scattering amplitudes and sum rules for massive spin-2 states• a compactiﬁed 5d theory of...

Scattering Amplitudes and Sum Rules for

Massive Spin-2 StatesElizabeth H. SimmonsR. Sekhar Chivukula

RSC, Dennis Foren, Kirtimaan Mohan Dipan Sengupta, EHS

Dark Matter

High-Mass DM Search Prospects

Spin-2?

(mediator)

DM

DM

Spin-2 Mediator

The Hierarchy Problem: Why do we have a light Higgs?

Randall Sundrum (RS1) Model

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y

x,t→

“TeV” Brane

“Planck” Brane

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Kaluza—Klein Gravitons

⇤ = e�k⇡rcMPl<latexit sha1_base64="h29wQ2p6KoT5XrmCT9drNWE5HJk=">AAACB3icbVDLSsNAFJ34rPUVdSnIYBHcWJIq6EYounGhUME+oIlhMpm0QyeTMDMRSsjOjb/ixoUibv0Fd/6N0zYLbT0wcDjnXO7c4yeMSmVZ38bc/MLi0nJppby6tr6xaW5tt2ScCkyaOGax6PhIEkY5aSqqGOkkgqDIZ6TtDy5HfvuBCEljfqeGCXEj1OM0pBgpLXnmnnOtwwGC55DcZ0cDJ6FQeDiHN17WYLlnVqyqNQacJXZBKqBAwzO/nCDGaUS4wgxJ2bWtRLkZEopiRvKyk0qSIDxAPdLVlKOISDcb35HDA60EMIyFflzBsfp7IkORlMPI18kIqb6c9kbif143VeGZm1GepIpwPFkUpgyqGI5KgQEVBCs21ARhQfVfIe4jgbDS1ZV1Cfb0ybOkVavax9Xa7UmlflHUUQK7YB8cAhucgjq4Ag3QBBg8gmfwCt6MJ+PFeDc+JtE5o5jZAX9gfP4ASxqYUQ==</latexit>

Gravity propagates in the finite bulk

Lint =1

⇤

X

i

ciXµ⌫i (TSM

µ⌫ + TDMµ⌫ )

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⇠ ⇤<latexit sha1_base64="u2EafGlG8uvPMQ71dPF7o9R5HOo=">AAAB83icbVDLSgMxFL1TX7W+qi7dBIvgqsxUQZdFNy5cVLAP6Awlk8m0oUlmSDJCGfobblwo4tafceffmLaz0NYDgcM553JvTphypo3rfjultfWNza3ydmVnd2//oHp41NFJpghtk4QnqhdiTTmTtG2Y4bSXKopFyGk3HN/O/O4TVZol8tFMUhoIPJQsZgQbK/m+ZgL59zYf4UG15tbdOdAq8QpSgwKtQfXLjxKSCSoN4VjrvuemJsixMoxwOq34maYpJmM8pH1LJRZUB/n85ik6s0qE4kTZJw2aq78nciy0nojQJgU2I73szcT/vH5m4usgZzLNDJVksSjOODIJmhWAIqYoMXxiCSaK2VsRGWGFibE1VWwJ3vKXV0mnUfcu6o2Hy1rzpqijDCdwCufgwRU04Q5a0AYCKTzDK7w5mfPivDsfi2jJKWaO4Q+czx+E45FY</latexit>

Potential LHC Resonances…

Discrete KK modes: RS Gravitons

[SM, DM]

DM w/ RS Graviton Mediator

What are the constraints on the RS low-energy theory?

Constrained to higher masses…

(massless radion) (massive radion)

Scattering of Massive Spin-2 Particles

at High Energy

Toward an Effective Theory for Massive Spin-2 Particles• Four-D spin-2 particles with a Mass

• Fierz/Pauli (1939), Boulware/Deser (1972)

• van Dam/Veltman/Zakharov (1970), Vainshtein (1972)

• “Gravity in Theory Space”: Arkani-Hamed/Georgi/Schwartz (2003)

• |Amp|∝ E10/(MPl m4)2 generically, but E6/(MPl m2)2 with tuning

• de Rham/Gabadadze (2010) - found an E6/(MPl m2)2 theory without ghosts…

• Our goals:

• Understand how these results are modified if the massive spin-2 particles arise from KK compactification of a 5D theory - and explore consequences.

• Understand validity of EFT incorporating massive spin-2 KK resonances.

Graviton/Spin-2 dofs• Massless graviton represented by a symmetric traceless tensor with

redundant degrees of freedom (local diffeomorphism invariance)

• In general a massless graviton in d dimensions has d(d-3)/2 physical degrees of freedom

• Massless 4D graviton has 2 degrees of freedom, while a massless 5D graviton has 5 degrees of freedom.

• Mass term breaks diffeomorphism invariance - A 4D massive spin-2 state has 5 degrees of freedom (2 spin-2 helicity states, 2 spin-1 helicity states and 1 spin-0 helicity state)

• A compactified 5D theory of gravity, therefore, can provide the states needed for the 4D massive KK spin-2 modes.

