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Home , CryoEDM, Free neutron decay, Neutral particle, Particle

The and its role in and particle

Dirk Dubbers* Physikalisches Institut, Universität Heidelberg, Philosophenweg 12, D-69120 Heidelberg, Germany

Michael G. Schmidt † Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, D-69120 Heidelberg, Germany arXiv 18 May 2011, extended version of: Reviews of , in print.

Experiments with cold and ultracold have reached a level of precision such that problems far beyond the scale of the present of become accessible to experimental investigation. Due to the close links between particle physics and cosmology, these studies also permit a deep look into the very first instances of our . First addressed in this article, both in theory and experiment, is the problem of baryogenesis, the mechanism behind the evident dominance of over in the universe. The question how baryogenesis could have happened is open to experimental tests, and it turns out that this problem can be curbed by the very stringent limits on an of the neutron, a quantity that also has deep implications for particle physics. Then we discuss the recent spectacular observation of neutron quantization in the ’s gravitational and of resonance transitions between such gravitational states. These measurements, together with new evaluations of neutron data, set new constraints on deviations from Newton’s gravitational law at the picometer scale. Such deviations are predicted in modern theories with extra-dimensions that propose unification of the Planck scale with the scale of the Standard Model. These experiments start closing the remaining “ window” on new - dependent forces in the submillimeter range. Another main topic is the weak-interaction parameters in various fields of physics and astrophysics that must all be derived from measured neutron decay data. Up to now, about 10 different neutron decay observables have been measured, much more than needed in the electroweak Standard Model. This allows various precise tests for new physics beyond the Standard Model, competing with or surpassing similar tests at high-energy. The review ends with a discussion of neutron and nuclear data required in the synthesis of the elements during the “first three minutes” and later on in stellar nucleosynthesis.

DOI: xx,xxxx/RevModPhys.xx.xxx PACS number(s): 12.15.-y, 12.60.-i, 14.20.Dh, 98.80.-k

*[email protected] † [email protected]

1 CONTENTS

Sections marked with an asterisk* contain complementary material not included in the RMP article.

I. Introduction 3 C. Neutron constraints on new forces and on

A. Background 3 dark matter candidates 38

B. Scope of this review 4 1. Constraints from UCN gravitational levels 38

II. History Of The Early Universe 7 2. Constraints from neutron scattering 40 III. Baryogenesis, CP Violation, and the Neutron 3. Constraints on spin-dependent new forces 41

Electric Dipole Moment 9 D. More lessons from the neutron 42

A. Baryogenesis 9 1. Gravitational vs. inertial mass 42 1. The dominance of matter over antimatter 9 2. Neutron mass, and test of E= mc 2 42 2. Some quantities of the Standard Model 10 3. The fine structure constant 43 3. First Sakharov criterion: Violation of 4. More experiments on free falling neutrons 43 number 11 VI. The Electroweak Standard Model and the

4. Second criterion: CP Violation 12 β-Decay Of Free Neutrons 44

5. Third criterion: Nonequilibrium 13 A. Neutron decay theory 44 6. Baryogenesis in the Standard Model (SM) 13 1. The example of 44 7. Baryogenesis beyond the Standard Model 14 2. Weak interactions of and 45 B. Electric dipole moments (EDMs) 16 3. Weak interactions of 46 1. EDMs and symmetry violations 16 4. The 20 and more observables accessible 2. Neutron EDM experiments 17 from neutron decay 48 3. *Upcoming neutron EDM experiments 20 5. Rare allowed neutron decays 50 4. Limits on atomic and particle EDMs 23 B. Experiments in neutron decay 51 C. EDMs in particle physics and cosmology 24 1. The neutron lifetime 51 1. EDMs in the Standard Model 24 2. Correlation coefficients 55 2. EDMs in 26 3. *Upcoming neutron decay experiments 59 3. Other approaches 29 4. Comparison with decay data 60 4. Conclusions on EDM constraints 30 C. results and the Standard D. The other electromagnetic properties of the Model 60 neutron 30 1. Results from neutrons and other particles 60 1. 30 2. Results from nuclear β -decays 61 2. moment 30 3. Unitarity of the CKM matrix 62 3. Magnetic dipole moment 31 4. *Induced terms 63

4. Mean square charge radius 31 D. Symmetry tests beyond the SM 64 5. Electromagnetic 31 1. Is the electroweak interaction purely IV. Baryon Number Violation, Mirror , V− A ? 64 and Neutron Oscillations 32 2. Was the universe left-handed from the A. Neutron- oscillations 32 beginning? 67 B. Neutron-mirrorneutron oscillations 32 3. reversal invariance 68 V. Extra Dimensions, New Forces, and the Free VII. The Role of the Neutron in Element Formation 69 Fall of the Neutron 33 A. Primordial nucleosynthesis 69 A. Large extra dimensions 33 1. The neutron in the first three 1. The hierarchy problem 33 minutes 69 2. Bringing down the Planck mass 34 2. Nuclear reactions in the expanding universe 71 3. Competing models 35 3. Neutron lifetime, particle families, and B. Gravitational quantum levels 35 baryon density of the universe 71 1. Neutron states in the gravitational potential 35 4. -element abundances 72

2. Ultracold neutron (UCN) transmission B. Stellar nucleosynthesis 73 experiments 36 1. Stellar nucleosynthesis and the s-process 73 3. Observation of UCN gravitational wave 2. Explosive nucleosynthesis and the r- functions and resonance transitions 37 process 74

VIII. Conclusions 75

2 I. INTRODUCTION By pure coincidence, the size of the neutron’s mass is such that, at thermal velocities, their de-Broglie In the morning, a weight watcher sees the mass of wavelengths are in the Ångstrom range and thus match typically 40 kilograms of , 32 kilograms of typical atomic distances as well. Hence, neutrons can neutrons, and 22 grams of . Neutrons are as give both structural information (by elastic diffraction) ubiquitous on earth as are protons and electrons. In the and dynamical information (by inelastic spectroscopy). laboratory, we can produce free electrons simply by Neutrons are the only penetrating particles with this heating a wire and free protons by igniting a feature, which enables them to tell us both “where the discharge, but things are not so easy for neutrons. are”, and, at the same time, “what the atoms Neutrons are tightly bound in nuclei and free neutrons do”. Dynamical information is provided separately for are unstable with a lifetime of about 15 minutes, so any length scale as chosen by the experimenter via there are very few free neutrons in our environment. momentum transfer (selected, for example, by the To set neutrons free we need a , a MeV scattering angle). In the specialist’s language: rather than an eV process, and to obtain neutron fluxes Neutrons provide us with the full atomic -time competitive for modern scientific experimentation we correlation functions of the material under study, both need big nuclear installations. for self-correlations of the atoms (from incoherent As its name says, the outstanding property of the neutron scattering) and for their pair-correlations (from neutron is its electric neutrality. In fact, the neutron is coherent neutron scattering). the only long-lived with zero charge By another unique coincidence (also linked to the (if we neglect the small mass). This makes neutron mass via the nuclear magneton), the size of the neutrons difficult to handle, as one cannot easily neutron’s is such that amplitudes of accelerate, guide, or focus them with electric fields. magnetic scattering and nuclear scattering are Furthermore, one can detect slow neutrons only in a of similar magnitude. This makes neutrons useful also destructive way, via nuclear reactions that produce for structural and dynamical studies of the magnetic secondary ionizing reaction products. Once detected, properties of matter. On the instrumentation side, the neutrons are gone, in contrast to charged particles, magnetic interaction enables powerful techniques for which leave an ionization track that we can detect in a spin-polarization, magnetic guidance, or confinement destructive-free manner. Another disadvantage of of neutrons. Furthermore, slow neutrons generally neutrons is that fluxes available are much lower than distinguish easily between elements of similar atomic , electron, or fluxes. number, and often between isotopes of the same So why take the pain and with neutrons when element, the latter feature making powerful contrast they are so difficult to produce and to handle? The variation techniques applicable. All these are answer is, although the neutron’s neutrality is a properties that neutrons have and x-rays do not. burden, it is also its greatest asset (we find a similar Neutron-particle physicists should be well acquainted situation for the neutrino: its neutrality to all charges with these particularities of neutron-material science if except the weak charge is both its greatest asset and, they want to exploit fully the potential of the field. for the experimenter, a heavy burden). When the usually dominant long-range Coulomb interaction is not present, other more subtle effects come to the A. Background forefront. Being (i.e., particles composed of quarks), neutrons respond to all known interactions: Neutron-particle physics is part of the growing field the gravitational, the weak, the electromagnetic, and of low-energy particle physics. However, is not high- the strong interaction. They, presumably, also respond energy physics, which strives for the largest to any other “fifth” interaction that might emanate, for in colliding beams of electrons, protons, or up to example, from the dark matter that pervades the naked , the synonym for particle physics? universe. These non-Coulombic interactions are all Certainly, the preferred method for the study of a new either weak, short-ranged, or both. In addition, with no phenomenon is direct particle production at beam Coulomb barriers to overcome, neutrons can easily energies above the threshold where the process really penetrate condensed matter (including nuclear matter), happens. If an unknown process is mediated by a with often negligible multiple scattering, and for our heavy particle of mass M, then this particle can be purposes are essentially point-like probes with produced if the energy supplied in a reaction exceeds negligible polarizability. Mc 2. A well-known example is the discovery of In condensed matter research, which is a main electroweak unification in the 1980s when proton application of slow neutrons, neutrons have other collision energies of hundreds of GeV became decisive advantages. By definition, thermal and cold available for the first time to produce the photon’s neutrons have kinetic energies that match the typical heavy brethrens Z0 and W± that mediate the weak energies of thermal excitations in the condensed state. interaction. 3 If the available beam energy is not sufficient to ● changes in neutron spin polarization of 10 −6 and make the process happen that we are after, then we better, as realized in the experiments on neutron- must use a different method. We can see particles that optical parity violation. herald a new process as “virtual” particles already below their threshold for direct production. The In most cases, the small signals searched for in low- appearance of virtual particles of mass M then is energy particle physics violate some basic symmetry signalled, in accordance with the Heisenberg like parity or time reversal symmetry. In other cases, , by a propagator of the type one searches for the breaking of some empirical “law” like baryon number conservation, which, as we shall 1 Propagator ∝ , (1.1) see, would instead signal the restoration of a higher 2− 2 (p / c ) M symmetry. In any case, such “exotic” footprints would with four-momentum transfer p, for the case of help us distinguish the true signals from much larger spinless (“scalar”) exchange particles. We then see the mundane backgrounds, and, furthermore, could help particle as a virtual precursor of the real process, via an identify the high-energy process responsible for it. The energy-dependent rise of the relevant cross section. task of the experimenter then is to suppress all For example, the top , a particle that was still accidental asymmetries of the measurement apparatus missing in an otherwise complete multiplet, was seen that may mimic the effect under investigation. For a virtually before it was actually produced more than a discussion of the relative merits of high-energy and decade ago. Such virtual effects, however, are large low-energy physics, see Ramsey-Musolf and Su only if one is not too far away from threshold, so high- (2008). energy experiments see new processes through their Historically, neutron-particle physics on a large virtual effects only when they are almost there. scale started in the middle of the past century at However, we want to explore processes also at scales dedicated research reactors in the USA and in Russia. From the mid-seventies onwards, many experiments in M2>> ( p / c ) 2 far beyond the Tera-eV scale neutron-particle physics were done at ILL, France, by accessible today, which is probably necessary to various international collaborations. In recent years, understand the deepest problems of and the several new neutron facilities with strong neutron- universe. These virtual effects then are very small and particle physics programs came into operation, which in our example become brought many newcomers to the field. 1 Propagator ∝ , (1.2) M 2 B. Scope of this review which is independent of reaction energy. This is where low-energy particle physics comes Excellent recent reviews on experiments in into play. For these processes far beyond the present neutron-particle physics were written by Abele (2008), Standard Model of particle physics, the signal and by Nico and (2005). Holstein (2009a) edited size ∝1/ M 2 is the same, no matter whether we work at a series of five “Focus” articles on this topic. Earlier the high-energy frontier of order Tera-eV or at the general reviews on neutron-particle physics were low-energy frontier of order pico-eV. Of course, we written by Pendlebury (1993), and Dubbers (1991a, should search where discovery potential is highest. 1999). Conference proceedings on neutron-particle The precision reached in high-energy experiments physics were edited by Soldner et al . (2009), and is impressive, but the sensitivity for small signals earlier by Arif et al . (2005), Zimmer et al . (2000a), reached in low-energy experiments is almost Dubbers et al . (1989), Desplanques et al . (1984), and incredible. In slow-neutron physics, as we shall see, von Egidy (1978). Books on the subject include one can detect: Byrne’s opus magnum “Neutrons, Nuclei and Matter” (1994), Alexandrov’s “Fundamental Properties of the ● changes in energy of 10 −22 eV and better, or of Neutron” (1992), and Krupchitsky’s “Fundamental about one Bohr cycle per year, as realized in the Research with Polarized Slow Neutrons” (1987). searches for a neutron electric dipole moment and for We will cite other books and review articles on neutron-antineutron oscillations; more special topics in the respective chapters of this ● changes in relative momentum of 10 −10 and better, review. When existing literature is too extensive to be via deflections of neutrons by several Å over a 10 m cited without omissions, we refer to recent reviews or length of beam, as realized in the search for a neutron papers and the references therein, with no focus on charge, as well as in neutron interferometry; historical priorities. Particle data quoted without a reference are from the listings and review articles of the Particle Data Group, Nakamura et al . (2010),

4 shortly PDG 2010. Astrophysical data are mostly from experimental side, we concentrate on recent the review Bartelmann (2010), and fundamental developments, and in addition discuss ongoing or constants from the CODATA compilation Mohr et al . planned experiments with no published results yet. The (2008). Instrumental data quoted without a reference present arXiv article is an extended version of the are from the website of the respective instrument. Reviews of Modern Physics article, with more The good existing coverage of past experiments information on upcoming experiments, and derivations allows us to focus on the relevance of neutron studies of some formulae that may be difficult to find in the in cosmology and astrophysics, where lately many new literature. and exciting topics have come to the forefront. On the

TABLE I. The successive transitions of the universe; MPl , TPl, and tPl are Planck mass, temperature, and time.

Transition Time Temperature = × −44 =2 = × 19 tPl 5 10 s kTPl M Pl c 1.2 10 GeV Inflation, GUT transitions t ≥10−38... − 36 s kT ≤1015− 16 GeV Electroweak → Electromagn. and weak 10 −11 s 246 GeV ↓ Standard Quark- plasma → Nucleons 3× 10−5 s 170 MeV Model Nucleons → Nuclei 150 s 78 keV Plasma of nuclei + electrons → Atoms 374,000 yr 0.3 eV Atoms → and ~5× 108 yr T ~ 30 K Today 13.7(1)× 109 yr T = 2.728(4) K

The present review is organised in such a way as to elements, both during the first minutes after the big follow the cosmological evolution of the universe (as bang and in the course of stellar evolution. outlined in Table I), and to look at what role neutrons Although much of scientific progress is based on play at a given cosmological time/energy scale and in progress in instruments and methods, in this review we the sector of particle physics relevant at this scale. In cannot adequately cover all the important Sec. II, we first give an unsophisticated outline of developments made in recent years in the field of Standard Big Bang theory. The chapters then sources, transport, and detection of neutrons. For following are rather self-contained and can be read in references to technical papers, see Abele (2008) and any order. This reflects the fact that with each new Dubbers (1991a). chapter, the universe enters a new phase, and after Furthermore, scientific topics where there was little such a transition, the new phase usually does not progress in the past decade will be treated rather remember much of the old phase. In Sec. III, we shortly. This is the case for most of the neutron’s discuss what neutron experiments can tell us about electromagnetic properties listed in Sec. III.D, for baryogenesis and the accompanying symmetry neutron-antineutron oscillations shortly covered in violations. Here the discussion of a neutron EDM Sec. IV.A, and for the neutron-nuclear weak plays a particular role both because it might signal new interactions mentioned at the end of Sec. VI.A.3. sources of CP violation beyond the Standard Model Neutron optics, well covered in the book by Utsuro (SM) of particle physics and because this is a most and Ignatovich (2010), will be treated here mainly in a important ingredient of baryogenesis in cosmology. technical context, and the beautiful experiments on Section IV is a short résumé on neutron oscillations. In neutron interferometry, which are covered in the book Sec. V, we present new results on neutron searches for by Rauch and Werner (2000), and which had paved the new forces, and, in particular, on searches for way for much of modern optics, are not part of deviations from Newton’s law of gravitation, due to, this review. for instance, extra spatial dimension. Going down the Below we give a list, in alphabetical order, of some timeline in Table I we reach the electroweak scale of research centers that have neutron beam stations the SM in Sec. VI and then discuss results and dedicated to neutron-particle physics. relevance of weak interaction parameters derived from neutron decay. Section VII closes our review with a description of the neutron’s role in creation of the

5 ESS: European Spallation Source Project (under TRIUMF: National Laboratory for Particle and design), Lund, Sweden (Vettier et al ., 2009). , with ultracold neutron (UCN) FRM-II: National neutron source Heinz-Maier- Source Project (design phase), Vancouver, Canada Leibnitz, Technical University Munich, Germany. (Martin et al ., 2008). ILL: European neutron source Institute Max von Laue - Paul Langevin, Grenoble, France. Tables II and III give, where available, some basic J-PARC: Japan Proton Accelerator Research Complex, data for these reactor and spallation neutron sources, with Japanese Spallation Neutron Source (JSNS, taken from the facilities’ web sites, unless cited under construction), Tokai, Ibaraki, Japan. otherwise. Of course, other beam quantities like LANSCE: Los Alamos Neutron Science Center at Los neutron brightness, i.e., neutrons/cm 2/s/nm/sr, neutron Alamos National Laboratory (LANL), USA. temperature, beam profile, divergence, time structure, NIST: National Institute of Science and Technology, or degree of spin-polarization are also important with NCNR neutron source, Gaithersburg, USA. parameters that may vary from place to place. Most PNPI: Petersburg Nuclear Physics Institute, with facilities listed have projects on upgrades, either by an WWR-M neutron source, Gatchina, Russia, increase in source power, or by the installation of PSI: Paul Scherrer Institut, with SINQ quasi- better neutron guides, see for instance Cook (2009). continuous spallation source, Villigen, Switzerland. We shall discuss neutrons fluxes and the specific SNS: Spallation Neutron Source at Oak Ridge properties of cold and in the context National Laboratory (ORNL), USA. of the individual experiments.

TABLE II. Main neutron reactor sources having cold neutron beamline(s) for particle physics. Given are the reactor power, the in-pile neutron flux density, and the capture flux density available at the cold neutron beam station. Reactor Start/ Power n-capture flux (cm –2 s –1) References on n-flux facility upgrade (MW) in-source in-beam ILL 1971/94 58 1.5× 10 15 2× 10 10 Häse et al ., 2002 (upgrade 2009) FRM-II 2004 20 8× 10 14 1× 10 10 NIST 1967/97 20 4× 10 14 2× 10 9 Nico and Snow, 2005 PNPI 1959 16 1× 10 14 6× 10 8

TABLE III. Main neutron spallation sources having cold neutron beamline(s) for particle physics. Listed are the power of the proton beam (a + sign indicates: in the process of ramping up), the neutron flux density within the source (time-average and peak, where applicable), and the capture flux density available at the cold neutron beam station. n-capture flux (cm –2 s –1) Spallation Start/ Power References on n-flux facility upgrade (MW) in-source in-beam peak average average ~1.3× 10 10 Ferguson, Greene, 2010, SNS 2008 ~1+ 5× 10 16 1× 10 14 @1.4 MW private communication PSI 1998 1.3+ n.a. 1× 10 14 10 9 1.7× 10 16 LANSCE 1985 0.1 -- 1× 10 8 Nico and Snow, 2005

J-PARC 2008 0.4+ -- -- ~ 10 9 Mishima et al. , 2009

6 II. HISTORY OF THE EARLY UNIVERSE “grand-unified” interaction of fields and particles, there appear various interactions with different Particle physics, the physics of the very small, and properties. In case of a phase transition, the critical cosmology, the physics of the very large, are temperature sets the energy scale of the process of intimately interwoven in standard big bang theory (or spontaneous symmetry breaking. We call the hot phase variants thereof). Hence, for each phenomenon seen in above the critical temperature with vanishing order particle physics at some high-energy scale, we find an parameter the unbroken or symmetric phase, and the epoch in the evolution of the universe when phase below the critical temperature broken or temperature and density were so high that the unsymmetrical. phenomenon in question played a dominant role. To We start with a look on the “old” big bang theory set the scene, we shall therefore first give a short (Weinberg’s “first three minutes”), whose overall history of the early universe, and then discuss the evolution is very simple so we retrace it in the various experiments done in neutron physics that help following few lines. For a modern view see shed light on the corresponding epoch of the universe. Bartelmann (2010) and Mukhanov (2005), and the Even when a scale is not accessible with neutron books recommended there. Let R(t) be the experiments (the inflation scale, for example), it must cosmological scale factor at a given time t (the still be shortly discussed in order to understand the “radius” of the universe), then the expansion rate of the subsequent processes of baryogenesis, etc. universe at time t is the Hubble function defined as We start at the earliest age of the universe, at which H(t )= Rɺ / R . The evolution of R(t), H(t ) , and of our present understanding of the laws of physics may temperature T(t) (see Table I) are derived from the be sufficient to describe the evolution of the universe, Friedmann equation, which reads, for a curvature using the standard quantum field theory leading to the parameter k = 0 of the flat universe, and a Standard Model (SM) of physics, cosmological constant Λ = 0 , and general relativity leading to the Standard Model of 1/2 cosmology. This starting time is definitely later than Rɺ 1 8 π  − H = = ρ the Planck time t =5 × 1044 s , which latter G  . (2.1) Pl R c 3  corresponds to an energy scale of E =1.2 × 1019 GeV . At this scale, quantum effects of The gravitational constant G is related to the Planck Pl mass in Table I as G= ℏ c/ M 2 . are dominant, and today’s theories are not Pl developed enough to cope with in any detail. The energy density of the universe is divided into ρ After this time, the history of the young universe the mass density m and the relativistic radiative ρ ρ= ρ + ρ becomes a succession of phase transitions (either in the energy density r , such that r m . In the strict sense, or less so), starting with an inflationary -dominated era, which lasted for the first period, explained below, and (perhaps simultaneously) about 400 000 years of the universe, the energy density with a breaking of grand unification. Above the scale varies with scale R as ρ≈ ρ ∝ 1/ R4 . The extra factor of grand unification, the electroweak and the strong r ρ ∝ 3 interactions are unified, below this scale, they describe of 1/ R as compared to the mass density m 1/ R separate processes. Later on, we reach the scale of the stems from the red-shift of all wavelengths λ ∝ R in present SM, at which electroweak unification is the expanding universe, such that radiative energy of ≈ =λ ∝ broken, and below which electromagnetic and weak relativistic particles varies as Er pchc/ 1/ R , interactions are separated. These transitions occur at where we have used the de Broglie relation. With specific transition temperatures during the cool-down ρ ∝1/ R4 entered in Eq. (2.1) we obtain Rɺ ∝1/ R of the expanding universe, as listed in Table I, see, for r example, the books by Kolb and Turner (1990) or by and hence, as verified by insertion, Mukhanov (2005), and later on are followed by a R( t ) ∝ t , (2.2) series of (nucleons to nuclei, etc.). H =ɺ = In each of the early transitions new properties of the ()t RR / 1/2 t , (2.3) system pop up spontaneously, which were not present T() t∝ 1/ t . (2.4) in the system before the transition. In this way, the universe successively acquired more and more The last equation follows from the Stefan-Boltzmann structure, which is a precondition for filling it with ; law the initial high symmetry, although of great beauty, in π 2 N the end would be rather sterile. Such a transition could ρ = (kT ) 4 , (2.5) r ℏ 3 be, for example, from a single type of particle field 30 (c ) possessing high internal symmetry to multiplets of fields of lower symmetry. In this way, instead of one

7 ρ ∝ 4 ρ ∝ 4 became transparent at temperature T ≈ 3000 K . The when we compare r T with r 1/ R , which gives T∝1/ R , and insert Eq. (2.2). The numerical directional fluctuations of this background reflect the − − quantum fluctuations of the inflationary epoch. They relation is kT/MeV= 1.55 Nt1/4 ( /s) 1/2 , by insertion are found to be nearly scale invariant, the relevant of Eq. (2.5) into Eq. (2.1), and use of Eq. (2.3). To take measure for this being the “spectral index” ns, expected into account the transitions between the different to be equal to one for everlasting inflation, and slightly stages of the universe, we note that the number of less for a finite inflation period. Indeed, CMB satellite degrees of freedom of highly-relativistic particles N in measurements find n =0.963 ± 0.015 . The parameter Eq. (2.5) decreases with every freeze-out process in s describing the amplitudes of these quantum Table 1, so N= N( T ) as shown in Fig. 19.3 in Olive fluctuations can be measured precisely, too, so and Peacock (2009), which leads to a small correction inflation is very accessible to observation today. These of Eq. (2.4). The number of degrees of freedom finally data are compatible with the simplest inflationary = 1 reached the stable value N 7 4 before model, while some other more complicated models are nucleosynthesis set in, see Sec. VII. in trouble, see Hinshaw et al . (2009), and references In the later matter-dominated universe, the energy therein. In spite of these successes, we have to admit ρ≈ ρ ∝ 3 density becomes m 1/ R , which, when that we still do not understand the real nature of inserted into Eq. (2.1), changes the solutions from inflation. Eqs. (2.2) and (2.3) to R∝ t 2/3 , and H = 2/3 t . At all When inflation ends, all ordinary matter and , temperature changes and scale changes are radiation fields will have their diluted by up to 100 orders of magnitude, so the universe ends up related as TTɺ /=− RR ɺ / =− H . Both rates diverge at extremely cold and essentially empty. The the “origin of time”, the unphysical big bang field φ , however, is about the same as before, but its singularity, and so need to be substituted by a φ forthcoming theory of gravity. kinetic energy then dominates over the potential V ( ) , However, to our present understanding, for a so the field may oscillate rapidly and decay into certain period of time in the very early universe, the ordinary matter and radiation. By this refilling or expansion rate of the universe is determined by an “reheating” process, all the particles are created from effective cosmological constant Λ > 0 representing the the inflaton energy liberated in the transition. energy density of an “inflaton field”. Instead of Due to the continuing expansion of the universe, Eq. (2.1), we then have Rɺ / R ≈ ( Λ / 3) 1/2 , which is the the temperature reached in the reheating process after inflation is much lower than the about 10 14-16 GeV equation for exponential growth with the “inflationary” involved in inflation. Grand unification, however, must solution 16 take place at about 10 GeV, and therefore cannot R( t )∝ exp Λ / 3 t . (2.6) occur after the end of inflation. Thus, Grand Unified Theory (GUT) can only play a role if GUT breaking If inflation lasts for a time such that the exponent comes together with inflation as done in “hybrid” reaches at least 50 to 60, it can explain the observation inflation models. Therefore inflation and GUTs form that the universe is both homogeneous and isotropic, one entry in Table I. which, without inflation, would lead to problems with The cosmological constant after inflation and GUT causality. Furthermore, inflation explains how the transitions is near zero, but evaluation of recent universe is flat on the one percent level, which would observations on the CMB and on supernovae showed otherwise require an immense fine-tuning, but is now, that, besides “dark” unknown matter, again, like in instead, substituted by a set of initial conditions for the inflation, a kind of (now “small” compared to the inflaton field ϕ , a scalar field with a self-interaction previous value!) cosmological constant Λ is required, potential V (ϕ ) . For access to the vast literature on this leading to an accelerated expansion of the universe Λ subject, we again refer to Bartelmann (2010) and today. If, in particular, this “constant ” is a time- Mukhanov (2005). dependent field, it is called dark energy It is also a wonderful effect of inflation that it (“quintessence”, “cosmon”), see Wetterich (1988), and automatically creates fluctuations, first as quantum Ratra and Peebles (1988). During the past few years, fluctuations, which then turn into classical fluctuations. new astrophysical data from different sources This effect is absolutely necessary for the structure (structure formation, redshifts, and others) confirmed formation (galaxies, stars) in the late history of our consistently the presence of large amounts of dark universe. In recent years the inflationary scenario has matter and of dark energy, see, for example, become accessible to observation via the precision Mukhanov (2005). measurements of the cosmic microwave background To conclude, cosmology and astrophysics have (CMB), offering a snapshot of the universe when it become accessible to experimental tests up to the very

8 first instances of the big bang, and we shall see what corresponding state, P stands for the parity neutron physics has to offer in this endeavor. operation that inverts the spatial coordinates r→ − r of the particle’s state, and T stands for the time reversal operation that inverts the time coordinate III. BARYOGENESIS, CP VIOLATION, AND t→ − t of the state, and, simultaneously, interchanges THE NEUTRON ELECTRIC DIPOLE MOMENT the ingoing and outgoing states. A system is called C, P, or T-symmetric, when it is invariant under these In cosmology, the most obvious fact still lacking operations, respectively. In nature, each of these explanation is the existence of large amounts of matter discrete symmetries is violated in some system or that form the galaxies, the stars, and interstellar matter, other. However, in standard local quantum field with essentially no antimatter, in a universe that started theory, our basis, the interactions are invariant under with no particle content right after inflation. This the combined operations CPT , with the consequence problem of baryogenesis turns out to be closely linked that particles and have identical to the possible existence of electric dipole moments of properties, except for the signs of their various particles. All explanations of baryogenesis in the early charges, and have equal distributions at thermal universe necessarily require a significant violation of equilibrium. CPT invariance implies that CP and T are what is called “CP symmetry”. In this case, particles either both good symmetries, or are both violated. like the neutron or the electron should carry a CP Given C and CP invariance, equal numbers of violating electric dipole moment, where specific particles and antiparticles were created in the early models of baryogenesis may be tested in ongoing universe after inflation. Later on, particles and EDM experiments. To substantiate this claim we first antiparticles should have almost completely discuss the models of baryogenesis in some detail. We annihilated each other, falling out of equilibrium only then give a survey of ongoing and planned neutron at a time when the density of the expanding and EDM experiments, and, finally, give a review of the cooling universe had become so low that the remaining theoretical implications of the neutron and atomic particles and antiparticles in question did not find each EDM limits. other anymore. For the nucleons and antinucleons this happened when the universe had an age of about 2 ms, with the corresponding freeze-out temperature of A. Baryogenesis kT ≈ 20 MeV . We shall discuss in more detail the mechanism of from thermal equilibrium in 1. The dominance of matter over antimatter another context in Sec. VII.A. The number of and antibaryons left over The matter content of the universe usually is after this “carnage” should be a minuscule quantified by the ratio of the number density of −18 n/ nγ= n / n γ ~10 , which, for the baryons, is baryons (i.e., hadrons composed of three quarks, which b b today are essentially the proton and the neutron) to the more than eight orders of magnitude below the number density of , which is observed value, Eq. (3.1). Hence, one would expect the η = − so-called (nb n ) / n γ to = ± × −10 b nb / n γ (6.08 0.14) 10 , (3.1) ≈ vanish, whereas observation gives nb 0 , so as derived from the recent CMB satellite data and from η ≈ ≈ × −10 nb / n γ 610 . The process responsible for this primordial nucleosynthesis data, see Sec. VII.A.4. At unexpectedly large baryon asymmetry of the universe the present temperature T =(2.728 ± 0.004) K of the (acronym BAU), generated from an initially symmetric = −3 universe the photon number density is nγ 405 cm . configuration, is called baryogenesis. Baryogenesis There is strong evidence that the universe consists must have taken place after inflation, otherwise entirely of matter, as opposed to antimatter, see Cohen particle density would have been diluted to zero. et al . (1998), and the few n/ n ~10 −4 There are three necessary conditions for p p baryogenesis called the three Sakharov criteria: seen in cosmic rays can all be explained by secondary pair production processes. This observed dominance of (1) Baryon number B, which is the number of baryons matter over antimatter is not what we expect from the minus the number of antibaryons, should not be a standard big bang model discussed above, in conserved quantity, that is, there must be combination with the known laws of physics as elementary processes that violate baryon number, subsumed in the SM of particle physics. in order that baryogenesis can proceed from an = > A cornerstone of particle physics is the CPT initial total B 0 to a universe with B 0 . theorem. Here, C stands for the operation of charge (2) Charge conjugation symmetry C and the combined conjugation, which carries a particle state into its symmetry CP should both be violated, the latter of

9 which, under the CPT theorem, is equivalent to a (3) Baryon-asymmetry generating processes must take violation of time reversal symmetry T. Indeed, one place far from thermal equilibrium, because can show that if C or CP were exact symmetries, otherwise, even for processes violating baryon then the reactions that generate an excess of number B as well as C and CP symmetry, both baryons would occur at the same rate as the particles and antiparticles, due to CPT symmetry, conjugate reactions that generate an excess of would still have the same thermal distribution. antibaryons, such that the baryon asymmetry

would retain its initial value η = 0 .

FIG. 1. The three Sakharov criteria, necessary for baryogenesis, and their possible realizations (SM = Standard Model).

Our mere existence requires that all of these three acquires its characteristic symmetry breaking necessary conditions for generating a baryon “Mexican hat” shape when µ 2 (T ) turns negative at asymmetry have been satisfied at some time in the the critical temperature T= T , see the dashed line in early universe. As is visualized in Fig. 1, there are c various ways to satisfy these conditions, as we shall Fig. 4 below. < discuss in the following. However, even for a given In Landau theory, for T T c the classical nonzero set of realizations, it will turn out that the construction expectation value of the Higgs of models is certainly restricted but is in no way υ= 2〈Φ 〉 =2( − µ2 / λ ) 1/2 appears as an order unique. parameter. Near the new potential minimum, the still unobserved moves with a mass 2 1/2 1/2 m =(2 −µ ) = υ (/2) λ . The W boson, which 2. Some quantities of the Standard Model H mediates the electroweak interaction, obtains its mass = 1 υ Let us first recall some basic quantities of the mW2 g w from the Higgs process, with the ≈ = electroweak Standard Model that will be needed in the observed value mW 80 GeV at T 0 , where discussions to follow. In the SM, all particles obtain υ = 246 GeV . The range of the electroweak their mass through the Higgs process. The Landau interaction is the of the W boson “quartic” effective potential of the Higgs field Φ λ= π = × −17 C2ℏ /m W c 1.510 m . The weak SU (2) =µ2 2 + 1 λ 4 V (Φ) | Φ|4 | Φ| , (3.2) gauge coupling constant in the SM is

10 =θ ≈ gw e / sin W 0.65 , (3.3) in large amounts in the present universe changes an initial B+ L = 0 to B+ L = 2 while leaving B− L = 0 1/2 where e =(4πα ) ≈ 0.31 is the (dimensionless) unchanged.) elementary electric charge. The fine structure constant For temperature T = 0 this is a tiny effect, therein is taken at the scale of MZ, the mass of the presumably unmeasurable in laboratory experiments. neutral electroweak Z0 exchange boson ( α ≈1/128 This is because it is induced by “instanton” gauge instead of its usual value 1/137). At scale MZ, the field configurations, quantum mechanical tunneling weak or Weinberg angle has been measured to solutions in Euclidean space with imaginary time that θ ≈ =θ ≈ mediate tunneling between the different topologically sinW 0.48 , and MZ M W/ cos W 91 MeV . inequivalent vacua. They have an exponential transition probability, typical for tunneling, 3. First Sakharov criterion: Violation of baryon Γ ~exp( –8π 2 / g 2 ) , (3.4) number instanon w

with the electroweak gauge coupling gw from Eq. (3.3) Violation of baryon number can have several 2 2 With a “Gamow factor” of 8π /g ≈ 190 , this sources. w transition probability is extremely small. a. Grand unification : GUT models violate baryon Figure 2 gives a typical graph of the effective number B because in their “unbroken” phase they action for chiral left-handed quarks and leptons with + = unify quark, , and gauge interactions, as we initially B L 2 for the simplified case of just one + = shall exemplify in the simplest GUT, based on SU (5) generation. This demonstrates that (B L ) 2 and symmetry. In this model, quarks (baryons) and leptons (B− L ) = 0 . occupy the same particle multiplet such that they can transform into each other, which changes both baryon number B and lepton number L. The proton (B = 1) , for instance, may decay, p+→π 0 + e + , by changing a quark into a (lepton number L = − 1), with B =1 − 0 = 1 equal to L =0 −− ( 1) = 1 , so (B+ L ) = 2 and (B− L ) = 0 , thus violating B+ L but conserving B− L .

