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Nuclear dependence at large transverse momentum
Physics Letters B 279 (1992) 377-383 North-Holland
PHYSICS LETTERS B
Nuclear dependence at large transverse momentum M a L u o a, J i a n w e i Q i u b a n d G e o r g e S t e r m a n a a Institute for Theoretical Physics. State University of New York, Stony Brook, NY 11794-3840, USA b Department of Physics andAstronomy, Iowa State University, Ames, IA 50011, USA Received 5 January 1992
We treat the anomalous nuclear dependence of high-pr jet and single-particle photoproduction cross sections in perturbative QCD. We show that the Aa dependence is associated with multi-patton matrix elements in nuclei. Assuming a natural model for these correlation functions, we derive predictions for the xv, xT and energy dependence of the enhancement. Significant enhancements are found without introducing scales other than AQco. The methods described may bc applied to other high-Pr nuclear cross sections.
A m o n g the most influential experiments o f the '70's were those that detected a power-law falloffin transverse m o m e n t u m for particles produced in h a d r o n - h a d r o n scattering [ 1 ]. This behavior was quickly recognized as the result o f an underlying p~4 hard-scattering process, convoluted with p a t t o n distributions and jet decay, or fragmentation, functions [ 2 ]. Further observation detected an " a n o m a l o u s " dependence on nuclear target atomic number, A, in which the cross section at fixed transverse m o m e n t u m grows [3,4] a p p r o x i m a t e l y as A " with c~> 1. This effect is most often described as due to multiple interactions o f p a r t o n s in nuclear matter, primarily double scattering [ 5,6 ]. O u r aim in this p a p e r is to show that nuclear enhancement can be described naturally in perturbative QCD, in terms o f a nonleading power, or "higher twist" formalism [ 7,8 ]. We will argue that a n o m a l o u s nuclear enh a n c e m e n t data provides information on multi-parton matrix elements in nuclei. It is therefore not a secondary phenomenon, to be described in terms o f known nuclear and particle parameters, but a fundamental process in its own right, which tells us something new about nuclear m a t t e r and its interaction with high-energy probes. To set the stage for our argument, we first briefly review the perturbative Q C D ( p Q C D ) description o f high-px j e t and single-particle inclusive cross sections. We then discuss factorization theorems at nonleading power, the isolation o f contributions to nuclear enhancement, and some representative predictions for cross sections. F o r simplicity, we work with p h o t o p r o d u c t i o n cross sections at lowest order, although the methods introduced here are more general. The success o f the hard scattering picture for high-pv processes in h a d r o n - h a d r o n scattering can be understood in terms o f the p a r t o n model or, m o r e generally, in terms o f the so-called factorization theorems o f p Q C D [9]. F o r instance, the cross section for hadrons h and h' to produce an outgoing j e t with large transverse mom e n t u m l T c a n be written to leading power as the convolution o f parton distributions, ~,/h (x, l 2 ), times a perturbative hard scattering function, H0, dcr(l) ¢oi d3---~ = ~
dx, dxjO,/h(x,, 12)H,j(x,, xj, l)Oj/h. (X;, l 2) .
( 1)
Parton distributions may be defined in terms o f hadronic matrix elements. An example is the distribution o f a quark in a h a d r o n m o v i n g in the + z direction, p~, = ( n ' p h ) a u, where we define r / u = ~ , + , nU=6¢,_.In this case we have 0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
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exp[i
(n.p)xy- l" ½(Ph I q ( 0 ) n ' 7 q ( y - )lPh ) ,
16 April 1992
(2)
with q(x) the quark field, u appears as the scale in the rcnormalization of the operator involving two quark fields separated by a light-like distance ( y - ) . In this form, the matrix element in eq. (2) is evaluated in the light cone A + = 0 gauge [ 10 ]. In a similar fashion, the single-particle inclusive cross section for hadron M with transverse m o m e n t u m lfr is determined by convolutions of exprcssions likc eq. ( 1 ) with fragmentation functions, DM/k (~, l~"), which describe the likclihood of finding hadron M in the decay products of parton k, with fraction r o f t h e parton's m o m e n t u m [2]. Most studies of nuclear enhancement have described it by means of two sequential, independent scattering processes, each of which is a leading-power partonic cross section like eq. ( 1 ) [ 5 ]. Nuclear enhancement enters as an extra factor of the nuclear radius, which results from integrating over the relative positions of the scatterings. Such cross sections require an additional a~, relative to lowest order. They arc infrared "unstable", in the limit that one of the scattcrings is responsible for the entire m o m e n t u m transfer, and one becomes soft [ 11 ]. Finally, even though each scattering in the sequence is leading power, nuclear enhancement is a nonleadingpower effect [ 6 ], suppressed by a factor of I F 2 compared to a single-scattering cross section like eq. ( 1 ). Thcse considerations suggest that we reanalyzc nonleading power corrections to nuclear scattering in the context of p Q C D factorization [6,8,12,13 ]. What we will sec is that