Experimental study of rare charged pion decays

The ${{\pi }^{-}}\to \ell {{\bar{\nu }}_{\ell }}$ (or, equivalently, ${{\pi }^{+}}\to \bar{\ell }{{\nu }_{\ell }}$) decay connects a pseudoscalar ${{0}^{-}}$ state (the pion) to the vacuum. At the lowest, tree level, the ratio of the $\pi \to e\bar{\nu }$ to $\pi \to \mu \bar{\nu }$ decay widths is given by [3, 21]

$$\begin{eqnarray}&&R_{e/\mu ,0}^{\pi }=\frac{\Gamma (\pi \to e\bar{\nu })}{\Gamma (\pi \to \mu \bar{\nu })}=\frac{m_{e}^{2}}{m_{\mu }^{2}}\times \frac{{{\left( m_{\pi }^{2}-m_{e}^{2} \right)}^{2}}}{{{\left( m_{\pi }^{2}-m_{\mu }^{2} \right)}^{2}}}\simeq 1.283\times {{10}^{-4}}.\end{eqnarray} \tag{ 1 }$$

The first factor in the above expression, the ratio of squared lepton masses for the two decays, comes from the helicity suppression by the V–A lepton-W boson weak couplings. If, instead, the decay could proceed directly through the pseudoscalar current, the ratio $R_{e/\mu }^{\pi }$ would reduce to the second, phase-space factor, or approximately 5.5. More complete treatment of the process includes $\delta R_{e/\mu }^{\pi }$, the radiative and loop corrections, and the possibility of lepton universality violation, i.e., that g_e and ${{g}_{\mu }}$, the electron and muon couplings to the W, respectively, may not be equal:

$$\begin{eqnarray}&&R_{e/\mu }^{\pi }=\frac{\Gamma (\pi \to e\bar{\nu }(\gamma ))}{\Gamma (\pi \to \mu \bar{\nu }(\gamma ))}=\frac{g_{e}^{2}}{g_{\mu }^{2}}\frac{m_{e}^{2}}{m_{\mu }^{2}}\frac{{{\left( m_{\pi }^{2}-m_{e}^{2} \right)}^{2}}}{{{\left( m_{\pi }^{2}-m_{\mu }^{2} \right)}^{2}}}\left( 1+\delta R_{e/\mu }^{\pi } \right),\end{eqnarray} \tag{ 2 }$$

where the '$(\gamma )$' indicates that radiative decays are fully included in the branching fractions. Steady improvements of the theoretical description of the π_e2 decay since the 1950s have recently culminated in a series of calculations that have refined the SM prediction to a precision of eight parts in 10⁵:

$$\begin{eqnarray}{{\left( R_{e/\mu }^{\pi } \right)}^{{\rm SM}}}=\frac{\Gamma (\pi \to e\bar{\nu }(\gamma ))}{\Gamma (\pi \to \mu \bar{\nu }(\gamma ))}{{|}_{{\rm calc}}}=\left\{ \begin{array}{ccccccccccccccc} 1.2352(5)\times {{10}^{-4}} & [22], \\ 1.2354(2)\times {{10}^{-4}} & [23], \\ 1.2352(1)\times {{10}^{-4}} & [24]. \\ \end{array} \right.\end{eqnarray} \tag{ 3 }$$

A comparison with equation (1) reveals that the radiative and loop corrections amount to almost 4% of $R_{e/\mu }^{\pi }$. However, as discussed below, the experimental precision at this time lags behind the theoretical one by more than an order of magnitude.

Because of the large helicity suppression of the π_e2 decay, its branching ratio is highly susceptible to small non-V–A contributions from new physics, making this decay a particularly suitable subject of study, as discussed in, e.g., [25–30]. This prospect provides the primary motivation for the ongoing PEN³ and PiENu⁴ experiments. Of the possible 'new physics' contributions in the Lagrangian, π_e2 is directly sensitive to the pseudoscalar one. At the precision of 10⁻³, $R_{e/\mu }^{\pi }$ probes the pseudoscalar and axial vector mass scales up to 1000 and 20 TeV, respectively [29, 30]. For comparison, Cabibbo–Kobayashi–Maskawa (CKM) matrix unitarity and precise measurements of several superallowed nuclear beta decays constrain the non-SM vector contributions to $\gt 20\;$ TeV, and scalar to $\gt 10\;$ TeV [35]. Although scalar interactions do not directly contribute to $R_{e/\mu }^{\pi }$, they can do so through loop diagrams, resulting in sensitivity to new scalar interactions up to 60 TeV [29, 30]. The subject was recently reviewed at length in [31]. In addition, ${{(R_{e/\mu }^{\pi })}^{{\rm exp} }}$ provides limits on masses of certain SUSY partners [28], on neutrino sector anomalies and more generally on massive sterile neutrinos [25, 27].

Even though $R_{e/\mu }^{\pi }$ is small, of order 10⁻⁴, it would be relatively straightforward to measure precisely because of the large difference in decay ejectile energies, 69.79 MeV versus 4.12 MeV for the e and μ channels, respectively. However, the subsequent decay of the muon, $\mu \to e{{\bar{\nu }}_{e}}{{\nu }_{\mu }}$, with the endpoint energy of 52.83 MeV, effectively brings the two processes kinematically closer, making the clean identification of the direct $\pi \to e$ events challenging in a 'tail' overlap region discussed below. This complication is partly mitigated by the $\sim 85$ times longer lifetime of the muon than that of the pion. Selecting only early decays, say, within a couple of pion lifetimes, suppresses the sequential $\pi \to \mu \to e$ events by more than an order of magnitude, thus effectively enhancing the apparent $\pi \to e\bar{\nu }$ branching ratio. Nearly all experiments to date have used a variant of the stopped pion decay technique to measure $R_{e/\mu }^{\pi }$.

The technique based on detecting and normalizing $\pi \to e$ to $\pi \to \mu \to e$ decays at rest in a nonmagnetic spectrometer is inherently independent of several important sources of systematic uncertainty that are shared by the two processes, and that consequently cancel in the ratio. Prime examples of shared uncertainties that cancel include the relative placement of the target and the positron detector and the resulting acceptance (on the other hand, using a magnetic spectrometer removes this insensitivity, as it would introduce a strong dependence of the solid angle on ${{e}^{+}}$ momentum.) Further cancellations occur for the efficiency of positron detection (largely independent of ${{E}_{{{e}^{+}}}}$, although corrections at the 10⁻³–10⁻² level are typically needed), various beam properties, efficiency of pion stop identification in the target, overall trigger efficiency etc. All of these quantities would have to be determined with higher precision in an experiment that would normalize the detected π_e2 decay events to the corresponding number of beam pions stopped in the target. Additional quantities such as the time displacement of the first detection interval from the arrival time of the pion and duration of the detection time gate, affect the extraction of $R_{e/\mu }^{\pi }$ in well understood ways.

The first notable measurement of $R_{e/\mu }^{\pi }$ with 6% uncertainty (rather than an upper limit or detection of a small number of π_e2 decay events), was performed by the University of Chicago group with positive pions stopped inside a double-focusing magnetic spectrometer [10]. This work was soon followed by a $\sim 2$% measurement [11] at the Columbia Nevis synchrocyclotron laboratory; decay positrons from π⁺ʼs stopped in an active target (AT) were detected in a ø 23 cm × 24 cm NaI(Tl) detector. Given the NaI detector size and acceptance for photons, the analysis included low-${{E}_{\gamma }}$ inner bremsstrahlung (IB) radiative ${{\pi }^{+}}\to {{e}^{+}}{{\nu }_{e}}\gamma$ events in the branching fraction, in accordance with the theoretical definition of $R_{e/\mu }^{\pi }$, as given in (2).

