EW Precision Tests

Abstract

The composite Higgs dynamics gives rise to a rich set of new-physics effects that can be used to probe this scenario through a comparison with the experimental data. One of the most distinctive phenomena is the presence of composite resonances around the TeV scale, which can be straightforwardly tested in collider experiments. A second important signature is the peculiar pattern of distortions of the Higgs couplings, which constitutes a direct manifestation of the non-linear Nambu–Goldstone structure. In addition to these features, the composite dynamics gives also rise to many indirect effects. These noticeably include a set of corrections to the Electro-Weak (EW) observables that describe the physics of the light Standard Model (SM) fermions and of the gauge fields. The importance of these observables comes from the fact that they can be measured in high-precision experiments and thus can be used to test even tiny corrections coming from a new-physics dynamics. Obvious examples are the Z-pole observables measured at the LEP experiment and the properties of the bottom quark easily accessible at b-factories. All these measurements agree with the SM and have been extensively used to set stringent constraints on beyond the Standard Model (BSM) scenarios. They constitute the so-called EW Precision Tests (EWPT) of the SM.

Appendix

7.1.1 The Custodial Symmetries

In the analysis of the constraints from EWPT we encountered two “custodial” symmetries that are of fundamental importance in keeping under control the corrections to the EW parameters. The first symmetry is the standard custodial group SO(3) _c , which forbids corrections to the $\hat{T}$ parameter. The second one is the discrete P _LR invariance which protects the couplings of the Z boson to the SM fields and, in particular, to the bottom quark. These symmetries are also responsible for a number of peculiar properties and selection rules on the mass-spectrum and on the couplings of the composite sector resonances, some of which we encountered in the main text. The way in which these protections work is explained below.

7.1.1.1 The SO(3) _c Symmetry

We start from assuming an SO(4) global symmetry of the sector responsible for EWSB, under which the four real Higgs field components $\Pi ^{i}$ form a fourplet or, equivalently, a (2, 2) pseudo-real matrix (see Eq. (2.144The SO(4) Algebra))

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

(7.68)

Under $\text{SO}(4) \simeq \text{SU}(2)_{L} \times \text{SU}(2)_{R}$, $\Sigma$ transforms as

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

(7.69)

In composite Higgs models, this SO(4) symmetry is part of (or coincides with, in the minimal coset SO(5)∕SO(4)) the unbroken subgroup $\mathcal{H}$ of the $\mathcal{G}/\mathcal{H}$ coset and it is by assumption an exact symmetry of the composite sector. In the SM, SO(4) is instead an accidental symmetry of the Higgs doublet Lagrangian, if considered in isolation. Both in composite Higgs and in the SM, the SO(4) symmetry is broken by the gauge fields and fermions couplings.

The Higgs VEV, $\langle \Pi ^{i}\rangle = v\,\delta _{i4}$, breaks SO(4) spontaneously to an SO(3) subgroup, realizing the symmetry breaking pattern

$$\displaystyle{ \text{SO}(4) \rightarrow \text{SO}(3)\,. }$$

(7.70)

The unbroken SO(3) is what we call the custodial SO(3) _c . Its action can be either viewed as rotations of the first three $\vec{\Pi }$ vector components (with the physical Higgs in $\Pi ^{4}$ being a scalar) or, equivalently, as the vector subgroup of $\text{SU}(2)_{L} \times \text{SU}(2)_{R}$, SU(2) _V , defined by equal left and right transformations $g_{L} = g_{R} = g_{V }$. Indeed the Higgs VEV in the matrix notation

$$\displaystyle{ \langle \Sigma \rangle = \frac{v} {\sqrt{2}}\mathbb{1}_{2}\,, }$$

(7.71)

is invariant under the vector transformations.

Both in the SM and in composite Higgs the $W_{\mu }^{\alpha }$ and B _μ fields weakly gauge the $\text{SU}(2)_{L} \times \text{U}(1)_{Y }$ subgroup of SO(4). Actually when dealing with fermions (see Sect. 2.4.2Higgs Couplings to Fermions) an extra unbroken U(1) _X group needs to be introduced, but this will play no role in what follows. The W ^α fields, which fully gauge SU(2) _L , preserve SO(4) provided we assign them to the (3, 1) representation of the group. The effect of the hypercharge gauging, which instead breaks SO(4), will be discussed later on.

