Asymptotic expansion of the lattice scalar propagator in coordinate space

18 February 1999

Physics Letters B 448 Ž1999. 76–84

Asymptotic expansion of the lattice scalar propagator in coordinate space Beatrice Paladini a b

a,1

, James C. Sexton

a,b,2

School of Mathematics, Trinity College Dublin, Dublin 2, Ireland Hitachi Dublin Laboratory, 16, Westland Row, Dublin 2, Ireland Received 19 May 1998; revised 17 November 1998 Editor: P.V. Landshoff

Abstract The asymptotic expansion of the massive scalar field propagator on a n- dimensional lattice is derived. The method used is based on the evaluation of the asymptotic expansion of the modified Bessel function In Ž n 2b . as the order n grows to infinity. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Lattice propagator; Asymptotic methods; Bessel function In Ž n 2b .

1. Introduction In perturbation theory, the advantage of a coordinate space description as a way of studying the divergences arising in Feynman diagrams has been discussed by many authors Že.g. w1,2x.. In the continuum, the analytic expression for the scalar field propagator in position space is well-known. On the lattice, however, the standard representation for the scalar propagator involves integrals over Bessel functions and has proved to be very difficult to analyse in the continuum limit. Several attempts have been made to derive a suitable expansion for the lattice scalar propagator in the limit where the lattice spacing goes to zero. In particular, we mention the analysis performed by Luscher and Weisz for the massless propagator w3x as ¨ 1 2

E-mail: [email protected] E-mail: [email protected]

well as the research carried out by Burgio, Caracciolo and Pelissetto in Ref. w4x. In the first of these studies, the x- dependence of the massless propagator is asymptotically derived. In the second, the m-dependence is obtained for the propagator at x s 0. The propagator at any other point x / 0 is then expressed, through a set of recursion relations, in terms of the value which its massless counterpart assumes on an element of the unit hypercube Ži. e, xm either 0 or 1.. In the present paper we wish to tackle the most general case of both m and x non-vanishing. The procedure that we adopt attacks the question directly at its core. We derive an asymptotic expansion of the modified Bessel function of the first kind which appears in the expression of the lattice propagator. The order and argument of this Bessel function go to infinity at different rates as the lattice spacing decreases to zero. We have found no tabulated asymptotic expansion for this particular case.

0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 1 5 4 9 - 4

B. Paladini, J.C. Sexton r Physics Letters B 448 (1999) 76–84

Expansions for the modified Bessel function of the first kind, In Ž x ., are, indeed, available for the cases where either the argument x or the order n becomes large, or for the case where both x and n grow to infinity keeping the value of their respective ratio constant Žapproximation by tangents. w11x. Unfortunately, none of the cases just mentioned characterizes the modified Bessel function at hand, and we have had to develop, as a result, a new asymptotic expansion. As a perturbative application of the technique developed in the analysis of the continuum limit expansion of the lattice scalar propagator, in the closing section of this paper we consider the mass renormalization of the discrete lf 4 theory in coordinate space. We introduce now briefly the notations that will be assumed in the following sections. Throughout this paper, we shall work on a n-dimensional lattice of finite spacing a. The convention

the Schwinger’s representation for the propagator in momentum space. Re-expressing, then, the integral over the four-momentum as a product of four one-dimensional integrals and completing the square in the resulting exponential, we can finally perform the integration and obtain our final formula. Adopting the notational convention given by Eq. Ž1., we write the n-dimensional expression as follows

x s

xmp ,

Ý

with p an integer ,

will also be used. The Euclidean free scalar propagators in a n-dimensional configuration space will be denoted by D C Ž x;n. and D L Ž x;n. with dnp

q` C

D Ž x ;n . s

Hy`

n Ž 2p . m 2 q p 2

p q

D L Ž x ;n . s

eı p x

dnp

Hy pa Ž 2p .

eı p x n

a

m 2 q pˆ 2

,

4 a

2

n

Ý ms1

sin2

pm a

ž / 2

.

ym 2 a

s Ž 2p .

Ž x2 .

yn r2

exp  ypm2 a q ıpm xm 4 1y nr2

1r2

m 1r2

.

Ž 5.

