R -Counterterm and E7(7) -Symmetry in Maximal Supergravity

4 E R -counterterm and 7(7)-symmetry in maximal supergravity

Johannes Brodel¨ in collaboration with Lance Dixon

Max-Planck-Institut f¨ur Gravitationsphysik (AEI), Potsdam and Leibniz Universitat¨ Hannover

Hidden Structures in Field theory Amplitudes Copenhagen, August 14th 2009 Counterterms in Theories of Gravity

Pure gravity • is nonrenormalizable by power counting in four dimensions, because the coupling 1 (Newtons constant) is dimensionful GNewton = 2 . MPlanck

• does not suffer from divergences at the one loop level (Rµν = 0 onshell, µνρσ µν 2 't Hooft RµνρσR − 4Rµν R + R → total divergence) [Veltman] • exhibits a two loop divergence related to the possible counterterm λρ στ µν Goroff van de Rµν Rλρ Rστ [Sagnotti][ Ven ]

N = 8 supergravity • one loop: no divergence in pure gravity, so no divergence in supergravity λρ στ µν • possible candidate Rµν Rλρ Rστ from pure gravity produces amplitudes Grisaru Tomboulis forbidden by supersymmetry: h− + + + · · · +i [ 1977 ][ 1977 ] • admissible counterterm R4 exists, which is the supersymmetric extension of

4 µ1ν1...µ4ν4 ρ1σ1...ρ4σ4 R = t t Rµ1ν1ρ1σ1 Rµ2ν2ρ2σ2 Rµ3ν3ρ3σ3 Rµ4ν4ρ4σ4

Deser, Kay Howe, Stelle Kallosh [ Stelle ][ Townsend ][ 1981 ]

R4 E August 14th, 2009, Johannes Brodel:¨ -counterterm and 7(7)-symmetry in maximal supergravity 2 / 13 N = 8 supergravity

DeWit Cremmer, Julia DeWit Maximal supersymmetric gravity theory in four dimensions . [Freedman][ Scherk ][Nicolai] • (selected) symmetries A • supersymmetry with operators QA, Q˜ , A = 1 . . . 8 SU(8) 63 compact generators • E7(7) → E7(7) ( SU(8) 70 noncompact generators

• Particle spectrum: 1 8 28 56 70 56 28 8 1 3 1 1 3 -2 - 2 -1 - 2 0 2 1 2 2 B− F − B− Λ− X ΛABC BAB F A B SPACE. A AB ABC ABCD + + + +

R4 E August 14th, 2009, Johannes Brodel:¨ -counterterm and 7(7)-symmetry in maximal supergravity 3 / 13 N = 8 supergravity

• Particle spectrum: unitary gauge 1 8 28 56 70 56 28 8 1 3 1 1 3 -2 - 2 -1 - 2 0 2 1 2 2 B− F − B− Λ− X ΛABC BAB F A B SPACE. A AB ABC ABCD + + + +

• noncompact part of E7(7) connects different, physically indistinguishable vacua of the theory, which are labelled by the expectation values of the 70 scalars. (analogy to the Goldstone situation)

• in order to maintain unitary gauge, any noncompact E7-rotation has to be accom- panied by a compensating SU(8) rotation

R4 E August 14th, 2009, Johannes Brodel:¨ -counterterm and 7(7)-symmetry in maximal supergravity 3 / 13 R4 counterterm

R: pure quantity, R: supersymmetrized quantity • suggested in 1981 as a counterterm for N = 8 supergravity, transforms covariantly Kallosh Deser, Kay Howe, Stelle under supersymmetry and SU(8) [ ][ Stelle ][ Townsend ] • no explicit expression for supersymmetric extension of R4 known, use of nonlin- earized SUSY might be necessary for realization • correponds to a divergence at three-loops, far beyond the results of recent consid- Bossard, Howe Green, Russo erations [ Stelle ][ Vanhove ]

Bern, Dixon, Carrasco • miraculously, the coefﬁcient determined by renormalization vanishes [ Johansson, Kosower ]

E7(7) • any symmetry not yet accounted for ? Candidate: SU(8) • usually: symmetry forbids the occurrence of the counterterm in the Lagrangian E7(7) Here: SU(8) is not a symmetry of the lagrangian, but of the equations of motion.

