Soft Wall holographic model for the minimal Composite Higgs

We will consider a theory where in addition to the SM $\mathcal{L}_{SM}$ there is a new strongly interacting sector $\mathcal{L}_{str.int.}$, presumed to be conformal in the UV. A global symmetry of this sector is spontaneously broken following the pattern $\mathcal{G}\rightarrow\mathcal{H}$. There are Goldstone bosons in the coset space $\mathcal{G}/\mathcal{H}$, and some of them have the quantum numbers of the Higgs doublet. As the $SU(2)_{L}\times U(1)$ global group is necessarily included in $\mathcal{H}$ we can couple the EW sector of the SM to the composite sector

$$\mathcal{L}=\mathcal{\widetilde{L}}_{str.int.}+\mathcal{L}_{SM}+\widetilde{J}_{L}^{\alpha\ \mu}W_{\mu}^{\alpha}+\widetilde{J}^{Y\ \mu}B_{\mu}.$$

(1)

There only appear the conserved currents of the strongly interacting sector $J_{L}^{\alpha\ \mu}$ and $J^{Y\ \mu}$ that contain the generators of the EW group. Moreover, we have to denote the misalignment between the $\mathcal{H}$ subgroup of the new sector and the actual $\mathcal{H}^{\prime}$ containing the $W_{\mu}^{\alpha}$ and $B_{\mu}$ EW gauge bosons. In Eqn. (1) everything related to the new composite sector is marked with tildes.

Let us specify to the case of MCHM, where the global symmetry breaking pattern is $SO(5)\rightarrow SO(4)$ and there are exactly four Goldstones. We denote by $T^{A},\ A=1,...,10$ the generators of $SO(5)$, represented by $5\times 5$ matrices, which are traceless $\operatorname{Tr}T^{A}=0$ and normalized as $\operatorname{Tr}(T^{A}T^{B})=\delta^{AB}$. They separate naturally into two groups:

• •

  The unbroken generators, in the case of MCHM those of $SO(4)\backsimeq SU(2)_{L}\times SU(2)_{R}$, we will cal $T^{a},\ a=1,...,6$. They are specified as

  $$T^{\alpha}_{L}=\begin{pmatrix}t^{\alpha}_{L}&0\\ 0&0\\ \end{pmatrix},\ T^{\alpha}_{R}=\begin{pmatrix}t^{\alpha}_{R}&0\\ 0&0\\ \end{pmatrix},\ \alpha=1,2,3,$$

  (2)

  where $t^{\alpha}_{L}$, $t^{\alpha}_{R}$ are $4\times 4$ matrices given by $(t^{\alpha}_{L/R})_{jk}=-\frac{i}{2}(\varepsilon_{\alpha\beta\gamma}\delta^{\beta}_{j}\delta^{\gamma}_{k}\pm(\delta_{j}^{\alpha}\delta_{k}^{4}-\delta_{k}^{\alpha}\delta_{j}^{4}))$, $j,k=1,...,4$.

• •

  The broken generators, corresponding to the coset $SO(5)/SO(4)$, are labeled as $\widehat{T}^{i},\ i=1,2,3,4$ and defined as follows

  $$\widehat{T}^{i}_{IJ}=-\frac{i}{\sqrt{2}}(\delta^{i}_{I}\delta^{5}_{J}-\delta^{i}_{J}\delta^{5}_{I}),\ \ I,J=1,...,5.$$

  (3)

A quantity parametrizing the vacuum misalignment and responsible for the EW symmetry breaking is the rotation angle $\theta$ that relates the linearly-realized global group $\mathcal{H}=SO(4)$ and the gauged group $\mathcal{H}^{\prime}=SO(4)^{\prime}$. It is natural to assign the value $\theta=0$ to the SM, hence we denote the generators of $SO(5)\rightarrow SO(4)^{\prime}$ as $\{T^{a}(0),\ \widehat{T}^{i}(0)\}$ and those of $SO(5)\rightarrow SO(4)$ as $\{T^{a}(\theta),\ \widehat{T}^{i}(\theta)\}$. We choose a preferred direction for the misalignment and the following connection between the generators holds

$$T^{\alpha}(\theta)=r(\theta)T^{\alpha}(0)r^{-1}(\theta),\ \text{with}\ r(\theta)=\begin{pmatrix}1_{3\times 3}&0&0\\ 0&\cos\theta&\sin\theta\\ 0&-\sin\theta&\cos\theta\\ \end{pmatrix}.$$

(4)

Compositness implies that some fundamental degrees of freedom are bound together by the new “color” force (hyper-color is usually used in the CH framework). MCHM does not admit complex Dirac fermions as fundamental fields at the microscopic level due to the nature of the global “flavor” symmetry group. The anomaly-free UV complete fundamental fermion theory should have $\mathcal{G}$ equivalent to $SU(n_{1})\times\ldots\times SU(n_{p})\times U(1)^{p-1}$, where $n_{i}$ is the number of fermions in a given irreducible representations and $p$ counts the number of different irreps Ferretti and Karateev (2014). The simplest UV-completable theory will be the next-to-minimal CH with $SO(6)\rightarrow SO(5)$, featuring five Goldstone bosons (other next-to-minimal patterns are mentioned, for instance, in Ref. Cacciapaglia and Sannino (2014)). Nevertheless, we choose to work with MCHM because of its simplicity that serves to illustrate the general procedure.

