Beyond the Sigma-Model

Abstract

In the previous chapter we restricted our attention to a specific class of new physics effects, that we can classify as “non-linear $\sigma$-model effects”. These are modifications of the SM driven by the pNGB nature of the Higgs and the associated non-linear $\sigma$-model structure of the effective Lagrangian.

Appendix

3.1.1 Discrete Symmetries

Discrete symmetries are often useful in the study of composite Higgs theory. In the case of the minimal coset SO(5)∕SO(4) the relevant ones are space–time parity P, charge conjugation C (often combined with P to form CP) and a $\mathbb{Z}_{2}$ external automorphism of the algebra called P _LR .

Concerning parity, there is not much to say. It corresponds to ordinary spatial coordinate reflection under which the Goldstone boson Higgs transforms like a scalar and the gauge fields like vectors. Notice that the action of parity does not flip L and R SO(4) generators given that the L–R labeling does not refer here to fermion chirality. The CCWZ d and e symbols are vectors under parity and thus the $\mathcal{O}(p^{2})$ bosonic Lagrangian (2.184) is accidentally P-invariant even if parity is not imposed as a symmetry of the composite sector. Composite sector breaking of P can emerge at $\mathcal{O}(p^{4})$ through the operators discussed in Sect. 3.2.1. Parity is obviously broken by the elementary fermion couplings to the SM gauge fields, and the same holds for charge conjugation. Nevertheless, one might still want to impose them as symmetries of the composite sector. Even if we will not consider this possibility here, we mention that in this case the chiral fermionic operators $\mathcal{O}_{F}^{L,R}$ that realize partial compositeness (see Sect. 2.4.2) would be supplemented by their opposite chirality P- and C-conjugate counterparts, with the same scaling dimensions.

Charge conjugation is less trivial. It acts as $H \rightarrow H^{{\ast}}$ on the complex Higgs field, which in the real fourplet notation (2.28) means

$$\displaystyle{ \vec{\Pi }\; \rightarrow \;\mathcal{C}_{\mathbf{4}}\,\vec{\Pi }\,,\;\;\;\;\text{where}\;\;\mathcal{C}_{\mathbf{4}} = \text{diag}(-1,+1,-1,+1)\,. }$$

(3.130)

Notice that $\mathcal{C}_{\mathbf{4}}$ is a unit-determinant orthogonal matrix and as such it is a proper element of the unbroken group SO(4). Namely, it is

$$\displaystyle{ \mathcal{C} = e^{i\,\pi [T_{L}^{2}+T_{ R}^{2}] }\,, }$$

(3.131)

which, with the suitable generator matrices, can be expressed in any representation of the complete group SO(5) or of the unbroken subgroup SO(4). Given that the charge conjugation operation happens to act on the Goldstone fields like an element of the unbroken symmetry group, we can simply use the results of Sect. 2.3 to derive its action on the Goldstone matrix, which is

$$\displaystyle{ U_{\mathbf{r}}[\Pi ]\; \rightarrow \;\mathcal{C}_{\mathbf{r}}\,U_{\mathbf{r}}[\Pi ]\,\mathcal{C}_{\mathbf{r}}^{-1}\,, }$$

(3.132)

for a generic representation r.

On the SM gauge fields, C acts as $W^{\alpha } \rightarrow (-)^{1-\delta _{\alpha,2}}W^{\alpha }$ and $B_{\mu } \rightarrow -B_{\mu }$. This can be uplifted to the transformation rule

$$\displaystyle{ A_{\mu } = A_{\mu,\,A}T^{A} \rightarrow \;\mathcal{C}\cdot A_{\mu } \cdot \mathcal{C}^{-1}\,, }$$