R4 E August 14th, 2009, Johannes Brodel:¨ -counterterm and 7(7)-symmetry in maximal supergravity 4 / 13 E7(7) Accessing the hidden SU(8) -symmetry

E7(7) Commutation relations between SU(8) generators T and SU(8) -generators X:

[T, T ] ∼ T, [X, T ] ∼ X, [X,X] ∼ T

Double soft scalar limit:

n 1 +2 p · (p − p ) M (X ,X ,...) −−−−−→ i 2 1 R(η )M (3, 4,...) n+2 1 2 → i n p1,p2 0 2 pi · (p + p ) i=3 1 2 X R ∼ [X1,X2] Arkani-Hamed • relation proven for pure N = 8 supergravity [Cachazo, Kaplan]

• as expected: compensating SU(8)-rotation after noncompact E7(7)-rotation

Main idea: test the validity of the above relation for amplitudes derived from an action of the form R + cR4 R •SPACE.How to obtain the corresponding amplitudes? Direct calculation in N = 8 supergravity seems impossible.

R4 E August 14th, 2009, Johannes Brodel:¨ -counterterm and 7(7)-symmetry in maximal supergravity 5 / 13 Incorporating the R4-term

String theory provides access to R4-corrected amplitudes: . . . • closed string theory reproduces N = 8 supergravity after compactiﬁcation to 4 dimensions ′ Green and taking the point particle limit α = 0. [Schwarz] closed • ﬁrst correction to the effective action: α′3R4 Gross . [Witten] • expanding string amplitudes around α′ = 0 and keeping terms up to O(α′3) generates the same α′ →0 amplitudes as would be obtained from an action of the form R + α′3R4 R + α′3R4 + · · ·

R4 E August 14th, 2009, Johannes Brodel:¨ -counterterm and 7(7)-symmetry in maximal supergravity 6 / 13 However ...

... not any string theory amplitude is ready available or easy to calculate.

1 6 p · (p − p ) M (XABCD,X , 3, 4, 5, 6) −−−−−→ i 2 1 RD (η )ME (3, 4, 5, 6) 6 1 2 ABCE → E i 4 D p1,p2 0 2 pi · (p + p ) i=3 1 2 X D I1...I4 I1I2I3I4E where R [X ,X ] = ε × η ∂ iD . I5...I8 E I5I6I7I8D iE η Constraints • amplitude has to be six-point at least (rhs nontrivial) • amplitude needs NMHV signature: h−−− + ++i (rhs at least MHV)

• scalars have to agree in three indices (from E7(7) commutation relations / proof of double soft scalar equality) • remaining particles need to have two open indices

Additional tools required to calculate the six-point amplitude: • supersymmetric Ward identities (SWI) and Kawai-Lewellen-Tye relations (KLT) • Strategy: available open string amplitudes −−→ certain class of open string ampli- SWI tudes −−→ closed string amplitude amplitude KLT

R4 E August 14th, 2009, Johannes Brodel:¨ -counterterm and 7(7)-symmetry in maximal supergravity 7 / 13 Supersymmetric Ward identities (SWI) I

0 = h[Q, β1β2 · · · βn]i = h0| [Q, β1β2 · · · βn] |0i = hβ1β2 · · · [Q, βi] · · · βni i=1 X N = 1 supersymmetry with particles g+, g−, λ+, λ− and supersymmetry relations:

Q(η), g+(p) = [pη] λ+(p), Q(η),λ+(p) = −hpηig+(p), − − − − Q(η), g (p) = hpηiλ (p), Q(η),λ (p) = − [pη] g (p),

− − − − − − MHV: h[Q(η), g g λ+ g+ g+ g+]i =h1ηihλ g λ+ g+ g+ g+i + h2ηihg λ λ+ g+ g+ g+i − − −h3ηihg g g+ g+ g+ g+i − − − − − − − − − NMHV: h[Q(η), g g g λ+ g+ g+]i =h1ηihλ g g λ+ g+ g+i + h2ηihg λ g λ+ g+ g+i − − − − − − + h3ηihg g λ λ+ g+ g+i−h4ηihg g g g+ g+ g+i

