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Large-PT jet production in QCD
Volume 77B, number 3
PHYSICS LETTERS
14 August 1978
LARGE-P T JET PRODUCTION IN QCD W. FURMANSKI
Institute of Physics, Jagellonian University, Cracow, Poland Received 3 May 1978
The contribution from qq ~ qq + X scattering to the large-PT jet cross section in hadronic collisions is calculated in quantum chromodynamics to order O (g6). The result finds a suggestive interpretation in the Parisi-Altarelli parton model, which yields also the general formula, including all quark-gluon subprocesses.
It has been pointed out recently by Sterman and Weinberg [1 ] that the production of jets can be calculated perturbatively in quantum chromodynamics. Following the method proposed in ref. [1] we derived the inclusive cross section for production of a large-PT jet in h a d r o n - h a d r o n collisions. From the large number of various elementary q u a r k - g h i o n subprocesses we have chosen the simplest one, i.e. the qq -+ qq + X scattering (initial quarks with different flavours). T h e relevant diagrams, contributing to the order O(g 6) are presented in fig. 1. The calculation was done in massless QCD with logarithmic accuracy, i.e. we neglect terms O ( a " const), retaining only those which may be dangerous for the perturbation expansion at large PT: O ( a in PT)' O ( a in 6) (a = g2/4n is the running coupling constant of QCD, 6 is the halfangle of the jet detector). The essential steps of the calculation are the following (a more detailed discussion is presented in ref. [2]): (a) Ultraviolet renormalization is done at/~ = PTThis enables us to drop the UV logarithms and to use the effective coupling constant a(PT). (b) Infrared divergences are regularized by putting quarks a little bit off-shell. (c) All double-logarithmic terms cancel after the real and virtual gluon corrections o f the same topology are added up. (d) The single-logarithmic terms generated by the final state quark cancel after all jet configurations are included (jet = quark, jet = gluon, jet = quark + gluon). Jet is defined in a similar way as in the Sterman-Wein312
L--v-
E_ i .....
Fig. 1. berg calculation, i.e. as a group of particles with total momentum p j and with an angular spread of less than 28,1 #1 The solid angle detector is not a very convenient device for the large-PT calculations, since its opening angle changes when we boost from hadron-hadron to parton-parton frame. However, the effect can be neglected with logarithmic accuracy. The invariant jet geometry, more appropriate for large-PT jets is discussed in ref. [2].
Volume 77B, number 3
PHYSICS LETTERS
(e) The infrared divergences connected with the initial state quarks can be absorbed into the hadronic structure functions, generating the PT dependence of the latter (see refs. [3] where this point was discussed and ref. [2] for details in our case). The final result for initial hadrons HI, H 2 and quarks with flavours A, B can be written as follows ,2.
14 August 1978
The hadronic structure functions G obey the renormalization group equation:
PT( d/ dp T) GN(p T) = (o~(PT)/27r)P~qqGN (pT) , where 1
1
GN(pT)=f dxxNa(x, PT)' pN= qq f
Ej do/d3pj
0 1
1
=f dxi f aX2aH,(Xl, PTj)GH2IB(X2 ' P T j ) o
o 1
1
×f
f
o
o
drBPo(rA)6( 1 - r A - r B ) (1)
1
X ; dx x 4 6 (7"1 --
XX2TA)5(T 2 -- xx1T B)
o × [ 5 ( 1 - x ) + ff~rrrPq(X)(-21n 6)] + (A "~ B) , where
.r1 = (pTj/X/~)e-YJ ,
r 2 = (pTj/Nfs)eYJ ,
P0(r) = [a(PTj)/p2j]2(C2/N) × [r2(1 + r 2) + (1 - "/')2(1 + (1 -- "/')2)1 ,
N = number of colours, C 2 = (N 2 - 1)/2N, Pq(X) = Pqq(X) + PqG(X),
[_:
1 +x2]
Pqq(X) = C2 5(1 - x ) + 0~-~+_l,
The distribution 1/(1 - x ) + is defined by the integration prescription: 1
f0
1
f(x) 0
f(x) - f(1) 1-x
,2 We neglect the transverse momenta of initial quarks.
dxxNpqq(X).
