- Email: [email protected]
Renormalon chain contributions to non-singlet evolution kernels in QCD
16 July 1998
Physics Letters B 431 Ž1998. 387–394
Renormalon chain contributions to non-singlet evolution kernels in QCD S.V. Mikhailov
1
Joint Institute for Nuclear Research, BogoliuboÕ Laboratory of Theoretical Physics, 141980 Dubna, Moscow Region, Russia Received 20 April 1998 Editor: P.V. Landshoff
Abstract Contributions to QCD non-singlet forward evolution kernels P Ž z . for the DGLAP equation and to V Ž x, y . for nonforward ŽER-BL. evolution equation are calculated for a certain class of diagrams which include renormalon chains. Closed expressions are presented for the sums of these contributions that dominate for a large number of flavors Nf 4 1. Calculations are performed in covariant j-gauge, in the MS scheme. The assumption of ‘‘naive nonabelianization’’ approximation for kernel calculations is discussed. The partial solution to the ER-BL evolution equation is obtained. q 1998 Published by Elsevier Science B.V. All rights reserved. PACS: 12.38.Cy; 12.38.-t; 13.60.Hb Keywords: DGLAP; Forward and nonforward evolution kernels; Anomalous dimensions; Multiloop calculation
1. Introduction Evolution kernels are the main ingredients of the well-known evolution equations for parton distribution of DIS processes and for parton wave functions in hard exclusive reactions. These equations describe the dependence of parton distribution functions and parton wave functions on the renormalization parameter m2 . Here, I continue to discuss the diagrammatic analysis and multiloop calculation of the forward DGLAP evolution kernel P Ž z . w1x and nonforward Efremov–Radyushkin–Brodsky–Lepage ŽER-BL. kernel V Ž x, y . w2x in a class of ‘‘all-order’’ approxi-
1
E-mail: [email protected].
mation of the perturbative QCD that has been started in w3x. There, the regular method of calculation and resummation of certain classes of diagrams for these kernels has been suggested. These diagrams include the chains of one-loop self-energy parts Žrenormalon chains. into the one-loop diagrams Žsee Fig. 1.. In this letter, the results for both the kinds of kernels, obtained earlier in the framework of a scalar model in six dimensions with the Lagrangian L i n t s Ý iN f Ž c i)c i w .Ž6. with the scalar ‘‘quark’’ flavours Ž c i . and ‘‘gluon’’ Ž w ., are extended to the non-singlet QCD kernels. For the readers convenience some important results of the previous paper w3x would be reminded. The insertion of the chain into ‘‘gluon’’ line Žchain-1. of the diagram in Fig. 1 a,b and resumma-
0370-2693r98r$ – see frontmatter q 1998 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 5 8 5 - 1
S.V. MikhailoÕr Physics Letters B 431 (1998) 387–394
388
Fig. 1. The diagrams in Figs. 1a – 1c are the ‘‘triangular’’ diagrams for the QCD DGLAP kernel; dashed line for gluons, solid line for quarks; black circle denotes the sum of all kinds of the one-loop insertions Ždashed circle., both quark and gluon Žghost. or mixed chains; MC denotes the mirror-conjugate diagram.
tion over all bubbles lead to the transformation of the one-loop kernel P0 Ž z . s az ' aŽ1 y z . into the kernel P Ž1. Ž z; A. chain 1
P0 Ž z . s az ™ P Ž1. Ž z ; A . s az Ž z .
where
yA
Ž 1 y A.
A s aNf gw Ž 0 . ,
gw Ž 0 . gw Ž A . as
; g2
Ž 4p .
3
.
Ž 1.
Here, gc Ž w .Ž ´ . are the one-loop coefficients of the anomalous dimensions of quark Žgluon at Nf s 1. fields in D-dimension Ž D s 6 y 2 ´ . discussed in w3x; for the scalar model gc Ž ´ . s gw Ž ´ . s B Ž2 y ´ ,2 y ´ .C Ž ´ ., and C Ž ´ . is a scheme-dependent factor corresponding to a certain choice of an MS-like scheme. The argument A of the function gw Ž A. in Ž1. is the standard anomalous dimension ŽAD. of a gluon field. So, one can conclude that the ‘‘all-order’’ result in Ž1. is completely determined by the single quark bubble diagram. The resummation of this ‘‘chain-1’’ subseries into an analytic function in A shouldn’t be taken by surprise. Really, the considered problem can be connected with the calculation of large Nf asymptotics of the AD’s in order of 1rNf . An approach was suggested by Vasil’ev and
collaborators at the beginning of 80’ w4x to calculate the renormalization-group functions in this limit, they used the conformal properties of the theory at the critical point g s g c corresponding to the non-trivial zero g c of the D-dimensional b-function. This approach has been extended by J. Gracey for calculation of the AD’s of the composite operators of DIS in QCD in any order n of PT, w5,6x. I have used another approach, which is close to w7,8x; contrary to the large Nf asymptotic method it does not appeal to the value of parameters Nf TR , CAr2 or CF , associated in QCD with different kinds of loops. Following this way, the ‘‘improved’’ QCD kernel P Ž1. Ž z; A. has been obtained in w3x for the case of quark or gluon bubble chain insertions in the Feynman gauge. In this paper, we present the QCD results similar to Eq. Ž1., for each type of diagrams appearing in the covariant j-gauge for the DGLAP non-singlet kernel P Ž z; A.. The analytic properties of the function P Ž z; A. in variable A are analyzed. The assumption of ‘‘Naive Nonabelianization’’ ŽNNA. approximation w9x for the kernel calculation w10x is discussed and its deficiency is demonstrated. The ER-BL evolution kernel V Ž x, y . is obtained in the same approximation as the DGLAP kernel, by using the exact relations between P and V kernels w11,3x for a class of ‘‘triangular diagrams’’ in Fig. 1. The considered class of diagrams represents the leading Nf contributions to both kinds of kernels. At the end, a partial solution for the ER-BL equation is presented Žcompare with w10,12x.. The obtained results are certainly useful for an independent check of complicated computer calculations in higher orders of perturbation theory ŽPT., similar to w15x; they may be a starting point for further approximation procedures.
2. Triangular diagrams for the DGLAP evolution kernel in QCD Here, the results of the bubble chain resummation for QCD diagrams in Fig. 1 a,b,c for the DGLAP kernel are discussed. These diagrams generate contributions ; a s Ž a s lnw1rz x. n in any order n of PT. Based on the resummation method of Ref. w3x in the QCD version, one can derive the kernels P Ž1 a, b,c.
S.V. MikhailoÕr Physics Letters B 431 (1998) 387–394
corresponding to the diagrams in Fig. 1 in the covariant j y gauge
gg Ž 0 .
2
P Ž1 a. Ž z ; A . s a s CF 2 z P Ž 1 y A . zyA
gg Ž A .
y a s CF P d Ž 1 y z .
ž
=
gg Ž 0 .
1
Ž 1 y A . gg Ž A .
P Ž1 b. Ž z ; A . s a s CF 2 P
ž
/ /
yj ,
2 z 1y A gg Ž 0 . 1 y z gg Ž A .
Ž 2.