The derivation of the standard representation for the lattice propagator in configuration space is carried out in a much similar fashion. Indeed, we have p

dnp

q L

D Ž x ;n . s

Hy

a p a

n

Ž 2p . m2 q pˆ 2

`

ym 2 a

½

n

2a

Łe

a2

m s1

xm

ž /q

m

a

`

1

p

a

yp

ž /H

y

=cos s

eı p x

H0 d a e =

dqm

Ž 2p .

2a

e a2

cosqm

5

ym 2 a

H0 d a e n

=

It is a well- known result that the continuum scalar field propagator in a 4-dimensional configuration space can be written in terms of the modified Bessel function K 1. The technique consists in using

Ž 2p .

=K 1y n r2 m Ž x 2 .

Ž 3.

2. The propagator on the lattice

dpm

q`

ŁH

m s1 y`

s

Ž 4.

n Ž 2p . m 2 q p 2

n

referring to the propagator evaluated in the continuum and on the lattice, respectively w8x. Note that in Eq. Ž3. we have introduced the short-hand notation pˆ 2 s

eı p x

H0 d a e =

Ž 2. ,

Hy` `

s

Ž 1.

ms1

dnp

q`

D C Ž x ;n . s

n p

77

2a

1

2a

½ Ł ž / ž /5 ey a2

ms1

a

I xm a

a2

Ž 6.

with I xm r aŽ2 ara2 . corresponding to the modified Bessel function of the first kind. Unfortunately, the integral appearing in Eq. Ž6. cannot be trivially solved. As a consequence, we are not able in this case to express the propagator in

B. Paladini, J.C. Sexton r Physics Letters B 448 (1999) 76–84

78

closed form. What we really wish to do here, though, is to show that in the continuum limit, i.e. when a ™ 0, D L Ž x;n. is given by the sum of D C Ž x;n. plus a series of correction terms depending on increasing powers of the lattice spacing a. As already mentioned in the introduction, the most direct way to proceed in order to achieve our goal is trying to derive the asymptotic expansion for the modified Bessel function I xm r aŽ2 ara2 . as a ™ 0. The strategy to adopt for the actual derivation of the expansion is determined by the fact that the order and the argument of the modified Bessel function become large at different rates. The standard techniques of global analysis Že.g, the steepest descents method. are not of much use in this case. As a consequence, we are forced to choose here a local w7x analysis approach. This implies beginning our study by examining the differential equation satisfied by the modified Bessel function at hand. With the purpose of determining uniquely the asymptotic behaviour of the solution, we shall also impose in the end the condition that the series representation derived reproduce Žthrough its leading term. the continuum result D C Ž x;n.. We first commence by setting

b'

2a xm2

and n '

xm a

,

Ž 7.

i.e. I xm r aŽ2 ara2 . ™ In Ž n 2b .. Thus, we wish to find an expansion for In Ž n 2b . as n ™ `. We observe now that In Ž n 2b . satisfies a differential equation of the form: x 2 d 2 yrdx 2 qx dyrdx y Ž x 2 q n 2 . y s 0. Hence, performing the change of variable x ™ n 2b

Ž 8.

we obtain

E 2 In Eb

2

Ž n 2b . q

yn 2

1

1 E In

b Eb

Ž n 2b .

C represents here the free parameter whose value shall be fixed later on according to the prescription made at the beginning of this paragraph. By using Eq. Ž10., we now get a new differential equation for YŽ b , n ., namely

E 2 Y Ž b ,n . Eb 2

Y Ž b ,n . ; e n

Eb

Eq. Ž9. simplifies slightly if we make the substitution

; 0.

In Ž n 2b . s

C

'b

Y Ž b ,n . .

Ž 10 .

b2

Y Ž b , n . s 0.

Ž 11 .

2

bq w Ž b ,n . d b

H

.

Ž 12 .

Actually, the form which we have used here for YŽ b , n . differs slightly from the standard Carlini’s substitution since it implicitly assumes that not only the leading term, but also all the subsequent terms in the expansion are expressible as exponentials. The reason which led us to adopt Eq. Ž12. as a viable option is essentially twofold: on one hand, it is dictated by the necessity of recovering the wellknown expression for the free scalar propagator in the continuum; on the other, the idea spurred in analogy to the method adopted in Watson w5x and originally due to Meissel Ž1892. in order to deduce the asymptotic expansion for the Bessel function of the first kind, Jn Ž n z . with n large. Carrying out the substitution in the differential equation for YŽ b , n ., we end up with a first-order inhomogeneous differential equation for the new function w Ž b , n ., i.e.

