DCPT-13/31 The Sakai-Sugimoto soliton

Dcpt-13/31 The Sakai-Sugimoto soliton

Abstract

The Sakai-Sugimoto model is the preeminent example of a string theory description of holographic QCD, in which baryons correspond to topological solitons in the bulk. Here we investigate the validity of various approximations of the Sakai-Sugimoto soliton that are used widely to study the properties of holographic baryons. These approximations include the flat space self-dual instanton, a linear expansion in terms of eigenfunctions in the holographic direction and an asymptotic power series at large radius. These different approaches have produced contradictory results in the literature regarding properties of the baryon, such as relations for the electromagnetic form factors. Here we determine the regions of validity of these various approximations and show how to relate different approximations in contiguous regions of applicability. This analysis clarifies the source of the contradictory results in the literature and resolves some outstanding issues, including the use of the flat space self-dual instanton, the detailed properties of the asymptotic soliton tail, and the role of the UV cutoff introduced in previous investigations. A consequence of our analysis is the discovery of a new large scale, that grows logarithmically with the ’t Hooft coupling, at which the soliton fields enter a nonlinear regime. Finally, we provide the first numerical computation of the Sakai-Sugimoto soliton and demonstrate that the numerical results support our analysis.

1 Introduction

There are still some unresolved puzzles regarding aspects of bulk solitons in holographic models of QCD. These include the validity of the flat space self-dual instanton used in the Sakai-Sugimoto model and the large distance behaviour of the electromagnetic form factors of the baryon. In this paper we shall address these issues and provide analytic resolutions that are confirmed by numerical investigations.

The cornerstone of all models of baryons in holographic QCD is that solitons in the bulk correspond to Skyrmions on the boundary. This correspondence was first observed by Atiyah and Manton [1] in four-dimensional Euclidean space, where the bulk soliton is the self-dual Yang-Mills instanton. The correspondence can be formulated as a flat space version of holography [2]. Holographic QCD differs from the Atiyah-Manton approach in that spacetime is curved with AdS-like behaviour and a five-dimensional Chern-Simon term is included that generates an abelian electric charge for the soliton. Here AdS-like means that the curvature is negative and there is a conformal boundary. The combination of the curvature of spacetime and the electromagnetic repulsion provides a stability that fixes the size of the soliton. These features are common to all models of holographic QCD, whether bottom-up or top-down.

Top-down approaches are derived from a string embedding and the Sakai-Sugimoto model [3, 4] is the prototypical example for top-down AdS/QCD models. In these models, the validity of the supergravity approximation requires working with a large number of colours and a large value of the ’t Hooft coupling . Although is just an overall multiplicative factor in the action, and thus irrelevant at the classical level, plays a vital role for the classical soliton, as it controls the ratio between the Yang-Mills and Chern-Simons terms. In particular, for large the size of the soliton becomes parametrically small with respect to the curvature scale. This suggests that most of the energy density of the soliton is concentrated in a small region of space, where the effect of the curvature has little influence on the fields of the soliton. This motivates the approach used in [5, 6], where the soliton is approximated by the flat space self-dual Yang-Mills instanton, with a size determined by minimization of the energy function on the instanton moduli space that results by restricting the full energy functional to the space of self-dual instanton fields. Note that this approximation is based on the assumption that the curvature and Chern-Simons term do not significantly alter the soliton fields, even though they are crucial in determining its size. We shall put this assumption to the test by numerically computing the Sakai-Sugimoto soliton and comparing it to the self-dual instanton. Furthermore, we show how to improve the self-dual instanton approximation via a simple generalization that maintains the symmetry of the instanton but introduces a more general profile function.