Ž5. the d Ž1 y z . - terms are exactly accumulated in the form of the w . . . xq prescription, and the j - terms successfully cancel. This is due to the evident current conservation for the case of quark bubble insertions, including the gluon bubbles into consideration merely modifies the effective AD, gg Ž A, j . ™ ggŽ q . Ž A., conserving the structure of the result Ž5., see w3x. Substituting the well-known expressions for gg Ž ´ . from the quark or gluon Žghost. loops Žsee, e.g., w13x.
ggŽ q . Ž ´ . s y8 Nf TR B Ž Dr2, Dr2 . C Ž ´ . , ,
Ž 3. ggŽ g . Ž ´ , j . s
q
P Ž1 c. Ž z ; A . s a s CF P d Ž 1 y z . =
ž
gg Ž 0 .
AŽ 3 y 2 A .
Ž 2 y A . Ž 1 y A . gg Ž A .
where a s s
CA 2
=
/
yj ,
Ž 4. as
389
, CF s Ž Nc2 y 1.r2 Nc , CA s Nc
q
Ž 7.
B Ž Dr2 y 1, Dr2 y 1 .
žž ž
3Dy2 Dy1
1yj 2
/
q Ž 1 y j . Ž D y 3.
2
/ /
´ CŽ ´ . ,
Ž 8.
and TR s 12 are the Casimirs of SUŽ Nc . group, and A s ya sgg Ž0.. The function gg Ž ´ . is the one-loop coefficient of the anomalous dimension of gluon field in D-dimension, here D s 4 y 2 ´ . In other words, it is the coefficient Z1Ž ´ . of a simple pole in the expansion of the gluon field renormalization constant Z, that includes both a finite part and all the powers of the ´-expansion. Eqs. Ž2. – Ž4. are valid for any kind of insertions, i.e., gg s ggŽ q . for the quark loop, gg s ggŽ g . for the gluon Žghost. loop, or for their sum
into the general formulae Ž2. – Ž4., and Ž5. one can obtain P Ž1. Ž z; A, j . for both the quark and the gluon loop insertions simultaneously. Here, the coefficient C Ž ´ . s G Ž1 y ´ . G Ž1 q ´ . implies a certain choice of the MS scheme where every loop integral is multiplied by the scheme factor G Ž Dr2 y 1.Ž m2r4p . ´. The renormalization scheme dependence of P Ž1. Ž z; A. is accumulated by the factor C Ž ´ . 2 . Of course, the final result Ž5. will be gaugedependent in virtue of the evident gauge dependence of the gluon loop contribution ggŽ g . Ž ´ , j ., in this case, e.g.,
gg Ž A, j . s ggŽ q . Ž A . q ggŽ g . Ž A, j . ;
A Ž j . s ya sgg Ž 0, j . s ya s Ž ggŽ q . Ž 0 . q ggŽ g . Ž 0, j . .
Ž 4p .
when both kinds of insertions are taken into account. The sum of contributions Ž2., Ž3., Ž4. results in P Ž1. Ž z; A, j . which has the expected ‘‘plus form’’ P
Ž1.
yA
Ž z ; A, j . s a s CF 2 P zz =
gg Ž 0, j . gg Ž A, j .
a s P0 Ž z . s a s C F 2 P z q
2
Ž 1 y A. q
,
1yz
q
Ž 5.
2z 1yz
2 z 1y A
,
Ž 6.
q
where, in comparison with Eq. Ž5., the one-loop result a s P0 Ž z . is written down, the latter can be obtained as the limit P Ž1. Ž z; A ™ 0, j .. Note that in
s ya s
ž
5 q 3
Ž1yj . 2
/
CA y
4 3
Nf TR ,
Ž 9.
is the contribution to the one-loop renormalization of the gluon field. The positions of zeros of gg Ž A, j . in A, i.e., the poles of P Ž z; A, j ., also depend on j . The kernel P Ž1. Ž z; A. became gauge-invariant in the case when only the quark insertions are involved,
2 For another popular definition of a minimal scheme, when a scheme factor is chosen as expŽ cP´ ., csyg E q . . . instead of G Ž Dr2y1., the coefficient C Ž ´ . does not contain any scheme ‘‘traces’’ in final expressions for the renormalization-group functions.
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S.V. MikhailoÕr Physics Letters B 431 (1998) 387–394
Fig. 2. The curve of the factor gg Ž0.rgg Ž A.; the arrow on the picture corresponds to the point A s 1rp ; the first singularity appears at A s A 0 s 5r2.
i.e., gg s ggŽ q . ; A s AŽ q . s ya sggŽ q . Ž0. s a s 43 TR Nf , and P Ž1. Ž z; AŽ q . . ™ P Ž1. Ž z; A. as it was presented in w3x. It is instructive to consider this case in detail. To this end, let us choose the common factor ggŽ q . Ž0.rggŽ q . Ž A. in formula Ž5. for the crude measure of modification of the kernel in comparison with the one-loop result a s P0 Ž z .. Considering the curve of this factor in the argument A in Fig. 2, one may conclude: Ži. the range of convergence of PT series corresponds to the left zero of the function ggŽ q . Ž A. and is equal to A 0 s 5r2, that corresponds to a s0 s 15prNf , so, this range looks very broad 3 , a s - 5p at Nf s 3; Žii. in spite of a wide range of PT fidelity, the resummation into PqŽ1. Ž z; A. is substantial – two zeros of the function PqŽ1. Ž z; A. in A appear within the range of convergence Žit depends on a certain MS scheme.; Žiii. the factor ggŽ q . Ž0.rggŽ q . Ž A. decays quickly with the growth of the argument A. Really, if we
3
Here we consider the evolution kernel P Ž z, A. Ž V Ž A.. by itself. We take out of the scope that the factorization scale m2 of hard processes would be chosen large enough, m2 G mr2 , where the r-meson mass represents the characteristic hadronic scale. Following this reason, the used coupling a s Ž m2 . could not be too large.
take the naive boundary of the standard PT applicability, a s s 1 Žat Nf s 3, AŽ q . s 1rŽ2p .., then this factor falls approximately to 0.7 Žat Nf s 6, AŽ q . s 1rp it falls to 0.5, see arrow in Fig. 2.; thus, the resummation is numerically important in this range. Note at the end that Eq. Ž5. could not provide valid asymptotic behavior of the kernels for z ™ 0. A similar z-behavior is determined by the doublelogarithmic corrections which are most singular at zero, like a s Ž a s ln2 w z x. n w14x. These contributions appear due to renormalization of the composite operator in the diagrams by ladder graphs, etc.rather than by the triangular ones.
3. Analysis of the NNA assumption for kernel calculations The expansion of PqŽ1. Ž z; A. in A provides the leading a s Ž a s Nf lnw1rz x. n dependence of the kernels with a large number Nf in any order n of PT w3x. But these contributions do not numerically dominate for real numbers of flavours Nf s 4, 5, 6. That may be verified by comparing the total numerical results for the 2- and 3-loop AD’s of composite operators ŽADCO. in w15x with their Nf-leading terms Žsee ADCO in Table 1.. Therefore, to obtain a satisfactory agreement at least with the second order results,
S.V. MikhailoÕr Physics Letters B 431 (1998) 387–394
391
Table 1 The results of gŽ1,2.Ž n. calculations performed in different ways, exact numerical results from w15x, approximations obtained from P Ž z, A, j . with j s 0 and j s y3; both numerical and analytical exact results are emphasized by boldface print
gŽ1.Ž n.
gŽ2.Ž n.
