Precise Measurements of the Decay of Free Neutrons

The impact of new and highly precise neutron \b{eta} decay data is reviewed. We focus on recent results from neutron lifetime, \b{eta} asymmetry, and electron-neutrino correlation experiments. From these results, weak interaction parameters are extracted with unprecedented precision, possible also due to progress in effective field theory and lattice QCD. Limits on New Physics beyond the Standard Model derived from neutron decay data are sharper than those derived from high-energy experiments, except for processes involving right-handed neutrinos.


Introduction
The β decay of the neutron into proton, electron and electron-antineutrino → + − ̅ is a fascinating field of study. It addresses many basic issues of contemporary physics, relevant at the smallest scales of elementary particle physics up to the largest scales of space-time in cosmology, astrophysics, and solar physics. Most experiments in nuclear and particle physics use accelerators with energies from MeV up to the TeV scale. In contrast, slow, i.e., thermal, cold, or ultracold neutrons (UCN), see left side of Table 1 below, have energies from meV down to peV, 24 orders of magnitude below the TeV scale, which is a world apart, unfamiliar to the majority of nuclear or particle physicists.
Due to their neutrality, neutrons cannot easily be accelerated, deflected, or focused. This is the burden of experimental neutron physics, but at the same time, it is its main asset: Neutrons react to all known forces of nature but not to the usually overwhelming electrostatic force. This makes neutrons highly sensitive to the subtlest effects.
In Section 2 we first want to convey the special flavor of experimental work with slow neutrons. The theory of neutron decay, including effective field theory (EFT) and lattice gauge theory, is treated in Section 3. Recent neutron decay experiments are reviewed in Sections 4.
Section 5 discusses applications of neutron decay data. EFT and lattice gauge theory make it possible to compare limits on New Physics obtained from low and high-energy experiments, as shown in Section 6.
Previous reviews on neutron decay include Refs. (1)(2)(3)(4)(5), a series of articles introduced by Ref. (6), plus a recent neutron conference (7). For reviews on beta decay in general, see (8)(9)(10), and references therein. In view of these previous reports, the present review will focus on the exciting developments of the past few years.
Neutron β decay is a subfield of neutron particle physics, which covers in addition searches for a neutron electric dipole moment (EDM) (11), of neutron-antineutron (12) and neutronmirror neutron oscillations (13), all sensitive to neutron energy shifts on the 10 -24 eV scale.
Furthermore, there exist studies on hadronic parity violation (14), on gravitational quantum states and limits on dark energy (15), and on the foundations of quantum mechanics via neutron interferometry (16), together with firsthand observations of geometric phases (17) and dressed particle effects (18).
the receding blades of a turbine (21), or by having neutrons lose their energy to local excitations in superfluid helium or in solid deuterium (22). Comparative measurements on different UCN sources are reported in (23).

Spin-polarization of neutrons.
Thermal or cold neutrons can be polarized via the spindependence of their magnetic or strong interaction. This is achieved by total reflection from magnetized supermirrors (32), or by transmission through nuclear spin-polarized 3 He gas, achieving >99% polarization with 10 -4 precision for beams of up to decimeter diameter, see (33,34). Neutron spins are inverted in-flight with near 100% efficiency by the adiabatic fastpassage method. To polarize a beam of UCN, one simply blocks one spin component by a magnetic barrier.

Detection of neutrons.
Slow neutrons can be only detected destructively, via various neutron capture reactions. For a long time, 3 He filled Geiger-Müller tubes were standard neutron detectors. Due to the 3 He-shortage of recent years, several new neutron detection systems were developed, see (35,36). To detect UCN, one must let them fall down by ~1 m to gain sufficient energy to overcome the reflective potential of the detector surface.
The components of the relativistic four-vectors ψ are reshuffled by multiplication with the γ matrices.
The unification of the weak and the electromagnetic interaction requires the weak vector current to be conserved in the same way as the electromagnetic vector current (CVC) is conserved. Weak CVC holds across the three particle families, but only if we take into account that, again for reasons unknown, the weak eigenstates of the quarks are slightly rotated away from their mass eigenstates, with angles given by the Cabibbo-Kobayashi-Maskawa (CKM) quark-mixing matrix (38,39), whose first (up-down) matrix element Vud enters Equation 1. The CKM matrix must be unitary, which means | | 2 + | | 2 + | | 2 = 1 for its first row, where Vus describes the mixing of the u quark with the strange s quarks of the second particle family.
The third term |Vub| 2 , which involves the bottom b quarks of the third family, is negligible in this context.

