Contents

RIKEN-MP-96, RIKEN-QHP-171, MAD-TH-14-10


Effective field theory for spacetime symmetry breaking

[5mm]

Yoshimasa Hidaka, Toshifumi Noumi and Gary Shiu
Theoretical Research Devision, RIKEN Nishina Center, Japan

[3mm] Department of Physics, University of Wisconsin, Madison, WI 53706, USA

[3mm] Center for Fundamental Physics and Institute for Advanced Study,

Hong Kong University of Science and Technology, Hong Kong

[5mm] hidaka@riken.jp, toshifumi.noumi@riken.jp, shiu@physics.wisc.edu


Abstract


We discuss the effective field theory for spacetime symmetry breaking from the local symmetry point of view. By gauging spacetime symmetries, the identification of Nambu-Goldstone (NG) fields and the construction of the effective action are performed based on the breaking pattern of diffeomorphism, local Lorentz, and (an)isotropic Weyl symmetries as well as the internal symmetries including possible central extensions in nonrelativistic systems. Such a local picture distinguishes, e.g., whether the symmetry breaking condensations have spins and provides a correct identification of the physical NG fields, while the standard coset construction based on global symmetry breaking does not. We illustrate that the local picture becomes important in particular when we take into account massive modes associated with symmetry breaking, whose masses are not necessarily high. We also revisit the coset construction for spacetime symmetry breaking. Based on the relation between the Maurer-Cartan one form and connections for spacetime symmetries, we classify the physical meanings of the inverse Higgs constraints by the coordinate dimension of broken symmetries. Inverse Higgs constraints for spacetime symmetries with a higher dimension remove the redundant NG fields, whereas those for dimensionless symmetries can be further classified by the local symmetry breaking pattern.

1 Introduction

Symmetry and its spontaneous breaking play an important role in various areas of physics. In particular, the low-energy effective field theory (EFT) based on the underlying symmetry structures provides a powerful framework for understanding the low-energy dynamics in the symmetry broken phase [1].

For internal symmetry breaking in Lorentz invariant systems, the EFT based on coset construction had been established in 1960’s [2, 3]. When a global symmetry group is broken to a residual symmetry group , the corresponding Nambu-Goldstone (NG) fields are introduced as the coordinates of the coset space and the general effective action can be constructed from the Maurer-Cartan one form,

(1.1)

Such a coset construction was also extended to spacetime symmetry breaking [4, 5] accompanied by the inverse Higgs constraints [6] and has been applied to various systems (see, e.g., [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19] for recent discussions). Although the coset construction captures certain aspects of spacetime symmetry breaking, its understanding seems incomplete compared to the internal symmetry case and as a result generated a lot of recent research activities [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. It would then be helpful to revisit the issue of spacetime symmetry breaking based on an alternative approach, providing a complementary perspective to the coset construction.

For this purpose, let us first revisit the identification of NG fields for spacetime symmetry breaking. As in standard textbooks, symmetry breaking structures are classified by the type of order parameters and their local transformations generate the corresponding NG fields (we refer to this as the local picture). In Lorentz invariant systems, since only the condensation of scalar fields is allowed, we need not pay much attention to the type of order parameters. However, when Lorentz symmetry is broken or does not exist, the type of order parameters becomes more important. For example, when the order parameter is a non-abelian charge density, there appear NG modes with a quadratic dispersion different from that in Lorentz invariant systems [30, 31, 32, 33, 34, 35, 36]. In addition, if the charge density and the other order parameter that break the same symmetry coexist, some massive modes associated with the symmetry breaking appear [11, 37, 38, 39].1

For spacetime symmetry breaking, the standard coset construction based on global symmetry (refer to as the global picture) does not distinguish the types of order parameters. As is well-known in the case of conformal symmetry breaking [5, 6], a naive counting of broken spacetime symmetries based on the global picture contains redundant fields and causes a wrong counting of NG modes (see, e.g., Refs. [20, 24, 25]). The inverse Higgs constraints are introduced to compensate such a mismatch of NG mode counting. As discussed in Refs. [11, 13, 14], the inverse Higgs constraints eliminate not only the redundant fields but also the massive modes. Thus, to identify the physical NG fields, we should take into account the massive modes associated with the symmetry breaking in addition to the massless modes. Such massive modes often play an important role, e.g., the smectic-A phase of liquid crystals near the smectic-nematic phase transition, in which the rotation modes are massive [40]. In this paper, we would like to construct the effective action including these modes based on the local picture.

