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A method for an improved measurement of the electron–antineutrino correlation in free neutron beta decay
ARTICLE IN PRESS
Nuclear Instruments and Methods in Physics Research A 545 (2005) 181–193 www.elsevier.com/locate/nima
A method for an improved measurement of the electron–antineutrino correlation in free neutron beta decay F.E. Wietfeldta,, B.M. Fishera, C. Trulla, G.L. Jonesb, B. Colletb, L. Goldinc, B.G. Yerozolimskyc, R. Wilsonc, S. Balashovd,1, Yu. Mostovoyd, A. Komivese, M. Leuschnerf, J. Byrneg, F.B. Batemanh, M.S. Deweyh, J.S. Nicoh, A.K. Thompsonh a
Department of Physics, Tulane University, New Orleans, LA 70118, USA b Physics Department, Hamilton College, Clinton, NY 13323, USA c Physics Department, Harvard University, Cambridge, MA 02139, USA d Kurchatov Institute, Moscow, Russian Federation e Physics Department, DePauw University, Greencastle, IN 46135, USA f Indiana University Cyclotron Facility, Bloomington, IN 47408, USA g University of Sussex, UK h National Institute of Standards and Technology, Gaithersburg, MD 20899, USA Received 6 July 2004; received in revised form 24 January 2005; accepted 31 January 2005 Available online 18 April 2005
Abstract The angular correlation between the beta electron and antineutrino in nuclear beta decay is characterized by the dimensionless parameter a. The value of a for free neutron decay, when combined with other neutron decay parameters, can be used to determine the weak vector and axial vector coupling constants gV and gA and test the validity and selfconsistency of the Electroweak Standard Model. Previous experiments that measured a in neutron decay relied on precise proton spectroscopy and were limited by systematic effects at about the 5% level. We present a new approach to measuring a for which systematic uncertainties promise to be much smaller. r 2005 Elsevier B.V. All rights reserved. PACS: 23.40.2; 23.40.Bw; 14.20.Dh Keywords: Beta decay; Neutron decay; CKM unitarity
Corresponding author. Present address: Particle Physics Department, Rutherford Appleton Laboratory, Oxon, UK.
E-mail address: [email protected] (F.E. Wietfeldt). 0168-9002/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2005.01.339
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1. Introduction The beta decay of the free neutron, n ! p þ eþ ne þ 0:783 MeV, is the prototype semileptonic weak interaction and simplest nuclear beta decay. It is the only nuclear beta decay system that can be analyzed theoretically without a need to calculate matrix elements between wave functions of nuclei. The decay energy is small compared to the neutron mass, so the momentum-transfer dependence of the weak form factors enters at only the 0.1% level. This simplicity makes neutron decay a suitable system for precise measurements of beta decay observables. The observables of neutron decay are its lifetime and the various angular correlations between the neutron, beta electron, antineutrino, and recoil proton momenta and spins. Typically we observe the neutron spin, as a prepared polarization state, and average over the spins of the final state particles. These observables are unambiguously related to fundamental parameters in the electroweak theory, with relatively little theoretical uncertainty. For this reason, improved precision on neutron decay observables can provide important tests of the Standard Model and possibly show signs of new physics. We note that pion beta decay is sensitive essentially to the same physics as neutron decay and is theoretically an even simpler system. Its main disadvantage is a very small branching ratio (about 108 ). The least well-known of this important group of neutron decay observables is the electron–antineutrino correlation coefficient a (often called ‘‘little a’’). It characterizes the angular correlation between the beta electron and antineutrino. Previous experiments that measured a relied on precise proton spectroscopy and were limited by systematic effects at about the 5% level. We propose a new approach to measuring a that uses an experimental asymmetry for which systematic uncertainties promise to be much smaller. Such a measurement would lead to an improved determination of the weak vector and axial vector coupling constants gV and gA , improved tests of the Standard Model and limits on new physics, and an improved test of the unitarity of the Cabbibo–Kobayashi–Maskawa
(CKM) system.
matrix
using
the
neutron
decay
2. Theoretical discussion The most important features of neutron decay are described by the formula of Jackson et al. [1], which gives the neutron decay probability as a function of the emitted electron (pe ) and antineutrino (pn ) momenta, and the neutron spin polarization (P): 1 p p N / E e jpe jðE 0 E e Þ2 1 þ a e n tn EeEn pe pn ðpe pn Þ þP A þ B þD . ð1Þ EeEn Ee En The neutron decay lifetime is tn , E e and E n are the electron and antineutrino energies, and E 0 is the endpoint energy that 1293 keV. The parameters a, A, B, and D are the asymmetry coefficients that are measured by experiment. In the Standard Electroweak Model, neglecting recoil order corrections, the values of these correlation coefficients and the lifetime are related to two basic parameters in the theory: the weak vector and axial vector coupling constants gV and gA . If we write their ratio as gA =gV ¼ l, we have [1] 3 7 2p _ 1 1 l2 a ¼ tn ¼ m5e c4 f g2V þ 3g2A 1 þ 3l2 Reflg þ l2 1 þ 3l2 Imflg D¼2 . 1 þ 3l2 A¼ 2
B ¼ 2
Reflg l2 1 þ 3l2 ð2Þ
In the expression for tn , f is a known phase space factor (f ¼ 1:71489ð2Þ [2]) and me is the electron mass. The triple correlation coefficient D is proportional to the (very small) imaginary part of l which violates time-reversal symmetry. For our present purpose we may assume that l is real. A measurement of tn plus any one of a, A, or B determines gA and gV uniquely. Additional measurements overconstrain the system and test the self-consistency of the Standard Electroweak Model. New physical forces or phenomena can
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add additional parameters to Eqs. (2), and so can change the relationships between tn , a, A, and B slightly; such effects could be detected by sufficiently precise experiments. The current Particle Data Group (PDG) recommended values of the neutron decay coefficients are [3]:
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CKM matrix must be unitary, which sets a number of constraints on its matrix elements. In particular, the sum of the squares of the elements in the first row must equal one: X jV ui j2 ¼ jV ud j2 þ jV us j2 þ jV ub j2 ¼ 1. (5) i
A ¼ 0:1173 0:0013 B ¼ 0:983 0:004.
ð3Þ
The least precisely known is the electron–antineutrino correlation coefficient a. A useful model-independent test of the selfconsistency of the Standard Model is obtained by noting that Eqs. (2) lead to [4] F1 1 þ A B a ¼ 0 2
F 2 aB A A ¼ 0.
and ð4Þ
Inserting the accepted values we have F 1 ¼ 0:0027 0:0058 and F 2 ¼ 0:0023 0:0041, so at the present level of uncertainty the Standard Model is validated. The uncertainties in F 1 and F 2 are dominated by the experimental uncertainties in the a coefficient. Recoil order corrections will cause F 1 and F 2 to differ from zero at the 104 level, but those corrections are calculable. Otherwise, a departure of F 1 or F 2 from zero could indicate new physics beyond the Standard Model. A number of important model-dependent tests for new physics can be made using the neutron decay parameters. For example, the values of a, A, and B can be related to the strength of hypothetical right-handed weak forces and scalar and tensor forces [5,6]. A precise comparison of the energy-dependence of a and A can place sharp limits on possible conserved-vector-current (CVC) violation and second-class currents in neutron decay [7,8]. There are no signs of such forcesat present, but popular speculative extensions to the Standard Model, such as left–right symmetric models and supersymmetry, could lead to observable effects of this type. There is currently an important problem in electroweak physics concerning the unitarity of the CKM mixing matrix. This matrix transforms the mass states of the down-type quarks ðd; s; bÞ into the states that couple to the up-type quarks ðu; c; tÞ via the weak force. To conserve probability the
The elements V us and V ub are measured in highenergy accelerator experiments while V ud is best measured in low-energy systems. Because V ud is much larger than the others, it dominatesthe uncertainty of the quadrature sum. The CVC principle requires gV ¼ G F jV ud jð1 þ DR Þ, where GF ¼ 1:11639ð1Þ 105 Ge V2 is the Fermi weak coupling constant determined from the muon lifetime, and the last factor is a calculated radiative correction. Thus, one can obtain jV ud j by measuring gV in nuclear beta decay experiments. The best experimental determinations of gV are summarized in Fig. 1. The most precise comes from an evaluation of nine superallowed (0þ ! 0þ ) beta decay systems [9,10].The value of gV obtained is less than that required for the CKM matrix to be unitary by 2.3 standard deviations.The uncertainty is dominated by theoretical corrections. We point out that the proper interpretation of this result is not that the CKM matrix is nonunitary, but rather that there is something missing from the theoretical analysis (if the experiments are correct). The most interesting possibility is that it is a sign of new physics beyond the Standard Model. A more mundane possibility is that the accepted value of V us is wrong. There have been some recent -14.40 Ft (0+→ 0+)
-14.45 gA (GeV-2)
a ¼ 0:103 0:004
tn ¼ 885:7 0:8 s
Unitary CKM
-14.50 neutron lifetime
-14.55 -14.60 -14.65
A (PERKEO 2002)
-14.70x10-6 11.40
11.45
A (PDG ave)
11.50
11.55
11.60x10-6
gA (GeV-2)
Fig. 1. An experimental summary of the weak nuclear force coupling constants gA and gV .
