Transcript Superfiniteness of N=8 supergravity at three

Superfiniteness of N = 8
Supergravity at Three Loops and
Beyond
Julius Wess Memorial
November 6, 2008
Zvi Bern, UCLA
Based on following papers:
ZB, N.E.J. Bjerrum-Bohr, D.C. Dunbar, hep-th/0501137
ZB, L. Dixon , R. Roiban, hep-th/0611086
ZB, J.J. Carrasco, H. Johansson and D. Kosower, arXiv:0705.1864 [hep-th]
ZB, J.J. Carrasco, D. Forde, H. Ita and H. Johansson, arXiv:0707.1035 [hep-th]
ZB, J.J. Carrasco, H. Johansson, arXiv:0805.3993 [hep-ph]
ZB, J.J. Carrasco, L.J. Dixon, H. Johansson, R. Roiban , arXiv:0808.4112 [hep-th]
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Outline
Will present concrete evidence for non-trivial UV cancellations
in N = 8 supergravity, and perhaps UV finiteness.
• Review of conventional wisdom on UV divergences in quantum
gravity.
• Remarkable simplicity of gravity amplitudes.
• Calculational method – reduce gravity to gauge theory:
(a) Kawai-Lewellen-Tye tree-level relations.
(b) Modern unitarity method (instead of Feynman diagrams).
• All-loop arguments for UV finiteness of N = 8 supergravity.
• Explicit three-loop calculation and “superfiniteness”.
• Progress on four-loop calculation.
• Origin of cancellation -- generic to all gravity theories.
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N = 8 Supergravity
The most supersymmetry allowed for maximum
particle spin of 2 is N = 8. Eight times the susy of
N = 1 theory of Ferrara, Freedman and van Nieuwenhuizen
We consider the N = 8 theory of Cremmer and Julia.
256 massless states
Reasons to focus on this theory:
• With more susy expect better UV properties.
• High symmetry implies technical simplicity.
• Recently conjectured by Arkani-Hamed, Cachazo
and Kaplan to be “simplest” quantum field theory.
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Finiteness of N = 8 Supergravity?
We are interested in UV finiteness of N = 8
supergravity because it would imply a new symmetry
or non-trivial dynamical mechanism.
The discovery of either should have a fundamental
impact on our understanding of gravity.
• Non-perturbative issues and viable models of Nature
are not the goal for now.
• Here we only focus on order-by-order UV finiteness
and to identify the mechanism behind them.
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Power Counting at High Loop Orders
Dimensionful coupling
Gravity:
Gauge theory:
Extra powers of loop momenta in numerator
means integrals are badly behaved in the UV.
Non-renormalizable by power counting.
Much more sophisticated power counting in
supersymmetric theories but this is the basic idea.
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Divergences in Gravity
One loop:
Vanish on shell
vanishes by Gauss-Bonnet theorem
Pure gravity 1-loop finite, but not with matter
‘t Hooft, Veltman (1974)
Two loop: Pure gravity counterterm has non-zero coefficient:
Goroff, Sagnotti (1986); van de Ven (1992)
Any supergravity:
is not a valid supersymmetric counterterm.
Produces a helicity amplitude
forbidden by susy.
Grisaru (1977); Tomboulis (1977)
The first divergence in any supergravity theory
can be no earlier than three loops.
squared Bel-Robinson tensor expected counterterm
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Deser, Kay, Stelle (1977); Kaku, Townsend, van Nieuwenhuizen (1977), Ferrara, Zumino (1978)
Opinions from the 80’s
If certain patterns that emerge should persist in the higher
orders of perturbation theory, then … N = 8 supergravity
in four dimensions would have ultraviolet divergences
Green, Schwarz, Brink, (1982)
starting at three loops.
There are no miracles… It is therefore very likely that all
supergravity theories will diverge at three loops in four
dimensions. … The final word on these issues may have to await
Marcus, Sagnotti (1985)
further explicit calculations.
The idea that all D = 4 supergravity theories diverge at
3 loops has been the accepted wisdom for over 25 years
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Where are the N = 8 Divergences?
Depends on whom you ask and when you ask.
3 loops: Conventional superspace power counting.
