# Talk:Nonlinear Sigma model

## Contents |

## Reviewer A

I think the article is fine. The only comment I would make is that the model does NOT have to be defined in Minkowski space (ie just after (1) where we have \eta I would add normally though often the model is also considered in Euclidean space-time).

Otherwise it is fine.

## Reviewer B

### comment 1

The purpose of a Scholarpedia article should in principle be that of giving undergraduate students some synthetic ideas concerning, possibly advances, scientific subjects. These ideas should start from the basic reasons that have induced the development of the subject, the particular scientific and technical reasons that have induced the development of the subject, the achieved results and applications, the connections between the presented subject and other subjects of scientific interest.

The (linear) Sigma Model was first introduced by Gell-Mann and Levy in 1960 in strict relation with the development of chiral theories, those justifying the V-A structure of weak currents. Its 4 dimensional gauged version was then studied in connection with the current algebra of and its renormalization. The non-linear model was also studied in four dimensions, in spite of its lack of renormalizabilty, as an effective model of chiral field theory and hence for the construction of chiral perturbation theory and many phenomenological applications. More recently the two-dimensional non-linear model became one of the major ingredients of string theory quantization and related subjects. Indeed the field is interpreted as the string (target space) coordinate while the two dimensional manifold (word sheet) represents for any string point the proper time and the coordinate along the string line. Friedan, in his thesis, has shown as one can get informations on string perturbative dynamics using the two dimensional non-linear sigma model renormalization properties.

Technically, from the field theory point of view, the two-dimensional non-linear sigma model appears as a special very interesting kind of renormalizable model due to the reparametrization invariance and the reparametrization freedom of the fields understood as coordinated of a Riemannian manifold.

The connections between the non-linear sigma model and conformal field theory is, however related to the former aspects, an important character of the model which has required a remarkable number of investigations.

Here I stop, even if the list of different interesting applications of the sigma model in both its linear and non-linear version could continue, since I do not want to write a report longer than the paper. I just make my conclusion explicit. What I have listed above is just a short indication for the History section, this should be followed by sketchy descriptions of the different aspects, their inter-relations and possible developments. In this light the paper needs a radical revision.

By the way, I have not understood in which sense gauging a model would imply spontaneous symmetry breaking.

### comment 2

Concerning his answer to my report I do not understand two points:

1) Where the strict limits on the number of pages come from? As a matter of fact the Author can verify that almost all the accepted articles are much longer than 2 pages.

2) Why the Author considers “historical” the needed reference to the Physical applications of the Sigma Model. The article the Author should write concerns Physics, even Quantum Physics, and not Mathematics.

Thus I think that the article in its present form could be accepted
under the title:
Non-Linear-Sigma-Model and the subtitle: The classical field theory
and Mathematical aspects.

Indeed sentences such as: “There are no kinetic terms for the gauge field A in the action” have no meaning at the fully quantized level since, even if those terms are not present at the classical level, they reappear, depending on space-time dimension, after the quantum corrections. There are also many other quantum effects that depend on the space- time dimension, such as e.g. infra-red effects in two dimensions that should be specified. Since this does not seem to fit the aims of the Author I think that the revision of the title with insertion of a suitable subtitle could be sufficient for acceptance.

## Author

### Author's comments to the Reviewer B comment 1

First, I would like to thank the reviewer B for taking his time and efforts in reviewing my very short contribution about (Non-linear) Sigma-Models to the Scholarpedia.

I would completely agree with all of his comments, if my contribution were either a review, a paper, or even a letter. It is not. According to the Scholarpedia guidelines, a single subject contribution is supposed to be limited to 1 or 2 pages. My first submission about NLSM as a field theory, was just about 1-page-long.

More importantly, the concept I have in mind, to meet the criteria of the Scholarpedia, is as follows. Firstly, my contribution should contain the most general and actual definition of the subject and the basic specific tools, in the application-INDEPENDENT and historically-INDEPENDENT (modern) way. Secondly, it should mention a few basic applications, and the references addressing details. It is already a challenge to acomplish that within two pages.

I am not in favor of including any historical details (they are inevitably biased, and consume a lot of space) and emphasizing linear sigma models (they do not share the main feature of the NLSM, namely, its geometrical nature, but a global symmetry only). I even find the name `linear Sigma Model' to be confusing and outdated, though I have to admit its historical origin and its occasional use in the current literature.

Still, thanks to the reviewers reports, I decided to expand my contribution to the Scholarpedia by mentioning some additional NLSM applications and connections, such as (i) four-dimensional pion effective action, (ii) renormalization, (iii) conformal field theory, and (iv) supersymmetry. It was done within my concept outlined above.

By the way, given a NLSM on a group manifold G, and a normal subgroup H of G, its global H-symmetries can be gauged (i.e. made local) by introducing the (YM) gauge fields, though without adding their kinetic term to the NLSM action. Then the gauge fields can be eliminated via their algebraic equations of motion, thus defining the NLSM on a coset G/H. Such NLSM describes the Goldstone bosons associated with the spontaneous symmetry breaking of G to H.

Should you have more comments, I would be happy to discuss them with you.

Best regards!

### Author's comments to the Reviewer B comment 2

OK, I agree that there are some important issues to be added. So, I wrote more about the NLSM renormalization at various places in my article, mentioned more physical applications, and inserted more equations and references, while keeping the article short.

