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Production of K+ and K− as a test on models for large pT hadronic reactions
Volume 81B, number 1
PHYSICS LETTFRS
29 January 1979
PRODUCTION O F K + A N D K - AS A TEST ON MODELS FOR LARGE PT HADRONIC REACTIONS G6sta GUSTAFSON and Olle MA,NSSON
Department of Theoretical Physics, University of Lund, Sweden Received 7 November 1978
It is shmvn that the difference between K÷ and K- production is a good test on models for large PT hadronic reactions. If large PT events are due to hard quark-quark scattering, the scalebreaking effects expected from QCD are much smaller for tile difference K÷ - K- than for the individual K or n spectra in pp collisions at 90 °. A fit for data from ISR and FNAL to the form f(xT)/pq gives n = 5.4. Compared to the values n ~ 8 obtained for single particle spectra, this is much closer to n = 4 which is expected from scaling. We regard this as a support for the hard quark--quark scattering model.
In models for large PT hadronic reactions based on a hard scattering between elementary constituents, scaling arguments imply that the inclusive hadron spectra should have the form (for fixed angle 0) [1 ] E ( d 3 o/dp 3)
= f(xT)/p 4
x T = 2PT/X,'Cs-~
(1) (2)
In the special case o f q u a r k - q u a r k scattering the simple parton model gives the result (if transverse momenta o f the incoming quarks are neglected) [2,3] E d3°dp 3~
ff d x 1 dx2 Ul (x 1)u2(x2) D(z) 17r do($,d;i) (3)
(The summation over quark flavours is suppressed.) Here uz(xi) are the structure functions o f the incoming hadrons, D(z) is the fragmentation function for an outgoing quark and do/d i is the q u a r k - q u a r k scattering cross section, x 1 and x 2 denote the Bjorken-x variables o f the quarks before scattering, XT Fc__o_t._O!2+ t_g 0_/2 l z =-2- k Xl X2 3 is the fraction o f the outgoing quark m o m e n t u m carfled by the observed large-PT hadron, and g and [ are the kinematic invariants o f the q u a r k - q u a r k scattering. However, in QCD many different scalebreaking effects are expected to modify this result [4]. u(x, Q2),
D(z, Q2) and the coupling constant a ( Q 2) all depend on Q2. Non-vanishing transverse momenta for the incoming quarks also have a scalebreaking effect [5]. In the work by Feynman, Field and Fox [6] it has been shown that, if all possible scalebreaking effects are taken into account, QCD is consistent with the observed behaviour .d3o
1
for PT ~< 8 GeV/c. However, the result is sensitive to the unknown structure and fragmentation functions for the gluons and the transverse momentum distribution of the constituents. The result is therefore no compelling argument for QCD and hard scattering of quarks and gluons. In this letter we show that in pp collisions at 90 ° scalebreaking effects should be much less important in the difference K ÷ - K - than in the individual K or 7r spectra. The difference K + - K is thus expected to show the basic hard scattering much clearer and is therefore a suitable test for hard scattering models in general and QCD in particular. As discussed in detail below, the fact that QCD gives large scaling violating effects is connected to the fact that sea quarks with low energies, which are scattered in 90 ° as in fig. la, give very large contributions to the inclusive cross sections. These reactions corre75
Volume 81 B, number 1
PItYSICS LETTERS
29 January 1979
a
s
Iv
./. -
V
v
j
f V
.A-. "S
a b Fig. 1. (a) A sea quark, s, is scattered against a valence quark, v, and goes out in 90°. This corresponds to small values oft" and Q2. (b) The valence quark is scattered in 90° and the sea quark is scattered backwards. This corresponds to large t" and Q2 and a small cross section. spond to small values of [ and Q2 even if the energy is large. If the sea is assumed to be invariant under charge conjugation (and if the hard scattering does not contam flavour exchange) jets from sea quarks give as many K + as K - . This is also the case for gluon jets and for contributions from fusion processes like q?:l gluon ~ q'q'. We thus conclude that only valence quark jets contribute to the difference K ÷ - K - . To see that sea quarks give very large contributions to the inclusive cross sections we study the reaction in fig. la where a sea quark and a valence quark collide and the sea quark is scattered at 90 ° . First we note that small values o f x I imply small values of Iil and Q2. If the transverse momenta of the incoming quarks can be neglected the momentum transfer [ is given by
- -~s XlXT,
(4)
where qT = ~?T vtS-/2 is the transverse momentum of the outgoing quark. For ep collisions _ Q 2 is the mass of the virtual photon. The correct kinematic quantity to use for the process in fig. la is not quite clear, but it ought to be related to the momentum of the exchanged gluon, i.e. Q2 ~ _ [ .