2 Strong coupling scale for massive gravity in 4 dimensions

2.1 The Stuckelberg mechanism for gravity

We start by briefly reviewing massive graviton amplitudes in 4 dimensions, as this

will be key to understanding the precise nature of interactions that gives rise to the

dangerous growth of matrix elements. A detailed account can be found in []. The

Einstein-Hilbert(EH) action (neglecting the cosmological constant term) is ,

SG =

Zd4xpgR (2.1)

where g is the determinant of the background metric, R is the Ricci scalar, given by

R = gµ⌫Rµ⌫ , where g

µ⌫ is the metric inverse and Rµ⌫ is the Ricci tensor. For a flat

background metric, we can expand the Ricci scalar using the weak field expansion

gµ⌫ = ⌘µ⌫ + hµ⌫ , where ⇠ 1/MP l, is the weak field expansion paramter and ⌘µ⌫ is

the usual Minkowski metric. The action is invariant under the local di↵eormorphism,

hµ⌫ ! hµ⌫ + @(µ⇠⌫) (2.2)

The 10 degrees of freedom in the symmetric tensor hµ⌫ can be split into a transverse

part with 6 degrees of a freedom, and a vector mode ⇢, with 4 degrees of freedom,

hµ⌫ ⌘ hTµ⌫ + @(µ⇢⌫). (2.3)

Out of the 10 degrees of freedom, the di↵eomorphism removes 2 ⇥ 4 degrees, thus

leaving behind two physical transverse degrees of freedom for the massless graviton3.

A Fierz-Pauli mass term can be added to this action4,

Sm = m2((hµ⌫)

2� h

2). (2.4)

where h = ⌘µ⌫hµ⌫ , the trace of the field. The mass term explicitly breaks the

di↵eomorphism invariance. It can be restored by introducing Stuckelberg fields Aµ,

with the replacement hµ⌫ ! hµ⌫ + @(µA⌫) such that the action SG + Sm is invariant

under,

hµ⌫ ! hµ⌫ + @(µ⇠⌫)

Aµ ! Aµ �1

2⇠µ. (2.5)

3It is easy to see that in d dimensions there are d(d + 1)/2 � 2d = d(d � 3)/2 degrees of

polarization(physical states) for a massless graviton4Note that this is the only consistent mass term that can be written down in 4 dimension. Any

other term would lead to ghost degrees of freedom, leading to the Ostrogradsky instability, a feature

of constrained lagrangian systems.

– 4 –

2 Strong coupling scale for massive gravity in 4 dimensions

2.1 The Stuckelberg mechanism for gravity

We start by briefly reviewing massive graviton amplitudes in 4 dimensions, as this

will be key to understanding the precise nature of interactions that gives rise to the

dangerous growth of matrix elements. A detailed account can be found in []. The

Einstein-Hilbert(EH) action (neglecting the cosmological constant term) is ,

SG =

Zd4xpgR (2.1)

where g is the determinant of the background metric, R is the Ricci scalar, given by

R = gµ⌫Rµ⌫ , where g

µ⌫ is the metric inverse and Rµ⌫ is the Ricci tensor. For a flat

background metric, we can expand the Ricci scalar using the weak field expansion

gµ⌫ = ⌘µ⌫ + hµ⌫ , where ⇠ 1/MP l, is the weak field expansion paramter and ⌘µ⌫ is

the usual Minkowski metric. The action is invariant under the local di↵eormorphism,

hµ⌫ ! hµ⌫ + @(µ⇠⌫) (2.2)

The 10 degrees of freedom in the symmetric tensor hµ⌫ can be split into a transverse

part with 6 degrees of a freedom, and a vector mode ⇢, with 4 degrees of freedom,

hµ⌫ ⌘ hTµ⌫ + @(µ⇢⌫). (2.3)

Out of the 10 degrees of freedom, the di↵eomorphism removes 2 ⇥ 4 degrees, thus

leaving behind two physical transverse degrees of freedom for the massless graviton3.

A Fierz-Pauli mass term can be added to this action4,

Sm = m2((hµ⌫)

2� h

2). (2.4)

where h = ⌘µ⌫hµ⌫ , the trace of the field. The mass term explicitly breaks the

di↵eomorphism invariance. It can be restored by introducing Stuckelberg fields Aµ,

with the replacement hµ⌫ ! hµ⌫ + @(µA⌫) such that the action SG + Sm is invariant

under,

hµ⌫ ! hµ⌫ + @(µ⇠⌫)

Aµ ! Aµ �1

2⇠µ. (2.5)

3It is easy to see that in d dimensions there are d(d + 1)/2 � 2d = d(d � 3)/2 degrees of

polarization(physical states) for a massless graviton4Note that this is the only consistent mass term that can be written down in 4 dimension. Any

other term would lead to ghost degrees of freedom, leading to the Ostrogradsky instability, a feature

of constrained lagrangian systems.

