General Relativity as an Effective Field Theoryfaller/talks/THSeminar_2006.pdf · General Relativity as an Effective Field Theory Sven Faller Theoretical Physics 1 University of Siegen

IntroductionQuantum Gravity

Effective Field Theory of GravityLeading Quantum Corrections

Evaluation of the Vertex CorrectionsGravitational Potential

Potential DefinitionsSummary

General Relativity as an Effective Field Theory

Sven Faller

Theoretical Physics 1University of Siegen

Theory Seminar 18.12.2006

Sven Faller General Relativity as an Effective Field Theory





table of contents

1 Introduction

2 Quantum Gravity

3 Effective Field Theory of Gravity

4 Leading Quantum Corrections

5 Evaluation of the Vertex Corrections

6 Gravitational Potential

7 Potential Definitions

8 Summary






FundamentalsGeneral Relativity

Motivation

all known field theories: quantum field theoriesgravity quantization - Feynman´s Gedankenexperiment

gravity must be a quantum field theoryproblem: consistent quantization method unknown

Quantum Gravity: De Witt, Feynman, ’t Hooft and Veltmanpresent energies, quantum gravity non-renormalizable

low-energy predictions independent of high-energyinfluenceDonoghue:possible solution Effective Field Theory of Gravity







Motivation











Motivation











Motivation











Motivation











Motivation











Motivation











Motivation











Motivation of Quantization

Feynman´s Gedankenexperiment: two-slit diffractionexperiment with gravity detector

characteristic for a quantum field⇒ should be describedby an amplitude rather than a probability







Motivation of Quantization

Feynman´s Gedankenexperiment: two-slit diffractionexperiment with gravity detector

characteristic for a quantum field⇒ should be describedby an amplitude rather than a probability







Newton’s Laws (1687)

law of inertia

no external force : d~rdt = ~v = const .

⇒ inertial frame of reference (IS)

second law

force ∝ inertia mass mi ⇒ ~F = mi · ~a.

third lawactio est reactio








law of inertia



second law










law of inertia



second law









Newton’s Relativity

three different masses: inertia mass mi , passivegravitational mass mG and active gravitational mass MG

third law: passive and active gravitational mass equalforce of gravity

~F12(~r) = −G m1 m2~r1 −~r2

|~r1 −~r2|3

problem: equality of inertia and passive massesexperimental measurements: verification of equality,bases for Einstein’s Principle of Equivalence










~F12(~r) = −G m1 m2~r1 −~r2

|~r1 −~r2|3











~F12(~r) = −G m1 m2~r1 −~r2

|~r1 −~r2|3











~F12(~r) = −G m1 m2~r1 −~r2

|~r1 −~r2|3











~F12(~r) = −G m1 m2~r1 −~r2

|~r1 −~r2|3








Einstein’s Special Relativity

Newton: Galilei transformations between ISEinstein 1905: Newton’s Theory must be specialized byuniversality of the velocity of light in all frames

x 7−→ x ′ = ΛΛΛx + a (Lorentz transformation)

Postulategeneral transformation for the line element must satisfy

ds2 = ηαβ dxµ dxν = c2 dt2 − d~x2





























General Relativity

Einstein (1916):

Die Grundlagen der allgemeinen Relativitätstheorie. Ann. d. Physik,

322(10):891-921

Newton: space R3 and parameter time Rt

Einstein : new relations between space-time and mass⇒ curved space-time mannifoldcurvature of space = measure for mass:

„matter tells space how to curve, and space tells matterhow to move“ (Miesner, 1973)







General Relativity

Einstein (1916):


322(10):891-921










General Relativity

Einstein (1916):


322(10):891-921










General Relativity

Einstein (1916):


322(10):891-921










General Relativity

Einstein (1916):


322(10):891-921










Principle of Equivalence

„At every space-time point in an arbitrary gravitational field it is possible tochoose a „locally inertial coordinate system“ such that, within sufficientlysmall region of the point in question, the laws of nature take the same form asin unaccelerated Cartesian coordinate systems in absence of gravitation.“

relation between accelerated local IS xα and static frameof reference xµ described by metric tensor, which leavesline element ds2 invariant:

gµν = ηαβ∂xα

∂xµ∂xβ

∂xν











∂xµ∂xβ

∂xν











∂xµ∂xβ

∂xν







Principle of General Covariance

General laws of nature should be expressed in terms ofequations which are true in all frames of reference andtransform covariantly by arbitrary substitutions.

general coordinate transformation: x 7−→ x ′ = f (x)

Principle of General Covariance is not an invarianceprinciple like Principle of Galilean or Special Relativity



























Recall: Lorentz Invariance

global coordinate change: xµ 7−→ x ′µ = Λµν xν

Minkowski metric ηµν invariantfields transform as scalars, vectors, etc.

φ(x) 7−→ φ′(x ′) = φ(x)

Aµ(x) 7−→ A′µ(x) = Λµν(x) Aν(x)










φ(x) 7−→ φ′(x ′) = φ(x)











φ(x) 7−→ φ′(x ′) = φ(x)








Covariant Derivative

local coordinate changes require covariant derivative:

Dµ Aν = ∂µAν + ΓνµλAλ = Aν,µ + ΓνµλAλ ≡ Aν;µ

affine connection Γλµν (geometric interpretation)for scalar fields: Φ;µ ≡ Φ,µ = ∂µΦ



























Riemann Space (R4)

metric definition: ds2 = gµν(x) dxµ dxν

affine connection: Γλµν = 12gλσ

(∂µgνσ + ∂νgνσ − ∂σgµν

)Ricci identity: Aµ;σν − Aµ;νσ ≡ Aλ Rλ

µσν

Riemann curvature tensor:

Rλµσν = Γλµν,σ − Γλµσ,ν + ΓλτσΓτµν − ΓλτνΓτµσ

Ricci tensor: Rµν = Rλµλν ≡ Rνµ

Ricci scalar: R = gµν Rµν







Riemann Space (R4)





µσν











Riemann Space (R4)





µσν











Riemann Space (R4)





µσν











Riemann Space (R4)





µσν











Riemann Space (R4)





µσν










Quantization

General relativity as a gauge theory (Sabbata 1985)

Poincaré group is non abeliancf. Yang-Mills theory

Lgauge = −14

F aµν F aµν = −1

2trF 2

gravity: introduction of vierbein- or tetrad fields e λµ

Lgauge = − e2g

eµλeνσR λσ

µν (ωωω) ≡ 2κ2

√−g R

with g = det[gµν ] and κ2 = 32πG.






