henrik johansson cern march 26, 2013 buds workshop infn frascati henrik johansson cern march 26,...

Henrik JohanssonCERN

March 26, 2013

BUDS workshopINFN Frascati

1201.5366, 1207.6666, 1210.7709

Based on work with: Zvi Bern, John Joseph Carrasco, Lance Dixon, Michael Douglas, Radu Roiban,

Matt von Hippel

Towards Determining the UV Behavior of Maximal Supergravity

H. Johansson, Frascati 2013

SUGRA status in one slide

After 35 years of supergravity, we can only now make

very precise statements about the D=4 ultraviolet

structure.

No D=4 divergence of pure SG has been found to date.

Susy forbids 1,2 loop div., R2, R3 c.t. incompatible with

susy

Pure gravity 1-loop finite, 2-loop divergent Goroff &

Sagnotti

With matter: 1-loop divergent ‘t Hooft & Veltman

Naively susy allows 3-loop div. R4

N=8 SG and N=4 SG 3-loop finite!

N=8 SG: no divergence before 7 loops

7-loop div. in D=4 implies a 5-loop div.

in D=24/5 -- calculation in progress!

UFinite?

N=8 SG


Why is it interesting ?

If N=8 SG is perturbatively finite, why is it interesting ?

It better be finite for a good reason!

Hidden new symmetry, for example

Understanding the mechanism might open a host

of possibilities

Any indication of hidden structures yet?

Gravity is a double copy of gauge theories

Color-Kinematics: kinematics = Lie algebra

Constraints from E-M duality Kallosh,….

Hidden superconformal N=4 SUGRA ? Symmetry?

Gravity

Bern, Carrasco, HJ

Ferrara, Kallosh, Van Proeyen

Henrik Johansson

Gauge Theory Analogy

Gauge theory in D>4 have same problem as D=4

gravity

Non-renormalizable due to dimensionful coupling

However, D=5 SYM has a UV completion: (2,0) theory

in D=6

Is D=5 SYM perturbatively UV finite ? Douglas;

Lambert et al.

If yes, how does it work ?

If no, what do we need to add ?

Solitons, KK modes ? Douglas; Lambert et al.

Understanding D=5 SYM might (or might not)

give clues to how to understand D=4 gravity.

(2,0) theory ?

D=5 SYM


Review UV status N=8 SUGRA and N=4 SYM

4pt amplitudes and UV divergences3,4-loop N=8 SUGRA & N=4 SYM5-loop nonplanar SYM6-loop planar D=5 SYM

5pt amplitudes and UV divergences1,2,3-loop N=8 SUGRA & N=4 SYM

Current 5-loop progress

Conclusion

Outline


UV properties N =8 SG

• N =8 SG: conventional superspace power counting forbids L=1,2

divergences Deser, Kay, Stelle; Howe and Lindström; Green,

Schwarz, Brink; Howe, Stelle; Marcus, Sagnotti

• Three-loop divergence ruled out by calculation: Bern, Carrasco,

Dixon, HJ, Kosower, Roiban, (2007), Bern, Carrasco, Dixon, HJ, Roiban (2008)

• L<7 loop divergences ruled out by counterterm analysis,

using E7(7) symmetry and other methods, but a L=7

divergence is still possibleBeisert, Elvang, Freedman, Kiermaier, Morales, Stieberger; Björnsson, Green, Bossard, Howe, Stelle, Vanhove, Kallosh, Ramond, Lindström, Berkovits, Grisaru, Siegel, Russo, and more….

In D=4 dimensions:

In D>4 dimensions:Through four loops N =8 SG and N =4 SYM diverge in exactly the same dimension:

Marcus and Sagnotti; Bern, Dixon, Dunbar, Perelstein, Rozowsky;Bern, Carrasco, Dixon, HJ, Kosower, Roiban


UV divergence trendPlot of critical dimensions of N = 8 SUGRA and N = 4 SYM

Known bound for N = 4 Bern, Dixon, Dunbar, Rozowsky, Perelstein; Howe, Stelle

current trend for N = 8If N = 8 div. at L=7

calculations:

L = 7 lowest loop order for possible D = 4 divergenceBeisert, Elvang, Freedman, Kiermaier, Morales, Stieberger; Björnsson, Green, Bossard, Howe, Stelle, Vanhove Kallosh, Ramond, Lindström,

Berkovits, Grisaru, Siegel, Russo, and more….

