On-shell recursion relations for nonrelativistic effective field theories

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Physics Letters B

www.elsevier.com/locate/physletb

On-shell recursion relations for nonrelativistic effective field theories

Martin A. Mojahed

^a,∗

, Tomáš Brauner

^b

aDepartmentofPhysics,NorwegianUniversityofScienceandTechnology,Hoegskoleringen5,N-7491Trondheim,Norway bDepartmentofMathematicsandPhysics,UniversityofStavanger,N-4036Stavanger,Norway

a rt i c l e i n f o a b s t r a c t

Articlehistory:

Received13August2021

Receivedinrevisedform17September 2021

Accepted29September2021 Availableonline4October2021 Editor: A.Volovich

We deriveon-shell recursionrelations fornonrelativistic effectivefield theories (EFTs)withenhanced softlimits. The recursion relations are illustrated through analyticcalculation of tree-level scattering amplitudesintheorieswithacomplexSchrödinger-typefield,realscalarwithlineardispersionrelation, and realscalar withLifshitz-type dispersionrelation. Ourresults show that the landscape ofgapless nonrelativisticEFTswithlocal S-matrixcanbeconstrainedbysofttheoremsandtheconsistencyofthe low-energyS-matrixsimilarlytomasslessrelativisticEFTs.

©^{2021TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

1. Introduction

On-shell recursion is a procedure to determine all scattering amplitudesinatheoryrecursively fromafinitesetof“seed”am- plitudes. It plays a central partinthe modern S-matrix program where physical andmathematical properties of scatteringampli- tudes are used to construct the S-matrix directly without the aid ofa Lagrangian.Originally developedin thecontext ofgauge theory by Britto, Cachazo, Feng and Witten (BCFW) [1], on-shell recursionwas soongeneralizedtogravitytheories [2],stringthe- ory [3], generic renormalizable andsome nonrenormalizablethe- ories [4].More recently,there hasalsobeenprogress towardsan on-shellformulationofscatteringamplitudesineffectivefieldthe- ories(EFTs) [5–8].

Beyondprovidinganefficienttoolforcalculatingscatteringam- plitudes, recursion relations have also been successfully utilized asa framework to explore andclassify the landscapeof possible EFTs [8–11]. This connectsto the newly emerging paradigm that seeksto definequantum(effective)fieldtheory withoutreference to a Lagrangian. While the basic principles underlying this pro- gramaremerelocalityandunitarity,thebulkofworkdonesofar hasfocusedon thesectorofLorentz-invariantfieldtheories.¹ Yet, recentyears havewitnessedthe EFT frameworkclaiming a much larger territory than originally conceived.The rangeof novel ap-

*

Correspondingauthor.

E-mailaddresses:[email protected](M.A. Mojahed), [email protected](T. Brauner).

1 Theonlyexceptionsweareawareofinclude severalrecentworksinacos- mologicalcontext,limitedtoEFTswithLorentz-invariantkinematicsbutLorentz- breaking interactions [12], and aspecific application ofrecursion techniquesto scatteringofphononsinNavier-Stokesfluids [13].

plicationsofquantumfieldtheorywithoutLorentzinvariancenow stretchesfrom nonrelativistic gravity [14] and spacetime geome- try [15] topreviouslyunthinkableexoticphasesofquantummatter (seee.g. Refs. [16,17] andreferencestherein).

Shouldthemodernscatteringamplitudeprogram providenew fundamentalinsightintotheverynatureofquantumfield theory, itthereforeseemsmandatorytoextendthescopeofdiscussionby givinguponLorentzinvariancealtogether.Theaimofthepresent letteristoinitiatetheexplorationofthisnewterraincognita.Our mainresultisthattheexistingon-shellrecursionapproachtoEFT canbe modifiedto nonrelativisticEFTswithrotationally-invariant gaplesskinematics,whereenergyisproportionaltoaninprinciple arbitrary(integer)power ofmomentum. Wedemonstrate thisby explicitexamples ofEFTs foracomplex Schrödinger scalaranda realLifshitzscalar.

