Higher-order threshold resummation for semi-inclusive e+e− annihilation

Physics Letters B 680 (2009) 239–246

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Physics Letters B www.elsevier.com/locate/physletb

Higher-order threshold resummation for semi-inclusive e +e − annihilation S. Moch a , A. Vogt b,∗ a b

Deutsches Elektronensynchrotron DESY, Platanenallee 6, D-15738 Zeuthen, Germany Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, United Kingdom

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 24 August 2009 Received in revised form 31 August 2009 Accepted 3 September 2009 Available online 6 September 2009 Editor: A. Ringwald

The complete soft-enhanced and virtual-gluon contributions are derived for the quark coefficient functions in semi-inclusive e + e − annihilation to the third order in massless perturbative QCD. These terms enable us to extend the soft-gluon resummation for the fragmentation functions by two orders to the next-to-next-to-next-to-leading logarithmic (N3 LL) accuracy. The resummation exponent is found to be the same as for the structure functions in inclusive deep-inelastic scattering. This finding, together with known results on the higher-order quark form factor, facilitates the determination of all soft and virtual contributions of the fourth-order difference of the coefficient functions for these two processes. Unlike the previous (N2 LL) order in the exponentiation, the numerical effect of the N3 LL contributions turns out to be negligible at LEP energies. © 2009 Elsevier B.V. All rights reserved.

Semi-inclusive e + e − annihilation (SIA) via a virtual photon or Z -boson, e + e − → γ / Z → h + X , is a classic process probing Quantum Chromodynamics (QCD), √the theory of the strong interaction. A wealth of precise measurements have been performed, at various centerof-mass (CM) energies s, of the total fragmentation function

1 dσ h

σtot dx

  = F h x, Q 2 ,

(1)

where h stands for a specific hadron species or the sum over all (charged) light hadrons, see Ref. [1] for a general overview. In the CM frame the scaling variable x is the fraction of the beam energy carried by the hadron h, and Q 2 = s is the square of the four-momentum q of the intermediate gauge boson. In perturbative QCD, the total (angle-integrated) fragmentation function F hI ≡ F h , as well as the transverse (F T ), longitudinal (F L ) and asymmetric (F A ) fragmentation functions for the double-differential cross section dσ h /dx d cos θh [2], are given by

F ah



x, Q

2



=

1   dz f=q,¯q,g x

z





C a,f z, αs Q

2



 D hf

x z

 ,Q

2

 +O

1 Q

 .

(2)

Here D hf are the parton fragmentation functions, the final-state (timelike, Q 2 = q2 ) analogue of the initial-state (spacelike, Q 2 = −q2 ) parton distribution functions in deep-inelastic scattering (DIS). Without loss of information in the present context, the renormalization scale of αs and the factorization scale of D hf have been identified with the physical hard scale Q 2 in Eq. (2). The coefficient functions C a,f are defined via expansions in the strong coupling as ≡ αs /(4π ). SIA, especially the observable (1), is an indispensable ingredient in fit analyses of the universal distributions D hf , for recent studies see Refs. [3–5]. The scale dependence of the fragmentation functions (and, in principle, the ratio F L / F T ) can also be employed to constrain αs , cf. Ref. [6]. Related quantities are relevant to studies of polarization transfer in fragmentation, see Ref. [7]. Here we are interested in the dominant (anti-)quark contributions to F hI , F Th and F hA ,



(1 )

(2 )

(3 )



C a,q (x, αs ) = σew δ(1 − x) + as ca,q (x) + a2s ca,q (x) + a3s ca,q (x) + · · · .

*

Corresponding author. E-mail address: [email protected] (A. Vogt).

0370-2693/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2009.09.001

(3)

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S. Moch, A. Vogt / Physics Letters B 680 (2009) 239–246