The soliton properties at large distance, and consequently the baryon electromagnetic form factors of the dual theory, have been calculated by expanding the self-dual instanton tail at the linear level and then extending this linear solution into the curved space at large distance from the core [7, 8, 9]. This approach relies on the fact that, for a small soliton, there is a region from the soliton core to the curvature scale in which the soliton is essentially in a linear regime and the curvature effects remain negligible. The result of this linear analysis is that the baryon density, and consequently all the electromagnetic form factors (including those of exited baryons obtained from a zero mode quantization) are exponentially suppressed at large distance. This is in contrast to the situation for other models, including the standard Skyrme model with massless pions, where the baryon density has an algebraic decay.

Bottom-up approaches are equally good toy-models for AdS/QCD, as long as they incorporate the features of confinement and chiral symmetry breaking. These models are relieved of the requirement of a string theory embedding, so there is a free choice of any AdS-like metric (provided it has a conformal boundary in the UV) and need not be small. The Pomarol-Wulzer model [10] is an example in this category, where the metric is a slice of AdS with a finite IR boundary at which left and right gauge fields are subject to matching conditions that mimic the salient features of the Sakai-Sugimoto model. Numerical computations of the Pomarol-Wulzer soliton have been performed at a value of the coupling that is of order one (where the soliton size is comparable to the curvature scale of the AdS slice), together with an asymptotic power series at large radius [11]. These results show that the baryon form factors have an algebraic decay, as in the Skyrme model, and not an exponential decay.

Cherman, Cohen and Nielsen [12] have described model independent relations for the baryon form factors at large distance. These relations are satisfied by the baryon form factors computed in the Skyrme model and the Pomarol-Wulzer model but not by those of the Sakai-Sugimoto model obtained from the linear analysis. The exponential decay of the soliton fields in the Sakai-Sugimoto model lies at the heart of this failure. Later, Cherman and Ishii [13] adapted the large radius expansion in [11] to the Sakai-Sugimoto model and found that the form factors have an algebraic decay and indeed satisfy the model independent relations, contradicting the earlier result of the linear analysis. However, their approach required the introduction of a UV cutoff and problems arise in attempting to remove this cutoff, so it is not clear which of the conflicting results is correct. Very recently, a preprint has appeared in which the large radius expansion has been applied to a general metric [14] and a conclusion drawn regarding the UV cutoff introduced into the Sakai-Sugimoto model. We shall comment on this conclusion in section 4.7, where we derive the correct procedure for removing the UV cutoff.

The contradictory conclusions described above raise a number of issues and questions concerning the use and validity of the various approximations and approaches. In fact, several candidates have been suggested for the source of the disagreement. One possibility is that the use of the flat space self-dual instanton in the Sakai-Sugimoto model is at the root of the problem. The validity of this approximation has never been tested, either numerically or analytically, and one may worry about a mechanism that allows the curvature and Chern-Simons term to stabilize the instanton size without altering the form of its fields. We shall test the use of the instanton approximation, firstly by introducing a generalization that allows some deformation of the instanton fields, and secondly via direct numerical computation of the Sakai-Sugimoto soliton. Our results strongly support the validity of the self-dual instanton approximation for large ’t Hooft coupling.

Another possibility is that either the linear expansion in [7] or the large radius expansion in [13] are not valid in the Sakai-Sugimoto model. In fact, we shall show that both approaches are valid but they are applicable in different regions of space. The contradictory results concerning the soliton tail, and consequently the baryon form factors, is a result of applying the linear expansion in an inappropriate region. The resolution of all the discrepancies in the literature resides in the existence of a new scale. This is a large scale that grows logarithmically with the ’t Hooft coupling and is therefore much larger than both the radius of curvature and the size of the small instanton. The linear expansion should be thought of as an expansion in , where the first term solves the linearised field equations. However, higher order terms are larger than the first order term both at the small instanton scale, which is of order and crucially at the new large scale of order . The fundamental property of the system is that there is a transition from a linear to a nonlinear regime at large distance. The existence of this new large scale explains the discrepancy over the form factor computations, which depend on the fall-off of the soliton tail. The vital observation is that the large and large radius limits do not commute. The crucial terms with algebraic decay are suppressed by additional powers of in comparison to the terms with exponential decay, so the algebraic decay is only evident at the large scale of order .