If one chooses to avoid the particularities of the microscopic structure of the new composite states (that seems advisable on the grounds of being as general as possible), it is impossible to treat the holographic MCHM completely in the AdS/QCD fashion of Ref. Erlich et al. (2005); Karch et al. (2006). To some extent, due to affecting directly the operator scaling dimension $\Delta$, the microscopic substructure sets the prescriptions for the bulk masses and UV boundary conditions, which in their turn influence all other holographic derivations. In our holographic model describing the minimal CH we only use a single entry from the list of field-operator correspondences Aharony et al. (2000)

$$A_{\mu}^{A}(x,z=\varepsilon)=1\cdot\phi_{\mu}^{A}(x)\ \leftrightarrow\ \mathcal{O}_{\mu}^{A}(x)\ \text{with}\ \Delta=3,$$

(5)

where $\mathcal{O}_{\mu}^{A}(x)$ are the unspecified conserved currents of the fundamental theory containing $SO(5)$ generators $T^{A}$, and $A_{\mu}^{A}(x,z)$ are dual $5D$ fields restricted to provide the sources $\phi_{\mu}^{A}(x)$ for the corresponding operators on the UV brane ($\varepsilon$ is an UV regulator). We take $\Delta=3$ (and zero bulk mass of the vector fields) as a universal feature for the conserved vector currents, because it should be so both in the case of fermionic ($\overline{\Psi}\gamma_{\mu}T^{A}\Psi$) and bosonic ($\partial_{\mu}s^{\top}T^{A}s$) fundamental degrees of freedom. The introduction of the scalar operator is indispensable in order to generate the breaking towards $SO(4)$. However, following a line similar to the vector case would mean inferring too much on the nature of the fundamental theory. Hence, we intend to construct the model so that this part of duality is realized in an alternative way.

The operators $\mathcal{O}_{\mu}^{A}(x)$ define the currents of Eqn. (1):

• •

  for $A=\alpha$ (left): $\frac{g}{\sqrt{2}}\mathcal{O}^{\alpha}_{L\mu}(x)=g_{V}J^{\alpha}_{L\mu}$;

• •

  for hypercharge realized as $Y=T^{3}_{R}$: $\frac{g^{\prime}}{\sqrt{2}}\mathcal{O}^{3}_{R\mu}(x)=g_{V}J^{Y}_{\mu}$.

The coupling coefficients are not fully established because the operators are taken with an abstract normalization $g_{V}$ that will be determined to provide agreement with the common MCHM notations. Introduction of $g_{V}$ is also substantiated by the discussion of Ref. Espriu and Katanaeva (2020), where it is argued that a degree of arbitrariness in the field-operator holographic correspondence is a necessary piece of AdS/QCD constructions.

In this subsection we put forward the details of the holographic 5D model realizing the 4D MCHM concept. We settle upon the idea that there are two composite operators, a vector and a scalar one, that define the theory, and hence we have spin one and spin zero fields on the 5D side. These fields live in the 5D AdS bulk with a metric given by

$$g_{MN}dx^{M}dx^{N}=\frac{R^{2}}{z^{2}}(\eta_{\mu\nu}dx^{\mu}dx^{\nu}-d^{2}z),\quad\eta_{\mu\nu}=\text{diag}(1,-1,-1,-1).$$

(6)

The dynamics is governed by the following $SO(5)$ gauge invariant action

	$\displaystyle S_{5D}=$	$\displaystyle\frac{1}{4g_{5}^{2}}\int d^{5}x\sqrt{-g}e^{-\Phi(z)}\operatorname{Tr}F_{MN}F_{KL}g^{MK}g^{LN}$		(7)
		$\displaystyle+\frac{1}{k_{s}}\int d^{5}x\sqrt{-g}e^{-\Phi(z)}\bigg{[}\operatorname{Tr}g^{MN}(D_{M}H)^{\top}(D_{N}H)-M^{2}_{H}\operatorname{Tr}HH^{\top}\bigg{]}.$

This 5D effective action includes matrix-valued scalar and vector fields and, as mentioned, is inspired by generalized sigma models used in the context of strong interactions. A similar starting action was used in the AdS/QCD study of Ref. Espriu and Katanaeva (2020). The dimensionality of the normalization constants $g_{5}^{2}$ and $k_{s}$ is set to compensate that of the additional dimension: $[g_{5}^{2}]=[k_{s}]=E^{-1}$. To have the gravitational background of a smoothly capped off AdS spacetime we introduce a SW dilaton function $\Phi(z)=\kappa^{2}z^{2}$ in the common inverse exponent factor.

