The 2ómega-dimensional light-cone integrals with momentum shift

The 2dimensional lightcone integrals with momentum shift

A. T. Suzuki

Citation: Journal of Mathematical Physics 29, 1032 (1988); doi: 10.1063/1.528018 View online: http://dx.doi.org/10.1063/1.528018

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The 2ro-dimensionallight-cone integrals with momentum shift

A. T. Suzuki

Instituto de Fzsica Teorica, Universidade Estadual Paulista, Rua Pamplona, 145, 01405-800 Paulo, Brazil (Received 22 July 1987; accepted for publication 16 September 1987)

A class of light-cone integrals typical to one-loop calculations in the two-component formalism is considered. For the particular cases considered, convergence is verified though the results cannot be expressed as a finite sum of elementary functions.

I. INTRODUCTION

In the evaluation of one-loop diagrams (such as the

"swordfish" diagrams) in the two-component formalism of the light-cone gauge, one finds integrals of the type

K(p,q) =

f

r(r-q)2 (p+ +r+) ^{dr 1} (1.1 ) and

KI( )

f

^dr (/+r1), 1=1,2, (1.2) p,q

=

r(r - q)2 (p+

+

^r+)

where the measure dr according to the dimensional regular- ization technique is defined over an analytically continued space-time of 2m dimensions.

Right from the start one should be aware that naive shifts of integration variable are not permissible in light-cone integrals that are linearly divergent by power counting as- sessment.' Happily none of the above integrals falls into this category and for convenience I consider the shifted versions

- f

^{dr 1}

K(p,q) =

(r - p)2(r _ p _ q)2 r+ (1.1') and

K- I( ^{p,q -}) -

f

(r_p)2(r_p_q)2 r+ ^{dr r1} ( 1.2') instead of Eqs. (1.1) and (1.2). Here I treat the singularity at r+

=

0 according to the prescription first suggested by Mandelstam,2 namely,

lIr+

=

lim [lI(r+ +iEr-)]. (1.3 )

E_O+

II. EVALUATION OF K(p,q)

Using the standard procedure of exponentiating propa- gators, Eq. (1.1') becomes

K(p,q)

= - LX>

^{da d/3} ei(ap'+pq'J

:~

^eix(r'+2r-R),

(2.1 ) where

x=a +/3, xR =/3q - ap.

(2.2) (2.3 ) Resolving the momentum integral through the employ- ment of the Mandelstam prescription [Eq. (1.3)], one ob- tains³

where

y

=

^a/(a

+

/3) , (2.5)

u=q+/(p+ +q+), (2.6)

f/

= qlqI

+

q2q2 = 2q+q- _ q2, (2.7)

t

^{= ~ [(1}

⁺

^{v - p)}

⁺

^{~ (1}

⁺

^{v - p)2 - 4v] , (2.8)}

t

^{= ~ [ (1}

+

^{v - p) - ~ (1}

+

^{v - p) 2 - 4v ] , (2.9)}

v = [2(p+

+

^{q+ )(p-}

+

q- )]/q2 , (2.10)

p=2p+p-/q2, (2.11)

p± = (po ±p3)/.J2. (2.12)

In order to carry out the y-integration, first I expand the denominator of the integrands in power series

1 ~ k - \

(1 - oy) =

k~\

^{(oy) (2.13 )}

so that now, the y-integrations can be expressed in terms of beta functions and hypergeometric functions of two vari- ables⁴

- ( - 1T)"T(2 -m)

K(p,q) = ( + +)

P

+q

00

X

2:

^~-\[ (i),"-2B(k

+

^{m - 2,m - 1)}

k=\

- (vq2),,,-2B(1,k)

XF\ (k,2 - m,2 - m;k

+

1;5'

-l,t -\)] .