Theplanofthetext isasfollows.Intheremainderofthissec- tion,we first briefly overviewthe BCFW recursion approach and itsmodificationapplicabletoEFTs,andthenoutlinethelandscape ofnonrelativisticEFTsrelevanttoourdiscussion.Sections2and3 constitutethecoreofthisletter,showinghowtosetuptherecur- sionprocedureforEFTswithnonrelativistickinematics.Anintegral partofthetextissection4whereweworkoutthreeexamples.

1.1. BCFWon-shellrecursion

Acentral idea ofthe on-shell recursion technology isto pro- moten-particleon-shellamplitudes An tomeromorphicfunctions bycomplexifying externalmomentaina waythatpreservesboth on-shellness and conservation of energy and momentum. In the BCFW recursion,two selected external momenta, p_i and p_j, are shifted,

https://doi.org/10.1016/j.physletb.2021.136705

0370-2693/©^{2021TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

ˆ

pi

≡

pi

+

zq

,

p

ˆ

j

≡

pj

−

zq

,

z

∈ C.

(1) (Shiftedquantitiesaredenotedwithahat.)Theauxiliarymomen- tumq mustsatisfy theon-shellconditionsq²=^pi·^q=^pj·^q=^0.

In fourspacetime dimensions, it isthus fixed up to rescaling.At treelevel,thecomplexifiedamplitude Aˆ_n(z)isarationalfunction ofz.Theoriginal,physicalamplitudeAn= ˆ^An(0)canberecovered by

An

=

¹ 2

π

i

dz_A

ˆ

_n

(

z

)

z

,

(2)

where the integrationcontour is an infinitesimal circleenclosing the originofthe complexplane. Cauchy’stheoremandfactoriza- tion then relate thephysical amplitude An tolower-point ampli- tudesinthefollowingway,

An

= −

I zRes=zI

A

ˆ

n

(

z

)

z

+

^Bn

=

I

A

ˆ

⁽_L^I)

(

z_I

)

A

ˆ

⁽_R^I)

(

z_I

)

P²_I

+

^Bn

.

(3)

The sumruns over all factorizationchannels I where thelower- point amplitudes Aˆ⁽_L^I) and Aˆ⁽_R^I) containone of ^pˆi, ^pˆj each. More- over, PI istheintermediatemomentumevaluatedatz=^0,andzI isfixedbytheon-shellcondition Pˆ²_I(zI)=^{0 to z}I= −^P2I/(2PI·^q). Finally, Bn denotesthe contributionofthe residueofthe poleat z= ∞^{.Thevalidityoftherecursionreliesonthelattereithervan-} ishingorbeingcalculable.²

Theabove approachdoesnotextendstraightforwardlytolow- energy EFTs. Technically, the problem is that the derivative cou- plings of EFTs imply polynomial growth ofscattering amplitudes atlarge z,andthusprecludethestandard recursionprocedure.A different kindofcomplexification ofthe kinematicalphase space isneeded.

1.2. On-shellrecursionforEFTs

The deeper reasonwhy BCFW recursion fails for EFTs is that factorizationaloneisnotsufficienttorelatehigher-pointEFT am- plitudes to lower-point ones; more information is needed. Since the formofanEFT is largelydictatedby symmetries,itishardly surprisingthattheadditionalinputcomesfromsymmetry(break- ing).

Spontaneoussymmetry breakingconstrains the scatteringam- plitudesof theassociatedNambu-Goldstone(NG) boson(s)in the

“(single) softlimit,” in whichthe momentum of one of thepar- ticles participating in the scattering process vanishes. This limit can be probed by rescaling the momentum of the chosen parti- cle, p_i,as p_i→

p_i,and takingthe scalingparameter to zero.

TheasymptoticbehavioroftheamplitudeAn ischaracterizedbya singlescalingexponent,

A_n

∝

^σi

, →

⁰

.

(4)

Asarule,albeitnotwithoutexceptions [11],spontaneoussymme- try breaking ensures that

σ

i≥^{1; this fact is known as “Adler’s} zero.” Theories where

σ

i is larger than naively expected from counting derivatives in the Lagrangian are dubbed “exceptional.”

