View metadata,citationandsimilarpapersatcore.ac.uk ) ving UV ours. HEP-PH-9505224 v -ratio and the quark a R f N e expansion. Remark- + e f N = ts. The ultra-violet ( heme dep endence of these par- ) QCD with N -function, the e co ecien D ), for SU( f ) p erformed. Resummed results are also ob- 1 N t in comparisons with the exact next-to-leading IR ts to all orders in the 1 {2 N (11 6 . The renormalization sc  R tities related to the y ratio, alue regulated for wski quan o y including more exact details of the Borel transform are considered. ) renormalon singularities in the Borel transform are isolated and the IR ts -lepton deca  for e note that this is already apparen ,w e co ecien analogous tial resummations is discussed and[1{3] they and are with compared exact with xed the order results from p erturbation other theory groups at NNLO. Prosp ects for impro the resummation b tained for the Mink and infra-red ( Borel sum (principal v order (NLO) and next-to-NLO (NNLO) p erturbativ ably These terms corresp ond toasymptotically renormalon to singularities dominate in the the co ecien Borel plane and are exp ected DTP/95/36 April 1995 ell provided byCERNDocumentServer ables brought toyouby Resummations -function and Deep Inelastic Scattering D CORE Observ ory, University of Durham urner and C.J.Maxw 1 = b QCD ett-T v ad, Durham, DH1 3LE, England o er of w Renormalon e for Particle The C.N.Lo some results for the QCD Adler South R f N Centr Exact large- taining the highest p o All-Orders con sum rules are used to resum to all orders the p ortion of QCD p erturbativ

1 Intro duction

In several recent pap ers [1{3] the p ossibility of resummation to all orders of the part of

p erturbative corrections contributed by QCD renormalons has b een explored.

The QCD p erturbative corrections to some generic QCD Green's function or current

correlator (to the Adler D -function of QCD vacuum p olarization for instance) can b e

written:

2 3 k+1

D = a + d a + d a +


+ d a +


; (1)

1 2 k

where a  = is the renormalization group (RG) improved coupling; and the p erturba-

s

tive co ecients d can themselves b e written as p olynomials of degree k in the number

k

of quark avours, N ;we shall assume massless quarks.

f

[k ] [k 1] [0]

k k 1

d = d N + d N +


+ d : (2)

k

f

k k f k

[k r ]

The \N -expansion" co ecients, d , will consist of sums of multinomials in the ad-

f

k

2

joint and fundamental Casimirs, C =N , C =(N {1)=2N , of SU(N ) QCD; and will have

A F

k r s

s

the structure C C . The terms in this \N -expansion" will corresp ond to Feynman

f

A F

diagrams with di ering numb ers of vacuum p olarization lo ops. By explicit evaluation of

[k ]

diagrams with chains of such lo ops inserted, it has b een p ossible to obtain the leading d

k

co ecient exactly to all orders for the Adler D -function [4{6] (and hence its Minkowski

+

continuations, the e e QCD R-ratio and the  -decay ratio, R ); the Gross Llewellyn-



Smith (GLS) sum rule corrections [6]; and heavy quark decay widths and p ole masses

[k ]

[7]. A general pro cedure enabling d to b e obtained from knowledge of the one-lo op

k

correction with a ctitious gluon mass has b een develop ed [1, 2].

In a recent pap er [8]we p ointed out that the large order b ehaviour of p erturbative

co ecients is most transparently discussed in terms of an expansion of the p erturbative

co ecients in p owers of b=(11C {2N )=6, the rst QCD b eta-function co ecient:

A f

(k ) (k 1) (0)

k k 1

d = d b + d b +


+ d : (3)

k

k k k

11

This \b-expansion" is uniquely obtained by substituting N =(

C 3b) in equation (2).

f

A

2

[k ] (k ) [k ]

k

d =(1=3) d and so exact knowledge of the leading-N d to all orders implies exact

f

k k k

(k )

knowledge of the d .

k

In QCD one exp ects the large-order growth of p erturbative co ecients to b e driven by

Borel plane singularities at z =z =2`=b, with `=1; 2; 3;:::. The singularities on the

`

negative real axis are the so-called ultra-violet renormalons, UV , and those on the p ositive

`

real axis are the infra-red renormalons, IR . These singularities result in the large-order

`

k

b ehaviour of the co ecients d  b k !. Indeed we showed in reference [8] that, given a set

k

of renormalon singularities at the exp ected p ositions in the Borel plane, the leading terms

(k )

k

in the b-expansion, d b , should, if expanded in p owers of N , asymptotically repro duce

f

k

[k r ]

the d co ecients of equation (2) up to O (1=k ) accuracy.We conversely checked that

k

(k )

k

the exact d b results corresp onded to a set of renormalon singularities at the exp ected

k

p ositions. We shall demonstrate in section 2 of the present pap er that, for the Adler

D -function and the GLS sum rule, the N -expansion co ecients obtained by expanding

f

(1) (2)

2 3

d b and d b are in go o d (10{20% level) agreement with those of the exact O (a ) next-

1 2

to-next-to-leading order (NNLO) p erturbative calculations for these quantities, so that 2

the anticipated asymptotic dominance of the leading-b term is already apparentinlow

orders.

Given the dominance of the leading-b terms, an obvious prop osal is to sum them to

all orders. That is to split D into two comp onents:

(L) (NL)

D = D + D (4)

where `L' and `NL' sup erscripts refer to leading and non-leading terms in the b-expansion.

1

X

(k )

(L) k k +1

D  d b a (5)

k

k =1

and

1 k 1

X X

(`)

(NL) k +1 `

D  a d b : (6)

k

k=1 `=0

The summation of terms can b e achieved by using the Borel sum. The Borel integral can

itself b e split into two comp onents and is well de ned for the UV singularities on the

`

(L)

negative axis, which contribute p oles to the Borel transform of D . The integral can

b e p erformed explicitly in terms of exp onential integral functions and other elementary

(L)

functions. The piece of the Borel integral for D involving the IR singularities on

`

the p ositive real axis is formally divergent; but a principal value or other prescription

can b e used to go around the p oles. The sp eci cation of this prescription is intimately

linked to the pro cedure needed to combine the non-p erturbativevacuum condensates in

the op erator pro duct expansion (OPE) with the p erturbation theory in order to arriveat

awell-de ned result for D [9].

