Gapped dilatons in scale invariant superfluids

Gapped dilatons in scale invariant superfluids Riccardo Argurio,1,* Carlos Hoyos ,2,†Daniele Musso ,3,‡and Daniel Naegels 1,§ 1Physique Th´eorique et Math´ematique and International Solvay Institutes, Universit´e Libre de Bruxelles, C.P. 231, B-1050 Brussels, Belgium 2Department of Physics and Instituto de Ciencias y Tecnologías Espaciales de Asturias (ICTEA), Universidad de Oviedo, c/ Federico García Lorca 18, E-33007 Oviedo, Spain 3Departamento de Física de Partículas and Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de Santiago de Compostela, E-15782 Santiago de Compostela, Spain Inovalabs Digital S.L. (TECHEYE), E-36202 Vigo, Spain (Received 9 July 2020; accepted 25 September 2020; published 19 October 2020) We study a paradigmatic model in field theory where a global Uð1Þand scale symmetries are jointly and spontaneously broken. At zero density the model has a noncompact flat direction, which at finite density needs to be slightly lifted. The resulting low-energy spectrum is composed by a standard gapless Uð1Þ Nambu-Goldstone mode and a light dilaton whose gap is determined by the chemical potential and corrected by the couplings. Even though Uð1Þand scale symmetries commute, there is a mixing between the Uð1ÞNambu-Goldstone and the dilaton that is crucial to recover the expected dynamics of a conformal fluid and leads to a phonon propagating at the speed of sound. The results rely solely on an accurate study of the Ward-Takahashi identities and are checked against standard fluctuation computations. We extend our results to a boosted superfluid, and comment the relevance of our findings to condensed matter applications. DOI: 10.1103/PhysRevD.102.076011 I. INTRODUCTION Scale invariance plays a special role in many-body and high-energy physics. It underlies the emergence of universality in many instances, such as critical phenomena, Landau-Fermi liquids or cold atoms at unitarity, to name a few. Scale transformations are a symmetry either at very low or very high energies compared to the intrinsic scales. In most cases they represent only an approximate symmetry valid in a restricted regime, requiring typically a certain degree of fine-tuning in the interactions, the thermodynamic variables, the external parameters, or the support of additional symmetries. When a scale invariant system is considered at non-zero particle number or at finite charge density, scale symmetry is spontaneously broken; such breaking is directly relevant to characterize the dynamics of the system mentioned above but it can also be useful to extract properties of large charge operators of a CFT via the state-operator correspondence [1–4]. Whenever an internal symmetry is spontaneously broken in a relativistic system, one expects to encounter gapless excitations in the form of Nambu-Goldstone (NG) modes [5–7], one for each broken symmetry. If instead the breaking involves spacetime symmetries, the counting of modes becomes more complicated [8–11], yet the presence of a Nambu-Goldstone mode associated to scale invariance, commonly known as dilaton, is still a possibility. Similarly, the counting of NG modes deviates from the standard Goldstone theorem expectation when the system is not Lorentz invariant [10,12–15].1 Even in relativistic systems, the presence of a nonzero charge density breaks the boosts spontaneously, thus the NG modes may show some features similar to those emerging in nonrelativistic systems. In the case of several internal symmetries, there can be additional gapped modes besides the gapless NG modes [17–22]. More specifically, in the presence of a chemical potential μfor a conserved charge Q, gapped modes emerge when the effective Hamiltonian2˜ H¼H−μQdoes not commute with the broken generators. The gap is fixed by group theory considerations and is proportional to the chemical potential. *[email protected] †[email protected] ‡[email protected] §[email protected] Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3. 1For a recent review on NG counting rules we refer to [16]. 