr) A F ( x ) ] - ~ : ( z ) [AS(x) - ((ts/2~) A F ( x ) ]
½ y~qeZq(x) - ~ c ( z ) s ( x ) gl(x) -(1-~:(z)[6(x)-?(x)]'}g , ( x ) {1 + [ e ( x ) - 6 ( x )
+7(x)]~c(z) +...}
(7)
where the quantities 7(x):= -~(x):-
(c~,/2zr)AF(x) << I, 18gl (x)
AS(x) - << 1, 18gl (x)
s(x)
e(x).--
- << 1, 18El (y)
~(z) :=
D ( z ) + D ( z ) - 2Ds(z) D ( z ) + -D(z)
have been introduced and 4 " + + " - has been expanded around 7 fi = e 0. Note that the EMC data suggest that As is negative. Then all correction add up. The difference between A~ ÷ + ' - and A~ reads
A~t++lt"-p
( S gl - At =
18Fx Fl
~s-(o~s/27~)AI')(O"~O-2Ds) 18F1
D +=~
+ "'" "
(8)
The favoured and unfavoured fragmentation functions have been measured and can be parametrised by [ 14 ] 168
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5 August 1993
D(z) = 0.7 (1 - z) 1.75 Z
-D(z) = q~--~D(Z),z
(9)
with q(z) =
l+z l-z"
(10)
The strange fragmentation function is not measured yet and is assumed to be of the order of the unfavoured one. If one sets m
Ds(z) ~ D(z),
(11)
one obtains
[z/ (1
+
~c(z)~ [1/(1 +
z)]O(z) z)]D(z) = z
(12)
and
P
18Fl(X) FI (x~ -
- 1 ~
[1/(1 + zi]D(z)
"
Hence the fragmentation suppresses the signal roughly by a factor of z. N u m e r a t o r and denominator on the right hand side drop rapidly as z approaches unity. In practice one cannot evaluate the difference at high values o f z, because fragmentation into z ~ 1 pions is very improbable and one gets very high statistical errors. To increase the statistic and reduce thereby the experimental error we integrate the semi-inclusive cross sections over all z's between Zmin and 1 first and then calculate the pionic asymmetry. Here Zmin has to be chosen such that the leading pions are selected. Taking the difference between the asymmetries constructed in this way one gets A~++~ 1gEl FI
18FI
fzlmi, d z [ 1 / ( 1 + ~ i ] D ( z )
This quantity is o f course rather sensitive to the actual value of Zmin. Varying Zmin between 0.I and 0.4, fzii, d z l z l (1 + z ) l D ( z ) l f:m d z [ 1 / (1 + z) ]O(z) varies from 0.24 to 0.53. Naturally a larger Zrnin leads to in a higher signal, but is unfortunately accompanied by less statistic and therefore higher statistical errors. Thus we have taken Zmin = 0.2 which seems to be a reasonable cut to select the leading pions [15]. One can try to discriminate experimentally against pions from resonance decay, but this is not really necessary. In fact one can think o f the D ( z ) ' s as generalized fragmentation functions including also such resonance decays, i.e. as total probabilities to obtain finally a pion from a parton without regard to possible intermediate states. F o r such a broader interpretation the forms given in eqs. (9), (10) and ( 11 ) are not really applicable and one would have to deduce generalized fragmentation functions from experiments. As the additional contributions from resonance decay are not that large it is, however, possible that the resulting functions would be hardly different from the ones we use. The only requirements which are really vital for our argument are that the fragmentation functions are spin-independent, isospin-symmetric, and different for s-quarks and light quarks. To calculate the signal we used the parametrisation EHLQ set 1 (1986 updated version) [16] for the spin independent quantities and two different parametrisations of [13], namely set a) and set d) (in the following RRa and R R d resp.), given at Q2 = 4 GeV 2 and evolve them via GLAP-evolution up to the appropriate Q2 values present at H E R M E S [ 11,12 ]. We have taken into account three active flavours. The first parametrisation 169
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+ +
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RRa
0,1
RR(I
f
0,08
0,06
0.04
0,02
0
Fig. 1. The asymmetry-difference A~ - A~++n- for two parameter sets from [ 13], assuming that the anomaly is not suppressed for semi-inclusive reactions. The parameter set RRa contains no large polarized gluon distribution but a large polarized strange quark distribution, while for RRb AG is large and As is small. The z-cut is 0.2.
