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The perturbative pion form factor with Sudakov suppression
NUCLEAR
P H VS I C S B
Nuclear Physics B 381 (1992) 129—140 North-Holland
________________
The perturbative pion form factor with Sudakov suppression Hsiang-Nan Li and George Sterman Institutefor Theoretical Physics, State University of New York, Stony Brook, NY 11794-3840, USA Received 23 March 1992 Accepted for publication 15 April 1992 We give a modified perturbative expression for the pion electromagnetic form factor in perturbative QCD, which includes Sudakov suppression of the elastic scattering of quarks that 2OAQCD the perturbative calculation are isolated in space. Beyond momentum transfers of becomes relatively insensitive to soft gluon exchange. Our analysis involves no phenomenological parameters such as a gluon mass, and may be applied to the proton form factor.
1. Introduction The applicability of perturbative QCD (pQCD) to exclusive processes is a matter of controversy [1,21.Although there is general agreement that pQCD can successfully make predictions for exclusive reactions as momentum transfers Q go to infinity, it remains unclear whether experimentally accessible energy scales are large enough to verify these predictions. In particular, perturbative expressions for electromagnetic form factors have been brought under searching criticism by Isgur and Llywellyn Smith [1,31 and Radyushkin [31.They pointed out that calculations based on phenomenologically acceptable wave functions [4—61 are typically dominated by soft virtual gluon exchange, and violate the assumption that the momentum transfer proceeds perturbatively. Their estimates [1] of the approach to asymptotic behavior were so pessimistic, as to bring the entire enterprise into doubt. In this paper, we will derive a modified perturbative expression for the pion electromagnetic form factor, one that takes into account partonic transverse momenta. Our expression reduces to the standard one as Q but at lower momentum transfers it takes into account an infinite summation of higher-order effects associated with the elastic scattering of the valence partons of the pion (quark and antiquark). The underlying physical principle is that the elastic scattering of an isolated colored parton, such as a quark, is suppressed at high energy by radiative corrections. The theoretical methods are those that were developed to study the quark form factor in perturbation theory [7] (the “Sudakov” form factor). For a quark—antiquark pair separated by a transverse distance b, we shall see that —~ ~,
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Elsevier Science Publishers B.V. All rights reserved
130
H.-N. Li, G. Sterman
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The pion form factor
(a)
(b) Fig. 1. (a) Diagrams for the pion form factor. (b) Lowest-order examples.
this suppression behaves, in the leading logarithmic approximation, as exp[ const. In Q ln(ln Q/ln b)], which falls for fixed b faster than any power of Q. The effect of this suppression is to select components of the pion wave function with small spatial extent, just as is assumed in predictions of “transparency” in nuclear scattering [8]. This approach has already been applied to elastic hadron— hadron scattering [9]. Our conclusion is that, once these effects are taken into account, pQCD remains an appropriate tool for the study of the exclusive processes. We shall argue that, at least at the high end of experimentally accessible energies, our modified perturbative calculation is self-consistent, in the sense that numerical answers come predominantly from momentum regions in which the effective coupling is relatively small. We will be able to quantify these arguments by studying dependence on a cutoff in b, the transverse distance between the scattered valence particles. Such perturbative calculations should be thought of as complementing other derivations of the form factors at moderate energy, based on QCD sum rules [10]. In this paper, we will discuss our formalism as it applies to the pion electromagnetic form factor. A brief review of the standard asymptotic expression for the form factor [11—131may help to motivate our viewpoint. The pion form factor can be written as the convolution, illustrated in fig. la, of a hard-scattering amplitude T~and wave functions 4(x) [11,12] —
F~(Q2) f’dxi dx2 ~(x2, ~2)T~(x1,
2/~,a~(~2))th(xi,~2)
=
x2,
Q
(1)
H-N. Li, G. Sterman
where
Q2
=
/
The pion form factor
131
2P
1 P2, and ~ is the renormalization and factorization To the 2(~t2)/x scale. 2.There is lowest order, as shown in fig. ib, TH is proportional to g 1x2Q no singularity at x 1 or x2 0, because the wave functions vanish at the endpoints 0, 1 [4,12,131. In this formalism, transverse momenta kT that flow from the wave functions through the hard scattering are neglected as power-suppressed 2 + k?~) corrections. Thus,k?,~/(x the gluon propagator in term fig. lbis isthe of asymptotic the form 1/(x1x2Q l/(x 2) 2)2.The first hard scattering, 1x2Q while 1x2Q the second is suppressed by Q~2at fixed x. On the other hand, the .
=
=
—
x-integral is now more singular, indicating that power-suppressed corrections, such as transverse momentum dependence, become more important at the endpoints. Let us focus on the technical reason that soft gluon exchange in the endpoint region is a difficulty in the usual perturbative treatment. The most natural choice of the factorization scale is ji Q, which eliminates logarithms of Q/~iin T 11. In endpoint regions, however, higher-order corrections in TH will produce logarithms 2/~t2).These logarithms can, in principle, be eliminated by choosing like=xln(x1x2Q ~2 2, but then the running coupling a~(x 2)diverges at the endpoints. As a 1x2Q result, the perturbative calculation looses its1x2Q self-consistency as a weak-coupling expansion. Fortunately, methods are available for reorganizing perturbation theory in these dangerous regions, by analyzing the amplitude in the Fourier transform (b) space of kT [9]. It has been pointed out that the incorporation of the transverse structure of hadronic wave functions [2,14,15], the introduction of an effective gluon mass [16] and a running coupling “frozen” beyond an infrared cutoff [16,171 are natural methods of stabilizing the pQCD predictions for form factors. In effect, the inclusion of Sudakov corrections produces numerical effects very similar to these approaches, but without the introduction of scales other than AQCD. The pion form factor in b-space is studied in sect. 2, where we derive a formula for the form factor that takes into account leading logarithms of b. In sect. 3 we exhibit the relevant b-dependence. In sect. 4, we use asymptotic [12,131 and Chernyak and Zhitnitsky [4] wave functions to obtain representative numerical results. These results are analyzed to show how much of the contribution comes from the perturbative region. In sect. 5 we give our conclusions. =
2. Modified factorization formula To understand the factorization procedure that leads to eq. (1), and how it may be generalized, we refer to the diagrams of fig. 2. These diagrams, describing quark—antiquark elastic scattering, contain logarithms due to the 0(a) radiative corrections shown. TH labels the hard scattering, all of whose lines we assume to be far off-shell for the purposes of this discussion, while the external fermion lines are all near or on the mass shell. (We imagine that they had just emerged from a
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The pion form factor
(a)
(b)
Cc)
(d)
Fig. 2. Radiative corrections to the basic scattering diagram.
