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We summarize recent progress on the symmetric subtraction of the Non-Linear Sigma Model in

The purpose of this paper is to provide an introduction to the recent advances in the study of the renormalization properties of the SU(2) Non-Linear Sigma Model (NLSM) and of the quantum deformation of the underlying non-linearly realized classical SU(2) local symmetry. The results reviewed here are based mainly on References [

The linear sigma model was originally proposed a long time ago in [_{2}. If one considers instead the model on the manifold defined by
_{π} (to be identified with the pion decay constant), while the pions are massless. Despite the fact that this is only an approximate description (since in reality the pions are massive and chiral SU(2) × SU(2) is not exact, even before being spontaneously broken), the approach turned out to be phenomenologically quite successful and paved the way to the systematic use of effective field theories as a low energy expansion.

The first step in this direction was to obtain a phenomenological lagrangian directly, by making use of a pion field with non-linear transformation properties dictated by chiral symmetry from the beginning. After the seminal work of

Modern applications involve, e.g., Chiral Perturbation Theory [

Effective field theories usually exhibit an infinite number of interaction terms, that can be organized according to the increasing number of derivatives. By dimensional arguments, the interaction terms must then be suppressed by some large mass scale M (so that one expects that the theory is reliable at energies well below M) (For a modern introduction to the problem, see e.g., [

The problem of the mathematically consistent evaluation of quantum corrections in this class of models has a very long history. On general grounds, the derivative couplings tend to worsen the ultraviolet (UV) behavior of the theory, since UV divergent contributions arise in the Feynman amplitudes that cannot be compensated by a multiplicative renormalization of the fields and a redefinition of the mass parameters and the coupling constants in the classical action (truncated at some given order in the momentum expansion). Under these circumstances, one says that the theory is non-renormalizable (A compact introduction to renormalization theory is given in [

It should be stressed that the key point here is the instability of the classical action: no matter how many terms are kept in the derivative expansion of the tree-level action, there exists a sufficiently high loop order where UV divergences appear that cannot be reabsorbed into the classical action. On the other hand, if in a non-anomalous and non-renormalizable gauge theory one allows for

Sometimes symmetries are so powerful in constraining the UV divergences that the non-linear theory proves to be indeed renormalizable (although not by power-counting), like for instance the NLSM in two dimensions [

In four dimensions the situation is much less favorable. It has been found many years ago that already at one loop level in the four-dimensional NLSM there exists an infinite number of one-particle irreducible (1-PI) divergent pion amplitudes. Many attempts were then made in the literature in order to classify such divergent terms. Global SU(2) chiral symmetry is not preserved already at one loop level [_{a}

More recently it has been pointed out [

At every order in the loop expansion there is only a finite number of divergent ancestor amplitudes. They uniquely fix the divergent amplitudes involving the pions. Moreover, non-renormalizability of this theory in four dimensions can be traced back to the instability of the classical non-linear local symmetry, that gets deformed by quantum corrections. These results hold for the full off-shell amplitudes [

A comment is in order here. In

The four dimensional SU(2) NLSM provides a relatively simple playground where to test the approach based on the LFE, that can be further generalized to the SU(N) case (and possibly even to a more general Lie group).

Moreover, when the background vector field becomes dynamical, the SU(2) NLSM action allows one to generate a mass term for the gauge field

The classical SU(2) NLSM in _{a}, a_{D} = ^{D}^{2−1} is the mass scale of the theory _{a}_{a}

_{L} × SU(2)_{R} chiral transformation

We notice that such a global transformation is non-linearly realized, as can be easily seen by looking at its infinitesimal version. E.g., for the left transformation one finds:
_{0} is given by _{a}

Perturbative quantization of the NLSM requires to carry out the path-integral
_{a}_{a}_{a}_{a}

The presence of two derivatives in the interaction term is the cause (in dimensions greater than 2) of severe UV divergences, leading to the non-renormalizability of the theory.

Some years ago it was recognized that the most effective classification of the UV divergences (both for on-shell and off-shell amplitudes) of the NLSM cannot be achieved in terms of the quantized fields _{a}_{a}

The LFE stems from the _{L}-symmetry that can be established from the gauge transformation of the flat connection _{μ}_{L}(2)-transformation of Ω
_{0} in _{L} transformations; however it is easy to made it invariant, once one realizes that it can be written as
_{μ}_{0} with
_{aμ}_{L} group, _{L} symmetry given by
_{a}

In order to implement the classical local SU(2)_{L} invariance at the quantum level, one needs to define the composite operator _{0} in _{0} through the term
_{0} is invariant under

The important observation now is that the variation of full one-particle irreducible (1-PI) vertex functional Γ^{(0)} = _{ext}_{a}, i.e_{a}^{(0)}:
_{0}-term, entering in the variation of the _{a}_{0} by taking functional derivatives w.r.t. its source _{0}, one is able to control its renormalization, once radiative corrections are included [

In the following Section we are going to give a compact and self-contained presentation of the algebraic techniques used to deal with bilinear functional equations like the LFE in

We are going to discuss in this Section the consequences of the LFE for the full vertex functional. The imposition of a quantum symmetry in a non-power-counting renormalizable theory is a subtle problem, since in general there is no control on the dimensions of the possible breaking terms as strong as the one guaranteed by the Quantum Action Principle (QAP) in the renormalizable case. Let us discuss the latter case first.