CF CA
Nf P CF
CA2 CF
Nf P CF CA
Nf2 P CF
ns2 Exact j s y3 js0
13.9 11.3 7.6
I 64 27
86.1 H 21.3 z ( 3 ) y42.0 y13.2
I12.9 I 21.3 z ( 3 ) 12.9 7.5
I 224 243
ns4 Exact j s y3 js0
23.9 23.5 15.8
I 13271 2700
140.0 H 19.2 z ( 3 ) y76.0 y23.5
I18.1 I 41.9 z ( 3 ) 23. 12.4
I 384277 243000
ns6 Exact j s y3 js0
29.7 31.1 20.7
I 428119 66150
173 H 19.01 z ( 3 ) y95.6 y29
I20.4 I 54.0 z ( 3 ) 28.5 15.2
I 80347571 41674500
ns8 Exact j s y3 js0
33.9 36.3 24
I 36241943 4762800
196.9 H 18.98 z ( 3 ) y109.0 y33.0
I21.9 I 62.7 z ( 3 ) 32.3 17.2
I2.1619
n s 10 Exact j s y3 js0
37.27 41.00 27.29
I8.5095
216.0 H 18.96 z ( 3 ) y119.28 y36.0
I23.2 I 69.6 z ( 3 ) 35.24 18.68
I2.3366
one should take into account the contribution from subleading Nf-terms. As a first step, let us consider the contribution from the completed renormalization of the gluon line – it should generate a part of subleading terms. Below we shall examine two special choices of the gauge parameter j . To facilitate the diagrammatic analysis, it is instructive to inspect first the Landau gauge j s 0. Indeed, the self-energy one-loop insertions into the quark lines as well as a certain part of vertex corrections to triangular diagrams are proportional to j ; therefore, they disappear in the Landau gauge. Moreover, one should not consider the renormalization of parameter j . The analytic properties of the function P Ž1. Ž z; A, j s 0. in the variable A s AŽ0. are modified – the function has no singularities in A until the ‘‘asymptotic freedom’’ exists, i.e., A - 0 Žat 13CA ) 4 Nf .. In spite of all these profits the kernel P Ž1. Ž z; A,0. generates the partial kernels a2s PŽ1.Ž z ., a 3s PŽ2.Ž z ., . . . which are rather far from the real ones. The ADCO gŽ1,2.Ž n.
corresponding to these kernels Žhere g Ž n . s H01dzz n P Ž z . . are presented in Table 1. Another exceptional gauge is j s y3. For this gauge the coefficient of one-loop gluon AD gg Ž0,y 3. coincides with the coefficient b 0 of the b-function 4 . Therefore this gauge may be used for a reformulation of the so-called w9x NNA proposition to kernel calculations. To obtain the NNA result in a usual way, one should substitute the coefficient b 0 for ggŽ q . Ž0. into the expression for AŽ q . by hand Žsee, e.g., w10x.. Note, the use of such an NNA procedure to improve PqŽ1. Ž z; A. leads to poor results even for a 2s P1Ž z . term of the expansion; a similar observation was also done in w16x. The NNA trick expresses common hope that the main logarithmic contribution may follow from the renormalization of the coupling
4
Here, for the b Ž a s .-function we adapt b Ž a s . sy b 0 a 2s q . . . , b 0 s 113 CA y 43 Nf TR
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S.V. MikhailoÕr Physics Letters B 431 (1998) 387–394
constant. This renormalization appears as a sum of contributions from all the sources of renormalization of a s , corresponding diagrammatic analysis for two-loop kernels is presented in w11,18x. In the case of the j s y3 gauge the one-loop gluon renormalization ‘‘imitates’’ the contributions from these other sources and the coefficient b 0 appears naturally. The elements of expansion of the ADCO g Ž n; A,y 3. Žthat corresponds to P Ž1. Ž z; A,y 3.. in a power series in a s , a 2s gŽ1.Ž n.; a 3s gŽ2.Ž n.; . . . and a few numerical exact results from w15x are collected in Table 1, let us compare them: Ži. we consider there the contribution to the coefficient gŽ1.Ž n. which is generated by the gluon loops and associated with Casimirs CF CAr2, the CF2-term is missed, but its contribution is insignificant. It is seen that in this order the CF CA-terms are rather close to exact values Žthe accuracy is about 10% for n ) 2. and our approximation works rather well; Žii. in the next order the contributions to gŽ2.Ž n. associated with the coefficients Nf P CF CA and CA2 CF are generated, while the terms with the coefficients CF3 , Nf P CF2 , CF2 CA are missed. In the third order, contrary to the previous item, all the generated terms are opposite in sign to the exact values, and the ‘‘j s y3 approximation’’ doesn’t work at all. So, we need the next step to improve the agreement – to obtain the subleading Nf-terms by the exact calculation. In any case, it seems rather difficult to collect the renormalization constant required by the NNA approximation in the kernel calculations. It is because different sources of renormalization of a s provide different z-dependent contributions, compare, e.g., Exp.Ž1. with Eq. Ž10. in w3x, the latter being generated by the insertions of self-energy quark parts into the quark line Žchain 2.. For this reason, necessary cancellation between the terms from different sources looks unlikely.
as it was done for the scalar model in w3x. We shall use again the exact relations between the V and P kernels established in any order of PT w11x for triangular diagrams. These relations were obtained by comparing counterterms for the same triangular diagrams considered in ‘‘forward’’ and ‘‘nonforward’’ kinematics. Let the diagram in Fig. 1a have a contribution to the DGLAP kernel in the form P Ž z . s pŽ z . q d Ž1 y z . P C; then its contribution to the ER-BL kernel is x
ž
V Ž x, y. sC u Ž y)x.
H0 y
pŽ z . z
dz
/
q d Ž y y x . P C,
Ž 10 .
where C ' 1 q Ž x ™ x, y ™ y .. From relation Ž10. and Eqs. Ž2., Ž4. for P Ž1 a,c. we immediately derive the expression for the sum of contributions V Ž1 aq1 c., V Ž1 aq1 c. Ž x , y ; A, j . s a s CF 2 P C u Ž y ) x . Ž 1 y A .
1y A
x
ž / y
gg Ž 0, j .
1 Ž 1 y A. y d Ž yyx. 2 Ž 2 y A.
gg Ž A, j .
,
Ž 11 .
that may naturally be represented in the ‘‘plus form’’. Expression Ž11. can be independently verified by other relations reducing any V to P w11,17x Žsee formulae for the V ™ P reduction there. and we came back to the same Eqs. Ž2., Ž4. for P Ž1 a,c.. Moreover, the first terms of the Taylor expansion of V Ž1 a,c. Ž x, y; A. in A coincide with the results of the two-loop calculation in w11x. The relation P ™ V similar to Eq. Ž10. has also been derived for the diagram in Fig. 1b V Ž1 b. Ž x , y . s C u Ž y ) x .
1 2y
P Ž1 b.
x
ž / y
;
Ž 12 .
q
therefore, substituting Eq. Ž3. into Ž12. we obtain 4. Triangular diagrams for the nonforward ERBL evolution kernel
V Ž1 b. Ž x , y ; A, j . s a s CF 2 PC u Ž y)x.
Here we present the results of the bubble resummation for the ER-BL kernel V Ž x, y .. It can be obtained as a ‘‘byproduct’’ of the previous results for the kernel DGLAP P Ž z ., i.e., in the same manner
=
gg Ž 0, j . gg Ž A, j .
.
x
ž / y
1y A
1 yyx
q
Ž 13 .
S.V. MikhailoÕr Physics Letters B 431 (1998) 387–394
Collecting the results in Ž11. and Ž13. we arrive at the final expression for V Ž1. in the ‘‘main bubbles’’ approximation V Ž1. Ž x , y ; A, j . sa s CF 2 PC u Ž y)x.
gg Ž 0, j .