pion in the initial state, and were about to combine to produce a pion in the final state.) On a diagram-by-diagram basis, the radiative corrections will produce logarithmic enhancements in the momentum transfer Q, and corresponding logarithmic infrared divergences. These enhancements are of two kinds, collinear, when the extra gluon is parallel to either the incoming or outgoing pair, and soft, when it carries a momentum much smaller than the momentum transfer. The two regions, soft and collinear, may overlap, producing double-logarithmic, or “Sudakov” enhancements, which are characteristic of gauge theory radiative corrections. Which diagram in fig. 2 gives which enhancements depends on the gauge. For the formal arguments that follow, it is simplest to work in a physical gauge, such as axial. In this case, the two-particle reducible diagrams, like figs. 2a,b, have the double logs, while the two-particle irreducible corrections, figs. 2c,d, contain only single soft enhancements. This distinction generalizes to all orders in perturbation theory, so that even complicated corrections that are one-particle irreducible contain at most one logarithm per loop in axial gauge, while Sudakov corrections come entirely from diagrams that are two-particle reducible in the channels of the external pions. To be explicit, we note that our reasoning is carried out in the set of axial gauges for which the gauge-fixing parameter n~is a linear combination of dimensionless vectors L’~, i 1, 2, which are light-like and in the directions of the external pions. In the classic reasoning that leads to eq. (1) [11—13],we use the fact that soft divergences cancel between fig. 2a and 2b, as well as between 2c and 2d, at least in the approximation that the momenta of the soft gluons is negligible compared to the total momentum carried by the hard part. This is a good approximation asymptotically, as long as the wave functions 4(x, ~2) vanish at the endpoints x 0, 1. Once this approximation is made, two-particle irreducible corrections may be absorbed into T~because, once their soft enhancements have cancelled, they =
=
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The pion form factor
133
are dominated by momenta of order Q2. Similarly, the Sudakov logarithms present in each of figs. 2a and 2b reduce, after we combine them, to a single collinear logarithm, that may be summed according to renormalization group methods, and grouped into the wave functions çb(x, ~.t2) of eq. (1), giving them their scale-dependence. Our alternate approach to factorization, inspired by the considerations referred to in the introduction, makes use of this same analysis, but does not involve the step in which transverse momenta are neglected in the hard scattering. That is, we shall not assume at the outset that x 2>> k~,with kT the transverse momen1x2Q tum carried in a gluonic correction like fig. 2b. This leads us to a factorization form with two wave functions ~/i~(x~, kT) corresponding to the external pions, combined with a new hard-scattering function, T 11(x1, x2, Q, kT, kT), which depends in general on transverse as well as longitudinal momenta, 2) f’dxi dx 2kT ~i(x F~(Q 2fd2kT d 2, kT, P2) =
x TH(xl, x2, Q, KT)~J(xl,kT, P1).
(2)
At kT 0, the new T~coincides with the standard function in eq. (1). Eq. (2) may be derived more formally using the methods described in detail in ref. [91,but is already familiar as an intermediate step in deriving eq. (1) [12]. There is an implicit factorization in matrix structures, the same as in the standard factorization, eq. (1). In this form, both soft and collinear logarithmic enhancements are factorized into the functions i/i, including those that will cancel in the sum over diagrams as Q —* cc• Indeed, for fixed x, ~ 0, the two expressions, eqs. (1) and (2), are equivalent in this limit, up to corrections that fall off as a power of Q. On the other hand, eq. (2) retains much more information on the limit x1 0 for fixed Q, where the cancellations that allow us to derive eq. (1) fail. As mentioned above, the resulting dependence remains power-suppressed, but is potentially important. The next step in our analysis is to re-express eq. (2) in terms of Fourier transform variables in the transverse configuration space, 2b 2b 2) dx d 1 d 2 F~(Q 1 dx2 2 2~(x2, b2, P2, ~) o (2ir) (2ir) =
—‘
=f
XTH(xl,x2,bI,b2,Q,~)~~(xI,bl,Pl,~),
(3)
where the functions ~ in eq. (3) may be defined in terms of explicit matrix elements, which complement the diagrammatic descriptions given above [9], ~9’(x,
b, P,
~)
=
fd2kT etkTbl/J(x,
=
dy f~e~KO
kT, P) T(~(0)y~q(y~, 0~,b)) I ~(P)).
(4)
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The pion form factor
In eq. (3) the ~9~’sinclude all leading logarithmic enhancements at large b, on a diagram-by-diagram basis, regardless of the nonleading b-dependence of TH. At lowest order, T~is given in momentum space by
TH(x,,
Q, kT)
2 =
(x1Q
+
4 g2 (~)~F x1 k~ 2+ (kTl + 1)(x1x2Q
kT 2) 2)
4 g2 ~‘F x
2+
(kTI + kT
1x2Q
(5)
2’ 2)
where we neglect transverse momentum in the numerator. In the second form, we neglect that part of the transverse momentum dependence associated with virtual fermion lines, which are linear rather than quadratic in the x’s. Because the second form for T~depends on only a single combination of transverse momenta, the expression for the form factor now involves only a single b-integral,
f dx1 dx2 1
F,~(Q2)
=
o
db 2~(x2,b, P2, (2~-)
~)
xT11(x1,x2,b,Q,~),~(x1,b,P1,~).
