On amplitudes, resonances and the ultraviolet completion of gravity
Rodrigo Alonso Alfredo Urbano




IPMU 19-0091







rodrigo.alonso@ipmu.jp
Kavli Institute for the Physics and Mathematics of the Universe (WPI) 
 University of Tokyo, Kashiwa, Chiba, 277-8583 Japan.

alfredo.urbano@sissa.it
INFN, sezione di Trieste, SISSA, via Bonomea 265, 34136 Trieste, Italy; 
 
IFPU, Institute  for  Fundamental Physics  of  the  Universe,  Via  Beirut  2,  34014 Trieste, Italy.





This letter constructs, making use of the on-shell 
spinor-helicity formalism, a possible  ultraviolet completion of gravity following a “bottom-up” approach. 
 The assumptions of locality, unitarity and causality  i) require an infinite tower of 
resonances with increasing spin and quantized mass, ii) introduce  a  duality relation among crossed scattering channels, and iii) 
dress all gravitational amplitudes in the Standard Model with a form factor that closely resembles either the Veneziano or the Virasoro-Shapiro amplitude in string theory.
As a consequence
of unitarity, the theory predicts leading order  deviations from General Relativity 
in the coupling of gravity to fermions 
  that 
  could be explained
   if space-time has torsion in addition to curvature.













On amplitudes, resonances and the ultraviolet completion of gravity
    Alfredo Urbano
    December 30, 2023
===================================================================







§ INTROUDUCTION


Prior to the Large Hadron Collider turn on, our tested theory of Nature was a gauge effective field theory (EFT) with a scale v at which unitarity was lost perturbatively. 
Unitarity, however, is sacred and its guardian – as present data seem to indicate – is the Higgs boson.
Despite all the apparent differences, the categorization above applies word for word to gravity when thought of as an EFT with scale M_ Pl≡ G_N^-1/2 and diffeomorphisms as gauge transformations, even though as for who its `Higgs(es)' is (are), there is no experimental evidence at present.


Exposing the need for a ultraviolet (UV) completion of gravity by means of its similarities with the Standard Model (SM)
 is preeminently a particle physicist “bottom-up” approach, and it is not to say that this is the sole issue in gravity that needs addressing 
  (
  

 open problems range from a non-perturbative formulation to the understanding of singularities relevant in cosmology and astrophysics <cit.>). 
   
 It is, nonetheless, the point of view that rules the course of this letter.


One can further elaborate that the above formulation, even within particle physics and EFT, is obscured by (of course fundamental) differences in the gauge symmetry, treatment of the massless mediator, the universality of gravity etc. 
In order to sidestep these differences, the best suited subject of study are on-shell amplitudes; they circumvent gauge redundancies, field redefinitions and gauge fixing present at the Lagrangian level while making ostensible the high-energy behavior. 
Furthermore, as for practical implications, the derivation of these amplitudes with conventional Feynman rules is greatly involved (the three- and four-graviton vertex derived from the Einstein-Hilbert action 
have, respectively, about 100 and 2500 terms when fully expanded <cit.>)  to finally collapse in a 


single-term remarkably simple amplitude <cit.>.

This is part of the evidence that supports postulating a theory with amplitudes as the starting and building blocks.
This approach is the on-shell amplitude program, see <cit.> for reviews. 
An important part of it is the on-shell spinor-helicity formalism, which seeks to exploit helicity (little group) transformation properties to determine the shape of amplitudes

while providing a common framework to formulate scattering amplitudes for all spins and,
 more recently, all masses <cit.>. Combined with  a recursive method to build higher point amplitudes from lower point ones (BCFW, CSW <cit.>), this program aims at a self-contained formulation of quantum field theories. It is not without its own challenges, as determination of off-shell contributions prevents  these methods to extend to arbitrary theories.


If one is to sidestep the Lagrangian formulation, however, care should be taken to ensure that the properties that are naturally implemented in it are satisfied by the formulated amplitudes. Since they are of particular importance in this work, let us briefly review them.















 Locality. Interactions are either point-like or mediated by the exchange of particles propagating between two space-time points.  
This elementary principle dictates the non-analytic 
structure of scattering amplitudes: 
the only singularities occur  when one or more intermediate particles go on-shell. 

Unitarity and Causality. The scattering matrix is a unitary operator as a consequence of probability conservation, and 
causality implies that local observables must commute outside the light-cone in position space <cit.>. Positivity, derived from unitarity, will play a central role in the analysis of this letter.
These principles impose, 
in addition,
stringent constraints on the high-energy behavior of scattering amplitudes <cit.>;
of relevance in this work will 
 be the extension of the Froissart bound <cit.> to a theory with a massless graviton <cit.> in the forward limit and the Cerulus-Martin bound <cit.> in the hard scattering region.
Furthermore, the property of causality results in analyticity of the scattering amplitude thus making possible to represent the latter by means of a dispersion relation via Cauchy's theorem <cit.>. 

In line with the “bottom-up” perspective mentioned above, we shall explore in this work whether it is possible to obtain a UV completion of gravity by adding massive resonances, and, if so, what are the properties required of them. 
To this end, we shall combine the on-shell spinor-helicity formalism with the above-mentioned fundamental properties.
In addition, the discussion will be restricted to weak coupling and tree level, which is to say in typical EFT language that the new resonances should lay below .










§ AMPLITUDES FOR GRAVITY
  
In this section we construct the elementary components of this study, these are three- and four-point amplitudes mediated by gravitational and massive spin J resonance interactions (section <ref> and <ref>, respectively). It will also serve as introduction to the formalism for the unacquainted reader.




 §.§ Graviton mediated amplitudes

On-shell Dirac or Weyl spinors, polarization vectors and polarization tensors are objects that interpolate between the Lorentz group SO(4)∼ SU(2)_L× SU(2)_R and the little group – that is U(1)_LG for massless or SU(2)_LG for massive particles – and, as such, transform under representations of both. The on-shell spinor-helicity formalism in essence seeks to use group theory in both groups to determine the shape of amplitudes. The simplest case is that of massless fermions; denote as _αp (^α̇p) an on-shell momentum p_μ left-handed (right-handed) spinor with SU(2)_L (SU(2)_R) index α (α̇) and anti-symmetric metric ϵ^αβ (ϵ^α̇β̇), ϵ^12=-ϵ^21=1. The spinor p (p) represents a helicity h=-1/2 (h=1/2) particle and hence transforms under U(1)_LG as p→pe^-iϕ/2 (pe^iϕ/2). Amplitudes are Lorentz invariant quantities but they comprise little group representations, and so a valid amplitude for two left-handed fermions is ϵ^αβ_βp_1_αp_2≡p_1p_2. The massive case is obtained upgrading the spinor to the fundamental representation p^I with I an SU(2)_LG  index which again has anti-symmetric metric ϵ_IJ.
Here we will typically omit the index I but use boldface p to separate it from the massless case following <cit.>.  The reader accustomed to Dirac spinors will find these variables demystified by the relation u^I(p)=(p^I,p^I) in Weyl's basis for γ^μ. 

One has then that higher spin is simply built out of the fundamental representations; the familiar polarization vectors read, e.g. ϵ_μ^-(p)=ξσ_μp/√(2)ξp with ξ an auxiliary spinor and [σ_μ]_αα̇ =(1,σ⃗) with σ⃗ the Pauli matrices. 
Objects like polarization vectors or tensors, however, will not appear in the formalism since one rather starts from amplitudes and demands proper little group scaling; 
for instance an amplitude describing a particle with momentum p_1 and helicity -1, 𝒜_p_1^-1, scales as e^-iϕ𝒜_p_1^-1. 
As conventional in amplitude methods we will derive amplitudes with all particles coming in, and we summarize our conventions about 
kinematics in fig. <ref>. Given the scattering of particles 1,2 with momenta p_1,p_2 and helicities h_1,2 into particles 3,4 with momenta p̅_3,4 and helicities h̅_3,4, denoted here 1^h_12^h_2→3̅^h̅_34̅^h̅_4, the all-incoming amplitude, 1^h_12^h_23^h_34^h_4, is obtained changing the sign of the momenta (and with it the helicities) of the outgoing particles

 p_3,4 = -p̅_3,4, h_3,4 = -h̅_3,4. 
Equivalently starting from an all incoming amplitude and taking some of the legs out going, s_ij≡(p_i+p_j)^2 will turn into one of the three Mandelstam variables s,t,u as given in fig. <ref> whereas 
 an incoming left-handed spinor p will turn into -p̅=e^iφp̅ 
 representing an outgoing momentum p̅ helicity +1/2  particle (our convention for the phase φ is in appendix <ref>).
Here we find it useful to write amplitudes in terms of s_ij, i,j since they make clear the connection between different physical processes related by crossing transformations, symmetries under particle exchange (eg. 1↔2) and can be viewed as a function with support on 3 different disconnected regions (r), (b), (g)  as shown in fig. <ref>.
 Finally, for energies close to M_ Pl the approximation of massless matter (i.e. SM particles), which we shall adopt in the following, is excellent.

 The first step in 
 our quest for the “bottom-up” UV completion of gravity with amplitude methods 
 is to build the on-shell three-point amplitudes 
 describing interactions of SM particles with gravity 
  by means of Lorentz-invariant combinations of appropriate powers of spinor variables. 
  As stated in the introduction, we restrict to tree-level amplitudes, and our theory of gravity is General Relativity (GR).
  A graviton of helicity h_g = 2 and momentum q coupling to  
a particle of momentum p_1, helicity h and a particle 2 of helicity ± h reads by little group scaling:

    12^h± h-21q^2+h∓ h2q^2-(h∓ h) ,



















with the same and opposite h sign yielding, respectively,  mass-dimension of 2h+2 or 2, and where the short-hand notation p_1=1 etc. is implied. 
Given that an n-point amplitude has mass-dimension 4-n and gravity's coupling is κ=√(8π)/ with mass-dimension -1, we find that
 only one of the two amplitudes is generated. Explicitly

    -9.5mm [thick]  (-1,1) node [anchor=north east] 1,h–(-0.45,0.15);
    	 [thick] (-1,-1) node [anchor= south east]  2,-h– (-.45,-.15);
    	 [style=decorate, decoration=snake,thick] (-1.1,0.034) node [anchor=east] q, h_g– (-0.4,0.034);
    	 [style=decorate, decoration=snake,thick] (-1.1,-0.034) – (-0.4,-.034);
    	 [black,fill=tangerineyellow,thick] (-0.48,0) circle (9pt);
    	= √(8π)/{[ 1q^2h+22q^-2h+2/12^2 ,                h_g = 2;                                              ;  1q^2-2h2q^2h+2/12^2 ,               h_g = -2 ].

This on-shell amplitude has p_1+p_2+q=0 and is gauge invariant, as can be checked by shifting q→q+ξ and projecting in terms of the  1,2,q spinors.
The absence of a (+h,+h) amplitude means that gravity conserves helicity which at this level coincides with any quantum number that the particle might have. As we shall see in the rest of our analysis, this observation plays an important role. 

The four-point amplitude of order O(κ^2)  is generated by graviton exchange, and hence contains a pole. 
The residue of this pole factorizes into the product of two local three-point on-shell amplitudes, given above. 
One can, therefore, reconstruct the on-shell part of the four-point amplitude 
1^h2^-h 3^-h^'4^h^'
as

    -18mm [thick] (-0.6,1.6) node[anchor= east] 1,h– (-.3,1.3); [thick] (-.3,1.3)–(0,1);  
    	 [thick] (-0.6,-.6+1) node[anchor= east] 2,-h– (-.3,-.3+1); [thick] (-.3,-.3+1)–(0,1);  
    	 [style=thick,decorate, decoration=snake] (0.035,1)  – (0.035,-1);
    	 [style=thick,decorate, decoration=snake] (-.035,1) – (-.035,-1);
    	 [thick] (-0.6,-1.6) node[anchor= east] 4,  h'–(-.3,-1.3); [thick] (-.3,-1.3)–(0,-1); 
    	 [thick] (-0.6,-.4) node[anchor= east] 3,-h'–(-.3,-0.7); [thick] (-.3,-.7)–(0,-1); 
    	[black,fill=tangerineyellow,thick] (0,1) circle (7pt);
    	[black,fill=tangerineyellow,thick] (0,-1) circle (7pt);
    	
              √(8π)/1q^2h+22q^2-2h/12^2
       1/q^2
       √(8π)/3q̅^2h'+24q̅^2-2h'/34^2






where q̅≡ -q given that the helicity h_g = +2 graviton with momentum q enters the second vertex with momentum -q and helicity h_g = -2. 
Two currents of possibly different helicity with h'≥ h are considered for arbitrary SM external states. 
Manipulation of the expression above leads, using the on-shell conditions, to (for brevity let us denote 𝒜_1^h2^-h3^-h^'4^h^'≡𝒜_h,h')

    𝒜_h,h'^ GR=
    s_13^1-h'-rs_14^1-h'+r/(8π)^-1M_ Pl^2s_12 (1423)^2h(3P̂_124)^2h'-2h ,

where r 
 is an integer for integer h' or semi-integer is so is h',
and P_ij≡ (p_i - p_j)/2, P̂_ij=P_ij^μσ_μ. The appearance of this parameter is related to the extension of the amplitude off-shell; whereas the dependence on spinor variables is fully fixed by the little group, one has that on-shell (i.e. when s_12=0) s_13=-s_14, and so amplitude methods alone cannot determine r.[Locality bounds r to range in the interval -1+h<r<1-h in order to avoid double poles. This is because we have the 
scaling 1423∼ s_14, 3P̂_12	4∼√(s_13s_14), and the condition above ensures that no negative powers of s_13,14 are present.] 
From the scaling with energy of the energy-momentum tensor, we find -1+h'≤ r≤1-h'. 
Furthermore when a scalar current is present (h=0) gravity has no handle to distinguish between particles 1 and 2, and the amplitude must be 1↔ 2 symmetric. 
The ambiguity in r, therefore, reduces to two cases only, h=h'=0, 1/2 each characterized by one parameter a,b, as displayed explicitly
 in table <ref>.[Knowledge of the full amplitude in GR can be attained
  through a Feynman rule computation. Here however we keep the contact terms arbitrary. 
  In this sense, note that experimentally we have only tested the pole terms, i.e. the long range interaction] 
In this table we collect all tree-level SM scattering amplitudes mediated by gravity constructed explicitly by means of eq. (<ref>).
We also display gravitational Compton scattering (i.e. scattering among gravitons and SM particles) and graviton-graviton scattering. 
These cases have their helicity structure dictated by the same formula of eq. (<ref>) but now there are poles in all three Mandelstam variables and we find the denominator s_12s_13s_14 (i.e. r=0).

