REVIEW
The Electroweak Chiral Lagrangian and New Precision Measurements
High Energy Physics - Phenomenology · Submitted 8 Apr 1993 · 19 references
Pith's verdict: A solid, standard reference: the operator list and the coefficient-to-observable dictionary are the durable parts, and the one-loop numbers are illustrative rather than predictive.
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Referee report
The paper
The paper claims
A complete operator list for a Higgsless electroweak theory, plus what a heavy-fermion doublet would make of it
The reading
What carries the argument
The carrying object is the unitary matrix field $U(x)$ transforming as a $(2,2)$ under $SU(2)_L times SU(2)_C$, out of which three building blocks are assembled: the field $T = U tau_3 U^dagger$, the vector $V_mu = (D_mu U) U^dagger$, and the gauge field strength $W_{mu nu}$. Every operator in the list is a trace structure built from these, with the equations of motion and the identities $D_mu T = [V_mu, T]$ and $D_mu V_nu - D_nu V_mu = i g W_{mu nu} - i g' B_{mu nu} T + [V_mu, V_nu]$ used to remove redundant terms. The second piece of machinery is the dictionary: by going to unitary gauge and reading off the two-point and three-point functions, the coefficients map onto $S$, $T$, $U$ and onto the general triple-gauge-vertex parameterization, so that a single set of numbers connects the effective theory to both oblique and direct measurements.
Extended reading notes
Core claim
The paper establishes three linked results. First, the complete set of dimension-four operators consistent with $SU(2)_L times U(1)_Y$ is given: eleven CP-conserving terms, of which the last, $L_{11} = alpha_{11} g epsilon^{mu nu rho lambda} Tr(T V_mu) Tr(V_nu W_{rho lambda})$, is new and corresponds to a CP-conserving but $C$- and $P$-violating piece of the triple gauge vertex; and, in the appendix, the CP-violating set, of which five had not been listed before. Second, the coefficients are tied to observables through $S = -16 pi alpha_1$, $alpha T = 2 g^2 beta_1$, $U = -16 pi alpha_8$, and through Eq. (13), which expresses $g^Z_1 - 1$, $kappa^Z - 1$, $kappa^gamma - 1$, $g^Z_5$ and $g^gamma_5$ in terms of $alpha_1, alpha_2, alpha_3, alpha_8, alpha_9, alpha_{11}$ and $beta_1$. Third, in a model of one heavy fermion doublet $U$, $D$ with technicolor interactions switched off, the one-loop coefficients are all finite constants of order $N/96pi^2$ — $alpha_1 = alpha_2 = -N/96pi^2$, $alpha_3 = -(N/96pi^2)(1 - (2/5)delta^2)$, $alpha_8$ and $alpha_9$ of order $delta^2$, $alpha_{11}$ linear in the mass splitting $delta$ — and the resulting triple gauge parameters are of order $10^{-3}$ for $N=4$ and $m_U simeq m_D simeq 1.5$ TeV.
Load-bearing premise
The numerical estimates of the triple gauge couplings assume that a one-loop calculation in a single heavy-fermion doublet with the strong technicolor interactions switched off is a reliable guide to the real strongly coupled theory, an assumption the authors state explicitly and do not test.
Editorial extensions
If this is right
- Any dimension-four operator consistent with $SU(2)_L times U(1)_Y$ that is absent from the CP-conserving list, or from the CP-violating list in the appendix, would falsify the completeness claim; the authors offer the trace-structure counting as the proof that nothing is missing.
- A measured nonzero $g^Z_5$ would signal the new operator $L_{11}$, since that parameter receives contributions only from the three-point function and vanishes in any weak-isospin symmetric theory.
- Because $U(1)_{em}$ gauge invariance forces $g^gamma_5 = 0$ and $g^gamma_1 - 1 = 0$, any experimental deviation in those two quantities would not be attributable to this class of heavy physics.
- In the one-doublet model the triple gauge deviations scale with the number of fermion doublets, so a multi-doublet technicolor model could push them to $10^{-2}$, within reach of a 500 GeV $e^+e^-$ collider.
- The linear dependence of $alpha_{11}$ and $g^Z_5$ on the $U$-$D$ mass splitting, versus the quadratic dependence of $beta_1$, $alpha_8$ and $alpha_9$, gives a diagnostic handle: a small splitting suppresses the oblique corrections while leaving a $g^Z_5$ signal intact.
Reading between the lines
- The completeness argument rests on a counting of trace structures rather than on a constructive proof that no other invariant can be formed, so an independent algebraic check — enumerating invariants by computer, for instance — would be the natural way to test the claim.
- The paper's own numbers suggest a sharp experimental dividing line: LEP-II cannot see these effects, a 500 GeV linear collider can, which makes the triple gauge couplings a stronger motivation for the higher-energy machine than the oblique parameters alone.
- Since the CP-violating operators are listed but never computed, the same one-doublet model could be run for them, and the resulting electric dipole moment estimates would give a second, complementary probe of the same heavy sector.
