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JHEP11(2020)136 Springer July 17, 2020 4 : a, October 17, 2020 : November 25, 2020 : , Received Accepted Published and Frank Saueressig Published for SISSA by 3 https://doi.org/10.1007/JHEP11(2020)136 c, [email protected] Chris Ripken , 2 b, [email protected] , . 3 -ﬁniteness, unitarity, and causality are analysed in detail and it is shown 2007.04396 Benjamin Knorr, The Authors. 1 UV Models of Quantum Gravity, Nonperturbative Eﬀects, Renormalization c a,

We employ the curvature expansion of the quantum effective action for gravity- , [email protected] https://orcid.org/0000-0002-6895-894X. https://orcid.org/0000-0001-6700-6501. https://orcid.org/0000-0003-2545-5047. https://orcid.org/0000-0002-2492-8271. 1 2 3 4 31 Caroline St. N., Waterloo,Institute ON of N2L Physics 2Y5, (THEP), Canada Staudingerweg University 7, of 55128 Mainz, Mainz, Germany E-mail: [email protected] Institute for Mathematics, AstrophysicsRadboud and University Particle Nijmegen, Physics (IMAPP), Heyendaalseweg 135, 6525 AJ Nijmegen,Perimeter The Institute Netherlands for Theoretical Physics, b c a Open Access Article funded by SCOAP ArXiv ePrint: infinite derivative gravity, and Asymptotic Safety. Keywords: Group, Scattering Amplitudes resulting from by explicit construction that thethese quantum structural effective action requirements provides without sufficientfreedom. introducing room non-localities Our to framework meet or provides higher-spin aseeking degrees bottom-up for of approach the to quantisation allscope of quantum gravity gravity is programs within illustrated the framework by of specific quantum examples, field theory. including Its effective field theory, Stelle gravity, Abstract: matter systems to constructcoupled scalar graviton-mediated fields in scattering a Minkowski amplitudes background.quantum for corrections By design, to non-minimally the these formalism parameterises processes all and is manifestly gauge-invariant. The conditions Tom Draper, Graviton-mediated scattering amplitudes fromquantum the effective action JHEP11(2020)136 1 ]. 8 ]. – 11 1 8 15 11 3 5 18 21 3 ] as well as the recent review [ 10 8 , 9 12 have confirmed all particles in the Standard 11 – 1 – LHC -channels u 14 22 19 - and t 21 5 6 13 scattering scattering 1 23 χχ φφ → → φφ φφ For related ideas in the context of string theory see [ A.1 Graviton propagator A.2 Gravity-matter vertices 5.2 Stelle gravity 5.3 Infinite derivative5.4 gravity Renormalisation group5.5 improvements from Asymptotic Form Safety factors realising Lorentzian Asymptotic Safety 4.1 s-channel scattering4.2 for distinct particles Scattering including 5.1 Effective field theory and IR properties 2.1 Construction of the generic gravity-scalar action 3.1 3.2 1 Model. In order tothe relate one experiment hand, to they theory, form scatteringas the amplitudes differential basis cross play for sections a computing and keysuch experimentally role. decay as accessible rates. unitarity, On quantities causality On such and the positivity other put hand, constraints theoretical on requirements the amplitudes [ Quantum field theories haveenergy been physics. extremely successful Theyparticle in physics, constitute which providing the has predictions been frameworkery in tested for of with high- the great formulating Higgs precision. boson, the experiments In Standard at particular, the with Model the discov- of B Conventions 1 Introduction 6 Summary and discussion A Explicit expressions for propagators and vertices 4 The physics of graviton-mediated scattering — general case 5 The physics of graviton-mediated scattering — examples 3 Gravity-mediated scalar-scalar scattering amplitude Contents 1 Introduction 2 The quantum effective action including form factors JHEP11(2020)136 ] ], 43 37 , 36 ], Causal 31 – 25 . Although the com- Γ and we summarise our ] constituting only some ) yields a perturbatively , we present the effective including form factors [ A 42 is an effective field theory 2 GR Γ ], Group Field Theory [ , we compute the most general GR 3 35 , 34 ], Asymptotic Safety [ . The explicit form of the propaga- ]. By now, there is a significant num- 24 6 – 19 – 22 16 . These sections contain the main novel results – 2 – 4 GeV [ into a fundamental theory valid on all scales with 19 GR 10 ' ]. This is reflected in the amplitude of gravity-mediated ], Causal Set Theory [ ], and weakly non-local gravity [ 14 Pl – 33 41 , M – 12 , we specialise our result to various quantum gravity models 32 38 5 ], Loop Quantum Gravity [ 21 , 20 ]. Therefore, it is commonly accepted that is generically difficult, we can study parameterisations capturing all contri- . The scattering amplitudes are then calculated from the tree-level Feynman 15 ] for a detailed discussion). The latter can be regarded as the operator equiv- Γ ]. Γ 44 45 The rest of this paper is structured as follows. In section Taking such a general approach allows us to study different quantum gravity theories In order to systematically study scattering amplitudes in a way that is agnostic to At this stage, it is an open debate under which conditions gravity may be formu- erties of our result areof analysed this in section paper.studied In in the section literature givingpaper their with description in a terms summarytors of and and form a factors. vertices entering discussion We conclude the in the computation section is given in appendix specific model where the amplitudestal become requirements scale-free without and introducing compatibledetail new with in particle all [ resonances. fundamen- This model is described in action that generates the scatteringfour-scalar amplitudes. scattering In amplitude section and the resulting differential cross section. The key prop- by specifying the corresponding formity, factors. infinite-derivative gravity, Asymptotic In Safety, this and way, renormalisation effectivecan group field improvements be theory, Stelle treated grav- in abehaviour, uniform we language. investigate their By considering compatibility their with pole unitarity structure and and causality. high-energy This leads to a to curved spacetime. Theinvariant) main scattering result amplitude of for this a gravity-mediated paperin two-to-two is scalar a particle the Minkowski scattering most background. generalmatter On (manifestly form this gauge- factors basis, in we orderidentify derive to classes conditions obey of for unitarity, quantum causality, effective the and actions gravitational positivity. compatible and This with enables these us fundamental properties. to putation of butions to a physicalamplitudes process. can be In analysed based particular,(also on the a see curvature [ high-energy expansion of behaviouralent of of the momentum-dependent scattering interactions which generalise Minkowski-space interactions selected examples. the underlying microscopic physics,tive we action build on adiagrams parameterisation constructed of from the the quantum propagators effec- and vertices encoded in corrections [ valid up to the Planckber scale of proposals on howString to Theory complete [ Dynamical Triangulations [ infinite derivative gravity [ lated as athe quantum standard field model theory. ofnon-renormalisable particle theory physics [ Applying to thescattering general of quantisation relativity matter; ( techniques forquadratically successful with a the for two-to-two centre-of-mass energy. particle This process, divergence is the aggravated tree-level by adding amplitude loop diverges JHEP11(2020)136 are (2.1) (2.2) CC . f ) 3 and which intro- R ( RR O f + QED µνρσ , which corresponds to . This expansion is ex- C 2 / R 1 − N (∆) G CC ≡ f . Pl µνρσ M C matter 1 2 + is the d’Alembertian and + Γ R ν gf D µ (∆) – 3 – + Γ D RR µν ]. Form factors arise naturally in the computation grav g Rf 44 , e.g., for near-flat spacetime. Up to second order in − 2 1 6 − Γ = Γ R ∆ = − R R g 2 We begin with the purely gravitational part. In this case, an . Throughout this work we study the scattering of scalar matter − D B √ necessary that flat spacetime is a solution to the equations of motion. x 4 d not Z N . 1 πG = 1 16 is Newton’s constant, N G = N G A few remarks are in order here. For simplicity, we have set the cosmological constant grav Γ of motion. While thetechniques) Minkowski as background well is as motivated conceptual for reasonsbased (flat technical physics) it spacetime (momentum-space is is a goodSince approximation we for discuss earth- scalar scatteringthus only, remain the off-shell. only gravitons that appear are virtual and can tion is chosen foradditional convenience, and poles signs appear are in the suchquantities propagator. that are if measured In the the insetting form remainder units factors of of this are the work, positive, Planck all no mass dimensionful to zero. This ensures that flat Minkowski spacetime is an on-shell solution to the equations Here the form factors which determine the flat-spacetime graviton propagator. The normalisa- Gravitational action. efficient expansion scheme ispected in to powers be of accurate thethe if curvature curvature, the tensor purely gravitational part reads We will now use form factorseffective to action construct a systematic curvature expansion of the quantum This will provide theprocesses basis in for a computing flat background. the amplitude for two-to-two scalar scattering ing functions depend ona partial natural derivatives. generalisation Thederivatives. to correspondence curved principle space, then provides replacing the2.1 partial derivatives by Construction covariant of the generic gravity-scalar action of loop corrections.duces A a textbook non-trivial example momentum isfactors dependence are the in functions electron of the momentum self-energy electron invariantstices. entering in propagator. the In propagators flat Generically, and spacetime, form interactionrepresentation the ver- of Fourier the transform form allows factors to to switch a from position-space the representation where momentum-space the correspond- only. The generalisation to other matter fields is left2 for future work. The quantum effectiveWe action start including our investigation form with factors adetailed brief discussion introduction can to form be factors found in in curved spacetime. [ A conventions in appendix JHEP11(2020)136 (2.8) (2.5) (2.6) (2.7) (2.3) (2.4) . ) ), respec- Z 3 . . 2.8 n # ) 4 h ν ) (∆ , φ ∂ Y 2 2 β ), and ( R n -symmetry, and that ( , 4 2 2 2.9 O 1 + Z , χ ) (∆ 2 η + , − ), ( φ X 2 φ 1 ) χ ρν ) n R φ φ 2.6 2 ]. h φ ) φ i