If the tree-level functional Γ^{(0)} is power-counting renormalizable, the renormalization procedure [

This procedure is a recursive one, since it allows to construct Γ^{(}^{n}^{)} once Γ^{(}^{j}^{)},

A desirable feature of power-counting renormalizable theories is that the dependence of 1-PI Green's functions under an infinitesimal variations of the quantized fields and of the parameters of the model is controlled by the so-called Quantum Action Principle (QAP) [

Let us now consider a certain symmetry ^{(0)} classical action. Under the condition that the symmetry ^{(}^{n}^{)} in _{0} according to
_{0} is also nilpotent, as a consequence of the nilpotency of _{0} on both sides of

The problem of establishing whether the functional identity
^{′(}^{n}^{)} = Γ^{(}^{n}^{)} + Ξ^{(}^{n}^{)} will fulfill _{0}) of the operator
_{0} in the space of integrated local polynomials in the fields, the external sources and their derivatives. Two
_{0}-invariant integrated local polynomials
_{1} and
_{2} belong to the same cohomology class in _{0}) if and only if
_{0}) is empty if the only cohomology class is the one of the zero element, so that the condition that
_{1} is
_{0}-invariant implies that
_{0} is empty in the space of breaking terms, then ^{(}^{n}^{)}. Moreover it must be checked that the UV dimensions of the possible counterterms Ξ^{(}^{n}^{)} are compatible with the action-like condition, so that renormalizability of the theory is not violated. An extensive review of BRST cohomologies for gauge theories is given in [

The QAP does not in general hold for non-renormalizable theories. This does not come as a surprise, since the appearance of UV divergences with higher and higher degree, as one goes up with the loop order, prevents to characterize the induced breaking of a functional identity in terms of a polynomial of a given finite degree (independent of the loop order).

Moreover for the NLSM another important difference must be stressed: the basic Green's functions of the theory are not those of the quantized fields _{a}_{aμ}_{0} (coupled to _{0}). This result follows from the invertibility of
_{0}|_{ϕa}_{=0} = _{D}_{a}_{a}_{0}] = Γ[_{0}]|_{ϕa}_{=0}.

Γ[_{0}] is the generating functional of the so called ancestor amplitudes, _{0} legs.

It is therefore reasonable to assume the LFE in

From a path-integral point of view, _{a}

As we have already noticed, in four dimensions the NLSM is non power-counting renormalizable, since already at one loop level an infinite number of divergent _{K}_{0} external _{0}-legs and _{J̃}

For instance, in _{μ}_{0}-

By taking into account Lorentz-invariance and global su(2)_{r} symmetry, the list of UV divergent amplitudes reduces to

It should be emphasized that the model is not power-counting renormalizable, even when ancestor amplitudes are considered, since according to

A special case is the 2-dimensional NLSM. For

A comment is in order here. In _{μ}_{0} is the unique scalar source required, in the special case of the SU(2) group, in order to control the renormalization of the non-linear classical SU(2) transformation of the _{a}

In order to study the properties of the LFE, it is very convenient to introduce a fictious BRST operator _{a}_{a}_{0}. Moreover, the BRST transformation of _{a}^{2} = 0.

The introduction of the ghosts allows to define a grading w.r.t. the conserved ghost number.

In terms of the operator _{a}

Suppose now that all divergences have been recursively subtracted up to order ^{(n)} in ^{(n)}:
_{0}). This subtraction procedure has been shown to be symmetric [

By the nilpotency of _{0} and their derivatives with ghost number zero. This can be achieved by using the techniques developed in [

One first builds invariant combinations in one-to-one correspondence with the ancestor variables _{aμ}_{0}. For that purpose it is more convenient to switch back to matrix notation. The difference _{μ}_{μ}_{μ}_{ba}_{aμ}_{aμ}

One can also prove that the following combination
_{a}_{0} → _{0} is also invertible.

In terms of the new variables _{0} and _{μ}_{a}_{b} = ω_{a}Θ_{ab}, i.e.,
_{ab}_{0} and _{aμ}_{0} and _{aμ}

The subtraction strategy is thus the following. One computes the divergent part of the properly normalized ancestor amplitudes that are superficially divergent at a given loop order according to the WPC formula in _{aμ}_{aμ}_{0} → _{0} is carried out. This gives the full set of counterterms required to make the theory finite at order

As an example, we give here the explicit form of the one-loop divergent counterterms for the NLSM in _{a}_{a}_{1} − ℐ_{2} − ℐ_{3} is zero, while the remaining invariants give

The invariants in the combination ℐ_{6} + 2ℐ_{7} generate the counterterms in the first line between square brackets; these counterterms are globally SU(2) invariant. The other terms are generated by invariants involving the source _{0}. In [_{a}_{a}_{μ}_{0} can be found in [

The correspondence with the linear sigma model in the large coupling limit has been studied in [

The massive NLSM in the LFE formulation has been studied in [

In the SU(2) NLSM just one scalar source _{0} is sufficient in order to formulate the LFE. For an arbitrary Lie group _{I}_{I}_{J}_{J}_{J}_{J}

At orders ^{(n)} is an inhomogeneous equation

For that purpose it is convenient to redefine the ghost according to
_{ab}_{0} and _{aμ}_{a}, ω̄_{a}

By the nilpotency of ^{(n)}:
^{(n)} is linear in _{a}_{a}_{a}_{a}_{aμ}_{0}).

