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The W and Z scattering as a probe of physics beyond the Standard Model: Effective Field Theory approach

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Pith's one-line read This thesis claims that same-sign WW scattering at the HL-LHC and HE-LHC can discover new physics described by single dimension-8 EFT operators, provided the EFT is used only up to the energy scale where perturbative unitarity breaks down.

desk verdict A careful, self-contained doctoral thesis built from the author's own prior papers, whose central non-empty discovery-region claim rests on an on-shell unitarity cutoff that is plausible but never quantitatively validated for off-shell W's. read the letter →

arxiv 1908.07596 v1 pith:K32DXCSY submitted 2019-08-20 hep-ph

Pawe{l} Kozów This is my paper · ORCID
classification hep-ph
keywords vectorbosonscatteringsame-signWWeffectivefieldtheorySMEFTHEFTdimension-8operatorspartialwaveunitarityHL-LHCdiscoverypotential
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This dissertation asks whether the effective-field-theory program can discover new physics in vector-boson scattering before the EFT itself loses predictive power. The author studies the process pp -> 2 jets + W* W* -> 2 jets + l nu_l l' nu'_l at the HL-LHC and HE-LHC, taking each EFT 'model' to be the Standard Model plus a single dimension-8 operator (SMEFT) or single dp=8 operator (HEFT) that modifies quartic gauge couplings only. The central claim is that when the EFT is used only up to the energy scale sqrt(sU) at which tree-level partial-wave unitarity is violated, every one of the studied models still has a non-empty discovery region: there are Wilson-coefficient values for which the expected significance exceeds 5 sigma. This matters because it shows the EFT approach to WW scattering is not self-defeating: the new-physics signal appears at energies where the EFT is still a valid description, and the method gives a procedure for defining search regions consistently. The thesis also provides a comparison of SMEFT and HEFT experimental signatures and shows how discovery regions change when the pp collision energy is increased.

What carries the argument

The central object is the EFT 'model': the Standard Model Lagrangian plus one genuine quartic-gauge-coupling operator (dimension-8 in SMEFT, primary dimension dp=8 in HEFT), with a Wilson coefficient f_i. The argument runs through the on-shell elastic W+W+ -> W+W+ helicity amplitudes and their partial-wave projections: tree-level amplitudes that grow with energy violate perturbative unitarity at a scale $\sqrt$(sU), and this scale sets the upper end of the EFT's validity, $Lambda^{2}$ <= sU. The polarization completeness identity (Eq. 7.7) connects these on-shell amplitudes to the off-shell process pp -> 2j + l nu_l l' nu'_l, justifying the transfer of qualitative and quantitative conclusions. For each operator, the paper derives analytic leading-energy forms of helicity amplitudes and partial waves, identifies which helicity configuration violates unitarity first, and uses that to define $\sqrt$(sU), then computes expected significances at HL-LHC and HE-LHC with the EFT truncated at that scale.

What would settle it

Compute the full off-shell matrix elements for pp -> 2j + l nu_l l' nu'_l keeping the auxiliary polarization state in Eq. (7.7) and check whether the on-shell approximation reproduces the high-MWW region; if off-shell contributions are not suppressed by the propagators, the sqrt(sU) cutoff is wrong. Alternatively, measure the WW invariant mass distribution at the HL-LHC: if a 5-sigma excess attributed to a single dimension-8 operator appears only for MWW above the sqrt(sU) of the best-fit Wilson coefficient, the EFT is being used beyond its claimed validity and the non-empty discovery-region conclusion fails.

Watch

Extended reading notes

Core claim

Using the EFT only in its region of validity, the expected significance of the dimension-8 (or dp=8) genuine quartic-gauge-coupling effects in same-sign WW scattering can exceed 5 sigma at the HL-LHC or HE-LHC for every single-operator EFT model considered, regardless of whether the SMEFT or HEFT basis is chosen. The region of validity is defined by the perturbative partial-wave unitarity bound sqrt(sU), computed from on-shell W+W+ and W+W- scattering amplitudes, including the stronger of the same- and opposite-sign limits and, in most cases, a helicity-space diagonalization of partial waves. The off-shell LHC process is related to the on-shell amplitudes through the polarization completeness identity, so that far off-shell contributions are suppressed and the qualitative behavior is governed by on-shell helicity amplitudes. The thesis presents the resulting discovery regions in the Wilson-coefficient space, compares SMEFT and HEFT signatures, and shows that raising the pp collision energy from 14 TeV (HL-LHC) to 27 TeV (HE-LHC) enlarges the discovery regions.

