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XI. The Borel transform and the renormalization group
PHYSICS REPORTS (Review Section of Physics Letters) 49, No. 2 (1979) 215-219. North-Holland Publishing Company
Xl. The Borel Transform and the Renormalization Group
G. PARISI* Laboratoire de Physique Thdorique de rEcole Normale Supdrieure*, Paris, France
Abstract" We study the consequences of the renormalization group on the large momenta behaviour of the Borel transform of the Green's functions. We discuss the implications on the structure of the ultraviolet singularities of the Borel transform.
* On leave of absence from INFN, Frascati (Italy). t Laboratoire Propre du C.N.R.S., associ~ ~ rEcole Normale Sup~rieure et ~ rUniversit6 de Paris-Sud. Postal address: 24, rue Lhomond, 75231 Paris Cedex 05, France.
Non-perturbative aspects in quantum fieid theory
216
It is quite difficult to study directly the solutions of the Dyson-Schwinger (D-S) equations of motion in a renormalizable field theory. Most of our non perturbative information come from the use of the renormalization group equations. apply the reno~alization g r o u p equations to the study of the large momenta ~ h a v i o ~ 0f the Borel transform of the Green's functions, and we c.heck that the results we find are consistent with the Borel transformed D-S equations. As a consequence the Borel transform is ultraviolet singular on the positive real axis in non asymptotically free field theories. The Borel summability of the Green's functions is no more valid. Let us consider the 0(02) 2 theory in dimension D - 4, where ~ is an N component scalar field. We define the Borel transform B(b) of the function A(O) by" at)
00
A(g) = K~ AKgK+ ',
B(b) = ~ (Ar/K!)b K.
=0
(I)
K=0
If A K grow like K! B(b)is analytic in the neighbourhood of the origin. If the function A(g) has "good" analyticity properties, the following relation holds [1] oo
A(g) = / exp (-b/g)B(b) db.
(2)
el
0
Vice versa, if the function B(b) can be analytically continued on the positive real axis up to infinity. the function defined by eq. (2) (provided that the integral converges) is the Borcl sum of the asymptotic expansion, eq. (1). The renormalization group equations for the Green's functions are
+ ~(0)--~ "4- dN "~ (~N(~) FN(P,O) =
0.
(3)
Let us work in the approximation in which fl(g) -- fl2g 2"
(~N(g) -- O.
(4)
The renormalization group equations for the Borel transform CN(P,b) of FN(p,g ) are:
-P-~p + fl2b + dN CN(P,b)= 0.
(5)
The solution is simply given by"
CN(p, b) = rN(b)pdr~(p/kt)tJ2b,
(6)
p being a normalization point. The deviation (proportional to (p/la)~2b)of the large momenta behaviour of the Green's function from the canonical value will play a crucial role in the rest of the paper. It is easy to check that. if we remove the approximation, eq. (4), we find, in the large p limit, the same result as eq. (6), apart from logarithmic corrections. Indeed the Bore| transform of eq. (3) is b
[B(b -
-p-~p + fl2b + dN CN(p, b) + 0
b')b' + ~N(b')] CN(P, b') db' = 0,
(7)
G. Parisi, T h e Bore! translorm and the renormalization group
217
where (13
[fl(g)- flz(~2]/g 2 =
-gila + O(g2) = f exp(-b/g)fl(b)db, 0 oo
6N(O)=
-6~g
(8)
+ 0(0 2) = I exp (-b/O)6s(bidb.
In the large momenta region, eqs. (7) and (8) imply t h a t
CN(p, b) ---* p"~ oO Ys(b)pa"(p/#)tJ2b(ln p/g){-(B,b+6;,m2}
(9)•
The large momenta behaviour of CN(p, b) given by eq. (9) is in agreement with the arguments of refs. [2-3]. Eq. (9)does not imply the presence of a Landau ghost for positive real values of g, e.g. the function fl(g) has a zero at g = fla/fl2 if fl(b) = fla; in this case the large p limit can be studied using the standard fixed point renormalization group analysis [4]. It L, :rucial to realize that the behaviour of FN(P, g) for large p depends on the analytic properties of CN(P,b) as function of both variables and not on the behaviour of CN(p, b) in the limit p -, :¢, considered as function of p only. Also, eq. (6) is compatible with the absence of a Landau ghost" e.g.. if rN(b) = exp (-b2), we find"
p--j
oo
FN(p,g) =
exp[ -b2 -
b/g + fiEb ln(p/#)]db.
