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Asymptotic freedom with infinitely many quarks?
Volume 84B, number 2
PHYSICS LETTERS
18 June 1979
ASYMPTOTIC FREEDOM WITH INFINITELY MANY QUARKS? H. NICOLAI [nstltut f~r Theoretische Physik, Universitat HeMelberg, Germany Received 3 January 1979
Relying on a recent recalculation of the &function in massive QCD, we show that a theory with infinitely many quark flavors is asymptotically free if the quark masses grow as rn2n = n °Mg, 1 < a < 3.6; the precise defimtion of the coupling constant (via the three-gluon vertex) enters in an essential way. In these models the running coupling constant decays hke an inverse power of Q2 as Q2 ~ 0o. Some phenomenologlcal implications of this curious result are discussed.
In a recent publication [ 1], Nachtmann and Wetzel recalculated the/3-function in massive QCD and obtained a result at variance with the conventional one [2] ; the discrepancy is explained by the fact that in ref. [1] the three-gluon vertex instead of the ghost-gluon vertex is used to define the renormalized coupling constant. In both cases, the result may be expressed as follows (the notation is the same as in ref. [1] ; f i s the number of flavors) 13(g, m, (x, M) -
(1)
g3 { 2 1 f/ :1 [ (rrt2~7} -11-3~ -h + O ( g 5) 167r2 "= \ M 2 lJ '
which makes the coefficient o f g 3 in eq. (1) stay negative for all M (choose M sufficiently large). The same is true for h(~) ---hNW(~) and 17 < f < oo but n o t for h (~) = hNW (~) and an infinite number of flavors if the quark masses are suitably arranged as will be exemplified below. The result shows that the question of which coupling constant is "best" may be not merely one of calculational convenience in a theory with an infinite number of massive quarks. Taking now f = oo, it is not difficult to prove that, at least up to the one-loop approximation, the condition oo
n~l 1 < o o, = rn 2
(4)
where, convennonally, hconv.(~) = 6~ 12~2
In (1 + 4 0 1 / 2 + 1
(1 + 4~) 1/2
(1 + 4~) 1/2 -- 1 '
(2)
ensures the convergence of all renormalized quantities of the theory * x, in fact, up to this order, (4) is necessary and sufficient. To be more explicit, we will assume m2=n°M
2,
o>l,
(5)
whereas Nachtmann and Wetzel obtain
1
hmv(~) = 18~ f dx 0
1/3 [ +2fdy.O,)
xO -x) x(1 - x ) + ~
~
3~2 ]
(3)
0 From these equations it may be seen that for f~> 17 and h (~) -- hconv" (~) there is no possible choice o f m 2
where n numbers the flavours and M 0 is an arbitrary mass. Because o > 1, (4) is evidently satisfied. The main result o f this paper is contained in the Theorem: If the quark masses are given by (5), the following estimate is optimal ¢ 1 This need not be true for unrenormalized quantitms. In the nonperturbative lattice regularizatlon of the model, the condition (4) emerges quite naturally. 219
Volume 84B, number 2
PHYSICS LETTERS
1/°
il < II - hNW \ M 2 ]_j
5
n=l
(6)
where the function ~0(o) is defined by
{'
ref. [3] ). In the hmlt M ~ ~o, the nonleadlng terms stay bounded and the estimate (6) holds. Note that for o = 2 m , m C hi, explicit calculations may be carried out by means of [3] oo
1 _ n=l x 2 + n 2
~0(o)= 18 f [x(1 -x)] l+l/°dx 0
18 June 1979
1 + ~-cothrrx 2x 2
(11)
oo
- 2(2 + 3/o)
;
p(y)yl/°dy
o
/; o
1 at l+t°
(7)
On the interval 1 < o <~ 3.6 *2 , ¢(o) < 0 and the coefficient o f g 3 in eq. (1) becomes arbitrarily negative as
_
n = l X4 + n 4
l
+ n__im~
2x 4
2x 3
1 cothTr,vF~xl
Lx/-i
and similar formulas derivable from eqs. (1 1). These may serve to check (6). For M >>M0, 13(g(M)) ,., 24rr2 (.~202)~0(o) "M 2"1/0 g3(M)+ O(g5(M)).
