Transcript 5 dimensional supergravity and the superconformal

On Recent Developments
in N=8 Supergravity
Renata Kallosh
Stanford university
EU RTN Varna, September 15, 2008
Outline
• Talks of Dixon, Cachazo, Green, RK at Strings
2008
• UV properties of gravity and supergravity:
• Light-cone counterterms versus the known
covariant ones, equivalence theorem
• New results on E 7(7) symmetry of N=8 SG,
Noether current
• Status of the triangle anomaly
• Comment to the next talk by K. Stelle
A possibility of UV finite N=8 SG
RK, arXiv:0808.2310
•
Recent work
•
RK The Effective Action of N=8 SG, arXiv:0711.2108
•
RK, Soroush, Explicit Action of E7,7 on N=8 SG Fields, arXiv:0802.4106
•
RK, T. Kugo, A footprint of E7,7 symmetry in tree diagrams of N=8 SG, work in progress
•
RK +… Counterterms in helicity formalism, work in progress
•
Most relevant work
•
Bern, Carrasco, Dixon, Johansson, Kosower, Roiban,
Three-Loop Superfiniteness of N=8 SG, hep-th/0702112, 0808.4112
•
Bianchi, Elvang, Freedman, Generating Tree Amplitudes in N=4 SYM and N = 8 SG
•
Drummond, Henn, Korchemsky and Sokatchev, Dual superconformal symmetry of
scattering amplitudes in N=4 SYM, arXiv:0807.1095
In N=8 SG
(h), (i) graphs have 3and 4-particle cuts!
Counterterms in gravity
On-shell counterterms in gravity should be generally
covariant, composed from contractions of Riemann
tensor
.
Terms containing Ricci tensor
and scalar
removable by nonlinear field redefinition in Einstein action
Since
has mass dimension 2, and the loop-counting parameter
GN = 1/MPl2 has mass dimension -2, every additional
requires another loop, by dimensional analysis
One-loop 
However,
is Gauss-Bonnet term, total derivative in four dimensions.
So pure gravity is UV finite at one loop (but not with matter)
‘t Hooft, Veltman (1974)
Pure gravity diverges at two loops
Relevant counterterm,
RK 1974, van Nieuwenhuizen, Wu, 1977
is nontrivial. By explicit Feynman diagram calculation
it appears with a nonzero coefficient at two loops
Goroff, Sagnotti (1986); van de Ven (1992)
Pure supergravity (N ≥ 1):
Divergences
deferred to at least three loops
cannot be supersymmetrized
– it produces a helicity amplitude (-+++)
forbidden by supersymmetry
Grisaru (1977); Tomboulis (1977)
However, at three loops, there is a perfectly acceptable
counterterm, for N=1 supergravity: The square of
the Bel-Robinson tensor, abbreviated
, plus (many)
other terms containing other fields in the N=8 multiplet.
Deser, Kay, Stelle (1977)
RK (1981); Howe, Stelle, Townsend (1981) : N=8 only linearized form
2007 N=8 SG is finite at the 3-loop order
A3=0
Bern, Carrasco, Dixon, Johansson, Kosower, Roiban
Maximal
supergravity
DeWit, Freedman (1977); Cremmer, Julia, Scherk (1978); Cremmer, Julia (1978,1979)
• Most supersymmetry allowed, for maximum
particle spin of 2
• Theory has 28 = 256 massless states.
• Multiplicity of states, vs. helicity, from coefficients in
binomial expansion of (x+y)8 – 8th row of Pascal’s triangle
SUSY charges
Qa, a=1,2,…,8
shift helicity by
1/2
• Ungauged theory, in flat spacetime
Basic Strategy
N=4
Super-Yang-Mills
Tree Amplitudes
KLT
Bern, Dixon, Dunbar, Perelstein
and Rozowsky (1998)
N =8
N =8
Unitarity
Supergravity
Supergravity
Loop Amplitudes
Tree Amplitudes
Divergences
• Kawai-Lewellen-Tye relations: sum of products of gauge
theory tree amplitudes gives gravity tree amplitudes.
• Unitarity method: efficient formalism for perturbatively
quantizing gauge and gravity theories. Loop amplitudes
from tree amplitudes.
Key features of this approach:
• Gravity calculations mapped into much simpler gauge
theory calculations.