(10)
v
In this case the perturbative expansion is asymptotic to the FN(P,O) function (as given by eq. (10)) only in the sector Re g >i 0. The theory is not asymptotically free for negative g" in this case we cannot apply the arguments of refs. [2, 5] on the existence of a Landau ghost in a theorx which is asymptotically free only for negative g. The large momenta behaviour of the Green's functions recalls non-renormalizable theories. In order to study the effects on the D - S equations, it may be convenient to write directly the Borel transformed D - S equations, using the convolution theorem for the Borel transform" b
b
B[Fl...FN] = f dbl.., f dbNfl(bl)...fN(bN)6(bl +... + bN-b), o
(11}
B[F,] = A(b), where B stands for "Borel transform of". Using dimensional ana ly sis it can b e -~.h~,.~,.u ~..,...a ,h.., ,,,,,, ,h~ ,,,,. 8 ~',,,.,.,;..., ,u,,,.,,,, . . . i,~ . . . ~.,, ,t. ¢1 ,. !) assures th,. . . . . ,~elf... consistent reproduction ofthe factor (p/#)a2b in the D - S equations. However. for large fl2b ~ 2). we find ultraviolet divergences, which, according to the B.H.P. theorem [6] are proportional to the insertion of local operators. More precisely, when fl2b = 2, there are logarithmically, divergent diagrams* whose divergence may be compensated by adding the operators of dimension 6 as * The existence of ultraviolet singularities for integer fl2b has been pointed out in ref. [7].
N on-perturbative aspects in quantum field theory
218
counterterms in the Lagrangian, the coefficients of the counterterms being proportional to exp(-1//~20). From this point of view, non asymptotically free field theories differ from non ~ld theories in that, in the first case, the coefficients of the counterterms are expohile in the second case they are proporti0n~ to a power of the coupling constant; Using the renormalization group, a direct analysis [8] s h o ~ that the singularities at b/~2 = 2 are cuts, e.g., if for simplicity we assume that 0 6 is the only operator of dimension 6, (i.e. we neglect operator mixing)~ we find:
Cc,(p,b) ,~ F(1 - o0(2 - fl2b)~'-',
fl2b ~, 2,
0c = 2//3///2 + ~/f12,
(12)
where C6(p, b) is the Borel transform of the 6-point vertex function F e(p, g) and the anomalous dimension of the operator ~6 is ~6(0) = ~ 0 + O(f/2) • The contribution of this ultraviolet singularity to the large order behaviour of the perturbative expansion is given by r6(P, O) "~ ~'. K ! K -1 + ~Or(½fl2) r.
(13)
K
Let us compare these results with the semi-classical arguments [9-12]. The instantons induce a singularity of the Borel transform at b = b~ = - 16~ 2. The ultraviolet singularities of the Borel transform are at b = bu = 2/fl2 = -6bt/(N + 8), the theory being O(N) invariant. For ,V > --2, Ibm! < Ib~[ and the ultraviolet singularities dominate the large order behaviour of the perturbative expansion; for N < - 8 the ultraviolet singularities are on the negative axis; for N = - 14, they are on the top of the instantons; while for N < - 14, the instanton is on the cut produced by the ultraviolet singularities. In this region (N < - 14) the instanton contribution computed in ref. [11] is ultraviolet divergent: the ultraviolet singularities are the nearest to the origin and they shield the instanton singularities. In this case it would be nonsense to try to consider the effects of the instantons, while neglecting the contributions of the ultraviolet singularities. Let us verify the correctness of eq. (13) in the large N limit. For convenience, we rescale the coupling constant 0 in such a way that f12 = 1. In this case the instantoa contribution is negligeable. It is convenient to indicate by a wavy line the geometrical sum of all the bubble diagrams; in fig. 1, we show the relevant diagrams contributing to F6(p, O) in the large N limit. In the massive theory at p -- 0, we find: ]"6(0, 0) --
f
d4q (1 + gF(q)) 3 (q2 -F m2)3 (1 -F all(q)) 3'
(14)
A simple computation shows that: H(p) ---, - I n p + R:L,
F(p) -, - I n p +
\
R 2,
Rt
-- R2
~ 0.
/
Fig. I. Relevant diagram for the large order behaviour in the limit N ~ ~ .
(15)
219
G. Parisi, The Borel transform and the renormalisation group
RI and R2 depend separately on the way the coupling constant is defined, but R1 pendent from the renormalization scheme. Erpanding (14) in powers of g, we find: F6(0,g) ~, ~ Kt K - 4 2 - r g K ( R '
1 - R2) 3.