E w Ž b ,n . q n 2 In Ž n 2b . s 0.

y n q

n 2 y 14

We consider at this stage the limit n ™ `. We now need to make some kind of assumption on the form of leading term governing the expansion of the solution to Eq. Ž11. as n becomes large. With this purpose, we consider a substitution originally suggested by Carlini Ž1817., Liouville Ž1837. and Green Ž1837. and whose underlying idea is that the controlling factor in the asymptotic expansion of a function is usually in the form of an exponential. Hence, we assume the asymptotic relation

Ž 9.

b2

4

q w 2 Ž b ,n . q 2 n 2 w Ž b ,n . y

n 2 y 14 b2

Ž 13 .

Our aim, at this point, is to look for a suitable series representation for the solution to this equation. With this purpose, we make the following observations.

B. Paladini, J.C. Sexton r Physics Letters B 448 (1999) 76–84

First of all, if we suppose that 2 n 2 w Ž b , n . y n 2rb 2 ; 0 represents a good approximation to Eq. Ž13. when n ™ `, then it is straightforward to derive the leading behaviour of w Ž b , n . as Ž2 b 2 .y1 . We note that this gives exactly the continuum result D C Ž x;n. once the overall solution to Eq. Ž9. is normalized by fixing the arbitrary parameter C to be equal to Ž2pn 2 .y1 r2 . Secondly, we observe that all the nonleading terms in the expansion of w Ž b , n . should feature only even powers of n , since only even powers of this variable appear in Eq. Ž13.. As a result of the remarks just made, we assume that w Ž b , n . admits an asymptotic series representation for n large of the type y2 n

w Ž b ,n . s

The coefficients a nŽ b . can be, at this stage, computed iteratively leading to w Ž b ,n . 1 s 2b

1 2

1

y 8b

8b

Ý an Ž b . n

.

Ž 14 .

q 2b

1 q

5

ny4

16 b 6

115 y

4b 5

5 y

1

2b

7 q

ny2

8b 4

y

4

16 b

128 b 4

7

q

13 q

3

25 q y

3

2b

1 q y

1

y

2

1

`

79

64b 6

ny6 q O Ž ny8 . .

128 b 8

Ž 18 .

ns0

This is consistent with the approximation that we considered when we deduced the form of the leading term of the solution. Substituting now this formula into Eq. Ž13., we get 2

`

2

a nq 1 Ž b . ny2 n;

Ý

n y

y

Ý ns0

In Ž n 2b . ; n ™` Ž 2pn 2b .

1 4

b2

nsy1 `

a m Ž b . a nym Ž b .

Ý

E Eb

8b

5

2b 2

a1 Ž b . s y

y

96 b

480 b

q

1

½

q 2b

1 3

ny2

24b 3

192 b 4

1024b

y

8b 4

;

Ž 16 .

751 3

y 1536 b

1381 4

q

3072 b 5

4943 6

1 q

ny4

82944b 9

q 322560 b

17 7

y

5

23040 b 8

ny6 q O Ž ny8 . .

Ž 19 .

n

Ý

a m Ž b . a nym Ž b .

ms0

E Eb

q

1152 b 6

75 q

11520 b

a nq 1 Ž b . s y 12

3

1 5

y

8b

/

31

y

1513

;

2

q

4b

11

1

2b

Ž 15 .

Matching, order by order, the coefficients corresponding to the same power of n , we are now able to deduce the set of relations defining the coefficients a nŽ b .. We have 1

2

1

1 q

2

29

128 b

a n Ž b . ny2 n .

a0 Ž b . s

1

9

ms0

ž

exp n 2b y

y

q q

y 12

1

½

= 1q

n

½

The asymptotic expansion of In Ž n 2b . as n ™ ` can be now easily derived. The result reads as follows

an Ž b .

5

nG1.

Ž 17 .

