Scattering Amplitudes of Massive Nambu–Goldstone Bosons

Scattering amplitudes of massive Nambu–Goldstone bosons

1, 2, Tom´aˇsBrauner ∗ and Martin F. Jakobsen † 1Department of Mathematics and Physics, University of Stavanger, 4036 Stavanger, Norway 2Department of Physics, Norwegian University of Science and Technology, 7491 Trondheim, Norway Massive Nambu–Goldstone (mNG) bosons are quasiparticles whose gap is determined exactly by symmetry. They appear whenever a symmetry is broken spontaneously in the ground state of a quantum many-body system, and at the same time explicitly by the system’s chemical potential. In this paper, we revisit mNG bosons and show that apart from their gap, symmetry also protects their scattering amplitudes. Just like for ordinary gapless NG bosons, the scattering amplitudes of mNG bosons vanish in the long-wavelength limit. Unlike for gapless NG bosons, this statement holds for any scattering process involving one or more external mNG states; there are no kinematic singularities associated with the radiation of a soft mNG boson from an on-shell initial or ﬁnal state. M. F. Jakobsen would like to dedicate this article to his parents (Bjørnar & Jannicke), grandparents (Frode & Marit and Per-Gunnar & Solbjørg), grandaunt (Turid) and sister (Katrine). Thank you for your unwavering support and encouragement during my studies.

PACS numbers: 11.30.Qc, 14.80.Va Keywords: Massive Nambu–Goldstone boson, spontaneous symmetry breaking, Adler’s zero

I. INTRODUCTION The story is further complicated by the fact that not all pNG bosons stemming from explicit breaking of a Spontaneous symmetry breaking is one of the most im- symmetry by a chemical potential are mNG bosons [2,5]. portant concepts in modern quantum physics. It is re- Examples of such states are somewhat exotic but not too sponsible for a vast range of phenomena, ranging from difficult to construct, the simplest one perhaps appearing superfluidity and ferromagnetism to the generation of in a system where a global SO(3) symmetry is completely masses of elementary particles. As a rule, it is associated spontaneously broken. Unlike the true mNG bosons, the with the presence of gapless quasiparticles in the spec- presence of such states in a given system is, however, not trum of the system: the Nambu–Goldstone (NG) bosons. guaranteed. We will revisit this case in AppendixA. Exact global symmetries are, however, rare in nature. The goal of this paper is to investigate further proper- When the spontaneously broken symmetry is not exact ties of mNG bosons beyond the sole fact that their gap but merely approximate, the associated soft mode ac- is fixed by symmetry. It is well known that ordinary NG quires a gap. Such modes are referred to as pseudo-NG bosons interact weakly at low energies. More precisely, (pNG) bosons. As a rule, the gap of a pNG boson de- barring special circumstances leading to a kinematic sin- pends not just on symmetry alone, but also on the details gularity, the scattering amplitude for a process involving of the dynamics of the system. a NG boson vanishes in the limit where the momentum of It turned out only recently that under certain circum- this NG boson goes to zero. This fact is usually referred stances, the gap of a pNG boson is determined exactly by to as Adler’s zero, and has recently been re-investigated symmetry [1]. Namely, breaking an otherwise exact sym- intensively in the context of a constructive approach to metry by coupling a chemical potential to one of its gen- scattering amplitudes; see Refs. [6–8] for some relevant erators leads to pNG-like modes with a gap fixed by the publications on the subject [9]. symmetry algebra and the chemical potential alone, inde- Here we show that mNG bosons share this property pendently of the details of the underlying dynamics. Such despite their gap. In fact, the nonzero gap protects them arXiv:1709.01251v2 [hep-th] 30 Jan 2018 modes have been dubbed massive NG (mNG) bosons [2]. against the mentioned kinematic singularities so that the The list of currently known mNG bosons covers a range of scattering amplitude for any process involving a mNG systems from condensed-matter to high-energy physics, external state vanishes as its momentum goes to zero. and includes (anti)ferromagnetic magnons in an exter- The plan of the paper is as follows. In Sec.II, we re- nal magnetic field, the neutral pion in a pion super- view the basic facts about mNG bosons. We also discuss fluid in dense quark matter, certain excitations of rel- to some extent how current conservation, which is crucial ativistic Bose–Einstein condensates [2], and the Kohn for establishing the existence of Adler’s zero, is modified mode [3] corresponding to center-of-mass oscillations of in the presence of a chemical potential. In the next two Bose–Einstein condensates in harmonic traps [4]. sections, we then warm up by analyzing in detail two concrete examples of systems featuring a mNG boson. The system described in Sec.III captures the behavior of antiferromagnetic magnons in an external magnetic field. ∗ [email protected] Its key advantage is that its relativistic kinematics is un- † [email protected] affected by the chemical potential, which only modifies 2 the perturbative interactions of magnons. The example Consider rather generally a class of theories defined by studied in Sec.IV, known from certain scenarios for kaon their classical action S[φ, A], depending on a set of scalar a i condensation in dense quark matter [10], features fully fields φ and gauge fields Aµ. Suppose that this action nonrelativistic kinematics despite its relativistic origin. is invariant under a set of simultaneous local transforma- It thus brings to light most of the subtleties that we will tions with infinitesimal parameters i(x), have to deal with in Sec.V, where a general argument for a i a i i i j k Adler’s zero in scattering amplitudes of mNG bosons is δφ = Fi (φ, A), δAµ = ∂µ + fjkAµ , (4) presented. Finally, in Sec.VI we summarize our findings i and give some concluding remarks. We also discuss to where fjk are the structure constants of the symmetry a some extent the limit in which the momenta of two NG group and Fi some local functions of the fields and pos- or mNG bosons, participating in a scattering process, are sibly of their derivatives. The requirement of gauge in- taken to zero simultaneously. variance implies the condition