Quantization



Lgauge = −14


2trF 2


Lgauge = − e2g

eµλeνσR λσ


√−g R







Quantization



Lgauge = −14


2trF 2


Lgauge = − e2g

eµλeνσR λσ


√−g R







Quantization

Background Field Method

introduced by ’t Hooft and Veltmann (1974)gravitational field expanded about smooth backgroundmetric gµν

gµν = gµν + κhµνgµν = gµν − κhµν + κ2hµλhλν + . . .

classical equations of motion: gµνquantum field hµν : all dynamical information






Quantization










Quantization










Quantization










Quantization

Gravitational Action

Einstein-Hilbert action Svac =∫

d4x√−g 2

κ2 R

matter action Sm =∫

d4x Lm

Sgr = Svac + Sm =

∫d4x

[√−g

2κ2 R + Lm

]further gauge invariant terms

L =√−g{λ+

2κ2 R + c1R

2 + c2RµνRµν + . . .

}+ Lm

upper bound: constants c1, c2 < 1074 (Stelle 1978)λ ≡ −8πGΛ, cosmological constant Λ ≡ 0






Quantization



d4x√−g 2

κ2 R


d4x Lm

Sgr = Svac + Sm =

∫d4x

[√−g

2κ2 R + Lm


L =√−g{λ+

2κ2 R + c1R

2 + c2RµνRµν + . . .

}+ Lm







Quantization



d4x√−g 2

κ2 R


d4x Lm

Sgr = Svac + Sm =

∫d4x

[√−g

2κ2 R + Lm


L =√−g{λ+

2κ2 R + c1R

2 + c2RµνRµν + . . .

}+ Lm







Quantization



d4x√−g 2

κ2 R


d4x Lm

Sgr = Svac + Sm =

∫d4x

[√−g

2κ2 R + Lm


L =√−g{λ+

2κ2 R + c1R

2 + c2RµνRµν + . . .

}+ Lm







Quantization



d4x√−g 2

κ2 R


d4x Lm

Sgr = Svac + Sm =

∫d4x

[√−g

2κ2 R + Lm


L =√−g{λ+

2κ2 R + c1R

2 + c2RµνRµν + . . .

}+ Lm







Quantization



d4x√−g 2

κ2 R


d4x Lm

Sgr = Svac + Sm =

∫d4x

[√−g

2κ2 R + Lm


L =√−g{λ+

2κ2 R + c1R

2 + c2RµνRµν + . . .

}+ Lm







Quantization

Expansion: Vacuum Lagrangian

metric expansion:p−g =

p−g

1−κ

2hαα −

κ2

4hαβhβα +

κ2

8

`hαα´2

+O(h3)

ffLagrangian expansion

2κ2

p−gR =

p−g»

2κ2

R + L(1)gr + L(2)

gr + . . .

–,

L(1)gr =

1κ

hµνˆgµνR − 2Rµν

˜,

L(2)gr =

12

Dαhµν Dαhµν −12

Dαh Dαh + Dαh Dβhαβ − Dαhµβ Dβhµα

+ R

„14

h2 −12

hµνhµν«

+ Rµν`2hλµhνλ − h hµν

´.






Quantization

Expansion: Vacuum Lagrangian

metric expansion:p−g =

p−g

1−κ

2hαα −

κ2

4hαβhβα +

κ2

8

`hαα´2

+O(h3)

ffLagrangian expansion

2κ2

p−gR =

p−g»

2κ2

R + L(1)gr + L(2)

gr + . . .

–,

L(1)gr =

1κ

hµνˆgµνR − 2Rµν

˜,

L(2)gr =

12

Dαhµν Dαhµν −12

Dαh Dαh + Dαh Dβhαβ − Dαhµβ Dβhµα

+ R

„14

h2 −12

hµνhµν«

+ Rµν`2hλµhνλ − h hµν

´.






Quantization

Expansion: Matter Lagrangian

e.g. scalar particle: Lm =√−g[1

2gµν∂µφ∂νφ− 12m2φ2]

Lagrangian expansion:

Lm =p−g˘L(0)

m + L(1)m + L(2)

m + . . .¯

L(0)m =

12

`∂µφ∂

µφ−m2φ2´L(1)

m = −κ

2hµνTµν

Tµν ≡ ∂µφ∂µφ−12

gµν`∂λφ∂

λφ−m2φ2´ (energy-momentum-tensor)

L(2)m = κ2

„12

hµλhνλ −14

hhµν«∂µφ∂νφ−

κ2

8

„hλσhλσ −

12

hh«ˆ∂µφ∂

µφ−m2φ2˜






Quantization

Expansion: Matter Lagrangian

e.g. scalar particle: Lm =√−g[1

2gµν∂µφ∂νφ− 12m2φ2]

Lagrangian expansion:

Lm =p−g˘L(0)

m + L(1)m + L(2)

m + . . .¯

L(0)m =

12

`∂µφ∂

µφ−m2φ2´L(1)

m = −κ

2hµνTµν

Tµν ≡ ∂µφ∂µφ−12

gµν`∂λφ∂

λφ−m2φ2´ (energy-momentum-tensor)

L(2)m = κ2

„12

hµλhνλ −14


κ2

8

„hλσhλσ −

12

hh«ˆ∂µφ∂

µφ−m2φ2˜






Quantization

Einstein Equation

Rµν −12

gµν R =κ2

4Tµν

gµν satisfies Einstein equationLagrangian terms linear in quantum field hµν vanishone-loop order:

L0 =p−g

2 R

κ2 + L(0)m

ffLquad =

p−g˘L(2)

g + Lgf + Lghost + L(2)m¯






Quantization

Einstein Equation

Rµν −12

gµν R =κ2

4Tµν


L0 =p−g

2 R

κ2 + L(0)m

ffLquad =

p−g˘L(2)







Quantization

Einstein Equation

Rµν −12

gµν R =κ2

4Tµν


L0 =p−g

2 R

κ2 + L(0)m

ffLquad =

p−g˘L(2)







Quantization

Einstein Equation

Rµν −12

gµν R =κ2

4Tµν


L0 =p−g

2 R

κ2 + L(0)m

ffLquad =

p−g˘L(2)







Quantization

Quantization Problems

field equations non linearcoupling constant κ has mass dimensioncoupling grows with energypossible solution: Effective Field Theoryseparate high enery fluctuations from small quantumfluctuations at ordinary energies






Quantization








Quantization








Quantization








Introduction

Effective Lagrangian

low-energy d.o.f.: hµν + ghost fields + matter fields

Z[J] =

Z[dφ][dhµν ]eiSeff(φ,g,h,J)

Seff =∫

d4x√−g Leff, Leff = Lgr + Lm

effective Lagrangian = expansion in powers of hµν

Lgr = L(0)gr + L(2)

gr + L(4)gr + . . .

