Goldstone Boson Higgs

Abstract

This chapter provides a first illustration of the composite Higgs scenario and a first characterization of its phenomenology. In particular of those aspects of the phenomenology that robustly follow from the Nambu–Goldstone Boson (NGB) nature of the Higgs in a model-independent way. Interestingly enough, this includes a specific pattern of Higgs coupling modifications with respect to the SM predictions. The basic concept behind the formulation of the composite Higgs scenarios is “vacuum misalignment”, a mechanism by which the composite Higgs boson can effectively behave as an elementary one.

Appendix

2.1.1 The SO(4) Algebra

The Lie algebra of SO(4) is the six-dimensional space of traceless Hermitian imaginary 4 × 4 matrices that define the fundamental (the 4) representation of the group. For applications to composite Higgs the most convenient choice of the Lie algebra basis is the one that makes explicit its connection with the algebra of the chiral group SU(2) _L ×SU(2) _R , which has also dimension 6. The two groups are indeed locally isomorphic, i.e.

$$\displaystyle{ \text{SO}(4) \simeq \text{SU}(2)_{L} \times \text{SU}(2)_{R}\,, }$$

(2.143)

which means that they have the same algebra. In order to prove the isomorphism and to derive the SO(4) basis we proceed as follows. Be $\vec{\Pi }$ a real vector in the 4 of SO(4), its four components are in one-to-one correspondence with the elements of a 2 × 2 pseudo-real matrix

$$\displaystyle{ \Sigma = \frac{1} {\sqrt{2}}\left (i\,\sigma _{\alpha }\Pi ^{\alpha }+ \mathbb{1}_{2}\Pi ^{4}\right ) = \frac{1} {\sqrt{2}}\overline{\sigma }_{i}\Pi ^{i}\,, }$$

(2.144)

where α = 1, 2, 3, $\sigma _{\alpha }$ are Pauli matrices and

$$\displaystyle{ \overline{\sigma }_{i} =\{ i\,\sigma _{\alpha }, \mathbb{1}_{2}\}\,. }$$

(2.145)

The $\overline{\sigma }$’s obey the following normalization, completeness and reality conditions

$$\displaystyle\begin{array}{rcl} \text{Tr}[\overline{\sigma }_{i}^{\dag }\overline{\sigma }_{ j}]& =& 2\,\delta _{ij}\,,\;\;\;\;\;\sum \limits _{i=1}^{4}(\overline{\sigma }_{ i}^{\dag })_{ a}^{\;\;b}\left (\overline{\sigma }_{ i}\right )_{c}^{\;\;d} = 2\,\delta _{ a}^{d}\delta _{ c}^{b}\,, \\ (\overline{\sigma }_{i})^{{\ast}}& =& \sigma _{ 2}\overline{\sigma }_{i}\sigma _{2}\,,\;\;\;\;\;\overline{\sigma }_{i}\overline{\sigma }_{j}^{\dag }-\overline{\sigma }_{ j}\overline{\sigma }_{i}^{\dag } = 2\,\overline{\sigma }_{ i}\overline{\sigma }_{j}^{\dag }- 2\,\delta _{ ij} \mathbb{1}_{2}\,,{}\end{array}$$

(2.146)

from which $\Sigma$ is immediately seen to be pseudo-real, i.e.

$$\displaystyle{ \Sigma ^{{\ast}} =\sigma _{ 2}\Sigma \sigma _{2}\,. }$$

(2.147)

The chiral group acts on $\Sigma$ by matrix multiplication,

$$\displaystyle{ \Sigma \; \rightarrow \; g_{L}\Sigma g_{R}^{\dag }, }$$

(2.148)

and it preserves the pseudo-reality condition ( 2.147). Therefore the matrix $\Sigma$ offers a consistent representation of the chiral group, which we call a pseudo-real bidoublet (2,2) with a self-explanatory notation. In order to demonstrate the local isomorphism among the two groups we consider an infinitesimal chiral transformation on $\Sigma$ and we show that it has the same effect as an SO(4) rotation on the $\vec{\Pi }$ vector. This is because

$$\displaystyle{ \text{Tr}\left [\Sigma ^{\dag }\Sigma \right ] = \vert \vec{\Pi }\vert ^{2}\,. }$$

(2.149)

The trace is invariant under Eq. ( 2.148), which means that the norm of $\vec{\Pi }$ is unchanged by the chiral transformations. Since SO(4) contains the most general norm-preserving infinitesimal transformation of a four-component vector, this demonstrates that any chiral transformation is an element of SO(4) and therefore the chiral group algebra is contained in the SO(4) one. However no sub-algebra exists, aside from the full algebra itself, with the same dimensionality of the original one. The isomorphism ( 2.143) is thus proven.

