Resummation Approach to Evolution Equations

Transcription

1 Proc. Natl. Sci. Counc. ROC(A) Vol. 23, No. 5, 999. pp (Invited Review Paper) Resummation Approach to Evolution Equations HSIANG-NAN LI Department of Physics National Cheng-Kung University Tainan, Taiwan, R.O.C. (Received December 8,998; Accepted April 9,999) ABSTRACT We derive the evolution equations of parton distribution functions appropriate in different kinematic regions in a unified and simple way using the Collins-Soper (CS) resummation technique. These equations include the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equation for a large momentum transfer Q, the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation for a small Bjorken variable, and the Ciafaloni- Catani-Fiorani-Marchesini (CCFM) equation, which unifies the above two equations. We propose a modified Balitsky-Fadin-Kuraev-Lipatov equation in the resummation framework, which possesses an intrinsic Q dependence, and show that its predictions for the structure function F 2 (,Q 2 ) are consistent with eperimental data. Key Words: perturbative QCD, resummation technique, parton distribution functions, evolution equations I. Introduction Perturbative quantum chromodynamics (PQCD), as a gauge field theory, involves large logarithms from radiative corrections at each order of the coupling constant α s, such as lnq in the kinematic region with a large momentum transfer Q and ln(/) in the region with a small Bjorken variable. These logarithms, which spoil the perturbative epansion, must be organized. To organize the logarithmic corrections to a parton distribution function, various evolution equations have been derived. For eample, the Dokshitzer- Gribov-Lipatov-Altarelli-Parisi (DGLAP) equation sums single logarithms lnq for an intermediate to all orders (Gribov and Lipatov, 972; Altarelli and Parisi, 977; Dokshitzer, 977), the Balitsky-Fadin-Kuraev- Lipatov (BFKL) equation sums ln(/) for a small (Kuraev et al., 977; Balitsky and Lipatov, 978; Lipatov, 986), and the Ciafaloni-Catani-Fiorani- Marchesini (CCFM) equation (Ciafaloni, 988; Catani et al., 990; Marchesini, 995), appropriate for both intermediate and small, unifies the above two equations. The conventional derivation of the evolution equations is usually complicated. The idea is to locate the region of loop momenta flowing through the rungs (radiative gluons) of a ladder diagram in which the leading logarithmic corrections are produced. For the DGLAP, BFKL, and CCFM equations, the important regions are those with strong transverse momentum ordering, strong rapidity ordering, and strong angular ordering, respectively. Summing the ladder diagrams with different kinematic orderings to all orders, one obtains the corresponding evolution equations. In this paper, we shall propose an alternative approach to all-order summation of the various logarithms. This approach is based on the Collins- Soper (CS) resummation technique (Collins and Soper, 98), which was developed originally for the organization of double logarithms ln 2 Q. Recently, we applied this technique to several hard QCD processes, such as deep inelastic scattering (DIS), Drell-Yan production, and inclusive heavy meson decays, and demonstrated how to resum the double logarithms contained in parton distribution functions into Sudakov form factors (Li, 997). Here we shall further show that it can also deal with the single-logarithm cases, i.e., the evolution equations mentioned above. It is known that the BFKL equation is independent of Q; thus, its predictions are insensitive to variation of the momentum transfer. However, recent eperimental data for the structure function F 2 (,Q 2 ) of deep inelastic scattering ehibit a stronger Q dependence (Ahmed et al., 995; Derrick et al., 995). It is possible that eperiments have not yet eplored the region in which the value of is low enough for the BFKL equation to be applicable. Hence, to eplain the data, the Q dependent DGLAP equation is incorporated in some way. One employs either the improved splitting 567