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indications that this may be the case [11] but the matter has yet to be settled. We see from Eqs. (2) that a measurement of the neutron lifetime tn and either a or A (B is much less sensitive) will provide two independent expressions for gA and gV , so an absolute determination of gV can be obtained from neutron decay alone. The most precise measurement of A was the recent result A ¼ 0:1189 0:0007 by thePERKEO collaboration at the Institut Laue-Langevin in France [12]. Using this, thevalue of gV obtained from neutron decay also disagrees with CKM unitarity and is even lower than that determined from the 0þ ! 0þ decays. However, there is considerable experimental disagreement over the value of A. The current PDG recommended value A ¼ 0:1173 0:0013 is a combination of four precision experiments, including Ref. [12], and the uncertainty contains a scale factor of 2.3 to account for this disagreement. Using the PDG value of A the uncertainty in gV is large enough to be consistent with both the 0þ ! 0þ result and CKM unitarity. It is widely believed that the problem with the A experiments comes from the difficulty with neutron polarimetery at this level of precision. The current situation is clearly unsatisfactory. An improved measurement of the a-coefficient would help here. It has a similar sensitivity to l as A, and would produce a similar band in Fig. 1. The nature of the a and A experiments are very different, and therefore systematically independent. An important advantage of the a coefficient is that it is measured using unpolarized neutrons. The neutron polarimetry question is entirely avoided. The present accepted value of a has an uncertainty of 4%,which is too large, but if a can be measured by a new experiment to less than 1%, the precision of gV determined from a and tn would be comparable to that from the PERKEO 2002 result. The current recommended value, a ¼ 0:103 0:004 [3] is based on experiments by Grigor’ev et al. (1968) [13]; Stratowa et al. (1978) [14] and Byrne et al. (2002) [15]. Each of these determined a by measuring the precise shape of the recoil proton energy spectrum. The angular correlation between the electron and antineutrino emitted in beta decay influences the average momentum carried by the
proton, and therefore alters the shape of its differential kinetic energy spectrum. This approach is statistically the most advantageous because it does not require a coincidence measurement. The neutron lifetime is long, and high counting statistics are always a challenge in these experiments, so it is not surprising that this has been the preferred method. If the a-coefficient is so important, one may wonder why its precision has not been improved since the most precise previous measurement in 1978 [14]. We believe it may be because of the systematic difficulty of measuring the low-energy proton spectral shape to high precision. The fractional effect on its shape is approximately equal to a, so this is a 10% effect. A determination of a to 5% precision using this approach requires a correct measurement of the recoil proton spectral shape to 0.5%. Both the Stratowa et al. and Byrne et al. experiments were limited by systematics at the 5% level. We suggest that a new approach is needed to break this 5% barrier in precision.
3. A new approach to a We propose a new method that promises reduced systematic uncertainties, and perhaps a factor of five improvement in precision of a. The idea behind this method was first suggested a decade ago by Yerozolimsky and Mostovoy [16–18]. It relies on the construction of an asymmetry that directly yields a without requiring precise proton spectroscopy. The basic scheme is shown in Fig. 2. A proton detector and electron detector are positioned on either side of a cold
neutron decay region solenoid B proton detector
+V
electron detector
neutron beam
Fig. 2. The basic scheme of the proposed method to measure the a coefficient.
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neutron beam. A long solenoid, aligned to the axis of the detectors, is located between the beam and proton detector. Coincidence detection of the electron and proton is possible for neutrons that decay in the indicated region. The solenoid produces a uniform central magnetic field B. Within the solenoid are a series of precisely aligned circular apertures of radius r. A proton’s trajectory inside the solenoid is helical, with radiusR proportional to its transverse momentum R ¼ p? c=eB [19]. Only those decay protons with transverse momentum below a threshold value (which depends on the position of the decay vertex) can pass unobstructed through the apertures and be detected. A pair of fine wire grids produces an electric field around the decay region, directing all decay protons toward the proton detector regardless of their initial axial momenta. Decay electrons are energetic enough to pass through this electric field. The determination of a from this scheme is best illustrated by the momentum-space diagram shown in Fig. 3. The cold neutron’s kinetic energy is very small (about 0.003 eV) so it can be treated as decaying at rest. Let us assume the decay vertex was on axis. A typical momentum vector for a detected beta electron is shown as pe . The solenoid and aperture arrangement will allow any proton whose transverse momentum is less than eBr=2c to be detected. The electric field guarantees that any value of the proton’s axial momentum will be accepted. Therefore, the proton’s momentum acceptance is described by the lower circular cylinder shown in the figure. Any proton whose momentum vector lies inside this cylinder is detected, and an electron–proton coincidence is obtained. Now consider the antineutrino. It is not detected, but conservation of momentum requires
eBr 2c
II
I
pv
pv
pe
-pe
that the antineutrino momentum vector (pn ) lie in a second cylinder (shaded in the figure), identical to the lower cylinder but translated by pe? . If we neglect the decay proton’s kinetic energy, conservation of energy fixes the length of the antineutrino pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi momentum vector to be jpn j ¼ Q=c p2e þ m2e c2 , where Q is the total decay energy. For any neutron decay where both the electron and proton are detected, the possible antineutrino momenta must fall into one of two kinematically distinct groups, indicated in the figure by the regions labeled I and II. These two regions are formed by the intersections of the upper cylinder with a spherical shell of radius jpn j.The detection solid angles subtended by the two regions are equal in size; so if a is zero, there is equal probability for the antineutrino to be in either group.A nonzero value of a is associated with an average correlation between the electron and antineutrino momenta, and will cause an asymmetry between coincidence events in groups I and II. To simplify this illustration we have assumed the decay vertex to be on axis, but in general it will be off axis. In that case the cylinders in Fig. 3 would have elliptical, rather than circular, cross-sections, but the above analysis is similar and its conclusions are the same. The two groups of coincidence events, corresponding to regions I and II, can be experimentally distinguished by time-of-flight (TOF). The beta electron is detected a few nanoseconds after the neutron decays. The proton is much slower and takes microseconds to reach the proton detector. The time between electron and proton detection can be easily measured using standard TOF methods. The protons associated with group I events have greater axial momentum than those of group II events, so they reach the proton detector more quickly. By recording for each event the electron energy and proton time-of-flight, we obtain, after many decays, N I events in group I and N II events in group II for each electron energy. It can be shown that
pe
Fig. 3. A momentum space diagram of the determination of a from the coincidence measurement.
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aðEÞ ¼
1 2X ðEÞ . ve ðfI ðEÞ fII ðEÞÞ X ðEÞðfI ðEÞ þ fII ðEÞÞ
(6)
ARTICLE IN PRESS F.E. Wietfeldt et al. / Nuclear Instruments and Methods in Physics Research A 545 (2005) 181–193
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Here X ðEÞ is the experimental asymmetry: X ðEÞ ¼
N I N II N I þ N II
with (7)
for some slice of electron energy E and ve is the beta electron speed (fraction of c). The parameters fI ðEÞ and fII ðEÞ are defined by R R dOe I dOn cos yen I f ðEÞ ¼ Oe OIn R R dOe II dOn cos yen fII ðEÞ ¼ . ð8Þ Oe OII n Looking at Fig. 3, fI can be understood to be the cosine of the angle between the electron and neutrino momentum vectors, averaged over the cone that defines group I, and then averaged again over the accepted beta momentum directions for a particular beta energy, assuming that there is no angular correlation (i.e. calculated assuming a ¼ 0). It is effectively just a geometry calculation; it contains no physics. The parameter fII is the corresponding average for neutrino group II. Note that fI is a positive number and fII is negative. These parameters, which are both functions of electron energy, depend on the transverse momentum acceptances of the proton and beta, and so they can be calculated from the measured axial magnetic field and the collimator geometries. The second term in the denominator of Eq. (6) is small, about 1% of the first term, so as a convenient approximation we can write: aðEÞ
1 KðEÞX ðEÞ ve
(9)
KðEÞ ¼
2 . f ðEÞ fII ðEÞ I
(10)
To a good approximation aðEÞ is proportional to the experimental asymmetry X ðEÞ at each slice of electron energy. For discussion purposes, Eq. (9) is easier to work with, but of course there is no difficulty in using the full expression, Eq. (6), when needed to analyze the data precisely. Using a ¼ 0:1, the asymmetry X ðEÞ has an average value of about 0:05 for the electron energies used in our experiment. Our calculations show that to determine fI ðE e Þ and fII ðE e Þ, and hence KðE e Þ, to a precision of o0:5%, it is sufficient to know the magnetic field to 1.5% and the collimator diameter and alignment to 1 mm. There is a small correction that comes from our neglect of the proton’s kinetic energy in the momentum space discussion. If we account for this energy, the neutrino momentum acceptances for groups I and II (see Fig. 3) differ by 0.1%. It would cause a slight asymmetry even if a were zero. This effect is a 2% correction to the measured a. While Fig. 2 shows the minimum configuration for this scheme to work, it would be more practical and efficient to extend the magnetic field from the proton detector to the electron detector and select both protons and beta electrons by their transverse momenta. This is indicated in Fig. 4, which shows a sketch of a realistic experimental apparatus. Instead of a continuous solenoid, the axial magnetic field is produced by an array of precisely
Fig. 4. Sketch of a realistic instrument to measure a using the proposed method.
