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Renormalization character and quantum S-matrix for a classically integrable theory
Volume 101B, number 5
PHYSICS LETTERS
21 May 1981
RENORMALIZATION CHARACTER AND QUANTUM S-MATRIX FOR A CLASSICALLY INTEGRABLE THEORY
H.J. de VEGA and J.M. MAILLET Laboratoire de Physlque Theonque et Hautes Energt s , Universitd Pierre and Marie Cuttle, 75230 Paris Cddex 05, France •
"
"
ie
1
Received 13 February 1981
The quantum theory of a N-component generalization of the sine-Gordon model is investigated. We find at the one-loop order that the model is renormalizable only when the corresponding classical theory is completely integrable: N = 1 (sineGordon model) and N = 2 (reduced 0(4) a-model). Moreover the coupling constant does nor renormalize in these two cases. Although the S-matrix for N = 2 is factorizable at the tree level, an anomaly appears at the oneqoop order. Its effect is like a local quartic coupling.
An impressive amount o f results are known at present for the quantum sine-Gordon (SG) model where the classical theory is completely integrable [1,2]. In this context, it is indeed appealing to investigate the quantum theories associated to completely integrable classical models. One would like to know which of the nice classical features of a relativistic model (spectrum, infinite number of conservation laws) have a quantum counterpart. Also, what are their influences (if any) on the renormalization group properties o f the quantum theory. The reduction procedure [3 ] allows to construct new classical integrable models which are related by gauge equivalence [4]. In this way the 0(3) o-model leads to SG theory. More generally, the O(N) and the CP N - 2 nonlinear o-models (with N t> 4) can be reduced to generalizations of classical sine-Gordon [3,5,6] or Gordon [7] models• In this letter we investigate the quantum theory of a first model of this class: the 0(4) reduced o-model [also called complex sine-Gordon (CSG) [5,8]. We start from the lagrang~an = ½ ((3~*) 2 / [1 - g 2 ( , ) 2 1 - m 2 ( ~ ) 2 }
(1)
at two space-time dimensions. There ~ = (qJl, ~02..... qJN) is a real field. F o r N = 1 this model reduces to SG through the transformation ~O1 (x)= g - ' l sin~sa(x)/2). For N = 2 it is gauge equivalent to the 0 ( 4 ) sigma-model with non-zero boundary conditions at infinity [3]. We quantize model (1) in the usual way by splitting .t2into free and interacting parts cQ
./9=./90+Zl,
Zo(X)=½[(au,)2_m2(,)2],
./91(x)=~1 k~_j = 1 ( 0 u , ) 2 g 2k ( , 2 ) k .
(2)
We note that there is an infinite number o f interaction vertices. We use dimensional regularization [9] and i Laboratoire associ6 au CNRS.
302
0 0 3 1 - 9 1 6 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.50 © North-Holland Publishing Company
Volume 101 B, number 5
PHYSICS LETTERS
21 May 1981
and hence no measure term is needed because 8 1 ~ : 0 (where v is the number of space-time dimensions). Power counting tells us that the general f o r m 6 f the counterterms will be
ANZ?(x) = m 2 G N ( ~ 2) + ( a . ~ ) 2 H N ( ~ 2) + ( ~ " ~).~)2IN(~2),
(3)
where G, H and ! are Taylor expandable functions. It is not clear solely by power counting if all ultraviolet divergences can be adsorbed in the renormalization of m 2 , g 2 and ~ . We understand that a quantum field theory is renormalizable if all divergences can be absorbed in the parameters and fields already present in the classical lagrangian (this means, without the freedom of introducing new arbitrary coupling constants). In order to elucidate the renormalization properties of this model we have computed all the relevant quantifies to one-loop order. For the two-point function we get: Pa(21)(bare)(p, _ p , m 2, g) = i~ala2 [p2 _ m 2 + (Ng2/aTr) (p2 + m 2 ) ( 1 / e _ .y + In 47r) + O(e, g4)l ,
(4)
where e = 1 - v/2 and ~ stands for Euler's constant. We then find for the mass and wave function renormalization: 8m 2 = (m2Ng2/2zr) (1/e - "y + In 4rr) + 0(e, g 4 ) ,
Z = 1 - (Ng2/41r)
(lie - "r + In 4 , )
(5)
+ O(e, g4) .