Z δS δS dx iF a + (∂ i + f i Aj k) = 0. (5) II. MASSIVE NAMBU–GOLDSTONE BOSONS a i i µ jk µ δφ δAµ AND CURRENT CONSERVATION By using the equation of motion for the scalar field, Following Ref. [2], consider a quantum system defined δS/δφa = 0, we infer immediately that the Noether cur- µ by its Hamiltonian H. Suppose that we pick one of the rents, defined by J (x) δS/δAi (x) up to a conven- i ≡ µ generators Q of its symmetry group G and assign it a tional sign, satisfy the covariant conservation law chemical potential, µ. The excitation spectrum of the D J µ ∂ J µ + f k Aj J µ = 0. (6) system is then determined by the many-body Hamilto- µ i ≡ µ i ij µ k nian H˜ H µQ. This Hamiltonian generally does not commute≡ with− the full group G; let us denote the Note the generality of our argument. First, we did not subgroup of G commuting with H˜ as G˜. assume any particular form of the transformation rule a By the standard Cartan decomposition of Lie algebras, for the scalar fields: the function F (φ, A) need not be the symmetry generators not commuting with H˜ can be linear, and it may even depend on field derivatives. Sec- organized into Hermitian-conjugate pairs Q± such that ond, we did not make any specific assumptions on the i Lagrangian density: it may depend on higher derivatives

[Q, Qi±] = qiQi±, (1) of the fields, and it may change upon the transforma- ± i tion (4) by a surface term. Finally, the gauge field Aµ in where qi are the roots of the Cartan subalgebra. It Eq. (6) is treated as a non-dynamical background, but it ˜ then follows that acting with Qi± on an eigenstate of H may take an arbitrary coordinate-dependent value. changes its energy (eigenvalue of H˜ ) by µqi. As a con- What we are actually interested in is the situation in ∓ Q sequence, once both µ and qi are chosen without loss of which the background gauge field Aµ for the generator generality to be positive, the many-body ground state Q equals (µ, 0); all the other background gauge fields Ai + µ 0 satisfies Q 0 = 0. On the other hand, Q− 0 can can be set to zero upon taking the functional derivative in | i i | i i | i be nonzero, and if it is (which signals spontaneous sym- order to obtain the Noether currents. It follows that the µ metry breaking), it represents a mNG state with energy currents J associated with the generators Q±, satisfying ± µqi. Eq. (1) (we drop for the sake of simplicity the index i), The total number of mNG states in the spectrum can fulfill the conservation law be determined as follows [2]. Define the real antisymmet- µ 0 ric matrix of commutators, ∂µJ iµqJ = 0. (7) ± ± ± 1 Consider now the one-particle state of a mNG boson ρij i lim 0 [Qi,Qj] 0 (2) ≡ − Ω Ωh | | i →∞ carrying momentum p, denoted as G(p) . By the argument following Eq. (1), this state| can bei created from (Ω denotes the spatial volume of the system), and the the many-body vacuum 0 by Q−. The matrix element analogous matrixρ ˜ , composed of generators of G˜ only. µ ij G(p) J (x) 0 is therefore| i nonzero. Spacetime transla- The number of mNG bosons is then given by htion invariance| − | i and spatial rotation invariance constrain 1 it to take the form nmNG = (rank ρ rankρ ˜). (3) 2 − µ ip x µ µ0 G(p) J (x) 0 = e · ip F1( p ) + iδ F2( p ) , (8) To provide a somewhat different perspective on the h | − | i | | | | spectrum of mNG bosons, we now discuss the conserva- where F1( p ) and F2( p ) are a priori unknown functions tion laws for Noether currents in the presence of a chem- of the mNG| | boson momentum.| | Applying the conserva- ical potential. We use the fact that in the Lagrangian tion law (7) to the current J µ then gives formalism, the chemical potential can be introduced as a − constant background temporal gauge field [11]. ω2F + ω(F µqF ) p2F µqF = 0, (9) 1 2 − 1 − 1 − 2 3 where ω(p) is the dispersion relation of the mNG mode. k k0 k k k k0 It is easy to see that ω(0) = µq is a solution of this equa- 0 tion for arbitrary F and F , which provides yet another 1 2 ++ k + p p0 k derivation of the mass of the mNG boson. −

We shall utilize the conservation law (7) and the matrix p p0 p p0 p p0 element (8) in our discussion of the mNG boson scattering amplitudes in the next sections. FIG. 1. Feynman diagrams for the scattering amplitude for the πG → πG process. The dashed line stands for the π field and the solid line for G. The four-momenta of the NG boson III. CASE STUDY: ANTIFERROMAGNET IN in the initial and final state are denoted respectively as k and EXTERNAL MAGNETIC FIELD k0, whereas those of the mNG boson are denoted as p and p0. The arrows indicate the flow of momentum. Let us start our discussion of scattering amplitudes of mNG bosons by looking at a concrete example. It is clear from Eq. (1) that the presence of a mNG boson up to constant and surface terms. We can see that the requires non-Abelian symmetry. We therefore choose to model contains one exactly massless mode and one mode study the simplest non-Abelian relativistic model with with the mass equal to µ, which is our mNG boson. This the symmetry-breaking pattern SO(3) SO(2). At the corresponds to the well-known fact that out of the two leading order of the derivative expansion,→ its low-energy magnons in antiferromagnets, only one becomes gapped effective Lagrangian is just the nonlinear sigma model, when an external magnetic field is turned on. Our notation then is: π for the truly massless (NG) mode, and G 1 L = (D φ~)2, (10) for the gapped (mNG) mode. 2 µ where the vector field φ~ has a fixed modulus, φ~ = v. The A. Scattering amplitude: direct calculation | | covariant derivative includes a background gauge field A~µ of SO(3) via In order to get insight in the properties of the scattering amplitudes in our model, let us perform a sample ~ ~ ~ ~ Dµφ ∂µφ + Aµ φ. (11) calculation and inspect the four-particle process ≡ × For future reference, we take note of the Noether currents πG πG, (15) arising from the SO(3) symmetry, → see Fig.1 for the corresponding Feynman diagrams and δS the explanation of our notation. A simple calculation J~µ = = φ~ Dµφ.~ (12) δA~µ × leads to the tree-level result for the on-shell amplitude with amputated external legs [12], This model can be thought of as describing the low- 2 energy dynamics of spin waves in antiferromagnets in an 2 2µ k0k0 1 1 ~ M = 2 k k0 + 2 . (16) external magnetic field, represented by A0. v · v p k − p k0 We choose the magnetic field to point along the z-axis, · · Let us first inspect the properties of this amplitude as that is, set A~µ = δµ0(0, 0, µ). In the classical ground the momentum of one of the NG states, say the incoming state, the field φ~ is then oriented in the xy plane, and we one, goes to zero. Naively the amplitude vanishes thanks can choose it to point in the x-direction, φ~ = (v, 0, 0). to the presence of the factors of k in the numerators. The fluctuations above this ground state areh i parameter- However, since p k = p k, both terms in the parentheses ized by two scalar fields, which we denote as π and G 0 0 in Eq. (16) are· singular· in this limit. A more careful for a reason that will be clear shortly. We shall use the evaluation leads to following nonlinear parameterization that automatically 2 takes account of the constraint on the length of φ~, 2µ k0 1 1 lim M = 0 , k 0 v2 p p cos α − p p cos β p T → 0 0 0 φ~ = v2 π2 G2, π, G . (13) − | | − | | (17) − − where α and β are the angles between k and p and p0, Inserting this into the Lagrangian, it acquires a form that respectively. The absence of Adler’s zero in such a scat- is suitable for a perturbative analysis of the model, tering process is a well-known issue, which arises from the presence of cubic interaction vertices in the model [13], 1 2 1 2 1 2 2 L = (∂µπ) + (∂µG) µ G (14) and thus from the latter two Feynman diagrams in Fig.1: 2 2 − 2 as the momentum k goes to zero, the internal propagator 2 p 2 2 2 1 (π∂µπ + G∂µG) in these diagrams approaches the mass shell, leading to + 2µ(∂0π) v π G + , − − 2 v2 π2 G2 a kinematic singularity. − − 4

