REVIEW
The Electroweak Chiral Lagrangian for the Standard Model with a Heavy Higgs
High Energy Physics - Phenomenology · Submitted 14 Aug 1993 · 30 references
Pith's verdict: A careful one-loop matching paper whose real content is the finite constants, and those survive the scheme worry because the physical combinations that use them are cross-checked.
T0 · AI review
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Referee report
The paper
The paper claims
One-loop matching fixes the finite, non-logarithmic chiral couplings that a heavy Higgs leaves behind in electroweak gauge Green's functions.
The reading
What carries the argument
The load-bearing device is the matching condition $Gamma_{SM}^R(mu) = Gamma_{EChL}^R(mu)$ for $mu$ less than or equal to $M_H$, applied to the complete basis of $SU(2)_L$ times $U(1)_Y$ invariant, CP-conserving operators up to dimension four. The basis separates the universal non-linear $\sigma$-model Lagrangian from the chiral operators; only the first six are needed as counterterms, but all are retained to parametrize finite heavy-Higgs effects. The calculation uses dimensional regularization, a generic $R_xi$ gauge, and large-mass techniques to take $M_H$ large inside the renormalized one-light-particle-irreducible integrals. Solving the matching equations fixes the chiral coefficients and the differences between the Standard Model and effective-theory counterterms.
Extended reading notes
Core claim
The central claim is that the heavy-Higgs limit of the Standard Model is not just logarithmic: after matching one-loop one-light-particle-irreducible gauge Green's functions, the electroweak chiral Lagrangian acquires definite finite coefficients. In the modified minimal subtraction scheme the nonzero ones are $a_0 = (g'^2/16pi^2)(3/8)(5/6 - log(M_H^2/mu^2))$, $a_1 = (1/16pi^2)(1/12)(5/6 - log(M_H^2/mu^2))$, $a_2 = (1/16pi^2)(1/24)(17/6 - log(M_H^2/mu^2))$, $a_3 = -(1/16pi^2)(1/24)(17/6 - log(M_H^2/mu^2))$, and $a_{11} = -(1/16pi^2)(1/24)$, with $a_8 = a_9 = a_{12} = a_{13} = 0$. The paper presents this as the complete one-loop non-decoupling Higgs contribution to the two- and three-point gauge functions, with the leading logarithm and the next-to-leading constant both included.
Load-bearing premise
The matching assumes that, for momenta well below $M_H$, the difference between the Standard Model and chiral-Lagrangian Green's functions is analytic and can be replaced by a polynomial of local chiral operators, with the chiral-operator loops themselves negligible at the order computed.
Editorial extensions
If this is right
- The heavy-Higgs values give a benchmark for the chiral coefficients; a technicolor or other strong symmetry-breaking sector would generally produce different finite constants, so precision electroweak data can discriminate among underlying theories.
- The paper derives the Higgs contributions to the oblique parameters: $epsilon_1 = alpha T = Delta rho = (g'^2/16pi^2)(15/24 - (3/4) log(M_H^2/M_Z^2))$, $epsilon_2 = 0$, and $epsilon_3 = (alpha/4s_W^2) S = (g^2/16pi^2)(-5/72 + (1/12) log(M_H^2/M_Z^2))$ at this order.
- It predicts the scale-independent combination $epsilon_1 + 9(g'^2/g^2)epsilon_3 = 0$ from the Higgs sector at this order, so a measured nonzero value would point to physics beyond the Standard Model Higgs.
- It gives a scale-independent trilinear coupling prediction $kappa_gamma - 1 = (1/16pi^2)(1/6)$ plus non-Higgs loops, measurable in $gamma gamma$ to $W^+_L W^-_L$ at a photon collider.
- The counterterm relations for $Delta Z_W$, $Delta Z_B$, $Delta xi_W$, and $Delta xi_B$ fix how the effective theory's renormalization must be chosen once the Standard Model prescription is fixed.
Reading between the lines
- The same matching program can be turned around: if future precision data determine the finite chiral constants, their pattern can test whether the symmetry-breaking sector is a heavy scalar, a strongly coupled sector, or something else, provided the non-Higgs loop contributions are subtracted.