The outline of this paper is as follows. In section 2 we review the main aspects of the Sakai-Sugimoto model. In section 3 we discuss the flat space self-dual instanton approximation and our radial generalization. Section 4 is devoted to the calculation of the tail properties of the soliton, and in particular a determination of the regions of validity of alternative approximations. By comparing these different approximations we are able to relate them to each other and hence predict the emergence of the new large scale. A numerical computation of the Sakai-Sugimoto soliton is described in section 5, where the numerical results are shown to support our analytic findings. Finally, some concluding remarks are made in section 6.

2 The Sakai-Sugimoto model

Consider a five-dimensional spacetime with a warped metric of the form

 ds2=H(z)dxμdxμ+1H(z)dz2. (2.1)

Here with are the coordinates of four-dimensional Minkowski spacetime and is the spatial coordinate in the additional holographic direction. The signature is .

A class of spacetimes that are particularly relevant for holographic baryons corresponds to the choice

 H=(1+z2L2)p, (2.2)

where and are positive constants, with the former setting a curvature length scale. In this paper we focus on the Sakai-Sugimoto model [3, 4], which corresponds to the choice . For general the scalar curvature of the metric, after setting the length scale , is

 R=−4H−3/4(H3/4H′)′=−4p(2+(7p−2)z2)(1+z2)2−p. (2.3)

This formula shows that the value of is crucial in determining the qualitative features of the spacetime. For the curvature is finite as For the spacetime is asymptotically AdS with constant negative curvature . For the curvature is negative for all and for the theory has a conformal boundary. In the case of a conformal boundary it is often useful to introduce conformal coordinates

 ds2=H(z(u))(dxμdxμ+du2), (2.4)

where solves the equation For large the asymptotic behaviour is for some constants and Thus as , revealing the conformal boundary.

Given the above properties, we refer to the metric as AdS-like if since there is then a conformal boundary and the curvature is negative and finite. The Sakai-Sugimoto model is a generic example with Unless otherwise specified, from now on we will fix the values and though occasionally we will reintroduce these constants to indicate the more general dependence.

The Sakai-Sugimoto model is a gauge theory in the five-dimensional spacetime introduced above. Our index notation is that uppercase indices include the holographic direction whilst lowercase indices exclude this additional dimension. Furthermore, greek indices include the time coordinate whilst latin indices (excluding ) run over the spatial coordinates. Thus, for example,

 Γ,Δ,…=0,1,2,3,z,μ,ν,…=0,1,2,3,I,J,…=1,2,3,z,i,j,…=1,2,3. (2.5)

To fix conventions, the gauge potential is hermitian and under a gauge transformation, it transforms as The associated field strength is and the covariant derivative is The action is the sum of a Yang-Mills term and a Chern-Simons term

 S=−Ncλ216π3∫√−g 12tr(FΓΔFΓΔ)d4xdz+Nc24π2∫ ω5(A)d4xdz, (2.6)

where is the earlier warped metric with The factors and are respectively the number of colours and the ’t Hooft coupling of the dual theory. Note that the number of colours acts just as a multiplicative factor and therefore plays a trivial role in the classical physics in the bulk. In particular, by keeping fixed and taking the limit we can always make any quantum corrections negligible.