In recent pap ers by Neub ert [3] and by Ball, Beneke and Braun [1, 2] a summation

of the leading-b terms has also b een considered. In these pap ers it has b een motivated

as a generalisation of the BLM scale xing prescription [10 ] and termed \nave non-

ab elianization" [11]. The Neub ert pro cedure uses weighted integrals over a running cou-

pling. For the Euclidean Adler D -function this representation is equivalent to splitting

the Borel integral into ultra-violet renormalon and infra-red renormalon singularities and

principal value regulating the latter. When one continues to Minkowski space to obtain

+

the e e R-ratio and the  -decay ratio, R , there are several inequivalentways to p erform



the continuation of the running coupling representation; and hence apparent additional

non-p erturbativeambiguities are claimed. In our view only a consideration of the singu-

larities in the Borel integral provides a satisfactory way of combining p erturbative e ects

with non-p erturbative condensates, along the lines discussed in reference [9]; and these

extra uncertainties are spurious.

+

Our intention in this pap er is to fo cus on the Adler D -function, the e e R-ratio, R



and the GLS sum rule (the latter was not considered in references [1{3]). For all of these

quantities there exist exact NNLO xed order p erturbative calculations and our interest is

in comparing the leading-b resummation with these exact xed order results. The large-b

results provide partial information ab out the Borel transform and the question is how

this can b est b e utilised. We discuss the renormalization scheme (RS) dep endence of

(L) (NL) (L)

the split b etween D and D in equation (4), the relative contribution of D b eing

RS-dep endent. This RS uncertainty needs to b e kept in mind and is carefully discussed. 3

The organisation of the pap er is as follows. In section 2 we shall discuss the b-expansion

for the Adler D -function and the GLS sum rule and will consider the extent to which the

dominance of the leading-b term is RS-dep endent. In section 3 the exact large-b results

are used to determine partially the Borel transforms for these quantities; and, having split

them into UV and IR renormalon pieces, a resummation along the lines discussed ab ove

is p erformed.

In section 4 the RS dep endence of the resummed results is considered, the numerical

results are presented and a comparison with the results of reference [3] and with the exact

NNLO p erturbative results is made. A discussion of the uncertainties and the prosp ects

for improving the resummation by including more of the exact structure of the Borel

transform is then undertaken. Section 5 contains overall conclusions.

~ ~

2 The b-expansions for D and K

We b egin by de ning the Adler D -function and the GLS sum rule, the quantities with

whichwe shall b e concerned in this pap er.

2 2

D (Q ) is related to the vacuum p olarization function, the correlator, (Q ), of two

vector currents in the Euclidean region,

Z

4

2 2 2 iq:x

(q q g q )(Q )=16 i d xe h0jTfJ (x)J (0)gj0i ; (7)

    

by

3

d

2

2 2

D (Q )=

Q (Q ) ; (8)

2

4 dQ

2 2

with Q =q > 0.

In p erturbation theory one has

0 1

2

 

X X

3

~

2 2

~ ~

@ A

D (Q )=d(R) Q 1+ D + D: (9) C Q

F f

f

4

f f

Here d(R) is the dimension of the quark representation of the colour group, d(R)=N for

SU(N ) QCD, and Q denotes the quark charges, summed over the accessible avours at a

f

~

3

~

given energy. D denotes corrections of the \light-by-light" typ e which rst enter at O (a )

~

and will b e subleading in N ; they are, it is to b e hop ed, small. D represents the QCD

f

corrections to the zeroth order parton mo del result and has the form

2 3 k+1

~

D = a + d a + d a +


+ d a +


: (10)

1 2 k

We shall next consider two QCD deep inelastic scattering sum rules. The rst is the

p olarized Bjorken sum rule (PBjSR):

Z

1

ep-en

2

K  g (x; Q )dx

PBj

1

0

 

1

g 3

A

~

=

1 C K : (11)

F

3 g 4

V 4

~

Here K denotes the p erturbative corrections to the zeroth order parton mo del sum rule,

2 3 k+1

~

K = a + K a + K a +


+ K a +


: (12)

1 2 k

We can also consider the GLS sum rule,

Z

1

1

 p+ p

2

K  (x; Q )dx F

GLS

3

6

0

 

3

~

~ ~

= 1 C K + K : (13)

F

4

~

The p erturbative corrections, K , are the same as for the PBjSR; but there are additional

~ ~

~ ~

corrections of \light-by-light" typ e, K , analogous to D of equation (9). These will similarly

3

enter at O (a ) and b e subleading in N and we shall assume once again that they are

f

small.

~ ~

For b oth D and K the rst two p erturbative co ecients, d , d and K , K , are known

1 2 1 2

from the exact p erturbative calculations [12{15]. We shall assume

MS renormalization

with renormalization scale =Q for the present; but will later discuss RS dep endence

more generally.

Writing d and d expanded in N as in equation (2), the exact calculations give:

1 2 f

   

11 2 41 11 1

d = +  N + C  C ; (14)

1 3 f A 3 F

12 3 8 3 8

   

151 19 970 224 5

2

d =  N + C +  +  N

2 3 A 3 5 f

f

162 27 81 27 9

   

29 19 10 90445 2737 55

2

+C +   N + C  

F 3 5 f 3 5

A

96 6 3 2592 108 18

   

127 143 55 23

2

+C C  +  + C : (15)

A F 3 5

F

48 12 3 32

Here  and  are Riemann  -functions. For later comparisons it will b e useful to write

3 5

these results numerically for SU(N ) QCD.

 

:063

d = :115N + :655N + ; (16)

1 f

N

   

:024 :180

2 2

d = :086N + N 1:40N + 2:10N :661 : (17)

2 f

f

2

N N

Expanding d and d in b as in equation (3) gives

1 2

 

11 C C

F A

d = 2 ; (18) b +

1 3

4 12 8

   

151 19 5 5 31

2

d =    b + C b

2 3 3 5 A

18 3 6 3 3

   

29 19 799

2

+C  +10  b+C

F 3 5 3

A

32 2 288

   

827 23 11

2

C +C +  + C : (19)

A F 3

F

192 2 32

As observed in reference [8], the b-expansion exhibits certain simpli cations relativeto

that in N . In particular the  , present in all orders of the N -expansion for d , is present

f 3 f 1 5

only in the leading term in the b-expansion. For d the  , present in all but the leading

2 5

(1)

term in the N -expansion, is now present only in d . In b oth cases the highest  -function

f

2

(0)

present cancels and is absent in the `conformal' b ! 0 limit [16]; so d do es not involve

1

(0)

 and d do es not involve  . Since the MS b eta-function co ecients do not involve

3 5

2

 -functions, this is presumably not an artefact of the particular RS chosen but maywell

b e of more fundamental signi cance. It may ultimately b e connected with the fact that

in the b ! 0 limit the z =2`=b U V and IR singularities in the Borel plane moveo to

` `

in nity, leaving only instanton singularities. As discussed in reference [8] the  -functions

are intimately linked with the presence of renormalon singularities. It is amusing to notice

that in the original NNLO result for d [17], whichwas subsequently found to b e in error

2

(0)

20

[13], d do es contain a non-vanishing

 term for SU(3) QCD. If a fundamental result

5

2

9

ab out the absence of  -functions in the conformal limit could b e established it would have

enabled the incorrect result to have b een dismissed at once.

