doi:10.1016/j.physletb.2021.136705

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Physics Letters B 822 (2021) 136705Contents lists available atScienceDirectPhysics Letters Bwww.elsevier.com/locate/physletbOn-shell recursion relations for nonrelativistic effective field theoriesMartin A. Mojaheda,∗, Tomáš BraunerbaDepartment of Physics, Norwegian University of Science and Technology, Hoegskoleringen 5, N-7491 Trondheim, NorwaybDepartment of Mathematics and Physics, University of Stavanger, N-4036 Stavanger, Norwaya r t i c l e i n f oa b s t r a c tArticle history:Received 13 August 2021Received in revised form 17 September 2021Accepted 29 September 2021Available online 4 October 2021Editor: A. VolovichWe derive on-shell recursion relations for nonrelativistic effective field theories (EFTs) with enhanced soft limits. The recursion relations are illustrated through analytic calculation of tree-level scattering amplitudes in theories with a complex Schrödinger-type field, real scalar with linear dispersion relation, and real scalar with Lifshitz-type dispersion relation. Our results show that the landscape of gapless nonrelativistic EFTs with local S-matrix can be constrained by soft theorems and the consistency of the low-energy S-matrix similarly to massless relativistic EFTs.©2021 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.1. IntroductionOn-shell recursion is a procedure to determine all scattering amplitudes in a theory recursively from a finite set of “seed” am-plitudes. It plays a central part in the modern S-matrix program where physical and mathematical properties of scattering ampli-tudes are used to construct the S-matrix directly without the aid of a Lagrangian. Originally developed in the context of gauge theory by Britto, Cachazo, Feng and Witten (BCFW) [1], on-shell recursion was soon generalized to gravity theories [2], string the-ory [3], generic renormalizable and some nonrenormalizable the-ories [4]. More recently, there has also been progress towards an on-shell formulation of scattering amplitudes in effective field the-ories (EFTs) [5–8].Beyond providing an efficient tool for calculating scattering am-plitudes, recursion relations have also been successfully utilized as a framework to explore and classify the landscape of possible EFTs [8–11]. This connects to the newly emerging paradigm that seeks to definequantum (effective) field theory without reference to a Lagrangian. While the basic principles underlying this pro-gram are mere locality and unitarity, the bulk of work done so far has focused on the sector of Lorentz-invariant field theories.1Yet, recent years have witnessed the EFT framework claiming a much larger territory than originally conceived. The range of novel ap-*Corresponding author.E-mail addresses:martinmoja96@gmail.com(M.A. Mojahed), tomas.brauner@uis.no(T. Brauner).1The only exceptions we are aware of include several recent works in a cos-mological context, limited to EFTs with Lorentz-invariant kinematics but Lorentz-breaking interactions [12], and a specific application of recursion techniques to scattering of phonons in Navier-Stokes fluids [13].plications of quantum field theory without Lorentz invariance now stretches from nonrelativistic gravity [14] and spacetime geome-try [15]to previously unthinkable exotic phases of quantum matter (see e.g. Refs. [16,17] and references therein).Should the modern scattering amplitude program provide new fundamental insight into the very nature of quantum field theory, it therefore seems mandatory to extend the scope of discussion by giving up on Lorentz invariance altogether. The aim of the present letter is to initiate the exploration of this new terra incognita. Our main result is that the existing on-shell recursion approach to EFT can be modified to nonrelativistic EFTs with rotationally-invariant gapless kinematics, where energy is proportional