For the proof of the theorem, use is made of the formula (o > 1) oo
1 n=l
if
_
X ° + t1°
X°-I
_ _ d r +O(x_O) ,
0
(8)
dt 0
1
1 +to
1
G
F(x)
x n =1 1 +(n/x) °
x
(9)
with F@ C ~, 0 < F ( x ) < 1 and F ' ( x ) = O ( x - 1 ) . Differentiating eq. (8) with respect to x and observing that F is smooth, one gets
n = 10C °
+
n°) 2
o- 1 a
1
T
dt
x2°-I
0J
1 +t°
+
O(x-2°),
oo
n °
n = l (X ° + n O ) 2
_ 1 1 ax°-I
; o
(10)
Inserting these formulas on the left-hand side of (6), we may interchange summation and integration I f M o / M > 0 (e.g. by the dominated convergence theorem, see ,2 The numerical error on this number Is about 0.1. Note that 13 h m ° ~ l {...} = - ~-d < 0 and h m o ~ ~ {...} = 1 > 0.
220
g2(M) "-' [const. + oltp(o)l [M 2 ~/°1-1 247r 2 \M02 ]
J
(13) '
and the running coupling constant drops to zero much more rapidly in the limit M ~ oo than in the case o f f <~ 16 zero mass quarks ,3. This feature persists if a finite number of quarks with arbitrary masses is added to our models. Of course, one could easily construct in this manner a model where the coupling constant g2(M) decays logarithmically at low (= present day) energies as observed in experiment whereas at higher energies the faster decay of eq. (13) sets m. The rapid decrease ofg2(M) also brings down the "meeting point" of strong and electromagnetic Interactions by several orders of magnitude which may be welcome to "grand synthesizers". Until now, we have been slightly cavalier about the runmng quark masses treating them as if they were constant. However, since they evolve according to (m the gauge ~ = 0, cf. ref. [ 1 ] ) dmn(M) _
dt +O(x_O) " ] + t°
(12)
Thus, on 1 < o <~3.6,
1 +t°
which follows from
J
dM
g2(M) mn(M) 27r2
+ O(g4(M)),
M
f1
xax
0 x + m 2 (M)/M 2 (14)
4:3 Of course, if 3(g) = g3 {13o(M) + 31 (M)g 2 + -..}, this argument is valid only if, in the limit M ~ 0% 31 (M)g 2 (M) = o(30(214)). Although this Is always tacitly assumed to be true, only higher-order computations can justify such a procedure.
Volume 84B, number 2
PHYSICS LETTERS
we have, to lowest order,
g2(M) mn(M) 2rr 2
M
dmn(M) t
~<
d)l//
~<0.
(15)
Inserting (13), one easily verifies that mn(M ) approaches a (non-vanishing) constant quite rapidly in the hmlt M-+ oo, by the same token, the condition (4) is preserved in this limit. This shows at least, if nothing else, that the foregoing considerations are not Inconsistent (if one wants to avoid this comphcatlon, one may define the masses Independently of the normalization point, e.g. as starting points of branch cuts [4] ). To justify the use of the 3-gluon coupling constant g(M) in the considerations below, we have to establish the connection between g(M) and the quark-gluon coupling constant ~(M) (or, equivalently, the ghost-gluon couphng constant). To lowest order we find / f ( M ) g (M) - { C O
we might be led to the erroneous conclusion that g(M) does not converge to zero in this hmlt. This difficulty does not occur for finitely many quarks since, in that case, one may treat all quarks as massless if M >2> max
l ~<1<)'
m1
In the models of this paper, the electromagnetic quark current takes the form
ju = ~ enqn T*tqn "
(21)
n=l
Assuming that the charges en of the quarks stay bounded (e.g. with alternating charges 2/3 and - 1 / 3 ) , we find for Adler's vacuum polarization function D(Q 2) [5], using (5) and (8),
R(s)
[02 ~lJo
D ( Q 2 ) = Q 2 0 ds ( s + Q 2 ) 2 ' Q - * ~ > const.l~ \M02]/
1 +SI12fdxx(1-x)ln{l+M2x(1-x)) 1=1 c
18 June 1979
' (22)
m2 where
lj, dyp(y) ln (1 +,M21m?f
-4If
R (s) = e(e+e-- -+ h adrons) o(e+e - -+ ~+bl-)
,Lj
?/3
-6 0
"M2
II
(22) lmphes, for s ~ ~,
g3(M)
R(s) ~_ const.(s/M2) l/° ,
dyp(y) rnf + y M 2 j j 3;I~2
+O(gS(M)),
IC0l<~,
(16)
and, by (8) and (13)
g(V)~-g(V)+O(1)(V2/M2)l/°g3(M)-~O,
(17)
as M-+ ~ (this IS also a consequence of the WardIdentity relating the 3-gluon and the ghost gluon vertex). (17) seems to contradict what one would obtain from a direct computation via eq. (I) and h = hconv.. To see that there IS no paradox involved, suppose that
Mdg(M)/dM= -A(M2/M2)g3(M),
A >0,
(24)
and therefore, in (22), no extra subtractions are necessary as 1/o < 1. (24) may be understood Intuitively since, by (5), the number of quarks up to a given s grows like (s/M2) 1/~. The rapid decay o f g ( M ) also leads to some Interesting predictions for deep inelastic structure functions, the moments of which are related to coefficient functions occurring In the Wilson expansion of the product of two currents Typically, these coefficient functions F (n) satisfy a renormahzatlon group equation of the form [6]
(18)
(M ~
were exact and define a new couphng constant by
~(M) =g(M) + A (M2/M2)g3(M) .
