The Peskin–Takeuchi parameterization is based on the following assumptions about the nature of the new physics: • The electroweak
gauge group is given by SU(2)L x U(1)Y, and thus there are no additional electroweak gauge bosons beyond the
photon,
Z boson, and
W boson. In particular, this framework assumes there are no
Z' or
W' gauge bosons. If there are such particles, the
S, T, U parameters do not in general provide a complete parameterization of the new physics effects. • New physics couplings to light
fermions are suppressed, and hence only oblique corrections need to be considered. In particular, the framework assumes that the
nonoblique corrections (i.e.,
vertex corrections and box corrections) can be neglected. If this is not the case, then the process by which the
S, T, U parameters are extracted from the precision electroweak data is no longer valid, and they no longer provide a complete parameterization of the new physics effects. • The energy scale at which the new physics appears is large compared to the
electroweak scale. This assumption is inherent in defining
S, T, U independent of the momentum transfer in the process. With these assumptions, the oblique corrections can be parameterized in terms of four vacuum polarization functions: the self-energies of the photon, Z boson, and W boson, and the mixing between the photon and the Z boson induced by loop diagrams. Assumption number 3 above allows us to expand the vacuum polarization functions in powers of q2/M2, where M represents the heavy mass scale of the new interactions, and keep only the constant and linear terms in q2. We have, \Pi_{\gamma\gamma}(q^2) = q^2 \Pi_{\gamma\gamma}^{\prime}(0) + ... \Pi_{Z \gamma}(q^2) = q^2 \Pi_{Z \gamma}^{\prime}(0) + ... \Pi_{ZZ}(q^2) = \Pi_{ZZ}(0) + q^2 \Pi_{ZZ}^{\prime}(0) + ... \Pi_{WW}(q^2) = \Pi_{WW}(0) + q^2 \Pi_{WW}^{\prime}(0) + ... where \Pi^{\prime} denotes the derivative of the vacuum polarization function with respect to q2. The constant pieces of \Pi_{\gamma\gamma} and \Pi_{Z \gamma} are zero because of the
renormalization conditions. We thus have six parameters to deal with. Three of these may be absorbed into the renormalization of the three input parameters of the electroweak theory, which are usually chosen to be the
fine structure constant \alpha, as determined from
quantum electrodynamic measurements (there is a significant running of α between the scale of the mass of the electron and the electroweak scale and this needs to be corrected for), the
Fermi coupling constant GF, as determined from the
muon decay which measures the weak current coupling strength at close to zero
momentum transfer, and the Z boson mass MZ, leaving three left over which are measurable. This is because we are not able to determine which contribution comes from the Standard Model proper and which contribution comes from physics
beyond the Standard Model (BSM) when measuring these three parameters. To us, the low energy processes could have equally well come from a pure Standard Model with redefined values of e, GF and MZ. These remaining three are the Peskin–Takeuchi parameters
S, T and
U, and are defined as: \alpha S = 4 s_w^2 c_w^2 \left[ \Pi_{ZZ}^{\prime}(0) - \frac{c_w^2 - s_w^2}{s_w c_w} \Pi_{Z \gamma}^{\prime}(0) - \Pi_{\gamma \gamma}^{\prime}(0) \right] \alpha T = \frac{\Pi_{WW}(0)}{M_W^2} - \frac{\Pi_{ZZ}(0)}{M_Z^2} \alpha U = 4 s_w^2 \left[ \Pi_{WW}^{\prime}(0) - c_w^2 \Pi_{ZZ}^{\prime}(0) - 2 s_w c_w \Pi_{Z \gamma}^{\prime}(0) - s_w^2 \Pi_{\gamma\gamma}^{\prime}(0) \right] where sw and cw are the sine and cosine of the
weak mixing angle, respectively. The definitions are carefully chosen so that • Any BSM correction which is indistinguishable from a redefinition of e, GF and MZ (or equivalently, g1, g2 and ν) in the Standard Model proper at the
tree level does not contribute to S, T or U. • Assuming that the
Higgs sector consists of electroweak doublet(s) H, the
effective action term \left|H^\dagger D_\mu H\right|^2/\Lambda^2 only contributes to T and not to S or U. This term violates
custodial symmetry. • Assuming that the
Higgs sector consists of electroweak doublet(s) H, the effective action term H^\dagger W^{\mu\nu}B_{\mu\nu}H/\Lambda^2 only contributes to S and not to T or U. (The contribution of H^\dagger B^{\mu\nu}B_{\mu\nu}H/\Lambda^2 can be absorbed into g1 and the contribution of H^\dagger W^{\mu\nu}W_{\mu\nu}H/\Lambda^2 can be absorbed into g2). • Assuming that the Higgs sector consists of electroweak doublet(s) H, the effective action term \left(H^\dagger W^{\mu\nu}H\right)\left(H^\dagger W_{\mu\nu}H\right)/\Lambda^4 contributes to U. ==Uses==