The cancellation of the $\hat{T}$ parameter immediately follows from this symmetry structure. Indeed $\hat{T}$ is defined (see Eq. (7.3)) in terms of the amputated two-point W field correlators at zero transferred momentum, and thus it should correspond to a non-derivative mass-term operator in the effective action. However the only such term which is compatible with the unbroken SO(3) _c (and also happens to respect the full SO(4)) is¹²

$$\displaystyle{ \mathcal{L}_{\text{mass}} = \frac{g^{2}v^{2}} {8} W_{\mu }^{\alpha }W_{\alpha }^{\mu }\,. }$$

(7.72)

This term contributes in the same way to $\Pi _{W^{3}W^{3}}$ and to $\Pi _{W^{1}W^{1}}$, thus it does not contribute to $\hat{T}$, which is proportional to the difference between the two. The custodial SO(3) _c symmetry thus implies $\hat{T} = 0$.

A non-vanishing $\hat{T}$ would correspond to the presence, in the effective Lagrangian, of an operator of the form

$$\displaystyle{ \frac{v^{2}} {8} \mathcal{S}_{\alpha \beta }W_{\mu }^{\alpha }W^{\beta,\mu }\,, }$$

(7.73)

where $\mathcal{S}$ is a symmetric traceless (since the trace component does not contribute) tensor in the (5, 1) representation of $\text{SU}(2)_{L} \times \text{SU}(2)_{R}$. In terms of $\mathcal{S}$, $\hat{T}$ can be expressed as $\hat{T} = \mathcal{S}_{33} -\mathcal{S}_{11}$. In a perfectly invariant theory no parameter exists with non-trivial SO(4) transformation properties. A non-vanishing $\mathcal{S}$, and thus in turn a non-vanishing $\hat{T}$, can only be constructed in terms of spurions, whose presence signals the explicit breaking of the symmetry.

Both in the SM and in composite Higgs, explicit SO(4) breaking emerges from the hypercharge gauging and from the coupling to fermions. The contributions to $\hat{T}$ from the latter are described extensively in Sect. 7.1.3, here we briefly discuss the effects of the former breaking. The breaking appears because only one of the three SU(2) _R generators, the third one, is gauged by the hypercharge field B _μ , i.e. only one of the three SU(2) _R gauge sources W _R is a truly dynamical field while the other components are set to zero. This breaking corresponds to a spurion $\mathcal{G}^{{\prime}}$ in the (1, 3), which can be inserted in the relation between the W _R source and the physical field B, namely

$$\displaystyle{ W_{R,\mu }^{\alpha } = \mathcal{G}^{{\prime}}{}^{\alpha }B_{\mu }\,. }$$

(7.74)

By two powers of this spurions, plus four powers of the Higgs VEV, which transforms in the (2, 2), a non-vanishing $\mathcal{S}$ tensor can be constructed and a contribution to $\hat{T}$ is generated. Notice however that the lack of symmetry in the hypercharge gauging becomes visible only at the loop level because it is only in the presence of at least one B _μ field propagator that we can distinguish the case in which all the three W _R fields are dynamical, and the symmetry is preserved, from the one in which only B _μ is dynamical and the symmetry is broken. Therefore $\hat{T}$ remains zero at tree-level and Eq. (7.72) gets generalized in the only possible way compatible with the unbroken electromagnetic U(1) symmetry, namely

$$\displaystyle{ \mathcal{L}_{\text{mass}} = \frac{v^{2}} {8} \left [(g\,W^{1})^{2} + (g\,W^{2})^{2} + (g\,W^{3} - g^{{\prime}}B)^{2}\right ]\,. }$$

(7.75)

The term above, which just coincides with the habitual SM one, gives masses to the W and to the Z that obey the ρ = 1 relation.

7.1.1.2 The P _LR Symmetry

We now turn to the P _LR symmetry and describe how it can protect the coupling of the Z boson to fermions. The right starting point, even before introducing the P _LR symmetry itself, is to remind ourselves how the SM gauge fields are introduced in the theory. The $W_{\mu }^{\alpha }$ fields gauge the SU(2) _L group while B _μ gauges the U(1) _Y hypercharge generator, defined in Sect. 2.4.2Higgs Couplings to Fermions as the sum of $t_{R}^{3}$ in $\text{SU}(2)_{R}$ and the X charge of the U(1) _X group, namely

$$\displaystyle{ Y = t_{R}^{3} + X\,. }$$

(7.76)