(3.133)

which we assign to the whole set of dynamical and non-dynamical sources that gauge SO(5). Therefore C coincides with $\mathcal{C}\in \; SO(4)$ even when acting on the A _μ sources. This makes very easy to work out the transformation rules of the d and the e symbols. They are just a fourplet and an adjoint of SO(4) and thus

$$\displaystyle{ d_{\mu }^{i} \rightarrow (\mathcal{C}_{\mathbf{ 4}})_{\;j}^{i}d_{\mu }^{j}\,,\;\;\;\;\;e_{ L,R}^{\alpha }{}_{ \mu } \rightarrow (-)^{1-\delta _{\alpha }^{2} }e_{L,R}^{\alpha }{}_{ \mu }\,. }$$

(3.134)

Furthermore, the $\mathcal{C}$ operation is automatically a symmetry of our Lagrangian and thus charge conjugation invariance is guaranteed for all the composite sector operators involving d and e only. This includes the $\mathcal{O}(p^{2})$ Lagrangian and the $\mathcal{O}(p^{4})$ operators of Sect. 3.2.1. Notice that charge conjugation coincides with $\mathcal{C}$ only for the Goldstones and for the A _μ sources, not for the U(1) _X source X _μ . Given that we embed the hypercharge gauge boson B _μ in it, it must transform with a minus sign

$$\displaystyle{ X_{\mu } \rightarrow -X_{\mu }\,. }$$

(3.135)

This sign flip needs not to be a symmetry of the theory, therefore C can be broken, but only through terms with odd powers of the U(1) _X source. Given that the latter can only enter through its field-strength tensor $\partial _{[\mu }X_{\nu ]}$ because of local invariance, C breaking is postponed to high orders in the derivative expansion and it does not emerge at $\mathcal{O}(p^{4})$ in the bosonic sector.

Let us now turn to the fermionic sector. Given that both P and C are broken by the SM couplings it is not worth trying to define their actions on the fermionic source fields. This makes sense instead for the product of the two symmetries, CP, which is preserved by the SM matter quantum number assignment. We take, as is normally done in the SM, the CP action to be $\chi (\vec{x},t) \rightarrow \chi ^{c}(\vec{x},t) = -i\gamma ^{2}\gamma ^{0}\chi ^{{\ast}}(-\vec{x},t)$, where χ denotes any of the elementary SM fields. Given the definition, the action on the fermionic sources in the various representations introduced in Sect. 2.4.2 is immediately worked out. In all cases where the elementary SM fermions are embedded in a real SO(5) representation such as the 5, the 10 or the 14, it is easy to verify that CP acts as the global $\mathcal{C}$ rotation in the appropriate representation times the $\chi \rightarrow \chi ^{c}$ operation. For instance, in the case of the 5 we have

$$\displaystyle{ (F)_{I} \rightarrow (\mathcal{C}_{\mathbf{5}})_{I}^{\;\;J}(F^{c})_{ I}\,,\;\text{with}\;\;\;\mathcal{C}_{\mathbf{5}} = \text{diag}(-1,+1,-1,+1,+1)\,, }$$

(3.136)

where F denotes in general the top ($Q_{t_{L}}$ and T _R ) or bottom ($Q_{b_{L}}$ and B _R ) sector fermionic sources. In order to construct the CCWZ invariants, as explained in Sect. 2.4.2, it is useful to define dressed sources by acting with the inverse of the Goldstone matrix. Given how the latter transforms, as in Eq. (3.132), their CP transformation reads

$$\displaystyle{ F_{\mathbf{r}} \rightarrow \mathcal{C}_{\mathbf{r}}F_{\mathbf{r}}^{c}\,, }$$

(3.137)

where r is the SO(4) representation where the dressed source lives. We see that CP acts as the SO(4) transformation $\mathcal{C}$, under which all the operators are automatically invariant, times the “intrinsic” CP operation $\chi \rightarrow \chi ^{c}$. Since the same holds for the bosonic fields, with the only exception of X _μ , this makes very easy to establish the CP quantum numbers of the operators. For instance all the $\mathcal{O}(p^{0})$ operators in Sect. 2.4.2 are CP-even, once their coefficients are set to a real value to obtain a real mass, while some of those of $\mathcal{O}(p)$ in Sect. 3.2.2 break CP.

