RESEARCH ARTICLE by using nuclear-spin magnetometers with spin- polarized 3He used as a source (13, 14), whereas the e–-e– spin interactions have been investigated Using the Earth as a Polarized Electron by using a spin-polarized torsion pendulum with SmCo5-Alnico toroids as sources (15). To search for long-range (l > ~1 m) spin-spin interactions, Source to Search for Long-Range the polarized source is placed near the appara- tus and the direction of its polarization is al- Spin-Spin Interactions tered. The spin associated with optically pumped nuclear sources can be reversed simply by 1 1 1 1 2 Larry Hunter, * Joel Gordon, Stephen Peck, Daniel Ang, Jung-Fu Lin changing the circular polarization of the pumping radiation. The solid-state electron-spin sources Many particle-physics models that extend the standard model predict the existence of long-range can be easily rotated in the laboratory. One then spin-spin interactions. We propose an approach that uses the Earth as a polarized spin source looks for a response in the magnetometer or to investigate these interactions. Using recent deep-Earth geophysics and geochemistry results, torsion pendulum that is synchronous with the we create a comprehensive map of electron polarization within the Earth induced by the source modulation. Great efforts are made in geomagnetic field. We examine possible long-range interactions between these spin-polarized these experiments to minimize the ordinary geoelectrons and the spin-polarized electrons and nucleons in three laboratory experiments. By magnetic-dipole coupling between the source combining our model and the results from these experiments, we establish bounds on torsion and apparatus. gravity and possible long-range spin-spin forces associated with the virtual exchange of either The concept of the experiment. Here we sug- spin-one axial bosons or unparticles. gest an alternative approach where, instead of the modulated laboratory spin source, one uses the dvances in particle physics have contrib- and spin directions, s%. The interaction range of polarized electrons within the Earth. The advan- uted to our understanding of the deep the force is denoted by l = ℏ/mz′ c where ℏ is tage of this approach is simply one of numbers. AEarth, most recently through the obser- Planck’s constant (h) divided by 2p and c is There are ~1049 unpaired electron spins in the vation of geoneutrinos (1). By contrast, the con- the speed of light. Earth. On average, about one extra electron out of on February 21, 2013 tributions to particle physics resulting from our Another interesting entity that could produce every 10 million will become polarized antipar- understanding of the Earth have largely been lim- long-range spin-spin interactions is the “unpar- allel to Earth’s magnetic field. Hence, there are ited to gravitational interactions or using the Earth ticle” (4). Unlike ordinary particles, unparticles on the order of 1042 polarized electrons in the as a large mass or baryonic source. Here we do not have well-defined masses but can be Earth, compared with ~1022 polarized neutrons suggest that our knowledge of the magnetic fields characterized in terms of an energy scale L,a or ~1025 polarized electrons in a typical labora- and electron-spin behavior within the Earth is scaling dimension d, and a dimensionless coupling tory source. Hence, the number of polarized geo- sufficiently developed that we can use the Earth’s constant cA. In the long-range limit, the virtual electrons exceeds the number in a laboratory polarized electrons to study anomalous long-range exchange of an axial-vector unparticle results in source by a factor of at least 1017. Laboratory spin-spin interactions. an effective potential (5) sources are usually located a few tenths of a me- www.sciencemag.org Many extensions of the standard model of pffiffiffi ter from the detection apparatus while a typical pGð þ = ÞGð ð − ÞÞ particle physics predict the existence of new par- 2 4 d 1 2 2 d 1 % % geoelectron is a few thousand kilometers away. Vu ¼ −c ðs1 ⋅ s2Þ ticles. The virtual exchange of these particles be- A ð2pÞ2dGðd − 1ÞGð2dÞ For ordinary electromagnetism and for the spin- tween ordinary fermions can result in spin-spin spin potentials associated with pseudoscalar bos- interactions that look quite different from those − ons, the dipole-dipole interaction falls off as the ℏc l 2d 2 expected from electromagnetism. The spin-  u ð3Þ cube of the distance, r, and the increased distance dependent forces that result from the exchange of r r of the geoelectrons reduces the interaction po- a pseudoscalar boson like the axion were orig- tential by ~21 orders of magnitude. For such po- Downloaded from inally investigated by Moody and Wilczek (2). where G is the gamma function, and lu ¼ ℏc=L tentials there is no net advantage in considering The exchange of a vector boson with mass mz′ is the characteristic length associated with the an Earth source. However, many of the anom- can lead to the “spin-dot-spin” and “spin-cross- unparticle. The unparticle potential does not ex- alous spin-spin potentials (e.g., Eqs. 1 to 3) fall spin” potentials (3) hibit the usual exponential decay with distance off as 1/rn,wheren is between 1 and 2. For these that is associated with the exchange of ordinary potentials, the suppression of the sensitivity by g1 g2 ¼ A A ðs% ⋅ s% Þ −r=l ð Þ massive particles. the distance will be between 7 and 14 orders of V1 p 1 2 e 1 4 r Ramsey conducted the first experiment that magnitude. Here, even with additional losses due looked for anomalous spin-spin couplings (6). to poor geometry and lower experimental sensi- ℏ2 g1 g2 g1 g2 r V ¼ V A þ A V ðs%  s% Þ ⋅ r% 1 þ Early work in the field placed limits on such tivity, a geoelectron source can result in sub- 2 p 1 2 l – – 4 2M1 2M2 couplings between electrons (e -e )(7, 8) and stantially improved constraints. 1 − =l between electrons and nucleons (9). Recently, The greatest disadvantage to searching for  e r ð(2)1Þ r2 constraints have been placed on short-range spin interactions with the Earth is that one can- (atomic scale) anomalous spin-spin forces by not modulate the spin source. To extract the where g denotes the vector (V) or axial (A) cou- considering hyperfine structure in hydrogen-like geoelectron spin-spin signal, one must reverse pling constants of fermions 1 or 2 with mass M atoms (10), spin-exchange collisions (11), and or modulate the contribution it makes to the magnetic resonance in deuterated molecular hy- experimental signal. This can be accomplished 1 Physics Department, Amherst College, Amherst, MA 01002, drogen (12). New searches for long-range inter- by mounting the detection apparatus on a ro- USA. 2Department of Geological Sciences, Jackson School of Geosciences, The University of Texas at Austin, Austin, TX actions have been quite successful in placing tating table. Such systems have been developed 78712, USA. bounds on the coupling constants associated with for searches for violations of local Lorentz in- *To whom correspondence should be addressed. E-mail: the potentials discussed above. Neutron-neutron variance (LLI) (16, 17). Although most contri- [email protected] (n-n) spin couplings have been investigated butions to the experimental signal are independent 928 22 FEBRUARY 2013 VOL 339 SCIENCE www.sciencemag.org RESEARCH ARTICLE of the table orientation, the “effective” spin as- The experiment is mounted on a table that ro- netic field angle with respect to vertical (qB = sociated with the geoelectrons creates a preferred tates between two data-collection positions sep- 63.8°) and the factor of 4 accounts for the dif- axis in the laboratory that couples to the table arated by 180°. The table rotation changes the ferences between the two table positions and the position via the various spin-spin interactions direction of the applied magnetic field vector two orientations of the nuclear spin with respect (Eqs. 1 to 3). from Bapp1 to Bapp2. This inverts the horizontal to the applied field. We follow the notation of Energy bounds on spins oriented relative to component of the magnetic field while leaving (19)inusinga“hat” over b to indicate that a Earth. We use the measurements reported from the vertical component fixed. The Cs magne- correction for Earth’s gyroscopic frequency has two recent LLI experiments [one in Amherst, tometer is used to hold the magnitude of the been applied. A random geoelectron with a spin 199 % MA (qlat = 42.37°N, 72.53°W) (18) and another magnetic field constant. The change in the Hg s2, separated from the apparatus by a distance r, in Seattle, WA (47.658°N, 122.3°W) (19)], and nuclear precession frequency with the table po- is also shown in the plot. To place bounds on the DnHg < : m s bounds from an earlier Seattle experiment that sition is measured to be N 1 1 Hz (2 coupling strengths in Eqs. 1 and 3, we sum the searched for the coupling of nuclear spin to Earth’s bounds will be quoted throughout this paper). interaction potentials over all of Earth’sspin- gravitational field (20), to extract limits on the The resulting bound on the energy of a 199Hg polarized electrons and require that the resulting s% b%Hg associated spin-spin couplings. nuclear spin ( 1 in Fig. 1) oriented north (N)is energy not exceed N . 199 b%Hg < DnHg=ð q Þ¼ :  −21 The geometry of the Amherst Hg-Cs co- N h N 4sin B 1 3 10 eV, where The coupling of the geoelectron spins by the magnetometer experiment is shown in Fig.