Let us turn to the determination of the SO(4) generators. In light of the discussion above we can split them into two sets $t^{a} =\{ t_{L}^{\alpha },t_{R}^{\alpha }\}$ with α = 1, 2, 3. Each set obeys SU(2) commutation relations and the two sets commute in accordance with the SU(2) _L ×SU(2) _R algebra, namely

$$\displaystyle\begin{array}{rcl} \left [t_{L}^{\alpha },t_{ L}^{\beta }\right ]& =& i\epsilon ^{\alpha \beta \gamma }t_{ L}^{\gamma }\,,\;\;\;\;\;\left [t_{ R}^{\alpha },t_{ R}^{\beta }\right ] = i\epsilon ^{\alpha \beta \gamma }t_{ R}^{\gamma }\,, \\ \left [t_{L}^{\alpha },t_{ R}^{\beta }\right ]& =& 0\,. {}\end{array}$$

(2.150)

Those generators, in the fundamental 4 representation, are easily extracted from the infinitesimal variations

$$\displaystyle\begin{array}{rcl} \delta _{L}\Sigma & =& i\,\delta _{\alpha }^{L} \frac{\sigma ^{\alpha }} {2}\Sigma \,, \\ \delta _{R}\Sigma & =& -\,i\,\delta _{\alpha }^{R}\Sigma \frac{\sigma ^{\alpha }} {2}\,,{}\end{array}$$

(2.151)

under chiral transformations $g_{L,R} \simeq \mathbb{1} + i\delta _{\alpha }^{L,R}\sigma ^{\alpha }/2$. The corresponding variations of $\vec{\Pi }$ have the form

$$\displaystyle\begin{array}{rcl} \delta _{L}\vec{\Pi }& =& i\,\delta _{\alpha }^{L}t_{ L}^{\alpha }\vec{\Pi }\,, \\ \delta _{R}\vec{\Pi }& =& i\,\delta _{\alpha }^{R}t_{ R}^{\alpha }\vec{\Pi }\,,{}\end{array}$$

(2.152)

from which, by matching with Eq. ( 2.151), we obtain

$$\displaystyle\begin{array}{rcl} (t_{L}^{\alpha })_{ ij}& =& \frac{1} {4}\text{Tr}[\overline{\sigma }_{i}^{\dag }\sigma ^{\alpha }\overline{\sigma }_{ j}] = -\frac{i} {2}\left [\varepsilon _{\alpha \beta \gamma }\delta _{i}^{\beta }\delta _{ j}^{\gamma } + \left (\delta _{ i}^{\alpha }\delta _{ j}^{4} -\delta _{ j}^{\alpha }\delta _{ i}^{4}\right )\right ]\,, \\ (t_{R}^{\alpha })_{ ij}& =& \frac{1} {4}\text{Tr}[\overline{\sigma }_{i}\sigma ^{\alpha }\overline{\sigma }_{ j}^{\dag }] = -\frac{i} {2}\left [\varepsilon _{\alpha \beta \gamma }\delta _{i}^{\beta }\delta _{ j}^{\gamma } -\left (\delta _{ i}^{\alpha }\delta _{ j}^{4} -\delta _{ j}^{\alpha }\delta _{ i}^{4}\right )\right ]\,.{}\end{array}$$

(2.153)

The generators obey the commutation relations in Eq. ( 2.150) and they are subject to the normalization and completeness relations

$$\displaystyle\begin{array}{rcl} \text{Tr}\left [t_{L}^{\alpha }t_{ L}^{\beta }\right ]& =& \text{Tr}\left [t_{ R}^{\alpha }t_{ R}^{\beta }\right ] =\delta ^{ab}\,,\;\;\;\;\;\text{Tr}\left [t_{ L}^{\alpha }t_{ R}^{\beta }\right ] = 0 \\ \sum \limits _{\alpha =1}^{3}\left [\left (t_{ L}^{\alpha }\right )_{ ij}\left (t_{L}^{\alpha }\right )_{ kl} + \left (t_{R}^{\alpha }\right )_{ ij}\left (t_{R}^{\alpha }\right )_{ kl}\right ]& =& -\frac{1} {2}\left (\delta _{ik}\delta _{jl} -\delta _{il}\delta _{jk}\right )\,, \\ \sum \limits _{\alpha =1}^{3}\left [\left (t_{ L}^{\alpha }\right )_{ ij}\left (t_{L}^{\alpha }\right )_{ kl} -\left (t_{R}^{\alpha }\right )_{ ij}\left (t_{R}^{\alpha }\right )_{ kl}\right ]& =& -\frac{1} {2}\epsilon _{ijkl}\,, {}\end{array}$$

(2.154)

where $\epsilon _{ijkl}$ is the anti-symmetric Levi-Civita tensor in four dimensions.

In composite Higgs models the SU(2) _L group is identified with the SM left-handed group and the hypercharge U(1) _Y is the third SU(2) _R generator up to the U(1) _X charge (see Sect. 2.4.2), which however vanishes for the Higgs field. In this case the four real components of the (2,2) representation defined in Eq. ( 2.144) form one complex SM-like Higgs doublet with 1∕2 hypercharge. This is immediately verified by noticing that $\Sigma$, thanks to pseudo-reality, can be written as

$$\displaystyle{ \Sigma = \left (H^{c},\,H\right )\,, }$$

(2.155)

in terms of the doublet H and of its conjugate $H^{c} = i\sigma _{2}H^{{\ast}}$. By remembering that H ^c is also a doublet but with $-1/2$ hypercharge it is immediate to verify that the action of the chiral group in Eq. ( 2.148) matches the expected Higgs transformation rules under the SU(2) _L ×U(1) _Y . By the definition ( 2.144) the H components are expressed as

$$\displaystyle{ H = \left [\begin{array}{c} h_{u} \\ h_{d} \end{array} \right ] = \frac{1} {\sqrt{2}}\left [\begin{array}{c} \Pi ^{2} + i\Pi ^{1} \\ \Pi ^{4} - i\Pi ^{3} \end{array} \right ]\,, }$$

(2.156)

in terms of the fourplet fields $\Pi _{i}$. Conversely, one real SO(4) fourplet or, equivalently, one pseudo-real (2,2), can be rewritten in terms of one complex Higgs doublet as in Eq. ( 2.28). This is to say that the real SO(4) fourplet decomposes as

$$\displaystyle{ \mathbf{4} = \mathbf{(2,2)}\; \rightarrow \;\mathbf{2}_{1/2}, }$$

(2.157)

under the SU(2) _L ×U(1) _Y subgroup.