ThelandscapeofLorentz-invariantexceptionalEFTsisverystrongly constrained [8,19,20]. Single-flavor scalar exceptional EFTs were the first effectivetheories shownto be on-shellconstructible [6]

2 Calculating Bn isa challengingproblem thathas beenconsideredinseveral contexts [18].

byamodificationoftheBCFWrecursionprocedureknownas“soft recursion.”

In the soft recursion procedure, all external momenta are shifted,

ˆ

pi

≡

^pi

(

1

−

^aiz

),

z

∈ C,

(5)

n

i=1

aipi

=

0

,

(6)

whereEq. (6) isimposedbyenergyandmomentumconservation.

Nontrivial solutions for the coefficientsa_i exist forgeneric kine- maticalconfigurationswhenn≥^D+^{2,whereD isthespacetime} dimension.Thesoftlimitforthei-thparticlecanthenbeaccessed bytakingz→¹/a_i.

InordertobeabletoapplyCauchy’stheorem,onemodifiesthe behaviorofthecomplexifiedamplitudeAˆn(z)atlargezbydividing itbythefactor

F_n

(

z

) ≡

n

i=¹

(

1

−

^aiz

)

^σi

.

(7)

Forexceptional EFTs, thisis sufficient to ensure vanishing ofthe boundary term Bn [6]. At the same time, the scaling (4) of the amplitudeinthesoftlimitguaranteesthat addingF_n(z)doesnot createanynewpolesin Aˆn(z).Onecanthenreconstructthephys- icalamplitude An= ˆ^An(0)similarlytotheBCFWrecursion,

A_n

=

¹ 2

π

i

dz _A

ˆ

_n

(

z

)

z F_n

(

z

) = −

I

Res

z=^z±I

A

ˆ

n

(

z

)

z F_n

(

z

) ,

(8) where each factorization channel I now gives rise to two poles z^±_I corresponding to solutions of the shifted on-shell condition Pˆ²_I(z)=^{0.Thesearegivenexplicitlyby}

z^±_I

=

¹ Q_I²

P_I

·

^QI

±

(

P_I

·

^QI

)

²

−

^P2_I^Q2_I

,

(9)

where PI≡

i∈^I

p_i and QI≡

i∈^I

a_ip_i. Factorization together with Eq. (8) thenimplytherecursionformula [6]

A_n

=

I

A

ˆ

⁽_L^I)

(

z⁻_I

)

A

ˆ

⁽_R^I)

(

z⁻_I

)

P²_I 1

−

^z_z^−I+

I

Fn

(

z⁻_I

)

+ (

z⁻_I

↔

^z+_I

).

(10)

1.3.NonrelativisticEFTs

Thetheorieswewillfocusoninthisletterliveinaflatspace- timeofD≡^d+1 dimensions.Theyenjoyinvarianceunderspace- time translations and d-dimensional spatial rotations. This is a fairlygeneralsetupthatadmits,ifdesired,avarietyofkinematical algebras [21].Thelatterincludethestatic(orAristotelian)algebra containingnoboostswhatsoever,andthePoincaré,Galilei(andits centralextension,Bargmann)andCarrollalgebrasfeaturingdiffer- entimplementationsoftherelativityprinciple.

The NG modes stemming from spontaneous breakdown of globalsymmetryinsuch theoriescan beclassifiedintotwo fam- ilies, referred to astype Am andtype B2m with positive integer m [22]. A NG mode fromthe first family is described by a real scalar field withdispersion relation

ω

²_∝p^2m. 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- jugatedpairwithdispersionrelation

ω

_∝p^2m.

WhetherornotNGmodesbelongingtothe A_m andB_2m fami- liescan existinagivenspatialdimension disconstrainedbythe

nonrelativisticversionoftheColeman-Hohenberg-Mermin-Wagner (CHMW)theorem [23,24].Inshort,atzerotemperature,aNGbo- son oftype Am mayexist onlyifm<d.Forfixedm,thisinturn givesalowerboundonthedimensionofspaced.Onthecontrary, type B_2m NG modes are not constrainedat all and can exist, at zerotemperature,foranypositivedandm.

