- Email: [email protected]
Polarization and the parton model: Factorization of mass singularities
Volume 82B, number 1
PHYSICS LETTERS
12 March 1979
POLARIZATION AND THE PARTON MODEL: FACTORIZATION OF MASS SINGULARITIES
Jochen KRIPFGANZ
Sektion Physik, Karl.Marx-Universitdt,Leipzig, DDR Received 27 July 1978
Factorization of mass singularities is shown for polarized parton legs in low order QCD perturbation theory.
Quantum chromodynamics (QCD) is the favoured candidate of a theory of strong interactions. This is mainly because of its property of asymptotic freedom. On the basis of light-cone and renormalization group techniques predictions can be made for a few processes. A much larger class of inclusive large momentum transfer processes is successfully described by the parton model. One example of such a process which is not light-cone dominated is large-mass lepton pair production in hadron-hadron scattering. It has become clear recently that the parton model approach to those processes may actually be justified (in the sense of a consistency check) by QCD perturbation theory. For hadron-induced reactions the parton model starts from the expression dOAB = a~b f d x a d x b aa/A(Xa)Gb/B(Xb)dOab,
(1)
where Ga/A(Xa) is a soft parton density describing the number of partons of type a (quark q, gluon g) of momentum fraction x a in hadron A. This primordial patton density cannot be calculated with present techniques, dOab is the "hard scattering" p a r t o n parton cross section, It is usually taken in lowest order (Born term). However, in trying to estimate higher order corrections one encounters large logarithms (mass singularities). Fhey arise from loop corrections and integration over non-observed partons (in collinear configurations). In this way we end up with a power series in g21og Q2/m2, where Q2 is a typical large momentum transfer, and g2 is renormalized at/~2 = Q2, i.e. it is of order 1/log Q2.
In this case successive terms are not small, and a perturbative approach is apparently spoiled. A perturbative treatment of deep inelastic leptonhadron scattering also leads to mass singularities, however. Therefore the experimentally observed quark density Gq/A(X , Q2) is not the primordial quark density of eq. (1). The two are related by a redefinition procedure absorbing all mass singularities. Politzer [ 1] observed that mass singularities arising from O(g 2) corrections to the Drell-Yan formula may be absorbed in completely the same way. Only finite correction terms O(g2(Q2)) remain. Similar low order perturbation theory studies have been carried out for other processes [2-4]. In leading order the usual patton model expressions are always reobtained with universal (i.e. process independent) Q2-dependent parton density and fragmentation functions and a running coupling constant g2(Q2). At the one-loop level a general proof of the process independence of the redefinition procedure is given in ref. [5]. In this way relations between several hard scattering processes are established. In the present letter we show that an analogous procedure is correct for polarized external particles (hadrons or leptons), too. Instead of hadrons we actually study quarks and gluons since those are the only particles available in perturbation theory. We explicitly calculate some low order contributions to lepton pair production from the scattering of polarized partons. From the way of deriving our results it is obvious, however, that the obtained expressions for various parton helicity densities are in fact process independent. 79
Volume 82B, number 1
PHYSICS LETTERS
12 March 1979
We study the reactions quark + antiquark -+ 3'* + X ,
(B)
quark + gluon -+ 7* + X ,
(C)
gluon + gluon -+ 3'* + X ,
(D)
quark + quark -+ 3,* + X .
,
/
/
/
o
b
Fig. 1. Lowest order Feynman diagrams for process (B).
Q2/m2
Leading contributions of the order o f g 2n logn are expected to factorize and to be covered by the parton model expression
G
?,a?~b,,~2 = d Orb /U~ a.q ?,q f
P2
-k 1 dXl dx2
Pckl Pl
X {Gq~)a(Xl,O2)G~bb(x2,Q2)+q+--+q} × do~'-q~'q/dO qq " ~ 2 ,
(2)
where d°qqa'q - 4rra~me2q dQ2 9Q2
(1-Xq~'q)8(XlX2S-Q2)
(3)
2
a.qa a is the q~ lower-order cross section. Q2) is the helicity density function of a quark q in a parton a. For the unpolarized case eq. (2) has been shown [1,2] to O(g21og for processes (A), (B), and to for processes (C), (D). We essentially repeat these calculations for polarized initial state partons. We work in the Coulomb gauge since we are anyway interested in the scattering of physical gluons in given helicity states. It turns out, however, that compared to the original Feynman gauge calculations the Coulomb gauge also leads to a significant simplification in the spin-average case. For the octet part of process (A) (7" couples to the original quark line) the spin structure of eq. (2) is trivial since the quark-gluon vertex 7 u is helicity conserving. The corresponding spin flip density is zero:
Gq/a (x,
Q2/m2) O(g41ogZQZ/m2)
+ Gq/q(X, Q2) = 0 .
