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Factorization in the massive SU(N) thirring model
Volume 79B, number 4,5
PHYSICS LETTERS
4 December 1978
F A C T O R I Z A T I O N IN THE M A S S I V E S U ( N ) T H I R R I N G M O D E L
S. MEYER and R. TROSTEL
lnstitut fiir Theoretische Physik, Freie Universitift Berlin, D-IO00 Berlin 33, Fed. Rep. Germany Received 7 September 1978
We study various massive SU(N) Thirring models including the broken chiral model in the 1/N-expansion. We show that the S-matrix factorizes when the renormalized coupling constant for the scalar and the chiral interaction is fixed aI a special value.
The recently proposed exact S-matrices for certain two dimensional relativistic theories have a different structure depending on the symmetry group, under which the massive particles transform (for a review and references see [1]). If the symmetry group is 0 ( 2 ) m U(1) the solutions of elastic unitarity, crossing symmetry and the factorisation equations depend on a free parameter, which plays the r61e of a coupling strength. In the case of O(N) symmetry (N > 2) there is no free parameter and the solution is analytic in N. The associated quantum field theories discovered so far are the non linear o-model and the Gross-Neveu model for O(2N) symmetry. This coordination is possible via the mechanism of dimensional transmutation [2] in the l/N-expansion which also serves as a method to check the proposed exact S-matrices [3]. In the present note we start with conventional massive theories e.g. the SU(N) invariant Thirring interaction, which involves in the general case, a massparameter and three independent coupling constants. Standard perturbation theory in these couplings gives particle production the tree approximation. To obtain a relation between the general SU(N) invariant Thirring model and the exact factorizing Smatrices we therefore use the 1/N-expansion. We find that factorization is possible only for the scalar and the chiral interaction at a special fixed value of the renormalized coupling constant. For the scalar case this is of course expected from the known S-matrix
of the Gross-Neveu elementary fermions [4]. It can also be shown that the non-linear o-model is a special point of the linear o-model, see Brezin and Zinn-Justin [5]. They claim the point is special as far as the tenorrealization group is concerned it is a fixed point. The chiral interaction is known to posses higher conservation laws as a classical field theory [6] and the exact S-matrix of the model has been proposed by Berg and Weisz [7]. The most general SU(N) invariant Thirring model is given by the lagrangian 1 2-
"~= ~ i i ~ i -
l q ~ i ~ i + 2gs (~i~i)
1_2(t ~ i~5
+2gp
i l
2
~ i ) 2 + ~1g v2( ~- i 7 ~z~i ) 2 + c . t . ,
(1)
in a basis which does not involve the X-matrices of the representation of SU(N) [8], with real g2(i = s, p, v) and the limitation of positive signs for the interaction terms. We consider first the case of a pure scalar coupling, which is the Gross-Neveu model [9] with the mass-term added. For the 1/N-expansion we define Xs = g 2 / N and use the equivalent lagrangian
('~o = i ~ i ~ i
-- m ~ i~i _ 1 02 _gsO~it~ i + c.t.
with the auxilary quantity o = - g s ~ i ~ i . The o-propagator is normalized by requiring that
Dro(p 2 = 0) = - i ,
(2)
with D o ( p 2) = - i / [ 1 + iII~(p2)] . 429
Volume 79B, number 4,5
PHYSICS LETTERS
The leading order in the 1IN expansion is given by the sum of all chains with simple fermion loops and reads Dor(p 2) = - i ( I + iIlro(p2)) -1 .
(3)
In this approximation the o-propagator is known explicitely, if the basic fermion loop is calculated: /-. d 2 k tr ((k" + m)(4~ - p + rn)} FI°(p2) =)ts a (27r)2 (k 2 - m 2 + - i O ~ - p)}- -m-2 + ie)'
(4) The ultraviolet divergent part is compensated, when the normalization condition (2) is satisfied and the renorrealized o-propagator turns out to be
Dr(p 2) = - i {
4 December 1978
and liver(p2)= x /" d 2 k tr{T**(k~+m)Vv(k--4~r+rn)} vd (2rr)2 (k 2 - m 2 + ie)((k - p)2 _ m 2 + ie)
(s) The propagators read (after renormalization)
D rrr(P2) = - i ( 2rr )'p p2 , x/_~p2 +4m 2_ _~p2 I-1. x 1-p2-_-4m2m~+4mZ+x/_pZ [
, (7')
and
Dr **u(P2) = _ i (g**u - P**Pu/p2) {
Xv
4m2
, ~/-pZ+4m2+
-x/~ _}-1.