• Only on-shell states appear.
Kawai-Lewellen-Tye relations
KLT, 1986
Derive from relation between
open & closed string amplitudes.
Low-energy limit gives N=8 supergravity amplitudes
as quadratic combinations of N=4 SYM amplitudes
consistent with product structure of Fock space,
,
Bern, Dixon, Dunbar and Kosower
Unitarity Method
Two-particle cut:
Three- particle cut:
Generalized
unitarity:
Apply decomposition of cut amplitudes in terms of product of tree
amplitudes.
UV divergent diagrams in unitarity cut method
Integral in D dimensions scales as
 Critical dimension Dc for log divergence (if no cancellations) obeys
N=8
BDDPR (1998)
N=4 SYM
Current Summary of Dixon et al
• Old power-counting formula from iterated 2-particle cuts predicted
N=8
• At 3 loops, new terms found from 3- and 4-particle cuts reduce the
overall degree of divergence, so that, not only is N=8 finite at 3 loops,
but
Dc = 6 at L=3, same as for N=4 SYM!
• Will the same happen at higher loops, so that the formula
N=4 SYM
continues to be obeyed by N=8 supergravity as well?
• If so, N=8 supergravity would represent a perturbatively finite,
pointlike theory of quantum gravity
• Not of direct phenomenological relevance, but could it point the way
to other, more relevant, finite theories?
3-loop manifestly supersymmetric linearized
counterterm ( known since 1981)
RK; Howe, Stelle, Townsend SUPERACTION
(Integral
over a submafifold of the full superspace)
• 2007 N=8 SG is finite at the 3-loop order
A3=0
Bern, Carrasco, Dixon, Johansson, Kosower, Roiban
L-loop counterterms
• Howe, Lindstrom, RK, 1981 : Starting from 8-loop order
infinite # of non-linear counterterms is available:
• 8-loop example with manifest E7,7 symmetry
• Analogous candidates for the L-loop divergences,
integrals over the full superspace
• What could be the reason for
AL=0
to provide all-loop order UV finite N=8 SG?
A possible explanation of the 3-loop
computation with UV finite answer
• A miracle!
• Howe, Stelle, 2002: if a harmonic superspace of the type of Galperin,
Ivanov, Kalitsyn, Ogievetsky,Sokatchev exists for N=8SG with manifest
>16 supersymmetries, the onset of divergences will start at L=5
• Our new explanation: if valid for the 3–loop order, is
also valid for all-loop orders.
• Main idea: to compare the candidate counterterms in 4+32 covariant onshell superspace with those in 4+16 light-cone superspace.
• Transform both results for the S-matrix into helicity formalism, easy to
compare
• We have found a mismatch between the covariant and light-cone UV
divergent answers for the S-matrix. If gauge anomalies are
absent, we may apply the S-matrix equivalence theorem for different
gauges. All-loop finiteness of N=8 SG follows.
Light-cone superspace
Brink, Lindgren, Nilsson, 1983
Brink, Kim, Ramond, 2007
• Light-cone chiral superfield for the CPT invariant N=8
supermultiplet
Only helicity states! Most suitable for helicity formalism
amplitudes!
Nobody ever constructed the light-cone superfield
counterterms.
N=8 SG divergences in helicity formalism
• 4-graviton
• Covariant superfields / helicity formalism
We were not able to match a complete answer for the
4-scalar UV divergent amplitude in the light-cone
gauge together with with the superfield partners,
including a complete SU(8) structure and a 4-graviton
amplitude deduced from the covariant superfield
counterterms.
• If the equivalence theorem for the S-matrix in lightcone versus covariant gauges is valid (if there are
no anomalies in the BRST symmetry of N=8 SG) we
consider this as a possible explanation of the recent
3-loop computation as well as a prediction for the
all-loop finitness of the perturbative theory.
Important note: the number of scalars before gauge-fixing a local SU(8) is
133, the number of independent parameters in the group element of E 7(7)
After gauge-fixing SU(8) the number of physical scalars is 70!
A hope that a better understanding of symmetries may
help to understand the situation with higher loops.
• N=8 local supergravity has a gauge SU(8) chiral
symmetry and a global E7,7 (R) symmetry
• This is a continuos symmetry in perturbative
supergravity (as different from string theory where it is
discrete)
conserved Noether current!