=
gg Ž A, j .
x
1y A
ž / ž y
1yAq
1 yyx
/
1
Ž1.
2.
q
,
Ž 14 .
Ž x , y ; A . cn Ž y ; A . dy s g Ž n; A . cn Ž x ; A . , Ž 15 . d Ž A .y
and dc Ž A. is the effective dimension of the quark field when the AD A is taken into account; CnŽ a . Ž z . are the Gegenbauer polynomials of an order of a . The partial solutions F Ž x;a s ,l . of the original ERBL-equation Žwhere l ' lnŽ m2rm20 ..
Ž m2Em q b Ž a s . Ea . F Ž x ;a s ,l . s
s
1
H0 V
Ž1.
Ž x , y ; A . F Ž y ;a s ,l . dy
Ž 17 .
are proportional to these eigenfunctions cnŽ x; A. for the special case b Ž a s . s 0, see, e.g. w3x. In the general case b Ž a s . / 0 let us start with an ansatz for the partial solution of Eq. Ž17., FnŽ x;a s ,l . ; xnŽ a s ,l . P cnŽ x; A., and the boundary condition is xnŽ a s ,0. s 1; FnŽ x;a s ,0. ; cnŽ x; A.. For this ansatz Eq. Ž17. reduces to
Ž m2Em q b Ž a s . Ea . ln ŽFn Ž x ;a s ,l . . s g Ž n; A . . 2
Ž1yA .
,
Ž 19 .
2.
where A s ya s Ž m g Ž0, j . and a s Ž m is the running coupling corresponding to b Ž a s .. A similar solution has been discussed in w10x in the framework of the standard NNA approximation. Solving simultaneously Eq. Ž18. and the renormalization-group equation for the coupling constant a s we arrive at the partial solution FnŽ x;a s ,l . in the form
Fn Ž x ,a s . ; xn Ž m2 . P cn Ž x ; A . ;
½
2
HaaŽ Žmm . .
where xn Ž m2 . s exp y
s
s
2 0
g Ž n, A . b Ž a.
da
5 Ž 20 .
Recently, a form of the solution ; cnŽ x; A. with A s ya s b 0 has been confirmed in w12x by the consideration of conformal constraints w19x on the meson wave functions in the limit Nf 4 1.
1
d Ž A. 2 Cn c cn Ž y ; A . ; Ž yy . c Ž yyy. , here dc Ž A . s Ž DA y 1 . r2, DA s 4 y 2 A, Ž 16 .
2
In the case n s 0 the AD of the vector current g Ž0; A. s 0, and the solution of the homogeneous equation in Ž18. provides the ‘‘asymptotic wave function’’
F 0 Ž x ;a s ,l . s c 0 Ž x ; A . ; Ž Ž 1 y x . x .
which has a ‘‘plus form’’ again due to the vector current conservation. The contribution V Ž1. in Ž14. should dominate for Nf 4 1 in the kernel V. Besides, the function V Ž1. Ž x, y; A, j . possesses an important symmetry of its arguments x and y. Indeed, the function V Ž x, y; A, j . s V Ž1. Ž x, y; A, j . P Ž yy .1yA is symmetric under the change x l y, V Ž x, y . s V Ž y, x .. This symmetry allows us to obtain the eigenfunctions cnŽ x . of the ‘‘reduced’’ evolution equation w18x
H0 V
393
s
Ž 18 .
5. Conclusion In this paper, I present closed expressions in the ‘‘all order’’ approximation for the DGLAP kernel P Ž z . and ER-BL kernel V Ž x, y . appearing as a result of the resummation of a certain class of QCD diagrams with the renormalon chain insertions. The contributions from these diagrams, P Ž1. Ž z; A. and V Ž1. Ž z; A., give the leading Nf dependence of the kernels for a large number of flavours Nf 4 1. These ‘‘improved’’ kernels are generating functions to obtain contributions to partial kernels like aŽsnq1. PŽ n.Ž z . in any order n of perturbation expansion. Here A ; a s is a new expansion parameter that coincides Žin magnitude. with the anomalous dimension of the gluon field. On the other hand, the method of calculation suggested in w3x does not depend on the nature of self-energy insertions and does not appeal to the value of the parameters Nf TR , CAr2 or CF associated with different loops. This allows us to obtain contributions from chains with different kinds of self-energy insertions, both quark and gluon Žghost. loops. The prize for this generalization is gauge
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S.V. MikhailoÕr Physics Letters B 431 (1998) 387–394
dependence of the final results for P Ž1. Ž z; A. and V Ž1. Ž z; A. on the gauge parameter j . The result for the DGLAP non-singlet kernel P Ž1. Ž z; AŽ j ., j . is presented in Ž5. in the covariant j-gauge. The analytic properties of this kernel in the variable a s are discussed for quark bubble chains only, and in the general case for two values of the gauge parameter j s 0;y 3. The insufficiency of the NNA proposition for the kernel calculation is demonstrated by the evident calculation in the third order in a s Žsee Table 1.. The contribution V Ž1. Ž x, y; AŽ j ., j . to the nonforward ER-BL kernel Ž14. is obtained for the same classes of diagrams as a ‘‘byproduct’’ of the previous technique w18x. A partial solution Ž20. to the ER-BL equation is derived.
Acknowledgements The author is grateful to Dr. M. Beneke, Dr. A. Grozin, Dr. D. Mueller, Dr. A. Kataev, Dr. M. Polyakov and Dr. N. Stefanis for fruitful discussions of the results, to Dr. A. Bakulev for help in calculation and to Dr. R. Ruskov for careful reading of the manuscript and useful remarks. This investigation has been supported in part by the Russian Foundation for Basic Research ŽRFBR. 98-02-16923 and INTAS 2058.
w2x
w3x w4x w5x w6x
w7x w8x w9x w10x
w11x w12x w13x w14x w15x
w16x w17x
References w1x V.N. Gribov, L.N. Lipatov, Sov. J. Nucl. Phys. 15 Ž1972. 438, 675; L.N. Lipatov, Sov. J. Nucl. Phys. 20 Ž1975. 94;
w18x w19x
Y.L. Dokshitser, JETP 46 Ž1977. 641; G. Altarelli, G. Parisi, Nucl. Phys. 126 Ž1977. 298. S.J. Brodsky, G.P. Lepage, Phys. Lett. B 87 Ž1979. 359; Phys. Rev. D 22 Ž1980. 2157; A.V. Efremov, A.V. Radyushkin, Phys. Lett B 94 Ž1980. 245; Theor. Math. Phys. 42 Ž1980. 97. S.V. Mikhailov, Phys. Lett. B 416 Ž1998. 421. A.N. Vasil’ev, Yu.M. Pis’mak, J.R. Honkonen, Theor. Math. Phys. 46 Ž1981. 157; 47 Ž1981. 291. J.A. Gracey, Phys. Lett. B 322 Ž1994. 141; Nucl. Phys. B 480 Ž1996. 73. J.A. Gracey, Renormalization group functions of QCD in large-Nf , talk presented at Third International Conference on the Renormalization Group, JINR, Dubna, Russia, 26–31 August 1996, hep-thr9609164. A. Palanques-Mestre, P. Pascual, Comm. Math. Phys. 95 Ž1984. 277. M. Beneke, V.M. Braun, Nucl. Phys. B 426 Ž1994. 301. D.J. Broadhurst, A.G. Grozin, Phys. Rev. D 52 Ž1995. 4082. P. Gosdzinsky, N. Kivel, Resummation of Žy b 0 a s . n corrections to the photon-meson transition form factor g ) qg ™ p 0 , hep-phr9707367. S.V. Mikhailov, A.V. Radyushkin, Nucl. Phys. B 254 Ž1985. 89. A.V. Belitsky, D. Muller, Phys. Lett. B 417 Ž1998. 129. ¨ C. Itzykson, J.-B. Zuber, Quantum field theory, McGraw-Hill. Inc., 1995. J. Blumlein, A. Vogt, Phys. Lett. B 370 Ž1996. 149; R. Kirschner, L.N. Lipatov, Nucl. Phys. 213 Ž1983. 122. S.A. Larin, T. van Ritbergen, J.A.M. Vermaseren, Nucl. Phys. B 427 Ž1994. 41; S.A. Larin, P. Nogueira, T. van Ritbergen, J.A.M. Vermaseren, Nucl. Phys. B 492 Ž1997. 338. L. Mankiewicz, M. Maul, E. Stein, Phys. Lett. 404 Ž1997. 345. F.M. Dittes, D. Muller, D. Robaschik, B. Geyer, J. Horejsi, ¨ Phys. Lett. B 209 Ž1988. 325; Fortschr. Phys. 42 Ž1994. 101. S.V. Mikhailov, A.V. Radyushkin, Nucl. Phys. B 273 Ž1986. 297. D. Muller, Z. Phys. C 49 Ž1991. 293; Phys. Rev. D 49 Ž1994. ¨ 2525.
