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Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions

Yiannis Constantinou

Physics Department University of Crete

December 9, 2011

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 1 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

Outline

1

Motivation

2

Introduction

3

General setup The action Equations of motion Iterative solution

Φ radiation

4

The radiation amplitude The local amplitude The non-local amplitude The part jn

z (k) of the radiation amplitude and destructive interference

The part jn

z′(k) of the amplitude

Summary

5

The emitted Energy - Spectral and Angular Distribution

6

Summary of Results

7

Future work

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 2 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

Motivation

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 3 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

Motivation

In the context of large extra dimensions, it is possible to have TeV scale gravity [ADD].

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 3 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

Motivation

In the context of large extra dimensions, it is possible to have TeV scale gravity [ADD]. Black holes can be formed in colliders for M > M∗ and b ≤ Rs [Argyres et. al. 1998].

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 3 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

Motivation

In the context of large extra dimensions, it is possible to have TeV scale gravity [ADD]. Black holes can be formed in colliders for M > M∗ and b ≤ Rs [Argyres et. al. 1998]. Various models have been proposed to compute the gravitational bremstrahlung[ D’Eath and Payne, Eardley and Giddings] (Colliding waves model, ACV model). Alternative methods are desirable.

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 3 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

Introduction

Based on work with Dmitry Gal’tsov, Pavel Spirin and Theodore Tomaras Gravitational bremstrahlung two massive particles → complicated. Emission of scalar radiation of gravitationally interacting particles

→ intermediate step.

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 4 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

Introduction

Based on work with Dmitry Gal’tsov, Pavel Spirin and Theodore Tomaras Gravitational bremstrahlung two massive particles → complicated. Emission of scalar radiation of gravitationally interacting particles

→ intermediate step.

Two massive particles, one of which is coupled to a scalar field. The interaction is purely gravitational. Non-linear problem, because of the graviton - graviton - scalar interaction

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 4 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup

The action Equations of motion Iterative solution

Φ radiation The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

The action

We assume the ADD scenario and a space-time M4 × Td

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 5 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup

The action Equations of motion Iterative solution

Φ radiation The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

The action

We assume the ADD scenario and a space-time M4 × Td Two classical massive particles with mass m and m′ that interact

• nly gravitationally.

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 5 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup

The action Equations of motion Iterative solution

Φ radiation The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

The action

We assume the ADD scenario and a space-time M4 × Td Two classical massive particles with mass m and m′ that interact

• nly gravitationally.

A massless scalar field Φ that interacts with m but not m′.

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 5 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup

The action Equations of motion Iterative solution

Φ radiation The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

The action

We assume the ADD scenario and a space-time M4 × Td Two classical massive particles with mass m and m′ that interact

• nly gravitationally.

A massless scalar field Φ that interacts with m but not m′. The action will be: S = Sg + SΦ + Sm + Sm′ S =

• dDx
• |g|
• − R

κ2

D

+ 1

2 gMN∂MΦ ∂NΦ

• − 1

2 e gMN˙ zM ˙ zN+ (m+fΦ)2 e

• dτ − 1

2 e′gMN˙ z′M ˙ z′N+ m′2 e′

• dτ ′

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 5 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup

The action Equations of motion Iterative solution

Φ radiation The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

Equations of motion

The equations of motion for the two particles are produced by varying the action: d dτ

• eg′

MN ˙

zN

= e

2 g′

LR,M ˙

zL˙ zR, d dτ

• e′gMN ˙

z′N

= e′

2 gLR,M ˙ z′L˙ z′R,

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 6 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup

The action Equations of motion Iterative solution

Φ radiation The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

Equations of motion

The equations of motion for the two particles are produced by varying the action: d dτ

• eg′

MN ˙

zN

= e

2 g′

LR,M ˙

zL˙ zR, d dτ

• e′gMN ˙

z′N

= e′

2 gLR,M ˙ z′L˙ z′R, while variation with respect to the einbeins gives e−2 = g′

MN˙

zM ˙ zN

(m + fΦ)2

e′−2 = gMN˙ z′M ˙ z′N m′2 Plugging these back into the action and varying with respect to

Φ, we obtain the scalar field EOM: DΦ = −κD

2 h′DΦ+κDh′

MNΦ,MN+f

• (g′

MN˙

zM ˙ zN)1/2δD(x−z(τ)) dτ,

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 6 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup

The action Equations of motion Iterative solution

Φ radiation The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

Equations of motion

It is sufficient to restrict ourselves to linearized gravity and the EOM are Dh

MN = −κD

• T

MN − η MN

T D − 2

• , T

MN =

• e˙

z

M˙

z

N δD(x − z(τ))

√−g′

dτ,

Similarly, Dh′MN = −κD

• T ′MN − η

MN

T ′ D − 2

• , T ′MN =
• e′˙

z′M˙ z′N δD(x − z′(τ))

√−g

dτ.