(5)
(Ref. [4], which uses the same value of Q2 for t-, uand s-channel exchanges, chooses the symmetric quantity
0 2 = 2~ial(g
+
i + a).
(6))
Many factors favour low values o f x 1 , I[I and Q2: (1) The structure function is proportional to 1/x 1 . (2) The hard scattering cross section do~d[ contains a factor 1/[2 from gluon exchange. (3) The coupling constant a(Q 2) becomes large ~ ( a 2 ) ~x l/(ln(Q2/A 2) +_C), 76
(7)
d o / d / i s proportional to a(Q2). From energy-momentum conservation we obtain, if we again neglect the transverse momenta of the incoming quarks i
^
t
~
i
~
2
(X 1 -- ~XT)(X 2 -- ~XT) = (~XT) .
(8)
Thus a low value o f x 1 has to be compensated by a large value of x 2. Although this is suppressed by a factor ~(1 - x2)3 the above enhancement factors make low x 1-values very important. From eq. (8) follows that x 1 has a lower limit ~ ? T / 2 , but if the quarks may have non-zero transverse momenta,/~T, before the scattering, even smaller values o f x I are possible. The result is that low values of Q2 give very important contributions to the total inclusive cross section. This means that scalebreaking effects from the Q2-clependence of u(x, Q2), D(z, Q2) and a(Q 2) become large. Also for small values of Q2 lowest order perturbation theory is not reliable. The result becomes sensitive to the parameters A and C in eq. (8) as well as a cut off at low values of [. In ref. [6] 1/[ 2 is replaced by 1 / ( [ - M02)2 with M 0 = 1 GeV. The effects of a broad k T distribution is much larger than one might immediately think, due to the fact that lower values of Q2 are possible. This means that in order to study the scalebreaking effects expected in QCD it is necessary to take all effects into account at the same time, as is done in ref. [6]. The situation is very similar if the sea quark is replaced by a gluon. The structure function for gluons is also supposed to behave ~ l / x for small x-values. However, we note that to obtain the described effect it is necessary that the sea quark goes out at 90 ° , whereas the valence quark is scattered in a small angle. In the situation shown in fig. lb, where the valence quark goes out at 90 ° , whereas the sea quark is scattered backwards, we have a very large value of [ and thus this process is very much suppressed compared to the one in fig. 1a.
Volume 81B, number 1
PHYSICS LETTERS
Thus, if w, aand to have a valence quark at 90 ° we do not obt .,~Llarge contributions if there is a sea quark or a soft gluon on the opposite side. Instead, collisions between two valence quarks are more important. (Collisions between a valence quark and a hard gluon might also contribute. The structure function for the gluons is not known but it is often assumed to be less suppressed at larze x-values than that for the sea quarks. Ref. [6] assumes it to be (1 + 9x)(1 - x)4/x at Q2 = 4 GeV2.) We conclude that for a valence quark jet at 90 ° we do not obtain important contributions from low Q2. or low i-values. The region of important Q2-values is smaller and thus the variation with Q2 of u(x, Q2), D(x, Q2) and a(Q 2) is much smaller. Also the magnitude o f a ( Q 2) is smaller which makes lowest order perturbation theory more reliable. Large k T do not get enhanced in the way described above and the result is thus less sensitive to the kT-distribution. Finally the cut-off parameter M 0 in the propagator 1/([ - M 2) is not so important. Our conclusion is thus that for the difference K ÷ - K - , which does not obtain contributions from sea quarks jets or gluon jets, the scalebreaking effects should be much smaller than for the inclusive K ÷- or K--spectra. The scalebreaking effects that remain should also be more reliably calculable within QCD. Besides being less sensitive to the kT-distribution and cut-off parameters, the difference K + - K - is also less sensitive to the unknown gluon structure- and fragmentation functions. We have fitted pp-scattering data from ISR [7] and Fermilab [8] for PT ~> 0.5 GeV/c to the form E d3°(K+) dp3
E d 3 ° ( K - ) - f(xT) dp3
f(XT) : AXT(I -- XT)m.