CP -violating GUT breaking must have taken place FIG. 2. In instanton or transition between after inflation, however, there are the problems two vacuum states of Fig. 3, a baryon B number mentioned to return, after inflation, to the unbroken violating effective instanton interaction (1, 2, 3 = GUT-symmetry (see end of the preceding Sec. II). color indices) is induced between u, d quarks and the Therefore GUT-baryogenesis, which was most electron (one generation). important in the early days of cosmology, is not attractive anymore, unless combined with inflation in hybrid inflation models. Furthermore, in the case of If, however, the temperature T is high, then there − B L conserving SU (5), even if a baryon asymmetry are similar transitions, but now we have thermal had been generated, it would have been washed out transitions over an unstable three-dimensional later on in an equilibrium epoch before the sphaleron (gauge-Higgs) configuration, see Fig. 3. electroweak phase transition due to (“hot”!) sphaleron This transition again is exponentially suppressed, but transitions, outlined next. now only by a Boltzmann factor

− b. *Axial anomaly : The SM permits baryon number Γsphaleron~exp(E sph / kT ) , (3.5) violation and, in principle, baryogenesis, too (Kuzmin = π 2 et al ., 1985). Baryon number violation then is due to where ETsph() fmT W () (8/ g w ) is the sphaleron the so-called axial anomaly in the electroweak sector. energy (setting c =ℏ = 1 ), with mW the mass of the W- In this effect of quantum field theory, the total number boson, and a numerical factor f (Klinkhammer and B+ L of quarks and leptons is violated if the Manton, 1984). topological winding number of a certain weak gauge As already mentioned in Sec. 2, the value of υ , field configuration that they are located in does not = 1 υ and with it the W mass mW2 g W , is temperature equal zero (“change in the Chern-Simons number”), υ= υ − dependent, (T ) : For increasing temperature T, while B L remains conserved in this process. 2 (Example: The creation of a as found the Higgs field, besides a negative (mass) , acquires

11 = − = an additional positive mass term that can be shown to leads to B 0 if originally B0 L 0 0 , and in be ~T 2 | Φ| 2 , and its classical value υ(T ) is reduced, thermal nonequilibrium (as in a strong electroweak being zero in the hot phase above the electroweak phase transition to be discussed later) it can produce a transition temperature. The sphaleron energy Esph of baryon asymmetry via a hot sphaleron process. Eq. (3.5) is due to a classical solution of the gauge- Higgs system. In the hot phase, with the Higgs expectation value vanishing, one has a much more 4. Second criterion: CP Violation complicated three-dimensional nonperturbative gauge configuration, which leads to unsuppressed B+ L The violation of C (and P) is no problem as it is violation (Bödeker et al ., 2000), the “hot sphaleron large for the electroweak theory, so the violation of effect”. CP is the real problem. In the SM there are two kinds of possible CP violations, and we will discuss others V beyond the SM. Sphaleron a. Strong CP violation : In the strong sector of the SM, one expects CP violation by an additional gauge kinetic term

2 CPVgs ɶ aµν a L= − θ Gµν G , (3.6) 32 π 2 ∑ a=1...8 T= 0: Instanton the so-called θ -term in the quantum chromodynamic Winding no. (QCD) Lagrangian, with the strong coupling µν λσ 1 2 3 a ɶ a= 1 ε a constant gs, field tensors G and Gµν2 µν G λσ ,

and color index a. This is equivalent to a complex FIG. 3. In the SU(2) gauge theory, there are w phase in the quark mass determinant: both are related degenerate ground states with nonzero topological by an anomalous axial U(1) symmetry winding number. Baryon number violating transitions A transformation. This “strong CP violation”, see Peccei between two such SM vacua are possible, either by (2008) and references therein, could be large in tunnelling (instanton transition at T = 0 ), or by principle, which is called the strong CP problem, but thermal excitation over the barrier (sphaleron up to now, there is no experimental indication for this. transition). Even small θ -values would produce large EDMs of

θ the neutron. Introducing as an “axion” field and a c. Majorana mass : Very interesting for baryogenesis U(1) PQ - Peccei-Quinn symmetry, as in, for example, would be the existence of very massive Majorana the Minimal Supersymmetric Standard Model θ ≡ giving a mass to ordinary neutrinos via the (MSSM, see below), 0 is a minimum of the so-called seesaw mechanism. A massive Majorana classical action. Equally, perturbative effects mixing lepton (i.e., a lepton that is identical to its antiparticle) the strong CP problem (present case a) and CP has a mass term in the Lagrangian that violates lepton violation in the Cabibbo-Kobayashi-Maskawa quark number L conservation. The out-of-equilibrium decay mixing matrix (or CKM-matrix, next case b) only lead of massive Majorana neutrinos (“leptogenesis”, see to tiny values of θ. Buchmüller et al ., 2005, and references therein) can violate both CP and lepton number. A lepton b. Electroweak CP violation : In the electroweak asymmetry then is partly transformed into a baryon sector of the three-generation SM, there is the

B-asymmetry via a hot sphaleron process in thermal experimentally well confirmed CP violation in K and equilibrium above the electroweak scale. Starting with B decays. This CP violation is naturally integrated δ a lepton asymmetry, B = 0 and L ≠ 0 , one ends up into one complex phase of the left-handed down 0 0 quarks in the CKM matrix (described in more detail in with a baryon asymmetry B=(8 N + 4)/(22 N + 13) Sec. VI.A). As we shall see below, the Jarlskog × ≠ = 28 = |B0 – L 0 | 0 , that is, B79 | L 0 | for N 3 particle determinant as a measure of the strength of generations. electroweak CP violation would indicate that this − = If instead one starts with B0 L 0 0 , as in SU (5) effect is much too small for baryogenesis, but one has GUTs, then one finds that B+ L =2 L ≠ 0 is to inspect that very carefully. Both sources (a) and (b) 0 0 0 of CP violations are genuine in the SM. washed out to B+ L ≈ 0 , and with conserved

B− L = 0 this leads to B = 0 . Thus, the hot sphaleron transition has two faces: In thermal equilibrium, it 12 < H < H c. Massive neutrinos : If we admit massive neutrinos in 1/ treaction or Γdecay , respectively. Still, this is the SM, as dictated by the recent discovery of neutrino a quantitative question if it comes to a successful oscillations, a Majorana type mass leading to the model of baryogenesis. This source of nonequilibrium seesaw mechanism generates not only lepton number is particularly effective at the very early times soon violation, as mentioned in Sec. 3.b above, but also after inflation, for leptogenesis, for example, with a free complex phases in the mass matrix that violate slow out of equilibrium decay of Majorana neutrinos CP . Such CP violation in the lepton sector may be (Buchmüller et al ., 2005). interesting for baryogenesis (then called leptogenesis, ν see Buchmüller et al ., 2005, and for the “ MSM” b. Phase transitions occurring either after the extension of the SM with three right-handed neutrinos inflationary phase or at the electroweak scale are more see Shaposhnikov and Tkachev, 2006), but it would vigorous as sources of nonequilibrium. In the first have very tiny effects on the neutron EDM. case (high scale), as mentioned before, one can also construct “hybrid inflation models” where inflation d. In models beyond the SM there are more CP and the breaking of a GUT symmetry come together. violating field phases that cannot be transformed By introducing a CP violating term in the effective away, and which could be large. This already happens interaction of such hybrid models, one can study an in models with two Higgs doublets, see Bernreuther asymmetry in the production of particles after (2002), but is most prominent in supersymmetric inflation using (quantum) transport equations models (SSM). Supersymmetry (SUSY) is a (Garbrecht et al ., 2006). This asymmetry then is symmetry between and . In the transformed into a lepton asymmetry via Majorana minimal SUSY model (MSSM), each particle of the fermions, which appear quite naturally in, for SM has a with equal mass and with equal example, SO (10) GUTs, and into a baryon asymmetry couplings – the quarks share a supermultiplet with the by the (hot) sphaleron transition. squarks and the gauge bosons with the . As The second case of a (rather low scale) these are not observed in experiment, electroweak transition is most interesting because the supersymmetry has to be broken, explicitly by soft present experiments at CERN (the European center for breakings in an effective theory, or spontaneously in a particle physics in Geneva, Switzerland) approach the more fundamental theory. Furthermore, instead of one TeV energy scale, and variants of the SM can be Higgs doublet, two doublets are required, Hu coupling tested. In contrast, the succeeding QCD transition for to the up and Hd to the down quarks. In the MSSM, the quark gluon plasma to the hadronic phase (Table free phases in the - and the stop R – I) is very mild (“crossover” transition), judging from stop L mass matrices violate CP ( stop R and stop L are QCD lattice calculations (unless there are strong SUSY-partners of the right and left-handed top effects of the baryonic chemical potential, see Boeckel quark). Extensions of the (minimal!) MSSM have an and Schaffner-Bielich, 2010). additional SM-neutral singlet chiral superfield (NMSSM, nMSSM, see also Sec. 7 below). 6. Baryogenesis in the Standard Model (SM) e. Spontaneous breaking of CP : Whereas in the model(s) with two Higgs doublets we have Indeed, at first look the SM seems to fulfill all spontaneous CP violation in the Higgs sector, the three conditions (Kuzmin et al ., 1985): breaking of CP symmetry is explicit in most SUSY models. There is also the exciting possibility of a (1) There is baryon number B violation from the spontaneous breaking of CP symmetry just around the sphaleron transition. phase transition. In the MSSM, unfortunately, this (2) There is CP violation in the electroweak sector does not happen (Huber et al ., 2000, Laine and (CKM matrix), and, possibly, in the strong sector Rummukainen, 1999), but might happen in the (θ -term). NMSSM (Huber and Schmidt, 2001). (3) Far off equilibrium, there could be a strong

electroweak phase transition, with 〈H 〉 = 0 above the

5. Third criterion: Nonequilibrium electroweak scale, and a classical neutral Higgs-field H value υ()2T= 1/2 〈 H () T 〉 below the electroweak Thermal nonequilibrium in the early cosmos, in scale going to υ = 246 GeV at T = 0 . principle, poses no problem: Naively, within the SM, in a Higgs effective a. Expansion of the universe : Some reactions or potential, Eq. (3.2), a temperature dependent decays fall out of equilibrium if they become too slow additional positive Higgs mass term proportional to T2 as compared to the Hubble expansion rate, 13 (mentioned in Sec. 3.b) turns around the unstable criterion would not be fulfilled. Looking at the SM as situation at Φ = 0 for T = 0 (see dashed and an effective theory, one can speculate that further continuous lines in Fig. 4). If one adds to this a one- (nonrenomalizable) terms like Φ 6 could appear in the loop contribution (integrating out the gauge bosons in SM Lagrangian. Their origin would need explanation, the loop) proportional to −Φ3 , these effects together but in this way one can enforce a strong first order produce the typical action (free energy) of a first order phase transition (see Sec. C.3). phase transition. The two minima corresponding to the The CP violation through the CKM matrix has a two phases, with a critical temperature Tc, at which experimental basis. However, if one wants to both phases coexist, give the typical picture of create a CP violating effect in the phase transition condensing droplets or boiling bubbles of a first-order discussed before, naively, it would be proportional to phase transition. the Jarlskog determinant

J =(m2 − mm 22 )( − mm 22 )( − m 2 ) CPυ12 ucctt d ×−2 22 − 22 − 2 (md mm ss )( mm bb )( m d ) (3.7)

with quark mq, and with the invariant measure of CP violation =† † = 2 δ JIm( VVVVud dc cb bu ) sssccc1 23123 sin , (3.8) where V is the CKM-matrix (in the KM version), the θ si and ci are the generalized Cabibbo angles sin i and θ δ cos i , and is the CP violating phase. Evaluating Eq. (3.8) with the experimental data gives = ± × −5 ≤ −19 J (3.05 0.20) 10 , and CP 10 . Even if we substitute the electroweak scale υ in

Eq. (3.7) by the critical temperature Tc of the < transition, CP remains very small – much too small FIG. 4. The Higgs potential at T T c (dashed line) has the usual “Mexican hat shape”. With additional T² for successful baryogenesis. This estimate is based on and Φ3 terms, the Higgs potential resembles the free the assumption that the quark masses enter perturbatively in loop calculations that lead to a CP energy of a first-order phase transition. violation. However, in the case of strong nonequilibrium there may be effective interactions CP violation can then take place far off containing derivatives suppressed not by CP , but by equilibrium in the walls of the expanding bubbles. the Jarlskog factor only, of order J ~ 10 −5 (Tranberg This would produce a chiral asymmetry between left- and Smit, 2003). Indeed, very recently such an handed and right-handed quarks, which, in the hot effective interaction was derived in the next-to- plasma in front of the expanding bubble, is leading order of a derivative expansion of a one-loop transformed into a baryon asymmetry by the hot effective action, with all quarks in the loop integrated sphaleron transition. Due to the exponential sphaleron out (Hernandez et al ., 2009). Still, in order to obtain suppression Eq. (3.5), this asymmetry then has to strong nonequilibrium one needs a mechanism beyond freeze out rapidly in the Higgs phase when the the SM. Low-scale inflation with an additional expanding bubble takes over. This leads to the inflaton field has been suggested. Indeed, one can postulate υ(T *)≥ T * , which makes the Boltzmann obtain a sizable baryon asymmetry this way (Tranberg factor Eq. (3.5)(with exponent ~−υ /T * ) sufficiently et al ., 2009), but this is rather speculative. small, where T* is the transition temperature (with T* < T because there is a delay transition time vs. c 7. Baryogenesis beyond the Standard Model expansion time).

Later on it turned out in lattice simulations Hence, in the SM, both CP violation and deviation (Kajantie et al ., 1996) that at Higgs-masses m above H from thermal equilibrium are too weak to permit the W-mass m ≈ 80 GeV (current Higgs limit W baryogenesis. Therefore presumably we must go > mH 114 GeV , 95% C.L. = confidence level) the beyond the SM, and more general variations of the transition is very smooth, a crossover rather than a electroweak theory are mandatory if we want to stick strong first order phase transition, such that the third to electroweak baryogenesis. Today, physics at the

14 electroweak scale is under experimental investigation CP violations in the MSSM enter preferably both at CERN and in low-energy experiments. Thus, through a complex phase factor in the gaugino- this type of baryogenesis can be tested and therefore is Higgsino mass matrix particularly attractive. A simple way to enlarge the m g〈 H 〉  SM is to introduce two Higgs doublets, instead of the M = 2 w d (3.9) µ  one Higgs doublet of the SM (Bernreuther, 2002, gw〈 H u 〉  Cline, 2006, Fromme et al ., 2006). µ The most prominent model beyond the SM is the with the W-ino mass m2 and the of the two-Higgs µ Minimal Supersymmetric Standard Model (MSSM). mass term Hu H d in the superpotential carrying one Besides having two Higgs doublets, the MSSM has free CP violating phase, and with the classical the field content of the SM just enlarged by scalar expectation values 〈H 〉 and 〈H 〉 of the two neutral (squarks, sleptons) and fermionic (Higgsino, u d Higgs fields. Substituting m in Eq. (3.9) by the B-ino gauginos) superpartners, the gaugino being the 2 mass m with a second CP violating phase helps much superpartner “ W-ino” to the SM gauge particle W in 1 in avoiding EDM constraints (Li et al ., 2010). The our case . The superpartners couple with the same slowly moving “thick” bubble wall (L> 1/ T ) of the strength in the case of unbroken SUSY. As no SUSY partners to the known particles have been detected so first-order phase transition allows for a quasiclassical 2 far, (soft) SUSY breaking is mandatory in a realistic WKB type treatment of order ℏ , but close to the µ model but requires further couplings. degeneracy m2 ~ in Eq. (3.9) there is a dominating In the MSSM, we can obtain a strong first-order oscillating contribution of order ℏ . The (quantum) phase transition if a light stop R-particle in the loop transport equations for the Higgsino-gaugino system strengthens the −Φ3 term in the effective Higgs in front of the expanding Higgs-phase bubble have action (Fig. 4) (Carena et al ., 1996, 2003a, Laine and been discussed in much detail (Carena et al ., 2003b, Rummukainen, 1998). As already seen in Bödecker Lee et al ., 2005, Konstandin et al ., 2006) and it turned et al . (1997), who thoroughly discuss perturbative out that one can get successful baryogenesis only after contributions in the MSSM, the Φ3 term argument is squeezing the parameters of the model. rather sketchy and also slightly gauge dependent. More recently, also the diffusive processes in front Taking into account thermal mass contributions, of the bubble wall in the hot plasma have been the conditions for such a strong transition are roughly reanalyzed very carefully, introducing also b quarks, ≤ τ leptons, and SUSY nondegenerate gauge mH 118 GeV for the lightest Higgs mass, just barely consistent with the present experimental limit interactions in the diffusion chain that lead from the Higgsino-Gaugino (also B-ino!) system to the left- m ≥114 GeV , and a bound on the “light” stop - H R handed quarks needed for the sphaleron process ≤ ≤ = mass 110 GeVmstopR m top 170 GeV (narrowing (Chung et al ., 2009, 2010, and references therein). As we will see in Sec. C.2, neutron EDM constraints are to mstopR ~ 140 GeV at the upper limit for mH). In case very restrictive. of a very strong mixing of the two-level Higgs boson Baryogenesis is much easier to achieve in models system due to CP violation, the lower limit on the that introduce, besides the MSSM superfields, a Higgs can be a considerably weakened new limit further scalar gauge singlet field N (sometimes called (m ≥ 70 GeV?) (Carena et al ., 2003b, and references H S), which also can obtain a classical expectation therein). value 〈N 〉 of the order of 〈H 〉 in an enlarged Higgs These conditions for a strong phase transition potential. These models are called NMSSM (next to β= υ υ depend on the ratio tanu / d of the vacuum MSSM) (Huber and Schmidt, 2001) and nMSSM expectation values of the two Higgs fields Hu and Hd (nearly MSSM) (Menon et al ., 2004, Huber et al ., (for up and down quarks), and on the average SUSY 2006). In these models, a Φ3 type term is already breaking mass. They can be weakened a bit by present in the effective potential at the tree level. allowing that the Higgs phase be only metastable In Profumo et al . (2007a), and in a recent paper by (against decay into a colored phase), i.e., stable for the Patel and Ramsey-Musolf (2011), also a reduction of time of our universe, which leads roughly to the effective scalar quartic self-coupling at the ≤ mH, m stopR 125 GeV (Carena et al ., 2009). The transition temperature can do the job. This alleviates present experimental limit for a right-handed stop much the search for a strong first-order phase ≥ transition, and this remains true in more general mass is mstopR 100 GeV . The additional requirement models (Noble and Perlstein, 2008). This allows the of baryogenesis then forces the lowest lying Higgs conditions on the Higgs mass m ≥114 GeV and on boson to be of the SM type. Light Higgs boson H ≤ ≤ scenarios are not successful (Funakubo and Senaha, the “light” stop R-mass 110 GeV mstopR m top 2009). 15 =170 GeV quoted above to be relaxed, furthermore, B. Electric dipole moments (EDMs) a “low”-mass stop R is not needed. There still is a low lying Higgs, but now Higgs masses up to Next, we discuss EDMs and their symmetries, ≤ describe experimental searches for the neutron EDM, mH 125 GeV are allowed for successful baryogenesis, causing a better overlap with the give current limits for EDMs of the neutron and other experimental lower limit. particles, and give a survey of upcoming neutron Recently it was proposed to use the three MSSM EDM experiments. superpartners of the right-handed neutrinos as singlets mentioned just before (“ µν MSSM ”, Chung and 1. EDMs and symmetry violations Long 2010). This offers new interesting possibilities.

Models without SUSY going beyond the SM Before discussing the electric dipole moment, we mentioned at the beginning of the present section and recall that the magnetic dipole moment operator of a discussed in Sec. 4.d above offer more freedom but particle is = gµ j / ℏ , with spin operator j. Its size is are less motivated. N defined as µ= g µ j , measured in units of the nuclear In conclusion, the necessary conditions for N µ =ℏ = × −8 baryogenesis certainly limit our choice of models. magneton Ne/2 m p 3.15 10 eV/T , with the When we go through the possible realizations listed in Landé factor g, and proton mass m p. Magnetic Fig. 1, baryogenesis probably does not happen in the moments can be measured with high precision by cases 1a): B-violation from the GUT transitions, and observing the spin precession of the polarization 2a) and 2b): CP violation from the SM electroweak vector P= 〈 j 〉 / ℏj about an external magnetic field B, transition. Still, this allows for quite a variety of ɺ = × different models. The electroweak baryogenesis – if P ω0 P , (3.10) realized in models beyond the SM, in particular, in with the precession frequency vector supersymmetric models – is very attractive because its ω =µB/ jℏ = γ B , and gyromagnetic ratio ingredients may be tested experimentally in the near 0 γ= µ = µ future in high energy physics at TeV energies and in /jℏ g N / ℏ . N.B.: We do not have to EDM experiments at neV energies. CP violation in apologize, as is frequently done, for using the these models being in the gaugino-Higgsino sector is “semiclassical” Larmor precession Eq. (3.10), which completely different from the CKM-matrix CP is just the Heisenberg equation violation of the SM, and might have a big CP iℏd〈j 〉 / d tH= 〈 j 〉 − 〈 j 〉 H for the interaction of a violating phase. This raises hopes that a large EDM of particle at rest with any classical field B, with the neutron could be present at the current Hamiltonian H = −γ j ⋅ B . (Semiclassical here only experimental limit – different from the situation for means that field B is unaffected by the quantum the SM where the neutron EDM is tiny. We shall transitions.) More generally, Eq. (3.10) is the discuss this in Sec. C. One also can face another irreducible representation of the Liouville equation for possibility: CP is broken in some model the density operator, and can, for j > 1 , be spontaneously (Huber and Schmidt, 2001) just at the 2 high temperatures of the electroweak phase transition generalized to higher multipole interactions, see Fano but not at T = 0 . In this case, there would be no trace (1964). = 1 in the EDMs. For the neutron with j 2 one usually writes One has to admit that baryogenesis via = µ σ and P = 〈σ 〉 , with the Pauli spin operator leptogenesis (Buchmüller et al ., 2005, and references = = − σ 2j / ℏ . The neutron’s g-factor gn 3.826 gives therein) with just heavy Majorana neutrinos added to − the magnetic moment µ = −0.603 × 107 eV/T and the content of the SM is also highly interesting and n γ π= µ π ℏ convincing in view of its connection to neutrino the gyromagnetic ratio n/2g n N /2 physics. The physics of the decay of the presumptive = 29.16 MHz/T . Majorana neutrinos is at much higher scales If the particle also has an electric dipole moment d (∼ 10 10 GeV) and much harder to test. The relevant then, like any vector operator in , CP violation in this case would have to be in the it is connected to the spin operator as d= d j / j ℏ , or, = 1 = Majorana neutrino mass matrix. Being in the lepton for j 2 , as d dσ , where d gives the size of the sector it may influence the EDM of the electron but EDM, usually in units of e cm. (N.B.: In quantum practically not that of the neutron. physics, isotropy of space requires that, in a rotationally symmetric problem, any operator is a multiple of a so-called spherical tensor; that is, all tensors of the same rank are proportional to each

16 other, and the vector operators , d, and σ all have and Ramsey, 1950) with a beam of thermal neutrons rank one.) If we expose the particle also to an electric at the reactor of ORNL, USA. The result < × −20 field E, then we have to replace in Eq. (3.10) the dn 5 10 e cm at 95% C.L. was published by = µ precession frequency ω0 2B / ℏ by Smith et al . only in 1957, the year of the discovery of parity violation (we write d for | d |, as long as no =µ + ℏ n n ω+ (Bd E ) / j . (3.11) nonzero EDM is found). Since then the sensitivity of The electric field is a polar vector, which, under these experiments has improved by more than six parity transformation P : r→ − r , transforms as orders of magnitude to today’s limit P : E→ − E , whereas the magnetic field B is an axial < × −26 dn 2.9 10 e cm (90% C.L., neutron) (3.14) vector, which transforms as P : B→ B . Therefore the precession frequency Eq. (3.11) transforms under obtained at ILL by a Sussex-RAL-ILL collaboration parity as (Baker et al ., 2006), using ultracold neutrons (UCN) stored in a “neutron bottle”. The energy sensitivity of → =µ − P: ω+ ω − 2(B d E )/ ℏ . (3.12) the apparatus required to reach this level of accuracy −26 Hence, for a nonvanishing EDM the frequency of spin is ~ 10 eV per 1 V/cm electric field unit, or − precession changes under parity operation P, so P 1022 eV for an electric field of E ~ 104 V/cm . symmetry is violated. Under P, the axial vector j This EDM limit was preceded by results changes sign on both sides of Eq. (3.10), so the < × −26 dn 6.3 10 e cm at 90% C.L. from ILL (Harris et precession equation remains the same, but with a −26 al ., 1999), d<9.7 × 10 e cm at 90% C.L. from shifted frequency. n An EDM d ≠ 0 would also violate time reversal PNPI (Altarev et al ., 1992, 1996), and < × −26 T: t→ − t , under which axial vectors like B and j dn 12 10 e cm at 95% C.L. again from ILL change their “sense of rotation”, T : B→ − B and (Smith et al ., 1990). There exist several review T : j→ − j , whereas E does not change, T : E→ E , articles on the experimental searches for a neutron nor does d〈j 〉 / dt on the left of Eq. (3.10). Therefore EDM. For the early experiments see Ramsey (1978, 1990), and for more recent developments, Golub and

T: ω+→ ω − = 2(µB − d E )/ ℏ , (3.13) Lamoreaux (1994), and Lamoreaux and Golub (2009). UCN physics in general is well described in the books hence, time reversal symmetry and CP symmetry by Golub et al . (1991), and by Ignatovitch (1990). (under the CPT theorem) are violated for a particle with nonzero EDM. A riddle: The neutron is a composite particle with TABLE IV. Some useful wall materials for UCN traps quark content dud , and we showed that a nonzero or neutron guides, their calculated Fermi potential V, EDM requires a violation of P and T symmetry. The and critical neutron velocity υ . The critical angle of water H O is a composite particle, too, with c 2 total reflection θ for cold neutrons is also given, in atom content HOH, and has a huge EDM. So why c search an EDM on the neutron dud when we have mm per m. υ −1 θ −3 found it already on HOH? For a solution, see Golub Material V (neV) c (ms ) c (10 ) and Pendlebury (1972) or Golub and Lamoreaux 58 Ni 343 8.1 11.6 (1994). Diamond-like 260 7.1 10.1 carbon Ni, Be, 65 Cu ~245 6.8 9.8 2. Neutron EDM experiments Stainless steel 188 6.0 8.6 Cu 169 5.7 8.1 One can measure particle EDMs with the highest Al 2O3 sapphire 149 5.3 7.6 precision by applying Ramsey’s method of separate SiO (quartz), 2 ~108 4.6 6.5 oscillatory fields (as used in all atomic clocks) to an Fomblin ensemble of spin-polarized particles. The Ramsey SiO 2 (glass) 91 4.2 6.0 method compares the internal “Larmor clock” of the particle with the external clock of a radiofrequency (RF) generator. We first discuss free neutron EDM b. Principles of nEDM experiments : Since 1984, all results, and at the end quote EDM results for other neutron-EDM searches in external electric fields particles. practised the storage of ultracold neutrons. UCN are defined as neutrons with velocities so low that they a. History and present status : The first experiment on are totally reflected by many materials under all the neutron EDM started many decades ago (Purcell

17 angles of incidence. The average nuclear “Fermi” analyser. The additional spin-flipper shown in Fig. 5 pseudopotential of materials seen by neutrons can induce a π -flip so one can start the measurement typically is of order V ~ 100 neV , see Table I; for thin cycle also with an inverted initial polarization –Pz0, coatings with reduced density, the values of V can be which helps to eliminate systematic errors. 10 to 20% lower. (The low value of V can be understood by considering that the long-wavelength UCN see the nuclear potential of some 10 MeV diluted in by the ration of atomic to nuclear volume (1 Å /1 fm)3= 10 15 .) Total reflection of UCN, and hence their storage in closed vessels, occurs up to a material dependent critical neutron velocity υ = 1/2 c (2V / m ) , which is also listed in Table IV. N.B.: The neutron guides mentioned on several occasions rely on the same principle and will be discussed in Sec. VI.B.1. In the laboratory, UCN can be manipulated equally well by gravitational, magnetic, and average nuclear forces, as the neutron experiences a gravitational force = mn g 102 neV/m , and has a magnetic moment of µ = n 62 neV/T . Hence, UCN of kinetic energy E =100 neV rise in the earth’s gravitational field to a = = maximum height of h E/ mgn 1.0 m , and a magnetic field totally repels neutrons of one spin- component if the field surpasses the value =µ = B E /n 1.6 T . Figure 5 schematically shows the EDM apparatus used at ILL. The neutron bottle of volume V = 20 ℓ , consisting of two diamond-like-carbon covered FIG. 5. Sketch of the Ramsey apparatus used at ILL to electrodes separated by a cylindric mantle of quartz, is search for a neutron electric dipole moment ( nEDM). filled with N ≈10 4 polarized UCN from ILL’s UCN The block arrow shows the neutron polarization right turbine source. Originally, this source had been after filling of the UCN trap (Port A and B open). The developed for the FRM reactor in Munich, see Steyerl dashed line represents a UCN trajectory in the UCN et al . (1986). After closure of port A in Fig. 5, two bottle. The sources of the E and B fields are not short Ramsey π / 2 -resonance flips are applied to the shown. UCN at the beginning and end of the storage period T of typically two minutes duration. The first RF pulse flips the initial “longitudinal” neutron polarization To monitor the magnetic field, atoms of the spin-½ 199 P ≈1 into the x-y plane, causing it to lie at right isotope Hg were stored together with the UCN as a z0 “comagnetometer” at about 10 –6 millibar pressure. angles to both B and B . Between the two flips, while 1 199Hg nuclei are known to have an EDM <10 –28 e cm, the now “transversally” polarized neutrons are freely see Sec. 4 below. The nuclei were polarized by atomic precessing about the magnetic field B = 1 T , the optical pumping, and their precession frequency was radiofrequency field (with rotating component B1) is monitored optically. gated off, such that its phase coherence is maintained between the two spin-flips. c. Sensitivity of EDM experiments : The Ramsey ω If the frequency of the RF field exactly equals method under nonperfect conditions (neutron ω= γ ≈ π × the Larmor frequency 0 B 2 30 Hz of the polarization P <1 and/or nonzero background B ≠ 0 ) neutron, the second flip will turn the polarization by is most sensitive to an external perturbation when ω another angle π / 2 into –P . The neutron bottle is ω z0 is slightly detuned from 0 such that, at the end of the then emptied onto the UCN detector, with ~ 2 m of free precession period T, and before the second RF free fall to accelerate the UCN sufficiently so they can pulse is applied, the spins have accumulated an overcome the repulsive potential of the aluminium ω− ω = π additional phase (0 )T /2 in the x-y plane. window of the gas detector for UCN. On their way = = down, the neutrons pass the magnetized polarizer foil (N.B.: For P 1 and B 0 , this sensitivity does no (see Fig. 5) a second time, the foil then acts as a spin longer dependent on such detuning.) The neutron 18 polarization P⊥ in this case no longer lies at a right error has surfaced lately (Pendlebury et al. , 2004), which was already known in earlier atomic EDM angle to B1, but is parallel to B1, such that the second π / 2 -flip is no longer effective. However, if an work (Commins, 1991). Once recognized as such, this electric field is superimposed (in the ILL experiment source of error is not difficult to understand. It is due of size E =10 kV/cm ), for d ≠ 0 an additional to unavoidable residual field inhomogeneities. The z n UCN in their rest frame see these inhomogeneities as ϕ = ℏ phase shift 2dn E z T / will be induced in the x-y a time dependent magnetic field B that continuously plane, after which the second π / 2 RF pulse will then changes direction. =−ϕ ≈− ϕ produce a longitudinal polarization Pz sin We know from the work of Berry (1984) that the = exposure of a spin-magnetic moment to a magnetic (for Pz0 1). field of slowly varying direction during time T leads How well can such a small rotation out of the x-y γ plane through ϕ << 1 be measured, in principle? Let to a geometric phase , in addition to the usual dynamic Larmor phase γ BT (with the neutron’s the direction of B1 define the quantization axis. For a n γ single particle, the polarization Pz (“transverse” with gyromagnetic ratio n ). This extra phase develops respect to the quantization axis) remains completely continuously and reaches the size γ = 1 after the >> 2 uncertain. Let, for N 1 particles, a spin analyzer be field has come back to its original orientation, ≈ − ϕ aligned along Pz . The analyzer transmits the where is the solid angle of the cone formed when ≈1 + ϕ ≈1 − ϕ number N+ 2 N (1 ) or N− 2 N (1 ) of the B-field vector as seen by the moving neutron particles when oriented parallel or antiparallel to sweeps through its changing orientations. In contrast to the dynamical phase, this geometric phase is axis z, respectively. With N= N+ + N − >> 1, the ϕ = − independent of the magnetic properties of the neutron. statistical error of (N+ N − ) / N is then γ = 1 Berry’s formula 2 was checked experimentally ϕ ≈ 1/2 1/2 ϕ ≈ calculated as 1/ N , or N 1 , which with a beam of polarized cold neutrons by Bitter and connects the number of neutrons under study to the Dubbers (1987), and with stored polarized UCN by uncertainty of their spin phase. Richardson et al . (1988). When the EDM-induced phase is measured with To quantify this geometric effect we approximate the E-field successively set parallel and antiparallel to the UCN trajectory in the trap by a horizontal circle of the B-field, the difference of both measurements gives radius r, so the neutrons move with angular velocity ℏ = ω= υ the overall phase shift of 4dn E z T / , with |E | Ez . r / r at right angles to the local field B oriented The measured EDM and its statistical error then are vertically as shown in Fig. 5. If B has a small radial − ℏ ℏ component B⊥ of constant amplitude everywhere on ± =(N+ N − ) ± dn d n , (3.15) this circle, the solid angle of the magnetic-field sweep 2NE T 2 N E T z z ≈ π 2 2 = ω π is B⊥ / B . After nr T / 2 turns, the extra where the + and − subscripts refer to measurements phase of the UCN accumulates to with + E z and – E z, and N is the total number of UCN γ= ω π × 1 n(r T /2) 2 . If solely the transversal stored during a measurement campaign. magnetic field B⊥ entered our argument, this The figure of merit for a given value of dn then geometric phase would cancel in an EDM search, with is ∝ N1/2 E T . The free-precession period T is z signals taken for ±E , because it is independent of size τ ≈ naturally limited by the neutron lifetime n 880 s . and direction of the electric field E. However,