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 7 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup

The action Equations of motion Iterative solution

Φ radiation The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

Perturbation theory

To solve the EOM, we will apply perturbation theory, with respect to f, the scalar charge and with respect to the gravitational coupling constant.

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 8 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup

The action Equations of motion Iterative solution

Φ radiation The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

Perturbation theory

To solve the EOM, we will apply perturbation theory, with respect to f, the scalar charge and with respect to the gravitational coupling constant. Zeroth order: the two particles simply move in straight lines, gravitational and scalar field are absent.

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 8 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup

The action Equations of motion Iterative solution

Φ radiation The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

Perturbation theory

To solve the EOM, we will apply perturbation theory, with respect to f, the scalar charge and with respect to the gravitational coupling constant. Zeroth order: the two particles simply move in straight lines, gravitational and scalar field are absent. First order: deviation from straight trajectory caused by the gravitational field produced by each particle in the zeroth order. Leading contribution to the radiation due to the acceleration of the charged particle, m.

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 8 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup

The action Equations of motion Iterative solution

Φ radiation The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

Perturbation theory

To solve the EOM, we will apply perturbation theory, with respect to f, the scalar charge and with respect to the gravitational coupling constant. Zeroth order: the two particles simply move in straight lines, gravitational and scalar field are absent. First order: deviation from straight trajectory caused by the gravitational field produced by each particle in the zeroth order. Leading contribution to the radiation due to the acceleration of the charged particle, m. Note: our solution is valid only if the deviations from straight trajectories are small and the iterative solution is convergent.

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 8 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup

The action Equations of motion Iterative solution

Φ radiation The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

Validity of the approximation(small deviation from flat metric)

With the two masses of the particles taken at the same order and eventually equal, the model is characterized by three classical parameters: The classical radius of the charge: rf =

• f 2

m

• 1

d+1

,

the D-dimensional gravitational radius of the mass m at rest rg =

• κ2

Dm

• 1

d+1 ,

the Schwarzschild radius of the black hole, associated with the collision energy √ s : rS = 1

√π

• 8Γ

d+3

2

• d + 2
• 1

d+1

GD

√

s c4

• 1

d+1

.

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 9 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup

The action Equations of motion Iterative solution

Φ radiation The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

Validity of the approximation

In the rest frame, √ s = 2mm′γ, so rs ∼ rgγν,

ν =

1 2 (d + 1) We assume that rg ∼ rf ≪ bγ−2ν,

• r equivalently

b ≫ rsγν Under this condition the deviation of the metric from unity in the rest frame of m′ is small, i.e. κDhMN ˙ z′M ˙ z′N ≪ 1

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 10 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup

The action Equations of motion Iterative solution

Φ radiation The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

The formal expansion

The world line of the two particles are: zM = 0zM +

1zM + . . . , 0zM = uMτ + bM,

z′M = 0z′M + 1z′M + . . . ,

0z′M = u′Mτ,

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 11 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup

The action Equations of motion Iterative solution

Φ radiation The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

The formal expansion

The world line of the two particles are: zM = 0zM +

1zM + . . . , 0zM = uMτ + bM,

z′M = 0z′M + 1z′M + . . . ,

0z′M = u′Mτ,

The scalar field is expanded:

Φ = 0Φ + 1Φ + . . . .

With EOM at zeroth order

D

0Φ = f

• δD(x − uτ − b) dτ.

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 11 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup

The action Equations of motion Iterative solution

Φ radiation The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

The formal expansion

The world line of the two particles are: zM = 0zM +

1zM + . . . , 0zM = uMτ + bM,

z′M = 0z′M + 1z′M + . . . ,

0z′M = u′Mτ,

The scalar field is expanded:

Φ = 0Φ + 1Φ + . . . .

