Peskin-Takeuchi parameter
Encyclopedia
In particle physics
, the Peskin–Takeuchi parameters are a set of three measurable quantities, called S, T, and U, that parameterize potential new physics
contributions to electroweak radiative corrections
. They are named after physicist
s Michael Peskin
and Tatsu Takeuchi, who proposed the parameterization in 1990; proposals from two other groups (see References below) came almost simultaneously.
The Peskin–Takeuchi parameters are defined so that they are all equal to zero at a reference point in the Standard Model
, with a particular value chosen for the (as yet unmeasured) Higgs boson
mass. The parameters are then extracted from a global fit to the high-precision electroweak data from particle collider experiments (mostly the Z pole data from the CERN
LEP collider) and atomic parity violation.
The measured values of the Peskin–Takeuchi parameters agree with the Standard Model. They can then be used to constrain models of new physics beyond the Standard Model. The Peskin–Takeuchi parameters are only sensitive to new physics that contributes to the oblique correction
s, i.e., the vacuum polarization
corrections to four-fermion
scattering processes.
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,
where
denotes the derivative of the vacuum polarization function with respect to q2. The constant pieces of 
and 
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
, 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:
where sw and cw are the sine and cosine of the weak mixing angle, respectively. The definitions are carefully chosen so that
Particle physics
Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...
, the Peskin–Takeuchi parameters are a set of three measurable quantities, called S, T, and U, that parameterize potential new physics
Beyond the Standard Model
Physics beyond the Standard Model refers to the theoretical developments needed to explain the deficiencies of the Standard Model, such as the origin of mass, the strong CP problem, neutrino oscillations, matter–antimatter asymmetry, and the nature of dark matter and dark energy...
contributions to electroweak radiative corrections
Renormalization
In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
. They are named after physicist
Physicist
A physicist is a scientist who studies or practices physics. Physicists study a wide range of physical phenomena in many branches of physics spanning all length scales: from sub-atomic particles of which all ordinary matter is made to the behavior of the material Universe as a whole...
s Michael Peskin
Michael Peskin
Michael Peskin is an American theoretical physicist. He was an undergraduate at Harvard University and obtained his Ph.D. in 1978 at Cornell University studying under Kenneth Wilson. He was a Junior Fellow at the Harvard Society of Fellows from 1977–1980.He is currently a professor in the theory...
and Tatsu Takeuchi, who proposed the parameterization in 1990; proposals from two other groups (see References below) came almost simultaneously.
The Peskin–Takeuchi parameters are defined so that they are all equal to zero at a reference point in the Standard Model
Standard Model
The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...
, with a particular value chosen for the (as yet unmeasured) Higgs boson
Higgs boson
The Higgs boson is a hypothetical massive elementary particle that is predicted to exist by the Standard Model of particle physics. Its existence is postulated as a means of resolving inconsistencies in the Standard Model...
mass. The parameters are then extracted from a global fit to the high-precision electroweak data from particle collider experiments (mostly the Z pole data from the CERN
CERN
The European Organization for Nuclear Research , known as CERN , is an international organization whose purpose is to operate the world's largest particle physics laboratory, which is situated in the northwest suburbs of Geneva on the Franco–Swiss border...
LEP collider) and atomic parity violation.
The measured values of the Peskin–Takeuchi parameters agree with the Standard Model. They can then be used to constrain models of new physics beyond the Standard Model. The Peskin–Takeuchi parameters are only sensitive to new physics that contributes to the oblique correction
Oblique correction
In particle physics, an oblique correction refers to a particular type of radiative correction to the electroweak sector of the Standard Model. Oblique corrections are defined in four-fermion scattering processes, at the CERN LEP collider. There are three classes of radiative corrections to...
s, i.e., the vacuum polarization
Vacuum polarization
In quantum field theory, and specifically quantum electrodynamics, vacuum polarization describes a process in which a background electromagnetic field produces virtual electron–positron pairs that change the distribution of charges and currents that generated the original electromagnetic...
corrections to four-fermion
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....
scattering processes.
Definitions
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 photonPhotonIn physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
, 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 fermionFermionIn particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....
s are suppressed, and hence only oblique corrections need to be considered. In particular, the framework assumes that the nonoblique correctionNonoblique correctionIn particle physics, a nonoblique correction, also called a direct correction, refers to a particular type of radiative correction to the electroweak sector of the Standard Model. Nonoblique corrections are defined in four-fermion scattering processes at the CERN LEP collider...
s (i.e., vertex correctionsVertex functionIn quantum electrodynamics, the vertex function describes the coupling between a photon and an electron beyond the leading order of perturbation theory...
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,
where
denotes the derivative of the vacuum polarization function with respect to q2. The constant pieces of and are zero because of the renormalization conditionsRenormalization
In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
. 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
, 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 ModelBeyond the Standard Model
Physics beyond the Standard Model refers to the theoretical developments needed to explain the deficiencies of the Standard Model, such as the origin of mass, the strong CP problem, neutrino oscillations, matter–antimatter asymmetry, and the nature of dark matter and dark energy...