The electric field strength Ez is limited to several tens besides B⊥ , there is the velocity-induced magnetic of kV/cm by the requirement that leakage currents and 2 field B =υ × E / c , which in our simple example is their magnetic fields must be strictly avoided, υ because, like the EDM interaction, they change sign collinear with B⊥ , so the geometric phase becomes γ=1 ≈ 1 π + 2 2 when Ez is inverted to –Ez,. Therefore for both T 2 2 (B⊥ B υ ) / B . Now the cross term and Ez one can expect improvements of at most a 2B⊥ ⋅ B υ , which is linear in the electric field E, no factor of two or three. UCN density ρ and bottle longer cancels, and the extra geometric phase volume V then are the only parameters that can bring nγ'(= BB /2 B2 ) ω T cannot be distinguished from significant statistical improvement in future EDM ⊥ υ r experiments via the total number N= V ρ of bottled a true EDM signal. In a real experiment, one must average over all UCN. neutron trajectories in the neutron bottle, although one

finds from numerical calculations (Pendlebury et al ., d. *False effects from geometric phases : In addition to 2004) that the rate of accumulation of the geometric the various systematic errors since long known in phase is nearly independent of the specific trajectory neutron EDM searches, a rather dangerous systematic 19 of the trapped neutron. In the ILL experiment, the Data Tables by Kostelecký and Russell (2011), in dominant geometric phase was that of the particular, their Table VII for the neutron. In addition, 199 Hg comagnetometer. Future EDM searches aim at EDM instruments were used for searches for neutron- an EDM down to 10 –28 e cm, in which case the mirror neutron oscillations, as described in Sec. IV.B, geometric-phase induced error may well become the and searches for spin-dependent “fifth forces” as leading systematic error. reported in Sec. V.C.3. e. The crystal-EDM experiment relies on a completely different method. It does not use trapped UCN, but a 3. *Upcoming neutron EDM experiments beam of monochromatic and polarized cold neutrons of velocity υ = 800 m/s . The neutrons interact with During the past few years, several new the internal electric field E ≈108 V/cm of a non- experimental neutron EDM projects were started centrosymmetric crystal. This field is four orders of worldwide, each aiming at sensitivities for a neutron –27 magnitude stronger than the laboratory fields used in EDM of well below 10 e cm. (We do not quote the stored-UCN EDM experiments. On the other individual EDM sensitivities aimed at, still too hand, the neutron’s interaction time uncertain at the present stage.) To reach this aim, all T= l /υ ≈ 2 ⋅ 10−4 s in a quartz crystal of length projects rely on significantly improved UCN densities, estimated to lie between 10 3 and l =14 cm is six orders of magnitude smaller than for 10 4 UCN/cm 3, as compared to ILL’s 40 UCN/cm 3 trapped UCN. The cold-neutron count rate of 60/s is measured at the exit of the source, and ~1 UCN/cm 3 similar to the corresponding rate N/ T≈ 100/ s measured in the ILL trap. In the following, we shall reached with UCN. give a survey on these upcoming EDM projects, in The experiment uses an elegant and sophisticated the order of the foreseen date of commissioning. two-crystal backscattering arrangement, with a temperature-induced switch from positive to negative a. CryoEDM is a follow-up project by a Sussex-RAL- crystal electric fields. It will be difficult for this Kure-Oxford-ILL collaboration located at ILL. The experiment to compete once the UCN-EDM projects “superthermal” UCN source, which is integrated in listed below come into operation. Still, this the CryoEDM apparatus, relies on a method that had experiment is interesting because systematic effects been proposed by Golub and Pendlebury (1975, seem to be well under control, and are completely 1977), and that was shown to work as expected for different from the systematic effects in the UCN- CryoEDM by Baker et al. (2003). It uses a genuine EDM experiments. Using this method, Fedorov et al . cooling process, in which Liouville’s law of = ±stat ± syst (2010) reached dn (2.5 6.5 5.5 ) conserved phase-space density does not limit UCN ×10−24 e cm . Their EDM sensitivity could be densities, as it does in ILL’s turbine UCN source, improved by a factor of about 65 by using better which relies on purely conservative forces. This neutron equipment, and so possibly could catch up method uses the coherent excitation of superfluid with the previous ILL experiment by Baker et al . helium (He-II) at low temperature, induced by the (2006). inelastic scattering of cold neutrons, which in the process lose all their energy and become ultracold. To f. Other uses of nEDM instruments : There have been excite superfluid helium, the neutrons must have a other uses of the UCN-EDM instruments. Altarev kinetic energy of 1.0 meV, which corresponds to a = et al . (2009) used the previous ILL-EDM apparatus to neutron temperature Tn 12 K and to a neutron de compare the spin precession of the neutron with that Broglie wavelength λ = 0.89 nm . At a bath 199 of the comagnetometer atoms of Hg, and searched temperature of T = 0.5 K , the inverse process (energy for daily variations of their frequency ratio. Such gain of UCN) is strongly suppressed, because at this variations are allowed in recent Lorentz and CPT temperature there are not enough or other symmetry violating extensions of the SM, induced by excitations present in the fluid. the electric term of a cosmic spin anisotropy field. Its CryoEDM uses cold neutrons from one of ILL’s interaction with the neutron (or rather with its neutron guides, spin-polarized to above 90% at component vertical to the earth’s rotation axis) was 0.89 nm, with polarized capture flux of − < × 20 − − constrained by this measurement to 2 10 eV 1.5× 109 cm 2 s 1 . The UCN produced at a rate at 95% C.L., for an improved evaluation see Altarev − − of 1 cm3 s 1 in the ≈10 ℓ source volume of the He-II et al . (2010a), where the energy scale for Lorentz converter are then guided to a Ramsey precession violation is tested on the level of 10 10 GeV. With double cell of volume 2× 22 ℓ . The cells are placed in polarized atoms, Brown et al . (2010) reached a 2σ a common uniform static field of B = 5.0 T . All bound of <3 × 10−24 eV for the neutron, see also the parts of the apparatus are immersed in He-II, 20 including the EDM precession cell, the SQUID phase up to 2012, the experiment will use the previous , the superconducting magnetic shield, EDM apparatus of the Sussex-RAL-ILL collaboration, and the solid-state neutron detectors. This is possible with N =3 × 10 5 UCN in the measurement cell, 4 because He has zero neutron-absorption cross instead of N =10 4 at ILL, with a sensitivity goal section, it can support strong electric fields, and, as a − somewhat below 1027 e cm . In a second phase from further advantage, 4He has a huge thermal 2012 on, a new EDM apparatus, presently under conductivity. In the present setup, CryoEDM will search for a neutron EDM at a sensitivity similar as in development, will be used that has a double-cell with opposite E-fields and improved magnetometry for the previous ILL experiment. The apparatus completed commissioning, for a better systematics, with about 10 times better sensitivity aim. status report see Baker et al . (2010). With an improved setup, the experiment will move to a new dedicated very-cold neutron beam-line at ILL in about d. The TUM-nEDM project: Technical University of Munich, Germany, is developing an nEDM apparatus, 2014, where the proposers expect a considerably higher final sensitivity. to be connected to an in-pile UCN source under development at the new FRM-II high-flux reactor.

This source is based on a 1 cm thick solid-D b. A PNPI EDM experiment is installed at ILL’s 2 converter of 170 cm 3 volume (“Mini-D2”) in a turbine UCN source and uses the upgraded double cell tangential beam tube, for a design study see Trinks et of the previous PNPI experiment, filled with 5 UCN/cm 3. At present, this experiment aims at a al . (2000), and Frei et al . (2009). The source is intended to become operational by 2014. PSI and sensitivity similar to that of the 2006 ILL experiment. It is intended to use this setup later at the existing FRM II collaborate, and both profit from detailed UCN studies done at the TRIGA reactor of Mainz PNPI reactor, with a new 35 ℓ He-II UCN source University, Germany. In its pulsed mode, the power of with about 10 3 times higher UCN density, installed in this TRIGA reactor goes up to 250 MW during 30 ms the thermal column of the reactor, see Serebrov et al . every 5 minutes. This makes it well suited for trapped (2009a, b). UCN experiments: To compare, ILL runs

(continuously) at 58 MW. A number of 10 5 measured c. The PSI nEDM project will use the 1.3 MW proton UCNs per reactor pulse are reported, see Frei et al . beam (2.2 mA, 590 MeV) from the PSI quasi- (2007), and an UCN density of 4 UCN/cm 3 was continuous cyclotron in a parasitic mode, with 1% achieved at 8 m distance from the source, comparable duty cycle. Every 800 s the experiment receives the to that achieved at present at ILL (T. Lauer, private full proton beam for 8 s, just the time needed to fill communication 2009). the UCN bottle, which is followed by a measurement period of about one neutron lifetime. The protons hit a e. The SNS nEDM project at ORNL follows a solid lead/zirconium target, which acts as a small, radically novel concept developed by Golub (1983), dedicated pulsed spallation neutron source with and Golub and Lamoreaux (1994). The experiment 10 neutrons liberated per proton collision. This source will use a superthermal UCN source, though will start feeds an UCN source, which consists of a 30 ℓ solid with a cold neutron flux lower than that of CryoEDM. ortho-deuterium D converter at 5 K, located on top of 2 The EDM precession cell is inside the UCN source, the spallation source. Both spallation and UCN which avoids the notorious problems of UCN sources are immersed in a common –10 extraction and transport. A ~10 fraction of polarized of 3.6 m 3 heavy water D O at room temperature. The 2 3He atoms (nuclear spin j = 1 ) is added to the UCN source is linked to a 2 m 3 storage reservoir, 2 4 = coated with diamond like carbon, from which the superfluid He (j 0) bath, which simultaneously UCN are drawn to fill the EDM cell. Several users can serves several purposes. be served in parallel from this UCN storage vessel. (1) The 3He nuclei act as a comagnetometer, like the The use of small pulsed-neutron sources with low 199 Hg nuclei in the ILL experiment, but with a repetition rates for UCN storage experiments had been magnetic moment similar to that of the neutron proposed by Pokotilovski (1995), and the “parasitic” (+11%). The diffusive of 3He in the 4He use of spallation neutron sources for UCN production bath in part averages out its large geometric phase in a solid D 2 converter by Serebrov et al . (1997). effect described inSec. 2.d. The PSI UCN source, described in Anghel et al . (2) The 3He nuclei also act as a neutron spin analyzer: (2009), was tested successfully under full power end For neutron spin opposite to the 3He spin, the of 2010 (though with still imperfect para to ortho- neutron capture cross section (~Megabarn for hydrogen conversion), for a status report on the whole UCN) is almost 200 times larger than for parallel EDM project see Altarev et al . (2010b). In a first spins, so when neutrons and 3He precess at a

21 ± slightly different frequency, then the neutron EDM, the application of an electric field Ez should capture rate is modulated at the difference of these then make the signal precess again. frequencies. (3) This brings us to the third role of the 3He nuclei: They also act as a counting gas for neutron detection. Neutron capture in 3He creates a fast proton and a fast triton with combined energy of 764 keV. In He-II, these fast charged particles produce scintillation light in the far ultraviolet (~80 nm), which a wavelength shifter makes detectable in photomultiplier tubes. The stray field of the polarized 3He is detected in a SQUID , which allows to measure the 3He precession frequency. The difference of neutron and 3He precession frequencies is detected in the phototubes via the modulated UV photon signal. This difference signal is sensitive only to a neutron EDM, because a 3He EDM signal is suppressed by Schiff’s rule mentioned in Sec. B.4. The neutron-3He difference signal is only one tenth of the 3He frequency, and hence is 10 times less sensitive to magnetic inhomogeneities than are the separately measured precession frequencies. With another elegant trick, one can make even this difference frequency disappear, in which case the pure EDM effect with no magnetic disturbance will be left: (4) Both neutrons and 3He nuclei can be forced to

precess at the same frequency about B by “dressing” them appropriately with radio FIG. 6. The “dressed neutron” effect can be used in frequency (RF) quanta.. EDM searches to suppress unwanted magnetic effects. Indeed, one can at will suppress the Larmor Spin precession about a static magnetic field B0 can be precession of a polarized ensemble by irradiating it suppressed when the neutrons are irradiated with a with a strong RF field of arbitrary frequency ω ' . For strong radio frequency field B1 of arbitrary frequency. neutrons, this was shown by Muskat et al . (1987), see With RF amplitude increasing from the top to the Fig. 6, who used a second-quantization treatment of bottom of the figure, spin-precession of the dressed the RF field. This method was developed in atomic neutrons slows down, then comes to a full stop, and spectroscopy by Cohen-Tannoudji and Haroche reverses its sign. From Muskat et al . (1987). (1969), but it is more transparent when applied to “point-particles” like neutrons, with no optical excitations involved. Of course, one can derive The SNS EDM project profits from other UCN dressed-particle effects (i.e., dressed- activities in the US, done at the solid D 2 source at diagrams, dressed g-factors, dressed-level crossings, LANL (Saunders et al ., 2004), at the solid D 2 source or multiple quantum transitions) from the Larmor at the PULSTAR reactor of the North Carolina State precession equation (3.10) (see Dubbers, 1976), but a University (Korobnika et al ., 2007), and at the Low “dressed neutron” description gives better physical Energy Neutron Source (LENS) at the Indiana insight to these phenomena. University Cyclotron Facility (Leuschner et al ., 2005). In any case, in the SNS experiment, the free 3He The project is in its construction phase, see Alarcon et nuclei precess faster than the free neutrons, but al . (2007) for an updated version of the EDM dressing both with RF quanta shifts the 3He precession proposal, and Ito (2007) for a status report. In 2009, frequency more strongly than the neutron precession the SNS accelerator surpassed a proton beam power of frequency. Therefore at a critical RF field amplitude 1 MW, and at the moment is heading for 1.4 MW. The = × ω γ current best estimate for data taking is 2017. B1 1.19 '/ n (with neutron gyromagnetic ratio γ ) both precession frequencies coincide, and n f. The TRIUMF EDM proposal foresees to transfer the the difference signal comes to a stop. For a nonzero UCN source developed at the Research Center for

22 Nuclear Physics (RCNP), Osaka, Japan (Masuda et quark, electron, quark-quark and electron-quark al ., 2002), to TRIUMF, see Martin et al . (2008). At amplitudes contributing. For the diamagnetic mercury RCNP the UCN source is installed at a continuous atom 199 Hg, Huber et al. (2007) find as the leading neutron spallation source driven by a 1 A , 390 MeV ≈×−3ɶ − ɶ + − 2 terms dHg 710( ddu d )10 d e , combining the proton beam, and the He-II converter at present contributions from Schiff momentum delivers a measured density of 15 UCN/cm 3. The (ρˆ ɶ ɶ ) ɶ cold-neutron moderator tank at RCNP is made of Sπ NN( d u , d d ) and the electron EDM, where du, d solid D2O at 20 K. When installed at the TRIUMF are color EDMs. There is a large expected overall proton beam of 40 A and 500 MeV, and equipped uncertainty due to significant cancellations. 199 with a 20 K D 2 cold-neutron moderator and a The experimental limit for Hg is < × −29 better UCN extraction system, this source is expected dHg 3.1 10 e cm (95% C.L.), from Griffith et al . to deliver up to a thousand times the UCN density (2009). These authors point out, citing Dmitriev and measured at RCNP. First UCN experiments at Sen’ov (2003), that under certain theoretical TRIUMF are foreseen for 2013. assumptions this translates into a proton EDM limit of

< × −25 dp 7.9 10 e cm (95% C.L., atoms), (3.17) 4. Limits on atomic and particle EDMs and a neutron EDM limit of Particle EDMs are available also for other neutral < × −26 dn 5.8 10 e cm (95% C.L., atoms). (3.18) systems, namely, de of electrons bound in paramagnetic atoms, and dN of heavy nuclei in To make the picture complete we also quote EDM diamagnetic atoms. These EDMs probe CP violation limits of short-lived particles . For the muon (we recall in different sectors, de in the leptonic sector, dn in the that all nonreferenced particle data are from PDG hadronic sector, and dN, in addition to the previous, is 2010), sensitive to nuclear effects. For a well-written account < × −19 of atomic EDMs see Sandars (2001). dµ 1.8 10 e cm , (95% C.L., muon) (3.19) The interaction of atomic EDMs with external measured with in a storage ring, and for the electric fields should vanish in the nonrelativistic limit neutral Λ0 for neutral atoms with point-like nuclei (according to < × −16 0 the Schiff screening theorem, see Liu et al ., 2007, and dΛ 1.5 10 e cm (95% C.L., Λ ). (3.20) references therein). However, the heavier atoms do have relativistic electrons and extended nuclei. These results are by-products of magnetic moment In heavy paramagnetic atoms, relativistic measurements on highly relativistic particles via spin- corrections can lead to measurable EDMs that are rotation in flight. When the particles fly through a growing with atomic number as Z3, and that are transverse magnetic field, in their rest frame, they also mainly sensitive to the electron EDM, plus to some see a strong electric field that would act on a hypothetic EDM. small scalar CP -odd electron- amplitude CS. For the paramagnetic thallium atom 205 Tl this In all, there is a worldwide competition of − ambitious and challenging neutron and dependence is d=−585 d +⋅ 51016 eC cm × , Tl e S EDM projects, and at present it is an open question, which is precise to about 10–20%, where the which of the different routes chosen in the various amplitude CS is composed of different quark-electron projects will first lead to success. In the next section, ≈ + + terms Cqe as CCS4 de 2 C se 0.01 C be , from Huber we discuss the conclusions that we can draw from the et al. (2007). The 205 Tl experiment gives a limit on the constraints obtained for the EDM of neutrons and electron EDM of other particles. ≈ < × −27 Note added in proof : Very recent experimental de dTl /585 1.6 10 e cm (90% C.L., atoms), (3.16) results on an anomalous like-sign di-muon charge asymmetry, Abazov et al. (2010a, b) indicate a CP from Regan et al. (2002), who neglect the contribution violation in the B0q- B 0 q mixing ( q = d, s) that is 40 from C S. Another promising candidate is the YbF molecule, which so far reached times larger than estimated in the SM from the CKM −26 matrix, see Lenz and Nierste (2007), plus numerous d<−( 0.2 ± 3.2) × 10 e cm , from Hudson et al . e recent papers on the subject. If confirmed, this has (2002). serious consequences for models beyond the SM. In heavy diamagnetic atoms, EDMs become measurable due to the finite size of the nucleus, with

23 C. EDMs in particle physics and cosmology proportional to θ and to a proton matrix element of a ≈ θ quark operator. Evaluation of gπ NN 0.038 | | tells us In the long history of searches for particle EDMs, − that from current algebra d~ 5× 1016 θ e cm , and so many theories on the origin of CP violation have been n −10 excluded by the increasingly tighter experimental θ must be really tiny ≤10 in view of the constraints, see for instance the history plot, Fig. 1 of experimental neutron EDM bound < × −26 Pendlebury and Hinds (2000). For reviews on theories dn 2.9 10 e cm . This is the so-called “strong CP on EDMs in general, see Pospelov and Ritz (2005), on problem” mentioned before, see also Baluni (1979). the neutron EDM, Khriplovich and Lamoreaux The θ term also receives a contribution from (1997), and on the electron EDM, Bernreuther and imaginary quark masses that can be calculated from Suzuki (1991). the chiral anomaly. Thus, the determinant of the 6× 6 quark mass matrix M can be taken to be real with a

“physical” θ in front of the strong-interaction field 1. EDMs in the Standard Model ɶ θ= θ + tensors GG in Eq. (3.6), argdet Mu M d . a. Quark models : A neutron EDM would be a clear One can promote θ to become a space-time sign of CP violation. As discussed before, CP dependent field (with a 1/ fa factor for canonical violation in flavor changing K and B decays can be normalization). When we enlarge the SM such that δ nicely explained by the phase in the CKM matrix there is a Peccei-Quinn symmetry U(1) PQ (as of the SM. In our present understanding, the neutron mentioned before) acting on colored fields, then the is a composite object made of three “valence” spontaneous breaking of this symmetry will have quarks (udd ) with three different colors, of , this θ field, the axion, as a . Further and of quark-antiquark pairs (“sea quarks”), if one explicit breaking by the chiral anomaly fixes the θ to looks closer on varying scales. This compositeness zero at a classical minimum and produces a small can be taken into account using quark models in axion mass. Further CP violating terms in the various ways for the constituent (dressed valence) effective action can lead to small “induced” quarks forming a color singlet with spin ½. In the θ simplest case, by using the old SU (6) contributions to (Pospelov and Ritz, 2005). with nonrelativistic wave function of the neutron, one obtains b. Effective QCD Lagrangian : When discussing the neutron EDM today, the cleanest procedure is to start =4 − 1 dn3 d d 3 d u (constituent quark model) (3.21) from an effective QCD Lagrangian at the 1 GeV scale that contains CP violating effects beyond the CKM for the neutron EDM dn, given the EDMs of d and u matrix in the tree action of the SM in an integrated-out quarks, see also Eq. (3.36) below for the neutron form. This can be obtained in the SM by considering magnetic dipole moment and its derivation. More loop effects (see later), but – very important – also in refined models for hadronic compositeness obtain extensions of the SM with further massive fields that roughly the same results by using a perturbative chiral are still not observed. The additional CP violating part quark model that includes a of Goldstone of the effective QCD Lagrangian for quarks, gluons, π bosons ( , K), see, for example, Borasoy (2000), Dib and photon-gluon couplings (corresponding to quark (2006), and Ottnad et al . (2010). EDMs d and quark color EDMs dɶ , respectively) is However, a CP violating θ -vacuum of QCD as i i described in Sec. A.4.a would most strongly influence added to the ordinary QCD-Lagrangian, Eq. (3.6), a neutron EDM. This was first worked out in the 2 g µν CPV= θ s aɶ , a framework of current algebra by Crewther et al . L Gµν G 32 π 2 (1979) who obtained µν µν −1 ψ σ + ɶ a a σγψ 2 i∑ ii( dFµν dGt i µν ) 5 i . (3.24) e Λ i= uds, , ,... d= gπ g π ln n4π 2m NNNN m n π The second term actually corresponds to dimension 6 (current algebra) (3.22) terms induced by a Higgs coupling to chiral quarks, although it looks like dimension 5. Indeed there are as the leading log contribution with a Λ (∼ mn ) cut π further dimension 6 operators not given above, a off, and with a CP violating NN coupling 3-gluon coupling (Weinberg operator) and 4-quark θ couplings, which also play a role in the extensive mu m d 3 2 2 gπ = 〈 pqqp|τ | 〉 (1− mmπ / η ) (3.23) NN f m+ m literature (Pospelov and Ritz, 2005) – and of course π u d one can imagine numerous higher-dimensional terms

24 suppressed by some power of a scale factor in the J m2 M 2 d= emm2α G 2 ln 2 b ln W denominator. d dcsF 108 π 5m 2 m 2 The effective Lagrangian only covers small- c b distance effects. Still, in a calculation of the neutron (via CKM matrix), (3.27) EDM large-scale strong interaction/ confinement effects have to be included. This is a difficult problem where J is the Jarlskog measure of CKM-CP violation not solved in full generality yet. Constituent quark in the SM given in Eq. (3.8) (for the weak and strong models based on chiral symmetry and its breaking are coupling strengths the Fermi coupling constant ∝ 2 α= 2 π a practical way to produce results. GF g w , Eq. (6.4), and sg s / 4 are used here). Together with other contributions, this leads to the d c. QCD sum rules (Novikov, Shifman, Vainshtein, ≈ −34 quark EDM estimate in the SM dd 10 e cm . This and Zakharov, or “NSVZ”) have a higher ambition is much too small to produce (via relation Eq. (3.25)) and have been used with high virtuosity (Pospelov and a neutron EDM that could be measured even in the far Ritz, 2005). Still they contain quite a few “technical” future. Larger results can be obtained if not just one assumptions. Particularly for strong-interaction quark in the neutron is responsible for the CP problems, a lattice simulation should be more reliable violation. Figure 7 shows typical SM perturbation in the end (Shintani et al ., 2005, 2008, Alles et al ., theory graphs with two or three quarks involved. 2007). Unfortunately, it will be rather agnostic “Strong penguin” diagrams together with ordinary compared to analytical calculations, but lattice data weak interaction (Fig. 7) turned out to be most can control these analytical calculations. QCD sum important (Gavela et al ., 1982, Khriplovich and rules lead to a neutron EDM in terms of quark Zhitnitsky, 1982, Eeg and Picek, 1984), with typical ɶ 2 2 EDMs dq and color EDMs dq at the 1 GeV scale π GIM cancellation ~(as / )log ( mmO t / c )~ (1) , (Pospelov and Ritz, 2005) π −22− 2 2 rather than the usual (mt m c ) / m W | | ~O (10−2 ...10 − 3 ) . The weak interactions are short d( d , d ɶ )= (1 ± 0,5) n q q (225 MeV) 3 range and can be handled with perturbation theory, × +ɶ + ɶ but the quarks in the infrared, inside the diagrams and [1.4(dd –0.25 d u ) 1.1( ed du 0.5 d )] (3.25) 1 at the outer lines, get bound into hadronic spin- /2 in the case of a Peccei-Quinn (PQ) symmetry states with both positive and negative parity ∗ suppressing θ . (namely, N ,Σ, Λ ), and and baryon Without PQ symmetry there is in addition the usual intermediate states play a role. The outcome of this θ term of size discussion is an estimate (McKellar et al ., 1987) − 〈 〉 d≈1032 e cm (via CKM matrix), (3.28) θ= ± θ |qq | × −16 n dn () (1 0.5) 3 2.510 e cm (225 MeV) still much too small to be measured. (3.26) Returning to CP violation by a possible θ -term, and a further term due to an induced θ term we have the opposite situation: Even a tiny θ ~ 10 −10 proportional to the quark color-EDMs dɶ . In these gives a neutron EDM close to the measured bounds. equations, 〈qq 〉 is the quark condensate, and As we already mentioned, this EDM can be calculated in current algebra, in chiral perturbation theory, in ɶ θ the di, d i , and are defined in the effective QCD sum rules, and in constituent quark models. The Lagrangian Eq. (3.24). Sea quark contributions turn values obtained are all in the same range, for a out to be suppressed. given θ ,

θ × −17 d. CP violation via CKM matrix : In the SM, a dn ~ f 10 e cm (strong CP violation), (3.29) nonvanishing quark EDM due to CP violation via the ≤ ≤ CKM matrix can be calculated if one goes to three- where 4f 40 , and each of the methods has big loop perturbation theory. A leading log term turns out error bars (Narison, 2008). It is also interesting to to be most important, and the dominant EDM of the d compare the hadronization procedures in a calculation quark results as (Czarnecki and Krause, 1997) of the nucleon magnetic moments, whose experimental value is well known, see Sec. D.3 below.

25

FIG. 7. Typical graphs in the Standard Model contributing to a neutron EDM. The vertical lines indicate hadronic intermediate states.

2. EDMs in supersymmetry CPV 2 consideration then results in dq~ ef | | m q / M for CPV There are good reasons to believe that the SM is scale M of the process, where f describes the CP just an effective theory up to the scale of weak violating process, eventually also comprising interaction of ∼100 GeV, and that in the TeV range 1/ (16π 2 ) loop factors. there are new particles and new couplings “beyond With d≤4 × 10−26 e cm respecting the current the SM” that will become visible in the forthcoming q accelerator experiments in the TeV range. Let us just neutron EDM bound (using Eq. (3.25), for example), mention some of the reasons. and with mq ~ 10 MeV one arrives at a bound

1/2 ● the mysterious hierarchy between the weak scale CPV 1/2 mq  M≥| f | ×  × 70 TeV . (3.30) and the Planck scale (see Sec. V.A.1); 10 MeV  ● the attractive picture of a grand unification of forces; Thus, even without measuring new particles at the ● the very small neutrino masses presumably required energy scale the neutron EDM limit (or requiring a further large scale (for the seesaw value) might tell us something about this scale M in mechanism); concrete “beyond” models – of course the detailed ● the dark matter component of our universe; and, mechanism of CP violation still . last not least, ● the up to now experimentally unexplored Higgs b. Minimal supersymmetry MSSM, one loop : The most sector. prominent model beyond the SM these days is the minimal supersymmetric SM (MSSM, introduced a. Dimensional analysis : When constructing models above in Sec. A.4.d). SUSY offers a solution or at beyond the SM one has to take care that they reduce least an improvement for the problems mentioned at to the SM in the well explored “low energy” region. the beginning of this section. In the MSSM with More technically speaking, the new structure should universal SUSY breaking parameters there are two CP be “integrated out” and should produce corrections to violating phases that can not be defined away: The θ µ the couplings of the SM, and should have further phase µ of the -parameter of the supersymmetric higher dimensional terms in the interaction Higgs mass (appearing as µH H in the Lagrangian, as in Eq. (3.24). These corrections then u d superpotential, see Eq. (3.9)) redefining the m phase have to carry, explicitly or implicitly, some power of a 2 to zero and the phase θ of the A parameter of the new mass scale factor M in the denominator. For A quark EDMs we also envisage new CP violating SUSY-breaking coupling of three scalar fields. The θ phases that could produce large values. These EDMs phase µ induces CP violation to the weak gaugino- also require a chiral “turner” changing quarks from Higgsino mass matrix and is most important for left-handed to right-handed, which must be electroweak baryogenesis in the MSSM. The proportional to some quark mass mq. Dimensional

26 θ θ θ phase A induces CP violation to the stop R-stop L mass parameters µ and A are heavily restricted. This matrix. becomes even more true if one also considers the restrictions by atomic data on the EDMs for 205 TI 199 θ θ and Hg. This forces µ and A to be very close to zero, see Fig. 8, much too small for electroweak baryogenesis in the MSSM.

c. MSSM, two loops : However, in the discussion above, the assumption of universal SUSY breaking and of a small SUSY breaking scale, and the restriction to one loop contributions in Eq. (3.31) was very (too) restrictive. If the squark masses are in the multi-TeV range, one-loop terms are suppressed, and two-loop contributions to the EDMs are dominant

(Pilaftsis 2002, Chang et al . 2002, Li et al., 2008, FIG. 8. (Color online) In supersymmetric (SUSY) 2010, Ellis et al ., 2008). In particular, restricting to extensions of the Standard Model, the EDMs of B-ino mass-induced CP violation, the influence particles are naturally large. In the simplest version of on EDM’s is much suppressed (Li et al ., 2009). a “minimal SUSY model” (MSSM), particle EDMs In Fig. 9, one can see that MSSM baryogenesis is depend only on two phase angles θ and θ . For a seriously constrained by the present neutron (also A µ electron) EDM data and will be conclusively tested in small SUSY breaking scale, experimental atomic and the near future by the LHC as well as the next neutron EDM limits strongly constrain these phases to generation of EDM measurements (Cirigliano et al ., θ= θ = near the origin A µ 0 where the three bands of 2010b, see also Ramsey-Musolf, 2009). In the figure, β= υ υ the figure cross – much too small for electroweak mA is the mass of the CP odd scalar, tanu / d is baryogenesis. From Pospelov and Ritz (2005). the ratio of the two Higgs expectation values, and reasonable values for the W-ino and the B-ino mass parameters m1 and m2, and the SUSY-Higgs mass In this constrained MSSM (CMSSM) one can parameter µ are assumed as indicated in the figure. calculate a one-loop EDM of the d quark (Ibrahim and Nath, 1998) d. SUSY with additional superfield : Data on EDMs m and electroweak baryogenesis can be much easier = −2 2 θ β − θ d dd eg 3 (sinµ tan sinA ) 2 2 accommodated in supersymmetric models with an 27 16 π M SUSY additional gauge singlet superfield S coupling to + 2 2 O( g2 , g 1 ) (SUSY). (3.31) the Hu and Hd doublets as SH uHd in the superpotential. The field value 〈S 〉 then plays the role Here g= g is the QCD coupling, g1 and g2 are the 3 s of the µ -parameter in the MSSM, introduced in Eq. electroweak couplings, md the quark mass at SUSY breaking mass scale M , and tanβ= υ / υ is the (3.9). In the “next to minimal” SSM (NMSSM), there SUSY u d is an S3 coupling in the superpotential (Huber and ratio of up/down Higgs expectation values mentioned CPV Schmidt, 2001). In the “nearly minimal” SSM before. This leads to an f in (3.30) of (nMSSM, Menon et al ., 2004, Huber et al ., 2006), the θ β− θ + CPV sinµ tan sin A −1 most important further term is ts S c.c. appearing in fMSSM ~ ( m d /10 GeV) , 2000 the soft SUSY breaking potential Vsoft , where ts is a (3.32) complex parameter introducing CP violation in the S- system. The superfield S finds its temperature leading roughly to dependent minimum 〈S 〉 together with the two Higgs ≥θ β − θ 1/2 × MSUSY (sinµ tan sinA ) 1.5 TeV . (3.33) fields. There are now more parameters than in the MSSM, and one has to inspect a great variety of Indeed, Eq. (3.25) was derived at a scale of 1 GeV. configurations. As indicated in Fig. 10 (Huber et al ., Therefore Eq. (3.31) must be continued by the 2006), it is easy to obtain a predicted baryon renormalization group. asymmetry higher than the observed baryon A careful analysis shows (Falk and Olive, 1998, asymmetry, while simultaneously fulfilling EDM Pospelov and Ritz, 2005) that for a small SUSY bounds (here the most restrictive is the electron breaking scale MSUSY (500 GeV, for example) the EDM).