Neutron decay at the nucleon level.
Neutron decay is observed not on the quark level but on the nucleon level. The complicated internal structure of the nucleons must be accounted for by introducing form factors, with ̅ [ V ( 2 ) + A ( 2 ) 5 ] replacing ̅ (1 − 5 ) . In neutron decay, the four-momentum transfer q can be neglected so the form factors can be taken at q 2 →0, and are then called coupling constants. Analogous to the anomalous magnetic moment component of the electromagnetic current (with which it forms a weak isospin triplet), a term called weak magnetism (40) must be added, proportional to the difference of neutron and proton anomalous magnetic moments, κpκn = 3.7 in nuclear magnetons, leading to a ~1% extra decay amplitude.
In contrast to the weak vector current, electroweak unification does not require the conservation of the weak axial-vector current, which has no analog in electromagnetism. In principle, the axial coupling gA could be very large in the strongly interacting environment of nucleons. Yet, all the complications due to the interior structure of the neutron can been reduced to a simple numerical factor λ ≡ gA/gV ≈ -1.275, surprisingly close to -1. Although required by the partially conserved axial current hypothesis (PCAC) (41), an induced pseudoscalar term, though present, is still negligible in neutron decay, in spite of a large coupling gP ≈ 230 from Lattice QCD (42). With Hamiltonian density ℋ = −ℒ, the nucleon transition matrix element then is In the decay probability, proportional to | → | 2 , the density of final states ρ determines the spectra and angular distributions of the outgoing particles, for reviews see (43,44).

The observables
The neutron's β spectrum, shown in Figure 1a, has an endpoint energy E0 = 0.782 MeV, while the proton's endpoint energy is only 751 eV. Integration of the spectrum over electron energy E gives the neutron decay rate with a phase-space statistical factor f, and radiative corrections R ′ and ΔR, to be explained in shown as seen from above, see text.
Besides the neutron lifetime τn, there are many other observables, which can be described by so-called correlation coefficients, see Table 3.
Six two-fold, four triple, four 4-fold, and one 5-fold scalar products can be formed from these vectors. Of the 16 possible correlation coefficients (including the Fierz term), Table 3 lists  Neglecting small radiative and other corrections, the dependence of the β-ν correlation a, the β asymmetry A, and the neutrino asymmetry B, to name just a few, on = A V ⁄ is (45),

Effective Field Theory
The physics of elementary particles is based on quantum field theory (QFT). In QFT one first postulates a set of symmetries, which include the symmetries of the SM group SU(3) × SU(2) × U(1) and the Poincaré group of special relativity. Then one writes down the most general Lagrangian ℒ 0 that obeys these symmetries and that is renormalizable (has ) + dim(1 Λ 2 ⁄ ) = 6 − 2 = 4, see Table 4.