– 4 –

Fierz-Pauli Mass term

SG = 2

Zd4x

pgR

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Quantizing Gravity 101 • Start with the Einstein - Hilbert action

• Expand about the flat Minkowski background up to order

than n, that takes part in the scattering amplitude, and therefore there is incomplete

cancellation at every power s. Therefore the amplitude will grow like s5/(M2

P lm8),

and therfore the strong coupling scale is set by ⇤5.

4 Compactified extra dimensions in RS

5 Conclusions

Acknowledgements

A Metrics and determinants

The four dimensional metric gµ⌫ is expanded around the flat Minkowski background

⌘µ⌫ as gµ⌫ ! ⌘µ⌫ + hµ⌫ . We use the metric convention ⌘µ⌫ = Diag(1,�1,�1,�1).

We require gµ↵g⇢⌫ = �

⇢⌫ where g = g

�1. The inverse is thus given by

gµ⌫ =

+1X

n=0

(�)n[⌘(⌘h)n]µ⌫ (A.1)

and

p� det g =

+1Y

n=1

exp

�(�)n

2ntr [(⌘h)n]

�(A.2)

=+1Y

n=1

+1X

mn=0

1

mn!

�(�)n

2ntr [(⌘h)n]

�mn

=+1Y

n=1

+1X

mn=0

(�1)mn

mn!(2n)mn(�)n·mntr[(⌘h)n]mn

The overall brackets imply contractions to leave the free indices wherever they ap-

pear. The first few terms in the expansion of the inverse and the determinant is,

g↵� = ⌘

↵�� h

↵� + 2[hh]↵� �

3[hhh]↵� ++O(4) (A.3)p� det g = 1 +

2h+

2

8

�h2� 2[hh]

�+

3

48

�h3� 6h[hh] + 8[hhh]

�+O(4)(A.4)

The 5-D metric for the orbifolded torus is,

GMN =

gµ⌫ �

2e+2�

⇢µ⇢⌫ e+2�

⇢µ

e+2�

⇢⌫ �e+2�

!(A.5)

– 16 –

than n, that takes part in the scattering amplitude, and therefore there is incomplete

cancellation at every power s. Therefore the amplitude will grow like s5/(M2

P lm8),

and therfore the strong coupling scale is set by ⇤5.

4 Compactified extra dimensions in RS

5 Conclusions

Acknowledgements

A Metrics and determinants

The four dimensional metric gµ⌫ is expanded around the flat Minkowski background

⌘µ⌫ as gµ⌫ ! ⌘µ⌫ + hµ⌫ . We use the metric convention ⌘µ⌫ = Diag(1,�1,�1,�1).

We require gµ↵g⇢⌫ = �

⇢⌫ where g = g

�1. The inverse is thus given by

gµ⌫ =

+1X

n=0

(�)n[⌘(⌘h)n]µ⌫ (A.1)

and

p� det g =

+1Y

n=1

exp

�(�)n

2ntr [(⌘h)n]

�(A.2)

=+1Y

n=1

+1X

mn=0

1

mn!

�(�)n

2ntr [(⌘h)n]

�mn

=+1Y

n=1

+1X

mn=0

(�1)mn

mn!(2n)mn(�)n·mntr[(⌘h)n]mn

The overall brackets imply contractions to leave the free indices wherever they ap-

pear. The first few terms in the expansion of the inverse and the determinant is,

g↵� = ⌘

↵�� h

↵� + 2[hh]↵� �

3[hhh]↵� ++O(4) (A.3)p� det g = 1 +

2h+

2

8

�h2� 2[hh]

�+

3

48

�h3� 6h[hh] + 8[hhh]

�+O(4)(A.4)

The 5-D metric for the orbifolded torus is,

GMN =

gµ⌫ �

2e+2�

⇢µ⇢⌫ e+2�

⇢µ

e+2�

⇢⌫ �e+2�

!(A.5)

– 16 –

Conventionally the expansion parameter = 2/M3/2 where M is the 5 dimensional Planck

mass, with

M2P l =

M3

k(1� e

�2krc⇡) (A7)

For krc ⇠ 10 or more we can drop the exponential on the right hand side.