to an in principle arbitrary (integer) power of momentum. We demonstrate this by explicit examples of EFTs for a complex Schrödinger scalar and a real Lifshitz scalar.The plan of the text is as follows. In the remainder of this sec-tion, we first briefly overview the BCFW recursion approach and its modification applicable to EFTs, and then outline the landscape of nonrelativistic EFTs relevant to our discussion. Sections2and3constitute the core of this letter, showing how to set up the recur-sion procedure for EFTs with nonrelativistic kinematics. An integral part of the text is section4where we work out three examples.1.1. BCFW on-shell recursionA central idea of the on-shell recursion technology is to pro-mote n-particle on-shell amplitudes Anto meromorphic functions by complexifying external momenta in a way that preserves both on-shellness and conservation of energy and momentum. In the BCFW recursion, two selected external momenta, piand pj, are shifted,https://doi.org/10.1016/j.physletb.2021.1367050370-2693/©2021 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

Martin

M.A. Mojahed and T. BraunerPhysics Letters B 822 (2021) 136705ˆpi≡pi+zq,ˆpj≡pj−zq,z∈C.(1)(Shifted quantities are denoted with a hat.) The auxiliary momen-tum qmust satisfy the on-shell conditions q2=pi·q =pj·q =0. In four spacetime dimensions, it is thus fixed up to rescaling. At tree level, the complexified amplitude ˆAn(z)is a rational function of z. The original, physical amplitude An=ˆAn(0)can be recovered byAn=12πi∮dzˆAn(z)z,(2)where the integration contour is an infinitesimal circle enclosing the origin of the complex plane. Cauchy’s theorem and factoriza-tion then relate the physical amplitude Anto lower-point ampli-tudes in the following way,An=−∑IResz=zIˆAn(z)z+Bn=∑IˆA(I)L(zI)ˆA(I)R(zI)P2I+Bn.(3)The sum runs over all factorization channels Iwhere the lower-point amplitudes ˆA(I)Land ˆA(I)Rcontain one of ˆpi, ˆpjeach. More-over, PIis the intermediate momentum evaluated at z=0, and zIis fixed by the on-shell condition ˆP2I(zI) =0to zI=−P2I/(2PI·q). Finally, Bndenotes the contribution of the residue of the pole at z=∞. The validity of the recursion relies on the latter either van-ishing or being calculable.2The above approach does not extend straightforwardly to low-energy EFTs. Technically, the problem is that the derivative cou-plings of EFTs imply polynomial growth of scattering amplitudes at large z, and thus preclude the standard recursion procedure. A different kind of complexification of the kinematical phase space is needed.1.2. On-shell recursion for EFTsThe deeper reason why BCFW recursion fails for EFTs is that factorization alone is not sufficient to relate higher-point EFT am-plitudes to lower-point ones; more information is needed. Since the form of an EFT is largely dictated by symmetries, it is hardly surprising that the additional input comes from symmetry (break-ing).Spontaneous symmetry breaking constrains the scattering am-plitudes of the associated Nambu-Goldstone (NG) boson(s) in the “(single) soft limit,” in which the momentum of one of the par-ticles participating in the scattering process vanishes. This limit can be probed by rescaling the momentum of the chosen parti-cle, pi, as pi→pi, and taking the scaling parameter to zero. The asymptotic behavior of the amplitude Anis characterized by a single scaling exponent,An∝σi,→0.