2We are proceeding in analogy to [19,20,22,23]. PHYSICAL REVIEW D 102, 076011 (2020) 2470-0010=2020=102(7)=076011(13) 076011-1 Published by the American Physical Society In general, there can also be additional modes whose gap, although proportional to μ, is not protected by symmetry [15,21]. Such analysis was generalized in [20] to cases where μQin the effective Hamiltonian is replaced by some other deformation involving a symmetry generator, for instance the magnetic field times the spin in a ferromagnet ˜ H¼H−gBzSz. The breaking of scale invariance presents some similarities with the story above due to the fact that the generator of dilatations Ddoes not commute with the Hamiltonian ½D; H¼iH.3Accordingly, the commutator with the effective Hamiltonian is ½˜ H;D¼−iH: ð1:1Þ For simplicity let us assume that Qis the generator of an Abelian Uð1Þsymmetry. If this symmetry is spontaneously broken, the ground state is not an eigenstate of Q. However, it must be by definition an eigenstate of ˜ H, so time translations generated by Hare spontaneously broken too. In fact time translations and the Uð1Þsymmetry are broken to a diagonal subgroup and there is just a single NG mode associated to both generators. Equation (1.1) implies that there is a mixing between the Uð1ÞNG and the dilaton, then—even though Qcommutes with ˜ H—the state produced by the corresponding charge density J0applied to the vacuum at some initial time is not an eigenstate of time evolution. Borrowing an analogy from high-energy physics, the “flavor”eigenstates defined by the symmetry generators are not aligned with the “mass” eigenstates. If the dilaton were not dynamical, or if it were integrated out, the mixing implied by (1.1) would be manifested in the form of an inverse Higgs constraint. Although interesting, one might wonder whether it is sensible to discuss the physics of a dilaton in the first place, since the energy density is in general nonzero at nonzero charge density. In that case, a scale transformation would change the vacuum energy density (as determined by the temporal component of the energy-momentum tensor Tμν) by an amount proportional to itself δhT00i∼h−i½D; T00i ¼ ðdþ1ÞhT00i;ð1:2Þ where dþ1is the number of spacetime dimensions. Both here and henceforth, we assume a relativistic theory, thus there cannot be a NG mode associated to the spontaneous breaking of scale invariance unless hT00i¼0.4This is quite restrictive. Since a gapless mode requires a degeneracy of ground states, the theory needs to have a moduli space of vacua in addition to scale invariance: these are flat directions in the potential, supposing we refer to a field theory with a Lagrangian.5 Maybe contrary to expectations, the situation at finite density is similar despite the fact that the energy density is nonvanishing. If the ground state is homogeneous and isotropic, the expectation value of the components of the energy-momentum tensor correspond to constant energy density and pressure hT00i¼ε;hTiji¼pδij:ð1:3Þ Scale invariance implies that the expectation value of the trace of the energy-momentum tensor will vanish hTμμi¼0, which fixes the equation of state ε¼dp, where dis the number of spatial dimensions. In addition, we have the usual relation between thermodynamic potentials at zero temperature, εþp¼μρ, where ρ¼hJ0iis the Uð1Þ charge density. Combining the two, the energy density of the scale invariant theory is ε¼d=ðdþ1Þμρ. At finite density the relevant quantity is not the energy density, but the free energy (density) given by the effective Hamiltonian T00 −μJ0. A scale transformation changes the expectation of the effective energy density as follows: δhT00 −μJ0i∼h−i½D; T00i−μh−i½D; J0i ¼ðdþ1Þε−dμρ ¼0:ð1:4Þ Then, under quite general assumptions, scale transformations do not shift the free energy of a finite density state in a scale invariant theory and it is legitimate to discuss the physics of a dilaton mode, at least at zero temperature. The observation above does not imply directly the existence of a gapless (or gapped) mode. In the absence of a general argument that would allow us to fix the properties of a dilaton mode, we study a concrete model of spontaneous breaking of scale invariance at nonzero density. We restrict the analysis to a relativistic theory in 3þ1dimensions, and keep the analysis classical. Such simple model is informative because it can be interpreted as an effective action `alaGinzburg-Landau for the order parameter. The principal highlights of the present study are two. On one side is the characterization of the dilaton dispersion relation and particularly its gap. This concerns mainly the effects of the chemical potential and its role in defining the effective low-energy spectrum. On the other side, we propose and check a method based uniquely on the study of Ward-Takahashi identities, that in our setup just correspond to classical conservation equations. 