I also like the Reviewer B proposal to make the title more specific, e.g. `Non-linear Sigma Model'.

## Editor note

- I'm not against articles from 2 to 15 pages, provided that they are well written and useful. --RG

## Comments by J Zinn-Justin

In my opinion, the article being rather concise, could be much improved by adding a historical review or at least adding the corresponding references with some comments (as reviewers sindicated). Let me suggest the following references:

The non-linear sigma model was introduced in

- Gell-Mann, M and Lévy, M (1960).
*Nuovo Cimento*16: 705.

The general group theoretical problem of non-linear realizations is discussed in

- Coleman, S; Wess, J and Zumino, B (1969). Structure of Phenomenological Lagrangians. I.
*Phys. Rev.*177: 2239-2247.

The quantization was discussed in

- Gerstein, I S; Jackiw, R; Lee, B W and Weinberg, S (1971). Chiral loops.
*Physical Review D*3: 2486-2492 . - Honerkamp, J and Meetz, K (1971). Chiral-Invariant Perturbation Theory.
*Physical Review D*3: 1996-1998. - Faddeev, L D and Slavnov, A A (1971). Invariant perturbation theory for nonlinear chiral Lagrangians.
*Theor. Math. Phys.*8: 843-850.

The renormalization group properties were discussed in

- Polyakov , A M (1975). Interaction of Goldstone particles in two dimensions. Applications to ferromagnets and massive Yang-Mills fields.
*Physics Letters B*59: 79-81. - Brézin, E and Zinn-Justin, J (1976). Renormalization of the Nonlinear σ Model in 2+ε Dimensions—Application to the Heisenberg Ferromagnets.
*Phys. Rev. Lett.*36 : 691-694. - Brézin, E and Zinn-Justin, J (1976). Spontaneous breakdown of continuous symmetries near two dimensions.
*Phys. Rev. B*14 : 3110-3120. - Bardeen, W A; Lee, B W and Shrock, R E (1976). Phase transition in the nonlinear σ model in a (2+ε)-dimensional continuum.
*Phys. Rev. D*14: 985-1005.

The renormalizability in two dimensions was proved in

- Brézin, E; Le Guillou, J C and Zinn-Justin, J (1976). Renormalization of the nonlinear σ model in 2+ε dimensions.
*Phys. Rev. D*14: 2615-2621.

For symmetric spaces different properties are described and RG functions calculated in

- Pisarski, R D (1979). Nonlinear σ models of symmetric spaces.
*Phys. Rev. D*20: 3358-3371. - Brézin, E; Hikami, S and Zinn-Justin, J (1980). Generalized non-linear σ-models with gauge invariance.
*Nucl. Phys. B*165 : 528-544.

The proof of the renormalizability for general homogeneous spaces is reproduced in

Zinn-Justin, J (2002). Quantum Field Theory and Critical Phenomena (4th edition). Oxford University Press, Oxford. ISBN 0198509235

Calculations of the RG beta-functions up to three and four loops for a class of non-linear models can be found in

- Hikami, S (1981). Three-loop ß-functions of non-linear σ models on symmetric spaces
*Phys. Lett. B*98: 208-210. - Bernreuther, W and Wegner, F J (1986). Four-Loop-Order β Function for Two-Dimensional Nonlinear Sigma Models.
*Phys. Rev. Lett.*57: 1383-1385. - Wegner, F (1989). Four-loop-order β-function of nonlinear σ-models in symmetric spaces.
*Nucl. Phys. B*316: 663-678.

For a discussion of general two-dimensional models on Riemannian manifolds see

- Friedan, D H (1980). Nonlinear Models in 2+ε Dimensions.
*Phys. Rev. Lett.*45: 1057-1060 . - Friedan, D H (1985). Nonlinear Models in 2+ε Dimensions.
*Annals of Physics*163: 318-419.

Then one can refer to SUSY generalizations, for example,

- Alvarez-Gaumé, L; Freedman, D Z and Mukhi, S (1981). The background field method and the ultraviolet structure of the supersymmetric nonlinear σ-model.
*Annals of Physics*134: 85-109. - M T, van de Ven; A E M, Zanon and D, FORENAME3 (409-428). Nuclear Physics B
*277*1986: Grisaru.

Jean Zinn-Justin

### Author's answer:

OK, I added today 37 new references on some original papers, with my short comments, to the REFERENCES part of my article, and moved all my earlier Refs (books and reviews only) + 1 new reference (the last book) to FURTHER READING. Most of the refs suggested by Prof. Jean Zinn-Justin were included too. It resulted in merely one extra page. There were no changes in the text. I am grateful to Jean Zinn-Justin for his kind suggestions, especially as regards his references on some old original papers on the subject. -- SK

## Humble Student

I know this isn't a forum, but I'm trying to understand the origin of the rotation in the section 19.5 of Weinberg's Quantum Theory of Fields, Vol II. I might be bias, but I think that is the starting point for someone who actually wants to start researhing the physics of the subject. Thanks a lot for your time, If I'm being arrogant I apologize in advance.

### Author's answer:

Thanks for viewing my article! The pion NLSM is just a phenomenological model, and its symmetries are the guide to construct it as the effective field theory of pions. Its origin follows from the fundamental theory, which is QCD. The symmetry considerations are the simplest way to get it.