The Nevis measurement brought to light an important experimental uncertainty, related to how accurately the NaI(Tl) energy response function is known. Positrons in the few to a 100 MeV energy range generate electromagnetic showers that cannot be completely contained in a finite-sized physical detector volume; there is a non-zero probability that some energy will escape the volume, primarily in the form of photons, inducing a low-energy 'tail' in the response function to a monoenergetic positron (use of a magnetic spectrometer would eliminate this problem, though at the cost of significantly more complicated acceptance systematics). This intrinsic instrumental tail coexists with the physical 'tail' of radiative decay events for which the accompanying photon escaped detection, primarily due to limited detector solid angle. We note in passing that in a nearly hermetic detector, radiative decays will generate a high energy 'tail' extending above the ∼ 69.8 MeV two-body decay positron energy; this was not a feature of the early measurements. In a sufficiently segmented detector such events can be properly treated by analyzing $(E+pc)/2$ rather than E alone.

The tail correction for $\pi \to e\nu$ events falling below the μ decay endpoint energy in the Nevis experiment was 9.1% with a ∼ 0.7% combined systematic and statistical uncertainty. While this correction diminishes for larger detectors, especially with external active enclosures, it remains a dominant source of systematic uncertainty for $R_{e/\mu }^{\pi }$, especially as experiments approach the theoretical precision of equation (3).

The TRIUMF group of Bryman and collaborators [32] used a similar technique to that used in the Nevis measurement, with improvements. TINA, the TRIUMF NaI(Tl) detector was larger at ø 46 cm × 51 cm long, and subtended a solid angle of $\simeq$ 0.7% of $4\pi$ sr, with energy resolution of about 4% full width at half-maximum (FWHM) at 70 MeV, almost four times better than the Nevis detectorʼs ∼15%. The experiment collected approximately $3.2\times {{10}^{4}}$ π_e2 decay events. The resulting branching ratio had a 1.2% overall uncertainty. The single largest contribution to the uncertainty, at 0.75%, arose from the tail correction.

TINA was used again in a follow-up measurement (figure 1) by the same group [33], with numerous improvements in experimental design including an increased solid angle coverage of 2.9% of $4\pi$ sr and an order of magnitude more detected π_e2 decay events. The measured branching fraction

$$\begin{eqnarray}&&R_{e/\mu }^{\pi }=[1.2265\pm 0.0034({\rm stat})\pm 0.0044({\rm syst})]\times {{10}^{-4}}\;\end{eqnarray} \tag{ 4 }$$

is in excellent agreement with SM predictions.

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At the Paul Scherrer Institute (PSI) a Bern–PSI group performed a measurement of $R_{e/\mu }^{\pi }$ at a similar level of precision using a radically different apparatus [34], essentially simultaneously with the TRIUMF measurement. The Bern–PSI setup is shown in figure 2. The key component of the apparatus was a nearly hermetic $\sim 4\pi$ sr calorimeter consisting of 132 bismuth germanium oxide detectors in the shape of hexagonal prisms, each 20 cm or 18 radiation lengths long, and with inscribed radius of 5.5 cm. The detectors were arranged so that a narrow opening allowed the beam (almost equal parts pions and muons) to stop in a small cylindrical active plastic scintillator target, whose light was read out by a central calorimeter crystal. The resulting rms resolution was sub-2% at 70 MeV. Because of the hermetic nature of the calorimeter, radiative muon decays presented a significant background under the $\pi \to e\bar{\nu }(\gamma )$ signal peak in the energy spectrum. A total of $3\times {{10}^{5}}$ $\pi \to e$, and $1.2\times {{10}^{6}}$ $\pi \to \mu$ events were recorded. The resulting branching fraction was

$$\begin{eqnarray*}&&R_{e/\mu }^{\pi }=[1.2346\pm 0.0035({\rm stat})\pm 0.0036({\rm syst})]\times {{10}^{-4}}\end{eqnarray*}$$

with the systematic uncertainty coming primarily from corrections for photonuclear reactions, radiative μ decay background, and electromagnetic losses.

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Table 1 summarizes the available experimental results on $R_{e/\mu }^{\pi }$. Combined, the 1992/93 TRIUMF and PSI results define the present experimental precision for the decay [35], as the earlier measurements do not significantly affect the current global average:

$$\begin{eqnarray}&&{{\left( R_{e/\mu }^{\pi } \right)}^{{\rm exp} }}=(1.230\pm 0.004)\times {{10}^{-4}}.\end{eqnarray} \tag{ 5 }$$

This experimental world average lags behind the theoretical precision of (3) by more than an order of magnitude. The precision gap, already notable in 1993, has only increased with the passage of time, and has motivated a new generation of experiments currently under way, and briefly discussed below.

Table 1.  Summary of published experimental results on $R_{e/\mu }^{\pi }$, and the Particle Data Group (PDG) average.

Bern–PSI 120 k $1.2346\pm 0.0050$ [34]

Experiment	πe2 events	${{10}^{4}}\times R_{e/\mu }^{\pi }$	References
U. Chicago/EFINS	1.2 k	$1.21\pm 0.07$	[10]
Columbia/Nevis	10.8 k	$1.247\pm 0.028$	[11]
TRIUMF/TINA	32 k	$1.218\pm 0.014$	[32]
TRIUMF/TINA	190 k	$1.2265\pm 0.0056$	[33]
PDG average	342 k	$1.230\pm 0.004$	[35]

In 2006 a new measurement of $R_{e/\mu }^{\pi }$ was proposed at the PSI by a collaboration of seven institutions from the USA and Europe (see footnote 3), with the aim to reach

$$\begin{eqnarray}&&\frac{\Delta R_{e/\mu }^{\pi }}{R_{e/\mu }^{\pi }}\simeq 5\times {{10}^{-4}}.\end{eqnarray} \tag{ 6 }$$

The PEN experiment uses an upgraded version of the PIBETA detector system, described in detail in [36], and previously used in a series of rare pion and muon decay measurements [37–40]. The main component of the PEN apparatus, shown in figure 3, is a spherical large-acceptance ($\sim 3\pi$ sr) electromagnetic shower calorimeter. The calorimeter consists of 240 truncated hexagonal and pentagonal pyramids of pure CsI, 22 cm or 12 radiation lengths deep. The inner and outer diameters of the sphere are 52 cm and 96 cm, respectively. Beam particles entering the apparatus with $p\simeq 75{\rm MeV}\;{{c}^{-1}}$ are first tagged in a thin upstream beam counter (BC) and refocused by a triplet of quadrupole magnets. After a $\sim 3$ m long flight path they pass through a 5 mm thick active degrader (AD) and a low-mass mini time projection chamber (mTPC), finally to reach a 15 mm thick AT where the beam pions stop. Decay particles are tracked non-magnetically in a pair of concentric cylindrical multiwire proportional chambers (MWPC1,2) and an array of twenty 4 mm thick plastic hodoscope (PH) detectors, all surrounding the AT. The BC, AD, AT and PH detectors are all made of fast plastic scintillator material and read out by fast photomultiplier tubes (PMTs). Signals from the beam detectors are sent to waveform digitizers, running at 2 GS s $^{-1}$ for BC, AD, and AT, and at $250\;{\rm MS}\;{{{\rm s}}^{-1}}$ for the mTPC.

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As discussed in the preceding section, a key source of systematic uncertainty in previous π_e2 experiments has been the hard to measure low energy tail of the detector response function, caused by electromagnetic shower leakage from the calorimeter, mostly in the form of photons. PEN is no exception in this respect. If not properly identified and suppressed, other physical processes also contribute events to the low energy part of the spectrum; unlike shower leakage they can also produce high energy events. One process is ordinary pion decay into a muon in flight, before the pion is stopped, with the resulting muon decaying within the time gate accepted in the measurement. Another is the unavoidable physical process of radiative decay. The latter is well measured and properly accounted for in the PEN apparatus, as discussed below in the section on radiative decay. Shower leakage and pion decays in flight can only be well characterized if the $\pi \to \mu \to e$ chain can be well separated from the direct $\pi \to e$ decay in the target. Therefore much effort has been devoted to digitization, filtering and analysis of the target waveforms [41], as illustrated in figure 4. The method used for separating the 2-peak (π_e2) and 3-peak ($\pi \to \mu \to e$) events is illustrated and explained in figure 5. The key to the method is provided by the beam and MWPC detectors which are used to predict the pion and positron energy deposition in the target, and the times of their signals. Once the predicted waveform is subtracted, the net waveform is scanned for the presence of a 4.1 MeV muon peak. The difference between the minimum χ² values with and without the muon peak is reported as $\Delta ({{\chi }^{2}})$, constructed so that clean 2- and 3-peak fits return values of $+1$ and $-1$, respectively. The scan is fast and returns a $\Delta ({{\chi }^{2}})$ value for every event, as illustrated in the figure.