Load-bearing premise

The whole discovery-region calculation assumes that the on-shell W+W+ scattering amplitudes, cut off at the energy where tree-level perturbative unitarity fails, faithfully describe the full off-shell LHC process pp -> 2j + l nu_l l' nu'_l; if far off-shell W's contribute significantly, or if loop effects extend the EFT's range, the discovery regions are miscalibrated.

Editorial extensions

If this is right

  • For each of the single-operator SMEFT and HEFT models studied, there is at least one Wilson-coefficient value for which the expected significance at the HL-LHC is above 5 sigma even though the EFT is used only up to its unitarity bound.
  • The correct validity scale for WW-scattering EFT analyses is the stronger unitarity limit from both same-sign and opposite-sign WW scattering, not the same-sign process alone; for M-type operators the opposite-sign channel provides the limiting bound.
  • Raising the pp collision energy from 14 TeV (HL-LHC) to 27 TeV (HE-LHC) moves and generally enlarges the discovery regions in the Wilson-coefficient space.
  • SMEFT and HEFT give different experimental signatures in same-sign WW scattering; in particular, the HEFT operators T42 and T44 enhance the --00 and -+00 polarization fractions, which is not seen for the SMEFT dimension-8 operators.
  • The proposed method for defining discovery regions is in principle not limited to the single-operator truncation and can be applied to models with several operators of arbitrary dimension, although the thesis works out the single-operator case.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the off-shell-to-on-shell mapping holds, the same discovery-region method transfers to other VBS channels such as WZ and ZZ scattering, and to future lepton colliders, because the polarization completeness identity is process-independent, though the amplitude set must be re-derived.
  • A measurement of final-state W polarizations in same-sign WW events could discriminate not only SMEFT from HEFT but also the S, M, and T operator classes within each basis, since the thesis shows distinct saturating helicity configurations for each class.
  • The single-operator truncation becomes more defensible if positivity bounds hold, because they imply dimension-8 gQGC coefficients can dominate dimension-6 ones; the thesis cites these bounds as motivation but does not rely on them for the main claim.
  • A null result at the HL-LHC at 5 sigma would exclude the studied single-operator models only within their unitarity-limited parameter space, leaving open the possibility of effects at larger Wilson coefficients where the EFT description is no longer valid.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. This doctoral thesis develops an EFT analysis of same-sign W pair production in pp→2j+W*W*→2j+lνl'ν' at HL-LHC and HE-LHC, using both SMEFT and HEFT bases. It defines EFT 'models' as the SM plus one dimension-8 (SMEFT) or dp=8 (HEFT) genuine quartic gauge coupling operator. In Ch. 4 it derives perturbative partial-wave unitarity bounds, and in Ch. 7.1 it computes on-shell W+W+→W+W+ and W+W−→W+W− amplitudes to define for each model a validity scale √sU(f_i) from the diagonalized |Re T|≤1/2 bound. The discovery-region analysis of Sec. 7.2 then restricts the full off-shell process to m_WW<√sU and finds, for every operator studied, values of the Wilson coefficient for which the expected significance exceeds 5σ. The thesis also compares SMEFT versus HEFT polarization signatures, identifying e.g. the T42/T44 enhancement of −−00 and −+00 fractions as a distinct non-linear feature, and studies the gain from HE-LHC.