(23)
0
0 ~F(n) + 3g ~g +]~1 3miami ,
(19)
From eqs (18) and (19), it follows thatg(M),~(M) 0 as M-+ ~, whereas by neglecting higher orders in
Mdf(M)/dM = +.4 (MZ/M2)f3(M) + O(~5(M)) , (20)
+ T(n)F (n) = 0 ,
(25)
where T (n) is the anomalous dimension of the corresponding operator in the Wilson expansion, also, we only consider nonsinglet operators with flavor struc221
Volume 84B, number 2
PHYSICS LETTERS
ture ~u - dd, for instance. If the canonical scahng has already been factored out o f F (n), eq. (25) is solved by
F(n)(O A ml(M) ) \M 2 ' M ,g(M)
(26)
[ ml(Q),g(Q)) r- log0/,_ (n) t =F(") [1, 0 - e x p / f-rogV/. 3' ( )
dtn /"
Now, 7(n)(t) = c(n) g 2 (ge t) + .... where C (n) is one of the coefficients computed in ref. [7], because o f (13), the integral In the exponent of (26) remains finite In the limit Q ~ co and the logarithmic deviations from Bjorken scaling are absent at least in this case. Thus, the above models bear some resemblance to super-renormahzable theories (we are dealing with a model in four dimensions!). In conclusion, the models investigated In this paper exhibit some unexpected properties. To find out whether these survive, one must eventually face up to the laborious task o f computing higher order corrections m perturbation theory since, as has been demonstrated in this letter, the usual stability assumptions may no longer hold in presence of infinitely many particles so the computation of the lowest order coefficient of the/3-function might not suffice to tell whether a model is asymptotically free or not. It is also worthwhile to point out that in higher orders there is an increasing variety o f coupling constants to choose from. For instance, by making a redefinition
~(M) = g(M) + C(M) g5(M) +
222
....
(27)
18 June 1979
the second order coefficient o f the/3-function may be changed. It is quite possible that by performing such redefinitions to arbitrary orders one is led to non-equivalent theories. Of course, these and other possibilities remain a subject for future studies. In view of the growlng number o f quark flavors being discovered, models with infinitely many pamcles (quarks or otherwise) certainly should not yet be excluded from the game and might even become relevant experimentally one day. This work would not have been possible without the help and encouragement by Professor O. Nachtmann whom I also want to thank for carefully reading the manuscript. My thanks are also due to Professor B. Stech for a valuable discussion and to Dr. W. Wetzel for helping me with the numerical analysis of eq. (7).
References [1 ] O. Nachtmann and W. Wetzel, Nucl Phys. B146 (1978) 273. [2] A de Rulula and H. Georgi, Phys Rev. D13 (1976) 1296; E.C. Pogglo, H.R Qumn and S. Weinberg, Phys. Rev. D13 (1976) 1958; H. Georgl and H.D. Politzer, Phys. Rev D14 (1976) 1829. [3] E.C. Tltchmarsh, The theory of functions (Oxford, 1939). [4] R. Oehme and W. Zlmmermann, Phys. Lett. 79B (1978) 93. [5] S.L. Adler, Phys. Rev. D10 (1974) 3714. [6] N. Christ, B. Hasslacher and A.H. Mueller, Phys. Rev D6 (1972) 3543. [7] H Georgl and H.D. Pohtzer, Phys Rev. D9 (1974) 417, D.J. Gross and F. Wflczek, Phys. Rev. D8 (1973) 3633; D9 (1974) 980.
