Scattering of Spinning Black Holes from Amplitudes based on work - - PowerPoint PPT Presentation
Scattering of Spinning Black Holes from Amplitudes based on work with Alfredo Guevara and Justin Vines arXiv:1812.06895, 1906.10071 [hep-th] Alexander Ochirov ETH Z urich QCD Meets Gravity 2019, UCLA, December 12 1 / 28 Motivation 2 / 28
based on work with Alfredo Guevara and Justin Vines arXiv:1812.06895, 1906.10071 [hep-th]
Alexander Ochirov
ETH Z¨ urich
QCD Meets Gravity 2019, UCLA, December 12
1 / 28
2 / 28
“photo” by Event Horizon Telescope Collaboration ’19
Artist’s impression of BH merger. Credit: SXS
◮ BH merger GW150914 seen by LIGO+Virgo
Streckung (10-21)
Hanford, Washington (H1) Livingston, Louisiana (L1)
H1 gemessen Numerisch (Relativitätstheorie) Rekonstruiert (Elementarwelle) Rekonstruiert (Vorlage) Numerisch (Relativitätstheorie) L1 gemessen Rekonstruiert (Elementarwelle) H1 gemessen (verschoben, invertiert) Rekonstruiert (Vorlage)
0,5 1,0 0,5 1,0 0,5 0,0
0,0
◮ EOB Hamiltonian from PM scattering instead of
from PN 2-body bound-state dynamics
Buonanno, Damour ’98 → Damour ’16
◮ On-shell amplitude methods:
quantum gravity scattering easier than GR dynamics
e.g. 3PM 0-spin Hamiltonian by Bern, Cheung, Roiban, Shen, Solon, Zeng ’19 talks by Shen, Solon and Zeng 5 / 28
◮ BH merger GW150914 seen by LIGO+Virgo
Streckung (10-21)
Hanford, Washington (H1) Livingston, Louisiana (L1)
H1 gemessen Numerisch (Relativitätstheorie) Rekonstruiert (Elementarwelle) Rekonstruiert (Vorlage) Numerisch (Relativitätstheorie) L1 gemessen Rekonstruiert (Elementarwelle) H1 gemessen (verschoben, invertiert) Rekonstruiert (Vorlage)
0,5 1,0 0,5 1,0 0,5 0,0
0,0
◮ EOB Hamiltonian from PM scattering instead of
from PN 2-body bound-state dynamics
Buonanno, Damour ’98 → Damour ’16
◮ On-shell amplitude methods:
quantum gravity scattering easier than GR dynamics
e.g. 3PM 0-spin Hamiltonian by Bern, Cheung, Roiban, Shen, Solon, Zeng ’19 talks by Shen, Solon and Zeng
This talk:
◮ 1PM and 2PM BH scattering with spin from amplitudes
5 / 28
6 / 28
7 / 28
Want: extract classical spin dependence (Sµ ∈ R4) from quantum spin amplitudes (s ∈ Z+)
7 / 28
Arkani-Hamed, Huang, Huang ’17
p1 p2 k
M(s,+)
3
= −κ 2 12⊙2s m2s−2 x2, M(s,−)
3
= −κ 2 [12]⊙2s m2s−2 x−2, e.g. M(0,±)
3
= −κ(p1 · ε±)2 x = −
√ 2 m (p1 · ε+)
= √
2 m (p1 · ε−)
−1
8 / 28
Arkani-Hamed, Huang, Huang ’17
p1 p2 k
M(s,+)
3
= −κ 2 12⊙2s m2s−2 x2, M(s,−)
3
= −κ 2 [12]⊙2s m2s−2 x−2, e.g. M(0,±)
3
= −κ(p1 · ε±)2 x = −
√ 2 m (p1 · ε+)
= √
2 m (p1 · ε−)
−1 Angular-momentum structure inside: M(s,+)
3
= M(0,+)
3
12⊙2s m2s = M(0,+)
3
m2s [2|⊙2sexp
ν ¯
σµν p1 · ε+
M(s,−)
3
= M(0,−)
3
[12]⊙2s m2s = M(0,−)