2. KK-graviton self couplings

The RS model (after neglecting Cosmological constant like terms) can be separated into

a pure gravity and a pure matter/matter-gravity Lagrangian. In five dimensions, the pure

gravity part read as,

Lgrav ⌘ e�2k|y|

L4D,curv[, h] +1

r2c

e�4k|y|

Q (A8)

(A9)

These are connected to the 4D phenomenology via KK expansion and integration over the

extra dimension. In de Donder gauge, repeated for ease of reference,

L4D,curv =1

4

2(@µh⌫⇢)

2� (@µh)

2

�

+

8

2h(@µh⌫⇢)

2� h(@µh)

2� 4hµ⌫(@µh

⇢�)(@⌫h⇢�)

� 8hµ⌫(@�h⌫⇢)(@�h⇢µ) + 4hµ⌫(@⇢hµ⌫)(@

⇢h) + 8hµ⌫(@⌫h

⇢�)(@⇢h�µ)

�

(A10)

Q =1

2

⇢(h0

h0)� (h0)2

�+

4

⇢h⇥(h0

h0)� (h0)2

⇤+ 4 [(hh0)h0

� (hh0h0)]

�(A11)

During KK expansion, each hµ⌫(x, y) is replaced with a sum of 4D KK modes h(n)µ⌫ (x) mul-

tiplying fifth-dimensional wavefunctions �(n)(�),

hµ⌫(x, y) =1

prc

+1X

n=0

h(n)µ⌫ (x)�

(n)(�) (A12)

where,

�(0)(�) =

pkrc �

(n)(�) =e+2krc|�|

Nn[J2(zn) + ↵nY2(zn)]

low tower⇠=

e+2krc|�|

NnJ2(zn) (A13)

and zn = xne�krc(⇡�|�|). Each wavefunction satisfies the following di↵erential equation,

d

d�

e�4krc|�|d�

(n)

d�

�= x

2n(krc)

2e�2krc(|�|+⇡)

�(n) (A14)

19

κ2

1 Short review of General RelativityEinstein & Hilbert (1915):

Gravity is curvature of spacetime, caused by matter−→ Interaction between gµν and Tµν

S =1

κ

!

d4x"

|g|#

R + Lmatter

$

Rµν − 12gµνR = κ Tµν

with

Rµν = ∂µΓλνλ − ∂λΓ

λµν + Γλ

µσΓσλν − Γλ

µνΓσλσ

Γρµν = 1

2gρσ

%

∂µgσν + ∂νgµσ − ∂σgµν

&

Tµν =1

"

|g|δLmatter

δgµν

−→ Very geometrical way to understand gravityB. Janssen (UGR) Granada, 14 dec 2006 5

(as motivated by Weinberg’s theorem)

=p

32⇡GN =2

Mpl=

1

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June 3, 2015 0:19 World Scientific Review Volume - 9in x 6in carrascoTASI page 11

11

1.3. Coming to grips with gravity as a double copy

I claimed that, once gauge-theory graph numerators are written in thecorrect way, one can express numerators for a related gravity theory asa product of gauge-theory numerators. This result can be proven recur-sively at tree level [7]. Loop-level relations are then expected to follow viaunitarity. But where does this notion come from?

First let us travel back in time to 1967 when Bryce DeWitt wrote hisseminal papers on the perturbative quantization of gravity, of which histhird [10] will be the most relevant here. Here we find the three-pointFeynman rule for Yang-Mills theory (eq. (2.1) of ref. [10]),

δS3

δAaµδAb

σδAcρ→ ifabc ((k1

ρ − k2ρ) ηµσ + (k2

µ − k3µ) ησρ + (k3

σ − k1σ) ηρµ)

(19)where particle 1 comes in with momenta k1 and color index a, particle 2comes in with momenta k2 and color-index b and particle 3 comes in withmomenta k3 and color-index c. Also, and more to the point, DeWitt givesus the 171 termb three-point Feynman rule for three gravitons (eq. (2.3) ofref. [10]),