(4)
The lower order diagrams for process (B) are shown in fig. 1. Performing a covariant spin sum, logarithmic contributions come from the square of diagram la and the interference term. With a physical polarization vector the interference term is finite, however. The logarithnric contribution is now entirely given by the 80
Fig. 2. The imaginary part of this diagram gives all the logarith mic quark-gluon contribution O(g21og if physical polarization vectors are used.
Q2/p2)
imaginary part of fig. 2. This is easily seen in the following way. The logarithmic contribution comes from the integration region where the outgoing quark of momentum p l - k l is about collinear t o P l , i.e. the momentum transfer k 2t is small. Contributions with a small denominator k 2 also contain a piece /~1~(/~1
/~1)'
(5)
as part of the trace, however. This expression vanishes in the limit k I -+ a "Pl since 4 and/b 1 anticommute (e "Pl = 0). Therefore only the contribution shown in fig. 2 yields a logarithmic singularity since it contains two small denominators. It is convenient to introduce Sudakov variables [6] Pl - kl = (l - a l ) P l +t31P2 + l l ,
(6)
P l'll =p2"ll =0. We work with zero quark mass and regularize the singularity by keeping the external gluon off-shell (p2 < 0). In the interesting limit 121 small we may use (-]~I)~*(Pl, ~.l)(/b 1 - ~l)~(pl, Xl)(--~ 1) 2 [/21/(1 - % ) ] [o~2(1 - ~,l-/S)/2 +(1 -- al)2(1 + Xl'rS)/2]~l ,
(7)
with X1 the gluon helicity. In this way eq. (2) is found to be satisfied provided that the antiquark density of
Volume 82B, number 1
PHYSICS LETTERS
12 March 1979
a gluon is given by
Gxqxgl-,. ~2, gig ~ ' s z
1 g2 p~?tg(x )I°g Q2/p~
I ='~ 4rr2
(8) -kg
The helicity transition functions 1
3
Pq/g(X) = gx- , ++
P~lg(X)= ½(1 --+
f
,I kT_kz II x) 2 ,
(9)
are those found in ref. [7] in the context o f an intuitive patton approach to scale violations in deep inelastic lepton scattering. The process independence of eq. (8) is also clear from the fact that it merely relies on the approximation eq. (7) which does not know about the particular hard scattering process. In the case of gluon-gluon scattering the leading logarithm contribution of order g41og Q2/p2 log Q2/p2, can be studied in exactly the same way. The lowest order amplitude is given by a sum of eight diagrams. In the Coulomb gauge, however, the entire leading log term comes from the ladder-type contribution shown in fig. 3. Eq. (2) is verified by applying eq, (7) twice. The Coulomb gauge also considerably simplifies the discussion of the q u a r k - q u a r k contribution (process D). Interference terms where the photon couples to both quark lines are immediately seen to be finite. In the Feynman gauge such terms are individually singular but singularities cancel for the gauge invariant sum. Leading contributions of order g4 log2 Q2/p~, arise from terms containing two powers of both of the denominators k 2 and k 2 (fig. 4) that may possibly be small. There is only one such contribution which is shown in fig. 4 (a similar one with Pl + * P2 is understood). We again end up with a ladder diagram. This may have been expected from previous perturbation theory studies of deep inelastic scattering [8]. In order to calculate the leading log contribution we use another low-/2 approximation for the trace
I
Fig. 4. Leading logarithm contribution for process (D). corresponding to quark line 1 : Tr ½(1 + ~.l@)/b 1¢(k 1 , X')(/b I - ~l)¢*(kl, ~.") 121 ~2 l_al
I I I
I I 1
I I f
Fig. 3. Leading logarithm contribution for process (C).
