# Asymptotic safety and quantum gravity amplitudes

Jan H. Kwapisz<EMAIL_ADDRESS>Krzysztof A. Meissner  Faculty of
Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland
<EMAIL_ADDRESS>
###### Abstract

Application of the Weinberg’s conditions of asymptotic safety to amplitudes
and not to couplings of the effective action can help to uniquely define the
running in the UV and avoid some of the asymptotic safety program problems. As
it turns out the requirement for a quantum gravity to be defined by S-matrix,
that obeys the Weinbergs asymptotic safety criteria is a very restrictive one.
The idea is illustrated by the 4-graviton amplitude in string theory and its
symmetries.

###### keywords:

Asymptotic safety , renormalisation group , string theory , graviton
interaction
_PACS_ : 04.60.-m 11.25.-w 14.80.Bn

††journal: Nuclear Physics B

##  1 Introduction.

In physics there are four known fundamental interactions: strong, weak,
electromagnetic and gravitational. The first three can be quantised by means
of perturbative quantum field theory approach and these theories are
predictive since a finite number of experiments is sufficient to fix all of
the couplings in the action. On the other hand if one quantises gravity in a
similar way, one gets a theory which requires infinite number of experiments
to specify its predictions. Therefore one has to find a new way to quantise
gravity to make it predictive again.

In order to do so one can either relax (some of the) assumptions of general
relativity and/or quantum theory in order to make it predictive or propose a
new quantisation scheme (non-perturbative one). In the first category there
are for example higher curvature gravity theories [1, 2, 3, 4, 5] or Hořava-
Lifschitz gravity [6]. There are also discrete spacetime approaches such as
loop quantum gravity [7, 8] or causal dynamical triangulations [9, 10]. In
string theory one abandons the whole concept of quantum field theory, see for
example [11, 12] and [13, 14, 15].

On the other hand one can accept that gravity indeed can possess an infinite
number of couplings, but the theory is nevertheless predictive as long as the
values of (almost) all couplings can be calculated theoretically and they are
functions of only finite number of experimental values. This is for example
the case of asymptotic safety in quantum gravity (ASQG) [16, 17, 18], where
one requires that all of the couplings reach a (non)-trivial fixed point in
the UV, which fixes almost all the couplings. Using the Functional
Renormalisation Group (FRG) [19, 20] techniques such a tentative fixed point
was found [21, 22, 23, 24] (previously conjectured by other considerations
[25, 26, 27, 28, 29]). Furthermore this approach gained much attention in the
recent years due to more compelling evidence for the fixed point [30, 31] and
rich phenomenological behaviour, such as prediction of Higgs mass in Standard
Model [32] and in various extensions [33, 34], prediction of masses of various
quarks [35, 36] and properties of dark matter [37, 38].

Despite these successes, recently the whole programme was challenged by some
issues [39] of fundamental importance, see also [40] and [41]. Namely if one
considers the quantum gravity in the effective field theory (EFT) approach
below Planck scale, then due to lack of _crossing_ and _universality_
properties in the amplitudes [39, 40] one cannot introduce a single running of
the Newtonian coupling $G_{N}(\mu)$ to describe their energy dependence. It
also happens in the pion model, which is the EFT for quantum chromodynamics,
where the pion amplitudes doesn’t posses the crossing symmetry [42, 43] in the
Mandelstam variables (not the crossing LSZ symmetry: $s->u$, which is the one
discussed in the article and required for unitarity). On the other hand the
QCD amplitudes does have in the massless (high energy) limit. In this letter
we discuss this problem and propose a possible cure. The idea is to use a
specific amplitude related to the coupling we are interested in, try to find a
formula valid at all scales, and only then ask a question about the asymptotic
safety. Such an approach requires a reformulation of the Weinberg’s criteria
of asymptotic safety. In what follows we concentrate on the 4-graviton
amplitude which is directly related to the Newtonian coupling. To have a well
defined amplitude for all energies without encountering the usual difficulties
we have chosen to use the known form of this amplitude calculated within the
framework of string theory, which is finite also at the loop level, to show
that the concrete definition proposed in this letter can have an explicit
realization and can solve also other issues posed in [39] (we would like to
emphasize that we don’t claim that the string theory is the theory of quantum
gravity, but we use it as an illustration since it is the only framework
existing at present to consistently calculate the graviton amplitudes).

