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Perturbative Ultraviolet Calculations in Supergravity

Tristan Dennen (NBIA)Based on work with: Bjerrum-Bohr, Monteiro, O’Connell

Bern, Davies, Huang, A. V. Smirnov, V. A. Smirnov

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PART I: ULTRAVIOLET DIVERGENCES VIA DOUBLE COPY

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Naively, two derivative coupling in gravity makes it badly ultraviolet divergent

Non-renormalizable by power counting

But: extra symmetry enforces extra cancellations Supersymmetry to the rescue? Added benefit: makes calculations simpler

UV Divergences in Supergravity

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Naturally, the theory with the most symmetry is the best bet for ultraviolet finiteness

N = 8 supergravity

Half-maximal supergravity has also attracted attention recently N = 4 supergravity

This is the primary focus of my talk

UV Divergences in Supergravity

Cremmer, Julia (1978)

Das (1977);Cremmer, Scherk, Ferrara (1978)

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1970’s-1980’s: Supersymmetry delays UV divergences until three loops in all 4D pure supergravity theories

Expected counterterm is R4

In N=8, SUSY and duality symmetry rule out couterterms until 7 loops

Expected counterterm is D8R4

See talk from Johansson I will discuss N = 4

supergravity in this talk See also talks from Bossard,

Carrasco, Vanhove

Arguments about FinitenessGrisaru; Tomboulis; Deser, Kay, Stelle;

Ferrara, Zumino; Green, Schwarz, Brink; Howe, Stelle; Marcus, Sagnotti; etc.

Bern, Dixon, Dunbar; Perelstein, Rozowsky (1998); Howe and Stelle (2003, 2009);

Grisaru and Siegel (1982); Howe, Stelle and Bossard (2009);

Vanhove; Bjornsson, Green (2010); Kiermaier, Elvang, Freedman (2010);

Ramond, Kallosh (2010); Beisert et al (2010); Kallosh; Howe and Lindström (1981);

Green, Russo, Vanhove (2006)Bern, Carrasco, Dixon, Johansson, Roiban (2010)

Beisert, Elvang, Freedman, Kiermaier, Morales, Stieberger (2010)

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Analogs of E7(7) for lower supersymmetry

Can help UV divergences in these theories Still have candidate counterterms at L = N - 1

(1/N BPS) Nice analysis for N = 8 counterterms

Duality Symmetries

N=8: E7(7)

N=6: SO(12)N=5: SU(5,1)N=4: SU(4) x SU(1,1)

Bossard, Howe, Stelle, Vanhove (2010)

Beisert, Elvang, Freedman, Kiermaier, Morales, Stieberger (2010)

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N=8 Supergravity Fantastic progress See Johansson’s talk

N=4 Supergravity Four points, L = 3

Unexpected cancellation of R4 counterterm Counterterm appears valid under all known symmetries

See Bossard’s talk for latest developments Four points, L = 2, D = 5

Valid non-BPS counterterm D2R4 does not appear

Recent Field Theory Calculations

Bern, Carrasco, Dixon, Johansson, Kosower, Roiban (2007)Bern, Carrasco, Dixon, Johansson, Roiban (2009)

Carrasco, Johansson (2011)

Bern, Davies, Dennen, Huang

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But how are the calculations done?

1. Find a representation of SYM that satisfies color-kinematics duality.

2. Construct the integrand for a gravity amplitude using the double copy method.

3. Extract the ultraviolet divergences from the integrals.

Recent Field Theory Calculations

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Color-kinematics duality provides a construction of gravity amplitudes from knowledge of Yang-Mills amplitudes

In general, Yang-Mills amplitudes can be written as a sum over trivalent graphs

Color factors Kinematic factors

Duality rearranges the amplitude so color and kinematics satisfy the same identities (Jacobi)

Color-Kinematics Duality

Bern, Carrasco, Johansson (2008)

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Four Feynman diagrams Color factors based on a Lie algebra

Color factors satisfy Jacobi identity: Numerator factors satisfy similar identity:

Color and kinematics satisfy the same identity!

Example: Four Gluons

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At higher multiplicity, rearrangement is nontrivial But still possible

Claim: We can always find a rearrangement so color and kinematics satisfy the same Jacobi constraint equations.

Five gluons and more

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How are the calculations done?

1. Find a representation of SYM that satisfies color-kinematics duality.

2. Construct the integrand for a gravity amplitude using the double copy method.

3. Extract the ultraviolet divergences from the integrals.

Recent Field Theory Calculations

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Once numerators are in color-dual form, “square” to construct a gravity amplitude

Gravity numerators are a double copy of gauge theory ones!

Proved using BCFW on-shell recursion

The two copies of gauge theory don’t have to be the same theory.

Gravity from Double Copy

Bern, Carrasco, Johansson (2008)

Bern, Dennen, Huang, Kiermaier (2010)

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The two copies of gauge theory don’t have to be the same theory.