(9)
n
PT (10)
The result of the fit is n = 5.4, m : 13.4,A : 0.59, with X2 = 148 for 65 degrees of freedom. The quality of the fit is seen in fig. 2, where p5.4 [o(K +) _ o ( K - ) ] is plotted against x T. We stress that the points are not expected to fall exactly on the same curve, because the effects o f scalebreaking is not just to add extra powers o f 1/PT. However, the difference between 4 and the value of n, which gives the best fit, ought to be a measure of the scalebreaking effects.
29 January 1979
(K'- K-)
p5.4 , ~ IGeV) ;~ 19.4} • 23.8 F N A L ,'. 27.4
! I
• 45.01 5 3 . 0 ~ ISR
10-1] , :l @," : ' •
~ 63.0"
%
%
10-a.l
.
.
4 IO'L
10 5~
\\ •
0'.5
o.o
x,
Fig. 2. p~4 (E d3°(K+)dp 3
E d3°(K-)~dp3--/
is plotted against x T. The data are from refs. [7] and [8]. For the ISR data only the higher energy points are shown (x/s = 45, 53 and 63 GeV). llowever, also the data at x/s = 23 and 31 GeV have been used in the fit, which is shown by the solid line. Because there is no background from ordinary low PT particle (the central plateau is almost charge symmetric), we do not need a large cut-off in PT" The value of x2/DOF is not improved if we only use data for PT /> 1 GeV/c. Different forms for f(xT) were tried, but none gave a significantly better fit. A test with the function f(XT) = A(1 - XT) m did not give a good fit. In fact, a fit with f(XT) = AXlT(1 --XT )m with l free, gave the result l = 1.0. In our opinion the fact that n is so close to 4 is a strong indication that the difference K + - K - at 90 ° is due to a hard scattering in which a valence quark produces a jet of hadrons. In principle it should be possible to analyze the difference 7r+ - ~r- in the same way. However, because 7r- can easily be produced in a d-quark jet whereas K has no valence quark in common with a proton, the relative difference between n + and lr- is smaller than between K ÷ and K - . Furthermore, the fact that the fragmentation functions for pions are more peaked at low z-values than the fragmentation functions for kaons, implies that the function f(xT) in eq. (9) becomes more steeply falling for pions and the separation 77
Volume 81 B, number 1
PHYSICS LETTERS
into the factors 1/p~. and f ( x T ) becomes more difficult. In fact, in the ISR data the errors are larger than the difference rr÷ - ~r-. The difference K ÷ - K - also gives an opportunity to study the opposite side jet in a fairly clean sample of collisions between two valence quarks. Thus for the difference between the results with a K ÷ or a K - trigger (weighted with their inclusive cross sections) we expect that the distribution for the opposite side jets should be more concentrated around 90 ° . Furthermore, it should be interesting to study e.g. the transverse m o m e n t u m balance between the jets and q u a n t u m number and Pout distributions in this way.
References [1] S.M. Berman, J.D. Bjorken and J.B. Kogut, Phys. Rev. D4 (1971) 3388.
78
29 January 1979
[2] S.D. Ellis and M.B. Kisslinger, Phys. Rev. D9 (1974) 2027. [3] R.D. Field and R.P. Feynman, Phys. Rev. D15 (1977) 2590. [4] A.P. Contogouris, R. Gaskell and A. Nicolaidis, Phys. Rev. D17 (1978) 839; A.P. Contogouris, R. Gaskell and S. Papadopoulos, Phys. Rev. DI7 (1978) 2314; B.L. Combridge, J. Kripfganz and J. Ranft; Phys. Lett. 70B (1977) 234. [5] M. Fontannaz and D. Sehiff, Nucl. Phys. B132 (1978) 457; R. Baier and B. Petersson, Bielefeld preprint 77/10. [6] R.P. Feynman, R.D. Field and G.C. Fox, CalTech preprint CALT-68-651 (1978). [71 B. Alper et al., Nucl. Phys. B100 (1975) 237. [8] D. Antreasyan et al., Phys. Rev. Lett. 38 (1977) 112 and 115.