( p " φφ 2 χ π + 3 and 2 + 4 1 φ → p f φφ = f = = ( φφ s φχ u φχ 4 By crossing symmetry, the scattering amplitude for the process A A the identifications The other diagrams are then obtainedare by obtained crossing symmetry; from the the 3.2 Let us briefly discussprocess, the there scattering amplitude are for threegraviton. a different These single diagrams Feynman scalar are diagrams shown field, The involving in amplitude figure the of the exchange of a virtual by interchanging fixing parameters three-point vertices together with theinvariant on-shell part. conditions projects This thefeature establishes former of the on any its gauge observable. gauge- invariance The of amplitude our related result, to the which matter is self-interactions an reads essential Kinematically we have to assume that thefinal energy transfer and is at squared least the initialfactors difference of mass, squared if theevaluated on-shell, former and thus is either vanishes larger. ifof the one form by factor Notably, the itself normalisation the appears, condition or scalar ( gives a kinetic factor form The Mandelstam variables are subject to the relation momentum conventions, we refer the readeris to computed appendix bycalculation combining was the performed graviton withyielding propagator the ( help of the Mathematica package suite and JHEP11(2020)136 φφ (3.8) (3.6) (3.7) → , φφ # 3 4 ) p p

s .

− (

4 1 u ) p + p ( = q ! scattering. The RR 2 φ ). The black dots -legs, the internal G m χ 2 χχ u-channel 2.1 2 2 ) − 1 2 2 φ → p p (c) s , , m φφ o 2 φ 2 φ 2 u m 2 2 , + s, m ( − φφ 4 t tu ). 4 3 4 , A p p Rφφ 2 φ 2.6 − + sf m ), and the black dots indicate the three- 2 t 2 -legs, the internal double line corresponds 2 12 φφ u n φ 2.1 4 3 )

− p p

) A 2 −

p −

u 3 1 t ) p + p ( = q ( + + , ) 1 2 φ 2 φ p CC φφ t ( m − G t-channel A – 7 – , m 2 2 2 2 φ = + s − 1 2 ) q p p (b) 2 φ u φφ s s, m , ( A 2 φ -vertices encoded in ( , m 2 φ m = 1 2 -legs, external dashed lines correspond to p p 2 2 φ hχχ Ricφφ φφ ). − s, m sf ( A t 2.6 , - and 2 φ 1 + 4 3 p p Ricφφ m 2 ) 2 hφφ sf 2 φ − ) m 2 s p 1 + +

1 + 2 4 p ( φ s − − ( f s-channel = " Feynman diagrams encoding the graviton-mediated contribution to the s The s-channel Feynman diagram that contributes to the q = π 3 + 4 (a) -vertex encoded in ( φφ 4 = 1 2 A p p hφφ φφ s A and This completes our computation of the scattering amplitudes. As an alternative check, weresults have in performed the explicit computation of all channels. This with the building blocks being Figure 2. scattering amplitude. The externalto lines the correspond gauge-fixed to gravitonpoint propagator obtained from ( Figure 1. external solid lines corresponddouble line to corresponds to theindicate gauge-fixed the graviton three-point propagator obtained from ( JHEP11(2020)136 . 3 (4.2) (4.3) (4.4) (4.5) (4.1) , ) in the in terms # 3.2 o u ) , o gives rise to 2 χ is studied in ) 2 φ and ,m 4 φχ ) φφ 2 χ t 2 χ ,m A 2 φ → , ,m s,m 2 χ ( , -channel contributions s,m φφ ) ( s = 2 θ , are problematic for the , Rχχ j s,m ) 0 ( ) 2 φ Rφφ θ sf cos is not affected by the form < . sf ,m and 12 s, j 2 φ ( φχ s Ricχχ α φχ s 12 (cos − j , we re-express φχ s A A − . . The angular dependence of the )) sf = 0 P s,m j s χ 2 χ A B )) ( j i ) 2 φ m 1+ − θ 2 ,m res φχ s 2 χ ,m 2 res A × M Ricφφ 2 φ . The self-interaction h and (cos . Since it involves 3 M j sf s,m φ ( 2 s,m res χχ ≥ Res m ( θ P 1+ M θ j – 8 – → gravitons. → Ricχχ s ) n Ricφφ s d cos φφ sf = lim ( and d cos . Intuitively, this corresponds to the contribution of 1 1 sf i 1 j 1 − CC − Z φχ s Z G )(1+ = 1 π ) A 2 χ π )(1+ 1 h 2 χ 1 j 2 φ 32 m partial-wave amplitude via 32 m ]. m 4 j and the masses ≡ Res +2 57 for ≡ ) − s θ – +2 j s s ( s ( α n 55 ( )( j φχ 2 φ n = 0 × a " m ) comprises only partial waves with 4 s φχ j ). The differential cross section for the process ( a − φχ s s 3.2 RR ( A G , we consider the process 1 denotes the Legendre polynomial of order 60 1 ) 12 − . x 4.1 ( ). The angular dependence of the amplitude j ) = ) = 4.2 P s s 3.2 ( ( It is now instructive to perform the partial-wave decomposition of the general re- In the cases where the scattering amplitude admits resonances associated with the , the scattering angle φχ 2 φχ 0 s a a whereas one has partial-wave amplitudes for all even “ladder diagrams” with an exchange of sult ( factors. Thus, where In general, massive polesunitarity coming of with the a theory negative [ residue, presence of additional massive degrees of freedom,of it the is resonance also convenient in to define the the spin- strength amplitude can then readily be separated by performing a partial-wave decomposition where 4.1 s-channel scatteringIn for order distinct to particles compute observablescentre-of-mass such frame. as differential cross Using sections, theof we relations evaluate ( of appendix In this section, we studyIn the properties section of the scatteringonly, amplitudes it computed in allows section forcontent a of eq. partial-wave analysis. ( section We will use this tool to analyse the physics 4 The physics of graviton-mediated scattering — general case JHEP11(2020)136 . GR (4.9) (4.6) (4.7) (4.8) (4.12) (4.10) (4.11) . This makes φχ 2 a . 2 ,