3
m2s 2|⊙2sexp
ν σµν
p1 · ε−
Guevara, AO, Vines ’18 inspired by soft theorems, e.g. Cachazo, Strominger ’14
8 / 28
Vines ’17
Stress-energy tensor (eff. source) for lin. Kerr BH:* T µν
BH(x) = 1
m
ρpρδ(4)(x − uτ),
pµ = muµ T µν
BH(k) = ˆ
δ(p · k)p(µ exp(−ia ∗ k)ν)ρ pρ, Sµ = maµ
*Hat notation absorbs straightforward powers of 2π. 9 / 28
Vines ’17
Stress-energy tensor (eff. source) for lin. Kerr BH:* T µν
BH(x) = 1
m
ρpρδ(4)(x − uτ),
pµ = muµ T µν
BH(k) = ˆ
δ(p · k)p(µ exp(−ia ∗ k)ν)ρ pρ, Sµ = maµ Couple to on-shell graviton hµν(k) → ˆ δ(k2)εµεν: hµν(k)T µν
BH(−k) = ˆ
δ(k2)ˆ δ(p · k)(p · ε)2 exp
p · ε
where Sµν = ǫµνρσpρaσ
*Hat notation absorbs straightforward powers of 2π. 9 / 28
Guevara, AO, Vines ’18
hµν(k)T µν
BH(−k) = ˆ
δ(k2)ˆ δ(p · k)(p · ε)2 exp
p · ε
Guevara, AO, Vines ’18
hµν(k)T µν
BH(−k) = ˆ
δ(k2)ˆ δ(p · k)(p · ε)2 exp
p · ε
M(s,+)
3
= M(0,+)
3
m2s [2|⊙2sexp
ν ¯
σµν p1 · ε+
M(s,−)
3
= M(0,−)
3
m2s 2|⊙2sexp
ν σµν
p1 · ε−
p1 p2 k
10 / 28
Guevara, AO, Vines ’18
hµν(k)T µν
BH(−k) = ˆ
δ(k2)ˆ δ(p · k)(p · ε)2 exp
p · ε
M(s,+)
3
= M(0,+)
3
m2s [2|⊙2sexp
ν ¯
σµν p1 · ε+
M(s,−)
3
= M(0,−)
3
m2s 2|⊙2sexp
ν σµν
p1 · ε−
p1 p2 k
Matching spin-induced multipole structure!
complementary picture: 1-body EFT of Kerr by Levi, Steinhoff ’15 match to Wilson coeffs by Chung, Huang, Kim, Lee ’18
10 / 28
Covariant formulation:
Bautista, Guevara ’19
M(s)
3
= M(0)
3 ε2 · exp
p1 · ε
Lorentz generators: (Σµν)σ1...σs
τ1...τs = Σµν,σ1 τ1δσ2 τ2 . . . δσs τs
+ . . . + δσ1
τ1 . . . δσs−1 τs−1 Σµν,σs τs,
Σµν,σ
τ = i[ηµσδν τ − ηνσδµ τ ]
Polarization tensors:
Guevara, AO, Vines ’18, Chung, Huang, Kim, Lee ’18
εa1...a2s
pµ1...µs = ε(a1a2 pµ1
. . . εa2s−1a2s)
pµs
, εab
pµ = ip(a|σµ|pb)]
√ 2m
11 / 28
Covariant formulation:
Bautista, Guevara ’19
M(s)
3
= M(0)
3 ε2 · exp
p1 · ε
Lorentz generators: (Σµν)σ1...σs
τ1...τs = Σµν,σ1 τ1δσ2 τ2 . . . δσs τs
+ . . . + δσ1
τ1 . . . δσs−1 τs−1 Σµν,σs τs,
Σµν,σ
τ = i[ηµσδν τ − ηνσδµ τ ]
Polarization tensors:
Guevara, AO, Vines ’18, Chung, Huang, Kim, Lee ’18
εa1...a2s
pµ1...µs = ε(a1a2 pµ1
. . . εa2s−1a2s)
pµs
, εab
pµ = ip(a|σµ|pb)]
√ 2m Spinor-helicity formulation:
Guevara, AO, Vines ’19
M(s,+)
3
= M(0)
3
m2s [2|⊙2sexp
ν ¯
σµν p1 · ε+
3
m2s [2|⊙2sexp(−2k · a)|1]⊙2s
M(s,−)
3
= M(0)
3
m2s 2|⊙2sexp
ν σµν
p1 · ε−
3
m2s 2|⊙2sexp(2k · a)|1⊙2s
aµ, β
α
= 1 2m2 ǫµνρσpaνσ
β ρσ,α ,
aµ, ˙
α ˙ β =
1 2m2 ǫµνρσpaν ¯ σ
˙ α ρσ, ˙ β
σµν = i
2σ[µ¯
σν], ¯ σµν = i
2 ¯
σ[µσν] (and tensor generalizations)
11 / 28
Define Pauli-Lubanski vector operator Σλ = 1 2mǫλµνρΣµνpρ Its 1-particle matrix elements are S{a}{b}
pµ
= (−1)sε{a}
p
· Σµ· ε{b}
p
= − s 2m
σµ|p(b1
12 / 28
Define Pauli-Lubanski vector operator Σλ = 1 2mǫλµνρΣµνpρ Its 1-particle matrix elements are S{a}{b}