nuclear enhancement can be described by a generalization of eqs. ( 1 ) and (2), with factorization integrals involving matrix elements of four, rather than two ficlds. The resulting hard-scattering corrections begin at order a~, rather than a~2, compared to lowest order, and are infrarcd-safe, all long-distance effects having been absorbed into the new matrix elements. The extension of p Q C D to double-scattering requires factorization at nonleading power. For unpolarized cross sections, the f r s t power correction is at the order o f l ~ 2, relative to leading behavior. Factorization at this level is a nontrivial result, which has bcen justificd in ref. [13]. To cut down on calculational complexity, we first discuss jet (parton) photoproduction cross sections, where l" is the m o m e n t u m of the produced parton. We will take the photon to have m o m e n t u m qU= nUx/s-~, and thc (nuclear) target to have m o m e n t u m Ap ~, with p" = ~'v/ss~. We dcfine s = 2p.q and t = - 2p-l. After factorization, the gencral I-F2 cross section is, as in eq. ( 1 ), separated into a sum over convolutions of long-distance matrix elements, T{,l, characterized by some set of fields {i}, and hard-scattering functions ,~*i. The convolution integrals are now over three independent fractional m o m e n t a x , of the form
2
d°4 = f cL~tdx2dx3 (L~ (l~ T~a(xl, x2, x3),~i}(x,, x2, x3, 1, s) , oh~5~ \4zr/ks/~{,}
(3)
as illustrated in fig. 1. (For this discussion, we will neglect the hadronic parton structure of the photon.) In the full expansion, many matrix elements will appear, even at Born approximation in the hard part. They will be of the general form [ 8,12,13 ],
TDI(XI,X2, X3)--
if
(27~)3
dyFdyf dy~ exp[in.p(x3yf +x2y[ + x l y i - ) ]
X taara(PA IT [B~(0)B~(y~-)/3r (y~-)Ba(y~ )]IPA ) ,
(4)
where the B's and/~'s are quark fields, covariant derivatives, i 0--gA, or gluonic field strengths [ 8,12 ], and where t~/~a is a constant matrix that produces a scalar from these fields. The state IP,~) represents the "target" hadron, in our case a nucleus. Typical contributions to .~ at the Born level, are shown in fig. 2, with an "observed" parton of m o m e n t u m I. ,~¢~must be calculated by summing over "cuts" of the diagram, corresponding to different final states. It is only after this sum that the answer turns out to be power suppressed, through cancellation of the leading power contribution from each diagram [ 14,15 ]. The calculable hard scattering function ,~ is independent of the identity of the target; indeed, this is the 378
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'\))t<'/ (a)
q
~:
Fig. 1. Graphical representation of factorization at first nonleading power.
(b)
(c) Fig. 2. Typical contributions to hard scattering at tree level. (a) Quark-gluon, (b) gluon-gluon, (c) quark-quark.
essence of factorization. Therefore, if we are to find nuclear enhancement, it must reside in the matrix elements, eq. (4), and we need criteria to identify such dependence in muitilocal operators of this type. Actually, this is not difficult. The integrals over y, in eq. (4) have a range that we expect to grow with the nuclear radius, that is, as A 1/3. On the other hand, these integrals include explicit oscillating exponentials, exp(in'pxiy7 ), which will damp growth in A unless one or more of the xi vanishes. This, in turn, requires delta functions, d(xi), in the hard scattering function .~. We will see in a moment how they may come about. Even with such delta functions, however, we expect a number of restrictions on growth with nuclear radius. Before describing them, let us emphasize that setting the momentum fraction o f a gluon line to zero does not produce a long-distance contribution in the hard scattering. A pole in the hard scattering may be thought of as a prescription for evaluating an imaginary part, which could always have been computed by a contour integral at a finite distance from the pole. Such deformations are possible because the hard scattering diagrams involve additional partons, with finite fractional momenta [ 15 ]. The same reasoning would not apply to leading-power gluon scattering at x = 0 . We now make two simple assumptions about expectation values ofmultilocal operators in nuclear states with large A. To avoid inessential overall factors, we normalize nuclear states to a fixed energy scale, (PA IP'A) = 2 (~opA/A)8 3(PA --P~ )- Our assumptions then are: (i) that the expectation value of a product of operators separated by a distance small compared to the nuclear radius is proportional to the nuclear volume, i.e., to A, and (ii) that the expectation value of a product of operators separated by a distance comparable to the nuclear radius can be independent of the distance only if the operators are individually Lorentz scalars and color singlets. According to the first assumption, the nuclear dependence [ 16 ] of leading-power patton distributions like eq. (2) is not the source of anomalous nuclear enhancement [ 17]. For instance, at fixed x # 0 in eq. (2), the quark fields arc kept close together by the oscillating exponential, so we expect the (suitably normalized) nuclear matrix element to grow like A for large A. (It need not be a simple multiple of the corresponding matrix element at small A, however. ) According to the second assumption, there can be at most a single extra power of A 1/3 at first nonleading power from the yi integrals, because the four operators in eq. (4) must pair into two color and spin singlets. Clearly, then, matrix elements that produce nuclear enhancement involve pairs of quark fields, a n d / o r pairs ofgluon fields. The above considerations place strong restrictions on contributions in .~ that can match with nuclear enhancement from their corresponding matrix elements. Without going into details, the evaluation of the hardscattering is carried out by a "collinear expansion", about the leading power [ 8,12,15 ]. We find it most convenient to carry out this expansion in Feynman gauge [ 12,18]. At the order to which we work, A 4/3 behavior comes entirely from hard-scattering diagrams with poles which set two of the x,'s in eq. (4) to zero. In this case, 379
