t PHY 91-23780 oundation, Gran y the National Science F View metadata,citationandsimilarpapersatcore.ac.uk

HEP-TH-9406081 ork supp orted in part b W 1 h ts t gauges. ed, whic EFI 93-71 1 ts, the func- t. Exp onen arian v ULES provided byCERNDocumentServer ysics TORS brought toyouby vior is obtained in all tofPh GA tao Xu A en OP CORE y of Chicago Abstract ersit -plane, and for general, linear, co Univ 2 k ergence relations obtained in the Landau gauge. y one-lo op expressions. Sum rules are deriv Chicago, Illinois 60637, USA v UGE FIELD PR ermi Institute and Departmen Reinhard Oehme and W Enrico F OR GA F ASYMPTOTIC LIMITS AND SUM R or gauge eld propagators, the asymptotic b eha F generalize the sup ercon are determined exactly b Asymptotically free theories are considered.tional Except form for of co ecien the leading asymptotic terms is gauge-indep enden directions of the complex

Interesting sum rules for the structure functions of propagators can b e

derived on the basis of their analytic prop erties, together with the asymptotic

b ehavior for large momenta as obtained with the help of the renormalization

group. For systems with a limited numb er of matter elds, one obtains su-

p erconvergence relations for the gauge eld propagator in the Landau gauge

[1, 2 ]. These relations are of interest in connection with the problem of con-

nement [4, 3]. Other results are dip ole representations, and information

ab out the discontinuity of the gauge eld structure functions. They indicate

the existence of an approximately linear quark{antiquark p otential [6,5],

and are imp ortant for understanding the structure of the theory in the state

space with inde nite metric [7].

It is the purp ose of this note to present results for the gauge eld prop-

agator in general, covariant, linear gauges. We obtain the asymptotic terms

2

for large momenta, and for all directions in the complex k {plane. Sum rules

are derived, which generalize the sup erconvergence relations of the Landau

gauge. An imp ortant asp ect of our results is the gauge{indep endence of the

functional form of the essential asymptotic terms. Only the co ecients of

these terms dep end up on the gauge parameter.

We consider a non{Ab elian gauge theory like QCD, with the gauge{ xing



part of the Lagrangian given by B
(@ A )+ B
B, where B is the usual



2



auxiliary eld. For 6=0, B can b e eliminated by B =(@ A ). In order



to de ne the structure function of the transverse gauge eld propagator, we

write

Z

%

ik x  2

dxe h0jTA (x)A (0)j0i = i D (k + i0)

ab

a b

 %    %   %  % 

 (k k g k k g + k k g k k g ) (1)

    

with A  @ A @ A . We assume the general p ostulates of covariant

gauge theories. Imp ortant are Lorentz covariance and simple sp ectral con-

ditions, as formulated in references like [8, 9] for state spaces with inde nite

metric. Exact Green's functions should b e connected with the formal p er-

2

turbation series in the coupling parameter g for g ! +0, at least as far as

the rst few terms are concerned. Top ological asp ects of the gauge theory

are not exp ected to in uence the asymptotic b ehavior we consider here.

As a consequence of Lorentz covariance, and the sp ectral conditions men-

2

tioned ab ove, it follows that the function D (k + i0) is the b oundary value of 1

2

an analytic function, which is regular in the cut k {plane, with a branch line

along the p ositive real axis. In contrast to the situation for higher Green's

functions [10] , explicite use of lo cal commutativity is not required for the

two-p oint functions [1]. Using renormalization group metho ds, together with

2

analyticity,we obtain the asymptotic b ehavior for k !1 in al l directions

of the complex plane. We present rst the essential leading terms, leaving

derivation and details for later.

2 2

For the analytic structure function D (k ), we nd for k !1in all

directions:

!

=

00 0

2

k

2 2 2 2

k D (k ; ;g; ) ' ln +


+ C (g ; )

0 R

2



0

!