This was for real representations. When the elementary SM fermions are in the complex spinorial 4, instead, no imaginary phase is introduced in the embeddings, see Eq. (2.133) and therefore CP is just

$$\displaystyle{ (F)_{i} \rightarrow (F^{c})_{ i}\,, }$$

(3.138)

when acting on the sources. We actually need the transformation property of the dressed sources, obtained by acting with the inverse Goldstone matrix and splitting the fourplet into two doublets, namely

$$\displaystyle{ \left [\begin{array}{c} F^{\mathbf{2_{L}}} \\ F^{\mathbf{2_{R}}} \end{array} \right ] = U_{\mathbf{4}}^{-1}F\,. }$$

(3.139)

The Goldstone matrix transformation is immediately obtained from Eq. (3.131)

$$\displaystyle{ U_{\mathbf{4}}\; \rightarrow \;\hat{\mathcal{C}}_{\mathbf{4}}\cdot U_{\mathbf{4}}\cdot \hat{\mathcal{C}}_{\mathbf{4}}^{-1}\,,\;\text{where}\;\;\;\hat{\mathcal{C}}_{\mathbf{ 4}} = \left [\begin{array}{cc} i\sigma _{2} & 0 \\ 0 &i\sigma _{2} \end{array} \right ]\,. }$$

(3.140)

This can be rewritten, using the symplectic condition in Eq. (2.171), in a seemingly more complicated way

$$\displaystyle{ U_{\mathbf{4}}\; \rightarrow \;\hat{\mathcal{C}}_{\mathbf{4}}\cdot \Omega ^{-1}\cdot U_{\mathbf{ 4}}^{{\ast}}\cdot \Omega \cdot \,\hat{\mathcal{C}}_{\mathbf{ 4}}^{-1}\,,\;\text{with}\;\;\;\Omega \cdot \hat{\mathcal{C}}_{\mathbf{ 4}}^{-1} = \left [\begin{array}{cc} \mathbb{1}_{2} & 0 \\ 0 &- \mathbb{1}_{2} \end{array} \right ]\,, }$$

(3.141)

in terms of the complex conjugate of the Goldstone matrix. This becomes useful if we take into account that the physical fields are always embedded in the source F in either the first two components of the fourplet or in one of the two last (we denote by F _± the two cases) but never in both at the same time. Therefore the matrix $\Omega \cdot \hat{\mathcal{C}}_{\mathbf{4}}$ reduces to either an overall plus or minus sign when acting on them, leading eventually to the following result

$$\displaystyle{ \left [\begin{array}{c} F^{\mathbf{2_{L}}} \\ F^{\mathbf{2_{R}}} \end{array} \right ]\; \rightarrow \;\left [\begin{array}{c} \pm (F_{\pm }^{\mathbf{2_{L}}})^{c} \\ \mp (F_{\pm }^{\mathbf{2_{R}}})^{c} \end{array} \right ]\,. }$$

(3.142)

Notice that the “^c” operation acts now on the dressed sources and thus it entails taking the complex conjugate of U ₄ , which is where Eq. (3.141) comes into play.

We now discuss P _LR . As the name suggests, it is a $\mathbb{Z}_{2}$ transformation that interchanges L and R generators of the SO(4) group in SO(5). It corresponds to parity in the SO(5) space and it is represented, in the fundamental, by the matrix

$$\displaystyle{ P_{LR}^{\mathbf{5}} = \text{diag}(-1,-1,-1,+1,+1)\,. }$$

(3.143)

It acts on the generators as

$$\displaystyle\begin{array}{rcl} P_{LR}^{\mathbf{5}}T_{ L}^{\alpha }P_{ LR}^{\mathbf{5}}& =& T_{ R}^{\alpha }\,,\;\;\;\;P_{ LR}^{\mathbf{5}}T_{ R}^{\alpha }P_{ LR}^{\mathbf{5}} = T_{ L}^{\alpha }\,, \\ P_{LR}^{\mathbf{5}}\hat{T}^{i}P_{ LR}^{\mathbf{5}}& =& \left (P_{ LR}^{\mathbf{4}}\right )_{\;\; j}^{i}\hat{T}^{j}\,, {}\end{array}$$