What is more, one has that the formula in eq. (<ref>), comprises all tree-level four-point amplitudes generated in GR. The amplitudes can be split into matter-matter, matter-graviton and graviton-graviton scattering (here and in the following `matter' generically refers to scalars, fermions and vectors). The fact that gravity does not change the helicity of matter implies that the amplitudes in table <ref> are the only non-vanishing matter-matter cases, one can see this diagrammatically and derive the helicity conservation rule ∑ h_i=0. 
This is not clear, however, for scattering with gravity where the three point vertex with structure

    -9.5mm [style=decorate, decoration=snake,thick]  (-0.95,1.05) –(-0.07,0.17);
    	 [style=decorate, decoration=snake,thick]  (-1,1) node [anchor=north east] 1, +2–(-0.12,0.12);
    	 [style=decorate, decoration=snake,thick] (-1,-1) node [anchor= south east]  2,-2– (-0.12,-0.12);
    	 [style=decorate, decoration=snake,thick] (-0.95,-1.05) – (-0.07,-0.17);
    	 [style=decorate, decoration=snake,thick] (-1,0.034) node [anchor=east] q,+2– (0.,0.034);
    	 [style=decorate, decoration=snake,thick] (-1,-0.034) – (0.,-.034);
    	[black,fill=tangerineyellow,thick] (0,0) circle (9pt);
    	 =      √(8π)/1q^6/12^22q^2

produces a diagram in which the helicity -2 is exchanged thus leading to a situation where  ∑ h_i≠0. 
The result of summing all diagrams, however, yields a vanishing amplitude in this case (not only for the pole terms but the full amplitude as can be obtained with Feynman diagrams). The same occurs for graviton scattering which makes table <ref> complete at tree level.










 §.§ Massive spin J mediated amplitudes

Consider now the exchange of a massive spin J resonance. Massive spinning particles in the spinor-helicity formalism are represented by symmetric 2J tensors on the spinor variables, i.e. for a particle with momentum q_μ, q_I_1×…× q_I_2J.
The coupling of this spin J resonance with mass M to (massless) matter with helicities h_1,h_2 is given by the following
 three-point amplitude, completely determined by little group scaling

    -4.5mm [thick]  (-1,1) node [anchor=north east] 1,h_1–(-0.5,0.5); [thick] (-0.5,0.5)–(-0.12,0.12);
    	 [thick] (-1,-1) node [anchor= south east]  2,h_2– (-.5,-.5); [thick] (-0.5,-0.5)–(-0.12,-0.12);
    	 [style=decorate, decoration=snake,thick] (-1,0.034) node [anchor=east] q, J, M– (0.,0.034);
    	 [style=decorate, decoration=snake,thick] (-1,-0.034) – (0.,-.034);
    	 [black,fill=tangerineyellow,thick] (0,0) circle (9pt);
    	 = ig_J   12^J-h_1-h_2 q^2J
       ×1^J+h_1-h_22^J-h_1+h_2/M^3J-h_1 -h_2 -1

where we introduced the coupling constant g_J, omitted Lorentz indexes, and we note that, to avoid inverse powers of 1,2, J≥ |h_1-h_2|  since otherwise we would get a vanishing amplitude. 
We now move to construct on-shell four-point amplitudes, and we divide our analysis in three steps.



  §.§.§ Legendre polynomials


In order to isolate the differences with massive mediators and for exposition purposes we consider first the case with external scalar particles. 
For a conventional diagrammatic derivation of the following results see, e.g. <cit.>.
 A simple manipulation of eq. (<ref>) in this case brings the spinor structure into the form

    12^J  q^2J1^J2^J    =     ( q1122 q)^J
       =    M^J ( q(p̂_1-p̂_2)/2 q)^J
       =     M^J( qP̂_12 q)^J ,

where  we use matrix notation to omit Lorentz indices. 
We next construct the amplitude contribution from the on-shell exchange of the massive J particle, which decomposes into 

    -14.5mm [thick, dashed] (-1,.5) node[anchor= east] 1– (-0,0); 
    	 [thick, dashed] (-1,-.5) node[anchor= east] 2–(0,0); 
    	 [style=thick,decorate, decoration=snake] (0.025,-0.17)  – (0.025,-2);
    	 [style=thick,decorate, decoration=snake] (-.045,-0.17) – (-.045,-2);
    	 [thick, dashed] (-1,-1.5) node[anchor= east] 3– (0,-2); 
    	 [thick,dashed] (-1,-2.5) node[anchor= east] 4– (0,-2); 
    	[black,fill=tangerineyellow,thick] (-.05,0) circle (9pt); 
    	[black,fill=tangerineyellow,thick] (-0.05,-2) circle (9pt);
    	        ig_J M ( qP̂_12 q/M^2)^J 
         i/s_12-M^2
        ig_J M( -qP̂_34 -q/ M^2)^J

The complication lies in performing the sum over spinor I-indexes.
An example of such a configuration can be depicted as (the minus sign in (-q) can be pulled out of the spinors to contribute a (-1)^J factor)



    q_ I_1P̂_12 q_ I_2 ×  q_ I_3P̂_12 q_ I_4 ×  q_ I_5P̂_12 q_ I_6×...
          [thick, dashed,SeaGreen] (0.125,0) – (1.375,0);
     [thick,SeaGreen, rounded corners=4pt] (1.375,0)–(1.5,0) – (1.5,1) – (1.375,1);
     [thick,SeaGreen, dashed] (0.125,1) – (1.375,1);
     [thick,SeaGreen, rounded corners=4pt] (.125,1)–(0,1) – (0,0) – (0.175,0);
    
     [thick,SeaGreen, rounded corners=4pt] (2.75,1)–(2.625,1) – (2.625,0) – (2.75,0);
     [thick,SeaGreen, dashed] (2.75,1) – (4,1);
     [thick,SeaGreen, dashed] (2.75,0) – (4,0);
     [thick,SeaGreen, rounded corners=4pt] (4,1)–(4.125,1) – (5.25,0)– (5.375,0);
     [thick,SeaGreen, rounded corners=4pt] (4,0)–(4.125,0) – (5.25,1)– (5.375,1);
     [thick,SeaGreen, dashed] (5.375,1) – (6.625,1);
     [thick,SeaGreen, dashed] (5.375,0) – (6.625,0);
     [thick,SeaGreen, rounded corners=4pt] (6.625,0)–(6.75,0) – (6.75,1) – (6.625,1);
    
        q^ I_1P̂_34 q^ I_2× q^ I_3P̂_34 q^ I_5× q^ I_4P̂_34 q^ I_6×...

Where the solid green lines signal the summed little group indexes while the dotted follows the matrix multiplication in Lorentz indexes.
Using the completion relations q^I q_I=q̂ and q^I q_I=M we can reduce any of the above terms to a product of traces[Implicit in our notation for  products of P̂'s is the proper contraction of Lorentz indices, P̂K̂=P̂_αα̇ϵ^α̇β̇K̂_ββ̇ϵ^γβ≡P̂ σ̅^μ K_μ with [σ̅_μ]^α̇α =(1,-σ⃗)] over slashed momenta as ∏[(P̂_12P̂_34)^n_i] with ∑ n_i=2J and n_i an ordered array. For instance with J=3 one such configuration is [(P̂_12P̂_34)^2](P̂_12P̂_34) and its coefficient is given by counting the ways to accommodate a single lasso `o' and double one `∞' in 3 slots; o∞, ∞o, etc; examples for J=2,3,4 are given in the appendix, eqs. (<ref>, <ref>, <ref>). In a second step we collapse the traces (P̂_12·P̂_34)^n into polynomials in P_12· P_34, P_12(34)^2, using the relation in eq. (<ref>) in the appendix.
 
All in all we get the following expression for the four-point amplitude

    𝒜_1234^J   =g^2_J(2J)!!/(2J-1)!!M^2/s_12-M^2×
       ∑_mJ  m2J-2mJ(-P_12^2P_34^2)^m(P_12· P_34)^J-2m/2^JM^2J
         =g^2_J(2J)!! /4^J(2J-1)!!M^2/s_12-M^2
     P_J[x(s_13)] ,

where P_J are Legendre polynomials and 

    x(s_13)=-P_12· P_43/M^2/4=P_12· P_34/M^2/4=1+2s_13/M^2 ,


where we have used the on-shell condition in  P_12^2=P_34^2=-M^2/4 and -s_14=M^2+s_13.

We consider next the case with equal helicity h_1 = h_2 ≡ h in the three-point amplitude. 
The spinor structure takes the form 

    12^J-2h q^2J1^J2^J   =   1/12^2h( q1122 q)^J 
       =   12^2h/M^4h( q1122 q)^J
       =   12^2h/M^4h - J( qP̂_12 q)^J .

This is the same structure we already found in the scalar case, eq. (<ref>). 
This means that in the computation of the four-point amplitude the case with equal helicity h_1 = h_2 ≡ h 
reduces to the previous result in eq. (<ref>) with Legendre polynomials, with an overall factor 
12^2h34^2h/M^4h that takes into account the helicity of the external particles. 
As an important difference, however, notice that in this type of coupling the interaction with the massive spin J resonance changes any quantum number that matter might have.



  §.§.§ Jacobi polynomials


 Consider now the case with opposite helicity h_1 = - h_2 ≡ h which corresponds to a helicity and
 quantum number conserving interaction as is the case for the graviton coupling.
The dependence on spinor variables reads

    12^J    q^2J   1^J+2h2^J-2h
       =   12^J-2h1 q^J-2h2 q^J-2h1 q^4h12^2h
       =   M^J ( qP̂_12 q)^J-2h ( q21 q)^2h .

The derivation of the four-point amplitude in this case is not a mere rescaling of the scalar case but one can follow the same steps in the computation,

    -18mm [thick] (-0.6,1.6) node[anchor= east] 1,h– (-.3,1.3); [thick] (-.3,1.3)–(0,1);  
    		 [thick] (-0.6,-.6+1) node[anchor= east] 2,-h– (-.3,-.3+1); [thick] (-.3,-.3+1)–(0,1);  
    		 [style=thick,decorate, decoration=snake] (0.035,1)  – (0.035,-1);
    		 [style=thick,decorate, decoration=snake] (-.035,1) – (-.035,-1);
    		 [thick] (-0.6,-1.6) node[anchor= east] 4,  h'–(-.3,-1.3); [thick] (-.3,-1.3)–(0,-1); 
    		 [thick] (-0.6,-.4) node[anchor= east] 3,-h'–(-.3,-0.7); [thick] (-.3,-.7)–(0,-1); 
    		[black,fill=tangerineyellow,thick] (0,1) circle (7pt);
    		[black,fill=tangerineyellow,thick] (0,-1) circle (7pt);
    		        ig_J  ( qP̂_12 q/M^2)^J-2h( q21 q/M^2)^2h
         i M^2/s_12-M^2
        ig_J (q̅P̂_43q̅/ M^2)^J-2h'(q̅34q̅/M^2)^2h'

	with q̅=-q. As in the computation that lead to eq. (<ref>) one can use the completion relations for the sum in little group indices of q, q to obtain traces over SU(2)_L,R indices.
 The computational difference is that traces are not just over chains of P̂_12 and P̂_34 but factors of 12 and 43 might replace each one of the two factors, e.g. in
tr(P̂_12P̂_3412P̂_34). The number of possible traces grows much steeply now so instead one can take a faster approach building on the scalar result.
 Take eq. (<ref>) with J P̂_12's contracted with J P̂_34's as obtained after working out combinatorics and expanding traces in Lorentz scalar products of momenta.
Then one can substitute P̂_12^J→P̂_12^J-2h(12)^2h to get the result for polarized states (note that longitudinal pieces drop
  out, q· P_12=2q̂1=q· P_34=0). This means that we need to unfold the sum in eq. (<ref>) to insert 2h factors of 12 and 2h' of 43; this is again a combinatorics problem that results in, denoting 𝒜^J_1^h2^-h3^-h'4^h'≡𝒜_h,h'^J

    𝒜_h,h'^J     =
    g^2_JM^2(2J)!!/4^J(2J-1)!!×
       (1432/M^2)^2h(3P̂_124/M^2)^2h'-2h×
       J2h^-1J2h'^-1∑_ℓ,mc_ℓ,m (x-1)^ℓ-2h'x^J-2m-ℓ/s_12-M^2 ,

with x as in eq. (<ref>) and c_ℓ,m as

    c_ℓ,m=
       (-1)^m+2h'/2^J-2h'Jm2J-2mJJ-2mℓℓ2h2hℓ-2h' ,












where the sum runs over values of ℓ,m with 
non-negative-entry binomials.
Although written in an unusual form,  we find that the polynomial in x

    ∑_ℓ,m c_ℓ,m   (x-1)^ℓ-2h' x^J-2m-ℓ = 
       J+2h'JJ 2h P_J-2h'^(2h'-2h,2h'+2h)(x) ,

is proportional to the Jacobi polynomials P_n^(a,b)(x)

    P^(a,b)_n=∑_kn+an-kn+bk(x-1/2)^k(x+1/2)^n-k.

All in all, eq. (<ref>) reads

    𝒜_h,h'^J=   g^2_J (2J)!!/4^J(2J-1)!!
     (1432/M^2)^2h(3P̂_124/M^2)^2h'-2h×
       M^2  /s_12-M^2J+2h'JJ2h'^-1
    P_J-2h'^(2h'-2h,2h'+2h)(x) ,

and it does contain the scalar and the same helicity  cases as the Jacobi polynomials 
reduce to Legendre polynomials in both limits. Let us also note that the measure for Jacobi polynomials is ∝(1-x)^a(1+x)^b, in our case a=2h'-2h, b=2h'+2h, and it is related to the helicity scaling of the amplitude as we shall see next.





The appearance of Legendre and Jacobi polynomials is indicative of an angular analysis that is, in turn, related to unitarity as we shall make explicit next.



  §.§.§ Wigner d-functions and unitarity


In order to touch on unitarity and angular analysis, we turn to the red region, (r), in fig. <ref>, 
that is the kinematic domain where the spin J resonances are kinematically accessible. 
We indicate the corresponding amplitude for the generic process 
1^h2^-h→3̅^h'4̅^h' in red, 𝒜_h,h', and
 we use the explicit substitutions in fig. <ref> which, in terms of the center-of-mass (CM) scattering angle θ, read, cf. appendix <ref>,




    s_13=-s s^2_θ/2,    s_14=-s c^2_θ/2,  
    x=1-s/M^2(1-c_θ)
    31=s_θ/2√(s) ,    14=32=c_θ/2√(s) ,
 
where c_θ≡cosθ, s_θ≡sinθ. We see that, on the mass-shell of the resonance, we have the identification  x(s=M^2)=c_θ. 