1

2

k

+ +


(2) C ln

1 0

2



0

The corresp onding asymptotic terms for the discontinuity along the p ositive,

2

real k {axis are then given by

!

= 1

00 0

2

k

00

2 2 2 2

k (k ; ;g; ) ' +


C (g ; ) ln

R 0

2

j j

0

!

2

2

k

+ +


(3) C ln

1 0

2

j j

0

The parameters and their limitations are as follows: The function D is nor-

2 2

malized at the real p oint k =  < 0, where

2 2 2

2 2

k D (k ; ;g; )j =1: (4)

k =

2 2

The anomalous dimension of the gauge eld is given by (g ; )= ( )g +

0

4 2

( )g +


, ( )= + ; ( )= + + , etc., and

1 0 00 01 1 10 11 12

2 4

 = . The renormalization group function is (g )= g +

0 00 01 0

13 2 3

6 2 1 2 1

g +


.For QCD, wehave = (16 ) ( N ); = (16 ) ;

1 00 F 01

2 3 2

2

2 1

= (16 ) (11 N );N =number of avors. We assume < 0

0 F F 0

3

corresp onding to asymptotic freedom. Consequently, the exp onent = in

00 0

eqs. (2) and (3) varies from 13=22 for N = 0 to 1/10 for N = 9, and from

F F

1=16 for N =10to15=2 for N = 16. Wehave0 < = < 1 for

F F 00 0

N  9 and = < 0 for 10  N  16; for = = 1, our relations (2)

F 00 0 F 00 0

and (3) would require mo di cations. 2

The essential asymptotic term in eqs.(2) and (3) is the one with the

2

co ecient C (g ; ), which is not identically zero, although there maybe

R

2

zero surfaces = (g ). In case C should vanish, the C { term b ecomes

0 R 1

relevant. Its co ecient is given by

( )

1 0 00

C = for 0 < < 1;

1

0 00 0

+

00 0 11 00

C = for < 0: (5)

1

0 00 0

For = 0, the co ecient C is p ositive and given by

R

Z

0

2 2 =

00 0

exp dx (x); C (g ; 0) = (g )

0 R

2

g

(x; 0)

00

 (x)  : (6)

0

(x) x

0

Eq. (6) follows from the exact solution for = 0, with the normalization

2 2 2

 D ( ; ;g;0) = 1 [1, 2]. This solution can b e written in the form

!

=

Z

2

00 0

2

g

g

2 2 2

k D (k ; ;g;0) = dx (x): (7) exp

0

2

2

g

g

2

k

2 2

Hereg  =g (u; g ), u = j j is the e ective gauge coupling, with the prop er-

2



ties

Z

2

g (u;g )

1

dx (x); ln u =

2

g

2

g (u; g ) ' 1=( ln u)+


(8)

0

for u !1, in the case of asymptotic freedom with < 0.

0

2

In general, C (g ; ) satis es a partial di erential equation. In an ap-

R

proximation, where only terms linear in are kept in the expression for the

2

anomalous dimension (g ; ), we obtain

 

Z

0

2 2 =

00 0

C (g ; ) = (g ) exp dx (x)

R 0

2

g

 

Z

0

2 = 2 =

00 0 00 0

f (x; g ) : (9)  1 dxx + (g )

2

g

0 0 3

Here

Z

2

g

2

f (x; g ) = f (x)+  (x)gexp dy  (y );

0 0 1 0

x

 (x) =  (x)= (x) = x;

1 1 01 0

2

(x; ) =  (x)+  (x)+ O( ): (10)

0 1

In eq. (9), the normalization for the full solution in the the {linear approx-

imation has b een used in order to x an otherwise undetermined co ecient

of C . This approximation consists of replacing the general anomalous di-

R

2 2 2

mension (g ; )by the -linear form  (g )+  (g ). The corresp onding

0 1

solution is given by

!

Z

2

g

 (x)

0

2 2 2

k D (k ; ;g; ) = exp dx

2

(x)

g

( !)