2 H.N. Li function, which embeds the BFKL summation, in the DGLAP equation (Askew et al., 994; Forshaw et al., 995), or the CCFM equation directly (Kwiecin ski et al., 996). In this paper, we shall not rely on the ln Q summation, but propose a modified BFKL equation in our framework, which contains an intrinsic Q dependence from the longitudinal cutoff of rung gluon momenta. It will be shown that the resultant gluon distribution function leads to predictions of F 2 (,Q 2 ), that are very consistent with the data. We shall derive the DGLAP, BFKL and CCFM equations in Sections II, III, and IV, respectively, using the CS resummation technique. The modified BFKL equation along with its analytical solution of the gluon distribution function is presented in Section V. Section VI is a conclusion. II. The DGLAP Equation Consider DIS of a hadron with a light-like momentum p µ = δ µ+ in the intermediate region, where = q 2 /(2p q) Q 2 /(2p q) is the Bjorken variable with q the momentum transfer to the hadron through a virtual photon. A quark distribution function φ(ξ,,µ), absorbing nonperturbative dynamics involved in DIS, is associated with the hadron. This distribution function describes the probability at the factorization or renormalization scale µ that a valence quark, interacting with the photon, carries the fractional momentum ξ. The definition of φ in the covariant gauge A=0 is given by φ(ξ,, µ) = dy 2π e iξ y p q (y ) 2 γ + Pe i dzn A(zn) q(0) p, () as shown in Fig. (a), with γ + being a Dirac matri and p the incoming hadron. Averages over color and spin are understood. The path-ordered eponential Pe i dzn A( represents an eikonal line in the direction n that collects collinear gluons. In the aial gauge n A=0, the pathordered eponential in Eq. () is equal to the identity, implying that collinear gluons are decoupled from the eikonal line as shown in Fig. (b). The n dependence then goes into the gluon propagator, ( i/l 2 )N µν (l), with N µν = g µν n µ l ν + n ν l µ + n n l 2 l µ l ν (n l). (2) 2 We have shown that the resummation results derived in the covariant and aial gauges are the same Fig.. Definition of a parton distribution function in (a) the covariant gauge and in (b) the aial gauge. (Li, 997). In this section, we shall adopt the aial gauge. The essential step in CS resummation is to obtain the derivative dφ/d. Because of the scale invariance of φ in n as indicated by Eq. () or Eq. (2), φ depends on p and n through the ratio (p n) 2 /n 2. We have the following chain rule relating d/d to d/dn α : d 2 dφ = n υ n υ d α φ (3) dn α with υ α =δ α+ being a vector along p. Note that we have allowed n to vary away from the light cone. If n lies on the light cone, i.e., n 2 =0, the ratio (p n) 2 /n 2 will not eist. φ in Eq. (), independent of, then coincides with the conventional definition of the quark distribution function. Therefore, the argument of φ denotes the additional logarithms ln( /µ) that come along with the arbitrary vector n. The definition of φ, with such an arbitrary n, is gauge dependent. However, it will be observed that the evolution kernel turns out to be n-independent. After the derivation, we bring n back to the light cone. That is, the vector n appears only at the intermediate stage of the derivation and as an auiliary tool of the resummation technique. In the aial gauge, d/dn α applies to the gluon propagator, giving d dn α N µν = n l (l µ N αν + l ν N µα ). (4) The loop momentum l µ (l ν ) contracts with the verte the differentiated gluon attaches, which is then replaced by a special verte υ α = n 2 υ α υ nn l. (5) This special verte can be easily read off from the combination of Eqs. (3) and (4). The contraction of l µ (l ν ) hints at the application of the Ward identities. Summing the diagrams with different differentiated gluons, and employing the Ward identities for the contraction of l µ (l ν ) with various types of vertices, we 568

3 Resummation Approach to Evolution Equations obtain the formula d dφ(,, µ)=2φ (,, µ), (6) shown in Fig. 2(a). The new function φ contains one special verte at the outer end of a valence quark line, which is represented by a square. The coefficient 2 comes from the equality of the new functions with the special verte on either side of the final-state cut. Equation (6) is an eact consequence of the Ward identities without any approimation. An approimation will be introduced when the subdiagram containing the special verte is factorized out of φ in order to obtain a differential equation of φ. The factorization of the subdiagram is possible only in the leading regions of the loop momentum flowing through the special verte. These regions are soft and hard, since the vector n does not lie on the light cone, and collinear enhancements are suppressed. In the soft and hard regions, φ is factorized into the convolution of the subdiagram with the original distribution function φ: φ (,, µ)= dξ[k(, ξ,, µ) + G(, ξ,, µ)]φ(ξ,, µ). (7) The function K, absorbing the soft divergences of the subdiagram, corresponds to Fig. 2(b), where the eikonal approimation for the valence quark propagator has been made. The function G, absorbing the ultraviolet divergences, corresponds to Fig. 2(c), where subtraction of the second diagram ensures a hard momentum flow in G. According to Fig. 2(b) and (c), K and G are written as K = ig 2 C F µ n d 4 n l υ µ υ ν δ(ξ ) [ (2π) 4 n υ l l 2 +2πiδ(l 2 )δ(ξ l + )]N µν δk, (8) G = ig 2 C F µ n N µν d 4 n l (2π) 4 n υ ξ(p l) µ[ (ξp l) 2γ ν + υ ν υ l ] δ(ξ ) δg, (9) 2 l with C F =4/3 being a color factor, and δk and δg additive counterterms. As will be eplained later, the approimation equivalent to the strong transverse momentum Fig. 2. (a) The derivative dφ/d in the aial gauge. (b) The O(α s ) function K. (c) The O(α s ) function G. ordering assumed in the conventional derivation has been employed to obtain Eq. (8). A straightforward calculation gives K = α s (µ) πξ C F [ +ln ν ( /ξ) µ δ( /ξ)], + G = α s (µ) + πξ C ξνp F ln µ δ( /ξ), (0) where the constants of order unity have been dropped, and ν= (υ n) 2 n 2 is the gauge factor. The symbol + in the function K is defined, in terms of a convolution with an arbitrary function f, by f(z) dz ( z) + = dz f(z) f() z + f() ln( ). () Equation (0) confirms our argument that φ depends on p and n through the ratio (p n) 2 /n 2 =(ν ) 2. Since an ultraviolet pole eists only in the soft virtual correction, we have δk= δg. Note that the first term in K contains ln( ), implying the characteristic scale ( ) of the soft function K. In the considered region with intermediate, ln(ξν /µ) in G is approimated by ln(ν /µ). That is, the hard function G is characterized by the larger scale. We then treat K and G using the renormalization group (RG) methods: 569

4 H.N. Li µ d dµ K = λ K = µ d dµ G. (2) The anomalous dimension of K is defined by λ K = µdδk/dµ, whose eplicit epression is not essential here. The RG solution of K+G is given by K(, ξ,, µ)+g(, ξ,, µ) =K(, ξ,, )+G(, ξ,, ) = α s ( ) πξ C F d µ λ µ K (α s ( µ )),, (3) ( /ξ) + where the initial conditions K(,ξ,, ) and G(,ξ,, ) do not contain large logarithms after choosing µ as, and the source of double logarithms, i.e., the integral containing λ K, vanishes. For intermediate, ln( ) is not large and does not need to be summed. This is the reason why the lower bound of the integration variable µ is set to. If approaches unity, ln( ) should be summed by choosing the lower bound of µ as ( ), which then leads to the threshold resummation. Inserting Eq. (3) into Eq. (7) and solving Eq. (6), we obtain φ(, µ)=φ(, Λ, µ) + Λ µ d µ µ α s ( µ ) π C F dξ 2 φ(ξ, µ, µ) ξ ( /ξ) + (4) with Λ being an arbitrary cutoff. We have shown that φ(, = µ,µ), in which the large logarithms ln( /µ) vanish, coincides with the conventional φ(,µ) as stated before. Differentiating Eq. (4) with respect to µ, and substituting the RG equations, µ d dµ φ((ξ), Λ( µ ), µ)= 2λ qφ((ξ), Λ( µ ), µ), (5) with λ q = α s /π being the quark anomalous dimension, we arrive at µ d dµ φ(, µ)=α s (µ) π C F dξ ξ 2 φ(ξ, µ) ( /ξ) + 2λ q (µ)φ(, µ). (6) Setting µ to Q, the above equation can be reepressed as Q dq d φ(, Q)=α s (Q) π dξ P(/ξ)φ(ξ, Q), (7) ξ with the kernel P(z)=C F [ +z2 + ( z) 3 δ( z)], (8) + 2 where the piece (z 2 )/( z) +, finite as z, has been added by hand. Equation (7) is the DGLAP equation. III. The BFKL Equation In this section, we shall derive the BFKL equation for the gluon distribution function in the small region using the CS resummation technique. It is convenient to adopt the covariant gauge A=0, in which the unintegrated gluon distribution function F(,p T ) is defined by F(, p T )= dy 2π d 2 y T 4π e i(p+ y p T y T ) 2 p F µ + (y, y T )Pe i dzn A(zn) F µ + (0) p, (9) with F µ + being the field tensor. F(,p T ) describes the probability that a gluon, interacting with the photon through a quark bo, carries a longitudinal momentum fraction and transverse momenta p T. Note that F depends on p T, instead of the large hadron momentum which appears in the DGLAP case. It will be shown later that the dependence disappears as 0. Though F does not depend on eplicitly, it varies implicitly with through the momentum fraction, which is proportional to ( ) for a fied parton momentum. For a similar reason, F depends on the ratio (p n) 2 /n 2 ; thus, Eq. (3) holds. In the covariant gauge, the operator d/dn α applies to the Feynman rules associated with the eikonal line, giving d n µ dn α n l = n l (g µα n µ l α ). (20) n l Combining Eqs. (3) and (20), we find that differentiation with respect to generates a special verte on the eikonal line: n α = n 2 υ n (υ l n l n α υ α ). (2) The derivative of F can then be epressed as d df(, p T ) d d F(, p T )=4F (, p ), (22) T 570