Effective field theory of neutron decay.
The conventional description of β decay in Section 3.1 turns out to be a low-energy EFT-approximation to the SM. Indeed, the dimension of the operator Table 3 Table 2, operating on the same fermion fields of , , , ̅ as ℒ . To give an example, the EFT tensor T operator is with Wilson coefficients εT. These V, A, S, T, P-operators had already been included in Lee and Yang's seminal article on parity violation (53). A simple one-to-one correspondence exists between Lee and Yang's coefficients Ci, Ci' and the Wilson coefficients εi. For our example of a tensor T interaction, this correspondence reads T + T ′ = For a complete list, see (10). One can also add to the left-handed (L = V-A) terms right-handed (R = V + A) terms, by replacing in the quark sector 1 -γ5 by 1 + γ5, and one preferably replaces the i = V, A, S, T, P scheme by the i = L, R, S, T, P scheme, which strongly reduces correlations between the various observables. One then speaks of left-and right-handed currents with Wilson coefficients εL and εR. If, instead, 1 ∓ 5 appears in the leptonic sector, one refers to the left-and right-handed neutrinos, with Wilson coefficients ̃ in the notation of Reference (10) and others.
In order to implement EFT at the nucleon level, one adds to Equation 2 all L, R, S, T, P nucleon matrix elements (54,55), like the above tensor T element T ̅ . The first EFT formulation of neutron decay (56) succeeded in reproducing the order 10 -3 radiative and recoil corrections, derived before with conventional methods, see (57) and references therein. Another useful result of this EFT calculation is that the next-order corrections can be limited to be 10 -5 or smaller. However, at present low-energy EFT universality cannot be tested in neutron decay because data for other processes with the same particle content like pion β decay are not yet accurate enough. Other uses of neutron decay EFT are discussed in Sections 5 and 6. in (10).

Standard Model Effective Field
A primary aim of contemporary particle physics is to find the EFT Wilson coefficients, be it from low-or high-energy experiments. The SMEFT approach permits comparison of limits from high-energy experiments with those from low-energy experiments, as discussed in Section 6. Experiments provide limits of the products gi εi or gj wj, so to determine the Wilson coefficients one needs to know the various couplings gi.

Lattice Gauge Theory
The weak couplings gA, gS, gT, and gP are dominated by strong-interaction effects. QCD at low energy cannot be calculated with perturbative methods because the running coupling constant of the strong interaction is no longer a small number. Today, calculations of the weak coupling constants are obtained primarily with lattice QCD theory, formulated on a large lattice of points in space and time, with periodic boundary conditions. The quark fields are placed on the lattice sites and are connected to their neighboring sites by the gluon fields. Lattice QCD relies on Monte Carlo methods and requires large computing resources. The axial coupling is calculated with a precision of ~1% to gA = 1.271 (13) in Reference (62) and to ~3% in References (63)(64)(65).
The numbers in parentheses give the one standard deviation error in units of the least significant digit. The results are compiled in the FLAG Review 2019 (66), which averages them to gA = 1.251 (33). Scalar and tensor couplings are calculated to be gS = 1.02 (10) and gT = 0.99(4).

Neutron Decay Experiments
Many clever approaches exist to measure neutron decay parameters. The format of the present review allows us to cover only a small number of these, and we concentrate on results from the past few years, whose statistics dominate earlier experiments. Only briefly listed are other experiments from the past decade and running experiments, both well covered in Reference (10), while projected experiments are covered in Reference (68). If in the following a result is given with no reference, then it is from the Particle Data Group (PDG-2020), Reference (69). For an overview of results that enter the PDG-2020 average we refer to Figure 4 in Section 5.

Neutron Lifetime
In the past few years, the precision of neutron lifetime measurements has considerably improved. We treat two recent lifetime measurements in more detail, and list shortly other ongoing lifetime experiments.
Contemporary neutron bottles are filled during several minutes with UCN, and then the door of the bottle is closed. For several further minutes, the spectrum is cleaned from "quasi-stable" UCNs whose kinetic energy is (usually only slightly) higher than the repulsive potential of the confining walls. To avoid quasi-stable orbits, all UCN traps have some curved or corrugated surfaces that favor unstable chaotic trajectories. After a holding time T, the door is reopened and the number N(T) of surviving UCNs is counted. This is repeated for two or more different holding times, with T ranging from a few minutes up to some fraction of an hour. If no UCNs are lost, τn is obtained from an exponential fit to ( ) ∝ exp (− ⁄ ), and no absolute measurement is required. For a recent review, see (75).
Lifetime experiments "in-bottle", on which we focus, may suffer from uncontrolled UCN losses. For material storage, these are due to residual inelastic wall interactions, for magnetic storage they are due to uncontrolled neutron spin flips. The energy spectrum of the stored UCN is extremely sensitive to small perturbations: Even the slow closure of the entrance port of an UCN bottle can visibly shift the lifetime result (76).
In the experiments described in the next two subsections, the UCNs are guided into the trap from below and are vertically confined by gravitation. At the end of a storage cycle, the UCNs leave the bottle through the same port to fall onto a UCN detector that is installed ~1 m below the trap. The largest systematic error of one quarter second is due to microphonic heating.