3Nevertheless, it is a conserved charge because ∂tD¼H,so its total time derivative in the Heisenberg picture vanishes. 4Note also that the combination of Lorentz invariance (which fixes the expectation value of the energy-momentum tensor to hTμνi¼Λημν) and the Ward-Takahashi identity for scale invariance, hTμμi¼0, fixes hT00i¼0. 5A related discussion about fine-tuning the cosmological constant to zero in order to have a flat dilatonic direction is contained in [24–27]. ARGURIO, HOYOS, MUSSO, and NAEGELS PHYS. REV. D 102, 076011 (2020) 076011-2 The paper is structured as follows. Section II introduces the model at zero density, where we emphasize the need for flat directions in the potential. This condition is relaxed in Sec. III, where we study the model at nonzero density. In Sec. IV the analysis is extended to allow for nonzero superfluid velocity. Each section has a subsection dedicated to the analysis of the Ward-Takahashi identities, together with a check of the latter method against standard Lagrangian computations for the fluctuations. We conclude the paper in Sec. Vwith further comments on the results, their interpretation, their applications and possible extensions. II. THE MODEL Consider the standard Goldstone model for a global Uð1Þsymmetry in four spacetime dimensions S¼Zd4x½∂μψ∂μψ−λðjψj2−v2Þ2;ð2:1Þ where ψis a scalar complex field charged under the Uð1Þ symmetry, which acts as ψ→eiαψ, while λand v represent—respectively—a dimensionless and a dimensionful coupling. Given the presence of a dimensionful coupling, the model (2.1) does not enjoy scale invariance. We can nonetheless make it scale invariant if we replace v with a dynamical real scalar field ξacting as a compensator: S¼Zd4x∂μψ∂μψþ1 2∂μξ∂μξ−λðjψj2−ξ2Þ2:ð2:2Þ The equations of motion are given by ∂2ψþ2λðjψj2−ξ2Þψ¼0;∂2ξ−4λðjψj2−ξ2Þξ¼0; ð2:3Þ and the generic stationary solution is ξ¼v; jψj2¼v2:ð2:4Þ The space of solutions (2.4) has two moduli, ξitself and the phase of ψ. Consider the fluctuations around (2.4), parametrized as follows ψ¼eiϑ ffiffi2 pvve τ ffiffi3 pvþρ ffiffiffi 6 p≃vþτ ffiffiffi 3 pþρ ffiffiffi 6 pþiϑ ffiffiffi 2 p; ξ¼ve τ ffiffi3 pv−2ρ ffiffiffi 6 p≃vþτ ffiffiffi 3 p−2ρ ffiffiffi 6 p;ð2:5Þ where τ,ρand θare real. The quadratic action for the fluctuations is given by Squad ¼Zd4x1 2∂μτ∂μτþ1 2∂μρ∂μρþ1 2∂μϑ∂μϑ−6λv2ρ2: ð2:6Þ We thus see that ρgets a mass 12λv2while τand ϑare massless. We identify the latter two with the Goldstone bosons for broken scale invariance, the dilaton and, for broken Uð1Þsymmetry, the Uð1ÞNG. The dispersion relations are trivially relativistic, since Lorentz symmetry is preserved. In order to study the low-energy modes about (2.4), one can alternatively rely entirely on symmetry considerations and, specifically, on the Ward-Takahashi identities. As we will show in the next subsection, such symmetryaware approach permits to obtain the equations of motion for the low-energy modes in a direct way, which is usually more transparent than the standard Lagrangian study of the fluctuations. A. WARD-TAKAHASHI IDENTITIES AND LOW-ENERGY MODES Model (2.2) features a conserved Uð1Þcurrent given by Jμ¼ið∂μψψ−ψ∂μψÞ;∂μJμ¼0;ð2:7Þ while the improved energy-momentum tensor is Tμν ¼2∂ðμψ∂νÞψþ∂μξ∂νξ−ημνL þ1 3ðημν∂2−∂μ∂νÞ1 2ξ2þjψj2:ð2:8Þ This expression satisfies on-shell the following Ward- Takahashi identities6 T½μν¼0;∂μTμν ¼0;T μμ¼0:ð2:9Þ We expand around the vacuum (2.4) by considering the fluctuation parametrization (2.5). Up to linear order in the fields, the Uð1Þcurrent is given by Jμ≃ffiffiffi 2 pv∂μϑ;ð2:10Þ so that its conservation equation gives the equation of motion for the Uð1ÞNG mode 0¼∂μJμ≃ffiffiffi 2 pv∂2ϑ:ð2:11Þ The energy-momentum tensor expanded to linear order is 6The trace Ward-Takahashi identity requires the improvement introduced in (2.8). GAPPED