(3.144)

where $P_{LR}^{\mathbf{4}}$ is

$$\displaystyle{ P_{LR}^{\mathbf{4}} = \text{diag}(-1,-1,-1,+1)\,. }$$

(3.145)

The P _LR operation belongs to O(4) ⊂  O(5), therefore it is not an element of the symmetry group and thus it is not automatically a symmetry of the composite sector. It could be imposed or more interestingly, as in the case encountered in Sect. 3.2.2, emerge as an accidental symmetry. Notice that the fourth real Higgs component, $\Pi ^{4}$, is P _LR -even. Therefore P _LR , provided it was a symmetry of some sector of the theory, will not be broken spontaneously by the Higgs VEV. On the Goldstone fourplet and on the Goldstone matrix in the fundamental, P _LR acts, respectively, as

$$\displaystyle{ \vec{\Pi } \rightarrow P_{LR}^{\mathbf{4}}\vec{\Pi }\,,\;\;\;\;\;\;U[\Pi ] \rightarrow P_{ LR}^{\mathbf{5}}U[\Pi ] \cdot P_{ LR}^{\mathbf{5}}\,, }$$

(3.146)

out of which the d and e symbols transformation rules (including the terms with the gauge sources, whose transformation rule is defined below) are found to be

$$\displaystyle{ d_{\mu,\,i} \rightarrow \left (P_{LR}^{\mathbf{4}}\right )_{ i}^{\;\;j}d_{\mu,\,j}\,,\;\;\;\;\;e_{\mu,\,\alpha }^{L} \leftrightarrow e_{\mu,\,\alpha }^{R}\,. }$$

(3.147)

By following this logic, P _LR can be defined also on the elementary gauge and fermionic source fields. Of course P _LR , differently from CP discussed above, is not a symmetry of the elementary sectors, therefore the SM field embedding into the sources will normally break it completely. Nevertheless we can assign transformation properties, for instance

$$\displaystyle{ A_{\mu } \rightarrow \; P_{LR} \cdot A_{\mu } \cdot P_{LR}^{-1}\,,\;\;\;\;\;F_{ I} \rightarrow (P_{LR}^{\mathbf{5}})_{ I}^{\;\;J}F_{ J}\,, }$$

(3.148)

to gauge and to fermions in the fundamental, respectively. The dressed fermion sources transformation rules, given how the Goldstone matrix transforms, are at this point completely obvious.

The action of P _LR on all the representations obtainable as tensor products of SO(5) fiveplets are immediately inferred from Eq. (3.143): it will be sufficient to act with the $P_{LR}^{\mathbf{5}}$ parity on each index. Some work is instead needed to obtain the representation on the spinorial, which turns out to be

$$\displaystyle{ \hat{P}_{LR}^{\mathbf{4}} = \left [\begin{array}{cc} 0 & \mathbb{1} \\ \mathbb{1}& 0 \end{array} \right ]\,, }$$

(3.149)

not to be confused with $P_{LR}^{\mathbf{4}}$ acting on the SO(4) fourplets. It is easy to verify, given the generators of the spinorial reported in Eq. (2.170), that $\hat{P}_{LR}^{\mathbf{4}}$ correctly acts on them like for the ones in the fundamental representation in Eq. (3.144). Not surprisingly, being a L–R interchange, P _LR flips the (2, 1) and (1, 2) components of the fourplet. On the Goldstone matrix we obviously have

$$\displaystyle{ U_{\mathbf{4}}[\Pi ] \rightarrow P_{LR}^{\mathbf{4}} \cdot U_{\mathbf{ 4}}[\Pi ] \cdot P_{LR}^{\mathbf{4}}\,, }$$

(3.150)

and thus the dressed sources in Eq. (3.139) simply get interchanged

$$\displaystyle{ F^{\mathbf{2_{L}} }\; \leftrightarrow \; F^{\mathbf{2_{R}} }\,. }$$

(3.151)