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A particularly telling figure regarding the PEN data quality are the decay time histograms of the $\pi \to e\nu$ decay and $\pi \to \mu \to e$ sequence, shown in figure 6 for a subset of data recorded in 2010. The $\pi \to e\nu$ data follow the exponential decay law over more than three orders of magnitude, and perfectly predict the measured $\pi \to \mu \to e$ sequential decay data once the latter are corrected for random (pile-up) events. Both event ensembles were obtained with minimal requirements (cuts) on detector observables, none of which bias the selection in ways that would affect the branching ratio. The probability of random $\mu \to e$ events originating in the target can be controlled in the data sample by making use of multihit time to digital converter data that record early pion stop signals. With this information one can strongly suppress events in which an 'old' muon was present in the target by the time of the pion stop that triggered the readout.

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The 'intrinsic' low energy tail of the PEN response function below ∼50 MeV, due to shower losses for π_e2 decay events for pions at rest, amounts to approximately 2% of the full yield. Events with $\pi \to \mu$ decays in flight, with subsequent ordinary Michel decay of the stopped muon in the target, add a comparable contribution to the tail. The two contributions can be simulated accurately, with the respective detector responses independently verified through comparisons with measured data in appropriately selected processes and regions of phase space. These response functions are entered into the maximum likelihood analysis (MLA), used to describe all measured processes simultaneously. The quantity $R_{e/\mu }^{\pi }$ is evaluated in the MLA as the ratio of the magnitudes found for the $\pi \to e(\gamma )$ and $\pi \to \mu \to e$ processes. Although verified through comparisons with Monte Carlo simulations, the intrinsic tail itself is not directly measurable at the required precision because of the statistical uncertainties arising in the tail data selection procedure. Radiative decay processes are directly measurable and accounted for in the MLA procedure. More information about the PEN/PIBETA detector response functions is given below, in the section on π_e3 decays, and in [36].

During the 2008–10 production runs the PEN experiment accumulated some $2.3\times {{10}^{7}}$ $\pi \to e\nu$, and more than $1.5\times {{10}^{8}}$ $\pi \to \mu \to e$ events, as well as significant numbers of pion and muon radiative decays. A comprehensive blinded MLA is under way to extract a new experimental value of $R_{e/\mu }^{\pi }$. As of this writing, there appear no obstacles that would prevent the PEN collaboration to reach a precision of $\Delta R/R\lt {{10}^{-3}}$. The competing PiENu experiment at TRIUMF, discussed below, has a similar precision goal. The near to medium future will thus bring about a substantial improvement in the limits on e–μ lepton universality, and the attendant SM limits.

The new PIENU experiment at TRIUMF builds on the earlier measurements at the same laboratory [33], aiming at a significant improvement in precision through refinements of the technique used. The branching fraction will again be obtained from the ratio of positron yields from the $\pi \to e\nu$ decay and from the $\pi \to \mu \to e$ decay chain. As in other experiments that detect decay positrons in a nonmagnetic spectrometer, many normalization factors, such as the solid angle of positron detection, cancel to first order in PiENu, leaving only corrections for small energy-dependent effects. Major improvements in precision in PiENu over the earlier TRIUMF TINA measurement derive from improved geometry and beamline, a superior calorimeter, as well as high-speed digitizing of all detector signals.

Figure 7 shows a schematic rendition of the PiENu experimental apparatus. ${\rm A}75\;{\rm MeV}\;{{c}^{-1}}$ π⁺ beam from the improved TRIUMF M13 beam line [42] is tracked in wire chambers, identified by plastic scintillators, and stopped in a 0.8 cm thick scintillator target. Fine tracking near the target is provided by two sets of single-sided silicon strip detectors located immediately upstream and downstream of the target assembly. The positrons from $\pi \to e\nu$ and $\pi \to \mu \to e$ decays are detected in the positron telescope, which consists of a silicon strip counter, two thin plastic counters, and an acceptance-defining wire chamber that covers the front of the crystal calorimeter. The solid-angle acceptance of the telescope counters is 20% of 4π sr.

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The primary energy measurement is performed using BINA, a cylindrical, single-crystal ø 48 cm × 48 cm long NaI(Tl) detector [43]. BINAʼs energy resolution of approximately 2.2% FWHM for incident positrons at 70 MeV [44] is better by a factor of approximately two than the resolution observed in the previous measurement with TINA [33]. The NaI detector is surrounded by two layers consisting of 97 pure CsI crystals, 8.5 cm thick, 2 cm wide and 25 cm long [45, 46] to capture electromagnetic shower leakage from BINA, thus helping to suppress the instrumental low-energy 'tail'.

As in PEN, analog signals from plastic scintillators, silicon strip, NaI, and CsI detectors are recorded as waveforms, using appropriately fast digitizers. To suppress background arising from old muon decay signals in the target region and to reduce possible distortions in the time spectrum due to pileup, the incident pion rate is kept at $6\times {{10}^{4}}\;{{{\rm s}}^{-1}}$.

The PiENu collaboration has accumulated upwards of $2\times {{10}^{7}}$ $\pi \to e\nu$ events through 2012. Combining the corresponding statistical uncertainty with reduced systematic uncertainties, the collaboration expects to reach $\Delta R_{e/\mu }^{\pi }/R_{e/\mu }^{\pi }\lt 0.1$%. The data acquisition phase of the experiment ended in 2012, and as of this writing, the collaboration is working on data analysis.

The decay ${{\pi }^{+}}\to {{e}^{+}}{{\nu }_{e}}\gamma$ proceeds via a combination of QED (IB) and direct, structure-dependent (SD) amplitudes [20, 21]. Under normal circumstances, as in $\pi \to \mu \nu \gamma$ decay, the direct amplitudes are hopelessly buried under an overwhelming IB background. However, the strong helicity suppression of the primary non-radiative process, $\pi \to e\nu$, also suppresses the bremsstrahlung terms, making the direct SD amplitudes measurable in certain regions of phase space [20, 47] (we recall that the same helicity suppression made possible sensitive searches for non-V–A interaction terms in precision measurements of the primary $\pi \to e\nu$ decay, discussed in section 2.) This result is of great use to effective low-energy theories of the strong interaction, primarily ChPT, which rely on the SD amplitudes to provide important input parameters. Whereas the IB amplitude is completely described by QED, the SD amplitude can be parametrized in terms of the pion form factors. As indicated in the tree-level Feynman diagrams in figure 8, standard V–A electroweak theory requires only two pion form factors F_A, axial vector, and F_V, vector (or polar-vector), to describe the SD amplitude. The amplitudes (form factors) F_A and F_V in principle depend on the four-momentum transfer to the e-ν pair (or to the W boson), $s={{\left( {{p}_{e}}+{{p}_{\nu }} \right)}^{2}}\equiv {{q}^{2}}$. In $\pi \to e\nu \gamma$ decay s remains low, $s\lt m_{\pi }^{2}$, so that it is a good approximation to evaluate F_V and F_A at $s=0$, often referred to as the soft pion limit. It is useful at this point to consider the tree-level double differential branching ratio for the $\pi \to e\nu \gamma$ decay which, in the usual parametrization, takes the form first worked out in detail by De Baenst and Pestieau [48]:

$$\begin{eqnarray}\begin{array}{rcl} \frac{{{{\rm D}}^{2}}{{\Gamma }_{\pi e2\gamma }}}{{\rm D}x\;{\rm D}y} & = & \frac{\alpha }{2\pi }{{\Gamma }_{\pi e2}}\left\{ {\rm IB}\left( x,y \right)+{{\left( \frac{{{F}_{V}}m_{\pi }^{2}}{2{{f}_{\pi }}{{m}_{e}}} \right)}^{2}}\left[ {{\left( 1+\gamma \right)}^{2}}{\rm s}{{{\rm D}}^{+}}(x,y)+{{\left( 1-\gamma \right)}^{2}}{\rm s}{{{\rm D}}^{-}}\left( x,y \right) \right] \right. \\ {} & {} & +\left( \frac{{{F}_{V}}{{m}_{\pi }}}{{{f}_{\pi }}} \right)\left[ \left( 1+\gamma \right)S_{\operatorname{int}}^{+}\left( x,y \right)+\left( 1-\gamma \right)S_{\operatorname{int}}^{-}\left( x,y \right) \right]\}, \\ \end{array}\end{eqnarray} \tag{ 7 }$$