To select the resonant contribution while being general we note here that, given our tree-level approximation, the imaginary part of the amplitude comes solely from poles, being explicit

    1/s - M^2 + iϵ = 𝒫𝒱[1/s - M^2] - iπδ(s - M^2) ,

with 𝒫𝒱 the Cauchy's principal value, and the delta function explicitly showing that the imaginary part only has support on-shell.
Therefore we can write for the imaginary part

    Im[ 𝒜^J_h,h']
    
         = 
    
     C_J δ(s-M^2) c_θ/2^μ_i+μ_fs_θ/2^μ_f-μ_i
    P_J-μ_f^(μ_f-
    μ_i,μ_f+μ_i)(c_θ) 
        =
      C_J δ(s-M^2) d^J_μ_i,μ_f(θ) ,

where C_J is a constant, μ_i = 2h, μ_f = 2h' and d^J_μ_i,μ_f are the Wigner d-functions, in full generality
given by <cit.>

    d_μ_i,μ_f^J(θ) = [(J+μ_i)!(J-μ_i)!(J+μ_f)!(J-μ_f)!]^1/2×
          ∑_a(-1)^μ_i - μ_f +a(cosθ/2)^2S+μ_f-μ_i-2a
    	(sinθ/2)^μ_i-μ_f+2a/(J+μ_f-a)!a!(μ_i-μ_f+a)!(J-μ_i-a)! ,
 
 with the sum taken over values such that the factorials are non-negative. 
 
 Wigner d-functions offer a simple generalization for the amplitude generated by the exchange of a massive particle with  spin J 
 in the scattering process 1^h_12^h_2→3̅^h̅_34̅^h̅_4 with μ_i=h_1-h_2, μ_f=h̅_3-h̅_4 where we recall that helicities of out-going particles are minus those of incoming h̅_3,4=-h_3,4. 
 Useful and physically meaningful relations for Wigner d-functions are d_-a,-b^J=d_b,a^J (related to in-out exchange) and d_a,-b^J(θ)=(-1)^J+ad_a,b^J(π-θ)  (related to  3↔ 4 (u,t) exchange).



Given the relations of eq. (<ref>), one can then work backwards to get the expression in terms of spinor variables and Jacoby polynomials P^(μ_f-μ_i,μ_f+μ_i)_ Min(J-|μ_i,f|) for the general case. Indeed, in generality, a Wigner d-function can be expressed as a Jacobi polynomial times and overall factor of powers of c_θ/2,s_θ/2, in the present case given by the helicity scaling factors in eq. (<ref>), through substitutions in (<ref>).

It is indeed in terms of the very same Wigner d-functions that the general partial wave expansion is defined as <cit.> 

    [thick]  (-1,1) node [anchor=north east] 1,h_1–(-0.5,0.5); [thick] (-0.5,0.5)–(-0.12,0.12);
    	 [thick] (-1,-1) node [anchor= south east]  2,h_2– (-.5,-.5); [thick] (-0.5,-0.5)–(-0.12,-0.12);
    	 [thick]  (1,1) node [anchor=north west] 3̅, h̅_3–(0.12,0.12);
    	 [thick] (1,-1) node [anchor= south west] 4̅, h̅_4– (-.12,.12);
    	[black,fill=Dandelion,thick] (0,0) circle (15pt);
    	
    𝒜_1^h_12^h_2→3̅^h̅_34̅^h̅_4 =  16π
    ∑_J⩾ Max(μ_i,μ_f)^∞(2J+1)a_if^J(s)d^J_μ_i,μ_f(θ)

with partial-wave amplitudes a_if^J(s).
The crucial difference compared to eq. (<ref>) is that this decomposition is general, and it applies to  both real and imaginary parts.
In practice, this means that the exchange of a  massive spin J resonance contributes to the imaginary part of the J^ th partial wave only,  but to the real part of partial waves with ≤ J.



Finally, as anticipated, let us connect with unitarity, for completeness and reference. By means of the optical theorem, unitarity of the scattering matrix implies

    2 Im[𝒜_i→ f] = ∑_n∫ dΠ_n 𝒜_f→ n^*𝒜_i→ n ,

where on the right-hand side we take the sum over all possible intermediate states n with phase-space measure dΠ_n. In the case of an elastic process i=f the right-hand side is a sum of moduli squared, and we have the positivity constraint 2 Im[𝒜_i→ i] ⩾ 0. 
As a simple special case it follows that in elastic scattering Im[a^J_ii(s)]⩾ 0, as one can infer by taking the forward limit in which d^J_a,b(0) = 1. 
Given that we have built our amplitude out of three-point amplitudes consequences of unitarity like positivity are implemented, but the unitarity relation does provide a way to rewrite our amplitude in terms of more straightforwardly observables quantities. If the initial state 12 can convert into 
a single-particle state denoted J, the r.h.s. of eq. (<ref>) reads
 
    Im[𝒜_12→ 34] =π∑_J 𝒜_34→ J^* 𝒜_12 → Jδ(s_12-M^2) ,

 where we neglected quantum corrections.
Furthermore when  particles 1,2 and 3,4 are the same species we have a positive imaginary part that is related to the mediator partial width

    Im[𝒜(s_12, θ=0)] = 16π (2J+1)MΓ_J→ 12δ(s_12-M^2) ,

which in particular implies in our amplitude (<ref>) for the exchange a spin J

    16π(2J+1) Γ_J → 1^h2^-h/M =g_J^2 (2J)!!/4^J(2J-1)!!J+2hJJ2h^-1 .

This is to say that the factorial factors and powers of 2 that relate the three- and four-point amplitudes disappear when rewriting in terms of the mediator width. Let us then define α^J=Γ_J/M and write eq. (<ref>) for h^'≠ h as

    𝒜^J_h,h'
    =   16π(2J+1) C_J,h,h^'α^J_h',h M^2  /s_12-M^2(1423/M^2)^2h×
       (3P̂_124/M^2)^2h'-2hP_J-2h'^(2h'-2h,2h'+2h)(x) ,

where we introduced a constant C_J,h,h^' for normalization

    C_J,h,h^'≡√((J-2h')!(J+2h')!/(J-2h)!(J+2h)!) ,

and a slight abuse of notation was committed since only for h'=h and 1^h 2^-h = 3^-h 4^h we have that α^J_h,h is a positive quantity with the interpretation of a decay rate. Aside from the obvious simplifications,  this notation encodes explicitly the strength of the interaction in α and, therefore, its range of validity. 





§ COMPLETING GRAVITY IN THE UV


In the previous section we computed all four-point amplitudes generated by graviton exchange and
the on-shell contribution to the amplitude due to the exchange of a massive spin J particle.
This section aims at putting these two results together for a consistent theory of gravity in the UV.
The goal is to tame the growth with CM energy by introducing resonances while  satisfying the general properties outlined in the introduction. In this road, we start with scalar particles, in particular
consider the 2-to-2 scattering of distinct scalar particles.[Similar arguments in the context of the four-graviton scattering were presented in <cit.>.]



 §.§ Towards a  “bottom-up” UV completion








The scattering amplitude 𝒜_ϕϕφφ for distinct scalar particles ϕ and φ in GR is

    𝒜_ϕϕφφ = 8π/M_ Pl^2(s_13s_14/s_12 - as_12) ,

and describes two different scattering processes that are related by crossing: 
ϕ(p_1)ϕ(p_2) →φ(p̅_3)φ(p̅_4) ((r) in fig. <ref>),
 ϕ(p_1)φ(p_3) →ϕ(p̅_2)φ(p̅_4) (b), (we have (g)=(b) from crossing symmetry).

In the r.h.s. of eq. (<ref>) the first term represents the pole contribution, the second term accounts for a possible contact term.
Furthermore, given that we are scattering different particles, we assume that there is only one channel open  in each one of the three scattering processes (as it happens in GR) so that all poles lie in the positive real s_12 axis (otherwise resonances with ϕ-φ number would be present in the spectrum).
Finally, notice that the amplitude is symmetric under 1↔ 2 exchange (equivalently, t↔ u exchange in (r)). 

In the following, we would like to emphasize the difficulties that one faces when trying to unitarize 
the amplitude 𝒜_ϕϕφφ in all the above scattering processes, and the consequences that 
solving these issues imply. 
A word of warning is pertinent before proceeding;
the solutions that we shall construct and present in this section are not unique, and a number of assumptions are required to obtain them. 
To be pristine, we have enumerated them as  (i)-(iii).







Take the annihilation process ϕ(p_1)ϕ(p_2) →φ(p̅_3)φ(p̅_4) 
in the red region (r) of fig. <ref> where, in terms of the familiar Mandelstam variables, we have s_12 = s, s_13 = t 
and 𝒜_ϕϕφφ^ GR = 8π(tu/s - as)/M_ Pl^2.
The GR contribution is problematic in the hard-scattering region 
where it grows with energy as s/M_ Pl^2, eventually violating perturbative unitarity. 
To formalize this aspect, one can compute the partial-wave amplitudes, cf. eq. (<ref>)

    a_ϕφ^J(s) = 1/32π∫_-1^+1d(cosθ)P_J(cosθ)𝒜_ϕϕφφ(s,cosθ) ,

and impose that they must lie inside the unitarity circle in the Argand plane.
One finds two non-vanishing partial-wave amplitudes

    a^0_ϕφ(s) = s(1-6a)/12 M_ Pl^2 ,
         a^2_ϕφ(s) =
    -s/60 M_ Pl^2 ,

where the last term corresponds to J= 2 exchange, while the first corresponds to J= 0. 
The J= 0 component arises from the coupling of the virtual graviton to the trace of the energy-momentum tensor of the scalar field which is 
indeed non-vanishing for a minimally coupled massless scalar field.[The a^0_ϕφ(s) partial wave vanishes if a= 1/6. This special value of a corresponds to a conformally coupled scalar field described (in four space-time dimensions, with ℛ the Ricci scalar) by the action

    S=∫ d^4x √(-g)[
    g^μν/2(∂_μϕ)(∂_νϕ) + ℛ/2aϕ^2
    ] ,

which has, for a=1/6, vanishing trace T^μ_ μ = 0.]
In eq. (<ref>) we see that the presence of a contact term could cancel the growth in energy 
of the J=0 partial wave amplitude but cannot change the bad high-energy behavior of the highest partial wave a^2_ϕφ(s).
To cure this problem, a first simple guess is the introduction of a spin J=2 resonance with mass M_2 so that the amplitude results into[The presence of the 
minus sign in eq. (<ref>) can be traced back to the minus sign in eq. (<ref>).]


    𝒜_ϕϕφφ   =   𝒜_ϕϕφφ^ GR -
     g_ϕ,2g_φ,2/s-M_2^2[M_2^2P_2(1+2t/M_2^2)._ on-shell
       +   . 
    (s-M_2^2)𝒫_(1,1)(
    s/M_2^2,t/M_2^2)
    ]_ off-shell ,

where the notation 𝒫_(i,j)(x,y) indicates a generic polynomial of degree i in the variable x
(j in the variable y) while g_ϕ,2 and g_φ,2 are the couplings of the two scalar fields with the spin J=2 resonance.
Compared with eq. (<ref>), the on-shell contribution is accompanied by an off-shell term (just as the GR amplitudes in table <ref> and eq. (<ref>)), and the relevant remark is that the off-shell term reduces to a polynomial 
of degree J-1 in s and t for a spin J mediator. 
We can now compute the partial-wave amplitudes. 
The contact term does not contribute to the highest partial-wave amplitude a^2_ϕφ(s) while the 
inclusion of the pole term gives 

    a^2_ϕφ(s) =
    -s/60 M_ Pl^2 - g_ϕ,2g_φ,2s/80π M_2^2 ,

from which we see that it is possible to compensate gravity if one takes 
g_ϕ,2g_φ,2 = -4π M_2^2/3M_ Pl^2, thus 
implying that the sign of the couplings are opposite. 
However, this possibility – although not a priori incorrect – does not extend to more fields. If 
we introduce a third field χ, from the ϕχ scattering we would deduce that χ has opposite-sign charge compared to ϕ, 
but if both χ and φ have opposite charge w.r.t. ϕ they must have the same sign w.r.t. each other, and the φχ scattering is not unitarized.








This discussion makes clear that a massive spin J=2 resonance alone cannot provide a viable UV completion of gravity.
We turn then to introduce a spin J=3 resonance with mass M_3.
Similarly to eq. (<ref>), it will contribute to the scattering amplitude with both an on-shell and an off-shell piece, 
the latter being a polynomial of order 2 in s and t

    𝒜_ϕϕφφ^J=3 = -g_ϕ,3g_φ,3M_3^2/s-M_3^2   
    P_3   (1+2t/M_3^2)  
       +    
    g_ϕ,3g_φ,3𝒫_(2,2)(
    s/M_3^2,t/M_3^2) .

The contribution to the partial-wave amplitude a^2_ϕφ(s) from the pole term has the form 
g_ϕ,3g_φ,3s^2/16π M_3^4. 
It means that one can compensate the graviton contribution
with same sign couplings g_ϕ,3, g_φ,3 and in particular universal g_ϕ=g_φ, in line with the coupling of the graviton  to matter and as opposed to the J=2 case.
However, it is clear that a massive spin J=3 resonance cannot suffice since 
its contribution to a^2_ϕφ(s) has a different scaling in s compared to gravity and, more worrisome, 
because it introduces an unbalanced contribution to the partial-wave amplitude a^3_ϕφ(s) that grows with energy as s^2.
One then faces the same problematic growth in a_ϕφ^3(s), and iterates the procedure with a massive spin J=4 resonance which, in turn, would require a
massive spin J=5 state in a domino effect that yields an infinite tower of increasing massive higher spins. 

This is a sketch of the well-known result that gravity requires an infinite set of resonances, here we would like to underline that the argument was elaborated for the process in region (r) ϕϕ→φφ but region (b) for ϕφ→ϕφ presents separate problems.
Indeed the balance of consecutive spins with opposite sign contributions in (r) does not immediately translate to the region (b)  since
one has s ↔ t and all Legendre polynomials, P_L(1+2s/M^2), and poles, 1/(t-M^2), have the same sign in the physical region s>0, -s<t<0. 
One could try to address this issue by adjusting the contact terms but here instead we attempt a summed expression for the amplitude which we write in the form

    𝒜_ϕϕφφ = 8π/M_ Pl^2(tu/s - as)N(s,t)/∏_k^∞(s-M_k^2) ,

where  the numerator  of the highest common denominator  is, at this stage, a polynomial N(s,t) of arbitrary high degree in s and t.
Notice that, without loss of generality, we defined it by factoring out the GR amplitude. 
Let us write N(s,t) as the product of its zeros in t

    𝒜_ϕϕφφ =  8π/M_ Pl^2(tu/s - as)∏_n^∞[
    t - f_n(s)
    ]
    /∏_k^∞(s-M_k^2) .