To proceed in this direction, it is convenient to recall the relation between the coset construction and gauge symmetry breaking for internal symmetry. When a gauge symmetry is broken, the NG fields are eaten by the gauge fields, and the dynamics is captured by the unitary gauge action for the massive gauge boson . Since the gauge boson mass is given by with the gauge coupling and the order parameter , the unitary gauge is not adequate to discuss the global symmetry limit , which corresponds to the singular massless limit. Rather, it is convenient to introduce NG fields by the Stückelberg method as

(1.2)

where and are the original and residual symmetry groups, respectively, and describes the NG fields. In this picture, we can take the global symmetry limit smoothly to obtain the same effective action constructed from the Maurer-Cartan one form (1.1). As this discussion suggests, the unitary gauge is convenient for constructing the general effective action. Indeed, it is standard to begin with the unitary gauge in the construction of the dilaton effective action and the effective action for inflation [41]. Based on this observation, we apply the following recipe of effective action construction to spacetime symmetry breaking in this paper:

  1. gauge the (broken) global symmetry,

  2. write down the unitary gauge effective action,

  3. introduce NG fields by the Stückelberg method and decouple the gauge sector.

relativistic symmetry diffeomorphism local Lorentz isotropic Weyl

 

translation
isometry
conformal
Table 1: Embedding of relativistic spacetime symmetries. In relativistic systems, spacetime symmetries can be classified into isometric and conformal transformations, by requiring spacetime isotropy. They can then be embedded into diffeomorphism, local Lorentz, and isotropic Weyl transformations.
nonrelativistic symmetry foliation preserving local rotation (an)isotropic Weyl internal

 

translation
Galilean
Schrödinger
Galilean conformal
Table 2: Embedding of nonrelativistic spacetime symmetries. One difference from the relativistic case is that nonrelativistic spacetime symmetries should preserve the spatial slicing. The corresponding coordinate transformations are then foliation preserving diffeomorphisms. Another difference is that nonrelativistic systems admit central extensions of spacetime symmetry algebras. Correspondingly, we included the internal gauge symmetry in the above table. See Appendix. A for details.

Our starting point is that any spacetime symmetry can be locally generated by Poincaré transformations and (an)isotropic rescalings. Correspondingly, we can embed any spacetime symmetry transformation into diffeomorphisms (diffs), local Lorentz transformations, and (an)isotropic Weyl transformations (see Tables 2 and 2 for concrete embedding of global spacetime symmetry). We then would like to gauge the original global symmetry to local ones. First, diffeomorphism invariance and local Lorentz invariance can be realized by introducing the curved spacetime action with the metric and the vierbein .2 On the other hand, there are two typical ways to realize isotropic Weyl invariance: Weyl gauging and Ricci gauging. In general, we can gauge the Weyl symmetry by introducing a gauge field and defining the covariant derivatives appropriately (Weyl gauging), whereas we can introduce a Weyl invariant curved space action if the original system is conformal (Ricci gauging). The anisotropic Weyl symmetry can be also gauged in a similar way. These procedures for spacetime symmetry allow us to gauge all the global symmetries together with the internal symmetries.

Once gauging global symmetry, we identify the broken local symmetry from the condensation pattern

(1.3)

and construct the effective action based on symmetry breaking structures. Here and in what follows, we use capital Latin letters for internal symmetry indices and the spin indices are implicit unless otherwise stated. When the condensation is spacetime dependent, diffeomorphism invariance is broken. On the other hand, local Lorentz invariance, (an)isotropic Weyl invariance, and internal gauge invariance are broken when the condensation has the Lorentz charge (spin), scaling dimension, and internal charge, respectively. If the symmetry breaking pattern is given, it is straightforward to take the unitary gauge and construct the effective action following the recipe. We will first apply our approach to some concrete examples to illustrate importance of the local viewpoint of spacetime symmetry breaking. We will then revisit the coset construction from such a local perspective. One important difference from the EFT for the internal symmetry breaking case in Lorentz invariant systems is that the EFT constructed from the unitary gauge contains not only massless modes but also massive modes associated with spacetime symmetry breaking. These massive modes transform nonlinearly under the broken symmetries, i.e., they are NG fields.

