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Unraveling conformal gravity amplitudesbased on [1806.05124] with Henrik Johansson & Fei Teng

Gustav Mogullgustav.mogull@physics.uu.se

Department of Physics and Astronomy, Uppsala University, Sweden

September 8, 2018

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Introduction

In this talk we’ll discuss various conformal (super)gravities.

Simplest example is the 4D Weyl theory:

S = − 1

κ2

∫d4x√|g | (Wµνρσ)2

Coupling κ is dimensionless; the Weyl tensor Wµνρσ is

Wµνρσ = Rµνρσ + gν[ρRσ]µ − gµ[ρRσ]ν +1

3gµ[ρgσ]νR,

(Wµνρσ)2 = (Rµνρσ)2 − 2(Rµν)2 +1

3R2,

Action has local scale invariance:

gµν → e−2λ(x)gµν =⇒ Wµνρσ → e−2λ(x)Wµνρσ

But... conformal (super)gravities are non-unitary! So why are weinterested?

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Conformal supergravities

Main motivation is power-counting renormalizability.

Classified as minimal or non-minimal [Fradkin & Tseytlin ‘85]:

e−1L = − 1

κ2

[F(φ)(W+

µνρσ)2 + h.c.]

+ · · ·

W±µνρσ =1

2

(Wµνρσ ± W̃µνρσ

), W̃µνρσ =

i

2

√|g |εµνκλW κλ

ρσ,

Several reasons to revisit these theories:1 Twistor string: Conformal supergravity amplitudes arise in the

twistor string [Witten ‘03]. Can a field theory description be found?2 Double copy: (DF )2 theory [Johansson, Nohle ‘17] gives bosonic,

heterotic string amplitudes by double copy [Azevedo, Chiodaroli,Johansson, Schlotterer ‘18] using Z-theory [Carrasco, Mafra,Schlotterer ‘16]. Conformal (super)gravity amplitudes arise fromdouble copy with (S)YM.

3 U(1) anomaly: R2 operators play a role in cancelling N = 4 SGanomaly [Carrasco, Kallosh, Roiban, Tseytlin ‘13],[Bern, Parra-Martinez, Roiban ‘17].

4 Loops: New amplitudes technology, including the double copy,extends our ability to probe the UV structure.

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Four-derivative scalar toy model

Conformal supergravities are 4-derivative theories.

To illustrate the difficulties involved in these theories, consider a4-derivative massless scalar toy model:

L = −1

2φ�2φ+ · · ·

Free equation of motion (EOM) �2φ = 0 has 2 on-shell solutions:

φpw(x) = e ip·x , φ��pw(x) = i (α · x) e ip·x , α · p 6= 0.

Non-planewave modes grow linearly away from the origin!

States cannot naturally be diagonalized.

The propagator does not distinguish between these states:

∆(p) = − i

p4

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Mass deformation

Usual solution: deform the Lagrangian with a mass parameter m:

L = −1

2φ�(� + m2)φ+ · · · =⇒ ∆(p) = −

i

p2(p2 −m2)=

1

m2

(i

p2−

i

p2 −m2

)Poles correspond to solutions of new EOM, �(� + m2)φ = 0:

φ0(x) = e ip·x , p2 = 0 ,

φm(x) = e ipm·x , p2m = m2

The massive state is ghostlike. This gives rise to non-unitary nature.For small m,

φm(x) = e ipm·x = e ip·x(

1 +im2

2p · qq · x +O(m4)

)The non-planewave mode φ��pw emerges as m→ 0:

φ��pw

(x) = limm2→0

φm(x)− φ0(x)

m2= i(α · x)e ip·x

=⇒ An(1�pw, 2, . . . , n) = limm2→0

An(1m, 2, . . . , n)−An(1, 2, . . . , n)

m2

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States in Weyl gravity

Mass deform Weyl gravity using Einstein gravity, with κ = κ/m:

e−1L = − 1

κ2(Wµνρσ)2 − 2

κ2m2R

Presence of m2R breaks local scale invariance gµν → e−2λ(x)gµν .

A new feature is state counting: states transform under little groups

m2 = 0 =⇒ SO(2) : h++, h−−

m2 6= 0 =⇒ SO(3) : h++m , h+

m, h0m, h

−m , h

−−m

The pairs [h++, h++m ], [h−−, h−−m ] are known as dipoles.

State counting is reflected in the propagator:

∆µν;ρσ(p) =i

2m2

(ηµρηνλ+ηµληνρ−ηµνηρλ

p2−ηµρηνλ+ηµληνρ− 2

3ηµνηρλ

p2 −m2

)

When m = 0, the massive h0m state decouples.