PHYSICS LETTERS
18 June 1979
ASYMPTOTIC FREEDOM WITH INFINITELY MANY QUARKS? H. NICOLAI [nstltut f~r Theoretische Physik, Universitat HeMelberg, Germany Received 3 January 1979
Relying on a recent recalculation of the &function in massive QCD, we show that a theory with infinitely many quark flavors is asymptotically free if the quark masses grow as rn2n = n °Mg, 1 < a < 3.6; the precise defimtion of the coupling constant (via the three-gluon vertex) enters in an essential way. In these models the running coupling constant decays hke an inverse power of Q2 as Q2 ~ 0o. Some phenomenologlcal implications of this curious result are discussed.
In a recent publication [ 1], Nachtmann and Wetzel recalculated the/3-function in massive QCD and obtained a result at variance with the conventional one [2] ; the discrepancy is explained by the fact that in ref. [1] the three-gluon vertex instead of the ghost-gluon vertex is used to define the renormalized coupling constant. In both cases, the result may be expressed as follows (the notation is the same as in ref. [1] ; f i s the number of flavors) 13(g, m, (x, M) -
(1)
g3 { 2 1 f/ :1 [ (rrt2~7} -11-3~ -h + O ( g 5) 167r2 "= \ M 2 lJ '
which makes the coefficient o f g 3 in eq. (1) stay negative for all M (choose M sufficiently large). The same is true for h(~) ---hNW(~) and 17 < f < oo but n o t for h (~) = hNW (~) and an infinite number of flavors if the quark masses are suitably arranged as will be exemplified below. The result shows that the question of which coupling constant is "best" may be not merely one of calculational convenience in a theory with an infinite number of massive quarks. Taking now f = oo, it is not difficult to prove that, at least up to the one-loop approximation, the condition oo
n~l 1 < o o, = rn 2
(4)
where, convennonally, hconv.(~) = 6~ 12~2
In (1 + 4 0 1 / 2 + 1
(1 + 4~) 1/2
(1 + 4~) 1/2 -- 1 '
(2)
ensures the convergence of all renormalized quantities of the theory * x, in fact, up to this order, (4) is necessary and sufficient. To be more explicit, we will assume m2=n°M
2,
o>l,
(5)
whereas Nachtmann and Wetzel obtain
1
hmv(~) = 18~ f dx 0
1/3 [ +2fdy.O,)
xO -x) x(1 - x ) + ~
~
3~2 ]
(3)
0 From these equations it may be seen that for f~> 17 and h (~) -- hconv" (~) there is no possible choice o f m 2
where n numbers the flavours and M 0 is an arbitrary mass. Because o > 1, (4) is evidently satisfied. The main result o f this paper is contained in the Theorem: If the quark masses are given by (5), the following estimate is optimal ¢ 1 This need not be true for unrenormalized quantitms. In the nonperturbative lattice regularizatlon of the model, the condition (4) emerges quite naturally. 219
Volume 84B, number 2
PHYSICS LETTERS
1/°
il < II - hNW \ M 2 ]_j
5
n=l
(6)
where the function ~0(o) is defined by
{'
ref. [3] ). In the hmlt M ~ ~o, the nonleadlng terms stay bounded and the estimate (6) holds. Note that for o = 2 m , m C hi, explicit calculations may be carried out by means of [3] oo
1 _ n=l x 2 + n 2
~0(o)= 18 f [x(1 -x)] l+l/°dx 0
18 June 1979
1 + ~-cothrrx 2x 2
(11)
oo
- 2(2 + 3/o)
;
p(y)yl/°dy
o
/; o
1 at l+t°
(7)
On the interval 1 < o <~ 3.6 *2 , ¢(o) < 0 and the coefficient o f g 3 in eq. (1) becomes arbitrarily negative as
_
n = l X4 + n 4
l
+ n__im~
2x 4
2x 3
1 cothTr,vF~xl
Lx/-i
and similar formulas derivable from eqs. (1 1). These may serve to check (6). For M >>M0, 13(g(M)) ,., 24rr2 (.~202)~0(o) "M 2"1/0 g3(M)+ O(g5(M)).
