Geometry of the Scalar Sector

Geometry of the Scalar Sector


The -matrix of a quantum field theory is unchanged by field redefinitions, and so only depends on geometric quantities such as the curvature of field space. Whether the Higgs multiplet transforms linearly or non-linearly under electroweak symmetry is a subtle question since one can make a coordinate change to convert a field that transforms linearly into one that transforms non-linearly. Renormalizability of the Standard Model (SM) does not depend on the choice of scalar fields or whether the scalar fields transform linearly or non-linearly under the gauge group, but only on the geometric requirement that the scalar field manifold is flat. We explicitly compute the one-loop correction to scalar scattering in the SM written in non-linear Callan-Coleman-Wess-Zumino (CCWZ) form, where it has an infinite series of higher dimensional operators, and show that the -matrix is finite.

Standard Model Effective Field Theory (SMEFT) and Higgs Effective Field Theory (HEFT) have curved , since they parametrize deviations from the flat SM case. We show that the HEFT Lagrangian can be written in SMEFT form if and only if has a invariant fixed point. Experimental observables in HEFT depend on local geometric invariants of such as sectional curvatures, which are of order , where is the EFT scale. We give explicit expressions for these quantities in terms of the structure constants for a general symmetry breaking pattern. The one-loop radiative correction in HEFT is determined using a covariant expansion which preserves manifest invariance of under coordinate redefinitions. The formula for the radiative correction is simple when written in terms of the curvature of and the gauge curvature field strengths. We also extend the CCWZ formalism to non-compact groups, and generalize the HEFT curvature computation to the case of multiple singlet scalar fields.




1. Department of Physics, University of California at San Diego, La Jolla, CA 92093, USA \affiliation2. CERN TH Division, CH-1211 Geneva 23, Switzerland

1 Introduction

Current experimental data is consistent with the predictions of the Standard Model (SM) with a light Higgs boson of mass  GeV. The measured properties of the Higgs boson agree with SM predictions, but the current experimental accuracy of measured single-Higgs boson couplings is only at the level of , and no multi-Higgs boson couplings have been measured directly. It is important to consider generalizations of the SM with additional parameters in order to quantify the accuracy to which the SM is valid or to detect deviations from SM predictions.

Over the past 40 years, many theoretical ideas have been proposed for the underlying mechanism of electroweak symmetry breaking. Theories that survive must be consistent with the currently observed pattern of electroweak symmetry breaking, which is well-described by the SM. A general model-independent analysis of electroweak symmetry breaking can be performed using effective field theory (EFT) techniques. Assuming there are no additional light particles beyond those of the SM at the electroweak scale  GeV, the EFT has the same field content as the SM. There are two main EFTs used in the literature, the Standard Model Effective Field Theory (SMEFT) and Higgs Effective Field Theory (HEFT). In this paper, we make the relationship between these two theories precise.

The Higgs boson of the SM is a neutral scalar particle. In the SM Lagrangian, it appears in a complex scalar field , which transforms as under the electroweak gauge symmetry. An oft-stated goal of the precision Higgs physics program is to test whether (a) the Higgs boson transforms as part of a complex scalar doublet which mixes linearly under with the three “eaten” Goldstone bosons , or (b) whether the Higgs field is a singlet radial direction which does not transform under the electroweak symmetry. In case (b), the three Goldstone modes transform non-linearly amongst themselves under the electroweak symmetry, in direct analogy to pions in QCD chiral perturbation theory, and do not mix with the singlet Higgs field. In case (a), there are relations between Higgs boson and Goldstone boson (i.e. longitudinal gauge boson) interactions, whereas in case (b), no relations are expected in general. An objective of this paper is to explore the distinction between these two pictures for Higgs boson physics.

The properties of the scalar sector of the SM and its EFT generalizations can be clarified by studying it from a geometrical point of view [1]. The scalar fields define coordinates on a scalar manifold . The geometry of is invariant under coordinate transformations, which are scalar field redefinitions. The quantum field theory -matrix also is invariant under scalar field redefinitions, so it depends only on coordinate-independent properties of . Consequently, experimentally measured quantities depend only on the geometric invariants of , such as the curvature. Formulating physical observables geometrically avoids arguments based on a particular choice of fields. It also allows us to correctly pose and answer the question of whether the Higgs boson transforms linearly or non-linearly under the electroweak gauge symmetry. Further, a geometric analysis gives a better understanding of the structure of the theory and its coordinate-invariant properties.