computation. Should we be able to prove the existence k k0 of Adler’s zero for mNG bosons on general grounds, we need a more robust approach. To that end, recall that f J µ (p) i = = + + h | − | i p + k p k0 the presence of Adler’s zero for true, massless NG bosons − is usually proved as a direct consequence of conservation

p p0 of the Noether current associated with the spontaneously broken symmetry [15]. We shall now therefore imagine + + + that the incoming mNG state in the process shown in p + k p k0 µ − Fig.1 is created by the current operator J , and investigate the matrix element − FIG. 2. Feynman diagrams contributing to the matrix ele- µ ment hf|J−(p)|ii. The dot on the external line carrying momentum p represents the current operator, otherwise the notation is the same as in Fig.1. µ µ f J (p) i k0, p0 J (p) k . (19) h | − | i ≡ h | − | i If, on the other hand, one of the mNG bosons in the process becomes soft [14], no such a kinematic singularity appears due to the non-vanishing mass of the mNG boson. A simple manipulation using the kinematics of the process shows that Note that the kinematics corresponding to this matrix element is different than that of the scattering amplitude lim M = lim M = 0. (18) in Fig.1: whereas the four-momenta k, k0, p0 label one- p 0 p0 0 → → particle asymptotic states and therefore are on-shell, the four-momentum p is created by the local current operator This is our first piece of evidence that the interactions of and thus can be off-shell. Keeping this momentum off, if mNG bosons are weak at low momentum in spite of their close, the mass shell is of course all-important for under- nonzero mass. standing the analytical structure of the matrix element and extracting from it the physical scattering amplitude. B. Scattering amplitude from current conservation As the first step, we write down the perturbative ex- So far, we have found Adler’s zero in a single scatter- pansion of the Noether currents (12), just as we previ- ing amplitude of the mNG state in our model by a direct ously did for the Lagrangian,

π2 + G2 J µ = π∂µG G∂µπ + δµ0µG v + , 1 − 2v − ··· 1 1 J µ = δµ0µπG v∂µG ∂µG(G2 π2) πG∂µπ + , (20) 2 − − − 2v − − v ··· 1 1 J µ = δµ0µ(v2 G2) + v∂µπ + πG∂µG ∂µπ(G2 π2) + , 3 − v − 2v − ···