where $x\equiv 2{{E}_{\gamma }}/{{m}_{\pi }}$ and $y\equiv 2{{E}_{e}}/{{m}_{\pi }}$ are the appropriately normalized photon and electron (positron) energies, respectively, $\gamma \equiv {{F}_{A}}/{{F}_{V}}$ is the ratio of the axial and the vector pion form factors, ${{f}_{\pi }}$ is the familiar pion decay constant and 'int' denotes interference terms between the IB and SD amplitudes. The functional dependence of the terms IB, SD⁺, SD⁻, ${\rm SD}_{\operatorname{int}}^{+}$ and ${\rm SD}_{\operatorname{int}}^{-}$ on x and y is well established and is given in the literature, e.g., in [21, 48]. We note that the SD⁺ and SD⁻ terms, which multiply the ${{\left( {{F}_{V}}+{{F}_{A}} \right)}^{2}}$ and ${{({{F}_{V}}-{{F}_{A}})}^{2}}$ form factor terms, respectively, map very different portions of the three-body phase space in the final state. The SD⁺ term peaks for large values of positron energies, $y\gtrsim 0.9$ and moderate photon energies, $x\gtrsim 0.5$, where there is relatively little background from the IB terms. The SD⁻ term, on the other hand, peaks for low values of $x+y\simeq 1$, where the IB amplitude is comparatively strong, as shown in figure 9; this kinematic region is also susceptible to background from muon radiative decays. It is not surprising then, that most of the measurements of the ${{\pi }_{e2\gamma }}$ decay to date have focused on determining the SD⁺ amplitude.

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The conserved vector current (CVC) hypothesis [3, 4] relates the vector form factor F_V to the ${{\pi }^{0}}\to \gamma \gamma$ decay amplitude [49–51]:

$$\begin{eqnarray}&&{{\Gamma }_{{{\pi }^{0}}\to \gamma \gamma }}=\frac{1}{2}\pi {{m}_{\pi }}{{\alpha }^{2}}{{\left| {{F}_{V}} \right|}^{2}}.\end{eqnarray} \tag{ 8 }$$

During a long period of time ending in 2012, then current Particle Data Group (PDG) values for the π⁰ lifetime and the ${{\pi }^{0}}\to \gamma \gamma$ branching fraction led to a CVC prediction of $F_{V}^{{\rm CVC}}=0.0259\pm 0.0009$. As a rule, radiative pion decay measurements had access only to one SD amplitude, the SD⁺. Treating the value of F_V as known, predicted by the CVC hypothesis, allowed analyses of ${{\pi }_{e2\gamma }}$ data to extract F_A, or, more specifically the ratio $\gamma \equiv {{F}_{A}}/{{F}_{V}}$. Recent measurements of the π⁰ lifetime, primarily that of the PrimEx collaboration [52], have led to a change in the PDG accepted value of the pion lifetime, along with a reduced uncertainty. Furthermore, Bernstein and Holstein have recently pointed out the necessary correction to the CVC expression in (8) arising from isospin breaking effects [47]. Combined, the two effects lead to a new isospin corrected CVC prediction for the pion vector form factor:

$$\begin{eqnarray}&&F_{V}^{{\rm CVC}}=0.0255\pm 0.0003\end{eqnarray} \tag{ 9 }$$

which modifies the previously reported values for F_A or $\gamma \equiv {{F}_{A}}/{{F}_{V}}$.

Before discussing individual measurements of the radiative ${{\pi }^{+}}\to {{e}^{+}}{{\nu }_{e}}\gamma$ decay, we note that the related decay ${{\pi }^{+}}\to {{e}^{+}}{{\nu }_{e}}{{e}^{+}}{{e}^{-}}$, where the photon is virtual, adds R, a second axial vector amplitude or form factor, to the SD terms. The R form factor is proportional to $\langle r_{\pi }^{2}\rangle$, the electromagnetic (charge) radius of the pion squared [20].

3.2.1. Ordinary radiative decay ${{\pi }_{e2\gamma }}$

The first determination of the SD amplitudes in ${{\pi }_{e2\gamma }}$ decay was performed by Depommier and collaborators. In their apparatus pions were stopped in a plastic scintillator target, and the decay positrons and photons detected in opposing NaI(Tl) and lead glass detectors, each preceded by a pair of thin plastic scintillator detectors in front [53]. Signals from all detectors were recorded in oscilloscope screen images which were subsequently analyzed. A total of 148 ± 15 ${{\pi }_{e2\gamma }}$ events resulted after background subtraction; some 110 were due to the SD emission. In their analysis, the authors used F_V = 0.0245 and obtained two solutions for γ, 0.4 and $-2.1$; the data did not favor one over the other. The authors also reported a branching ratio for ${{E}_{e}},{{E}_{\gamma }}\gt 48$ MeV with 17% relative uncertainty, where, however, the detector resolution had not been taken into account, so the result could not be compared with theoretical calculations.

The Berkeley–UCLA experiment by Stetz and collaborators [54] detected the positrons in a magnetic spectrometer and the photons in a lead glass Čerenkov hodoscope. The experimenters observed $226\pm 22.4$ events, and reported a branching ratio of $(5.6\pm 0.7)\times {{10}^{-8}}$ for ${{E}_{e}}\gt 56$ MeV and ${{\theta }_{e\gamma }}\gt 132{}^\circ$. To determine the SD terms, the authors used ${{F}_{V}}=0.0261\pm 0.0009$, based on the 1977 average value for the π⁰ lifetime. They too found two solutions for γ: $0.44\pm 0.12$ or $-2.36\pm 0.12$, neither of which was clearly favored by the data.

In the mid-1980s at the Swiss Institute for Nuclear Research (SIN, now part of PSI) a Lausanne–Zurich collaboration [55] set up an improved version of the Berkeley–UCLA apparatus to study the ${{\pi }_{e2\gamma }}$ decay and determine F_A. They used an intense pion beam with the stopping rate in the target of $2.5\times {{10}^{7}}\;{{{\rm s}}^{-1}}$, an 8 × 8 array of NaI(Tl) crystals for photon detection, and a large-acceptance magnetic spectrometer for the positron. An MWPC near the target provided additional tracking information. Data were taken in two detector geometries, centered around ${{\theta }_{e\gamma }}=180{}^\circ$ and $135{}^\circ$, respectively. The former configuration accepts negligible contributions from the IB and SD⁻ amplitudes, while the latter accepts a sizable SD⁻ contribution still with minimal IB background. Using this method, the collaborators collected 653 ± 29 ${{\pi }_{e2\gamma }}$ events, as well as 801 ± 34 π_e2 events for normalization. Thanks to the expanded phase space coverage compared to previous experiments, Bay et al were able to resolve the quadratic ambiguity inherent in the SD⁺ analysis only, and eliminate the negative-sign solution by a confidence factor of 8.5. Using the CVC value of ${{F}_{A}}=0.0255(5)$, incidentally the same central value as the current best isospin-corrected CVC prediction in (9), the authors obtained $\gamma =0.52\pm 0.06$, or ${{F}_{A}}=0.0133\pm 0.0015$.

