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Non-compact sigma models and composite gauge bosons
Volume 135B, number 5,6
PHYSICS LETTERS
16 February 1984
NONICOMPACT SIGMA MODELS AND COMPOSITE GAUGE BOSONS J.W. VAN HOLTEN
Physics Department, Universityof Wuppertal, D-5600 Wuppertal 1, West Germany Received 15 July 1983
It is shown that composite gauge bosom can arise in non-compact sigma models in four-dimensional space-time, provided there is a potential giving masses to the fields. In the case of the SU(1,1 )[U(1 ) model this breaks the non-compact symmetry and reproduces the scalar sector of gauged N = 4 supergravity. Non-abelian and supersymmetric generalizations are possible.
All the known interactions o f elementary particles can be described by gauge theories. It may however be questioned whether these interactions are truly fundamental, or whether they are only the long-distance manifestation o f some more intricate type o f interactions at very short ranges. This last point of view provides the basis for all attempts at unification o f the elementary particles and forces. If it is correct, it could well be that some or all o f the vector bosons transmitting the electroweak and strong interactions are composite. For example, if supergravity is the correct theory o f nature at the level o f the Planck mass, some gauge bosom must be composite, because the maximal gauge group that can be admitted with elementary vector fields is the local SO(8) symmetry o f gauged N = 8 supergravity. It has been known for quite some time that this does not have an SU(3) X SU(2) × U(1) subgroup, as required by present day phenomenology [ 1 ]. Although dynamical generation of composite particles is known to occur frequently in two- and three-dimensional models [2,3], the results cannot be carried over directly to four space-time dimensions [4]. In fact, apart from QCD, where color confinement is widely accepted to be the source o f the binding o f quarks inside hadrons, no dynamical models o f composite particles in four dimensions have been constructed satisfying all the requirements o f relativistic field theory, such as unitarity and renormalizability. This is closely linked to the observation that the only theories known to exhibit asymptotic freedom in four dimensions are non-abelian gauge theories. It has been speculated that composite states, notably composite gauge bosons, could arise in supergravity theories [5 ]. Though these theories are not asymptotically free (they are not even renormalizable), they may be finite. It is known that this suffices for the generation o f bound states [3]. In this paper I show, that non-compact omodels o f the kind occurring in extended supergravity can exhibit dynamical generation o f gauge bosom. For this to occur the scalar fields must get a mass [6]. A potential giving mass to the scalars is therefore introduced in the model. It is the same kind o f potential as is found in gauged extended supergravity. The theory to be discussed here is the theory o f a single complex scalar field z with the lagrangian -
=
au-~ auz [1 - (g/m2)-~z] 2
+m 2
-~z
(1)
1 - (g/m2)2z
Here m is a mass parameter and g a dimensionless coupling constant. The fields are restricted to lie inside the circle
2z < m2/g. In the limit m 2 ~ 0 , g / m 2 constant, the SU(1,1)/U(1) o-model is reproduced [5,7,8]. The hamiltonian corresponding to (1) is V~" Vz
~( = H [ 1 - ( g / m Z ) - ~ z ] 2 I I + [1 -(g/mZ)2z] 2
m2
2z
1-(g/mZ)-~z
0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
-,
II =
i
(2)
[1-(g/m2)-~z] 2" 427
Volume 135B, number 5,6
PHYSICS LETTERS
16 February 1984
F o r g > 0 and m 2 ~< 0 the theory is manifestly unitary and positive definite. We will be interested here mainly in the case m 2 > 0, whence the classical potential is unbounded from below. However, I will show that quantum corrections stabilize the potential and do make it bounded from below. Therefore the quantum theory corresponding to eqs. (1), (2) is well-defined and consistent. A more convenient formulation o f the theory for studying dynamical aspects is obtained by introducing two complex scalar fields u, o spanning a linear representation o f SU(1,1): v=
z [l_(g/m2)-~z]l/2 '
u =
m gl/2
e ia
[1-(g/m2)-~z] a/2
(3)
'
and satisfying the constraint
~u --6o = m2 /g .