• It was poorly understood until recently (not acidentally
called “hidden”)
We have constructed the conserved Noether current
Based on R.K. and M. Soroush,
We have found an explicit, closed form exact in all orders
in k E7,7 transformations of all fields in the theory.
of the work with M. Soroush
What is known about N=8
anomalies?
SU(8) is chiral, anomalies possible
Gauge theory anomalies make QFT
inconsistent!
1984
Computed the contribution to SU(8) anomaly from fermions:
8 gravitini and 56 gaugini, found a non-vanishing answer
Later in 1985 N. Marcus computed the contribution from vectors
to triangle anomaly: Found an exact cancellation!
During Strings 2008 many discussions. The main point is: can we trust
this computation? Preliminary conclusion, most likely the claim about the
cancellation of SU(8) anomaly is correct.
• In 4d the consistent anomaly is associated
with the 6-form
(10)
T
SU(8) subgroup
X
Orthogonal to SU(8)
Implications of the one-loop SU(8)
anomaly cancellation
• E7(7) is not anomalous due to consistency
condition for anomalies
• Supersymmetry is not anomalous since
the algebra of two local supersymmetries
has a local SU(8) symmetry
• Maybe all this will help to understand the
status of UV divergences in N=8
supergravity.
Stelle et al, work in progress
• Proposal: take the known 3-loop or L-loop
covariant counterterm and convert it into the
light-cone one
This should give a candidate counterterm
responsible for the UV divergences in N=8 SG in
the light-cone gauges
We agree that as of now the light-cone
counterterms have not been constructed so far
as the explicit functions of the light-cone
superfields
The light-cone counterterm should agree
with the known form of divergent amplitudes in
covariant gauges
4-graviton divergent amplitude
4-vector divergent amplitude
2-vectors, 2-scalars divergent amplitude
4-scalar divergent amplitude
If the light-cone superfield counterterms (to be
constructed) will reproduce the known covariant
answer for the divergent amplitudes, we will have to
use other tools to study UV divergences in N=8 SG
All p+ and 1/p+ should disappear and all helicity brackets should appear!
• If they will not agree with expected structures,
this will prove
• A3=0 and eventually AL=0 and N=8 SG
finiteness
Back up slides
Compare spectra
28 = 256 massless states, ~ expansion of (x+y)8
SUSY
24 = 16 states
~ expansion
of (x+y)4
Light-cone superspace
Brink, Lindgren, Nilsson, 1983
Brink, Kim, Ramond, 2007
• Light-cone chiral superfield for the CPT invariant N=8
supermultiplet
New results: Simplest example of the Lorentz covariant 4-scalar UV
divergent amplitude
•Incomplete SU(8) structure, the second SU(8) structure is not covariant
N = 8 Supergravity from N = 4 Super-Yang-Mills
KLT only valid at tree level.
To answer questions of divergences in quantum gravity we
need loops.
Unitarity method provides a machinery for turning tree
amplitudes into loop amplitudes.
Apply KLT to unitarity cuts:
Unitarity cuts in gravity theories can be reexpressed as
sums of products of unitarity cuts in gauge theory.
Allows advances in gauge theory to be carried over to gravity.
N=8 supergravity in four dimensions during the last 25
years was believed to be UV divergent. During the last
few years studies of multi-particle amplitudes in QCD
were simplified using N=4 super Yang-Mills theory.
This, in turn, led to significant progress in computation
of amplitudes in N=8 supergravity. Some
spectacular cancellations of UV divergences were
discovered.
Is N=8 supergravity UV finite? If the answer is
"yes" what would this mean for Quantum Gravity?
UV finite or not?
• Quantum gravity is nonrenormalizable by power
counting, because the coupling, Newton’s constant,
GN = 1/MPl2 is dimensionful
• String theory cures the divergences of quantum
gravity by introducing a new length scale, the string
tension, at which particles are no longer pointlike.
• Is this necessary? Or could enough symmetry, e.g.
supersymmetry, allow a point particle theory of
quantum gravity to be perturbatively ultraviolet finite?
• If the latter is true, even if in a “toy model”, it would
have a big impact on how we think about quantum
gravity.
If N=8 massless QFT will be found UV finite one would compare
it with early days of massless Yang-Mills theory before one
knew how to add a Higgs mechanism and to describe the
realistic particle physics.