##  2 Running of the couplings.

In order to understand the criticism expressed in [39] we review the notion of
running couplings in the quantum field theories (QFT). In experimental physics
one investigates the properties of matter by colliding the particles prepared
in a given state (momenta etc), measure the outcome of the collisions. and
compare it with the theoretical amplitudes calculated in the QFT framework.
These amplitudes are encoded in the $S$-matrix, which is an array of
probabilities between all the possible non-interacting in-states in the far
past and all the possible non-interacting out-states in the far future. Tree
level values of amplitudes get (infinitely) corrected by loop diagrams.
Therefore, to have correct definitions of in/out states, such that they are
not-interacting, one redefines (renormalizes) the splitting of the Hamiltonian
into free and interaction parts, see [18, 44]. Then all the infinities are
absent in the amplitudes. The renormalisation procedure introduces another
source of scale (energy) dependence into the theory on top of the classical
one by introducing counter-terms to absorb the infinities and defining the
theory in terms of renormalized couplings $g_{R}=g_{B}-\delta g$, where
$g_{B}$ are the bare couplings.

In principle one should write all possible counter-terms respecting the
symmetries of the theory, however if a classical Lagrangian has only couplings
of non-negative dimension $\Delta_{i}\geq 0$, then by power counting arguments
one can show that only the $n$-point interactions with $n\leq 4$ can be
divergent. In that case the counterterms will have the same structure as the
original terms of the bare lagrangian.

This fact has two further consequences. First, for dimensionless couplings (we
neglect the masses in our discussion due to near conformal structure of the
Standard Model [45, 46, 47, 48, 49, 50, 33]) the corrections are proportional
to the couplings itself and the sign of the correction will not be channel
dependent (_crossing_). Furthermore the 4-point diagrams (interactions with 4
external legs) exhibit the crossing symmetry, i.e. permutation symmetry in the
Mandelstam variables $s,t,u$ [51] (_universality_). These two facts mean that
the divergent and finite parts of the amplitudes can be described by the same,
universal behaviour [39, 40, 52, 53, 54]. Then the entire momenta dependence
of the theory is contained in a couple of energy dependent functions which in
case of perturbatively renormalisable theories are exactly couplings in the
Lagrangian. Then the scale dependence of the coupling is given by the
renormalisation group equations:

$\mu\frac{\partial g_{i}}{\partial\mu}=\beta_{i}.$ (1)

Obviously this change of couplings with scale is not physical in itself and
the coupling has to be tied to observable quantities at each scale. The
smaller the coupling the better the process is described by the tree-level
diagrams only. Standard Model possesses only marginal couplings and one
relevant (Higgs mass) and can be quantised using the quantum field theory loop
expansion due to the couplings lying in the perturbative range.

##  3 Quantising General Relativity.

On the other hand the action of General Relativity is given by

$S=\frac{1}{8\pi G_{N}}\int d^{4}x\sqrt{-g}R,$ (2)

where the dimension of $[R]=2$, then $[G_{N}]=-2$. By the power-counting
argument the quantum gravity is the theory with infinite number of
counterterms (if there are no cancellations as in supergravity [55, 56, 57]),
however one cannot make infinite number of experiments to determine the
predictions of the theory. The lack of cancellations was confirmed by the
explicit Feynman diagrams calculations showing that General Relativity
requires counterterms not proportional to $R$ at one-loop level when coupled
to matter[58, 59] (which disappear in the matter free case by the equations of
motions) and at two loops in the matter free case [60, 61].