Physics Letters B 431 Ž1998. 387–394
Renormalon chain contributions to non-singlet evolution kernels in QCD S.V. Mikhailov
1
Joint Institute for Nuclear Research, BogoliuboÕ Laboratory of Theoretical Physics, 141980 Dubna, Moscow Region, Russia Received 20 April 1998 Editor: P.V. Landshoff
Abstract Contributions to QCD non-singlet forward evolution kernels P Ž z . for the DGLAP equation and to V Ž x, y . for nonforward ŽER-BL. evolution equation are calculated for a certain class of diagrams which include renormalon chains. Closed expressions are presented for the sums of these contributions that dominate for a large number of flavors Nf 4 1. Calculations are performed in covariant j-gauge, in the MS scheme. The assumption of ‘‘naive nonabelianization’’ approximation for kernel calculations is discussed. The partial solution to the ER-BL evolution equation is obtained. q 1998 Published by Elsevier Science B.V. All rights reserved. PACS: 12.38.Cy; 12.38.-t; 13.60.Hb Keywords: DGLAP; Forward and nonforward evolution kernels; Anomalous dimensions; Multiloop calculation
1. Introduction Evolution kernels are the main ingredients of the well-known evolution equations for parton distribution of DIS processes and for parton wave functions in hard exclusive reactions. These equations describe the dependence of parton distribution functions and parton wave functions on the renormalization parameter m2 . Here, I continue to discuss the diagrammatic analysis and multiloop calculation of the forward DGLAP evolution kernel P Ž z . w1x and nonforward Efremov–Radyushkin–Brodsky–Lepage ŽER-BL. kernel V Ž x, y . w2x in a class of ‘‘all-order’’ approxi-
1
E-mail: [email protected].
mation of the perturbative QCD that has been started in w3x. There, the regular method of calculation and resummation of certain classes of diagrams for these kernels has been suggested. These diagrams include the chains of one-loop self-energy parts Žrenormalon chains. into the one-loop diagrams Žsee Fig. 1.. In this letter, the results for both the kinds of kernels, obtained earlier in the framework of a scalar model in six dimensions with the Lagrangian L i n t s Ý iN f Ž c i)c i w .Ž6. with the scalar ‘‘quark’’ flavours Ž c i . and ‘‘gluon’’ Ž w ., are extended to the non-singlet QCD kernels. For the readers convenience some important results of the previous paper w3x would be reminded. The insertion of the chain into ‘‘gluon’’ line Žchain-1. of the diagram in Fig. 1 a,b and resumma-
0370-2693r98r$ – see frontmatter q 1998 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 5 8 5 - 1
S.V. MikhailoÕr Physics Letters B 431 (1998) 387–394
388
Fig. 1. The diagrams in Figs. 1a – 1c are the ‘‘triangular’’ diagrams for the QCD DGLAP kernel; dashed line for gluons, solid line for quarks; black circle denotes the sum of all kinds of the one-loop insertions Ždashed circle., both quark and gluon Žghost. or mixed chains; MC denotes the mirror-conjugate diagram.
tion over all bubbles lead to the transformation of the one-loop kernel P0 Ž z . s az ' aŽ1 y z . into the kernel P Ž1. Ž z; A. chain 1
P0 Ž z . s az ™ P Ž1. Ž z ; A . s az Ž z .
where
yA
Ž 1 y A.
A s aNf gw Ž 0 . ,
gw Ž 0 . gw Ž A . as
; g2
Ž 4p .
3
.
Ž 1.
Here, gc Ž w .Ž ´ . are the one-loop coefficients of the anomalous dimensions of quark Žgluon at Nf s 1. fields in D-dimension Ž D s 6 y 2 ´ . discussed in w3x; for the scalar model gc Ž ´ . s gw Ž ´ . s B Ž2 y ´ ,2 y ´ .C Ž ´ ., and C Ž ´ . is a scheme-dependent factor corresponding to a certain choice of an MS-like scheme. The argument A of the function gw Ž A. in Ž1. is the standard anomalous dimension ŽAD. of a gluon field. So, one can conclude that the ‘‘all-order’’ result in Ž1. is completely determined by the single quark bubble diagram. The resummation of this ‘‘chain-1’’ subseries into an analytic function in A shouldn’t be taken by surprise. Really, the considered problem can be connected with the calculation of large Nf asymptotics of the AD’s in order of 1rNf . An approach was suggested by Vasil’ev and
collaborators at the beginning of 80’ w4x to calculate the renormalization-group functions in this limit, they used the conformal properties of the theory at the critical point g s g c corresponding to the non-trivial zero g c of the D-dimensional b-function. This approach has been extended by J. Gracey for calculation of the AD’s of the composite operators of DIS in QCD in any order n of PT, w5,6x. I have used another approach, which is close to w7,8x; contrary to the large Nf asymptotic method it does not appeal to the value of parameters Nf TR , CAr2 or CF , associated in QCD with different kinds of loops. Following this way, the ‘‘improved’’ QCD kernel P Ž1. Ž z; A. has been obtained in w3x for the case of quark or gluon bubble chain insertions in the Feynman gauge. In this paper, we present the QCD results similar to Eq. Ž1., for each type of diagrams appearing in the covariant j-gauge for the DGLAP non-singlet kernel P Ž z; A.. The analytic properties of the function P Ž z; A. in variable A are analyzed. The assumption of ‘‘Naive Nonabelianization’’ ŽNNA. approximation w9x for the kernel calculation w10x is discussed and its deficiency is demonstrated. The ER-BL evolution kernel V Ž x, y . is obtained in the same approximation as the DGLAP kernel, by using the exact relations between P and V kernels w11,3x for a class of ‘‘triangular diagrams’’ in Fig. 1. The considered class of diagrams represents the leading Nf contributions to both kinds of kernels. At the end, a partial solution for the ER-BL equation is presented Žcompare with w10,12x.. The obtained results are certainly useful for an independent check of complicated computer calculations in higher orders of perturbation theory ŽPT., similar to w15x; they may be a starting point for further approximation procedures.