) θ , 2 s . ) s 2 N ( θ ). This form of the cross (cos G 2 2 s . φχ 2 for clarity. The amplitude is . In this case all gravity and π P a 4.5 15 2 . ) N cos N s 2 60 GR ( G = ) gives the diﬀerential and total s G + 5 − . − 2 φχ 2 1 2 N 2 ) φχ 4.8 s a s G ( ) and ( s |A| ) = ∝ N s φχ 0 s 4.4 ( ) + 5 a 2   θ 1 π φχ 2 ) and ( π G π GR CM 64 s θ s , σ ! 4.7 (cos 16 4 = 2 0 = – 9 – φχ s P Ω = ) s sin d t u s dσ CM s , a ! ( 2 N N

N 16 φχ 0 φχ G s 12 σ s G a σ d dΩ dΩ  

π G d = s 4

) = →∞ lim s = GR CM = 8 s ( dΩ ! . We also reinstate powers of and ﬁve transverse-traceless modes related to φχ 0 GR CM Z a φχ s Ω ! φχ 0 d = 0 φχ s = dσ a φχ χ A s

σ φχ dΩ s m d σ =

φ m In order to illustrate this result, we first specialise to The partial-wave decomposition gives a particularly simple form for the cross section. In the high-energy limit, andquadratically measured with in the the energy process transfer energy s, itself, the cross section scales that the scattering is solelyInserting mediated the by the partial-wave amplitudes masslesscross degrees into sections of ( freedom described by plitudes in the spin-zero and spin-two channels are These partial-wave amplitudes do not exhibit any poles at non-zero momentum, indicating massless, i.e., given by the well-known result where the second equality holds in the centre-of-mass frame. The two non-vanishing am- scalar mode related to it clear that internalonly lines two also independent depend polarisations. on off-shell modes, sincegravity-matter on-shell form the graviton factors has are zero. In addition, we assume that the scalar fields are with the partial-wave amplitudessection given illustrates in the eqs. physical ( number of (off-shell) graviton modes. There is a single Inserting the partial-wave expansion of the amplitude then gives Due to orthogonality of the Legendre polynomials, this gives for the total cross section For brevity, we will notmass frame, discuss the the cross contribution section of reads the self-interaction. In the centre-of- JHEP11(2020)136 1 (4.15) (4.16) (4.14) (4.13) should σ a positive ) s ( , . Z S S c f , 2 1+ g 1+ : the cross section, − − − with s s s GR h . ) ∝ ∝ ∝ while the graviton prop- behaviour of the partial- ) s ) ) ) ( R . ) s s 2 φ 2 χ ,c ) s ( Z ( S S . p , m ,c , m ,c Z 1 CC , and faster if the vertex form ) scales as 2 φ 2 χ 2 − p ] though, stating that G 7→ 4.5 ≥ 2 > h , s, m s, m ) ( 2 ( 1 →∞ lim )+max(0 )+max(0 s s ( S R , g 0 ,f ,f g S RR Ricφφ Ricχχ ,f f f . Since the scattering amplitudes are linear is to have appropriate factors of the energy. 2 +max(0 . Hence rescalings cancel and the amplitude N →∞ →∞ − lim lim 2 s 1 s g G − which maps a rank-(0,2) tensor to a rank-(0,2) tensor, +max(0 – 10 – , sf − 0 ) 2 g 1 s s ρσ ( − − 2 µν Z ∝ s > , ) Z , , 0 ∝ s ) g ( √ s R R ) ( − f c s 2 φχ s s s ( a CC . The growth of the cross section is directly linked to the fact ∝ ∝ ∝ φχ 0 s a sf ) ) ) 2 s →∞ 2 φ 2 χ lim s ( log →∞ lim , m , m s RR 2 φ 2 χ G s, m s, m ( ( →∞ lim s Rφφ Rχχ f f each vertex receives an additional contribution . Requiring that the dimensionless cross section stays finite at all energies gives 2 s →∞ →∞ lim lim s s by construction, the cross section must depend on it quadratically. The only way to N Based on the diagrammatic structure of the scattering amplitudes visualised in figure Including form factors introduces sufficient freedom to tame the growth of the ampli- Z is commonly referred to as the wave function renormalisation. In gravity it can be tensor-valued, 2 G agator picks up anremains additional invariant. factor where the definingincluding object appropriate is symmetries. actually at least quadratically infactors the contribute. squared momentum, it is clear thatfield this redefinition analysis of is the actually gravitonfunction, fluctuations: independent replacing of a potential momentum-dependent Assuming that theconclude self-interaction that is boundedness sub-leading of for the these total partial-wave amplitude amplitudes, requires we that the propagators fall off wave amplitudes. The spin-two partial-wave amplitude ( Similarly, the spin-zero partial-wave amplitude behaves as On this basis one can distinguish various scenarios for the large Requiring a bounded crossbehaviour section of at the high propagatorsintroduce energies the and asymptotic gives form scaling a factors laws condition in on the the vertices. asymptotic It is then convenient to balance the mass dimension introduced by tudes in conditions on the form factors. In particular, to exclude additional graviton modes we need measured in the relevant energy scale, divergestheories quadratically it with the should energy. be Fornot “healthy” subject increase faster to than thethat Froissart Newton’s bound constant [ has mass-dimension in This points at one of the problems related to the quantisation of JHEP11(2020)136 ] 4 ), φ φφ t 60 f : we A – s 5.3 , (4.19) (4.17) (4.18) 58 ) is ill- . Since φφ s 5.5 real A 4.1 RR f . and φφ 4 CC A . f + R ∈ φφ u . In this case the t- and . s A ) 2 φφ + s ( ). Finally, in section -channel contributions do not o → φφ t u for all = A 5.4 ) needs to extend for all