pµ
= (−1)sε{a}
p
· Σµ· ε{b}
p
= − s 2m
σµ|p(b1
Spin quantized explicitly: εp{a}· Σµ· ε{a}
p
εp{a}· ε{a}
p
= ssµ
p,
a1 = . . . = a2s = 1, (s − 1)sµ
p,
2s
j=1aj = 2s + 1,
(s − 2)sµ
p,
2s
j=1aj = 2s + 2,
. . . −ssµ
p,
a1 = . . . = a2s = 2, in terms of unit spin vector sµ
p = − 1
2m
σµ|p1
1 2m ¯ up1γµγ5u1
p = − 1
2m ¯ up2γµγ5u2
p
p · sp = 0 s2
p = −1
12 / 28
Puzzle: two reps of M(s,+)
3
= M(0)
3
m2s 21⊙2s = M(0)
3
m2s [2|⊙2se−2k·a|1]⊙2s
seem to depend differently on aµ
13 / 28
Puzzle: two reps of M(s,+)
3
= M(0)
3
m2s 21⊙2s = M(0)
3
m2s [2|⊙2se−2k·a|1]⊙2s
seem to depend differently on aµ Fix:
Guevara, AO, Vines ’18
“divide” by lim
s→∞ε2 · ε1 = lim s→∞ 1 m2s 2|⊙2sek·a|1⊙2s = lim s→∞ 1 m2s [2|⊙2se−k·a|1]⊙2s
13 / 28
Puzzle: two reps of M(s,+)
3
= M(0)
3
m2s 21⊙2s = M(0)
3
m2s [2|⊙2se−2k·a|1]⊙2s
seem to depend differently on aµ Fix:
Guevara, AO, Vines ’18
“divide” by lim
s→∞ε2 · ε1 = lim s→∞ 1 m2s 2|⊙2sek·a|1⊙2s = lim s→∞ 1 m2s [2|⊙2se−k·a|1]⊙2s
Hint:
Levi, Steinhoff ’15
“spin-induced higher multipoles should naturally be considered in the body-fixed frame”
13 / 28
Puzzle: two reps of M(s,+)
3
= M(0)
3
m2s 21⊙2s = M(0)
3
m2s [2|⊙2se−2k·a|1]⊙2s
seem to depend differently on aµ Fix:
Guevara, AO, Vines ’18
“divide” by lim
s→∞ε2 · ε1 = lim s→∞ 1 m2s 2|⊙2sek·a|1⊙2s = lim s→∞ 1 m2s [2|⊙2se−k·a|1]⊙2s
Hint:
Levi, Steinhoff ’15
“spin-induced higher multipoles should naturally be considered in the body-fixed frame” Solution:
Bautista, Guevara ’19 Guevara, AO, Vines ’19 also in Arkani-Hamed, Huang, O’Connell ’19
must only compare states of same momentum!
13 / 28
Bautista, Guevara ’19 Guevara, AO, Vines ’19 also in Arkani-Hamed, Huang, O’Connell ’19
Consider p1 → p2 boost: pρ
2 = exp
i
m2 pµ 1kνΣµν
ρ
σpσ 1
|2b = U
b 12 a exp
i
m2 pµ 1kνσµν
|2b] = U
b 12 a exp
i
m2 pµ 1kν¯
σµν
k2 = (p2 − p1)2 = 0 U12 ∈ SU(2)
14 / 28
Bautista, Guevara ’19 Guevara, AO, Vines ’19 also in Arkani-Hamed, Huang, O’Connell ’19
Consider p1 → p2 boost: pρ
2 = exp
i
m2 pµ 1kνΣµν
ρ
σpσ 1
|2b = U
b 12 a exp
i
m2 pµ 1kνσµν
|2b] = U
b 12 a exp
i
m2 pµ 1kν¯
σµν
k2 = (p2 − p1)2 = 0 U12 ∈ SU(2) Self-duality of σµν, ¯ σµν implies i m2 pµ
1kνσ β µν,α
= k · a β
α ,
i m2 pµ
1kν¯
σ
˙ α µν, ˙ β = −k · a ˙ α ˙ β
in terms of left- and right-handed reps of Pauli-Lubanski vector aµ, β
α =
1 2m2 ǫµνρσpaνσ
β ρσ,α ,
aµ, ˙
α ˙ β =
1 2m2 ǫµνρσpaν¯ σ
˙ α ρσ, ˙ β
14 / 28
Arbirary-spin reps boost as |2⊙2s = ek·a U12|1
|2]⊙2s = e−k·a U12|1]
2|⊙2s =
[2|⊙2s =
*m2s cancels due to papb = −[papb] = −mǫab. 15 / 28
Arbirary-spin reps boost as |2⊙2s = ek·a U12|1
|2]⊙2s = e−k·a U12|1]
2|⊙2s =
[2|⊙2s =
Back to spin dependence of 3-pt amplitude:* M(s,+)
3
= M(0)
3