5 Resummation Approach to Evolution Equations with the triple-gluon verte for vanishing l: Γ µνλ = g µν υ λ g νλ υ µ +2g λµ υ ν. (24) Fig. 3. (a) The derivative df/d in the covariant gauge. (b) The soft structure and (c) the ultraviolet structure of the O(α s ) subdiagram containing the special verte. described by Fig. 3(a), where the new function F contains one special verte denoted by the symbol. Note that the coefficient 4 in front of F is twice the corresponding coefficient in the DGLAP case. Since a gluon interacts with the virtual photon through a quark bo, two quark lines, and thus two eikonal lines after factorizing out the gluon distribution function, are adjacent to the parton verte. The leading regions of the loop momentum l flowing through the special verte are also soft and hard. If l is collinear to p, the first term υ l in n α vanishes, and the second term υ α, as contracted with a verte in the distribution function which is dominated by momenta parallel to p, gives a small contribution. The soft divergences of the subdiagram are shown in Fig. 3(b), and the ultraviolet divergences in Fig. 3(c). We employ the relation f abc t b t c =(i/2)nt a for the color structure, with t being the color matrices and N=3 the number of colors. Absorbing t a, into the parton verte, Fig. 3(b) leads to F s (, p T )g λµ = 2 i d 4 l Γ µνλ n ν Ng2 (2π) 4 (2υ l)n l [2πiδ(l 2 )F(, p T + l T ) The θ function sets the upper bound of l T to p T to ensure a soft momentum flow. Therefore, it is not necessary to introduce a renormalization scale µ into F. There is no ξ integration in Eq. (23) since we have included the transverse degrees of freedom of the parton as an infrared cutoff and made the approimation δ(ξ l + / ) δ(ξ ). It can be easily shown that υ λ in Eq. (24), contracted with a verte in the quark bo diagram, leads to a contribution suppressed by a power /s, s= (p+q) 2, compared with the contribution from the last term υ ν. Similarly, the term υ µ, contracted with a verte in the distribution function, which is dominated by momenta parallel to p, also makes a negligible contribution. Hence, we neglect the first two terms of Γ µνλ and the tensors g λµ drop from both sides of Eq. (23). Evaluating the integral in a straightforward manner, Eq. (23) reduces to F s (, p T )= α s 4 d 2 l T πl T 2 [F(, p T + l T ) θ(p T 2 l T 2 )F(, p T )] (25) with α s =Nα s /π. It has been argued that when the fractional momentum of a parton vanishes, the associated collinear enhancements are suppressed (Li, 997). The vanishing of the contribution from the first diagram of Fig. 3(c), G () = α s 4 d 2 l T π [ l T 2 l T 2 +( ν) 2 ν l 2 T +( ν) 2 ν 2 [l 2 T +( ν) 2 ] 3/2ln l 2 T +( ν) 2 + ν ] (26) at 0, supports this argument. Hence, F does not acquire a dependence on the large scale, which differs from the DGLAP case for a large value of. Neglecting G () along with its soft subtraction (the second diagram in Fig. 3(c)), that is, adopting F = F s, Eq. (22) becomes + θ(p 2 l 2 T T ) F(, p l 2 T )], (23) 57 F(, p T ) d ln (/) = α s d 2 l T πl T 2 [F(, p T + l T )

6 H.N. Li function in the DGLAP equation does not involve transverse degrees of freedom. IV. The CCFM Equation Fig. 4. (a) The subdiagram containing the special verte for the CCFM equation. (b) The subdiagram for the CCFM equation after resumming the double logarithms in J. θ(p T 2 l T 2 )F(, p T )], (27) which is eactly the BFKL equation. It is then understood that the subdiagram containing the special verte plays the role of the BFKL kernel. Obviously, the BFKL kernel is independent of n as in the DGLAP case. Comparing the derivations of the DGLAP and BFKL equations, it is easy to observe that the difference resides mainly in their treatment of the soft real gluon emission. A complete loop integral associated with the real gluon emission involves a parton distribution function φ(+l + /, p T +l T ) (28) if integration over ξ is performed first in Eq. (7). Equation (28) indicates that the parton coming out of the hadron carries a longitudinal momentum +l + and transverse momenta p T +l T such that a real gluon of momentum l can be radiated. Since the BFKL equation is appropriate for the multi-regge