The UV theory can have additional states, such as massive meson excitations in the case of theories with strong dynamics. At low energies, the EFT interactions in the electroweak symmetry breaking sector are described by a Lagrangian with scalar degrees of freedom on some manifold , with the Lagrangian expanded in gradients of the scalar fields. The geometric description captures the features of the UV dynamics needed to make predictions for experiments at energies below the scale of new physics.

The geometrical structure of non-linear sigma models has been worked out over many years, mainly in the context of supersymmetric sigma models (see e.g. [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]). The applications to the SM Higgs sector presented here are new, and they provide a better understanding of the structure of HEFT and the search for signals of new physics through the couplings of the Higgs boson.

Some of the results in this paper have already been given in Ref. [1]. Here we provide more explanation of the results presented there, as well as details of explicit calculations in that work. These calculations include the proof of renormalizability of the SM written in non-linear form, and the derivation of the one-loop effective action for a curved scalar manifold . For most of the paper, we will assume that the scalar sector has an enlarged global symmetry, known as custodial symmetry. Also note that we will usually treat the scalar sector in the ungauged case, referring to the scalar fields as Higgs and Goldstone bosons. The gauged version of the theory follows immediately by replacing ordinary derivatives by gauge covariant derivatives. In the gauged case, the Goldstone bosons are eaten via the Higgs mechanism, becoming the longitudinal polarization states of the massive electroweak gauge bosons. Thus, the Higgs-Goldstone boson relations we refer to are in fact relations between the couplings of the Higgs boson and the three longitudinal gauge boson states and  [13, 14, 15].

The organization of the paper is as follows. The relationship between the SM, SMEFT and HEFT is discussed in Sec. 2 from a geometrical point of view. It is shown that SMEFT is a special case of HEFT when is expanded about an invariant fixed point. Further, it is shown that the existence of such an invariant fixed point is a necessary and sufficient condition for the existence of a choice of scalar fields such that the Higgs field transforms linearly under the electroweak gauge symmetry. In Sec. 3, a scalar field redefinition is performed on the SM Lagrangian to write it in terms of the non-linear exponential scalar field parametrization of Callan, Coleman, Wess and Zumino (CCWZ) [16, 17]. In this non-linear parametrization, the SM contains an infinite series of terms with arbitrarily high dimension, but it nonetheless remains renormalizable. We demonstrate renormalizability of the CCWZ form of the Lagrangian by an explicit calculation of the one-loop correction to . The -matrix is finite, even though Green’s functions are divergent. The one-loop calculations in the linear and non-linear parameterizations only differ by equation-of-motion terms. Both parameterizations have a divergence-free -matrix at one loop after including the usual counterterms computed in the unbroken phase. Sec. 4 presents the covariant formalism for curved scalar field space. We discuss global and gauge symmetries in terms of Killing vectors of the scalar manifold, and we derive the one-loop correction to the effective action for curved . In Sec. 5, the geometric formulation of theories is connected with the standard coordinates of CCWZ. We give formulæ for the curvature tensor in terms of field strengths for a general sigma model. We also discuss the extension of the CCWZ standard coordinates to non-compact groups. As shown in Ref. [1], the sign of deviations from SM values of Higgs boson-longitudinal gauge boson scattering amplitudes is controlled by sectional curvatures in HEFT. For theories based on compact groups, these sectional curvatures are typically positive. We compute the sectional curvature, and show that in certain cases, it can be negative. In Sec. 6, we briefly discuss the SM and custodial symmetry violation, and the relation between the SM scalar manifold and the configuration space of a rigid rotator. Sec. 7 generalizes HEFT to the case of multiple singlet Higgs bosons. Finally, Sec. 8 provides our conclusions. Additional formulae are provided in the appendices, including intermediate steps in the computation of the one-loop correction to HEFT given in Refs. [18, 1], and discussion of the complications for non-reductive cosets.