(6)
Using these counterterms we get for the higher 1PI Green's functions
I"(21n.!.,a2n(Pl ..... P 2 n ; m 2 ; g ) - Z
n r(2n) al .....(bare) a2n (Pl .... , P2n ; m2 + 5 m 2 ; g )
(n/> 2),
(7)
The divergent part of
(2n) , ;m2;g) r'~al..... a2n (P l .... P2n 2n-1 =i (2g2)n(n87re- 1)! ( N - 2)((n - 1) /~1"= odd i
Pi'Pi+l -- 12 l ~ i < ~j ¢ 2 n - 1 odd i and j
XSala 2 .... ~2n- 1,2n + permutations.
(Pi + P i + I ) " (Pj + P / + I ) )
(8)
We see that for N = 2 mass and wave function renormalization are enough to eliminate the one-loop poles at two dimensions. The theory can be renormalized in this case without changing the form of Z?. Moreover g does not renormalize. I f N = 1, we have a i = 1 for all i and all permutations add in eq. (8). We get in this way a zero residue at v = 2 which is in agreement with the fact that there is no coupling-constant renormalization in the SG model ~[1 ]. For N i> 3 the model is not renormalizable in the sense precised before because counterterms of the type (g'*/e) X (~" DU~)2 are needed. In order to examine the nature o f the renormalization f o r N = 2 and by analogy with SG theory (N = 1) we try to cast the CSG lagrangian in the form: .6? = (Ouct) 2 + (~/3) 2 + V(t~,/3) + total divergence.
(9)
If this would be possible, normal ordering will eliminate all divergences. We consider the following transformation
~1 =~1 ('~, 3),
~2 = ¢'2('~, 3). 303
Volume 101B, number 5
PHYSICS LETTERS
21 May 1981
Then if for simplicity we assume a zero total divergence in eq. (9) we get that ~ = ~1 + i@2 must be an analytic function o f z = ~ + i/3 in some domain and must verify Id~0/dz 12 + I~k(z)12 = l/g 2. By power expanding qJ in its domain of analyticity it follows that the unique solution is the trivial one (~ = const.). Of course we cannot exclude that a more general transformation could bring &?to the form of eq. (9). Although the resemblance of the one-loop results in real and complex sine-Gordon models, the explanation of the renormalization appears to be more involved in CSG. The O (iV) symmetry is manifestly not enough to explain that, because the theory is O(N) symmetric but not-renormalizable for N t> 3. It must be noted that our model exhibits the property of classical complete integrability precisely for N = 1 and 2 [for N >i 3 the S-matrix is not factorizable already at the tree level (see below) which suggests that the classical theory is not completely integrable]. All this leads to a natural question: what is the general connection between classical integrability and the non-renormalization of coupling constants? Other models that share both properties are the generalized Toda field theories [10]. In these models associated to Lie algebras the only divergent diagram is the tadpole. Hence we can renormalize by simple normal ordering the lagrangian. This renormalizes the mass(es) but not the coupling constant. Also, a related phenomenon is the non-universal critical behaviour exhibited by the exactly integrable eightvertex model [11]. The renormalization group equation (RGE) for the complex sine-Gordon model follows from the above computations
[mO/Om - e g O / O g - 2nT(g)] I"(2n) a (x 1 .... X2n;m2;g) = APa~(2n) 1 ..... a2n(Xl, We find from eq. (6) 1 ...
2n
'
"'"
X2n;m2;g)
"
7(g) = g2121r + O(g4),
(10) (11)
and here A stands for the soft-mass insertion. Of course we cannot ensure the validity of eq. (10) beyond the one-loop level but it is nevertheless interesting to explore its consequences. The solution of the RGE implies a non-universal short-distance behaviour for the Green's functions at v = 2.