The gauging is conveniently described, as we saw in Sect. 2.3.2Gauge Sources and Local Invariance, by introducing external sources associated to all the group generators and identifying part of them as dynamical fields only at a late stage of the calculation. We thus consider three SU(2) _L , three SU(2) _R and one U(1) _X sources, namely

$$\displaystyle\begin{array}{rcl} W_{L,\mu }& =& W_{L,\mu }^{\alpha }t_{ L}^{\alpha } = g\,W_{\mu }^{\alpha }t_{ L}^{\alpha }\,, \\ W_{R,\mu }& =& W_{R,\mu }^{\alpha }t_{ R}^{\alpha } = g^{{\prime}}B_{\mu }t_{ R}^{3}\,, \\ \mathcal{X}_{\mu }& =& g^{{\prime}}B_{\mu }\,, {}\end{array}$$

(7.77)

where $t_{L,R}^{\alpha }$ are the $\text{SU}(2)_{L} \times \text{SU}(2)_{R}$ generators, for which a normalized explicit representation is provided in Eq. (2.153The SO(4) Algebra). The physical value of the source fields, in terms of three W’s and B, is also reported in the equation above.

The full $\text{SU}(2)_{L} \times \text{SU}(2)_{R} \times \text{U}(1)_{X}$ global group can be formally promoted to a local symmetry by regarding the W _L , W _R and $\mathcal{X}$ sources as the gauge fields associate to the three semi-simple factors. However only the unbroken (or linearly-realized) subgroup $\text{SO}(3)_{c} \times \text{U}(1)_{X}$ will be relevant in what follows. This subgroup acts on the sources as

$$\displaystyle\begin{array}{rcl} W_{L,\mu }& \rightarrow & g_{V } \cdot \left (W_{L,\mu } + i\partial _{\mu }\right ) \cdot g_{V }^{\dag }\,, \\ W_{R,\mu }& \rightarrow & g_{V } \cdot \left (W_{R,\mu } + i\partial _{\mu }\right ) \cdot g_{V }^{\dag }\,, \\ \mathcal{X}_{\mu }& \rightarrow & \mathcal{X}_{\mu } + \partial _{\mu }\alpha _{X}\,, {}\end{array}$$

(7.78)

where α _X denotes the U(1) _X transformation parameter. What is peculiar in the expression above is that $W_{L}^{\mu }$ and $W_{R}^{\mu }$, in spite of being two distinct fields, both transform as if they were gauge connections associated to the SO(3) _c local group. These local symmetry transformations will be very effective in constraining the fermion couplings.

We now introduce P _LR , which is defined in section “Discrete Symmetries” in Appendix in Chap. 3Beyond the Sigma-Model as the discrete $\mathbb{Z}_{2}$ transformation that interchanges the SU(2) _L and SU(2) _R generators inside SO(4). Therefore it acts on the gauge sources as

$$\displaystyle{ W_{L,\mu }^{\alpha } \leftrightarrow W_{ R,\mu }^{\alpha }\,, }$$

(7.79)

while it leaves $\mathcal{X}_{\mu }$ invariant. On the Higgs fourplet, P _LR acts like a parity reflection of the first three components

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

(7.80)

therefore the Higgs VEV is even and P _LR survives as an unbroken symmetry after EWSB. In the presence of P _LR , the unbroken group $\text{SO}(3)_{c} \times \text{U}(1)_{X}$ is enlarged to $\text{O}(3)_{c} \times \text{U}(1)_{X}$.

Let us now analyze the implications of the gauge symmetry transformations in Eq. (7.78) on the zero-momentum, i.e. non-derivative, couplings of the gauge fields to the fermions. Since we are interested in the Z boson couplings, we restrict our attention to the interactions of the neutral sources $W_{L}^{3}$ and $W_{R}^{3}$ to one charge-eigenstate chiral fermion ψ. The only interactions allowed by the symmetries are

$$\displaystyle\begin{array}{rcl} & & i\,\overline{\psi }\gamma ^{\mu }\left (\partial _{\mu } - i\,t_{L}^{3}W_{ L,\mu }^{3} - i\,t_{ R}^{3}W_{ R,\mu }^{3} - i\,X\mathcal{X}_{\mu }\right )\psi \\ & & \quad +\; c\,(W_{L,\mu }^{3} - W_{ R,\mu }^{3})\overline{\psi }\gamma ^{\mu }\psi \,. {}\end{array}$$

(7.81)