With EOM at zeroth order

D

0Φ = f

• δD(x − uτ − b) dτ.

Finally for the metrics: hMN = 0hMN + 1hMN + . . . , and similarly for h′

• MN. The sources:

0T MN = m

• δD(x− 0z(τ)) uMuNdτ,

0T ′MN = m

• δD(x− 0z′(τ)) u′Mu′Ndτ

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 11 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup

The action Equations of motion Iterative solution

Φ radiation The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

First order EOM

By choosing an appropriate gauge, the EOM of the two particles read:

ΠMN 1 ¨

zN = −κDΠMN

• h′

NL,R − 1

2h′

LR,N

• uLuR,

(1)

Π′MN 1¨

z′N = −κDΠ′MN

• hNL,R − 1

2hLR,N

• u′Lu′R,

where the projectors onto the space transverse to the world-lines are

ΠMN = ηMN − uMuN, Π′MN = ηMN − u′Mu′N ,

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 12 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup

The action Equations of motion Iterative solution

Φ radiation The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

Second order equation for Φ radiation

As mentioned before, the radiation comes from the first order in Φ, the first order EOM of Φ is

D

1Φ(x, y) = j(x, y) ≡ ρ(x, y) + σ(x, y),

where the first term is localized on the world-line of the radiating particle m

ρ(x, y) = −f

• 1zµ(τ) ∂µδ4(x − uτ − b) δd(y)dτ,

while the second is the non-local current

σ(x, y) = κD ∂M

• h′MN ∂N

0Φ − 1

2h′ ∂M 0Φ

• .

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 13 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup

The action Equations of motion Iterative solution

Φ radiation The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

Φ radiation

The momentum radiated due to scalar bremstrahlung: Pµ =

• V

ddy

• Ω

∂NT Nµd4x =

• V

ddy

• Ω

(∂µΦ) DΦ d4x.

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 14 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup

The action Equations of motion Iterative solution

Φ radiation The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

Φ radiation

The momentum radiated due to scalar bremstrahlung: Pµ =

• V

ddy

• Ω

∂NT Nµd4x =

• V

ddy

• Ω

(∂µΦ) DΦ d4x.

Once we substitute Φ, Pµ = 1 16π3V

• n
• d3k

k0 kµ |jn(k)|2

• k0=√

k2+k2

T

,

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 14 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup

The action Equations of motion Iterative solution

Φ radiation The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

Φ radiation

The momentum radiated due to scalar bremstrahlung: Pµ =

• V

ddy

• Ω

∂NT Nµd4x =

• V

ddy

• Ω

(∂µΦ) DΦ d4x.

Once we substitute Φ, Pµ = 1 16π3V

• n
• d3k

k0 kµ |jn(k)|2

• k0=√

k2+k2

T

,

Finally the energy emitted is:

dE d|k|dΩ2 = 1 16π3V

• n

k

2|j n(k)| 2,

dΩ2 = sin θ dθ dϕ.

Assuming that the impact parameter b ≪ R, the compactification radius, the sum over the KK modes can be approximated as an integral and can be computed easily: dE dωdΩd+2

= ωd+2

2(2π)d+3 |j(k)|2.

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 14 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude

The local amplitude The non-local amplitude The part j

n z (k) of the radiation

amplitude and destructive interference The part j

n z′ (k) of the

amplitude Summary

The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

The source

What remains now is to find ρn(k) and σn(k) and substitute in the previous formula. We need to Fourrier transform the expression we saw before:

ρ(x, y) = −f

• 1zµ(τ) ∂µδ4(x − uτ − b) δd(y)dτ,

σ(x, y) = κD ∂M

• h′MN ∂N

0Φ − 1

2h′ ∂M 0Φ

• ,

and substitute z(k) and h in them.