The explicit dependence on _{a}

The solution of ^{(n)} will be the sum of a ^{(n)}, depending only on _{aμ}_{0}, plus a lower order term:
_{t}_{a}, ω̄_{a}, K̅_{0}, _{aμ}_{a}, ϕ_{a}

An important remark is in order here. The theory remains finite and respects the LFE if one adds to Γ^{(n)} some integrated local monomials in _{aμ}_{0} and ordinary derivatives thereof (with finite coefficients), compatible with Lorentz symmetry and global SU(2) invariance, while respecting the WPC condition in

This observation suggests that these finite parameters cannot be easily understood as physical free parameters of the theory, since they cannot appear in the tree-level action. It was then proposed to define the model by choosing the symmetric subtraction scheme discussed in Section 5 and by considering as physical parameters only those present in the classical action plus the scale of the radiative corrections Λ [

When the vector source _{aμ}

The subtraction procedure based on the LFE has been used to implement a mathematically consistent formulation of non-linearly realized massive Yang-Mills theory. SU(2) Yang-Mills in the LFE formalism has been formulated in [_{a}

This is a very powerful (and somehow surprising) result. Indeed all possible monomials constructed out of _{aμ}

Otherwise said, the peculiar structure of the Yang-Mills action
_{aμv}_{aμ}

The approach based on the LFE can also be used for non-perturbative studies of Yang-Mills theory on the lattice. The phase diagram of SU(2) Yang-Mills has been considered in [

An analytic approach based on the massless bound-state formalism for the implementation of the Schwinger mechanism in non-Abelian gauge theories has been presented in [

A very important physical application of non-linearly realized gauge theories is the formulation of a non-linearly realized electroweak theory, based on the group SU(2) × U(1). The set of gauge fields comprises the SU(2) fields _{aμ}_{μ}_{μ}_{3}_{μ}_{μ}_{W}_{W}_{μ}_{μ}_{1}_{μ}_{2}_{μ}^{±}

The inclusion of physical scalar resonances in the non-linearly realized electroweak model, while respecting the WPC, yields some definite prediction for the Beyond the Standard Model (BSM) sector. Indeed it turns out that it is impossible to add a scalar singlet without breaking the WPC condition. The minimal solution requires a SU(2) doublet of scalars, leading to a CP-even physical field (to be identified with the recently discovered scalar resonance at 125.6 GeV) and to three additional heavier physical states, one CP-odd and neutral and two charged ones [

The WPC and the symmetries of the theory select uniquely the tree-level action of the non-linearly realized electroweak model. As in the NLSM case, mathematically additional finite counterterms are allowed at higher orders in the loop expansion. In [

The question remains open of whether a Renormalization Group equation exists, involving a finite change in the higher order subtractions, in such a way to compensate the change in the sliding scale Λ of the radiative corrections. We notice that in this case the finite higher order counterterms would be a function of the tree-level parameters only (unlike in the conventional effective field theory approach, where they are treated as independent extra free parameters). This issue deserves further investigation, since obviously the possibility of running the scale Λ in a mathematically consistent way would allow to obtain physical predictions of the same observables applicable in different energy regimes.

The LFE makes it apparent that the independent amplitudes of the NLSM are not those of the quantum fields, over which the path-integral is carried out, but rather those of the background connection _{μ}_{0}, coupled to the solution of the non-linear constraint _{0}. The WPC can be formulated only for these ancestor amplitudes; the LFE in turn fixes the descendant amplitudes, involving at least one pion external leg. Within this formulation, the minimal symmetric subtraction discussed in Section 5 is natural, since it provides a way to implement the idea that the number of ancestor interaction vertices, appearing in the classical action and compatible with the WPC, must be finite.

However, it should be stressed that the most general solution to the LFE, compatible with the WPC, does not forbid to choose different finite parts of the higher order symmetric counterterms (as in the most standard view of effective field theories, where such arbitrariness is associated with extra free parameters of the non-renormalizable theory), as far as they are introduced at the order prescribed by the WPC condition and without violating the LFE.

In this connection it should be noticed that the addition of the symmetric finite renormalizations in

It is a pleasure to acknowledge many enlightening discussions with R. Ferrari. Useful comments and a careful reading of the manuscript by D. Bettinelli are also gratefully acknowledged.

The author declares no conflict of interest.

We report here the invariants controlling the one-loop divergences of the NLSM in _{aμ}