PHYSICS LETTFRS
29 January 1979
PRODUCTION O F K + A N D K - AS A TEST ON MODELS FOR LARGE PT HADRONIC REACTIONS G6sta GUSTAFSON and Olle MA,NSSON
Department of Theoretical Physics, University of Lund, Sweden Received 7 November 1978
It is shmvn that the difference between K÷ and K- production is a good test on models for large PT hadronic reactions. If large PT events are due to hard quark-quark scattering, the scalebreaking effects expected from QCD are much smaller for tile difference K÷ - K- than for the individual K or n spectra in pp collisions at 90 °. A fit for data from ISR and FNAL to the form f(xT)/pq gives n = 5.4. Compared to the values n ~ 8 obtained for single particle spectra, this is much closer to n = 4 which is expected from scaling. We regard this as a support for the hard quark--quark scattering model.
In models for large PT hadronic reactions based on a hard scattering between elementary constituents, scaling arguments imply that the inclusive hadron spectra should have the form (for fixed angle 0) [1 ] E ( d 3 o/dp 3)
= f(xT)/p 4
x T = 2PT/X,'Cs-~
(1) (2)
In the special case o f q u a r k - q u a r k scattering the simple parton model gives the result (if transverse momenta o f the incoming quarks are neglected) [2,3] E d3°dp 3~
ff d x 1 dx2 Ul (x 1)u2(x2) D(z) 17r do($,d;i) (3)
(The summation over quark flavours is suppressed.) Here uz(xi) are the structure functions o f the incoming hadrons, D(z) is the fragmentation function for an outgoing quark and do/d i is the q u a r k - q u a r k scattering cross section, x 1 and x 2 denote the Bjorken-x variables o f the quarks before scattering, XT Fc__o_t._O!2+ t_g 0_/2 l z =-2- k Xl X2 3 is the fraction o f the outgoing quark m o m e n t u m carfled by the observed large-PT hadron, and g and [ are the kinematic invariants o f the q u a r k - q u a r k scattering. However, in QCD many different scalebreaking effects are expected to modify this result [4]. u(x, Q2),
D(z, Q2) and the coupling constant a ( Q 2) all depend on Q2. Non-vanishing transverse momenta for the incoming quarks also have a scalebreaking effect [5]. In the work by Feynman, Field and Fox [6] it has been shown that, if all possible scalebreaking effects are taken into account, QCD is consistent with the observed behaviour .d3o
1
for PT ~< 8 GeV/c. However, the result is sensitive to the unknown structure and fragmentation functions for the gluons and the transverse momentum distribution of the constituents. The result is therefore no compelling argument for QCD and hard scattering of quarks and gluons. In this letter we show that in pp collisions at 90 ° scalebreaking effects should be much less important in the difference K ÷ - K - than in the individual K or 7r spectra. The difference K + - K is thus expected to show the basic hard scattering much clearer and is therefore a suitable test for hard scattering models in general and QCD in particular. As discussed in detail below, the fact that QCD gives large scaling violating effects is connected to the fact that sea quarks with low energies, which are scattered in 90 ° as in fig. la, give very large contributions to the inclusive cross sections. These reactions corre75
Volume 81 B, number 1
PItYSICS LETTERS
29 January 1979
a
s
Iv
./. -
V
v
j
f V
.A-. "S
a b Fig. 1. (a) A sea quark, s, is scattered against a valence quark, v, and goes out in 90°. This corresponds to small values oft" and Q2. (b) The valence quark is scattered in 90° and the sea quark is scattered backwards. This corresponds to large t" and Q2 and a small cross section. spond to small values of [ and Q2 even if the energy is large. If the sea is assumed to be invariant under charge conjugation (and if the hard scattering does not contam flavour exchange) jets from sea quarks give as many K + as K - . This is also the case for gluon jets and for contributions from fusion processes like q?:l gluon ~ q'q'. We thus conclude that only valence quark jets contribute to the difference K ÷ - K - . To see that sea quarks give very large contributions to the inclusive cross sections we study the reaction in fig. la where a sea quark and a valence quark collide and the sea quark is scattered at 90 ° . First we note that small values o f x I imply small values of Iil and Q2. If the transverse momenta of the incoming quarks can be neglected the momentum transfer [ is given by
- -~s XlXT,
(4)
where qT = ~?T vtS-/2 is the transverse momentum of the outgoing quark. For ep collisions _ Q 2 is the mass of the virtual photon. The correct kinematic quantity to use for the process in fig. la is not quite clear, but it ought to be related to the momentum of the exchanged gluon, i.e. Q2 ~ _ [ .