The corresp onding results for the N -expansion for the deep inelastic sum rules are:

f

 

1 23 7

K = N + C C ; (20)

1 f A F

3 12 8

!

   

115 3535  5 133 5

3

2

K = N + N +  C + N +  C

2 f 5 A f 3 F

f

648 1296 2 9 864 18

     

5437 55 1241 11 1

2 2

+C  + C C +  + C : (21)

5 A F 3

A F

648 18 432 9 32

In numerical form, for SU(N ) QCD,

 

:438

K = :333N + 1:48N + ; (22)

1 f

N

   

:244 :008

2 2

K = :177N + N 2:51N + 4:53N + :686 + : (23)

2 f

f

2

N N

Expanding in p owers of b as in equation (3),

 

C 7

A

K = b + C ; (24)

1 F

12 8

     

115 335 3 15 133 5

2

K = b + b +   C + b  C

2 3 5 A 3 F

72 144 2 9 288 6

     

179 11 11 389 1

2 2

+C  +  + C C + C : (25)

3 3 A F

A F

144 4 192 4 32

(0)

Notice that K do es not contain  ; but a similar remark ab out the absence of  in K

1 3 5

2

holds.

Wenow wish to demonstrate that the leading term in the b-expansion, when expanded

in N , approximates the N -expansion co ecients well, even in rather low orders.

f f

For d and d wehave

1 2

(1)

d b = :345b = :115N + :634N; (26)

f

1

(2)

2 2 2 2

d b = :776b = :086N :948N N +2:61N : (27)

f

2

f

2

The subleading, N , N N and N , co ecients approximate well in sign and magnitude

f

2

those in the exact expressions in equations (16) and (17). The leading, N and N ,

f

f

co ecients of course agree exactly. 6

For K and K wehave

1 2

(1)

K b = b = :333N +1:83N; (28)

f

1

(2)

2 2 2 2

K b = 1:59b = :177N 1:95N N +5:37N : (29)

f

2

f

2

The agreement with the exact N , N N and N co ecients in equations (22) and (23) is

f

again rather go o d.

Wenow turn to a consideration of the RS dep endence of the N and b expansions. In

f

variants of minimal subtraction, where the 1= p ole in dimensional regularization is sub-

tracted along with an N -indep endent nite part, K , the QCD p erturbative co ecients

f

will have the form of p olynomials in N as in equation (2). Mo di ed minimal subtrac-

f

tion (

MS), corresp onding to K =(ln 4 { ), with =0:5722 :::, Euler's constant, is most

E E

u

commonly employed. We can consider MS with renormalization scale =e Q, where u

is an N -indep endentnumb er. The most general subtraction pro cedure which will result

f

in p erturbative co ecients p olynomial in N ,however, can b e regarded as MS with scale

f

u+v=b

=e Q, where v is again N -indep endent. We shall refer to such renormalization

f

schemes as `regular' schemes. Of course the renormalization scheme is not sp eci ed by

the scale and subtraction pro cedure alone but by higher order b eta-function co ecients

as well. Anyvariant of minimal subtraction with an N -indep endent renormalization

f

scale will have v =0. Momentum space subtraction (MOM) based on the ggg vertex at a

2 2

symmetric subtraction p oint  =Q [18] corresp onds to u=2:56 and v =C f ( ), where f

A

is a cubic p olynomial in the gauge parameter  .For the Landau gauge,  =0, v =2:49C .

A

For other versions of MOM based on the qqg or ghost vertices, v will involve C and C .

A F

Let us denote the p erturbative co ecients in the

MS scheme with =Q (u=v =0) by

0

d ; and those with general u and v by d . Then

k

k

(1) (0)

0

d = (d + u)b +(d + v )

1 1

1

= d + bu + v: (30)

1

(1)

Changing the value of u shifts only the leading-b co ecient d and so do es not change the

1

(0)

relative contribution of the leading-b term. Changing v ,however, one can make the d

1

co ecient as large as one pleases and hence destroy the dominance of the leading-b term

noted ab ove for the D -function and the sum rules in low orders; although the leading-b

[k r ]

term should still repro duce asymptotically the d co ecients to O (1=k ) accuracy.

k

For d and higher co ecients the sp eci cation of the RS will involve higher b eta-

2

function co ecients as well as the scale and subtraction pro cedure. The RG-improved

2

coupling, a( ), will evolve with renormalization scale according to the b eta-function

equation

da

2 2 k

= ba (1 + ca + c a +


+ c a +


) : (31)

2 k

dln

Here b and c are universal with [19]

1

b = (11C 2N ) ;

A f

6

" #

2

7 C C C 3 11 5

A F

A

c = + : (32) + C C

A F

8 b 8 b 4 4 7

Integrating equation (31) with a suitable choice of b oundary condition [20], one obtains

a transcendental equation for a:

" #

Z

a

 1 1 ca 1

b ln dx = + c ln + + ; (33)

2 2

 a 1+ca x B (x) x (1 + cx)

0

2 3 k

where B (x)=(1 + cx + c x + c x +


+ c x +


). The b eta-function co ecients,

2 3 k



c , c , ::: together with b ln lab el the RS. In a xed order p erturbative calculation one

2 3



would truncate the b eta-function. For the all-orders resummations of the next section,

however, one requires an all-orders de nition of the coupling. In the MS scheme the higher

MS MS MS

b eta-function co ecients, c , c , :::, presumably exhibit factorial growth, c  k !;

2 3 k

and the `a' coupling in the Borel integral would not b e de ned, since B (x)would itself

need to b e de ned by a Borel integral or other summation. One therefore needs to use

a nite scheme [21] where B (x) has a nite radius of convergence and can b e summed.