Decomposing the gauge potential into a sum of non-abelian and abelian components

 AΓ=AΓ+12ˆAΓ,FΓ=FΓ+12ˆFΓ, (2.7)

the Chern-Simons term, up to a total derivative, is

 Nc24π2∫(38ˆAΓtr(FΔΣFΞΥ)+116ˆAΓˆFΔΣˆFΞΥ)εΓΔΣΞΥd4xdz. (2.8)

The action, conveniently rescaled, becomes

 S = 216π3NcλS (2.9) = ∫{−14H1/2ˆFμνˆFμν−H3/22ˆFμzˆFμz−12H1/2tr(FμνFμν)−H3/2tr(FμzFμz)}d4xdz +1Λ∫(ˆAΓtr(FΔΣFΞΥ)+16ˆAΓˆFΔΣˆFΞΥ)εΓΔΣΞΥd4xdz,

where the indices are now raised using the flat 5-dimensional Minkowski metric tensor . For convenience, in the above we have introduced the rescaled ’t Hooft coupling

 Λ=8λ27π. (2.10)

As we are concerned with the static soliton solution of the theory, from now on we shall restrict to the case of time independent fields. The appropriate static ansatz is

 A0=0,AI=AI(xJ),ˆA0=ˆA0(xJ),ˆAI=0, (2.11)

so that the abelian potential generates an electric field . The action restricted to static fields is then

 S = ∫{12H1/2(∂iˆA0)2+H3/22(∂zˆA0)2−12H1/2tr(F2ij)−H3/2tr(F2iz)}d4xdz (2.12) +1Λ∫ˆA0 tr(FIJFKL)εIJKLd4xdz.

The static field equations that follow from the variation of this action are

 1H1/2DjFji+Dz(H3/2Fzi)=1ΛεiJKLFKL∂JˆA0 (2.13) H3/2DjFjz=1ΛεijkFjk∂iˆA0 (2.14) 1H1/2∂i∂iˆA0+∂z(H3/2∂zˆA0)=1Λtr(FIJFKL)εIJKL. (2.15)

Baryon number is identified with the instanton number of the soliton

 B=−132π2∫tr(FIJFKL)εIJKLd3xdz, (2.16)

and the Chern-Simons coupling implies that the instanton charge density sources the abelian electric field.

For later computational purposes, it will be convenient to rewrite the action by rearranging the terms as

 S = Missing or unrecognized delimiter for \bigg (2.17) +1Λ∫ˆA0 tr(FIJFKL)εIJKLd4xdz.

As we shall see, the static soliton solution of the field equations that follow from (2.12) is quite complicated. Even for the single static soliton, symmetry reduction can only reduce the field equations to coupled partial differential equations for five functions of two variables, which then need to be solved numerically. This approach will be described in detail in section 5, where we present the results of the first numerical computation of the Sakai-Sugimoto soliton.

The lack of an exact solution has motivated various approximate descriptions of the soliton, some of which we shall discuss later. First we consider a new approximation, in which the fields are assumed to have spherical symmetry. Because of the warp factor in the metric, such an assumption is clearly incompatible with the true solution of the field equations, so no exact solutions can be obtained in this way. However, we can certainly restrict the functional space to such a set of symmetric trial fields and determine the fields that are stationary points of the restricted action. The advantage of this approach is that it reduces the problem to a single ordinary differential equation, which is much easier to deal with than the full coupled partial differential equations. Furthermore, the radial approximation is a generalization of the self-dual flat space instanton approximation that has been used heavily in previous studies, so we are able to further investigate this approximation by examining how the radial approximation compares to the self-dual approximation in the large limit. The obvious disadvantage of the radial approximation is that it is unclear whether the approximate fields provide a reasonable description of the true solution. Fortunately, our later numerical solution will allows us to investigate this aspect too.

To specify the fields within the radial approximation we define the coordinates and by

 ρ=√x21+x22+x23+z2,z=ρcosθ. (3.1)

The radial approximation involves two real profile functions and and is given by

 ˆA0=a(ρ),AI=−σIJxJb(ρ), (3.2)

where is the anti-symmetric ’t Hooft tensor defined in terms of the Paul matrices by

 σij=εijkσk,σzi=σi. (3.3)

The non-abelian field has the same symmetry as the self-dual instanton, but has a more general radial profile function.