where terms of higher order in the fields are omitted. the pole contributions to the matrix element f J µ (p) i . µ µ µ µ Using only the knowledge of the propagator ofh the| −G-field| i The matrix element f J (p) i with J J1 iJ2 µ can now be evaluated perturbativelyh | − | i similarly− ≡ to the− pre- and of the linear pieces of the current J , that is without − vious direct calculation of the scattering amplitude. The having to evaluate the scattering amplitude explicitly, Feynman diagrams that contribute to it are shown in the pole part of the matrix element of the current can be Fig.2 and fall into two distinct classes. The first three expressed as diagrams arise from the part of the current linear in G, µ0 µ 2 2 µ iv(µδ + p ) and contain a pole at p = µ . The last three arise from f J (p) i pole = ( iMoff-shell), (21) h | − | i − p2 µ2 − the parts of the current quadratic and cubic in the fields, − 2 and do not have a simple pole in the p variable. where the subscript “off-shell” refers to the fact that only It is obvious from Figs.1 and2 that there is a one- the momenta k, k0, p0 but not p are now on-shell in the to-one correspondence between diagrams contributing to scattering amplitude. the scattering amplitude for the process πG πG and The non-pole part of the current matrix element is → 5 likewise evaluated straightforwardly, The full symmetry group of the model in the limit µ = 0 is G = SO(4) SU(2) SU(2), which is most easily ' × µ 1 µ0 µ µ µ seen by thinking of φ as a collection of four real scalar f J (p) i non-pole = (µδ + k + p0 k0 ) h | − | i v − fields. The non-Abelian nature of this symmetry creates µk0 a convenient setting for the presence of mNG bosons in 0 (µδµ0 + pµ + 2kµ) (22) − vp0 k0 the spectrum. When µ > m, the classical ground state of · the model carries a nonzero expectation value of φ, and µk0 µ0 µ µ (µδ + p 2k0 ), can be chosen as − vp0 k − · r 1 0 µ2 m2 where we used momentum conservation and the on-shell φ = , v − . (26) condition for k, k0, p0 to simplify the result. h i √2 v ≡ λ We shall now see that the scattering amplitude for the πG πG process is actually completely determined by The symmetry-breaking pattern then reads the→ non-pole diagrams in Fig.2. Indeed, the operator G = SU(2) SU(2) SU(2)0, momentum conservation condition (7) implies that L × R → (27) ˜ 0 µ G = SU(2)L U(1)R U(1) . (pµ µδµ0) f J (p) i = 0. (23) × → − h | − | i Here the primes refer to the fact that the generators of This leads to a cancellation of the pole in the pole part the unbroken SU(2) and U(1) subgroups are linear com- of the current matrix element, upon which the off-shell binations of generators of the SU(2) and U(1) factors in amplitude Moff-shell can be expressed as G and G˜, respectively. We can see that two of the sym- 2 metry generators are broken spontaneously and at the 2 2µ k0k0 1 1 M = k k0 + . (24) same time explicitly by the chemical potential, and thus off-shell v2 · v2 p k − p k 0 · 0 0 · expect a single mNG boson in the spectrum. To check this, we parameterize the doublet φ as Once the momentum p is set on the mass shell, this is seen to be equivalent to the previously found result (16). 1 ϕ The moral of this exercise is that we do not need to φ , (28) √ v + ψ3 + iψ4 calculate the scattering amplitude explicitly: it can be ≡ 2 extracted from the non-pole contributions to the matrix where ϕ is a complex field, whereas ψ3,4 are real. Insert- element of the broken current upon using current con- ing this into the model Lagrangian and dropping constant servation. This is a major step towards proving that the terms, it becomes scattering amplitude vanishes in the limit of zero momentum of the mNG boson. Before proceeding to the µ 2 2 L = ∂µϕ∗∂ ϕ + iµ(ϕ∗∂0ϕ ϕ∂0ϕ∗) λv ψ3 (29) general argument, we will however work out in detail an- − − 1 2 1 2 other example. In the calculation above, we have namely + (∂µψ3) + (∂µψ4) + µ(ψ4∂0ψ3 ψ3∂0ψ4) used heavily the relativistic kinematics to simplify the 2 2 − expressions. We want to see to what extent the situation 2 2 λ 2 2 2 λvψ3(2ϕ∗ϕ + ψ3 + ψ4) (2ϕ∗ϕ + ψ3 + ψ4) . complicates in systems where not only the interactions, − − 4 but also the kinematics are not Lorentz-invariant. It is easy to see that the ϕ field excites a pair of states with the dispersion relations IV. CASE STUDY: RELATIVISTIC MODEL p ω (p) = p2 + µ2 µ. (30) FOR KAON CONDENSATION ± ± These can be thought of as a genuine particle–antiparticle Following Ref. [10], we introduce the linear sigma pair thanks to the fact that they carry the charge of the model, defined by the Lagrangian unbroken exact U(1)0 symmetry. The lighter of the two is µ 2 2 gapless and represents a so-called type-B NG boson [16]. L = D φ†D φ m φ†φ λ(φ†φ) , (25) µ − − The heavier of the two, on the other hand, has gap 2µ. This is the mNG boson of the extended symmetry group where φ is a doublet of complex scalars and the covariant G, broken both explicitly and spontaneously [2]. It has derivative incorporates a chemical potential via D φ 0 been shown by an explicit calculation that its gap does (∂ iµ)φ. The Lagrangian has a manifest G˜ = SU(2) ≡ 0 not receive radiative corrections at one loop [17]. U(1)− symmetry, corresponding to unitary rotations of the× The ψ sector of the model likewise contains two ex- φ doublet. The chemical potential µ is then associated 3,4 citations with the nonrelativistic dispersion relations with the U(1) factor of the symmetry group. This model has been used to describe kaon condensation in dense q 2 2 2 p 2 2 2 2 2 quark matter, where the SU(2) stands for isospin and ω3,4(p) = p + 3µ m (3µ m ) + 4µ p . − ± − U(1) for strangeness. (31) 6

plitude with amputated external legs,

p p0 M = 4λ (34) p p ψ 3 4λ2v2(p + k)2 p + k p0 p0 + 2 2 2 2 2 p p0 (p + k) [(p + k) 2λv ] 4µ (p + k ) + + − 0 0 ψ3 ψ3 − − k0 k0 2 2 2 4λ v (p p0) + − . k k ψ3 2 2 2 2 2 (p p0) [(p p0) 2λv ] 4µ (p0 p0) k k0 − − − − − Adler’s zero is not manifest in this case, which is com- mon for linear sigma models: a cancellation between two or more Feynman diagrams is usually required in order FIG. 3. Feynman diagrams for the scattering amplitude for to ascertain the vanishing of the scattering amplitude in the NG + mNG → NG + mNG process. All quasiparticles the soft limit. To that end, note that the dispersion re- participating in the process are excited by the ϕ field. The NG mode is treated as a particle and thus corresponds to an lations (30) for the NG and mNG mode can be encoded incoming line in the initial state and an outgoing line in the in the kinematic relations final state. The mNG mode is treated as an antiparticle and 2 2 thus corresponds to an outgoing line in the initial state and an p = 2µp0, k = 2µk0, (35) − incoming line in the final state. The dashed line represents the matrix propagator D; only the Dψ3ψ3 component is needed and analogously for p0 and k0. It is then easy to see that here since there are no cubic interaction vertices linear in p 0 ψ4 in the model. The notation for the four-momenta of the 2 (p + k) → 2µ(p0 + k0), gapless and the gapped state is the same as in Fig.1. The −−−→ (36) 0 p 0 arrows on the field lines indicate the flow of the U(1) charge. 2 → (p p0) 2µ(p0 p0), The flow of momentum is indicated by the arrows next to the − −−−→ − momentum labels. which immediately leads to the expected result