Dividing this equation by In Ž n 2b . and plotting w9x the result against b for a fixed Žlarge. value of n , we see a fluctuation of the ratio around unity Žsee Fig. 1.. This reproduces exactly the behaviour which ought to be expected by the ratio of two functions

B. Paladini, J.C. Sexton r Physics Letters B 448 (1999) 76–84

80

m. The resulting a-integrals are all well-defined and finite. We can, therefore, proceed to their evaluation and obtain yn r2

Ž 4p . G Ž q.

D L Ž x ;n;q . ; a™ 0

 D0L Ž x ;n;q .

qa2D2L Ž x ;n;q . q a 4D4L Ž x ;n;q . qa6D6L Ž x ;n;q . q O Ž a8 . 4 . Fig. 1. Behaviour of the ratio of the asymptotic expansion of In Ž n 2b . with In Ž n 2b . as a function of b with n equal to 100.

asymptotic to each other. Furthermore, we observe that the fluctuation around 1 is a feature of the whole positive b-axis, therefore implicitly suggesting the validity of our expansion even for large values of b . Recalling the expressions of b and n in terms of a , xm and a, we are now able to derive the formula for the n- dimensional propagator up to sixth order in the lattice spacing. In fact, we can do more than that. We consider at this point a generalisation of D L Ž x;n. by introducing an arbitrary exponent q in the denominator of Eq. Ž6.; that is, we now look at p

dnp

q L

D Ž x ;n;q . s

Hy

a p a

Ž 2p .

1

s

m

H d aa G Ž q. 0 =

½

2a y

Łe

ms1

Pr Ž m; x . s

2

q pˆ

r

2m

Ž x2 .

D0L Ž x ;n;q . s 2P n yq Ž m; x .

a2

n D2L Ž x ;n;q . s P1q n yq Ž m; x . 8 2

q

8

4

ž / ž /5 a

a

2

96

a

2

D4L Ž x ;n;q . s

.

Ž 20 . The main reason for considering this generalised quantity rests in the fact that Eq. Ž20. represents a key element in the expression of general one-loop lattice integrals with bosonic propagators and zero external momenta w4x. Having reached a formula for the asymptotic expansion of the modified Bessel function as the lattice spacing vanishes, the study of the continuum limit of D L Ž x;n;q . is not technically more difficult than the analysis of the same limit for the n- dimensional lattice propagator D L Ž x;n. ' D L Ž x;n;1.. Indeed, we simply have now to substitute Eq. Ž19. into Eq. Ž20. and carry out the product over the dimensional index

P2q n yq Ž m; x .

4

q

e

I xm

Ž 23 .

2

y

2a

Ž 22 .

1 2

x2 2

1 2

with K r m Ž x 2 . representing, as usual, the modified Bessel function of the second w12x kind and r a real number. Therefore, we have

qy 1 ym 2 a

1

Kr mŽ x 2 .

1 2

x

`

n

The full expression of each of the coefficients D2Li Ž x;n;q .Ž i s 0,1,2,3. is given in terms of the new function Pr Ž m; x . defined as

eı p x n

Ž 21 .

P3q n yq Ž m; x .

n Ž n q 8. 256 n y

ž

P2q n yq Ž m; x . 2

128

13 q 192 x 2 P3q n yq Ž m; x .

/

n q 24 q 1536

q

Ž x2 .

x4

P4q n yq Ž m; x . 2

x6

x2 x4

y

q 1280

Ž x4.

2

2

256

q

Ž 24 .

2

1536

P5q n yq Ž m; x . 2

2

36864

P6q n yq Ž m; x . 2

Ž 25 .

B. Paladini, J.C. Sexton r Physics Letters B 448 (1999) 76–84

D6L Ž x ;n;q . s

n Ž n y 1. Ž n y 2. 12288 q

3n Ž 3n q 22 .

P3q n yq Ž m; x .

