Applications of the Reduction of Couplings ∗

Applications of the principle of reduction of couplings to the standard model and supersymmetric grand unified theories are reviewed. Phenomenological applications of renormalization group invariant sum rules for soft supersymmetry-breaking parameters are also reviewed. ∗ To appear in the Proceedings of the Ringberg Symposium on Quantum Field Theory, Tegernsee, Germany, June 1998. 1 Application to the Standard Model High energy physicists have been using renormalizability as the predictive tool, and also to decide whether or not a quantity is calculable. As we have learned in the previous talk by Professor Oehme, it is possible, using the method of reduction of couplings [1, 2, 3], to renormalize a theory with fewer number of counter terms then usually counted, implying that the traditional notion of renormalizability should be generalized in a certain sense . Consequently, the notion of the predictability and the calculability [5] may also be generalized with the help of reduction of couplings. Of course, whether the generalizations of these notions have anything to do with nature is another question. The question can be answered if one applies the idea of reduction of couplings to realistic models, make predictions that are specific for reduction of couplings, and then wait till experimentalists find positive results . In 1984 Professor Zimmermann, Klaus Sibold and myself [8] began to apply the idea of reduction of couplings to the SU(3)C × SU(2)L × U(1)Y gauge model for the strong and electroweak interactions. As it is known, this theory has a lot of free parameters, and at first sight it seemed there exists no guiding principle how to reduce the couplings in this theory. There were two main problems associated with this program. The one was that it is not possible to assume a common asymptotic behavior for all couplings, and the other one is how to increase the predictive power of the model without running into the contradiction with the experimental knowledge (of that time) such as the masses of the known fermions. Professor Zimmermann suggested to use asymptotic freedom as a guiding principle, and assumed that QCD is most fundamental among the interactions of the standard model (SM). Since pure QCD is asymptotically free, we tried to switch on as many SM interactions as possible while keeping asymptotic freedom and added them to QCD. The result was almost unique: There exist two possibilities (or two asymptotically free (AF) surfaces in the space of couplings). It turned out that on the first surface, the SU(2)L gauge coupling α2 is bigger than the QCD Earlier references related to the idea of reduction of couplings are given in [4]. Professor Shirkov who is present here was one of the authors who considered such theoretical possibilities. For reviews, see [6]. Here I would like to restrict myself to phenomenological applications of reduction of couplings. See [7, 9] for the other applications. Professor Oehme reminded me that in his seminar talk given at the Max-PlanckInstitute early 1984, professor Peccei suggested phenomenological applications of this idea .