1-2 loops: Green, Schwarz, Brink; Marcus and Sagnotti 3-5 loops: Bern, Carrasco, Dixon, HJ, Kosower, Roiban6 loops: Bern, Carrasco, Dixon, Douglas, HJ, von Hippel

Finite?

Divergent


UV divergence trendPlot of critical dimensions of N = 8 SUGRA and N = 4 SYM



calculations:




26/5 or 24/5 ?

Finite?

Divergent


N =8 Amplitude and Counter Term Structure

4pt amplitude form (any dimension)

divergence occurs in

Counter term

D = 8

D = 6

D = 7

D = 5.5

Looporder

1

2

3

4

If amplitude for L 4 has at least 8 derivatives then by dimensional analysis: no divergence before L = 7 !

D = 24/5 ?5 ??


N =8 Amplitude and Counter Term Structure

4pt amplitude form (any dimension)

divergence occurs in

Counter term

D = 8

D = 6

D = 7

D = 5.5

Looporder

1

2

3

4

If amplitude for L 5 has at least 10 derivatives then by dimensional analysis: no divergence before L = 8 !

D = 26/5 ?5 ??


Earliest appearance of N = 8 Divergence

3 loops Conventional superspace power counting Green, Schwarz, Brink (1982)Howe and Stelle (1989)Marcus and Sagnotti (1985)

5 loops Partial analysis of unitarity cuts; If N = 6 harmonic superspace exists; algebraic renormalisation

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

6 loops If N = 7 harmonic superspace exists Howe and Stelle (2003)

7 loops If N = 8 harmonic superspace exists; string theory U-duality analysis;lightcone gauge locality arguments;E7(7) analysis, unique 1/8 BPS candidate

Grisaru and Siegel (1982);Green, Russo, Vanhove; Kallosh; Beisert, Elvang, Freedman, Kiermaier, Morales, Stieberger; Bossard, Howe, Stelle, Vanhove

8 loops Explicit identification of potential susy invariant counterterm with full non-linear susy

Howe and Lindström; Kallosh (1981)

9 loops Assume Berkovits’ superstring non-renormalization theorems can be carried over to N = 8 supergravity

Green, Russo, Vanhove (2006)

Finite Identified cancellations in multiloop amplitudes; lightcone gauge locality and E7(7),inherited from hidden N=4 SC gravity

Bern, Dixon, Roiban (2006),Kallosh (2009–12), Ferrara, Kallosh, Van Proeyen (2012)


3, 4, 5, 6-Loop Amplitudes

3-loop N =8 SG & N =4 SYMColor-kinematics dual form: Bern, Carrasco, HJ

UV divergent in D=6:Bern, Carrasco, Dixon, HJ, Roiban

4-loops: 85 integral types

H. Johansson, QMUL 2013 15

4-loops N =4 SYM and N =8 SGBern, Carrasco, Dixon, HJ, Roiban 1201.5366

• 85 diagrams • Power counting manifest both N =4 and N =8• Both diverge in D=11/2

up to overall factor, divergence same as for N=4 SYM part


N =4 SYM 5-loop AmplitudeN=4 SYM important stepping stone to N=8 SG1207.6666 [hep-th]

Bern, Carrasco, HJ, Roiban

• 416 integral topologies:

• Used maximal cut method Bern, Carrasco, HJ, Kosower

• Maximal cuts: 410• Next-to-MC: 2473• N2MC: 7917• N3MC: 15156

Unitarity cuts done in D dimensions...integrated UV div. in D=26/5


N =4 SYM 5-loop UV divergence

Non-Planar UV divergence in D=26/5:

Double traces and single-trace NNLC finite in D=26/5,only single-trace LC and NLC are divergent

Vanish!