The scalar degrees of freedom are collected in the matrix-valued field $H$. Let us denote the group transformations $g\in SO(5)$ and $h\in SO(4)$. The matrix of the Goldstone fields $\xi$ transforms under $SO(5)$ as: $\xi\rightarrow\xi^{\prime}=g\xi h^{\top}$. The other scalar degrees of freedom with the quantum numbers of SO(4) are collected in the matrix $\Sigma$ transforming as $\Sigma\rightarrow\Sigma^{\prime}=h\Sigma h^{\top}$. The breaking from $SO(5)$ to $SO(4)$ also appears there and is parametrized by a function $f(z)$. From these components we can construct a proper combination leading to $H\rightarrow H^{\prime}=gHg^{\top}$

$$H=\xi\Sigma\xi^{\top},\ \Sigma=\begin{pmatrix}0_{4\times 4}&0\\ 0&f(z)\\ \end{pmatrix}+\sigma^{a}(x,z)T^{a},\ \xi=\exp\left(\frac{i\pi^{i}(x,z)\widehat{T}^{i}}{\chi_{\pi}}\right),$$

(8)

where $[\chi_{\pi}]=[f(z)]=E^{1}$. The minutiae of the scalar fields, introduced in $\Sigma$ as $\sigma^{a}$, will be further omitted in this study. It follows then that in this representation: $H=H^{\top}$, the $\operatorname{Tr}HH^{\top}$ quadratic piece of Eqn. (7) brings no field interactions and the value of $M^{2}_{H}$ is of no consequence.

Holography prescribes that every global symmetry of the 4D model comes as a gauge symmetry of its 5D dual. Thus, to make the Lagrangian invariant under the gauge transformation $A_{M}\rightarrow A^{\prime}_{M}=gA_{M}g^{-1}+ig\partial_{M}g^{-1}$ the covariant derivative is introduced in the 5D action (7), defined as

$$D_{M}H=\partial_{M}H+[A_{M},H],\quad D_{M}H\rightarrow gD_{M}Hg^{-1}.$$

(9)

The field strength tensor that produces the vector field kinetic term in Eqn. (7) is

$$F_{MN}=\partial_{M}A_{N}-\partial_{N}A_{M}+[A_{M},A_{N}].$$

(10)

Generally, we take $A_{M}=-iA_{M}^{A}T^{A}$, where the upper index runs through both broken and unbroken indices $A_{M}^{a}T^{a}+A_{M}^{i}\widehat{T}^{i}$. These 5D vector fields are unrelated to the $W_{\mu}^{\alpha}$ or $B_{\mu}$ gauge bosons of the EW interactions, but for their eventual mixing.

The $A_{\mu}^{A}$ fields are connected by duality to the $\mathcal{O}_{\mu}^{A}$ vector composite operators with the same generators and have the boundary condition (5). For the fifth component of the vector field we assume that

$$A_{z}^{A}(x,\varepsilon)=0,$$

(11)

because there is no $4D$ source for it to couple to. The common holographic gauge $A_{z}^{A}\equiv 0$ fulfills this condition trivially, but this is not the only possibility. On the other hand, the dual counterpart of $H$ and the value of $M^{2}_{H}$ remain unspecified. The near-boundary behavior of the Goldstone fields $\pi^{i}(x,z)$ will be eventually determined in Section III.2 from considerations of another type. The treatment of the Goldstone is an essential aspect of the model because they correspond to the four components of the Higgs doublet.

The basic principle of AdS/CFT correspondence states that the partition function of the 4D theory and the on-shell action of its 5D holographic dual coincide in the following sense Witten (1998); Gubser et al. (1998):

$$Z_{4D}[\phi]=\operatorname{Exp}iS_{5D}^{on-shell}|_{\phi(x,z)\rightarrow\phi(x,z=\varepsilon)}.$$

(12)

Essentially, all bulk fields $\phi(x,z)$ are set to their boundary values $\phi(x,z=\varepsilon)$, which could be identified with the sources $\phi(x)$ as in the case of Eqn. (5).

