Improved constraints on monopole-dipole interaction mediated by pseudo-scalar bosons

We present a more stringent upper limit on long-range axion-mediated forces obtained by the QUAX-g$_p$g$_s$ experiment, located at the INFN - Laboratori Nazionali di Legnaro. We investigate the possible coupling between the electron spins of a paramagnetic GSO crystal and unpolarized nucleons of lead disks by measuring variations of GSO magnetization with a dc-SQUID magnetometer. Such an induced magnetization can be interpreted as the effect of a long-range spin dependent interaction mediated by axions or Axion Like Particles (ALPs). The corresponding coupling strength is proportional to the CP violating term $g_p^eg_s^N$, i.e. the product of the pseudoscalar and scalar coupling constants of electron and nucleon, respectively. Previous upper limits are improved by one order of magnitude, namely $g_p^eg_s^N/(\hbar c) \le 4.3\times10^{-30}$ at 95% confidence level, in the interaction range $10^{-3}$ m $<\lambda_a<0.2$ m. We eventually discuss our plans to improve the QUAX-g$_p$g$_s$ sensitivity by a few orders of magnitude, which will allow us to investigate the $\vartheta\simeq 10^{-10}$ range of CP-violating parameter and test some QCD axion models.


Introduction
The signature of symmetry breaking at extremely high energies can be highlighted by the presence of long-range ultraweak forces mediated by pseudo-Goldstone bosons [1]. In particular, the pseudo-boson can be either the QCD axion or an axion-like-particle (ALP), which involves P and T violating forces with strength proportional to the product of the couplings at the pseudo-boson vertices [2]. There are two options for coupling pseudo-scalar bosons with fundamental fermions: i) the spin-dependent pseudoscalar vertex, and ii) the scalar vertex that becomes spin-independent in the non relativistic limit. Thus, in a multipole expansion, the two fields are described by the "dipole" (pseudo-scalar coupling g p ) and "monopole" (scalar coupling g s ) moments, respectively. For instance, exchange of virtual axions -a possible solution of the strong CP problem -mediates a monopole-dipole force where g s is proportional to the QCD vacuum angle ϑ 10 −10 ÷ 10 −14 . In Fig.(1) we report the Feynman diagram of the g p g s interaction between an electron e − and a nucleus N mediated by an axion or ALPs that we investigate in this paper. The monopole-dipole coupling of the spin of a polarized electron with an unpolarized nucleus mediated by axions is described by the potential [2] V md (r) = g e p g N s 8πm e c (σ ·r) 1 rλ a + 1 where λ a is the Compton wavelength of the axion (interaction range), g e p and g  Figure 1: Interaction diagram of a scalar-pseudoscalar coupling between a nucleus N and an electron e − . N is unpolarized and interacts at the scalar vertex with the coupling constant g N s , whereas e − is polarized and interacts at the pseudoscalar vertex with the coupling constant Here the mediator is the axion a and the interaction strength is proportional to g N s g e p .
Pauli spin matrices, and r andr are the distance and unit vector between the nucleon and the electron, respectively [1,2,3]. It is worth noticing that for the axion the expected values of g e p g N s coupling strength as a function of its mass m a is [2] where m u and m d are the masses of the up and down quarks. In the conservative Kobayashi-Maskawa model, the predicted value of the vacuum angle is ϑ ∼ 10 −14 , the pion-nucleon σ term is taken to be 60 MeV [4], f a is the breaking energy scale of the Peccei-Quinn symmetry, and so the coupling strength approaches, for instance dc-SQUIDs and paramagnetic salts [5], high precision torsion balances [6,7,8,9], atomic magnetometers [10, 11] and stored ion spectroscopy [12]. New experiments have also been proposed [13,14] that should be able to reach better sensitivities.
µ ≡ µ Bσ in the effective magnetic field where µ B is Bohr's magneton and e is the electron charge. Clearly, this field is not a genuine magnetic field, as the interaction potential is generated by pseudoscalar exchange rather than by photon exchange, and so it does not satisfy the Maxwell's equations. Once Eq.
(3) is integrated over a macroscopic monopole source of volume V S with N O(10 23 ) nuclei and ρ N nucleon density, the resulting total effective magnetic field can have a measurable amplitude. In fact, the macroscopic magnetization in- The plan of the paper is as follows. In Section 2 we describe the QUAXg p g s apparatus and the effect of the long range, spin dependent interaction on our observable, i.e. the magnetization induced on the detector by an effective magnetic field source. In Section 3 we discuss our data analysis and the results we obtained with the present experimental set-up. Conclusions and plans to 30 improve the QUAX-g p g s sensitivity are presented in Section 4. To detect the variation of magnetization we use the most sensitive magnetometer available, namely a dc-SQUID operated at ∼4 K. As shown in Fig.(3), from the pick-up coil to the SQUID loop. As B eff is not a true magnetic field, we can reduce the environmental magnetic disturbances around the GSO with magnetic shields. In particular, we make use of two concentric Bi-2223 cylindrical superconducting shields at liquid helium temperature, and a µ-metal external shield at room temperature to reduce the field trapped in the inner shields.