1 (1--al)26M, ~2 [l~?'l,Y+
x ' l f K h ''
(10)
A non-diagonal piece in X', X" vanishes after angular integration. X1 is the helicity of quark 1. The Sudakov parametrization (6) has been used again. e(kl, X) appears in eq. (10) since we may explicitly express the Coulomb gauge gluon propagator in terms of transverse polarization vectors (the static potential has to be subtracted but this does not show up at the leading log level). Using eq. (10), and an expression of the type of eq. (7) for the next rung of the ladder we are now ready to verify eq. (2). As the corresponding antiquark density of a quark we find
Gqqqq(x, 0 2 ) =
~ ( g 2 ' ~ 2 log2 \47r 2 ]
Q2/p2
dalpXg~.q, ,[email protected], , , g/q [ ~ 1 ) q / g [X/Oq),
XE ~ , hg
(ll)
with Pg/n(X) = ( ( N 2 --+
I 1 I
I
Pg/q(X) = ((N 2 -
D/2N)x-1 02)
1)/2N)(I - X)2X -1
and N = 3 for SU(3)c. The process independence of this expression is again obvious. Details of this calculation are given in ref. [9]. There we also show that non-leading q u a r k - q u a r k and gluon-gluon contributions (of the order of g41og Q2/p2) factorize, too. In this way the finite O(g 2) quark-gluon terms are reproduced. This shows that correction terms to parton model formulae may 81
Volume 82B, number 1
PHYSICS LETTERS
be calculated consistently. After completing this work we received a number of papers [ 1 0 - 1 4 ] extending the proof of factorization to all orders in perturbation theory. All this work (see also ref. [15]) is done in a physical gauge (axial or Coulomb gauge) for the same reason we have discussed in some detail. The restriction to transverse gluons explicitly eliminates a large number of spurious contributions. There seems to be no major difficulty in generalizing this higher order proof to polarized external particles. In the notation of ref. [12] this corresponds to the introduction of additional helicity indices in their matrix integral equation. However, in a Coulomb gauge it may be rather more complicated than in an axial gauge to actually carry out the whole program. This is essentially the question of cancellation of singularities for 2PI contributions. This problem is being studied. I would like to thank W. Furmanski, K. Gaemers, C. Sachrajda and T.M. Yan for interesting discussions.
82
12 March 1979
References [1] H.D. Politzer, Nucl. Phys. B129 (1977) 301. [2] C. Sachrajda, Phys. Lett. 73B (1978) 185. [3] K.H. Craig and C.H. Llewellyn-Smith, Phys. Lett. 72B (1978) 349. [4] C. Sachrajda, Phys. Lett. 76B (1978) 100; W. Furmanski, Phys. Lett. 77B (1978) 312. [5] D. Amati, R. Petronzio and G. Veneziano, preprint CERN-TH 2470 (1978). [6] V.V. Sudakov, Zh. Eksp. Teor. Fiz. 30 (1956) 87. [7] G. Altarelli and G. Parisi, Nucl. Phys. B126 (1977) 298. [8] V.N. Gribov and L.N. Lipatov, Yad. Fiz. 15 (1972) 781. [9] J. Kripfganz, preprint KNU-HEP 78-12. [ 10] Yu.L. Dokshitser, D.I. Dyakonov and S.I. Troyan, Proc. XIII Winter School of LNPI (Leningrad, 1978). [ 11] C.H. Llewellyn-Smith, Lectures presented at the XVII Intern. Universit~/tswochenffir Kernphysik (Schladming, 1978). [12] R.K. Ellis, H. Georgi, M. Machacek, H.D. Politzer and G.G. Ross, Phys. Lett. 78B (1978) 281. [13] D. Amati, R. Petronzio and G. Veneziano, preprint CERN-TH-2527 (1978). [14] S.B. Libby and G. Sterman, Stony Brook preprint ITP-SB-78-41. [15] J. Frenkel, M.J. Shailer and J.C. Taylor, Oxford Univ. preprint 65/78.