Around the same time as the SIN experiment, a group at the Los Alamos Meson Physics Facility (LAMPF) made a measurement of ${{F}_{A}}/{{F}_{V}}$ [56] using the Crystal Box detector which consisted of 396 NaI(Tl) crystals, 36 plastic scintillation hodoscope counters, and a cylindrical drift chamber surrounding the stopping target [57]. Unlike the other experiments discussed above, the Crystal Box detectorʼs single-particle acceptance for events originating in the target was about 45% of $4\pi$ sr. Such a large acceptance enabled the experimenters to cover a broad portion of the decay phase space, collecting 2364 coincidence events that included both decay signals and random coincidences. However, only $71\;\pm \;11$ high-energy nearly back-to-back photon–positron pairs were used to determine γ, as they were largely background free and strongly dominated by the SD⁺ amplitude. The low statistics is reflected in the large relative uncertainty in $\gamma =0.25\pm 0.12$; however the remaining ∼2300 coincident events were included in a likelihood analysis which preferred the positive γ solution by a factor of 2175 : 1. In all, the likelihood analysis found a total of $179\;\pm \;18$ $\pi \to e\nu \gamma$ events, in good agreement with the integral of the e–γ timing peak. Thus, although a low statistics measurement, this experimentʼs main result is the strongly favored selection of the $\gamma \gt 0$ solution, decisively removing the prior longstanding quadratic ambiguity.

Although the subject of this review are measurements of rare decays of the pion, it is worthwhile to consider the 1988 analysis of Dominguez and Solà [58] who extracted a soft-pion value for ${{F}_{A}}(0)$ from semileptonic tau lepton decays, $\tau \to {{\nu }_{\tau }}+n\pi$. Analysis of decays with odd (even) values of n gives access to the vector (axial-vector) hadronic spectral functions up to the kinematical limit $t\sim 3\;{\rm Ge}{{{\rm V}}^{2}}$. Dominguez and Solà studied and refined such fits to existing τ decay data, and extracted the soft-pion value of ${{F}_{A}}(0)=0.017\pm 0.004$, or $\gamma (0)={{F}_{A}}(0)/{{F}_{V}}(0)=0.67\pm 0.17$, with $F_{V}^{{\rm CVC}}\simeq 0.0254$. We note that this result is independent of pion radiative decay data.

The lone measurement of $\pi \to e\nu \gamma$ decays in flight was performed by the Moscow Institute for Nuclear Research (INR) group, using the ISTRA apparatus and a 17 GeV pion beam at the IHEP Protvino U70 accelerator [59]. The experimental technique and systematics are radically different from all of the stopped pion decay measurements discussed so far. Since decays occurred in flight, the experimenters were free to use negative pions and observe ${{\pi }^{-}}\to {{e}^{-}}\nu \gamma$ decays, making this the only experiment to date to do so. The wide acceptance of the apparatus enabled the experimenters to study the pion rest frame kinematical region defined by

$$\begin{eqnarray}&&{{E}_{\gamma }}\gt 21\;{\rm MeV},\quad {{E}_{e}}\gt 70\;{\rm MeV}-0.8{{E}_{\gamma }},\quad {\rm and}\quad {{\theta }_{e\gamma }}\gt 60{}^\circ .\end{eqnarray} \tag{ 10 }$$

The results of this work, obtained in a likelihood analysis, can be summarized as follows: (a) $\gamma \equiv {{F}_{A}}/{{F}_{V}}=0.41\pm 0.23$ (with no corresponding value for F_V quoted explicitly), (b) negative solution for γ disfavored by a likelihood factor of $5\times {{10}^{9}}$ or 6.7 standard deviations, (c) an independently determined ${{F}_{V}}=0.014\pm 0.009$, and (d) the branching ratio for the kinematic limits of equation (10), $B=(1.61\pm 0.23)\times {{10}^{-7}}$. Although not explicitly given, the total number of $\pi \to e\nu \gamma$ decay events appears to be approximately 90 after background subtraction, which would explain the large quoted uncertainties. Besides the strong preference for the $\gamma \gt 0$ solution, this work presented another notable result: a deficit of SD⁻ events, forcing a non-physical negative SD⁻ amplitude. The authors speculated that the deficit may be due to a destructive interference with a tensor term of the size ${{F}_{T}}=0.0056\pm 0.0017$. We will revisit the tensor hypothesis below.

3.2.2. The decay ${{\pi }^{+}}\;\to \;{{e}^{+}}{{\nu }_{e}}{{e}^{+}}{{e}^{-}}$ (or π_e2ee)

This pion decay mode merits special attention, as it contributes information not accessible through the regular ${{\pi }_{e2\gamma }}$ radiative decay channel. Following unsuccessful attempts [60], the π_e2ee decay was first observed by the SINDRUM I collaboration in a 1985 experiment [61]; the same data were reanalyzed more carefully and the results reported in 1989 [62].

The SINDRUM detector system, which became known as 'SINDRUM I' after the construction of 'SINDRUM II', was built primarily to search for the forbidden $\mu \to 3e$ decay [63]. The instrument, schematically depicted in figure 10, was capable of recording the trajectories of electrons and positrons in a solenoidal magnetic field with the help of five concentric cylindrical MWPCʼs of very similar design and construction to those used later in the PIBETA/PEN experiments. Decay particles were required to reach a 64 element scintillating hodoscope situated outside the tracking detectors, resulting in a lower threshold on transverse momentum around $17\;{\rm MeV}\;{{c}^{-1}}$. A cone-shaped 12-element segmented scintillating target was used, enabling a reduction of random coincidences between beam pions and decay particles by an order of magnitude.

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In their first paper Egli et al [61] reported merely the first observation of the decay, with 79 recorded events. The authors also reported values of form factor ratios $\gamma ={{F}_{A}}/{{F}_{V}}$ and $R/{{F}_{V}}$ with large uncertainty limits. They were, however, able to exclude by a ratio of 9:1 the previously reported large negative value of $\gamma =-2.49\pm 0.06$ from ${{\pi }_{e2\gamma }}$ [54] in favor of a smaller positive solution $\gamma =0.7\pm 0.5$, thus anticipating the soon to follow ${{\pi }_{e2\gamma }}$ results of the Lausanne–Zurich [55] and LAMPF [56] groups.

In a second article [62], the SINDRUM I collaboration reported results of a more careful and comprehensive analysis of the same 1985 data set. By applying a series of cuts designed to reduce the backgrounds, the authors arrived at a data set of 98 events with one background event, based on $\sim 4\times {{10}^{12}}$ pions stopped in the target. Using this event set, the detector acceptance, the CVC value F_V = 0.0255, the Bay et al value ${{F}_{A}}=0.0133\pm 0.015$ [55] and the PCAC value for the second axial form factor ${{R}_{{\rm PCAC}}}=0.068\pm 0.004$, the authors derive the branching ratio for the full phase space:

$$\begin{eqnarray}&&B({{\pi }^{+}}\to {{e}^{+}}{{\nu }_{e}}{{e}^{+}}{{e}^{-}})=(3.2\pm 0.5\pm 0.2)\times {{10}^{-9}},\end{eqnarray} \tag{ 11 }$$

where the second uncertainty is propagated from ${{R}_{{\rm PCAC}}}$. Of additional interest are the values for the three pion form factors derived in a likelihood analysis of the data:

$$\begin{eqnarray}&&{{F}_{V}}=0.023_{-0.013}^{+0.015},\qquad {{F}_{A}}=0.021_{-0.013}^{+0.011},\qquad R=0.059_{-0.008}^{+0.009}.\end{eqnarray} \tag{ 12 }$$

It is worth noting that fixing the vector form factor value to $F_{V}^{{\rm CVC}}=0.0255$ does not significantly alter the maximum likelihood values of F_A and R.

A subsequent Dubna experiment [64] which detected only seven events of π_e2ee decay did not improve the precision of the SINDRUM I branching ratio for the decay.

3.2.3. Summary of the early measurements:

Study of the radiative pion decay has been a major component in the long term program of the PIBETA/PEN experiments at PSI. Firstly, the strong intrinsic physics significance for low-energy QCD and ChPT as well as the sensitivity to non-V–A interaction terms such as tensor, and its relatively poor experimental quantification prior to 2000 discussed in the preceding section, clearly placed radiative pion decay at high priority. Secondly, ${{\pi }^{+}}\to {{e}^{+}}\nu \gamma$ events for which the positron annihilates externally create a significant background in the measurement of the pion beta decay ${{\pi }^{+}}\to {{\pi }^{0}}{{e}^{+}}\nu$, π_e3, the primary goal of the PIBETA experiment, discussed in section 4 below. Thirdly, π_e3 events for which one of the photons in the subsequent π⁰ decay converts into an asymmetric ${{e}^{+}}{{e}^{-}}$ pair, with only one detected electromagnetic shower, create a background for the ${{\pi }_{e2\gamma }}$ process. Furthermore, a precise measurement of $R_{e/\mu }^{\pi }$ of equation (2) is impossible without a precise knowledge of the radiative decay width. In fact, because of the overlapping nature of the instrumental response functions to the various decay processes, precision measurement of any one of them requires simultaneous detection and characterization of all pion and muon decay processes present in the data sample.