The instanton charge density is

 −132π2tr(FIJFKL)εIJKL = 3π2b(1−ρ2b)(2b+ρb′) (3.4) = 1π2ρ3(32(ρ2b)2−(ρ2b)3)′

yielding the instanton number

 B=c2(3−2c),wherec=limρ→∞(ρ2b). (3.5)

The requirement that therefore determines that giving the large behaviour

 b=1ρ2+O(1ρ4). (3.6)

In evaluating the action density of the radial field, the first term to consider is

 tr(F2IJ)=12(2b+ρb′)2+48b2(1−bρ2)2. (3.7)

The remaining term that is required is

 tr(F2ij)=8ρ(6ρb2+2b3ρ(ρ2+2z2)(bρ2−2)+b′(b′ρ+4b)(ρ2−z2)). (3.8)

Substituting these expressions into the action (2.17), writing and performing the angular integration over gives

 S2π2 = ∫{(P1−P2+P3)a′2−12P1((2b+ρb′)2+4b2(1−bρ2)2) (3.9) +8(6b2P2+2b3ρ2(P2+2P3)(bρ2−2)+b′(b′ρ+4b)ρ(P2−P3))}ρ3dρdt −16Λ∫a(3(ρ2b)2−2(ρ2b)3)′dρdt,

where the three functions are defined by the following angular integrals

 P1(ρ) = 1π∫π0H(ρcosθ)3/2sin2θdθ=12+18ρ2 P2(ρ) = 1π∫π0H(ρcosθ)2−1H(ρcosθ)1/2sin2θdθ=16ρ2−172ρ4+O(ρ6) P3(ρ) = 1π∫π0H(ρcosθ)2−1H(ρcosθ)1/2sin2θcos2θdθ=112ρ2−5576ρ4+O(ρ6). (3.10)

The field equation for , that follows from the variation of (3.9), may be integrated once to yield

 a′=−8ρb2(3−2ρ2b)Λ(P1−P2+P3), (3.11)

where the constant of integration has been set to zero in order to have a vanishing electric field at the origin . Integration by parts of the Chern-Simons term in (3.9), together with the solution (3.11), produces the following energy functional, that depends only on the profile function

 E2π2 = ∫{64ρ2b4(3−2ρ2b)2Λ2(P1−P2+P3)+12P1((2b+ρb′)2+4b2(1−bρ2)2) −8(6b2P2+2b3ρ2(P2+2P3)(bρ2−2)+b′(b′ρ+4b)ρ(P2−P3))}ρ3dρ.

Minimization of this energy gives a second order ordinary differential equation for that must be solved subject to the boundary conditions

 b′(0)=0andρ2b→1asρ→∞. (3.13)

Given this profile function, can be obtained by integrating (3.11). We shall present this numerical solution at the end of this section, but first we see how the flat space self-dual instanton approximation fits within this formalism.

In the case of large ’t Hooft coupling (which is required in top-down approaches) the Chern-Simons term is parametrically suppressed with respect to the Yang-Mills term. The role of the Chern-Simons coupling is to provide an electric contribution that stabilize the soliton against the shrinking induced by the spacetime curvature. Large should therefore correspond to a small soliton size, so that space is approximately flat in the soliton core. This motivates the use of the flat space self-dual instanton to approximate the soliton [5, 6].

To investigate the large limit it is useful to first introduce the rescaled coordinate The boundary condition as determines that the appropriate associated rescaling of the profile function is In terms of these variables the energy (3) can be written as where the first two terms are

 E0=12π2∫{((2˜b+˜ρ˜b′)2+4˜b2(1−˜b˜ρ2)2)}˜ρ3d˜ρ, (3.14)
 E1=π23∫{4˜b2(192˜b2(2˜ρ2˜b−3)2+˜ρ4˜b2−2˜ρ2˜b+6)+5˜ρ˜b′(˜ρ˜b′+4b)}˜ρ5d˜ρ. (3.15)

Restricting to the leading order term, the energy is minimized by the profile function of the flat space self-dual instanton

 ˜b=1˜ρ2+˜μ2, (3.16)

where is the rescaled arbitrary size of the instanton. The leading order term in the energy is and is independent of the size of the instanton.