Z Z

2

g x

 (x)  (y )

1 0

dx dy  1+ exp ; (11)

2 2

(x) (y )

g g

with the notation as de ned in eqs. (8) and (10).

An imp ortant asp ect of the leading asymptotic terms for

2 2 2 2 2 2

k D (k ; ;g; ) and k (k ; ;g; )

0

is their indep endence of the gauge parameter , except for the co ecient

2

C (g ; ). In addition, the exp onents of the logarithms in eqs. (2) and

R

(3) are completely and exactly determined by one{lo op co ecients of the

2

anomalous dimension (g ; 0) and of the {function.

2 2

In view of the asymptotic b ehavior of D (k ; ;g; ) as given in eq. (2),

2

whichisvalid for all directions in the k {plane, we can write the usual un-

substracted disp ersion representation

Z

02 2

1

(k ; ;g; )

2 2 02

D (k ; ;g; ) = dk (12)

02 2

k k

0

We also have sucient b oundedness for the discontinuity  in order to write

a dip ole representation

Z

02 2

1

 (k ; ;g; )

2 2 02

; D (k ; ;g; ) = dk

02 2 2

k ) (k

0 4

Z

2

k

2 2 02 02 2

 (k ; ;g; ) = dk (k ; ;g; ): (13)

0

For = 0, the dip ole representation has b een discussed in refs. [5] and [6]

in connection with an approximately linear quark{antiquark p otential.

Of particular interest is the situation for 0 < = < 1, corresp onding

00 0

2 2

to N  9 in QCD. There, the function D + k vanishes faster than k

F

0

2

for k !1, and hence wehave the sum rule

Z

1

2 2 2

dk (k ; ;g; ) = : (14)

0

0

This is the generalization of the sup erconvergence relation [1, 2 ]

Z

1

2 2 2

dk (k ; ;g;0) = 0;

0

whichwas obtained previously in the Landau gauge. The relation (14) ex-

2

presses the fact that the co ecient of the asymptotic term prop ortional to k

in the representation (12) is given by = .Itisnot valid for = < 0.

0 00 0

The distribution asp ects of sum rules like eq. (14), and of the related disp er-

sion representations, have b een discussed in refs. [1, 2 ].

In order to derive the asymptotic prop erties of the structure function

D ,we consider the renormalization group equation for the dimensionless

2 2 2 2 2

function R(k = ;g; ) k D(k ; ;g; ). We obtain

! ! !

2 02 2

k  k

R ;g; = R ;g; R ;g;  ; (15)

2 2 02

  

where wehave used the relation

1 02 2 02 2

R ( = ;g; )=Z ( = ;g; );

3

2

which follows from the normalization condition (4): R(1;g; ) =1at k =

2

 < 0 , and where Z is the square of the conventional renormalization

3

02



factor for the gauge eld. Further, in eq. (15),g  =g( ;g) is the e ective

2



 

02



1

(running) gauge coupling parameter, and  is given by = R ;g; .

2

 5

We use a mass indep endent renormalization scheme [11 ], which is appropriate

for the study of asymptotic limits.

2 2

Let us rst consider the limit k !1 along the negative, real k {axis,

2 2 2 2

where R(k = ;g; ) is analytic and real. We set u = jk = j and de ne the

2 2

function R(g ; g ; )  R(u; g ; ), withg (u; g ) b eing the e ective coupling

de ned in eq. (8). From eq. (15), we then obtain the di erential equation

2 2

@R(g ; g ; )

2 2 2 2

(g ) = (g ;  )R(g ; g ; );

2

@g

2 2 1 2 2

 = (g ; g ; )  R (g ; g ; ): (16)

For 6= 0, it is more convenienttowork with the equation

@ 

2 2

=  (g ; ); (17) (g )

2

@ g

with  as de ned in eq. (16). From the general solution (7) of eq. (16) for

=0,aswell as the solution (11) for 6= 0 in the {linear approximation,

2 2

we know that R and  have a branch p oint as a function ofg  atg  =0,

13 2

N

F

2 

2 3

which is of the form (g  ) ;  = . For QCD, wehave = .