- The heavy-fermion estimate is perturbative while the motivating theory is strongly coupled, so a lattice calculation of the same coefficients in a technicolor theory would be the decisive check on whether the $10^{-3}$ numbers survive.
Original abstract
A revised and complete list of the electroweak chiral lagrangian operators up to dimension-four is provided. The connection of these operators to the $S$, $T$ and $U$ parameters and the parameters describing the triple gauge boson vertices $WW\gamma$ and $WWZ$ is made, and the size of these parameters from new heavy physics is estimated using a one flavor-doublet model of heavy fermions. The coefficients of the chiral lagrangian operators are also computed in this model.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
A revised and complete list of the electroweak chiral lagrangian operators up to dimension-four is provided. The connection of these operators to the $S$, $T$ and $U$ parameters and the parameters describing the triple gauge boson vertices $WW\gamma$ and $WWZ$ is made, and the size of these parameters from new heavy physics is estimated using a one flavor-doublet model of heavy fermions. The coefficients of the chiral lagrangian operators are also computed in this model.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | A revised and complete list of the electroweak chiral lagrangian operators up to dimension-four is provided. | contribution 0.96 | 0.90 | 0.31 | asserted 1.00 |
| s1 | The connection of these operators to the $S$, $T$ and $U$ parameters and the parameters describing the triple gauge boson vertices $WW\gamma$ and $WWZ$ is made, and the size of these parameters from new heavy physics is estimated using a one flavor-doublet model of heavy fermions. | contribution 0.81 | 0.86 | 0.24 | asserted 0.99 |
| s2 | The coefficients of the chiral lagrangian operators are also computed in this model. | result 0.70 | 0.91 | 0.11 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v1
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- references and citing sentences
- relations to other papers
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Figures
Showing 1 of the paper's 2 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:hep-ph/9304240 by the authors.
References
How the literature uses it
36 arXiv papers cite this in 62 sentences
Papers citing it · 41 sentences without a confident label
Uses its method or tool 3 papers
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We can make one-to-one correspondences between the chiral coefficients $_1$, $_8$, $_0$ and the oblique parameters $S$, $T$, $U$ as in Ref. [ref] [equation] Here the magnitudes of $_i(m_Z)$ are fixed by $S$, $T$, and $U$ at $μ=m_Z$, where $_EM^=1/137.36$ is the fine structure constant to conform with the $T$ parameter definition of by Peskin and Takeuchi [ref] .
Constraints on the electroweak chiral Lagrangian from the precision data · Constraints from the two-point vertices · 0.81
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At this stage, one integrates out the heavy $W'$ fields and matches the theory to the electroweak chiral lagrangian [ref] .
Z to b bbar and Chiral Currents in Higgsless Models · RGE Analysis · 0.93
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Here we will follow the formalism in Ref. [ref] where the spontaneous symmetry breaking pattern is assumed to be,[See Ref. [ref] for an alternative formulation with the spontaneous symmetry breaking pattern, $SU(2)_L×U(1)_Y→U(1)_em$.] [equation] $SU(2)_R$ and $SU(2)_C$ being approximate symmetries.
Towards the HEFT-hedron: the complete set of positivity constraints at NLO · HEFT parametrization · 0.89
Compares with it 2 papers
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Since $ρ-1=α T=2β_1$, we finally obtain [equation] Eqs.[ref] and [ref] exactly coincide with the Feynman-diagram calculated results in previous works [ref] .
A nondiagrammatic calculation of the Rho parameter from heavy fermions · Introduction · 1.00
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This translates into the estimate $|Δκ|∼6.2×10^-3$ and $|ΔQ|∼2×10^-5$, which agrees with the estimate of the size of the $W$ form factors that was obtained within the framework of technicolor theories in Ref. [ref] , where the authors considered the contribution to the $WWγ$ vertex from a heavy fermion $SU(N)$ doublet.
On-shell $WW\gamma$ vertex in the T-Parity and non T-Parity Littlest Higgs model · Estimate of the contribution of the ultraviolet completion of the LHM to the $W$ form factors · 0.98
Cites it as background 14 papers
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In model-independent studies the $ρ$ parameter is related to the $T$ parameter in the Peskin-Takeuchi's formalism [ref] , and is also related to the $β_1$ parameter in the electroweak chiral Lagrangian [ref] .
A nondiagrammatic calculation of the Rho parameter from heavy fermions · Introduction · 0.97
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The electroweak chiral Lagrangian is a general low-energy description for electroweak symmetry breaking patterns [ref] , especially for those strong dynamical electroweak symmetry breaking mechanisms [ref] .
New Terms for the Compact Form of Electroweak Chiral Lagrangian · Introduction · 1.00
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A first analysis of the NLO counterterms for the gauge boson and Goldstone sectors was given in [ref] , which was followed by a more systematic treatment of the operator basis in the CP-conserving [ref] and CP-violating [ref] sectors.
Effective Theory of a Dynamically Broken Electroweak Standard Model at NLO · Introduction · 0.83
Discussion
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