The organization of this paper is as follows. In Sec. 2, we explain our basic strategy in more detail. After reviewing the EFT for internal symmetry breaking, we discuss how global spacetime symmetry can be gauged. We then summarize how to construct the effective action based on the local symmetry breaking pattern. In Sec. 3, we apply our approach to codimension one branes to illustrate the difference between the global and the local picture of spacetime symmetry breaking. In the global picture, one may characterize the branes by the spontaneous breaking of translation and Lorentz invariance. In the local picture, on the other hand, such a symmetry breaking pattern can be further classified by the spin of the condensation forming the branes. We see that the spectra of massive modes associated with symmetry breaking depend on the spin of the condensation and the mass of massive modes is not necessarily high. If the masses are small compared with the typical energies of the system, the modes play a role as low energy degrees of freedom. Therefore, the local picture becomes important in following the dynamics of such massive modes appropriately. In Sec. 4 we discuss a system with one-dimensional periodic modulation, i.e., a system in which the condensation is periodic in one direction, by applying the effective action constructed in Sec. 3. We find that the dispersion relations of NG modes for the broken diffeomorphism are constrained by the minimum energy condition, in contrast to the codimension one brane case. In Sec. 5 such a discussion is extended to the breaking of a mixture of spacetime and internal symmetries. In Sec. 6 we revisit the coset construction from the local symmetry picture. We first show that the parameterization of NG fields in the coset construction is closely related to the local symmetry picture. We then discuss the relation between the Maurer-Cartan one form and the connections for spacetime symmetries. We finally classify the physical meanings of the inverse Higgs constraints based on the coordinate dimension of the broken symmetries. In Sec. 7 we make a brief comment on applications to gravitational systems. The final section is devoted to a summary. Details on the nonrelativistic case are summarized in Appendix A. A derivation of the Nambu-Goto action is given in Appendix B.

2 Basic strategy

In this section, we outline our basic strategy to construct the effective action for symmetry breaking including spacetime ones. In Sec. 2.1, we first review the relation between coset construction and gauge symmetry breaking for internal symmetry, and explain how the local picture can be used to construct the effective action for global symmetry breaking. To extend this discussion to spacetime symmetry, in Secs. 2.2 and  2.3, we discuss how global spacetime symmetries can be embedded into local ones. We then present our recipe for the effective action construction in Sec. 2.4.

2.1 EFT for internal symmetry breaking

Let us first review how the coset construction for internal symmetry breaking [2, 3] can be reproduced from the effective theory for gauge symmetry breaking. Suppose that a global symmetry group is spontaneously broken to a subgroup and the coset space satisfying

(2.1)

where and are the Lie algebras of and , respectively. represent the broken generators. In the coset construction, we introduce representatives of the coset space as with , whose left transformation is given by

(2.2)

where and . In general, depends on both and . The transformation from to is nonlinear, so that it is called the nonlinear realization. If is the element of the unbroken symmetry , , linearly transforms: . Here, it is useful to introduce the Maurer-Cartan one form to construct the effective Lagrangian, which is defined as

(2.3)

If we decompose into the broken component and the unbroken component as , each component transforms as

(2.4)

under transformation. Here note that the broken component transforms covariantly. In general, is reducible under transformation, and it can be decomposed into direct sums, . At the leading order in the derivative expansion, the effective Lagrangian for Lorentz invariant systems is given by3

(2.5)

where the trace is defined in a -invariant way. and are the component of the Maurer-Cartan one form and the decay constant for each irreducible sector, respectively. By construction, this Lagrangian is invariant under transformation.