DILATONS IN SCALE INVARIANT SUPERFLUIDS PHYS. REV. D 102, 076011 (2020) 076011-3 Tμν ≃v ffiffiffi 3 pðημν∂2−∂μ∂νÞτ;ð2:12Þ and the trace Ward-Takahashi identity yields the equation of motion for the dilaton 0¼Tμμ≃ffiffiffi 3 pv∂2τ:ð2:13Þ From (2.11) and (2.13) we can observe that we recover the two massless modes of (2.6). The Ward-Takahashi computation, however, descends directly from symmetry arguments, being therefore more convenient (and easier) to apply, especially when dealing with models more complicated than (2.2). In particular, this approach allows to identify immediately and without ambiguities the nature of each Goldstone boson, simply by associating every (gapless) mode to the Ward-Takahashi identity that yields its equation of motion. It is important to stress that the model (2.2) is fine-tuned. Indeed, (classical) scale invariance dictates that the potential should contain only quartic terms in the scalars, but the fact that the potential is a perfect square constitutes a finetuning, specifically considered to the purpose of having a flat direction. The latter is of course a necessary condition for the presence of a low-energy dilaton mode. The simple argument is as follows. In such a relativistic setup, scale invariance implies the absence of any reference scale in the (effective) Lagrangian. If scale invariance is to be broken spontaneously by a vacuum expectation value (VEV), then the latter must be arbitrary. Hence this VEV parametrizes a noncompact flat direction. Moreover the absence of any reference scale means that the flat direction must also correspond to a vanishing vacuum energy. The particle which corresponds to moving along this flat direction is the dilaton. We conclude that any effective theory that aims at describing spontaneous scale symmetry breaking (among others), must allow for a noncompact flat direction in its potential. For instance, if we added a generic term preserving scale invariance but breaking the exchange symmetry between jψjand ξ, namely (without loss of generality) V¼λðjψj2−ξ2Þ2þλ0ðjψj2Þ2;ð2:14Þ the equations extremizing the potential would become λψðjψj2−ξ2Þ¼−λ0jψj2ψ;ð2:15Þ λξðjψj2−ξ2Þ¼0:ð2:16Þ Considering λ0>0for Vto be bounded from below, the only solution is ξ¼0¼ψ, i.e., the flat direction is completely lifted, even though scale invariance is respected. III. SPONTANEOUS SYMMETRY BREAKING AT FINITE DENSITY In this section we depart from the Lorentz-invariant setup discussed above, by introducing a nonzero chemical potential μfor the charge associated to the global Uð1Þ symmetry. As we will see, we will still be able to identify the dilaton and the Uð1ÞNG, though their dispersion relations will be modified in an interesting way. We start with a scale-invariant theory defined by the action S¼Zd4x∂μψ∂μψþ1 2∂μξ∂μξ−λðjψj2−ξ2Þ2−λ0ðjψj2Þ2; ð3:1Þ whose potential corresponds to the extension already introduced in (2.14). We are going to switch on a chemical potential μfor the Uð1Þsymmetry. As discussed before, at finite chemical potential, the ground state is no longer determined by the Hamiltonian Hbut by the effective Hamiltonian ˜ H¼H−μQ, where Qis the Uð1Þcharge operator. As we will discuss, this modifies the effective potential of the theory and allows the fields to acquire a nonzero value. Notably, one can recover the zero chemical potential symmetry breaking case described by (2.2) by means of an appropriate limit for both μand λ0. The main result of the present section is to show that the dilatonic mode acquires a gap, which depends on μand λ0. A nonzero chemical potential can be implemented by extracting a time-dependent phase from the complex field ψ¼eiμtϕ;ψ¼e−iμtϕ:ð3:2Þ The equations of motion then read ∂2ϕþ2iμ∂0ϕ−μ2ϕþ∂ϕVðjϕj;ξÞ¼0; ∂2ξþ∂ξVðjϕj;ξÞ¼0;ð3:3Þ where Vðjϕj;ξÞ≡Vðjψj;ξÞis given by (2.14). Note that these equations can equivalently be obtained introducing (3.2) in (3.1), identifying a new effective potential Vϕðjϕj;ξÞ¼Vðjϕj;ξÞ−μ2jϕj2and taking the variation with respect to ϕ,ξ. Although Vϕis not the true potential (indeed, the energy density is E∼Vðjϕj;ξÞþμ2jϕj2), the extrema of Vϕcorrespond to solutions of the equations of motion of the original action (3.1). We will show in the following that Vϕdetermines the ground state for the effective