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 15 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude

The local amplitude The non-local amplitude The part j

n z (k) of the radiation

amplitude and destructive interference The part j

n z′ (k) of the

amplitude Summary

The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

The local amplitude

After some calculations the local part of the source can be expressed in terms of the MacDonald functions:

ρn(k) = − κ2 Dm′f 4πvV ei(kb) l        2− γ2 ∗ v2γ2   z′ z − 2 γ   1 d + 2 − γ2 ∗ 2v2γ2    K0(zl )−i γ2 ∗ γ2 (kb) v2γz2 ˆ K1(zl )    .

where: z ≡ (ku)b

γv ,

z′ ≡ (ku′)b

γv ,

zl ≡ (z2 + p2

T b2)1/2 ,

the hatted Macdonald functions defined by ˆ Kν(x) ≡ xνKν(x) and γ2

∗ ≡ γ2 − (d + 2)−1

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 16 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude

The local amplitude The non-local amplitude The part j

n z (k) of the radiation

amplitude and destructive interference The part j

n z′ (k) of the

amplitude Summary

The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

The local amplitude

After some calculations the local part of the source can be expressed in terms of the MacDonald functions:

ρn(k) = − κ2 Dm′f 4πvV ei(kb) l        2− γ2 ∗ v2γ2   z′ z − 2 γ   1 d + 2 − γ2 ∗ 2v2γ2    K0(zl )−i γ2 ∗ γ2 (kb) v2γz2 ˆ K1(zl )    .

where: z ≡ (ku)b

γv ,

z′ ≡ (ku′)b

γv ,

zl ≡ (z2 + p2

T b2)1/2 ,

the hatted Macdonald functions defined by ˆ Kν(x) ≡ xνKν(x) and γ2

∗ ≡ γ2 − (d + 2)−1

After taking the ultra-relativistic limit and accounting for the effective number of interaction modes,

ρn(k) ≃ − λei(kb) v

• z′

z ˆ Kd/2(z)−i (kb) γz2 ˆ Kd/2+1(z) + 1 (d + 2)γ

• d −

(d + 1)z′ γz

• ˆ

Kd/2(z) + ...

• ,

where λ ≡ κ2

Dm′f

2(2π)d/2+1bd .

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 16 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude

The local amplitude The non-local amplitude The part j

n z (k) of the radiation

amplitude and destructive interference The part j

n z′ (k) of the

amplitude Summary

The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

The non-local amplitude

We write the non-local amplitude as a sum of two terms, one depending on z and one depending on z’ as follows:

σn(k) ≡ σn

0(k) + σn 1(k) ,

σn 0 (k) = λ ei(kb) γvz′2 a2 ξ2   βˆ Kd/2(z)−i(kb)ˆ Kd/2+1(z)− (d +1)β a2 ˆ Kd/2+1(z)+ β sin2φ a2 ˆ Kd/2+2(z)   ,

and

σn

1(k) ≃ λγvz′2

a2 ξ2

• (ξ2 − β) ˆ

Kd/2(z′) + i(kb) ˆ Kd/2+1(z′)

• ,

Similarly we will separate the total amplitude in a z and z’ part.

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 17 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude

The local amplitude The non-local amplitude The part j

n z (k) of the radiation

amplitude and destructive interference The part j

n z′ (k) of the

amplitude Summary

The emitted Energy - Spectral and Angular Distribution Summary of Results Future work Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 18 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude

The local amplitude The non-local amplitude The part j

n z (k) of the radiation

amplitude and destructive interference The part j

n z′ (k) of the

amplitude Summary

The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

jn

z in the frequency range ω ≫ γ/b In the regime with ϑ ∼ 1, i.e. z ∼ γ the amplitude decays exponentially with γ because of the MacDonald functions.

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 19 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude

The local amplitude The non-local amplitude The part j

n z (k) of the radiation

amplitude and destructive interference The part j

n z′ (k) of the

amplitude Summary

The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

jn

z in the frequency range ω ≫ γ/b In the regime with ϑ ∼ 1, i.e. z ∼ γ the amplitude decays exponentially with γ because of the MacDonald functions. In the case of ϑ ∼ 1/γ, in which z ∼ 1. Adding ρn(k) and σn

0(k)

and using the ultra-relativistic expansions we obtain in leading

• rder:

jn z (k) ≃ λ (d + 1)ei(kb) γψ   2ψ − γ−2 d + 2 ˆ Kd/2(z) − cos2 α ψ2ω2b2 sin2θ + tan2α

• ˆ

Kd/2+1(z) − sin2 θ sin2ϕ + tan2α d + 1 ˆ Kd/2+2(z)     . Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 19 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude

The local amplitude The non-local amplitude The part j

n z (k) of the radiation

amplitude and destructive interference The part j

n z′ (k) of the

amplitude Summary

The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

jn

z in the frequency range ω ≫ γ/b In the regime with ϑ ∼ 1, i.e. z ∼ γ the amplitude decays exponentially with γ because of the MacDonald functions. In the case of ϑ ∼ 1/γ, in which z ∼ 1. Adding ρn(k) and σn

0(k)

and using the ultra-relativistic expansions we obtain in leading

• rder:

jn z (k) ≃ λ (d + 1)ei(kb) γψ   2ψ − γ−2 d + 2 ˆ Kd/2(z) − cos2 α ψ2ω2b2 sin2θ + tan2α

• ˆ

Kd/2+1(z) − sin2 θ sin2ϕ + tan2α d + 1 ˆ Kd/2+2(z)     .

The terms in this expression are of order γ−1. The terms of order γ and 1 in the two ultra-relativistic expressions have opposite signs and cancel in the sum.

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 19 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude

The local amplitude The non-local amplitude The part j

n z (k) of the radiation

amplitude and destructive interference The part j

n z′ (k) of the

amplitude Summary

The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

jn

z in the frequency range ω ≫ γ/b In the regime with ϑ ∼ 1, i.e. z ∼ γ the amplitude decays exponentially with γ because of the MacDonald functions. In the case of ϑ ∼ 1/γ, in which z ∼ 1. Adding ρn(k) and σn

0(k)

and using the ultra-relativistic expansions we obtain in leading

• rder:

jn z (k) ≃ λ (d + 1)ei(kb) γψ   2ψ − γ−2 d + 2 ˆ Kd/2(z) − cos2 α ψ2ω2b2 sin2θ + tan2α

• ˆ

Kd/2+1(z) − sin2 θ sin2ϕ + tan2α d + 1 ˆ Kd/2+2(z)     .

The terms in this expression are of order γ−1. The terms of order γ and 1 in the two ultra-relativistic expressions have opposite signs and cancel in the sum. This is a general phenomenon of destructive interference related to the gravitational interaction. The two leading powers in the ultra-relativistic expansion of the direct Φ emission are cancelled by the indirect emission due to Φ − Φ-h interaction. The consequence is that the z-type frequency is highly suppressed in this limit, contrary to what one would expect naively.

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 19 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude

The local amplitude The non-local amplitude The part j

n z (k) of the radiation

amplitude and destructive interference The part j

n z′ (k) of the

amplitude Summary

The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

jn

z in the frequency range ω γ/b For ω ≪ γ/b and ϑ ∼ 1/γ |ρn| ≫ |σn| and, therefore, jn(k)

• ω≪γ/b ≃ ρn(k) ≃ −λ
• 1

γψ ˆ

Kd/2(z) + i sin ϑ cosφ

γψ2ωb ˆ

Kd/2+1(z)

• .

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 20 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude

The local amplitude The non-local amplitude The part j

n z (k) of the radiation

amplitude and destructive interference The part j

n z′ (k) of the

amplitude Summary

The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

jn

z in the frequency range ω γ/b For ω ≪ γ/b and ϑ ∼ 1/γ |ρn| ≫ |σn| and, therefore, jn(k)

• ω≪γ/b ≃ ρn(k) ≃ −λ
• 1

γψ ˆ

Kd/2(z) + i sin ϑ cosφ

γψ2ωb ˆ

Kd/2+1(z)

• .

For ϑ ∼ 1, on the other hand, ρn, σn

0 and σn 1 are all of the same

• rder, but suppressed compared to the previous case. In addition,

the contribution of this regime to the emitted energy is further suppressed by the integration measure.

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 20 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude

The local amplitude The non-local amplitude The part j

n z (k) of the radiation

amplitude and destructive interference The part j

n z′ (k) of the

amplitude Summary

The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

jn

z in the frequency range ω γ/b For ω ≪ γ/b and ϑ ∼ 1/γ |ρn| ≫ |σn| and, therefore, jn(k)

• ω≪γ/b ≃ ρn(k) ≃ −λ
• 1

γψ ˆ

Kd/2(z) + i sin ϑ cosφ

γψ2ωb ˆ

Kd/2+1(z)

• .