2. Triangular diagrams for the DGLAP evolution kernel in QCD Here, the results of the bubble chain resummation for QCD diagrams in Fig. 1 a,b,c for the DGLAP kernel are discussed. These diagrams generate contributions ; a s Ž a s lnw1rz x. n in any order n of PT. Based on the resummation method of Ref. w3x in the QCD version, one can derive the kernels P Ž1 a, b,c.
S.V. MikhailoÕr Physics Letters B 431 (1998) 387–394
corresponding to the diagrams in Fig. 1 in the covariant j y gauge
gg Ž 0 .
2
P Ž1 a. Ž z ; A . s a s CF 2 z P Ž 1 y A . zyA
gg Ž A .
y a s CF P d Ž 1 y z .
ž
=
gg Ž 0 .
1
Ž 1 y A . gg Ž A .
P Ž1 b. Ž z ; A . s a s CF 2 P
ž
/ /
yj ,
2 z 1y A gg Ž 0 . 1 y z gg Ž A .
Ž 2.
Ž5. the d Ž1 y z . - terms are exactly accumulated in the form of the w . . . xq prescription, and the j - terms successfully cancel. This is due to the evident current conservation for the case of quark bubble insertions, including the gluon bubbles into consideration merely modifies the effective AD, gg Ž A, j . ™ ggŽ q . Ž A., conserving the structure of the result Ž5., see w3x. Substituting the well-known expressions for gg Ž ´ . from the quark or gluon Žghost. loops Žsee, e.g., w13x.
ggŽ q . Ž ´ . s y8 Nf TR B Ž Dr2, Dr2 . C Ž ´ . , ,
Ž 3. ggŽ g . Ž ´ , j . s
q
P Ž1 c. Ž z ; A . s a s CF P d Ž 1 y z . =
ž
gg Ž 0 .
AŽ 3 y 2 A .
Ž 2 y A . Ž 1 y A . gg Ž A .
where a s s
CA 2
=
/
yj ,
Ž 4. as
389
, CF s Ž Nc2 y 1.r2 Nc , CA s Nc
q
Ž 7.
B Ž Dr2 y 1, Dr2 y 1 .
žž ž
3Dy2 Dy1
1yj 2
/
q Ž 1 y j . Ž D y 3.
2
/ /
´ CŽ ´ . ,
Ž 8.
and TR s 12 are the Casimirs of SUŽ Nc . group, and A s ya sgg Ž0.. The function gg Ž ´ . is the one-loop coefficient of the anomalous dimension of gluon field in D-dimension, here D s 4 y 2 ´ . In other words, it is the coefficient Z1Ž ´ . of a simple pole in the expansion of the gluon field renormalization constant Z, that includes both a finite part and all the powers of the ´-expansion. Eqs. Ž2. – Ž4. are valid for any kind of insertions, i.e., gg s ggŽ q . for the quark loop, gg s ggŽ g . for the gluon Žghost. loop, or for their sum
into the general formulae Ž2. – Ž4., and Ž5. one can obtain P Ž1. Ž z; A, j . for both the quark and the gluon loop insertions simultaneously. Here, the coefficient C Ž ´ . s G Ž1 y ´ . G Ž1 q ´ . implies a certain choice of the MS scheme where every loop integral is multiplied by the scheme factor G Ž Dr2 y 1.Ž m2r4p . ´. The renormalization scheme dependence of P Ž1. Ž z; A. is accumulated by the factor C Ž ´ . 2 . Of course, the final result Ž5. will be gaugedependent in virtue of the evident gauge dependence of the gluon loop contribution ggŽ g . Ž ´ , j ., in this case, e.g.,
gg Ž A, j . s ggŽ q . Ž A . q ggŽ g . Ž A, j . ;
A Ž j . s ya sgg Ž 0, j . s ya s Ž ggŽ q . Ž 0 . q ggŽ g . Ž 0, j . .
Ž 4p .
when both kinds of insertions are taken into account. The sum of contributions Ž2., Ž3., Ž4. results in P Ž1. Ž z; A, j . which has the expected ‘‘plus form’’ P
Ž1.
yA
Ž z ; A, j . s a s CF 2 P zz =
gg Ž 0, j . gg Ž A, j .
a s P0 Ž z . s a s C F 2 P z q
2
Ž 1 y A. q
,
1yz
q
Ž 5.
2z 1yz
2 z 1y A
,
Ž 6.
q
where, in comparison with Eq. Ž5., the one-loop result a s P0 Ž z . is written down, the latter can be obtained as the limit P Ž1. Ž z; A ™ 0, j .. Note that in
s ya s
ž
5 q 3
Ž1yj . 2
/
CA y
4 3
Nf TR ,
Ž 9.
is the contribution to the one-loop renormalization of the gluon field. The positions of zeros of gg Ž A, j . in A, i.e., the poles of P Ž z; A, j ., also depend on j . The kernel P Ž1. Ž z; A. became gauge-invariant in the case when only the quark insertions are involved,
2 For another popular definition of a minimal scheme, when a scheme factor is chosen as expŽ cP´ ., csyg E q . . . instead of G Ž Dr2y1., the coefficient C Ž ´ . does not contain any scheme ‘‘traces’’ in final expressions for the renormalization-group functions.
390
S.V. MikhailoÕr Physics Letters B 431 (1998) 387–394
Fig. 2. The curve of the factor gg Ž0.rgg Ž A.; the arrow on the picture corresponds to the point A s 1rp ; the first singularity appears at A s A 0 s 5r2.
i.e., gg s ggŽ q . ; A s AŽ q . s ya sggŽ q . Ž0. s a s 43 TR Nf , and P Ž1. Ž z; AŽ q . . ™ P Ž1. Ž z; A. as it was presented in w3x. It is instructive to consider this case in detail. To this end, let us choose the common factor ggŽ q . Ž0.rggŽ q . Ž A. in formula Ž5. for the crude measure of modification of the kernel in comparison with the one-loop result a s P0 Ž z .. Considering the curve of this factor in the argument A in Fig. 2, one may conclude: Ži. the range of convergence of PT series corresponds to the left zero of the function ggŽ q . Ž A. and is equal to A 0 s 5r2, that corresponds to a s0 s 15prNf , so, this range looks very broad 3 , a s - 5p at Nf s 3; Žii. in spite of a wide range of PT fidelity, the resummation into PqŽ1. Ž z; A. is substantial – two zeros of the function PqŽ1. Ž z; A. in A appear within the range of convergence Žit depends on a certain MS scheme.; Žiii. the factor ggŽ q . Ž0.rggŽ q . Ž A. decays quickly with the growth of the argument A. Really, if we
3
Here we consider the evolution kernel P Ž z, A. Ž V Ž A.. by itself. We take out of the scope that the factorization scale m2 of hard processes would be chosen large enough, m2 G mr2 , where the r-meson mass represents the characteristic hadronic scale. Following this reason, the used coupling a s Ž m2 . could not be too large.
take the naive boundary of the standard PT applicability, a s s 1 Žat Nf s 3, AŽ q . s 1rŽ2p .., then this factor falls approximately to 0.7 Žat Nf s 6, AŽ q . s 1rp it falls to 0.5, see arrow in Fig. 2.; thus, the resummation is numerically important in this range. Note at the end that Eq. Ž5. could not provide valid asymptotic behavior of the kernels for z ™ 0. A similar z-behavior is determined by the doublelogarithmic corrections which are most singular at zero, like a s Ž a s ln2 w z x. n w14x. These contributions appear due to renormalization of the composite operator in the diagrams by ladder graphs, etc.rather than by the triangular ones.