Supergravity states are a tensor product of Yang-Mills states

And more! Relatively compact expressions for gravity amplitudes

Inherited from Yang-Mills simplicity In a number of interesting theories

Gravity from Double Copy

N = 8 sugra: (N = 4 SYM) x (N = 4 SYM)N = 6 sugra: (N = 4 SYM) x (N = 2 SYM)N = 4 sugra: (N = 4 SYM) x (N = 0 SYM)N = 0 sugra: (N = 0 SYM) x (N = 0 SYM)

Damgaard, Huang, Sondergaard, Zhang (2012)Carrasco, Chiodaroli, Gunaydin, Roiban (2013)

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What we really want is multiloop gravity amplitudes Color-kinematics duality at loop level

Consistent loop labeling between three diagrams Non-trivial to find duality-satisfying sets of numerators

Double copy gives gravity

Loop Level

Bern, Carrasco, Johansson (2010)

Just replace c with n

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How are the calculations done?

1. Find a representation of SYM that satisfies color-kinematics duality.

2. Construct the integrand for a gravity amplitude using the double copy method.

3. Extract the ultraviolet divergences from the integrals.

Recent Field Theory Calculations

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PART II: OBTAINING COLOR-DUAL NUMERATORS

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Known Color-Dual NumeratorsN = 4 SYM 1 Loop 2 Loops 3 Loops 4 Loops L Loops

4 point trivial trivial ansatz ansatz

5 point construction ansatz ansatz

6 point construction

7 point construction

n point construction

Pure YM 1 Loop 2 Loops L Loops

4 point ansatz (All-plus)

n point (All-plus and single minus)

Boels, Isermann, Monteiro, O’Connell (2013) Bern, Davies, Dennen, Huang, Nohle (2013)

Bern, Carrasco, Johansson (2010)Carrasco, Johansson (2011)

Bern, Carrasco, Dixon, Johansson, Roiban (2012)Yuan (2012)

Bjerrum-Bohr, Dennen, Monteiro, O’Connell (2013)

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Strategy: Write down an ansatz for a master numerator All possible terms

Subject to power counting assumptions Symmetries of the graph

Take unitarity cuts of the ansatz and match against the known amplitude

Gives a set of constraints Very powerful, but relies on having a good ansatz

Not always possible to write down all possible terms

Numerators by Ansatz

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Jacobi identity moves one leg past another

Strategy: Keep moving leg 1 around the loop Obtain a finite difference equation for pentagon

One-Loop Construction

Depends on the leg we cycle around

Yuan (2012)Bjerrum-Bohr, Dennen, Monteiro, O’Connell (2013)

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Assume we know the box numerators (more on this in a minute) They are independent of loop momentum

Solution to difference equation is linear in loop momentum

Get projections of linear term

4 different projections give us the entire linear term in terms of box numerators

One-Loop Construction

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More generally: given all scalar parts, we can rebuild everything else

Two options for scalar parts: Match unitarity cuts Guess boxes (and hexagons,

etc.), and use reflection identities

Boxes from self-dual YM Good for MHV amplitudes

One-Loop Construction

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Reduce integrals to pentagons Partial fraction propagators Purely algebraic

Doesn’t care about loop momentum in numerators

Key assumption: loop momentum drops out after reducing the amplitude to pentagons

Then reduce pentagons to boxes and match unitarity cuts

Obtain equations like:

Integrand Reduction

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hexagon, via Jacobi This is a linear equation for

hexagon numerators Do the same for all cuts and all

permutations Get 719 equations for 6!=720

unknown numerators The final unknown is fixed by the

condition that loop momentum drops out

Integrand Reduction

Box cut coefficient Factors from integrand reduction 25 M

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Seven point amplitudes work the same way

7! = 5040 heptagon scalar numerators 5019 equations 19 extra constraints from our assumptions

2 degrees of freedom in color-dual form Works for any R sector

Reconstruct the rest of the numerators

Conjecture: This procedure will work for arbitrary multiplicity at 1 loop

Higher Multiplicity

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Some of this might extend to higher loops

Exploit global properties of the system of Jacobi identities

Express loop-dependence in terms of scalar parts of master numerators

Two loops looks promising

Integrand reduction might not extend so easily

Hybrid approach? Use ansatz techniques for scalar parts, then reconstruct loop dependence, match to cuts

Prospects for Higher Loops

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How are the calculations done?

1. Find a representation of SYM that satisfies color-kinematics duality.

2. Construct the integrand for a gravity amplitude using the double copy method.

3. Extract the ultraviolet divergences from the integrals.

Recent Field Theory Calculations

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PART III: EXTRACTING ULTRAVIOLET DIVERGENCES

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Bern, Davies, Dennen, Huang (2012)

N = 4 SYM copy Use BCJ representation

Pure YM copy Use Feynman diagrams

in Feynman gauge Only one copy needs to

satisfy the duality Double copy gives N = 4

supergravity Power counting gives

linear divergence Valid counterterm under

all known symmetries

Three Loop Construction

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Numerators satisfy BCJ duality

Factor of pulls out of every graph

Graphs with triangle subdiagrams have vanishing numerators

N = 4 SYM Copy

Bern, Carrasco, Johansson (2010)

