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Introduction to the Functional RG and Applications to Gauge Theories

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Renormalization Group and Effective Field Theory Approaches to Many-Body Systems

Part of the book series: Lecture Notes in Physics ((LNP,volume 852))

Abstract

This lecture course is intended to fill the gap between graduate courses on quantum field theory and specialized reviews or forefront-research articles on functional renormalization group approaches to quantum field theory and gauge theories.

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Notes

  1. 1.

    In this short introduction, we use but make no attempt at fully explaining the standard QFT nomenclature; for the latter, we refer the reader to any standard QFT textbook, such as [22, 23].

  2. 2.

    Now, only the “sup” part of \(\varGamma _k\) is convex. For finite \(k\), any non-convexity of \(\varGamma _k\) must be of the form of the last regulator term of Eq. (6.20).

  3. 3.

    In case of fermionic Grassmann-valued fields, the following \(\phi \) derivative should act on Eq. (6.22) from the right.

  4. 4.

    Also the double-well potential with \(\omega ^2<0\) can be studied with RG techniques, see [30, 31].

  5. 5.

    I am grateful to A. Wipf for analytically determining the 2nd-order coefficient.

  6. 6.

    One may wonder whether a gauge-invariant flow can be set up with a gauge-invariant regularization procedure. In fact, this is an active line of research, and various promising formalisms have been developed so far [40–42]. However, the price to be paid for the resulting simple gauge constraints comes in the form of nontrivial Nielsen identities, non-localities or extensive algebraic constructions. For practical application, we thus consider the standard formulation described here as the most efficient approach so far.

  7. 7.

    An alternative option could be to use only the flow equation together with a regulator that does automatically suppress artificial relevant operators. In fact, this is conceivable in the framework of optimization [50].

  8. 8.

    In the Dyson-Schwinger literature, the gluon and ghost propagator behavior is often characterized by dressing functions \(Z_{\text{ DSE}},G_{\text{ DSE}}\) which are related to the wave function renormalizations by \(Z_A(p^2)=Z_{\text{ DSE}}^{-1}(p^2)\) and \(Z_{\text{ gh}}(p^2)=G_{\text{ DSE}}^{-1}(p^2)\) for \(k\rightarrow 0\).

  9. 9.

    Be aware of footnote 3 on p. xxx.

  10. 10.

    The occurrence of \(\gamma _5\) in the fermion mass term arises from our fermion conventions [99]; these are related to more standard conventions by a discrete chiral rotation.

  11. 11.

    Of course, in order to avoid any ambiguity with respect to possible Fierz rearrangements of the four-fermion interactions in the point-like limit, all possible linearly-independent four-fermion interactions, in principle, have to be included in the truncation. For simplicity, we confine ourselves here just to the scalar–pseudo-scalar channel, where chiral condensation is expected to occur. For the four-fermion interactions that will be generated by the flow, we use the Fierz decomposition as proposed in [100].

  12. 12.

    The momentum-independent part can, for instance, be fixed such that \(\partial _tZ_\phi (q=k)=0\), ensuring that the approximation of a momentum-independent \(Z_\phi \) is self-consistent.

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Acknowledgments

It is a great pleasure to thank A. Schwenk and J. Polonyi for organizing the ECT\(\ast \) school and for creating such a stimulating atmosphere. I am particularly grateful to the students for their active participation, critical questions and detailed discussions which have left their traces in these lecture notes. I would like to thank J. Braun, C.S. Fischer, J. Jaeckel, J.M. Pawlowski, and C. Wetterich for pleasant and fruitful collaborations on some of the topics presented here, and for numerous intense discussions, some essence of which has condensed into these lecture notes. Critical remarks on the manuscript by J. Braun, M. Ghasemkhani, J.M. Pawlowski, and A. Wipf are gratefully acknowledged. This work was supported by the DFG Gi 328/1-3 (Emmy-Noether program).

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  1. Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743, Jena, Germany

    Holger Gies

  2. Institute for Theoretical Physics, Heidelberg University, Philosophenweg 16, 69120, Heidelberg, Germany

    Holger Gies

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  1. Holger Gies
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  1. Institut für Kernphysik, und Extreme Matter Institute EMMI, Technische Universität Darmstadt, Schlossgarten 2, Darmstadt, 64289, Germany

    Achim Schwenk

  2. Lab. de Physique Therique, Université Louis Pasteur, rue de l'Université 3, Strasbourg, 67084, France

    Janos Polonyi

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Gies, H. (2012). Introduction to the Functional RG and Applications to Gauge Theories. In: Schwenk, A., Polonyi, J. (eds) Renormalization Group and Effective Field Theory Approaches to Many-Body Systems. Lecture Notes in Physics, vol 852. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27320-9_6

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