(4)
The phase factor in u can be removed by making a gauge transformation on u. This gauge invariance is necessary if both real and imaginary part o f u are to be unphysical degrees o f freedom, the modulus o f u being eliminated by the constraint (4). The auxiliary gauge field is constructed as
Au = (2ig/m2)(-6~uv _ d~uu ) _ 2ig
-z~uz 1 -- ~ / m 2 )2z
(5)
In terms o f u, v, A u and a Lagrange multiplier D for imposing the constraint (4), the action reads (in d dimensions):
S=f ddx[-DuoDuv+DuuDuu-D(~v-Ku+m2/g)-m2-fo+m2fiu+ie(fo+~u)],
D u = O u - i A u.
(6)
The ie-prescription has been added in such a way as to make the path integral in Minkowski space converge [9,6]. It is equivalent to adding a small negative imaginary part to m 2 in (1): m 2 ~ m 2 - ie, which gives the correct ieprescription for the propagator o f the field z. The masses in eq. (6) must satisfy m
2 - m2=_m 2
(7)
O
Using the constraint (4) it is possible to write the mass term simply as
m2~u ,
(8)
with no explicit mass for v. It is convenient to do this in the following. However, it should be noted that the starting point was eq. (6), and that it may turn out in the end, that the auxiliary field D gets a vacuum expectation value which shifts back the mass terms to this form. Of course the fields u, ~ are non-physical, having a propagator with negative residue and anti-causal ie-prescription. However, they only occur as internal lines inside diagrams and never as external legs o f S-matrix elements. Thus'they are like Faddeev-Popov ghosts in gauge theories. Unitarity o f the theory is guaranteed by the equivalent form o f the lagrangian (1), as discussed above. I now proceed to analyse the theory as defined by eqs. (6), (8). Introducing sourcesj, h for the scalars u, o one can construct an effective action for Au, D by integrating out the scalar fields and performing a Legendre transformation with respect to the sources. With
exp(iW[Au,O , h,j]) = f DoD~DuD~exp(i [S + f-ho +-6h +-]u + ~/']),
o c = 6 W/6h, u c = 6 W/6T,
(9)
one obtains the result
F [Au, D, Oc, Uc] = i Trlog ( - D 2 + D - ie) + i T r l o g , ( - D 2 + m 2 + D + i e) -f~c(-D 428
2 + D ) v c - ~c(D 2 + m 2 + D ) u c + (m2/g)D].
(lO)
Volume 135B, number 5,6
PHYSICS LETTERS
16 February 1984
This effective action represents a sum o f one-loop diagrams for the correlation functions of the auxiliary fields, but they correspond to infinite sums o f bubble-diagrams in terms o f the original fields z, 2 [ 10]. The effective potential can now be obtained as minus the effective action for constant fields, per unit of vohime. With ( A > = 0 as a result o f Lorentz invariance, one finds
Neff
= F- drip log ' ] ~ p2+m 2 +D
+ D(bcO c - EcUc + m2/g) - m2~cUc.
(11)
The momenta have been rotated to euclidean values using the opposite ie-prescriptions in (10). Notice that for m 2 -+ 0 all quantum corrections vanish and one gets back the classical potential. Differentiating the potential twice with respect to D gives
Neff OD
1
- m 2 f d x f - driP 1 +~cOc - KcUc + m2/g 0 J (2r[)d (p2 + D + x m 2 ) 2
(12)
and
fdxfdap 1
____a2Veff= _ 2 m 2 OD2 0
1 (2¢r)d (p2 + D + x m 2 ) 3
(13)
The following observations can now be made. (a) For D < 0 the potential has an imaginary part, reflecting instability o f the corresponding vacuum state. Thus we require D ~> 0. (b) The second derivative is finite in d = 4 and negative definite f o r D ~> 0. Hence Veff as a function of D has negative curvature and at most one stationary point, a maximum. This maximum occurs for D > 0 if and only if
BVeff/ODID= 0 > O,
(14)
as is also clear from fig. 1. Otherwise it is nowhere stationary but has a maximum at D = 0. (c) The effective potential has a field4ndependent quadratic divergence, which can be absorbed in a redefinition o f the constant part o f the potential; and a logarithmic divergence linear in D, which is absorbed by a renormalization o f the coupling constant g. Defining the renormalized coupling constant gR(//) by
Veff
"D Fig. 1. Veffas a function of D, (A) with aVeff/aDIo < O, (B): with aVeff/aDIo > O. 429
Volume 135B, number 5,6
PHYSICS LETTERS
16 February 1984
(15)
1/g = # - e [ 1 / g R ~ ) -- 1/87r2e _ (1 -- "),)/16,21, one l~mds ~geff
OD
-
Id-eIO 167r2
log
D ~-2
- (D + m2)log
(+D) m2 167r2m2"] _ 1 - m21og +----T-v--~l+OcVc-KcUc. m2 47r/a2 gR~.la) J
(16)
Here/~ is the renormalization mass scale which can be chosen independently of m. Note that eq. (16) is infrared finite for D ~ 0. Because the bare coupling g is independent of/a, this must also be true for the rhs o f e q . (15). Compared to a fixed mass/a0, gR(/.t) must therefore vary as 1/gR(/~ ) = 1/gR(/10) -- (1/167r 2) log (/~2//~02) .