The dynamics of holographic fields is governed by a set of second order equations of motion (EOMs). Thus, a 5D field can be attributed with two solutions. According to the usual AdS/CFT dogma, the leading mode at small $z$ corresponds to the bulk-to-boundary propagator. It connects a source at the boundary and a value of a field in the bulk and should exhibit enough decreasing behavior in the IR region to render the right-hand side of Eqn. (12) finite. The subleading mode represents an infinite series of normalizable solutions, known as the Kaluza-Klein (KK) decomposition. There, the 4D and $z$ dependencies are separated: the $z$-independent functions are identified with a tower of physical states at the 4D boundary that are further promoted into the bulk with the $z$-dependent profiles.

From consideration of the KK solutions one gets knowledge about the spectra of the composite 4D resonances. While from Eqn. (7), evaluated on the bulk-to-boundary solutions, one can extract the $n$-point correlation functions of the composite operators Witten (1998); Gubser et al. (1998); Freedman et al. (1999). The 4D partition function is given by the functional integral over the fundamental fields $\varphi$ contained in the selected operators (e.g. $\mathcal{O}^{A}_{\mu}$) and in the fundamental Lagrangian $\mathcal{L}_{str.int.}$

	$\displaystyle Z_{4D}[\phi]$	$\displaystyle=$	$\displaystyle\int[\mathcal{D}\varphi]\operatorname{Exp}i\int d^{4}x[\mathcal{L}_{str.int.}(x)+\phi_{\mu}^{A}(x)\mathcal{O}^{A\mu}(x)+\ldots]$		(13)
		$\displaystyle=$	$\displaystyle\operatorname{Exp}\sum\limits_{q}\frac{1}{q!}\int\prod\limits_{k=1}^{q}d^{4}x_{k}\langle\mathcal{O}_{1}(x_{1})...\mathcal{O}_{q}(x_{q})\rangle i\phi^{1}(x_{1})...i\phi^{q}(x_{q}).$

From the schematic definition in Eqn. (13) and the correspondence postulate (12), it is clear that the Green functions can be obtained by the variation of the 5D effective action with respect to the sources. Diagrammatically we can represent the correlation functions by the left panel of Fig. 1, where in general the number of legs could be equal to the number $n$ of operators in the correlator. At the same time, couplings involving the composite resonances can be estimated taking the proper term in the 5D Lagrangian, inserting the KK modes for the interacting 5D fields and integrating over the $z$-dimension. Due to $\ln Z_{4D}=iS^{eff}_{4D}$ a calculation of this kind brings an effective vertex.

Interaction of a given composite state with the SM gauge bosons happens through the mixing of the latter with other composite particles. Due to the misalignment the EW bosons couple to a variety of resonances, because the rotated currents $\widetilde{J}^{\alpha}_{\mu}$ overlap with different types of vectorial currents that are holographically connected to vector composite fields. Besides, all radial excitations in a KK tower should generally be included in the internal propagation. The procedure in this case is the following: calculate the $n$-point correlation function, build the effective 4D Lagrangian via attaching $W_{\mu}^{\alpha}$ or $B_{\mu}$ fields as physical external sources, and reduce the legs where the composite resonances become physical and put on-shell (substituted with their KK modes). This is shown in the right panel of Fig. 1.

In the unbroken sector with ${a}=1,..,6$

	$\displaystyle\partial_{z}\frac{e^{-\Phi(z)}}{z}\partial_{z}A_{\mu}^{a}-\frac{e^{-\Phi(z)}}{z}\Box A_{\mu}^{a}-\partial_{z}\frac{e^{-\Phi(z)}}{z}\partial_{\mu}A_{z}^{a}=0,$		(15)
	$\displaystyle\Box A_{z}^{a}=\partial^{\mu}\partial_{z}A_{\mu}^{a}.$		(16)

If we act with $\partial^{\mu}$ on the first equation and substitute $\Box A_{z}^{a}$ from the second one, we would get the third term equal to the first one. Then, the result is

$$\Box\partial^{\mu}A_{\mu}^{a}=0,$$

(17)

that implies either $\partial^{\mu}A_{\mu}^{a}=0$ (transversality) or $q^{2}_{A^{\parallel}}=0$ (longitudinal mode), where

$$A_{\mu}^{a}=A_{\mu}^{a\bot}+A_{\mu}^{a\parallel},$$

(18)

with $A_{\mu}^{a\bot}=\mathcal{P}_{\mu\nu}A^{a\nu}$, $\mathcal{P}_{\mu\nu}=\left(\eta_{\mu\nu}-\frac{q_{\mu}q_{\nu}}{q^{2}}\right)$, and $A_{\mu}^{a\parallel}=\frac{q_{\mu}q_{\nu}}{q^{2}}A^{a\nu}$.