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aligned coils with unequal spacing. This minimizes magnetic field inhomogeneities in the proton transport region (see Section 4.3). The proton and beta collimators each consist of a series of knife-edge apertures designed to minimize scatter and energy loss. The electron detector is a plastic scintillator with a backscatter-suppressing veto detector (see Section 4.1). The proton detector is a cooled silicon surface barrier detector with focussing and acceleration electrodes. Surrounding the neutron beam region is a +2.5 kV electrostatic mirror. It provides four distinct benefits tothe experiment: (1) it improves the coincidence rate by allowing protons emitted in both directions from the decay region to be detected; (2) it significantly narrows the time-of-flight spread of the slow proton group, making it into a sharp peak and thus improving the signal/background ratio; (3) it causes the velocities of all protons, fast and slow, to be approximately the same in the proton collimator, mitigating the false asymmetry due to transversemagnetic fields; and (4) by making the proton velocities more uniform, it reduces the significance of any velocity-dependence of the proton detector efficiency. Most of the time-of-
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flight difference between the two proton groups comes from the different time spent in the electrostatic mirror; the time-of-flight in the collimator (about 2 ms) is similar for both groups. A vacuum of 105 Pa will be needed in the neutron beam and proton transport regions to avoid problems from protons scattering from residual gas. Simulated data from a Monte Carlo simulation of the experiment, in the configuration of Fig. 4, are presented in Fig. 5. These data are equivalent to 18 beam days at the NIST Center for Neutron Research (NCNR) fundamental physics end station NG-6 [20]. In the plot of proton TOF verses beta kinetic energy, the neutron decay events form a characteristic wish bone shape. The lower leg is the fast proton group (group I in Fig. 3) and the upper leg is the slow proton group (group II). For beta electron kinetic energies less than 320 keV the two proton groups are distinct and the asymmetry in Eq. (9) can be simply calculated for each energy slice. At higher energies, the two groups overlap. When analyzed, the data in the electron energy range 48–320 keV give the result a ¼ 0:1044 0:0049. A single experimental run at the NCNR of about 200 beam days (a typical
4µs Monte Carlo Data proton TOF
3
2
48-320 keV
1
0
Counts
0
2500 2000 1500 1000 500 0
48-62 keV
1.5
2.0 2.5 3.0 3.5 proton TOF (µs)
200
2500 2000 1500 1000 500 0
400 600keV beta kinetic energy
192-208 keV
1.5
2.0 2.5 3.0 3.5 proton TOF (µs)
2500 2000 1500 1000 500 0
304-320 keV
1.5
2.0 2.5 3.0 3.5 proton TOF (µs)
Fig. 5. Simulated data from a Monte Carlo simulation of the a experiment. For beta kinetic energies less than 320 keV, the two proton groups are well separated. The lower plots are time-of-flight distributions for narrow slices of beta energy.
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beta electron max. transverse momentum (keV/c)
run for a neutron decay experiment) would yield a measurement of a with a statistical precision of 1.5%. Additional runs at the NCNR or a higher flux source could reduce the statistical uncertainty to 0.5–1.0%. The optimal parameters of the experiment were determined by Monte Carlo simulation. If the transverse momentum acceptances for electrons and protons are too small, then too few coincidence events are obtained. As the acceptances are increased the coincidence efficiency improves, but the electron energy at which the two proton groups begin to overlap (see Fig. 5) decreases, which reduces the useful fraction of data. The optimization of this trade-off is illustrated in Fig. 6. The statistical figure of merit, the relative inverse statistical variance in a, is plotted as a contour vs. the maximum transverse momentum acceptance for both the beta electrons and recoil protons. The broad diagonal lines are the cutoff electron kinetic energy in kiloelectron-volts, i.e. the electron kinetic energy above which the two proton groups overlap. The dashed line indicates the total momentum associated with the cutoff 600 500 400 400
300
0.9
200 600
0.7
0.8 0.5
100 500 700
0 0
0.2
0.6 0.3 0.1
electron energy for a given proton momentum acceptance. For points on this line all electrons below the cutoff energy are accepted, so points above the line are not meaningful. We see that the kinematics favor a 20% larger proton (compared to electron) transverse momentum acceptance. The peak statistical sensitivity corresponds to a cutoff energy of 250 keV, well below the peak of the beta spectrum. It should be noted that the analysis shown in Fig. 6 considers on-axis decays only. The conclusions are similar when off-axis decays are also considered. This optimization fixes the product of the axial magnetic field and collimator diameter for both electrons and protons. For systematic reasons it may be desirable to run the experiment slightly off the peak, for example with proton (electron) transverse momentum acceptance of 270 (225) keV/c and a cutoff electron energy of 320 keV. This would allow a larger region of electron energy to be used so that systematic effects that depend on electron energy can be better studied. It is better for statistics to make the magnetic field as weak as possible so that the collimator diameters will be large. More of the neutron beam will lie in the effective decay region. However the collimator regions must be sufficiently long so that, for a given magnetic field, both the proton and electron complete at least one full orbit of cyclotron motion inside the collimator. This is needed for the momentum acceptances to be uniform. For this reason, the maximum practical separation of the proton and electron detectors from the neutron beam sets the minimum axial magnetic field.
100
0.4
300
200
100 200 300 400 500 600 recoil proton max. transverse momentum (keV/c)
Fig. 6. Statistical optimization of the transverse momentum acceptance. The statistical figure of merit, the inverse statistical variance in a (1=s2a ) normalized to the maximum, is plotted as a contour vs. the maximum transverse momentum acceptance for both the beta electrons and recoil protons. The broad diagonal lines are the cutoff electron kinetic energy (the electron energy above which the two proton groups overlap) in kiloelectronvolts. The dashed line indicates the total electron momentum associated with the cutoff electron energy for a given proton momentum acceptance. Points above the dashed line are not meaningful.
4. Systematic effects An important goal of this method is to reduce all systematic effects to less than 0.5% of a. To accomplish this we have anticipated and analyzed a number of effects, discussed in the following subsections. 4.1. Backscatter from the electron detector In beta spectroscopy there is usually some probability for an electron to backscatter out of
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A sketch of this idea is shown in Fig. 7. A beta electron enters the detection chamber from the left, transported by the 400 G magnetic field. All electrons that are accepted by the electron collimator will enter the detection chamber. The magnetic field drops off rapidly close to the energy detector at the rear of the chamber. This is accomplished using an additional field-shaping coil with opposite current. As the electron approaches the energy detector, its trajectory diverges from the axis, and some of its transverse energy is converted into longitudinal energy. All electrons will strike the plastic scintillator energy detector, including those whose original momenta were mostly transverse. The magnetic field at the energy detector is low, so if the electron backscatters it has little chance to be transported back through the entrance of the chamber. It is much more likely to strike the array of veto detectors: a hexagonal cone formed from six trapezoid-shaped plastic scintillators. The veto array covers the full 2p of solid angle for backscattered electrons, except for the entrance hole, which is less than 1% of 2p. A Monte Carlo simulation of this scheme, using electron transport in realistic magnetic fields and the ETRAN [22] electron transport code, showed that the fraction of events in the backscatter tail is beta collimator neutron beam
β
veto array
energy detector
50 Bz on axis (mT)
the detector without depositing its full energy. This will cause the beta electron to register at an energy lower than its original decay energy. Looking at the plot in Fig. 5, one can see that when a highenergy electron does this it will tend to fill in the gap between the two proton groups at low energies and confound their separation.With a plastic scintillator electron detector, for example, the backscatter probability is about 4%. We have studied this problem in detail using our Monte Carlo Model of the experiment. The main effect is to cause protons from the fast group to appear close in time to the slow group at lower electron energy. By excluding events that kinematically cannot belong to the slow group, most, but not all, of these events can be removed. Our analysis has shown that even with the most careful analysis this problem can lead to an error as large as 3% in a, so backscattered-electron events must be removed from the data. It is important to note that removing electron events cannot produce a bias in the determination of a since for each detected electron the acceptances for corresponding group I and group II neutrinos are equal. We require an experimental technique that will suppress electron backscatter events, while collecting a large phase space volume of low-energy (o400 keV) electrons with high efficiency. Backscatter is a long-standing problem in beta spectroscopy and many different methods have been used to control it. We studied past techniques carefully and concluded that none are suitable for this experiment. Magnetic and electrostatic spectrometers necessarily have narrow phase space acceptance. Schemes that rely on a ‘‘magnetic mirror’’ [21] are ineffective in the weak magnetic field needed for this experiment; we are not in the adiabatic regime. The use of a thin transmission detector or a time projection chamber to discriminate backscattered electrons is not sensitive enough when the electron energy is this low. A gas ionization detector could also make a backscatter-free measurement, but a detector thick enough to fully stop these electrons requires a window that would cause energy loss issues as bad as or worse than backscattering. We devised a new scheme for backscatter suppression and demonstrated that it works well.
189
40 30 20 10 0 0
0.2
0.4 0.6 0.8 distance from neutron beam (m)
1.0
Fig. 7. A sketch of the backscatter-suppressed electron detector concept. The magnetic field shown was calculated from the coil configuration in Fig. 4.