(6)
This is the form whose behavior we will study below. Its essential advantage, compared to eq. (1) is the extra b-dependence in the hard scattering. Radiative corrections in higher orders will generate logarithms of the form ln(t/~) in TH, where t is the largest mass scale appearing in TH, t=max(~~Q, 1/b).
(7)
2 =x 2 only as long as x 2> 1/b2 for Thus, the natural choicetoinzero, T11 isthe ~.t 1x2Q 1x2Q values of the x’s closer scale of radiative corrections is dominated by the transverse distance bridged by the exchanged gluon. If this distance is small, radiative corrections will be small, regardless of the values of the x’s. Of course, when b is large and x 2 is small, radiative corrections are still large in TH. We 1x2Q that the wave functions ~ in eq. (3) strongly suppress are about to see, however, this region. Note that if we do not make the approximation of eq. (5) above, there will be additional independent dimensionless scales in the hard scattering, such as x 2b2, whose logarithms may be important at higher orders. We shall not attempt a1Qcomplete catalogue of such effects, which remain suppressed by the wave functions.
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The pion form factor
3. Asymptotic behavior in b and
135
Q
The large-b asymptotic behavior of the wave functions was analyzed in ref. [91.Here we shall simply quote the results, commenting only on such minor differences and simplifications that apply. The essential result we will be using is the exponentiation of Sudakov logarithms in b and Q referred to above. The result may be summarized as, ,~
~(x,
b, P,
~) =exp
—s(x, b,
Q) —s(1 —x,
b,
Q)
~ diii _2f
~yq(g(~)) 1/b
/.L
x~(x,~) +O(a~(1/b)).
(8)
a~/~is the quark anomalous dimension in axial gauge. The explicit form for the Sudakov exponent s(~,b, Q), with ~ x or 1 x, including all leading and next to leading logarithms may be conveniently expressed in terms of the variables Yq
=
—
=
—
ln[~Q/(I~A)], /~ln(bA).
(9)
In these terms, we have ~ s(~,b,
Q)
=
A~
~ _i)_
ln(_~)+ ~
~
A~’~i32 In( —2b)
+
1
___
ln(2c~)+ 1 —
2) —
At -~
—
~I3~
Atl)
_____
A~1~ —ln(~e2~’)ln 4f3~
—b
A~
1),3
-
32~3~ [ln2(2c~)
-
1n2(
-2b)].
(10)
In this expression, the coefficients A(t) and /3~ are ______
12 At1~=~, At2
‘
I~2
lS3—l9n~ 24
~—r2—~n 1+~f3l
where
flf
ln(~-e~’),
(11)
(= 3 below) is the number of quark flavors and y is the Euler constant.
136
H. -N. Li, G. Sterman
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The pion form factor
Similarly, the renormalization group applied to TH gives TH(xl, x2, b,
4f
Q, ~) = exp
~d~i yq(g(~i))
tL~L
X
TH(xl, x2, b,
Q,
(12)
t),
where t is given by eq. (7). The anomalous dimensions help take into account the strong couplings of quarks when t is small. Note that it is not sufficient to simply match the scale of the running coupling in the hard scattering to the gluon’s momentum, without including the quark anomalous dimension. Using the results of eqs. (9)—(12) in eq. (6), we derive our lowest-order expression for the pion form factor 2)= 16~T~’Ff1dxl dx F~(Q
2 4(xi)4(x2)j xexp[—5~’(x1, x2, b,
b db a~(t)Ko(~~Qb)
Q)],
(13)
with
21
2
9’(x1, x2, b,
Q) =
~ (s(x~,b, i=1
Q)
+s(1 —xe, b,
Q))
—
—ln——--~-, (14) /3~ —b
where 1 = ln(t/A), A A0c~. K0 in eq. (13) is the modified Bessel function of order zero, which is the Fourier transform to b-space of the gluon propagator. Notice in particular the argument of the running coupling, defined in eq. (7) above, which cuts off the growth of radiative corrections in the hard scattering, except in the b —s 1/A limit, where the entire integrand is suppressed anyway. In eq. (13), we have neglected the evolution of the wave function ~ with 1/b, which is a nonleading effect that does not pertain directly to the mechanisms that we are discussing in this paper. Now let us return to the question of the self-consistency of the perturbative calculation for the pion form factor, in terms of our reformulated expression, eq. (13). We propose to analyze this expression, not by cutting off the x-integrals near their endpoints, but by testing the sensitivity of the integral to the large b-region. Of course, b is strictly limited to be less than 1/A, where the (negative) Sudakov exponent diverges. We expect that, because eq. (1) is the true asymptotic behavior, as Q increases, the b-integral will become more and more dominated by small b, where the effective coupling is small, regardless of the momentum fraction carried by the gluon. This requires a numerical study of the integrals in eq. (13), to which we now turn.
H-N. Li, G. Sterman
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The pion form factor
137
4. Numerical results The typical behavior of Sudakov suppression with momentum transfer Q is shown in fig. 3. There is no suppression for small b, where the two fermion lines are close to each other. For simplicity of analysis, we set the Sudakov exponential to unity in the small-b region where it induces a (small) enhancement, since in this region it is dominated by low orders in perturbation theory, and should be thought of as a part of the higher-order corrections to T~.In this vein, we have also set any factor ~ to unity whenever ~ < v~/(bQ).As b increases, however, exp S in eq. (13) decreases, reaching zero at b = 1/A. Suppression in the large-b region is weaker for smaller Q, where Sudakov effects are mild. To illustrate the qualitative effects of Sudakov corrections, eq. (13) is evaluated for two models of distribution amplitudes: the asymptotic wave function [12] —
=
~as(X)
~x(1
-x),
(15)
and the Chernyak—Zhitnitsky wave function [4] __~ix(1_x)(1_2x)2, ~2N~
~CZ(x)
where N~= 3 is the number of colors and
1.2
f~= 0.133
(16)
GeV the pion decay constant.
exp(—S)
l.O/A
Fig. 3. Behavior of Sudakov suppression in the transverse separation b for energy
Q2 /fl
=
20 and
Q3/fl = 50 with
x1 =
=
0.5.