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two fields, for instance the gluon lines carrying momenta x2p and x3p in fig. 2a, have vanishing momentum fraction, while all nonzero plus momentum is carried into the hard part by quark lines. Then two of the Y7 integrals in eq. (4) have no oscillatory suppression, although, in accordance with our assumptions above, we expect color confinement to keep the two gluon fields close together. Otherwise, we would predict A 5/3 nuclear dependence. Double-pole contributions at x2 = x3 = 0 are present in each of the three possible cuts of fig. 2a, as well as those of fig. 2b, in which a gluon supplies the large momentum from the nucleus. For each cut, the "observed" momentum is always fixed at l u. Explicit calculation shows that all leading power contribution cancel in the sum over cuts for any fixed graph with two extra poles. The remainder is an expansion in the transverse momentum carried by the gluons. This cancellation applies to any graph with the same topology as in figs. 2a, 2b. We neglect here "initial-state" poles (xl = 0 in fig. 2a, for example), which turn out to give small effects [15,18 ]. If the leading contribution is already nonleading power (/.F 2 ), as with four-quark diagrams [8,12] (fig. 2c), or diagrams with two extra transverse-polarized gluons, then the sum over final states suppresses the nuclear dependence by an extra power of l~:2 compared to the first nonleading power. To get an overall dependence A4/3l~ 2 from a matrix element of the form in eq. (4), we must begin with cut diagrams that are individually of leading power. The only possibility is to take the (n.A) component for the two gluons whose longitudinal momenta will be set to zero by the extra poles from the hard scattering. As mentioned above, the sum over cuts will produce an expansion in their transverse momenta, and will consequently produce matrix elements of the fields 0-r (n'A). In summary, by isolating contributions to the hard scattering that can match with A 4/3 behavior from the nucleus, we find factorized forms at first nonleading power that involve the matrix elements Tq(x,A) = f
~f~-exp[i(n'p)xyT ] ~ dy~ dy-------f-~O(y~ - y f )O(yy
× ½(PA Iq(0) (n'y)n~F'~(y~)n~,V',dyT_ )q(Yi- ) [PA ) T~(x,A)= ; ~ X~
1
•
exp[i(n.p)xy?] fd dy~dy~ 2re O(y~-yf)O(y~)
,
(5)
with DPA>normalized as above. All fields are on the light-cone in the n u direction. In these expressions, we have already set x 2 = x 3 = 0 , and have suppressed ordered exponentials that ensure gauge invariance. We have also modified normalizations relative to eq. (4). A factor of 1/27r has been absorbed into the hard scattering, while a factor of ½ has been introduced into the quark-gluon matrix element and 1/(xn.p) into the gluon-gluon matrix element, to bring them into closer correspondence with leading-power parton distributions like eq. (2) [ 10]. These changes are purely formal, but they influence our guess for the sizes of the matrix elements in eq. (5), which we will describe shortly. A physical interpretation for these matrix elements follows by recognizing that classically, fdy-F + T ( y - ) measures the integral of the transverse force experienced by a unit color charge as it passes through the nucleus along a light-like trajectory in the minus direction. It may then be useful to consider these matrix elements as leading-power patton distributions, modified by an average squared transverse momentum transfer. The calculation of the hard scattering functions corresponding to the two matrix elements in eq. (5) is not difficult. The result involves derivatives of a phase-space delta function. For numerical estimates, it is therefore convenient to integrate by parts, producing an expression with the same content as eq. (3), in which derivatives with respect to x act on the matrix elements T o and Tg. Because the first nonvanishing corrections are second order in transverse momenta, we find up to two derivatives in our expression. For jet photoproduction, we have ¢DIdo'4/3
380
c~Er~(4noq)2 [ ( - -
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where x = - u ~ (s+ t), and where el is the charge of quark flavor f i n units of the electron charge. ~i represents an effective "partonic flux", given in terms of the matrix elements in eq. (5) by
~Ax'A)-LkOx2,,,,
x
/\-t(s+
In view of our interpretation for the matrix elements, the derivatives in eq. (7) reflect the kinematic influence of transverse momentum transfer on the parton flux. The hard-scattering functions 11, in eq. (6), summed over outgoing parton type, are given by ( N = 3 is the number of colors) _ _
2N e
,
+
+
s
+
=
±[/-,L(,+ql
.