Similar considerations hold for the complex SO(4) fourplet, which we will encounter in the main text when dealing with the SM matter fermions. Its complex components ψ ⁱ can be traded for the elements of a generic 2 × 2 matrix

$$\displaystyle{ \Psi = \frac{1} {\sqrt{2}}\left (\psi ^{4} + i\,\sigma _{\alpha }\psi ^{\alpha }\right ) = \frac{1} {\sqrt{2}}\overline{\sigma }_{i}\psi ^{i}\,, }$$

(2.158)

which transforms in the (2,2) representation as in Eq. ( 2.148). Since it does not obey the pseudo-reality condition we dub it a complex bidoublet (2,2) _c. Under the SU(2) _L ×U(1) _Y subgroup the two columns of $\Psi$ form two doublets with opposite ± 1∕2 Y charge, namely

$$\displaystyle{ \Psi = \frac{1} {\sqrt{2}}\left [\begin{array}{cc} \psi ^{4} + i\,\psi ^{3} & \psi ^{2} + i\,\psi ^{1} \\ -\psi ^{2} + i\,\psi ^{1} & \psi ^{4} - i\,\psi ^{3} \end{array} \right ] \equiv \left (\Psi _{-},\,\Psi _{+}\right )\,. }$$

(2.159)

This corresponds to the decomposition

$$\displaystyle{ \mathbf{4_{c}} = \mathbf{(2,2)_{c}}\; \rightarrow \;\mathbf{2}_{1/2} \oplus \mathbf{2}_{-1/2}\,. }$$

(2.160)

From Eq. ( 2.159) we can easily read the up and down components of the two doublets in terms of the fourplet fields. Conversely, the fourplet components are written in terms of $\Psi _{\pm }^{u,d}$ as

$$\displaystyle{ \vec{\psi }= \frac{1} {\sqrt{2}}\{ - i\,\Psi _{+}^{u} - i\,\Psi _{ -}^{d},\;\Psi _{ +}^{u} - \Psi _{ -}^{d},\;i\,\Psi _{ +}^{d} - i\,\Psi _{ -}^{u},\;\Psi _{ +}^{d} + \Psi _{ -}^{u}\}^{T}\,. }$$

(2.161)

The above equation is often referred to as the embedding of the two doublets in the complex 4.

Other relevant representations are the (2,1) and the (1,2). As the notation suggests these are doublets under one of the chiral SU(2) factors and they are invariant under the other one. Their SU(2) _L ×U(1) _Y decomposition is obviously

$$\displaystyle\begin{array}{rcl} \mathbf{(2,1)}& \rightarrow & \mathbf{2}_{0}\,, \\ \mathbf{(1,2)}& \rightarrow & \mathbf{1}_{1/2} \oplus \mathbf{1}_{-1/2}\,.{}\end{array}$$

(2.162)

The adjoint of SO(4), the 6, also deserves some comment. Given that the algebra splits into the tensor product of two SU(2)’s, the adjoint is a reducible representation and it is represented, in SU(2) _L ×SU(2) _R notation, as

$$\displaystyle{ \mathbf{6} = (\mathbf{3},\mathbf{1}) \oplus (\mathbf{1},\mathbf{3}) }$$

(2.163)

where two terms correspond to the generators $t_{L}^{\alpha }$ and $t_{R}^{\alpha }$, respectively. The decomposition reads

$$\displaystyle\begin{array}{rcl} \mathbf{(3,1)}& \rightarrow & \mathbf{3}_{0}\,, \\ \mathbf{(1,3)}& \rightarrow & \mathbf{1}_{0} \oplus \mathbf{1}_{1} \oplus \mathbf{1}_{-1}\,.{}\end{array}$$

(2.164)

The last representation which is worth mentioning is the 9 = (3,3). It corresponds to a real 3 × 3 matrix with the chiral group acting in the spin one representation on the two sides, or to the symmetric traceless tensor product of two fourplets. It decomposes as

$$\displaystyle{ \mathbf{9} = \mathbf{(3,3)}\; \rightarrow \;\mathbf{3}_{0} \oplus \mathbf{3}_{1} \oplus \mathbf{3}_{-1}\,. }$$

(2.165)

2.1.2 Explicit CCWZ for SO(5)∕SO(4)

The abstract definitions of Sect. 2.3, where the CCWZ construction is illustrated for a generic $\mathcal{G}/\mathcal{H}$ coset, become concrete and fully explicit in the particular case of the minimal coset SO(5)∕SO(4).