Lm = L(0)m + L(2)

m + . . .






Introduction



Z[J] =


Seff =∫



Lgr = L(0)gr + L(2)

gr + L(4)gr + . . .

Lm = L(0)m + L(2)

m + . . .






Introduction



Z[J] =


Seff =∫



Lgr = L(0)gr + L(2)

gr + L(4)gr + . . .

Lm = L(0)m + L(2)

m + . . .






Feynman RulesScatteringPotentialVertex Corrections

Graviton Progpagator

second order Lagrangian Lgr

harmonic gauge→ gauge fixing Lagrangian Lgf

quantum field hµν bilinear Lagrangian Lfreegr = − 1

2 hαβ ∆−1αβγδ hγδ

graviton propagator in harmonic gauge

�qαβ µν =

12

iq2 + iε

(ηαµηβν + ηανηβµ − ηαβηµν

)













�qαβ µν =

12

iq2 + iε


)













�qαβ µν =

12

iq2 + iε


)













�qαβ µν =

12

iq2 + iε


)







Vertex Factors

vertex factors at one-loop order

�−→q

p

p′

�`′ ↗ p

`↖ p′

�−→k

↘ q

↗ `







Scalar-Graviton-Vertex

vertex factor

τµν = iZ

d4x d4x1 d4x2 d4x3 ei(px1−p′x2+qx3) · ∂

∂φ(x1)

∂

∂φ(x2)

∂

∂hµν(x3)

·−κ

2hαβ ·

»∂αφ(x)∂βφ(x)− 1

ηαβ`∂γφ(x)∂γφ(x)−m2φ(x)2´–ff

scalar-graviton-vertex

�−→q

p

p′

µν = − iκ2

{pµp′ν + pνp′µ − ηµν

[(p · p′

)−m2]}







Scalar-Graviton-Vertex

vertex factor

τµν = iZ

d4x d4x1 d4x2 d4x3 ei(px1−p′x2+qx3) · ∂

∂φ(x1)

∂

∂φ(x2)

∂

∂hµν(x3)

·−κ

2hαβ ·

»∂αφ(x)∂βφ(x)− 1

ηαβ`∂γφ(x)∂γφ(x)−m2φ(x)2´–ff

scalar-graviton-vertex

�−→q

p

p′

µν = − iκ2

{pµp′ν + pνp′µ − ηµν

[(p · p′

)−m2]}







S-Matrix

Feynman diagrams→ invariant matrix element iMlong range interaction:Mfull = Aq2(1+ακ2q2 +βκ2q2 ln(−q2)+γκ2q2 m√

−q2+ . . . )

R-matrix: R = S− 1p, pT incoming, p′T , p′ outgoing momentum:

〈p′|R|p〉 = (2π)4δ4(p′ + p′T − p − pT ) iM

Born approximation: nonrelativistic limit position-space potential

V (~r) = − 12m1

12m2

∫d3~q

(2π)3 ei~q·~r M







S-Matrix


−q2+ . . . )


〈p′|R|p〉 = (2π)4δ4(p′ + p′T − p − pT ) iM


V (~r) = − 12m1

12m2

∫d3~q

(2π)3 ei~q·~r M







S-Matrix


−q2+ . . . )


〈p′|R|p〉 = (2π)4δ4(p′ + p′T − p − pT ) iM


V (~r) = − 12m1

12m2

∫d3~q

(2π)3 ei~q·~r M







S-Matrix


−q2+ . . . )


〈p′|R|p〉 = (2π)4δ4(p′ + p′T − p − pT ) iM


V (~r) = − 12m1

12m2

∫d3~q

(2π)3 ei~q·~r M







S-Matrix


−q2+ . . . )


〈p′|R|p〉 = (2π)4δ4(p′ + p′T − p − pT ) iM


V (~r) = − 12m1

12m2

∫d3~q

(2π)3 ei~q·~r M







Expansion: Gravitational Potential

lowest order: V (r) = −G m1·m2r (Newton)

higher order effects: O(v2/c2), O(Gm/rc2)general form:

V (r) = −G m1 m2

r

[1 + a · G(m1 + m2)

r c2 . . .

]dimensional analysis: loop diagrams→ extra power ofκ2 ∼ G, factor ~

gravitational potential: general form

V (r) = −Gm1m2

r

(1 + α

G(m1 + m2)

rc2 + βG~

r2c3 + . . .

)Sven Faller General Relativity as an Effective Field Theory









V (r) = −G m1 m2

r

[1 + a · G(m1 + m2)

r c2 . . .



V (r) = −Gm1m2

r

(1 + α

G(m1 + m2)

rc2 + βG~

r2c3 + . . .










V (r) = −G m1 m2

r

[1 + a · G(m1 + m2)

r c2 . . .



V (r) = −Gm1m2

r

(1 + α

G(m1 + m2)

rc2 + βG~

r2c3 + . . .










V (r) = −G m1 m2

r

[1 + a · G(m1 + m2)

r c2 . . .



V (r) = −Gm1m2

r

(1 + α

G(m1 + m2)

rc2 + βG~

r2c3 + . . .










V (r) = −G m1 m2

r

[1 + a · G(m1 + m2)

r c2 . . .



V (r) = −Gm1m2

r

(1 + α

G(m1 + m2)

rc2 + βG~

r2c3 + . . .