hs ~/p2-4m21n-~=-~...~-~f-p2+4m2Z -~P2+21 -1
X l-~
p2
~/_pZ +4m + , , / 2 ~
(8')
j (5)
If we take for the renormalized coupling constant Xs = rr the o-propagator is the same as in the Gross-Neveumodel. We can now check that in the lowest nontrivial order of the I/iV expansion the scattering amplitudes factorize and no particle production occurs for the above special value of the coupling constant. This is done just in the same way as in ref. [4]. We now turn to the general SU(N) case given by the lagrangian (l). We restrict our analysis here to subcases either Xp or Xv = 0. Introducing the auxilary quantities 7r= --gpi~i'g5~i and V**= -gvfiT**t~i we obtain the equivalent lagrangian
~ l = ~ i ( i ~ - m ) ~ i --~-al 2 _gl 7r2 _
There is a relevant difference between the fermion loop contributions as far as the limit p2 _~ 0 is concerned. For scalar and vector coupling this limit is nonzero, whereas the pseudoscalar part goes to zero. Only in the afore mentioned cases the loop contribution can be made proportional to the inverse boson propagator by fixing the coupling constant. This is a necessary condition to obtain factorizing scattering amplitudes and vanishing of production amplitudes corresponding to the arguments of ref. [4]. We indicate for later reference how this can be shown. The scattering amplitude for the process 3 --, 3 is given by the set of diagrams up to the order l/N2:
g1# g V** laerm.
-gsO~i~i -- gprr t~iiT 5 ffi -- gv V**~iTg ~i
+ c.t., (l ')
which includes the broken chiral model with lagrangian . ~ = ~i(i~ -
m)t~ i -
1
~(o 2 + lr2)
- - g ( o ~ i ~ i - n~iiy5~;i) + c.t.
(6)
The leading contributions to the various boson propagators in the 1/N expansion are obtained by taking the basic fermion loops with scalar, pseudoscalar and vector vertices, besides eq. (4), we have
x f d2k llTr(P2)= Pd (2r0 2
430
tr{yS(~+m)yS(e-g+m)) , (ki~_~m 2 ~e)((k~p)2--m2+]e) (7)
with vertices P = 1,3 '5, "}," and all boson propagators indicated by the same wavy lines. Different combinatorial factors multiply the three and loop diagrams. The cancellation between the two sets of diagrams shows up when the fermion loops are calculated with the help of the cutting rule [10], namely the different trace contributions from the fermion loops can be rearranged to become identical to the tree graphs. This is the case for the pure scalar coupling [4]. One immediately realizes that for fermion loops with all vertices being either 3`5 or 7** the traces of the fermion triangle cancel among themselves as a consequence of C-parity invariance. We than calculate the tree graph amplitudes and find non factorising contributions to the S-matrix.
Volume 79B, number 4,5
PHYSICS LETTERS
Let us now consider the 3-particle scattering in the broken chiral model given by eq. (6). A careful analysis of all diagrams shows that fermion loops vanish, when two o-lines and one n-line are attached to the vertices of the triangle again as a consequence of C-parity invariance. The remaining diagrams are trees with one o- and one n-propagator and with two n-propagators and fermion triangles with one o- and two n-lines. Factorization only occurs, when the renormalized coupling constant is fixed at g = n and the n-meson mass is zero, which shows up in the pole of the n-propagator at p2 = 0. This is a puzzling result, since it indicates that an exact S-matrix for the fermions exists irrespective of the infrared problems of the model. Our conclusion is therefore that factorization of the S-matrix in the massive SU(N) Thirring models fixes the renonnalized coupling for the scalar and the broken chiral model, whereas no infinite value can be found where the pseudoscalar - and the vector - theory can be associated with an exact factorizing S-matrix.
4 December 1978
We would like to thank B. Berg and P. Weisz for illuminating discussions.
References [ 1 ] A. Zamolodchikov and A. Zamolodchikov, Factorised S-matrices in two dimensions as the exact solutions of certain relativistic quantum field models, preprint ITEP35 Moscow (1978). [2] S. Coleman and E. Weinberg, Phys. Rev. D7, (1973) 1888. [3] B. Berg, M. Karowski, V. Kurak and P. Weisz, Phys. Lett. 76B (1978) 502. [4] A.B. Zamolodchikov and A.B. Zamolodchikov, Phys. Lett. 72B (1978) 481. [5] E. Brezin and J. Zinn-Justin, Phys. Rev. D14 (1976) 2615. [6] A. Neveu and N. Papanicolaou, Commun. Math. Phys. 58 (1978) 31. [7] B. Berg and P. Weisz, FUB-HEP preprint 78/18. [8] P.K. Mitter and P. Weisz, Phys. Rev. D8 (1973) 4410. [9] D.J. Gross and A. Neveu, Phys. Rev. D10 (1974) 3235. [10] See e.g.B. Berg, Nuovo Cimento 41A (1977) 58.