The condition (17) modifies the second equation in the system into

$$\Box^{2}A_{z}^{a}=0.$$

(19)

While acting with $\Box^{2}$ on Eqn. (15) and taking into account $q^{2}_{A^{\bot}}\neq 0$ we get the following equation for the transversal mode

$$\partial_{z}\frac{e^{-\Phi(z)}}{z}\partial_{z}A_{\mu}^{a\bot}-\frac{e^{-\Phi(z)}}{z}\Box A_{\mu}^{a\bot}=0.$$

(20)

However, the result for the longitudinal mode with $q^{2}_{A^{\parallel}}=0$ turns out trivial, meaning that the remaining system for $A_{\mu}^{a\parallel}$ and $A_{z}^{a}$ is underdefined. We choose to work in a class of solutions where Eqn. (19) is fulfilled with the gauge

$$A_{z}^{a}(x,z)\equiv 0.$$

(21)

As a result the EOM for the longitudinal mode simplifies to

$$\partial_{z}A_{\mu}^{a\parallel}=0.$$

(22)

The following boundary terms are left in the on-shell action (14)

$$\frac{1}{2g_{5}^{2}}\int d^{4}x\left.e^{-\Phi(z)}\frac{R}{z}A^{a\mu}(\partial_{z}A^{a}_{\mu}-\partial_{\mu}A^{a}_{z})\right|_{\varepsilon}^{\infty}=-\frac{1}{2g_{5}^{2}}\int d^{4}x\left.\frac{R}{z}A^{a\bot\mu}\partial_{z}A^{a\bot}_{\mu}\right|_{z=\varepsilon}.$$

(23)

Only the transversal term remains, giving rise to the two-point function studied in Section IV.1.

Let us perform a 4D Fourier transform $A_{\mu}^{a}(x,z)=\int d^{4}qe^{iqx}A_{\mu}^{a}(q,z)$ and let us focus on finding solutions of the EOMs. First, the transverse bulk-to-boundary propagator, which we denote $V(q,z)$, is defined by

$$A_{\mu}^{a\bot}(q,z)=\phi_{\mu}^{a\bot}(q)\cdot V(q,z),\quad\ V(q,\varepsilon)=1,$$

(24)

where $\phi_{\mu}^{a\bot}$ should be understood as a projection of the original source $\phi_{\mu}^{A\bot}=\mathcal{P}_{\mu\nu}\phi^{A\nu}$. The analogous longitudinal projection will be denoted by $\phi_{\mu}^{A\parallel}$.

From Eqn. (20), changing to the variable $y=\kappa^{2}z^{2}$, we arrive to the following EOM

$$yV^{\prime\prime}(q,y)-yV^{\prime}(q,y)+\frac{q^{2}}{4\kappa^{2}}V(q,y)=0$$

(25)

It is a particular case of the confluent hypergeometric equation (see Appendix A for a review of the properties and solutions of this equation), and the dominant mode at small $z$ is

$$V(q,z)=\Gamma\left(-\frac{q^{2}}{4\kappa^{2}}+1\right)\Psi\left(-\frac{q^{2}}{4\kappa^{2}},0;\kappa^{2}z^{2}\right).$$

(26)

The subdominant solution (see Eqn. (120)) gives us the tower of massive states, identified with vector composite resonances at the boundary. Normalizable solutions can only be found for discrete values of the 4D momentum $q^{2}=M_{V}^{2}(n)$ and we may identify $\left.V(q,z)\right|_{q^{2}=M_{V}^{2}(n)}=V_{n}(z)$. The KK decomposition is set as follows

$$A^{a\bot}_{\mu}(q,z)=\sum\limits_{n=0}^{\infty}V_{n}(z)A_{\mu(n)}^{a\bot}(q).$$

(27)

The $z$ profile and the spectrum can be expressed using the discrete parameter $n=0,1,2,...$

$$V_{n}(z)=\kappa^{2}z^{2}\sqrt{\frac{g_{5}^{2}}{R}}\sqrt{\frac{2}{n+1}}L_{n}^{1}(\kappa^{2}z^{2}),\quad\ M_{V}^{2}(n)=4\kappa^{2}(n+1),$$

(28)

where $L_{n}^{m}(x)$ are the generalized Laguerre polynomials. The profiles $V_{n}(z)$ are subject to the Dirichlet boundary condition and are normalized to fulfill the orthogonality relation

$$\frac{R}{g_{5}^{2}}\int\limits_{0}^{\infty}dze^{-\kappa^{2}z^{2}}z^{-1}V_{n}(z)V_{k}(z)=\delta_{nk}.$$

(29)

For the longitudinal mode, $A_{\mu}^{a\parallel}(q,z)$, the bulk-to-boundary solution is similarly defined. Its EOM (22), however, admits only trivial continuation into the bulk

$$A_{\mu}^{a\parallel}(q,z)=\phi_{\mu}^{a\parallel}(q)\cdot V^{\parallel}(q,z),\quad V^{\parallel}(q,z)=1.$$

(30)

The previous results are well known. Let us now see the equivalent derivation in the broken sector.