δS3

δϕµνδϕστ δϕρλ

→ 2ηµτ ηνσk1λk1

ρ + 2ηµσηντk1λk1

ρ − 2ηµνηστ k1λk1

ρ +

2ηλτ ηµνk1σk1

ρ + 2ηλσηµνk1τk1

ρ + ηµτ ηνσk2λk1

ρ + ηµσηντk2λk1

ρ + ηλτ ηνσk2µk1

ρ +

ηλσηντk2µk1

ρ + ηλτηµσk2νk1

ρ + ηλσηµτ k2νk1

ρ + ηλτ ηνσk3µk1

ρ + ηλσηντk3µk1

ρ −

ηλνηστ k3µk1

ρ + ηλτηµσk3νk1

ρ + ηλσηµτ k3νk1

ρ − ηλµηστ k3νk1

ρ + ηλνηµτ k3σk1

ρ +

ηλµηντk3σk1

ρ + ηλνηµσk3τk1

ρ + ηλµηνσk3τk1

ρ + 2ηµνηρτ k1λk1

σ + 2ηµνηρσk1λk1

τ −

2ηλρηµνk1σk1

τ + 2ηλνηµρk1σk1

τ + 2ηλµηνρk1σk1

τ + ηµτ ηνρk1σk2

λ + ηµρηντ k1σk2

λ +

ηµσηνρk1τk2

λ + ηµρηνσk1τk2

λ + ηντ ηρσk1λk2

µ + ηνσηρτk1λk2

µ + ηλτ ηνρk1σk2

µ −

ηλρηντk1σk2

µ + ηλνηρτk1σk2

µ + ηλσηνρk1τk2

µ − ηλρηνσk1τ k2

µ + ηλνηρσk1τk2

µ +

2ηνρηστ k2λk2

µ + ηµτ ηρσk1λk2

ν + ηµσηρτ k1λk2

ν + ηλτηµρk1σk2

ν − ηλρηµτ k1σk2

ν +

ηλµηρτk1σk2

ν + ηλσηµρk1τk2

ν − ηλρηµσk1τk2

ν + ηλµηρσk1τ k2

ν + 2ηµρηστ k2λk2

ν +

2ηλτ ηρσk2µk2

ν + 2ηλσηρτ k2µk2

ν − 2ηλρηστ k2µk2

ν + ηµτ ηνσk1λk2

ρ + ηµσηντk1λk2

ρ +

ηλνηµτ k1σk2

ρ + ηλµηντk1σk2

ρ + ηλνηµσk1τk2

ρ + ηλµηνσk1τk2

ρ + 2ηµτ ηνσk2λk2

ρ +

2ηµσηντk2λk2

ρ −2ηµνηστ k2λk2

ρ +2ηλνηστ k2µk2

ρ +2ηλµηστ k2νk2

ρ +ηντηρσk1λk3

µ +

ηνσηρτ k1λk3

µ − ηνρηστ k1λk3

µ + ηλτηνρk1σk3

µ + ηλνηρτ k1σk3

µ + ηλσηνρk1τk3

µ +

ηλνηρσk1τk3

µ + ηντηρσk2λk3

µ + ηνσηρτ k2λk3

µ + ηλτηρσk2νk3

µ + ηλσηρτ k2νk3

µ +

ηλτηνσk2ρk3

µ + ηλσηντk2ρk3

µ + ηµτ ηρσk1λk3

ν + ηµσηρτ k1λk3

ν − ηµρηστ k1λk3

ν +

ηλτηµρk1σk3

ν + ηλµηρτ k1σk3

ν + ηλσηµρk1τk3

ν + ηλµηρσk1τ k3

ν + ηµτ ηρσk2λk3

ν +

ηµσηρτ k2λk3

ν + ηλτ ηρσk2µk3

ν + ηλσηρτ k2µk3

ν + ηλτηµσk2ρk3

ν + ηλσηµτ k2ρk3

ν +

2ηλτ ηρσk3µk3

ν + 2ηλσηρτ k3µk3

ν − 2ηλρηστ k3µk3

ν + ηµτ ηνρk1λk3

σ + ηµρηντk1λk3

σ +

ηλνηµρk1τk3

σ + ηλµηνρk1τ k3

σ + ηµτ ηνρk2λk3

σ + ηµρηντk2λk3

σ − ηµνηρτk2λk3

σ +

ηλτηνρk2µk3

σ + ηλνηρτk2µk3

σ + ηλτ ηµρk2νk3

σ + ηλµηρτ k2νk3

σ − ηλτ ηµνk2ρk3

σ +

ηλνηµτ k2ρk3

σ + ηλµηντk2ρk3

σ + 2ηλρηντk3µk3

σ + 2ηλρηµτ k3νk3

σ + ηµσηνρk1λk3

τ +

ηµρηνσk1λk3

τ + ηλνηµρk1σk3

τ + ηλµηνρk1σk3

τ + ηµσηνρk2λk3

τ + ηµρηνσk2λk3

τ −

ηµνηρσk2λk3

τ + ηλσηνρk2µk3

τ + ηλνηρσk2µk3

τ + ηλσηµρk2νk3

τ + ηλµηρσk2νk3

τ −

ηλσηµνk2ρk3

τ + ηλνηµσk2ρk3

τ + ηλµηνσk2ρk3

τ + 2ηλρηνσk3µk3

τ + 2ηλρηµσk3νk3

τ −

bDeWitt actually introduces shorthand for symmetrization and permutation so the ex-pression takes up less space on the page — for pedagogy I have expanded it out so thatstudents can play with the full expression without fear of misinterpreting which symbolsshould be permuted over.


12 J. J. M. Carrasco

2ηλρηµνk3σk3

τ +2ηλνηµρk3σk3

τ +2ηλµηνρk3σk3

τ − ηλτ ηµσηνρk1 · k2 − ηλσηµτ ηνρk1 ·

k2 − ηλτηµρηνσk1 · k2 + ηλρηµτ ηνσk1 · k2 − ηλσηµρηντ k1 · k2 + ηλρηµσηντ k1 · k2 +

2ηλτ ηµνηρσk1 · k2 − ηλνηµτ ηρσk1 · k2 − ηλµηντηρσk1 · k2 + 2ηλσηµνηρτ k1 · k2 −

ηλνηµσηρτ k1 · k2 − ηλµηνσηρτ k1 · k2 − 2ηλρηµνηστ k1 · k2 + 2ηλνηµρηστ k1 · k2 +