Howe and Lindstrom (1981)
Green, Schwarz and Brink (1982)
Howe and Stelle (1989)
Marcus and Sagnotti (1985)
5 loops: Partial analysis of unitarity cuts. ZB, Dixon, Dunbar, Perelstein,
and Rozowsky (1998)
If harmonic superspace with N = 6 susy manifest exists
Howe and Stelle (2003)
6 loops: If harmonic superspace with N = 7 susy manifest exists
Howe and Stelle (2003)
7 loops: If a superspace with N = 8 susy manifest were to exist.
Grisaru and Siegel (1982)
8 loops: Explicit identification of potential susy invariant counterterm
Kallosh; Howe and Lindstrom (1981)
with full non-linear susy.
9 loops: Assume Berkovits’ superstring non-renormalization
theorems can be naively carried over to N = 8 supergravity.
Also need to extrapolate to higher loops. Green, Vanhove, Russo (2006)
Superspace gets here with additional speculations. Stelle (2006)
Note: none of these are based on demonstrating a divergence. They
are based on arguing susy protection runs out after some point. 8
Reasons to Reexamine This
1) The number of established UV divergences for any pure
supergravity theory in D = 4 is zero!
2) Discovery of novel cancellations at 1 loop –
the “no-triangle integral property”. ZB, Dixon, Perelstein, Rozowsky;
ZB, Bjerrum-Bohr, Dunbar; Bjerrum-Bohr, Dunbar, Ita, Perkins, Risager; Bjerrum-Bohr, Vanhove
Arkani-Hamed, Cachazo, Kaplan
3) Every explicit loop calculation to date finds N = 8 supergravity
has identical power counting as N = 4 super-Yang-Mills theory,
which is UV finite. Green, Schwarz and Brink; ZB, Dixon, Dunbar, Perelstein, Rozowsky;
Bjerrum-Bohr, Dunbar, Ita, Perkins Risager; ZB, Carrasco, Dixon, Johanson, Kosower, Roiban.
4) Interesting hint from string dualities. Chalmers; Green, Russo, Vanhove
– Dualities restrict form of effective action. May prevent
divergences from appearing in D = 4 supergravity, although
issues with decoupling of towers of massive states.
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Gravity Feynman Rules
Propagator in de Donder gauge:
Three vertex:
About 100 terms in three vertex
An infinite number of other messy vertices.
Naive conclusion: Gravity is a nasty mess.
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Gravity vs Gauge Theory
Consider the gravity Lagrangian
Infinite number of
complicated interactions
+…
Compare to Yang-Mills Lagrangian
Only three and four
point interactions
Gravity seems so much more complicated than gauge theory.
Multiloop calculations appear impossible.
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Standard Off-Shell Formalisms
In graduate school you learned that scattering amplitudes need
to be calculated using unphysical gauge dependent quantities:
off-shell Green functions
Standard machinery:
– Fadeev-Popov procedure for gauge fixing.
– Taylor-Slavnov Identities.
– BRST.
– Gauge fixed Feynman rules.
– Batalin-Fradkin-Vilkovisky quantization for gravity.
– Off-shell constrained superspaces.
For all this machinery relatively few calculations in quantum
gravity to check assertions on UV properties.
Explicit calculations from ‘t Hooft and Veltman;
Goroff and Sagnotti; van de Ven
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Why are Feynman diagrams clumsy for
high loop calculations?
• Vertices and propagators involve
gauge-dependent off-shell states.
Origin of the complexity.
unphysical states
propagate
• To get at root cause of the trouble we need to do things
differently.
• All steps should be in terms of gauge invariant
on-shell states.
On-shell formalism.
• Radical rewrite of quantum field theory needed.
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Simplicity of Gravity Amplitudes
On-shell three vertices contains all information:
gauge theory:
gravity:
“square” of
Yang-Mills
vertex.
Any gravity scattering amplitude constructible solely from
on-shell 3 vertex.
• BCFW on-shell recursion for tree amplitudes.
Britto, Cachazo, Feng and Witten; Brandhuber, Travaglini, Spence; Cachazo, Svrcek;
Benincasa, Boucher-Veronneau, Cachazo; Arkani-Hamed and Kaplan, Hall
• Unitarity method for loops.
ZB, Dixon, Dunbar and Kosower; ZB, Dixon, Kosower; Buchbinder and Cachazo;
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ZB, Carrasco, Johansson, Kosower; Cachzo and Skinner.
On-Shell Recursion
Britto, Cachazo, Feng and Witten
Consider tree amplitude under complex deformation
of the momenta.
complex momenta
A(z) is amplitude with shifted momenta
If
Poles in z come from
kinematic poles in
amplitude.
• Remarkably, gravity is as well behaved at
• We just need three vertex to start the recursion!
Sum over residues
gives the on-shell
recursion relation
as gauge theory.
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KLT Relations
A remarkable relation between gauge and gravity
amplitudes exist at tree level which we exploit.
At tree level Kawai, Lewellen and Tye derived a
relationship between closed and open string amplitudes.
In field theory limit, relationship is between gravity and gauge theory
Gravity
amplitude
where we have stripped all coupling constants
Full gauge theory
amplitude
Strongly suggests a unified description
of gravity and gauge theory.
Color stripped gauge
theory amplitude
Holds for any external states.
See review: gr-qc/0206071
Progress in gauge
theory can be imported
into gravity theories
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Gravity and Gauge Theory Amplitudes
Berends, Giele, Kuijf; ZB, De Freitas, Wong
gravity
gauge
theory
• Holds for all states appearing in a string theory.
• Holds for all states of N = 8 supergravity.
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Bern, Dixon, Dunbar and Kosower
Onwards to Loops: Unitarity Method
Two-particle cut:
Three-particle cut:
Generalized
unitarity:
Bern, Dixon and Kosower
Allows us to systematically
Construct loop amplitudes
from on-shell tree amplitudes.
Generalized cut interpreted as cut propagators not canceling.
A number of recent improvements to method
Britto, Cachazo, Feng; Buchbinber, Cachazo; ZB, Carrasco, Johansson, Kosower; Cachazo and Skinner;
Ossola, Papadopoulos, Pittau; Forde; Berger, ZB, Dixon, Forde, Kosower.
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Gravity vs Gauge Theory
Consider the gravity Lagrangian
Infinite number of irrelevant
(for S matrix) interactions!
+…
Only three-point
interactions needed
for on-shell recursion
Compare to Yang-Mills Lagrangian
Only three-point
interactions needed.
no
Gravity seems so much more complicated than gauge theory.
Multiloop calculations appear impossible.
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N = 8 Supergravity from N = 4 Super-Yang-Mills
Using unitarity method and KLT we express cuts of N = 8
supergravity amplitudes in terms of N = 4 amplitudes.
Key formula for N = 4 Yang-Mills two-particle cuts:
Key formula for N = 8 supergravity two-particle cuts:
Note recursive structure!
Generates all contributions 2
with s-channel cuts.
1
3
1
3
2
4 1
4
4
2
4
1
3 2
3 20
Iterated Two-Particle Cuts to All Loop Orders
ZB, Rozowsky, Yan (1997); ZB, Dixon, Dunbar, Perelstein, Rozowsky (1998)
constructible from
iterated 2 particle cuts
not constructible from
iterated 2 particle cuts
Rung rule for iterated two-particle cuts
N = 4 super-Yang-Mills
N = 8 supergravity
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Power Counting To All Loop Orders
From ’98 paper:
• Assumed rung-rule contributions give
the generic UV behavior.
• Assumed no cancellations with other
uncalculated terms.
Elementary power counting from rung rule gives
finiteness condition:
In D = 4 finite for L < 5.
L is number of loops.
counterterm expected in D = 4, for
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No-Triangle Property
ZB, Dixon, Perelstein, Rozowsky; ZB, Bjerrum-Bohr and Dunbar; Bjerrum-Bohr, Dunbar, Ita, Perkins,
Risager.
One-loop D = 4 theorem: Any one loop amplitude is a linear
combination of scalar box, triangle and bubble integrals
Passarino and Veltman, etc
with rational coefficients:
• In N = 4 Yang-Mills only box integrals appear. No
triangle integrals and no bubble integrals.
• The “no-triangle property” is the statement that
same holds in N = 8 supergravity.