27

FIG. 9. (Color online) Also in a careful reanalysis taking more freedom (see text) the EDMs strongly constrain β baryogenesis. The tan - mA plane, with colored exclusion regions due to the Higgs-mass, electroweak = × −26 baryogenesis (EWB), and branching ratios for b-decays. The blue line at dn 2.9 10 e cm indicates the current experimental limit for the n-EDM: the viable parameter space lies below this curve. From Cirigliano et al . 2010b.

e. *Left-right symmetric SUSY : In supersymmetric models with two Higgs doublets, the Peccei-Quinn mechanism for suppressing a QCD θ ≠ 0 vacuum can be realized. Another possibility to suppress θ ≠ 0 is offered by models that are parity P and CP symmetric in the ultraviolet (i.e., at very high energies), which are the left-right symmetric models with an enlarged × weak gauge sector SU(2)L SU (2) R . The left-right symmetry in the Yukawa sector enforces Hermitian Yukawa matrices and therefore θ ~ argdet M = 0 (Mohapatra and Senjanovic, 1978, Ecker and Grimus, 1985, Frere et al ., 1992). As the last possible axion window begins to close, see Sec. V.C.3, alternatives to the Peccei-Quinn mechanism are most welcome. Of course, in order to arrive at the SM at “low FIG. 10. Severe EDM constraints exist also when an energies”, the SU (2) left-right symmetry has to be additional scalar superfield is added to the MSSM, broken at some scale. This enforces spontaneous with more free parameters. The model is investigated breaking of the CP symmetry. θ then should stay by counting the (randomly chosen) points in parameter space that fulfil various constraints. The small, but a large CKM phase is required by the number # of results is shown for the analysis of the present data. Supersymmetry helps to achieve such a low scale (Aulakh et al ., 1997, Dvali et al ., 1998): baryon asymmetry of the universe for large θ = 0 is protected by a SUSY nonrenormalization M =1 TeV , in dependence of the baryon-entropy 2 theorem, whereas strong interactions can increase the η s = × 10 abundance parameter 10 (nB / s ) 10 . CKM phase already below the L–R breaking scale, Approximately 50% of the parameter sets predict a which is assumed to be much above the SUSY value of the baryon asymmetry higher than the breaking scale (Hiller and Schmaltz, 2001, Pospelov η s = and Ritz, 2005). observed value 10 0.87 . The lower line corresponds to the small number of parameter sets (4.8%) that also In a discussion of CP violating effects in SUSY fulfill current bounds on the electron EDM with 1 theories with the focus on EDMs and baryogenesis, of TeV . From Huber et al . (2006). course one also has to keep under control the

28 consequences for flavor changing processes, the CP sector of the SM and of the problems with violating K and B decays, which at the present electroweak baryogenesis in the SM, such a simple accuracy can already be well explained by the unitary modification of the SM is phenomenologically CKM matrix of the SM. This is a very broad subject, attractive (Bochkarev et al ., 1990). The general form and we refer to a few reviews (Jarlskog and Shablin, of the 2-Higgs doublet model and possible CP 2002, Masiero and Vives, 2003, Chung et al ., 2005, violations are well-known (Gunion and Haber, 2005). and Ramsey-Musolf and Su, 2008) It is possible to obtain viable baryogenesis with a lightest Higgs mass as large as about 200 GeV and extra Higgs states of mass above 200 GeV. The 3. Other approaches neutron EDM is predicted to be larger than about 10 -2 e cm close to the present bound (Fromme et al ., a. Two Higgs doublets : Without supersymmetry, there 2006). is no particular reason for two Higgs doublets. However, in view of our ignorance about the Higgs

FIG. 11. (Color online) When the Standard Model is augmented not by SUSY, but by a higher dimensional term in the effective Higgs sector, strong EDM constraints exist as well. Shown are the Higgs masses mh and scales Λ of new physics excluded (shaded areas) by the present EDM limits for neutron dn, thallium dTl , and η = × −10 mercury dHg . (a) In the minimal scenario, the observed baryon asymmetry 1 6 10 is excluded by the η present limit on the neutron EDM dn (and so is one tenth and ten times 1 ). (b) Therefore additional CP -odd sources must be introduced to lower the exclusion limits. From Huber et al. (2007).

b. Higher-dimensional operators : Trying to stay as Φ6/ Λ 2 term, and λΦ4 plays the role of the close to the SM as possible but still to obtain a strong perturbative −Φ3 term discussed in Sec. A.6 above, first order phase transition it has been proposed see Fig 4. If the two cut offs Λ and Λ are identified (Bödeker et al ., 2005, Grojean et al ., 2005, Huber CP et al ., 2007, Grinstein and Trott, 2008) to augment (and taken to be quite low), and if only a single CP the SM with dimension 6 terms in the Higgs violating phase in the top-Higgs vertex is retained, Lagrangian. In particular, one can have a term then Huber et al. (2007) obtain successful † 3 2 baryogenesis. (H H ) / Λ , but also further CP violating terms of However, this causes, in particular, the neutron the type (usual Higgs coupling) ×(H† H ) / Λ 2 in the − CP EDM bound to exclude almost all of the mh Λ simplest case. In order to understand such parameter space, as shown in Fig. 11. On the other nonrenormalizable terms one can imagine that they are hand, one can easily fulfill the requirement that the full obtained by integrating out a further heavy Higgs set of dimension-6 Higgs-quark couplings with SM doublet (as discussed before). The role of the Φ4 term flavor structure as indicated above does not violate the in the Higgs potential Eq. (3.2) is now played by the EDM bounds − a nice example how the EDMs 29 constrain model building. Higher-dimensional scales or close to the inflationary scale are more operators in an effective theory can also be considered probable. in SUSY theories (B(beyond)MSSM, Blum and Nir, 2008, Blum et al ., 2010). D. The other electromagnetic properties of the neutron 4. Conclusions on EDM constraints A multipole (2 L-pole) operator M(L) of a quantum In particle physics, a neutron EDM is a clear signal systems with quantum number j is of CP and T violation in a flavor conserving system. In determined by its reduced matrix element 〈j|| M(L ) || j 〉 , the SM, CP violation induced via the electroweak with L= j + j , so it obeys the triangle rule L≤ 2 j . For CKM matrix is discussed thoroughly in theoretical the neutron with j = 1 , there are no multipole papers, but it turns out to be extremely small, nothing 2 to be tested by experiments in foreseeable future. On moment observables beyond the dipole with L =1.We the contrary, strong CP violation via a QCD θ -term is keep the following discussion on the neutron's −10 multipole moments rather short, as there have been severely restricted to θ ≤10 by present EDM few new experimental results during the past decade. measurements. Here, theoretical uncertainties are related to hadronization of matter. In theories beyond the SM, desirable for various reasons listed at the 1. Electric charge beginning of the preceding section, there are further sources of CP violation, and the measured bound on The electric charge of the neutron is known to be the neutron EDM constitutes a serious limitation on zero to a high precision, but the Standard Model such violations, in particular, if it is combined with containing an abelian U(1) gauge symmetry does not other EDM measurements. This is an extremely Y require this, even after taking into account quantum interesting connection between neutron physics and anomaly cancellations (Foot et al ., 1993). Majorana “beyond” models. It would be exciting to finally masses of the neutrinos and, of course, a nonabelian measure a concrete value and thus to be able to further gauge group of grand unification support charge restrict the parameters of such models in particle neutrality of atoms and neutrons, but there remain theory. problems (Witten, 1979). Concerning cosmology, we found that already Overall, only few hints exist for physics beyond the present EDM results strongly constrain possible routes Standard Model, and the neutrality of neutrons for baryogenesis as well as models beyond the SM, (Baumann et al. , 1988) and of atoms (Dylla and King, and we presented prominent examples: 1973), both of order 10 –21 e, is such a hint, since in the ● The experimental neutron and atomic EDM limits SM this would require an incredible fine-tuning. Some rule out electroweak baryogenesis in the MSSM with a models beyond the SM that violate B− L symmetry low SUSY breaking scale and in a most simple setting, could accommodate a nonzero neutron charge Fig. 8. =ε − ≠ qn ( B L ) 0 , too, with the interesting signature ● The present nEDM limit also strongly constrains that the charge of the hydrogen atom (which has more refined MSSM results on baryogenesis, Fig.. 9. B= L ) would remain zero (Foot et al, 1993). ● For the nearly minimal SUSY model nMSSM, The elegant experiments quoted above were done which has an additional scalar superfield and further decades ago, and, at the time, were rather one-man free parameters, the measured EDM limit reduces shows. This is a pity in view of the efforts invested in heavily the parameter space viable for baryogenesis, other searches beyond the SM. See Unnikrishnan and see Fig. 10. Gillies (2004) for a review on neutron and atom ● When the SM is extended by additional higher- charges and on astrophysical limits, and Fedorov et al . dimensional terms, a strong first-order phase transition (2008) and Plonka-Spehr (2010) for recent proposals can be achieved, but baryogenesis is excluded by the for tests of neutron and Arvanitaki et al . (2008) for neutron EDM in some versions, see Fig. 11. tests of atom neutrality. If in future EDM experiments with sensitivities down to 10−28 e cm a nonzero EDM is found, then this will be a strong hint that the creation of the baryon 2. Magnetic monopole moment asymmetry has occurred near the electroweak scale. If no EDM is seen, other ways of baryogenesis like The neutron was also tested for a magnetic leptogenesis or other models at intermediate energy monopole moment, that is, for a free magnetic charge gn residing on the neutron. Magnetic monopoles play an important role in grand unified theories. When 30 such a magnetic monopole gn flies through a magnetic Generally, the accuracy of theoretical predictions field B, it follows a flight parabola due to the for the magnetic moments of hadrons is very limited = acceleration a(gn / m n ) B . Using neutrons flying and cannot compete at all with the high precision from through a crystal with “zero effective mass” experimental result. Given some uncertainties/free m<< m , see Sec. V.D.4, in a superimposed strong parameters, these model calculations hardly allow one n eff n to draw definitive and precise conclusions on the magnetic field, Finkelstein et al. (1986) obtained for hadronization in the infrared for the neutron and the neutron magnetic monopole moment intermediate states. The test of nonstandard = ± × −20 contributions then is in much worse shape than for the gn(0.85 2.2) 10 g D , (3.34) neutron EDM, where CP violation is filtered out. = α where gD is the Dirac magnetic charge gD ce / = ε2 π 0c(2ℏ / e ) . This is the best limit for a free 4. Mean square charge radius particle, although a tighter limit g<10 −26 g is n D obtained from SQUID measurements on bulk matter The internal structure of the neutron is mostly studied (Vant-Hull, 1968). in high-energy experiments. In recent years, the field has seen much progress, see Grabmayr (2008) for a review. At low energy, the neutron mean-square 3. Magnetic dipole moment charge radius and with it the initial slope of the neutron’s electric form factor can be measured in The proton and the neutron magnetic moments are neutron-electron scattering on heavy atoms. High and known very precisely, the latter as a by-product of a low-energy data on this quantity are now consistent neutron EDM search (Greene et al ., 1979) with with each other, see also the discussion in Sec. V.C.2. µ= − 1.913 042 73(45) µ , and n N µ/ µ = − 0.684 979 35(16) , (3.35) n p 5. Electromagnetic polarizability where the numbers in parentheses are the standard error in the last two digits, with the nuclear magneton The neutron electric polarizibility was measured by − the energy dependent neutron transmission though lead µ =3.152 451 2326(45) × 108 eV/Tesla . In the N (Schmiedmayer et al ., 1991), with an error of same simplified quark model, the neutron magnetic moment size as for the result obtained from deuteron break-up is given by γ d→ np . The PDG 2010 average is µ=4 µ − 1 µ ± × −4 3 n3 d 3 u (constituent quark model), (3.36) (11.6 1.5) 10 fm . The neutron electric see also Eq. (3.21). In lowest order of perturbation polarizibility can be thought of as measuring the slope theory, fundamental particles like the quarks (or the of the confining strong potential acting between the electron), have the Dirac magnetic moment quarks. The neutron magnetic polarizibility was not measured with free neutrons, the value obtained from µ =e/(2 mcon ) ≈ 3 e /(2 m ) for a constituent quark qq q q N deuteron break-up is (3.7± 2.0) × 10−4 fm 3 . A more con ≈ 1 + 2 mass mq3 m N . The with charge 3 e has thorough discussion of these quantities is beyond the µ= µ − 1 scope of this article. 2 N , the with charge 3 e has In heavy nuclei, one also finds a P violating µ= − µ , and from this N amplitude due to what is called an anapole moment. µ=−4 µ − 1 µ =− µ We shall leave this topic aside, because in the SM the 3( ) 3 (2) 2 , (3.37) n N N N anapole moment of the elementary is gauge which is in reasonable agreement with the measured dependent and is not truly a physical observable. value in Eq. (3.35). As we already discussed in the context of the neutron EDM, the detailed description of a neutron in terms of quarks and gluons is much more involved. One can use chiral perturbation theory (Puglia and Ramsey-Musolf, 2000, Berger et al ., 2004, and references therein), QCD sum rules (Ioffe and Smilga, 1984, Balitzky and Young, 1983, Aliev et al ., 2002) or lattice gauge theory (Pascalutsa and Vanderhaeghen, 2005). There is also the interesting related question of form factors (Alexandrou et al ., 2006).

31 IV. BARYON NUMBER VIOLATION, MIRROR magnitude to arrive, for instance, from the Soudan-2 UNIVERSES, AND NEUTRON OSCILLATIONS τ > × 31 result on iron from 2002 nn (bound) 7.2 10 yr (90% C.L.) at In the preceding Sec. III we had seen that in τ > × 8− 1 the GUT era of the universe, see Table I, baryon nn 1.3 10 s number B is violated. This chapter is on baryon (90% C.L., from bound nuclei, ∆B = 2 ). (4.3) number violating processes of the neutron. We have seen that the proton can be unstable in grand unified In a conference contribution, Ganezer et al . (2008), the 32 + + 0 values τ (bound)> 1.77 × 10 yr and theories, for example, p→ e + π , a process that nn τ > × 8− 1 has been searched for intensively, with a limit on the nn (free) 2.36 10 s at 90% C.L. are given for the τ > × 33 Super-Kamiokande-I measurements on 16 O. partial lifetime of pe π 8.2 10 years (90% C.L.), Mohapatra (2009) reviewed the theoretical rationale obtained in Super-Kamiokande by watching 50 000 m 3 of nn oscillations, and emphasized that B− L of water over many years. The neutron can undergo breaking by the seesaw mechanism in neutrino similar B− L conserving ∆B =∆ L = 1 processes, e.g., oscillations (∆L = 2) may be related to a seesaw n→ e + + π − , which has a limit of B− L breaking in nn oscillations (∆B = 2) , which τ > × 32 ne π 1.6 10 years could bring nn oscillations into experimental reach. In (90% C.L., bound nuclei, ∆B = 1). (4.1) particular, a mechanism was proposed, in which baryogenesis occurs via a baryon-number carrying scalar, whose asymmetric decays into 6 q and 6q A. Neutron-antineutron oscillations could provide a source for baryon asymmetry and baryogenesis, see Babu et al . (2009). This scenario In some models beyond the SM also ∆B = 2 easily satisfies the three Sakharov conditions, and has processes are allowed, in which case a single baryon the remarkable feature that it does not need the with B =1 changes into an antibaryon B = − 1 . sphaleron mechanism, as do many of the models indeed do “oscillate” into antimesons, as seen in discussed before, because the decays are allowed well 0↔ 0 0 ↔ the CP violating K K and B B 0 processes, below the sphaleron decoupling temperature of about but mesons and antimesons are quark-antiquark pairs 100 GeV. both with B = 0 . What oscillates is the meson’s Since the review of Dubbers (1991a), no new data quantum number S=+1 ↔ S =− 1 , with have been taken on free neutron oscillations, and ∆S = 2 . following the criteria given at the end of the Protons p+ cannot oscillate into antiprotons p− introductory Sec. I.B, we would not have taken up this because this violates charge conservation, but nothing subject again, had there not been new results on in principle forbids neutron-antineutron oscillations another type of neutron oscillations, not from neutrons n↔ n . A search for such oscillations was done years to , but from neutrons to so-called mirror ago at ILL, see Baldo-Ceolin et al . (1994). A cold neutrons, treated in the following. neutron beam of intensity 10 11 s−1 freely propagated in vacuum over a length of 80 m with a mean flight time B. Neutron-mirrorneutron oscillations of t ≈ 0.1 s . During this time, a fraction P≈ ( t /τ ) 2 nn nn of antineutrons would develop and annihilate in a The existence of a mirror world that would carbon foil spanned across the beam at the end of the compensate for the P violating mirror asymmetry of flight path. This annihilation would liberate the weak interaction is an old idea that resurfaced in 2 ≈ 2mn c 2 GeV , mostly in form of that would recent years in the context of searches for dark-matter have been detected in a large-volume tracking detector. candidates, see Berezhiani and Bento (2006), and The limit of Mohapatra et al . (2005). Mirror particles could be present in the universe and would not interact with τ > × 8 nn 0.86 10 s ordinary particles except for their gravitational (90% C.L., free neutron, ∆B = 2 ) (4.2) interaction, and except for some mixing between the was obtained for the oscillation period after one year two worlds via appropriate gauge particles. of running time. For a report on new plans for free nn Due to such mixing, neutrons n could possibly ′ searches, see Snow (2009). oscillate into mirror neutrons n , and thereby cease to One can also derive a limit on free-neutron- interact with ordinary matter, that is they simply would antineutron oscillations from the stability of nuclei, but disappear from our world. Pokotilovski (2006a) to this end one has to extrapolate over 30 orders of discussed various schemes for experiments on neutron-

32 mirror neutron oscillations (both in “appearance” and deviations at short distances. In addition, these “disappearance” mode, in the language of neutrino experiments have triggered a number of other neutron physicists). studies on this subject. It seems hard to overestimate One would think that the disappearance of our the importance of discovering these new forces, which cherished neutrons into nowhere would not have gone would “provide us with a rare window into Planckean unnoticed, and that experimental limits on nn ′ physics and the scale of supersymmetry breaking” oscillation times should be far longer than the (Dimopoulos and Giudice, 1996). We end this section measured neutron lifetime. However, this is not so, with a review of some earlier experiments on the because all previous neutron lifetime experiments were neutron’s mass and gravitational interaction. done in the presence of the earth’s magnetic field, invisible for the mirror neutron. If we assume that the mirror magnetic field, to which the mirror neutrons A. Large extra dimensions would react is negligible on earth, the requirement of energy conservation would lead to the suppression 1. The hierarchy problem of nn ′ oscillations. Therefore suppression of the magnetic field at the site of the experiment is required Our universe is known to be flat to high precision, for a nn ′ search. with three spatial and one temporal dimension. Is there For the old neutron-antineutron experiment, a more to it? The compatibility of general relativity and 100 m³ large magnetic shield had been constructed quantum mechanics/quantum field theory (QFT) today around the free-flight volume, with residual fields is still not well established. The most prominent B <10 nT (Bitter et al ., 1991), small enough that the version, superstring theory, requires 10 space-time neutron’s magnetic interaction µB< ℏ / t was below dimensions for consistency. This revived the old ideas of Kaluza and Klein who introduced a fifth dimension the Heisenberg uncertainty limit. An equivalent in order to unify gravity and electromagnetism. If such requirement is that the neutron spin precession along extra dimensions of number n are compactified (that is, the flight path has to be negligible, and indeed the “curled up” to a radius R), the 4-dimensional precession angle had been measured to description contains a whole “tower” of Kaluza-Klein ϕ=( µ B /ℏ ) t < 0.03 , Schmidt et al . (1992). However, particles, for each particle with mass2∝ n/ R 2 . This is in the neutron lifetime experiments, there was no such ′ too heavy to be seen in present experiments if 1/ R is of magnetic shield, and therefore nn oscillations would the order of the Planck mass ( ℏ =c = 1 ), but have been highly suppressed in these experiments. measurable if in the TeV range. Ban et al. (2007) and Serebrov et al. (2008a, Forgetting about for a while, 2009c) used their respective EDM experiments at ILL, theoretical ideas about the role of “extra” dimensions which are well shielded magnetically, to obtain limits in gravity have been put forward in the last ten years. on neutron-mirror neutron oscillations and reached The extra n dimensions are not visible in everyday life τ > because they are compactified to a radius so small that nn ′ 103 s (95% C.L.), and wavelengths of objects in these extra dimensions τ > 448 s (90% C.L.), (4.4) nn ′ become very short, and the next “Kaluza-Klein level” respectively. This was an elegant low-cost becomes so high that it can no longer be excited in experimental response to an interesting physics today’s low-energy world. question. Introduction of extra dimensions may help with To conclude, neutron oscillation experiments test another problem, namely, the vastly different energy basic symmetries of the universe, although the scales existing in the universe, see Table I. The scale underlying theories are more speculative than for the of the gravitational interaction is given by the Planck ≈ 19 2 other neutron tests beyond the Standard Model. mass MPl 10 GeV / c and defines the strength of the gravitational coupling constant G=1/ M 2 in Pl Newton’s law. V. EXTRA DIMENSIONS, NEW FORCES, AND The scale of the electroweak interaction is given by THE FREE FALL OF THE NEUTRON υ = the Higgs expectation value 246 GeV . The ratio of This chapter deals with hidden spatial dimensions gravitational to weak coupling strengths then is υ 2 2≈ × − 34 of the universe, which are required in superstring /M Pl 410 , and this is called the hierarchy theory for reasons of self-consistency, and which can problem of the SM: If, in a quantum system, one scale lead to deviations from Newton’s gravitational law. is very small compared to another scale, then Recent experiments on neutron-state quantization in unavoidable radiative corrections should make both the earth’s gravitational field are sensitive to such scales of comparable size, unless a plausible reason is

33 known for its smallness, such as an additional ∗ ∗= ∗2 + n Planck mass M Pl as G1/( M Pl ) . We only want symmetry. Otherwise, some incredible fine-tuning is to have one single scale, and therefore we set this true υ 2 2 required, here of / M Pl to nearly zero, which seems Planck mass equal to the electroweak scale very unlikely and calls for some deeper explanation. ∗ ≡υ = M Pl 246 GeV (in SI units, ∗ + +− G= ℏn1/ (υ n 2 c n 1 ) ). In particular, at the outer

2. Bringing down the Planck mass range R of the compactified dimensions, we write Eq. (5.2) as An elegant way to solve this hierarchy problem is to postulate that, in reality, there is no hierarchy because, basically, only one single energy scale exists, namely, the electroweak scale. Gravity then indeed is of the same strength as the other interactions, but only at very short distances (Arkani-Hamed et al ., 1998, 1999, called the ADD model). At larger distance, gravity looks weak only because it is “diluted” in the 3 + n spatial dimensions of the “bulk”, with n ≥1 extra dimensions, whereas the three other forces of the SM act only in the three spatial dimensions of the “brane” and are not diluted. Gravitational interaction has infinite range, because the exchanged is massless. Its field lines are not lost but merely diluted in space, which is the essence of Gauss’ law, which states that the field’s flux through any spherical shell around a body is constant, from which follows the 1/ r² force law in the usual way. Correspondingly, in 3 + n dimensions Gauss’ law, applied to a 2 + n -dimensional shell, leads to a 1/ r2+ n force law, which decays faster with distance r than FIG. 12. The ADD model has n extra dimensions, does Newton’s 1/ r² law. If the n extra dimensions are compactified to distances r< R , and only one single compactified to radius R, this fast fall-off occurs only energy scale. In this model, gravitation is diluted for small distances r< R , while for r>> R the in 3 + n dimensions, while the other interactions are gravitational force falls off with the usual 1/r². On the other hand, the electromagnetic force is assumed to act active only in three dimensions. At the electroweak λ ≈ −3 ∝ only in three dimensions, and for all distances r falls scale w 10 fm , the Coulomb potential Vel 1/ r ∝ n+1 off as 1/ r². We follow ADD and postulate that at very equals the gravitational potential Vgrav 1/ r . At short distances r<< R all forces are unified and have large distance, r>> R , both have the usual 1/ r the same strength. As gravitation initially falls off potential, but with coupling strengths differing by a much faster than electromagnetism, in our visible ∼ 34 world with r>> R , gravitation looks much weaker factor 10 . than electromagnetism. To quantify these statements we first write down Newton’s gravitational potential for two masses m, M ∗ = −G mM = in three spatial dimensions VRgrav ( ) , for rR , (5.3) Rn R = −mM >> under the assumption that all extra dimensions of VrGgrav ( ) , for rR . (5.1) r number n have the same range. >> At very short distances, we write the force law At large distances r R , we continue this with in 3 + n dimensions as the 1/ r potential, so G∗ mM * mM = − >> VrG( )= − , for rR << , (5.2) Vrgrav ( )n , for rR . (5.4) grav r n+1 R r as shown in Fig. 12. This must be identical to Newton’s gravitational ∗ The “true” gravitational constant G* in this equation potential, Eq. (5.1), i.e., G= G/ R n . Insertion of G for dimensional reasons must depend on the “true” and G* leads to the compactification radius of the extra dimensions 34 − π ∗ 1/ n 2/ n 2= 3 − 2kr c = π G  M  − MMPl (1 e ) / k , where the boundary y r c R = =Ż Pl ≈×1018 m 10 34/ n , (5.5)   w   of the warped dimension can be extended to infinity in G  υ  the special model, such that M can be lowered to =υ = × −18 the TeV/ c2 range. The effective gravitational potential where Ż w 1/ 0.80 10 m is the reduced Compton wavelength of the electroweak scale. for masses m1 and m2 on our 4-dimensional brane then The larger the number n of extra dimensions, the is − shorter is their effective range R in Eq. (5.5): mm∞ G mme mr m VrG( )=12 + 12 d m , (5.6) R ≈ 0.1 m for n = 2 ; r∫0 krk ≈ = R 0.2 m for n 3 ; where the second term comes from the exchange of R ≈ 0.3 nm for n = 4 ; massive Kaluza-Klein particles of the R ≈ 0.5 pm for n = 6 , i.e., for 4+n = 10 dimensions. graviton/gravitational field.

= ≈ 16 One extra dimension n 1, with R 10 m , is excluded because astronomical observation confirm B. Gravitational quantum levels ≥ Newton’s law at this scale. For n 2 extra dimensions and their shorter compactification radii, the exclusion 1. Neutron states in the gravitational potential plot Fig. 16 below gives present bounds, as will be discussed in Sec. C. Ultracold neutrons (UCN) moving at distance z closely The Coulomb potential Vel ( r ) , on the other hand, above the surface of a flat horizontal neutron mirror falls off as 1/ r for all distances r. In the ADD model of see a triangular potential, which is a superposition of large extra dimensions, the electroweak and the the linear gravitational potential V ( z ) = mgz for z > 0 gravitational interaction are unified at the electroweak ≈ and the repulsive potential V0 100 neV of the scale υ , at which point their coupling strengths neutron glass mirror for z ≤ 0 . n = ∗ coincide. This happens at r G/ G , or, with Neutrons can be bound in this triangular potential *= υ n+ 2 = υ 2 =υ = G 1/ and G 1/ , at r 1/ Ż w . Hence, and form standing waves. In a simplified treatment, > → ∞ th ψ both potentials, Coulomb’s Vel (r) and the gravitational for z 0 and V0 , the n eigenfunction n (z ) is Vgrav (r) meet at the electroweak length scale the (rescaled) Airy function Ai(z− z ) , shifted by a − n Ż ≈1018 m . At large distances r>> R , the ratio of th w distance zn such that its n zero-crossing coincides gravitational to electrostatic interaction is ψ ≡ with the surface of the mirror, and n (z ) 0 υ 2 2≈ × − 34 /M Pl 410 as given above. Figure 12 for z ≤ 0 , see Fig. 13 for n =1 and n = 2 . In other indicates the distance dependence of both potentials. words, all ψ (z ) look the same up to their nth node, for = n+1 n Near r R , of course, a smooth transition from 1/ r ψ 2 ψ 2 instance |2 (z )| in Fig. 13 looks like |1 (z )| up to to 1/ r is expected. st its 1 node. These solutions of the Schrödinger equation are 3. Competing models well known from the two-dimensional electron gas in a semi-conductor heterostructure. In the case of the The ADD model of large extra dimensions is only electrons, eigenenergies are in the eV range, while for one of many possible models on the unification of neutrons, the first few energy levels En lie at (1.44, = forces. It is an attractive model as it solves deep 2.53, 3.42, 4.21, …) peV, for n 1, 2, 3, 4, ... , with = −12 >> problems of contemporary physics and, as is always 1 peV 10 eV , so V0 E n is fulfilled, and setting welcome, predicts measurable signals both at the LHC → ∞ V0 is justified. The corresponding classical and in low-energy physics. Randall and Sundrum turning points z= E/ mg are at distances (1999), for instance, in their pioneering (RS) model n n have only one extra spatial dimension (but with (13.7, 24.1, 32.5, 40.1, …) m above the surface of possible relations to string theory). They obtain a the mirror. hierarchy of scales by introducing a 5-dimensional anti de Sitter space with one “warped” extra dimension y, such that the metric of our 4-dimensional subspace decreases along y as exp(−k|y|). The Planck mass MPl is then related to the 5-dimensional Planck mass M as

35

FIG. 13. Scheme of the experiment on UCN gravitational quantum levels. UCN plane waves enter from the left, and form standing waves in the triangular potential formed by the mirror potential V ≈ ∞ and the gravitational = = ψ potential V mgz . For the absorber height h 20 m shown, only neutrons in the first eigenstate 1 with energy = ψ E1 1.4 peV pass the device and are detected, while the higher eigenstates n>1 are removed from the beam. Adapted from Luschikov and Frank (1978).

FIG. 14. Ultracold neutron transmission of the device Fig. 13, measured in dependence of the height h of the absorber. Dotted line: classical expectation; solid line: quantum calculation. From Westphal et al . (2007b).

2. Ultracold neutron (UCN) transmission Figure 14 shows the number N(h) of transmitted experiments UCN as a function of the height h of the absorber. The dotted line is the classical expectation, N( h ) ∝ h 3/2 . Nesvizhevsky et al . (2002, 2005) observed the The solid line is a quantum mechanical calculation quantization of gravitational neutron states in an with three free parameters: the overall normalization, experiment at ILL. The UCN entered a device the neutron loss rate on the absorber, and the consisting of a horizontal neutron mirror and a population coefficient for the n =1 ground state. For a neutron absorber/scatterer placed at a variable height h different theoretical approach to the problem with above the mirror, see Fig. 13, and were detected after similar results, see Adhikari et al . (2007). Up to a leaving this slit system. height h ≈ 12 m , no neutrons can pass the system

36 because the n =1 ground state has sufficient overlap When the roles of the absorber and the mirror were with the absorber and is blocked (although visible exchanged, with the mirror above the absorber, the light with its much longer wavelength easily passes). transmission dropped to 3%, as predicted by theory, At first sight, one would expect a stepwise increase of see Westphal et al . (2007b). This proves that the neutron transmission, but this is almost completely effect is really due to gravitation, and not to simple washed out. For a recent review, see Baeßler (2009). “box-state” quantization between two mirrors.

FIG. 15. (Color online) Position sensitive detection of the transmitted UCN after their “fall” over a step of 30 m height at the exit of the slit system, measured at two different distances from the slit exit, x = 0 cm , and x = 6 cm . This differential phase sensitive measurement of neutron transmission in principle is much more sensitive to fifth forces than is the integral transmission curve, Fig. 14. Adapted from Jenke et al . (2009).

3. Observation of UCN gravitational wave the detailed phase structure of |ψ (z )| 2 at a fixed functions and resonance transitions height h is more sensitive to the details of the potential V(z) than is the integral transmission In the meantime, both, the wavefunctions of UCN measurement Nh()= |ψ ()|d zz2 of the first gravitational quantum states, and resonance ∫ h transitions between such states have been measured. experiment shown in Fig. 14. Figure 15 shows both The vertical distribution |ψ (z )| 2 of the UCN the calculated and measured UCN distributions for distances x = 0 cm and x = 6 cm behind the exit of leaving the gravity spectrometer was measured in a the slit system, see Abele et al . (2009). specially developed position sensitive detector of Jenke et al . (2011) report the first observation of spatial resolution 1.5 m . After leaving the slit vibration-induced resonance transitions between region, with fixed height of 40 m , the UCN were UCN gravitational levels. In principle such dropped onto a second mirror, installed a resonance transitions between quantized distance 30 m below the first mirror, to enhance gravitational levels can be detected in an apparatus the short wavelengths, as indicated in Fig. 15. While consisting of three sections (like a Rabi resonance- the UCN move along the second mirror their vertical in-flight apparatus). The first preparatory section distribution rapidly changes shape. It turns out that singles out and transmits only the gravitational ground state n =1. In the second, the resonance

37 transition section, the neutron sees a time dependent quantum states states. Voronin et al . (2011) study the vertical vibrational perturbation that induces a possibility to extend such gravitational experiments resonant π -flip to the n = 3 level. The third section to antineutrons. is the neutron state analyser, which only transmits Gudkov et al . (2011) investigate the possibility to those UCN that remained in the n =1 ground state. study short-range interactions via Fabry-Perot In the experiment quoted, the three sections are interferometry between narrowly spaced plates, realized in a single device consisting of a mirror on using parametric enhancement of the phase shifts of the bottom and a rough scatterer on the top. The slow neutrons. As the resonance signals would be scatterer suppresses the population of higher levels very narrow, one must take care that the signals are by scattering and absorption, and only allows the not washed out by limited resolution. n =1 ground state to pass. The state-selector on top introduces an asymmetry, because the ground state passes the system with higher probability than the C. Neutron constraints on new forces and on dark excited state. Resonance transitions are induced by matter candidates vibrating the entire system. The UCN detector is installed behind this device and registers the 1. Constraints from UCN gravitational levels occurrence of a resonance via a sizeable (20%) reduction in UCN transmission. Meanwhile, the The gravitational neutron experiments described statistics of the resonance signals has been above are very sensitive to the form of the potential considerably improved over that presented in the near the mirror. They therefore can test the existence Jenke et al . (2011) paper. of any short-range “fifth force” that couples to the Nesvizhevsky et al . (2010a and 2010b), in what neutron via some new charge q. Such new forces can they call a “neutron whispering gallery”, observed be mediated by the exchange of bosons of different the quantization of cold neutrons that are subject to types (scalar, pseudoscalar, vector, etc.), originating centrifugal forces near the surface of a curved from different sectors of models beyond the SM neutron guide, dependent on neutron wavelength, (supersymmetry, extra dimensions, etc.), or by the and found results closely following theoretical exchange of multiple bosons, see the review by expectations. This may open the road for new studies Adelberger et al . (2003). The new interaction with of short-range forces using cold neutrons of meV coupling constant g is not necessarily linked to kinetic energy instead of UCN, see also gravitation, and one can make the general ansatz of Nesvizhevsky and Petukhov (2008), and Watson an additional Yukawa potential as, (2003). ℏcg2 exp(− r λ ) The experiments on quantized gravitational levels V( r ) = , (5.7) π suffer from low neutron count rates. Like the EDM 4 r experiments, they would profit a lot if one of the new where λ = h/ m c is the Compton wavelength of the UCN sources under construction would deliver the B exchanged boson of mass m . high fluxes envisaged. Furthermore, the instruments B This Yukawa ansatz also covers the ADD model used for the searches of new interactions described ∝ n+1 above were not optimized for this purpose. Therefore with potential V() r 1/ r for n large extra with new instruments on new sources improvements dimensions discussed above, if it is seen as an for all these measurements by several orders of effective 4-dimensional model with towers of magnitude are expected. Kaluza-Klein particle exchanges (recurrencies), For the experiment on quantized gravitational which are summed over. As a demonstration, for one levels, a new instrument GRANIT is under extra dimension with n +1 = 2 this gives Σ ∞ − ≈ 2 << construction, for details see Nesvizhevsky et al . k=0 exp(kr / R ) / r Rr / for r R , via power (2007), Pignol (2009), and Kreuz et al . (2009). In the series expansion. For r>> R , this becomes 1/ r GRANIT project, UCN will be stored for long times multiplied by exp(−r / R ) for k =1, which gives Eq. up to the neutron lifetime. It is intended to induce (5.7) again. However, the precise shape of the radiofrequency transitions between the gravitational deviations from Newton’s law are likely to depend levels, with whatever fields couple to them. In its on the details of the Kaluza-Klein spectrum (Callin first version, GRANIT will run not with stored UCN, and Burgess, 2006). but in transmission mode, like the previous The range λ is the Compton wavelength of spectrometers at ILL. 2 Abele et al . (2010) discuss the use of a Ramsey Kaluza-Klein modes. Their strength is g~ nG , separate oscillatory field apparatus with vibration- with the number of extra dimensions n , and induced transitions between neutron-gravitational gravitational coupling constant G, see Kehagias and

38 Sfetsos (2000). However, g 2 may reach values six to upcoming dedicated instruments mentioned above eight orders of magnitude higher than this whenever will further push these limits. For scalar interactions with a Yukawa potential, Fig. 16 shows an exclusion there are additional gauge bosons freely propagating 2 in the bulk, for instance when these are linked to a plot in the plane of interaction strength g versus the 2 global B− L symmetry, see Arkani-Hamed et al . range λ of the interaction. The top line g =1 of (1999), and, for a more recent review, Antoniadis this figure corresponds to the strong interaction, and (2007). the bottom line g 2=10 − 37 to the gravitational The apparatus shown in Fig. 13 was built to interaction. The regions above the curves in Fig. 16 detect neutron gravitational quantization, but is not are excluded by the measurements. specially optimized for detecting new forces. Still, the measurement shown in Fig. 14 gave interesting limits on such extra forces. It is expected that the

FIG. 16. Exclusion plot on new short-range interactions. Shown are the strength g2 of the interaction versus its range λ . The upper scale gives the corresponding mass m= h/ λ c of the exchanged boson. Values above the curves shown are excluded by experiment. The constraints from neutron scattering in the subatomic range are combined from several different neutron measurements, see Nesvizhevsky et al . (2004, 2008). Most curves are adapted from Kamyshkov et al . (2008)

The straight line named “UCN quantized experiments with small objects for ranges between 2 gravitational levels” in Fig. 16 gives the constraints nm and 1 m , see Klimchitskaya et al. (2009) and derived from the UCN transmission curve Fig. 14 on references therein. The force on the cantilever is an attractive Yukawa interaction of the neutron with dominated by the Casimir force, which must be the underlying mirror. The curve falls off known very well in order to extract limits on new − as g 2∝ λ 4 , from Nesvizhevsky and Protasov forces. The cantilever limits follow a curve 2∝ λ λ 3 (2004), see also Abele et al . (2003), and Bertolami gexp( d 0 / ) / , so the bounds diverge and Nunes (2003). The latter authors also discuss exponentially as soon as λ falls below the distance d0 implications for the weak equivalence principle. between probe and surface. Calculations of the Figure 16 also shows bounds from torsion balance Casimir effect between microscopic and macroscopic experiments with large objects for ranges λ > 2 m , bodies are very demanding and not yet completely from Heckel et al . (2008), and from cantilever under control, see Canaguier-Ourand et al. (2010), 39 and references therein. The UCN limit is not yet gives the limit g 2λ 2≤ 0.016 fm 2 . In the past, it was competitive with the published Casimir limits, but the often asked who would need these neutron scattering neutron has the advantage that it has no background lengths to such high precision, now we know. from Casimir forces. ● Scattering lengths from Bragg scattering and those Below the nanometer scale, Fig. 16 shows other from neutron interferometry have different responses limits from high-energy experiments with antiprotonic to new forces. Comparison of both sets of data for 13 atoms at CERN ( Ap , from Pokotilovski, 2006b), and nuclei gives the asymptotic limit g 2λ 2≤ 0.0013 fm 2 . from small-angle scattering of ~100 GeV neutrons on ● Comparison of scattering lengths at neutron protons ( n-p, from Kamyshkov and Tithof, 2008). energies of 1 eV with those at thermal neutron

energies gives the asymptotic limit

2λ 2≤ 2 2. Constraints from neutron scattering g 0.0008 fm (all values with 95% C.L.). In Fig. 16, the “n-scattering” curve shows this last, Also conventional neutron scattering experiments most constraining result, the linear part of the curve turn out to be sensitive to extra short-range forces. In being the above asymptotic value. It is comforting that Fig. 16, the new limits termed "n-scattering" reach far several independent data and evaluation methods all into the subatomic range. They are from a recent give very similar constraints. For completeness, we densely written paper by Nesvizhevsky et al . (2008), add the result of another evaluation of energy which presents several novel and independent dependent total neutron scattering on 208 Pb by methods to extract the following limits on new forces Pokotilovski (2006b). A similar evaluation had from existing neutron scattering data (see also Pignol, already been made by Leeb and Schmiedmayer 2009): (1992). ● A global fit of a nuclear random-potential model to more than 200 measured neutron scattering lengths

FIG. 17. Exclusion plot on new short-range spin-dependent interactions. The neutron measurements constrain axion interactions with nucleons in the axion window, left open by the astrophysical constraints (dashed vertical lines). The methods have room for improvement by several orders of magnitude. Black lines: coupling to nucleon spins; gray lines: coupling to electron spins. Most curves are adapted from Baeßler et al . (2009).