0
We wish to add the following remarks concerning formula (1): (a) The angular distribution of the jet cross section is determined by P0(r), i.e. by the Born cross section for qq -~ qq elastic scattering. Thus, the double-scattering diagrams do not contribute at large PT : they give only double-logarithmic terms which cancel with corresponding gluon bremsstrahlung corrections. Physically, it means that the large momentum is transferred by one gluon, the other one being predominantly soft. (b) The functions Pqq(X), PqG(X) are known from the parton model analysis of the Bjorken scaling violation in QCD, as done by Parisi and Altarelli [4]. Thus, the radiative corrections in our calculation generate exactly the same PT dependence of the structure functions as derived for the non-singlet sector in deep inelastic electroproduction. (c) The O(a) correction term can be also expressed by the Parisi-Altarelh functions. This suggests that the parton model rederivation of formula (1) should be possible. We discuss this point in more detail in ref. [5], here we limit ourselves to a brief argument. The quark, after being scattered at wide angle, may do nothing else with probability P0, may emit a soft gluon (with momentum fraction less than e) with probability Ps, may emit a parallel gluon (with quark-gluon opening angle 0 less than 25) with probability Pit and, finally, it may emit a hard, non-paralM gluon with probability P2" The last probability is free of infrared divergences and can be derived explicitly using the Parisi-Altarelli functions. We get [5]: P2(x) = ~ P q q ( X ) ( - 2 In 5)
'
/~q(X) = C 2
1 + x 2 1--x'
for the final quark with fraction x of the parent momentum. For sufficiently small e and 6 the single-jet inclusive distribution in the parent quark is 313
Volume 77B, number 3
PHYSICS LETTERS
dij dx -- (Po +e, + e)6(1 -x) +_~_~ 2n" [Pq(X) + P q ( 1 - x ) ] ( - 2 In 6)0 (1 - x - e). ~
Using the m o m e n t u m sum rule we may determine P0 +etl + P = 1 -
(a/27r)C2(41neln8
+ 3 ln6).
The e cut-off may be replaced by the (1 - x ) + prescription and we get
dNj dx = 6(1
- x ) + (a/27r)eq(X)(-2
in 6 ) .
Taking the qq ~ qq Born term and using the p a r e n t child relation we arrive at formula (1). The Sterman-Weinberg formula can be rederived even more directly: since the process considered in ref. [1] is almost exclusive, only t h e P 0 +P~ +Ps terms in both jets contribute and we get at once ,3.
o(E, O, ~2, e, 6) = (do/d~2)Born'~2"(PO +e[i +Ps )2
:
""'E
~2 (31n6+41n6
(d) The argument sketched above can be repeated for any other large-PT subprocess. This allows to suggest the general expression for the jet cross section without further tedious calculations: one has simply to replace Pq(x) in formula (1) by Pc(X) = ~FPcF(x) and sum over all QCD Born subprocesses A + B -~ C + X (sums go over all allowed q u a r k - g l u o n configurations, the functions FCF(X) should be taken from the Parisi-Altarelli analysis, the full PT dependence of the structure functions, derived from electroproduction, should be included). (e) Recently, the O ( g 6) calculation of the singleparticle inclusive distribution at large PT was done by Sachrajda [6] with the conclusion that all radiative corrections can be translated into the PT dependence
,s We have to replace e by 2e in order to conform to the normalization used in ref. [1 ].