2

00 0

11 N

F

3

We see that j j = n=d is rational, with n and d b eing relative primes. It

is then convenient to uniformize the algebraic branch p ointbyintro ducing

2 1=d

x =(g ) as a uniformization variable.

We consider rst the case 0 < <1, corresp onding to N  9 for QCD.

F

n

Here it is convenient to use eq. (16). We de ne y (x)=( )x , and

0

obtain the di erential equation

dy n

n1 2 dn1 n d n

= x y dx ( +x y)(x ; + x y); (18)

0 0

dx

0

where

2

(g ; ) ( )

0

2 2

=  ( )+g  ( )+


(g ; ) 

0 1

2 2

(g ) g

0

( )

1 1

 ( ) = ( ); etc.; (19)

0 0

2

0

0

with the de nitions given b elow eq. (4). 6

2

In an appropriate nite domain including g = 0, and excluding p ossi-

2

ble, nontrivial xed p oints corresp onding to zero es of (g ), it is reasonable

2 2

to assume that (g ; ) is continuously di erentiable. As far as (g ) and

2 2 2

(g ; ) are represented bypower series expansions for g ! +0;(g ; )is

2

also a p ower series in g and . Under these circumstances, the r.h.s. of

eq. (18) satis es the Lipschitz condition for x = 0, and wehave exactly one

solution through every p oint x =0;y = C. In as far as the r.h.s. of eq. (18)

is also a p ower series, we obtain the solution in the form of a series:

!

2

C  +1

n 0 d

y (x) = C + x + C  ( )  ( ) x +

0 0 0 0

0

 1

0

0

dn

 ( )x +


; (20) +

0 0

 1

where wehave separated the terms prop ortional to C .

2 2 2

For the asymptotic expansion of R(g ; g ; ) forg  ! +0, eq. (20) implies

!

1

2 2 2  1 2 2 +1

R(g ; g ; ) ' + C (g ) + C (g ) +

R R 10 12 00

0 0

0 0

( ) 1

1 0

2

+ g +


; (21)

1 

0 0

2

with C = C = , and 0 < <1. This formula is also valid for C =0.

R R

0

2

The term prop ortional to C in eq.(20) cancels in the inversion leading to

eq.(21).

For <0, corresp onding to 10  N  16 for QCD, it is more convenient

F

n

to use eq. (16). With  = , n and d b eing p ositiveintegers which are

d

2

relative primes, we uniformize the branch p ointatg =0byintro ducing

2 d n d 2

again a new variable x so thatg  = x . Then we de ne z (x)= x R(x ;g ; ),

and obtain the di erential equation

n

dz x

n1 d1 d

= nx + dx z(x ; ) (22)

0

dx z

n 1

As long as x z remains b ounded around x =0,wehave again the Lipschitz

condition satis ed and obtain the p ower series solution

d

z (x) = C + C  (0)x +

R R 0

d

0 d+n n

( (0) +  (0))x +


; (23) + x +

0 0

0

d + n

0 0 7

2 2

with C 6=0. In terms of R(g ; g ; ), eq. (23) leads to the asymptotic

R

2

expansion for <0 andg  ! +0:

1

1

2 2 2  2  +1

R(g ; g ; ) ' C (g ) + C ( )(g  ) +

R R 10 00

0 0

( + )

10 0 11

2

+ + g +


(24)

(1  )

0 0 0

For C =0,we cannot use eq. (22), but obtain the asymptotic expression

R

directly from eqs. (20) and (21) :

( ) 1

1 0

2 2 2

R(g ; g ; ) ' g +


: (25) +

1+jj

0 0 0

This relation corresp onds to eq. (21) with C = 0 and <0.