The SO(5) generators, reported explicitly in Eq. ( 2.25) for the fundamental 5 representation, can be split into an unbroken subset T ^a which represents the SO(4) subgroup and obeys the commutation relations in Eq. ( 2.150) and a broken one $\hat{T}^{i}$ associated to the four Goldstone bosons, with commutation relations

$$\displaystyle\begin{array}{rcl} \left [T^{a},\hat{T}^{i}\right ]& =& i\,f_{\;\;\;\; j}^{ai}\hat{T}^{j} =\hat{ T}^{j}\left (t^{a}\right )_{ j}^{\;\;i}\,, \\ \left [\hat{T}^{i},\hat{T}^{j}\right ]& =& i\,f_{\;\;\;\; a}^{ij}T^{a} = \left (t_{ a}\right )^{ji}T^{a}\,,{}\end{array}$$

(2.166)

where $t^{a} =\{ t_{L}^{\alpha },t_{R}^{\alpha }\}$ are the SO(4) generators in the 4 as in Eq. ( 2.153). The generators in the 5, defined in Eq. ( 2.25), obey normalization and completeness conditions

$$\displaystyle\begin{array}{rcl} \text{Tr}\left [T^{A}T^{B}\right ]& =& \delta ^{AB}\,, \\ \sum \limits _{A=1}^{10}\left (T^{A}\right )_{ IJ}\left (T^{A}\right )_{ KL}& =& -\frac{1} {2}\left (\delta _{IK}\delta _{JL} -\delta _{IL}\delta _{JK}\right )\,.{}\end{array}$$

(2.167)

Given the generators, it is not hard to compute the Goldstone matrix in the fundamental representation

$$\displaystyle{ U = e^{i\frac{\sqrt{2}} {f} \Pi _{i}(x)\hat{T}^{i} } = \left [\begin{array}{cc} \mathbb{1} -\Big (1 -\cos \frac{\Pi } {f} \Big)\frac{\ \vec{\Pi }\,\vec{\Pi }^{T}} {\Pi ^{2}} & \ \ \sin \frac{\Pi } {f} \frac{\vec{\Pi }} {\Pi }\ \\ -\sin \frac{\Pi } {f} \frac{\ \vec{\Pi }^{T}} {\Pi } & \cos \frac{\Pi } {f} \end{array} \right ]\,, }$$

(2.168)

in terms of the four real Higgs field components. The complex Higgs doublet notation can be reached afterwards by substituting Eq. ( 2.28). The Goldstone matrix considerably simplifies in the unitary gauge ( 2.34) and thus it is worth reporting it

$$\displaystyle{ U\mathop{ =}\limits_{ \text{UG}}\left [\begin{array}{ccc} \mathbb{1}_{3} & \vec{0} & 0 \\ \vec{0}^{T} & \cos \frac{V + h} {f} &\sin \frac{V + h} {f} \\ 0 & -\sin \frac{V + h} {f} &\cos \frac{V + h} {f} \end{array} \right ]\,, }$$

(2.169)

where $\mathbb{1}_{3}$ is the 3 × 3 identity matrix and $\vec{0}$ is the three-dimensional null vector. The Goldstone matrix in the unitary gauge is a rotation in the 4–5 plane of the five-dimensional space.

As explained in the main text, the Goldstone matrix can be defined in any representation of the group as the exponential of the appropriate generator matrices. Above we computed the one in the fundamental and one should worry of how to obtain the others. For all the representations constructed as tensor product of fundamentals, which can thus be expressed as tensors with fiveplet indices, this is completely straightforward and does not require any additional calculation: the Goldstone matrix acts by rotating each index with the 5 × 5 matrix U. However not all the SO(5) representations are tensor product of fundamentals, the simplest counterexample is the spinorial, for which the Goldstone matrix needs to be recomputed. The spinorial has dimension 4 and its generators are

$$\displaystyle\begin{array}{rcl} T_{\mathbf{4}}{}_{L}{}^{\alpha }& =& \frac{1} {2}\left [\begin{array}{cc} \sigma ^{\alpha } &0 \\ 0&0 \end{array} \right ]\,,\;\;\;\;\;T_{\mathbf{4}}{}_{R}^{\alpha } = \frac{1} {2}\left [\begin{array}{cc} 0&0\\ 0 & \sigma ^{\alpha } \end{array} \right ]\,, \\ \hat{T}_{\mathbf{4}}^{i}& =& \frac{1} {2\sqrt{2}}\left [\begin{array}{cc} 0 & \overline{\sigma }_{i} \\ \overline{\sigma }_{i}^{\dag }&0 \end{array} \right ]\,, {}\end{array}$$

(2.170)

where $\sigma ^{\alpha }$ denotes the three Pauli matrices and $\overline{\sigma }$ is defined in Eq. ( 2.145). The spinorial can be also regarded as the fundamental of the symplectic group Sp(4), which is isomorphic to SO(5). The generators indeed obey the symplectic condition

$$\displaystyle{ \Omega \cdot T_{\mathbf{4}}^{A} + \left (T_{\mathbf{ 4}}^{A}\right )^{T} \cdot \Omega = 0\,, }$$

(2.171)

with the antisymmetric unitary matrix

$$\displaystyle{ \Omega = e^{i\,\pi \left [T_{\mathbf{4}}{}_{L}^{2}-T_{\mathbf{ 4}}{}_{R}^{2}\right ] } = \left [\begin{array}{cc} i\,\sigma ^{2} & 0 \\ 0 & - i\,\sigma ^{2} \end{array} \right ]\,. }$$

(2.172)

For completeness, we report normalization and completeness relations also for the spinorial

$$\displaystyle\begin{array}{rcl} \text{Tr}\left [T_{\mathbf{4}}^{A}T_{\mathbf{ 4}}^{B}\right ]& =& \frac{1} {2}\delta ^{AB}\,, \\ \sum \limits _{A=1}^{10}\left (T_{\mathbf{ 4}}^{A}\right )_{ I}^{\;\;J}\left (T_{\mathbf{ 4}}^{A}\right )_{ K}^{\;\;L}& =& \frac{1} {4}\left (\delta _{I}^{L}\delta _{ K}^{J} - \Omega _{ IK}\Omega ^{JL}\right )\,.{}\end{array}$$