"m2;g) •
Fa(2tn,!..,a2n(.~1 ..... Xx2n,"m 2"~'* = ;~2n'r~)r'(2n) ,.sJh_.,.O at .... ,a2n (x ~ 1, "",X2n,
(12)
We pursue the study of our model by computing the S-matrix. Lorentz and O(N) invariance imply the following form for the 2 -* 2 S-matrix (a3P3, a4P 4 out lalPl, a2P 2 in) = (21r)22p02p 0
X {6(p'1 - p'3)f(p'2 - P'4) [6a,a2 6a3a4S1(O) + 6ala3fia2a 4 $2(0) + 6ala46a2a3S3(O)] + (1 # 2)}. Here 0 stands for the rapidity (Pl + P2) 2 = 4m2 c°sh2(0/2) • It is convenient for the subsequent discussion to define the ratios h(O) = $2(0)[$3(0),
g(O) = S1(0)/$3(0 ) .
(13)
(14)
We find at leading order (tree diagrams) for N arbitrary: $1(0 ) = -$3(0 ) = - i g 2 coth 0 + O(g4),
$2(0 ) = 1 + ig2/sinh 0 + O(g4).
(15)
These tree amplitudes identically verify the factorization equations when N = 2. Moreover the 2 --*4 amplitude vanishes at the tree level in that case [12]. We then find: g(0)=-I 304
+O(g2),
h(O)=tanhO/ig 2 + O ( g 0 ) .
Volume 101B, number 5
PHYSICS LETTERS
21 May 1981
This implies that g(O) ~ O. In that case, the factorization equations can be justified only if the non-trivial condition g(O) = - 1 holds to all orders [2]. That would imply a vanishing reflection coefficient between chargedparticle states. For N / > 3 the factorization equations are not satisfied at the tree level. This strongly suggests that the classical theory is not completely integrable in that case. For N = 1 and 2 we find for the renormalized amplitudes at one-loop order: SI(0 ) : S3(iff - 0) = - i g 2 coth 0 + (iga/2rr sinh 0) X ~1 + cosh 0 + [(N - 2)/sinh 0] (irr - 0) cosh20 - 2rri coth 0} + O(g6), $2(0 ) = 1 + ig2/sinh 0 + (ig4/2rr sinh 0) [N - 2 + (2~/sinh 0) (1 + ½sinh20)l
(16)
+ Ofg6),
N = 1 or 2 .
For N = 1 the 2 ~ 2 S-matrix reduces to S = S 1 + S 2 + S 3 S = 1 + ig2/sinh 0 + ig4/27r sinh 0 - g4[2 sinh 2 0 + O ( g 6 ) . This expression exactly agrees with the expansion o f the SGS-matrix [2] :
S(O) = (sinh 0 + i sin lr3,)/(sinh 0 - i sin 7r7), where 1, = (27r/g 2 - 1) - 1 .
(17)
We have for N = 2
g(O) = -1 + g217r cosh 0 + O(g4). Although the S-matrix is factorizable at the tree level we see that this is no more true at one-loop order. This can be considered as a quantum anomaly and implies that the infinite number o f classically conserved charges is no more conserved in the quantum theory for N = 2. Also the reflection coefficient is non-zero:
R(O) = S 1 + S 3 = ig4/rr sinh 0 + O(g6) .
(18)
It should be noted that R (0)has a purely kinematical dependence in 0. A tree diagram corresponding to a ( , 2 ) 2 coupling will give this 0 dependence. Moreover ( , 2 ) 2 is the only local interaction with this property. So, if we add to the lagrangian
AZ? = (m2g4 /47r) ( , 2 ) 2 ,
(19)
the CSGS-matrix turns to be factorizable at the one-loop level. This term has no influence on the one-loop renormalization properties but it will be relevant yet at the twoloop renormalization. In conclusion, our results suggest several issues for the CSG theory. First, is there any hidden (super) symmetry that explains that cancellations we have found? Moreover, this symmetry may be simply be the classical complete integrability. A link could exist between the one-loop renormalization properties and the tree level factorizability. More generally factorizability at the nth order could ensure renormalizability at the (n + 1)th order. In particular for n = 1, this would imply that a coupling precisely as the one given by eq. (19) is needed for two-loop renormalizability.