Itwas observedearly on [19] thattheenhanced scaling (4) of scatteringamplitudesinexceptionalEFTsisaconsequenceofhid- den symmetry. Motivatedby this observation, one ofus mapped inRef. [25] thelandscapeofnonrelativisticEFTsthatadmitsucha hiddensymmetry. Wewillshow inaforthcoming paperthatun- like in the Lorentz-invariantcase, this is infact not sufficient to guaranteethat agivenEFT isexceptional.Thecatalogueofcandi- dateEFTscompiledinRef. [25] willneverthelessserveasauseful guideforconstruction ofexplicitexamples ofnonrelativisticEFTs viarecursioninsection4.Wewillthusbeabletogiveexamplesof theoriesofthe A1, A2 and B2 type.Beforedoing so,wehowever firstneedtoestablishthesoftrecursionprocedurefornonrelativis- ticEFTs.Thisisthesubjectofthenexttwosections.

2. MomentumdeformationinnonrelativisticEFTs

Inthissection,we introducethemomentumshiftsneededfor soft recursion. In contrary to the relativisticmomentum shift in Eq. (5),wefirstshiftthespatialmomentap_ionly,andthenusethe on-shellconditiontodefineanappropriateshiftoftheenergies.

2.1. SoftshiftsfortypeB2mtheories

Thefollowingshiftsrespecttheon-shellconditionfortypeB_2m theories,

ˆ

p_i

≡

^pi

(

1

−

^aiz

),

(11)

ˆ

p⁰_i

≡ ˆ

p^2m_i

=

p^2m_i

(

1

−

aiz

)

^2m

.

(12) Momentumandenergyconservationthenimposerespectivelythe followingconstraintsontheai coefficients,

n

i=¹

a_ie_ip_i

=

⁰

,

(13)

n

i=¹

(

1

−

^zai

)

^2me_ip^2m_i

=

⁰

.

(14)

Hereeidenotesasign,chosensothatei= +^{1 forparticlesinthe} final state andei= −^{1 for particles in the initial state. Similarly} to the relativistic case reviewed in section 1.2, the existence of nontrivial solutions to Eq. (13) requires n≥^d+^2. Equation (14) then imposes 2m additional constraints. Only amplitudes with n≥^d+²+^{2m maytherefore be} reconstructed using softrecur- sion.Forgivend andm,thistells ushowmanyseed amplitudes weneedtoinitiatetherecursionprocedure.

2.2. SoftshiftsfortypeA_mtheories

Fortype Am theorieswedefineanalogously

ˆ

p_i

≡

^pi

(

1

−

^aiz

),

(15)

ˆ

p⁰_i

≡ | (

p^2m_i

)

^1/2

| (

1

−

^aiz

)

^m

,

(16) whichpreserveson-shellness andyieldsthe followingconstraints frommomentumandenergyconservation,

n

i=1

aieip_i

=

0

,

(17)

n

i=¹

(

1

−

^zai

)

^me_i

| (

p^2m_i

)

^1/2

| =

⁰

.

(18)

AnalogouslytothetypeB_2m case,theexistenceofnontrivialsolu- tionsfora_irequiresn≥^d+²+^m>2+^{2m,wherethelastinequal-} ityfollowsfromthenonrelativisticCHMWtheorem.Forthespecial caseofm=^{1,whichincludesthefamilyof}Lorentz-invariantthe- ories,theabove constraintsbecome equivalent to Eq. (6) and we recovertherelativisticboundn≥^d+³=^D+^2.

Notethatforbothtype AmandtypeB2mtheories,themanifold ofsolutionsfortheai coefficientsisinvariantunderoverallrescal- ing,a_i→λa_i,andoverallshift,a_i→^ai+^{c.Thisguaranteesthatin} thespecialcaseoftype A1 theorieswherealltheconstraintsonai arelinear,possiblesolutionsforai spananaffinespace.

3. Softrecursion

Weargued in section 1.2that forrelativisticexceptional EFTs, recursionrelationsamongscatteringamplitudesmaybesetupus- ingEq. (8).Sincetheargumentonlydependsontheassumedsoft behaviorof An,factorizationandvanishingoftheboundaryterm, itcanbegeneralizedtoanytheorywiththeseproperties.Specifi- cally,fortheoriesoftype A_m andB_2mweobtain

A_n

= −

I

2m

i=¹ Res

z=^ziI

A

ˆ

_n

(

z

)

z F_n

(

z

) .