Hamiltonian ˜ H. A. Effective Hamiltonian and ground state In order to determine the effective Hamiltonian and the associated ground state we need to find expressions for the Uð1Þcharge Qand Hamiltonian. We will use the usual ARGURIO, HOYOS, MUSSO, and NAEGELS PHYS. REV. D 102, 076011 (2020) 076011-4 definitions in terms of the temporal components of the energy-momentum tensor Tμν and Uð1Þcurrent Jμ H¼Zd3xT00;Q¼Zd3xJ0:ð3:4Þ Then, the effective Hamiltonian at finite chemical potential is determined by the temporal component of an effective energy-momentum tensor tμν ˜ H¼Zd3xðT00 −μJ0Þ≡Zd3xt00:ð3:5Þ The Uð1Þcurrent can be written as follows J0¼2μjϕj2þj0;J i¼ji;ð3:6Þ where jμ¼ið∂μϕϕ−ϕ∂μϕÞ:ð3:7Þ Similarly, for the energy-momentum tensor7 T00 ¼μJ0þt00;ð3:8Þ T0i¼Ti0¼μJiþt0i¼μjiþt0i;ð3:9Þ Tij ¼tij þδijðμJ0−2μ2jϕj2Þ¼tij þδijμj0;ð3:10Þ where tμν ¼2∂ðμϕ∂νÞϕþ∂μξ∂νξ−ημνLϕ þ1 3ðημν∂2−∂μ∂νÞ1 2ξ2þjϕj2;ð3:11Þ and Lϕ¼∂μϕ∂μϕþ1 2∂μξ∂μξ−λðjϕj2−ξ2Þ2 −λ0ðjϕj2Þ2þμ2jϕj2:ð3:12Þ Notice that from (3.9) we have that T0i¼Ti0implying that the Ward-Takahashi identities for boost transformations are satisfied, so that the full Lorentz symmetry is still preserved in the presence of a nonvanishing chemical potential. The effective potential for Lϕis the one we had identified previously in the equations of motion (3.3) Vϕ¼λðjϕj2−ξ2Þ2þλ0ðjϕj2Þ2−μ2jϕj2;ð3:13Þ Since t00 determines the effective Hamiltonian (3.5), we see that the ground state will correspond to the minimum of the effective potential. The effective potential has three extrema8 ξ¼ϕ¼0; ξ¼0;jϕj2¼v2¼μ2 2ðλþλ0Þ; ξ2¼jϕj2¼v2¼μ2 2λ0:ð3:14Þ Out of the three extrema (3.14), the first two are saddle points and only the last is a minimum, which is the true ground state of the system. Note that for the true minimum to exist, and for Vϕto be bounded from below, we need to have λ0>0. In other words, we need to lift the flat direction that we had at μ¼0in order to have a minimum, and symmetry breaking, when μ≠0. We now proceed to investigate the low-energy spectrum around this (degenerate) minimum. B. Nambu-Goldstone dynamics from Ward-Takahashi identities We perturb the fields around the ground state ξ2¼jϕj2¼v2¼μ2 2λ0. We use the same parametrization as in (2.5), though adapted to the field ϕ ϕ¼eiϑ ffiffi2 pvve τ ffiffi3 pvþ1 ffiffiffi 6 pρ;ξ¼ve τ ffiffi3 pv−2 ffiffiffi 6 pρ:ð3:15Þ As before, the kinetic terms are diagonal and canonically normalized for ϑ,τand ρ. We still identify ϑas the fluctuation of the phase of the condensate and τas a fluctuation of its magnitude, while ρcorresponds to an orthogonal direction of increasing potential energy. For μ¼0,ϑand τare naturally associated to the Uð1ÞNG and dilaton, while ρenters as a Higgs fluctuation. This simple picture is a bit complicated when μ≠0, as the would-be Goldstones undergo some mixing and also a nonvanishing gap for one linear combination. We will study this effect in some approximation here and in more detail in the next section. When the perturbation (3.15) is introduced in the effective potential (3.13) and expanded to quadratic order, one finds no term for ϑand the following mass matrix for ðτ;ρÞ M¼4v2 32λ0ffiffiffi 2 pλ0 ffiffiffi 2 pλ0λ0þ9λ:ð3:16Þ In principle both perturbations are massive and mixed, but in the limit λ0≪λin which there is an almost flat direction 7The notations are such that the capital letters (T00 etc.) refer to the dynamics of ðψ;ξÞ(and by extension, of ϕ) dictated by (3.1). The lowercase letters refer instead to the dynamics given by (3.12) which is not the Lagrangian for ðϕ;ξÞbut it shares the same potential. 8Note that these uniform and static solutions are extrema of the effective potential (3.13), but not of the energy (3.8). GAPPED DILATONS IN SCALE INVARIANT SUPERFLUIDS PHYS. REV. D 102, 076011 (2020) 076011-5 in the original potential (2.14), the mixing becomes very small and there is a large hierarchy between the mass of τ, m2 τ∼λ0v2∼μ2, and the mass of ρ,m2 ρ∼λv2. In the following we will assume that we are in this situation, in which case the Higgs fluctuation ρcan be set to zero in the low energy description to a good approximation. The dynamical equations for the remaining fluctuations can be derived from the Ward-Takahashi identities. When evaluated on shell the Uð1Þcurrent should be