The electroweak prefactors σew can be found in Ref. [2]. The first- and second-order coefficient functions have been calculated long ago in Refs. [8] and [9], respectively. More recently the latter results have been confirmed (and some typos corrected) in two independent ways (3) in Refs. [10,11]. The three-loop corrections ca (x) have not been derived so far. The coefficient functions in Eq. (3) include large-x (threshold) double-logarithmic enhancements of the form ans (1 − x)−1 lnk (1 − x) with k = 0, . . . , 2n − 1. Such contributions, which spoil the convergence of the perturbation series at sufficiently large values of x, can be resummed by the soft-gluon exponentiation [12,13]. For the process at hand this resummation has been worked out to next-to-leading logarithmic (NLL) accuracy in Ref. [14]. The inclusion of this resummation has led to improvements in the most recent global fit of fragmentation functions [5]. Hence an extension of the soft-gluon exponentiation for e + e − → γ / Z → h + X to a higher accuracy is not only of theoretical but also of phenomenological interest. In this Letter we employ the analytic continuation approach of Ref. [10] to derive the soft and virtual contributions to the thirdorder coefficient functions in Eq. (3). These results are then used to extend the results of Ref. [14] to the next-to-next-to-next-to-leading logarithmic (N3 LL) accuracy reached before for inclusive deep-inelastic scattering [15] and the total cross sections for lepton-pair and Higgs-boson production in proton–(anti-)proton collisions [16,17]. A substantial intermediate step towards the present extension has been taken before in Ref. [18]. Recently progress on resummation in perturbative QCD has also been achieved in the framework of soft-collinear effective theory (SCET), see, e.g., Ref. [19] for applications of SCET to the massless-parton processes mentioned above. Up to small contributions from higher-order group invariants entering at the third and higher orders, the soft plus virtual contributions are the same for the DIS quark coefficient functions for F 1 , F 2 and F 3 [20,21]. The same holds for the corresponding (in this order) SIA coefficient functions for F T , F I and F A . Hence we will drop the index a from now on, and refer to the former coefficient functions (l) (l) collectively as c S (x), and the latter as c T (x). In this limit the bare (unrenormalized and unfactorized) partonic DIS (spacelike) structure function F bS is given by [22,23]

F Sb



b s,

α Q

2



= δ(1 − x) +



 b l s



α

l =1

Q2

μ2

−lε

F Sb,l

(4)

with

F Sb,1 = 2F1 δ(1 − x) + S1 ,





F Sb,2 = 2F2 + (F1 )2 δ(1 − x) + 2F1 S1 + S2 ,





F Sb,3 = (2F3 + 2F1 F2 )δ(1 − x) + 2F2 + (F1 )2 S1 + 2F1 S2 + S3 .

(5)

Here μ is the scale of dimensional regularization with D = 4 − 2ε , and abs the bare strong coupling. Fl represents the l-loop quark form factor [22–27]. The x-dependence of the real-emission functions Sk is given by the D-dimensional +-distributions

 (−kε )i  1 f kε (x) = (1 − x)−1−kε + = − δ(1 − x) + Di kε i!

(6)

i =0

where we have introduced the abbreviation Dk = [(1 − x)−1 lnk (1 − x)]+ . The transition to the bare SIA (timelike) fragmentation functions F Sb is performed as follows: In Eq. (5) the factors 2Fl are replaced everywhere by 2 Re FlT and all products Fk Fl by |FkT FlT |, where FlT is the complex l-loop timelike form factor which can be obtained from the spacelike Fl by Eq. (3.3) of Ref. [22]. The analytic continuation of the real-emission terms Sk is carried out as discussed in Ref. [10]. In fact, these functions turn out to be the same for the spacelike and timelike cases (this holds only in the present large-x limit, not for the full real emission contributions). Finally the standard renormalization and mass factorization is performed to the third order (3) for the resulting timelike analogue of Eq. (5), yielding the Dk and δ(1 − x) terms of ca (x) in Eq. (3). For the convenience of the reader, we include also the large-x limits of the well-known first- and second-order MS coefficient functions [8,9]. As expected from the above discussion, these and the third-order coefficient function share all non-ζ2 terms with their spacelike (n) (n) counterparts, hence we will present them via the corresponding differences δTS cn = c T − c S . The results read

δTS c 1 (x) = 12ζ2 C F δ(1 − x),





δTS c 2 (x) = 48ζ2 C 2F D1 − 36ζ2 C 2F D0 + (−108 + 24ζ2 )C 2F +

466 3



 − 24ζ2 C A C F −

76 3



(7)

 C F n f ζ2 δ(1 − x),



(8)



3332 536 2 δTS c 3 (x) = 96ζ2 C 3F D3 − 216C 3F + 88C A C 2F − 16C 2F n f ζ2 D2 − (324 + 96ζ2 )C 3F − − 192ζ2 C A C 2F + C n f ζ2 D1 3 3 F   