(19)

Here zⁱ_I, i=¹,. . ., 2m are solutions to the on-shell condition, whichisofalgebraicorder2minz,

P

ˆ

⁰_I

2

− ˆ

^P2m_I

=

^{0 forA}m

,

(20)

P

ˆ

⁰_I

− ˆ

^P2m_I

=

^{0 forB}2m

,

(21) foragivenfactorizationchannelI,wherecomparedtoEq. (10), P_I isnowdefinedwiththeappropriatesignse_iwherenecessary.Fac- torizationthen impliesthat theamplitude (19) canbe expressed intermsoflower-pointamplitudes,

A_n

= −

I

2m

i=¹ Res

z=^ziI

A

ˆ

⁽_L^I)

(

z

)

A

ˆ

⁽_R^I)

(

z

)

z F_n

(

z

)

D^(I)

(

z

) ,

(22) where

D^(I)

(

z

) =

P

ˆ

⁰_I

2

− ˆ

^P2m_I ^forAm

,

(23)

D^(I)

(

z

) = ˆ

^P0_I

− ˆ

^P2m_I ^forB2m

.

(24) Notice that the contribution from factorization channel I in Eq. (22) matchestheresidueatz=^zi_I ^{ofthefollowingmeromor-} phicfunction

A

ˆ

⁽_L^I)

(

z

)

A

ˆ

⁽_R^I)

(

z

)

z Fn

(

z

)

D^(I)

(

z

) .

(25)

Thisfunctioncanalsohavenonvanishingresiduesatz=¹/a_i and z=^{0. This follows from the fact that the} intermediate propaga- tor D^(I)(z), hence also the subamplitudes Aˆ⁽_L^I)(z) and Aˆ⁽_R^I)(z), is off-shellfor z=^zi_I^{.The on-shellargument implyingthat thesoft} behavioroftheamplitudesdictatedbyEq. (4) cancelsthezerosof Fn(z)isthennolongervalid.InthespecialcasewhereAˆ⁽_L^I)(z)and Aˆ⁽_R^I)(z)arebothlocalfunctionsofmomenta(thatis,theyhaveno

poles)we can applyCauchy’s theoremtothe meromorphic func- tion inEq. (25) and recastthe amplitude (22) intermsof asum overresiduesatz=^{0 andz}=¹/ai,

A_n

=

I

A

ˆ

⁽_L^I)

(

0

)

_A

ˆ

^(I)

R

(

0

)

D^(I)

(

0

)

+

I

n

i=1 z=Res¹/ai

A

ˆ

⁽_L^I)

(

z

)

A

ˆ

⁽_R^I)

(

z

)

z Fn

(

z

)

D^(I)

(

z

)

≡

A_n^ch

+

A^ct_n

.

(26)

This expressionis particularlyuseful forconcreteapplications. In termsofFeynmandiagrams,thefirsttermcorrespondstothesum over diagrams with an internal propagator, whereas the second (double) sum encodes contributions fromn-point contact opera- tors.Thetwodifferenttypesofcontributionsaredistinguishedby thenotationintroducedinthelastlineofEq. (26).

3.1. Validitycriterion

Thus far we have simply assumed that the boundary term Bn vanishes. A sufficient condition for this to happen is that Aˆn(z)/Fn(z)→^{0 asz}→ ∞^{.Acriterionforthelatterwas inturn} givenbyElvangetal. inRef. [9].Theirargumentonlyreliesondi- mensional analysis,thesoftbehaviorof A_n,theanalytic structure oftree-levelamplitudes,andthefreedomtoshiftalla_ibyanover- all constant.Sincethelatterpropertysurvivesinall type Am and B2m theories,asshowninsection 2,itiseasy toadapttheargu- mentofRef. [9] forourpurposes.