An extreme example is the so-called 't Ho oft scheme [21 ] where c =c =


=c =


=0,

2 3 k

B (x)=1 + cx. This results in the all-orders de nition of the coupling,



1 ca

b ln

= + c ln : (34)

 a 1+ca

In such a nite scheme, where c , c , ::: are N -indep endent, the b-expansion of equa-

2 3 f

(1)

tion (3) will contain an extra d =b term (for k>1) and the d will strictly no longer b e

k

k

p olynomials in N . bd ,however, is a p olynomial in N of degree k + 1. The leading-b co-

f k f

(k )

ecients, d , in regular schemes are indep endentof c ,c ,:::, since these b eta-function

2 3

k

co ecients, c , are O (1=N ) relativeto d .

k f k

3 Leading-b Resummations

Let us b egin by recalling the de nition of the Borel transform. If D has a series expansion

in `a' as in equation (1), we can write

Z

1

z=a

D = dz e B [D](z ) ; (35)

0

where B [D ](z ) is the Borel transform of D , de ned by

1

m

X

z d

m

B [D ](z )=

: (36)

m!

m=0

~ ~

For the Euclidean quantities, D and K , de ned earlier we shall deduce from the exact

5=6

large-N results that, in the MS scheme with =e Q, B [D ](z ) is of the form

f

1

2

X

A (`)+A (`)z + A (`)z +A (`)z +

0 1 1 2

B [D ](z ) =

 

+

` `

z

`=1

1+

z

`

1

2

X

B (`)+B (`)z + (`)z + (`)z +


B B

0 1 1 2

+ +


; (37)

 

+

`

`

z

`=1

1

z

` 8

where z =2`=b. The two terms corresp ond to a summation over the ultra-violet renor-

`

malons, UV , and infra-red renormalons, IR , resp ectively. A (`), A (`), and B (`),

` ` 0 1 ` 0

B (`), will b e obtained from the large-N results. The barred terms are sub-leading in

1 ` f

5=6

N and remain unknown. The use of the so-called `V-scheme' [10 ], MS with =e Q,

f

means that only the constant and O (z ) terms in the numerator p olynomials are leading

u bz (u+5=6)

in N .For a general MS scale, =e Q,anoverall factor e should multiply the

f

unbarred leading-N terms in the numerator. The presence of this exp onential factor,

f

when it is expanded in p owers of z , can mask the presence of the UV and IR renormalons

in low orders of p erturbation theory.

Notice that, whilst the residue at each renormalon singularity is only known to leading

order in N , the A (`) and B (`) constant terms in the numerator p olynomials are known

f 0 0

P

1

exactly; indeed (A (`)+B (`))=1, as is required to repro duce the unit co ecientof

0 0

`=1

the O (a) term in D in equation (1). Wenow turn to the explicit determination of the

~ ~

co ecients and exp onents for D and K .

~

For D the leading-N terms in the QED Gell-Mann{Low function are generated by[5]

f

!

n2

2n

3

d

[n]

=

; P(x)

n

2 dx

x=1

where

1

k

X

32 (1) k

P (x)= : (38)

2 2 2

3(1 + x) (k x )

k =2

5=6

In the V-scheme, MS with =e Q, one then has

 

n

3

[n+2]

(n)

d = 2 : (39)

n+2

n

2

It is then straightforward to deduce that the co ecients and exp onents in equation (37)

~

for B [D](z ) are

3

`+1

`+1 2

) b(1) (` +

8 (1) (3` +6` +2) 8

2

A (`) = ; A (`)=

0 1

2 2 2 2 2 2

3 ` (`+1) (` +2) 3` (` +1) (` +2)

` = 1;2;3;:::

B (1)=0; B (2)=1; B (`)=A (`) `3

0 0 0 0

B (1)=0; B (2)=0; B (`)=A (`) `3

1 1 1 1

=2 ` =1;2;3;::: ; =1; =2 ` 3 : (40)

` 2 `

So IR is absent, as required from the absence of a dimension two condensate in the

1

OPE [4, 8]. IR is a single p ole. All the other singularities are double p oles. Not only

2

are the co ecients for the UV and IR singularities related by the curious symmetry

` `

B (`)=A (`); but the form of A (`) means that there is an additional relation,

0;1 0;1 0

A (`)=B (` + 2), so that the constant term in the numerator p olynomial for UV exactly

0 0 `

cancels that for IR . This ensures that

`+2

1

X

(`)+B (`)) = B (2)=1; (A

0 0 0

`=1 9

(n)

n d UV UV IR

1

n

0 1 .8148148 .7198242 .28018

1 -.4874471 -.6296296 -.5921448 .10470

2 .8938293 .8518519 .8258924 .06794

3 -1.525257 -1.611111 -1.586113 .06086

4 3.927235 3.888889 3.858335 .06890

5 -11.24973 -11.38889 -11.34378 .09405

6 39.23893 39.16667 39.08871 .15022

7 -154.1541 -154.5833 -154.4291 .27499

8 688.5574 688.3333 687.9894 .56801

9 -3410.339 -3412.500 -3411.647 1.3079

10 18638.50 18637.50 18635.17 3.3243

(n)

~

D , compared with the Table 1: Leading-b co ecients, d , for the Adler D -function,

n

contribution of the rst UV renormalon, `UV ', (equation (41)). The V-scheme, MS with

1

5=6

=e Q, is assumed. `UV ' and `IR' denote the separate sums over the UV and IR

` `

singularities.

which, as noted earlier, is required to repro duce the unit co ecient of the O (a) term in

the p erturbative expansion. The precise origin of these relations b etween UV and IR

renormalons remains unclear and deserves further study. They have also b een noted and

discussed in reference [22].

For the D -function the singularity nearest the origin is UV and from the A (1), A (1)

1 0 1

in equation (40) this should corresp ond to

 

n

12n +22

1

(n)

d j =

n! : (41)

UV

n 1

27 2

(n)

In Table 1 we compare the exact leading-b, d , co ecients with the contribution

n

from UV of equation (41). UV and IR denote the separate sums over the UV and IR

1 ` `

5=6

singularities. The MS scheme with =e Q (V-scheme) is assumed. With this choice

of scheme UV dominates even in low orders and the alternating factorial b ehaviour is

1

apparent.

~

Wenow consider the co ecients and exp onents in equation (37) for B [K ](z ). The

generating function for the leading-b co ecient in the V-scheme is [6]

 

n

n

1

1 d (3 + x)

(n)

K =

: (42)

2

n

x

n

2

3 2 dx

(1 x )(1 )

x=0

4

This results in

4 1 8 5

9 18 9 18

~

B [K ](z )= + : (43)

bz bz bz bz

(1 + (1 + (1 (1 ) ) ) )

2 4 2 4

, UV , IR , IR resp ectively. Eachnumerator and The terms corresp ond to UV

1 2 1 2

exp onent will contain in addition O (1=N ) corrections corresp onding to the barred terms

f 10

(n)

n K UV + IR UV IR

1 1

n

0 1 1.333333 .3888889 .6111111

1 .1666667 .2222222 -.2083333 .3750000

2 .6250000 .6666667 .2152778 .4097222

3 .3125000 .3333333 -.3281250 .6406250

4 1.968750 2.000000 .6614583 1.307292

5 1.640625 1.666667 -1.660156 3.300781

6 14.94141 15.00000 4.990234 9.951172

7 17.43164 17.50000 -17.48291 34.91455

8 209.7949 210.0000 69.96582 139.8291

9 314.6924 315.0000 -314.9231 629.6155

10 4723.846 4725.000 1574.808 3149.039

(n)

~

K , K Table 2: As for Table 1 but for the Deep Inelastic Scattering sum rules, .

n

`UV +IR ' denotes the contribution of the singularities nearest the origin, the rst two

1 1

terms of equation (44).

in equation (37). The constant terms in the numerators sum to 1, again ensuring a unit

~

O (a) co ecientin K.Itwould b e interesting to try to understand the fact that only the

rst two UV and IR renormalons are leading in N in the context of the OPE for the

f

deep inelastic sum rules, a topic discussed in reference [23].