  10504 1672 + (306 + 216ζ2 − 96ζ3 )C 3F − − 248ζ2 − 480ζ3 C A C 2F + − 32ζ2 C 2F n f ζ2 D0 9 9  

  993 13457 220 108 2 + + 180ζ2 − 936ζ3 + 72ζ22 C 3F − + ζ2 − 1616ζ3 + ζ2 C A C 2F 2 6 3 5     74728 528 2 667 136 + − 196ζ2 − 1056ζ3 + ζ C 2A C F + + ζ2 − 80ζ3 C 2F n f 27 5 2 3 3      23504 16 1624 16 2 − + ζ2 − 96ζ3 C A C F n f + + ζ2 C F n f ζ2 δ(1 − x). (9) 27

3

27

3

S. Moch, A. Vogt / Physics Letters B 680 (2009) 239–246

241

Here C A and C F are the standard group invariants, with C A = 3 and C F = 4/3 in QCD, and n f the number of light flavours. ζk denotes Riemann’s ζ -function. The third-order SIA coefficient functions can be obtained by adding the corresponding DIS results given in Eqs. (4.14)–(4.19) and Appendix B of Ref. [20], see also Eq. (3.8) of Ref. [21]. The first half of Eq. (9) agrees with the result of Ref. [18], the δ(1 − x) contribution in the second half has not been presented before. Below we will need the N-independent parts δTS g 0k ≡ δTS ck ( N )| N 0 of the Mellin transforms of Eqs. (7)–(9) obtained via

1





dx x N −1 − 1 a(x)+

N

a =

(10)

0

together with δ(1 − x) → 1. These contributions are given by (γe is the Euler–Mascheroni constant)

ζ2−1 δTS g 01 = 12C F , (11)     466 76 ζ2−1 δTS g 02 = C A C F − 24ζ2 − C 2F 108 − 48ζ2 − 36γe − 24γe2 − (12) CFn f , 3 3   993 768 2 ζ2−1 δTS g 03 = C 3F + 18ζ2 − 792ζ3 + ζ2 − 306γe + 288γe ζ3 − 162γe2 + 96γe2 ζ2 + 72γe3 + 24γe4 2 5   13457 5024 588 2 10504 1666 2 88 3 2 2 + CACF − + 482ζ2 + ζ3 − ζ + γe − 160γe ζ2 − 480γe ζ3 + γe − 96γe ζ2 + γe 6 3 5 2 9 3 3    74728 528 2 667 272 1672 − 196ζ2 − 1056ζ3 + ζ + C 2F n f − 44ζ2 − ζ3 − γe + 16γe ζ2 + C 2A C F 27 5 2 3 3 9      268 2 16 3 23504 16 1624 16 − γe − γe + C A C F n f − − ζ2 + 96ζ3 + C F n2f + ζ2 . (13) 3

3

27

3

27

3

The corresponding DIS coefficients can be found in Eqs. (4.6)–(4.8) of Ref. [15]. For processes such as DIS and SIA, the dominant large-x/large-N contributions to the MS coefficient functions C N can be resummed by a single exponential in Mellin space [12]













C N Q 2 = g 0 Q 2 · exp G N Q 2



  + O N −1 lnn N .

(14)

The prefactor g 0 collects, order by order in the strong coupling constant αs , all N-independent contributions. The exponent G N contains terms of the form lnk N to all orders in αs . Besides the physical hard scale Q 2 (= ∓q2 in DIS/SIA, with q the four-momentum of the exchanged gauge boson), both functions depend on the renormalization scale μr and the mass-factorization scale μ f . The exponential in Eq. (14) is build up from universal radiative factors for each initial- and final-state parton p, Δp and J p , together with a process-dependent contribution Δint . The resummation exponents for DIS and SIA [14] take the very similar form N G DIS = ln Δq + ln J q + ln Δint DIS ,

N G SIA = ln Δq + ln J q + ln Δint SIA .