Significance. The paper has real strengths: the helicity-amplitude calculations are detailed and cross-checked against VBFNLO, FeynRules, and FeynCalc; the unitarity analysis includes both same- and opposite-sign WW scattering and uses helicity-space diagonalization; and the main output is a set of falsifiable predictions (non-empty discovery regions) rather than only exclusion limits. If the validity-cutoff prescription survives scrutiny, the non-emptiness result is an important, non-obvious statement: it indicates that EFT searches in VBS can be sensitive to BSM physics even when the EFT is used only up to its unitarity bound, and the SMEFT/HEFT signature comparison provides a useful discriminator between linear and non-linear electroweak symmetry breaking. The thesis is largely a compilation of published papers [a-d], but the self-contained derivation of the unitarity framework gives it added value as a reference.

major comments (2)
  1. [Sec. 7.1.3, Eq. (7.7)-(7.10); Sec. 7.2] The central non-emptiness claim depends on using the on-shell unitarity scale √sU as a hard upper cutoff on the WW invariant mass in the off-shell process. The argument in Sec. 7.1.3 is asymptotic and does not quantitatively control the decomposition in Eq. (7.10): the propagator suppression of large k_i^2 is not the same as a bound on the individual virtualities, and the auxiliary-polarization amplitudes in Eq. (7.10) have no on-shell counterpart whose partial waves were constrained by Eqs. (4.34)/(4.36). Events with one W far off shell can therefore contribute to the selected sample even when m_WW<√sU, and if the EFT expansion parameter is controlled by the largest k_i^2 rather than by m_WW, the true breakdown scale can lie below √sU, changing or emptying the 5σ regions. The manuscript provides no numerical validation of the equivalence, such as distributions of k_i^2 in the signal region or a comparison of the full off-shell amplitudes with their on-shell approximation as a function of m_WW. Please add such a check, or restrict the significance calculation to a phase-space region where the on-shell mapping is demonstrated.
  2. [Sec. 4, Eqs. (4.34)/(4.36); Sec. 7.2] The paper identifies the EFT validity limit with the scale at which the tree-level partial waves violate |Re T|≤1/2. This is a standard and useful criterion, but it is a necessary rather than a sufficient condition: loop corrections in a non-renormalizable EFT can become large before the tree-level bound is saturated, particularly for dimension-8 operators whose amplitudes grow as s^2/Λ^4 and for which the loop expansion parameter near √sU can be O(1); the √sU values in Tables 7.1-7.2 are a few TeV, so s/(16π^2v^2) is not parametrically small. Since the discovery regions are defined by integrating up to √sU, an earlier actual breakdown would shrink them. Please state explicitly that √sU is an upper bound on the validity region and provide an estimate of the size of one-loop corrections, or an NDA-based argument, showing that the bound is not over-optimistic.
minor comments (4)
  1. [Eq. (7.8)] The printed formula for the auxiliary polarization is malformed: the denominator as typeset, k^2−m_W^2 k^2 m_W^2, is not a well-formed expression and should be replaced by a single square-root formula, presumably √(k^2(k^2−m_W^2)) or the intended equivalent.
  2. [Chapter 1 (Introduction)] The cross-references to 'Sec. 1', 'Sec. 2', ..., 'Sec. 7' do not match the actual chapter numbering (the Standard Model is Ch. 2, the conclusions are Ch. 8, etc.). Please correct the section cross-references.
  3. [Tables 7.1-7.2 and appendix tables] The units quoted for the Wilson coefficients are inconsistent: the captions write 'TeV4' where TeV^{-4} is meant for the dimension-8 SMEFT coefficients, while the HEFT operators have TeV^{-2} or dimensionless coefficients. Please standardize the notation for f_i and c_i throughout.
  4. [Sec. 7.2 and figures] A short summary table in the main text listing the final discovery regions (operator, sign of f_i, f_i range, √sU range, and maximum significance) would greatly improve readability, since the numerical results are otherwise distributed over long appendices.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: the unitarity cutoff and the discovery significance are computed from the same EFT Lagrangian as a consistency condition, not as a fitted prediction; the thesis reproduces its prior-work results and does not rely on load-bearing self-citations.