For ϑ ∼ 1, on the other hand, ρn, σn

0 and σn 1 are all of the same

• rder, but suppressed compared to the previous case. In addition,

the contribution of this regime to the emitted energy is further suppressed by the integration measure. More interesting is the case with ω ∼ γ/b. If ϑ ∼ 1, then z ∼ γ, jn

z

is exponentially suppressed because of the Macdonald functions.

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 20 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude

The local amplitude The non-local amplitude The part j

n z (k) of the radiation

amplitude and destructive interference The part j

n z′ (k) of the

amplitude Summary

The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

jn

z in the frequency range ω γ/b For ω ≪ γ/b and ϑ ∼ 1/γ |ρn| ≫ |σn| and, therefore, jn(k)

• ω≪γ/b ≃ ρn(k) ≃ −λ
• 1

γψ ˆ

Kd/2(z) + i sin ϑ cosφ

γψ2ωb ˆ

Kd/2+1(z)

• .

For ϑ ∼ 1, on the other hand, ρn, σn

0 and σn 1 are all of the same

• rder, but suppressed compared to the previous case. In addition,

the contribution of this regime to the emitted energy is further suppressed by the integration measure. More interesting is the case with ω ∼ γ/b. If ϑ ∼ 1, then z ∼ γ, jn

z

is exponentially suppressed because of the Macdonald functions. However, for ϑ ∼ 1/γ, ρn ∼ γ and σn

0 ∼ γ.

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 20 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude

The local amplitude The non-local amplitude The part j

n z (k) of the radiation

amplitude and destructive interference The part j

n z′ (k) of the

amplitude Summary

The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

The part jn

z′(k) of the amplitude Using the angles ϑ and φ the expression can be written as: jn

z′ ≃ − λ

γψ

• 1

γ2ψ − 1

• ˆ

Kd/2 (z′) + i sin ϑ cosφ

γz′ψ ˆ

Kd/2+1 (z′)

• Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC)

21 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude

The local amplitude The non-local amplitude The part j

n z (k) of the radiation

amplitude and destructive interference The part j

n z′ (k) of the

amplitude Summary

The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

The part jn

z′(k) of the amplitude Using the angles ϑ and φ the expression can be written as: jn

z′ ≃ − λ

γψ

• 1

γ2ψ − 1

• ˆ

Kd/2 (z′) + i sin ϑ cosφ

γz′ψ ˆ

Kd/2+1 (z′)

• For ω ≫ γ/b one has z′ ≫ 1 and, consequently, jn

z′ is

exponentially suppressed.

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 21 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude

The local amplitude The non-local amplitude The part j

n z (k) of the radiation

amplitude and destructive interference The part j

n z′ (k) of the

amplitude Summary

The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

The part jn

z′(k) of the amplitude Using the angles ϑ and φ the expression can be written as: jn

z′ ≃ − λ

γψ

• 1

γ2ψ − 1

• ˆ

Kd/2 (z′) + i sin ϑ cosφ

γz′ψ ˆ

Kd/2+1 (z′)

• For ω ≫ γ/b one has z′ ≫ 1 and, consequently, jn

z′ is

exponentially suppressed. For (ω ∼ γ/b, ϑ ∼ 1) jn

z′ it is dominated by its real part which is of

• rder O(1/γ).

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 21 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude

The local amplitude The non-local amplitude The part j

n z (k) of the radiation

amplitude and destructive interference The part j

n z′ (k) of the

amplitude Summary

The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

The part jn

z′(k) of the amplitude Using the angles ϑ and φ the expression can be written as: jn

z′ ≃ − λ

γψ

• 1

γ2ψ − 1

• ˆ

Kd/2 (z′) + i sin ϑ cosφ

γz′ψ ˆ

Kd/2+1 (z′)

• For ω ≫ γ/b one has z′ ≫ 1 and, consequently, jn

z′ is

exponentially suppressed. For (ω ∼ γ/b, ϑ ∼ 1) jn

z′ it is dominated by its real part which is of

• rder O(1/γ).

For (ω ≪ γ/b, ϑ ∼ 1) one obtains jn

z′ ∼ σn 0 ∼ 1/γ, however, this

region contributes negligibly little to the emitted energy.