(17)
The natural choice for/l 0 is of course ta0 = m. In the limit e -+ 0 one gets ~Veff_ 1 FDlog D - D - - - ( D + m 2 ) l o g ( 1 + D ) 167r2m 2 1 ~D 167r2 I. m2 ~ - , + gR(m------~+mZl°g47r +VcOc--UcUc"
(18)
As expected, # has completely disappeared from this expression. A non-trivial stationary point of Vef f is found by solving the equations
OVeff/~D= 0 ,
aVeff/Ou c = - ( D + mZ)uc = 0 ,
0 Veff/Oo c = Do c = 0 .
(19)
The second equation has only one solution, u c = 0. The third equation has two solutions in principle. The ambiguity can be resolved by adding a source term for v c and studying the limit in which the source approaches zero. One easily discovers, that for any finite source, no matter how small, the effective potential has a singularity 1/D. Therefore the only stable solution is v c = 0. The first equation (19) has an acceptable solution only if condition (14)is satisfied. From eq. (18) one obtains upon substitution of v c = u c = 0:
OVeff/OOlD= 0 = m 2 [1/gR(m ) + (1/167r2)log 41r] > 0 .
(20)
It is satisfied for all positive gR(m). As a result the auxiliary field D gets a non-vanishing expectation value _D and both the physical and unphysical scalar field acquire a mass: 2 = D,
m 0
m2u = D + m 2
(21)
It may also be checked, that elimination o f the auxiliary field D leads to a potential in terms of the physical field v c which is bounded from below. The proof is simple: eliminating D by solving the first eq. (19), one gets: 8 Veff/~O c = ~JVeff/~o c ID,uc=eonst" + ~ Veff/ODIvc,Uc=const" ~D/O-6c [Uc=COnst. = Do c ,
(22)
where the second term is zero as a result o f e q . (19). Since D is positive, Veff is an increasing function o f v c. The above results imply that the vector-boson propagator gets a pole at k 2 = 0, indicating that the gauge field A u becomes dynamical at the quantum level. To see this, consider the quadratic term for A u in the effective action, obtained by expanding the trace-log terms in eq. (10). Equivalently they are obtained by computing the diagrams of fig. 2. Since the UV divergences cancel between physical and unphysical scalar loops, one finds the result
62F/6 AugAv IA=0 = - ( k 2 6 u v - kukv)I(k2;g, m 2 ) ,
(23)
with
I(k 2 ;g, m 2) = (1/48u 2) log(1 + m2/D) + O ( k 2 / m 2 ) .
