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Renormalisability and nonproduction in complex sine Gordon model
Volume 133B, number 5
PHYSICS LETTERS
22 December 1983
RENORMALISABILITY AND NONPRODUCTION IN COMPLEX SINE GORDON MODEL Guy BONNEAU Laboratoire de Physique Th~orique et Hautes Energies 1, Universit~ Pierre et Marie Curie, Tour 16.15, 4, place Jussieu, 75230 Paris Cedex 05, France
Received 9 Augustus 1983
Two-loop renormalisability of the complex sine Gordon model is studied. We then show how the nonproduction properties of the classical theory can be maintained to one loop order if a finite "counterterm" is added to the lagrangian: it happens to be precisely the one needed to maintain to two-loop order the previously found one loop vanishing of the CaUanSymanzik ~3function.
1. Introduction. The study o f two dimensional models with special classical properties such as nonproduction, infinite number o f conserved currents, nonlocal charges .... offers nice applications o f quantum field theory methods and renormalisation program. In this letter we give some new results for one o f these completely integrable models, the so-called complex sine Gordon model [1] :
1 ~ * £- 2 1 _g2~ff*
m2~*--£1
+m2£2,
(1)
(respectively 4 * ) is a complex field of charge +1 (respectively - 1). The one-loop properties o f this model (which possesses an infinite number o f classically conserved currents and has a factorizable S matrix), have been studied by Maillet and de Vega [2] who showed that: (i) It is multiplicatively renormalisable, with a vanishing ~ function in its C a l l a n - S y m a n z i k equation (to one-loop order). (ii) The factorization condition for the 2 ~ 2 S matrix is no longer satisfied but can be restored i f a O(h) (ff~*)2 " c o u n t e r t e r m " is added to the lagrangian (1): they also made the conjecture that this "counterterm" should be necessary for higher order i Laboratoire Associ6 au CNRS no LA 280. 0.031-9163/83/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
renormalisability. The study o f this model to all orders o f perturbation theory supposes the knowledge o f its intrinsic definition. One should first answer the question: which symmetry defines the theory? We have not been able ,1 to find any local symmetry for £1- So, in order to understand the model we present here a "phenomenological" approach based on two-loop calculations o f Green functions and 3 ~ 3 S matrix element at one-loop order. 2. Green f u n c t i o n s ( t w o loop-order). We have computed (with dimensional regularisation) to two-loop order the divergent parts o f 1"2(/9,-p) and F4(p 1 ,P2, P3,P4). The corresponding graphs being drawn on figs. 1 and 2, the results confirm perturbative renormalisability ,2 with the following expressions for the renormalisation constants ,3
,1 The nonproduction property is not sufficient to define the model since it allows a field redefinition which changes the renormalisability character of a theory [ 3]. Morever, model (1) is not the only one with 0(2) symmetry and nonproduction character [4]. ,2 Details of the calculations will be published elsewhere [4]. ,3 1/~ = 1/(1 - d/2) - ~ + log 4n where d is the space-time dimension and 0' the Euler constant (we used the modified minimal substraction scheme MS). 341
Volume 133B, number 5
PHYSICS LETTERS
22 December 1983
43 Fig. 1. Two loops contributions to p2 (19, --p). Blobs represent insertion of one loop counterterms ( ~ : mass and wavefunction,~8(: four particles coupling).
O ;?
Fig. 3. Two loops contributions to l?2N(p1,P2 ""P2N)" Blobs represent insertion of one-loop counterterms ( ~ : mass and wavefunction,-;~: 2n particles coupling).
All these calculations made us confident in the multiplicative renormalisability of the CSG model (1) to two-loop order. Notice also that Zg2 is now different from 1, so the/3 function no longer vanishes. This can be cured, in a unique manner ,4 by addition of a finite oneloop "counterterm" Fig. 2. Two loops contributions to I`4 (Pl ,P,P3,P4)"Blobs represent insertion of one loop counterterms ( ~ : mass and wavefunction;~(,--~: four and six particles couplings).