(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:
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 sectorHiggs sectorIn particle physics, the Higgs sector is the collection of quantum fields and/or particles that are responsible for the Higgs mechanism i.e. for the spontaneous symmetry breaking. The word "sector" refers to a subgroup of the total set of fields and particles...
consists of electroweak doublet(s) H, the effective action term
only contributes to T and not to S or U. This term violates custodial symmetryCustodial symmetryIn particle physics with one or more electroweak Higgs doublets in the Higgs sector, the effective action term \left|H^\dagger D_\mu H\right|^2/\Lambda^2 which generically arises whenever we have new physics beyond the Standard Model at the scale Λ contributes to the Peskin-Takeuchi T parameter....
. - Assuming that the Higgs sectorHiggs sectorIn particle physics, the Higgs sector is the collection of quantum fields and/or particles that are responsible for the Higgs mechanism i.e. for the spontaneous symmetry breaking. The word "sector" refers to a subgroup of the total set of fields and particles...
consists of electroweak doublet(s) H, the effective action term
only contributes to S and not to T or U. (The contribution of 
can be absorbed into g1 and the contribution of 
can be absorbed into g2). - Assuming that the Higgs sectorHiggs sectorIn particle physics, the Higgs sector is the collection of quantum fields and/or particles that are responsible for the Higgs mechanism i.e. for the spontaneous symmetry breaking. The word "sector" refers to a subgroup of the total set of fields and particles...
consists of electroweak doublet(s) H, the effective action term
contributes to U.
Uses
- The S parameter measures the difference between the number of left-handed fermions and the number of right-handed fermions that carry weak isospin. It tightly constrains the allowable number of new fourth-generationGeneration (particle physics)In particle physics, a generation is a division of the elementary particles. Between generations, particles differ by their quantum number and mass, but their interactions are identical....
chiral fermions. This is a problem for theories like the simplest version of technicolor (physics)Technicolor (physics)Technicolor theories are models of physics beyond the standard model that address electroweak symmetry breaking, the mechanism through which elementary particles acquire masses...
that contain a large number of extra fermion doublets. - The T parameter measures isospinIsospinIn physics, and specifically, particle physics, isospin is a quantum number related to the strong interaction. This term was derived from isotopic spin, but the term is confusing as two isotopes of a nucleus have different numbers of nucleons; in contrast, rotations of isospin maintain the number...
violation, since it is sensitive to the difference between the loop corrections to the Z boson vacuum polarization function and the W boson vacuum polarization function. An example of isospin violation is the large mass splitting between the top quarkTop quarkThe top quark, also known as the t quark or truth quark, is an elementary particle and a fundamental constituent of matter. Like all quarks, the top quark is an elementary fermion with spin-, and experiences all four fundamental interactions: gravitation, electromagnetism, weak interactions, and...
and the bottom quarkBottom quarkThe bottom quark, also known as the beauty quark, is a third-generation quark with a charge of − e. Although all quarks are described in a similar way by the quantum chromodynamics, the bottom quark's large bare mass , combined with low values of the CKM matrix elements Vub and Vcb, gives it a...
, which are isospin partners to each other and in the limit of isospin symmetry would have equal mass. - The S and T parameters are both affected by varying the mass of the Higgs bosonHiggs bosonThe Higgs boson is a hypothetical massive elementary particle that is predicted to exist by the Standard Model of particle physics. Its existence is postulated as a means of resolving inconsistencies in the Standard Model...
(recall that the zero point of S and T is defined relative to a reference value of the Standard Model Higgs mass). The Higgs boson of the Standard Model has not been discovered, and experiments at the CERNCERNThe European Organization for Nuclear Research , known as CERN , is an international organization whose purpose is to operate the world's largest particle physics laboratory, which is situated in the northwest suburbs of Geneva on the Franco–Swiss border...
LEP collider have set a lower bound of 114 GeVGEVGEV or GeV may stand for:*GeV or gigaelectronvolt, a unit of energy equal to billion electron volts*GEV or Grid Enabled Vehicle that is fully or partially powered by the electric grid, see plug-in electric vehicle...
on its mass. However, if we assume that the Standard Model is correct, a best fit value of the Higgs mass can be extracted from the S, T fit. The best fit is near the current lower bound, and the 95% confidence level upper bound is around 200 GeV.http://lepewwg.web.cern.ch/LEPEWWG/ If there is new physics, the corrections to S and T from the new physics can compensate for the effects of a heavier Higgs boson, relaxing this constraint. - The U parameter tends not to be very useful in practice, because the contributions to U from most new physics models are very small. This is because U actually parameterizes the coefficient of a dimension-eight operatorClassical scaling dimensionIn theoretical physics, namely quantum field theory, the classical scaling dimension of an operator O is the power of mass of an operator determined by dimensional analysis from the Lagrangian...
, while S and T can be represented as dimension-six operators.