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 21 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude

The local amplitude The non-local amplitude The part j

n z (k) of the radiation

amplitude and destructive interference The part j

n z′ (k) of the

amplitude Summary

The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

The part jn

z′(k) of the amplitude Using the angles ϑ and φ the expression can be written as: jn

z′ ≃ − λ

γψ

• 1

γ2ψ − 1

• ˆ

Kd/2 (z′) + i sin ϑ cosφ

γz′ψ ˆ

Kd/2+1 (z′)

• For ω ≫ γ/b one has z′ ≫ 1 and, consequently, jn

z′ is

exponentially suppressed. For (ω ∼ γ/b, ϑ ∼ 1) jn

z′ it is dominated by its real part which is of

• rder O(1/γ).

For (ω ≪ γ/b, ϑ ∼ 1) one obtains jn

z′ ∼ σn 0 ∼ 1/γ, however, this

region contributes negligibly little to the emitted energy. Similarly, for (ω ≪ γ/b, ϑ ∼ 1/γ) the amplitude is jn

z′ ∼ σn 0 ≪ ρn ∼ γ2.

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 21 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude

The local amplitude The non-local amplitude The part j

n z (k) of the radiation

amplitude and destructive interference The part j

n z′ (k) of the

amplitude Summary

The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

The part jn

z′(k) of the amplitude Using the angles ϑ and φ the expression can be written as: jn

z′ ≃ − λ

γψ

• 1

γ2ψ − 1

• ˆ

Kd/2 (z′) + i sin ϑ cosφ

γz′ψ ˆ

Kd/2+1 (z′)

• For ω ≫ γ/b one has z′ ≫ 1 and, consequently, jn

z′ is

exponentially suppressed. For (ω ∼ γ/b, ϑ ∼ 1) jn

z′ it is dominated by its real part which is of

• rder O(1/γ).

For (ω ≪ γ/b, ϑ ∼ 1) one obtains jn

z′ ∼ σn 0 ∼ 1/γ, however, this

region contributes negligibly little to the emitted energy. Similarly, for (ω ≪ γ/b, ϑ ∼ 1/γ) the amplitude is jn

z′ ∼ σn 0 ≪ ρn ∼ γ2.

Finally, based on numerical study and previous results in D = 4

• ne obtains that this expression is valid also in the regime

(ω ∼ γ/b, ϑ ∼ 1/γ) and gives jn

z′ ∼ γ.

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 21 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude

The local amplitude The non-local amplitude The part j

n z (k) of the radiation

amplitude and destructive interference The part j

n z′ (k) of the

amplitude Summary

The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

Summary

The behaviour of the local and non-local currents in all characteristic frequency and angular regimes is summarized in the following Table.

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 22 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

The emitted energy

Now that we have computed and studied j, we have to integrate to get the total energy radiated. We have split j in two parts, jn

z

and jn

z′ so it useful to split the energy in three pieces proportional

to |jn

z (k)|2, |jn z′(k)|2 and jn z jn z′ + jn z jn z′

dE = dEz + dEz′ + dEzz′ ,

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 23 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

The emitted energy

Now that we have computed and studied j, we have to integrate to get the total energy radiated. We have split j in two parts, jn

z

and jn

z′ so it useful to split the energy in three pieces proportional

to |jn

z (k)|2, |jn z′(k)|2 and jn z jn z′ + jn z jn z′

dE = dEz + dEz′ + dEzz′ , The general expression of the emitted energy will be E ∼ 1 8(2π)2d+5

κ4

Dm′2f 2

b3d+3

γ# , Φ radiation is emitted in well-defined, relatively narrow windows,

with the amplitudes shown in the previous table. Thus it is straightforward to compute the powers of γ, since the range of integration does not introduce further powers of γ.