The self-dual approximation involves restricting the profile function to the self-dual form (3.16) and using the next order term in the energy, as an energy function on the moduli space of instanton sizes. Explicitly, substituting (3.16) into (3.15) and performing the integration yields

 E1=2π2(23˜μ2+2565˜μ2), (3.17)

which is minimized when

 ˜μ=4(310)1/4. (3.18)

Returning to unscaled variables, with the size of the instanton, the self-dual approximation gives

 E=2π2(4+23μ2+2565Λ2μ2)+O(1Λ2), (3.19)

where

 μ=4√Λ(310)1/4. (3.20)

A similar scaling analysis of equation (3.11) shows that the leading order result for simply corresponds to replacing the term in (3.11) by its flat space limit After substituting the self-dual approximation and integrating, the result is

 a=8(ρ2+2μ2)Λ(ρ2+μ2)2. (3.21)

Note that is independent of within this self-dual approximation.

In summary, the first term in the energy (3.19) is independent of the instanton size and is simply the flat space self-dual Yang-Mills result of in our units. The second term is and also derives from the Yang-Mills functional but from the leading order correction to the the metric expansion around flat space. This gravitational contribution drives the instanton towards zero size. The third term is and is the first contribution from the electrostatic abelian field. This term resists the shrinking of the instanton size. These competing effects combine to produce the finite size (3.20), which is small for large with the energy dominated by the flat space self-dual contribution. The correction from the size stabilizing terms is subleading and is .

Returning to the radial approximation, the profile function that minimizes the energy (3) subject to the boundary conditions (3.13), was obtained using a shooting method with a fourth order Runge-Kutta algorithm to solve the second order ordinary differential equation obtained from the variation of the energy. The results are displayed in Figure 1 for two values of the coupling These plots illustrate the flow of the radial approximation to the self-dual approximation as For finite the main difference between the radial and self-dual approximations is that the self-dual approximation overestimates the value at the origin. As we shall see later, the full numerical solution confirms this overestimation, with the radial approximation being an improvement that reduces, but does not eliminate, this error.

The above rescaling to the self-dual instanton in the limit is a radial restriction of the following rescaling used in [5, 6]

 ˜xI=√ΛxI,˜t=t,˜AI=AI/√Λ,˜ˆA0=ˆA0. (3.22)

Defining then in the rescaled variables the action becomes

 S = ∫⎧⎨⎩−˜H3/22tr(˜F2IJ)−1−˜H22˜H1/2tr(˜F2ij) (3.23) +1Λ⎛⎝˜H3/22(˜∂I˜ˆA0)2+1−˜H22˜H1/2(˜∂i˜ˆA0)2⎞⎠⎫⎬⎭d4˜xd˜z +1Λ∫ ˜ˆA0 tr(˜FIJ˜FKL)εIJKLd4˜xd˜z.

Using the metric (2.2), with a general value of and expanding in gives

 S=∫{ −12tr(˜F2IJ) (3.24) +1Λ(−34p˜z2tr(˜F2IJ)+p˜z2tr(˜F2ij)+12(˜∂I˜ˆA0)2+˜ˆA0 tr(˜FIJ˜FKL)εIJKL) +O(1Λ2)}d4˜xd˜z,

which highlights the convenience of the rescaling (3.22). The leading order term is scale invariant and is simply the Yang-Mills action in flat space. The next term is of order and contains the size stabilizing contributions from both the abelian field and the curvature (due to the positive value of ). The action of the leading order term is minimized by the self-dual instanton and the term of order defines an action on the self-dual instanton moduli space that fixes the size of the instanton.