-
R2
is inde-
(16)
In the limit N --, ~ , ~3/flzz - - , 0 and V~/f12 ~ 3; eq. (16) is in perfect agreement with eq. (13). In order to get the correct result it is crucial to consider all the diagrams relevant at this order, the contribution of a single diagram being proportional to eq. (13) with c~ set equal to zero. The techniques here described can be extended also to the study of the infrared divergences [3]. However, at the present moment, we are far from understanding how to use the perturbative expansion to compute the Green's functions for the coupling constant not too small in the cases when the Borei summability is no more valid and the Borel transform is singular on the real axis. The author is grateful to Prof. K. Symanzik for his collaboration in the study of the limit N ~ ~o. He also acknowledges many illuminating discussions with E. Br6zin, C. ltzykson, J. ZinnJustin and J.B. Zuber.
References [I] [2] [3] [4]
S. Gram, V. Grecchi and B. Simon, Phys. Lett. 32B (1970) 631. G. Parisi, Lectures given at the Carg6se Summer Institute (1977). G. Parisi, On infrared singularities, Nucl. Phys. B, to be published. K. Symanzik, Comm. Math. Phys. 23 (1971) 61; K. Wilson, Phys. Rev. D3 (1971) 1818. [5] N.N. Khuri, Phys. Rev. D12 (1975) 2298. [6] N.N. Bogoliubov, O.S. Parasiuk, Acta Math. 97 (1957) 227: K. Hepp, Comm. Math. Phys. 2 (1966~ 301. [7] G. 't Hooft, Lectures given at Erice (1977}: B. Lautrup, Phys. Lett. 69B (1977) 109: P. Olesen, Phys. Lett. 73B (1977) 327. [8] G. Parisi, Singularities of the Borel transform in renormalizable theories, Phys. Lett. 76B (1978) 65. [9] C.S. Lam, Nuovo Cimento 55 (1968) 258. [10] L.N. Lipatov, JETP 72 (1977) 41 i. [11] E. Br6zin, J.C. Le GuiUou and J. Zinn-Justin, Phys. Rev. DI5 (1977) 1544, 1558. [12] C. ltzykson, G. Parisi and J.B. Zuber, Phys. Rev. Lett. 38 (1977) 306.
Xl. The Borel Transform and the Renormalization Group
G. PARISI* Laboratoire de Physique Thdorique de rEcole Normale Supdrieure*, Paris, France
Abstract" We study the consequences of the renormalization group on the large momenta behaviour of the Borel transform of the Green's functions. We discuss the implications on the structure of the ultraviolet singularities of the Borel transform.
* On leave of absence from INFN, Frascati (Italy). t Laboratoire Propre du C.N.R.S., associ~ ~ rEcole Normale Sup~rieure et ~ rUniversit6 de Paris-Sud. Postal address: 24, rue Lhomond, 75231 Paris Cedex 05, France.
Non-perturbative aspects in quantum fieid theory
216
It is quite difficult to study directly the solutions of the Dyson-Schwinger (D-S) equations of motion in a renormalizable field theory. Most of our non perturbative information come from the use of the renormalization group equations. apply the reno~alization g r o u p equations to the study of the large momenta ~ h a v i o ~ 0f the Borel transform of the Green's functions, and we c.heck that the results we find are consistent with the Borel transformed D-S equations. As a consequence the Borel transform is ultraviolet singular on the positive real axis in non asymptotically free field theories. The Borel summability of the Green's functions is no more valid. Let us consider the 0(02) 2 theory in dimension D - 4, where ~ is an N component scalar field. We define the Borel transform B(b) of the function A(O) by" at)
00
A(g) = K~ AKgK+ ',
B(b) = ~ (Ar/K!)b K.
=0
(I)
K=0
If A K grow like K! B(b)is analytic in the neighbourhood of the origin. If the function A(g) has "good" analyticity properties, the following relation holds [1] oo
A(g) = / exp (-b/g)B(b) db.
(2)
el
0
Vice versa, if the function B(b) can be analytically continued on the positive real axis up to infinity. the function defined by eq. (2) (provided that the integral converges) is the Borcl sum of the asymptotic expansion, eq. (1). The renormalization group equations for the Green's functions are
+ ~(0)--~ "4- dN "~ (~N(~) FN(P,O) =
0.
(3)
Let us work in the approximation in which fl(g) -- fl2g 2"
(~N(g) -- O.