φφ + φφ 4 - and 4.13 φφ s 1 t A − + ), inﬁnite derivative gravity (section ≡ A > φφ u ) s φφ A 5.2 ( A + – 11 – RR φφ t -channels A u , 2 + , we encounter a quadratic divergence in the forward

, sf φφ u φφ s φφ , we discuss a realisation of a model where this requirement 1 A A - and A

− t . Requiring that the 5.5 s > ]. At this point the form factor of the four-point interaction 2 2 6 and ) fixed →∞ 1 π fixed. While this will not yield a diverging cross section, it does lim s s t t ( 64 φφ t , CC A = . We will focus on scattering in the effective field theory framework sf . → ∞ CM 1 = 0 s ! φ φφ Ω m d ), classical Stelle gravity (section dσ = improvement from Asymptotic Safety (section

, encoded in figure are negative and not bounded from below, ( 5.1 χ φφ u u : ], summarised in section m 4 RG A When including Let us first focus on the graviton-mediated part of the amplitude, comprising The resulting differential cross section reads φ 45 f and -channel only, since this allows to perform a partial-wave decomposition of the ampli- and discuss a settrans-Planckian of energies form without factors introducinga any that poles result, lead for satisfy to realmarised all squared scattering in momenta constraints amplitudes table and, regarding which as unitarity are and scale-free causality. at Our results are sum- specific examples corresponding todiscussion distinct to quantum the gravity programs. scatterings of We will distinguishable limit particles,tude. the receiving Furthermore, contributions we from will the less, restrict ourselves to the(section case where the scalar fields are mass- is explicitly met. 5 The physics ofHaving graviton-mediated studied scattering the — general examples properties of the scattering amplitude, we will now consider growth of the amplitude in theof forward-scattering limit has to be tamed by the asymptotics In [ scattering limit violate causality, which requires thatthe centre-of-mass the energy amplitude [ grows slowerbecomes than crucial. quadratically with Since the asymptotics of the graviton-mediated diagrams are fixed, the and introduce new poles for realt momenta puts more stringent bounds on for further discussions. We now consider theu-channel scattering also of contribute to identical the particles, defined amplitude owing and to the the poles partial-wave decompositionare in entirely the ( due forward to and the backward massless scatteringbehaviour nature limit. of of the These the graviton. divergences amplitudes, Since we our will focus not is on investigate the these high-energy divergences and refer to [ 4.2 Scattering including JHEP11(2020)136 , s S RR f and log (5.1) (5.2) ch.) 72 / . s t/u # N µνρσ 4 + / C ! = 1 ] for a pedagogical being the Einstein- falloff (s ch.), behaviour -divergent ( 1 2 UV ∆ c 18 ]. S Λ const scale-free const scale-free exp exp const scale-free UV

64 , log 49 , UV Λ µνρσ ∞ ], also see [ C ] while the Einstein-Hilbert action . Starting with a bare action R (2) 2 64 c + S UV – 65 , Λ + (2) 48 61 [ S R , ! 20 49 / , 2 UV ∆ tr log inﬁnite tower at imaginary squared momentum ghost d.o.f. essential singularity at ghost d.o.f. pole at cutoﬀ scale n/a pole structure Λ – 12 – = 7 17 1 2 ,

2 cutoff scale c − to regularise the trace. For 16 ]. In this approach one takes the viewpoint that a log S 17 UV UV R and = Λ 1 4 c R / " g 1-loop − = 1 Γ tanh const massive spin-two exp log form factors . The column “gravity form factors” gives the functional form of 1 √ c 5 x 4 d minimally coupled, massless scalar fields yields Z . Matter form factors have been given in [ s 2 N ]. π 1 120 / 32 s 17 , N = is the second variation with respect to the fluctuation fields. Furthermore, we 16 [ Characteristic features of the scattering amplitudes obtained from the quantum gravity 2 in the corresponding theory. The columns “pole structure” and “UV behaviour” describe (2) / 20 + 1 S / − improvement CC s Alternatively, the one-loop form factors can be calculated directly from the one-loop non-local f Γ = 7 Asymptotic Safety: form factor model Asymptotic Safety: RG gravity infinite derivative Stelle gravity const massive spin-two effective field theory theory gravity 2 -channel amplitude for large centre-of-mass energy, respectively. For pure gravity one has supplemented by c where introduced the cutoff operator Hilbert action supplemented bygravitational minimally form coupled factors scalar is fields, the universal part of the and effective action obtained by effective fieldquantum theory theory [ is validone may up then to compute a to the cross the section tree-level byintroduction. Feynman including diagrams These perturbative [ corrections (one-loop) comprise corrections non-analytic contributions proportional to s 5.1 Effective field theoryThe and differential IR cross section properties calculated in the previous section readily incorporates results Table 1. models discussed in section and the pole structure of the graviton propagator for complexified momenta and the asymptotics of the JHEP11(2020)136 ] ] ]. to 74 80 75 (5.4) (5.5) (5.3) , CC f 55 . Thus, 2 UV and . Λ ) we obtain , respectively. # 1 RR & 1 4.5 f − C c − C s c 1 = − s s ) and ( . − 0 and 4.4 s 1 < 1 " − R 2 C c s c 1 . ] for recent reinterpretations of = 1 60 C 60 s c 79 − − – − = = 76 = properties of amplitudes and cross sec- ] CC is associated with a negative-energy state. Stelle 2 Stelle 2 UV 57 , ) by setting the form factors CC ) diverge at 56 G 2.2 – 13 – , f 5.4 , a , α R 0 # c ]. Formally, this can be shown by calculating the from non-perturbative Monte Carlo simulations [ 1 − > 55 effects. Generically non-localities appear in the form − R c = R 1 RR c f IR 1 − s 12 RR f . While the additional pole in the scalar sector corresponds − = 3 s 1 ], potentially providing an explanation of dark energy. A recent " 73 2 Stelle 0 – s α 1 71 12 = ]. Some non-local terms may have interesting consequences for the uni- 70 – Stelle 0 is negative, the massive spin-two pole corresponds to an Ostrogradski ghost a 66 control the scale where the modifications owed to the higher-derivative terms set 0 Stelle 2 > α C While this work is mostly concerned with By construction, effective field theory becomes invalid at energies , c R Since signalling that the theorythis degree is of problematic freedom (also asfor virtual see a particle more [ leading detailed to discussion the onamplitudes). violation the of role microcausality of and the [ ghost modes in gravity-mediated scattering This is known toresidues violate associated with unitarity the [ massive poles [ amplitudes at high energies,derivatives rendering acting them on finite. thedegrees Since metric, of quadratic freedom. it gravity In contains isat the four expected finite partial-wave decomposition, energy. that this Indeed, this is theThis reflected expressions theory is by ( gives poles illustrated appearing rise into to figure a additional healthy degree of freedom, the pole in As expected, the presence of the quadratic curvature terms changes the behaviour of the a constant: For the gravity-matter interaction wec implement minimal coupling.in. The Inserting free the parameters form factors into the partial-wave expressions ( 5.2 Stelle gravity As a second applicationThis of our theory general possesses results severalStelle we gravity attractive consider is classical features, obtained Stelle from such gravity the as [ action perturbative ( renormalisability. and phenomenological aspects ofliterature more general [ non-local termsverse have at been large discussed scales in [ the reconstruction of the form factor lends support for this idea also from first principles. these form factors onlyenergy limit capture of the the low-energy form behaviour. factors, different approaches In havetions, order to let be to us used. access brieflyfactors the comment if high- on a theory contains massless modes, e.g., the logarithms above. Both fundamental JHEP11(2020)136 ] ) χχ GR and 40 2.5 , (5.6) (5.7) → 39 = 1 φφ R c . ) , triggering the 2 φ R c )) for m is obtained by the = − s 5.14 = 1 1000 (∆ . ) C 2 φ s c m C c − for − 100 (∆ 2 S a s e c e 1 60 = 10 − associated with the process , and expressed in terms of the non- 0 φφ = S a c ) includes form factors already at the level = IDG 2 s 1 , f C c IDG 1 . The amplitude – 14 – = − , a 10 ) carries over to the quantum effective action, the s ∆ / R ∆ 0.100 1 c R C c c 5.6 e . − s = s e )) and from the RG improvement (eqs. ( 0.010 set the scale where the form factors give relevant contribu- 1 12 . The exponentials are designed to regulate the high-energy CC 5.4 S 1 c = − ) ) appear at the level of the bare action and are thus subject N 0.001 1 IDG 0 5.6 G -4 , f ]. Studies at the level of infinite derivative scalar theories [ 10 a , and 100 R 1 10