conserved and the trace of the energy momentum tensor should vanish ∂μJμ¼0;T μμ¼0:ð3:17Þ This gives two equations, which is sufficient to determine the dynamics of ϑand τ. The trace of the energymomentum tensor, to linear order in the fluctuations, is Tμμ≃ffiffiffi 3 pv∂2τþ4 3μ2τ−2ffiffiffi 2 3 rμ∂0ϑ;ð3:18Þ whereas the divergence of the current is ∂μJμ≃ffiffiffi 2 pv∂2ϑþ2ffiffiffi 2 3 rμ∂0τ:ð3:19Þ This translates into the set of coupled equations ∂2τþ4 3μ2τ−2ffiffiffi 2 3 rμ∂0ϑ≃0; ∂2ϑþ2ffiffiffi 2 3 rμ∂0τ≃0:ð3:20Þ As suggested by the general analysis in the introduction, the chemical potential introduces a mixing between the Uð1ÞNG and the dilaton. The equations can be diagonalized using expansions in Fourier modes τðx0;xÞ¼Zdωd3q ð2πÞ4e−iωx0þiq·x˜τðω;qÞ; ϑðx0;xÞ¼Zdωd3q ð2πÞ4e−iωx0þiq·x ˜ ϑðω;qÞ:ð3:21Þ Expanding at low momentum q2=μ2≪1, the equations have solutions when the modes satisfy the dispersion relations ω2≃q2 3;ω2≃4μ2þ5 3q2:ð3:22Þ Therefore, there is a gapless mode πand a gapped mode σ, which at low momentum correspond respectively to the combinations ˜π≃ ˜ ϑ−isignðω=qÞq ffiffiffi 2 pμ ˜τ; ˜σ≃˜τ−iffiffiffi 2 3 rsignðω=μÞ1þq2 24μ2˜ ϑ:ð3:23Þ A few comments are in order. In the first place, the dispersion relation of πin (3.22) is such that it moves at the speed of sound as fixed by conformal invariance c2 s¼1=3, i.e., it can be identified as a conformal superfluid phonon, while σis the gapped dilaton. This identification is consistent with an effective field theory approach, see e.g., [4]. Note that the mixing is necessary for this to happen, otherwise the phonon would move at the speed of light due to relativistic invariance of the rest of the terms. The second observation is that the gap of σis fixed by the chemical potential mσ¼2μ, and independent of the couplings λand λ0in this approximation. This is very reminiscent of the massive Goldstone bosons appearing when internal symmetries are spontaneously broken in the presence of a chemical potential. A last observation is that because of the mixing, it is no longer true that each Ward-Takahashi identity is tied to one specific mode. Indeed reexpressing τ and ϑin terms of πand σ, one can easily see that both fields appear in both equations (3.20). C. Exact dispersion relations The results obtained from the Ward-Takahashi identities are easy to interpret physically but we had to introduce several approximations to derive them, in particular we used the hierarchy between the masses of the Higgs fluctuation and the dilaton to freeze out the first. In order to go beyond this approximation we need to include the Higgs mode in the analysis, whose dynamics is not captured by the Ward-Takahashi identities. This can be more simply done using the effective Lagrangian. Consider again the vacuum ξ2¼jϕj2¼v2¼μ2 2λ0and the fluctuations (3.15) around it. The quadratic Lagrangian for the fluctuations is Lquad ¼1 2∂μρ∂μρþ1 2∂μϑ∂μϑþ1 2∂μτ∂μτ þ2ffiffiffi 2 3 rμτ∂tθþ2 ffiffiffi 3 pμρ∂tθ−2 3ffiffiffi 2 pμ2τρ −2 3μ2τ2−μ29λþλ0 3λ0ρ2:ð3:24Þ By going to Fourier space we get Lquad ¼1 2yTð−ω;−qÞ·Mðω;qÞ·yðω;qÞ;y¼ðϑ;ρ;τÞ; ð3:25Þ where ARGURIO, HOYOS, MUSSO, and NAEGELS PHYS. REV. D 102, 076011 (2020) 076011-6 M¼0 B B B @ ω2−q2i2 ffiffi3 pμω i2ffiffi2 p ffiffi3 pμω −i2 ffiffi3 pμω ω2−q2−2ð9λþλ0Þ 3λ0μ2−2ffiffi2 p 3μ2 −i2ffiffi2 p ffiffi3 pμω −2ffiffi2 p 3μ2ω2−q2−4 3μ2 1 C C C A : ð3:26Þ Studying the zeros of the determinant of M, one finds one massless mode, the Uð1ÞNG boson, and two gapped modes: ω2 1jq¼0¼0; ω2 2;3jq¼0¼3μ2 λ0λþλ0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ2−2 3λλ0þλ02 r:ð3:27Þ Expanding for low momentum qand for λ0≪λ, we get ω2 1≃1 3q2;ð3:28Þ ω2 2≃6μ2λ λ01þλ0 3λþ1þ2λ0 9λq2;ð3:29Þ ω2 3≃4μ21−λ0 3λþ5 3−2λ0 9λq2:ð3:30Þ Comparing with the dispersion relations in (3.22),we observe that the speed of the phonon is not modified by corrections depending on λ0, while the mass of the gapped dilaton is corrected, though mildly. Indeed, contrary to massive NG bosons associated to internal symmetries, the mass of the gapped dilaton is not protected by the symmetry. For λ0≪λ,ω2 1and ω2 3reduce to the dispersion relations obtained in (3.22) from the study of the Ward-Takahashi identities, and we have a hierarchy between the two massive modes. Furthermore, in the limit