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reduced by 90% compared to a simple, unvetoed, plastic scintillator detector. A full-scale prototype electron spectrometer, based on this concept, was recently built and tested using a 1 MeV electron beam at the NIST Van de Graaff accelerator. Details can be found in a recent paper [23]. These experimental results support the conclusions of the Monte Carlo simulation. Other essential studies of electron backscatter from scintillator detectors were carried out recently by Goldin and Yerozolimsky [24]. This spectrometer design will reduce the error on a due to electron backscatter to less than 0.5%. 4.2. Electron scatter from collimators If a beta electron scatters from any material between the neutron decay vertex and the detector it will lose energy, leading to the same problem as detector backscatter. In our apparatus, the only objects the beta electrons can scatter from and still reach the electron detector are the collimators. By making the collimators very thin, with a knife edge on their inner surface, the area available for scattering to reach the detector will be effectively zero. It will be possible for an electron to penetrate the knife edge and lose energy. Use of a very highZ material such as tungsten for the collimators will minimize this effect. The Monte Carlo electron transport routine PENELOPE [25] was used to calculate the trajectories and interactions of 600 keV electrons through a series of tungsten collimators in the geometery of the experiment (Fig. 4) with a 0.04 T axial magnetic field. Several collimator shapes were evaluated, and for each the fraction of electrons that lost energy in the collimator and were subsequently detected in the beta detector was calculated. Using 1-mm-thick collimators with 60 knife edges this fraction was less than 0.05%. 4.3. Transverse magnetic fields The transverse momentum acceptance of both the fast and slow proton groups must be equal, otherwise a false asymmetry will result. Detailed proton transport Monte Carlo simulations have shown that transverse magnetic field components
in the electrostatic field region and the proton collimation region must be less than 104 of the axial field, or o4 mT, near the axis within the radius of the proton collimator, so that the false asymmetry will be less than 0.5% of the true asymmetry due to a. It will be necessary to wind some additional anti-Helmholtz transverse coils or sine coils to cancel local external magnetic fields or gradients, and the magnetic fields in and around the experiment will be carefully monitored throughout the run. One obvious problem is that an opening in the solenoid is needed to admit the neutron beam to the center of the electrostatic field region, and any gap will tend to create a large transverse magnetic field there.Our solution is to use an array of coils spaced about 7 cm apart as depicted in Fig. 4, instead of a solenoid, to produce the axial magnetic field. The resulting field differs from a continuous solenoid only by the presence of a small ripple, the effect of which tends to average out. Furthermore, by varying the spacing of the coils, we can ‘‘flatten’’ the shape of the field, largely eliminating the end effect, and so reduce the length of the coil system. The current in most of these coils is the same so we can connect them in series and power them with a single current supply. The magnetic field shape of this system, compared to a single solenoid and a set of uniformly spaced coils, is shown in Fig. 8. In our simulations, the ‘‘false a’’ due to transverse magnetic fields using
axial magnetic field (mT)
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40.05 40.00 39.95 39.90 -200
-150
-100 -50 0 axial position (cm)
50
100
Fig. 8. The axial magnetic field in the proton and electron transport regions using a single 5-m-long solenoid (dashed line), a 5-m-long array of equally spaced 7-cm-long coils (dotted line), and a 2.6-m-long array of unequally spaced 7-cm coils (solid line). We plan to use the latter configuration. The neutron beam is admitted at z ¼ 0
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this coil configuration is less than 103 . A possible concern is trapping of electrons and protons in magnetic mirrors formed by the ripples in the axial magnetic field. In our scheme, the electrostatic mirror provides a sufficient minimum axial energy (41 keV) to all decay particles to guarantee they will not be trapped. 4.4. Transverse electric fields The electrostatic mirror must create a highly uniform axial electric field in the decay region. It must also let the protons and electrons pass freely though the field generating electrodes and thus must use wire grids. Moreover, the mirror region must be screened from the grounded vacuum chamber by a cylindrical grid or film, which will maintain a linear potential gradient at the circumference of the active volume. The planar end grids create strongly non-uniform axial and transverse electric fields in the region near the wires. This field is confined to a region within a few times the wire spacing of the grid and thus has limited time to alter the momentum of the protons. (The effect from the electron-end grid is much smaller both because of the greater velocity of the electrons and because it cannot alter the asymmetry of the proton populations.) Detailed Monte Carlo simulations of the complete electron–proton transport system using grids consisting of parallel wires running in a single direction have shown that a grid of 20 mm diameter wires on a 2-mm grid introduce a false asymmetry of ð1:1 0:5Þ 104 , or a systematic uncertainty in a of about 0.2%, which is acceptable. Finite-element field models constructed using FEMM [26] have shown that two co-axial grids of wires with 2 mm axial separation and 10 mm radial separation limit transverse electric fields to less than 104 of the axial field within the radius of the proton collimator. 4.5. Electron detector energy resolution To extract a from the data using Eq. (9), the asymmetry must be measured as a function of electron energy and then divided by the electron speed to a precision of 0.5% or better. This means
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that the energy calibration of the electron detector must be known absolutely to within 1%. Note that this calibration precision refers to the centroid and shape of the energy response function, not its width. If a plastic scintillator is used, the width (energy resolution) of the energy detector will be much greater than 1%. This is not a serious problem as long as the energy response is well understood. 4.6. Proton detector efficiency Our preferred proton detector is a thin, cooled silicon surface barrier detector and front-end preamp package surrounded by wire grids that create an acceleration potential of 25 kV for the protons. We have used a similar scheme in past neutron decay experiments to detect recoil protons [27,28] and it works quite well. At 25 keV, the protons are well-separated from the noise peak, and the thin detector is relatively insensitive to gamma and X-ray backgrounds. The concept for this detector is shown in Fig. 9. The two groups of protons, fast and slow, differ in their initial kinetic energies by typically a few hundred electron-volts. However, all protons are accelerated by about 2 keV in the electrostatic mirror, which mitigates the difference in the velocity distributions entering the detector acceleration region. The angular distribution of protons entering this region will
surface barrier detector
- 25 kV
proton
hemispherical electrode
Fig. 9. A sketch of the proton detector concept.
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be slightly different for the two groups. While this is not expected to be a serious issue, the equality of proton detector efficiency for the two groups must be verified by calculation and by a separate experiment using a low-energy proton source. Unlike backscatter from the electron detector, backscatter from the proton detector is not a problem; the small fraction (about 103 ) that backscatter and do not deposit enough energy to register a signal simply remove events from the data without any systematic effect on a, provided that the backscatter probability is the same for both proton groups, as we expect.
Acknowledgements
4.7. Beam-related background
References
Gamma and beta radiation created by neutron capture in the vicinity of the beam will cause backgrounds in the electron and proton detectors. This is a delayed coincidence experiment, so only accidental coincidences from background radiation will appear as background events in the data. The geometry here is advantageous for minimizing background. Both detectors are well-separated from the neutron beam and can be mostly surrounded by shielding. The areas of the apparatus closest to where the neutron beam passes can be covered with neutron absorbers containing enriched 6 Li to absorb scattered neutrons without prompt gamma radiation.
[1] J.D. Jackson, S.B. Treiman, H.W. Wyld, Nucl. Phys. 4 (1957) 206. [2] I.S. Towner, J.C. Hardy, in: E.M. Henley, W.C. Haxton (Eds.), Symmetries and Fundamental Interactions in Nuclei, World Scientific, Singapore, 1995, p. 183. [3] Particle Data Group, Phys. Lett. B 592 (2004) 1. [4] Y. Mostovoy, A. Frank, JETP Lett. 24 (1976) 38. [5] D. Dubbers, Nucl. Phys. A 527 (1991) 239c. [6] B. Yerozolimsky, Y. Mostovoy, Sov. J. Nucl. Phys. 53 (1991) 260. [7] B.R. Holstein, Rev. Mod. Phys. 46 (1974) 789. [8] S. Gardner, C. Zhang, Phys. Rev. Lett. 86 (2001) 5666. [9] I.S. Towner, J.C. Hardy, J. Phys. G 29 (2003) 197. [10] D.H. Wilkinson, Nucl. Instr. and Meth. A 488 (2002) 654. [11] A. Sher, et al., Phys. Rev. Lett. 91 (2003) 261802 T. Spadero, KLOE Collaboration, in: Proceedings Recontre de la Valle D’Aoste, 2004; KLOE Collaboration, Contract No. 8-0813, ICHEP04, 2004; T. Alexopoulos, KTeV Collaboration, hep-ex/0406001, -002, -003, 2004. [12] H. Abele, et al., Phys. Rev. Lett. 88 (2002) 211801. [13] V.K. Grigor’ev, et al., Sov. J. Nucl. Phys. 6 (1968) 239. [14] C. Stratowa, R. Dobrozemsky, P. Weinzierl, Phys. Rev. D 18 (1978) 3970. [15] J. Byrne, et al., J. Phys. G 28 (2002) 1325. [16] B. Yerozolimsky, et al., Workshop on Fundamental Neutron Physics at the Advanced Neutron Source OakRidge, Tennessee, November 9–10, 1993. [17] B. Yerozolimsky, et al., arXiv:nucl-ex/0401014, 2004. [18] S. Balashov, Yu. Mostovoy, Russian Research Center Kurchatov Institute, Preprint IAE-5718/2, Moscow, 1994. [19] J.D. Jackson, Classical Electrodynamics, second ed., Wiley, New York, 1975, p. 581. [20] P.R. Huffman, et al., in: J.L. Duggan, I.L. Morgan (Eds.), Application of Accelerators in Research and Industry: 17th International Conference, CAARI 2002, November
5. Conclusions We present a new approach to the measurement of the electron–antineutrino correlation coefficient in free neutron decay. It relies on the measurement of an asymmetry in the coincident detection of beta electrons and recoil protons that is proportional to a. Unlike previous experiments, it does not require precise spectroscopy of low energy protons. We have shown that a measurement of a to a precision of less than 1% is possible, five times smaller than the best previous experiment. A wide range of systematic effects has been considered and we have shown that all can be controlled at the 0.5% level.