Tree Level

iM =�kpk ′

q

p′

m1 m2 = ταβ(k , k ′) ·(

iPαβγδ

q2 + iε

)· τγδ(p,p′)

nonrelativitstic position space potential

V (~r) = −κ2

8m1m2

∫d3~q

(2π)3 ei~q·~r 1~q2 = −κ

2

8m1m2

14πr







Tree Level

iM =�kpk ′

q

p′

m1 m2 = ταβ(k , k ′) ·(

iPαβγδ

q2 + iε

)· τγδ(p,p′)

nonrelativitstic position space potential

V (~r) = −κ2

8m1m2

∫d3~q

(2π)3 ei~q·~r 1~q2 = −κ

2

8m1m2

14πr







Vertex Corrections - Overview

� =�(a)

+�(b)

+�(c)

+�(d)

+�(e)

+�(f)







General Form

QED: Ward identiy⇔ vertex: energy conservation∂µTµν = 0momentum conservation: qµVµν ≡ 0

general vertex form

�q k1

k2 Vµν = 〈 k2 |Tµν | k1 〉 (1)

= F1(q2)

[kµ1 kν2 + kν1 kµ2 +

12

q2 gµν]

(2)

+ F2(q2)[qµqν − gµνq2] . (3)







General Form


general vertex form

�q k1

k2 Vµν = 〈 k2 |Tµν | k1 〉 (1)

= F1(q2)


12

q2 gµν]

(2)

+ F2(q2)[qµqν − gµνq2] . (3)







General Form


general vertex form

�q k1

k2 Vµν = 〈 k2 |Tµν | k1 〉 (1)

= F1(q2)


12

q2 gµν]

(2)

+ F2(q2)[qµqν − gµνq2] . (3)







Form Factors

�tree-level limit−−−−−−−−→�

normalization condition: F1 ≡ 0F2 no normalization conditionform factors dimensionless







Form Factors









Form Factors









Form Factors








First Loop Diagram

�←−q

k2

k1

mµν

x`

= iPσραβ iPγδλκi∫

d4`

(2π)41

`2(`− q)2[(`− k2)2 −m2]

· τρσ(k2 − `, k2,m) τλκ(k1, k2 − `,m) τµναβγδ(`,q) .






Results

form factor F1(q2)

result from ln(−q2) π2 m√−q2

our - 3/4 1/16Donoghue - 3/4 1/16Akhundov et al. - 5/4 - 1/16

form factor F2(q2)


our 7/3 7/8Donoghue 3 7/8Akhundov et al. -7/3 - 7/8






Results

form factor F1(q2)


our - 3/4 1/16Donoghue - 3/4 1/16Akhundov et al. - 5/4 - 1/16

form factor F2(q2)


our 7/3 7/8Donoghue 3 7/8Akhundov et al. -7/3 - 7/8






Second Loop Diagram

�k1

k2

µν

x` m

= iPαβλκiPγδρσVαβγδ∫

d4`

(2π)4

τµνλκρσ(`,q)

`2(`− q)2






Results

form factor F1(q2)


our 0 0Donoghue 0 0Akhundov et al. 0 0

form factor F2(q2)


our -13/3 0Donoghue -13/3 0Akhundov et al. 7/3 0






Results

form factor F1(q2)


our 0 0Donoghue 0 0Akhundov et al. 0 0

form factor F2(q2)


our -13/3 0Donoghue -13/3 0Akhundov et al. 7/3 0






Result: Vertex Correctionform factors

F1(q2) = 1 +κ2q2

32π2

»−

34

ln(−q2) +1

16π2 mp−q2

–,

F2(q2) =κ2 m2

32π2

»−2 ln(−q2) +

78

π2 mp−q2

–.

tree-level normalized→ factor κ/2i

Vµν = −iκ2

»1 +

κ2 q2

32π2

„−

34

ln(−q2) +1

16π2 mp−q2

«–„kµ1 kν2 + kν1 kµ2 +

12

q2ηµν«

−κ3 m2

64π2

»−2 ln(−q2) +

78

π2 mp−q2

–„qµqν −

12

q2ηµν«.






Result: Vertex Correctionform factors

F1(q2) = 1 +κ2q2

32π2

»−

34

ln(−q2) +1

16π2 mp−q2

–,

F2(q2) =κ2 m2

32π2

»−2 ln(−q2) +

78

π2 mp−q2

–.

tree-level normalized→ factor κ/2i

Vµν = −iκ2

»1 +

κ2 q2

32π2

„−

34

ln(−q2) +1

16π2 mp−q2

«–„kµ1 kν2 + kν1 kµ2 +

12

q2ηµν«

−κ3 m2

64π2

»−2 ln(−q2) +

78

π2 mp−q2

–„qµqν −

12

q2ηµν«.






Vacuum Polarisation - Diagrams

�

=

�(a)

+

�(b)

+

�(c)

vacuum polarisation tensor Παβγδ → graviton propagatorcorrection

∆αβγδ + ∆αβµν iΠµνρσ∆ρσγδ






Vacuum Polarization

counter-term, graviton self energy and ghost→ Veltman and ’t Hooft (1974)matter loop

Παβγδ =i

32π2

κ2

4

»−15

`qαqβ − q2ηαβ

´`qγqδ − q2ηγδ

´−

130

`qαqγ − q2ηαγ

´·`qβqδ − q2ηβδ

´−

130

`qαqδ − q2ηαδ

´`qβqγ − q2ηβγ

´–− 2m4

3ηαβηγδ

−23

m23ηαβ`ηγδq2 − qγqδ

´−

23

m23ηγδ`ηαβq2 − qαqβ

´ffln(−q2)






Vacuum Polarization

counter-term, graviton self energy and ghost→ Veltman and ’t Hooft (1974)matter loop

Παβγδ =i

32π2

κ2

4

»−15

`qαqβ − q2ηαβ

´`qγqδ − q2ηγδ

´−

130

`qαqγ − q2ηαγ

´·`qβqδ − q2ηβδ

´−

130

`qαqδ − q2ηαδ

´`qβqγ − q2ηβγ

´–− 2m4

3ηαβηγδ

−23

m23ηαβ`ηγδq2 − qγqδ

´−

23

m23ηγδ`ηαβq2 − qαqβ

´ffln(−q2)






One Particle Irreduzible Diagrams

�k1

k ′1k2

k ′2

q=�+�

+�+�q






Gravitational Potential

1PI-diagrams→ S-matrix contributionsiM = Vµν1 (k1, k2, q,m1)

ˆ444µνρσ+444µναβ iΠαβγδ444γδρσ

˜Vρσ2 (k ′1, k

′2,−q,m)

position space gravitational potential

V (r) = −G m1 m2

r

[1− G(m1 + m2)

r c2 − 16730π

G ~r2c3

]include massless Neutrino-loop

gravitational potential

V (r) = −G m1 m2

r

[1− G(m1 + m2)

r c2 −(

16730π

+Nν

40π

)G ~r2c3

]Sven Faller General Relativity as an Effective Field Theory








˜Vρσ2 (k ′1, k

′2,−q,m)