431
PHYSICS LETTERS
4 December 1978
F A C T O R I Z A T I O N IN THE M A S S I V E S U ( N ) T H I R R I N G M O D E L
S. MEYER and R. TROSTEL
lnstitut fiir Theoretische Physik, Freie Universitift Berlin, D-IO00 Berlin 33, Fed. Rep. Germany Received 7 September 1978
We study various massive SU(N) Thirring models including the broken chiral model in the 1/N-expansion. We show that the S-matrix factorizes when the renormalized coupling constant for the scalar and the chiral interaction is fixed aI a special value.
The recently proposed exact S-matrices for certain two dimensional relativistic theories have a different structure depending on the symmetry group, under which the massive particles transform (for a review and references see [1]). If the symmetry group is 0 ( 2 ) m U(1) the solutions of elastic unitarity, crossing symmetry and the factorisation equations depend on a free parameter, which plays the r61e of a coupling strength. In the case of O(N) symmetry (N > 2) there is no free parameter and the solution is analytic in N. The associated quantum field theories discovered so far are the non linear o-model and the Gross-Neveu model for O(2N) symmetry. This coordination is possible via the mechanism of dimensional transmutation [2] in the l/N-expansion which also serves as a method to check the proposed exact S-matrices [3]. In the present note we start with conventional massive theories e.g. the SU(N) invariant Thirring interaction, which involves in the general case, a massparameter and three independent coupling constants. Standard perturbation theory in these couplings gives particle production the tree approximation. To obtain a relation between the general SU(N) invariant Thirring model and the exact factorizing Smatrices we therefore use the 1/N-expansion. We find that factorization is possible only for the scalar and the chiral interaction at a special fixed value of the renormalized coupling constant. For the scalar case this is of course expected from the known S-matrix
of the Gross-Neveu elementary fermions [4]. It can also be shown that the non-linear o-model is a special point of the linear o-model, see Brezin and Zinn-Justin [5]. They claim the point is special as far as the tenorrealization group is concerned it is a fixed point. The chiral interaction is known to posses higher conservation laws as a classical field theory [6] and the exact S-matrix of the model has been proposed by Berg and Weisz [7]. The most general SU(N) invariant Thirring model is given by the lagrangian 1 2-
"~= ~ i i ~ i -
l q ~ i ~ i + 2gs (~i~i)
1_2(t ~ i~5
+2gp
i l
2
~ i ) 2 + ~1g v2( ~- i 7 ~z~i ) 2 + c . t . ,
(1)
in a basis which does not involve the X-matrices of the representation of SU(N) [8], with real g2(i = s, p, v) and the limitation of positive signs for the interaction terms. We consider first the case of a pure scalar coupling, which is the Gross-Neveu model [9] with the mass-term added. For the 1/N-expansion we define Xs = g 2 / N and use the equivalent lagrangian
('~o = i ~ i ~ i
-- m ~ i~i _ 1 02 _gsO~it~ i + c.t.
with the auxilary quantity o = - g s ~ i ~ i . The o-propagator is normalized by requiring that
Dro(p 2 = 0) = - i ,
(2)
with D o ( p 2) = - i / [ 1 + iII~(p2)] . 429
Volume 79B, number 4,5
PHYSICS LETTERS
The leading order in the 1IN expansion is given by the sum of all chains with simple fermion loops and reads Dor(p 2) = - i ( I + iIlro(p2)) -1 .
(3)
In this approximation the o-propagator is known explicitely, if the basic fermion loop is calculated: /-. d 2 k tr ((k" + m)(4~ - p + rn)} FI°(p2) =)ts a (27r)2 (k 2 - m 2 + - i O ~ - p)}- -m-2 + ie)'
(4) The ultraviolet divergent part is compensated, when the normalization condition (2) is satisfied and the renorrealized o-propagator turns out to be
Dr(p 2) = - i {
4 December 1978
and liver(p2)= x /" d 2 k tr{T**(k~+m)Vv(k--4~r+rn)} vd (2rr)2 (k 2 - m 2 + ie)((k - p)2 _ m 2 + ie)
(s) The propagators read (after renormalization)
D rrr(P2) = - i ( 2rr )'p p2 , x/_~p2 +4m 2_ _~p2 I-1. x 1-p2-_-4m2m~+4mZ+x/_pZ [
, (7')
and
Dr **u(P2) = _ i (g**u - P**Pu/p2) {
Xv
4m2
, ~/-pZ+4m2+
-x/~ _}-1.