Q1 /A
tO,
H. -N. Li, G. Sterman
138
0.16
Q2F(Q2)
/
The pion form factor
as
0
______________
1.0/A
(a) 0.32
~
2F(Q2)
Q
CZ
___________________________________
1.0/A
(b) Fig. 4. Dependence of Q2F~,(Q2)on the cutoff b~with (a) the asymptotic wave function and (b) the wave function of Chernyak and Zhitnitsky. The energies Q 1, Q2 and Q3 are as in fig. 3.
In order to see how the contribution to eq. (13) is distributed in b-space the integration is done with a variable cut off in b, b~.Typical numerical results are shown in fig. 4a 2F,.(Q2) and 4b for and 4~, respectively. The showing the on ~as b~,increase from zero at b~ = 0, curves, and reach their full dependence of Q height at b~= 1/A, beyond which we consider any remaining contributions as truly nonperturbative. As anticipated, we observe a faster rise as Q increases. At the largest value shown, Q/A = 50, the curves are quite flat for large b, indicating little contribution from large distances. To be quantitive, we may consider the cutoff up to which half of the whole contribution has been accumulated. For the wave function cb~,50% of Q2F~(Q2)comes from the region with b ~ 0.39/A for Q/A = 10 and b ~ 0.25/A for Q/A = 20. Similarly, for the wave function 4~, 50% is due to the region with b ~ 0.46/A for Q/A = 10 and b ~ 0.32/A for Q/A = 20. A liberal standard to judge the relevance of the perturbative method would be that 50% of the result come from the region where the coupling constant is no larger than, say, 0.7. From this point of view, pQCD begins to be self-consistent at
H.-N. Li, G. Sterman
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The pion form factor
139
about Q 20A (2 GeV if A is taken to be 0.1 GeV). Note that this applies to our model CZ wave function, as well as to the asymptotic wave function, even though the two differ dramatically when analyzed from the point of view of cutoffs in the x’s [1]. As the momentum transfer increases, the perturbative calculation improves. By the time Q/A reaches 50, the perturbative region in b, defined as above, gives 90% of the form factor for and 80% for qYS
~
5. Conclusions We have shown explicitly that, with the help of Sudakov suppression, pQCD enlarges its range of applicability down to accessible energies. The transverse momenta flowing through the pion distribution amplitudes and hard-scattering subdiagram play an essential role in our discussion. This effect is common in most exclusive processes and we cannot have a complete perturbative description for them without including it. At the same time, the approach to perturbative behavior is relatively slow and we cannot expect the leading order perturbative predictions to be precise even at the highest available energies. We should note that our predictions for Q2F~(Q2)from the asymptotic wave function are of the order 0.1, which is about the experimental data [18] (but see ref. [21]). The use of the CZ wave function can increase the results by a factor of 2, which still falls short. This is because we ignore the evolution of the wave function. We have taken this simplified approach, because we are interested here primarily in isolating the role of radiative corrections. In future work on the proton form factor [4,5,19], the full set of evolution phenomena should be included [6,20]. Additionally one-loop corrections [13] may be studied in our formalism. We emphasize again, that there is no parameter to be adjusted in our calculation beyond the input wave functions. The suppression of the non-perturbative region as Q increases results from including a formally higher power-suppressed transverse momentum dependence in the hard scattering, rather than introducing a gluon or quark mass [2,3]. This scheme can be extended to calculate more complicated processes like the nucleon magnetic form factor [22]. ~-
This work was supported in part by the National Science Foundation under Grant NSF 9108054, by the Department of Energy under Grant DE-FGO287ER40371 and by the Texas National Research Laboratory Commission. References [1] N. Isgur and C.H. Liewellyn Smith, NucI. Phys. B317 (1989) 526 [2] T. Huang and Q.-X. Shen, Z. Phys. C50 (1991) 139 [3] N. Isgur and C.H. Llewellyn Smith, Phys. Rev. Lett. 52 (1984) 1080 A.V. Radyushkin, Acta Phys. Polonica, B15 (1984) 403;
A.P. Bakulev and A.P. Radyushkin, Phys. Lett. B271 (1991) 223
140 [4) [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
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The pion form factor
V.L. Chernyak and AR. Zhitnitsky, Phys. Rep. 112 (1984) 173; NucI. Phys. B246 (1984) 52 M. Gari and N.G. Stefanis, Phys. Lett. Bl75 (1986) 462 ID. King and C.T. Sachrajda, NucI. Phys. B279 (1987) 785 J.C. Collins, in Perturbative quantum chromodynamics, ed. A.H. Mueller (World Scientific, Singapore, 1989) S.J. Brodsky and A.H. Mueller, Phys. Lett. B206 (1988) 685 J. Butts and G. Sterman, NucI. Phys. B325 (1989) 62 V.A. Nesterenko and A.V. Radyushkin, Phys. Lett. BuS (1982) 410; B.L. loffe and A.V. Smilga, NucI. Phys. B216 (1983) 373 G.R. Farrar and D.R. Jackson, Phys. Rev. Lett. 43 (1979) 246 G.P. Lepage and S.J. Brodsky, Phys. Lett. B87 (1979) 359 A.V. Efremov and A.V. Radyushkin, Phys. Lett. B94 (1980) 245 Z. Dziembowski and L. Mankiewicz, Phys. Rev. Lett. 58 (1987) 2175 L.S. Kisslinger and SW. Wang, Univ. of Maryland preprint, PP No. 91-233 (1991) C.-R. Ji, A.F. Sill and R.M. Lombard-Nelsen, Phys. Rev. D36 (1987) 165 J.M. Cornwall, Phys. Rev. D26 (1982) 1453 C.J. Bebek et al., Phys. Rev. D17 (1978) 1693 C.E. Carlson and F. Gross, Phys. Rev. D36 (1987) 2060 M. Gari and N.G. Stefanis, Phys. Rev. D35 (1987) 1074 C.E. Carlson and J. Milana, Phys. Rev. Lett. 65 (1990) 1717 H-N. Li, Stony Brook preprint, ITP-513-92-25 (1992) F.-M. Dittes and A.V. Radyuskin, Yad. Fiz. 34 (1981) 529 (Soy. J. Nuci. Phys. 34 (1981) 293); RD. Field, R. Gupta, S. Otto and L. Chang, NucI. Phys. B186 (1981) 429; M.H. Sarmadi, Ph.D. thesis, Univ. of Pittsburgh (1982); E. Braaten and S.-M. Tze, Phys. Rev. D35 (1987) 2255