References [ 1 ] V.E. Zajarov, L.A. Takhtadzhyan and L.D. Faddeev, Dokl. Akad. Nauk SSSR 219 (1974) 1334 [Soy. Phys.Dokl. 19 (1975) 824] ; L.A. Takhtadzhvan and L.D. Faddeev, Teor. Mat. Fiz. 21 (1974) 160 [Theor. Math. Phys. (USSR) 21 (1975) 1046] ; 305
Volume 101B, number 5
PHYSICS LETTERS
21 May 1981
S. Coleman, Phys. Rev. D l l (1975) 2088; S. Mandelstam, Phys. Rev. D l l (1975) 3027; T. Dashen, B. Hasslacher and A. Neveu, Phys. Rev. D l l (1975) 3424; E.K. Sklyanin, L.A. Takhtadzhyan and L.D. Faddeev, Teor. Mat. Fiz. 40 (1979) 194 [Theor. Math. Phys. (USSR) 40 (1980) 688]. [2] A.B. Zamolodchikov and AI.B. Zamolodchikov, Ann. Phys. 80 (1979) 253. [3] K. Pohlmeyer, Commun. Math. Phys. 46 (1976) 207. [4] J. Honerkamp, Freiburg preprint (1979). [5 ] K. Pohlmeyer and K.H. Rehren, J.Math. Phys. 20 (1979) 2628. [6] H. Eichenherr and J. Honerkamp, Freiburg preprint 12/(1979). [7] V.L. Golo and B.A. Putko, Math. Phys. Lett. 4 (1980) 195. [8] F. Lund, Phys. Rev. Lett. 38 (1977) 1175; B.S. Getmanov, JETP Lett. 25 (1977) 119 [Pis'ma Zh. Eksp. Teor. Fiz. 25 (1977) 132] ; Teor. Mat. Fiz. 38 (1979) 186 [Theor. Math. Phys. 38 (1979) 124]. [9] C.G. Bollini and J.J. Giambiagi, Phys. Lett. 40B (1972) 566; G. 't Hooft and M. Veltman, Nucl. Phys. B44 (1972) 189. [10] AN. Mijailov,JETP Lett. 30 (1980) 414 [Pis'ma Zh. Eksp. Teor. Fiz. 30 (1979) 443] ; A.N. Leznov and M.V. Saveliev,Math. Phys. Lett. 3 (1979) 489; S.A. Bulgadaev, Phys. Lett. 96B (1980) 151. [11] R.J. Baxter, Ann. Phys. 70 (1972) 193,323. [12] J. Kaplan and G. Valent, private communication.
306
PHYSICS LETTERS
21 May 1981
RENORMALIZATION CHARACTER AND QUANTUM S-MATRIX FOR A CLASSICALLY INTEGRABLE THEORY
H.J. de VEGA and J.M. MAILLET Laboratoire de Physlque Theonque et Hautes Energt s , Universitd Pierre and Marie Cuttle, 75230 Paris Cddex 05, France •
"
"
ie
1
Received 13 February 1981
The quantum theory of a N-component generalization of the sine-Gordon model is investigated. We find at the one-loop order that the model is renormalizable only when the corresponding classical theory is completely integrable: N = 1 (sineGordon model) and N = 2 (reduced 0(4) a-model). Moreover the coupling constant does nor renormalize in these two cases. Although the S-matrix for N = 2 is factorizable at the tree level, an anomaly appears at the oneqoop order. Its effect is like a local quartic coupling.