(24)
Hence the inverse propagator (23) has a gauge-invariant zero at k 2 = 0 with positive coefficient; this coefficient is 430
PHYSICS LETTERS
Volume 135B, number 5,6
<>
~
~
16 February 1984
Fig. 2. One-loop contributions to the vector-field two-point function.
the induced inverse charge squared and is a calculable function o f the ratio m2/D. Note that I(k2;g, m 2) vanishes identically if m 2 -+ 0. In this case no bosons are generated. Two possible ways to extend the results above present themselves readily. First, non-abelian vector bosons are generated by using an SU(N, N ) / U ( N ) model with N / > 2 rather than N = 1. Secondly fermions may be introduced by supersymmetrization o f the models discussed. This preserves the cancellations necessary for renormalizability anti dynamical generation o f the vector bosons. The fermions may be in chiral representations o f the gauge group, since the anomahes cancel between positive- and negative-norm fermions. Finally the application o f the results to supergravity deserves comment. Because the supersymmetrized version o f the model described here is an N = 1 truncation o f the N = 4 extended supergravity theory, and the potential term is required if the vector field is to become dynamical, one may draw the tentative conclusion, that gauging o f the internal SO(N) symmetry is necessary for the generation o f bound-state gauge bosons in supergravity [ 11 ]. Whether the result can be extended to the case o f N = 8 supergravity is an open problem. The most serious difficulty is, that an E 7/SU(8) e-model has a mismatch o f 7 scalar fields between the numbers o f physical and unphysical degrees o f freedom, 70 and 63 respectively. A possible solution could be to describe 7 physical scalars b y antisymmetric tensors which do not contribute to the two-point function o f the gauge fields. At present I do not know whether this possibility is a realistic one. I thank Paolo Di Vecchia for stimulating discussions and encouragement.
R e.ference: [l ] M. Gell-Mann, Washington Meeting of the A.P.S. (1977), unpublished. [2] A. d'Adda, P. Di Vecchia and M. LIlscher, Nucl. Phys. B146 (1978) 63; B152 (1979) 125; E. Witten, Nucl. Phys. B149 (1979) 285; H.E. Haber, I. Hincliffe and E. Rabinovici, Nucl. Phys. B172 (1980) 458; O. Alvarez, Phys. Rev. D17 (1978) 1123. [3] A.C. Davis, P. Solomonson and J.W. van Holten, Phys. Lett. 113B (1982) 472; Nucl. Phys. B208 (1982) 1894. [4] K.M. Bitar and R. Raja, Phys. Rev. D27 (1983) 1894. [5 ] E. Cremmer and B. Julia, Nuel. Phys. B159 (1979) 141 ; E. Cremmer, Lectures at the Spring School on Supergravity, Trieste (1981 ), and references therein. [6 ] J.W. van Holten, Wuppertal preprint WU B 83-13. [7 ] E. Cremmer, S. Ferrara and J. Scherk, Phys. Lett. 74B (1978) 61. [8] A.C. Davis, A.J. Macfarlane and J.W. van Holten, Phys. Lett. 125B (1983) 151 ; Wuppertal preprint WU B 83-6 (1983). [9] D.J. Amit and A.C. Davis, Princeton preprint IAS (1983). [10] W. Bardeen, B.W. Lee and R.E. Shrock, Phys. Rev. D14 (1976) 985. [ll ] B. de Wit and H. Nicolai, Nucl. Phys. B208 (1982) 323.
431
PHYSICS LETTERS
16 February 1984
NONICOMPACT SIGMA MODELS AND COMPOSITE GAUGE BOSONS J.W. VAN HOLTEN
Physics Department, Universityof Wuppertal, D-5600 Wuppertal 1, West Germany Received 15 July 1983
It is shown that composite gauge bosom can arise in non-compact sigma models in four-dimensional space-time, provided there is a potential giving masses to the fields. In the case of the SU(1,1 )[U(1 ) model this breaks the non-compact symmetry and reproduces the scalar sector of gauged N = 4 supergravity. Non-abelian and supersymmetric generalizations are possible.