A£A =
--½(hg2/4rr)g2[(ffOu ~*)2 + c.c.].
Then, to maintain the renormalisability of the model to two-loop order, a complete series
(hg2/4rr)Xn(g2~t~*)n-lg2 2hg21+(hg2~2(2..+ Z~o = l -
-47~ ~
Zrn: = 1 + 4 ~
\4rr]\e-"2"
1.
g
/
g
1 hg2 g2 [ ( ~ (2)
The striking cancellations that happens in these long calculations make this renormalisability not trivial and indicates that nonlinear symmetries are hidden. For a full two-loop analysis, one needs the divergent part o f all proper Green functions p2N(pl ,P2,.-., P2N). For N 1> 3 we have computed these one only at zero external momenta (fig. 3) and found that they arefinite [4]. As a consequence, no m2(ffff*) N counterterms are needed up to two-loop order (N>~ 2). 342
(4)
is needed with definite values for the coefficients Xn (if no new coupling constant is introduced). One can show [4] that the unique "counterterm" is
[hg212[ 6
Z 2 = 1 + 4(fig2] z 1
[ ( ~ u if*)2 + c.c.],
n = 2,3 ....
4),
(3)
S£A-
2 47r
~ , ) 2 + ( ~ , ~ ~)2] (1 _ g 2 f f f f , ) 2
(5)
In conclusion our two-loop results strongly favour the conjecture of multiplicative renormalisability of the CSG model (1). Moreover, one-loop properties (asymptotic scale invariance and renormalisability) are maintained if, and only if, the series (5) of finite "counterterms" is added to the lagrangian (1). #4 A dimension zero term m 2 (~07;*)2 is of no use and the other dimension two monomial, qJqJ* auqJ au~ * only rescaled the coupling constant.
Volume 133B, number 5
PHYSICS LETTERS
3. S matrix elements (one-loop order). As shown in ref. [2], the factorization equations for 2 ~ 2 S matrix elements are no longer satisfied at one-loop order. This can be cured, in two ways ,s, by addition of an appropriate O(~) finite "counterterm". The first one was considered in ref. [2] A£ B = --m 292 (Bg2/47r) (4 4 *)2
(6)
and, in order not to spoil the two-loop multiplicative renormalisability, has to be accompanied by an infinite series which appears to be (ref. [4] ) oo
A£B = - m 2 g 2 - fig2 ~ (44*)2 ~ 2(n+3) r.2.,..,.*,n n=0 3(n+l)(n+2) ~ ~,~u ) .
(7) The second one ,6 happens to be precisely the A£ A ofeqs. (3)-(5) introduced to maintain the vanishing of the t3 function. Then one has a prejudice in favor of this second issue apparently missed by the authors of ref. [2]. We check our indication by a calculation of the 3 -+ 3 S matrix. The connected amputated on shell Green function ~c ~61amp 'on shell which vanishes at the tree level has been computed with the help o f a computer (ref. [4] ) in the following situations: (i) lagrangian (1) alone, (ii) lagrangian (1) + A.~A eq. (3), (iii) lagrangian (1) + A£ B eq. (6), (iv) lagrangian (1) + [aA£ A + (1 - a ) A £ B ] . The results are:
(i)
i
~hg2 g4 I_2s +
sin2 1 ,
(ii) i hg2 g4 [-12(s - 3m2)] q7~
22 December 1983
~_2 (iv) i ~ g4 [(1 -- a) 12m 2 -- 2(1 -- 5a) (s -- 3m2)1. As factorization o f S 2 ~ 2 is a necessary condition for nonproduction it is not surprising that G 6c amp on shell can only be made to vanish via a local finite O(h) "counterterm" when this condition is satisfied [cases (ii),