+++
-0.02
10 -1 X
describes a large sea polarisation and no gluon polarisation at the reference point. The gluon polarisation enters only due to evolution and its contribution remains therefore small. In the second parametrisation the sea is weakly polarised and the gluons strongly, with the unpolarised distribution as an upper bound. We show the signal resulting from the two parametrisations (RRa and RRd) at Q2 = 10 GeV 2 in fig. 1. It was evaluated at the HERMES x-bins and the averaged Q2-value in each x-bin for HERMES kinematics. The error bars reflect the projected statistical error in a standard HERMES run (600h, 40mA). The error bars and the averaged Q2 values were estimated [15] with PEPSI [9,10]. It is obvious that the signal is of the same order for both parametrisations and is hardly detectable with the accuracy reachable in a standard HERMES run. To resolve it, the statistical error would have to be reduced at least by a factor of 2. (II) No anomalous contribution present in A ~++~-. Without any anomalous gluon contribution in the semiinclusive scattering and with the same assumptions about fragmentation functions as above one obtains for a proton target now --
A~++,- = [ D ( z ) + D ( z ) ] [ 4 A u ( x ) [D(z) +-D(z)]([4u(x)
+ ~Ad(x)] +2Ds(z)~as(x)
+ ~d(x)) + 2Ds(z)~s(x)
= ~ q e 2 A q ( x ) - l~C(z)as(x) ½y~qe~q(x)- ~c(z)s(x) 1
½ ~ q eqz [Aq (x) - (~s/Zn) AF (x) ] + 6 (~s/2n) AF (x) - ~ c (z)As (x)
-
F,(x)gl(x)(1
½Zq e2qq(x) - ~c(z)s(x) +67(x)-Jc(z)6(X))l___~(z_~(.~ ~ g[ l ( 1x ) + (e(x)-5(x))x(z)+67(x)+...]
where the same notation as above is used and a ' + + ' the difference between ~.p~++ ' - and A~ becomes
A.++~p
s g, ([z/(l+z)]D(z)'~ -A~-
18FIF~
[1/(l+z)]D(z)]-
,
has been expanded again around 7
as ([z/(l+z)]D(z)'~
1--~
[1/(l+~)--~(~i]
(15) 5 = e
0. Now
6(c~dZn)aF. ÷
18F~
'
(16)
or taking the asymmetries for the z-integrated quantities, An+ + , -
.., +
s
g,
- A ~ - 1 8 F , F] 6 (C~s/2n)AF 18El
fz]mi. fzlmi dz [ l / ( 1 -{- ~- i ]O~z)
As - ~
dz[1/(l+z)]n(z)/ (17)
The first two terms are the same as in the previous scenario eq. (14), whereas the third has to be compared with the gluon contribution above. One sees immediately that now the gluon contribution is substantial larger, 170
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+ + 0.!
* I~IL,
+ A=0
i
0.08 p
A = [),; A= I
0.08
0.06 0.04
~- ,,
___~__. ,,
0.04
0.02 0.02 0 -0.02
-o.o
10 - I
lo -1
X
Fig. 2. The asymmetry-difference At - A ~ + +n- as in fig. 1, assuming that the polarised gluons do not contribute at all to semi-inclusive pion production.
Fig. 3. The scenario with a large gluon polarisation (RRd) and a suppression factor 2 for the anomalous contribution.
namely by the factor 6 f~m~. d z [ 1/ ( 1 + z ) ] D ( z ) / f~mi, dz [ z~ ( 1 + z ) ] D ( z ) ~ 18, when the quantities are again integrated over z with a Zmin = 0.2 which l e a d s t o fZ~mi,dz[z/(1 +
z)]D(z)/ flmm d z [ 1 /
(1 +
z)]D(z)
0.34.
