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Dynamical generation of interaction in an exactly solvable model
Volume 145B, number 5,6
PHYSICS LETTERS
27 September 1984
DYNAMICAL GENERATION OF INTERACTION IN AN EXACTLY SOLVABLE MODEL L.V. AVDEEV and M.V. CHIZHOV Joint Institute for Nuclear Research, Dubna, USSR
Received 21 May 1984
The dynamical generation of interaction in the chiral-invariant Gross-Neveu model leads to an asymptotically free charge behaviour and a correlation between coupling constants. The known exact solution possesses similar properties.
In refs. [1,2] a hypothesis about the dynamical origin of the Green functions in the framework of the Bogotubov compensation method has been put forward. The perturbation theory solution of the compensation equations leads to a definite choice of particle multiplets and their interactions. The previously independent coupling constants prove to be related to each other. Then, an effective action arising from radiative corrections to the Green functions depends on a single arbitrary constant. In the present letter we would like to pay attention to the motivation of the dynamical generation hypothesis. Therefore, it was natural to study a theory with a known exact solution, the chiral-invariant GrossNeveu model'in two space-time dimensions. As has been established [3], to obtain finite expressions for the energy and momentum of the one-hole states over the vacuum, it is necessary to require that the coupling constants have an asymptotically free dependence on the ultraviolet momentum cut-off and satisfy a special correlation condition. A derivation of both these requirements in perturbation theory would confirm the hypothesis made. The usual perturbation theory and renormalization procedure allows the use of arbitrary coupling constants in the initial lagrangian. After elimination of divergences and renormalizations, if no symmetry constraints are set, this ambiguity is left also in the final expressions. The hypothesis of the dynamical origin of the Green functions provides us with the compensation equations which restrict the arbitrariness. All ambiguity of the regularization and renormalization 0 3 7 0 - 2 6 9 3 / 8 4 / $ 0 3 . 0 0 © Elsevier Science Publishers B.V. (North-HoUand Physics Publishing Division)
2ig4 IoI
- 2 ig~ "~So~"s
2ig~ ~r® ~1,
Fig. 1.
procedure is absorbed by the only arbitrary coupling constant remaining. We use the following initial lagrangian in two spacetime dimensions for a multiplet ~a (a = 1,2 . . . . . n) of spinor fields in the fundamental representation of the global SU(n) isotopic group: (1)
£ = i t~aTuo u ~ a .
Suppose, further, that the interaction terms, £1=gl(~ad/a) 2 ,
£2=--g2(~a'~st~a) 2 ,
£3 = g3( ~a"[ts ¢a)2 '
(2)
pictured in fig. 1, are to be generated dynamically from the radiative corrections. The generation self-consistency is expressed by the compensation equations, represented graphically in fig. 2. On the •ht-hand side the complete Green function without the initial interaction term is involved. To leading order in perturbation theory, the self-consis-
Fig. 2. 397
Volume 145B, number 5,6
PHYSICS LETTERS
From eqs. (7) and (8) we obtain two solutions:
tency equations take the form: 1 = I'](gl,g2,g3)
(f = 1, 2, 3).
(3)
The Green functions P], corresponding to the terms (2), with the dynamically generated interactions, can be determined by summing up the regularized logarithmically ultraviolet-divergent integrals of the form C = Reg (i/n) f d2p/p 2 from the radiative corrections. To perform the summation of the leading logarithms, we use the renormalization-group method: Pl(gl,g2,g3)
=exp(-f dx"/j[gl(X),g2(x),¢3(x)l ) -1 o
(/=
1,2, 3).
27 September 1984
(4)
g2 = g l ,
g3=gl/n,
and g2 = - [(n2 - n + 2 ) / ( r / - 1)(n + 2)]g 1 , g3 = [(n2 - n + 2)/4n]g 1 .
£eff = i ~a'y~0/~Ca + gl [( t~a ~a)2 _ (~a,), 5 ~a)2 ]
0 In $j/Ox = --T/(ffl, g2, g3),
glC=rr/2n.
(/= 1,2, 3).
(5)
In eq. (5) we have taken into account the fact that the anomalous dimension of the one-loop propagator equals zero. Now, after substituting the expressions (4), the compensation equations (3) imply that C
exp(-f
dx3,1[ffl(X),e2(x),e3(x)])=2,
(6)
0
and "Yl(gl, g2, g3) = ")'2(gl, g2, g3) = "1'3(g1,g2, g3).