(4)As a rule, albeit not without exceptions [11], spontaneous symme-try breaking ensures that σi≥1; this fact is known as “Adler’s zero.” Theories where σiis larger than naively expected from counting derivatives in the Lagrangian are dubbed “exceptional.” The landscape of Lorentz-invariant exceptional EFTs is very strongly constrained [8,19,20]. Single-flavor scalar exceptional EFTs were the first effective theories shown to be on-shell constructible [6]2Calculating Bnis a challenging problem that has been considered in several contexts [18].by a modification of the BCFW recursion procedure known as “soft recursion.”In the soft recursion procedure, allexternal momenta are shifted,ˆpi≡pi(1−aiz),z∈C,(5)n∑i=1aipi=0,(6)where Eq. (6)is imposed by energy and momentum conservation. Nontrivial solutions for the coefficients aiexist for generic kine-matical configurations when n ≥D +2, where Dis the spacetime dimension. The soft limit for the i-th particle can then be accessed by taking z→1/ai.In order to be able to apply Cauchy’s theorem, one modifies the behavior of the complexified amplitude ˆAn(z)at large zby dividing it by the factorFn(z)≡n∏i=1(1−aiz)σi.(7)For exceptional EFTs, this is sufficient to ensure vanishing of the boundary term Bn[6]. At the same time, the scaling (4)of the amplitude in the soft limit guarantees that adding Fn(z)does not create any new poles in ˆAn(z). One can then reconstruct the phys-ical amplitude An=ˆAn(0)similarly to the BCFW recursion,An=12πi∮dzˆAn(z)zFn(z)=−∑IResz=z±IˆAn(z)zFn(z),(8)where each factorization channel Inow gives rise to two poles z±Icorresponding to solutions of the shifted on-shell condition ˆP2I(z) =0. These are given explicitly byz±I=1Q2I[PI·QI±√(PI·QI)2−P2IQ2I],(9)where PI≡∑i∈Ipiand QI≡∑i∈Iaipi. Factorization together with Eq. (8)then imply the recursion formula [6]An=∑IˆA(I)L(z−I)ˆA(I)R(z−I)P2I(1−z−Iz+I)Fn(z−I)+(z−I↔z+I).(10)1.3. Nonrelativistic EFTsThe theories we will focus on in this letter live in a flat space-time of D ≡d +1 dimensions. They enjoy invariance under space-time translations and d-dimensional spatial rotations. This is a fairly general setup that admits, if desired, a variety of kinematical algebras [21]. The latter include the static (or Aristotelian) algebra containing no boosts whatsoever, and the Poincaré, Galilei (and its central extension, Bargmann) and Carroll algebras featuring differ-ent implementations of the relativity principle.The NG modes stemming from spontaneous breakdown of global symmetry in such theories can be classified into two fam-ilies, referred to as type Amand type B2mwith positive integer m[22]. A NG mode from the first family is described by a real scalar field with dispersion relation ω2∝p2m. A NG mode from the second family, on the other hand, is described by two real scalar fields (or one complex scalar) forming a canonically con-jugated pair with dispersion relation ω∝p2m.Whether or not NG modes belonging to the Amand B2mfami-lies can exist in a given spatial dimension dis constrained by the 2

M.A. Mojahed and T. BraunerPhysics Letters B 822 (2021) 136705nonrelativistic version of the Coleman-Hohenberg-Mermin-Wagner (CHMW) theorem [23,24]. In short, at zero temperature, a NG bo-son of type Ammay exist only if m <d. For fixed m, this in turn gives a lower bound on the dimension of space d. On the contrary, type B2mNG modes are not constrained at all and can exist, at zero temperature, for any positive dand m.It was observed early on [19] that the enhanced scaling (4)of scattering amplitudes in exceptional EFTs is a consequence of hid-den symmetry. Motivated by this observation, one of us mapped in Ref. [25]the landscape of nonrelativistic EFTs that admit such a hidden symmetry. We will show in a forthcoming paper that un-like in the Lorentz-invariant case, this is in fact not sufficient to guarantee that a given EFT is exceptional. The catalogue of candi-date EFTs compiled in Ref. [25]will nevertheless serve as a useful guide for construction of explicit examples of nonrelativistic EFTs via recursion in section4. We will thus be able to give examples of theories of the A1, A2and B2type. Before doing so, we however first need to establish the soft recursion procedure for nonrelativis-tic EFTs. This is the subject of the next two sections.2. Momentum deformation in nonrelativistic EFTsIn this section, we introduce the momentum shifts needed for soft recursion. In contrary to the relativistic momentum shift in Eq. (5), we first shift the spatial momenta pionly, and then use the on-shell condition to define an appropriate shift of the energies.2.1. Soft shifts for type B2mtheoriesThe following shifts respect the on-shell condition for type B2mtheories,ˆpi≡pi(1−aiz),(11)ˆp0i≡ˆp2mi=p2mi(1−aiz)2m.