P H VS I C S B
Nuclear Physics B 381 (1992) 129—140 North-Holland
________________
The perturbative pion form factor with Sudakov suppression Hsiang-Nan Li and George Sterman Institutefor Theoretical Physics, State University of New York, Stony Brook, NY 11794-3840, USA Received 23 March 1992 Accepted for publication 15 April 1992 We give a modified perturbative expression for the pion electromagnetic form factor in perturbative QCD, which includes Sudakov suppression of the elastic scattering of quarks that 2OAQCD the perturbative calculation are isolated in space. Beyond momentum transfers of becomes relatively insensitive to soft gluon exchange. Our analysis involves no phenomenological parameters such as a gluon mass, and may be applied to the proton form factor.
1. Introduction The applicability of perturbative QCD (pQCD) to exclusive processes is a matter of controversy [1,21.Although there is general agreement that pQCD can successfully make predictions for exclusive reactions as momentum transfers Q go to infinity, it remains unclear whether experimentally accessible energy scales are large enough to verify these predictions. In particular, perturbative expressions for electromagnetic form factors have been brought under searching criticism by Isgur and Llywellyn Smith [1,31 and Radyushkin [31.They pointed out that calculations based on phenomenologically acceptable wave functions [4—61 are typically dominated by soft virtual gluon exchange, and violate the assumption that the momentum transfer proceeds perturbatively. Their estimates [1] of the approach to asymptotic behavior were so pessimistic, as to bring the entire enterprise into doubt. In this paper, we will derive a modified perturbative expression for the pion electromagnetic form factor, one that takes into account partonic transverse momenta. Our expression reduces to the standard one as Q but at lower momentum transfers it takes into account an infinite summation of higher-order effects associated with the elastic scattering of the valence partons of the pion (quark and antiquark). The underlying physical principle is that the elastic scattering of an isolated colored parton, such as a quark, is suppressed at high energy by radiative corrections. The theoretical methods are those that were developed to study the quark form factor in perturbation theory [7] (the “Sudakov” form factor). For a quark—antiquark pair separated by a transverse distance b, we shall see that —~ ~,
0550-3213/92/$05.00 © 1992
—
Elsevier Science Publishers B.V. All rights reserved
130
H.-N. Li, G. Sterman
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The pion form factor
(a)
(b) Fig. 1. (a) Diagrams for the pion form factor. (b) Lowest-order examples.
this suppression behaves, in the leading logarithmic approximation, as exp[ const. In Q ln(ln Q/ln b)], which falls for fixed b faster than any power of Q. The effect of this suppression is to select components of the pion wave function with small spatial extent, just as is assumed in predictions of “transparency” in nuclear scattering [8]. This approach has already been applied to elastic hadron— hadron scattering [9]. Our conclusion is that, once these effects are taken into account, pQCD remains an appropriate tool for the study of the exclusive processes. We shall argue that, at least at the high end of experimentally accessible energies, our modified perturbative calculation is self-consistent, in the sense that numerical answers come predominantly from momentum regions in which the effective coupling is relatively small. We will be able to quantify these arguments by studying dependence on a cutoff in b, the transverse distance between the scattered valence particles. Such perturbative calculations should be thought of as complementing other derivations of the form factors at moderate energy, based on QCD sum rules [10]. In this paper, we will discuss our formalism as it applies to the pion electromagnetic form factor. A brief review of the standard asymptotic expression for the form factor [11—131may help to motivate our viewpoint. The pion form factor can be written as the convolution, illustrated in fig. la, of a hard-scattering amplitude T~and wave functions 4(x) [11,12] —
F~(Q2) f’dxi dx2 ~(x2, ~2)T~(x1,
2/~,a~(~2))th(xi,~2)
=
x2,
Q
(1)
H-N. Li, G. Sterman
where
Q2
=
/
The pion form factor
131
2P
1 P2, and ~ is the renormalization and factorization To the 2(~t2)/x scale. 2.There is lowest order, as shown in fig. ib, TH is proportional to g 1x2Q no singularity at x 1 or x2 0, because the wave functions vanish at the endpoints 0, 1 [4,12,131. In this formalism, transverse momenta kT that flow from the wave functions through the hard scattering are neglected as power-suppressed 2 + k?~) corrections. Thus,k?,~/(x the gluon propagator in term fig. lbis isthe of asymptotic the form 1/(x1x2Q l/(x 2) 2)2.The first hard scattering, 1x2Q while 1x2Q the second is suppressed by Q~2at fixed x. On the other hand, the .