For all these reasons, the PIBETA collaboration⁵ performed a series of measurements at the PSI, focused on improving the experimental precision of the ${{\pi }_{e2\gamma }}$ branching ratio as well as the form factors F_A and F_V. The apparatus is essentially the same as in the PEN configuration shown in figure 3. The only differences were the absence of the beam mTPC, the use of a segmented target in the early runs (section 4), a thicker AT and degrader detectors to accommodate the higher pion beam momentum of $\sim 114\;{\rm MeV}\;{{c}^{-1}}$, and custom electronics for one- and two-arm triggers. The arrangement of the central detectors is shown in figure 11. The data were collected in two distinct sets of runs.

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The first PIBETA radiative pion decay measuring period took place during 1999–2001, in a configuration optimized for recording pion beta decay, ${{\pi }^{+}}\to {{\pi }^{0}}{{e}^{+}}{{\nu }_{e}}$, events with a high stopping rate of pions in the target, averaging approximately $8\times {{10}^{5}}\;{{{\rm s}}^{-1}}$. The experiment trigger logic was based on shower energy summed over a cluster of adjacent CsI calorimeter detector modules, with two energy thresholds: a high one of $\sim 51\;{\rm MeV}$, and a low one of $\sim$5 MeV. The experiment recorded every event containing two showers, both with energy above the high threshold (HT), and spatially separated by an opening angle ${{\theta }_{12}}\;\gtrsim \;90{}^\circ$. Two-shower events with one exceeding the high and the other the low energy threshold were prescaled by a factor of at least an order of magnitude, while events with two low-threshold showers were prescaled even more. One-arm events, consisting of one shower were also recorded, with the purpose of collecting significant samples of ${{\pi }^{+}}\to {{e}^{+}}\nu$ and ${{\mu }^{+}}\to {{e}^{+}}\nu \bar{\nu }$ events for normalization and systematic studies. Detector response to the $\pi \to e\nu$ single-arm events, shown in figure 12, best illustrates the intrinsic performance of the spectrometer. Over $4\times {{10}^{4}}$ ${{\pi }_{e2\gamma }}$ events were collected and analyzed in this set of runs. The results, bringing about an order of magnitude improvement in precision of the branching ratio and ${{F}_{A}}/{{F}_{V}}$, were published in [38]. However, the authors found a deficit of events in one kinematic region, corresponding to high-${{E}_{\gamma }}$ and low-E_e events, that was well outside the limits of the statistical uncertainties of the fit. The authors concluded that further study was needed in a dedicated measurement, as the first run had been optimized for measurement of ${{\pi }^{+}}\to {{\pi }^{0}}{{e}^{+}}{{\nu }_{e}}$ decays.

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The second run, dedicated to radiative ${{\pi }_{e2\gamma }}$ decays took place in 2004, with the pion stopping rate lowered to $1.5\times {{10}^{5}}\;{{{\rm s}}^{-1}}$ and the trigger electronics modified to accept all HT one-arm events. The first half of the run was carried out with the same detectors as in 1999–2001, while in the second half the nine-element AT was replaced with a single unit. Approximately $2.4\times {{10}^{4}}$ additional events were collected over significantly broader kinematic regions. Most importantly, the lower beam rate resulted in large improvements in the ratio of peak signal to accidental background (P/B) in $\Delta {{t}_{e\gamma }}$ spectra, and sufficient low-energy data to perform an independent energy calibration for the charged and neutral showers in the HT trigger. The combined data sets were carefully analyzed, and the results published in [39]. The key results are shown in figures 13–15.

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With the improved calibrations, the previously observed event deficit vanished, as seen in figure 13, which shows the measured yield in eight different kinematic regions, compared with the best-fit standard description of equation (7) with the addition of radiative corrections by Bystritsky et al [67]. No statistically significant deviations were observed.

The wider kinematic coverage of the new measurement enabled a much improved analysis, including an independent determination of the polar- and axial-vector form factors F_V and F_A and for the first time, a determination of the four-momentum transfer ${{q}^{2}}={{\left( {{p}_{e}}+{{p}_{\nu }} \right)}^{2}}$ dependence of the polar vector amplitude F_V, figure 14:

$$\begin{eqnarray}&&{{F}_{V}}({{q}^{2}})={{F}_{V}}(0)+a{{q}^{2}}\qquad {\rm with}\qquad a=0.10\pm 0.06.\end{eqnarray} \tag{ 15 }$$

The uncertainties on the slope parameter are wide enough to accommodate the resonance chiral perturbation calculation of Mateu and Portolés [68].

Even after the inclusion of the lower-energy phase space regions, the PIBETA ${{\pi }_{e2\gamma }}$ decay data remained dominated by the SD⁺ amplitude, leading to a stringent constraint on ${{F}_{V}}+{{F}_{A}}$, and a lax one on ${{F}_{V}}-{{F}_{A}}$. As in previous measurements, the probability of the negative solution for $\gamma ={{F}_{A}}/{{F}_{V}}$ is disfavored by a large factor, about 880. The narrow elliptically shaped locus of best-fit values for F_A and F_V is shown in figure 15. The corresponding equation of the major axis of the ellipse is given by

$$\begin{eqnarray}&&{{F}_{A}}=(-1.0286\times {{F}_{V}}+0.038\;53)\pm 0.000\;14.\end{eqnarray} \tag{ 16 }$$

We note that, for a fixed value of F_V, the uncertainty in F_A has been reduced 16-fold compared to the prior world average. In a fully unconstrained fit, the authors report

$$\begin{eqnarray}&&{{F}_{V}}=0.0258\pm 0.0017\qquad {\rm and}\qquad {{F}_{A}}=0.0117\pm 0.0017.\end{eqnarray} \tag{ 17 }$$

The PIBETA result for F_V represents a five-fold improvement over the prior average. The correlated nature of the PIBETA form factor results allows for a posteriori improvement in precision of F_A as the precision of the determination of the π⁰ lifetime improves.

Finally, thanks to the experimentʼs broad kinematic coverage, the PIBETA authors have reported a branching ratio

$$\begin{eqnarray}&&{{B}_{{{\pi }_{e2\gamma }}}}({{E}_{\gamma }}\gt 10\;{\rm MeV},{{\theta }_{e\gamma }}\gt 40{}^\circ )=73.86(54)\times {{10}^{-8}},\end{eqnarray} \tag{ 18 }$$

with more than an order of magnitude improved precision over the previous result.

The PEN experiment will add new data to the existing ${{\pi }_{e2\gamma }}$ data set. Since the PEN beam stopping rate is significantly lower than that used in the 2004 PIBETA run, the impact on the statistical uncertainties will not be great. However, the cleaner running conditions of PEN present greater access to the kinematic region ${{E}_{\gamma }},{{E}_{e}}\lt 50$ MeV, previously strongly contaminated by muon decay background and thus, holding the prospect for a better constraint of the SD⁻ amplitude which, in turn would lead to improved precision in the direct determination of F_V. These results will be forthcoming in the near future.

Unlike the ${{\pi }_{e2(\gamma )}}$ decays discussed in the preceding sections, the extremely rare $\mathcal{O}({{10}^{-8}})$ pion beta decay is not suppressed by any factor apart from the restricted phase space of final states due to the small difference between the charged and neutral pion masses, $\Delta ={{m}_{\pm }}-{{m}_{0}}$. As a pure vector ${{0}^{-}}\to {{0}^{-}}$ transition, it is completely analogous to the superallowed Fermi nuclear beta decays. In fact, it can be regarded as the simplest realization of the latter, fully free of complications arising from nuclear structure corrections. Superallowed Fermi decays have historically led to the formulation of the CVC hypothesis, and have played a critical role in testing the unitarity of the CKM quark mixing matrix through evaluations of the V_ud element [69–71].