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The overall rejection factor of the two superconducting shields is expected to be ∼ 10 12 [16], which is sufficient to make environmental magnetic disturbances negligible. We calibrated the apparatus using a solenoid with a diameter of 5 cm, coaxial to the pick-up coil, providing a uniform magnetic field over L p . The resulting 60 conversion factor between the output voltage of the SQUID electronics and the magnetic field at the pick-up is 4.25×10 −11 T/V. More details about the SQUID readout can be found in Ref. [15]. The SQUID output is fed to a band-pass filter having lower and upper cutoff frequencies of 0.1 Hz and 25 Hz.Measurements are taken at 10 Hz, well above the 1/f noise knee of the SQUID [17]. In our experimental setup, the dominant noise source is the additive flux noise of the SQUID, which therefore represents the sensitivity limit of the magnetometer.

Data analysis and results
In Fig.(4) we report the measured noise, which is compatible with the additive flux noise of the SQUID. Currently this fixes the sensitivity limit of the 70 magnetometer [15]. We tested the hypothesis whether the wheel rotation may introduce an excess noise by comparing measurements obtained with rotating or non rotating wheel. We found no modification of the magnetic noise level in the frequency band of the measurement. In addition, multiple measurements were taken and no time dependence of the output has been found.

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To obtain an optimal estimate of the amplitude of the modulated effective magnetic field, we performed a phase sensitive detection with a digital lockin and the reference phase of the rotating wheel. To estimate this phase, we  and so we conclude that we have observed no induced magnetization in the GSO crystal due to monopole-dipole interaction mediated by axions or ALPs.

Results
Using numerical integration of Eq.(4) over the volume of the sources and taking into account the geometry of the apparatus, we can convert our mea-100 surements of the effective magnetic field in a upper limit on g e p g N s . Since the modulation and intensity of the signal both depend on λ a , a correction curve g(λ a ) was estimated to obtain the actual measured limit. Using this procedure we get our best upper limit on the coupling g e p g N s /( c) ≤ 4.3 × 10 −30 at 95% C.L. in the range 1 cm < λ a < 20 cm. Above the upper end of this range, the 105 effective magnetic field no longer depends on λ a and so the sensitivity of our experiment to axion or ALP mediators decreases. At the lower end the sensitivity is limited by the exponential decay e −r/λa of the monopole dipole interaction.  [7,18,10,8,11,19] in terms of the strength of the monopole-110 dipole interaction. It is worth noticing that the g e p g N s coupling can be also strongly constrained by star cooling processes [20].

Conclusions and perspectives
We have reported on a measurement of the g e p g N s coupling obtained with the QUAX-g p g s experiment that places an upper limit for this type of interaction.

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Currently, our limit is the best for spin-dependent forces mediated by axions or ALPs, as we obtained an enhancement of magnetic field sensitivity of one order of magnitude with respect to other experiments reported in the literature.
Despite the experimental approach in this paper is similar to that of Ref. [5], we succeeded in improving the sensitivity by using a SQUID with an intrinsic 120 lower noise and a paramagnetic crystal with a higher susceptibility.
The sensitivity of the QUAX-g p g s experiment can be further improved by using a resonant electrical LC circuit with a high quality factor Q [23]. In fact, we can add a coil of inductance L 100 mH wound around the GSO and =10 ϑ Figure 6: Exclusion plots of monopole-dipole coupling vs. λa. The upper limit on the g e p g N s coupling is lowered of more than one order of magnitude in respect to the previous measurements for λa ∼ 10 cm (red line). We also show in transparent grey the g e p g N s limit derived from Eq.(2) with ϑ ≤ 10 −10 resulting from neutron EDM [21,22], and upper limits already reported in the literature: Hammond [7], Youdin [10], Ni [18] and Adelberger [19].
connected to a low loss capacitor with C 20 µF, to form an LC circuit with 125 resonance frequency f LC 110 Hz. The pickup coil L p is also wound on the crystal and connected to the SQUID input coil L i as described in this paper.
The coupling with the SQUID reduces the value of L and, when L and L p are perfectly coupled, the resonance frequency increases up to the maximum value √ 2f LC ∼160 Hz. By increasing f w to 6.5 Hz and N s to 24, the signal frequency 130 will coincide with the LC resonance frequency. In this case, the SNR at the SQUID input coil could increase up to a factor of Q with respect to the present configuration. Such sensitivity improvement is effective as long as the Johnson and crystal magnetization noises exceed the additive noise of the SQUID.
Our improved experimental apparatus could eventually invade the gray ex-135 clusion region in the g p g s − λ a plane of Fig.(6), which has been established by the limit of the anomalous neutron EDM d n ≤ 3 × 10 −26 e cm [21,22]. As a final remark, we mention that QUAX-g p g s can also explore the dipole-dipole coupling g e p g N p . To this aim, it is sufficient to replace the unpolarized sources of effective magnetic field mounted on the wheel with spin-polarized sources.

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Conceptually new designs for spin-polarized masses based on permanent magnets are reported in the literature [24]. These and other related issues will be the subject of a forthcoming paper.