In summary, the way to extract the self-dual instanton limit is to convert to the rescaled coordinates (3.22) and then perform the limit

 limΛ→∞˜A(˜x)=˜Aself−dual(˜x), (3.25)

to converge to a self-dual instanton with a size in rescaled coordinates given by(3.18). For large but finite the small unscaled instanton size is

It is important to note that the self-dual limit has nothing to say about the asymptotic fields of the soliton at large distance. This is because the rescaling performed in (3.22) involves zooming in to a scale of order . To study the fields of the soliton at distances greater than requires alternative approaches that we describe in the next section.

4 The soliton tail

4.1 A linear expansion in flat space

In this subsection we consider a linear expansion that we shall see is valid in the region where we recall that we have set This region is far enough from the soliton core that a linear expansion is possible but is close enough to the origin that the curvature of the metric can be neglected by setting

To derive this expansion we still use as the small parameter of the expansion, but now we keep the length scale fixed rather than zooming in to the core. In this limit

 limΛ→∞ΛA(x)=Atail(x), (4.1)

where is a finite term that solves the linearised field equations. The task is to compute and to confirm its region of applicability.

We define the expansion

 AI=A(1)I+A(2)I+…,ˆA0=ˆA(1)0+ˆA(2)0+… (4.2)

in which

 A(n)I,ˆA(n)0∝1Λn. (4.3)

The limit (4.1) picks up only the first term in this expansion

 Atail(x)=ΛA(1)(x). (4.4)

As the space is now taken to be flat, the calculation in this subsection will involve expanding the self-dual instanton to provide the leading order contribution. This result will then be used in the next subsection to match to a linear analysis in curved space.

To perform the analysis it is convenient to write the self-dual instanton in the gauge in which it has the ’t Hooft form

 AI=12σIJ∂Jlog(1+μ2ρ2). (4.5)

Given that then the first term in the expansion is

 A(1)I=−σIJxJμ2ρ4=μ22σIJ∂J1ρ2∝1Λ, (4.6)

which satisfies the field equations ((2.13) and (2.14) with ) at the linear level since

 ∂IA(1)I=0and∂J∂JA(1)I=0. (4.7)

These equations are simply those of an abelian gauge potential: the first is the condition of Coulomb gauge and the second is the vanishing of the Laplacian.

The term gives the dominant contribution to the field strength

 F(1)IJ = ∂IA(1)J−∂JA(1)I=2μ2ρ4(σIJ+2ρ2(σJKxKxI−σIKxKxJ)). (4.8)

From (3.21) the abelian gauge potential at linear order is

 ˆA(1)0=8Λρ2, (4.9)

which satisfies the final field equation ((2.15) with ) at linear order.

Defining at second order the field equations are

 ∂IF(2)IJ+i[A(1)I,F(1)IJ]=0and∂I∂IˆA(2)0=0, (4.10)

with solution

 A(2)I=σIJxJμ4ρ6∝1Λ2,ˆA(2)0=0. (4.11)

The next non-zero term in is at third order, where the field equation gives

 ∂I∂IˆA(3)0=1Λtr(F(1)IJF(1)KL)εIJKL, (4.12)

and is solved by

 ˆA(3)0=−8μ4Λρ6∝1Λ3. (4.13)

For this expansion to be reliable requires and These conditions correspond to the requirement that which means far from the soliton core. The use of the flat space metric approximation, required , so combining these constraints results in the region of validity as claimed at the start of this subsection.

4.2 A linear expansion in curved space

We now extend the linear expansion of the previous subsection to distances beyond the restriction This requires that the curvature of the metric is now taken into account and the approximation can no longer be used. The linear analysis in this subsection is equivalent to that in [7] and produces the same result. However, the derivation is a little different as we wish to elucidate the aspects that will play a role in our additional analysis later in the paper.