(iii), (iv)]. Moreover, Born contributions of ~ m 2 ( 4 4 " ) 3 and h ( 4 4 " ) [ ( 4 ~ u 4*) 2 + c.c.] being respectively proportional to m 2 and (s - 3m2), one immediately finds that the contribution of the second term of the series A£A [eq. (5)] exactly cancels G6 L~. This is not the case for series Ad~B and otA£A (1 -- a ) A £ B. As a conclusion nonproduction can be maintained at the one-loop level (at least factorization for $2_~ 2 and $ 3 ~ 3 ) with addition of a finite "counterterm" ~£A which is just the one needed to maintain to two-loop order asymptotic scale invariance (/3 = 0) and multiplicative renormalisability. From this "phenomenological" approach to the CSG model we make two conjectures: (i) If dimensionally regularised, lagrangian (1) is all-order renormalisable. (ii) There exist local finite "counterterms" which, when added to lagrangian (1) insures the vanishing of the 13function as well ,7 as nonproduction properties to all orders. The first one is 1~ A£A = - - 2 47r
g 2 [ ( 4 0 ~ 4 ~ )2 + ( 4 ~ 4 0 (1
2
* 2
- g 4040)
)2 ] '
where 4^u = Z q;1/2 4 is the bare field We think that these terms would also appear in the quantization o f the infinite local conserved currents of CSG theory [4]. ,7 This link between nonproduction and asymptotic scale invariance appears also in other completely integrable models.
(iii) i hg2 g4 [--2s + 18m 2 ]
References 4:5 Addition of a term qJ~ * ~g~ ~laO* only changes the definition of the couplingg2 or can be absorbed through a field redefinition. ,6 Of course any combination a~£A + (1 - a)Lx£B is also allowed I
[1] K. Pohlmeyer, Commun. Math. Phys. 46 (1976) 207; F. Lund and T. Regge, Phys. Rev. D14 (1976) 1524; [2] H.J. de Vega and J.M. MaiUet, Phys. Lett. 101B (1981) 302. [3] G. Bormeau, Nucl. Phys. B221 (1983) 178. [4] G. Bonneau, in preparation.
343
PHYSICS LETTERS
22 December 1983
RENORMALISABILITY AND NONPRODUCTION IN COMPLEX SINE GORDON MODEL Guy BONNEAU Laboratoire de Physique Th~orique et Hautes Energies 1, Universit~ Pierre et Marie Curie, Tour 16.15, 4, place Jussieu, 75230 Paris Cedex 05, France
Received 9 Augustus 1983
Two-loop renormalisability of the complex sine Gordon model is studied. We then show how the nonproduction properties of the classical theory can be maintained to one loop order if a finite "counterterm" is added to the lagrangian: it happens to be precisely the one needed to maintain to two-loop order the previously found one loop vanishing of the CaUanSymanzik ~3function.
1. Introduction. The study o f two dimensional models with special classical properties such as nonproduction, infinite number o f conserved currents, nonlocal charges .... offers nice applications o f quantum field theory methods and renormalisation program. In this letter we give some new results for one o f these completely integrable models, the so-called complex sine Gordon model [1] :
1 ~ * £- 2 1 _g2~ff*
m2~*--£1
+m2£2,
(1)
(respectively 4 * ) is a complex field of charge +1 (respectively - 1). The one-loop properties o f this model (which possesses an infinite number o f classically conserved currents and has a factorizable S matrix), have been studied by Maillet and de Vega [2] who showed that: (i) It is multiplicatively renormalisable, with a vanishing ~ function in its C a l l a n - S y m a n z i k equation (to one-loop order). (ii) The factorization condition for the 2 ~ 2 S matrix is no longer satisfied but can be restored i f a O(h) (ff~*)2 " c o u n t e r t e r m " is added to the lagrangian (1): they also made the conjecture that this "counterterm" should be necessary for higher order i Laboratoire Associ6 au CNRS no LA 280. 0.031-9163/83/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