(12)Momentum and energy conservation then impose respectively the following constraints on the aicoefficients,n∑i=1aieipi=0,(13)n∑i=1(1−zai)2meip2mi=0.(14)Here eidenotes a sign, chosen so that ei=+1for particles in the final state and ei=−1for particles in the initial state. Similarly to the relativistic case reviewed in section1.2, the existence of nontrivial solutions to Eq. (13)requires n ≥d +2. Equation (14)then imposes 2madditional constraints. Only amplitudes with n ≥d +2 +2mmay therefore be reconstructed using soft recur-sion. For given dand m, this tells us how many seed amplitudes we need to initiate the recursion procedure.2.2. Soft shifts for type AmtheoriesFor type Amtheories we define analogouslyˆpi≡pi(1−aiz),(15)ˆp0i≡|(p2mi)1/2|(1−aiz)m,(16)which preserves on-shellness and yields the following constraints from momentum and energy conservation,n∑i=1aieipi=0,(17)n∑i=1(1−zai)mei|(p2mi)1/2|=0.(18)Analogously to the type B2mcase, the existence of nontrivial solu-tions for airequires n ≥d +2 +m >2 +2m, where the last inequal-ity follows from the nonrelativistic CHMW theorem. For the special case of m =1, which includes the family of Lorentz-invariant the-ories, the above constraints become equivalent to Eq. (6) and we recover the relativistic bound n ≥d +3 =D +2.Note that for both type Amand type B2mtheories, the manifold of solutions for the aicoefficients is invariant under overall rescal-ing, ai→λai, and overall shift, ai→ai+c. This guarantees that in the special case of type A1theories where all the constraints on aiare linear, possible solutions for aispan an affine space.3. Soft recursionWe argued in section1.2that for relativistic exceptional EFTs, recursion relations among scattering amplitudes may be set up us-ing Eq. (8). Since the argument only depends on the assumed soft behavior of An, factorization and vanishing of the boundary term, it can be generalized to any theory with these properties. Specifi-cally, for theories of type Amand B2mwe obtainAn=−∑I2m∑i=1Resz=ziIˆAn(z)zFn(z).(19)Here ziI, i =1, ..., 2mare solutions to the on-shell condition, which is of algebraic order 2min z,(ˆP0I)2−ˆP2mI=0forAm,(20)ˆP0I−ˆP2mI=0forB2m,(21)for a given factorization channel I, where compared to Eq. (10), PIis now defined with the appropriate signs eiwhere necessary. Fac-torization then implies that the amplitude (19)can be expressed in terms of lower-point amplitudes,An=−∑I2m∑i=1Resz=ziIˆA(I)L(z)ˆA(I)R(z)zFn(z)D(I)(z),(22)whereD(I)(z)=(ˆP0I)2−ˆP2mIforAm,(23)D(I)(z)=ˆP0I−ˆP2mIforB2m.(24)Notice that the contribution from factorization channel Iin Eq. (22)matches the residue at z=ziIof the following meromor-phic functionˆA(I)L(z)ˆA(I)R(z)zFn(z)D(I)(z).(25)This function can also have nonvanishing residues at z=1/aiand z=0. This follows from the fact that the intermediate propaga-tor D(I)(z), hence also the subamplitudes ˆA(I)L(z)and ˆA(I)R(z), is off-shell for z =ziI. The on-shell argument implying that the soft behavior of the amplitudes dictated by Eq. (4) cancels the zeros of Fn(z)is then no longer valid. In the special case where ˆA(I)L(z)and ˆA(I)R(z)are both local functions of momenta (that is, they have no 3