(7)
The explicit one-loop calculations of the vertex Feynman graphs done by the method of ref. [4] give the following results: ")'1(gl, g2, g3) = r¢-1 [ - n g l + (g¢1 - g 2 ) ( 1 + 2g3/g1)] ,
398
(11)
Solving eqs. (5) and (6) yields the coupling-constant renormalization condition: (12)
In the case of ultraviolet cut-off regularization, eq. (12) results in a nonanalytic dependence, A 2 = la2exp(rr/2ngl),
(13)
implying thatg 1 ~ +0 when A ~ oo. The relations obtained coincide with the conditions in the exact solution of the model, to ensure the finiteness of the energy and momentum for the hole states. Thus, the physical solutions imply a certain correlation between the couplings in the effective lagrangian and their particular dependence on the regularization parameter. This fact substantiates the dynamical origin of the in. teraction lagrangian. Also, it is worth mentioning that the exact solution allows us to find the full perturbation series for the renormalization-group ~ function of the model. We differentiate the exact charge-renormalization condition [3], like (13), with respect to In A 2 and then express A 2 in terms of the coupling constant. In the SU(2) case, n = 2, the exact result looks like (14)
Here
= rr-1 I--rig2 + (gl - g2)( 1 + 2g3/g2)] , 73(gl, g2, g3) = - - r r - l g l g 2 / g 3 .
+ (,gl/n)(~aTut)a) 2 .
[3(h ) = - h 2 /2rr .
")'2(gl, g2, g3)
(10)
Eqs. (9) and (10) determine the straight-line singular solutions of the renormalization-group equations. As argued in ref. [5 ], the existence of those solutions in the one-loop approximation guarantees their existence for the exact equations. Consider the solution (9) which corresponds to a chiral-invariant effective lagrangian of the GrossNeveu model with the connected coupling constants:
Here, the - 1 term subtracts the initial interaction. Changing sign of the anomalous dimensions 7j (gl, g2, g3) of the relevant vertices allows us to sum up divergences rather than counterterms. The effective charges gj(x) obey the equations:
gi (0) =gl
(9)
(8)
h = 4gl/(1 +g2 _ g 2 ) ,
(15)
Volume 145B, number 5,6
PHYSICS LETTERS
1 + O(g3), according to eq. (9). It is interwithg3 = ggl esting to point out the absence (in the renormalization scheme o f ref. [3]) of the higher-order corrections in eq. (14). The other solution (10) o f the compensation equations breaks explicitly the chiral invariance of the effective lagrangian, and we do not know a way to obtain it in the framework of the exact solution with the use of the Bethe Ansatz. Nevertheless, it is a consequence of the dynamical equations and should be considered on an equal footing with (9). The one-loop renormalization condition in the case (10) has the form
gl C = - [ ( n - 1)(n + 2)/8n] ft.
(16)
It is self-consistent for g l < 0 when asymptotic freedom is not lost. The dynamical generation hypothesis is most essential when studying quantum field theories in real fourdimensional space-time, since it allows us to find correlations between different coupling constants and masses, which are to be tested experimentally. The consideration o f the two-dimensional model was only
27 September 1984
necessary for us to show the substantiation of the dynamical mechanism proposed. However, the mass generation for the hole states, which occurs in the exact solution, cannot be described in perturbation theory without taking collective excitations into account. A generation of the mass terms m t~a ~ka or m t~a75 ~ba in the effective lagrangian in two dimensions turns out to be incompatible with the dynamical origin o f the vertex parts. This is also in agreement with the result of the exact solution that the initial fermions (pseudoparticles) should be massless. We are grateful to A.D. Donkov for useful discussions.
References [1] A.V. Chizhov and M.V. Chizhov, Phys. Lett. 125B (1983) 190. [2] M.V. Chizhov, JINR preprint P2-84-172 (1984). [3] J.H. Lowenstein, Surv. High Energy Phys. 2 (1981) 207. [4] A.A. Vladimirov, Teor. Mat. Fiz. 43 (1980) 210 [Theor. Math. Phys. (USSR) 43 (1980) 417]. [5] I.V. Tyutin, Sov. Phys. Lebedev Inst. Rep. No. 8 (1978) 3.