full rationale

The central claims are not circular. The discovery regions are defined by requiring expected significance above 5 sigma while restricting the EFT to energies below the tree-level partial-wave unitarity bound sqrt(sU). Both quantities are computed from the same Lagrangian and Wilson coefficient, but this is a self-consistency requirement, not an input-output identity: the significance is an event-counting prediction and the unitarity bound is an independent theoretical constraint. The paper does not fit sqrt(sU) to the significance, nor does it define the EFT model in terms of the discovery region. The Sec. 7.1.3 polarization-completeness argument (Eq. 7.7) is an approximate matching between on-shell and off-shell kinematics; even if that approximation were quantitatively inadequate, that would be a validity/correctness concern, not circularity, because the full pp significance is not constructed to equal the on-shell unitarity bound by definition. The reliance on the author's prior papers [a-d] is bibliographic: the thesis includes the derivations, analytic amplitudes, and cross-checks with VBFNLO, FeynRules, and FeynCalc, so the cited results are reproduced rather than imported as unverified premises. No uniqueness theorem or ansatz is smuggled in via self-citation. The only mild caveat is the thesis's opening statement that it is 'based on the results of [a-d]', which is a normal self-reference and is not load-bearing for the derivation presented in the thesis.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The thesis does not introduce new particles, forces, or dimensions. The 'EFT models' are parameter choices for existing operators, and the 'discovery region' is a methodological concept, not a new physical entity. The free parameter is the Wilson coefficient, and the key assumptions are the unitarity cutoff, the on-shell approximation, and the operator truncation.

free parameters (1)
  • Wilson coefficients f_i of dimension-8/dp=8 gQGC operators = Scanned over ranges such as |f| = 0.01 to 10 TeV^-4 (SMEFT) and corresponding HEFT coefficients
    These coefficients define each EFT 'model' and are varied by hand to map discovery regions. They are not fitted to data but are arbitrary inputs, making them free parameters in the phenomenological projection.
assumptions (3)
  • ad hoc to paper Tree-level partial wave unitarity bounds (|Re T| <= 1/2, or the diagonalized version) mark the maximum energy at which the EFT is valid.
    The thesis uses this criterion to define sqrt(sU) as a hard cutoff for EFT validity. This is a reasonable but non-unique choice; loop corrections or unitarization could shift the true breakdown scale, so the exact cutoff is an assumption of the analysis.
  • domain assumption On-shell WW scattering amplitudes provide a valid proxy for the off-shell process pp -> WWjj at high WW invariant mass.
    Sec 7.1.3 uses the polarization completeness identity (Eq. 7.7) to argue that off-shell amplitudes reduce to on-shell ones when MWW >> mW. This justifies applying the on-shell unitarity bound to the full hadronic process, but the approximation is not exact and is load-bearing for the discovery region method.
  • domain assumption Dimension-8 (or dp=8) genuine quartic-gauge-coupling operators dominate over dimension-6 operators in same-sign WW scattering.
    The thesis argues that dimension-6 operators are strongly constrained by other measurements and that positivity bounds suggest dimension-8 dominance. This justifies restricting the analysis to single dimension-8 operators as EFT models, which is a significant truncation choice.

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Pith. "Pith review of The W and Z scattering as a probe of physics beyond the Standard Model: Effective Field Theory approach." pith.science (2026). https://pith.science/paper/K32DXCSY

@misc{pith2026190807596,
  author       = {Pith},
  title        = {Pith review of: The W and Z scattering as a probe of physics beyond the Standard Model: Effective Field Theory approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K32DXCSY}},
  note         = {Machine review of arXiv:1908.07596}
}
abstract

In this work the vector boson scattering process is investigated through the reaction $pp\rightarrow 2 \mathrm{jets} + W^\ast W^\ast \rightarrow 2 \mathrm{jets} + l\nu_l l'\nu_l'$, where $W^\ast$ denote in general off-shell $W^+$, in the EFT approach with the HL-LHC and HE-LHC experiments in mind. We have investigated the discovery potential of certain classes of the EFT "models" of both the SMEFT and HEFT bases with the particular emphasis on using the EFT "models" in their region of validity. A novel method has been proposed for determining the discovery regions of physics beyond the SM, described by the EFT "models". Independent of the basis chosen, the discovery regions are found to be non-empty. We then compare differences in experimental signatures between SMEFT and HEFT, which is an important step for distinguishing between the two hypotheses in the future data. Finally, we investigated what the effect on the discovery regions is when increasing the $pp$ collision energy.