For the proof of the theorem, use is made of the formula (o > 1) oo
1 n=l
if
_
X ° + t1°
X°-I
_ _ d r +O(x_O) ,
0
(8)
dt 0
1
1 +to
1
G
F(x)
x n =1 1 +(n/x) °
x
(9)
with F@ C ~, 0 < F ( x ) < 1 and F ' ( x ) = O ( x - 1 ) . Differentiating eq. (8) with respect to x and observing that F is smooth, one gets
n = 10C °
+
n°) 2
o- 1 a
1
T
dt
x2°-I
0J
1 +t°
+
O(x-2°),
oo
n °
n = l (X ° + n O ) 2
_ 1 1 ax°-I
; o
(10)
Inserting these formulas on the left-hand side of (6), we may interchange summation and integration I f M o / M > 0 (e.g. by the dominated convergence theorem, see ,2 The numerical error on this number Is about 0.1. Note that 13 h m ° ~ l {...} = - ~-d < 0 and h m o ~ ~ {...} = 1 > 0.
220
g2(M) "-' [const. + oltp(o)l [M 2 ~/°1-1 247r 2 \M02 ]
J
(13) '
and the running coupling constant drops to zero much more rapidly in the limit M ~ oo than in the case o f f <~ 16 zero mass quarks ,3. This feature persists if a finite number of quarks with arbitrary masses is added to our models. Of course, one could easily construct in this manner a model where the coupling constant g2(M) decays logarithmically at low (= present day) energies as observed in experiment whereas at higher energies the faster decay of eq. (13) sets m. The rapid decrease ofg2(M) also brings down the "meeting point" of strong and electromagnetic Interactions by several orders of magnitude which may be welcome to "grand synthesizers". Until now, we have been slightly cavalier about the runmng quark masses treating them as if they were constant. However, since they evolve according to (m the gauge ~ = 0, cf. ref. [ 1 ] ) dmn(M) _
dt +O(x_O) " ] + t°
(12)
Thus, on 1 < o <~3.6,
1 +t°
which follows from
J
dM
g2(M) mn(M) 27r2
+ O(g4(M)),
M
f1
xax
0 x + m 2 (M)/M 2 (14)
4:3 Of course, if 3(g) = g3 {13o(M) + 31 (M)g 2 + -..}, this argument is valid only if, in the limit M ~ 0% 31 (M)g 2 (M) = o(30(214)). Although this Is always tacitly assumed to be true, only higher-order computations can justify such a procedure.
Volume 84B, number 2
PHYSICS LETTERS
we have, to lowest order,
g2(M) mn(M) 2rr 2
M
dmn(M) t
~<
d)l//
~<0.
(15)
Inserting (13), one easily verifies that mn(M ) approaches a (non-vanishing) constant quite rapidly in the hmlt M-+ oo, by the same token, the condition (4) is preserved in this limit. This shows at least, if nothing else, that the foregoing considerations are not Inconsistent (if one wants to avoid this comphcatlon, one may define the masses Independently of the normalization point, e.g. as starting points of branch cuts [4] ). To justify the use of the 3-gluon coupling constant g(M) in the considerations below, we have to establish the connection between g(M) and the quark-gluon coupling constant ~(M) (or, equivalently, the ghost-gluon couphng constant). To lowest order we find / f ( M ) g (M) - { C O
we might be led to the erroneous conclusion that g(M) does not converge to zero in this hmlt. This difficulty does not occur for finitely many quarks since, in that case, one may treat all quarks as massless if M >2> max
l ~<1<)'
m1
In the models of this paper, the electromagnetic quark current takes the form
ju = ~ enqn T*tqn "
(21)
n=l
Assuming that the charges en of the quarks stay bounded (e.g. with alternating charges 2/3 and - 1 / 3 ) , we find for Adler's vacuum polarization function D(Q 2) [5], using (5) and (8),
R(s)
[02 ~lJo
D ( Q 2 ) = Q 2 0 ds ( s + Q 2 ) 2 ' Q - * ~ > const.l~ \M02]/
1 +SI12fdxx(1-x)ln{l+M2x(1-x)) 1=1 c
18 June 1979
' (22)
m2 where
lj, dyp(y) ln (1 +,M21m?f
-4If
R (s) = e(e+e-- -+ h adrons) o(e+e - -+ ~+bl-)
,Lj
?/3
-6 0
"M2
II
(22) lmphes, for s ~ ~,
g3(M)
R(s) ~_ const.(s/M2) l/° ,
dyp(y) rnf + y M 2 j j 3;I~2
+O(gS(M)),
IC0l<~,
(16)
and, by (8) and (13)
g(V)~-g(V)+O(1)(V2/M2)l/°g3(M)-~O,
(17)
as M-+ ~ (this IS also a consequence of the WardIdentity relating the 3-gluon and the ghost gluon vertex). (17) seems to contradict what one would obtain from a direct computation via eq. (I) and h = hconv.. To see that there IS no paradox involved, suppose that
Mdg(M)/dM= -A(M2/M2)g3(M),
A >0,
(24)
and therefore, in (22), no extra subtractions are necessary as 1/o < 1. (24) may be understood Intuitively since, by (5), the number of quarks up to a given s grows like (s/M2) 1/~. The rapid decay o f g ( M ) also leads to some Interesting predictions for deep inelastic structure functions, the moments of which are related to coefficient functions occurring In the Wilson expansion of the product of two currents Typically, these coefficient functions F (n) satisfy a renormahzatlon group equation of the form [6]
(18)
(M ~
were exact and define a new couphng constant by
~(M) =g(M) + A (M2/M2)g3(M) .