µ αα˙ αα˙ αα˙ µ αα˙ α α˙ Conventions: pµp = det(p ), p = pµ(σ ) , p = π π˜ , Q|0i = 0 α α˙ ′ ′ A hiji = πi πjα, [ij] =π ˜iα˙ π˜j , sij = α hiji[ij] = 2α ki · kj Q(η) = Q ηA

R4 E August 14th, 2009, Johannes Brodel:¨ -counterterm and 7(7)-symmetry in maximal supergravity 8 / 13 Supersymmetric Ward identities (SWI) I

0 = h[Q, β1β2 · · · βn]i = h0| [Q, β1β2 · · · βn] |0i = hβ1β2 · · · [Q, βi] · · · βni i=1 X N = 1 supersymmetry with particles g+, g−, λ+, λ− and supersymmetry relations:

Q(η), g+(p) = [pη] λ+(p), Q(η),λ+(p) = −hpηig+(p), − − − − Q(η), g (p) = hpηiλ (p), Q(η),λ (p) = − [pη] g (p),

− − − − − − MHV: h[Q(η), g g λ+ g+ g+ g+]i =h11ihλ g λ+ g+ g+ g+i+h21ihg λ λ+ g+ g+ g+i − − −h31ihg g g+ g+ g+ g+i =0 − − − − − − − − − NMHV: h[Q(η), g g g λ+ g+ g+]i =h11ihλ g g λ+ g+ g+i+h2ηihg λ g λ+ g+ g+i − − − − − − +h3ηihg g λ λ+g+g+i−h4ηihg g g g+g+g+i =0

R4 E August 14th, 2009, Johannes Brodel:¨ -counterterm and 7(7)-symmetry in maximal supergravity 8 / 13 Supersymmetric Ward identities (SWI) II

hg− g− g− g+ g+ g+i

hλ− g− g− λ+ g+ g+i ··· hg− λ− g− g+ λ+ g+i ··· hg− g− λ− g+ g+ λ+i

hλ− λ− g− λ+ λ+ g+i ··· hλ− g− λ− λ+ g+ λ+i ··· hg− λ− λ− g+ λ+ λ+i

hλ− λ− λ− λ+ λ+ λ+i • linear system of equations: 20 amplitudes and 30 = 6 + 18 + 6 source terms equations → rank 18 • two amplitudes have to be known, in order to determine all others • imagine the same consideration for N = 8 supergravity ... algebraically not feasi- ble. Second technique necessary: KLT

R4 E August 14th, 2009, Johannes Brodel:¨ -counterterm and 7(7)-symmetry in maximal supergravity 9 / 13 Kawai-Lewellen-Tye-relation (KLT) 2

closed open

” “ ′ ′ α →0 α →0

+ α′3R4 + · · · +α′2F 4 + α′3 · · ·

1 8 2856705628 8 1 14641 3 1 1 3 1 1 -2 - 2 -1 - 2 0 2 1 2 2 -1 - 2 0 2 1

Kawai, Tye [N = 8] = [N = 4] ⊗ [N = 4] [ Levellen ]

1234 5+ − + − − 4+ + − + − hX X1235 F F4 B B i ⇔ hg λ g λ4 g g iL + − 5+ − + − SPACE. ×hg λ5 λ g g g iR

Bianchi, Elvang • unique decomposition of any N = 8 state into two N = 4 states [ Freedman ]

R4 E August 14th, 2009, Johannes Brodel:¨ -counterterm and 7(7)-symmetry in maximal supergravity 10 / 13 Matching orders of α′

• String theory relation:

M5(1, 2, 3, 4, 5) = iπ sin(πs12) sin(πs34) A5(1, 2, 3, 4, 5) A5(2, 1, 4, 3, 5) + P(2, 3)