μ→0;λ0→0with μ2 2λ0→v2;ð3:31Þ we recover the masses (2.6) of the relativistic model (2.2) ω2 1jq¼0¼0; ω2 2jq¼0¼12v2λ; ω2 3jq¼0¼0;ð3:32Þ and ω3describes the massless dilaton. This suggests a connection between the corrections to the mass of the gapped dilaton at finite chemical potential and the lack of a flat direction in the potential at zero chemical potential. The masses of gapped NGs might be protected only if there are flat directions associated to them, of course this will always be the case for internal symmetries. IV. BOOSTED SUPERFLUID Since the chemical potential breaks Lorentz invariance, it is interesting to study the effect on the NG modes when the superfluid is set on motion relative to the frame determined by the effective Hamiltonian induced by the chemical potential, that one can identify as the “laboratory”frame. We consider again (2.2) and introduce both a chemical potential and a superfluid velocity ψ¼eiμ0uμxμϕ;ψ¼e−iμ0uμxμϕ:ð4:1Þ Where uμ¼γð1;−  βÞ,γ¼1=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1−jβj2 pis a timelike fourvelocity uμuμ¼þ1. The chemical potential is μ¼γμ0, and the time direction in the laboratory frame is x0. The background plane wave (4.1) is the same as (3.2) seen by a boosted observer, compared to the laboratory frame. Since (2.2) is Lorentz invariant, the dispersion relations for the gapless low-energy modes can be obtained by boosting those obtained from (3.1) (i.e., the case with just a chemical potential). For the sake of providing an explicit check, we repeat the exercise of computing them directly through the Ward-Takahashi identities and through the perturbative Lagrangian approach. A. Effective Hamiltonian and ground state We proceed in a similar fashion to the case of zero velocity. The Hamiltonian and the charge are still determined by the energy-momentum tensor and the current as in (3.4), and the effective Hamiltonian at nonzero chemical potential by (3.5). Because of the boost, the expressions for the current and the energy-momentum tensor are slightly modified. Jμ¼2μ0jϕj2uμþjμ; Tμν ¼2μ2 0uμuνjϕj2þμ0ðuμjνþuνjμÞ−ημνμ0uαjαþtμνðμ0Þ: ð4:2Þ Where jμand tμν take the same form as before (3.7) and (3.11), replacing μby μ0. Recalling that the chemical potential is μ¼μ0u0¼μ0γ, the effective Hamiltonian is H−μQ¼Zd3xðT00 −μJ0Þ¼Zd3xðt00ðμ0Þ−μ  β·  jÞ: ð4:3Þ Since jivanishes for constant ϕ, the extrema of the effective potential are the same as before (3.14) replacing μby the effective chemical potential in the rest frame of the fluid μ0. The ground state is thus ξ2¼jϕj2¼v2 0¼μ2 0 2λ0. GAPPED DILATONS IN SCALE INVARIANT SUPERFLUIDS PHYS. REV. D 102, 076011 (2020) 076011-7 B. Nambu-Goldstone dynamics from Ward-Takahashi identities We can use the same parametrization for perturbations of the ground state as in (3.15), replacing vby v0. The same considerations about the mass hierarchy of τand ρapply, so in this analysis we will assume λ0≪λand freeze ρ. The dynamics of the low energy modes are determined by the conservation equations for the current and the energymomentum tensor. For the boosted superfluid they take the form ∂μJμ¼2μð∂0þ  β·  ∇Þjϕj2þ∂μjμ; Tμμ¼2μ2 0jϕj2−2μ0uμjμþtμμðμ0Þ ¼2μ2 0jϕj2−2μðj0þ  β·  jÞþtμμðμ0Þ:ð4:4Þ Therefore, we should just replace the terms with a single time derivative by the material derivative μ∂0→μD0¼ μð∂0þ  β·  ∇Þand otherwise change μby the effective μ0: ∂μJμ≃ffiffiffi 2 pv0∂2ϑþ2ffiffiffi 2 3 rμD0τ; Tμμ≃ffiffiffi 3 pv0∂2τþ4 3μ2 0τ−2ffiffiffi 2 3 rμD0ϑ:ð4:5Þ From this, we obtain the equations ∂2τþ4 3μ2 0τ−2ffiffiffi 2 3 rμD0ϑ≃0; ∂2ϑþ2ffiffiffi 2 3 rμD0τ≃0:ð4:6Þ The dispersion relation for the gapless mode can be more easily found by noting that μ¼γμ0and using comoving coordinates. Taking  βparallel to the x3direction, we introduce x0¼γðx0 βþβx3 βÞ;x 3¼γðx3 βþβx0 βÞ; x1¼x1 β;x 2¼x2 β:ð4:7Þ Then ∂ ∂x0 β¼γð∂0þβ∂3Þ;∂ ∂x3 β¼γð∂3þβ∂0Þ;∂2¼∂2 β: ð4:8Þ The equations become ∂2 βτþ4 3μ2 0τ−2ffiffiffi 2 3 rμ0∂x0 βϑ≃0; ∂2 βϑþ2ffiffiffi 2 3 rμ0∂x0 βτ≃0:ð4:9Þ These are the same as before (3.20), replacing μby μ0.We introduce an expansion of the modes in the rest frame in plane waves τðx0 β;xβÞ¼Zdωβd3qβ ð2πÞ4e−iωβx0 βþiqβ·xβ˜τðωβ;qβÞ; ϑðx0 β;xβÞ¼Zdωβd3qβ ð2πÞ4e−iωβx0 βþiqβ·xβ ˜ ϑðωβ;qβÞ:ð4:10Þ We recover the expected low momentum dispersion relations