C c 38

0.001 0.100 0.010

0 ( | | ), we observe that these are the partial-wave amplitudes found in a − . , c -channel scattering, we will set the corresponding form factors to zero 5 s ≡ R ∆ ∆ / c 0 2 R 4.10 c a e M − Illustration of the partial-wave amplitude = = 2 ] a RR f , respectively. The dashed red line highlights the position of the pole 81 , The form factors ( 40 = 1 ∗ resulting partial-wave amplitudes are Comparing to ( multiplied by an exponential factor. In the s-channel, this results in an amplitude that the corresponding terms have notgravity-mediated been worked out. Asand these work are with not minimally critical coupled for scalar analysing fields. to renormalisation. Assuming that ( locality scale behaviour of loop diagrams,free yielding at a theory high which energyfurthermore is [ renormalisable indicate and that asymptotically tamingducing the form growth factors of for all the scattering matter amplitudes vertices. requires For intro- the gravity-matter sector given in ( is [ The parameters tions. Typically, these are identified, rescaling 5.3 Infinite derivativeBy gravity construction, infinite derivative gravityof ( the bare action. A typical choice for the functions determining the flat-space propagators Figure 3. in classical Stelle gravityg (eqs. ( transition to the regimeis where visualised the by amplitude the isindicating scale-free. dashed that horizontal The the line. asymptotic results value The of agree amplitude the for for amplitude GR is depicted by the orange line, JHEP11(2020)136 . - ], t r for 83 2 (5.8) (5.9) a are negative, u ]. In a similar 82 and t , ]. The key idea stems # by the running gauge 1000 ) 82 4 e e . Since ( u Notably, the partial waves O 3 100 . + and ), resulting from infinite derivative -improve, i.e., where one wants 4 t , crossing symmetry entails that the 0 , the two amplitudes coincide while r 5.7 r 1 RG φφ 10 . → . s improvement” [ log , eq. ( 0 φφ 2 ) a πr 1 e RG s 1 4 2 − 0 π r − ( 6 2 . e 5 , indicating that the theory does not give rise to – 15 – ) = / 0 r 0 ( a 1 + 0.100 V − " s > ) = 1 , as is pictured in figure scale with the relevant physical scale, here the radius 2 − 0 a πr r ( 4 0.010 2 RG e → ∞ s 7→ − 0.001 -5 -7 -9 10 ) r 10 10 10