PHYSICS LETTERS
12 March 1979
POLARIZATION AND THE PARTON MODEL: FACTORIZATION OF MASS SINGULARITIES
Jochen KRIPFGANZ
Sektion Physik, Karl.Marx-Universitdt,Leipzig, DDR Received 27 July 1978
Factorization of mass singularities is shown for polarized parton legs in low order QCD perturbation theory.
Quantum chromodynamics (QCD) is the favoured candidate of a theory of strong interactions. This is mainly because of its property of asymptotic freedom. On the basis of light-cone and renormalization group techniques predictions can be made for a few processes. A much larger class of inclusive large momentum transfer processes is successfully described by the parton model. One example of such a process which is not light-cone dominated is large-mass lepton pair production in hadron-hadron scattering. It has become clear recently that the parton model approach to those processes may actually be justified (in the sense of a consistency check) by QCD perturbation theory. For hadron-induced reactions the parton model starts from the expression dOAB = a~b f d x a d x b aa/A(Xa)Gb/B(Xb)dOab,
(1)
where Ga/A(Xa) is a soft parton density describing the number of partons of type a (quark q, gluon g) of momentum fraction x a in hadron A. This primordial patton density cannot be calculated with present techniques, dOab is the "hard scattering" p a r t o n parton cross section, It is usually taken in lowest order (Born term). However, in trying to estimate higher order corrections one encounters large logarithms (mass singularities). Fhey arise from loop corrections and integration over non-observed partons (in collinear configurations). In this way we end up with a power series in g21og Q2/m2, where Q2 is a typical large momentum transfer, and g2 is renormalized at/~2 = Q2, i.e. it is of order 1/log Q2.
In this case successive terms are not small, and a perturbative approach is apparently spoiled. A perturbative treatment of deep inelastic leptonhadron scattering also leads to mass singularities, however. Therefore the experimentally observed quark density Gq/A(X , Q2) is not the primordial quark density of eq. (1). The two are related by a redefinition procedure absorbing all mass singularities. Politzer [ 1] observed that mass singularities arising from O(g 2) corrections to the Drell-Yan formula may be absorbed in completely the same way. Only finite correction terms O(g2(Q2)) remain. Similar low order perturbation theory studies have been carried out for other processes [2-4]. In leading order the usual patton model expressions are always reobtained with universal (i.e. process independent) Q2-dependent parton density and fragmentation functions and a running coupling constant g2(Q2). At the one-loop level a general proof of the process independence of the redefinition procedure is given in ref. [5]. In this way relations between several hard scattering processes are established. In the present letter we show that an analogous procedure is correct for polarized external particles (hadrons or leptons), too. Instead of hadrons we actually study quarks and gluons since those are the only particles available in perturbation theory. We explicitly calculate some low order contributions to lepton pair production from the scattering of polarized partons. From the way of deriving our results it is obvious, however, that the obtained expressions for various parton helicity densities are in fact process independent. 79
Volume 82B, number 1
PHYSICS LETTERS
12 March 1979
We study the reactions quark + antiquark -+ 3'* + X ,
(B)
quark + gluon -+ 7* + X ,
(C)
gluon + gluon -+ 3'* + X ,
(D)
quark + quark -+ 3,* + X .
,
/
/
/
o
b
Fig. 1. Lowest order Feynman diagrams for process (B).
Q2/m2
Leading contributions of the order o f g 2n logn are expected to factorize and to be covered by the parton model expression
G
?,a?~b,,~2 = d Orb /U~ a.q ?,q f
P2
-k 1 dXl dx2
Pckl Pl
X {Gq~)a(Xl,O2)G~bb(x2,Q2)+q+--+q} × do~'-q~'q/dO qq " ~ 2 ,
(2)
where d°qqa'q - 4rra~me2q dQ2 9Q2
(1-Xq~'q)8(XlX2S-Q2)
(3)
2
a.qa a is the q~ lower-order cross section. Q2) is the helicity density function of a quark q in a parton a. For the unpolarized case eq. (2) has been shown [1,2] to O(g21og for processes (A), (B), and to for processes (C), (D). We essentially repeat these calculations for polarized initial state partons. We work in the Coulomb gauge since we are anyway interested in the scattering of physical gluons in given helicity states. It turns out, however, that compared to the original Feynman gauge calculations the Coulomb gauge also leads to a significant simplification in the spin-average case. For the octet part of process (A) (7" couples to the original quark line) the spin structure of eq. (2) is trivial since the quark-gluon vertex 7 u is helicity conserving. The corresponding spin flip density is zero:
Gq/a (x,
Q2/m2) O(g41ogZQZ/m2)
+ Gq/q(X, Q2) = 0 .