For the purposes of this subsection it will be sufficient to consider only the first order terms and . As these terms satisfy the linearised field equations we can perform a separation of variables in and , expand in eigenfunctions of the linear operator in flat space, and then extend each eigenfunction separately into the curved region beyond . The existence of an overlap region in which the linear flat space approximation and the linear curved space approximation are both valid, allows the computation of the coefficients of the eigenfunction expansion in curved space.

The easiest case is that of the abelian potential which satisfies the linearized field equation (2.15) given by

 ∂i∂iˆA(1)0+H1/2∂z(H3/2∂zˆA(1)0)=0. (4.14)

We can therefore extend (4.9) to the curved regime by writing

 ˆA(1)0 = 8Λξ(xI) (4.15)

where is a harmonic function in the four-dimensional curved space, which in the flat regime is

 ξ(xi,z)≃1ρ2forρ≲1. (4.16)

We now separate variables and write for the three-dimensional radius. The harmonic function can be expanded in a Laplace-Fourier expansion (Laplace expansion in , Fourier expansion in ). In flat space there is the exact identity

 1ρ2=1r2+z2=∫∞0e−krrcos(kz)dk. (4.17)

Note that all the momentum modes must appear in this expansion in order to reconstruct the function exactly. Now we extend this expansion into the curved region by replacing it with

 ξ(xi,z)=∫∞0e−krrψ+(k)(z)dk, (4.18)

where are defined as the eigenfunctions satisfying the linear equation

 H1/2∂z(H3/2∂zψ±(k))+k2ψ±(k)=0, (4.19)

with the superscript referring to even and odd parity with respect to . The boundary conditions for are

 ψ+(k)(0)=1,∂zψ+(k)(0)=0. (4.20)

Only the even eigenfunctions appear in the expansion for , but later we shall need the odd eigenfunctions which satisfy the boundary conditions

 ψ−(k)(0)=0,∂zψ−(k)(0)=1. (4.21)

The expression (4.15) with defined in (4.18) gives the exact extension of in the curved region and reduces to (4.16) in the almost flat region since, for every value of ,

 ψ+(k)(z)≃cos(kz)forz≪1, (4.22)

as in this region.

Next we consider the non-abelian field given by (4.6) in the flat regime. First we decompose into parity components

 A(1)i=A(1+)i+A(1−)i,A(1)z=A(1+)z (4.23)

where the superscript again stands for the parity with respect to . The odd component vanishes in the chosen gauge where

In the flat regime (4.6) gives the parity components

 A(1+)i=μ22εijkσk∂j1ρ2,A(1−)i=−μ22σi∂z1ρ2,A(1+)z=μ22σi∂i1ρ2. (4.24)

Applying the parity decomposition to the linearized field equations (2.13) and (2.14) yields

 ∂j∂jA(1+)i+H1/2∂z(H3/2∂zA(1+)i)=0, (4.25) ∂i(∂iA(1+)z−∂zA(1−)i)=0, (4.26) ∂j(∂jA(1−)i−∂iA(1−)j)+H1/2∂z(H3/2∂zA(1−)i)−∂i(H1/2∂z(H3/2A(1+)z))=0. (4.27)

The easiest component to deal with is as this decouples from the other components and satisfies the same equation as the abelian potential The first component in (4.24) is therefore extended to curved space as

 A(1+)i = μ22εijkσk∂jξ(xI), (4.28)

where is the same harmonic function defined in (4.18). Note that here, as for , only the even eigenfunctions appear in the expansion.

The remaining two components and are slightly more complicate as their equations (4.26) and (4.27) are coupled together. We see from (4.26) that if is expanded using the eigenfunctions then must be expanded in terms of their derivatives which then gives a consistent expansion for (4.27). We therefore define the functions

 ϕ±(k)(z)=∂zψ∓(k)(z). (4.29)

In the almost flat region so we have that

 ψ−(k)(z)≃sin(kz)kandϕ+(k)(z)≃cos(kz)forz≪1. (4.30)

The flat space results (4.24) may be rewritten as

 A(1−)i=μ22σi∫∞0e−k