m2s 21⊙2s = M(0)
3
m2s
= M(0)
3
m2s [2|⊙2se−2k·a|1]⊙2s = M(0)
3
m2s
− − − →
s→∞ M(0) 3 e−k·a lim s→∞(U12)⊙2s
unambigiously! aµ is now classical (C-number) spin of Kerr BH
*m2s cancels due to papb = −[papb] = −mǫab. 15 / 28
16 / 28
Kosower, Maybee, O’Connell ’18 Maybee, O’Connell, Vines ’19 Kosower’s talk
LO impulses: ∆pµ
a =
d4k ˆ δ(2pa· k)ˆ δ(2pb· k)kµe−ik·b/iM4(k)
a =
d4k ˆ δ(2pa· k)ˆ δ(2pb· k)e−ik·b/ ×
m2
a
pµ
aSν a kνM4(k) +
a , iM4(k)
2(p1 + p2)
p1 p2 p3 p4 k
pb = 1 2(p3 + p4)
17 / 28
Kosower, Maybee, O’Connell ’18 Maybee, O’Connell, Vines ’19 Kosower’s talk
LO impulses: ∆pµ
a =
d4k ˆ δ(2pa· k)ˆ δ(2pb· k)kµe−ik·b/iM4(k)
a =
d4k ˆ δ(2pa· k)ˆ δ(2pb· k)e−ik·b/ ×
m2
a
pµ
aSν a kνM4(k) +
a , iM4(k)
2(p1 + p2)
p1 p2 p3 p4 k
pb = 1 2(p3 + p4) Net effect of
kµ = ¯ kµ → 0, pµ
1, pµ 2 → mauµ a,
pµ
3, pµ 4 → mbuµ b
Sµ
1 , Sµ 2 → maaµ a,
Sµ
3 , Sµ 4 → mbaµ b
17 / 28
Cachazo, Guevara ’17 Guevara ’17
p1 p2 p3 p4 k
Idea: Replace kµ = ¯ kµ → 0 by non-zero on-shell t = k2 → 0 Indeed, k2 = 0 ⇒ pi · k = O(t) = 0
18 / 28
Cachazo, Guevara ’17 Guevara ’17
p1 p2 p3 p4 k
Idea: Replace kµ = ¯ kµ → 0 by non-zero on-shell t = k2 → 0 Indeed, k2 = 0 ⇒ pi · k = O(t) = 0 M(sa,sb)
4
(p1,−p2, p3,−p4) = −1 t
M(sa)
3
(p1,−p2, k±)M(sb)
3
(p3,−p4,−k∓) + O(t0)
18 / 28
Guevara, AO, Vines ’19
p1 p2 p3 p4 k
γ = 1 √ 1 − v2 = pa· pb mamb → ua· ub M4 = −(κ/2)2γ2 m2sa−2
a
m2sb−2
b
t
U121|
U34[3|
+(1+v)2 U12[1|
U343|
Guevara, AO, Vines ’19
p1 p2 p3 p4 k
γ = 1 √ 1 − v2 = pa· pb mamb → ua· ub M4 = −(κ/2)2γ2 m2sa−2
a
m2sb−2
b
t
U121|
U34[3|
+(1+v)2 U12[1|
U343|
k · aa,b = ik · w ∗ aa,b, [w ∗ aa,b]µ = ǫµνρσaν
a,bpρ apσ b
mambγv
19 / 28
Guevara, AO, Vines ’19
p1 p2 p3 p4 k
γ = 1 √ 1 − v2 = pa· pb mamb → ua· ub M4 = −(κ/2)2γ2 m2sa−2
a
m2sb−2
b
t
U121|
U34[3|
+(1+v)2 U12[1|
U343|
k · aa,b = ik · w ∗ aa,b, [w ∗ aa,b]µ = ǫµνρσaν
a,bpρ apσ b
mambγv M4(k) = − κ 2
am2 b
k2 γ2
±
(1 ± v)2 exp[±i(k · w ∗ a0)], aµ
0 = aµ a + aµ b
19 / 28
Guevara, AO, Vines ’19
from momentum transfer/mismatch kµ M4(k) = − κ 2
am2 b
k2 γ2
±
(1 ± v)2 exp[±i(k · w ∗ a0)]
20 / 28
Guevara, AO, Vines ’19
from momentum transfer/mismatch kµ M4(k) = − κ 2
am2 b
k2 γ2
±
(1 ± v)2 exp[±i(k · w ∗ a0)] to impact parameter bµ M4(b) =
d4k ˆ δ(2pa· k)ˆ δ(2pb· k)e−ik·bM4(k) = −Gmamb γ v
(1 ± v)2 log
pa = p pb = −p b ∆pa ∆pb
b2 < 0 b · pa = 0 b · pb = 0
eikonal Fourier transform e.g. in Bjerrum-Bohr, Damgaard, Festuccia, Plant´ e, Vanhove ’18 20 / 28
Guevara, AO, Vines ’19
M4(b) = −Gmamb γ v
(1 ± v)2 log
∆pµ
a =
d4k ˆ δ(2pa· k)ˆ δ(2pb· k)kµe−ik·b/iM4(k)
∂bµ M4(b)
*Relied on little-group so(3) algebra of Sµ a in rest frame of pa, i.e.
[Sµ
a , Sν a ] =
i ma ǫµνρσpaρSaσ ⇒ [Sµ
a , M4] =
i ma ǫµνρσpaνSaρ ∂M4 ∂Sσ
a
.