M.A. Mojahed and T. BraunerPhysics Letters B 822 (2021) 136705poles) we can apply Cauchy’s theorem to the meromorphic func-tion in Eq. (25) and recast the amplitude (22)in terms of a sum over residues at z=0 and z=1/ai,An=∑IˆA(I)L(0)ˆA(I)R(0)D(I)(0)+∑In∑i=1Resz=1/aiˆA(I)L(z)ˆA(I)R(z)zFn(z)D(I)(z)≡Achn+Actn.(26)This expression is particularly useful for concrete applications. In terms of Feynman diagrams, the first term corresponds to the sum over diagrams with an internal propagator, whereas the second (double) sum encodes contributions from n-point contact opera-tors. The two different types of contributions are distinguished by the notation introduced in the last line of Eq. (26).3.1. Validity criterionThus far we have simply assumed that the boundary term Bnvanishes. A sufficient condition for this to happen is that ˆAn(z)/Fn(z) →0as z→∞. A criterion for the latter was in turn given by Elvang et al. in Ref. [9]. Their argument only relies on di-mensional analysis, the soft behavior of An, the analytic structure of tree-level amplitudes, and the freedom to shift all aiby an over-all constant. Since the latter property survives in all type Amand B2mtheories, as shown in section2, it is easy to adapt the argu-ment of Ref. [9]for our purposes.We start with a generic expression for the n-point tree-level amplitude,An=∑j(∏kgnjkk)Mj,(27)where Mjare functions of momenta and gkare coupling constants associated with fundamental operators in the Lagrangian. Funda-mental operators are defined in turn as the lowest-dimension op-erators whose on-shell matrix elements are needed to derive, at the leading-order in the low-energy expansion, any tree-level am-plitude in the theory by recursion. Following the line of reasoning of Ref. [9]then leads to the generalized validity criterion[An]−minj(∑knjk[gj])−n∑i=1σi<0,(28)where square brackets indicate scaling dimension with respect to a uniform rescaling of all the momenta pi. It is easy to check that the criterion (28)is satisfied by all the example theories presented in the next section.4. Example calculationsWe will now work out three simple analytical examples of re-cursive reconstruction of scattering amplitudes in theories of type B2, A1and A2, respectively. All three sample theories feature tree-level amplitudes with soft scaling σi=2. Yet, each of the theories possesses Lagrangian representations with less than two deriva-tives per field, which means that they possess enhanced soft limits. We will show in a forthcoming paper that the enhanced scaling of scattering amplitudes in these theories is a consequence of an interplay of spontaneously broken symmetry and dispersion rela-tions of NG bosons. Each of the three theories contains just one physical NG mode. Since we no longer have to distinguish differ-ent σifor different particles participating in the scattering process, we introduce a shorthand notation replacing Eq. (7),F(σ)n(z)≡n∏i=1(1−aiz)σ.(29)4.1. B2: Schrödinger-DBI theoryOur first example features a complex scalar field endowed with the actionS=∫dtddx(†i∂0+√G−1),(30)G≡1−2∇·∇†+(∇·∇†)2−(∇·∇)(∇†·∇†).(31)This is a minimal nonrelativistic modification of one of the very few relativistic single-flavor exceptional theories [19]: the Dirac-Born-Infeld (DBI) theory. We therefore name it the “Schrödinger-DBI” (SDBI) theory.Our SDBI theory can be interpreted as describing fluctuations of a d-dimensional brane embedded in a (d +2)-dimensional Eu-clidean space. The symmetry of the SDBI action (30)is accordingly R ×ISO(d +2), with the first factor of Rcorresponding to time translations [25]. This symmetry is spontaneously broken down to R ×ISO(d) ×SO(2)by the presence of the brane, and the real and imaginary parts of correspond to NG fields of sponta-neously broken translations in the two extra dimensions. The term in Eq. (30)with a single time derivative is only invariant under the full symmetry up to a surface term. It is thus an example of a Wess-Zumino-Witten (WZW) term.The action (30)fixes all tree-level amplitudes. We will now demonstrate that the recursion formula (26) correctly reproduces the six-point amplitude starting from the seed four-point ampli-tude. In fact, the argument of section2.1limits the validity of the recursion for n =6to d ≤2spatial dimensions. However, the amplitudes Anas functions of the momenta pido not depend ex-plicitly on d. Whatever analytic relations between the amplitudes we find will therefore be independent of das well. One may think of this as carrying out the recursive step from A4to A6in d =2di-mensions, and then analytically continuing the result to any value of dof interest.To make the calculation transparent, we first explicitly list the relevant parts of the Lagrangian,L2=†(i∂0+∇2),(32)L4=−12(∇·∇)(∇†·∇†),(33)L6=−12(∇·∇)(∇·∇†)(∇†·∇†).