Closing in on the radiative weak chiral couplings
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Abstract
We point out that, given the current experimental status of radiative kaon decays, a subclass of the \(\mathcal{O} (p^4)\) counterterms of the weak chiral lagrangian can be determined in closed form. This involves in a decisive way the decay \(K^\pm \rightarrow \pi ^\pm \pi ^0 l^+ l^\), currently being measured at CERN by the NA48/2 and NA62 collaborations. We show that consistency with other radiative kaon decay measurements leads to a rather clean prediction for the \(\mathcal{{O}}(p^4)\) weak couplings entering this decay mode. This results in a characteristic pattern for the interference Dalitz plot, susceptible to be tested already with the limited statistics available at NA48/2. We also provide the first analysis of \(K_S\rightarrow \pi ^+\pi ^\gamma ^*\), which will be measured by LHCb and will help reduce (together with the related \(K_L\) decay) the experimental uncertainty on the radiative weak chiral couplings. A precise experimental determination of the \(\mathcal{{O}}(p^4)\) weak couplings is important in order to assess the validity of the existing theoretical models in a conclusive way. We briefly comment on the current theoretical situation and discuss the merits of the different theoretical approaches.
1 Introduction
As opposed to their analogs in the bottom and charm sector, radiative kaon decays are generically dominated by Bremsstrahlung, while resonance effects are suppressed. This makes the extraction of the weak counterterms quite challenging. The best strategy to determine the chiral couplings is to study the interference term between the Bremsstrahlung and resonance contributions, which is substantially bigger than the resonance piece. Since the interference term is linear in the counterterms, this has the additional advantage that one is sensitive to both their magnitude and sign.
This strategy has been applied, e.g., to the decay modes \(K^{\pm }\rightarrow \pi ^{\pm }\gamma ^*\), \(K_S\rightarrow \pi ^0\gamma ^*\), \(K^{\pm }\rightarrow \pi ^{\pm }\pi ^0\gamma \) or \(K^{\pm }\rightarrow \pi ^{\pm }\gamma \gamma \). This way three independent combinations of counterterms are known. In order to fully determine the set of weak counterterms \(N_{14},..., N_{18}\) an extra decay mode has to be measured. From this perspective, \(K^+\rightarrow \pi ^+\pi ^0\gamma ^*\) stands out as the most promising candidate: besides counterterm combinations already present in \(K^{\pm }\rightarrow \pi ^{\pm }\gamma ^*\) and \(K^{\pm }\rightarrow \pi ^{\pm }\pi ^0\gamma \), it contains an additional combination. Furthermore, it has already been measured by NA48/2, which collected of the order of 5000 events thereof, statistics that will be improved by the new data currently being collected at NA62.
The main observation of this note is that the undetermined counterterm in \(K^+\rightarrow \pi ^+\pi ^0\gamma ^*\) turns out to be sizeable. Our observation is not based on model estimates, but arises from combining the different experimental results on radiative kaon decays. If this expectation on the size of the counterterm is combined with the strategy laid out in [1, 2], which shows how cuts in the dilepton invariant mass can diminish the dominance of Bremsstrahlung effects, a measurement could be performed already with the NA48/2 data. In this note we provide a detailed analysis of the interference term to aid the counterterm determination.
In the last part of the paper we comment on the decay mode \(K_S\rightarrow \pi ^+\pi ^ e^+e^\). In spite of the challenges that it poses, there are prospects to measure it at LHCb in the near future. For this decay mode the relevant counterterm combinations turn out to be entirely predicted from the already measured radiative kaon decay modes, so that one can have a rather precise estimate of how the interference term should look. LHCb could therefore provide a very relevant consistency check of the counterterm structure, and potentially improve the precision of the weak chiral couplings.
We conclude this note with some remarks on the present theoretical understanding of the radiative weak chiral couplings.