399
PHYSICS LETTERS
27 September 1984
DYNAMICAL GENERATION OF INTERACTION IN AN EXACTLY SOLVABLE MODEL L.V. AVDEEV and M.V. CHIZHOV Joint Institute for Nuclear Research, Dubna, USSR
Received 21 May 1984
The dynamical generation of interaction in the chiral-invariant Gross-Neveu model leads to an asymptotically free charge behaviour and a correlation between coupling constants. The known exact solution possesses similar properties.
In refs. [1,2] a hypothesis about the dynamical origin of the Green functions in the framework of the Bogotubov compensation method has been put forward. The perturbation theory solution of the compensation equations leads to a definite choice of particle multiplets and their interactions. The previously independent coupling constants prove to be related to each other. Then, an effective action arising from radiative corrections to the Green functions depends on a single arbitrary constant. In the present letter we would like to pay attention to the motivation of the dynamical generation hypothesis. Therefore, it was natural to study a theory with a known exact solution, the chiral-invariant GrossNeveu model'in two space-time dimensions. As has been established [3], to obtain finite expressions for the energy and momentum of the one-hole states over the vacuum, it is necessary to require that the coupling constants have an asymptotically free dependence on the ultraviolet momentum cut-off and satisfy a special correlation condition. A derivation of both these requirements in perturbation theory would confirm the hypothesis made. The usual perturbation theory and renormalization procedure allows the use of arbitrary coupling constants in the initial lagrangian. After elimination of divergences and renormalizations, if no symmetry constraints are set, this ambiguity is left also in the final expressions. The hypothesis of the dynamical origin of the Green functions provides us with the compensation equations which restrict the arbitrariness. All ambiguity of the regularization and renormalization 0 3 7 0 - 2 6 9 3 / 8 4 / $ 0 3 . 0 0 © Elsevier Science Publishers B.V. (North-HoUand Physics Publishing Division)
2ig4 IoI
- 2 ig~ "~So~"s
2ig~ ~r® ~1,
Fig. 1.
procedure is absorbed by the only arbitrary coupling constant remaining. We use the following initial lagrangian in two spacetime dimensions for a multiplet ~a (a = 1,2 . . . . . n) of spinor fields in the fundamental representation of the global SU(n) isotopic group: (1)
£ = i t~aTuo u ~ a .
Suppose, further, that the interaction terms, £1=gl(~ad/a) 2 ,
£2=--g2(~a'~st~a) 2 ,
£3 = g3( ~a"[ts ¢a)2 '
(2)
pictured in fig. 1, are to be generated dynamically from the radiative corrections. The generation self-consistency is expressed by the compensation equations, represented graphically in fig. 2. On the •ht-hand side the complete Green function without the initial interaction term is involved. To leading order in perturbation theory, the self-consis-
Fig. 2. 397
Volume 145B, number 5,6
PHYSICS LETTERS
From eqs. (7) and (8) we obtain two solutions:
tency equations take the form: 1 = I'](gl,g2,g3)
(f = 1, 2, 3).
(3)
The Green functions P], corresponding to the terms (2), with the dynamically generated interactions, can be determined by summing up the regularized logarithmically ultraviolet-divergent integrals of the form C = Reg (i/n) f d2p/p 2 from the radiative corrections. To perform the summation of the leading logarithms, we use the renormalization-group method: Pl(gl,g2,g3)
=exp(-f dx"/j[gl(X),g2(x),¢3(x)l ) -1 o
(/=
1,2, 3).
27 September 1984
(4)
g2 = g l ,
g3=gl/n,
and g2 = - [(n2 - n + 2 ) / ( r / - 1)(n + 2)]g 1 , g3 = [(n2 - n + 2)/4n]g 1 .
£eff = i ~a'y~0/~Ca + gl [( t~a ~a)2 _ (~a,), 5 ~a)2 ]
0 In $j/Ox = --T/(ffl, g2, g3),
glC=rr/2n.
(/= 1,2, 3).
(5)
In eq. (5) we have taken into account the fact that the anomalous dimension of the one-loop propagator equals zero. Now, after substituting the expressions (4), the compensation equations (3) imply that C
exp(-f
dx3,1[ffl(X),e2(x),e3(x)])=2,
(6)
0
and "Yl(gl, g2, g3) = ")'2(gl, g2, g3) = "1'3(g1,g2, g3).