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Minimal 4-derivative φ3 theory

L = −1

2φ�2φ+

g

2φ2�φ− g2

8φ4 + m2

(1

2(∂µφ)2 +

g

3!φ3

)This theory is special as its EOM factorizes:(

� + m2 − gφ) (

�φ− g

2φ2)

= 0

Any solution to the φ3 EOM �φ = g2φ

2 also works here.

Tree-level amplitudes follow from perturbative solutions to EOM. So,

An(1, 2, . . . , n) = m2Aφ3

n (1, 2, . . . , n)

And, amplitudes with a single massive planewave state vanish:

An(1m, 2, . . . , n) = 0

=⇒ An(1��pw, 2, . . . , n) = limm2→0

An(1m, 2, . . . , n)−An(1, 2, . . . , n)

m2

= −Aφ3

n (1, 2, . . . , n)

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Minimal (DF )2 theory

The minimal (DF )2 (Lee-Wick) theory is described by

L =1

2(DµF

µν,a)2 − m2

4(F aµν)2

EOM solved by Yang-Mills equations DµFµν,a = 0:

DλDλDρF

ρµ,a − DλDµDρF

ρλ,a + [Dµ,Dλ]DρF aρλ + m2DρF

ρµ,a = 0

The amplitudes are therefore given by

An(1, 2, . . . , n) = m2AYMn (1, 2, . . . , n)

An(1m, 2, . . . , n) = 0

An(1��pw, 2, . . . , n) = −AYMn (1, 2, . . . , n)

Checked by explicit calculation up to n = 7.

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Weyl gravity amplitudes

e−1L = − 1

κ2(Wµνρσ)2 − 2

κ2m2R

EOM Bµν + m2Rµν = 0, where Rµν = 0 =⇒ Bµν = 0:

Bµν = −2∇ρ∇(µRν)ρ + �Rµν +2

3∇µ∇νR −

1

6gµν�R

+ 2RρµRρν −

2

3RµνR −

1

2gµν(

(Rρλ)2 − 1

3R2)

So Weyl gravity is a minimal theory (EG=Einstein Gravity):

Mn(1, 2, . . . , n) = m2MEGn (1, 2, . . . , n)

Mn(1m, 2, . . . , n) = 0

Mn(1��pw, 2, . . . , n) = −MEGn (1, 2, . . . , n)

Similar argument made in AdS[Maldacena ‘11], [Adamo, Mason ‘13]

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Factorization of four-derivative trees

Return to minimal 4-derivative φ3 theory.Massive/massless amplitudes factorize as one would expect:

An(1, . . . , n) =∑

channels

[i

m2P2

φ0

P+

−im2(P2 −m2)

φm

P

]

One can show in the m→ 0 limit

An(1, . . . , n) =∑

channels

[−iP4

φpw

P

+−iP2

φpw φ�pw

P+−iP2

φ�pw φpw

P

]

So leading poles 1/P4 always imply an exchange of planewaves;

Planewave and non-planewave modes mix in subleading 1/P2 poles.

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One-Loop Cuts

Because the all-planewave amplitude An(12 · · · n) = 0, only twos-channel cuts contribute:

φpw

φpw

φpw

φpw

φ��pw

φ��pw

φpw

φpw

φpw

φpw

φpw

φpw

φpw

φpw

φ��pw

φ��pw

Also, because An(1��pw2 · · · n) = −Aφ3

n (12 · · · n),

AL=1n (12 · · · n) = 2Aφ

3,L=1n (12 · · · n)

Checked explicitly using Feynman diagrams in minimal φ3 theory.

In Weyl theory, implies no 1-loop UV divergence.

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Einstein supergravity

Minimal Weyl “N = 0” supergravity inherits Einstein supergravity(τ = χ+ ie−ϕ complex string coupling):

e−1L = −R

2+

1

4(∂µϕ)2 +

1

4e2ϕ(∂µχ)2 = −R

2+∂µτ̄ ∂

µτ

4(Im τ)2

Amplitudes generalize to N = 1, 2, 4 using the Ward identities.

Well-known double copy structure:

(N = 0, 1, 2, 4 Einstein SG) = YM⊗ (N = 0, 1, 2, 4 SYM)

At the level of individual states:

V = A+ + ηIλ+I +

1

2ηIηJSIJ +

1

3!εIJKLη

IηJηKλL− + η1η2η3η4A−

H+ ≡A+ ⊗ V = h++ + ηIψ+I + · · ·+ η1η2η3η4 C̄ ,

H− ≡A− ⊗ V = C + ηIΛ−I + · · ·+ η1η2η3η4 h−−

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Covariant SU(1,1)/U(1) formalism

Theory has a global nonlinear SL(2,R)∼=SU(1,1) symmetry:

τ → aτ + b

cτ + d, det

(a bc d

)= 1, a, b, c , d ∈ R

Global SU(1,1) realized linearly as φα → Uαβφβ :

φαφα = 1, φ1 = φ̄1, φ2 = −φ̄2.