##  4 Asymptotic safety programme.

Gravity treated as quantum field theory has infinite number of (counter)-terms
in Lagrangian. However this doesn’t necessary mean that the theory is un-
predictive at arbitrary scales. After all, the quantum effective action of
$\phi^{4}$ theory also possesses an infinite number of terms but the value of
all the couplings can be calculated from two quantities $\lambda_{ph}$ and
$m_{ph}$, closely related to observations. Steven Weinberg [16, 17] noticed
that quantum gravity (QG) could be predictive if all the irrelevant
($\Theta_{i}<0$) couplings could be determined in terms of the relevant and
marginal ones. The so called asymptotic safety mechanism could be operative if
all the couplings of the theory reached a fixed point (such that the right
hand side of Eq. 1 vanishes) and then the theory will not diverge when moving
with the scale towards UV [16, 17, 62] and get closer to conformal invariance.
If all the couplings of a theory reach zero at the fixed point then we call
the theory asymptotically free, otherwise when $g_{i}^{\ast}\neq 0$ we call it
asymptotically safe. In particular if the gravity is asymptotically safe, then
the dimensionless $\tilde{G}_{N}=G_{N}\mu^{2}$ should have a fixed point.

In the early days the running of $G_{N}$ was found in $2$-dimensions and its
vicinity [25, 26, 27, 28] by means of the $2+\epsilon$ dimensional expansion.
Similar claims were also made with the $1/N$ expansion approach [29]. However
only after applying the Wilsonian inspired exact Wetterich-Morris-Ellwanger
equation [19, 20, 63] to tackle this problem the approximate fixed point was
found in seminal Reuter article [21, 23, 24] and then explored in various
truncations.

Applying this equation to quantum gravity, even though it is exact, raises
many problems of both fundamental and technical type, see [39, 31, 64, 65,
41], but one could hope that even problematic, these issues can be solved in
the future. However, recently in [39, 40] the whole notion of the running
gravitational coupling was challenged. The argument is the following. The
running coupling is merely a description of a collective behaviour of
amplitudes and should be reflected in them. Since $[G_{N}]=-2$ the corrections
to the amplitudes should be proportional to $Gq^{2}$, where $q^{2}$ is one of
the Mandelstam variables [66, 40], see also [67, 68]. Since in different
reactions $q^{2}$ can be positive or negative, then depending on the reaction
the gravitational correction can go into different directions. Therefore the
notion of the universal running cannot be introduced for the Newtonian
coupling in four dimensions.

##  5 Running in terms of amplitudes.

The asymptotic safety criteria are very appealing theoretically for testing
the consistency of the theory. For this purpose we can rephrase them as
follows: the theory should approach the scale invariant regime in the UV,
requires finite number of experimental input to be fully predictive and all
the observables should be finite. Furthermore Weinberg conjectures that
approaching the UV limit should be universal and channel independent, hence
(in terms of gravity) we only consider $G_{N}(\mu)$ rather than
$G_{N}(p_{i})$.

The amplitudes are the main observables in QFT (by their direct connection to
the cross sections) therefore we propose to identify the “running” with the
change of the amplitude with scale and not of any particular term in the
(effective) lagrangian.

The importance of the amplitudes approach rather than the effective action
treatment can be clearly seen in string theory. In string theory one can
calculate the low-energy effective action using the the non-linear sigma model
approach [69, 70], but the string theory amplitudes cannot be reproduced by
any of this effective actions since finiteness of string amplitudes is tied to
the modular invariance that cannot be reproduced by means of a lagrangian of
any conventional QFT.

One should note that there may be other symmetries of the amplitude (and of
the corresponding effective action) like $O(d,d)$ symmetry in string theory
[71, 72] that tie together many terms in the lagrangian – for example a
4-graviton amplitude involves many higher curvature terms that are uniquely
fixed by the symmetry [73, 74].