V (r) = −G m1 m2

r

[1− G(m1 + m2)

r c2 − 16730π

G ~r2c3



V (r) = −G m1 m2

r

[1− G(m1 + m2)

r c2 −(

16730π

+Nν

40π

)G ~r2c3









˜Vρσ2 (k ′1, k

′2,−q,m)


V (r) = −G m1 m2

r

[1− G(m1 + m2)

r c2 − 16730π

G ~r2c3



V (r) = −G m1 m2

r

[1− G(m1 + m2)

r c2 −(

16730π

+Nν

40π

)G ~r2c3






Gravitational Potential (so far)

our result:

V (r) = −G m1 m2

r

[1− G(m1 + m2)

r c2 −(

16730π

+Nν

40π

)G ~r2c3

]Donoghue

V (r) = −G m1 m2

r

[1− G(m1 + m2)

r c2 − (135 + 2Nν)

30π2G ~r c3

].

Akhundov et al.

V (r) = −G m1 m2

r

[1 +

G(m1 + m2)

r c2 − 10730π2

G ~r2 c2






Further Potential Definition

Hamber and Liu (1995)

��m1

�m2�m1 m2�q






Further Contributions

double-seagull diagramm:

�V4 = −G m1 m2

r

[2G(m1 + m2)

r c2 − 14π

G ~r2 c3

]

triangle diagrams:

�V〉◦〈 = −G m1 m2

r

[112π

G ~r2 c3

]







double-seagull diagramm:

�V4 = −G m1 m2

r

[2G(m1 + m2)

r c2 − 14π

G ~r2 c3

]

triangle diagrams:

�V〉◦〈 = −G m1 m2

r

[112π

G ~r2 c3

]







cross-boxed diagramms (Bjerrum-Bohr 2003)

�+�V�(r) = −473

m1 m2 G2

π r3 .


V (r) = −G m1 m2

r

[1 +

G(m1 + m2)

r c2 − 64 + Nν

40πG ~r2c3

]







cross-boxed diagramms (Bjerrum-Bohr 2003)

�+�V�(r) = −473

m1 m2 G2

π r3 .


V (r) = −G m1 m2

r

[1 +

G(m1 + m2)

r c2 − 64 + Nν

40πG ~r2c3

]






Effective Gravity

Summary

full theory of quantum gravity unknowneffective field theory of gravity

low energy effects separated from high-energy effectsone-loop order quantum predictions

evaluate leading quantum corrections→ effective potentialand Schwarzschild-metricunderstanding quantum nature of gravity






Effective Gravity

Summary









Effective Gravity

Summary









Effective Gravity

Summary









Effective Gravity

Summary









Effective Gravity

Summary









Effective Gravity

Thanks

supervisor: Prof. Dr. T. Mannelsecond supervisor: Dr. A. Khodjamirianfor usefull tips and discussions

Dr. Th. FeldmannDr. E. Bjerrum-BohrProf. Dr. F. DonoghueDipl.-Phys. M. JungDipl.-Phys. N. OffenDipl.-Phys. K. Grybel






Effective Gravity

Expansion - Affine Connection

background field method: metric expansion


affine connection: Γλµν = Γλµν +−Γλµν +

=Γλµν

with

Γλµν =12

gλσ(∂µgσν + ∂ν gσµ − ∂σgµν

)(O(h0)) ,

−Γλµν =

κ

2gλσ(Dµhσν + Dνhσµ − Dσhµν

)(O(h1)) ,

=Γλµν = −κ

2

2hλγ(Dµhγν + Dνhµγ − Dγhµν

)(O(h2)) .






Effective Gravity





=Γλµν

with

Γλµν =12


)(O(h0)) ,

−Γλµν =

κ


)(O(h1)) ,

=Γλµν = −κ

2


)(O(h2)) .






Effective Gravity





=Γλµν

with

Γλµν =12


)(O(h0)) ,

−Γλµν =

κ


)(O(h1)) ,

=Γλµν = −κ

2


)(O(h2)) .






Effective Gravity

Expansion: CurvatureRiemann curvature tensor:Rβ

αµν = DµΓβαν − DνΓβαµ + ΓλανΓβλµ − ΓλαµΓβλν ≡ Rβαµν +−Rβαµν +

=Rβαµν

Ricci scalar:R = gαµ

=Rαµ − κhαµ

−Rαµ + κ2hαγ hγµRαµ

= κ2−

12

Dµ`hβγDµhγβ

´+

12

Dβˆhβν`2Dµhνµ − Dνhµµ

´˜+

14

`Dµhνβ + Dβhνµ − Dνhµβ

´`Dµhβν + Dνhβµ − Dβhµν

´−

14

`2Dµhνµ − Dνhµµ

´Dνhββ −

12

hαµDµDαhββ

+12

hµαDβ`Dαhβµ + Dµhβα − Dβhαµ

´+ κ2hβµhαβ Rµ

α

ff






Effective Gravity

Expansion: CurvatureRiemann curvature tensor:Rβ

αµν = DµΓβαν − DνΓβαµ + ΓλανΓβλµ − ΓλαµΓβλν ≡ Rβαµν +−Rβαµν +

=Rβαµν

Ricci scalar:R = gαµ

=Rαµ − κhαµ

−Rαµ + κ2hαγ hγµRαµ

= κ2−

12

Dµ`hβγDµhγβ

´+

12

Dβˆhβν`2Dµhνµ − Dνhµµ

´˜+

14

`Dµhνβ + Dβhνµ − Dνhµβ

´`Dµhβν + Dνhβµ − Dβhµν

´−

14

`2Dµhνµ − Dνhµµ

´Dνhββ −

12

hαµDµDαhββ

+12

hµαDβ`Dαhβµ + Dµhβα − Dβhαµ

´+ κ2hβµhαβ Rµ

α

ff






Effective Gravity

Gauge Fixing and Ghost Field

Yang-Mills field theorygauge fixingintroduce Fadeev-Popov ghost fields

gauge fixing Lagrangian (’t Hoof and Veltman 1974)

Lgf =√−g{(

Dνhµν −12

Dµh)(

Dλhµλ − 12

Dµh)]

ghost field Lagrangian (ebd.)