(15)

The radiation factors are given by integrals over functions of the running coupling. Specifically, the effects of collinear soft-gluon radiation off an initial-state or ‘observed’ final-state quark are collected by



2

2 f

ln Δq Q , μ



1 =

dz

z N −1 − 1

0

1−z

(1−z)2 Q 2

dq2   2  A αs q . q2

(16)

μ2f

Collinear emissions from an ‘unobserved’ final-state quark lead to the so-called jet function,



ln J q Q

 2

1 =

dz

z N −1 − 1

0

1−z



2 (1 − z ) Q



   dq2   2  + B αs [1 − z] Q 2 . A αs q q2

(17)

(1 − z )2 Q 2

Finally the process-dependent contributions from large-angle soft gluons are resummed by



1



ln Δint Q 2 =

dz 0

 z N −1 − 1   D αs [1 − z]2 Q 2 . 1−z

(18)

The functions g 0 in Eq. (14) and A, B and D in Eqs. (16)–(18) are given by the expansions

F (αs ) =

 l=l0

Fl

αsl  l ≡ F l as , 4π

(19)

l=l0

where l0 = 0 with g 00 = 1 for F = g 0 , and l0 = 1 else. The known expansion coefficients of the cusp anomalous dimension (the coefficients of D0 ≡ 1/(1 − x)+ in the MS quark–quark splitting functions) read [28,29]

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S. Moch, A. Vogt / Physics Letters B 680 (2009) 239–246

A 1 = 4C F ,



A 2 = 8C F



67 18

  5 − ζ2 C A − n f ,



A 3 = 16C F C 2A

9

245 24

−

67 9

ζ2 +

11 6

11

ζ3 +

5

       55 209 10 7 1 . ζ22 + C F n f − + 2ζ3 + C A n f − + ζ2 − ζ3 + n2f − 24

108

9

3

27

(20)

The first three coefficients of the jet function (17) are given by [12,15,30]

B 1 = −3C F ,



B 2 = C 2F −



3 2





+ 12ζ2 − 24ζ3 + C F C A −

3155 54

+

44 3





ζ2 + 40ζ3 + C F n f

 247 8 − ζ2 , 27

3

   288 2 712 272 2 − 18ζ2 − 68ζ3 − ζ2 + 32ζ2 ζ3 + 240ζ5 + C A C 2F −46 + 287ζ2 − ζ3 − ζ2 − 16ζ2 ζ3 − 120ζ5 2 5 3 5     599375 32126 21032 652 176 5501 32 + ζ2 + ζ3 − ζ22 − ζ2 ζ3 − 232ζ5 + C 2F n f − 50ζ2 + ζ3 + C 2A C F − 729 81 27 15 3 54 9     8714 232 32 160906 9920 776 208 + ζ2 − ζ3 + C A C F n f − ζ2 − ζ3 + ζ22 . + C F n2f −

B 3 = C 3F −

(21) (22)

29

729

27

27

729

81

9

15

(23)

Together with Eqs. (11)–(13), all functions but D in Eqs. (16)–(18) are known to order αs3 . Consequently the first three coefficients of D SIA can by determined by comparing the αs -expansion of Eq. (14) with the fixed-order results (7)–(9). This procedure yields

D kSIA = 0

(24)

3 for k = 1, 2, 3, hence Δint SIA = 1 to at least N LL accuracy. D 1 = 0 was, of course, included in the NLL resummation of Ref. [14]. However, B 2 was unknown at that time, and only B 2 + D 2 could be extracted from the two-loop results of Ref. [9] alone. As expected from the identity of the DIS and SIA soft-emission functions Sk in Eq. (5), there is a strong similarity between the respective coefficient functions also in the framework of the soft-gluon exponentiation — recall that

D kDIS = 0,

Δint DIS = 1

(25)

was proven to all orders in αs in Refs. [31,32]. We expect that such a proof can also be derived for SIA. For the time being assuming the all-order validity of Eq. (24), the difference between the SIA (timelike, T) and DIS (spacelike, S) large-N coefficient functions exponentiates as

         δTS C N Q 2 = δTS g 0 Q 2 · exp G N Q 2 + O N −1 lnn N

(26)

where, after performing the integrations in Eqs. (16)–(18), the function G N takes the form