2ηλµηνρηστ k1 · k2 − ηλτ ηµσηνρk1 · k3 − ηλσηµτ ηνρk1 · k3 − ηλτηµρηνσk1 · k3 +

2ηλρηµτ ηνσk1 · k3 − ηλσηµρηντ k1 · k3 + 2ηλρηµσηντk1 · k3 + 2ηλτ ηµνηρσk1 · k3 −

ηλνηµτ ηρσk1 · k3 − ηλµηντηρσk1 · k3 + 2ηλσηµνηρτ k1 · k3 − ηλνηµσηρτ k1 · k3 −

ηλµηνσηρτ k1 · k3 − 2ηλρηµνηστ k1 · k3 + ηλνηµρηστ k1 · k3 + ηλµηνρηστ k1 · k3 −

ηλτηµσηνρk2 · k3 − ηλσηµτ ηνρk2 · k3 − ηλτ ηµρηνσk2 · k3 + 2ηλρηµτ ηνσk2 · k3 −

ηλσηµρηντk2 · k3 + 2ηλρηµσηντk2 · k3 + ηλτ ηµνηρσk2 · k3 − ηλνηµτ ηρσk2 · k3 −

ηλµηντηρσk2 · k3 + ηλσηµνηρτ k2 · k3 − ηλνηµσηρτ k2 · k3 − ηλµηνσηρτk2 · k3 −

2ηλρηµνηστ k2 · k3 + 2ηλνηµρηστ k2 · k3 + 2ηλµηνρηστ k2 · k3

Problem 3. Using symbolic manipulation software, derive these Feynmanrules starting from their respective actions.

Recall that Feynman rules for off-shell Green’s functions depend on thegauge, and hence are unphysical. To talk about physical observables weneed to take the external particles to be physical — “on shell”. So thismeans particles k1, k2, and k3 have to be massless if we want to talk aboutthe three-particle scattering amplitudes of either gluons or gravitons. Notethat this means that ki · kj = 0. This also means, for 3-point amplitudes,that ki have to be complex. This will not be a bother to us, but it issomething to pay attention to if you’re going to plug in numbers. To get aphysical amplitude for external gluons we must contract eq. (20) with threetransverse polarization vectors. Gluons can have either positive or negativehelicity polarization states in four dimensions, satisfying ε±i · ki = 0. Inpure Yang-Mills at three-points, in four dimensions, we can only have twonegative-helicity gluons scattering into a positive helicity gluon or vice-versa, either choice will restrict some terms in our expressions. We arefreec to choose our polarization vectors such that ε±i · ε±j = 0. Considering,without loss of generality, gluons 1 and 2 to be negative helicity, and gluon3 to be positive helicity we have our on-shell Feynman rule for Yang-Millstheory (YM):

!δS3

δA−aµ δA−b

σ δA+cρ

"

on-shell

→ −2ifabc (k1σηµρ − k2

µηρσ) . (20)

Here I also used conservation of momentum to express k3 = −k1 − k2.Now gravitons are also massless and also must be in one of two states in

four dimensions, which can be taken to have helicity ±2. Their polarizationtensors then factorize into a product of spin-1 polarization vectors: ε±±

iµν =

ε±iµε±i

ν . The constraints from dotting the graviton Feynman rule intocSee, e.g., eqs. (31)–(35) of ref. [1].

Graviton triple vertex Source:arXiv 1506.00974


76 J. J. M. Carrasco

References

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[3] Z. Bern, J. S. Rozowsky and B. Yan, “Two loop four gluon amplitudes inN = 4 super-Yang-Mills,” Phys. Lett. B 401, 273 (1997) [hep-ph/9702424].

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[7] Z. Bern, T. Dennen, Y.-t. Huang and M. Kiermaier, “Gravity as the Squareof Gauge Theory,” Phys. Rev. D 82, 065003 (2010) [arXiv:1004.0693 [hep-th]].

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[10] B. S. DeWitt, Phys. Rev. 162, 1239 (1967).[11] R. Britto, F. Cachazo, B. Feng, “New recursion relations for tree amplitudes

of gluons,” Nucl. Phys. B715, 499-522 (2005) [hep-th/0412308]; R. Britto,F. Cachazo, B. Feng and E. Witten, “Direct proof of tree-level recursionrelation in Yang-Mills theory,” Phys. Rev. Lett. 94, 181602 (2005) [hep--th/0501052].

[12] V. P. Nair, “A current algebra for some gauge theory amplitudes,” Phys.Lett. B 214, 215 (1988).

[13] G. Georgiou, E. W. N. Glover and V. V. Khoze, “Non-MHV tree amplitudesin gauge theory,” JHEP 0407, 048 (2004) [hep-th/0407027];Y.-t. Huang, “N = 4 SYM NMHV loop amplitude in superspace,” Phys.Lett. B 631, 177 (2005) [hep-th/0507117];H. Feng and Y.-t. Huang, “MHV lagrangian for N = 4 super Yang-Mills,”JHEP 0904, 047 (2009) [hep-th/0611164].