(4)
The lower order diagrams for process (B) are shown in fig. 1. Performing a covariant spin sum, logarithmic contributions come from the square of diagram la and the interference term. With a physical polarization vector the interference term is finite, however. The logarithnric contribution is now entirely given by the 80
Fig. 2. The imaginary part of this diagram gives all the logarith mic quark-gluon contribution O(g21og if physical polarization vectors are used.
Q2/p2)
imaginary part of fig. 2. This is easily seen in the following way. The logarithmic contribution comes from the integration region where the outgoing quark of momentum p l - k l is about collinear t o P l , i.e. the momentum transfer k 2t is small. Contributions with a small denominator k 2 also contain a piece /~1~(/~1
/~1)'
(5)
as part of the trace, however. This expression vanishes in the limit k I -+ a "Pl since 4 and/b 1 anticommute (e "Pl = 0). Therefore only the contribution shown in fig. 2 yields a logarithmic singularity since it contains two small denominators. It is convenient to introduce Sudakov variables [6] Pl - kl = (l - a l ) P l +t31P2 + l l ,
(6)
P l'll =p2"ll =0. We work with zero quark mass and regularize the singularity by keeping the external gluon off-shell (p2 < 0). In the interesting limit 121 small we may use (-]~I)~*(Pl, ~.l)(/b 1 - ~l)~(pl, Xl)(--~ 1) 2 [/21/(1 - % ) ] [o~2(1 - ~,l-/S)/2 +(1 -- al)2(1 + Xl'rS)/2]~l ,
(7)
with X1 the gluon helicity. In this way eq. (2) is found to be satisfied provided that the antiquark density of
Volume 82B, number 1
PHYSICS LETTERS
12 March 1979
a gluon is given by
Gxqxgl-,. ~2, gig ~ ' s z
1 g2 p~?tg(x )I°g Q2/p~
I ='~ 4rr2
(8) -kg
The helicity transition functions 1
3
Pq/g(X) = gx- , ++
P~lg(X)= ½(1 --+
f
,I kT_kz II x) 2 ,
(9)
are those found in ref. [7] in the context o f an intuitive patton approach to scale violations in deep inelastic lepton scattering. The process independence of eq. (8) is also clear from the fact that it merely relies on the approximation eq. (7) which does not know about the particular hard scattering process. In the case of gluon-gluon scattering the leading logarithm contribution of order g41og Q2/p2 log Q2/p2, can be studied in exactly the same way. The lowest order amplitude is given by a sum of eight diagrams. In the Coulomb gauge, however, the entire leading log term comes from the ladder-type contribution shown in fig. 3. Eq. (2) is verified by applying eq, (7) twice. The Coulomb gauge also considerably simplifies the discussion of the q u a r k - q u a r k contribution (process D). Interference terms where the photon couples to both quark lines are immediately seen to be finite. In the Feynman gauge such terms are individually singular but singularities cancel for the gauge invariant sum. Leading contributions of order g4 log2 Q2/p~, arise from terms containing two powers of both of the denominators k 2 and k 2 (fig. 4) that may possibly be small. There is only one such contribution which is shown in fig. 4 (a similar one with Pl + * P2 is understood). We again end up with a ladder diagram. This may have been expected from previous perturbation theory studies of deep inelastic scattering [8]. In order to calculate the leading log contribution we use another low-/2 approximation for the trace
I
Fig. 4. Leading logarithm contribution for process (D). corresponding to quark line 1 : Tr ½(1 + ~.l@)/b 1¢(k 1 , X')(/b I - ~l)¢*(kl, ~.") 121 ~2 l_al
I I I
I I 1
I I f
Fig. 3. Leading logarithm contribution for process (C).
