REVIEW
Radiative Semileptonic Kaon Decays
High Energy Physics - Phenomenology · Submitted 21 Sep 1992 · 35 references
Pith's verdict: A careful, complete O(p4) CHPT calculation of radiative semileptonic kaon decays; the only real soft spot is a forward-looking Kl3γ sentence that outruns the numbers.
T0 · AI review
Why Pith says this
Highlight any text to comment on it.
Referee report
The paper
The paper claims
The paper computes complete next-to-leading-order chiral perturbation theory amplitudes for three families of radiative semileptonic kaon decays, supplying the one-loop and anomaly contributions that earlier treatments omitted.
The reading
What carries the argument
The engine is the order-$p^4$ decomposition of the chiral generating functional into three pieces: one-loop diagrams built from the lowest-order Lagrangian, a local action of order $p^4$ carrying the renormalized couplings $L_4$, $L_5$, $L_9$, $L_{10}$, and the Wess-Zumino-Witten functional that encodes the chiral anomaly. The first two are chirally invariant; the third is not, and is used in Bardeen's form so that vector gauge invariance survives. Two structural devices carry the argument. The first is the Ward identity $q_mu A^{mu nu} = -sqrt{2} F_K p^nu$, derived in Appendix C without formal manipulation of time-ordered products, which guarantees that the full amplitude is invariant under $epsilon_mu$ to $epsilon_mu + q_mu$ and that the infrared-finite tensor amplitude is transverse. The second is the split of each rate into inner bremsstrahlung, structure-dependent, and interference terms with explicitly different Dalitz shapes, so that the two unknown form factors $V$ and $A$ can in principle be separated. That separation matters because measuring the structure-dependent terms alone leaves a fourfold ambiguity, mapping $(V,A)$ to $-(V,A)$ or to $(A,V)$; only the interference term breaks it. The same machinery produces the loop functions collected in Appendix A and the invariant-amplitude decompositions of Appendix B.
Extended reading notes
Core claim
The claim is that the matrix elements for $K^+$ to $l^+$ neutrino $l$ photon, $K^+$ to $l^+$ neutrino $l$ plus $l'^+l'^-$, and $K$ to pion $l$ neutrino photon are now known to order $p^4$ in chiral perturbation theory, loops included, and that the resulting rates agree with existing measurements. The three channels behave very differently, and the paper's structure follows that. For the single-photon decay there is no one-loop correction beyond the renormalization of the kaon decay constant, so the whole rate is fixed by the ratio $L_9+L_{10}$ of low-energy constants, giving $M_K(V+A)=-0.097$, $M_K(V-A)=-0.037$, and $A/V=0.45$; the measured structure-dependent rate translates to $M_K|V+A|=0.105$ with an eight-per-mille error, so the two agree. For the decays with a lepton pair, the entire one-loop effect collapses into the kaon electromagnetic form factor, which is dominated at low energies by $L_9$; the authors show this collapse is special to order $p^4$ and fails at order $p^6$. For the radiative $K_{l3}$ decays the loop amplitude is comparable in size to the local counterterm and interferes with it destructively, so the order-$p^4$ branching ratios differ from tree level by amounts that should be visible in a year of running. The authors also report a numerical disagreement of about an order of magnitude with an earlier calculation of $K^+$ to $e^+$ neutrino $e$ plus a muon pair, and note that the extrapolation of existing data to full phase space rests on that earlier result.
Load-bearing premise
The predictions assume that corrections of order $p^6$ and higher are small, which the authors explicitly say cannot be guaranteed, and the photon-energy dependence of the form factors is estimated from resonance exchange rather than derived.
Editorial extensions
If this is right
- The single-photon decay becomes a parameter-free prediction once $L_9+L_{10}$ is fixed elsewhere, and the quoted structure-dependent branching ratios of $1.30\times10^{-5}$ for the electron mode and $9.24\times10^{-6}$ for the muon mode sit inside the experimental bounds.
- Decays with a $\mu^+\mu^-$ pair are dominated by structure-dependent terms, so their rates and spectra test the form factors directly; the authors predict $1.35\times10^{-8}$ for $K^+\to\mu^+\nu_\mu\mu^+\mu^-$ and $1.12\times10^{-8}$ for $K^+\to e^+\nu_e\mu^+\mu^-$, the latter almost entirely structure-dependent.
- In the radiative $K_{l3}$ channels the loop and counterterm amplitudes are comparable and interfere destructively, so the order-$p^4$ branching ratios differ from tree level by effects reaching more than $4\times10^5$ events per year in $K^0_{e3}\gamma$.
- Existing data on $K^+\to\mu^+\nu_\mu e^+e^-$ above a 140 MeV pair mass agree with the form-factor-included calculation, so the extrapolation of that measurement to full phase space should be revisited.
Reading between the lines
- Because the photon-energy dependence of the form factors enters only at order $p^6$, a measurement of the low-$x$ end of the photon spectrum in $K^+\to e^+\nu_e\gamma$ is effectively a measurement of resonance couplings rather than of the chiral constants, which makes it a cheap handle on physics that modern lattice calculations attack from first principles.
- The fourfold ambiguity that appears when only the squares of $V$ and $A$ are measured is not special to kaons; any radiative semileptonic decay analysed through rate-squared terms alone should show the same degeneracy, which suggests looking for it in radiative $\eta$ and charm semileptonic channels.
- The observation that the one-loop contribution to the pair-producing decays collapses into the kaon electromagnetic form factor at order $p^4$ but not at order $p^6$ turns the pair-mass spectrum into a clean order separator: a shape measurement, not a rate, is what would tell the two orders apart.