hs ~/p2-4m21n-~=-~...~-~f-p2+4m2Z -~P2+21 -1
X l-~
p2
~/_pZ +4m + , , / 2 ~
(8')
j (5)
If we take for the renormalized coupling constant Xs = rr the o-propagator is the same as in the Gross-Neveumodel. We can now check that in the lowest nontrivial order of the I/iV expansion the scattering amplitudes factorize and no particle production occurs for the above special value of the coupling constant. This is done just in the same way as in ref. [4]. We now turn to the general SU(N) case given by the lagrangian (l). We restrict our analysis here to subcases either Xp or Xv = 0. Introducing the auxilary quantities 7r= --gpi~i'g5~i and V**= -gvfiT**t~i we obtain the equivalent lagrangian
~ l = ~ i ( i ~ - m ) ~ i --~-al 2 _gl 7r2 _
There is a relevant difference between the fermion loop contributions as far as the limit p2 _~ 0 is concerned. For scalar and vector coupling this limit is nonzero, whereas the pseudoscalar part goes to zero. Only in the afore mentioned cases the loop contribution can be made proportional to the inverse boson propagator by fixing the coupling constant. This is a necessary condition to obtain factorizing scattering amplitudes and vanishing of production amplitudes corresponding to the arguments of ref. [4]. We indicate for later reference how this can be shown. The scattering amplitude for the process 3 --, 3 is given by the set of diagrams up to the order l/N2:
g1# g V** laerm.
-gsO~i~i -- gprr t~iiT 5 ffi -- gv V**~iTg ~i
+ c.t., (l ')
which includes the broken chiral model with lagrangian . ~ = ~i(i~ -
m)t~ i -
1
~(o 2 + lr2)
- - g ( o ~ i ~ i - n~iiy5~;i) + c.t.
(6)
The leading contributions to the various boson propagators in the 1/N expansion are obtained by taking the basic fermion loops with scalar, pseudoscalar and vector vertices, besides eq. (4), we have
x f d2k llTr(P2)= Pd (2r0 2
430
tr{yS(~+m)yS(e-g+m)) , (ki~_~m 2 ~e)((k~p)2--m2+]e) (7)
with vertices P = 1,3 '5, "}," and all boson propagators indicated by the same wavy lines. Different combinatorial factors multiply the three and loop diagrams. The cancellation between the two sets of diagrams shows up when the fermion loops are calculated with the help of the cutting rule [10], namely the different trace contributions from the fermion loops can be rearranged to become identical to the tree graphs. This is the case for the pure scalar coupling [4]. One immediately realizes that for fermion loops with all vertices being either 3`5 or 7** the traces of the fermion triangle cancel among themselves as a consequence of C-parity invariance. We than calculate the tree graph amplitudes and find non factorising contributions to the S-matrix.
Volume 79B, number 4,5
PHYSICS LETTERS
Let us now consider the 3-particle scattering in the broken chiral model given by eq. (6). A careful analysis of all diagrams shows that fermion loops vanish, when two o-lines and one n-line are attached to the vertices of the triangle again as a consequence of C-parity invariance. The remaining diagrams are trees with one o- and one n-propagator and with two n-propagators and fermion triangles with one o- and two n-lines. Factorization only occurs, when the renormalized coupling constant is fixed at g = n and the n-meson mass is zero, which shows up in the pole of the n-propagator at p2 = 0. This is a puzzling result, since it indicates that an exact S-matrix for the fermions exists irrespective of the infrared problems of the model. Our conclusion is therefore that factorization of the S-matrix in the massive SU(N) Thirring models fixes the renonnalized coupling for the scalar and the broken chiral model, whereas no infinite value can be found where the pseudoscalar - and the vector - theory can be associated with an exact factorizing S-matrix.
4 December 1978
We would like to thank B. Berg and P. Weisz for illuminating discussions.
References [ 1 ] A. Zamolodchikov and A. Zamolodchikov, Factorised S-matrices in two dimensions as the exact solutions of certain relativistic quantum field models, preprint ITEP35 Moscow (1978). [2] S. Coleman and E. Weinberg, Phys. Rev. D7, (1973) 1888. [3] B. Berg, M. Karowski, V. Kurak and P. Weisz, Phys. Lett. 76B (1978) 502. [4] A.B. Zamolodchikov and A.B. Zamolodchikov, Phys. Lett. 72B (1978) 481. [5] E. Brezin and J. Zinn-Justin, Phys. Rev. D14 (1976) 2615. [6] A. Neveu and N. Papanicolaou, Commun. Math. Phys. 58 (1978) 31. [7] B. Berg and P. Weisz, FUB-HEP preprint 78/18. [8] P.K. Mitter and P. Weisz, Phys. Rev. D8 (1973) 4410. [9] D.J. Gross and A. Neveu, Phys. Rev. D10 (1974) 3235. [10] See e.g.B. Berg, Nuovo Cimento 41A (1977) 58.
431