Other lifetime experiments.
-The PNPI-magnetic trap at ILL (84) is built from an upright cylinder of permanent magnets.
A special feature of the experiment is that the UCNs are transported into the decay volume from

Neutron Decay Correlations
In recent years, the precision of neutron correlation measurements has strongly improved as well. We treat a few recent correlation measurements in more detail, and list shortly other ongoing correlation experiments.

General considerations.
Within the SM, all correlations depend only on λ = gA/gV, as in Equation 4. The parity-violating β asymmetry A and the e-̅ correlation coefficient a measure the deviation of |λ| from unity with (about equal) high sensitivity for λ. We limit our discussion to two recent β asymmetry A and one β-ν correlation a measurement, and quote first neutron limits on the possible presence of a Fierz interference term b.
In these experiments, the charged decay particles are spiraling adiabatically about a magnetic guide field, which connects the neutron decay volume with the detector(s). In the two asymmetry experiments, PERKEO III and UCNA, two energy-sensitive electron detectors are positioned symmetrically at both ends of the apparatus. These are fast plastic scintillators, which have low sensitivity to gamma ray background and offer fast timing, as required for electron backscatter detection in the opposite detector. The photons of the scintillators are read out by photomultiplier tubes; an alternate scheme for photon readout with higher energy resolution is proposed in Reference (93). A time-of-flight method for detector characterization was developed in Reference (94). In this context, the detection of synchrotron radiation from gyrating β particles may also become interesting, see (95) and references therein.
Magnetic transport has the advantage that for a decay event that occurs at an arbitrary position x of the decay volume, the solid angle of initial emission along or opposite to the field direction is always exactly 2π, which makes such asymmetry measurements independent of the precise local orientation B(x) of the field and of the detector position. In both β asymmetry experiments, the guide field decreases toward the detector positions to exclude glancing incidence of the electrons on the detector, by means of the inverse magnetic mirror effect. The field decrease also avoids local field minima within the decay volume, in which the electrons could be temporarily trapped and could be lost or assigned to the wrong hemisphere. The spatial distribution of charged decay particles on the detector surface after magnetic transport was calculated and tested with electrons in Reference (96). Both experiments on A use a blinding scheme for data evaluation to eliminate potential bias.

Correlation coefficients B, C, D, R, N:
-The neutrino asymmetry B must be detected via e-p coincidence. The PDG-2020 average is

Rare Standard Model decay channels.
Radiative neutron decay → + − ̅ γ is accompanied by the emission of a bremsstrahlung photon. Its branching ratio of ~1% was precisely measured with the NIST in-beam lifetime apparatus (115). The photon's polarized spectrum was calculated by EFT (116).
Another rare neutron decay channel is bound β decay → H + ̅ , where the emitted electron ends up in a bound S-state of hydrogen H with a branching ratio of 4×10 -6 . For an experiment in preparation, see Reference (117).

Upcoming correlation experiments.
The following instruments are running or are being planned.
-Nab at SNS (118,119) will use the time of flight of electrons and protons to determine the electron-antineutrino coefficient a and the Fierz term b from cuts to the Dalitz plot.
-The proton asymmetry C can be measured without coincidence at high count rates by detection of protons only. Recent results from PERKEO III are still blinded.
-PERC is a beam station that will deliver not neutrons but an intense beam of neutron decay products, extracted from inside a neutron guide, for measurements of decay correlations by interested experimental groups (120,121).
-NoMoS is an R × B drift momentum spectrometer to be installed at PERC (122).
-ANNI will be a cold beam station at the ESS dedicated to neutron-particle physics (123). By exploiting the time-structure of ESS, a successor to PERC could provide more than an order of magnitude improved statistics.
-BRAND generalizes the nTRV concept of electron polarization measurement, see Section 4.2.6, and proposes to measure simultaneously 11 correlation coefficients (124) in one single run.

