# Quantum gravity and amplitudes

Quantum gravity and amplitudes

# It has been well established since a long time now that quantum field theory approach to quantum gravity fails. It gives results which are not consistent with perturbative unitarity and leads to loss of predictability. There are several ways to understand this. The perturbative expansion of the Einstein-Hilbert Lagrangian yields terms which are second order in derivatives. Thus, the coupling constant in gravity has negative mass dimensions. The tree amplitude for four graviton scattering grows with the energy scale and indicates power counting non-renormalizability. Further, if one computes loop corrections, it is not hard to see that one would encounter more and more divergent integrals at each loop order because of increasing number of derivatives, which is an implication of negative mass dimension coupling.

It has been well established since a long time now that quantum field theory approach to quantum gravity fails. It gives results which are not consistent with perturbative unitarity and leads to loss of predictability. There are several ways to understand this. The perturbative expansion of the Einstein-Hilbert Lagrangian yields terms which are second order in derivatives. Thus, the coupling constant in gravity has negative mass dimensions. The tree amplitude for four graviton scattering grows with the energy scale and indicates power counting non-renormalizability. Further, if one computes loop corrections, it is not hard to see that one would encounter more and more divergent integrals at each loop order because of increasing number of derivatives, which is an implication of negative mass dimension coupling.

# Thus, one can say that perturbative quantum gravity does not make sense. However, this argument is too naive as it stands and it will be inappropriate to immediately arrive at this conclusion without explicitly computing physical quantities of interest. It may happen that there are cancellations between the divergent parts of the diagrams and the final result does make sense. Indeed, as it turns out, pure gravity with zero cosmological constant is finite at one-loop. This can be understood by analysing the arising counter-terms. If one assumes a non-zero cosmological constant, divergences do arise but they can be absorbed into the tree level action. Despite behaving nicely at one-loop, gravity does diverge at two-loops. This was first observed explicitly by Goroff and Sagnotti. Quantum gravity at two-loops is said to be non-renormalizable. This fact led to abolishing perturbative treatments of quantum gravity and motivated interest in building new frameworks like string theory, loop quantum gravity and causal sets to name a few. However, each of these frameworks have their own technical limitations and this is why the problem of consistently quantizing gravity in four dimensions remains open to this day.

Thus, one can say that perturbative quantum gravity does not make sense. However, this argument is too naive as it stands and it will be inappropriate to immediately arrive at this conclusion without explicitly computing physical quantities of interest. It may happen that there are cancellations between the divergent parts of the diagrams and the final result does make sense. Indeed, as it turns out, pure gravity with zero cosmological constant is finite at one-loop. This can be understood by analysing the arising counter-terms. If one assumes a non-zero cosmological constant, divergences do arise but they can be absorbed into the tree level action. Despite behaving nicely at one-loop, gravity does diverge at two-loops. This was first observed explicitly by Goroff and Sagnotti. Quantum gravity at two-loops is said to be non-renormalizable. This fact led to abolishing perturbative treatments of quantum gravity and motivated interest in building new frameworks like string theory, loop quantum gravity and causal sets to name a few. However, each of these frameworks have their own technical limitations and this is why the problem of consistently quantizing gravity in four dimensions remains open to this day.

# In recent years, and in particular over the last two decades, there has been tremendous development in the field of scattering amplitudes. On one hand, new techniques such as recursion relations and on-shell methods are implemented to simplify computations in Yang-Mills and gravity. While on the other, conceptual developments are made in both these theories and their supersymmetric counterparts. Several works in the past have outlined the detailed computations of multi-photon/gluon/graviton scattering amplitudes. A corollary of these developments indicate that the UV behaviour of gravity is in some sense, better than Yang-Mills. Indeed, the BCFW recursion relations imply that the fall-off behaviour of gravity tree amplitudes at large values of complex momenta goes like 1/z^2, as opposed to Yang-Mills which goes like 1/z, where z is the complex momentum parameter. This is related to the fact that the group of diffeomorphisms is at play in gravity. However, the real surprise of the developments is that gravity is rather linked to Yang-Mills. This has been realized in the so called double copy relations where gravity amplitudes are a certain square of the Yang-Mills ones.