renormalisability. The study o f this model to all orders o f perturbation theory supposes the knowledge o f its intrinsic definition. One should first answer the question: which symmetry defines the theory? We have not been able ,1 to find any local symmetry for £1- So, in order to understand the model we present here a "phenomenological" approach based on two-loop calculations o f Green functions and 3 ~ 3 S matrix element at one-loop order. 2. Green f u n c t i o n s ( t w o loop-order). We have computed (with dimensional regularisation) to two-loop order the divergent parts o f 1"2(/9,-p) and F4(p 1 ,P2, P3,P4). The corresponding graphs being drawn on figs. 1 and 2, the results confirm perturbative renormalisability ,2 with the following expressions for the renormalisation constants ,3
,1 The nonproduction property is not sufficient to define the model since it allows a field redefinition which changes the renormalisability character of a theory [ 3]. Morever, model (1) is not the only one with 0(2) symmetry and nonproduction character [4]. ,2 Details of the calculations will be published elsewhere [4]. ,3 1/~ = 1/(1 - d/2) - ~ + log 4n where d is the space-time dimension and 0' the Euler constant (we used the modified minimal substraction scheme MS). 341
Volume 133B, number 5
PHYSICS LETTERS
22 December 1983
43 Fig. 1. Two loops contributions to p2 (19, --p). Blobs represent insertion of one loop counterterms ( ~ : mass and wavefunction,~8(: four particles coupling).
O ;?
Fig. 3. Two loops contributions to l?2N(p1,P2 ""P2N)" Blobs represent insertion of one-loop counterterms ( ~ : mass and wavefunction,-;~: 2n particles coupling).
All these calculations made us confident in the multiplicative renormalisability of the CSG model (1) to two-loop order. Notice also that Zg2 is now different from 1, so the/3 function no longer vanishes. This can be cured, in a unique manner ,4 by addition of a finite oneloop "counterterm" Fig. 2. Two loops contributions to I`4 (Pl ,P,P3,P4)"Blobs represent insertion of one loop counterterms ( ~ : mass and wavefunction;~(,--~: four and six particles couplings).
A£A =
--½(hg2/4rr)g2[(ffOu ~*)2 + c.c.].
Then, to maintain the renormalisability of the model to two-loop order, a complete series
(hg2/4rr)Xn(g2~t~*)n-lg2 2hg21+(hg2~2(2..+ Z~o = l -
-47~ ~
Zrn: = 1 + 4 ~
\4rr]\e-"2"
1.
g
/
g
1 hg2 g2 [ ( ~ (2)
The striking cancellations that happens in these long calculations make this renormalisability not trivial and indicates that nonlinear symmetries are hidden. For a full two-loop analysis, one needs the divergent part o f all proper Green functions p2N(pl ,P2,.-., P2N). For N 1> 3 we have computed these one only at zero external momenta (fig. 3) and found that they arefinite [4]. As a consequence, no m2(ffff*) N counterterms are needed up to two-loop order (N>~ 2). 342
(4)
is needed with definite values for the coefficients Xn (if no new coupling constant is introduced). One can show [4] that the unique "counterterm" is
[hg212[ 6
Z 2 = 1 + 4(fig2] z 1
[ ( ~ u if*)2 + c.c.],
n = 2,3 ....
4),
(3)
S£A-
2 47r
~ , ) 2 + ( ~ , ~ ~)2] (1 _ g 2 f f f f , ) 2
(5)
In conclusion our two-loop results strongly favour the conjecture of multiplicative renormalisability of the CSG model (1). Moreover, one-loop properties (asymptotic scale invariance and renormalisability) are maintained if, and only if, the series (5) of finite "counterterms" is added to the lagrangian (1). #4 A dimension zero term m 2 (~07;*)2 is of no use and the other dimension two monomial, qJqJ* auqJ au~ * only rescaled the coupling constant.