=
=
—
x-integral is now more singular, indicating that power-suppressed corrections, such as transverse momentum dependence, become more important at the endpoints. Let us focus on the technical reason that soft gluon exchange in the endpoint region is a difficulty in the usual perturbative treatment. The most natural choice of the factorization scale is ji Q, which eliminates logarithms of Q/~iin T 11. In endpoint regions, however, higher-order corrections in TH will produce logarithms 2/~t2).These logarithms can, in principle, be eliminated by choosing like=xln(x1x2Q ~2 2, but then the running coupling a~(x 2)diverges at the endpoints. As a 1x2Q result, the perturbative calculation looses its1x2Q self-consistency as a weak-coupling expansion. Fortunately, methods are available for reorganizing perturbation theory in these dangerous regions, by analyzing the amplitude in the Fourier transform (b) space of kT [9]. It has been pointed out that the incorporation of the transverse structure of hadronic wave functions [2,14,15], the introduction of an effective gluon mass [16] and a running coupling “frozen” beyond an infrared cutoff [16,171 are natural methods of stabilizing the pQCD predictions for form factors. In effect, the inclusion of Sudakov corrections produces numerical effects very similar to these approaches, but without the introduction of scales other than AQCD. The pion form factor in b-space is studied in sect. 2, where we derive a formula for the form factor that takes into account leading logarithms of b. In sect. 3 we exhibit the relevant b-dependence. In sect. 4, we use asymptotic [12,131 and Chernyak and Zhitnitsky [4] wave functions to obtain representative numerical results. These results are analyzed to show how much of the contribution comes from the perturbative region. In sect. 5 we give our conclusions. =
2. Modified factorization formula To understand the factorization procedure that leads to eq. (1), and how it may be generalized, we refer to the diagrams of fig. 2. These diagrams, describing quark—antiquark elastic scattering, contain logarithms due to the 0(a) radiative corrections shown. TH labels the hard scattering, all of whose lines we assume to be far off-shell for the purposes of this discussion, while the external fermion lines are all near or on the mass shell. (We imagine that they had just emerged from a
132
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The pion form factor
(a)
(b)
Cc)
(d)
Fig. 2. Radiative corrections to the basic scattering diagram.
pion in the initial state, and were about to combine to produce a pion in the final state.) On a diagram-by-diagram basis, the radiative corrections will produce logarithmic enhancements in the momentum transfer Q, and corresponding logarithmic infrared divergences. These enhancements are of two kinds, collinear, when the extra gluon is parallel to either the incoming or outgoing pair, and soft, when it carries a momentum much smaller than the momentum transfer. The two regions, soft and collinear, may overlap, producing double-logarithmic, or “Sudakov” enhancements, which are characteristic of gauge theory radiative corrections. Which diagram in fig. 2 gives which enhancements depends on the gauge. For the formal arguments that follow, it is simplest to work in a physical gauge, such as axial. In this case, the two-particle reducible diagrams, like figs. 2a,b, have the double logs, while the two-particle irreducible corrections, figs. 2c,d, contain only single soft enhancements. This distinction generalizes to all orders in perturbation theory, so that even complicated corrections that are one-particle irreducible contain at most one logarithm per loop in axial gauge, while Sudakov corrections come entirely from diagrams that are two-particle reducible in the channels of the external pions. To be explicit, we note that our reasoning is carried out in the set of axial gauges for which the gauge-fixing parameter n~is a linear combination of dimensionless vectors L’~, i 1, 2, which are light-like and in the directions of the external pions. In the classic reasoning that leads to eq. (1) [11—13],we use the fact that soft divergences cancel between fig. 2a and 2b, as well as between 2c and 2d, at least in the approximation that the momenta of the soft gluons is negligible compared to the total momentum carried by the hard part. This is a good approximation asymptotically, as long as the wave functions 4(x, ~2) vanish at the endpoints x 0, 1. Once this approximation is made, two-particle irreducible corrections may be absorbed into T~because, once their soft enhancements have cancelled, they =
=
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The pion form factor
133
are dominated by momenta of order Q2. Similarly, the Sudakov logarithms present in each of figs. 2a and 2b reduce, after we combine them, to a single collinear logarithm, that may be summed according to renormalization group methods, and grouped into the wave functions çb(x, ~.t2) of eq. (1), giving them their scale-dependence. Our alternate approach to factorization, inspired by the considerations referred to in the introduction, makes use of this same analysis, but does not involve the step in which transverse momenta are neglected in the hard scattering. That is, we shall not assume at the outset that x 2>> k~,with kT the transverse momen1x2Q tum carried in a gluonic correction like fig. 2b. This leads us to a factorization form with two wave functions ~/i~(x~, kT) corresponding to the external pions, combined with a new hard-scattering function, T 11(x1, x2, Q, kT, kT), which depends in general on transverse as well as longitudinal momenta, 2) f’dxi dx 2kT ~i(x F~(Q 2fd2kT d 2, kT, P2) =
x TH(xl, x2, Q, KT)~J(xl,kT, P1).
(2)
At kT 0, the new T~coincides with the standard function in eq. (1). Eq. (2) may be derived more formally using the methods described in detail in ref. [91,but is already familiar as an intermediate step in deriving eq. (1) [12]. There is an implicit factorization in matrix structures, the same as in the standard factorization, eq. (1). In this form, both soft and collinear logarithmic enhancements are factorized into the functions i/i, including those that will cancel in the sum over diagrams as Q —* cc• Indeed, for fixed x, ~ 0, the two expressions, eqs. (1) and (2), are equivalent in this limit, up to corrections that fall off as a power of Q. On the other hand, eq. (2) retains much more information on the limit x1 0 for fixed Q, where the cancellations that allow us to derive eq. (1) fail. As mentioned above, the resulting dependence remains power-suppressed, but is potentially important. The next step in our analysis is to re-express eq. (2) in terms of Fourier transform variables in the transverse configuration space, 2b 2b 2) dx d 1 d 2 F~(Q 1 dx2 2 2~(x2, b2, P2, ~) o (2ir) (2ir) =
—‘
=f
XTH(xl,x2,bI,b2,Q,~)~~(xI,bl,Pl,~),
(3)
where the functions ~ in eq. (3) may be defined in terms of explicit matrix elements, which complement the diagrammatic descriptions given above [9], ~9’(x,
b, P,
~)
=
fd2kT etkTbl/J(x,
=
dy f~e~KO
kT, P) T(~(0)y~q(y~, 0~,b)) I ~(P)).
