Light Higgs channel of the resonant decay of magnon condensate in superfluid 3He-B

In superfluids the order parameter, which describes spontaneous symmetry breaking, is an analogue of the Higgs field in the Standard Model of particle physics. Oscillations of the field amplitude are massive Higgs bosons, while oscillations of the orientation are massless Nambu-Goldstone bosons. The 125 GeV Higgs boson, discovered at Large Hadron Collider, is light compared with electroweak energy scale. Here, we show that such light Higgs exists in superfluid 3He-B, where one of three Nambu-Goldstone spin-wave modes acquires small mass due to the spin–orbit interaction. Other modes become optical and acoustic magnons. We observe parametric decay of Bose-Einstein condensate of optical magnons to light Higgs modes and decay of optical to acoustic magnons. Formation of a light Higgs from a Nambu-Goldstone mode observed in 3He-B opens a possibility that such scenario can be realized in other systems, where violation of some hidden symmetry is possible, including the Standard Model.


SUPPLEMENTARY NOTE 1 Spin waves in 3 He-B
Spin waves in 3 He-B correspond to motions of the rotation matrix R aj . The matrix can be represented by means of the rotation axisn and the rotation angle θ as R aj = cos θ δ aj + (1 − cos θ) n a n j − sin θ e ajk n k . (1) The motion is affected by the energy of the spin-orbit interaction F so and the gradient energy F ∇ : where χ B is the spin susceptibility of the 3 He-B, γ the gyromagnetic ratio for the 3 He atom, Ω B the Leggett frequency, ∆ the superfluid gap, and K 1 , K 2 and K 3 are parameters of the gradient energy. The spin-orbit interaction energy has a simple form in terms ofn and θ with a minimum at θ = arccos(−1/4): The equation of small spin oscillations near the equilibrium value In a texture withn ∥ H or in a high magnetic field ω 2 L /Ω 2 B ≫ 1 one can separate transverse and longitudinal oscillations. In the case of short wavelengths (when the spin changes on a much shorter distance than the texture) one can write the quasiclassical spectra for plane waves: where β n is an angle betweenn and H, the orbital anisotropy axisl j = R aj S 0 a and the spin wave velocities are introduced as The spin wave velocities are anisotropic, they have different values if the wave propagates in the direction ofl or in the perpendicular direction. The second line of Supplementary equation (6) describes a longitudinal wave, the light Higgs mode with a "relativistic" spectrum The first line of Supplementary equation (6) describes two modes of transverse waves, optical and acoustic magnons, with spectra of the form: In Supplementary equations (8) and (9) the effects of anisotropy and of the spin-orbit interaction are omitted for simplicity.

Little Higgs field for spin waves in 3 He-B
Let us introduce a vector field n =n sin θ/2 .
The spin-orbit interaction (4) provides a "Mexican Hat" potential for the n-field where n 2 0 = 5/8 and parameter Λ = 32 15 χB γ 2 Ω 2 B . In the terminology of particle physics the n-vector serves as the "little Higgs" field. In the vacuum states the amplitude of the field is fixed, |n| = n 0 , while they are degenerate with respect to the orientation ofn. The broken SU (2) symmetry leads to two Nambu-Goldstone modes (propagating oscillations of the orientation ofn), and one light Higgs mode (propagating oscillations of the amplitude |n| around n 0 ). These three modes comprising the little Higgs field are similar to the bosonic sector of Standard Model, where also the SU (2) symmetry is instrumental. This low energy sector of Standard Model contains the NG modes (the gauge bosons) and one "light Higgs" (the 125 GeV Higgs boson). Our two Nambu-Goldstone spin-wave modes correspond to the doublet of the W-bosons. The spectrum of the spin wave modes in 3 He-B splits in magnetic field into acoustic and optical modes. The similar splitting is discussed for the spectrum of the W-bosons in magnetic field (see e.g. Ref. [3]). Moreover, in strong magnetic fields the Bose condensation of the W-bosons is expected, which is similar to the Bose condensation of optical magnons.
where k r is the classical trajectory along the cell diameter and n is integer quantum number. This quantization corresponds to the wave modes in cylinder with high radial and small azimuthal quantum numbers. The effect of the spin-orbit interaction on the spectrum of short-wave acoustic magnons can be neglected, but anisotropy of wave velocity is important. The ratio of the velocities for k ∥l and k ⊥l is c ∥ /c ⊥ ≈ √ 4/3. Substituting the transverse magnon spectrum (6) without the spin-orbit term into (12) and taking into account that ω = N ω L we get where c is a harmonic mean velocity in the non-uniform texture: and β l is an angle betweenl and H. The distance between the resonances is: Note that the spin wave spectra (6) have been obtained with the assumption of zero coupling between transverse and longitudinal modes (ω 2 /Ω 2 B ≫ 1 or β n ≪ 1). In our experiment this condition is approximately valid for directly excited magnons with ω > 2Ω B . We use the same approximation also for parametrically excited magnons with ω ≈ Ω B . This is probably the reason why the agreement of the experimental data with Eq. (15) is much better for the directly excited magnons.

SUPPLEMENTARY NOTE 3 Trap for magnon quasiparticles
In the case of optical magnons with ω ≈ ω L , localized in the center of the cell, wheren is almost parallel to H, equation (5) can be rewritten in a form of a Schrödinger equation for magnon quasiparticles, where complex value s + = 1 √ 2 (S x + iS y ) plays role of the wave function and precession frequency ω plays role of the energy. Effect of texture on the gradient terms is neglected here because it adds only a small correction to the total gradient energy. [ Non-uniform potential for magnons is formed by the order parameter texture and the magnetic field (β n and ω L parameters).
In our setup the potential has a quadratic minimum in the center of the sample: in the flare-out texture angle β n near the sample axis is linear, β n = β ′ n r and magnetic field of the longitudinal coil has also quadratic profile near the center.
We use pulsed NMR to populate a few lowest levels in this harmonic trap. If the number of magnons in the system is small enough, interaction between the levels is negligible and the excited states can be resolved in the measurements independently. If the magnon population is above a certain threshold, they collapse to the ground state and form a Bose-Einstein condensate.
From the spectra of the magnon levels in the trap we can find values of spin-wave velocities c ⊥ and c ∥ . This work is presented in Ref. 4.