(n)

K is then given by (in the V-scheme)

n

       

n n n n

8

1 4 1 5 1 1 1

(n)

K =

n! + n! n! n! : (44)

n

9 2 9 2 18 4 18 4

(n)

Table 2 shows that K is dominated, even in low orders, by the combined UV + IR

1 1

n

contributions of the two singularities nearest the origin, the rst two terms of equation

(44).

Wenow wish to use the Borel integrals to p erform the leading-b resummation de ned

in equations (5,6). For the Adler D -function wehave

Z

1

1

X

A (`)+A (`)z

0 1

(L) F(a)z

~

D (a) = dze

z

2

(1 + )

0

`=1

z

`

0 1

Z

1

1

X

B (2) B (`)+B (`)z

0 0 1

F (a)z

@ A

+ dz e + : (45)

z z

2

(1 (1 ) )

0

`=3

z z

2

`

The co ecients A , A , B , B are summarised in equation (40). We assume that the

0 1 0 1

resummation has b een p erformed in a nite RS corresp onding to MS subtraction with

u+v=b

=e Q.`a', the coupling, is then de ned by the integrated b eta-function equation of

equation (33). One then nds that the exp onent in equation (45) is

# "

Z

a

Q ca 1 1 5

F (a)=bln : (46) b c ln + v + dx

2 2

 6 1+ca x B (x) x (1 + cx)

0

MS 11

(L)

~

The RS dep endence of D re ects the fact that only a subset of the p erturbation series

has b een resummed, hence violating the exact RS-invariance whichwould apply to the

full series. We shall return to this RS dep endence in a moment.

The rst, UV renormalon, term in equation (45) is a completely well-de ned inte-

gral. It may b e p erformed in terms of the exp onential integral function (with negative

argument),

Z

t

1

e

Ei(x)= dt

: (47)

t

x

The rst term yields

1

i h

X

2 2 F (a)z (L)

`

~

A (`) (A (`) bz A (`)) bz Ei(F (a)z ) F (a)z fe D (a)j =

1 0 ` 1 ` UV

` `

`=1

+z (A (`) bz A (`))g : (48)

` 0 ` 1

Toevaluate the second, IR renormalon, term we shall use a principal value prescription;

corresp ondingly we need to de ne Ei(x) with a p ositive argument as a principal value.

We nd

(L) F (a)z

2

~

D (a)j = e z B (2)Ei(F (a)z )

IR 2 0 2

1

h i

X

F (a)z 2 2

`

+ fe Ei(F (a)z ) F (a)z (B (`)+bz B (`)) bz B (`)

` 0 ` 1 1

` `

`=3

z (B (`)+bz B (`))g : (49)

` 0 ` 1

Finally

(L) (L) (L)

~ ~ ~

D (a)=D (a)j + D (a)j : (50)

UV IR

For the Deep Inelastic Scattering sum rules we will have, analogously, using equation

(43),

!

Z

4

1

1

(L) F (a)z

9 18

~

K (a) = dz e

z z

(1 + (1 + ) )

0

z z

1 2

!

Z

8 5

1

F (a)z

9 18

+ dz e : (51)

z z

(1 (1 ) )

0

z z

1 2

These integrals may b e expressed once again in terms of Ei(x):

 

4 1

(L) F (a)z F (a)z

1 2

~

K (a)j = e e z Ei(F (a)z )+ z Ei(F (a)z ) (52)

UV 1 1 2 2

9 18

and

 

8 5

(L) F (a)z F (a)z

1 2

~

K (a)j = e e z Ei(F (a)z ) z Ei(F (a)z ) : (53)

IR 1 1 2 2

9 18

Similarly

(L) (L) (L)

~ ~ ~

K (a)= K (a)j + K (a)j : (54)

UV IR

Before wenumerically evaluate these results and comment further on RS dep endence,

we shall derive the analogous resummations for the Minkowski continuations of the D -

+

function, the e e annihilation R-ratio and the analogous quantityin-decay, R . The



R-ratio is related to D by a disp ersion relation,

Z

2

s+i

1

D (Q )

2

R(s)=

dQ : (55)

2

2i Q

si 12

Here s is the physical timelike Minkowski squared momentum transfer. A p erturbative

~

result for R of the form of equation (9) can b e written down involving a quantity R

with p erturbative co ecients r . The r are directly related to the d via the disp ersion

k k k

2 2 2

relation (55): r =d , r =d  b =12. The  term arises due to analytical continuation.

1 1 2 2

In the Borel plane one nds (to leading order in N )

f

sin(bz=2)

~

~

B [R](z )=

B [D](z ) : (56)

bz=2

The leading-b resummation is then obtained from equation (45) simply by adding an extra

sin(bz=2)

factor in the integrand.

bz=2

Z

1

1

X

sin(bz=2) A (`)+A (`)z

0 1

F (a)z (L)

~

dz e R (a) =

z

2

bz=2 (1 + )

0

`=1 z

`

0 1

Z

1

1

X

sin(bz=2) B (2) B (`)+B (`)z

0 0 1

F (a)z

@ A

dz e + + : (57)

z z

2

bz=2 (1 (1 ) )

0

z z `=3

2

`

Writing the `sin' as a sum of complex exp onentials and using partial fractions, the UV

integrals can b e explicitly p erformed in terms of the generalised exp onential integral

functions Ei(n; x),

Z

xt

1

e

Ei(n; x)= dt : (58)

n

t

1

One also needs

!

Z

1

sin(bz=2) b

F (a)z

dz e = arctan : (59)

z 2F (a)

0

One nds

! !

2 8 11 b

2

(L)

~

R (a)j = arctan

UV

b 3 3 2F (a)



1

h i

X

A (`)i

0

F z F z

+

` `

+ e Ei(1;F z ) e Ei(1;F z )

` + `

b

`=1



h i

(A (`) A (`)z )i

0 1 `

F z F z

+

` `

+ e Ei(2;F z ) e Ei(2;F z ) ; (60)

` + `

b

i b

where F =F (a)  .



2

By analytically continuing Ei(n; x) from Re x> 0toRex<0, one can obtain the

principal value IR contribution,

! !