Recent proofs by Bjerrum-Bohr and Vanhove; Arkani-Hamed, Cachazo and Kaplan
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L-Loop Observation
2
ZB, Dixon, Roiban
3
..
numerator factor
1
4
From 2 particle cut:
1 in N = 4 YM
numerator factor
From L-particle cut:
Using generalized unitarity and
no-triangle property all one-loop
subamplitudes should have power
counting of N = 4 Yang-Mills
Above numerator violates no-triangle property.
Too many powers of loop momentum in
one-loop subamplitude.
There must be additional cancellation with other contributions!
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Full Three-Loop Calculation
ZB, Carrasco, Dixon,
Johansson, Kosower, Roiban
Besides iterated 2 particle cuts need:
For second cut have:
reduces everything to
product of tree amplitudes
Use KLT
supergravity
super-Yang-Mills
N = 8 supergravity cuts are sums of products of
N = 4 super-Yang-Mills cuts
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Complete Three Loop Result
ZB, Carrasco, Dixon, Johansson, Kosower, Roiban; hep-th/0702112
ZB, Carrasco, Dixon, Johansson, Roiban arXiv:0808.4112 [hep-th]
N = 8 supergravity
amplitude manifestly has
diagram-by-diagram power
counting of N = 4 sYM!
By integrating this we
have demonstrated D = 6
divergence.
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Superfinite: UV cancellations beyond those needed for finiteness
Finiteness Conditions
Through L = 3 loops the correct finiteness condition is (L > 1):
“superfinite”
in D = 4
same as N = 4 super-Yang-Mills
bound is saturated at L = 3
not the weaker result from iterated two-particle cuts:
finite
in D = 4
for L = 3,4
(old prediction)
Beyond L = 3, as already explained, from special cuts we have
strong evidence that cancellations continue to all loop orders.
All one-loop subdiagrams
should have same UV
power-counting as N = 4
super-Yang-Mills theory.
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No known susy argument explains all-loop cancellations
N = 8 Four-Loop Calculation in Progress
Some N = 4 YM contributions:
ZB, Carrasco, Dixon, Johansson, Roiban
50 planar and non-planar diagrammatic topologies
N = 4 super-Yang-Mills case is complete.
N = 8 supergravity still in progress – so far looks good.
Four loops will teach us a lot:
1. Direct challenge to a potential superspace explanation:
existence of N = 6 superspace suggested by Stelle.
2. Study of cancellations will lead to better understanding.
3. Need 16 not 14 powers of loop momenta to come out
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of integrals to get power counting of N = 4 sYM
Origin of Cancellations?
There does not appear to be a supersymmetry
explanation for observed cancellations, especially as
the loop order continues to increase.
If it is not supersymmetry what might it be?
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Tree Cancellations in Pure Gravity
Unitarity method implies all loop cancellations come from tree
amplitudes. Can we find tree cancellations?
ZB, Carrasco, Forde, Ita, Johansson
You don’t need to look far: proof of BCFW tree-level on-shell
recursion relations in gravity relies on the existence such
Britto, Cachazo, Feng and Witten;
cancellations!
Susy not required
Bedford, Brandhuber, Spence and Travaglini;
Cachazo and Svrcek; Benincasa, Boucher-Veronneau and Cachazo;
Arkani-Hamed and Kaplan; Arkani-Hamed, Cachazo and Kaplan
Consider the shifted tree amplitude:
How does
?
behave as
Proof of BCFW recursion relies on
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Loop Cancellations in Pure Gravity
ZB, Carrasco, Forde, Ita, Johansson
Powerful new one-loop integration method due to Forde makes
it much easier to track the cancellations. Allows us to directly link
one-loop cancellations to tree-level cancellations.
Observation: Most of the one-loop cancellations
observed in N = 8 supergravity leading to “no-triangle
property” are already present even in non-supersymmetric
gravity. Susy cancellations are on top of these.
n
legs
Maximum powers of
loop momenta
Cancellation generic
to Einstein gravity
Cancellation from N = 8 susy
Key Proposal: This continues to higher loops, so that most of the
observed N = 8 multi-loop cancellations are not due to susy, but
in fact are generic to gravity theories! If N = 8 is UV finite
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suspect also N = 5, 6 is finite.
Schematic Illustration of Status
Same power count as N=4 super-Yang-Mills
UV behavior unknown
from feeding 2 and 3 loop
calculations into iterated cuts.
loops
No known susy
explanation for allloop cancellations.
no triangle
property.
explicit 2- and 3-loop
computations
terms
behavior unknown
4-loop calculation
in progress.
