Renormalization group analysis of a vector Gürsey model

Abstract

To find a non-trivial field theoretical model is one of the outstanding problems in theoretical high energy physics. The perturbatively nontrivial φ 4 theory in four dimensions was shown to go to a free theory as the cut-off was lifted a while ago. The Nambu-Jona- Lasinio model, which is non-renormalizable perturbatively, was also shown to go to a trivial model. There are claims that the gauged Nambu-Jona- Lasinio model may give a non-trivial theory for a non realistically large number of flavors. All these examples show that the search for non-trivial models may be an interesting endeavor. On the other hand, there are two-dimensional models with infinite number of local and non-local charges. These models were shown to give scattering matrices without particle production indicating a behavior not expected from fully interacting theories. When the symmetry present in these models is broken, these models give rise to particle production in perturbative calculations. Another class of models are those with zero scattering amplitudes for the constituent fields, even though their Lagrangians seem to have non-zero interaction terms. These interactions arise from constraints. An example of them is taking a product of the constituent field equal to a power of the auxiliary field, i.e. forming composites of the constituent fields. The interaction of the constituent field with the composite field is defined. The Lagrangian contains the kinetic part of only the constituent spinor fields and the kinetic terms for the composites are formed as a result of vacuum polarizations of the composite field due to its interaction with the constituent field. Such models simulate the models with only spinors. The first work on models with only spinors goes back to the work of Heisenberg. Gürsey proposed his model as a substitute for the Heisenberg model in the fifties. This spinor model is important since it is conformally invariant classically and has classical solutions which may be interpreted as instantons and merons, similar to the solutions of pure Yang- Mills theories in four dimensions. This original model can be generalized to include vector, pseudovector and pseudoscalar interactions. A while ago, Akdeniz et al. claimed to have written a polynomial Lagrangian equivalent to the one given by Gürsey. Now we know that the models studied are only naively equivalent to these spinor models. He, with his collaborators, has also shown that in some of these models, the interaction of the spinor field with other spinors goes to zero. The model is not only asymptotically free, it is actually free, meaning that the coupling constant goes to zero as the cut-off is removed. We could not find physical processes which give results indicative of an interacting theory. With this motivation we want to give a new interpretation of the old work of Akdeniz et al.. First of all we see that the polynomial form of the original Gürsey model really does not correspond to it in the exact sense. The two versions obey different symmetries. Then we go to higher orders in our calculation in the new version, beyond the one loop for the scattering processes. We see that while the non-trivial scattering of the fundamental fields is not allowed, bound states can scatter from each other with non-trivial amplitudes. In our model we find that we do not need an infinite renormalization in any of the diagrams. We need to renormalize the coupling constants by a finite amount. The contributions from even number of diagrams are finite, hence require no infinite renormalization. The scattering of two vectors to four, or to any higher even number of vectors is finite, while the productions of spinors from the scattering of vectors go to zero as the cut-off is removed. A further point would be to couple an elementary scalar field to the model similar to the work of Bardeen et al. We see that some properties of the previous model are changed drastically. In our new model spinors can scatter from each other and some interactions are needed infinite renormalization. We need to renormalize the coupling constants by an infinite amount. Using renormalization group analysis techniques we construct the renormalization group equations for the model. When the renormalization group equations are solved, we encounter the "Landau Pole" problem which is well known in quantum electro dynamics. It means that the model is trivial. To find non-trivial model we can add scalar and vector fields, which have different symmetries, to the polynomial Lagrangian. We see that only one of them gives non-trivial model result in the constructed model.

Özet

Fermiyonlar doğanın vazgeçilmez yapıtaşlarıdır. Sadece Fermiyonları kullanarak doğayı açıklayan bir model kurmak tekrar eden bir fikirdir. Bu doğrultuda bir deneme de Gürsey'in çalışmasıdır ve bu çalışmada polinom olmayan Lagrange fonksiyonu kendi kendine etkileşen spinörleri tanımlamak için yazılmıştır. Bu modelin önemi konformal değişmez olması ve klasik çözümlere sahip olmasıdır. Kortel bu teori için sonradan instantonik ve mezonik çözümler olduğu gösterilen bir çözüm bulmuştur. Akdeniz ve diğerleri (1982, 1983) tarafından vektör Gürsey modeline eşdeğer polinom Lagrange fonksiyonu yazılması ve kuantize edilmesi için bir kaç çalışma yapılmıştır. Eşdeğer modelde temel alanların birbirleriyle saçılmaları entegral sınırı sonsuza götürüldüğünde yani cut-off kaldırıldığında sıfır olmaktadır. Diğer alanlardan oluşturulan, diğer bir deyişle kompozit olan alanların birbirleriyle saçılmaları sonlu kalmaktadır ve kompozit alanlarla spinör alanların etkileşmeleri sıfır saçılma genliğine sahiptir. Sonuç olarak temel spinör alanlar için sonlu renormalizasyona sahip model elde edilmiştir. Cut-off kaldırıldığında etkileşmelerin sıfırdan farklı olduğu model, başka bir deyişle trivial olmayan model arayışlarına modele skaler alan eklenerek devam edilmiştir. Önceki modele ait bazı özelliklerin değiştiği görülmüştür. Modelde bazı etkileşmeler için sonsuz renormalizasyona ihtiyaç vardır. Model için renormalizasyon grubu denklemleri oluşturulup çözümler elde edildiğinde kuantum elektro dinamiğinde çok iyi bilinen "Landau Kutbu" problemi ile karşılaşılmıştır. Landau kutbu probleminin kaldırılması için polinom Lagrange fonksiyonuna değişik gruplarda skaler ve/veya vektör alanlar eklenmiştir ve altı tane yeni model oluşturulmuştur. Oluşturulan modellerin sadece bir tanesinde Landau kutbu bulunmadığı görülmüştür ve trivial olmayan bir model elde edilmiştir.