(2.173)

The Goldstone matrix in the spinorial is straightforwardly obtained by exponentiating the broken generators and it turns out to be most easily expressed in the complex doublet Higgs notation rather than in terms of the real fourplet $\vec{\Pi }$. It reads

$$\displaystyle{ U_{\mathbf{4}} = e^{i\frac{\sqrt{2}} {f} \Pi _{i}(x)\hat{T}_{\mathbf{4}}^{i} } = \left [\begin{array}{cc} \cos \frac{\vert H\vert } {\sqrt{2}f} \mathbb{1}_{2} & i\,\sin \frac{\vert H\vert } {\sqrt{2}f} \frac{\Sigma } {\vert H\vert } \\ i\,\sin \frac{\vert H\vert } {\sqrt{2}f} \frac{\Sigma ^{\dag }} {\vert H} &\cos \frac{\vert H\vert } {\sqrt{2}f} \mathbb{1}_{2} \end{array} \right ]\,, }$$

(2.174)

where $\Sigma$ is the pseudo-real bidoublet representation of the Higgs as defined in Eq. ( 2.155). The result further simplifies in the unitary gauge

$$\displaystyle{ U_{\mathbf{4}}\mathop{ =}\limits_{ \text{UG}}\left [\begin{array}{cc} \cos \frac{V + h} {2f} \mathbb{1}_{2} & i\,\sin \frac{V + h} {2f} \mathbb{1}_{2} \\ i\,\sin \frac{V + h} {2f} \mathbb{1}_{2} & \cos \frac{V + h} {2f} \mathbb{1}_{2} \end{array} \right ]\,. }$$

(2.175)

Any SO(5) representation, including the 5, is the tensor product of spinorials (the conjugate $\overline{\mathbf{4}}$ is equivalent to the spinorial itself and its Goldstone matrix is $U_{\overline{\mathbf{4}}} = U_{\mathbf{4}}^{{\ast}} =\varOmega U_{\mathbf{4}}\varOmega ^{\dag }$). The knowledge of U ₄ thus allows to derive the Goldstone matrix in any representation.

Let us now turn to the determination of the d _μ and e _μ symbols. Those are defined in Eq. ( 2.69) in the presence of non-dynamical source gauge fields A _μ, A , one for each of the 10 SO(5) generators. However, only a subset of those sources will be made physical by giving them a kinetic term, all the others will be eventually set to zero. The physical sources are the ones in the SM subgroup, which is embedded in the unbroken SO(4).²⁰ We can thus split the A _μ, A ’s in unbroken and broken components

$$\displaystyle{ A_{\mu,\,A} = \left \{A_{\mu,\,a} =\{ A_{\mu,\,\alpha }^{L},\,A_{\mu,\,\alpha }^{R}\},\,A_{\mu,\,i} = 0\right \}\,, }$$

(2.176)

and already set the latter ones to zero while retaining, for the moment, all the unbroken generator sources. The unbroken sources have been further split in the two sets that correspond to the two SU(2) factors of $\text{SO}(4) \simeq \text{SU}(2)_{L} \times \text{SU}(2)_{R}$. The only truly dynamical sources are the ones associated with the four SM gauge fields, namely we will eventually set

$$\displaystyle\begin{array}{rcl} A_{\mu,\,\alpha }^{L}& =& \left \{g\,W_{\mu }^{1},\,g\,W_{\mu }^{2},\,g\,W_{\mu }^{3}\right \}\,, \\ A_{\mu,\,\alpha }^{R}& =& \left \{0,\,0,\,g^{{\prime}}B_{\mu }\right \}\,, {}\end{array}$$

(2.177)

in accordance with Eq. ( 2.68).

The d and e symbols can be straightforwardly computed from the definition ( 2.69), or obtained in a somewhat faster way by first classifying the possible structures which they can contain compatibly with the SO(4) symmetry. The result is

$$\displaystyle\begin{array}{rcl} d_{\mu }^{i}& =& \sqrt{2}\left ( \frac{1} {\Pi }\sin \frac{\Pi } {f} - \frac{1} {f}\right )\frac{\vec{\Pi }^{T}D_{\mu }\vec{\Pi }} {\Pi ^{2}} \Pi ^{i} -\frac{\sqrt{2}} {\Pi } \sin \frac{\Pi } {f} D_{\mu }\Pi ^{i}\,, \\ e_{\mu }^{L}{}^{\alpha }& =& A^{L}_{ \mu }{}^{\alpha } - \frac{4} {\Pi ^{2}}\sin ^{2} \frac{\Pi } {2f}\vec{\Pi }^{T}i\,t_{ L}^{\alpha }D_{\mu }\vec{\Pi }\,, \\ e_{\mu }^{R}{}^{\alpha }& =& A^{R}_{ \mu }{}^{\alpha } - \frac{4} {\Pi ^{2}}\sin ^{2} \frac{\Pi } {2f}\vec{\Pi }^{T}i\,t_{ R}^{\alpha }D_{\mu }\vec{\Pi }\,, {}\end{array}$$