40 In the same paper, as a by-product, Nesvizhevsky et al . reconfirmed the Garching-Argonne value of the neutron-electron scattering length =− ± × −3 bne ( 1.31 0.03) 10 fm , which significantly differed from the JINR Dubna value =− ± × −3 bne ( 1.59 0.04) 10 fm , and which determines 2 the neutron’s mean square charge radius 〈rn 〉 , see Isgur (1999), Kopecky et al . (1997), and references therein. The value of bne is also of high interest because it determines the initial slope of the neutron form factor GE(q²), and for a long time was the only thing known about this form factor. Because they needed a precision value of bne for their evaluation, the authors turned the problem around and extracted bne from recent precise measurements of GE(q²), finding =− ± × −3 bne ( 1.13 0.08) 10 fm , which again excludes the Dubna value. FIG. 18. Expected effect of a spin-dependent extra interaction on the UCN transmission curve from the 3. Constraints on spin-dependent new forces gravitational level experiment. One neutron spin- component would be attracted, the other repelled at Of high interest in modern particle theories are the short distances (dashed lines). Even unpolarized limits on spin-dependent short-range interactions neutrons are sensitivity to such spin-dependent mediated by light pseudoscalar bosons like the axion. interactions (solid line). From Baeßler (2009). As discussed in Sec. III.A.4.a, we need the axion to solve the strong CP problem, and, in addition, the axion is a favorite dark-matter candidate. However, the Recently, Zimmer (2008, 2010) proposed that UCN axion is experimentally excluded over a wide range of trapped in rather large neutron bottles could also be masses and coupling strengths g2. As a historical used to obtain tight constraints on short-range spin- = ⋅ remark, neutron reactions were also helpful in the hunt dependent interactions of type V ()r σnB eff () r , with for the axion. Döhner et al . (1988) did an early axion a suitable pseudo-magnetic field Beff . Every wall search, using the PERKEO neutron decay encounter of the UCN would slightly shift the spin spectrometer, and set limits in the MeV mass range phase of neutron polarization 〈σ 〉 , and would also (and, using the auxiliary calibration spectrometer n depolarize the UCN to some degree. PERKINO, the then virulent 17 keV neutrino signal Indeed, Serebrov (2009) used the existing data on could be explained as an ordinary backscattering the depolarization of stored UCN to derive the effect, Abele et al ., 1993). Today, are excluded 2− 21 2 –6 –5 constraint |g g |λ < 210 × cm for λ between 10 except for a narrow window between 10 eV and S P –2 10-2 eV, which corresponds to a range λ and 10 m, which covers the whole “axion window”, between 20 m and 2 cm (dashed vertical lines in see also Serebrov et al . (2010), and further 2− 22 2 –6 – 2 λ < × λ Fig. 17), with unknown interaction strength g . |gS g P | 410 cm for between 10 m and 10 Constraints on the spin-dependent axion-nucleon 5 m, as is also shown in Fig. 17. interaction strength within this “axion window” were Voronin et al . (2009) took their existing crystal- derived from neutron data, as shown in Fig. 17. The EDM neutron data, as discussed in Sec. III.B.2.e, and potential V(r) for the interaction of one fermion with looked for a possible additional phase-shift due to the spin of another fermion is attractive for one extra forces. This enabled them to set new constraints < −12 neutron spin direction, and repulsive for the other. Its |gs g p | 10 , valid down to the atomic range effect on the UCN transmission curve, Fig. 14, would − λ ≃ 1010 m= 1 Å , which then turns into be as shown in Fig. 18, from Baeßler et al . (2007), see λ 2< × − 29 2 λ ∼ −13 also Westphal et al . (2007a). This leads to the upper |gS g P | 510 cm , valid down to 10 m . exclusion curve in Fig. 17. These very short ranges had already been excluded by the Supernova SN 1987A neutrino events, see Hagmann et al . (2010), but such an independent check is very useful.

41 3 = ± Petukhov et al . (2010) used their He neutron mgrav/ m inertial 1.00011 1.00017 . (5.8) 3 polarizer cell to measure T1 of He as a function of an external magnetic field. By comparison of their For macroscopic pieces of matter, the weak –13 improved relaxation theory with these data (plus T2 equivalence principle is tested to a 10 level, see data for 3He from other sources) they obtained Adelberger et al. (2009). The neutron’s 10 –4 limit is |g g |λ 2< 310 × − 23 cm 2 , shown also in Fig. 17; see derived from a free matter wave, i.e., a quantum S P object. It is very likely, but not guaranteed that bulk also Petukhov et al . (2011) and Fu et al . (2011) for matter and matter waves obey the same law of recent reevaluations. References to the other curves in gravitation. Furthermore, the neutron is a pure t = 1 Fig. 17 numbered 1. and 2. are given in Baeßler et al . 2 (2009). isospin object, and gravitation possibly depends on Hence, in recent years, it was shown that the isospin. neutron is a unique tool to search for new forces at very short ranges. Such forces are predicted by powerful new theories that advance the unification of 2. Neutron mass, and test of E= mc 2 physics and that may shed light on unsolved problems like quantum gravity or the nature of dark matter in the To precisely determine the neutron’s inertial mass universe. the neutron capture reaction p+ n → d + γ with

Eγ = 2.22 MeV is used. The masses of the proton and

of the deuteron in atomic mass units u are known D. More lessons from the neutron mass −10 to 10 or better. To be competitive, Eγ needs to be 1. Gravitational vs. inertial mass measured with ~10 –7 accuracy. A precision

measurement of Eγ was done at the GAMS facility of The neutron has no charge and so its inertial mass the ILL by a NIST-ILL collaboration. At GAMS the cannot be measured the usual way in a e/m-sensitive hydrogen target (a Kapton foil) was installed in-pile mass spectrometer. One could measure the neutron and looked at by nearly perfect, flat diffraction crystals mass in nuclear recoil experiments, but not to a high installed at about 10 m distance outside the reactor precision. One could also measure the ratio of the µ vessel. From the result for Eγ Kessler et al. (1999) neutron magnetic moment to neutron mass n/ m n in a Stern-Gerlach experiment. Indeed, the gravitational obtained mass of the neutron was derived from the “sagging” of = mn 1.008 664 916 37(82) u , (5.9) an UCN beam in a magnetic storage ring (Paul, 1990) − = ± 2 with 8× 10 10 relative error. to mgrav (914 34) MeV/c . This agrees within the Neutron capture γ -ray energies were used to test 4% error margin with the precisely known inertial 2 mass of the neutron the Einstein energy-mass relation E= mc to an m = 939.565 346(23) MeV/c 2 , as is expected unprecedented precision. For a neutron capture inertial 28+ → 29 + γ from the weak equivalence principle, which requires reaction like Sin Si multiple one can = write mgrav m inertial . A better test of the weak equivalence principle was ∆Mc2 ≡( M +−− mmMc ) 2 = E . (5.10) 28d p 29 ∑ i γ i done by Schmiedmayer (1989), based on the following argument. The most precise measurements of neutron The masses M28 and M29 had been measured before scattering lengths in condensed matter are done with with two silicon atoms stored simultaneously in a gravitational spectrometers, see the compilation by single , one of the isotope 28 Si, the other Koester et al . (1991). In these experiments, one drops of 29 Si. All relevant deuteron and 29 Si γ -transition neutrons from a given height onto a horizontal surface energies Ei entering the sum on the right-hand side of and determines the critical height, above which the Eq. (5.10) (with appropriate signs) then were measured neutrons are no longer totally reflected from the at GAMS. With these data, one arrives at a test of the material. These measurements of neutron scattering most famous formula of physics to an accuracy lengths depend on the gravitational neutron mass, of 4× 10 −7 , more than 50 times better than previous whereas the conventional measurements via neutron tests, see Rainville et al. (2005), and Jentschel et al. scattering depend on the neutron’s inertial mass. A (2009). comparison of the scattering lengths measured with both methods gives the ratio

42 3. The fine structure constant validity of quantum electrodynamics, whereas the composed values from h/m measurements, although The precise value of the neutron mass was also less accurate, rely on different assumptions. used to determine the electromagnetic coupling constant, i.e., the fine-structure constant α=2 πε ≈ 4. More experiments on free falling neutrons e/40ℏ c 1/137 , to a high precision. At first sight, this is surprising, as the neutron has no electric charge. Sometimes the value of a fundamental constant Many other experiments on the neutron’s mass and can be calculated from some other constants known gravitational interaction were done over the years, with higher accuracy than its directly measured value. some of which are listed as follows. The fine-structure constant can be written in terms ● McReynolds (1951) at Brookhaven National = 1 α 2 of the Rydberg constant R∞ 2 mche / as Laboratory (USA) first observed the free fall of the α 2 = neutron. (2R∞ /)(/ c h m e ) , where R∞ is much better α ● Collela et al . (1975) at the Ford Nuclear Reactor in known than is . The quantity h/me can be determined Michigan (USA) first studied gravitation in the = λ υ from the de Broglie relation h/ m e e e by quantum regime by measuring the gravitationally υ measuring both the electron’s velocity e and de induced phase shift of a neutron matter wave with a λ neutron interferometer. Broglie wavelength e . For a like the electron, however, this cannot be done with sufficient ● The effective mass of a particle in a periodic precision, but for a neutral particle like the neutron this potential can have either sign, and vanishes at the inflection point of the dispersion curve ω(k ) . The is possible. We use hm/= (/ hmmm )( / )( m / m ) e nnppe same is true for a neutron moving through the periodic and write the fine-structure constant as potential inside a crystal. When the neutron’s effective 2R m m mass becomes near zero it becomes more sensitive to α2 = ∞ n p υ λ n n , (5.11) external perturbations by up to six orders of c m m p e magnitude, as was shown experimentally by Zeilinger where R is known to 7× 10 −12 , the mass ratios to et al . (1986). This effect had been applied in a search ∞ for a magnetic monopole moment of the neutron, as × −10 better than 5 10 , and the velocity of light c is as was discussed in Sec. III.D.2. defined in the SI system and has no error. ● In another elegant experiment, Frank et al . (2009) For the neutron, Krüger et al . (1999) measured measured the inelastic “diffraction in time” effect =υ λ × −8 h/ m n n n at ILL to 7 10 , and used the precisely induced by an optical phase grating moving known value mn/mp from the preceding section. The transversally through a beam of free-falling υ monochromatic UCNs (Frank et al ., 2003). In the neutron velocity n was measured via the beat frequency of the neutron polarization signal after experiment, parameters were chosen such that the ℏ double passage through a magnetic undulator (in the energy loss of the UCN, induced by the phase = πυ form of a current carrying meander coil) with 10 m grating rotating with frequency 2 / a , exactly flight path between the two passages. The neutron compensated the gain mngh in UCN kinetic energy υ wavelength λ was measured by diffraction on a after a fall through height h (grating’s velocity , n lattice constant a = 5 m ). Two identical neutron perfect silicon crystal in backscattering geometry, for which the Bragg condition is nearly independent of the Fabry-Perot monochromators at distance h above each angle of incidence. Today the result other monitored this compensation. The energy loss − was varied between ℏ =10 neV and 40 neV by α 1 =137.036 0114(50) , with better knowledge of the varying the grating’s velocity from υ =15 m/s to 60 silicon lattice constants (see Mohr et al. , 2008), m/s. The height of fall was adjusted accordingly in the translates to range h =10 cm to 40 cm. As a by-product, the α −1 =137.036 0077(28) (neutron). (5.12) experiment tested the weak equivalence principle to 2× 10 −3 . In the same way, one can derive α −1 from the velocities and matter-wavelengths of free atoms, with, Nowadays, atomic physics experiments have at the present time, three times lower error than for the become very competitive in testing the gravitational neutron. The current world average of α is dominated interaction, for references see Cronin et al . (2009). by the electron g − 2 measurement leading to Experiments with atoms have a number of advantages, among them high intensity, ease of manipulation, and −1 −10 α =137.035 999 070(98) with 7× 10 accuracy portability. The corresponding neutron experiments (Gabrielse et al ., 2007). This value is based on the have the advantage that neutrons, in this context, are

43 structureless and “inert”, can easily penetrate matter occurrence is only a question of observational (or are totally reflected from it), and have well evidence. calculable interactions. In the language of particle physics, neutron decays are “rare events” with a low-energy signature in a noisy environment. Therefore the experimental study VI. THE ELECTROWEAK STANDARD MODEL of neutron decay is challenging, and error margins are AND THE β -DECAY OF FREE NEUTRONS easily underestimated. The precision of neutron decay data has dramatically improved over past decades. We now turn to the epoch at a scale of ≤1 TeV , However, there are still inconsistencies between see Table I, which is very well described by the various neutron decay data that need to be resolved, present Standard Model of particle physics, based on but at a much higher level of precision than in the past. the U(1)× SU (2) × SU (3) gauge group. This scale There is a high demand for better neutron data for Y L C physics both within and beyond the SM, which is the is much better understood than the earlier epochs of reason why, in recent years, more and more neutron the universe discussed in the preceding chapters. To decay “hunters” have joined the party. describe the β -decay of the free neutron References to earlier reviews on experiments and n→ p+ + e − + v , (6.1) on the theory of neutron decay are given in Sections B e and C, respectively. with half-life of ~ 10 minutes, and β endpoint energy of 782.3 keV, we shall need only a small number of parameters in the SM. A. Neutron decay theory There are two main reasons for our interest in precise neutron decay data. First, most “semileptonic” Before we turn to the description of the weak weak interaction processes (i.e., involving both quarks interaction, we recall the corresponding expressions and leptons) occurring in nature must be calculated for the electromagnetic interaction, in the limit of low today from measured neutron decay parameters. energy. Examples range from proton or neutron weak cross sections needed in cosmology, astrophysics, and particle physics, over decay and muon capture, to 1. The example of electromagnetism such mundane problems as precise neutrino detector efficiencies. Particles interact with each other via their currents. Second, in neutron decay, many tests on new The interaction ⋅B between a magnetic moment and physics beyond the SM can be made, relevant again a magnetic field, for instance, is the interaction both in particle physics and in studies of the very early between the molecular and macroscopic universe. According to conventional wisdom, the SM electromagnetic currents in the sources of and B. In and its vector-axialvector ( V− A ) structure are Dirac’s relativistic theory, the electromagnetic four- extremely well tested. However, if one looks closer current is a vector current with components then one finds that there is still much room for elmag ′ jµ= e ψ γ µ ψ , with µ =1,...,4 . Here, γ µ is a four- couplings with other symmetries, like scalar S or × tensor T couplings, or for additional right-handed vector with the 4 4 Dirac matrices as elements, ψ ψ ′ V+ A couplings, with incomplete parity violation. and and are four-component free-particle wave Precision bounds on such amplitudes beyond the SM functions ψ =up( )exp( − ipx ⋅ ) for the initial and final all come from low-energy physics, and progress in this states of the relativistic charged fermion. field slowed down years ago when low energy physics The electromagnetic interaction between two became unfashionable. different point-like charged fermions (denoted by the Neutron tests beyond the SM are possible because, subscripts 1 and 2) proceeds via exchange of virtual as we shall see, there are many more observables photons, which are massless spin-1 (or “vector”) accessible in neutron decay (more than 20) than there bosons. The transition matrix element, in first-order are SM parameters (three), which makes the problem perturbation theory, then contains the scalar product of strongly over-determined. When we go beyond the SM ′γ ′γ µ the two vector currents eu1µ u 1 and eu2 u 2 , and a there are up to 10 allowed complex coupling constants. propagator –i/q² for the massless photon, These couplings are allowed in the sense that they obey basic symmetries like Lorentz invariance Melmag = − 2′γ− 2 ′ γ µ ieu(11µ uq )( u 22 u ) , (6.2) (symmetries that in the age of quantum gravity and dark energy are no longer sacrosanct), and that their

44 = − with four-momentum transfer q p2 p 1 . One calls mirror. 50 years after its discovery, the origin of this this a neutral-current interaction, because the photon symmetry violation is still an open question, as will be exchanged in the process has no electric charge. discussed in Sec. D.2. If an electron interacts with an extended object like For ψ and ψ ′ one can insert any of the six known a proton, then there is an additional induced-tensor quark fields q= (, udscbt ,,,,) or lepton fields interaction with the proton’s anomalous magnetic = ν µν τν l(, e e ,,µ ,, τ ) (or the antiparticle fields q, l ). moment κµ=(1 g − 1) µ = 1.793 µ , given in pN2 p N N If two lepton currents interact, then one calls this a µ = units of the nuclear magneton Neℏ / 2 m p with g- purely leptonic interaction; if a lepton and a quark = current interact, a semileptonic interaction; and if two factor g p 5.586 . This interaction involves currents quark currents interact, a purely hadronic interaction. anom = − κ µ σ ν of the form jµ ip N ( pqp µν ) , where p stands µ→ + ν + ν for the proton wavefunction, and b. For purely leptonic muon decay , e e µ , σ=1 γγ − γγ µν2 i( µ ν ν µ ) . for instance, the transition matrix element is (we omit the “weak” superscript)

M =(G / 2)[νγ (1 −µ− γ ) ][ e γµ (1 γν ) ] , 2. Weak interactions of leptons and quarks muon F µ µ 5 5 e (6.3) We first regard the weak interaction of structureless where e,ν , etc., are the single particle Dirac spinors quarks q and leptons l before we turn to the weak e interactions of composite neutrons and protons. for the electron, , etc., and summation µ = over repeated indices 1,...,4 is implied (the a. Charged-current weak interactions , on the other upright in this formula stands for the muon). hand, like neutron decay in Eq. (6.1), are mediated by The strength of the weak interaction is given by the the exchange of massive vector bosons W ± . Their 3 = Fermi constant GF /(ℏ c ) 1.166 378 8(7) mass is M ≈ 80 GeV , and they carry both electric −5 − 2 W ×10 GeV , newly determined by Webber et al . and weak charges. In most nuclear and particle (2011), which is related to the weak coupling constant β -decays, the exchanged momenta are low, gw, Eq. (3.3), as ()pc2<< ( Mc 2 ) 2 , and the corresponding propagator W G/2= 1 gmc2 /( 2 ) 2 . (6.4) ∝ 2 F8 w W becomes 1/ MW , see Eq. (1.2). As the range of the interaction is given by the Compton wavelength of the = c. Semileptonic neutron decay on the quark level reads exchanged boson Ż ℏ / MW c , this leads to a point- d→ u + e − + ν . The corresponding matrix element for like interaction between two weak currents. e The weak current in the “V− A ” electroweak SM the point-like quarks is = ψ′ γ ψ µ is composed of a vector part Vµ g w µ , the same M =γ − γ γ − γν quark(G F / 2)[ uµ (15 ) de ][ (1 5 )e ] . as for the electromagnetic interaction, and an (6.5) = ψ′ γ γ ψ additional axial-vector part Aµ g w µ 5 of equal Here two complications arise. The first is due to = − amplitude, but of opposite sign, so jµ V µ A µ “quark mixing”. In the early 1960’s, it was observed =ψγ′ − γψ gw µ (15 ) , with weak coupling constant gw. that, while purely leptonic muon decay proceeds with a 2 − γ γ≡ γγγγ strength given by GF , semileptonic neutron or nuclear The factor (15 ) (with 5i 1234 ) projects out × 2 the left-handed part of the spinor ψ , so the weak decays proceed only with 0.95 GF , while strangeness-changing decays of strange particles current jµ is completely left-handed. As vectors Vµ (like Σ, Λ , Ξ -baryons or K-mesons) proceed and axial-vectors Aµ have different signs under a × 2 with 0.05 GF . Cabibbo then postulated that the down parity transformation P, the parity-transformed current quark state d ' that participates in the weak interaction Pjg=−ψγ′ (1 + γψ ) =−+ ( VA ) is completely µw µ5 µµ is not the ordinary mass eigenstate d but has a small right-handed. admixture of state s, and the strange In nature so far, only left-handed weak currents quark has a small admixture of d, such that have been observed, and therefore parity symmetry is ′ =θ + θ ′ = −θ + θ d dcosC s sin C , s dsinC s cos C . The said to be maximally violated in the weak interaction. 2 Cabibbo angle θ has cosθ ≈ 0.95 and This means that if one looked into a mirror, one would C C 2 θ ≈ see processes that are not present in nature without the sinC 0.05 , so the probabilities of strangeness

45 conserving and strangeness changing transitions sum in the limit of T-invariance. We adopt a notation that is up to unity, and together have the same strength as consistent with Holstein, 1974. muon decay (“universality of the weak interaction”). Hence, the hadronic weak interaction acts on quark b. Forbidden induced terms : The SM excludes “second states that are obtained by a rotation in flavor space class” currents, which involve the terms (the properties down, strange, etc., being called the + − σν γ f3 (0) qµ /2 M and ig2(0)µν q 5 / 2 M (with flavors of the quarks). Later on, this concept was 2M= m + m ), because of their weird properties extended to three families of particles (even before the p n detection of the third family in experiment), in order to under combined charge conjugation and isospin take into account time-reversal T violating amplitudes, transformations, though, of course, these parameters via an additional complex phase factor exp(iδ ) in the should be tested if possible. Second class currents are CKM quark-mixing matrix forbidden in the SM, but can be induced by isospin- violating effects due to the differences in mass and ′ d VVVud us ub  d charge of the u and d quarks. In neutron decay, such ′ =  isospin breaking effects are expected within the SM s VVVcd cs sb s . (6.6)   − ′  only on the 4× 10 5 scale, see Kaiser (2001), and their b VVVtd ts tb  b detection at the present level of accuracy is unlikely The CKM matrix V depends on three angles and and would require exotic processes beyond the SM. one complex phase δ , which are free parameters of We can furthermore safely neglect the “induced − γ the SM, though there are also other ways of pseudoscalar” term g3(0) 5 q µ at the low energies of parameterizing the matrix (after the detection of neutron decay. In the nuclear current we therefore only neutrino flavor oscillations, a similar mixing matrix is retain f1, f2, and g1, which amounts to replacing postulated for weak interactions of the leptons). uγ(1− γ ) d in Eq. (6.5) by p( f (0)γ+ g (0) γ γ Rotations in complex are described by unitary µ 5 1µ 1 µ 5 − σ ν rotation matrices, hence the CKM matrix V acting in a if2 (0)µν q /2 M ) n , where p is the proton and n is Hilbert space should obey VV † =1 , where V † is the the neutron wave function. conjugate transpose of V. This means that quark mixing is a zero-sum game: every quark gives as much c. Conserved Vector Current : We can simplify this as it takes. More on this will be discussed in Sec. C.3. further. The electromagnetic vector current of hadrons (i.e., their isospin current) is known to be conserved (i.e., it is divergence-free), which means that the 3. Weak interactions of nucleons electric charge of the proton, for instance, is simply the sum of electric charges of its constituents, and is not Nucleon structure introduces a second renormalized by “dressed nucleon” strong-interaction complication: In semileptonic decays, here neutron effects. The SM can unify the weak and the decay, it is not free quark currents that interact with the electromagnetic interactions only if the weak hadronic leptonic current, but the currents of quarks bound vector current has the same property, i.e., if the weak within a neutron. As already mentioned, the neutron is charges remain unaltered by the strong interaction. a complicated object made up not only of three valence Therefore conservation of the electroweak vector quarks but also of so-called sea quarks (virtual qq current (“CVC”) requires that the vector coupling be pairs of all flavors) and gluons, all coupled to each unaffected by the intrinsic environment of the nucleon, = other by the exchange of strongly self-interacting i.e., f1 (0) 1 . gluons. This makes a neutron state difficult to calculate. d. *Weak magnetism : With the same reasoning we can attack f2(0). Under CVC, not only the charges but also a. Form factors : To account for the internal structure the higher multipoles of the electromagnetic and weak of the nucleons one introduces empirical form factors hadronic couplings should remain unaffected by fi(q²) and gi(q²), with i =1,2,3 , the requirement of dressed-nucleon effects. Therefore under CVC the Lorentz invariance limiting the number of form-factors hadronic vector current in an electromagnetic to six. Furthermore, transition energies in neutron transition and the hadronic vector currents in weak β − decay are so low compared to the nucleon mass that + or β decays should form the (+ 1, 0, − 1) - these form factors only need to be taken at zero components of an isovector triplet, linked to each other momentum transfer q2 → 0 . They therefore reduce to by the Wigner-Eckart theorem. Hence, there should be pure numbers fi(0) and gi(0), whose quotients are real a simple link between the ordinary magnetism of the nucleons and the “weak magnetism” term f2 in β -

46 decay. This hypothesis on the connection between f. Partially Conserved Axialvector Current : We saw weak and electromagnetic multipole interaction that the electroweak hadronic vector current is = parameters, made in the early 1960’s, was an important conserved (CVC), which led to f1 (0) 1 . For the precursor of electroweak unification. axial-vector coupling, things are different. The pion is The relation between the weak magnetism a pseudoscalar particle that can weakly decay into two β amplitude f2(0) in neutron-to-proton -decay and leptons that are vector particles, hence the hadronic ordinary magnetism of protons and neutrons is derived axial (= pseudovector) current cannot be conserved from the general relation between the spin-½ matrix under the weak interaction. Without a conservation law elements of a vector operator V= ( V+ , VV0 , − ) , for the axial current one would expect that strong- + 0 0 interaction radiative effects would make the namely, 〈+1|V | − 1 〉 = 〈+1||V + 1〉 − 〈− 1 || V − 1 〉 . 2 2 2 2 2 2 electroweak axial coupling of hadrons comparable in In our case of isospin-½ and the electroweak hadronic size to the strong interaction, i.e., very large as vector current Vµ , this reads compared to the weak coupling. However, , the theory of + = z− z 〈pVn||µ 〉 〈 pVp || µ 〉 〈 nVn || µ 〉 . (6.7) the strong interaction of the SM, is based on a symmetry between left-handed and right-handed The anomalous magnetic moments of the proton and hadronic currents called chiral symmetry. This the neutron are described by the currents symmetry would also require the conservation of the − κ σ ν − κ σ ν ip( e / 2 m p )( pµν qp ) , in( e / 2 m p )( nµν qn ) , = λ = − axial vector current, i.e. |g1 (0)| 1 , and 1. respectively, with nuclear magneton e/2 mp, and with Though, at low energies, chiral symmetry is = θ the weak charge gw e / sin W absorbed in the Fermi spontaneously broken, i.e., the symmetry of the coupling constant. The “weak magnetism” term dynamic system is broken, while the symmetry of the − σ ν underlying effective Lagrangian remains intact. if(2 (0)/2 Mp )(µν qn ) is given by the difference Therefore the axial current is expected to be at least between proton and neutron anomalous magnetic partially conserved (PCAC), specifically in the limit of moments, f (0)=κ − κ = 3.706 , with κ =1 g − 1 2 p n p2 p vanishing pion mass (connected to vanishing quark = + κ =1 = − 1.793 and n2 g n 1.913 . masses, which break the symmetry explicitly), with λ = |g1 (0)| near 1, that is, g1(0)/ f 1 (0) near –1. e. Neutron decay matrix element : Hence, in V− A Historically, the value of λ measured in neutron electroweak SM the β -decay matrix element is decay, now at λ ≈ − 1.27 , gave a first hint to chiral κ− κ symmetry and its spontaneous breaking. As, within G ν M =F γ() + λγ + p n σ PCAC, λ is linked to the axial coupling of the neutronVp ud [(1µ5 µν qn )] 2 2M pseudoscalar pion, one can make a connection ×γ − γ ν between λ and the measured strong and weak [e µ (15 )e ] . (6.8) interaction parameters of the pion, called the In the end, the complicated interior of the neutron is Goldberger-Treiman relation. In this way, the taken care of by one free parameter untractable parts in the calculation of λ are shifted ϕ into measured parameters, though this procedure has λ =g(0) = g (0) e i , (6.9) 1 1 an accuracy of only several percent. The deviation λ which becomes λ = − g (0) in the case of time of from –1, due to the strong interaction, can also be 1 derived from QCD calculations on the lattice, with the reversal invariance. Indeed, from searches for T result λ = −1.26 ± 0.11 , for references see Abele violation in neutron decay we know that ϕ≈ π , within (2008). error, see Sec. B.2.f below. λ = Conventionally one writes gA/ g V , with g. Neutron-nuclear weak interactions : The weak = ggA/ V g1 (0)/ f 1 (0) , though one should not confuse interaction between neutrons and nuclei is of high interest, exploring otherwise inaccessible features of these zero-momentum form factors gA and gV with the elementary coupling constants g ≈ 0 and g = − 1/ 2 the strong interaction, though this topic is beyond the V A scope of this review. Experiments on slow-neutron of the electroweak neutral leptonic current in the SM. transmission through bulk matter, for instance, produce As stated above, the Fermi constant G is known very F the most blatant manifestations of parity violation. In precisely from the measured muon lifetime. Therefore one version of these beautiful experiments, when we neglect T violation and take G from muon F unpolarized neutrons, upon transmission through an decay, the matrix element Eq. (6.8) has only two free “isotropic” crystal (nonmagnetic, cubic, etc.), become parameters: the first element V of the CKM matrix ud longitudinally polarized. Even nonscientists feel that and the ratio λ of axial-vector to vector amplitudes.

47 ν something is wrong. The solution is that all bulk matter a. Electron-antineutrino coefficient a : The e- e is left-handed with respect to its weak interaction with correlation parameter a describes the angular the neutron. See Snow et al . (2011) for an experiment correlation between electron and antineutrino three- on P violating spin rotation in 4He, and Gericke et al . momenta pe and pν . This correlation is parity P (2011) for an experiment on P violation in the conserving and, for a given λ , is determined by n+ p → d + γ reaction. momentum conservation of the three spin-½ particles There exist several reviews on the theoretical emitted in unpolarized neutron decay. It leads to an implications of these measurements, in particular, in angular distribution of ν emission with respect to the the context of effective field theory, see Holstein e (2009b), Ramsey-Musolf and Page (2006), and direction of electron emission references therein. Both experiment and theory on cp⋅ c p υ d2Γ∝+ (1ae v ) d Ω=+ (1 a e cos)θ d Ω , neutron-nuclear weak interactions are very W We c e challenging, in particular for the more interesting low- e ν mass nuclei. (6.12) with total electron energy W=( pc22 + mc 241/2 ) e e e = γ m c 2 , electron velocity υ /c= cp / W , neutron 4. The 20 and more observables accessible from e e e e = θ neutron Wν p ν c , and the angle of electron emission with respect to the direction of neutrino In free neutron β − -decay, there are astonishingly emission. In the V− A model, the coefficient a many observables accessible to experiment. Precise depends on λ as values for all these observables are needed in order to 2 1− λ obtain all the coupling constants g and g′ , with a = . (6.13) i i 1+ 3 λ 2 i= SVT, , , AP , (S for scalar, T for tensor, and P for pseudoscalar interaction, see Sec. D.1). In the SM with A simple scheme for deriving the dependence of = ′ this and other correlation coefficients on λ was given purely left-handed currents, one has gi g i for λ 2 i= V, A , and g= g ′ = 0 for i= S, T , and g and g′ in Dubbers (1991a). As a measures the deviation of i i P P from 1, it is highly sensitive to the violation of axial are negligible. vector current conservation (PCAC), with With the matrix element of Eq. (6.8) one arrives at ∂a/ a = − 2.8 at a ≈ − 0.10 (for λ ≈ − 1.27 , where ∂ a neutron decay rate λ λ stands for ∂/ ∂ λ ). 2 5 − c (m c ) τ1=e G 222 | V |(1 + 3 λ ) f , (6.10) The coefficient a is P conserving because it n2π 3 (ℏc ) 7 Fud involves the product of two polar vectors, and was already searched for before the discovery of parity with the phase space factor f =1.6887(2) (from violation. The neutrino momentum cannot be Towner and Hardy, 2010), which is the integral over measured in experiment, but must be reconstructed as β = − + the -spectrum (for the precise shape of the electron pν ( pe p p ) from the measured electron and and proton spectra see, for instance, Glück, 1993). proton momenta. Though, up to now, coefficient a has After corrections for radiative effects and weak always been determined from the shape of the proton magnetism, the lifetime becomes (Marciano and Sirlin, spectrum, see Sec. B.2.a, with the electron remaining 2006) undetected. The decay protons have an endpoint (4908.7± 1.9) s energy of 750 eV and for detection must be accelerated τ = , (6.11) to typically 30 keV. n |V |2 (1+ 3λ 2 ) ud where the error in the numerator reflects the b. β -decay asymmetry A : The β -asymmetry uncertainty of electroweak radiative corrections due to parameter A is the correlation coefficient between hadronic loop effects. neutron spin σn and electron momentum pe. This λ = The ratio gA/ g V , on the other hand, can be asymmetry is due to the parity violating helicity of the −υ obtained from the measurements of one of the many emitted electrons (of helicity e / c , with spins neutron-decay correlation coefficients discussed next. pointing preferentially against the direction of flight), In the SM, these coefficients are insensitive to in conjunction with angular momentum conservation. Coulomb corrections of order α . The correlation leads to an angular distribution of the e– emitted in the decay of spin-polarized neutrons

48 cp υ 4λ d2Γ ∝ (1 + A〈σ 〉 ⋅e ) d Ω=+ (1 AP e cos)θ d Ω Cx= =− xAB( + ) . (6.18) n en e C+ λ 2 C We c 1 3 (6.14) The kinematic factor xC depends on the total energy = where θ is the angle of electron emission with respect release W0, and for the neutron is xC 0.27484 , see = to the direction of neutron polarization Pn〈σ n 〉 . Glück (1996), and references therein. The sensitivity λ ∂ = ≈ The β -asymmetry is P violating because it to is λC/ C 0.52 at C 0.24 . contains the scalar product of a vector pe and an − axialvector σn . Within the V A model it depends on λ as λ( λ + 1) A = − 2 . (6.15) 1+ 3 λ 2 Like coefficient a, the β -asymmetry A directly measures the deviation of λ from –1 and is very sensitive to it, with ∂λ A/ A = − 3.2 at A ≈ − 0.12 .