We thank D.M. Gilliam for helpful discussions and support. This work was supported by the National Science Foundation (PHY-0420851, PHY-0420361, PHY-0420716, PHY-0420563), the NIST Physics Laboratory and Center for Neutron Research, and the US Department of Energy Interagency Agreement DE-AI0293ER40784. C.T. gratefully acknowledges support from the Louisiana Board of Regents BoRSF, agreement NASA/LEQSF(2001-2005)-LaSPACE and NASA/LaSPACE Grant NGT5-40115.
ARTICLE IN PRESS F.E. Wietfeldt et al. / Nuclear Instruments and Methods in Physics Research A 545 (2005) 181–193 12–16, 2002, Denton, TX, AIP Conference Proceedings, Melville, New York, 2003, pp. 275–278. [21] J.L. Mortara, et al., Phys. Rev. Lett. 70 (1993) 394. [22] S.M. Seltzer, Int. J. Appl. Radiat. Isotopes 42 (1991) 917. [23] F.E. Wietfeldt, et al., Nucl. Instr. and Meth. Phys. Res. A 538 (2005) 574.
[24] [25] [26] [27]
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L. Goldin, B. Yerozolimsky, arXiv:nucl-ex/0401015, 2004. F. Salvat, et al., University of Barcelona Preprint, 1996. URL: http://femm.foster-miller.com/ L.J. Lising, et al., emiT collaboration, Phys. Rev. C 62 (2000) 055501. [28] M.S. Dewey, et al., Phys. Rev. Lett. 91 (2003) 152302.
Nuclear Instruments and Methods in Physics Research A 545 (2005) 181–193 www.elsevier.com/locate/nima
A method for an improved measurement of the electron–antineutrino correlation in free neutron beta decay F.E. Wietfeldta,, B.M. Fishera, C. Trulla, G.L. Jonesb, B. Colletb, L. Goldinc, B.G. Yerozolimskyc, R. Wilsonc, S. Balashovd,1, Yu. Mostovoyd, A. Komivese, M. Leuschnerf, J. Byrneg, F.B. Batemanh, M.S. Deweyh, J.S. Nicoh, A.K. Thompsonh a
Department of Physics, Tulane University, New Orleans, LA 70118, USA b Physics Department, Hamilton College, Clinton, NY 13323, USA c Physics Department, Harvard University, Cambridge, MA 02139, USA d Kurchatov Institute, Moscow, Russian Federation e Physics Department, DePauw University, Greencastle, IN 46135, USA f Indiana University Cyclotron Facility, Bloomington, IN 47408, USA g University of Sussex, UK h National Institute of Standards and Technology, Gaithersburg, MD 20899, USA Received 6 July 2004; received in revised form 24 January 2005; accepted 31 January 2005 Available online 18 April 2005
Abstract The angular correlation between the beta electron and antineutrino in nuclear beta decay is characterized by the dimensionless parameter a. The value of a for free neutron decay, when combined with other neutron decay parameters, can be used to determine the weak vector and axial vector coupling constants gV and gA and test the validity and selfconsistency of the Electroweak Standard Model. Previous experiments that measured a in neutron decay relied on precise proton spectroscopy and were limited by systematic effects at about the 5% level. We present a new approach to measuring a for which systematic uncertainties promise to be much smaller. r 2005 Elsevier B.V. All rights reserved. PACS: 23.40.2; 23.40.Bw; 14.20.Dh Keywords: Beta decay; Neutron decay; CKM unitarity
Corresponding author. Present address: Particle Physics Department, Rutherford Appleton Laboratory, Oxon, UK.
E-mail address: [email protected] (F.E. Wietfeldt). 0168-9002/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2005.01.339
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1. Introduction The beta decay of the free neutron, n ! p þ eþ ne þ 0:783 MeV, is the prototype semileptonic weak interaction and simplest nuclear beta decay. It is the only nuclear beta decay system that can be analyzed theoretically without a need to calculate matrix elements between wave functions of nuclei. The decay energy is small compared to the neutron mass, so the momentum-transfer dependence of the weak form factors enters at only the 0.1% level. This simplicity makes neutron decay a suitable system for precise measurements of beta decay observables. The observables of neutron decay are its lifetime and the various angular correlations between the neutron, beta electron, antineutrino, and recoil proton momenta and spins. Typically we observe the neutron spin, as a prepared polarization state, and average over the spins of the final state particles. These observables are unambiguously related to fundamental parameters in the electroweak theory, with relatively little theoretical uncertainty. For this reason, improved precision on neutron decay observables can provide important tests of the Standard Model and possibly show signs of new physics. We note that pion beta decay is sensitive essentially to the same physics as neutron decay and is theoretically an even simpler system. Its main disadvantage is a very small branching ratio (about 108 ). The least well-known of this important group of neutron decay observables is the electron–antineutrino correlation coefficient a (often called ‘‘little a’’). It characterizes the angular correlation between the beta electron and antineutrino. Previous experiments that measured a relied on precise proton spectroscopy and were limited by systematic effects at about the 5% level. We propose a new approach to measuring a that uses an experimental asymmetry for which systematic uncertainties promise to be much smaller. Such a measurement would lead to an improved determination of the weak vector and axial vector coupling constants gV and gA , improved tests of the Standard Model and limits on new physics, and an improved test of the unitarity of the Cabbibo–Kobayashi–Maskawa
(CKM) system.
matrix
using
the
neutron
decay
2. Theoretical discussion The most important features of neutron decay are described by the formula of Jackson et al. [1], which gives the neutron decay probability as a function of the emitted electron (pe ) and antineutrino (pn ) momenta, and the neutron spin polarization (P): 1 p p N / E e jpe jðE 0 E e Þ2 1 þ a e n tn EeEn pe pn ðpe pn Þ þP A þ B þD . ð1Þ EeEn Ee En The neutron decay lifetime is tn , E e and E n are the electron and antineutrino energies, and E 0 is the endpoint energy that 1293 keV. The parameters a, A, B, and D are the asymmetry coefficients that are measured by experiment. In the Standard Electroweak Model, neglecting recoil order corrections, the values of these correlation coefficients and the lifetime are related to two basic parameters in the theory: the weak vector and axial vector coupling constants gV and gA . If we write their ratio as gA =gV ¼ l, we have [1] 3 7 2p _ 1 1 l2 a ¼ tn ¼ m5e c4 f g2V þ 3g2A 1 þ 3l2 Reflg þ l2 1 þ 3l2 Imflg D¼2 . 1 þ 3l2 A¼ 2
B ¼ 2
Reflg l2 1 þ 3l2 ð2Þ
In the expression for tn , f is a known phase space factor (f ¼ 1:71489ð2Þ [2]) and me is the electron mass. The triple correlation coefficient D is proportional to the (very small) imaginary part of l which violates time-reversal symmetry. For our present purpose we may assume that l is real. A measurement of tn plus any one of a, A, or B determines gA and gV uniquely. Additional measurements overconstrain the system and test the self-consistency of the Standard Electroweak Model. New physical forces or phenomena can
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add additional parameters to Eqs. (2), and so can change the relationships between tn , a, A, and B slightly; such effects could be detected by sufficiently precise experiments. The current Particle Data Group (PDG) recommended values of the neutron decay coefficients are [3]:
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CKM matrix must be unitary, which sets a number of constraints on its matrix elements. In particular, the sum of the squares of the elements in the first row must equal one: X jV ui j2 ¼ jV ud j2 þ jV us j2 þ jV ub j2 ¼ 1. (5) i
A ¼ 0:1173 0:0013 B ¼ 0:983 0:004.