Figures

Figures reproduced from arXiv: 1908.07596 by the authors.

Figure 2.1
Figure 2.1. Shape of the scalar potential for µ 2 > 0 (left) and µ 2 < 0 (right). In the second case there is a continuous set of degenerate vacua, corresponding to different phases θ, connected through a massless field excitation ξ. For details see the text. i.e. as a function of the scalar fields. The classical scalar field values at the minimum of V are equal to vacuum expectation values of the scalar field operators. Hence … view at source ↗
Figure 2.2
Figure 2.2. World data on the ratio Re+e− [35]. The broken lines show the naive quark model approximation with NC = 3. The solid curve is the 3-loop perturbative QCD prediction. A simple estimate on the value of gs can be made studying the ratio between 3-jet and 2-jet e +e − annihilation – a quark can emit a gluon that can hadronize to a jet. This estimate results in αs ≡ g 2 s 4π ∼ 0.12. (2.115) Accounting for QCD loop correc… view at source ↗
Figure 2.3
Figure 2.3. Combined LEP and SLD measurements of sin 2θef f (averaged over leptons e, µ, τ ); Γl denotes the width Γ(Z → l ¯l) (averaged over the three leptonic modes). The shaded region shows the SM prediction. The arrows point in the direction of increasing values of mt and mh. The point shows the predicted values if, among the electroweak radiative corrections, only the photon vacuum polarization is included. Its arrow indic… view at source ↗
Figures from the paper (28 more)
Figure 2
Figure 2. Figure 2: indicated that the Higgs mass would be in the lower part of the considered region [PITH_FULL_IMAGE:figures/full_fig_p032_2.png]
Figure 2.4
Figure 2.4. Figure 2.4: Measured energy dependence of cross sections: [PITH_FULL_IMAGE:figures/full_fig_p033_2_4.png]
Figure 3.1
Figure 3.1. Figure 3.1: Representative Feynman diagrams for single, triple, and quartic gauge couplings [PITH_FULL_IMAGE:figures/full_fig_p037_3_1.png]
Figure 4.1
Figure 4.1. Figure 4.1: Argand circles: if inelastic channels are closed, i.e. if [PITH_FULL_IMAGE:figures/full_fig_p043_4_1.png]
Figure 5
Figure 5. Figure 5: are the only ones whose scattering amplitude is asymptotically constant in energy. [PITH_FULL_IMAGE:figures/full_fig_p050_5.png]
Figure 5.1
Figure 5.1. Figure 5.1: Contributions of different helicities (multiplicity taken into account) to the total [PITH_FULL_IMAGE:figures/full_fig_p051_5_1.png]
Figure 6.1
Figure 6.1. Figure 6.1: One loop contributions to the anomalous lepton magnetic moment in the SM. [PITH_FULL_IMAGE:figures/full_fig_p053_6_1.png]
Figure 7.1
Figure 7.1. Figure 7.1: Energy dependence of the total unpolarized elastic on-shell [PITH_FULL_IMAGE:figures/full_fig_p076_7_1.png]
Figure 7.2
Figure 7.2. Figure 7.2: Contributions of the polarized cross sections (multiplicity taken into account) [PITH_FULL_IMAGE:figures/full_fig_p077_7_2.png]
Figure 7.3
Figure 7.3. Figure 7.3: For description see the caption of Fig. 7.1; the [PITH_FULL_IMAGE:figures/full_fig_p077_7_3.png]
Figure 7.4
Figure 7.4. Figure 7.4: For description see the caption of Fig. 7.2; the [PITH_FULL_IMAGE:figures/full_fig_p078_7_4.png]
Figure 7
Figure 7. Figure 7: and 7.6 shows the polarised cross section fractions for [PITH_FULL_IMAGE:figures/full_fig_p081_7.png]
Figure 7.5
Figure 7.5. Figure 7.5: Contributions of the polarized cross sections (multiplicity taken into account) [PITH_FULL_IMAGE:figures/full_fig_p081_7_5.png]
Figure 7.6