We can now use unitarity and locality as a guideline.
Unitarity and locality, as explicitly shown in the previous section, imply that on a pole in s the residue of the amplitude must be a polynomial of finite degree in t corresponding to particle exchange. 
This is not the case if  (i)   the functions f_n(s) in eq. (<ref>) are assumed analytic around s=M_ℓ^2, and one is forced to introduce inverse powers of t

    𝒜_ϕϕφφ =  8π/M_ Pl^2(tu/s - as)∏_n^∞[
    t - f_n(s)
    ]
    /∏_k^∞(s-M_k^2)∏_m^∞(t-M̂^2_m) ,

such that the poles in t cancel against the zeros when evaluating at s=M_ℓ^2, explicitly

    {M̂_n^2}⊂   {f_n(M_ℓ^2) } ,     ∀ ℓ

where both sets (poles and zeros) are infinite.
Remarkably, 
even if the starting point of our construction only required s-channel resonances, unitarity and our assumption (i) led us to resonances in the 
dual t-channel. 
The same analysis therefore applies to residues when evaluating in t =M̂^2_ℓ, which should be polynomials in s; in particular  to avoid double poles we have now that

    { M_n^2}⊂    {f_n^-1(M̂_ℓ^2) } ,     ∀ ℓ

with f^-1_n the inverse function and which, as eq. (<ref>), means that the set of zeroes contains the set of poles. The complication in satisfying eq. (<ref>) is that it must hold for all ℓ=0,...,∞. 
Here eq. (<ref>) will be satisfied by simply assuming that
  (ii) the set of elements given by f_n evaluated in the ℓ^th mass M_ℓ^2, i.e. {f_n(M_ℓ^2)}, contains the set {M̂_n^2} and ℓ more elements  that is

    f_n(M_ℓ^2)≡M̂_n-ℓ^2 ,

which, is worth pointing out, means  that also eq. (<ref>) is automatically satisfied as

    f_n^-1(M̂_ℓ^2)≡  M_n-ℓ^2 ,

 while the spectrum in M_n^2,M̂_n^2 is still arbitrary.[The function can be constructed explicitly given G,Ĝ such that G(M_n^2)=n,Ĝ(M̂_n^2)=n, as f_n(x)=Ĝ^-1(n-G(x)).]
Complying with eqs. (<ref>, <ref>) nevertheless only ensures the absence of double poles;  one should also demand finite degree polynomials to respect unitarity and locality. 
Given  (i), f_n have a Taylor expansion 

f_n(s)=f_n(s_ℓ) + f_n^' (s_ℓ) (s-s_ℓ)+O(f_n^'').
If there are second derivatives each function f_n will contribute with one power of s, and one has n=0,..,∞. 
One, therefore, is led to impose then that only a finite number of f_n(s) have second derivatives (of course, the same argument applies to higher derivatives). 
The simplest way to address this is to  (iii) assume that all f_n(s) are linear functions, f_n(s)=f_n^1s+f_n^0. This reduces the problem, and in particular eqs. (<ref>, <ref>), 
to a linear system whose solution is strongly over-constrained since we have

    f_n^1=    - M̂^2_n-ℓ-M̂^2_ℓ/ M^2_n-ℓ- M^2_ℓ ,     f_n^0=   M̂^2_n-ℓM^2_n-ℓ-M̂^2_ℓ M^2_ℓ/M^2_n-ℓ-M^2_ℓ ,

where the r.h.s. should be the same for  all ℓ. This clearly is not true for a general spectrum. 
It is true however for linear spectrum in both s,t
 
    M_n^2    = nM^2 ,    M̂_n^2    = nM̂^2+M̂_0^2 ,    
    n    =1,2,…∈ℕ
     ,

where the spectrum in s has no M_0^2 due to the pole in s=0, as we shall see shortly. 
In this case f_n^1 in eq. (<ref>) reduces to the ratio f_n^1 = -M̂^2/M^2 (independent from n) while 
f_n^0 = nM̂^2 + M̂_0^2, and 
the amplitude now  reads 

    𝒜_ϕϕφφ =    8π/M_ Pl^2(tu/s -as )𝒞×
       ∏_n^∞[
    M^2t +M̂^2s - M^2(nM̂^2+M̂_0^2)
    ]
    /∏_k^∞(s-kM^2)
    ∏_m^∞(t-mM̂^2-M̂_0^2)
     ,

where 𝒞 is a normalization constant.
The above solution presents an interplay between resonances in the two channels with the t-channel spectrum that 
determines the s-channel couplings and vice versa. 
Let us make this explicit and evaluate the second line at t=ℓM̂^2+M̂_0^2 to obtain the couplings of the ℓ^th t-channel resonance

    ∏_n^∞M̂^2[s- M^2(n-ℓ)]/∏^∞_k(s-kM^2)∏^∞_m (m-ℓ)M̂^2∝∏_r=0^ℓ (s+rM^2) ,

which depends on the s-channel spectrum by means of a finite-degree polynomial in s, as dictated by unitarity and locality.
 Note in particular that there is always a power of s to cancel against the pole in s=0 from graviton exchange in the first line of eq. (<ref>), and 
 this is the reason for the absence of the M^2_0 term that we anticipated before. 
 If we evaluate eq. (<ref>) at s=ℓ M^2 the product above would be on a finite number of terms of the form 
 (t+rM̂^2-M̂^2_0), with no zero at t=0 since, in this case, there is no GR pole to cancel.
 As for the normalization factor 𝒞 we can address this if we use the Euler definition of the Γ function 

    Γ(z) = 1/z∏_n=1^∞(1 + 1/n)^z/1 + z/n ,

to write 

    Γ(1-s̃)Γ(1-t̂)/Γ(1-t̂-s̃)   
    = 
    𝒞   ∏_n^∞[
    	M^2t +M̂^2s - M^2(nM̂^2+M̂_0^2)
    	]
    /∏_k^∞(s-kM^2)
    	∏_m^∞(t-mM̂^2-M̂_0^2)
     ,

with 

    s̃ =s/M^2 ,     t̂=t-M̂_0^2/M̂^2  .

This is the Veneziano amplitude with a linear homogeneous transformation in s and a linear inhomogeneous transformation in t. 
It is good to pause at this point, and define

    𝒜_ VZ^η,γ_0(s,t)    ≡   Γ(1-s̃)Γ(1+ηγ_0-ηt̃)/Γ(1+ηγ_0-ηt̃-s̃) ,
    η   ≡   M^2/M̂^2 ,
    γ_0   ≡   M̂^2_0/M^2 ,

where t̃=t/M^2, and we see the expression is not symmetric in s,t but is compatible with locality. 
Let us now validate this amplitude by checking the high-energy behavior in the hard-scattering limit.
Stirling's approximation for large s yields

    𝒜_ VZ^η,γ_0   ∼ e^Rs̃,     R   =log[(1-η s_θ/2^2)^1-η s_θ/2^2(η s_θ/2^2)^η s_θ/2^2] ,

so provided η≤1 there is an exponential decrease for large s which makes up for the growth with s of GR.
One has, therefore, that the factor in eq. (<ref>) does yield a valid UV behavior;  what is more, the exponential fall-off is a faster decrease with s than one can obtain with any finite number of resonances in QFT. 

The non-symmetric behavior in s↔ t (or t↔ u) of eq. (<ref>) makes it suitable for the UV completion of
 distinguishable particle scattering. This is the case, for instance, of the fermion-vector scattering that we write as

    𝒜_1/2,1=
    8π23143P̂_124/M_ Pl^2 s𝒜_ VZ^η,γ_0(s,t) ,

whereas if we have s-t symmetry enforced by the external states as in the same-fermion scattering we simply set η=1,γ_0=0 in order to get a s,t symmetric result

    𝒜_1/2,1/2   =
    8π2314/M_ Pl^2 (s/t+t/s+b)
    𝒜_ VZ^η=1,γ_0=0(s,t) .








For simplicity of notation, let us denote 𝒜_ VZ^η=1,γ_0=0≡𝒜_ VZ in what follows.

The scalar amplitude 𝒜_ϕϕφφ, on the contrary, is t↔ u symmetric (this is a property of the external states and, as such, it must be respected by the full amplitude), in particular this means that there will be u-poles and zeroes as well. The previous amplitude, therefore, must be modified to account for this property.
 Let us start again from the Veneziano factor in eq. (<ref>)

    ∏_n^∞(s̃+t̂-n)(s̃+û-n)/(s̃-n)(t̂-n)(û-n) ,

naïvely augmented by extra factors to guarantee the t↔ u symmetry. To check the validity of this expression, we can use again unitarity and locality as a guideline.
When evaluating at û=ℓ, the zeroes (s̃+ℓ-n) cancel against the poles (s̃-k) but those in  (s̃+t̂-n)=(1-η^-1)t̂-γ_0-ℓ-n  do not contain the poles  (t̂-m)
 so we are forced to introduce extra terms in the numerator. 
 Furthermore, given that the factors s̃+t̂-n=(1-η^-1)t̂-γ_0-ℓ-n do not cancel against terms in the denominator they will yield an infinite degree polynomial in t when taking all terms in the product unless η=1 (M^2=M̂^2) when they reduce to a constant. Therefore we set η=1 in the following. 
 The extra factors in the numerator can be found by noting that poles in s are all simple and hence we can extend this to t,u by symmetrizing

    ∏_n^∞(s̃+t̂-n)(s̃+û-n)(t̂+û-n)/(s̃-n)(t̂-n)(û-n) .

Although the condition M^2=M̂^2 was enforced there is still the M̂^2_0 parameter which is unconstrained so far; let us make it explicit by writing  (this will 
also serve as a definition for the constant 𝒞 in front of the amplitude in eq. (<ref>))

    𝒞∏_n^∞   (s̃+t̂-n)(s̃+û-n)(t̂+û-n)/(s̃-n)(t̂-n)(û-n)
       =Γ(1+2γ_0)Γ(1+γ_0-ũ)Γ(1+γ_0-t̃)Γ(1-s̃)/Γ(1+ũ+γ_0)Γ(1+t̃+γ_0)Γ(1+s̃+2γ_0)
       ≡𝒜_ VS^γ_0(s,t,u) ,


which is a solution symmetric in t↔ u but asymmetric in s↔ t for γ_0≠0.
Consequently, it is well-suited for the UV completion  of the scalar-fermion  scattering amplitude where we have a general 𝒜_ VS^γ_0 factor while for scattering of identical scalars we expect

    𝒜_ϕϕϕϕ=8π/M_ Pl^2(tu/s+su/t+ts/u) 𝒜_ VS^γ_0=0(s,t,u) .

Let us denote 𝒜_ VS^γ_0=0≡𝒜_ VS in what follows. 
If one is to reconstruct the scattering of indistinguishable scalars by symmetrizing the amplitude for distinct scalars as

    𝒜_ϕϕϕϕ=𝒜_ϕϕφφ+𝒜_ϕϕφφ(s↔ t)+𝒜_ϕϕφφ(s↔ u) ,

then the same factor 𝒜_ VS^γ_0=0 should also appear in the case of distinct scalar, and we get to our final result for the ϕφ
amplitude
 
    𝒜_ϕϕφφ   =   8π/M_ Pl^2(tu/s - as) 𝒜_ VS(s,t,u) .

The same factor is needed for gravitational Compton scattering this time given that there are poles in all s,t,u=0 which – as shown after eq. (<ref>) – requires γ_0=0. 
















Graviton-graviton scattering does also have infrared poles in all three channels and it reads then

    𝒜_g^+2g^-2g^-2g^+2 = 
     8π( 2314)^4/M_ Pl^2 stu𝒜_ VS(s,t,u)
    
     .

Notice  that once the helicity structure of the scattering amplitude ( 2314)^4 is factorized, the rest of the four-graviton 
amplitude – stripped of any knowledge about the helicity of the external particles – is completely symmetric in the three Mandelstam variables s,t,u.



The construction carried out in these examples can be repeated for all the SM scattering amplitudes studied in 
section <ref>. 
Before analyzing in detail the properties of our UV completion and its actual validity, therefore, let us summarize our results.



 §.§ Generalization and emergent properties

In the previous section we have derived what are in practice deformations of Veneziano and Virasoro-Shapiro amplitudes (which reduce to these in a certain limit η=1,γ_0=0) and used them to unitarize GR amplitudes by multiplication.
We also provided a number of examples to illustrate which type of factor is adequate for a given scattering process. In the s_ij notation 
our “bottom-up” UV completion of gravity  can then be summarized as

    𝒜_1^h_12^h_23^h_34^h_4 = 𝒜_1^h_12^h_23^h_34^h_4^ GR×{[    𝒜_ VZ^η,γ_0(s_12,s_13);                          ; 𝒜_ VS^γ_0(s_12,s_13,s_14) ].





















As for the assignment or 𝒜_ VZ or 𝒜_ VS we have noted that invariance under particle exchange (i.e crossing symmetry) for certain processes determines the factor, 
and so for identical fermion-fermion and vector-vector  scattering we have 𝒜_ VZ^η=1,γ_0=0 whereas for identical scalar-scalar and graviton-graviton 𝒜_ VS^γ_0=0. 
Gravitational Compton scattering has IR poles in all s,t,u channels, and this led us to choose 𝒜_ VS^γ_0=0. 
Scalar-fermion and scalar-vector, on the contrary, only have an IR pole in s_12 and must be symmetric in 1↔ 2 (s_13↔ s_14), and 𝒜_ VS^γ_0(s_12,s_13,s_14) provides a valid UV completion. Equivalently, fermion-vector has a factor 𝒜_ VZ^η,γ_0(s_12,s_13). 
Finally same-spin matter scattering of distinct particles need not be symmetric under 2↔3; however, if we expect to obtain undistinguishable same-particle scattering by symmetrization of distinguishable, we also need 𝒜_ VZ^η=1,γ_0=0 for distinguishable fermion-fermion and vector-vector and 𝒜_ VS^γ_0=0 for distinguishable scalar-scalar. All this is summarized in table <ref>.
	

The main properties that emerge from our “bottom-up” construction are the following.



  ∘ Infinite resonances with quantized mass.  
In the “bottom-up” approach, these properties follow directly from unitarity and locality. 
We find that the spectrum in eq. (<ref>) provides a consistent solution.
From a “top-down” perspective, the presence of an infinite tower of states with increasing quantized mass is typically the outcome of 
 compactification.


  ∘  Veneziano or Virasoro-Shapiro form factor. The UV completion of gravity is realized by dressing the GR amplitude with either the 
Veneziano or the Virasoro-Shapiro form factor, as discussed in eq. (<ref>). These factors change the UV growth with s of GR into an exponential fall-off.


  ∘  Resonance duality. 
As a consequence of unitarity, massive higher-spin resonances are always present at least in two of the three scattering regions in fig. <ref>, and they are therefore related by the corresponding crossing transformation. This remains true even for the processes in which the GR scattering  proceeds via the exchange of gravitons in one single channel,
 and hence dual channel resonances will carry SM charge. One also has that an expansion in terms of resonances in a given process, e.g. (r), is not a good expansion in the dual crossing related process which has its own expansion in pole terms, (b).



Some of these properties may be familiar for readers expert in string theory. 
For instance, it is well known that in type-II superstring theory the scattering of four massless bosons is described by an amplitude with 
 a Γ-structure that  matches the factor 𝒜_ VS in eq. (<ref>) <cit.>.
 

Nonetheless, let us stress that our results here are a mere consequence of the “bottom-up” approach that only obeys to 
the fundamental properties of unitarity, locality and causality, and is not committed to any particular model.
It is, therefore, important to understand what can be learned from this complementary point of view.
To this end, after outlining in this section the general structure of our result, we shall now move to analyze it in more detail.






