(4)
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The pion form factor
In eq. (3) the ~9~’sinclude all leading logarithmic enhancements at large b, on a diagram-by-diagram basis, regardless of the nonleading b-dependence of TH. At lowest order, T~is given in momentum space by
TH(x,,
Q, kT)
2 =
(x1Q
+
4 g2 (~)~F x1 k~ 2+ (kTl + 1)(x1x2Q
kT 2) 2)
4 g2 ~‘F x
2+
(kTI + kT
1x2Q
(5)
2’ 2)
where we neglect transverse momentum in the numerator. In the second form, we neglect that part of the transverse momentum dependence associated with virtual fermion lines, which are linear rather than quadratic in the x’s. Because the second form for T~depends on only a single combination of transverse momenta, the expression for the form factor now involves only a single b-integral,
f dx1 dx2 1
F,~(Q2)
=
o
db 2~(x2,b, P2, (2~-)
~)
xT11(x1,x2,b,Q,~),~(x1,b,P1,~).
(6)
This is the form whose behavior we will study below. Its essential advantage, compared to eq. (1) is the extra b-dependence in the hard scattering. Radiative corrections in higher orders will generate logarithms of the form ln(t/~) in TH, where t is the largest mass scale appearing in TH, t=max(~~Q, 1/b).
(7)
2 =x 2 only as long as x 2> 1/b2 for Thus, the natural choicetoinzero, T11 isthe ~.t 1x2Q 1x2Q values of the x’s closer scale of radiative corrections is dominated by the transverse distance bridged by the exchanged gluon. If this distance is small, radiative corrections will be small, regardless of the values of the x’s. Of course, when b is large and x 2 is small, radiative corrections are still large in TH. We 1x2Q that the wave functions ~ in eq. (3) strongly suppress are about to see, however, this region. Note that if we do not make the approximation of eq. (5) above, there will be additional independent dimensionless scales in the hard scattering, such as x 2b2, whose logarithms may be important at higher orders. We shall not attempt a1Qcomplete catalogue of such effects, which remain suppressed by the wave functions.
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The pion form factor
3. Asymptotic behavior in b and
135
Q
The large-b asymptotic behavior of the wave functions was analyzed in ref. [91.Here we shall simply quote the results, commenting only on such minor differences and simplifications that apply. The essential result we will be using is the exponentiation of Sudakov logarithms in b and Q referred to above. The result may be summarized as, ,~
~(x,
b, P,
~) =exp
—s(x, b,
Q) —s(1 —x,
b,
Q)
~ diii _2f
~yq(g(~)) 1/b
/.L
x~(x,~) +O(a~(1/b)).
(8)
a~/~is the quark anomalous dimension in axial gauge. The explicit form for the Sudakov exponent s(~,b, Q), with ~ x or 1 x, including all leading and next to leading logarithms may be conveniently expressed in terms of the variables Yq
=
—
=
—
ln[~Q/(I~A)], /~ln(bA).
(9)
In these terms, we have ~ s(~,b,
Q)
=
A~
~ _i)_
ln(_~)+ ~
~
A~’~i32 In( —2b)
+
1
___
ln(2c~)+ 1 —
2) —
At -~
—
~I3~
Atl)
_____
A~1~ —ln(~e2~’)ln 4f3~
—b
A~
1),3
-
32~3~ [ln2(2c~)
-
1n2(
-2b)].
(10)
In this expression, the coefficients A(t) and /3~ are ______
12 At1~=~, At2
‘
I~2
lS3—l9n~ 24
~—r2—~n 1+~f3l
where
flf
ln(~-e~’),
(11)
(= 3 below) is the number of quark flavors and y is the Euler constant.
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The pion form factor
Similarly, the renormalization group applied to TH gives TH(xl, x2, b,
4f
Q, ~) = exp
~d~i yq(g(~i))
tL~L
X
TH(xl, x2, b,
Q,
(12)
t),
where t is given by eq. (7). The anomalous dimensions help take into account the strong couplings of quarks when t is small. Note that it is not sufficient to simply match the scale of the running coupling in the hard scattering to the gluon’s momentum, without including the quark anomalous dimension. Using the results of eqs. (9)—(12) in eq. (6), we derive our lowest-order expression for the pion form factor 2)= 16~T~’Ff1dxl dx F~(Q
2 4(xi)4(x2)j xexp[—5~’(x1, x2, b,
b db a~(t)Ko(~~Qb)
Q)],
(13)
with
21
2
9’(x1, x2, b,
Q) =
~ (s(x~,b, i=1
Q)
+s(1 —xe, b,
Q))
—
—ln——--~-, (14) /3~ —b
where 1 = ln(t/A), A A0c~. K0 in eq. (13) is the modified Bessel function of order zero, which is the Fourier transform to b-space of the gluon propagator. Notice in particular the argument of the running coupling, defined in eq. (7) above, which cuts off the growth of radiative corrections in the hard scattering, except in the b —s 1/A limit, where the entire integrand is suppressed anyway. In eq. (13), we have neglected the evolution of the wave function ~ with 1/b, which is a nonleading effect that does not pertain directly to the mechanisms that we are discussing in this paper. Now let us return to the question of the self-consistency of the perturbative calculation for the pion form factor, in terms of our reformulated expression, eq. (13). We propose to analyze this expression, not by cutting off the x-integrals near their endpoints, but by testing the sensitivity of the integral to the large b-region. Of course, b is strictly limited to be less than 1/A, where the (negative) Sudakov exponent diverges. We expect that, because eq. (1) is the true asymptotic behavior, as Q increases, the b-integral will become more and more dominated by small b, where the effective coupling is small, regardless of the momentum fraction carried by the gluon. This requires a numerical study of the integrals in eq. (13), to which we now turn.