Applications Other than Particle Physics
The rates of all weak processes that involve both leptons and quarks must be calculated from measured neutron data. We first mention some applications outside of particle physics. In big bang nucleosynthesis, the neutron lifetime enters twice: in the neutrino cross section ∝ 1/ , which determines at what time the early Universe falls out of equilibrium (~1 s); and thereafter in the decaying number of neutrons ∝ exp (− ⁄ ) available for element production. The neutron lifetime contributes the largest error in the calculation of the primordial 4 He mass fraction, see table II in (125). However, this theory error is markedly smaller than the observational error of stellar 4 He abundance, see Review 24 "Big Bang Nucleosynthsesis" in PDG-2020. Hence, for the time being, the experimental lifetime τn is precise enough for this application. The same holds for solar and stellar temperatures, which depend on neutron weak interaction data, and also for the efficiencies of solar neutrino detectors based on inverse neutron decay. Therefore, we refer the reader to our earlier review on these topics (3), and treat in the following applications in particle physics only.

Results within the Standard Model
In this Section, we list results on SM weak interaction parameters derived from neutron, nuclear, and pion β decay experiments. Figure 4a shows the PDG-2020 data for the neutron lifetime τn and λ (updated), whose averages and scale factors are In Equation 3 on τn, the radiative correction R ′ and Δ R are needed. The so-called outer radiative correction R ′ depends only on the electron energy and charge number Z of the daughter nucleus. For the neutron, the phase factor is f = 1.6887 (2), and R ′ = 1.014902 (2) from Reference (126), hence (1 + R ′ ) = 1.7139 (2). Δ R is the transition independent part of the radiative corrections, which is the same for all nuclei, including the neutron. Δ R has recently been reevaluated using dispersion relation techniques (127,128), leading to a reduced error and a 3σ shift from the previous value to Δ R = 0.02477 (24). Even better limits are obtained from a global fit (133) with up to 14 parameters using EFT, based on 6 neutron and more than 30 nuclear observables. This global fit, with a reduced chisquare χ 2 = 0.8, sharpens the neutron lifetime such that its additional dependence on λ leads to a 2.4 times better value for λ: total = −1.27529 (45), and | | total = 0.97370 (25). 10.

Limits beyond the Standard Model
With EFT, large quantities of weak interaction data can be combined within a single fit, see (10,133,134), so why should we bother to evaluate neutron and other data separately? The answer is that each data source has its specific strengths and weaknesses, both from theory and experiment, and it is good practice to have a separate look on them. Besides, overly tight guidance by perspective outlooks may block serendipitous discoveries.
In the following subsections, limits on exotic processes and the corresponding Wilson coefficients are mostly taken from the reviews in References (9) and (10), some of which are updated in Reference (133).  12.

Cabibbo-Kobayashi
An advantage of this SM link between neutron lifetime and λ is that it shows explicitly the which we combine to T = −0.009 (7). As pointed out previously (3), much better limits on the Fierz terms are expected from neutron decay in a joint fit with B(E) and C(E) data, whose strongly elongated χ 2 -contours are nearly orthogonal to those of A(E) and a(E).
T invariance requires coupling constants to be real, with relative phases being either 0 or 180°.
With sin AV = (1 + 3 2 )/(2| |), this D value corresponds to a phase AV = 180.017 (26) figure 3 in (143). These limits can be translated into Wilson coefficients, the mixing angle ζ corresponding to -εR, and the right-to-left mass ratio squared corresponding to R (54). A nonzero ζ would change gA by a factor 1 -2εR, and Vud by 1 -εR (10). The new neutron data do not change these limits significantly, and therefore we do not reopen this topic.