We next move on to the effective action construction for gauge symmetry breaking. In this case, it is convenient to take the unitary gauge, where the NG fields are eaten by the gauge field and do not fluctuate. The general effective action can then be constructed only from the massive gauge field in an gauge invariant way. In relativistic systems, the effective Lagrangian takes the form

(2.6)

where and are the gauge coupling and the order parameters for symmetry breaking, respectively, and is the gauge field in the broken symmetry sector. Note that since we are considering the unitary gauge, Eq. (2.6) is not invariant under gauge transformation. Because the gauge boson mass is given by , the global symmetry limit for fixed corresponds to the singular massless limit, so that the unitary gauge is not appropriate to discuss the global symmetry limit. In order to take the global symmetry limit, it is convenient to introduce the NG fields by performing a field-dependent gauge transformation (the Stückelberg method):

(2.7)

The gauge invariance can be recovered by assigning a nonlinear transformation rule on the NG fields and we can take the global symmetry limit smoothly in this picture. Since the gauge sector decouples from the NG fields in the global symmetry limit, the effective action for the NG fields can be obtained by the replacements:

(2.8)
(2.9)

The latter is nothing but the Maurar-Cartan one form. As this discussion suggests, the unitary gauge is useful to construct the general ingredients needed to obtain the effective action for global symmetry breaking. Note that Wess-Zumino terms in the coset construction are reproduced by Chern-Simons terms in the unitary gauge action.

2.2 Local properties of spacetime symmetries

In the previous subsection, we saw that the unitary gauge action for gauge symmetry breaking can be used to construct the general effective action for global symmetry breaking. We now would like to extend such a discussion to spacetime symmetry breaking. For this purpose, let us recall the local properties of (infinitesimal) spacetime symmetry transformations in this subsection.4 Any spacetime symmetry transformation has an associated coordinate transformation

(2.10)

and its local properties around a point can be read off by expanding the parameter covariantly as

(2.11)

The first term is the zeroth order in and describes translations of the coordinate system. On the other hand, deformations of the coordinate system are encoded in the second term (the linear order in ), which can be decomposed as

(2.12)

The trace part and the symmetric traceless part are local isotropic rescalings (dilatations) and local anisotropic rescalings, respectively. The antisymmetric part corresponds to local Lorentz transformations. Any spacetime symmetry can therefore be locally decomposed into Poincaré transformations and isotropic/anisotropic rescalings. Correspondingly, the symmetry transformations of local fields are specified by their Lorentz charges and isotropic/anisotropic scaling dimensions.

As is suggested by Eqs. (2.11) and (2.12), we can embed global spacetime symmetry transformations into diffeomorphisms, local Lorentz transformations, and local isotropic/anisotropic Weyl transformations. For simplicity, let us consider the case of relativistic systems in this section (see Appendix A for extension to the nonrelativistic case). In relativistic systems, any spacetime symmetry transformation can be locally decomposed into the Poincaré part and the dilatation part because anisotropic rescalings are incompatible with the Lorentz symmetry:

(2.13)

Note that we have conformal transformations for general functions and isometric transformations for because the metric field transforms as

(2.14)

The transformation rules of local fields are then determined by their spin and scaling dimension. When a field follows a representation of the Lorentz algebra and has a scaling dimension , its symmetry transformation is given by

(2.15)

Here, the curved spacetime indices (Greek letters) and the local Lorentz indices (Latin letters) are converted by the vierbein as . The covariant derivative is defined in terms of the spin connection as

(2.16)

with the Christoffel symbols defined by

(2.17)

To identify the transformation (2.15) as local symmetries, it is convenient to rewrite it in the form,

(2.18)

We then notice that the latter three terms can be thought of as local Weyl transformations, local Lorentz transformations, and diffeomorphisms, respectively. Since the transformation rule of , , and under each local transformation is given by

(2.19)
(2.20)
(2.21)

we can reproduce the transformation (2.18) by the parameter choice

(2.22)

Note that the metric and the vierbein are invariant under the original global transformation (2.22), although they are not invariant under general local ones. Any global spacetime symmetry in relativistic systems can therefore be embedded into local Weyl transformations, local Lorentz transformations, and diffeomorphisms.

2.3 Gauging spacetime symmetry

In the previous subsection we discussed that any spacetime symmetry transformation in relativistic systems is locally generated by Poincaré and Weyl transformations. Isometric transformations can be embedded in diffeomorphisms and local Lorentz transformations, whereas conformal transformations require local Weyl transformations as well. Since NG fields correspond to local transformations of the order parameters for broken symmetries, we would like to construct the effective action from this local symmetry point of view. For this purpose, let us summarize how we can gauge global spacetime symmetry to those local symmetries.