An impressive amount o f results are known at present for the quantum sine-Gordon (SG) model where the classical theory is completely integrable [1,2]. In this context, it is indeed appealing to investigate the quantum theories associated to completely integrable classical models. One would like to know which of the nice classical features of a relativistic model (spectrum, infinite number of conservation laws) have a quantum counterpart. Also, what are their influences (if any) on the renormalization group properties o f the quantum theory. The reduction procedure [3 ] allows to construct new classical integrable models which are related by gauge equivalence [4]. In this way the 0(3) o-model leads to SG theory. More generally, the O(N) and the CP N - 2 nonlinear o-models (with N t> 4) can be reduced to generalizations of classical sine-Gordon [3,5,6] or Gordon [7] models• In this letter we investigate the quantum theory of a first model of this class: the 0(4) reduced o-model [also called complex sine-Gordon (CSG) [5,8]. We start from the lagrang~an = ½ ((3~*) 2 / [1 - g 2 ( , ) 2 1 - m 2 ( ~ ) 2 }
(1)
at two space-time dimensions. There ~ = (qJl, ~02..... qJN) is a real field. F o r N = 1 this model reduces to SG through the transformation ~O1 (x)= g - ' l sin~sa(x)/2). For N = 2 it is gauge equivalent to the 0 ( 4 ) sigma-model with non-zero boundary conditions at infinity [3]. We quantize model (1) in the usual way by splitting .t2into free and interacting parts cQ
./9=./90+Zl,
Zo(X)=½[(au,)2_m2(,)2],
./91(x)=~1 k~_j = 1 ( 0 u , ) 2 g 2k ( , 2 ) k .
(2)
We note that there is an infinite number o f interaction vertices. We use dimensional regularization [9] and i Laboratoire associ6 au CNRS.
302
0 0 3 1 - 9 1 6 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.50 © North-Holland Publishing Company
Volume 101 B, number 5
PHYSICS LETTERS
21 May 1981
and hence no measure term is needed because 8 1 ~ : 0 (where v is the number of space-time dimensions). Power counting tells us that the general f o r m 6 f the counterterms will be
ANZ?(x) = m 2 G N ( ~ 2) + ( a . ~ ) 2 H N ( ~ 2) + ( ~ " ~).~)2IN(~2),
(3)
where G, H and ! are Taylor expandable functions. It is not clear solely by power counting if all ultraviolet divergences can be adsorbed in the renormalization of m 2 , g 2 and ~ . We understand that a quantum field theory is renormalizable if all divergences can be absorbed in the parameters and fields already present in the classical lagrangian (this means, without the freedom of introducing new arbitrary coupling constants). In order to elucidate the renormalization properties of this model we have computed all the relevant quantifies to one-loop order. For the two-point function we get: Pa(21)(bare)(p, _ p , m 2, g) = i~ala2 [p2 _ m 2 + (Ng2/aTr) (p2 + m 2 ) ( 1 / e _ .y + In 47r) + O(e, g4)l ,
(4)
where e = 1 - v/2 and ~ stands for Euler's constant. We then find for the mass and wave function renormalization: 8m 2 = (m2Ng2/2zr) (1/e - "y + In 4rr) + 0(e, g 4 ) ,
Z = 1 - (Ng2/41r)
(lie - "r + In 4 , )
(5)
+ O(e, g4) .
(6)
Using these counterterms we get for the higher 1PI Green's functions
I"(21n.!.,a2n(Pl ..... P 2 n ; m 2 ; g ) - Z
n r(2n) al .....(bare) a2n (Pl .... , P2n ; m2 + 5 m 2 ; g )
(n/> 2),
(7)
The divergent part of
(2n) , ;m2;g) r'~al..... a2n (P l .... P2n 2n-1 =i (2g2)n(n87re- 1)! ( N - 2)((n - 1) /~1"= odd i
Pi'Pi+l -- 12 l ~ i < ~j ¢ 2 n - 1 odd i and j
XSala 2 .... ~2n- 1,2n + permutations.