(8)
Expressions for single-particle cross sections may be found by convoluting partonic cross sections with fragmentation functions, exactly as for the leading power. Once we have information on the nuclear matrix elements Tq and Tg, we can use the factorized expressions in cq. (6) to predict the behavior of cross sections. Conversely, the measurement of nuclear dependence in photoproduction cross sections may be used to determine the matrix elements. To get an idea of what is involved, we have evaluated jet and single-particle pion and proton cross sections as functions ofxv and IT by assuming the forms
T,( x, A) =A4/3A~cDO,/N(X) ,
(9)
for i a quark or gluon, with O,/N(x) the simplified (scaling) distributions for partons in a proton that are specified in ref. [ 18 ]. Nuclear cross sections are computed assuming isospin invariance and Z/A = 0.4. For dcfiniteness, we take AOCD=200 McV. Modifications due to evolution and to nuclear effects on single-parton distributions should not change our qualitative results, and we have neglected them. Similarly, we have used fragmentation functions given in ref. [ 19 ], fixed at a scale of 25 GcV 2, without evolution. The resulting predictions are, in any case, meant to be taken seriously only for their kinematic dependence, which is not very sensitive to the precise choices of distributions and fragmentation functions. In fig. 3 we show a representative kinematic dependence of the parameter a - 1, computed from the definition
del (A) A~ +
den/3(A) = A , d3/
de1 (A) d3/ ,
(10)
evaluated at A = 200, with a, the leading power nucleon Born cross section. Remarkably, a ' s computed with different values of A in the range 50
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--
0.6
(a) /
I
1-
,
,
,
,--1
,
,
,
,
1-,
.
.
0.6
/ 0.4
I
I ~3
0.4
,,;r 4
0.2
0.2
\ /
0.0
. . . . -0.5
0.0 0
0.5
.... 2
I .... 4
t .... 6
I .... 8
10
XF
Fig. 3. Dependence of a - 1, calculated by eq. ( 10 ) wit h the mat rix elements of eq. ( 9 ), ( a ) on xv for lr--- 6 GeV at s = 300 (dashed) and 600 GeV 2 (solid) in jet production, (b) on 11 at x~.=0.5 and s=600 GeV 2 for n + and proton. p r o v i d e a m u c h l a r g e r f r a m e w o r k t o t e s t a n d d e v e l o p t h e f o r m a l i s m we h a v e o u t l i n e d here. H a d r o n - n u c l c u s i n t e r a c t i o n s will i n v o l v e i n i t i a l as well as f i n a l s t a t e i n t e r a c t i o n s , so t h e a n a l y s i s will b e m o r e c o m p l e x , a l t h o u g h q u a l i t a t i v e l y s i m i l a r . S u c h a n a n a l y s i s m a y allow u s t o c o m p a r c t h e s a m e m a t r i x e l e m e n t s in d i f f e r e n t p r o c e s s e s , w h i c h w o u l d r e s u l t i n m u c h s t r o n g e r p r e d i c t i v e p o w e r . Issues o f t h i s sort, a l o n g w i t h t h e q u e s t i o n s o f h i g h e r o r d e r c o r r e c t i o n s , e v o l u t i o n , a n d a p p l i c a t i o n s to n u c l e u s - n u c l e u s s c a t t e r i n g , r e m a i n to b e s t u d i e d . W e t h a n k S. G a v i n , T. H a n , R.L. M c C a r t h y , J. M i l a n a a n d A.H. M u e l l e r for useful c o n v e r s a t i o n s . T h i s w o r k was s u p p o r t e d in p a r t b y t h e N a t i o n a l S c i e n c e F o u n d a t i o n u n d e r G r a n t N o . N S F 9 1 0 8 0 5 4 , b y t h e D e p a r t m e n t o f Energy u n d e r G r a n t No. D E - F G 0 2 - 8 7 E R 4 0 3 7 1 a n d b y t h e T e x a s N a t i o n a l R e s e a r c h L a b o r a t o r y C o m m i s s i o n .
References
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A. Zieminski, in: Proc. Fourth Meeting of the Division of Particles and Fields (Storrs, CT, 1988), ed. K. Hailer et al. (World Scientific, Singapore, 1989). [ 5 ] P.M. Fishbane and J.S. Trefil, Phys. Rev. D 12 ( 1975 ) 2113; J.H. Kuhn, Phys. Rev. D 13 (1976) 2948; M.J. Longo, Nucl. Phys. B 134 (1978) 70; V.V. Zmushko, Yad. Fiz. 32 (1980) 246 [Soy. J. Nucl. Phys. 32 (1980) 127]; A. Krzywicki, J. Engels, B. Petersson and U. Sukhatme, Phys. Lett. B 85 (1979) 407; G.T. Bodwin, S.J. Brodsky and G.P. Lepage, Phys. Rev. Lctt. 47 ( 1981 ) 1799; M. Lev and B. Petersson, Z. Phys. C 21 ( 1983 ) 155. [6] K. Kastella, J. Milana and G. Sterman, Phys. Rev. D 39 (1989) 2586. [ 7 ] A. de Rfijula, H. Georgi and H.D. Politzer, Ann. Phys. (NY) 103 ( 1977 ) 315; S. Gottlieb, Nucl. Phys. B 139 (1978) 125; M. Okawa, Nucl. Phys. B 187 ( 1981 ) 71; E.V. Shuryak and A.L. Vainshtein, Phys. Left. B 105 ( 1981 ) 65; R.L. Jaffe and M. Soldate, Phys. Lett. B 105 ( 1981 ) 467; Phys. Rev. D 26 (1982) 49; S.P. Luttrell