Figure 7.6. Figure 7.6: For description see Fig. 7.5; the [PITH_FULL_IMAGE:figures/full_fig_p081_7_6.png]
Figure 7.7
Figure 7.7. Figure 7.7: Cartoon plot which shows the regions in fi and Λ (for an arbitrary higher￾dimension operator Oi) in terms of BSM signal observability and applicability of EFT ”models” based on the choice of a higher-dimension operator in an analysis of the same-sign VBS process with…
Figure 7.8
Figure 7.8. Figure 7.8: Typical examples of kinematic distributions used for the assessment of BSM [PITH_FULL_IMAGE:figures/full_fig_p088_7_8.png]
Figure 7.9
Figure 7.9. Figure 7.9: Typical examples of BSM signal significances computed as a function of [PITH_FULL_IMAGE:figures/full_fig_p089_7_9.png]
Figure 7.10
Figure 7.10. Figure 7.10: Regions in the Λ vs f (positive f values) space for dimension-8 operators in which a 5σ BSM signal can be observed and the EFT is applicable. The unitarity limit is shown in blue. Also shown are the lower limits for a 5σ signal significance from Eq. (7.25) (dashed l…
Figure 7.11
Figure 7.11. Figure 7.11: Regions in the Λ vs f (negative f values) space for dimension-8 operators in which a 5σ BSM signal can be observed and the EFT is applicable. For the meaning of curves see caption of [PITH_FULL_IMAGE:figures/full_fig_p092_7_11.png]
Figure 7.12
Figure 7.12. Figure 7.12: Maximum value p c max i of the coupling constants related to individual dimension-8 operators, calculated at the energy where the unitarity limit is reached, as a function of the relevant f value [PITH_FULL_IMAGE:figures/full_fig_p093_7_12.png]
Figure 7.13
Figure 7.13. Figure 7.13: Regions in the Λ vs. ci or fi (with ci , fi > 0) space for dp = 8 HEFT operators in which a 5σ BSM signal can be observed and the EFT is applicable. The unitarity limit is shown in blue, the lower limits for a 5σ signal significance from Eq. (7.25) is in yellow, the…
Figure 7.14
Figure 7.14. Figure 7.14: Same description as in Fig. 7.13. From top to bottom and from left to right, [PITH_FULL_IMAGE:figures/full_fig_p096_7_14.png]
Figure 7.15
Figure 7.15. Figure 7.15: Same description and operators as in Fig. 7.13. Negative values for [PITH_FULL_IMAGE:figures/full_fig_p097_7_15.png]
Figure 7.16
Figure 7.16. Figure 7.16: Same description and operators as in Fig. 7.14. Negative values for [PITH_FULL_IMAGE:figures/full_fig_p098_7_16.png]
Figure 7
Figure 7. Figure 7: shows the results for the individual operators [PITH_FULL_IMAGE:figures/full_fig_p100_7.png]
Figure 7.17
Figure 7.17. Figure 7.17: Regions in the Λ vs f (positive f values) space for dimension-8 operators in which a 5σ BSM signal can be observed and the EFT is applicable. The unitarity limit is shown in blue; the lower limits for a 5σ signal significance from Eq. (7.25) (red) and the upper limi…
Figure 7.18
Figure 7.18. Figure 7.18: Regions in the Λ vs f (negative f values) space for dimension-8 operators in which a 5σ BSM signal can be observed and the EFT is applicable. The unitarity limit is shown in blue; the lower limits for a 5σ signal significance from Eq. (7.25) (red) and the upper limi…
Figure 7.19
Figure 7.19. Figure 7.19: Regions in the Λ vs f (negative f values) space for M1 operator in which a 5σ BSM signal can be observed and the EFT is applicable. The unitarity limit is shown in blue; the lower limits for a 5σ signal significance from Eq. (7.25) (red) and the upper limit on 2σ EF…

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Reviewed August 14, 2026 · model on record in the stance chip above.