3. Analysis of the NNA assumption for kernel calculations The expansion of PqŽ1. Ž z; A. in A provides the leading a s Ž a s Nf lnw1rz x. n dependence of the kernels with a large number Nf in any order n of PT w3x. But these contributions do not numerically dominate for real numbers of flavours Nf s 4, 5, 6. That may be verified by comparing the total numerical results for the 2- and 3-loop AD’s of composite operators ŽADCO. in w15x with their Nf-leading terms Žsee ADCO in Table 1.. Therefore, to obtain a satisfactory agreement at least with the second order results,
S.V. MikhailoÕr Physics Letters B 431 (1998) 387–394
391
Table 1 The results of gŽ1,2.Ž n. calculations performed in different ways, exact numerical results from w15x, approximations obtained from P Ž z, A, j . with j s 0 and j s y3; both numerical and analytical exact results are emphasized by boldface print
gŽ1.Ž n.
gŽ2.Ž n.
CF CA
Nf P CF
CA2 CF
Nf P CF CA
Nf2 P CF
ns2 Exact j s y3 js0
13.9 11.3 7.6
I 64 27
86.1 H 21.3 z ( 3 ) y42.0 y13.2
I12.9 I 21.3 z ( 3 ) 12.9 7.5
I 224 243
ns4 Exact j s y3 js0
23.9 23.5 15.8
I 13271 2700
140.0 H 19.2 z ( 3 ) y76.0 y23.5
I18.1 I 41.9 z ( 3 ) 23. 12.4
I 384277 243000
ns6 Exact j s y3 js0
29.7 31.1 20.7
I 428119 66150
173 H 19.01 z ( 3 ) y95.6 y29
I20.4 I 54.0 z ( 3 ) 28.5 15.2
I 80347571 41674500
ns8 Exact j s y3 js0
33.9 36.3 24
I 36241943 4762800
196.9 H 18.98 z ( 3 ) y109.0 y33.0
I21.9 I 62.7 z ( 3 ) 32.3 17.2
I2.1619
n s 10 Exact j s y3 js0
37.27 41.00 27.29
I8.5095
216.0 H 18.96 z ( 3 ) y119.28 y36.0
I23.2 I 69.6 z ( 3 ) 35.24 18.68
I2.3366
one should take into account the contribution from subleading Nf-terms. As a first step, let us consider the contribution from the completed renormalization of the gluon line – it should generate a part of subleading terms. Below we shall examine two special choices of the gauge parameter j . To facilitate the diagrammatic analysis, it is instructive to inspect first the Landau gauge j s 0. Indeed, the self-energy one-loop insertions into the quark lines as well as a certain part of vertex corrections to triangular diagrams are proportional to j ; therefore, they disappear in the Landau gauge. Moreover, one should not consider the renormalization of parameter j . The analytic properties of the function P Ž1. Ž z; A, j s 0. in the variable A s AŽ0. are modified – the function has no singularities in A until the ‘‘asymptotic freedom’’ exists, i.e., A - 0 Žat 13CA ) 4 Nf .. In spite of all these profits the kernel P Ž1. Ž z; A,0. generates the partial kernels a2s PŽ1.Ž z ., a 3s PŽ2.Ž z ., . . . which are rather far from the real ones. The ADCO gŽ1,2.Ž n.
corresponding to these kernels Žhere g Ž n . s H01dzz n P Ž z . . are presented in Table 1. Another exceptional gauge is j s y3. For this gauge the coefficient of one-loop gluon AD gg Ž0,y 3. coincides with the coefficient b 0 of the b-function 4 . Therefore this gauge may be used for a reformulation of the so-called w9x NNA proposition to kernel calculations. To obtain the NNA result in a usual way, one should substitute the coefficient b 0 for ggŽ q . Ž0. into the expression for AŽ q . by hand Žsee, e.g., w10x.. Note, the use of such an NNA procedure to improve PqŽ1. Ž z; A. leads to poor results even for a 2s P1Ž z . term of the expansion; a similar observation was also done in w16x. The NNA trick expresses common hope that the main logarithmic contribution may follow from the renormalization of the coupling
4
Here, for the b Ž a s .-function we adapt b Ž a s . sy b 0 a 2s q . . . , b 0 s 113 CA y 43 Nf TR
392
S.V. MikhailoÕr Physics Letters B 431 (1998) 387–394
constant. This renormalization appears as a sum of contributions from all the sources of renormalization of a s , corresponding diagrammatic analysis for two-loop kernels is presented in w11,18x. In the case of the j s y3 gauge the one-loop gluon renormalization ‘‘imitates’’ the contributions from these other sources and the coefficient b 0 appears naturally. The elements of expansion of the ADCO g Ž n; A,y 3. Žthat corresponds to P Ž1. Ž z; A,y 3.. in a power series in a s , a 2s gŽ1.Ž n.; a 3s gŽ2.Ž n.; . . . and a few numerical exact results from w15x are collected in Table 1, let us compare them: Ži. we consider there the contribution to the coefficient gŽ1.Ž n. which is generated by the gluon loops and associated with Casimirs CF CAr2, the CF2-term is missed, but its contribution is insignificant. It is seen that in this order the CF CA-terms are rather close to exact values Žthe accuracy is about 10% for n ) 2. and our approximation works rather well; Žii. in the next order the contributions to gŽ2.Ž n. associated with the coefficients Nf P CF CA and CA2 CF are generated, while the terms with the coefficients CF3 , Nf P CF2 , CF2 CA are missed. In the third order, contrary to the previous item, all the generated terms are opposite in sign to the exact values, and the ‘‘j s y3 approximation’’ doesn’t work at all. So, we need the next step to improve the agreement – to obtain the subleading Nf-terms by the exact calculation. In any case, it seems rather difficult to collect the renormalization constant required by the NNA approximation in the kernel calculations. It is because different sources of renormalization of a s provide different z-dependent contributions, compare, e.g., Exp.Ž1. with Eq. Ž10. in w3x, the latter being generated by the insertions of self-energy quark parts into the quark line Žchain 2.. For this reason, necessary cancellation between the terms from different sources looks unlikely.
as it was done for the scalar model in w3x. We shall use again the exact relations between the V and P kernels established in any order of PT w11x for triangular diagrams. These relations were obtained by comparing counterterms for the same triangular diagrams considered in ‘‘forward’’ and ‘‘nonforward’’ kinematics. Let the diagram in Fig. 1a have a contribution to the DGLAP kernel in the form P Ž z . s pŽ z . q d Ž1 y z . P C; then its contribution to the ER-BL kernel is x
ž
V Ž x, y. sC u Ž y)x.
H0 y
pŽ z . z
dz
/
q d Ž y y x . P C,
Ž 10 .
where C ' 1 q Ž x ™ x, y ™ y .. From relation Ž10. and Eqs. Ž2., Ž4. for P Ž1 a,c. we immediately derive the expression for the sum of contributions V Ž1 aq1 c., V Ž1 aq1 c. Ž x , y ; A, j . s a s CF 2 P C u Ž y ) x . Ž 1 y A .
1y A
x
ž / y
gg Ž 0, j .