2 Sm Smeft Heft

In this section, we discuss the scalar sector of the SM and its EFT generalizations, SMEFT and HEFT, as well as the relationship between these three theories. We begin with a summary of the scalar sector of the SM.

The SM scalar Lagrangian (with the gauge fields turned off) is


This scalar Lagrangian is the most general invariant Lagrangian with terms of dimension built out of a Higgs doublet that transforms as under . As is well-known, the SM scalar sector has an enhanced global custodial symmetry group . This global symmetry can be made manifest by writing the SM complex scalar doublet field in terms of four real scalar fields,


Substitution in Eq. (1) yields the Lagrangian


where . Lagrangian Eq. (3) is invariant under global symmetry transformations


The scalar field transforms linearly as the four-dimensional vector representation of the global symmetry group . The minimum of the potential is the three-sphere of radius ,


which is the Goldstone boson vacuum submanifold of the SM. The radius of the sphere, GeV, is fixed by the gauge boson masses. It is conventional to choose the vacuum expectation value


and expand the Lagrangian about this vacuum state in the shifted fields and , ,


The vacuum expectation value spontaneously breaks the global symmetry group to the unbroken global symmetry group . The Goldstone bosons , , transform as a triplet under the unbroken global symmetry, whereas transforms as a singlet. We will refer to both the enlarged global symmetries and as custodial symmetries. The unbroken global symmetry group leads to the relation , which is a successful prediction of the SM. The experimental success of this gauge boson mass relation implies that custodial symmetry is a good approximate symmetry of the SM.

The Lagrangian Eq. (3) in terms of shifted fields Eq. (7) becomes


The singlet is the physical Higgs field with mass


whereas the Goldstone bosons are strictly massless. In the gauged theory, the three Goldstone bosons of the global symmetry breakdown are “eaten” via the Higgs mechanism, becoming the longitudinal polarization states of the massive and gauge bosons. Note that the -invariant potential depends on an -invariant combination of both and .

Equating the scalar kinetic energy term in Eq. (8) with


defines the scalar metric for the SM scalar manifold with coordinates given by the scalar fields . Distances on are determined by .

The four-dimensional SM scalar manifold is shown in Fig. 1. The symmetry acts by rotations. The minimum of the potential is the solid red curve, and forms the three-dimensional Goldstone boson submanifold of radius . The parameterization Eq. (7) is a Cartesian coordinate system for centered on the vacuum (black dot), where is the horizontal direction, and , , are the three other directions orthogonal to . The angular coordinates of are . The symmetry acts linearly on .

Figure 1: Two-dimensional depiction of the four-dimensional scalar manifold of the SM. The SM vacuum is the black dot shown in the figure. The origin (green dot) is an invariant fixed point. The left and right diagrams show the fields in Cartesian and polar coordinates, respectively. symmetry acts linearly on the Cartesian coordinates. In polar coordinates, is -invariant, and the angular coordinates transform non-linearly under the symmetry. The scalar manifold is flat, so the scale setting the curvature is formally infinite.

In Cartesian coordinates, it seems intuitively clear that and interactions are related, given that the four scalar fields belong to the same Higgs doublet Eq. (2). However, the precise relation is subtle. In order to understand this point better, it is instructive to express the SM Lagrangian Eq. (3) in polar coordinates as well.

In polar coordinates,1


where is the magnitude of , and is a four-dimensional unit vector. The four shifted scalar fields consist of the three dimensionless angular coordinates (the direction of on ), and the radial coordinate . The SM Lagrangian in polar coordinates is


An advantage of expressing the SM Lagrangian in polar coordinates is that the three Goldstone boson fields of are derivatively coupled. In addition, the scalar potential in polar coordinates only depends on the radial coordinate , whereas in Cartesian coordinates it depends on all four scalar fields.

The symmetry transformations of in polar coordinates are


so the Higgs field is invariant under transformations, and transforms linearly by an orthogonal transformation that preserves the constraint . Due to the constraint, however, only three of the four components of are independent. Without loss of generality, one can take the first three components of to be the independent components. Then, the fourth component is a non-linear function of the independent components . The non-linear constraint turns the linear transformation on into a non-linear transformation when written in terms of unconstrained fields. Thus, the transformation on the three independent angular coordinates is a non-linear transformation.