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Open Issues
• Will 4 loops be superfinite? Will be answered soon!
• Physical origin of cancellations not understood. Clear
link to high energy behavior of tree amplitude.
• Link to N = 4 super-Yang-Mills? So far link is mainly
a technical trick. But KLT relations surely much
deeper.
• Can we construct an all orders proof of finiteness?
• Can we get a handle on non-perturbative issues?
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Summary
• Modern unitarity method gives us means to calculate at high
loop orders. Allows us to unravel the UV structure of gravity.
• Exploit KLT relations at loop level. Map gravity to gauge theory.
• Observed novel cancellations in N = 8 supergravity
– No-triangle property implies cancellations strong enough
for finiteness to all loop orders, but in a limited class of terms.
– At four points three loops, established that cancellations are
complete and N = 8 supergravity has the same power counting
as N = 4 Yang-Mills theory.
– Key cancellations appear to be generic in gravity.
• Four-loop N = 8 calculation in progress.
N = 8 supergravity may well be the first example of a
unitary point-like perturbatively UV finite theory of
quantum gravity. Proving this remains a challenge.
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Extra transparancies
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Basic Strategy
N=4
Super-Yang-Mills
Tree Amplitudes
KLT
ZB, 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
ZB, Dixon, Dunbar, Kosower (1994)
from tree amplitudes.
Key features of this approach:
• Gravity calculations equivalent to two copies of much
simpler gauge-theory calculations.
• Only on-shell states appear.
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Feynman Diagrams for Gravity
Suppose we want to put an end to the speculations by explicitly
calculating to see what is true and what is false:
Suppose we wanted to check superspace claims with Feynman diagrams:
If we attack this directly get
terms in diagram. There is a reason
why this hasn’t been evaluated using
Feynman diagrams..
In 1998 we suggested that five loops is where the divergence is:
This single diagram has
terms
prior to evaluating any integrals.
More terms than atoms in your brain!
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N = 8 All-Orders Cancellations
5-point
1-loop
known
explicitly
must have cancellations between
planar and non-planar
Using generalized unitarity and no-triangle property
any one-loop subamplitude should have power counting of
N = 4 Yang-Mills
But contributions with bad overall power counting yet no
violation of no-triangle property might be possible.
One-loop
hexagon
OK
Total contribution is
worse than for N = 4
Yang-Mills.
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No-Triangle Property Comments
• NTP not directly a statement of improved UV behavior.
— Can have excellent UV properties, yet violate NTP.
— NTP can be satisfied, yet have bad UV scaling at
higher loops.
• Really just a technical statement on the type
of analytic functions that can appear at one loop.
• Used only to demonstrate cancellations of loop momenta
beyond those observed in 1998 paper, otherwise wrong
analytic structure.
ZB, Dixon, Roiban
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Method of Maximal Cuts
ZB, Carrasco, Johansson, Kosower
A refinement of unitarity method for constructing complete
higher-loop amplitudes is “Method of Maximal Cuts”.
Systematic construction in any theory.
To construct the amplitude we use cuts with maximum number
tree amplitudes
of on-shell propagators:
on-shell
Maximum number of
propagator placed
on-shell.
Then systematically release cut conditions to obtain contact
terms:
Fewer propagators
placed on-shell.
Related to leading singularity method.
Cachazo and Skinner; Cachazo; Cachazo, Spradlin, Volovich; Spradlin, Volovich, Wen
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ZB, Dixon, Dunbar and Kosower
Applications of Unitarity Method
1. Now the most popular method for pushing
state-of the art one-loop QCD for LHC physics.
Berger, ZB, Dixon, Febres Cordero, Forde, Kosower, Ita, Maitre;
Britto and Feng; Ossola, Papadopoulos, Pittau;
Ellis, Giele, Kunzst, Melnikov, Zanderighi
2. Planar N = 4 Super-Yang-Mills
amplitudes to all loop orders. Spectacular
verification by Alday and Maldacena at
four points using string theory.
Anastasiou, ZB, Dixon, Kosower; ZB, Dixon, Smirnov
3. Study of UV divergences in gravity.
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