and S. Wada, Nucl. Phys. B 197 (1982) 290. [8] R.K. Ellis, W. Furmanski and R. Petronzio, Nucl. Phys. B 207 (1982) 1; B 212 (1983) 29; R.L. Jaffe, Nucl. Phys. B 229 (1983) 205; J. Qiu, Phys. Rev. D 42 (1990) 30. [9 ] J.C. Collins, D.E. Soper and G. Sterman, in: Perturbative quantum chromodynamics, ed. A.H. Mueller (World Scientific, Singapore, 1989). [ 10] G. Curci, W. Furmanski and R. Petronzio, Nucl. Phys. B 175 (1980) 27; J.C. Collins and D.E. Soper, Nucl. Phys. B 194 ( 1982 ) 445. [ 11 ] K. Kastella, Phys. Rev. D 36 (1987) 2734. [ 12] J. Qiu and G. Sterman, Nucl. Phys. B 353 ( 1991 ) 105. [ 13 ] J. Qiu and G. Sterman, Nucl. Phys. B 353 ( 1991 ) 137. [ 141 J.M.F. Labastida and G. Stcrman, Nucl. Phys. B 254 ( 1985 ) 425. [ 15] M. Luo, J. Qiu and G. Sterman, in preparation. [ 16 ] J.J. Auberl et al., Phys. Lett. B 123 ( 1983 ) 275. [ 17 ] T. Ochiai, S. Date and H. Sumiyoshi, Prog. Theor. Phys. 75 ( 1986 ) 288. [ 18 ] J. Qiu and G. Sterman, Phys. Rcv. Lett. 67 ( 1991 ) 2264; Stony Brook preprint ITP-SB-91-9 ( 1991 ). [ 19] R. Baier, J. Engels and B. Petcrsson, Z. Phys. C 2 (1979) 265. [20] S. Gavin and J. Milana, Brookhaven and William and Mary, preprint BNL 46793, WM-91-116 ( 1991 ); S.J. Brodsky, P. Hoyer, A.H. Mueller and W.-K. Tang, SLAC preprint SLAC-PUB-5585 ( 1991 ).
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Nuclear dependence at large transverse momentum M a L u o a, J i a n w e i Q i u b a n d G e o r g e S t e r m a n a a Institute for Theoretical Physics. State University of New York, Stony Brook, NY 11794-3840, USA b Department of Physics andAstronomy, Iowa State University, Ames, IA 50011, USA Received 5 January 1992
We treat the anomalous nuclear dependence of high-pr jet and single-particle photoproduction cross sections in perturbative QCD. We show that the Aa dependence is associated with multi-patton matrix elements in nuclei. Assuming a natural model for these correlation functions, we derive predictions for the xv, xT and energy dependence of the enhancement. Significant enhancements are found without introducing scales other than AQco. The methods described may bc applied to other high-Pr nuclear cross sections.
A m o n g the most influential experiments o f the '70's were those that detected a power-law falloffin transverse m o m e n t u m for particles produced in h a d r o n - h a d r o n scattering [ 1 ]. This behavior was quickly recognized as the result o f an underlying p~4 hard-scattering process, convoluted with p a t t o n distributions and jet decay, or fragmentation, functions [ 2 ]. Further observation detected an " a n o m a l o u s " dependence on nuclear target atomic number, A, in which the cross section at fixed transverse m o m e n t u m grows [3,4] a p p r o x i m a t e l y as A " with c~> 1. This effect is most often described as due to multiple interactions o f p a r t o n s in nuclear matter, primarily double scattering [ 5,6 ]. O u r aim in this p a p e r is to show that nuclear enhancement can be described naturally in perturbative QCD, in terms o f a nonleading power, or "higher twist" formalism [ 7,8 ]. We will argue that a n o m a l o u s nuclear enh a n c e m e n t data provides information on multi-parton matrix elements in nuclei. It is therefore not a secondary phenomenon, to be described in terms o f known nuclear and particle parameters, but a fundamental process in its own right, which tells us something new about nuclear m a t t e r and its interaction with high-energy probes. To set the stage for our argument, we first briefly review the perturbative Q C D ( p Q C D ) description o f high-px j e t and single-particle inclusive cross sections. We then discuss factorization theorems at nonleading power, the isolation o f contributions to nuclear enhancement, and some representative predictions for cross sections. F o r simplicity, we work with p h o t o p r o d u c t i o n cross sections at lowest order, although the methods introduced here are more general. The success o f the hard scattering picture for high-pv processes in h a d r o n - h a d r o n scattering can be understood in terms o f the p a r t o n model or, m o r e generally, in terms o f the so-called factorization theorems o f p Q C D [9]. F o r instance, the cross section for hadrons h and h' to produce an outgoing j e t with large transverse mom e n t u m l T c a n be written to leading power as the convolution o f parton distributions, ~,/h (x, l 2 ), times a perturbative hard scattering function, H0, dcr(l) ¢oi d3---~ = ~
dx, dxjO,/h(x,, 12)H,j(x,, xj, l)Oj/h. (X;, l 2) .