(2.178)

where $D_{\mu }\vec{\Pi }$ is the SO(4) covariant derivative

$$\displaystyle{ D_{\mu }\vec{\Pi } = \left (\partial _{\mu } - iA_{\mu,\,\alpha }^{L}t_{ L}^{\alpha } - iA_{\mu,\,\alpha }^{R}t_{ R}^{\alpha }\right )\vec{\Pi }\,, }$$

(2.179)

not to be confused with the CCWZ covariant derivative introduced in Sect. 2.3.3. We have split the e _μ, a symbol in two components associated with the decomposition $\mathbf{6} = \mathbf{(3,1)} \oplus \mathbf{(1,3)}$ of the adjoint in irreducible representations. In the absence of additional symmetries, the two objects can be employed separately in the construction of invariants. For instance, it is possible to define two independent field-strength tensors following Eq. ( 2.78)

$$\displaystyle\begin{array}{rcl} E_{\mu \nu }^{L}{}^{\alpha }& =& \partial _{\mu }e^{L}_{ \nu }{}^{\alpha } - \partial _{\nu }e^{L}_{ \mu }{}^{\alpha } +\epsilon ^{\alpha \beta \gamma }e^{L}_{ \mu,\,\beta }e^{L}_{ \nu,\,\gamma }\,, \\ E_{\mu \nu }^{R}{}^{\alpha }& =& \partial _{\mu }e^{R}_{ \nu }{}^{\alpha } - \partial _{\nu }e^{R}_{ \mu }{}^{\alpha } +\epsilon ^{\alpha \beta \gamma }e^{R}_{ \mu,\,\beta }e^{R}_{ \nu,\,\gamma }\,,{}\end{array}$$

(2.180)

in the (3,1) and (1,3), respectively. In the following we will also occasionally employ a collective notation $E_{\mu \nu }^{a} =\{ E^{L}{}_{\mu \nu }^{\alpha },E^{R}{}_{\mu \nu }^{\alpha }\}$ for the six field-strength tensor components.

For some practical calculation, especially when willing to switch to the complex Higgs doublet notation, the d and e objects in Eq. ( 2.178) are conveniently expressed in terms of 2 × 2 matrices obtained by contracting them with $\overline{\sigma }$ and $\sigma$, namely

$$\displaystyle\begin{array}{rcl} d_{\mu }^{(2)}& =& d_{\mu }^{i}\overline{\sigma }_{ i} = \left ( \frac{1} {\sqrt{2}\vert H\vert }\sin \frac{\sqrt{2}\vert H\vert } {f} - \frac{1} {f}\right )\frac{\partial _{\mu }\vert H\vert ^{2}} {\vert H\vert ^{2}} \Sigma - \frac{\sqrt{2}} {\vert H\vert }\sin \frac{\sqrt{2}\vert H\vert } {f} D_{\mu }\Sigma \,, \\ e_{L}^{(2)}{}_{ \mu }& =& e_{\mu }^{L}{}^{\alpha } \frac{\sigma _{\alpha }} {2} = A_{\mu }^{L} + \frac{i} {2\vert H\vert ^{2}}\sin ^{2} \frac{\vert H\vert } {\sqrt{2}f}\left [\Sigma \,D_{\mu }\Sigma ^{\dag }- D_{\mu }\Sigma \,\Sigma ^{\dag }\right ]\,, \\ e_{R}^{(2)}{}_{ \mu }& =& e_{\mu }^{R}{}^{\alpha } \frac{\sigma _{\alpha }} {2} = A_{\mu }^{R} + \frac{i} {2\vert H\vert ^{2}}\sin ^{2} \frac{\vert H\vert } {\sqrt{2}f}\left [\Sigma ^{\dag }D_{\mu }\Sigma - D_{\mu }\Sigma ^{\dag }\Sigma \right ]\,, {}\end{array}$$

(2.181)

where the Higgs matrix covariant derivative, in accordance with ( 2.179), is

$$\displaystyle{ D_{\mu }\Sigma = \partial _{\mu }\Sigma - \frac{i} {2}A_{\mu }^{L}\,\Sigma + \frac{i} {2}\Sigma \,A_{\mu }^{R}\,, }$$

(2.182)

with $A_{\mu }^{L,R} = A_{\mu,\,\alpha }^{L,R}\sigma ^{\alpha }/2$. Notice that the d _μ symbol matrix representation is pseudo-real and those of $e_{\mu }^{L,R}$ are Hermitian and traceless, as obvious from the definition. In the chiral notation, where the SO(4) rotation gets split into two SU(2) _L ×SU(2) _R transformations g _L and g _R , d ⁽²⁾ and $e_{L,R}^{(2)}$ transform as

$$\displaystyle\begin{array}{rcl} d_{\mu }^{(2)}& \rightarrow & g_{ L} \cdot d_{\mu }^{(2)} \cdot g_{ R}^{\dag }\,, \\ e_{L}^{(2)}{}_{ \mu }& \rightarrow & g_{L} \cdot (e_{L}^{(2)}{}_{ \mu } + i\partial _{\mu }) \cdot g_{L}^{\dag }\,, \\ e_{R}^{(2)}{}_{ \mu }& \rightarrow & g_{R} \cdot (e_{R}^{(2)}{}_{ \mu } + i\partial _{\mu }) \cdot g_{R}^{\dag }\,,{}\end{array}$$

(2.183)

i.e. respectively like one bidoublet and two gauge fields.