ν c. Antineutrino asymmetry B : The e -asymmetry parameter B is the correlation coefficient between σ neutron spin n and antineutrino momentum pν . Nonzero B leads to a P violating angular distribution

ν FIG. 19. Two-fold correlations between momenta pe, of the e (of helicity one, with spins all pointing into pν and spins σ , σ in the β -decay of slow neutrons, the direction of flight) emitted from polarized neutrons n e with the correlation coefficients: β -asymmetry A, cp 2Γ ∝ + ⋅v Ω=+θ Ω neutrino-asymmetry B, electron-antineutrino d (1B〈σn 〉 ) d e (1 BP n cos) d e Wν correlation a, electron helicity G, and spin-spin and (6.16) spin-momentum coefficients N and H. with neutrino energy Wν , and Under V− A , all correlation coefficients depend λ( λ − 1) B = 2 . (6.17) only on the one parameter λ , and therefore any + λ 2 1 3 correlation coefficient can be expressed by any other correlation coefficient, and by various combinations of With ∂λ B/ B = 0.077 at B ≈1.0 , the parameter B such coefficients, for instance a= 1 − B + A , etc. is about 40 times less sensitive to variations of λ than Among such relations (which usually must be are the parameters A and a. This makes B valuable for corrected for weak magnetism), Eq. (6.18) has the searches of decay amplitudes beyond the SM, as we special feature of being model independent, holding as shall see in Sec. D. well under non-zero scalar, tensor, or right-handed Correlation B is related to correlation A by a rule amplitudes. given by Weinberg (1959): If in V− A leptonic decays one interchanges the roles of electrons and neutrinos, e. Twofold correlations involving electron spin : β - for instance by measuring the neutrino asymmetry instead of the β -asymmetry, then even powers of λ decay of slow neutrons involves four vector quantities accessible to experimental investigation: the in the numerator of the coefficient change sign, while momenta p and p of the electron and the proton, and the odd interference terms do not. e p the spins σ and σ of the neutron and the electron. In n e d. Proton asymmetry C : The proton asymmetry principle, proton spin σ p is also accessible, but in parameter C is the correlation coefficient between neutron decay, proton helicity 〈σ p 〉 is extremely small neutron spin σ and proton momentum p . It leads to n p and difficult to measure. a P violating angular distribution of the protons Angular distributions and correlations are (pseudo-) emitted from polarized neutrons β = 1 scalars, and for a -transition from an initial j 2 to W()θ= 1 + 2 CP cos θ , with n ′ = 1 a final j 2 , we can construct 17 different scalars or pseudoscalars from these four vectors. These correlation coefficients use up most of the letters of the 49 alphabet, see Jackson et al. (1957a, 1957b), and Ebel g. Four and fivefold correlations : Next, we look for and Feldman (1957). products involving four vectors. When we multiply The simplest such correlations are given by the two scalar products we obtain five more independent scalar products of any two of these four vectors, which scalars and their respective coupling constants, gives the six correlation coefficients shown in Fig. 19. K(σ ⋅p )( p ⋅ p ) , Q(σ ⋅p )(σ ⋅ p ) , They are all conserved under time-reversal T operation e e e ν e e n e ⋅ ⋅ ⋅ ⋅ ( p→ − p , and simultaneously σ→ − σ ). Scalar S(σe σ n )(p e p ν ) , T (σep e )(σ n pν ) , ⋅ ⋅ products involving two vectors (momenta) or two axial U (σepν )(σ np e ) (6.22) vectors (spins), i.e., a and N, conserve parity P = − = − = = − ( p→ − p , σ→ σ ), while the other four, A, B, G, In the SM, K A , Q A , S 0 , T B , = and H, are P violating. In the SM with its maximally and U 0 (the latter when we neglect a T violating parity violating V− A structure, one finds the electron Coulomb correction). helicity coefficient G = − 1, and (when neglecting Scalar products of two vector products add nothing weak magnetism) new, as they can be decomposed into products of scalar products. Finally, we can construct a T violating m m five-fold product W (σ ⋅p ) σ ⋅ (p × p ), , with H= − e a , and N= − e A (6.19) e e n e ν = − We We W D , corresponding to the D-coefficient with electron helicity analysis. This, however, will not play for the other bilinear correlations involving the a role soon. Hence, indeed, within the SM, the problem electron spin. Beyond the SM, these relations no is heavily over-determined, and we shall see in Sec. D longer hold, as we shall see in Sec. D. how to profit from this. f. Threefold correlations : From any three different vectors out of p , p , σ , and σ one can form four e p n e 5. Rare allowed neutron decays triple products, all T violating: Dσ ⋅(p × p ) , Lσ ⋅(p × p ) , There are two further allowed but rare neutron n e ν e e v decay channels, radiative β -decay under emission of Rσ⋅( σ × p ) , Vσ⋅( σ × p ) . (6.20) e n e n e v an inner photon, and bound β -decay Triple products involving two momenta are P of a neutron into a hydrogen atom. conserving, and those involving two spins are P violating. Triple correlations involving neutrino momentum are measured via the proton momentum as a. Radiative neutron decay : The rare decay mode σ ⋅(p × p ) =−⋅σ (p × p ) , etc. nep ne ν n→ p+ + e − +ν + γ (6.23) In the V− A SM, such T violating amplitudes e should be immeasurably small, with has a bremsstrahlung photon in the final state. This process is overwhelmingly due to bremsstrahlung of |λ |sin ϕ D = − 2 . (6.21) the emitted electron, which is a well-understood 1+ 3 λ 2 process (Glück, 2002), although in effective field Should one ever find a D-coefficient with nonzero ϕ , theory tiny deviations from the SM photon energy spectrum are possible (Bernard et al ., 2004). then one may try to corroborate it by measuring the To first order, the photon spectrum varies as 1/ ω electron energy dependence of the V-coefficient, which and ends at the β -endpoint energy under V− A is V= − ( mc2 / WD ) . e e ℏω =E = 782 keV . Therefore every “octave” in In the SM, L= (α m / pa ) and R= − (α mpA / ) max 0 e e e e the spectrum contains the same number of photons. α are non-zero only from order Coulomb corrections Angular momentum conservation transforms part of (plus imaginary parts beyond the SM that depend on the electron helicity into photon polarization. This scalar and tensor amplitudes only). These triple parity violating effect could be measurable in the products can be used for searches for new physics visible part of the spectrum, but is diluted by having beyond the SM. One must keep in mind that triple four particles in the final state. For references, see correlations can also be induced by final state effects, Cooper et al . (2010). which, however, are far below (two and more orders of magnitude) the present experimental limits for the correlation coefficients, and can well be calculated.

50 b. In bound β -decay of a neutron, the emitted electron For reviews focussing on neutron decay ends up in a of hydrogen, experiments, see Nico (2009), Paul (2009), Byrne (1995), Yerozolimsky (1994), Schreckenbach and → + ν n H e , (6.24) Mampe (1992), Dubbers (1991b), Byrne (1982), which must be an atomic S-state for lack of orbital Robson (1983) for an historical review, and Wietfeldt angular momentum. Due to the small hydrogen and Greene (2011) for a discussion of the status of the binding energy of 13.6 eV, as compared to the kinetic neutron lifetime. For drawings of instruments used in these experiments, see the reviews by Abele (2008), energy release E = 782 keV , this process has a tiny 0 Nico (2009), and Wietfeldt and Greene (2011). −6 branching ratio of 4× 10 , of which 10% go to the excited n = 2 metastable S-state of hydrogen, and another 6% to higher S-states with n > 2 , where n is 1. The neutron lifetime the main quantum number. The populations of the four n = 2 hyperfine states Over past decades, published neutron lifetimes have = =− + τ = ± with total angular momentum F1, M F 1, 0, 1 , kept decreasing from n (1100 160) s at the end of = = − and F0, M F 0 can in the V A SM be written as the 1950’s to the PDG 2010 average of τ = ± n (885.7 0.8) s . Published error bars have W=1 a + 2 B , W=2( B − A ) , 1 2 2 decreased by a factor 200 over this period. At all times, = − − = W31 W 1 W 2 , W4 0 (6.25) however, adopted lifetime values were larger than the present PDG average by about three standard with the λ -dependent correlation coefficients a, A, B deviations. To underestimate error bars is self- from above, for details and references see Dollinger destructive, because, instead of applauding the et al . (2006), and Faber et al . (2009). Of course, one progress made, the physics community will regret the will not measure λ “the hard way” via bound β - continuing lack of consistency. It now seems, though, decay, but one can search for interesting deviations that neutron decay data converge to consistent values, from V− A theory, as we shall see in Sec. D. This though again well below the PDG 2010 averages. gives us, together with the branching ratio for bound The neutron lifetime τ can be measured by two , four more observables in neutron decay. n principally different methods: with cold neutrons “in- To the list of 17 correlation coefficients discussed beam”, or with UCN “in-trap”. above we add the neutron lifetime, the four quantities in bound β -decay, the radiative decay observables, a. The “in-beam” method uses electrons and/or plus several small but interesting terms that enter the protons emitted from a certain neutron beam volume spectral shapes (weak magnetism f , the SM-forbidden 2 filled with an average number Nn of neutrons. The second class amplitudes g2, f3, and the Fierz charged decay particles are counted in detectors interference terms b and b′ discussed in Sec. D). installed near the n-beam at a rate of Hence, altogether, the number of observables = = τ accessible in neutron decay is well above 20. ne n p N nn/ . (6.26)

To derive τ one compares the rate n (or n ) to the n e p B. Experiments in neutron decay rate nn in a neutron detector at the end of the beam. The neutron detection efficiency depends on the σ Experimental results exist on ten of the 20 or more neutron cross section n and the effective thickness of τ neutron decay parameters listed above: namely, n , a, the detector material. b, b′ ,A, B, C, N, D, and R. For the Fierz amplitudes b Over the years, the in-beam method has seen many and b′ (vanishing in the SM), and for the T violating improvements: triple-correlation coefficients D and R, only upper σ∝ υ ● The use of thin neutron detectors with n1/ n limits have been derived. In this chapter we shall compensates for variations in neutron flight time discuss the present experimental status for each of T ∝1/ υ through the decay volume. these parameters, describe the experiments that n ● Magnetic guidance of the decay electrons to the contribute significantly to the PDG 2010 world π average, and add new experimental results to this detectors permits effective 4 detection of the average. The evaluation of these parameters within and charged decay products (Christensen et al. , 1972). beyond the SM will be the subject of the subsequent ● The use of neutron guides permits measurements Sections C and D. far away from the neutron source in a low-background environment (Byrne et al ., 1980; Last et al .; 1988; Nico et al ., 2005).

51 ● In-beam trapping of decay protons and their sudden cold or thermal neutrons over up to ~100 m distance. release onto a detector leads to considerable For neutrons with velocities above the critical velocity, υ> υ background reduction (Byrne et al ., 1980; Nico et al ., c , see Sec. III.B.2, the critical angle of total 2005). neutron reflection is θ≈sin θυυ = / . Table IV ● Use of a proton trap of variable length eliminates c c c lists θ for cold neutrons of effective temperature edge effects (Byrne et al ., 1990; Nico et al ., 2005). c T ≈ 30 K at their most probable velocity n υ ≈ υ = 0 700 m/s . Thermal neutrons with 0 2200 m/s θ have three times lower c . So-called neutron supermirrors are coated with about 100 to several 1000 double layers of Ni/Ti of continuously varying thickness and have typically two to three times the critical angle of Ni coated mirrors ( 'm = 2' or 'm = 3' ). The first neutron guide made entirely from such supermirrors was installed at ILL’s fundamental physics facility as a so-called ballistic guide, whose cross section varies over its length such as to minimize losses, see Häse et al . (2002) for its design, and Abele et al . (2006) for its performance. Figure 20 displays all neutron lifetimes that enter the PDG 2010 average, plus two other measurements discussed below. The open diamonds ( ◊) show the in- beam neutron lifetimes from Spivak (1988), Byrne et al . (1996), and Nico et al. (2005), see also Dewey et al . (2009) for future plans. The gray horizontal bar is τ = ± the average n (881.9 1.3) s of these measurements. The width of the gray bar gives this error, enlarged by a scale factor S=[χ 2 / ( n − 1)] 1/2 = 2.5 , as discussed below.

b. The UCN “in-trap” method for neutron lifetime measurement uses stored ultracold neutrons (full squares ( ■) in Fig. 20). 25 years ago, sufficiently

strong UCN sources were installed at the ILL (Steyerl FIG. 20. The mean neutron lifetime, horizontal gray et al ., 1986) and at Gatchina (Altarev et al ., 1986) to bar, as derived from cold neutron decay “in-beam” (◊), be used for neutron lifetime measurements with and from the decay of ultracold neutrons trapped in trapped UCN. In these experiments, one counts the “bottles” ( ■). The width of the gray bar gives the number of UCN that survived in the neutron bottle upscaled error. For the UCN measurements, the over successive storage periods of variable lengths T , vertical arrows show the extrapolation from the but constant initial neutron number Nn(0), measured storage times ( □) to the derived neutron life = − τ NTNn() n (0)exp( T / storage ) , (6.27) times ( ■), see Eq. (6.28). For details and references, see text. with the UCN disappearance rate

1 1 1 = + . (6.28) τ τ τ The in-beam method involves absolute neutron and storage n loss electron or proton counting, and for the NIST group now involved in these experiments, development of After each such measurement, the neutron trap is absolute neutron calibration is one of its professional emptied and refilled with UCN. The UCN loss rate τ duties. 1/ loss is due to neutron interactions with the walls of A word on neutron guides, used nowadays for all the trap, whereas collisions with atoms of the rest gas such “in-beam” measurements. Neutron guides usually can be neglected. typically are rectangular glass tubes of cross section With the trapped-UCN lifetime method, no absolute ~ 6× 12 cm 2 coated inside with a thin layer of totally particle counting is needed, because the mean τ reflecting material. They permit low-loss transport of residence time storage can be obtained from a fit to the

52 exponential neutron decay law, with Nn(T) measured ● The UCN storage experiment by Nesvizhevsky et successively for several different storage times T. The al . (1992) used a neutron container with an open roof, τ important task is to eliminate loss from Eq. (6.28). in which the UCN were confined both by the walls and Different techniques have been developed for this by gravity. The rate of wall encounters in the trap, and purpose, as discussed below, with the results shown in with it the UCN loss rate, was varied in two different Fig. 20. ways. First, the UCN energy spectrum, and with it the wall encounter rate, was varied by gravitational In principle, the rates ne, np, nn of decay electrons, protons, and of lost UCN can also be measured with selection. To this end, the trap was tilted under a suitable detectors during the whole storage interval, all certain angle, and the UCN with the highest energies obeying the same decay law (ideally, i.e., when there (dependent on tilt angle) were poured out through the are no changes in the UCN spectral shape) upper opening of the trap. Second, two different bottle of about 60 ℓ and up to 240 ℓ were used, nt()n( t ) nt () e=p = n =− τ with largely different wall encounter rates. The UCN exp(t /storage ) . (6.29) ne(0) n p (0) n n (0) bottles were kept at temperatures of around 15 K and were coated inside with a thin layer of solid oxygen. Experience shows that experimenters should aim at τ = The longest storage time reached was storage 876 s , constructing their apparatus such that large corrections ± to the data are avoided, because data subject to large and to obtain the neutron lifetime of (888.4 3.3) s an corrections are more prone to hidden errors. Certainly, extrapolation over a mere ∆t = 12 s was needed. large corrections can be precise, and small corrections ● The experiment by Arzumanov et al. (2000) at ILL are no guarantee that the data are free of error. With used a trap of two concentric cylinders of slightly this in mind, we shall give the size and type of the different diameters. UCN were stored either in the largest correction applied for each of the UCN lifetime large inner volume with small surface-to-volume ratio experiments. and low losses, or in the small outer volume between the two cylinders with large surface-to-volume ratio ● The first lifetime experiment with trapped UCN by and high losses. The surviving UCN were measured Mampe et al . (1989) at ILL’s UCN turbine source both right after the filling of the trap, of number N , used a rectangular glass vessel, covered inside with a i and after a longer storage time, of number Nf. special UCN-compatible oil (“Fomblin”), as a trap, see Furthermore, a “jacket” of thermal-neutron detectors also Bates (1983). The trap had a movable back wall, covered the outside of the UCN storage vessel, which such that it was possible to vary the surface-to-volume measured the integral number J of inelastically ratio of the trap, and with it, the loss rate. In UCN upscattered UCN during the storage interval T. The storage experiments, the velocity spectrum of the neutron lifetime was then derived as stored UCN changes with time, mainly because fast τ =(885.4 ± 1.0) s solely from the three directly UCN have more wall encounters and hence higher n losses than have slow UCN. A judicious choice of measured numbers J, Ni, Nf, with all detector storage time intervals ensured that the total number of efficiencies cancelling. Among the lifetimes wall encounters during measurement was the same for contributing to the PDG 2010 average, this is the one all sizes of the UCN trap used. In this case, UCN with the smallest error. The method used in this spectral effects should cancel, except for a small experiment is very elegant, but requires an ∆ correction for a vertical UCN density profile due to extrapolation t over more than a hundred seconds. ± gravity, and for other effects unknown at the time that The quoted systematic error of 0.4 s is problematic we shall discuss later on. The longest storage time in view of the later discovery that velocity spectra of τ = 730 s was reached for the maximum bottle UCN suffer minute changes during storage, even for a storage constant number of wall collisions, see Lamoreaux and volume of 140 ℓ , see the corresponding open square in Golub (2002), Barabanov and Belyaev (2006), and Fig. 20. To arrive at the neutron lifetime references therein. The authors Arzumanov et al . τ = ± n (887.6 3.0) s (filled square), the data were (2009) now acknowledge that this systematic error can extrapolated by ∆t ≈ 150 s (vertical arrow) to infinite optimistically only be arrived at in a future experiment trap volume. A problem common to all such UCN after considerable additional improvements of their storage experiments is the removal of UCN with apparatus (now in progress). elevated energy, “marginally trapped” on quasistable The PDG 2010 neutron lifetime average closed trajectories. This was achieved by making the τ = ± movable back wall of the trap from glass with a n (885.7 0.8) s (PDG 2010) (6.30) corrugated surface that made the UCN follow chaotic trajectories. is based on the above experiments, with no scale factor needed.

53 ● A neutron lifetime experiment installed at NIST (Huffman et al ., 2000) uses a superthermal 4He bath both for in-situ UCN production and for e– scintillation detection. The 4He bath is inside a magnetic UCN trap of about 1 m length. In this trap, lateral UCN confinement is assured by a long superconducting quadrupole field with transverse magnetic potential V ()ρ∝ || ρ , for lateral distance ρ to the field axis z. Axial confinement is achieved by two “humps” in the longitudinal potential V(z) at the ends of the trap, produced by two solenoids with opposite windings. A prototype of this instrument had produced a value of τ = +74 n 833−63 s , see Dzhosyuk et al . (2005). With a new trap of 3.1 T depth and 8 ℓ volume 4.5× 10 4 UCN can now be stored at density 5 cm –3. A sensitivity of ∆τ ≈ ± 2 s is expected after one reactor cycle n (O’Shaughnessy et al. , 2009). FIG. 21. Neutron lifetime from the latest UCN storage ● The experiment by Serebrov et al . (2005, 2008b) at experiment. Shown is the UCN disappearance rate ILL is an improved version of the experiment by τ γ∝ τ 1/ storage versus the (normalized) loss rate 1/ loss , Nesvizhevsky et al. (1992) described above. Instead of see Eq. (6.28). For γ = 0 , the linear fit gives solid oxygen, the authors used a new type of synthetic τ= τ = ± oil as a wall coating, which has lower inelastic UCN storage n (878.5 0.8) s . Open dots are for the losses than Fomblin oil (used by Mampe et al ., 1989), larger trap, full dots for the smaller trap, each for five and a higher coating efficiency than solid oxygen (used different UCN energy bands. The extrapolation is over by Nesvizhevsky et al ., 1992). With a mean UCN loss ∆τ = 5 s . Shown is also the PDG 2010 average. From τ ≈ time of loss 2 days extrapolation over less than Serebrov et al. (2005, 2008b). ∆t = 5 s was required to reach the lifetime τ = ± n (878.5 0.8) s in the limit of zero wall-collision rate γ , see Fig. 21. The authors do not exclude that ● Pichlmaier et al . (2010) used an improved version σ of the first UCN lifetime apparatus by Mampe et al . the 2.9 discrepancy to their Nesvizhevsky et al . (1989). In their experiment, the UCN spectrum was (1992) result was due to some small imperfection in first shaped in a large prestorage volume, the UCN trap the oxygen coating in the earlier experiment. If this was proven to be N2-gas-tight over several days, and had occurred in the smaller of the two traps, this would its Fomblin oil coating was continuously being shift up the dark points in Fig. 21 (or rather those in refreshed. The result τ =(880.7 ± 1.8) s , with the corresponding 1992 figure), and would shift n σ the γ = 0 intersection to a longer neutron lifetime. The statistical and systematic errors of equal size, is 2.5 below the PDG 2010 value, and 1.1 σ above the Serebrov et al . (2005) experiment was excluded from Serebrov et al . (2005) value. To arrive at this number the PDG 2010 average, from which it differs by an extrapolation over rather large intervals of 6.4 standard deviations. ∆t = 110 s and more was needed. However, the Serebrov and Fomin (2010) have reanalyzed the UCN lifetime experiments of Mampe et al . (1989) dominant part (about 90%) of this extrapolation is and of Arzumanov et al . (2000) for effects affecting based on geometric quantities like the length of the the UCN spectra that where not known at the time of trap that where sufficiently well controlled. measurement. They find the first lifetime shifted We update the PDG 2010 average by −6 s , from (887± 3.0) s to (881.6± 3.0) s , and τ = ± = n (885.7 0.8) s (scale factor S 1) with the last the second lifetime by −5.5 s , from (885.4± 1.0) s to two results to (879.9± 2.6) s , which makes them both consistent τ =(881.9 ± 1.3) s , (our average), (6.31) with the average Eq. (6.31) below. Steyerl et al . (2010) n point out that surface roughness effects may be another with the error blown up by S = 2.5 , following PDG source of error in such measurements. procedures (see Introductory Text in Reviews, Tables, Plots, PDG 2010). Both averages differ by 2.5 σ .

54 A caveat : When applying, as we have done, scale sight onto the reactor core or onto the moderator, and factors to widely differing results, one must assume hence with rather low background. The protons that all errors entering the average are underestimated emerging from this beam hole, detected at a rate of by the same scale factor. A posteriori , this assumption 90 s–1, were analyzed in a spherical electrostatic holds for most past neutron decay measurements (see spectrometer. the neutron history plots in PDG 2010). However, it is ● Byrne et al . (2002) measured the proton spectrum not certain whether it also holds for the latest UCN at ILL “in-beam”, using a modified version of their τ lifetime measurements, on which the average of n proton-trap lifetime apparatus mentioned above. In this mostly depends. experiment, the protons adiabatically emerged from a Fortunately, the tests beyond the SM done in Sec. D high magnetic field to a low field region. This below mostly depend on the correlation coefficient B, transformed their transverse energy into longitudinal and less on the lifetime τ and the β -asymmetry A. energy, which is easier to analyse with a superimposed n electric field. The result was a = −0.1054 ± 0.0055 , Nevertheless, for quantities depending directly on τ n with 5% error. like neutrino cross sections (see Sec. VII.A) we must ● aSPECT (Zimmer et al ., 2000b, Glück et al ., 2005) keep this caveat in mind. measured the energy spectrum of the decay protons, Note added in proof : In the 2011 updated web after adiabatic expansion from high into low magnetic version of PDG 2010, a new lifetime average is given field, too, in a retardation spectrometer of the type used for neutrino mass measurements in tritium decay. τ = ± n (881.7 1.4) s (PDG 2011), (6.32) To be detected, the protons must overcome the potential of an electric counter field, which can be with a scale factor S = 2.6 , in close agreement with varied from zero to beyond the proton endpoint energy our average Eq. (6.31), and with similar caveats . of 750 eV. The derivative of this transmission

spectrum then gives the proton’s energy spectrum. The

apparatus was tested at FRM II and ILL at a proton 2. Correlation coefficients count rate 460 s –1 and a background rate of 0.2 s –1, and

reached a statistical error of a = ± 0.004 (stat) , but Next, we discuss the various neutron-decay correlation experiments. All correlation coefficients these runs had problems in quantifying the quoted are corrected for weak magnetism. Radiative background, see Baeßler et al . (2008), problems likely corrections of order α are not necessary for the to be avoided in a new scheduled run. correlation coefficients, as they only affect the T The PDG 2010 average for the electron- violating imaginary parts of the form factors, see antineutrino correlation is Jackson et al . (1957b). For further radiative = − ± corrections, see Glück (1998). a 0.103 0.004 , (6.33) with a relative error of 4%. a. Electron-antineutrino coefficient a : In principle, the correlation between electron and antineutrino momenta b. β -decay asymmetry A : At present, the correlation can be measured directly from the angular dependence between neutron spin and electron momentum is the of electron-proton coincidences, but such main source for λ = g/ g . The β -asymmetry is measurements so far have not been successful. Instead, A V the coefficient a is derived from the shape of the measured by counting the electrons from polarized proton spectrum, which is sensitive to the electron- neutron decay, alternately for neutrons spin up (+) and neutrino correlation a, and, for polarized neutron spin down (–) with respect to the holding field. The decay, to A and B as well. In contrast, the shape of the measured asymmetry is electron spectrum from unpolarized neutron decay N+− N − υ =e e = e θ depends very little on the relative sizes of the coupling Am+ − AP n 〈cos 〉 , (6.34) N+ N c constants. The following experiments contribute to the e e PDG 2010 world average of a. θ=1 + θ θ with 〈cos 〉 2 (1 cos0 ) , where 0 is the detector ● Stratowa et al . (1978) measured angle of admission about the axis of neutron = − ± ∝β = υ a 0.1017 0.0051 from the shape of the proton polarization. The dependence of Am e / c on electron spectrum to 5%, a result that was not electron energy W ∝γ =(1 − β 2 ) 1/2 has the familiar surpassed for 25 years. In this elegant experiment, the e shape β vs. γ known from special relativity, plus a neutron decay volume was “in-pile” near the core of a small 7 MW Austrian research reactor. A through- small (∼ 1%) almost linear energy dependence from going tangential beam tube was used, with no direct weak magnetism, omitted in Eq. (6.34).

55 Early neutron correlation experiments used neutron field B(x), hence there is a clean division between ⋅ > beam holes right outside a reactor’s biological shield. electrons emitted along (pe B 0) and against To suppress background, they required multiple (p⋅ B < 0) the local field direction. In this way, coincidences (between proton signals and E plus E e e e electron emission with respect to the field direction is electron signals), which allowed only small decay θ ≡ 1 always well defined, with 〈cos 〉 2 , and no volumes. This, in turn, led to low event rates of a few precision alignment of the detectors relative to the counts per hour, for a review see Robson (1983). decay volume is necessary. Lee and Yang (1956) had The following experiments contribute to the present already advocated such a scheme for their proposed world average of A. search for parity violation. The full energy dependence ● Erozolimskii et al . (1990, 1991) measured A on the Am(Ee) was measured and A was extracted on the basis vertical polarized cold neutron beam from the liquid of Eq. (6.34). The 1986 result was hydrogen (LH 2) cold source that is installed in the A = −0.1146 ± 0.0019 , with 2% relative error. The center of the PNPI reactor core (Altarev et al ., 1986). largest correction of +13% accounted for electrons The polarized beam has capture flux density counted in the wrong detector due to magnetic-mirror 1.8× 109 s− 1 cm − 2 and polarization reversals of electron momenta induced by B-field P =0.7867 ± 0.0070 . The decay electrons emitted inhomogeneities over the neutron decay volume. vertical to the neutron beam axis were counted, ● A new version of the PERKEO instrument by alternately for neutron spin up and down, in a small Abele et al . (2002), used with a neutron polarizer with = ± = − ± plastic scintillator with 〈cosθ 〉 ≈ 1 , in coincidence with Pn 0.989 0.003 , gave A 0.1189 0.0007 , with decay protons detected under 4π solid angle. The a statistics-dominated 0.6% relative error. In this result was A = −0.1135 ± 0.0014 (Yerozolimsky et al ., measurement, the correction to the magnetic mirror 1997), with 1.3% relative error. The largest correction effect mentioned above had been suppressed applied was +27% for neutron polarization. to +0.09%, the largest correction then became ● Schreckenbach et al . (1995) measured the β - the +1.1% correction for neutron polarization. The PDG 2010 average derived from the above asymmetry A with a cold neutron beam passing a gas- = − ± filled time projection chamber, in which the tracks of measurements is A 0.1173 0.0013 , where the the decay electrons were measured. This chamber was error is increased by a scale factor of 2.3. backed by plastic scintillators to measure electron ● A later measurement with PERKEO was done at energy. The experiment profited from ILL’s low- the new fundamental physics cold neutron facility background cold neutron guides and supermirror at ILL. The station’s dedicated “ballistic” supermirror cold neutron guide at the time had a capture flux neutron polarizers with, at the time, P =0.981 ± 0.003 n density of 1.3× 1010 cm− 21 s − (unpolarized). The (as did the instruments discussed in the following). experiment used an improved neutron polarizer with The result A =0.1160 ± 0.0015 had 1.3% relative P =0.997 ± 0.001 (Kreuz et al. , 2005a). In this error, with the largest correction of +18% on the n β effective solid angle 〈cos θ 〉 of electron detection, see measurement, the error of the -asymmetry A was also Liaud et al . (1997). again cut in half, and the corrections diminished ● The PERKEO instrument in its first version (Bopp to 0.4%. The combined result covering the period 1997 et al. , 1986, 1988), got rid of the need for coincident to 2008 is A = −0.11933 ± 0.00034 , with a 0.3% detection in the measurement of A, due to a drastically relative error, as reported by Abele (2008, 2009). enlarged neutron decay volume and reduced Publication of this result is underway, the delay being background. With this the measured neutron decay rate due to changes of affiliation of several of the main increased by several orders of magnitude to 100 s –1 per collaborators. This new result for A differs from the detector (for polarized neutron decay), with PERKEO 1986 result by +2.5 standard errors. = ± ● A new β -asymmetry result with trapped ultracold Pn 0.974 0.005 . Neutron density in-beam at ILL was 1500 cm –3 (unpolarized) over several dm 3. In the neutrons, reported by Liu et al . (2010), used the 5 K instrument, a B =1.6 T field from superconducting solid deuterium UCN source installed at the LANSCE pulsed proton beam. UCN density in the trap was coils magnetically coupled two plastic scintillators to –3 the neutron decay volume and to each other. The 0.2 cm . The initial UCN polarization, created by spin electrons, spiralling about B, reached the detectors selection via a 7 T magnetic barrier, was = +0 with 4π effective solid angle. Electrons that Pn 1.00 −0.0052 , and depolarization during storage was backscattered on one detector deposited their below 0.0065. The decay electrons were counted in ∆ remaining energy as delayed events in the other a Ee- E e arrangement of multiwire chambers and detector. In the PERKEO approach, the neutron plastic scintillators, magnetically coupled to the UCN polarization always adiabatically follows the local trap volume and to each other by a B =1 T field. UCN

56 generated backgrounds were negligible. The result was The PDG 2010 average derived from these and = − ± +0.00123 some earlier, statistically no longer significant A 0.11966 0.00089(stat)−0.00140 (syst) . measurements, is To update the PDG 2010 average = ± A = −0.1173 ± 0.0013 , which had a scale factor B 0.9807 0.0030 , (6.36) S = 2.3 , we replace the Abele et al . (2002) result with 0.3% relative error. A = −0.1189 ± 0.0007 by the Abele (2008) combined ● An interesting measurement of the two quantities = − ± = − ± = ± result A 0.11933 0.00034 , and add the Liu et al . Pn A 0.1097 0.0016 and Pn B 0.9233 0.0037 (2010) result given above, which latter, for further was done by Mostovoi et al . (2000), using the same analysis, we simplify to A = −0.1197 ± 0.0016 . This PNPI apparatus at ILL for both measurements. From gives the new world average these quantities λ =(AB − )/( A + B ) A = −0.1188 ± 0.0008 , (6.35) = −1.2686 ± 0.0046 was derived, where neutron polarization has shortened out, which often is the with a scale factor S = 2.5 on the error. A history plot largest correction in such measurements. The authors of A showing the corrections applied to the raw data then calculated both A and B from this value of λ , (similar to Fig. 20 for τ ) is given in Abele (2009). n using Eqs. (6.15) and (6.17), which, of course is not an The comments made above on the scale factor of the independent derivation of these quantities, so only λ neutron lifetime average apply equally to A. In from this measurement is included in the compilations, particular, we note what happens when to an old but not A and B. From this calculated B and the measurement with error σ that is a number of m = ± 1 measured PnB the authors derive Pn 0.935 0.005 . standard deviations away from the “true value”, we add a new and more precise measurement with smaller d. Proton asymmetry C : Schumann et al . (2008) for the σ< σ error 2 1 : The scale factor S on the total error of first time measured the correlation between neutron both measurements does not decrease, but keeps polarization and proton momentum, using the same = + σ σ 2 PERKEO setup as used for the B-measurement. The increasing as S m /(1 (2 / 1 )) for decreasing proton asymmetry was measured in coincidence with error σ of the new measurement, to a perennial 2 all electrons as a function of electron energy, see Fig. → S m . 22. The result c. Antineutrino asymmetry B : The correlation between C = −0.2377 ± 0.0026 (6.37) neutron polarization and antineutrino momentum is with 1% relative error agrees with reconstructed from electron-proton coincidences in C = −0.2369 ± 0.0009 as calculated from Eq. (6.18) polarized neutron decay. The following experiments with the measured A and B values. In principle, the contribute to the present world average of B. proton asymmetry could also be measured with single ● Kuznetsov et al. (1995) measured B at PNPI and protons as a function of proton energy. From measured later at ILL in a setup similar to the one used in the A- energy spectra of C one can derive A and B separately. measurement by Erozolimskii et al . (1990) cited For instance, at low electron energy, C(Ee) in Fig. 22 above. The proton time-of-flight, measured in reflects the neutrino asymmetry B ≈ + 1 , and it reflects coincidence with the energy-resolved electrons, is the electron asymmetry A ≈ − 0.1 at endpoint sensitive to the direction of antineutrino emission with energy E0. respect to neutron spin direction. The neutron = ± e. Correlation N: This describes the correlation polarization was Pn 0.9752 0.0025 . The final result B =0.9821 ± 0.0040 (Serebrov et al ., 1998) had between neutron spin and electron spin. Kozela et al . a 0.4% error. (2009) for the first time made the electron spin σ ● Schumann et al . (2007), improving on Kreuz et al . variable e accessible in neutron decay experiments. (2005b), measured B with PERKEO by detecting both Their main goal was the T violating parameter R electrons and time-delayed protons from polarized described below, and the N-coefficient was a by- neutron decay with plastic scintillators like in the A- product of this research. The experiment used the = ± fundamental physics beam-line of the continuous measurement, with Pn 0.997 0.001 . The protons were detected via secondary electrons released during spallation neutron source SINQ at PSI. The apparatus their passage through a thin carbon foil at a high consisted of multiwire proportional counters for negative electric potential in front of the scintillator. electron track reconstruction on both sides of the cold The result B =0.9802 ± 0.0050 had an error similar to neutron beam. These track detectors were backed by the PNPI experiment, all corrections applied were thin Pb-foils for electron spin analysis, followed by smaller than this error. plastic scintillators for electron energy measurement.