When the system is isometry invariant before symmetry breaking, we gauge the Poincaré symmetry by introducing the curved spacetime action with the metric and the vierbein . For example, an action in a non-gravitational system on Minkowski space,

(2.23)

can be reformulated as

(2.24)

where the covariant derivative is given by Eq. (2.16). From the viewpoint of the curved space action (2.24), the original non-gravitational system can be reproduced by taking the metric and the vierbein as the Minkowski ones with the gauge choice,

(2.25)

The original global Poincaré symmetry can be also understood as the residual symmetries under the gauge conditions (2.25). The same story holds for non-gravitational systems on curved spacetimes.

On the other hand, there are two typical ways to gauge the Weyl symmetry: Weyl gauging and Ricci gauging (see, e.g., Ref. [42]). When the system is conformal, we can introduce a local Weyl invariant curved spacetime action, essentially because the local Weyl invariance is equivalent to the traceless condition of the energy-momentum tensor. Such a procedure is called the Ricci gauging and we need not introduce additional fields in this case. When the Ricci gauging is not applicable, we need to introduce a gauge field for Weyl symmetry and the covariant derivative defined by

(2.26)

where and the local Weyl transformation rule is given by5

(2.28)

If the curved spacetime action is global Weyl invariant, a local Weyl invariant action can be obtained by replacing the covariant derivative with the Weyl covariant derivative . For example, Eq. (2.24) is reformulated as

(2.29)

Note that the original action (2.23) can be reproduced by imposing the gauge condition

(2.30)

and symmetries of the action are reduced to the original global ones. Also, while the first two conditions in Eq. (2.30) are always invariant under the original global symmetries, the condition is not necessarily invariant when the original system is conformal. Indeed, it is not invariant under the special conformal transformation. Correspondingly, the Weyl gauge field appears in a particular combination in the action. For example, the action of a massless free scalar can be gauged as

(2.31)

where appears in a special conformal invariant combination .

2.4 EFT recipe

diffeomorphism local Lorentz local Weyl internal gauge

 

  spacetime dependence spin scaling dimension   internal charge
metric vierbein   Weyl gauge field gauge field
Table 3: Broken symmetries, condensation patterns, and gauge fields.

As we have discussed, all the global symmetries in relativistic systems can be embedded into diffeomorphisms, local Lorentz symmetries, local Weyl symmetry, and internal gauge symmetries. We can also gauge the global symmetry by the use of the procedures in the previous subsection and the standard internal gauging. Similar discussions hold for nonrelativistic systems accompanied by local anisotropic Weyl symmetries and internal symmetry associated with the possible central extension, as we illustrate in Appendix A. We now extend the discussion in Sec. 2.1 for internal symmetry to spacetime symmetry. First, the symmetry breaking patterns are classified by the condensation patterns:

(2.32)

When the condensation is spacetime dependent, diffeomorphism invariance is broken. On the other hand, local Lorentz invariance, local isotropic/anisotropic Weyl invariance, and internal gauge invariance are broken when the condensation has a Lorentz charge (spin), scaling dimension, and internal charge, respectively. Once we identify the symmetry breaking pattern, we can construct the effective action based on the following recipe just as in the case of internal symmetry breaking:

  1. gauge the (broken) global symmetry,

  2. write down the unitary gauge effective action,

  3. introduce NG fields by the Stückelberg method and decouple the gauge sector.

The first step can be performed by introducing gauge fields based on the procedure in Sec. 2.3 (see also table 3). We then take the unitary gauge, where the NG fields do not fluctuate. Using the dynamical degrees of freedom in the unitary gauge, we construct the general unitary gauge effective action invariant under the residual symmetries. Finally, we perform the Stückelberg method to introduce the NG fields and restore the full gauge symmetry. By decoupling the gauge sector, we obtain the effective action for the NG fields. In the following sections, we apply this recipe to concrete examples for spacetime symmetry breaking.

We emphasize that the condensation pattern rather than the breaking pattern of global symmetries plays an important role to identify the NG fields, unlike the case for internal symmetry breaking in Lorentz invariant systems. The breaking pattern of global symmetries itself cannot distinguish the breaking of diffeomorphism, local Lorentz, and Weyl symmetries. As will be seen in the next section, this difference becomes important when we discuss the massive modes originating from the symmetry breaking, although the existence of massless modes does not depend on the condensation pattern.