21 / 28
Guevara, AO, Vines ’19
M4(b) = −Gmamb γ v
(1 ± v)2 log
∆pµ
a =
d4k ˆ δ(2pa· k)ˆ δ(2pb· k)kµe−ik·b/iM4(k)
∂bµ M4(b) ∆aµ
a =
1 ma
d4k ˆ δ(2pa· k)ˆ δ(2pb· k)e−ik·b/ ×
m2
a
pµ
aSν a kνM4(k) +
a , iM4(k)
= 1 m2
a
aaν a
∂ ∂bν − ǫµνρσpaνaaρ ∂ ∂aσ
a
*Relied on little-group so(3) algebra of Sµ a in rest frame of pa, i.e.
[Sµ
a , Sν a ] =
i ma ǫµνρσpaρSaσ ⇒ [Sµ
a , M4] =
i ma ǫµνρσpaνSaρ ∂M4 ∂Sσ
a
.
21 / 28
Guevara, AO, Vines ’19
M4(b) = −Gmamb γ v
(1 ± v)2 log
∆pµ
a =
d4k ˆ δ(2pa· k)ˆ δ(2pb· k)kµe−ik·b/iM4(k)
∂bµ M4(b) ∆aµ
a =
1 ma
d4k ˆ δ(2pa· k)ˆ δ(2pb· k)e−ik·b/ ×
m2
a
pµ
aSν a kνM4(k) +
a , iM4(k)
= 1 m2
a
aaν a
∂ ∂bν − ǫµνρσpaνaaρ ∂ ∂aσ
a
Complete match to 1PM classical solution!
Vines ’17
*Relied on little-group so(3) algebra of Sµ a in rest frame of pa, i.e.
[Sµ
a , Sν a ] =
i ma ǫµνρσpaρSaσ ⇒ [Sµ
a , M4] =
i ma ǫµνρσpaνSaρ ∂M4 ∂Sσ
a
.
21 / 28
Vines ’17
Linear and angular impulses ∆pµ
a = GmambℜZµ
∆aµ
a = −Gmb
ma
a(aa·ℜZ) + ǫµνρσ(ℑZν)paρaaσ
Zµ = γ v
2[ηµν ∓ i(∗w)µν](b ∓ w ∗ a0)ν (b ∓ w ∗ a0)2 Zµ automatic from scattering function M4(b) ∂ ∂bµ M4(b) = −GmambℜZµ, ∂ ∂aµ M4(b) = GmambℑZµ
22 / 28
23 / 28
◮ Th: classical from 2-massive-p. irreducible graphs
with 1 massive prop. per loop
Neill, Rothstein ’13 24 / 28
◮ Th: classical from 2-massive-p. irreducible graphs
with 1 massive prop. per loop
Neill, Rothstein ’13
◮ 1 loop: triangles with massive propagators
p1 p2 p3 p4 ℓ k3 k4 24 / 28
◮ Th: classical from 2-massive-p. irreducible graphs
with 1 massive prop. per loop
Neill, Rothstein ’13
◮ 1 loop: triangles with massive propagators
p1 p2 p3 p4 ℓ k3 k4
◮ 1 loop: boxes contribute to ∆pa,b
Kosower, Maybee, O’Connell ’18 Kosower’s talk
but not to scattering angle θ
Bjerrum-Bohr, Damgaard, Festuccia, Plant´ e, Vanhove ’18 24 / 28
◮ Th: classical from 2-massive-p. irreducible graphs
with 1 massive prop. per loop
Neill, Rothstein ’13
◮ 1 loop: triangles with massive propagators
p1 p2 p3 p4 ℓ k3 k4
◮ 1 loop: boxes contribute to ∆pa,b
Kosower, Maybee, O’Connell ’18 Kosower’s talk
but not to scattering angle θ
Bjerrum-Bohr, Damgaard, Festuccia, Plant´ e, Vanhove ’18
◮ 2 loops: topologies with more massive props. contribute
Bern, Cheung, Roiban, Shen, Solon, Zeng ’19 24 / 28
Guevara, AO, Vines ’18
◮ Incoming spins ⊥ to scattering plane
⇒ outgoing spins stay aligned, ∆aa,b = 0, scattering within plane ⇒ scattering angle θ implies ∆pa,b
25 / 28
Guevara, AO, Vines ’18
◮ Incoming spins ⊥ to scattering plane
⇒ outgoing spins stay aligned, ∆aa,b = 0, scattering within plane ⇒ scattering angle θ implies ∆pa,b
◮ Use known non-spinning formula from eikonal
2 sin θ 2 = −E (2mambγv)2 ∂ ∂b
(2π)2 e−ik·b lim
sa,sb→∞M(sa,sb) 4
+O(G3)
Kabat, Ortiz (1992); Akhoury, Saotome ’13 Bjerrum-Bohr, Damgaard, Festuccia, Plant´ e, Vanhove ’18 25 / 28
Guevara, AO, Vines ’18
◮ Incoming spins ⊥ to scattering plane