57 The Pb-foil acted as a Mott polarimeter, via the spin- Within errors, this is in agreement with the expected dependent angular distribution of the electrons N =0.0439 ± 0.0003 from Eq. (6.19), backscattered by the foil. From the asymmetry of = − N( mWAe / e ) , taken at the electron endpoint backscattered events upon reversal of the neutron spin energy W with the measured A from Eq. (6.35). followed 0

N =0.056 ± 0.011( stat) ± 0.005(syst) . (6.38)

FIG. 22. The negative of the experimental proton asymmetry –C as a function of electron energy Ee. The bold line indicates the fit region. From Schumann et al . (2008).

f. Triple correlation D: This is the coefficient for of the delicate symmetry conditions. The experiment ⋅ × =−± ± × −4 the T violating triple product σn(p e p ν ) between reached D ( 2.8 6.4(stat) 3.0(syst)) 10 . neutron polarization, electron momentum, and proton The PDG 2010 average from the above momentum. measurements

● For a long time this coefficient was dominated by −4 −4 D =−( 4 ± 6) × 10 (6.39) the result D =−( 11 ± 17) × 10 of Steinberg et al. (1974, 1976), done at ILL. In this experiment, the is consistent with zero. This translates into a phase longitudinal neutron polarization, the electron between gA and gV detectors and the proton detectors were arranged at ϕ =−180.06 °± 0.07 ° , (6.40) right angles to each other in order to maximise the above triple product. The neutron spin was inverted in which means that, within error, λ is real, and T is regular intervals. Erozolimskii at PNPI did similar conserved. measurements in the 1970’s with a slightly larger error. g. The coefficient R of the triple product σ⋅( σ × p ) ● Later on, Lising et al. (2000) realized that an e n e between electron spin, neutron spin, and electron octagonal detector arrangement with 135° between the momentum was measured for the first time by Kozela electron and proton detector axes would give a three et al . (2009) with the apparatus described above for times higher quality factor NS 2, due to much higher coefficient N. To derive R the authors regarded the count rate N at a slightly reduced signal S. Their electron spin component perpendicular to the decay experiment, done at the fundamental physics beamline plane spanned by σ and p . R-events were separated of the NIST reactor, gave D =( − 6 ± 12(stat) n e from N-events by their differing kinematic factors, ±5(syst)) × 10 −4 . which depend on the angle between the σ -p decay ● Soldner et al. (2004) used an arrangement with n e plane and the Mott scattering plane, with the result electron track detection in a multiwire proportional chamber, which, together with the orthogonally R =0.008 ± 0.015(stat) ± 0.005(syst) . (6.41) arranged proton detectors, permitted off-line control

58 A nonzero R requires scalar and tensor b. Upcoming correlation experiments : Several couplings gS and gT not present in the V− A Standard correlation experiments are on the way. Model, as will be discussed in Sec. D. ● aCORN is built for the measurement of electron- antineutrino correlation a directly from electron- proton coincidences (Wietfieldt et al ., 2009). Decay 3. *Upcoming neutron decay experiments electrons originating in a small volume of a cold neutron beam and emitted vertically to the beam axis z Several new neutron decay instruments are in the are detected under small solid angle in an electron construction or in the test phase. detector installed in + y directions. Decay protons are guided electrostatically to a small proton detector a. Upcoming lifetime experiments : Several lifetime installed in –y direction. Precise circular apertures experiments are underway. limit the divergence of the electron and proton fluxes. ≤ ≤ ● Ezhov et al . (2009) use a magnetic UCN trap built For low electron energies (80 keVEe 300 keV) , from rare-earth permanent magnets. The trap has the the proton time of flight spectrum, measured in delay form of a bucket of 20 cm inner diameter, open at the to the prompt electron signal, splits into two well top. Near the surface of its inner walls is a repulsive defined groups: The early proton arrivals are those magnetic potential, produced by a magnetic field up emitted in direction of the proton detector, opposite to to 1.2 T , whose direction changes sign every few electron emission, i.e., directionally anticorrelated centimeters in the azimuthal direction. UCN from with the electrons. The late proton arrivals are those the ILL turbine source are brought into this trap from first emitted in the same direction as the electrons, so above, via a cylindrical “elevator” box, which is filled they had to make a detour through the electrostatic with UCN, then slowly lowered into the trap, emptied mirror to arrive late at the proton detector. These there, and withdrawn. In the prototype apparatus protons are correlated with the direction of electron 1800 UCN per filling were stored at ILL in an emission. When installed at NIST, a coincidence effective volume of 10 ℓ . UCN lost by spin-flip were count rate of 0.1 s –1 is expected in this experiment. counted in the same detector as the surviving UCN, so ● The PERKEO program continued with a new their detection efficiencies do not enter the result. A instrument for neutron correlation measurements new 90 ℓ version of this apparatus is under test. (Märkisch et al ., 2009). In the new apparatus, the ● At FRM II, a superconducting trap of toroidal form neutron decay volume was further enlarged, with a called PENeLOPE is under development. It has a measured decay electron rate of 5⋅ 104 s − 1 for a depth of 1.8 T, a volume of 700 ℓ , and a proton continuous polarized cold neutron beam. In view of detector above the active volume, see Zimmer (2000), the size of the new instrument, conventional magnet and Materne et al . (2009). The trap will be filled from coils had to be used. The rather small magnetic field the FRM II solid D 2 UCN source that is under of B = 0.15 T is acceptable because the up to 2 cm construction. It is expected to hold up to 2⋅ 10 8 UCN. radii of gyration of protons and electrons have an Prototype testing is under way. astonishingly small effect on the detector response ● A superthermal 4He UCN source placed within a function (Dubbers et al ., 2008b). In a recent second magnetic Ioffe trap is proposed by Leung and Zimmer period of beam time, this instrument was used with a (2009), with a long octupole field with transverse pulsed neutron beam. The decay electrons and their magnetic potential V ()ρ∝ || ρ 3 made from permanent asymmetry were observed from a cloud of neutrons magnets, plus two superconducting coils as magnetic moving through the 2.5 m length of the instrument’s end caps. After filling, the magnetic trap, and with it active volume. At present the results of this run are the trapped UCN, are withdrawn from the stationary under evaluation. 4He bath in axial direction. In the process, some UCN ● The Nab instrument is built for the measurement of are lost as they must penetrate the end window of the correlation a and Fierz interference b (Po čani ć et al ., 4He vessel, but this is largely compensated by the 2009). It aligns the proton and electron momenta advantage that the decay protons can be detected in a emerging from unpolarized neutron decay in a high low-background environment. Prototype testing has magnetic field region by adiabatic magnetic expansion started. from 4 T to 0.1 T into both the + y and –y direction, at ● A conceptual design study for a novel magneto- right angles to the neutron beam. Both protons and gravitational UCN trap at LANSCE was presented by electrons are detected, and their relative time-of-flight Walstrom et al . (2009). is measured in two silicon detector arrays under 4π solid angle. The method is based on the observation (Bowman, 2005) that for a given electron energy, the proton momenta squared have a probability distribution with a slope proportional to the 59 correlation coefficient a. From a 20 cm 3 neutron decay 4. Comparison with muon decay data volume, a signal rate of 400 s –1 is expected for full µ→ + ν + ν power of SNS. The Fierz term b is to be extracted Muon decay e e µ tests the purely from the electron energy spectrum. A new asymmetric leptonic part of the weak interaction. There are three version of the instrument is described in Alarcon et al . observable vectors p , σ , σ in muon decay, besides (2010). Later on, the project will evolve into the abBA e e µ instrument, where the electron and antineutrino the muon lifetime. From these, four correlations can asymmetries A and B will also be measured in a be constructed (three scalar and one triple product) configuration similar to the previous PERKEO II that have been measured over the years with precision instrument. similar to that obtained for the neutron correlation ● At SNS the PANDA project intends to measure the coefficients. In muon decay, also the shape of the proton asymmetry C in coincidence with the electrons, electron spectrum depends sensitively on the coupling and dependent on the proton energy, see Alarcon constants involved. This is in contrast to neutron et al ., 2008. decay, where the shapes of the unpolarized or ● The PERC project (Dubbers et al ., 2008a) is based polarized electron spectra from neutron decay depend on the belief that the highest precision in neutron very little on the relative sizes of the coupling decay can be reached with single-particle detection. constants involved; only the shapes of the proton PERC, short for Proton and Electron Radiation spectra depend measurably on the coupling constants. ρ = 3 Channel, will not be another neutron decay The Michel parameters with SM values 4 , spectrometer, but a facility that delivers well-defined η = 0 (from the shape of unpolarized muon decay beams of electrons and protons from neutron decay spectra), plus the four correlation coefficients ξ =1, taking place inside a long section of a cold neutron δ = 3 , and two transverse electron spin components guide at a rate of 10 6 decays per meter of guide. With 4 PERC, one can measure the shapes and magnitudes of (from polarized muon decay), have absolute errors −4 electron or proton energy spectra from polarized or (4, 34, 35, 6, 80, 80)× 10 , respectively. The unpolarized neutron decay; with or without electron absolute errors for recent neutron measurements of a, spin analysis. The neutron decay coefficients a, b, b′ , A, B, C, D, N, R, are of comparable magnitude, A, B, C, N, G, Q will be accessible from such single namely, (40, 4, 30, 9, 6, 120, 80)× 10 −4 respectively. particle spectra. Our primary aim, however, are not The relative error of the muon lifetime, on the other these correlation coefficients but the weak amplitudes hand, is two orders of magnitude smaller than that of that the spectra depend on (within and beyond the the neutron lifetime, from which we profit when using SM), like V , g , g , g , ζ , m , f , f , g , etc.. The ud A S T R 2 3 2 GF. For muon results beyond the SM, see Sec. D. PERC construction period starts in 2011, and the facility will be open for dedicated user instruments at FRM-II several years later. C. Weak interaction results and the Standard ● At FRM II a project on neutron bound-beta decay Model is underway for the study of physics beyond the SM in the scalar/tensor and right-handed current sectors In this section, we discuss implications for the (Dollinger et al . 2006), see also end of Sec. D below. Standard Model of the neutron decay data presented in The neutron decay volume will be “in-pile”, next to the preceding section.Global analyses of neutron and the reactor core inside a tangential through-going nuclear β -decay data were done by Severijns et al . beam-tube. The hydrogen atoms from neutron decay (2006), who expand and supersede the analyses of leave the beam tube with a recoil energy of 326 eV. Gaponov and Mostovoy (2000), and Glück et al . Selection of the various n = 2 hyperfine states of the (1995). For a discussion of neutron decay in the atoms will be done by judicious Stark mixing and framework of effective field theory, see Ando et al . quenching via the unstable 2 P state, plus by additional (2004), and Gudkov et al . (2006). Reviews on low- induced hyperfine RF transitions along the hydrogen energy weak interactions in a broader context were flight path. Excited states with n > 2 will be depleted written by Herczeg (2001), and Erler and Ramsey- by laser ionization. Finally, the fast hydrogen atoms in Musolf (2005). a selected hyperfine state are ionized, velocity selected with a magnetic field, and accelerated onto a CsI scintillator. A first version of the apparatus aims 1. Results from neutrons and other particles at the detection of bound neutron decay without the analysis of the hyperfine states. The neutron decay parameters relevant for the SM

are λ = g/ g and the upper left element Vud of the A V CKM matrix. The PDG 2010 value for gA/gV was 60 derived from the β -asymmetry A and the Mostovoi et of semileptonic hyperon decays by Cabibbo et al . al . (2001) value for λ to λ = −1.2694 ± 0.0028 , with (2004) gives a scale factor S = 2.0 . With the updated value of A, |λ |=+=F D 1.2670 ± 0.0035 (). (6.43) Eq. (6.35), this becomes λ = −1.2734 ± 0.0021 , with a scale factor S = 2.6 . Inclusion of the results for a, B, Combination of this with the neutron value Eq. (6.42) and N does not change this result significantly, giving causes a shift to λ = −1.2734 ± 0.0019 (neutron) (6.42) λ = −1.2719 ± 0.0017 (baryons), (6.44) with a scale factor of 2.3 (we leave out the proton with S = 2.2 . asymmetry C, which in part was based on the same Next, we derive the CKM matrix element |Vud | data set as one of the B measurements). Figure 23FIG. from neutron decay. The neutron lifetime τ from λ n 23 shows the values derived from various Eq. (6.31) and λ from Eq. (6.42) inserted into + τ measurements. For the values labelled F D and n Eq. (6.11) gives on the abscissa of Fig. 23 see Eq. (6.43) and text to = ± Eq. (6.52) below. Vud 0.9742 0.0012 (neutron), (6.45)

which within error equals the value = ± Vud 0.9747 0.0015 derived from the PDG 2010 data.

|Vud | can also be obtained from the rare decay π+→ π0 + + + ν × −8 e e with 1.0 10 branching ratio, which gives (Po čani ć et al ., 2004) = ± |Vud | 0.9728 0.0030 (pion). (6.46) Combination of this with the neutron value Eq. (6.45) causes a marginal shift to = ± Vud 0.9740 0.0010 (particles). (6.47)

2. Results from nuclear β -decays

Altogether, the SM parameters λ and |V | have ud FIG. 23. The ratio of axial vector to vector coupling the following main sources: λ = constants gA/ g V (horizontal gray bar) as derived (1) Neutron decay correlations, in particular, the β - from various sources: from the neutron β -asymmetry asymmetry A, are functions of λ only; τ λ A, neutrino asymmetry B, proton-asymmetry C, from (2) The neutron lifetime n is a function of both the electron-neutrino correlation a, the hyperon and |V | . correlations F+ D , and indirectly with Eq. (6.10), ud +→ + from the neutron lifetime τ combined with (3) The nuclear ft values for superallowed 0 0 n β 2 -transitions are a function of |Vud | only. nuclear |Vud | . The width of the horizontal gray bar gives the upscaled error. In the nuclear superallowed transitions, only the vector matrix element enters π 2ℏ 7 = 2 (c ) ln 2 The nucleons are members of the flavor-SU (3) ft 25 2 2 , (6.48) baryon spin-½ octet, which they join with the strange c( mc ) GF || V ud hyperons (the Σ,Λ,Ξ -baryons with various states of which conforms with Eq. (6.10) for half-life t =τ ln 2 β electric charge). The -decay parameters of this and λ = 0 . From any two of the three observables multiplet are linked to each other via the SU (3) τ λ A, n , and ft one can calculate the parameters transformation properties. The neutron’s λ = g/ g A V and |V | plus the remaining third observable, as is β ud should be related to the hyperons’ F and D -decay frequently done in all three combinations. Of course, parameters as |λ | =F + D . Indeed, a global evaluation

61 it is preferable to use all measured observables and decay, which is obtained from Eqs. (6.11) and (6.48) make a global fit. as,

F =R 1 +λ2 τ =± tn−Vector f 2 (1 3 ) n ln2 (3068 3.8) s (neutron), (6.51) with the radiatively corrected phase space factor f R =1.71465 ± 0.00015 . To demonstrate the potential sensitivity of neutron decay measurements we have τ used in Eq. (6.51) only the last two lifetime n results (Serebrov et al ., 2005, and Pichlmaier et al , 2010) and the last two β -asymmetry results (Abele, 2008, and Liu et al , 2010). This neutron Ft value is inserted in Fig. 24 at Z =1 . To compare, the corresponding value derived from the PDG 2010 neutron data is F = ± tn-Vector (3071 9) s . For the neutron decay data the experimental error dominates, while for the nuclear decay data the error from theory dominates. Sometimes it is argued that the neutron derived value for |V | is more indirect ud because it needs input from the two measured τ λ FIG. 24. : Measured Ft values both from nuclear quantities n and . However, the nuclear derivation + + superallowed 0→ 0 β -transitions (Z ≥ 5) , and of |Vud | also requires input from various sources: from from the measured vector part of neutron decay nuclear lifetimes, branching ratios, Q-values, plus (Z = 1) . The superallowed data are shown before ( ◊) nuclear theory. and after ( ♦) corrections for nuclear and radiative Finally, using Eq. (6.11) one obtains from the τ effects, as well as their mean value after correction neutron lifetime n and the nuclear |Vud | an indirect (horizontal gray bar). For comparison, the value for λ (see the last three entries in Fig. 23), F neutron tn-Vector value is inserted as derived from which we can include in the world average to obtain Eq. (6.51) using only the latest two results for the λ =g/ g =− 1.2735 ± 0.0012 τ A V neutron lifetime n and the latest two results (all neutron data plus nuclear |Vud | ), (6.52) for λ = g/ g . A V = with a scale factor S 3.1 . A few years ago, this value was 2.5 standard deviations lower and had a λ = =− ± For the nuclear decays, besides the rather well smaller error, gA/ g V 1.2699 0.0007 , with known radiative corrections, nuclear structure and S =1.1 , see Severijns et al . (2006). This is because isospin corrections must be applied, and much recent neutron decay data disagree significantly with progress has been made in recent years. Figure 24 ≈ the former world average. With |Vud | 0.97 fixed, the shows the nuclear data from Table IX of Hardy and sensitivity of τ to λ is ∂τ/ τ = 1.3 at τ ≈ 880 s Towner (2009), before ( ft , ◊), and after corrections n λ n n lower than for a and A, which was −2.8 and − 3.2 , ( Ft , ♦). From these data the mean value respectively. Ft =(3071.81 ± 0.83) s (nuclear Ft ) (6.49) is derived with an error given by the width of the gray 3. Unitarity of the CKM matrix horizontal band in Fig. 24. From Eq. (6.48) then = ± F 2 Vud 0.97425 0.00022 (nuclear t ) (6.50) The measured value of |Vud | can be used to test the unitarity of the first row of the CKM matrix with an error five times smaller than the error from 2+ 2 + 2 =− neutron plus pion. Notice, however, that in this |Vud |||| V us V ub | 1 , (6.53) evaluation the data point with the smallest error = at Z =12 is also the one with the largest correction. where unitarity in the SM requires 0 . The third 2≈ − 5 For comparison, we also derive an Ft -value for term, |Vub | 10 as derived from B-decays, is the neutron, due solely to the vector part of neutron negligible, so mainly tests the old Cabibbo theory

62 ≈ θ with Vudcos C . Some years ago, a positive and f3(0) and g2(0) for second-class currents. As some = 0.0083(28) with +3σ standard deviations from of the formulae on these terms are hard to find in the literature, we shall discuss them in more detail. unitarity was observed for the neutron (Abele et al ., One should test the SM prediction for the weak 2002), and similar deviations on the +2σ and +1σ magnetism term f (0) because it is at the heart of level were observed from nuclear and pion decays, see 2 electroweak unification. Best suited for a the survey of Abele et al . (2004), which summarizes a measurement of weak magnetism is the typically 1% workshop on this topic edited by Abele and Mund electron-energy dependence of the correlation (2002). coefficients. Past experiments on the β -asymmetry A Today it seems that the culprit for the deviation from unitarity was the V value (with V ≈ sin θ in with PERKEO gave f2(0) as a free parameter only us us C with an error of about seven times the expected effect. Cabibbo theory). Recent remeasurements of , , To discuss the influence of such small corrections and hyperon weak transitions shifted the measured let us write an arbitrary correlation coefficient X from |V | f (0) by as much as six standard errors. When us + Sec. A.4 within the SM as X= s / ξ , with uncertainties from theory in the form factor f (0) are + normalization ξ=1 + 3 λ 2 . For the β -asymmetry A, included, this shifts the PDG 2004 value = ± for instance, s = −2(λ2 + λ ) from Eq. (6.15). When |Vus | 0.2200 0.0026 to the PDG 2010 value = ± we have additional terms ε << s in the numerator s, |Vus | 0.2252 0.0009 . + + and ε′ << ξ in the denominator ξ , then the With the value |V | from 0→ 0 decays, Eq. ud correlation coefficient is frequently written in one of (6.50), this re-establishes unitarity on the level of one the forms part per thousand, see the review Blucher and Marciano (2009), and the update by Towner and XX'1/≈( +ε s − εξ′ /) =+− X ( ξεεξ s ′ )/ 2 . Hardy (2010), which give (6.59) st 2++ 2 2 = ± 1 row: |Vud | | V us | | V ub | 0.9999 0.0006 For the β -asymmetry A, for instance, the weak (with nuclear Vud), (6.54) magnetism correction is

−1 2 Using |Vud | alone from the neutron gives ε=2 λλµλµ +−− wm 3 M[(2 2 ) W 0 st 2++ 2 2 = ± −(11λ2 + 7 λµ −− 5)] λµ W , (6.60) 1 row: |Vud | | V us | | V ub | 1.0000 0.0026 ε′ =−1 −+ λλµ 2 + 2 (with neutron Vud). (6.55) wm 2M [( )( WmW0 / ) High-energy results for the other rows and +2(15 +λ2 − 2 λµ )W ] , (6.61) columns give (Ceccucci et al ., 2010) see Holstein et al . (1972), and, for more details, nd 2+ 2 + 2 = ± 2 row: |Vcd | | V cs | | V cb | 1.101 0.074 , (6.56) Holstein (1974). For the moment we write shortly 2 2 1st col.: |V |2+ | V | 2 + | V | 2 = 1.002 ± 0.005 , (6.57) M, m, W for Mc , mec , We. In these equations, ud cd td µ≡ κ − κ +=1 4.706 , and the total electron nd 2+ 2 + 2 = ± p n 2 col.: |Vus | | V cs | | V ts | 1.098 0.074 . (6.58) =1 − − 2 endpoint energy is W0 2 ( mn ( mmm p )/) n . The The sum of the three angles of the so-called unitarity solution Eq. (6.59) then agrees with the result in α++= β γ ± o triangle (183 25) is also consistent with Appendix 3 of Wilkinson (1982). the SM, as is the Jarlskog invariant To take into account g2(0) and f3(0) one must apply =+0.19 × − 5 β J (3.05−0.11 ) 10 , see Eq. (3.8). the following corrections to the -asymmetry A From a model-independent analysis by Cirigliano ε=−(2 / 3M )[(1 + 2 λ ) W +− ( λ 1) Wg ] (0) , (6.62) et al . (2010a) follows that precision electroweak scc 0 2 ε′ = λ +2 + 2 constraints alone would allow violations of unitarity scc (2/M )[( WmWg0 / ) 2 (0) ( mWf / ) 3 (0)] , as large as ∆ = 0.01 , while the bound (6.63) ∆=− ± × −4 ( 1 6) 10 from Eqs. (6.54) and (6.55) also from Holstein (1974), in agreement with Gardner constrains contributions to the low-energy effective and Zhang (2001), who also present the analytic Lagrangian to Λ >11 TeV at 90% C.L. solution for the electron-neutrino correlation a. Analytic solutions for the proton asymmetry C are given by Sjue (2005, Erratum 2010). Numerical weak 4. *Induced terms magnetism corrections for various neutron decay correlation coefficients are tabulated in Glück (1998). For neutron decay there are no results yet on the If one wants to measure both weak magnetism f2 and small induced terms f2(0) for weak magnetism second-class amplitude g2 without assuming the 63 validity of CVC, then it is best to measure the precise V− A . We then discuss whether the universe was spectral shapes for two different correlation created with a left-right asymmetry from the coefficients. beginning, or whether parity violation arose as a For nuclei, the weak magnetism prediction is spontaneous symmetry breaking in the course of one confirmed on the ±15% level, see Minamisono et al . of the many phase and other transitions of the (1998) and references therein. In particle physics, the universe. We investigate only the occurence of one −→ − ν type of such exotic couplings at a time, and derive best test of weak magnetism is from Σ ne e decay, also with error ±15% . For the proton, weak limits for the respective coupling parameters, which is reasonable as long as no such exotic signals are in magnetism is tested at q2= 0.1 (GeV/c) 2 to sight. ± about 30% (Mueller et al ., 1997, see also Acha We base our studies on the PDG 2010 data, recent et al ., 2007). For a feature of the SM as basic as CVC shifts in then neutron lifetime τ and β -asymmetry A this precision is not very impressive and should be n are only a minor problem here, because the limits improved whenever possible. derived depend mainly on the antineutrino and proton →+ + − +ν + γ Radiative neutron decay n p e e was asymmetries B and C. measured at NCNR-NIST by Nico et al . (2006), see also Cooper et al . (2010), and Beck et al . (2002) for an earlier search. The experiment used the NIST in- 1. Is the electroweak interaction purely V− A ? beam lifetime apparatus described above, complemented with a γ detector. The bremsstrahlung The most general Lorentz invariant weak probability was measured in triple γ e− p + -coincidence Hamiltonian can be constructed from five types of at a signal-to-noise ratio of about unity, in the scalar products, γ -energy interval from 15 to 313 keV. The result was Scalar×Scalar (S), a branching ratio Vector×Vector (V), Tensor×Tensor (T), = ± × −3 BR (3.09 0.32) 10 , Axial vector×Axial vector (A), (radiative neutron decay), (6.64) Pseudoscalar×Pseudoscalar (P), in agreement with the theoretical value with 10 complex coupling constants. Neutron decay is BR= 2.85 × 10 −3 . not sensitive to pseudoscalar P amplitudes, due to its In particle physics, the most well-studied such low energy release, so we limit discussion to possible radiative decay is radiative kaon decay scalar S and tensor T amplitudes. In fact, no one → π± ∓ ν γ = knows why nature seems to have chosen the V− A KL e e with branching ratio BR (9.35 − − variety, and, contrary to common belief, tests ±0.15) × 10 3 for an expected BR= (9.6 ± 0.1) × 10 3 . excluding S and T amplitudes are not very stringent. + + The radiative decay of the pion π→ µνγµ occurs The effect of S-T admixtures on neutron decay with BR= (2.0 ± 0.25) × 10 −4 for an expected correlation coefficients due to supersymmetry were investigated by Profumo et al . (2007b), under the = × −4 µ−→ − ννγ BR 2.5 10 , and that of the muon e e µ condition that simultaneously the tight bounds from with BR= (0.014 ± 0.004) for an expected the SM-forbidden lepton-flavor violating µ→ e γ BR= 0.013 . decay (B.R.< 1.2 × 10−11 ) and the results from the muon g − 2 measurements (precision of g: 6× 10 −10 )

are obeyed. These S-T admixtures generally are D. Symmetry tests beyond the SM ≤ −4 α π ≈ −3 10 , but can reach order /2 10 in the limit Precision measurements of SM parameters are at of maximal left-right mixing. the same time tests for physics beyond the SM. In the From a global fit to all available nuclear and present section, we shall discuss limits on amplitudes neutron data, Severijns et al. (2006) obtain the limits beyond the SM derived from neutron decay, and shall g/ g < 0.0026 and g/ g < 0.0066 compare them with limits derived from other nuclear S V T A or particle reactions. We shall not do this in a (left-handed, 95% C.L., all data 2006) (6.65) systematic way, as this was done recently by Severijns for a possible left-handed S-T sector. For a right- et al . (2006) for beta decay in general, but we limit the handed S-T sector the limits are less stringent, discussion to some basic issues. < < We first turn to the question whether the basic gS/ g V 0.067 and gT/ g A 0.081 symmetry of the electroweak interaction is really (right-handed, 95% C.L., all data 2006). (6.66)

64 There should be a strong interest for better constraints on gS and gT, in particular, in the right- handed S and T sectors, Eq. (6.66). However, as it sometimes happens, the lack of rapid progress over past decades led to a of interest in the problem, the caravan moved on although the problem is still with us. a. Left-handed S or T coupling : One mode of access to scalar and tensor amplitudes is given by the so- called Fierz interference term b in β -decay. The Fierz term enters unpolarized energy spectra as Γ ∝ + d (1 (me / Wb e )) d W e , with total electron energy = + We E e m e . It describes gS-gV and gT-gA interference, i.e., b is linear in both gS and gT. For a purely left-handed weak interaction,

S+ 3λ 2 T g g b = 2 , with S = S , T = T . (6.67) 1+ 3 λ 2 g g FIG. 25. Exclusion plot for the SM-forbidden Fierz V A interference terms b, derived from electron-energy As there are few neutron investigations on the Fierz dependence of the neutron β -asymmetry A(Ee) term, we shall treat this in more detail. It is difficult to with 1σ (dark gray), 2σ (gray), and 3σ contours determine b from the shape of unpolarized β -spectra, (light gray) contours. The Standard Model prediction because experimental spectra are deformed by the is b = 0 . The vertical line indicates λ from Eq. (6.42) differential nonlinearity of the apparatus, usually ten times larger than the typically one percent integral nonlinearity of the calibration curve of a β -detector. Often one does not have the full energy The Fierz b coefficient necessarily also enters all dependence Xm(Ee) at hand, but only the published measured decay correlations Xm in the form values of the correlation coefficients X. In this case the parameters b and λ can be derived from two or X X ′ = m . (6.68) more such coefficients, see Glück et al . (1995). One m + 1(me / W e ) b can take for instance the measured values a and A and apply the appropriate energy averages m〈 W −1 〉 in This can be seen, for instance, in Eq. (6.34) for the e e Eq. (6.68), for numerical values see Appendix 6 of measured neutron β -asymmetry Am, where a b-term Wilkinson (1982). However, a and A are very in the count rates would cancel in the numerator and λ add up in the denominator. Limits on b can then be collinear in b and , too, and they constrain b only − < < σ obtained from a fit to the measured β -asymmetry to 0.3b 0.7 at 2 . Nor does inclusion of the neutron lifetime τ , with |V | 2 as additional free spectrum A′ ( E ) , which, to our knowledge, has not n ud m e parameter, give much better constraints. been done before. This fit should be less sensitive to Things are better if one includes the neutrino instrumental nonlinearities than the unpolarized asymmetry B and the proton asymmetry C, which are spectrum, as only ratios of count rates are involved. more “orthogonal” with respect to the Fierz terms than For demonstration, we have made a fit to some λ are A and a. However, the neutrino asymmetry B older Am(Ee) data of PERKEO, with and the Fierz ′ term b as free parameters, which gave contains a second Fierz term b in the numerator, + ′ < B( me / Wb e ) |b | 0.19 (neutron, 95% C.L., from Am(Ee)), (6.69) B′ = , (6.70) 1(+ m / W ) b see the exclusion plot Fig. 25. Such evaluations e e should be done systematically for all available with correlation spectra simultaneously, which may avoid 2λT− S − T the strong correlation between b and λ seen in b′ = 2λ (6.71) + λ 2 Fig. 25. 1 3 from Jackson et al . (1957) , with S and T defined as for Eq. (6.67). Therefore a third parameter is needed if one wants to extract the two basic quantities gS and gT

65 in Eq. (6.67), for instance the newly measured proton asymmetry C, using Glück (1996). A fit to the coefficients A, B, C, with the χ 2 minimum projected onto the b− b ′ plane, gives −0.3

= − ± F 2 2 2 b 0.0022 0.0026 (nuclear t0→ 0 ), (6.75) ε= ε′ = + λ For the electron helicity G, ST ST S3 T . In while b′ is not accesssible from nuclear data. both cases, imaginary Coulomb corrections linear in S Making use of Eqs. like (6.19) for further neutron and T have been neglected. decay correlations, one can isolate additional left- Limits on right-handed gS and gT from neutron decay experiments are given in Schumann (2007) as handed amplitudes linear in gS or gT, and finds relations like < < |gS / g V | 0.15 , |gT / g A | 0.10 m S S− λ 2 T (95% C.L., neutron right-handed). (6.78) H+e a = − 2 , K+ A = 2 . (6.76) + λ 2 + λ 2 We 1 3 1 3 In the purely leptonic sector, bounds on S and T couplings from muon decay are of similar quality as the neutron bounds, with < ≡ |gS / g V | 0.55 , |gT / g V |0 (90% C.L., muon, left-handed), (6.79) and

66 < < depend differently on these parameters. For the β - |gS / g V | 0.074 , |gT / g V | 0.021 (90% C.L., muon, right-handed), (6.80) asymmetry A, one finds ε= λδζδζ − ++ λδζ2 + 2 see Fetscher and Gerber (2009). rhc 2( )( )2( ) , ε′ =−( δζ )3(2 + λδζ 2 + ) , (6.82) c. Bound β -decay : Neutron decay into a hydrogen rhc atom is one of the two rare allowed decay modes and similarly for other correlation coefficients, see, for discussed in Sec. A.5.b above. In principle, one can instance Appendix C.3. of Severijns et al . (2006). also derive limits on scalar and tensor amplitudes from bound β -decay. The Wi from Eq. (6.25) are linearly sensitive to scalar S and tensor T amplitudes − + in the combinations gS g T , and gS3 g T , see Faber et al . (2009).

2. Was the universe left-handed from the beginning?

Today, many models beyond the SM start with a left-right symmetric universe, and the left-handedness of the electroweak interaction arises as an “emergent property” due to a spontaneous symmetry breaking in the course of a phase transitions of the vacuum in the early universe. In the simplest case, the gauge group × SU (2) L of the SM is replaced by SU(2)L SU (2) R , which then is spontaneously broken. This should lead to a mass splitting of the corresponding , FIG. 27. Exclusion plot for the parameters of the namely, the left-handed W1 and the right-handed W2, manifest left-right symmetric model (beyond the SM), δ = 2 with masses mL, mR, and mass ratio squared namely, the mass ratio squared (mR / m L ) of left δ =2 << (mL / m R ) 1 . and right-handed W bosons, and their relative If, by additional symmetry breaking, the mass phase ζ . The exclusion contours, defined as in Fig. eigenstates W1 and W2 do not coincide with the 25, are based on the neutron decay correlation electroweak eigenstates WL and WR, then coefficients A, B, and C. The Standard Model prediction is δ= ζ = 0 . W= Wcosζ + W sin ζ , L 1 2 W= − Wsinζ + W cos ζ , (6.81) R 1 2 with left-right mixing angle ζ (where we omit an overall phase factor). When right-handed currents are admitted in the weak Hamiltonian, then neutron decay parameters depend not only on | Vud | and λ , but also on δ and ζ , each parameter bilinear in δ and ζ , but each in a different way. For references, see Severijns et al . (2006), Glück et al . (1995), and Dubbers et al . (1990) for earlier work. a. *Limits on right-handed currents from neutron decay : We use the framework of so-called manifest left-right symmetry, Beg et al. (1977) and Holstein and Treiman (1977). Right-handed V+ A currents −γ +γ there are obtained simply by replacing 5 by 5 in the formulae for the left-handed V− A currents. We δ= 2 ζ λ derive (mL / m R ), , and from three measurements, for instance from A, B, C, which each FIG. 28. Exclusion plot as in the preceding figure, but F F based on the coefficients A, B, C, and tn / t 0→ 0 .