In recent years, and in particular over the last two decades, there has been tremendous development in the field of scattering amplitudes. On one hand, new techniques such as recursion relations and on-shell methods are implemented to simplify computations in Yang-Mills and gravity. While on the other, conceptual developments are made in both these theories and their supersymmetric counterparts. Several works in the past have outlined the detailed computations of multi-photon/gluon/graviton scattering amplitudes. A corollary of these developments indicate that the UV behaviour of gravity is in some sense, better than Yang-Mills. Indeed, the BCFW recursion relations imply that the fall-off behaviour of gravity tree amplitudes at large values of complex momenta goes like 1/z^2, as opposed to Yang-Mills which goes like 1/z, where z is the complex momentum parameter. This is related to the fact that the group of diffeomorphisms is at play in gravity. However, the real surprise of the developments is that gravity is rather linked to Yang-Mills. This has been realized in the so called double copy relations where gravity amplitudes are a certain square of the Yang-Mills ones.

# Alongside these developments, there has been a resurgence of interest during the last few years in probing the ultraviolet structure of gravity, using on-shell techniques. In recent years, concrete results by Bern and collaborators have shown that the UV behaviour of gravity is much more subtle and interesting than was thought earlier. In particular, the two-loop divergence was analyzed and it was observed that its coefficient is sensitive to off-shell degrees of freedom in the theory. In their work, they have added non-dynamical three-forms to gravity and found that the coefficient of the two-loop divergence changes. Also, when pseudo-scalar fields are replaced by their duals, i.e anti-symmetric two-forms, the divergence once again changes. The Gorof and Sagnotti computation of the two-loop divergence, although gives a direct result for the coefficient, falls short of giving any understanding of the particular number. Thus, it is not very surprising that the coefficient depends on off-shell degrees of freedom. However, in a subsequent paper, what they found is that the coefficient of the renormalization scale dependence remains unchanged if one changes the off-shell contents of the theory. So, while the coefficient of the divergence does change, the renormalization scale dependence is not sensitive to the unphysical contents of the theory. Thus, it comes as a rather surprise that there is no direct relation between the coefficients of the divergent part of the two-loop amplitude and that of the renormalization scale dependence in gravity, unlike in conventional quantum field theories.

Alongside these developments, there has been a resurgence of interest during the last few years in probing the ultraviolet structure of gravity, using on-shell techniques. In recent years, concrete results by Bern and collaborators have shown that the UV behaviour of gravity is much more subtle and interesting than was thought earlier. In particular, the two-loop divergence was analyzed and it was observed that its coefficient is sensitive to off-shell degrees of freedom in the theory. In their work, they have added non-dynamical three-forms to gravity and found that the coefficient of the two-loop divergence changes. Also, when pseudo-scalar fields are replaced by their duals, i.e anti-symmetric two-forms, the divergence once again changes. The Gorof and Sagnotti computation of the two-loop divergence, although gives a direct result for the coefficient, falls short of giving any understanding of the particular number. Thus, it is not very surprising that the coefficient depends on off-shell degrees of freedom. However, in a subsequent paper, what they found is that the coefficient of the renormalization scale dependence remains unchanged if one changes the off-shell contents of the theory. So, while the coefficient of the divergence does change, the renormalization scale dependence is not sensitive to the unphysical contents of the theory. Thus, it comes as a rather surprise that there is no direct relation between the coefficients of the divergent part of the two-loop amplitude and that of the renormalization scale dependence in gravity, unlike in conventional quantum field theories.

# Using unitarity cuts, they have analyzed the divergent contribution and the associated renormalization scale dependence of the identical helicity four graviton scattering amplitude at two loops. It is quite clear from their computation that this coefficient gets a non-vanishing contribution from the two-particle cut, where the identical helicity one-loop four graviton amplitude appears on one side of the cut and a tree level four graviton amplitude appears on the other side. The three particle cuts do not contribute and vanish identically. Thus, overall the coefficient of the divergent part of the four graviton two-loop amplitude reduces to the same helicity one-loop amplitude.

Using unitarity cuts, they have analyzed the divergent contribution and the associated renormalization scale dependence of the identical helicity four graviton scattering amplitude at two loops. It is quite clear from their computation that this coefficient gets a non-vanishing contribution from the two-particle cut, where the identical helicity one-loop four graviton amplitude appears on one side of the cut and a tree level four graviton amplitude appears on the other side. The three particle cuts do not contribute and vanish identically. Thus, overall the coefficient of the divergent part of the four graviton two-loop amplitude reduces to the same helicity one-loop amplitude.

# Thus, the divergence of quantum gravity at two-loops is related to the non-vanishing of the all plus four point amplitude at one-loop. It is therefore very important to understand this non-vanishing and its origin.

Thus, the divergence of quantum gravity at two-loops is related to the non-vanishing of the all plus four point amplitude at one-loop. It is therefore very important to understand this non-vanishing and its origin.