Lghost =√−g η?µ

[DλDλgµν − Rµν

]ην

complex ghostfield η: only contribution from vacuumpolarization to the graviton propagator






Effective Gravity




Lgf =√−g{(

Dνhµν −12

Dµh)(

Dλhµλ − 12

Dµh)]




]ην







Effective Gravity




Lgf =√−g{(

Dνhµν −12

Dµh)(

Dλhµλ − 12

Dµh)]




]ην







Effective Gravity




Lgf =√−g{(

Dνhµν −12

Dµh)(

Dλhµλ − 12

Dµh)]




]ην







Effective Gravity




Lgf =√−g{(

Dνhµν −12

Dµh)(

Dλhµλ − 12

Dµh)]




]ην







Effective Gravity




Lgf =√−g{(

Dνhµν −12

Dµh)(

Dλhµλ − 12

Dµh)]




]ην







Effective Gravity

Field Theories - Overview

renormalizabel and non-renormalizable field theoriesgeneral form of the Lagrangian

L = L(c1, c2, . . . , cn)

low energy structure determined by finite parametersc1, c2, . . . cntwo different typs of quantum field theories

asympotically free theories - ultraviolet stable theoriesultraviolet unstable theories

ultraviolet unstable theories low energy limit offundamental theory - no difference between renormalizableand effective non-renormalizable theory






Effective Gravity



L = L(c1, c2, . . . , cn)









Effective Gravity



L = L(c1, c2, . . . , cn)









Effective Gravity



L = L(c1, c2, . . . , cn)









Effective Gravity



L = L(c1, c2, . . . , cn)









Effective Gravity



L = L(c1, c2, . . . , cn)









Effective Gravity



L = L(c1, c2, . . . , cn)









Effective Gravity

Effective Field Theory

two different types of effecive field theoriesdecoupling effective field theories

heavy degrees of freedom integrated outeffective level no light particlesLagrangian general form

Leff = LD≤4 +∑D>4

1ΛD−4

∑iD

giD OiD

non-decoupling effective field theoriesfundamental to effective level by phase transistionspontaneously broken symmetry→ light pseudo-Goldstonebosons






Effective Gravity





1ΛD−4

∑iD

giD OiD







Effective Gravity





1ΛD−4

∑iD

giD OiD







Effective Gravity





1ΛD−4

∑iD

giD OiD







Effective Gravity





1ΛD−4

∑iD

giD OiD







Effective Gravity





1ΛD−4

∑iD

giD OiD







Effective Gravity





1ΛD−4

∑iD

giD OiD







Effective Gravity





1ΛD−4

∑iD

giD OiD







Effective Gravity

Effective General Relativity

quantization process: first order curvaturerenormalization higher order neededone-loop Feynman rules→ higher order curvature nocontributiongravitational action

S = Svac + Sm + Sgf + Sghost

=

∫d4x√−g{

2R

κ2 + Lm + Lgf + Lghost

}quantum degrees of freedom: gravitation field hµν ,ghostfields ηµ, η∗µ






Effective Gravity




=

∫d4x√−g{

2R








Effective Gravity




=

∫d4x√−g{

2R








Effective Gravity




=

∫d4x√−g{

2R








Effective Gravity




=

∫d4x√−g{

2R








Effective Gravity

Effective Quantum Gravity

Sgrav =∫

d4x√−g{ 2κ2 R + c1R

2 + c2RµνRµν + · · ·+ Lm}

parameter cifinite numbervalues unknownfree parameters

low-energy limit: effective quantum gravity

Sgrav =

∫d4x√−g{

2κ2 R + c1R

2 + c2RµνRµν + Lm

}higher energy: theory renormalizationvalue of c1, c2 shifted→ renormalization effects absorbedby parameters






Effective Gravity


Sgrav =∫

d4x√−g{ 2κ2 R + c1R

2 + c2RµνRµν + · · ·+ Lm}



Sgrav =

∫d4x√−g{

2κ2 R + c1R








Effective Gravity


Sgrav =∫

d4x√−g{ 2κ2 R + c1R

2 + c2RµνRµν + · · ·+ Lm}



Sgrav =

∫d4x√−g{

2κ2 R + c1R








Effective Gravity


Sgrav =∫

d4x√−g{ 2κ2 R + c1R

2 + c2RµνRµν + · · ·+ Lm}



Sgrav =

∫d4x√−g{

2κ2 R + c1R








Effective Gravity


Sgrav =∫

d4x√−g{ 2κ2 R + c1R

2 + c2RµνRµν + · · ·+ Lm}



Sgrav =

∫d4x√−g{

2κ2 R + c1R








Effective Gravity


Sgrav =∫

d4x√−g{ 2κ2 R + c1R

2 + c2RµνRµν + · · ·+ Lm}



Sgrav =

∫d4x√−g{

2κ2 R + c1R








Effective Gravity


Sgrav =∫

d4x√−g{ 2κ2 R + c1R

2 + c2RµνRµν + · · ·+ Lm}



Sgrav =

∫d4x√−g{

2κ2 R + c1R








Effective Gravity


Sgrav =∫

d4x√−g{ 2κ2 R + c1R

2 + c2RµνRµν + · · ·+ Lm}



Sgrav =

∫d4x√−g{

2κ2 R + c1R








Effective Gravity

Contributions

gravitational contributions

L(0)gr (Λ) , L(2)

gr (R) , L(4)gr (R2)

matter fieldscalar matter field:

L(0)m (φ,m) , L(2)

m (φ,m,R)

massless matter field:

L(0)m = 0 , L(2)

m (φ,R) , L(0)m (φ,R,R2)






Effective Gravity

Contributions


L(0)gr (Λ) , L(2)

gr (R) , L(4)gr (R2)


L(0)m (φ,m) , L(2)

m (φ,m,R)


L(0)m = 0 , L(2)

m (φ,R) , L(0)m (φ,R,R2)






Effective Gravity

Contributions


L(0)gr (Λ) , L(2)

gr (R) , L(4)gr (R2)


L(0)m (φ,m) , L(2)

m (φ,m,R)


L(0)m = 0 , L(2)

m (φ,R) , L(0)m (φ,R,R2)






Effective Gravity

Contributions


L(0)gr (Λ) , L(2)

gr (R) , L(4)gr (R2)


L(0)m (φ,m) , L(2)

m (φ,m,R)