lim M = 0. (37) p 0 One of these modes is gapless and corresponds to a so- → called type-A NG boson [16]. The other one is gapped It is easy to check that in this case, the Adler zero prop- and represents a Higgs-like mode. In the calculation be- erty also holds for the gapless NG mode; there is no low, we actually do not need these dispersion relations, kinematic singularity present in this model. That is be- but only the propagator in the ψ sector, which takes a 3,4 cause of the structure of the cubic interaction vertices: matrix form and can be extracted from the bilinear part the internal propagator in the diagrams in Fig.3 carries of the Lagrangian (29), a different mode than the external legs, and thus remains off-shell in the limit k 0. 2 i p 2iµp0 → D(p) = 2 2− 2 . p2(p2 2λv2) 4µ2p +2iµp0 p 2λv − − 0 − (32) B. Scattering amplitude from current conservation All the other Feynman rules of the model can be read off the Lagrangian (29) trivially. As the next step, we shall now again see how to reproduce this result without evaluating the scattering amplitude explicitly, using only current conservation. To that A. Scattering amplitude: direct calculation end, we first need to identify the Noether current that excites the mNG boson of the model. Adding the chemical potential to the theory explicitly breaks two of the gen- Let us now, as in the previous section, evaluate the erators of the right SU(2) factor in the symmetry group scattering amplitude for a sample scattering process. For G. The corresponding currents take the form the sake of convenience, we choose the process µ T J = φ τ2∂µφ + φ†τ2∂µφ∗, R1 − (38) NG + mNG NG + mNG, (33) µ T → J = iφ τ ∂ φ iφ†τ ∂ φ∗, R2 − 2 µ − 2 µ where “NG” refers to the type-B NG mode of the model, where τ2 is the second Pauli matrix. In this case, it is which is the antiparticle of the mNG mode [18]. The dia- more convenient to define the “ladder currents” with an grams contributing to this process at tree level are shown additional factor of √2, in Fig.3, which also explains all the notation required.

A straightforward application of Feynman rules leads µ 1 µ µ J (JR1 iJR2). (39) to the following intermediate result for the on-shell am- ± ≡ √2 ± 7

J µ − p p0 f | i f J µ (p) i = h | − | i

k k0

µ ψ3 J p + k − = + + p p0 ψ3 ψ3 − ψ3 Propagator

p + k p + k + + ψ3 ψ3 ψ4 ψ3 i | i

FIG. 5. A generic scattering process involving a mNG boson.

ψ3 ψ4 The initial and ﬁnal state |ii and |fi can include an arbitrary + + p p0 p p0 − − number of mNG and non-mNG modes. ψ3 ψ3

2µ, the current conservation condition (7) implies

µ (pµ 2µδµ0) f J (p) i = 0, (43) FIG. 4. Feynman diagrams contributing to the matrix ele- − h | − | i µ ment hf|J−(p)|ii. The dot on the external line carrying mo- as opposed to Eq. (23). The prefactor (pµ 2µδµ0) clearly mentum p represents the current operator, otherwise the no- µ − tation is the same as in Fig.3. The ψ3,4 labels on the internal cancels the pole in f J (p) i pole, although the propaga- h | − | i propagators indicate that mixing has to be taken into account. tor of the mNG boson now takes a nonrelativistic form. Upon canceling the pole, the off-shell scattering amplitude can be expressed in terms of the non-pole contribu- µ Only the current J µ is needed as it excites the mNG tions N as boson. Using the parameterization− (28), it becomes − 1 µ 0 µ µ µ µ µ µ Moff-shell = (pµN 2µN ). (44) J = iv∂ ϕ i(ψ3∂ ϕ ϕ∂ ψ3) + (ψ4∂ ϕ ϕ∂ ψ4). −v − − − − − − − − (40) As in the previous section, we now want to evaluate Upon using some kinematics for the initial and the final the matrix element (19). The Feynman diagrams that state of the scattering process, it is straightforward to contribute to it are displayed in Fig.4. The pole part show that in the on-shell limit, this result coincides with of the matrix element is again related to the scattering the previously derived Eq. (34). Even more importantly, amplitude of interest by a simple expression, however, Eq. (44) makes the presence of Adler’s zero in the limit p 0 manifest as long as N µ is not singular µ → − µ ivp in this limit, which it is not by construction. [It does f J (p) i pole = ( iMoff-shell), (41) h | − | i p2 2µp − not include the contribution of the one-particle pole at − 0 p0 = ω+(p).] This is the last crucial ingredient that we where the subscript off-shell indicates that only the four- need for a general proof of the existence of Adler’s zero momenta k, k0 and p0 are on-shell. The non-pole part of µ in scattering amplitudes of mNG bosons. the matrix element, f J (p) i non-pole, which we will for brevity call simply Nh µ|, is− now| i given by a larger number − of diagrams as a result of the mixing of the ψ3,4 fields. V. GENERAL ARGUMENT Evaluating all the contributions explicitly yields

µ µ 2 µ 2λv(p + 2k )[(p + k) + 2µ(p0 + k0)] We would now like to generalize our argument from N = 2 2 2 2 2 the previous section so that it: − − (p + k) [(p + k) 2λv ] 4µ (p0 + k0) µ µ − 2− 2λv(p 2p0 )[(p p0) + 2µ(p0 p0)] Applies to any (compact) symmetry group and 2 − 2 − 2 2 − 2 . • − (p p0) [(p p0) 2λv ] 4µ (p0 p0) symmetry breaking pattern G H. − − − − − (42) → Does not require the evaluation of speciﬁc Feynman In the present case where the gap of the mNG mode is • diagrams, but only relies on current conservation. 8