67 Figure 27 shows the result of such a fit, with We see that the limits on the mass of the right- λ δ ζ , , and as free parameters, projected onto handed WR derived from low-energy experiments are the (δ , ζ ) plane, using the PDG 2010 values of A, B, somewhat lower but of the same order as present limits from high-energy experiments. The limits from and C (for B, as above, only the value of Serebrov both sources complement each other, for instance the et al . (1998) was used). If in addition we use the ratio neutron tests the hadronic sector, while the muon tests of neutron to nuclear superallowed Ft values, the leptonic sector. The high-energy limits for direct Ft/ F t from Eqs. (6.49) and (6.51), we obtain the n 0→ 0 production are valid for right-handed neutrinos ν exclusion plot of Fig. 28, both in good agreement with R δ= ζ = with masses up to the mass of WR, whereas low- the SM expectation 0 . The limits on the ν energy limits require the R mass to be below the mass mR of the right-handed WR and the mixing β angle ζ are derived as relevant -endpoint energy (the sum of the masses of the ordinary left-handed neutrinos being at most of > −<<ζ mR 250 GeV, 0.23 0.06 order eV). (95% C.L., neutron). (6.83) d. *Right-handed currents from bound β -decay : In The neutron limits on the right-handed neutron decay into a hydrogen atom, a unique parameters δ and ζ , impressive as they are, have not experiment on right-handed currents may become improved during past years. Again, the picture looks feasible, see Byrne (2001) and references therein. The somewhat better if one uses only the latest results population of the hyperfine level F=1, M = − 1 , is for τ and λ , but this we leave to future evaluations. F n predicted to be W = 0 , Eq. (6.25), for any left-handed 4 b. Limits on right-handed currents from nuclear interaction, simply from angular momentum ≠ physics : Severijns et al. (2006) reviewed searches for conservation. If a population W4 0 is detected, then right-handed currents, which are a traditional topic this would be a unique sign for the existence of right- also in nuclear physics. As Ft'= F t / (1 − 2ζ ) is linear handed currents, so to speak a “ g − 2 type” in ζ , tight limits are obtained for ζ from experiment for V+ A amplitudes. This, however, + + > = nuclear 0→ 0 transitions, requires that optical transitions from n 2 to n 2 atomic states are suppressed. −0.0006 ≤ζ ≤ 0.0018 (90% C.L., nuclear Ft ). Let us estimate the sensitivity of this method. In a (6.84) left-right symmetric V+ A model as discussed above, δ = 2 ζ From the longitudinal polarization of β -particles with left-right mass ratio (WL / W R ) and phase , emitted from unpolarized and polarized nuclei, limits one expects a population –4 =1 λδλζ ++−2 + λ 2 δ of order several times 10 are derived for the bilinear W4 2[( 1) ( 1) ] / (1 3 ) , where 2 forms δζ and (δ+ ζ ) , respectively, from which a and ζ are linked to the antineutrino helicity Hν as 22 2 2 lower limit on the mass of a right-handed W is 1−=−+Hν 2[(δζ ) 3( λδζ + )]/(13) + λ . obtained of comparable size as that of the neutron, 2 2 2 2 For either ζ<< δ or ζ>> δ , one finds Eq. (6.83) + λ 2 > (1 ) −3 mR 320 GeV W=(1 −=×− H ) 3.1 10 (1 H ) , (6.88) 4 + λ 2 ν ν (90% C.L., nuclear β -helicity), (6.85) 4(1 3 ) for details see Severijns et al. (2006), and references hence, the sensitivity to deviations of neutrino helicity therein. from unity is weak, but, hopefully, background-free. c. Limits on right-handed currents from high energy physics : We can compare these results with those 3. Time reversal invariance from direct high-energy searches for WR. The limit for In Sec. III, we extensively discussed CP violation. the mass of WR is Under CPT invariance, CP violation goes hand in > mR 715 GeV (90% C.L., high energy). (6.86) hand with T violation, and we need not discuss this topic again in any depth. Limits on time reversal The limits derived from muon decay are invariance exist both from nuclear and particle >ζ < mR 549 GeV, 0.022 physics, but none better than the neutron limit −4 (68% and 90% C.L., muon). (6.87) D =−( 4 ± 6) × 10 from Eq. (6.39).

68 In the SM, merely a D <10 −12 would enter via the VII. THE ROLE OF THE NEUTRON IN CKM phase ϕ , hence D is sensitive to effects beyond ELEMENT FORMATION the SM. could induce a D of size up to the present experimental limit. Left-right symmetry The next epoch in the history of the universe where could produce a D one order of magnitude, and neutrons play a dominant role is reached when the supersymmetry a D two orders of magnitude below temperature has dropped to about 1 MeV. In Sec. A, the present bound, as cited in Lising et al . (2000). we discuss the role of neutrons in the production of Final state effects begin only two orders of magnitude the first light elements during the first few minutes of below the present neutron bound. the universe, a process called primordial The T-violating R-coefficient for the neutron is nucleosynthesis. We adopt a simplified standard − description of the process, for a more elegant R =(8 ± 16) × 10 3 , from Eq. (6.41), while for Z0- derivation see the book by Kolb and Turner. In the decay it is subsequent Sec. B, the formation of the heavier −0.022

=− ± × −3 R ( 1.7 2.5) 10 (muon), (6.90) A. Primordial nucleosynthesis from Abe et al . (2004). The best R-coefficient from nuclear β -decay is that from the β -decay of 1. The neutron density in the first three minutes polarized 8Li, Up to a time t ~ 1 s , corresponding to a R =−( 0.9 ± 2.2) × 10 −3 ( 8Li ), (6.91) temperature kT ∼1 MeV , the expanding universe was in thermal equilibrium. It was mainly filled with from Huber et al . (2003). photons, three kinds of neutrinos/antineutrinos, For most theoretical models, EDM experiments are −10 electrons and , and a tiny ~ 10 fraction of more sensitive to CP and T reversal violations than baryons from CP violating baryogenesis in the very are nuclear and neutron correlation experiments. The early universe, frozen out as protons and neutrons neutron R-coefficient is discussed by Yamanaka et al . during the quark-gluon phase transition (2010) in effective field theory within the framework × −5 < of the MSSM. They find interesting relations (their at t ~3 10 s . At t 1 sec , protons and neutrons Table 2) between the expected size of R and the sizes were coupled to each other by the weak neutrino of other T violating neutron decay correlation reactions coefficients L, S, U, and V, as well as corrections to n+ e+ ↔ p + ν , B, H, K, N , Q, and W. e n+ν ↔ p + e − (7.1) Other tests beyond the Standard Model are e possible in neutron decay, for instance on accompanied by neutron decay exchange, exotic fermions or sterile neutrinos, and we → +− + ν refer to the theory reviews cited above. Thus, neutron n p e e . (7.2) β -decay provides a wealth of weak interaction These processes involve all members d, u, e, ν of parameters that are very useful to investigate e extensions of the Standard Model up to high energy the first generation of particles. One can show that scales. dark matter and dark energy did not influence primordial nucleosynthesis, and that the hot photon environment did not change the phase space available for free-neutron decay significantly. The neutron mass exceeds the proton mass by 2= − 2 = mc( mp m n ) c 1.293 MeV . Therefore the reactions in Eq. (7.1) are exothermic when read from left to right, and endothermic in the opposite sense. Free neutron decay, Eq. (7.2), proceeded only to the right because three-particle encounters are unlikely at this stage. These reactions kept neutrons and protons in chemical equilibrium. Their number ratio (≈ mass ratio) at temperature T is approximately given by the Boltzmann law n / p= exp( − m/ kT ) ,

69 = where for simplicity we use the same symbol for the where Tf T( t f ) is the freeze-out temperature. As we number of protons and neutrons as for the particles shall see below, this ratio is n/ p ≈ 1/6 , with themselves. ≈ The dashed line in Fig. 29 gives the neutron mass kT f 0.7 MeV . At this time and temperature (arrow fraction n/ ( n+ p ) from the Boltzmann law as a A in Fig. 29), the expanding system fell out of thermal function of temperature and time, see Eq . (2.4). On equilibrium because nucleons and neutrinos did not the right of this figure, for t ≪1 s , n≈ p and the find each other anymore, and the universe became transparent to neutrinos. neutron mass fraction goes to one half. If, for t >1 s , The strong-interaction cross sections are much neutrons and protons were kept in equilibrium, the larger than the weak neutrino cross sections in neutrons would disappear with falling temperature Eq. (7.1), so the nucleons in the expanding universe within a few seconds, and Boltzmann’s law would still had sufficient time to find each other and fuse. predict that the universe would consist entirely of However, nuclear fusion of protons and neutrons only protons, and no element other than hydrogen would started about three minutes after freeze-out, because populate the universe today. the first nuclei formed in the hot universe were

unstable in the universe’s strong photon field. (Nucleosynthesis was delayed mainly by the high entropy of the thermal γ radiation, due to radiation dominance, and less by the low binding energy of the deuteron.) This suppression of nucleosynthesis was effective until the temperature had dropped to below 0.1 MeV (arrow B in Fig. 29). By that time, the neutron to proton ratio Eq. (7.3) had dropped to a new value n′/ p ′ , with n+ p = n′ + p ′ , due to free-neutron β − -decay and to continuing neutrino interactions, which latter have a phase space available that is much larger than for neutron decay. The time interval ≈ between the arrows A and B in Fig. 29 is td 150 s (Weinberg’s famous “first three minutes”). The process of 4He production stopped shortly after when almost all neutrons had ended up bound in 4He. We can, in a rough approximation, determine the early neutron-to-proton fraction Eq. (7.3), and with it

the freeze-out temperature Tf, directly from the FIG. 29. Neutron mass fraction n/ ( n+ p ) in the early 4 observed primordial He mass fraction Yp. We just universe, as a function of temperature T and time t have to take into account that essentially all neutrons (upper scale) after the big bang. For the dashed line it finally wound up in 4He nuclei (which is due to the is assumed that neutrons and protons are permanently large binding energy of 4He). As about half of the 4He in thermal equilibrium, with n/p following mass is from neutrons, and all neutrons are bound = − Boltzmann’s law n / p exp( m/ kT ) . For the dash- in 4He, the 4He mass fraction is twice the neutron dotted line it is assumed that neutrons decouple from =′ ′ + ′ = mass fraction, Yp 2/( nn p ) at t 150 s . Using the protons near arrow A and decay freely. ′ ≈ − τ Nucleosynthesis sets in near arrow B as soon as Eq. (7.3), and the approximation n nexp( t d / n ) , deuteron break-up by photons ceases. The solid line is this gives from a full network calculation with all relevant He-4 mass 2n′ 2 nuclear reactions. From Coc (2009). Y = ≈ = p ′′+ + ′′ total massnp 1 pn / 2exp(−t /τ ) ≈ d n . (7.4) Fortunately, the weak neutrino cross sections in + 1 exp( m/ kT f ) Eq. (7.1) are small enough that, in the expanding and 4 cooling universe, these reactions were stopped early The He mass fraction Yp can be observed in ≈ regions of the universe with very low metallicity, i.e., enough at freeze-out time t f 2 s when still a sizable very low density of elements beyond A = 4 , which number of neutrons was present, with points to a very old age of the population observed. = − The measured values then are extrapolated to zero n/ p= exp( m/ kT f ) (7.3) t t f metallicity. (Astrophysicists call metals what nuclear 70 physicists call heavy ions – everything beyond Γ ≈ σνn ν c . The neutrino number density is ≈ 3 helium.) The observed value is Yp 25% . At the =ρ ∝ nνr / E ν T from Eq. (2.5), and we have set beginning of nucleosynthesis the neutron to proton E∝ T , dropping all thermal averages and ′ ′ = − ≈ ν ratio becomes npY/p /(2 Y p )1/7 , from writing Eν for 〈Eν 〉 . The neutrino-nucleon cross Eq. (7.4), from which one can calculate back section σ is too small to be measured precisely in the n/ p ≈ 1/6 at the time of decoupling of the weak ν laboratory. Instead we must calculate σ∝1/ τ from interaction, which gives kT ≈ 0.7 MeV from ν n f the measured neutron lifetime τ , which is based on Eq. (7.3) that we have mentioned above. n the fact that the processes in Eqs. (7.1) and (7.2) all

have the same . Furthermore, for

E>> m c 2 , σ is known to grow with the square of 2. Nuclear reactions in the expanding universe ν e ν σ∝ 2 τ neutrino energy Eν , so ν T / n , and altogether Altogether, the four main isotopes formed in Γ ∝ T 5 /τ . primordial nucleosynthesis via various fusion n The expansion rate H on the right-hand side of reactions are 2H, 3He, 4He, and 7Li. From the known H ∝ 1/2 2 nuclear reaction data for these fusion processes one Eq. (7.5) is, from Eqs. (2.1) and (2.5), N T . 2 3 The freeze-out condition (7.5) then gives can calculate the mass ratios H/H, He/H, Yp, 7 5τ ∝ =H ∝ 1/2 2 and Li/H. The path to heavier nuclides is blocked by Tf/ nΓ N T f . At freeze-out time the the absence of stable A = 5 and A = 8 nuclei and by number of relativistic degrees of freedom (dof) is the entropy-driven time delay in nucleosynthesis =1 + N4 (22 7 N ν ) . This contains 2 dof for the 2 mentioned above (at later time temperature is lower helicity states of the photon, 4× 7 dof for the 2 and Coulomb barriers are more difficult to overcome). 8 + – × 7 For numerical and observational results on helicity states of e and e , and 2Nν 8 dof for the 1 primordial abundances, we use the recently published helicity state of Nν types of neutrinos and analyses by Iocco et al . (2009) and Cyburt et al . antineutrinos. The relativistic weight factors employed (2008). A simplified but still rather long analytical 7 are 1 for bosons and 8 for fermions. This gives derivation of the nucleosynthesis process, precise to ≈ × + 1/6τ 1/3 several percent, was given by Bernstein et al . (1989). Tf const(22 7 N ν ) n , (7.6) In the following, we shall present an even simpler description that will help us understanding the main where const is some known combination of natural 4 parameter dependences of the He abundance Yp, with constants and masses. This equation is not meant to main emphasis on its dependence on the neutron provide us with precise values of Tf and Yp. Instead, lifetime. we use it to investigate the sensitivity of Yp to the τ Let Γ be the rate, at which the reactions Eq. (7.1) parameters n and Nν . occur. As it will turn out, the reaction rate Γ(t ) decreases with time much faster than the expansion rate, given by the Hubble function H(t ) . As a rule of 3. Neutron lifetime, particle families, and baryon thumb, which works surprisingly well, the system density of the universe = H falls out of equilibrium at crossover Γ()tf () t f . 4 To find the sensitivity of the He abundance Yp to − (At crossover, the time Γ 1 between successive the value of the neutron lifetime, we form H−1 ∂ ∂ τ τ reactions equals the momentary age t ~ of the (Ypp / Y )/( nn / ) from Eq. (7.4), with Tf from universe, and less than one reaction is expected Eq. (7.6). At time t =150 s , with τ = 882 s and within t.) At the time of crossover, the freeze-out d n ≈ = Yp 25% , this sensitivity becomes temperature is Tf T( t f ) , so we derive Tf from the ∆ =+ + ∆ τ τ condition Ypp/ Y (0.50 0.17) nn / , where the first number +0.50 is mainly due to the dependence of the Γ()T= H () T . (7.5) f f neutrino cross section on the neutron lifetime, and the To calculate the reaction rate Γ in the expanding second number +0.17 is due to free-neutron decay up universe on the left-hand side, we must carefully to time td. This result is just several percent below the integrate the appropriate rate equations over time and elaborate results of Iocco et al . (2009) and Bernstein et al . (1989), who both find over neutrino energy Eν . Instead, we make the = + τ τ simplified ansatz of a momentary total neutrino Ypp/ Y 0.72 nn/ . (7.7) reaction rate proportional to the local neutron density

71 The plus sign in Eq. (7.7) is as expected, because a baryon-to-photon ratio obtained from the cosmic longer neutron lifetime gives a smaller neutrino cross microwave background satellite data (WMAP 5-year section and an earlier freeze-out time with more result, Dunkley et al ., 2009) is neutrons and hence more helium. Furthermore, in the η = ± × −10 ∆η η = (6.225 0.170) 10 , or ( / )obs 0.027 . first three minutes free-neutron decay is slower for From Eq. (7.9) we then find ∆Y = ± 0.00025 due to longer τ and leads to more helium. p n the error in the baryon density of the universe. From Eq. (7.7) we can derive how well we need to To sum up, big bang nucleosythesis is a parameter- know the neutron lifetime, when we require that the free theory because the three parameters τ , N , error due to the uncertainty in τ is small compared to n ν n and η are all known from measurement. The error in the observational error of Yp. As discussed in Sec. VI.B.1, measurements of the neutron lifetime the neutron lifetime dominates the error in the over past years differ by almost one percent. Hence, in calculated Yp, and may require a shift of ∆ ≈ − the extreme case that the PDG 2010 average would Yp 0.0015 . Iocco et al . adopted the observed need to be shifted by ∆τ/ τ = − 0.008 , this would = ± n n value Yp obs 0.250 0.003 , and the reanalysis by = ± shift the calculated value Yp 0.2480 0.0003 , as Cyburt et al . (2008) gives a very similar value, = − Y =0.252 ± 0.003 (while earlier authors prefer adopted by Iocco et al . (2009), by Yp 0.0015 . p obs From Eq. (7.6), we also find the sensitivity to wider error margins). Hence, the theoretical changes in the number Nν of light species. Taking the uncertainty from the neutron lifetime may be about half the observational error of Y . This is not dramatic, derivative of Y in Eq. (7.6) with respect to N we p p ν but the field keeps progressing, and if we want the ∆ =+ ∆ = obtain Yp 0.010 N ν for Nν 3 neutrino species errors due to nuclear inputs to remain negligible, then ≈ 1 the problems with the neutron lifetime should be and YP 4 . Bernstein et al . find the same value +0.010, to which they add +0.004 to account for fixed. a small N -dependence of the deuterium bottleneck, ν ∆ =+ ∆ so Yp 0.014 N ν , or 4. Light-element abundances

∆YY/ =+ 0.17 ∆ NN / . (7.8) p p ν ν There is also good agreement between the Again, the sign is positive because energy density, predicted abundance of deuterium 2 = ± × −5 Eq. (2.5), increases with the number of light ( H/H)calc (2.53 0.12) 10 , calculated with = η species Nν , which speeds up expansion, Eq. (2.1), Nν 3 and the from WMAP, and the observed 2 = ± × −5 and leads to earlier freeze-out at higher neutron abundance ( H/H)obs (2.87 0.22) 10 , see, for number. instance, Iocco et al . (2009). For 3He the calculated For the number of light neutrino species we use the 3 = ± × −5 abundance is ( He/H)calc (1.02 0.04) 10 , while value Nν = 3 , with ∆Nν/ N ν = 0.0027 , when we 0 only an upper bound can be given for the observed take Nν and Nν from the Z -resonance, which 3< × − 5 abundance ( He/H)obs 1 10 . The agreement is not gives N =2.9840 ± 0.0082 . From Eq. (7.8) we then − ν as good for 7Li, with (7 Li/H)= (4.7 ± 0.5) × 10 10 find ∆Y = ± 0.0001 . calc p 7 = ± × −10 vs. ( Li/H)obs (1.9 1.3) 10 , which is possibly η = 7 As already mentioned, the ratio nb / n γ of due to Li fusion and consumptions in stars. The baryon (i.e., nucleon) to photon density, Eq. (3.1), is update by Cyburt et al . (2008) comes, within errors, to another important parameter because a high photon the same results. The overall concordance of the data density delays nucleosynthesis. Bernstein et al . find is astonishingly good, and constitutes a strong pillar of ∆ =+ ∆ η η ≈ 1 ∆η η Yp 0.009 / at YP 4 , for ≪ , in the standard big bang model. agreement with Iocco et al. , who find One can also use the number of neutrino species Nν and/or the baryon content η as free ∆ =+ ∆ η η Yp/ Y p 0.039 / (7.9) parameters in primordial nucleosynthesis calculations. When both N and η are free parameters, from at η =6.22 × 10 −10 . (The relation η = 273 h2 can be ν b Fig. 12 of Iocco et al. one finds used to compare results from different sources). The η = = ± η = ± × −10 sign in Eq. (7.9) is positive because a larger nb / n γ Nν 3.18 0.22 and (5.75 0.55) 10 means a lower number of photons, hence less delay in (from nucleosynthesis), (7.10) nucleosynthesis and more helium formation. The compatible with Nν = 3 and the WMAP values

72 −10 2 = Nν =3.18 ± 0.44 and η =(6.22 ± 0.17) × 10 outgoing deuteron H is known to have spin j 1 . (from microwave background). (7.11) Therefore p-p fusion is a |∆j | = 1 Gamow-Teller 2 The two independent values for η can be transition, whose cross section is proportional to g A , combined to η =(6.08 ± 0.14) × 10 −10 as used in where gA is the axial vector weak-coupling constant as defined in Sec.VI.A.3. Eq. (3.1). These studies are interesting because with The most frequently studied solar neutrino process = Eqs. (7.10) and (7.11) we compare Nν at time t 1 s β + 8→ 8 ++ + ν is the -decay B Be e e , whose = × 10 with Nν measured today at t 1.4 10 y , and we continuous neutrino spectrum reaches up to 14 MeV. compare η and N at time t =1 s with η at a 8 ν The B neutrino flux Φ8 is the dominant component time t =3.7 × 105 y when the universe became in the original Homestake neutrino experiment transparent to photons, and find that the numbers have (detection threshold 0.8 MeV) and is the only not changed considerably in between. component seen in the Kamiokande experiments 8 From these data on primordial nucleosynthesis (threshold 4.5 MeV). The B neutrino flux decreases ∝ −5.2 many stringent and often unique constraints on new with gA as Φ8 g A , see for instance Eq. (6c) in physics can be derived, for instance on extra Adelberger et al . (1998), so dimensions, on the time dependence of fundamental ∆ =− ∆ constants, extra light species, and others, as is also Φ8/ Φ 8 5.2 gA / g A (7.12) discussed by Iocco et al. (2009), and references (in a coupled multi-component reaction system, it is therein. To sum up, the early universe is a rich not unusual that the partial flux of one reaction laboratory for the study of the fundamental component decreases with increasing coupling). interactions with a continuing need for precise nuclear The cross sections of the weak reactions in the sun and neutron data. are too small to be measured directly, so again g=( − 1.2734 ± 0.0019) g must be taken from free- A V B. Stellar nucleosynthesis neutron decay data, Eq. (6.42). Neutrino detection in solar neutrino experiments also needs input from More than half a billion years later, the first experimental neutron decay data, in particular, when it heavier elements were created in protostars: The first relies on the inverse neutron decay reaction +ν → + + conglomerates of primordial hydrogen and helium pe n e . Neutrino fluxes are both calculated atoms as formed under the catalytic action of dark and measured with typically 10% precision, therefore matter. As we shall see, in the ensuing stellar the present quality of neutron decay data is quite processes of element formation, neutron reactions are sufficient, in spite of the strong sensitivity of Φ8 to dominant, too. To understand these processes of the error of gA. stellar nucleosynthesis more and better neutron- After the p-p chain and the CNO cycle, other nuclear data are needed. exothermic fusion reactions take place, for instance 12C+ 12 C → 20 Ne + 4 He , followed by successive α - captures along the N= Z diagonal of the nuclear 56 1. Stellar nucleosynthesis and the s-process chart, up to the doubly magic isotope Ni, which 56 decays back to stable Fe within several days via two Under gravitational contraction, the heats up β + until nuclear fusion sets in. This nuclear burning, successive -transitions. 56 which prevents further contraction of the star, starts When Fe is reached, nuclear burning stops 56 with the breeding of 4He via the p-p chain in the sun, because Fe has the highest mean binding energy per = or via the CNO-cycle in heavier stars. nucleon of all isotopes (EB / A 8.6 MeV) , and hence In past years, nuclear processes within the sun for 56 Fe both fission and fusion processes are have received much attention, in particular, in the endothermic. Iron, with an atomic ratio Fe/H≈ 10 −4 , context of the so-called solar-neutrino problem: Only therefore is one of the most abundant heavier species one third to one half of the neutrinos generated by in the universe, see the insert of Fig. 30. Beyond iron, weak-interaction processes in the sun arrive as such almost all elements up to the heaviest actinides on earth, the others are converted into other neutrino originate in stellar processes by successive neutron flavors by so-called neutrino-oscillation processes. captures, followed by β − -decays back towards the The p-p chain starts with the weak-interaction stable valley of isotopes. The capture of neutrons is fusion reaction p+→ p2 H + e + + ν . To come close e the preferred reaction because neutrons see no to each other, the Pauli principle requires the Coulomb barrier. So here, too, neutrons play a incoming protons to have antiparallel spins, while the 73 decisive role, and a great amount of input data is 2. Explosive nucleosynthesis and the r-process required from experimental neutron and nuclear physics. We give only a short overview of this rich The stellar s-process cannot reach all known stable and interesting field because many excellent reviews isotopes, see Fig. 30, and stops when bismuth is exist on the subject, for instance Langanke and reached, so other processes of nucleosynthesis must Martínez-Pinedo (2003), Kratz et al. (2007), and exist. The rapid or r-process takes place in regions of references therein. extremely high neutron densities, many orders of Several different nucleosynthesis processes must magnitude higher than for the s-process. In these high exist in order to explain the observed nuclear fluxes, multiple successive neutron captures take abundances beyond 56 Fe. The first such process is place, until extremely neutron rich isotopes with very called the slow or s-process, other processes will be short β − lifetimes are reached, up to 30 neutrons discussed in the following section. In the interior of away from the valley of stability. The path of this r- stars like the sun, late in their as red giants, process is not far away from the so-called neutron 83± − 21 − neutron fluxes of 10 cm s are created in (α ,n ) drip-line, beyond which nuclei are no longer bound reactions like 13C+ 4 He → 18 O + n , at temperatures of systems. tens to hundreds of keV, and over time scales of The r-process must be due to an explosive event thousands of years. As the neutron capture rate is and takes no longer than about one second. When it small compared to the typical β − -decay rate, the s- stops, the entire r-process path is populated with neutron-rich nuclides that decay back to the stable process usually leads to no more than one additional β − neutron away from the stable valley, as shown by the valley under emission, see the dashed diagonal dark zigzag line along the valley of stability in arrows in Fig. 30. The r-process path reaches up to the Fig. 30, which shows an excerpt of the nuclear chart. heaviest transuranium elements, which undergo To understand the s-process, we must know the spontaneous fission into two lighter nuclei located far neutron capture cross sections and their energy from stability in the intermediate nuclear mass region. dependence from kT = 0.3 keV up to several The precise location of the r-process path on the 100 keV. Such neutron cross sections are studied at nuclear chart depends on the prevailing neutron dedicated pulsed neutron sources, mostly based on the density and temperature. The vertical double line in (p , n ) reactions at small proton accelerators, or by Fig. 30 shows nuclear shell closure at magic neutron number N = 50 . At shell closure, neutron cross (γ ,n ) reactions with up to 100 MeV bremsstrahlung sections are very small, and hence neutron absorption γ 's produced at large linear electron accelerators. − rates become smaller than the corresponding β - Today the most prolific pulsed sources of fast decay rates. When this happens, no more neutrons are neutrons are spallation sources with annexed time-of- added, and the r-process proceeds along the isotone flight facilities. To obtain the dependence of the (thick vertical line in Fig. 30) towards the valley of capture cross section on neutron energy, prompt β − capture γ -rates are measured in dependence of the stability, up to the kink where lifetimes are again neutron’s time-of-flight to the probe, details are given long enough for neutron captures to resume (thick in the review by Käppeler et al . (2010). Besides the diagonal line). As isotopes crowd at shell closure neutron cross sections one must also know the (“waiting point”), a peak in the element abundance = properties and decay modes of low-lying excited appears near A 80 , marked “r” in the inset to nuclear levels. For instance, the isotope 176 Lu plays an Fig. 30. important role as a nuclear thermometer for the s- A similar abundance peak, marked “s” in the inset, process, and spectroscopic and lifetime measurements appears at somewhat higher Z where the = at LOHENGRIN at ILL were used to constrain magic N 50 isotone reaches the valley of stability, temperatures and neutron fluxes during the s-process where neutron shell closure affects the s-process. (Doll et al ., 1999). Such measurements then can be More such double peaks are seen near A =130 used to constrain stellar models. and A = 200 , due to the respective N = 82 and N =126 neutron shell closures. From these observed abundance peaks one can reconstruct geometrically the location of the waiting point regions on the nuclear chart. They are found to lie around 80 Zn, 130 Cd, and 195 Tm for magic neutron numbers N = 50, 82, and 126 , respectively.

74

FIG. 30. Pathways of nucleosynthesis in the nuclear chart Z vs. N. The slow or s-process proceeds near the valley of stability via neutron captures (horizontal lines) and subsequent β − -decays (diagonal lines). The rapid or r-process takes place far-off stability after multiple neutron captures (hatched area). s-only stable isotopes are marked s, r-only isotopes are marked r. The isolated open squares are proton-rich stable isotopes reached only by the so-called rp or p-processes. The inset shows relative element abundances in the solar system, for the small peaks marked r and s see text. Adapted from Käppeler et al . (2010).

To give an example, the isotope 80 Zn has 10 more data on isotopes far-off stability, which come from neutrons than the last stable zinc isotope. Baruah et al . radioactive beam and from neutron facilities. (2008) measured the masses of short-lived isotopes up At ILL, the fission product spectrometer to 81 Zn with a precision of m/ m ≈ 10 −8 , after LOHENGRIN gives both the fission yields (Bail deceleration of the neutron rich isotopes and capture et al ., 2008) as well as spectroscopic information on a in a Penning trap. From the measured masses of multitude of neutron-rich fission products. Bernas neighboring isotopes, the neutron separation energies et al . (1991) did an early study of the start of the 68 68,69 were derived. From this one calculates the boundaries r-process via the Fe and Co isotopes, which are T ≥109 K and n ≥1020 g/cm 3 for the neutron 9 and 10 neutrons away from stability. More recently, Rz ąca-Urban et al . (2007) studied the level scheme of temperature and density at the location of the r- 138 the iodine isotope I, 11 neutrons away from the process. stable isotope 127 I. For a review, see also Bernas Some nuclides can only be reached by the r- (2001). With all these data on neutron rich isotopes process, others only by the s-process, and many by one can hope that, in the course of time, one will be both, see Fig. 30. Some nuclides on the proton rich able to elucidate the nature of element formation and side of the stable valley are reached neither by the r- recycling processes going on in the universe. process nor by the s-process. Their existence must be To conclude, element-production in the universe, due either to a rapid proton capture, called an rp - both primordial and stellar, is full of interesting and process, or to (γ ,n ) or (γ , α ) photodisintegration in a often unexplored neutron physics, and requires a great very hot photon bath, called the p-process. amount of precision data from nuclear accelerators While the s-process is well understood, the r-, rp -, and from fast and slow-neutron sources. and p-processes are much less so. A possible site of these processes is in supernova explosions, where core collapse may cause extreme neutron fluxes in the VIII. CONCLUSIONS expanding envelope. An alternative site is in binary systems that consist of a normal star feeding matter Neutrons play an important role in the history of onto a companion neutron star. To clarify the nature the universe for two reasons. Firstly, there is a high of the r-process, responsible for about half the heavy neutron abundance in the universe, and neutrons play nuclei in the solar system, one needs better nuclear a dominant role in the formation of the chemical

75 elements, both shortly after the big bang, and during construction. In particular, transitions between the ongoing stellar evolution. The second, more subtle gravitational quantum levels were measured in a reason is that today’s experiments with cold and recent experiment. In addition, both EDM and ultracold neutrons, together with high-energy collider gravitational experiments will gain considerably once experiments, shed light on processes going on at the new UCN sources under construction at several extremely high energy scales or temperatures in the places will come into operation. early universe. These processes often take place at In the β -decay of free neutrons, many weak- times much earlier than the time when neutrons made interaction parameters are accessible to experiment, their first appearance, which was several ten of which at present are under study, many of microseconds after the start of the universe in a which are measured to a precision that competes well (modified) big bang model. with the precision reached in muon decay. Neutron Limits on the electric dipole moment of the decay data are needed in several fields of science neutron, together with those of the electron and the because today they are the only source to predict proton, severely constrain CP and time reversal precisely semileptonic weak interaction parameters nonconserving theories beyond the Standard Model, needed in particle physics, astrophysics, and in particular, supersymmetric models as the Minimal cosmology. Neutron data further permit precise tests Supersymmetric Standard Model (MSSM), as is of several basic symmetries of our world. Figures 25 shown in Figs. 8 and 9. At the same time, these limits and 26 show exclusion plots derived from neutron also severely constrain the ways how the process of decay on the so-called Fierz interference term in β - baryogenesis could have happened, which is decay, which would signal interactions with responsible for the evident dominance of matter over underlying scalar and tensor symmetries. Processes antimatter in the universe. How tight these constraints with these symmetries are not foreseen in the Standard are today is shown in Figs. 9, 10, and 11. Model, which has pure V− A symmetry, dominant in In the next generation of EDM experiments, the the universe for unknown reasons. Figures 27 and 28 experimental sensitivity will have passed the region show that from neutron decay data one can exclude a where most models on electroweak baryogenesis right-handed W-boson up to masses of rather high predict a nonzero EDM. If these upcoming energy. The existence of such right-handed experiments find a neutron EDM, this will be a strong interactions would help to understand how the evident hint that the creation of matter-antimatter asymmetry but still unexplained left-right asymmetry of the has occurred around the electroweak scale. If they do universe came into being. Still higher effective scales not see an EDM, other ways of baryogenesis, as Λ >10 TeV can be excluded from the present model- leptogenesis or other models at intermediate energy independent limits on violation of unitarity in CKM scales or close to the inflationary scale, are more quark mixing. probable. The precision of main neutron decay parameters The last few years have seen the emergence of a has strongly increased in recent years. Some errors new field of neutron studies, the measurement of seem to have been underestimated in the past, though quantum states of ultracold neutrons in the earth’s with the recent advent of new results, in particular, on gravitational field, as shown in Figs. 14 and 15. These the neutron lifetime τ and the β -asymmetry A, data and other neutron investigations make it possible to n test Newton’s gravitational law down to atomic and now seem to consolidate. Four new neutron decay subatomic distances. Deviations from Newton’s law observables, namely, C, N, R, and the branching ratio → ν γ are required in new models with “compactified” extra for the radiative decay n p e e , have become spatial dimensions that elegantly solve the so-called accessible recently, and more will follow. Constraints hierarchy problem, the riddle of why the gravitational on models beyond the V− A Standard Model (left- force is so much weaker than all other known forces. right symmetric, scalar, tensor, and other couplings) These neutron measurements constrain new forces today all come from low-energy experiments, though down to the sub-picometer range, see Fig. 16. Other are still less stringent than many people think. These neutron measurements constrain spin-dependent models beyond the SM will be investigated at a much exotic forces down to the micrometer range, see higher level of precision than is possible today, and Fig. 17. In this way, they start closing the last multi-TeV energy scales become accessible with “window” where one could still find the axion, a beams of milli-eV to nano-eV neutrons. particle needed to constrain strong CP violation, and a Ultracold neutrons, used in recent neutron-mirror favorite candidate for dark matter. What will come neutron oscillation experiments, could further signal next? The first experiments on neutron gravitational the presence of a mirror world, invoked to solve the levels have just exploited the first prototype UCN still unexplained left-right asymmetry of the universe. instruments, and much progress can be expected from Neutron processes were also used to test Einstein’s the next generation of instruments now under 76 mass-energy relation to an unprecedented accuracy, Neutron β -Decay?”, Phys. Rev. Lett. 88 , 211801 50 times better than before. Furthermore, neutron- (4). nucleon and neutron-nuclear weak interaction studies Abele, Hartmut, Stefan Baeßler, and Alexander will provide another angle of attack onto the strong Westphal, 2003, “Quantum States of Neutrons in interaction. the Gravitational Field and Limits for Non- Finally, in the last chapter, the role of the neutron Newtonian Interaction in the Range between 1 m in the formation of the elements was investigated, and 10 m ”, Lect. Notes Phys. 631 , 355-366. both during the “first three minutes”, and, later on, during stellar processes. Here neutron and nuclear Abele, H., E. Barberio, D. Dubbers, F. Glück, J.C. physics experiments provide whole networks of data Hardy, W.J. Marciano, A. Serebrov, and N. needed to understand these processes of Severijns, 2004, “Quark Mixing, CKM Unitarity”, nucleosynthesis, with a continued strong need for Eur. Phys. J. C 33 , 1-8. better data. Our excursion led us from the first Abele, H., G. Helm, U. Kania, C. Schmidt, J. 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