All the known interactions o f elementary particles can be described by gauge theories. It may however be questioned whether these interactions are truly fundamental, or whether they are only the long-distance manifestation o f some more intricate type o f interactions at very short ranges. This last point of view provides the basis for all attempts at unification o f the elementary particles and forces. If it is correct, it could well be that some or all o f the vector bosons transmitting the electroweak and strong interactions are composite. For example, if supergravity is the correct theory o f nature at the level o f the Planck mass, some gauge bosom must be composite, because the maximal gauge group that can be admitted with elementary vector fields is the local SO(8) symmetry o f gauged N = 8 supergravity. It has been known for quite some time that this does not have an SU(3) X SU(2) × U(1) subgroup, as required by present day phenomenology [ 1 ]. Although dynamical generation of composite particles is known to occur frequently in two- and three-dimensional models [2,3], the results cannot be carried over directly to four space-time dimensions [4]. In fact, apart from QCD, where color confinement is widely accepted to be the source o f the binding o f quarks inside hadrons, no dynamical models o f composite particles in four dimensions have been constructed satisfying all the requirements o f relativistic field theory, such as unitarity and renormalizability. This is closely linked to the observation that the only theories known to exhibit asymptotic freedom in four dimensions are non-abelian gauge theories. It has been speculated that composite states, notably composite gauge bosons, could arise in supergravity theories [5 ]. Though these theories are not asymptotically free (they are not even renormalizable), they may be finite. It is known that this suffices for the generation o f bound states [3]. In this paper I show, that non-compact omodels o f the kind occurring in extended supergravity can exhibit dynamical generation o f gauge bosom. For this to occur the scalar fields must get a mass [6]. A potential giving mass to the scalars is therefore introduced in the model. It is the same kind o f potential as is found in gauged extended supergravity. The theory to be discussed here is the theory o f a single complex scalar field z with the lagrangian -
=
au-~ auz [1 - (g/m2)-~z] 2
+m 2
-~z
(1)
1 - (g/m2)2z
Here m is a mass parameter and g a dimensionless coupling constant. The fields are restricted to lie inside the circle
2z < m2/g. In the limit m 2 ~ 0 , g / m 2 constant, the SU(1,1)/U(1) o-model is reproduced [5,7,8]. The hamiltonian corresponding to (1) is V~" Vz
~( = H [ 1 - ( g / m Z ) - ~ z ] 2 I I + [1 -(g/mZ)2z] 2
m2
2z
1-(g/mZ)-~z
0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
-,
II =
i
(2)
[1-(g/m2)-~z] 2" 427
Volume 135B, number 5,6
PHYSICS LETTERS
16 February 1984
F o r g > 0 and m 2 ~< 0 the theory is manifestly unitary and positive definite. We will be interested here mainly in the case m 2 > 0, whence the classical potential is unbounded from below. However, I will show that quantum corrections stabilize the potential and do make it bounded from below. Therefore the quantum theory corresponding to eqs. (1), (2) is well-defined and consistent. A more convenient formulation o f the theory for studying dynamical aspects is obtained by introducing two complex scalar fields u, o spanning a linear representation o f SU(1,1): v=
z [l_(g/m2)-~z]l/2 '
u =
m gl/2
e ia
[1-(g/m2)-~z] a/2
(3)
'
and satisfying the constraint
~u --6o = m2 /g .
(4)
The phase factor in u can be removed by making a gauge transformation on u. This gauge invariance is necessary if both real and imaginary part o f u are to be unphysical degrees o f freedom, the modulus o f u being eliminated by the constraint (4). The auxiliary gauge field is constructed as
Au = (2ig/m2)(-6~uv _ d~uu ) _ 2ig
-z~uz 1 -- ~ / m 2 )2z
(5)
In terms o f u, v, A u and a Lagrange multiplier D for imposing the constraint (4), the action reads (in d dimensions):
S=f ddx[-DuoDuv+DuuDuu-D(~v-Ku+m2/g)-m2-fo+m2fiu+ie(fo+~u)],
D u = O u - i A u.