G N Q 2 = ln N · g 1 (λ) + g 2 (λ) + as g 3 (λ) + a2s g 4 (λ) + · · ·

(27)

with λ = β0 as ln N. The first three expansion coefficients of δTS g 0 for μr = μ f = Q have been given above in Eqs. (11)–(13). We will address the fourth-order coefficient below. The functions g 1 to g 4 have been derived in Refs. [12,15,33,34]. For completeness we include these functions, also here restricting ourselves to choice μr = μ f = Q of the scales:





g 1DIS (λ) = A 1 1 − ln(1 − λ) + λ−1 ln(1 − λ) ,





1

g 2DIS (λ) = ( A 1 β1 − A 2 ) λ + ln(1 − λ) + g 3DIS (λ) =

and

2

(28)

A 1 β1 ln2 (1 − λ) − ( A 1 γe − B 1 ) ln(1 − λ),

    ln(1 − λ) 1 ln2 (1 − λ) + A 1 β12 + A 1 β2 − A 1 β12 ln(1 − λ) + 2 1−λ 1−λ 2 1−λ   1 ln(1 − λ) + ( A 1 β1 γ e + A 2 β1 − B 1 β1 ) 1 − − 1−λ 1−λ     1  2 1 , − A 1 β 2 + A 1 γ e + ζ2 + A 2 γ e − B 1 γ e − B 2 1 − (30) 2 1−λ 1



A 1 β2 − A 1 β12 + A 2 β1 − A 3



1+λ−

1



(29)

S. Moch, A. Vogt / Physics Letters B 680 (2009) 239–246

243

 ln2 (1 − λ) 1 ln3 (1 − λ) 1 g 4DIS (λ) = − A 1 β13 + A 1 β12 γe + A 2 β12 − B 1 β12 2 6 2 (1 − λ) (1 − λ)2    ln(1 − λ) 1 + A 1 β13 − A 1 β1 β2 − A 1 β1 γe2 + ζ2 + A 2 β12 − 2 A 2 β1 γe − A 3 β1 + 2B 1 β1 γe + 2B 2 β1 2 (1 − λ)2

   ln(1 − λ)  1 1 − A 1 β13 − A 1 β1 β2 A 1 β13 − A 1 β1 β2 + A 1 β3 ln(1 − λ) + 1−λ 2 2    1  1 1 1 − + A 1 β13 − A 1 β1 β2 − A 1 β12 γe + A 1 β2 γe − A 2 β12 + A 2 β2 + B 1 β12 − B 1 β2 + 2 1−λ 2 (1 − λ)2   1 1 1 1 1  + A 1 β13 − A 1 β1 β2 − A 1 β3 − A 1 3γe ζ2 + γe3 + 2ζ3 2

3

6





+ A 2 β1 γe − A 2 γe2 + ζ2 −

6 5 6

  + B 1 γe2 + ζ2 + 2B 2 γe + B 3

A 2 β12 +



1−

3 1 3

5

A 2 β2 + 1

6



(1 − λ)2

+

A 3 β1 − A 3 γ e − 1 3

1 3

A 4 − B 2 β1



A 1 β13 − 2 A 1 β1 β2 + A 1 β3 + A 2 β2 − A 2 β12 + A 3 β1 − A 4 λ.

(31)

Factors of β0 = 11/3C A − 2/3n f have been suppressed in Eqs. (28)–(31) for brevity. The dependence on β0 is recovered by A k → A k /β0k ,

B k → B k /β0k , βk → βk /β0k+1 and multiplication of g 3 and g 4 by β0 and β02 , respectively. Note that Eq. (31) includes all known coefficients of the beta function of QCD, see Ref. [35] and references therein. All parameters entering Eqs. (28)–(31) are known except for the four-loop cusp anomalous dimension A 4 . The small (see below) impact of this quantity — which first occurs in the αs5 ln3 N contribution to δTS C N — can be included by a Padé estimate as in Ref. [15], backed up by a recent calculation of one Mellin moment of the fourth-order quark–quark splitting function [36], cf. also Ref. [37]. E.g., for n f = 5 one may use A 4 ≈ 1550 (recall our small expansion parameter as = αs /(4π )) and assign a conservative uncertainly of 50% to this value. Due to the vanishing of δTS g 00 the two highest logarithms, αsl ln2l N and αsl ln2l−1 N, are the same for the SIA and DIS structure functions to all orders in αs . The expansion of Eq. (26) with Eqs. (11)–(13) provides the six highest logarithms, cf. Ref. [15], of the coefficient-function difference δTS C N , αsl ln2l−a N with a = 2, . . . , 7, at all orders from the fourth. In particular, all ln N enhanced terms are thus fixed at order αs4 . After transformation to x-space these contributions read