Local U(1) realized using composite gauge field aµ = iφα∂µφα.

Introduce U(1)-covariant derivative ∇̃µ ≡ ∇µ − qU(1)aµ.

Pµ = φαεαβ∇̃µφβ P̄µ = −φαεαβ∇̃µφβ

N = 0 Einstein supergravity Lagrangian becomes

e−1L = −R

2+ P · P̄

For scattering, a scalar parametrization is required.

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Minimal conformal supergravity

We seek a bosonic extension to the Weyl Lagrangian.

EOM should be implied by Einstein supergravity:

Rµν −1

2gµνR = Tµν , ∇̃µPµ = 0

Tµν =2√|g |

δ(√|g |P ·P̄)

δgµν= 2P(µP̄ν) − gµνP ·P̄

It should also have

1 Global SU(1,1) symmetry acting on the (constrained) scalars φα;2 Local U(1) symmetry acting on scalars and other fields;3 Local scale invariance gµν → e−2λ(x)gµν ;4 Overall mass dimension 4.

Excluding total derivatives, GB, the 4 allowed combinations are

(Wµνρσ)2, P̄µ∇̃µ∇̃νPν + 2(Rµν−1

3gµνR)P̄µPν , P2P̄2, (P ·P̄)2,

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Minimal conformal supergravity (2)

The unique mass-deformed Lagrangian is [Ciceri, Sahoo ‘15]

e−1L = −1

4(Wµνρσ)2 + P̄µ∇̃µ∇̃νPν + 2(Rµν−

1

3gµνR)P̄µPν − P2P̄2 −

1

3(P ·P̄)2

+ m2

(−R

2+ P · P̄

),

Therefore, checked up to n = 7:

Mn(1, 2, . . . , n) = m2MESGn (1, 2, . . . , n)

Mn(1m, 2, . . . , n) = 0

Mn(1��pw, 2, . . . , n) = −MESGn (1, 2, . . . , n)

Implies same behavior of 1-loop amplitudes:

ML=1n (12 · · · n) = 2MESG,L=1

n (12 · · · n)

Amplitudes arise from double copy with minimal (DF )2:

(N = 0, 1, 2, 4 min.CSG) =(min.(DF )2

)⊗ (N = 0, 1, 2, 4 SYM)

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Covariant description

L agrees with the supergravity literature [Ciceri & Sahoo ‘15]:

e−1L = −1

2

[1

2R(M)abcdR(M)+

abcd + P2P̄2 +1

3(P ·P̄)2

− 2P̄a[DaDbPb +D2Pa]− 2DaPbDaP̄b −DaPaDbP̄b

]+ h.c.

R(M)+abcd is the self-dual covariant curvature.

Da = eaµDµ covariant under local U(1)/conformal transformations.

Covariance achieved using new gauge fields bµ, fµa associated with

dilatations and conformal boosts respectively.

To reproduce previous form

1 Gauge fix special conformal symmetry =⇒ bµ = 02 Solve for fµ

a as a composite field using R(M)µνabeνb = 0

=⇒ fµa = −1

4

(Rµ

a − 1

6eµ

aR

)

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The Berkovits-Witten theory

Historically seen as a “contamination” of the twistor string in themulti-trace gluon sector by gravitational modes.N = 4 MHV amplitudes [Berkovits, Witten ‘04]:

Mn(H+1 , · · · ,H

+k ,H

−k+1, · · · ,H

−n ) = (−1)niδ8(Q)

k∏i=1

n∑j=1,j 6=i

[ij]〈jq〉2

〈ij〉〈iq〉2

Product of factors occurring in Hodges matrix [Hodges ‘11].

Also obtained from (DF )2 double copy [Johansson, Nohle ‘17]:

(N = 0, 1, 2, 4 BW CSG) = (DF )2 ⊗ (N = 0, 1, 2, 4 SYM)

Until now, no Lagrangian formulation of the underlying theory(some hints provided in [Berkovits, Witten ‘04]).

Theory believed to derive from unique 10D Lagrangian [de Roo ‘91].

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Non-minimal conformal supergravity

Full class of Lagrangians [Butter, Ciceri, de Wit & Sahoo ‘17]:

e−1L = −F2

[1

2(W+

µνρσ)2−P̄µ∇̃µ∇̃νPν−2(Rµν−1

3gµνR)P̄µPν+P2P̄2 +

1

3(P ·P̄)2

]+ h.c.

Main result: the Berkovits-Witten theory is given by

F = i τ̄

Reminiscent to coupling with vector multiplets:

e−1L = −R

2+∂µτ̄∂

µτ

4(Im τ)2− 1

4

[i τ̄(F+,A

µν )2 + h.c.]