§ 
 ANALYSIS AND RESULTS


We start our analysis by investigating the high-energy behavior of the amplitudes in eq. (<ref>) that are UV-completed by  the Veneziano and 
Virasoro-Shapiro form factors 𝒜_ VZ and 𝒜_ VS. The asymmetric factors 𝒜_ VZ^η,γ_0 and 𝒜_ VS^γ_0 are discussed in section <ref>. These amplitudes have one or at most two parameters whereas the set of constraints from unitarity are infinitely many so it is non-trivial step to satisfy them.


We consider two bounds in different kinematic regimes.
For a generic  elastic amplitude 𝒜(s,t), at large s and fixed t the Froissart bound <cit.> does not apply in gravitational theories with a massless graviton since there is no mass gap. However, in this limit causality still implies polynomial boundedness with 
the amplitude that can grow with s but more slowly than s^2 <cit.>. 
At large s and fixed scattering angle, causality implies that the amplitude can not fall faster than e^-√(s)ln s (the Cerulus-Martin bound <cit.>) which recenlty has been extended to the more general case <cit.>.
We shall start with causality examining the Regge limit s→∞ with t fixed  (and hence forward scattering θ→ 0)  we find the high-energy behavior 

    𝒜_1^h2^-h3^-h^'4^h^'^ GR×{[        𝒜_ VZ  ∼s̃^2+t̃/t̃;                          ;        𝒜_ VS ∼s̃^2+2t̃/t̃ ].        ∀   h,h' .

In the physical region s>0, t<0 one has a power-law milder than s^2 and hence compatible with causality as phrased in <cit.>.
In the hard-scattering limit s→∞ with t/s fixed (and hence θ fixed) we find 

    𝒜_1^h2^-h3^-h^'4^h^'^ GR×{[                                 𝒜_ VZ  ∼ s_θ/2^2s_θ/2^2 s̃c_θ/2^2c_θ/2^2 s̃;                                                                            ;                                 𝒜_ VS ∼  s_θ/2^4s_θ/2^2 s̃c_θ/2^4c_θ/2^2 s̃ ].    ∀   h,h' ,

with an exponential fall-off controlled by s^2_θ/2ln(s^2_θ/2)+c^2_θ/2ln (c^2_θ/2)  < 0 <cit.>.  
This is a fall-off that is much faster than in any QFT, and the amplitude indeed violates the Cerulus-Martin bound. 
This is due to the presence of an infinite number of resonances, and indeed the lower bound applicable in this case has been extended recently <cit.> in agreement with the scaling above. 

Next, we turn to inspecting the properties of the resonances. This is a process that, given the results and conventions of section <ref>, can be made automatic.
Let us sketch the generic procedure.
 There can be resonances in all three channels s_12,s_13,s_14; in order  to identify them, 
 we first turn to the domain of the amplitude where they are kinematically accessible (r), (b), (g) via the respective substitutions in 
 eqs. (<ref>, <ref>, <ref>),
  and make explicit the CM scattering angle dependence in t=-ss^2_θ/2 and the spinor brackets. 
 One then identifies the poles and evaluates the residue as a function of θ only.
  Finally, the different resonances can be extracted via a projection in the orthogonal set of Wigner d-functions. 
  This is to say, when condensed in equations
 





    α^n,J_2h,2h'   =1/32π∫_-1^+1s=M^2_n Res[𝒜_h,h']d^J_2h,2h'(θ)dc_θ ,
    α^n,J_h+h',h+h'   =1/32π∫_-1^+1 s=M^2_n Res[𝒜_h,h']d^J_h+h',h+h'(θ)dc_θ ,
    α^n,J_h-h',h-h'   =1/32π∫_-1^+1 s=M^2_n Res[𝒜_h,h']d^J_h-h',h-h'(θ)dc_θ ,

where the residue Res rescaled as

    .[ s=M^2_n Res[𝒜_h,h']; s=M^2_n Res[𝒜_h,h']; s=M^2_n Res[𝒜_h,h'] ]}≡lim_s→ M_n^2M_n^2 - s/M_n^2{[ 𝒜_1^h2^-h3^-h'4^h';                   ; 𝒜_1^h3^-h'2^-h4^h';                   ; 𝒜_1^h4^h'2^-h3^-h' ].

 is a function of (s,θ), and with α given in eq. (<ref>) and generalized to the h'± h case. Let us note that regions (b), (g) correspond to elastic scattering and as such the resonances in s_13,s_14 variables have couplings subject to positivity; explicitly one has

    α^J_2h,2h , α^J_h+h',h+h' , α^J_h-h',h-h'≥ 0 .

Wigner d-functions d^J_a,b are defined only for J≥Max(a,b) which implies in particular that the lowest spin resonance will have 
J_ min=h'-h if resonances are present in the s_14 channel. Therefore, gravitational Compton scattering, scattering with scalars and graviton scattering have resonances of spin h'-h; scalar resonances are to be found only for scalar scattering and graviton scattering whereas fermion resonances only for scalar-fermion scattering. The rest of the amplitudes have Venenziano-like factors and a minimum spin h'+h=1,3/2,2 for fermion-fermion, fermion-vector and vector-vector.

 Let us be explicit about the angle dependence of the minimum spin Wigner d-function

    d^J_min_h_1-h_2,h_3-h_4(θ) 
     [very thick, blue, ->] (0,0)–(0,-1);
     [very thick, red, ->] (-0.5,0)–(-2.5,-1);
     [very thick, ForestGreen, ->] (0.5,0)–(2.5,-1);
    
    
    s_θ/2^2h'-2hc_θ/2^2h'+2h      c_θ/2^2h'+2h         c_θ/2^2h'-2h

This angular dependence has to be compared with that of the amplitudes in eq. (<ref>) which we separate in two factors, namely the 
GR part containing the overall s_ij and the helicity factor (cf. eq. (<ref>) and appendix <ref>)

    s_13^1-h'-rs_14^1-h'+r/s_12(2314)^2h(3P̂_124)^2h'-2h
     [very thick, blue, ->] (0,0)–(0,-1);
     [very thick, red, ->] (-0.5,0)–(-3,-1);
     [very thick, ForestGreen, ->] (0.5,0)–(3,-1);
    
    
    ss_θ/2^2(1-h+r)c_θ/2^2(1+h-r)    ss_θ/2^-2c_θ/2^2(1+h-r)   ss_θ/2^-2c_θ/2^2(1-h+r)

and the residue of either the Veneziano form factor 𝒜_ VZ or the Virasoro-Shapiro form factor 𝒜_ VS. 
For both 𝒜_ VZ and 𝒜_ VS the physically accessible resonances are located at the poles of the factor Γ(1-s̃), that are 
s̃_n = 1+n, i.e. M_n^2 = (1+n)M^2 with n=0,1,…∈ℕ.




For brevity we denote s=(1+n)M^2 as 1+n in the subscript of Res, and we get for the Veneziano form factor

    1+n Res[𝒜_ VZ] =    -1/(1+n)!∏_k=0^n(k+t̃)
     =   -1/(1+n)!∏_k=0^n[k-(1+n)s_θ/2^2] ,

while in the Virasoro-Shapiro case

    1+n Res[𝒜_ VS]   =(-1)^n/(1+n)!^2∏_k=0^n(k+t̃)(k+ũ)
    
    
       = s_θ/2^2c_θ/2^2/(n!)^2∏_k=1^n[k-(1+n)s_θ/2^2] .

This means that, on top of the angular powers coming from the helicity factor in eq. (<ref>), we expect 
at least another factor of s_θ/2^2c^2_θ/2 or s_θ/2^2 depending on whether we have the Virasoro-Shapiro or Veneziano. Take e.g. the former and restrict to (b) and n=0 where we have 𝒜_ 𝒱𝒮∼ c_θ/2^4+2h; for this amplitude to admit a decomposition in terms of Wigner d-functions d^J_h'+h which themselves scale with at least 2h+2h' powers of c_θ/2 (cf. eq. (<ref>)) one has that h'≤ 2.
In the same fashion for Veneziano, comparing eq. (<ref>) and eq. (<ref>), we obtain  h'≤1.
Consequently,  we have that this procedure does not apply to higher spin as external states. 

Finally and before moving on, it is pertinent to discuss the quantum numbers of the resonances. 
This can be done by simply combining the representation of the two external states  that annihilate into a resonance. 
It seems one would have all possible combinations of two particles but it is not always the case that we have all s,t,u channels open (this is the case for all the GR amplitudes UV-completed by means of the Veneziano form factor). 
Is there a rule to tell when is a channel open? 
 If we assign helicity charge 𝔥 = +1 to all positive helicity matter particles and 𝔥 = -1 to all negative ones, then 
 we find that an inherited consequence of helicity conservation in our scenario is that there are no resonances with helicity charge  𝔥 = ± 2. 
 This selection rule, for instance, is responsible for the absence of massive higher spin resonances in the u-channel for the process 
  q^+1/2q̅^-1/2→ V^+1V^-1.






 §.§ Resonance analysis

After having laid out the tools to extract a resonance spin and couplings the following subsection explores the spectrum within amplitudes in all three classes of matter-matter, matter-graviton and graviton-graviton scattering. Within each category, rather than covering all possibilities we focus on representative cases the emphasis being on positivity.


  §.§.§ Graviton-graviton scattering


After the generalities let us turn to a few cases to illustrate the analysis. Consider pure gravity and the process in region (r) g^+2g^-2→ g^+2g^-2, which we note is the same process as in (b) due to crossing symmetry; one has

    1+n Res[𝒜_2,2]=   8π M^2(1+n)c_θ/2^8/M_ Pl^2 (n!)^2∏_k=1^n[k-(1+n)s_θ/2^2]^2 .

We note that resonances start at J_ min=4. Our definition of α reads now

    α_4,4^n,J≡   M^2/M_ Pl^2 N_4,4^n,J≡α_ Pl N_4,4^n,J ,

where 

    N_4,4^n,J=   (1+n)/4n!∫_-1^+1c_θ/2^8 d^J_4,4(θ)∏_k=1^n[k-(1+n)s_θ/2^2]^2 dc_θ .

With these rewriting of the pseudo-widths α we have separated a universal coupling α_ Pl≡ M^2/M_ Pl^2 related to the first new resonance mass and a rational number N, which is tabulated in table <ref>.


Consider next the other channel; g^+2g^+2→ g^+2g^+2, with resonances now starting at J_ min=0 and therefore our basis for the projection being Legendre polynomials. The very same steps lead now to

    1+n Res[𝒜_2,2]=   8π M^2(1+n)/M_ Pl^2(n!)^2∏_k=1^n[k-(1+n)s_θ/2^2]^2 ,

and couplings

    α_0,0^n,J≡α_ Pl N_0,0^n,J ,

where 

    N_0,0^n,J=   (1+n)/4(n!)^2∫_-1^+1P_J(c_θ)∏_k=1^n[k-(1+n)s_θ/2^2]^2 dc_θ ,

with the corresponding numerical values tabulated in table <ref>.


The lack of odd spins can be traced to the amplitude being even under θ→π-θ.
The positivity of all entries in this tables is in accordance with unitarity and indeed the α's correspond to tree-level decay widths. 

For illustrative purposes, we show in fig. <ref> the Chew-Frautschi  plot 
yielding the spin of the resonances (on the y-axes) versus the square of their quantized mass (on the x-axes, labeled with the integer n) for the two channels g^+ 2g^+ 2→ g^+ 2g^+ 2 (left panel) and 
g^+ 2g^- 2→ g^+ 2g^- 2 (right panel).  Each resonance is colored according to the corresponding value of tree-level decay width (α_0,0^n,J and α_4,4^n,J, respectively) as indicated in the legend. Resonances are more and more weakly coupled for increasing values of spin.









  §.§.§ Matter-matter scattering


The analysis carried out in the previous section presents no difference when applied to matter as external states. 
There is, however,  a difference in the amplitudes themselves since some of them present an independent parameter other than α_ Pl. 
These are fermion-fermion and distinguishable scalar scattering (see table <ref>). 
In the following, we shall focus on these cases to show what can the analysis reveal about these parameters.

 
 
∘  Consider first distinct scalar in the elastic channel so that

    1+n Res[𝒜_0,0]=   8π M^2(1+n)/M_ Pl^2(n!)^2(c_θ/2^2/s_θ/2^2+as_θ/2^2)s_θ/2^2c_θ/2^2×
       ∏_k=1^n[k-(1+n)s_θ/2^2]^2 ,

 and again 
 
    N_0,0^n,J=   (1+n)/4(n!)^2∫_-1^+1( c_θ/2^2/s_θ/2^2+as_θ/2^2  )
     s_θ/2^2c_θ/2^2P_J(c_θ)×
       ∏_k=1^n[k-(1+n)s_θ/2^2] dc_θ .

 Positivity demands α_0,0^n,J, N_0,0^n,J≥ 0, and as table <ref> shows this is not the case for all values of a but only those in the interval

    0≤ a≤ 2 ,

which is compatible with 0 (that is with the case of a scalar field minimally coupled to Einstein-Hilbert gravity) and 1/6 for the conformal case. It is interesting to remark that the parameter a does not appear in same-scalar scattering since we symmetrize to find a(s_12+s_13+s_14)=0.
  

∘ We now move to consider the 2-to-2 fermion scattering.
Consider first the distinguishable case, e.g. a fermion and a lepton q^1/2ℓ^-1/2→ q^1/2ℓ^-1/2. Residues in the s-channel have quantum number under the SM gauge group
 (3,2,1/6), (3,1,5/3), (3̅,2,5/6), etc. or consider q_R^1/2q_L^-1/2→ q_R^1/2q_L^-1/2 with more elaborate color (3⊕6,2,5/6), (3⊕6,2,-1/6) and baryon number 2/3. Each of these will have a residue at M^2(1+n) as

    1+n Res[𝒜̃_1/2,1/2]=   8π  M^2/M_ Pl^2n!(-c_θ/2^2/s_θ/2^2+bc^2_θ/2/2)×
       ∏_k=0^n[k-(1+n)s_θ/2^2] ,

with minimum spin J_ min=1 and coefficients
 
    Ñ_1,1^n,J=   1/4n!∫_-1^+1(-c_θ/2^2/s_θ/2^2 + bc^2_θ/2/2)d^J_1,1(θ)×
       ∏_k=0^n[k-(1+n)s_θ/2^2] dc_θ ,

that are listed in table <ref>.
 


The residue for same fermion species scattering is instead

    1+n Res[𝒜_1/2,1/2]=
       8π M^2/M_ Pl^2n!c_θ/2^2(-1/s_θ/2^2-s_θ/2^2+b) ×
       ∏_k=0^n[k-(1+n)s_θ/2^2] ,

with minimum spin J_ min=1 and coefficients

    N_1,1^n,J=   1/4n!∫_-1^+1c_θ/2^2(-1/s_θ/2^2-s_θ/2^2+b)d^J_1,1(θ)×
       ∏_k=0^n[k-(1+n)s_θ/2^2] dc_θ .

We remark that the coefficient b affects the residues of the massive resonances whereas it does not alter the graviton pole where it enters as a free parameter. 
All the coefficients N_1,1^n,J  (as well as the coefficients Ñ_1,1^n,J) are subject to positivity.
 