314
14 August 1978
of the initial and final state hadronic structure functions. As is known, however, the formula: Born cross section plus scaling violating corrections to order a gives a poor fit to the large-PT data below pT = 10 GeV [7]. Thus the QCD analysis o f the single particle spectra at large PT has not solved, up to now, the old puzzle: why Bjorken scaling starts at 2 GeV whereas the large PT scaling fails even at 10 GeV? We expect that a detailed numerical analysis of the jet cross-section formulas may provide some new information about what happens at large PT. The freedom in the description of the final state is reduced to a minimum in the jet case: we get a well-defined series in a with coefficients determined by the jet geometry and the magnitude of the O(a) corrections can be compared in a direct way with the Born term. 1 am very grateful to Professor A. Biatas and Professor K. Zalewski for encouragement in the course of this work, for useful comments, remarks and discussions and for the critical reading of the manuscript. I am also indebted to T.R. Taylor for an illuminating discussion.
References [1 ] G. Sterman and S. Weinberg, Phys. Rev. Lett. 39 (1977) 1436; G. Sterman, Stony Brook preprints ITP-SB-77-69, ITP-SB77-72. [ 2] W. Furmafiski, Jagellonian Univ. preprint TPJU. [3] J. Kogut and J. Shigemitsu, Cornell Univ. preprint (1977); H.D. Politzer, Nucl. Phys. B129 (1977) 301; A.V. Radushkin, Phys. Lett. 69B (1977) 245; G. AltareUi, G. Parisi and R. Petronzio, Phys. Lett. 76B (1978) 351; K.H. Craig and C.H. LleweUyn-Smith, Phys. Lett. 72B (1978) 349; C.T. Sachrajda, Phys. Lett. 73B (1978) 185. [4] G. Parisi, Proc. ll~me Rencontre de Moriond (1976), ed. J. Tran Thanh Van; G. Altarelli and G. Parisi, Nucl. Phys. B126 (1977) 298. [5 ] W..Furmafiski, Jagellonian Univ. preprint TPJU. [6] C.T. Sachrajda, Phys. Lett. 76B (1978) 100. [7] B.L. Combridge, J. Kripfganz and J. Ranft, Phys. Lett. 70B (1977) 234.
PHYSICS LETTERS
14 August 1978
LARGE-P T JET PRODUCTION IN QCD W. FURMANSKI
Institute of Physics, Jagellonian University, Cracow, Poland Received 3 May 1978
The contribution from qq ~ qq + X scattering to the large-PT jet cross section in hadronic collisions is calculated in quantum chromodynamics to order O (g6). The result finds a suggestive interpretation in the Parisi-Altarelli parton model, which yields also the general formula, including all quark-gluon subprocesses.
It has been pointed out recently by Sterman and Weinberg [1 ] that the production of jets can be calculated perturbatively in quantum chromodynamics. Following the method proposed in ref. [1] we derived the inclusive cross section for production of a large-PT jet in h a d r o n - h a d r o n collisions. From the large number of various elementary q u a r k - g h i o n subprocesses we have chosen the simplest one, i.e. the qq -+ qq + X scattering (initial quarks with different flavours). T h e relevant diagrams, contributing to the order O(g 6) are presented in fig. 1. The calculation was done in massless QCD with logarithmic accuracy, i.e. we neglect terms O ( a " const), retaining only those which may be dangerous for the perturbation expansion at large PT: O ( a in PT)' O ( a in 6) (a = g2/4n is the running coupling constant of QCD, 6 is the halfangle of the jet detector). The essential steps of the calculation are the following (a more detailed discussion is presented in ref. [2]): (a) Ultraviolet renormalization is done at/~ = PTThis enables us to drop the UV logarithms and to use the effective coupling constant a(PT). (b) Infrared divergences are regularized by putting quarks a little bit off-shell. (c) All double-logarithmic terms cancel after the real and virtual gluon corrections o f the same topology are added up. (d) The single-logarithmic terms generated by the final state quark cancel after all jet configurations are included (jet = quark, jet = gluon, jet = quark + gluon). Jet is defined in a similar way as in the Sterman-Wein312
L--v-
E_ i .....