δTS c 4 (x) =

96ζ2 C 4F D5

880

243

3

+

3

9

C A C 3F

160





ζ2 D4 − (432 + 576ζ2 )C 4F − (3552 − 576ζ2 )C A C 3F + 576C 3F n f 

1936 2 2 704 64 2 2 − CACF + C A C 2F n f − C F n f ζ2 D3 + (1674 + 2160ζ2 + 192ζ3 )C 4F 9 9 9     25238 4100 − − 2800ζ2 − 2880ζ3 C A C 3F + − 352ζ2 C 3F n f 3 3      9616 3248 256 2 2 − − 528ζ2 C 2A C 2F + − 96ζ2 C A C 2F n f − C F n f ζ2 D2 3 3 3  

  1248 2 22916 23120 4368 2 4 + 1122 + 936ζ2 − 4320ζ3 − ζ CF − + ζ2 − 3584ζ3 − ζ C A C 3F 5 2 3 3 5 2     488 4592 224230 17176 5184 2 + + ζ2 + 64ζ3 C 3F n f + − ζ2 − 7392ζ3 + ζ2 C 2A C 2F 3 3 9 3 5      69728 3056 4888 64 − − ζ2 − 576ζ3 C A C 2F n f + − ζ2 C 2F n2f ζ2 D1 9 3 9 3 

 3003 − + 3312ζ2 − 3288ζ3 + 792ζ22 + 192ζ2 ζ3 − 5184ζ5 C 4F 2   24507 78428 2 − + ζ2 − 8816ζ3 − 1452ζ2 − 1728ζ2 ζ3 − 1440ζ5 C A C 3F 2 9   6620501 243752 168560 5952 2 + − ζ2 − ζ3 + ζ2 + 1664ζ2 ζ3 + 2784ζ5 C 2A C 2F 243 27 9 5     3551 13568 688 1983208 66392 1152 2 + + ζ2 + ζ3 C 3F n f − − ζ2 − 2336ζ3 + ζ2 C A C 2F n f 9 9 3 243 27 5    135020 464 128 + − ζ2 + ζ3 C 2F n2f ζ2 D0 + · · · . −

360C 4F

−

3

C 3F n f

(32)

The first four terms correspond to a NNLO + NLL accuracy as first obtained for DIS in Ref. [38]. For the present case these terms have been presented, in a different notation, already in Ref. [18]. The coefficients of D1 and D0 (recall the definition below Eq. (6)) are new results of the present study. The latter coefficient depends on our assumption that Eq. (24) extends to k = 4.

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S. Moch, A. Vogt / Physics Letters B 680 (2009) 239–246

The fourth-order result (32) can be verified, and extended to the δ(1 − x) contribution, in the following manner. Eq. (5) is extended to the fourth order,









F Sb,4 = 2F4 + 2F1 F3 + (F2 )2 δ(1 − x) + (2F3 + 2F1 F2 )S1 + 2F2 + (F1 )2 S2 + 2F1 S3 + S4 ,

(33)

and is subtracted from its timelike counterpart obtained as discussed above. Assuming that also S4 is identical in the two cases, the only unknown in δTS F 4b to order ε 0 is the four-loop anomalous dimension A 4 . All other unknown quantities, such as the ε 1 and ε 2 contributions to the spacelike three-loop form factor [23,26,27] (also the latter new result is not needed in the present context), drop out in this difference. Also the four-loop form factor is known from its exponentiation [39] to a sufficient accuracy in ε [23]. The soft and virtual contributions to δTS c 4 are then extracted from the fourth-order mass factorization formula (here given in terms of the bare coupling)

    1 4 7 2 1 11 1 δTS F 4b = δTS c 4 + [β2 − P 2 ]δTS a1 + β0 β1 − P 1 β0 − P 0 β1 + P 0 P 1 δTS b1 + β03 − P 0 β02 + P 02 β0 − P 03 δTS d1 3 3 6 3 2 6 6     1 5 1 + β1 − P 1 δTS a2 + 3β02 − P 0 β0 + P 02 δTS b2 + [3β0 − P 0 ]δTS a3 + ε -terms. 2

2

2

(34)