- The vanishing Higgs contribution to the scale-invariant combination $epsilon_1 + 9(g'^2/g^2)epsilon_3$ is a clean null test; a nonzero measurement would be evidence for additional heavy states rather than a minimal Higgs.
- Because the finite constants depend on the matching scale only through logarithmic running, a measurement at one scale can be run to another and compared with the heavy-Higgs benchmark, making the coefficients useful in global fits beyond leading-log approximations.
- The method extends to four-point gauge Green's functions, which the paper explicitly leaves for future work; doing so would fix the remaining dimension-four coefficients and complete the heavy-Higgs fingerprint.
Original abstract
The most general chiral Lagrangian for electroweak interactions with the complete set of $SU(2)_L\times U(1)_Y$ invariant operators up to dimension four is considered. The two-point and three-point functions with external gauge fields are derived from this effective chiral Lagrangian to one-loop order in a generic $R_\xi$-gauge. The same set of Green's functions are paralelly studied in the renormalizable standard model to one-loop order, in a $R_\xi$-gauge and in the large Higgs mass limit. An appropriate set of matching conditions connecting the Green's functions of the two theories allows us to derive, systematically, the values of the chiral Lagrangian coefficients corresponding to the large Higgs mass limit of the standard model. These chiral parameters represent the non-decoupling effects of a heavy Higgs particle and incorporate both the leading logarithmic dependence on $\mh$ and the next to leading constant contributions. Some phenomenological implications are also discussed.
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
The most general chiral Lagrangian for electroweak interactions with the complete set of $SU(2)_L\times U(1)_Y$ invariant operators up to dimension four is considered. The two-point and three-point functions with external gauge fields are derived from this effective chiral Lagrangian to one-loop order in a generic $R_\xi$-gauge. The same set of Green's functions are paralelly studied in the renormalizable standard model to one-loop order, in a $R_\xi$-gauge and in the large Higgs mass limit. An appropriate set of matching conditions connecting the Green's functions of the two theories allows us to derive, systematically, the values of the chiral Lagrangian coefficients corresponding to the large Higgs mass limit of the standard model. These chiral parameters represent the non-decoupling effects of a heavy Higgs particle and incorporate both the leading logarithmic dependence on $\mh$ and the next to leading constant contributions. Some phenomenological implications are also discussed.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The most general chiral Lagrangian for electroweak interactions with the complete set of $SU(2)_L\times U(1)_Y$ invariant operators up to dimension four is considered. | contribution 0.63 | 0.72 | 0.04 | asserted 0.95 |
| s1 | The two-point and three-point functions with external gauge fields are derived from this effective chiral Lagrangian to one-loop order in a generic $R_\xi$-gauge. | method 0.70 | 0.93 | 0.04 | asserted 1.00 |
| s2 | The same set of Green's functions are paralelly studied in the renormalizable standard model to one-loop order, in a $R_\xi$-gauge and in the large Higgs mass limit. | method 0.64 | 0.88 | 0.04 | asserted 1.00 |
| s3 | An appropriate set of matching conditions connecting the Green's functions of the two theories allows us to derive, systematically, the values of the chiral Lagrangian coefficients corresponding to the large Higgs mass limit of the standard model. | method 0.72 | 0.93 | 0.61 | asserted 1.00 |
| s4 | These chiral parameters represent the non-decoupling effects of a heavy Higgs particle and incorporate both the leading logarithmic dependence on $\mh$ and the next to leading constant contributions. | interpretation 0.92 | 0.73 | 0.42 | asserted 1.00 |
| s5 | Some phenomenological implications are also discussed. | contribution 0.52 | 0.75 | 0.03 | asserted 0.71 |
What was examined, and what was not
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References
How the literature uses it
32 arXiv papers cite this in 48 sentences
Papers whose main result relates to this one's
States the same result 1 paper
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The values of the chiral parameters representing the non-decoupling effects of the heavy Higgs are presented.