(Pi + P i + I ) " (Pj + P / + I ) )
(8)
We see that for N = 2 mass and wave function renormalization are enough to eliminate the one-loop poles at two dimensions. The theory can be renormalized in this case without changing the form of Z?. Moreover g does not renormalize. I f N = 1, we have a i = 1 for all i and all permutations add in eq. (8). We get in this way a zero residue at v = 2 which is in agreement with the fact that there is no coupling-constant renormalization in the SG model ~[1 ]. For N i> 3 the model is not renormalizable in the sense precised before because counterterms of the type (g'*/e) X (~" DU~)2 are needed. In order to examine the nature o f the renormalization f o r N = 2 and by analogy with SG theory (N = 1) we try to cast the CSG lagrangian in the form: .6? = (Ouct) 2 + (~/3) 2 + V(t~,/3) + total divergence.
(9)
If this would be possible, normal ordering will eliminate all divergences. We consider the following transformation
~1 =~1 ('~, 3),
~2 = ¢'2('~, 3). 303
Volume 101B, number 5
PHYSICS LETTERS
21 May 1981
Then if for simplicity we assume a zero total divergence in eq. (9) we get that ~ = ~1 + i@2 must be an analytic function o f z = ~ + i/3 in some domain and must verify Id~0/dz 12 + I~k(z)12 = l/g 2. By power expanding qJ in its domain of analyticity it follows that the unique solution is the trivial one (~ = const.). Of course we cannot exclude that a more general transformation could bring &?to the form of eq. (9). Although the resemblance of the one-loop results in real and complex sine-Gordon models, the explanation of the renormalization appears to be more involved in CSG. The O (iV) symmetry is manifestly not enough to explain that, because the theory is O(N) symmetric but not-renormalizable for N t> 3. It must be noted that our model exhibits the property of classical complete integrability precisely for N = 1 and 2 [for N >i 3 the S-matrix is not factorizable already at the tree level (see below) which suggests that the classical theory is not completely integrable]. All this leads to a natural question: what is the general connection between classical integrability and the non-renormalization of coupling constants? Other models that share both properties are the generalized Toda field theories [10]. In these models associated to Lie algebras the only divergent diagram is the tadpole. Hence we can renormalize by simple normal ordering the lagrangian. This renormalizes the mass(es) but not the coupling constant. Also, a related phenomenon is the non-universal critical behaviour exhibited by the exactly integrable eightvertex model [11]. The renormalization group equation (RGE) for the complex sine-Gordon model follows from the above computations
[mO/Om - e g O / O g - 2nT(g)] I"(2n) a (x 1 .... X2n;m2;g) = APa~(2n) 1 ..... a2n(Xl, We find from eq. (6) 1 ...
2n
'
"'"
X2n;m2;g)
"
7(g) = g2121r + O(g4),
(10) (11)
and here A stands for the soft-mass insertion. Of course we cannot ensure the validity of eq. (10) beyond the one-loop level but it is nevertheless interesting to explore its consequences. The solution of the RGE implies a non-universal short-distance behaviour for the Green's functions at v = 2.
"m2;g) •
Fa(2tn,!..,a2n(.~1 ..... Xx2n,"m 2"~'* = ;~2n'r~)r'(2n) ,.sJh_.,.O at .... ,a2n (x ~ 1, "",X2n,
(12)
We pursue the study of our model by computing the S-matrix. Lorentz and O(N) invariance imply the following form for the 2 -* 2 S-matrix (a3P3, a4P 4 out lalPl, a2P 2 in) = (21r)22p02p 0
X {6(p'1 - p'3)f(p'2 - P'4) [6a,a2 6a3a4S1(O) + 6ala3fia2a 4 $2(0) + 6ala46a2a3S3(O)] + (1 # 2)}. Here 0 stands for the rapidity (Pl + P2) 2 = 4m2 c°sh2(0/2) • It is convenient for the subsequent discussion to define the ratios h(O) = $2(0)[$3(0),
g(O) = S1(0)/$3(0 ) .