1 (1--al)26M, ~2 [l~?'l,Y+
x ' l f K h ''
(10)
A non-diagonal piece in X', X" vanishes after angular integration. X1 is the helicity of quark 1. The Sudakov parametrization (6) has been used again. e(kl, X) appears in eq. (10) since we may explicitly express the Coulomb gauge gluon propagator in terms of transverse polarization vectors (the static potential has to be subtracted but this does not show up at the leading log level). Using eq. (10), and an expression of the type of eq. (7) for the next rung of the ladder we are now ready to verify eq. (2). As the corresponding antiquark density of a quark we find
Gqqqq(x, 0 2 ) =
~ ( g 2 ' ~ 2 log2 \47r 2 ]
Q2/p2
dalpXg~.q, ,[email protected], , , g/q [ ~ 1 ) q / g [X/Oq),
XE ~ , hg
(ll)
with Pg/n(X) = ( ( N 2 --+
I 1 I
I
Pg/q(X) = ((N 2 -
D/2N)x-1 02)
1)/2N)(I - X)2X -1
and N = 3 for SU(3)c. The process independence of this expression is again obvious. Details of this calculation are given in ref. [9]. There we also show that non-leading q u a r k - q u a r k and gluon-gluon contributions (of the order of g41og Q2/p2) factorize, too. In this way the finite O(g 2) quark-gluon terms are reproduced. This shows that correction terms to parton model formulae may 81
Volume 82B, number 1
PHYSICS LETTERS
be calculated consistently. After completing this work we received a number of papers [ 1 0 - 1 4 ] extending the proof of factorization to all orders in perturbation theory. All this work (see also ref. [15]) is done in a physical gauge (axial or Coulomb gauge) for the same reason we have discussed in some detail. The restriction to transverse gluons explicitly eliminates a large number of spurious contributions. There seems to be no major difficulty in generalizing this higher order proof to polarized external particles. In the notation of ref. [12] this corresponds to the introduction of additional helicity indices in their matrix integral equation. However, in a Coulomb gauge it may be rather more complicated than in an axial gauge to actually carry out the whole program. This is essentially the question of cancellation of singularities for 2PI contributions. This problem is being studied. I would like to thank W. Furmanski, K. Gaemers, C. Sachrajda and T.M. Yan for interesting discussions.
82
12 March 1979
References [1] H.D. Politzer, Nucl. Phys. B129 (1977) 301. [2] C. Sachrajda, Phys. Lett. 73B (1978) 185. [3] K.H. Craig and C.H. Llewellyn-Smith, Phys. Lett. 72B (1978) 349. [4] C. Sachrajda, Phys. Lett. 76B (1978) 100; W. Furmanski, Phys. Lett. 77B (1978) 312. [5] D. Amati, R. Petronzio and G. Veneziano, preprint CERN-TH 2470 (1978). [6] V.V. Sudakov, Zh. Eksp. Teor. Fiz. 30 (1956) 87. [7] G. Altarelli and G. Parisi, Nucl. Phys. B126 (1977) 298. [8] V.N. Gribov and L.N. Lipatov, Yad. Fiz. 15 (1972) 781. [9] J. Kripfganz, preprint KNU-HEP 78-12. [ 10] Yu.L. Dokshitser, D.I. Dyakonov and S.I. Troyan, Proc. XIII Winter School of LNPI (Leningrad, 1978). [ 11] C.H. Llewellyn-Smith, Lectures presented at the XVII Intern. Universit~/tswochenffir Kernphysik (Schladming, 1978). [12] R.K. Ellis, H. Georgi, M. Machacek, H.D. Politzer and G.G. Ross, Phys. Lett. 78B (1978) 281. [13] D. Amati, R. Petronzio and G. Veneziano, preprint CERN-TH-2527 (1978). [14] S.B. Libby and G. Sterman, Stony Brook preprint ITP-SB-78-41. [15] J. Frenkel, M.J. Shailer and J.C. Taylor, Oxford Univ. preprint 65/78.