Many different parameterizations of in terms of the independent unconstrained coordinates are possible. Two natural non-linear parameterizations are the square root parameterization and the exponential parameterization, which are defined by




respectively. For most of this paper, we use the exponential parameterization for since it corresponds to the standard coordinates of CCWZ.

Rotations in the , and planes act linearly on , and leave invariant. However, rotations in the , and planes mix and . For example, a rotation gives


In terms of the independent unconstrained coordinates of the square root parameterization, , and rotations act linearly, but a 14 rotation gives


The transformation Eq. (17) is non-linear. Consequently, Eq. (13) is called a non-linear transformation, since it is non-linear when written in terms of unconstrained coordinates .

In polar coordinates, and are very different objects, and it is not at all obvious that and interactions are related. Nevertheless, all we have done is switch from Cartesian coordinates to polar coordinates while keeping the Lagrangian fixed. This change of coordinates does not affect physical observables such as -matrix elements. Any relations that exist amongst physical observables must be present irrespective of the choice of coordinates.

We have summarized the standard analysis of the SM in Cartesian and polar coordinates. In Cartesian coordinates, the Higgs field and the three Goldstone fields form a four-dimensional representation which transforms linearly under . In polar coordinates, the Higgs field is an singlet or invariant, and the three Goldstone bosons parameterizing the unit vector transform among themselves under the non-linear transformation law Eq. (13). The Higgs boson field in polar coordinates is not the same field as the Higgs boson field in Cartesian coordinates. The relation between the two Higgs boson fields is


so that


By the Lehmann-Symanzik-Zimmermann (LSZ) reduction formula, and give the same -matrix, and both are perfectly acceptable choices for the Higgs boson field.2

2.1 Fixed Point

We now return to the question of whether the Higgs field transforms linearly or non-linearly under the electroweak gauge symmetry, and whether interactions of the Higgs boson and the three Goldstone bosons (i.e. longitudinal gauge boson polarizations) are related. As we have just seen, this question is not well-posed in the SM, since the answer depends on the choice of coordinates. However, it is intuitively clear that there is an underlying relationship between the couplings of the Higgs and Goldstone bosons in the SM that does not remain valid in the general context of HEFT. We need to formulate any coupling relations in a coordinate-invariant way. There are two conditions which make the SM special — (i) there is a point (or ) of which is an invariant fixed point, and (ii) the scalar manifold is flat, i.e. it has a vanishing Riemann curvature tensor.3 As we now see, relations in the SM between the couplings of the Higgs boson and the three Goldstone bosons arise from these two conditions which are no longer true in HEFT in general.

We first analyze whether the Higgs field is part of a multiplet that transforms linearly under the symmetry. Even in the SM, the answer to this question depends on the choice of coordinates. The coordinate-invariant formulation of the question is: Does there exist a choice of coordinates for such that the Higgs field is part of a multiplet that transforms linearly under the symmetry? We now show that the answer is yes if and only if has an invariant fixed point.4

It is clear from the transformation law Eq. (4) for that the origin is an invariant fixed point. Any other theory that can be formulated using fields which transform linearly under the symmetry also must have an invariant fixed point at . Thus, if there exists a choice of coordinates which transform linearly under the symmetry, then the scalar manifold has an invariant fixed point.

Now, we prove the converse statement. Consider a general scalar manifold , which is described by coordinates which transform under transformations and which contains an invariant fixed point . Is there a choice of coordinates such that the scalar fields transform linearly under the symmetry? The key result we need for the proof in this direction is the linearization lemma of Coleman, Wess and Zumino [16], which states that if is an invariant fixed point, there exists a set of coordinates in a neighborhood of which transform linearly under transformations in some (possibly reducible) representation of . If this representation contains the four-dimensional vector representation of , then the four coordinates , , which transform as a vector, can be combined into a Higgs doublet , as in Eq. (2). Thus, the Higgs field is part of a linear representation if and only if there is an invariant fixed point whose tangent space transforms under in a representation that contains the vector representation. In most of our examples, the scalar manifold is four-dimensional, and the tangent space of automatically transforms as the vector representation, so we will omit the condition that the tangent space transforms as the vector representation.