[14] M. Bianchi, H. Elvang and D. Z. Freedman, “Generating Tree Amplitudesin N = 4 SYM and N = 8 SG,” JHEP 0809, 063 (2008) [arXiv:0805.0757[hep-th]];

Naive Expectation

→s5

m8M2pl

ϵ0μν →

kμkν

m2

⇥O(s7) ! O(s5)

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+

Unitarity is violated at a scale Λ5 = (Mplm4)1/5 ≪ Mpl

This expectation is realized if one simply adds mass to the 4D Graviton

Cheung and Remmen, arXiv:1601.04068

{Four-point terms can likewise reduce divergence to O(s3); it has been claimed that this is the best one can do in massive spin-2 theories…

However, one can add nonlinear (3-point) terms to moderate divergence

Compactified 5D Theory

This growth cannot occur in a compactified 5-D model … Compactification is an IR phenomenon, UV

behavior determined by power-counting is |Amp| ∝E3/(M5)3!

Diagrams Relevant to Longitudinal h(n)h(n) ! h(n)h(n)

There are ten diagrams relevant to aforementioned process, sevenof which diverge as O(s5):

26 / 38SEH

5D / 1

M35

Zd5x

p�gR5D

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What happens in the compactified theory?

Boson Scattering in Compactified Theories:

A Flat Extra Dimension

Yang-Mills Theory:

5-D

(This construction led to the development of “Higgsless

Models…”)

4D Vector KK ScatteringNote

“KK momentum conservation”

Orbifolding a Compact Extra Dimension, like RS

Suppose there exists a compact extra dimension between two 4Dbranes (at y = 0 and ⇡rc) with the metric GMN equal to...

⌘MN = Diag(+1,�1,�1,�1,�1) = ⌘µ⌫ ⌦ ⌘55

at one of those branes (flat 4D = our 4D). Next, orbifold:

Choose even solutions on [�⇡rc ,⇡rc ] =) massless particles

8 / 38

Mode expansion for gravity in a toroidal extra dimension

We will denote latin alphabets as the 5-D coordinates while greek indices are reserved

for 4-D coordinates. The 4-D part of the metric gµ⌫ , is expanded in the weak field

approximation around the flat Minkowski metric ⌘µ⌫ as gµ⌫ = ⌘µ⌫ + 4Dhµ⌫ , where

4D is a small coupling(weak field expansion parameter). The fields r and ⇢µ are the

radion and the graviphoton respectively. The 5-D massless symmetric tensor hMN ,

that satisfies the full 5-D di↵eomorphism invariance, with 5 transverse degrees of

freedom is split into a tensor hµ⌫ with two transverse degrees of freedom, a vector

⇢µ with two transverse degrees of freedom, and a scalar degree of freedom in r. The

5-D action is given by,

S = M35

Zd4xdy

p

GR(5). (3.2)

We denote the extra dimensional co-ordinate by y. Additionally, G is the deterimant

of the 5-D metric and R(5) is the 5-D Ricci scalar expanded as G

MNRMN , where

GMN is the 5-D metric inverse, and RMN the 5-D Ricci tensor. We denote the 5-

D Planck mass by M5, which is related to the 4-D Planck mass by M35 = M

2P l/L.

The expansion of the metric inverse and determinant is detailed in Appendix A1.

The compactification allows us to expand the fields in fourier modes over the 5-D

co-ordinates. These read,

hµ⌫(x, y) =1X

n=�1hµ⌫,n(x)e

i!ny

hµ5(x, y) =1X

n=�1⇢µ(x, n)e

i!ny

h55(x, y) =1X

n=�1r(x, n)ei!ny (3.3)

where n is an integer that characterizes the mode number with frequencies !n ⌘

2⇡nL . The orthogonality of the fourier expansion imposes,

R L

0 dye(i!my)⇤

ei!ny = L�mn.

The field hMN is broken to a tensor, vector, and a scalar as hµ,⌫(x), ⇢µ(x), r(x)

respectively. Additionally, reality of the 5-D fields impose,

h⇤µ⌫,n = hµ⌫,�n, ⇢

⇤µ,n = ⇢µ,�n, r

⇤n = ��n. (3.4)

– 7 –










S = M35

Zd4xdy

p

GR(5). (3.2)



MNRMN , where



2P l/L.




hµ⌫(x, y) =1X

n=�1hµ⌫,n(x)e

i!ny

hµ5(x, y) =1X

n=�1⇢µ(x, n)e

i!ny

h55(x, y) =1X

n=�1r(x, n)ei!ny (3.3)



R L

0 dye(i!my)⇤

ei!ny = L�mn.



h⇤µ⌫,n = hµ⌫,�n, ⇢

⇤µ,n = ⇢µ,�n, r

⇤n = ��n. (3.4)

– 7 –

Graviphoton (massless spin-1 state from gμν) has odd parity in y and does not play a role in this model

“graviphoton”

“radion”

Restating the Problem

Recall: we’re interested in how the high-energy limits of...