Motivated by the original Weinberg notion of asymptotic safety we propose
three criteria in terms of UV behaviour of amplitudes. First, specifying
finite number of amplitudes is sufficient to derive all of the theory
predictions. Second, all the amplitudes should possess scale invariant regime
for very large energies (near the cutoff) [75]:

$A(p_{1},p_{2},\ldots)=\lambda^{-d}A(\lambda p_{1},\lambda p_{2},\ldots),$ (3)

where $d$ is the dimension of $A$. In terms of quantum field theory this
corresponds to the massless limit. Third, we demand the $E\to+\infty$ limit to
be unambiguous for the ’running’ part of amplitude $A^{\prime}$, i.e. the
amplitude without the polarizations’ contractions. In particular for the 4
identical particle elastic scattering this means that

$\lim_{s\to\infty,t-\textrm{fixed}}A^{\prime}(s,t,u)=\lim_{t\to\infty,s-\textrm{fixed}}A^{\prime}(s,t,u).$
(4)

We call this asymptotic crossing symmetry. In particular first and third
conditions ensure finiteness and UV-fundamentality of the theory and the
second condition ensures the scale invariance of the UV-phase. Furthermore if
we do not assume the asymptotic crossing symmetry then there is no universal
notion of running governing the high energy behaviour of the theory. It is no
longer described by any universal “running” and there is no notion of fixed
points of this running, which was the main objection of [39]. In the context
of quantum gravity one should require that the theory has a graviton in its
spectrum and the four graviton amplitude is not trivial. Another reason for
the asymptotic crossing symmetry is that then the low energy limit is
described by a single coupling $G_{N}$, which we then match to high energy
physics. The amplitude defined in that way satisfies both _crossing_ and
_universality_ properties by construction. Given these criteria we identify
the “running” in quantum gravity with the change of ’scalar’ part (multiplying
the polarization tensors kinematics part) of the 4-graviton amplitude in the
full theory. Finally let us note that in this picture there is no notion of
problematic Wick rotation because we do not have to calculate the amplitude in
the Euclidian signature and then analytically continue into Lorentz one.
Finally unitarity is manifest by the optical theorem and correct structure of
the poles. As we discuss below string theory in the gravitational sector
satisfies our criteria.

##  6 String theory amplitudes example.

Actually the crossing symmetry was the main motivation for the Veneziano
amplitude [76], which gave birth to the string theory [77]. In Type II
superstring theory the Virasoro-Shapiro amplitude (closed string analog for
Veneziano amplitude for four gravitons) takes the following form [11]

$A(p_{1},p_{2},p_{3},p_{4})=2g_{D}^{2}K_{cl}C(s,t,u),$ (5)

where $C(s,t,u)$ is given by

$C(s,t,u)=-\pi\frac{\Gamma(-s/8)\Gamma(-t/8)\Gamma(-u/8)}{\Gamma\left(1+\frac{s}{8}\right)\Gamma\left(1+\frac{t}{8}\right)\Gamma\left(1+\frac{u}{8}\right)}$
(6)

and $K_{cl}$ is a lengthy kinematic (cross-symmetric) and polarization factor
polynomial in $s,t,u$. If we vary $s$ and keep $t$ fixed, then this amplitudes
has infinite number of simple poles [78] on a real line at $s=8n$, with $n\geq
0$. This can be understood as summing over infinite number of particles with
growing masses in the $s$-channel. The similar conclusion can be drawn if we
keep $s$ fixed and vary $t$. This property is called duality. Hence we cannot
take large energy limit ($s\to\infty$, $s/t$ fixed) along the real line so we
shift $s$ slightly in the imaginary direction. Ultimately, the poles are
anyway shifted off the real axis making the $s\to\infty$ limit well defined
because massive states are unstable in the interacting theory, hence their
amplitudes possess non-trivial imaginary part. As a result we get the
exponential fall-off:

$C(s,t,u)\varpropto\exp\left(-\frac{1}{4}(s\log s+t\log t+u\log u)\right),$
(7)

satisfying our criteria. Our conditions are also satisfied beyond the tree
level amplitude, since this behavior is also present for the resumed loop
amplitude in this limit [79, 80, 81] (string theory preserves generally the
crossing symmetry at the loop level [82]). On the other hand in the low-energy
limit the Veneziano amplitude can be matched with tree-level graviton
amplitude, since $C(s,t,u)\sim\frac{1}{stu}$ with
$\kappa=\frac{1}{2}\alpha^{\prime}g^{2}$ for heterotic strings [11], where
$g\approx 1$ and $\alpha^{\prime}\approx M_{P}^{2}$. Furthermore the second
term in the expansion of the Virasoro-Shapiro amplitude has the opposite sign
as the first term (in the expansion of the beta function) which is a hint for
asymptotically safe anti-screening behaviour as an intermediate step between
classical gravity and string theory [83] but the issue deserves further
investigation. In the future work we are planning to check whether the bounds
discussed in [84] are satisfied and hence whether GR EFT can be UV completed
by string theory.

It should be emphasized that the fact that the amplitude (7) asymptotically
vanishes does not say anything about the individual contributions from the
interaction vertices in the lagrangian, in particular whether the Newton’s
constant $G_{N}$ asymptotically vanishes or not. However, this asymptotic
behavior points to an important and far reaching change of perspective with
respect to the usual approach where only the Newton’s constant and its running
are taken into account.

##  7 Remarks

Before coming to the conclusions let us make a few remarks. Our approach can
be complementary to the other studies of asymptotic safety phenomena such as
vertex expansion [85, 86, 87, 88] of the $\Gamma_{k}$ or causal dynamical
triangulations [9, 10], yet the crossing symmetry is not manifest there. It
would be interesting to compare those in the future. On the other hand the
recent proposals of redefining the FRG approach to quantum gravity [89, 90]
puts the whole momentum dependence into the the form factors for higher
curvature terms in the lagrangian. Actually the amplitudes for scalar
particles scattering including the graviton propagators were recently
calculated and they also satisfy our criteria [91, 92]. However the
coefficients of $\Lambda$ and $R$ doesn’t run in any sense in that approach,
since the form factors are attached to the $C^{2}$ and $R^{2}$ terms in the
effective action. This can make the physical running of higher curvature
couplings essential to the Weinberg proposal, rather than the Einstein-Hilbert
truncation see [1, 2, 3, 4, 5]. Furthermore due to lack of crossing symmetry
in the crucial 4-graviton amplitude in this approach [5, 92] one cannot also
rely on the RG-improvement and identify the ”running” scale of the Wetterich
equation with some physical scale, which is usually done. These recent
developments suggests that maybe some multi-scale generalisation of Wetterich
equation could be in order to capture the dependence of the amplitudes, but it
seems a long way ahead.

In this letter we focus only on the amplitudes with gravitons on external
lines, since running of the Newton’s constant is the crucial issue in the
whole AS scenario. If we include matter there will be obviously other
amplitudes with gravitons. Yet with a proper treatment of the polarisation
contractions (on top of the kinematical part) the whole amplitude can still
satisfy the _crossing_ and _universality_ properties. The argument behind this
conjecture relies on the double copy ’equivalence’ (for $N>4$ supergravities)
gravity and Yang-Mills squared [93] if we properly identify gauge with
polarisation contractions and gauge with gravitational couplings. In the
string theory the situation is much clearer due to the KLT relations [94] and
the linear relations between amplitudes conjectured by Gross and Mende [95,
96, 97] (for proof see [98] and references therein) where all the closed
strings amplitudes stripped from kinematics share the Gamma function behaviour
and posses an exponential fall-off [99, 100] at tree level. This actually
allow for a single coupling running interpretation in context of all the
interaction in the string theory (at least at tree level).