Does not assume any particular form of the propa- the two four-momenta, it is straightforward to see that • gator of the mNG field. current conservation leads to a complete cancellation of the pole in Eq. (47), upon which the off-shell scattering We will follow rather closely the usual proof of existence amplitude can be expressed as of Adler’s zero for exact spontaneously broken symme- µ tries [15]. A generic scattering process involving a mNG i(pµ µqδµ0)N (p) boson in the initial state can be represented by the dia- Moff-shell(p) = − − . (48) (2π)3ω(p)F ( p ) + F ( p ) gram in Fig.5. Just like in our above analysis of specific 1 | | 2 | | examples, the mNG state is created by a local Noether As the final step, we can bring the four-momentum p current operator, and the diagram therefore corresponds on-shell and take the soft limit. It is now obvious that to the matrix element f J µ (p) i , where p is the mNG h | − | i the scattering amplitude for the process involving a mNG boson four-momentum. boson vanishes in the limit p 0 provided that N µ is To understand the analytic structure of this matrix el- not singular in this limit (which→ it is not by construction),− ement, we will need the Källén–Lehmannspectral repre- and that the denominator in Eq. (48) does not vanish in sentation. Its general nonrelativistic version for a time- this limit. That latter requirement is equivalent to the ordered Green’s function of two local fields, A(x) and statement that the coupling of the broken charge to the B(x), takes the form [17] mNG state does not vanish in the soft limit, which is " actually one of the hallmarks of mNG bosons [2]. This X 0 A(0) n, p n, p B(0) 0 D (p) = i(2π)3 h | | ih | | i concludes our general proof of the existence of Adler’s AB p ω(p) + i n 0 − zero in scattering amplitudes of mNG bosons. # 0 B(0) n, p n, p A(0) 0 h | | − ih − | | i , (45) − p + ω(p) i 0 − VI. CONCLUSIONS where the Hamiltonian eigenstates n, p are assumed to In this paper, we have analyzed the low-energy prop- be normalized according to m, p n,| q i= δ δ3(p q) mn erties of scattering amplitudes for processes involving and ω (p) is their energy. Noteh that| thei index n is− dis- n one or more mNG bosons. We showed that as a concrete for one-particle states and continuous for multipar- sequence of exact symmetry constraints, these scattering ticle states. Only the former are relevant for us here. µ amplitudes exhibit Adler’s zero just like those of ordi- We now set A φ and B J , where φ is an interpo- nary (gapless) NG bosons. When the momentum of the lating field for the→ mNG state,→ that− is a field for which the mNG boson is tuned to zero (and the momenta of the matrix element 0 φ(0) G(p) between the many-body other participating particles are modified accordingly to vacuum 0 andh the| one-particle| i mNG state G(p) is maintain energy and momentum conservation, but oth- nonzero.| Thei pole part of the two-point function| ofi the erwise tend to nonzero limits), the scattering amplitude current and the interpolating field φ then reads vanishes. There are no kinematic singularities associated µ with radiation of soft mNG bosons from the initial or mNG pole 3 0 φ(0) G(p) G(p) J (0) 0 µ (p) i(2π) h | | ih | − | i, DφJ− final state due to the nonzero gap of the mNG boson. −−−−−−→ p0 ω(p) − (46) This result, in fact, ensures that mNG bosons are well- where ω(p) now denotes the dispersion relation of the defined quasiparticles in spite of their nonzero gap: due mNG state. The matrix element 0 φ(0) G(p) can be to their weak interactions at low momentum, their width naturally absorbed into the definitionh | of| the scatteringi necessarily goes to zero in the long-wavelength limit. amplitude M of the process, which apart from the initial The examples analyzed explicitly in this paper in- state i and the final state f , also includes a mNG state. clude antiferromagnetic spin waves in an external mag- Altogether,| i the matrix element| i for the process depicted netic field, and a model for kaon condensation in dense in Fig.5 has the following representation, quark matter, where the mNG mode is one of the gapped kaons. However, our conclusions hold equally well for 3 µ µ i(2π) other known examples of mNG bosons such as ferromag- f J (p) i = G(p) J (0) 0 ( iMoff-shell) h | − | i h | − | ip0 ω(p) − netic spin waves in an external magnetic field, or the − neutral pion in the pion superfluid phase of quantum + N µ(p), (47) − chromodynamics. where N µ(p) is the non-pole contribution. As the next step, we− use the parameterization of the current matrix element G(p) J µ (0) 0 , following from Eq. (8), and the A. Double soft limits of scattering amplitudes current conservationh | − | conditioni (7). Some caution is required here: while the four-momentum in Eq. (8) is on- Given the fact that mNG bosons respect the Adler zero shell, that is, the frequency therein equals ω(p), the four- property, it is interesting to consider what happens in the momentum in Eq. (47) is off-shell and its temporal com- limit where the momenta of two NG or mNG bosons tend ponent is denoted simply as p0. Distinguishing carefully to zero simultaneously [19]. The behavior of scattering 9 amplitudes of true, gapless NG bosons in this limit has re- p˜0 f f f cently attracted considerable attention, see, for instance, | i k˜ | i p˜0 | i