L(0)m = 0 , L(2)

m (φ,R) , L(0)m (φ,R,R2)






Effective Gravity

Counter Terms

loop-diagrams→ UV-divergenciesUV-divergencies separated non-divergent parts byconstant ci and di regulationone-loop order: Veltman and ’t Hoof (1974)

L(1)M =

p−g

8π2ε

1

120R2 +

720

RµνRµν

ffwith ε = 4− D

MS-scheme

c(r)1 = c1 +

1960π2ε

and c(r)2 = c2 +

7160π2ε

two-loop order

L(2)M =

209κ2880(16π2)2

1ε

p−g Rαβ

γδRγδρσRρσ

αβ






Effective Gravity

Counter Terms


L(1)M =

p−g

8π2ε

1

120R2 +

720

RµνRµν

ffwith ε = 4− D

MS-scheme

c(r)1 = c1 +

1960π2ε

and c(r)2 = c2 +

7160π2ε

two-loop order

L(2)M =

209κ2880(16π2)2

1ε

p−g Rαβ

γδRγδρσRρσ

αβ






Effective Gravity

Counter Terms


L(1)M =

p−g

8π2ε

1

120R2 +

720

RµνRµν

ffwith ε = 4− D

MS-scheme

c(r)1 = c1 +

1960π2ε

and c(r)2 = c2 +

7160π2ε

two-loop order

L(2)M =

209κ2880(16π2)2

1ε

p−g Rαβ

γδRγδρσRρσ

αβ






Effective Gravity

Counter Terms


L(1)M =

p−g

8π2ε

1

120R2 +

720

RµνRµν

ffwith ε = 4− D

MS-scheme

c(r)1 = c1 +

1960π2ε

and c(r)2 = c2 +

7160π2ε

two-loop order

L(2)M =

209κ2880(16π2)2

1ε

p−g Rαβ

γδRγδρσRρσ

αβ






Effective Gravity

Counter Terms


L(1)M =

p−g

8π2ε

1

120R2 +

720

RµνRµν

ffwith ε = 4− D

MS-scheme

c(r)1 = c1 +

1960π2ε

and c(r)2 = c2 +

7160π2ε

two-loop order

L(2)M =

209κ2880(16π2)2

1ε

p−g Rαβ

γδRγδρσRρσ

αβ






Effective Gravity

Evaluation Vertex Factors

momentum space vertex factors

Vµ1ν1,...,µmνn = +iZ

d4x d4x1 . . . d4xn d4y1 . . . d4ym ei(p1x1+···+pnxn+q1y1+···+qmym)

· δ

δJ1(x1)· . . . · δ

δJn(xn)· δ

δHν1µ11 (y1)

· . . . · δ

δHµmνmm (ym)

· Lint`φ1, . . . , φn,H1, . . .Hm

´(x)

sources of gravity: J1, . . . , Jnexternal and internal gravity field: Hµ1ν1

1 , . . . ,Hµmνmm

incoming p1, . . . ,pn, outgoing q1, . . .qm momentumSven Faller General Relativity as an Effective Field Theory





Effective Gravity





· δ

δJ1(x1)· . . . · δ

δJn(xn)· δ

δHν1µ11 (y1)

· . . . · δ

δHµmνmm (ym)

· Lint`φ1, . . . , φn,H1, . . .Hm

´(x)


1 , . . . ,Hµmνmm






Effective Gravity





· δ

δJ1(x1)· . . . · δ

δJn(xn)· δ

δHν1µ11 (y1)

· . . . · δ

δHµmνmm (ym)

· Lint`φ1, . . . , φn,H1, . . .Hm

´(x)


1 , . . . ,Hµmνmm






Effective Gravity





· δ

δJ1(x1)· . . . · δ

δJn(xn)· δ

δHν1µ11 (y1)

· . . . · δ

δHµmνmm (ym)

· Lint`φ1, . . . , φn,H1, . . .Hm

´(x)


1 , . . . ,Hµmνmm






Effective Gravity

Graviton-Graviton-Scalar-Vertex

Lagrangian O(h2)

L(2)m = κ2

„12

hµνhνλ−14


κ2

8

„hλσhλσ−

12

hh«ˆ∂µφ∂

µφ−m2φ2˜vertex factor

Vηλρσ = +iZ

d4x d4x1 d4x2 d4x3 d4x4 ei(px1−p′x2+kx3−kx4)

· ∂

∂φ(x1)· ∂

∂φ(x2)· ∂

∂hηλ(x3)· ∂

∂hρσ(x4)

· κ2

2hηλ»

1ηλαδ1δ

ρσβ −14`ηηλ1ρσαβ + ηρσ1ηλαβ

´–∂αφ(x)∂βφ(x)

− 14

„1ηλρσ −

12ηηλ −

12ηηληρσ

«ˆ∂γφ(x)∂γφ(x)−m2φ(x)2˜ffhρσ






Effective Gravity


Lagrangian O(h2)

L(2)m = κ2

„12

hµνhνλ−14


κ2

8

„hλσhλσ−

12

hh«ˆ∂µφ∂

µφ−m2φ2˜vertex factor

Vηλρσ = +iZ

d4x d4x1 d4x2 d4x3 d4x4 ei(px1−p′x2+kx3−kx4)

· ∂

∂φ(x1)· ∂

∂φ(x2)· ∂

∂hηλ(x3)· ∂

∂hρσ(x4)

· κ2

2hηλ»

1ηλαδ1δ

ρσβ −14`ηηλ1ρσαβ + ηρσ1ηλαβ

´–∂αφ(x)∂βφ(x)