We start with a generic expression for the n-point tree-level amplitude,

A_n

=

j

k

gⁿ_k^jk

M_j

,

(27)

whereM_jarefunctionsofmomentaandg_karecouplingconstants associated withfundamental operators inthe Lagrangian. Funda- mentaloperatorsaredefinedinturnasthelowest-dimensionop- erators whoseon-shell matrix elements are neededto derive, at theleading-orderinthelow-energyexpansion,anytree-levelam- plitudeinthetheoryby recursion.Followingthelineofreasoning ofRef. [9] thenleadstothegeneralizedvaliditycriterion

[

^An

] −

^min

j

k

n_jk

[

^gj

]

−

n

i=¹

σ

_i

<

0

,

(28)

wheresquare bracketsindicatescaling dimensionwithrespectto auniformrescalingofallthemomenta p_i.Itiseasytocheckthat thecriterion (28) issatisfiedbyalltheexampletheoriespresented inthenextsection.

4. Examplecalculations

Wewillnow workoutthreesimpleanalytical examplesofre- cursivereconstructionofscatteringamplitudesintheoriesoftype B2,A1 andA2,respectively.Allthreesampletheoriesfeaturetree- levelamplitudeswithsoftscaling

σ

i=^{2.Yet,eachofthetheories} possesses Lagrangian representations with less than two deriva- tivesperfield,whichmeansthattheypossessenhancedsoftlimits.

We will show in a forthcoming paper that the enhanced scaling ofscatteringamplitudes inthesetheoriesisa consequenceofan interplayofspontaneously brokensymmetryanddispersion rela- tions of NG bosons. Each ofthe three theories contains just one

physicalNGmode. Sincewe nolonger havetodistinguishdiffer- ent

σ

ifordifferentparticlesparticipatinginthescatteringprocess, weintroduceashorthandnotationreplacingEq. (7),

F_n^(σ)

(

z

) ≡

n

i=¹

(

1

−

^aiz

)

^σ

.

(29)

4.1. B2:Schrödinger-DBItheory

Ourfirst example features a complexscalar field endowed withtheaction

S

=

dtd^dx

^†i

∂

0

+ √

G

−

¹

,

(30)

G

≡

¹

−

²

∇ · ∇

^†

+

∇ · ∇

^†

2

−

∇ · ∇

∇

^†

· ∇

^†

.

(31)

Thisis a minimal nonrelativistic modificationof one of the very few relativistic single-flavor exceptional theories [19]:the Dirac- Born-Infeld(DBI) theory.We thereforename itthe “Schrödinger- DBI”(SDBI)theory.

OurSDBI theory can be interpreted as describing fluctuations ofad-dimensionalbraneembedded ina(d+²)-dimensionalEu- clideanspace.ThesymmetryoftheSDBIaction (30) isaccordingly R×^ISO(d+²),withthe first factorof R corresponding to time translations [25]. This symmetry is spontaneously broken down to R×^ISO(d)×^SO(2) by the presence of the brane, and the realandimaginaryparts ofcorrespondtoNGfieldsofsponta- neouslybrokentranslationsinthetwoextradimensions.Theterm in Eq. (30) with a single time derivative is only invariant under thefullsymmetryuptoasurfaceterm.Itisthusanexampleofa Wess-Zumino-Witten(WZW)term.

The action (30) fixes all tree-level amplitudes. We will now demonstratethat therecursionformula (26) correctlyreproduces the six-pointamplitudestarting fromthe seed four-point ampli- tude. In fact, the argument of section 2.1 limits the validity of therecursionforn=^{6 to d}≤^{2 spatial}dimensions. However,the amplitudes An asfunctionsofthemomenta p_idonotdependex- plicitlyond.Whateveranalytic relationsbetweentheamplitudes wefindwillthereforebeindependentofdaswell.Onemaythink ofthisascarryingouttherecursivestepfromA4to A6ind=^{2 di-} mensions,andthenanalyticallycontinuingtheresulttoanyvalue ofdofinterest.

Tomakethecalculationtransparent,we firstexplicitlylistthe relevantpartsoftheLagrangian,

L2

=

^†

(

i

∂

0

+ ∇

²

),

(32) L4

= −

¹

2

∇ · ∇

∇

^†

· ∇

^†

,

(33)

L6

= −

¹ 2

∇ · ∇

∇ · ∇

^†

∇

^†

· ∇

^†

.