The electroweak chiral parameters for a heavy Higgs in the Standard Model · 0.90
Papers citing it · 25 sentences without a confident label
Builds on it 2 papers
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In order to find the connection between $S(m_Z)$ and $S(Λ)$, we extend the previous one-loop RGE [ref] of the EWCL used by Bagger et. al.
Constraints on the electroweak chiral Lagrangian from the precision data · Introduction · 0.98
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These results build upon previous results obtained in the absence of the $h$ degree of freedom [ref] . * Structures that are not strictly required as counterterms, but that receive finite loop contributions from $ℒ_0$ and/or have a similar field composition or NDA suppression as the counterterms, should also be retained.
The Standard Model as an Effective Field Theory · HEFT counting · 0.95
Uses its method or tool 1 paper
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From the above gauge-fixing functions, $F_±$, $F_Z$ and $F_A$, we derive the corresponding Faddeev-Popov Lagrangian [ref] (see also, [ref] ), by: _FP = ∑_i,j=+,-,Z,A c^i † δF_i/δα_j c^j , where $c^j$ are the ghost fields and $α_j$ are the corresponding gauge transformation parameters ($j=+,-, Z,A$).
Anatomy of Higgs decays into $\gamma \gamma$ and $\gamma Z$ within the EChL in the $R_\xi$ gauges · The Electroweak Chiral Lagrangian: relevant interactions · 0.83
Uses its data or values 1 paper
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With the combined LEP data, provided in [ref] , a constraint on this oblique correction of $-0.12<S<0.16$ can be imposed at the 95% C.L., and with the relations presented in [ref] , which imply $S=-4πa_1$, the EChL parameter $a_1$ is restrained to be $-0.013<a_1<0.009$.
New Physics Signals of the Electroweak Chiral Lagrangian in Vector Boson Scattering at the LHC · The Effective Electroweak Chiral Lagrangian with a light Higgs · 0.99
Compares with it 2 papers
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The relevant terms for the description of EW gauge boson scattering amplitudes are[Our notation is taken from [ref] and compares: 1) with [ref] as, $a_1=(g/g') α_1$, $a_2=(g/g') α_2$, $a_3= -α_3$, $a_4= α_4$, $a_5= α_5$; 2) with [ref] as, $ℓ_1=4 a_5$, $ℓ_2= 4 a_4$, $ℓ_5= a_1$, $ℓ_6= 2(a_2 -a_3)$; and with [ref] as, $L_1=a_5$, $L_2=a_4$, $L_9=a_3-a_2$, $L_10=a_1$.]: [equation] Regarding the present experimental constraints on the previous EW chiral coefficients, we have summarized in fig:constraints the most recent available set from the literature [ref] .
Production of vector resonances at the LHC via WZ-scattering: a unitarized EChL analysis · The Effective Electroweak Chiral Lagrangian · 0.86
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The latter combination is renormalization group invariant, as it happened in the case of the Higgsless Electroweak Chiral Lagrangian (ECL) [ref] .
One-loop $\gamma\gamma\to W^+_L W^-_L$ and $\gamma\gamma\to Z_L Z_L$ from the Electroweak Chiral Lagrangian with a light Higgs-like scalar · Discussion · 0.82
Cites it as background 13 papers
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Two types of EFTs are commonly used to model the BSM physics in high-energy physics community - the SM Effective Field Theory (SMEFT) [ref] and the Higgs Effective Field Theory (HEFT) [ref] .
Matching the real Higgs triplet extension of Standard Model to HEFT · Introduction · 1.00
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The electroweak chiral Lagrangian and the extension to one Higgs scalar have been treated to a large extend [ref] .
One-Loop Renormalization of the Higgs Sector of the Electroweak Chiral Lagrangian extended by N Scalar Singlets · Motivation · 0.97
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[ref] , and also listed in other works like [ref] and references therein.
One-loop $\gamma\gamma\to W^+_L W^-_L$ and $\gamma\gamma\to Z_L Z_L$ from the Electroweak Chiral Lagrangian with a light Higgs-like scalar · The Electroweak Chiral Lagrangian with a light Higgs · 0.86
Discussion
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