Original abstract
We evaluate the matrix elements for the radiative kaon decays $K^+ \to l^+\nu_l\gamma$, $l^+ \nu_l l'^+ l'^-$ and $K \to \pi l \nu_l\gamma$ \ ($l,\ l' = e,\mu$) to next-to-leading order in chiral perturbation theory. We calculate total rates and rates with several kinematical cuts and confront the results with existing data. Measurements at future kaon facilities will allow for a more detailed comparison between theory and experiment.
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
We evaluate the matrix elements for the radiative kaon decays $K^+ \to l^+\nu_l\gamma$, $l^+ \nu_l l'^+ l'^-$ and $K \to \pi l \nu_l\gamma$ \ ($l,\ l' = e,\mu$) to next-to-leading order in chiral perturbation theory. We calculate total rates and rates with several kinematical cuts and confront the results with existing data. Measurements at future kaon facilities will allow for a more detailed comparison between theory and experiment.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We evaluate the matrix elements for the radiative kaon decays $K^+ \to l^+\nu_l\gamma$, $l^+ \nu_l l'^+ l'^-$ and $K \to \pi l \nu_l\gamma$ \ ($l,\ l' = e,\mu$) to next-to-leading order in chiral perturbation theory. | contribution 0.89 | 0.96 | 0.10 | asserted 0.99 |
| s1 | We calculate total rates and rates with several kinematical cuts and confront the results with existing data. | contribution 0.71 | 0.95 | 0.14 | asserted 1.00 |
| s2 | Measurements at future kaon facilities will allow for a more detailed comparison between theory and experiment. | future work 0.99 | 0.04 | 0.10 | hedged possibility 0.82 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
Figures
Figures reproduced from arXiv:hep-ph/9209261 by the authors.
References
How the literature uses it
32 arXiv papers cite this in 60 sentences
Papers citing it · 33 sentences without a confident label
Uses its method or tool 1 paper
-
The contribution to the $K_e2$ acceptance due to the radiative $K^+→e^+νγ$ inner bremsstrahlung process was taken into account following [ref] .
New physics limits from kaon decays · Measurement of $R_K$ with NA48/2 · 0.98
Uses its data or values 4 papers
-
[ref] and the simulation point meson masses and decay constants, together with $F_A^π$ and $F_A^K$ from $χ$PT at $(p^4)$ and $(p^6)$, respectively [ref] .
Isospin-breaking corrections to light-meson leptonic decays from lattice simulations at physical quark masses · Results · 0.85
-
With both form factors positive, the IB-SD interference in $$ is predicted to be net destructive, dominated by the $INT^- ∝(- )$ term: Table 4 of Ref. [ref] gives $INT^- = -3.83×10^-5$ and $INT^+ = +1.44×10^-5$ over the full phase space.
Comment on "Radiative corrections to tau -> pi(K) nu_tau[gamma]: A reliable new physics test" · The physical signs · 0.90
-
The theoretical prediction for $K^±→μ^±ν_μe^+ e^-$ branching ratio is $3.15×10^-6$ for $m_ee>20$ MeV [ref] .
Displaced Searches for Axion-Like Particles and Heavy Neutral Leptons at Mu3e · Heavy neutral leptons · 1.00
Compares with it 4 papers
-
In this work, we evaluate leading in the electromagnetic coupling constant $α$ corrections ($O ( α)$) to the (anti)neutrino energy spectra and compare our calculations to previous results for the muon decay [ref] in the approximation of vanishing electron mass as well as with well-known results for the total decay widths [ref] .
Radiative (anti)neutrino energy spectra from muon, pion, and kaon decays · Introduction · 1.00
-
The $O(p^4)$ ChPT predictions from Ref. [ref] are systematically larger than our results by about $8%–17%$, depending on the channel, with the largest difference observed for $K_μ2e$.
Rare kaon decays $K^- \to \ell^- \bar{\nu}_\ell \ell'^{+} \ell'^{-}$: Standard Model predictions from lattice QCD · Results · 0.99
-
It is noted that somewhat different decompositions of the axial-vector amplitudes are used in Refs. [ref] .
Triple-product asymmetry in the radiative two-pion tau decay · Resonance chiral Lagrangians and relevant amplitudes of the radiative two-pion tau decay · 0.87
Cites it as background 11 papers
-
The hadronic tensor $H^μν_W$ can be decomposed in terms of a “point-like" contribution $H_pt^μν$ (i.e. the expression obtained by treating the $D_s$ meson as a point-like particle) and four structure-dependent (SD) scalar form factors, $F_V, F_A, H_1$ and $H_2$ [ref] [Here, we use the dimensionless definitions of $H_1,2$ introduced in Ref. [ref] which differ by simple factors from those used in our earlier papers [ref] .
Lattice calculation of the $D_{s}$ meson radiative form factors over the full kinematical range · Definition of the form factors · 0.87
-
The radiative amplitude is the sum of the inner bremsstrahlung ($T_IB$) component of $𝒪(e p)$ and a structure dependent ($T_SD$) component of $𝒪(e p^3)$ [ref] .
The Standard model prediction for K(e2)/K_(mu2) and pi(e2)/pi(mu2) · The Standard Model prediction · 0.81
-
The Ward identity $k_μH_M^μν(k,p)=f_P p^ν$ implies that the hadronic matrix element $H_M^μν(k, p)$ can be expressed in terms of four structure-dependent form factors and the decay constant as follows [ref] [equation] where the convention of the Levi-Civita symbol in Minkowski space is chosen to be $ε^0123=1$ and $ε_0123=-1$; note that different conventions of the Levi-Civita symbol can lead to the opposite sign in front of $F_V$ in the literature. $R_1, R_2, F_V, F_A$ denote four dimensionless form factors that depends on $k^2$ and $p·k$.
Lattice Calculation of Light Meson Radiative Leptonic Decays · The Decay Amplitude · 0.85
Discussion
Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.