The EOMs for the broken sector with ${i}=1,..,4$ are more complicated due to the appearance of mixing with $\pi^{i}$

	$\displaystyle\partial_{z}\frac{e^{-\Phi(z)}}{z}\left(\partial_{z}A^{i}_{\mu}-\partial_{\mu}A^{i}_{z}\right)-\frac{e^{-\Phi(z)}}{z}\Box A^{i}_{\mu}-\frac{2g_{5}^{2}f^{2}(z)R^{2}}{k_{s}}\frac{e^{-\Phi(z)}}{z^{3}}\left(A^{i}_{\mu}-\frac{\partial_{\mu}\pi^{i}}{\chi_{\pi}}\right)=0$		(31)
	$\displaystyle\frac{e^{-\Phi(z)}}{z}\left(\partial^{\mu}\partial_{z}A^{i}_{\mu}-\Box A^{i}_{z}\right)-\frac{2g_{5}^{2}f^{2}(z)R^{2}}{k_{s}}\frac{e^{-\Phi(z)}}{z^{3}}\left(A^{i}_{z}-\partial_{z}\frac{\pi^{i}}{\chi_{\pi}}\right)=0$		(32)
	$\displaystyle\partial_{z}\frac{f^{2}(z)R^{2}e^{-\Phi(z)}}{z^{3}}\left(A^{i}_{z}-\partial_{z}\frac{\pi^{i}}{\chi_{\pi}}\right)-\frac{f^{2}(z)R^{2}e^{-\Phi(z)}}{z^{3}}\left(\partial^{\mu}A^{i}_{\mu}-\Box\frac{\pi^{i}}{\chi_{\pi}}\right)=0$		(33)

Combining $\partial^{\mu}\times$ (31) with other two equations we arrive again at the condition

$$\Box\partial^{\mu}A^{i}_{\mu}=0,$$

(34)

with the same options $\partial^{\mu}A_{\mu}^{i}=0$ and $q^{2}_{A^{\parallel}}=0$ as in the unbroken case. The condition on $A^{i}_{z}$ is different though

$$\partial_{z}\frac{e^{-\Phi(z)}}{z}\Box^{2}A^{i}_{z}-\frac{2g_{5}^{2}f^{2}(z)R^{2}e^{-\Phi(z)}}{k_{s}z^{3}}\Box^{2}\frac{\pi^{i}}{\chi_{\pi}}=0.$$

(35)

The system of equations obeyed by $A_{\mu}^{i\parallel}$, $A_{z}^{i}$ and $\pi^{i}$ is insufficient to determine them and we can only solve the problem with the help of an appropriate gauge condition. There are various possibilities, but we find the option explained below most useful for the physics we aspire to describe. We impose

$$A^{i}_{z}(x,z)=\xi\partial_{z}\frac{\pi^{i}(x,z)}{\chi},$$

(36)

where the parameter $\xi$ is arbitrary.

The fact that $\pi^{i}(x,z)$ appears both in the scalar part of the model Lagrangian and in this gauge condition makes it distinct from other $5D$ fields in the model. To analyze the Goldstone solution we assume that the corresponding EOM defines the $z$-profile $\pi(x,z)$ that couples to the physical mode $\pi^{i}(x)$ on the boundary. The Neumann boundary condition, $\partial_{z}\pi(x,z)|_{z=\varepsilon}=0$, is imposed due to Eqn. (11).

Now both parts of Eqn. (35) have the same $x$-dependence, and $\Box^{2}$ can be taken out of the bracket. It results in the following equation on $\pi(x,z)$

$$\partial_{z}\frac{e^{-\Phi(z)}}{z}\partial_{z}\pi(x,z)-\frac{2g_{5}^{2}f^{2}(z)R^{2}}{\xi k_{s}}\frac{e^{-\Phi(z)}}{z^{3}}\pi(x,z)=0.$$

(37)

At the same time it allows to get rid of $A^{i}_{z}$ and $\partial_{\mu}\frac{\pi^{i}}{\chi_{\pi}}$ in Eqn. (31). Then,