2 Experimental status of the radiative chiral weak counterterms
Values of the counterterm combinations together with the decay mode from which they can most precisely be extracted
Decay mode  Counterterm combination  Expt. value 

\(K^\pm \rightarrow \pi ^{\pm } \gamma ^*\)  \(N_{14}N_{15}\)  \(\,0.0167 (13)\) 
\(K_{S}\rightarrow \pi ^{0} \gamma ^*\)  \(2 N_{14}+N_{15} \)  \(+\,0.016 (4)\) 
\(K^{\pm }\rightarrow \pi ^{\pm }\pi ^{0}\gamma \)  \(N_{14}N_{15}N_{16}N_{17}\)  \(+\,0.0022 (7)\) 
\(K^{\pm }\rightarrow \pi ^{\pm }\gamma \gamma \)  \(N_{14}N_{15}2N_{18}\)  \(\,0.0017 (32)\) 
3 Prospects for \(K^+\rightarrow \pi ^+\pi ^0e^+e^\)
It should be noted that Eq. (14) differs from a NLO computation using chiral perturbation theory (see e.g. [1]): the Bremsstrahlung contribution is here evaluated using Low’s theorem, which is an exact current algebra result, whereas the weak and strong countertems are evaluated at the classical level, neglecting loop corrections. Since \(\mathcal{{M}}_K\) is taken from experiment, we are including the effects of final state interactions (pion rescattering), which are known to be important.
The decay rate is overwhelmingly dominated by the Bremsstrahlung component. However, as initially pointed out in [1], this dominance decreases as the dilepton invariant energy \(q^2\) is increased. Judicious cuts in \(q^2\) can therefore allow experimental analyses to reach information on the weak counterterms. This strategy was developed in detail in [2], suggesting an optimal cut at \(q_c\sim 50\) MeV.^{1}
Branching ratios for the Bremsstrahlung and the relative weight of the electric and electric interference terms for different cuts in q, starting at \(q_{min}\) (first row) and ending at 180 MeV. To highlight the role of the different counterterms, the last columns show how the interference term changes when they are switched off one at a time
\(q_c\) (MeV)  \(10^8\times \Gamma _\mathcal{{B}}\)  \(\displaystyle \left[ \frac{\Gamma _\mathcal{{E}}}{\Gamma _\mathcal{{B}}}\right] ^{1}\)  \(\displaystyle \left[ \frac{\Gamma _{\textsc {int}}}{\Gamma _\mathcal{{B}}}\right] ^{1}_{(1,1,1)}\)  \(\displaystyle \left[ \frac{\Gamma _{\textsc {int}}}{\Gamma _\mathcal{{B}}}\right] ^{1}_{(1,0,1)}\)  \(\displaystyle \left[ \frac{\Gamma _{\textsc {int}}}{\Gamma _\mathcal{{B}}}\right] ^{1}_{(1,1,0)}\)  \(\displaystyle \left[ \frac{\Gamma _{\textsc {int}}}{\Gamma _\mathcal{{B}}}\right] ^{1}_{(0,1,1)}\) 

\(2m_l\)  418.27  1100  \(\) 253  \(\) 225  \(\) 115  216 
2  307.96  821  \(\) 265  \(\) 226  \(\) 98  159 
4  194.74  529  \(\) 363  \(\) 264  \(\) 78  101 
8  109.60  304  1587  \(\) 850  \(\) 59  58 
15  56.12  161  102  156  \(\) 43  31 
35  15.50  50  18  21  \(\) 26  11 
55  5.62  22  7  9  \(\) 18  5 
85  1.37  8  3  4  \(\) 13  3 
100  0.67  5  2  3  \(\) 11  2 
120  0.24  3  1.6  2  \(\) 10  1.4 
140  0.04  2  1.0  1.1  \(\) 8  0.9 
180  0.003  1  0.7  0.8  \(\) 7  0.7 
Notice that \(\mathcal{{N}}_E^{(0)}\) determines the overall sign of the interference term, while \(\mathcal{{N}}_E^{(2)}\) is important to revert this sign already at low \(q_c\) values. \(\mathcal{{N}}_E^{(1)}\) has a marginal effect. It is interesting to note that a sizeable \(\mathcal{{N}}_E^{(2)}\) tends to make the interference term smaller, though not dramatically. As the table shows, different patterns for the counterterm values lead to distinct behaviors of the interference term with \(q_c\). This could be used to extract a value for \(\mathcal{{N}}_E^{(2)}\) or, more generally, to fit for \(\mathcal{{N}}_E^{(0)}\), \(\mathcal{{N}}_E^{(1)}\) and \(\mathcal{{N}}_E^{(2)}\).