(6)
The ie-prescription has been added in such a way as to make the path integral in Minkowski space converge [9,6]. It is equivalent to adding a small negative imaginary part to m 2 in (1): m 2 ~ m 2 - ie, which gives the correct ieprescription for the propagator o f the field z. The masses in eq. (6) must satisfy m
2 - m2=_m 2
(7)
O
Using the constraint (4) it is possible to write the mass term simply as
m2~u ,
(8)
with no explicit mass for v. It is convenient to do this in the following. However, it should be noted that the starting point was eq. (6), and that it may turn out in the end, that the auxiliary field D gets a vacuum expectation value which shifts back the mass terms to this form. Of course the fields u, ~ are non-physical, having a propagator with negative residue and anti-causal ie-prescription. However, they only occur as internal lines inside diagrams and never as external legs o f S-matrix elements. Thus'they are like Faddeev-Popov ghosts in gauge theories. Unitarity o f the theory is guaranteed by the equivalent form o f the lagrangian (1), as discussed above. I now proceed to analyse the theory as defined by eqs. (6), (8). Introducing sourcesj, h for the scalars u, o one can construct an effective action for Au, D by integrating out the scalar fields and performing a Legendre transformation with respect to the sources. With
exp(iW[Au,O , h,j]) = f DoD~DuD~exp(i [S + f-ho +-6h +-]u + ~/']),
o c = 6 W/6h, u c = 6 W/6T,
(9)
one obtains the result
F [Au, D, Oc, Uc] = i Trlog ( - D 2 + D - ie) + i T r l o g , ( - D 2 + m 2 + D + i e) -f~c(-D 428
2 + D ) v c - ~c(D 2 + m 2 + D ) u c + (m2/g)D].
(lO)
Volume 135B, number 5,6
PHYSICS LETTERS
16 February 1984
This effective action represents a sum o f one-loop diagrams for the correlation functions of the auxiliary fields, but they correspond to infinite sums o f bubble-diagrams in terms o f the original fields z, 2 [ 10]. The effective potential can now be obtained as minus the effective action for constant fields, per unit of vohime. With ( A > = 0 as a result o f Lorentz invariance, one finds
Neff
= F- drip log ' ] ~ p2+m 2 +D
+ D(bcO c - EcUc + m2/g) - m2~cUc.
(11)
The momenta have been rotated to euclidean values using the opposite ie-prescriptions in (10). Notice that for m 2 -+ 0 all quantum corrections vanish and one gets back the classical potential. Differentiating the potential twice with respect to D gives
Neff OD
1
- m 2 f d x f - driP 1 +~cOc - KcUc + m2/g 0 J (2r[)d (p2 + D + x m 2 ) 2
(12)
and
fdxfdap 1
____a2Veff= _ 2 m 2 OD2 0
1 (2¢r)d (p2 + D + x m 2 ) 3
(13)
The following observations can now be made. (a) For D < 0 the potential has an imaginary part, reflecting instability o f the corresponding vacuum state. Thus we require D ~> 0. (b) The second derivative is finite in d = 4 and negative definite f o r D ~> 0. Hence Veff as a function of D has negative curvature and at most one stationary point, a maximum. This maximum occurs for D > 0 if and only if
BVeff/ODID= 0 > O,
(14)
as is also clear from fig. 1. Otherwise it is nowhere stationary but has a maximum at D = 0. (c) The effective potential has a field4ndependent quadratic divergence, which can be absorbed in a redefinition o f the constant part o f the potential; and a logarithmic divergence linear in D, which is absorbed by a renormalization o f the coupling constant g. Defining the renormalized coupling constant gR(//) by
Veff
"D Fig. 1. Veffas a function of D, (A) with aVeff/aDIo < O, (B): with aVeff/aDIo > O. 429
Volume 135B, number 5,6
PHYSICS LETTERS
16 February 1984
(15)
1/g = # - e [ 1 / g R ~ ) -- 1/87r2e _ (1 -- "),)/16,21, one l~mds ~geff
OD
-
Id-eIO 167r2
log
D ~-2
- (D + m2)log
(+D) m2 167r2m2"] _ 1 - m21og +----T-v--~l+OcVc-KcUc. m2 47r/a2 gR~.la) J
(16)
Here/~ is the renormalization mass scale which can be chosen independently of m. Note that eq. (16) is infrared finite for D ~ 0. Because the bare coupling g is independent of/a, this must also be true for the rhs o f e q . (15). Compared to a fixed mass/a0, gR(/.t) must therefore vary as 1/gR(/~ ) = 1/gR(/10) -- (1/167r 2) log (/~2//~02) .