region, where the transverse momenta flowing through the rungs of a ladder diagram are of the same order, i.e., l T p T, the l T dependence of φ is not negligible. On the other hand, the strong rapidity ordering corresponds to the soft approimation φ(+l + /, p T +l T ) φ(, p T +l T ). (29) That is, the loop integral is dominated by the value of φ at a small value of, an approimation which will be justified for the gluon distribution function in Sect. V. On the other hand, the DGLAP equation is appropriate for the strong transverse momentum ordered region, which corresponds to the alternative soft approimation φ(+l + /, p T +l T ) φ(+l + /, p T ). (30) The p T dependence of the parton distribution function is then integrated out from both sides of the evolution equation. This is the reason why a parton distribution Based on the above discussion, it is not difficult to derive the CCFM equation using CS formalism. Start with Eq. (22) but with the unintegrated gluon distribution function F depending on p T and : d df(, p T, )=4F (, p T, ), (3) which manifests the attempt to unify the DGLAP and BFKL equations. Following the standard procedures used in CS resummation, one should factorize the subdiagram containing the special verte by absorbing its soft and hard contributions into the functions K and G, respectively. This approach leads to a new unified evolution equation, which has been studied elsewhere. To reproduce the CCFM equation, however, the absorption of virtual corrections must be performed in a different way: we embed the virtual gluons into K instead of into G. That is, the subdiagram is factorized according to Fig. 4(a), where the two jet functions J group the virtual corrections and the real gluon between them is soft. First, we resum the double logarithms contained in J by considering its derivative: d dj(p T, )=2[K J (p T, µ)+g J (, µ)]j(p T, ). (32) At lowest order, the function K J comes from the first diagram in Fig. 3(b), and G J from Fig. 3(c). The coefficient 2 counts the two eikonal lines adjacent to the parton verte. The relation between K J +G J and J is simply multiplicative since J groups only virtual gluons. We have set the infrared cutoff of K J to p T, as indicated by its argument. This cutoff is necessary here due to the lack of a corresponding real gluon emission, which serves as a soft regulator. The one-loop K J can be easily obtained by working out the second integral in Eq. (25) without the θ function, and the anomalous dimension of K J is found to be γ J = α s /2. The function G J can also be computed, but its eplicit epression is not important. Standard RG analysis gives K J (p T, µ)+g J (, µ)= d µ µ γ J (α s ( µ )) (33) with the initial conditions K J (p T,p T )=G J (, )=0. We have neglected the constants of order unity in K J and G J, which generate the net-to-leading-logarithm p T 572

7 Resummation Approach to Evolution Equations summation. Substituting Eq. (33) into Eq. (32), we can solve for J(p T,Q)= (Q,p T )J (0) (34) with the double-logarithm eponential (Q, p T )=ep[ α s d Q p T d µ ]. (35) µ We have set the upper bound of to be Q, and ignored the running of α s. The initial condition J (0), regarded as a tree-level gluon propagator, will appear in the integrand for the real gluon emission shown in Fig. 4 (b). We split the above eponential into / (Q, p T )= 2 / S (Q, zq) 2 NS (z, q, p T ) (36) with z=/ξ and q=l T /( z), where ξ is the momentum fraction entering J from the bottom and l T is the transverse loop momentum carried by the real gluon. The so-called Sudakov eponential S and the non- Sudakov eponential NS are given by Q 2 S (Q, zq)=ep[ α s dp 2 (zq) 2 p 2 p T 0 p T /p dz ], z the integral for the real gluon emission follow from the previous discussion of Eq. (28). The θ function requires Q>zq such that the Sudakov eponential S is meaningful, which comes from the angular ordering of the radiative gluons Q/( )>l T /((ξ ) ). Compared with the transverse momentum ordering for the DGLAP equation, Q (l T ) has been divided by the gluon energy ((ξ ) ). Hence, the inserted scale zq reflects the special ordering for the CCFM equation. Performing integration over l + and l, we obtain F (, p T, ) = α s 4 d 2 l dξ T 2n 2 (ξ )2 π (Q, p [n + l 2 T +2n (ξ ) 2 2 ] 2 2 T ) θ(q zq)f(ξ, p T + l T, ), (39) where we have assumed that n=(n +,n,0) for convenience. The above epression can then be substituted into Eq. (3) to obtain the solution of F. We adopt the variable changes ξ=/z and l T =( z)q, and