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Pros and cons of Feynman diagrams😃 Straightforward to write down😃 Analysis is relatively easy to pipeline😃 D-dimensional😡 Lots of diagrams😡 Time and memory constraints

Many of the N=4 SYM BCJ numerators vanish If one numerator vanishes, the other is irrelevant

Power counting for divergences – can throw away most terms very quickly

Pure YM Copy

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To extract ultraviolet divergences from integrals:1. Series expand the integrand and select the logarithmic terms2. Reduce all the tensors in the integrand3. Regulate infrared divergences (uniform mass)4. Subtract subdivergences5. Evaluate vacuum integrals

Ultraviolet Analysis

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Counterterms are polynomial in external kinematics Count up the degree of the

polynomial using dimension operator

Derivatives reduce the dimension of the integral by at least 1. Apply again to reduce further… all the way down to

logarithmic. Now can drop dependence on external momenta.

1. Series Expansion

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Simpler: equivalent to series expansion of the integrand

Terms less than logarithmic have no divergence Terms more than logarithmic vanish as the IR regulator is set to

zero Left with logarithmically divergent terms

1. Series Expansion

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The tensor integral knows nothing about external vectors

Must be proportional to metric tensor

Contract both sides with metric to get

Generalizes to arbitrary rank – Need rank 8 for 3 and 4 loops

2. Tensor Reduction

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Integrals have infrared divergences (in 4 dimensions) One strategy: Use dimensional regulator for both IR and UV

Subdivergences will cancel automatically, becausethere are no 1- or 2-loop divergences in this theory

But, integrals will generally start at Very difficult to do analytically

Another strategy: Uniform mass regulator for IR Integrals will start at -- much easier! Regulator dependence only enters through subleading terms

Now we have a sensible integral!

3. Infrared Regulator

Marcus, Sagnotti (1984)Vladimirov

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What about higher loops? At three loops, the integrals have logarithmic subdivergences

Integrals are Mass regulator can enter subleading terms!

Recursively remove all contributions from divergent subintegrals Regulator dependence drops out

4. Subdivergences

Marcus, Sagnotti (1984)Vladimirov

Regulator dependent

Regulator independent Reparametrize subintegral

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Get about 600 vacuum integrals containing UV information

Evaluation: MB: Mellin Barnes integration FIESTA: Sector decomposition FIRE: Integral reduction using integration by parts identities

5. Vacuum Integrals

cancelledpropagator doubled

propagator

Czakon

A.V. Smirnov & Tentyukov

A.V. Smirnov

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Series expand the integrand and select the logarithmic terms Reduce all the tensors in the integrand Regulate infrared divergences Subtract subdivergences Evaluate vacuum integrals

The sum of all 12 graphs is finite!

Three Loop Result

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If R4 counterterm is allowed by supersymmetry, why is it not present?

R4 nonrenormalization from heterotic string

Violates Noether-Gaillard-Zumino current conservation

Hidden superconformal symmetry

Existence of off-shell superspace formalism

Different proposals lead to different expectations for four loops

Opinions on the Result

Tourkine, Vanhove (2012)

Kallosh (2012)

Bossard, Howe, Stelle (2012)

Kallosh, Ferrara, Van Proeyen (2012)

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Allowed counterterm D2R4, non-BPS Will it diverge? If yes: first example of a divergence in any pure supergravity

theory in 4D If no: hidden symmetry enforcing cancellations?

Same approach as three loops.

N = 4 SYM numerators: 82 nonvanishing (comp. 12)

Pure YM numerators: ~30000 Feynman diagrams (comp. ~1000)

Integrals are generally quadratically divergent Requires a deeper series expansion

Four Loop Setup

Bern, Davies, Dennen, A. V. Smirnov, V. A. Smirnov (in progress)

Bern, Carrasco, Dixon, Johansson, Roiban (2012)

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Four loops is significantly more complicated than three loops. (unsurprisingly)

Can we avoid subtracting subdivergences? Ultimately, there are no subdivergences in the theory so far.

Any subdivergences that appear must be a result of our IR regulator.

Observation: in all cases we’ve looked at, subdivergences cancel manifestly with a uniform mass regulator.

2 loops: D = 4,5,6 3 loops: D = 4 Consistent with four-loop QCD beta function calculations

Analysis of subintegrals for 4 loops suggests the same should be true here.

Vast Simplifications?

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Series expand the integrand and select the logarithmic terms Reduce all the tensors in the integrand Regulate infrared divergences Subtract subdivergences Evaluate vacuum integrals

Calculation nearly done Overall cancellation of and Now working on showing the necessary cancellation of

Stay tuned!

Four Loop Status

Czakon (2004)

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Expectations for ultraviolet divergences in supergravity

BCJ color-kinematics duality

Construction of gravity integrand via double copy How to obtain color-dual numerators

Progress with one-loop construction

UV analysis of gravity integrals 3 loop, N = 4 supergravity Status of four loop

Summary

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