( 1)
Parton distributions may be defined in terms o f hadronic matrix elements. An example is the distribution o f a quark in a h a d r o n m o v i n g in the + z direction, p~, = ( n ' p h ) a u, where we define r / u = ~ , + , nU=6¢,_.In this case we have 0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
377
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exp[i
(n.p)xy- l" ½(Ph I q ( 0 ) n ' 7 q ( y - )lPh ) ,
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(2)
with q(x) the quark field, u appears as the scale in the rcnormalization of the operator involving two quark fields separated by a light-like distance ( y - ) . In this form, the matrix element in eq. (2) is evaluated in the light cone A + = 0 gauge [ 10 ]. In a similar fashion, the single-particle inclusive cross section for hadron M with transverse m o m e n t u m lfr is determined by convolutions of exprcssions likc eq. ( 1 ) with fragmentation functions, DM/k (~, l~"), which describe the likclihood of finding hadron M in the decay products of parton k, with fraction r o f t h e parton's m o m e n t u m [2]. Most studies of nuclear enhancement have described it by means of two sequential, independent scattering processes, each of which is a leading-power partonic cross section like eq. ( 1 ) [ 5 ]. Nuclear enhancement enters as an extra factor of the nuclear radius, which results from integrating over the relative positions of the scatterings. Such cross sections require an additional a~, relative to lowest order. They arc infrared "unstable", in the limit that one of the scattcrings is responsible for the entire m o m e n t u m transfer, and one becomes soft [ 11 ]. Finally, even though each scattering in the sequence is leading power, nuclear enhancement is a nonleadingpower effect [ 6 ], suppressed by a factor of I F 2 compared to a single-scattering cross section like eq. ( 1 ). Thcse considerations suggest that we reanalyzc nonleading power corrections to nuclear scattering in the context of p Q C D factorization [6,8,12,13 ]. What we will sec is that nuclear enhancement can be described by a generalization of eqs. ( 1 ) and (2), with factorization integrals involving matrix elements of four, rather than two ficlds. The resulting hard-scattering corrections begin at order a~, rather than a~2, compared to lowest order, and are infrarcd-safe, all long-distance effects having been absorbed into the new matrix elements. The extension of p Q C D to double-scattering requires factorization at nonleading power. For unpolarized cross sections, the f r s t power correction is at the order o f l ~ 2, relative to leading behavior. Factorization at this level is a nontrivial result, which has bcen justificd in ref. [13]. To cut down on calculational complexity, we first discuss jet (parton) photoproduction cross sections, where l" is the m o m e n t u m of the produced parton. We will take the photon to have m o m e n t u m qU= nUx/s-~, and thc (nuclear) target to have m o m e n t u m Ap ~, with p" = ~'v/ss~. We dcfine s = 2p.q and t = - 2p-l. After factorization, the gencral I-F2 cross section is, as in eq. ( 1 ), separated into a sum over convolutions of long-distance matrix elements, T{,l, characterized by some set of fields {i}, and hard-scattering functions ,~*i. The convolution integrals are now over three independent fractional m o m e n t a x , of the form
2
d°4 = f cL~tdx2dx3 (L~ (l~ T~a(xl, x2, x3),~i}(x,, x2, x3, 1, s) , oh~5~ \4zr/ks/~{,}
(3)
as illustrated in fig. 1. (For this discussion, we will neglect the hadronic parton structure of the photon.) In the full expansion, many matrix elements will appear, even at Born approximation in the hard part. They will be of the general form [ 8,12,13 ],
TDI(XI,X2, X3)--
if
(27~)3
dyFdyf dy~ exp[in.p(x3yf +x2y[ + x l y i - ) ]
X taara(PA IT [B~(0)B~(y~-)/3r (y~-)Ba(y~ )]IPA ) ,
(4)
where the B's and/~'s are quark fields, covariant derivatives, i 0--gA, or gluonic field strengths [ 8,12 ], and where t~/~a is a constant matrix that produces a scalar from these fields. The state IP,~) represents the "target" hadron, in our case a nucleus. Typical contributions to .~ at the Born level, are shown in fig. 2, with an "observed" parton of m o m e n t u m I. ,~¢~must be calculated by summing over "cuts" of the diagram, corresponding to different final states. It is only after this sum that the answer turns out to be power suppressed, through cancellation of the leading power contribution from each diagram [ 14,15 ]. The calculable hard scattering function ,~ is independent of the identity of the target; indeed, this is the 378
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'\))t<'/ (a)
q
~:
Fig. 1. Graphical representation of factorization at first nonleading power.
(b)
(c) Fig. 2. Typical contributions to hard scattering at tree level. (a) Quark-gluon, (b) gluon-gluon, (c) quark-quark.