integrate Eq. (39) from =0 to Q. To work out the integration, F(/z, p T +l T, ) is approimated by F(/z, p T +l T, l T ). We obtain F(, p T, Q) NS (z, q, p T )=ep[ α s z p T /q dz z p T 2 (z q) 2 dp 2 ], (37) p 2 = F (0) + α s dz d 2 q πq 2θ(Q zq) S (Q, zq) NS (z, q, p T ) where the variable changes µ =( z )p and =p for S, and µ =p and =z q for NS have been employed. Picking up the last term of the triple-gluon verte in Eq. (24), F can be written, according to Fig. 4(b), as F (, p T, ) = i 2 Ng2 dξ d 4 l υ n (2π) 4 υ ln l 2πiδ(l 2 ) 2 (Q, p T ) δ(ξ l + / )θ(q zq)f(ξ, p T + l T, ). (38) It is easy to write down the above epression by referring to the first term of Eq. (23), which corresponds to the real gluon emission. The propagator /υ l comes from the eikonalized tree-level J (0) on the right-hand side of Fig. 4(b). J (0) on the left-hand side has been absorbed into F. The arguments ξ and p T +l T of F in z( z) F(/z, p T +( z)q, l T ), (40) where the nonperturbative initial condition F (0) corresponds to the lower bound of and the term suppressed by /Q 2 in the integral has been dropped. Equation (40) can be rewritten as F(, p T, Q) = F (0) + dz d 2 q πq 2θ(Q zq) S (Q, zq)p(z, q, p T ) F(/z, p T +( z)q, l T ) (4) with the splitting function P = α s [ z + NS (z, q, p T ) z 2+z( z)], (42) 573

8 H.N. Li which is eactly the CCFM equation. To arrive at Eq. (42), we have employed the identity /(z( z)) /( z)+/z. The last term 2+z( z) has been put in by hand. This term, finite at z 0 and at z also, can not be obtained using the conventional approach. Note that only the non-sudakov form factor NS in front of /z is retained. Because NS vanishes when the upper bound zq of approaches zero, as shown in Eq. (37), it smears the z 0 pole of the splitting function P. V. Modified BFKL Equation In this section, we shall propose a modified BFKL equation based on CS formalism. After we perform contour integration over l, Eq. (23) becomes F s (, p T, Q) = α s 4 d 2 l T π dl + 2l + n 2 (2n l +2 + n + l T 2 ) 2 [F(, p T + l T, Q) θ(p T 2 l T 2 )F(, p T, Q)], (43) where the Q dependence of F has been made eplicit. To derive the BFKL equation, we must etend the loop momentum l + to infinity in Eq. (43). As stated before, the real gluon emission involves the distribution function F(+l + /, p T +l T ) as part of the integrand, which is then approimated by F(, p T +l T ) according to strong rapidity ordering. Therefore, the behavior of F, which vanishes at the momentum fraction equal to unity, should introduce an upper bound of l +. Following this argument, we truncate l + at some scale, and a plausible choice for this scale is Q. This cutoff of the longitudinal loop momenta can be regarded as a correction to the assumption of rapidity ordering. Evaluating the integral in a straightforward manner, and substituting the result into Eq. (22), we derive a modified BFKL equation with a Q dependent evolution kernel: df(, p T, Q) d ln (/) = α s d 2 l T πl T 2 [F(, p T + l T, Q) θ(p T 2 l T 2 )F(, p T, Q)] α s d 2 l T π F(, p T + l T, Q) l T 2 + Q 2, (44) where the first and last terms correspond to the lower and upper bounds of l +, respectively. The gauge vector n has been chosen to render the coefficient of Q 2 equal to unity. It can be shown that our predictions are insensitive to this coefficient as long as it is of order unity. Obviously, Eq. (44) approaches Eq. (27) in the Q limit. In fact, the Q-dependent correction term also appears in the integration as deriving CCFM equation in Eq. (40), but has been dropped there. It is then clear that the Q dependences of the CCFM and modified BFKL equations arise from different sources: the former is attributable to the all-order lnq summation while the latter results from truncation of the phase space for radiative corrections. To etract the behavior of F from Eq. (44), we truncate the loop momentum l T in the virtual gluon emission at a scale Q 0 of order GeV. This simplification is reasonable since the virtual gluon contribution acts as a soft regulator for the real gluon emission only and setting the cutoff to Q 0 serves the same purpose. The replacement of p T with Q 0 allows us to solve Eq. (44) analytically by means of Fourier transform, and the solution of F retains the essential BFKL features. It will be shown that the first term of the integral is responsible for the rise of F, and that the last term acts to slow the rise. We then epect F to increase faster for a larger value of Q, where the effect of the last