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The pion form factor
137
4. Numerical results The typical behavior of Sudakov suppression with momentum transfer Q is shown in fig. 3. There is no suppression for small b, where the two fermion lines are close to each other. For simplicity of analysis, we set the Sudakov exponential to unity in the small-b region where it induces a (small) enhancement, since in this region it is dominated by low orders in perturbation theory, and should be thought of as a part of the higher-order corrections to T~.In this vein, we have also set any factor ~ to unity whenever ~ < v~/(bQ).As b increases, however, exp S in eq. (13) decreases, reaching zero at b = 1/A. Suppression in the large-b region is weaker for smaller Q, where Sudakov effects are mild. To illustrate the qualitative effects of Sudakov corrections, eq. (13) is evaluated for two models of distribution amplitudes: the asymptotic wave function [12] —
=
~as(X)
~x(1
-x),
(15)
and the Chernyak—Zhitnitsky wave function [4] __~ix(1_x)(1_2x)2, ~2N~
~CZ(x)
where N~= 3 is the number of colors and
1.2
f~= 0.133
(16)
GeV the pion decay constant.
exp(—S)
l.O/A
Fig. 3. Behavior of Sudakov suppression in the transverse separation b for energy
Q2 /fl
=
20 and
Q3/fl = 50 with
x1 =
=
0.5.
Q1 /A
tO,
H. -N. Li, G. Sterman
138
0.16
Q2F(Q2)
/
The pion form factor
as
0
______________
1.0/A
(a) 0.32
~
2F(Q2)
Q
CZ
___________________________________
1.0/A
(b) Fig. 4. Dependence of Q2F~,(Q2)on the cutoff b~with (a) the asymptotic wave function and (b) the wave function of Chernyak and Zhitnitsky. The energies Q 1, Q2 and Q3 are as in fig. 3.
In order to see how the contribution to eq. (13) is distributed in b-space the integration is done with a variable cut off in b, b~.Typical numerical results are shown in fig. 4a 2F,.(Q2) and 4b for and 4~, respectively. The showing the on ~as b~,increase from zero at b~ = 0, curves, and reach their full dependence of Q height at b~= 1/A, beyond which we consider any remaining contributions as truly nonperturbative. As anticipated, we observe a faster rise as Q increases. At the largest value shown, Q/A = 50, the curves are quite flat for large b, indicating little contribution from large distances. To be quantitive, we may consider the cutoff up to which half of the whole contribution has been accumulated. For the wave function cb~,50% of Q2F~(Q2)comes from the region with b ~ 0.39/A for Q/A = 10 and b ~ 0.25/A for Q/A = 20. Similarly, for the wave function 4~, 50% is due to the region with b ~ 0.46/A for Q/A = 10 and b ~ 0.32/A for Q/A = 20. A liberal standard to judge the relevance of the perturbative method would be that 50% of the result come from the region where the coupling constant is no larger than, say, 0.7. From this point of view, pQCD begins to be self-consistent at
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The pion form factor
139
about Q 20A (2 GeV if A is taken to be 0.1 GeV). Note that this applies to our model CZ wave function, as well as to the asymptotic wave function, even though the two differ dramatically when analyzed from the point of view of cutoffs in the x’s [1]. As the momentum transfer increases, the perturbative calculation improves. By the time Q/A reaches 50, the perturbative region in b, defined as above, gives 90% of the form factor for and 80% for qYS
~
5. Conclusions We have shown explicitly that, with the help of Sudakov suppression, pQCD enlarges its range of applicability down to accessible energies. The transverse momenta flowing through the pion distribution amplitudes and hard-scattering subdiagram play an essential role in our discussion. This effect is common in most exclusive processes and we cannot have a complete perturbative description for them without including it. At the same time, the approach to perturbative behavior is relatively slow and we cannot expect the leading order perturbative predictions to be precise even at the highest available energies. We should note that our predictions for Q2F~(Q2)from the asymptotic wave function are of the order 0.1, which is about the experimental data [18] (but see ref. [21]). The use of the CZ wave function can increase the results by a factor of 2, which still falls short. This is because we ignore the evolution of the wave function. We have taken this simplified approach, because we are interested here primarily in isolating the role of radiative corrections. In future work on the proton form factor [4,5,19], the full set of evolution phenomena should be included [6,20]. Additionally one-loop corrections [13] may be studied in our formalism. We emphasize again, that there is no parameter to be adjusted in our calculation beyond the input wave functions. The suppression of the non-perturbative region as Q increases results from including a formally higher power-suppressed transverse momentum dependence in the hard scattering, rather than introducing a gluon or quark mass [2,3]. This scheme can be extended to calculate more complicated processes like the nucleon magnetic form factor [22]. ~-
This work was supported in part by the National Science Foundation under Grant NSF 9108054, by the Department of Energy under Grant DE-FGO287ER40371 and by the Texas National Research Laboratory Commission. References [1] N. Isgur and C.H. Liewellyn Smith, NucI. Phys. B317 (1989) 526 [2] T. Huang and Q.-X. Shen, Z. Phys. C50 (1991) 139 [3] N. Isgur and C.H. Llewellyn Smith, Phys. Rev. Lett. 52 (1984) 1080 A.V. Radyushkin, Acta Phys. Polonica, B15 (1984) 403;
A.P. Bakulev and A.P. Radyushkin, Phys. Lett. B271 (1991) 223
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The pion form factor
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