Also reminiscent of U(1) anomaly counterterm in N = 4 SYM:

Γlocalanom. =

1

2(4π)2

∫ddx

(i τ̄(R+)2 + h.c.

)+ SUSY

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Mass-deformed amplitudes

Expose ghostlike states using mass deformation:

e−1L = −i τ̄

2

[1

2(W+

µνρσ)2−P̄µ∇̃µ∇̃νPν−2(Rµν−1

3gµνR)P̄µPν+P2P̄2 +

1

3(P ·P̄)2

]+ h.c. + m2

(−R

2+ P · P̄

)For example, at n = 4: one-plus

MMHV4 (H−1 H

−2 H−3 H−4 ) = iδ8(Q)

stu + 2m6

(s −m2)(t −m2)(u −m2)

MMHV4 (H+

1 H+2 H−3 H−4 ) = iδ8(Q)

[12]4

st

(t

s −m2+

m2

u

)Smooth interpolation between Berkovits-Witten as m→ 0, Einsteinsupergravity as m→∞.

Checked amplitudes using numeric Berends-Giele up to n = 7.

Also checked 6-point NMHV against the double copy.

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Coupling to vector multiplets

Finally, we can couple N = 4 vector multiplets to the theory:

e−1L = −i τ̄

2

[1

2(W+

µνρσ)2 +1

2(F+,Aµν )2 − P̄µ∇̃µ∇̃νPν + P2P̄2 +

1

3(P ·P̄)2

− 2(Rµν−1

3gµνR)P̄µPν

]+ h.c. + m2

(−R

2+ P · P̄

)Confirmed up to n = 7 that amplitudes match Berkovits-Wittensingle-trace amplitudes:

Mn(H+1 · · ·H

+k H−k+1 · · ·H

−r−1Vr · · · Vn) = i

(−1)r−1δ8(Q)

〈r , r+1〉 · · · 〈nr〉

k∏i=1

n∑j=1j 6=i

[ij]〈jq〉2

〈ij〉〈iq〉2

Have also calculated double-trace amplitudes:

Mn(V1 · · · Vr−1|Vr · · · Vn) = ip2r,n

p2r,n −m2

δ8(Q)

〈12〉 · · · 〈r−1, 1〉〈r , r+1〉 · · · 〈nr〉

Again, checked by explicit calculation up to n = 7.

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Conclusions

In this work we have

1 Starting with a simple φ3 toy model, developed new techniques fordescribing amplitudes from their Lagrangians in 4-derivative theories;

2 Identified both the Lagrangian and double-copy origins of a variety ofconformal (super)gravity amplitudes;

3 Found a Lagrangian whose tree-level amplitudes match those of theBerkovits-Witten theory, and the (DF )2 ⊗ SYM double copy.

Although we have described non-planewave states to some degree,much work remains to be done.Non-dipole states of N = 4 CSG arrange into gravitino multiplets:

ΨA = Ψ+A + ΦABη

B +ABC−A ηDηE εBCDE + ΨB−−

A ηCηDηE εBCDE + Ψ−−−A η1η2η3η4

Need to describe double-copy origin of these states.

Further studies of loops, UV structure and conformal anomalies alsoto follow!

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Thanks for listening!

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Scalar parametrizations

To scatter, we need a scalar parametrization satisfying φαφα = 1.

For instance, using Pµ = φαεαβ∇̃µφβ , P̄µ = −φαεαβ∇̃µφβ ,

φ1 =1

2√

Im τ(1− iτ), φ2 =

1

2√

Im τ(1 + iτ)

=⇒ e−1L = −R

2+ P · P̄ = −R

2+∂µτ̄ ∂

µτ

4(Im τ)2

A more convenient choice is

φ1 =1√

1− |C |2, φ2 = − C√

1− |C |2

=⇒ e−1L = −R

2+

∂µC̄∂µC

(1− |C |2)2

Scalar C has a global U(1) symmetry, natural double copy structure:

C = A− ⊗ A+, C̄ = A+ ⊗ A−

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Scattering amplitudes from classical solutions

Tree-level amplitudes can be obtained from perturbative solution toclassical EOMs [Berends & Giele ‘88]:

φ(p) =∞∑n=0

φ(n)(p), φ(n)(p) ∼ gn

In our 4-derivative theory, �(� + m2)φ(0) = 0 =⇒ φ(0) is either amassive or massless planewave.

J(p1, p2, . . . , pn) ≡ δn−1φ(n−2)(−p1)

δφ(0)(p2)δφ(0)(p3) . . . δφ(0)(pn).

Amplitude then extracted using LSZ:

An(1, 2, . . . , n) = i limp2

1→0p2

1(p21 −m2) J(p1, p2, . . . , pn)

An(1m, 2, . . . , n) = i limp2

1→m2p2

1(p21 −m2) J(p1, p2, . . . , pn)

The same procedure applies in conformal gravity.