3 Codimension one brane

Figure 1: Scalar vs nonzero spin branes. While the diffeomorphism symmetry is broken for both types of branes, the local Lorentz symmetry is broken only for the nonzero spin case.

In this section, we apply our approach to codimension one branes on the Minkowski space to illustrate the difference between the global and the local picture of spacetime symmetry breaking. In the global picture, one may characterize the branes by the spontaneous breaking of translation and Lorentz invariance. In the local picture, on the other hand, such a symmetry breaking pattern can be further classified by the spin of the condensation forming the brane (see also Fig. 1):

  1. Scalar brane

    When a scalar field forms a codimension one brane, the only broken symmetry is the diffeomorphism invariance in the -direction orthogonal to the brane. In particular, the local Lorentz symmetry is not broken although the global one is.

  2. Nonzero spin branes

    When a nonzero spin field forms a codimension one brane and the condensation is aligned to the -direction, the local Lorentz invariance associated with the -direction is broken as well as the -diffeomorphism invariance.

Since those two cases are classified into the same category in the global picture, the local picture is necessary to distinguish them. In the rest of this section, we discuss in which situation the difference becomes important, if we take into account the massive modes associated with symmetry breaking.

In Sec. 3.1 we first perform the tree-level analysis of NG fields around scalar brane backgrounds, to illustrate our strategy in the previous section concretely. In Secs. 3.2 and 3.3, we construct the general effective action for the diffeomorphism symmetry breaking and apply it to single scalar brane backgrounds. In Secs. 3.4 and 3.5, we include local Lorentz symmetry breaking in the effective action construction and apply it to single nonzero spin brane backgrounds. For single brane backgrounds, it tuns out that the dynamics in the low-energy limit results in the same action regardless of the spin of the field that condenses. However, we find that the degeneracy is resolved beyond the low-energy limit and the resolving scale is not necessarily high. We see that the effective action based on the local picture can be used to investigate such an intermediate scale.

Though discussion in this section is only for single brane backgrounds, the effective action constructed in Secs. 3.2 and 3.4 is applicable to more general setups. In Sec. 4, we discuss periodic modulation and clarify the difference from the single brane case.

3.1 Real scalar field model for scalar brane

In order to illustrate our strategy, let us begin with a real scalar field model,

(3.1)

and perform the tree-level analysis of NG fields around domain-wall configurations. Here and are constant parameters and the potential has two minima at . The equation of motion

(3.2)

has the following domain-wall solution with the boundary conditions :

(3.3)

where characterizes the thickness of the brane. Note that there exists a one-parameter family of domain-wall solutions parameterized by the brane position  because of the translation invariance. The domain-wall configuration Eq. (3.3) breaks the translation invariance and the corresponding NG field can be obtained by promoting to a field as6

(3.4)

The action for the NG field is then given by

(3.5)

Using the integrated version of the equation of motion,7

(3.6)

we can further reduce the action (3.5) to the form

(3.7)

where we dropped total derivative terms at the second equality.

Let us then reproduce the action (3.7) along the line of our strategy. Following the EFT recipe in the previous section, we first gauge the translation symmetry to the diffeomorphism symmetry by introducing the curved coordinate action

(3.8)

We next consider fluctuations around the domain-wall background (3.3). Since -coordinate transformations of generate fluctuations of , we can take the unitary gauge at least as long as fluctuations are small. In other words, we can choose a coordinate frame such that the constant- slices coincide with the constant- slices. In this coordinate frame, the action is given by

(3.9)

where we used Eq. (3.6) in the second equality. Note that the action (3.9) enjoys only the -dimensional diffeomorphism symmetry along the -directions

(3.10)

and the NG field is eaten by the metric in this gauge. We then restore the -diffeomorphism invariance by the field-dependent coordinate transformation (the Stückelberg method)

(3.11)

After the transformation, the action (3.9) takes the form

(3.12)

where we dropped the tilde for simplicity. The action (3.12) is now invariant under the full diffeomorphism symmetry by assigning the following nonlinear transformation rule on the NG field :

(3.13)

Finally, we remove gauge degrees of freedom by taking the Minkowski coordinate. Since we are working on the Minkowski space, the full diffeomorphism invariance, nonlinearly realized by the NG fields, allow us to set the metric field as . The action (3.12) is then reduced to Eq. (3.7).