Fig. 1. berg calculation, i.e. as a group of particles with total momentum p j and with an angular spread of less than 28,1 #1 The solid angle detector is not a very convenient device for the large-PT calculations, since its opening angle changes when we boost from hadron-hadron to parton-parton frame. However, the effect can be neglected with logarithmic accuracy. The invariant jet geometry, more appropriate for large-PT jets is discussed in ref. [2].
Volume 77B, number 3
PHYSICS LETTERS
(e) The infrared divergences connected with the initial state quarks can be absorbed into the hadronic structure functions, generating the PT dependence of the latter (see refs. [3] where this point was discussed and ref. [2] for details in our case). The final result for initial hadrons HI, H 2 and quarks with flavours A, B can be written as follows ,2.
14 August 1978
The hadronic structure functions G obey the renormalization group equation:
PT( d/ dp T) GN(p T) = (o~(PT)/27r)P~qqGN (pT) , where 1
1
GN(pT)=f dxxNa(x, PT)' pN= qq f
Ej do/d3pj
0 1
1
=f dxi f aX2aH,(Xl, PTj)GH2IB(X2 ' P T j ) o
o 1
1
×f
f
o
o
drBPo(rA)6( 1 - r A - r B ) (1)
1
X ; dx x 4 6 (7"1 --
XX2TA)5(T 2 -- xx1T B)
o × [ 5 ( 1 - x ) + ff~rrrPq(X)(-21n 6)] + (A "~ B) , where
.r1 = (pTj/X/~)e-YJ ,
r 2 = (pTj/Nfs)eYJ ,
P0(r) = [a(PTj)/p2j]2(C2/N) × [r2(1 + r 2) + (1 - "/')2(1 + (1 -- "/')2)1 ,
N = number of colours, C 2 = (N 2 - 1)/2N, Pq(X) = Pqq(X) + PqG(X),
[_:
1 +x2]
Pqq(X) = C2 5(1 - x ) + 0~-~+_l,
The distribution 1/(1 - x ) + is defined by the integration prescription: 1
f0
1
f(x) 0
f(x) - f(1) 1-x
,2 We neglect the transverse momenta of initial quarks.
dxxNpqq(X).
0
We wish to add the following remarks concerning formula (1): (a) The angular distribution of the jet cross section is determined by P0(r), i.e. by the Born cross section for qq -~ qq elastic scattering. Thus, the double-scattering diagrams do not contribute at large PT : they give only double-logarithmic terms which cancel with corresponding gluon bremsstrahlung corrections. Physically, it means that the large momentum is transferred by one gluon, the other one being predominantly soft. (b) The functions Pqq(X), PqG(X) are known from the parton model analysis of the Bjorken scaling violation in QCD, as done by Parisi and Altarelli [4]. Thus, the radiative corrections in our calculation generate exactly the same PT dependence of the structure functions as derived for the non-singlet sector in deep inelastic electroproduction. (c) The O(a) correction term can be also expressed by the Parisi-Altarelh functions. This suggests that the parton model rederivation of formula (1) should be possible. We discuss this point in more detail in ref. [5], here we limit ourselves to a brief argument. The quark, after being scattered at wide angle, may do nothing else with probability P0, may emit a soft gluon (with momentum fraction less than e) with probability Ps, may emit a parallel gluon (with quark-gluon opening angle 0 less than 25) with probability Pit and, finally, it may emit a hard, non-paralM gluon with probability P2" The last probability is free of infrared divergences and can be derived explicitly using the Parisi-Altarelli functions. We get [5]: P2(x) = ~ P q q ( X ) ( - 2 In 5)
'
/~q(X) = C 2
1 + x 2 1--x'
for the final quark with fraction x of the parent momentum. For sufficiently small e and 6 the single-jet inclusive distribution in the parent quark is 313
Volume 77B, number 3
PHYSICS LETTERS
dij dx -- (Po +e, + e)6(1 -x) +_~_~ 2n" [Pq(X) + P q ( 1 - x ) ] ( - 2 In 6)0 (1 - x - e). ~
Using the m o m e n t u m sum rule we may determine P0 +etl + P = 1 -
(a/27r)C2(41neln8
+ 3 ln6).