(7)
The explicit one-loop calculations of the vertex Feynman graphs done by the method of ref. [4] give the following results: ")'1(gl, g2, g3) = r¢-1 [ - n g l + (g¢1 - g 2 ) ( 1 + 2g3/g1)] ,
398
(11)
Solving eqs. (5) and (6) yields the coupling-constant renormalization condition: (12)
In the case of ultraviolet cut-off regularization, eq. (12) results in a nonanalytic dependence, A 2 = la2exp(rr/2ngl),
(13)
implying thatg 1 ~ +0 when A ~ oo. The relations obtained coincide with the conditions in the exact solution of the model, to ensure the finiteness of the energy and momentum for the hole states. Thus, the physical solutions imply a certain correlation between the couplings in the effective lagrangian and their particular dependence on the regularization parameter. This fact substantiates the dynamical origin of the in. teraction lagrangian. Also, it is worth mentioning that the exact solution allows us to find the full perturbation series for the renormalization-group ~ function of the model. We differentiate the exact charge-renormalization condition [3], like (13), with respect to In A 2 and then express A 2 in terms of the coupling constant. In the SU(2) case, n = 2, the exact result looks like (14)
Here
= rr-1 I--rig2 + (gl - g2)( 1 + 2g3/g2)] , 73(gl, g2, g3) = - - r r - l g l g 2 / g 3 .
+ (,gl/n)(~aTut)a) 2 .
[3(h ) = - h 2 /2rr .
")'2(gl, g2, g3)
(10)
Eqs. (9) and (10) determine the straight-line singular solutions of the renormalization-group equations. As argued in ref. [5 ], the existence of those solutions in the one-loop approximation guarantees their existence for the exact equations. Consider the solution (9) which corresponds to a chiral-invariant effective lagrangian of the GrossNeveu model with the connected coupling constants:
Here, the - 1 term subtracts the initial interaction. Changing sign of the anomalous dimensions 7j (gl, g2, g3) of the relevant vertices allows us to sum up divergences rather than counterterms. The effective charges gj(x) obey the equations:
gi (0) =gl
(9)
(8)
h = 4gl/(1 +g2 _ g 2 ) ,
(15)
Volume 145B, number 5,6
PHYSICS LETTERS
1 + O(g3), according to eq. (9). It is interwithg3 = ggl esting to point out the absence (in the renormalization scheme o f ref. [3]) of the higher-order corrections in eq. (14). The other solution (10) o f the compensation equations breaks explicitly the chiral invariance of the effective lagrangian, and we do not know a way to obtain it in the framework of the exact solution with the use of the Bethe Ansatz. Nevertheless, it is a consequence of the dynamical equations and should be considered on an equal footing with (9). The one-loop renormalization condition in the case (10) has the form
gl C = - [ ( n - 1)(n + 2)/8n] ft.
(16)
It is self-consistent for g l < 0 when asymptotic freedom is not lost. The dynamical generation hypothesis is most essential when studying quantum field theories in real fourdimensional space-time, since it allows us to find correlations between different coupling constants and masses, which are to be tested experimentally. The consideration o f the two-dimensional model was only
27 September 1984
necessary for us to show the substantiation of the dynamical mechanism proposed. However, the mass generation for the hole states, which occurs in the exact solution, cannot be described in perturbation theory without taking collective excitations into account. A generation of the mass terms m t~a ~ka or m t~a75 ~ba in the effective lagrangian in two dimensions turns out to be incompatible with the dynamical origin o f the vertex parts. This is also in agreement with the result of the exact solution that the initial fermions (pseudoparticles) should be massless. We are grateful to A.D. Donkov for useful discussions.
References [1] A.V. Chizhov and M.V. Chizhov, Phys. Lett. 125B (1983) 190. [2] M.V. Chizhov, JINR preprint P2-84-172 (1984). [3] J.H. Lowenstein, Surv. High Energy Phys. 2 (1981) 207. [4] A.A. Vladimirov, Teor. Mat. Fiz. 43 (1980) 210 [Theor. Math. Phys. (USSR) 43 (1980) 417]. [5] I.V. Tyutin, Sov. Phys. Lebedev Inst. Rep. No. 8 (1978) 3.
399