(4)
The renormalization group equations for the Borel transform CN(P,b) of FN(p,g ) are:
-P-~p + fl2b + dN CN(P,b)= 0.
(5)
The solution is simply given by"
CN(p, b) = rN(b)pdr~(p/kt)tJ2b,
(6)
p being a normalization point. The deviation (proportional to (p/la)~2b)of the large momenta behaviour of the Green's function from the canonical value will play a crucial role in the rest of the paper. It is easy to check that. if we remove the approximation, eq. (4), we find, in the large p limit, the same result as eq. (6), apart from logarithmic corrections. Indeed the Bore| transform of eq. (3) is b
[B(b -
-p-~p + fl2b + dN CN(p, b) + 0
b')b' + ~N(b')] CN(P, b') db' = 0,
(7)
G. Parisi, T h e Bore! translorm and the renormalization group
217
where (13
[fl(g)- flz(~2]/g 2 =
-gila + O(g2) = f exp(-b/g)fl(b)db, 0 oo
6N(O)=
-6~g
(8)
+ 0(0 2) = I exp (-b/O)6s(bidb.
In the large momenta region, eqs. (7) and (8) imply t h a t
CN(p, b) ---* p"~ oO Ys(b)pa"(p/#)tJ2b(ln p/g){-(B,b+6;,m2}
(9)•
The large momenta behaviour of CN(p, b) given by eq. (9) is in agreement with the arguments of refs. [2-3]. Eq. (9)does not imply the presence of a Landau ghost for positive real values of g, e.g. the function fl(g) has a zero at g = fla/fl2 if fl(b) = fla; in this case the large p limit can be studied using the standard fixed point renormalization group analysis [4]. It L, :rucial to realize that the behaviour of FN(P, g) for large p depends on the analytic properties of CN(P,b) as function of both variables and not on the behaviour of CN(p, b) in the limit p -, :¢, considered as function of p only. Also, eq. (6) is compatible with the absence of a Landau ghost" e.g.. if rN(b) = exp (-b2), we find"
p--j
oo
FN(p,g) =
exp[ -b2 -
b/g + fiEb ln(p/#)]db.
(10)
v
In this case the perturbative expansion is asymptotic to the FN(P,O) function (as given by eq. (10)) only in the sector Re g >i 0. The theory is not asymptotically free for negative g" in this case we cannot apply the arguments of refs. [2, 5] on the existence of a Landau ghost in a theorx which is asymptotically free only for negative g. The large momenta behaviour of the Green's functions recalls non-renormalizable theories. In order to study the effects on the D - S equations, it may be convenient to write directly the Borel transformed D - S equations, using the convolution theorem for the Borel transform" b
b
B[Fl...FN] = f dbl.., f dbNfl(bl)...fN(bN)6(bl +... + bN-b), o
(11}
B[F,] = A(b), where B stands for "Borel transform of". Using dimensional ana ly sis it can b e -~.h~,.~,.u ~..,...a ,h.., ,,,,,, ,h~ ,,,,. 8 ~',,,.,.,;..., ,u,,,.,,,, . . . i,~ . . . ~.,, ,t. ¢1 ,. !) assures th,. . . . . ,~elf... consistent reproduction ofthe factor (p/#)a2b in the D - S equations. However. for large fl2b ~ 2). we find ultraviolet divergences, which, according to the B.H.P. theorem [6] are proportional to the insertion of local operators. More precisely, when fl2b = 2, there are logarithmically, divergent diagrams* whose divergence may be compensated by adding the operators of dimension 6 as * The existence of ultraviolet singularities for integer fl2b has been pointed out in ref. [7].
N on-perturbative aspects in quantum field theory
218
counterterms in the Lagrangian, the coefficients of the counterterms being proportional to exp(-1//~20). From this point of view, non asymptotically free field theories differ from non ~ld theories in that, in the first case, the coefficients of the counterterms are expohile in the second case they are proporti0n~ to a power of the coupling constant; Using the renormalization group, a direct analysis [8] s h o ~ that the singularities at b/~2 = 2 are cuts, e.g., if for simplicity we assume that 0 6 is the only operator of dimension 6, (i.e. we neglect operator mixing)~ we find:
Cc,(p,b) ,~ F(1 - o0(2 - fl2b)~'-',
fl2b ~, 2,
0c = 2//3///2 + ~/f12,
(12)
where C6(p, b) is the Borel transform of the 6-point vertex function F e(p, g) and the anomalous dimension of the operator ~6 is ~6(0) = ~ 0 + O(f/2) • The contribution of this ultraviolet singularity to the large order behaviour of the perturbative expansion is given by r6(P, O) "~ ~'. K ! K -1 + ~Or(½fl2) r.