4096 y

2

Ž n y 1. Ž n y 2. 4096 85n q 666

x 2 P4q n yq Ž m; x .

q 12288 q

49152 3n q 52

2

2

3n q 160 y 61440 2

73728 8

x q

n q 48 q 589824 2

x x

x q

x x

q

q

6

Ž x2 .

p

3

Pr reŽ m; x . ;

12288

4

P6q

x4Ž x2 .

n yq 2

Ž m; x .

while, for mŽ x 2 .1r2 Ž r in / 0. 3 , we find

2

Pr inŽ m; x . ;

2 2 r in y1

Ž x2 .

r in

2

q Ž y1 .

P7q n yq Ž m; x .

20480

½

½

= ln

2

1

G Ž 1 y rre . Ž x 2 .

5

G Ž 1 q rre .

6

q

2 2 r re

m2 r re

49152

Ž x4.

1

2 sin rre p

y

3n q 98

5 57344

x4

Ž x 2 . P5q n yq Ž m; x .

12288

analytic functions of the mass and space coordinate times a modified Bessel function of the second kind of increasing order r . We intend now to demonstrate how Eq. Ž21. is in perfect agreement with both the studies performed in Ref. w3x and w4x. With this aim, we analyse D L Ž x;n;q . in the limit mŽ x 2 .1r2 ™ 0. Given the functional dependence of Eq. Ž21. on Pr Ž m; x . and given the definition in Eq. Ž22., this translates into considering the appropriate expansion for the Bessel function K r m Ž x 2 . . We wish to recall at this point that the series representation of the modified Bessel function of the second kind assumes different forms depending on whether the order r is a real Ž rre . or integer Ž r in .. In particular, for mŽ x 2 .1r2 small and r s rre we have 1 2

2

Ž n y 1 . Ž n y 2 . q 59n q 1102

q

81

r re

Ž 27 .

still small and

r s r in

G Ž r in . m2 r in

r in q1

G Ž 1 q r in .

mŽ x 2 .

1r2

2

4 2

y

x2 Ž x .

y 12 c Ž 1 . y 12 c Ž 1 q r in .

589824 x4 x6 q 245760

q

Ž x4.

P8q n yq Ž m; x . 2

3

21233664

P9q n yq Ž m; x . . 2

Ž 26 .

The expansion obtained for D L Ž x;n;q . clearly shows how the finite corrections introduced by formulating the theory on a lattice can be analytically expressed by a series of increasing Ževen. powers of the lattice spacing a with coefficients given by

5

Ž 28 .

with c denoting the c- function w6x. Using the relation G Ž 1 y r re . G Ž r re . s prsin rre p , we find that Pr reŽ m; x . and Pr inŽ m; x . assume the same functional form 2 ry1G Ž r .r Ž x 2 . r Ž r ) 0. as m ™ 0. The limits m ™ 0 and rre ™ r in are, therefore, uniform. After performing the final mechanical analysis to set n s 4 and q s 1, we 3

ln

For r in s 0 the correct expansion reads P0 Ž m; x . ; c Ž1.y

m Ž x 2 .1r2 2

.

B. Paladini, J.C. Sexton r Physics Letters B 448 (1999) 76–84

82

find that the massless limit of our continuum expansion for D L Ž x;n;q . reproduces exactly the expansion obtained by Luscher and Weisz for the massless ¨ 4-dimensional propagator w3x. The limit x ™ 0 proves to be more difficult to analyse. The short-distance behaviour is, indeed, singular and the limit not uniform in this case. Note that this observation matches the analogous remark made in Ref. w1x about the behaviour of the dimensionally regularised propagator. Observe also that the logarithmic mass-behaviour described in Ref. w4x by Burgio, Caracciolo and Pelissetto is recovered, in our formulation, for integral values of r .

Eq. Ž29., the lattice propagators have replaced their continuous counterparts and the summation over the lattice sites has taken the place of the continuum integral. Our present goal is to examine Eq. Ž29. as a ™ 0. Using the asymptotic expansion of the propagator as derived in Eq. Ž21., it is straightforward to obtain up to fourth order in the lattice spacing M Ltad ; d n z  f Ž x , y, z . D0L Ž 0;n .

H

qa2 g Ž x , y, z . D0L Ž 0;n . qf Ž x , y, z . D2L Ž 0;n .