Now that the basic objects are known we can straightforwardly apply the general CCWZ machinery and derive some useful formulas. First, we compute the 2-derivative non-linear $\sigma$-model Lagrangian of Eq. ( 2.72) and verify that it agrees with the expression reported in the main text. After setting the gauge sources to their physical value ( 2.177) we obtain

$$\displaystyle\begin{array}{rcl} \mathcal{L}^{(2)}& =& \frac{f^{2}} {4} d_{\mu,\,i}d^{\mu,\,i} = \frac{f^{2}} {8} \text{Tr}[(d_{\mu }^{(2)})^{\dag }d^{(2),\,\mu }] \\ & =& \frac{f^{2}} {2\vert H\vert ^{2}}\sin ^{2}\frac{\sqrt{ 2}\vert H\vert } {f} D_{\mu }H^{\dag }D^{\mu }H + \frac{f^{2}} {8\vert H\vert ^{4}}\left (2\frac{\vert H\vert ^{2}} {f^{2}} -\sin ^{2}\frac{\sqrt{2}\vert H\vert } {f} \right )\left (\partial _{\mu }\vert H\vert ^{2}\right )^{2}\,,{}\end{array}$$

(2.184)

in accordance with the result in Eq. ( 2.32) obtained for the linear $\sigma$-model.

We would also like to compute the E _{μ ν} field-strength components and the antisymmetric part of the two-derivative tensor D ⋅ d defined in Eq. ( 2.75), which we will need in the following chapter. These objects can be obtained directly from their definitions in Eqs. ( 2.180) and ( 2.75), or derived in fast way by employing the identity ( 2.82) proven in Sect. ( 2.3.3). In this second case we proceed by first computing the “dressed” field-strength tensors $\mathcal{F}$ defined in Eqs. ( 2.79), ( 2.80), which in our case consist of 3 CCWZ multiplets in the (3,1), (1,3) and (2,2) representations. Those are rather simple because they contain no derivatives of the Goldstone fields and read

$$\displaystyle\begin{array}{rcl} & & \mathcal{F}_{}{L}_{\mu \nu }^{\alpha } =\cos ^{2} \frac{\Pi } {2f}A^{L}_{ \mu \nu }{}^{\alpha } - \frac{4} {\Pi ^{2}}\sin ^{2} \frac{\Pi } {2f}\,\vec{\Pi }^{T}t_{ L}{}^{\alpha }(A^{R}_{ \mu \nu,\,\beta }\,t_{R}^{\beta })\vec{\Pi }\,, \\ & & \mathcal{F}_{}{R}_{\mu \nu }^{\alpha } =\cos ^{2} \frac{\Pi } {2f}A^{R}_{ \mu \nu }{}^{\alpha } - \frac{4} {\Pi ^{2}}\sin ^{2} \frac{\Pi } {2f}\,\vec{\Pi }^{T}t_{ R}{}^{\alpha }(A^{L}_{ \mu \nu,\,\beta }\,t_{L}^{\beta })\vec{\Pi }\,, \\ & & \mathcal{F}_{\mathbf{4}}{}_{\mu \nu }^{i} = \frac{\sqrt{2}} {\Pi } \sin \frac{\Pi } {f} \left (A^{L}_{ \mu \nu,\,\alpha }\,i\,t_{L}^{\alpha } + A^{R}_{ \mu \nu,\,\alpha }\,i\,t_{R}{}^{\alpha }\right )^{ij}\Pi _{ j}\,,{}\end{array}$$

(2.185)

where $A_{\mu \nu }^{L,R}$ denote the field-strengths associated with the gauge sources

$$\displaystyle{ A_{\mu \nu }^{L,R}{}^{\alpha } = \partial _{\mu }A_{\nu }^{L,R}{}^{\alpha } - \partial _{\nu }A_{\mu }^{L,R}{}^{\alpha } +\epsilon ^{\alpha \beta \gamma }A_{\mu }^{L,R}{}_{ \beta }A_{\nu }{}^{L,R}_{ \gamma }\,. }$$

(2.186)

After setting the sources to their physical values in Eq. ( 2.177), they reduce to the familiar W _{μ ν} and B _{μ ν} SM tensors.

The last object we need in order to apply Eq. ( 2.82) (since $d_{\mathbf{r}_{\pi }}^{2} = 0$ for a symmetric coset) is $d_{\mathbf{Ad}}^{2}$, the adjoint tensor formed out of two d-symbols defined in Eq. ( 2.74). In our case it splits in two components

$$\displaystyle{ d_{L}^{2}{}_{ \mu \nu }{}^{\alpha } = d_{\mu }^{i}(i\,t_{ L}{}^{\alpha })_{ ij}d_{\nu }^{j}\,,\;\;\;\;\;d_{ R}^{2}{}_{ \mu \nu }^{\alpha } = d_{\mu }^{i}(i\,t_{ R}{}^{\alpha })_{ ij}d_{\nu }^{j}\,. }$$

(2.187)

The explicit form of $d_{L,R}^{2}$ in terms of $\Pi$ can be easily worked out, however the expression in terms of the d-symbol provided by the equation above is already the simplest one for practical calculations. The field-strengths E ^L, R _{μ ν} and D ⋅ d _[μ, ν] are, finally

$$\displaystyle\begin{array}{rcl} E_{L}{}_{\mu \nu }^{\alpha }& =& \mathcal{F}_{ L}{}_{\mu \nu }^{\alpha }\, -\, d_{ L}^{2}{}_{ \mu \nu }^{\alpha }\,, \\ E_{R}{}_{\mu \nu }^{\alpha }& =& \mathcal{F}_{ R}{}_{\mu \nu }^{\alpha }\, -\, d_{ R}^{2}{}_{ \mu \nu }^{\alpha }\,, \\ (D \cdot d)_{[\mu,\nu ]}^{i}& =& D_{\mu }d_{\nu }^{i} - D_{\nu }d_{\mu }^{i} = \mathcal{F}_{\mathbf{ 4}}{}_{\mu \nu }^{i}\,.{}\end{array}$$