⇒ outgoing spins stay aligned, ∆aa,b = 0, scattering within plane ⇒ scattering angle θ implies ∆pa,b
◮ Use known non-spinning formula from eikonal
2 sin θ 2 = −E (2mambγv)2 ∂ ∂b
(2π)2 e−ik·b lim
sa,sb→∞M(sa,sb) 4
+O(G3)
Kabat, Ortiz (1992); Akhoury, Saotome ’13 Bjerrum-Bohr, Damgaard, Festuccia, Plant´ e, Vanhove ’18
◮ Triangle contributions encode θ
p1 p2 p3 p4 ℓ k3 k4
◮ Compute triangle coeffs in HCL
Cachazo, Guevara ’17; Guevara ’17 25 / 28
Guevara, AO, Vines ’18
◮ Incoming spins ⊥ to scattering plane
⇒ outgoing spins stay aligned, ∆aa,b = 0, scattering within plane ⇒ scattering angle θ implies ∆pa,b
◮ Use known non-spinning formula from eikonal
2 sin θ 2 = −E (2mambγv)2 ∂ ∂b
(2π)2 e−ik·b lim
sa,sb→∞M(sa,sb) 4
+O(G3)
Kabat, Ortiz (1992); Akhoury, Saotome ’13 Bjerrum-Bohr, Damgaard, Festuccia, Plant´ e, Vanhove ’18
◮ Triangle contributions encode θ
p1 p2 p3 p4 ℓ k3 k4
◮ Compute triangle coeffs in HCL
Cachazo, Guevara ’17; Guevara ’17
◮ Extract angular-momentum dependence from spin exponentials
25 / 28
Guevara, AO, Vines ’18
θ⊳ = πG2E mb 2v4 ∂ ∂b
dz 2πi (1 − vz)4 (z2 − 1)3/2
1 − vz aa
26 / 28
Guevara, AO, Vines ’18
θ⊳ = πG2E mb 2v4 ∂ ∂b
dz 2πi (1 − vz)4 (z2 − 1)3/2
1 − vz aa
θ1-loop = θ⊳ + θ⊲ = −πG2E ∂ ∂b
where E =
a + m2 b + 2mamb
f(σ, a) = 1 2a2
( + κ − 2a)5 4vκ
= vb + σ + a, κ =
26 / 28
Guevara, AO, Vines ’18
θ⊳ = πG2E mb 2v4 ∂ ∂b
dz 2πi (1 − vz)4 (z2 − 1)3/2
1 − vz aa
θ1-loop = θ⊳ + θ⊲ = −πG2E ∂ ∂b
where E =
a + m2 b + 2mamb
f(σ, a) = 1 2a2
( + κ − 2a)5 4vκ
= vb + σ + a, κ =
true at least through O(a2), possibly wrong beyond O(a4)
Bini, Damour ’18 Vines, Steinhoff, Buonanno ’18 26 / 28
p1 p2 p3 p4 ℓ k3 k4
M(s)
4 (p1, −p2, k+ 3, k− 4 ) = −
κ 2
2|⊙2s exp
4νσµν
p1 · ε−
4
= − κ 2
[2|⊙2s exp
3ν ¯
σµν p1 · ε+
3
= κ 2
[13]42 + 14[32] ⊙2s (2p1 · k4)(2p2 · k4)(2k3 · k4)
◮ first appeared in
Arkani-Hamed, Huang, Huang ’17
◮ problematic at s ≥ 4
double-copy aspects in Johansson, AO ’19
◮ alternatives proposed in
Chung, Huang, Kim, Lee ’18 27 / 28
◮ Spin exponentiation pattern inherent to Kerr BH
Guevara, AO, Vines ’18 Bautista, Guevara ’19 Guevara, AO, Vines ’19 Arkani-Hamed, Huang, O’Connell ’19
◮ 1PM match for general spin (to all orders in spin)
Vines ’17
◮ 2PM results for aligned spins:
◮ match at presently known orders in spins Bini, Damour ’18 Vines, Steinhoff, Buonanno ’18 ◮ conjectured results for higher orders in spins consistent with Siemonsen, Vines ’19
◮ Used HCL,
need better connections between classical-limit approaches
Guevara ’17 Bjerrum-Bohr, Damgaard, Festuccia, Plant´ e, Vanhove ’18 Cheung, Rothstein, Solon ’18 Kosower, Maybee, O’Connell ’18 Koemans Collado, Di Vecchia, Russo ’19 Bjerrum-Bohr, Cristofoli, Damgaard, Vanhove ’19 Maybee, O’Connell, Vines ’19 K¨ alin, Porto ’19
◮ Open questions at higher orders in G and spin
More new results to come!