• α′-expansion

(α′πh12i[12])3 sin(πs ) = α′πh12i[12] − + · · · 12 3! ′2 ′3 SYM O(α ) O(α ) A5(12345) = A5 (12345) + A5 (12345) + A5 (12345) + · · · ,

′ • any particular order in α of M5 can be determined by multiplying all suitable com- binations of orders of the right hand side:

′ ′ ′ O(α 2) 3 SYM O(α 2) O(α 2) SYM M5 (1, 2, 3, 4, 5) = iπ s12s34 A5 (12345) A5 (21435) + A5 (12345) A5 (21435) h π2 2 2 SYM SYM − 6 (s12 + s34)A5 (12345) A5 (21435) + P(2, 3)=0 . i

R4 E August 14th, 2009, Johannes Brodel:¨ -counterterm and 7(7)-symmetry in maximal supergravity 11 / 13 How does everything combine?

b • try to ﬁnd a N = 8 amplitude sat-

isfying all constraints, whose KLT- b b b b b b b b b decomposition consists of N = 1 amplitudes only b b b b b b b b b 1234 5+ − + − hX X1235 F F4 B B i = b N = 1 − 4+ + − + − SPAKLT hg λ g λ4 g g iL

h b b b b b b b b b + − 5+ − + − SPACE ×hg λ5 λ g g g iR i b b b b b b b b b

N = 4

R4 E August 14th, 2009, Johannes Brodel:¨ -counterterm and 7(7)-symmetry in maximal supergravity 12 / 13 How does everything combine?

b • try to ﬁnd a N = 8 amplitude sat-

N = 4

R4 E August 14th, 2009, Johannes Brodel:¨ -counterterm and 7(7)-symmetry in maximal supergravity 12 / 13 How does everything combine?

b • try to ﬁnd a N = 8 amplitude sat-

N = 4

R4 E August 14th, 2009, Johannes Brodel:¨ -counterterm and 7(7)-symmetry in maximal supergravity 12 / 13 How does everything combine?

b • try to ﬁnd a N = 8 amplitude sat-

h b b b b b b b b b + − 5+ − + − SPACE ×hg λ5 λ g g g iR i • consider the available open string b b b b b b b b b calculations hg− g− g− g+ g+ g+i, b hφ−,φ−,φ−,φ+,φ+,φ+i, − − − + + + Stieberger hφ ,φ ,λ ,λ ,φ ,φ i [ Taylor ] • determine in a ﬁrst step the amplitude hλ− λ− λ− λ+ λ+ λ+i • Having now two amplitudes from the ﬁrst N = 1 diamond on the disposal, calculate all permutations of amplitudes of type hλλggggi N = 4 • apply KLT-relations in order to determine the desired N = 8 result R4 E August 14th, 2009, Johannes Brodel:¨ -counterterm and 7(7)-symmetry in maximal supergravity 12 / 13 Result and Outlook

6 1234 5+ − + − 1 pi · (p2 − p1) 4 5+ − + − hX X1235 F F B B i −−−−−→ R (ηi)hF F B B i 4 ′ → 5 4 ′ ˛O(α 3) p1,p2 0 2 pi · (p1 + p2) ˛O(α 3) ˛ Xi=3 ˛ ˛ ˛ 1 p · (p − p ) − − p · (p − p ) − − 3 2 1 hF 4+F B+ B i − 4 2 1 hF 5+F B+ B i = 4 ′ 5 ′ 2 » p3 · (p1 + p2) ˛O(α 3) p4 · (p1 + p2) ˛O(α 3)– ˛ ˛ ˛ ˛

• Double soft scalar limit relation does indeed hold for amplitudes in an α′3R4- corrected N = 8 supergravity → R4-term not restricted by onshell symmetry E7(7)/SU(8) • vanishing coefﬁcient still awaits explanation

R4 E August 14th, 2009, Johannes Brodel:¨ -counterterm and 7(7)-symmetry in maximal supergravity 13 / 13 Result and Outlook

Thanks!

R4 E August 14th, 2009, Johannes Brodel:¨ -counterterm and 7(7)-symmetry in maximal supergravity 13 / 13