R

With eqs. (21) and (24), wehave obtained the asymptotic expressions

2

k

2

for R( ;g; ) in the limit k !1along the real axis, provided we can

2



2 2

useg  (u; g ) '1=( ln u)+


. It remains to consider the limit k !1

0

2

in all directions of the complex k {plane. We return to eq. (15). Setting

02 2 2 2 2 i'

 = jk j,we nd, with  < 0 and k = jk je for all j'j :

! !

2 2

k k

i'

R ;g; = R ;g; R(e ; g; ): (26)

2 2

 

2 2

k k

1

Hereg  =g(j j;g) and  = R (j j;g; ). For < 0, the e ective

2 2 0

 

2

k

2

j!1, and  remains b ounded in this limit, couplingg  vanishes for j

2



as may b e seen from eqs. (21) and (24). Because R is analytic in the

2

cut complex k {plane, we can then use the p erturbation expansion for the

structure function,

! !

2 2

k k

2 4

R ;g; ' 1+g ( )ln + O(g ); (27)

0

2 2

 

2

and write forg  ! +0:

i' 2 4

R(e ; g;  ) ' 1+g ( )i' + O (g ): (28)

0

2

Eq. (26) expresses the asymptotic limit for k !1in all directions in terms

2

of the limit along the negative real k {axis. With eqs. (26), (28), (21) and

(24), we nally obtain the limits given in eqs. (2) and (3). 8

A priori, the co ecients C or C app earing in the solutions of the non-

R

linear, ordinary di erential equations are undetermined constants. However,

2 2 2 2

b ecause of the normalization condition R(g ; g ; )=1 or (g ;g ; )= ,

2

the co ecients b ecome functions of g and , satisfying partial di erential

2

equations in these variables. For C (g ; ), we nd the equation

R

2 02 2 02 0

C (g ; ) = R(g ;g ; )C (g ; )

R R

0 1 02 2

= R (g ; g ; ); (29)

and the corresp onding di erential equation is:

@C @C

R R

2 2 2

(g ) = (g ; ) (g ; )C : (30)

R

2

@g @

For = 0, and in the {linear approximation, wehave given C in eqs.

R

2 2 2

(6) and (9), which satisfy eq. (30) with = 0 and (g ; )=  (g )+  (g )

0 1

resp ectively.

Several of the results presented in this pap er have b een obtained by one

of us (R.O.) in collab oration with W. Zimmermann [12 ]. It is a pleasure to

thank Wolfhart Zimmermann for his contribution and for many discussions.

References

[1] R. Oehme and W. Zimmermann, Phys. Rev. D21 (1980) 475, 1661.

[2] R. Oehme, Phys. Lett. B252 (1990) 641.

[3] K. Nishijima, in Symmetry in Nature,Festschrift for Luigi A. Radiacati

di Brozolo (Scuola Normale Sup eriore, Pisa, 1989) pp. 627{655.

[4] R. Oehme, Phys. Rev. D42 (1990) 4209.

[5] R. Oehme, Phys. Lett. B232 (1989) 498.

[6] K. Nishijima, Prog. Theor. Phys. 77 (1987) 1035. 9

[7] R. Oehme, Mo d. Phys. Lett. A8 (1993) 1533;

N - Newsletter No. 7 (1992) 1.

[8] T. Kugo and I. Ojima, Prog. Theor. Phys. Suppl. No. 66 (1978) 1;

N. Nakanishi and I. Ojima, Covariant Operator Formalism of Gauge

Theories and Quantum Gravity (World Scienti c, Singap ore, 1990) p.

382.

[9] F. Stro cchi, Comm. Math. Phys. 56 (1977) 57;

Phys. Rev. D17 (1978) 2010.

[10] H.J. Bremermann, R. Oehme and J.G. Taylor,

Phys. Rev. 109 (1958) 2178.

[11] S. Weinb erg, Phys. Rev. D8 (1973) 3497;

G. t'Ho oft, Nucl. Phys. B61 (1973) 455;

J.C. Collins and A.J. MacFarlane, Phys. Rev. D10 (1974) 1201.

[12] R. Oehme and W. Zimmermann, (1983) unpublished. 10