We show the first few values of the decomposition in table <ref>.
Clearly, unitarity provides a non-trivial condition since not all coefficients are positive for any value of b.

Numerically, we find the situation illustrated in fig. <ref>.
For each value of n on the y-axis we find an allowed range of values for b (dotted green lines, while solid red lines indicate intervals excluded by unitarity).
What is non-trivial is that all these constraints must be satisfied simultaneously, all the way up to n →∞. 
This is true in the region shaded in green, that is set by the first resonance of the spectrum with n=0.
We note, in particular, that the value b=0 is not compatible with unitarity.
Hence unitarity bounds the a priori free parameter to lie in the range

    2/3≤ b ≤22/5 .

The bound that we get from the positivity of the coefficients Ñ_1,1^n,J is milder, 0 ⩽ b⩽ 8.
This very simple result, despite being a trivial consequence of unitarity, has profound implications for the structure of the theory. 
We shall discuss this important issue in section <ref>.




  §.§.§ Matter-graviton scattering


For completeness let us treat here the mixed gravity-matter case. 
This case is different in the sense that only in gravitational Compton scattering the intermediate state can be classified as `matter' in the following sense. 
In all previous processes resonances had helicity charge 𝔥 = 0. Furthermore, as already discussed,
 channels with charge 𝔥 = ± 2  are closed since no resonance has such charge. The case that is left is the one in which the exchanged resonances have charge 𝔥 = ± 1.
 This situation is realized in gravitational Compton scattering, where resonances have the charge of the matter we scatter.

Let us select fermion-graviton scattering to discuss semi-integer resonances and focus on region (g) to obtain the lowest spin resonance at J_ min=3/2. We have


    1+n Res[𝒜_1/2,2]=   8π M^2 (1+n)/M_ Pl^2(n!)^2 c_θ/2^3
    ∏_k=1^n[k-(1+n)s_θ/2^2]^2 .


The couplings are given by

    N   _-3/2,-3/2^n,J=(1+n)/4(n!)^2×
       ∫_-1^+1 c_θ/2^3d^J_-3/2,-3/2(θ) ∏_k=1^n[k-(1+n)s_θ/2^2]^2 dc_θ ,

and the first few of them are listed in table <ref>. We find all positive coefficients, as demanded by unitarity, and we have a tower of infinite resonances with increasing half-integer spin.




 §.§ Asymmetric Veneziano and Virasoro-Shapiro form factor

We now move to investigate the special cases of the scattering amplitudes that are completed by the form factors 𝒜_ VZ^η,γ_0 and 𝒜_ VS^γ_0. 
The goal is to check whether the positivity constraint coming from unitarity  provides  non-trivial input on the free parameters η and γ_0.
To this end, let us consider first fermion-vector scattering in the elastic channel, q^1/2V^-1→ q^1/2V^-1 in  region  (b), that is UV-completed by the form factor (notice the exchange s↔ t w.r.t. eq. (<ref>))

    𝒜_ VZ^η,γ_0 =Γ(1-t̃)Γ(1+ηγ_0-ηs̃)/Γ(1+ηγ_0-ηs̃-t̃) .

The values in s for which the Γ hits the pole Γ(-n) read

    s̃_n=η^-1(n+1+ηγ_0)=(n+1)η^-1+γ_0 .

We can  compute the residues 

    s̃_n Res[𝒜^η,γ_0_1/2,1] =    8π M^2 s̃_n/M_ Pl^2 n!ηc_θ/2^3
    ∏_k=1^n(
    k-s̃_n s_θ/2^2
    ) ,

and couplings 

    . N_3/2,3/2^n,J|_η,γ_0= s̃_n/4n!η∫_-1^+1c_θ/2^3
    ∏_k=1^n(
    k-s̃_n s_θ/2^2
    )dc_θ ,

with minimal spin given in this case by J_ min = 3/2. The first few values are listed in table <ref>. 
As clear from eqs. (<ref>, <ref>),  the parameters η, γ_0 only enter the decomposition thorough the combination η^-1(n+1)+γ_0 which we define to be γ̃_0.
















It is easy to see that the first Regge trajectory automatically satisfies positivity since it has the form γ̃_0^n. The sub-leading trajectory however requires for positive entries that η,γ_0 satisfy

    (η^-1-1)+ γ_0/n+1≤3/2n .

Where one can see that whereas the bound on γ_0 converges onto 3/2, the value of η has to be ever closer to 1. Hence one has that positivity not only restricts deformations from Veneziano in the 
 `η -direction', it forces η onto the value of Veneziano amplitude, i.e.

    Unitarity    enforces   η=1 ,    γ_0<3/2 .

The analysis can be now repeated for 𝒜_ VS^γ_0 again in region (b) for the scalar-fermion and scalar-vector scattering. We have the Veneziano-Shapiro form factor

    𝒜_ VS^γ_0=Γ(1+2γ_0)Γ(1-t̃)Γ(1+γ_0-s̃)Γ(1+γ_0-ũ)/Γ(1+t̃+2γ_0)Γ(1+γ_0+s̃)Γ(1+γ_0+ũ) .

The residues at s̃_n=γ_0+1+n read

    s̃_n Res[𝒜_ VS^γ_0]   =Γ(1+2γ_0)s_θ/2^2(s̃_nc^2_θ/2+γ_0)/Γ(2+n+2γ_0)n!×
       ∏_k=1^n(k-s̃_ns^2_θ/2)(k+2γ_0-s̃_ns^2_θ/2) ,

to be supplemented by the GR amplitudes for (h,h') = (0,1/2), (0,1) evaluated at s̃_n

    𝒜^ GR_0,1/2   =8π M^2 s̃_n/M_ Pl^2(c_θ/2+c^3_θ/2/2s_θ/2^2) ,
    𝒜^ GR_0,1   =8π M^2 s̃_n c^2_θ/2/M_ Pl^2 s_θ/2^2 .

The product of eq. (<ref>) and (<ref>) is projected onto Wigner d-functions d^J_1/2,1/2 (with J_ min = 1/2) and d_1,1^J (with J_ min = 1) respectively for the coefficients N_h',h'^n,J. One again has that the sub-leading Regge trajectory is not guaranteed positive and it sets upper limits on γ_0 the most stringent of which comes from the first term and yields, respectively

    Scalar-Fermion ,  N^1,1/2_1/2,1/2      γ_0<2.5215 ,
        Scalar-Vector ,  N^1,1_1,1      γ_0<3.1169 ,

where the values are the positive solution to a third degree polynomial approximated here, but with exact values as given in the appendix (cf. eqs. (<ref>, <ref>)) and they imply a squared mass for the first (n=0) state lighter than 3.5215 M^2 and 4.1169 M^2 respectively.


 §.§ Low energy limit and matching to GR


In the low energy limit, s_ij≪ M^2, the expansion in energy yields an EFT that should naïvely reproduce the results of  GR at the leading order.

This limit for the amplitudes in eq. (<ref>) is straightforward to take, and one finds that the corrections  enter at order M^-2=α_ Pl^-1M_ Pl^-2

    𝒜_ VZ^γ_0=1+[Γ^'(1+γ_0)/Γ^'(1+γ_0)-Γ^'(1)/Γ(1)]s_12/α_ PlM_ Pl^2 ,
       𝒜_ VS^γ_0=1+[2Γ^'(1+γ_0)/Γ(1+γ_0)-Γ^'(1)/Γ(1)-Γ^'(1+2γ_0)/Γ(1+2γ_0)]s_12/α_ PlM_ Pl^2 .

This in particular means that the entries in table <ref> 
should match the tree-level computation of Einstein-Hilbert gravity coupled to matter to order s/M^2. This is the case for most of the amplitudes but crucially not all. 
All the amplitudes do reproduce the on-shell contribution of the graviton pole (which is after all the only way we have tested gravity) 
but we recall 
that there are unspecified
off-shell pieces in the scalar-scalar and fermion-fermion scattering, cf. table <ref>. Let us compare these amplitudes  with the result in GR minimally coupled to matter obtained by means of Feynman rules. For the scattering of distinguishable scalars and identical fermions we find

    8π/M_ Pl^2(s_13s_14/s_12) ,
       8π2314/M_ Pl^2(s_13/s_12 +s_12/s_13 + 1/2) ,

and so a=0, b=1/2. The rather unexpected finding is that whereas a is compatible with the positivity bounds, (<ref>,<ref>), b in the fermion case is not. This means that the current UV completion differs from Einstein-Hilbert GR at the tree level in the infrared s/M^2→0. 

The difference is however in b, a contact term, which does not modify gravity at long distance; it does nevertheless require modifying Hilbert-Einstein gravity as our IR theory. Let us put forward a possible modification.


The value b=1/2 in eq. (<ref>) arises from the Lagrangian density  of a right-handed fermion minimally coupled to gravity of the form 

    ℒ = i√(-g)/2[
    ψ̅σ^ae^μ_a(∇_μψ) - 
    (∇_μψ)σ^a e^μ_aψ] ,

where 
the vierbein e^μ_a links global coordinates with those in a locally flat space, 
g^μν = e^μ_ae^ν_bη^ab. 
The torsion-free  spin connection ω_μ^ab enters via the covariant derivative 
∇_μψ = ∂_μψ + ω_μ^abσ_abψ/4, with σ_ab≡ (σ̅_aσ_b - σ̅_bσ_a)/2, and it can be derived in terms of vierbein e^μ_a and the Christoffel symbols as 
ω_μ^ab = e_ν^aΓ^ν_σμg^σρe_ρ^b - g^νρe_ρ^b(∂_μe_ν^a). 

Similarly to the scalar case (in which a minimally coupled scalar field has a=0 but a non-zero value can be obtained by adding a non-minimal coupling), modifying the value b=1/2 requires going beyond this minimal picture. 
A non-minimal coupling can be obtained by generalizing eq. (<ref>) to 

    ℒ = i√(-g)/2[(1-iα)
    ψ̅σ^ae^μ_a(∇̃_μψ) - h.c.
    ] ,

where α is a real parameter (note that choosing a left-handed fermion makes α→-α) and the covariant derivative 
∇̃_μψ = ∂_μψ + ω̃_μ^abσ_abψ/4 is now written in terms of a 
modified spin connection  that takes into account the presence of a non-zero torsion 
by means of the definition ω̃_μ^ab≡ω_μ^ab + 𝒦_μ^ab  
where ω_μ^ab is the torsion-free spin connection defined above and 𝒦_μ^ab is the so called contorsion tensor <cit.>.
The effect of the real constant α introduced in eq. (<ref>) reduces to a total derivative if the theory is torsion-free but it becomes 
 non-trivial in the presence of non-zero torsion. Torsion is encoded in the term
 
    S_γ = M_ Pl^2/16πγ∫ ( dω̃^ab  +ω̃^ac∧ω̃_c^   b)∧ e_a ∧ e_b ,

where d and ∧ are, respectively,  the external derivative (dx^μ∂_μ∧) and product, γ is the Immirzi parameter (that is, in full generality, a complex number) and ω̃^ab=ω̃^ab_μ dx^μ, e^a=e^a_μ dx^μ.


The relevant property about eq. (<ref>) is that in the limit γ→ 0 the torsion vanishes. 
To obtain the contribution of torsion to fermion scattering we
integrate out the contorsion[It is indeed only in the case of fermions coupled to gravity that the effects of torsions become manifest.
	As all other scattering processes with fermions (fermion-scalar, fermion-vector and gravitational Compton scattering with fermions) do not leave any room for contact terms, as explained in section <ref>.] tensor (by means of the equation of motion of the connection ω̃_μ^ab).










This procedure generates the four-fermion effective operators <cit.>

    ℒ= 
    3π G_N/8(
    γ^2/γ^2 + 1)(1-α^2+2α/γ)(ψ^†σ_μψ)^2 ,












and the parameter b  becomes

    b =  1/2
    +3γ/8(1+γ^2)[
     γ - 
    α(2+αγ)
    ] ,

and the bound in eq. (<ref>) can be now satisfied. For instance, if one takes α = 0 and restricts the analysis to real and positive values of the Immirzi parameter, the bound on b is satisfied if γ⩾ 2/√(5). 




An equivalent way to obtain a modification of b slightly more familiar to the working class particle theorist is through the introduction of a Kalb-Ramond field, a three form H≡ 3dB with B=B_μνdx^μ∧dx^ν. This field occurs in string theory and its presence is associated with gravitational torsion <cit.>. Let it couple to a fermion as

    S_ int=-g/  M∫ H ∧ (ψ^†σ_μψ dx^μ) ,











 
 where we can identify M as the mass scale in our UV completion and the coupling g is real and dimensionless.
 
 
 
 One expects in string theory g∼ M/M_ Pl but the precise value of g depends on the expectation values of the moduli fields and, therefore, on the details of the compactification procedure.
Consequently, it is  important to remark that the coupling g in the low-energy limit of string theory is in principle computable but, on balance, a free parameter.





Integrating out the two-form B  one obtains a contact interaction (the equation of motion for H is algebraic or, alternatively,  the propagator pole of B cancels against the derivative coupling) which reads
 
    b_ KR = 18 g^2 M_ Pl^2/π M^2 ,

and translates through eq. (<ref>) the Immirzi parameter into a ratio of scales. In order to satisfy the unitarity bound in eq. (<ref>) we obtain an upper and lower bound on the 
 mass ratio gM_  Pl/M

    1/108≤g^2 M_ Pl^2/π M^2≤13/60 .



 
 This simple discussion teaches us an important prerogative of the “bottom-up” approach to the UV completion of gravity pursued in this paper. 
 From a typical “top-down” perspective like that of string theory, 
 it is difficult to make firm statements about what we expect in our phenomenological four-dimensional world. 
 In this sense, the possible presence of space-time torsion is a prototypical example, with deviations from GR well motivated theoretically but difficult to compute without introducing free parameters 
 encompassing  the complicated step of compactification.
 Our  “bottom-up” perspective, on the contrary, predicts a deviation from GR, and gives a precise indication about what we should expect for it.
 The above analysis exposes in plain sight the true complementarity of the two approaches. 
   







§ GRAND FINALE

The on-shell amplitude program has paved the road for the formulation and
 analysis of amplitudes featuring  resonances of arbitrary spin and mass,  clearing the computational obstacles of the conventional Lagrangian approach.
 
 
 In this letter, this formalism was applied to study the UV completion of gravity following a “bottom-up” approach. 
The derivation of a viable UV completion started from the need to tame the growth with energy of the scattering amplitudes mediated by gravity, which at energies comparable to the Planck scale  grow above the unitarity bound. 
With typical particle physicist demeanor, we approached the problem by introducing massive resonances and the solution obtained was validated against unitarity, locality and causality. 
In this regard, we do not put forth in this work a full quantum theory of gravity 
but rather a UV-complete formulation of tree-level amplitudes. These amplitudes nonetheless yield a great deal of information on the full theory, 
and have non-trivial implications in the infrared.