(5)
(Ref. [4], which uses the same value of Q2 for t-, uand s-channel exchanges, chooses the symmetric quantity
0 2 = 2~ial(g
+
i + a).
(6))
Many factors favour low values o f x 1 , I[I and Q2: (1) The structure function is proportional to 1/x 1 . (2) The hard scattering cross section do~d[ contains a factor 1/[2 from gluon exchange. (3) The coupling constant a(Q 2) becomes large ~ ( a 2 ) ~x l/(ln(Q2/A 2) +_C), 76
(7)
d o / d / i s proportional to a(Q2). From energy-momentum conservation we obtain, if we again neglect the transverse momenta of the incoming quarks i
^
t
~
i
~
2
(X 1 -- ~XT)(X 2 -- ~XT) = (~XT) .
(8)
Thus a low value o f x 1 has to be compensated by a large value of x 2. Although this is suppressed by a factor ~(1 - x2)3 the above enhancement factors make low x 1-values very important. From eq. (8) follows that x 1 has a lower limit ~ ? T / 2 , but if the quarks may have non-zero transverse momenta,/~T, before the scattering, even smaller values o f x I are possible. The result is that low values of Q2 give very important contributions to the total inclusive cross section. This means that scalebreaking effects from the Q2-clependence of u(x, Q2), D(z, Q2) and a(Q 2) become large. Also for small values of Q2 lowest order perturbation theory is not reliable. The result becomes sensitive to the parameters A and C in eq. (8) as well as a cut off at low values of [. In ref. [6] 1/[ 2 is replaced by 1 / ( [ - M02)2 with M 0 = 1 GeV. The effects of a broad k T distribution is much larger than one might immediately think, due to the fact that lower values of Q2 are possible. This means that in order to study the scalebreaking effects expected in QCD it is necessary to take all effects into account at the same time, as is done in ref. [6]. The situation is very similar if the sea quark is replaced by a gluon. The structure function for gluons is also supposed to behave ~ l / x for small x-values. However, we note that to obtain the described effect it is necessary that the sea quark goes out at 90 ° , whereas the valence quark is scattered in a small angle. In the situation shown in fig. lb, where the valence quark goes out at 90 ° , whereas the sea quark is scattered backwards, we have a very large value of [ and thus this process is very much suppressed compared to the one in fig. 1a.
Volume 81B, number 1
PHYSICS LETTERS
Thus, if w, aand to have a valence quark at 90 ° we do not obt .,~Llarge contributions if there is a sea quark or a soft gluon on the opposite side. Instead, collisions between two valence quarks are more important. (Collisions between a valence quark and a hard gluon might also contribute. The structure function for the gluons is not known but it is often assumed to be less suppressed at larze x-values than that for the sea quarks. Ref. [6] assumes it to be (1 + 9x)(1 - x)4/x at Q2 = 4 GeV2.) We conclude that for a valence quark jet at 90 ° we do not obtain important contributions from low Q2. or low i-values. The region of important Q2-values is smaller and thus the variation with Q2 of u(x, Q2), D(x, Q2) and a(Q 2) is much smaller. Also the magnitude o f a ( Q 2) is smaller which makes lowest order perturbation theory more reliable. Large k T do not get enhanced in the way described above and the result is thus less sensitive to the kT-distribution. Finally the cut-off parameter M 0 in the propagator 1/([ - M 2) is not so important. Our conclusion is thus that for the difference K ÷ - K - , which does not obtain contributions from sea quarks jets or gluon jets, the scalebreaking effects should be much smaller than for the inclusive K ÷- or K--spectra. The scalebreaking effects that remain should also be more reliably calculable within QCD. Besides being less sensitive to the kT-distribution and cut-off parameters, the difference K + - K - is also less sensitive to the unknown gluon structure- and fragmentation functions. We have fitted pp-scattering data from ISR [7] and Fermilab [8] for PT ~> 0.5 GeV/c to the form E d3°(K+) dp3
E d 3 ° ( K - ) - f(xT) dp3
f(XT) : AXT(I -- XT)m.