28 / 28
29 / 28
30 / 28
Covariant spin exponentiation: M(s)
3
= M(0)
3 ε2 · exp
p1 · ε
ε2 = exp i m2 pµ
1kνΣµν
ε1, ˜ ε1 = U (s)
12 ε1
— LG transform M(s)
3
= M(0)
3 ˜
ε1 exp
m2 pµ
1kνΣµν
p1 · ε
= M(0)
3 ˜
ε1 exp
⊥
p1 · ε
Σµν
⊥ = Σµν + 2
m2 p[µ
1 Σν]ρp1ρ
⇒ p1µΣµν
⊥ = 0
— SSC
31 / 28
◮ Quantum fields
⇐ reps of SO(1, 3) ⊂ SL(2, C)
◮ Quantum states
⇐ reps of little group’s dbl cover
◮ massless states ⇐ SO(2)
⊂ U(1)
◮ massive states
⇐ SO(3) ⊂ SU(2)
Minor complication: spinorial reps use groups’ double covers U(1) and SU(2) arise naturally in spinor helicity
32 / 28
Arkani-Hamed, Huang, Huang ’17
massless massive det{pα ˙
β} = 0
det{pα ˙
β} = m2
pα ˙
β = λpα˜
λp ˙
β ≡ |pα[p| ˙ β
pα ˙
β = λ a pαǫab˜
λ b
p ˙ β ≡ |paα[pa| ˙ β
det{λ
a pα} = det{˜
λ
a p ˙ α} = m
pµ = 1
2p|σµ|p]
pµ = 1
2pa|σµ|pa]
pα ˙
β˜
λ ˙
β p = 0
pα ˙
β˜
λa ˙
β p
= mλ
a pα
pq = −qp ⇒ pp = 0 paqb = −qbpa e.g. papb = −mǫab [pq] = −[qp] ⇒ [pp] = 0 [paqb] = −[qbpa] e.g. [papb] = mǫab pq[qp] = 2p·q paqb[qb pa] = 2p·q
33 / 28
Consider Lorentz transform pµ → Lµ
νpν
↔ Lµ
ν ∈ S O ( 1 , 3 )
= 1
2 tr
σµS
∈ S L ( 2 , C )
σνS† Massless: |p → S|p = eiφ/2|Lp p| → p|S−1 = eiφ/2Lp| |p] → S†−1|p] = e−iφ/2|Lp] [p| → [p|S† = e−iφ/2[Lp|
eihφ ∈ U(1) encode 2d rotations in frame where p = (E, 0, 0, E)
Massive: |pa → S|pa = ωa
b|Lpb
|pa → |paS−1 = ωa
b|Lpa
|pa] → S†−1|pa] = ωa
b[Lpb|
[pa| → [pa|S† = ωa
b[Lpb|
ω ∈ SU(2) encode 3d rotations in rest frame where p = (m, 0, 0, 0)
34 / 28
Massless: εµ
p+ = q|σµ|p]
√ 2qp εµ
p− = p|σµ|q]
√ 2[pq] ⇒ ε±
p ·p = ε± p ·q = 0
εµ
p+εν p−+ εµ p−εν p+ = −ηµν + pµqν + qµpν
p·q εh1
p ·εh2 p = −δh1(−h2)
Massive: εab
pµ = ip(a|σµ|pb)]
√ 2m ⇒ p · εab
p = 0
εab
pµεpνab = −ηµν + pµpν
m2 εab
p · εpcd = −δ(a (c δb) d)
Guevara, AO, Vines ’18 Chung, Huang, Kim, Lee ’18
and (symmetrized) tensor products thereof
35 / 28
Arkani-Hamed, Huang, Huang ’17
Take pµ = (E, P cos ϕ sin θ, P sin ϕ sin θ, P cos θ) |pa = λ a
pα =
√ E−P cos θ
2
− √ E+P e−iϕsin θ
2
√ E−P eiϕsin θ
2
√ E+P cos θ
2
λ a
p ˙ α =
√ E+P eiϕsin θ
2
− √ E−P cos θ
2
√ E+P cos θ
2
− √ E−P e−iϕsin θ
2
sµ(ua
p) =
1 2m ¯ upa γµγ5ua
p = (−1)a−1sµ p
sµ
p = 1
m(P, E cos ϕ sin θ, E sin ϕ sin θ, E cos θ)
36 / 28
gµν = ηµν + κhµν ⇒ gµν = ηµν + κhµν + O(κ2) Spin 0:
Lscalar = gµν(∂µϕ)†(∂νϕ) − m2ϕ†ϕ Lϕϕh = −κhµν(∂µϕ†)(∂νϕ) ⇒ hµν
3
ϕ†
1
ϕ2 ≃ iκp(µ
1 pν) 2
Spin 1:
LProca = −1 2V †
µνV µν + m2V † µ V µ