 In more depth, the main results of our analysis are:
 
(i) After enumerating our working assumptions – locality, unitarity and causality, weak coupling limit – in section <ref>, in section <ref> we constructed the most general tree-level
 scattering amplitude mediated by gravitational interactions  in the context of GR with  SM particles (including gravitons) on the external states, eq. (<ref>).
 We considered not only the contributions to the scattering amplitudes coming from the poles but also, in full generality, the possible presence of contact (polynomial) terms compatible with our working assumptions.

(ii) In section <ref> we computed the contribution due to the exchange of a massive resonance with arbitrary spin J, eq. (<ref>). 
The result of this computation was already quoted in <cit.> in terms of `spinning Gegenbauer polynomials'.
Here we derived an equivalent expression in terms of Jacobi polynomials that makes more transparent the helicity structure of the amplitude.
Furthermore, by combining the Jacobi polynomials with the helicity factor, we related the scattering amplitudes to Wigner d-functions. 

Finally, the study puts in  the foreground the role of unitarity that, in the case of elastic processes, relates the residue of the amplitude to the decay width of resonances.  

(iii) In section <ref> we combined the results of (i) and (ii) to obtain a valid UV completion of GR amplitudes. 
To this end, we only used as guidelines unitarity, locality and causality.
A number of properties – i.e. quantization of the mass spectrum and  duality relations among resonances exchanged in channels related by crossing transformations – elegantly arises  from the mathematical consistency dictated by these three fundamental principles. The obtained solutions follow from three ansatz, clearly displayed in section <ref>, put forward to solve 
the mathematical consistency conditions that follow from locality and unitarity.
The UV completion dresses  the GR amplitudes with form factors that are product of Euler Γ functions, thus closely resembling the Veneziano and Virasoro-Shapiro amplitudes as summarized in section <ref>. 







 
(iv) Section <ref> is devoted to the analysis of the physical properties of the resonances. By performing an angular decomposition in terms of Wigner d-functions, 
we extracted couplings and spin of the resonances. Unitarity enforces positivity constraints on the  couplings of resonances kinematically accessible in elastic scattering processes.
The  “bottom-up” approach showcases the power of this basic principle. 
As a consequence of unitarity, for instance, we find that the proposed UV completion predicts leading order deviations from GR minimally coupled to fermions. 
These deviations needed to restore unitarity are present if space-time has torsion in addition to curvature. 
 Unitarity also imposes non-trivial constraints on the contact terms introduced in point (ii) 
 as well as on possible deformations of the Veneziano and Virasoro-Shapiro amplitudes found at point (iii). 
  
 
The ambitious and overarching question lurking in the background is uniqueness: are fundamental principles like unitarity and locality so demanding as to select one theory of gravity only?

We believe that the “bottom-up” approach may help in shedding light on this question,  see <cit.> for a related discussion. 
In this regard, our findings have glaring similarities with string theory,



tantalizingly yet inconclusively in line with the belief that string theory is the only consistent quantum theory of gravity.
A more detailed  exploration of this connection is left for future work since it would have diluted the main aim of this letter.
The main aim being the model-independent “bottom-up” and unbiased approach to the UV completion of gravity with fundamental principles like unitarity and locality as sole reference.


This road has been shown here to lead to relevant results in complementarity to the “top-down” prevailing trend, an outstanding example
are the constraints obtained from unitarity in  (iv) 

given that studying the same subject in the  “top-down” perspective of string theory requires overcoming the obstacle of compactification to make contact with our observable
 four-dimensional Universe, losing predictivity  along the way.

  

 
 




  
*Acknowledgments: 
We thank Marco Fabbrichesi and Simeon Hellerman for discussions and Yu-tin Huang for discussions and for bringing to our attention the work in <cit.>.
A.U. thanks the Kavli IPMU, 
where  this  project  was  completed, and the city of Tokyo for  the  warm  and kind hospitality. R.A. acknowledges Kendrick Lamar for reminding him with his lyrics of Compton scattering. A.U. acknowledges financial support from the H2020-MSCA-RISE project “InvisiblesPlus” and from the INFN grant SESAMO. 
This work was supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.











§ BASICS OF MASSLESS KINEMATICS AND SPINOR-HELICITY FORMALISM



We consider the massless 2-to-2 scattering 1(p_1) + 2(p_2) →3̅(p̅_3) + 4̅(p̅_4) in the CM frame with four-momenta

    p_1 = (E,0,0,E) ,   p_2 = (E,0,0,-E) ,   p̅_3 = (E,Esinθ,0,Ecosθ) ,   p̅_4 = (E,-Esinθ,0,-Ecosθ) ,

where θ is the scattering angle. The physical region for this process is the red region (r) in fig. <ref>.
The four-momenta in eq. (<ref>) describe ingoing initial-state and outgoing finale-state particles.
The Mandelstam variables are given by 

    s    ≡    (p_1 + p_2)^2 = (p̅_3 + p̅_4)^2 = 4E^2 ,
    
    t   ≡    (p_1 - p̅_3)^2 = (p_2 - p̅_4)^2 = -4E^2/2(1-cosθ)= -s/2(1-cosθ) = -ssin^2θ/2 ,
    
    u   ≡    (p_1 - p̅_4)^2 = (p_2 - p̅_3)^2 = -4E^2/2(1+cosθ)= -s/2(1+cosθ) = -scos^2θ/2 = -s-t ,

and, consequently, we have for the scattering angle cosθ = 1+2t/s. For the on-shell production of a resonance with mass M we have s=4E^2 = M^2.

For a massless particle of momentum p^μ≡(p^0,p^1,p^2,p^3), p^2=0, the map into a (1/2,1/2) representation of the Lorentz group of the momentum can be written as the outer product of two two-component spinors (we use the mostly-minus flat Minkowski metric)

    p̂_αα̇≡ p_μ(σ^μ)_αα̇ = 
    (
    [   p^0 - p^3 -p^1 + ip^2;  -p^1 -ip^2   p^0 + p^3 ]) =_αpp_β̇

given that det(p̂)=p^2 = 0. For real momenta, p_αα̇ is Hermitian and we have the reality condition p_α̇ = ±(_αp)^*. 
Indices are raised and lowered as ε^αβ_βp = p^α and p_α̇=ε_α̇β̇^β̇p, 
with the Levi-Civita symbol in two dimensions ε^αβ = ε^α̇β̇ = ((0,1),(-1,0)) = -ε_αβ = -ε_α̇β̇.

In the main text, we used explicitly the spinor-helicity formalism following the convention according to which all momenta are ingoing.
This means that, compared to eq. (<ref>), we have p̅_3,4→ p_3,4 =-p̅_3,4 (notice that these flipped momenta 
enter in fig. <ref> in the definition of s_ij).
In this case, we find that one possible explicit choice of spinors is

    (p̂_1)_αα̇   =   (
    [  0  0;  0 2E ]) = _αp_1p_1_β̇
       ⇒   p_1 =p_1=  (
    [     0; √(2E) ]) ,
    
    (p̂_2)_αα̇   =   (
    [ 2E  0;  0  0 ]) = _αp_2p_2_β̇   ⇒   p_2 =p_2=  (
    [ √(2E);     0 ]) ,

and

    (p̂_3)_αα̇   = (
    [ -2Esin^2θ/2       Esinθ;       Esinθ -2Ecos^2θ/2 ]) = _αp_3p_3_β̇   ⇒   p_3 = - p_3 = √(2E)(
    [ -sinθ/2;  cosθ/2 ]) ,
    
    (p̂_4)_αα̇   = (
    [ -2Ecos^2θ/2      -Esinθ;      -Esinθ -2Esin^2θ/2 ]) =_αp_4p_4_β̇   ⇒   p_4= - p_4 = √(2E)(
    [ cosθ/2; sinθ/2 ]) .

One can now compute explicitly the angle/square brackets that enter in the helicity structure of the scattering amplitudes studied in the main text (see table <ref>) for region (r) whereas the case for regions (b) and (g) is obtained by a reshuffling of the 1-2-3-4 indexes.
In summary we have, in each of the regions,

    2314
     [very thick, blue, ->] (0,0)–(0,-1);
     [very thick, red, ->] (-0.5,0)–(-2.5,-1);
     [very thick, ForestGreen, ->] (0.5,0)–(2.5,-1);
    
    
    -scos^2θ/2       scos^2θ/2        -s

and

    3P̂_124
     [very thick, blue, ->] (0,0)–(0,-1);
     [very thick, red, ->] (-0.5,0)–(-2.5,-1);
     [very thick, ForestGreen, ->] (0.5,0)–(2.5,-1);
    
    s/2sinθ       -scosθ/2       scosθ/2

from which it is possible to write explicitly all amplitudes in terms of conventional Mandelstam variables.
 


Let us just illustrate a couple of examples.
In the four-graviton scattering in the red region (r) we have 

    𝒜_g^2g^-2→ g^2g^-2= 𝒜_g^2g^-2g^-2g^2= 
    8π23^414^4/M_ Pl^2 stu .

An explicit computation gives

    2314 = (
    ε^βα_αp_2_βp_3)(
    p_1_α̇ε^α̇β̇p_4_β̇) =  -4E^2/2(1+cosθ) = u = 1423 ,

and the amplitude is 𝒜_g^2g^-2g^-2g^2 = 8π u^3/M_ Pl^2 st.

Similarly, as a second benchmark example, we can compute the gravitational Compton scattering with fermions whose amplitude in the red region (r) is  (see table <ref>)

    𝒜_q^1/2q̅^-1/2→ g^2g^-2 = 
    𝒜_q^1/2q̅^-1/2g^-2g^2 = 
    8π2314(3P̂_124)^3/M_ Pl^2 stu .

An explicit computation gives

    3P̂_124 =
     p_3^α
     ( P̂_12)_αβ̇
     
     
     
    
    
    
    
    
    
    
    
    
    
    
     
     
     
     
     
     
     
     
    
    
    
    
    Finally, useful formulas used but not quoted in the text are the following.
    
    
      ∘ In section <ref> an example of the contractions that give rise to the spin J mediated four-point amplitude with external scalars 
     is given by (for J = 2, 3, 4)
    
        J=2 ,     1/4![8 -1.5mm [thick, rounded corners=2pt] (0.125,0)–(.25,0)–(0.25,.5)–(0,.5)–(0,0)–(0.125,0);
        	 [thick, rounded corners=2pt] (0.125,0)–(.25,0)–(0.25,.5)–(0,.5)–(0,0)–(0.125,0);
        	+16 -1.5mm [thick, rounded corners=2pt] (0.175,0)–(.25,0)–(0.5,.5) – (0.625,.5)–(.625,0) – (.5,0) – (.25,.5)– (0.125,.5)– (.125,0) –(0.2,0);
        	]
        
        J=3 ,     3!2^3/6![(-1.5mm [thick, rounded corners=2pt] (0.125,0)–(.25,0)–(0.25,.5)–(0,.5)–(0,0)–(0.125,0);
        	)^3+ 6 -1.5mm [thick, rounded corners=2pt] (0.125,0)–(.25,0)–(0.25,.5)–(0,.5)–(0,0)–(0.125,0);
        	 [thick, rounded corners=2pt] (0.175,0)–(.25,0)–(0.5,.5) – (0.625,.5)–(.625,0) – (.5,0) – (.25,.5)– (0.125,.5)– (.125,0) –(0.2,0);
        	
        +8 -1.5mm [thick, rounded corners=2pt] (0.175,0)–(.25,0)–(0.5,.5) – (0.625,.5)–(.875,0) – (1,0) – (1,.5)– (.875,.5)–(.625,0)–(.5,0)–(.25,.5)–(.125,.5)–(.125,0)–(.175,0);
        	]
        
        J=4 ,     4!2^4/8![
        (-1.5mm [thick, rounded corners=2pt] (0.125,0)–(.25,0)–(0.25,.5)–(0,.5)–(0,0)–(0.125,0);
        	)^4+12(
        -1.5mm [thick, rounded corners=2pt] (0.175,0)–(.25,0)–(0.5,.5) – (0.625,.5)–(.625,0) – (.5,0) – (.25,.5)– (0.125,.5)– (.125,0) –(0.2,0);
        	)^2+12 (-1.5mm [thick, rounded corners=2pt] (0.125,0)–(.25,0)–(0.25,.5)–(0,.5)–(0,0)–(0.125,0);
        	)^2
        -1.5mm [thick, rounded corners=2pt] (0.175,0)–(.25,0)–(0.5,.5) – (0.625,.5)–(.625,0) – (.5,0) – (.25,.5)– (0.125,.5)– (.125,0) –(0.2,0);
        	+32 -1.5mm [thick, rounded corners=2pt] (0.125,0)–(.25,0)–(0.25,.5)–(0,.5)–(0,0)–(0.125,0);
        	-1.5mm [thick, rounded corners=2pt] (0.175,0)–(.25,0)–(0.5,.5) – (0.625,.5)–(.875,0) – (1,0) – (1,.5)– (.875,.5)–(.625,0)–(.5,0)–(.25,.5)–(.125,.5)–(.125,0)–(.175,0);
        	+48 -1.5mm [thick, rounded corners=2pt] (0.175,0)–(.25,0)–(0.5,.5) – (0.625,.5)–(.875,0) – (1,0) – (1.25,.5)– (1.375,.5) – (1.375,0)–(1.25,0) – (1,.5)–(.875,.5)– (.625,0)–(.5,0)–(.25,.5)–(.125,.5)–(.125,0)–(.15,0);
        	]
    
    Each term schematically represents a product of traces over slashed momenta, which itself one writes in terms of Lorentz scalar products as
    
        (σ̅^μ p_μσ^μ k_μ)^n=2^n∑_m
        n2m (p· k)^n-2m[(p· k)^2-p^2k^2]^m .
    
      ∘ In section <ref> we extracted in eqs. (<ref>, <ref>) an upper bound on the coefficient γ_0 of the modified 
    Virasoro-Shapiro form factor 𝒜_ VS^γ_0 as a consequence of unitarity applied to the scalar-fermion and scalar-vector scattering processes.
    In more detail, in the scalar-fermion case the bound follows from the positivity condition imposed on the coefficient N^1,1/2_1/2,1/2 while in
     the scalar-vector the same happens for N^1,1_1,1. We find
    
        Scalar-fermion:        N^1,1/2_1/2,1/2 >0      30 + γ_0(
        101  + 46γ_0 -36γ_0^2
        ) >0      ⟹   γ_0 < 2.5215 , 
         Scalar-vector:        N^1,1_1,1>0      14 +γ_0[
        47-γ_0(
        -24 + 13γ_0
        )
        ] > 0      ⟹   γ_0 < 3.1169 .
    