term is weaker. The Fourier transform of Eq. (44) leads to with df(, b, Q) d ln (/) = S(b, Q)F(, b, Q) (45) S(b,Q)=2 α s (/b)[ln(q 0 b)+γ ln 2+K 0 (Qb)], (46) where K 0 is the Bessel function and γ the Euler constant. The argument of α s has been chosen as /b since we are working in the conjugate b space. Equation (45) can be trivially solved to give F(, b, Q)=F( 0, b)ep[ S(b, Q)ln ( 0 /)], (47) where 0 is the initial momentum fraction below which F begins to evolve according to the BFKL summation. Transforming Eq. (47) back to the momentum space, we derive the analytical solution to the modified BFKL equation, F(, p T, Q)= bdbj 0 (p T b)f(, b, Q), (48) and the gluon density g by integrating Eq. (48) over p T : 574

9 Resummation Approach to Evolution Equations Fig. 5. The dependence of F 2 on obtained using the modified BFKL equation (solid lines) and using the conventional BFKL equation (dashed lines). The eperimental data are also shown. g(, Q 2 )= 0 Q d 2 p T π F(, p T, Q) =2Q dbj (Qb)F(, b, Q). (49) In the above epressions, J 0 and J are the zeroth and first order Bessel functions, respectively. To proceed with numerical analysis, we assume a flat gluon distribution function (Collins and Gault, 982; Donnachie and Landshoff, 986): F (0) (, b)=3n g ( ) 5 ep( Q 0 2 b 2 /4), (50) for 0, where N g is a normalization constant. We set Q 0 = GeV and 0 =0. arbitrarily. It can be shown that the predictions vary only slightly for other choices of the parameters Q 0 and 0 of the same order and for other choices of the initial condition. N g will be determined by the data of the structure function F 2 (,Q 2 ) at a specific value of Q 2 and can then be employed to make predictions at other values of Q 2. We compute the structure function F 2, whose epression, according to the p T -factorization theorem (Catani et al., 99; Catani and Hautmann, 994), is given by 575

10 H.N. Li F 2 (, Q 2 )= dξ ξ 0 p c d 2 p T π H(/ξ, p T, Q)F(ξ, p T, Q), (5) where p c is the upper bound of p T, which will be specified later. The hard scattering subamplitude H denotes the contribution from the quark bo diagrams, where both the incoming photon and gluon are off shell with q 2 = Q 2 and p 2 = p 2 T, respectively. We assume that a charm quark is massless, and that a bb quark pair is not involved in the bo diagram; i.e., the active flavor number n f in the running coupling constant α s is equal to 4. Following these assumptions, we concentrate on a range of Q 2 which is between 8 and 20 GeV 2. A simple calculation gives H(z, p T, Q) 2 = e α s q 2π z{[z 2 +( z) 2 2z( 2z) p 2 T Q +2z p 4 2 T 2 Q 4] 4z 2 p 2 T /Q ln+ 4z2 p 2 T /Q 2 2 4z 2 p 2 T /Q 2}. 2 (52) with e q being the electric charge of the quark q. To obtain a meaningful H, we choose the upper bound of p T in Eq. (5) as p c = min(q, ξ Q). (53) 2 We evaluate the integral in Eq. (5) in a straightforward manner for Q 2 =5 GeV 2, and then determine the normalization constant N g =3.656 based on data fitting. When Q 2 varies, we adjust N g such that g has a fied normalization gd. F2 for Q 2 =8.5, 2, and 0 20 GeV 2 can then be computed, and our results along with the eperimental data are displayed in Fig. 5. It is obvious that our predictions agree with the data well. The curves have a steeper rise at larger values of Q, which is a consequence of the Q dependent modified BFKL equation. For comparison, we also present the results obtained using the conventional BFKL equation, which can be obtained by simply substituting l 2 T +M 2 for the denominator l 2 T +Q 2 in Eq. (44), or equivalently, Mb for the argument Qb of the Bessel function K 0 in Eq. (46) with an etremely large M= GeV. In this case, the normalization constant obtained using the best fit to the data for Q 2 =5 GeV 2 is N g = It is found that the shapes of the curves are almost independent of Q; thus, the match with the data is not very Fig. 6. The dependence of g on obtained using the modified BFKL equation for, from bottom to top, Q 2 =8.5,2,5, and 20 GeV 2. satisfactory. Finally, we show in Fig. 6 the behavior of the gluon density g computed by Eq. (49). The variation of g with for different Q can be understood from the combination of ln(q 0 b) and K 0 (Qb) in the b 0 limit, written as (/ 0 ) 2α s ln(q/q 0 ). (54) The above epression indicates that the eponent λ, characterizing the rise of g~ λ at small values of, increases with Q. The values λ 0.36 for Q 2 =8.5 GeV 2 and λ 0.5 for Q 2 =20 GeV 2 are deduced from Fig. 6, and are consistent with