(

0.100 0.001

]. 0 a V (blue line). The corresponding result from GR is given by the orange line for reference scale. This gives the correct one-loop Uehling potential [ 41 IR = 1 R c Illustration of the partial-wave amplitude is an the infinite derivative gravity amplitude decreases exponentially. The amplitude improvement step then consists in replacing the charge 0 1 is obtained by the rescaling r & -channels contain exponentials whose arguments are given by RG ) flows. First, we will look at the process of “ u When considering the scattering of identical particles s = 1 3 C RG which can also be obtained by more conventional perturbativeand methods [ this results in awith cross quadratic section arguments, that i.e., Gaussian divergesexponential, form exponentially. see factors. also This For [ can a be discussion of amended even by and choosing odd exponentials polynomials in the In this way, at one-loop order, where The coupling while identifying the We will now turn our attention to( form factors from non-perturbative renormalisation group from particle physics, and we willOne quickly starts illustrate with it for a theto physical well-known include quantity Uehling the that potential. leading one order wants quantum to corrections, for example the Coulomb potential, decreases exponentially as are regular on the positiveadditional real massive axis, degrees of freedom. 5.4 Renormalisation group improvements from Asymptotic Safety Figure 4. gravity with comparison. For energiesfor below the non-locality scale, c JHEP11(2020)136 . 1 − ω (5.15) (5.12) (5.13) (5.14) (5.10) (5.11) ), one in the ≡ 2 ∗ 4.9 g B = ]. Neglecting . k . g 0 ∗ ] and black hole g 102 , ≥ 88 1 →∞ − 2 – k k s 101 ), 84 lim . − includes quantum fluc- 4.5 s 1 . ) . ] ∗ k g ( g 1 , appearing in the effective 2 2 89 improvement does not intro- 1 g B G s N − B 1 1 2 ) and ( s G 1 B 60 − RG − − 4.4 1 − = s 1 ] computes the dependence of cou- . = = s all quantum fluctuations are included ω . N 98 , − π tu 65 s . = can be computed from projecting the Wet- , ω RG-imp 2 is regular for all values ) 2 where = 8 ) , η k k CC 2 ( 7→ − 0 k f k s N – 16 – , a g G 2 t u ( ) N = k → . Neglecting the terms proportional to G N ∗ G improvement can then be determined by comparing -improved amplitude ω G η g k ω s π RR (1) 1 f 8 − . The effective coupling − then ensures that the RG O RG 1 + k s 1 0 . The Newton constant 2 , − = ] for discussions on the limitations of this strategy. k ≥ = (2 + ) ) = s 1 k 2 2 k improvement procedure to the classical amplitude ( 97 ( ≥ g k k – ( k 2 G 2 -dependence of 2 s G 95 ) gives two non-vanishing partial-wave amplitudes for spin zero p RG k for k∂ k RG-imp s . The function 1 12 ] onto the (Euclidean) Einstein-Hilbert action [ ) N A and subsequently identifies the coarse-graining scale with a physical ≡ 2 5.13 G ) k = k 2 ( 101 g then interpolates between the fixed point value k – G ( ) are numerical coefficients which, in general, depend on the regularisation ) = 2 99 2 G 2 k k ], also see [ ( ( RG-imp 0 B → G a 94 G improvements in gravity have been used by various groups to study potential – ), is obtained in the limit N 0 and G → 89 2.2 k 1 RG B lim The amplitude ( When applying the Typically, the asymptotic safety program [ The form factors “created” bythe the partial-wave amplitudes to the general expressions ( The regularity of duce unphysical poles situated at negative values of and spin two: As a result one then arrives at the replaces momentum scale. For s-channelcarried scattering by the the graviton natural which cuts choice offEuclidean is the propagator background the in admits momentum the a infrared. transfer standard s Exploiting that Wick the rotation flat suggests the cutoff-identification denominator, the flow equation admits the analytic solution [ The flow of and Here procedure. Their precise valuesto are not note important that in they the are present numbers discussion and of it suffices action ( In the simplest case the terich equation [ the effect of aNewton’s cosmological coupling constant, this leads to a flow equation for the dimensionless phenomenological consequences of Asymptotic Safetyphysics in [ cosmology [ plings on a coarse-grainingtuations scale with momenta spirit, JHEP11(2020)136 . 3 RG RG (5.16) re-derived improvement which ensure ]. While these FRG RG 101 – RG-imp 2 99 improvement already [ a RG improvement at the level and -dependent coupling cannot RG k , . Thus we conclude that the RG-imp 0 µν a F are fixed by the position of the ]. The first non-perturbative calcu- 1 → ∞ − ] within a conformally reduced setting, ∗ 1 (∆) 104 s g 2 Euclidean signature e ) actually manages to tame the growth of 105 = improvement actually gives rise to a specific µν C F 5.11 c scale with a physical momentum scale, and the RG – 17 – 7→ = improvement procedure which will not hold in a RG R µν c F RG µν improvement assigns the same functional dependence to F 2 1 e RG , incorporating a key feature of Asymptotic Safety. At the same s ) indicates that the ]. In fact, we can see this limitation of the 5.4 96 , 95 ]. ] defined a running Newton’s constant from the form factor related to the 103 106 Currently, there are two paths towards studying momentum-dependent correlation We close this subsection stressing that there has been a significant effort in determining At this stage the following cautious remark related to applying the culations. On the background side,the the Polyakov first effective study action oflation form in of factors two a with dimensions the formwhereas [ factor [ has been performedlinear in term, [ that is in terms of total derivatives. Form factors have also been considered based on solutions of thecomputations Wetterich equation have for not reached theinvestigated in level this work where can the bethe studied gauge-invariant in scattering status detail, amplitudes of it these is nevertheless efforts worthwhile to in summarise order to identifyfunctions the followed missing in links. the literature, so-called background calculations and fluctuation cal- a similar strategy seemsconstant not [ feasible in the case ofthe a momentum running dependence Newton’s, of or propagators cosmological and vertices from first-principle computations the same in a morefundamental general point calculation of with view individual thatof form whereas the factors. action in It can electrodynamics, is be also straightforwardly clear carried from out, a be one-to-one. As aconsider consequence, multi-particle scatterings it and is higher-order expected vertices.put that forward This the is in in procedure [ line breaksin with down the arguments when case discussed we here:both the the spin-two and spin-zero sector of the off-shell graviton, which generically are not procedure to more generalwith scattering multiple processes arguments, is indicating that in theythere order. depend is Generically, on no form several, natural independent factors identification scales.map come of Thus the between the several-parameter form factors and the improvement based on the simplethe formula amplitude ( at large time the ansatz does thisdegree in of the freedom simplest in possibleactually the way, signalling introducing spin-zero one a and additional deficitfirst-principle spin-two massive computation. in channel. the We expect that this feature is realisation of Stelle gravityfixed in point. which Hence, the partial-waveFurthermore, amplitudes all again take remarks the madeas form illustrated well. in in the figure Inthat particular, context the one of amplitudes again Stelle approach has gravity constant massive values carry poles as over in to this case The comparison to ( JHEP11(2020)136 0 ≥ (5.17) (5.18) (5.19) C , c R c sets the scale t c . ∆) C c , ) ], as well as gravity-matter 2 u tanh ( + 110 C , . 2 c t ] x 109 ] + t x c 2 t s c ( (∆) = int tanh[ CC f ) x tanh[ 2 t u x c t c + – 18 – , f 2 1 + ] for a comprehensive overview of different matter t )( 64 ∆) s ) = ( R is invariant under crossing symmetry and that the self- c x ]. Follow-up works also investigated partial momentum C ( ] performed a calculation of momentum-dependent prop- φχ 4 int πG 107 f A ] and four-point function [ tanh ( 113 = 4 , R ) shows that those alone are not sufficient to make statements 108 c φχ 4 112 A 4.14 -plane which exhibits a Regge-type scaling behaviour asymptotically. ), whose contribution to the amplitude is given by s ), the form factors of the quadratic curvature terms can be mapped (∆) = 3.4 RR A.2 ) ensures that f ] for a calculation of the propagator of a scalar field coupled to gravity. On ]. Notably, [ 5.19 44 111 ) and ( ] this question was answered in the affirmative. The physics ingredient underlying A.1 Most of the calculations have in common that they only resolve the propagators. The 45 The choice ( interaction does not contributecontrol at the position low of energy. the imaginary Thewhere poles two the in numerical the self-interactions graviton parameters propagator start while contributing to the amplitude. The forward-scattering with the interpolation function being A concrete realisation of this mechanism is provided by the gravitational form factors The construction is accompaniedinteractions, eq. by ( a four-point vertex associated with the scalar self- at Planckian energy.accommodate This raises amplitudes the which questionIn are whether [ bounded the everywhere quantumsuch and effective models scale-free action is an can at infinite toweraxis high of of massless energy. the (Lee-Wick complex type) poles located on the imaginary is currently missing and constitutes an area for5.5 future research. Form factors realisingThe Lorentzian discussion Asymptotic of the Safety gravity, previous the subsections resulting revealed scattering that amplitudes in exhibit many additional approaches massive to degrees quantum of freedom definition to bring themomentum dependence propagator in into the standard vertices.fluctuation calculations, form, This observation and at has the thedependence been general of extensively cost observation used the of was in three- introducing thatlators the and discussed four-graviton the extra in vertex remaining these seems works momentum to onto be their diffeomorphism weak. invariant form A factor map expressions of the corre- dependences of the three- [ systems [ agators on a background with constant curvature. discussion around eq. ( about scattering amplitudes. One can always perform a momentum-dependent field re- fields, and [ the side of fluctuationeqs. calculations, ( most worksone-to-one employ to a the flat graviton propagator. background.graviton was The first As fully calculated momentum-dependent shown in propagator in [ of the for gravity coupled to matter, see e.g. [ JHEP11(2020)136 2 ), a 2.3 (blue (5.20) = 1 R c . ) s C ) with . The form factors . The partial-wave c 5 GR 1000 5.20 so that the partial-wave s tanh( s s C 100 c 1 + 1 10 60 resulting from ( − . This behaviour is triggered by the 0 a and does not require the introduction of = 2 s s 1 . → ∞ 5 / s 0 a – 19 – , a − ) 0.100 s = R 2 c a only and it is illustrated in figure R c 0.010 s tanh( ) are s ) is that it leads to a modified gravitational propagator which and R is made finite by the self-interaction. In the present discussion, c C 5.17 c φφ 5.17 0.001 1 -4 10 1 + A 10

