Volume 148B, number 1,2,3 PHYSICS LETTERS 22 November 1984 SPONTANEOUS SYMMETRY BREAKING IN SIX-DIMENSIONAL EINSTEIN-YANG-MILLS THEORY '~" S. RANDJBAR-DAEMI and C. WETTERICH Institute for Theoretical Physics, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland Received 8 June 1984 We investigate all possible solutions of the six-dimensionalEinstein-Yang-Mills equations with four-dimensional Poinear~ invariance and geometryC~ 4 × B. (B is a two-dimensionalinternal space with finite volume.) Excluding the trivial case of B being flat, we find that for all solutions the internal space is a sphere and the gauge fields are in a monopole configuration on S 2 . Therefore, the isometry group SO(3) is not spontaneously broken. Field theories in more than four space-time di- mensions [1] are considered as models for the unifi- cation of all interactions. In particular, they may give an explanation [2--45] why we observe several genera- tions of quarks and leptons only distinguished by their masses. In this context, six-dimensional mode/s of gauge fields coupled to gravity [2,4] have interest- ing properties: The six-dimensional Einstein-Yang- Mills action admits solutions of the field equations where the geometry is a direct product of four- dimensional Minkowski space ~ 4 and a two-dimen- sional sphere S 2. The gauge fields are in a monopole configuration on S 2. Chiral four-dimensional fermions can be obtained by dimensional reduction from six- dimensional Weyl spinors in appropriate representa- tions of the gauge group. For example, dimensional reduction of a six-dimensional SO(12) gauge theory leads to a four-dimensional theory with an even num- ber of fermion generations belonging to the sixteen dimensional spinor representation of the unification group SO(IO) [5]. However, it has been argued [6] that for non- abelian gauge groups all these monopole solutions of the six-dimensional Einstein-Yang-Milis theory are classically unstable. In the case of a six-dimensional SU(3) gauge theory, the excitation spectrum above the monopole solution contains a four-dimensional Work supported by the SchweizerischerNationalfonds. 48 scalar field with negative (mass) 2. This tachyonic mode indicates classical instability. It generalizes to other monopole solutions with different six-dimen- sional gauge groups. What is the meaning of this tachyon? Is the negative mass 2 of the scalar a sign of spontaneous symmetry breaking in the four-dimen- sional theory? Or does it indicate that ~ 4 carl never be a stable solution for the six-dimensional theory? In order to answer these questions, we did a sys- tematic investigation of a// possible solutions of the six-dimensional Einstein-Yang-Mills theory con- sistent with four-dimensional Poincar6 invariance. In particular, the geometry is ~ 4 X B, where B is an arbitrary two-dimensional space with t'mite volume. (To simplify the discussions, we exclude the trivial case of B being flat.) With these assumptions, we Fred the following results: (i) For all solutions, the internal space B must be a sphere S 2. (i 0 For all solutions, the gauge field is in a mono- pole configuration on S 2 preserving the rotation sym- metry SO(3) of S 2. Every monopole configuration on S 2 leads indeed to a solution with direct product from c'~4 X 82, pro- tided we choose a suitable Fine tuning of the cosmo- logical constant. Since all solutions are rotation in- variant, the corresponding induced four-dimensional gauge group SO(3) is not spontaneously broken! The geometry of internal space is uniquely determined by 0370-2693/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)