The condition that contains an fixed point divides theories into those which can and cannot be written in a form where the Higgs boson is part of a multiplet that transforms linearly under the electroweak gauge symmetry group (or the larger global custodial symmetry group ). There are theories which satisfy the condition that contains an invariant fixed point, but which do not have relations between the couplings of the Higgs boson and the Goldstone bosons. To understand this point better, we now introduce SMEFT and HEFT.

2.2 Smeft

SMEFT is an effective theory with the most general Lagrangian written in terms of SM fields, including all independent higher dimension operators with dimension greater than four, suppressed by an EFT power counting scale . The independent operators at dimension six, and their renormalization [19, 20], has been worked out in detail  [21, 22, 23, 24, 25, 26, 27, 28].

In SMEFT, all operators involving scalar fields are written in terms of the Higgs doublet field . For simplicity, at present we assume that the custodial symmetry group of SMEFT is . The SMEFT scalar kinetic energy term, which consists of all operators built out of Higgs doublet fields with two derivatives, is


where the sum in the first line is over all independent mass dimension operators built out of two derivatives and Higgs doublet fields and , and the second line gives the explicit expression including the leading operator. Using Eq. (2) to write the Higgs doublet in terms of four real scalars , yields a scalar kinetic energy term of the form


where the arbitrary functions and are defined by power series expansions in their argument . In the limit, the kinetic energy term of SMEFT reduces to the SM kinetic energy term, so the functions and satisfy and . Comparison of Eq. (21) with Eq. (10) yields the SMEFT scalar metric


The Riemann curvature tensor of the curved scalar manifold in SMEFT can be calculated from the above metric. The SM is a special case of the SMEFT in which all higher dimension operators with are set to zero, or equivalently, one takes the limit . From Eq. (22), we see that in this limit the SMEFT metric yields the SM scalar metric in Cartesian coordinates, and becomes flat with vanishing Riemann curvature tensor.

Most composite Higgs models [29, 30] can be written in SMEFT form. A simple example is the composite Higgs model [31]. The symmetry breaking field lives on a sphere of radius in five dimensions, and can be written as


is the SMEFT field, and the Lagrangian can be written in SMEFT form. In general, composite Higgs theories solve the hierarchy problem by vacuum misalignment. There is a field configuration where the vacuum is “aligned,” so that the electroweak symmetry is unbroken. This is the point of SMEFT, and measures deviations from this point, as in Eq. (23). In the neighborhood of , gives a linear representation of . For HEFT to reduce to SMEFT form, this representation must transform as the vector of . Composite Higgs models which are consistent with experimental data are of this type [32, 33].

The SMEFT is the EFT generalization of the SM where the scalar manifold has an invariant fixed point, so that the Lagrangian can be written in terms of the Higgs doublet field or the four-dimensional vector field on which the symmetry acts linearly. This restriction is not enough to give the same scattering amplitudes of Higgs bosons and Goldstone bosons (longitudinal gauge bosons) as the SM, which can be verified by explicit computation using Eq. (21). In Refs. [1, 34], it was shown that the high energy behavior of the cross sections for and scattering depend on two sectional curvatures which can be obtained from the Riemann curvature tensor . The one-loop radiative correction in the scalar sector also depends on the Riemann curvature tensor  [1]. The details of these calculations are presented later in this paper. The important point is that the scattering cross sections and the one-loop radiative correction in SMEFT are equal to the SM values if and only if is flat, i.e. the Riemann curvature tensor of SMEFT vanishes. This statement is a coordinate-independent condition, which is true in the SM using either Cartesian or polar coordinates. Thus, the intuitive idea that the Goldstone boson and Higgs boson directions in Fig. 1 are related in the SM can be formulated precisely as the condition that in the SM is a four-dimensional flat Euclidean space.

2.3 Heft

Figure 2: The HEFT scalar manifold. There is for each value of . An invariant fixed point exists if there is a value of for which the radius of vanishes. The fixed point at is shown in a dotted region of since it need not exist. There is no boundary at the transition between the solid and dotted regions, if the dotted region does not exist. Instead, the manifold can extend to infinity, or is smoothly connected without a point where . SMEFT has a scalar manifold where is an invariant fixed point that always exists, and are like the HEFT manifold including the dotted section.