In this subsection we illustrated our approach by performing the tree-level analysis of NG fields around domain-wall backgrounds in the model (3.1). As we have seen, the introduction of the curved coordinate action (3.9) allows us to impose the unitary gauge condition , which breaks the -diffeomorphism invariance. The scalar is then eaten by the metric field . By performing the Stückelberg method and removing the gauge degrees of freedom, we obtained the action for NG fields. More generally, the action (3.1) can be modified with higher derivative terms due to quantum corrections for example. In the next subsection we construct the general effective action for NG fields by introducing the general unitary gauge action consistent with the symmetry.

3.2 Effective action for -diffeomorphism symmetry breaking

We then construct the general effective action for the -diffeomorphism symmetry breaking, by introducing the general unitary gauge action consistent with the symmetry. Just as in the previous real scalar model, let us introduce the metric field and work in the general coordinate system. We can then impose the unitary gauge condition, which prohibits fluctuations of the NG field and breaks the -diffeomorphism symmetry. In such a unitary gauge, the dynamical degrees of freedom are the metric field only (if there are no additional matter degrees of freedom) and there remains the -dimensional diffeomorphism symmetry. Schematically, we write this unitary gauge setup as

(3.14)

The general effective action is then constructed from the metric field in a -dimensional diffeomorphism invariant way. This setup is essentially the same as the one in single-field inflation [41]. Following the results there, ingredients of the unitary gauge effective action are given by

(3.15)

The lowest few terms of the expansion in fluctuations around the Minkowski metric, , and derivatives are given by

(3.16)

Here, ’s are scalar functions of , which depend on the details of the microscopic theory. Note that arises from in the previous real scalar model. One may then identify with higher derivative interactions in the real scalar model.

We next introduce the NG field for the -diffeomorphism by the Stückelberg method. Just as we did in the previous subsection, we perform a field-dependent coordinate transformation (3.11). Practically, this transformation can be realized by replacing a function with  [41], where we dropped the tilde for simplicity. Correspondingly, the unitary gauge action (3.16) is transformed as

(3.17)

where we used

(3.18)

The full diffeomorphism symmetry is now restored accompanied by the nonlinearly transformation rule (3.13). We then would like to remove the gauge degrees of freedom and construct the effective action for the NG field only. Since we are working on the Minkowski space, the full diffeomorphism invariance, nonlinearly realized by the NG fields, allow us to set the metric field as . In this Minkowski coordinate system, the ingredients (3.15) are given by

(3.19)

and the effective action (3.17) can be expressed as

(3.20)

where Latin indices indicate that we use the Minkowski coordinate.

So far, we have not taken into account the background equation of motion. Indeed Eq. (3.20) contains linear order terms in :

(3.21)

To remove such tadpole terms, we impose the background equation of motion . In the following we use because the constant shift of does not change the action for . Then, we obtain the action,

(3.22)

where and the second term is a total derivative. Note that the derivative with respect to in the last term of Eq. (3.22) acts not only explicitly on but also on , i.e., . Up to the second order in , the bulk action (the first term) can be expanded as

(3.23)

where and can be interpreted as the propagating speed in the -direction. On the other hand, the total derivative term can be expanded as

(3.24)

where note that there can arise linear order terms in from the total derivative term. We will revisit its physical meaning in the next section.