0.001 0.100 0.010

1 0 a 12 ), respectively. They exhibit the following properties: = 3.7 0 while being well-defined on the entire real axis apart from a first order pole a 2 − Illustration of the partial-wave amplitude s is obtained by the rescaling . Hence, the model has the same degrees of freedom as ) and ( = 1 demonstrating the gauge invariance of our result. Gauge independence is recovered 3.2 = 0 The amplitudes are independent of the gauge parameters introduced in eq. ( The key feature of ( C c s 1) to construct the most general amplitude for a two-scalar-to-two-scalar process mediated vertices and scalar self-interactions. Thethe explicit scattering expressions of for distinguishable theeqs. particle amplitudes ( describing species and a single particle species are given in In this paper, we have usedΓ the form factor parameterisation of theby quantum effective gravitons action in adependence Minkowski background. for Our the result graviton covers and the scalar most propagators general as momentum well as the scalar-scalar-graviton amplitudes approach a constantpoles located value at for imaginarymassive squared gravitational momentum modes. 6 Summary and discussion Their shape depends on in the denominators tame the growth of the amplitude for large this contribution is not needed and thus was notfalls included of in as the analysis. at amplitudes arising from ( Figure 5. line). The corresponding result fromvalue GR of is given the by partial-wave thefor amplitude orange line is for displayed comparison. by The the asymptotic horizontal dashed line. The amplitude limit of the amplitude JHEP11(2020)136 ], including the ]. The quadratic 17 , 55 -cutoff scale. 16 UV improvement motivated ), including all interaction RG improvement scheme is not ). Inserting specific choices 2.1 4.5 RG ]. These form factors capture the 64 ) and ( , 4.4 49 , 48 – 20 – and performing the simplest GR -scattering are given in eqs. ( behaviour, valid at energy scales well below the χχ amplitude comes with a negative residue, illustrating a violation of uni- IR → = 2 φφ j tarity. In this context,amplitudes it computed in is a ratherby remarkable Asymptotic observation Safety that recoversparameters starting the in from form terms the of factors theSafety. of renormalisation Stelle group At fixed the gravity point and same underlying time, fixes Asymptotic we the have free shown that the universal curvature terms introduce additional massive polestrum. (ghosts) At in energies the partial-wave abovethe spec- these poles, the amplitudes become scale-free. The pole in from the effective fielduniversal theory framework logarithmic of correction general terms relativity [ [ the level of thefrom amplitude the this vertices. factor is cancelled by identical contributions coming Wick rotation is required. and vector modes from theical propagator. scalar Secondly, and any tensor gauge modes dependence are in projected the away phys- by evaluating the amplitude on-shell. nitions of internalmomentum-dependent lines. inverse wave Any function renormalisation redefinition in of the the propagator. graviton At fluctuations leads to a in a two-step process: first, the three-point functions project out the unphysical scalar We gave a form factor description for classical Stelle gravity [ The form factor formulation naturally includes the one-loop effective action arising The amplitudes are derived in Lorentzian signature. Hence, no prescription for a The scattering amplitude is invariant under (momentum-dependent) field redefi- The general expression for the partial-wave amplitudes associated with gravity- Notably, our construction is based on the parameterisation of quantum effective action. 2) 1) 3) 2) for the form factorsquantum gravity then models. allowed The us resulting to insights contrast can be the summarised amplitudes as resulting follows: from different fields. This requires themonomials extension which of contribute the topaper parameterisation these and ( processes. will be This addressed is in beyond future work. themediated scope of the present This entails that the formalismthe readily scattering processes describes studied one-loop in andform this multi-loop work factors by contributions obtained using to at theis corresponding any expressions given systematic for order the in ingeneralisation the perturbation to theory. sense scattering Moreover, that processes our it related approach to allows amplitudes for with a more straightforward and other (but external technically tedious) In particular, properties 1)ables. and 2) This ensure that allowscausality we directly us are at to the dealing level implement withthe of valid fundamental quantum these effective physical requirements observables action. observ- which related then to translate into unitarity constraints and on JHEP11(2020)136 - ) t ]. 2.2 ]. 119 (A.1) 96 , improve- 95 , RG ) completion of ) 2 2 p p ( ( 4 UV ). p pgp that the Lorentzian -channel amplitudes G 2.1 s G σ ) does not include mas- behaviour is generated p σ ρ p completions such as the 4 p without introducing new ) ρ 2 ν ν p ( UV p δ p µ µ UV ( UV p p divergences. ) + ) + ], exhibiting scattering amplitudes 2 UV 2 p ], which provide a p ( 45 ( proof of principle Tr 118 gpp G – G ρσ η ]. This construction ρσ 116 ). This yields η µν ν η p 2.3 – 21 – 115 4 1 µ , p ] introduces form factors in such a way that the 2 + p ) + 114 81 2 σ , p p ( ρ are dressed by exponential prefactors without introducing 40 1 p G µν ) GR ν σ η δ . This is in contrast to stringy 1 2 µ ( ρ δ + ) = p ( ρσ µν G i 1 which are boundedmassive everywhere degrees and of scale-free freedom or inby ghosts. the the collective The interplay tamingthe of of propagator an the and infinite suitable tower formconstruction of factors obeys massless in unitarity (Lee-Wick the and matter type) causality sector. poles conditions. This in ensures that the additional massive poles atfall finite momentum. of As exponentially a forchannel result, centre-of-mass amplitude the energy obtained by above crossingunclear the whether symmetry non-locality this grows scale. class exponentially, so of The that models it is is free from ment is limited togravity as a well single as the scale formbe identification, factors described related the in to two a vertices with one-to-one different multiple way. off-shell arguments This cannot modes also of resolves the puzzlepropagators raised obtained in in [ able to capture the full information encoded in the form factors: since Finally, we reviewed the model introduced in [ Infinite derivative gravity [ In conclusion, we have