that obtained by Martin et al. (995) using a phenomenological fit to the eperimental data (λ 0.3 for Q 2 =4 GeV 2 ) and with that obtained by Askew et al. (994) and Forshaw et al. (995), who solved the conventional BFKL equation numerically (λ 0.5 for a wide range of Q 2 ). VI. Conclusion In this paper, we have shown that the CS resummation technique provides a unified and simple means of organizing of the various large logarithms and reduces to the DGLAP, BFKL, and CCFM equations in different kinematic regions. The main idea is to relate the derivative of a parton distribution function to a new function involving a special verte. When we epress the new function as a factorization of the subdiagram containing the special verte with the original parton distribution function, we obtain the evolution equation. The subdiagram, evaluated under 576

11 Resummation Approach to Evolution Equations the soft approimation corresponding to the specific ordering of radiative gluons, leads to the appropriate evolution kernel. When we use the resummation technique, it is not necessary to consider the orderings of radiative gluons at the beginning since all possible orderings have been embedded in the new function. We have proposed a modified BFKL equation which possesses an intrinsic Q dependence resulting from truncation of the phase space for radiative corrections instead of from lnq summation in the CCFM equation. We have also computed the DIS structure function F 2 (,Q 2 ) using the analytical solution of the unintegrated gluon distribution function. The predictions obtained ehibit a stronger Q dependence, and agree better with the eperimental data than those obtained using the conventional BFKL equation. Encouraged by this success, we shall apply the above formalism to the study of various topics in small- physics. Acknowledgment This work was supported by the National Science Council of the Republic of China under grant NSC M References Ahmed, T. et al. (H Collaboration) (995) A measurement of the proton structure function F 2 (,Q 2 ). Nucl. Phys., B439, Altarelli, G. and G. Parisi (977) Asymptotic freedom in parton language. Nucl. Phys., B26, Askew, A. J., J. Kwiecin ski, A. D. Martin, and P. J. Sutton (994) Properties of the BFKL equation and structure function predictions for DESY HERA. Phys. Rev. D, 49, Balitsky, Y. Y. and L. N. Lipatov (978) The Pomeranchuk singularity in quantum chromodynamics. Sov. J. Nucl. Phys., 28, Catani, S. and F. Hautmann (994) High-energy factorization and small- deep inelastic scattering beyond leading order. Nucl. Phys., B427, Catani, S., F. Fiorani, and G. Marchesini (990) Small- behaviour of initial state radiation in perturbative QCD. Nucl. Phys., B336, Catani, S., M. Ciafaloni, and F. Hautmann (99) High-energy factorization and small- heavy flavour production. Nucl. Phys., B366, Ciafaloni, M. (988) Coherence effects in initial jets at small Q 2 /s. Nucl. Phys., B296, Collins, J. C. and D. E. Soper (98) Back-to-back jets in QCD. Nucl. Phys., B93, Collins, P. D. B. and F. Gault (982) Large- t elastic pp, p p elastic scattering and the triple-scattering mechanism. Phys. Lett. B, 2, Derrick, M. et al. (ZEUS Collaboration) (995) Measurement of the parton structure function F 2 from the 993 HERA data. Z. Phys. C, 65, Dokshitzer, Y. L. (977) Calculation of structure functions of deepinelastic scattering and e + e annihilation by perturbation theory in quantum chromodynamics. Sov. Phys. JETP, 46, Donnachie, A. and P. V. Landshoff (986) Dynamics of elastic scattering. Nucl. Phys., B267, Forshaw, J. R., R. G. Roberts, and R. S. Thorne (995) Analytical approach to small structure functions. Phys. Lett. B, 356, Gribov, V. N. and L. N. Lipatov (972) Deep inelastic ep scattering in perturbation theory. Sov. J. Nucl. Phys., 5, Kuraev, E. A., L. N. Lipatov, and V. S. Fadin (977) The Pomeranchuk singularity in nonabelian gauge theories. Sov. Phys. JETP, 45, Kwiecin ski, J., A. D. Martin, and P. J. Sutton (996) Description of F 2 at small incorporating angular ordering. Phys. Rev. D, 53, Li, H. N. (997) Resummation in hard QCD processes. Phys. Rev. D, 55, Lipatov, L. N. (986) The bare pomeron in quantum chromodynamics. Sov. Phys. JETP, 63, Marchesini, G. (995) QCD coherence in the structure function and associated distributions at small. Nucl. Phys., B445, Martin, A. D., R. G. Roberts, and W. J. Stirling (995) Pinning down the glue in the proton. Phys. Lett. B, 354,

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