2 14 8 b

2

(L)

~

R (a)j = arctan

IR

b 3 3 2F (a)

h i

B (2)i

0

F z F z

+

2 2

+ e Ei(1; F z ) e Ei(1; F z )

2 + 2

b



1

h i

X

(`)i B

0

F z F z

+

` `

+ e Ei(1; F z ) e Ei(1; F z )

` + `

b

`=3



i h

(`)+B (`)z )i (B

0 1 `

F z F z

+

` `

+ :(61) e Ei(2; F z ) e Ei(2; F z )

` + `

b 13

Then

(L) (L) (L)

~ ~ ~

R (a)=R (a)j + R (a)j : (62)

UV IR

The  -decay analogue of the R-ratio, R , can b e de ned in terms of the R-ratio by



the integral representation [24]

Z

2

M



ds

2 2 2

^

R = 2

(1 s=M ) (1 + 2s=M )R(s)



 

2

M

0



 

3

2 2

~

= d(R)(jV j + jV j ) 1+ C R : (63)

ud us F 

4

P

2 2 2

^

Here R (s) denotes R(s) with the Q replaced by jV j + jV j  1, where the V 's are

ud us

f

f

~

KM mixing matrix elements. R has the form



 2  3  k+1

~

R = a + r a + r a +


+ r a +


(64)



1 2 k

It is then straightforward to show that

" #

sin(bz=2)

2 2 1

~

~

B [R ](z )=

+ B [D ](z ) : (65)



bz bz bz

bz=2

(1 (1 (1 ) ) )

2 6 8

Pro ceeding in a manner analogous to that for the R-ratio and de ning

F z F z

+ q q

 (p; q )=e Ei(p; F z ) e Ei(p; F z ) ;

 + q q

we nd

! !

2

8 11 b

2

(L)

~

R (a)j =

arctan

UV



b 3 3 2F (a)

1

X

2i

[(A (`)(G(`)+H(`)) z A (`)G(`)) (1;`)

0 ` 1 +

b

`=1

+H(`)(A (`) z A (`)) (2;`)] (66)

0 ` 1 +

and

! !

2 14 8 b

2

(L)

~

R (a)j = arctan

IR



b 3 3 2F (a)

  

2i 64 32 14 23

 (1; 1) + ln 2 8 + z ln 2

3 1

b 3 3 3 3

  

6i 2i 703 245 32

+  (1; 2) +64ln2+z ln 2  (1; 3)

3

b b 18 36 3

  

2i 1627 128 2035 16

ln 2 + z + ln 2  (1; 4)

4

b 972 81 7776 81

   

2i 11 z 2i 247 5z

3 4

+  (2; 3)  (2; 4)

b 27 6 b 648 162

   

4i 13 11 z 4i 5z

3 4

+ +  (3; 3)  (3; 4)

b 27 18 b 432 1728

1

X

2i

[(B (`)(G(`)+H(`)) + z B (`)G(`)) (1;`)

0 ` 1

b

`=5

(`)+z B (`)) (2;`)] ; (67) +H(`)(B

0 ` 1 14

where

2

6`(3` +16` + 19) 6

G(`)= ; H(`)= :

2 2 2

(` +1) (` +3) (`+4) (`+ 1)(` + 3)(` +4)

Similarly again to the R-ratio,

(L) (L) (L)

~ ~ ~

R (a)=R (a)j + R (a)j : (68)

UV IR

  

4 RS-dep endence of the resummed results

Armed with these resummed expressions, wenow return to the question of the RS dep en-

(L)

dence of D (a) b efore presenting the numerical results.

For a generic quantitywe can write

(L) (NL)

D = D (a)+ D (a) : (69)

`D ' on the left of the equation denotes the full Borel sum, i.e. equation (37) with barred

and unbarred terms included and the IR singularities principal value regulated; and we

assume that this exists. Crucially, the full Borel sum is RS indep endent and so will not

dep end on `a'. The L and NL comp onents do dep end on `a', however. We can consider `a'

varying b etween a=0 and a=+1, lab elling p ossible RS's. From equation (46), as a ! 0

ca

so F (a) ! +1, resulting from the c ln

term, and wehave assumed c>0, whichis

1+ca

(L)

true for N  8 in SU(3) QCD. One then has D (0)=0. Corresp ondingly, from equation

f

(NL)

(69), D (0)=D and so, as a ! 0, the NL comp onent contributes the whole resummed

ca

D . As `a' increases the c ln term in equation (46) decreases and as a !1it

1+ca

vanishes, resulting in a nite limit F (1):

" #

Z

1

Q 1 1 5

F (1)= bln b + v dx + : (70)

2 2

 6 x B (x) x (1 + cx)

0

MS

(L)

We assume that B (x) is such that the integral exists. Thus D (a) increases from

(L) (L) (NL)

D (0)=0 to a nite maximum value, D (1), as `a' increases. Corresp ondingly D (a),

which provides the whole resummed D at a=0, decreases as `a' increases.

(L)

This RS dep endence of D (a) is clearly problematic. It is monotonic and hence there

is no basis for cho osing a particular scheme. There is also a dep endence on the particular

nite scheme, characterised by the choice of B (x), and on the parameter v . The maximum

(L) (NL)

value, D (1), do es p erhaps minimize the relative contribution of the unknown D

(NL)

comp onent but there is no guarantee that D (1) is p ositive; and it is entirely p ossible

(L)

that D (1)overestimates D .

We shall cho ose v =0, corresp onding to a variant of minimal subtraction, a choice

which one can motivate by the observed dominance of the leading-b term in

MS noted in

section 2. For reasons of simplicitywe shall cho ose the 't Ho oft scheme, corresp onding to

B (x)=1 + cx. With these choices one has

Q 5

F (1)=bln b ; (71)

 6

MS 15 0.045

0.040 IR 0.035 UV UV+IR 0.030

0.025 (a) (L) ~

D 0.020

0.015

0.010

0.005

0 0 0.2 0.4 0.6 0.8 1

~a

(L) a

~

Figure 1: The leading-b resummation, D (a), plotted versusa ~ =1 e ('t Ho oft scheme

v =0), Q=M =91GeV and  (N =5) is as in the text. The solid curve is the overall

Z f

MS

(L) (L)

~ ~

result split into D (a)j (dashed) and D (a)j (dashed-dot) contributions.