(2.188)

The above formulas can be also obtained by computing E and D ⋅ d directly from their definitions. This provides a non-trivial cross-check of Eq. ( 2.82). In the matrix notation, E and D ⋅ d become

$$\displaystyle\begin{array}{rcl} E_{L}^{(2)}{}_{ \mu \nu }& =& \cos ^{2} \frac{\vert H\vert } {\sqrt{2}f}A_{\mu \nu }^{L} + \frac{1} {\vert H\vert ^{2}}\sin ^{2} \frac{\vert H\vert } {\sqrt{2}f}\left (\Sigma A_{\mu \nu }^{R}\Sigma ^{\dag }-\frac{1} {2}\text{Tr}[\Sigma A_{\mu \nu }^{R}\Sigma ^{\dag }]\right ) \\ & & + \frac{i} {8}\left (d_{\mu }^{(2)}d_{\nu }^{(2)}{}^{\dag }- d_{\nu }^{(2)}d_{\mu }^{(2)}{}^{\dag }\right )\,, \\ E_{R}^{(2)}{}_{ \mu \nu }& =& \cos ^{2} \frac{\vert H\vert } {\sqrt{2}f}A_{\mu \nu }^{R} + \frac{1} {\vert H\vert ^{2}}\sin ^{2} \frac{\vert H\vert } {\sqrt{2}f}\left (\Sigma ^{\dag }A_{\mu \nu }^{L}\Sigma -\frac{1} {2}\text{Tr}[\Sigma ^{\dag }A_{\mu \nu }^{L}\Sigma ]\right ) \\ & & + \frac{i} {8}\left (d_{\mu }^{(2)}{}^{\dag }d_{\nu }^{(2)} - d_{\nu }^{(2)}{}^{\dag }d_{\mu }^{(2)}\right )\,, \\ D \cdot d_{[\mu,\nu ]}^{(2)}& =& \frac{\sqrt{2}\,i} {\vert H\vert }\sin \frac{\sqrt{ 2}\vert H\vert } {f} \left (A_{\mu \nu }^{L}\Sigma - \Sigma A_{\mu \nu }^{R}\right )\,, {}\end{array}$$

(2.189)

where $A_{\mu \nu }^{L,R} = A_{\mu \nu }^{L,R}{}^{\alpha }\sigma _{\alpha }/2$.

All the formulas above greatly simplify in the unitary gauge, in which

$$\displaystyle{ \Sigma = \frac{V + h} {\sqrt{2}} \, \mathbb{1}_{2}\,. }$$

(2.190)

For the d and e symbols we have

$$\displaystyle\begin{array}{rcl} d_{\mu }^{(2)}& & \mathop{=}\limits_{ \text{UG}} -\frac{\sqrt{2}} {f} \partial _{\mu }h\, \mathbb{1}_{2} + \sqrt{2}\,i\sin \frac{V + h} {f} (A_{\mu }^{L} - A_{\mu }^{R})\,, \\ e_{L}^{(2)}{}_{ \mu }& & \mathop{=}\limits_{ \text{UG}}A_{\mu }^{L} -\sin ^{2}\frac{V + h} {2\,f} (A_{\mu }^{L} - A_{\mu }^{R})\,, \\ e_{R}^{(2)}{}_{ \mu }& & \mathop{=}\limits_{ \text{UG}}A_{\mu }^{R} +\sin ^{2}\frac{V + h} {2\,f} (A_{\mu }^{L} - A_{\mu }^{R})\,, {}\end{array}$$

(2.191)

while for E and D ⋅ d one finds

$$\displaystyle\begin{array}{rcl} E_{L}^{(2)}{}_{ \mu \nu }& & \mathop{=}\limits_{ \text{UG}}\cos ^{2}\frac{V + h} {2f} A_{\mu \nu }^{L} +\sin ^{2}\frac{V + h} {2f} A_{\mu \nu }^{R} + \frac{i} {8}\left (d_{\mu }^{(2)}d_{\nu }^{(2)}{}^{\dag }- d_{\nu }^{(2)}d_{\mu }^{(2)}{}^{\dag }\right )\,, \\ E_{R}^{(2)}{}_{ \mu \nu }& & \mathop{=}\limits_{ \text{UG}}\cos ^{2}\frac{V + h} {2f} A_{\mu \nu }^{R} +\sin ^{2}\frac{V + h} {2f} A_{\mu \nu }^{L} + \frac{i} {8}\left (d_{\mu }^{(2)}{}^{\dag }d_{\nu }^{(2)} - d_{\nu }^{(2)}{}^{\dag }d_{\mu }^{(2)}\right )\,, \\ D \cdot d_{[\mu,\nu ]}^{(2)}& & \mathop{=}\limits_{ \text{UG}}\sqrt{2}\,i\sin \frac{V + h} {f} \left (A_{\mu \nu }^{L} - A_{\mu \nu }^{R}\right )\,. {}\end{array}$$

(2.192)