On the other hand it is conjectured that the high energy gravitons scatterings
are dominated by the productions of black holes, see [101, 102]. Yet it raises
a question whether such an amplitude can be tested and not whether it exists
at all, which is a slightly different question than the one we are answering.
Actually this a question for a non-perturbative definition of quantum gravity,
a programme which didn’t succeed ultimately in any approach. The closest to
such a formulation is an AdS/CFT conjecture [103] which is however incomplete
from the theoretical point of view [104, 105] and works only in AdS spacetime.
Yet due to the anti-screening character of the asymptotic safety one can hope
that these issues can be resolved in our language and we leave that for future
work, see also a recent resolution of that problem proposed in [106], and the
study of this problem in Ho$\rm\check{r}$ava gravity [107].

##  8 Conclusions.

Four lessons can be learnt from our investigation. First, we emphasized the
importance of crossing symmetry in the Weinberg’s asymptotic safety proposal
(with a concrete example of string theory graviton amplitudes). Second, we
propose to associate the running with the amplitudes and not with couplings in
the lagrangian (which is actually done in the conventional QFTs since there
both notions coincide) - they have very definite UV-behaviour avoiding the
ambiguities in the running emphasized in [39]. It seems that if one wants to
apply Weinbergs criteria and requires physical, single-scale running of the
4-graviton amplitude such that in low energy it is tied to $G_{N}$ is a very
restrictive criteria. This fact points to an important conclusion that the
running of the Newton’s constant is not sufficient to predict or model the UV
behavior of gravity and the other degrees of freedom may play an important or
even decisive role.

Thirdly our criteria shows the importance of matter-gravitational interactions
and their study in terms of quantum gravity. For example the amplitudes for
pure gravity lack the crossing symmetry at one-loop [66] and two-loop level
[108]. On the other hand the $N=8$ supergravity amplitudes have the crossing
symmetry [66] also at higher loops [109], which is by the way another example
satisfying our criteria. Furthermore the result [66] still has to be
unitarised and it was shown [110] that under certain assumptions in order to
preserve causality one requires an infinite tower of higher spins states, such
that they form the Veneziano amplitude and the EFT result in UV completed by
string theory. This might be the hint that Einstein-Hilbert truncation is only
the low energy approximation and hence $L\sim\frac{1}{G_{N}}(R+\ldots)$ and
only after taking the dotted part into account one can introduce the
meaningful “running ”of $G_{N}$ as the non-linear sigma models hints. It would
be extremely interesting if such Lagrangian could be obtained without the
supersymmetric degrees of freedom and without a tachyon. Let us also note that
the amplitudes of linear-sigma models satisfy our claims in the non-
gravitational context [84] and have a non-vanishing value in the UV. Also
using the amplitudes picture one can check the effects of quantum gravity
interactions on matter and understand the asymptotic safety conditions on the
particle properties in the Standard Model and beyond [32, 33, 34, 35, 36, 37,
38, 111, 87] (one could also check whether this conditions agree or disagree
with the stringy inspired swampland conditions [112, 113, 114, 115, 116]).

Finally the $N\geq 5$ supergravities and string theory have a vanishing
conformal and chiral anomalies [117, 118, 119], which allows for a conformal
UV limit. On the other hand the effects of this anomalies can be disastrous on
the Early Universe cosmology [120]. Hence it seems that there is a connection
between asymptotic crossing symmetry (and asymptotic safety), finiteness of
the amplitudes in 4 dimensions [121, 122, 123] and absence of conformal
anomaly. It would be extremely interesting to check whether amplitudes
calculated in the framework of the proposed $E_{10}$ symmetry [124, 125, 126,
127] also satisfy the criteria proposed in this paper.

## Acknowledgements

We thank P. Chankowski, J. Donoghue, S. Giddings, S. Melville, H. Nicolai and
M. Reichert for valuable discussions and for comments on the manuscript.
K.A.M. and J.H.K. were partially supported by the Polish National Science
Center grant DEC-2017/25/B/ST2/00165. J.H.K. was supported by the Polish
National Science Centre grant 2018/29/N/ST2/01743.

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