essence of factorization. Therefore, if we are to find nuclear enhancement, it must reside in the matrix elements, eq. (4), and we need criteria to identify such dependence in muitilocal operators of this type. Actually, this is not difficult. The integrals over y, in eq. (4) have a range that we expect to grow with the nuclear radius, that is, as A 1/3. On the other hand, these integrals include explicit oscillating exponentials, exp(in'pxiy7 ), which will damp growth in A unless one or more of the xi vanishes. This, in turn, requires delta functions, d(xi), in the hard scattering function .~. We will see in a moment how they may come about. Even with such delta functions, however, we expect a number of restrictions on growth with nuclear radius. Before describing them, let us emphasize that setting the momentum fraction o f a gluon line to zero does not produce a long-distance contribution in the hard scattering. A pole in the hard scattering may be thought of as a prescription for evaluating an imaginary part, which could always have been computed by a contour integral at a finite distance from the pole. Such deformations are possible because the hard scattering diagrams involve additional partons, with finite fractional momenta [ 15 ]. The same reasoning would not apply to leading-power gluon scattering at x = 0 . We now make two simple assumptions about expectation values ofmultilocal operators in nuclear states with large A. To avoid inessential overall factors, we normalize nuclear states to a fixed energy scale, (PA IP'A) = 2 (~opA/A)8 3(PA --P~ )- Our assumptions then are: (i) that the expectation value of a product of operators separated by a distance small compared to the nuclear radius is proportional to the nuclear volume, i.e., to A, and (ii) that the expectation value of a product of operators separated by a distance comparable to the nuclear radius can be independent of the distance only if the operators are individually Lorentz scalars and color singlets. According to the first assumption, the nuclear dependence [ 16 ] of leading-power patton distributions like eq. (2) is not the source of anomalous nuclear enhancement [ 17]. For instance, at fixed x # 0 in eq. (2), the quark fields arc kept close together by the oscillating exponential, so we expect the (suitably normalized) nuclear matrix element to grow like A for large A. (It need not be a simple multiple of the corresponding matrix element at small A, however. ) According to the second assumption, there can be at most a single extra power of A 1/3 at first nonleading power from the yi integrals, because the four operators in eq. (4) must pair into two color and spin singlets. Clearly, then, matrix elements that produce nuclear enhancement involve pairs of quark fields, a n d / o r pairs ofgluon fields. The above considerations place strong restrictions on contributions in .~ that can match with nuclear enhancement from their corresponding matrix elements. Without going into details, the evaluation of the hardscattering is carried out by a "collinear expansion", about the leading power [ 8,12,15 ]. We find it most convenient to carry out this expansion in Feynman gauge [ 12,18]. At the order to which we work, A 4/3 behavior comes entirely from hard-scattering diagrams with poles which set two of the x,'s in eq. (4) to zero. In this case, 379
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two fields, for instance the gluon lines carrying momenta x2p and x3p in fig. 2a, have vanishing momentum fraction, while all nonzero plus momentum is carried into the hard part by quark lines. Then two of the Y7 integrals in eq. (4) have no oscillatory suppression, although, in accordance with our assumptions above, we expect color confinement to keep the two gluon fields close together. Otherwise, we would predict A 5/3 nuclear dependence. Double-pole contributions at x2 = x3 = 0 are present in each of the three possible cuts of fig. 2a, as well as those of fig. 2b, in which a gluon supplies the large momentum from the nucleus. For each cut, the "observed" momentum is always fixed at l u. Explicit calculation shows that all leading power contribution cancel in the sum over cuts for any fixed graph with two extra poles. The remainder is an expansion in the transverse momentum carried by the gluons. This cancellation applies to any graph with the same topology as in figs. 2a, 2b. We neglect here "initial-state" poles (xl = 0 in fig. 2a, for example), which turn out to give small effects [15,18 ]. If the leading contribution is already nonleading power (/.F 2 ), as with four-quark diagrams [8,12] (fig. 2c), or diagrams with two extra transverse-polarized gluons, then the sum over final states suppresses the nuclear dependence by an extra power of l~:2 compared to the first nonleading power. To get an overall dependence A4/3l~ 2 from a matrix element of the form in eq. (4), we must begin with cut diagrams that are individually of leading power. The only possibility is to take the (n.A) component for the two gluons whose longitudinal momenta will be set to zero by the extra poles from the hard scattering. As mentioned above, the sum over cuts will produce an expansion in their transverse momenta, and will consequently produce matrix elements of the fields 0-r (n'A). In summary, by isolating contributions to the hard scattering that can match with A 4/3 behavior from the nucleus, we find factorized forms at first nonleading power that involve the matrix elements Tq(x,A) = f
~f~-exp[i(n'p)xyT ] ~ dy~ dy-------f-~O(y~ - y f )O(yy
× ½(PA Iq(0) (n'y)n~F'~(y~)n~,V',dyT_ )q(Yi- ) [PA ) T~(x,A)= ; ~ X~
1
•
exp[i(n.p)xy?] fd dy~dy~ 2re O(y~-yf)O(y~)
(5)
with DPA>normalized as above. All fields are on the light-cone in the n u direction. In these expressions, we have already set x 2 = x 3 = 0 , and have suppressed ordered exponentials that ensure gauge invariance. We have also modified normalizations relative to eq. (4). A factor of 1/27r has been absorbed into the hard scattering, while a factor of ½ has been introduced into the quark-gluon matrix element and 1/(xn.p) into the gluon-gluon matrix element, to bring them into closer correspondence with leading-power parton distributions like eq. (2) [ 10]. These changes are purely formal, but they influence our guess for the sizes of the matrix elements in eq. (5), which we will describe shortly. A physical interpretation for these matrix elements follows by recognizing that classically, fdy-F + T ( y - ) measures the integral of the transverse force experienced by a unit color charge as it passes through the nucleus along a light-like trajectory in the minus direction. It may then be useful to consider these matrix elements as leading-power patton distributions, modified by an average squared transverse momentum transfer. The calculation of the hard scattering functions corresponding to the two matrix elements in eq. (5) is not difficult. The result involves derivatives of a phase-space delta function. For numerical estimates, it is therefore convenient to integrate by parts, producing an expression with the same content as eq. (3), in which derivatives with respect to x act on the matrix elements T o and Tg. Because the first nonvanishing corrections are second order in transverse momenta, we find up to two derivatives in our expression. For jet photoproduction, we have ¢DIdo'4/3
380
c~Er~(4noq)2 [ ( - -
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where x = - u ~ (s+ t), and where el is the charge of quark flavor f i n units of the electron charge. ~i represents an effective "partonic flux", given in terms of the matrix elements in eq. (5) by
~Ax'A)-LkOx2,,,,
x
/\-t(s+
In view of our interpretation for the matrix elements, the derivatives in eq. (7) reflect the kinematic influence of transverse momentum transfer on the parton flux. The hard-scattering functions 11, in eq. (6), summed over outgoing parton type, are given by ( N = 3 is the number of colors) _ _
2N e
,
+
+
s
+
=
±[/-,L(,+ql
.