ð3Þ
The least precisely known is the electron–antineutrino correlation coefficient a. A useful model-independent test of the selfconsistency of the Standard Model is obtained by noting that Eqs. (2) lead to [4] F1 1 þ A B a ¼ 0 2
F 2 aB A A ¼ 0.
and ð4Þ
Inserting the accepted values we have F 1 ¼ 0:0027 0:0058 and F 2 ¼ 0:0023 0:0041, so at the present level of uncertainty the Standard Model is validated. The uncertainties in F 1 and F 2 are dominated by the experimental uncertainties in the a coefficient. Recoil order corrections will cause F 1 and F 2 to differ from zero at the 104 level, but those corrections are calculable. Otherwise, a departure of F 1 or F 2 from zero could indicate new physics beyond the Standard Model. A number of important model-dependent tests for new physics can be made using the neutron decay parameters. For example, the values of a, A, and B can be related to the strength of hypothetical right-handed weak forces and scalar and tensor forces [5,6]. A precise comparison of the energy-dependence of a and A can place sharp limits on possible conserved-vector-current (CVC) violation and second-class currents in neutron decay [7,8]. There are no signs of such forcesat present, but popular speculative extensions to the Standard Model, such as left–right symmetric models and supersymmetry, could lead to observable effects of this type. There is currently an important problem in electroweak physics concerning the unitarity of the CKM mixing matrix. This matrix transforms the mass states of the down-type quarks ðd; s; bÞ into the states that couple to the up-type quarks ðu; c; tÞ via the weak force. To conserve probability the
The elements V us and V ub are measured in highenergy accelerator experiments while V ud is best measured in low-energy systems. Because V ud is much larger than the others, it dominatesthe uncertainty of the quadrature sum. The CVC principle requires gV ¼ G F jV ud jð1 þ DR Þ, where GF ¼ 1:11639ð1Þ 105 Ge V2 is the Fermi weak coupling constant determined from the muon lifetime, and the last factor is a calculated radiative correction. Thus, one can obtain jV ud j by measuring gV in nuclear beta decay experiments. The best experimental determinations of gV are summarized in Fig. 1. The most precise comes from an evaluation of nine superallowed (0þ ! 0þ ) beta decay systems [9,10].The value of gV obtained is less than that required for the CKM matrix to be unitary by 2.3 standard deviations.The uncertainty is dominated by theoretical corrections. We point out that the proper interpretation of this result is not that the CKM matrix is nonunitary, but rather that there is something missing from the theoretical analysis (if the experiments are correct). The most interesting possibility is that it is a sign of new physics beyond the Standard Model. A more mundane possibility is that the accepted value of V us is wrong. There have been some recent -14.40 Ft (0+→ 0+)
-14.45 gA (GeV-2)
a ¼ 0:103 0:004
tn ¼ 885:7 0:8 s
Unitary CKM
-14.50 neutron lifetime
-14.55 -14.60 -14.65
A (PERKEO 2002)
-14.70x10-6 11.40
11.45
A (PDG ave)
11.50
11.55
11.60x10-6
gA (GeV-2)
Fig. 1. An experimental summary of the weak nuclear force coupling constants gA and gV .
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indications that this may be the case [11] but the matter has yet to be settled. We see from Eqs. (2) that a measurement of the neutron lifetime tn and either a or A (B is much less sensitive) will provide two independent expressions for gA and gV , so an absolute determination of gV can be obtained from neutron decay alone. The most precise measurement of A was the recent result A ¼ 0:1189 0:0007 by thePERKEO collaboration at the Institut Laue-Langevin in France [12]. Using this, thevalue of gV obtained from neutron decay also disagrees with CKM unitarity and is even lower than that determined from the 0þ ! 0þ decays. However, there is considerable experimental disagreement over the value of A. The current PDG recommended value A ¼ 0:1173 0:0013 is a combination of four precision experiments, including Ref. [12], and the uncertainty contains a scale factor of 2.3 to account for this disagreement. Using the PDG value of A the uncertainty in gV is large enough to be consistent with both the 0þ ! 0þ result and CKM unitarity. It is widely believed that the problem with the A experiments comes from the difficulty with neutron polarimetery at this level of precision. The current situation is clearly unsatisfactory. An improved measurement of the a-coefficient would help here. It has a similar sensitivity to l as A, and would produce a similar band in Fig. 1. The nature of the a and A experiments are very different, and therefore systematically independent. An important advantage of the a coefficient is that it is measured using unpolarized neutrons. The neutron polarimetry question is entirely avoided. The present accepted value of a has an uncertainty of 4%,which is too large, but if a can be measured by a new experiment to less than 1%, the precision of gV determined from a and tn would be comparable to that from the PERKEO 2002 result. The current recommended value, a ¼ 0:103 0:004 [3] is based on experiments by Grigor’ev et al. (1968) [13]; Stratowa et al. (1978) [14] and Byrne et al. (2002) [15]. Each of these determined a by measuring the precise shape of the recoil proton energy spectrum. The angular correlation between the electron and antineutrino emitted in beta decay influences the average momentum carried by the
proton, and therefore alters the shape of its differential kinetic energy spectrum. This approach is statistically the most advantageous because it does not require a coincidence measurement. The neutron lifetime is long, and high counting statistics are always a challenge in these experiments, so it is not surprising that this has been the preferred method. If the a-coefficient is so important, one may wonder why its precision has not been improved since the most precise previous measurement in 1978 [14]. We believe it may be because of the systematic difficulty of measuring the low-energy proton spectral shape to high precision. The fractional effect on its shape is approximately equal to a, so this is a 10% effect. A determination of a to 5% precision using this approach requires a correct measurement of the recoil proton spectral shape to 0.5%. Both the Stratowa et al. and Byrne et al. experiments were limited by systematics at the 5% level. We suggest that a new approach is needed to break this 5% barrier in precision.
3. A new approach to a We propose a new method that promises reduced systematic uncertainties, and perhaps a factor of five improvement in precision of a. The idea behind this method was first suggested a decade ago by Yerozolimsky and Mostovoy [16–18]. It relies on the construction of an asymmetry that directly yields a without requiring precise proton spectroscopy. The basic scheme is shown in Fig. 2. A proton detector and electron detector are positioned on either side of a cold
neutron decay region solenoid B proton detector
+V
electron detector
neutron beam
Fig. 2. The basic scheme of the proposed method to measure the a coefficient.
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neutron beam. A long solenoid, aligned to the axis of the detectors, is located between the beam and proton detector. Coincidence detection of the electron and proton is possible for neutrons that decay in the indicated region. The solenoid produces a uniform central magnetic field B. Within the solenoid are a series of precisely aligned circular apertures of radius r. A proton’s trajectory inside the solenoid is helical, with radiusR proportional to its transverse momentum R ¼ p? c=eB [19]. Only those decay protons with transverse momentum below a threshold value (which depends on the position of the decay vertex) can pass unobstructed through the apertures and be detected. A pair of fine wire grids produces an electric field around the decay region, directing all decay protons toward the proton detector regardless of their initial axial momenta. Decay electrons are energetic enough to pass through this electric field. The determination of a from this scheme is best illustrated by the momentum-space diagram shown in Fig. 3. The cold neutron’s kinetic energy is very small (about 0.003 eV) so it can be treated as decaying at rest. Let us assume the decay vertex was on axis. A typical momentum vector for a detected beta electron is shown as pe . The solenoid and aperture arrangement will allow any proton whose transverse momentum is less than eBr=2c to be detected. The electric field guarantees that any value of the proton’s axial momentum will be accepted. Therefore, the proton’s momentum acceptance is described by the lower circular cylinder shown in the figure. Any proton whose momentum vector lies inside this cylinder is detected, and an electron–proton coincidence is obtained. Now consider the antineutrino. It is not detected, but conservation of momentum requires
eBr 2c
II
I
pv
pv
pe
-pe
that the antineutrino momentum vector (pn ) lie in a second cylinder (shaded in the figure), identical to the lower cylinder but translated by pe? . If we neglect the decay proton’s kinetic energy, conservation of energy fixes the length of the antineutrino pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi momentum vector to be jpn j ¼ Q=c p2e þ m2e c2 , where Q is the total decay energy. For any neutron decay where both the electron and proton are detected, the possible antineutrino momenta must fall into one of two kinematically distinct groups, indicated in the figure by the regions labeled I and II. These two regions are formed by the intersections of the upper cylinder with a spherical shell of radius jpn j.The detection solid angles subtended by the two regions are equal in size; so if a is zero, there is equal probability for the antineutrino to be in either group.A nonzero value of a is associated with an average correlation between the electron and antineutrino momenta, and will cause an asymmetry between coincidence events in groups I and II. To simplify this illustration we have assumed the decay vertex to be on axis, but in general it will be off axis. In that case the cylinders in Fig. 3 would have elliptical, rather than circular, cross-sections, but the above analysis is similar and its conclusions are the same. The two groups of coincidence events, corresponding to regions I and II, can be experimentally distinguished by time-of-flight (TOF). The beta electron is detected a few nanoseconds after the neutron decays. The proton is much slower and takes microseconds to reach the proton detector. The time between electron and proton detection can be easily measured using standard TOF methods. The protons associated with group I events have greater axial momentum than those of group II events, so they reach the proton detector more quickly. By recording for each event the electron energy and proton time-of-flight, we obtain, after many decays, N I events in group I and N II events in group II for each electron energy. It can be shown that
pe
Fig. 3. A momentum space diagram of the determination of a from the coincidence measurement.