This large difference is due to the fact that the gluons now contribute to At but not to ..pan++"- . Obviously this leads to a much higher signal for parametrisation with a strong gluon polarisation than for those with a large strange quark polarisation and no or a small gluon polarisation only. Fig. 2 shows the result for the two scenarios of [13], with large and small gluon contribution respectively. Obviously a large gluon component ( R R d ) would now be easy to detect, while the signal for RRa is hardly statistically significant. Let us also note that the pure gluon signal is z-independent, a property which would be checked by the experiment. At this point it is important to note that the expected absolute experimental errors in the relevant m e d i u m x region (0.04 ~< x ~< 0.3) are smaller than for A~ alone as the average pion multiplicity is there close to 1 (1.0 for Z m i n = 0.1 and 0.6 for Zmin = 0.2) such that A~ and A ~++"- count the same events (up to those o f interest). The statistical errors are thus not independent and when the difference is taken they cancel partially. Probably none of the extreme scenarios described above is realistic. A part of the gluon anomaly will be present also in the semi-inclusive channels. The actual fraction will certainly depend on the dynamics, i.e. x, Q2 and z values. To estimate to what extend the gluon anomaly has to be suppressed "on average" in the semi-inelastic channel such that the signal is still visible and for sake of simplicity we take a constant fraction of the anomaly 2 into account in the following. Then the signal build up out o f the z-integrated pionic asymmetry is given by
A•+ p
q-r~-
-t-
-- A t --
18FI
S ( lridZ' ,'l lff-FiF1 fZlin d z [ i / ( 1
+ z~3"]
- ~
f lZmin d z [ 1 / ( 1 +
6(1 - 2) + 2
. fzlmin d z [ 1 / (1 + z ~ ( ~ , ]
z)]O(z) (18)
1
The signal for RRd-like parametrisations is shown in fig. 3 for different values of 2. It is clear from the figure that a 50% suppresion is sufficient to keep the signal detectable in a standard HERMES-run in the case of large gluon polarisation ( R R d - p a r a m e t r i s a t i o n ) . Finally we want to mention that if the anomalous contribution to pion production is not suppressed it should be seen in experiments like those of ref. [ 17]. Such ( i m p r o v e d ) experiments are therefore complementary to the 171
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proposed analyses from HERMES. Together they should be able to decide whether there is a large anomalous gluon contribution or not for not too small x-values, i.e. (x >/ 0.01 ). L.M. thanks A. von Humboldt Stiflung and A.S. thanks the MPI f/Jr Kernphysik for support. This work was partly supported by the KBN grant 2-0224-91-01. We also acknowledge helpful discussions with S. Bass.
References [1] [2] [3] [4] [5] [6] [7] [8]
EMC Collab,, J. Ashman et al., Nucl. Phys. B 328 (1990) 1. SMC Collab., B. Adeva et al., Phys. Lett. B 302 (1993) 533. W. Meyer (El41 Collab.), talk given at the Meeting of the German Physical Society (Mainz, 1993), and to be published. R.D. Carlitz, J.C. Collins and A.H. Mueller, Phys. Lett B 214 (1988) 229. G. Altarelli and G.G. Ross, Phys. Lett. B 212 (1988) 391. L. Mankiewicz and A. Schfifer, Phys. Lett. B 242 (1990) 455. A.E. Dorokhov, N.I. Kochelev and Yu.A. Zubov, Int. J. Mod. Phys. A 8 (1993) 603. L.L. Frankfurt, M.I. Strikman, L. Mankiewicz, A. Sch~ifer, E. Rondio, A. Sandacz and V. Papavassiliou, Phys. Lett. B 230 (1989) 141. [9] L. Mankiewicz, A. Sch~ifer and M. Veltri, Comput. Phys. Commun. 71 (1992) 360. [10] M. Veltri, M. Diiren, L. Mankiewicz, K. Rith and A. Sch/ifer, Physics at HERA Workshop, Vol. 1, (DESY, 1992) p. 447. [11 ] K. Coulter et al., HERMES proposal, DESY/PRC 90/1 (1990). [12] M. D/iren and K. Rith, Physics at HERA Workshop, Vol. 1 (DESY, 1992) p. 427. [13] G.G. Ross and R. G. Roberts, The gluon contribution to polarized nucleon structure functions, preprint RAL-90-062 (1990); there are several misprints in this paper, please note that B should be 200 instead of 2 for the scenario with a large gluon contribution. [14] J.J. Aubert et al., Phys. Lett. B 160 (1985) 417. [ 15 ] M. Veltri, PhD Thesis, University of Heidelberg (1992) MPIH-V 18-1992. [16] E. Eichten, I. Hinchliffe, K. Lane and C. Quigg, Rev. Mod. Phys. 56 (1984) 579; 58 (1985) 1065; updated version (1986). [17] FNAL E581/704 Collab., D.L. Adams et al., Phys. Lett. B 261 (1991) 197.
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