4

3. The tadpole diagram in lf theory qa4 h Ž x , y, z . D0L Ž 0;n . As a direct implementation of the result obtained for the continuum expansion of the scalar propagator, we now wish to derive the one-loop renormalization mass counterterm of lf 4 theory through the study of the lattice tadpole diagram. In a n-dimensional continuum space, the contribution to the full propagator associated with the graph in Fig. 2 is well-known and proportional to the integral Hd n z D C Ž x y z . D C Ž0. D C Ž z y y .. The ultraviolet behaviour of the diagram is entirely due to the divergence in D C Ž0. w1x. In terms of our notational conventions, the lattice version of the tadpole graph in n-dimensions is immediately written as M Ltad s ym4yn

l

ž / 2

an

= Ý D L Ž x y z ;n . D L Ž z y y ;n . D L Ž 0;n . , z

Ž 29 . 4y n

with l the coupling constant and m a multiplicative dimensional factor introduced to preserve the dimensional correctness of the theory. Note that, in

Fig. 2. Self-energy tadpole diagram in lf 4 theory.

qg Ž x , y, z . D2L Ž 0;n . qf Ž x , y, z . D4L Ž 0;n . q . . . 4 ,

Ž 30.

with f Ž x, y, z ., g Ž x, y, z . and hŽ x, y, z . denoting functions which are associated with products of the type D0L D0L , D2L D0L and D4L D0L plus D2L D2L , respectively. We observe that the n-dimensional coefficients which appear in the expressions of f, g and h are exclusively evaluated at x y z and z y y and correspond, hence, to infinities which are, ultimately, integrable. As a result, the singularities in M Ltad are fundamentally generated by the poles in D0L Ž0;n., D2L Ž0;n. and D4L Ž0;n. only. It is now of paramount importance to remark that each of the latter quantities scales, implicitly, with the lattice spacing a. This scaling needs to be made explicit in the analysis by considering the mass as physical, i.e. m R s ma, and by recalling that, in a discrete formulation, the space coordinate is also defined in terms of the lattice spacing. At this stage, the isolation of the UV-divergences can take place along the lines of the method introduced in Collins for the study of the continuum case. Indeed, performing a point-splitting of the tadpole z-vertex through the introduction of an arbitrary variable ´ R s ´ra, such that ´ R ™ 0 Ža fixed., the investigation of the divergent behaviour in each of the D2Li Ž0;n.Ž i s 0,1,2. translates now into examin-

B. Paladini, J.C. Sexton r Physics Letters B 448 (1999) 76–84

ing the divergences in D2Li Ž ´ R ;m R ;n. 4 for vanishing ´ R and n fixed and non-integral. We note that each of the series expansions obtained splits, naturally, into two sub-series, with the poles in ´ R all contained in the first. In terms of ´ R , the second sub-series is, in fact, analytic. Thus, focusing on the singular contribution, we take the limit n ™ 4 and extract the infinities in ´ R through the singularities of the G-function which appears in the series. The procedure leads to a renormalization mass counterterm of the type

d L m2R s

1 a2

m 2R

½

½ž

a

2

8p 2 Ž n y 4 .

½ž

m4R 64p

q....

a

2

/

m 2

1

½ž /

4p 2

a2

m 2R

q

4 a2

y

m 2R 8p

2

m Ž ny4 .

64p 2 Ž n y 4 .

y

5

m Ž ny4 .

q

Ž n y 4.

Ž ny4 .

q

Ž n y 4.

m4R

m6R 1536p

m 2

Ž ny4 .

Ž n y 4.

5

2 R 2

Ž 31.

q

4

4p

a2 .

ž /½Ž m 2R

ž / 4

½ž / a2

m2R

y

m2R

2

ž / a2

q

m2R 4 a2

55

log Ž m2R .

4 q c Ž 1. q c Ž 2.

4 a2 1

4

4p

a2 .

ž /½Ž m 2R

ž / 4

~°

ž /¢Ž 4p

= log

2

1 8

y c Ž 1. y c Ž 3.

1

1

q

The explicit presence of the mass in the argument of the propagator coefficients has been included, in this case, simply to recall that it is m R rather than the bare m which now figures in the calculation.

1

= log

log Ž m2R .

4 q c Ž 1. q c Ž 2.

4 a2

qa4

y

2

m2R

= log

log Ž m 2R .