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aðEÞ ¼
1 2X ðEÞ . ve ðfI ðEÞ fII ðEÞÞ X ðEÞðfI ðEÞ þ fII ðEÞÞ
(6)
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186
Here X ðEÞ is the experimental asymmetry: X ðEÞ ¼
N I N II N I þ N II
with (7)
for some slice of electron energy E and ve is the beta electron speed (fraction of c). The parameters fI ðEÞ and fII ðEÞ are defined by R R dOe I dOn cos yen I f ðEÞ ¼ Oe OIn R R dOe II dOn cos yen fII ðEÞ ¼ . ð8Þ Oe OII n Looking at Fig. 3, fI can be understood to be the cosine of the angle between the electron and neutrino momentum vectors, averaged over the cone that defines group I, and then averaged again over the accepted beta momentum directions for a particular beta energy, assuming that there is no angular correlation (i.e. calculated assuming a ¼ 0). It is effectively just a geometry calculation; it contains no physics. The parameter fII is the corresponding average for neutrino group II. Note that fI is a positive number and fII is negative. These parameters, which are both functions of electron energy, depend on the transverse momentum acceptances of the proton and beta, and so they can be calculated from the measured axial magnetic field and the collimator geometries. The second term in the denominator of Eq. (6) is small, about 1% of the first term, so as a convenient approximation we can write: aðEÞ
1 KðEÞX ðEÞ ve
(9)
KðEÞ ¼
2 . f ðEÞ fII ðEÞ I
(10)
To a good approximation aðEÞ is proportional to the experimental asymmetry X ðEÞ at each slice of electron energy. For discussion purposes, Eq. (9) is easier to work with, but of course there is no difficulty in using the full expression, Eq. (6), when needed to analyze the data precisely. Using a ¼ 0:1, the asymmetry X ðEÞ has an average value of about 0:05 for the electron energies used in our experiment. Our calculations show that to determine fI ðE e Þ and fII ðE e Þ, and hence KðE e Þ, to a precision of o0:5%, it is sufficient to know the magnetic field to 1.5% and the collimator diameter and alignment to 1 mm. There is a small correction that comes from our neglect of the proton’s kinetic energy in the momentum space discussion. If we account for this energy, the neutrino momentum acceptances for groups I and II (see Fig. 3) differ by 0.1%. It would cause a slight asymmetry even if a were zero. This effect is a 2% correction to the measured a. While Fig. 2 shows the minimum configuration for this scheme to work, it would be more practical and efficient to extend the magnetic field from the proton detector to the electron detector and select both protons and beta electrons by their transverse momenta. This is indicated in Fig. 4, which shows a sketch of a realistic experimental apparatus. Instead of a continuous solenoid, the axial magnetic field is produced by an array of precisely
Fig. 4. Sketch of a realistic instrument to measure a using the proposed method.
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aligned coils with unequal spacing. This minimizes magnetic field inhomogeneities in the proton transport region (see Section 4.3). The proton and beta collimators each consist of a series of knife-edge apertures designed to minimize scatter and energy loss. The electron detector is a plastic scintillator with a backscatter-suppressing veto detector (see Section 4.1). The proton detector is a cooled silicon surface barrier detector with focussing and acceleration electrodes. Surrounding the neutron beam region is a +2.5 kV electrostatic mirror. It provides four distinct benefits tothe experiment: (1) it improves the coincidence rate by allowing protons emitted in both directions from the decay region to be detected; (2) it significantly narrows the time-of-flight spread of the slow proton group, making it into a sharp peak and thus improving the signal/background ratio; (3) it causes the velocities of all protons, fast and slow, to be approximately the same in the proton collimator, mitigating the false asymmetry due to transversemagnetic fields; and (4) by making the proton velocities more uniform, it reduces the significance of any velocity-dependence of the proton detector efficiency. Most of the time-of-
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flight difference between the two proton groups comes from the different time spent in the electrostatic mirror; the time-of-flight in the collimator (about 2 ms) is similar for both groups. A vacuum of 105 Pa will be needed in the neutron beam and proton transport regions to avoid problems from protons scattering from residual gas. Simulated data from a Monte Carlo simulation of the experiment, in the configuration of Fig. 4, are presented in Fig. 5. These data are equivalent to 18 beam days at the NIST Center for Neutron Research (NCNR) fundamental physics end station NG-6 [20]. In the plot of proton TOF verses beta kinetic energy, the neutron decay events form a characteristic wish bone shape. The lower leg is the fast proton group (group I in Fig. 3) and the upper leg is the slow proton group (group II). For beta electron kinetic energies less than 320 keV the two proton groups are distinct and the asymmetry in Eq. (9) can be simply calculated for each energy slice. At higher energies, the two groups overlap. When analyzed, the data in the electron energy range 48–320 keV give the result a ¼ 0:1044 0:0049. A single experimental run at the NCNR of about 200 beam days (a typical
4µs Monte Carlo Data proton TOF
3
2
48-320 keV
1
0
Counts
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2500 2000 1500 1000 500 0
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2.0 2.5 3.0 3.5 proton TOF (µs)
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2500 2000 1500 1000 500 0
400 600keV beta kinetic energy
192-208 keV
1.5
2.0 2.5 3.0 3.5 proton TOF (µs)
2500 2000 1500 1000 500 0
304-320 keV
1.5
2.0 2.5 3.0 3.5 proton TOF (µs)
Fig. 5. Simulated data from a Monte Carlo simulation of the a experiment. For beta kinetic energies less than 320 keV, the two proton groups are well separated. The lower plots are time-of-flight distributions for narrow slices of beta energy.
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beta electron max. transverse momentum (keV/c)
run for a neutron decay experiment) would yield a measurement of a with a statistical precision of 1.5%. Additional runs at the NCNR or a higher flux source could reduce the statistical uncertainty to 0.5–1.0%. The optimal parameters of the experiment were determined by Monte Carlo simulation. If the transverse momentum acceptances for electrons and protons are too small, then too few coincidence events are obtained. As the acceptances are increased the coincidence efficiency improves, but the electron energy at which the two proton groups begin to overlap (see Fig. 5) decreases, which reduces the useful fraction of data. The optimization of this trade-off is illustrated in Fig. 6. The statistical figure of merit, the relative inverse statistical variance in a, is plotted as a contour vs. the maximum transverse momentum acceptance for both the beta electrons and recoil protons. The broad diagonal lines are the cutoff electron kinetic energy in kiloelectron-volts, i.e. the electron kinetic energy above which the two proton groups overlap. The dashed line indicates the total momentum associated with the cutoff 600 500 400 400
300
0.9
200 600
0.7
0.8 0.5
100 500 700
0 0
0.2
0.6 0.3 0.1
electron energy for a given proton momentum acceptance. For points on this line all electrons below the cutoff energy are accepted, so points above the line are not meaningful. We see that the kinematics favor a 20% larger proton (compared to electron) transverse momentum acceptance. The peak statistical sensitivity corresponds to a cutoff energy of 250 keV, well below the peak of the beta spectrum. It should be noted that the analysis shown in Fig. 6 considers on-axis decays only. The conclusions are similar when off-axis decays are also considered. This optimization fixes the product of the axial magnetic field and collimator diameter for both electrons and protons. For systematic reasons it may be desirable to run the experiment slightly off the peak, for example with proton (electron) transverse momentum acceptance of 270 (225) keV/c and a cutoff electron energy of 320 keV. This would allow a larger region of electron energy to be used so that systematic effects that depend on electron energy can be better studied. It is better for statistics to make the magnetic field as weak as possible so that the collimator diameters will be large. More of the neutron beam will lie in the effective decay region. However the collimator regions must be sufficiently long so that, for a given magnetic field, both the proton and electron complete at least one full orbit of cyclotron motion inside the collimator. This is needed for the momentum acceptances to be uniform. For this reason, the maximum practical separation of the proton and electron detectors from the neutron beam sets the minimum axial magnetic field.
100
0.4
300
200
100 200 300 400 500 600 recoil proton max. transverse momentum (keV/c)
Fig. 6. Statistical optimization of the transverse momentum acceptance. The statistical figure of merit, the inverse statistical variance in a (1=s2a ) normalized to the maximum, is plotted as a contour vs. the maximum transverse momentum acceptance for both the beta electrons and recoil protons. The broad diagonal lines are the cutoff electron kinetic energy (the electron energy above which the two proton groups overlap) in kiloelectronvolts. The dashed line indicates the total electron momentum associated with the cutoff electron energy for a given proton momentum acceptance. Points above the dashed line are not meaningful.
4. Systematic effects An important goal of this method is to reduce all systematic effects to less than 0.5% of a. To accomplish this we have anticipated and analyzed a number of effects, discussed in the following subsections. 4.1. Backscatter from the electron detector In beta spectroscopy there is usually some probability for an electron to backscatter out of
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A sketch of this idea is shown in Fig. 7. A beta electron enters the detection chamber from the left, transported by the 400 G magnetic field. All electrons that are accepted by the electron collimator will enter the detection chamber. The magnetic field drops off rapidly close to the energy detector at the rear of the chamber. This is accomplished using an additional field-shaping coil with opposite current. As the electron approaches the energy detector, its trajectory diverges from the axis, and some of its transverse energy is converted into longitudinal energy. All electrons will strike the plastic scintillator energy detector, including those whose original momenta were mostly transverse. The magnetic field at the energy detector is low, so if the electron backscatters it has little chance to be transported back through the entrance of the chamber. It is much more likely to strike the array of veto detectors: a hexagonal cone formed from six trapezoid-shaped plastic scintillators. The veto array covers the full 2p of solid angle for backscattered electrons, except for the entrance hole, which is less than 1% of 2p. A Monte Carlo simulation of this scheme, using electron transport in realistic magnetic fields and the ETRAN [22] electron transport code, showed that the fraction of events in the backscatter tail is beta collimator neutron beam
β
veto array
energy detector
50 Bz on axis (mT)
the detector without depositing its full energy. This will cause the beta electron to register at an energy lower than its original decay energy. Looking at the plot in Fig. 5, one can see that when a highenergy electron does this it will tend to fill in the gap between the two proton groups at low energies and confound their separation.With a plastic scintillator electron detector, for example, the backscatter probability is about 4%. We have studied this problem in detail using our Monte Carlo Model of the experiment. The main effect is to cause protons from the fast group to appear close in time to the slow group at lower electron energy. By excluding events that kinematically cannot belong to the slow group, most, but not all, of these events can be removed. Our analysis has shown that even with the most careful analysis this problem can lead to an error as large as 3% in a, so backscattered-electron events must be removed from the data. It is important to note that removing electron events cannot produce a bias in the determination of a since for each detected electron the acceptances for corresponding group I and group II neutrinos are equal. We require an experimental technique that will suppress electron backscatter events, while collecting a large phase space volume of low-energy (o400 keV) electrons with high efficiency. Backscatter is a long-standing problem in beta spectroscopy and many different methods have been used to control it. We studied past techniques carefully and concluded that none are suitable for this experiment. Magnetic and electrostatic spectrometers necessarily have narrow phase space acceptance. Schemes that rely on a ‘‘magnetic mirror’’ [21] are ineffective in the weak magnetic field needed for this experiment; we are not in the adiabatic regime. The use of a thin transmission detector or a time projection chamber to discriminate backscattered electrons is not sensitive enough when the electron energy is this low. A gas ionization detector could also make a backscatter-free measurement, but a detector thick enough to fully stop these electrons requires a window that would cause energy loss issues as bad as or worse than backscattering. We devised a new scheme for backscatter suppression and demonstrated that it works well.