Tests of Lorentz invariance.
Attempts to unify the SM with general relativity often lead to a spontaneous breaking of Lorentz invariance, a process parametrized in Reference (144).
Such a violation can be observed as a sidereal variation of many possible observables; for a recent list of experiments, see (145). Searches for daily variations of neutron and nuclear β decay observables (9) have obtained sub-percent limits on such Lorentz violation (146).
In particle physics, the number of new models is almost unlimited, and so is the number of constraints that may be derived from neutron decay, such as lepton flavor universality (147), so we end our discussion at this point.  (134). However, being universal, SMEFT studies can make global fits to even larger numbers of experimental data, ranging from atomic CP-violating effects (148) through nuclear and particle β decays all the way to the large trove of high-energy data. We have stated that lowenergy EFTs are universal and model independent, but it must be kept in mind that SMEFTs require several assumptions: -The energy gap between Λ and Λ0 must be large, as it is guaranteed in low-energy EFTs with Λ ~ mW >> E ~ Λ0.

Comparison with High-Energy Limits
-No exotic particles must exist with masses below Λ0 (an exception is right-handed neutrinos).
-Exotic particles must be weakly coupled so that electroweak symmetry is linearly realized; -At high energies, dimension 8 operators with their even larger number of field configurations may not be negligible. Furthermore, the LHC limits quoted below assume that only one operator at a time is present ("sole source" vs. "global analysis").
EFT techniques permit comparison of limits for processes with the same Feynman diagram.
An example are pp collisions with ̅ → ̅ , with the neutrino seen as missing transverse energy, which have the same diagrammatic representation as neutron decay → ̅ . When comparing low-and high-energy Wilson coefficients εi and wj, the running of the coupling constants with energy must be taken into account, and the high-energy coefficients wj must be translated to low-energy coefficients εi and ̃, or vice versa, via appropriate matching conditions, see Section 3.3.3. The subscript L and R stand for left-and right-handed quark currents, the tilde stands for right-handed neutrinos. A, D, and R, referred to in the third column, are correlation coefficients. T indicates time reversal violation. Errors in equalities are given at 1σ or 68% CL; errors in inequalities are given at 90% CL or 1.64σ. Table 5 lists the best results on the EFT Wilson coefficients εi and ̃, both from low-energy and high-energy experiments (deduced at the renormalization scale μ = 2 GeV in the minimal Scalar εS = 0.0000(10) Superallowed Ft
The LHC limits given in Table 5  to energy limits of order 10 TeV, but this discussion is beyond the scope of the present article.

Summary
This article has reviewed the present status of neutron β decay experiments with emphasis on new data for the neutron lifetime, the neutron β asymmetry, the β-ν correlation, and the Fierz term. Their impact both within and beyond the standard model of particle physics is considerable, in particular on CKM unitarity, Equation 11, on putative dark neutron decays, Section 5.3.2, and on limits on tensor and other exotic couplings, Equation 13. Altogether, deviations from the V-A structure of the SM are excluded well below the 10 -3 level. Fifteen years ago, these limits were still on the 10% level (8). New developments in EFT and lattice QCD theory have made it possible to compare constraints on New Physics via the appropriate Wilson coefficients from low-and high-energy experiments, see Table 5. It turns out that neutron and other β decay experiments compare well with and are in part complementary to limits derived from LHC experiments.

SUMMARY POINTS
4. Deviations from the SM are parametrized by appropriate Wilson coefficients. Progress in effective field theory permits to compare limits on Wilson coefficients from neutron decay with corresponding limits from high-energy proton-proton collisions.
5. Limits on Wilson coefficients from low-energy experiments are generally more precise and require fewer assumptions than the corresponding high-energy limits.
6. High-energy experiments, by contrast, are more sensitive to non-SM right-handed neutrinos, and this higher sensitivity makes them complementary to the low-energy experiments.