UV IR

and we shall use this in the resummations. It corresp onds simply to taking `a' as the

5=6

one-lo op coupling in the MS scheme with =e Q (the V-scheme):

1

a = ; (72)

1-lo op

Q

b ln



V

5=6

where  =e  . This is in fact the same choice for a as in references [1{3], where it is

V

MS

2

motivated by noting that using the one-lo op form for a( ) makes the leading-b summation

-indep endent. We stress once again that in our view its signi cance is that it maximizes

(L)

D (a) for a given choice of nite scheme, B (x), and parameter v . In Figure 1 we show

(L)

~

D (a) plotted versus `a' ('t Ho oft scheme with v =0). In the gures wehave plotted

a

versus ~a =1e , so that the full RS variation can b e tted in a unit interval ina ~ .We

have taken Q=M =91GeV and 

(N =5)=111MeV. The solid curve gives the overall

Z

f

MS

(L) (L) (L)

~ ~ ~

D (a), split into D (a)j (dashed) and D (a)j (dashed-dot) contributions. Similar

UV IR

2

(L) 2

~

curves for K (a) with Q =2:5GeV and  (N =3)=177MeV are given in Figure 2; and

f

MS

(L) (L)

~ ~

for R (a) with Q=91GeV in Figure 3. The corresp onding curve for R (a)isgiven in



Figure 4, Q=M =1:78GeV and  (N =3)=177MeV . The qualitative b ehaviour is as we

 f

MS

describ ed earlier. The relative sizes of the UV and IR contributions re ect the disp osition

of the UV and IR singularities describ ed ab ove for the di erent quantities. 16 0.250 IR UV 0.200 UV+IR

0.150 (a) (L) ~ K 0.100

0.050

0 0 0.2 0.4 0.6 0.8 1

~a

2

(L) 2

~

Figure 2: As for Figure 1 but for K (a) with Q =2:5GeV .

The a !1 limits obtained using F (1) as in equation (71) are tabulated in Table 3.

(L) (L) (L)

~ ~ ~

Sp eci cally, the `Resummed' column contains 1 + D (1), 1 + R (1), 1 + R (1) and



(L)

~

1 K (1) for each observable and wehave added some extra energies, Q =20GeV and

0

2

2

Q =2:5GeV . The values of 

for N =5; 3 are as noted ab ove and have b een chosen to

f

1

MS

corresp ond to the values used in reference [3]. In each case wehave taken care to include

sucient terms in the summation over UV and IR singularities to guarantee accuracy

` `

to the quoted numb er of signi cant gures.

In the columns headed `Neub ert' we compare with the results of Table 1 in reference

[3]. As noted, using F (1) is equivalent to the one-lo op de nition of the coupling used in

reference [3]; and so a direct comparison of the results can b e made. Neub ert's running

coupling representation should b e equivalent to our principal value regulated UV=IR

~ ~

Borel transform split for the Euclidean quantities D and K and indeed, notwithstanding

the minor disagreement in the nal signi cant digit of D (M ), such agreement is observed



~ ~

for D (Q ). For the Minkowski quantities R and R Neub ert nds up to three di erent

0 

inequivalentways to continue his representation from the Euclidean region. These are

lab elled `1,2,3' as in reference [3]. Our resummed result in each case lies in the middle of

the range of results he obtains.

There is clearly an ambiguity asso ciated with the IR renormalons or, corresp ondingly,

with vacuum condensates in the OPE. For a leading IR singularity in which there is a

`

z =a

~ ~

`

single p ole one would exp ect an ambiguity in the principal value  B e , where B is the

~

residue of the renormalon. For D the leading IR singularityisIR .For the Minkowski

2 17 0.045

0.040 IR 0.035 UV UV+IR 0.030

0.025 (a) (L) ~

R 0.020

0.015

0.010

0.005

0 0 0.2 0.4 0.6 0.8 1

~a

(L)

~

Figure 3: As for Figure 1 but for R (a) with Q=M =91GeV. Z

0.140

0.120 IR UV 0.100 UV+IR

0.080 (a) τ (L) ~ R 0.060

0.040

0.020

0 0 0.2 0.4 0.6 0.8 1

~a

(L)

~

Figure 4: As for Figure 1 but for R (a) with Q=M =1:78GeV .



 18

Observable Energy Neub ert Resummed FOPT Expt

1 2 3 (NNLO)

D m 1.139 | | 1.138 1.082 |



Q 1.055 | | 1.055 1.045 |

0

m | | | 1.042 1.035 |

Z

R m 1.103 1.101 | 1.102 1.076 |



Q 1.053 1.053 | 1.053 1.044 |

0

m | | | 1.041 1.035 1:040  0:004

Z

R m 1.144 1.101 1.184 1.132 1.102 1:183  0:010

 

K Q | | | .797 .897 :768  0:09

1

Table 3: Comparison of resummed results (`Resummed') of section 3 using F (1), equa-

2

2 2 2

tion (71) (see text). Q =(20GeV) , Q =2:5GeV . The columns headed `Neub ert' compare

0 1

these values with table 1 of reference [3]. `FOPT (NNLO)' gives the exact NNLO p ertur-

bative results with =Q MS scheme and the  values noted in the text. `Expt' gives

MS

the exp erimentally deduced values for some of these quantities [25{27].

sin(bz=2)

~ ~

quantities, R and R , the factor apparently removes the single p ole at z =z

 2

bz=2

but there is presumably still a branch p oint singularityat z=z beyond the leading-N

2 f

~

approximation [4]. The determination of the residue B would require a resummation of

~

the numerator p olynomial and so it is only known to leading-N (for DB (2)=1). Taking

f 0

~

B =1 and putting N =3; 5values for b and `a'values corresp onding to the energies and 

f

MS

4

~ ~ ~

<

considered in Table 3 yields an ambiguity 10 for D , R , R , so the signi cant gures





~

quoted in the resummed result do not change. For K ,however, the leading singularityis

2

2 2

IR and one would estimate the IR ambiguity  10 for N =3 and Q =2:5GeV , which

1 f

is clearly signi cant.

The column lab elled `FOPT' gives the xed order p erturbation theory results ob-

3

tained at NNLO (up to O (a )) using the exact p erturbative co ecients in the MS scheme

with =Q [12{15]. The coupling `a' is de ned using the NNLO truncated b eta-function,

MS 2

B (x)=1 + cx + c x , in equation (33), with the values of  as ab ove.

2

MS

The column lab elled `Expt' gives the values determined from exp erimental data for

~ ~ ~

(1+R ) [25 ], (1+R ) [26 ] and (1{K ) [27 ]. The results of adjusting  to t the FOPT and



MS

resummed predictions to these exp erimental values are summarised in Table 4. Both xed

order p erturbation theory and the leading-b resummed results exhibit RS-dep endence and

it is not at all obvious which pro cedure gives the closest approximation to the all-orders

sum. We will defer a more detailed discussion of this question, including a consideration

of how the resummed results compare with use of the e ectivecharge formalism [28 , 29 ],

until a future work.