	$\displaystyle\partial_{z}\frac{e^{-\Phi(z)}}{z}\partial_{z}A_{\mu}^{i\bot}-\frac{e^{-\Phi(z)}}{z}\Box A_{\mu}^{i\bot}-\frac{2g_{5}^{2}f^{2}(z)R^{2}}{k_{s}}\frac{e^{-\Phi(z)}}{z^{3}}A_{\mu}^{i\bot}=0,$		(38)
	$\displaystyle\partial_{z}\frac{e^{-\Phi(z)}}{z}\partial_{z}A_{\mu}^{i\parallel}-\frac{2g_{5}^{2}f^{2}(z)R^{2}}{k_{s}}\frac{e^{-\Phi(z)}}{z^{3}}A_{\mu}^{i\parallel}=0.$		(39)

At the boundary we have the following terms in the effective $4D$ action:

	$\displaystyle\int d^{4}x$	$\displaystyle\left.\bigg{[}e^{-\Phi(z)}\frac{R}{z}\frac{1}{2g_{5}^{2}}A^{i\mu}(\partial_{z}A^{i}_{\mu}-\partial_{\mu}A^{i}_{z})+e^{-\Phi(z)}\frac{f^{2}(z)R^{2}}{z^{3}}\frac{R}{k_{s}}\frac{\pi^{i}}{\chi_{\pi}}\left(A^{i}_{z}-\partial_{z}\frac{\pi^{i}}{\chi_{\pi}}\right)\bigg{]}\right|_{0}^{\infty}$		(40)
		$\displaystyle\xrightarrow{\xi=1}-\frac{1}{2g_{5}^{2}}\int d^{4}x\left.\frac{R}{z}\left(A^{i\bot\mu}\partial_{z}A^{i\bot}_{\mu}+A^{i\parallel\mu}\partial_{z}A^{i\parallel}_{\mu}-A^{i\mu}\partial_{\mu}\partial_{z}\frac{\pi^{i}}{\chi_{\pi}}\right)\right|_{z=\varepsilon}$		(41)

The two-point function of the longitudinal mode is non-zero and that is the crucial difference from the previous sector. The choice $\xi=1$ is explained in a minute. For now we observe that it makes identical the bulk EOMs for $\pi^{i}$ and $A_{\mu}^{i\parallel}$ and eliminates the Goldstone mass term from the boundary: for $\xi=1$ all the Goldstones (including the component associated to the Higgs) are massless. It is also instructive to justify the system of EOMs (37)–(39) by deriving them in the model where $\xi=1$ is set from the start in Eqn. (36). That exercise is worked out in Appendix B.

As in the unbroken case we perform the 4D Fourier transformation and establish the propagation between the source and the bulk for the transverse solution

$$A_{\mu}^{\bot}(q,z)=\phi_{\mu}^{i\bot}(q)\cdot A(q,z),\qquad A(q,\varepsilon)=1.$$

(42)

Changing variables to $y=\kappa^{2}z^{2}$ we arrive at the following EOM

$$yA^{\prime\prime}(q,y)-yA^{\prime}(q,y)+\left(\frac{q^{2}}{4\kappa^{2}}-\frac{g_{5}^{2}(f(y)R)^{2}}{2yk_{s}}\right)A(q,y)=0.$$

(43)

An analytical solution of this EOM exists either for $f^{2}(y)\sim y$ or $f^{2}(y)\sim$ const. The last option taken together with the boundary condition on $A(q,z)$ leads to the implausible conclusion: $f(y)=0$. Therefore we turn to the linear ansatz

$$f(z)=f\cdot\kappa z,$$

(44)

where the constant $f$ has the dimension of mass. We also introduce a convenient parameter

$$a=\frac{g_{5}^{2}(fR)^{2}}{2k_{s}}.$$

(45)

The bulk-to-boundary mode of the confluent hypergeometric equation above is specified as

$$A(q,\kappa^{2}z^{2})=\Gamma\left(-\frac{q^{2}}{4\kappa^{2}}+1+a\right)\Psi\left(-\frac{q^{2}}{4\kappa^{2}}+a,0;\kappa^{2}z^{2}\right).$$

(46)

The other mode for discrete values of $q^{2}$ and $\left.A(q,z)\right|_{q^{2}=M_{A}^{2}(n)}=A_{n}(z)$ gives the $z$-profiles and masses of the eigenstates

$$A_{n}(z)=\kappa^{2}z^{2}\sqrt{\frac{g_{5}^{2}}{R}}\sqrt{\frac{2}{n+1}}L_{n}^{1}(\kappa^{2}z^{2}),\quad M_{A}^{2}(n)=4\kappa^{2}\left(n+1+a\right),\quad n=0,1,2....$$

(47)

The orthogonality relation is completely equivalent to that of Eqn. (29). In fact, the only difference is that the intercept of the Regge trajectory is larger than in the unbroken case, though the pattern is identical. These states are heavier than their unbroken counterparts just as in QCD axial vector mesons are heavier than the vector ones.