LV V h = κhµν V †
µσV σ ν
− m2V †
µ Vν
hνρ
3
V †λ
1
V µ
2
≃ −iκ
ηλ(νηρ)µ + ηλµp(ν
1 pρ) 2 − ηλ(νpρ) 2 pµ 1 − pλ 2p(ν 1 ηρ)µ
37 / 28
Spin 0: 3± 1 2 = −iκ(p1 · ε3)2 = −iκ 2 m2x2
±
where x = √ 2 p1 · ε3 m Spin 1: 3+ 1
{a}
2{b} = −iκ 2 x2
+1(a1 2(b11a2)2 b2)
3− 1
{a}
2{b} = −iκ 2 x2
−[1(a1 2(b1][1a2)2 b2)]
38 / 28
Arkani-Hamed, Huang, Huang ’17
M3(1{a}, 2{b}, 3+) = −iκ 2 1a2b⊙2s m2s−2 x2
+
M3(1{a}, 2{b}, 3−) = −iκ 2 [1a2b]⊙2s m2s−2 x2
−
x = √ 2 p1 · ε3 m : x+ = r|1|3] mr3, x− = −[r|1|3 m[r3] = − 1 x+ NB! Independent of ref. momentum r p2
2 − m2 = 2p1· p3 = 3|1|3] = 0
⇒ ∃ x ∈ C : |1|3 = −mx|3]
39 / 28
Guevara, AO, Vines ’18
1a2b = [1a2b] + 1 mx+ [1ak][k2b], mx+ = √ 2(p1· ε+) [1a2b]
1 mx− 1akk2b, mx− = √ 2(p1· ε−) [1a2b]⊙2s =
mx− ⊙2s = 1a|⊙2s 2s
2s j
mx− j |2b⊙2s = 1a|⊙2sexp
ν Jµν 2
p2 · ε−
ν Jµν 1
p1 · ε−
M(s)
3 (1, 2, k−) =
x2
−
m2s−2 [12]⊙2s = M(0)
3
m2s 2|2sexp
ν Jµν
p · ε−
M(s)
3 (1, 2, k+) =
x2
+
m2s−2 12⊙2s = M(0)
3
m2s [2|2sexp
ν Jµν
p · ε+
40 / 28
Witten ’03 used in Cachazo, Strominger ’14
Massless momentum kµ Jµν =
kσµν, β α
∂ ∂λβ
k
+ ˜ λk ˙
α¯
σµν, ˙
α ˙ β
∂ ∂˜ λk ˙
β
α,β ˙ β = 2i
∂ ∂λβ)
k
ǫ ˙
α ˙ β + ǫαβ˜
λk( ˙
α
∂ ∂˜ λk ˙
β)
α ˙ ασν β ˙ βJµν
Conde, Joung, Mkrtchyan ’16
Massive extension Jα ˙
α,β ˙ β = 2i
p(α
∂ ∂λβ)a
p
ǫ ˙
α ˙ β + ǫαβ˜
λ a
p( ˙ α
∂ ∂˜ λ
˙ β)a p
∀ spin-0 function f(p) Jµνf(p) = Lµνf(p), Lµν = 2ip[µ ∂ ∂pν] Jµνpρ = pσΣµν,σ
ρ,
Jµνερ = εσΣµν,σ
ρ,
Σµν,ρ
σ = i[ηµρδν σ − ηνρδµ σ]
41 / 28
Guevara, AO, Vines ’18
Differentiation action: kµε+
ν Jµν|pa = kµε− ν Jµν|pa] = 0
kµε−
ν Jµν
p · ε−
|pa⊙2s = (2s)! (2s − j)!|pa⊙(2s−j)⊙ |kkpa mx−
kµε+
ν Jµν
p · ε+
|pa]⊙2s = (2s)! (2s − j)!|pa]⊙(2s−j)⊙ |k][kpa] mx+
Algebraic realization on |p⊙2s and |p]⊙2s:
ν Jµν
p · ε−
= (2s)! (2s − j)! |kk| mx−
⊙ I⊗2s−j , j ≤ 2s , j > 2s
ν Jµν
p · ε+
= (2s)! (2s − j)! |k][k| mx+
⊙ I⊗2s−j , j ≤ 2s , j > 2s
connects to intuitive “spin-operator” terminology in Guevara ’17
42 / 28
Cachazo, Strominger ’14
Soft theorem:*
Mn+1 =
n
(pi · ε)2 pi · k − i(pi · ε)(kµενJµν
i )
pi · k − 1 2 (kµενJµν
i )2
pi · k
=
n
(pi · ε)2 pi · k
i
pi · ε − 1 2 kµενJµν
i
pi · ε
Mn + O(k2)
Vines ’17
Energy tensor of Kerr BH: T µν(k) = δ(p · k)p(µ exp(−ia ∗ k)ν)
ρ pρ + O(G)
⇒ εµν
k Tµν(k) = δ(k2)δ(p · k)(p · ε)2
p · ε − 1 2 kµενSµν p · ε
+ O(k3)
where pµ = muµ, Sµν = ǫµνρσpρaσ
*Omitting prefactors of −i(κ/2)n−2 in Mn, where κ =
√ 32πG.
43 / 28