Refs. [20, 21]. The limit of the scattering amplitude in p˜ p˜0 k p˜ − general turns out to be nonzero, and it reflects the non- ˜ k +p ˜ p˜0 p˜ + k Abelian nature of the underlying symmetry. − p˜ i i i As explained in detail in Ref. [21], this effect arises p˜ | i | i | i solely from Feynman diagrams where the two NG bosons in question, and another external leg, are attached to the FIG. 6. Topologies of Feynman diagrams that can potentially same quartic interaction vertex; see the first diagram in lead to a nonzero scattering amplitude in the limit where the Fig.6. The reason is that when two of the momenta at- momenta of two of the participating particles are sent to zero tached to the quartic vertex go to zero, the propagator simultaneously. We use the same notation for lines and ver- attached to it becomes on-shell, and the resulting singu- tices as in Sec.III. larity may cancel the suppression of the amplitude due to the presence of derivatives in the vertex. In order to see a singularity in processes involving two the radiation of the soft gapless NG boson from the ex- (m)NG bosons interacting through such a quartic vertex, ternal mNG boson line makes the scattering amplitude it is essential that both momenta and energies of the two nonzero at low momentum. modes add up to zero in the soft limit. This excludes To see a truly new effect, only existing in presence of a nontrivial double soft limit in processes involving one mNG bosons, consider finally the last diagram in Fig.6. NG and one mNG boson, and in processes involving two As in the case of the diagram with a quartic vertex, we mNG bosons in the initial or final state. The only possi- assume that one of the mNG bosons shown in the figure bility seems to be processes where one of the mNG bosons is incoming, while the other is outgoing. We then get is in the initial and the other in the final state. 2 2 2µ i( iM ) M p p0 2 For illustration, let us recall the effective theory for (˜p p˜0 ) − = − + (z ). (52) v 0 − 0 (˜p p˜ )2 − µv (p p )2 O antiferromagnets, discussed in Sec.III. Following the no- − 0 − 0 tation introduced therein, we write the four-momenta of This kind of nonzero double soft limit arising from a cu- the incoming and outgoing mNG boson including a scal- bic interaction vertex cannot appear in Lorentz-invariant ing factor z as theories for gapless NG bosons, as such cubic vertices can p 2 2 be removed from the theory altogether by a field redefi- p˜µ = ( µ2 + z2p2, zp) = (µ + z p , zp) + , 2µ ··· nition [8]. 2 02 (49) µ p 2 2 2 z p Altogether, we have identified three different mecha- p˜0 = ( µ + z p0 , zp0) = (µ + , zp0) + , 2µ ··· nisms whereby a nontrivial double soft limit of scatter- where the ellipsis stands for terms of order z4 or ing amplitudes may be realized in theories with mNG smaller. Using the Feynman rules following from the La- bosons. The first one appears when a NG boson and a grangian (14), the first diagram in Fig.6 evaluates to mNG boson in the initial or final state are attached to the same cubic interaction vertex. This case accompanies the 2 violation of the Adler zero property for the gapless NG i 2 i( iM ) izM (p p0) 2 2 (˜p p˜0) − 2 = 2 − + (z ), boson alone. The second and third mechanism are both v − (k +p ˜ p˜0) 2v k (p p0) O − · − (50) associated with a pair of mNG bosons, one in the initial where iM is the amplitude corresponding to the blob in and the other in the final state of the scattering process. the diagram.− We can see that in this concrete example, Whether they are attached to a cubic or a quartic ver- the double soft limit of the full scattering amplitude is tex, their presence leads to a singular propagator in the safe. However, in general we expect diagrams with this Feynman diagram and thus potentially a nonvanishing topology to give a nontrivial limit when the momenta of soft limit of the scattering amplitude. one incoming and one outgoing mNG boson go to zero simultaneously. B. Scattering amplitudes of pseudo-NG bosons Next, let us have a look at the second diagram in Fig.6. This type of kinematics was already observed in Sec.III to lead to a violation of the Adler zero property for the What we have not touched upon so far was the scatter- gapless NG boson. What if now the momentum of the ing amplitudes of pNG bosons that are not mNG bosons, incoming mNG boson goes to zero as well? Setting k˜µ = yet their mass also arises from the chemical potential in zkµ, a simple calculation gives for this diagram, the system. As mentioned in the introduction, this is a somewhat more exotic, yet perfectly viable possibility. 2µ i( iM ) M One might expect equally strong constraints on the scat- k˜0 − = + (z). (51) − v (˜p + k˜)2 µ2 − v O tering amplitudes in this case, since after all, we still have − the exact conservation law (7). However, it is known that In this case, we do get a nonzero double soft limit. That the properties of such pNG bosons differ from those of is, however, not so surprising given the fact that already the mNG bosons. Apart from the obvious fact that their 10 gap is not determined by the symmetry and chemical po- Finally, the object gab(π) in Eq. (A1) is a G-invariant tential alone, they also couple differently to the broken metric on the coset space, which is determined uniquely current: unlike for the true mNG bosons, this coupling up to a set of a priori unknown parameters, which repre- vanishes in the limit of low momentum [2], which invali- sent the low-energy couplings of the effective theory. dates our proof of the existence of Adler’s zero in Sec.V The invariant metric can be determined directly in i for the case of pNG bosons. terms of the Maurer–Cartan form ωa(π), defined by Based on this observation, we conjecture that the scattering amplitudes of pNG bosons whose mass arises from i 1 ∂U(π) ω (π)T iU(π)− , (A3) the chemical potential do not have the Adler zero prop- a i ≡ − ∂πa erty, just like the amplitudes of any other pNG bosons. In order to test this conjecture, we have analyzed to some where U(π) is a representative element of the coset space a extent a model where a global SO(3) symmetry is com- G/H, which encodes the NG fields π . Imposing the pletely broken. It is known that in presence of a chemical G-invariance of the Lagrangian, we obtain potential for one of the generators, this system has one c d NG, one mNG, and one pNG boson [2,5]. To our sur- gab(π) = gcd(0)ωa(π)ωb (π), (A4) prise, the scattering amplitude for the process we chose to analyze still exhibits Adler’s zero. However, our gen- where gcd(0) is a set of constants that play the role of eral argument given in Sec.V does not apply to this case, the low-energy effective couplings; their values are con- and a further, more detailed investigation is therefore re- strained by the requirement that gab(0) be a symmetric quired. We leave this issue to the future. For the sake of invariant tensor of the unbroken subgroup H. Eq. (A4) makes it clear that we do not really need to know the convenience, we provide some details of our preliminary a full Killing vectors hi (π), but only their projections of analysis in the appendix. c a c the form ωa(π)hi (π) = νi (π), where the rotation matrix i νj(π) is defined by ACKNOWLEDGMENTS i 1 ν (π)T U(π)− T U(π). (A5) j i ≡ j The authors would like to express their thanks to Jens The above relations determine completely the structure Oluf Andersen for numerous discussions of the subject. of the leading-order effective Lagrangian for an arbitrary This work has been supported in part by a grant within symmetry-breaking pattern G/H. the ToppForsk-UiS program of the University of Sta- vanger and the University Fund. 1. Effective Lagrangian and the spectrum Appendix A: Example of non-mNG-type pNG boson Let us now see how the above general formalism applies In this appendix, we shall analyze a low-energy effec- to the case where the continuous SO(3) rotation symme- tive theory for a complete spontaneous breaking of an try is completely broken. Without loss of generality, we SO(3) symmetry. To that end, we shall use the effec- can assume that the matrix gab(0) of effective couplings tive Lagrangian formalism, developed in Ref. [22], whose has a diagonal form, notation we closely follow. g (0) diag(g , g , g ). (A6) The leading-order effective Lagrangian for NG bosons ab ≡ 1 2 3 in a relativistic system in presence of background gauge We will turn on a chemical potential for the third gener- fields reads ator of SO(3), that is, set