(17)
The natural choice for/l 0 is of course ta0 = m. In the limit e -+ 0 one gets ~Veff_ 1 FDlog D - D - - - ( D + m 2 ) l o g ( 1 + D ) 167r2m 2 1 ~D 167r2 I. m2 ~ - , + gR(m------~+mZl°g47r +VcOc--UcUc"
(18)
As expected, # has completely disappeared from this expression. A non-trivial stationary point of Vef f is found by solving the equations
OVeff/~D= 0 ,
aVeff/Ou c = - ( D + mZ)uc = 0 ,
0 Veff/Oo c = Do c = 0 .
(19)
The second equation has only one solution, u c = 0. The third equation has two solutions in principle. The ambiguity can be resolved by adding a source term for v c and studying the limit in which the source approaches zero. One easily discovers, that for any finite source, no matter how small, the effective potential has a singularity 1/D. Therefore the only stable solution is v c = 0. The first equation (19) has an acceptable solution only if condition (14)is satisfied. From eq. (18) one obtains upon substitution of v c = u c = 0:
OVeff/OOlD= 0 = m 2 [1/gR(m ) + (1/167r2)log 41r] > 0 .
(20)
It is satisfied for all positive gR(m). As a result the auxiliary field D gets a non-vanishing expectation value _D and both the physical and unphysical scalar field acquire a mass: 2 = D,
m 0
m2u = D + m 2
(21)
It may also be checked, that elimination o f the auxiliary field D leads to a potential in terms of the physical field v c which is bounded from below. The proof is simple: eliminating D by solving the first eq. (19), one gets: 8 Veff/~O c = ~JVeff/~o c ID,uc=eonst" + ~ Veff/ODIvc,Uc=const" ~D/O-6c [Uc=COnst. = Do c ,
(22)
where the second term is zero as a result o f e q . (19). Since D is positive, Veff is an increasing function o f v c. The above results imply that the vector-boson propagator gets a pole at k 2 = 0, indicating that the gauge field A u becomes dynamical at the quantum level. To see this, consider the quadratic term for A u in the effective action, obtained by expanding the trace-log terms in eq. (10). Equivalently they are obtained by computing the diagrams of fig. 2. Since the UV divergences cancel between physical and unphysical scalar loops, one finds the result
62F/6 AugAv IA=0 = - ( k 2 6 u v - kukv)I(k2;g, m 2 ) ,
(23)
with
I(k 2 ;g, m 2) = (1/48u 2) log(1 + m2/D) + O ( k 2 / m 2 ) .
(24)
Hence the inverse propagator (23) has a gauge-invariant zero at k 2 = 0 with positive coefficient; this coefficient is 430
PHYSICS LETTERS
Volume 135B, number 5,6
<>
~
~
16 February 1984
Fig. 2. One-loop contributions to the vector-field two-point function.
the induced inverse charge squared and is a calculable function o f the ratio m2/D. Note that I(k2;g, m 2) vanishes identically if m 2 -+ 0. In this case no bosons are generated. Two possible ways to extend the results above present themselves readily. First, non-abelian vector bosons are generated by using an SU(N, N ) / U ( N ) model with N / > 2 rather than N = 1. Secondly fermions may be introduced by supersymmetrization o f the models discussed. This preserves the cancellations necessary for renormalizability anti dynamical generation o f the vector bosons. The fermions may be in chiral representations o f the gauge group, since the anomahes cancel between positive- and negative-norm fermions. Finally the application o f the results to supergravity deserves comment. Because the supersymmetrized version o f the model described here is an N = 1 truncation o f the N = 4 extended supergravity theory, and the potential term is required if the vector field is to become dynamical, one may draw the tentative conclusion, that gauging o f the internal SO(N) symmetry is necessary for the generation o f bound-state gauge bosons in supergravity [ 11 ]. Whether the result can be extended to the case o f N = 8 supergravity is an open problem. The most serious difficulty is, that an E 7/SU(8) e-model has a mismatch o f 7 scalar fields between the numbers o f physical and unphysical degrees o f freedom, 70 and 63 respectively. A possible solution could be to describe 7 physical scalars b y antisymmetric tensors which do not contribute to the two-point function o f the gauge fields. At present I do not know whether this possibility is a realistic one. I thank Paolo Di Vecchia for stimulating discussions and encouragement.
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