− 14

„1ηλρσ −

12ηηλ −

12ηηληρσ

«ˆ∂γφ(x)∂γφ(x)−m2φ(x)2˜ffhρσ






Effective Gravity


�`↗ ρσ

p

`′ ↖ p′ηλ

Vηλρσ =iκ2

2

»1ηλαδ1

δρσβ −

14

`ηηλ1ρσαβ + ηρσ1ηλαβ

´–`pαp′β + pβp′α

´−

12

»1ηλρσ −

12ηηληρσ

–`(p · p′)−m2´ff






Effective Gravity

Three-Graviton-Vertex

τµναβγδ

(k, q) = −iκ

2

Pαβγδ

»kµkν + (k − q)µ(k − q)ν + qµqν +

3

2η

µν q2–

+ 2qλqσˆ1

σλαβ

1µν

γδ+ 1

σλγδ

1µν

αβ− 1

µσαβ

1νλ

γδ− 1

µσγδ

1νλ

αβ

˜+

ˆqλqµ`

ηαβ1νλ

γδ+ ηγδ1

νλαβ

´+ qλqν `

ηαβ1µλ

γδ+ ηγδ1

µλαβ

´− q2`

ηαβ1µν

γδ+ ηγδ1

µναβ

´− ηµν qλqσ

`ηαβ1

σλγδ

+ ηγδ1σλ

αβ

´˜+

ˆ2qλ

˘1

λσαβ

1ν

γδσ(k − q)µ + 1

λσαβ

1µ

γδσ(k − q)ν

− 1λσ

γδ1

ναβσ

kµ − 1λσ

γδ1

µαβσ

kν ¯+ q2`

1µ

αβσ1

νσγδ

+ 1νσ

αβ1

µγδσ

´+ η

µν qσqλ`1

λραβ

1σ

γδρ+ 1

λργδ

1σ

αβρ

´˜+

`k2 + (k − q)

´»1

µσαβ

1ν

γδσ+ 1

νσγδ

1µ

αβσ−

1

2η

µνPαβγδ

–−

`1

µνγδ

ηαβk2 − 1µν

αβηγδ(k − q)2´ffffffff






Effective Gravity

Born approximation

covariant normalized: 〈p′|p〉 = 2Epδ3(~p − ~p′), Ep =

√m2 + ~p2

Born approximation: 〈p′|p〉 = −i V (~q)(2π)δ(E~p′ − E~p)nonrelativistic limit: interaction potential

−V (~q) =M 12m1

12m2

∫d3~pT

(2π)3 δ3(~p′ + ~p)

nonrelativistic limit: V (~q) = − 12m1·2m2

MFourier transformation to position-space:

nonrelativistic limit: position-space potential

V (~r) = − 12m1

12m2

∫d3~q

(2π)3 ei~q·~r M






Effective Gravity

Born approximation


√m2 + ~p2


−V (~q) =M 12m1

12m2

∫d3~pT

(2π)3 δ3(~p′ + ~p)




V (~r) = − 12m1

12m2

∫d3~q

(2π)3 ei~q·~r M






Effective Gravity

Born approximation


√m2 + ~p2


−V (~q) =M 12m1

12m2

∫d3~pT

(2π)3 δ3(~p′ + ~p)




V (~r) = − 12m1

12m2

∫d3~q

(2π)3 ei~q·~r M






Effective Gravity

Born approximation


√m2 + ~p2


−V (~q) =M 12m1

12m2

∫d3~pT

(2π)3 δ3(~p′ + ~p)




V (~r) = − 12m1

12m2

∫d3~q

(2π)3 ei~q·~r M






Effective Gravity

Born approximation


√m2 + ~p2


−V (~q) =M 12m1

12m2

∫d3~pT

(2π)3 δ3(~p′ + ~p)




V (~r) = − 12m1

12m2

∫d3~q

(2π)3 ei~q·~r M






Effective Gravity

Expansion Form Factors

dimensionless combinations: κ2m2, κ2q2

expansion:

F1(q2) = 1 + d1q2 + κ2q2„

l1 + l2 ln(−q2)

µ2 + l3

sm2

−q2 + . . .

«,

F2(q2) = −4(d2 − d3)m2 + κ2m2„

l4 + l5 ln(−q2)

µ2 + l6

sm2

−q2 + . . .

«di : L(2)

m contributionsli : one-loop contributionsl1, l4 : divergent high enery contributionsl2, l3, l5, l6 : finite non-analytic low energy contributions






Effective Gravity



expansion:

F1(q2) = 1 + d1q2 + κ2q2„

l1 + l2 ln(−q2)

µ2 + l3

sm2

−q2 + . . .

«,

F2(q2) = −4(d2 − d3)m2 + κ2m2„

l4 + l5 ln(−q2)

µ2 + l6

sm2

−q2 + . . .

«di : L(2)







Effective Gravity



expansion:

F1(q2) = 1 + d1q2 + κ2q2„

l1 + l2 ln(−q2)

µ2 + l3

sm2

−q2 + . . .

«,

F2(q2) = −4(d2 − d3)m2 + κ2m2„

l4 + l5 ln(−q2)

µ2 + l6

sm2

−q2 + . . .

«di : L(2)







Effective Gravity



expansion:

F1(q2) = 1 + d1q2 + κ2q2„

l1 + l2 ln(−q2)

µ2 + l3

sm2

−q2 + . . .

«,

F2(q2) = −4(d2 − d3)m2 + κ2m2„

l4 + l5 ln(−q2)

µ2 + l6

sm2

−q2 + . . .

«di : L(2)







Effective Gravity



expansion:

F1(q2) = 1 + d1q2 + κ2q2„

l1 + l2 ln(−q2)

µ2 + l3

sm2

−q2 + . . .

«,

F2(q2) = −4(d2 − d3)m2 + κ2m2„

l4 + l5 ln(−q2)

µ2 + l6

sm2

−q2 + . . .

«di : L(2)







Effective Gravity



expansion:

F1(q2) = 1 + d1q2 + κ2q2„

l1 + l2 ln(−q2)

µ2 + l3

sm2

−q2 + . . .

«,

F2(q2) = −4(d2 − d3)m2 + κ2m2„

l4 + l5 ln(−q2)

µ2 + l6

sm2

−q2 + . . .

«di : L(2)







Effective Gravity

Renormalization

combination l1, l4 and di → renormalized values

d (r)1 (µ2) = d1 + κ2l1

d (r)2 (µ2) + d (r)

3 (µ2) = d2 + d3 − κ2 l44

experiments: measure renormalized values

d (r)i (µ2)→ measured values depend on µ2 choice in

logarithms but all physics independent of µ2






Effective Gravity

Renormalization


d (r)1 (µ2) = d1 + κ2l1

d (r)2 (µ2) + d (r)

3 (µ2) = d2 + d3 − κ2 l44









Effective Gravity

Renormalization


d (r)1 (µ2) = d1 + κ2l1

d (r)2 (µ2) + d (r)

3 (µ2) = d2 + d3 − κ2 l44





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General Relativity as an Effective Field Theoryfaller/talks/THSeminar_2006.pdf · General Relativity as an Effective Field Theory Sven Faller Theoretical Physics 1 University of Siegen