(13)
(14)
We find at leading order (tree diagrams) for N arbitrary: $1(0 ) = -$3(0 ) = - i g 2 coth 0 + O(g4),
$2(0 ) = 1 + ig2/sinh 0 + O(g4).
(15)
These tree amplitudes identically verify the factorization equations when N = 2. Moreover the 2 --*4 amplitude vanishes at the tree level in that case [12]. We then find: g(0)=-I 304
+O(g2),
h(O)=tanhO/ig 2 + O ( g 0 ) .
Volume 101B, number 5
PHYSICS LETTERS
21 May 1981
This implies that g(O) ~ O. In that case, the factorization equations can be justified only if the non-trivial condition g(O) = - 1 holds to all orders [2]. That would imply a vanishing reflection coefficient between chargedparticle states. For N / > 3 the factorization equations are not satisfied at the tree level. This strongly suggests that the classical theory is not completely integrable in that case. For N = 1 and 2 we find for the renormalized amplitudes at one-loop order: SI(0 ) : S3(iff - 0) = - i g 2 coth 0 + (iga/2rr sinh 0) X ~1 + cosh 0 + [(N - 2)/sinh 0] (irr - 0) cosh20 - 2rri coth 0} + O(g6), $2(0 ) = 1 + ig2/sinh 0 + (ig4/2rr sinh 0) [N - 2 + (2~/sinh 0) (1 + ½sinh20)l
(16)
+ Ofg6),
N = 1 or 2 .
For N = 1 the 2 ~ 2 S-matrix reduces to S = S 1 + S 2 + S 3 S = 1 + ig2/sinh 0 + ig4/27r sinh 0 - g4[2 sinh 2 0 + O ( g 6 ) . This expression exactly agrees with the expansion o f the SGS-matrix [2] :
S(O) = (sinh 0 + i sin lr3,)/(sinh 0 - i sin 7r7), where 1, = (27r/g 2 - 1) - 1 .
(17)
We have for N = 2
g(O) = -1 + g217r cosh 0 + O(g4). Although the S-matrix is factorizable at the tree level we see that this is no more true at one-loop order. This can be considered as a quantum anomaly and implies that the infinite number o f classically conserved charges is no more conserved in the quantum theory for N = 2. Also the reflection coefficient is non-zero:
R(O) = S 1 + S 3 = ig4/rr sinh 0 + O(g6) .
(18)
It should be noted that R (0)has a purely kinematical dependence in 0. A tree diagram corresponding to a ( , 2 ) 2 coupling will give this 0 dependence. Moreover ( , 2 ) 2 is the only local interaction with this property. So, if we add to the lagrangian
AZ? = (m2g4 /47r) ( , 2 ) 2 ,
(19)
the CSGS-matrix turns to be factorizable at the one-loop level. This term has no influence on the one-loop renormalization properties but it will be relevant yet at the twoloop renormalization. In conclusion, our results suggest several issues for the CSG theory. First, is there any hidden (super) symmetry that explains that cancellations we have found? Moreover, this symmetry may be simply be the classical complete integrability. A link could exist between the one-loop renormalization properties and the tree level factorizability. More generally factorizability at the nth order could ensure renormalizability at the (n + 1)th order. In particular for n = 1, this would imply that a coupling precisely as the one given by eq. (19) is needed for two-loop renormalizability.
References [ 1 ] V.E. Zajarov, L.A. Takhtadzhyan and L.D. Faddeev, Dokl. Akad. Nauk SSSR 219 (1974) 1334 [Soy. Phys.Dokl. 19 (1975) 824] ; L.A. Takhtadzhvan and L.D. Faddeev, Teor. Mat. Fiz. 21 (1974) 160 [Theor. Math. Phys. (USSR) 21 (1975) 1046] ; 305
Volume 101B, number 5
PHYSICS LETTERS
21 May 1981
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