1 a µ b L = gab(π)Dµπ D π . (A1) i i3 2 Aµ = δµ0δ µ. (A7) Here πa are the NG fields that parameterize the coset This determines the effective Lagrangian completely via space of broken symmetry, G/H. The Latin indices Eq. (A1). For the moment, we will only need the part of a, b, . . . label broken generators from this coset space. the Lagrangian bilinear in the NG fields πa, which is, up In contrast, the Latin indices i, j, . . . will denote generic to a rescaling of the fields, independent of the choice of generators of the whole symmetry group G. There is one parameterization of the matrix U(π), i external gauge field Aµ assigned to each generator Ti, 1 2 2 2 and it enters the covariant derivative of the NG field via L = g (∂ π ) + g (∂ π ) + g (∂ π ) bilin 2 1 µ 1 2 µ 2 3 µ 3 D πa ∂ πa Ai ha(π), (A2) 1 µ ≡ µ − µ i + µ(g + g g )(π π˙ π π˙ ) (A8) 2 1 2 − 3 1 2 − 2 1 a where hi (π) are the Killing vectors that realize the ac- 1 1 µ2(g g )π2 µ2(g g )π2. tion of the symmetry group G on the coset space G/H. − 2 3 − 2 1 − 2 3 − 1 2 11

The form of the mass terms indicates that the ground By looking in turn at the poles at ω = ω (p) and using state is stable under the perturbation caused by the the spectral representation (45), we then± find chemical potential provided that g3 is larger than both s g3 g1 and g2, which we will from now on assume. 1 (Ωp + 1) 1 p 2g1 − The excitation spectrum of the theory is easy to work 0 π1(0) +, = 3/2 , h | | i (2π) 2g3Ωpω+(p) out. First, the π3 mode does not feel the presence of s g3 the chemical potential, and thus behaves as an ordinary i (Ωp + 1) 1 p 2g2 − gapless NG boson: its dispersion relation reads 0 π2(0) +, = 3/2 , h | | i (2π) 2g3Ωpω+(p) (A15) ω3(p) = p . (A9) s g3 | | 1 (Ωp 1) + 1 p 2g1 − 0 π1(0) , = 3/2 , The π1,2 modes mix and their dispersion relations there- h | |− i (2π) 2g3Ωpω (p) fore take a more complicated form, − s g3 i (Ωp 1) + 1 p 2g2 − 2 2 2 g3(g3 g1 g2) 2 0 π2(0) , = −3/2 . ω (p) = p + µ + − − µ (1 Ωp), (A10) h | |− i (2π) 2g3Ωpω (p) ± 2g1g2 ± − where s 3. Evaluation of scattering amplitudes 2 4g1g2 p Ωp 1 + 2 2 . (A11) ≡ g3 µ The evaluation of the scattering amplitude for a given process proceeds according to the following steps: From here, we can in turn extract the mass spectrum in the π1,2 sector, Choose a specific parameterization of the matrix • s U(π) and expand the Lagrangian up to the desired (g3 g1)(g3 g2) order in the fields πa. m+ = µ − − , m = µ. (A12) g1g2 − Extract the interaction vertices from the expanded • Whereas we find one mNG mode as predicted by Eq. (3), Lagrangian. there is also another pNG mode which is not of the mNG Construct all tree-level Feynman diagrams con- type, although its mass comes from the chemical poten- • tial alone. It is this mode that is of interest to us. tributing to the given process. Note that as a result of the mixing in the π1,2 sector, diagrams with different fields attached to the external legs may con- 2. Coupling of fields to states tribute to the same process, since different fields couple to the same one-particle state [17].

The analysis of scattering amplitudes in the present Test scaling of the scattering amplitude in the long- model is complicated by the mixing in the π1,2 sector. In • wavelength limit numerically. such a situation, it is mandatory to use the Lehmann– Symanzik–Zimmermann formalism to extract the physi- The last point deserves a more detailed comment. Al- cal scattering amplitude from the off-shell Green’s func- ready for four-particle scattering, a relatively large num- tion of the fields. To that end, we need to know how the ber of Feynman diagrams may contribute as a result of fields couple to the asymptotic one-particle states in the the mixing, which makes testing the asymptotic behavior scattering process. of the scattering amplitude in the long-wavelength limit Such coupling can be extracted from the propagators analytically difficult. It is more convenient to perform of the fields using the Källén–Lehmannspectral repre- a numerical “experiment” [6]. All one needs to do is to sentation (45). The propagator of π3 in the interaction generate a set of random kinematical variables that sat- picture is just that of a free massless scalar field, and we isfy the energy and momentum conservation conditions readily obtain for a given process. One then introduces a scaling param- 1 eter z into the momentum of the particle whose soft limit 0 π3(0) 3, p = p . (A13) is to be investigated, and makes sure that the momenta h | | i (2π)32g p 3| | of all other participating particles are modified so that the on-shell and conservation conditions are satisfied for To extract the couplings between the fields π1,2 and the states , p with the dispersion relations ω (p), we first any value of z. Finally, one simply plots the value of the |± i ± scattering amplitude as a function of z as z tends to zero. write down the matrix inverse propagator in the π1,2 sec- In this way, we have verified that the scattering am- tor, following from the Lagrangian Lbilin, plitudes of the mNG boson (ω ) of the model exhibit 2 2 − 1 g1p µ (g3 g2) iµω(g1 + g2 g3) Adler’s zero as expected, using the NG + mNG NG + D − (ω, p) = − − − 2 2 − . → +iµω(g1 + g2 g3) g2p µ (g3 g1) mNG process as an example. Then we analyzed analo- − − − (A14) gously the NG+pNG NG+pNG process. Surprisingly, → 12 the scattering amplitude still vanishes as the momentum While we do not show the details of our evaluation of the of one of the pNG bosons tends to zero. This might be scattering amplitudes as they are specific for the chosen a special property of the process that we chose to study, parameterization of U(π) and the chosen set of random or due to some hidden symmetry of the model at hand kinematical variables, we do hope that the details pre- that we are not aware of. sented in this appendix will enable others to reproduce This issue would definitely deserve a more careful look. our results, and go beyond.

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