In particular, the terms on the first line are enforced by the covariant derivative structure of the kinetic term and thus their coefficient is uniquely determined by the $t_{L,R}^{3}$ and X eigenvalue of ψ. The one on the second line is instead separately gauge-invariant given that the shift term in the local transformation of Eq. (7.78) cancels when we take the difference $W_{L} - W_{R}$. Therefore it has an arbitrary coefficient “c”. Not surprisingly, since they are rigidly fixed by the gauge symmetries, the interactions on the first line reproduce the SM W ₃ and B vertices, which after the weak angle rotation reduce to the standard photon and Z boson couplings. This is immediately verified by substituting the explicit value of the sources in Eq. (7.77), obtaining

$$\displaystyle{ t_{L}^{3}W_{ L,\mu }^{3} + t_{ R}^{3}W_{ R,\mu }^{3} + X\mathcal{X}_{\mu } = g\,t_{ L}^{3}W_{\mu }^{3} + g^{{\prime}}(t_{ R}^{3} + X)B_{\mu }\,, }$$

(7.82)

and noticing that $t_{R}^{3} + X = Y$ as in Eq. (7.76). Since the ones on the first line match with the SM, the only deviation comes from the term on the second line. Using the explicit value of the sources and performing the weak angle rotation to the Z and photon field basis one immediately finds that $W_{L}^{3} - W_{R}^{3} = g/\cos \theta _{w}Z$. No corrections to the zero-momentum photon coupling are thus generated, as an obvious consequence of the unbroken electromagnetic gauge group, while the Z boson interaction can be distorted by an amount

$$\displaystyle{ \delta g_{\psi } = c\,, }$$

(7.83)

having adopted the standard convention of normalizing the coupling deviation by the g∕cosθ _w factor (see Eq. (7.24)).

However $W_{L}^{3} - W_{R}^{3}$ is odd under P _LR , therefore if ψ is a P _LR eigenstate, no matter if even or odd, P _LR enforces c = 0 and no corrections to the Z couplings can occur. This result can be easily extended to multiple fermionic fields with definite P _LR parity. In particular the Z boson couplings to a set of eigenstates with the same P _LR parity are necessarily canonical and flavor-diagonal. Flavor-violating Z interactions, indeed, can only involve eigenstates with opposite parity.

When we consider the SM fermions, in order for the Z couplings to be protected, P _LR must be a symmetry not only of the composite sector (possibly an accidental one as we saw happening in some cases in the main text), but also of the partial compositeness mixing of the elementary SM field we are interested in. The left-handed bottom quark coupling to the Z is particularly relevant, let us thus discuss under which condition it benefits of the P _LR protection. We start from the case in which the q _L doublet mixes with a fundamental of SO(5), i.e. with the (2, 2) fourplet SO(4) representation inside the fundamental. The embedding of the doublet, provided in Eq. (2.116Higgs Couplings to Fermions), immediately shows that the Z coupling to b _L is protected in this case, compatibly with what we found in Sects. 3.2.2Order p Fermionic and 7.2. Indeed the b _L field only appears in the first and in the second component of the fourplet, which are both P _LR -odd according to Eq. (7.80). Therefore P _LR is preserved by the mixing, provided b _L is regarded as an odd field, and the b _L coupling is protected. We also see from the same equation that the t _L fields appear instead both in the third component of the multiplet, which is odd, and in the fourth one which is even. The t _L mixing thus breaks P _LR and no protection is present for its coupling with the Z. This was for the first mixing of the q _L , the one with the 5 _2∕3 multiplet that participate in the generation of the top quark mass. The situation is reversed for the mixing with the $\mathbf{5}_{-1/3}$, for which the embedding is reported in Eq. (2.127Higgs Couplings to Fermions). The t _L coupling is protected in that case, while the b _L one is not, given that the b _L mixes with both an even and an odd component. The modifications of the $Zb_{L}\overline{b}_{L}$ coupling is thus induced only by the second mixing parameter, $\lambda _{b_{L}}$, and not by $\lambda _{t_{L}}$. It should be rather obvious, at this point, that no protection is instead present when the q _L mixes with operators in the spinorial representation. Indeed the mixing occurs in this case with a (2, 1) representation of SO(4) and the P _LR symmetries interchanges the (2, 1) and the (1, 2) components of the spinorial. Since the mixing occurs with the former and not with the latter, it breaks P _LR and no protection is found. This is in accordance with the results of Sect. 3.2.2Order p Fermionic.