    
    
    
    
    
    ∘ Finally, we discuss the case with external massless gauge bosons V with helicity h=± 1. Let us consider first the scattering in the helicity 
    configuration V^+1V̅^-1→ V^+1V̅^-1, where in full generality V̅ is the complex conjugate of V. 
    This allows us to describe both the scattering with real external gauge fields (photons, gluons or the Z boson) and charged electroweak gauge bosons W.
    The amplitude for this process is symmetric under the s-t exchange.[On the contrary, notice that 
    	in the u-channel it describes a different helicity configuration, that is V^+1V^+1→ V^+1V^+1. 
    	In light of the selection rule discussed in section 3, 
    	the absence of the pole in u in eq. (<ref>) can be explained by the fact that in the u-channel process 
    	V^+1V^+1→ V^+1V^+1 we have same helicities h_1 = h_2 = 1.]
    As customary, from eq. (<ref>) (with the only difference that the angle/square brackets factor is 14^223^2 = u^2) we get 
    
        𝒜^ GR_V^+1V̅^-1→ V^+1V̅^-1(s,t)    =   κ^2 u^2/4A_VV̅(s,t) 
            GR=   κ^2 u^2/4(
        1/s+1/t) .
    
    Our proposal for the UV completion reads  
    
        A_VV̅(s,t) = (
        1/s+1/t)Γ(1-s̃)Γ(1-t̃)/Γ(1-s̃-t̃) .
    
    Notice that in this case we do not have any free parameter.[Of course, one could write
    	
        A_VV̅(s,t) = [
        	1/s+1/t
        	+a/M_ Pl^2 + b(s+t)/M_ Pl^4+ …]Γ(1-s̃)Γ(1-t̃)/Γ(1-s̃-t̃) ,
    
    	but we discard these terms in agreement with the fact that we look for a tree-level UV completion, and in our approximation they all vanish.
    ]
    We have μ_i = μ_f = 2, and we can decompose the residues in terms of d^J⩾ 2_2,2(θ).
    We find
    
        R_s̃ = 1+n^(+1,-1;+1,-1)   ≡   lim_s̃→ 1 +n(1-s̃+n)
        κ^2 u^2/4A_VV̅(s,t)
           =   -8π M^2/M_ Pl^2(n+1)n!ũ^2(t̃ + s̃)
        ∏_j=1^n(t̃ + j) ,
    
    and we consider the decomposition
    
        -ũ^2(t̃ + s̃)
        ∏_j=1^n(t̃ + j) =
        ∑_J=2^n+3
        C_2,2^n,J d_2,2^J(θ) .
    
    We find all positive coefficients, in agreement with unitarity.
    We collect the first few of them in table <ref>.
    
    
    
    
    The previous section gave us the contribution to massless particle scattering mediated by a spin J particle. With these pieces one can now hope to formulate a general amplitude with the first term being graviton exchange yet possessing a well-behaved high energy limit as well as all sanity properties outlined in the introduction.
    
    In this road let us first consider scattering of distinguishable particles, for simplicity let us take scalar particles ϕ , φ (we could also take say a lepton and a quark). This means also there is no `crossing symmetry' but rather flipping particle momenta we obtain a different physical process. Making use of the s^2 bound on the amplitude to use a dispersion relation we write our general expression as:
    
        𝒜_ϕφ=s_14^2/M_PL^2s_13+∑g_ϕ g_φ P_J[x(s_14)]/s_13/M_n^2-1+As_13+B
    
    where A, B are funtions of s_14 and the couplings g_φ ,  g_ϕ are dimensionless. In this relation, given that we are scattering different particles we assume there is only one channel open (as it happens in Hilbert-Einstein gravity) so that all poles and in the positive real s_13 axis; otherwise resonances with ϕ-φ number would be present in the spectrum.
    
    Let us now illustrate the difficulty in unitarizing all processes that the above amplitude describes. First take ϕ(p_1) ϕ̅(p_3)→φ(p_4) φ̅(p_2), and s_13→ s,  s_14→ t, this is region blue in fig. <ref>. One first guess, given that the graviton's contribution grows with CM energy is simply to cancel this out with a spin J=2 resonance, so we write:
    
        𝒜'=t^2/M_PL^2s+g_ϕ g_φ/M^2t^2+t+1/6/s-M^2
    
    This resonance could do the job if g_φ g_ϕ=-(M/M_PL)^2 in particular implying that the sign of the couplings or `charges' are opposite. However this possibility does not extend to more fields; say we introduce a third field χ, from ϕχ scattering we would deduce χ has opposite-sign charge compared to ϕ, but if both χ and φ have opposite charge w.r.t. ϕ they have  same sign w.r.t. each other and χφ scattering is not unitarized.
    
    How about the contact terms A(t) s+ B(t)? To answer this questoin let us make use of the partial wave decomposition 
    
        𝒜_L≡1/32π∫ d(cosθ) 𝒜(s,cosθ)P_L(cosθ)
    
    to best exploit unitarity constraints. One has that the graviton contribution to the highest partial wave is
    
        𝒜^(h_μν)_2   =4 s/15M_PL^2
    
    for A(t) s+ B(t) to scale with s the same one should have A(t)=A(0) or B(t)=t, none of which however project onto 𝒜_2.
    Hence one turns now to spin J=3, such a spin particle contribution to 𝒜_2 reads:
    
        𝒜^(J=3)_2   =-2g_φg_ϕs^2/M^4
    
    and now we can have sign(g_φ)=sign(g_ϕ).[One could try to use a spin 4 resonance instead and then the signs again would have to be opposite, but we can again discard this case since it would not work for a third species χ.] This is indeed more in line with the expectation that the UV completion of gravity will share the latter's universality in coupling to all matter equally. However one now has introduced a contribution to the partial wave 𝒜_3 which grows faster with s than that of the graviton. We are then advocated to a spin J=4 resonance which will require itself a J=5 and it follows that this series in spin cannot end.
    
    One can make this statement more precise demanding the  growth with s of the partial wave for the highest spin accessible at some energy s be cancelled by a one more unit of spin resonance (therefore this only concerns the Regge trajectory), we obtain:
    
        M_n+1^2≤[4(J-1)/α_n+1,J]^1/J^2-J[J(2J+1)α_n+1,J/(J-1)α_n,J]^1/J-1M_n^2
    
    with α_n=g_ϕ,ng_φ,n/4π.
    
    Regardless of the location of poles or their coupling strength however we can encounter a greater challenge to unitarization by turning to a different process by crossing.
    Take ϕ(p_1)φ̅(p_4)→ϕ(p_3)φ̅(p_2) now so s_14→ s and s_13→ t, which is the green region in fig. <ref>.
    
        𝒜=s^2/M_PL^2t+∑g_ϕ,n g_φ,n P_J[x(s)]/t/M_n^2-1+At+B
    
    which in practice has swapped s and t. This change has severe consequences; for one the argument of the Legendre polynomials 1+2s/M_n^2 is now always ≥1 which implies P_J≥ 0, while the poles all contribute the same negative sign for all values of t. This means that the cancellations in 𝒜 cannot longer occur since all terms in our sum have the same sign! This gives a hopelessly bad convergence and posses a serious problem.  Discuss, could At+B help here?.
    
    There is a potential issue with the derivation here nevertheless, we have extrapolated the on-shell result from a physical region where resonances are indeed accessible (the s-channel process) to another region where these resonances are in the t-channel and hence outside the kinematically allowed space. This extrapolation is stricly only valid for the full amplitude with off-shell pieces as well.
    
    To address this let us turn back to region (b) and attempt to sum the full series of resonances and hopefully capture too off-shell pieces once we demand all sanity checks outlined in the introduction.
    
    
        𝒜= t^2/s+∑g^ϕ_n g^φ_n P_J[x(t)]/s-s_n+As+B
        =𝒫(s,t)/∏ (s-s_n)
    
    where for simplicity we have set our units of mass to the Planck scale and s_n=M_n^2 and 𝒫 is a polynomial of arbitrarily high powers of t,s.
    Let us write it as
    
        𝒫(s,t)=C(t) ∏_n^∞ (t-t_n(s))
    
    From the l.h.s. of eq. (<ref>), this function, when evaluated at the poles must yield a constant times a finite polynomial in t. In particular the ratio of two such residues:
    
        𝒫(s_n,t)/𝒫(s_m,t)=∏^∞ (t-t_i(s_n))/∏^∞ (t-t_j(s_m))=∑^ finite c_n,kt^k/∑^ finite c_n,lt^l
    
    Finiteness then must be achieved because an infinite (but countable) number of factors drop out in this ratio i.e. the series t_0(s_n), t_1(s_n), t_2(s_n),... and t_0(s_m),t_1(s_m),... have an infinite overlap in their elements:
    
        dim({t_i(s_n) }∩{t_i(s_m) }) =  dim(IN)
    
    One can take a cue from this condition that a shift in the argument of the functions t_n still leaves a lot of common elements in the series to `generate' these functions by a shift of some original function.
    
    This goes as follows: take t_0(s) to be an arbitrary function of s, t_0=F(-s) (the reason for the sign will be clear momentarily) while the spectrum of poles is the natural numbers s_n=n. Then generate the remaining functions as t_n(s)=F(n-s) which shifts the function F by n units to the right.
    This ansatz satisfies the above condition, take n=3, n=2 for concreteness and clarity
    
        {t_i(3)=F(i-3) }/{t_j(2)=F(j-2)} =
        {F(-3),F(-2),... }/{F(-2),F(-1),...}= {F(-3)}
    
    or in the case of n,m general the seires would differ by |n-m| elements.
    One can further deform this solution to avoid assumptions about the spacing of the spectrum:
    
        t_i(s)   =F(i-f(s))     f(s_n)   =n ,
    
    with e.g. f(s)=s^2, s_n=√(n).
    Let us put this ansatz into a single residue now instead of a ratio to find:
    
        _s_n(𝒜)∝ C(t)∏_j(t-F(j-f(s_n))
    
    which has to be a finite polynomial. We are forced then by our ansatz to take C(y) as inverse powers of zeros on t, in particular we arrive at the final form:
    
        𝒜=C(0)t^2∏_j(t-F(j-f(s)))/∏_k(t-F(k))∏_n (s-s_n)
    
    where the overal t is inserted to reproduce the graviton contribution at s=0. This ansatz returns as residues
    
        _s_n(𝒜)=C(0)t^2∏_j=1^n (t-F(-j))/∏^∞ (s_m-s_n)
    
    and hence the nth pole has a spin J=2+n resonance.
    More remarkably our ansatz requires `poles' in t located at F(k); these when turning to region 𝒜 (s⇔ t) will be accessible if F(k)∈ IR^+. One could try and move these poles outside of all physical regions by, for example, taking the zeros to be imaginary:
    
        t^2∏_n^∞ (t-(n-s))(t+ (n-s))/∏_m^∞(t- m)(t+ m))∏_k^∞ (s-k)
    
    While these amplitude admits a resonance decomposition in s we find it not to be unitary and badly behaved in 𝒜.
    
    If one instead accepts that this type of solutions bring along new resonances one realizes this ansatz, by dropping our assumption of no t channel resonances, might be able to unitarize region 𝒜 too. In particular this ansatz now must have a pole decomposition in t and so
    
        _s=F(n)(𝒜)∝∏_j (F(n)-F(j-f(s)))/∏ (s-s_m)
    
    This having to be a finite dimensional polynomial imposes constraints on our functions F and f, in particular one would have to solve for s in F(n)=F(j-f(s)) for a given F,f to write the above as a product of zeros and then have these cancel against the factors (s-s_n) in the denominator. Needles to say this gets more complicated the higher degree polynomial one selects of F(j-f(s)); what's more the presence of multiple zeros for each factor seems to conflict with a finite degree polynomial for final result.
    
    One is therefore left with the `trivial' solution with F(j-f(s))=(j-s) which returns:
    
        𝒜=t^2∏_n^∞ (t+s-n)/∏_m^∞(t-m)∏_k^∞ (s-k)
    
    which is Veneziano's amplitude times t^2. It is clear now that this procedure extends to any external helicity, given that our formulas in  () have factored out the spinor structure. And so for scattering of two distinguishable helicity h particles we read:
    
        𝒜=12^2h34^2ht^2-2h∏_n^∞ (t+s-n)/∏_m^∞(t-m)∏_k^∞ (s-k)
    
    
    
    
     §.§ s,t,u duality
     
    In GR the 2-to-2 graviton scattering is described at the tree level by the amplitude 
    
        𝒜_1^-22^+2→ 3^+24^-2(s,t)    ≡   κ^2/4
         (1324)^4 
        A(s,t)
            GR=   κ^2/4stu(1324)^4 ,
    
    with massless external gravitons (u=-s-t) and fixed helicities.   The other two non-vanishing amplitudes can be obtained from eq. (<ref>) by crossing (t ↔ s and t ↔ u, respectively <cit.>
        𝒜_1^+22^+2→ 3^+24^+2(s,t)     GR=   κ^2/4stu(3412)^4 ,
        𝒜_1^-22^+2→ 3^-24^+2(s,t)     GR=   κ^2/4stu(1423)^4 .
    
    The 1/stu factor is completely fixed in GR by requiring  that only simple poles are allowed <cit.>, 
    and the rest of the amplitude depends on the fixed helicities (see eq. (<ref>) and the appendix). 
    We look for a UV completion. In the following, we focus on the function A(s,t) defined in eq. (<ref>). 
    We remark that, once the helicity structure of the scattering amplitude is factorized, the function A(s,t)–stripped of any knowledge about the helicity of the external particles–must be completely general and symmetric in the three Mandelstam variables s,t,u since it basically describes scattering of identical scalar particles.
    
    The most general amplitude of this sort is 
    
        A(s,t) = N(s,t)/∏_k=1^n (s-M_k^2)(t-M_k^2)(u-M_k^2) ,
    
    where we assumed the presence of n resonances with mass M_k. The task is to determine whether a numerator N(s,t) that is consistent with our assumptions can be defined.
    The simplest guess is to organize the amplitude as the familiar sum of s-, t-, and u-channel diagrams. 
    However, this simple construction leads to unitarity/causality violation at high energies <cit.>. A more promising ansatz is
    
        A(s,t) = ∏_i ⩽ j=1^n ∏_x=s,t,u (x + M_i^2 + M_j^2)
          /
          stu∏_k(s-M_k^2)(t-M_k^2)(u-M_k^2)
           ,
    
     where the structure of the numerator is chosen in such a way that all the poles are simple. 
     We further impose the correct IR behavior as s→ 0 (that is eq. (<ref>)).
     It is simple to see that decoupling all resonances in the IR implies that for every {i,j}  there must exist a k such that 
     M_i^2 + M_j^2 = M_k^2. This relation has  two important implications on the mass spectrum of the resonances. First,  
    it must run over infinitely many resonances in order not to spoil the cancellation of the massive IR residues. Second, the linear dependence 
     M_k^2 = kM^2 with k ∈ℤ^+ represents a trivial solution. The amplitude, therefore, becomes
        
        A(s,t)    =   1/ stu ∏_k=1^∞
          (s + M_k^2)(t + M_k^2)(u + M_k^2)
          /
         (s - M_k^2)(t-M_k^2)(u-M_k^2)
          
           =   1/stu Γ(1 - s̃)Γ(1 - t̃)Γ(1 - ũ)/Γ(1+s̃)Γ(1+t̃)Γ(1+ũ)
             ,
      
    where we used the Euler definition of the Γ function and we introduced the notation x̃≡ x/M^2 for the three Mandelstam variables.