(8)
Expressions for single-particle cross sections may be found by convoluting partonic cross sections with fragmentation functions, exactly as for the leading power. Once we have information on the nuclear matrix elements Tq and Tg, we can use the factorized expressions in cq. (6) to predict the behavior of cross sections. Conversely, the measurement of nuclear dependence in photoproduction cross sections may be used to determine the matrix elements. To get an idea of what is involved, we have evaluated jet and single-particle pion and proton cross sections as functions ofxv and IT by assuming the forms
T,( x, A) =A4/3A~cDO,/N(X) ,
(9)
for i a quark or gluon, with O,/N(x) the simplified (scaling) distributions for partons in a proton that are specified in ref. [ 18 ]. Nuclear cross sections are computed assuming isospin invariance and Z/A = 0.4. For dcfiniteness, we take AOCD=200 McV. Modifications due to evolution and to nuclear effects on single-parton distributions should not change our qualitative results, and we have neglected them. Similarly, we have used fragmentation functions given in ref. [ 19 ], fixed at a scale of 25 GcV 2, without evolution. The resulting predictions are, in any case, meant to be taken seriously only for their kinematic dependence, which is not very sensitive to the precise choices of distributions and fragmentation functions. In fig. 3 we show a representative kinematic dependence of the parameter a - 1, computed from the definition
del (A) A~ +
den/3(A) = A , d3/
de1 (A) d3/ ,
(10)
evaluated at A = 200, with a, the leading power nucleon Born cross section. Remarkably, a ' s computed with different values of A in the range 50
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--
0.6
(a) /
I
1-
,
,
,
,--1
,
,
,
,
1-,
.
.
0.6
/ 0.4
I
I ~3
0.4
,,;r 4
0.2
0.2
\ /
0.0
. . . . -0.5
0.0 0
0.5
.... 2
I .... 4
t .... 6
I .... 8
10
XF
Fig. 3. Dependence of a - 1, calculated by eq. ( 10 ) wit h the mat rix elements of eq. ( 9 ), ( a ) on xv for lr--- 6 GeV at s = 300 (dashed) and 600 GeV 2 (solid) in jet production, (b) on 11 at x~.=0.5 and s=600 GeV 2 for n + and proton. p r o v i d e a m u c h l a r g e r f r a m e w o r k t o t e s t a n d d e v e l o p t h e f o r m a l i s m we h a v e o u t l i n e d here. H a d r o n - n u c l c u s i n t e r a c t i o n s will i n v o l v e i n i t i a l as well as f i n a l s t a t e i n t e r a c t i o n s , so t h e a n a l y s i s will b e m o r e c o m p l e x , a l t h o u g h q u a l i t a t i v e l y s i m i l a r . S u c h a n a n a l y s i s m a y allow u s t o c o m p a r c t h e s a m e m a t r i x e l e m e n t s in d i f f e r e n t p r o c e s s e s , w h i c h w o u l d r e s u l t i n m u c h s t r o n g e r p r e d i c t i v e p o w e r . Issues o f t h i s sort, a l o n g w i t h t h e q u e s t i o n s o f h i g h e r o r d e r c o r r e c t i o n s , e v o l u t i o n , a n d a p p l i c a t i o n s to n u c l e u s - n u c l e u s s c a t t e r i n g , r e m a i n to b e s t u d i e d . W e t h a n k S. G a v i n , T. H a n , R.L. M c C a r t h y , J. M i l a n a a n d A.H. M u e l l e r for useful c o n v e r s a t i o n s . T h i s w o r k was s u p p o r t e d in p a r t b y t h e N a t i o n a l S c i e n c e F o u n d a t i o n u n d e r G r a n t N o . N S F 9 1 0 8 0 5 4 , b y t h e D e p a r t m e n t o f Energy u n d e r G r a n t No. D E - F G 0 2 - 8 7 E R 4 0 3 7 1 a n d b y t h e T e x a s N a t i o n a l R e s e a r c h L a b o r a t o r y C o m m i s s i o n .
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