(34)Charge conservation dictates that the numbers of incoming and outgoing Schrödinger scalars must match in any scattering pro-cess. We use the convention that the particles labeled 1, ..., n/2are incoming, whereas the particles n/2, ..., nare outgoing. The seed on-shell four-point amplitude then follows immediately from Eq. (33)asA4=2(p1·p2)(p3·p4).(35)We are now ready to derive the six-point amplitude by recur-sion. We will use the indices a, b, cto label a permutation of the incoming particles and d, e, fa permutation of the outgoing par-ticles such that a, b, fare on the same side of the factorization 4

M.A. Mojahed and T. BraunerPhysics Letters B 822 (2021) 136705channel. We can then identify the nine factorization channels in terms of cand falone,I={(c,f)}={(14), (15), (16), (24), (25), (26),(34), (35), (36)}.(36)Energy and momentum conservation fix the parameters of the in-termediate propagator for each factorization channel,PI≡pa+pb−pf=pd+pe−pc,(37)12(P0I−P2I)=−pa·pb−pf·pf+pa·pf+pb·pf=−pd·pe−pc·pc+pd·pc+pe·pc.The channel contribution Ach6as defined by Eq. (26)readsAch6=4∑I(pa·pb)(pd·pe)(pc·PI)(pf·PI)P0I−P2I(38)=∑σ,ρ∈S3(pσ(1)·pσ(2))(pρ(4)·pρ(5))(pσ(3)·kσρ)(pρ(6)·kσρ)k0σρ−k2σρ,where σand ρdenote respectively permutations of {1, 2, 3}and {4, 5, 6}, and we have used the shorthand notationkσρ≡pσ(1)+pσ(2)−pρ(6).(39)The second line of Eq. (38)is manifestly equal to the Feynman diagram expression one obtains from Eq. (33).Similarly, the contact contribution to the six-point amplitude follows from Eq. (26)asAct6=4∑I6∑i=1Resz=zi(ˆpa·ˆpb)(ˆpd·ˆpe)(ˆpc·ˆPI)(ˆpf·ˆPI)zF(2)6(z)(ˆP0I−ˆP2I)≡−2∑I6∑i=1f(zi).(40)The residues at zi≡1/aifor a given factorization channel can be rewritten asf(za)=Resz=za(ˆpa·ˆpb)(ˆpd·ˆpe)(ˆpc·ˆpf−ˆpc·ˆpb)zF(2)6(z),f(zb)=Resz=zb(ˆpa·ˆpb)(ˆpd·ˆpe)(ˆpc·ˆpf−ˆpc·ˆpa)zF(2)6(z),f(zc)=Resz=zc(ˆpa·ˆpb)(ˆpc·ˆpd+ˆpc·ˆpe)(ˆpd·ˆpf+ˆpe·ˆpf)zF(2)6(z),f(zd)=Resz=zd(ˆpa·ˆpb)(ˆpd·ˆpe)(ˆpc·ˆpf−ˆpe·ˆpf)zF(2)6(z),f(ze)=Resz=ze(ˆpa·ˆpb)(ˆpd·ˆpe)(ˆpc·ˆpf−ˆpd·ˆpf)zF(2)6(z),f(zf)=Resz=zf(ˆpd·ˆpe)(ˆpa·ˆpc+ˆpb·ˆpc)(ˆpa·ˆpf+ˆpb·ˆpf)zF(2)6(z).After substituting the expressions above into Eq. (40), collecting the contributions to the residue at each zifrom all factorization channels, and using (shifted) momentum conservation, we obtainAct6=−126∑i=1Resz=zi1zF(2)6(z)(41)×∑σ,ρ∈S3(ˆpσ(1)·ˆpσ(2))(ˆpσ(3)·ˆpρ(4))(ˆpρ(5)·ˆpρ(6)).A final application of Cauchy’s theorem yieldsAct6=12∑σ,ρ∈S3(pσ(1)·pσ(2))(pσ(3)·pρ(4))(pρ(5)·pρ(6)),(42)which is manifestly equal to the contribution from the contact term in Eq. (34).4.2. A1: spatial GalileonOur second example includes a whole class of Lagrangians of a real scalar field φ,L=12(∂μφ)2+d+1∑n=3cnφGn−1,(43)where cnare real coupling constants and Gnis a polynomial of order nin the second spatial derivatives of φ,Gn≡1(d−n)!i1···inkn+1···kdj1···jnkn+1···kd×(∂i1∂j1φ)···(∂in∂jnφ).(44)This is a nonrelativistic version of another type of a relativis-tic single-flavor exceptional theory [19]: the Galileon. As opposed to the usual, Lorentz-invariant Galileon theory [26], the interac-tion part of Eq. (43) contains only spatial derivatives of φ. We therefore dub it “spatial Galileon.” The action (43)is invariant un-der polynomial shifts of φof first order in spatial coordinates, φ→φ+α+β·x. This spatial version of the usual Galileon sym-metry is a special case of a class of “multipole algebras” that have recently attracted attention in the context of fracton physics [16]. All interaction