The bulk-to-boundary solution of the longitudinal EOM (39) is

$$A_{\mu}^{i\parallel}(q,z)=\phi_{\mu}^{i\parallel}(q)\cdot A^{\parallel}(q,z),\ A^{\parallel}(q,z)=\Gamma\left(1+a\right)\Psi\left(a,0;\kappa^{2}z^{2}\right).$$

(48)

That is equivalent to the transverse propagator of Eqn. (46) but with $q^{2}=0$.

The Goldstone EOM (37) is the same as Eqn. (39). However, $\frac{1}{2}(\partial_{\mu}\pi^{i}(x))^{2}$ is the correct normalization of the Goldstone kinetic term in the 4D effective Lagrangian appearing after the integration over the $z$-dimension, and that fixes the constant factor differently

$$\pi(x,z)=F^{-1}\chi_{\pi}\Gamma\left(1+a\right)\Psi\left(a,0;\kappa^{2}z^{2}\right),$$

(49)

where

$$F^{2}=-\frac{2R\kappa^{2}a}{g_{5}^{2}}\left(\ln\kappa^{2}\varepsilon^{2}+2\gamma_{E}+\psi\left(1+a\right)\right).$$

(50)

In Section IV.1 we will find the same $F^{2}$ in the residue of the massless pole of the broken vector correlator. The exact accordance is only possible for $\xi=1$. Furthermore, solution (49) fixes the due boundary interaction

$$\int d^{4}x(-F)\partial^{\mu}\pi^{i}(x)\phi^{i}_{\mu}(x).$$

(51)

As a result of $W^{\alpha}_{\mu}$ and $B_{\mu}$ couplings in Eqn.(1) the mixing in Eqn.(51) for $i=1,2,3$ implies that the three Goldstones would be eaten by the SM gauge bosons to provide them masses proportional to $F$. Notice that there is no physical source to mix with the fourth Goldstone, it remains in the model as the physical Higgs particle $\pi^{4}(x)=h(x)$. The phenomenological discussion of its properties are postponed to a latter section.

To end the section, we introduce a convenient expression for the bulk-to-boundary propagators as the sums over the resonances (one should utilize Eqns. (121) and (123):

	$\displaystyle V(q,z)=\sum\limits\limits_{n=0}^{\infty}\frac{F_{V}(n)V_{n}(z)}{-q^{2}+M^{2}_{V}(n)},\quad A(q,z)=\sum\limits\limits_{n=0}^{\infty}\frac{F_{A}(n)A_{n}(z)}{-q^{2}+M^{2}_{A}(n)},$		(52)
	$\displaystyle F_{A}^{2}(n)=F_{V}^{2}(n)=\frac{8R\kappa^{4}}{g_{5}^{2}}(n+1).$		(53)

Here $F_{V/A}(n)$ are the decay constants related to the states with the corresponding quantum numbers. The longitudinal broken and Goldstone solutions could be represented by infinite sums too.

The holographic prescriptions given in Eqns. (13) and (12) allow us to define the correlation function as

$$\langle\mathcal{O}_{\mu}^{a/i}(q)\mathcal{O}_{\nu}^{b/j}(p)\rangle=\delta(p+q)\int d^{4}xe^{iqx}\langle\mathcal{O}_{\mu}^{a/i}(x)\mathcal{O}_{\nu}^{b/j}(0)\rangle=\frac{\delta^{2}iS_{boundary}^{5D}}{\delta i\phi^{a/i}_{\mu}(q)\delta i\phi^{b/j}_{\nu}(p)},$$

(54)

where the boundary remainders of the on-shell action were established in Eqns. (23) and (41). We further define the correlators

	$\displaystyle i\int d^{4}xe^{iqx}\langle\mathcal{O}_{\mu}^{a/i}(x)\mathcal{O}_{\nu}^{b/j}(0)\rangle_{\bot}=\delta^{ab/ij}\left(\frac{q_{\mu}q_{\nu}}{q^{2}}-\eta_{\mu\nu}\right)\Pi_{unbr/br}(q^{2}),$		(55)
	$\displaystyle i\int d^{4}xe^{iqx}\langle\mathcal{O}_{\mu}^{i}(x)\mathcal{O}_{\nu}^{j}(0)\rangle_{\parallel}=\delta^{ij}\frac{q_{\mu}q_{\nu}}{q^{2}}\Pi^{\parallel}_{br}(q^{2}).$		(56)

We should take into account that $\Pi_{unbr/br}(q^{2})$ are subject to short distance ambiguities of the form $C_{0}+C_{1}q^{2}$ (see e.g. Refs. Reinders et al. (1985); Afonin and Espriu (2006)).