Within the framework of the V–A theory of the weak interactions, the pion beta decay rate can be expressed in terms of the leading-order width Γ₀ and the radiative and loop corrections ${{\delta }_{\pi }}$ as [72, 73]

$$\begin{eqnarray}&&\Gamma ={{\Gamma }_{0}}(1+{{\delta }_{\pi }})=\frac{G_{F}^{2}|{{V}_{ud}}{{|}^{2}}}{30{{\pi }^{3}}}{{\Delta }^{5}}f(\epsilon ,\Delta ){{\left( 1-\frac{\Delta }{2{{m}_{+}}} \right)}^{3}}(1+{{\delta }_{\pi }}),\end{eqnarray} \tag{ 19 }$$

where G_F is the Fermi coupling constant and $\epsilon =m_{e}^{2}/{{\Delta }^{2}}\simeq 0.012\;375$. Finally, up to leading order in ${{\Delta }^{2}}/{{\left( {{m}_{+}}+{{m}_{0}} \right)}^{2}}\;\simeq \;2.8\times {{10}^{-4}}$, the function $f(\epsilon ,\Delta )$ has the value of $f(\epsilon ,\Delta )\;\simeq \;0.941\;04$. The overall uncertainty of the rate in (19) is dominated by two comparable contributions, one from the Δ⁵ factor, and the other from the ${{\delta }_{\pi }}$ radiative/loop corrections, each uncertainty contribution being in the range of 0.05–0.1% of the full rate [35, 70, 74, 75]. Thus, the pion beta decay rate provides a direct means to determine the CKM matrix element $|{{V}_{ud}}|$. In fact, being free of nuclear structure corrections present in superallowed nuclear beta decays, and free of tree-level axial corrections present in neutron beta decay, π_e3 decay offers the theoretically cleanest path to measuring V_ud and hence, testing quark-lepton universality. However, the extremely low branching ratio for the process has so far limited the experimental precision.

Experimentally, ${{\pi }^{+}}\to {{\pi }^{0}}{{e}^{+}}{{\nu }_{e}}$ decay is observed primarily through detection of the final decay particles produced in the near-instantaneous neutral pion decay ($\tau \simeq 8.5\times {{10}^{-17}}$) s. The positron is generally harder to detect with precise efficiency, except for π⁺ decays in flight, since its kinetic energy ranges from 0 to only 4 MeV. The key π⁰ decay modes and their branching ratios are:

$$\begin{eqnarray}\begin{array}{rcl} {{\pi }^{0}}\to & \gamma \gamma , & \quad B\simeq 0.988 \\ {} & \gamma {{e}^{+}}{{e}^{-}}, & \quad B\simeq 0.012. \\ \end{array}\end{eqnarray} \tag{ 20 }$$

All experiments to date have focused on the $2\gamma$ channel rather than the $\gamma {{e}^{+}}{{e}^{-}}$, Dalitz mode. Due to the small charged to neutral pion mass difference, the maximum kinetic energy of the π⁰ is low, about 75 keV. Thanks to the correspondingly low π⁰ velocity, the directions of the two emitted photons deviate from $180{}^\circ$ by no more than $3.8{}^\circ$, on average by $\sim 3.2{}^\circ$. Furthermore, the Doppler broadening of the photon energies results in a narrow boxlike spectrum between $\sim$ 65.6 MeV and $\sim$ 69.4 MeV. This kinematics provides a robust signal that is additionally separated in time from the dominant prompt hadronic background events thanks to the 26 ns pion lifetime.

The first observation [76] of the decay at CERN in 1962 using a stopped π⁺ beam and a combination of lead glass and NaI(Tl) detectors, was followed by a quick succession of early measurements at CERN [77], Dubna (total absorption Pb glass Čerenkov counters) [78], Columbia University [79], Lawrence Berkeley Lab [80] and Carnegie Tech (now Carnegie Mellon) [81]. The Berkeley, Columbia and Carnegie Tech experiments used combinations of spark chambers and scintillation detectors. The five measurements achieved approximately 20% uncertainties on the branching ratio, based on samples of between 30 and 50 events each.

To date only three measurements of the ${{\pi }^{+}}\to {{\pi }^{0}}{{e}^{+}}{{\nu }_{e}}$ decay branching ratio have been made with precision better than 10%, approximately one for every 20 years, which reflects the challenges of the task.

The CERN group of Depommier et al [82] was first to break the 10% uncertainty level in 1967, using a lead glass Čerenkov detector array along with plastic scintillator detectors, as shown in figure 16. The CERN apparatus had an acceptance of $\sim$22.4%, and enabled the experimenters to record $332\pm {{10}_{{\rm bgd}}}\pm {{23}_{{\rm stat}}}$ pion beta decays from $\sim 1.5\times {{10}^{11}}$ pion stops in the target. The reported branching ratio value was

$$\begin{eqnarray}&&B({{\pi }_{e3}})\equiv \frac{\Gamma ({{\pi }^{+}}\to {{\pi }^{0}}{{e}^{+}}\nu )}{\Gamma ({{\pi }^{+}}\to {{\mu }^{+}}\nu )}=\left( 1.00_{-0.10}^{+0.08} \right)\times {{10}^{-8}}\end{eqnarray} \tag{ 21 }$$

in very good agreement with CVC theory predictions, but not precise enough to test the radiative corrections at $\sim$4%. The authors increased the lower-side uncertainty of their result in order to account for the possible influence of nuclear reaction products in their measurement of the π⁰ detection efficiency of the lead glass detector array.

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In the early 1980s a Temple–Los Alamos group performed a radically different measurement of the pion beta decay rate at LAMPF [83]. Unlike previous experiments which detected decays of π⁺ʼs brought to rest in a target, McFarlane and collaborators detected decays of an intense pion beam in flight in the apparatus shown in figure 17. At $2\times {{10}^{8}}$ pions s $^{-1}$ and with a momentum of $522.1\pm 0.8\;{\rm MeV}\;{{c}^{-1}}$, the LAMPF pion beam was orders of magnitude more intense and an order of magnitude higher in energy than in the previous experiments. The decay volume was evacuated to $\sim {{10}^{-7}}$ torr in order to limit the pion hadronic interactions with material in the path, primarily single charge exchange reactions which would be hard to distinguish from pion beta decay events. Because of the low recoil in the π⁺ rest frame, the π⁰ had transverse momentum of less than $5\;{\rm MeV}\;{{c}^{-1}}$, or 1% of the longitudinal momentum. The average π⁰ momentum of about $505\;{\rm MeV}\;{{c}^{-1}}$ implied a short mean π⁰ laboratory flight length of about 100 nm before decay. The acceptance was defined by the combination of an upstream collimator, minimum opening angle of the two photons, and the solid angle of the active area of the two photon detectors. The latter were the two arms of the LAMPF π⁰ spectrometer [84], appropriately modified [83] for the pion beta decay measurement. Each photon detector arm consisted of a front plastic scintillator veto detector followed by three successive lead glass converter layers (0.56 radiation length thick, each). Each converter layer was followed by two scintillation hodoscopes for position and time measurement. Finally, behind the forward crate containing the converter-hodoscope layers was a rear crate with an array of 15 lead glass blocks, forming a 14 radiation lengths thick calorimeter. The cross sectional area of the detector package as seen by the photons was $45\;{\rm cm}\times 75\;{\rm cm}$. The arms were well matched to detecting the final-state photons that ranged between 175 and 350 MeV in energy. At ${{\sigma }_{t}}\simeq 250$ ps per arm, the time resolution was excellent for such large detectors. At ${{\sigma }_{E}}\simeq 1.53\sqrt{E}$ (in MeV), the energy resolution was adequate for the task.

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This measurement apparatus has several advantages over the stopped decay technique. First, the hadronic background can be effectively eliminated in an evacuated decay region. Second, the high pion beam momentum focuses the decay photons into a solid angle significantly smaller than $4\pi$ sr. Finally, in principle, the measurement can be done with both charged pion states, although in practice the intensity of π⁻ beams is significantly lower than that of π⁺.

