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Submitted on 1 Jan 1982
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DYNAMICAL GENERATION OF GAUGE BOSONS IN CPn-1 MODELS
A. Davis
To cite this version:
A. Davis. DYNAMICAL GENERATION OF GAUGE BOSONS IN CPn-1 MODELS. Journal de
Physique Colloques, 1982, 43 (C3), pp.C3-266-C3-268. �10.1051/jphyscol:1982353�. �jpa-00221908�
JOURNAL DE PHYSIQUE
CoZZoque C 3 , suppZ6ment au n o 12, Tome 43, de'cembre 1982
DYNAMICAL GENERATION OF GAUGE BOSONS IN &pn-l MODELS
A.C. Davis
CERN, Geneva, SwitzerZand
The two-dimensional CP"-' models have proved useful to study many theo- retical ideas like chiral symmetry breaking and the U(1) problem2)p3) and non- perturbative methods. They have recently been applied to stu y dynamical gauge sym- metry breaking by fermion condensate&) and vacuum alignment57. In all these cases the model displays a remarkable property: classically the model has a local gauge invariance, but no explicit gauge field; quantum mechanically the gauge field is generated dynamically.
The tpn-l model consists of n complex scalar fields zi, i = 1,.
. . ,n, subject to the constraint zlzi = n/2f, where f is a dimensionless constant. The Lagrangian is
where
and h is a Lagrange multiplier imposing the constraint in the path integral. The Lagrangian (1) is invariant under global SUin) and local U(1) symmetry. However, the gauge field A has no kinetic terms and can be eliminated by the equation of motion 1-1
By integrating over z in the path integral (1) can be solved in the l/n expansion.
h acquires a non-zero vacuum expectation value (VEV). Expanding around the VEV in powers of l/n we find that kinetic terms are generated for A
.
A,, has become a dynamical gauge field via quantum corrections. Since gauge fieyds are confining in two-dimensions, the effective theory consists of interactions between z and AF, to give z z bound states. This is a remarkable result.The model can be generalized to include fermions2) and to include a non-Abelian (auxiliary) gauge invariance6). Further, we can construct a model where the left- and right-handed fermions are in different representations of the gauge group4). The model has a U(n) global and U(Q) local invariance with n > %, and can be used to study gauge symmetry breaking. If we put IJR in R and IJL in Q of the U(R) gauge group, then the generalized cpn-l Lagrangian is
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982353
A.C. Davis
(4) cont
.
where i = l,.. .,n and is the index for global U(n) and cx, = 1,
...,
R is the index for local U(&). In two-dimensions bR(qL,) couples only to D+(D-) wherea a B
nr
= ;n
1-1a =a
s b L ~ ( r ) ~! - l a
t G v
-ra being U(R) matrices and
gS(gA) is a coupling constant for the symmetric (antisymmetric) channel. The fermion composite, (I is a scalar of U(n), allowing the l/n expansion to be performed, but is not a scalar of U(R). Again, A is an auxiliary field and can be eliminated by the equation of motion, which is mope complicated than in the case of (1). To solve (4) we integrate over z and $ in the path integral and solve for the remaining fields in a saddle-point approximation. We find that X and Q acquire non-zero VEV. Since
$,B is not a singlet of the U(R) gauge group, the group is broken by the VEV, I%-&.
In particular, detailed analysis in the l/n expansion shows that if gs > g~ then U(R) is broken to an O(R) gauge group and if gA > gs then U(R) is broken to an Sp(R) gauge group. Specializing to the case g~ > g ~ , the results can be summarized in terms of the effective action
where a
p]
runs over components in O(L) [not in o(R)] and we have projected all fields into the unbroken O(R) and broken components, represented by the matrices T~and .tl respectively,
and E is l / g ~
-
l/gA. BI does not appear in the physical spectrum, having been eaten in a dynamical Higgs mechanism to give mass to WU. Fuv and Gyv are thekinetic terms for AP and Wv: All fields except the 0(R) gauge field A,, acquire mass, the mass obtained by dimensional transmutation. There is also a U(1) anomaly coupling A
: to the pseudoscalar Aa. Similar results hold for g~ > gs. For gA : gS there is a vacuum alignment problem, discussed in detail elsewhere4 ) 95).
C3-268 JOURNAL DE PHYSIQUE
In this model we have generalized the CP"-' mechanism. That is, we have dyna- mically generated a U(L) gauge boson via the l/n expansion and, by a fermion composite acquiring non-zero VEV, dynamically broken the gauge group to O(R). For both cases we have discussed, quantum corrections have induced dynamics for an auxiliary gauge field at the classical level.
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