The e cut-off may be replaced by the (1 - x ) + prescription and we get
dNj dx = 6(1
- x ) + (a/27r)eq(X)(-2
in 6 ) .
Taking the qq ~ qq Born term and using the p a r e n t child relation we arrive at formula (1). The Sterman-Weinberg formula can be rederived even more directly: since the process considered in ref. [1] is almost exclusive, only t h e P 0 +P~ +Ps terms in both jets contribute and we get at once ,3.
o(E, O, ~2, e, 6) = (do/d~2)Born'~2"(PO +e[i +Ps )2
:
""'E
~2 (31n6+41n6
(d) The argument sketched above can be repeated for any other large-PT subprocess. This allows to suggest the general expression for the jet cross section without further tedious calculations: one has simply to replace Pq(x) in formula (1) by Pc(X) = ~FPcF(x) and sum over all QCD Born subprocesses A + B -~ C + X (sums go over all allowed q u a r k - g l u o n configurations, the functions FCF(X) should be taken from the Parisi-Altarelli analysis, the full PT dependence of the structure functions, derived from electroproduction, should be included). (e) Recently, the O ( g 6) calculation of the singleparticle inclusive distribution at large PT was done by Sachrajda [6] with the conclusion that all radiative corrections can be translated into the PT dependence
,s We have to replace e by 2e in order to conform to the normalization used in ref. [1 ].
314
14 August 1978
of the initial and final state hadronic structure functions. As is known, however, the formula: Born cross section plus scaling violating corrections to order a gives a poor fit to the large-PT data below pT = 10 GeV [7]. Thus the QCD analysis o f the single particle spectra at large PT has not solved, up to now, the old puzzle: why Bjorken scaling starts at 2 GeV whereas the large PT scaling fails even at 10 GeV? We expect that a detailed numerical analysis of the jet cross-section formulas may provide some new information about what happens at large PT. The freedom in the description of the final state is reduced to a minimum in the jet case: we get a well-defined series in a with coefficients determined by the jet geometry and the magnitude of the O(a) corrections can be compared in a direct way with the Born term. 1 am very grateful to Professor A. Biatas and Professor K. Zalewski for encouragement in the course of this work, for useful comments, remarks and discussions and for the critical reading of the manuscript. I am also indebted to T.R. Taylor for an illuminating discussion.
References [1 ] G. Sterman and S. Weinberg, Phys. Rev. Lett. 39 (1977) 1436; G. Sterman, Stony Brook preprints ITP-SB-77-69, ITP-SB77-72. [ 2] W. Furmafiski, Jagellonian Univ. preprint TPJU. [3] J. Kogut and J. Shigemitsu, Cornell Univ. preprint (1977); H.D. Politzer, Nucl. Phys. B129 (1977) 301; A.V. Radushkin, Phys. Lett. 69B (1977) 245; G. AltareUi, G. Parisi and R. Petronzio, Phys. Lett. 76B (1978) 351; K.H. Craig and C.H. LleweUyn-Smith, Phys. Lett. 72B (1978) 349; C.T. Sachrajda, Phys. Lett. 73B (1978) 185. [4] G. Parisi, Proc. ll~me Rencontre de Moriond (1976), ed. J. Tran Thanh Van; G. Altarelli and G. Parisi, Nucl. Phys. B126 (1977) 298. [5 ] W..Furmafiski, Jagellonian Univ. preprint TPJU. [6] C.T. Sachrajda, Phys. Lett. 76B (1978) 100. [7] B.L. Combridge, J. Kripfganz and J. Ranft, Phys. Lett. 70B (1977) 234.