(34)

Charge conservation dictates that the numbers of incoming and outgoing Schrödinger scalars must match in any scattering pro- cess. We usethe convention that the particleslabeled 1,. . . ,n/2 are incoming, whereas the particles n/2,. . . ,n are outgoing. The seedon-shellfour-pointamplitudethenfollowsimmediatelyfrom Eq. (33) as

A4

=

²

(

p₁

·

^p2

)(

p₃

·

^p4

).

(35) Wearenow readytoderivethe six-pointamplitudeby recur- sion.Wewillusetheindicesa,b,c tolabelapermutation ofthe incomingparticlesandd,e, f apermutationoftheoutgoingpar- ticlessuch that a,b, f are on thesame side ofthe factorization

channel. We can then identifythe nine factorizationchannels in termsofcand f alone,

I

= {(

c

,

f

)} = {(

14

), (

15

), (

16

), (

24

), (

25

), (

26

),

(

34

), (

35

), (

36

)}.

⁽³⁶⁾

Energyandmomentumconservationfixtheparameters ofthein- termediatepropagatorforeachfactorizationchannel,

PI

≡

^pa

+

^pb

−

^pf

=

^pd

+

^pe

−

^pc

,

(37) 1

2

P⁰_I

−

^P2I

= −

^pa

·

^pb

−

^pf

·

^pf

+

^pa

·

^pf

+

^pb

·

^pf

= −

p_d

·

p_e

−

p_c

·

p_c

+

p_d

·

p_c

+

p_e

·

p_c

.

ThechannelcontributionA^ch₆ asdefinedbyEq. (26) reads

A^ch₆

=

⁴

I

(

p_a

·

^pb

)(

p_d

·

^pe

)(

p_c

·

^PI

)(

p_f

·

^PI

)

P⁰_I

−

^P2_I ⁽³⁸⁾

=

σ,ρ∈^S3

(

p_σ(1)

·

^pσ(2)

)(

p_ρ(4)

·

^pρ(5)

)(

p_σ(3)

·

^kσρ

)(

p_ρ(6)

·

^kσρ

)

k⁰_σρ

−

^k2_σρ

,

where

σ

and

ρ

denoterespectivelypermutationsof {¹,2,3}^and {⁴,5,6}^{,andwehaveusedtheshorthandnotation}

k_σρ

≡

^pσ(1)

+

^pσ(2)

−

^pρ(6)

.

(39) The second line of Eq. (38) is manifestly equal to the Feynman diagramexpressiononeobtainsfromEq. (33).

Similarly, the contact contribution to the six-point amplitude followsfromEq. (26) as

A^ct₆

=

⁴

I

6

i=1

Resz=zi

(

p

ˆ

_a

· ˆ

^pb

)(

p

ˆ

_d

· ˆ

^pe

)(

p

ˆ

_c

· ˆ

^PI

)(

p

ˆ

_f

· ˆ

^PI

)

z F₆⁽²⁾

(

z

)

P

ˆ

⁰_I

− ˆ

^P2I

≡ −

²

I

6

i=1

f

(

z_i

).

(40)

The residuesatzi≡¹/ai fora givenfactorizationchannel canbe rewrittenas

f

(

za

) =

Res

z=za

(

p

ˆ

_a

· ˆ

p_b

)(

p

ˆ

_d

· ˆ

p_e

)(

p

ˆ

_c

· ˆ

p_f

− ˆ

p_c

· ˆ

p_b

)

z F₆⁽²⁾

(

z

) ,

f

(

z_b

) =

Res

z=zb

(

p

ˆ

_a

· ˆ

p_b

)(

p

ˆ

_d

· ˆ

p_e

)(

p

ˆ

_c

· ˆ

p_f

− ˆ

p_c

· ˆ

p_a

)

z F₆⁽²⁾

(

z

) ,

f

(

zc

) =

Res

z=zc

(

p

ˆ

_a

· ˆ

p_b

)(

p

ˆ

_c

· ˆ

p_d

+ ˆ

p_c

· ˆ

p_e

)(

p

ˆ

_d

· ˆ

p_f

+ ˆ

p_e

· ˆ

p_f

)

z F⁽₆²⁾

(

z

) ,

f

(

z_d

) =

Res

z=zd

(

p

ˆ

_a

· ˆ

p_b

)(

p

ˆ

_d

· ˆ

p_e

)(

p

ˆ

_c

· ˆ

p_f

− ˆ

p_e

· ˆ

p_f

)