In Fig. 2 we plot the results of the last four columns of Table 2, including also the error band, estimated by the 1 sigma shift of the experimental input.
4 Longdistance contributions to \(K_S\rightarrow \pi ^+\pi ^e^+e^\)
The branching ratio for \(K_S\rightarrow \pi ^+\pi ^e^+e^\) has already been measured by the NA48 collaboration [17, 18]. In order to probe the chiral \(\mathcal{{O}}(p^4)\) structure of this decay one needs to reach the percent precision, which might be possible at LHCb [19]. Here we provide a first analysis of this decay mode, which bears parallelisms with \(K_L\rightarrow \pi ^+\pi ^e^+e^\), already studied in [1, 20].

In Eq. (19) the Bremsstrahlung piece is the dominant contribution while the magnetic piece is a CPviolating effect. For \(K_L\) one finds the reverse situation, which makes this latter decay mode especially suited to determine the magnetic counterterm \(N_{19}+N_{31}\) combination.

An interesting feature of Eq. (19) is the SU(3)violating pole structure that comes with \(L_9\). In \(K_L\) such a structure is also present but violates CP and is completely negligible. Such a term is also present in \(K^+\rightarrow \pi ^+\pi ^0\gamma ^*\), but there it represents a tiny \(\mathcal{{O}}(m_{\pi ^+}^2m_{\pi ^0}^2)\) isospin violation, which we neglected altogether in the previous section. In contrast, for \(K_S\rightarrow \pi ^+\pi ^\gamma ^*\) it is a nonnegligible contribution.

The weak electric counterterm combinations are the same in both decay modes. However, \(\mathcal{{N}}_E^{(3)}\) and \(\mathcal{{N}}_E^{(4)}\) exchange their roles: for \(K_S\) the impact of \(\mathcal{{N}}_E^{(4)}\) is negligible, while for \(K_L\) it is \(\mathcal{{N}}_E^{(3)}\) that can be safely dismissed.
Using the same formalism employed in the previous section for \(K^+\rightarrow \pi ^+\pi ^0\gamma ^*\), in Fig. 3 we show the Dalitz plot for the interference differential decay rate of the decay \(K_S\rightarrow \pi ^+\pi ^e^+e^\) in the \((E_{\gamma },T_c)\) plane for fixed dilepton invariant masses \(q=20\) MeV and \(q=50\) MeV. Note that, in contrast to \(K^+\rightarrow \pi ^+\pi ^0e^+e^\), the interference term is positive all over the physical region.
5 Concluding remarks
Predictions of different counterterm combinations by two sets of models. On the third column we show the results of the weak deformation model (WDM), factorization model (FM) and holographic electroweak model (HEW), which yield the same predictions. On the fourth column, we list the estimates for the resonance model of ref. [28], keeping the parameters \(f_V\), \(g_V\) and \(f_A\) and setting \(\alpha _V=0=\alpha _A\), for a meaningful comparison
Counterterm combinations  Decay mode  WDM/FM/HEW  \(R^\mu \) 

\(N_{14}N_{15}\)  \(K^{\pm }\rightarrow \pi ^{\pm }\gamma ^*\)  \(\,3L_9L_{10}2H_1\)  \(\,0.02\eta _V\) 
\(2N_{14}+N_{15} \)  \(K_S\rightarrow \pi ^0\gamma ^*\)  \(\,2L_{10}4H_1\)  \(0.08\eta _V\) 
\(N_{14}N_{15}N_{16}N_{17}\)  \(K^{\pm }\rightarrow \pi ^{\pm }\pi ^0\gamma \)  \(\,2(L_9+L_{10})\)  \(\,0.01\eta _A\) 
\(N_{14}N_{15}2N_{18}\)  \(K^{\pm }\rightarrow \pi ^{\pm }\gamma \gamma \)  \(\,3(L_9+L_{10})\)  \(\,0.01\eta _A\) 
\(N_{14}+2N_{15}3(N_{16}N_{17})\)  \(K^{\pm }\rightarrow \pi ^{\pm }\pi ^0\gamma ^*\)  \(6L_94L_{10}+4H_1\)  \(0.16\eta _V+0.01\eta _A\) 
\(N_{14}N_{15}3(N_{16}N_{17})\)  \(K_L\rightarrow \pi ^+\pi ^\gamma ^*\)  \(\,4L_{10}+4H_1\)  \(0.04\eta _V+0.01\eta _A\) 
\(N_{14}N_{15}3(N_{16}+N_{17})\)  \(K_S\rightarrow \pi ^+\pi ^\gamma ^*\)  \(\,4L_{10}+4H_1\)  \(0.04\eta _V0.04\eta _A\) 
\(7(N_{14}N_{16})+5(N_{15}+N_{17})\)  \(K_S\rightarrow \pi ^+\pi ^\pi ^0\gamma \)  \(10L_914L_{10}\)  \(0.48\eta _V+0.01\eta _A\) 
On the right panel of Fig. 4 we show estimates for \(N_{16}\) and \(N_{17}\), where the external yellow band corresponds to a value of \(N_E^{(2)}=0.089(11)\), obtained assuming that \(N_{17}\) is negligible. The internal yellow band corresponds to the same central value but assuming that a precision of \(4\times 10^{3}\) can be reached. The experimental determination would fall out of the yellow band if \(N_{17}\) turns out to be sizeable, roughly \(N_{17}\gtrsim 2\times 10^{3}\).