HEFT is a generalization of the SM using the polar coordinate form of the SM Lagrangian, Eq. (12). The theory is written in terms of three angular coordinates that parametrize a unit vector , and one or more coordinates . As in the SM, the unit vector parametrizes the Goldstone bosons directions [35, 36, 37, 38, 39]. Here we restrict to one additional field. The case of multiple is considered in Sec. 7. The coordinate is chosen so that is the ground state. The HEFT Lagrangian is


where is an arbitrary dimensionless function with a power series expansion in  [40], normalized so that


since the radius of in the vacuum is fixed to be by the gauge boson masses. The HEFT manifold is shown schematically in Fig. 2. has a coordinate , with an fiber at each value of . While is often called the radial direction by analogy with the polar coordinate form of the SM, in HEFT, is simply a scalar field, and need not be the radius of anything. HEFT power counting is discussed in [41], and is a combination of chiral power counting [42, 43] and naive dimensional analysis [44]. The terms omitted in Eq. (24) are the NLO operators [45, 46, 47, 48, 49].

The transformation laws for and are given in Eq. (13), so is invariant and transforms non-linearly. The SM and SMEFT are both special cases of HEFT. In the SM, the radial function is


The SMEFT kinetic energy term Eq. (21) yields the polar coordinate kinetic energy term


This kinetic energy term can be put into the standard form of HEFT by performing a field redefinition on to make the coefficient of the term equal to . Thus, the HEFT scalar metric for one singlet Higgs field is


where the function is parametrized by coefficients , ,


The coefficient is already constrained by experiment to be equal to its SM value to a precision of about . The coefficient is not constrained at present. The HEFT scalar metric reduces to the SM scalar metric when .

In the SMEFT, the functions and in Eq. (22) are expanded out in powers of , whereas in the HEFT literature, they are treated as arbitrary (unexpanded) functions.

When is it possible to rewrite HEFT in SMEFT form? We have seen that a necessary and sufficient condition is that there must exist an invariant fixed point on . One can then define as coordinates around and write the Lagrangian in terms of . The general HEFT manifold consists of and a sequence of spheres of radius fibered over each point of . The HEFT manifold is depicted in Fig. 2. acts on the point on the surface of by rotation, so that maps points on the the red curve onto itself. No point of is invariant under the full group, so the only way to have an invariant fixed point is if the sphere has zero radius, i.e. if for some . Such a point may not exist; its existence depends on the structure of the HEFT manifold. For example, if the HEFT manifold has no invariant fixed point. In the SM, is given by Eq. (26), and at . If there is an fixed point, the HEFT can be written as a SMEFT. Some examples are given in Refs. [50, 51, 49].

To summarize, HEFT with no invariant point, i.e. no point where , cannot be written in SMEFT form, and hence cannot be written using a doublet field (or equivalently, a four-dimensional vector field which transforms linearly under the electroweak gauge symmetry. This statement answers the question posed in the introduction: when do the scalar fields of HEFT transform linearly or non-linearly under the gauge symmetry? They transform linearly if and only if for some , so that there is a fixed point.

Thus, we have shown that the relationship of the SM, SMEFT and HEFT is described by the hierarchy SM SMEFT HEFT. SMEFT is a special case of HEFT when there is a value of the Higgs field where . The SM is the special case of SMEFT (and HEFT) when there are no higher dimension operators in the theory, and so is flat.

One can convert the SMEFT Lagrangian to HEFT form using Eq. (11) to switch from Cartesian and polar coordinates. One can attempt to convert from HEFT to SMEFT form using


with some function of . This substitution gives a Lagrangian that need not be analytic in . However, if there is an fixed point, then there is a suitable change of variables such that the resulting Lagrangian is analytic in .

Scattering amplitudes are evaluated in perturbation theory by expanding the action in small fluctuations about the vacuum (the black dot) in Fig. 2. The curvature of is a local quantity, given by the metric and its derivatives up to second order, evaluated at the vacuum state. Scattering amplitudes, and hence experimentally measurable cross sections depend directly on the curvature [1, 34], so the curvature of the EFT scalar manifold can be determined experimentally.