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40 30 20 10 0 0
0.2
0.4 0.6 0.8 distance from neutron beam (m)
1.0
Fig. 7. A sketch of the backscatter-suppressed electron detector concept. The magnetic field shown was calculated from the coil configuration in Fig. 4.
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reduced by 90% compared to a simple, unvetoed, plastic scintillator detector. A full-scale prototype electron spectrometer, based on this concept, was recently built and tested using a 1 MeV electron beam at the NIST Van de Graaff accelerator. Details can be found in a recent paper [23]. These experimental results support the conclusions of the Monte Carlo simulation. Other essential studies of electron backscatter from scintillator detectors were carried out recently by Goldin and Yerozolimsky [24]. This spectrometer design will reduce the error on a due to electron backscatter to less than 0.5%. 4.2. Electron scatter from collimators If a beta electron scatters from any material between the neutron decay vertex and the detector it will lose energy, leading to the same problem as detector backscatter. In our apparatus, the only objects the beta electrons can scatter from and still reach the electron detector are the collimators. By making the collimators very thin, with a knife edge on their inner surface, the area available for scattering to reach the detector will be effectively zero. It will be possible for an electron to penetrate the knife edge and lose energy. Use of a very highZ material such as tungsten for the collimators will minimize this effect. The Monte Carlo electron transport routine PENELOPE [25] was used to calculate the trajectories and interactions of 600 keV electrons through a series of tungsten collimators in the geometery of the experiment (Fig. 4) with a 0.04 T axial magnetic field. Several collimator shapes were evaluated, and for each the fraction of electrons that lost energy in the collimator and were subsequently detected in the beta detector was calculated. Using 1-mm-thick collimators with 60 knife edges this fraction was less than 0.05%. 4.3. Transverse magnetic fields The transverse momentum acceptance of both the fast and slow proton groups must be equal, otherwise a false asymmetry will result. Detailed proton transport Monte Carlo simulations have shown that transverse magnetic field components
in the electrostatic field region and the proton collimation region must be less than 104 of the axial field, or o4 mT, near the axis within the radius of the proton collimator, so that the false asymmetry will be less than 0.5% of the true asymmetry due to a. It will be necessary to wind some additional anti-Helmholtz transverse coils or sine coils to cancel local external magnetic fields or gradients, and the magnetic fields in and around the experiment will be carefully monitored throughout the run. One obvious problem is that an opening in the solenoid is needed to admit the neutron beam to the center of the electrostatic field region, and any gap will tend to create a large transverse magnetic field there.Our solution is to use an array of coils spaced about 7 cm apart as depicted in Fig. 4, instead of a solenoid, to produce the axial magnetic field. The resulting field differs from a continuous solenoid only by the presence of a small ripple, the effect of which tends to average out. Furthermore, by varying the spacing of the coils, we can ‘‘flatten’’ the shape of the field, largely eliminating the end effect, and so reduce the length of the coil system. The current in most of these coils is the same so we can connect them in series and power them with a single current supply. The magnetic field shape of this system, compared to a single solenoid and a set of uniformly spaced coils, is shown in Fig. 8. In our simulations, the ‘‘false a’’ due to transverse magnetic fields using
axial magnetic field (mT)
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40.05 40.00 39.95 39.90 -200
-150
-100 -50 0 axial position (cm)
50
100
Fig. 8. The axial magnetic field in the proton and electron transport regions using a single 5-m-long solenoid (dashed line), a 5-m-long array of equally spaced 7-cm-long coils (dotted line), and a 2.6-m-long array of unequally spaced 7-cm coils (solid line). We plan to use the latter configuration. The neutron beam is admitted at z ¼ 0
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this coil configuration is less than 103 . A possible concern is trapping of electrons and protons in magnetic mirrors formed by the ripples in the axial magnetic field. In our scheme, the electrostatic mirror provides a sufficient minimum axial energy (41 keV) to all decay particles to guarantee they will not be trapped. 4.4. Transverse electric fields The electrostatic mirror must create a highly uniform axial electric field in the decay region. It must also let the protons and electrons pass freely though the field generating electrodes and thus must use wire grids. Moreover, the mirror region must be screened from the grounded vacuum chamber by a cylindrical grid or film, which will maintain a linear potential gradient at the circumference of the active volume. The planar end grids create strongly non-uniform axial and transverse electric fields in the region near the wires. This field is confined to a region within a few times the wire spacing of the grid and thus has limited time to alter the momentum of the protons. (The effect from the electron-end grid is much smaller both because of the greater velocity of the electrons and because it cannot alter the asymmetry of the proton populations.) Detailed Monte Carlo simulations of the complete electron–proton transport system using grids consisting of parallel wires running in a single direction have shown that a grid of 20 mm diameter wires on a 2-mm grid introduce a false asymmetry of ð1:1 0:5Þ 104 , or a systematic uncertainty in a of about 0.2%, which is acceptable. Finite-element field models constructed using FEMM [26] have shown that two co-axial grids of wires with 2 mm axial separation and 10 mm radial separation limit transverse electric fields to less than 104 of the axial field within the radius of the proton collimator. 4.5. Electron detector energy resolution To extract a from the data using Eq. (9), the asymmetry must be measured as a function of electron energy and then divided by the electron speed to a precision of 0.5% or better. This means
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that the energy calibration of the electron detector must be known absolutely to within 1%. Note that this calibration precision refers to the centroid and shape of the energy response function, not its width. If a plastic scintillator is used, the width (energy resolution) of the energy detector will be much greater than 1%. This is not a serious problem as long as the energy response is well understood. 4.6. Proton detector efficiency Our preferred proton detector is a thin, cooled silicon surface barrier detector and front-end preamp package surrounded by wire grids that create an acceleration potential of 25 kV for the protons. We have used a similar scheme in past neutron decay experiments to detect recoil protons [27,28] and it works quite well. At 25 keV, the protons are well-separated from the noise peak, and the thin detector is relatively insensitive to gamma and X-ray backgrounds. The concept for this detector is shown in Fig. 9. The two groups of protons, fast and slow, differ in their initial kinetic energies by typically a few hundred electron-volts. However, all protons are accelerated by about 2 keV in the electrostatic mirror, which mitigates the difference in the velocity distributions entering the detector acceleration region. The angular distribution of protons entering this region will
surface barrier detector
- 25 kV
proton
hemispherical electrode
Fig. 9. A sketch of the proton detector concept.
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be slightly different for the two groups. While this is not expected to be a serious issue, the equality of proton detector efficiency for the two groups must be verified by calculation and by a separate experiment using a low-energy proton source. Unlike backscatter from the electron detector, backscatter from the proton detector is not a problem; the small fraction (about 103 ) that backscatter and do not deposit enough energy to register a signal simply remove events from the data without any systematic effect on a, provided that the backscatter probability is the same for both proton groups, as we expect.
Acknowledgements
4.7. Beam-related background
References
Gamma and beta radiation created by neutron capture in the vicinity of the beam will cause backgrounds in the electron and proton detectors. This is a delayed coincidence experiment, so only accidental coincidences from background radiation will appear as background events in the data. The geometry here is advantageous for minimizing background. Both detectors are well-separated from the neutron beam and can be mostly surrounded by shielding. The areas of the apparatus closest to where the neutron beam passes can be covered with neutron absorbers containing enriched 6 Li to absorb scattered neutrons without prompt gamma radiation.
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5. Conclusions We present a new approach to the measurement of the electron–antineutrino correlation coefficient in free neutron decay. It relies on the measurement of an asymmetry in the coincident detection of beta electrons and recoil protons that is proportional to a. Unlike previous experiments, it does not require precise spectroscopy of low energy protons. We have shown that a measurement of a to a precision of less than 1% is possible, five times smaller than the best previous experiment. A wide range of systematic effects has been considered and we have shown that all can be controlled at the 0.5% level.
We thank D.M. Gilliam for helpful discussions and support. This work was supported by the National Science Foundation (PHY-0420851, PHY-0420361, PHY-0420716, PHY-0420563), the NIST Physics Laboratory and Center for Neutron Research, and the US Department of Energy Interagency Agreement DE-AI0293ER40784. C.T. gratefully acknowledges support from the Louisiana Board of Regents BoRSF, agreement NASA/LEQSF(2001-2005)-LaSPACE and NASA/LaSPACE Grant NGT5-40115.
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