For brevity we have suppressed the ε −3 , . . . , ε −1 terms which form a consistency check but do not provide new information. The functions an , bn and dn are the ε 1 , ε 2 and ε 3 contributions, respectively, to the D-dimensional coefficient functions at order αsn , cf. Ref. [21], and P n denotes the Nn LO quark–quark splitting functions. In x-space obviously all products of these functions in Eq. (34) have to be read as Mellin-convolutions. The determination of δTS c 4 from Eqs. (33) and (34) reproduces the result in Eq. (32) — hence D kSIA = D kDIS (= 0) in Eq. (18) corresponds to δTS Sk = 0 in Eqs. (5), (33) and their higher-order generalizations — and includes the final large-x coefficient,

   7255 13896 2 31856 3 ζ2−1 δTS c 4 δ(1−x) = − − 3779ζ2 − 3816ζ3 − ζ2 + 4080ζ2 ζ3 + 14880ζ5 + ζ2 − 1216ζ32 C 4F 2 5 105   191411 153802 62452 2 8128 67328 102472 3 2 + + ζ2 − 42808ζ3 + ζ2 + ζ2 ζ3 − ζ5 − ζ + 4064ζ3 C 3F C A 12 9 9 3 3 105 2   14817221 63347 1856680 5306 2 2584 3 + − − ζ2 + ζ3 + ζ2 − 2032ζ2 ζ3 + 6256ζ5 + ζ2 − 992ζ32 C 2F C 2A 324 3 27 45 21   13294462 206162 416032 + + ζ2 − ζ3 − 1100ζ22 + 1936ζ2 ζ3 + 8976ζ5 C F C 3A 243 27 9   409 23350 55592 2 2272 6272 + − ζ2 + 6840ζ3 − ζ2 − ζ2 ζ3 + ζ5 C 3F n f 6 9 45 3 3   706405 187834 416384 6932 2 + + ζ2 − ζ3 + ζ + 320ζ2 ζ3 − 1408ζ5 C 2F C A n f 81 27 27 45 2   2109553 106168 127000 1088 2 − + ζ2 − ζ3 + 352ζ2 ζ3 − ζ2 + 1632ζ5 C F C 2A n f 81 27 9 5     3233 14824 20656 2464 2 305917 17504 8336 16 − + ζ2 − ζ3 + ζ2 C 2F n2f + + ζ2 − ζ3 − ζ22 C F C A n2f  −

81

27

39352

304

243

+

9

27

ζ2 +

64 9

45





81

ζ3 C F n3f + 768 + 1920ζ2 + 896ζ3 −

+ 3 A4.

27

384 5

9



ζ22 − 5120ζ5 f l11 C F

5

dabc dabc nc (35)

See Ref. [20] for the f l11 diagram class leading to the term with dabc dabc /nc = 5/18n f in QCD. The numerical effect of this contribution is very small and will be disregarded in the following. The Mellin transform of these equations provides the αs4 prefactor δTS g 04 in Eq. (26), and hence (up to the residual uncertainty due to A 4 ) the seventh tower of large-x logarithms from order αs5 for this difference. For n f = 5 quark flavours, the numerical expansion of δTS g 0 is given by

  δTS g 0 (αs )  2.094αs 1 + 1.463αs + 2.749αs2 + {6.659 + 0.094 A 4 /1000}αs3 + · · · .

(36)

Thus the two new terms form a correction of almost 5% at αs = 0.12, with a negligible uncertainty from the missing exact value of A 4 , and the fourth-order contribution is less than half of the previous term for αs < 0.2. It is well known that the coefficients in Eq. (36) are due to ζ2 -terms (i.e., powers of π 2 ) from the analytic continuation of the form factor which are subject to a separate exponentiation (see, e.g., Ref. [39]). The corresponding results for the SIA and DIS cases read

g T,0 (αs ) = 1 + 1.045αs + 2.266αs2 + 4.703αs3 + · · · , g S,0 (αs ) = 1 − 1.050αs − 0.797αs2 − 1.056αs3 + · · · . The pattern of the corrections in Eq. (37) and the size of the g T,0 amounts to less than 0.5% for αs = 0.12.