However, there are also significant disadvantages compared to the stopped decay measurement. The proper time that beam pions spend in the decay region is very short, about 10⁻³ pion lifetimes, requiring enormously higher beam intensities to achieve high statistical precision for the rare pion beta decays. Normalization, i.e., counting the number of pions passing through the apparatus is very challenging, as the readily available processes such as the ${{\pi }_{\mu 2}}$ decay have radically different kinematics and detection systematics, and require separate detectors. Finally, detection efficiency (including detector acceptance), as well as trigger and event selection efficiency must be determined in absolute terms with high precision.

Given all the above considerations, the Temple–LAMPF group reported the experimental partial rate, $1/{{\tau }_{\pi \beta }}$, and branching ratio, ${{B}_{\pi \beta }}$:

$$\begin{eqnarray}&&\frac{1}{{{\tau }_{\pi \beta }}}=(0.394\pm 0.015){{{\rm s}}^{-1}}\qquad {\rm or}\qquad {{B}_{\pi \beta }}=(1.026\pm 0.039)\times {{10}^{-8}}\end{eqnarray} \tag{ 22 }$$

based on 1124 ± 36 pion beta decay events, and $2.14\times {{10}^{14}}$ beam pions. The overall 3.8% uncertainty breaks down to 3.1% statistical and 2.0% systematic contributions. The largest systematic uncertainty contributions arose from the number of beam pions, time in decay region, and the various efficiencies. The authors also extracted a new value of $0.964\pm 0.019$ for the cosine of the Cabibbo angle, or V_ud. While not as precise as the contemporary values of V_ud derived from superallowed Fermi nuclear beta decays, it was consistent with them.

The primary goal of the PIBETA experiment conducted at PSI and introduced in sections 2 and 3, was to improve by an order of magnitude the existing experimental precision of the pion beta decay branching ratio. The apparatus is described schematically in figure 3, while the central detector region is shown in figure 11. A $\sim 114\;{\rm MeV}\;{{c}^{-1}}$ pion beam of the PSI πE1 beamline was focused onto the segmented nine-element AT which stopped an average of $0.8-1\times {{10}^{6}}$ ${\rm pions}\;{{{\rm s}}^{-1}}$. The PIBETA approach was essentially similar to the 1967 CERN experiment of Depommier et al [82], but with larger acceptance, much improved energy resolution of the electromagnetic shower calorimeter, MWPC tracking of charged particles between the central region and the calorimeter, a more intense pion beam, and faster electronics and data acquisition system. The key detector response functions to pion beta decay events are shown in figures 18–20. The corresponding spectra for the normalizing π_e2 decay, and for the ${{\pi }_{e2\gamma }}$ decay which can contribute background events for the pion beta decay, are shown in sections 2.3 and 3.3. Since a subset of pion beta decays can also be misidentified as radiative ${{\pi }_{e2\gamma }}$ decays, it was essential that all three processes be simultaneously accounted for and understood at the few parts per thousand level. In fact, ordinary ('Michel') and radiative muon decays, though not discussed here, were included as well in the comprehensive PIBETA analysis, since they comprise the majority of actual events occurring in the apparatus, and dominate the accidental background to all pion decay channels. Figures 18–20 demonstrate that the pion beta decay event sample was clean and well described in terms of the relevant instrumental resolutions.

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During three runs in 1999–2001, the PIBETA collaboration acquired over 64 000 pion beta decay events, which led to an improvement of the experimental precision of the π_e3 branching ratio of about an order of magnitude [37]. The PIBETA result was evaluated in two ways, (a) normalizing to the average experimental π_e2 branching ratio of (5) $R_{e/\mu }^{\pi }=(1.230\pm 0.004)\times {{10}^{-4}}$ ('exp. norm.'), and (b) normalizing to the established theoretical value of (3), $R_{e/\mu }^{\pi }=(1.2352\pm 0.0005)\times {{10}^{-4}}$ ('theo. norm.'). The resulting values were:

$$\begin{eqnarray}&&B_{\pi \beta }^{{\rm exp} .\;{\rm norm}.}=\left[ 1.036\pm 0.004({\rm stat})\pm 0.004({\rm syst})\pm 0.003({{\pi }_{e2}}) \right]\times {{10}^{-8}},\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&B_{\pi \beta }^{{\rm theo}.\;{\rm norm}.}=\left[ 1.040\pm 0.004({\rm stat})\pm 0.004({\rm syst}) \right]\times {{10}^{-8}}.\end{eqnarray} \tag{ 24 }$$

The external experimental uncertainty included the uncertainties of the branching ratios $R_{e/\mu }^{\pi \;{\rm exp} }$, $B_{{{\pi }^{0}}\to \gamma \gamma }^{{\rm exp} }$ and that of $\tau _{{{\pi }^{+}}}^{{\rm exp} }$, the pion lifetime, and were dominated by the former. The combined internal systematic uncertainty was dominated by the uncertainties related to the ratio of the 'pion gate' live fractions for the π_e3 and π_e2 processes, and the number of the π_e2 events used for normalization, followed by the ratio of acceptances for the same two processes.

This result represents the most stringent test of CVC and Cabibbo universality in a meson, as well as of the proper treatment of the radiative corrections, which, combined, predict $B_{\pi \beta }^{{\rm SM}}=(1.038-1.040)\times {{10}^{-8}}$ at 90% confidence limits [65]. Excluding radiative corrections gives the range $B_{\pi \beta }^{{\rm SM}}=(1.005-1.007)\times {{10}^{-8}}$, a $\gt 4\sigma$ discrepancy from the PIBETA result. The PIBETA collaborators also extracted

$$\begin{eqnarray}&&V_{ud}^{\pi \beta /{\rm exp} .{\rm norm}.}=0.9728\pm 0.0030,\quad {\rm and}\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&V_{ud}^{\pi \beta /{\rm theo}.{\rm norm}.}=0.9748\pm 0.0025.\end{eqnarray} \tag{ 26 }$$

These values are in excellent agreement with the PDG recommended value for V_ud, although five times less precise.

Following in the same vein, it is tempting to go one step further and turn the PIBETA determination of the π_e3 branching ratio around, using it instead to evaluate $R_{e/\mu }^{\pi }$. This is accomplished by fixing V_ud to its extraordinarily precise PDG 2013 recommended value of $0.974\;25\pm 0.000\;22$ and adjusting $R_{e/\mu }^{\pi }$ until the extracted value of $V_{ud}^{\pi \beta }$ agrees. This intriguing exercise yields

$$\begin{eqnarray}&&{{(R_{e/\mu }^{\pi })}^{{\rm PIBETA}}}=(1.2366\pm 0.0064)\times {{10}^{-4}},\end{eqnarray} \tag{ 27 }$$

in good agreement with direct measurements reviewed in section 2. Appropriately averaging this value of $R_{e/\mu }^{\pi }$ with those listed in table 1, one obtains a slightly higher value than the current PDG average: ${{(R_{e/\mu }^{\pi })}^{{\rm new}\;{\rm avg}}}=(1.2317\pm 0.0031)\times {{10}^{-4}}$.

Given the advantage in lower theoretical uncertainties compared to nuclear decays, there was every incentive to pursue a higher precision result in the pion beta decay rate or branching ratio. However, the urgency of a further improvement was considerably reduced by the Brookhaven National Laboratory experiment BNL E865 result [85] and the subsequent renormalization of V_us that removed a longstanding 2–3σ shortfall in CKM matrix unitarity. In light of the considerable experimental challenges that an improved measurement would pose, the present full agreement of available data on V_ud with CKM unitarity means that a major new project on improving the precision of pion beta decay is not currently planned. We note, though, that the status of neutron beta decay is in considerable flux, with major discrepancies in the available data sets on both the beta asymmetry and the neutron lifetime [86, 87]. At present there are a number of projects addressing these deficiencies. While not as theoretically clean as the pion beta decay, neutron decay is free of the nuclear structure and Coulomb corrections that affect nuclear beta decays, including the superallowed Fermi ${{0}^{+}}\to {{0}^{+}}$ transitions [88].