z F₆⁽²⁾

(

z

) ,

f

(

ze

) =

Res

z=ze

(

p

ˆ

_a

· ˆ

p_b

)(

p

ˆ

_d

· ˆ

p_e

)(

p

ˆ

_c

· ˆ

p_f

− ˆ

p_d

· ˆ

p_f

)

z F₆⁽²⁾

(

z

) ,

f

(

z_f

) =

Res

z=zf

(

p

ˆ

_d

· ˆ

p_e

)(

p

ˆ

_a

· ˆ

p_c

+ ˆ

p_b

· ˆ

p_c

)(

p

ˆ

_a

· ˆ

p_f

+ ˆ

p_b

· ˆ

p_f

)

z F₆⁽²⁾

(

z

) .

After substituting the expressions above into Eq. (40), collecting the contributions to the residue ateach z_i from all factorization channels,andusing(shifted)momentumconservation,weobtain

A^ct₆

= −

¹ 2

6

i=¹ Resz=^zi

1 z F₆⁽²⁾

(

z

)

(41)

×

σ,ρ∈^S3

(

p

ˆ

_σ(1)

· ˆ

^pσ(2)

)(

p

ˆ

_σ(3)

· ˆ

^pρ(4)

)(

p

ˆ

_ρ(5)

· ˆ

^pρ(6)

).

AfinalapplicationofCauchy’stheoremyields

A^ct₆

=

¹ 2

σ,ρ∈^S3

(

p_σ(1)

·

p_σ(2)

)(

p_σ(3)

·

p_ρ(4)

)(

p_ρ(5)

·

p_ρ(6)

),

(42)

which is manifestly equal to the contribution from the contact terminEq. (34).

4.2. A₁:spatialGalileon

OursecondexampleincludesawholeclassofLagrangiansofa realscalarfieldφ,

L

=

¹

2

(∂

_μ

φ)

²

+

d+1

n=3

c_n

φ

G_n−1

,

(43)

where cn are real coupling constants and Gn is a polynomial of orderninthesecondspatialderivativesofφ,

G_n

≡

¹

(

d

−

ⁿ

)!

^{i1···inkn+1···kd}

^j1···jn_k

n+1···kd

× (∂

_i1

∂

_j1

φ) · · · (∂

_in

∂

_jn

φ).

(44) This is a nonrelativistic version of another type of a relativis- ticsingle-flavorexceptionaltheory [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 thereforedubit“spatialGalileon.”Theaction (43) isinvariantun- der polynomial shifts of φ of first order in spatial coordinates, φ→φ+

α

₊β·^{x.Thisspatialversion oftheusual Galileonsym-} metryisaspecialcaseofaclassof“multipolealgebras”thathave recentlyattractedattentioninthecontextoffractonphysics [16].

AllinteractiontermsinEq. (43) aswell asthespatialpartofthe kinetictermareoftheWZWtype [27].

Since the spatial Galileon is a type A1 theory,the validity of therecursion islimitedto n-point amplitudeswithn≥^d+^{3, as} showninsection2.2.Forillustration,wewillnowrestrictEq. (43) tothe quarticinteraction termandshow how to reconstructthe six-pointamplitude.Thisrequiressettingd=^3,sinceford<3 the quarticspatialGalileoninteractiondoesnotexist.

It is convenient to express the Feynman rule for the n-point spatialGalileonvertexas [28]

Vn

(

p₁

, . . . ,

p_n

) =

^cn

σ∈Zn

G

(

p_σ(1)

, . . . ,

p_σ(n−1)

),

(45)

where G(p₁,. . . ,p_n−1) is the Gram determinant, that is the de- terminant of the (n−¹)×(n−¹) matrix with entries p_i·^pj. Importantly,theGram determinantisa symmetric,homogeneous polynomialofordertwoinallitsarguments,

G

(λ

p₁

, . . . ,

p_n−1

) = λ

²G

(

p₁

, . . . ,

p_n−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=^ncnG(p₁,. . . ,p_n−1).

The six-point amplitude is now determined in terms of the four-pointseedamplitudebyEq. (26),