Whether there is an invariant fixed point where is a non-perturbative question, since in the ground state. One has to move a distance of at least away from the ground state to probe the existence of a fixed point where vanishes.

3 Renormalization of the Model

One of the main points of Refs. [1, 34] and this paper is that the scalar sector can be studied in a coordinate-invariant way. Thus, the SM written in the linear Cartesian coordinates Eq. (8), and the SM written in non-linear polar coordinates Eq. (12), are completely equivalent formulations of the same theory. In particular, even though Eq. (12) is a non-linear formulation of the SM, where the Lagrangian contains operators of arbitrarily high dimension, it is still renormalizable. In this section, we demonstrate this result by explicit computation of the one-loop scattering amplitude. It is instructive to see how the theory is renormalizable even when written in non-linear form — we find that Green’s functions can be divergent but the -matrix is finite. We compute the scattering amplitude in the theory in the linear and non-linear formulations. The SM is the special case . Our results are related to the well-known calculations by Longhitano [35, 36] and by Appelquist and Bernard [37, 38] in the non-linear sigma model with no Higgs field, and by Gavela et al. [52] in HEFT.

3.1 Preliminaries

The sigma model has an -component real scalar field with Lagrangian


which is invariant under transformations


where is a real orthogonal matrix. The global symmetry group of the theory is , which has generators. We are mainly interested in the broken phase . The minimum of the potential in Eq. (31) is at , so the set of minima form the surface , the sphere in -dimensions, with radius . All points on are equivalent vacua. One can make an transformation so that


where is a unit vector pointing to the North pole of the sphere. The global symmetry group of the theory is spontaneously broken to the subgroup , the rotations that leave invariant. The vacuum manifold is . The number of broken generators is , so the theory has Goldstone bosons.

The generators of are


where the non-zero entries of have in row , column , and in row , column . It is often convenient to consider without the restriction , which includes each unbroken generator twice, since . The matrices have been normalized so that


The broken generators are


The unbroken generators are , , which are the generators of the subgroup.

The unbroken transformations with the vacuum choice are rotations that leave the North pole fixed, i.e. rotations among the first components of . Of course, one could have picked any other vacuum state , a vector of length pointing in some direction , which is invariant under , transformations that leave fixed. Since can be rotated to by a transformation, the two vacua are equivalent and is conjugate to , , where is the transformation that maps to , .

In the linear realization, one expands about the classical vacuum in Cartesian coordinates


The Lagrangian Eq. (31) with this field parametrization is


The unbroken global symmetry subgroup under which is a vector is manifest in this coordinate system, but the original global symmetry group of the underlying theory is not obvious. From Eq. (38), we see immediately that all are massless, and is massive with


The masses and couplings in Eq. (38) are given in terms of two parameters and , which is a reflection of the hidden invariance of the theory.

We now parameterize the model in a different way, following the non-linear realization of CCWZ. Let




Eq. (40) is a polar coordinate system in field space with radial coordinate and dimensionless angular coordinates of the sphere . The field is a real orthogonal matrix, so


The Lagrangian Eq. (31) with this field parameterization is


The potential only depends on the radial coordinate ; it is independent of the Goldstone boson fields , which are massless and derivatively coupled. Expanding the exponential in a power series gives the leading terms


The full expression is given in Appendix A. The Lagrangian Eq. (44) naively looks like a non-renormalizable theory with an infinite set of higher dimension operators. However, it is simply the renormalizable Lagrangian Eq. (31) written using a different parametrization of the fields.

The Lagrangians Eq. (31) and Eq. (44) correspond to different choices of coordinates for the scalar manifold , and they are related by a field redefinition. Since the -matrix is invariant under a field redefinition, the two theories have the same -matrix. Renormalizability of Lagrangian Eq. (44) is hidden, as is invariance. Treating Eq. (44) as an EFT with the usual power counting rules (for a pedagogical review, see [53]) gives the same -matrix as Eq. (31). In particular, Eq. (44) is a renormalizable theory with a finite number of renormalization counterterms even though it looks superficially non-renormalizable.