(37)

α

4 s -term

in Eq. (36) strongly suggests that the fourth-order contribution to

S. Moch, A. Vogt / Physics Letters B 680 (2009) 239–246

245

Table 1 N Numerical values of the five-flavour coefficients cka of the aks ln2k−a+1 N contributions to the coefficient function C SIA . The first six columns are exact up to the numerical truncation, and the same for F I , F T and F A . The seventh column neglects the tiny (and non-universal) f l11 contributions, and uses the estimate A 4 = 1550 for the four-loop cusp anomalous dimension. k

ck1

ck2

ck3

ck4

ck5

ck6

ck7 /10

1 2 3 4 5 6 7 8 9 10

2.66667 3.55556 3.16049 2.10700 1.12373 0.49944 0.19026 0.06342 0.01879 0.00501

7.0785 25.6908 43.3408 46.6020 36.4525 22.3131 11.1933 4.7503 1.7455 0.5652

– 105.621 309.335 514.068 577.143 481.110 315.972 170.251 77.500 30.470

–

– –

– –

2306.0 11774.1 32365.2 55037.7 65426.2 58765.0 41980.1 24725.4

2090 23741 110255 293931 506294 618949 574684 425171

– – –

104.34 1016.50 3125.96 5393.82 6314.54 5515.83 3808.07 2160.26 1035.7

4664 29009 119399 294105 487117 589591 551698

Fig. 1. Left: the LL, NLL, N2 LL and N3 LL results for the threshold resummation (14) of the SIA coefficient functions (3) in N-space. Terms to order αsn are included in g T,0 for the Nn LL curves. Right: the convolutions of these results with a schematic large-x shape for the quark fragmentation functions, using the standard ‘minimal prescription’ contour [13] for the Mellin inversion.

The coefficients of the known lnk N terms are given in Table 1 to the tenth order in αs , using the notation cka for the coefficient of N aks ln2k−a+1 N in C SIA . Hence, as in Ref. [15] for the DIS case, the coefficients of the leading (next-to-leading, etc.) logarithms are denoted by ck1 (ck2 , etc.). The qualitative pattern of these coefficients is similar to the DIS case (where all numbers ck,a>2 are smaller). The higherorder coefficients rise very rapidly, by about an order of magnitude or more, with a until a = k − θk4 without showing the larger-a turnover of the DIS coefficients, cf. Table 1 of Ref. [15]. Indeed, the coefficient for the two cases are very similar for a k, but the SIA coefficient are more than double their DIS counterparts at a > k where the numbers are large. Consequently the higher-order soft plus virtual contributions are qualitatively similar, but larger in the timelike case. The numerical size of its resummed coefficient function (14) is illustrated in Fig. 1 for a value of αs corresponding to LEP1, s = M 2Z . Obviously the size of the coefficient function, as well as the relative impact of the new N2 LL and N3 LL corrections, increases towards lower CM energies. Nevertheless one can conclude from Fig. 1 that the accuracy now reached for the dominant large-x/large-N contributions should be sufficient for the foreseeable future. To summarize, we have first employed the close relation between the perturbative corrections to the structure functions in deepinelastic scattering (DIS) and the fragmentation functions in semi-inclusive e + e − annihilation (SIA), see also Ref. [40], to derive the complete soft and virtual corrections to the third-order quark coefficient functions for the latter observables. This result then made it possible to extend the soft-gluon exponentiation in SIA from the next-to-leading logarithmic (NLL) contributions [14] by two orders to N3 LL accuracy (we confirm the intermediate results in Ref. [18]). It turns out that the resummation exponents are the same, presumably to all orders, for the DIS and SIA coefficient functions. Hence the threshold enhancement is structurally identical in the two cases, and the same thus holds for the class of large-x 1/ Q 2 power corrections associated with the renormalon ambiguity of its perturbation series [32,41]. The N3 LL exponentiation fixes the seven highest large-x logarithms at the fourth and all higher orders in αs . The especially simple connection between the soft and virtual contributions to the DIS and SIA coefficient functions also facilitates a full N3 LL resummation of the SIA–DIS difference, including the next-to-next-to-next-to-leading order αs4 δ(1 − x) contribution to this difference. Since the prefactor of the resummation exponential is larger in SIA than in DIS, the soft-gluon enhancement is numerically larger in the former case. However, while the N2 LL contributions are still significant at LEP energies, the N3 LL corrections are practically negligible, indicating that a sufficient perturbative accuracy in the large-x limit has been reached with the present results.

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