3.3 Physical spectra for single scalar domain-wall

We next take a close look at the effective action (3.22) and discuss physical spectra for a single scalar brane case. For simplicity, let us consider the case in this subsection.8 Then, the brane profile is characterized by the free function in the effective action. Generically, is related to the order parameter as

(3.25)

because the operator in the unitary gauge action typically arises from

(3.26)

To illustrate the physical spectra, it is convenient to take the well-studied domain-wall profile obtained in Sec. 3.1,

(3.27)

and the corresponding function of the form

(3.28)

Here, the constants and characterize the domain-wall profile and, in particular, specifies the thickness of the brane. We then determine the physical spectra for the NG field . From the bulk action (3.23), the linear order equation of motion follows as9

(3.29)

in the coordinate space. Here, the prime denotes the derivative with respect to and . By the Fourier transformation in the directions along the brane, we rewrite it as

(3.30)

whose linearly independent solutions are given by

(3.31)

and

(3.32)

We notice that only the constant mode, , has a finite value throughout the space, whereas the other modes diverge outside the brane. Since the field corresponds to the translational transformation parameter, the constant mode, , generates a shift of brane position without changing the brane profile and can be interpreted as the standard gapless NG mode propagating along the brane. It is also convenient to express the solutions in terms of the canonically normalized field :

(3.33)

and

(3.34)

This normalization provides how the energy of each mode distributes in the -direction. For example, it is clear that the energy of the gapless NG mode localizes on the brane. We also notice that the solutions in Eq. (3.34) have a finite energy density for as well as the first solution in Eq. (3.33). More concretely, the two modes in Eq. (3.34) behave like massive modes with the mass outside the brane ,

(3.35)

Also, gauge transformation parameters corresponding to the two modes diverge outside the brane , as is suggested by Eq. (3.32). We therefore interpret them as bulk propagations of the original scalar field , rather than standard NG modes. In Appendix B we show that the low-energy effective action after integrating out those gapped modes is nothing but the Nambu-Goto action.

To summarize, there exist two types of physical modes around the single scalar brane background: the standard massless NG mode localizing on the brane and the massive modes propagating in the bulk direction. In particular, only the standard localized NG mode is relevant in the low-energy scale and the standard coset construction takes into account this degrees of freedom only. Conversely, if is much smaller than a typical scale of excitation energy, the massive modes are not negligible and they should be included in the low-energy effective theory.

3.4 Inclusion of local Lorentz symmetry breaking

We then discuss the case when a nonzero spin field has a space-dependent condensation. To illustrate the degrees of freedom and residual symmetries in the unitary gauge, let us consider a (space-like) vector on the Minkowski space as a concrete example. Suppose that a vector field has a space-dependent condensation of the form

(3.36)

Here and in what follows, we use integers to denote the -directions of the local Lorentz index. Since has a Lorentz charge, the local Lorentz symmetry is broken as well as the -diffeomorphism invariance. Following the EFT recipe, we then introduce the vierbein to gauge the Lorentz symmetry. Schematically, we write the degrees of freedom and symmetries after introducing the vierbein as

(3.37)

In order to take the unitary gauge, it is convenient to note the decomposition

(3.38)

where specifies the direction of and corresponds to the NG field for the local Lorentz symmetry. On the other hand, specifies the amplitude of and corresponds to the NG field for the -diffeomorphism. Using the local Lorentz and diffeomorphism invariance, we can remove those NG fields to set and , at least as long as the fluctuations are small. In such a unitary gauge, the only dynamical degrees of freedom are the vierbein and the residual symmetries are the -dimensional local Lorentz and diffeomorphism invariance along the -directions. Schematically, we write this setup as

(3.39)

We then construct the effective action based on these degrees of freedom and residual symmetries. Schematically, let us decompose the effective action into the three types of contributions as

(3.40)

Here, is the effective action (3.16) and breaks the diffeomorphism invariance only:

(3.41)

On the other hand, and break the local Lorentz invariance and, in particular, represents terms that may exist even if the diffeomorphism is unbroken (the following ’s are constants when the diffeomorphism is unbroken). At the lowest order with respect to fluctuations and derivatives, they are given by10

(3.42)
(3.43)

where and are scalar functions depending on and the unit vector breaks the -diffeomorphism invariance explicitly. We next introduce the NG fields by the Stückelberg method and decouple the gauge degrees of freedom. As in Sec. 3.2, we first introduce the NG fields for the -diffeomorphism by the field-dependent gauge transformation (3.11). Similarly, we introduce NG fields ’s for the local Lorentz transformation in the - plane () as

(3.44)

where ’s are generators of . In particular, is transformed as

(3.45)

Since the full diffeomorphism and local Lorentz invariance can be restored by assigning nonlinear transformation rules on the NG fields and , we can set using the full gauge degrees of freedom. After these procedures, and take the form