Bern, Carrasco, HJ, Roiban


Testing D=5 interceptPlot of critical dimensions of N = 8 SUGRA and N = 4 SYM



calculations:





6-Loop Planar D=5 SYM

•68 planar diagrams•Given by dual conformal invariance (up to integer 0,1,-1,2,-2... prefactors)

• Independently constructed by: Eden, Heslop, Korchemsky, Sokatchev;

Bourjaily,

DiRe, Shaikh, Spradlin, Volovich

Bern, Carrasco, Dixon, Douglas, HJ, von Hippel

6-Loop Planar D=5 SYM

The Parking Spot Escalation

our secret collaborator…

6-Loop D=5 SYM divergence

•Using integration by parts identities, div. simplifies to 3 integrals

•Numerical integration – modified version of FIESTA 1000 node cluster at Stony Brook

•Result: divergence is nonzero.

•What cancels this divergence ? Solitons/KK modes ? Douglas;

Lambert et al.

Bern, Carrasco, Dixon, Douglas, HJ, von Hippel

Divergence to all loop orders ?Bern, Carrasco, Dixon, Douglas, HJ, von Hippel

Intriguing pattern of UV divergence in critical dimension of maximal susy YM

• Accurate to <1%

• Why do the UV divergences follow this approximate curve?

• Is it asymptotically exact ?


5pt N=8 SUGRA calculations


C-K amplitudes at 1 loop

2

41

3Green, Schwarz,Brink (1982)

Duality-satisfying loop amplitudes:

2

41

3

N=4 SYM:

All-plus QCD:

N=4 SYM andAll-plus QCD:

1106.4711 [hep-th]Carrasco, HJ

SYM UV div in D=8:

Carrasco, HJ 1106.4711 [hep-th]

SU(8) violating SG UV div in D=8:

SG UV div in D=8:

1-loop 5-pts UV divergences

counterterms:


2-loop 5-pts N =4 SYM and N =8 SG

The 2-loop 5-point amplitude with Color-Kin. duality

N = 8 SG obtained from numerator double copies

Carrasco, HJ

1106.4711 [hep-th]


SYM UV div in D=7:

SG UV div in D=7:

Carrasco, HJ 1106.4711 [hep-th]

SU(8) violating SG UV div in D=8:


3-loop 5-point SYM and N =8 SG

(“ladder-like” diagrams)

Carrasco, HJ(to appear)

Non-planar D-dimensional amplitude with manifest color-kinematics duality

(N = 8 SG obtained from squaring the numerators)


3-loop 5-point SYM and N =8 SG

some “Mercedes-like” diagrams… Carrasco, HJ(to appear)


3-loop 5-point SYM and N =8 SGCarrasco, HJ(to appear)

…in total 42 diagrams.

For SYM the UV divergent diagrams (in D=6) are very simple:

(for SG the UV div. comes from the other diagrams as well)

N =8 SG 5-loop StatusWorking on reorganizing 5-loop N=4 SYM

Bern, Carrasco, HJ, Roiban (in progress)

• 416 + 336 = 752 integral topologies• BCJ: 2500 functional Jacobi eqns

Relaxing ansatz:• Non-manifest crossing symmetry• Allow for non-local numerators• Gauge variant numerators• Relax power counting• Allow for triangle diagrams• …

Once we have integrand, integration will take ~ 1 day:• No subdivergences in D=24/5 • No IR divergences since D>4, and absence of bubbles• No more difficult than IBP:s for N=4 SYM < 1 day• If needed, numerical integration ~ few days

SYM SG ?

3 loops, D=6:

4 loops, D=11/2:SYM SG

SYM SG


Fortune-telling from pattern

5 loops, D=26/5:

related to diagrams in thequartic Casimir

Summary

Explicit calculations in N = 8 SUGRA up to four loops show that the power counting exactly follows that of N = 4 SYM -- a finite theory

5 loop calculation in D=24/5 probes the potential 7-loop D=4 counterterm -- will provide critical input to the N = 8 question !

D=5 SYM have a 6-loop UV divergence, showing that the standard perturbative expansion misses some of the (2,0) theory contributions.

Color-Kinematics duality allows for gravity calculations for multiloop multipoint amplitudes -- greatly facilitating UV analysis in gravity.

Numbers in UV divergences of N =8 SUGRA and N =4 SYM coincide, suggesting a deeper connection between the theories

Stay tuned for the 5-loop SUGRA result…

THANK YOU!

henrik johansson cern march 26, 2013 buds workshop infn frascati henrik johansson cern march 26,...

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