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 23 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

Frequency and angular distribution for d=0 and

γ = 105

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 24 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

Frequency and angular distribution for d=0 and

γ = 105

The powers of γ are shown in the following table

❍❍❍❍❍

ϑ ω ω ≪ γ/b ω ∼ γ/b ω ∼ γ2/b ω ≫ γ2/b γ−1 negligible (phase space) Ed ∼ γ3 , from jn

z and jn z′

Ed ∼ γd+2, from jn

z

negligible radiation 1 negligible (phase space) Ed ∼ γd+1, from jn

z′

negligible radiation negligible radiation

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 24 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

Let us now look at the energy emitted from the important cells of the table: According to Table II, the z-type radiation (due to |jn

z |2) is always

beamed inside ϑ ∼ 1/γ. Furthermore, for d 2 it is dominant with characteristic frequency ω ∼ γ2/b. The cases d = 0 and d = 1 will be treated separately. Ez = Cd

κ4

Dm′2f 2

b3d+3 γd+2 with C2 = 1.42 × 10−6, C3 = 6.02 × 10−7, C4 = 3.45 × 10−7, C5 = 2.67 × 10−7 and C6 = 2.76 × 10−7.

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 25 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

Let us now look at the energy emitted from the important cells of the table: According to Table II, the z-type radiation (due to |jn

z |2) is always

beamed inside ϑ ∼ 1/γ. Furthermore, for d 2 it is dominant with characteristic frequency ω ∼ γ2/b. The cases d = 0 and d = 1 will be treated separately. Ez = Cd

κ4

Dm′2f 2

b3d+3 γd+2 with C2 = 1.42 × 10−6, C3 = 6.02 × 10−7, C4 = 3.45 × 10−7, C5 = 2.67 × 10−7 and C6 = 2.76 × 10−7. Wide angle radiation (ϑ ∼ 1) is mainly z′-type (due to |jn

z′|2) in all

dimensions and has characteristic frequency ω ∼ γ/b. Also, for d 3 radiation with ω ∼ γ/b is predominantly emitted in wide

• angles. For d 3 the emitted energy is given by

Ez′ = C′

d

κ4

Dm′2f 2

b3d+3 γd+1 , C′

d =

2d−8Γ

3d+3

2

• Γ2 2d+3

2

• Γ

d+3

2

• Γ

d−2

2

• π3d/2+4Γ(2d + 3)Γ(d)

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 25 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

While for d = 2 one obtains Ez′ = 105κ4

6m′2f 2

216(2π)7b9 γ3 ln γ .

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 26 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

While for d = 2 one obtains Ez′ = 105κ4

6m′2f 2

216(2π)7b9 γ3 ln γ . According to Table II the emitted energy in 4D is concentrated in the region ω ∼ γ/b, θ ∼ γ−1. The total emitted energy is E0 = C0

κ4

4m′2f 2

b3

γ3 ,

C0 ≈ 8.3 × 10−5 . and E1 = C1

κ4

5m′2f 2

b6

γ3 ln γ ,

C1 ≈ 1.64 × 10−5 .

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 26 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

The total radiated energy Ed depends on the phase-space in an intricate way, so that the resulting radiation does not have a simple universal expression. Specifically, it was found that in the absence of extra dimensions one obtains E0 = C0 m

rg

b

2 rf

b

• γ3,

C0 ≈ 8.3 × 10−5, with the ‘‘basic’’ relativistic enhancement factor γ3. For one extra dimension one has E1 = C1 m

rg

b

4 rf

b

2 γ3 ln γ,

C1 ≈ 1.64 × 10−5 , with almost the same (up to the logarithm) enhancement factor. For d 2 one finds Ed = Cd m

rg

b

2(d+1) rf

b

d+1 γd+2 ,

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 27 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

Future work

Extend to vector radiation

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 28 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

Future work

Extend to vector radiation Extend to full gravitational problem

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 28 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

Future work

Extend to vector radiation Extend to full gravitational problem Include back - reaction

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 28 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

THANK YOU!

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 29 / 29

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions arXiv:1106.3509 Motivation Introduction General setup The radiation amplitude The emitted Energy - Spectral and Angular Distribution Summary of Results Future work

Transplanckian gravity is classical

λB = c √

s , rS = 1

√π

• 8Γ

d+3

2

• d + 2
• 1

d+1

GD

√

s c4

• 1

d+1

.

(2) l∗ =

• GD/c31/(d+2) = /M∗c

(3) The system is classical if:

→ 0, λB ≪ l∗ ≪ rs

Or equivalently if √ s ≫ M∗

Scalar Bremsstrahlung in Gravity-Mediated Ultrarelativistic Collisions : arXiv:1106.3509 (Physics Department, UoC) 30 / 29