4 q c Ž 1. q c Ž 2.

4 a2

°1 m ¢ž a / q 4 a

As expected, the counterterm evaluated through the study of the lattice continuum limit in n-dimensions contains a divergence in n for n ™ 4 as well as a quadratic divergence for a ™ 0. However, multiplying Eq. Ž31. by a 2 , the second order pole in a disappears completely leaving the equation, now expressed in lattice units, finite Žfor n / 4.. The lattice derivation of the mass counterterm can be also performed directly in four-dimensions. Due to the important differences existing between the expansions in Eq. Ž27. and Eq. Ž28., a few changes

4

m 2R

1

qa ~

/ 3

d L4 m2R s

2

m Ž ny4 .

1

take now place in the investigation, though. The isolation of the infinities is no longer accomplished through the extraction of the poles in the G-functions, but resides ultimately in the observation that the divergences in the lattice spacing mirror exactly the analogue divergent behaviours in ´ R . Operationally, the latter remark produces a four-dimensional lattice counterterm of the type

y

2

1

m2R

q a4

y

5

8p 2 Ž n y 4 .

q a2

=

m Ž ny4 .

83

m 2R

ž / 4

2

y 18

m2R

2

ž / a2

y c Ž 1. y c Ž 3. 12 a2 .

3

q

3 64

m2R

5

3

ž / a2

y c Ž 1. y c Ž 4.

¶• ¶• ßq . . .ß

5

Ž 32 .

84

B. Paladini, J.C. Sexton r Physics Letters B 448 (1999) 76–84

We notice that, in agreement with standard results w10x, in this case both a quadratic and a logarithmic counterterm is needed at every order in the calculation. Nevertheless, in terms of lattice units, Eq. Ž32. is, again, finite.

Acknowledgements Beatrice Paladini wishes to thank both the European Human and Capital Mobility Program and Hitachi Dublin Laboratory for their financial support towards the completion of this work.

4. Conclusions References In the present work, we derived an asymptotic expansion for the modified Bessel function In Ž n 2b . as n ™ `. The expansion obtained was of vital importance to analytically evaluate the continuum expansion of both the lattice scalar propagator in a n-dimensional configuration space and its related generalised quantity D L Ž x;n;q .. The study of the small mŽ x 2 .1r2-behaviour of D L Ž x;n;q . in the limit a ™ 0 was shown to involve only the standard series expansion of modified Bessel functions of either real Ž rre . or integer Ž r in . order. The uniformity of the limits m ™ 0 and rre ™ r in was observed and the Luscher and Weisz expansion for the massless prop¨ agator in four-dimensions recovered. The result obtained in Ref. w4x for the mass-dependence was also reproduced for integral values of r . Finally, as a perturbative application of the results obtained, the one-loop mass counterterm in lf 4 lattice theory was evaluated for both the n y and 4 y dimensional case.

w1x J. Collins, Renormalization, Cambridge University Press, 1984. w2x D.Z. Freedman, K. Johnson, J.I. Latorre, Nucl. Phys. B 371 Ž1992. 353. w3x M. Luscher, P. Weisz, Nucl. Phys. B 445 Ž1995. 429. ¨ w4x G. Burgio, S. Caracciolo, A. Pelissetto, Nucl. Phys. B 478 Ž1996. 687. w5x G.N. Watson, A treatise on the theory of Bessel functions, Second Edition, Cambridge University Press, 1962. w6x I.S. Gradshteyn, I.M. Ryzhik, Tables of integrals, series and products, Fourth Edition, Academic Press, 1965. w7x C.M. Bender, S.A. Orszag, Advanced mathematical methods for scientists and engineers, McGraw-Hill, 1978. w8x M. Creutz, Quarks, gluons and lattices, Cambridge University Press, 1983. w9x S. Wolfram, Mathematica, Addison-Wesley, 1988. w10x I. Montvay, G. Munster, Quantum fields on a lattice, Cambridge Monographs on Mathematical Physics, Cambridge University Press, 1994. w11x Z.X. Wang, D.R. Guo, Special functions, World Scientific, 1989. w12x A.F. Nikiforov, V.B. Uvarov, Special functions of mathematical physics, Birkhauser, 1988. ¨