(23)
0
0 ~F(n) + 3g ~g +]~1 3miami ,
(19)
From eqs (18) and (19), it follows thatg(M),~(M) 0 as M-+ ~, whereas by neglecting higher orders in
Mdf(M)/dM = +.4 (MZ/M2)f3(M) + O(~5(M)) , (20)
+ T(n)F (n) = 0 ,
(25)
where T (n) is the anomalous dimension of the corresponding operator in the Wilson expansion, also, we only consider nonsinglet operators with flavor struc221
Volume 84B, number 2
PHYSICS LETTERS
ture ~u - dd, for instance. If the canonical scahng has already been factored out o f F (n), eq. (25) is solved by
F(n)(O A ml(M) ) \M 2 ' M ,g(M)
(26)
[ ml(Q),g(Q)) r- log0/,_ (n) t =F(") [1, 0 - e x p / f-rogV/. 3' ( )
dtn /"
Now, 7(n)(t) = c(n) g 2 (ge t) + .... where C (n) is one of the coefficients computed in ref. [7], because o f (13), the integral In the exponent of (26) remains finite In the limit Q ~ co and the logarithmic deviations from Bjorken scaling are absent at least in this case. Thus, the above models bear some resemblance to super-renormahzable theories (we are dealing with a model in four dimensions!). In conclusion, the models investigated In this paper exhibit some unexpected properties. To find out whether these survive, one must eventually face up to the laborious task o f computing higher order corrections m perturbation theory since, as has been demonstrated in this letter, the usual stability assumptions may no longer hold in presence of infinitely many particles so the computation of the lowest order coefficient of the/3-function might not suffice to tell whether a model is asymptotically free or not. It is also worthwhile to point out that in higher orders there is an increasing variety o f coupling constants to choose from. For instance, by making a redefinition
~(M) = g(M) + C(M) g5(M) +
222
....
(27)
18 June 1979
the second order coefficient o f the/3-function may be changed. It is quite possible that by performing such redefinitions to arbitrary orders one is led to non-equivalent theories. Of course, these and other possibilities remain a subject for future studies. In view of the growlng number o f quark flavors being discovered, models with infinitely many pamcles (quarks or otherwise) certainly should not yet be excluded from the game and might even become relevant experimentally one day. This work would not have been possible without the help and encouragement by Professor O. Nachtmann whom I also want to thank for carefully reading the manuscript. My thanks are also due to Professor B. Stech for a valuable discussion and to Dr. W. Wetzel for helping me with the numerical analysis of eq. (7).
References [1 ] O. Nachtmann and W. Wetzel, Nucl Phys. B146 (1978) 273. [2] A de Rulula and H. Georgi, Phys Rev. D13 (1976) 1296; E.C. Pogglo, H.R Qumn and S. Weinberg, Phys. Rev. D13 (1976) 1958; H. Georgl and H.D. Politzer, Phys. Rev D14 (1976) 1829. [3] E.C. Tltchmarsh, The theory of functions (Oxford, 1939). [4] R. Oehme and W. Zlmmermann, Phys. Lett. 79B (1978) 93. [5] S.L. Adler, Phys. Rev. D10 (1974) 3714. [6] N. Christ, B. Hasslacher and A.H. Mueller, Phys. Rev D6 (1972) 3543. [7] H Georgl and H.D. Pohtzer, Phys Rev. D9 (1974) 417, D.J. Gross and F. Wflczek, Phys. Rev. D8 (1973) 3633; D9 (1974) 980.