To conclude this section let us consider how the leading-b resummation mightbe

improved. An obvious improvementwould b e to include the full branch p oint structure

of the renormalon singularities by incorp orating the subleading in N ,

and , pieces

f

`

` 19

Observable N  /MeV tted to exp erimental data

f

MS

Resummed NNLO FOPT

+93 +226

R 5 98 287

45 145

+65 +25

R 3 404 435



57 28

+20 +15

K 3 375 495

33 16

Table 4: Values of  adjusted to t the predictions of NNLO FOPT and the resummed

MS

~ ~ ~

results to the exp erimental data for (1+R) [25], (1+R ) [26 ] and (1{K ) [27 ].



of the exp onents in equation (37) into the resummations. One exp ects

= cz + ;

` ` `

0

= cz + : (73)

`

`

`

0

The rst term can b e deduced from RG considerations but the and are the one-lo op

`

`

~

anomalous dimensions of the relevant op erators [9, 30]. For D it is known that IR has a

2

0

corresp onding OPE op erator with vanishing one-lo op anomalous dimension, =0 [31], so

2

0

=cz is known. To the b est of our knowledge the remaining and 's are not known.

2 `

2

`

One could nonetheless include the rst RG-predictable terms in equation (73) in the

resummations and see byhowmuch the results change. The problematic RS-dep endence

(L)

of D (a)would b e qualitatively unchanged, however.

Further improvement could b e achieved by including some of the subleading in N ,

f

barred, co ecients in the numerator p olynomials in equation (37). A complete xed

n

order p erturbative calculation for D up to O (a )would enable the series co ecients

n

of B [D ](z )uptoO(z ) to b e determined exactly; but to obtain the co ecients of the

n

numerator p olynomials for each singularityupto O(z ), even given knowledge of the full

branch p oint exp onents discussed ab ove, would still b e very dicult. The improvement

of p erturbation series by developing a representation of the form of equation (37) with

truncated numerator p olynomials was suggested in reference [31].

One further caveat concerns the structure of the UV singularities. It has b een sug-

`

gested in reference [30] that diagrams with more than one renormalon chain will mo dify

the form of the ultraviolet renormalon singularities [8]; this would lead to additional UV

terms in equation (37) with O (z ) or higher, subleading in N ,numerator co ecients but

f

with increasing leading-N exp onents . The presence of such terms would destroy the

f `

asymptotic dominance of the leading-b co ecient to all-orders in N [8], which seemed

f

to b e already evident in the comparisons with the exact NLO and NNLO co ecients 20

discussed in section 2. The motivation for leading-b resummation would hence disapp ear

if this UV structure is correct. Further clari cation of this p ointisobviously required.

5 Conclusions

In this pap er wehaveinvestigated the p ossibility of resummation to all orders of the

leading term in the `b-expansion' of QCD p erturbative co ecients in equation (3). This

expansion was intro duced and motivated in reference [8] by a consideration of renormalon

(k )

k

singularities in the Borel plane; and, if such singularities are present, then the d b term

k

should, when expanded in N , repro duce the N -expansion co ecients of equation (2)

f f

to all orders in N with asymptotic accuracy O (1=k ). Wechecked explicitly in section 2

f

~ ~

that for the QCD Adler D -function (D ) and Deep Inelastic Scattering sum rules (K ) this

asymptotic dominance of the leading-b terms was already evident in comparisons with

the exact NLO and NNLO p erturbative co ecients for those quantities. The interesting

(0)

absence of  -functions from the d co ecient which survives in the b ! 0 limit was also

k

noted. The RS dep endence of the leading-b co ecient and the need to give an all-orders

de nition of the coupling in order to p erform resummations was discussed.

In section 3 we used exact large-N results [4{6] to obtain partial information ab out

f

~ ~

the Borel transforms B [D ](z ) and B [K ](z ). Ultra-violet and infra-red renormalon singu-

larities are present and we obtained the constant co ecients in the numerator p olynomials

exactly; and the exp onents, single and double p oles, to leading-N (equations (40), (43)).

f

5=6 (n)

We showed that in the V-scheme (

MS with =e Q) the leading-b co ecients, d

n

(n)

and K , are dominated, even in low orders, by the renormalon singularities nearest the

n

origin, resp ectively UV and UV +IR combined (see Tables 1 & 2).

1 1 1

For each quantitywe split the leading-b Borel sum into UV and IR p oles. The rst

contribution could b e evaluated exactly in terms of the exp onential integral function

Ei(x) (equation (47)) and elementary functions; and the second, IR, contribution could

b e obtained as a principal value, in terms of a principal value of Ei(x). We showed how

+

~

to mo dify the Borel transform for the Minkowski continuations of D , the e e R-ratio

~ ~

(R ) and the R-ratio for  -decay(R ), and a similar resummation was p erformed for these



Minkowski quantities in terms of a generalised Ei(n; x) function (equation (58)).

(L)

In this waywe obtained the D comp onent of the split de ned in equation (4) for

the ab ove quantities. Unfortunately the result obtained by summing the leading-b terms

(L) (L)

is RS-dep endent, D (a). In section 4 we showed that one can maximize D (a) and

(NL)

hence p erhaps minimize D (a) for any given choice of nite scheme and subtraction

(L)

pro cedure. Maximizing D (a) whilst cho osing the 't Ho oft scheme and a variantof

minimal subtraction (v =0) was equivalent to using the one-lo op coupling in the V-scheme,

the choice also made in references [1{3].

We compared our resummed results with those of reference [3](Table 3) and found

~ ~

agreement for D (K was not discussed in reference [3]). For the Minkowski quantities

~

derived from D the running coupling representation used to resum leading-b terms in [3]

cannot b e uniquely continued to the Minkowski region. Our unique resummed result layin

the middle of the range of values obtained from the inequivalent continuations. We stress

that the Borel sum with IR singularities identi ed and principal value regulated provides 21

a unique result and, in our view, a rm foundation for combining vacuum condensates in

the OPE with IR renormalons to achievea well-de ned overall result (see reference [9]).

4

<

10 for There is, of course, ambiguity due to the IR p oles and we estimated this to b e



~ ~ ~

the D , R , R resummed results in Table 3, so the displayed signi cant gures should b e



2

~

valid; but much larger,  10 , for K due to the presence of an IR singularity.

1

We also compared with xed order p erturbation theory up to NNLO for these quan-

(L)

tities. Since b oth D (a) and the NNLO xed order result su er from RS dep endence it

is unclear which is the more reliable and further investigation of this question is required.

We nally considered how the leading-b resummation might b e improved by including

more exact information ab out the Borel transform.

Acknowledgements

Wewould like to thank David Broadhurst for a numb er of stimulating discussions and

Martin Beneke for useful corresp ondence ab out the curious connections b etween IR and

UV renormalon residues for vacuum p olarization, noted in section 2.

CL-T gratefully acknowledges receipt of a P.P.A.R.C. U.K. Studentship. 22

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