We will close this note with some comments on the theoretical understanding of the weak counterterm values, i.e., on the electroweak interactions at low energies. The situation is not as solid as for the strong sector, where the bulk of the nonperturbative effects entering the GasserLeutwyler coefficients \(L_i\) is due to resonance exchange [23], with vector meson dominance (VDM) as a solid guiding principle. The weak counterterms, as opposed to the strong ones, are sensitive to the whole range of energies. Resonance models with VDM are therefore based on the assumption that the lowenergy region is dominant. Given the large number of parameters to fit, additional simplifying assumptions are used in order to end up with predictive schemes. Failure to reproduce the experimental numbers can therefore be attributed to (a) a nonnegligible contribution from short distances inside the weak counterterms; and (b) assumptions on the longdistance contributions which are not supported phenomenologically.
While the predictions of the different models should be taken with caution, they have merits that sometimes are not fully appreciated. In table 3 we have listed a number of counterterm combinations relevant for different radiative kaon decays (see [24] for a more comprehensive list), together with the predictions of a number of models: the weak deformation model (WDM) [25], factorization model (FM) [26], holographic electroweak model (HEW) [27] and the resonance model studied in Ref. [28].
In contrast to the WDM model, the resonance model of ref. [28] allows one to gauge the relative weight of vector and axial resonances as longdistance components of the weak chiral couplings and thereby test how robust is the VDM assumption. From Table 3 it is clear that if VMD is assumed, then there is a reasonable qualitative agreement with the conclusions of Eq. (29). In particular, notice that the cancellations of the form \((L_9+L_{10})\) in the third column of Table 3 are interpreted as vectorial cancellations in the fourth column. Notice also that the counterterms entering \(K_{L,S}\rightarrow \pi ^+\pi ^\gamma ^*\) are predicted to be the same in the third column but have different axialvector contributions in the fourth column. A determination of \(N_{16}\) and \(N_{17}\) through \(K^+\rightarrow \pi ^+\pi ^0\gamma ^*\) should clarify the importance of axialvectors in those counterterm combinations.
Footnotes
 1.
\(q_c\) is defined as the lower integration limit of the integral in q. Accordingly, \(q_c=\sqrt{2}m_e\) corresponds to performing the whole integral in q, while increasing \(q_c\) progressively eliminates the lowq region, where Bremsstrahlung dominates.
 2.
 3.
A similar situation is encountered in \(K\rightarrow 3\pi \) decays. See ref. [27] for a detailed discussion.
Notes
Acknowledgements
We thank Brigitte BlochDevaux for very stimulating discussions and for her comments on a first version of this manuscript. O.C. thanks the University of Naples for a pleasant stay during the completion of this work. L.C. and G.D. were supported in part by MIUR under Project No. 2015P5SBHT and by the INFN research initiative ENP. The work of O.C. is supported in part by the Bundesministerium for Bildung und Forschung (BMBF FSP105), and by the Deutsche Forschungsgemeinschaft (DFG FOR 1873).
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