(9)
n
PT (10)
The result of the fit is n = 5.4, m : 13.4,A : 0.59, with X2 = 148 for 65 degrees of freedom. The quality of the fit is seen in fig. 2, where p5.4 [o(K +) _ o ( K - ) ] is plotted against x T. We stress that the points are not expected to fall exactly on the same curve, because the effects o f scalebreaking is not just to add extra powers o f 1/PT. However, the difference between 4 and the value of n, which gives the best fit, ought to be a measure of the scalebreaking effects.
29 January 1979
(K'- K-)
p5.4 , ~ IGeV) ;~ 19.4} • 23.8 F N A L ,'. 27.4
! I
• 45.01 5 3 . 0 ~ ISR
10-1] , :l @," : ' •
~ 63.0"
%
%
10-a.l
.
.
4 IO'L
10 5~
\\ •
0'.5
o.o
x,
Fig. 2. p~4 (E d3°(K+)dp 3
E d3°(K-)~dp3--/
is plotted against x T. The data are from refs. [7] and [8]. For the ISR data only the higher energy points are shown (x/s = 45, 53 and 63 GeV). llowever, also the data at x/s = 23 and 31 GeV have been used in the fit, which is shown by the solid line. Because there is no background from ordinary low PT particle (the central plateau is almost charge symmetric), we do not need a large cut-off in PT" The value of x2/DOF is not improved if we only use data for PT /> 1 GeV/c. Different forms for f(xT) were tried, but none gave a significantly better fit. A test with the function f(XT) = A(1 - XT) m did not give a good fit. In fact, a fit with f(XT) = AXlT(1 --XT )m with l free, gave the result l = 1.0. In our opinion the fact that n is so close to 4 is a strong indication that the difference K + - K - at 90 ° is due to a hard scattering in which a valence quark produces a jet of hadrons. In principle it should be possible to analyze the difference 7r+ - ~r- in the same way. However, because 7r- can easily be produced in a d-quark jet whereas K has no valence quark in common with a proton, the relative difference between n + and lr- is smaller than between K ÷ and K - . Furthermore, the fact that the fragmentation functions for pions are more peaked at low z-values than the fragmentation functions for kaons, implies that the function f(xT) in eq. (9) becomes more steeply falling for pions and the separation 77
Volume 81 B, number 1
PHYSICS LETTERS
into the factors 1/p~. and f ( x T ) becomes more difficult. In fact, in the ISR data the errors are larger than the difference rr÷ - ~r-. The difference K ÷ - K - also gives an opportunity to study the opposite side jet in a fairly clean sample of collisions between two valence quarks. Thus for the difference between the results with a K ÷ or a K - trigger (weighted with their inclusive cross sections) we expect that the distribution for the opposite side jets should be more concentrated around 90 ° . Furthermore, it should be interesting to study e.g. the transverse m o m e n t u m balance between the jets and q u a n t u m number and Pout distributions in this way.
References [1] S.M. Berman, J.D. Bjorken and J.B. Kogut, Phys. Rev. D4 (1971) 3388.
78
29 January 1979
[2] S.D. Ellis and M.B. Kisslinger, Phys. Rev. D9 (1974) 2027. [3] R.D. Field and R.P. Feynman, Phys. Rev. D15 (1977) 2590. [4] A.P. Contogouris, R. Gaskell and A. Nicolaidis, Phys. Rev. D17 (1978) 839; A.P. Contogouris, R. Gaskell and S. Papadopoulos, Phys. Rev. DI7 (1978) 2314; B.L. Combridge, J. Kripfganz and J. Ranft; Phys. Lett. 70B (1977) 234. [5] M. Fontannaz and D. Sehiff, Nucl. Phys. B132 (1978) 457; R. Baier and B. Petersson, Bielefeld preprint 77/10. [6] R.P. Feynman, R.D. Field and G.C. Fox, CalTech preprint CALT-68-651 (1978). [71 B. Alper et al., Nucl. Phys. B100 (1975) 237. [8] D. Antreasyan et al., Phys. Rev. Lett. 38 (1977) 112 and 115.