in the rest frame ω2 β≃c2 sq2 β;ω2 β≃4μ2 0þ5 3q2 β;ð4:11Þ where c2 s¼1=3is the speed of sound of the scale invariant theory. These expressions can be translated to frequency and momentum in the laboratory frame using that ω¼γðωβþβqβ3Þ;q 3¼γðqβ3þβωβÞ; q1¼qβ1;q 2¼qβ1:ð4:12Þ Note that the dispersion relations (4.11) are valid for low momentum in the rest frame of the fluid jqβj≪jμ0j. For the gapless modes they can be matched with a low momentum expansion in the laboratory frame jqj≪jμj, however for the gapped modes this is not possible, as for generic β, q3∼ωβ∼μ. Therefore, finding the dispersion relations of the gapped modes at low momentum in the laboratory frame requires solving (4.6) directly. We classify the dispersion relations of the gapless modes taking as reference the direction of the superfluid velocity in the laboratory frame. The dispersion relations for the longitudinal modes is ωk¼csβ 1βcs q3;q 1¼q2¼0;ð4:13Þ while the dispersion relation for the transverse modes is ω2 ⊥¼c2 s q2 1þq2 2 γ2ð1−β2c2 sÞ;q 3¼0:ð4:14Þ These expressions agree with the ones obtained by relativistic addition of velocities. Note that for jβj>c sboth (positive frequency) longitudinal modes (4.13) propagate in the same direction as the superfluid velocity. This is the reason why we expressed linearly the dispersion relations. ARGURIO, HOYOS, MUSSO, and NAEGELS PHYS. REV. D 102, 076011 (2020) 076011-8 For the gapped modes the low momentum dispersion relations are ωk¼2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1−β2c2 s qμ−2 3 βq3 1−β2c2 sþ5þβ2c2 s 12γ2ð1−β2c2 sÞ5=2 q2 3 μ; q1¼q2¼0; ω2 ⊥¼4ð1−β2c2 sÞ2μ2þ5−β2 3ð1−β2c2 sÞðq2 1þq2 2Þ;q 3¼0; ð4:15Þ where again the longitudinal dispersion relation is expressed linearly. The gap is reduced by the superfluid velocity, but in this approximation remains finite even in the limit β→1, where the condensate vanishes (i.e., at fixed μ). Note also that, at leading order for momenta in the same direction of the flow, the frequency is reduced. C. Exact dispersion relations We now study the effects of including the Higgs fluctuation ρ, and the corrections for finite λ0=λ. We thus resort to expanding the full Lagrangian. According to (4.1), we switch on a chemical potential μ¼μ0γand a background wave vector k3¼μ0γβ. The effective potential is now: V¼λðjϕj2−ξ2Þ2þλ0jϕj4−ðμ2−k2 3Þjϕj2;ð4:16Þ For stationary solutions, the situation is not very different from the case with just μ, in fact one just needs to replace μ2 by ðμ2−k2 3Þ¼μ2 0in (3.13). Therefore, we consider the solution ξ2¼jϕj2;jϕj2¼μ2 0 2λ0:ð4:17Þ The fluctuation around (4.17) are still given by (3.15) where however v¼v0¼μ0 ffiffiffiffiffi 2λ0 p. Writing k3¼βμ, the quadratic Lagrangian for the fluctuations is Lquad ¼1 2∂μρ∂μρþ1 2∂μϑ∂μϑþ1 2∂μτ∂μτ þ2ffiffiffi 2 3 rμτð∂tþβ∂3Þθþ2 ffiffiffi 3 pμρð∂tþβ∂3Þθ −2 3ffiffiffi 2 pμ2 0ρτ −2 3μ2 0τ2−μ2 0 9λþλ0 3λ0ρ2:ð4:18Þ In analogy to (3.25) and (3.26), by going to Fourier space we get the kinetic matrix: 0 B B B @ ω2−q2i2 ffiffi3 pμðω−βq3Þi2ffiffi2 p ffiffi3 pμðω−βq3Þ −i2 ffiffi3 pμðω−βq3Þω2−q2−2ð9λþλ0Þ 3λ0μ2 0−2ffiffi2 p 3μ2 0 −i2ffiffi2 p ffiffi3 pμðω−βq3Þ−2ffiffi2 p 3μ2 0ω2−q2−4 3μ2 0 1 C C C A :ð4:19Þ From the determinant of (4.19), one can find the exact dispersion relations. First of all, setting the momenta q¼0one finds that there is a massless mode corresponding to the Uð1ÞNG boson and two gapped modes: ω2 1jq¼0¼0;ω2 2;3jq¼0¼3 λ0λμ2 0þλ0μ2ð1−c2 sβ2Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ2μ4 0−2 3λμ2 0λ0μ2ð1−c2 sβ2Þþλ02μ4ð1−c2 sβ2Þ2 r;ð4:20Þ where cs¼1=3as before. Now, expanding at low frequencies and momenta, one can extract analytically the dispersion relation for the Uð1ÞNG mode: ω1¼cs 1−c2 sβ2ð2csβq3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1−β2Þ2q2 3þð1−β2Þð1−c2 sβ2Þðq2 1þq2 2Þ qÞ:ð4:21Þ Notice that the above expression is independent of the ratio λ0=λ. Indeed one can check that in the longitudinal and transverse case, it reproduces correctly the expressions (4.13) and (4.14), respectively. For the massive modes, one has to expand the frequencies around the respective gaps. To first order in momenta and in λ0=λ, the dispersion relations for the gapped dilaton are: GAPPED DILATONS IN SCALE INVARIANT SUPERFLUIDS PHYS. REV. D 102, 076011 (2020) 076011-9