3.2 Renormalization

The linear model including renormalization counterterms is


In dimensional regularization in dimensions, the one-loop counterterms , and are given by


These renormalization counterterms can be computed using perturbation theory in the unbroken phase, where . The combinations and are the counterterm renormalizations of the invariant operators and , and they are gauge independent.

Field theory divergences arise from the short distance structure of the theory. Thus the renormalization counterterms do not depend on whether the symmetry is unbroken or spontaneously broken; the same counterterms Eq. (45) also renormalize the broken theory.5 In the broken phase, one uses Eq. (37) with the replacement ,


The tadpole shift has a perturbative expansion in powers of , and it is computed order by order in perturbation theory by cancelling the tadpole graphs to maintain . At tree-level, . The Lagrangian Eq. (38) including renormalization counterterms is


The Lagrangian Eq. (38) gives finite Green’s functions and finite -matrix elements in the broken phase. The underlying -symmetry of the theory ensures that the counterterms in Eq. (48) are given in terms of , and of the unbroken theory Eq. (46), plus a tadpole shift . The term in Eq. (48) is a pure counterterm, and keeps the Goldstone bosons massless in the presence of radiative corrections. The Higgs mass is .

In the non-linear realization, one uses


with given by Eq. (41). Since Eq. (49) is simply a different choice of field coordinates in comparison to Eq. (47), the renormalization constants and tadpole shift are the same. Note that no factor is needed in the exponent of . The in Eq. (41) are periodic variables, since a rotation about some axis is equivalent to the identity transformation, and cannot be multiplicatively renormalized.

The renormalized Lagrangian in the non-linear parameterization is


with given by Eq. (46). This Lagrangian can be expanded in a power series in and used in perturbation theory. The claim which we wish to prove is that Eq. (50) gives finite -matrix elements (but not necessarily Green’s functions), since it is a field redefinition of Eq. (48).

3.3 Scattering

The finiteness of the -matrix using Lagrangian Eq. (50) seems surprising, and is worth explaining in some detail. The Lagrangian Eq. (50) contains vertices with an arbitrary number of fields. For example, it contains the vertices in Fig. 3 which involve five and six scalar fields.

Figure 3: Some vertices in the Lagrangian Eq. (50). Solid lines are and dashed lines are .

We will use Eq. (50) to compute the infinite part of scattering to one loop.6 In the non-linear case, we will only give the explicit results for the amplitude to , but we have checked that the -matrix is finite to all orders in . The skeleton graphs that contribute to the -matrix for are shown in Fig. 4. The tree-level amplitude is given by the skeleton graphs in Fig. 4 with the blobs replaced by tree vertices, and the one-loop correction to the amplitude is given by using the one-loop irreducible vertex for one blob in each graph, and tree vertices for the rest. We will give the results of the various contributions using the linear parameterization, Eq. (48), and the non-linear one, Eq. (50), which will be denoted by subscripts and , respectively. In this subsection, we only compute the infinite parts of the graphs, and omit an overall factor of .

Figure 4: Skeleton graphs for the scattering -matrix. The shaded blobs are irreducible vertices and two-point functions. There is also a pion wavefunction correction to the amplitude.

The tadpole vanishes by invariance. The tadpole graphs are shown in Fig. 5 and give the one-point function

Figure 5: tadpole graphs. Graph includes counterterm and tadpole vertices.

where the three terms are the infinite contributions from the three diagrams. The linear and non-linear parameterizations give the same result. Using the counterterms from Eq. (46), , and requiring that vanishes gives


Note that the tadpole shift is finite in the non-gauged case, but it develops an infinite piece when gauge interactions are turned on.

The infinite contribution to the two-point function is

Figure 6: propagator graphs. Graph includes counterterm and tadpole vertices.

from the individual graphs in Fig. 6. The two forms of the Lagrangian give a different result. In the non-linear parameterization, is derivatively coupled, so graph (d) is ; in the linear parameterization, the graph is since there is a coupling in the potential. In the non-linear parameterization, the one-loop corrected propagator in Fig. 4(b) is


on expanding out the correction Eq. (53), and does not have a double pole in because . This feature is important for the cancellation of divergences.

The propagator graphs give the infinite contribution to the two-point function