terms in Eq. (43)as well as the spatial part of the kinetic term are of the WZW type [27].Since the spatial Galileon is a type A1theory, the validity of the recursion is limited to n-point amplitudes with n ≥d +3, as shown in section2.2. For illustration, we will now restrict Eq. (43)to the quartic interaction term and show how to reconstruct the six-point amplitude. This requires setting d =3, since for d <3the quartic spatial Galileon interaction does not exist.It is convenient to express the Feynman rule for the n-point spatial Galileon vertex as [28]Vn(p1,...,pn)=c′n∑σ∈ZnG(pσ(1),...,pσ(n−1)),(45)where G(p1, ..., pn−1)is the Gram determinant, that is the de-terminant of the (n −1) ×(n −1)matrix with entries pi·pj. Importantly, the Gram determinant is a symmetric, homogeneous polynomial of order two in all its arguments,G(λp1,...,pn−1)=λ2G(p1,...,pn−1).(46)Due to momentum conservation in the vertex, all the contribu-tions to the sum in Eq. (45)are then equal and we can write Vn=nc′nG(p1, ..., pn−1).The six-point amplitude is now determined in terms of the four-point seed amplitude by Eq. (26),5

M.A. Mojahed and T. BraunerPhysics Letters B 822 (2021) 136705A6=∑I{A(I)4LA(I)4R(P0I)2−P2I+6∑i=1Resz=1/aiˆA(I)4L(z)ˆA(I)4R(z)zF(2)6(z)[(ˆP0I)2−ˆP2I]}.(47)For a generic permutation σof the external momenta, the numer-ator in the last term can be cast asV4(ˆpσ(1),ˆpσ(2),ˆpσ(3),ˆPI)V4(ˆpσ(4),ˆpσ(5),ˆpσ(6),ˆPI)=(4c′4)2G(ˆpσ(1),ˆpσ(2),ˆpσ(3))G(ˆpσ(4),ˆpσ(5),ˆpσ(6)).(48)The scaling property (46)of the Gram determinant then ensures that the denominator factor F(2)6(z)in Eq. (47)is canceled. Thus, all the residues inside the second sum in Eq. (47) vanish and only the first, “channel” term therein survives. This is manifestly equal to the expression for A6one obtains using Feynman diagrams.4.3. A2: Lifshitz scalar with polynomial shift symmetryOur final example is a so-called z=2 Lifshitz theory, which possesses the following kinetic term,L2=12(∂0φ)2−12(∇2φ)2.(49)This Lagrangian is strictly invariant under the spatial Galileon symmetry.3We can thus add the spatial Galileon interactions in Eq. (43)to it. The ensuing theory can be viewed as a fine-tuned version of the spatial Galileon where the usual kinetic term pro-portional to (∇φ)2is set to zero.This is a type A2theory, so the validity of the recursion is lim-ited to n-point amplitudes with n ≥d +4as shown in section2.2. At the same time, the CHMW theorem requires that d >2. We thus cannot reconstruct the six-point amplitude by recursion. We can however consider a seed five-point vertex and use recursion to reconstruct the eight-point amplitude. This requires setting d =4, since for d <4the quintic spatial Galileon does not exist.Following the same steps as in the previous example, Eq. (26)then gives the following result for the eight-point amplitude,A8=∑IA(I)5LA(I)5R(P0I)2−P4I,(50)which agrees with the Feynman diagram expression.5. OutlookWe have derived recursion relations for nonrelativistic EFTs with enhanced soft limits. To the best of our knowledge, this is the first time that on-shell constructibility for theories without Lorentz invariance has been shown.Beyond providing a new tool for calculating explicit tree-level amplitudes in specific field theories, soft recursion is a key ingre-dient in the “soft bootstrap” program, which explores and classifies the space of possible EFTs. In a paper soon to appear, we will carry out a more detailed classification of possible seed ampli-tudes. When combined with soft recursion, this will allow us to perform a scan of the landscape of nonrelativistic EFTs, improving on our previous symmetry-based study [25].Our recursion relations can also be applied to theories with uni-versal albeit not necessarily vanishing soft behavior by following 3Lifshitz scalars with polynomial shift symmetries have been classified in Refs. 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