Volume 133B, number 5
PHYSICS LETTERS
3. S matrix elements (one-loop order). As shown in ref. [2], the factorization equations for 2 ~ 2 S matrix elements are no longer satisfied at one-loop order. This can be cured, in two ways ,s, by addition of an appropriate O(~) finite "counterterm". The first one was considered in ref. [2] A£ B = --m 292 (Bg2/47r) (4 4 *)2
(6)
and, in order not to spoil the two-loop multiplicative renormalisability, has to be accompanied by an infinite series which appears to be (ref. [4] ) oo
A£B = - m 2 g 2 - fig2 ~ (44*)2 ~ 2(n+3) r.2.,..,.*,n n=0 3(n+l)(n+2) ~ ~,~u ) .
(7) The second one ,6 happens to be precisely the A£ A ofeqs. (3)-(5) introduced to maintain the vanishing of the t3 function. Then one has a prejudice in favor of this second issue apparently missed by the authors of ref. [2]. We check our indication by a calculation of the 3 -+ 3 S matrix. The connected amputated on shell Green function ~c ~61amp 'on shell which vanishes at the tree level has been computed with the help o f a computer (ref. [4] ) in the following situations: (i) lagrangian (1) alone, (ii) lagrangian (1) + A.~A eq. (3), (iii) lagrangian (1) + A£ B eq. (6), (iv) lagrangian (1) + [aA£ A + (1 - a ) A £ B ] . The results are:
(i)
i
~hg2 g4 I_2s +
sin2 1 ,
(ii) i hg2 g4 [-12(s - 3m2)] q7~
22 December 1983
~_2 (iv) i ~ g4 [(1 -- a) 12m 2 -- 2(1 -- 5a) (s -- 3m2)1. As factorization o f S 2 ~ 2 is a necessary condition for nonproduction it is not surprising that G 6c amp on shell can only be made to vanish via a local finite O(h) "counterterm" when this condition is satisfied [cases (ii),
(iii), (iv)]. Moreover, Born contributions of ~ m 2 ( 4 4 " ) 3 and h ( 4 4 " ) [ ( 4 ~ u 4*) 2 + c.c.] being respectively proportional to m 2 and (s - 3m2), one immediately finds that the contribution of the second term of the series A£A [eq. (5)] exactly cancels G6 L~. This is not the case for series Ad~B and otA£A (1 -- a ) A £ B. As a conclusion nonproduction can be maintained at the one-loop level (at least factorization for $2_~ 2 and $ 3 ~ 3 ) with addition of a finite "counterterm" ~£A which is just the one needed to maintain to two-loop order asymptotic scale invariance (/3 = 0) and multiplicative renormalisability. From this "phenomenological" approach to the CSG model we make two conjectures: (i) If dimensionally regularised, lagrangian (1) is all-order renormalisable. (ii) There exist local finite "counterterms" which, when added to lagrangian (1) insures the vanishing of the 13function as well ,7 as nonproduction properties to all orders. The first one is 1~ A£A = - - 2 47r
g 2 [ ( 4 0 ~ 4 ~ )2 + ( 4 ~ 4 0 (1
2
* 2
- g 4040)
)2 ] '
where 4^u = Z q;1/2 4 is the bare field We think that these terms would also appear in the quantization o f the infinite local conserved currents of CSG theory [4]. ,7 This link between nonproduction and asymptotic scale invariance appears also in other completely integrable models.
(iii) i hg2 g4 [--2s + 18m 2 ]
References 4:5 Addition of a term qJ~ * ~g~ ~laO* only changes the definition of the couplingg2 or can be absorbed through a field redefinition. ,6 Of course any combination a~£A + (1 - a)Lx£B is also allowed I
[1] K. Pohlmeyer, Commun. Math. Phys. 46 (1976) 207; F. Lund and T. Regge, Phys. Rev. D14 (1976) 1524; [2] H.J. de Vega and J.M. MaiUet, Phys. Lett. 101B (1981) 302. [3] G. Bormeau, Nucl. Phys. B221 (1983) 178. [4] G. Bonneau, in preparation.
343