Massless 5D Gravity: M2!2 ! O(s)

Massive 4D Gravity: M2!2 ! O(s5)

are consistent despite massless 5D gravity’s KK expansioninvolving infinitely many massive spin-2 modes.

To see how this is resolved, we focus on the h(n)

h(n)

! h(n)

h(n)

pure longitudinal helicity amplitude Mnn!nn at O(2).

Mnn!nn = =X

k

M(k)nn!nn s

k

25 / 38

Diagrams Relevant to Longitudinal h(n)h(n) ! h(n)h(n)

There are ten diagrams relevant to aforementioned process, sevenof which diverge as O(s5):

26 / 38

The Radion-Mediated Diagrams

The remaining three diagrams are radion-mediated and diverge like

O(s3), thereby contributing at M(3)nn!nn and lower.

Important to Note: We didn’t include a radion in our previousdark matter calculation!

29 / 38

Intermediate massless graviton

Intermediate radion

“Seagull”

Expectations for Massive KK Spin-2 Modes: Torus

Amplitude cancels through order s2 - same true for all helicity states

Radion diagrams start contributing at order s3

Order s persists, for elastic scattering of each KK mode. Sum will reproduce expected E3 behavior

Spin-2 Scattering in Randall-Sundrum Model

Randall Sundrum Model

No analog of KK momentum conservation…

RS Fields and Interactions

RS Analysis

Longitudinal RS-KK Scattering

Cancellations in RS Model

Cancellations in RS Model (cont’d)

Why do the needed cancellations persist in RS?

RS Mode Expansion










S = M35

Zd4xdy

p

GR(5). (3.2)



MNRMN , where



2P l/L.




hµ⌫(x, y) =1X

n=�1hµ⌫,n(x)e

i!ny

hµ5(x, y) =1X

n=�1⇢µ(x, n)e

i!ny

h55(x, y) =1X

n=�1r(x, n)ei!ny (3.3)



R L

0 dye(i!my)⇤

ei!ny = L�mn.



h⇤µ⌫,n = hµ⌫,�n, ⇢

⇤µ,n = ⇢µ,�n, r

⇤n = ��n. (3.4)

– 7 –

A Sturm-Liouville Problem

RS Coupling Relations

Integration-by-Parts:

O(s5) & O(s4) Sum Rules

Sturm-Liouville IdentitiesCompleteness: O(s5) ✔

Integration-by-Parts:

Completeness: O(s4) ✔

Same sum-rules as in Yang-Mills case! Expected to hold for arbitrary compactifications….

(Explicitly true in RS)

YM “Higgsless” models - Csaki, et. al. 2004

O(s3): Radion Coupling

RS: Checked numerically, no analytic proof! Can show, for any compactification, radion

coupling needed to cancel O(s3) growth.

⇒

O(s2)

Same combination of radio and (massless) graviton coupling as in O(s3) sum-rule!

Assuming O(s3) sum-rule, implies:

Proved for RS form of metric … no general proof.

Implications &

Applications

All Helicities: NO sum-rules imposed

All Helicities: after sum-rules imposed

Application: Massive Gravity

Consider 11 → 11, can satisfy first two sum-rules with:

Cancels O(s5) and O(s4) growth… agrees with Arkani-Hamed/Georgi/Schwartz (2003)

“b-terms” correspond to a choice of the “non-linear” O(h3) and O(h4) couplings… do not have form arising from higher-D

Two massive spin-2 states• Consider 11→11 with two massive states, R=m2/m1.

• Can cancel O(s5) and O(s4) growth, with

• Can cancel O(s3) with a radion coupling

• May lead to a O(s2) Theory can have a distinct light state, and then a tower of KK states with typical spacing….

“Bi-Gravity”

• Consider a (massless) graviton and one massive spin-2 particle.

• Sum-Rules show O(s5) growth in elastic spin-2 scattering can be cancelled, but not O(s4) with these interactions … need additional “non-linear” interactions of massive spin-2 field.

Summary• Spin-2 mediators for DM are natural in RS models - we are exploring

the limits of RS effective field theory by analyzing scattering amplitudes for massive spin-2 states.

• In KK theories, we have discovered that cancellations due to different diagrams reduce O(s5) growth to O(s).

• We have uncovered sum rules enforcing this cancellation; the first two generalize to arbitrary compactifications, others are more subtle.

• Our results extend & are consistent with literature on massive gravity.

• Much remains to be explored … including — given a valid EFT, exploring DM parameter space more completely, — connection of results to modern “amplitudology”…

scattering amplitudes and sum rules for massive spin-2 states• a compactiﬁed 5d theory of...

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