(2.14)

Isolating the k

=

1 term of the sum and employing the following functional relations for the hypergeometric func- tions^5:

F] (a,/3,/3';r;x,y)

= F\ (a

+

1,/3,/3';r;X,y)

- (/3 /r)xF\ (a

+

1,/3

+

1,/3';r

+

l;x,y)

- (/3'/r)yF\(a

+

^1,/3,/3'

+

^l;r

+

^{1;X,Y), (2.15)}

1032 J. Math. Phys. 29 (4). April 1988 0022-2488/88/041032-02$02.50 © 1988 American Institute of PhySics 1032 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

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FI (a,/3,/3';a;x,y) = (1 - x) -P(1 - y) -P',

F

I

(a,/3,O;y;x,y) =

zPI

(a,/3;y;x),

FI(a,O,/3';y;x,y)

=

²Fl(a,/3';y;y),

(2.16) (2.17) (2.18 ) and expanding, wherever necessary, for w-2, one gets

K(p,q)

=

t? {In(2P+P -)_sln(S-I)

(p+ +q+)

i S

-€ln(€i I)}

+

(p+

~q+)

,., u" { 2 (2P+P-)

XL

^{+ I n - -}

k=l(k+l) (k+I)

i

~ 1 2FI(1,k+2;k+3;s-l)

+ "- - +

-=-.!..:--'--"'--:...-"'--=-~

m=1 m (k+2)s

+

2F I(I,k

+

2;k1=.3;€-I) } +0(2-w).

(k

+

^2)s

(2.19) Furthermore, using the expansion⁴

2Fl (l,k

+

^2;k

+

^3;z)

(k+2)

In(1-z) k ~-k-l

= - Z(k+2) -

m~o

^{(I +m) , (2.20)}

the final expression for

K

is written as

K(p,q) = t?

{In(2P +f-)-51 n

( s - l )

(p+ +q+) q

S

-€ln(€i I)}

+

^(p+

~q+)

^{k t l}

(k:

¹⁾

X { 2

+

^In(2P+P-)

(k+ 1)

i

- S

k + I

In( ^S ""i I) - ^€

^{k + I}

In( ^€ i I)

k 1 k

(s

^{k - m}

+ €

k - m) }

+ L - - L

...:::--~---'- m=lm m=O (1+m)

+0(2-w). (2.21 )

III. EVALUATION OF j(I (p,q)

The evaluation of Ki (p,q) follows in a completely anal- ogous way. After the y-integrations are performed the result is

1033 J. Math. Phys., Vol. 29, No.4, April 1988

i

K/(p,q)=(l+qi)K(p,q)-t? q r(2-w) (p+ +q+)

X i:u"-I[( _1ri)W-2 k=1

XB(k+w-l,w-l) - ( - ~2)w-2B(l,k

+

¹⁾

XFI(k

+

1,2 - w,2 - w;k

+

2;S-I,€ -1)], (3.1) which yields

IV. CONCLUDING REMARKS

The explicit calculation showed that the integral defined in Eq. (1.2) has its pole part canceled out. It suggests thl\t in the Mandelstam prescription the transversal components of the vector momentum, ,J, over the longitudinal component, r+ , yield n

< °

as far as power counting is concerned. There remains to be seen whether the lower limit for n can be deter- mined accurately. In any case, it is interesting to note that this pattern of convergence for momentum integrals in the light-cone gauge

a

fa Mandelstam is characteristic of this gauge.

ACKNOWLEDGMENT

This work was supported by F APESP-Siio Paulo, under Contract No. Proc. 86/2.376-9.

'R. B. Mann and L. V. Tarasov, Mod. Phys. Lett. A 1, 525 (1986).

2S. Mandelstam, Nuc!. Phys. B213, 149 (1983).

3D. M. Capper, J. J. Dulwich, and M. J. Litvak, Nuc!. Phys. B 241, 463 (1984).

41. S. Gradshteyn and I. M. Ryzhik, Tableo/Integra/s, Series, and Products (Academic, New York, 1980).

5p. Appell and J. Kampe de Feriet, Fonctions Bypergeometriques, et By- perspheriques-Polynomes D 'Hermite (Gauthier-Villars, Paris, 1926).

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