1 Ž 1 y A. y d Ž yyx. 2 Ž 2 y A.
gg Ž A, j .
,
Ž 11 .
that may naturally be represented in the ‘‘plus form’’. Expression Ž11. can be independently verified by other relations reducing any V to P w11,17x Žsee formulae for the V ™ P reduction there. and we came back to the same Eqs. Ž2., Ž4. for P Ž1 a,c.. Moreover, the first terms of the Taylor expansion of V Ž1 a,c. Ž x, y; A. in A coincide with the results of the two-loop calculation in w11x. The relation P ™ V similar to Eq. Ž10. has also been derived for the diagram in Fig. 1b V Ž1 b. Ž x , y . s C u Ž y ) x .
1 2y
P Ž1 b.
x
ž / y
;
Ž 12 .
q
therefore, substituting Eq. Ž3. into Ž12. we obtain 4. Triangular diagrams for the nonforward ERBL evolution kernel
V Ž1 b. Ž x , y ; A, j . s a s CF 2 PC u Ž y)x.
Here we present the results of the bubble resummation for the ER-BL kernel V Ž x, y .. It can be obtained as a ‘‘byproduct’’ of the previous results for the kernel DGLAP P Ž z ., i.e., in the same manner
=
gg Ž 0, j . gg Ž A, j .
.
x
ž / y
1y A
1 yyx
q
Ž 13 .
S.V. MikhailoÕr Physics Letters B 431 (1998) 387–394
Collecting the results in Ž11. and Ž13. we arrive at the final expression for V Ž1. in the ‘‘main bubbles’’ approximation V Ž1. Ž x , y ; A, j . sa s CF 2 PC u Ž y)x.
gg Ž 0, j .
=
gg Ž A, j .
x
1y A
ž / ž y
1yAq
1 yyx
/
1
Ž1.
2.
q
,
Ž 14 .
Ž x , y ; A . cn Ž y ; A . dy s g Ž n; A . cn Ž x ; A . , Ž 15 . d Ž A .y
and dc Ž A. is the effective dimension of the quark field when the AD A is taken into account; CnŽ a . Ž z . are the Gegenbauer polynomials of an order of a . The partial solutions F Ž x;a s ,l . of the original ERBL-equation Žwhere l ' lnŽ m2rm20 ..
Ž m2Em q b Ž a s . Ea . F Ž x ;a s ,l . s
s
1
H0 V
Ž1.
Ž x , y ; A . F Ž y ;a s ,l . dy
Ž 17 .
are proportional to these eigenfunctions cnŽ x; A. for the special case b Ž a s . s 0, see, e.g. w3x. In the general case b Ž a s . / 0 let us start with an ansatz for the partial solution of Eq. Ž17., FnŽ x;a s ,l . ; xnŽ a s ,l . P cnŽ x; A., and the boundary condition is xnŽ a s ,0. s 1; FnŽ x;a s ,0. ; cnŽ x; A.. For this ansatz Eq. Ž17. reduces to
Ž m2Em q b Ž a s . Ea . ln ŽFn Ž x ;a s ,l . . s g Ž n; A . . 2
Ž1yA .
,
Ž 19 .
2.
where A s ya s Ž m g Ž0, j . and a s Ž m is the running coupling corresponding to b Ž a s .. A similar solution has been discussed in w10x in the framework of the standard NNA approximation. Solving simultaneously Eq. Ž18. and the renormalization-group equation for the coupling constant a s we arrive at the partial solution FnŽ x;a s ,l . in the form
Fn Ž x ,a s . ; xn Ž m2 . P cn Ž x ; A . ;
½
2
HaaŽ Žmm . .
where xn Ž m2 . s exp y
s
s
2 0
g Ž n, A . b Ž a.
da
5 Ž 20 .
Recently, a form of the solution ; cnŽ x; A. with A s ya s b 0 has been confirmed in w12x by the consideration of conformal constraints w19x on the meson wave functions in the limit Nf 4 1.
1
d Ž A. 2 Cn c cn Ž y ; A . ; Ž yy . c Ž yyy. , here dc Ž A . s Ž DA y 1 . r2, DA s 4 y 2 A, Ž 16 .
2
In the case n s 0 the AD of the vector current g Ž0; A. s 0, and the solution of the homogeneous equation in Ž18. provides the ‘‘asymptotic wave function’’
F 0 Ž x ;a s ,l . s c 0 Ž x ; A . ; Ž Ž 1 y x . x .
which has a ‘‘plus form’’ again due to the vector current conservation. The contribution V Ž1. in Ž14. should dominate for Nf 4 1 in the kernel V. Besides, the function V Ž1. Ž x, y; A, j . possesses an important symmetry of its arguments x and y. Indeed, the function V Ž x, y; A, j . s V Ž1. Ž x, y; A, j . P Ž yy .1yA is symmetric under the change x l y, V Ž x, y . s V Ž y, x .. This symmetry allows us to obtain the eigenfunctions cnŽ x . of the ‘‘reduced’’ evolution equation w18x
H0 V
393
s
Ž 18 .
5. Conclusion In this paper, I present closed expressions in the ‘‘all order’’ approximation for the DGLAP kernel P Ž z . and ER-BL kernel V Ž x, y . appearing as a result of the resummation of a certain class of QCD diagrams with the renormalon chain insertions. The contributions from these diagrams, P Ž1. Ž z; A. and V Ž1. Ž z; A., give the leading Nf dependence of the kernels for a large number of flavours Nf 4 1. These ‘‘improved’’ kernels are generating functions to obtain contributions to partial kernels like aŽsnq1. PŽ n.Ž z . in any order n of perturbation expansion. Here A ; a s is a new expansion parameter that coincides Žin magnitude. with the anomalous dimension of the gluon field. On the other hand, the method of calculation suggested in w3x does not depend on the nature of self-energy insertions and does not appeal to the value of the parameters Nf TR , CAr2 or CF associated with different loops. This allows us to obtain contributions from chains with different kinds of self-energy insertions, both quark and gluon Žghost. loops. The prize for this generalization is gauge
394
S.V. MikhailoÕr Physics Letters B 431 (1998) 387–394
dependence of the final results for P Ž1. Ž z; A. and V Ž1. Ž z; A. on the gauge parameter j . The result for the DGLAP non-singlet kernel P Ž1. Ž z; AŽ j ., j . is presented in Ž5. in the covariant j-gauge. The analytic properties of this kernel in the variable a s are discussed for quark bubble chains only, and in the general case for two values of the gauge parameter j s 0;y 3. The insufficiency of the NNA proposition for the kernel calculation is demonstrated by the evident calculation in the third order in a s Žsee Table 1.. The contribution V Ž1. Ž x, y; AŽ j ., j . to the nonforward ER-BL kernel Ž14. is obtained for the same classes of diagrams as a ‘‘byproduct’’ of the previous technique w18x. A partial solution Ž20. to the ER-BL equation is derived.
Acknowledgements The author is grateful to Dr. M. Beneke, Dr. A. Grozin, Dr. D. Mueller, Dr. A. Kataev, Dr. M. Polyakov and Dr. N. Stefanis for fruitful discussions of the results, to Dr. A. Bakulev for help in calculation and to Dr. R. Ruskov for careful reading of the manuscript and useful remarks. This investigation has been supported in part by the Russian Foundation for Basic Research ŽRFBR. 98-02-16923 and INTAS 2058.
w2x
w3x w4x w5x w6x
w7x w8x w9x w10x
w11x w12x w13x w14x w15x
w16x w17x
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