introduced a new perspective on gravity-mediated particle 4) 3) supplemented by the gauge fixing action ( In this appendix, wescalar-graviton vertex, collect computed explicit from expressions the for quantum effective the action gravitonA.1 ( propagator and Graviton scalar- propagator We first list the expression for the graviton propagator, which we compute from ( Veneziano and Virasoro-Shapiro amplitudes [ general relativity by introducing an infinite tower of higher-spin degreesA of freedom [ Explicit expressions for propagators and vertices of quantum gravity startingity from model-building. a microscopic In descriptionquantum particular, as effective they well action as serve is for asrequirements able quantum a of grav- to Asymptotic accommodate Safetysive scattering [ higher-spin amplitudes resonances which obey the scattering within the frameworkguidelines of for relativistic the quantum construction fieldand theory. of causality well-behaved Our constraints. scattering results amplitudes provide We satisfying expect unitarity that these are useful for top-down constructions JHEP11(2020)136 . 1 2 (A.3) (A.2) ]. We ν φ ν φ ) ) p p are the , 2 2 1 2 44 ) φ µ φ µ φ µ φ 2 2 , p p p p 1 p 2 h 2 h 1 2 ν h φ ( + + µ φ µ φ p p p p 1 1 p p µ h φ µ φ + + RR ν h ν h p p p p p ( ( G µν µν − · ν h 2 + + η η h p 1 2 µν 2 2 p 3) ν φ ν φ η 2 ]. p p 2 1 2 h µν )+ β − µ h µ h 2 φ 2 φ 2 p η p p ν φ p p β 1 2 120 p ( − − − ) + 1 2 π 2 1 2 1 2 φ 2 φ 1 2 φ 2 φ 3 ν φ 2 φ ν φ p p p p p p 256 2 ,p + + ( − − µ φ 2 µ h 2 h 2 h 1 2 − 2 φ p p p p 2 φ 2 φ ) p p + ,p 2 , 1 2 h 1 2 1 4 1 4 p + + ) ν φ p ( is differentiable so that the vertex is 2 2 h 2 h ( p + + + p p p 1 , ( CC µ φ φφ ) ) p f Rφφ 2 2 G 1 2 1 2 RR 2 φ f p π ) ( p ) ) G 3 2 ( 2 1 2 φ )+ 3 64 2 φ 2 φ RR 2 p φφ ( 2 φ β − – 22 – f ,p ,p + G 2 1 2 ,p 2 φ φφ β . π 2 φ 2 φ 1 f p )+ 3 5) ) , 2 φ π 1 2 64 ,p ,p . This choice is related to an incomplete gauge fixing − − 2 φ p 3 − 2 h 2 h ,p )) 2 ) 1 p ( p p − 2 1 2 h 2 φ ( 128 β ( ( 3) 2 φ p p p = 3 ) ( ( p CC 2 φφ ( − 1)( β p , f ) + ( ) β 2 XX φφ πG Ricφφ Ricφφ − Rφφ 2 ( f p f f f f p 1 µν CC ( ). First, we state the gravity-matter three-point vertex: 2 β 64 ( 1 2 1 2 1 2 η ( p G 1 4 α − CC + − − − 2.5 CC π corresponds to the momentum of the graviton and π G α 2 3 p (1 + ) = π π πG h 2 128 64 2 3 2 p p φ 64 64 p − − ,p 1 φ ) = 32 ) = ) = ) = ) = ) = 2 2 2 2 2 2 ,p p p p p p p h ( ( ( ( ( ( p 4 1 ( p Tr gpp pgp G XX G G µν G G ) G hφφ ( φ Γ In this expression, momenta of the scalars.variation of The form finite factors differencewill and in assume takes the that the the second placefinite scalar line of everywhere. the kinetic is naively form characteristic expected for factor derivative the [ Here we list thevertex explicit originating formulas from of ( the gravity-matter vertex and the matter four-point We can see that thesingularity identity for and the trace gauge component choice — are in gauge that invariant, and case, that the there gauge is fixing a A.2 operator is actually a Gravity-matter projector vertices [ with the scalar propagator functions JHEP11(2020)136 and with with QFT (B.8) (B.3) (B.4) (B.5) (B.6) (B.7) (B.1) (B.2) (A.4) 2 θ p χ ↔ and 1 , p . ) φ . 4 , ) p 4 q · p 3 − p · , − 3 2 , p , 4 q 2 , p p 4 p · + reads p 2 · 2 χ + m 2 , p 2 φ m 3 , p p m q 3 · p q − 2 . · 2 . , p 1) . 4 = = . , p 2 p − θ . 2 µ µ 4 , · p p 2 4 p 1 , m 1 · 2 . + cos − 1 , p p = , 2 2 χ 3 1 q 2 , p p + m 2 = 0 3 − p · 2 i p , p 1 − p · furthermore defines the scattering angle − m is symmetric under interchanging (1 q 2 φ – 23 – 1 , p 2 0 i q 2 2 q p m X = , p χ p 2 2 · diag = q φ = p , p 1 · = f · 2 2 = p p , p 1 µ ( p q 2 p p q µν ( χ , 2 4 η p 2 φ φ , q f f 2 p + 2 χ + ) = ) = 4 4 2 φ m , p , p m q is assumed to be symmetric in all its arguments. 3 3 4 q − in an s-channel scattering, we choose a frame in which , p , p φ 2 2 χ f = = , p , p m 1 1 µ µ p p 3 1 ( ( via p p ) ) 2 4 and p φ χ φ ( φ 2 φ Γ , whereas ( φ m 4 Γ p Finally, the four-scalar vertices read ↔ 3 This must bementum positive transfer. and The gives three-momentum restrictionsrespect to on the kinematically allowed minimal mo- By four-momentum conservation, werelated find by that the magnitudes of the three-momenta are Finally, we define themasses centre-of-mass frame. For two different scalar fields scalar field reads At any vertex, all momenta are defined as ingoing, When no indices arebold-face used, vectors vectors correspond in to standard three-vectors. font In correspond this to way, the four-vectors, four-momentum whereas of a free literature, we employ away, mostly-minus the convention Minkowski metric for reads the signature of the metric. In this With this, the on-shell condition for a free scalar field of mass p B Conventions In this appendix we summarise our conventions. To make contact with standard Let us recall that we assumed that JHEP11(2020)136 Phys. (B.9) , (B.13) (B.10) (B.11) (B.12) , . ! ! θ θ , cos cos θ , θ . 2 χ 2 χ 2 χ m m m cos cos , 4 4 + 2 2 . 2 χ 2 φ q q − − 2 2 m s s m p p − q q 2 s 2 2 φ 2 φ , 1 4 ≡ ) m m − + 2 2 2 4 = 4 4 2 2 p p 2 (2010) 10353 q q − − q + + 5 s s − − 2 3 2 φ 2 2 p , m p p r r – 24 – 2 φ + 1 2 1 2 − − 2 2 m p + − = 4( = = − 2 φ 2 φ 2 2 2 + s ) ) ) 2 1 m m 4 2 3 ]. 1 4 Scholarpedia p p p p , − − ), which permits any use, distribution and reproduction in = = + + + 2 χ 2 χ 2 1 1 1 u m m SPIRE p p p p + IN − − t [ = ( = ( = ( s s 2 2 t s + u

s CC-BY 4.0 − − Froissart bound Asymptotic behavior and subtractions in the Mandelstam representation This article is distributed under the terms of the Creative Commons = = t u (1961) 1053 123 . We can also check the well-known identity θ Rev. M. Froissart, M. Froissart, [1] [2] References Research on Matter (FOM) grant 13VP12. Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. knowledges support by Perimeter InstituteInstitute for Theoretical is Physics. supported Researchof at in Innovation, Perimeter part Science and byMinistry Industry the of Canada Colleges Government and and Universities. of bylands F.S. the Canada acknowledges Organisation financial Province for through support of Scientific from the Research the Ontario Nether- (NWO) through Department within the the Foundation for Fundamental Acknowledgments We would like toRachwał, thank Martin Wim Reuter, Beenakker, and Anupam Melissa Mazumdar, van Beekveld Alessia for Platania, interesting Lesław discussions. B.K. ac- In this way weangle can write the scattering amplitudes as a function of s and the scattering indicating the Mandelstam variables satisfy a linear constraint equation. 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