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Unraveling conformal gravity amplitudesbased on [1806.05124] with Henrik Johansson & Fei Teng
Gustav [email protected]
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.