(13)
K
Let us compare these results with the semi-classical arguments [9-12]. The instantons induce a singularity of the Borel transform at b = b~ = - 16~ 2. The ultraviolet singularities of the Borel transform are at b = bu = 2/fl2 = -6bt/(N + 8), the theory being O(N) invariant. For ,V > --2, Ibm! < Ib~[ and the ultraviolet singularities dominate the large order behaviour of the perturbative expansion; for N < - 8 the ultraviolet singularities are on the negative axis; for N = - 14, they are on the top of the instantons; while for N < - 14, the instanton is on the cut produced by the ultraviolet singularities. In this region (N < - 14) the instanton contribution computed in ref. [11] is ultraviolet divergent: the ultraviolet singularities are the nearest to the origin and they shield the instanton singularities. In this case it would be nonsense to try to consider the effects of the instantons, while neglecting the contributions of the ultraviolet singularities. Let us verify the correctness of eq. (13) in the large N limit. For convenience, we rescale the coupling constant 0 in such a way that f12 = 1. In this case the instantoa contribution is negligeable. It is convenient to indicate by a wavy line the geometrical sum of all the bubble diagrams; in fig. 1, we show the relevant diagrams contributing to F6(p, O) in the large N limit. In the massive theory at p -- 0, we find: ]"6(0, 0) --
f
d4q (1 + gF(q)) 3 (q2 -F m2)3 (1 -F all(q)) 3'
(14)
A simple computation shows that: H(p) ---, - I n p + R:L,
F(p) -, - I n p +
\
R 2,
Rt
-- R2
~ 0.
/
Fig. I. Relevant diagram for the large order behaviour in the limit N ~ ~ .
(15)
219
G. Parisi, The Borel transform and the renormalisation group
RI and R2 depend separately on the way the coupling constant is defined, but R1 pendent from the renormalization scheme. Erpanding (14) in powers of g, we find: F6(0,g) ~, ~ Kt K - 4 2 - r g K ( R '
1 - R2) 3.
-
R2
is inde-
(16)
In the limit N --, ~ , ~3/flzz - - , 0 and V~/f12 ~ 3; eq. (16) is in perfect agreement with eq. (13). In order to get the correct result it is crucial to consider all the diagrams relevant at this order, the contribution of a single diagram being proportional to eq. (13) with c~ set equal to zero. The techniques here described can be extended also to the study of the infrared divergences [3]. However, at the present moment, we are far from understanding how to use the perturbative expansion to compute the Green's functions for the coupling constant not too small in the cases when the Borei summability is no more valid and the Borel transform is singular on the real axis. The author is grateful to Prof. K. Symanzik for his collaboration in the study of the limit N ~ ~o. He also acknowledges many illuminating discussions with E. Br6zin, C. ltzykson, J. ZinnJustin and J.B. Zuber.
References [I] [2] [3] [4]
S. Gram, V. Grecchi and B. Simon, Phys. Lett. 32B (1970) 631. G. Parisi, Lectures given at the Carg6se Summer Institute (1977). G. Parisi, On infrared singularities, Nucl. Phys. B, to be published. K. Symanzik, Comm. Math. Phys. 23 (1971) 61; K. Wilson, Phys. Rev. D3 (1971) 1818. [5] N.N. Khuri, Phys. Rev. D12 (1975) 2298. [6] N.N. Bogoliubov, O.S. Parasiuk, Acta Math. 97 (1957) 227: K. Hepp, Comm. Math. Phys. 2 (1966~ 301. [7] G. 't Hooft, Lectures given at Erice (1977}: B. Lautrup, Phys. Lett. 69B (1977) 109: P. Olesen, Phys. Lett. 73B (1977) 327. [8] G. Parisi, Singularities of the Borel transform in renormalizable theories, Phys. Lett. 76B (1978) 65. [9] C.S. Lam, Nuovo Cimento 55 (1968) 258. [10] L.N. Lipatov, JETP 72 (1977) 41 i. [11] E. Br6zin, J.C. Le GuiUou and J. Zinn-Justin, Phys. Rev. DI5 (1977) 1544, 1558. [12] C. ltzykson, G. Parisi and J.B. Zuber, Phys. Rev. Lett. 38 (1977) 306.